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+{"text":"\\section{Introduction}\\label{sec:intro} \nExpander graphs, graphs in which all small subsets exhibit good expansion properties, are intriguing objects of study that arise in such diverse fields as number theory, computer science and discrete geometry.  As Hoory, Linial and Wigderson remark in their wide ranging survey on expanders \\cite{HLW}, one reason for their ubiquity is that they may be defined in at least three languages: combinatorial\/geometric, probabilistic and algebraic.  We refer the reader to this survey and to the expository article of Sarnak \\cite{S} for more background on expander graphs and their applications.\n\nWe shall be concerned, almost exclusively, with {\\em $d$-regular} graphs, in which every vertex has exactly $d$ neighbours.  \nFrom the algebraic viewpoint, which we take throughout, a $d$-regular graph $G$ is an expander if there is a significant gap between $\\lambda(G)$, the largest eigenvalue of the adjacency matrix of $G$ (for a $d$-regular graph this value is always $d$) and $\\lambda_2(G)$, the largest modulus of any other eigenvalue.  \nClassic results of Dodziuk, Alon--Milman and Alon \\cite{D84,AM85,A} show that the difference $d-\\lambda_2(G)$ controls the combinatorial expansion of $G$. More precisely, writing $h(G) = \\min_{S} |E(S,S^c)|\/|S|$, where $E(S,S^c)$ is the number of edges from $S$ to $S^c$, its complement in $V$, and where the minimum is taken over subsets $S$ of the vertices of $G$ with $|S| \\leq |S^c|$, we have \n\\[\n\\frac{d-\\lambda_2(G)}{2} \\leq h(G) \\leq \\sqrt{2d(d-\\lambda_2(G))}.\n\\]\nGiven the theorem of Alon and Boppana \\cite{A,N} that $\\lambda_2(G) \\ge 2\\sqrt{d-1}-o_{n}(1)$ for every $d$-regular graph with $n$ vertices, it is particularly significant if $\\lambda_2(G)\\le 2\\sqrt{d-1}$, in which case the graph is said to be {\\em Ramanujan}.  A major open problem is to prove the existence of infinite families of $d$-regular Ramanujan graphs for all $d\\ge 3$.   Explicit constructions coming from number theory given by Lubotzky, Phillips and Sarnak \\cite{LPS} and Margulis \\cite{Ma} for the case that $d-1$ is prime represented a major breakthrough.  Morgenstern \\cite{Mo} gives examples of such families whenever $d-1$ is a prime power.  However, it seems unlikely that number theoretic approaches will be successful in resolving the problem in its full generality.\n\nA combinatorial approach to the problem, initiated by \\citet{F1}, is to prove that one \nmay obtain new (larger) Ramanujan graphs from smaller ones. In this approach one starts with a base graph $H$ which one ``lifts'' to obtain a larger graph $G$ which covers the original graph $H$ in the sense that there is a homomorphism from $G$ to $H$ such that all fibres in $G$ of vertices of $H$ are of equal size. If $G$ is a cover of $H$ and the fibres in $G$ of vertices in $H$ have size $k$, then $G$ is called a $k$-lift of $H$. \n\nIt is easily observed that the lift $G$ inherits all the eigenvalues of the base graph $H$.  Indeed, let $\\mu$ be an eigenvalue of $H$ with eigenvector $x$, and define a vector $y$ with entries indexed by $V(G)$ by setting, for each $i$, $y_{v}=x_{i}$ for all vertices $v\\in V_{i}$; \nthen $y$ is an eigenvector of $G$ with eigenvalue $\\mu$.  In fact these lifted eigenvectors of $H$ span the space of all vectors that are constant on each of the fibres $V_{i}$ of the lift.  The remaining eigenvalues of $G$ are referred to as the {\\em new eigenvalues} of the lift (note however that it is possible for some new eigenvalues to be equal to ``old'' eigenvalues).  Since eigenvectors of symmetric real matrices corresponding to distinct eigenvalues are orthogonal, these are exactly the set of eigenvalues which have an eigenvector $x$ which is balanced on each fibre (i.e. for which $\\sum_{v\\in V_{i}}x_{v}=0$ for all $i\\in V(H)$).  \nSince the base graph is given it suffices to concentrate our study on the new eigenvalues of $G$.  We denote by $\\lambda^{*}(G)$ the largest absolute value of a new eigenvalue of $G$. (For the remainder of the paper, \nfor any graph $F$ we denote by $\\lambda(F)$ the largest eigenvalue of $F$.) \n\nA {\\em random $n$-lift} $G$ of a graph $H$ is obtained by assigning to each vertex $i$ of $H$ a distinct set $V_{i}$ of $n$ vertices, and placing a random matching (i.e. one chosen uniformly at random from the $n!$ possibilities) between $V_{i}$ and $V_{j}$ for each edge $ij$ of $H$.  Random lifts were introduced by Amit, Linial, Matou\\v sek and Rozenman \\cite{ALMR}. In that article a variety of properties of random lifts are discussed, related to connectivity, expansion, independent sets, colouring and perfect matchings; the proofs of these results, and others, were developed in several subsequent papers \\cite{AL1,AL2,AL3,LR}. \nAs remarked in \\cite{F1}, any finite cover $F$ of $H$ in which fibres have size $n$ has a positive probability of appearing as $G$ (in fact this probability is precisely $(n!)^{|V(H)-E(H)|}(\\mathrm{Aut}(F\/H))^{-1}$, where $\\mathrm{Aut}(F\/H)$ is the group of automorphisms of \n$F$ over $H$), and so such random lifts form a ``seemingly reasonable model of a probabilistic space of finite quotients of [the infinite \n$d$-ary tree]''. \n\nAlthough very few graphs are {\\em known} to be Ramanujan, it is conjectured that a positive proportion of regular graphs are in fact Ramanujan, and \nAlon's conjecture\/Friedman's theorem states that for any $\\epsilon > 0$, only an asymptotically negligible proportion of $d$-regular graphs have $\\lambda_2(G) > 2\\sqrt{d-1}+\\epsilon$. In this spirit, Friedman \\cite{F1} studied the eigenvalues of random lifts of regular graphs, and Lubetzky, Sudakov and Vu \\cite{LSV} conjectured that a random lift of a Ramanujan graph has a positive probability of being Ramanujan.  Since for all $d$ the complete graph $K_{d+1}$ is a $d$-regular Ramanujan graph, this would imply the existence of arbitrarily large $d$-regular Ramanujan graphs.  In the terminology from \\cite{F1}, the main result of this paper implies that with extremely high probability the lifts of Ramanujan graphs are $O(\\sqrt{d})$-weakly Ramanujan, in that all non-trivial eigenvalues are $O(\\sqrt{d})$.\n\nIn \\cite{F1}, Friedman used the trace method of Wigner to prove results which in particular imply that if $H$ is $d$-regular then $\\lambda^{*}(G)= O(d^{3\/4})$ whp\\footnote{If a statement holds with probability which tends to $1$ as $n$ tends to infinity, we say that it occurs `with high probability', or `whp' for short.}.  This was later tightened to $O(d^{2\/3})$ by Linial and Puder, by a careful analysis of the trace method.  They also made a conjecture concerning word maps, which if verified would prove $\\lambda^{*}(G)=O(d^{1\/2})$ whp.  Two initial cases of the conjecture were proved, a third has been proven more recently by Lui and Puder \\cite{LP}.  \nBilu and Linial \\cite{BL} then showed that every $d$-regular graph $H$ has {\\em some} $2$-lift $G$ with $\\lambda^*(G) = O(d^{1\/2}\\log^{3\/2}{d})$.\nThe next major step in this area was taken by Lubetzky, Sudakov and Vu \\cite{LSV}, who proved that $\\lambda^{*}(G)=O(\\max(\\lambda_2(H),d^{1\/2})\\log{d})$ whp.  In particular, in the case that $\\lambda_2(H)=O(d^{1\/2})$ this result gives that $\\lambda^{*}(G)=O(d^{1\/2}\\log{d})$. \n\nIn this article we prove that whp $\\lambda^*(G)=O(d^{1\/2})$, a result which is best possible up to the constant.  \n\\begin{thm}\\label{main}\nLet $H$ be any $d$-regular graph and let $G$ be a random $n$-lift of $H$.\nFor all $n$ sufficiently large, \nwith probability at least $1-n^{-2d^{1\/2}}$, \n$\\lambda^*(G) \\leq 430656 \\sqrt{d}$. \n\\end{thm}\nFurthermore, we are able to {\\em explain} the likely cause of large eigenvalues should they occur.  This cause is, with very high probability, a small (i.e.~of size not depending on $n$) subgraph of $G$.\n\\begin{thm}\n\\label{explain}\nLet $H$ be any $d$-regular graph, write $h=|V(H)|$, and let $G$ be a random $n$-lift of $H$.\nFor all $n$ sufficiently large, with probability at least $1-n^{-hd}$, $G$ contains an induced subgraph $G'$ with at most $hd$ vertices, \nsuch that $\\lambda^*(G) \\leq 1189248 \\lambda(G')$. \n\\end{thm}\nOne might protest that the eigenvalue of $G'$ is not necessarily a new eigenvalue of $G$, and so $G'$ is not the `cause' of a new eigenvalue of large modulus in $G$.  However, the following approximate converse to \\refT{explain} justifies our use of such an epithet for $G'$.\n\\begin{prop}\\label{protest}\nFor any induced subgraph $G'$ of $G$ with $|V(G')| \\leq n-h\\sqrt{n}$, we have $\\lambda^*(G) \\geq \\lambda(G') - 7\/2$. \n\\end{prop}\nThe short proof of \\refP{protest} appears in \\refS{sec:zbound}.\n\nCostello and Vu \\cite{CV} remark that ``[t]he main intuition that underlies many problems concerning the rank of a random matrix is that dependency should come from small configurations,'' and their paper can be seen as confirmation of this intuition for a rather broad class of random matrices. In this spirit, \\refT{explain} should be viewed as stating that for random lifts, {\\em any exceptionally large eigenvalues come from small configurations.}\nIt would be very interesting to know whether our ``exceptionally large'' can be replaced by ``slightly large''. A rather ambitious question one could ask in this direction is the following.\n\\begin{question}\nIs there a constant $C$ not depending on $n$ (perhaps depending on $H$) such that for any \n$\\epsilon > 0$, given that $\\lambda^*(G) \\geq (2+\\epsilon)\\sqrt{d-1}$, with probability $1-o_n(1)$, $G$ contains \na subgraph $G'$ with at most $C$ vertices such that $\\lambda(G') \\geq (2+\\epsilon\/2)\\sqrt{d-1}$?\n\\end{question}\n\nWe note that the probability bound in \\refT{explain} is extremely strong.  Indeed, the failure probability, $n^{-hd}$  is much smaller than the probability that $G$ contains $H$ as a subgraph -- the probability \nof the latter event is greater than $n^{-hd\/2}$ -- in which case $\\lambda^{*}(G)=d$.\n\nOur proof, like that of Lubetzky, Sudakov and Vu \\cite{LSV} and many others, relies on reducing an uncountable collection of possible `reasons' for a large eigenvalue to a finite (and hopefully relatively small) sub-collection which still express all ways in which a large eigenvalue can occur.  They used the well-known method of $\\epsilon$-nets to make this reduction. (Amit and Linial \\cite{AL2} \nalso used $\\epsilon$-nets to prove a lower bound on edge expansion for random lifts of connected graphs which need not necessarily be regular.)  However, the number of events (points of the $\\epsilon$-net) one is required to consider is $\\exp(\\Theta(nh\\log{d}))$.  The appearance of the $\\log{d}$ here is a major obstacle to proving that $\\lambda^*(G)=O(\\sqrt{d})$ by the $\\epsilon$-net approach.  Our approach is based on a convexity argument that allows us to reduce to a smaller, $\\exp(\\Theta(nh))$-sized family of events.  Furthermore, the events in this collection are concerned with vectors with dyadic entries.  These are easier to deal with than general vectors; in particular, direct combinatorial arguments may be applied and we need not appeal to martingale inequalities to obtain probability bounds.\n\nModulo a few trivial changes to our proof (e.g. changing ``precisely'' to ``at most'' in the proof of Proposition \\ref{zbound}, below) replacing $d$ by $\\Delta$ throughout gives a proof of the following generalisation to the case that the base graph $H$ is not regular. \n\n\\begin{thm} \\label{thm:omit} Let $H$ be any graph of maximum degree $\\Delta$ and let $G$ be a random $n$-lift of $H$.  For all $n$ sufficiently large, \nwith probability at least $1-n^{-2\\Delta^{1\/2}}$, \n$\\lambda^*(G) \\leq 430656 \\sqrt{\\Delta}$, and with probability at least $1-n^{-h\\Delta}$, \n$G$ contains an induced subgraph $G'$ with at most $h\\Delta$ vertices, \nsuch that $\\lambda^*(G) \\leq 1189248 \\lambda(G')$. \\end{thm}  \n\nSince $G$ will always contain two edge disjoint stars whose centres have degree $\\Delta$ and lie in the same fibre, $\\lambda^*(G)\\ge \\sqrt{\\Delta}$.  Thus this result is also tight up to the constant.\n\n\nWhile we focus here on the case of an $n$-lift where $n$ is large we would like to bring to the reader's attention the \nrecent result of Oliveira \\cite{Olive} that a random $n$-lift $G$ of a graph $H$ on $h$ vertices with maximum degree $\\Delta$ satisfies $\\lambda^{*}(G)=O(\\Delta^{1\/2}\\log^{1\/2}(hn))$ whp.\n\n\\section{Notation}\n\nAll logarithms in this paper are natural logarithms unless otherwise specified. \nFor positive integers $k$ we write $[k]=\\{1,\\ldots,k\\}$. We write ${\\mathbb N}_0 = \\{0,1,2,\\ldots\\}$ and ${\\mathbb N}=\\{1,2,\\ldots\\}$. \nFor any graph $F=(V,E)$ and $u,v \\in V$ we write $u \\sim_F v$ if $uv \\in E$. \nFor the remainder of the paper, $d \\geq 2$ is a positive integer, $H=([h],E(H))$ is a fixed $d$-regular graph, \nand $G=(V(G),E(G))$ is the random $n$-lift of $H$, where $V(G)=\\{(i,j),i \\in [h],j \\in [n]\\}$. \nFor $i$ in $[h]$ we write $V_i =  \\{(i,j): j \\in [n]\\}$, and call $V_i$ the {\\em fibre} of $i$ in $G$. \nLet $M$ be the adjacency matrix of $G$. \n\nFor any $m$-by-$m$ matrix $A=(a_{uv})_{u,v \\in [m]}$ and vectors $x,y \\in \\mathbb{R}^m$, we write \n$\\ip{x}{y}_A = \\sum_{u,v \\in [m]} x_u a_{uv} y_v$. Also, for a set $E \\subset \\mathbb{R}^2$, we write \n$\\ip{x}{y}_{A,E}$ for the restricted sum\n\\[\n\\sum_{\\{u,v \\in [m]: (x_u,y_v) \\in E\\}} x_u a_{uv} y_v. \n\\]\nFinally, we use the Vinogradov notation $f \\ll g$ to mean that  $f=O(g)$, i.e.~$f$ is bounded by a constant times $g$, \nindependent of $n$. We write $f \\asymp g$ to mean that $f=O(g)$ and $g=O(f)$. \n               \n\\section{An overview of the proof} \\label{sec:duck}  \nAs in \\cite{KS} and much subsequent work, we will bound the eigenvalues of $G$ using the \nRayleigh quotient principle, which is to say by bounding $\\ipo{x}_M$ for suitable vectors~$x$. \nMore precisely, writing $X = \\{x \\in \\mathbb{R}^{V(G)}:\\|x\\|_2^2\\leq 1\\}$, Rayleigh's quotient principle tells us the following. \n\\begin{fact}\\label{rayleigh}\n$\\lambda^*(G)  = \\sup\\{ |\\ipo{x}_M|:  x \\in X, \\forall~i \\in [h],~\\sum_{v \\in V_i} x_v = 0\\}$. \n\\end{fact}\n\\refF{rayleigh} forms the basis for our study of $\\lambda(G)$. \nHowever, to make use of it we first need to have an idea \nof the diversity of possible ways in which $G$ could admit a balanced vector \n(a vector satisfying \n$\\sum_{v \\in V_i} x_v = 0$ for all $i \\in [h]$) having a large value for $\\ipo{x}_M$.\nTo begin to get a feel for this, we now give two rather different examples of how $\\lambda^*(G)$ can be large.  For simplicity, for the examples we assume that $H$ is the complete graph $K_{d+1}$.  We also provide bounds on the probability of such examples occurring in $G$.  These bounds give something of the flavour of the bounds we shall be required to prove for the general case.\n\n\\textbf{Example 1: $G$ contains a large-ish clique.}\n\nFix $s\\in{\\mathbb N}$ and vertices $v_1,\\ldots,v_s$ of $G$ in distinct fibres. If $G[\\{v_1,\\ldots,v_s\\}]$ happens to be a clique, then setting $x_{u}$ to be $1$ if $u \\in \\{v_1,\\ldots,v_s\\}$, $-1\/(n-1)$ for all other vertices in the same fibres as $v_1,\\ldots,v_s$, and zero on all other vertices, we obtain a vector $x$, balanced on each fibre of $H$ and for which $Mx=(s-1)x$. Thus, if $G$ contains a clique of order at least $K\\sqrt{d}+1$ then $\\lambda^*(G) \\geq K\\sqrt{d}$. \n\nTo bound the probability that $G$ contains such a clique, for each possible choice of $v_1,\\ldots,v_s$ the probability that $G[\\{v_{1},\\dots ,v_{s}\\}]$ is a clique is $n^{-\\binom{s}{2}}$ (since each edge $v_{i}v_{i'}$ is present in $G$ independently with probability $n^{-1}$).  Since there are only ${(d+1) \\choose s} n^{s}$ choices of the $s$-tuple $(v_{1},\\dots ,v_{s})$ the probability that $G$ contains a clique of size $s$ is at most ${(d+1) \\choose s}n^{s-\\binom{s}{2}}$, which is $o(1)$ for any $s\\ge 4$. \n\n\\textbf{Example 2: uneven edge densities all over $G$.} \n\nSuppose that there exist sets $(A_{i})_{i \\in V(H)}$, with $A_{i}\\subset V_{i}$ and $|A_{i}|=n\/2$ for all $i\\in V(H)$, such that $e(A_{i},A_{j})\\ge n\/4+Kn\/\\sqrt{d}$ for each $i \\neq j$, $i,j \\in V(H)$.  Then setting $x_{u}$ to be $(nh)^{-1\/2}$ if $u$ is in $\\bigcup_{i}A_{i}$ and $-(nh)^{-1\/2}$ otherwise, we obtain a balanced vector $x$ with $\\|x\\|_2^2 = 1$ and $\\ipo{x}_M \\geq 2K\\sqrt{d}$, so $\\lambda^*(G) \\geq 2K\\sqrt{d}$.\n\nHere, for each choice of sets $(A_{i})_{i\\in V(H)}$, with $A_{i}\\subset V_{i}$ and $|A_{i}|=n\/2$ for all $i\\in V(H)$, the probability that for all $i,j \\in V(H)$, $i\\neq j$ we have $e(A_{i},A_{j})\\ge n\/4+Kn\/\\sqrt{d}$, is of order at most $\\exp(-c\\binom{d+1}{2}K^{2}n\/d)$, for some constant $c > 0$ (this is not hard to derive by hand; it can also be obtained straightforwardly from \\refP{bigbound} in Section~\\ref{sec:prob}). If $K$ is sufficiently large, this bound is strong enough for a union bound to show\nthat with high probability, there is no such choice of sets $(A_i)_{i \\in V(H)}$. \n\nThe preceding examples present two rather different structures within $G$, both of which give rise to large new eigenvalues, \nand show how the new eigenvectors are also rather different. \nWe may also switch our point of view, first fixing a vector $x$ (for the first example a vector taking the value $1$ on a single vertex of each fibre $V_{i}, \\, i=1,\\dots,s$ and taking the value $-\\I{u\\in V_{[s]}}\/(n-1)$ on other vertices $u$; for the second example a vector taking the value $(nh)^{-1\/2}$ on $n\/2$ vertices in each fibre and $-(nh)^{-1\/2}$ on the rest) then asking what structure in $G$ is required if $|\\ipo{x}_M|$ is to be large for this specific vector $x$. This is essentially the perspective we will take for most of the rest of the paper. \n\nFrom this viewpoint, a possible cause for $\\lambda^{*}(G)$ being large consists of a vector $x \\in X$ together with evidence, in the form of specified edge counts between subsets of vertices of $G$, that $|\\ipo{x}_M|$ is large. \nFor a vector $x$, $i \\in [h]$ and $w \\in \\mathbb{R}$ and let\n\\[\n A_{i,w}(x) = \\{v \\in V_i: x_{v} = w\\}, \\qquad a_{i,w}(x)=|A_{i,w}(x)|. \n\\]\nWe write $A_{i,w}=A_{i,w}(x)$ and $a_{i,w}=a_{i,w}(x)$ when the dependency on $x$ is clear. \nWe say that the collection $\\{a_{i,w}:i \\in [h],w \\in \\mathbb{R}\\}$ is the {\\em type} of the vector $x$, and denote this collection ${\\bf a}(x)$. By the symmetry of the model, the probability that $|\\ipo{x}_M|$ \nis large should be a function only of the type of $x$. We will seek sets of constraints on the edge densities between sets corresponding to distinct $a_{i,w}$ and $a_{i',w'}$, which \ncodify all possible ways in which a large new eigenvalue can appear. A type, together with a particular such set of constraints, will be called a {\\em pattern}; the precise definition of patterns will appear later in the section. \n\nWhile this initial concept of a pattern is useful for demonstrating the idea we have in mind, a number of changes are needed before we can \nput it into play. The most fundamental issue is that we are trying to bound $\\sup_{x \\in X} \\ipo{x}_M$, the supremum being over an uncountable collection. \nWe will shortly show that for a moderate cost, the uncountable \nsupremum in \\refF{rayleigh} can be replaced by a supremum over a more tractable collection. \nAlso, it turns out that for {\\em any} vector $x \\in X$, the contribution to $\\ipo{x}_M$ made by entries $x_{i,j},x_{i',j'}$ whose \nweights differ by more than a factor of $\\sqrt{d}$ is negligible, which will allow us to further restrict the collection of types we need to consider. \nThis is not a complete list of the required changes, but before proceeding too far into the argument it is useful to fill in some of these initial steps. \n\nRecall that $M$ is the adjacency matrix of $G$, \nand let $\\overline{M} = (\\overline{m}_{(i,j),(i',j')})_{(i,j),(i',j') \\in V(G)}$ be the matrix with \n\\[\n\\overline{m}_{(i,j),(i',j')} = \\begin{cases}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\/n\t& \\mbox{if } i \\sim_{H} i' \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\t\t\t\t\t\t\t& \\mbox{otherwise.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{cases}\n\\]\nIn other words, $\\overline{m}_{(i,j),(i',j')} = \\p{(i,j) \\sim_G (i',j')}$, so \nfor all $x \\in \\mathbb{R}^{V(G)}$, we have $\\ipo{x}_{\\overline{M}} = \\E{\\ipo{x}_M}$. \nIf $x$ is balanced on each fibre, i.e., satisfies $\\sum_{v \\in V_i} x_v = 0$ for all $i \\in [h]$, \nthen $\\ipo{x_i}_{\\overline{M}} =x_i^t (\\overline{M}x_i) = x_i^t \\cdot \\overline{0}=0$. \nThus, letting $N=M-\\overline{M}$, for such $x$ we have \n\\[\n\\ipo{x}_N = \\ipo{x}_M - \\ipo{x}_{\\overline{M}} = \\ipo{x}_M. \n\\]\nIf, on the other hand, $x$ is constant on fibres of $G$, then $\\ipo{x}_M = \\ipo{x}_{\\overline{M}}$ and so \n$\\ipo{x}_N=0$. \nFrom this we easily obtain the following fact. \n\\begin{fact}\\label{uncount}\n$\\lambda^*(G) = \\sup_{x \\in X} |\\ipo{x}_N|$.\n\\end{fact}\n\\begin{proof}\nThe inequality $\\lambda^*(G) \\leq \\sup_{x \\in X} |\\ipo{x}_N|$ follows from \\refF{rayleigh} \nsince $\\ipo{x}_N=\\ipo{x}_M$ if $x$ is balanced on fibres of $G$. \nOn the other hand, any vector in $x \\in X$ can be expressed as $y+z$, where $y \\in X$ is balanced on fibres of $H$ \nand $z$ is constant on fibres of $H$. Since $\\langle y,z \\rangle_N=0$, we then have $\\ipo{x}_N=\\ipo{y}_N=\\ipo{y}_M$, which proves the other inequality. \n\\end{proof}\nBy using $N$ instead of $M$, \\refF{uncount} allows us to maintain the property of only considering ``new'' eigenvectors of $M$, without \ninsisting that the vectors we consider remain balanced on each fibre of $G$. This turns out to be remarkably helpful. \n\nNext, let $D^+ = \\{0\\} \\cup \\{2^i\/(nh)^{1\/2}: i \\in {\\mathbb N}\\}$, let \n\\begin{align}\nZ \t& = \\left\\{x \\in \\mathbb{R}^{V(G)}:\\|x\\|_2^2 \\leq 10,\\forall~v \\in V(G),x_v \\in D^+, \\sup_{v \\in V(G)} x_v \\leq d \\cdot \\inf_{v \\in V(G): x_v \\ne 0} x_v\\right\\}\\, . \\label{zdef}\n\\end{align}\nand let \n\\[\nE^* = \\{(x_1,x_2) \\in \\mathbb{R}^2: x_1,x_2 > 0, x_1\/x_2 \\in (d^{-1\/2},d^{1\/2})\\}. \n\\]\nThe following proposition, which is proved in \\refS{sec:zbound}, formalizes our discussion on ``restricting the collection of types we need to consider''. \n\\begin{prop}\\label{zbound}\n$\\lambda^*(G) \\leq 96 \\sup_{x \\in Z} |\\ipo{x}_{N,E^*}| + 480\\sqrt{d}.$\n\\end{prop}\n\nA relatively straightforward step in the proof of \\refP{zbound}, that will also turn out to be useful in another part of the paper, is to show that not much is lost when we restrict our attention to pairs of vertices whose weights in $x$ differ by at most a multiplicative factor of $\\sqrt{d}$. This is the content of the following lemma, which is proved in \\refS{sec:zbound}. \n\\begin{lem}\\label{ipoe}\nFor all $y \\in \\mathbb{R}^{V(G)}$, \n$|\\ipo{y}_{N,\\mathbb{R}^2\\setminus E^{*}}|\\le 4\\sqrt{d}\\|y\\|_2^2.$\n\\end{lem}\n\nWe are now in a position to further elaborate on what will constitute a pattern of a large eigenvalue. \nRecall that for $y \\in \\mathbb{R}^{V(G)}$, \nfor each $i \\in [h]$ and each $w \\in \\mathbb{R}$, we have $A_{i,w}(y) = \\{v \\in V(G):y_{v} = w\\}$, \nand $a_{i,w}(y) = |A_{i,w}(y)|$, and that ${\\bf a}(y)$ is called the type of $y$.  \nWe remark that if $y \\in Z$ then ${\\bf a}(y)$ satisfies the following properties.\n\n\\begin{enumerate}\n\\item[{\\bf 1.}] For all $w \\in \\mathbb{R}$, $w \\not \\in D^+$, we have $a_{i,w} = 0$.\n\\item[{\\bf 2.}] For all $i \\in V(H)$, $\\sum_{w\\in D^+} a_{i,w} \\le n$. \n\\item[{\\bf 3.}] There exists $w_{0}\\in D^+$, $w_0>0$, such that $a_{i,w}=0$ unless $w_{0}\\le w\\le w_{0}d$. \n\\item[{\\bf 4.}] $\\sum_{i \\in [h], w \\in D^+} w^2 a_{i,w} \\le 10$. \n\\end{enumerate}\nWe call a collection ${\\bf a} = \\{a_{i,w}: i \\in V(H), w \\in \\mathbb{R}\\}$ a {\\em $Z$-type} if it satisfies properties {\\bf 1}-{\\bf 4}, so that \nif $y \\in Z$ then the type ${\\bf a}(y)$ of $y$ is a $Z$-type. \nNext, given $y \\in \\mathbb{R}^{V(G)}$ and $ii' \\in E(H)$, $w,w' \\in \\mathbb{R}$, let $e_{i,w,i',w'}(y,G) = |e_G(A_{i,w}(y),A_{i',w'}(y)|$, and write \n\\[\n{\\bf e}(y,G) = \\{e_{i,w,i',w'}(y,G): ii' \\in E(H), w,w' \\in \\mathbb{R}\\}.\n\\]\nFor all $ii' \\in E(H)$ and all $w,w' \\in \\mathbb{R}$, we necessarily have \n\\begin{equation*}\\label{eq:edef}\ne_{i,w,i',w'}(y,G) \\in \\{0,1,\\ldots, \\min(a_{i,w}(y),a_{i',w'}(y))\\}.\n\\end{equation*}\nA {\\em pattern} is a pair $({\\bf a},{\\bf e})$, where ${\\bf a}$ is a $Z$-type and \n\\[\n{\\bf e}= \\{e_{i,w,i',w'}: ii' \\in E(H), w,w' \\in \\mathbb{R}\\}\n\\]\nis a collection with $e_{i,w,i',w'} \\in \\{0,1,\\ldots, \\min(a_{i,w},a_{i',w'})\\}$ for all $ii' \\in E(H)$ and all $w,w' \\in \\mathbb{R}$. \n\nWrite $D^{>0}=D^+\\setminus\\{0\\}=\\{2^i\/\\sqrt{nh}:i \\in {\\mathbb N}\\}$, and \nlet $\\Gamma$ be the graph on vertex set $\\{(i,w):i\\in V(H), w\\inD^{>0}\\}$ with an edge $(i,w)(i',w')$ if $i\\sim_H i'$ and $w\/w'\\in (d^{-1\/2},d^{1\/2})$. \n\n\n\nFor a given pattern, $({\\bf a},{\\bf e})$, we write \n\\begin{align*}\np({\\bf a},{\\bf e}) = \\left|\\sum_{(i,w)\\sim_{\\Gamma} (i',w')}w w' \\left(e_{i,w,i',w'}- \\frac{a_{i,w}a_{i',w'}}{n} \\right)  \\right| ,\n\\end{align*}\nand call $p({\\bf a},{\\bf e})$ the \\emph{potency} of $({\\bf a},{\\bf e})$. We remark that for any $y \\in Z$, \n\\begin{align}\n|\\langle y,y \\rangle_{N,E^*}| & = 2\\left|\\sum_{(i,w)\\sim_{\\Gamma} (i',w')}w w' \\left(e_{i,w,i',w'}(y,G)- \\frac{a_{i,w}(y)a_{i',w'}(y)}{n} \\right)\\right| \\nonumber\\\\\n& =  2\\,  p({\\bf a}(y),{\\bf e}(y,G)), \\label{eq:potconnect}\n\\end{align}\nthe factor $2$ arising from the symmetry of the matrix $N$. \nWe say that a pattern $({\\bf a},{\\bf e})$ {\\em can be found in $G$} if there exists $y \\in Z$ such that \n${\\bf a}(y)={\\bf a}$ and ${\\bf e}(y,G)={\\bf e}$. \nIn this case we say that $G$ {\\em contains} the pattern $({\\bf a},{\\bf e})$, \nand call the collection $\\{A_{i,w}(y)\\}_{(i,w) \\in V(\\Gamma)}$ a {\\em witness} for the pattern $({\\bf a},{\\bf e})$. \n\n\\begin{prop}\\label{findwit}\nFor $K > 0$, if $\\lambda^*(G) \\geq 192(K+3) \\sqrt{d}$ then a pattern $({\\bf a},{\\bf e})$ with potency at least $K\\sqrt{d}$ can be found in $G$.\n\\end{prop}\n\\begin{proof}\nBy \\refP{zbound}, in this case there exists $y \\in Z$ with $|\\langle y,y \\rangle_{N,E^*}| \\ge 2K\\sqrt{d}$, and the result follows from (\\ref{eq:potconnect}). \n\\end{proof}\nOur main aim, then, is to show that with high probability, no high-potency pattern can be found in $G$. \nTo do so, we shall use a reduction which is conceptually analogous to how one bounds the probability that a fixed graph $H$ appears as a subgraph of the Erd\\H{o}s--R\\'enyi random graph $G(n,p)$: rather than proving the bound directly for $H$, one instead considers a strictly balanced (maximally edge-dense) subgraph of $H$. \n\nGiven a pattern $({\\bf a},{\\bf e})$ and a set of vertices $S \\subset V(\\Gamma)$, we define the \n{\\em sub-pattern of $({\\bf a},{\\bf e})$ induced by $S$} to be the pattern $({\\bf a}',{\\bf e}')$ obtained from $({\\bf a},{\\bf e})$ by \nsetting \n\\[\na'_{i,w} = \\begin{cases} \n\t\t\t\t\ta_{i,w} \t& \\mbox{if}~(i,w) \\in S \\\\\n\t\t\t\t\t0\t\t\t\t& \\mbox{otherwise}. \n\t\t\t\t\\end{cases}\n\\]\nand setting \n\\[\ne_{i,w,i',w'}' = \\begin{cases} \n\t\t\t\t\te_{i,w,i',w'} \t& \\mbox{if}~(i,w),(i',w') \\in S \\\\\n\t\t\t\t\t0\t\t\t\t& \\mbox{otherwise}. \n\t\t\t\\end{cases}\n\\]\nWe write $({\\bf a},{\\bf e})_S$ for the sub-pattern of $({\\bf a},{\\bf e})$ induced by $S$.  \n\nWe also require the following variant of potency, which \nallows us to consider a ``maximally potent'' subgraph of $\\Gamma$ rather than $\\Gamma$ itself. For a pattern $({\\bf a},{\\bf e})$ we define\n\\[\n\\tilde{p}(({\\bf a},{\\bf e})):=\\max_{E\\subset E(\\Gamma)} \\left|\\sum_{(i,w)(i',w')\\in E}w w' \\left(e_{i,w,i',w'}- \\frac{a_{i,w}a_{i',w'}}{n}\\right) \\right|\n\\]\nWe will prove the following theorem.\n\\begin{thm}\\label{redbound}\nFix $L \\ge 20$. \nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold.\n\\begin{align*}\n\\tilde{p}(({\\bf a},{\\bf e})_{S}) \t& \\geq \\frac{p({\\bf a},{\\bf e})}{2}-55L\\sqrt{d}\\\\\n\\p{({\\bf a},{\\bf e})_S~\\mbox{can be found in G}} & \\ll \\prod_{(i,w) \\in S} a_{i,w}^{d\/4}\\pran{\\prod_{(i,w) \\in S} {n \\choose a_{i,w}\\wedge \\lfloor n\/2 \\rfloor}}^{1-L\/10}. \n\\end{align*}\n\\end{thm}\n\nWe prove this theorem in \\refS{sec:key}. \n\\refT{main} will follow straightforwardly from \\refP{findwit} and \\refT{redbound}.\nMore precisely, \\refP{findwit} ensures that the number of events whose probability we must bound is not too large, and \\refT{redbound} ensures that the probability of each event is sufficiently small that we can simply apply a union bound to prove \\refT{main}. \nBefore {\\em providing} the proof, we state one additional, easy bound which we will require, on \nthe total number of patterns satisfying a constraint on the sizes of the $a_{i,w}$.\n\\begin{lem}\\label{patcount}\nFor fixed $A \\in {\\mathbb N}$, the number of patterns with $a_{i,w} < A$ for all $(i,w) \\in V(\\Gamma)$ \nis at most $\\log_2(nh) \\cdot A^{2h d\\log_2 d}$. \n\\end{lem}\n\nAssuming this lemma, which is proved in \\refS{sec:zbound}, we are now ready to give our proof of \\refT{main}. (The proof of \\refT{explain}, while conceptually almost identical to that of \\refT{main}, ends up requiring a somewhat more technical development and we therefore defer it to \\refS{sec:explain}.)\n\n\\begin{proof}[Proof of Theorem~\\ref{main}]\nFix $M\\, \\ge\\, 430656\\,\\, = \\,\\, 192(2240\\, +\\, 3)$, let $K=K(M)=(M\/192)-3 \\ge 2240$, and let $L(M)=K\/112\\ge 20$.\n\nBy \\refP{findwit}, if $\\lambda^*(G) \\geq M\\sqrt{d}$ then there is a pattern $({\\bf a},{\\bf e})$ with potency at least \n$K\\sqrt{d}=112 L\\sqrt{d}$ such that $({\\bf a},{\\bf e})$ can be found in $G$.   Let $({\\bf a},{\\bf e})_{S(({\\bf a},{\\bf e}),L)}$ be the sub-pattern of $({\\bf a},{\\bf e})$ obtained by applying \\refT{redbound} to $({\\bf a},{\\bf e})$, it follows that $\\tilde{p}(({\\bf a},{\\bf e})_{S(({\\bf a},{\\bf e}),L)}) \\ge (112\/2-55)L\\sqrt{d} =L\\sqrt{d}$ and that $({\\bf a},{\\bf e})_{S(({\\bf a},{\\bf e}),L)}$ can be found in $G$.\n\nWe say a pattern $({\\bf a},{\\bf e})$ is an {\\em $L$-reduction} if there is a pattern $({\\bf a}',{\\bf e}')$ and \n$S=S(({\\bf a}',{\\bf e}')),L)$ as in \\refT{redbound}, such that $({\\bf a}',{\\bf e}')_{S(({\\bf a}',{\\bf e}'),L)}=({\\bf a},{\\bf e})$. It follows from the preceding paragraph that for any $M \\geq 430656$, \n\\[\n\\p{\\lambda^*(G) \\geq M\\sqrt{d}} \\leq \\p{E},\n\\]\nwhere $E=E(M)$ is the event that $G$ contains an $L(M)$-reduction $({\\bf a},{\\bf e})$ with $\\tilde{p}({\\bf a},{\\bf e})\\ge L(M)\\sqrt{d} \\geq 20\\sqrt{d}$.  We use \\refT{redbound} to bound this probability for each fixed pattern (reduction) $({\\bf a},{\\bf e})$, the proof is then completed with a union bound.  We now give the details.  We split into two cases, depending on whether the pattern (reduction) $({\\bf a},{\\bf e})$ has $a_{i,w}\\geq 4hd\\log_{2}{d}$ for some $(i,w) \\in V(\\Gamma)$.  Let $E_{1}=E_1(M)$ be the event that $G$ contains an $L$-reduction $({\\bf a},{\\bf e})$ with $\\tilde{p}({\\bf a},{\\bf e})\\ge L\\sqrt{d}$ in which at least one $a_{i,w}\\geq 4hd\\log_{2}{d}$ and $E_{2}=E_2(M)$ the event that $G$ contains an $L$-reduction $({\\bf a},{\\bf e})$ with $\\tilde{p}({\\bf a},{\\bf e})\\ge L\\sqrt{d}$ in which $a_{i,w}< 4hd\\log_{2}{d}$ for all $i,w$.  We note that $E_1$ and $E_2$ need not be disjoint. To prove \\refT{main} it suffices to show that both $\\p{E_1}$ and $\\p{E_2}$ are less than $n^{-2d^{1\/2}}\/2$ for all $n$ sufficiently large. \n\nFirst note that for {\\em any} $L$-reduction $({\\bf a},{\\bf e})$, writing $\\alpha=\\sum_{(i,w) \\in V(\\Gamma)} a_{i,w}$, since $1 - L\/10 \\le -1$, by \\refT{redbound} we have \n\\[\n\\p{({\\bf a},{\\bf e})_S~\\mbox{can be found in}~G} \\ll \\prod_{(i,w) \\in S} a_{i,w}^{d\/4}\\cdot {n \\choose \\alpha \\wedge \\lfloor n\/2 \\rfloor}^{-1}. \n\\]\nFor any $L$-reduction $({\\bf a},{\\bf e})$ considered in $E_1$, we have $\\alpha \\geq 4hd \\log_2(d)$. \nWe note that since non-zero entries in any $Z$-type differ by a factor of at most $d$, for any $i \\in V(H)$ there are at most $\\log_2(2d)$ values $w\\ne 0$ for which $a_{i,w} \\ne 0$. It follows that in total there are at most $h\\log_2(2d)$ vertices $(i,w) \\in V(\\Gamma)$ with $a_{i,w} \\ne 0$, so\n\\begin{align*}\n\\p{({\\bf a},{\\bf e})_S~\\mbox{can be found in}~G} \n& \\ll n^{(d\/4) h\\log_2(2d)} \\cdot {n \\choose 4hd\\log_2(d)}^{-1} \\\\\n& \\leq n^{(hd\\log_2 d)\/2} \\cdot \\pran{\\frac{8hd \\log_2 d}{n}}^{4hd \\log_2 d}.\n\\end{align*}\nWe next take a union bound over all $L$-reductions considered in $E_1$. \nClearly the number of such reductions is at most the total number of patterns, which by \\refL{patcount} applied with $A=n$ is at most $\\log_2(nh) n^{2hd\\log_2 d}$. \nIt follows that \n\\[\n\\p{E_1} \\ll (8hd \\log_2 d)^{4hd \\log_2 d} \\frac{\\log_2(nh)}{n^{3(hd\\log_2 d)\/2}},\n\\]\nand so $\\p{E_1}\\le n^{-2d^{1\/2}}\/2$ for $n$ sufficiently large. \n\nNext, for any $L$-reduction $({\\bf a},{\\bf e})$ considered in $E_2$, we have \n\\[\n\\prod_{(i,w) \\in S} a_{i,w}^{d\/4} < (4hd\\log_2 d)^{(d\/4)h\\log_2 (2d)}.\n\\]\nAlso, for any pattern $({\\bf a},{\\bf e})$, it follows straightforwardly from the definition of a pattern that \n\\begin{align*}\n\\tilde{p}({\\bf a},{\\bf e}) \t& \\leq  \\sum_{(i,w)\\sim_{\\Gamma} (i',w')}w w' \\left|e_{i,w,i',w'}- \\frac{a_{i,w}a_{i',w'}}{n} \\right|\\\\\n \t\t\t& \\leq \\sum_{(i,w)(i',w') \\in E(\\Gamma)} ww' \\min(a_{i,w},a_{i',w'}) \\\\\n\t\t\t& \\leq \\sum_{(i,w) \\in V(\\Gamma)} \\mathop{\\sum_{(i',w')\\sim_{\\Gamma}(i,w)}}_{w' \\leq w,a_{i,w} >0} w^2 a_{i,w} \\\\\n\t\t\t& \\leq \\sum_{(i',w') \\in V(\\Gamma)} a_{i',w'} \\cdot \\sum_{(i,w) \\in E(\\Gamma)} w^2 a_{i,w} \\\\\n\t\t\t& \\leq \\sum_{(i,w) \\in V(\\Gamma)} a_{i,w}. \n\\end{align*}\nIt follows that $\\alpha=\\sum_{(i,w) \\in S} a_{i,w} \\geq Ld^{1\/2}> 3d^{1\/2}$ for every $L$-reduction $({\\bf a},{\\bf e})$ with $\\tilde{p}({\\bf a},{\\bf e}) \\ge L\\sqrt{d}$.  So, for any pattern considered in $E_2$, for $n \\geq 6d^{1\/2}$ we have \n\\[\n{n \\choose \\alpha \\wedge \\lfloor n\/2 \\rfloor}^{-1} \\leq \\frac{(6d^{1\/2})^{3d^{1\/2}}}{n^{3d^{1\/2}}}.\n\\]\nFurthermore, the total number of $L$-reductions considered in $E_2$ is at most the total number of patterns \nwith all $a_{i,w}$ less than $4hd \\log_2d$, which by \\refL{patcount} \nis at most $\\log_2(nh) (4hd\\log_2 d)^{2hd \\log_2 d}$. \nThus, by a union bound, \n\\[\n\\p{E_2} \\ll (4hd\\log_2d)^{3hd\\log_2 d}\\cdot (6d^{1\/2})^{2d^{1\/2}}\\cdot \\frac{\\log_2(nh)}{n^{3d^{1\/2}}}, \n\\]\nwhich implies that $\\p{E_2}\\le n^{-2d^{1\/2}}\/2$ for $n$ sufficiently large. This completes the proof\n\\end{proof}\n\n\\section{Two tools from probability} \\label{sec:prob}\nIn this section, we establish all the probability bounds we will require in the remainder of the paper. \\refP{bigbound}, below, \nbounds the probability that a random matching has certain prescribed edge counts between given pairs of sets. \n\\refL{lem:measure} gives a lower bound on the integral of a function when its value is not too small relative to another function. \nWe proceed immediately to details. \n\nLet $V=\\{v_1,\\ldots,v_n\\}$, $W=\\{w_1,\\ldots,w_n\\}$, and let $\\mathbf{M}$ be a uniformly random matching of $V$ and $W$. \nLet $A_0,\\ldots,A_s$ and $B_0,\\ldots,B_t$ be partitions of $V$ and $W$ respectively, and write $a_i=|A_i|$ and $b_j=|B_j|$. \n(Here $s$ and $t$ are constants not depending on $n$.) \nWriting $e_{\\mathbf{M}}(A_i,B_j)$ for the number of edges from $A_i$ to $B_j$ in $\\mathbf{M}$, it is straightforward that $\\mu_{ij} = \\E{e(A_i,B_j)} = a_ib_j\/n$. \n\nNow fix integers $\\{e_{ij}\\}_{0 \\leq i \\leq s,0 \\leq j \\leq t}$ with $\\sum_{j=0}^t e_{ij}=a_i$ for each $0 \\leq i \\leq s$ and $\\sum_{i=0}^s e_{ij} = b_j$ for each $0 \\leq j \\leq t$. (From now on, summations and products over $i$ (resp.~$j$) should be understood to have $0 \\leq i \\leq s$ (resp.~$0 \\leq j \\leq t$).) We wish to bound \n\\[\n\\p{\\bigcap_{ij} \\{e(A_i,B_j) = e_{ij}\\}}.\n\\]\nOur aim is to prove a bound not too different from what we would obtain were the $e(A_i,B_j)$ independent with Binomial$(a_ib_j,1\/n)$ distribution. \nBy direct enumeration, $\\p{\\bigcap_{ij} \\{e(A_i,B_j) = e_{ij}\\}}$ equals \n\\[\n\\frac{1}{n!} \\cdot \\prod_i {a_i \\choose e_{i0},\\ldots,e_{it}} \\prod_j {b_j \\choose e_{0j},\\ldots,e_{sj}} \\prod_{ij} e_{ij}! = \\frac{1}{n!}\\prod_i a_i! \\cdot \\prod_j b_j! \\cdot \\pran{\\prod_{ij} e_{ij}!}^{-1},\n\\]\nwith the convention that $0!=1$. Applying Stirling's formula, this is of the same order as \n\\begin{equation}\\label{jast}\n\\pran{\\frac{\\prod_i a_i \\prod_j b_j}{\\prod_{ij:e_{ij}\\neq 0} e_{ij}}}^{1\/2} \\cdot n^{-(n+1\/2)} \\prod_{i} a_i^{a_i} \\prod_{j} b_j^{b_j} \\prod_{ij:e_{ij}\\neq 0} e_{ij}^{-e_{ij}}\n\\end{equation}\nNow write $e_{ij}=(a_ib_j\/n)(1+\\epsilon_{ij}) = \\mu_{ij} (1+\\epsilon_{ij})$. \nWe then have \n\\begin{align*}\n\\prod_{ij:e_{ij}\\neq 0} e_{ij}^{e_{ij}} & = \\prod_{ij:e_{ij} \\neq 0} \\mu_{ij}^{\\mu_{ij}(1+\\epsilon_{ij})} \\prod_{ij:e_{ij}\\neq 0} (1+\\epsilon_{ij})^{\\mu_{ij}(1+\\epsilon_{ij})} \\\\\n\t\t\t\t\t\t\t\t\t\t& = \\prod_{ij} \\mu_{ij}^{\\mu_{ij}(1+\\epsilon_{ij})} \\prod_{ij:e_{ij}\\neq 0} (1+\\epsilon_{ij})^{\\mu_{ij}(1+\\epsilon_{ij})} \\\\ \n\t\t\t\t\t\t\t\t\t\t& = \\prod_{ij} \\mu_{ij}^{\\mu_{ij}} \\prod_{ij} \\mu_{ij}^{\\mu_{ij}\\epsilon_{ij}} \\prod_{ij:e_{ij}\\neq 0} (1+\\epsilon_{ij})^{\\mu_{ij}(1+\\epsilon_{ij})}  \n\\end{align*}\nFor fixed $i$, since $\\sum_{j} e_{ij}=a_i$ we must have $\\sum_j \\epsilon_{ij} \\mu_{ij}=0$. \nLikewise for fixed $j$ we have $\\sum_i \\epsilon_{ij} \\mu_{ij} = 0$, and it follows that \n\\begin{align*}\n\\prod_{ij} \\mu_{ij}^{\\epsilon_{ij} \\mu_{ij}} & = \\prod_{i} \\pran{\\prod_{j} \\pran{\\frac{a_i}{n}}^{\\epsilon_{ij} \\mu_{ij}} \\prod_j b_j^{\\epsilon_{ij} \\mu_{ij}}} \\\\\n\t\t\t\t\t\t\t\t\t\t\t& = \\prod_{ij} b_j^{\\epsilon_{ij} \\mu_{ij}} \\\\\n\t\t\t\t\t\t\t\t\t\t\t& = \\prod_j \\pran{ \\prod_i b_j^{\\epsilon_{ij} \\mu_ij} } = 1.\n\\end{align*}\nA similar calculation shows that \n\\begin{align*}\n\\prod_{ij} \\mu_{ij}^{\\mu_{ij}} = n^{-n} \\cdot \\prod_i a_i^{a_i} \\prod_j b_j^{b_j}, \n\\end{align*}\nand so \\refQ{jast} equals \n\\[\nn^{-1\/2}\\pran{\\frac{\\prod_i a_i \\prod_j b_j}{\\prod_{ij:e_{ij}\\neq 0} e_{ij}}}^{1\/2} \\cdot \\prod_{ij:e_{ij}\\neq 0} (1+\\epsilon_{ij})^{-\\mu_{ij}(1+\\epsilon_{ij})}. \n\\]\nWe now focus our attention on the latter product of the preceding equation. \nIt is helpful to multiply and divide by $1= \\prod_{ij} e^{\\epsilon_{ij} \\mu_{ij}}$, to obtain the equivalent expression \n\\begin{align*}\n\t& \\prod_{ij:e_{ij}\\neq 0} \\exp\\pran{-\\mu_{ij}(1+\\epsilon_{ij})\\log(1+\\epsilon_{ij})} \\prod_{ij} \\exp\\pran{\\mu_{ij}\\epsilon_{ij}} \\\\\n= \t& \\prod_{ij:e_{ij}\\neq 0} \\exp\\pran{-\\mu_{ij}[(1+\\epsilon_{ij})\\log(1+\\epsilon_{ij})-\\epsilon_{ij}]} \\prod_{ij:e_{ij}=0} \\exp\\pran{-\\mu_{ij}}.\n\\end{align*}\n(For the second product we use that when $e_{ij}=0$, $\\epsilon_{ij}=-1$.)\nSince $(1+\\epsilon)\\log(1+\\epsilon)-\\epsilon$ approches $1$ as $\\epsilon \\downarrow -1$, taking $0\\log0-0=1$ by convention then allows us to \ncombine the two preceding products.\nIn sum, we have established the following proposition. \n\\begin{prop}\\label{bigbound}\nLet $e_{ij} = \\mu_{ij}(1+\\epsilon_{ij})$ and let $E_{ij} = e_{\\mathbf{M}}(A_i,B_j)$. Writing $\\chi = n^{-1\/2}\\pran{\\frac{\\prod_i a_i \\prod_j b_j}{\\cdot \\prod_{ij:e_{ij}\\neq 0} e_{ij}}}^{1\/2}$, \nwe have \n\\[\n\\p{\\bigcap_{ij} \\{E_{ij} = e_{ij}\\}} \\asymp\n\\chi \\cdot e^{-\\sum_{ij} \\mu_{ij}[(1+\\epsilon_{ij})\\log(1+\\epsilon_{ij}) - \\epsilon_{ij}]}.\n\\]\n\\end{prop}\nIn fact, it is a weakening of \\refP{bigbound} that will be useful later in the paper. \n\\begin{cor}\n\\label{cor:bb}\nUnder the conditions of \\refP{bigbound}, we have \n\\[\n\\p{\\bigcap_{ij} \\{E_{ij} = e_{ij}\\}} \\ll\n\\pran{\\prod_{1 \\leq i \\leq s}a_i \\prod_{1 \\leq j \\leq t} b_j}^{1\/4} \\cdot e^{-\\sum_{ij} \\mu_{ij}[(1+\\epsilon_{ij})\\log(1+\\epsilon_{ij}) - \\epsilon_{ij}]}.\n\\]\n\\end{cor}\n\\begin{proof}\nFor each $0 \\leq i \\leq s$ we have $\\prod_{0 \\leq j \\leq t: e_{ij} \\neq 0} e_{ij} \\geq a_i$, \nso $\\prod_{ij:e_{ij} \\neq 0} e_{ij} \\geq \\prod_i a_i$. \nWe likewise have $\\prod_{ij:e_{ij} \\neq 0} e_{ij} \\geq \\prod_j b_j$, and so \n\\[\n\\prod_{ij:e_{ij} \\neq 0} e_{ij} \\geq \\prod_i a_i^{1\/2} \\prod_j b_j^{1\/2}. \n\\]\nAlso, $n^{-1\/2}a_0^{1\/2}b_0^{1\/2} \\leq 1$. The result follows. \n\\end{proof}\nWe will also use a lemma which is essentially a version of the following basic fact: \nif $X$ and $Y$ are non-negative random variables and $E$ is the event that $X \\leq cY$, then $\\E{X \\I{E}} \\le c\\E{Y}$. \n(For generality we shall state the lemma in terms of a measure space, although we shall only apply it to finite measure spaces, and in this case the reader may think of the integrals as simply weighted sums.)\n\n\\begin{lem} \\label{lem:measure} \nLet $(\\Omega, \\mathcal{E},\\nu)$ be a measure space and $g,h:\\Omega\\to \\mathbb{R}$ positive measurable functions, \nand fix $0 < c < 1$. \nWriting \n\\[\nE = \\left\\{ \\omega \\in \\Omega: \\frac{h(\\omega)}{g(\\omega)} \\geq c\\frac{\\int h d\\mu}{\\int g d\\mu}\\right\\},\n\\]\nwe have \n\\[\\int_E h d\\mu \\ge (1-c) \\int_{\\Omega} h d\\mu.\n\\] \n\\end{lem}\n\\begin{proof} \\begin{equation*} \\int_{\\Omega \\setminus E}hd\\mu =\\int _{\\Omega\\setminus E}g \\cdot \\frac{h}{g}d\\mu \n\\le c \\int_{\\Omega\\setminus E}g \\cdot \\frac{\\int h}{\\int g}  d\\mu\n\\le c\\int_{\\Omega} h d\\mu \\end{equation*}\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reduced patterns: a proof of \\refT{redbound}} \\label{sec:key}\n\\refT{redbound} requires us to find, given a high potency pattern $({\\bf a},{\\bf e})$, a sub-pattern $({\\bf a},{\\bf e})_S$ which is unlikely to occur in $G$. In fact, our aim will be slightly more exigent. We will show the existence of sub-pattern $({\\bf a},{\\bf e})_S$ which is `locally unlikely' at all $(i,w)\\in S$. Given a pattern $({\\bf a},{\\bf e})$, for $i\\sim_{H} i'$ and $w,w' \\in \\mathbb{R}$ write $\\mu_{i,w,i',w'} = a_{i,w}a_{i',w'}\/n$ and write \n\\[\n\\epsilon_{i,w,i',w'}=\\epsilon_{i,w,i',w'}({\\bf a},{\\bf e}) = \n\\begin{cases}\n1\t& \\mbox{ if }a_{i,w}=0\\mbox{ or }a_{i',w'}=0,\\\\\ne_{i,w,i',w'} \/ \\mu_{i,w,i',w'} -1 & \\mbox{ if } a_{i,w}\\neq 0\\mbox{ and }a_{i',w'} \\neq 0. \n\\end{cases}\n\\]\nThe quantity $\\epsilon_{i,w,i',w'}$ encodes the deviation from the expected number $\\mu_{i,w,i',w'}$ of edges between sets $A \\subset V_i$, $A' \\subset V_{i'}$ with $|A|=a_{i,w}$ and $|A'|=a_{i',w'}$ if $|E_G(A,A')|=e_{i,w,i',w'}$. We will bound the likelihood of such deviations using \\refC{cor:bb}, and to that end define a function $b:(-1,\\infty) \\to [0,\\infty)\n\\[ b(\\epsilon)= (1+\\epsilon)\\log(1+\\epsilon) - \\epsilon \\qquad \\epsilon\\in (-1,\\infty).\\]\nWe then have the following. \n\\begin{prop}\\label{prop:redbound} Fix $L \\ge 20$. \nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold.\n\\begin{align*}\n\\tilde{p}(({\\bf a},{\\bf e})_{S}) \t& \\geq \\frac{p({\\bf a},{\\bf e})}{2}-55L\\sqrt{d}\\qquad \\text{and}\\\\\n\\sum_{(i',w')\\in N_{\\Gamma}(i,w)\\cap S} \\frac{a_{i,w}a_{i',w'}}{n} b(\\epsilon_{i,w,i',w'})& \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right) \\qquad \\text{for all} \\, (i,w)\\in S\\, .\\end{align*}\n\\end{prop}\nThe second inequality in \\refP{prop:redbound} encodes the idea that the pattern is everywhere locally unlikely.\nBefore continuing to the proof of \\refP{prop:redbound} let us show that \\refT{redbound} does indeed follow from \\refP{prop:redbound} and the probability bound \\refC{cor:bb}.\n\n\\begin{proof}[Proof of \\refT{redbound}]  \nGiven $L\\ge 20$ and a pattern $({\\bf a},{\\bf e})$, let $S=S(({\\bf a},{\\bf e}),L)$ be the subset of $V(\\Gamma)$ given by \\refP{prop:redbound}.  Since it is immediate from the statement of \\refP{prop:redbound} that \n\\[ \\tilde{p}(({\\bf a},{\\bf e})_{S})  \\geq \\frac{p({\\bf a},{\\bf e})}{2}-55L\\sqrt{d},\\]\nall that is left to prove is that\n\\[\\p{({\\bf a},{\\bf e})_S~\\mbox{can be found in G}}  \\ll \\prod_{(i,w) \\in S}\\!\\! a_{i,w}^{d\/2}\\, \\pran{\\prod_{(i,w) \\in S} {n \\choose a_{i,w}\\wedge \\lfloor n\/2 \\rfloor}}^{1-L\/10}. \\]\nWe recall that finding a copy of $({\\bf a},{\\bf e})_S$ in $G$ corresponds exactly to finding a witness, i.e. a collection of sets $\\{A_{i,w}\\}_{(i,w) \\in S}$, such that for each $(i,w)\\in S$ the set $A_{i,w}\\subseteq V_i$ has cardinality $a_{i,w}$, and such that $e(A_{i,w},A_{i',w'})=e_{i,w,i',w'}$ whenever $(i,w)\\sim_{\\Gamma} (i',w')$.\n\nSince there are at most\n\\[ \\prod_{(i,w)\\in S}\\binom{n}{a_{i,w}\\wedge \\lfloor n\/2\\rfloor}\\]\nchoices of the collection $\\{A_{i,w}\\}_{(i,w) \\in S}$ it suffices to prove that\n\\[ \n\\p{\\bigcap_{(i,w)\\sim_{\\Gamma} (i',w')} \\{e(A_{i,w},A_{i',w'})=e_{i,w,i',w'}\\}}\\ll \\prod_{(i,w) \\in S} \\!\\! a_{i,w}^{d\/4}\\, \\pran{\\prod_{(i,w) \\in S} {n \\choose a_{i,w}\\wedge \\lfloor n\/2 \\rfloor}}^{-L\/10}\n\\] \nfor each such collection $\\{A_{i,w}\\}_{(i,w) \\in S}$.  Furthermore, since the inequality $(en\/a)^a \\ge {n \\choose a\\wedge \\lfloor n\/2 \\rfloor}$ holds for all $1\\le a\\le n$, it suffices to prove that \n\\begin{align*}\n& \\quad \\p{\\bigcap_{(i,w)\\sim_{\\Gamma} (i',w')} \\{e(A_{i,w},A_{i',w'})=e_{i,w,i',w'}\\}} \\\\\n\\ll & \\quad \\prod_{(i,w) \\in S}\\!\\! a_{i,w}^{d\/4}\\, \\cdot \\, \\exp\\left(-\\frac{L}{10}\\, \\cdot \\sum_{(i,w) \\in S} a_{i,w}\\, \\log\\left(\\frac{en}{a_{i,w}}\\right)\\right)\n\\end{align*}\nfor each such collection $\\{A_{i,w}\\}_{(i,w) \\in S}$.\n\nThe required bound now follows by applying \\refC{cor:bb} for each matching, then using the independence of the matchings and the second bound of \\refP{prop:redbound}. \\end{proof}\n\nThe rest of the section is dedicated to proving \\refP{prop:redbound}.  We divide into two cases based on whether the potency of $({\\bf a},{\\bf e})$ derives mostly from large deviation terms (terms in which $\\epsilon_{i,w,i',w'}$ is particularly large) or small deviation terms.  The reason for splitting the proof in this way is that the expression $b(\\epsilon_{i,w,i',w'})$ behaves rather differently in the two cases, resembling $\\epsilon\\log{\\epsilon}$ and $\\epsilon^2$ respectively.\n\nWe shall partition $\\Gamma$ into two subgraphs $\\Gamma_{\\mathrm{LD}}=\\Gamma_{\\mathrm{LD}}({\\bf a},{\\bf e})$ and $\\Gamma_{\\mathrm{SD}}=\\Gamma_{\\mathrm{SD}}({\\bf a},{\\bf e})$ by setting\n\\[ V(\\Gamma_{\\mathrm{LD}}) \\, =\\, V(\\Gamma_{\\mathrm{SD}}) \\, = \\, V(\\Gamma)\\]\nand defining\n\\begin{align*}\nE(\\Gamma_{\\mathrm{LD}}) & \\, =\\, \\{(i,w)\\sim_{\\Gamma} (i',w')\\, :\\, \\epsilon_{i,w,i',w'}({\\bf a},{\\bf e})> e^2 -1\\}\\, ,\n\\\\  \nE(\\Gamma_{\\mathrm{SD}}) & \\, =\\, \\{(i,w)\\sim_{\\Gamma} (i',w')\\, :\\, -1 \\le \\epsilon_{i,w,i',w'}({\\bf a},{\\bf e})\\le e^2 -1\\} \\, .\n\\end{align*}\nWe have chosen $e^2-1$ as the cutoff between the large and small deviations regimes as a matter of technical convenience to do with the details of the proof.\nNote that the potency $p({\\bf a},{\\bf e})$ of a pattern $({\\bf a},{\\bf e})$ may be expressed as\n\\[ p({\\bf a},{\\bf e}) \\, = \\,  \\left| \\sum_{(i,w)\\sim_{\\Gamma} (i',w')}\\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}  \\right|\\, .\\] \nWe define now two variants of potency.\n\\[ p_{\\mathrm{LD}}({\\bf a},{\\bf e})\\, = \\, \\left|\\sum_{(i,w)\\sim_{\\Gamma_{\\mathrm{LD}}} (i',w')}\\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'} \\right|   \\, .\\] \nand\n\\[ p_{\\mathrm{SD}}({\\bf a},{\\bf e})\\, = \\,  \\left|\\sum_{(i,w)\\sim_{\\Gamma_{\\mathrm{SD}}} (i',w')}\\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}  \\right|\\, .\\] \nThe latter expressions respectively capture the ``large deviations potency'' and ``small deviations potency.'' \nSince $p({\\bf a},{\\bf e}) \\le p_{\\mathrm{LD}}({\\bf a},{\\bf e})+p_{\\mathrm{SD}}({\\bf a},{\\bf e})$, for every pattern $({\\bf a},{\\bf e})$ we have that \\[ \\max\\{p_{\\mathrm{LD}}({\\bf a},{\\bf e}),p_{\\mathrm{SD}}({\\bf a},{\\bf e})\\}\\ge \\frac{p({\\bf a},{\\bf e})}{2}\\, .\\] \nWe shall prove the following propositions.\n\n\\begin{prop}\\label{prop:rbld} Fix $L \\ge 20$.\nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold: \n\\[\np_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S})\\geq p_{\\mathrm{LD}}({\\bf a},{\\bf e})-30L\\sqrt{d} \\,  ,\n\\\nand, for all $(i,w) \\in S$, \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap S} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'})  \\ge  \\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\n\\] \n\\end{prop}\n\\begin{prop}\\label{prop:rbsd} Fix $L \\ge 20$.\nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold.\n\\begin{align*}\np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S}) \t& \\geq p_{\\mathrm{SD}}({\\bf a},{\\bf e})-55L\\sqrt{d}\\qquad \\text{and}\\\\\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap S} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\frac{\\epsilon_{i,w,i',w'}^2}{15} & \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right) \\qquad \\text{for all} \\, (i,w)\\in S\\, .\\end{align*}\n\\end{prop}\n\nSince \n\\begin{equation}\\label{eq:b} \nb(\\epsilon) \\, \\ge  \\begin{cases}   \\frac{\\epsilon^2}{15}\t&  \\mbox{ if } \\epsilon \\leq e^2-1 \\\\\n\t\t\t\t\\pran{1+\\frac{\\epsilon}{2}} \\log(1+\\epsilon) & \\mbox{ if } \\epsilon > e^2 -1\\, ,\n\t\t\t\\end{cases}\n\t\t\t\\end{equation}\n\\refP{prop:redbound} follows immediately from Propositions~\\ref{prop:rbld} and \\ref{prop:rbsd} by applying the appropriate proposition to the pattern $({\\bf a},{\\bf e})$ (i.e. applying \\refP{prop:rbld} if $p_{\\mathrm{LD}}({\\bf a},{\\bf e}) \\ge p({\\bf a},{\\bf e})\/2$ and applying \\ref{prop:rbsd} if $p_{\\mathrm{SD}}({\\bf a},{\\bf e}) \\ge p({\\bf a},{\\bf e})\/2$). The required bound on $\\tilde{p}(({\\bf a},{\\bf e})_S)$ is obtained by considering the set $E\\subset E(\\Gamma)$ given by $E=\\Gamma_{\\mathrm{LD}}|_S$ or by $E=\\Gamma_{\\mathrm{SD}}|_S$, as appropriate.\n\nThe proofs of \\refP{prop:rbld} and \\refP{prop:rbsd} are similar.  In both cases the set $S$ is found by repeatedly applying some straightforward reduction rules. In spirit, these rules can be understood by analogy with the following procedure. \nSuppose we are given a graph $F=(V,E)$ with average degree $\\mu$. To find an induced subgraph with {\\em minimum} degree at least $\\mu\/2$, we may simply repeatedly throw away vertices of degree less than $\\mu\/2$ until no such vertices remain. Throwing away vertices of such small degree can only increase the average degree in what remains, so this procedure must terminate with a non-empty subgraph satisfying the desired global minimum degree requirement. In this example the measure of `importance' of a vertex is its degree. When we apply a similar style of argument, the analogue of degree will be ``local potency'', suitably defined. Also, our rules for throwing away vertices will be more involved, and so it will take some work to verify that in throwing away vertices we do not decrease the overall potency by too much. \n\n\nWe now proceed to the proofs of Propositions~\\ref{prop:rbld} and ~\\ref{prop:rbsd}.   We prove \\refP{prop:rbld} first since the reductions required for its proof are simpler.\n\n\\subsection{Large deviation patterns: A proof of \\refP{prop:rbld}}\\label{sec:ldps}\nNow and for the remainder of Section~\\ref{sec:ldps}, we fix $L\\ge 20$ and a pattern $({\\bf a},{\\bf e})$.\nOur aim is to find $S \\subset V(\\Gamma)$ with $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S})  \\geq p_{\\mathrm{LD}}({\\bf a},{\\bf e})-30L\\sqrt{d}$ and such that the inequality \n\\[ \\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap S} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon}{2}} \\log(1+\\epsilon) \\ge  \\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right) \\]\nholds for all $(i,w)\\in S$.\n\nAs alluded to above, we shall choose $S$ by repeatedly removing vertices of $V(\\Gamma)$ that correspond to ``inconsequential'' parts of the pattern, so that in the resulting sub-pattern $({\\bf a},{\\bf e})_S$ there is a significant contribution to the overall potency from every vertex.  We write  \n\\[ p_{\\mathrm{LD}}(({\\bf a},{\\bf e});(i,w)) = \\left|\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right| \\, ,\\]\nfor the `local' large deviations potency at $(i,w)$, \nand write \n\\[ \\widehat{N}_{i,w}({\\bf a},{\\bf e}) = \\sum_{(i',w')\\in N_{\\Gamma}(i,w)} (w')^2 a_{i',w'} \\left(\\frac{w'}{wd^{1\/2}}\\right)\\]\nfor the amount of ``$L_2$-squared weight'' of the type ${\\bf a}$ that appears on $\\Gamma$-neighbours of $(i,w)$ weighted by the factor $w'\/wd^{1\/2}$.\n\n\nThe intuition behind this weighting factor is that in the large deviations regime, the most significant contributions to potency should be made by edges $(i,w)(i',w') \\in E(\\Gamma_{\\mathrm{LD}})$ with $w$ and $w'$ close to a factor of $d^{1\/2}$ apart. Note that this is {\\em exactly} what happens for the eigenfunctions of the universal cover (the infinite $d$-ary tree) with near-supremal eigenvalues.\n\nWe say that a set $U \\subset V(\\Gamma)$ satisfies condition (LD1) if for each $(i,w) \\in U$ we have \n\\begin{itemize}\n\\item[(LD1)] $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge L a_{i,w}w^2 d^{1\/2}$,\n\\end{itemize}\nand that $U$ satisfies condition (LD2) if for each $(i,w) \\in U$ we have \n\\begin{itemize}\n\\item[(LD2)] $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_S;(i,w)) \\ge L \\widehat{N}_{i,w}({\\bf a},{\\bf e}) \/d^{1\/2}$.\n\\end{itemize}\nCondition (LD1) asks that $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_S;(i,w))$ is large relative to the size of $a_{i,w}$; the factor $a_{i,w}w^2$ approximately measures the proportion of $L^2$-squared weight of ${\\bf a}$ used by $a_{i,w}$.  In Condition (LD2) the factor $a_{i,w}w^2$ is replaced by $\\widehat{N}_{i,w}\/d$ which (in some sense) measures the average level of opportunity available to $a_{i,w}$.  (In the case of large deviations a good opportunity corresponds to $(i',w') \\in N_{\\Gamma_{\\mathrm{LD}}}(i,w) \\cap U$ with large $a_{i',w'}$ large and such that the ratio $w'\/w$ is close to $d^{1\/2}$.) \n\nWe then let $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ be the maximal subset of $V(\\Gamma)$ satisfying both (LD1) and (LD2). \nThe monotonicity of $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w))$ in $U$ guarantees that $S$ is unique and that $S$ can be determined by starting from $V(\\Gamma)$ by repeatedly throwing away vertices that violate at least one of the two conditions, until no such vertices remain. \nIn other words, writing $k = |V(\\Gamma)|-|S|$, \nwe may order the vertices of $V(\\Gamma) \\setminus S$ as $(i_1,w_1),\\ldots,(i_k,w_k)$ in such a way that \nfor each $1 \\leq j \\leq k$, letting $S_j = V(\\Gamma) \\setminus \\{(i_1,w_1),\\ldots,(i_{j-1},w_{j-1})\\}$, \nwe have \n\\[\np_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))<L \\max\\left( a_{i,w}w^2 d^{1\/2}, \\widehat{N}_{i,w} \/d^{1\/2}\\right)\\]\nIn other words, this ordering ``verifies'' that $(i_j,w_j) \\not \\in S$, for each $1 \\leq j \\leq k$. \n\nProving \\refP{prop:rbld} now reduces to establishing the following two lemmas.\n\n\\begin{lem}\\label{lem:potboundLD} Let $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ be as defined above.  Then $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S})  \\geq p_{\\mathrm{LD}}({\\bf a},{\\bf e})-30L\\sqrt{d}$.\\end{lem}\n\n\\begin{lem}\\label{lem:suffLD} \nGiven any subset $U$ of $V(\\Gamma)$, if $U$ satisfies (LD1) and (LD2) then for all $(i,w) \\in U$, \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\ge  \\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, .\n\\]\n\\end{lem}\n\n\\begin{proof}[Proof of \\refL{lem:potboundLD}.]  With $(i_1,w_1),\\ldots,(i_k,w_k)$ and $S_1,\\ldots,S_k$ as above, \nwrite\n\\[\nW_1 = \\{ (i_j,w_j): 1 \\le j \\le k, p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))< L a_{i,w}w^2 d^{1\/2}\\}\n\\]\nand write \n\\[\nW_2= \\{ (i_j,w_j), 1 \\le j \\le k, p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))< L \\widehat{N}_{i,w} \/d^{1\/2}\\}\\, .\n\\]\nWe note that $W_1 \\cup W_2 =\\{(i_1,w_1),\\dots ,(i_k,w_k)\\}$.\nSince\n\\[\np_{\\mathrm{LD}}(({\\bf a},{\\bf e})_S) \\ge p_{\\mathrm{LD}}(({\\bf a},{\\bf e})) - \\sum_{1 \\leq j \\leq k} p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j)),\n\\]\nto prove the proposition it suffices to show that \n\\begin{equation}\\label{thirty}\n\\sum_{1 \\leq j \\leq k} p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j)) \\leq 30Ld^{1\/2}. \n\\end{equation}\nFirst, for any $1 \\leq j \\leq k$ with $(i_j,w_j) \\in W_1$, \nwe have \n$p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\\le L a_{i,w}w^2 d^{1\/2}$, and so \n\\[\n\\mathop{\\sum_{1 \\leq j \\leq k}}_{(i_j,w_j) \\in W_1}p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j)) \n\\leq \n \\mathop{\\sum_{1 \\leq j \\leq k}}_{(i_j,w_j) \\in W_1} L a_{i,w} w^2 d^{1\/2}\\le 10 Ld^{1\/2}\\, ,\\] \nthe last inequality holding since ${\\bf a}$ is a Z-type so $\\sum_{(i',w') \\in V(\\Gamma)} w'^2 a_{i',w'} \\le 10$. \n \nNow, for $(i_j,w_j)\\in W_2$, we have that $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\\le L \\widehat{N}_{i,w}\/d^{1\/2}$, where (we recall) \n\\[\n\\widehat{N}_{i,w}=\\sum_{(i',w')\\in N_{\\Gamma}(i,w)}a_{i',w'}w'^2\\left(\\frac{w'}{wd^{1\/2}}\\right)\\] \nNote that for any $(i,w) \\in V(\\Gamma)$ and any neighbour $(i',w')$ of $(i,w)$ we have \n$w'\/wd^{1\/2} \\leq 1$. Since all weights $w$ are in $D^{>0}$, it follows that  \n\\begin{equation}\\label{twod}\n\\sum_{(i,w)\\in N_{\\Gamma}(i',w')} \\frac{w'}{wd^{1\/2}}\n= \\sum_{i \\in N_H(i')} \\sum_{w \\in D^{>0}, w \\le d^{1\/2} w'} \\frac{w'}{wd^{1\/2}}\n\\le \\sum_{i \\in N_H(i')} \\sum_{j \\in {\\mathbb N}} 2^{-j}\n= 2d\n\\end{equation}\nfor each $(i',w')\\in V(\\Gamma)$.  Thus\n\\begin{align*}\n\\sum_{(i,w) \\in W_2} p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j)) & \n\\le \\frac{L}{d^{1\/2}}\\, \\cdot \\, \\sum_{(i,w)\\in W_2} \\sum_{(i',w')\\in N_{\\Gamma}(i,w)}a_{i',w'}w'^2\\left(\\frac{w'}{wd^{1\/2}}\\right) \\\\\n&\n\\le \\frac{L}{d^{1\/2}}\\, \\cdot \\, \\sum_{(i',w')\\in V(\\Gamma)} a_{i',w'}w'^2\\, \\cdot \\sum_{(i,w)\\in N_{\\Gamma}(i',w')} \\frac{w'}{wd^{1\/2}}\\\\\n&\n\\le 20 Ld^{1\/2}\\, ,\\end{align*}\nwhere the final inequality follows from \\eqref{twod} and the fact that ${\\bf a}$ is a $Z$-type.\n\nSince $W_1\\cup W_2=\\{(i_1,w_1),\\dots ,(i_k,w_k)\\}$, the above bounds establish \\eqref{thirty}, completing the proof of the proposition.\\end{proof}\n\nIn preparation for the proof of \\refL{lem:suffLD}, we first record the following fact. \nGiven $U \\subset V(\\Gamma)$ and $(i,w) \\in V(\\Gamma)$, define \n\\[\nU_{\\mathrm{LD}}(i,w) = \\left\\{(i',w') \\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U: \\frac{\\epsilon_{i,w,i',w'}w^2 d}{w'^2}\\,\\ge \\, \\frac{Ln}{2a}\\right\\} \\, .\n\\]\nThe set $U_{\\mathrm{LD}}(i,w)$ contains the vertices we shall view as making an important contribution in our forthcoming use of \\refL{lem:measure}.\n\\begin{lem}\\label{lem:pointwise} If $U \\subset V(\\Gamma)$ satisfies (LD2) then \nfor all $(i,w)\\in U$, \n\\[\n\\sum_{(i',w')\\in U_{\\mathrm{LD}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'} \\ge \\frac{1}{2} p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w))\\, \n\\]\n\\end{lem}\n\\begin{proof}\nFix $(i,w) \\in U$, let \n$\\Omega = N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U$, and let $\\cE = 2^{\\Omega}$. \nThen for $(i',w') \\in \\Omega$, set \n\\[\n\\mu(i',w') = a_{i,w}a_{i',w'}, \\quad h(i',w') = \\frac{ww'\\epsilon_{i,w,i',w'}}{n}, \\quad g(i',w')=\\frac{w'^2}{nd^{1\/2}}\\frac{w'}{wd^{1\/2}}. \n\\]\nUsing (LD2), we then have \n\\[\n\\int h d\\mu =  p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w))\\ge L\\frac{\\widehat{N}_{i,w}}{d^{1\/2}}\n\\]\nand\n\\[ \n\\int g\\, d\\mu = \\frac{a_{i,w}}{nd^{1\/2}} \\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U}  a_{i',w'}w'^2\\, \\frac{w'}{wd^{1\/2}} \\ge\\frac{a_{i,w}\\widehat{N}_{i,w}}{nd^{1\/2}}\\, .\n\\]\nIt follows that \n\\[\nU_{\\mathrm{LD}}(i,w) \\supseteq \\left\\{(i',w') \\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U: \\frac{h(i',w')}{g(i',w')} \\geq \\frac{1}{2}\\frac{\\int h d\\mu}{\\int g d\\mu}\\right\\},\n\\]\nand so by \\refL{lem:measure} applied with $c=1\/2$, \n\\[\n\\sum_{(i',w')\\in U_{\\mathrm{LD}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\ge \\int_{\\frac{h}{g} \\geq \\frac{1}{2}\\frac{\\int h}{\\int g}} h \\, d\\mu \\ge \n\\frac{1}{2} \\int_{\\Omega} h \\, d\\mu = \\frac{1}{2} p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w)).\n\\]\n\\end{proof}\n\nWe now prove \\refL{lem:suffLD}, completing the proof of \\refP{prop:rbld}.\n\n\\begin{proof}[Proof of \\refL{lem:suffLD}]\nWe are given $U \\subset V(\\Gamma)$ satisfying (LD1) and (LD2), and aim to show that \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\ge  \\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, .\n\\]\n\nNote that $c\\log{x}\\ge \\log(c^2 x)$ for $x\\ge e^2$ and $c\\ge 1$.  Since $1+\\epsilon_{i,w,i',w'}\\ge e^2$ for all $(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)$ we have that \n\\[\n\\frac{w d^{1\/2}}{w'} \\log(1+\\epsilon_{i,w,i',w'}) \\ge \\log\\left((1+\\epsilon_{i,w,i',w'})\\left(\\frac{w^2 d}{(w')^2}\\right)\\right)\\ge \\log\\left(\\frac{\\epsilon_{i,w,i',w'}w^2 d}{(w')^2}\\right)\\, .\n\\]\nIt follows that\n\\begin{align}\n& \\quad \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'})  \\nonumber\\\\\n= & \\quad \n\\frac{1}{w^2 d^{1\/2}} \\frac{ww'a_{i,w}a_{i',w'}}{n} \\, \\cdot \\left(\\frac{wd^{1\/2}}{w'}\\right)\\, \\cdot \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\nonumber\\\\\n\\ge & \\quad \n\\frac{1}{2w^2 d^{1\/2}}\\frac{ww'a_{i,w}a_{i',w'}}{n} \\epsilon_{i,w,i',w'} \\log\\left(\\frac{\\epsilon_{i,w,i',w'}w^2 d}{(w')^2}\\right)\\, \\label{eq:rbld1}.\n\\end{align}\n\nNote that the expression inside the preceding logarithm is precisely the expression included in the definition of $U_{\\mathrm{LD}}(i,w)$.  \nApplying first (\\ref{eq:rbld1}), then \\refL{lem:pointwise} and finally (LD1), we obtain that \n\\begin{align*}\n& \\quad \\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\\\\n\\ge & \\quad\n\\frac{1}{2w^2 d^{1\/2}}\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U} \\frac{ww'a_{i,w}a_{i',w'}}{n} \\epsilon_{i,w,i',w'} \\log\\left(\\frac{\\epsilon_{i,w,i',w'}w^2 d}{w'^2}\\right)\\\\\n\\ge & \\quad \n\\frac{1}{4w^2 d^{1\/2}}p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w))\\log\\left(\\frac{Ln}{2a}\\right)\\\\\n\\ge & \\quad \n\\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, ,\n\\end{align*}\ncompleting the proof of the lemma.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Small deviation patterns: A proof of \\refP{prop:rbsd}}\\label{sec:rbsd}\nOur proof of \\refP{prop:rbsd} is similar to our proof of \\refP{prop:rbld} given above.  In that spirit, throughout Section~\\ref{sec:rbsd} we fix $L\\ge 20$ and a pattern $({\\bf a},{\\bf e})$. We will find the required set $S$ by repeatedly removing vertices of $V(\\Gamma)$ that make a small contribution to potency.\n\nGiven a pattern $({\\bf a},{\\bf e})$ and $(i,w) \\in V(\\Gamma)$, we write \n\\[ p_{\\mathrm{SD}}(({\\bf a},{\\bf e});(i,w)) = \\left|\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right| \\, ,\\]\nfor the `local' small deviations potency at $(i,w)$, \nand write \n\\[\nN_i = N_i ({\\bf a},{\\bf e}) = \\sum_{(i',w') \\in V(\\Gamma): i' \\sim_H i} w'^2 a_{i',w'}.\n\\]\nfor the proportion of the $L_2$-squared weight of ${\\bf a}$ that appears on fibres $V_{i'}$ that neighbour $V_i$.\nOur next aim is to state small deviations analogues of Lemmas~\\ref{lem:potboundLD} and~\\ref{lem:suffLD}. However, our reduction rules are slightly more involved in this case, and require one additional definition. \nLet \n\\[\nM_{i,w} = M_{i,w}({\\bf a},{\\bf e}) = \\max\\left\\{\\frac{N_i}{a_{i,w}w^2 d},\\frac{en}{a_{i,w}}\\right\\}\\,, \n\\]\nand let $m_{i,w} = (\\log M_{i,w})\/M_{i,w}$. Note that we always have $M_{i,w} \\ge en\/a_{i,w} \\ge e$. Since the function $x \\log x^{-1}$ is increasing on $(0,e^{-1}]$ it follows that we may equivalently write \n\\[\nm_{i,w} = \t\\begin{cases}\n\t\t\t\\frac{a_{i,w} w^2 d}{N_i} \\log\\left(\\frac{N_i}{a_{i,w} w^2 d}\\right) & \\mbox{ if } \n\t\t\t\\frac{w^2 d}{N_i} \\le \\frac{1}{en} \\\\\n\t\t\t\\frac{a_{i,w}}{en} \\log\\left(\\frac{en}{a_{i,w}}\\right) & \\mbox{ if }\n\t\t\t\\frac{1}{en} \\le \\frac{w^2 d}{N_i} \n\t\t\t\\end{cases}\n\\]\nIn what follows we will use that $m_{i,w} \\le 1.18 M_{i,w}^{-2\/3}$ always holds. \nThis follows from the fact that $\\log x \\le 1.18 x^{1\/3}$ on $(0,\\infty)$.\n\n \nWe say that a set $U \\subset V(\\Gamma)$ satisfies (SD1), (SD2), and (SD3), respectively, if\n\\begin{itemize}\n\\item[(SD1)] $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge L a_{i,w}w^2 d^{1\/2}$\n\\item[(SD2)] $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge L N_i({\\bf a},{\\bf e})  a_{i,w}\/(nd^{1\/2})$, or \n\\item[(SD3)] $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge L N_i({\\bf a},{\\bf e}) m_{i,w}({\\bf a},{\\bf e})\/d^{1\/2} \\, $\n\\end{itemize}\nfor all $(i,w)\\in U$.\n\nWe remark that (SD1) is identical to (LD1), and asks that $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))$ is large relative to $a_{i,w}w^2$.\nIn (SD2), $a_{i,w}\/n$ is the proportion of the fibre $V_i$ consumed by a set of size $a_{i,w}$, \nand $N_i \/ d$ represents the `average opportunity' in the neighbourhood of $(i,w)$. \nNotice that unlike in the large deviations case, no factor of the form $(w'\/wd^{1\/2})$ appears. This corresponds to the intuition that in the small deviations case, large potency is most likely to come from large sets of roughly equal weight. \nFinally, (SD3), in which $m_{i,w}$ appears, is a slight strengthening of either (SD1) or (SD2) -- by a logarithmic factor -- depending on which value $m_{i,w}$ takes. \n\nLet $S=S(({\\bf a},{\\bf e}),L)$ be the maximal subset of $V(\\Gamma)$ such that every $(i,w) \\in S$ satisfies (SD1), (SD2), and (SD3). As in \\refS{sec:ldps}, writing $k = |V(\\Gamma)|-|S|$, \nwe may order the vertices of $V(\\Gamma) \\setminus S$ as $(i_1,w_1),\\ldots,(i_k,w_k)$ in such a way that \nfor each $1 \\leq j \\leq k$, letting $S_j = V(\\Gamma) \\setminus \\{(i_1,w_1),\\ldots,(i_{j-1},w_{j-1})\\}$, \nwe have \n\\[\np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))<L \\max\\left( a_{i,w}w^2 d^{1\/2}, N_{i}a_{i,w} \/(nd^{1\/2}),N_i m_{i,w}\/d^{1\/2})\\right)\\]\nIn other words, this ordering verifies that $(i_j,w_j) \\not \\in S$, for each $1 \\leq j \\leq k$. \n\\refP{prop:rbsd} is an immediate consequence of the following two lemmas. \n\\begin{lem}\\label{lem:potboundSD}\nLet $S=S(({\\bf a},{\\bf e}),L)$ be as defined above. Then we have $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_S) \\ge p_{\\mathrm{SD}}(({\\bf a},{\\bf e}))-55L\\sqrt{d}$. \n\\end{lem}\n\\begin{lem}\\label{lem:suffSD}\nGiven any subset $U$ of $V(\\Gamma)$, if $U$ satisfies (SD1), (SD2), and (SD3) then for all $(i,w) \\in U$, \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\frac{\\epsilon_{i,w,i',w'}^2}{15}  \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, .\n\\]\n\\end{lem}\n\\begin{proof}[Proof of \\refL{lem:potboundSD}]\nWrite \n\\begin{align*}\nW_1 & = \\{ (i_j,w_j): 1 \\le j \\le k, p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))< L a_{i,w}w^2 d^{1\/2}\\} \\\\\nW_2  & = \\{ (i_j,w_j), 1 \\le j \\le k, p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))< L N_{i}a_{i,w} \/(nd^{1\/2})\\} \\\\\nW_3  & = \\{ (i_j,w_j), 1 \\le j \\le k, p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))< L N_{i}m_{i,w} \/d^{1\/2}\\}\\, .\n\\end{align*}\nAlso, let $W_3^-= W_3 \\setminus W_1$. We note that for $(i,w) \\in W_3^{-}$ we have $m_{i,w} \\ge a_{i,w} w^2 d\/N_i$.\nSince $M_{i,w}\\ge e$ in all cases, we have that $m_{i,w}\\le e^{-1}$ and so\n\\begin{equation}\\label{eq:forclaim}\n\\frac{a_{i,w} w^2 d}{N_i}  \\le e^{-1},\n\\end{equation}\na bound we will use later.\n\n\nSince $W_1 \\cup W_2 \\cup W_{3}^{-} = \\{(i_1,w_1),\\dots ,(i_k,w_k)\\}$ \nand \n\\[\np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_S) \\ge p_{\\mathrm{SD}}(({\\bf a},{\\bf e})) - \\sum_{1 \\le j \\le k} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\\, , \n\\]\nit suffices to prove that \n\\[ \n\\sum_{(i_j,w_j) \\in W_1 \\cup W_2 \\cup W_{3}^{-}} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\\le 55Ld^{1\/2}\\, .\n\\]\nThe proof that \n\\begin{equation}\\label{eq:wone}\n\\sum_{(i,w) \\in W_1} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\\le 10 Ld^{1\/2}\\,\n\\end{equation}\nis identical to that given in the proof of \\refL{lem:potboundLD}.  \n\nFor $W_2$ we use the fact that for all $i' \\in V(H)$, \n\\[\n\\sum_{(i,w): i\\in N_H (i')} a_{i,w}\\le nd\\, ,\n\\]\nand proceed as follows.\n\\begin{align}\n\\sum_{(i_j,w_j) \\in W_2} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\n&\n\\le \n\\frac{L}{nd^{1\/2}} \\sum_{(i_j,w_j) \\in W_2} a_{i_j,w_j} \\sum_{(i',w'):i'\\in N_H (i_j)}w'^2 a_{i',w'}\n\\nonumber\\\\\n& \n\\le\n\\frac{L}{nd^{1\/2}} \\sum_{(i',w')\\in V(\\Gamma)} w'^2 a_{i',w'} \\sum_{(i,w): i\\in N_H (i')} a_{i,w}\n\\nonumber\\\\\n&\n=\n10 L d^{1\/2},\\label{eq:wtwo}\n\\end{align}\nthe last inequality holding since ${\\bf a}$ is a Z-type so $\\sum_{(i',w') \\in V(\\Gamma)} w'^2 a_{i',w'} \\le 10$. \n\nThe argument for $W_3^-$ is slightly more involved.  \nWe shall deduce our bound on $\\sum_{(i_j,w_j) \\in W_3^-} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))$ from the following claim.\n\n\\textbf{Claim:} For each $i\\in V(H)$ and each $j\\ge 1$ there are at most $(j-1)$ vertices $(i,w)\\in W_3^-$ for which $M_{i,w}\\le 2^j$.\n\n\\textbf{Proof.}\nFix $j \\ge 1$. First, by (\\ref{eq:forclaim}) we have that $w^2 \\le N_i\/(ea_{i,w}d)$. If \n$M_{i,w} \\le 2^j$ then by the definition of $M_{i,w}$ we also have \n$a_{i,w} \\ge en\/2^j$, and so \n\\[\nw^2 \\le \\frac{2^jN_i}{e^2 nd} < \\frac{2^{j-2}N_i}{nd}.\n\\]\nUsing the other term in the maximum which defines $M_{i,w}$, we see that if $M_{i,w} \\le 2^j$ then we also have \n\\[\nw^2 \\ge \\frac{N_i}{2^ja_{i,w}d} \\ge \\frac{N_i}{2^j n d}.\n\\] \nSince $w$ is a dyadic multiple of $(nd)^{-1\/2}$ and our upper and lower bounds for $w^2$ are a factor of less than $2^{2j-2}$ apart, the claim follows. \\qed\n\nBy the above claim and the fact, noted earlier, that $m_{i,w} \\le 1.18 M_{i,w}^{-2\/3}$ always holds, we have \n\\begin{equation}\\label{nine}\n\\sum_{w:(i,w)\\in W_3^-} m_{i,w} \\le \\sum_{w:(i,w)\\in W_3^-} 1.18 M_{i,w}^{-2\/3} \\le \\sum_{j \\ge 1} \\frac{(j-1)}{2^{2j\/3}}< 3.5.\n\\end{equation}\nWe are now ready to complete the proof of the lemma.  We have \n\\begin{align*}\n\\sum_{(i_j,w_j) \\in W_3^-} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_{S_j};(i_j,w_j))\n&\n\\le\n\\frac{L}{d^{1\/2}} \\sum_{(i_j,w_j) \\in W_3^-}  m_{i_j,w_j} \\sum_{(i',w'):i'\\in N_H (i_j)}w'^2 a_{i',w'}\\\\\n&\n\\le\n\\frac{L}{d^{1\/2}} \\sum_{(i',w')\\in V(\\Gamma)} w'^2 a_{i',w'} \\sum_{(i,w)\\in W_3^-: i\\in N_H (i')} m_{i,w}\\\\\n&\n\\le\n\\frac{L}{d^{1\/2}} \\sum_{(i',w')\\in V(\\Gamma)} w'^2 a_{i',w'} \\sum_{(i,w)\\in W_3^-: i\\in N_H (i')} 1.18M_{i,w}^{-2\/3}\\\\\n& \n< \n35Ld^{1\/2}\\, ,\n\\end{align*}\nthe first inequality following from the definition of $W_3$, and the final inequality holding from (\\ref{nine}) and since ${\\bf a}$ is a Z-type. Combining this bound with (\\ref{eq:wone}) and (\\ref{eq:wtwo}) completes the proof.\n\\end{proof}\n\nThe last step before proving \\refL{lem:suffSD} and thereby completing the proof of \\refP{prop:redbound}, is to establish an analogue of \\refL{lem:pointwise} for the small deviations case. In this case it turns out to be more straightforward to argue directly rather than to use \\refL{lem:measure}.\n\nGiven $U \\subset V(\\Gamma)$ and $(i,w) \\in V(\\Gamma)$, let \n\\[\nU_{\\mathrm{SD}}(i,w)=\\left\\{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w) \\cap U:\\frac{|\\epsilon_{i,w,i',w'}|}{ww'n}\\ge \\frac{p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))}{2w^2 a_{i,w} N_i}\\right\\}\\, .\n\\]\n\\begin{lem}\\label{easylemma} \nFor all $(i,w) \\in U$, we have \n\\[\n\\left|\\sum_{(i',w')\\in U_{\\mathrm{SD}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right| \\ge \\frac{1}{2} p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))\\, .\n\\]\n\\end{lem}\n\n\\begin{proof} Fix $(i,w) \\in U$. Since \n\\[\np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w)) \n= \\left|\\sum_{(i',w') \\in N_{\\Gamma_{\\mathrm{SD}}}(i,w) \\cap U}\n\\frac{ww'a_{i,w}a_{i',w'}}{n}\\epsilon_{i,w,i',w'}\\right|, \n\\]\nby the triangle inequality it suffices to show that \n\\[\n\\sum_{(i',w')\\in \\overline{U}_{\\mathrm{SD}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, |\\epsilon_{i,w,i',w'}|  \\le\n\\frac{p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))}{2}\\, ,\n\\]\nwhere $\\overline{U}_{\\mathrm{SD}}(i,w)$ denotes the set $(N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap U)\\setminus U_{\\mathrm{SD}}(i,w)$.\nWe observe that the above sum may be re-expressed as\n\\[\nw^2 a_{i,w} \\sum_{(i',w')\\in \\overline{U}_{\\mathrm{SD}}(i,w)} w'^2 a_{i',w'}\\, \\left(\\frac{|\\epsilon_{i,w,i',w'}|}{ww'n} \\right)\\, ,\n\\]\nand that the bound \n\\[\n\\frac{|\\epsilon_{i,w,i',w'}|}{ww'n}\\le \\frac{p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))}{2w^2 a_{i,w} N_i}\n\\]\nholds for all $(i',w')\\in \\overline{U}_{\\mathrm{SD}}(i,w)$.  The required bound now follows immediately by noting that\n\\[\n\\sum_{(i',w')\\in \\overline{U}_{\\mathrm{SD}}(i,w)} w'^2 a_{i',w'}\\le \\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)}w'^2 a_{i',w'}\\le N_i\\, ,\n\\]\nthe last inequality holding by the definition of $N_i$. \n\\end{proof}\n\\begin{proof}[Proof of \\refL{lem:suffSD}] \nFix $U \\subset V(\\Gamma)$ satisfying (SD1), (SD2), and (SD3), and fix $(i,w) \\in U$. To prove the lemma, we must show that \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\frac{\\epsilon_{i,w,i',w'}^2}{15}  \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right).\n\\]\nThe proof divides into two cases depending on which value $m_{i,w}$ takes.  Before considering these cases separately, we note the following inequality: for all $(i,w) \\in S$, \n\\begin{align*}\n& \\quad \\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} \\, \\cdot \\, \\frac{\\epsilon_{i,w,i',w'}^2}{15} \\\\\n= & \\quad \n\\frac{n}{15}\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)\\cap U} \\frac{ww'a_{i,w}a_{i',w'}|\\epsilon_{i,w,i',w'}|}{n} \\, \\cdot \\, \\left(\\frac{|\\epsilon_{i,w,i',w'}|}{ww'n}\\right)\\\\\n\\ge & \\quad \n\\frac{n}{15}\\sum_{(i',w')\\in U_{\\mathrm{SD}}(i,w)} \\frac{ww'a_{i,w}a_{i',w'}|\\epsilon_{i,w,i',w'}|}{n} \\, \\cdot \\, \\left(\\frac{|\\epsilon_{i,w,i',w'}|}{ww'n}\\right)\\\\ \n\\ge & \\quad \n\\frac{n}{15} \\frac{p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))}{2w^2 a_{i,w} N_i}\\, \\cdot \n\\sum_{(i',w')\\in U_{\\mathrm{SD}}(i,w)} \\frac{ww'a_{i,w}a_{i',w'}|\\epsilon_{i,w,i',w'}|}{n}\\\\\n\\ge & \\quad \n\\frac{n\\, p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))^2}{60 w^2 a_{i,w} N_i}\\, .\n\\end{align*}\nthe second inequality holding by the definition of $U_{\\mathrm{SD}}(i,w)$ and the third holding by \\refL{easylemma}. \nIt therefore suffices to prove that for all $(i,w) \\in S$, \n\\begin{equation}\\label{STP}  \n\\frac{n\\, p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))^2}{60 w^2 a_{i,w} N_i} \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, .\n\\end{equation}\n\nFirst, if $w^2 d\/N_i \\le 1\/en$, then $m_{i,w}=\\frac{w^2 a_{i,w} d}{N_i}\\log\\left(\\frac{N_i}{a_{i,w} w^2 d}\\right)$. In this case, by (SD3), we have that\n\\[  p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))\\ge L w^2 a_{i,w} d^{1\/2} \\log\\left(\\frac{N_i}{w^2 a_{i,w} d}\\right)\\ge L w^2 a_{i,w} d^{1\/2} \\log\\left(\\frac{en}{a_{i,w}}\\right)\\, ,\n\\] \nwhere the final inequality follows since $a_{i,w} w^2 d\/N_i \\le a_{i,w}\/en$.  Combining this bound with (SD2) yields that \n\\[  p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))^2  \n\\ge \\frac{L^2 w^2 a_{i,w}^2N_i}{n} \\log\\left(\\frac{en}{a_{i,w}}\\right)\\, ,\n\\] \nwhich verifies \\eqref{STP} in this case, since $L \\ge 20 > 6$.\n\nIn the remaining case we have $1\/en \\le w^2 d\/N_i$, from which it follows that $m_{i,w}=a_{i,w}\\log(en\/a_{i,w})\/en$. So, by (SD3), we have that\n\\[  \np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))\\ge \\frac{La_{i,w}N_i}{end^{1\/2}}\\log(en\/a_{i,w})\\, .\n\\]\nCombining this bound with (SD1), \nwe obtain that \n\\[\np_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))^2 \\ge \\frac{L^2a_{i,w}^2 w^2 N_i}{en}\\log(en\/a_{i,w}),\n\\]\nwhich verifies \\eqref{STP} in this case since $L\\ge 20\\ge 6e$.This completes the proof. \n\\end{proof}\n\n\n\\section{Proof of \\refT{explain}}\\label{sec:explain}\n\nOur proof of \\refT{explain} is similar in nature to our proof of \\refT{main}.  As in that proof we shall use \\refP{findwit}, which tells us that if $\\lambda^*(G)$ is large then $G$ contains a pattern $({\\bf a},{\\bf e})$ of large potency.  We require two other results.  The first, which is easily proved, is an approximate converse to \\refP{findwit}.  The second, whose proof is more involved, is a variant of \\refT{redbound}.\n\nGiven a pattern $({\\bf a},{\\bf e})$ and a witness $\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}$ for the pattern, write $\\alpha=\\alpha({\\bf a},{\\bf e})=\\sum_{(i,w) \\in V(\\Gamma)} a_{i,w}$, and write $G[\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}]$ for the subgraph of $G$ induced by $\\bigcup_{(i,w) \\in V(\\Gamma)} A_{i,w}$. \n\\begin{prop}\\label{witfind}\nIf $G$ contains a pattern $({\\bf a},{\\bf e})$ with $p({\\bf a},{\\bf e})=p\\sqrt{d}$ then $\\lambda^*(G) \\geq (2p-40)\\sqrt{d}$. \nFurthermore, $\\lambda(G[\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}]) \\geq (2p-40)\\sqrt{d} - \\frac{\\alpha\\sqrt{10}}{n}$. \n\\end{prop}\n\\begin{proof}\nLet $\\{A_{i,w}: (i,w) \\in V(\\Gamma)\\}$ be a witness for $({\\bf a},{\\bf e})$, \nand let $y \\in \\mathbb{R}^{nh}$ be defined by, for each \n$(i,w) \\in V(\\Gamma)$, setting $y(v)=w$ for all $v \\in A_{i,w}$ (and setting $y(v)=0$ for all remaining $v$). \nSince \n\\[\n\\|y\\|_2^2 = \\sum_{(i,w) \\in V(\\Gamma)} w^2 a_{i,w} \\le 10, \n\\]\n\nby \\refL{ipoe} we have \n\\[\n|\\ipo{y}_N| = |\\ipo{y}_{N,E^*} + \\ipo{y}_{N,\\mathbb{R}^2\\setminus E^*}| \\geq |\\ipo{y}_{N,\\mathbb{R}^2\\setminus E^*}| - 40\\sqrt{d}.\n\\]\nBut $2p({\\bf a},{\\bf e}) = |\\ipo{y}_{N,\\mathbb{R}^2\\setminus E^*}|$, so $|\\ipo{y}_N| \\geq (2p-40)\\sqrt{d}$, and the first result follows from \\refF{uncount}. \n\nFor the second bound, write $G'=G[\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}]$, and write $M'$ for the adjacency matrix of $G'$. \nAlso, let $y' \\in \\mathbb{R}^{a}$ be the vector obtained from $y$ by retaining only coordinates corresponding to vertices of $G'$ \n(recall that all other coordinates of $y$ are equal to zero). Then \n\\[\n|\\ipo{y}_N| = |\\ipo{y}_M- \\ipo{y}_{\\overline{M}}| = |\\ipo{y'}_{M'}- \\ipo{y}_{\\overline{M}}|,\n\\]\nand since all entries of $\\overline{M}$ are either zero or $1\/n$, \n\\[\n\\ipo{y}_{\\overline{M}} \\leq \\frac{1}{n} (\\sum_{v \\in [nh]} y_v)^2 \\leq \\frac{\\alpha\\sqrt{10}}{n}, \n\\]\nwhere in the last inequality we used that, given the constraints that $y$ has $\\alpha$ non-zero entries and $\\|y\\|_2^2 \\leq 10$, the sum $(\\sum_{v \\in [nh]} y_v)^2$ is maximized by taking all nonzero entries equal to $10^{1\/2}\\alpha^{-1\/2}$. \nThus \n\\[\n\\frac{|\\ipo{y'}_{M'}|}{\\|y'\\|_2^2} \\geq \\ipo{y'}_{M'} \\ge |\\ipo{y}_N|-|\\ipo{y'}_M| \\geq (2p-40)\\sqrt{d} - \\frac{\\alpha\\sqrt{10}}{n}.\\qedhere\n\\]\n\\end{proof}\n\n\\begin{thm}\\label{redboundtwo}\nFix $L \\ge 20$. \nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold.\n\\begin{align*}\np(({\\bf a},{\\bf e})_{S}) \t& \\geq p({\\bf a},{\\bf e})- 150 L\\sqrt{d}\\\\\n\\p{({\\bf a},{\\bf e})_S~\\mbox{can be found in G}} & \\ll \\prod_{(i,w) \\in S} a_{i,w}^{d\/4}\\pran{\\prod_{(i,w) \\in S} {n \\choose a_{i,w}\\wedge \\lfloor n\/2 \\rfloor}}^{1-L\/10}. \n\\end{align*}\n\\end{thm}\n\nIn the same way as \\refT{redbound} was deduced from \\refP{prop:redbound} (together with \\refC{cor:bb}) \\refT{redboundtwo} may be deduced from the following proposition (\\refP{prop:redboundtwo}).  We recall from \\refS{sec:key} the definition of $b:(-1,\\infty) \\to [0,\\infty)$,\n\\[ \nb(\\epsilon)= (1+\\epsilon)\\log(1+\\epsilon) - \\epsilon \\qquad \\epsilon\\in (-1,\\infty).\n\\]\n\n\\begin{prop}\\label{prop:redboundtwo} Fix $L \\ge 20$. \nFor any pattern $({\\bf a},{\\bf e})$, there exists $S=S(({\\bf a},{\\bf e}),L) \\subset V(\\Gamma)$ such that the following properties hold.\n\\begin{align*}\np(({\\bf a},{\\bf e})_{S})\t& \\geq p({\\bf a},{\\bf e}) - 150 L\\sqrt{d}\\qquad \\text{and}\\\\\n\\sum_{(i',w')\\in N_{\\Gamma}(i,w)\\cap S} \\frac{a_{i,w}a_{i',w'}}{n} b(\\epsilon_{i,w,i',w'})& \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right) \\qquad \\text{for all} \\, (i,w)\\in S\\, .\\end{align*}\n\\end{prop}\n\\refT{redboundtwo} follows from \\refP{prop:redboundtwo} in exactly the same way that \\refT{redbound} follows from \\refP{prop:redbound}.  Rather than repeat this proof we refer the reader to the proof of \\refT{redbound} given in \\refS{sec:key}.\n\nWe recall that our proof of \\refP{prop:redbound} divided into two cases depending on whether $p_{\\mathrm{LD}}({\\bf a},{\\bf e}) \\ge p({\\bf a},{\\bf e})\/2$  or $p_{\\mathrm{SD}}({\\bf a},{\\bf e}) \\ge p({\\bf a},{\\bf e})\/2$.  These two cases were dealt with separately by \\refP{prop:rbld} and \\refP{prop:rbsd} respectively.  However, in \\refP{prop:redboundtwo} we seek a lower bound on $p(({\\bf a},{\\bf e})_S)$ rather than on $\\tilde{p}(({\\bf a},{\\bf e})_S)$, and for this reason we can not treat the large and small deviations cases separately. In other words, we must work with the graph $\\Gamma$ rather than exclusively focussing on one of the subgraphs $\\Gamma_{\\mathrm{LD}},\\Gamma_{\\mathrm{SD}}$.  That having been said, our proof of \\refP{prop:redboundtwo} resembles the proofs of Propositions~\\ref{prop:rbld} and \\ref{prop:rbsd} in almost all other respects.\n\nWe begin by establishing the following sufficient condition for the second bound of \\refP{prop:redboundtwo} to hold.  For the remainder of the section fix a pattern $({\\bf a},{\\bf e})$ and a constant $L \\ge 20$. In what follows we denote by\n\\[\np(({\\bf a},{\\bf e});(i,w)) = \\left| \\sum_{(i',w')\\in N_{\\Gamma}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right|\n\\]\nthe `local' potency at $(i,w)$, and we recall the large and small deviations variants defined in \\refS{sec:key}:\n\\[\np_{\\mathrm{LD}}(({\\bf a},{\\bf e});(i,w)) = \\left|\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right|\n\\]\nand  \n\\[\np_{\\mathrm{SD}}(({\\bf a},{\\bf e});(i,w)) = \\left|\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{SD}}}(i,w)} \\frac{w w' a_{i,w} a_{i',w'}}{n} \\, \\epsilon_{i,w,i',w'}\\right|\\, .\n\\]\nWe say that a set $U \\subset V(\\Gamma)$ satisfies (G1), (G2), (G3), and (G4), respectively, if \n\\begin{itemize}\n\\item[(G1)] $p(({\\bf a},{\\bf e})_U;(i,w)) \\ge 2L a_{i,w}w^2 d^{1\/2}$,\n\\item[(G2)] $p(({\\bf a},{\\bf e})_U;(i,w)) \\ge 2L \\widehat{N}_{i,w} \/d^{1\/2}$, \n\\item[(G3)] $p(({\\bf a},{\\bf e})_U;(i,w)) \\ge 2L N_i a_{i,w}\/(nd^{1\/2})$, or \n\\item[(G4)] $p(({\\bf a},{\\bf e})_U;(i,w)) \\ge 2L N_i m_{i,w}\/d^{1\/2} \\, .$\n\\end{itemize}\nfor all $(i,w)\\in U$, where $N_{i}=N_{i}({\\bf a},{\\bf e})$, $\\widehat{N}_{i,w}=\\widehat{N}_{i,w}({\\bf a},{\\bf e})$ and $m_{i,w}=m_{i,w}({\\bf a},{\\bf e})$ are as defined in \\refS{sec:key}.\n\n\\begin{lem}\\label{lem:suff} If $U$ satisfies (G1),(G2),(G3), and (G4) then\n\\[\n\\sum_{(i',w')\\in N_{\\Gamma}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n} b(\\epsilon_{i,w,i',w'}) \\ge  \\frac{L}{10} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right) \\, .\n\\]\n\\end{lem}\n\n\\begin{proof} Since $E(\\Gamma)=E(\\Gamma_{\\mathrm{LD}})\\cup E(\\Gamma_{\\mathrm{SD}})$ it follows from the triangle inequality that $p(({\\bf a},{\\bf e})_S;(i,w))\\le p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w)) + p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))$, and so\n\\[\n\\max\\{p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w)),p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w))\\}\\ge \\frac{p(({\\bf a},{\\bf e})_U;(i,w))}{2}\\, .\n\\]\nFirst suppose that $p_{\\mathrm{LD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge p(({\\bf a},{\\bf e})_U;(i,w))\/2$. \nIn this case, (G1) and (G2) imply that $U$ satisfies conditions (LD1) and (LD2), and so, by an application of \\refL{lem:suffLD} we then have \n\\[\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n}\n\\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\ge  \\frac{L}{4} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, .\n\\]\nThe lemma now follows since (by \\eqref{eq:b}) $b(\\epsilon_{i,w,i',w'}) \\ge (1+\\epsilon_{i,w,i',w'}\/2)\\log(1+\\epsilon_{i,w,i',w'})$.\n\n\nBy \\refL{lem:suffLD} we then have\n\\[\n\\sum_{(i',w')\\in N_{\\Gamma_{\\mathrm{LD}}}(i,w)\\cap U} \\frac{a_{i,w}a_{i',w'}}{n}\n\\, \\cdot \\, \\pran{1+\\frac{\\epsilon_{i,w,i',w'}}{2}} \\log(1+\\epsilon_{i,w,i',w'}) \\ge  \\frac{L}{2} a_{i,w}\\log\\left(\\frac{en}{a_{i,w}}\\right)\\, ,\n\\]\n(note that the $L$ of conditions (LD1) and (LD2) became $2L$ in conditions (G1) and (G2)), which proves the result in this case.\n\nOtherwise, we must have that $p_{\\mathrm{SD}}(({\\bf a},{\\bf e})_U;(i,w)) \\ge p(({\\bf a},{\\bf e})_U;(i,w))\/2$ and in this case the result follows similarly from \\eqref{eq:b} and \\refL{lem:suffSD}.\n\\end{proof}\n\nNow let $S\\subset V(\\Gamma)$ be the maximal subset of $V(\\Gamma)$ satisfying all of (G1),(G2),(G3), and (G4). The proof of \\refP{prop:redboundtwo} is then completed by the following lemma.\n\n\\begin{lem}\\label{lem:Stwo} Let $S$ be the subset defined above.  Then\n\\[\np(({\\bf a},{\\bf e})_S)\\ge p({\\bf a},{\\bf e}) - 150L\\sqrt{d}\\, .\n\\]\n\\end{lem}\n\n\\begin{proof} This is proved exactly as Lemmas~\\ref{lem:potboundLD} and~\\ref{lem:potboundSD} were proved; we skip the details. \\qedhere \\end{proof}\n\nWe now have all the necessary preliminaries in place and we may turn to our proof of \\refT{explain}.\n\n\\begin{proof}[Proof of \\refT{explain}] We begin by noting that the star graph $F$ consisting of a single vertex of degree $d$ attached to $d$ vertices of degree one is always a subgraph of $G$ and has $\\lambda(F)=\\sqrt{d}$, which proves the result when $\\lambda^*(G) \\leq 1189248 \\sqrt{d}$. \n\nWe may now suppose that $\\lambda^*(G) = M\\sqrt{d}$ for some $M \\geq 1189248$.  It follows by \\refP{findwit} that $G$ contains a pattern $({\\bf a},{\\bf e})$ of potency at least $K\\sqrt{d}$, where $K=K(M)=(M\/192) -3\\ge 6191$. By \\refT{redboundtwo}, applied with $L=L(M)=K\/151\\ge 41$, we have that $G$ contains a reduced pattern $({\\bf a},{\\bf e})$ with $p({\\bf a},{\\bf e}) \\ge L\\sqrt{d}$ for which\n\\begin{equation}\\label{eq:probbound}\n\\p{({\\bf a},{\\bf e})~\\mbox{can be found in G}}  \\ll \\prod_{(i,w) \\in S} a_{i,w}^{d\/4}\\pran{\\prod_{(i,w) \\in V(\\Gamma)} {n \\choose \\alpha \\wedge \\lfloor n\/2 \\rfloor}}^{-1}. \n\\end{equation}\nwhere $\\alpha=\\sum_{(i,w) \\in V(\\Gamma)} a_{i,w}$.\n\nLet $\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}$ be any witness for $({\\bf a},{\\bf e})_S$, and write \n$G'=G[\\{A_{i,w}\\}_{(i,w) \\in V(\\Gamma)}]$. By \\refP{witfind} we have $\\lambda(G') \\geq (2L-40)\\sqrt{d}-\\frac{\\alpha\\sqrt{10}}{n}$, \nso for $n \\ge \\sqrt{10}hd$, if it happens that $\\alpha \\leq hd$ then $\\lambda(G') \\geq L\\sqrt{d} \\ge \\lambda^{*}(G)\/1189248$, as required. \n\nTo complete the proof we must bound by $n^{-hd}$ the probability that any reduced pattern $({\\bf a},{\\bf e})$ of potency at least $L\\sqrt{d}$ with $\\alpha>hd$ may be found in $G$.  \nWe remind the reader that for reduced patterns $({\\bf a},{\\bf e})$ we have the inequality (\\ref{eq:probbound}). \nWe split the proof of the required bound into two steps. First, recall the event $E_1$ from the proof of \\refT{main}, which was the event \nthat any reduced pattern $({\\bf a},{\\bf e})$ with $p({\\bf a},{\\bf e}) \\ge L\\sqrt{d}$ and for which $a_{i,w}\\ge 4hd\\log_2 d$ for some $(i,w) \\in V(\\Gamma)$, can be found in $G$. In proving \\refT{main} we showed that $\\p{E_1}\\le n^{-hd}\/2$\n\nSecond, let $E_3=E_3(M)$ be the event that \n$G$ contains a reduced pattern $({\\bf a},{\\bf e})$ with $p({\\bf a},{\\bf e}) \\ge L\\sqrt{d}$, with $\\alpha=\\sum_{(i,w) \\in S} a_{i,w} > hd$ and with $a_{i,w} \\leq 4hd\\log_2 d$ for all $(i,w) \\in V(\\Gamma)$. We complete the proof by proving that $\\p{E_3} \\leq n^{-hd}\/2$ for all $n$ sufficiently large. \nAs in the proof of \\refT{main}, by a union bound we obtain that\n\\[\n\\p{E_3(M)} \\leq \\log_2(nh) (4hd\\log_2 d)^{3hd \\log_2 d}  {n \\choose \\alpha \\wedge \\lfloor n\/2 \\rfloor}^{-1}.\n\\]\nSince $hd+1 \\leq \\alpha= \\sum_{(i,w) \\in S} a_{i,w}< (4hd \\log_2 d)(h \\log_2 d)$, the following bound holds for all $n \\geq 2\\alpha$:\n\\[\n\\p{E_3(M)} \\leq \\log_2(nh) (4hd\\log_2 d)^{3hd \\log_2 d} \\frac{(2(hd+1))^{hd+1}}{n^{hd+1}},\n\\]\nwhich is less than $n^{-hd}\/2$ for $n$ large enough. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n \n\\section{The remaining proofs}\\label{sec:zbound}\nBefore proving \\refP{zbound} we establish the following, intermediate result, \nwhich contains our key convexity argument. For this step, it is notationally convenient to consider vectors $x \\in \\mathbb{R}^{V(G)}$ with $\\|x\\|_2^2 = O(nh)$ rather than $O(1)$. \nTo this end, \ndefine \n\\begin{align*}\nD & = \\{0 \\} \\cup \\{2^i: i \\in {\\mathbb N}\\} \\cup \\{-2^i: i \\in {\\mathbb N}\\} \\\\\n\\hat{D} & = \\{0 \\} \\cup \\{2^i: i \\in {\\mathbb N}\\},\n\\end{align*}\nand let \n\\begin{align*}\nX^* & = \\{x \\in \\mathbb{R}^{V(G)}:\\|x\\|_2^2 \\le nh\\}\\, , \\\\\nY  & = \\{y \\in \\mathbb{R}^{V(G)}:\\|y\\|_2^2 \\leq 5nh,\\forall~v \\in V(G),y_v \\in D\\}\\, , \\nonumber\\\\\nY^+& = \\{y \\in \\mathbb{R}^{V(G)}:\\|y\\|_2^2 \\leq 10nh,\\forall~v \\in V(G),y_v \\in \\hat{D}\\}\\, . \n\\nonumber\n\\end{align*}\nNote that by Fact~\\ref{uncount}, $nh\\lambda^*(G) = \\sup_{x \\in X^*} |\\ipo{x}_N|$. \n\\begin{prop}\\label{dyprop}\n$nh \\lambda^*(G) \\leq 12 \\sup_{y \\in Y^+} |\\ipo{y}_N|.$\n\\end{prop}\n\\begin{proof}[Proof of \\refP{dyprop}] \nThe proof is rather straightforward; it consists of first an averaging argument, and second a polarization argument. \nThe polarization argument becomes a little delicate only because we are trying to maintain the property that the \nentries of all vectors remain in $D$. \n\nFor $r \\in \\mathbb{R}$ let $\\sigma(r)=r\/|r|$ if $r \\neq 0$, and $\\sigma(r)=0$ if $r=0$. We call $\\sigma(r)$ the sign of $r$. \nAlso, let $\\ell(r)$ be the greatest element of $D$ which is less than or equal to $r$, and let $u(r)$ be the least element of \n$D$ which is greater than or equal to $r$. \n\nFor $x \\in \\mathbb{R}^{V(G)}$, write \n\\[S_x = \\{ y \\in \\mathbb{R}^{V(G)}: \\forall v \\in V(G), \\sigma(y_v)=\\sigma(x), y_v \\in \\{\\ell(x_v),u(x_v)\\}\\}.\n\\] \nNow fix any $x \\in X^*$ and let $\\by \\in Y$ be randomly chosen as follows. Independently for each $v \\in V(G)$: \n\\begin{itemize}\n\\item if $0 \\leq |x_v| < 1$ let $\\by_i=\\sigma(x_i)$ with probability $|x_i|$ and $\\by_i=0$ otherwise.\n\\item if $2^j \\leq |x_v| < 2^{j+1}$ for some $j \\geq 1$, then let $\\by_v = \\sigma_v 2^{j+1}$ with probability $(|x_v|-2^j)\/2^{j+1}$ and \nlet $\\by_v = \\sigma_v 2^j$, otherwise. \n\\end{itemize}\nBy definition, all entries of $\\by$ have values in $\\{0,\\pm 1,\\pm 2,\\pm 4,\\ldots\\}$. Furthermore, \nfor each $v$ with $|x_v| \\geq 1$ we have $\\by_v^2 \\leq (2x_v)^2$, and for $i$ with $|x_v| < 1$ we have $\\by_v^2 \\leq 1$. \nIt follows that \n\\[\n\\sum_{v \\in V(G)} \\by_v^2 \\leq \\sum_{v \\in V(G)} (2x_v)^2+1 = 5nh,\n\\]\nso $\\by$ is a random vector in $Y \\cap S_x$. Since, for each $v \\in V(G)$, $\\E{\\by_v}=x_v$ and the coordinates of $\\by$ \nare chosen independently, we then have\n\\[\nx = \\sum_{y \\in Y \\cap S_x} y \\p{\\by=y},\n\\]\nso \n\\[\n\\ipo{x}_N = \\sum_{y,z \\in Y\\cap S_x} \\ip{y}{z}_N \\p{\\by=y}\\p{\\by=z}.\n\\]\nIn other words, $\\ipo{x}_N$ is a convex combination of elements of $\\{\\ip{y}{z}_N:y,z \\in Y\\}$. \nChoosing $x \\in X$ such that $|\\ipo{x}_N|=nh\\lambda^*(G)$, we then have\n\\begin{equation}\\label{dyprop1}\nnh \\lambda^*(G) = |\\ipo{x}{x}_N| \\leq \\sup_{y,z \\in Y \\cap S_x} |\\ip{y}{z}_N|.\n\\end{equation}\nNote that for all $y,z \\in S_x$ and all $v \\in V(G)$, \neither $y_v=z_v$ or else $\\{y_v,z_v\\}=\\{\\ell(x_v),u(x_v)\\}$. In particular, $y_v$ and $z_v$ are either both non-negative \nor both non-positive. \n\nChoose $y,z \\in Y \\cap S_x$ for which the supremum in \\refeq{dyprop1} is achieved, and write \n$y=y_+-y_-$, $z=z_+-z_-$, where e.g.~$y_+$ is the vector obtained from $y$ by replacing \nall negative entries of $y$ by zeros. \nThen for all $v \\in V(G)$, $\\sigma(y^+_v)=\\sigma(z^+_v)$ and $\\sigma(y^-_v)=\\sigma(z^-_v)$ \nWe then have \n\\[\nnh\\lambda^*(G) \\leq |\\ip{y}{z}_N| = |\\ip{y_+}{z_+}_N-\\ip{y_+}{z_-}_N - \\ip{y_-}{z_+}_N +\\ip{y_-}{z_-}_N|,\n\\]\nso either one of $|\\ip{y_+}{z_-}_N|$,$|\\ip{y_-}{z_+}_N|$ is at least $nh\\lambda^*(G)\/6$ or else \none of $|\\ip{y_+}{z_+}_N|$, $|\\ip{y_-}{z_-}_N|$ is at least $nh\\lambda^*(G)\/3$. \n\nFirst suppose $|\\ip{y_+}{z_-}_N| \\geq nh\\lambda^*(G)\/6$. Since \n\\begin{equation}\\label{polarize}\n\\ipo{y_++z_-}_N = 2 \\ip{y_+}{z_-}_N + \\ipo{y_+}_N + \\ipo{z_-}_N,\n\\end{equation}\nit follows that either $\\ipo{y_++z_-}_N| \\geq nh\\lambda^*(G)\/9$ or else \none of $|\\ipo{y_+}_N|$ or $|\\ipo{z_-}_N|$ is at least $nh \\lambda^*(G)\/9$. \nAlso, since $y,z \\in S_x$, the non-zero coordinates of $y_+$ correspond to zeros in $z_-$, we have $y_++z_- \\in Y^+$. \nSince $\\|y_+\\|_2^2,\\|z_-\\|_2^2$, and $\\|y_++z_-\\|_2^2$ are all at most $\\|y\\|_2^2+\\|z\\|_2^2 \\leq 10nh$, \nin this case we have proved the proposition. This argument \nalso handles the case $|\\ip{y_-^}{z_+}_N| \\geq nh\\lambda^*(G)\/6$, by symmetry. \n\nThe other case is that $|\\ip{y_+}{z_+}_N| \\geq nh\\lambda^*(G)\/3$ \n(the proof in the case that $|\\ip{y_-}{z_-}_N| \\geq nh\\lambda^*(G)\/3$ is symmetric). \nSince \n\\[\n\\ipo{y_+-z_+}_N= \\ipo{y_+}_N + \\ipo{z_+}_N -2\\ip{y_+}{z_+}_N,\n\\]\nit follows that either one of $\\ipo{y_+}_N, \\ipo{z_+}_N$ is at least $nh\\lambda^*(G)\/12$, \nor else \n\\[\n\\ipo{y_+-z_+}_N \\geq nh\\lambda^*(G)\/2.\n\\] \nIn the former case we have proved the proposition. In the latter case, note that \nsince for all $v \\in V(G)$, either $y_v = z_v$ or else $\\{y_v,z_v\\}=\\{\\ell(x_v),u(x_v)\\}$, \nwe have that all all entries of $y_+-z_+$ are in $D$, and \nfurthermore, for each $v$, either $|y_{v,+}-z_{v,+}| = 0$ \nor $|y_{v,+}-z_{v,+}|=1$ or $|y_{v,+}-z_{v,+}| = |y_{v,+}| \\wedge |z_{v,+}|$. It follows that $\\|y_+-z_+\\|^2 \\leq 6nh$. \nNow write $w=y_+-z_+$, and then separate $w$ into its positive and negative \nparts: $w=w_+-w_-$. \nWe then have \n\\[\nnh\\lambda^*(G)\/2 \\leq |\\ipo{w}_N| = | \\ipo{w_+}_N- 2\\ip{w_+}{w_-}_N + \\ipo{w_-}_N|. \n\\]\nBut we also have \n\\[\n\\ipo{w_++w_-}_N= 2\\ip{w_+}{w_-}_N + \\ipo{w_+}_N+ \\ipo{w_-}_N,\n\\]\nand it follows that either one of $|\\ipo{w_+}_N|$ or $|\\ipo{w_-}_N|$ is at least $nh\\lambda^*(G)\/10$, \nor else $|\\ipo{w_++w_-}_N| \\geq nh\\lambda^*(G)\/10$. \nSince all of $\\|w_+\\|_2^2,\\|w_-\\|_2^2$ and $\\|w_++w_-\\|_2^2$ are at most $\\|w\\|_2^2 \\leq 6nh$, \nthey are all in $Y^+$ and so the proof is complete. \n\\end{proof}\n\n\\begin{proof}[Proof of \\refL{ipoe}]\nFor this proof write $N=(n_{vw})_{v,w \\in V(G)}$. \nFor a given vertex $v \\in V(G)$, there are precisely $d$ vertices $w \\in V(G)$ \nwith $n_{vw}=1-1\/n$, and $(n-1)d$ vertices $w \\in V(G)$ with \n$n_{vw}=-1\/n$ (the remaining entries in row $v$ of $N$ are all zero). Thus, for any $v \\in V(G)$, \n\\begin{align*}\n|y_v \\sum_{w \\in V(G)} n_{vw} y_w \\I{y_v\\geq \\sqrt{d}y_w}| & \\leq d(1-1\/n) \\cdot \\frac{y_v^2}{\\sqrt{d}}+ (n-1)d(1\/n) \\frac{y_v^2}{\\sqrt{d}} \\\\\n\t& < 2 y_v^2 \\sqrt{d}.\n\\end{align*}\nWriting $A= \\{(x,y) \\in \\mathbb{R}^2:x \\geq y\\sqrt{d}\\}$, it follows that\n\\begin{align*}\n|\\ipo{y}_{N,\\mathbb{R}^2 \\setminus E^*}| = 2|\\ipo{y}_{N,A}|\n \\leq 2 \\sum_{v \\in V(G)} 2y_v^2 \\sqrt{d} \n\t\t= 4\\sqrt{d} \\|y\\|_2^2.\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{zbound}]\nChoose $z \\in Y^+$ for which $|\\ipo{z}_N| = |\\sup_{y \\in Y^+} \\ipo{y}_N| \\geq nh\\lambda^*(G)\/12$, possible by Proposition \\ref{dyprop}.\nSince $\\|z\\|_2^2 \\le 10nh$, \nby \\refL{ipoe} we have \n\\[\n|\\ipo{z}_{N,E^*}| \\geq \\ipo{z}_N - 40nh\\sqrt{d}. \n\\]\nNext, for each $m \\in {\\mathbb N}_0$ let $I_m = E^* \\cap [d^{m\/2},d^{(m+2)\/2})\\times[d^{m\/2},d^{(m+2)\/2}) \\subset \\mathbb{R}^2$, \nand note that $E^* = \\bigcup_{m \\in {\\mathbb N}} I_m$, so by the triangle inequality \n\\[\n|\\ipo{z}_{N,E^*}| \\leq \\sum_{m =0}^{\\infty} |\\ipo{z}_{N,I_m}|. \n\\]\nSet \n\n\\[\np_m = \\sum_{\\{v \\in V(G): z_v \\in [d^{m\/2},d^{(m+2)\/2})\\}} \\frac{z_v^2}{\\|z\\|_2^2}\\, , \n\\quad \\alpha_m = \\frac{|\\ipo{z}_{N,I_m}|}{(|\\ipo{z}_{N,E^*}|p_m)}\\, ,\n\\]\nso that \n\\[\n|\\ipo{z}_{N,I_m}| = \\alpha_m p_m |\\ipo{z}_{N,E^*}|. \n\\]\nSince no point in $\\mathbb{R}^2$ lies in more than two of the $I_m$, we have \n$\\sum_{m\\in {\\mathbb N}} p_m \\leq 2$, and so there must be $m^* \\in {\\mathbb N}$ for which $\\alpha_{m^*} \\geq 1\/2$. \nLetting $z^*$ the the vector whose entry in position $v$ is $z_v\\I{z_v \\in I_{m^*}}$, \nwe then have that \n\\[\n|\\ipo{z^*}_{N,E}| = |\\ipo{z}_{N,I_{m^*}}| \\geq p_{m^*} |\\ipo{z}_{N,E^*}|\/2, \n\\]\nand furthermore, $\\|z^*\\|_2^2 = p_m^* \\|z\\|_2^2$. \nLet $y$ be the vector obtained from $z^*$ by multiplying all entries of $z^*$. \nBy $2^{\\lfloor -(\\log_2 p_m)\/2 \\rfloor}$, we have \n\\[\n|\\ipo{y}_{N,E}| \\geq \\frac{|\\ipo{z}_{N,E^*}|}{8} \\geq \\frac{|\\ipo{z}_N|}{8} - 5nh\\sqrt{d} \\geq \\frac{nh\\lambda^*(G)}{96}-5nh\\sqrt{d}, \n\\]\nand $\\|y\\|_2^2 \\leq \\|z\\|_2^2 \\leq 10nh$.  \nFinally, recall the definition of $Z$ from (\\ref{zdef}). Letting $x$ be the vector obtained \nfrom $y$ by dividing all entries by $(nh)^{1\/2}$, we obtain a vector $x \\in Z$ with \n$|\\ipo{x}_{N,E}| \\ge \\lambda^*(G)\/96- 5\\sqrt{d}$, which completes the proof. \n\\end{proof}\n\nWe now give the promised proofs of \\refP{protest} and Lemma~\\ref{patcount}.\n\n\\begin{proof}[Proof of \\refP{protest}]\nLet $a \\leq n-hn^{1\/2}$ be the number of vertices in $G'$, and let $x' \\in \\mathbb{R}^{V(G')}$ be an eigenvector of $G'$ with eigenvalue $\\lambda$, chosen so that $\\|x'\\|_2^2 = 1$. For each $i \\in [h]$, let $t_i = \\sum_{v \\in V_i \\cap V(G')} x_v$. \nWe define a vector $y \\in \\mathbb{R}^{V(G)}$, for all $i \\in [h]$ and each $v \\in V_i \\setminus V(G')$, setting $y_v = - t_i\/|V_i \\setminus V(G')|$, and taking all other $y_v$ equal to zero. Taking $x=x'+y$, it is immediate that $x$ is balanced. \nAlso, for $v \\in V(G)\\setminus V(G')$, we have $|x_v|=|y_v| \\leq 1\/(n-a)$, and it follows that $\\|x\\|_2^2 \\leq 1+nh\/(n-a)^2 \\leq 1+1\/\\lambda$, since $(n-a) \\geq hn^{1\/2} > (nd^2)^{1\/2} \\geq (nd\\lambda)^{1\/2}$. Now, \n\\begin{align*}\n\\ipo{x}_N \t& = \\ipo{x}_M \\\\\n\t\t\t& = \\ipo{x'}_M + \\ipo{y}_M + 2\\ip{x'}{y}_M \\\\\n\t\t\t& = \\lambda + \\ipo{y}_M + 2\\ip{x'}{y}_M.\n\\end{align*}\nSince all entries of $y$ have modulus at most $(n-a)^{-1}$, it follows that $|\\ipo{y}_M| \\leq |E(G)|\/(n-a)^2 = nhd\/(2(n-a)^2) \\leq n^{1\/2}d\/(2(n-a))$. \nSimilarly, since $\\sum_{v \\in V(G')} |x_v| \\leq a^{1\/2}$, we have $2|\\ip{x'}{y}_M| \\leq 2a^{1\/2}d\/(n-a)$. \nIt follows that \n\\[\n|\\ipo{x}_N| \\geq \\lambda - \\frac{d(n^{1\/2}+4a^{1\/2})}{2(n-a)} \\geq \\lambda - \\frac{5dn^{1\/2}}{2(n-a)} > \\lambda - \\frac{5}{2},\n\\]\nand recalling that $\\|x\\|_2^2 \\leq 1+1\/\\lambda < \\lambda\/(\\lambda-1)$ completes the proof. \n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{patcount}]\nWe may specify a pattern by first choosing $w_0$, then choosing $a_{i,w}$ for each $(i,w)$ with $i \\in [h]$ \nand $w \\in D^{>0} \\cap [w_0,w_0d]$, and finally choosing $f_{i,w,i',w'}$ for each pair $(i,w),(i',w') \\in V(\\Gamma)$ \nwith $i\\sim_{H} i'$. \n\nIf the pattern is to be non-empty then there is some $(i,w) \\in V(\\Gamma)$ for \nwhich $\\alpha_{i,w} nh\/w^2$ is a positive integer. Since $\\alpha_{i,w} \\leq 1$ it follows that $w_0^2 \\leq w^2 \\leq nh$ \nand so there are at most $|D^{>0} \\cap [0,\\sqrt{nh}]| \\leq \\log_2(nh)$ choices for $w_0$. \n\nHaving chosen $w_0$, since $|D^{>0} \\cap [w_0,w_0d]| \\leq \\log_2 (2d)$, there are at most \n$A^{h\\log (2d)}$ choices for the $a_{i,w}$. \nFinally, for each pair $(i,w),(i',w')$ with $i \\sim_{H} i'$, there are at most $1+\\min(a_{i,w},a_{i',w'}) \\leq A$ choices \nfor $f_{i,w,i',w'}$. There are at most $hd \\log_2(2d)\/2$ such pairs, so the \ntotal number of choices for the $f_{i,w,i',w'}$ is at most $A^{hd \\log_2(2d)\/2}$. \n\nCombining the bounds of the two preceding paragraphs, we obtain that the total number of patterns \nwith $a_{i,w} < A$ for all $(i,w) \\in V(\\Gamma)$ is at most \n\\[\n\\log_2(nh) \\cdot A^{h \\log_2(2d)} A^{hd \\log_2(2d)\/2}< \\log_2(nh) \\cdot A^{2h d\\log_2 d}. \n\\]\n\\end{proof}\n\n\n\\section{Acknowledgements} {\\Small S.G. heard of this problem in a seminar of Benny Sudakov at the IPAM long program on ``Combinatorics: Methods and Applications in Mathematics and Computer Science''.  He would like to thank IPAM for providing such an excellent program.  Most of the research for this article took place while he was a postdoctoral fellow at McGill University, he would like to thank them for their support and facilities.  He is currently supported by CNPq (Proc. 500016\/2010-2).  L.A-B. was supported during this research by an NSERC Discovery Grant. Both authors would like to thank two anonymous referees for many valuable comments and suggestions.} \n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\n\n\\subsection{A. Conformal symmetry in one dimension}\n\n\nLet us briefly recap some basic features of conformal invariance in one dimension. It is helpful to keep in mind that, in interacting systems like the SYK models, this conformal invariance is only an approximate symmetry of certain large $N$ theories.\n\nConsider coupling a CQM to an external metric $h_{\\mu\\nu}$ and a source $\\lambda$ for a scalar operator $\\mathcal{O}_{\\Delta}$ of dimension $\\Delta$. Define the generating functional of connected correlation functions,\n\\begin{equation}\nW = - i \\ln \\mathcal{Z}\\,.\n\\end{equation}\nThe connected one-point functions of the stress tensor $t^{\\mu\\nu}$ and $\\mathcal{O}_{\\Delta}$ are defined by functional variation,\n\\begin{equation}\n\\label{E:deltaW}\n\\delta W = \\int dt \\sqrt{-h} \\left( \\frac{1}{2}\\langle t^{\\mu\\nu} \\rangle \\delta h_{\\mu\\nu} - \\langle \\mathcal{O}_{\\Delta}\\rangle \\delta \\lambda\\right)\\,.\n\\end{equation}\nBy assumption, $W$ is invariant under infinitesimal reparameterizations, under which $h_{\\mu\\nu}$ and $\\lambda$ vary as\n\\begin{equation}\n\\delta_{\\xi} h_{\\mu\\nu} =D_{\\mu}\\xi_{\\nu}+D_{\\nu}\\xi_{\\mu}\\,, \\qquad \\delta_{\\xi}\\lambda = \\xi^{\\mu}D_{\\mu}\\lambda\\,,\n\\end{equation}\nwith $D_{\\mu}$ the covariant derivative. Plugging these variations into~\\eqref{E:deltaW} and demanding $\\delta_{\\xi}W=0$ leads to the diffeomorphism Ward identity\n\\begin{equation}\nD_{\\nu}\\langle t^{\\mu\\nu} \\rangle = - (D^{\\mu}\\lambda)\\langle \\mathcal{O}_{\\Delta}\\rangle\\,.\n\\end{equation}\nIn one dimension with $h = - dt^2$, this becomes equation~\\eqref{E:eom3} from the main text,\n\\begin{equation}\n\\label{E:diffWard}\n\\dot{E} =  \\dot{\\lambda} \\langle \\mathcal{O}_{\\Delta}\\rangle\\,.\n\\end{equation}\n\nOn the other hand, we ought to couple the CQM to the metric in a Weyl-invariant way. Then $W$ is invariant under an infinitesimal Weyl rescaling under which the metric and source $\\lambda$ vary as\n\\begin{equation}\n\\delta_{\\omega} h_{\\mu\\nu} = 2 \\omega h_{\\mu\\nu}\\,, \\qquad \\delta_{\\omega}\\lambda = (\\Delta - 1)\\omega \\lambda\\,,\n\\end{equation}\nand $\\delta_{\\omega}W=0$ gives the Weyl Ward identity (in terms of $E = - h_{\\mu\\nu}\\langle t^{\\mu\\nu}$)\n\\begin{equation}\n\\label{E:weylWard}\nE = (1-\\Delta)\\lambda \\langle \\mathcal{O}_{\\Delta}\\rangle\\,.\n\\end{equation}\n\nOne way of stating Polcinski's paradox~\\cite{Jensen:2011su} is that the diffeomorphism Ward identity~\\eqref{E:diffWard} is not compatible with the Weyl Ward identity~\\eqref{E:weylWard} in one dimension. Actually there is a loophole: the two are compatible if the correlation functions of $\\mathcal{O}$ are topological, as they are for operators in the theory of $2N$ free Majorana fermions.\n\nSo there is a conflict between reparameterization invariance and conformal symmetry for interacting systems in one dimension.\n\nBefore going on, we should reiterate what is in some sense the main point of this Letter, namely how gravity solves this paradox near AdS$_2$. The boundary dual is not a CQM on its own, but instead a particular hydrodynamics coupled to the correlation functions of a CQM. \n\nThose conformal correlators are strongly constrained by the conformal symmetry, as we now discuss. In any dimension, conformal transformations are the combination of a coordinate transformation $x^{\\mu}=x^{\\mu}(y^{\\nu})$ and Weyl rescaling $h_{\\mu\\nu} \\to e^{2\\Omega}h_{\\mu\\nu}$ which leave the metric invariant. In one dimension with the flat metric $h=-dt^2$, any coordinate transformation gives a conformal transformation: the combined action of\n\\begin{equation}\nt = t(w)\\, \\qquad \\Omega = - \\ln t'(w)\\,,\n\\end{equation} \nsends the metric to itself, $-dt^2 \\to -dw^2$. Wick-rotating to Euclidean signature and compactifying Euclidean time, the modes of $t(w)$ generate a Virasoro algebra with $c=0$. \n\nThere are two ways to think about the vanishing central charge. The first is simply that the stress tensor vanishes by the Weyl Ward identity. The second is that there is no Weyl anomaly possible in one dimension.\n\nAs is familiar from two-dimensional CFT, the conformal transformations which are regular everywhere generate the global conformal group $SL(2;\\mathbb{R})$. There is a single special conformal generator $K$, and in the usual way one defines primary operators as those annihilated by $K$. The vacuum two-point function of a primary operator $\\mathcal{O}_{\\Delta}$ of dimension $\\Delta$ is, in Euclidean signature\n\\begin{equation}\n\\langle \\mathcal{O}_{\\Delta}(\\tau)\\mathcal{O}_{\\Delta}(0)\\rangle = \\frac{1}{|\\tau|^{2\\Delta}}\\,,\n\\end{equation}\nand similarly for three-point functions of $\\mathcal{O}_{\\Delta}$.\n\nWe conclude this Subsection with a few comments on thermodynamics. Consider the thermal partition function of a CQM\n\\begin{equation}\n\\mathcal{Z}_E = \\text{tr}\\left( e^{-\\beta H}\\right)\\,,\n\\end{equation}\nwhich as usual is the partition function of the Euclidean theory on a circle of size $\\beta$ with thermal boundary conditions. There is only one local counterterm which can be used to redefine the theory,\n\\begin{equation}\n\\label{E:CT}\n\\ln \\mathcal{Z}_E \\to \\ln\\mathcal{Z}_E + m\\int d\\tau \\sqrt{h_E}\\,,\n\\end{equation}\nwhere $\\tau$ is Euclidean time, $h_E$ the Euclidean metric, and $m$ is some mass scale. This counterterm is not scale-invariant. Thus, in a CQM, the thermal partition function $\\mathcal{Z}_{CQM}$ is an unambiguous observable. This should not be a surprise; in odd dimension, the logarithm of the partition function of a CFT on a Euclidean sphere -- the ``sphere free energy'' -- is a useful and unambiguous (up to a quantized imaginary ambiguity which may exist in $d=4k-1$ dimensions) CFT observable. In one dimension, this partition function is the extremal entropy,\n\\begin{equation}\nS_0 = \\frac{\\partial (T\\ln \\mathcal{Z}_{CQM})}{\\partial T} = \\ln \\mathcal{Z}_{CQM}\\,.\n\\end{equation} \n\nNow consider a non-conformal QM which realizes an emergent conformal invariance at low energies. Suppose that the low-energy description is a(n approximate) CQM with extremal entropy $S_0$ deformed by a dimension $\\Delta$ operator. The low-temperature partition function is\n\\begin{equation}\n\\ln \\mathcal{Z}_E = - \\beta E_0 + S_0 + P_1 T^{\\Delta-1} + \\hdots\\,,\n\\end{equation}\nwhere $E_0$ is the ground state energy. Observe that $E_0$ is unphysical: it is redefined by the counterterm~\\eqref{E:CT}. This partition function leads to a low-temperature entropy\n\\begin{equation}\nS = S_0 +\\Delta  P_1 T^{\\Delta-1} + \\hdots\\,.\n\\end{equation}\nComparing this with the entropy~\\eqref{E:nearExtremalS} of near-extremal black holes in dilaton gravity, we see that the black hole entropy is that of a CQM deformed by a $\\Delta = 2$ operator.\n\n\n\\subsection{B. Holographic renormalization}\n\n\nIn this Appendix we fill in various details on the near-AdS$_2$ solutions described in the main text.\n\nWe begin with the perturbative solution near the endpoint of a holographic RG flow, with a small amount of infalling dust $T_{ww}(w)$. The full perturbative solution is\n\\begin{align}\n\\nonumber\n\\varphi & = \\varphi_0 + \\ell r + \\mathcal{O}(\\ell^2 r^2)\\,,\n\\\\\n\\nonumber\ng & = - \\left(r^2 + 2 \\{ t(w),w\\} +  \\frac{\\ell r^3}{6}U''[\\varphi_0] + \\mathcal{O}(\\ell^2r^2)\\right)dw^2 \n\\\\\n\\label{E:perturbativeSolution}\n& \\qquad + 2dw dr\\,,\n\\end{align}\nwith\n\\begin{equation}\n\\label{E:bulkWard}\n\\ell\\partial_w \\{ f(w),w\\}  = -T_{ww}(w)\\,.\n\\end{equation}\n\nWe account for the dust with a massless scalar field $\\chi$\n\\begin{equation*}\nS_{\\rm matter} = - \\frac{1}{2}\\int d^2x \\sqrt{-g} Z_0[\\varphi] (\\partial\\chi)^2\\,,\n\\end{equation*}\nwith $Z[\\varphi_0]=1$. The infalling solutions are $\\chi(w) = \\lambda(w)$, and we take the perturbative limit $\\lambda^2 \\sim\\ell$. To first order in $\\ell$, the stress tensor evaluated on this solution is $T_{ww} = \\kappa^2\\dot{\\lambda}^2$ and the dilaton source vanishes $\\Phi = 0$.\n\nWe proceed to holographically renormalize the dilaton gravity~\\eqref{E:dilatonS} on backgrounds of the form~\\eqref{E:perturbativeSolution}. To proceed we define the dual theory on a constant-$r$ slice at asymptotically large $r$, subject to the constraint that $\\ell r$ is still asymptotically small. In physical terms, we are taking the almost zero energy limit. In practice we drop the corrections to the background~\\eqref{E:perturbativeSolution} with two or more powers of $\\ell$ and then proceed in the usual way.\n\nThe bulk action is divergent as one integrates to large $r$, so we introduce a cutoff slice at $r=\\Lambda$, add covariant boundary terms on the cutoff slice to eliminate divergences, and then remove the cutoff by sending $\\Lambda\\to\\infty$.\n\nThe authors of~\\cite{Grumiller:2007ju} have studied the problem of holographic renormalization in a general dilaton gravity. For the case at hand, the renormalized action is simply\n\\begin{align}\nS_{ren} &= \\lim_{\\Lambda\\to\\infty} \\frac{1}{2\\kappa^2}\\left\\{ \\int d^2x \\sqrt{-g} \\left( \\varphi R + U\\right)\\right.\n\\\\\n\\nonumber\n&  \\qquad\\qquad + \\left. \\int dt \\sqrt{-\\gamma} \\left( 2\\varphi K - U \\right) \\right\\} + S_{\\rm matter} + \\mathcal{O}(\\ell^2)\\,,\n\\end{align}\nwhere $\\gamma$ is the induced metric on the cutoff slice and $K$ the trace of its extrinsic curvature.\n\nThe variation of with on-shell action with respect to the metric and scalar is\n\\begin{align}\n\\begin{split}\n\\label{E:deltaSren}\n\\delta S_{ren} =& \\lim_{\\Lambda\\to\\infty} \\int dt \\sqrt{-\\gamma}\\left\\{ - (n^{\\mu}\\partial_{\\mu}\\chi)\\delta\\chi\\right.\n\\\\\n& \\qquad \\quad \\left. +\\frac{1}{2\\kappa^2} \\left( n^{\\rho}\\partial_{\\rho}\\varphi - \\frac{U}{2}\\right)\\gamma^{\\mu\\nu}\\delta g_{\\mu\\nu}\\right\\}\\,,\n\\end{split}\n\\end{align}\nwith $n^{\\mu}$ the outward pointing normal vector to the cutoff slice. Both $n^{\\mu}\\partial_{\\mu}\\varphi$ and $U$ are $\\mathcal{O}(\\ell)$ for the perturbative solution~\\eqref{E:perturbativeSolution}, so we require the on-shell variation of the $g_{\\mu\\nu}$ in response to a variation of the boundary metric to $\\mathcal{O}(\\ell^0)$, which is simply $\\delta g_{\\mu\\nu} = r^2 \\delta h_{\\mu\\nu} + \\mathcal{O}(r^0,\\ell)$. Plugging this variation into $\\delta S_{ren}$ and evaluating it on the solution~\\eqref{E:perturbativeSolution} gives the boundary stress tensor\n\\begin{equation}\n\\langle t^{\\mu\\nu}\\rangle = \\frac{2}{\\sqrt{-h}}\\frac{\\delta S_{ren}}{\\delta h_{\\mu\\nu}} = \\frac{\\ell h^{\\mu\\nu}}{\\kappa^2}\\{t(w),w\\}\\,,\n\\end{equation}\nso that the energy is $E = - h_{\\mu\\nu} \\langle t^{\\mu\\nu}\\rangle = - (\\ell\/\\kappa^2) \\{ t(w),w\\}$, which derives the result~\\eqref{E:energy} in the main text. We also obtain the expectation value of the dimension$-1$ operator $\\mathcal{O}$ dual to $\\chi$,\n\\begin{equation}\n\\langle \\mathcal{O}\\rangle = - \\frac{1}{\\sqrt{-h}}\\frac{\\delta S_{ren}}{\\delta \\lambda} = \\dot{\\lambda}\\,,\n\\end{equation}\nso that~\\eqref{E:bulkWard} becomes\n\\begin{equation*}\n\\dot{E} = \\dot{\\lambda}\\langle \\mathcal{O}\\rangle\\,.\n\\end{equation*}\n\nNow we consider the perturbative solution in the presence of massive matter,\n\\begin{equation}\nS_{\\rm matter} = - \\frac{1}{2}\\int d^2x \\sqrt{-g} \\left( Z_0[\\varphi] (\\partial\\chi)^2 + Z_1[\\varphi]m^2 \\chi^2\\right)\\,,\n\\end{equation}\nwith $Z_0[\\varphi_0]=Z_1[\\varphi_0]=1$. The field $\\chi$ is now dual to an operator $\\mathcal{O}_{\\Delta}$ of dimension $\\Delta(\\Delta - 1)=m^2$. We take $\\chi$ to be a free field, but it is easy to allow for self-interactions.\n\nAs above we take the perturbative limit with $\\chi \\ll \\ell$. In this limit,\n\\begin{align}\n\\begin{split}\nT_{\\mu\\nu} & = \\kappa^2 \\left( \\partial_{\\mu}\\chi\\partial_{\\nu}\\chi - \\frac{ (\\partial\\chi)^2 +m^2}{2}g_{\\mu\\nu}\\right)\\,,\n\\\\\n\\Phi & = \\kappa^2 \\left( Z_0'[\\varphi_0](\\partial\\phi)^2 + Z_1'[\\varphi_0]m^2 \\chi^2\\right)\\,.\n\\end{split}\n\\end{align}\nThe dilaton and metric are perturbed as\n\\begin{align}\n\\nonumber\n\\varphi &= \\varphi_0 + \\ell  \\psi(w,r) + \\mathcal{O}(\\ell^2 r^2)\\,,\n\\\\\n\\nonumber\ng & = -\\Big(r^2 + 2 \\{ t(w),w\\} + \\ell  f(w,r)+ \\mathcal{O}(\\ell^2 r^2)\\Big)dw^2 \n\\\\\n\\label{E:perturbativeSolution2}\n& \\qquad + 2 dw dr \\,.\n\\end{align}\nTo leading order in $\\ell$, the solution for $\\chi$ is given by the most general solution of\n\\begin{equation}\n(\\Box - m^2) \\chi = 0\\,,\n\\end{equation}\non the AdS$_2$ background~\\eqref{E:perturbativeSolution2}, setting all of the $\\mathcal{O}(\\ell)$ corrections to vanish. We take $\\Delta$ to be general but not half-integer, and pick the standard quantization for $\\chi$ so that $\\Delta>1\/2$. Then near the boundary $r\\to\\infty$ $\\chi$ is\n\\begin{equation}\n\\label{E:bdyChi}\n\\chi = r^{\\Delta - 1}  \\sum_{n=0}\\frac{a_n(w)}{r^n}+ r^{-\\Delta} \\sum_{m=0}\\frac{b_m(w)}{r^m}\\,,\n\\end{equation}\nwhere the $a_{n}$ with $n>0$ are determined by $a_0$ and similarly for the $b_n$, for example\n\\begin{equation}\na_1 = \\dot{a}_0\\,, \\qquad b_1 = \\dot{b}_0\\,.\n\\end{equation}\nWe identify the source for the dual operator as\n\\begin{equation}\n\\lambda = \\lim_{r\\to\\infty} r^{1-\\Delta} \\chi = a_0(t)\\,.\n\\end{equation}\n\nThe stress tensor and dilaton source have three distinct sets of terms. Near the boundary, they are of the schematic form\n\\begin{equation}\nT_{\\mu\\nu} = r^{2\\Delta}\\Sigma^{a}_{\\mu\\nu}(r,a^2) + \\Sigma_{\\mu\\nu}(r,ab) + r^{-2\\Delta} \\Sigma^b_{\\mu\\nu}(r,b^2)\\,,\n\\end{equation}\nwhere $\\Sigma^a_{\\mu\\nu}(r,a^2)$ is some power series in $1\/r$ with coefficients built out of two powers of the $a_n$ and their derivatives, and similarly for the other two series. Because $\\Delta\\cnot \\in \\mathbb{Z}\/2$ by assumption, these three series never mix.\n\nSolving the $rr$ and $rw$ components of Einstein's equations, we obtain the correction to the dilaton to be\n\\begin{align}\n\\label{E:psi}\n&\\psi = r + \\frac{\\kappa^2}{\\ell}r^{2(\\Delta -1)}\\left( \\frac{(\\Delta-1)a_0^2}{2(3-2\\Delta)} + \\mathcal{O}(r^{-1})\\right)\n\\\\\n\\nonumber\n&+ \\frac{\\kappa^2}{\\ell r}\\left(\\Delta(\\Delta-1) a_0 b_0 +  \\frac{(\\Delta^2-1)\\dot{(a_0b_0)}-\\Delta \\dot{a}_0b_0}{3r} + \\mathcal{O}(r^{-2})\\right)\n\\\\\n\\nonumber\n& \\qquad - \\frac{\\kappa^2}{\\ell^2}r^{-2\\Delta}\\left(  \\frac{\\Delta b_0^2}{2(2\\Delta+1)} + \\mathcal{O}(r^{-1})\\right)\\,,\n\\end{align}\nand, while the correction to the metric is calculable, we do not require it for what follows. The $ww$ component imposes exactly one condition,\n\\begin{equation}\n\\label{E:generalHydro}\n\\frac{\\ell}{\\kappa^2} \\partial_w \\left( \\{t(w),w\\}\\right) =(2\\Delta-1) \\left( \\Delta \\dot{\\left(a_0 b_0\\right)}  -a_0\\dot{b}_0\\right)\\,.\n\\end{equation}\n\nAs for a massless $\\chi$, we have succeeded in rewriting the dilaton equations of motion in terms of an equation~\\eqref{E:generalHydro} which only involves the boundary time. Let us rewrite it in terms of boundary variables.\n\nThere are additional $\\mathcal{O}(\\chi^2)$ boundary counterterms required to renormalize the action. The leading divergence is removed by the counterterm,\n\\begin{equation}\n\\label{E:leadingCT}\nS_{\\rm CT} = \\frac{\\Delta -1}{2}\\int dt \\sqrt{-\\gamma}\\,\\chi^2 + \\mathcal{O}(\\partial^2 \\chi^2)\\,,\n\\end{equation}\nand there are subleading counterterms with at least two boundary derivatives which remove subleading divergences. Varying the on-shell action with respect to the boundary metric $h$ and source $\\lambda$ gives\n\\begin{align}\n\\begin{split}\n\\label{E:constitutiveGeneral}\n\\langle \\mathcal{O}_{\\Delta}\\rangle &= (1- 2\\Delta) b_0(t)\\,,\n\\\\\nE &  = - \\frac{\\ell}{\\kappa^2}\\{ t(w),w\\} + (2\\Delta-1)(\\Delta-1)a_0b_0\\,,\n\\\\\n& = - \\frac{\\ell}{\\kappa^2}\\{t(w),w\\} + (1-\\Delta)\\lambda \\langle \\mathcal{O}_{\\Delta}\\rangle\\,.\n\\end{split}\n\\end{align}\n(The additional $a_0b_0$ terms in the energy come from (i.) the $\\mathcal{O}(1\/r)$ correction to the dilaton~\\eqref{E:psi}, inserted into the metric variation of $S_{ren}$ in~\\eqref{E:deltaSren}, and (ii.) the metric variation of the leading counterterm~\\eqref{E:leadingCT}.)\nIn terms of $E$, $\\langle \\mathcal{O}_{\\Delta}\\rangle$, and $\\lambda=a_0$, the equation of motion~\\eqref{E:generalHydro} is simply\n\\begin{equation}\n\\label{E:diffAgain}\n\\dot{E} = \\dot{\\lambda}\\langle \\mathcal{O}_{\\Delta}\\rangle\\,.\n\\end{equation}\n\nIt is straightforward to allow for half-integer $\\Delta$. When $\\Delta\\in \\mathbb{Z}\/2$ there are logarithmic terms in the near-boundary solution for bulk fields as well as logarithmic boundary counterterms. It is similarly straightforward to consider self-interacting matter, e.g. $\\chi^4$ theory. The final result is the same: the Einstein's equations can be rewritten as the diffeomorphism Ward identity in the dual quantum mechanics for \\emph{any} value of $\\Delta$.\n\nAs in the main text, we have shown that the gravitational dynamics near AdS$_2$ can be rewritten as the diffeomorphism Ward identity~\\eqref{E:diffAgain} with a ``constitutive relation'' for the energy given by~\\eqref{E:constitutiveGeneral}. This equation of motion follows from the same effective action~\\eqref{E:Seff} presented in the main text.\n\nWe see that the hydrodynamic effective action~\\eqref{E:Seff} (equivalently,~\\eqref{E:Shydro}) encodes the gravitational dynamics for dilaton gravity coupled to general scalar matter.\n\n\n\\subsection{C. Electric AdS$_2$}\n\nIn this main text, we considered dilaton gravities with AdS$_2$ vacua at the roots of the dilaton potential. There is another way to realize AdS$_2$ solutions: the AdS$_2$ may be supported by an electric field. In this Appendix we work out various aspects of these ``electric'' AdS$_2$ vacua of dilaton gravities with a Maxwell field, Einstein-Maxwell-Dilaton gravity.\n\nOur motivation for studying this problem is somewhat indirect. Hartman and Strominger~\\cite{Hartman:2008dq} have claimed that the dual to gravity on electric AdS$_2$ vacua is invariant under a Virasoro algebra with \\emph{nonzero} central charge, a claim which was supported by way of holographic renormalization in~\\cite{Castro:2008ms} (although see also~\\cite{Castro:2014ima}). \n\nIn the setting of two-dimensional CFT, Roberts and Stanford~\\cite{Roberts:2014ifa} have shown that large central charge,  a large gap for higher-spin Virasoro primary operators, and a ``low'' density of light states is enough to guarantee a maximal Lyapunov exponent $\\lambda_L = 2\\pi T$. Their argument would go through more or less unaltered for a CQM invariant under a Virasoro symmetry with large central charge, assuming that such a thing existed.\n\nThus we revisit electric AdS$_2$ vacua and the claim of a Virasoro algebra with $c \\neq 0$. We will find that the central charge of the Virasoro symmetry vanishes.\n\nConsider the most general Einstein-Maxwell-Dilaton gravity,\n\\begin{equation}\nS_{\\rm bulk} = \\frac{1}{2\\kappa^2}\\int d^2x \\sqrt{-g} \\left( \\varphi R + U[\\varphi]  - \\frac{W[\\varphi]}{4}F^2\\right)\\,.\n\\end{equation}  \nThe authors of~\\cite{Castro:2008ms} consider the case\n\\begin{equation}\nU=8\\varphi\\,, \\qquad W = 1\\,.\n\\end{equation}\nThe equations of motion are now\n\\begin{align}\n\\nonumber\n0 & = D_{\\mu}D_{\\nu}\\varphi - g_{\\mu\\nu}\\Box \\varphi +\\frac{g_{\\mu\\nu}}{2} \\left(U - \\frac{W}{4}F^2\\right) + \\frac{W}{2}F_{\\mu\\rho}F_{\\nu}{}^{\\rho}\\,,\n\\\\\n\\label{E:einstein2}\n0 & = D_{\\nu}\\left( WF^{\\mu\\nu}\\right)\\,,\n\\\\\n\\nonumber\n0 & = R + U'  - \\frac{W'}{4}F^2\\,.\n\\end{align}\nThey admit ``electric'' AdS$_2$ solutions with constant dilaton and constant electric field,\n\\begin{equation}\n\\varphi = \\varphi_0\\,, \\qquad F_{\\mu\\nu} = E \\varepsilon_{\\mu\\nu}\\,, \\qquad R = - \\frac{2}{L^2}\\,,\n\\end{equation}\nprovided the dilaton $\\varphi_0$, electric field $E$, and AdS radius $L$ satisfy the two relations\n\\begin{equation}\n\\label{E:familyOfAdS2}\nU[\\varphi_0] = \\frac{W[\\varphi_0]}{2}E^2\\,, \\quad \\frac{2}{L^2} = U'[\\varphi_0] + \\frac{W'[\\varphi_0]E^2}{2}\\,.\n\\end{equation}\nSo, adjusting the electric field simultaneously adjusts the dilaton and AdS radius. Observe that, for smooth potentials $U$ and $W$, there is generally a one-parameter family of AdS$_2$ solutions controlled by the strength of the electric field. The most general such solution is, in a radial gauge,\n\\begin{align}\n\\begin{split}\n\\label{E:electricAdS2}\n\\varphi & = \\varphi_0\\,,\n\\\\\nA & =  - EL^2 r \\left( f_1(t) - \\frac{f_2(t)}{r^2}\\right)dt + a(t) dt\\,,\n\\\\\ng & = L^2 \\left( - r^2 \\left( f_1(t) + \\frac{f_2(t)}{r^2}\\right)^2 dt^2 + \\frac{dr^2}{r^2}\\right)\\,.\n\\end{split}\n\\end{align}\nUsing the defining function $1\/(r^2L^2)$ the conformal boundary $r\\to\\infty$ is endowed with a metric\n\\begin{equation}\nh = \\lim_{r\\to\\infty} \\frac{\\gamma}{r^2L^2} = - f_1(t)dt^2\\,,\n\\end{equation}\nwhere $\\gamma$ is the induced metric on a constant-$r$ slice.\n\nNow let us holographically renormalize the bulk theory. Consider the following definition of the renormalized theory:\n\\begin{align}\n\\begin{split}\n\\label{E:S1}\nS_{1} &= \\lim_{\\Lambda\\to\\infty}  \\frac{1}{2\\kappa^2}\\left\\{\\int d^2x \\sqrt{-g} \\Big( \\varphi R + U - \\frac{W}{4}F^2\\Big) \\right.\n\\\\\n& \\qquad  \\quad\\quad  +\\left. \\int dt \\sqrt{-\\gamma} \\Big( 2\\varphi K - LU + \\frac{W}{2L}A^2\\Big)\\right\\},\n\\end{split}\n\\end{align}\nwhere $A^2 = \\gamma^{\\mu\\nu}A_{\\mu}A_{\\nu}$. This matches the renormalization scheme of~\\cite{Castro:2008ms} for the particular dilaton theory they study. It is easy to verify that this renormalized action is finite on-shell. Under a variation of the metric and gauge field, keeping the dilaton fixed and using that $\\partial_{\\mu}\\varphi = 0$ for the solutions at hand, $S_1$ varies as\n\\begin{align}\n\\nonumber\n\\delta S_1 =& \\lim_{\\Lambda\\to\\infty} \\frac{1}{2\\kappa^2}\\int dt \\sqrt{-\\gamma} \\left\\{ -\\frac{L}{2}\\left(   U+\\frac{W}{2L^2}A^2\\right)\\gamma^{\\mu\\nu} \\delta g_{\\mu\\nu} \\right.\n\\\\\n\\label{E:deltaS1}\n& \\qquad \\qquad \\left. + W \\left( F^{\\mu\\nu} n_{\\nu} + \\frac{\\gamma^{\\mu\\nu}}{L}A_{\\nu}\\right) \\delta A_{\\mu}\\right\\}\\,.\n\\end{align}\nOn-shell, a variation of the boundary metric $h = - f_1(t)^2 dt^2$ induces a variation of both the bulk metric and gauge field~\\eqref{E:electricAdS2} according to $\\delta f_1 = - \\frac{\\delta h_{tt}}{2f_1}$,\n\\begin{align}\n\\begin{split}\n\\delta g_{tt} &= r^2 L^2 \\delta h_{tt} + \\mathcal{O}(r^0) \\,, \n\\\\\n\\delta A_t &= r \\frac{EL^2}{2}\\frac{\\delta h_{tt}}{f_1} + \\mathcal{O}(r^0)\\,.\n\\end{split}\n\\end{align}\nInserting this variation into~\\eqref{E:deltaS1} and evaluating it on an AdS$_2$ solution~\\eqref{E:electricAdS2} gives the dual stress tensor. This however identically vanishes,\n\\begin{equation}\n\\langle t^{tt}\\rangle \\equiv  \\frac{2}{\\sqrt{-h}}\\frac{\\delta S_1}{\\delta h_{tt}} = 0\\,,\n\\end{equation}\nso the Virasoro symmetry at play has zero central charge.\n\nHowever, this is not the end of the story. First, it is more natural to work in an alternative quantization, as one commonly does for Maxwell theory in AdS$_3$~\\cite{Marolf:2006nd,Jensen:2010em}. The growing mode of $A_t$ ought to be fixed as a boundary condition, allowing the constant term to fluctuate. Second, the Cvetic and Papadimitriou~\\cite{Cvetic:2016eiv} have argued that the renormalization scheme~\\eqref{E:S1} is not quite correct. They advocate for the prescription\n\\begin{align}\n\\begin{split}\n\\label{E:SrenElectric}\nS_{ren} &= \\lim_{\\Lambda\\to\\infty} \\frac{1}{2\\kappa^2}\\left\\{ \\int d^2x \\sqrt{-g} \\Big( \\varphi R + U - \\frac{W}{4}F^2\\Big) \\right.\n\\\\\n& \\qquad \\quad+ \\int dt \\sqrt{-\\gamma} \\Big( 2\\varphi K -L U \\Big)\n\\\\\n& \\qquad \\quad - \\left.\\int dt\\sqrt{-\\gamma}\\,W\\Big( A_{\\mu}F^{\\mu\\nu}n_{\\nu}+ \\frac{L}{4}F^2\\Big)\\right\\}.\n\\end{split}\n\\end{align}\nIn alternate quantization we fix the canonical momentum conjugate to $A_t$, which in the bulk is\n\\begin{equation}\n\\pi^t \\equiv \\frac{\\delta S_{bulk}}{\\delta (\\partial_r A_t)} = - \\frac{WE}{2\\kappa^2}\\,.\n\\end{equation}\nWe identify this conjugate momentum as a fixed, external charge $\\mathcal{Q}$ in the dual CQM.\n\nThis definition~\\eqref{E:SrenElectric} has the virtue that it conserves the symplectic structure on the boundary.\n\nVarying it with respect to the boundary metric gives\n\\begin{equation}\n\\langle t^{tt}\\rangle = 0\\,.\n\\end{equation}\nThe charge $\\mathcal{Q}$ is conjugate to a $U(1)$ gauge field $\\mathcal{A}$,\n\\begin{equation}\n\\langle \\mathcal{A}\\rangle =- \\frac{\\delta S_{ren}}{\\delta \\mathcal{Q}}\\,.\n\\end{equation}\nThe charge $\\mathcal{Q}$ is time-independent, and so $\\langle \\mathcal{A}\\rangle$ is determined up to a total derivative, as is appropriate for a gauge field. That is, $\\langle \\mathcal{A}\\rangle$ contains no local information. However, its holonomy around the Euclidean time circle is physical, encoding the chemical potential $\\mu$.\n\nTo deduce the holographic formula for $\\langle \\mathcal{A}\\rangle$ we need to deduce the variation of $S_{ren}$ with respect to $W[\\varphi_0]E$. In order to remain on-shell, this variation induces a variation in the AdS radius and dilaton in accordance with~\\eqref{E:familyOfAdS2}, with\n\\begin{equation}\n\\label{E:deltaQ}\n\\delta \\mathcal{Q} = - \\frac{1}{\\kappa^2 E L^2}\\delta \\varphi\\,.\n\\end{equation} \nThe final result is that\n\\begin{equation}\n\\label{E:chemical}\n\\langle \\mathcal{A}(t)\\rangle = a(t)\\,,\n\\end{equation}\nfor $a(t)$ the $\\mathcal{O}(r^0)$ term in the Maxwell field~\\eqref{E:electricAdS2}.\n\nNow consider an electric AdS$_2$ black hole:\n\\begin{align}\n\\begin{split}\n\\varphi &= \\varphi_0\\,,\n\\\\\nA &= - EL^2 \\left(r  - 2r_h + \\frac{r_h^2}{r}\\right)dt\\,,\n\\\\\ng & =  L^2 \\left( - r^2 \\left( 1 - \\frac{r_h^2}{r^2}\\right)^2dt^2 + \\frac{dr^2}{r^2}\\right)\\,.\n\\end{split}\n\\end{align}\nWe have chosen the $\\mathcal{O}(1)$ term of the gauge field so that $A$ is regular at the Euclideanized horizon. The dual CQM is at a temperature $T = r_h\/\\pi$ and charge $Q = - WE\/(2\\kappa^2)$. The chemical potential is given by~\\eqref{E:chemical} to be $\\mu = 2\\pi T E L^2$.\n\nThe Bekenstein-Hawking entropy is\n\\begin{equation}\nS =\\frac{2 \\pi \\varphi_0}{\\kappa^2}\\,.\n\\end{equation}\nIt can also be computed from the on-shell, Euclidean, holographically renormalized action, which is\n\\begin{equation}\n\\label{E:electricAdS2Sren}\nS_{E} = \\frac{2\\pi \\varphi_0}{\\kappa^2} \\,.\n\\end{equation}\nThe Euclidean action computes the canonical ensemble free energy $G(T,Q) = - T S_{E}$, whose variation gives the entropy and chemical potential as\n\\begin{equation}\n\\delta G = - S \\delta T + \\mu \\delta Q\\,.\n\\end{equation}\nUsing~\\eqref{E:deltaQ} and taking thermodynamic derivatives, we indeed recover the Bekenstein-Hawking entropy and chemical potential $\\mu = 2\\pi T E L^2$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nLinear programming is a fundamental problem in many areas, such as operations research, network, machine learning, business analysis and finance \\citep{von1947theory,dantzig1963linear,luenberger1984linear,boyd2004convex}.\nIn this paper, we consider the \\emph{maximum support} of the linear feasibility problem\n\\begin{equation}\n\\label{feasibility:primal}\n\\begin{aligned}\n\\mathrm{find} ~~&x \\in {\\mathbb R}^n \\\\\n\\mathrm{subject~to}~~ &Ax=0, ~x\\geq 0,~ x\\neq 0,\n\\end{aligned}\n\\end{equation}\nwith its dual problem\n\\begin{equation}\n\\label{feasibility:dual}\n\\begin{aligned}\n\\mathrm{find} ~~&u \\in {\\mathbb R}^m \\\\\n\\mathrm{subject~to}~~ &A^T u> 0,\n\\end{aligned}\n\\end{equation}\nwhere $A\\in {\\mathbb R}^{m\\times n}$ is an integer (or rational) matrix and $\\mathrm{rank}(A) = m$.\nThe \\emph{maximum support} means that the set of positive coordinates of the returned solution of (\\ref{feasibility:primal}) should be inclusion-wise maximum.\nActually, for the solution $\\hat{x}$ returned by our algorithm, any coordinate $\\hat{x}_i=0$ if and only if this coordinate equals to 0 for all feasible solutions of (\\ref{feasibility:primal}).\nThus, our algorithm can be directly used to test the feasibility of the general linear system $Ax=b,x\\geq 0$ with the same time complexity, i.e., given the maximum support solution $(\\bar{x},\\bar{x}')$ to the system $Ax-bx'=0, (x,x')\\geq 0$, if $\\bar{x}'>0$ then the original problem  $Ax=b,x\\geq0$ has a solution $\\widetilde{x} = \\bar{x}\/\\bar{x}'$, otherwise it is infeasible.\n\nThere are many (polynomial-time) algorithms for solving linear programming problems, e.g., \\citep{karmarkar1984new}, \\citep{wright1997primal} and \\citep{renegar1988polynomial}.\nRecently, \\cite{chubanov2015polynomial} proposed a polynomial-time projection and rescaling algorithm for solving problem \\eqref{feasibility:primal}.\nDue to its simplicity and efficiency, this kind of algorithms has become a potentially \\emph{practical} class of polynomial algorithms. See e.g., \\citep{dadush2016rescaled}, \\citep{roos2018improved} and \\citep{pena2018computational}.\n\nChubanov's algorithm \\citep{chubanov2015polynomial} and its variants typically consist of two procedures. The key part is \\emph{basic procedure} (BP) and the other part is \\emph{main algorithm} (MA).\nThe BP returns one of the following three results:\n\\begin{enumerate}[(i)]\n  \\item a feasible solution of \\eqref{feasibility:primal};\n  \\item a feasible solution of the dual problem \\eqref{feasibility:dual};\n  \\item a cut for the feasible region of \\eqref{feasibility:primal}.\n\\end{enumerate}\nNote that exactly one of \\eqref{feasibility:primal} and \\eqref{feasibility:dual} is feasible according to Farkas' lemma. Thus \\eqref{feasibility:primal} is infeasible if BP returns (ii).\nIf BP returns (iii), the other procedure MA rescales the matrix $A$ by using this cut and call BP again on the rescaled matrix $A$.\nAccording to \\citep{khachian1979polynomial} which gives a positive lower bound on the entries of a solution of a linear system, after a certain number of rescalings, one can conclude that there is no feasible solution for \\eqref{feasibility:primal}. So the number of rescaling operations can be bounded, i.e., the number of MA calls can be bounded.\nConsequently, the algorithm can terminate in finite time no matter whether problem \\eqref{feasibility:primal} is feasible or infeasible.\n\nTo be more precise, we quantify the time complexity.\nThe total time complexity of these Chubanov-type algorithms are typically $O(\\mathsf{T_{MA}}*\\mathsf{T_{BP}})$,\nwhere $\\mathsf{T_{MA}}$ denotes the number of MA calls (rescaling operations), and $\\mathsf{T_{BP}}$ denotes the time required by the basic procedure BP.\nAccording to the classic lower bound \\citep{khachian1979polynomial}, $\\mathsf{T_{MA}}$ can simply be bounded by $O(nL)$ for these Chubanov-type algorithms, where $L$ denotes the bit size of $A$.\nHowever, $\\mathsf{T_{BP}}$ is the most important and tricky part.\nTheoretical results and practical performances vary for different BP procedures.\nThe typical BP procedures include the perceptron method, von Neumann's method, and their variants (see e.g., \\citep{dantzig1992varepsilon,dunagan2008simple,dadush2016rescaled,pena2017solving}).\nWe review more details of these BP in Section \\ref{sec:bp}.\nUsually, $\\mathsf{T_{BP}}$ equals to $O(n^4)$ or $O(n^3m)$ in these BP procedures.\nIn this work, we improve $\\mathsf{T_{BP}}$ by a factor of $\\sqrt{n}$ if (\\ref{feasibility:primal}) or (\\ref{feasibility:dual}) is well-conditioned (measured by \\eqref{eq:rho}), but in the worse case, $\\mathsf{T_{BP}}$ still equals to $O(n^3m)$ in our algorithm.\n\n\\topic{Our Motivation}\nIn practice, these Chubanov-type projection and rescaling algorithms usually run much faster on the primal infeasible instances (i.e., \\eqref{feasibility:primal} is infeasible) than on the primal feasible instances (i.e., dual infeasible) no matter what basic procedure (von Neumann, perceptron or their variants) we use (also see Table \\ref{table:comparison} in Section \\ref{sec:exp}).\nIn this paper, we try to explain this phenomenon theoretically. Moreover, we try to provide a new algorithm to address this issue.\n\n\\topic{Our Contribution} Concretely, we make the following technical contributions:\n\\begin{enumerate}\n  \\item First, for the theoretical explanation, we provide Lemma \\ref{lm:infeasible} which shows that\n  the time complexity $\\mathsf{T_{BP}}$ can be $O(n^{2.5}m)$ rather than $O(n^3m)$ (see Lemma \\ref{lm:tbp}) in certain situations if \\eqref{feasibility:primal} is infeasible.\n  This gives an explanation of why these Chubanov-type algorithms usually run much faster if \\eqref{feasibility:primal} is infeasible.\n  \\item Then,  we explicitly develop the \\emph{dual algorithm} (see Section \\ref{sec:dual}) to improve the performance when \\eqref{feasibility:primal} is feasible.\n      Our dual algorithm is the first algorithm which rescales the row space of $A$ in MA (see Table \\ref{tab:comp}).\n      As a result, we provide a similar Lemma \\ref{lm:infeasibled} which shows that the time complexity $\\mathsf{T_{BP}}$ of our dual algorithm can be $O(n^{2.5}m)$ rather than $O(n^3m)$ in certain situations if \\eqref{feasibility:dual} is infeasible (i.e. \\eqref{feasibility:primal} is feasible).\n\n        Naturally, we obtain a new fast polynomial primal-dual projection algorithm (called $\\mathsf{PPDP}$) by integrating our primal algorithm (which runs faster on the primal infeasible instances) and our dual algorithm (which runs faster on the primal feasible instances). See Section \\ref{sec:ppdp}.\n  \\item Finally, the numerical results validate that our primal-dual $\\mathsf{PPDP}$ algorithm is quite balanced between feasible and infeasible instances, and it runs significantly faster than other algorithms (see Table \\ref{table:comparison} in Section \\ref{sec:exp}).\n\\end{enumerate}\n\n\\topic{Remark}\nOur algorithms are based on Dadush-V{\\'e}gh-Zambelli  algorithm \\citep{dadush2016rescaled} and the improvements of Roos's algorithm \\citep{roos2018improved} (see Section \\ref{sec:relatealg} and Table \\ref{tab:comp}).\nBesides, we introduce a new step-size term $c$ for practical consideration (see Line 13 and 14 of Algorithm \\ref{alg:bp} and \\ref{code:dual Dadush}).\nFor the maximum support problem \\eqref{feasibility:primal}, $\\mathsf{T_{BP}}=O(n^4)$ for Chubanov's algorithm and Roos's algorithm, and  $\\mathsf{T_{BP}}=O(n^3m)$ for Dadush-V{\\'e}gh-Zambelli  algorithm.\nNote that in the worst case $\\mathsf{T_{BP}}=O(n^3m)$ for our algorithms, but it can be improved by a factor of $\\sqrt{n}$ in certain situations.\nRecall that $\\mathsf{T_{MA}}=O(nL)$ for these Chubanov-type algorithms as we discussed before.\nThus the time complexity of our algorithms (in the worst case) match the result of Dadush-V{\\'e}gh-Zambelli  algorithm \\citep{dadush2016rescaled}, i.e., $O(\\mathsf{T_{MA}}*\\mathsf{T_{BP}})=O(n^4mL)$ (see our Theorems \\ref{thm:primal}--\\ref{thm:primal_dual}).\nHowever, we point out that the total time complexity of Chubanov's algorithm and Roos's algorithm are $O(n^4L)$, and hence is faster than ours. They speed up it from $O(n^5L)$ to $O(n^4L)$ by using an amortized analysis while we currently do not use. We leave this speedup as a future work.\n\n\\topic{Organization}\nIn Section \\ref{sec:pre}, we introduce some useful notations and review some related algorithms.\nThe details and results for our primal algorithm and dual algorithm are provided in Section \\ref{sec:primal} and Section \\ref{sec:dual}, respectively.\nThen, in Section \\ref{sec:ppdp}, we propose the efficient primal-dual $\\mathsf{PPDP}$ algorithm.\nFinally, we conduct the numerical experiments in Section \\ref{sec:exp} and include a brief conclusion in Section \\ref{sec:con}.\n\n\n\n\\section{Preliminaries}\n\\label{sec:pre}\n\nIn this section, we first review some classic basic procedures and then introduce some notations to review some related algorithms at the end of this section.\n\\subsection{Classic Basic Procedures} \\label{sec:bp}\nRecall that BP returns one of the following three results:\n(i) a feasible solution of \\eqref{feasibility:primal};\n(ii) a feasible solution of the dual problem \\eqref{feasibility:dual};\n(iii) a cut for the feasible region of \\eqref{feasibility:primal}.\nHere we focus on the first two outputs, the last one is controlled by an upper bound lemma (similar to Lemma \\ref{lm:roosbound}).\n\nLetting $y=Ax$, when solving \\eqref{feasibility:dual}, we know that there is at least an index $k$\nsuch that $a_k^Ty\\leq 0$, where $a_k$ is the $k$th-column of $A$ (otherwise $y$ is already a feasible solution for \\eqref{feasibility:dual}).\nOn the other hand, to solve \\eqref{feasibility:primal}, we want to minimize $\\n{y}$. The goal is to let $y$ go to 0 (in which case $x$ is a feasible solution for \\eqref{feasibility:primal}).\nWe review some classic update methods as follows:\n\n\\topic{von Neumann's algorithm}\nIn each iteration, find an index $k$ such that $a_k^Ty\\leq 0$, and then update $x$ and $y$ as\n\\begin{equation}\\label{eq:update}\n\\red{y'}=\\alpha y+\\beta a_k, \\quad x'=\\alpha x+\\beta e_k \\quad (\\mathrm{note~ that~}y=Ax \\mathrm{~and~} \\red{y'}=Ax'),\n\\end{equation}\nwhere $\\alpha,\\beta>0$ are chosen such that $\\|\\red{y'}\\|$ is smallest and $\\alpha+\\beta=1$ \\citep{dantzig1992varepsilon}.\n\\begin{figure}[!h]\n\\centering\\includegraphics[scale=0.8]{von.pdf}\n\\end{figure}\n\n\\topic{Perceptron}\nChoose $\\alpha=\\beta=1$ in \\eqref{eq:update} at every iteration. See e.g. \\citep{rosenblatt1957perceptron,novikoff1963convergence}.\n\n\\topic{Dunagan-Vempala}\nFix $\\alpha=1$ and choose $\\beta$ to minimize $\\n{\\red{y'}}$ \\citep{dunagan2008simple}.\n\n\\subsection{Notations}\nBefore reviewing the related algorithms (in the following Section \\ref{sec:relatealg}), we need to define\/recall some useful notations.\nWe use $P_A$ and $Q_A$ to denote the projections of ${\\mathbb R}^n$ onto the null space ($\\mathcal{N}_A$) and row space ($\\mathcal{R}_A$) of the $m\\times n$ matrix $A$, respectively:\n\\begin{equation*}\nP_A \\triangleq I-A^T (A A^T)^{\\dag}A,\\qquad Q_A \\triangleq A^T (A A^T)^{\\dag}A.\n\\end{equation*}\nwhere $(\\cdot)^{\\dag}$ denotes the Moore-Penrose pseudoinverse. Particularly, $(A A^T)^{\\dag}=(A A^T)^{-1}$ if $\\mathrm{rank}(A)=m$.\n\n\nWe further define the following notations:\n\\begin{equation}\n\\label{eq:vec}\nv=Q_Ay\\in \\mathcal{R}_A,\\quad z=P_Ay\\in \\mathcal{N}_A,\\quad y=v+z\\in {\\mathbb R}^n.\n\\end{equation}\nUsually, $z$ is used to denote the feasible solution of \\eqref{feasibility:primal} and $v$ indicates the feasibility of \\eqref{feasibility:dual}.\n\nTo analyze case (iii) of BP, we note that \\eqref{feasibility:primal} is feasible if and only if the system\n\\begin{equation}\n\\label{feasibility:normalized primal}\n\\begin{aligned}\nAx=0, ~x\\in [0,1]^n,~ x\\neq 0\n\\end{aligned}\n\\end{equation}\nis feasible since \\eqref{feasibility:primal} is a homogeneous system.\nFrom now on, we will consider problem \\eqref{feasibility:normalized primal} instead of \\eqref{feasibility:primal}. Similarly, we use a normalized version \\eqref{eq:bounded dual} to replace \\eqref{feasibility:dual}.\nNow, we recall a useful lemma which gives an upper bound for the coordinates of any feasible solution. This upper bound will indicate a cut for case (iii).\n\\begin{lemma}[\\citep{roos2018improved}]\n\\label{lm:roosbound}\nLet $x$ be any feasible solution of \\eqref{feasibility:normalized primal}, $y$ and $v$ are defined as in (\\ref{eq:vec}), then every non-zero coordinate $v_j$ of $v$ gives rise to an upper bound for $x_j$, according to\n\\begin{equation*}\nx_j\\leq \\mathrm{bound}_j(y)\\triangleq \\mathbf{1}^T\\Big[\\frac{v}{-v_j}\\Big]^+,\n\\end{equation*}\nwhere $x^+\\triangleq \\max\\{0,x\\}$ and $\\mathbf{1}$ denotes the all-ones vector.\n\\end{lemma}\nThis means that we can scale the column $j$ of $A$ by a factor $\\mathrm{bound}_j(y)$ to make the feasible solutions of (\\ref{feasibility:normalized primal}) closer to the all-ones vector $\\mathbf{1}$.\nSimilarly to $x^+$, we denote $x^-\\triangleq -(-x)^+$.\nFurthermore, we need the definition of condition number $\\rho(Q)$ for a matrix $Q$ \\citep{goffin1980relaxation}:\n\\begin{equation}\\label{eq:rho}\n\\rho(Q)\\triangleq \\max_{x:\\|x\\|_2=1}\\min_{i} \\langle x, \\frac{q_i}{\\|q_i\\|_2} \\rangle,\n\\end{equation}\nwhere $q_i$ is the $i$th-column of $Q$.\n\n\\subsection{Related Algorithms}\\label{sec:relatealg}\nNow, we are able to review some related algorithms for solving \\eqref{feasibility:normalized primal} based on the BP procedures introduced in Section \\ref{sec:bp}.\n\n\n\\topic{Chubanov's algorithm}\nInstead of updating in the original space $y=Ax$, \\cite{chubanov2015polynomial} updates in the projection space $z=\\blue{P_A}y$, where $P_A = I-A^T (A A^T)^{-1}A$ is a null space projection of $A$.\nIn each BP iteration, it updates $y$ and $z$ in the same way as von Neumann's update (just replacing $A$ by $P_A$).\nIntuitively, BP either finds a feasible solution $x^*$ of \\eqref{feasibility:normalized primal} or finds a cut (i.e., an index $j$ such that $x_j^*\\leq 1\/2$ for any feasible solution $x^*$ of \\eqref{feasibility:normalized primal} in $[0,1]^n$).\nThen the main algorithm MA rescales the null space of $A$ by dividing the $j$th-column of $A$ by 2.\nAccording to [Khachian, 1979], there is a lower bound for the feasible solutions of \\eqref{feasibility:normalized primal}.\nThus the number of rescaling operations can be bounded.\nFinally, the algorithm terminates in polynomial-time, where either BP returns a feasible solution or MA claims the infeasibility according to the lower bound.\n\n\\topic{Roos's algorithm} \\cite{roos2015chubanov,roos2018improved} provided two improvements of Chubanov's algorithm:\n\\begin{enumerate}\n  \\item A new cut condition was proposed, which is proved better than the one used by Chubanov.\n  \\item The BP can use \\emph{multiple indices} to update $z$ and $y$ ($z=P_Ay$) in each iteration, e.g., a set of indices satisfying $(P_A)_i^Tz\\leq 0$. Recall that von Neumann's update only uses one index $k$ satisfying $a_k^Ty\\leq 0$.\n\\end{enumerate}\n\n\\topic{Dadush-V{\\'e}gh-Zambelli}\nCompared with Chubanov's algorithm, \\cite{dadush2016rescaled} used the Dunagan-Vempala update instead of von Neumann's update as its BP, along with Roos' new cut condition. Besides, the updates are performed in the orthogonal space $v=\\blue{Q_A}y$, where $Q_A = A^T (A A^T)^{-1}A$ is a row space projection matrix of $A$.\nBut the rescaling space in MA is the same, i.e., the null space of $A$.\n\n\\topic{Comparison}\nTo demonstrate it clearly, we provide a comparison of our algorithms with other algorithms in Table \\ref{tab:comp}. Note that our primal-dual $\\mathsf{PPDP}$ algorithm is the integration of our primal algorithm and dual algorithm.\n\n\\begin{table}[!ht]\n  \\caption{Comparison of our algorithms with other algorithms}\n  \\vspace{1mm}\n  \\label{tab:comp}\n \\centering\n \\begin{tabular}{ccccc}\n  \\toprule\n    Algorithms  & Update method & Update space & Rescaling space&\n    \\#indices\\\\\n  \\midrule\n  Chubanov's algorithm &  von Neumann & Null space & Null space &One\\\\\n  Roos' algorithm     & von Neumann & Null space & Null space&Multiple\\\\\n  Dadush-V{\\'e}gh-Zambelli  & Dunagan-Vempala &Row space & Null space&One\\\\\n  Our primal algorithm& Dunagan-Vempala  & Row space& Null space&Multiple\\\\\n  Our dual  algorithm & Dunagan-Vempala  & Null space& Row space&Multiple\\\\\n  \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\newpage\n\\section{Our Primal Algorithm}\n\\label{sec:primal}\n\nIn this section, we introduce our primal algorithm which consists of the basic procedure BP and the main algorithm MA.\nThe details of BP and MA are provided in Section \\ref{sec:bpprimal} and Section \\ref{sec:maprimal} respectively.\n\n\\subsection{Basic Procedure (BP)}\\label{sec:bpprimal}\nOur BP is similar to \\citep{dadush2016rescaled} (or \\citep{dunagan2008simple}) (see Table \\ref{tab:comp}). The details are described in Algorithm \\ref{alg:bp}.\nThe main difference is that we use multiple indices $K$ to update (see Line 9 of Algorithm \\ref{alg:bp}) and introduce the step-size $c$ for practical consideration (see Line 13 and 14 of Algorithm \\ref{alg:bp}).\n\n\n\n\n\\begin{algorithm}[!htb]\n\t\\caption{Basic Procedure for the Primal Problem}\n\t\\label{alg:bp}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE $Q_A$\n        \\ENSURE\n            $y, z, J, \\mathrm{case}.$\n        \\STATE $r=size(Q_A), \\mathrm{threshold}=1\/2r^{3\/2}, c\\in (0,2), \\mathrm{case}=0$\n        \\STATE $y=\\mathbf{1}\/r, v=Q_Ay, z=y-v=P_Ay$\n\t\t\\WHILE{$\\mathrm{case}=0$}\n\t\t\\IF {$z>0$}\n\t\t    \\STATE $\\mathrm{case} = 1$ ($z$ is primal feasible); \\textbf{return}\n\t\t\\ELSIF {$v> 0\\; and\\; r==n$}\n            \\STATE $\\mathrm{case} = 2$ ($v$ is dual feasible); \\textbf{return}\n\t\t\\ELSE\n            \\STATE find $K=\\{k: v_k \\leq 0\\}$\n\t\t    \\STATE $q_K=Q_A \\sum_{k\\in K}e_k$\n\t\t    \\STATE $\\alpha=\\inner{\\frac{q_K}{\\|q_K\\|_2}}{v}$\n\t\t    \\IF {$\\alpha \\leq -\\mathrm{threshold}$}\n                \\STATE $y=y-c(\\frac{\\alpha}{\\|q_K\\|_2}\\sum_{k\\in K} e_k)$\n\t\t        \\STATE $v=v-c(\\frac{\\alpha}{\\|q_K\\|_2} \\sum_{k\\in K} q_k)$\n\t\t    \\ELSE\n\t\t        \\STATE find a nonempty set $J$ such that $J\\subseteq \\{j: bound_j(y)\\leq \\frac{1}{2} \\}$ (a cut); \\textbf{return}\n\t\t    \\ENDIF\n\t\t\\ENDIF\n\t\t\\ENDWHILE\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn the BP (Algorithm \\ref{alg:bp}), the norm of the iterated vector $v=Q_A y$ is decreasing, while each coordinate of $y$ is increasing.\nThus, after a certain number of iterations, we will obtain a feasible solution $z=y-v=P_Ay>0$.\nOtherwise, it is always possible to find a cut $J$ (Line 16), along with some rescaling operations for the matrix $A$, to make the feasible solutions of (\\ref{feasibility:normalized primal}) closer to the all-ones vector.\nThe cut is guaranteed by the following lemma.\n\\begin{lemma}\\label{lm:cut}\nLet $Q_A$ be the projection matrix at a given iteration of BP (Algorithm \\ref{alg:bp}).\nSuppose that $\\alpha=\\inner{\\frac{q_K}{\\|q_K\\|_2}}{v}> -\\mathrm{threshold}$, then the set $J=\\{j: \\mathrm{bound}_j(y)\\leq \\frac{1}{2} \\}$ is nonempty and every solution $x$ of problem (\\ref{feasibility:normalized primal}) satisfies $x_j\\leq \\frac12$ for all $j\\in J$.\n\\end{lemma}\n\nThis lemma is proved with Lemma \\ref{lm:roosbound} and we defer the proof to Appendix \\ref{app:lmcut}.\n\nFor the time complexity of Algorithm \\ref{alg:bp}, i.e. $\\mathsf{T_{BP}}$, we give the following lemma (the proof is in Appendix \\ref{app:tbp}).\n\\begin{lemma}\\label{lm:tbp}\nThe time complexity of Algorithm \\ref{alg:bp} $\\mathsf{T_{BP}}=O(n^3m)$. Concretely, it uses at most $O(n^2)$ iterations and each iteration costs at most $O(mn)$ time.\n\\end{lemma}\n\nNote that Lemma \\ref{lm:tbp} holds regardless \\eqref{feasibility:normalized primal} is feasible or infeasible. However, as we discussed before, the algorithm usually performs much better on the infeasible instances than on the feasible instances. To explain this phenomenon, we give the following lemma. The proof is deferred to Appendix \\ref{app:infeasible}.\n\\begin{lemma}\\label{lm:infeasible}\nIf \\eqref{feasibility:normalized primal} is infeasible,\nthe time complexity of Algorithm \\ref{alg:bp} $\\mathsf{T_{BP}}=O(n^2m\/\\rho(Q_A))$, where $\\rho(Q_A)$ is the condition number defined in (\\ref{eq:rho}).\nIn particular, $\\rho(Q_A)$ equals to $1\/\\sqrt{n}$ under well-condition (e.g., $A$ is an identity matrix), then $\\mathsf{T_{BP}}=O(n^{2.5}m)$ if problem \\eqref{feasibility:normalized primal} is infeasible.\n\\end{lemma}\n\n\n\n\\subsection{Main Algorithm (MA)}\\label{sec:maprimal}\nThe details of our MA are described in Algorithm \\ref{alg:ma}.\nParticularly, we rescale the null space of $A$ in Line 8.\n\\begin{algorithm}[!htb]\n\t\\caption{Main Algorithm for the Primal Problem}\n\t\\label{alg:ma}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE\n            $A\\in{\\mathbb R}^{m\\times n}, d=\\mathbf{1}, \\tau=2^{-L}, \\mathrm{case}=0, H=\\varnothing$.\n\t\t\\WHILE{$\\mathrm{case}=0$}\n            \\STATE $Q_A=A^T (A A^T)^{\\dag}A$\n            \\STATE $(y,z,J,\\mathrm{case})\\leftarrow$ Basic Procedure for Primal Problem$(Q_A)$\n\t\t\\IF {$\\mathrm{case}=0$}\n\t\t    \\STATE $d_J=d_J \/2$\n\t\t\n\t\t   \\STATE $H=\\{i:d_i\\leq \\tau\\}$\n\t\t   \\STATE $d_H=0$\n\t\t  \n\t\t   \\STATE $A_{J}=A_{J}\/2$\n\t\t   \\STATE $A=A_{\\overline{H}}$\n\t\t\n\t\t\\ENDIF\n\t\t\\ENDWHILE\n        \\IF{$\\mathrm{case}=1$}\n            \\STATE $d=d_{\\overline{H}}$\n            \\STATE $D=\\mathrm{diag}(d)$\n            \\STATE Define $x$ as $x_{\\overline{H}}=Dz, x_H=0$\n        \\ENDIF\n\t\\end{algorithmic}\n\\end{algorithm}\n\nNow, we state the complexity of our primal algorithm in the following theorem.\nThe proof is deferred to Appendix \\ref{app:thmprimal}\n\\begin{theorem}\\label{thm:primal}\nThe time complexity of the primal algorithm is $O(n^4mL)$.\n\\end{theorem}\n\n\n\n\n\\section{Our Dual Algorithm}\n\\label{sec:dual}\nThe Chubanov-type algorithms all focus on the primal problem (\\ref{feasibility:primal}) (or the normalized version \\eqref{feasibility:normalized primal}), i.e.,\ntheir MA always rescale the null space of $A$ (see Table \\ref{tab:comp}).\nWe emphasize that these algorithms usually perform much better on the infeasible instances than on the feasible ones (see our Lemma \\ref{lm:infeasible} which gives an explanation).\nNow, we want to address this unbalanced issue by providing a dual algorithm.\nOur dual algorithm explicitly considers the dual problem (\\ref{feasibility:dual}) and rescales the \\emph{row space} of $A$, unlike the previous algorithms.\nWe already know that the primal algorithm runs faster on the primal infeasible instances.\nThus we expect the dual algorithm runs faster on the dual infeasible instances (i.e., primal feasible instances).\nAs expected, our dual algorithm does work.\nTherefore, in Section \\ref{sec:ppdp}, we integrate our primal algorithm and dual algorithm to obtain a quite \\emph{balanced} primal-dual algorithm and its performance is also remarkably better than the previous algorithms.\n\nSimilar to our primal algorithm, the dual algorithm also consists of the basic procedure BP and the main algorithm MA.\nThe details of BP and MA are provided in Section \\ref{subsec:bpdual} and Section \\ref{subsec:madual} respectively.\nSimilar to \\eqref{feasibility:normalized primal}, we consider the normalized version of (\\ref{feasibility:dual}) due to the homogeneity:\n\\begin{equation}\n\\label{eq:bounded dual}\n\\mathrm{find }\\; u\\in {\\mathbb R}^m \\; \\mathrm{subject} \\; \\mathrm{to} \\; x=A^T u> 0, x\\in (0,1]^n.\n\\end{equation}\n\n\\subsection{Basic Procedure for the Dual Problem}\n\\label{subsec:bpdual}\n\nThe basic procedure for the dual problem is described in Algorithm \\ref{code:dual Dadush}.\n\\begin{algorithm}[!htb]\n\t\\caption{Basic Procedure for the Dual Problem}\n\t\\label{code:dual Dadush}\n\t\\begin{algorithmic}[1]\n        \\REQUIRE\n          $P_A$\n        \\ENSURE\n            $y, z, J, \\mathrm{case}.$\n        \\STATE $\\mathrm{threshold}=1\/2n^{3\/2}, c\\in (0,2), \\mathrm{case}=0$\n        \\STATE  $y = \\mathbf{1}\/n, z=P_Ay, v=y-z=Q_Ay$\n\t\t\\WHILE {$\\mathrm{case}=0$}\n\t\t\\IF {$v> 0$}\n            \\STATE $\\mathrm{case}=2$ (dual feasible); \\textbf{return}\n\t\t\\ELSIF {$z\\geq 0$}\n\t\t    \\STATE $\\mathrm{case}=1$ (primal feasible); \\textbf{return}\n\t\t\\ELSE\n\t\t    \\STATE find $K=\\{k:\\langle z, e_k\\rangle\\leq 0\\}$\n\t\t    \\STATE $p_K=P_A \\sum_{k\\in K}e_k$\n\t\t    \\STATE $\\alpha=\\langle \\frac{p_K}{\\|p_K\\|_2}, z\\rangle $\n\t\t    \\IF {$\\alpha \\leq -\\mathrm{threshold}$}\n\t\t        \\STATE $y=y-c(\\frac{\\alpha}{\\|p_K\\|_2}\\sum_{k\\in K} e_k)$\n                \\STATE $z=z-c(\\frac{\\alpha}{\\|p_K\\|_2} \\sum_{k\\in K} p_k)$\n                \\STATE $v=y-z$\n\t\t    \\ELSE\n\t\t        \\STATE find a nonempty set $J$ such that $J\\subseteq \\{j: \\mathrm{bound}'_j(y)\\leq \\frac{1}{2} \\}$ (a cut); \\textbf{return}\n\t\t    \\ENDIF\n\t\t\\ENDIF\n\t\t\\ENDWHILE\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn this basic procedure, either a feasible solution for the primal problem is found, or a dual feasible solution is found, or a cut of the bounded row space is found (which is denoted as $bound'_j(y)$ in Line 17).\nNow, we need to provide an upper bound in Lemma \\ref{lm:dualcut}, which shows that a cut of the bounded row space can be derived from $z$, instead of $v$ in the case of null space (see Lemma \\ref{lm:roosbound}).\n\\begin{lemma}\n\\label{lm:dualcut}\n\tLet $x$ be any feasible solution of \\eqref{eq:bounded dual} and $z=P_A y$ for some $y$. Then every non-zero coordinate $z_j$ of $z$ gives rise to an upper bound for $x_j$, according to\n\t\\[\n\tx_j \\leq \\mathrm{bound}'_j(y) \\triangleq \\bm{1}^T \\Big[\\frac{z}{-z_j}\\Big]^+.\n\t\\]\n\\end{lemma}\nThe proof of this lemma is deferred to Appendix \\ref{app:dualcut}.\nAccording to this lemma, we can obtain the following guaranteed cut in Lemma \\ref{lm:cutd}, which is similar to Lemma \\ref{lm:cut}.\nThe proof is almost the same as Lemma \\ref{lm:cut} just by replacing Lemma \\ref{lm:roosbound} with our Lemma \\ref{lm:dualcut}.\n\n\\begin{lemma}\\label{lm:cutd}\nLet $P_A$ be the projection matrix at a given iteration of BP (Algorithm \\ref{code:dual Dadush}).\nSuppose that $\\alpha=\\inner{\\frac{p_K}{\\|p_K\\|_2}}{z}> -\\mathrm{threshold}$, then the set $J=\\{j: \\mathrm{bound}'_j(y)\\leq \\frac{1}{2} \\}$ is nonempty and every solution $x$ of problem (\\ref{feasibility:normalized primal}) satisfies $x_j\\leq \\frac12$ for all $j\\in J$.\n\\end{lemma}\n\nSame to Algorithm \\ref{alg:bp}, for the time complexity $\\mathsf{T_{BP}}$ of the dual Algorithm \\ref{code:dual Dadush}, we have the following lemma.\n\\begin{lemma}\\label{lm:tbpd}\nThe time complexity of Algorithm \\ref{code:dual Dadush} $\\mathsf{T_{BP}}=O(n^3m)$. Concretely, it uses at most $O(n^2)$ iterations and each iteration costs at most $O(mn)$ time.\n\\end{lemma}\nNote that Lemma \\ref{lm:tbp} also holds regardless \\eqref{eq:bounded dual} is feasible or infeasible.\nNow, we want to point out that our dual algorithm can perform much better on the dual infeasible instances (primal feasible instances) under well-condition as we expected and discussed before.\nSimilar to Lemma \\ref{lm:infeasible}, we have the following lemma for the dual algorithm.\n\\begin{lemma}\\label{lm:infeasibled}\nIf \\eqref{eq:bounded dual} is infeasible,\nthe time complexity of Algorithm \\ref{alg:bp} $\\mathsf{T_{BP}}=O(n^2m\/\\rho(P_A))$, where $\\rho(P_A)$ is the condition number defined in (\\ref{eq:rho}).\nIn particular, $\\rho(P_A)$ equals to $1\/\\sqrt{n}$ under well-condition (e.g., $A=(I,-I)$, where $I$ is an identity matrix), then $\\mathsf{T_{BP}}=O(n^{2.5}m)$ if problem \\eqref{eq:bounded dual} is infeasible.\n\\end{lemma}\nNote that it is easy to see that \\eqref{feasibility:normalized primal} is feasible (i.e., \\eqref{eq:bounded dual} is infeasible) if $A=(I,-I)$.\n\n\nBesides,\nwhen the dual problem \\eqref{eq:bounded dual} is feasible,  we can also utilize the geometry of the problem to bound the iteration complexity instead of Lemma \\ref{lm:tbpd}.\nConsider the following kind of condition number of the set $Im(A)\\bigcap [0,1]^n$:\n\\[\n\\delta_{\\infty}(Im(A)\\bigcap [0,1]^n)\\triangleq \\max_{x}\\{\\prod_{i} x_i : x\\in Im(A)\\bigcap [0,1]^n\\}.\n\\]\n\nAs each rescaling in the basic procedure will at least enlarge the value of $\\delta_{\\infty}(Im(A)\\bigcap [0,1]^n)$ by two times, and the largest possible value of $\\delta_{\\infty}(Im(B)\\bigcap [0,1]^n)$ for all matrices $B$ is 1, it takes at most $-\\log_2 \\delta_{\\infty}(Im(A)\\bigcap [0,1]^n)$ basic procedures before getting a feasible solution. This means that the iteration complexity of the whole algorithm is $O(n^2 \\log \\frac{1}{\\delta_{\\infty}(Im(A)\\bigcap [0,1]^n)})$.\n\n\n\\subsection{Main Algorithm for the Dual Problem}\\label{subsec:madual}\n\nThe main algorithm for the dual problem is described in Algorithm \\ref{code:main procedure of dual Dadush}.\nParticularly, we rescale the row space of $A$ in Line 6.\n\n\\begin{algorithm}[!htb]\n\t\\caption{Main Algorithm for the Dual Problem}\n\t\\label{code:main procedure of dual Dadush}\n\t\\begin{algorithmic}[1]\n        \\REQUIRE\n            $A\\in{\\mathbb R}^{m\\times n}, d=\\mathbf{1}, \\tau=2^{-L}, \\mathrm{case}=0$.\n\t\t\\WHILE{$\\mathrm{case}=0$}\n            \\STATE $P_A=I-A^T (A A^T)^{\\dag}A$\n            \\STATE $(y,z,J,\\mathrm{case})\\leftarrow$ Basic Procedure for Dual Problem$(P_A)$\n\t\t\\IF {$\\mathrm{case}=0$}\n\t\t    \\STATE $d_J=2d_J$\n            \\STATE $A_J=2A_J$\n\t\t    \\STATE $H=\\{i:d_i\\geq 2^L\\}$\n            \\STATE $d_H=0$\n\t\t    \\STATE $A_H=0$\n\t\t\\ENDIF\n\t\t\\ENDWHILE\n        \\IF{$\\mathrm{case}=1$}\n            \\STATE $D=\\mathrm{diag}(d)$\n            \\STATE $x=Dz$\n        \\ENDIF\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\newpage\nNow, we have the following theorem for our dual algorithm.\nThe proof is provided in Appendix \\ref{app:thmdual}.\n\\begin{theorem}\\label{thm:dual}\nThe time complexity of the dual algorithm is $O(n^4mL)$.\n\\end{theorem}\n\n\n\\section{Our Primal-Dual $\\mathsf{PPDP}$ Algorithm}\n\\label{sec:ppdp}\nIn this section, we propose a new polynomial primal-dual projection algorithm (called $\\mathsf{PPDP}$) to take advantages of our primal algorithm and dual algorithm.\nSimilarly, the $\\mathsf{PPDP}$ algorithm also consists of two procedures (MA and BP).\nIntuitively, the BP solves problems (\\ref{feasibility:primal}) and (\\ref{feasibility:dual}) simultaneously.\nRecall that the primal algorithm runs faster on the infeasible instances and the dual algorithm runs faster on the feasible instances (see Table \\ref{table:comparison}).\nThe MA rescales the matrix (row space or null space) according to the output of BP.\nThe MA and BP are formally described in Algorithms  \\ref{code:main procedure of primal-dual} and \\ref{code:primal dual} respectively. The details are deferred to Appendix \\ref{app:algo}.\nThus, we have the following theorem.\n\n\\begin{theorem}\\label{thm:primal_dual}\nThe time complexity of our primal-dual $\\mathsf{PPDP}$ algorithm is $O(n^4mL)$.\n\\end{theorem}\n\nNote that the final output of our $\\mathsf{PPDP}$ algorithm is a feasible solution for either (\\ref{feasibility:primal}) or  (\\ref{feasibility:dual}).\nObviously, the algorithm will stop whenever it finds a solution of \\eqref{feasibility:primal} or \\eqref{feasibility:dual}, thus the time complexity of our $\\mathsf{PPDP}$ algorithm follows easily from Theorems \\ref{thm:primal} and \\ref{thm:dual}.\n\n\\section{Experiments}\n\\label{sec:exp}\nIn this section, we compare the performance of our algorithms with Roos' algorithm \\citep{roos2018improved} and Gurobi (one of the fastest solvers nowadays).\nWe conduct the experiments on the randomly generated matrices.\nConcretely, we generate $100$ integer matrices $A$ of size $625\\times 1250$, with each entry uniformly randomly generated in the interval $[-100,100]$. The parameter $c\\in (0,2)$ is the step-size which is a new practical term introduced in this work.\nThe average running time of these algorithms are listed in Table \\ref{table:comparison}.\n\n\\begin{table}[!htb]\n  \\caption{Running time (sec.) of algorithms wrt. (\\ref{feasibility:primal}) is feasible or infeasible}\n  \\vspace{1mm}\n \\label{table:comparison}\n \\centering\n \\begin{tabular}{ccc}\n  \\toprule\n  Algorithms  & \tfeasible instances & \tinfeasible instances \\\\\n  \\midrule\n  Gurobi (a fast optimization solver)  & 3.08  &1.58 \\\\\n  Roos's algorithm \\citep{roos2018improved}\n  &  10.75          & 0.83     \\\\\n  Our primal algorithm ($c=1.8$) & 9.93     &  \\red{0.48}  \\\\\n  Our dual algorithm ($c=1.8$) & \\red{0.35}          &  4.57    \\\\\n  Our $\\mathsf{PPDP}$ algorithm ($c=1.8$) & \\red{0.60}          & \\red{0.58}     \\\\\n  \\bottomrule\n \\end{tabular}\n\\end{table}\n\nTable \\ref{table:comparison} validates that our new primal-dual $\\mathsf{PPDP}$ algorithm is quite balanced on the feasible and infeasible instances due to the integration of our primal and dual algorithm.\nMoreover, it shows that our $\\mathsf{PPDP}$ algorithm can be a practical option for linear programming since it runs remarkably faster than the fast optimization solver Gurobi.\n\n\\section{Conclusion}\n\\label{sec:con}\nIn this paper, we try to theoretically explain why the Chubanov-type projection algorithms usually run much faster on the primal infeasible instances.\nFurthermore, to address this unbalanced issue, we provide a new fast polynomial primal-dual projection algorithm (called $\\mathsf{PPDP}$) by integrating our primal algorithm (which runs faster on the primal infeasible instances) and our dual algorithm (which runs faster on the primal feasible instances).\nAs a start, we believe more improvements (e.g., the amortized analysis speedup) can be made for the Chubanov-type projection algorithms both theoretically and practically.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank Jian Li (Tsinghua University), Yuanxi Dai (Tsinghua University) and Rong Ge (Duke University) for useful discussions.\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}  \\label{S:Intro}\n\nThe simple-multiplicity case of the BiTangential Nevanlinna-Pick \n(BTNP) Interpolation Problem  over the right half plane $\\Pi_{+} = \\{ \nz \\in {\\mathbb C} \\colon \\operatorname{Re} z > 0\\}$ can be formulated \nas follows. Let ${\\mathcal S}^{p \\times m}(\\Pi_{+})$ denote the Schur class of\n$\\mathbb C^{p\\times m}$-valued functions that are analytic and contractive-valued\non $\\Pi_+$:\n$$\n {\\mathcal S}^{p \\times m}(\\Pi_{+}) : = \\{ S \\colon \\Pi_{+} \\to {\\mathbb C}^{p\n \\times m} \\colon \\| S(\\lambda) \\| \\le 1 \\text{ for all } \\lambda \\in\n \\Pi_{+} \\}.\n$$\nThe data set ${\\mathfrak D}_{\\rm simple}$ for the problem consists of\na collection of the form\n\\begin{align}\n{\\mathfrak D}_{\\rm simple} =  \\{ & z_{i} \\in \\Pi_{+}, \\;  x_{i} \\in \n{\\mathbb C}^{1 \\times p},\\; y_{i} \\in {\\mathbb C}^{1 \\times m}\\;  \n\\text{ for }\\; i=1,\\ldots,N, \\notag \\\\\n&w_j\\in \\Pi_{+}, \\;  u_{j} \\in {\\mathbb C}^{m \\times 1},\\, v_{j} \\in {\\mathbb C}^{p\n  \\times 1}\\;  \\text{ for } \\; j = 1, \\dots, N',\\notag \\\\\n  & \\rho_{ij} \\in {\\mathbb C} \\text{ for } (i,j) \\text{ such that }\n  z_{i} = w_{j} =: \\xi_{ij}\\}. \\label{babydata}\n  \\end{align}\n  The problem then is to find a function $S\\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ \nthat satisfies the collection of interpolation conditions\n\\begin{align}\n  &  x_{i} S(z_{i}) = y_{i} \\;  \\text{ for } \\; i = 1, \\dots, N, \n    \\label{inter1'} \\\\\n &  S(w_{j}) u_{j} = v_{j}\\; \\text{ for } \\; j = 1, \\dots, N', \n \\label{inter2'} \\\\\n & x_{i} S'(\\xi_{ij}) y_{j} = \\rho_{ij} \\; \\text{ for } (i,j) \\; \\text{ \n such that } \\; z_{i} = w_{j} =: \\xi_{ij}.  \\label{inter3'}\n\\end{align}\nWe note that the existence of a solution $S$ \nto interpolation conditions \\eqref{inter1'}, \\eqref{inter2'}, \n\\eqref{inter3'} forces the data set \\eqref{babydata} to satisfy \nadditional compatibility equations; indeed, if $S$ solves \n\\eqref{inter1'}--\\eqref{inter3'}, and if $(i,j)$ \nis a pair of indices where $z_{i} = w_{j} =: \\xi_{ij}$, then the \nquantity $x_{i} S(\\xi_{ij}) u_{j}$ can be computed in two ways:\n\\begin{align*}\n    & x_{i} S(\\xi_{ij}) u_{j} = (x_{i} S(\\xi_{ij})) u_{j} = y_{i} \n    u_{j}, \\\\\n& x_{i} S(\\xi_{ij}) u_{j} = x_{i} (S(\\xi_{ij}) u_{j}) = x_{i} v_{j}\n\\end{align*}\nforcing the compatibility condition\n\\begin{equation}   \\label{babycomp}\n    x_{i} v_{j} = y_{i} u_{j} \\; \\text{ if } \\; z_{i} = w_{j}.\n\\end{equation}\nMoreover, there is no loss of generality in assuming that each row \nvector $x_{i}$ and each column vector $u_{j}$ in \\eqref{babydata} is nonzero;\nif $x_{i}= 0$ for some $i$, existence of a solution $S$ then forces also that \n$y_{i}=0$ and then the interpolation condition $x_{i} S_{i}(z_{i}) = \ny_{i}$ collapses to $0 = 0$ and can be discarded, with a similar \nanalysis in case some $u_{j} = 0$. \n\n\\smallskip\n\nThe following result gives the precise solution criterion.  The \nresult actually holds even without the normalization conditions on the data \nset discussed in the previous paragraph.\n\n\n\\begin{theorem}  \\label{T:BT-NP}\n{\\rm (See \\cite[Section 4]{LA} for the case where $z_{i} \\ne w_{j}$ for all $i,j$).} \nGiven a data set ${\\mathfrak  D}_{\\rm simple}$ as in \\eqref{babydata}, there \nexists a solution $S$ of the associated problem {\\rm BTNP} if and only if the \nassociated Pick matrix\n   \\begin{equation}  \\label{Pick-simple}\n       P_{{\\mathfrak D}_{\\rm simple}} : = \\begin{bmatrix} \n    P_{11} & P_{12} \\\\ P_{12}^{*} & P_{22} \\end{bmatrix}\n    \\end{equation}\n    with entries given by\n    \\begin{align*}\n&\t[P_{11}]_{ij} = \\frac{x_{i} x_{j}^{*} - y_{i} \ny_{j}^{*}}{z_{i} + \\overline{z}_{j}} \\; \\text{ for } \\; 1 \\le i,j \\le N, \\\\\n&  [P_{12}]_{ij} = \\left\\{\\begin{array}{cl}\n {\\displaystyle\\frac{x_{i}v_{j} - y_{i}u_{j}}{ w_{j}-z_{i}}} & \\text{ if } \\; \n z_{i} \\ne w_{j},  \\\\ \\rho_{i j} & \\text{ if } z_{i} = w_{j}, \\end{array}\\right. \n \\text{ for } 1  \\le i \\le N, 1 \\le j \\le N', \\\\\n& [ P_{22}]_{ij} = \\frac{ u_{i}^{*} u_{j} - \nv_{i}^{*} v_{j}}{  \\overline{w}_{i} +  w_{j} } \\; \\text{ for } \\; 1 \\le \ni, j \\le N',\n\\end{align*}\nis positive semidefinite.\n\\end{theorem}\n\nGiven a data set ${\\mathfrak D}_{\\rm simple}$ as above, \nit is convenient to repackage it in a more aggregate form \nas follows (see \\cite{BGR}). With data as in  \\eqref{babydata}, \nform the septet of matrices\n$(Z,X,Y, W, U, V, \\Gamma)$ where:\n\\begin{align}\n & Z = \\begin{bmatrix} z_{1} & & 0 \\\\ & \\ddots & \\\\ 0 & & z_{N} \n\\end{bmatrix}, \\; \\; \n X = \\begin{bmatrix} x_{1} \\\\ \\vdots \\\\ x_{N} \\end{bmatrix}, \\; \\;  \n Y = \\begin{bmatrix} y_{1} \\\\ \\vdots \\\\  y_{N} \\end{bmatrix}, \\notag \\\\\n& W = \\begin{bmatrix} w_{1} & & 0 \\\\ & \\ddots & \\\\0 & & w_{N'} \n\\end{bmatrix}, \\; \\; \nU = \\begin{bmatrix} u_{1} & \\cdots & u_{N'} \\end{bmatrix}, \\; \\;\nV = \\begin{bmatrix} v_{1} & \\cdots & v_{N'} \\end{bmatrix}, \\notag \\\\\n& \\Gamma = \\left[ \\gamma_{ij} \\right]_{i=1,\\ldots N}^\n{j=1,\\ldots,N^\\prime} \\; \\text{ where } \\; \n\\gamma_{ij} =  \\left\\{\\begin{array}{cl}\n {\\displaystyle\\frac{x_{i}v_{j} - y_{i}u_{j}}{ w_{j}-z_{i}}} & \\text{ if } \\;\n z_{i} \\ne w_{j},  \\\\ \\rho_{i j} & \\text{ if } z_{i} = w_{j}. \n\\end{array}\\right.\n\\label{babydata'}\n\\end{align}\nNote that the compatibility condition \\eqref{babycomp} translates to \nthe fact that $\\Gamma$ satisfies the Sylvester equation\n$$\n\\Gamma W-Z\\Gamma   = \\begin{bmatrix} X & -Y \\end{bmatrix} \n    \\begin{bmatrix} V \\\\ U \\end{bmatrix}.\n$$\nThe normalization requirements ($x_{i} \\ne 0$ for all $i$ and $u_{j} \n\\ne 0$ for all $j$ together with $z_{1}, \\dots, z_{N}$ all distinct \nand $w_{1}, \\dots, w_{N'}$ all distinct)  translate to the conditions\n$$\n (Z,X) \\text{ is controllable}, \\quad (U, W) \\text{ is observable.}\n$$\nThen it is not hard to see that the interpolation conditions \n\\eqref{inter1'}, \\eqref{inter2'}, \\eqref{inter3'} can be written in \nthe more aggregate form \n \\begin{align} \n\t& \\sum_{z_{0} \\in \\sigma(Z)} {\\rm Res}_{\\lambda = z_{0}} (\\lambda I \n\t- Z)^{-1} X  S(\\lambda) = Y, \\label{resint1} \\\\\n& \\sum_{z_{0} \\in \\sigma(W)} {\\rm Res}_{\\lambda = z_{0}} S(\\lambda) U (\\lambda \nI - W)^{-1} = V, \\label{resint2} \\\\\n& \\sum_{z_{0} \\in \\sigma(Z)\\cup \\sigma(W)} {\\rm Res}_{\\lambda = z_{0}} \n(\\lambda I - Z)^{-1} \nX S(\\lambda) U (\\lambda I - W)^{-1} = \\Gamma. \\label{resint3}\n\\end{align}\n\nSuppose that $(Z,X)$ is any controllable input pair and that $(U,W)$ \nis an observable output pair.  Assume in addition that $\\sigma(Z) \n\\cup \\sigma(W) \\subset \\Pi_{+}$ and that $S$ is an analytic matrix \nfunction (of appropriate size) on $\\Pi_{+}$.  We define the \n\\textbf{Left-Tangential Operator Argument (LTOA) point evaluation} \n$(XS)^{\\wedge L}(Z)$ of $S$ at $Z$ in left direction $X$ by\n$$\n  (X S)^{\\wedge L}(Z) = \\sum_{z_{0}\\in\\sigma(Z)} {\\rm Res}_{\\lambda = z_{0}}\n  (\\lambda I - Z)^{-1} X  S(\\lambda).\n$$\nSimilarly we define the \\textbf{Right-Tangential Operator Argument \n(RTOA) point evaluation} $(S U)^{\\wedge R}(W)$ of $S$ at $W$ in right \ndirection $U$ by\n$$\n  (S U)^{\\wedge R}(W) = \\sum_{z_{0} \\in \\sigma(W)} {\\rm Res}_{\\lambda = z_{0}}\n  S(\\lambda) U (\\lambda I - W)^{-1}.\n$$\nFinally the \\textbf{BiTangential Operator Argument (BTOA) point \nevaluation} $(X S U)^{\\wedge L,R}(Z, W)$ of $S$ at left argument $Z$ \nand right argument $W$ in left direction $X$ and right direction $U$ \nis given by\n$$\n(X S U)^{\\wedge L,R}(Z, W) =\n\\sum_{z_{0} \\in \\sigma(Z)\\cup \\sigma(W)} {\\rm Res}_{\\lambda = z_{0}}\n(\\lambda I - Z)^{-1} X S(\\lambda) U (\\lambda I - W)^{-1}.\n$$\nWith this condensed notation, we write the interpolation conditions \n\\eqref{resint1}, \\eqref{resint2}, \\eqref{resint3} simply as\n\\begin{align} \n    & (X S)^{\\wedge L}(Z) = Y, \\label{BTOAint1} \\\\\n    & (S U)^{\\wedge R} (W) = V, \\label{BTOAint2} \\\\\n    & (X S U)^{\\wedge L,R}(Z,W) = \\Gamma. \\label{BTOAint3}\n\\end{align}\nLet us say that the data set \n\\begin{equation}   \\label{data}\n    {\\mathfrak D} = (Z,X,Y; U,V,W; \\Gamma)\n\\end{equation}\nis a {\\em $\\Pi_{+}$-admissible BiTangential Operator Argument (BTOA)\ninterpolation data set} \nif the following conditions hold:\n\\begin{enumerate}\n    \\item Both $Z$ and  $W$ have spectrum inside $\\Pi_{+}$:  \n    $\\sigma(Z) \\cup \\sigma(W) \\subset \\Pi_{+}$.\n    \\item $(Z,X)$ is controllable and $(U,W)$ is observable.\n    \\item $\\Gamma$ satisfies the Sylvester equation\n \\begin{equation}   \\label{Sylvester'}\n    \\Gamma W - Z \\Gamma = X V - Y U.\n \\end{equation}\n  \\end{enumerate}\n Then it makes sense to consider the collection of interpolation \n conditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3} for \n {\\em any} $\\Pi_{+}$-admissible BTOA interpolation data set $(Z,X,Y; \n U,V,W; \\Gamma)$.  It can be shown that these interpolation \n conditions can be expressed equivalently as a set of \n higher-order versions of the  interpolation conditions \n \\eqref{inter1'}, \\eqref{inter2'}, \\eqref{inter3'}\n (see \\cite[Theorem 16.8.1]{BGR}), as well as a representation of $S$ in \n the so-called Model-Matching form (see \\cite[Theorem 16.9.3]{BGR}, \n \\cite{Francis})\n $$\n   S(\\lambda) = T_{1}(\\lambda) + T_{2}(\\lambda) Q(\\lambda) T_{3}(\\lambda),\n $$\n where $T_{1}$, $T_{2}$, $T_{3}$ are rational matrix functions \n analytic on $\\Pi_{+}$ with $T_{2}$ and $T_{3}$ square and analytic \n and invertible along the imaginary line, and where $Q$ is a \n free-parameter matrix function analytic on all of $\\Pi_{+}$. \n \n\\smallskip\n\n It is interesting to note that the Sylvester equation \n \\eqref{Sylvester'} is still necessary for the existence of a \n $p \\times m$-matrix function $S$ analytic on $\\Pi_{+}$ satisfying the \n BTOA interpolation conditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \n \\eqref{BTOAint3}.  Indeed, note that\n  \\begin{align*}\n &   \\left( (\\lambda I - Z)^{-1} X S(\\lambda) \n  U (\\lambda I - W)^{-1}\\right) W -\n  Z \\left( (\\lambda I - Z)^{-1} X S(\\lambda) U (\\lambda I - W)^{-1} \\right) \\\\\n  & = (\\lambda I - Z)^{-1} X S(\\lambda) \n  U (\\lambda I - W)^{-1} (W - \\lambda I + \\lambda I) \\\\\n  &  \\quad \\quad + (\\lambda I - Z - \\lambda I)\n  (\\lambda I - Z)^{-1} X S(\\lambda) U (\\lambda I - W)^{-1}  \\\\\n  & = -(\\lambda I - Z)^{-1} X S(\\lambda)  U  +\n  \\lambda \\cdot  (\\lambda I - Z)^{-1} X S(\\lambda)  U (\\lambda I - W)^{-1} \\\\\n  & \\quad \\quad + X S(\\lambda) U (\\lambda I - W)^{-1} - \n  \\lambda \\cdot  (\\lambda I - Z)^{-1} X S(\\lambda)  U (\\lambda I - W)^{-1} \\\\\n  & =  -(\\lambda I - Z)^{-1} X S(\\lambda)  U  + X S(\\lambda) U (\\lambda I - \n  W)^{-1}.\n \\end{align*}\n If we now take the sum of the residues of the first and last \n expression in this chain of equalities over points $z_{0} \\in \n \\Pi_{+}$ and use the interpolation conditions \n \\eqref{resint1}--\\eqref{resint3}, we arrive at\n $$\n  \\Gamma W - Z \\Gamma = -Y U + X V\n $$ \n and the Sylvester equation \\eqref{Sylvester'} follows.\n \n\\smallskip \n \n \n We now pose the \\textbf{BiTangential Operator Argument Nevanlinna-Pick \n (BTOA-NP) Interpolation Problem}:  Given a $\\Pi_{+}$-admissible BTOA \n interpolation data set \\eqref{data}, find $S$ in the \n matrix Schur class over the right half plane ${\\mathcal S}^{p \\times \n m}(\\Pi_{+})$ which satisfies the BTOA interpolation conditions \n \\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3}.  \n \n\\smallskip\n\n Before formulating the solution, we need some additional \n notation.  Given a $\\Pi_{+}$-admissible BTOA interpolation data set \n \\eqref{data},  introduce two additional matrices $\\Gamma_{L}$ and \n $\\Gamma_{R}$ as the unique solutions of the respective Lyapunov \n equations\n \\begin{align}\n& \\Gamma_{L} Z^{*} + Z \\Gamma_{L} = X X^{*} - Y Y^{*}, \\label{GammaL}\\\\\n& \\Gamma_{R} W + W^{*} \\Gamma_{R} = U^{*} U - V^{*}V.      \\label{GammaR} \n \\end{align}\n We define the  BTOA-Pick matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ associated \n with the data set \\eqref{data} by\n \\begin{equation}   \\label{GammaData}\n     \\boldsymbol{\\Gamma}_{\\mathfrak D} = \\begin{bmatrix} \\Gamma_{L} & \\Gamma\n     \\\\ \\Gamma^{*} & \\Gamma_{R} \\end{bmatrix}.\n \\end{equation}\n The following is the canonical generalization of Theorem \\ref{T:BT-NP} \n to this more \n general situation.\n \n \\begin{theorem}  \\label{T:BTOA-NP}  Suppose that \n $$\n {\\mathfrak D}  = ( X,Y,Z; U,V, W; \\Gamma)\n $$\n is a $\\Pi_{+}$-admissible BTOA interpolation data set.   Then there \n exists a solution $S \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ of the BTOA-NP \n interpolation problem associated with data set ${\\mathfrak D}$ if \n and only if the associated  BTOA-Pick matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$\n defined by \\eqref{GammaData} is positive semidefinite.\n \n In case $\\Gamma_{\\mathfrak D}$ is strictly positive definite \n ($\\Gamma_{\\mathfrak D} \\succ 0$), the set of all solutions is \n parametrized as follows.  Define a $(p+m) \\times (p+m)$-matrix \n function \n $$\n \\Theta(\\lambda) =  \\begin{bmatrix} \\Theta_{11}(\\lambda) & \n \\Theta_{12}(\\lambda) \\\\ \\Theta_{21}(\\lambda) & \\Theta_{22}(\\lambda) \n\\end{bmatrix}\n$$ via\n\\begin{equation}  \\label{Theta}\n\\Theta(\\lambda) = \\begin{bmatrix} I_{p} & 0 \\\\ 0 & I_{m} \\end{bmatrix} +\n\\begin{bmatrix}-X^{*} & V  \\\\ -Y^{*} & U  \\end{bmatrix}\n    \\begin{bmatrix} \\lambda I + Z^{*} & 0 \\\\ 0 & \\lambda I - W \n    \\end{bmatrix}^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1}  \n\\begin{bmatrix} X & -Y \\\\-V ^{*} & U^{*}  \\end{bmatrix}. \n\\end{equation}\nThen $S$ is a solution of the BTOA-NP interpolation problem if and \nonly if $S$ has a representation as\n\\begin{equation}   \\label{LFTparam}\nS(\\lambda) = (\\Theta_{11}(\\lambda) G(\\lambda) + \\Theta_{12}(\\lambda))\n( \\Theta_{21}(\\lambda) G(\\lambda) + \\Theta_{22}(\\lambda))^{-1}\n\\end{equation}\nwhere $G$ is a free-parameter function in the \nSchur class ${\\mathcal S}^{p \\times m}(\\Pi_{+})$.\n\\end{theorem}\nNote that the first part of Theorem \\ref{T:BTOA-NP} for the special \ncase where the data set ${\\mathfrak D}$ has the  \nform \\eqref{babydata'} coming from the data set \\eqref{babydata} for a \nBT-NP problem amounts to the content of Theorem \\ref{T:BT-NP}.\n\n\\smallskip\n\nThe BTOA-NP interpolation problem and closely related problems have \nbeen studied and analyzed using a variety of methodologies by  number \nof authors, especially in the 1980s and 1990s, largely inspired by \nconnections with the then emerging $H^{\\infty}$-control theory (see \n\\cite{Francis}).  We mention in \nparticular the Schur-algorithm approach in \\cite{LA, DD, AD}, the method of \nFundamental Matrix Inequalities by the \nPotapov school (see e.g., \\cite{KP})) and the related formalism of the \nAbstract Interpolation Problem of Katsnelson-Kheifets-Yuditskii \n(see \\cite{KKY, Kh}), the Commutant Lifting approach of \nFoias-Frazho-Gohberg-Kaashoek (see \\cite{FF, FFGK}, and the \nReproducing Kernel approach of Dym and collaborators (see \\cite{Dym, \nDymOT143}).  Our focus here is to revisit two other approaches:  \n(1) the Grassmannian\/Kre\\u{\\i}n-space-geometry \napproach of Ball-Helton \\cite{BH}, and (2) the state-space implementation \nof this approach due to Ball-Gohberg-Rodman (\\cite{BGR}).  The first \n(Grassmannian) approach  relies on Kre\\u{\\i}n-space geometry to arrive at the \nexistence of a solution; the analysis is constructive only after one \nintroduces bases to coordinatize various subspaces and operators. The \nsecond (state-space) approach has the same starting point as the first \n(encoding the problem in terms of the graph of the sought-after solution rather \nthan in terms of the solution itself), but finds state-space \ncoordinates in which to coordinatize the $J$-inner function \nparametrizing the set of solutions and then verifies the \nlinear-fractional parametrization by making use of intrinsic \nproperties of $J$-inner functions together with an explicit \nwinding-number argument, thereby bypassing any appeal to general \nresults from Kre\\u{\\i}n-space geometry.  This second approach \nproved to be more accessible to users (e.g., engineers) who were not comfortable with \nthe general theory of Kre\\u{\\i}n spaces.\n\n\\smallskip\n\nIt turns out that the solution criterion  $\\boldsymbol{\\Gamma}_{\\mathfrak D}\\succeq 0$\narises more naturally in the second (state-space) approach.  \nFurthermore, when $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$ \n($\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is strictly positive \ndefinite), one gets a linear-fractional parametrization for the set \nof all Schur-class solutions of the interpolation conditions. The \nmatrix function $\\Theta$ generating the linear-fractional map also \ngenerates a matrix kernel function $K_{\\Theta,J}$ which is a positive \nkernel exactly when $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$.  We can then \nview the fact that the associated reproducing kernel space \n${\\mathcal H}(K_{\\Theta,J})$ is a Hilbert space as also a solution criterion \nfor the BTOA-NP interpolation problem in the nondegenerate case.\n\n\\smallskip\n\nIn the first (Grassmannian\/Kre\\u{\\i}n-space-geometry) \napproach, on the other hand, the immediate solution criterion is in terms of the \npositivity of a certain finite-dimensional subspace  $({\\mathcal M}_{\\mathfrak \nD}^{[\\perp {\\mathcal K}]})_{0}$ of a Kre\\u{\\i}n space constructed \nfrom the interpolation data ${\\mathfrak D}$.  In the Left Tangential \ncase, one can identify $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ as the Kre\\u{\\i}n-space \ngramian matrix with respect to a natural basis for  \n$({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})_{0}$, thereby confirming directly \nthe equivalence of the two seemingly distinct solution criteria.  For the general \nBiTangential case, the connection between $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ and  \n$({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})_{0}$ is not so direct, but nevertheless, \nusing ideas from \\cite{BHMesa}, we present here a direct proof as to why \n$\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succeq 0$ is equivalent to Kre\\u{\\i}n-space positivity \nof $({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})_{0}$ which is interesting in its own right. \nAlong the way, we also show how the Fundamental Matrix Inequality approach \nto interpolation of the Potapov school \\cite{KP} can be incorporated into this \nBTOA-interpolation formalism to give an alternative derivation of the linear-fractional \nparametrization which also bypasses the winding-number argument, at \nleast for the classical Schur-class setting.  We also sketch how all \nthe results extend to the more general problem where one seeks \nsolutions of the BTOA interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3} \nin the Kre\\u{\\i}n-Langer generalized Schur class ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$ \nwith the integer $\\kappa$ as small as possible.\n\n\\smallskip\n\nThe plan of the paper is as follows.  In Section \\ref{S:proof1} we \nsketch the ideas of the second (state-space) approach, with the \nFundamental Matrix Inequality approach and the reproducing-kernel \ninterpretation dealt with in succeeding subsections.  In Section \n\\ref{S:proof2} we sketch the somewhat more involved ideas behind the \nfirst (Grassmannian\/Kre\\u{\\i}n-space-geometry) approach.  In Section \n\\ref{S:synthesis} we identify the connections between the two \napproaches and in particular show directly that the two solution \ncriteria are indeed equivalent.  In the final Section \n\\ref{S:negsquares} we indicate how the setup extends to interpolation \nproblems for the generalized Schur class ${\\mathcal S}_{\\kappa}^{p \\times \nm}(\\Pi_{+})$.\n\n\n\n\n\\section{The state-space approach to the BTOA-NP interpolation problem}  \\label{S:proof1}\n\nIn this section we sketch the analytic proof of Theorem  \\ref{T:BTOA-NP} from \\cite{BGR}.\nFor ${\\mathcal U}$ and ${\\mathcal Y}$ Hilbert spaces, we let ${\\mathcal L}({\\mathcal U}, {\\mathcal Y})$ denote the\nspace of bounded linear operators mapping ${\\mathcal U}$ into ${\\mathcal Y}$,\nabbreviated to ${\\mathcal L}({\\mathcal U})$ in case ${\\mathcal U} = {\\mathcal Y}$.  We then define the\noperator-valued version of the Schur class ${\\mathcal S}_{\\Omega}({\\mathcal U}, {\\mathcal Y})$\nto consist of holomorphic functions $S$ on $\\Omega$\nwith values equal to contraction operators between ${\\mathcal U}$ and ${\\mathcal Y}$. \n\n\\smallskip\n\nWe first recall some standard facts concerning positive kernels and \nreproducing kernel Hilbert spaces (see e.g., \\cite{BB-HOT}).\nGiven a point-set $\\Omega$  and coefficient Hilbert space ${\\mathcal Y}$ along \nwith a function $K \\colon \\Omega \\times \\Omega \\to {\\mathcal L}({\\mathcal Y})$, we say \nthat $K$ is a {\\em positive kernel on $\\Omega$} if\n\\begin{equation}   \\label{posker}\n    \\sum_{i,j=1}^{N}\\langle K(\\omega_{i}, \\omega_{j}) y_{j}, y_{i} \n    \\rangle_{{\\mathcal Y}} \\ge 0\n\\end{equation}\nfor any collection of $N$ points $\\omega_{1}, \\dots, \\omega_{N} \\in \n\\Omega$ and vectors $y_{1}, \\dots, y_{N} \\in {\\mathcal Y}$ with arbitrary \n$N\\ge 1$. It is well known  that \nthe following are equivalent:\n\\begin{enumerate}\n    \\item $K$ is a positive kernel on $\\Omega$.\n    \\item $K$ is the {\\em reproducing kernel} for a {\\em reproducing kernel \n    Hilbert space} ${\\mathcal H}(K)$ consisting of  functions $f\\colon \\Omega \n    \\to {\\mathcal Y}$ such that, for each $\\omega \\in \\Omega$ and $y \\in {\\mathcal Y}$ the \n\tfunction $k_{\\omega,y} \\colon \\Omega \\to {\\mathcal Y}$ defined by\n\\begin{equation}  \\label{RKHSa}\nk_{\\omega,y}(\\omega') = K(\\omega', \\omega)y\n\\end{equation}\nis in ${\\mathcal H}(\\Omega)$ and has the reproducing property: for each $f \\in \n{\\mathcal H}(K)$, \n\\begin{equation}   \\label{RKHSb}\n\\langle f, k_{\\omega,y}\\rangle_{{\\mathcal H}(K)} = \\langle f(\\omega), y \n\\rangle_{{\\mathcal Y}}.\n\\end{equation}\n\\item $K$ has a {\\em Kolmogorov decomposition}:  there is a Hilbert \nspace ${\\mathcal X}$ and a function $H \\colon \\Omega \\to {\\mathcal L}({\\mathcal X}, {\\mathcal Y})$ so that\n\\begin{equation}   \\label{Kolmogorov}\n    K(\\omega', \\omega) = H(\\omega') H(\\omega)^{*}.\n\\end{equation}\n\\end{enumerate}\n\n\\begin{proof}[Proof of Theorem \\ref{T:BTOA-NP}]  We first illustrate the \n    proof of necessity for the easier \n     simple-multiplicity case as formulated \n    in Theorem \\ref{T:BT-NP}; the idea is essentially the same as the necessity proof \n    in Limebeer-Anderson \\cite{LA}.\n\n\\smallskip\n \nIt is well known that a Schur-class function $F \\in{\\mathcal S}_{\\mathbb D}({\\mathcal U},{\\mathcal Y})$ \non the unit disk can be characterized not only by the positivity of \nthe de Branges-Rovnyak kernel \n$$\n  {\\mathbf K}_{F}(\\lambda, w) = \\frac{ I - F(\\lambda) F(w)^{*}}{1 - z \n  \\overline{\\zeta}}\n$$\non the unit disk ${\\mathbb D}$, but also by positivity of the\nblock $2 \\times 2$-matrix kernel defined on $({\\mathbb D} \\times \n    {\\mathbb D}) \\times ({\\mathbb D} \\times {\\mathbb D})$ by\n$$\n   \\widetilde  {\\mathbf K}_{F}(\\lambda, \\lambda_{*}; w, w_{*}): = \\begin{bmatrix} \n   {\\displaystyle\\frac{ I -  F(\\lambda) \n    F(w)^{*}}{1 - \\lambda \\overline{w}}} & {\\displaystyle\\frac{F(\\lambda) - \n    F(\\overline{w}_{*})}{\\lambda - \\overline{w}_{*}}}\\vspace{1mm}\\\\\n   {\\displaystyle\\frac{F(\\overline{\\lambda}_{*})^{*} - \n   F(w)^{*}}{\\lambda_{*} - \\overline{w}}} &\n   {\\displaystyle\\frac{I - F(\\overline{\\lambda}_{*})^{*} F(\\overline{w}_{*})}{1 - \\lambda_{*} \n   \\overline{w}_{*}}} \n\\end{bmatrix}.\n$$\nMaking use of the linear-fractional change of variable from ${\\mathbb D}$ to $\\Pi_{+}$\n$$\n    \\lambda \\in {\\mathbb D} \\mapsto z = \\frac{ 1+ \\lambda}{1- \\lambda} \\in \\Pi_{+}\n$$\nwith inverse given by\n$$\n  z \\in \\Pi_{+} \\mapsto \\lambda = \\frac{z -1}{z +1} \\in {\\mathbb D},\n$$\nit is easily seen that the function $S$ defined on $\\Pi_{+}$ is in the \nSchur class ${\\mathcal S}_{\\Pi_+}({\\mathcal U},{\\mathcal Y})$ over $\\Pi_{+}$ if and only if, not \nonly the $\\Pi_{+}$-de Branges-Rovnyak kernel\n\\begin{equation}   \\label{dBRkerPi+}\nK_{S}(z, \\zeta) = \\frac{I - S(z) S(\\zeta)^{*}}{z + \\overline{\\zeta}}\n\\end{equation}\nis a positive kernel on $\\Pi_{+}$, but also the ($2 \\times 2$)-block de \nBranges-Rovnyak kernel\n\\begin{equation}  \\label{ker22}\n  {\\mathbf K}_{S}(z, z_{*}; \\zeta, \\zeta_{*}): = \\begin{bmatrix} \n{\\displaystyle\\frac{ I - S(z) S(\\zeta)^{*}}{ z + \\overline{\\zeta}}} &\n{\\displaystyle\\frac{S(z) - S(\\overline{\\zeta}_{*})}{z - \\overline{\\zeta}_{*}}} \n\\vspace{1mm}\\\\  {\\displaystyle\\frac{ S(\\overline{z}_{*})^{*} - S(\\zeta)^{*}}\n{ z_{*} - \\overline{\\zeta}}} &\n{\\displaystyle\\frac{I - S(\\overline{z}_{*})^{*} S(\\overline{\\zeta}_{*})}\n{ z_{*} + \\overline{\\zeta}_{*}}}\n\\end{bmatrix}\n\\end{equation}\nis a positive kernel on $(\\Pi_{+}\\times\\Pi_+) \\times (\\Pi_+\\times \\Pi_{+})$.  Specifying the \nlatter kernel at the points $(z, z_{*}), (\\zeta, \\zeta_{*}) \\in \\Pi_{+} \\times \\Pi_{+}$ \nwhere  $z, \\zeta = z_{1}, \\dots, z_{N}$ and $z_{*}, \\zeta_{*} = \\overline{w}_{1}, \\dots, \n\\overline{w}_{N'}$, leads to the conclusion that the block matrix\n\\begin{equation}  \\label{Pick-pre}\n\\begin{bmatrix} \\left[ {\\displaystyle\\frac{I - S(z_{i}) \n    S(z_{j})^{*}}{z_{i} + \\overline{z}_{j}}} \\right]  &\n   \\left[ {\\displaystyle\\frac{S(z_{i}) - S(w_{j'})}{ z_{i} - w_{j'}}} \\right] \n   \\vspace{1mm}\\\\\n   \\left[ {\\displaystyle\\frac{S(w_{i'})^{*} - S(z_{j})^{*}}{\\overline{w}_{j'} - \n   \\overline{z}_{i}}} \\right] & \\left[ {\\displaystyle\\frac{ I - S(w_{i'})^{*} \n   S(w_{j'})}{\\overline{w}_{i'} + w_{j'}}} \\right] \\end{bmatrix},\n\\end{equation}\nwhere $1 \\le i,j \\le N$ and $1 \\le i', j' \\le N'$, is positive \nsemidefinite. Note that the entry  $\\frac{S(z_{i}) - S(w_{j'})}{ \nz_{i} - w_{j'}}$ in the upper right corner is to be interpreted as \n$S'(\\xi_{i j'})$ in case $z_{i} = w_{j'} =: \\xi_{i,j'}$ for some pair \nof indices $i,j'$.  \n\n\\smallskip\n\nSuppose now that $S \\in {\\mathcal S}_{\\Pi_{+}}({\\mathcal U}, {\\mathcal Y})$ is a \nSchur-class solution of the interpolation conditions \\eqref{inter1'}, \n\\eqref{inter2'}, \\eqref{inter3'}.  When we multiply the matrix \n\\eqref{Pick-pre} on the left by the block diagonal matrix\n$$\n  \\begin{bmatrix} \\operatorname{diag}_{1 \\le i \\le N}[ x_{i}] & 0 \\\\ \n      0 & \\operatorname{diag}_{1 \\le i' \\le N'} [ u_{i'}^{*}] \n  \\end{bmatrix}\n$$\nand on the right by its adjoint, we arrive at the matrix \n$P_{{\\mathfrak D}_{\\rm simple}}$ \\label{Psimple}.  This verifies the \nnecessity of the condition $P_{{\\mathfrak D}_{\\rm simple}} \\succeq 0$ \nfor a solution of the BT-NP interpolation problem to exist.\n\n\\smallskip\n\nWe now consider the proof of necessity for the general case.\nWe note that the proof of necessity in \n    \\cite{BGR} handles explicitly only the case where the Pick matrix \n    is invertible and relies on use of the matrix-function $\\Theta$ \n    generating the linear-fractional parametrization (see \n    \\eqref{oct1} below).  We give a proof \n    here which proceeds directly from the BTOA-interpolation formulation; it \n   amounts to a specialization of the proof of necessity for the more \n   complicated multivariable interpolation problems in the Schur-Agler \n   class done in \\cite{BB05}.      \n\n\\smallskip\n\nThe starting point is the observation that the positivity of the \nkernel ${\\mathbf K}_{S}$ implies that it has a Kolmogorov \ndecomposition \\eqref{Kolmogorov};  furthermore the extra structure of the arguments of \nthe kernel ${\\mathbf K}_{S}$ implies  that the Kolmogorov \ndecomposition can be taken to have the form\n\\begin{equation}   \\label{KS2}\n {\\mathbf K}_{S}(z, z_{*}; \\zeta, \\zeta_{*}) =\n \\begin{bmatrix} H(z) \\\\ G(z_{*})^* \\end{bmatrix}\n     \\begin{bmatrix} H(\\zeta)^{*} & G(\\zeta_{*}) \\end{bmatrix}\n\\end{equation}\nfor holomorphic operator functions\n$$  \nH \\colon \\Pi_{+} \\to {\\mathcal L}({\\mathcal X}, {\\mathcal Y}), \\quad G \\colon \\Pi_{+} \\to {\\mathcal L}({\\mathcal U}, {\\mathcal X}).\n$$\nIn the present matricial setting of $\\mathbb C^{p\\times m}$-valued functions, \nthe spaces ${\\mathcal U}$ and ${\\mathcal Y}$ are finite dimensional and can be identified\nwith $\\mathbb C^p$ and $\\mathbb C^m$, respectively.  \nIn particular we read off the identity\n\\begin{equation}\\label{Gamma'12}\n H(z)G(\\zeta) =  \\frac{ S(z) - S(\\zeta)}{z - \\zeta}\n \\end{equation}\nwith appropriate interpretation in case $z = \\zeta$. Observe that \nfor a fixed $\\zeta\\in \\Pi_+\\backslash\\sigma(Z)$, we have from \\eqref{Gamma'12}\n\\begin{align}\n(XH)^{\\wedge L}(Z)\\cdot G(\\zeta)\n&=\\sum_{z_{0} \\in \\sigma(Z)} {\\rm Res}_{\\lambda \n= z_{0}} (\\lambda I- Z)^{-1} X H(\\lambda)G(\\zeta)\\notag\\\\ \n&=\\sum_{z_{0} \\in \\sigma(Z)} {\\rm Res}_{\\lambda = z_{0}} (\\lambda I- Z)^{-1} X  \n\\frac{ S(\\lambda) - S(\\zeta)}{\\lambda - \\zeta}\\notag\\\\\n&=\\sum_{z_{0} \\in \\sigma(Z)} {\\rm Res}_{\\lambda = z_{0}} (\\lambda I- Z)^{-1}\n\\frac{Y - XS(\\zeta)}{\\lambda - \\zeta}\\notag\\\\\n&=(\\zeta I-Z)^{-1}(XS(\\zeta)-Y)\\label{aug1}\n\\end{align}\nwhere we used the interpolation condition \\eqref{resint1} for the third equality.\nSince the function $g(\\zeta)=(\\zeta I-Z)^{-1}YU(\\zeta I-W)^{-1}$ \nsatisfies an estimate of the form\n$ \\|g(\\zeta \\| \\le \\frac{M}{ |\\zeta|^{2}}\\; $ as $\\; |\\zeta| \\to \\infty$, it follows that\n$$\n\\sum_{z_{0} \\in \\sigma(Z)\\cup\\sigma(W)}{\\rm Res}_{\\zeta= z_{0}}\n(\\zeta I-Z)^{-1}YU(\\zeta I-W)^{-1}=0.\n$$\nOn the other hand, due to condition \\eqref{resint1}, the function on the \nright hand side of \\eqref{aug1}\nis analytic (in $\\zeta$) on $\\Pi_+$, so that \n\\begin{align*}\n&\\sum_{z_{0} \\in \\sigma(W)} {\\rm Res}_{\\zeta=z_0}\n(\\zeta I-Z)^{-1}(XS(\\zeta)-Y)U(\\zeta I-W)^{-1}\\\\\n&=\\sum_{z_{0} \\in \\sigma(Z)\\cup\\sigma(W)} \n{\\rm Res}_{\\zeta=z_0}(\\zeta I-Z)^{-1}(XS(\\zeta)-Y)U(\\zeta I-W)^{-1}.\n\\end{align*}\nWe now apply the {\\bf RTOA} point evaluation to both sides in \\eqref{aug1} and\nmake use of the two last equalities and the interpolation condition \\eqref{resint3}:\n\\begin{align}\n&(XH)^{\\wedge L}(Z)(GU)^{\\wedge R}(W)\\notag \\\\\n&=\\sum_{z_{0} \\in \\sigma(W)} {\\rm Res}_{\\zeta=z_0}(\\zeta I-Z)^{-1}\n(XS(\\zeta)-Y)U(\\zeta I-W)^{-1}\\notag\\\\\n&=\\sum_{z_{0} \\in \\sigma(Z)\\cup\\sigma(W)} {\\rm Res}_{\\zeta=z_0}\n(\\zeta I-Z)^{-1}(XS(\\zeta)-Y)U(\\zeta I-W)^{-1}\\notag\\\\\n&=\\sum_{z_{0} \\in \\sigma(Z)\\cup\\sigma(W)} \n{\\rm Res}_{\\zeta=z_0}(\\zeta I-Z)^{-1}XS(\\zeta)U(\\zeta I-W)^{-1}=\\Gamma.\n\\label{aug2}\n\\end{align}\nLet us now introduce the block $2 \\times 2$-matrix ${\\mathbf \\Gamma}'_{\\mathfrak \nD}$ by\n\\begin{equation} \n  \\boldsymbol{\\Gamma}'_{\\mathfrak D} = \\begin{bmatrix} (XH)^{\\wedge L}(Z) \n  \\\\ (GU)^{\\wedge R}(W)^{*} \\end{bmatrix}\n  \\begin{bmatrix} ((XH)^{\\wedge L}(Z) )^{*} & (GU)^{\\wedge R}(W) \n  \\end{bmatrix}.\n\\label{aug4}\n\\end{equation}\nWe claim that $\\boldsymbol{\\Gamma}'_{\\mathfrak D} = \\boldsymbol{\\Gamma}_{\\mathfrak D}$.  Note \nthat equality of the off-diagonal blocks follows from \n\\eqref{aug2}. It remains to show the two equalities\n\\begin{align}\n\\Gamma'_{L}&:=(XH)^{\\wedge L}(Z) ((XH)^{\\wedge L}(Z))^{*}= \\Gamma_{L}, \\label{verifyGamma}\\\\\n\\Gamma'_{R}&:=((G^{*}U)^{\\wedge R}(W) )^{*} (G^{*}U)^{\\wedge R}(W)=\n   \\Gamma_{R}.\\label{verifyGamma1}\n\\end{align}\nTo verify \\eqref{verifyGamma}, we note \nthat $\\Gamma_{L}$ is defined as the unique solution of the Lyapunov \nequation \\eqref{GammaL}.  Thus it suffices to verify that $\\Gamma'_{L}$\nalso satisfies \\eqref{GammaL}.  Toward this end, the two expressions \\eqref{ker22} and \n\\eqref{KS2} for ${\\mathbf K}_{S}$ give us equality of the \n$(1,1)$-block entries:\n$$\n  H(z) H(\\zeta)^{*} = \\frac{ I - S(z) S(\\zeta)^{*}}{ z + \n  \\overline{\\zeta}}\n$$\nwhich we prefer to rewrite in the form\n\\begin{equation}  \\label{identity}\n z \\cdot H(z) H(\\zeta)^{*} + H(z) H(\\zeta)^{*} \\cdot \n \\overline{\\zeta} = I - S(z) S(\\zeta)^{*}.\n\\end{equation}\nTo avoid confusion, let us introduce the notation $\\chi$ for the \nidentity function $\\chi(z) = z$ on $\\Pi_{+}$.  Then it is \neasily verified that\n\\begin{equation}  \\label{identity'}\n ( X \\chi \\cdot H)^{\\wedge L}(Z) = Z (X H)^{\\wedge L}(Z).\n\\end{equation}\nMultiplication on the left by $X$ and on the right by $X^{*}$ and \nthen plugging in the left operator argument $Z$ for $\\lambda$ in \\eqref{identity} then gives\n\\begin{align*}\n& Z (X H)^{\\wedge L}(Z) H(\\zeta)^{*}X^{*} + (X H)^{\\wedge L}(Z) ( \\zeta \n \\cdot H(\\zeta)^{*}X^{*} \\\\\n & \\quad = XX^{*} - (X S)^{\\wedge L}(Z) S(\\zeta)^{*}  = XX^{*} - Y S(\\zeta)^{*}X^{*}.\n\\end{align*}\nReplacing the variable $\\zeta$ by the operator argument $Z$ and \napplying the adjoint of the identity \\eqref{identity'} then brings us to\n$$\nZ (XH)^{\\wedge L}(Z)( (XH)^{\\wedge L}(Z))^{*} Z^{*} = \nX X^{*} - Y \\left( X S^{\\wedge L}(Z) \\right)^{*} = X X^{*} - Y Y^{*},\n$$\ni.e., $\\Gamma'_{L}$ satisfies \\eqref{GammaL} as wanted.  The proof \nthat $\\Gamma'_{R}$ (see \\eqref{verifyGamma1}) \nsatisfies \\eqref{GammaR} proceeds in a similar way.\n \n\\smallskip\n\nFor the sufficiency direction, for simplicity we shall assume \n    that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is strictly positive definite rather \n    than just positive semidefinite. We then must show that \n    solutions $S$ of the BTOA-NP problem exist and in fact the set of \n    all solutions is given by the linear-fractional parametrization\n    \\eqref{LFTparam}.  The case where the Pick matrix is \n    positive-semidefinite then follows by perturbing the semidefinite \n    Pick matrix to a definite Pick matrix and using an approximation \n    and normal families argument.  The ideas follow \\cite{BGR}.\n    \n\\smallskip\n\nLet us therefore assume that $\\Gamma_{\\mathfrak D}$ is positive \ndefinite.  Then we can form the rational matrix function $\\Theta$ \ngiven by \\eqref{Theta}.  Let us write $\\Theta$ in the more condensed \nform \n\\begin{equation}\n\\Theta(\\lambda) = I_{p+m} - {\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak \nD}^{-1} {\\mathbf C}^{*} J\n\\label{oct1}\n\\end{equation}\nwhere we set\n\\begin{equation}\n{\\mathbf A} = \\begin{bmatrix} -Z^{*} & 0 \\\\ 0 & W \\end{bmatrix}, \\quad\n{\\mathbf C} = \\begin{bmatrix} -X^{*} & V \\\\ -Y^{*} & U \\end{bmatrix},\\quad \nJ = \\begin{bmatrix} I_{p} & 0 \\\\ 0 & -I_{m} \\end{bmatrix}.\n\\label{oct2}\n\\end{equation}\nRecall that $\\Gamma_{L}$, $\\Gamma_{R}$, $\\Gamma$ satisfy the \nLyapunov\/Sylvester equations \\eqref{GammaL}, \\eqref{GammaR}, \n\\eqref{Sylvester'}.  Consequently one can check that \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}$ satisfies the $(2 \\times 2)$-block \nLyapunov\/Sylvester  equation\n\\begin{align*}\n  &  \\begin{bmatrix} \\Gamma_{L} & \\Gamma \\\\ \\Gamma^{*} & \\Gamma_{R} \n\t\\end{bmatrix} \\begin{bmatrix} -Z^{*} & 0 \\\\ 0 & W \\end{bmatrix}\n+ \\begin{bmatrix} -Z & 0 \\\\ 0 & W^{*} \\end{bmatrix} \\begin{bmatrix} \n\\Gamma_{L} & \\Gamma \\\\ \\Gamma^{*} & \\Gamma_{R} \\end{bmatrix} \\\\\n& \\quad = \\begin{bmatrix} Y Y^{*} - X X^{*} & XV - YU \\\\ V^{*} X^{*} - U^{*} \nY^{*} & V^{*} U - V^{*} V \\end{bmatrix},\n\\end{align*}\nor, in more succinct form,\n\\begin{equation}   \\label{bigLySyl}\n  \\boldsymbol{\\Gamma}_{\\mathfrak D} {\\mathbf A} + {\\mathbf A}^{*} \\boldsymbol{\\Gamma}_{\\mathfrak D} = - {\\mathbf C}^{*} J {\\mathbf C}.\n\\end{equation}\nUsing this we compute\n\\begin{align*}\n &   J - \\Theta(\\lambda) J \\Theta(\\zeta)^{*}  = \\\\\n&  J - \\left( I - {\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} \n {\\mathbf C}^{*} J \\right) J \\left( I - J {\\mathbf C} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} \n (\\overline{\\zeta} I - {\\mathbf A}^{*})^{-1} {\\mathbf C}^{*} \\right) \\\\\n & ={\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} {\\mathbf C}^{*}\n + {\\mathbf C} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} (\\overline{\\zeta} I -{\\mathbf A}^{*})^{-1} \n {\\mathbf C}^{*} \\\\\n & \\quad \\quad \\quad \\quad - {\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak \n D}^{-1} {\\mathbf C}^{*} J {\\mathbf C} \\Gamma_{\\mathfrak D}^{-1} (\\overline{\\zeta} I \n - {\\mathbf A}^{*})^{-1} {\\mathbf C}^{*} \\\\\n & = {\\mathbf C} (\\lambda I - A)^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} \\Xi(\\lambda,\\zeta) (\\overline{\\zeta} I - \n {\\mathbf A}^{*})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} {\\mathbf C}^{*}\n \\end{align*}\n where\n$$\n     \\Xi(\\lambda,\\zeta) =(\\overline{\\zeta} I - {\\mathbf A}^{*}) \\boldsymbol{\\Gamma}_{\\mathfrak D} \n     + \\boldsymbol{\\Gamma}_{\\mathfrak D} (\\lambda I - {\\mathbf A}) - {\\mathbf C}^{*} J {\\mathbf C}= (\\lambda + \\overline{\\zeta}) \n     \\boldsymbol{\\Gamma}_{\\mathfrak D},\n$$\n where we used \\eqref{bigLySyl} in the last step. We conclude that \n \\begin{equation}\n K_{\\Theta,J}(\\lambda.\\zeta):= \\frac{J - \\Theta(\\lambda) J \\Theta(\\zeta)^{*}}\n {\\lambda + \\overline{\\zeta}}=  \n{\\mathbf C}(\\lambda I - {\\mathbf A})^{ -1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} \n   (\\overline{\\zeta}I - {\\mathbf A}^{*})^{-1} {\\mathbf C}^{*}.\n \\label{ad1}\n\\end{equation}\n By assumption, $\\sigma(Z)\\cup\\sigma(W)\\subset \\Pi_+$, so   \nthe matrix ${\\mathbf A} = \\sbm{ -Z^{*} & 0 \\\\ 0 & W }$ has no \n eigenvalues on the imaginary line, and hence $\\Theta$ is analytic and \n invertible on $i {\\mathbb R}$.  As a consequence of \\eqref{ad1}, \n we see that  $\\Theta(\\lambda)$ is $J$-coisometry for $\\lambda \n \\in i{\\mathbb R}$.  As $J$ is a finite matrix we actually have (see \\cite{AI}): \n \\begin{itemize}\n     \\item {\\em for $\\lambda \\in i{\\mathbb R}$, $\\Theta(\\lambda)$ is \n     $J$-unitary:}\n \\begin{equation}  \\label{ThetaJunitary}\n     J - \\Theta(\\lambda)^{*} J \\Theta(\\lambda) = J - \\Theta(\\lambda) J \n     \\Theta(\\lambda)^{*} = 0 \\; \\text{ for } \\; \\lambda \\in i{\\mathbb R}.\n \\end{equation}\n  \\end{itemize}\n The significance of the assumption that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is \n not only invertible but also positive definite is that\n \\begin{itemize}\n     \\item {\\em for $\\lambda \\in \\Pi_{+}$ a point of analyticity for \n     $\\Theta$, $\\Theta(\\lambda)$ is $J$-bicontractive:}\n \\begin{equation} \\label{ThetaJcontr}\n\tJ - \\Theta(\\lambda)^{*} J \\Theta(\\lambda) \\succeq 0, \\quad\n\tJ - \\Theta(\\lambda) J \\Theta(\\lambda)^{*} \\succeq 0 \\text{ for } \n\t\\lambda \\in \\Pi_{+}.\n\\end{equation}\n \\end{itemize}\n Here we make use of the fact that $J$-co-contractive is equivalent \n to $J$-contractive in the matrix case (see \\cite{AI}). \nThese last two observations have critical consequences.  Again \n   writing out $\\Theta$ and $J$ as\n$$ \\Theta(\\lambda) = \\begin{bmatrix} \\Theta_{11}(\\lambda) & \n\\Theta_{12}(\\lambda) \\\\ \\Theta_{21}(\\lambda) & \\Theta_{22}(\\lambda) \n\\end{bmatrix}, \\quad J = \\begin{bmatrix} I_{p} & 0 \\\\ 0 & -I_{m} \n\\end{bmatrix},\n$$\nrelations \\eqref{ThetaJunitary} and \\eqref{ThetaJcontr}) \ngive us (with the variable $\\lambda$ suppressed)\n $$\n   \\begin{bmatrix} \\Theta_{11} \\Theta_{11}^{*} - \\Theta_{12} \n       \\Theta_{12}^{*} & \\Theta_{11} \\Theta_{21}^{*} - \\Theta_{12} \n       \\Theta_{22}^{*} \\\\\n  \\Theta_{21} \\Theta_{11}^{*} - \\Theta_{22} \\Theta_{12}^{*} &\n  \\Theta_{21} \\Theta_{21}^{*} - \\Theta_{22} \\Theta_{22}^{*} \n\\end{bmatrix} \\preceq \\begin{bmatrix} I_{p} & 0 \\\\ 0 & - I_{m} \n\\end{bmatrix}\n$$\nfor $\\lambda$ a point of analyticity of $\\Theta$ in $\\Pi_{+}$ with equality for \n$\\lambda$ in $i {\\mathbb R} = \\partial \\Pi_{+}$ (including the point \nat infinity). In particular,\n$$\n  \\Theta_{21} \\Theta_{21}^{*} - \\Theta_{22} \\Theta_{22}^{*} \\preceq \n  -I_{m}\n$$\nor equivalently,\n\\begin{equation}   \\label{Theta22inv}\n  \\Theta_{21} \\Theta_{21}^{*} + I_{m} \\preceq \\Theta_{22} \n  \\Theta_{22}^{*}.\n\\end{equation}\nHence, $\\Theta_{22}(\\lambda)$ is invertible at all points \n$\\lambda$ of analyticity in $\\Pi_+$, namely, $\\Pi_{+} \\setminus \n\\sigma(W)$, and then, since multiplying \non the left by $\\Theta_{22}^{-1}$ and on the right by its adjoint \npreserves the inequality, we get \n\\begin{equation} \\label{Theta22invTheta21}\n  \\Theta_{22}^{-1} \\Theta_{21} \\Theta_{21}^{*} \\Theta_{22}^{*-1} + \n  \\Theta_{22}^{-1} \\Theta_{22}^{* -1} \\preceq I_{m}.\n\\end{equation}\nWe conclude: \n\\begin{itemize}\n    \\item {\\em $\\Theta_{22}^{-1}$ has analytic continuation to a \ncontractive $m \\times m$-matrix function on all of $\\Pi_{+}$ and \n$\\Theta_{22}^{-1} \\Theta_{21}$ has analytic continuation to an \nanalytic $m \\times p$-matrix rational function which is pointwise \nstrictly contractive on the closed right half \nplane $\\overline{\\Pi}_{+} = \\Pi_{+} \\cup i{\\mathbb R}$.}\n\\end{itemize}\n\nIt remains to make the connection of $\\Theta$ with the BTOA-NP interpolation \nproblem.  Let us introduce some additional notation.\nFor $N$ a positive integer, \n$ H^{2}_{N}(\\Pi_{+})$ is short-hand notation for the ${\\mathbb \nC}^{N}$-valued Hardy space $H^{2}(\\Pi_{+}) \\otimes {\\mathbb C}^{N}$ \nover the right half plane $\\Pi_{+}$.  Similarly $L^{2}_{N}(i \n{\\mathbb R}) = L^{2}(i {\\mathbb R}) \\otimes {\\mathbb C}^{N}$ is the \n${\\mathbb C}^{N}$-valued $L^{2}$-space over the imaginary line $i {\\mathbb R}$.  \n\n\\smallskip\n\nIt is well known (see e.g.\\ \\cite{Hoffman}) that the space \n$H^{2}_{N}(\\Pi_{+})$ (consisting of analytic functions on $\\Pi_{+}$) \ncan be identified with \na subspace of $L^{2}_{N}(i {\\mathbb R})$ (consisting of measurable \nfunctions on $i {\\mathbb R}$ defined only almost everywhere with \nrespect to linear Lebesgue measure) via the process of taking \nnontangential limits from $\\Pi_{+}$ to a point on $i {\\mathbb R}$.  \nSimilarly the Hardy space $H^{2}_{N}(\\Pi_{-})$ over the left half \nplane can also be identified with a subspace (still denoted as \n$H^{2}_{N}(\\Pi_{-})$) of $L^{2}_{N}(i {\\mathbb R})$, and, after these \nidentifications, $H^{2}_{N}(\\Pi_{-}) = H^{2}_{N}(\\Pi_{+})^{\\perp}$ \nas subspaces of $L^{2}_{N}(i {\\mathbb R})$:\n$$\n  L^{2}_{N}(i {\\mathbb R}) = H^{2}_{N}(\\Pi_{+}) \\oplus \n  H^{2}_{N}(\\Pi_{-}).\n$$\nWe shall use these identifications freely in the discussion to follow.\nGiven the  $\\Pi_{+}$-admissible interpolation data set\n\\eqref{data},  we define a  subspace of \n$L^{2}_{p+m}(i{\\mathbb R})$ by\n\\begin{align}\n    {\\mathcal M}_{\\mathfrak D} = &\\left \\{ \\begin{bmatrix} V \\\\ U  \\end{bmatrix} (\\lambda I - W)^{-1} x + \n    \\begin{bmatrix} f(\\lambda) \\\\ \ng(\\lambda) \\end{bmatrix}  \\colon x \\in {\\mathbb C}^{n_{W}} \\text{ and } \n\\begin{bmatrix} f \\\\ g \\end{bmatrix} \\in H^{2}_{p+m}(\\Pi_{+} )  \n    \\right. \\notag \\\\\n& \\left. \\text{ such that }  \n \\sum_{z_{0} \\in \\Pi_{+}} {\\rm Res}_{\\lambda = z_{0}} (\\lambda I - Z)^{-1} \n\\begin{bmatrix} X & -Y \\end{bmatrix}   \\begin{bmatrix} f(\\lambda) \\\\ \n    g(\\lambda) \\end{bmatrix}  = \\Gamma x \\right\\}.\n    \\label{cMrep}\n\\end{align}\nand a subspace of $L^{2}_{m}(i {\\mathbb R})$ by \n$$ \n{\\mathcal M}_{{\\mathfrak D}, -}=\\{ U (\\lambda I - W)^{-1} x \\colon x \\in {\\mathbb C}^{n_{W}} \\}\n    \\oplus H^{2}_{m}(\\Pi_{+}).\n$$\nUsing $\\Pi_{+}$-admissibility assumptions  on the data set ${\\mathfrak D}$ one can show\n(we refer to \\cite{BGR} for details, subject to the disclaimer in \nRemark \\ref{R:Amaya} below) that \n$$\n{\\mathcal M}_{{\\mathfrak D}, -}= P_{\\sbm{ 0 \\\\  L^{2}_{m}(i{\\mathbb R})}}{\\mathcal M}_{\\mathfrak D}.\n$$\nFurthermore, a variant of the Beurling-Lax Theorem assures us that there is a \n$m \\times m$-matrix \ninner function $\\psi$ on $\\Pi_{+}$ so that\n\\begin{equation}   \\label{cM-rep}\n {\\mathcal M}_{{\\mathfrak D},-} = \\psi^{-1} \\cdot H^{2}_{m}(\\Pi_{+}).\n\\end{equation}\nMaking use of \\cite[Theorem 6.1]{BGR} applied to the null-pole triple \n$(U,W; \\emptyset, \\emptyset; \\emptyset)$ over $\\Pi_{+}$,\none can see that such a \n$\\psi$ (defined uniquely up to a constant unitary factor on the left) \nis given by the state-space realization formula \n\\begin{equation}\n\\psi(z)=I_m-UP^{-1}(zI+W^*)^{-1}U^*,\n\\label{sep1}\n\\end{equation}\nwhere the positive definite matrix $P$ is uniquely defined from the Lyapunov equation \n$PW+W^*P=U^*U$, with $\\psi^{-1}$ given by\n\\begin{equation}   \\label{psi-inv}\n\\psi(z)^{-1}=I_m+U(zI-W)^{-1}P^{-1}U^*,\n\\end{equation}\ni.e., that $(U,W)$ is the right null pair of $\\psi$. \nFurthermore, a second application of  \\cite[Theorem 6.1]{BGR} \nto the null-pole triple $(\\sbm{ V \\\\ U}, W; Z, \\sbm{ X & -Y}; \\Gamma)$ \nover $\\Pi_{+}$ leads to:\n\\begin{itemize}\n    \\item {\\em ${\\mathcal M}_{\\mathfrak D}$ has the Beurling-Lax-type \n    representation} \n\\begin{equation}   \\label{cMBLrep}\n{\\mathcal M}_{\\mathfrak D} = \\Theta \\cdot \n    H^{2}_{p+m}(\\Pi_{+}).\n\\end{equation}\n\\end{itemize}\nBy projecting the identity \\eqref{cMBLrep}  onto the bottom component and recalling \nthe identity \n\\eqref{cM-rep}, we see that\n\\begin{equation}   \\label{Theta22-1}\n\\begin{bmatrix} \\Theta_{21} & \\Theta_{22} \\end{bmatrix} H^{2}_{p + \n    m}(\\Pi_{+}) = {\\mathcal M}_{{\\mathfrak D}, -} =\n    \\psi^{-1} H^{2}_{m}(\\Pi_{+}).\n \\end{equation} \n On the other hand, for any $\\sbm{ f_{+} \\\\ f_{-} } \\in \n H^{2}_{p+m}(\\Pi_{+})$, we have\n \\begin{align}\n \\begin{bmatrix} \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \n     \\begin{bmatrix} f_{+} \\\\ f_{-} \\end{bmatrix} &=\n\\Theta_{21} f_{+} + \\Theta_{22} f_{-} \\notag\\\\\n&= \\Theta_{22} \n(\\Theta_{22}^{-1} \\Theta_{21} f_{+} + f_{-}) \\in \\Theta_{22} \nH^{2}_{m}(\\Pi_{+})\\label{Theta22-2},\n\\end{align}\nsince $\\Theta_{22}^{-1} \\Theta_{21}$ is analytic on $\\Pi_+$.  Since \nthe reverse containment\n$$\n\\Theta_{22}\\cdot  H^{2}_{m}(\\Pi_{+}) \\subset \\begin{bmatrix} \\Theta_{21} & \n\\Theta_{22} \\end{bmatrix}\\cdot H^{2}_{p+m}(\\Pi_{+})\n$$\nis obvious, we may combine \\eqref{Theta22-1} and \\eqref{Theta22-2} to conclude that\n\\begin{equation}   \\label{Theta22-3}\n    \\Theta_{22}\\cdot  H^{2}_{m}(\\Pi_{+}) = \\begin{bmatrix} \\Theta_{21} & \n    \\Theta_{22} \\end{bmatrix} \\cdot H^{2}_{p+m}(\\Pi_{+}) = \\psi^{-1} \n   \\cdot H^{2}_{m}(\\Pi_{+}).\n \\end{equation}\nIt turns out that the geometry of ${\\mathcal M}_{\\mathfrak D}$ encodes the \ninterpolation conditions:\n\\begin{itemize}\n    \\item {\\em An analytic function $S \\colon \\Pi_{+} \\to {\\mathbb C}^{p \\times m}$ \n    satisfies the interpolation conditions \\eqref{BTOAint1}, \n    \\eqref{BTOAint2}, \\eqref{BTOAint3} if and only if}\n \\begin{equation}  \\label{sol-rep}\n  \\begin{bmatrix} S \\\\ I_m \\end{bmatrix} \\cdot {\\mathcal M}_{{\\mathfrak D}, -}\n      \\subset {\\mathcal M}_{\\mathfrak D}.\n \\end{equation}\n \\end{itemize}\n \n It remains to put the pieces together to arrive at the \n linear-fractional parametrization \\eqref{LFTparam} for the set of \n all solutions (and thereby prove that solutions exist).\n Suppose that $S \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ satisfies the \n interpolation conditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \n \\eqref{BTOAint3}.  As a consequence of the criterion \\eqref{sol-rep}\n combined with \\eqref{cM-rep} and \\eqref{cMBLrep}, we have\n $$\n   \\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\psi^{-1}  \\cdot\n       H^{2}_{m}(\\Pi_{+}) \\subset \\begin{bmatrix} \\Theta_{11} & \n       \\Theta_{12} \\\\ \\Theta_{21} & \\Theta_{22} \\end{bmatrix}  \\cdot\n       H^{2}_{p+m}(\\Pi_{+}).\n $$\n Hence there must be a $(p+m) \\times m$ matrix function\n $\\sbm{Q_{1} \\\\ Q_{2}} \\in \n H^2_{(p+m) \\times m}(\\Pi_{+})$ so that\n  \\begin{equation}  \\label{Sparam}\n \\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\psi^{-1} = \n\\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\ \n     \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \\begin{bmatrix} Q_{1}  \n     \\\\ Q_{2} \\end{bmatrix}.\n\\end{equation}\n \nWe next combine this identity with the $J$-unitary property of \\eqref{ThetaJunitary}: \nfor the (suppressed) argument $\\lambda \\in i{\\mathbb R}$ we have\n\\begin{align*}\n    0 \\succeq \\psi^{-1 *} (S^{*} S - I) \\psi^{-1}& = \\psi^{-1 *} \n\\begin{bmatrix} Q_{1}^{*} & Q_{2}^{*} \\end{bmatrix}\n    \\Theta^{*} J \\Theta \\begin{bmatrix} Q_{1} \\\\ Q_{2} \\end{bmatrix} \n    \\psi^{-1} \\\\\n    & =\\psi^{-1 *} \\begin{bmatrix} Q_{1}^{*} & Q_{2}^{*} \\end{bmatrix} \n    J \\begin{bmatrix} Q_{1} \\\\ Q_{2} \\end{bmatrix} \\psi^{-1}\\\\\n    & = \\psi^{-1 *} (Q_{1}^{*} Q_{1} - Q_{2}^{*} Q_{2}) \\psi^{-1}.\n\\end{align*}\nWe conclude that\n$$\n   \\| Q_{1}(\\lambda) x\\|^{2} \\le \\| Q_{2}(\\lambda) x \\|^{2} \\; \\text{ for all \n   } \\; x \\in {\\mathbb C}^{m} \\; \\text{ and }\\; \\lambda \\in i {\\mathbb R}.\n$$\nIn particular, if $Q_{2}(\\lambda) x = 0$ \nfor some $\\lambda \\in i{\\mathbb R}$ and $x \\in {\\mathbb C}^{m}$, then \nalso $Q_{1}(\\lambda) x = 0$ and hence \n$$\n \\psi(\\lambda)^{-1} x = \\begin{bmatrix} \\Theta_{21}(\\lambda) & \n \\Theta_{22}(\\lambda) \\end{bmatrix} \\begin{bmatrix} Q_{1}(\\lambda) \\\\ \n Q_{2}(\\lambda) \\end{bmatrix} x = 0,\n$$\nwhich forces $x=0$ since $\\psi$ is rational matrix inner.  We conclude:\n\\begin{itemize}\n    \\item {\\em for $\\lambda \\in i{\\mathbb R}$, $Q_{2}(\\lambda)$ is \n    invertible and $G(\\lambda) =  Q_{1}(\\lambda) Q_{2}(\\lambda)^{-1}$\nis a contraction.}\n\\end{itemize}\n\nThe next step is to apply a winding-number argument to get similar \nresults for $\\lambda \\in \\Pi_+$.  From the  bottom component of \\eqref{Sparam} we have, \nagain for the moment with $\\lambda \\in i {\\mathbb R}$,\n\\begin{equation}   \\label{bottom}\n    \\psi^{-1} = \\Theta_{21} Q_{1} + \\Theta_{22} Q_{2} \n= \\Theta_{22} (\\Theta_{22}^{-1} \\Theta_{21}  G + I_{m}) Q_{2}.\n\\end{equation}\nWe conclude that, for the argument $\\lambda \\in i {\\mathbb R}$, \n\\begin{equation}   \\label{wno'}\n \\operatorname{wno} \\det( \\psi^{-1}) = \n \\operatorname{wno}\\det(\\Theta_{22}) + \\operatorname{wno} \n \\det(\\Theta_{22}^{-1} \\Theta_{21} G + I_{m})  + \n \\operatorname{wno} \\det(Q_{2}) \n\\end{equation}\nwhere we use the notation $\\operatorname{wno}f$ to indicate {\\em \nwinding number} or {\\em change of argument} of the function $f$ as \nthe variable runs along the imaginary line. Since both $\\det \n\\Theta_{22}^{-1}$ and $\\det \\psi$ are analytic on \n$\\overline{\\Pi}_{+}$, a consequence of the \nidentity \\eqref{Theta22-3}  is that\n\\begin{equation}   \\label{wno''}\n\\operatorname{wno} \\det( \\psi^{-1}) = \\operatorname{wno}\\det(\\Theta_{22}).\n\\end{equation}\nCombining the two last equalities gives\n\\begin{equation}\n \\operatorname{wno}\n \\det(\\Theta_{22}^{-1} \\Theta_{21} G + I_{m})  + \\operatorname{wno} \\det(Q_{2})=0.\n\\label{wno}\n\\end{equation}\n\n\nWe have already observed that\n$$\n \\| \\Theta_{22}(\\lambda)^{-1} \\Theta_{21}(\\lambda)\\| < 1\\quad\\mbox{and}\\quad\n \\| G(\\lambda) \\| \\le 1 \\; \\text{ for } \\; \\lambda \\in i {\\mathbb R}.\n$$\nHence, for $0 \\le t \\le 1$ we have $\\| t\\Theta_{22}(\\lambda)^{-1} \n\\Theta_{21}(\\lambda)  G(\\lambda) \\| < 1$ and hence\n$t \\Theta_{22}(\\lambda)^{-1} \\Theta_{21}(\\lambda) G(\\lambda) + I$ is \ninvertible for $\\lambda \\in i {\\mathbb R}$ for all $0 \\le t \\le 1$.\nHence \n$$ \ni(t): = \\operatorname{wno} \\det(t \\Theta_{22}(\\lambda)^{-1} \n\\Theta_{21}(\\lambda) G(\\lambda) + I)\n$$\nis well defined and independent of $t$ for $0 \\le t \\le 1$.  \nAs clearly $i(0) = 0$, it follows that \n$$\ni(1) =  \\operatorname{wno} \\det( \\Theta_{22}(\\lambda)^{-1} \n\\Theta_{21}(\\lambda) G(\\lambda) + I) = 0\n$$\nwhich, on account of \\eqref{wno}, implies \n$  \\operatorname{wno} \\det(Q_{2})  = 0$.  As $Q_{2}$ is analytic \non $\\Pi_{+}$, we conclude that $\\det Q_{2}$ has no zeros in \n$\\Pi_{+}$, i.e., $Q_{2}^{-1}$ is analytic on $\\Pi_{+}$.  By the \nmaximum modulus theorem it then follows that\n$G(\\lambda) : = Q_{1}(\\lambda) Q_{2}(\\lambda)^{-1}$ is in the Schur class ${\\mathcal S}^{p \n\\times m}(\\Pi_{+})$.  Furthermore, from \\eqref{Sparam} we  have\n\\begin{equation}\n\\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} = \\begin{bmatrix} \n    \\Theta_{11} & \\Theta_{12} \\\\ \\Theta_{21} & \\Theta_{22} \n\\end{bmatrix} \\begin{bmatrix} G \\\\ I \\end{bmatrix} Q_{2} \\psi.\n\\label{ad3}\n\\end{equation}\nFrom the bottom component we read off that $Q_{2} \\psi = (\\Theta_{21} \nG + \\Theta_{22})^{-1}$.  From the first component we then get\n$$\n S = (\\Theta_{11} G + \\Theta_{12}) Q_{2} \\psi =  (\\Theta_{11} G + \n \\Theta_{12}) (\\Theta_{21} G + \\Theta_{22})^{-1}\n$$\nand the representation \\eqref{LFTparam} follows.\n\n\\smallskip\n\nConversely, if $G \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$, we can reverse  the \nabove argument (with $Q_{1}(\\lambda) = G(\\lambda)$ and  $Q_{2}(\\lambda) = I_{m}$) to see that \n$S$ of the form \\eqref{LFTparam} is a Schur-class solution of the interpolation \nconditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3}.\n\\end{proof}\n\n\\begin{remark} \\label{R:Amaya}\n    The theory from \\cite{BGR} is worked out explicitly only with \n$H^{2}_{m}(\\Pi_{+})$ replaced by its rational subspace $\\operatorname{\\rm \nRat} H^{2}_{m}$ consisting of elements of $H^{2}_{m}$ with \nrational-function column entries, and similarly $H^{2}_{m}(\\Pi_{-})$ and \n$L^{2}(i {\\mathbb R})$ replaced by  their respective rational subspaces $\\operatorname{Rat} \nH^{2}_{m}(\\Pi_{-})$ and $\\operatorname{Rat} L^{2}(i {\\mathbb R})$.\nNevertheless the theory is easily adapted to the $L^{2}$-setting here. \nSubspaces ${\\mathcal M}$ of $L^{2}_{p+m}(i {\\mathbb R})$ having a representation of \nthe form \\eqref{cMrep} (with $\\sbm{U \\\\ V}, W, \\begin{bmatrix} X & -Y \n\\end{bmatrix}, Z, \\Gamma$ all equal to finite matrices rather than infinite-dimensional \noperators) are characterized by the conditions:  (1) \n${\\mathcal M}$ is forward-shift invariant, i.e., ${\\mathcal M}$ is invariant under multiplication \nby the function $\\chi(\\lambda) = \\frac{ \\lambda -1}{ \\lambda +1}$, (2) the subspace\n$({\\mathcal M} + H^{2}_{p+m}(\\Pi_{+}))\/ H^{2}_{p+m}(\\Pi_{+})$ has finite dimension, and (3) the \nquotient space ${\\mathcal M}\/\\left({\\mathcal M} \\cap H^{2}_{p+m}(\\Pi_{+})\\right)$ has finite dimension.\nThe representation \\eqref{cM-rep} with $\\psi^{-1}$ of the form \n\\eqref{psi-inv} with finite matrices $U,W,P$ is roughly the special case \nof the statement above where ${\\mathcal M} = {\\mathcal M}_{{\\mathfrak D},-} \\supset\nH^{2}_{m}(\\Pi_{+})$.  The analogue of such representations \n\\eqref{cMrep} and \\eqref{cM-rep}--\\eqref{psi-inv} for more general \nfull-range pure forward shift-invariant subspaces of \n$L^{2}_{p+m}(\\Pi_{+})$ (or dually of full-range pure backward \nshift-invariant subspaces of $L^{2}_{p+m}(\\Pi_{+})$) involving infinite-dimensional \n(even unbounded) operators  $\\sbm{ U \\\\ V}, W, Z, \\sbm{X & -Y }, \\Gamma$   \nis worked out in the Virginia Tech dissertation of Austin Amaya \\cite{Amaya}.\n\\end{remark}\n\n\n\\subsection{The Fundamental Matrix Inequality approach of Potapov} \n\\label{S:FMI}\nThe linear fractional parametrization formula \\eqref{LFTparam} can be \nalternatively established by the Potapov's method of the Fundamental\nMatrix Inequalities. As we will see, this method bypasses the winding number argument.\n\n\n\\smallskip\n\nConsider a $\\Pi_{+}$-admissible BTOA interpolation data set ${\\mathfrak D}$ as in \n\\eqref{data} giving rise to the  collection  \\eqref{BTOAint1}, \n\\eqref{BTOAint2}, \\eqref{BTOAint3} of BTOA interpolation conditions \nimposed on a Schur-class function $S^{p \\times m}(\\Pi_{+})$.\nWe assume that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is positive definite.  We form \nthe matrix $\\Theta(\\lambda)$ as in \\eqref{oct1}--\\eqref{oct2}  and \nassume all knowledge of all the properties of $\\Theta$ falling out of \nthe positive-definiteness of $\\boldsymbol{\\Gamma}_{\\mathfrak D}$, specifically\n\\eqref{bigLySyl}--\\eqref{Theta22inv} above.\n\n\\smallskip\n\nThe main idea is to extend the interpolation data by one extra interpolation node \n$z\\in\\Pi_+$ with the corresponding full-range value $S(z)$, i.e., by \nthe tautological full-range interpolation condition\n\\begin{equation}  \\label{generic-int}\n S(z) = S(z)\n\\end{equation}\nwhere $z$ is a generic point in the right half plane.  \nTo set up this augmented problem as a BTOA problem, we have a choice \nas to how we incorporate the global generic interpolation condition \n\\eqref{generic-int} into the BTOA formalism:  (a) \nas a LTOA interpolation condition:\n\\begin{equation}  \\label{genericLTOA}\n   (X_{z} S)^{\\wedge L}(Z_{z}) = Y_{z} \\;  \\text{ where } \\; X_{z} = I_{p}, \n   \\, Y_{z} = S(z), \\, Z_{z} = z I_{p},\n\\end{equation}\nor  as a RTOA interpolation condition:\n\\begin{equation}   \\label{genericRTOA}\n    (S U_{z})^{\\wedge R}(W_{z}) = V_{z} \\; \\text{ where } \\;\n    U_{z} = I_{m}, \\, V_{z} = S(z), \\, W_{z} = z I_{m}.\n\\end{equation}\nWe  choose here to work with the left versions \\eqref{genericLTOA} exclusively; working \nwith the right version \\eqref{genericRTOA} will give seemingly different but in the end \nequivalent parallel results.  \n\n\\smallskip\n\nAs a first step, we wish to combine \\eqref{BTOAint1} and \n\\eqref{genericLTOA} into a single LTOA interpolation condition.\nThis is achieved by augmenting the matrices $(Z, X, Y)$ to the \naugmented triple $(Z_{\\rm aug}, X_{\\rm aug}, Y_{\\rm aug})$ given by\n$$\nZ_{\\rm aug} = \\begin{bmatrix} Z & 0 \\\\ 0 & z I_{p} \\end{bmatrix}, \n\\quad\nX_{\\rm aug} = \\begin{bmatrix} X \\\\ I_{p} \\end{bmatrix}, \\quad\nY_{\\rm aug} = \\begin{bmatrix} Y \\\\ S(z) \\end{bmatrix}.\n$$\nHere all matrices indexed by aug depend on the parameter $z$, but for \nthe moment we suppress this dependence from the notation.\nAs the RTOA-interpolation conditions for the augmented problem remain \nthe same as in the original problem (namely, \\eqref{BTOAint2}), we set\n$$\nU_{\\rm aug} = U, \\quad V_{\\rm aug} = V, \\quad W_{\\rm aug} = W.\n$$\nWe therefore take the \\textbf{augmented data set} ${\\mathfrak D}_{\\rm \naug}$ to have the form\n\\begin{equation} \\label{Daug}\n    {\\mathfrak D}_{\\rm aug} = (X_{\\rm aug}, Y_{\\rm aug}, Z_{\\rm aug}; \n    U_{\\rm aug}, V_{\\rm aug}, W_{\\rm aug}; \\Gamma_{\\rm aug})\n\\end{equation}\nwhere the coupling matrix $\\Gamma_{\\rm aug}$ is still to be \ndetermined.\n\n\\smallskip\n\nWe know that $\\Gamma_{\\rm aug}$ must solve the Sylvester equation\n\\eqref{Sylvester'} associated with the data set ${\\mathfrak D}_{\\rm \naug}$, i.e., $\\Gamma_{\\rm aug}$ must have the form $\\Gamma_{\\rm aug} \n= \\sbm{ \\Gamma_{{\\rm aug},1} \\\\ \\Gamma_{{\\rm aug},2} }$  with\n$$    \\begin{bmatrix} \\Gamma_{{\\rm aug},1} \\\\ \\Gamma_{{\\rm aug},2} \n    \\end{bmatrix} W - \\begin{bmatrix} Z & 0 \\\\ 0 & z I_{p} \n\\end{bmatrix} \\begin{bmatrix} \\Gamma_{{\\rm aug},1} \\\\ \\Gamma_{{\\rm aug},2} \n    \\end{bmatrix} = \n   \\begin{bmatrix} X \\\\ I_{p} \\end{bmatrix} V - \\begin{bmatrix} Y \\\\ \n       S(z) \\end{bmatrix} U.\n$$\nEquivalently, $\\Gamma_{\\rm aug} = \\sbm{ \\Gamma_{{\\rm aug},1} \\\\ \n\\Gamma_{{\\rm aug},2}}$ is determined by the decoupled system of equations\n\\begin{align}  \n   &  \\Gamma_{{\\rm aug}, 1} W - Z \\Gamma_{{\\rm aug}, 1} = X V - Y U, \n   \\notag \\\\\n& \\Gamma_{{\\rm aug},2} W - (z I_{p}) \\Gamma_{{\\rm aug}, 2} = V - S(z) U.\n\\label{aug-Sylvester''}\n\\end{align}\nIn addition, the third augmented interpolation condition takes the \nform\n$$\n \\left( \\begin{bmatrix} X \\\\ I_{p} \\end{bmatrix} S \\, U \n \\right)^{\\wedge L,R} \\left( \\begin{bmatrix} Z & 0 \\\\ 0 & zI_{p} \n\\end{bmatrix}, W \\right) = \\begin{bmatrix} \\Gamma_{{\\rm aug}, 1} \\\\\n\\Gamma_{{\\rm aug}, 2} \\end{bmatrix}\n$$\nwhich can be decoupled into two independent bitangential \ninterpolation conditions\n\\begin{equation}   \\label{coupledBTint}\n    ( X S U)^{\\wedge L,R}(Z,W) = \\Gamma_{{\\rm aug},1}, \\quad\n    (I_{p} S U)^{\\wedge L,R}(z I_{p}, W ) = \\Gamma_{{\\rm aug},2}.\n\\end{equation}\nFrom the first of the conditions \\eqref{coupledBTint} coupled with \nthe interpolation condition \\eqref{BTOAint3}, we are forced to take\n$  \\Gamma_{{\\rm aug}, 1} = \\Gamma$.\n\n\\smallskip\n\nSince the point $z \\in \\Pi_{+}$ is generic, we may assume as a first \ncase that $z$ is disjoint from the spectrum $\\sigma(W)$ of $W$.  Then \nwe can solve the second of the equations \\eqref{aug-Sylvester''} \nuniquely for $\\Gamma_{{\\rm aug}, 2}$:\n\\begin{equation} \\label{Gammaaug2}\n\\Gamma_{{\\rm aug},2} = (S(z) U - V) (zI_{n_{W}} - W)^{-1}.\n\\end{equation}\nA consequence of the RTOA interpolation condition \\eqref{BTOAint2} is \nthat the right-hand side of \\eqref{Gammaaug2} has analytic \ncontinuation to all of $\\Pi_{+}$.  It is not difficult to see that \n$(I_{p} S U)^{L,R}(z I_{p}, W)$ in general is just the value of this \nanalytic continuation at the point $z$; we conclude that the formula \n\\eqref{Gammaaug2} holds also at points $z$ in $\\sigma(W)$ with proper \ninterpretation.  In this way we have completed the computation of the \naugmented data set  \\eqref{Daug}:\n\\begin{equation}   \\label{Daug'}\n    {\\mathfrak D}(z):= {\\mathfrak D}_{\\rm aug} = \n \\left(   \\begin{bmatrix} Z & 0 \\\\ 0 & zI_{p} \\end{bmatrix}, \\,\n\\begin{bmatrix} X \\\\ I_{p} \\end{bmatrix}, \\, \n\\begin{bmatrix} Y \\\\ S(z) \\end{bmatrix}; \\, U,V, W; \\begin{bmatrix} \n    \\Gamma \\\\ T_{S,1}(z) \\end{bmatrix} \\right).\n\\end{equation}\nwhere we set\n\\begin{equation}   \\label{TS1}\n  T_{S,1}(z) =  (S(z) U - V) (zI_{n_{W}} - W)^{-1}.\n\\end{equation}\n\nWe next compute the Pick matrix $\\boldsymbol{\\Gamma}_{{\\mathfrak D}_{\\rm \naug}(z)}$ for the augmented data set ${\\mathfrak D}_{\\rm aug}$ \n\\eqref{Daug'} according to the recipe \\eqref{GammaL}--\\eqref{GammaData}.  \nThus \n$$\n\\boldsymbol{\\Gamma}_{{\\mathfrak D}_{\\rm aug}(z)} = \n\\begin{bmatrix}\\Gamma_{{\\rm aug}, L} & \\Gamma_{\\rm aug} \\\\ (\\Gamma_{\\rm \naug})^{*} & \\Gamma_{{\\rm aug}, R}\\end{bmatrix},\\quad\\mbox{where}\\quad\n\\Gamma_{\\rm aug} = \\begin{bmatrix} \\Gamma \\\\ T_{S,1}(z) \\end{bmatrix}, \n\\; \\Gamma_{{\\rm aug},R} = \\Gamma_{R},\n$$\nand where $\\Gamma_{{\\rm aug}, L} = \\sbm{\\Gamma_{{\\rm aug}, L11} & \n\\Gamma_{{\\rm aug}, L12} \\\\ \\Gamma_{{\\rm aug}, L21} & \\Gamma_{{\\rm \naug}, L22 }}$ is determined by the Lyapunov equation \\eqref{GammaL} \nadapted to the interpolation data set ${\\mathfrak D}(z)$:\n\\begin{align*}\n&\\begin{bmatrix} \\Gamma_{{\\rm aug}, L11} & \\Gamma_{{\\rm aug}, L12} \\\\ \n    \\Gamma_{{\\rm aug}, L21} & \\Gamma_{{\\rm aug}, L22} \\end{bmatrix}\n\\begin{bmatrix} Z^{*} & 0 \\\\ 0 & \\overline{z} I_{p} \\end{bmatrix} +\n\\begin{bmatrix} Z & 0 \\\\ 0 & z I_{p} \\end{bmatrix}\n\\begin{bmatrix} \\Gamma_{{\\rm aug}, L11} & \\Gamma_{{\\rm aug}, L12} \\\\ \n    \\Gamma_{{\\rm aug}, L21} & \\Gamma_{{\\rm aug}, L22} \\end{bmatrix} \\\\\n& \\quad = \\begin{bmatrix} X \\\\ I_{p} \\end{bmatrix} \\begin{bmatrix} X^{*} & \n I_{p} \\end{bmatrix} - \\begin{bmatrix} Y \\\\ S(z) \\end{bmatrix}\n\\begin{bmatrix} Y^{*} & S(z)^{*} \\end{bmatrix}.\n\\end{align*}\nOne can solve this equation uniquely for $\\Gamma_{{\\rm aug}, Lij}$ ($i,j=1,2$) \nwith the result\n\\begin{align*}\n & \\Gamma_{{\\rm aug}, L11} = \\Gamma_{L}, \\quad \\Gamma_{{\\rm aug}, L21} \n  = (\\Gamma_{{\\rm aug}, L12})^{*}  = T_{S,2}(z), \\notag \\\\\n&  \\Gamma_{{\\rm aug},L22} = \\frac{I - S(z) S(z)^{*}}{z + \\overline{z}}\n\\end{align*}\nwhere we set\n\\begin{equation}   \\label{TS2}\n   T_{S,2}(z): = (X^{*} - S(z) Y^{*})(zI_{n_{Z}} + Z^{*})^{-1}.\n\\end{equation}\nIn this way we arrive at the  Pick matrix for data set ${\\mathfrak \nD}(z)$, denoted for convenience as $\\boldsymbol{\\Gamma}_{\\mathfrak D}(z)$ \nrather than as $\\boldsymbol{\\Gamma}_{{\\mathfrak D}(z)}$:\n$$\n\\boldsymbol{\\Gamma}_{\\mathfrak D}(z) = \\begin{bmatrix} \\Gamma_{L} & \nT_{S,2}(z)^{*} & \\Gamma \\\\ T_{S,2}(z) & \\frac{ I - S(z) S(z)^{*}}{ z + \n\\overline{z}} & T_{S,1}(z) \\\\ \\Gamma^{*} & T_{S,1}(z)^{*} & \n\\Gamma_{R} \\end{bmatrix}.\n$$\nIf we interchange the second and third rows and then also the second \nand third columns (i.e., conjugate by a permutation matrix), we get a \nnew matrix having the same inertia; for simplicity from now on we use \nthe same notation $\\boldsymbol{\\Gamma}_{\\mathfrak D}(z)$ for this transformed \nmatrix:\n$$\n  \\boldsymbol{\\Gamma}_{{\\mathfrak D}}(z) = \\begin{bmatrix} \\Gamma_{L} & \\Gamma & \n  T_{S,2}(z)^{*} \\\\ \\Gamma^{*} & \\Gamma_{R} & T_{S,1}(z)^{*} \\\\\n  T_{S,2}(z) & T_{S,1}(z) & \\frac{ I - S(z) S(z)^{*}}{z + \n  \\overline{z}} \\end{bmatrix}.\n$$\n\nHad we started with a finite number ${\\mathbf z} = \\{z_{1}, \\dots, z_{N}\\}$ \nof generic interpolation nodes in $\\Pi_{+}$ rather than a single \ngeneric point $z$ and augmented the interpolation conditions \n\\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3} with the \ncollection of tautological interpolation conditions\n$$\n  S(z_{i}) = S(z_{i}) \\; \\text{ for } \\; i = 1, \\dots, N\n$$\nmodeled as the additional LTOA interpolation condition\n$$\n  (X_{{\\mathbf z}} S)^{\\wedge L}(Z_{{\\mathbf z}}) = Y_{{\\mathbf z}}\n$$\nwhere\n$$\nZ_{{\\mathbf z}} = \\sbm{ z_{1}I_{p} & & 0 \\\\ & \\ddots & \\\\ 0 & & z_{N} I_{p} }, \\quad\nX_{{\\mathbf z}} = \\sbm{I_{p} \\\\ \\vdots \\\\ I_{p} }, \n\\quad Y_{{\\mathbf z}} = \\sbm{ S(z_{1}) \\\\ \\vdots \\\\ S(z_{N})},\n$$\nthe same analysis as above would lead us to the following conclusion: \n{\\em there is a matrix function $S$ in the Schur class ${\\mathcal S}^{p \\times \nm}(\\Pi_{+})$ satisfying the interpolation conditions \n\\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3} if and only if, \nfor ${\\mathbf z} = \\{z_{1}, \\dots, z_{N}\\}$ any collection of $N$ distinct \npoints in $\\Pi_{+}$, the associated augmented Pick matrix \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}({\\mathbf z})$ is positive-semidefinite, where}\n$$\n\\boldsymbol{\\Gamma}_{\\mathfrak D}({\\mathbf z}) =\n\\begin{bmatrix} \\Gamma_{L} & \\Gamma & T_{S,2}(z_{1})^{*} & \\cdots & \n    T_{S,2}(z_{N})^{*} \\\\\n\\Gamma^{*} & \\Gamma_{R} & T_{S,1}(z_{1})^{*} & \\cdots & \nT_{S,1}(z_{N})^{*} \\\\\nT_{S,2}(z_{1}) & T_{S,1}(z_{1}) & \n\\frac{I - S(z_{1})S(z_{1})^{*}}{z_{1} + \\overline{z}_{1}} & \\cdots  &\n\\frac{I - S(z_{1}) S(z_{N})^{*}}{z_{1} + \\overline{z}_{N}}  \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nT_{S,2}(z_{N}) & T_{S,1}(z_{N}) & \n\\frac{ I - S(z_{N}) S(z_{1})^{*}}{z_{1} + \\overline{z}_{N}} & \\cdots & \n\\frac{ I - S(z_{N}) S(z_{N})^{*}}{z_{N} + \\overline{z}_{N}} \\end{bmatrix} \\succeq 0.\n$$\nAs the finite set of points ${\\mathbf z} = \\{z_{1}, \\dots, z_{N}\\}$ ($N=1,2, \n\\dots$) is an arbitrary finite subset of $\\Pi_{+}$, this condition in \nturn amounts to the assertion that the kernel \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}(z, \\zeta)$ defined by\n\\begin{equation}  \\label{sep8a}\n  \\boldsymbol{\\Gamma}_{\\mathfrak D}(z, \\zeta) = \\begin{bmatrix}  \\Gamma_{L} & \\Gamma & \n  T_{S,2}(\\zeta)^{*} \\\\\n \\Gamma^{*} & \\Gamma_{R} & T_{S,1}(\\zeta)^{*} \\\\\n T_{S,2}(z) & T_{S,1}(z) & \\frac{ I - S(z) S(\\zeta)^{*}}{z + \n \\overline{\\zeta}}  \\end{bmatrix}\n\\end{equation}\nis a positive kernel on $\\Pi_{+}$ (see \\eqref{posker}).\nObserve from \\eqref{TS2}, \\eqref{TS1} that\n\\begin{align*}\n\\begin{bmatrix}T_{S,2}(z) & T_{S,1}(z)\\end{bmatrix}&=\n- \\begin{bmatrix}I_p & -S(z)\\end{bmatrix}\\begin{bmatrix}\n -X^{*} & V \\\\ -Y^{*} & U \\end{bmatrix}\n \\begin{bmatrix}(zI+Z^*)^{-1} &0 \\\\ 0 & (zI-W)^{-1}\\end{bmatrix}\\notag\\\\\n&= - \\begin{bmatrix}I_p & -S(z)\\end{bmatrix}{\\bf C}(z I-{\\bf A})^{-1},\n\\end{align*}\nwhere ${\\bf C}$ and ${\\bf A}$ are defined as in \\eqref{oct2}. Taking the latter \nformula into account, we way write\n\\eqref{sep8a} in a more structured form as\n\\begin{equation} \\label{sep8c}\n\\boldsymbol{\\Gamma}_{\\mathfrak D}(z,\\zeta)=\\begin{bmatrix} \\Gamma_{\\mathfrak D} &\n-(\\overline\\zeta I-{\\bf A}^*)^{-1}\n{\\bf C}^*\\begin{bmatrix}I_p \\\\ -S(\\zeta)^*\\end{bmatrix}\\\\\n-\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}{\\bf C}(z I-{\\bf A})^{-1}&\n{\\displaystyle\\frac{I - S(z)S(\\zeta)^*}{z+\\overline{\\zeta}}}\\end{bmatrix}.\n\\end{equation}\nSince the matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is positive definite, the kernel \\eqref{sep8c} \nis positive if and only if the \nSchur complement of $\\Gamma_{\\mathfrak D}$ is a positive kernel on  \n$\\Pi_+\\backslash  \\sigma(W)$ \nand therefore, admits a unique positive extension to the whole $\\Pi_+$:\n$$\n\\frac{I - S(z)S(\\zeta)^*}{z+\\overline{\\zeta}}-\n\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}{\\bf C}(z I-{\\bf A})^{-1}\\Gamma_{\\mathfrak D}^{-1}\n(\\overline\\zeta I-{\\bf A}^*)^{-1}{\\bf C}^*\\begin{bmatrix}I_p \\\\ \n-S(\\zeta)^*\\end{bmatrix}\\succeq 0.\n$$\nThe latter can be written as\n$$\n\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}\\left\\{\\frac{J}{z+\\overline{\\zeta}}-\n{\\bf C}(z I-{\\bf A})^{-1}\\Gamma_{\\mathfrak D}^{-1}\n(\\overline\\zeta I-{\\bf A}^*)^{-1}{\\bf C}^*\\right\\}\n\\begin{bmatrix}I_p \\\\ -S(\\zeta)^*\\end{bmatrix}\\succeq 0,\n$$\nand finally, upon making use of \\eqref{ad1}, as\n\\begin{equation}\n\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}\\frac{\\Theta(z)J\\Theta(\\zeta)^*}\n    {z+\\overline{\\zeta}}\n\\begin{bmatrix}I_p \\\\ -S(\\zeta)^*\\end{bmatrix}\\succeq 0.\n\\label{sep8d}\n\\end{equation}\nWe next define two functions $Q_1$ and $Q_2$ by the formula\n\\begin{equation}\n\\begin{bmatrix}Q_2(z) & -Q_1(z)\\end{bmatrix}=\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}\n\\begin{bmatrix}\\Theta_{11}(z) & \\Theta_{12}(z)\\\\ \\Theta_{21}(z) & \\Theta_{22}(z)\n\\end{bmatrix},   \n\\label{sep8f}\n\\end{equation}\nand write \\eqref{sep8d} in terms of these functions as\n$$\n\\begin{bmatrix}Q_2(z) & -Q_1(z)\\end{bmatrix}\\frac{J}{z+\\overline{\\zeta}}\n\\begin{bmatrix}Q_2(\\zeta)^* \\\\ -Q_1(\\zeta)^*\\end{bmatrix}=\n\\frac{Q_2(z)Q_2(\\zeta)^*-Q_1(z)Q_1(\\zeta)^*}{z+\\overline{\\zeta}}\n\\succeq 0.\n$$\nBy Leech's theorem \\cite{leech}, there exists a Schur-class function \n$G\\in\\mathcal S^{p\\times m}(\\Pi_+)$ such that \n$$\nQ_2 G=Q_1,\n$$\nwhich, in view of \\eqref{sep8f} can be written as \n$$\n(\\Theta_{11}-S\\Theta_{21})G=S\\Theta_{22}-\\Theta_{12},\n$$\nor equivalently, as\n\\begin{equation}   \\label{sep8g}\n    S (\\Theta_{21} G + \\Theta_{22}) = \\Theta_{11} G + \\Theta_{12}.\n\\end{equation}\nNote that $\\Theta_{22}(z)$ is invertible and that \n$\\Theta_{22}(z)^{-1} \\Theta_{21}(z)$ is strictly contractive on all of \n$\\Pi_{+} \\setminus \\sigma(W)$ (and then on all of $\\Pi_{+}$ by \nanalytic continuation) as a consequence of the bullet immediately \nafter \\eqref{Theta22invTheta21} above.  As $G$ is in the Schur class \nand hence is contractive on all of $\\Pi_{+}$, \nit follows that $\\Theta_{22}(z)^{-1}\\Theta_{21}(z)G(z)+I_m$ is \ninvertible on all of $\\Pi_{+}$.\nHence \n$$\\Theta_{21}(z)G(z)+\\Theta_{22}(z)=\\Theta_{22}(z)(\\Theta_{22}(z)^{-1}\n\\Theta_{21}(z)G(z)+I_m)\n$$ \nis invertible for all $z\\in\\Pi_+$ and we can solve \\eqref{sep8g} for $S$ \narriving at at the formula \\eqref{LFTparam}. \n\n\\begin{remark} \\label{R:FMI}  Note that in this Potapov approach to \n    the derivation of the linear-fractional parametrization via the \n    Fundamental Matrix Inequality, the winding-number argument \n    appearing in the state-space approach never appears.  What \n    apparently replaces it, once everything is properly organized, is \n    the theorem of Leech.\n\\end{remark}\n\n\\subsection{Positive kernels and reproducing kernel Hilbert spaces}  \\label{S:RKHS}\nAssume now that we are given a $\\Pi_{+}$-admissible interpolation data \nset and that the PIck matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is invertible. \nThen one can define the matrix function $\\Theta(z)$ as in \n\\eqref{oct1} and then $K_{\\Theta,J}(z, \\zeta) = \n\\frac{J - \\Theta(z) J \\Theta(\\zeta)^{*}}{z + \\overline{\\zeta}}$ is \ngiven by \\eqref{ad1}. A straightforward computation then shows that, \nfor any $N=1,2,\\dots$ with points $z_{1}, \\dots, z_{N}$ in \n$\\Pi_{+}\\setminus \\sigma(W)$ and vectors ${\\mathbf y}_{1}, \\dots, {\\mathbf y}_{N}$ in \n${\\mathbb C}^{p+m}$, we have\n\\begin{align*}\n&\\sum_{i,j=1}^{N} \\langle K_{\\Theta,J}(z_{i}, z_{j}) {\\mathbf y}_{j}, {\\mathbf y}_{i} \n\\rangle_{{\\mathbb C}^{p+m}}  = \\\\\n& \\quad = \\sum_{i,j=1}^{N} \\bigg\\langle \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1}\n\\bigg( \\sum_{i=1}^{N} (\\overline{z}_{i} I - {\\mathbf A}^*)^{-1}{\\mathbf C}^{*} _{i} \n\\bigg), \\; \\sum_{j=1}^{N} (\\overline{z}_{j} I - {\\mathbf A}^*)^{-1}{\\mathbf C}^{*} _{j} \n \\bigg\\rangle, \n\\end{align*}\nand hence\n $K_{\\Theta,J}$ is a positive kernel on $\\Pi_{+} \\setminus \\sigma(W)$ \n if $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$.  More generally, if \n $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ has some number $\\kappa$ of negative \n eigenvalues, then for any choice of points $z_{1}, \\dots, z_{N} \\in \n \\Pi_{+} \\setminus \\sigma(W)$ the block Hermitian matrix\n\\begin{equation} \\label{KThetaJblock}\n \\left[ K_{\\Theta,J}(z_{i}, z_{j}) \\right]_{i,j=1, \\dots, N}\n \\end{equation}\nhas at most $\\kappa$ negative eigenvalues.  If we impose the controllability and observability \nassumptions on the matrix pairs $(U,W)$ and $(Z,X)$,  then there \nexist a choice of $z_{1}, \\dots, z_{N} \\in \\Pi_{+} \\setminus \n\\sigma(W)$ so that the matrix \\eqref{KThetaJblock} has exactly \n$\\kappa$ negative eigenvalues, in which case we say that $\\Theta$ is \nin the generalized $J$-Schur class ${\\mathcal S}_{J, \\kappa}(\\Pi_{+})$ \n(compare with the Kre\\u{\\i}n-Langer generalized Schur class discussed \nat the beginning of Section \\ref{S:negsquares} below).  In the case \nwhere $\\Theta \\in {\\mathcal S}_{J, \\kappa}(\\Pi_{+})$ with $\\kappa > 0$, there \nis still associated a space of functions ${\\mathcal H}(K_{\\Theta,J})$ as in \n\\eqref{RKHSa}--\\eqref{RKHSb}; the space ${\\mathcal H}(K_{\\Theta,J})$ is \nnow a {\\em Pontryagin space} with negative index equal to $\\kappa$\n(see Section \\ref{S:Kreinprelim} for background on Pontryagin and \nKre\\u{\\i}n spaces).  In any case, in this way we arrive at yet another \ninterpretation of the condition that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ be \npositive definite.\n\n\\begin{theorem} \\label{T:RKHS}\n    Assume that we are given a $\\Pi_{+}$-admissible interpolation \n    data set with $\\Gamma_{\\mathfrak D}$ is invertible (so  \n    $\\Theta$ and $K_{\\Theta,J}$ are defined).  Then \n    ${\\mathcal H}(K_{\\Theta,J})$ is a Hilbert space if and only if \n    $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$.\n\\end{theorem}\n\nIn Section \\ref{S:synthesis} below (see display \\eqref{HKThetaJ})\nwe shall spell this criterion out in more detail and arrive at another \ncondition equivalent to positive-definiteness \nof the Pick matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$.\n\n\\section{The Grassmannian\/Kre\\u{\\i}n-space-geometry approach to the \nBTOA-NP interpolation problem}  \n\\label{S:proof2}\nIn this section we sketch the Grassmannian\/Kre\\u{\\i}n-space geometry \nproof of Theorem \\ref{T:BTOA-NP} based on the work in \\cite{BH}---see \nalso \\cite{Ball-Indiana} for a more expository account and \\cite{BF} \nfor a more recent overview which also highlights the method in \nvarious multivariable settings.  These treatments work with the \nSarason \\cite{Sarason67} or Model-Matching \\cite{Francis} formulation \nof the Nevanlinna-Pick interpolation problem, while we work with the \nLTOA-interpolation formulation.\nThe translation between the two is given in \\cite[Chapter 16]{BGR}\n(where the Sarason\/Model Matching formulation is called \n{\\em divisor-remainder form}.\n\n\\subsection{Kre\\u{\\i}n-space preliminaries}  \\label{S:Kreinprelim}\nLet us first review a few preliminaries concerning Kre\\u{\\i}n spaces.\nA Kre\\u{\\i}n space by definition is a linear space ${\\mathcal K}$ endowed\nwith an indefinite inner product $[ \\cdot, \\cdot]$ which is\n{\\em complete} in the following sense:  there are two subspaces\n${\\mathcal K}_{+}$ and ${\\mathcal K}_{-}$ of ${\\mathcal K}$ such that the restriction of $[\n\\cdot, \\cdot ]$ to ${\\mathcal K}_{+} \\times {\\mathcal K}_{+}$ makes ${\\mathcal K}_{+}$ a\nHilbert space while the restriction of $-[ \\cdot, \\cdot]$ to\n${\\mathcal K}_{-} \\times {\\mathcal K}_{-}$ makes ${\\mathcal K}_{-}$ a Hilbert space, and \n\\begin{equation}\n{\\mathcal K} = {\\mathcal K}_{+} [\\dot +] {\\mathcal K}_{-}\n\\label{dec}\n\\end{equation}\nis a $[ \\cdot, \\cdot]$-orthogonal direct sum decomposition\nof ${\\mathcal K}$.  In this case the decomposition \\eqref{dec}\nis said to form a {\\em fundamental decomposition} for ${\\mathcal K}$.\nFundamental decompositions are never unique except in the trivial\ncase where one of ${\\mathcal K}_{+}$ or ${\\mathcal K}_{-}$ is equal to the zero space.\nIf $\\min(\\dim {\\mathcal K}_{+}, \\, \\dim {\\mathcal K}_{-})=\\kappa<\\infty$, then \n${\\mathcal K}$ is called a Pontryagin space of index $\\kappa$.\n\n\\smallskip\n\nUnlike the case of Hilbert spaces where closed subspaces all look\nthe same, there is a rich geometry for subspaces of a Kre\\u{\\i}n\nspace. A subspace ${\\mathcal M}$ of a Kre\\u{\\i}n space ${\\mathcal K}$ is said to be\n{\\em positive}, {\\em isotropic}, or {\\em negative} depending on\nwhether $[u, u] \\ge 0$ for all $u \\in {\\mathcal M}$, $[ u, u ] =0$ for all\n$u \\in {\\mathcal M}$ (in which case it follows that $[u,v] = 0$ for all\n$u,v \\in {\\mathcal M}$ as a consequence of the Cauchy-Schwarz inequality),\nor $[u,u] \\le 0$ for all $u \\in {\\mathcal M}$.  Given any subspace ${\\mathcal M}$,\nwe define the Kre\\u{\\i}n-space orthogonal complement\n${\\mathcal M}^{[\\perp]}$ to consist of all $v \\in {\\mathcal K}$ such that $[u,v] =\n0$ for all $u \\in {\\mathcal K}$.  Note that the statement that ${\\mathcal M}$ is\nisotropic is just the statement that ${\\mathcal M} \\subset {\\mathcal M}^{[ \\perp\n]}$.  If it happens that ${\\mathcal M} = {\\mathcal M}^{[ \\perp]}$, we say that ${\\mathcal M}$\nis a {\\em Lagrangian} subspace of ${\\mathcal K}$.  Simple examples show that \nin general, unlike the Hilbert space case, it can happen that ${\\mathcal M}$ \nis a closed subspace of the Kre\\u{\\i}n space ${\\mathcal K}$ yet the space \n${\\mathcal K}$ cannot be split at the ${\\mathcal K}$-orthogonal direct sum of ${\\mathcal M}$ and \n${\\mathcal M}^{[\\perp]}$ (e.g., this happens dramatically if ${\\mathcal M}$ is an \nisotropic subspace of ${\\mathcal K}$). If ${\\mathcal M}$ is a subspace of ${\\mathcal K}$ for which this does \nhappen, i.e., such that ${\\mathcal K} = {\\mathcal M} [+] {\\mathcal M}^{[\\perp]}$, we say \nthat ${\\mathcal M}$ is a {\\em regular subspace} of ${\\mathcal K}$.\n\n\\smallskip\n\nExamples of such subspaces arise from placing appropriate\nKre\\u{\\i}n-space inner products on the direct sum ${\\mathcal H}^\\prime \\oplus\n{\\mathcal H}$ of two Hilbert spaces and looking at graphs of operators of\nan appropriate class.\n\n\\begin{example}  \\label{E:unitary}   Suppose that ${\\mathcal H}'$ and\n    ${\\mathcal H}$ are two Hilbert spaces and we take ${\\mathcal K}$ to be the\n    external direct sum ${\\mathcal H}' \\oplus {\\mathcal H}$ with inner product\n$$\n    \\left[ \\begin{bmatrix} x \\\\ y \\end{bmatrix}, \\,\n    \\begin{bmatrix} x' \\\\ y' \\end{bmatrix} \\right] =\n    \\left\\langle \\begin{bmatrix} I_{{\\mathcal H}'} & 0 \\\\ 0 & -I_{{\\mathcal H}}\n    \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix}, \\,\n    \\begin{bmatrix} x' \\\\ y' \\end{bmatrix} \\right\\rangle_{{\\mathcal H}' \\oplus {\\mathcal H}}\n$$\nwhere $\\langle \\cdot, \\cdot \\rangle_{{\\mathcal H}' \\oplus {\\mathcal H}}$ is the\nstandard Hilbert-space inner product on the direct-sum Hilbert\nspace ${\\mathcal H}' \\oplus {\\mathcal H}$.  In this case it is easy to find a\nfundamental decomposition: take ${\\mathcal K}_{+} = \\sbm{ {\\mathcal H} \\\\ \\{0\\}}$\nand ${\\mathcal K}_{-} = \\sbm{ \\{0\\} \\\\ {\\mathcal H}' }$. Now let $T$ be a bounded\nlinear operator from ${\\mathcal H}$ to ${\\mathcal H}'$ and let ${\\mathcal M}$ be the graph of\n$T$:\n$$\n  {\\mathcal M} = {\\mathcal G}_{T} = \\left\\{ \\begin{bmatrix} T x \\\\ x \\end{bmatrix}\n  \\colon x \\in {\\mathcal H} \\right\\} \\subset {\\mathcal K}.\n  $$\n  Then a nice exercise is to work out the following facts:\n  \\begin{itemize}\n      \\item ${\\mathcal G}_{T}$ is negative if and only if $\\|T\\| \\le 1$, in \n      which case ${\\mathcal G}_{T}$ is {\\em maximal negative}, i.e.,  the subspace\n      ${\\mathcal G}_{T}$ is not contained in any strictly larger negative \n      subspace.\n\n      \\item ${\\mathcal G}_{T}$ is isotropic if and only if $T$ is isometric\n      ($T^{*} T = I_{{\\mathcal H}}$).\n\n      \\item ${\\mathcal G}_{T}$ is Lagrangian if and only if $T$ is unitary:\n      $T^{*} T = I_{{\\mathcal H}}$ and $T T^{*} = I_{{\\mathcal H}'}$.\n   \\end{itemize}\n\\end{example}\n\nLet ${\\mathcal M}$ be a fixed subspace of a Kre\\u{\\i}n space ${\\mathcal K}$ and ${\\mathcal G}$ a \nclosed subspace of ${\\mathcal M}$.  In order that ${\\mathcal G}$ be maximal negative as \na subspace of ${\\mathcal K}$, it is clearly necessary that ${\\mathcal G}$ be maximal \nnegative as a subspace of ${\\mathcal M}$.  The following lemma (see \\cite{BH} or \n\\cite{Ball-Indiana}\nfor the proof) identifies when the converse holds.\n\n\\begin{lemma}  \\label{L:maxneg}  Suppose that ${\\mathcal M}$ is a closed \n    subspace of a Kre\\u{\\i}n-space ${\\mathcal K}$ and ${\\mathcal G}$ is a negative \n    subspace of ${\\mathcal M}$.  Then a subspace ${\\mathcal G} \\subset {\\mathcal M}$ which is maximal-negative \n    as a subspace of ${\\mathcal M}$ is automatically also maximal negative as \n    a subspace of ${\\mathcal K}$  if and only if the Kre\\u{\\i}n-space \n    orthogonal complement\n    $$\n    {\\mathcal K} [-] {\\mathcal M} = \\{ k \\in {\\mathcal K} \\colon [k, m ]_{{\\mathcal K}} = 0 \\text{ for \n    all } m \\in {\\mathcal M}\\}\n    $$\n    is a positive subspace of ${\\mathcal K}$.\n\\end{lemma}\n\n\\subsection{The Grassmannian\/Kre\\u{\\i}n-space approach to \ninterpolation}\nSuppose now that we are given a $\\Pi_{+}$-admissible \nBTOA-interpolation data set as in \\eqref{data}.\nLet ${\\mathcal M}_{\\mathfrak D} \\subset L^{2}_{p+m}(i {\\mathbb R})$ be as in \n\\eqref{cMrep}. We view ${\\mathcal M}_{\\mathfrak D}$ as a subspace of the \nKre\\u{\\i}n space \n\\begin{equation}    \\label{defcK}  \n{\\mathcal K}  =\\begin{bmatrix} L^{2}_{p}(i {\\mathbb R}) \\\\ \\mathcal M_{\\mathfrak D,-} \n\\end{bmatrix}=\n\\begin{bmatrix} L^{2}_{p}(i {\\mathbb R}) \\\\ \\psi^{-1} \n    H^{2}_{m}(\\Pi_{+}) \\end{bmatrix}\n\\end{equation}\n(where we use the notation in \\eqref{cM-rep})\nwith Kre\\u{\\i}n-space inner product $[ \\cdot, \\cdot ]_{J}$ induced by the matrix $J = \n\\sbm{ \nI_{p} & 0 \\\\ 0 & -I_{m} }$:\n$$\n\\left[ \\begin{bmatrix} f_{1} \\\\ f_{2}  \\end{bmatrix}, \\, \n\\begin{bmatrix} g_{1} \\\\ g_{2}  \\end{bmatrix} \n\\right]_{J}: =\n\\langle f_{1}, \\, g_{1} \\rangle_{L^{2}_{p}(i {\\mathbb R})} -\n\\langle f_{2}, \\, g_{2} \\rangle_{ \\psi^{-1} H^{2}_{m}(\\Pi_{+})}.\n$$\nA key subspace in the Kre\\u{\\i}n-space geometry approach to the BTOA-NP problem is \nthe $J$-orthogonal complement of ${\\mathcal M}_{\\mathfrak D}$ inside ${\\mathcal K}$:\n\\begin{equation}  \\label{cPdata}\n {\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]} : = {\\mathcal K} [-]_{J} {\\mathcal M}_{\\mathfrak D} = \n \\{ f \\in {\\mathcal K} \\colon [ f, g ]_{J} = 0 \\text{ for all } g \\in \n {\\mathcal M}_{\\mathfrak D} \\}.\n\\end{equation}\nWe then have the following result.\n\n\\begin{theorem}  \\label{T:BTOA-NP'} \nThe BTOA-NP has a solution $S \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ if and \nonly if the subspace ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ \\eqref{cPdata} \nis a positive subspace of ${\\mathcal K}$ \\eqref{defcK}, i.e.,\n $$\n [ f, f ]_{_J} \\ge 0\\; \\text{ for all } \\; f \\in {\\mathcal M}_{\\mathfrak \n D}^{[\\perp {\\mathcal K}]}.\n $$\n If it is the case that ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is a Hilbert \n space in the ${\\mathcal K}$-inner product, then there is rational  \n $J$-inner function $\\Theta$ so that \n \\begin{enumerate}\n     \\item $\\Theta$ provides a \n Beurling-Lax representation for ${\\mathcal M}_{\\mathfrak D}$ \\eqref{cMBLrep}, \n and\n \\item the set of all Schur-class solutions $S \\in {\\mathcal S}^{p \\times \n m}(\\Pi_{+})$ of the interpolation conditions \\eqref{BTOAint1}, \n \\eqref{BTOAint2}, \\eqref{BTOAint3} is given by the linear-fractional \n parametrization formula \\eqref{LFTparam} with $G \\in {\\mathcal S}^{p \\times \n m}(\\Pi_{+})$.\n \\end{enumerate}\n \\end{theorem}\n \n \\begin{proof}[Sketch of the proof of Theorem \\ref{T:BTOA-NP'}]\nWe first argue the ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ being a positive \nsubspace of ${\\mathcal K}$ is necessary for the BTOA-NP to have a solution.\nLet $S\\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ be such a solution and let \n$M_{S} \\colon \\psi^{-1} H^{2}_{m}(\\Pi_{+}) \\to L^{2}_{p}(i {\\mathbb R})$ be the \noperator of multiplication by $S$:\n$$\n  M_{S} \\colon \\psi^{-1} h \\mapsto  S \\cdot \\psi^{-1} h.\n$$\nThe operator norm of $M_{S}$ is the same as the supremum norm of $S$\nover $i {\\mathbb R}$:\n$$\n \\| M_{S}\\|_{\\rm op} = \\|S\\|_\\infty:=\\sup_{\\lambda \\in i {\\mathbb R}} \\| S(\\lambda) \\|.\n$$\nLet us consider the graph space of $M_{S}$, namely\n\\begin{equation}\n  {\\mathcal G}_{S} = \\begin{bmatrix} M_{S} \\\\ I_m \\end{bmatrix} \\psi^{-1}  H^{2}_{m}(\\Pi_{+}) \n   = \\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} \n   H^{2}_{m}(\\Pi_{+}).\n\\label{ad2}\n\\end{equation}\nBy the first bullet in Example \\ref{E:unitary}, it follows that \n\\begin{itemize} \n    \\item {\\em $\\| S\\|_{\\infty} \\le 1$ if and only if ${\\mathcal G}_{S}$ \n    is a maximal negative subspace of ${\\mathcal K}$.}\n\\end{itemize}\nMoreover, as a consequence of the criterion \\eqref{sol-rep} for $S$ \nto satisfy the interpolation conditions, we have\n\\begin{itemize}\n    \\item {\\em $S$ satisfies the interpolation conditions if and only \n    if ${\\mathcal G}_{S} \\subset {\\mathcal M}_{\\mathfrak D}$.}\n    \\end{itemize}\nBy combining these two observations, we see that if $S$ is a \nsolution to the BTOA-NP, then the \nsubspace ${\\mathcal G}_{S}$ is contained in ${\\mathcal M}_{\\mathfrak D}$ and is maximal \nnegative in ${\\mathcal K}$.  It follows that ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is \na positive subspace in ${\\mathcal K}$  as a consequence of Lemma \\ref{L:maxneg}.\nThis verifies the necessity part in Theorem \\ref{T:BTOA-NP'}.\n\n\\smallskip\n\nConversely, suppose that ${\\mathfrak D}$ is a $\\Pi_{+}$-admissible \nBTOA-interpolation data set.  Then we can form the space \n$$\n{\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]} \\subset {\\mathcal K} = \\sbm{ L^{2}_{p}(i{\\mathbb \nR}) \\\\  \\psi^{-1} H^{2}_{m}(\\Pi_{+}) }.\n$$\nSuppose that ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is a positive subspace of ${\\mathcal K}$.  \nBy Lemma \\ref{L:maxneg}, a subspace ${\\mathcal G}$ of ${\\mathcal M}_{\\mathfrak \nD}$ which is maximal negative as a subspace of ${\\mathcal M}_{\\mathfrak D}$ is \nalso maximal negative as a subspace of ${\\mathcal K}$.  We also saw in the necessity \nargument that if the subspace ${\\mathcal G}$ has the form ${\\mathcal G}_{S}$ \\eqref{ad2} \nfor a matrix function $S$ \nand ${\\mathcal G}_{S}\\subset {\\mathcal M}_{\\mathfrak D}$, then $S$ satisfies \nthe interpolation conditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \n\\eqref{BTOAint3}.  However, not all maximal negative subspaces ${\\mathcal G} = \n\\sbm{ T \\\\ I} \\psi^{-1} H^{2}_{m}(\\Pi_{+})$ \nof ${\\mathcal K}$ have the form ${\\mathcal G} = {\\mathcal G}_{S}$ for a matrix function $S$; the \nmissing property is {\\em shift-invariance}, i.e.,  one must require \nin addition that ${\\mathcal G}$ is invariant under multiplication by the \ncoordinate function $\\chi(\\lambda) = \\frac{\\lambda - 1}{\\lambda +1}$.  Then \none gets that $T$ and $M_{\\chi}$ commute and one can conclude that $T$ \nis a multiplication operator:  $T = M_{S}$ for some multiplier \nfunction $S$.  Thus the issue is to construct maximal negative \nsubspaces of ${\\mathcal M}_{\\mathfrak D}$ (which are then also maximal \nnegative as subspaces of ${\\mathcal K}$ by Lemma \\ref{L:maxneg}) which are \nalso shift-invariant.\n\nTo achieve this goal, it is convenient to assume that ${\\mathcal M}_{\\mathfrak \nD}^{[\\perp {\\mathcal K}]}$ is strictly positive, i.e., that ${\\mathcal M}_{\\mathfrak \nD}^{[\\perp {\\mathcal K}]}$ is a Hilbert space.  It then follows in particular that \n${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is regular, i.e., ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ \nand its $J$-orthogonal complement (relative to ${\\mathcal K}$) ${\\mathcal M}_{\\mathfrak D}$ form\na $J$-orthogonal decomposition of ${\\mathcal K}$:\n$$\n  {\\mathcal K} = {\\mathcal M}_{\\mathfrak D}  [+]_{J} {\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}.\n$$\nOne can argue that one can use an approximation\/normal-families \nargument to reduce the general case to this special case, but we do \nnot go into details on this point here.  Then results from \\cite{BH} \nimply that there is a $J$-Beurling-Lax representer for \n${\\mathcal M}_{\\mathfrak D}$, i.e., there is a $J$-phase function \n$$\n\\Theta \\in L^{2}_{(p+m) \\times (p+m)}(i {\\mathbb R})\\quad\\mbox{with}\\quad \n\\Theta(\\lambda)^{*} J \\Theta(\\lambda) = J \\; \\text{ for a.e. } \\; \\lambda \\in \\Pi_{+}\n$$ \nsuch that \\eqref{cMBLrep} holds.  As both \n$$\n{\\mathcal M}_{\\mathfrak D} \\ominus ({\\mathcal M}_{\\mathfrak D} \\cap \nH^{2}_{p+m}(\\Pi_{+}))\\quad\\mbox{and}\\quad H^{2}_{p+m}(\\Pi_{+})\\ominus(\nH^{2}_{p+m}(\\Pi_{+}) \\cap {\\mathcal M}_{\\mathfrak D})\n$$ \nare finite-dimensional, in \nfact one can show that $\\Theta$ is rational and bounded on $i \n{\\mathbb R}$.  Then the multiplication operator $M_{\\Theta} \\colon k \n\\mapsto \\Theta \\cdot k$ is a Kre\\u{\\i}n-space isomorphism from \n$H^{2}_{p+m}(\\Pi_{+})$ (a Kre\\u{\\i}n space with inner product induced \nby $J = \\sbm{ I_{p} & 0 \\\\ 0 & -I_{m}}$) onto ${\\mathcal M}_{\\mathfrak D}$ \nwhich also intertwines the multiplication operator $M_{\\chi}$ on the \nrespective spaces.  It follows that shift-invariant ${\\mathcal M}_{\\mathfrak \nD}$-maximal-negative subspaces ${\\mathcal G}$ are exactly those of the form\n$$\n{\\mathcal G} = \\Theta \\cdot \\begin{bmatrix} G \\\\ I_{m} \\end{bmatrix} \\cdot \nH^{2}_{m}(\\Pi_{+}), \\;  \\text{ where } G \\in \\;  {\\mathcal S}^{p \\times m}(\\Pi_{+}).\n$$\nBy the preceding analysis, any such subspace ${\\mathcal G}$ also has the form\n$$\n  {\\mathcal G} = \\begin{bmatrix} S  \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} \n  H^{2}_{m}(\\Pi_{+})\n$$\nwhere $S \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ is a Schur-class solution of \nthe interpolation conditions \\eqref{BTOAint1}, \\eqref{BTOAint2}, \n\\eqref{BTOAint3}.  Moreover one can reverse this analysis to see that \nany solution $S$ of the BTOA-NP interpolation problem arises in this \nway from a $G \\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$.  From the subspace \nequality\n$$\n\\begin{bmatrix} S  \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} \n  H^{2}_{m}(\\Pi_{+}) = \\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\ \n  \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \\cdot \\begin{bmatrix} G \\\\ I_{m} \n\\end{bmatrix} \\cdot \nH^{2}_{m}(\\Pi_{+})\n$$\none can solve for $S$ in terms of $G$: in particular we have\n$$\n\\begin{bmatrix} S  \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} I_{m} \n    \\in \\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\ \n  \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \\cdot \\begin{bmatrix} G \\\\ I_{m} \n\\end{bmatrix} \\cdot  H^{2}_{m}(\\Pi_{+}),\n$$\nso there must be a function $Q \\in H^{\\infty}_{m \\times m}(\\Pi_{+})$ \nso that\n$$ \n\\begin{bmatrix} S  \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} I_{m} \n   = \\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\ \n  \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \\cdot \\begin{bmatrix} G \\\\ I_{m} \\end{bmatrix}\n  \\cdot Q.\n$$\nAs we saw in Section 2, the latter equality (which is the same as \\eqref{ad3}) implies the \nrepresentation formula \\eqref{LFTparam} for the set of solutions $S$.\nThis completes the proof of Theorem \\ref{T:BTOA-NP'}.\n\\end{proof}\n\n\\begin{remark}  \\label{R:no-wno}\nNote that in this Grassmannian\/Kre\\u{\\i}n-space approach\nwe have not even mentioned that the $J$-phase $\\Theta$ is actually $J$-inner \n(i.e., $\\Theta(\\lambda)$ is $J$ contractive at its points of analyticity \nin $\\Pi_{+}$);  this condition and the winding number argument in the \nproof via the state-space approach in Section \\ref{S:proof1} have been \nreplaced by the condition that ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is a \npositive subspace and consequences of this assumption coming out of \nLemma \\ref{L:maxneg}.\n\\end{remark}\n\n\\section{State-space versus Grassmannian\/Kre\\u{\\i}n-space-geometry \nsolution criteria}  \\label{S:synthesis}\n\nAssume that we are given a $\\Pi_{+}$-admissible interpolation data \nset ${\\mathfrak D}$ with $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ invertible.\nWhen we combine the results of Theorems \\ref{T:BTOA-NP}, \n\\ref{T:BTOA-NP'} and \\ref{T:RKHS}, we see immediately that $\\boldsymbol{\\Gamma}_{\\mathfrak D} \n\\succ 0$ if and only if the subspace ${\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ \nis positive as a subspace of the Kre\\u{\\i}n-space ${\\mathcal K}$ \n\\eqref{defcK}, since each of these two conditions is equivalent to \nthe existence of solutions for the BTOA-NP interpolation problem with \ndata set ${\\mathfrak D}$.  It is not too much of a stretch to \nspeculate that the strict positive definiteness of $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ \nis equivalent to strict positivity of ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$.\nFurthermore, in the case where $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is invertible, \nby the analysis in Section \\ref{S:RKHS} we know that \npositive-definiteness of $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is equivalent to \npositivity of the kernel $K_{\\Theta,J}$ \\eqref{ad1}, or to the \nreproducing kernel space ${\\mathcal H}(K_{\\Theta,J})$ being a Hilbert space.\nThe goal of this section is to carry out some additional geometric \nanalysis to verify these equivalences for the nondegenerate case \n($\\boldsymbol{\\Gamma}_{\\mathfrak D}$ invertible)  directly.\n\n\\begin{corollary}  \\label{C:BTOA-NP}  Suppose that ${\\mathfrak D}$ is \n    a $\\Pi_{+}$-admissible BTOA interpolation data set, let \n    $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ be the matrix given in \\eqref{GammaData} and let \n    ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]} \\subset {\\mathcal K}$ be the subspace defined in\n    \\eqref{cPdata}.  Then the following are equivalent:\n  \\begin{enumerate}\n \\item $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$.\n\\item ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is a strictly positive subspace \n      of ${\\mathcal K}$ (i.e., ${\\mathcal M}^{[ \\perp {\\mathcal K}]}$ is a Hilbert space in the \n      $J$-inner product).\n\\item The reproducing kernel Pontryagin space ${\\mathcal H}(K_{\\Theta,J})$ is \nactually a Hilbert space.\n\\end{enumerate}\n\\end{corollary}\n\n\\smallskip\n\n\\begin{proof} For simplicity we consider first the case where the data set \n    $\\mathfrak D$ has the form\n\\begin{equation}  \\label{dataL}\n{\\mathfrak D}_{L} = (Z, X, Y; \\emptyset, \\emptyset,  \\emptyset;\\emptyset),\n\\end{equation}\ni.e., there are only Left Tangential interpolation conditions \n\\eqref{BTOAint1}. \n\n\\smallskip\n\n\\noindent\n\\textbf{Case 1: The LTOA setting.}\nIn case $\\mathfrak D$ has the form ${\\mathfrak D} = {\\mathfrak \nD}_{L}$ as in \\eqref{dataL}, the matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ collapses down to \n$\\boldsymbol{\\Gamma}_{{\\mathfrak D}_{L}} = \\Gamma_{L}$ and ${\\mathcal M}_{{\\mathfrak D}_{L}}$ collapses down to\n$$\n    {\\mathcal M}_{{\\mathfrak D}_{L}} = \\left\\{ \\begin{bmatrix} f \\\\ g \n\\end{bmatrix} \n \\in H^{2}_{p+m}(\\Pi_{+}) \\colon\n\\left( \\begin{bmatrix}X & -Y \\end{bmatrix} \\begin{bmatrix} f \\\\ g \n\\end{bmatrix} \\right)^{\\wedge L}(Z) = 0 \\right\\}.\n$$\nFurthermore, in the present case, ${\\mathcal M}_{{\\mathfrak D}_{L},-}=H^2_m(\\Pi_+)$ and \ntherefore,  ${\\mathcal K}$ given by \\eqref{defcK} is simply\n ${\\mathcal K} = \\sbm{L^{2}_{p}(i {\\mathbb R}) \\\\ H^{2}_{m}(\\Pi_{+})}$.\n\n\\smallskip\n\n We view the map $\\sbm{ f \\\\ g} \\mapsto \\left(\\sbm{X & -Y } \\sbm{ f \n \\\\ g} \\right)^{\\wedge L}(Z)$ as an operator\n $$\n  {\\mathcal C}_{Z, \\sbm{ X & -Y}} \\colon H^{2}_{p+m}(\\Pi_{+}) \\to {\\mathbb \n  C}^{n_{Z}} \n $$\n which can be written out more explicitly as an integral operator \n along the imaginary line:\\footnote{We view operators of this form as \n {\\em control-like} operators; they and their cousins ({\\em observer-like} operators)\nwill be discussed in a broader context as part of the analysis of Case 2 to \ncome below.}\n$$ \n  {\\mathcal C}_{Z, \\, \\sbm{ X & -Y }} \\colon\n  \\begin{bmatrix} f_{+} \\\\ f_{-} \\end{bmatrix} \\mapsto\n  \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty}\n- (iyI - Z)^{-1} \\begin{bmatrix} X & -Y \\end{bmatrix} \\begin{bmatrix} \nf_{+}(iy) \\\\ f_{-}(iy) \\end{bmatrix} \\, dy.\n$$\nThen we can view ${\\mathcal M}_{{\\mathfrak D}_{L}}$ as an operator kernel:\n $$\n  {\\mathcal M}_{{\\mathfrak D}_{L}} = \\operatorname{Ker} \\,  {\\mathcal C}_{Z, \\sbm{ X & -Y}}.\n  $$\n We are actually interested in the $J$-orthogonal complement \n\\begin{align*}\n {\\mathcal M}_{{\\mathfrak D}_{L}}^{[\\perp {\\mathcal K}]}:={\\mathcal K} [-]_J {\\mathcal M}_{{\\mathfrak D}_{L}}\n&=\\sbm{L^{2}_{p}(i {\\mathbb R}) \\\\ H^{2}_{m}(\\Pi_{+})} [-]_J {\\mathcal M}_{{\\mathfrak D}_{L}}\\\\\n&=\\sbm{ H^{2}_{p}(\\Pi_{-}) \\\\ 0 } \\oplus\\left( H^{2}_{p+m}(\\Pi_+)\n [-]_{J} {\\mathcal M}_{{\\mathfrak D}_{L}}\\right).\n\\end{align*}\n As the subspace $\\sbm{ H^{2}_{p}(\\Pi_{-}) \\\\ 0 }$ is clearly \n positive, we see that   ${\\mathcal M}_{{\\mathfrak D}_{L}}^{[\\perp {\\mathcal K}]}$ is \n positive if and only if its subspace  \n$$\n{\\mathcal M}_{{\\mathfrak  D}_{L}}^{[\\perp H^{2}_{p+m}(\\Pi_{+})]}: = H^{2}_{p+m}(\\Pi_{+}) \n [-]_{J} {\\mathcal M}_{{\\mathfrak D}_{L}}\n$$\nis positive. By standard operator-theory duality, we \n can express the latter (finite-dimensional and hence closed) subspace as an operator range:\n $$\n {\\mathcal M}_{{\\mathfrak D}_{L}}^{[\\perp H^{2}_{p+m}(\\Pi_{+})]} = \\operatorname{Ran} J \\left({\\mathcal C}_{Z,\\sbm{ X & -Y}} \n \\right)^{*},\n $$\n where the adjoint is with respect to the standard Hilbert-space \n inner product on $H^{2}_{p+m}(\\Pi_{+})$ and the standard Euclidean \n inner product on ${\\mathbb C}^{n_{Z}}$. \n One can compute the adjoint $\\left({\\mathcal C}_{Z, \\sbm{ X & -Y}} \n \\right)^{*}: \\; {\\mathbb C}^{n_{Z}}\\to \\sbm{ H^{2}_{p}(\\Pi_+) \\\\\nH^{ 2}_{m}(\\Pi_+) }$ explicitly as \n$$\n({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} \\colon x \\mapsto \n\\begin{bmatrix} -X^{*} \\\\ Y^{*} \\end{bmatrix} (\\lambda I + Z^{*})^{-1} x.\n$$\nThen the Kre\\u{\\i}n-space orthogonal complement $H^{2}_{p+m}(\\Pi_{+}) [-]_{J} {\\mathcal M}_{{\\mathfrak D}_{L}}$ \ncan be identified with\n\\begin{align}  \n{\\mathcal M}_{{\\mathfrak D}_{L}}^{[\\perp H^{2}_{p+m}(\\Pi_{+}) ]} &=\nJ \\cdot {\\rm Ran} ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} \\notag\\\\\n&=\\left\\{ \\begin{bmatrix} -X^{*} \\\\ -Y^{*} \\end{bmatrix} (\\lambda I + \nZ^{*})^{-1} x \\colon x \\in {\\mathbb C}^{n_Z} \\right\\}. \\label{cMDL}\n\\end{align}\n\nTo characterize when ${\\mathcal M}_{{\\mathfrak D}_{L}}^{[\\perp \nH^{2}_{p+m}(\\Pi_{+})]}$ is a \npositive subspace, it suffices to compute the Kre\\u{\\i}n-space \ninner-product gramian matrix ${\\mathbb G}$ for ${\\mathcal M}_{{\\mathfrak \nD}_{L}}^{[\\perp H^{2}_{p+m}(\\Pi_{+})]}$ with respect to its parametrization by \n${\\mathbb C}^{n_{Z}}$ in \\eqref{cMDL}:\n\\begin{align*}\n& \\langle {\\mathbb G} x, x' \\rangle_{{\\mathbb C}^{n_{Z}}}  \\\\\n& \\quad = \n \\frac{1}{2 \\pi}\\left\\langle J \\begin{bmatrix} -X^{*} \\\\ -Y^{*} \n\\end{bmatrix} (\\lambda I + Z^{*})^{-1} x,\\, \n\\begin{bmatrix} -X^{*} \\\\ -Y^{*} \\end{bmatrix} (\\lambda I \n+ Z^{*})^{-1} x' \\right\\rangle_{H^{2}_{p+m}(\\Pi_{+})} \\\\\n&  \\quad  = \\frac{1}{2 \\pi}\\int_{-\\infty}^{\\infty}\\langle (-iy I + Z)^{-1} (X X^{*} -  \nY Y^{*}) (iyI + Z^{*})^{-1} x,\\, x' \\rangle_{{\\mathbb C}^{n_{Z}}}\\, dy. \n\\end{align*}\nThus ${\\mathbb G}$ is given by\n$$   \n{\\mathbb G} = \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} (-iyI + Z)^{-1} (X X^{*} - Y \nY^{*}) (iyI + Z^{*})^{-1}\\, dy.\n$$\nIntroduce the change of variable $\\zeta = i y$, $d\\zeta = i \\, dy$ to \nwrite this as a complex line \nintegral\n\\begin{align*}\n {\\mathbb G} & = \\frac{1}{2 \\pi i} \\lim_{R \\to \\infty} \\int_{\\Gamma_{R,1}} \n (-\\zeta I + Z)^{-1} (X X^{*} - \n  Y Y^{*}) (\\zeta I + Z^{*})^{-1}\\, d \\zeta \\\\\n  & =  \\frac{1}{2 \\pi i}  \\lim_{R \\to \\infty} \n  \\int_{-\\Gamma_{R,1}} (\\zeta I - Z)^{-1} (X  X^{*} - YY^{*}) (\\zeta I \n  + Z^{*})^{-1}\\, d\\zeta\n\\end{align*}\nwhere $\\Gamma_{R,1}$ is the straight line from $-iR$ to $iR$ and \n$-\\Gamma_{R,1}$ is the same path but with reverse orientation (the \nstraight line from $iR$ to $-iR$).  Since the integrand\n\\begin{equation}   \\label{fzeta}\n f(\\zeta) =\n(\\zeta I - Z)^{-1} (X  X^{*} - YY^{*}) (\\zeta I  + Z^{*})^{-1}\n\\end{equation}\nsatisfies an estimate of the form\n$ \\| f(\\zeta \\| \\le \\frac{M}{ |\\zeta|^{2}}\\; $ as $\\; |\\zeta| \\to \\infty$, \nit follows that\n$$\n\\lim_{R \\to \\infty} \\int_{\\Gamma_{R,2}}  (\\zeta I - Z)^{-1} (X  X^{*} - YY^{*}) (\\zeta I \n  + Z^{*})^{-1}\\, d\\zeta = 0\n$$\nwhere $\\Gamma_{R,2}$ is the semicircle of radius $R$ with counterclockwise \norientation starting at the point $-iR$ and ending at the point \n$iR$ (parametrization:  $\\zeta = R e^{i \\theta}$ with $-\\pi\/2 \\le \n\\theta \\le \\pi\/2$).  Hence we see that  \n$$\n{\\mathbb G} = \\frac{1}{ 2 \\pi i} \\lim_{R \\to \\infty} \\int_{\\Gamma_{R}} \n(\\zeta I - Z)^{-1} (X  X^{*} - YY^{*}) (\\zeta I \n  + Z^{*})^{-1}\\, d\\zeta\n$$\nwhere $\\Gamma_{R}$ is the simple closed curve $-\\Gamma_{R,1} + \n\\Gamma_{R,2}$.  By the residue theorem, this last expression is \nindependent of $R$ once $R$ is so large that all the RHP poles of the \nintegrand $f(\\zeta)$ \\eqref{fzeta} are inside the curve $\\Gamma_{R}$, \nand hence\n$$\n{\\mathbb G} =  \\frac{1}{2 \\pi i} \\int_{\\Gamma_{R}}\n(\\zeta I - Z)^{-1} (X X^{*} - Y Y^{*} ) (\\zeta I + Z^{*})^{-1}\\, \nd\\zeta \n$$\nfor any $R$ large enough. This enables us to compute ${\\mathbb G}$ via residues:\n\\begin{equation}   \\label{bbG}\n    {\\mathbb G} = \\sum_{z_{0} \\in\\Pi_+} {\\rm Res}_{\\zeta = z_{0}}\n(\\zeta I - Z)^{-1} (X X^{*} - Y Y^{*} ) (\\zeta I +  Z^{*})^{-1}.\n\\end{equation}\n\nWe wish to verify that ${\\mathbb G}$ satisfies the Lyapunov equation\n\\begin{equation}  \\label{LyapunovG}\n    {\\mathbb G} Z^{*} + Z {\\mathbb G} = X X^{*} - Y Y^{*}.\n\\end{equation}\nToward this end let us first note that\n\\begin{align*}\n&(\\zeta I - Z)^{-1}A(\\zeta I + Z^{*})^{-1}Z^*+Z(\\zeta I - Z)^{-1}A\n(\\zeta I + Z^{*})^{-1}\\\\\n&=(\\zeta I - Z)^{-1}A-A(\\zeta I + Z^{*})^{-1}\n\\end{align*}\nfor any $A\\in\\mathbb C^{n_Z\\times n_Z}$. Making use of the latter equality with \n$A=X X^{*} - YY^{*}$ \nwe now deduce from the formula \\eqref{bbG} for ${\\mathbb G}$ that\n\\begin{align*}\n    {\\mathbb G} Z^{*} + Z {\\mathbb G} & = \n  \\sum_{z_{0} \\in \\Pi_+} {\\rm Res}_{\\zeta = z_{0}} \n \\left( (\\zeta I - Z)^{-1} (XX^{*} - Y Y^{*})  \\right. \\\\\n & \\quad \\quad \\quad \\quad  \\quad \\quad \\left. - (XX^{*} - Y Y^{*}) \n (\\zeta I + Z^{*})^{-1} \\right) \\\\\n & =  I \\cdot (XX^{*} - Y Y^{*}) - (XX^{*} - Y Y^{*})  \\cdot 0\n  = XX^{*} - YY^{*}\n\\end{align*}\nwhere for the last step we use that $Z$ has all its spectrum in the \nright half plane while $-Z^{*}$ has all its spectrum in the left\nhalf plane; also note that in general the sum of the residues of any \nresolvent matrix $R(\\zeta) = ( \\zeta I - A)^{-1}$ is the identity \nmatrix, due to the Laurent expansion at infinity for $R(\\zeta)$:  \n$R(\\zeta) = \\sum_{n=0}^{\\infty} A^{n} \\zeta^{-n-1}$.\nThis completes the verification of \\eqref{LyapunovG}.\n\n\\smallskip\n\nSince both $\\Gamma_{L}$ and ${\\mathbb G}$ satisfy the same Lyapunov \nequation \\eqref{GammaL} which has a \nunique solution since $\\sigma(Z) \\cap \\sigma(-Z^{*}) = \\emptyset$, we \nconclude that ${\\mathbb G} = \\Gamma_{L}$.   This completes the direct proof \nof the equivalence of conditions (1) and (2) in Corollary \\ref{C:BTOA-NP}  for the case \nthat $\\mathfrak D = {\\mathfrak D}_{L}$.\n\nTo make the connection with the kernel $K_{\\Theta,J}$, we note that there \nis a standard way to identify a reproducing kernel Hilbert space ${\\mathcal H}(K)$ of a \nparticular form with an operator range (see e.g.\\ \\cite{Sarason} \nor \\cite{BB-HOT}).  Specifically, let $M_{\\Theta}$ be the \nmultiplication operator\n$$\n M_{\\Theta} \\colon f(\\lambda) \\mapsto \\Theta(\\lambda) f(\\lambda)\n$$\nacting on $H^{2}_{p+m}(\\Pi_{+})$, identify $J$ with $J \\otimes \nI_{H^{2}}(\\Pi_{+})$ acting on $H^{2}_{p+m}(\\Pi_{+})$, and define $W \n\\in {\\mathcal L}(H^{2}_{p+m}(\\Pi_{+}))$ by\n$$\n  W = J - M_{\\Theta} J (M_{\\Theta})^{*}.\n$$\nFor $w \\in \\Pi_{+}$ and ${\\mathbf y} \\in {\\mathbb C}^{p+m}$, let \n$k_{w,{\\mathbf y}}(z) = \\frac{1}{z - \\overline{w}}{\\mathbf y}$ by the kernel element \nassociate with the Szeg\\H{o} kernel $k_{\\rm Sz} \\otimes I_{{\\mathbb \nC}^{p+m}}$. One can verify\n$$\n W k_{w,{\\mathbf y}} = K_{\\Theta,J}( \\cdot, w) {\\mathbf y} \\in {\\mathcal H}(K_{\\Theta, J}),\n$$\nand furthermore,\n\\begin{align*}\n    \\langle W k_{w_{j},{\\mathbf y}_{j}}, \\, W k_{w_{i},{\\mathbf y}_{i}} \n    \\rangle_{{\\mathcal H}(K_{\\Theta,J})} & =\n    \\langle K_{\\Theta,J}((w_{i},w_{j}) {\\mathbf y}_{j},\\, {\\mathbf y}_{i} \n    \\rangle_{{\\mathbb C}^{p+m}} \\\\\n    & = \\langle W k_{w_{j}, {\\mathbf y}_{j}}, \\, k_{w_{i}, {\\mathbf y}_{i}} \n    \\rangle_{H^{2}_{p+m}(\\Pi_{+})}.\n\\end{align*}\nAs $\\Theta$ is rational and $M_{\\Theta}$ is a $J$-isometry, one can see that \n$\\operatorname{Ran} W$ is already closed. Hence we have \nthe concrete identification ${\\mathcal H}(K_{\\Theta,J}) = \\operatorname{Ran} \nW$ with lifted inner product\n$$\n\\langle W f, W g\\rangle_{{\\mathcal H}(K_{\\Theta,J})} = \\langle W f, g \n\\rangle_{H^{2}_{p+m}(\\Pi_{+})}.\n$$\nAs $M_{\\Theta}$ is a $J$-isometry, the operator \n$M_{\\Theta} J (M_{\\Theta})^{*} =: M_{\\Theta}(M_{\\Theta})^{[*]}$ is \nthe $J$-selfadjoint projection onto $\\Theta \\cdot \nH^{2}_{p+m}(\\Pi_{+})$ and $WJ = I - M_{\\Theta} (M_{\\Theta})^{[*]}$ is \nthe $J$-self-adjoint projection onto $H^{2}_{p+m} [-] \\Theta \\cdot \nH^{2}_{p+m}(\\Pi_{+}) = {\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$.  We then see \nthat, for all $f,g \\in H^{2}_{p+m}(\\Pi_{+})$,\n$$\\langle W J f, W J g \\rangle_{{\\mathcal H}(K_{\\Theta,J})} =\n \\langle  W J f, J g \\rangle_{H^{2}_{p+m}(\\Pi_{+})} =\n\\langle J \\cdot WJ f, WJg \\rangle_{H^{2}_{p+m}(\\Pi_{+})},\n$$\ni.e., the identity map is a Kre\\u{\\i}n-space isomorphism between \n${\\mathcal H}(K_{\\Theta,J})$ and ${\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]}$ with the \n$J$-inner product.\nIn particular, we arrive at the equivalence of conditions (2) and (3) \nin Corollary \\ref{C:BTOA-NP} for Case 1.\n\n\\smallskip\n\n\\noindent\n\\textbf{Case 2: The general BTOA setting:}  \nTo streamline formulas to come, we introduce two types of \ncontrol-like operators and two types of observer-like operators\nas follows (for fuller details and systems-theory motivation, we \nrefer to \\cite{BR} for the discrete-time setting and \\cite{Amaya} for \nthe continuous-time setting). Suppose that $(A,B)$ is an input pair of matrices \n(so $A$ has, say, \nsize $N \\times N$ and $B$ has size $N \\times n$).  We assume that either $A$ is \nstable ($\\sigma(A) \n\\subset \\Pi_{-}$) or $A$ is antistable ($\\sigma(A) \\subset \\Pi_{+}$).  \nIn case $\\sigma(A) \\subset \\Pi_{+}$, we define a \ncontrol-like operator as appeared in the Case 1 analysis\n$$\n{\\mathcal C}_{A,B} \\colon H^{2}_{n}(\\Pi_{+}) \\to {\\mathbb C}^{N}\n$$\nby\n\\begin{align*}\n  {\\mathcal C}_{A,B} \\colon g & \\mapsto (Bg)^{\\wedge L}(A):  = \\sum_{z \\in \\Pi_{+}} {\\rm \n  Res}_{\\lambda = z} (\\lambda I - A)^{-1} B g(\\lambda) \\\\\n  & = - \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} (iy I - A)^{-1} B \n  g(iy)\\, dy.\n\\end{align*}\nIn case $\\sigma(A) \\subset \\Pi_{-}$, we define a complementary \ncontrol-like operator \n$$\n{\\mathcal C}^{\\times}_{A,B} \\colon H^{2}_{n}(\\Pi_{-}) \\to {\\mathbb C}^{N}\n$$\nby \n\\begin{align*} \n {\\mathcal C}^{\\times}_{A,B} \\colon g & \\mapsto (Bg)^{\\wedge L}(A):  = \\sum_{z \n \\in \\Pi_{-}} {\\rm Res}_{\\lambda = z} (\\lambda I - A)^{-1} B g(\\lambda) \\\\\n  & =  \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} (iy I - A)^{-1} B \n  g(iy)\\, dy.\n\\end{align*}\nSuppose next that $(C,A)$ is an output-pair, say of respective sizes \n$n \\times N$ and $N \\times N$, and that $A$ is either stable or \nantistable.  In case $A$ is antistable ($\\sigma(A) \\subset \\Pi_{+}$), \nwe define the observer-like operator \n$$\n  {\\mathcal O}_{C,A} \\colon {\\mathbb C}^{N} \\to H^{2}_{n}(\\Pi_{-})\n$$\nby\n$$\n{\\mathcal O}_{C,A} \\colon x \\mapsto C (\\lambda I - A)^{-1} x.\n$$\nIn case $A$ is stable (so $\\sigma(A) \\subset \\Pi_{-})$, then the \ncomplementary observer-like operator  is given by the same formula \nbut maps to the complementary $H^{2}$ space:\n$$\n{\\mathcal O}^{\\times}_{C,A} \\colon {\\mathbb C}^{N} \\mapsto  H^{2}_{n}(\\Pi_{+})\n$$\ngiven again by\n$$\n  {\\mathcal O}^{\\times}_{C,A} \\colon x \\mapsto C (\\lambda I - A)^{-1} x.\n$$\nWe are primarily interested in the case where $A$ is antistable and \nwe consider the operators ${\\mathcal C}_{A,B} \\colon H^{2}_{n}(\\Pi_{+}) \\to \n{\\mathbb C}^{N}$ and ${\\mathcal O}_{C,A} \\colon {\\mathbb C}^{N} \\mapsto \nH^{2}_{n}(\\Pi_{-})$.  However a straightforward exercise is to show \nthat the complementary operators come up when computing adjoints:\nfor $A$ antistable, $-A^{*}$ is stable and we have the formulas\n$$\n({\\mathcal O}_{C,A})^{*} = -{\\mathcal C}^{\\times}_{-A^{*}, C^{*}} \\colon \nH^{2}_{n}(\\Pi_{-}) \\mapsto {\\mathbb C}^{N}, \\quad\n({\\mathcal C}_{A,B})^{*} = {\\mathcal O}^{\\times}_{B^{*}, -A^{*}} \\colon {\\mathbb C}^{N} \n\\mapsto H^{2}_{n}(\\Pi_{+}).\n$$\n\nAssume now that \n${\\mathcal M}_{\\mathfrak D} \\subset L^{2}_{p+m}(\\Pi_{+})$ is defined as in \n\\eqref{cMrep} for a $\\Pi_{+}$-admissible interpolation data set \n${\\mathfrak D} = (U,V,W; Z, X, Y; \\Gamma)$. Thus $(U,W)$ and $(V,W)$ are \noutput pairs with $\\sigma(W) \\subset \\Pi_{+}$ and $(Z,X)$ and $(Z,Y)$ \nare input pairs with $\\sigma(Z) \\subset \\Pi_{+}$.  We therefore have \nobserver-like and control-like operators\n\\begin{align*}\n&{\\mathcal O}_{V,W} \\colon {\\mathbb C}^{n_{W}} \\to H^{2}_{p}(\\Pi_{-}), \n\\quad {\\mathcal O}_{U,W} \\colon {\\mathbb C}^{n_{W}} \\to H^{2}_{m}(\\Pi_{-}), \\\\\n& {\\mathcal C}_{Z,X} \\colon H^{2}_{p}(\\Pi_{+}) \\to {\\mathbb C}^{n_{Z}}, \\quad\\quad\n{\\mathcal C}_{Z,Y} \\colon H^{2}_{m}(\\Pi_{+}) \\to {\\mathbb C}^{n_{Z}}\n\\end{align*}\ndefined as above, as well as the observer-like and control-like operators\n$$\n {\\mathcal O}_{\\sbm{ V \\\\ U}, W} : = \\begin{bmatrix} {\\mathcal O}_{V,W} \\\\ {\\mathcal O}_{U,W} \n \\end{bmatrix},\\quad \n {\\mathcal C}_{Z, [ X \\; \\;  -Y]} = \\begin{bmatrix} \n {\\mathcal C}_{Z,X} & - {\\mathcal C}_{Z,Y} \\end{bmatrix}.\n $$\nThen the adjoint  operators have the form\n \\begin{align*}\n&({\\mathcal O}_{V,W})^{*} = -{\\mathcal C}^{\\times}_{-W^{*}, V^{*}} \\colon H^{2}_{p}(\\Pi_{-}) \\to\n{\\mathbb C}^{n_{W}}, \\\\ \n& ({\\mathcal O}_{U,W})^{*} = -{\\mathcal C}^{\\times}_{-W^{*}, U^{*}} \\colon \nH^{2}_{m}(\\Pi_{-}) \\to {\\mathbb C}^{n_{W}}, \\\\\n& ({\\mathcal C}_{Z,X})^{*} = {\\mathcal O}^{\\times}_{X^{*}, -Z^{*}} \\colon {\\mathbb \nC}^{n_{Z}} \\to H^{2}_{p}(\\Pi_{+}), \\\\\n& ({\\mathcal C}_{Z,Y})^{*} = {\\mathcal O}^{\\times}_{Y^{*}, -Z^{*}} \\colon {\\mathbb C}^{n_{Z}}\n\\to H^{2}_{m}(\\Pi_{+})  \n\\end{align*}\nand are given explicitly by: \n\\begin{align*}\n  & ({\\mathcal O}_{V,W})^{*} \\colon g_{1} \\mapsto  \n  -\\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} (iy I + W^{*})^{-1} V^{*} \n  g_1(iy)\\, dy, \\\\\n  &  ({\\mathcal O}_{U,W})^{*} \\colon g_{2} \\mapsto  \n  -\\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} (iy I + W^{*})^{-1} \n  U^{*}g_2(iy)\\, dy, \\\\\n& ({\\mathcal C}_{Z,X})^{*} \\colon x \\mapsto  X^{*} (\\lambda I + Z^{*})^{-1} x,\n\\quad ({\\mathcal C}_{Z,Y})^{*} \\colon x \\mapsto  Y^{*} (\\lambda I + Z^{*})^{-1} x.\n  \\end{align*}\nFurthermore one can check via computations as in the derivation of \n\\eqref{bbG} above that the $J$-observability and  $J$-controllability gramians\n\\begin{align*}  & {\\mathcal G}^{J}_{Z,\\sbm{X & -Y}} : = \n    {\\mathcal C}_{Z,X} {\\mathcal C}_{Z,X}^{*} - {\\mathcal C}_{Y,Z} {\\mathcal C}_{Z,Y}^{*} =: {\\mathcal G}_{Z,X} - \n    {\\mathcal G}_{Z,Y}, \\\\\n  & {\\mathcal G}^{J}_{\\sbm{ V \\\\ U}, W} : = {\\mathcal O}^{*}_{V,W} {\\mathcal O}_{V,W} - \n  {\\mathcal O}_{U,W}^{*} {\\mathcal O}_{U,W} =: {\\mathcal G}_{V,W} - {\\mathcal G}_{U,W}\n\\end{align*}\nsatisfy the respective Lyapunov equations\n\\begin{align*}\n    & {\\mathcal G}^{J}_{Z,\\sbm{ X & -Y}} Z^{*} + Z {\\mathcal G}^{J}_{Z,\\sbm{ X & -Y}} \n    = X X^{*} - Y Y^{*}, \\\\\n    & {\\mathcal G}^{J}_{\\sbm{ V \\\\ U}, W} W + W^{*} {\\mathcal G}^{J}_{\\sbm{V \\\\ U}, W} = \n    V^{*}V - U^{*} U.\n\\end{align*}\nHence, by the uniqueness of such solutions and the characterizations \nof $\\Gamma_{L}$ and $\\Gamma_{R}$ in \\eqref{GammaL}, \\eqref{GammaR},\nwe get \n\\begin{equation} \\label{GammaLRid}\n    {\\mathcal G}^{J}_{\\sbm{X & -Y}, Z} = \\Gamma_{L}, \\quad\n    {\\mathcal G}^{J}_{\\sbm{V \\\\ U}, W} = -\\Gamma_{R}.\n\\end{equation}\nThen the representation \\eqref{cMrep} for ${\\mathcal M}_{\\mathfrak D}$ can be rewritten more \nsuccinctly as\n\\begin{align}  \n    {\\mathcal M}_{\\mathfrak D} = &\\left\\{ {\\mathcal O}_{\\sbm{V \\\\ U}, W} x + \\sbm{f_{1} \n    \\\\ f_{2}} \\colon x \\in {\\mathbb C}^{n_{W}} \\text{ and } \\sbm{ \n    f_{1} \\\\ f_{2} }  \\in H^{2}_{p+m}(\\Pi_{+})  \\right. \\notag \\\\\n& \\left. \\quad \\quad  \\text{such that } {\\mathcal C}_{Z,\\sbm{ X & -Y}} \\sbm{ f_{1} \\\\ f_{2}} = \n \\Gamma x \\right\\}.   \n\\label{cMrep'}\n\\end{align}\nIt is readily seen from the latter formula that \n    \\begin{align}\nP_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D}&=\\operatorname{Ran}{\\mathcal O}_{\\sbm{ V \\\\ U}, W},\n\\label{ad8} \\\\\n{\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+}) &=\n\\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}, \\notag\n\\end{align}\nand therefore, \n$$\n{\\mathcal M}_{\\mathfrak D} \\cap \\sbm{ H^{2}_{p}(\\Pi_{+}) \\\\ 0 } =\n\\sbm{ \\operatorname{Ker} {\\mathcal C}_{Z, X} \\\\ 0 }, \\quad\n{\\mathcal M}_{\\mathfrak D} \\cap \\sbm{0 \\\\ H^{2}_{m}(\\Pi_{+})} =\n\\sbm{ 0 \\\\ \\operatorname{Ker} {\\mathcal C}_{Z, Y} }.\n$$\n\n\\begin{lemma}  \\label{L:cMDperp} If ${\\mathcal M}_{\\mathfrak D}$ is given by \n    \\eqref{cMrep'}, then the $J$-orthogonal complement \n${\\mathcal M}_{\\mathfrak D}^{[\\perp]} = L^{2}_{p+m}(i {\\mathbb R}) [-]_{J} {\\mathcal M}_{\\mathfrak D}$\nwith respect to the space $L^{2}_{p+m}(i {\\mathbb R})$\nis given by\n\\begin{align} \n    {\\mathcal M}_{\\mathfrak D}^{[\\perp]} = &\\left\\{ J( {\\mathcal C}_{Z, \\sbm{ X & \n    -Y}})^{*} y + \\sbm{ g_{1} \\\\ g_{2}} \\colon y \\in {\\mathbb \n    C}^{n_{Z}} \\text{ and } \\sbm{ g_{1} \\\\ g_{2}} \\in \n    H^{2}_{p+m}(\\Pi_{-}) \\right. \\notag \\\\\n& \\left. \\quad \\quad \\text{such that } \\; ({\\mathcal O}_{V,W})^{*} g_{1} - ({\\mathcal O}_{U,W})^{*} g_{2} = \n-\\Gamma^{*} y \\right\\}.\n\\label{cMperprep'}\n\\end{align}\n    \\end{lemma}\n    \n\\begin{proof} Since ${\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ is\n    $J$-orthogonal to ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+}) =\n    \\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}$, it follows that \n$P_{H_{p+m}^{2}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ is also \n    $J$-orthogonal to $\\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}$.  Hence\n    $P_{H^{2}_{p+m}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \\subset J \n    \\operatorname{Ran}( ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}$ and \n    each ${\\mathbf g}\\in{\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ has the form \n$$\n{\\mathbf g} = J({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}y + \\sbm{ g_{1} \\\\ g_{2}}\\quad\\mbox{with}\\quad \n    y \\in {\\mathbb C}^{n_{Z}} \\; \\; \\mbox{and} \\; \\;  \\sbm{g_{1} \\\\ g_{2}} \\in H^{2 \n    \\perp}_{p+m}(\\Pi_{+}).\n$$\n  For such an element to be in \n    ${\\mathcal M}_{\\mathfrak D}^{[\\perp]}$, we compute the $J$-inner product of such an \n    element against a generic element of ${\\mathcal M}_{\\mathfrak D}$: for all \n    $\\sbm{f_{1} \\\\ f_{2}} \\in H^{2}_{p+m}(\\Pi_{+})$ and $x \\in \n    {\\mathbb C}^{n_{Z}}$ such that ${\\mathcal C}_{Z, X} f_{1} - {\\mathcal C}_{Z, Y} \n    f_{2} = \\Gamma x$, we must have\n \\begin{align*}\n     0 & = \\left\\langle J \\left( J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} y + \n     \\sbm{ g_{1} \\\\ g_{2}} \\right), {\\mathcal O}_{\\sbm{V \\\\ U}, W} x + \\sbm{ \n     f_{1} \\\\ f_{2}} \\right\\rangle_{L^{2}_{p+m}(i {\\mathbb R})} \\\\\n     & = \\langle y, {\\mathcal C}_{Z,X} f_{1} - {\\mathcal C}_{Z,Y} f_{2} \n     \\rangle_{{\\mathbb C}^{n_{_Z}}} +\n  \\langle ({\\mathcal O}_{V,W})^{*} g_{1} - ({\\mathcal O}_{U,W})^{*} g_{2}, x \n  \\rangle_{{\\mathbb C}^{n_{_W}}}  \\\\\n  & = \\langle y, \\Gamma x \\rangle_{{\\mathbb C}^{n_{Z}}} + \n   \\langle ({\\mathcal O}_{V,W})^{*} g_{1} - ({\\mathcal O}_{U,W})^{*} g_{2}, x \n  \\rangle_{{\\mathbb C}^{n_{_W}}} \n\\end{align*} \nwhich leads to the coupling condition\n$({\\mathcal O}_{V,W})^{*} g_{1} - ({\\mathcal O}_{U,W})^{*} g_{2} = - \\Gamma^{*} y$ in \\eqref{cMperprep'}.\n\\end{proof}\nAs a consequence of the representation \\eqref{cMperprep'} we see that\n\\begin{align}\n& P_{H^{2}_{p+m}(\\Pi_{+})} {\\mathcal M}_{\\mathfrak D}^{[\\perp]} =\n\\operatorname{Ran} J  ({\\mathcal C}_{Z,\\sbm{ X & -Y}})^{*},  \\notag\\\\\n& {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \\cap H^{2}_{p+m}(\\Pi_{-}) =\n\\operatorname{Ker} \\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & - \n({\\mathcal O}_{U,W})^{*} \\end{bmatrix}\n\\label{ad9}\n\\end{align}\nand therefore,\n$$\n {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \\cap \\sbm{ H^{2}_{p}(\\Pi_{-}) \\\\ 0 } =\n\\sbm{\\operatorname{Ker} ({\\mathcal O}_{V,W})^{*} \\\\ 0},\\quad \n {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \\cap \\sbm{ 0 \\\\ H^{2}_{m}(\\Pi_{-})} =\n\\sbm{ 0 \\\\ \\operatorname{Ker} ({\\mathcal O}_{U,W})^{*}}.\n$$\nIn this section we shall impose an additional assumption:\n\\smallskip\n\n\\noindent\n\\textbf{Nondegeneracy assumption:}\n{\\sl Not only ${\\mathcal M}_{\\mathfrak D}$ but also ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+})$ \nand ${\\mathcal M}_{\\mathfrak D}^{[\n\\perp]} \\cap H^{2}_{p+m}(\\Pi_{-})$ (see \\eqref{ad8} and \n\\eqref{ad9}) are regular subspaces (i.e., have good Kre\\u{\\i}n-space \northogonal complements---as explained in Section \\ref{S:Kreinprelim}) of the\nKre\\u{\\i}n space $L^{2}_{p+m}(\\Pi_{+})$ (with the $J$-inner product).}\n\n\\smallskip\n\nWe proceed via a string of lemmas.\n\n\\begin{lemma} \\label{L:decom}\n{\\rm (1)} The space ${\\mathcal M}_{\\mathfrak D}$ given in \\eqref{cMrep'} decomposes as\n\\begin{equation}   \\label{Mdecom}\n  {\\mathcal M}_{\\mathfrak D} = \\widehat{\\mathbb G}_{T} [+] {\\mathcal M}_{{\\mathfrak D},1} [+] \n  {\\mathcal M}_{{\\mathfrak D},2},\n\\end{equation}\nwhere \n\\begin{align}\n    \\widehat{\\mathbb G}_{T} &= {\\mathcal M}_{\\mathfrak D} [-]_{J} \n    \\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}, \\notag \\\\\n    {\\mathcal M}_{{\\mathfrak D},1} &= \\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}[-]_{J} \n\\left(\\sbm{\\operatorname{Ker} {\\mathcal C}_{Z, X}\\\\ 0} \\oplus  \n\\sbm{0\\\\ \\operatorname{Ker} {\\mathcal C}_{Z, Y}}\\right), \\notag \\\\\n     {\\mathcal M}_{{\\mathfrak D}, 2} &=\n\\sbm{\\operatorname{Ker} {\\mathcal C}_{Z, X}\\\\ 0} \\oplus  \\sbm{0\\\\ \\operatorname{Ker} {\\mathcal C}_{Z, Y}}.\n    \\label{def012}\n\\end{align}\nMore explicitly, the operator $T: \\operatorname{Ran}\n{\\mathcal O}_{\\sbm{ V \\\\ U}, W}\\to \\operatorname{Ran} J ({\\mathcal C}_{Z,\\sbm{X & -Y}})^{*}$ \nis uniquely determined by the identity\n\\begin{equation}  \\label{TGamma}\n    {\\mathcal C}_{Z, \\sbm{ X & -Y}} T {\\mathcal O}_{\\sbm{ V \\\\ U}, W} = -\\Gamma,\n\\end{equation}\nand  $\\widehat{\\mathbb G}_{T}$ is the graph space for $-T$ parametrized as\n\\begin{align}\n\\widehat{\\mathbb G}_{T} & = \\left\\{- {\\mathbf f} + T {\\mathbf f} \\colon \n{\\mathbf f}\\in \\operatorname{Ran}\n{\\mathcal O}_{\\sbm{V \\\\ U}, W}\\right\\} \\notag \\\\\n& =  \\left\\{- {\\mathcal O}_{\\sbm{V \\\\ U}, W} x + T {\\mathcal O}_{\\sbm{V \\\\ U}, W} x \\colon\nx \\in {\\mathbb C}^{n_{W}} \\right\\},\n\\label{graphT}\n\\end{align}\nwhile ${\\mathcal M}_{{\\mathfrak D},1}$ is given\nexplicitly by\n\\begin{equation}   \\label{cMD1}\n    {\\mathcal M}_{{\\mathfrak D},1} = \\operatorname{Ran}  \\begin{bmatrix}\n   ( {\\mathcal C}_{Z,X})^{*} ({\\mathcal G}_{Z,X})^{-1} {\\mathcal G}_{Z,Y} \\\\  ({\\mathcal C}_{Z,Y})^{*} \\end{bmatrix}.\n\\end{equation}\n\n{\\rm (2)} Dually, the subspace ${\\mathcal M}_{\\mathfrak D}^{[\\perp]} =\nL^{2}_{p+m}(i {\\mathbb R}) [-]_{J} {\\mathcal M}_{\\mathfrak D}$ \n   decomposes as\n  \\begin{equation}   \\label{Mperpdecom}\n    {\\mathcal M}_{\\mathfrak D}^{[\\perp]} = {\\mathbb G}_{T^{[*]}} [+]  \n    ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1} [+] ( {\\mathcal M}_{\\mathfrak  D}^{[\\perp]})_{2},\n  \\end{equation}\nwhere\n  \\begin{align}\n  & {\\mathbb G}_{T^{[*]}}  = {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \n   [-]_{J} \\operatorname{Ker} \\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & - \n   ({\\mathcal O}_{U,W})^{*} \\end{bmatrix},\\notag \\\\\n  & ({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})_{1} = \n  \\operatorname{Ker}\\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & - \n  ({\\mathcal O}_{U,W})^{*} \\end{bmatrix}\n[-]_{J} \\left( \\sbm{ \\operatorname{Ker} ({\\mathcal O}_{V,W})^{*} \\\\ 0} \n\\oplus \\sbm{ 0 \\\\ \\operatorname{Ker} \n({\\mathcal O}_{U,W})^{*} } \\right),\\notag\\\\ \n   & ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{2} =\n   \\sbm{ \\operatorname{Ker} ({\\mathcal O}_{V,W})^{*} \\\\ 0}  \\oplus \\sbm{ 0 \\\\ \\operatorname{Ker} \n   ({\\mathcal O}_{U,W})^{*}}.    \n\\label{def012perp}\n  \\end{align}\nMore explicitly,\n \\begin{align*}\n {\\mathbb G}_{T^{[*]}} & = \\left\\{ {\\mathbf g} + T^{[*]} {\\mathbf g} \\colon {\\mathbf g}\n  \\in \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} \\right\\}  \\\\\n  & =\n  \\left\\{ J({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} x +  T^{[*]} J ({\\mathcal C}_{Z, \\sbm{ X\n  & -Y}})^{*} x \\colon x \\in {\\mathbb C}^{n_{Z}}\\right\\}\n  \\end{align*}\nwhere  $T^{[*]} = J T^{*} J\\colon  \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}\n\\to \\operatorname{Ran} {\\mathcal O}_{\\sbm{ V \\\\ U}, W}$ is the $J$-adjoint  of $T$, and\n  \\begin{equation}   \\label{cMDperp1}\n   ({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})_{1} = \\operatorname{Ran} \n   \\begin{bmatrix} {\\mathcal O}_{V,W} \\\\ {\\mathcal O}_{U,W}({\\mathcal G}_{U,W})^{-1} {\\mathcal G}_{V,W}  \\end{bmatrix}.   \n   \\end{equation}\n\\end{lemma}\n\n\\begin{proof}  By the Nondegeneracy Assumption we can define subspaces\n\\eqref{def012} and \\eqref{def012perp}, so that ${\\mathcal M}_{\\mathfrak D}$\nand ${\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ decompose as in \\eqref {Mdecom} and \n\\eqref{Mperpdecom}, respectively.\n\n\\smallskip\n\nGiven an element ${\\mathbf g} \\in P_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D}$,\nthere is an ${\\mathbf f}\\in H^{2}_{p+m}(\\Pi_{+})$ so that $-{\\mathbf g} + {\\mathbf f} \\in {\\mathcal M}_{\\mathfrak D}$; \nfurthermore, one can choose \n$$\n{\\mathbf f}\\in H^{2}_{p+m}(\\Pi_{+}) [-]_{J} ({\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+}))=\nP_{H^{2}_{p+m}(\\Pi_{+})}{\\mathcal M}_{\\mathfrak D}^{[\\perp]}. \n$$\nIf ${\\mathbf f}'$ is another such choice, \n    then $(-{\\mathbf g} + {\\mathbf f}) - (-{\\mathbf g} + {\\mathbf f}') = {\\mathbf f} - {\\mathbf f}'$ is in\n    ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+})$ as well as in \n    $H^{2}_{p+m}(\\Pi_{+}) [-]_{J} ({\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+}))$.  By \n    the Nondegeneracy Assumption, we conclude that ${\\mathbf f} = {\\mathbf f}'$.  \n    Hence there is a well-defined map ${\\mathbf g} \\mapsto {\\mathbf f}$ defining a \n    linear operator $T$ from \n$$\nP_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D}=\\operatorname{Ran}{\\mathcal O}_{\\sbm{ V \\\\ U}, W}\n\\quad\\mbox{into}\\quad\nP_{H^{2}_{p+m}(\\Pi_{+})}{\\mathcal M}_{\\mathfrak D}^{[\\perp]}=\n\\operatorname{Ran} J ({\\mathcal C}_{Z,\\sbm{X & -Y}})^{*}\n$$\n(see \\eqref{ad8} and \\eqref{ad9}).\n    In this way we arrive at a well-defined operator $T$ so that \n    $\\widehat{\\mathbb G}_{T}$ as in \\eqref{graphT} is equal to the subspace \n    (see \\eqref{def012})\n$$\n{\\mathcal M}_{\\mathfrak D} [-]_{J} \\left({\\mathcal M}_{\\mathfrak D} \\cap \nH^{2}_{p+m}(\\Pi_{+})\\right)\n={\\mathcal M}_{\\mathfrak D} [-]_{J} \\operatorname{Ker} {\\mathcal C}_{Z, \\sbm{ X & -Y}}.\n$$\nTo check that $T$ is also given by \n    \\eqref{TGamma}, combine the fact that \n    $$\n     -{\\mathcal O}_{\\sbm{V \\\\ U}, W} x + T {\\mathcal O}_{\\sbm{V \\\\ U}, W} x\\in {\\mathcal M}_{\\mathfrak D}\n    $$\n    together with the characterization \\eqref{cMrep'} for \n    ${\\mathcal M}_{\\mathfrak D}$ to \n    deduce that\n    $$\n    {\\mathcal C}_{Z, \\sbm{ X & -Y}} \\cdot T {\\mathcal O}_{\\sbm{V \\\\ U}, W} x = \n    -\\Gamma x\n    $$\n    for all $x$ to arrive at \\eqref{TGamma}.  \n    \n\\smallskip\n\n    To get the formula \\eqref{cMD1}, we first note that\n\\begin{equation}  \\label{DecoupledCZXY}\n  H^{2}_{p+m}(\\Pi_{+}) [-]_{J}\\sbm{\\operatorname{Ker} {\\mathcal C}_{Z, X} \\\\  \n  \\operatorname{Ker} {\\mathcal C}_{Z, Y}}\n= \\sbm{ \\operatorname{Ran} \n  ({\\mathcal C}_{Z,X})^{*} \\\\ \\operatorname{Ran}  ({\\mathcal C}_{Z,Y})^{*} }.\n\\end{equation}\nThe space ${\\mathcal M}_{{\\mathfrak D}, 1}$ is the intersection of this space \nwith ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+})$.\nTherefore, it consists of elements of the form $\\sbm{ ({\\mathcal C}_{Z,X})^{*} y_{1} \\\\ \n({\\mathcal C}_{Z,Y})^{*} y_{2} }$ \nsubject to condition\n$$\n 0 = \\begin{bmatrix} {\\mathcal C}_{Z,X} & - {\\mathcal C}_{Z,Y} \\end{bmatrix}\n \\begin{bmatrix} ({\\mathcal C}_{Z,X})^{*} y_{1} \\\\ ({\\mathcal C}_{Z,X})^{*} y_{2} \n \\end{bmatrix} = {\\mathcal C}_{Z,X} ({\\mathcal C}_{Z,X})^{*} y_{1} - {\\mathcal C}_{Z,Y} \n ({\\mathcal C}_{Z,Y})^{*} y_{2}.\n$$\nBy the $\\Pi_{+}$-admissibility requirement on the data set ${\\mathfrak \nD}$, the gramian ${\\mathcal G}_{Z,X}: = {\\mathcal C}_{Z,X} ({\\mathcal C}_{Z,X})^{*}$ is \ninvertible and hence we may solve this last equation for $y_{1}$:\n$$\n  y_{1} = {\\mathcal G}_{Z,X}^{-1} {\\mathcal C}_{Z,Y} ({\\mathcal C}_{Z,Y})^{*} y_{2}.\n$$\nWith this substitution, the element $\\sbm{ ({\\mathcal C}_{Z,X})^{*} y_{1} \\\\\n({\\mathcal C}_{Z,Y})^{*} y_{2} }$ \nof the $J$-orthogonal complement space \\eqref{DecoupledCZXY} assumes \nthe form\n$$\n  \\begin{bmatrix} ({\\mathcal C}_{Z,X})^{*} {\\mathcal G}_{Z,X}^{-1} {\\mathcal C}_{Z,Y} \n      ({\\mathcal C}_{Z,Y})^{*} y_{2} \\\\\n      ({\\mathcal C}_{Z,Y})^* y_{2} \\end{bmatrix}\n$$\nand we have arrived at the formula  \\eqref{cMD1} for ${\\mathcal M}_{{\\mathfrak \nD},1}$.\n\n\\smallskip\n\nFor the dual case (2), similar arguments starting with the \nrepresentation \\eqref{cMperprep'} for ${\\mathcal M}^{[\\perp]}_{\\mathfrak D}$ show that \nthere is an operator $T^{\\times}$ from \n$\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}$ into \n${\\mathcal M}_{\\mathfrak D}^{[\\perp]} [-]_{J} \\left({\\mathcal M}_{\\mathfrak D}^{[\\perp]}  \n\\cap H^{2}_{p+m}(\\Pi_{-}) \\right)$ so that\n$$\n  {\\mathcal M}_{\\mathfrak D}^{[\\perp]} [-]_{J} \\left( {\\mathcal M}_{\\mathfrak D}^{[\\perp]} \n  \\cap H^{2}_{p+m}(\\Pi_{-})\\right) = (I + T^{\\times}) \n  \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}.\n$$\nFrom the characterization \\eqref{cMperprep'} of the space \n${\\mathcal M}_{\\mathfrak D}^{[\\perp]}$ we see that\nthe condition \n$$\n J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}y + T^{\\times} J ({\\mathcal C}_{Z, \\sbm{ X & \n-Y}})^{*}y  \\in {\\mathcal M}_{\\mathfrak D}^{[\\perp]}\n$$\nrequires that, for all $y \\in {\\mathbb C}^{n_{Z}}$,\n$$\n\\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & - ({\\mathcal O}_{U,W})^{*} \\end{bmatrix} \n   T^{\\times} \\begin{bmatrix} ({\\mathcal C}_{Z,X})^{*} \\\\ ({\\mathcal C}_{Z,Y})^{*} \n    \\end{bmatrix} y = - \\Gamma^{*}y.\n$$\nCancelling off the vector $y$ and rewriting as an operator equation \nthen gives:\n\\begin{align*}\n & \\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & - ({\\mathcal O}_{U,W})^{*} \\end{bmatrix} \n   T^{\\times} \\begin{bmatrix} ({\\mathcal C}_{Z,X})^{*} \\\\ ({\\mathcal C}_{Z,Y})^{*} \n    \\end{bmatrix} \\\\\n & \\quad \\quad   =\n\\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & ({\\mathcal O}_{U,W})^{*} \\end{bmatrix} J \n    T^{\\times} J  \\begin{bmatrix} ({\\mathcal C}_{Z,X})^{*} \\\\ (-{\\mathcal C}_{Z,Y})^{*} \n    \\end{bmatrix} \\\\\n  & \\quad \\quad =  ({\\mathcal O}_{\\sbm{V \\\\ U}, W})^{*} J T^{\\times} J ({\\mathcal C}_{Z, \\sbm{ X & \n    -Y}})^{*} = -\\Gamma^{*}.\n\\end{align*}\nTaking adjoints of both sides of the identity \\eqref{TGamma} \nsatisfied by $T$, we see that \n$$\n  ({\\mathcal O}_{\\sbm{V \\\\ U}, W} )^{*} T^{*} ( {\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} = \n  -\\Gamma^{*}.\n$$\nSince $({\\mathcal O}_{\\sbm{V \\\\ U}, W} )^{*}$ is injective on \nthe range space of $T^{\\times}$ or $JT^{*}J$ and $({\\mathcal C}_{Z, \\sbm{ X & \n-Y}})^{*}$ maps onto the domain space of $T^{\\times}$ or $T^{*}$, it \nfollows that $T^{\\times} = J T^{*} J = T^{[*]}$.\nThe remaining points in statement (2) of the Lemma follow in much the \nsame way as the corresponding points in statement (1).\n\\end{proof}\n\\begin{lemma}  \\label{L:MperpK}\n{\\rm (1)} With ${\\mathcal K}$ as in \\eqref{defcK}, the subspace \\eqref{cPdata} decomposes as\n\\begin{equation}   \\label{MperpK}\n    {\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]} =  {\\mathbb G}_{T^{[*]}} \n    [+] ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1} [+] \\sbm{ \n    \\operatorname{Ker} ({\\mathcal O}_{V,W})^{*} \\\\ 0 }.\n\\end{equation}\nIn particular, ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is $J$-positive if and only \nif its subspace \n$$\n    ({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})_{0}:=  {\\mathbb G}_{T^{[*]}} \n    [+] ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1}\n$$\nis $J$-positive.\n\n\\smallskip\n\n{\\rm (2)} Dually, define a space ${\\mathcal K}' \\subset L^{2}_{p+m}(i{\\mathbb R})$ \nby\n\\begin{equation}  \\label{defcK'}\n  {\\mathcal K}' = \\sbm{ H^{2}_{p}(\\Pi_{-}) \\oplus \\operatorname{Ran} \n  ({\\mathcal C}_{Z,X})^{*} \\\\ L^{2}_{m}(i {\\mathbb R}) }.\n\\end{equation}\nThen  ${\\mathcal M}_{\\mathfrak D}^{[ \\perp]} \\subset {\\mathcal K}'$ and the space\n$$\n({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})^{[ \\perp {\\mathcal K}']}: = {\\mathcal K}' [-]_{J} {\\mathcal M}_{\\mathfrak D}^{[ \\perp]} \n= {\\mathcal K}' \\cap {\\mathcal M}_{\\mathfrak D}\n$$\nis given by\n$$\n    ({\\mathcal M}^{[\\perp]})^{[ \\perp {\\mathcal K}']} = \\widehat{\\mathbb G}_{T} [+] \n{\\mathcal M}_{{\\mathfrak D},1} [+] \\sbm{ 0 \\\\ \\operatorname{Ker} {\\mathcal C}_{Z,Y} }.\n$$\n In particular,  $({\\mathcal M}_{\\mathfrak D}^{[\\perp]})^{[ \\perp {\\mathcal K}']}$ is \n $J$-negative if and only if its subspace\n$$ \n( ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})^{[ \\perp {\\mathcal K}']})_{0}: = {\\mathbb G}_{T} [+] \n{\\mathcal M}_{{\\mathfrak D}, 1}\n$$\nis $J$-negative.\n\n \\end{lemma}\n \n\\begin{proof}\n    By definition, ${\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]} = {\\mathcal K} \\cap \n    {\\mathcal M}_{\\mathfrak D}^{[ \\perp ]}$, where ${\\mathcal M}_{\\mathfrak D}^{[ \\perp\n    ]}$ is given by \\eqref{Mperpdecom} and where, due to \\eqref{defcK} and \n\\eqref{cMrep}, ${\\mathcal K} =\\sbm{ L^{2}(i {\\mathbb R})     \\\\ \\operatorname{Ran} {\\mathcal O}_{U,W} \n\\oplus H^{2}_{m}(\\Pi_{+})}$. Note that \n    $$\n{\\mathcal G}_{T^{[*]}} \\subset {\\mathcal K}, \\quad ({\\mathcal M}_{\\mathfrak D}^{[ \n\\perp]})_{1} \\subset H^{2}_{p+m}(\\Pi_{+}) \\subset {\\mathcal K},\n$$\nwhile\n$$\n    ({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})_{2} \\cap {\\mathcal K} =\n \\sbm{ \\operatorname{Ker} ({\\mathcal O}_{V,W})^{*} \\\\ \\operatorname{Ker} \n ({\\mathcal O}_{U,W})^{*}} \\cap \\sbm{ L^{2}_{p}(i{\\mathbb R}) \\\\ \n \\operatorname{Ran} {\\mathcal O}_{U,W} \\oplus H^{2}_{m}(\\Pi_{+}) } = \n\\sbm{ \\operatorname{ Ker } ({\\mathcal O}_{V,W})^{*} \\\\ 0 }.\n$$\nPutting the pieces together leads to the decomposition \\eqref{MperpK}.\nSince the $J$-orthogonal summand  $\\sbm{ \\operatorname{Ker} \n({\\mathcal O}_{V,W}))^{*} \\\\ 0 }$ is clearly $J$-positive, it follows \nthat ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is $J$-positive if and only if \n$\\widehat {\\mathbb G}_{T^{[*]}} [+] ({\\mathcal M}_{\\mathfrak D}^{[ \\perp \n]})_{1}$ is $J$-positive.\nStatement (2) follows in a similar way.\n\\end{proof}\n\n\\begin{lemma}  \\label{L:posnegsubspaces}\n{\\rm (1)} The subspace ${\\mathbb G}_{T^{[*]}}$ is $J$-positive if and only if \n$I + T T^{[*]}$ is $J$-positive on the subspace\n$P_{H^{2}_{p+m}(\\Pi_{+})}{\\mathcal M}_{\\mathfrak D}^{[\\perp]} = \n\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y }})^{*}$.\n\n{\\rm (2)} The subspace $({\\mathcal M}_{\\mathfrak D}^{[ \\perp ]})_{1}$ is \n$J$-positive if and only if \nthe subspace \\\\ $P_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D}  =  \n\\operatorname{Ran} {\\mathcal O}_{\\sbm{ V \\\\ U}, W}$\n is $J$-negative.\n\n{\\rm (3)} The subspace $\\widehat {\\mathbb G}_{T}$ is  $J$-negative  if and \nonly if $I + T^{[*]} T$ is a $J$-negative operator on the subspace \n$P_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D} = \\operatorname{Ran}  \n{\\mathcal O}_{\\sbm{ V \\\\ U},W}$.\n\n\\smallskip\n\n{\\rm (4)} The subspace ${\\mathcal M}_{{\\mathfrak D}, 1}$ is  $J$-negative  if and only \nif the subspace \\\\\n$P_{H^{2}_{p+m}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}^{[ \\perp ]}  = \n\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}$\nis  $J$-positive.\n\\end{lemma}\n     \n\\begin{proof}\nTo prove (1), note that ${\\mathbb G}_{T^{[*]}}$ being a $J$-positive \nsubspace means that\n$$\n \\left\\langle \\begin{bmatrix} I \\\\ T^{[*]} \\end{bmatrix} x, \\, \n \\begin{bmatrix} I \\\\ T^{[*]} \\end{bmatrix} x\\right\\rangle_{J \\oplus J} =\n     \\langle (I + T T^{[*]}) x, x \\rangle_{J} \\ge 0\n$$\nfor all $x \\in \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}$, i.e., that\n$I + T T^{[*]}$ is a $J$-positive operator.\n\n\\smallskip\n\nTo prove (2), use \\eqref{cMDperp1} to see that elements ${\\mathbf g}$ of \n$({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1}$ have the form \n$$\n{\\mathbf g} =\\sbm{ {\\mathcal O}_{V,W} \\\\ \n{\\mathcal O}_{U,W} ({\\mathcal G}_{U,W})^{-1} {\\mathcal G}_{V,W} } x\\quad\\mbox{for some} \\quad x \\in {\\mathbb \nC}^{n_{W}}.\n$$  \nThe associated $J$-gramian is then given by\n\\begin{align*}\n &   \\begin{bmatrix} ({\\mathcal O}_{V,W})^{*} & {\\mathcal G}_{V,W} ({\\mathcal G}_{U,W})^{-1} \n\t({\\mathcal O}_{U,W})^{*} \\end{bmatrix} \\begin{bmatrix} I_{p} & 0 \\\\ 0 \n\t& -I_{m} \\end{bmatrix} \\begin{bmatrix} {\\mathcal O}_{V,W} \\\\ {\\mathcal O}_{U,W} \n\t({\\mathcal G}_{U,W})^{-1} {\\mathcal G}_{V,W} \\end{bmatrix} \\\\\n& = {\\mathcal G}_{V,W} - {\\mathcal G}_{V,W} ({\\mathcal G}_{U,W})^{-1} {\\mathcal G}_{V,W}.\n\\end{align*}\nBy a Schur-complement analysis, this defines a negative semidefinite operator \n(in fact by our Nondegeneracy Assumption, a negative definite operator) if and only if\n\\begin{align*}\n    \\begin{bmatrix} {\\mathcal G}_{V,W} & {\\mathcal G}_{V,W} \\\\ {\\mathcal G}_{V,W} & {\\mathcal G}_{U,W} \n    \\end{bmatrix}  = \\begin{bmatrix} {\\mathcal G}_{V,W}^{\\frac{1}{2}} & 0 \\\\ 0 \n    & I \\end{bmatrix}\\begin{bmatrix} I_{\\operatorname{Ran} {\\mathcal G}_{V,W}} \n    & {\\mathcal G}_{V,W}^{\\frac{1}{2}} \\\\ {\\mathcal G}_{V,W}^{\\frac{1}{2}} & {\\mathcal G}_{U,W} \n    \\end{bmatrix} \\begin{bmatrix} {\\mathcal G}_{V,W}^{\\frac{1}{2}} & 0 \\\\ 0 & \n    I \\end{bmatrix} \\prec 0,\n\\end{align*}\nwhich in turn happens if and only if\n$$ \n\\begin{bmatrix} I_{\\operatorname{Ran} {\\mathcal G}_{V,W}} \n    & {\\mathcal G}_{V,W}^{\\frac{1}{2}} \\\\ {\\mathcal G}_{V,W}^{\\frac{1}{2}} & {\\mathcal G}_{U,W} \n    \\end{bmatrix} \\prec 0.\n$$\nYet another Schur-complement analysis converts this to the condition \n$$\n{\\mathcal G}_{U,W} - {\\mathcal G}_{V,W} \\prec 0\n$$ \nwhich is equivalent to \n$\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W}$ being a $J$-negative subspace.\n\nThe proofs of statements (3) and (4) are parallel to those  of (1) \nand (2) respectively.\n\\end{proof} \n\n\\begin{lemma}  \\label{L:GammaD-factored} The Pick matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ \n    \\eqref{GammaData} can be factored as follows:\n\\begin{equation}  \\label{GammaData-factored}\t\n\\boldsymbol{\\Gamma}_{\\mathfrak D}\n = \\begin{bmatrix} -{\\mathcal C}_{Z, \\sbm{ X & -Y}} & 0 \\\\ 0 & \n({\\mathcal O}_{\\sbm{V \\\\ U},W})^{*} J \\end{bmatrix}\n\\begin{bmatrix} I & T \\\\ T^{[*]} & -I \\end{bmatrix}\n    \\begin{bmatrix} -J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} & 0 \\\\ 0 & \n\t{\\mathcal O}_{\\sbm{ V \\\\ U}, W} \\end{bmatrix}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}  Multiplying out the expression on the right-hand \n    side in \\eqref{GammaData-factored}, we get\n$$\n\\begin{bmatrix}  {\\mathcal G}^{J}_{Z, \\sbm{ X & -Y}} & - {\\mathcal C}_{Z, \\sbm{ X & -Y}} \n    T {\\mathcal O}_{\\sbm{V \\\\ U}, W} \\\\\n-( {\\mathcal O}_{\\sbm{V \\\\ U}, W})^{*} J T^{[*]} J ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*} \n & - {\\mathcal G}^{J}_{\\sbm{V \\\\ U}, W} \\end{bmatrix},\n$$\nwhich is exactly $\\sbm{\\Gamma_{L} & \\Gamma \\\\ \\Gamma^{*} & \\Gamma_{R}}=:\n\\boldsymbol{\\Gamma}_{\\mathfrak D}$\nas we can see from the identities \\eqref{GammaLRid} and \\eqref{TGamma}.\n\\end{proof}\n\n\\begin{lemma} \\label{L:GammaDpos}\n The following conditions are equivalent:\n    \n\\begin{enumerate}\n    \\item The matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ \\eqref{GammaData} is \n    positive.\n    \n     \\item The subspace $P_{H^{2}_{p+m}(\\Pi_-)} {\\mathcal M}_{\\mathfrak D} = \n    \\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W}$ is $J$-negative and the \n    subspace ${\\mathbb G}_{T^{[*]}}$ is $J$-positive.\n    \n    \\item The subspace $P_{H^{2}_{p+m}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}^{[ \\perp]} \n    = \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}$ is \n    $J$-positive and the subspace $\\widehat {\\mathbb G}_{T}$ is \n    $J$-negative.\n    \n   \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n    From the factorization \\eqref{GammaData-factored} we see that \n    $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$ if and only if the Hermitian form \n    on the subspace $\\sbm{\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & \n    -Y}})^{*} \\\\ \\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W} }$\n    induced by the operator $\\sbm{I & T \\\\ T^{[*]} & - I}$ in the $J \n    \\oplus J$-inner product is positive. On the one hand we may consider the factorization\n$$\n \\begin{bmatrix} I & T \\\\ T^{[*]} & -I \\end{bmatrix} =\n     \\begin{bmatrix} I & 0 \\\\ T^{[*]} & I \\end{bmatrix} \n\t \\begin{bmatrix} I & 0 \\\\ 0 & -I - T^{[*]} T \\end{bmatrix}\n\\begin{bmatrix} I & T \\\\ 0 & I \\end{bmatrix}\n$$\nto deduce that $\\sbm{ I & T \\\\ T^{[*]} & -I  }$ is $(J \\oplus \nJ)$-positive if and only if \n\\begin{enumerate}\n\\item[(i)] the identity operator $I$ is $J$-positive on \n$\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}$ (i.e., the subspace\n$\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{X & -Y}})^{*}$ is $J$-positive), and \n\\item[(ii)]  $-I - T^{[*]}T$ is a $J$-positive operator on \n$\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W}$, i.e., $\\widehat{\\mathbb \nG}_{T}$ is a $J$-negative subspace.  \n\\end{enumerate}\nNote that this analysis \namounts to taking the $J$-symmetrized Schur complement of the matrix \n$\\sbm{ I & T \\\\ T^{[*]} & -I}$ with respect to the (1,1)-entry. This \nestablishes the equivalence of (1) and (3).\n\n\\smallskip\n\nOn the other hand we may take the $J$-symmetrized Schur complement of $\\sbm{ I & T \\\\ T^{[*]} & -I}$\nwith respect to the (2,2)-entry, corresponding to the factorization\n$$\n \\begin{bmatrix} I & T \\\\ T^{[*]} & -I \\end{bmatrix} = \n     \\begin{bmatrix} I & -T \\\\ 0 & I \\end{bmatrix}\n\\begin{bmatrix} I + T T^{[*]} & 0 \\\\ 0 & -I \\end{bmatrix}\n    \\begin{bmatrix} I & 0 \\\\ -T^{[*]} & I \\end{bmatrix}.\n$$\nIn this way we see that $(J \\oplus J)$-positivity of $\\sbm{ I & T \\\\ T^{[*]} & -I}$\ncorresponds to \n\\begin{enumerate}\n\\item[(i$^{\\prime}$)] $I + T T^{[*]}$ is a $J$-positive operator (i.e., \nthe subspace ${\\mathbb G}_{T^{[*]}}$ is $J$-positive), and \n\\item[(ii$^{\\prime}$)] minus the identity operator $-I$ is $J$ positive on $\\operatorname{Ran}  \n{\\mathcal O}_{\\sbm{ V \\\\ U}, W}$ (i.e., the subspace is $\\operatorname{Ran}  \n{\\mathcal O}_{\\sbm{ V \\\\ U}, W}$ is $J$-negative).  \n\\end{enumerate}\nThis establishes the  equivalence of (1) and (2).\n\\end{proof}\n\nTo conclude the proof of Corollary \\ref{C:BTOA-NP} for the general \nBiTangential case (at least with the Nondegeneracy Assumption in \nplace), it remains only to assemble the various pieces. By Lemma \n\\ref{L:MperpK} part (1), we see that ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ \nbeing $J$-positive is equivalent to \n\\begin{equation}  \\label{condition1}\n {\\mathbb G}_{T^{[*]}} \\text{ and } ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1} \n\\text{ are $J$-positive subspaces.}\n\\end{equation}\nBy Lemma \\ref{L:posnegsubspaces}, we see that $ ({\\mathcal M}_{\\mathfrak \nD}^{[\\perp]})_{1} $ being $J$-positive is equivalent to $\\operatorname{Ran} {\\mathcal O}_{\\sbm{ V \\\\ U}, W}$ \nbeing $J$-negative.  We therefore may amend \\eqref{condition1} to\n\\begin{equation}   \\label{condition2}\n{\\mathbb G}_{T^{[*]}} \\text{ is $J$-positive  and }  \nP_{H^{2 \\perp}_{p+m}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}  = \\operatorname{Ran} {\\mathcal O}_{\\sbm{ V \\\\ U}, W}\n\\text{ is $J$-negative}\n\\end{equation}\nwhich is exactly statement (2) in Lemma \\ref{L:GammaDpos}.  Thus (1) \n$\\Leftrightarrow$ (2) in Corollary \\ref{C:BTOA-NP} follows from (1) \n$\\Leftrightarrow$ (2) in Lemma \\ref{L:GammaDpos}.\n\n\\smallskip\n\nFor the general BTOA case, the reproducing kernel space \n${\\mathcal H}(K_{\\Theta,J})$ again can be identified with a range space, namely\n\\begin{equation}   \\label{HKThetaJid}\n {\\mathcal H}(K_{\\Theta,J}) = \\operatorname{Ran} (P^{J}_{H^{2}_{p+m}(\\Pi_{+})} - \n P^{J}_{{\\mathcal M}_{\\mathfrak D}})\n\\end{equation}\nwith lifted indefinite inner product, \nwhere $ P^{J}_{H^{2}_{p+m}(\\Pi_{+})}$ and $P^{J}_{{\\mathcal M}_{\\mathfrak D}}$ are the \n$J$-orthogonal projections of $L^{2}_{p+m}(i{\\mathbb R})$ onto $H^{2}_{p+m}(\\Pi_{+})$ and ${\\mathcal M}_{\\mathfrak D}$\nrespectively (see \\cite[Theorem 3.3]{BHMesa}). Due to $J$-orthogonal decompositions\n\\begin{align*}\n &   H^{2}_{p+m}(\\Pi_{+}) \n= \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} \\, [+] \n {\\mathcal M}_{{\\mathfrak D}, 1} [+] \\, {\\mathcal M}_{{\\mathfrak D}, 2}, \\\\\n& {\\mathcal M} = \\widehat{\\mathbb G}_{T} \\, [+]  {\\mathcal M}_{{\\mathfrak D}, 1} [+] \\, \n{\\mathcal M}_{{\\mathfrak D}, 2},\n \\end{align*}\n we can simplify the difference of $J$-orthogonal projections to\n$$\n P^{J}_{H^{2}_{p+m}} -  P^{J}_{{\\mathcal M}_{\\mathfrak D}}\n = P^{J}_{\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X &-Y}})^{*}} - \n P^{J}_{\\widehat {\\mathbb G}_{T}}.\n $$\n By a calculation as in the proof for Case 1, one can show that \n \\begin{equation}   \\label{HKThetaJ}\n {\\mathcal H}(K_{\\Theta,J}) = ( \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{X & \n -Y}})^{*})_{J} \\, [+] (\\widehat {\\mathbb G}_{T})_{-J}\n \\end{equation}\n with the identity map a Kre\\u{\\i}n-space isomorphism, \n where the subscripts on the right hand side indicating that one should use the $J$-inner \n product for the first component but the $-J$-inner product for the \n second component. We conclude that ${\\mathcal H}(K_{\\Theta, J})$ is a HIlbert \n space exactly when condition (3) in Lemma \\ref{L:GammaDpos} holds. We now see that \n (1) $\\Leftrightarrow$ (3) in Corollary \\ref{C:BTOA-NP} is an immediate consequence of (1) \n $\\Leftrightarrow$ (3) in Lemma \\ref{L:GammaDpos}.\n \\end{proof}\n \nThe above analysis actually establishes a bit more which we collect \nin the following Corollary.\n\n\\begin{corollary}  \\label{C:BTOA-NP'}  The following conditions are \n    equivalent:\n\\begin{enumerate}\n    \\item The subspace ${\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]}$ is $J$-positive.\n    \n    \\item The subspace $({\\mathcal M}_{\\mathfrak D}^{[ \\perp ]})^{[\\perp {\\mathcal K}']}$ is \n    $J$-negative.\n \\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n    We have seen in Lemma \\ref{L:MperpK} part (2) that \n    $({\\mathcal M}_{\\mathfrak D}^{[\\perp]})^{[ \\perp {\\mathcal K}']}$ being $J$-negative is \n    equivalent to\n\\begin{equation}   \\label{condition1'}\n   \\widehat{\\mathfrak G}_{T} \\text{ and } ({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1} \\text{ \n   are $J$-negative subspaces.}\n\\end{equation}\nLemma \\ref{L:posnegsubspaces} (4) tells us that $ {\\mathcal M}_{{\\mathfrak D},1}$ \nbeing $J$-negative is equivalent to \\\\\n$\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*}$  \nbeing $J$-positive.  Thus condition \\eqref{condition1'} can be amended to\n\\begin{equation}   \\label{condition2'}\n \\widehat{\\mathbb G}_{T} \\text{ is negative and }\n P_{H^2_{p+m}(\\Pi_+)} {\\mathcal M}_{\\mathfrak D}^{[ \\perp]} = \n\\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*} \\text{ is $J$-positive.}\n\\end{equation}\nWe next use the equivalence of (1) $\\Leftrightarrow$ (3) in Theorem \n\\ref{L:GammaDpos} to see that condition \\eqref{condition2'} is also \nequivalent to $\\boldsymbol{\\Gamma}_{\\mathfrak D} \\succ 0$.  We then use the \nequivalence (1) $\\Leftrightarrow$ (2) in Theorem \\ref{L:GammaDpos} \nto see that this last condition in turn is equivalent \nto ${\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]}$ being $J$-positive.\n\\end{proof}\n\n\n\n\n \n\\section{Interpolation problems in the generalized Schur class}  \\label{S:negsquares}\nMuch of the previous analysis extends from the Schur class  ${\\mathcal S}^{p \\times m}(\\Pi_{+})$ \nto a larger class  ${\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ (generalized Schur class)\nconsisting of $\\mathbb C^{p\\times m}$-valued\nfunctions that are meromorphic on $\\Pi_+$ with total pole multiplicity equal $\\kappa$ \nand such that their $L^\\infty$ norm (that is, $\\sup_{y\\in\\mathbb R}\\|S(iy)\\|$) does \nnot exceed one. The values \n$S(iy)$ are understood in the sense of non-tangential boundary limits that exist for \nalmost all \n$y\\in\\mathbb R$. The multiplicity of a pole $z_0$ for a matrix-valued function $S$ is \ndefined as the sum of absolute values of all negative partial multiplicities \nappearing in the Smith form of $S$\nat $z_0$ (see e.g.\\ \\cite[Theorem 3.1.1]{BGR}). Then the total pole multiplicity of $S$ \nis defined as the sum of multiplicities of all poles.  Let us \nintroduce the notation \n$$\nm_{P}(S) = \\text{ sum of all pole multiplicities of $S$ over all poles in } \\Pi_{+}.\n$$\nIt follows by the maximum modulus principle that  ${\\mathcal S}^{p \\times m}_0(\\Pi_{+})$ \nis just the classical Schur class. Generalized Schur functions appeared first \nin \\cite{T} in the interpolation context and \nwere comprehensively studied by Kre\\u{\\i}n and Langer in \\cite{kl,kl1}. Later\nwork on the classes ${\\mathcal S}_{\\kappa}$ include \\cite{dls1}, \\cite{J},\n\\cite{CG1}, and \\cite{ADRS}, as well as \\cite{Gol}, \\cite{nud}, \\cite{BH}, \n\\cite{BH86} and the book \\cite{BGR} in the context of interpolation.\n\n\\smallskip\n\nThe class ${\\mathcal S}_{\\kappa}(\\Pi_{+})$ can alternatively be characterized \nby any of the following conditions:\n\\begin{enumerate}\n    \\item  ${\\rm sq}_{-}(K_{S}) = \\kappa$ where the kernel $K_{S}$ is \n    given by \\eqref{dBRkerPi+}.\n    \n    \\item ${\\rm sq}_{-}({\\mathbf K}_{S}) = \\kappa$, where ${\\mathbf \n    K}_{S}$ is the $2 \\times 2$-block matrix kernel \\eqref{ker22}.\n    \n    \\item $S$ admits left and right \n(coprime) {\\em Kre\\u{\\i}n-Langer factorizations}\n$$\nF(\\lambda)=S_R(\\lambda)\\vartheta_R(\\lambda)^{-1}= \\vartheta_L(\\lambda)^{-1}S_L(\\lambda),\n$$\nwhere $S_L, \\, S_R\\in{\\mathcal S}^{p\\times m}(\\Pi_+)$ and $\\vartheta_L$ and $\\vartheta_R$ are matrix-valued finite\nBlaschke products of degree $\\kappa$ (see \\cite{kl1} for the scalar-valued case and \n\\cite{dls1} for the Hilbert-space operator-valued case).\nBy a $\\mathbb C^{n\\times n}$-valued finite Blaschke product we mean the product of $\\kappa$ Blaschke \n(or Blaschke-Potapov) factors \n$$\nI_n-P+\\frac{\\lambda-\\alpha}{\\lambda+\\overline{\\alpha}}P\n$$\nwhere $\\alpha\\in\\Pi_+$ and $P$ is an orthogonal projection in $\\mathbb C^n$. \n\\end{enumerate}\nThere is also an intrinsic characterization of matrix triples \n$(C,A,B)$ which can arise as the pole triple over the unit disk for a \ngeneralized Schur class function---see \\cite{bolleiba} for details.\n\n\\smallskip\n\nLet us take another look at the  BiTangential Nevanlinna-Pick problem \n\\eqref{inter1'}--\\eqref{inter3'}.\nIf the Pick matrix \\eqref{Pick-simple} is not positive semidefinite, \nthe problem has no solutions in the Schur class ${\\mathcal S}^{p\\times n}(\\Pi_+)$, \nby Theorem \\ref{T:BT-NP}. However, there always exist generalized Schur \nfunctions that are analytic at all interpolation nodes $z_i,w_j$ and satisfy \ninterpolation conditions \\eqref{inter1'}--\\eqref{inter3'}. One can show that \nthere exist such functions with only one pole of a sufficiently high multiplicity \nat any preassigned point in $\\Pi_+$. The question of interest is\nto find the smallest integer $\\kappa$, for which interpolation conditions  \n\\eqref{inter1'}--\\eqref{inter3'} \nare met for some function $S\\in {\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ and then \nto describe the set of all such functions. \n\n\\smallskip\n\nThe same question makes sense in the more general setting of the {\\bf BTOA-NP} \ninterpolation problem:\n{\\em given a $\\Pi_{+}$-admissible BTOA interpolation data set \\eqref{data},\nfind the  smallest integer $\\kappa$, for which interpolation conditions  \n\\eqref{BTOAint1}--\\eqref{BTOAint3} are satisfied for some function \n$S\\in {\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ which is analytic on \n$\\sigma(Z)\\cup\\sigma(W)$, and  describe the set of all such functions}. \n\n\\smallskip\n\nThe next theorem gives the answer to the question above in the so-called nondegenerate case.\n\n\\begin{theorem}  \\label{T:BTOA-NPkap}  Suppose that ${\\mathfrak D}  = ( X,Y,Z; U,V, W; \\Gamma)$\n is a $\\Pi_{+}$-admissible BTOA interpolation data set and let us \n assume that the BTOA-Pick matrix \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}$ defined by \\eqref{GammaData} is invertible. \nLet $\\kappa$ be the  smallest \ninteger for which there is a function $S\\in {\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ which is analytic on \n$\\sigma(W) \\cup \\sigma(Z)$ and satisfies the interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3}.\nThen $\\kappa$ is given by any one of the following three equivalent \nformulas:\n\\begin{enumerate}\n    \\item $\\kappa=\\nu_-(\\boldsymbol{\\Gamma}_{\\mathfrak D})$, the number of negative \n    eigenvalues of $\\boldsymbol{\\Gamma}_{\\mathfrak D}$.\n    \\item $\\kappa = \\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[\\perp] {\\mathcal K}})$, the negative signature \n   of the Kre\\u{\\i}n-space ${\\mathcal M}_{\\mathfrak D}^{[\\perp] {\\mathcal K}}$ \n    in the $J$-inner product.\n    \n    \\item $\\kappa =  \\nu_{-}({\\mathcal H}(K_{\\Theta, J}))$,  the negative signature  of the \nreproducing kernel Pontryagin space ${\\mathcal H}(K_{\\Theta,J})$, where $\\Theta$ is defined \nas in \\eqref{sep1} and $K_{\\Theta,J}$ as in \\eqref{ad1}.\n\\end{enumerate}\nFurthermore, the function $S$ belongs to the generalized Schur class \n${\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ and satisfies the interpolation conditions \n\\eqref{BTOAint1}--\\eqref{BTOAint3}\nif and only if it is of the form\n\\begin{equation}   \\label{LFTparamkap}\nS(\\lambda) = (\\Theta_{11}(\\lambda) G(\\lambda) + \\Theta_{12}(\\lambda))\n( \\Theta_{21}(\\lambda) G(\\lambda) + \\Theta_{22}(\\lambda))^{-1}\n\\end{equation}\nfor a Schur class function  $G\\in {\\mathcal S}^{p \\times m}(\\Pi_{+})$ such that\n\\begin{equation}   \\label{LFTparampar}\n\\det (\\psi(\\lambda)(\\Theta_{21}(\\lambda) G(\\lambda) + \\Theta_{22}(\\lambda)))\\neq 0,\\quad \\lambda\\in \\Pi_{+}\n\\backslash(\\sigma(Z)\\cup\\sigma(W))\n\\end{equation}\nwhere $\\psi$ is the $m \\times m$-matrix function defined in \n\\eqref{sep1}.\n\\end{theorem}\n\n\\subsection{The state-space approach}  \\label{S:statekap}\nThe direct proof of the necessity of condition (1) in Theorem \n\\ref{T:BTOA-NPkap} for the existence of class-${\\mathcal S}_{\\kappa}^{p \\times \nm}(\\Pi_{+})$ solution of the interpolation  conditions \n\\eqref{BTOAint1}--\\eqref{BTOAint3} relies on the\ncharacterization of the class ${\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$ in \nterms of the kernel \\eqref{ker22} mentioned above: {\\em a \n$\\mathbb C^{p\\times n}$-valued function meromorphic on $\\Pi_+$ belongs to ${\\mathcal S}_\\kappa^{p \\times m}(\\Pi_{+})$\nif and only if the kernel ${\\mathbf K}_{S}(\\lambda, \\lambda_{*}; \\zeta, \\zeta_{*})$ defined  as in \\eqref{ker22}\nhas $\\kappa$ negative squares on $\\Omega_S^4$}:\n\\begin{equation}\n{\\rm sq}_-{\\mathbf K}_{S}=\\kappa,\n\\label{aug3}\n\\end{equation}\nwhere $\\Omega_S\\subset\\Pi_+$ is the domain of analyticity of $S$. The latter equality means that the \nblock matrix $\\left[{\\mathbf K}_{S}(z_i, z_i; z_j, z_j)\\right]_{i,j=1}^N$ has at most $\\kappa$ \nnegative eigenvalues for any \nchoice of finitely many points $z_1.\\ldots,z_N\\in\\Omega_S$, and it has exactly $\\kappa$ negative eigenvalues \nfor at least one such choice.\n\n\\smallskip\nNow suppose that $S \\in {\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$ satisfies \nthe interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3}.\nThe kernel ${\\mathbf K}_{S}$ satisfying condition \\eqref{aug3} still admits the Kolmogorov decomposition\n\\eqref{KS2}, but this time the state space ${\\mathcal X}$ is a Pontryagin space of negative index $\\kappa$.\nAll computations following formula \\eqref{KS2} go through with $\\Pi_+$ replaced by $\\Omega_S$ showing \nthat the matrix  $\\boldsymbol{\\Gamma}'_{\\mathfrak D}$ defined in \\eqref{aug4} is equal to the Pick matrix \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}$ given in \\eqref{GammaData}. Note that the operations bringing the \nkernel ${\\mathbf K}_{S}$ to the matrix $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ amount \nto a sophisticated conjugation of the kernel ${\\mathbf K}_{S}$.  We \nconclude that  $\\nu_-(\\boldsymbol{\\Gamma}_{\\mathfrak D})=\\nu_-(\\boldsymbol{\\Gamma}'_{\\mathfrak D}) \\le \\kappa$.\nOnce one of the sufficiency arguments has been carried out (by whatever \nmethod) to show that $\\nu_-(\\boldsymbol{\\Gamma}_{\\mathfrak \nD})=\\nu_-(\\boldsymbol{\\Gamma}'_{\\mathfrak D}) < \\kappa$ implies that there is a \na function $S$ in a generalized Schur class ${\\mathcal S}^{p \\times \nm}_{\\kappa'}(\\Pi_{+})$ with $\\kappa' < \\kappa$ satisfying the \ninterpolation conditions, then $\\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D}) < \n\\kappa$ leads to a contradiction to \nthe minimality property of $\\kappa$.   We conclude \nthat $\\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D}) = \\kappa$ is necessary for \n$\\kappa$ to be the smallest integer so that there is a solution $S$ \nof class ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$ of the interpolation \nconditions \\eqref{BTOAint1}--\\eqref{BTOAint3}. \n\n\\smallskip\n\nWe now suppose that $\\nu_-(\\boldsymbol{\\Gamma}_{\\mathfrak D})=\\kappa$. The identity \\eqref{ad1} relies on \nequality \\eqref{bigLySyl} and on the assumption that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ \nis invertible. In particular, the matrix $\\Theta(\\lambda)$ still is $J$-unitary for each \n$\\lambda \\in i \\mathbb R$, i.e., equalities \\eqref{ThetaJunitary} hold for all $\\lambda\\in i\\mathbb R$.\nBy using the \ncontrollability\/observability assumptions on $(Z,X)$ and $(U,W)$, it \nfollows from the formula on the right hand side of \\eqref{ad1} that the kernel \n$K_{\\Theta,J}$ \\eqref{ad1} has $\\kappa$ negative squares on\n$\\Omega_\\Theta = \\Pi_{+} \\setminus \\sigma(W)$ (the points of \nanalyticity for $\\Theta$ in the right half plane $\\Pi_{+}$):\n$$\n {\\rm sq}_{-} K_{\\Theta,J} = \\kappa.\n$$\n\n\n\\smallskip\n\nWe shall have need of the {\\em Potapov-Ginsburg transform} $U = \\sbm{ \nU_{11} & U_{12} \\\\ U_{21} & U_{22}}$ of \na given block $2 \\times 2$-block matrix function $\\Theta = \\sbm{ \n\\Theta_{11} & \\Theta_{21} \\\\ \\Theta_{21} & \\theta_{22}} $(called  \nthe {\\em Redheffer transform} in \\cite{BGR}) defined by\n$$\n  U = \\begin{bmatrix} U_{11} & U_{12} \\\\ U_{21} & \n  U_{22} \\end{bmatrix} : = \\begin{bmatrix} \\Theta_{12} \n  \\Theta_{22}^{-1} & \\Theta_{11} - \\Theta_{12} \\Theta_{22}^{-1} \n  \\Theta_{21} \\\\ \\Theta_{22}^{-1} & - \\Theta_{22}^{-1} \\Theta_{21} \n  \\end{bmatrix}.\n$$\n This transform is the result of rearranging the inputs and outputs in the system of \n equations\n\\begin{equation}  \\label{chain}\n\\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\  \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \n\\begin{bmatrix} x_{2} \\\\ y_{2} \\end{bmatrix} = \\begin{bmatrix} y_{1} \\\\ x_{1} \n  \\end{bmatrix}\n\\end{equation}\nto have the form\n\\begin{equation}  \\label{scattering}\n  \\begin{bmatrix} U_{11} & U_{12} \\\\ U_{21} & U_{22} \\end{bmatrix} \n      \\begin{bmatrix} x_{1} \\\\ x_{2} \\end{bmatrix} = \\begin{bmatrix} \n\t  y_{1} \\\\ y_{2} \\end{bmatrix},\n\\end{equation}\nand in circuit theory has the interpretation as the change of \nvariable from the {\\em chain formalism} \\eqref{chain} to the {\\em scattering \nformalism} \\eqref{scattering}.  Based on this connection it is not hard to show that\n$$\n {\\rm sq}_{-} K_{U} = {\\rm sq}_{-} K_{\\Theta,J} = \\kappa\n$$\nwhere the notation $K_{U}$  is as in \\eqref{dBRkerPi+} and \n$K_{\\Theta,J}$ as in \\eqref{ad1}\n(see \\cite[Theorem 13.1.3]{BGR}).\nWe conclude that $U$ is in the generalized Schur class ${\\mathcal S}_{\\kappa}^{(p+m) \\times (m+p)}(\\Pi_{+})$.  \nBy the Kre\\u{\\i}n-Langer factorization result for the generalized Schur \nclass (see \\cite{kl1}), it follows that $\\kappa$ is also equal to the \ntotal pole multiplicity of $U$ over points in $\\Pi_{+}$:\n$$\nm_{P}(U) = \\kappa.\n$$\n\\smallskip\n\nWe would like to show next that\n\\begin{equation}   \\label{toshow1}\n    m_{P}(U_{22}) = m_{P}(\\Theta_{22}^{-1} \\Theta_{21}) = \\kappa.\n\\end{equation}\nVerification of this formula will take several steps and \nfollow the analysis in \\cite[Chapter 13]{BGR}.\nWe first note that the calculations \\eqref{cMrep}--\\eqref{cMBLrep} \ngo through unchanged so we still have the Beurling-Lax representation\n\\begin{equation}  \\label{BLrep}\n  {\\mathcal M}_{\\mathfrak D} = \\Theta \\cdot H^{2}_{p+m}(\\Pi_{+})\n\\end{equation}\nwhere ${\\mathcal M}_{\\mathfrak D}$ also has the representation \\eqref{cMrep}.\nThe observability assumption on the output pair $(U,W)$ translates to \nan additional structural property on ${\\mathcal M}_{\\mathfrak D}$:\n\\begin{itemize}\n    \\item $(U,W)$ {\\em observable implies}\n\\begin{equation}  \\label{obs-imp1}\n    {\\mathcal M}_{\\mathfrak D} \\cap \\begin{bmatrix} L^{2}(i {\\mathbb R}) \\\\ 0 \n    \\end{bmatrix} = {\\mathcal M}_{\\mathfrak D} \\cap \\begin{bmatrix} \n    H^{2}_{p}(\\Pi_{+}) \\\\ 0 \\end{bmatrix}.\n\\end{equation}\n\\end{itemize}\nMaking use of \\eqref{BLrep}, condition \\eqref{obs-imp1} translates to \nan explicit property of $\\Theta$, namely:\n$$\nf \\in H^{2}_{p}(\\Pi_{+}),\\, g \\in H^{2}_{m}(\\Pi_{+}),\\, \\Theta_{21}f \n+ \\Theta_{22}g = 0 \\Rightarrow \\Theta_{11} f + \\Theta_{12}g \\in \nH^{2}_{p}(\\Pi_{+}).\n$$\nSolving the first equation for $g$ gives $g = -\\Theta_{22}^{-1} \n\\Theta_{21} f$ and this last condition can be rewritten as\n$$\nf \\in H^{2}_{p}(\\Pi_{+}), \\, \\Theta_{22}^{-1} \n\\Theta_{21} f \\in H^{2}_{m}(\\Pi_{+}) \\Rightarrow\n(\\Theta_{11} - \\Theta_{12} \\Theta_{22}^{-1} \\Theta_{21}) f \\in \nH^{2}_{p}(\\Pi_{+}),\n$$\nor, more succinctly,\n$$\n f \\in H^{2}_{p}(\\Pi_{+}),\\, U_{22} f \\in H^{2}_{m}(\\Pi_{+}) \n \\Rightarrow U_{12} f \\in H^{2}_{m}(\\Pi_{+}).\n$$\nThis last condition translates to\n\\begin{equation}  \\label{obs-imp2}\n    m_{P}(U_{22}) = m_{P}\\left( \\sbm{ U_{12} \\\\ U_{22} } \\right).\n\\end{equation}\n\nSimilarly, the controllability assumption on the input pair $(Z,X)$ \ntranslates to an additional structural property on ${\\mathcal M}_{\\mathfrak \nD}$, namely:\n\\begin{itemize}\n    \\item $(Z,X)$ {\\em controllable implies}\n\\begin{equation}   \\label{control-imp1}\n    P_{\\sbm{ 0 \\\\ H^{2}_{m}(\\Pi_{+})}} \\left( {\\mathcal M}_{\\mathfrak D} \\cap \n    H^{2}_{p+m}(\\Pi_{+}) \\right) = \\begin{bmatrix} 0 \\\\ H^{2}_{m}(\\Pi_{+}) \n    \\end{bmatrix}.\n\\end{equation}\n\\end{itemize}\nIn terms of $\\Theta$, from the representation \\eqref{BLrep} we see \nthat this means that, given any $h \\in H^{2}_{m}(\\Pi_{+})$, we can \nfind $f \\in H^{2}_{p}(\\Pi_{+})$ and $g \\in H^{2}_{m}(\\Pi_{+})$ so that\n$$\n\\Theta_{11} f + \\Theta_{12} g \\in H^{2}_{p}(\\Pi_{+}), \\quad\n\\Theta_{21} f + \\Theta_{22} g = h.\n$$\nWe can solve the second equation for $g$\n$$\n  g = \\Theta_{22}^{-1} h - \\Theta_{22}^{-1} \\Theta_{21} f \\in \n  H^{2}_{m}(\\Pi_{+})\n$$\nand rewrite the first expression in terms of  $f$ and $h$:\n$$\n (\\Theta_{11} - \\Theta_{12} \\Theta_{22}^{-1} \\Theta_{21}) f + \n \\Theta_{12} \\Theta_{22}^{-1} h \\in H^{2}_{p}(\\Pi_{+}).\n$$\nPutting the pieces together, we see that an equivalent form of \ncondition \\eqref{control-imp1} is: {\\em for any $ h \\in H^{2}_{m}(\\Pi_{+})$, there exists \nan $f \\in H^{2}_{p}(\\Pi_{+})$ such that\n$$\n\\Theta_{22}^{-1} h - \\Theta_{22}^{-1} \\Theta_{21} f \\in \nH^{2}_{m}(\\Pi_{+}), \\quad  (\\Theta_{11} -  \\Theta_{12} \\Theta_{22}^{-1} \n\\Theta_{21}) f + \\Theta_{12} \\Theta_{22}^{-1} h \\in \nH^{2}_{p}(\\Pi_{+}).\n$$}\nMore succinctly,\n\\begin{align*}\n& h \\in H^{2}_{m}(\\Pi_{+}) \\Rightarrow \\exists f \\in \n H^{2}_{p}(\\Pi_{+}) \\text{ so that } \\\\\n& U_{21} h + U_{22} f \\in H^{2}_{m}(\\Pi_{+}), \\quad\nU_{12} f + U_{11} h \\in H^{2}_{p}(\\Pi_{+}),\n\\end{align*}\nor, in column form, for each $h \\in H^{2}_{m}(\\Pi_{+})$ there exists \n$f \\in H^{2}_{p}(\\Pi_{+})$ so that\n$$\n \\begin{bmatrix} U_{11} \\\\ U_{21} \\end{bmatrix} h + \\begin{bmatrix} \n     U_{12} \\\\ U_{22} \\end{bmatrix} f \\in H^{2}_{p+m}(\\Pi_{+}).\n$$\nThe meaning of this last condition is:\n\\begin{equation}   \\label{control-imp2}\n    m_{P}(U) = m_{P}\\left(\\sbm{ U_{12} \\\\ U_{22}}  \\right).\n    \\end{equation}\nCombining \\eqref{obs-imp2} with \\eqref{control-imp2} gives us \n\\eqref{toshow1} as wanted.\n\n\\smallskip\nSince $\\Theta$ is not $J$-contractive in $\\Pi_+$ anymore,\nwe cannot conclude that $\\Theta_{22}^{-1}\\Theta_{21}$ is contraction valued. However, \ndue to equalities \\eqref{ThetaJunitary}, the function $\\Theta_{22}(\\lambda)^{-1}\\Theta_{21}(\\lambda)$\nis a contraction for each $\\lambda\\in i\\mathbb R$.\nTherefore, $\\Theta_{22}^{-1}\\Theta_{21}$ belongs to the generalized Schur class \n$\\mathcal S^{m\\times p}_\\kappa(\\Pi_+)$. We next wish to argue that\n\\begin{equation}  \\label{toshow2}\n{\\rm wno} \\det \\Theta_{22}+{\\rm wno} \\det \\psi=\\kappa,\n\\end{equation}\nwhere $\\psi$ is given by \\eqref{sep1}.\nFrom the representation \\eqref{BLrep} and the form of ${\\mathcal M}_{\\mathfrak \nD}$ in \\eqref{BLrep} we see that\n$$\n\\Theta_{22} \\begin{bmatrix} \\Theta_{22}^{-1} \\Theta_{21} & I_{m}\n\\end{bmatrix} H^{2}_{p+m}(\\Pi_{+}) = \\begin{bmatrix} \\Theta_{21} & \n\\Theta_{22} \\end{bmatrix} H^{2}_{p+m}(\\Pi_{+}) = \\psi^{-1} \nH^{2}_{m}(\\Pi_{+}).\n$$\nWe rewrite this equality as\n\\begin{equation}   \\label{subspace-eq}\n  \\begin{bmatrix} \\Theta_{22}^{-1} \\Theta_{21} & I_{m} \\end{bmatrix} \n      H^{2}_{p+m}(\\Pi_{+}) = \\Theta_{22}^{-1} \\psi^{-1} \n      H^{2}_{m}(\\Pi_{+}).\n\\end{equation}\nIn particular,\n$$\n  \\Theta_{22}^{-1} \\psi^{-1} H^{2}_{m}(\\Pi_{+}) \\supset \n  H^{2}_{m}(\\Pi_{+})\n$$\nso the matrix function $\\Theta_{22}^{-1} \\psi^{-1}$ has no zeros (in \nthe sense of its Smith-McMillan form) in $\\Pi_{+}$.  As \n$\\Theta_{22}^{-1}$ and $\\psi^{-1}$ are invertible on the boundary $i \n{\\mathbb R}$, we see that ${\\rm wno} \\det(\\Theta_{22}^{-1} \n\\psi^{-1})$ is well-defined and by the Argument Principle we have\n\\begin{align}\n& -{\\rm wno} \\det \\Theta_{22} - {\\rm wno} \\det(\\psi) \n= {\\rm wno} \\det(\\Theta_{22}^{-1} \\psi^{-1}) \\notag \\\\\n& \\quad \\quad \\quad \\quad = \n m_{Z}(\\det(\\Theta_{22}^{-1} \\psi^{-1})) - m_{P}(\\det(\\Theta_{22}^{-1} \n \\psi^{-1})) \\notag \\\\\n & \\quad \\quad \\quad \\quad = -m_{P}(\\det(\\Theta_{22}^{-1} \\psi^{-1})) = \n -m_{P}(\\Theta_{22}^{-1} \\psi^{-1}) \\notag \\\\\n &\\quad \\quad \\quad \\quad  = -\\dim P_{H^{2}_{m}(\\Pi_{-})} \\Theta_{22}^{-1} \\psi^{-1} \n H^{2}_{m}(\\Pi_{+})\n \\label{know1}\n\\end{align}\nwhere $m_{Z}(S)$ is the total zero multiplicity of the rational matrix \nfunction $S$ over all zeros in $\\Pi_{+}$.\nOn the other hand we have\n\\begin{equation}  \\label{know2}\n    \\dim P_{H^{2}_{m}(\\Pi_{-})} \\begin{bmatrix} \\Theta_{22}^{-1} \\Theta_{21} & I_{m} \\end{bmatrix} \n      H^{2}_{p+m}(\\Pi_{+}) = m_{P}(\\Theta_{22}^{-1} \\Theta_{21}) = \n      \\kappa\n\\end{equation}\nwhere we make use of \\eqref{toshow1} for the last step.  Combining \n\\eqref{know1} and \\eqref{know2} with \\eqref{subspace-eq} finally \nbrings us to \\eqref{toshow2}.\n\n\\smallskip\n\nIn addition to the Beurling-Lax representation \\eqref{cMBLrep} or \n\\eqref{BLrep}, we also still have the Beurling-=Lax representation \n\\eqref{cM-rep} for ${\\mathcal M}_{{\\mathfrak D},-}$ with $\\psi, \\psi^{-1}$ \ngiven by \\eqref{sep1} and \\eqref{psi-inv}.\nHowever, the condition \\eqref{sol-rep} should be modified as follows:\n\\begin{itemize}\n    \\item {\\em A meromorphic function $S \\colon \\Pi_{+} \\to {\\mathbb C}^{p \\times m}$ \nhas total pole multiplicity at most $\\kappa$ over $\\Pi_+$ and \n    satisfies the interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3} if and only if there \n    is an $m\\times m$-matrix\nvalued function $\\Psi$ analytic on $\\Pi_+$ with $\\det \\Psi$ having no zeros on $\\sigma(Z)\\cup\\sigma(W)$ \nand with $\\kappa$ zeros in $\\Pi_+$ such that \n \\begin{equation}  \\label{sol-rep'}\n  \\begin{bmatrix} S \\\\ I_m \\end{bmatrix}\\psi^{-1}\\Psi H^2_m(\\Pi_+) \n      \\subset {\\mathcal M}_{\\mathfrak D}.\n \\end{equation}}\n \\end{itemize}\nNow instead of \\eqref{Sparam}, we have\n  \\begin{equation}  \\label{Sparam'}\n \\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\psi^{-1} \\Psi =\n\\begin{bmatrix} \\Theta_{11} & \\Theta_{12} \\\\\n     \\Theta_{21} & \\Theta_{22} \\end{bmatrix} \\begin{bmatrix} Q_{1}\n     \\\\ Q_{2} \\end{bmatrix}\n\\end{equation}\nfor some $(p+m) \\times m$ matrix function $\\sbm{Q_{1} \\\\ Q_{2}} \\in H^2_{(p+m) \\times m}(\\Pi_{+})$.\nThen we conclude from the $J$-unitarity of $\\Theta$ on $i\\mathbb R$ (exactly as in Section 2) that for \nalmost all \n$\\lambda\\in i\\mathbb R$, the matrix $Q_2(\\lambda)$ is invertible whereas the matrix \n$G(\\lambda)=Q_1(\\lambda)Q_2(\\lambda)^{-1}$\nis a contraction. The identity \\eqref{bottom} arising from looking at \nthe bottom component of \\eqref{Sparam'} must be modified to read\n$$\n    \\psi^{-1} \\Psi = \\Theta_{21} Q_{1} + \\Theta_{22} Q_{2} = \n    \\Theta_{22} (\\Theta_{22}^{-1} \\Theta_{21} G + I_{m}) Q_{2}\n$$\nleading to the modification of \\eqref{wno'}:\n\\begin{align*}   \n   & {\\rm wno} \\det \\psi^{-1} + {\\rm wno} \\det \\Psi = \\notag \\\\\n    & \\quad \\quad \\quad \\quad \n{\\rm wno} \\det \\Theta_{22} + {\\rm wno} \\det (\\Theta_{22}^{-1} \\Theta_{21} G + \n    I_{m}) + \\rm{ wno} \\det Q_{2}.\n\\end{align*}\nThe identity \\eqref{wno''} must be replaced by \\eqref{toshow2}.  \nUsing that ${\\rm wno} \\det \\Psi = \\kappa$, with all these adjustments \nin place we still arrive at ${\\rm wno} \\det Q_{2} = 0$ and hence \n$Q_{2}$ has no zeros in $\\Pi_+$ and $G$ extends inside $\\Pi_+$ as a \nSchur-class function.\nThe representation \\eqref{LFTparamkap} follows from \\eqref{Sparam'} as well as the equality\n$\\Psi = \\psi(\\Theta_{21}G+\\Theta_{22}) Q_{2}$. Since $\\Psi$ has no zeros \nin $\\sigma(Z) \\cap \\sigma(W)$ while $\\psi(\\Theta_{21}G+\\Theta_{22})$ \nand $Q_2$ are analytic on all of $\\Pi_+$, we see that $\\psi(\\Theta_{21}G+\\Theta_{22})$ has no zeros in  \n$\\sigma(Z) \\cap \\sigma(W)$ as well.\n\n\\smallskip\n\nConversely, for any $G\\in\\mathcal S^{p\\times m}(\\Pi_+)$ such that $\\psi(\\Theta_{21}G+\\Theta_{22})$ has \nno zeros on $\\sigma(Z)\\cup\\sigma(W)$, we let\n$$\n\\begin{bmatrix}S_1 \\\\ S_2\\end{bmatrix}=\\Theta\\begin{bmatrix}G \\\\ I_m\\end{bmatrix},\\quad \\Psi=\\psi S_2, \n    \\quad S=S_1S_2^{-1}, \n$$\nso that \n$$\n\\begin{bmatrix}S \\\\ I_m\\end{bmatrix}\\psi^{-1}\\Psi=\\Theta\\begin{bmatrix}G \\\\ I_m\\end{bmatrix}.\n$$\nSince $\\Theta$ is $J$-unitary on $i\\mathbb R$ and $G$ is a Schur-class, \nit follows that $S(\\lambda)$ is contractive for almost all $\\lambda\\in i\\mathbb R$.\nSince $\\det \\Psi$ has no zeros on $\\sigma(Z)\\cup\\sigma(W)$ and has $\\kappa$ zeros in $\\Pi_+$, due to \nthe equalities\n$$\n{\\rm wno}\\det \\Psi={\\rm wno}\\det \\psi+{\\rm wno}\\det\\Theta_{22}+\n{\\rm wno}\\det(\\Theta_{22}^{-1}\\Theta_{21}G+I)=\\kappa\n$$\nwe see that $S$  satisfies the interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3} \nby the criterion \\eqref{sol-rep'}\nand has total pole multiplicity at most $\\kappa$ in $\\Pi_+$. However, since  \n$\\nu_-(\\boldsymbol{\\Gamma}_{\\mathfrak D})=\\kappa$,\nby the part of the sufficiency criterion already proved we know that $S$ must \nhave at least $\\kappa$ poles in $\\Pi_+$.  Thus $S$ has exactly \n$\\kappa$ poles in $\\Pi_{+}$ and therefore is in the $\\mathcal \nS_\\kappa^{p\\times m}(\\Pi_{+})$-class.\n\n\\smallskip\n\n\\subsection{The Fundamental Matrix Inequality approach for the \ngeneralized Schur-class setting} \nThe Fundamental Matrix Inequality method extends to the present setting as follows. \nAs in the definite case,\nwe extend the interpolation data by an arbitrary finite set of \nadditional full-matrix-value interpolation conditions to conclude that the kernel \n$\\boldsymbol{\\Gamma}_{\\mathfrak D}(z,\\zeta)$ \ndefined as in \\eqref{sep8c} has at most $\\kappa$ negative squares in \n$\\Omega_S \\setminus  \\sigma(W)$. \nSince the constant block\n(the matrix $\\Gamma_{\\mathfrak D}$) has $\\kappa$ negative eigenvalues (counted with multiplicities), \nit follows that \n${\\rm sp}_-\\boldsymbol{\\Gamma}_{\\mathfrak D}(z,\\zeta)=\\kappa$ which holds if and only if the Schur complement \nof $\\Gamma_{\\mathfrak D}$ in \\eqref{sep8c} is a positive kernel on $\\Omega_S \n\\setminus \\sigma(W)$: \n$$\n\\frac{I_{p} - S(z)S(\\zeta)^*}{z+\\overline{\\zeta}}-\n\\begin{bmatrix}I_p & -S(z)\\end{bmatrix}{\\bf C}(z I-{\\bf A})^{-1}\\Gamma_{\\mathfrak D}^{-1}\n(\\overline\\zeta I-{\\bf A}^*)^{-1}{\\bf C}^*\\begin{bmatrix}I_p \\\\ -S(\\zeta)^*\\end{bmatrix}\\succeq 0.\n$$\nAs in Section \\ref{S:FMI}, the latter positivity condition can be written in the form \\eqref{sep8d} \n(all we need is formula \\eqref{ad1} which still holds true) and eventually, implies equality \n\\eqref{sep8g} for some $G\\in\\mathcal S^{p\\times m}(\\Pi_+)$, which in turn, implies \nthe representation \\eqref{LFTparamkap}. However, establishing the necessity of the condition \n\\eqref{LFTparampar} \nrequires a good portion of extra work. Most of the known proofs are still based the Argument Principle \n(the winding number computations \\cite{BGR} or the operator-valued \nversion of Rouch\\'e's theorem \\cite{GS}).\nFor example, it can be shown that if $K$ is a $p\\times m$ matrix-valued polynomial \nsatisfying interpolation conditions \n\\eqref{BTOAint1}--\\eqref{BTOAint3} and if $\\varphi$ is the inner function given \n(analogously to \\eqref{sep1}) by\n$$\n\\varphi(z)=I_p-X^*(zI+Z^*)^{-1}\\widetilde{P}^{-1}X,\n$$\nwhere the positive definite matrix $\\widetilde{P}$ is uniquely defined from the Lyapunov equation\n$\\widetilde{P}Z+Z^*\\widetilde{P}=XX^*$, then the matrix function \n\\begin{equation}\n\\Sigma:=\\begin{bmatrix}\\Sigma_{11} & \\Sigma_{12}\\\\ \\Sigma_{21} & \\Sigma_{22}\\end{bmatrix}=\n\\begin{bmatrix}\\varphi^{-1} & -\\varphi^{-1}K \\\\ 0 & \\psi\\end{bmatrix}\n\\begin{bmatrix}\\Theta_{11} & \\Theta_{12}\\\\ \\Theta_{21} & \\Theta_{22}\\end{bmatrix}\n\\label{nov2}\n\\end{equation}\nis analytic on $\\Pi_+$. Let us observe that by the formulas \\eqref{oct1}, \\eqref{bigLySyl} and well known\nproperties of determinants,\n\\begin{align}\n\\det\\Theta(\\lambda)&=\\det\\left(I- {\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1} {\\mathbf C} J\\right)\\notag\\\\\n&=\\det\\left(I- {\\mathbf C} J{\\mathbf C} (\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1}\\right)\\notag\\\\\n&=\\det(\\boldsymbol{\\Gamma}_{\\mathfrak D} (\\lambda I - {\\mathbf A})+\\boldsymbol{\\Gamma}_{\\mathfrak D}{\\mathbf A}+{\\mathbf A}^*\\boldsymbol{\\Gamma}_{\\mathfrak D})\\cdot\n\\det((\\lambda I - {\\mathbf A})^{-1} \\boldsymbol{\\Gamma}_{\\mathfrak D}^{-1})\\notag\\\\\n&=\\frac{\\det(\\lambda I+{\\mathbf A}^*)}{\\det(\\lambda I - {\\mathbf A})}=\\frac{\\det(\\lambda I-Z)\\det(\\lambda I+W^*)}\n{\\det(\\lambda I +Z^*)\\det(\\lambda I-W)}.\\notag\n\\end{align}\nSimilar computations show that \n$$\n\\det \\psi(\\lambda)=\\frac{\\det(\\lambda I-W)}{\\det(\\lambda I +W^*)},\\quad \n\\det \\varphi(z)=\\frac{\\det(\\lambda I-Z)}{\\det(\\lambda I +Z^*)}.\n$$\nCombining the three latter equalities with \\eqref{nov2} gives $\\det \\Sigma(\\lambda)\\equiv 1\\neq 0$. \nTherefore, for \n$G\\in\\mathcal S^{p\\times m}$, the total pole multiplicity of the function \n$$\n\\Upsilon=(\\Sigma_{11}G+\\Sigma_{12})(\\Sigma_{21}G+\\Sigma_{22})^{-1}\n$$\nis the same as the number of zeros of the denominator \n$$\\Sigma_{21}G+\\Sigma_{22}=\n\\psi(\\Theta_{21}G+\\Theta_{22}),\n$$\nthat is $\\kappa$, by the winding number argument.\nOn the other hand, since \n\\begin{equation}  \\label{Takagi-Sarason}\n S=K+\\varphi \\Upsilon\\psi,\n \\end{equation}\nas can be seen from \\eqref{LFTparamkap} and \\eqref{nov2},\nthe total pole multiplicity of $S$ equals $\\kappa$ if no poles of $\\Upsilon$ \noccur at zeros of $\\varphi$ and \n$\\Psi$, that is, in $\\sigma(Z)\\cup\\sigma(W)$.  We note that the form \n\\eqref{Takagi-Sarason} where $K,\\varphi,\\psi$ are part of the data \nand $\\Upsilon$ is a free meromorphic function with no poles on $i \n{\\mathbb R}$ but $\\kappa$ poles in $\\Pi_{+}$ (including possibly at \npoints of $\\sigma(W) \\cup \\sigma(Z)$) corresponds to a \nvariant of the interpolation problem \n\\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3} sometimes \ncalled the {\\em Takagi-Sarason problem} (see \\cite[Chapter 19]{BGR}, \\cite{Bolo04}).  It \nturns out that discarding the side-condition \\eqref{LFTparampar} on \nthe Schur-class free-parameter function $G$ leads to a \nparametrization of the set of all solutions of the Takagi-Sarason \nproblem.\n\n\\subsection{Indefinite kernels and reproducing kernel Pontryagin \nspaces}  \\label{S:RKHSkap}\nFrom the formula \\eqref{ad1} for $K_{\\Theta,J}$, we see from \nthe observability  assumption on $({\\mathbf C}, {\\mathbf A})$ (equivalently, the \nobservability and controllability assumptions on $(\\sbm{V \\\\ U}, W)$ \nand $(Z, \\begin{bmatrix} X & -Y \\end{bmatrix})$) that \n$$\n \\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D}) = {\\rm sq}_{-}(K_{\\Theta,J}).\n$$\nBy the general theory of reproducing kernel Hilbert spaces sketched \nin Section \\ref{S:RKHSkap}, it follows that ${\\mathcal H}(K_{\\Theta,J})$ is a Pontryagin space with \nnegative index $\\nu_{-}({\\mathcal H}(K_{\\Theta,J})$ equal to the number of \nnegative eigenvalues of $\\boldsymbol{\\Gamma}_{\\mathfrak D}$:\n$$\n  \\nu_{-}({\\mathcal H}(K_{\\Theta,J})) = \\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D}).\n$$\nWe conclude that the formula for $\\kappa$ in statement (1) agrees with that in statement (2)\nin Theorem \\ref{T:BTOA-NPkap}. \n\n\n\\subsection{The Grassmannian\/Kre\\u{\\i}n-space approach for the \ngeneralized Schur-class setting} \\label{S:Grass-kappa}\n\nThe Grassmannian approach extends to to the present setting as follows. \nThe suitable analog of Lemma \\ref{L:maxneg} is the following:\n\\begin{lemma}  \\label{L:maxnegkap}  \nSuppose that ${\\mathcal M}$ is a closed subspace of a Kre\\u{\\i}n-space ${\\mathcal K}$ \nsuch that the ${\\mathcal K}$-relative orthogonal \ncomplement ${\\mathcal M}^{[\\perp]}$ has negative signature equal $\\kappa$. If ${\\mathcal G}$ is a negative\nsubspace of ${\\mathcal M}$, then ${\\mathcal G}$ has codimension at least $\\kappa$ in any maximal negative \nsubspace of ${\\mathcal K}$. Moreover, the codimension of such a ${\\mathcal G}$ in any maximal negative \nsubspace of ${\\mathcal K}$ is equal to $\\kappa$ if and only if ${\\mathcal G}$ is a maximal negative\nsubspace of ${\\mathcal M}$. \n\\end{lemma}\n\nLet us now assume that we are given a $\\Pi_{+}$-admissible \ninterpolation data set $\\mathfrak D$ with $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ \ninvertible. Then ${\\mathcal M}_{\\mathfrak D}$  given by \\eqref{cMrep} is a regular subspace of the \nKre\\u{\\i}n space $L^{2}_{p+m}(i {\\mathbb R})$ with the $J (= \n\\sbm{I_{p} & 0 \\\\ 0 & - I_{m}}$)-inner product.  \n\nWith Lemma \\ref{L:maxnegkap} in hand, we argue that \n$\\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})  \\ge \\kappa$ is necessary for the existence of \n${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$-functions $S$ analytic on \n$\\sigma(Z) \\cup \\sigma(W)$ satisfying the interpolation conditions\n\\eqref{BTOAint1}, \\eqref{BTOAint2}, \\eqref{BTOAint3}.  \n\n\\smallskip\n\n\\noindent\n\\textbf{Proof of necessity for the generalized Schur-class setting.}\nIf $S \\in \n{\\mathcal S}_{\\kappa'}^{p \\times m}(\\Pi_{+})$ is a solution of the \ninterpolation conditions with $\\kappa' \\le \\kappa$, then as in Section \n\\ref{S:statekap}, there is a $m \\times m$-matrix \nfunction $\\Psi$ with $\\det \\Psi$ having no zeros in $\\sigma(Z) \\cup \n\\sigma(W)$ and having $\\kappa$ zeros in $\\Pi_{+}$ so that the \nsubspace ${\\mathcal G}_{S}: = \\sbm{ S  \\\\ I_{m} } \\psi^{-1} \\Psi H^{2}_{m}(\\Pi_{+})$ \nsatisfies the inclusion \\eqref{sol-rep'}.  We note that then \n${\\mathcal G}_{S}$ is a negative subspace of ${\\mathcal K}$ and the fact that $\\Psi$ has \n$\\kappa$ zeros means that ${\\mathcal G}_{S}$ has codimension $\\kappa$ in a \nmaximal negative subspace of ${\\mathcal K}: = \\sbm{ L^{2}_{p}(i {\\mathbb R}) \\\\ \nH^{2}_{m}(\\Pi_{+})}$.  As ${\\mathcal G}_{S}$ is also a subspace of \n${\\mathcal M}_{\\mathfrak D}$, it follows  by Lemma \\ref{L:maxnegkap} that \nthe negative signature of ${\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]}$ must be \nat least $\\kappa$.  Thus $\\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]}) \n\\ge \\kappa$ is necessary for the existence of a solution $S$ of the \ninterpolation problem in the class ${\\mathcal S}_{\\kappa}^{p \\times \nm}(\\Pi_{+})$.  As part of the sufficiency direction, we shall show \nthat conversely, if $\\kappa = \\nu_{-}({\\mathcal M}^{[ \\perp {\\mathcal K}]})$, then we \ncan always find solutions $S$ of the interpolation conditions in the \nclass ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$.  This establishes \nthe formula in statement (2) of Theorem \\ref{T:BTOA-NPkap} as the \nminimal $\\kappa$   such that solutions of the interpolation \nconditions can be found in class ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$.\n\n\\smallskip\n\n\\noindent\n\\textbf{Proof of sufficiency for the generalized Schur-class setting.}\nLet us suppose that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ is invertible and hence \nthat ${\\mathcal M}_{\\mathfrak D}$ is a regular subspace of the Kre\\u{\\i}n \nspace $L^{2}_{p+m}(i {\\mathbb R})$ with the $J = \\sbm{ I_{p} & 0 \\\\ 0 \n& -I_{m}}$-inner product.  By the results of \n\\cite{BH}, there is a $J$-phase function $\\Theta$ so that the \nBeurling-Lax representation \\eqref{cMBLrep} holds (we avoid using the \nformula \\eqref{oct1} for $\\Theta$ at this stage).\nWe now assume that ${\\mathcal M}_{\\mathfrak D}^{[ \\perp \n{\\mathcal K}]}$ has negative signature $\\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[ \\perp \n{\\mathcal K}]})$ equal to $\\kappa$. We wish to verify the linear-fractional \nparametrization \\eqref{LFTparamkap}--\\eqref{LFTparampar} for the set \nof all ${\\mathcal S}_{\\kappa}^{p\\times m}(\\Pi_{+})$-class solutions of the \ninterpolation conditions.\n\nSuppose first that $S$ is any ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$-class \nsolution of the interpolation conditions. By the graph-space criterion\nfor such solutions, there is a $m \\times m$-matrix valued function \n$\\Psi$ analytic on $\\Pi_{+}$ with $\\det \\Psi$ having $\\kappa$ zeros \nbut none in $\\sigma(Z) \\cup \\sigma(W)$, so that \\eqref{sol-rep'} holds.\nBut then \n$$\n{\\mathcal G}_{S}: = \\sbm{ S \\\\ I_{m} } \\psi^{-1} \\Psi \nH^{2}_{m}(\\Pi_{+})\n$$ \nis a shift-invariant negative subspace of ${\\mathcal K}$\ncontained in ${\\mathcal M}_{\\mathfrak D}$ and having codimension $\\kappa$ in a maximal \nnegative subspace of ${\\mathcal K}$.  It now follows from Lemma \n\\ref{L:maxnegkap} that ${\\mathcal G}_{S}$ is maximal negative as a subspace of \n${\\mathcal M}_{\\mathfrak D}$.  As ${\\mathcal G}_{S}$ is also shift-invariant and \nmultiplication by $\\Theta$ is a Kre\\u{\\i}n-space isomorphism from \n$H^{2}_{p+m}(\\Pi_{+})$ onto ${\\mathcal M}_{\\mathfrak D}$, it follows that\n${\\mathcal G}_{S}$ is the image under multiplication by $\\Theta$ of a \nshift-invariant $J$-maximal negative subspace of \n$H^{2}_{p+m}(\\Pi_{+})$, i.e.,\n\\begin{equation}  \\label{Srep'}\n  {\\mathcal G}_{S}: = \\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} \\Psi \n\\cdot  H^{2}_{m}(\\Pi_{+}) = \\Theta \\cdot \\begin{bmatrix} G \\\\ I_{m} \n\\end{bmatrix} \\cdot H^{2}_{m}(\\Pi_{+})\n\\end{equation}\nfor a ${\\mathcal S}^{p \\times m}(\\Pi_{+})$-class function $G$.  From the fact \nthat $\\Psi$ has no zeros in $\\sigma(Z) \\cup \\sigma(W)$ one can read \noff from \\eqref{Srep'} that $\\psi (\\Theta_{21} G + \\Theta_{22})$ has no zeros in \n$\\sigma(Z) \\cup \\sigma(W)$ and from the representation \\eqref{Srep'} \nthe linear-fractional representation \\eqref{LFTparamkap} follows as \nwell. From the subspace identity \\eqref{Srep'} one can also read off \nthat there is a $m \\times m$ matrix function $Q$ with $Q^{\\pm 1} \\in \nH^{\\infty}_{m \\times m}(\\Pi_{+})$ such that\n$$\nS \\psi^{-1} \\Psi = (\\Theta_{11} G + \\Theta_{12}) Q\\quad\\mbox{and}\\quad\n\\psi^{-1} \\Psi = (\\Theta_{21} G + \\Theta_{22})Q.\n$$\nSolving the second equation for $Q$ then gives\n$$\n  Q = (\\Theta_{22} G + \\Theta_{22})^{-1} \\psi^{-1} \\Psi.\n$$\nSubstituting this back into the first equation and then solving for \n$S$ leads to the linear-fractional representation \\eqref{LFTparamkap} \nfor $S$.\n\n\\smallskip\n\nLet now $G$ be any Schur-class function satisfying the additional \nconstraint \\eqref{LFTparampar}.  Since multiplication by $\\Theta$ is \na Kre\\u{\\i}n-space isomorphism from $H^{2}_{p+m}(\\Pi_{+})$ to \n${\\mathcal M}_{\\mathfrak D}$ and $\\sbm{ G \\\\ I_{m}} H^{2}_{m}(\\Pi_{+})$  is a \nmaximal negative shift-invariant subspace of ${\\mathcal M}_{\\mathfrak D}$, it follows that \n$\\Theta \\cdot \\sbm{ G \\\\ I_{m}} H^{2}_{m}(\\Pi_{+})$ is maximal \nnegative as a subspace of ${\\mathcal M}_{\\mathfrak D}$.\nBy Lemma \\ref{L:maxnegkap}, it follows that $\\Theta \\cdot \\sbm{ G \\\\ \nI_{m}} H^{2}_{m}(\\Pi_{+})$ has codimension $\\kappa = \n\\nu_{-}({\\mathcal M}_{\\mathfrak {\\mathcal D}}^{[ \\perp] {\\mathcal K}})$ in a maximal negative \nsubspace of ${\\mathcal K}$.  As $\\Theta \\cdot \\sbm{ G \\\\ \nI_{m}} H^{2}_{m}(\\Pi_{+})$ is also shift-invariant, it follows that\nthere must be a contractive matrix function $S$ on the unit circle \nand a  bounded analytic $m \\times m$-matrix function $\\Psi$ on \n$\\Pi_{+}$ such that $\\Psi$ has exactly $\\kappa$ zeros in $\\Pi_{+}$\nand $\\Psi$ is bounded and invertible on $i {\\mathbb R}$ so that\n\\begin{equation}  \\label{Srep}\n\\begin{bmatrix} S \\\\ I_{m} \\end{bmatrix} \\cdot \\psi^{-1} \\Psi \n   \\cdot H^{2}_{m}(\\Pi_{+}) = \\Theta \\cdot \\begin{bmatrix} G \\\\ I \n\\end{bmatrix} \\cdot  H^{2}_{m}(\\Pi_{+}).\n\\end{equation}\nIn particular, $\\psi^{-1} \\Psi\\cdot  H^{2}_{m}(\\Pi_{+}) \\subset \n(\\Theta_{21} G + \\Theta_{22}) \\cdot H^{2}_{m}(\\Pi_{+})$, so\nthere is a $Q \\in H^{\\infty}_{p \\times m}(\\Pi_{+})$ so that\n$ \\psi^{-1} \\Psi = (\\theta_{21} G + \\Theta_{22}) Q$, i.e., so that\n$$\n  \\Psi = \\psi (\\Theta_{21} G + \\Theta_{22}) Q.\n$$\nAs $\\psi (\\Theta_{21} G + \\Theta_{22})$ has no zeros in $\\sigma(Z) \n\\cup \\sigma(W)$ by assumption, it follows that none of the zeros of \n$\\Psi$ are in $\\sigma(Z) \\cup \\sigma(W)$.  By the criterion \n\\eqref{sol-rep'} for ${\\mathcal S}_{\\kappa'}^{p \\times m}(\\Pi_{+})$-class \nsolutions of the interpolation conditions with $\\kappa' \\le \\kappa$, \nwe read off from \\eqref{Srep} that $S$ so constructed is a \n${\\mathcal S}_{\\kappa'}^{p \\times m}(\\Pi_{+})$-class solution of the \ninterpolation conditions for some $\\kappa' \\le \\kappa$. However, from \nthe proof of the necessity direction already discussed, it follows \nthat necessarily $\\kappa' \\ge \\kappa$.  Thus $S$ so constructed is a \n${\\mathcal S}_{\\kappa'}^{p \\times m}(\\Pi_{+})$-class solution of the \ninterpolation conditions.  The subspace identity \\eqref{Srep} leads to\nthe formula \\eqref{LFTparamkap} for $S$ in terms of $G$ just as in \nthe previous paragraph.\n\n\n\\begin{remark} \\label{R:summary}\nWe conclude that the Grassmannian approach extends to the generalized \nSchur-class setting.  As in the classical Schur-class case,  \none can avoid the elaborate winding-number argument used in Section \n\\ref{S:statekap} by using Kre\\u{\\i}n-space geometry (namely, the fact \nthe a Kre\\u{\\i}n-space isomorphism maps maximal negative \nsubspaces to maximal negative subspaces combined with Lemma \n\\ref{L:maxnegkap}),  unlike the story for the Fundamental Matrix Inequality \nPotapov approach, which avoids the winding number argument in an \nelegant way for the definite case but appears to still require such \nan argument for the indefinite generalized Schur-class setting.\n\\end{remark}\n\n\\subsection{State-space versus Grassmannian\/Kre\\u{\\i}n-space-geometry solution criteria in the generalized \nSchur-class setting}  \\label{S:synthesiskap}\nThe work of the previous subsections shows that each of conditions \n(1) and (2) in Theorem \\ref{T:BTOA-NPkap} is equivalent to the \nexistence of ${\\mathcal S}_{\\kappa}^{p \\times m}(\\Pi_{+})$-class solutions f \nthe interpolation conditions \\eqref{BTOAint1}--\\eqref{BTOAint3}, \nand that condition (2) is equivalent to condition \n(1).  It follows that conditions (1), (2), (3) are all equivalent to \neach other.  Here we wish to see this latter fact directly in a more \nconcrete from, analogously to what is done in Section \n\\ref{S:synthesis} above for the classical Schur-class setting.\n\n\\smallskip\n\nAs in Section \\ref{S:synthesis}, we impose an assumption a little \nstronger than the condition that $\\boldsymbol{\\Gamma}_{\\mathfrak D}$ be \ninvertible, namely, the Nondegeneracy Assumption:  {\\em \n${\\mathcal M}_{\\mathfrak D}$, ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{+})$, \nand ${\\mathcal M}_{\\mathfrak D} \\cap H^{2}_{p+m}(\\Pi_{-})$ are all regular \nsubspaces of $L^{2}_{p+m}(i {\\mathbb R})$ (with the $J\\, (= \\sbm{ I_{p} \n& 0 \\\\ 0 & -I_{m} })$-inner product).}  Then Lemmas \\ref{L:cMDperp} and \n\\ref{L:decom} go through with no change.  Lemma \\ref{L:MperpK} goes \nthrough, but with the {\\em in particular} statement generalized to \nthe following (here $\\nu_{-}({\\mathcal L})$ refers to negative signature of \nthe given subspace ${\\mathcal L}$ of $L^{2}_{p+m}(i {\\mathbb R})$ with respect \nto the $J$-inner product):\n\\begin{itemize}\n    \\item {\\em In particular, $\\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[ \\perp \n    {\\mathcal K}]}) = \\kappa$ if and only if}\n  $$\n  \\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[\\perp {\\mathcal K}]})_{0}) = \\kappa\n  $$\n{\\em if and only if}\n$$\n  \\nu_{-}({\\mathbb G}_{T^{[*]}}) + \\nu_{-}(({\\mathcal M}_{\\mathfrak D}^{[ \n  \\perp]})_{1}) = \\kappa.\n$$\n    \\end{itemize}\nLemma \\ref{L:posnegsubspaces} has the more general form:\n\\begin{enumerate}\n    \\item $\\nu_{-}({\\mathbb G}_{T^{[*]}}) =\\kappa$ {\\em if and only if }\n    $\\nu_{-}(I + T T^{[*]}) = \\kappa$  ({\\em where $I + T T^{[*]}$ is \n    considered as an operator on } $\\operatorname{Ran} J ({\\mathcal C}_{Z, \n    \\sbm{ X & -Y}})^{*}$).\n    \n    \\item $\\nu_{-}(({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})_{1}) = \n   \\nu_{-}(\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W})$.\n   \n   \\item $\\nu_{-}(\\widehat {\\mathbb G}_{T}) = \\nu_{-}( I + T^{[*]} \n   T)$ ({\\em where } $I + T^{[*]}T$ {\\em is considered as an operator \n   on } $\\operatorname{Ran} {\\mathcal O}_{\\sbm{ V \\\\ U}, W}$).\n   \n   \\item $\\nu_{+}({\\mathcal M}_{{\\mathfrak D},1}) = \\nu_{-}(\\operatorname{Ran} \n   J ({\\mathcal C}_{Z, \\sbm{ X & -Y}})^{*})$.\n    \\end{enumerate}\n Lemma \\ref{L:GammaD-factored} is already in general form but its \n corollary, namely Lemma \\ref{L:GammaDpos}, can be given in a more \n general form:\n \\begin{itemize}\n     \\item {\\em The following conditions are equivalent:}\n\\begin{enumerate}\n    \\item $\\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D}) = \\kappa$.\n    \n     \\item $\\nu_{+}(\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\ U}, W}) + \n    \\nu_{-}({\\mathbb G}_{T^{[*]}}) = \\kappa.$\n    \n    \\item $\\nu_{-}( \\operatorname{Ran} J ({\\mathcal C}_{Z, \\sbm{ X & \n    -Y}})^{*} )  + \\nu_{+}(\\widehat{\\mathbb G}_{T}) = \\kappa.$\n \\end{enumerate}\n \\end{itemize}\n Putting the pieces together, we have the following chain of \n reasoning.  By the generalized version of Lemma \\ref{L:MperpK}, we \n have\n \\begin{equation}   \\label{1}\n \\nu_{-}({\\mathcal M}_{\\mathfrak D})^{[ \\perp {\\mathcal K}]} = \\nu_{-}({\\mathbb \n G}_{T^{[*]}}) + \\nu_{-}(({\\mathcal M}_{\\mathfrak D}^{[ \\perp]})_{1})\n \\end{equation}\n where, by the generalized version of Lemma \\ref{L:posnegsubspaces} \n part (2),\n $$\n \\nu_{-}(({\\mathcal M}_{\\mathfrak D}^{[\\perp]})_{1}) = \n \\nu_{-}(\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\U}, W}).\n $$\n Thus \\eqref{1} becomes\n $$\n \\nu_{-}({\\mathcal M}_{\\mathfrak D})^{[ \\perp {\\mathcal K}]} = \\nu_{-}({\\mathbb G}_{T^{[*]}}) \n + \\nu_{-}(\\operatorname{Ran} {\\mathcal O}_{\\sbm{V \\\\U}, W}).\n $$\n By (1) $\\Leftrightarrow$ (2) in the generalized Lemma \n \\ref{L:GammaD-factored}, we get\n $$\n \\nu_{-}({\\mathcal M}_{\\mathfrak D}^{[ \\perp {\\mathcal K}]}) = \n \\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D})\n $$\n which gives us (1) $\\Leftrightarrow$ (2) in Theorem \n \\ref{T:BTOA-NPkap}.\n \n\\smallskip\n\n To give a direct proof of (1) $\\Leftrightarrow$ (3) in Theorem \n \\ref{T:BTOA-NPkap}, we note the concrete identification \n \\eqref{HKThetaJid} of the space ${\\mathcal H}(K_{\\Theta,J})$ (with \n $J$-inner product on $\\operatorname{Ran} (P^{J}H^{2}_{p+m}(\\Pi_{+}) \n - P^{J}_{{\\mathcal M}_{\\mathfrak D}})$ which again leads to the more compact \n identification \\eqref{HKThetaJ} from which we immediately see that\n $$\n \\nu_{-}({\\mathcal H}(K_{\\Theta,J})) = \\nu_{-}(\\operatorname{Ran} J({\\mathcal C}_{Z, \n \\sbm{ X & -Y}})^{*}) + \\nu_{+}(\\widehat {\\mathbb G}_{T}).\n $$\n By (1) $\\Leftrightarrow$ (3) in the generalized Lemma \n \\ref{L:GammaDpos}, this last expression is equal to \n $\\nu_{-}(\\boldsymbol{\\Gamma}_{\\mathfrak D})$, and we have our more concrete \n direct proof of the equivalence of conditions (1) and (3) in Theorem \n \\ref{T:BTOA-NPkap}.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\n\\label{intro}\nThe notion of a ``Siegel zero\" was first introduced in the context of Dirichlet\n$L$-functions, $L(s,\\chi)$, where $\\chi$ is a quadratic Dirichlet character\nof conductor $D.$ \nRoughly speaking, a Siegel zero of $L(s,\\chi)$ is a zero\non the real axis close to 1.  More precisely, given \n$\\epsilon>0$, an $\\epsilon$-Siegel zero of $L(s,\\chi)$ is a real number\n$\\beta$ in the interval \n$(1-\\epsilon (\\log D)^{-1},1)$  such that $L(\\beta,\\chi)=0$.     \nThe idea of a \"Siegel zero\" has been generalized to\n$L$-functions associated with Maass forms by Hoffstein and \nLockhart ~\\cite{hofflock}.  In this context, the conductor $D$ is replaced by\n$(\\lambda N+1)$, where $\\lambda$ and $N$ are the Laplace eigenvalue\nand level of a Maass\nform, respectively.\n\nThe connection with Eisenstein series comes from a 1975 paper of Goldfeld\n~\\cite{dg} \nin which he proves that if one assumes that the class number, $h(-d)$, of \nthe imaginary quadratic field $k$ of discriminant $-d$\nis small, \nthen the Siegel zero is given by a certain asymptotic formula.  This \nasymptotic formula is computed by writing the Dedekind zeta function\n$\\zeta_k$ in \ntwo ways: \n\\begin{equation}\\label{dg}\n\\zeta(s)L(s,\\chi_{-d})=\\zeta_k(s)= \\frac{1}{|\\frak{o}_k^{\\times}|}\n\\sum_C 2^s d^{-\\frac{s}{2}}E(z_C,s),\n\\end{equation}\n where \n$E(z,s)$ is the nonholomorphic Eisenstein series on the upper half\nplane, the sum\nis over Heegner points $z_C$ corresponding to ideal classes $C$ of $k$,\nand $\\chi_{-d}$ is the unique quadratic character of conductor $d$ such that\n$\\chi_{-d}(-1)=-1.$\nIt is interesting, in this context, to \nnote that for any fixed $\\epsilon,$ and \n$d$ sufficiently large, a positive proportion of the functions\n$E(z_C,s)$ vanish for some $s$ \nin the interval $(1-\\epsilon (\\log d)^{-1},1)$.  \nTo prove this one uses the equidistribution of Heegner points \nand the following result:\n\n\\begin{thm} ($GL(2)$ Case, due to Bateman and Grosswald \\cite{bg}):  \nGiven $\\epsilon >0,$ there exists $Y >0$ such that\nif $y>Y$ then for each $x,$ $E(x+iy, \\beta(x,y)) =0$ for some \n$\\beta(x,y) \\in (1-\\epsilon(\\log y)^{-1},1),$ and for all $x,\\epsilon$,\nwe have\n$$1-\\beta(x,y)\\sim \\frac{3}{\\pi y}, \\mbox{ as } y \\rightarrow \\infty.$$ \n\\end{thm}\n\nHoffstein \\cite{jhoff} has proved an analogous result for Hilbert modular\nEisenstein series.  We extend it to the generality of\nMoeglin and Waldspurger \\cite{mw}.  However, our \nresult will be weaker than Hoffstein's in that we will not obtain a \nprecise error term, and in that we will not obtain any estimates for the \noptimal value of $Y$.  \n\nThe author wishes to thank Carlos Moreno, who suggested that the result, \noriginally proved only for $GL_n$ over $\\ensuremath{{\\mathbb Q}}$ would go through in this \ngenerality, as well as Dorian Goldfeld, Herv\\'e Jacquet, and David \nGinzburg, for help and advice along the way.\n\n\\subsection{Notation}\n\\label{notation}\n\nFor the most part, we follow the notation of Moeglin and\nWalspurger \\cite{mw}.  This entails \ncertain redundancies:  for example the symbol $k$ is used both for\nthe global field and an element of the maximal compact, while $M$ is \nboth a Levi subgroup and an intertwining operator.  All references\nare to \\cite{mw} until section 4\n\nThus, let $k$ be a global field, $\\ensuremath{{\\mathbb A}}$ the adeles of $k$,\n$G$ a\nconnected reductive algebraic group defined over $k$, and $\\mathbf{G}$\na finite central covering of $G(\\ensuremath{{\\mathbb A}})$ such that $G(k)$ lifts to a \nsubgroup of $\\ensuremath{\\mathbf{G}}$.\n\nFix once and for all a choice of minimal parabolic $P_0$ of $G$ \nand a Levi subgroup\n$M_0$ of $P_0,$ both defined over $k$.    \nThis also fixes a definition of ``standard'' for parabolics and Levis (p.4).\nLet $T_0$ be the maximal split torus in the center of $M_0$ and\n$\\Delta_0$ the set of simple positive roots for $T_0$ determined by\n$P_0.$  \nFix a maximal compact subgroup $\\mathbf{K}$ of $\\ensuremath{\\mathbf{G}}$,\nas in I.1.4.\n\nLet $P$ be a standard parabolic subgroup of $G$, and let $U$ be its \nunipotent radical, and $M$ its standard Levi. \nThen $U(\\ensuremath{{\\mathbb A}})$ lifts canonically into $\\ensuremath{\\mathbf{G}}$.  \nLet $\\ensuremath{\\mathbf{M}}$ denote the preimage of $M(\\ensuremath{{\\mathbb A}})$ in $\\ensuremath{\\mathbf{G}}$.  \nIf $\\chi$ is a rational character of $M$, we obtain a map \n$\\ensuremath{\\mathbf{M}}\\mapsto \\ensuremath{{\\mathbb R}}^{\\times}_+$ by projecting to $M(\\ensuremath{{\\mathbb A}})$, using $\\chi$ to \nget to $\\ensuremath{{\\mathbb A}}^{\\times}$, and then taking the absolute value.  \nLet $\\ensuremath{\\mathbf{M}}^1$ denote the intersection of the kernels of all such maps.\nLet $X_M^{\\ensuremath{\\mathbf{G}}}$ and $\\mbox{Re} X_M^{\\ensuremath{\\mathbf{G}}}$ \ndenote the groups of continuous homomorphisms of $\\ensuremath{\\mathbf{M}}$ to\n$\\ensuremath{{\\mathbb C}}^{\\times}$ and $\\ensuremath{{\\mathbb R}}^{\\times}_+$ respectively, which\nare trivial on $\\ensuremath{\\mathbf{M}}^1$ and on the center of $\\ensuremath{\\mathbf{G}}$.  When $k$ is a \nnumber field, $X_M^{\\ensuremath{\\mathbf{G}}}$ may be identified with a complex vector\nspace $(\\frak{a}_M^{\\ensuremath{\\mathbf{G}}})^*$.  When $k$ is a function field, it has \na finite number of connected components, and there is a natural \nprojection from $(\\frak{a}_M^{\\ensuremath{\\mathbf{G}}})^*$ to the identity\ncomponent, which still identifies $\\mbox{Re} X_M^\\ensuremath{\\mathbf{G}}$ with a real vector \nspace.  See I.1.4 and I.1.6.\n\nThere is unique map $m_P:\\ensuremath{\\mathbf{G}}\\longrightarrow \\ensuremath{\\mathbf{M}}^1\\ensuremath{\\backslash} \\ensuremath{\\mathbf{M}}$ defined by requiring that if $g=umk$ for\nsome $u\\in U(\\ensuremath{{\\mathbb A}}), m\\in \\ensuremath{\\mathbf{M}}$ and $k \\in \\ensuremath{\\mathbf{K}}$, then $m_P(g) =\\ensuremath{\\mathbf{M}}^1m.$  \nIf $\\varphi$ is a function $U(\\ensuremath{{\\mathbb A}})M(k)\\ensuremath{\\backslash} \\ensuremath{\\mathbf{G}} \\mapsto \\ensuremath{{\\mathbb C}}$ and $k\\in \\ensuremath{\\mathbf{K}}$, \nlet us define a function $\\varphi_{k}$ on $M(k)\\ensuremath{\\backslash} \\ensuremath{\\mathbf{M}}$ by \n$\\varphi_{k}(m)=m^{-\\rho_P}\\varphi(mk),$ where $\\rho_P$ is half the sum of the\nroots of $M$ in Lie $U$.  \nLet $\\phi_{\\pi}$ be an automorphic form on\n$U(\\ensuremath{{\\mathbb A}})M(k)\\ensuremath{\\backslash} \\ensuremath{\\mathbf{G}}$ such that\nfor each $k$, the function $\\phi_{\\pi,k}$ is a cusp form on $\\ensuremath{\\mathbf{M}}$ which \ngenerates a\nsemisimple isotypic submodule of type\n $\\pi,$  where $\\pi$ is an automorphic subrepresentation of \n$\\ensuremath{\\mathbf{M}}$ in the sense of \\cite{mw}, page 78.\n\nFor each \n$\\lambda \\in X_M^{\\ensuremath{\\mathbf{G}}}$, let \n$$\\lambda\\phi_{\\pi}(g)=m_{P}(g)^{\\lambda}\\phi_{\\pi}(g).$$  Then for each $k$, the function $\\lambda \\phi_{\\pi,k}$ is a cusp form on $\\ensuremath{\\mathbf{M}}$ which generates a semisimple isotypic submodule of type $\\pi\\otimes \\lambda.$\n\nFor $\\lambda$ in a suitable cone in $X_M^{\\ensuremath{\\mathbf{G}}}$, the Eisenstein series is defined by the following convergent sum:\n$$E(\\lambda \\phi_{\\pi},\\pi\\otimes \\lambda)(g)=\\sum_{\\gamma \\in P(k)\\ensuremath{\\backslash} G(k)}\\lambda \\phi_{\\pi}(\\gamma g).$$  It is holomorphic for $\\lambda$ in the domain of convergence, and extends to a meromorphic \nfunction on all of $X_M^{\\ensuremath{\\mathbf{G}}}$.  We may assume without loss of generality\nthat $\\pi$ is unitary.  This amounts to,\nat most, altering our choice of ``base point.''  (See  I.3.3)\n(We deviate from \\cite{mw} in viewing the \nEisenstein series as a function on $X_M^{\\ensuremath{\\mathbf{G}}}$, rather than their $\\frak{P}$,\nwhich is a principal homogenous space for the quotient of $X_M^{\\ensuremath{\\mathbf{G}}}$ by \na certain finite group.) \n\nSuppose that $\\phi_{\\pi}$ is real valued.  Then \n$E(\\lambda \\phi_{\\pi},\\pi\\otimes \\lambda)$ is real valued for \n$\\lambda \\in \\mbox{Re} X_M^{\\ensuremath{\\mathbf{G}}}.$\nLet us fix $P$ and $\\phi_{\\pi}$ once and for all.  \nFor each $\\alpha \\in \\Delta_0,$ let \n$P_{\\alpha}=M_{\\alpha}U_{\\alpha}$\ndenote the standard maximal parabolic such that every element of \n$\\Delta_0$ is a root of $M_{\\alpha}$ except $\\alpha.$ \nOur choice of $\\Delta_0$ determines a set of positive roots for \nthe action of $T_0$ on any Levi $M'$ which we denote by $R^+(T_0,M')$.\n\nLet $W$ denote the Weyl group of $G$, and $W_M$ that of $M$.  \nLet $W(M)$ be the set of $w\\in W$ of\nminimal length in their class $wW_M$, such that $wMw^{-1}$ is a \nstandard Levi of $G$.\nLet $W(M,M_{\\alpha})$ denote the set of $w \\in W$ such that $w^{-1}\\theta >0$ for every $\\theta \\in R^+(T_0,M_{\\alpha})$ and $wMw^{-1}$ is a standard \nLevi subgroup of $M_{\\alpha}.$ \n\nIf $k$ is a function field we fix once and for all a place $v_0$, \nand a uniformizing parameter $\\frak{w}$, and let $q=|\\frak{w}|^{-1}_{v_0}.$  \nLet\n$\\frak{m}=\\ensuremath{{\\mathbb R}}_+^{\\times}$ in the number\nfield case or $\\frak{w}^{\\ensuremath{{\\mathbb Z}}}$, in the function field case.  Thus $\\ensuremath{\\frak{m}}$ may be \nembedded in $\\ensuremath{{\\mathbb A}}^{\\times}$ either at $k_{v_0}$ or diagonally at the infinite \nplaces, as a subgroup on which  \n the absolute value is injective.  One then has a \nsubgroup of $T_0(\\ensuremath{{\\mathbb A}})$ isomorphic to $\\ensuremath{\\frak{m}}^R$, where $R$ is the rank of $T_0$, \nand\nin  I.2.1 of \\cite{mw} this is extended to a subgroup $A_{\\ensuremath{\\mathbf{M}}_0}$ of $\\ensuremath{\\mathbf{G}}$, \nstill isomorphic to $\\ensuremath{\\frak{m}}^R.$  We then define $A_{\\ensuremath{\\mathbf{M}}_{\\alpha}}=A_{\\ensuremath{\\mathbf{M}}_0}\\cap \nZ_{\\ensuremath{\\mathbf{M}}_{\\alpha}}.$  \n\nIf $\\lambda \\in \\mbox{Re} X_M^{\\ensuremath{\\mathbf{G}}}$, then $\\left. \\lambda \\right|_{A_{\\ensuremath{\\mathbf{M}}_{\\alpha}}}$\nfactors through the quotient $A_{\\ensuremath{\\mathbf{M}}_{\\alpha}}\/A_\\ensuremath{\\mathbf{G}}$, and is trivial on any \ntorsion in this quotient.  It follows that \nwe may fix a map\n$\\tilde{\\alpha}:\\ensuremath{\\frak{m}} \\rightarrow A_{\\ensuremath{\\mathbf{M}}_{\\alpha}}$ such that\n$\\left. \\lambda \\right|_{A_{\\ensuremath{\\mathbf{M}}_{\\alpha}}}$ is determined by $\\lambda \\circ \n\\tilde{\\alpha}.$  \nIn the function field case, we also denote by $\\tilde{\\alpha}$ the \ncomposition of $\\tilde{\\alpha}$ with  $|\\cdot |_{v_0}^{-1}:q^{\\ensuremath{{\\mathbb Z}}}\n\\rightarrow \\frak{w}^\\ensuremath{{\\mathbb Z}}.$\nThen $\\lambda \\circ \\ensuremath{\\tilde{\\alpha}}$ is a continuous homomorphism of a subgroup of \n$\\ensuremath{{\\mathbb R}}_+^{\\times}$ to $\\ensuremath{{\\mathbb R}}_+^{\\times}$, so \nwe may define $\\langle\\tilde{\\alpha},\\lambda\\rangle$ to be the \nunique real number such that\n$$\\lambda\\circ\\tilde{\\alpha}(y)=y^{\\langle\\tilde{\\alpha},\\lambda\\rangle}\n.$$\nReplacing $\\tilde{\\alpha}$ by the map $y\\mapsto \\tilde{\\alpha}(y^{-1})$ \nif necessary, we may add the stipulation that \n$\\langle \\tilde{\\alpha},\\alpha\\rangle >0.$\nWe will also need to refer to the Siegel set $S$, defined in section I.2.1 of \n\\cite{mw}, using a compact subset $\\omega$ of $\\ensuremath{\\mathbf{P}}_0.$\n\n{\\bf Definition:  } Let us say that a map $\\Lambda : \\ensuremath{{\\mathbb C}}\\rightarrow X_M^{\\ensuremath{\\mathbf{G}}}$ is \n\\emph{elementary} if it is holomorphic and the restriction to $\\ensuremath{{\\mathbb R}}$ is an \naffine map into the real vector space $\\mbox{Re} X_M^\\ensuremath{\\mathbf{G}}.$\n\nThe constant term of $E(\\lambda\\phi_\\pi,\\lambda\\otimes\\pi)$ along $P_\\alpha$\n(see I.2.6)\nis given in terms of Eisenstein series $E^{M_{\\alpha}},$ \ndefined analogously, with $M_{\\alpha}$ replacing $G$, and\nintertwining operators $M(w,\\pi)$ defined in  II.1.6:\n\\begin{equation}\\label{ctfrommw}\nE_{P_{\\alpha}}(\\lambda\\phi_{\\pi},\\lambda\\otimes\\pi)=\n\\sum_{w\\in W(M,M_{\\alpha})}E^{M_{\\alpha}}\n(M(w,\\lambda\\otimes \\pi)\\lambda\\phi_{\\pi},w(\\lambda \\otimes \\pi)).\n\\end{equation}\n(See II.1.7).\n\nSuppose that $E$ has a singularity along a root \nhyperplane $H$, associated to a root $\\theta$ as in IV.1.6.  \nOur theorem applies only to the case when $H_\\ensuremath{{\\mathbb R}}:=H\\cap \\mbox{Re} X_M^\\ensuremath{\\mathbf{G}}$ \nis non-empty.\nIn this case, $H_\\ensuremath{{\\mathbb R}}$ will be of the form\n$$\\{\\lambda \\in \\mbox{Re} X_M^{\\ensuremath{\\mathbf{G}}}: \\langle \\lambda, \\check{\\theta} \\rangle=c_H\\},$$ \nfor some $c_H\\in \\ensuremath{{\\mathbb R}}$.  See\nIV.1.6 and I.1.11.  \nHere $\\check{\\theta}$ may be taken to be the coroot associated to a positive\nroot $\\theta,$ or its projection to the dual of $\\mbox{Re} X_M^\\ensuremath{\\mathbf{G}}$:  the set $H$ is\nthe same either way.\nWe say that $\\lambda\\in H$ is \\emph{generic for $\\alpha$} \nif $\\lambda$ does not lie in \nany other hyperplane along which $E$ has a singularity, and\n$\\langle\\tilde{\\alpha},w_1\\lambda\\rangle = \n\\langle\\tilde{\\alpha},w_2\\lambda\\rangle$ for  $w_1,w_2\\in W(M,M_{\\alpha})$\niff $w_1=w_2$.\nIt follows that for $W(M,M_\\alpha)$ nonempty, and $\\lambda$ generic \nfor $\\alpha,$\nthere is a unique \n$\\ensuremath{w_{\\mbox{\\tiny max}}}(\\lambda,\\alpha)$ \nsuch that $\\langle\\tilde{\\alpha},w\\lambda\\rangle$ is maximal. \n\n\\subsection{Statement of Main Result}\n\\label{statement}\n\n\\begin{thm}\\label{mainthm}\nFix a root hyperplane $H$ along which the Eisenstein series is singular, \na root $\\alpha$,\nan elementary map $\\Lambda$ such that $\\Lambda(\\ensuremath{{\\mathbb C}})\\cap H=\\Lambda(0)$ is\ngeneric for $\\alpha$,  and elements $p\\in \\omega, k\\in \\ensuremath{\\mathbf{K}}.$ \nLet $$E(y,s)=E(\\Lambda(s)\\phi_{\\pi}, \\Lambda(s)\\otimes \\pi)\n(p\\tilde{\\alpha}(y)k).$$\nand for each $w\\in W(M,M_\\alpha),$ let \n$$E_w(y,s)=E^{M_\\alpha}(\nM(w,\\Lambda(s)\\otimes \\pi)\\Lambda(s)\\phi_{\\pi},w( \\Lambda(s)\\otimes \\pi))\n(p\\tilde{\\alpha}(y)k).$$\nWe say that $\\beta$ is an $\\epsilon$-Siegel zero of $E(y,\\sigma)$ if\n$\\beta \\in (-\\epsilon (\\log y)^{-1},\\epsilon (\\log y)^{-1})$ and\n$E(y,\\beta)=0.$\nLet $\\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk)=\\{w\\in W(M,M\\alpha): E_w(1,s) \\mbox{ has a \npole at }s=0\\}.$\nIf $H,\\alpha,p,k,$ and $\\Lambda$ satisfy\n\\begin{enumerate}\n\\item{The set $W(M,M_\\alpha)$ is nonempty,\nand $\\ensuremath{w_{\\mbox{\\tiny max}}}(\\Lambda(0),\\alpha)\\notin \\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk),$}\n\\item{$E_{\\ensuremath{w_{\\mbox{\\tiny max}}}(\\Lambda(0),\\alpha)}(1,0)\\neq 0,$}\n\\item{$E(y,s)$ has a simple pole at $s=0.$}\n\\end{enumerate}\nthen we have the following\n\\begin{description}\n\\item[A)]{ \nIf $\\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk)\\neq \\emptyset$, \nlet\n$\\ensuremath{w_{\\mbox{\\tiny ms}}}$  be the unique element of $\\ensuremath{W_{\\mbox{\\tiny sng}}}$ such that \n$\\langle \\tilde{\\alpha},\\ensuremath{w_{\\mbox{\\tiny ms}}}\\Lambda(0) \\rangle$ is maximal \n(among elements of \\ensuremath{W_{\\mbox{\\tiny sng}}}), and denote $\\ensuremath{w_{\\mbox{\\tiny max}}}(\\Lambda(0),\\alpha)$\nmore briefly by $\\ensuremath{w_{\\mbox{\\tiny max}}}.$  Then\n\\begin{description}\n\\item[i)]{There exists $Y>0$ \n(dependent on $\\epsilon,H,\\alpha,p,k,\\Lambda,$ and $\\phi_\\pi$), \nsuch that for all $y>Y$, $E(y,\\sigma)$\nhas an $\\epsilon$-Siegel zero.\n}\n\\item[ii)]{Let $\\beta:(Y,\\infty)\\rightarrow \\ensuremath{{\\mathbb R}}$ or $q^\\ensuremath{{\\mathbb Z}}\\cap\n(Y,\\infty)\\rightarrow \\ensuremath{{\\mathbb R}}$ in the function field case,\nbe any function such that for\neach $y$, $\\beta(y)$ is an $\\epsilon$-Siegel zero of $E(y,\\sigma).$  \nThen\n$\\beta(y) \\sim e y^{-\\langle \\ensuremath{\\tilde{\\alpha}}, \\ensuremath{w_{\\mbox{\\tiny max}}} \\Lambda(0)-\\ensuremath{w_{\\mbox{\\tiny ms}}} \\Lambda(0) \\rangle},$\nwhere \n$$e=\n \\frac{\\chi_\\pi(\\ensuremath{w_{\\mbox{\\tiny ms}}}^{-1}\\ensuremath{\\tilde{\\alpha}}(y)\\ensuremath{w_{\\mbox{\\tiny ms}}})\n\\left({{\\mbox{Res}}\\atop{s =0}}E^{M_\\alpha}(M(\\ensuremath{w_{\\mbox{\\tiny ms}}},\\Lambda(s)\\otimes \\pi)\\Lambda(s)\n\\phi_\\pi,\\ensuremath{w_{\\mbox{\\tiny ms}}}(\\Lambda(s)\\otimes \\pi))(pk)\\right)\n}{\\chi_\\pi(\\ensuremath{w_{\\mbox{\\tiny max}}}^{-1}\\ensuremath{\\tilde{\\alpha}}(y)\\ensuremath{w_{\\mbox{\\tiny max}}})\nE^{M_\\alpha}(M(\\ensuremath{w_{\\mbox{\\tiny max}}},\\Lambda(0)\\otimes \\pi)\\Lambda(0)\n\\phi_\\pi,\\ensuremath{w_{\\mbox{\\tiny max}}}(\\Lambda(0)\\otimes \\pi))(pk)\n}.\n$$\n}\n\\end{description}}\n\\item[B)]{\nIf $\\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk)=\\emptyset$,\nthen the conclusion is \nweaker:\nif\n$y>Y$ then either $E(y,\\sigma)$ has an $\\epsilon$-Siegel zero, or\n$E(y,\\sigma)$ is holomorphic at zero.\nIf $k$ is a function field, then it is always the latter.}\n\\end{description}\n\\end{thm}\nWe will also prove the following lemma, relevant to the problem \nof choosing $\\alpha,\\Lambda,p$ and $k$ so that conditions 1-3 above \nare satisfied.  \n\\begin{lem}\\label{condlem}\nFor $\\alpha \\in \\Delta_0$ such that $\\alpha \\notin R^+(T_0,M)$ and \n$\\theta\\notin R^+(T_0,M_\\alpha)$, we have\n\\begin{enumerate}\n\\item{the identity element, $1$, is in $W(M,M_\\alpha),$}\n\\item{for all $p,k$ and $\\Lambda$, $1 \\notin \\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk),$}\n\\item{if $c_H>0$, then \n$\\{\\lambda \\in H_\\ensuremath{{\\mathbb R}},\\mbox{ generic for }\\alpha: \\ensuremath{w_{\\mbox{\\tiny max}}}(\\lambda,\\alpha)=1\\}$ is a \nnonempty open subset of $H_\\ensuremath{{\\mathbb R}}$.}\n\\end{enumerate} \n\\end{lem}\nIt is also known (IV.1.11(c)) \nthat when $c_H>0,$ the singularity along $H$ is without\nmultiplicity.  For each $p,k$ the restriction of $E_1$ and the \ncontinuation of $h_H(\\lambda)E$ are meromorphic functions of $\\lambda\n\\in H$.  \nAs we see in an example below, either or both may be trivial \n(i.e., zero for all $\\lambda$) for \ncertain values of $p,k$.  But, for any $p,k$ such that both are \nnontrivial, $\\Lambda$ may\nthen be chosen so that all three conditions are satisfied.  \n\n\\section{ Basic Analytic Result}\n\\label{bar}\nOur approach to proving the existence of Siegel zeros of Eisenstein series is\nas follows.  First, we show that any function that can be written in a \ncertain form will have a zero on the real axis very close to its pole. Then, \nwe show that our Eisenstein series can always be put into that form.  \nThe lengthy definition of the function $F(y,s)$ in the following lemma should,\ntherefore, be taken as the ``mold'' into which we will fit the Eisenstein \nseries.  As we see here, any function which fits into this mold will have\nSiegel zeros.\\label{secwitlem}\n\n\\begin{lem}\\label{mainlemma}\nFix real numbers $a, b, c,$ and $d,$ such that \n$b > d$.  \nLet $\\sigma$ be a real variable, \nwhich we may think of as restricted to a small neighborhood of $0$ \nand\nlet $y$ be another real variable, which we think of as positive and large,\nkeeping in mind that when we apply this Lemma to the function field case, \n$y$ will only range over $q^{\\ensuremath{{\\mathbb Z}}}.$  Let \n$A(\\sigma)$ and $C(\\sigma)$ be two real valued \nfunctions that are both continuous and nonvanishing\nfor $\\sigma$ in a neighborhood of $0,$ \nand \nlet  $B(y,\\sigma)$ and $D(y,\\sigma)$ be two more real valued functions, \nwhich are \nboth continuous in $\\sigma$, and such that $B(y,\\sigma)y^{-(a\\sigma+b)}$, and \n$D(y,\\sigma)y^{-(c\\sigma+d)}$ tend to zero as $y$ tends to $\\infty$ for all values \nof $\\sigma$ in some \nneighborhood of $0$, and that convergence is uniform as $\\sigma$ ranges\nover this neighborhood.  \n\nDefine:\n$$F(y,\\sigma)=(A(\\sigma)y^{a\\sigma+b}+B(y,\\sigma))+\\frac{1}{\\sigma}\n(y^{c\\sigma+d}C(\\sigma)+D(y,\\sigma)).$$\n\nThen we have\n\\begin{description}\\item[i)]{\n For every $\\epsilon>0,$ there exists $Y(\\epsilon)>0,$ \nsuch that if $y>Y(\\epsilon),$ \nthen $F(y,\\sigma)$ has a zero \nin the interval $(-\\epsilon (\\log y)^{-1},\\epsilon (\\log y)^{-1})$.}\n\\item[ii)]{\nNow fix an $\\epsilon >0$, and \ntake $\\beta: (Y(\\epsilon),\\infty)\\rightarrow\\ensuremath{{\\mathbb R}}$ such that for each $y$ \nwe have \n $\\beta(y)\\in (-\\epsilon (\\log y)^{-1},\\epsilon (\\log y)^{-1}),$ and\n$F(\\beta(y),y)=0$.  For any such $\\beta$, \n$$ \n-\\beta(y) \\sim \\frac{C(0)}{y^{(b-d)}A(0)} \\mbox { as }\ny \\rightarrow \\infty.$$} \\end{description} \n\\end{lem}\n\n{\\bf Proof:  }  \nBy replacing \n$F(y,\\sigma)$ with \n$y^{-(c\\sigma+d)}F(y,\\sigma)$, which has the same zeros, we may assume $c=d=0.$\nBy considering the four functions $\\pm F(y,\\pm \\sigma)$ we may assume that\n$A(0)$ and $C(0)$ are positive.  \n\nNext, choose $\\delta, Y_1, m, M$ such that \n\\begin{equation} \\label{yano} \ny>Y_1, |\\sigma|<\\delta \\Rightarrow \\left\\{\\begin{array}{l}\n0<m<A(\\sigma)<M,\\\\0<m<C(\\sigma)<M, \\\\ |D(y,\\sigma)|<\\frac{m}{2}, \\\\ \n|B(y,\\sigma)|<\\frac{m}{2} y^{a\\sigma+b}.\n\\end{array}\\right.\n\\end{equation}\nThen, for each $y>Y_1,$ the function\n$A(\\sigma)y^{a\\sigma+b}+B(y,\\sigma)$ is bounded on \n$|\\sigma|<\\delta,$\nwhile $C(\\sigma)+D(y,\\sigma)$ is bounded away from zero.\nHence, for\nevery such $y$, there is a neighborhood of the form $(-l,0)$ on which \n$F(y,\\sigma)<0$.\nNow suppose that $y>Y_1$, and \n$\\epsilon (\\log y)^{-1} <\\delta.$  Then the bounds (\\ref{yano}) are valid at\n$\\sigma=-\\epsilon (\\log y)^{-1},$ yielding\n$$F(y,-\\epsilon (\\log y)^{-1})>\\frac{m}{2}e^{-a\\epsilon} y^{b}\n-\\epsilon^{-1} (M+\\frac{m}{2}) \\log y.$$ \nClearly, if we fix an $\\epsilon >0$, we may first choose $Y_2\\geq Y_1$ such\nthat  \nthe above is valid for all $y>Y_2$, and then\n choose $Y_3$ such that if $y>Y_3,$ the right side is\npositive.  We then let $Y=\\max(Y_2,Y_3)$, \nand the first assertion is proved. \nMoreover, if we fix and $\\epsilon'<\\epsilon$, we can choose $Y_2'$ \nsuch that for every $y>Y_2', \\epsilon'<\\varepsilon<\\epsilon,$ $\\varepsilon \n(\\log y)^{-1} <\\delta,$ and then choose $Y_3'$ such that for\n$y>Y_3', \\epsilon'<\\varepsilon<\\epsilon,$ \n$$\\frac{m}{2}e^{-a\\varepsilon} y^{b}\n-\\varepsilon^{-1} (M+\\frac{m}{2}) \\log y >0.$$  \nIt follows that any $\\beta(y),$ defined as above \\emph{with respect to}\n$\\epsilon$ satisfies\n$$\\beta(y) \\in (-\\epsilon'(\\log y)^{-1},0) \\mbox { for }y>Y'.$$\nSince this works for any $\\epsilon',$ we have\n$$\\lim_{y \\rightarrow \\infty} y^{-\\beta(y)}=1$$\nindependently of the choice of $\\epsilon,$ and independently of any possible \nchoice of $\\beta(y).$\nTo prove \\emph{ii)}, we note that \n$F(y,\\beta(y))=0$ iff  $$A(\\beta(y))y^{a \\beta(y) +b} +B(y,\\beta(y))=\n\\frac{-1}{\\beta(y)}(C(\\beta(y))+D(y,\\beta(y))).$$  We have seen that \nfor $\\beta(y)$ near $0$ and $y$ large the left side is nonzero, so we may \nput this into the form\n$$\\beta(y)=-\\frac{C(\\beta(y))+D(y,\\beta(y))}{A(\\beta(y))y^{a \\beta(y) +b} +B(y,\\beta(y))}.$$\nSo\n\\begin{eqnarray*}&&\n\\hskip -.2in \\lim_{y\\rightarrow \\infty}\n\\frac{C(0)\/(A(0)y^{b})}{-\\beta(y)}\n\\\\&&=\\lim_{y\\rightarrow \\infty} \\frac{C(0)}{C(\\beta(y))+D(y,\\beta(y))}\n\\frac{\\left(A(\\beta(y))y^{a\\beta(y)+b}+B(y,\\beta(y))\\right)y^{-b}}{A(0)}.\n\\end{eqnarray*}\nThe second limit is evidently 1, which proves the asymptotic formula.   \\begin{flushright}$\\blacksquare$\\end{flushright}\n\n{\\bf Remarks 1.} If, on the other hand, $F$ is in the same form, with \n$b < d,$ but all\nthe other assumptions are the same, then similar arguments show that \n$F(y,\\sigma)$ will be nonvanishing \nfor $\\sigma$ in a neighborhood of $0$ and $y$ sufficiently large.\n\n{\\bf 2.} If $F$ is as above, but $C(0)=0,$ then the asymptotic\nformula is no longer correct, but the Siegel zero still exists\nfor all values of $y>Y$ such that $D(y,\\sigma)\\neq 0.$ \n\n\\subsection{Necessary Fact}\n\\label{necessaryfacts}\nWe will need one more well known fact from the theory of Eisenstein series: \nthe Eisenstein series is well approximated by its constant term.  \n\\begin{prop}\n\\label{approx}\nIf $k$ is a function field, then there is a constant $c$ such that\n$E(\\lambda\\phi_{\\pi},\\pi\\otimes\\lambda)(g)-\nE_{P_{\\alpha}}(\\lambda\\phi_{\\pi},\\pi\\otimes\\lambda)(g)=0$,\nwhenever $g\\in S$ and $m_{P_0}(g)^{\\alpha}>c$, where $S$ is a Siegel domain\nas in I.2.1.  \nIn particular, \nfor $\\Lambda,p,k,\\ensuremath{\\tilde{\\alpha}}$ as in the main theorem,\n$$(E-E_{P_{\\alpha}})(\\Lambda(\\sigma)\\phi_{\\pi},\\pi\\otimes\\Lambda(\\sigma))\n(p\\tilde{\\alpha}(y)k)=0,$$\nfor $p\\in \\omega, k\\in K$ and $y$ sufficiently large.\nIf $k$ is a number field, then for $\\Lambda,p,k,\\ensuremath{\\tilde{\\alpha}}$ as in the main theorem,\n$$\n\\sigma\\left(\n(E-E_{P_{\\alpha}})(\\Lambda(\\sigma)\\phi_{\\pi},\\Lambda(\\sigma)\n\\otimes\\pi)(p\\tilde{\\alpha}(y)k)\n\\right)\n$$ \nis rapidly decreasing as a function of $y$ for $p\\in \\ensuremath{\\mathbf{G}}^1\\cap \\omega,k\\in K$, \nuniformly for $\\sigma$ in a neighborhood of zero.\n\\end{prop}\n{\\bf Proof:  }\nThe function field case is immediate from Lemma I.2.7.  \nFor the number field case we use Lemma I.2.10.  \nLemma I.4.4, in conjunction with I.2.5 yields the bounds required by\nthe hypotheses of Lemma I.2.10.\n\\hfill  $\\blacksquare$\n\\section{Proofs}\n\\subsubsection{Proof of Theorem \\ref{mainthm}}\nAs $H,\\alpha,\\Lambda,p$ and $k$ are fixed, we suppress them from the\nnotation, denoting $\\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,\\Lambda,pk)$ by $\\ensuremath{W_{\\mbox{\\tiny sng}}}$, etc.\nThe idea is to fit $E(y,\\sigma)$ into the mold described in section 3.\nIn the new notation we have introduced, equation \\eqref{ctfrommw} reads\n\\begin{equation}\\label{decomp}\nE_{P_{\\alpha}}(y,\\sigma)=\\sum_{w\\in W(M,M_{\\alpha})}E_w(y,\\sigma).\n\\end{equation}\nWe observe that\n\\begin{equation}\\label{athing}E_w(y,\\sigma)=\n\\chi_\\pi(w^{-1}\\tilde{\\alpha}(y)w)\ny^{\\langle\\tilde{\\alpha}, \\rho_{P_\\alpha}+w \\Lambda(\\sigma)\\rangle}\nE_w(1,\\sigma).\n\\end{equation}\nNow, $\\chi_\\pi$ is both real-valued and unitary.  When $k$ is a number\nfield, \n$\\chi_\\pi(w^{-1}\\tilde{\\alpha}(y)w)$\n is trivial, but when $k$ is a function field, \n$\\chi_\\pi(w^{-1}\\tilde{\\alpha}(q^n)w)$ may equal $-1$ for some $w$ when $n$ is\nodd.  In this case, one may consider restrictions to odd and even $n$ \nseperately, and combine the results.  We omit the details, and assume that\n$\\chi_\\pi(w^{-1}\\tilde{\\alpha}(y)w)$ is identically $1$, when $w$ is either\n$\\ensuremath{w_{\\mbox{\\tiny max}}}$ or $\\ensuremath{w_{\\mbox{\\tiny ms}}}.$\n\nFor each $w$, the map \n$\\sigma \\mapsto \\langle \\tilde{\\alpha},w \\Lambda(\\sigma) \\rangle$ \nis an affine map $\\ensuremath{{\\mathbb R}} \\rightarrow \\ensuremath{{\\mathbb R}}$, and so we may define \n$a,b,c,$ and $d$ by the conditions \n$\\langle \\tilde{\\alpha}, \\rho_{P_\\alpha}+\\ensuremath{w_{\\mbox{\\tiny max}}}\\Lambda(\\sigma)\\rangle\n=a\\sigma+b,$ and,\nif $\\ensuremath{W_{\\mbox{\\tiny sng}}}$ is nonempty,\n$\\langle \\tilde{\\alpha}, \\rho_{P_\\alpha}+\\ensuremath{w_{\\mbox{\\tiny ms}}}\\Lambda(\\sigma)\\rangle\n=c\\sigma+d.$\nIf $\\ensuremath{W_{\\mbox{\\tiny sng}}}$ is empty, we take $c=0$, and $d$ any real number less than \n$b$.\nWe take \n\\begin{eqnarray*}\nA(\\sigma)&=&E_{\\ensuremath{w_{\\mbox{\\tiny max}}}}(1,\\sigma),\\\\\nC(\\sigma)&=&\\sigma E_{w_{\\mbox{\\tiny ms}}}(1,\\sigma), \\mbox{ or }\n0 \\mbox{ if }\\ensuremath{W_{\\mbox{\\tiny sng}}}=\\emptyset,\\\\\nD(y,\\sigma)&=&\\sigma \\left(\n\\sum_{w \\in \\ensuremath{W_{\\mbox{\\tiny sng}}}-\\left\\{w_{\\mbox{\\tiny ms}}\\right\\}}E_w(y,\\sigma)\n+(E-E_{P_\\alpha})(y,\\sigma)\\right)\n,\\\\\nB(y,\\sigma)&=&\\sum_{w \\in W(M,M_{\\alpha})-\\ensuremath{W_{\\mbox{\\tiny sng}}}-\\left\\{\\ensuremath{w_{\\mbox{\\tiny max}}}\\right\\}}E_w(y,\\sigma),\n\\end{eqnarray*}\nWhere we extend $C$ and $D$ to $\\sigma=0$ by continuity.  \n\nWe check that all the hypotheses of the main lemma are satisfied.  \nAs, $\\Lambda(0)$ is generic, \nwe may choose an open ball containing it \nwhich intersects no other hyperplane along which $E$ is singular.  \nThe continuity of each function on the set of $\\sigma$ mapping into \nthis ball is clear.  The fact that all functions are real valued follows\nfrom the fact that $\\phi_\\pi$ is.  Condition 2 is precisely that \n$A(0)\\neq 0,$ and in the case when $\\ensuremath{W_{\\mbox{\\tiny sng}}} \\neq \\emptyset,$ the value \nof $C$ at $0$ is the residue of $E_{w_{\\mbox{\\tiny ms}}}(1,\\sigma)$\nat zero.  The bounds on $B$ and $D$ as $y \\rightarrow \\infty$ follow\neasily from \\eqref{athing} and Proposition \\ref{approx}.\\hfill  $\\blacksquare$\n\n\\subsubsection{Proof of Lemma \\ref{condlem}}\nThe first assertion is trivial.\nThe second and third follow from more general assertions, which we now\nprove.\nFor $w \\in W(M,M_\\alpha)$, let $W_{M_\\alpha}(wMw^{-1})$ be defined in the same\nway as $W(M),$ with $M_\\alpha$ replacing $G$ and $wMw^{-1}$ replacing $M$.\n\n\\begin{lem} \\label{hollemma}\n Let $H$ be a root hyperplane associated to a root $\\theta$,\nand fix $\\alpha.$\nIf there exist $p,k,\\Lambda$ such that \n$\\Lambda(0)\\in H$ is generic for $\\alpha$, and\n$w \\in \\ensuremath{W_{\\mbox{\\tiny sng}}}(H,\\alpha,p,k,\\Lambda)$, \nthen there exists $w'\\in W_{M_\\alpha}(wMw^{-1})$ such that \n$w'w\\theta<0.$\n\\end{lem}\n\n{\\bf Proof:  }  For some\n$w'\\in W_{M_\\alpha}(wMw^{-1}),$ write $w'w=s_{\\gamma_\\ell} \\dots\ns_{\\gamma_1}$, where for each $i$, $s_{\\gamma_i}$ is an ``elementary\nsymmetry'' as in section I.1.8.  See also section IV.4.1.  Let\n$w(j)=s_{\\gamma_j}\\dots s_{\\gamma_1}.$  Then\n$$M(w'w,\\pi \\otimes \\lambda)=M(s_{\\gamma_\\ell},w(\\ell-1)(\\pi \\otimes \\lambda))\n\\circ \\dots \\circ M(s_{\\gamma_1},\\pi\\otimes\\lambda).$$\nThe singularities of $M(s_\\gamma,\\tau \\otimes \\mu)$ are carried by a \nlocally finite set of hyperplanes associated to $\\gamma.$  Hence the \nsingularities of $M(s_{\\gamma_i},w(i-1)(\\pi\\otimes\\lambda))$ are carried by\na locally finite set of root hyperplanes associated to the root \n$w(i-1)^{-1}\\gamma_i,$ and, in general, the singularities of \n$M(w'w,\\pi\\otimes\\lambda)$ are carried by a locally finite set of root \nhyperplanes each of which is associated to a root $\\gamma$ such that \n$w'w\\gamma <0.$  \n\nSuppose that $w'w\\theta >0$ for all $w'\\in W_{M_\\alpha}(wMw^{-1}).$  \nThen there is a locally finite set of root hyperplanes, not containing\n$H$, that carries the singularities of $M(ww',\\pi \\otimes \\lambda)$ \nfor every $w'.$  Hence it carries the singularities of all the \ncuspidal components of all the constant terms of \n$E^{M_\\alpha}(M(w,\\pi\\otimes \\lambda)\\lambda \\phi_\\pi,w(\\pi\\otimes \\lambda))$, \n(equation \\eqref{ctfrommw} above, and  IV.1.9 (b)). By I.4.10 it\ncarries the singularities of\n$E^{M_\\alpha}(M(w,\\pi\\otimes \\lambda)\\lambda \\phi_\\pi,w(\\pi\\otimes \\lambda))$\nas well.  The result follows.\n\\hfill  $\\blacksquare$\n\nUnder the hypotheses of Lemma \\ref{condlem}, \n$w'\\theta >0$ for all $w'\\in W_{M_\\alpha}(M),$ and\n2 follows.  As for \n3, it's clear that the set in question is always open, and that it's \nempty iff $\\{\\lambda \\in H_\\ensuremath{{\\mathbb R}}:\\ensuremath{w_{\\mbox{\\tiny max}}}(\\lambda,\\alpha)=1\\}$ is.\nThus, we only need to prove that this latter set is nonempty.\n\n\\begin{lem}\nIf $c_H>0$, and $1\\in W(M,M_\\alpha)$, then there exists $\\lambda \\in H_\\ensuremath{{\\mathbb R}}$\nsuch that $\\ensuremath{w_{\\mbox{\\tiny max}}}(\\lambda,\\alpha)=1.$\n\\end{lem}\n{\\bf Proof:  } The linear functional on\n$\\Delta_0$ given by pairing with $\\tilde{\\alpha}$ corresponds to a \npositive multiple of the one given by pairing with the\nfundamental\ncoweight $\\hat{\\varpi}_{\\alpha}$, in the basis dual to $\\Delta_0.$\nGiven an element $\\lambda$ of $\\mbox{Re} X_{M_0}^\\ensuremath{\\mathbf{G}},$ which is \nirrational, i.e., has trivial kernel in the coweight lattice,\nwe can define\nnotions of $\\lambda$-positivity for coroots and $\\lambda$-dominance for\ncoweights.  If the definitions of positivity for coroots that come from\n$\\lambda$ and $\\Delta_0$ coincide, then the definitions of dominance \nfor coweights will as well.  Hence, whenever\n$$\\langle \\lambda ,\\check{\\gamma} \\rangle >0 \\hskip .5in \n \\forall \\gamma \\in \\Delta_0,$$\nand $\\lambda$ is irrational, \nwe have\n$$\\langle \\lambda, \\tilde{\\alpha} \\rangle \\geq \n\\langle w\\lambda ,\\tilde{\\alpha} \\rangle \n\\hskip .5in \\forall  w \\in W,$$\nwith equality only if $w^{-1}\\ensuremath{\\tilde{\\alpha}}=\\ensuremath{\\tilde{\\alpha}}.$\nNow project $\\lambda$ to $\\bar{\\lambda}\\in \\mbox{Re} X_M^\\ensuremath{\\mathbf{G}}.$  This projection \ncorresponds to restriction from $Z_{\\ensuremath{\\mathbf{M}}_0}$ to $Z_{\\ensuremath{\\mathbf{M}}}$ (see pp.7,11).  \nSince the image of $w^{-1}\\tilde{\\alpha}$ is contained in $Z_\\ensuremath{\\mathbf{M}}$ whenever \n$w\\in W(M,M_\\alpha)$, we find that \n$$\\langle \\bar{\\lambda},\\tilde{\\alpha}\\rangle \\geq \n\\langle w\\bar{\\lambda},\\tilde{\\alpha}\\rangle\n\\hskip .5in \\forall w\\in W(M,M_\\alpha),$$ \nwith equality only if $w=1.$  \nThus, for $\\bar{\\lambda}$ the projection of any $\\lambda$ as above, we have\n$\\ensuremath{w_{\\mbox{\\tiny max}}}(\\bar{\\lambda},\\alpha)=1.$  Now, let $\\gamma_\\theta$ be the root \nassociated to $\\theta$ as in section I.1.11 (so that $\\check{\\theta}$ is \ndefined as the projection of $\\check{\\gamma_\\theta}$ to $\\mbox{Re} \\frak{a}_M$).\nIt should be emphasized that $\\langle \\lambda,\\check{\\gamma_\\theta} \\rangle$\nand $\\langle \\bar{\\lambda},\\check{\\theta} \\rangle$ need not be equal.  \nHowever, $\\langle \\lambda, \\check{\\theta} \\rangle$ and \n$\\langle \\bar{\\lambda}, \\check{\\theta} \\rangle$ \\emph{are} always equal.  \nSo, it's enough to verify that for some irrational $\\lambda$, \nwith $\\langle \\lambda, \\check{\\gamma} \\rangle >0 \\forall \\gamma \\in \\Delta_0,$\nthe quantity\n$\\langle \\lambda, \\check{\\theta} \\rangle$ is also positive.  This is\nstraightforward, when $\\check{\\theta}$ is written in terms of the basis\n$\\Delta_0^\\vee$ of coroots $\\check{\\gamma}$,\nand $\\lambda$ in terms of the dual basis.\n\\hfill  $\\blacksquare$\n\n\\section{Examples}\n\\label{examples}\n\nLet us consider the example when $G=GL(n)$, the representation $\\pi$ is unramified, and we choose $\\phi_{\\pi}$ to be the spherical vector.  In this case, \nthe intertwining operators may be given explicitly in terms of automorphic $L$-functions, as in \\cite{ep}.  It follows that a pole of the Eisenstein series comes from either a zero or a pole of one of these $L$-functions.  \nThe zeroes are quite mysterious, but the poles of \nRankin-Selberg $L$-functions on $GL(n)$ have been completely determined by \nJacquet and Shalika \\cite{jsi},\\cite{jsii}, and referring back to the \n$GL(2)$ case, we see that it is these poles which provide the correct \nanalog of the pole of the $GL(2)$ Eisenstein series at 1. \n\nAs noted, the case $G=GL(2)$ is the simplest example of our theorem.  Let us consider two more which exhibit some of the features of the general case, but are not so complicated as to become unwieldy.  \n\\subsection{Example One:  $GL(4)$ over $\\ensuremath{{\\mathbb Q}}$}\n\nFirst let us consider the case $k=\\ensuremath{{\\mathbb Q}},$ and $\\ensuremath{\\mathbf{G}}=GL(4,\\ensuremath{{\\mathbb A}})$.  We take the \nusual Borel, consisting of all upper triangular elements, and let $P$ \nbe the standard maximal\nparabolic such that $M\\cong GL(2)\\times GL(2).$\n\nIn this case, $\\mbox{Re} X_M^{G}$ is one dimensional, consisting of all real powers\nof the modulus\n$$\\left(\\begin{smallmatrix} h_1 &*\\\\ & h_2\\end{smallmatrix}\\right)\n\\mapsto \\left|\\frac{\\det h_1}{\\det h_2}\\right|^2,$$\nso we identify it with $\\ensuremath{{\\mathbb R}}$.  In order to have good knowledge of the poles of the intertwining operators, we will want to make a suitable choice of $\\pi$ and $\\phi_{\\pi}$, which can also be described quite explicitly.\n\nLet $\\varphi$ be a real-valued Maass Hecke eigenform of level 1,\nright invariant by the center and the maximal compact, and define a \nfunction on $GL(4,\\ensuremath{{\\mathbb A}})$ by \n$$I\\left(\\left(\\begin{smallmatrix}h_1 &*\\\\&h_2\\end{smallmatrix}\\right) k\n,s,\\varphi\\right)=\n\\left|\\frac{\\det h_1}{\\det h_2}\\right|^{2s+1} \\varphi(h_1)\\varphi(h_2),$$\nfor $k$ now in the maximal compact of $GL(4,\\ensuremath{{\\mathbb A}}).$\nOur Eisenstein series is \n$$E(g,s,\\varphi)= \\sum_{\\gamma \\in P(\\ensuremath{{\\mathbb Q}})\\ensuremath{\\backslash}\nGL(4,\\ensuremath{{\\mathbb Q}})}I(\\gamma g, s,\\varphi).$$\nIn this case $W(M,M')=\\emptyset$, unless $M'=M,$ and\n$$E_P(g,s,\\varphi)=I(g,s,\\varphi)+\n\\frac{L(2s,\\varphi\\times\\varphi)}{L(2s+1,\\varphi\\times\\varphi)}\nI(g,-s,\\varphi).$$\nHere $L$ denotes the completed $L$ function-- including gamma factors.\n(This is essentially a special case of the computation in \\cite{ep}.)\nSince $\\mbox{Re} X_M^G$ is one dimensional, a codimension 1 subspace is a point.  \nThus our ``root hyperplane'' is just $s=\\frac{1}{2},$ or some zero of \n$L(2s+1,\\varphi\\times \\varphi).$  We consider the one at $\\frac{1}{2},$  \nand let\n$\\Lambda(\\sigma)=\\sigma+\\frac{1}{2},$ and $\\ensuremath{\\tilde{\\alpha}}(y)=diag(y,y,1,1),$ embedded\nat the infinite place.  Note that if $p=\\left(\\begin{smallmatrix} h_1&*\\\\\n&h_2\\end{smallmatrix}\\right)$ with $\\varphi(h_1)$ or $\\varphi(h_2)=0,$\nthen the constant term is zero for all $s$.  \nFor any other $p$,\n$\\ensuremath{W_{\\mbox{\\tiny sng}}}=\n\\left\\{\\left(\\begin{smallmatrix}&I\\\\ensuremath{\\mathcal{I}}&\\end{smallmatrix}\\right)\\right\\}.$\nFor $\\sigma$ positive, $\\ensuremath{w_{\\mbox{\\tiny max}}}=I,$  and the theorem applies.\n\nLet $A(\\sigma)=I(pk,\\sigma+\\frac{1}{2},\\varphi),\nC(\\sigma)=\\frac{L(2\\sigma+1,\\varphi\\times\\varphi)}\n{L(2\\sigma+2,\\varphi\\times\\varphi)}I(pk,-\\sigma-\\frac{1}{2},\\varphi)$ and \n$D(y,\\sigma)=E(p\\tilde{\\alpha}(y)k,\\sigma+\\frac{1}{2},\\varphi)-\nE_P(p\\tilde{\\alpha}(y)k,\\sigma+\\frac{1}{2},\\varphi).$  Let $a=4$, \n$b=4$, $c=-4$ and $d=0.$  The main analytic lemma applies.  \nThe asymptotic formula it produces is\n\n$$-\\beta \\sim \\frac{( \\varphi,\\varphi)}\n{2 L(2,\\varphi \\times \\varphi)}\n\\left|\\frac{\\det h_1}{\\det h_2}\\right|^{-2}\ny^{-4},$$\nwhere $(\\varphi,\\varphi)$ is the Petersson inner product of \n$\\varphi$ with itself.\n\\subsection{Example Two: $GL(3)$ over $\\ensuremath{{\\mathbb Q}}$}\n\nNow let us consider the case $k=\\ensuremath{{\\mathbb Q}}$, and $\\ensuremath{\\mathbf{G}}=GL(3,\\ensuremath{{\\mathbb A}})$.  We take $P=P_0$ to\nbe the Borel subgroup of upper triangular matrices and $M=T_0$ to be the\ntorus of diagonal matrices.  The associated set of simple positive roots\nis $\\{\\alpha,\\theta\\}$, defined by\n$$\\alpha\\left(\\left(\\begin{smallmatrix}t_1&&\\\\&t_2&\\\\&&t_3\\end{smallmatrix}\\right)\\right)\n=\\frac{t_1}{t_2}\\hskip .5in\n\\theta\\left(\\left(\\begin{smallmatrix}t_1&&\\\\&t_2&\\\\&&t_3\\end{smallmatrix}\\right)\\right)\n=\\frac{t_2}{t_3}.\n$$\nWe take $\\pi$ to be the trivial representation, and $\\phi_\\pi$ to be the\nconstant function 1.  We parametrize \n$\\mbox{Re} X_M^{\\ensuremath{\\mathbf{G}}}$, by associating the map\n$$\\ensuremath{\\mathcal{I}}_{\\lambda}\\left[\\left(\\begin{smallmatrix}t_1&&\\\\&t_2&\\\\&&t_3\\end{smallmatrix}\\right)\\right]=|t_1|^{\\lambda_1+1}|t_2|^{\\lambda_2}|t_3|^{\\lambda_3-1},$$ to the triple $\\lambda=(\\lambda_1,\\lambda_2,\\lambda_3) \\in \\ensuremath{{\\mathbb R}}^3$\nsuch that \n$\\lambda_1+\\lambda_2+\\lambda_3=0.$\n \nWe work with $\\alpha$, and define the map $\\tilde{\\alpha}$ by\n$\\tilde{\\alpha}(y)=\n\\left(\\begin{smallmatrix}y&&\\\\&1&\\\\&&1\\end{smallmatrix}\\right),$ embedded\nat the infinite place.\nThe Weyl group $W$ of $G$ may be identified with the group of permutation \nmatrices.  Then \n$$\nW(M,M_{\\alpha})=\\left\\{I,\n\\left(\\begin{smallmatrix}&1&\\\\1&&\\\\&&1\\end{smallmatrix}\\right)\n\\left(\\begin{smallmatrix}&&1\\\\1&&\\\\&1&\\end{smallmatrix}\\right)\n\\right\\}.$$\nWe compute the intertwining operators as usual, by reducing from the \ngeneral case to the relative rank one case, i.e., to a Levi isomorphic to $GL(2).$  As on $GL(2)$, each root contributes a ratio of Riemann\nzeta functions.  Specifically, if $\\zeta^*(s)=\\pi^{-\\frac{s}{2}}\\Gamma(\\frac{s}{2})\\zeta(s)$ is the ``completed'' Riemann zeta function, then we have\n\\begin{eqnarray*}\nM\\left(\n\\left(\\begin{smallmatrix}&1&\\\\1&&\\\\&&1\\end{smallmatrix}\\right),\n\\ensuremath{\\mathcal{I}}_{\\lambda}\\right)\\ensuremath{\\mathcal{I}}_{\\lambda}&=&\n\\frac{\\zeta^*(\\lambda_1-\\lambda_2)}{\\zeta^*(\\lambda_1-\\lambda_2+1)}\n\\ensuremath{\\mathcal{I}}_{(\\lambda_2,\\lambda_1,\\lambda_3)} \\\\\nM\\left(\n\\left(\\begin{smallmatrix}&&1\\\\1&&\\\\&1&\\end{smallmatrix}\\right),\n\\ensuremath{\\mathcal{I}}_{\\lambda}\\right)\\ensuremath{\\mathcal{I}}_{\\lambda}&=&\\frac{\\zeta^*(\\lambda_2-\\lambda_3)}{\\zeta^*(\\lambda_2-\\lambda_3+1)}\\frac{\\zeta^*(\\lambda_1-\\lambda_3)}{\\zeta^*(\\lambda_1-\\lambda_3+1)}\n\\ensuremath{\\mathcal{I}}_{(\\lambda_3,\\lambda_1,\\lambda_2)}\n\\end{eqnarray*}\nThe value of $E^{M_{\\alpha}}$ at a point\n$$g=\\left(\\begin{smallmatrix}1&x_2&x_3\\\\&1&x_1\\\\&&1\\end{smallmatrix}\\right)\\left(\\begin{smallmatrix}y_1y_2&&\\\\&y_1&\\\\&&1\\end{smallmatrix}\\right) k\n\\in GL(3,\\ensuremath{{\\mathbb R}})$$\nmay be given in terms of the Eisenstein series \n$$e(\\tau,s)=\\sum_{{(c,d)=1}\\atop{c>0}}\\frac{y^s}{|c\\tau+d|^{2s}}$$\nfor $\\tau=x+iy$ \nin the upper half\nplane.  Specifically, \nif we let $\\tau_1=x_1+iy_1$, then we find that\n$$\nE^{M_{\\alpha}}(\\ensuremath{\\mathcal{I}}_{\\lambda},\\ensuremath{\\mathcal{I}}_{\\lambda}\\otimes 1)(g)= \n(y_2\\sqrt{y_1})^{\\lambda_1+1} e(\\tau_1,\n\\frac{\\lambda_2-\\lambda_3+1}{2}).\n$$\nIt follows that the value of $E_{P_{\\alpha}}(\\ensuremath{\\mathcal{I}}_{\\lambda})$ is given by \n\\begin{eqnarray*}\n&&(y_2\\sqrt{y_1})^{\\lambda_1+1} e(\\tau_1,\n\\frac{\\lambda_2-\\lambda_3+1}{2})\\\\&&\n+(y_2\\sqrt{y_1})^{\\lambda_2+1} \\frac{\\zeta^*(\\lambda_1-\\lambda_2)}{\\zeta^*(\\lambda_1-\\lambda_2+1)}e(\\tau_1,\\frac{\\lambda_1-\\lambda_3+1}{2})\\\\&&\n+(y_2\\sqrt{y_1})^{\\lambda_3+1} \\frac{\\zeta^*(\\lambda_2-\\lambda_3)}{\\zeta^*(\\lambda_2-\\lambda_3+1)}\n\\frac{\\zeta^*(\\lambda_1-\\lambda_3)}{\\zeta^*(\\lambda_1-\\lambda_3+1)}e(\\tau_1,\\frac{\\lambda_1-\\lambda_2+1}{2}).\\end{eqnarray*}\nIf we multiply $E(\\ensuremath{\\mathcal{I}}_{\\lambda},\\ensuremath{\\mathcal{I}}_{\\lambda}\\otimes 1)(g)$ by\n$$\\zeta^*(\\lambda_1-\\lambda_2+1)\\zeta^*(\\lambda_2-\\lambda_3+1)\\zeta^*(\\lambda_1-\\lambda_3+1)$$ the resulting\nfunction is essentially the function $G_{\\nu_1,\\nu_2}$ appearing in \nBump \\cite{yellowbump}.  It is invariant under the six permutations of $\\lambda_1,\\lambda_2$, and $\\lambda_3.$\nIt's poles are along the lines $\\lambda_i-\\lambda_j=1,$ and are permuted \ntransitively by \nits functional equations.  Let us therefore restrict our attention to the\nplane $\\lambda_2-\\lambda_3=1.$\nNow, it's clear that of the three terms that make up $E_{P_{\\alpha}}$, the \nfirst and third will be singular along this hyperplane, while the second will\nnot.  It is thus clear that\n$\\{\\lambda \\mbox{ generic for }\\alpha:\\ensuremath{w_{\\mbox{\\tiny max}}}(\\lambda,\\alpha)\n=\\left(\\begin{smallmatrix}&1&\\\\1&&\\\\&&1\n\\end{smallmatrix}\\right)\\}$ \n is the union of\n$\\Omega_1:=\\{\\lambda\\in H: \\lambda_2>\\lambda_1>\\lambda_3\\}$ and\n$\\Omega_2:=\\{\\lambda\\in H: \\lambda_1<\\lambda_3,\\lambda_3\\neq \\lambda_1+1\\}$. \nThe function $e(\\tau,s)$ does not vanish identically in $s$ for any $\\tau,$\nand its residue is never zero.  Thus, we may use any $p,k.$  \nWe fix \n$\\lambda^0 \\in \\Omega_1\\cup \\Omega_2,$ such that \n$e(\\tau_1,\\frac{\\lambda_1^0-\\lambda_3^0+1}{2})\\neq 0$, and \n $\\lambda^1$ in the hyperplane $\\lambda_2-\\lambda_3=0$ and define $\\Lambda$\nby $\\Lambda(\\sigma)=(1+\\sigma)\\lambda^0-\\sigma \\lambda^1.$  There is \nno real loss of generality, since an arbitrary $\\Lambda$ may be obtained \nfrom this one by a simple scaling.  But this ensures that\n$\\Lambda(\\sigma)_2-\\Lambda(\\sigma)_3=\\sigma+1,$ so that the residue of\n$\\zeta^*(\\Lambda(\\sigma)_2-\\Lambda(\\sigma)_3)$ at zero is 1, while that of\n$e(\\tau, \\frac{\\Lambda(\\sigma)_2-\\Lambda(\\sigma)_3+1}{2})$ is \n$\\frac{1}{\\zeta^*(2)}=\\frac{6}{\\pi}$.\nIf $\\lambda^0\\in \\Omega_1$,  then\n$\\ensuremath{w_{\\mbox{\\tiny ms}}}=I$.  In this case, the asymptotic formula\nreads\n$$\\beta \\sim \\frac{6\\zeta^*(\\lambda_1^0-\\lambda_2^0+1)}\n{\\pi \\zeta^*(\\lambda_1^0-\\lambda_2^0)e(\\tau_1,\\frac{\\lambda_1^0-\\lambda_3^0+1}{2})(y_2\\sqrt{y_1})^{\\lambda_2^0-\\lambda_1^0}}.$$\nOn the other hand, if $\\lambda^0\\in \\Omega_\\alpha^2$, then \n$\\ensuremath{w_{\\mbox{\\tiny ms}}}=\\left(\\begin{smallmatrix}&&1\\\\1&&\\\\&1&\\end{smallmatrix}\\right),$\nand the asymptotic formula is \n$$\\beta \\sim \\frac{6\\zeta^*(\\lambda_1^0-\\lambda_2^0+1)\\zeta^*(\\lambda_1^0-\\lambda_3^0)e(\\tau_1,\\frac{\\lambda_1^0-\\lambda_2^0+1}{2})}\n{\\pi\\zeta^*(\\lambda_1^0-\\lambda_2^0)\\zeta^*(\\lambda_1^0-\\lambda_3^0+1)e(\\tau_1,\\frac{\\lambda_1^0-\\lambda_3^0+1}{2})(y_2\\sqrt{y_1})}.$$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe spectral crowding problem and the closely related problem of leakage to \nhigher-energy states have pestered the field of superconducting quantum computing \nsince its very inception \\cite{Fazio1999}. For almost twenty years a tremendous amount \nof effort has been made to resolve both of these problems \n(see., e.\\ g., \n\\cite{JM2003, Montangero2007, Safaei2008, Rebentrost2009, Safaei2009, Gambeta2009,\nForney2010, Stojanovic2012, Schutjens2013, Motzoi2013, Martinis2016}).\nThe effort was certainly worth making, because every time an operation lifting \nthe system off its ground state is performed (such as, e.\\ g., a strong Rabi pulse), leakage \nout of the computational subspace develops, which results in the degradation of the \ngate fidelity. One may argue that even if the other problems facing quantum computing, \nsuch as scalability and decoherence, are ever successfully resolved, the crowding \nand the leakage problems would still be with us, simply as a matter of principle. \nAs long as one insists on using Josephson junctions, one will have to deal with the matrix \nelements that kick the system all over the spectra of the corresponding \nanharmonic potentials.\n\nAs an example, consider the Greenberger-Horne-Zeilinger (GHZ) state \nprotocol proposed in Ref.\\ \\cite{AGJM2008} for a\n{\\it fully connected network of three identical} Josephson phase qubits. \nIn the rotating wave approximation (RWA) the system is described by the Hamiltonian, \n\\begin{align}\nH &= \n\\vec{\\Omega}_1 \\cdot \\vec{\\sigma}_1 + \\vec{\\Omega}_2\\cdot\\vec{\\sigma}_2\n+\\vec{\\Omega}_3 \\cdot \\vec{\\sigma}_3\n\\nonumber \\\\\n& \\quad \n+ \\frac{1}{2}\\bigg[g\\left(\\sigma_x^1 \\sigma_x^2+\\sigma_y^1 \\sigma_y^2\\right)\n\t\t\t\t+ \\tilde{g}\\sigma_z^1\\sigma_z^2\n\\nonumber \\\\\n& \\quad \\quad \\quad \\quad\n+ g\\left(\\sigma_x^2 \\sigma_x^3+\\sigma_y^2 \\sigma_y^3\\right)\n\t\t\t\t+ \\tilde{g}\\sigma_z^2\\sigma_z^3\n\\nonumber \\\\\n& \\quad \\quad \\quad \\quad\n+ g\\left(\\sigma_x^1 \\sigma_x^3+\\sigma_y^1 \\sigma_y^3\\right)\n\t\t\t\t+ \\tilde{g}\\sigma_z^1\\sigma_z^3\n \\bigg],\n\\end{align}\nwhere $\\Omega$'s are the Rabi frequencies, $g$ and $\\tilde{g}$ are the coupling \nconstants, and $\\sigma$'s are the Pauli matrices.\nThe protocol consist of the following sequence of {\\it symmetric} pulses:\n\\begin{equation}\nX_{\\pi\/2} U_{\\rm int} Y_{\\pi\/2}|000\\rangle = e^{-i\\alpha}e^{i(\\pi\/4)}|{\\rm GHZ}\\rangle,\n\\end{equation}\nwith the entangling time set to $t_{\\rm GHZ} = \\pi\/(2|g-\\tilde{g}|)$,\nand\n\\begin{align}\nX_{\\theta} &= X^{(3)}_{\\theta}X^{(2)}_{\\theta}X^{(1)}_{\\theta},\n\\\\\nY_{\\theta} &= Y^{(3)}_{\\theta}Y^{(2)}_{\\theta}Y^{(1)}_{\\theta},\n\\end{align}\nbeing the simultaneous single-qubit rotations (the unimportant overall \nphase $\\alpha$ depends on the values of $g$ and $\\tilde{g}$). \nThe crucial step in the protocol is the initial $Y$-rotation \nby $\\pi\/2$ performed on all the qubits in the circuit, which is supposed to \nresult in the {\\it fully uniform} superposition of the computational states,\n\\begin{align}\n\\label{eq:symmetricSTATE}\n{|\\psi\\rangle}_{\\rm unif.} \n&=  (1\/\\sqrt{8})\n\\left( |000\\rangle + |001\\rangle+ \\dots + |110\\rangle + |111\\rangle \\right).\n\\end{align}\nThe protocol is very fast, but, unfortunately, is impossible to implement \ndue to severe problems with spectral crowding \\cite{MN2010}. The energy levels,\nsuch as $|200\\rangle$, etc., quickly get populated during the initial rotations.\n\nSince the problem of leakage out of the computational subspace is not going away\nany time soon, one may try to tackle it by using the proverbial ``If you can't beat \nthem, lead them'' principle. Since the higher energy levels are here \nto stay, we should look for ways to minimize their influence or make them irrelevant \naltogether. It seems that Nature herself, by way of quantum mechanical trickery, \noffers a curious possibility of doing just that, in the form of the so-called quantum \nZeno effect \\cite{Khalfin1957, MS1977, Itano1990, Panov1999, MITexperiment2006, \nFacchi2009, Matsuzaki2010, Kakuyanagi2015}. \nBefore analyzing that effect in superconducting qubits, let us take a quick look at \nhow it works in the simplest possible scenario.\n\n\n\\section{Brief review of the quantum Zeno effect}\n\nConsider a two-level system, with basis states $|1\\rangle$ and $|2\\rangle$, whose \ndynamics is described by the Hamiltonian \n\\begin{align}\nH &= V (|1\\rangle\\bra2| + |1\\rangle\\bra2|) \n=\n\\begin{pmatrix}\n0 & V \\cr \nV & 0\n\\end{pmatrix},\n\\end{align}\nwhere the off-diagonal matrix elements are chosen to be real, for convenience.\nDenote the general time-dependent state of the system by\n\\begin{equation}\n|\\psi(t)\\rangle = a_1(t)|1\\rangle + a_2(t)|2\\rangle, \n\\end{equation}\nsubject to the normalization condition,\n\\begin{equation}\n\\label{eq:3}\n|a_1(t)|^2+|a_2(t)|^2=1.\n\\end{equation}\nThe evolution of the corresponding amplitudes, $a_1$ and $a_2$, \nis given by (here we use $\\hbar =1$)\n\\begin{align}\ni \\frac{da_1}{dt}=Va_2, \\quad\ni \\frac{da_2}{dt}=Va_1.\n\\end{align}\nAssume,\n\\begin{equation}\na_1(0)=1, \\quad a_2(0) = 0.\n\\end{equation}\nThen, in linear order, for small times, $t=\\Delta t$, such that $|V \\Delta t|\\ll 1$, \nthe amplitudes are given by\n\\begin{equation}\na_1(\\Delta t) \\approx 1 + {o}(\\Delta t^2), \\quad\na_2(\\Delta t) \\approx -iV\\Delta t,\n\\end{equation}\nwith the probability of finding the system in state $|2\\rangle$ being\n\\begin{equation}\nw_2(\\Delta t) = |a_1(\\Delta t)|^2 \\approx V^2 \\Delta t^2. \n\\end{equation}\nNormalization condition (\\ref{eq:3}) then gives the probability of finding the \nsystem in state\n$|1\\rangle$,\n\\begin{equation}\nw_1(\\Delta t)=1-w_2(\\Delta t)=1-V^2 \\Delta t^2.\n\\end{equation}\n\nAssume now that the total time, $T$, of system's evolution is divided into $n$ \nsmall equal time intervals, $\\Delta t$, such that $T=n\\Delta t$, with \n$|V\\Delta t|\\ll 1$. Assume additionally that at the end of the first time interval \nan ideal instantaneous measurement represented by the projection operator \n$P_2\\equiv |2\\rangle\\bra2|$ has been made -- typically implemented by some kind \nof tunneling process out of state $|2\\rangle$, -- that gave a {\\it negative} result. \nThen the state of the system immediately after the measurement is described \nby the same ket $|1\\rangle$ in which the system was initially prepared at $t=0$.\n\nWe now calculate the probability $W^{(n)}_{1}(T)$ of the {\\it complex event}, \n${\\cal E}$, consisting of a series of $n$ $P_2$-measurements, resulting in the \nsystem {\\it remaining} in the $|1\\rangle$ state at time $T$. For this to be possible, \neach of the intermediate measurements has to produce a {\\it negative} result. \nSince all intermediate ``negative'' events are independent and each occurs with \nprobability $w_1$, the overall probability of the complex event ${\\cal E}$ is \ngiven by\n\\begin{align}\n\\label{eq:9}\nW^{(n)}_{1}(T)&=[w_1(\\Delta t)]^n \n\\approx \\left(1-V^2 \\Delta t^2\\right)^n\n\\nonumber \\\\\n&= \\left(1- \\frac{V^2 T^2}{n^2}\\right)^n \n\\equiv \\left(1- \\frac{q}{n^2}\\right)^n,\n\\end{align}\nwhere $q \\equiv V^2 T^2$.\nIn the limit $n\\rightarrow \\infty$, we get\n\\begin{align}\nW^{(\\infty)}_{1}(T)\n&= \\lim_{n\\rightarrow \\infty}\n\\left[\\left(1- \\frac{q}{n^2}\\right)^{n^2\/q}\\right]^{q\/n}\n\\nonumber \\\\\n&=\\lim_{n\\rightarrow \\infty}e^{-q\/n}=1,\n\\end{align}\nwhich constitutes the so-called quantum Zeno effect. Essentially, and \ncounter-intuitively, by continually ``watching'' the system's $|2\\rangle$ state, \nwe ``freeze'' the system in state $|1\\rangle$. \n\n\\section{Application to a Rabi-driven superconducting qubit}\n\nIt is now clear how to use the quantum Zeno effect to restrict qubit's dynamics\nto the computational subspace. By continually measuring the qubit's higher \nstate(s), we will automatically confine the qubit to its computational subspace \nspanned by $|1\\rangle$ (ground state) and $|2\\rangle$, thus suppressing the leakage \nto states $|3\\rangle$, $|4\\rangle$, etc.\n\nTo be more specific, let us restrict consideration to a three-level system, \n\\{$|1\\rangle$, $|2\\rangle$, $|3\\rangle$\\}, with energies \\{$E_1$, $E_2$, $E_3$\\}, \nwhose Rabi oscillation dynamics is implemented by a microwave drive of \nfrequency, $\\omega = \\omega_{12}$, which is resonant with the \n$|1\\rangle \\leftrightarrow |2\\rangle$ transition. This is described by the Hamiltonian, \nwhich, in the rotating wave approximation, is given by \\cite{JM2003}\n\\begin{align}\nH &= \n\\begin{pmatrix}\n0 & \\Omega e^{i\\phi} & 0 \\cr \n\\Omega e^{-i\\phi} & 0 & \\sqrt{2} \\Omega e^{i\\phi} \\cr\n0 & \\sqrt{2} \\Omega e^{-i\\phi} & \\eta \\cr\n\\end{pmatrix},\n\\end{align}\nwhere $\\eta \\equiv E_3-2E_2<0$ is the qubit anharmonicity \n(assuming $E_1=0$), $\\phi$ is the phase of the drive, and $\\Omega$ is \nthe time-dependent (pulse-shaped) Rabi frequency, which we take to be \nconstant, for simplicity. \n\nFor future use, we will be interested in the case \n$\\phi = -\\pi\/2$, which implements qubit rotations around the $Y$-axis of \nthe Bloch sphere, with the corresponding Hamiltonian being\n\\begin{align}\n\\label{eq:12}\nH &= \n\\begin{pmatrix}\n0 & -i\\Omega & 0 \\cr \ni\\Omega & 0 & -i\\sqrt{2} \\Omega \\cr\n0 & i\\sqrt{2} \\Omega & \\eta \\cr\n\\end{pmatrix}.\n\\end{align}\nWriting the unitary evolution operator in the form\n\\begin{equation}\nU(\\Delta t) = 1 +(-i)H\\Delta t + \\frac{(-i)^2}{2!}H^2\\Delta t^2 + \\dots,\n\\end{equation}\nand assuming that the initial state of the system is\n\\begin{align}\n|\\psi(0)\\rangle = a_{1}(0)|1\\rangle + a_{2}(0)|2\\rangle,\n\\\\\n|a_{1}(0)|^2+|a_{2}(0)|^2=1,\n\\\\\na_3(0) = 0,\n\\end{align}\nwe find, in {\\it second} order, the state at a later time $\\Delta t$ to be\n\\begin{align}\n\\label{eq:15}\na_1(\\Delta t)&= a_{1}(0)\\left(1-\\frac{1}{2}{\\Omega}^2\\Delta t^2\\right) \n- a_{2}(0){\\Omega} \\Delta t,\n\\\\\n\\label{eq:16}\na_2(\\Delta t)&= a_{2}(0)\\left(1-\\frac{3}{2}{\\Omega}^2\\Delta t^2\\right) \n+ a_{1}(0) {\\Omega}\\Delta t,\n\\\\\n\\label{eq:17}\na_3(\\Delta t)&=\\sqrt{2}a_{2}(0) {\\Omega}\\Delta t +o(\\Delta t^2). \n\\end{align}\nCorrespondingly, the probabilities for finding the system in state $|3\\rangle$ and \nin the computational subspace are, respectively,\n\\begin{align}\nw_3(\\Delta t) &= |a_3(\\Delta t)|^2 = 2|a_{2}(0)|^2 {\\Omega}^2\\Delta t^2,\n\\\\\nw_{\\rm comp.}(\\Delta t) &\\equiv w_1(\\Delta t)+w_2(\\Delta t) \n= 1- 2|a_{2}(0)|^2 {\\Omega}^2\\Delta t^2.\n\\end{align}\nBy anology with the discussion around Eq.\\ (\\ref{eq:9}), in the case of a long \nsequence of $n$ $P_3$-measurements with negative outcomes, the probability \n$W^{(n)}_{{\\rm comp.}}(T)$ of the complex event ${\\cal E}_{\\rm comp.}$, \nin which the qubit {\\it remains in its computational subspace} at time $T$, can \nbe estimated as,\n\\begin{widetext}\n\\begin{align}\n\\label{eq:22}\nW^{(n)}_{{\\rm comp.}}(T)\n& \\approx \n\\left(1- 2|a_{2}|_{t=(n-1)\\Delta T}^2 {\\Omega}^2 \\Delta t^2\\right)\n\\dots\n\\left(1- 2|a_{2}|_{t=2\\Delta T}^2 {\\Omega}^2 \\Delta t^2\\right)\n\\left(1- 2|a_{2}|_{t=\\Delta T}^2 {\\Omega}^2 \\Delta t^2\\right)\n\\left(1- 2|a_{2}|_{t=0}^2 {\\Omega}^2 \\Delta t^2\\right)\n\\\\\n& \\geq \n\\left(1- 2 {\\Omega}^2 \\Delta t^2\\right)^n\n= \\left(1- \\frac{2{\\Omega}^2 T^2}{n^2}\\right)^n \\rightarrow 1, \n\\quad\nn\\rightarrow \\infty,\n\\end{align} \n\\end{widetext}\nsince $|a_2|^2\\leq 1$ at every step of system's evolution.\n\n\nAs far as the evolution of the {\\it amplitudes} is concerned, \nwe return to Eqs.\\ (\\ref{eq:15}), (\\ref{eq:16}), (\\ref{eq:17}), and notice that \nif at time $\\Delta t$, $|V\\Delta t|\\ll 1$, an ideal $P_3$-measurement is made \nwith the negative outcome (for the system to be found in its $|3\\rangle$-state), \nthe updated renormalized state of the system will be, in second order,\n\\begin{align}\na_1(\\Delta t)&= a_{1}-a_{2}{\\Omega}\\Delta t \n- a_1\\left(\\frac{1}{2}-|a_2|^2\\right){\\Omega}^2\\Delta t^2,\n\\\\\na_2(\\Delta t)&= a_{2}+a_{1}{\\Omega}\\Delta t \n- a_2\\left(\\frac{1}{2}+|a_1|^2\\right){\\Omega}^2\\Delta t^2,\n\\\\\na_3(\\Delta t) & = 0,\n\\end{align}\nwhere on the right hand side of each equation \nthe label $(0)$ has been dropped for notational simplicity.\nFor $a_1=1$, $a_2=0$, we get\n\\begin{align}\na_1(\\Delta t)&= 1-\\frac{{\\Omega}^2\\Delta t^2}{2}\\approx \\cos (V\\Delta t),\n\\\\\na_2(\\Delta t)&= {\\Omega}\\Delta t \\approx \\sin (V\\Delta t),\n\\\\\na_3(t) & = 0,\n\\end{align}\nand for $a_1=0$, $a_2=1$, we get\n\\begin{align}\na_1(\\Delta t)&= -{\\Omega}\\Delta t \\approx -\\sin ({\\Omega}\\Delta t),\n\\\\\na_2(\\Delta t)&= 1-\\frac{{\\Omega}^2\\Delta t^2}{2}\n\\approx \\cos ({\\Omega}\\Delta t),\n\\\\\na_3(\\Delta t) & = 0.\n\\end{align}\nThe above discussion indicates that in the presence \nof continuous higher-energy state mesurement the qubit will keep evolving \nas if the higher-energy state did not exist.\n\nFigures \\ref{fig:1}, \\ref{fig:2}, and \\ref{fig:3} show the time dependence of \nvarious state populations,\n\\begin{equation}\np_i(t) \\equiv |a_i(t)|^2, \\quad i = 1,2,3,\n\\end{equation}\nduring the execution of the Rabi pulse applied to the system initially prepared\nin its ground state, $|\\psi(0)\\rangle = (1,0,0)$), and interrupted by $n=25$, 50, \nand 100 ideal $P_3$-measurements, with the assumption that the detector \nnever clicked. We distinguish the following three cases:\n\nThe ideal case, shown just for comparison, corresponds to the evolution \nunder the action of the model Hamiltonian,\n\\begin{align}\nH_{\\rm ideal} &= \n\\begin{pmatrix}\n0 & -i{\\Omega} & 0 \\cr \ni{\\Omega} & 0 & 0 \\cr\n0 & 0 & \\eta \\cr\n\\end{pmatrix}.\n\\end{align}\nwithout any leakage out of the computational subspace.\n\nThe Zeno case corresponds to the evolution under the action of $H$ given \nin (\\ref{eq:12}) and subjected to $n$ projective $P_3$-measurements. The \nevolution was numerically calculated in accordance with the iterative scheme,\n\\begin{align}\n|\\psi_1\\rangle &= |1\\rangle ,\n\\nonumber \\\\\n|\\psi_k\\rangle &= \n\\frac{P_{\\rm comp.} |\\psi_{k-1}\\rangle}\n{\\sqrt{||P_{\\rm comp.} |\\psi_{k-1}\\rangle||}} ,\n\\nonumber \\\\\n|\\psi_{k+1}\\rangle &= \\exp\\{-i H \\Delta t\\} |\\psi_k\\rangle ,\n\\end{align}\nfor $k=2, 3, \\dots, n$, where $P_{\\rm comp.}$ is the projection operator \nonto the computational subspace,\n\\begin{equation}\nP_{\\rm comp.} \\equiv 1 - P_3 = 1-|3\\rangle\\bra3|.\n\\end{equation}\nAt the final time $T=n\\Delta t$ the qubit remains in its computational \nsubspace (that is, $p_{3}(T)\\equiv |a_3(T)|^2=0$), with the probability \n(let us call it the {\\it survival probability}) given by the product \n(compare to (\\ref{eq:22})),\n\\begin{align}\nW^{(n)}_{{\\rm comp.}}(T)\n& = \\prod_{k=1}^{n} \\left(1- p_3(t_k)\\right).\n\\end{align} \n\nFinally, the no-Zeno case corresponds to the exact evolution,\n\\begin{equation}\nU(t) = \\exp\\{-iHt\\},\n\\end{equation}\nwith the leakage to $|3\\rangle$ taken into account. The probability for the \nsystem to be found in the computational subspace in a single \nmeasurement performed at the end of the evolution is \n\\begin{equation}\nW_{\\rm comp.}(T)= 1 - p_3(T).\n\\end{equation} \n\nA quick glance at the figures shows the presence of the Zeno effect in our \nsystem. There is a critical number of the measuring steps, $n_{\\rm crit}$, \ndepending on the choice of the system parameters, \nabove which the survival probability $W^{(n)}_{{\\rm comp.}}(T)$ exceeds \n$W_{\\rm comp.}(T)$. In the limit $n\\rightarrow \\infty$, \n$W^{(n)}_{{\\rm comp.}}(T)$ approaches 1, as expected.\n\n\\begin{figure}[!ht]\n\\includegraphics[angle=0,width=1.00\\linewidth]{fig1}\n\\caption{ \\label{fig:1} \nPopulations of various qubit states during the Rabi pulse interrupted by \n$n=25$ projective measurements of state $|3\\rangle$.\n}\n\\end{figure}\n\\begin{figure}[!ht]\n\\includegraphics[angle=0,width=1.00\\linewidth]{fig2}\n\\caption{ \\label{fig:2} \nPopulations of various qubit states during the Rabi pulse interrupted by \n$n=50$ projective measurements of state $|3\\rangle$.\n}\n\\end{figure}\n\\begin{figure}[!ht]\n\\includegraphics[angle=0,width=1.00\\linewidth]{fig3}\n\\caption{ \\label{fig:3} \nPopulations of various qubit states during the Rabi pulse interrupted by \n$n=100$ projective measurements of state $|3\\rangle$.\n}\n\\end{figure}\n\n\\section{Taking a closer look at Eq.\\ (\\ref{eq:17}) and designing a \nmeasurement model}\n\n\\begin{figure\n\\includegraphics[angle=0,width=1.00\\linewidth]{fig4}\n\\caption{ \\label{fig:4} \nPopulations of various qubit states during the Rabi pulse with the state $|3\\rangle$\nsubjected to tunneling.\n}\n\\end{figure}\n\nWe now take a closer look at Eq.\\ (\\ref{eq:17}), in which the second order \nterm was ignored. Restoring that $o(\\Delta t^2)$ term, we find\n\\begin{align}\na_3(\\Delta t) &= \\sqrt{2}a_{2}(0) {\\Omega}\\Delta t \n+ \\frac{\\sqrt{2}}{2}\\left[a_1(0) \\Omega^2 - a_2(0)\\Omega\\eta i\\right] \n\\Delta t^2.\n\\end{align}\nAssume now that the $|3\\rangle$ level is allowed to tunnel out with the rate \n$\\Gamma$. The third diagonal matrix element of the Hamiltonian (\\ref{eq:12}) \nwould then get modified as\n\\begin{equation}\n\\eta \\rightarrow \\eta - i\\Gamma\/2,\n\\end{equation}\ngiving\n\\begin{align}\n\\label{eq:42}\na_3(\\Delta t) &= \\sqrt{2}a_{2}(0) {\\Omega}\\Delta t \n\\nonumber \\\\\n&+ \\frac{\\sqrt{2}}{2}\\left[a_1(0) \\Omega^2 \n- a_2(0) \\Omega \\left(\\frac{\\Gamma}{2}+\\eta i\\right)\\right]  \\Delta t^2.\n\\end{align}\nWe now make the following observation: if the tunneling rate is chosen to be\n\\begin{equation}\n\\Gamma = \\frac{4}{\\Delta t},\n\\end{equation}\nthe first order term, $\\sqrt{2}a_{2}(0) {\\Omega}\\Delta t$, in Eq.\\ (\\ref{eq:42}) \nwould get canceled, giving\n\\begin{align}\n\\label{eq:49}\na_3(\\Delta t) &=\n\\frac{\\sqrt{2}}{2}\\left[a_1(0) \\Omega^2 - a_2(0)\\Omega \\eta i\\right]\n\\Delta t^2,\n\\end{align}\nwhich would result in the {\\it fourth} order population of the unwanted state \n$|3\\rangle$.\n\nFigure \\ref{fig:4} shows the result of exact simulation for the system initially \nprepared in the ground state, using the Hamiltonian\n\\begin{align}\n\\label{eq:50}\nH &= \n\\begin{pmatrix}\n0 & -i\\Omega & 0 \\cr \ni\\Omega & 0 & -i\\sqrt{2} \\Omega \\cr\n0 & i\\sqrt{2} \\Omega & \\eta - i\\Gamma\/2 \\cr\n\\end{pmatrix},\n\\end{align}\nwith $\\Gamma = 40$ [ns$^{-1}$] and $\\Omega = 0.05$ [GHz] \n(this effectively corresponds to $n=50$ with $\\Delta t = 0.1$ [ns] of the Zeno \ncase). The survival probability is defined by\n\\begin{equation}\nW^{({\\rm tunnel.})}_{{\\rm comp.}}(t) = |a_1(t)|^2+|a_2(t)|^2.\n\\end{equation}\nThe corresponding dynamics can be viewed as a reasonable measurement model \nfor state $|3\\rangle$. We see that for the chosen system's parameters leakage out of the \ncomputational subspace is virtually nonexistent.\n\n\\newpage\n\\section{Conclusions}\n\nWorking with the RWA Hamiltonian of a three-level qubit driven by a Rabi \npulse, we showed that, by allowing the system to continuously tunnel out \nof its higher-energy state (the greater, $\\Gamma$ the better), we effectively \nconfine system's dynamics to the computational subspace. Such quantum \nZeno behavior can potentially be used to resolve the long standing spectral \ncrowding problem that plagued the field of superconducting quantum \ncomputing for almost two decades.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\vspace{-0.05in}}\n\\newcommand{\\mysubsection}[1]{\\vspace{-0.1in}\\subsection{#1}\\vspace{-0.05in}}\n\\newcommand{\\mysubsubsection}[1]{\\vspace{-0.1in}\\subsubsection{#1}\\vspace{-0.05in}}\n\\newcommand{\\myparagraph}[1]{\\par\\smallskip\\par\\noindent{\\bf{}#1:~}}\n\\newcommand{\\ceil}[1]{{\\left\\lceil#1  \\right\\rceil}}\n\\newcommand{\\comment}[1]{}\n\\newcommand\\numberthis{\\addtocounter{equation}{1}\\tag{\\theequation}}\n\\newcommand{\\textnormal{poly}}{\\textnormal{poly}}\n\n\n\\newcommand{{\\mathcal{A}}}{{\\mathcal{A}}}\n\\newcommand{{\\mathcal{B}}}{{\\mathcal{B}}}\n\\newcommand{{\\mathcal{G}}}{{\\mathcal{G}}}\n\\newcommand{{\\mathcal{I}}}{{\\mathcal{I}}}\n\\newcommand{{\\mathcal{S}}}{{\\mathcal{S}}}\n\\newcommand{{\\mathcal{M}}}{{\\mathcal{M}}}\n\\newcommand{{\\mathcal{L}}}{{\\mathcal{L}}}\n\\newcommand{{\\mathcal{J}}}{{\\mathcal{J}}}\n\\newcommand{{\\mathcal{I}_R}}{{\\mathcal{I}_R}}\n\n\\def {\\cal N}   {{\\cal N}}\n\\def {\\cal C}   {{\\cal C}}\n\n\n\\newcommand{{\\mathcal{C}}}{{\\mathcal{C}}}\n\\newcommand{{\\bar{x}}}{{\\bar{x}}}\n\\newcommand{{\\bar{y}}}{{\\bar{y}}}\n\\newcommand{{\\bar{z}}}{{\\bar{z}}}\n\\newcommand{{\\bar{c}}}{{\\bar{c}}}\n\\newcommand{\\textnormal{OPT}}{\\textnormal{OPT}}\n\\newcommand{\\textnormal{\\textsf{Greedy}}}{\\textnormal{\\textsf{Greedy}}}\n\\newcommand{\\textsc{Greedy+Singleton}}{\\textsc{Greedy+Singleton}}\n\\newcommand{\\textsc{EnumGreedy}}{\\textsc{EnumGreedy}}\n\\newcommand{{\\varepsilon}}{{\\varepsilon}}\n\\newcommand{{\\mu}}{{\\mu}}\n\\newcommand{{\\mathbb{E}}}{{\\mathbb{E}}}\n\\newcommand{\\floor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\\newcommand{{\\varphi}}{{\\varphi}}\n\\begin{document}\n\t\\newtheorem{thm}{Theorem}[section]\n\t\\newtheorem{prop}[thm]{Proposition}\n\t\\newtheorem{assm}[thm]{Assumption}\n\t\\newtheorem{lem}[thm]{Lemma}\n\t\\newtheorem{obs}[thm]{Observation}\n\t\\newtheorem{cor}[thm]{Corollary}\n\t\\newtheorem{lemma}[thm]{Lemma}\n\t\\newtheorem{definition}[thm]{Definition}\n\t\\newtheorem{theorem}[thm]{Theorem}\n\t\\newtheorem{proposition}[thm]{Proposition}\n\t\\newtheorem{observation}[thm]{Observation}\n\t\\newtheorem{claim}[thm]{Claim}\n\t\t\\newtheorem{example}[thm]{Example}\n\t\\newtheorem{defn}[thm]{Definition}\n\t\\newcommand{\\ariel}[1]{{\\color{red} (Ariel :#1)}}\n\t\\newcommand{\\ilan}[1]{{\\color{blue} Ilan :\\color{magenta}#1}}\n\t\\def {\\mathcal I}   {{\\mathcal I}}\n\t\\newcommand{\\mathbbm{1}}{\\mathbbm{1}}\n\t\n\n\t\\begin{titlepage}\n\t\\title{\n\t\t Bin Packing with Partition Matroid can be Approximated within $o(OPT)$ Bins\n\t}\n\t\\author{Ilan Doron-Arad\\thanks{Computer Science Department, \n\t\tTechnion, Haifa, Israel. \\texttt{idoron-arad@cs.technion.ac.il}}\n\t\\and\n\tAriel Kulik\\thanks{CISPA Helmholtz Center for Information Security, Saarland Informatics Campus, Germany. \\texttt{ariel.kulik@cispa.de}} \n\t\\and\n\tHadas Shachnai\\thanks{Computer Science Department, \n\t\tTechnion, Haifa, Israel. \\texttt{hadas@cs.technion.ac.il}}\n}\n\n\t\\maketitle\n\n\t\\input{abstract}\n\t\n\t\n\t\\thispagestyle{empty}\n\\end{titlepage}\n\n\t\\tableofcontents\n\\thispagestyle{empty}\n\t\\newpage\n\\setcounter{page}{1}\n\n\\input{introduction}\n\\input{preliminaries}\n\\input{Overview}\n\n\\input{eviction}\n\\input{shifting}\n\\input{FromPolytope}\n\\input{pack}\n\\input{Reconstruct}\n\\input{discussion}\n\n\\section{Algorithm $\\textsf{Partition}$}\n\\label{sec:FromPolytope}\n\n\nAlgorithm \\textsf{Partition} constructs an ${\\varepsilon}$-nice partition of small size based on the integrality properties of the polytope, as given in Lemma~\\ref{O(1)}. Let $\\bar{z}$ be \n a good prototype (satisfying the conditions in Definition~\\ref{def:goodprototype}).\nSince the values of entries in the support of $\\bar{z}$ are not necessarily integral, $\\bar{z}$ may not satisfy the conditions of Lemma~\\ref{O(1)}. Thus, we \nmodify $\\bar{z}$ to a prototype having integral entries.  Define the {\\em integralization} of $\\bar{z}$ as a prototype $\\bar{z}^* \\in \\mathbb{R}^{{\\mathcal{C}}}_{\\geq 0}$ such that, for all $C \\in {\\mathcal{C}}$, $\\bar{z}^*_C = \\ceil{\\bar{z}_C}$.\n\n\\begin{observation}\n\t\\label{ob:y}\n\tThe $\\bar{z}^*$-polytope is non-empty, $\\textnormal{\\textsf{supp}} (\\bar{z}^*) =\\textnormal{\\textsf{supp}} (\\bar{z})$, and $\\|\\bar{z}^*\\| \\leq \\|\\bar{z}\\|+|\\textnormal{\\textsf{supp}}(\\bar{z})|$. \n\\end{observation}\n\nBy Observation~\\ref{ob:y}, it follows that $\\bar{z}^*$ is a good prototype, and there is a vertex $\\bar{\\gamma}$ of the $\\bar{z}^*$-polytope that satisfies constraint \\eqref{F4} with equality. Let $F = \\{\\{\\ell\\}~|~ \\ell \\in I, \\exists t \\in I \\cup {\\mathcal{C}} \\text{ s.t. } \\bar{\\gamma}_{\\ell,t} \\in (0,1)\\}$ be the set of  \nitems that are fractionally assigned to some type by $\\bar{\\gamma}$. Since $\\bar{z}^*$ is a good prototype with integral entries, by Lemma~\\ref{O(1)} $|F|$ is small. Thus, \nan ${\\varepsilon}$-nice partition is obtained by assigning the integral items in $\\bar{\\gamma}$ and then adding the {\\em fractional} items,\nwith only a small increase in the size of the ${\\varepsilon}$-nice partition. \n\nWe start by finding a packing for items assigned integrally by $\\bar{\\gamma}$ to slot-types via bipartite matching. Specifically, let $$V = \\left\\{\\left(C,j,k\\right) ~|~C\\in \\textsf{supp}(\\bar{z}^*), j \\in C, k \\in \\left[\\bar{z}^*_C\\right] \\right\\}$$ where each vertex in $V$ represents a slot within a configuration $C \\in \\textsf{supp}(\\bar{z}^*)$, and an index for one of the $\\bar{z}^*_C$ bins associated with $C$. Also, let $U = \\{\\ell \\in I~|~ \\exists j \\in I \\text{ s.t. } \\bar{\\gamma}_{\\ell,j} = 1\\}$ be all items assigned integrally to some slot-type by $\\bar{\\gamma}$. Now, define the {\\em assignment graph} of $\\bar{\\gamma}$ as the bipartite graph $G = (U, V, E)$, where $E = \\{\\left(\\ell, \\left(C,j,k\\right)\\right) \\in U \\times V~|~ \\bar{\\gamma}_{\\ell,j} = 1\\}$. That is, there is an edge between any item $\\ell \\in U$ to $(C,j,k) \\in V$ if $\\ell$ is assigned integrally (i.e., completely) to $j$ by $\\bar{\\gamma}$. If the edge $(\\ell,(C,j,k))$ is taken to the matching, we replace the slot $j$ by $\\ell$ in the $k$-th bin associated with $C$. \n\n\\begin{lemma}\n\t\\label{lem:matchingU}\nThere is a matching in $G$ of size $|U|$. \n\\end{lemma}\n The proof of Lemma~\\ref{lem:matchingU} is based on finding a fractional matching in $G$, by taking $\\frac{1}{d(\\ell)}$ of an edge $(\\ell,v) \\in E$ to the matching, where $d(\\ell)$ is the degree of $\\ell$ in $G$. By constraint \\eqref{F3} of the $\\bar{z}^*$-polytope, this guarantees a feasible fractional matching of size $|U|$. Since $G$ is bipartite, there is also an integral matching of size $|U|$ in $G$ \\cite{schrijver2003combinatorial}. Let $M$ be a matching in $G$ of size $|U|$. \n\n\nWe construct below a packing based on $M$. For all $C \\in \\textsf{supp}(\\bar{z}^*)$ and $k \\in [\\bar{z}^*_C]$, define $$A_{C,k} = \\{\\ell \\in U ~|~ \\exists j \\in I \\text{ s.t. }  (\\ell,(C,j,k)) \\in M\\}$$  as the {\\em bin} of $C$ and $k$, which contains all items  coupled by the matching to the $k$-th bin associated with $C$.  The next lemma follows since $M$ is a matching in the assignment graph of $\\bar{\\gamma}$.\n\n\n\n\\begin{lemma}\n\t\\label{lem:allowed}\nFor all $C \\in \\textnormal{\\textsf{supp}} (\\bar{z}^*)$ and $k \\in [\\bar{z}^*_C]$, $A_{C,k}$ is allowed in $C$. \n\\end{lemma} \n\nWe now define a packing of all items.\nGiven two tuples $S,T$, let $S \\oplus T$ denote  the concatenation of $S$ and $T$.  The {\\em packing of} $\\bar{\\gamma}$ and $M$ is given by\n\n\\begin{equation}\n\\label{Astar}\nA^* = \\big(h~|~h \\in F\\big) \\oplus \\big(A_{C,k} ~|~ C \\in \\textsf{supp}(\\bar{z}^*), k \\in [\\bar{z}^*_C]\\big). \n\\end{equation} In words, $A^*$ contains a bin for every fractional item, as well as all the bins associated with configurations in the support of $\\bar{z}^*$. It follows that $A^*$ is a packing of $U$ and all fractional items. \n\nWe construct an ${\\varepsilon}$-nice partition below whose packing is $A^*$. Let $\\mathcal{H} = \\textsf{supp}(\\bar{z}^*) \\cup F$. By Lemma~\\ref{O(1)}, we get $|F| \\leq {\\varepsilon}^{-21}|\\textsf{supp}(\\bar{z}^*)|^2$, and it follows that $|\\mathcal{H}| \\leq {\\varepsilon}^{-22}Q^2({\\varepsilon})$ . For all $C \\in \\mathcal{H}$ define the {\\em category of} $C$ as \\begin{equation}\n\t\\label{BC}\n\tB_C = \\left\\{A_{C,k}~|~ k \\in [\\bar{z}^*_{C}] \\right\\}  \\cup \\big\\{ h \\in F ~|~ h = C \\big\\}\n\\end{equation} that contains all the bins associated with $C$, and possibly a singleton of a fractional item.\\footnote{If $C$ is a singleton of this item.} By \\eqref{Astar} and \\eqref{BC}, $\\{B_C\\}_{C \\in \\mathcal{H}}$ is a partition of the bins in $A^*$. In addition, for all $C \\in \\mathcal{H}$ define  $$D_{C} = \\{\\ell \\in I~|~\\bar{\\gamma}_{\\ell,C} = 1\\}$$ as all items assigned integrally to $C$ by $\\bar{\\gamma}$. Finally, define the {\\em division} of $\\bar{\\gamma}$ and $M$, as $A^*$, $\\mathcal{H}$, $(B_C)_{C \\in \\mathcal{H}}$, and $(D_C)_{C \\in \\mathcal{H}}$. Algorithm $\\textsf{Partition}$ computes the above ${\\varepsilon}$-nice partition. We give the pseudocode in Algorithm~\\ref{Alg:Assign}.\n\n\n\\begin{algorithm}[htb]\n\t\\caption{$\\textsf{Partition} (\\bar{z}$)}\n\t\\label{Alg:Assign}\n\t\n\tGenerate $\\bar{z}^*$, the integralization of $\\bar{z}$\n\t\n\tFind a vertex $\\bar{\\gamma}$ of the $\\bar{z}^*$-polytope satisfying \\eqref{F4} with equality \\label{assign:find_gamma}\n\t\n\tConstruct the assignment graph $G$ of $\\bar{\\gamma}$\n\t\n\tFind a maximum matching $M$ in $G$\n\t\n\tReturn the division of $\\bar{\\gamma}$ and $M$\n\t\n\\end{algorithm}\n\n  \\begin{claim}\n\t\\label{eqeq6}\n\tFor any $C \\in \\mathcal{H}$ and $G \\in {\\mathcal{G}}$ the following hold.\n\t\\begin{enumerate}\n\t\t\\item $D_C \\subseteq \\textnormal{\\textsf{fit}}(C)$. \n\t\t\\item $s(D_C) \\leq \\left(1-s(C)\\right) \\cdot |B_C|$. \n\t\t\\item  $|D_C \\cap G| \\leq |B_C| \\cdot \\left(     k(G) - |G \\cap C|  \\right)$.\n\t\\end{enumerate} \n\\end{claim} \n\n The properties of the $\\bar{z}^*$-polytope are the key to proving Claim~\\ref{eqeq6} and consequently Lemma~\\ref{FromPolytope}. Recall that $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope; thus, the first property of Claim~\\ref{eqeq6} follows from constraint \\eqref{F1}, the second by constraint \\eqref{F2}, and the third by constraint \\eqref{F5}. Moreover, using constraint \\eqref{F3}, we construct the matching $M$. Finally, we use constraint \\eqref{F4} to show that $\\{D_C\\}_{C \\in \\mathcal{H}}$ is a partition of items not packed in $A^*$.\n\\begin{lemma}\n\t\\label{lem:assign}\n\tThe division of $\\bar{\\gamma}$ and $M$ is an ${\\varepsilon}$-nice partition of size at most $\\|\\bar{z}\\|+{\\varepsilon}^{-22}|\\textsf{supp}(\\bar{z})|^2$.\n\\end{lemma}\n\n\\noindent{\\bf Proof of Lemma~\\ref{FromPolytope}}\nThe correctness of Algorithm~\\ref{Alg:Assign} follows from~\\ref{lem:assign}. Observe that the vertex~$\\bar{\\gamma}$ in Step~\\ref{assign:find_gamma} can be found via standard linear programing, thus the algorithm runs in polynomial time. \n\\qed\n\n\n\\section{Approximation Algorithm for ${\\varepsilon}$-Structured Instances}\n\\label{sec:overview}\n\nOur algorithm uses a configuration-LP  \nrelaxation of the given BPP instance.\nFor ${\\varepsilon} \\in (0,0.1)$ such that ${\\varepsilon}^{-1}\\in\\mathbb{N}$, let $\\mathcal{I} = (I,\\mathcal{G},s,k)$ be an ${\\varepsilon}$-structured BPP instance. Recall that a {\\em configuration} of $\\mathcal{I}$ is a subset of items $C \\subseteq I$ such that $\\sum_{\\ell \\in C} s(\\ell) \\leq 1$, and $|C \\cap G| \\leq k(G)$ for all $G \\in \\mathcal{G}$. Let $\\mathcal{C}({\\mathcal{I}})$ be the set of all configurations of $\\mathcal{I}$; we use ${\\mathcal{C}}$ when the instance ${\\mathcal{I}}$ is clear from the context. Given $S \\subseteq I$, a partition $(A_1, \\ldots, A_m)$ of $S$ is a {\\em packing} (or, a {\\em packing of} $S$) if $A_b \\in {\\mathcal{C}}$ for all $b \\in [m]$; if $S = I$ then we say that $A$ is a {\\em packing of} ${\\mathcal{I}}$. For $\\ell \\in I$, let $\\mathcal{C}[\\ell] = \\{C \\in \\mathcal{C}~|~\\ell \\in C\\}$ be the set of configurations of $\\mathcal{I}$ that contain $\\ell$. Our algorithm initially solves the following configuration-LP relaxation of the problem.\n\n\\begin{equation}\n\t \\label{C-LP}\n\t\\begin{aligned}\n\t\t~~~~~ \\min\\quad        & ~~~~~\\sum_{C \\in \\mathcal{C}} \\bar{x}_C                                                           \\\\\n\t\t\\textsf{s.t.\\quad} & ~~~\\sum_{~C \\in \\mathcal{C}[\\ell]} \\bar{x}_C = 1   & \\forall \\ell \\in I~~~~~\\\\  \n\t\t& ~~~~~\\bar{x}_C \\geq 0 ~~~~~~~~&~~~~~~~~~~~~~~~~~~~~ \\forall C \\in  \\mathcal{C}~~~~\n\t\\end{aligned}\n\\end{equation}\n\n\nA solution for the LP~(\\ref{C-LP}) assigns to each configuration  \n$C \\in  \\mathcal{C}$ a real number  $\\bar{x}_C \\in [0,1]$ which indicates the fractional selection of $C$ for the solution such that each item is fully {\\em covered}.  For simplicity, denote by $\\|\\bar{x}\\| = \\sum_{C \\in \\mathcal{C}} \\bar{x}_C  $ the $\\ell_1$-norm of $\\bar{x}$ (that is, the objective value in~(\\ref{C-LP})).  Note that the configuration-LP~(\\ref{C-LP}) has an exponential number of variables; thus, it cannot be solved in polynomial time by applying standard techniques. A common approach for solving such linear programs is to use a {\\em separation oracle} for the dual program.\n\n\nConsider the {\\em configuration maximization problem (CMP)} in which we are given a BPP instance $\\mathcal{I}=(I,\\mathcal{G},s,k)$ and a weight function $w:I\\rightarrow \\mathbb{R}_{\\geq 0}$; the objective is to find a configuration $C\\in \\mathcal{C}$ such that $\\sum_{\\ell \\in C} w(\\ell)$ is maximized. By a well known connection between separation and optimization, an FPTAS for CMP implies an FPTAS for the configuration-LP of $\\mathcal{I}$~\\cite{grigoriadis2001approximate,fleischer2011tight,grotschel2012geometric,plotkin1995fast}. CMP\ncan be solved via an easy reduction \nto {\\em knapsack with partition matroid}, which admits an FPTAS~\\cite{chakaravarthy2013knapsack}. Thus, we have\n\\begin{lemma}\n\\label{configurationLP}\nThere is an algorithm  $\\textnormal{\\textsf{SolveLP}}$ which given a BPP instance ${\\mathcal{I}}$ and ${\\varepsilon}>0$, returns  in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a solution for the configuration-LP of $\\mathcal{I}$ of value at most $(1+{\\varepsilon})\\textnormal{OPT}$, where $\\textnormal{OPT}$ is the value of an optimal solution for the configuration-LP of $\\mathcal{I}$.\n\\end{lemma}\n\nWe give the proof of Lemma~\\ref{configurationLP} in Appendix~\\ref{app:omitted}. A key component in our scheme is the construction of a {\\em prototype} of a packing. \n\n\\begin{definition}\n\t\\label{def:prototype}\n\tGiven a BPP instance ${\\mathcal{I}}$,  a {\\em prototype} is a vector $\\bar{x} \\in \\mathbb{R}_{\\geq 0}^{{\\mathcal{C}}}$. \n\\end{definition}\n\n\nIn particular, a solution for~\\eqref{C-LP} is a prototype. \nIn the context of a prototype, each configuration $C \\in \\mathcal{C}$ is considered as a set of {\\em slots}: each slot  $\\ell \\in C$ is a placeholder for an item, where items that {\\em fit} in the place of a slot $\\ell$ are those in the group of $\\ell$ of smaller or equal size. \nFor any $\\ell \\in I$\ndefine  $\\textsf{group}(\\ell) = G$, where $G\\in {\\mathcal{G}}$ is the unique group such that $\\ell\\in G$. Now, we define the subset of items that fit in place of a slot $\\ell \\in I$ by\n\n\\begin{equation}\n\t\\label{fitl}\n\t\\textsf{fit}(\\ell) = \\{\\ell' \\in \\textsf{group}(\\ell)~|~ s(\\ell') \\leq s(\\ell)\\}~~~~~~~~~~~~~~\\forall \\ell \\in I.\n\\end{equation}\n\n\n Each configuration is viewed as a set of slots, representing subsets of items that can \n replace the slots in the actual packing of the instance.\n We associate a polytope with the prototype, in which additional items (i.e, items which to not replace a slot) may be assigned to the unused capacity of a configuration $C$. To enable efficient packing of the instance, these additional items\n must be {\\em small} relative to the unused capacity of $C$. To this end, we define the set of items which {\\em fit with} a set of slots $C \\in \\mathcal{C}$ as \\begin{equation}\n\t\\label{fitS}\n\t\\textsf{fit}(C) = \\{\\ell \\in I~|~   s(\\ell) \\leq \\min \\{{\\varepsilon}^2, {\\varepsilon} \\cdot (1-s(C))\\}\\}~~~~~~~~~~~~~~\\forall C \\in \\mathcal{C}.\n\\end{equation} \n\n In words, $\\textsf{fit}(C)$ contains items of sizes smaller than ${\\varepsilon}^2$ and also at most ${\\varepsilon}$-fraction of the unused capacity of $C$. Given a prototype $\\bar{x}$, the {\\em $\\bar{x}$-polytope} is a relaxation for a packing of the instance that assigns items fractionally, either as replacement for slots or in addition to a set of slots. We define the set of {\\em types} of the instance $\\mathcal{I}$ to be $I\\cup \\mathcal {{\\mathcal{C}}}$. The set of types includes the {\\em slot-types}, i.e.,  slots (items) in $I$, and {\\em configuration-types}, i.e., configurations in $\\mathcal{C}$. A point in the $\\bar{x}$-polytope\n has an entry for each pair of an item $\\ell \\in I$ and a type $t \\in I \\cup {\\cal C}$ which\n  represents the fractional {\\em assignment} of the item to the type. Formally,\n\n\n \n \n \\begin{definition}\n \t\\label{def:polytope}\n \tGiven a BPP instance ${\\mathcal{I}}$, the set ${\\mathcal{C}}$ of configurations for ${\\mathcal{I}}$, and a prototype $\\bar{x}$ of ${\\mathcal{I}}$, the $\\bar{x}$-polytope is the set containing all points $\\bar{\\gamma} \\in [0,1]^{I \\times (I\\cup \\mathcal{C})}$ which satisfy the following constraints.\n\n\n\\begin{align}\n\t\t\\displaystyle \\bar{\\gamma}_{\\ell,t} = 0   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~\t~~~~~~~~~~\\forall \\ell \\in I , t \\in I \\cup \\mathcal{C}\\textnormal{ s.t. } \\ell \\notin \\textnormal{\\textsf{fit}}\\left(t \\right)~~~~ \\label{F1}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle \\sum_{\\ell \\in I} \\bar{\\gamma}_{\\ell,C} \\cdot  s(\\ell) \\leq \\left(1-s(C)\\right) \\cdot \\bar{x}_C ~~~~~~~~~~~~~~~~~~~\t~~~~~~\\forall C \\in \\mathcal{C} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\label{F2}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle    \\sum_{\\ell \\in G} \\bar{\\gamma}_{\\ell,C} \\leq \\bar{x}_C \\cdot \\left(k(G)-|C \\cap G|\\right)~~~~  ~~~~~~~~~~\t~~~~~~~~~~\\forall G \\in \\mathcal{G}, C \\in \\mathcal{C}~~~~~~~~~~~~~~~~~~~~~~~\\label{F5}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle \\sum_{\\ell \\in I} \\bar{\\gamma}_{\\ell,j} \\leq \\sum_{\\substack{C \\in \\mathcal{C}[j]}} \\bar{x}_C~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~\t~~~~~~~~~~\\forall j \\in I   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\label{F3}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle   \\sum_{t \\in I \\cup \\mathcal{C}} \\bar{\\gamma}_{\\ell,t} \\geq 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~\t\\forall \\ell \\in I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\label{F4}\n\t\\end{align}\n\t \\end{definition}\n\t\n\tConstraints~(\\ref{F1}) indicate that an item $\\ell \\in I$ cannot be assigned to type $t$ if $\\ell$ does not fit in~$t$. \n\tConstraints~(\\ref{F2}) set an upper bound on the total (fractional) size of items assigned to each configuration-type $C \\in \\mathcal{C}$. This bound is equal to the residual capacity of $C$ times the number of bins packed with $C$, given by $\\bar{x}_C$.\n\tConstraints~(\\ref{F5}) bound the number of items in each group $G$ assigned to configuration-type $C$; at most $k(G)-|C \\cap G|$ items in $G$ can be added to $C$ without violating the cardinality constraint of $G$. Constraints~(\\ref{F3}) bound the number of items assigned to slot-type $j \\in I$ by the total selection of configurations containing $j$ in $\\bar{x}$.\n\tFinally, constraints~(\\ref{F4}) guarantee that each item is fully assigned to the types.\n\tWe note that if a prototype $\\bar{x}$ is a solution for (\\ref{C-LP}) then\n\tthe $\\bar{x}$-polytope is non-empty;\n\tin particular, it contains the point \n\t$\\bar{\\gamma}$\n\twhere $\\bar{\\gamma}_{\\ell, j}=1$ $\\forall \\ell, j \\in I$ \n\tsuch that \n\t$\\ell=j$, and $\\bar{\\gamma}_{\\ell, t}=0$ otherwise.\n\t \nLet $\\textsf{supp}(\\bar{x}) = \\{C \\in \\mathcal{C}~|~ \\bar{x}_C>0\\}$ be the {\\em support} of $\\bar{x}$. Throughout this paper, we use prototypes $\\bar{x}$ for which $\\textsf{supp}(\\bar{x})$ is polynomial in the input size; thus, these prototypes have\nsparse representations.\nThe next lemma shows that if the prototype $\\bar{x}$ has a small support, and each configuration in the support contains a few items then the vertices of the $\\bar{x}$-polytope are almost integral.\nThus, given a vertex $\\bar{\\lambda}$ of such $\\bar{x}$-polytope, the items assigned fractionally by $\\bar{\\lambda}$ can be packed using only a small number of extra bins. \n\n\n \\begin{lemma}\n \t\\label{O(1)}\n \tLet  ${\\mathcal{I}}$ be a BPP instance, $k \\geq 1$ an integer and $\\bar{x}$ a prototype of $\\mathcal{I}$ such that $|C| \\leq k$\n \tfor all $C \\in \\textnormal{\\textsf{supp}}(\\bar{x})$, and $\\bar{x}_C \\in \\mathbb{N}$.\n \tThen given a vertex $\\bar{\\lambda}$\n \tof the $\\bar{x}$-polytope for which constraints \\eqref{F4} hold with equality, $$\\left|\\left\\{\\ell \\in I~|~ \\exists t \\in I \\cup {\\mathcal{C}} \\text{ s.t. } \\bar{\\lambda}_{\\ell,t} \\in (0,1)\\right\\}\\right| \\leq 8k^2 \\cdot |\\textnormal{\\textsf{supp}}(\\bar{x})|^2.$$ \n \\end{lemma}\n\nThe proof of Lemma~\\ref{O(1)} bears some similarity to a proof of~\\cite{DKS21_arXiv}, which shows the integrality properties of a somewhat different polytope. We give the complete proof in Appendix~\\ref{sec:poly}. We now describe the main components of our scheme, which converts an initial prototype $\\bar{x}$ (defined by a solution for the configuration-LP) into another prototype $\\bar{z}$, and then constructs a packing based on $\\bar{z}$. The necessary conditions for a prototype allowing to construct an efficient packing are given below.  Let $Q:(0,0.1)\\to \\mathbb{R}$ where $Q = \\exp({\\varepsilon}^{-17})$ for all ${\\varepsilon}\\in (0,0.1)$. \n\n\\begin{definition}\n\t\\label{def:goodprototype}\n\tGiven ${\\varepsilon} \\in (0, 0.1)$ and an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}}$, a prototype $\\bar{x}$ of ${\\mathcal{I}}$ is a {\\em good prototype} if the $\\bar{x}$-polytope is non-empty, $|\\textnormal{\\textsf{supp}}(\\bar{x})| \\leq Q({\\varepsilon})$, and $|C| \\leq {\\varepsilon}^{-10}$ for all $C \\in \\textnormal{\\textsf{supp}}(\\bar{x})$. \n\\end{definition}\nObserve that a solution $\\bar{x}$ for the configuration-LP of $\\mathcal{I}$\nis not necessarily a good prototype, since it may have a support of large size.\nGiven this initial prototype, our scheme generates a good prototype in two steps.\nIn the first step, algorithm {\\sf Evict} constructs an intermediate prototype $\\bar{y}$ \nwhich selects only configurations containing\na small number of items. This results in a small increase in the total number of bins used. Also, the $\\bar{y}$-polytope is non-empty. Given ${\\varepsilon}>0$, we say that an item $\\ell\\in I$ is ${\\varepsilon}$-large if $s(\\ell)\\geq {\\varepsilon}^2$. We use $L({\\varepsilon},{\\mathcal I})$ to denote the set of ${\\varepsilon}$-large items of an instance ${\\mathcal I}$. If ${\\mathcal I}$ and ${\\varepsilon}$ are known by context we simply use $L$ (instead of $L({\\varepsilon},{\\mathcal I})$). The properties of $\\bar{y}$ are summarized in the next lemma (see the details of algorithm {\\sf Evict} in Section~\\ref{sec:eviction}).\n \n\n\\begin{lemma}\n\t\\label{lem:eviction}\n\tThere is an algorithm $\\textsf{Evict}$ that given ${\\varepsilon} \\in (0,0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}}$,  and a solution $\\bar{x}$ for the configuration-LP~\\eqref{C-LP}, returns in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a prototype $\\bar{y}$ which satisfies the following. (i) There exists $\\bar{\\gamma}$ in the $\\bar{y}$-polytope such that $\\bar{\\gamma}_{\\ell,j} = 0$ for all $\\ell,j \\in I, \\ell \\neq j$; (ii) for all $C \\in \\textnormal{\\textsf{supp}}(\\bar{y})$ it holds that $|C| \\leq {\\varepsilon}^{-10}$, and $s(C \\setminus L) \\leq {\\varepsilon}$; (iii) $\\|\\bar{y}\\| \\leq (1+{\\varepsilon})\\|\\bar{x}\\| ; $ (iv) $\\sum_{C\\in {\\mathcal{C}}[\\ell]} {\\bar{y}}_C\\leq 2$ for every $\\ell \\in I$.\n\\end{lemma}\n\n\nObserve  that  property (i) of Lemma~\\ref{lem:eviction}\tallows  $\\bar{\\gamma}_{\\ell,C}>0$ for item $\\ell \\in  \tI$ and a set of slots $C\\in {\\mathcal{C}}$. Note that \\textsf{Evict} does not return the vector $\\bar{\\gamma}$ and only guarantees its existence. \nIn the second step, our scheme uses algorithm $\\textsf{Shift}$ to obtain a good prototype. This is done by a novel constructive use of fractional grouping over a small subset of carefully chosen groups. This step is formalized in the next lemma.\n\n\\begin{lemma}\n\\label{lem:Cnf}\t\nGiven ${\\varepsilon} \\in (0, 0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, let ${\\mathcal{I}}$ be an ${\\varepsilon}$-structured BPP instance. Furthermore, let $\\bar{y}$ be a prototype of $\\mathcal{I}$ with non-empty $\\bar{y}$-polytope which satisfies the following. $(i)$ For all $C \\in \\textsf{supp}(\\bar{y})$ it holds that $|C| \\leq {\\varepsilon}^{-10}$ and $s(C \\setminus L) \\leq {\\varepsilon}$;\n$(ii)$ there exists $\\bar{\\gamma}$ in the $\\bar{y}$-polytope such that \n$\\bar{\\gamma}_{\\ell,j} = 0$ for all $\\ell,j \\in I$ where $\\ell \\neq j$; (iii) $\\sum_{C\\in {\\mathcal{C}}[\\ell]} {\\bar{y}}_C\\leq 2$ for every $\\ell \\in I$. Then, given ${\\mathcal{I}}$, $\\bar{y}$, and ${\\varepsilon}$, Algorithm $\\textsf{Shift}$ returns in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a good prototype $\\bar{z}$ such that $\\|\\bar{z}\\| \\leq (1+5{\\varepsilon})\\|\\bar{y}\\|+Q({\\varepsilon})$.\n\\end{lemma}\n\nAlgorithm $\\textsf{Shift}$ is presented in Section~\\ref{sec:AlgPrototype}. Given a good prototype $\\bar{z}$, our scheme proceeds to find a partition of the items into slot-types and configuration-types using the following construction. \n\n\n For configurations $S,C \\in \\mathcal{C}$, we say that $S$ is {\\em allowed} in $C$ if each item $\\ell \\in S$ can be mapped to a distinct slot $j \\in C$, such that $\\ell \\in \\textsf{fit}(j)$. \n We consider a packing of a subset of $I$ such that the bins in the packing are partitioned into a bounded number of {\\em categories}. Each category is associated with \n $(i)$ a configuration $C \\in {\\mathcal{C}}$ such that all bins in the category are allowed in $C$, and $(ii)$ a {\\em completion}: a subset of (unpacked) items bounded by total size and number of items per group, where each item fits with $C$. Also, we require that each item $\\ell \\in I$ is either in this packing or in a completion of a category. The above constraints are analogous to constraints \\eqref{F1}-\\eqref{F4} of the $\\bar{x}$-polytope, that is used for finding  an assignment of the items to slots and configurations. Formally, \n \n \n \n \\begin{definition}\n \t\\label{def:partition}\n \t\n \tGiven ${\\varepsilon} \\in (0, 0.1)$ and an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}} = (I,{\\mathcal{G}},s,k)$, an ${\\varepsilon}$-{\\em nice partition} $\\mathcal{B}$ of $\\mathcal{I}$ is a packing $(A_1,..., A_m)$ of a subset of $I$, a subset of configurations $\\mathcal{H} \\subseteq {\\mathcal{C}}$, categories $\\left(B_C\\right)_{C \\in \\mathcal{H}}$, and completions $\\left(D_C\\right)_{C \\in \\mathcal{H}}$ such that the following conditions hold.\n \t\n \t\\begin{itemize}\n \t\t\\item $|\\mathcal{H}| \\leq {\\varepsilon}^{-22}Q^2({\\varepsilon})$.\n \t\t\n \t\t\\item $\\{B_C\\}_{C \\in \\mathcal{H}}$ is a partition of $\\{A_i~|~i \\in [m]\\}$\n \t\t\n \t\t\\item $\\{D_C\\}_{C \\in \\mathcal{H}}$ is a partition of $I \\setminus \\bigcup_{i \\in [m]} A_i$.\n \t\t\n \t\t\\item For any $C \\in \\mathcal{H}$ and $A \\in B_C$ it holds that $A$ is allowed in $C$\n \t\t\n \t\t\\item  For any $C \\in \\mathcal{H}$ and $G \\in {\\mathcal{G}}$ it holds that:\n \t\t\n \t\t\\begin{enumerate}\n \t\t\t\\item  $D_C \\subseteq \\textnormal{\\textsf{fit}}(C)$.\n \t\t\t\n \t\t\t\\item $s(D_C) \\leq \\left(1-s(C)\\right) \\cdot |B_C|$.\n \t\t\t\n \t\t\t\\item $|D_C \\cap G| \\leq |B_C| \\cdot \\left(k(G)-|C \\cap G|\\right)$.\n \t\t\\end{enumerate}\n \t\\end{itemize} \n \n The {\\em size} of $\\mathcal{B}$ is $m$, and for all $C \\in \\mathcal{H}$ we say that $B_C$ is the category of $C$ and $D_C$. \n \\end{definition}  \n \n  Algorithm {\\sf Partition} initially rounds up the entries of $\\bar{z}$ to obtain the prototype $\\bar{z}^*$.\n  It then finds a vertex $\\bar{\\lambda}$ of the $\\bar{z}^*$-polytope, which is almost integral by  Lemma~\\ref{O(1)}. Thus, with the exception of a small number of items, each item is fully assigned either to a slot or to a configuration. Algorithm {\\sf Partition} uses $\\bar{\\lambda}$ to construct an ${\\varepsilon}$-nice partition. We generate a category for each $C \\in \\textnormal{\\textsf{supp}}(\\bar{z}^*)$ and define $D_C = \\{ \\ell \\in I~| \\bar{\\lambda}_{\\ell,C}=1 \\}$ to be the set of all items assigned to $C$. We also generate $\\bar{z}^*_C$ copies (bins) of each configuration and replace its slots by items via matching.\n  \n\\begin{lemma}\n\t\\label{FromPolytope}\n\tThere is an algorithm $\\textsf{Partition}$ that given ${\\varepsilon} \\in (0, 0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}}$, and a good prototype $\\bar{z}$ of $\\mathcal{I}$, returns  in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ an ${\\varepsilon}$-nice partition of $\\mathcal{I}$ of size at most $\\|\\bar{z}\\|+{\\varepsilon}^{-22}Q^2({\\varepsilon})$.\n\\end{lemma}\n\nAlgorithm $\\textsf{Partition}$ is presented in Section~\\ref{sec:FromPolytope}. Given an ${\\varepsilon}$-nice partition of size $m$, a packing of the instance in roughly $m$ bins is obtained using the next lemma. \n\n\\begin{lemma}\n \\label{lem:GREEDY}\n There is a polynomial-time algorithm \\textsf{Pack} which given ${\\varepsilon} \\in (0, 0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}}$, and ${\\varepsilon}$-nice partition of $\\mathcal{I}$ of size $m$, returns  in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a packing of $\\mathcal{I}$ in at most $(1+2{\\varepsilon})m+2{\\varepsilon}^{-22}Q^2({\\varepsilon})$ bins. \n \\end{lemma} \n\n\nAlgorithm \\textsf{Pack} utilizes Algorithm \\textsf{Greedy} to add the items in a completion of a category to the bins of this category, possibly using a few extra bins. Algorithm \\textsf{Pack} is presented in Section~\\ref{sec:alg_pack} and Algorithm \\textsf{Greedy} is presented in Section~\\ref{sec:greedy}. Using the above components, we construct an AFPTAS for ${\\varepsilon}$-structured BPP instances. The pseudocode of the scheme is given in Algorithm~\\ref{alg:Fscheme}. We summarize in the next result.\n\n\\begin{algorithm}[h]\n\t\\caption{$\\textsf{AFPTAS}({{\\mathcal I}}, {\\varepsilon})$}\n\t\\label{alg:Fscheme}\n\t\n\t\\SetKwInOut{Input}{Input}\n\t\t\\SetKwInOut{Output}{Output}\n\t\\Input{An ${\\varepsilon}$-structured instance ${\\mathcal I}$ and ${\\varepsilon}\\in (0,0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$}\n\t\\Output{A packing of ${\\mathcal I}$}\n\t\n\t\n\tFind a solution for the configuration-LP of $\\mathcal{I}$; that is, $\\bar{x} = \\textsf{SolveLP}({\\varepsilon},\\mathcal{I})$\\label{LP}  \n\t\n\t\n\tLet $\\bar{y} = \\textsf{Evict}({\\varepsilon},{\\mathcal{I}},\\bar{x})$ \\label{step:evicAFPTAS}\n\t\n\tFind a good prototype $\\bar{z} = \\textsf{Shift}({\\varepsilon},{\\mathcal{I}},\\bar{y})$ \\label{GetPolytope}\n\t\n\t\n\tFind an ${\\varepsilon}$-nice partition $\\mathcal{B}$ of $\\mathcal{I}$ by $\\textsf{Partition}({\\varepsilon},{\\mathcal{I}},\\bar{z})$ \\label{step:BPPnice}\n\t\t\n\n\tReturn \n\t $\\Phi = \\textsf{Pack}({\\varepsilon},\\mathcal{I},\\mathcal{B})$  \\label{step:greedy1}\n\n\t\n\n\\end{algorithm}\n\n\n\n\n\n\n\n\n \n\\begin{lemma}\n\\label{lem:AFPTAS}\nGiven ${\\varepsilon} \\in (0, 0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, and an ${\\varepsilon}$-structured  BPP instance ${\\mathcal{I}}$, Algorithm~\\ref{alg:Fscheme} returns in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a packing of  ${\\mathcal I}$ in at most $(1+60{\\varepsilon})\\textnormal{OPT}({\\mathcal I})+Q^3({\\varepsilon})$ bins.\n\\end{lemma}\n\n\nThe proof of Lemma~\\ref{lem:AFPTAS} follows immediately from Lemmas~ \\ref{configurationLP},  ~\\ref{lem:eviction}, \\ref{lem:Cnf}, ~\\ref{FromPolytope}, and \\ref{lem:GREEDY}. \nA formal proof is given in Appendix~\\ref{app:omitted}. \nThe next lemma is an immediate consequence of Lemmas~\\ref{lem:AFPTAS} and~\\ref{lem:reductionReconstruction}. \n\\begin{lemma}\n\t\\label{lem:gen_afptas}\nThere is an algorithm $\\textnormal{\\textsf{Gen-AFPTAS}}$ which given a BPP instance ${\\mathcal{J}}$ and ${\\varepsilon}\\in (0,0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, returns in time $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$ a packing of ${\\mathcal{J}}$ using at most $(1+130{\\varepsilon})\\cdot \\textnormal{OPT}({\\mathcal{J}}) + 3\\cdot  Q^3({\\varepsilon})$  bins. \n\\end{lemma}\nWe give the proof of Lemma~\\ref{lem:gen_afptas} in Appendix~\\ref{app:omitted}. \nGiven a BPP instance $\\mathcal{J}$, Theorem~\\ref{thm:main} follows by applying $\\textnormal{\\textsf{Gen-AFPTAS}}$ to $\\mathcal{J}$ taking ${\\varepsilon} = \\left(  \\ln \\ln W \\right)^{-\\frac{1}{17}}$, where \n$W=s(I) + V({\\mathcal I}) +\\exp(\\exp(100^{17}))=\\Theta(\\textnormal{OPT}({\\mathcal{J}}))$. By the above selection of ${\\varepsilon}$, it holds that \n$Q^3({\\varepsilon})=o(\\textnormal{OPT}(\\mathcal{J}))$ and ${\\varepsilon}  \\textnormal{OPT}(\\mathcal{J})=o(\\textnormal{OPT}\\left( \\mathcal{J}) \\right)$. This yields the  $o(\\textnormal{OPT}({\\mathcal I})))$ additive approximation ratio. The proof of Theorem~\\ref{thm:main} is given in Appendix~\\ref{app:omitted}. \n\n\n\n\n\n\n\n\n\n\n\\section{Omitted Proofs of Section~\\ref{sec:FromPolytope}}\n\\label{app:omittedFromPolytope}\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:matchingU}:}\nFor all $\\ell \\in U$, let $d(\\ell) = |\\{v \\in V~|~ (\\ell,v) \\in E\\}|$ be the degree of $\\ell$ in $G$. For all $\\ell \\in U$ there is exactly one $j \\in I$ such that $\\bar{\\gamma}_{\\ell,j} = 1$ because $\\bar{\\gamma}$ satisfies constraint \\eqref{F4} of the $\\bar{z}^*$-polytope with equality; let $\\ell^* = j$. Therefore, for every $\\ell \\in U$ it holds:\n\n\n\\begin{equation}\n\t\\label{welldegree}\n\td(\\ell) =  |\\{ (C,\\ell^*,k) \\in V~|~ (\\ell,(C,\\ell^*,k)) \\in E\\}| = \\sum_{C \\in {\\mathcal{C}}[\\ell^*]~} \\sum_{k \\in [\\bar{z}^*_C]} 1 =  \\sum_{C \\in {\\mathcal{C}}[\\ell^*]} \\bar{z}^*_C \\geq \\sum_{\\ell' \\in I} \\bar{\\gamma}_{\\ell',\\ell^*} \\geq \\bar{\\gamma}_{\\ell,\\ell^*} = 1. \n\\end{equation} The first inequality is because $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope. Define a fractional matching as a vector $\\bar{M} \\in [0,1]^E$ such that for all $(\\ell,v) \\in E$ define $\\bar{M}_{(\\ell,v)} = \\frac{1}{d(\\ell)}$; note that $\\bar{M}$ is well-defined (i.e., for all $\\ell \\in U$ it holds that $d(\\ell)>0$) by \\eqref{welldegree}. We show below that $\\bar{M}$ is a fractional matching in $G$ of size~$|U|$. \n\n\\begin{enumerate}\n\t\\item $\\bar{M}$ is a feasible fractional matching in $G$. Let $\\ell' \\in U$. It holds that:\n\t\n\t$$\\sum_{(\\ell',v) \\in E} \\bar{M}_{(\\ell',v)} = \\sum_{(\\ell',v) \\in E}  \\frac{1}{d(\\ell)}  = d(\\ell) \\cdot \\frac{1}{d(\\ell)}=1.$$ In addition, let $(C,j,k) \\in V$ and let $v = (C,j,k)$. It holds that:\n\t\n\t\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\sum_{(\\ell,v) \\in E} \\bar{M}_{(\\ell,v)} ={} & \\sum_{(\\ell,v) \\in E}  \\frac{1}{d(\\ell)} = \\sum_{\\ell \\in U \\text{ s.t. } \\bar{\\gamma}_{\\ell,j} = 1}  \\frac{1}{d(\\ell)} \\leq \\sum_{\\ell \\in U \\text{ s.t. } \\bar{\\gamma}_{\\ell,j} = 1} \\bar{\\gamma}_{\\ell,j} \\cdot \\frac{1}{d(\\ell)}\\\\ \\leq{} &\n\t\t\t\\sum_{\\ell \\in U \\text{ s.t. } \\bar{\\gamma}_{\\ell,j} = 1}  \\bar{\\gamma}_{\\ell,j} \\cdot \\frac{1}{\\sum_{\\ell' \\in I} \\bar{\\gamma}_{\\ell',j}} \\leq  \\frac{1}{\\sum_{\\ell' \\in I} \\bar{\\gamma}_{\\ell',j}} \\cdot \\sum_{\\ell \\in I} \\bar{\\gamma}_{\\ell,j} = 1.\n\t\t\\end{aligned}\n\t\\end{equation*} The second inequality is by \\eqref{welldegree}. By the above, we conclude that $\\bar{M}$ is a feasible fractional matching in $G$.\n\t\n\t\\item $|\\bar{M}| = |U|$. $$|\\bar{M}| =  \\sum_{e \\in E}\\bar{M}_{e} =  \\sum_{\\ell \\in U} \\sum_{(\\ell,v) \\in E} \\bar{M}_{e} =  \\sum_{\\ell \\in U} \\sum_{(\\ell,v) \\in E} \\frac{1}{d(\\ell)}  = \\sum_{\\ell \\in U} \\frac{d(\\ell)}{d(\\ell)}=  |U|.$$\n\\end{enumerate} By the above, $\\bar{M}$ is a fractional matching in $G$ with size $|U|$. Since $G$ is a bipartite graph, there is an integral matching with size $|U|$ in $G$ \\cite{schrijver2003combinatorial}.  \\qed\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:allowed}:}\nFor all $\\ell \\in A_{C,k}$ there is exactly one $j \\in C$ such that $(\\ell,(C,j,k)) \\in M$ by the definition of $A_{C,k}$ and because $M$ is a matching; we define $\\ell^* = j$ to be the {\\em slot of } $\\ell$. We construct an injective function $\\sigma: A_{C,k} \\rightarrow C$ such that  for all $\\ell \\in A_{C,k}$ define $\\sigma(\\ell) = \\ell^*$. By the above, for all $\\ell \\in A_{C,k}$ it holds that $\\ell^*$ is well defined and it follows that $\\sigma$ is a function. Let $\\ell_1,\\ell_2 \\in A_{C,k}$ such that $\\ell_1 \\neq \\ell_2$. Since $M$ is a matching, it follows that $\\ell^*_1 \\neq \\ell^*_2$; hence, $\\sigma(\\ell_1) \\neq \\sigma(\\ell_2)$ and we conclude that $\\sigma$ is injective. Finally, let $\\ell \\in A_{C,k}$. Since $(\\ell,(C,\\ell^*,k)) \\in M$, by the definition of $E$ it follows that $\\bar{\\gamma}_{\\ell,\\ell^*} = 1$; thus, $\\ell \\in \\textsf{fit}(\\ell^*)$ by \\eqref{F1} since $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope. \\qed\n\n\n\n\n\n\n\n\n\n\n\\noindent{\\bf Proof of Claim~\\ref{eqeq6}:}\nWe prove the three properties of the claim below. \n\n\\begin{enumerate}\n\t\\item $D_C \\subseteq \\textsf{fit}(C)$: Let $\\ell \\in D_C$. By the definition of $D_C$ it holds that $\\bar{\\gamma}_{\\ell,C} = 1$, and in particular $\\bar{\\gamma}_{\\ell,C} >0$; thus, since $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope it follows by \\eqref{F1} that $\\ell \\in \\textsf{fit}(C)$. \n\t\n\t\\item $s(D_C) \\leq \\left(1-s(C)\\right) \\cdot |B_C|$:\n\t\\begin{equation*}\n\t\ts(D_C) = \\sum_{\\ell \\in I \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} s(\\ell) =    \\sum_{\\ell \\in I \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} \\bar{\\gamma}_{\\ell,C} \\cdot s(\\ell) \\leq   \\sum_{\\ell \\in I} \\bar{\\gamma}_{\\ell,C} \\cdot s(\\ell)  \n\t\t\\leq (1-s(C)) \\bar{z}^*_C \\leq  \\left(1-s(C)\\right) \\cdot |B_C|. \n\t\\end{equation*} The second inequality is because $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope; thus, the inequality follows by \\eqref{F2}. The last inequality is by \\eqref{BC}. \n\t\n\t\n\t\n\t\\item  $|D_C \\cap G| \\leq |B_C| \\cdot \\left(  k(G) - |G \\cap C|  \\right)$:\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t|D_C \\cap G| ={} & \\left|\\bigcup_{\\ell \\in I \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} \\{\\ell\\} \\cap G \\right| \\leq \\sum_{\\ell \\in I \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} \\left| \\{\\ell\\} \\cap G \\right| = \\sum_{\\ell \\in G \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} \\left| \\{\\ell\\} \\right|\\\\\n\t\t\t={} & \\sum_{\\ell \\in G \\text{ s.t. } \\bar{\\gamma}_{\\ell,C} = 1} \\bar{\\gamma}_{\\ell,C} \\leq \\sum_{\\ell \\in G} \\bar{\\gamma}_{\\ell,C} \\leq \\bar{z}^*_C \\cdot \\left(  k(G) - |G \\cap C|  \\right) \\leq |B_C| \\cdot \\left(  k(G) - |G \\cap C|  \\right).  \n\t\t\\end{aligned} \n\t\\end{equation*} The third inequality is because $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope; thus, the inequality follows by \\eqref{F5}. The last inequality is by \\eqref{BC}. \\qed\n\t\n\\end{enumerate} \n\n\n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:assign}:} \nWe divide the proof of Lemma~\\ref{lem:assign} to the following claims. For simplicity, let $A^* = (A_1, \\ldots, A_m)$.\n\n\\begin{claim}\n\t\\label{eqeq1}\n\t$A^*$ is a packing.\n\\end{claim} \n\n\\begin{proof}\n\tLet $i \\in [m]$. By \\eqref{Astar}, we split into two cases.  \n\t\n\t\\begin{enumerate}\n\t\t\\item $A_i \\in F$.  Then, by the definition of $F$ it holds that $A_i$ is a singleton, a set containing a single item. Thus, $s\\left(A_i\\right) \\leq 1$ and for all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap A^*_b| \\leq 1 \\leq k(G)$.\n\t\t\n\t\t\\item  There are $C \\in C^*$ and $k \\in [\\bar{z}^*_{C^*}]$ such that $A_i = A_{C,k}$. Therefore, by Lemma~\\ref{lem:allowed} it holds that $A^*_b$ is allowed in $C$. Hence, there is an injective function $\\sigma: A_i \\rightarrow C$ such that for all $\\ell \\in A_i$ it holds that $\\ell \\in \\textsf{fit}\\left(\\sigma(\\ell)\\right)$. Therefore, \n\t\t\n\t\t\\begin{enumerate}\n\t\t\t\\item $s(A_i) \\leq 1$. \t$$s(A_i) = \\sum_{\\ell \\in A_i} s(\\ell) \\leq  \\sum_{\\ell \\in A_i} s\\left(\\sigma(\\ell)\\right) \\leq \\sum_{\\ell \\in C} s(\\ell) = s(C) \\leq 1.$$ The first inequality is because for all $\\ell \\in A_i$ it holds that $\\ell \\in \\textsf{fit}\\left(\\sigma(\\ell)\\right)$; thus, the inequality follows by \\eqref{fitl}. The second inequality is because $\\sigma$ is injective. The last inequality is because $C$ is a configuration. \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\item For all $G \\in {\\mathcal{G}}$ it holds that $|A_i \\cap G| \\leq k(G)$. Let $G \\in {\\mathcal{G}}$. Now, $$|A_i \\cap G| = \\left|\\bigcup_{\\ell \\in A_i} \\{\\ell\\} \\cap G\\right| = \\bigcup_{\\ell \\in A_i} \\left|\\{\\ell\\} \\cap G\\right| \\leq \\bigcup_{\\ell \\in A_i} \\left|\\{\\sigma(\\ell)\\} \\cap G\\right|  \\leq    \\left|\\bigcup_{\\ell \\in C} \\{\\ell\\} \\cap G\\right| = |C \\cap G| \\leq k(G).$$ The first inequality is because $\\sigma$ is injective and that $G \\cap M = \\emptyset$; thus, by \\eqref{fitl} for all $\\ell \\in A_i \\cap G$ it holds that $\\sigma(\\ell) \\in A_i \\cap G$. The second inequality is because $\\sigma$ is injective. The last inequality is because $C$ is a configuration. \n\t\t\\end{enumerate}\n\t\t\n\t\t\n\t\t\n\t\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{eqeq2}\n\t$|\\mathcal{H}| \\leq {\\varepsilon}^{-22}Q^2({\\varepsilon})$ \n\\end{claim} \n\n\\begin{proof}\n\t$$|\\mathcal{H}| = \\left|\\textsf{supp}(\\bar{z}^*) \\cup F\\right| \\leq  \\left|\\textsf{supp}(\\bar{z}^*)\\right| + \\left| F\\right| \\leq |\\textsf{supp}(\\bar{z}^*)|+{\\varepsilon}^{-21}|\\textsf{supp}(\\bar{z}^*)|^2 \\leq {\\varepsilon}^{-22}|\\textsf{supp}(\\bar{z})|^2 \\leq  {\\varepsilon}^{-22}Q^2({\\varepsilon}).$$\n\t\n\tThe second inequality is because $\\bar{z}^*$ holds the conditions of Lemma~\\ref{FromPolytope} by Observation~\\ref{ob:y}; thus, the inequality follows by Lemma~\\ref{O(1)}. The third inequality is because $0<{\\varepsilon}<0.1$ and by Observation~\\ref{ob:y}. The last inequality is because $\\bar{z}$ holds the conditions of Lemma~\\ref{FromPolytope}, that is, $\\bar{z}$ is a good prototype.\n\\end{proof}\n\n\n\\begin{claim}\n\t\\label{eqeq3}\n\tThe size of $\\mathcal{B}$ is at most $\\|\\bar{z}\\| + {\\varepsilon}^{-22}|\\textnormal{\\textsf{supp}}(\\bar{z})|^2$.\n\\end{claim} \n\n\\begin{proof}\n\tThe size of $\\mathcal{B}$ is defined as the number of entries in $A^*$. Therefore, by \\eqref{Astar} the size of $\\mathcal{B}$ is at most:\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t|F|+\\sum_{C \\in \\textsf{supp}(\\bar{z}^*)} \\sum_{k \\in [\\bar{z}^*_C]}  ={} & |F|+\\sum_{C \\in \\textsf{supp}(\\bar{z}^*)} \\bar{z}^*_C  \\leq \\|\\bar{z}^*\\| + {\\varepsilon}^{-21}|\\textsf{supp}(\\bar{z}^*)|^2\\\\ \\leq{} & \\|\\bar{z}\\| +|\\textsf{supp}(\\bar{z})|+  {\\varepsilon}^{-21}|\\textsf{supp}(\\bar{z})|^2 \\leq  \\|\\bar{z}\\| + {\\varepsilon}^{-22}|\\textsf{supp}(\\bar{z})|^2 \\leq \\|\\bar{z}\\| + {\\varepsilon}^{-22}Q^2({\\varepsilon}).\n\t\t\\end{aligned}\n\t\\end{equation*} The first inequality is because $\\bar{z}^*$ holds the conditions of Lemma~\\ref{FromPolytope} by Observation~\\ref{ob:y}; thus, the inequality follows by Lemma~\\ref{O(1)}. The second inequality is by Observation~\\ref{ob:y}. The last inequality is because $\\bar{z}$ is a good prototype. \n\t\n\t\n\t\n\t\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{partition:BC}\n\t$\\{B_C\\}_{C \\in \\mathcal{H}}$ is a partition of $\\{A_i~|~i \\in [m]\\}$\n\\end{claim} \n\n\\begin{proof}\n\t\n\tLet $i \\in [m]$. By \\eqref{Astar}, we split into two cases.  \n\t\n\t\\begin{enumerate}\n\t\t\\item $A_i \\in F$.  Then, by the definition of $\\mathcal{H}$ it holds that $A_i \\in \\mathcal{H}$; thus, by \\eqref{BC} it follows that $A_i \\in B_{A_i}$ and for all $C \\in \\mathcal{H} \\setminus \\{A_i\\}$ it holds that $A_i \\notin B_{C}$.\n\t\t\n\t\t\\item  There are $C \\in \\mathcal{H}$ and $k \\in [\\bar{z}^*_{C}]$ such that $A_i = A_{C,k}$. Therefore, by \\eqref{BC} it follows that $A_i \\in B_{C}$ and for all $C' \\in \\mathcal{H} \\setminus \\{C\\}$ it holds that $A_i \\notin B_{C'}$.\n\t\\end{enumerate} \n\t\n\\end{proof}\n\n\nLet $f = \\{\\ell~|~\\{\\ell\\} \\in F\\}$ be all fractional items.  We use the following auxiliary claim. \n\n\n\\begin{claim}\n\t\\label{fU}\n\t$A^*$ is a packing of $f \\cup U$. \n\\end{claim} \n\n\\begin{proof}\n\tLet $\\ell \\in U \\cup f$. We split into two cases.\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\ell \\in f$. By \\eqref{Astar}, there is $i \\in [m]$ such that $A_i = \\{\\ell\\}$.\n\t\t\n\t\t\\item  $\\ell \\in U$. Since $M$ is a matching of size $|U|$ in $G$, there is $(C,j,k) \\in V$ such that $(\\ell,(C,j,k)) \\in M$. Therefore, by the definition of the bin of $C$ and $k$ it holds that $\\ell \\in A_{C,k}$. Hence, by \\eqref{Astar} there is $i \\in [m]$ such that $A_i = A_{C,k}$ and the claim follows.\n\t\\end{enumerate} Moreover, by \\eqref{Astar} it follows that for any $\\ell \\in I \\setminus (U \\cup f)$ and $i \\in [m]$ it holds that $\\ell \\notin A_i$.  \n\\end{proof}\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{eqeq4}\n\t$\\{D_C\\}_{C \\in \\mathcal{H}}$ is a partition of $I \\setminus \\bigcup_{i \\in [m]} A_i$.\n\\end{claim} \n\n\\begin{proof}\n\t\n\tLet $\\ell \\in I \\setminus \\bigcup_{i \\in [m]} A_i$. By Claim~\\ref{fU}, it follows that $I \\setminus \\bigcup_{i \\in [m]} A_i = I \\setminus (U \\cup f)$ and we get $\\ell \\in I \\setminus (U \\cup f)$. Thus, as $\\ell$ is not fractional and is not fully assigned to a slot type as it is not in $U$,  it follows that there is $C \\in \\textsf{supp}(\\bar{z}^*)$ such that $\\bar{\\gamma}_{\\ell,C} = 1$ by \\eqref{F4} since $\\bar{\\gamma}$ is in the $\\bar{z}^*$-polytope. In addition, since \\eqref{F4} holds with equality, we get that there is exactly one such $C$. Since $\\textsf{supp}(\\bar{z}^*) \\subseteq \\mathcal{H}$, it follows that $\\ell \\in D_C$ and for all  $C' \\in \\mathcal{H} \\setminus \\{C\\}$ it holds that $\\ell \\notin D_{C'}$. Hence, the claim follows. \n\t\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{eqeq5}\n\tFor any $C \\in \\mathcal{H}$ and $A \\in B_C$ it holds that $A$ is allowed in $C$\n\\end{claim} \n\n\\begin{proof}\n\t\n\tBy Claim~\\ref{partition:BC}, there is $i \\in [m]$ such that $A = A_i$. By \\eqref{Astar}, we split into two cases.  \n\t\n\t\\begin{enumerate}\n\t\t\\item $A_i \\in F$.  Then, by \\eqref{BC} it holds that $A_i = C$ and $C$ is a singleton. Then, it trivially follows that $A_i$ is allowed in $A_i$, since for each item $\\ell \\in A_i$ it holds that $\\ell \\in \\textsf{fit}(\\ell)$ by \\eqref{fitl}.  \n\t\t\n\t\t\\item  There are $C \\in \\mathcal{H}$ and $k \\in [\\bar{z}^*_{C}]$ such that $A_i = A_{C,k}$. Therefore, by Lemma~\\ref{lem:allowed} the claim follows. \n\t\\end{enumerate} \n\t\n\\end{proof}\n\n\n\nThe proof  of Lemma~\\ref{lem:assign} follows by Claim~\\ref{eqeq1}, Claim~\\ref{eqeq2}, Claim~\\ref{eqeq3}, Claim~\\ref{partition:BC}, Claim~\\ref{eqeq4}, Claim~\\ref{eqeq5}, and Claim~\\ref{eqeq6}. \\qed\n\n\n\\section{Omitted Proofs of Section~\\ref{sec:eviction}}\n\\label{app:omittedEvict}\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:evictionHelp}:}\n\tLet $\\ell_1, \\ldots, \\ell_{r}$ be the items in $C \\setminus L$ in decreasing order such that for all $a,b \\in [r], a<b$ it holds that $\\ell_a < \\ell_b$. In addition, let $U_C~=~\\{\\ell_1, \\ldots, \\ell_h\\}$ such that $h = \\alpha$. \tFor the simplicity of the notations, for any $j \\in [h]$ let $T_j =s \\left( C \\cap L \\cup \\{\\ell_1, \\ldots, \\ell_{j-1}\\} \\right)$ be the total size of the large items in $C$ and the first $j-1$ small items in $C$ according to the above sorted order of items.\\footnote{With a slight abuse of notation, assume that $\\{\\ell_1, \\ldots, \\ell_{j-1}\\} = \\emptyset$ for $j=1$.} We use the following auxiliary claims.\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{claim:C1}\n\t\t For all $i,k \\in [h]$ which satisfy that $k>i+{\\varepsilon}^{-4}$, it holds that $s(\\ell_i) > \\frac{s(\\ell_k)}{{\\varepsilon}^2}$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t Assume towards a contradiction that there are $i,k \\in [h]$ which satisfy that $k>i+{\\varepsilon}^{-4}$ and $s(\\ell_i) \\leq \\frac{s(\\ell_k)}{{\\varepsilon}^{2}}$. Thus, \n\t\t\n\t\t\\begin{equation}\n\t\t\t\\label{eq:sik}\n\t\t\ts(\\{\\ell_i, \\ell_{i+1}, \\ldots, \\ell_k\\}) \\geq (k-i) \\cdot s(\\ell_k) \\geq {\\varepsilon}^{-4} \\cdot s(\\ell_k) \\geq {\\varepsilon}^{-4}  \\cdot {\\varepsilon}^{2} \\cdot  s(\\ell_i) =  {\\varepsilon}^{-2}  s(\\ell_i) \n\t\t\\end{equation} The first inequality is because the items are in decreasing order that induces a non increasing order of item sizes. The second inequality is because $k>i+{\\varepsilon}^{-4}$. The third inequality is by the assumption that $s(\\ell_i) \\leq \\frac{s(\\ell_k)}{{\\varepsilon}^{2}}$. \n\t\t\n\t Note that it holds that $T_i<1$ since $T_i$ is the total size of items in a proper subset of $C$ and the sizes of the items are strictly larger than zero. Now, because $\\ell_i \\in U_C$ and $i < \\alpha$, we conclude by \\eqref{fitS}  and \\eqref{h} that \\begin{equation}\n\t\t\t\\label{eq:llli}\n\t\t\ts(\\ell_i) > {\\varepsilon} \\left( 1-T_i  \\right).\n\t\t\\end{equation}  Therefore, \n\t\t\\begin{equation}\n\t\t\t\\label{eq:contr}\n\t\ts(C) \\geq T_i  +s(\\{\\ell_i, \\ldots, \\ell_k\\}) \\geq  T_i  +{\\varepsilon}^{-1} \\left( 1-T_i  \\right) \\geq T_i  +\\left( 1-T_i  \\right)+({\\varepsilon}^{-1}-1)  ( 1-T_i) > 1.\n\t\t\\end{equation}\n\t\tThe first inequality is because $C \\cap L \\cup \\{\\ell_1, \\ldots, \\ell_{i-1}\\}$ is disjoint to $\\{\\ell_i, \\ldots, \\ell_k\\}$. The second inequality is by \\eqref{eq:sik} and \\eqref{eq:llli}. The last inequality follows since $0<{\\varepsilon}<0.1$ and $0\\leq T_i<1$. By \\eqref{eq:contr}, we reach a contradiction to the fact that $s(C) \\leq 1$ because $C$ is a configuration. \n\t\\end{proof}\n\t\n\t\n\t\t\\begin{claim}\n\t\t\\label{claim:C1*}\n\t\tFor any $i \\in [h]$ such that $i+{\\varepsilon}^{-4} < h-1$ it holds that $\\ell_i \\in \\mathcal{L}_C$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\\begin{equation}\n\t\t\\label{eq:LC}\n\t\ts(\\ell_i)> \\frac{s(\\ell_{h-1})}{{\\varepsilon}^{2}} > {\\varepsilon}^{-2} \\cdot {\\varepsilon} (1-T_{h-1}) = {\\varepsilon}^{-1} (1-T_{h-1}) \\geq {\\varepsilon}^{-1} (1-s(R(C))).\n\t\\end{equation}\n\t\n\tThe first inequality is by Claim~\\ref{claim:C1}. The second inequality is because  $h-1 < h \\leq \\alpha$; thus, similarly to \\eqref{eq:llli}, it holds that $s(\\ell_{h-1}) > {\\varepsilon} \\left( 1-T_{h-1}  \\right)$. The last inequality is because $T_{h-1}$ is the total size in a proper subset of $R(C)$; thus, $T_{h-1} \\leq s(R(C))$. By \\eqref{eq:LC}, we conclude that for any $i \\in [h]$ such that $i +{\\varepsilon}^{-4}< h-1$ it holds that $\\ell_i \\in \\mathcal{L}_C$.\n\t\\end{proof}\n\n\t\t\n\t Now, using the above claim: \n\t\n\t$$|\\mathcal{L}_C| \\geq h-2-{\\varepsilon}^{-4} \\geq {\\varepsilon}^{-5}- 2{\\varepsilon}^{-4} = {\\varepsilon}^{-4} ({\\varepsilon}^{-1}-2)> {\\varepsilon}^{-4}.$$\n\t\n\tThe first inequality is since by Claim~\\ref{claim:C1*} it holds that $\\ell_1, \\ldots, \\ell_{t} \\in \\mathcal{L}_C$ for $t = h-2-{\\varepsilon}^{-4}$. The second inequality is because $h =  \\alpha$. \\qed \n\t\n\\noindent{\\bf Proof of Lemma~\\ref{lem:wProperties}:}\n\t\tLet $C \\in \\mathcal{C}$. We split the proof into the following claims which together form the proof of the lemma.\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\begin{claim}\n\t\t\t\\label{claim:evic1}\n\t\t\t Condition~1 of Lemma~\\ref{lem:wProperties} holds for $|U_C| = \\alpha$.\n\t\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\t\n\t\t$$\\|\\bar{w}^C\\| = \\sum_{\\ell \\in \\mathcal{L}_C} \\bar{w}^C_{R(C) \\setminus \\{\\ell\\}} =   \\sum_{\\ell \\in \\mathcal{L}_C}  \\frac{1}{|\\mathcal{L}_C|-1}  = \\frac{|\\mathcal{L}_C|}{|\\mathcal{L}_C|-1} \\leq \\frac{{\\varepsilon}^{-4}}{{\\varepsilon}^{-4}-1} \\leq 1+2{\\varepsilon}^4. $$\n\t\t\n\t\tThe first equality is because by the definition $\\bar{w}^C$, for any other entry $C'$ that is not in the second summation it holds that $\\bar{w}^C_{C'} = 0$. The second equality is by the definition of $\\bar{w}^C$ in case that $|U_C| = \\alpha$. The first inequality is because $|U_C| = \\alpha$; therefore, by Lemma~\\ref{lem:evictionHelp} it holds that $|\\mathcal{L}_C| \\geq {\\varepsilon}^{-4}$. The last inequality is because $0<{\\varepsilon}<0.1$. \n\t\\end{proof}\n\t\t\n\t\t\n\t\t\n\t\t\t\\begin{claim}\n\t\t\t\\label{claim:evic2}\n\t\t\t\t Condition~2 of Lemma~\\ref{lem:wProperties} holds for $|U_C| = \\alpha$.\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\t  Let $\\ell \\in R(C)$. Then, \n\t\t\t\n\t\t\t$$\\sum_{C' \\in \\mathcal{C}[\\ell]} \\bar{w}^C_{C'} \\geq \\sum_{j \\in \\mathcal{L}_C \\setminus \\{\\ell\\}} \\bar{w}^C_{R(C) \\setminus \\{j\\}} = \\sum_{j \\in \\mathcal{L}_C \\setminus \\{\\ell\\}} \\frac{1}{|\\mathcal{L}_C|-1} \\geq \\frac{|\\mathcal{L}_C|-1}{|\\mathcal{L}_C|-1} = 1.$$ \n\t\t\t\n\t\t\tThe first inequality is since for any $j \\in \\mathcal{L}_C \\setminus \\{\\ell\\}$ it holds that  $R(C) \\setminus \\{j\\} \\in \\mathcal{C}[\\ell]$ and that all entries in $\\bar{w}^C$ are non-negative. The first equality is by the definition of $\\bar{w}^C$ in case that $|U_C| = \\alpha$. The last inequality is because $|\\mathcal{L}_C \\setminus \\{\\ell\\}| \\geq |\\mathcal{L}_C|-1$. \n\t\t\\end{proof}\n\t\t\n\t\t\t\\begin{claim}\n\t\t\t\\label{claim:evic3}\n\t\t\t\t Condition~3 of Lemma~\\ref{lem:wProperties} holds for $|U_C| = \\alpha$.\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\t  For the third condition of the lemma, let $C ' \\in \\textsf{supp}(\\bar{w}^C)$. By the definition of $\\bar{w}^C$ in case that $|U_C| = \\alpha$, there is $\\ell \\in \\mathcal{L}_C$ such that $C' = R(C) \\setminus \\{\\ell\\}$. Now,\n\t\t\t\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:1-RC}\n\t\t\t\t1-s(R(C)) = 1-s(R(C)\\setminus \\{\\ell\\})-s(\\ell) \\leq 1-s(R(C)\\setminus \\{\\ell\\}) -\\frac{1-s(R(C))}{{\\varepsilon}}.\n\t\t\t\\end{equation}\n\t\t\tThe first equality is because $\\ell \\in \\mathcal{L}_C$ and $\\mathcal{L}_C \\subseteq R(C)$. The first inequality is by the definition of $\\mathcal{L}_C$. Therefore, \n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:1-RC2}\n\t\t\t\t1-s(R(C)) \\leq \\left(\\frac{1}{1+\\frac{1}{{\\varepsilon}}}\\right)(1-s(R(C)\\setminus \\{\\ell\\})) \\leq {\\varepsilon}(1-s(R(C)\\setminus \\{\\ell\\})).\n\t\t\t\\end{equation}\n\t\t\tThe first inequality is by \\eqref{eq:1-RC}. Let $j \\in C \\setminus R(C)$. Now, \n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:fit1}\n\t\t\t\ts(j) \\leq 1-s(R(C)) \\leq  {\\varepsilon}(1-s(R(C)\\setminus \\{\\ell\\})).\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\tThe first inequality is because $j \\notin R(C)$ and $s(C) \\leq 1$. The second inequality is by \\eqref{eq:1-RC2}. Therefore, by \\eqref{eq:fit1} we get that $j \\in  \\textsf{fit}(C')$ by \\eqref{fitS} (because $j \\in C \\setminus R(C)$ it holds that $j \\notin L$). Because there are no restrictions over $j$ besides that $j \\in C \\setminus R(C)$, we conclude that $C \\setminus R(C) \\subseteq \\textsf{fit}(C')$. In addition, \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t$$s(C \\setminus R(C)) \\leq 1-s(R(C)) \\leq 1-s(R(C) \\setminus \\{\\ell\\}) = 1-s(C').$$\n\t\t\t\n\t\t\tThe first inequality is because $R(C) \\subseteq C$ and $s(C) \\leq 1$. The second  inequality is because $s(\\ell)\\geq 0$. The first equality is by the definition of $C'$. \n\t\t\\end{proof}\n\t\t\n\t\t\t\\begin{claim}\n\t\t\t\\label{claim:evic4}\n\t\t\t\t Condition~4 of Lemma~\\ref{lem:wProperties} holds.\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\tlet $C ' \\in \\textsf{supp}(\\bar{w}^C)$. Assume towards a contradiction that $s(C' \\setminus L)>{\\varepsilon}$. Therefore, \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:w4}\n\t\t\t\t{\\varepsilon}<s(C' \\setminus L) \\leq 1-s(C' \\cap L).\n\t\t\t\\end{equation}\n\t\t\tThe second inequality is because $C' \\in {\\mathcal{C}}$ is a configuration. Recall $\\ell_1$ from \\eqref{h} for $C'$; it also holds that \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:w4b}\n\t\t\t\ts(\\ell_1) \\leq {\\varepsilon}^2 = {\\varepsilon} \\cdot {\\varepsilon} < {\\varepsilon} \\cdot \\left(1-s(C' \\cap L)\\right).\n\t\t\t\\end{equation}\n\t\t\tThe first inequality is because $\\ell_1 \\in I \\setminus L$. The last inequality is by \\eqref{eq:w4}. Therefore, by~ \\eqref{eq:w4b} we get that $\\ell_1 \\in  \\textsf{fit}(C' \\cap L)$ by \\eqref{fitS}. By the definition of the relaxation $\\bar{w}^{C}$ it holds that $C'\\cap L=C\\cap L$, thus $\\ell_1 \\in \\textsf{fit}(C\\cap L)$. Then, by \\eqref{h} and the definition of $U_C$ we conclude that $U_C  = \\emptyset$. Hence,  by the definition of $\\bar{w}^C$ and that $C' \\in \\textsf{supp}(\\bar{w}^C)$, it holds that $C' = C \\cap L$; thus, $s(C' \\setminus L) = 0$. For any ${\\varepsilon}>0$, this is a contradiction that $s(C' \\setminus L) > {\\varepsilon}$. \n\t\t\\end{proof}\n\t\t\n\t\t\n\t\t\t\\begin{claim}\n\t\t\t\\label{claim:evic6}\n\t\t\tCondition~5 of Lemma~\\ref{lem:wProperties} holds.\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\tLet $C' \\in \\textsf{supp}(\\bar{w}^C)$ and $G \\in {\\mathcal{G}}$. Now,  $$\\left|G \\cap \\left(C \\setminus R(C)\\right)\\right| \\leq |G \\cap C|-|G \\cap R(C)| \\leq  k(G) -|G \\cap R(C)| \\leq k(G) - |G \\cap C'|.$$ The last inequality is because by the definition of $\\bar{w}^C$ it holds that $C' \\subseteq R(C)$\n \t\t\\end{proof}\n\t\n\t\t\n\t\t\n\t\t\t\\begin{claim}\n\t\t\t\\label{claim:evic5}\n\t\t\tConditions 1-3 of Lemma~\\ref{lem:wProperties} hold for $|U_C| < \\alpha$. \n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\tFor the first condition of the lemma: $$\\|\\bar{w}^C\\| = \\bar{w}^C_{R(C)} = 1 \\leq 1+2{\\varepsilon}^4. $$ The first equality is because by the definition $\\bar{w}^C$, for any other configuration $C' \\in \\mathcal{C} \\setminus \\{R(C)\\}$ it holds that $\\bar{w}^C_{C'} = 0$. The second equality is by the definition of $\\bar{w}^C$ in case that $|U_C| < \\alpha$.  \n\t\t\t\n\t\t\tFor the second condition of the lemma, let $\\ell \\in R(C)$. Then, $$\\sum_{C' \\in \\mathcal{C}[\\ell]} \\bar{w}^C_{C'} = \\bar{w}^C_{R(C)} = 1.$$ The first equality is because by the definition $\\bar{w}^C$, for any other configuration $C' \\in \\mathcal{C} \\setminus \\{R(C)\\}$ it holds that $\\bar{w}^C_{C'} = 0$. The second equality is bythe definition of $\\bar{w}^C$ in case that $|U_C| < \\alpha$.\n\t\t\t\n\t\t\tFor the third condition of the lemma, let $C ' \\in \\textsf{supp}(\\bar{w}^C)$. By the definition of $\\bar{w}^C$ in case that $|U_C| < \\alpha$, it holds that $C' = R(C)$. Let $j \\in C \\setminus R(C)$. It holds that $j \\in \\textsf{fit}(C')$ by \\eqref{h}, since $|U_C|<\\alpha$. Because there are no restrictions over $j$ besides that $j \\in C \\setminus R(C)$, we conclude that $C \\setminus R(C) \\subseteq \\textsf{fit}(C')$. In addition, \n\t\t\t\n\t\t\t$$s(C \\setminus R(C)) \\leq 1-s(R(C)) = 1-s(C'). $$ The first inequality is because $R(C) \\subseteq C$ and $s(C) \\leq 1$. The first equality is by the definition of $C'$. \n\t\t\\end{proof} The proof of Lemma~\\ref{lem:wProperties} follows by Claim~\\ref{claim:evic1}, Claim~\\ref{claim:evic2}, Claim~\\ref{claim:evic3}, Claim~\\ref{claim:evic4}, Claim~\\ref{claim:evic5}, and Claim~\\ref{claim:evic6}.\n\t\t\n\t\t\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\t\n\t\t\n\t\t\n\n\t\\noindent{\\bf Proof of Lemma~\\ref{lem:eviction}:}\nWe use in the proof of Lemma~\\ref{lem:eviction} the following claims. \n\n\\begin{claim}\n\t\\label{lem:evictionPolynomial}\n\tAlgorithm~\\ref{Alg:eviction} is polynomial.\n\\end{claim}\n\n\n\\begin{proof}\n\tThe number of elements over which the loop in Step~\\ref{step:forsupp} iterates is polynomial, since  $|\\textsf{supp}(\\bar{x})|$ is at most the encoding size of the input (in a sparse representation). Let $C \\in \\textsf{supp}(\\bar{x})$ be a configuration in the support of $\\bar{x}$. It follows by \\eqref{h} that $U_C$ can be easily computed in polynomial time by iteratively adding items from $C \\setminus L$ in decreasing order of the items. Therefore, given $U_C$, the relaxation of $C$ that is $\\bar{w}^C$ can be computed in polynomial time: by the definition of $\\bar{w}^C$, the number of nonzero entries in $\\bar{w}^C$ is at most $|\\mathcal{L}_C|$, which is polynomial in the size of the instance. This is since $\\mathcal{L}_C \\subseteq I$ is a subset of the items. Finally, Step~\\ref{step:return} is also polynomial as $\\bar{y}$ is a linear combination of a polynomial number (i.e., $|\\textsf{supp}(\\bar{x})|$) of vectors, each with a polynomial number of nonzero entries. By the above, the claim follows. \n\\end{proof}\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{lem:eviction1}\n\t\tFor any $0<{\\varepsilon}<0.1$, an ${\\varepsilon}$-structured BPP instance $\\mathcal{I}$, and a solution $\\bar{x}$ to the configuration LP in \\eqref{C-LP}, Algorithm~\\ref{Alg:eviction} returns a prototype $\\bar{y}$  of $\\mathcal{I}$ with $\\bar{\\lambda}$ in the $\\bar{y}$-polytope such that for all $\\ell,j \\in I, \\ell \\neq j$ it holds that $\\bar{\\lambda}_{\\ell,j} = 0$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tWe define below a point $\\bar{\\lambda} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ in the $\\bar{y}$-polytope. Thus, it follows that the $\\bar{y}$-polytope is non-empty. For any $C \\in \\mathcal{C}$, define the {\\em point of $C$} as a vector $\\bar{\\gamma}^C \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ as follows. For any $\\ell \\in R(C)$, define \n\t\t\n\t\t\\begin{equation}\n\t\t\t\\label{gammal}\n\t\t\t\\bar{\\gamma}^C_{\\ell,\\ell} = 1.\n\t\t\\end{equation}\n\t\n\tMoreover, for any $\\ell \\in C \\setminus R(C)$ and $C' \\in \\mathcal{C}$ define \n\t\t\\begin{equation}\n\t\t\\label{gammaC}\n\t\t\\bar{\\gamma}^C_{\\ell,C'} = \\bar{w}^C_{C'}.\n\t\\end{equation}\n\nFor any other $\\ell \\in I$ and $t \\in (I \\cup \\mathcal{C})$ define\n\n\t\\begin{equation}\n\t\t\\label{gammae}\n\t\\bar{\\gamma}^C_{\\ell,t} = 0.\n\\end{equation}\n\n\tFinally, define \n\n\\begin{equation}\n\t\\label{lambda}\n\t\\bar{\\lambda} = \\sum_{C \\in \\mathcal{C}} \\bar{x}_C \\cdot \\bar{\\gamma}^C.\n\\end{equation}\n\n\nWe show below that all constraints of the $\\bar{y}$-polytope are satisfied for $\\bar{\\lambda}$. Let  $\\ell \\in I$  and $t \\in I \\cup \\mathcal{C}$ such that  $\\ell \\notin \\textsf{fit}\\left(t \\right)$. Then, \n\n\\begin{equation}\n\t\\label{eq:ev1}\n\t\\bar{\\lambda}_{\\ell,t} = \\sum_{C \\in \\mathcal{C}} \\bar{y}_C \\cdot \\bar{\\gamma}^C_{\\ell,t} = \\sum_{C \\in \\mathcal{C}} \\bar{y}_C \\cdot 0 = 0.\n\\end{equation}\n\nThe first equality is by \\eqref{lambda}. Because $\\ell \\notin \\textsf{fit}\\left(t \\right)$, then $\\ell \\neq t$ by \\eqref{fitl}; moreover, for all $C \\in \\mathcal{C}$ such that $\\ell \\in C \\setminus R(C)$ and $C ' \\in \\textsf{supp}(\\bar{w}^C)$,  it holds that $t \\neq C'$ because $\\ell \\in \\textsf{fit}(C')$ by Lemma~\\ref{lem:wProperties}. Therefore, by \\eqref{gammae} the second equality follows. By \\eqref{eq:ev1} we conclude that constraint \\eqref{F1} is satisfied for $\\bar{\\lambda}$. \n\nLet $C \\in \\mathcal{C}$. Then, \n\n\n\n\n\n\\begin{equation}\n\t\t\\label{eq:ev2}\n\t\\begin{aligned}\n\t\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell,C} \\cdot  s(\\ell) &= \\sum_{\\ell \\in I~} \\sum_{C' \\in \\mathcal{C}} \\bar{x}_{C'} \\cdot \\bar{\\gamma}^{C'}_{\\ell,C} \\cdot  s(\\ell) =  \\sum_{C' \\in \\mathcal{C}~} \\sum_{\\ell \\in I} \\bar{x}_{C'} \\cdot \\bar{\\gamma}^{C'}_{\\ell,C} \\cdot  s(\\ell)\\\\\n\t={} & \\sum_{C' \\in \\mathcal{C}~} \\sum_{\\ell \\in C' \\setminus R(C')} \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C} \\cdot  s(\\ell) \n\t=   \\sum_{C' \\in \\mathcal{C}}  \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C} \\cdot s\\left(C' \\setminus R(C')\\right)\\\\\n\t\\leq{} & \\sum_{C' \\in \\mathcal{C}}  \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C} \\cdot \\left(1-s(C)\\right)   =(1-s(C)) \\cdot  \\sum_{C' \\in \\textsf{supp}(\\bar{x})}  \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C} = (1-s(C)) \\bar{y}_{C}. \n\t\\end{aligned}\n\\end{equation}\n\n\n\n\n\n\n\nThe first equality is by \\eqref{lambda}. The second equality is by changing the order of summation. The third equality is by \\eqref{gammal}, \\eqref{gammaC} and \\eqref{gammae}.  The inequality is because for all $C' \\in \\mathcal{C}$, if $C \\in \\textsf{supp}({\\bar{w}^{C'}})$ then  $s(C' \\setminus R(C')) \\leq 1-s(C)$ by Lemma~\\ref{lem:wProperties}. The last equality is by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}. Therefore, by \\eqref{eq:ev2} we conclude that constraint \\eqref{F2} is satisfied for $\\bar{\\lambda}$. \n\n\n\n\n\n\n\n\n\n\nLet $G \\in \\mathcal{G}$ and $C \\in \\mathcal{C}$. Then, \n\n\n\\begin{equation}\n\t\\label{eq:ev3}\n\t\\begin{aligned}\n\t \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C}  ={} &   \\sum_{\\ell \\in G~}  \\sum_{C' \\in \\mathcal{C}} \\bar{x}_{C'} \\cdot \\bar{\\gamma}^{C'}_{\\ell,C}  = \\sum_{C' \\in \\mathcal{C}~}  \\sum_{\\ell \\in G}  \\bar{x}_{C'} \\cdot \\bar{\\gamma}^{C'}_{\\ell,C} = \\sum_{C' \\in \\mathcal{C}~~}  \\sum_{\\ell \\in G \\cap \\left(C' \\setminus R(C') \\right)}  \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C}\\\\\n\t   ={} & \\sum_{C' \\in \\mathcal{C} \\text{ s.t. } C \\in \\textsf{supp}(\\bar{w}^{C'})~~}   \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C}  \\sum_{\\ell \\in G \\cap \\left(C' \\setminus R(C') \\right)} 1 \n\t   \\leq  \\sum_{C' \\in \\mathcal{C}} \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C} \\cdot \\left(k(G)-|G \\cap C|\\right) \\\\ ={} & \\left(k(G)-|G \\cap C|\\right) \\sum_{C' \\in \\textsf{supp}(\\bar{x})~}  \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_{C}  =\\bar{y}_C \\cdot \\left(k(G)-|G \\cap C|\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\n\n\n\nThe first equality is by \\eqref{lambda}. The second equality is by changing the order of summation. The third equality is by \\eqref{gammal}, \\eqref{gammaC} and \\eqref{gammae}.  The inequality is by Condition~5 of Lemma~\\ref{lem:wProperties}. The last equality is by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}. Therefore, by \\eqref{eq:ev3} we conclude that constraint \\eqref{F5} is satisfied for $\\bar{\\lambda}$. \n\n\nLet $j \\in I$. Then, \n\n\\begin{equation}\n\t\\label{eq:ev4}\n\t\\begin{aligned}\n\t \\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell,j} ={} &  \\sum_{\\ell \\in I~}  \\sum_{C \\in \\mathcal{C}} \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,j} =   \\sum_{C \\in \\mathcal{C}~} \\sum_{\\ell \\in I} \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,j} =  \\sum_{C \\in \\mathcal{C}[j] \\text{ s.t. } j \\in R(C)} \\bar{x}_{C}\\\\ \\leq{} &  \\sum_{C \\in \\mathcal{C}[j] \\text{ s.t. } j \\in R(C)} \\bar{x}_{C} \\cdot \\left(  \\sum_{C' \\in \\mathcal{C}[j]} \\bar{w}^{C}_{C'} \\right) \n\t\\leq  \\sum_{C \\in \\mathcal{C}[j]} \\sum_{C' \\in \\mathcal{C}[j]}  \\bar{x}_{C} \\cdot  \\bar{w}^{C}_{C'}\\\\   ={} &   \\sum_{C' \\in \\mathcal{C}[j]} \\sum_{C \\in \\mathcal{C}[j]}  \\bar{x}_{C} \\cdot  \\bar{w}^{C}_{C'}\n\t \\leq \\sum_{C' \\in \\mathcal{C}[j]} \\sum_{C \\in \\textsf{supp}(\\bar{x})}  \\bar{x}_{C} \\cdot  \\bar{w}^{C}_{C'}  =  \\sum_{\\substack{C' \\in \\mathcal{C}[j]}}  \\bar{y}_{C'} = \\sum_{\\substack{C \\in \\mathcal{C}[j]}}  \\bar{y}_{C}.\n\t\\end{aligned}\n\\end{equation}\n\n\n\n\nThe first equality is by \\eqref{lambda}. The second equality is by changing the order of summation. The third equality is by \\eqref{gammal}, \\eqref{gammaC} and \\eqref{gammae}.  The first  inequality is because for all $C \\in {\\mathcal{C}}$ and $j \\in R(C)$ it holds that  $ \\sum_{C' \\in \\mathcal{C}[j]} \\bar{w}^{C}_{C'}  \\geq 1$ by Lemma~\\ref{lem:wProperties}. The second inequality is because for all $C,C' \\in {\\mathcal{C}}$ it holds that $ \\bar{x}_{C} \\cdot  \\bar{w}^{C}_{C'} \\geq 0$.  The fourth equality is is by changing the order of summation. The third inequality is because for all $C \\in \\mathcal{C}[j] \\setminus \\textsf{supp}(\\bar{x})$ it holds that $\\bar{x}_{C} \\cdot  \\bar{w}^{C}_{C'} = 0$. The fifth equality is by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}. The sixth equality is by a change in the notation of a variable. Therefore, by \\eqref{eq:ev4} we conclude that constraint \\eqref{F3} is satisfied for $\\bar{\\lambda}$. \n\nLet $\\ell \\in I$. Then, we use the following equations. \n\n\\begin{equation}\n\t\\label{eq:ev5a}\n\t\\sum_{t \\in I} \\bar{\\lambda}_{\\ell,t} =   \\sum_{t \\in I~} \\sum_{C \\in \\mathcal{C}} \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,t}  =  \\sum_{C \\in \\mathcal{C}~} \\sum_{t \\in I}  \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,t} =  \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in R(C)} \\bar{x}_{C}.\n\\end{equation}\n\nThe first equality is by \\eqref{lambda}. The second equality is by changing the order of summation. The third equality is by \\eqref{gammal} and \\eqref{gammae}. In addition, \n\n\n\\begin{equation}\n\t\\label{eq:ev5b}\n\t\\begin{aligned}\n\t\t\\sum_{t \\in \\mathcal{C}} \\bar{\\lambda}_{\\ell,t} ={} &   \\sum_{t \\in \\mathcal{C}~} \\sum_{C \\in \\mathcal{C}} \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,t}  =  \\sum_{C \\in \\mathcal{C}~} \\sum_{t \\in \\mathcal{C}}  \\bar{x}_{C} \\cdot \\bar{\\gamma}^{C}_{\\ell,t} =  \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in C \\setminus R(C)~} \\sum_{t \\in \\mathcal{C}} \\bar{x}_{C} \\cdot \\bar{w}^{C}_{t}\\\\   ={} &  \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in C \\setminus R(C)~} \\bar{x}_{C} \\sum_{t \\in \\mathcal{C}}   \\bar{w}^{C}_{t} \\geq \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in C \\setminus R(C)~} \\bar{x}_{C} \\sum_{t \\in \\mathcal{C}[\\ell]}   \\bar{w}^{C}_{t} \\geq   \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in C \\setminus R(C)~} \\bar{x}_{C}.\n\t\\end{aligned}\n\\end{equation}\n\nThe first equality is by \\eqref{lambda}. The second equality is by changing the order of summation. The third equality is by \\eqref{gammaC} and \\eqref{gammae}. The first inequality is because $\\mathcal{C}[\\ell] \\subseteq \\mathcal{C}$. The second inequality is because $ \\sum_{t \\in \\mathcal{C}[\\ell]}   \\bar{w}^{C}_{t} \\geq 1$ by Lemma~\\ref{lem:wProperties}. Thus, \n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{equation}\n\t\\label{eq:ev5d}\n\\sum_{t \\in I \\cup \\mathcal{C}} \\bar{\\lambda}_{\\ell,t} = \t\\sum_{t \\in I} \\bar{\\lambda}_{\\ell,t}+ \t\\sum_{t \\in \\mathcal{C}} \\bar{\\lambda}_{\\ell,t}  \\geq \\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in R(C)} \\bar{x}_{C} +\\sum_{C \\in \\mathcal{C}[\\ell] \\text{ s.t. } \\ell \\in C \\setminus R(C)~} \\bar{x}_{C} = \\sum_{C \\in \\mathcal{C}[\\ell]} \\bar{x}_{C} \\geq 1.  \n\\end{equation}\n\n\nThe first equality is because $I \\cap \\mathcal{C} = \\emptyset$. The first inequality is by \\eqref{eq:ev5a} and \\eqref{eq:ev5b}. The second equality is because for all $C \\in \\mathcal{C}[\\ell]$ either $\\ell \\in R(C)$ or $\\ell \\in C\\setminus R(C)$. The last inequality is because $\\bar{x}$ is a solution for \\eqref{C-LP}. Therefore, by  \\eqref{eq:ev5d}, constraint \\eqref{F4} is satisfied for $\\bar{\\lambda}$. In summary, we conclude that $\\bar{\\lambda}$ is a point in the $\\bar{y}$-polytope; hence, the $\\bar{y}$-polytope is non-empty. \n\n\t\\end{proof}\n\t\n\t\n\n\t\n\t\t\t\\begin{claim}\n\t\t\\label{lem:eviction2}\n\t\tFor any $0<{\\varepsilon}<0.1$, an ${\\varepsilon}$-structured BPP instance $\\mathcal{I}$, and a prototype $\\bar{x}$ of $\\mathcal{I}$, Algorithm~\\ref{Alg:eviction} returns a prototype $\\bar{y}$ such that $||\\bar{y}|| \\leq (1+{\\varepsilon})||\\bar{x}||$.\n\t\\end{claim}\n\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\t\n\t\t\t\n\t\t\t\t\\begin{align*}\n\t\t\t\t\\|\\bar{y}\\| ={} & \\left\\|  \\sum_{C \\in \\textsf{supp}(\\bar{x})} \\bar{x}_{C} \\cdot \\bar{w}^{C} \\right\\| \\leq  \\sum_{C \\in \\textsf{supp}(\\bar{x})} \\bar{x}_{C} \\cdot \\| \\bar{w}^{C} \\|  \\leq  \\sum_{C \\in \\textsf{supp}(\\bar{x})} \\bar{x}_{C} \\cdot (1+2{\\varepsilon}^4) = (1+2{\\varepsilon}^4)\\|\\bar{x}\\| \\leq (1+{\\varepsilon})\\|\\bar{x}\\|.\n\t\t\t\\end{align*}\n\t\t\t\t\n\t\tThe first equality is by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}. The first inequality is by the triangle inequality. The second inequality is because $\\|\\bar{w}^{C}\\|  \\leq (1+2{\\varepsilon}^4)$ by Lemma~\\ref{lem:wProperties} for all $C \\in \\textsf{supp}(\\bar{x})$. The last inequality is because $0<{\\varepsilon}< 0.1$. \n\t\t\n\n\t\t\t\t\n\t\t\t\\end{proof}\n\t\t\t\n\t\t\t\\begin{claim}\n\t\t\t\t\\label{lem:eviction3}\n\t\t\t\tFor any $0<{\\varepsilon}<0.1$, an ${\\varepsilon}$-structured BPP instance $\\mathcal{I}$, and a prototype $\\bar{x}$ of $\\mathcal{I}$,  Algorithm~\\ref{Alg:eviction} returns a prototype $\\bar{y}$ such that for all $C \\in \\textsf{supp}(y)$ it holds that $|C| \\leq {\\varepsilon}^{-10}$.\n\t\t\t\\end{claim}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tLet $C \\in \\textsf{supp}(\\bar{y})$ be a configuration. Because $C \\in \\textsf{supp}(\\bar{y})$, then by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}, there is $C' \\in \\textsf{supp}(\\bar{x})$ such that $\\bar{w}^{C'}_C \\neq 0$.  Therefore,  $$|C| \\leq |R(C')| = |C' \\cap L \\cup U_{C'}| \\leq |C' \\cap L|+|U_{C'}| \\leq {\\varepsilon}^{-2}+\\alpha \\leq 2{\\varepsilon}^{-6} \\leq {\\varepsilon}^{-10}.$$ \n\t\t\t\t\n\t\t\t\tThe first inequality is because the maximal number of items in $C$ is bounded by $|R(C')|$ by the definition of the relaxation vector of a configuration. The first equality is by the definition of $R(C')$. The second inequality is by the union bound. The third inequality is since the size of each large item in $L$ is at least ${\\varepsilon}^2$ and there can be at most ${\\varepsilon}^{-2}$ such items in a configuration without exceeding the maximal total size of items in a configuration which is $1$; moreover, by \\eqref{h} it holds that $|U_C| \\leq \\alpha$. The fourth inequality is because $\\alpha = \\lceil {\\varepsilon}^{-5} \\rceil$. The last inequality is because $0<{\\varepsilon}<0.1$. \n\t\t\t\\end{proof}\n\t\t\t\n\t\t\t\t\\begin{claim}\n\t\t\t\t\\label{lem:eviction4}\n\t\t\t\tFor any $0<{\\varepsilon}<0.1$, an ${\\varepsilon}$-structured BPP instance $\\mathcal{I}$, and a prototype $\\bar{x}$ of $\\mathcal{I}$,  Algorithm~\\ref{Alg:eviction} returns a prototype $\\bar{y}$ such that for all $C \\in \\textsf{supp}(y)$ it holds that $s(C \\setminus L) \\leq {\\varepsilon}$. \n\t\t\t\\end{claim}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\tLet $C \\in \\textsf{supp}(\\bar{y})$ be a configuration. Because $C \\in \\textsf{supp}(\\bar{y})$, then by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}, there is $C' \\in \\textsf{supp}(\\bar{x})$ such that $\\bar{w}^{C'}_C \\neq 0$.  Therefore,  $C \\in \\textsf{supp}(\\bar{w}^{C'})$ and it follows that $s(C \\setminus L) \\leq {\\varepsilon}$ by Lemma~\\ref{lem:wProperties}. \n\t\t\t\\end{proof}\n\t\t\t\n\t\t\t\n\t\t\t\t\\begin{claim}\n\t\t\t\t\\label{lem:evictionF4}\n\t\t\tfor every $\\ell \\in I$ it holds that $\\sum_{C\\in {\\mathcal{C}}[\\ell]} {\\bar{y}}_C\\leq 2$. \n\t\t\t\\end{claim}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\t\\begin{align*}\n\t\t\\sum_{C\\in {\\mathcal{C}}[\\ell]} {\\bar{y}}_C ={} & \\sum_{C\\in {\\mathcal{C}}[\\ell]~} \\sum_{C' \\in \\textsf{supp}(\\bar{x})} \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_C =  \\sum_{C' \\in \\textsf{supp}(\\bar{x})~} \\sum_{C\\in {\\mathcal{C}}[\\ell]} \\bar{x}_{C'} \\cdot \\bar{w}^{C'}_C \\leq  \\sum_{C' \\in \\textsf{supp}(\\bar{x})~} \\bar{x}_{C'} \\cdot \\sum_{C\\in {\\mathcal{C}}}   \\bar{w}^{C'}_C \\\\ ={} &\\sum_{C' \\in \\textsf{supp}(\\bar{x})~} \\bar{x}_{C'} \\cdot \\|\\bar{w}^{C'}\\| \\leq \\sum_{C' \\in \\textsf{supp}(\\bar{x})~} \\bar{x}_{C'} \\cdot (1+2{\\varepsilon}^{4}) \\leq \\sum_{C' \\in \\textsf{supp}(\\bar{x})~} \\bar{x}_{C'} \\cdot 2 = 2 \\cdot \\|\\bar{x}\\| = 2. \n\t\t\t\\end{align*}\n\t\t\tThe first equality is by Step~\\ref{step:return} of Algorithm~\\ref{Alg:eviction}. The second inequality is because $\\|\\bar{w}^{C'}\\|  \\leq (1+2{\\varepsilon}^4)$ by Lemma~\\ref{lem:wProperties}. The third inequality is because $0<{\\varepsilon}< 0.1$. The last equality is by \\eqref{LP}. \n\t\t\t\\end{proof}\n\t\t\t\n\t\t\n\t\t\t\n\t\t\n\t\t\tThe proof of Lemma~\\ref{lem:eviction} follows by Claim~\\ref{lem:evictionPolynomial}, Claim~\\ref{lem:eviction1}, Claim~\\ref{lem:eviction2}, Claim~\\ref{lem:eviction3}, Claim~\\ref{lem:eviction4}, and Claim~\\ref{lem:evictionF4}. \\qed\n\t\n\t\n\t\n\t\n\n\\section{Omitted Proofs of Section~\\ref{sec:greedy}}\n\\label{sec:proofsGreedy}\n\n For this section as in Section~\\ref{sec:greedy}, we have a BPP instance ${\\mathcal{I}} = (I, {\\mathcal{G}},s,k)$ and $\\delta \\in (0,0.5)$ such that for all $\\ell \\in I$ it holds that $s(\\ell) \\leq \\delta$. Also, assume that $I \\neq \\emptyset$ otherwise all proofs here are trivial.  \n \n\\noindent{\\bf Proof of Lemma~\\ref{clm:confappend}:} We prove the claim by the definition of packing. For convenience, let $T = (S, X_1, \\ldots, X_m)$ and also $T = (T_1, \\ldots, T_{m+1})$.\n\n\\begin{enumerate}\n\t\\item It holds that $T$ is a partition of $I$. Let $\\ell \\in I$. If $\\ell \\in S$, then $\\ell \\in T_1$ and for all $i \\in [m+1] \\setminus \\{1\\}$ it holds that $\\ell \\notin T_i$ because $X_{i-1} \\subseteq I \\setminus S$. Otherwise, it holds that $\\ell \\in I \\setminus S$; therefore, because $X$ is a packing of ${\\mathcal{I}} \\setminus S$ there is exactly one $i \\in [m]$ such that $\\ell \\in X_i$ and it follows that $\\ell \\in T_{i+1}$ (only).\n\t\n\t\\item For all $i \\in [m+1]$ it holds that $s(T_i) \\leq 1$. If $i = 1$ then $s(T_1) = s(S) \\leq 1$ because $S \\in {\\mathcal{C}}({\\mathcal{I}})$. Otherwise, it holds that $s(T_i) = s(X_{i-1}) \\leq 1$ because $X$ is a packing of ${\\mathcal{I}} \\setminus S$. \n\t\n\t\\item For all $G \\in {\\mathcal{G}}$ and $i \\in [m+1]$ it holds that $|T_i \\cap G| \\leq k(G)$.  If $i = 1$ then $|T_i \\cap G| = |G \\cap S| \\leq k(G)$ because $S \\in {\\mathcal{C}}({\\mathcal{I}})$. Otherwise, it holds that $|T_i \\cap G| = |X_{i-1} \\cap G|\\leq k_S(G \\setminus S) = k(G)$. The inequality is because $X$ is a packing of ${\\mathcal{I}} \\setminus S$. The last equality is by \\eqref{IIS}.\\qed\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{bG}:} Let $n = |G|$; by \\eqref{promise} and \\eqref{bounding} it holds that $n \\geq 1$. Now, let $\\ell_1, \\ldots, \\ell_n$ be the items in $G$ in an increasing order of item sizes. Define\t\\begin{equation}\n\t\\label{bounding:psi}\n\t\\psi = \\argmin_{\\left\\{i \\in [n] ~\\big|~  \\ceil{  \\frac{n-i}{k(G)}} \\leq V({\\mathcal{I}})-1\\right\\}} i.\n\\end{equation} Observe that for $i =n$ it holds that $ \\ceil{  \\frac{n-i}{k(G)}}  \\leq V({\\mathcal{I}})-1$ because $n \\geq 1$; thus, it follows that $\\psi$ is well defined. Now, define $S^* = \\{\\ell_1, \\ldots, \\ell_{\\psi}\\}$. We show next that $S^* \\in b^*(G)$.\n\n\\begin{itemize}\n\t\\item $S^* \\in b(G)$. First, $S^* \\subseteq G$ by definition. Second, $$ \\ceil{ \\frac{|G \\setminus S^*|}{k(G)} } = \\ceil{  \\frac{n-\\psi}{k(G)}} \\leq V({\\mathcal{I}})-1 \\leq p({\\mathcal{I}})-1.$$ The first inequality is by \\eqref{bounding:psi}. The last inequality is by \\eqref{promise}. \n\t\n\t\n\t\n\t\n\t\n\t\\item $|S^*| \\leq k(G)$. \\begin{equation}\n\t\t\\label{S:psi}\n\t\t\\begin{aligned}\n\t\t\t|S^*| ={} & \\psi \\leq |G|-(V({\\mathcal{I}})-1 ) \\cdot k(G) \\leq |G|-\\left(\\frac{|G|}{k(G)}-1 \\right) \\cdot k(G) = k(G).\n\t\t\\end{aligned}\n\t\\end{equation} The first inequality is by \\eqref{bounding:psi}, since $G \\in {\\mathcal{B}}({\\mathcal{I}})$ and $\\ceil{\\frac{|G|-\\left( |G|-(V({\\mathcal{I}})-1) \\cdot k(G)\\right)}{k(G)}} \\leq (V({\\mathcal{I}})-1)$ (recall that $V({\\mathcal{I}}) \\in \\mathbb{N}$). \n\t\n\t\n\t\n\t\\item $s(S^*) \\leq \\frac{s(G)}{s(I)}$. It holds that: $$s(S^*) =\\psi \\cdot \\frac{ s(S^*)}{\\psi} \\leq \\psi \\cdot \\frac{s(G)}{n} \\leq \\frac{k(G)}{|G|} \\cdot s(G) \\leq \\frac{s(G)}{p({\\mathcal{I}})-2}  \\leq \\frac{s(G)}{(1+2\\delta) \\cdot s(I)+2-2} \\leq \\frac{s(G)}{ s(I)}.$$ The first inequality is because $S^*$ is a subset of items of $G$ where each item in $G \\setminus S^*$ has a larger or equal size compared to the size of any item in $S^*$ by the sorted order of the items; thus, the size of the average item in $S^*$ is smaller or equal to the size of the average item in $G$. The second inequality is by \\eqref{S:psi}. The third inequality is by \\eqref{bounding} and that $G \\in {\\mathcal{B}}({\\mathcal{I}})$:  it holds that $\\frac{|G|}{k(G)}+1 \\geq \\ceil{\\frac{|G|}{k(G)}} > p({\\mathcal{I}})-1$. The fourth inequality is by \\eqref{promise}.\\qed\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nGiven a BPP instance ${\\mathcal{J}}$, let $A^{{\\mathcal{J}}}$ be the first entry (from the left) of the returned tuple by $\\textsf{Greedy}({\\mathcal{J}},\\delta)$ and let $A^{{\\mathcal{J}}}_0,A^{{\\mathcal{J}}}_1$ be the object $A^{{\\mathcal{J}}}$ after Step~\\ref{gr:end0} and after Step~\\ref{gr:end1} in the computation of $\\textsf{Greedy}({\\mathcal{J}},\\delta)$, respectively. Note that $A^{{\\mathcal{J}}}_0$ is well defined (the algorithm is guaranteed to construct $A^{{\\mathcal{J}}}_0$) by Lemma~\\ref{bI}. For simplicity, we use $A, A_0,A_1$ instead of $A^{{\\mathcal{J}}}, A^{{\\mathcal{J}}}_0, A^{{\\mathcal{J}}}_1$, respectively, when ${\\mathcal{J}} = {\\mathcal{I}}$. Now, for the proof of Lemma~\\ref{thm:greedy}, we use the following auxiliary claims. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{clm:A00}\n\tFor any BPP instance ${\\mathcal{I}} = (I,{\\mathcal{G}},s,k)$ it holds that $A_0 \\in {\\mathcal{C}}({\\mathcal{I}})$ and for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A_0|}{k(G)}    } \\leq p({\\mathcal{I}})-1$.\n\\end{claim}\n\n\\begin{proof}\nObserve that $A_0 \\in b^*({\\mathcal{I}})$ by Step~\\ref{gr:end0}. In addition, note that $A_0$ is well defined by Lemma~\\ref{bI}. Then, by \\eqref{bounding:A} for each $G \\in {\\mathcal{B}}({\\mathcal{I}})$ there is $S_G \\in b^*(G)$ such that $A_0 = \\bigcup_{G \\in {\\mathcal{B}}({\\mathcal{I}})} S_G$. We show the conditions of the lemma as follows. \\begin{itemize}\n\t\t\\item $s(A_0) \\leq 1$. $$s(A_0) =  s\\left(  \\bigcup_{G \\in {\\mathcal{B}}({\\mathcal{I}})} S_G \\right)  \\leq \\sum_{G \\in {\\mathcal{B}}({\\mathcal{I}})} \\frac{s(G)}{s(I)} \\leq \\frac{s(I)}{s(I)} = 1.$$ The first equality is by \\eqref{bounding:A}. The first inequality is by the definition of a minimal bounding subset of a bounding group.\n\t\t\n\t\t\n\t\t\n\t\t\\item For all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap A_0| \\leq k(G)$. If $G \\in {\\mathcal{G}} \\setminus {\\mathcal{B}}({\\mathcal{I}})$ then $|G \\cap A_0| =0 < k(G)$ by \\eqref{bounding:A}. Otherwise, $|G \\cap A_0| = \\left|  S_G  \\right| \\leq  k(G)$ where the  equality is by \\eqref{bounding:A} and the inequality is by  the definition of a minimal bounding subset of a bounding group. \n\t\t\n\t\t\n\t\t\\item For each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $ \\ceil{ \\frac{|G \\setminus A_0|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. $$ \\ceil{ \\frac{|G \\setminus A_0|}{k(G)}    } = \\ceil{ \\frac{|G \\setminus S_G|}{k(G)}    }\\leq p({\\mathcal{I}})-1$$ The first equality is by \\eqref{bounding:A}. The first inequality is because $S_G \\in b^*(G)$ and in particular $S_G \\in b(G)$; thus, the inequality follows by the definition of a bounding subset of $G$.  \n\t\\end{itemize}\n\t\n\n\t\n\t\n\t\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{clm:A1}\n\tFor any BPP instance ${\\mathcal{I}} = (I,{\\mathcal{G}},s,k)$ it holds that $A_1 \\in {\\mathcal{C}}({\\mathcal{I}})$ and for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A_1|}{k(G)}    } \\leq p({\\mathcal{I}})-1$.\n\\end{claim}\n\n\\begin{proof}\n\tWe prove the claim using loop invariant for the while loop of Step~\\ref{gr:while2}. Let $A(s),A(t)$ be the object $A$ before and after an iteration of the while loop of Step~\\ref{gr:while2}, respectively. Now, assume that $A(s) \\in {\\mathcal{C}}({\\mathcal{I}})$ and for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. We show the conditions of the claim below for $A(t)$. By Step~\\ref{gr:swap2}, let $\\ell \\in I$ such that $A(t) = A(s) \\cup \\{\\ell\\}$. Therefore,  \\begin{itemize}\n\t\t\\item $s(A(t)) \\leq 1$. $$s(A(t)) =  s\\left(  A(s) \\right) + s(\\ell) \\leq (1-\\delta)+\\delta =1.$$ The first equality is by Step~\\ref{gr:swap2}. The inequality is because $s(A(s)) \\leq 1-\\delta$ by Step~\\ref{gr:while2}; in addition, for all items the size is at most $\\delta$. \n\t\t\n\t\t\n\t\t\n\t\t\\item For all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap A(t)| \\leq k(G)$. If $G \\in {\\mathcal{G}} \\setminus \\{\\textsf{group}(\\ell)\\}$; then, $|G \\cap A(t)| = |A(s) \\cap G| \\leq k(G)$. The equality is by Step~\\ref{gr:swap2} and the inequality is by the assumption that $A(s) \\in {\\mathcal{C}}({\\mathcal{I}})$. Otherwise, $|G \\cap A(t)| = \\left|  A(s) \\cap G \\right| +|\\{\\ell\\} \\cap G| = |A(s) \\cap G|+1 \\leq  k(G)$. The first equality is by Step~\\ref{gr:swap2} and the inequality is by Step~\\ref{gr:while2}. \n\t\t\n\t\t\n\t\t\\item For each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. If $G \\in {\\mathcal{G}} \\setminus \\{\\textsf{group}(\\ell)\\}$; then, $   \\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } =  \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. The equality is by Step~\\ref{gr:swap2} and the inequality is by the assumption on $A(s)$. Otherwise, $$ \\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } =  \\ceil{ \\frac{|G \\setminus A(s)|-|\\{\\ell\\}|}{k(G)}    } \\leq \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    } \\leq p({\\mathcal{I}})-1.$$ The first equality is by Step~\\ref{gr:swap2} and the last inequality is by the assumption on $A(s)$. Therefore, by Claim~\\ref{clm:A00} and the above loop invariant the claim follows.  \n\t\\end{itemize}\n\t\n\t\n\t\n\t\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{clm:A}\n\t$A \\in {\\mathcal{C}}({\\mathcal{I}})$ and for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $\\ceil{ \\frac{|G \\setminus A|}{k(G)}    } \\leq p({\\mathcal{I}})-1$.\n\\end{claim}\n\n\\begin{proof}\n\tWe prove the claim using loop invariant for the while loop of Step~\\ref{gr:while1}. Let $A(s),A(t)$ be the object $A$ before and after an iteration of the while loop of Step~\\ref{gr:while1}, respectively. Now, assume that $A(s) \\in {\\mathcal{C}}({\\mathcal{I}})$ and for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. We show the conditions of the claim below for $A(t)$.  By Step~\\ref{gr:swap}, let $\\ell,y \\in I$ such that $A(t) = \\left(A(s) \\setminus \\{\\ell\\} \\right) \\cup \\{y\\}$. Therefore,  \\begin{itemize}\n\t\t\\item $s(A(t)) \\leq 1$. $$s(A(t)) =  s\\left(  A(s) \\right) -s(\\ell)+s(y) \\leq s\\left(  A(s) \\right) +s(y) \\leq (1-\\delta)+\\delta =1.$$ The first equality is by Step~\\ref{gr:swap}. The second inequality is because $s(A(s)) \\leq 1-\\delta$ by Step~\\ref{gr:while1}; in addition, for all items the size is at most $\\delta$. \n\t\t\n\t\t\n\t\t\n\t\t\\item For all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap A(t)| \\leq k(G)$. If $G \\in {\\mathcal{G}} \\setminus \\{\\textsf{group}(\\ell)\\}$; then, $|G \\cap A(t)| = |A(s) \\cap G| \\leq k(G)$. The equality is by Step~\\ref{gr:swap} and the inequality is by the assumption that $A(s) \\in {\\mathcal{C}}({\\mathcal{I}})$. Otherwise, $|G \\cap A(t)| = \\left|  A(s) \\cap G \\right| -|\\{\\ell\\} \\cap G|+|\\{y\\} \\cap G| = |A(s) \\cap G| \\leq  k(G)$. The first equality is by Step~\\ref{gr:swap} and the inequality is by the assumption that $A(s) \\in {\\mathcal{C}}({\\mathcal{I}})$\n\t\t\n\t\t\n\t\t\\item For each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $\\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$. If $G \\in {\\mathcal{G}} \\setminus \\{\\textsf{group}(\\ell)\\}$; then, $\\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } = \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    }\\leq p({\\mathcal{I}})-1$. The equality is by Step~\\ref{gr:swap} and the inequality is by the assumption on $A(s)$. Otherwise, $$\\ceil{ \\frac{|G \\setminus A(t)|}{k(G)}    } = \\ceil{ \\frac{|G \\setminus A(s)|+|\\{\\ell\\} \\cap G|-|\\{y\\} \\cap G|}{k(G)}    }= \\ceil{ \\frac{|G \\setminus A(s)|}{k(G)}    } \\leq p({\\mathcal{I}})-1$$ The equalities are by Step~\\ref{gr:swap} and the inequality is by the assumption on $A(s)$. Therefore, by Claim~\\ref{clm:A1} and the above loop invariant the claim follows.  \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{itemize}\n\t\n\t\n\t\n\t\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{ABCV}\n\t$V({\\mathcal{I}} \\setminus A)\\leq p({\\mathcal{I}})-1$.\n\\end{claim}\n\n\\begin{proof}\n\t\\begin{equation*}\n\t\tV({\\mathcal{I}} \\setminus A) = \\max_{G \\in {\\mathcal{G}}_A} \\ceil{ \\frac{|G|}{k_A(G)}  } = \\max_{G \\in {\\mathcal{G}}}        \\ceil{ \\frac{|G \\setminus A|}{k(G)}    } \\leq  p({\\mathcal{I}})-1\n\t\\end{equation*} The second equality is by \\eqref{IIS}. The last inequality is because for all bounding groups $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A|}{k(G)}    } \\leq p({\\mathcal{I}})-1$ by Claim~\\ref{clm:A} and for all $G \\in {\\mathcal{G}} \\setminus {\\mathcal{B}}({\\mathcal{I}})$ it holds that $   \\ceil{ \\frac{|G \\setminus A|}{k(G)}    } \\leq p({\\mathcal{I}})-1$ by \\eqref{bounding}.\n\\end{proof}\n\n\n\\begin{lemma}\n\t\\label{delta<}\n\tIf $s(A) < 1-\\delta$, then Algorithm~\\ref{alg:GREEDY} returns a packing for ${\\mathcal{I}}$ with at most $p({\\mathcal{I}})$ bins.\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe use the following auxiliary claims. \n\t\n\t\\begin{claim}\n\t\t\\label{VD1}\n\t\tFor all $G \\in {\\mathcal{G}}$ it holds that $|A \\cap G| = \\min\\{k(G),|G|\\}$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tAssume towards a contradiction that there is $G \\in {\\mathcal{G}}$ such that $|A \\cap G| < \\min\\{k(G),|G|\\}$. Therefore, there is $\\ell \\in G \\setminus A$. Observe $A$ after Step~\\ref{gr:end0}. Because $s(A) < 1-\\delta$, it follows by Step~\\ref{gr:while2} and Step~\\ref{gr:swap2} that $\\ell$ can be added to $A$ in contradiction. \n\t\\end{proof}\n\t\n\t\\begin{claim}\n\t\t\\label{VD2}\n\t\tFor all $G \\in {\\mathcal{G}}$, $\\ell \\in A \\cap G$ and $y \\in G \\setminus A$ it holds that $s(y) \\leq s(\\ell)$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tAssume towards a contradiction that there are  $\\ell \\in A, y \\in I \\setminus A$ such that  $s(y) > s(\\ell)$ and $\\textnormal{\\textsf{group}}(\\ell)  = \\textnormal{\\textsf{group}}(y)$. Observe $A$ after Step~\\ref{gr:end1}.  Because $s(A) < 1-\\delta$, it follows by Step~\\ref{gr:while1} and Step~\\ref{gr:swap} that $y$ can be added to $A$ and $\\ell$ can be removed from $A$ in contradiction. \n\t\t\n\t\\end{proof}\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{ABCV2}\n\t\tIf $s(A) < 1-\\delta$ then $V({\\mathcal{I}} \\setminus A)\\leq V({\\mathcal{I}})-1$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\tV({\\mathcal{I}} \\setminus A) ={} & \\max_{G \\in {\\mathcal{G}}_A} \\ceil{ \\frac{|G|}{k_A(G)}  } = \\max_{G \\in {\\mathcal{G}}}        \\ceil{ \\frac{|G \\setminus A|}{k(G)}    } =   \\max_{G \\in {\\mathcal{G}}}        \\ceil{ \\frac{|G|-\\min\\{ |G|,k(G) \\} }{k(G)}} \\\\ \\leq{} & \\max_{G \\in {\\mathcal{G}}}        \\ceil{ \\frac{|G|}{k(G)} -1 } =  \\max_{G \\in {\\mathcal{G}}}        \\ceil{ \\frac{|G|}{k(G)}}-1 = V({\\mathcal{I}})-1.\n\t\t\t\\end{aligned}\n\t\t\\end{equation*} The second equality is by \\eqref{IIS}. The third equality is by Claim~\\ref{VD1}. The inequality is because if $\\min\\{ |G|,k(G) \\} = |G|$ then  it follows that $V({\\mathcal{I}} \\setminus A) = 0 \\leq V({\\mathcal{I}})-1$, assuming that $I \\neq \\emptyset$. \n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\tNow, back to the proof of Lemma~\\ref{delta<}. We prove the lemma by induction on $V({\\mathcal{I}})$. For $V({\\mathcal{I}}) = 1$, it follows that $V({\\mathcal{I}} \\setminus A) \\leq V({\\mathcal{I}})-1$ by Claim~\\ref{ABCV2} and it follows that $V({\\mathcal{I}} \\setminus A) = 0$. Therefore, by \\eqref{IIS} it follows that $I \\setminus A = \\emptyset$ and therefore $A = I$. Thus, $I$ is a configuration of ${\\mathcal{I}}$ by Claim~\\ref{clm:A}. Hence, by Step~\\ref{gr:if1} and Step~\\ref{gr:if} it holds that Algorithm~\\ref{alg:GREEDY} returns $(I)$, which is a packing of ${\\mathcal{I}}$ because $I = A$ and $A$ is a configuration of ${\\mathcal{I}}$. \n\t\n\tRecall that for any BPP instance ${\\mathcal{J}}$ we define $A^{{\\mathcal{J}}}$ as the first entry of the tuple returned by $\\textsf{Greedy}({\\mathcal{J}},\\delta)$. Now, for the assumption of the induction, assume that  $V({\\mathcal{I}}) \\geq 2$. Assume that for any BPP instance ${\\mathcal{J}} = (J,{\\mathcal{G}}_J,s_J,k_J)$ such that (i) $s_J(\\ell) \\leq \\delta ~\\forall \\ell \\in J$, (ii) $V({\\mathcal{J}}) \\leq V({\\mathcal{I}})-1$, and (iii) $s\\left(A^{{\\mathcal{J}}}\\right) < 1-\\delta$, it holds that $\\textsf{Greedy}({\\mathcal{J}},\\delta)$ returns a packing of ${\\mathcal{J}}$ of at most $V({\\mathcal{J}})$ bins. For the step of the induction, we use the following auxiliary claim. \n\t\n\t\n\t\\begin{claim}\n\t\t\\label{VD3}\n\t\t$s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) < 1-\\delta$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tAssume towards a contradiction that $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) \\geq 1-\\delta$. Therefore, it holds that $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) > s(A)$; thus, we reach a contradiction if one of the following conditions hold. \n\t\t\n\t\t\\begin{enumerate}\n\t\t\t\n\t\t\t\\item $|A^{{\\mathcal{I}} \\setminus A}| > |A|$. Therefore, $$|A^{{\\mathcal{I}} \\setminus A}| = \\sum_{G \\in {\\mathcal{G}}} |A^{{\\mathcal{I}} \\setminus A} \\cap G| \\leq \\sum_{G \\in {\\mathcal{G}}} \\min\\{k(G),|G|\\} = \\sum_{G \\in {\\mathcal{G}}} |A \\cap G| = |A|.$$ The inequality is because $A^{{\\mathcal{I}} \\setminus A} \\in {\\mathcal{C}}({\\mathcal{I}} \\setminus A)$ by Claim~\\ref{clm:A} and ${\\mathcal{C}}({\\mathcal{I}} \\setminus A) \\subseteq {\\mathcal{C}}({\\mathcal{I}})$ by \\eqref{IIS}.\\footnote{Claim~\\ref{clm:A} is stated and proven for $A$ and not for $A^{{\\mathcal{I}} \\setminus A}$ for simplicity; however, the exact arguments can be used for proving the claim also for $A^{{\\mathcal{I}} \\setminus A}$.} The second equality is by Claim~\\ref{VD1}. The above equation is a contradiction that $|A^{{\\mathcal{I}} \\setminus A}| > |A|$. \n\t\t\t\n\t\t\t\n\t\t\t\\item There are $G \\in {\\mathcal{G}}$, $\\ell \\in A^{{\\mathcal{I}} \\setminus A} \\cap G$ and $y \\in G \\setminus A^{{\\mathcal{I}} \\setminus A}$ such that $s(y) > s(\\ell)$. This is a contradiction to Claim~\\ref{VD2}. \n\t\t\\end{enumerate}\n\t\t\n\t\tOtherwise, if the two cases above are not satisfied:\n\t\t\n\t\t$$s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) = \\sum_{G \\in {\\mathcal{G}}~} s\\left(G\\cap A^{{\\mathcal{I}} \\setminus A}\\right) \\leq  \\sum_{G \\in {\\mathcal{G}}~} s\\left(G\\cap A \\right) = s(A).$$ The inequality is by Claim~\\ref{VD1}, Claim~\\ref{VD2} and that conditions 1. and 2. above are not satisfied. By the above equation, we reach a contradiction that  $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) > s(A)$. \n\t\t\n\t\\end{proof}\n\t\n\t\n\t\n\tBy Claim~\\ref{VD3} we can apply the assumption of the induction on ${\\mathcal{I}} \\setminus A$; thus, Algorithm~\\ref{alg:GREEDY} returns a packing for ${\\mathcal{I}} \\setminus A$ with at most $V({\\mathcal{I}} \\setminus A)$ bins. In addition, by Claim~\\ref{ABCV2} it holds that $V({\\mathcal{I}} \\setminus A) \\leq V({\\mathcal{I}})-1$. Hence, by Step~\\ref{gr:return}, the returned object by the algorithm is $X = (A) \\oplus \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ which contains at most $V({\\mathcal{I}})$ entries. Moreover, it holds that $X$ is a packing of ${\\mathcal{I}}$ by Claim~\\ref{clm:confappend} using the following arguments: (i) $A \\in {\\mathcal{C}}({\\mathcal{I}})$ by Claim~\\ref{clm:A} and (ii) $ \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ is a packing of ${\\mathcal{I}} \\setminus A$ by the assumption of the induction. \n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\t\\label{delta>>}\n\tIf $s(A) \\geq 1-\\delta$, then Algorithm~\\ref{alg:GREEDY} returns a packing for ${\\mathcal{I}}$ with at most $p({\\mathcal{I}})$ bins.\n\\end{lemma}\n\n\\begin{proof}\n\tWe prove the claim by induction on $p({\\mathcal{I}})$. For $p({\\mathcal{I}}) \\leq 1$ it holds by \\eqref{promise} that $I$ is a configuration of ${\\mathcal{I}}$: it follows that $V({\\mathcal{I}} \\setminus A) = 0$ by Claim~\\ref{ABCV} and therefore, by \\eqref{IIS} it follows that $I \\setminus A = \\emptyset$ and $A = I$.  Hence, by Step~\\ref{gr:if1} and Step~\\ref{gr:if} it holds that Algorithm~\\ref{alg:GREEDY} returns $(I)$, which is a packing of ${\\mathcal{I}}$ because $I$ is a configuration of ${\\mathcal{I}}$. Now, for the assumption of the induction, assume that for any BPP instance ${\\mathcal{J}} = (J,{\\mathcal{G}}_J,s_J,k_J)$ such that (i) $s(\\ell) \\leq \\delta ~\\forall \\ell \\in J$, (ii) $s\\left(A^{{\\mathcal{J}}}\\right) \\geq 1-\\delta$, and (iii) $p({\\mathcal{J}}) \\leq p({\\mathcal{I}})-1$, it holds that $\\textsf{Greedy}({\\mathcal{J}},\\delta)$ returns a packing of ${\\mathcal{J}}$ of at most $p({\\mathcal{J}})$ bins.\n\t\n\tFor the step of the induction, assume that $p({\\mathcal{I}}) > 1$. We use the following auxiliary claim. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{prom:2}\n\t\t$p\\left({\\mathcal{I}} \\setminus A\\right) \\leq  p({\\mathcal{I}})-1$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\tWe use the following inequality.  \\begin{equation}\n\t\t\t\\label{ABC}\n\t\t\t(1+2\\delta) \\cdot s(I \\setminus A) +2 \\leq (1+2\\delta) \\cdot (s(I)-(1-\\delta))+2 \\leq (1+2\\delta) \\cdot s(I) -1+2\\leq p({\\mathcal{I}})-1. \n\t\t\\end{equation} The first inequality is because $s(A) \\geq 1-\\delta$. The second inequality is because $\\delta \\in (0,0.5)$. The last inequality is by \\eqref{promise}. Now, $$p\\left({\\mathcal{I}} \\setminus A\\right) = \\max \\left\\{    (1+2\\delta) \\cdot s(I \\setminus A)+2, V({\\mathcal{I}} \\setminus A)    \\right\\} \\leq p({\\mathcal{I}})-1.$$ The first equality is by \\eqref{IIS} and \\eqref{promise}. The first inequality is by \\eqref{ABC} and Claim~\\ref{ABCV}\n\t\\end{proof}\n\t\n\tTo conclude the lemma, observe the following two complementary cases. \\begin{enumerate}\n\t\t\\item $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) \\geq 1-\\delta$. Then, by Claim~\\ref{prom:2} and that $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) \\geq 1-\\delta$, we can apply the assumption of the induction on ${\\mathcal{I}} \\setminus A$; thus, Algorithm~\\ref{alg:GREEDY} applied on ${\\mathcal{I}} \\setminus A$ returns a packing for ${\\mathcal{I}} \\setminus A$ with at most $p\\left({\\mathcal{I}} \\setminus A\\right)$ bins. In addition, by Claim~\\ref{prom:2} it holds that $p\\left({\\mathcal{I}} \\setminus A\\right) \\leq  p({\\mathcal{I}})-1$. Hence, by Step~\\ref{gr:return}, the returned object by the algorithm on ${\\mathcal{I}}$ is $X = (A) \\oplus \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ which contains at most $ 1+p({\\mathcal{I}})-1  = p({\\mathcal{I}})$ entries. Moreover, it holds that $X$ is a packing of ${\\mathcal{I}}$ by Claim~\\ref{clm:confappend} using the following arguments: (i) $A \\in {\\mathcal{C}}({\\mathcal{I}})$ by Claim~\\ref{clm:A} and (ii) $ \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ is a packing of ${\\mathcal{I}} \\setminus A$ by the assumption of the induction. \n\t\t\n\t\t\\item $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) < 1-\\delta$. Then, by Lemma~\\ref{delta<} and that $s\\left(A^{{\\mathcal{I}} \\setminus A}\\right) < 1-\\delta$, Algorithm~\\ref{alg:GREEDY} applied on ${\\mathcal{I}} \\setminus A$ returns a packing for ${\\mathcal{I}} \\setminus A$ with at most $V({\\mathcal{I}} \\setminus A)$ bins. In addition, by Claim~\\ref{ABCV} it holds that $V({\\mathcal{I}} \\setminus A) \\leq  p({\\mathcal{I}})-1$. Hence, by Step~\\ref{gr:return}, the returned object by the algorithm is $X = (A) \\oplus \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ which contains at most $ 1+p({\\mathcal{I}})-1  \\leq p({\\mathcal{I}})$ entries by \\eqref{promise}. Moreover, it holds that $X$ is a packing of ${\\mathcal{I}}$ by Claim~\\ref{clm:confappend} using the following arguments: (i) $A \\in {\\mathcal{C}}({\\mathcal{I}})$ by Claim~\\ref{clm:A} and (ii) $ \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ is a packing of ${\\mathcal{I}} \\setminus A$ by Lemma~\\ref{delta<}. \n\t\\end{enumerate}\n\\end{proof}\n\n\n\\begin{lemma}\n\t\\label{gr:running}\n\tThe running time of Algorithm~\\ref{alg:GREEDY} is $\\textnormal{poly} (|{\\mathcal{I}} |)$. \n\\end{lemma}\n\n\\begin{proof}\n\tStep~\\ref{gr:if} and Step~\\ref{gr:if1} can be trivially computed in linear time in $|{\\mathcal{I}}|$. Step~\\ref{gr:end0} can be computed in polynomial time in $|{\\mathcal{I}}|$ by (i) finding the bounding groups by \\eqref{bounding}, and (ii) compute a bounding subset of ${\\mathcal{I}}$ as follows. For each bounding group $G$ we compute a bounding subset of $G$ by sorting the items and choosing a minimal number of items according to the sorted order that form a bounding subset of $G$ (for more details see the proof of Lemma~\\ref{bG}).    \n\t\n\tthe while loop of Step~\\ref{gr:while2} runs at most $|I|$ times because in each iteration an item is added to the constructed bin and is not added again by Step~\\ref{gr:while2} and Step~\\ref{gr:swap2}. Finally, the while loop of Step~\\ref{gr:while1} runs at most ${|I| \\choose 2} = \\textnormal{poly} (|{\\mathcal{I}} |)$ times because in each iteration two items are replaced in Step~\\ref{gr:swap}, one is add to the constructed bin and the other one is removed; for any two items, this can happen at most once by Step~\\ref{gr:while1}. We conclude that the running time of Algorithm~\\ref{alg:GREEDY} is $\\textnormal{poly} (|{\\mathcal{I}} |)$. \n\\end{proof}\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{thm:greedy}:} If $s(A) < 1-\\delta$ it holds that $X = (A) \\oplus \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$ is a packing of ${\\mathcal{I}}$ of at most $V({\\mathcal{I}})$ bins by Lemma~\\ref{delta<}; therefore, the number of bins is at most $p({\\mathcal{I}})$ by \\eqref{promise}. Otherwise, it holds that $s(A) \\geq 1-\\delta$; thus, it holds that $X$ is a packing of ${\\mathcal{I}}$ of at most $p({\\mathcal{I}})$ bins by Lemma~\\ref{delta>>}. Therefore, the number of bins in the packing is at most $p({\\mathcal{I}}) \\leq (1+2\\delta) \\cdot \\max\\left\\{ s(I),~V(\\mathcal{I})\\right\\}+2$ by \\eqref{promise}. Finally, the running time is $\\textnormal{poly} (|{\\mathcal{I}} |)$ by Lemma~\\ref{gr:running}.\\qed\n\n\n\\section{Omitted Proofs of Section~\\ref{sec:overview}}\n\\label{app:omitted}\n\n\\newcommand{\\textnormal{LP}}{\\textnormal{LP}}\n\\newcommand{\\mathcal{D}}{\\mathcal{D}}\n\\newcommand{\\textnormal{Dual}}{\\textnormal{Dual}}\n\\newcommand{\\textnormal{\\textsf{Ellipsoid}}}{\\textnormal{\\textsf{Ellipsoid}}}\n\\noindent{\\bf Proof of Lemma~\\ref{configurationLP}:}\nThe approach presented here is considered as standard, and often the result is mentioned without a proof (see, e.g., Theorem~1.1 in \\cite{bansal2010new}). We include the proof \nfor completeness. We refer to terms such as separation oracle and  well-described polyhedron as defined in~\\cite{grotschel2012geometric}. \n\nLet ${\\mathcal I}=(I,{\\mathcal{G}}, s, k )$ be a BPP instance and ${\\varepsilon}\\in (0,0.1)$. \nFor any $\\mathcal{D}\\subseteq {\\mathcal{C}}$ we define the following linear program.\n\\begin{equation}\n\t\\label{C-LP-relaxed}\n\t\\textnormal{LP}(\\mathcal{D}):~~~~~\n\t\\begin{aligned}\n\t\t~~~~~ \\min\\quad        & ~~~~~\\sum_{C \\in \\mathcal{D}} \\bar{x}_C                                                           \\\\\n\t\t\\textsf{s.t.\\quad} & ~~~\\sum_{~C \\in \\mathcal{C}[\\ell]\\cap \\mathcal{D}} \\bar{x}_C \\geq 1   & \\forall \\ell \\in I~~~~~\\\\  \n\t\t& ~~~~~\\bar{x}_C \\geq 0 ~~~~~~~~&~~~~~~~~~~~~~~~~~~~~ \\forall C \\in  \\mathcal{C}~~~~\n\t\\end{aligned}\n\\end{equation}\nWhile $\\textnormal{LP}({\\mathcal{C}})$ is not identical to \\eqref{C-LP}, solving $\\textnormal{LP}({\\mathcal{C}})$ is equivalent to solving \\eqref{C-LP} and we will focus on this objective. \n\nFor any $\\mathcal{D}\\subseteq {\\mathcal{C}}$, the dual linear program of $\\textnormal{LP}(\\mathcal{D})$ is the following. \n\\begin{equation}\n\t\\label{dual}\n\t\\textnormal{Dual}(\\mathcal{D}):~~~~~\n\t\\begin{aligned}\n\t\t~~~~~ \\max\\quad        & ~~~~~\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell}                                   \\\\\n\t\t\\textsf{s.t.\\quad} & ~~~\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell} \\leq 1   & \\forall C\\in \\mathcal{D}~~~~~\\\\  \n\t\t& ~~~~~\\bar{\\lambda}_{\\ell} \\geq 0 ~~~~~~~~&~~~~~~~~~~~~~~~~~~~~ \\forall \\ell \\in I~~~~\n\t\\end{aligned}\n\\end{equation}\nWe note that $\\textnormal{Dual}({\\mathcal{C}})$ can be solved in polynomial time given a separation oracle (see Theorem 6.3.2 in~\\cite{grotschel2012geometric}). However, since no such separation oracle exists, we apply a technique dating back to~\\cite{karmarkar1982efficient}  in order to obtain an approximate solution.\n\nGiven $\\mathcal{D}\\subseteq {\\mathcal{C}}$ and $v\\in \\mathbb{R}$, define a polytope\n\\begin{equation}\n\t\\label{eq:F_def}\n\tF(\\mathcal{D}, v) = \\left\\{\\bar{\\lambda}\\in \\mathbb{R}^I_{\\geq 0} ~~\\middle|~~\\begin{aligned}\n\t\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell} \\geq v \\\\ \n\t\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell} \\leq 1 &~~~~~~~\\forall C\\in \\mathcal{D}\n\\end{aligned}\\right\\}\n\\end{equation}\nClearly, $\\textnormal{OPT}(\\textnormal{Dual}(\\mathcal{D})) \\geq v$ if and only if $F(\\mathcal{D},v) \\neq \\emptyset$. \n\nObserve that, for every $\\mathcal{D}\\subseteq {\\mathcal{C}}$ and $v\\in \\mathbb{R}_{\\geq 0}$, each of the inequalities in the definitions $F(\\mathcal{D}, v)$ can be represented using $\\varphi= O(|I| + \\|v\\|)$ bits, where $\\|v\\|$ is the size of the representation for the number~$v$; thus, $(F(\\mathcal{D}, v), n ,\\varphi)  $  is a well-described polyhedron (Definition 6.2.2 in~\\cite{grotschel2012geometric}). \nBy Theorem 6.4.1 in~\\cite{grotschel2012geometric} there is an algorithm $\\textnormal{\\textsf{Ellipsoid}}$ which given a separation oracle for $F(\\mathcal{D},v)$ determines if $F(\\mathcal{D},v)\\neq \\emptyset$ in time $\\textnormal{poly}(|I|, \\|v\\|)$. \n\nWe use $\\textnormal{\\textsf{Ellipsoid}}$ to determine if $F({\\mathcal{C}},v)\\neq \\emptyset$ with the following (flawed) separation oracle. Given $\\bar{\\lambda}\\in \\mathbb{R}^I_{\\geq 0}$, the oracle first checks if $\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell} <v$. If this is the case, the algorithm returns $\\sum_{\\ell \\in I} \\bar{\\lambda_{\\ell}} <v$ as a separating hyperplane.  Otherwise, the algorithm runs the FPTAS for CMP with the instance~${\\mathcal I}$, the weight function $w(\\ell)= \\bar{\\lambda}_{\\ell}$ and $\\frac{{\\varepsilon}}{10}$ as the error. If the FPTAS returned $C\\in {\\mathcal{C}}$ such that $\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell}>1$, the algorithm returns  $\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell}>1$ as a separating hyperplane. If the configuration returned by the FPTAS does not meet this  condition, then  the oracle aborts the execution of $\\textnormal{\\textsf{Ellipsoid}}$.\n\nObserve that the execution of the separation oracle runs in time $\\textnormal{poly}(|{\\mathcal{I}}|, \\frac{1}{{\\varepsilon}}, \\|v\\|)$. Thus, the execution of $\\textnormal{\\textsf{Ellipsoid}}$ with the oracle terminates in polynomial time. The execution can either end with a declaration that $F({\\mathcal{C}},v)=\\emptyset$ or be aborted by the separation oracle. Consider each of these two cases:\n\\begin{itemize}\n\t\\item\nThe execution terminated by declaring that $F({\\mathcal{C}},v)=\\emptyset$. Then, as the separating hyperplanes returned by the oracle are indeed separating hyperplanes, it follows that $F({\\mathcal{C}},v)=\\emptyset$  is a correct statement. Let $\\mathcal{D}$ be the set of configurations returned as separating hyperplanes throughout the execution. Then, as all the separating hyperplanes returned are separating hyperplanes for $F(\\mathcal{D}, v)$ we conclude that $F(\\mathcal{D},v)=\\emptyset$ as well. Furthermore, $|\\mathcal{D}|$ is polynomial in $|I|$ and $\\|v\\|$, as the running time of $\\textnormal{\\textsf{Ellipsoid}}$ is polynomial in these variables. \n\n\\item Otherwise, the execution of $\\textnormal{\\textsf{Ellipsoid}}$ has been aborted. Let $\\bar{\\lambda}$ be the value given to the separation oracle on its last call (the one which ended up with the abortion). It follows that $\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell}\\geq v$ and $\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell} \\leq \\frac{1}{1-\\frac{{\\varepsilon}}{10}}$ for all $C \\in {\\mathcal{C}}$ (otherwise the FPTAS must return a solution $C$ for which $\\sum_{\\ell \\in C} \\bar{\\lambda}_{\\ell} >1$). Then it holds that $\\left(1-\\frac{{\\varepsilon}} {10}\\right) \\bar{\\lambda} \\in F\\left({\\mathcal{C}},\\left(1-\\frac{{\\varepsilon}} {10}\\right)\\cdot v \\right)$, and consequently $F\\left({\\mathcal{C}},\\left(1-\\frac{{\\varepsilon}} {10}\\right)\\cdot v \\right)\\neq \\emptyset$.\n\\end{itemize}\n\nThus, using a binary search, we can find $v\\in [0, n]$ and $\\mathcal{D}\\subseteq {\\mathcal{C}}$ such that $F({\\mathcal{C}},v)=F(\\mathcal{D}, v)=\\emptyset$, $F\\left({\\mathcal{C}}, \\left( 1-\\frac{{\\varepsilon}}{2}\\right) \\cdot v\\right)\\neq \\emptyset$, and $|\\mathcal{D}|$ is polynomial in $|I|$.  As $F\\left({\\mathcal{C}}, \\left( 1-\\frac{{\\varepsilon}}{2}\\right) \\cdot v\\right)\\neq \\emptyset$ it follows that $\\textnormal{OPT}(\\textnormal{Dual}({\\mathcal{C}})) \\geq\\left( 1-\\frac{{\\varepsilon}}{2}\\right) \\cdot  v$, and by strong duality it holds that $\\textnormal{OPT}(\\textnormal{LP}({\\mathcal{C}})) \\geq \\left( 1-\\frac{{\\varepsilon}}{2}\\right) \\cdot v$.   Furthermore, as $F({\\mathcal{C}},v)=F(\\mathcal{D}, v)=\\emptyset$ it follows that $\\textnormal{OPT}(\\textnormal{Dual}(\\mathcal{D}))\\leq v$, and by strong duality $\\textnormal{OPT}(\\textnormal{LP}(\\mathcal{D}))\\leq v$. As $|\\mathcal{D}|$ is polynomial we can solve $\\textnormal{LP}(\\mathcal{D})$ in polynomial time and obtain a solution ${\\bar{x}}$ (which is also a solution for $\\textnormal{LP}({\\mathcal{C}})$), such that $$\\|{\\bar{x}}\\| =\\textnormal{OPT}(\\textnormal{LP}(\\mathcal{D})) \\leq v \\leq \\frac{\\textnormal{OPT}(\\textnormal{LP}({\\mathcal{C}}))}{1-\\frac{{\\varepsilon}}{2}} \\leq (1+{\\varepsilon})\\cdot \\textnormal{OPT}(\\textnormal{LP}({\\mathcal{C}})).$$\n\nOverall, we obtained a $(1+{\\varepsilon})$-approximate solution in $\\textnormal{poly}(|{\\mathcal{I}}|, \\frac{1}{{\\varepsilon}})$ time, as required. \n\n \\qed\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:AFPTAS}:} \nObserve that the optimum of \\eqref{C-LP} is at most $\\textnormal{OPT}({\\mathcal I})$, thus by Lemma~\\ref{configurationLP} it holds that $\\bar{x}$ is a solution to the configuration LP \\eqref{C-LP} of $\\mathcal{I}$ and  $\\|{\\bar{x}}\\|\\leq(1+{\\varepsilon})\\textnormal{OPT}(\\mathcal{I})$. Therefore, by Lemma~\\ref{lem:eviction}, Algorithm $\\textsf{Evict}({\\varepsilon},{\\mathcal{I}},\\bar{x})$ in Step~\\ref{step:evicAFPTAS} returns a prototype $\\bar{y}$ with $\\bar{\\gamma}$ in the $\\bar{y}$-polytope such that (i) for all $\\ell,j \\in I, \\ell \\neq j$ it holds that $\\bar{\\gamma}_{\\ell,j} = 0$; (ii) for all $C \\in \\textnormal{\\textsf{supp}}(\\bar{y})$ it holds that $|C| \\leq {\\varepsilon}^{-10}$ and $s(C \\setminus L) \\leq {\\varepsilon}$; (iii) $\\|\\bar{y}\\| \\leq (1+{\\varepsilon})\\|\\bar{x}\\|$; (iv) for all $\\ell \\in I$ it holds that $\\sum_{C \\in {\\mathcal{C}}[\\ell]} {\\bar{y}}_C \\leq 2$.  Then, in Step~\\ref{GetPolytope}, Algorithm $\\textsf{Shift}({\\varepsilon},{\\mathcal{I}},\\bar{y})$ returns a good prototype $\\bar{z}$ by Lemma~\\ref{lem:Cnf} such that\n\n\\begin{equation}\n\t\\label{final1}\n\t\\begin{aligned}\n\t\t\\|\\bar{z}\\| \\leq{} & (1+5{\\varepsilon})\\|\\bar{y}\\|+Q({\\varepsilon}) \\leq (1+5{\\varepsilon})\\cdot (1+{\\varepsilon})\\cdot \\|\\bar{x}\\|+Q({\\varepsilon}) \\\\ \\leq{} & (1+9{\\varepsilon})(1+{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+Q({\\varepsilon})\\leq (1+19{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+Q({\\varepsilon}).\n\t\\end{aligned}\n\\end{equation} The first inequality is by Lemma~\\ref{lem:Cnf} . The second inequality is by Lemma~\\ref{lem:eviction}. The third inequality is by   Lemma~\\ref{configurationLP} and because $0<{\\varepsilon} < 0.1$. Consequently, by Lemma~\\ref{FromPolytope}, in Step~\\ref{step:BPPnice} it holds that $\\mathcal{B}$ is an ${\\varepsilon}$-nice partition with size at most: \n\n\\begin{equation}\n\t\\label{final2}\n\t\\|\\bar{z}\\| + {\\varepsilon}^{-22}Q^2({\\varepsilon}) \\leq (1+19{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+Q({\\varepsilon})+ {\\varepsilon}^{-22}Q^2({\\varepsilon}) \\leq (1+19{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+{\\varepsilon}^{-23}Q^2({\\varepsilon})\n\\end{equation} The first inequality is \\eqref{final1}. The second inequality is because $0<{\\varepsilon}<0.1$. Then, by Lemma~\\ref{lem:GREEDY}, in Step~\\ref{step:greedy1}, a full packing $\\Phi$ of $\\mathcal{I}$ is constructed. The number of bins in $\\Phi$ is bounded by:\n\n\\begin{equation*}\n\t\\label{final3}\n\t\\begin{aligned}\n\t\t\\#\\textsf{bins}(\\Phi) \\leq{} & (1+2{\\varepsilon}) \\left( (1+19{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+{\\varepsilon}^{-23}Q^2({\\varepsilon}) \\right) +2{\\varepsilon}^{-22}Q^2({\\varepsilon}) \\leq (1+60{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+3{\\varepsilon}^{-23}Q^2({\\varepsilon})\\\\\n\t\t\\leq{} &(1+60{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+{\\varepsilon}^{-24}Q^2({\\varepsilon}) \\leq (1+60{\\varepsilon}) \\textnormal{OPT}(\\mathcal{I})+Q^3({\\varepsilon}).\n\t\\end{aligned}\n\\end{equation*} The first inequality is by \\eqref{final2} and Lemma~\\ref{lem:GREEDY}. The other inequalities hold as $0<{\\varepsilon}<0.1$. \\qed\n\n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:gen_afptas}}\n\n\n\\begin{algorithm}[h]\n\t\\caption{$\\textnormal{\\textsf{Gen-AFPTAS}}({{\\mathcal{J}}}, {\\varepsilon})$}\n\t\\label{alg:genalg}\n\t\n\t\\SetKwInOut{Input}{Input}\n\t\\SetKwInOut{Output}{Output}\n\t\\Input{An  BPP instance ${\\mathcal{J}}$ and ${\\varepsilon}\\in (0,0.1)$ such that  ${\\varepsilon}^{-1}\\in \\mathbb{N}$}\n\t\\Output{A packing of ${\\mathcal{J}}$}\n\t\n\t\n\tCompute an ${\\varepsilon}$-structured instance $\\mathcal{I}$ by $\\textsf{Reduce}({\\cal J},{\\varepsilon})$ \\label{step:reduction}\\;\n\t\n\tCompute $\\Phi =\\textsf{AFPTAS}({\\mathcal I},{\\varepsilon})$ (Algorithm~\\ref{alg:Fscheme}) \\label{step:solve_structured}\\;\n\t\n\t\n\tReturn a packing for ${\\cal J}$ by $A = \\textsf{Reconstruct}({\\mathcal{J}},{\\varepsilon},\\Phi)$ \\label{step:recon1}\n\\end{algorithm}\nThe pseudo-code of $\\textnormal{\\textsf{Gen-AFPTAS}}$ is given in Algorithm~\\ref{alg:genalg}. \nConsider the execution of $\\textnormal{\\textsf{Gen-AFPTAS}}$ with a BPP instance ${\\mathcal{J}}$ and ${\\varepsilon}\\in(0,0.1)$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$.\nBy Lemma~\\ref{lem:reductionReconstruction} it holds that the instance~${\\mathcal I}$ computed in Step~\\ref{step:reduction} is an ${\\varepsilon}$-structured instance and $\\textnormal{OPT}({\\mathcal I}) \\leq \\textnormal{OPT}({\\mathcal{J}})$. Thus by Lemma \\ref{lem:AFPTAS} it holds $\\Phi$ (calculated in Step~\\ref{step:solve_structured}) is a packing of ${\\mathcal I}$ which uses at most \n$$ (1+60{\\varepsilon}) \\cdot \\textnormal{OPT}({\\mathcal I}) + Q^3({\\varepsilon})\\leq  (1+60{\\varepsilon}) \\cdot \\textnormal{OPT}({\\mathcal{J}}) + Q^3({\\varepsilon})$$\nbins.  Thus, by Lemma~\\ref{lem:reductionReconstruction}, it holds that the packing $A$ returned by the algorithm is a packing of ${\\mathcal{J}}$ which uses at most \n$$\n(1+60{\\varepsilon}) \\cdot \\textnormal{OPT}({\\mathcal{J}}) + Q^3({\\varepsilon}) +13{\\varepsilon}\\cdot  \\textnormal{OPT}({\\mathcal{J}})  +1 \\leq (1+130) \\cdot \\textnormal{OPT}({\\mathcal{J}}) +3\\cdot Q^{3}({\\varepsilon}) \n$$\nbins.\n\nIt easily follows from Lemmas~\\ref{lem:AFPTAS} and~\\ref{lem:reductionReconstruction} that  the overall running time of the Algorithm~\\ref{alg:genalg} is $\\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$. \n\\qed\n\n\n\\noindent{\\bf Proof of Theorem~\\ref{thm:main}:}\nLet $\\mathcal{J} =  (I, \\mathcal{G},s,k)$ be a BPP instance. Recall that $V({\\mathcal{J}}) = \\max_{G \\in \\mathcal{G}} \\ceil{\\frac{|G|}{k(G)}}$ is the maximum cardinality of a group in $\\mathcal{J}$ divided by the cardinality constraint of the group and let $W =s(I)+ V({\\mathcal{J}})+c$ where $c = \\exp\\left(\\exp\\left({100^{17}}\\right)\\right)$ is a large constant. \n\\begin{claim}\n\t\\label{clm:Weps}\n$$ \n\\textnormal{OPT}(\\mathcal{J}) \\leq 2\\cdot W\n$$\n\\end{claim}\n\\begin{proof}\n\tLet $L_{\\frac{1}{2}}= \\{ \\ell \\in I~|~s(\\ell) > \\frac{1}{2}\\}$. By Lemma~\\ref{thm:greedy} it follows that $\\textnormal{\\textsf{Greedy}}$ finds a packing of the instance ${\\mathcal{J}} \\setminus L_{\\frac{1}{2}}$ using \n\tat most $$\\left(1+2\\cdot \\frac{1}{2}\\right) \\cdot\\max\\left\\{s(I\\setminus L_{\\frac{1}{2}}), V({\\mathcal{J}}\\setminus L_{\\frac{1}{2}}) \\right\\}+2\\leq 2 \\cdot\\max \\left\\{s(I\\setminus L_{\\frac{1}{2}}), V({\\mathcal{J}}) \\right\\}+2$$\n\tbins, where we define ${\\mathcal{J}} \\setminus L_{\\frac{1}{2}} = (I_{L_{\\frac{1}{2}}},{\\mathcal{G}}_{L_{\\frac{1}{2}}},s_{L_{\\frac{1}{2}}},k_{L_{\\frac{1}{2}}})$ as the BPP instance such that \\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tI_{L_{\\frac{1}{2}}} = I \\setminus L_{\\frac{1}{2}}, ~~~~~~~~~~~~~~~~~~~~ ~~~~~~\n\t\t\t\\mathcal{G}_{L_{\\frac{1}{2}}} = \\{G \\cap I_{L_{\\frac{1}{2}}} \\neq \\emptyset~|~ G \\in {\\mathcal{G}}\\}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\t\t\t\\\\\n\t\t\ts_{L_{\\frac{1}{2}}}(\\ell) = s(\\ell), ~~\\forall \\ell \\in I_{L_{\\frac{1}{2}}}, ~~~~~~~~~~~\n\t\t\tk_{L_{\\frac{1}{2}}}(G \\cap I_{L_{\\frac{1}{2}}}) = k(G), ~~\\forall G \\in {\\mathcal{G}} \\text{ s.t. } G \\cap I_{L_{\\frac{1}{2}}} \\neq \\emptyset.~~~~~~\n\t\t\\end{aligned}\n\t\\end{equation*} The items in $L_{\\frac{1}{2}}$ can be packed  into $|L_{\\frac{1}{2}}|\\leq 2 \\cdot s \\left( L_{\\frac{1}{2}}\\right) $ bins (that is, partitioned into $|L_{\\frac{1}{2}}|$ configurations) with  a single item per bin. Thus, set of items $I$ can be packed into number of bins bounded by\n\t$$\n\t2 \\cdot\\max \\left\\{s(I\\setminus L_{\\frac{1}{2}}), V({\\mathcal{J}}) \\right\\}+2 + 2 \\cdot s \\left( L_{\\frac{1}{2}}\\right)  \\leq \n\t2 \\left( s(I) + V({\\mathcal{J}})\\right)+2 \\leq 2 \\cdot W\n\t$$\n\\end{proof}\n Now, define ${\\varepsilon} = \\floor{\\left(  \\ln \\ln W \\right)^{\\frac{1}{17}}}^{-1}$. Since $W \\geq c$, it holds that  $0<{\\varepsilon} <0.1$.\nIn the following we show that the running time of $\\textnormal{\\textsf{Gen-AFPTAS}}$ on the input $\\mathcal{J}$  and ${\\varepsilon}$, as defined above, is polynomial in $|\\mathcal{J}|$, while the number of bins in the packing returned by the algorithm is $\\textnormal{OPT}(\\mathcal{J})+o(\\textnormal{OPT}(\\mathcal{J}))$.\n\n\nBy Lemma~\\ref{lem:gen_afptas}, there are polynomial functions $f,g$ such that the running time of $\\textnormal{\\textsf{Gen-AFPTAS}} \\left( \\mathcal{J}, {\\varepsilon} \\right)$ is at most $f\\left(\\frac{1}{{\\varepsilon}}\\right) \\cdot g(|\\mathcal{J}|)$. We assume without the loss of generality that $f$ is monotone. Therefore, \n\\begin{equation}\n\t\\label{eq:polymain0}\n\tf\\left(\\frac{1}{{\\varepsilon}}\\right)  \\cdot g(|\\mathcal{J}|) = f\\left(\\floor{ \\left(  \\ln \\ln W \\right)^{\\frac{1}{17}}}\\right)  \\cdot g(|\\mathcal{J}|) \\leq  f(W)  \\cdot g(|\\mathcal{J}|) \\leq f(2\\cdot |{\\mathcal{J}}|+c) \\cdot g(|{\\mathcal{J}}|)\n\\end{equation}\nThe first equality is by the definition of ${\\varepsilon}$. The first inequality is because $W>c>1$ and for all $x \\geq 1$ it holds that $\\ln x \\leq x$.  The last inequality hold as $V({\\mathcal{I}}), s(I)\\leq |I|\\leq |{\\mathcal{J}}|$. \n Since $c$ is a constant it follows from  the \\eqref{eq:polymain0} that the running time of $\\textnormal{\\textsf{Gen-AFPTAS}}({\\mathcal{J}},{\\varepsilon})$ is polynomial in $|{\\mathcal{J}}|$.\n\n\nBy Lemma~\\ref{lem:gen_afptas},  it holds that the packing $A$ returned by $\\textsf{Gen-AFPTAS} \\left( \\mathcal{J}, {\\varepsilon} \\right)$ is a packing of $\\mathcal{J}$ which uses at most  $ (1+130{\\varepsilon})OPT(\\mathcal{I})+3\\cdot Q^3({\\varepsilon})$ bins. \nNow,  \\begin{equation}\n\t\\label{eq:Q}\n\t\\begin{aligned}\n\t\t\tQ({\\varepsilon}) ={} & \\exp({\\varepsilon}^{-17}) =  \\exp \\left( \\left(   \\floor{\\left( \\ln \\ln W \\right)^{\\frac{1}{17}}}^{-1} \\right)^{-17} \\right) \\\\ \\leq{} & \\exp \\left(   \\ln \\ln W \\right) = \\ln  W \\leq \\ln  \\left( 2\\cdot\\textnormal{OPT}(\\mathcal{J}) +c \\right). \n\t\\end{aligned}\n\\end{equation}\n\nThe first equality is by the definition of $Q({\\varepsilon})$. The second equality is by the definition of ${\\varepsilon}$.  The last inequality holds by $\\textnormal{OPT}({\\mathcal{J}})\\geq s(I)$, $\\textnormal{OPT}({\\mathcal{J}})\\geq V({\\mathcal{I}})$  and the definition of $W$. Hence,  the number of bins used by $A$ is at most\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\label{eq:n1}\n\t& (1+130{\\varepsilon})\\textnormal{OPT}(\\mathcal{J})+3\\cdot Q^3({\\varepsilon}) \\\\\n\t\t\\leq& \\textnormal{OPT}(\\mathcal{J})+  130 \\floor{\\left(  \\ln \\ln W \\right)^{\\frac{1}{17}}}^{-1}\\textnormal{OPT}(\\mathcal{J})+3\\cdot\\ln^3  \\left( 2\\cdot\\textnormal{OPT}(\\mathcal{J}) +c \\right)\\\\\n\t\t\\leq & \\textnormal{OPT}(\\mathcal{J})+  130\\cdot \\frac{ \\textnormal{OPT}({\\mathcal{J}})}{\\left(\\ln \\ln\\left( \\frac{\\textnormal{OPT}({\\mathcal{J}})}{2}\\right) \\right)^{\\frac{1}{17}} -1} +3\\cdot \\ln^3  \\left( 2\\cdot\\textnormal{OPT}(\\mathcal{J}) +c \\right) \\\\  ={} &  \\textnormal{OPT}(\\mathcal{J})+O\\left(\\frac{\\textnormal{OPT}(\\mathcal{J})}{ \\left(\\ln \\ln \\textnormal{OPT}({\\mathcal{J}}) \\right)^{\\frac{1}{17}}}\\right).\n\t\\end{aligned}\n\\end{equation}\nThe  first inequality is by the definition of ${\\varepsilon}$ and \\eqref{eq:Q}. The second inequality holds since $W\\geq \\frac{\\textnormal{OPT}({\\mathcal{J}})}{2}$ by Claim~\\ref{clm:Weps}.\n\n  Overall, we showed that the algorithm returns a packing of ${\\mathcal{J}}$ using $ \\textnormal{OPT}(\\mathcal{J})+O\\left(\\frac{\\textnormal{OPT}(\\mathcal{J})}{ \\left(\\ln \\ln \\textnormal{OPT}({\\mathcal{J}}) \\right)^{\\frac{1}{17}}}\\right)$ bins in polynomial time in $|{\\mathcal{J}}|$. \\qed\n\n\\section{Omitted Proofs of Section~\\ref{sec:alg_pack}}\n\\label{sec:PackProofs}\n\n\nWe use the following auxiliary claim. \\begin{claim}\n\t\\label{clm:fitifit}\n\tFor all $C \\in \\mathcal{H}$ and $\\ell \\in D_C$ it holds that $s_C(\\ell) < 0.1$.\n\\end{claim}\n\n\\begin{proof}\n\t$$s_C(\\ell) = \\frac{s(\\ell)}{1-s(C)} \\leq \\frac{{\\varepsilon} \\cdot (1-s(C))}{1-s(C)} = {\\varepsilon} < 0.1.$$\n\t The first equality is by \\eqref{IC}. The first inequality is because $\\ell \\in \\textsf{fit}(C)$ by Definition~\\ref{def:partition}; hence, the inequality follows by \\eqref{fitS}. \n\\end{proof}\n\n\\noindent{\\bf Proof of Lemma~\\ref{clm:GREEDY}:} Let $X = \\textsf{tuple}(B_C),  Y = \\textsf{Greedy}({\\mathcal{I}}_C,{\\varepsilon})$, $X = (X_1, \\ldots, X_r)$, $Y = (Y_1, \\ldots, Y_t)$, $S = D_C \\cup \\bigcup_{B \\in B_C} B$, $Z = X+Y$, and $Z = (Z_1, \\ldots, Z_p)$. We prove the following.\n\n\\begin{enumerate}\n\t\\item $Z$ is a partition of $S$. Let $\\ell \\in S$. If $\\ell \\in D_C$, then there is $i \\in [r]$ such that $\\ell \\in X_i$; therefore, by the definition of $Z$ it holds that $\\ell \\in Z_i$. Otherwise, there is $i \\in [t]$ such that $\\ell \\in Y_i$; therefore, by the definition of $Z$ it holds that $\\ell \\in Z_i$.\n\t\n\t\\item For all $i \\in [p]$ it holds that $s(Z_i) \\leq 1$. If $D_C = \\emptyset$ then the claim is satisfied because $s(Z_i) = s(X_i) \\leq s(C) \\leq 1$; The first inequality is because $X_i \\in B_C$ and thus $X_i$ is allowed in $C$; The second inequality is because $C$ is a configuration. Otherwise, If $i \\leq t$ and $i \\leq r$: $$s(Z_i) = s(X_i)+s(Y_i) \\leq s(C)+s(Y_i) \\leq s(C)+(1-s(C)) \\cdot s_C(Y_i) \\leq s(C)+1-s(C) = 1.$$ The first inequality is because by Definition~\\ref{def:partition} it holds that $X_i$ is allowed in $C$. The second inequality is by \\eqref{IC}. The third inequality is by Lemma~\\ref{thm:greedy}; also, the conditions of Lemma~\\ref{thm:greedy} are indeed satisfied by Claim~\\ref{clm:fitifit}. Using similar arguments, we can show the claim for $i > t$ or $i >r$ by the definition of $Z$.\n\t\n\t\\item For all $i \\in [p]$ and $G \\in {\\mathcal{G}}$ it holds that $|G \\cap Z_i| \\leq k(G)$. If $|G\\cap D_C| = 0$ then the claim is trivially satisfied because $X_i$ is allowed in the configuration $C$. Otherwise, if $i \\leq t$ and $i \\leq r$: \t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t|G \\cap Z_i|  \\leq{} & |G \\cap X_i| +|G \\cap Y_i|  \\leq |G \\cap C|+|G \\cap Y_i| \\leq |G \\cap C|+k_C(G \\cap D_C) \\\\ \\leq{} & |G \\cap C|+\\left(k(G)-|G \\cap C|\\right) = k(C).\n\t\t\\end{aligned}\n\t\\end{equation*}  The second inequality is because by Definition~\\ref{def:partition} it holds that $X_i$ is allowed in $C$. The third inequality is by Lemma~\\ref{thm:greedy}; also, the conditions of Lemma~\\ref{thm:greedy} are indeed satisfied by Claim~\\ref{clm:fitifit}. The fourth inequality is by \\eqref{IC}. Using similar arguments, we can show the claim for $i > t$ or $i >r$ by the definition of $Z$.\n\n\\item It  holds that $p \\leq (1+2{\\varepsilon}) \\cdot |B_C|+2$. \t\\begin{equation*}\n\t\\begin{aligned}\n\tp ={} & \\max\\{r, t\\} \\leq \\max\\left\\{|B_C|, (1+2{\\varepsilon})\\cdot   \\max\\left\\{  s_C(D_C) , \\max_{G \\in {\\mathcal{G}}_{C}} \\frac{|G|}{k_C(G)}   \\right\\} +2     \\right\\} \\\\ \\\\ \\leq{} &  \\max\\left\\{|B_C|, (1+2{\\varepsilon})\\cdot   \\max\\left\\{  \\frac{s(D_C)}{1-s(C)} , \\max_{G \\in {\\mathcal{G}} \\text{ s.t. } G \\cap D_C \\neq \\emptyset} \\frac{|G\\cap D_C|}{k(G)-|G \\cap C|}   \\right\\} +2     \\right\\} \\\\ \\\\ \\leq{} &  \\max\\left\\{|B_C|, (1+2{\\varepsilon})\\cdot   \\max\\left\\{  \\frac{(1-s(C)) \\cdot |B_C|}{1-s(C)} , \\max_{G \\in {\\mathcal{G}} \\text{ s.t. } G \\cap D_C \\neq \\emptyset} \\frac{|B_C| \\cdot (k(G) - |C \\cap G|)}{k(G)-|G \\cap C|}   \\right\\} +2     \\right\\} \\\\={} & (1+2{\\varepsilon})\\cdot |B_C|+2.\n\t\\end{aligned}\n\\end{equation*} The first inequality is by Lemma~\\ref{thm:greedy}; also, the conditions of Lemma~\\ref{thm:greedy} are indeed satisfied by Claim~\\ref{clm:fitifit}. The second inequality is by \\eqref{IC}. The third inequality is by Definition~\\ref{def:partition}. \n\n \\qed\n\\end{enumerate} \n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:GREEDY}:} For all $C \\in \\mathcal{H}$ let $X^C = \\textsf{tuple}(B_C),  Y^C = \\textsf{Greedy}({\\mathcal{I}}_C,{\\varepsilon})$, $X^C = (X^C_1, \\ldots, X^C_{r(C)})$, $Y^C = (Y^C_1, \\ldots, Y^C_{t(C)})$, $S^C = D_C \\cup \\bigcup_{B \\in B_C} B$, $Z^C = X^C+Y^C$, and $P_C = (Z^C_1, \\ldots, Z^C_{p(C)})$. In addition, let $P = (P_1, \\ldots, P_a)$ be the tuple returned by Algorithm~\\ref{alg:pack}. We prove the following.\n\n\\begin{enumerate}\n\t\\item $P$ is a partition of $I$: Let $\\ell \\in I$. Then, by Definition~\\ref{def:partition} there is exactly one $C \\in \\mathcal{H}$ such that $\\ell \\in S^C$; then, by Lemma~\\ref{clm:GREEDY} there is exactly one $i \\in [p(C)]$ such that $\\ell \\in Z^C_i$; therefore, by Step~\\ref{step:pPack} of Algorithm~\\ref{alg:pack} there is exactly one  $j \\in [a]$ such that $\\ell \\in P_j$ and the claim follows. \n\t\n\t\\item For all $i \\in [a]$ it holds that $s(P_i) \\leq 1$: By Step~\\ref{step:pPack} of Algorithm~\\ref{alg:pack} there is $C \\in \\mathcal{H}$ and $j \\in [p(C)]$ such that $P_j = Z^C_j$. Therefore, $s(P_i) = s(Z^C_j) \\leq 1$ by Lemma~\\ref{clm:GREEDY}. \n\t\n\t\\item For all $i \\in [a]$ and $G \\in {\\mathcal{G}}$ it holds that $|G \\cap P_i| \\leq k(G)$:  By Step~\\ref{step:pPack} of Algorithm~\\ref{alg:pack} there is $C \\in \\mathcal{H}$ and $j \\in [p(C)]$ such that $P_i = Z^C_j$. Therefore, $|P_i \\cap G| = |Z^C_j \\cap G| \\leq k(G)$ by Lemma~\\ref{clm:GREEDY}. \n\t\n\t\\item The size of $P$ that is $a$ is at most $(1+2{\\varepsilon}) \\cdot m+  2 \\cdot {\\varepsilon}^{-22} \\cdot Q^2({\\varepsilon})$:\n\t%\n\t$$a = \\sum_{C \\in \\mathcal{H}} p(C) \\leq \\sum_{C \\in \\mathcal{H}} \\left((1+2{\\varepsilon})\\cdot |B_C|+2\\right) \\leq (1+2{\\varepsilon}) \\cdot m+  2 \\cdot {\\varepsilon}^{-22} \\cdot Q^2({\\varepsilon})$$ The first equality is by Step~\\ref{step:pPack} of Algorithm~\\ref{alg:pack}. The first inequality is by Lemma~\\ref{clm:GREEDY}. The second inequality is because $\\{B_C\\}_{C \\in \\mathcal{H}}$ is a partition of $\\{A_i~|~i \\in [m]\\}$ and that $|\\mathcal{H}| \\leq  {\\varepsilon}^{-22} \\cdot Q^2({\\varepsilon})$ by Definition~\\ref{def:partition}. \n\t\n\t\n\t \\qed\n\\end{enumerate} \n\n\n\n  \n\\section{Reduction and Reconstruction of the Instance}\n\\label{sec:reduction}\n\nIn this section we present algorithms \\textsf{Reduce} and \\textsf{Reconstruct}, which handle the structuring of the instance and \nthe transformation of the solution for the structured instance into a solution for the original one. For this section, fix ${\\mathcal{J}} = (I, \\mathcal{G}, s,k)$ to be a BPP instance, and let ${\\varepsilon} \\in (0, 0.1]$.\n\nAlgorithm \\textsf{Reduce} generates from a given instance ${\\mathcal{J}}$ a structured instance ${\\mathcal I}$, for which any optimal packing required at most the minimum number of bins required for packing ${\\mathcal{J}}$\n(see Section~\\ref{sec:F1}). Given a packing of ${\\mathcal I}$, algorithm \\textsf{Reconstruct} finds a packing  of the original instance ${\\mathcal{J}}$, with only a slight increase in the total number of bins used (see Section~\\ref{sec:RECONSTRUCTION}). The proofs of the results in this section are given in Appendix~\\ref{sec:reductionProofs}. \n\n\\subsection{The Reduction}\n\\label{sec:F1}\n\nTowards structuring the instance, we first classify the items and groups in ${\\mathcal{J}}$.\nWe note that similar classifications were used in prior work (see, e.g., \\cite{DW17,Jansen_et_al:2019,DKS21}).  \n For all $i \\in  \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}$ let \n \\begin{equation}\n\t\\label{Interval}\n\tI_i = \\big\\{\\ell \\in I~|~ s(\\ell) \\in \\big[{\\varepsilon}^{i+1}, {\\varepsilon}^{i}\\big)\\big\\}\n\\end{equation} be the $i$-th {\\em interval} of ${\\mathcal{J}}$, containing all items of sizes in the interval $[{\\varepsilon}^{i+1}, {\\varepsilon}^{i})$; we refer to $i$ is a {\\em pivot} of ${\\mathcal{J}}$. Now, for $w \\in \\{2,\\ldots,{\\varepsilon}^{-1}+1\\}$, we say that $w$ is a {\\em minimal pivot} of $\\mathcal{{\\mathcal{J}}}$ if the $w$-th interval of ${\\mathcal{J}}$ contains minimal total size of items among all intervals of ${\\mathcal{J}}$; that is,\n\n\\begin{equation}\n\t\\label{eq:k}\n\tw = \\argmin_{i \\in \\{2, \\ldots, {\\varepsilon}^{-1}+1\\}} s(I_i).\n\\end{equation}\n\n\nThe classification of the items depends on the selection of a pivot for ${\\mathcal{J}}$. Specifically, given $w \\in \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}$ we classify item $\\ell$ as $w$-{\\em  heavy} if $s(\\ell) \\geq \\varepsilon^{w}$, $w$-{\\em medium} if $s(\\ell) \\in [\\varepsilon^{w+1},\\varepsilon^{w})$ and $w$-{\\em light} otherwise. Let $H_w = \\{\\ell \\in I~|~s(\\ell) \\geq {\\varepsilon}^{w}\\}$ be the set of $w$-heavy items in ${\\mathcal{J}}$. \n\nFor the classification of groups, fix a pivot $w \\in \\{2, \\ldots, {\\varepsilon}^{-1}+1\\}$ of ${\\mathcal{J}}$. We sort the groups in $\\mathcal{G}$ in a non-increasing order by the total number of $w$-heavy and $w$-medium items of the group; then, we classify the first up to $ {\\varepsilon}^{-3w-5}$ groups according to the above order as the $w$-{\\em large groups}, and the remaining groups are called $w$-{\\em small}. More formally, let $g_w(G) = |G \\cap (H_w \\cup I_w)|~:~\\forall G \\in {\\mathcal{G}}$ and $n = |{\\mathcal{G}}|$. Now, let $G_1, \\ldots, G_{n}$ be a \nnon-increasing order of ${\\mathcal{G}}$ by $g_w$, where\ngroups having the same $g_w$ values are placed in fixed arbitrary order.\nNow, let $\\kappa_w = \\min\\left\\{{\\varepsilon}^{-3w-5},|{\\mathcal{G}}| \\right\\}$ and define the set of $w$-large groups as ${\\mathcal{G}}_L(w) = \\left\\{G_i~|~i \\in  \\left[\\kappa_w \\right] \\right\\}$ and the set of $w$-small groups as ${\\mathcal{G}} \\setminus {\\mathcal{G}}_L$.\n\n\n\\begin{lemma}\n\t\\label{lem:few_large_groups}\t\n\tFor all $w \\in \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}$ there are at most ${\\varepsilon}^{-3w-5}$ groups $G \\in {\\mathcal{G}}$ such that $$|G \\cap (H_w\\cup I_w) | \\geq {\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}).$$ \n\\end{lemma}\n\nThe proof of Lemma~\\ref{lem:few_large_groups} follows by noting that\nthe total size of items in a group having at least ${\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}})$ items that are $w$-medium  or $w$-heavy is at least ${\\varepsilon}^{3w+5} \\cdot \\textnormal{OPT}({\\mathcal{J}})$. \n\nBy Lemma~\\ref{lem:few_large_groups}, the number of $w$-heavy and $w$-medium items in a $w$-small group is at most ${\\varepsilon}^{2w+4}\\cdot \\textnormal{OPT}(\\mathcal{J})$. This is useful for the reduction, in which the matroid constraint is slightly relaxed for $w$-small groups, as described below. The $w$-large groups appear in the reduced instance with the original cardinality constraint. In contrast, for each $w$-small group we keep only the $w$-light and $w$-medium items with the original cardinality constraint of the group, whereas all $w$-heavy items from $w$-small groups are placed in a single {\\em union group} with unbounded cardinality constraint. Specifically, define the {\\em reduced $w$-small groups} containing only $w$-light and $w$-medium items as\n\n\n\n\\begin{equation}\n\\label{Is}\n{\\mathcal{G}}_S(w) = \\{G \\setminus H_w~|~G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w)\\}.\n\\end{equation}\n\nAlso, let the {\\em $w$-union group} containing all $w$-heavy items from $w$-small groups be \n\n\\begin{equation}\n\\label{Iu}\n\\Gamma_w = \\{\\ell \\in H_w~|~\\ell \\in G \\text{ s.t. } G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w)\\}.\n\\end{equation}\n\n\n\\begin{algorithm}[h]\n\t\\caption{$\\textsf{Reduce}({\\varepsilon},{\\mathcal{J}} = (I,{\\mathcal{G}},s,k))$}\n\t\\label{alg:reduction}\n\t\n\tFind a minimal pivot $w \\in \\{2,\\ldots,{\\varepsilon}^{-1}+1\\}$ of ${\\mathcal{J}}$.\\label{step:pivot}\n\t\n\tFind ${\\mathcal{G}}_L(w), {\\mathcal{G}}_S(w), \\Gamma_w$, the $w$-large groups, $w$-reduced small groups, and $w$-union of ${\\mathcal{J}}$.  \n\t\n\tLet ${\\mathcal{G}}_R = {\\mathcal{G}}_L(w) \\cup {\\mathcal{G}}_S(w) \\cup \\{\\Gamma_w\\}$.\\label{step:groups}\n\t\n\tDefine  the following new cardinality bounds for all $G \\in {\\mathcal{G}}_R$:\\label{kr} \\[\n\tk_R(G) =  \\label{kr} \n\t\\begin{cases}\n\t\tk(G) & G \\in {\\mathcal{G}}_L(w)\\\\\n\t\tk(G') & G = G' \\setminus H_w \\text{ s.t. } G' \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w) \\\\\n\t\t|\\Gamma_w|+1 & G = \\Gamma_w.\\\\\n\t\\end{cases}\n\t\\]\n\t\n\t\n\t\n\tReturn the reduced instance $\\left(I,{\\mathcal{G}}_R,s,k_R\\right)$. \\label{Ir}\n\t\n\\end{algorithm}\n\n\n\nAlgorithm \\textsf{Reduce}  constructs the above groups, together with the new cardinality constraints, and preserves the initial set of items and item sizes. The pseudocode of Algorithm \\textsf{Reduce} is given in Algorithm~\\ref{alg:reduction}. The first claim of Lemma~\\ref{lem:reductionReconstruction} holds since the only of groups containing items larger than ${\\varepsilon}^2$ are the $w$-large groups and the $w$-union group. Moreover, a packing of the original instance is also a feasible packing for the reduced instance, thus the optimum of the reduced instance can only be smaller. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The Reconstruction}\n\\label{sec:RECONSTRUCTION}\n\nWe now describe how a packing of the reduced instance is transformed to a packing for the original instance.  \nFor simplicity, assume that objects such as $w$ computed by algorithm $\\textsf{Reduce}({\\varepsilon},{\\mathcal{J}})$ are known and fixed for the reconstruction of ${\\mathcal{J}}$. In addition, for the remainder of this section, fix a packing $A = (A_1, \\ldots, A_m)$ for ${\\mathcal{I}} = \\textsf{Reduce}({\\varepsilon},{\\mathcal{J}})$. \n\nRecall that Step~\\ref{kr} of algorithm \\textsf{Reduce} relaxes the cardinality constraint for the $w$-heavy items in the set $\\Gamma_w$. The reconstruction algorithm redefines $A$ for items in $\\Gamma_w$ to ensure that the matroid constraint is satisfied for these items; this may require using a few extra bins. Then, $w$-medium and $w$-light items from $w$-small groups are discarded from the packing and packed separately using Algorithm \\textsf{Greedy} (see Lemma~\\ref{thm:greedy}). This results in a feasible packing of the original instance ${\\mathcal{J}}$. \nWe now describe how our reconstruction algorithm resolves violations among $w$-heavy items. This is done by rearranging the packing of $w$-heavy items in $A$ using a variant of the classic {\\em linear shifting} technique of~\\cite{fernandez1981bin}. \nLet $\\beta = {\\varepsilon}^{-w-2}$ be the {\\em shifting parameter}, and $P_1,\\ldots, P_q$ the $\\mathcal{{\\mathcal{I}}}$-{\\em partition} obtained by applying linear shifting to the items in $\\Gamma_w$. More specifically, given the set $\\Gamma_w$ in non-decreasing order of item sizes, for all $1\\leq i \\leq j<q$ and $\\ell \\in P_i, y \\in P_j$ it holds that $s(\\ell) \\geq s(y)$, $|P_i| = \\ceil{\\frac{|\\Gamma_w|}{\\beta}}$, and $P_{q}$ contains the remaining (possibly less than  $\\ceil{\\frac{|\\Gamma_w|}{\\beta}}$) items from $\\Gamma_w$. It follows that $q \\leq \\beta$ and let $q({\\mathcal{I}}) = q$ be the number of sets in the above partition of $\\Gamma_w$. \n\n\\begin{lemma}\n\\label{lem:Bgreedy}\nThere is an algorithm \\textnormal{\\textsf{Fill}} which given a BPP instance ${\\mathcal{J}}$ and a  packing $A = (A_1, \\ldots, A_m)$ of ${\\mathcal{I}} = \\textnormal{\\textsf{Reduce}}({\\varepsilon},{\\mathcal{J}})$ finds in time $\\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$ a partition $(B_1, \\ldots, B_m,R)$ of $\\Gamma_w$ such that the following hold. \n\n\\begin{enumerate}\n\\item For all $i \\in [m]$ and $j \\in \\{2, \\ldots,q({\\mathcal{I}})\\}$ it holds that $|B_i \\cap P_j| \\leq |A_i \\cap P_{j-1}|$ and $|B_i \\cap P_1| = 0$.\n\n\\item For all $i \\in [m]$ and $G \\in {\\mathcal{G}}$ it holds that \n$|B_i \\cap G| \\leq k(G)$.\n\n\\item $|R| \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})+1$. \n\\end{enumerate}   \n\\end{lemma}\n\n\nThe first condition of the lemma is essentially a shifting argument. The lemma is proved by constructing the following greedy algorithm \\textsf{Fill}. We start with empty sets for $B_1, \\ldots, B_m$ and with $R = \\Gamma_w$. In each iteration, we try to move an item from $R$ to some $B_i, i \\in [m]$ without violating the conditions of Lemma~\\ref{lem:Bgreedy}.\nIf no move is possible, we return $(B_1, \\ldots, B_m,R)$. The details are given in Appendix~\\ref{sec:fill}. \n\n\nWe now combine the output of algorithm $\\textsf{Fill}$ with $A$. We first remove from $A$ the items in $\\Gamma_w$ and all $w$-medium items from $w$-small groups; then, for all $i \\in [m]$ we add to $A_i$ the items in $B_i$. The remaining items in $R$ are packed using extra bins. Formally, let \\begin{equation}\n\t\\label{X:X} \\Omega_w = \\{\\ell \\in I_w~|~\\ell \\in G \\text{ s.t. } G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w)\\}\n\\end{equation} be the $w$-medium items from $w$-small groups. In addition, given two tuples $T_1, T_2$, let $T_1 \\oplus T_2$ be the concatenation of $T_1$ by $T_2$. Now, given $(B_1, \\ldots, B_m,R) = \\textsf{Fill}(A)$, define the tuple \n\n\\begin{equation}\n\\label{F_A}\nF_A = \\big((A_i \\setminus (\\Gamma_w\\cup \\Omega_w)) \\cup B_i~|~ i \\in [m]\\big) \\oplus \\big(\\{\\ell\\}~|~\\ell \\in R\\big).\t\n\\end{equation}\n\n\n\nIn the following, we transform $F_A$ into a feasible packing of ${\\mathcal{J}}$. Let $F_A = (U_1, \\ldots, U_{m'})$. Observe that Lemma~\\ref{lem:Bgreedy} does not guarantee that $F_A$\nsatisfies the cardinality constraints for all the $w$-small groups.\nSuch violations are resolved by discarding $w$-light items from $U_i, i \\in [m']$ to ensure  that the cardinality constraints of all groups in ${\\mathcal{G}}$ are satisfied.  More specifically, for any bin $i \\in [m']$ and group $G \\in {\\mathcal{G}}$ let $F_A(i,G)$ be an arbitrary exclusion-minimal subset of $w$-light items from $U_i$ such that $| (U_i \\setminus F_A(i,G) )  \\cap G| \\leq k(G)$. Note that such a subset exists, since by taking all $w$-light items to $F_A(i,G)$, the above condition is satisfied by Lemma~\\ref{lem:Bgreedy} and \\eqref{F_A}. Now, the discarded items are accumulated across all bins and groups to form the set $D(A)$, namely  \n\n\n\\begin{equation}\n\\label{D}\nD(A) = \\bigcup_{i \\in [m']} \\bigcup_{G \\in {\\mathcal{G}}} F_A(i,G).\n\\end{equation}\n\n\n\n\\begin{lemma}\n\\label{thm:D}\n\nFor any packing $A$ of ${\\mathcal{I}}$,\n\n\\begin{enumerate}\n\\item  $|D(A) \\cap G| \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})$ for all $G \\in {\\mathcal{G}}$. \n\n\\item $s(D(A)) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})$.\n\n\\end{enumerate}\n\n\\end{lemma}\n\n\n\nThe first condition in the lemma follows from Lemma~\\ref{lem:few_large_groups}, since we do not discard items from $w$-large groups. The second condition holds since any $w$-heavy item is larger than any $w$-light item by factor at least ${\\varepsilon}^{-1}$.  These properties are useful for packing the discarded items in only a few extra bins using algorithm \\textsf{Greedy}. \n\n\n\nGiven the partial packing of ${\\mathcal{J}}$, algorithm \\textsf{Reconstruct} proceeds to pack the remaining items, i.e., the $w$-medium items from $w$-small groups and items in $D(A)$. All  of these items are packed in a few extra bins.\nThis is done by algorithm \\textsf{Greedy}, for which we define a residual instance containing only the above remaining items, which preserves the item sizes and group cardinality constraints for the remaining items. Formally, define the {\\em discarded instance} of ${\\mathcal{J}}$ and $A$ as $\\mathcal{E} = (E,{\\mathcal{G}}_E,s_E,k_E)$, where  \n\n\\begin{equation}\n\\label{Ie}\n\\begin{aligned}\nE = \\Omega_w \\cup D(A), ~~~~~~~~~~~~~~~~~~~~ ~~~~~~\n\\mathcal{G}_E = \\{G \\cap E~|~ G \\in {\\mathcal{G}}\\}, ~~~~~~~~~~~~~~~~~~ \n\\\\\ns_E(\\ell) = s(\\ell), ~~\\forall \\ell \\in E, ~~~~~~~~~~~~~~~~~~~\nk_E(G \\cap E) = k(G), ~~\\forall G \\in {\\mathcal{G}}~~~~~~~~~~~~\n\\end{aligned}\n\\end{equation}\n\nGiven $S \\subseteq I$ and a tuple $T =  (T_1, \\ldots, T_n)$ such that $T_i \\subseteq I$ for all $i \\in [n]$, let $T \\setminus S = (T_1 \\setminus S, \\ldots, T_n \\setminus S)$  be the tuple induced by removing the items in $S$ from all entries of $T$. The pseudocode of algorithm \\textsf{Reconstruct} is given in Algorithm~\\ref{Alg:RECON}. By \\eqref{F_A} and \\eqref{Ie}, the output of algorithm \\textsf{Reconstruct} is a packing of ${\\mathcal{J}}$. Furthermore, this packing is of size at most $m+13{\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})+1$ by Lemmas~\\ref{lem:Bgreedy}, \\ref{thm:D}, and \\ref{lem:few_large_groups}. \n\n\n\\begin{algorithm}[h]\n\\caption{$\\textsf{Reconstruct}(\\mathcal{J}, {\\varepsilon}, A)$}\n\\label{Alg:RECON}\n\n\n\nGenerate $F_A$ using $\\textsf{Fill}(\\mathcal{J}, A)$ and \\eqref{F_A}.\\label{step:1}\n\nCompute the discarded instance $\\mathcal{E} = (E,{\\mathcal{G}}_E,s_E,k_E)$ of ${\\mathcal{J}}$ and $A$.\\label{step:2} \n\nReturn $\\left( F_A \\setminus E \\right) \\oplus \\textsf{Greedy}(\\mathcal{E})$.\\label{step:reconl}\\label{step:3}\n\n\\end{algorithm}\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this paper we presented \nan $o(\\textnormal{OPT})$ additive approximation algorithm for Bin Packing with Partition Matroid. While BPP is a natural generalization of Bin Packing variants that have been\nstudied in the past, to the best of our knowledge it is studied here for the first time.\nOur result improves upon the APTAS of \\cite{DKS21} for the special case of Group Bin Packing and  generalizes the AFPTAS of \\cite{epstein2010afptas} for the special case of Bin Packing with Cardinality Constraints. Our algorithm is based on rounding a solution for the configuration-LP formulation of the problem. The rounding process relies on the key notion of a {\\em prototype}, in which items are placeholders for other items, and the use of fractional grouping~\\cite{FKS21}. Our algorithm demonstrates the power of this fractional version of linear grouping in solving constrained packing problems; it also \nshows how fractional grouping can be used {\\em constructively}.\n\nWhile our algorithm outputs a solution which uses $\\textnormal{OPT}+o(\\textnormal{OPT})$ bins,  the function hidden by the little-o notation is of the form $\\frac{x}{(\\ln \\ln x)^{\\frac{1}{17}}}$.  We believe that a tighter analysis may lead to a better additive approximation, \nfor example, to an algorithm which returns a solution using $\\textnormal{OPT}+O\\left(\\frac{\\textnormal{OPT}}{\\ln \\textnormal{OPT}} \\right)$ bins. We leave the tighter analysis for the full version of this paper. The existence of approximation algorithms for BPP which return a solution using at most $\\textnormal{OPT}+O(\\textnormal{OPT}^{1-{\\varepsilon}})$ bins, for some constant ${\\varepsilon}>0$, remains open.\n\nThe techniques presented in this paper seem to be useful also in other settings.  \nOur preliminary study suggests we can apply these techniques \nto obtain a polynomial time approximation scheme for {\\em Multiple Knapsack with Partition Matroid}, a generalization of the Multiple Knapsack problem (see, e.g., \\cite{CK05, Ja10}) in which the items assigned to each bin form an independent set of a partition matroid. Another  application comes from the design of approximation algorithms for \nMachine Scheduling with Partition Matroid, \na generalization of the classic Machine Scheduling problem in which the jobs assigned to a machine must be an independent set of a given partition matroid. We note that the problem is a generalization of Machine Scheduling with Bag-Constraints studied in~\\cite{DW17,Jansen_et_al:2019}.\n\nThe problem of Bin Packing with Partition Matroid is a special case of Bin Packing with Matroid, for which the input is a set of items $I$, a size function $s:I\\to \\mathbb{R}$ and a matroid~${\\mathcal{M}}$. The objective is to partition $I$ into a minimal number of bins $A_1,\\ldots, A_m$ such that $A_b$ is an independent set of the matroid ${\\mathcal{M}}$, and $s(A_b)\\leq 1$ for all $b \\in [m]$. This problem is a natural generalization of both  Bin Packing and Matroid Partitioning; yet, we were unable to find any published results.\nWe note that our approach for solving BPP  heavily relies on the structure of the partition matroid, and therefore cannot be easily extended to handle a general matroid.  \n\n\n \n\\section{Algorithm \\textsf{Evict}}\n\\label{sec:eviction}\n\nOur scheme uses algorithm \\textsf{Evict} for reducing the number of items in each configuration in the support of a prototype. Let $\\bar{x}$ be a solution for~\\eqref{C-LP}. \n Given a configuration $C \\in \\mathcal{C}$, algorithm \\textsf{Evict} replaces $C$ by a set of slots consisting of items in $C$.\n Let $\\ell_1, \\ldots, \\ell_{r}$ be the items in $C \\setminus L$ \n sorted in non-increasing order by sizes. Denote by $h \\in [r]$ the minimum index of an item such that any item of larger index $\\ell \\in \\{\\ell_{h+1}, \\ldots, \\ell_{r}\\}$ \nhas size at most ${\\varepsilon}$ of the residual capacity from packing \n$C \\cap L \\cup \\{\\ell_1, \\ldots, \\ell_{h}\\}$.\\footnote{If $h=0$ then $\\{\\ell_1, \\ldots, \\ell_{h}\\} =\\emptyset$.}\n In case $h$ is larger than $\\alpha= {\\varepsilon}^{-5}$ set\n $h = \\alpha$. Formally, \n \\begin{equation}\n\t\\label{h}\nh= \\min \\bigg\\{ \\alpha, \\min \\big\\{0 \\leq h' \\leq r~|~ \\forall j \\in \\{h'+1, \\ldots,r\\} ~: \\ell_j \\in \n\\textsf{fit} (C \\cap L \\cup \\{\\ell_1, \\ldots, \\ell_{h'}\\} ) \\big\\}\\bigg\\}\n\\end{equation}\n\n\n Now, define $U_C = \\{\\ell_1, \\ldots, \\ell_h\\}$ as the set of the first $h$ items in the above order. \n Let $R(C) = C \\cap L \\cup U_C$ be the items in $C$ which are considered as slots by algorithm \\textsf{Evict}. Also, let $\\mathcal{L}_C = \\{\\ell \\in U_C~|~s(\\ell) \\geq \\frac{1-s(R(C))}{{\\varepsilon}}\\}$ be all items in $U_C$ of relatively large size. \n\nConsider two items $\\ell_i, \\ell_k \\in U_C$ such that $|U_C| > k > i+{\\varepsilon}^{-4}$. By \\eqref{h}, the distance between the items in the sorted order implies a considerable difference in their sizes; namely, $s(\\ell_i) > \\frac{s(\\ell_k)}{{\\varepsilon}^2}$.\nFurthermore, the smaller item cannot be too small: $s(\\ell_k) > {\\varepsilon}\\left(1-s(R(C))\\right)$, since otherwise $\\ell_k \\notin U_C$. It follows that $\\ell_i \\in \\mathcal{L}_C$, which suggests that $\\mathcal{L}_C$ is not much smaller than $U_C$.  \n\n\\begin{lemma}\n\t\\label{lem:evictionHelp}\nFor any configuration $C \\in \\mathcal{C}$ such that $|U_C| = \\alpha$, it holds that $|\\mathcal{L}_C| \\geq {\\varepsilon}^{-4}$.\n\\end{lemma}\n\nAlgorithm \\textsf{Evict} constructs from $\\bar{x}$ a prototype $\\bar{y}$ as follows. For any $C \\in \\textsf{supp}(\\bar{x})$, the value $\\bar{x}_C$ is split among at most $|\\mathcal{L}_C|$ new configurations, each generated by removing from $C$ a subset of items. Initially, all items in $C \\setminus R(C)$ are removed. Then, an item from $\\mathcal{L}_C$ may be removed, depending on whether $|U_C| = \\alpha$ or not.\n\n\nIf $|U_C| = \\alpha$, then the next largest item not in $U_C$ (that is, $\\ell_{h+1}$) does not necessarily belong to $\\textsf{fit}(R(C))$. However, because the size of items in $\\mathcal{L}_C$ is relatively large, removing an item $\\ell \\in \\mathcal{L}_C$ from $C$ guarantees that $C \\setminus R(C) \\subseteq \\textsf{fit}(R(C) \\setminus \\{\\ell\\})$. This is useful for proving that the prototype $\\bar{y}$ returned by algorithm \\textsf{Evict} has non-empty $\\bar{y}$-polytope. Otherwise, we have $|U_C| < \\alpha$. By \\eqref{h}, for any item  $\\ell \\in C \\setminus R(C)$ it holds that $\\ell \\in \\textsf{fit}(R(C))$. Hence, in this case, no need to remove an item from $\\mathcal{L}_C$. \nIn the above procedure we use a vector per configuration. Given a configuration $C\\in \\mathcal{C}$, define the {\\em relaxation of $C$} as the following vector $\\bar{w}^C \\in \\mathbb{R}_{\\geq 0}^{\\mathcal{C}}$.\n\n\\begin{enumerate}\n\t\\item  \t\\label{eq:y=a} If $|U_C| = \\alpha$, then for any $\\ell \\in \\mathcal{L}_C$ define $\\bar{w}^C_{R(C) \\setminus \\{\\ell\\}} = \\frac{1}{|\\mathcal{L}_C|-1}$. For any other configuration $C' \\in \\mathcal{C}$ such that there is no $ \\ell \\in \\mathcal{L}_{C'}$ satisfying $C=R(C') \\setminus \\{\\ell\\}$, set $\\bar{w}^C_{C'} = 0$. \n\n\\item \t\\label{eq:y<a} If $|U_C| < \\alpha$, then set $\\bar{w}^C_{R(C)} = 1$. For all $C' \\in \\mathcal{C} \\setminus \\{R(C)\\}$ set $\\bar{w}^C_{C'} = 0$.\n\\end{enumerate}\n\n\n\n\n\n The relaxation vector of a configuration \n satisfies several properties that are useful for our algorithm. In particular, \n the total size of $\\bar{w}^C$ is only slightly larger than $1$, all items in $R(C)$ are\n covered, and uncovered items fit for placement with the slots. Finally, small items are added only to configurations that are almost full. This is formalized in the next lemma. \n \\begin{lemma}\n\t\\label{lem:wProperties}\n\tFor any $C \\in \\mathcal{C}$, the following hold for $\\bar{w}^C$, the relaxation of $C$. \n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item $\\|\\bar{w}^C\\| \\leq 1+2{\\varepsilon}^4$.\n\t\t\n\t\t\\item For all $\\ell \\in R(C)$,\n\t\t it holds that \n\t\t$\\sum_{C' \\in \\mathcal{C}[\\ell]} \\bar{w}^C_{C'} \\geq 1$. \n\t\t\n\t\t\\item For all $C ' \\in \\textnormal{\\textsf{supp}}(\\bar{w}^C)$,\n\t\t it holds that \n\t\t$C \\setminus R(C) \\subseteq\\textnormal{\\textsf{fit}}(C')$ and $s(C \\setminus R(C)) \\leq 1-s(C')$. \n\t\t\n\t\t\\item  For all $C ' \\in \\textnormal{\\textsf{supp}}(\\bar{w}^C)$ it holds that $s(C' \\setminus L) \\leq {\\varepsilon}$. \n\t\t\n\t\t\\item  For all $C ' \\in \\textnormal{\\textsf{supp}}(\\bar{w}^C)$ and $G \\in {\\mathcal{G}}$ it holds that $\\left|G \\cap \\left(C \\setminus R(C)\\right)\\right| \\leq k(G) -|G \\cap C'|$. \n\t\\end{enumerate} \n\\end{lemma}\n\n\n\\begin{algorithm}[htb]\n\t\\caption{$\\textsf{Evict} (\\bar{x}$)}\n\t\\label{Alg:eviction}\n\t\n\t\n\tFor every $C\\in \\textsf{supp}(\\bar{x})$ compute $\\bar{w}^C$, the relaxation of $C$.\\label{step:forsupp}\\label{step:y}\n\t\n\t\n\treturn $\\bar{y} = \\sum_{C \\in \\textsf{supp}(\\bar{x})} \\bar{x}_C \\cdot \\bar{w}^C$.\\label{step:return}\n\t\\end{algorithm}\n\n\nAlgorithm \\textsf{Evict} computes a linear combination of the relaxations of all configurations $C \\in \\mathcal{C}$, where the coefficient of $\\bar{w}^C$ is $\\bar{x}_C$. The pseudocode of \\textsf{Evict} is given in Algorithm~\\ref{Alg:eviction}, and the proof of Lemma~\\ref{lem:eviction} is given in Appendix~\\ref{app:omittedEvict}. \n\n\n\n\n\n\n\n\n\n\\section{Algorithm \\textsf{Greedy}}\n\\label{sec:greedy}\n\nIn this section we give the proof of Lemma~\\ref{thm:greedy}.  Algorithm \\textsf{Greedy} is indeed a greedy algorithm, which sequentially adds a new bin and tries to (i) maximize the total size packed in the bin while (ii) packing items from groups which have more items w.r.t. the matroid constraint. Some of the proofs from this section are deferred to Section~\\ref{sec:proofsGreedy}.  \n\n For this section, fix a BPP instance ${\\mathcal{I}} = (I, {\\mathcal{G}},s,k)$ and let $\\delta \\in (0,0.5)$ such that for all $\\ell \\in I$ it holds that $s(\\ell) \\leq \\delta$.  We use ${\\mathcal{C}}({\\mathcal{J}})$ to denote the set of configurations of a BPP instance ${\\mathcal{J}}$. Also, recall that the operator $\\oplus$ denotes tuple (or, packing) concatenation. In addition, for any $S \\subseteq I$ we define ${\\mathcal{I}} \\setminus S = (I_S,{\\mathcal{G}}_S,s_S,k_S)$ as the BPP instance such that \\begin{equation}\n\t\t\\label{IIS}\n\t\\begin{aligned}\n\t\tI_S = I \\setminus S, ~~~~~~~~~~~~~~~~~~~~ ~~~~~~\n\t\t\\mathcal{G}_S = \\{G \\cap I_S\\neq \\emptyset~|~ G \\in {\\mathcal{G}}\\}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\t\t\\\\\n\t\ts_S(\\ell) = s(\\ell), ~~\\forall \\ell \\in I_S, ~~~~~~~~~~~\n\t\tk_S(G \\cap I_S) = k(G), ~~\\forall G \\in {\\mathcal{G}} \\text{ s.t. } G \\cap I_S\\neq \\emptyset.~~~~~~~~~~~\n\t\\end{aligned}\n\\end{equation}\n\\begin{claim}\n\t\\label{clm:confappend}\n\tFor $S \\in {\\mathcal{C}}({\\mathcal{I}})$ and a packing $X$ of ${\\mathcal{I}} \\setminus S$ it holds that $T = (S) \\oplus X$ is a packing of ${\\mathcal{I}}$.\n\\end{claim}\n\nUsing Claim~\\ref{clm:confappend}, the algorithm first finds a configuration  $S$ of ${\\mathcal{I}}$ and then packs recursively the instance ${\\mathcal{I}} \\setminus A$. Recall that $V(\\mathcal{I}) = \\max_{G \\in {\\mathcal{G}}} \\ceil{\\frac{|G|}{|k(G)|}}$ and let the {\\em promise} of ${\\mathcal{I}}$ be a maximum bound over $\\textnormal{OPT}({\\mathcal{I}})$ combining a bound over item sizes and over the matroid: \\begin{eqnarray}\n\t\\label{promise}\n\tp({\\mathcal{I}}) = \\max \\left\\{    (1+2\\delta) \\cdot s(I)+2, V({\\mathcal{I}})    \\right\\}. \n\\end{eqnarray} We show that the proposed algorithm \\textsf{Greedy} constructs a packing of ${\\mathcal{I}}$ in at most $p({\\mathcal{I}})$ bins. For doing so, in each bin we must pack enough items from groups $G \\in {\\mathcal{G}}$ which satisfy that $\\ceil{  \\frac{|G|}{k(G)} } > p({\\mathcal{I}})-1$; otherwise, the packing might require more bins than $p({\\mathcal{I}})$. Thus, we define the set of {\\em bounding groups} as \n\n\n\n\n\n \\begin{eqnarray}\n\t\\label{bounding}\n\t{\\mathcal{B}} ({\\mathcal{I}}) =\\left\\{    G \\in {\\mathcal{G}} ~\\bigg|~ \\ceil{ \\frac{|G|}{k(G)} } > p({\\mathcal{I}})-1   \\right\\}\n\\end{eqnarray} \n\nTo guarantee that after packing the first bin by the algorithm we remain with not too many items from some group, we must take certain number of items from each bounding group. Specifically, for each bounding group $G \\in {\\mathcal{B}}({\\mathcal{I}})$ we define a {\\em bounding subset} of $G$ as a subset of items $S \\subseteq G$ such that $ \\ceil{ \\frac{|G \\setminus S|}{k(G)} } \\leq p({\\mathcal{I}})-1  $. Also, let $b(G)$ be the set of all bounding subsets of $G$. Finally, define a {\\em minimal bounding subset} of $G$ as a bounding subset $S^* \\in b(G)$ of $G$ such that (i) $|S^*| \\leq k(G)$ and (ii) $s(S^*) \\leq \\frac{s(G)}{s(I)}$. For any $G \\in {\\mathcal{B}}({\\mathcal{I}})$ let $b^*(G)$ be the set of minimal bounding subsets of $G$.  \\begin{lemma}\n\t\\label{bG}\n\tFor any $G \\in {\\mathcal{B}}({\\mathcal{I}})$ it holds that $b^*(G) \\neq \\emptyset$.\n\\end{lemma}\n\n\n\n\n\nWe generalize the definition of minimal bounding subset for the entire instance. Specifically, define the set of  {\\em bounding subsets} of ${\\mathcal{I}}$ as all unions of minimal bounding subsets of all bounding groups: \\begin{equation}\n\t\\label{bounding:A}\n\tb^*({\\mathcal{I}}) = \\left\\{    S \\subseteq \\bigcup_{G \\in {\\mathcal{B}}({\\mathcal{I}})} G~\\bigg|~ \\forall G \\in {\\mathcal{B}}({\\mathcal{I}})~:~ S \\cap G \\in b^*(G) \\right\\}\n\\end{equation} \n\n\n\n\n\n\n\\begin{lemma}\n\t\\label{bI}\n\t $b^*({\\mathcal{I}}) \\neq \\emptyset$.\n\\end{lemma}\n\n\n\\begin{proof}\n\t\n\tFor all $G \\in {\\mathcal{B}}({\\mathcal{I}})$ let $S_G \\in b^*(G)$; there is such $S_G \\in b^*(G)$ by Lemma~\\ref{bG}. Now, define $S = \\bigcup_{G \\in {\\mathcal{B}}({\\mathcal{I}})} S_G$; it follows by \\eqref{bounding:A} that $S \\in b^*({\\mathcal{I}})$. \n\\end{proof}\n\n\n\n\n\n\\begin{algorithm}[h]\n\t\\caption{$\\textsf{Greedy}(\\mathcal{I} = (I,{\\mathcal{G}},s,k), \\delta)$}\n\t\\label{alg:GREEDY}\n\t\n\t\n\t\\If{$I \\in {\\mathcal{C}}({\\mathcal{I}})$\\label{gr:if1}}{\n\t\t\n\t\treturn $(I)$.\\label{gr:if}\n\t\t\n\t}\n\t\n\t\n\tChoose arbitrary $A \\in b^*({\\mathcal{I}})$ bounding subset of ${\\mathcal{I}}$.\\label{gr:end0}\n\t\n\t\n\t\t\\While{$s(A) \\leq 1-\\delta ~\\textnormal{and}~ \\exists \\ell \\in I \\setminus A \\textnormal{ s.t. } |\\textnormal{\\textsf{group}}(\\ell) \\cap A| < k\\left( \\textnormal{\\textsf{group}}(\\ell) \\right)$ \\label{gr:while2}}{\n\t\t\n\t\t\n\t\t$A \\leftarrow A \\cup \\{\\ell\\}$.\\label{gr:swap2}\n\t\t\n\t}\\label{gr:end1}\n\n\t\n\t\\While{$\\exists \\ell \\in A, y \\in I \\setminus A \\textnormal{ s.t. } s(y) > s(\\ell), \\textnormal{\\textsf{group}}(\\ell)  = \\textnormal{\\textsf{group}}(y), ~\\textnormal{and}~ s(A) \\leq 1-\\delta $ \\label{gr:while1}}{\n\t\t\n\t\t$A \\leftarrow \\left(A \\setminus \\{\\ell\\} \\right) \\cup \\{y\\}$.\\label{gr:swap}\n\t\t\n\t}\n\t\n\n\t\n\tReturn $(A) \\oplus \\textsf{Greedy}\\left(\\mathcal{I} \\setminus A, \\delta \\right)$.\\label{gr:return}\n\t\n\\end{algorithm}\n\n\n\n\n\nGiven the definitions of bounding groups and bounding subsets, we describe Algorithm \\textsf{Greedy}. The algorithm is recursive. The base case is when the set of items $I$ forms a configuration of ${\\mathcal{I}}$; in this case, we simply return a packing of one bin containing all items. Otherwise, we construct a bounding subset $A$ of ${\\mathcal{I}}$, which exists by Lemma~\\ref{bI}. Then, we try to increase the total size of items in $A$, without exceeding the capacity of the bin, in two ways. First, by adding items to $A$ from groups that do not meet the cardinality constraint. Second, by replacing items from the same group between $A$ and $I \\setminus A$ such that the item moved into $A$ is of larger size. Finally, we define the resulting bin $A$ as the first bin in the packing, and pack recursively the instance ${\\mathcal{I}} \\setminus A$ using Algorithm \\textsf{Greedy}. We give the pseudocode of Algorithm \\textsf{Greedy} in Algorithm~\\ref{alg:GREEDY}. \n\nThe proof of Lemma~\\ref{thm:greedy} is based on viewing two complementary cases. First, if $A$, the first bin in the packing, has {\\em small} total size of items, namely $s(A) < 1-\\delta$, then in this case $A$ is a {\\em basis} of maximum total size of the matroid over the items. This also holds for the following bins of the packing, or we would be able to increase the size of $A$. Thus, an inductive argument shows that the algorithm finds a packing of ${\\mathcal{I}}$ in at most $V({\\mathcal{I}})$ bins.\n\nThe complementary case is where $s(A) \\geq 1-\\delta$; then, the total size deducted from $s(I)$ by $A$ is significant, and there can be at most $(1+2\\delta) \\cdot s(I)+1$ bins satisfying this property. In addition, it also holds that $V({\\mathcal{I}} \\setminus A) \\leq p({\\mathcal{I}})-1$ because we take a bounding subset for $A$ with additional items; using the first two cases, it can be inductively deduced that the returned packing in case that $s(A) \\geq 1-\\delta$ is of at most $p({\\mathcal{I}})$ bins.   \n\n\n\n\n\\section{Introduction}\\label{Introduction}\n\\label{sec:intro}\nThe {\\em bin packing (BP)} problem involves \npacking a set of items in a minimum number of containers (bins) of the same (unit) size. Bin Packing is one of the most studied problems in combinatorial optimization. Indeed, \n in many real-life scenarios, a solution for BP is essential for optimizing the allocation of resources. In this paper, we consider the Bin Packing problem with a partition matroid constraint.\n The input is a set of items of sizes in $(0,1]$, and a partition matroid over the \n items. The goal is to pack all items in a minimum number of unit-size bins, such that each bin forms an independent set in the matroid. \n\n \n Formally, a {\\em bin packing with partition matroid (BPP)} instance is a tuple $\\mathcal{I} = (I,\\mathcal{G},s,k)$, where~$I$ is a set of items, $\\mathcal{G}$ is a partition of $I$ into groups,\n  $s:I \\rightarrow (0,1]$ gives the item sizes, and $k:{\\mathcal{G}} \\rightarrow \\mathbb{N}_{>0}$ is a cardinality constraint for the groups.  The {\\em instance matroid} of ${\\mathcal{I}}$ is the partition matroid ${\\mathcal{M}}=(I, {\\mathcal{S}})$ where ${\\mathcal{S}}=\\{S\\subseteq I\t~|~\\forall G\\in {\\mathcal{G}}:~|S\\cap G|\\leq k(G)\\}$. \n  A {\\em configuration} of the instance ${\\mathcal{I}}$ is \na subset of items $C \\subseteq I$ such that $\\sum_{\\ell \\in C} s(\\ell) \\leq 1$ and $C\\in {\\mathcal{S}}$. That is,  the total size of items in $C$ is at most one and for each {\\em group} $G \\in \\mathcal{G}$ the configuration~$C$ contains at most $k(G)$ items from $G$. \n A {\\em packing} of $\\mathcal{I}$ is a partition  of $I$ into $m$ subsets called {\\em bins} $(A_1, \\ldots, A_m)$ such that $A_b$ is a configuration\n for all $b \\in [m]$.\\footnote{For any $n \\in \\mathbb{N}$ we denote by $[n]$ the set $\\{1,2,\\ldots,n\\}$.}  \n     The objective is to find a packing of all items in a minimal number of bins. Indeed, the special case where each group consists of a {\\em single} item is the classic Bin Packing problem.\n \n Bin Packing with Partition Matroid has natural applications in resource allocation on the Cloud to ensure fault tolerance~\\cite{EAFR} and security~\\cite{guerine2019provenance},\nas well as in harvesting computing capacity in distributed systems~\\cite{boinc04,JOUR}.\nFor a simple example, consider a set $I$ of unit time jobs accessing shared memory of size $Q$. Each job $\\ell \\in I$ has some memory requirement, as well as the type $R \\in {\\cal R}$ of processor on which it can execute, where ${\\cal R}$ is the set of distinct\nprocessor types.\nAssume the number of available processors of type $R$ is $k(R)$, for all\n$R \\in {\\cal R}$. We seek a feasible schedule \nof $I$ that minimizes the maximum completion time of any job (or, the {\\em makespan}). Then each time unit can be viewed as a single bin of capacity $Q$, and  each job $\\ell$ as an item whose size is equal to the memory requirement of job $\\ell$.\nThe goal is to use as few bins (= time units) as possible, while ensuring that in each bin at most $k(R)$ jobs execute, for all $R \\in {\\cal R}$.\nBy setting the bin capacities to $1$ and scaling down the item sizes accordingly, we have an instance of BPP.\n \n\n Let $\\textnormal{OPT}=\\textnormal{OPT}({\\mathcal I})$ be the value of an optimal solution for an instance~ ${\\mathcal I}$ of a minimization problem~$\\mathcal{P}$. \n As in the Bin Packing problem, we distinguish between {\\em absolute} and {\\em asymptotic} approximation.\n For $\\alpha \\geq 1$, we say that ${\\mathcal{A}}$ is an absolute $\\alpha$-approximation algorithm for \n $\\mathcal{P}$ if for any instance ${\\mathcal I}$ of~$\\mathcal{P}$ it holds that  $ {\\mathcal{A}} ({\\mathcal I})\/\\textnormal{OPT}({\\mathcal I}) \\leq \\alpha$, where ${\\mathcal{A}}({\\mathcal I})$ is the value of the solution returned by ${\\mathcal{A}}$. Algorithm\n ${\\mathcal{A}}$  is an {\\em asymptotic} $\\alpha$-approximation algorithm for \n $\\mathcal{P}$ if  for any instance ${\\mathcal I}$ it holds that ${\\mathcal{A}} ({\\mathcal I}) \\leq \\alpha \\textnormal{OPT}({\\mathcal I}) +o(\\textnormal{OPT}({\\mathcal I}))$.\nAn  {\\em asymptotic polynomial-time\n\tapproximation scheme (APTAS)} is a family of algorithms $({\\mathcal{A}}_{{\\varepsilon}})_{{\\varepsilon}>0}$ such that, for every ${\\varepsilon}>0$, ${\\mathcal{A}}_{{\\varepsilon}}$ is a polynomial time asymptotic $(1+{\\varepsilon})$-approximation algorithm for $\\mathcal{P}$.  An {\\em asymptotic {\\bf fully} polynomial-time\n\tapproximation scheme (AFPTAS)} is an APTAS $({\\mathcal{A}}_{{\\varepsilon}})_{{\\varepsilon}>0}$  such that ${\\mathcal{A}}_{{\\varepsilon}} ({\\mathcal I})$ runs in time $\\textnormal{poly}(|{\\mathcal I}|, \\frac{1}{{\\varepsilon}})$, where $|{\\mathcal I}|$ is the encoding length of the instance ${\\mathcal I}$; that is, there is a bivariate polynomial  $p$ such that ${\\mathcal{A}}_{{\\varepsilon}}({\\mathcal I})$ runs in time $p(|{\\mathcal I}|,\\frac{1}{{\\varepsilon}})$ or less.\n\n As Bin Packing is known to admit an $o(OPT)$ \n {\\em additive} approximation~\\cite{karmarkar1982efficient,hoberg2017logarithmic}, this \n gives rise to the following question:\n Is Bin Packing with Partition Matroid harder to solve than classic BP?\n While key techniques for solving BP break down in the presence of a partition matroid constraint, we show that by applying different tools we can obtain  a result similar to \n the best known result for BP, up to the function hidden\n by the additive approximation.\n Our main result is the following.\n \n \\begin{theorem}\n \t\\label{thm:main}\n There is a polynomial-time algorithm that, given a BPP instance $\\mathcal{I}$, returns a packing of $\\mathcal{I}$ in $\\textnormal{OPT}({\\mathcal{I}}) +O\\left( \\frac{\\textnormal{OPT}({\\mathcal{I}})}{\\left(\\ln \\ln \\textnormal{OPT}({\\mathcal{I}}) \\right)^{1\/17}} \\right)=\\textnormal{OPT}(\\mathcal{I})+o\\left( \\textnormal{OPT}(\\mathcal{I})\\right)$ bins.\n \\end{theorem}\n\nWe obtain the result by initially deriving an AFPTAS for BPP. Then Theorem~\\ref{thm:main} follows by using the AFTPAS with~${\\varepsilon}$ depending on the input instance. \nAs a special case, \nTheorem~\\ref{thm:main} improves upon the recent APTAS of Doron-Arad et al.~\\cite{DKS21} for {\\em group bin packing (GBP)}, where the cardinality constraint of all groups is equal to one (i.e., $k(G)=1~\\forall G \\in {\\mathcal{G}}$). Theorem~\\ref{thm:main} also generalizes the AFTPAS of Epstein and Levin~\\cite{epstein2010afptas} for {\\em bin packing with cardinality constraints (BPCC)}, the special case of BPP where all items belong to a single group (i.e., ${\\mathcal{G}}=\\{I\\}$).\n\nWe remark that we did not attempt to optimize the function hidden by $o(\\textnormal{OPT}({\\mathcal{I}}))$ in Theorem~\\ref{thm:main}; namely, our techniques may be used to derive\nbetter additive factor. We leave such efforts for the journal version of this paper.\n\n\\subsection{Techniques}\n\\label{sec:technique}\n\nThe classic asymptotic approximation schemes for Bin Packing (see, e.g., \\cite{Vega1981BinPC, karmarkar1982efficient}) rely on the nice property that {\\em small} items can be added to \na partial \npacking of the instance with little overhead, using simple algorithms such as First-Fit (see, e.g., \\cite{Williamson_Shmoys}). For example, assume we are given a set of items $I$, a size function\t $s:I\\rightarrow [0,1]$ and a partition $A_1,\\ldots, A_m $ and $S$ of $I$. Furthermore, assume that  $s(A_b) =\\sum_{\\ell \\in A_b} s(\\ell) \\leq 1$ for any $b\\in [m]$,\nand $s(\\ell)\\leq \\delta$ for any $\\ell \\in S$, for some $0<\\delta <1$.\\footnote{For any function $f:I\\rightarrow \\mathbb{R}$ and $A\\subseteq I$ we define $f(A)= \\sum_{\\ell \\in A} f(\\ell)$.}  Then, using First-Fit one can easily find a partition $D_1,\\ldots, D_{m'}$ of $I$ such that $s(D_b)\\leq 1 $ for all $b\\in [m']$ and $m'\\leq \\max\\{ m , (1+2\\delta ) s(I)+1\\}$. \n Consequently, the main focus of \n these schemes is on packing efficiently items of sizes larger than $\\delta$, where $\\delta>0$  is some small parameter value used by the algorithm. \n\nWe note that packing small items in the presence of a partition matroid constraint is more involved. As we show below, an efficient packing of the small items can still be obtained using a relatively simple algorithm; however, the setting in which the algorithm can be applied is more restricted, and items cannot be easily added to a partial packing of the instance (i.e., a set of configurations). Furthermore, the quality of such a packing depends on the  {\\em cardinality bound} of the BPP instance   $\\mathcal{{\\mathcal{I}}} = (I,{\\mathcal{G}},s,k)$, defined by $V(\\mathcal{I}) = \\max_{G \\in {\\mathcal{G}}} \\ceil{\\frac{|G|}{|k(G)|}}$. Formally,\n\\begin{lemma}\n\\label{thm:greedy}\nGiven a BPP instance ${\\mathcal{I}} = (I, {\\mathcal{G}},s,k)$ and $\\delta\\in (0,0.5)$, such that $s(\\ell) \\leq \\delta$ for all $\\ell \\in I$, there is an algorithm  $\\textnormal{\\textsf{Greedy}}$ that\treturns in polynomial time a packing of ${\\mathcal{I}}$ in at most $(1+2\\delta) \\cdot \\max\\left\\{ s(I),~V(\\mathcal{I})\\right\\}+2$ bins.\n\\end{lemma} \n\nTo obtain a packing of the input instance ${\\mathcal I}$ of BPP, our algorithm partitions the set of items $I$ into subsets $I_1,\\ldots, I_r$; for each subset $I_j$  it generates a  partial packing $A'_1,\\ldots, A'_m$ (i.e., $A'_1,\\ldots ,A'_m$ are configurations, and $ \\bigcup_{b\\in [m]} A'_b\\subseteq I_j$ ) which can be extended to a packing  of $I_j$ (i.e, one which contains all the items in $I_j$) using $\\textnormal{\\textsf{Greedy}}$. For this procedure to work, a crucial property is\n that $s(\\ell) \\ll 1- s(A'_b)$ for every $\\ell \\in I_j\\setminus \\bigcup_{b' \\in [m]} A'_{b'}$ and $b\\in [m]$. \nFurthermore, for any $G\\in {\\mathcal{G}}$, $|A'_b\\cap G|$ has to be upper bounded by a uniform value\nfor all $b\\in [m]$,\n and $|G\\cap (I_j\\setminus \\bigcup_{b\\in [m]} A'_b)|$ must be small.\nThus, a main part of the algorithm deals with  \ngeneration of the above partition of $I$ into $I_1,\\ldots,I_r$ and the corresponding partial packings.\\footnote{See the details in Section~\\ref{sec:overview}.}\n\n\nLet ${\\mathcal{C}}$ denote the set of configurations of a BPP instance ${\\mathcal{I}}=(I,{\\mathcal{G}}, s,k)$. \nThe novelty of our algorithm lies in an interpretation of vectors ${\\bar{x}}\\in \\mathbb{R}_{\\geq 0}^{{\\mathcal{C}}}$ (that is, vectors which have a non-negative entry for each configuration $C\\in {\\mathcal{C}}$)  as  {\\em prototypes}.\nThe prototype ${\\bar{x}}\\in \\mathbb{R}_{\\geq 0}^{{\\mathcal{C}}}$ serves as a blueprint for a (fractional) packing. Within the context of prototypes, each item $\\ell \\in C$ of a configuration~$C$ is interpreted as a placeholder (or, ``slot'') for items which can replace it. Also, some items may be added, thus utilizing the available capacity of the  configuration~$C$ (given by $1-s(C)$). The value ${\\bar{x}}_C$ represents a solution in which ${\\bar{x}}_C$ configurations match the blueprint corresponding to\n$C$. We associate a polytope with each prototype; a non-empty polytope ensures the prototype can be used to obtain a solution for the given BPP instance.\n\nThe partition of $I$ into $I_1,\\ldots, I_r$ and $S$, and the generation of the partial packings ($A'_1,\\ldots, A'_m$) follow from integrality properties of the polytope (associated with a prototype). However, for the integrality properties to be useful,  both the number of configurations on the support of the prototype (i.e, $\\{C\\in {\\mathcal{C}}~|~{\\bar{x}}_C>0\\}$) and  the number of items in each configuration on the support must be small.\n\nWe obtain an initial prototype by solving a standard configuration-LP formulation of the problem. In this initial prototype, each item serves as a placeholder for itself.  The algorithm  modifies the prototype sequentially using two steps: {\\em eviction} and {\\em shifting}. The eviction step reduces the number of items per configuration on the support of the prototype. The shifting step reduces the overall number of distinct items used by configurations on the support. Consequently, our algorithm can be viewed as a multi-step rounding process for the solution of the configuration-LP. \n\nThe shifting step applies the {\\em fractional grouping} technique\nintroduced in~\\cite{FKS21}. \nThe technique, originally developed for solving a constrained submodular maximization problem, can be viewed as a {\\em fractional} version of\n{\\em linear shifting}~\\cite{fernandez1981bin}. Given the items sorted in non-increasing order by sizes, suppose that each item $\\ell \\in I$ is associated with a weight $w_\\ell \\in (0,1]$. For a subset of items $I' \\subseteq I$ and an integer $\\tau \\geq 1$, we define a partition of $I'$ into $\\tau$ classes $\\Delta_1, \\ldots , \\Delta_\\tau$ of roughly the same (fractional) weight, where the weight of class $i$ is given by \n$w(\\Delta_i) = \\sum_{\\ell \\in \\Delta_i} w_\\ell$.\\footnote{Our scheme applies fractional grouping separately to certain groups $G \\in \\mathcal{G}$\n\t(see the details in Section~\\ref{sec:AlgPrototype}).}\nIn our setting, the weights $w_{\\ell}$ are determined by the prototype (after the eviction). \nFollowing the fractional grouping, each of the items in a class $\\Delta_i$ can be replaced (in the prototype) by one of a few representatives from the same class, thus reducing the number of distinct items in configurations on the support of the prototype, while ensuring the polytope associated with the prototype remains non-empty. \n\nLinear shifting can be viewed as\nthe special case of fractional grouping in which the weight  of all  items is equal to one (i.e., $w_{\\ell}=1$). \nA key idea of the technique is that the size of each item in $\\Delta_{i}$ can be rounded up to the maximal size of an item in this set. By {\\em shifting} the items in $\\Delta_{i}$ to the bins used by items of $\\Delta_{i-1}$, it is shown that the increase in item sizes does not incur a significant increase in the number of bins required for packing the instance. The technique works well for bin packing problems in which the set $I'$ on which we apply  grouping is known in advance, as in classic Bin Packing~\\cite{fernandez1981bin}. In other applications~\\cite{DKS21,bansal2016improved}, the set of items $I'$ is unknown to the algorithm. To overcome this difficulty, the algorithms of~\\cite{DKS21, bansal2016improved}  guess properties of the sets $\\Delta_1,\\ldots, \\Delta_\\tau$ which suffice for efficient grouping. These guesses render the running times exponential in $1\/{\\varepsilon}$.\nThus, the running times of these schemes are polynomial only when ${\\varepsilon}$ is {\\em fixed}.\nFractional grouping overcomes this difficulty by avoiding the \nguessing step, thus eliminating the exponential dependence on $1\/{\\varepsilon}$. Hence, \nthe use of fractional grouping is a key to obtaining an asymptotic {\\em fully} polynomial-time approximation scheme (AFPTAS) from which we derive the result in Theorem~\\ref{thm:main}. Unlike \tprevious works~\\cite{FKS21, KMS22}, this paper gives the first constructive application of fractional grouping. \n\n\n\\subsection{Prior Work}\n\\label{sec:related}\n\nThe classic Bin Packing problem \nis known to be NP-hard and cannot be approximated within a ratio \nbetter than $\\frac{3}{2}$, unless P=NP. \nThis ratio is achieved by the simple First-Fit Decreasing algorithm~\\cite{S94}.\nThe paper~\\cite{Vega1981BinPC} \npresents an APTAS for Bin Packing, which uses at most $(1+{\\varepsilon})\\textnormal{OPT}+1$ bins, for any fixed ${\\varepsilon} \\in (0, 1\/2)$. The paper~\\cite{karmarkar1982efficient}  \ngives an AFPTAS and approximation algorithm that uses at most $\\textnormal{OPT}+O(\\log^2(\\textnormal{OPT}))$ bins. \nThe additive factor was improved in~\\cite{Ro13} to\n$O(\\log \\textnormal{OPT} \\cdot \\log \\log \\textnormal{OPT})$, and later to\n$O(\\log(\\textnormal{OPT}))$~\\cite{hoberg2017logarithmic}.\nFor comprehensive surveys of known results for BP see, e.g.,~\\cite{C+13,C+17}.\n\n\nAs the literature on Bin Packing and its variants is immense, we review below results that relate to packing under {\\em partition matroid constraint}.\nThe hardness of approximation of BPP (with respect to absolute approximation ratio) follows from the hardness of BP.\n\nThe special case of Group Bin Packing (in which $k(G)=1$ for all $G \\in {\\mathcal{G}}$) was first studied in~\\cite{JO97}. The approximation ratio of $2.7$ obtained in this paper  was later improved\nto better constants in several papers (e.g.,~\\cite{mccloskey2005approaches,allornothing}).\nThe best known result for general instances is an APTAS due to~\\cite{DKS21}. \nThe algorithm of~\\cite{DKS21}  is based on extensive guessing of  properties of an optimal solution which are then used as guidance  for the assignment of items to bins.\nIn particular, the algorithm  does not round a solution for a configuration-LP, and cannot be viewed as a rounding algorithm in general.  \nThe extensive guessing leads to running time that is exponential in $\\frac{1}{{\\varepsilon}}$.  \nFor a special case of GBP where the maximum cardinality of a group is some constant, an AFPTAS follows from a result of~\\cite{jansen1999approximation}.   \n\n\nGBP was studied also in the context of scheduling on identical machines. \nDas and Wiese~\\cite{DW17} introduced the problem of makespan minimization with bag constraints.\nIn this generalization of the classic makespan minimization problem, each job belongs to a {\\em bag}. The goal is to schedule the jobs on a set of $m$ identical machines, for some $m \\geq 1$, such that no two jobs in the same bag are assigned to the same machine, and the makespan is minimized. For the classic problem of makespan minimization with no bag constraints, there are known \n{\\em polynomial time approximation scheme (PTAS)}~\\cite{hochbaum1987using,MS:10} as well as\n{\\em efficient polynomial time approximation scheme (EPTAS)}~\\cite{hochba1997approximation,alon1998approximation,MS:5,JKV16}. Das and Wiese~\\cite{DW17} developed a PTAS for the problem with bag constraints. Later, Grage et al.~\\cite{Jansen_et_al:2019} obtained an EPTAS.\n\nAnother special case of BPP is Bin Packing with Cardinality Constraints, in which $|{\\mathcal{G}}|=1$, i.e., we have a single group $G$ with $k(G)=k$, for some integer $k \\geq 1$.\nBPCC has been studied since the 1970's~\\cite{KSS75,KSS77,KP99, CKP03}. The best known result is an AFPTAS due to~\\cite{epstein2010afptas}, which relies on rounding a non-standard configuration-LP formulation of the problem. \nThe techniques used in this work bear some similarities to the techniques of~\\cite{epstein2010afptas}.\nFor example, our interpretation for a configuration, \nin which unused capacity is an available capacity for additional items,\nis similar to the concept of {\\em windows} used in~\\cite{epstein2010afptas}.\nThe AFTPAS of~\\cite{epstein2010afptas} utilizes the property that if $k<\\frac{1}{{\\varepsilon}}$ then\nonly $\\frac{1}{{\\varepsilon}}$ items fit into a bin; thus, linear shifting can be applied to the whole instance.We note that in the presence of multiple  groups, \nthe cardinality bound for each group may be small (or even equal to $1$), yet the number of items that can be packed in a single bin may be arbitrarily large.  This is one of several\nhurdles encountered when attempting to adapt the algorithm of~\\cite{epstein2010afptas} to our setting of BP with a partition matroid constraint.\n\n\nIn the {\\em matroid partitioning} problem, we are given a ground set $U$ and a matroid $({{\\mathcal{M}}},{{\\mathcal{S}}})$, where ${{\\mathcal{S}}}$ is a family of subsets of $U$, known as the  \nthe {\\em independent sets} of the matroid. We seek a partition of $U$ to as few independent sets as possible. The problem is polynomially solvable for any matroid ${{\\mathcal{M}}}$ over $U$ using a combinatorial algorithm (see, e.g.,~\\cite{E65,GW92}). When ${\\mathcal{M}}$ is a partition matroid, the matroid partitioning problem can be viewed as a variant of BPP with unbounded bin capacities.\n\nTo the best of our knowledge, Bin Packing with Partition Matroid is studied here\nfor the first time.\n\n\\subsection{Organization of the Paper}\n\nIn Section~\\ref{sec:preliminaries} we give some definitions and notation.\n Section~\\ref{sec:overview} gives an overview of our scheme, that is applied to a {\\em structured} instance, and the main lemmas used in its analysis.\n In Sections~\\ref{sec:eviction}$-$\\ref{sec:alg_pack}\n we present the main components of the scheme: algorithm $\\textsf{Evict}$ (Section~\\ref{sec:eviction}), algorithm $\\textsf{Shift}$\n (Section~\\ref{sec:AlgPrototype}), algorithm $\\textsf{Partition}$ (Section~\\ref{sec:FromPolytope}), and algorithm $\\textsf{Pack}$ (Section~\\ref{sec:alg_pack}).\n In Section~\\ref{sec:reduction} we present algorithms \\textsf{Reduce} and \\textsf{Reconstruct}, which handle the structuring of the instance and \n the transformation of the solution for the structured instance into a solution for the original instance, respectively. We conclude in Section~\\ref{sec:discussion} with a summary and directions for future work. \n \n For clarity of the presentation, we defer the formal proofs to the Appendix. Also, algorithm $\\textsf{Greedy}$ is presented in Appendix~\\ref{sec:greedy}.\n\n\n\n\n\\section{Omitted Proofs of Section~\\ref{sec:AlgPrototype}}\n\\label{app:omittedShifting}\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:significantDiscard}:}\nWe prove the two properties of the lemma below. \n\n\n\\begin{enumerate}\n\t\\item \t\\begin{equation*}\n\t\t\\label{eq:ev5b}\n\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in I \\setminus L~} f_{\\bar{y}}(\\ell) \\cdot s(\\ell) ={} &   \\sum_{\\ell \\in I \\setminus L~}  \\sum_{C \\in \\mathcal{C}[\\ell]} \\bar{y}_C \\cdot s(\\ell)  =   \\sum_{C \\in \\mathcal{C}~}  \\sum_{\\ell \\in C \\setminus L} \\bar{y}_C \\cdot s(\\ell) =   \\sum_{C \\in \\mathcal{C} } \\bar{y}_C  \\sum_{\\ell \\in C \\setminus L}  s(\\ell)\\\\   ={} &  \\sum_{C \\in \\mathcal{C} } \\bar{y}_C  \\cdot s(C \\setminus L) \\leq \\sum_{C \\in \\mathcal{C}} \\bar{y}_C  \\cdot {\\varepsilon}  = {\\varepsilon}  \\sum_{C \\in \\mathcal{C}} \\bar{y}_C = {\\varepsilon} \\|\\bar{y}\\|.\n\t\t\\end{aligned}\n\t\\end{equation*} The first equality is by the definition of frequency. The second equality is by changing the order of summation. The inequality follows since $\\bar{y}$ holds the conditions of Lemma~\\ref{lem:Cnf}. \n\t\n\t\\item Assume towards a contradiction that there is $G \\in \\mathcal{G} \\setminus  \\mathcal{G}_{\\eta}$ such that $f_{\\bar{y}}(G \\setminus L) > {\\varepsilon} \\cdot \\|\\bar{y}\\|$. Therefore, by the definition of ${\\mathcal{G}}_{\\eta}$ as the set of groups with maximal frequncies of small items, it holds that\n\t\\begin{equation}\n\t\t\\label{eq:G'cap}\n\t\tf_{\\bar{y}}(G_{\\eta} \\setminus L)  > {\\varepsilon} \\cdot \\|\\bar{y}\\|.\n\t\\end{equation} We reach a contradiction by the following. \n\t\n\t\\begin{equation}\n\t\t\\label{eq:f1}\n\t\t\\sum_{\\ell \\in I} f_{\\bar{y}}(\\ell) = \\sum_{\\ell \\in I ~}  \\sum_{C \\in \\mathcal{C}[\\ell]} \\bar{y}_C =  \\sum_{C \\in \\mathcal{C}~}  \\sum_{\\ell \\in C} \\bar{y}_C \\leq  \\sum_{C \\in \\mathcal{C}~}  {\\varepsilon}^{-10} \\cdot \\bar{y}_C =  {\\varepsilon}^{-10} \\|\\bar{y}\\|.\n\t\\end{equation} The first equality is by the definition of the frequency. The second equality is by changing the order of summation. The inequality follows since $\\bar{y}$ holds the conditions of Lemma~\\ref{lem:Cnf}; thus, for all $C \\in \\textsf{supp}(\\bar{y})$ it holds that $|C| \\leq {\\varepsilon}^{-10}$. In addition, \n\t\n\t\\begin{equation}\n\t\t\\label{eq:f2}\n\t\t\\sum_{\\ell \\in I} f_{\\bar{y}}(\\ell) \\geq \\sum_{i \\in [\\eta]~} \t\\sum_{\\ell \\in G_i \\setminus L} f_{\\bar{y}}(\\ell) \\geq \\eta \\cdot f_{\\bar{y}}(G_{\\eta} \\setminus L) > \\eta \\cdot  {\\varepsilon} \\cdot \\|\\bar{y}\\| >  {\\varepsilon}^{-10} \\|\\bar{y}\\|.\n\t\\end{equation} The first inequality is because for all $\\ell \\in I$ it holds that $f_{\\bar{y}}(\\ell) \\geq 0$. The second inequality is because for all $G' \\in {\\mathcal{G}}_{\\eta}$ it holds that $f_{\\bar{y}}(G' \\setminus L) \\geq f_{\\bar{y}}(G_{\\eta} \\setminus L)$. The third inequality is by \\eqref{eq:G'cap}. The last inequality is since $\\eta =  {\\varepsilon}^{-12} $ or ${\\mathcal{G}}_{\\eta}  = {\\mathcal{G}}$. By \\eqref{eq:f1} and \\eqref{eq:f2} we reach a contradiction. \\qed\n\\end{enumerate}\n\n\n\n\n\n\nWe use the following auxiliary claim. \\begin{claim}\n\t\\label{AETA}\n\t$|{\\mathcal{A}} \\cup {\\mathcal{G}}_{\\eta}| \\leq 2K({\\varepsilon})$. \n\\end{claim}\n\n\\begin{proof}\n\t$$|{\\mathcal{A}} \\cup {\\mathcal{G}}_{\\eta}| \\leq |{\\mathcal{A}}|+|{\\mathcal{G}}_{\\eta}| \\leq K({\\varepsilon})+\\eta \\leq 2K({\\varepsilon}).$$\n\tThe second inequality is because $\\mathcal{I}$ is ${\\varepsilon}$-structured, there can be at most $K({\\varepsilon})$ massive groups by \\eqref{Agroups} and Definition~\\ref{def:structuring}. The third inequality is because $K({\\varepsilon}) = {\\varepsilon}^{-{\\varepsilon}^{-2}}$, $\\eta \\leq {\\varepsilon}^{-12}$, and ${\\varepsilon} < 0.1$. \n\\end{proof}\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:P(C)}:}\nLet $C \\in \\textsf{supp}(\\bar{y})$. We show the properties of the lemma below.\n\n\\begin{enumerate}\n\t\\item   \n\t\n\t\n\t\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\label{pcHelpL}\n\t\t\ts\\left(P(C)\\right) \\leq \\sum_{\\Phi \\in \\mathcal{Q}} s\\left(\\textsf{last}_{|C \\cap \\Phi|}(\\Phi)\\right) \\leq {} &   \\sum_{\\Phi \\in \\mathcal{Q}} s\\left(C \\cap \\Phi\\right) \\leq s(C). \n\t\t\\end{aligned}\n\t\\end{equation}\n\nThe second inequality is because for all $\\Phi \\in \\mathcal{Q}$ it holds that $\\textsf{last}_{|C \\cap \\Phi|}(\\Phi)$ contains the $|C \\cap \\Phi|$ items in $\\Phi$ with minimal size. The last inequality is because each item belongs to at most one class in $\\mathcal{Q}$.  \n\n\n\n\n\t\\item Let $\\ell \\in \\textsf{fit}(C)$. First, by \\eqref{fitS} it holds that $s(\\ell) \\leq {\\varepsilon}^{2}$. Second, we use the following inequalities.\t\\begin{equation}\n\t\t\\label{Ffit}\n\ts(\\ell) \\leq {\\varepsilon} \\cdot (1-s(C)) \\leq {\\varepsilon} \\cdot (1-s(P(C))).\n\t\\end{equation} The first inequality is because $\\ell \\in \\textsf{fit}(C)$. The second inequality is by \\eqref{pcHelpL}. Therefore, by \\eqref{Ffit} and that $s(\\ell) \\leq {\\varepsilon}^{2}$ we conclude that $\\ell \\in \\textsf{fit}(P(C))$ by \\eqref{fitS}. \n\n\t\\item \n\t\n\t$$|P(C)| = \\left|\\bigcup_{\\Phi \\in \\mathcal{Q}} \\textsf{last}_{|C \\cap \\Phi|}(\\Phi)\\right| \\leq  \\sum_{\\Phi \\in \\mathcal{Q}} |\\textsf{last}_{|C \\cap \\Phi|}(\\Phi)| = \\sum_{\\Phi \\in \\mathcal{Q}} |C \\cap \\Phi| \\leq |C| \\leq {\\varepsilon}^{-10}.$$ The first equality is by \\eqref{eq:mapS,eq:mapM}. The second equality is by the definition of \\textsf{last}. The second inequality is because each item in $C$ belongs to at most one class in $\\mathcal{Q}$. The last inequality is because $C \\in \\textsf{supp}(\\bar{y})$; thus, since $\\bar{y}$ satisfies the conditions of Lemma~\\ref{lem:Cnf} it holds that $|C| \\leq {\\varepsilon}^{-10}$. \n\t\n\t\n\t\\item First, there can be at most ${\\varepsilon}^{-\\upsilon-10}$ classes for each important group $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ by Observation~\\ref{ob:FG}. Thus, the number of classes is bounded by $|{\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}| \\cdot {\\varepsilon}^{-\\upsilon-10}$. Second, by Claim~\\ref{AETA} we have a bound of $2K({\\varepsilon}) $ on the number  for  important groups. By the above,\n\t\n\t\n\t\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\label{bound1111}\n\t\t\t|\\mathcal{Q}| \\leq{} & |{\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}| \\cdot {\\varepsilon}^{-\\upsilon-10} \\leq 2K({\\varepsilon})  \\cdot {\\varepsilon}^{-\\upsilon-10} = 2 {\\varepsilon}^{-{\\varepsilon}^{-2}} \\cdot {\\varepsilon}^{-3{\\varepsilon}^{-2}-10} \\leq {\\varepsilon}^{-6{\\varepsilon}^{-2}} \n\t\t\\end{aligned}\n\t\\end{equation}\n\n\nThe first inequality is by Claim~\\ref{AETA}. Therefore, \n\t\n\t\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\label{bound1}\n\t\t|\\{P(C')~|~C' \\in \\textsf{supp}(\\bar{y})\\}| \\leq{} & \\left( {\\varepsilon}^{-10} |\\mathcal{Q}| \\right)^{{\\varepsilon}^{-10}} \\leq  \\left( {\\varepsilon}^{-{\\varepsilon}^{-1}} \\cdot {\\varepsilon}^{-6{\\varepsilon}^{-2}} \\right)^{{\\varepsilon}^{-10}} \\leq    \\left( {\\varepsilon}^{-{\\varepsilon}^{-3}} \\right)^{{\\varepsilon}^{-10}} \\leq \\left( \\exp({\\varepsilon}^{-1})^{{\\varepsilon}^{-3}}\\right)^{{\\varepsilon}^{-10}}\\\\ \\leq{} &  \\left( \\exp({\\varepsilon}^{-4}) \\right)^{{\\varepsilon}^{-10}} \\leq  \\exp({\\varepsilon}^{-14}) \\leq  \\exp({\\varepsilon}^{-14}) +3K({\\varepsilon})-3K({\\varepsilon}) \\\\\n\t\t \\leq{} & 2 \\cdot \\exp({\\varepsilon}^{-14})-3K({\\varepsilon}) \\leq Q({\\varepsilon})-3K({\\varepsilon}).\n\t\t\\end{aligned}\n\t\\end{equation*} \tRecall that for each $C' \\in \\textsf{supp}(\\bar{y})$ it holds that $|C'| \\leq {\\varepsilon}^{-10}$; in addition, by \\eqref{eq:mapS,eq:mapM}, for each item $\\ell \\in P(C')$ there is a single $\\Phi \\in \\mathcal{Q}$ such that $\\ell \\in \\textsf{last}_{|C' \\cap \\Phi|} (\\Phi)$ and it follows that the number of distinct options for choosing $\\ell$ is bounded by $|C' \\cap \\Phi| \\cdot |\\mathcal{Q}| \\leq |C'| \\cdot |\\mathcal{Q}| \\leq {\\varepsilon}^{-10} \\cdot |\\mathcal{Q}|$. Thus, by choosing at most ${\\varepsilon}^{-10}$ items as described above, the first inequality follows. The second inequality is by \\eqref{bound1111}. The last inequalities follow since $0<{\\varepsilon}<0.1$, $K({\\varepsilon}) = {\\varepsilon}^{-{\\varepsilon}^{-2}}$ and $Q({\\varepsilon}) =   \\exp \\left({\\varepsilon}^{-17}\\right)$.\n\n\\item Let $G \\in {\\mathcal{G}}$. Now, \n\t\\begin{equation*}\n\t\\begin{aligned}\n\t|G \\cap P(C)| \\leq{} &  \\left|   \\bigcup_{\\Phi \\in \\mathcal{Q} \\text{ s.t. } \\Phi \\subseteq G} \\textsf{last}_{|C \\cap \\Phi|} (\\Phi)       \\right| = \\sum_{\\Phi \\in \\mathcal{Q} \\text{ s.t. } \\Phi \\subseteq G} \\left|        \\textsf{last}_{|C \\cap \\Phi|} (\\Phi)       \\right| \\\\ ={} & \\sum_{\\Phi \\in \\mathcal{Q} \\text{ s.t. } \\Phi \\subseteq G} |C \\cap \\Phi|  \\leq  |G \\cap C|\n\t\\end{aligned}\n\\end{equation*} \n\n\n\t\n\tThe first inequality is by \\eqref{eq:mapS,eq:mapM}. The second equality is because each item belongs to at most one class. The third equality is because for all $\\Phi \\in \\mathcal{Q}$ the number of items in $\\textsf{first}_{|C \\cap \\Phi|} (\\Phi)$ is the same as in $|C \\cap \\Phi|$ by the definition of $\\textsf{first}$. The last inequality is because by  \\eqref{eq:mapS,eq:mapM} every item in $C \\cap G$ belongs to at most one class and each class is a subset of some group. \n\\end{enumerate}\n\\qed\n\n\n\n\n\t\t\n\n\nIn the following we prove Lemma~\\ref{lem:Cnf}. The proof relies on the following claims and definitions and is given at the end of this section. \n\n\\begin{claim}\n\t\\label{lem:shiftingPolynomial}\n\tAlgorithm~\\ref{Alg:shifting} is runs in time $\\textnormal{poly}(\\frac{1}{{\\varepsilon}}, |{\\mathcal{I}}|)$.\n\\end{claim}\n\n\n\\begin{proof}\nFinding the significant groups can be done in polynomial time by sorting the groups according to the small item frequencies, where the the small item frequency of a group can be computed in polynomial time as $|\\textsf{supp}(\\bar{y})|$ is polynomial. In addition, finding the massive groups, all groups with large items can be computed in linear time by iterating over all groups. Thus, the construction of the important groups takes polynomial time and therefore the constructing the classes in Step~\\ref{step:classesQ} can be also achieved in polynomial time by \\eqref{eq:classS}. Furthermore, since $|\\textsf{supp}(\\bar{y})|$ is polynomial, computing the projections of all $C \\in \\textsf{supp}(\\bar{y})$ is polynomial. Finally, the linear combinations in Step~\\ref{step:u} and Step~\\ref{step:shiftreturn} are of a polynomial number of elements; thus, they can be computed in polynomial time. \n\\end{proof}\n\n\t\n\tFor convenience, we repeat the construction of $\\bar{\\lambda}$ as given in Section~\\ref{sec:AlgPrototype}. Recall that $\\bar{\\gamma}$ in the $\\bar{y}$-polytope such that for all $\\ell,j \\in I$, $\\ell \\neq j$ it holds that $\\bar{\\gamma}_{\\ell,j} = 0$. We define below a point $\\bar{\\lambda} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ and show that $\\bar{\\lambda}$ is in the $\\bar{z}$-polytope. Thus, it follows that the $\\bar{z}$-polytope is non-empty. For all $C \\in {\\mathcal{C}}$, let $P^{-1}(C) = \\{C' \\in {\\mathcal{C}}~|~ P(C') = C\\}$. We start with constructing  a point $\\bar{\\psi} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$, which is used in the construction of $\\bar{\\lambda}$. For all $\\ell \\in I$ and $C \\in {\\mathcal{C}}$ define\n\t\n\t\\begin{equation}\n\t\t\\label{lambda'lC}\n\t\t\\bar{\\psi}_{\\ell,C} = \\sum_{C' \\in P^{-1}(C)} \\bar{\\gamma}_{\\ell,C'}.\n\t\\end{equation}\n\t\n\t \n\t \n\t Moreover, for any $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$, $i \\in [\\tau(G)-1]$, $\\ell \\in \\Delta_{i+1}(G)$ and $j \\in \\Delta_{i}(G)$ define\n\t \n\t \n\t\\begin{equation}\n\t\t\\label{lambda'lDelta}\n\t\\bar{\\psi}_{\\ell,j} =   \\frac{f_{\\bar{u}}(j)}{f_{\\bar{u}}(\\Delta_{i}(G))} \\cdot \\bar{\\gamma}_{\\ell,\\ell}.\n\t\\end{equation}\n\t\n\t\n\nFor any other $\\ell,j \\in I$ define $\t\\bar{\\psi}_{\\ell,j} = 0$. We use the following auxiliary claims. \n\t\t\n\t\t\\begin{claim}\n\t\t\t\\label{psi5}\n\t\t$\\bar{\\psi}$ satisfies Constraint \\eqref{F1} of the $\\bar{u}$-polytope. \n\t\\end{claim}\n\n\n\\begin{proof}\nLet $\\ell \\in I$ and $t \\in I \\cup {\\mathcal{C}}$ such that $\\bar{\\psi}_{\\ell,t}>0$. We split into two cases as follows. \n\n\\begin{enumerate}\n\t\\item  $t \\in {\\mathcal{C}}$. By \\eqref{lambda'lC}, there exists $C \\in P^{-1}(t)$ such that $\\bar{\\gamma}_{\\ell,C}>0$.  Therefore, $\\ell \\in \\textsf{fit}(C)$ by \\eqref{F1} since $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope. Consequently, by Lemma~\\ref{lem:P(C)} it follows that $\\ell \\in  \\textsf{fit}(t)$. \n\t\n\t\\item There are $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ and $i \\in [\\tau(G)-1]$ such that $t \\in \\Delta_{i}(G)$. By \\eqref{lambda'lDelta} it follows that $\\ell \\in \\Delta_{i+1}(G)$. Hence, $\\ell \\in  \\textsf{fit}(t)$ by Observation~\\ref{ob:FG}. \n\t\n\\end{enumerate} By the definition of $\\bar{\\psi}$, for any $\\ell \\in I$, the above cases cover all possibilities for all $t \\in I \\cup {\\mathcal{C}}$ such that $\\bar{\\psi}_{\\ell,t}>0$. \n\\end{proof}\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\begin{claim}\n\t\t\t\\label{psi6}\n\t\t$\\bar{\\psi}$ satisfies constraint \\eqref{F2} of the $\\bar{u}$-polytope. \n\t\\end{claim}\n\t\n\t\n\t\\begin{proof}\n\t\t\tLet $C \\in \\mathcal{C}$. \n\t\t\n\t\t\t\\begin{equation*}\n\t\t\t\\label{C!E}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,C} \\cdot  s(\\ell) ={} &  \\sum_{\\ell \\in I~} \\sum_{C' \\in P^{-1}(C)} \\bar{\\gamma}_{\\ell,C'} \\cdot  s(\\ell)= \\sum_{C' \\in P^{-1}(C)~}  \\sum_{\\ell \\in I}  \\bar{\\gamma}_{\\ell,C'} \\cdot  s(\\ell)  \\\\\n\t\t\t\t\\leq{} & \\sum_{C' \\in P^{-1}(C)~} \\left(1-s(C')\\right) \\bar{y}_{C'} \\leq  \\sum_{C' \\in P^{-1}(C)~}  \\left(1-s(C)\\right) \\bar{y}_{C'} \\leq   \\left(1-s(C)\\right) \\bar{u}_C. \n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\t\n\t\t\n\t\t\n\t\tThe  first inequality is because $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope; thus, the inequality follows by \\eqref{F2}. The second inequality is because for all $C' \\in \\textsf{supp}(\\bar{y})$ it holds that $s(P(C')) \\leq s(C')$ by Lemma~\\ref{lem:P(C)}. The last inequality is by Step~\\ref{step:u} of Algorithm~\\ref{Alg:shifting}.\n\t\t\n\t\t\n\n\t\\end{proof}\n\t\n\t\n\t\n\t\t\\begin{claim}\n\t\t\t\\label{psi7}\n\t\t$\\bar{\\psi}$ satisfies constraint \\eqref{F5} of the $\\bar{u}$-polytope. \n\t\\end{claim}\n\t\n\t\n\t\\begin{proof}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\tLet $G \\in \\mathcal{G}$ and $C \\in \\mathcal{C}$. \n\t\t\n\t\t\n\t\t\t\\begin{equation*}\n\t\t\n\t\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in G} \\bar{\\psi}_{\\ell,C} ={} & \\sum_{\\ell \\in G~} \\sum_{C' \\in P^{-1}(C)} \\bar{\\gamma}_{\\ell,C'} = \\sum_{C' \\in P^{-1}(C)~} \\sum_{\\ell \\in G} \\bar{\\gamma}_{\\ell,C'} \\leq  \\sum_{C' \\in P^{-1}(C)} \\bar{y}_{C'} \\cdot \\left(k(G)-|C' \\cap G|\\right) \\\\ \\leq{} &  \\left(k(G)-|C \\cap G|\\right) \\sum_{C' \\in P^{-1}(C)} \\bar{y}_{C'}  \\leq \\bar{u}_C \\cdot \\left(k(G)-|C \\cap G|\\right) .\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\t\n\t\t The first inequality is because $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope; thus, the inequality follows by \\eqref{F5}. The second inequality is by Property~5 of Lemma~\\ref{lem:P(C)}. \n\t\t\n\t\t\n\t\t\n\t\t\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\t\n\t\tWe use the following auxiliary claims.\n\t\t\n\t\t\t\t\\begin{claim}\n\t\t\t\t\\label{aux:1}\n\t\t\t\tFor all $\\Phi \\in \\mathcal{Q}$ it holds that $f_{\\bar{y}}(\\Phi) \\leq \\frac{f_{\\bar{u}}(\\Phi)}{\\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)} $. \n\t\t\t\\end{claim}\n\t\t\t\n\t\t\t\\begin{proof}\n\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{a1jL}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\tf_{\\bar{y}}(\\Phi) ={} & \\sum_{\\ell \\in \\Phi} f_{\\bar{y}}(\\ell) = \\sum_{\\ell \\in \\Phi~} \\sum_{C \\in {\\mathcal{C}}[\\ell]} \\bar{y}_C = \\sum_{C \\in {\\mathcal{C}}~} \\sum_{\\ell \\in \\Phi \\cap C}  \\bar{y}_C = \\sum_{C \\in {\\mathcal{C}}~} |C \\cap \\Phi| \\cdot \\bar{y}_C\\\\ ={} & \n\t\t\t\t\t\t\\sum_{C \\in {\\mathcal{C}}~}  \\sum_{C' \\in P^{-1}(C)~} |C' \\cap \\Phi| \\cdot \\bar{y}_{C'} =  \\sum_{C \\in {\\mathcal{C}}~} |C \\cap \\Phi|  \\sum_{C' \\in P^{-1}(C)~}  \\bar{y}_{C'} \\\\ \\leq{} &  \\sum_{C \\in {\\mathcal{C}}~} |C \\cap \\Phi|  \\cdot \\frac{\\bar{u}_C}{\\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)} = \\frac{1}{\\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)} \\cdot \\sum_{C \\in {\\mathcal{C}}~} |C \\cap \\Phi|  \\cdot \\bar{u}_C.\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation} The sixth equality is because for all $C \\in {\\mathcal{C}}$ it holds that $|C \\cap \\Phi| = |P(C) \\cap \\Phi|$ by \\eqref{eq:mapS,eq:mapM}. The last inequality is by Step~\\ref{step:u} of Algorithm~\\ref{Alg:shifting}. Additionally, \t\\begin{equation}\n\t\t\t\t\t\\label{a2jL}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\tf_{\\bar{u}}(\\Phi) ={} & \\sum_{\\ell \\in \\Phi} f_{\\bar{u}}(\\ell) = \\sum_{\\ell \\in \\Phi~} \\sum_{C \\in {\\mathcal{C}}[\\ell]} \\bar{u}_C = \\sum_{C \\in {\\mathcal{C}}~} \\sum_{\\ell \\in \\Phi \\cap C}  \\bar{u}_C = \\sum_{C \\in {\\mathcal{C}}~} |C \\cap \\Phi| \\cdot \\bar{u}_C\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation} The claim follows by \\eqref{a1jL} and \\eqref{a2jL}. \n\t\t\t\\end{proof}\n\t\t\t\n\t\t\t\n\n\t\t\\begin{claim}\n\t\\label{jS}\n\tLet $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$, $i \\in [\\tau(G)-1]$ and let $j \\in \\Delta_{i}(G)$. It holds that $\\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,j} \\leq f_{\\bar{u}}(j)$.\n\\end{claim}\n\n\\begin{proof}\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,j} ={} & \\sum_{\\ell \\in  \\Delta_{i+1}(G)} \\frac{ f_{\\bar{u}}(j) }{f_{\\bar{u}}( \\Delta_{i}(G))} \\cdot \\bar{\\gamma}_{\\ell,\\ell} \\leq  \\sum_{\\ell \\in  \\Delta_{i+1}(G)} \\frac{ f_{\\bar{u}}(j) }{f_{\\bar{u}}( \\Delta_{i}(G))} \\cdot f_{\\bar{y}}(\\ell)  \\cdot  = \\frac{ f_{\\bar{u}}(j) }{f_{\\bar{u}}( \\Delta_{i}(G))} \\cdot f_{\\bar{y}}(\\Delta_{i+1}(G)) \\\\ \\leq{} &  \\frac{ f_{\\bar{u}}(j) }{f_{\\bar{y}}( \\Delta_{i}(G))\\cdot \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)} \\cdot   { f_{\\bar{y}}(\\Delta_{i+1}(G))}{} \n\t\t\t\\leq \n\t\t\t \\frac{ f_{\\bar{u}}(j) }{{\\varepsilon}^{\\upsilon}\\cdot\\|\\bar{y}\\| \\cdot \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)} \\cdot   {\\left({\\varepsilon}^{\\upsilon}\\cdot\\|\\bar{y}\\|+2\\right)}{}  \\\\\n\t\t\t  =& f_{\\bar{u}}(j) = \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{u}_C.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\t The first equality is by \\eqref{lambda'lDelta}. The second inequality is by Claim~\\ref{aux:1}. The third inequality is by Observation~\\ref{ob:FG}. \n\n\t\n\\end{proof}\n\nThe following observation follows from Claim~\\ref{jS}, that cover the only case where for some $\\ell,j \\in I$ it holds that $\\bar{\\psi}_{\\ell,j}>0$ by \\eqref{lambda'lDelta}. \n\t\\begin{observation}\n\t\t\\label{psi8}\n\t$\\bar{\\psi}$ satisfies constraint \\eqref{F3} of the $\\bar{u}$-polytope. \n\\end{observation}\n\n\n\n\n\\noindent{\\bf Proof of Claim~\\ref{ob:psi}:}\nClaim~\\ref{ob:psi} follows from Claim~\\ref{psi5}, Claim~\\ref{psi6}, Claim~\\ref{psi7}, and Observation~\\ref{psi8}. \\qed\n\n\n\nWe show below that $\\psi$ satisfies constraint \\eqref{F4} for a specific subset of the items, using the following auxiliary claims. \n\n\n\n\n\t\\begin{claim}\n\t\\label{aux:65}\n\tFor all $\\ell \\in I$ it holds that $\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C} = \\sum_{C \\in {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,C}$. \n\\end{claim}\n\n\\begin{proof}\n\t\\begin{equation*}\n\t\t\\label{1jL}\n\t\t\\begin{aligned}\n\t\t\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C} ={} & \\sum_{C \\in {\\mathcal{C}}~} \\sum_{C' \\in P^{-1}(C)} \\bar{\\gamma}_{\\ell,C'} = \\sum_{C' \\in {\\mathcal{C}}~~}  \\sum_{C \\in {\\mathcal{C}} \\text{ s.t. } P(C') = C}    \\bar{\\gamma}_{\\ell,C'} = \\sum_{C \\in {\\mathcal{C}}}    \\bar{\\gamma}_{\\ell,C}.\n\t\t\\end{aligned}\n\t\\end{equation*} \n\\end{proof}\n\n\n\t\\begin{claim}\n\t\\label{psi9S}\n\tFor all $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ and $\\ell \\in G \\setminus \\Delta_{1}(G)$ it holds that $\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t} \\geq 1$. \n\\end{claim}\n\n\\begin{proof}\n\t\n \n by \\eqref{lambda'lDelta}, there is $i \\in \\{2,3, \\ldots, \\tau(G)\\}$ such that $\\ell \\in \\Delta_{i}(G)$. Therefore,\n \n \t\\begin{equation*}\n \t\\begin{aligned}\n \t\t\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t} ={} & \\sum_{j \\in \\Delta_{i-1}(G)} \\frac{f_{\\bar{u}}(j)}{f_{\\bar{u}}(\\Delta_{i-1}(G))} \\cdot \\bar{\\gamma}_{\\ell,\\ell} +\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C} =  \\frac{\\bar{\\gamma}_{\\ell,\\ell} }{{f_{\\bar{u}}(\\Delta_{i-1}(G))}} \\sum_{j \\in \\Delta_{i-1}(G)} f_{\\bar{u}}(j) +\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C} \\\\ \n \t\t={} & \\frac{\\bar{\\gamma}_{\\ell,\\ell} }{{f_{\\bar{u}}(\\Delta_{i-1}(G))}} \\cdot f_{\\bar{u}}(\\Delta_{i-1}(G)) +\\sum_{C \\in {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,C} = \\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,t} \\geq 1.  \n \t\\end{aligned}\n  \\end{equation*} \n \nThe first equality is by \\eqref{lambda'lDelta}. The third equality is by Claim~\\ref{aux:65}. The fourth equality is because for all $\\ell \\neq j \\in I$ it holds that $\\bar{\\gamma}_{\\ell,j} = 0$. The last inequality is because $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope and thus satisfies property \\eqref{F4}.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe repeat the definition of $\\bar{\\mu}$ from Section~\\ref{sec:AlgPrototype}. We define a point $\\bar{\\mu} \\in \\mathbb{R}^{I}_{\\geq 0}$ as follows. For all $\\ell \\in I$, define \n\n\\begin{equation}\n\t\\label{mu}\n\t\\bar{\\mu}_{\\ell} = \\max \\left\\{0,1-\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t}\\right\\}.\n\\end{equation}\n\t\n\t\n\t\n\tWe prove below some properties of $\\bar{\\mu}$. \n\t\n\t\n\t\t\\begin{claim}\n\t\t\\label{mu:fit}\n\t\t$\\textnormal{\\textsf{supp}}(\\bar{\\mu}) \\setminus L \\subseteq \\textnormal{\\textsf{fit}}(\\emptyset)$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\tLet $\\ell \\in \\textsf{supp}(\\bar{\\mu})  \\setminus L$. Thus, it follows that $\\ell \\in I \\setminus L$. Hence, $\\ell \\in \\textsf{fit}(\\emptyset)$ by \\eqref{fitS}. \n\t\\end{proof}\n\t\n\t\t\\begin{claim}\n\t\t\\label{mu:fit2}\n\t\tFor all $\\ell \\in \\textnormal{\\textsf{supp}}(\\bar{\\mu})$ it holds that $\\ell \\in  \\textnormal{\\textsf{fit}} \\left(r\\left(\\textnormal{\\textsf{group}}(\\ell)\\right)\\right)$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\tLet $\\ell \\in \\textsf{supp}(\\bar{\\mu})$. Thus, by the definition of the maximal slot $r\\left(\\textnormal{\\textsf{group}}(\\ell)\\right)$ it holds that $s(\\ell) \\leq s\\left(r\\left(\\textnormal{\\textsf{group}}(\\ell)\\right)\\right)$ and that $\\textnormal{\\textsf{group}}(\\ell) = r\\left(\\textnormal{\\textsf{group}}(\\ell)\\right)$. Hence, the claim follows by \\eqref{fitl}. \n\t\\end{proof}\n\t\n\tWe use the following auxiliary claim. \n\t\t\\begin{claim}\n\t\t\\label{aux:mugroup}\n\tFor all $S \\subseteq I$ it holds that $\\sum_{\\ell \\in S} \\bar{\\mu}_{\\ell} \\leq f_{\\bar{y}}(S)$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in S} \\bar{\\mu}_{\\ell} ={} &  \\sum_{\\ell \\in S} \\max \\left\\{0,1-\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t}\\right\\} \\leq  \\sum_{\\ell \\in S} \\max \\left\\{0,1-\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C}\\right\\} = \\sum_{\\ell \\in S} \\max \\left\\{0,1-\\sum_{C \\in  {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,C}\\right\\}\\\\\n\t\t\t \\leq{} & \\sum_{\\ell \\in S} \\max \\left\\{0,\\sum_{j \\in I} \\bar{\\gamma}_{\\ell,j}\\right\\} = \\sum_{\\ell \\in S}  \\bar{\\gamma}_{\\ell,\\ell} \\leq  \\sum_{\\ell \\in S} f_{\\bar{y}}(\\ell) =  f_{\\bar{y}}(S). \n\t\t\t\\end{aligned}\n\t\t\\end{equation*}  The second equality is by Claim~\\ref{aux:65}. The second inequality is because $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope; thus, the inequality follows by \\eqref{F4}. \n\t\\end{proof}\n\t\n\n\n\n\t\\begin{claim}\n\t\\label{mu:group:ups}\n\tFor all $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ it holds that $\\sum_{\\ell \\in G} \\bar{\\mu}_{\\ell} \\leq {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2$.\n\\end{claim}\n\n\\begin{proof}\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in G} \\bar{\\mu}_{\\ell} = \\sum_{\\ell \\in \\Delta_{1}(G)} \\bar{\\mu}_{\\ell} \\leq{} & f_{\\bar{y}} \\left( \\Delta_{1}(G) \\right) \\leq  {\\varepsilon}^{\\upsilon}\\cdot \\|\\bar{y}\\|+2.\n\t\t\\end{aligned}\n\t\\end{equation*}  The first equality is by Claim~\\ref{psi9S} and \\eqref{mu}. The first inequality is by Claim~\\ref{aux:mugroup}. The second inequality is by Observation~\\ref{ob:FG}.\n\\end{proof}\n\n\t\t\\begin{claim}\n\t\t\\label{mu:group}\n\t\tFor all $G \\in {\\mathcal{G}}$ it holds that $\\sum_{\\ell \\in G} \\bar{\\mu}_{\\ell} \\leq {\\varepsilon} \\cdot \\|\\bar{y}\\|+2$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\tWe split the proof into two cases. \\begin{enumerate}\n\t\t\\item $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$. The claim follows by Claim~\\ref{mu:group:ups}. \n\t\t\n\t\t\\item  $G \\in {\\mathcal{G}} \\setminus ({\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}})$. Then, \t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\sum_{\\ell \\in G} \\bar{\\mu}_{\\ell} ={} &  \\sum_{\\ell \\in G \\setminus L} \\bar{\\mu}_{\\ell} \\leq f_{\\bar{y}} \\left( G \\setminus L \\right) \\leq {\\varepsilon}\\cdot \\|\\bar{y}\\| \\leq  {\\varepsilon}  \\cdot \\|\\bar{y}\\|+2.\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}  The first equality is because $G \\notin {\\mathcal{A}}$; thus, it follows that $L \\cap G = \\emptyset$ by \\eqref{Agroups}.  The first inequality is by Claim~\\ref{aux:mugroup}. The second inequality is by Lemma~\\ref{lem:significantDiscard}. \n\t\\end{enumerate}\n\t\\end{proof}\n\t\n\t\n\t\t\\begin{claim}\n\t\t\\label{mu:size}\n\t$\\sum_{\\ell \\in I \\setminus L} \\bar{\\mu}_{\\ell}  \\cdot s(\\ell) \\leq {\\varepsilon}\\cdot \\|\\bar{y}\\|$.\n\t\\end{claim}\n\t\n\t\n\t\\begin{proof}\n\t\tWe use the following inequalities. First, \n\t\t\t\\begin{equation}\n\t\t\t\t\\label{aux:size1}\n\t\t\t\\begin{aligned}\n\t\t \t\\sum_{\\ell \\in I \\setminus L} \\bar{\\mu}_{\\ell}  \\cdot s(\\ell)  ={} & \\sum_{\\ell \\in I \\setminus L}  \\max \\left\\{0,1-\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t}\\right\\} \\cdot s(\\ell)\\\\ \\leq{} & \\sum_{\\ell \\in I \\setminus L} \\max \\left\\{0,1-\\sum_{C \\in {\\mathcal{C}}} \\bar{\\psi}_{\\ell,C}\\right\\} \\cdot s(\\ell) = \\sum_{\\ell \\in I \\setminus L} \\max \\left\\{0,1-\\sum_{C \\in  {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,C}\\right\\} \\cdot s(\\ell) \n\t\t\\end{aligned}\n\t\t\\end{equation} The last equality is by Claim~\\ref{aux:65}. \tSecond, \n\t\n\t\t\\begin{equation}\n\t\t\t\\label{aux:size2}\n\t\t\\begin{aligned}\n\t\t\t\\sum_{\\ell \\in I \\setminus L} \\max \\left\\{0,1-\\sum_{C \\in  {\\mathcal{C}}} \\bar{\\gamma}_{\\ell,C}\\right\\} \\cdot s(\\ell) \n\t\t\\leq{} & \\sum_{\\ell \\in I \\setminus L} \\max \\left\\{0,\\sum_{j \\in I} \\bar{\\gamma}_{\\ell,j}\\right\\} \\cdot s(\\ell) = \\sum_{\\ell \\in I \\setminus L}  \\bar{\\gamma}_{\\ell,\\ell} \\cdot s(\\ell)\\\\ \\leq{} &  \\sum_{\\ell \\in I \\setminus L} f_{\\bar{y}}(\\ell) \\cdot s(\\ell) \\leq {\\varepsilon}\\|\\bar{y}\\|. \n\t\t\\end{aligned}\n\t\\end{equation} The first inequality is because $\\bar{\\gamma}$ is in the $\\bar{y}$-polytope; thus, the inequality follows by \\eqref{F4}. The last inequality is by Lemma~\\ref{lem:significantDiscard}. The claim follows by \\eqref{aux:size1} and \\eqref{aux:size2}.\n\n\t\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\noindent{\\bf Proof of Claim~\\ref{lem:mu}:}\n\tThe proof follows by Claim~\\ref{mu:fit}, Claim~\\ref{mu:group}, and Claim~\\ref{mu:size}. \\qed\n\t\n\tWe Define another point $\\bar{\\rho} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ based on $\\bar{\\mu}$. For all $\\ell \\in I \\setminus L$ define $\\bar{\\rho}_{\\ell,\\emptyset} = \\bar{\\mu}_{\\ell}$; in addition, for all $G \\in {\\mathcal{G}}, \\ell \\in G$ define $\\bar{\\rho}_{\\ell,r(G)} = \\bar{\\mu}_{\\ell}$; for any other entry $(\\ell,t)  \\in I \\times \\left(I \\cup {\\mathcal{C}}\\right)$ define $\\bar{\\rho}_{\\ell,t} = 0$. Finally, using the definitions of $\\bar{\\rho}$ and $\\bar{\\psi}$ we define:\n\n\t \n\t \\begin{equation}\n\t \t\\label{LA}\n\\bar{\\lambda} = \\bar{\\rho}+\\bar{\\psi}.\n\t \\end{equation}\n\t \n\t \n\t \t\\begin{claim}\n\t \t\\label{LA5}\n\t $\\bar{\\lambda}$ satisfies constraint \\eqref{F1} of the $\\bar{z}$-polytope. \n\t \\end{claim}\n\t \\begin{proof}\n\t \t Let $\\ell \\in I$ and $t \\in I \\cup {\\mathcal{C}}$ such that $\\bar{\\lambda}_{\\ell,t}>0$. We split into three cases.\n\t \t \n\t \t \\begin{enumerate}\n\t \t \t\\item $\\ell \\in I \\setminus L$ and $\\bar{\\rho}_{\\ell,t}>0$.  By the definition of $\\bar{\\rho}$ it follows that $t = \\emptyset$. In addition, $\\bar{\\rho}_{\\ell,\\emptyset} = \\bar{\\mu}_{\\ell}$ and it follows that $\\ell \\in \\textsf{supp}(\\bar{\\mu})$. Therefore, by Lemma~\\ref{mu:fit} it holds that $\\ell \\in \\textsf{fit}(t)$. \n\t \t \t\n\t \t \t\t\\item $\\ell \\in L$ and $\\bar{\\rho}_{\\ell,t}>0$.  By the definition of $\\bar{\\rho}$ it follows that $t = r\\left(\\textsf{group}(\\ell)\\right)$. In addition, $\\bar{\\rho}_{\\ell,r\\left(\\textsf{group}(\\ell)\\right)} = \\bar{\\mu}_{\\ell}$ and it follows that $\\ell \\in \\textsf{supp}(\\bar{\\mu})$. Therefore, by Lemma~\\ref{mu:fit2} it holds that $\\ell \\in \\textsf{fit}(t)$. \n\t \t \t\n\t \t \t\t\\item $\\bar{\\psi}_{\\ell,t}>0$.  It follows that $\\ell \\in \\textsf{fit}(t)$ by Claim~\\ref{ob:psi}. \n\t \t \\end{enumerate} By \\eqref{LA}, the above proves the claim. \n\t \\end{proof}\n\t\n\t \t\\begin{claim}\n\t \t\\label{LA6}\n\t \t$\\bar{\\lambda}$ satisfies constraint \\eqref{F2} of the $\\bar{z}$-polytope. \n\t \\end{claim}\n\t \\begin{proof}\n\t Let $C \\in {\\mathcal{C}}$. We split into two cases.\n\t \n\t \\begin{enumerate}\n\t \t\\item $C \\neq \\emptyset$. \t\\begin{equation*}\n\t \t\t\\label{LA6aux}\n\t \t\t\\begin{aligned}\n\t \t\t\t\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell,C} \\cdot s(\\ell) ={} & \\sum_{\\ell \\in I} \\left(\\bar{\\rho}_{\\ell,C}+\\bar{\\psi}_{\\ell,C}\\right) \\cdot s(\\ell) = \\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,C} \\cdot s(\\ell) \\leq (1-s(C))\\bar{u}_C \\leq (1-s(C))\\bar{z}_C. \n\t \t\t\\end{aligned}\n\t \t\\end{equation*} The second equality is because for all $\\ell \\in I$ and $C' \\in {\\mathcal{C}} \\setminus  \\{\\emptyset\\}$ it holds that $\\bar{\\rho}_{\\ell,C'} = 0$. The first inequality is because $\\bar{\\psi}$ satisfies constraint  \\eqref{F2} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The second inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n\t \t\n\t \t\\item $C = \\emptyset$. \t\\begin{equation*}\n\t \t\t\\begin{aligned}\n\t \t\t\t\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell, \\emptyset} \\cdot s(\\ell) ={}\n\t \t\t\t & \\sum_{\\ell \\in I} \\left(\\bar{\\rho}_{\\ell, \\emptyset}+\\bar{\\psi}_{\\ell, \\emptyset}\\right) \\cdot s(\\ell)  =  \n\t \t\t\t \\sum_{\\ell \\in I\\setminus L} \\bar{\\mu}_{\\ell} \\cdot s(\\ell)+\\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell, \\emptyset} \\cdot s(\\ell)\n\t \t\t\t \\\\ \\leq{} & {\\varepsilon}\\|\\bar{y}\\|+(1-s( \\emptyset))\\bar{u}_{\\emptyset} = {\\varepsilon}\\|\\bar{y}\\|+\\bar{u}_{\\emptyset} \\leq \\bar{z}_{\\emptyset} = (1-s( \\emptyset)) \\cdot \\bar{z}_{\\emptyset}.\n\t \t\t\\end{aligned}\n\t \t\\end{equation*}  The first inequality is by Claim~\\ref{mu:size} and because $\\bar{\\psi}$ satisfies constraint  \\eqref{F2} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The second inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n\t \\end{enumerate}\n\t \n\t \t \\end{proof}\n\t \n\t \n\t \n\t \t\\begin{claim}\n\t \t\\label{LA7}\n\t \t$\\bar{\\lambda}$ satisfies constraint \\eqref{F5} of the $\\bar{z}$-polytope. \n\t \\end{claim}\n\t \n\t \\begin{proof}\n\t \tLet $G \\in {\\mathcal{G}}$ and $C \\in {\\mathcal{C}}$. We split into two cases.\n\t \t\n\t \t\\begin{enumerate}\n\t \t\t\\item $C \\neq \\emptyset$. \t\\begin{equation*}\n\t \t\t\t\\label{LA6aux}\n\t \t\t\t\\begin{aligned}\n\t \t\t\t\\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C} = \\sum_{\\ell \\in G} \\left(\\bar{\\rho}_{\\ell,C}+\\bar{\\psi}_{\\ell,C}\\right) =   \\sum_{\\ell \\in G} \\bar{\\psi}_{\\ell,C} \\leq \\bar{u}_C \\cdot \\left(   k(G) - |G \\cap C|  \\right) \\leq \\bar{z}_C \\cdot \\left(   k(G) - |G \\cap C|  \\right).\n\t \t\t\t\\end{aligned}\n\t \t\t\\end{equation*} The second equality is because for all $\\ell \\in I$ and $C' \\neq \\emptyset$ it holds that $\\bar{\\rho}_{\\ell,C'} = 0$. The first inequality is because $\\bar{\\psi}$ satisfies constraint  \\eqref{F5} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The second inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n\t \t\t\n\t \t\t\\item $C = \\emptyset$. \t\\begin{equation*}\n\t \t\t\t\\begin{aligned}\n\t \t\t\t\t\\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,\\emptyset} ={} & \\sum_{\\ell \\in G} \\left(\\bar{\\rho}_{\\ell,\\emptyset}+\\bar{\\psi}_{\\ell,\\emptyset}\\right)   =  \\sum_{\\ell \\in G\\setminus L} \\bar{\\mu}_{\\ell}+ \\sum_{\\ell \\in G} \\bar{\\psi}_{\\ell,\\emptyset}  \\leq {\\varepsilon}  \\cdot \\|\\bar{y}\\|+2+\\bar{u}_{\\emptyset} \\cdot k(G)  \\\\ \\leq{} & \\left( {\\varepsilon}\\|\\bar{y}\\|+2 \\right) \\cdot k(G)+\\bar{u}_{\\emptyset} \\cdot k(G) \\leq \\bar{z}_{\\emptyset} \\cdot k(G).\n\t \t\t\t\\end{aligned}\n\t \t\t\\end{equation*}  The first inequality is by Claim~\\ref{mu:group} and because $\\bar{\\psi}$ satisfies constraint  \\eqref{F5} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The last inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n\t \t\\end{enumerate}\n\t \t\n\t \t\n\t \\end{proof}\n \n\t\n\n\t \t\\begin{claim}\n\t \t\\label{LA8}\n\t \t$\\bar{\\lambda}$ satisfies constraint \\eqref{F3} of the $\\bar{z}$-polytope. \n\t \\end{claim}\n \n \\begin{proof}\n \tLet $j \\in I$. We split into two cases, as follows. \\begin{enumerate}\n \t\t\\item If $\\textsf{group}(j) \\in {\\mathcal{G}}_{\\eta}\\cup {\\mathcal{A}}$ and $j = r\\left(   \\textsf{group}(j) \\right)$. Then, \t\t\\begin{equation*}\n \t\t\t\\begin{aligned}\n \t\t\t\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell,j} = \\sum_{\\ell \\in I} \\left(\\bar{\\rho}_{\\ell,j}+\\bar{\\psi}_{\\ell,j}\\right) =  \\sum_{\\ell \\in \\textsf{group}(j)} \\bar{\\mu}_{\\ell,j} + \\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,j} \\leq {\\varepsilon}^{\\upsilon}\\cdot\\|\\bar{y}\\|+2+\\sum_{C \\in {\\mathcal{C}}[j]} \\bar{u}_C \\leq \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{z}_C.\n \t\t\t\\end{aligned}\n \t\t\\end{equation*}  The first inequality is by Claim~\\ref{mu:group:ups} and because $\\bar{\\psi}$ satisfies constraint  \\eqref{F3} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The second inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n \t\t\n \t\t\\item Otherwise, \t$$\\sum_{\\ell \\in I} \\bar{\\lambda}_{\\ell,j} = \\sum_{\\ell \\in I} \\left(\\bar{\\rho}_{\\ell,j}+\\bar{\\psi}_{\\ell,j}\\right) = \\sum_{\\ell \\in I} \\bar{\\psi}_{\\ell,j} \\leq \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{u}_C \\leq \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{z}_C.$$ The first inequality is because $\\bar{\\psi}$ satisfies constraint  \\eqref{F3} of the $\\bar{u}$-polytope by Claim~\\ref{ob:psi}. The second inequality is by Step~\\ref{step:shiftreturn} in Algorithm~\\ref{Alg:shifting}. \n \t\\end{enumerate}\n \t\n \n \\end{proof}\n\t \n\t \n\t \n\t \\begin{claim}\n\t \t\\label{LA9}\n\t \t$\\bar{\\lambda}$ satisfies constraint \\eqref{F4} of the $\\bar{z}$-polytope. \n\t \\end{claim}\n\t \n\t \\begin{proof}\n\t \tLet $\\ell \\in I$. \n\t \t%\n\t \t$$\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\lambda}_{\\ell,t} = \\sum_{t \\in I \\cup {\\mathcal{C}}} \\left(\\bar{\\rho}_{\\ell,j}+\\bar{\\psi}_{\\ell,t}\\right) =  \\bar{\\mu}_{\\ell}+\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t} = \\max \\left\\{0,1-\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t}\\right\\}+\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t} \\geq 1.$$ \n\t \t\n\t \t \\end{proof}\n \t \n \t The following observation follows by Claim~\\ref{LA5}, Claim~\\ref{LA6}, Claim~\\ref{LA7}, Claim~\\ref{LA8}, and Claim~\\ref{LA9}.\n\n\t\n\t\\noindent{\\bf Proof of Claim~\\ref{lem:shifting1}:}\nThe proof follows by Claim~\\ref{LA5}, Claim~\\ref{LA6}, Claim~\\ref{LA7}, Claim~\\ref{LA8}, and Claim~\\ref{LA9}. \\qed\n\t\n\n\n\\begin{claim}\n\t\\label{lem:shifting2a}\n\tAlgorithm~\\ref{Alg:shifting} returns a prototype $\\bar{z}$  of $\\mathcal{I}$ such that $|\\textnormal{\\textsf{supp}}(\\bar{z})| \\leq Q({\\varepsilon})$.\n\\end{claim}\n\n\n\\begin{proof}\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t|\\textsf{supp}(\\bar{z})| \\leq{} & |\\{P(C)~|~ C \\in \\textsf{supp}(\\bar{y})\\}|+1+|{\\mathcal{G}}_{\\eta}\\cup {\\mathcal{A}}| \\\\ \\leq{} & Q({\\varepsilon})-3K({\\varepsilon})+ |{\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}|+1 \\leq Q({\\varepsilon})-3K({\\varepsilon})+2K({\\varepsilon})+1 \\leq Q({\\varepsilon}). \n\t\t\\end{aligned}\n\t\\end{equation*} \n The first inequality is because for all $C \\in \\textsf{supp}(\\bar{z})$ either $C  = \\emptyset$, there is $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ such that $C = r(G)$, or there is $C' \\in \\textsf{supp}(\\bar{y})$ such that $P(C') = C$ by Step~\\ref{step:u} and Step~\\ref{step:shiftreturn} of Algorithm~\\ref{Alg:shifting}. The second inequality is by Lemma~\\ref{lem:P(C)}. The third inequality is by Claim~\\ref{AETA}. \n\\end{proof}\n\n\\begin{claim}\n\t\\label{lem:shifting2}\n Algorithm~\\ref{Alg:shifting} returns a prototype $\\bar{z}$  of $\\mathcal{I}$ such that $\\|\\bar{z}\\| \\leq (1+5{\\varepsilon}) \\|\\bar{y}\\|+Q({\\varepsilon})$. \n\\end{claim}\n\n\t\t\n\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\\begin{proof}\nWe use the following inequalities.\n\n\\begin{equation}\n\t\\label{imp:1}\n\\begin{aligned}\n\\left\\|     \\sum_{G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}} \\left( {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2 \\right) \\cdot \\mathbb{I}^{r(G)}     \\right\\| \\leq {} & \\sum_{G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}} \\left( {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2 \\right) \\cdot \\|\\mathbb{I}^{r(G)}\\| \\\\ ={} &  |{\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}| \\cdot ({\\varepsilon}^{\\upsilon}\\cdot \\|\\bar{y}\\|+2) \\leq 2K({\\varepsilon}) \\cdot ({\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2) \\\\ \\leq{} & {\\varepsilon}^{-2{\\varepsilon}^{-2}} \\cdot {\\varepsilon}^{3{\\varepsilon}^{-2}} \\cdot \\|\\bar{y}\\|+ 4K({\\varepsilon}) \\\\ \\leq{} & {\\varepsilon} \\cdot \\|\\bar{y}\\|+ 4K({\\varepsilon}). \n\\end{aligned}\n\\end{equation} The first inequality is by the triangle inequality. The second inequality is by Claim~\\ref{AETA}. \n\n\n\n\\begin{equation}\n\t\\label{imp:2}\n\t\\begin{aligned}\n    {} & \\left\\| \\left(4{\\varepsilon} \\|\\bar{y}\\|+{\\varepsilon}^{-3} \\right)\\cdot \\mathbb{I}^{\\emptyset} +\\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right) \\sum_{C \\in \\textsf{supp}(\\bar{y})} \\bar{y}_C \\cdot \\mathbb{I}^{P(C)} \\right\\| \\\\\n     \\leq{}  &  \\left\\|\\left(4{\\varepsilon} \\|\\bar{y}\\|+{\\varepsilon}^{-3} \\right)\\ \\cdot \\mathbb{I}^{\\emptyset}\\right\\|+ \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)  \\left\\| \\sum_{C \\in \\textsf{supp}(\\bar{y})} \\bar{y}_C \\cdot \\mathbb{I}^{P(C)} \\right\\| \\\\ \\leq{} & \n     \\left( 4{\\varepsilon} \\|\\bar{y}\\| +{\\varepsilon}^{-3}\\right)\\cdot  \\| \\mathbb{I}^{\\emptyset}\\|+  \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)  \\sum_{C \\in \\textsf{supp}(\\bar{y})} \\bar{y}_C \\cdot \\| \\mathbb{I}^{P(C)} \\|  \\\\ \n     ={} & 4{\\varepsilon} \\|\\bar{y}\\| +{\\varepsilon}^{-3}+ \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)  \\|\\bar{y}\\| = (1+4{\\varepsilon})\\cdot \\|\\bar{y}\\|  +2{\\varepsilon}^{-\\upsilon} +{\\varepsilon}^{-3}\n\t\\end{aligned}\n\\end{equation} \tThe first inequality is by the triangle inequality. The third equality is since for all $C \\in {\\mathcal{C}}$ it holds that $\\|\\mathbb{I}^C\\| = 1$ by the definition of the indicator of $C$. Now, \n\n\t\t\t\\begin{align*}\n\t\t\t\t\\|\\bar{z}\\| \\leq{} & \\left\\|  \\sum_{G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}} \\left( {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2 \\right) \\cdot \\mathbb{I}^{r(G)}  +\n\t\t\t\t\\left(4{\\varepsilon} \\|\\bar{y}\\| +{\\varepsilon}^{-3}\\right)\\cdot \\mathbb{I}^{\\emptyset} +\\left(1+\\frac{2\\cdot {\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right) \\sum_{C \\in \\textsf{supp}(\\bar{y})} \\bar{y}_C \\cdot \\mathbb{I}^{P(C)} \\right\\| \\\\ \n\t\t\t\t\\leq{} &  {\\varepsilon} \\cdot \\|\\bar{y}\\|+ 4K({\\varepsilon})+(1+4{\\varepsilon})\\cdot \\|\\bar{y}\\|  +2{\\varepsilon}^{-v} +{\\varepsilon}^{-3}\\leq (1+5{\\varepsilon}) \\cdot \\|\\bar{y}\\|+7{\\varepsilon}^{-3{\\varepsilon}^{-2}} \\\\ \\leq{} & (1+5{\\varepsilon}) \\cdot \\|\\bar{y}\\|+Q({\\varepsilon}).  \n\t\t\t\\end{align*} The first equality is by Step~\\ref{step:u} and Step~\\ref{step:shiftreturn} of Algorithm~\\ref{Alg:shifting}. The second inequality is by the triangle inequality and by \\eqref{imp:1} and \\eqref{imp:2}. The last inequalities are because ${\\varepsilon} <0.1$, $K({\\varepsilon})  = {\\varepsilon}^{-{\\varepsilon}^{-2}}$ and $Q({\\varepsilon}) = \\exp({\\varepsilon}^{-17})$. \n\t\t\n\n\t\t\n\t\t\t\n\t\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{claim}\n\t\t\t\\label{lem:shifting5}\n\t\tAlgorithm~\\ref{Alg:shifting} returns a prototype $\\bar{z}$  of $\\mathcal{I}$ such that for all $C \\in \\textnormal{\n\t\t\\textsf{supp}}(\\bar{z})$ it holds that $|C| \\leq {\\varepsilon}^{-10}$. \n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\tLet $C \\in \\textsf{supp}(\\bar{z})$ be a configuration. If $C = \\emptyset$ or there is $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ such that $C = r(G)$ that contains a single item, then the claim is trivial. Otherwise, because $C \\in \\textsf{supp}(\\bar{z})$, then by Step~\\ref{step:u} of Algorithm~\\ref{Alg:shifting}, there is $C' \\in \\textsf{supp}(\\bar{y})$ such that $P(C') = C$.  Therefore,  $|C| = |P(C')| \\leq {\\varepsilon}^{-10}$ by Lemma~\\ref{lem:P(C)}. \n\t\t\\end{proof}\n\n\t\t\n\t\t\\noindent{\\bf Proof of Lemma~\\ref{lem:Cnf}:} The proof follows by Claim~\\ref{lem:shiftingPolynomial}, Claim~\\ref{lem:shifting1}, Claim~\\ref{lem:shifting2a}, Claim~\\ref{lem:shifting2}, and Claim~\\ref{lem:shifting5}. \\qed\n\t\t\n\n\\section{Algorithm \\textsf{Pack}}\n\\label{sec:alg_pack}\n\nIn this section we give the proof of Lemma~\\ref{lem:GREEDY}. \nLet $A$, $\\mathcal{H}$, $\\left(B_C\\right)_{C \\in \\mathcal{H}}$, and $\\left(D_C\\right)_{C \\in \\mathcal{H}}$ be an ${\\varepsilon}$-nice partition of $\\mathcal{I}$ of size $m$. For all $C \\in \\mathcal{H}$, algorithm \\textsf{Pack} uses algorithm \\textsf{Greedy} (see Lemma~\\ref{thm:greedy}) to add the items in $D_C$ to the bins in $B_C$, possibly by using a few extra bins.  This requires a definition of a residual instance with further restrictions so the bins of $B_C$ remain feasible as a packing of ${\\mathcal{I}}$. Specifically, given $C \\in \\mathcal{H}$, define ${\\mathcal{I}}_C = (D_C, {\\mathcal{G}}_C,s_C,k_C)$ as the {\\em residual instance} of $C$ and ${\\mathcal{I}}$, such that $\\mathcal{G}_C = \\{G \\cap D_C \\neq \\emptyset ~|~ G \\in {\\mathcal{G}}\\}$, and \\begin{equation}\n\t\\label{IC}\n\t\\begin{aligned}\n\t\t&s_C(\\ell) = \\frac{s(\\ell)}{1-s(C)}&~~~~~~~~~~~&\\forall \\ell \\in D_C\\\\ \n\t\t&k_C(G \\cap D_C) = k(G)-|G \\cap C| &&\\forall G \\in {\\mathcal{G}} \\text{ s.t. } G \\cap D_C \\neq \\emptyset~~\n\t\\end{aligned}\n\\end{equation} In words, we modify the size function such that a bin $B \\in B_C$ may \nreceive \nan additional subset of items $S \\subseteq D_C$ with the following attributes: the total size of items in $S$ is at most $1-s(C)$, and for each $G \\in {\\mathcal{G}}$ it holds that $|S \\cap B| \\leq k(G)-|C \\cap G|$. Thus, $B \\cup S$ is a configuration of ${\\mathcal{I}}$. Also, observe that if $D_C \\neq \\emptyset$ then by Definition~\\ref{def:partition} $s(C) < 1$; thus, ${\\mathcal{I}}_C$ is well defined.\n\n\n\\begin{algorithm}[h]\n\t\\caption{$\\textsf{Pack}\\left({\\varepsilon}, {\\mathcal{I}}, A, \\mathcal{H}, \\left(B_C\\right)_{C \\in \\mathcal{H}}, \\left(D_C\\right)_{C \\in \\mathcal{H}}\\right)$}\n\t\\label{alg:pack}\n\t\nInitialize $P \\leftarrow ()$.\n\n\\For{$C \\in \\mathcal{H}$}{\n\n\nCompute the residual instance ${\\mathcal{I}}_C$ of $C$ and ${\\mathcal{I}}$.\n\n$P_C \\leftarrow \\textsf{tuple}(B_C) + \\textsf{Greedy}({\\mathcal{I}}_C,{\\varepsilon})$.\n\n $P \\leftarrow P \\oplus P_C$.\\label{step:pPack}\n\n\n\n}\n\nReturn $P$.\n\n\\end{algorithm}\n\nWe use the following notation for the algorithm. Given two tuples $T = (T_1, \\ldots, T_t)$ and $R = (R_1, \\ldots, R_r)$ where $T_i, R_j \\subseteq I ~\\forall i \\in [t], j \\in [r]$, recall that $T \\oplus R$ is the concatenation of $T,R$. In addition,  if $r> t$ let $T+R = (T_1 \\cup R_1, \\ldots, T_t \\cup R_t, R_{t+1}, \\ldots, R_r)$ and if $r \\leq t$ let $T+R = (T_1 \\cup R_1, \\ldots, T_r \\cup R_r, T_{r+1}, \\ldots, T_t)$; in words, this is an element-wise addition of the items in $R$ to $T$. Finally, given $C \\in \\mathcal{H}$, let $\\textsf{tuple}(B_C)$ be the elements (bins) in $B_C$ ordered arbitrarily as a tuple.  For all $C \\in \\mathcal{H}$ algorithm \\textsf{Pack} calls algorithm \\textsf{Greedy} with parameter ${\\varepsilon}$ for the residual instance ${\\mathcal{I}}_C$ and creates the packing $\\textsf{tuple}(B_C) + \\textsf{Greedy}({\\mathcal{I}}_C,{\\varepsilon})$; then, algorithm \\textsf{Pack} concatenates all of these packings into a packing of ${\\mathcal{I}}$. We give the \npseudocode of \\textsf{Pack} in Algorithm~\\ref{alg:pack}. \nThe proof of the next lemma follows from~\\eqref{IC}, Definition~\\ref{def:partition}, and Lemma~\\ref{thm:greedy}. We give the full details and the proof of Lemma~\\ref{lem:GREEDY} in Appendix~\\ref{sec:PackProofs}. \n\n\\begin{lemma}\n\t\\label{clm:GREEDY}\n\tFor all $C \\in \\mathcal{H}$, $\\textnormal{\\textsf{tuple}}(B_C) + \\textnormal{\\textsf{Greedy}}({\\mathcal{I}}_C,{\\varepsilon})$ is a packing of $D_C \\cup \\bigcup_{B \\in B_C} B$ with respect to ${\\mathcal{I}}$ in at most $(1+2{\\varepsilon}) \\cdot |B_C|+2$ bins. \n\\end{lemma} \n\n\\section{Properties of the $\\bar{x}$-polytope}\n\\label{sec:poly}\n\\newcommand{\\bar{m}}{\\bar{m}}\n\\newcommand{\\bar{\\gamma}}{\\bar{\\gamma}}\n\\newcommand{\\bar{b}}{\\bar{b}}\n\n\n\n In this section we give the proof of Lemma~\\ref{O(1)}. Let $\\bar{x}$ be a prototype of $\\mathcal{I}$ which satisfy the conditions of Lemma~\\ref{O(1)}: there is an integer $k$ such that for all $C \\in \\textsf{supp}(\\bar{x})$ it holds that  $|C| \\leq k$ and $\\bar{x}_C \\in \\mathbb{N}$; also, let $\\bar{\\lambda}$ be a vertex of the $\\bar{x}$-polytope for which constraint \\eqref{F4} holds with equality.  We say that a constraint is {\\em tight} if it holds with equality. Define the {\\em active} types of $\\bar{x}$ as $$T = \\textsf{supp}(\\bar{x}) \\cup \\{j \\in I~|~ \\exists C \\in \\textsf{supp}(\\bar{x}) \\text{ s.t. } j \\in C\\}.$$  \n \n \n \nFor every $t \\in T$ and $G \\in {\\mathcal{G}}$, we define below a value $L_{t,G}$. Let $C \\in {\\mathcal{C}} \\cap T$,$j \\in I \\cap T$, and $G \\in {\\mathcal{G}}$. Define $L_{C,G} = \\bar{x}_C \\cdot \\left(   k(G)-|G \\cap C|  \\right)$ and $L_{j,G} = \\sum_{C' \\in {\\mathcal{C}}[j]} \\bar{x}_{C'}$. For every  $(\\ell,t) \\in I \\times (I \\cup {\\mathcal{C}})$, we say the entry $\\bar{\\lambda}_{\\ell,t}$ {\\em corresponds} to $\\ell$ and $t$. Note that all entries $\\bar{\\lambda}_{\\ell,t}$ such that $t \\in (I \\cup {\\mathcal{C}}) \\setminus T$ (i.e., not corresponding to an active type) are required to be zero by the definition of the $\\bar{x}$-polytope.\n \n \n \n  For the following, fix a group $G \\in {\\mathcal{G}}$. A {\\em movement} of $G$ is a vector $\\bar{m} \\in \\mathbb{R}^{I  \\times (I \\cup {\\mathcal{C}})}$ which is used as a relaxation of the constraints corresponding to items in $G$ in the $\\bar{x}$-polytope:\n\n \\begin{align}\n\t\t\\displaystyle \\bar{m}_{\\ell,t} = 0   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~\t~~~~~~~~~~\\forall \\ell\\in I, t \\in T \\text{ s.t. } \\ell \\notin G  \\textnormal{ or } \\bar{\\lambda}_{\\ell, t} \\in \\{0,1\\} ~ \\label{m0}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle  \\sum_{t \\in T} \\bar{m}_{\\ell,t} = 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\forall \\ell \\in I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\label{ml}\\\\\n\t\t\\rule{0pt}{1.8em}\n\t\t\\displaystyle    \\sum_{\\ell \\in G} \\bar{m}_{\\ell,t} = 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~  ~~~~~~~~~~\t~~~~~~~~~~\\forall t \\in T \\text{ s.t.} \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,t} =L_{t,G}~~~~~~~~~~~~~~~~~~\\label{mT}\n\t\\end{align}\n\n\n We say that $G$ is {\\em fractional} if there are $\\ell \\in G$ and $t \\in T$ such that $\\bar{\\lambda}_{\\ell,t} \\in (0,1)$.  \n\n\\begin{claim}\n\t\\label{lem:movement}\n\tFor every group $G \\in {\\mathcal{G}}$, If $G$ is fractional then $G$ has a movement $\\bar{m} \\neq 0$.\n\\end{claim}\nThe proof of Claim~\\ref{lem:movement} utilizes properties of {\\em totally unimodular} matrices. A matrix $A$ is totally unimodular if every square submatrix of $A$ has a determinant $1$, $-1$ or $0$. If $A\\in \\mathbb{R}^{n\\times m}$ is totally unimodular and $b\\in \\mathbb{Z}^{m}$ is an integral vector, it holds that the vertices of the polytope  $$P_A =\\{\\bar{v} \\in \\mathbb{R}_{\\geq 0}^n~|~A \\bar{v} \\leq \\bar{b}\\}$$ are integral. That is,  if $\\bar{v} \\in P_A$ is a vertex  of $P_A$ then $\\bar{v} \\in \\mathbb{Z}^n$ \\cite{HK56}.  We use the following criteria for total unimodularity, which is a simplified version of a theorem from~\\cite{HK56}.\n\\begin{lemma}\n\t\\label{lem:unimodular}\n\tLet $A\\in \\mathbb{R}^{n\\times m}$ be a matrix which satisfies the following properties.\n\t\\begin{itemize}\n\t\t\\item All the entries of $A$ are  in  $\\{-1,1,0\\}$.\n\t\t\\item Every column of $A$ has up to two non-zero entries.\n\t\t\\item If a column of $A$ has two non-zero entries, then these entries have opposite signs.\n\t\t\\end{itemize}\n\tThen, $A$ is totally unimodular.\n\t\\end{lemma}  \n\n\n\\vspace{0.08in}\n\\noindent{\\bf Proof of Claim~\\ref{lem:movement}:}\nLet $G \\in {\\mathcal{G}}$ be a fractional group. Define $X = \\{ (\\ell,t) \\in G \\times T~|~ \\ell \\notin \\textsf{fit}(t)\\}$  as all {\\em forbidden pairs} of $G$, where for all $(\\ell,t) \\in X$ it holds that $\\bar{\\lambda}_{\\ell,t} = 0$ by \\eqref{F1}.  We show the existence of the movement $\\bar{m} \\neq 0$ using the polytope $P$ defined as follows. \n\n\t\\begin{equation}\n\t\t\\label{eq:Pj_def}\n\tP= \\left\\{ {\\bar{y}}\\in \\mathbb{R}_{\\geq 0}^{G \\times T} ~\\middle |~ \\begin{array}{lcc}\n\t\t\\displaystyle \\forall (\\ell,t) \\in X&:& \\displaystyle{\\bar{y}}_{\\ell,t}\\leq 0\\\\\n\t\t\t\\rule{0pt}{1.8em}\n\t \\displaystyle\t\\forall \\ell\\in G&:& \\displaystyle  \\sum_{t \\in T \\textnormal{ s.t. } (\\ell,t) \\in  \\notin X} -{\\bar{y}}_{\\ell,t} \\leq -1\\\\\n\t \t\\rule{0pt}{1.8em}\n\t \\displaystyle \\forall t \\in T&:& \n\t \\displaystyle  \\sum_{\\ell\\in G \\textnormal{ s.t. } (\\ell,t)\\notin X } {\\bar{y}}_{\\ell,t} \\leq L_{t,G}\n\t\\end{array}\\right\\}.\n\t\\end{equation}\n\nDefine a vector $\\bar{\\phi} \\in \\mathbb{R}^{G \\times T}$ by $\\bar{\\phi}_{\\ell,t}=\\bar{\\lambda}_{\\ell,t}$ for all $(\\ell,t) \\in G \\times T$. It follows from the definition of the $\\bar{x}$-polytope that $\\bar{\\phi} \\in P$. We can represent the inequalities in \\eqref{eq:Pj_def} using a matrix notation as  $$P =\\left\\{{\\bar{y}} \\in \\mathbb{R}^{G \\times T}_{\\geq 0} ~\\middle|~ A{\\bar{y}} \\leq \\bar{b} \\right\\}.$$ It follows that $A$ contains only entries in $\\{-1,0,1\\}$ and the entries in $\\bar{b}$ are all integral. Furthermore, every column of $A$ contains at most $2$ non-zero entries, and if there are two non-zero entries in a column then they are of a different sign. By Lemma~\\ref{lem:unimodular}, it follows that $A$ is totally unimodular. It thus holds that all the vertices of the polytope $P$ are integral. As $\\bar{\\phi} \\in P$ is non-integral (since $G$ is fractional), it follows that $\\bar{\\phi}$ is not a vertex of $P$. Hence, there is a vector $\\bar{\\gamma}\\in \\mathbb{R}^{G \\times T}$, $\\bar{\\gamma} \\neq 0$  such that $\\bar{\\phi}+\\bar{\\gamma}, \\bar{\\phi}-\\bar{\\gamma} \\in P$. \n\nWe define $\\bar{m} \\in \\mathbb{R}^{I \\times (I \\cup {\\mathcal{C}})}$ by $\\bar{m}_{\\ell,t}=\\bar{\\gamma}_{\\ell,t}$ for all $(\\ell,t)\\in G \\times  T$ and $\\bar{m}_{\\ell,t} =0$ otherwise. Clearly, $\\bar{m} \\neq 0$ as $\\bar{\\gamma} \\neq 0$. Observe that for $\\ell \\in I \\setminus G$ and $t \\in (I \\cup {\\mathcal{C}})$ it holds that $\\bar{m}_{\\ell,t}=0$ by definition. For $\\ell\\in G$ and $t \\in (I \\cup {\\mathcal{C}})$ such that $\\bar{\\lambda}_{\\ell,t} =0$, as $\\bar{\\phi}_{\\ell,t}+\\bar{\\gamma}_{\\ell,t}, \\bar{\\phi}_{\\ell,t} -\\bar{\\gamma}_{\\ell,t}\\geq 0$ and $\\bar{\\phi}_{\\ell,t}= \\bar{\\lambda}_{\\ell,t}=0$,  it follows that $\\bar{m}_{\\ell,t} = \\bar{\\gamma}_{\\ell,t}=0$. \n For $\\ell \\in G$ and $t \\in I \\cup {\\mathcal{C}}$ such that $\\bar{\\lambda}_{\\ell,t} =1$,  for all $t'\\in I \\cup {\\mathcal{C}} \\setminus \\{t\\}$ it holds that $\\bar{\\phi}_{\\ell,t'}= \\bar{\\lambda}_{\\ell,t'} =0$ because constraint \\eqref{F4} in the $\\bar{x}$-polytope is tight for $\\bar{\\lambda}$. Thus, by the previous argument, we have $\\bar{\\gamma}_{\\ell,t'}=0$ for every $t'\\in I \\cup {\\mathcal{C}} \\setminus \\{t\\}$ in case that $\\bar{\\lambda}_{\\ell,t} =1$. Therefore, \n \n \n \\begin{equation}\n \t\\label{eq:gam_supp_first}\n-1 -\\bar{\\gamma}_{\\ell,t}=- \\sum_{t'\\in I \\cup {\\mathcal{C}} \\textnormal{ s.t } (\\ell,t') \\notin X} \\left(\\bar{\\phi}_{\\ell,t'}  +\\bar{\\gamma}_{\\ell,t'}\\right) \\leq -1.\n\\end{equation} \nThe equality is because $\\bar{\\lambda}$ and also $\\bar{\\phi}$ hold constraint \\eqref{F4} with equality.  The last inequality is due to $ \\bar{\\phi}+\\bar{\\gamma} \\in P$. \nSimilarly, as $\\bar{\\phi}-\\bar{\\gamma} \\in P$, we have\n\\begin{equation}\n\t \t\\label{eq:gam_supp_second}\n\t \t-1 +\\bar{\\gamma}_{\\ell,t}=- \\sum_{t'\\in I \\cup {\\mathcal{C}} \\textnormal{ s.t } (\\ell,t') \\notin  X} \\left(\\bar{\\phi}_{\\ell,t'}  -\\bar{\\gamma}_{\\ell,t'}\\right) \\leq -1. \n\\end{equation} \nBy~\\eqref{eq:gam_supp_first} and \\eqref{eq:gam_supp_second} we have $\\bar{\\gamma}_{\\ell,t}=0$. Overall, we have that $\\bar{m}$ satisfies \\eqref{m0}. \n\n\n\n\n\n\n\n\n\n\n\nFor any $\\ell\\in I \\setminus G$ it holds that $\\sum_{t \\in T} \\bar{m}_{\\ell,t} = 0 $ by \\eqref{m0}. For $\\ell\\in G$, since  $\\bar{\\phi}+\\bar{\\gamma}\\in P$ we have\n\\begin{equation}\n\t\\label{eq:t_sum_first}\n-1 -\\sum_{t \\in T} \\bar{\\gamma}_{\\ell,t} = \\sum_{t \\in T}-(\\bar{\\lambda}_{\\ell,t} +\\bar{\\gamma}_{\\ell,t}) =  \n \\sum_{t \\in T~\\textnormal{ s.t. }(\\ell,t)\\notin X}-(\\bar{\\phi}_{\\ell,t} +\\bar{\\gamma}_{\\ell,t}) \\leq -1,\n \\end{equation}\n\n where the first equality holds since $\\bar{\\lambda}$ satisfies with equality constraint \\eqref{F4} of the $\\bar{x}$-polytope and the second equality uses $\\bar{\\phi}_{\\ell,t} +\\bar{\\gamma}_{\\ell,t}=0$ for $(\\ell,t) \\in X$ because $\\phi+\\bar{\\gamma} \\in P$. Similarly, since $\\phi-\\bar{\\gamma}\\in P$, we have \n \\begin{equation}\n \t\t\\label{eq:t_sum_second}\n -1 +\\sum_{t \\in T} \\bar{\\gamma}_{\\ell,t} = \\sum_{t \\in T}-(\\bar{\\lambda}_{\\ell,t} -\\bar{\\gamma}_{\\ell,t}) =  \n \\sum_{t \\in T~\\textnormal{ s.t. }(\\ell,t)\\notin X}-(\\bar{\\phi}_{\\ell,t} -\\bar{\\gamma}_{\\ell,t}) \\leq -1.\n \\end{equation}\n\n\nBy \\eqref{eq:t_sum_first} and \\eqref{eq:t_sum_second} we have \n  $\\sum_{t \\in T} \\bar{m}_{\\ell,t}=\\sum_{t \\in T} \\bar{\\gamma}_{\\ell,t}=0$. Thus, $\\bar{m}$ satisfies \\eqref{ml}. Finally, let $t \\in T$ such that $\\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,t} = L_{t,G}$ . As before,\n  \n  \\begin{equation}\n  \t\\label{eq:gam_group_sum_first}\n  \tL_{t,G} +\\sum_{\\ell\\in G} \\bar{\\gamma}_{\\ell,t}= \\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,t}+\\sum_{\\ell\\in G} \\bar{\\gamma}_{\\ell,t} =  \\sum_{\\ell\\in G} \\left(\\bar{\\phi}_{\\ell,t} +\\bar{\\gamma}_{\\ell,t}\\right)\n  \t= \\sum_{\\ell\\in G \\textnormal{ s.t. } (\\ell,t)\\notin X} \\left(\\bar{\\phi}_{\\ell,t} +\\bar{\\gamma}_{\\ell,t}\\right)  \\leq L_{t,G}, \n  \t\\end{equation}\n  where the inequality follows from $\\bar{\\phi}+\\bar{\\gamma} \\in P$. Using a similar argument,\n  \n   \\begin{equation}\n   \t\t\\label{eq:gam_group_sum_second}\n  \tL_{t,G} -\\sum_{\\ell\\in G} \\bar{\\gamma}_{\\ell,t}= \\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,t}-\\sum_{\\ell\\in G} \\bar{\\gamma}_{\\ell,t} =  \\sum_{\\ell\\in G} \\left(\\bar{\\phi}_{\\ell,t} -\\bar{\\gamma}_{\\ell,t}\\right)\n  \t= \\sum_{\\ell\\in G \\textnormal{ s.t. } (\\ell,t)\\notin X} \\left(\\bar{\\phi}_{\\ell,t} -\\bar{\\gamma}_{\\ell,t}\\right)  \\leq L_{t,G}.\n  \\end{equation}\n  \nBy \\eqref{eq:gam_group_sum_first} and \\eqref{eq:gam_group_sum_second}, we have $\\sum_{\\ell\\in G} \\bar{m}_{\\ell,t} = \\sum_{\\ell\\in G} \\bar{\\gamma}_{\\ell,t} =0$. Thus, $\\bar{m}$ satisfies \\eqref{mT}. Overall, we show that $\\bar{m} \\neq 0$ is a movement of $G$.  \n\\qed\n\n\n\\begin{claim}\n\t\\label{lem:FewFractionalGroups}\n\tThere are at most $|T|$ fractional groups.\n\\end{claim}\n\n\n\\begin{proof}\n\tAssume towards a contradiction that there are $|T|+1$ fractional groups. Denote these groups by $G_1, \\ldots, G_{|T|+1} \\in {\\mathcal{G}}$. By Lemma~\\ref{lem:movement}, for all $j \\in [|T|+1]$ it holds that $G_j$ has a movement $\\bar{m}^j \\neq 0$. Consider the following set of equalities over $a_1, \\ldots, a_{|T|+1}$:\n\t\\begin{equation}\n\t\t\\label{eq:homegeneous} \\forall t \\in T: ~~~~ \\sum_{j=1}^{|T|+1} a_j \\sum_{\\ell \\in I} \\bar{m}^j_{\\ell,t} \\cdot s(\\ell) = 0.\n\t\\end{equation}\n\n\tThese are $|T|$ homogeneous linear equalities in $|T|+1$ variables. Thus, there exist $a_1, \\ldots, a_{|T|+1} \\in \\mathbb{R}$, not all zeros, for which  \\eqref{eq:homegeneous} holds.\tDefine \n\t\n\t$$K_1 = \\min_{(\\ell,t)\\in I \\times  T}\\min \\left\\{ \\bar{\\lambda}_{\\ell,t}, 1-\\bar{\\lambda}_{\\ell,t} ~\\middle|~0<\\bar{\\lambda}_{\\ell,t}<1\\right\\}, $$ \n\t$$\tK_2= \\min \\left\\{\n\tL_{t,G_j} - \\sum_{\\ell \\in G_j} \\bar{\\lambda}_{\\ell,t}~\\middle |~ j \\in [ |T|+1], t \\in T\\text{ s.t. } \\sum_{\\ell \\in G_j}\\bar{\\lambda}_{\\ell,t} < L_{t,G_j}  \\right\\},\n\t$$  \n\tand $K= \\min\\{K_1,K_2\\}$.  Additionally, define \n\t$$W=\\max\\left\\{ |\\bar{m}^{j}_{\\ell,t}|~|~1 \\leq j \\leq |T|+1,~ (\\ell,t)\\in I \\times  T\\right\\}.$$\n\t\n\t\n\t\n\tObserve that $W,K>0$. Using a scaling argument, we may assume that  for any $1\\leq j\\leq |T|+1$ it holds that $$a_j \\leq \\frac{K}{|I| \\cdot W}.$$ Define \n\t\n\t$$\\bar{\\phi} = \\bar{\\lambda} + \\sum_{j=1}^{|T|+1} a_j \\cdot \\bar{m}^j.$$ \n\t\n\t\n\tIn the following we show that $\\bar{\\phi}$ is in the $\\bar{x}$-polytope. For every $\\ell\\in I\\setminus (G_1\\cup \\ldots \\cup G_{|T|+1})$  and $t \\in T$ it holds that $\\bar{m}^{1}_{\\ell,t}= \\ldots = \\bar{m}^{|T|+1}_{\\ell,t} = 0$ due to \\eqref{m0}. Also, for all $\\ell \\in I$ and $t \\in I \\cup {\\mathcal{C}} \\setminus T$, it holds that $\\bar{m}^{1}_{\\ell,t}= \\ldots = \\bar{m}^{|T|+1}_{\\ell,t} = 0$ due to \\eqref{m0} as $\\bar{\\lambda}_{\\ell,t} = 0$.  Thus,\n\t$\\bar{\\phi}_{\\ell,t} = \\bar{\\lambda}_{\\ell,t}\\in  [0,1]$ in this case. Also, for $\\ell \\in I$ and $t \\in T$ such that $ \\bar{\\lambda}_{\\ell,t}\\in  \\{0,1\\}$ it holds that $\\bar{\\phi}_{\\ell,t} = \\bar{\\lambda}_{\\ell,t} = 0$ by \\eqref{m0}. Finally, for $j \\in [|T|+1]$, $\\ell \\in G_j$, and $t \\in T$ such that $ \\bar{\\lambda}_{\\ell,t} \\in (0,1)$, by \\eqref{m0} it holds that $\\bar{\\phi}_{\\ell,t} =  \\bar{\\lambda}_{\\ell,t} + a_j \\cdot \\bar{m}^j_{\\ell,t}$. Following the definitions  of $K$ and $W$, we have \n\t$$ 0\\leq K -W \\cdot \\frac{K}{|I|\\cdot W} \\leq \\bar{\\lambda}_{\\ell,t} + a_j \\cdot \\bar{m}^j_{\\ell,t} \\leq 1-K + W \\cdot \\frac{K}{|I|\\cdot W} \\leq 1.$$\n\tThus, $\\bar{\\phi} \\in [0,1]^{I \\times  (I \\cup {\\mathcal{C}})}$. \t\n\t\n\tFor every $(\\ell,t)\\in I \\times T$ such that $\\ell \\notin \\textsf{fit}(t)$, it holds that $\\bar{\\lambda}_{\\ell,t}=0$ by \\eqref{F1} since $ \\bar{\\lambda}$ is in the $\\bar{x}$-polytope. Thus by \\eqref{m0} we have $\\bar{m}^{j}_{\\ell,t} =0$ for all $1\\leq j \\leq |T|+1$. Therefore, $\\bar{\\phi}_{\\ell,t} =0$. We conclude that constraint \\eqref{F1} is satisfied for $\\bar{\\phi}$. \n\t\n\tFor every $t \\in {\\mathcal{C}}$ it holds that \n\t\n\t$$\n\t\\sum_{\\ell \\in I} \\bar{\\phi}_{\\ell,t} \\cdot s(\\ell) = \n\t\\sum_{\\ell\\in I} \\bar{\\lambda}_{\\ell,t}\\cdot s(\\ell)+ \\sum_{j=1}^{|T|+1} a_j \\sum_{\\ell\\in I}\n\t\\bar{m}^j_{\\ell,t} s(\\ell)= \\sum_{\\ell\\in I} \\bar{\\lambda}_{\\ell,t}\\cdot s(\\ell) \\leq (1-s(t)) \\bar{x}_t .$$\n\t\n\twhere the second equality is by \\eqref{eq:homegeneous}, and the inequality is due to $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope; hence, the inequality follows by \\eqref{F2}. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\nLet $G \\in \\mathcal{G}$ and $C \\in {\\mathcal{C}}$. If for all $1 \\leq j \\leq |T|+1$ it holds that $G \\neq G_j$,  then $\\bar{\\phi}_{\\ell,C} = \\bar{\\lambda}_{\\ell,C}$ for every $\\ell\\in G$ by \\eqref{m0}. Thus,  \n\t$\\sum_{\\ell\\in G } \\bar{\\phi}_{\\ell,C} = \\sum_{\\ell \\in G } \\bar{\\lambda}_{\\ell,C}\\leq \\bar{x}_C \\cdot (k(G)-|G \\cap C|)$, as $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope and thus satisfies constraint \\eqref{F5}. Otherwise, there is $1\\leq j \\leq |T|+1$ such that $G = G_j$.  Observe that for every $\\ell\\in G$ it holds that $\\bar{\\phi}_{\\ell,C} = \\bar{\\lambda}_{\\ell,C}+a_j \\bar{m}^j_{\\ell,C}$ by \\eqref{m0}; \n\tconsider the following cases.\n\t\n\t\\begin{itemize}\n\t\t\\item $\\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,C} <L_{C,G}$. Using the definitions of $K$ and $W$, we have\n\t\t\t\\begin{equation*}\n\t\t    \\label{eq:LGM1}\n\t\t    \t\\begin{aligned}\n\t\t       \\sum_{\\ell\\in G} \\bar{\\phi}_{\\ell,C}= \n\t\t    \\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,C} +a_j \\sum_{\\ell\\in G}  \\bar{m}^j_{\\ell,C}\\leq \n\t\t    L_{C,G} -K +  \\frac{K}{|I|\\cdot W  } \\cdot |I| \\cdot W = L_{C,G} = \\bar{x}_C \\cdot (k(G)-|G \\cap C|).\n\t\t    \\end{aligned}\n\t\t\\end{equation*}\n\t\n\t\t\\item $\\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,C} =L_{C,G}$. By \\eqref{mT} we have\n\t\t\n\t\n\t\t\n\t\t\t\\begin{equation}\n\t\t    \\label{eq:LGM2}\n\t\t    \\sum_{\\ell\\in G} \\bar{\\phi}_{\\ell,C}= \n\t\\sum_{\\ell\\in G} \\bar{\\lambda}_{\\ell,C} + a_j\\sum_{\\ell \\in G}  \\bar{m}^j_{\\ell,C}=L_{C,G}  = \\bar{x}_C  \\cdot (k(G)-|G \\cap C|).\n\t\t\\end{equation}\n\t\t\n\n\t\\end{itemize}\n\t\n\tWe showed  that $\\sum_{\\ell \\in G} \\bar{\\phi}_{\\ell,C} \\leq \\bar{x}_C$ in all cases. We conclude that constraint \\eqref{F5} is satisfied for $\\bar{\\phi}$. \n\t\n\tLet $j \\in I$. We split into two cases, as follows. \n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item For all $1 \\leq i \\leq |T|+1$ it holds that $ \\textsf{group}(j) \\neq G_i$. Then, by \\eqref{fitl},  for all $\\ell \\notin \\textsf{group}(j)$ it holds that $\\bar{\\lambda}_{\\ell,j} = 0$; thus, it holds that $\\bar{\\phi}_{\\ell,j} = \\bar{\\lambda}_{\\ell,j}$ for every $\\ell \\in I$ by \\eqref{m0}. By the above,  $\\sum_{\\ell\\in I } \\bar{\\phi}_{\\ell,j} = \\sum_{\\ell \\in I } \\bar{\\lambda}_{\\ell,j} \\leq \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{x}_C$, as $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope and thus satisfies constraint \\eqref{F3}. \n\t\t\n\t\t\n\n\t\n\t\n\t\t\t\\item There is $1\\leq i \\leq |T|+1$ such that $j \\in G_i$. Observe that for every $\\ell\\in I$ it holds that $\\bar{\\phi}_{\\ell,j} = \\bar{\\lambda}_{\\ell,j}+a_i \\bar{m}^i_{\\ell,j}$ by \\eqref{m0} since for items $\\ell \\notin G_i$ it holds that $\\bar{\\lambda}_{\\ell,j} = 0$; \n\t\t\tconsider the following cases. \n\t\t\t\n\t\t\t\\begin{itemize}\n\t\t\t\\item $\\sum_{\\ell\\in G_i} \\bar{\\lambda}_{\\ell,j} <L_{j,G_i}$. Using the definitions of $K$ and $W$, we have\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\\sum_{\\ell\\in I} \\bar{\\phi}_{\\ell,j} ={} & \t\\sum_{\\ell\\in I} \\bar{\\lambda}_{\\ell,j} +a_i \\sum_{\\ell\\in I}  \\bar{m}^i_{\\ell,j} = \n\t\t\t\t\t\\sum_{\\ell\\in G_i} \\bar{\\lambda}_{\\ell,j} +a_i \\sum_{\\ell\\in G_i}  \\bar{m}^i_{\\ell,j}\\leq \n\t\t\t\t\tL_{j,G_i} -K +  \\frac{K}{|I|\\cdot W  } \\cdot |I| \\cdot W \\\\ ={} & L_{j,G_i} = \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{x}_C.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\t\t\n\t\t\t\\item $\\sum_{\\ell\\in G_i} \\bar{\\lambda}_{\\ell,j} =L_{j,G_i}$. By \\eqref{mT} we have\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:LGM2}\n\t\t\t\t\\sum_{\\ell\\in I} \\bar{\\phi}_{\\ell,j} = \t\\sum_{\\ell\\in G_i} \\bar{\\phi}_{\\ell,j} = \n\t\t\t\t\\sum_{\\ell\\in G_i} \\bar{\\lambda}_{\\ell,j} + a_i \\sum_{\\ell \\in G}  \\bar{m}^i_{\\ell,j}=L_{j,G} = \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{x}_C.\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\t\n\t\t\\end{itemize}\n\t\n\t\n\t\\end{enumerate}\n\t\n  \n\n\n\t\n\t\n\tWe showed  that $\\sum_{\\ell \\in I} \\bar{\\phi}_{\\ell,j} \\leq L_{j,G}$ in all cases. We conclude that constraint \\eqref{F3} is satisfied for $\\bar{\\phi}$.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\tFor every $\\ell \\in I$, we have\n\t$$\\sum_{t \\in T} \\bar{\\phi}_{\\ell,t} = \n\t\\sum_{t \\in T} \\bar{\\lambda}_{\\ell,t} + \\sum_{j=1}^{|T|+1}\n\ta_j \\sum_{t \\in T} \\bar{m}^j_{\\ell,t} = \n\t\\sum_{t \\in T}  \\bar{\\lambda}_{\\ell,t} = \\sum_{t \\in I \\cup {\\mathcal{C}}}  \\bar{\\lambda}_{\\ell,t}\\geq 1,$$\n\twhere the second equality is by \\eqref{ml}, and the last inequality follows since $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope; thus, it satisfies constraint \\eqref{F4}. Overall, we have that $\\bar{\\phi}$ is in the $\\bar{x}$-polytope.\n\t\n\tWe can also define $\\bar{\\phi}' =  \\bar{\\lambda}-\\sum_{j=1}^{|T|+1} a_j\\cdot \\bar{m}^j$. By a symmetric argument we can show that $\\bar{\\phi}'$ is in the $\\bar{x}$-polytope as well. It also holds that $\\bar{\\phi},\\bar{\\phi}' \\neq \\bar{\\lambda}$ as there is $1\\leq j^* \\leq |T|+1$ such that $a_{j^*}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          \\neq 0$; also, there are $\\ell^* \\in G_{j^*}$  and $t^*\\in T$ such that $\\bar{m}^{j^*}_{\\ell^*,t^*}\\neq 0$. Thus, $\\bar{\\phi}_{\\ell^*,j^*}=  \\bar{\\lambda}_{\\ell^*,t^*} +a_{j^*} \\bar{m}^{j^*}_{\\ell^*,j^*}\\neq  \\bar{\\lambda}_{\\ell^*,t^*}$ and $\\bar{\\phi}'_{\\ell^*,j^*}=  \\bar{\\lambda}_{\\ell^*,t^*} -a_{j^*} \\bar{m}^{j^*}_{\\ell^*,j^*}\\neq  \\bar{\\lambda}_{\\ell^*,t^*}$. \n\tFurthermore, $ \\bar{\\lambda} =\\frac{1}{2} \\cdot \\bar{\\phi}+\\frac{1}{2}\\cdot \\bar{\\phi}'$, and we conclude that $ \\bar{\\lambda}$ is not a vertex of the $\\bar{x}$-polytope. Contradiction. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\n\t\\label{lem:slot-types}\n\nIf a group $G \\in {\\mathcal{G}}$ is fractional, then $$|\\{\\bar{\\lambda}_{\\ell,t} \\in (0,1) ~|~ \\ell \\in G, t \\in T\\}| \\leq 2|T|.$$ \n\n\\end{lemma}\n\n\\begin{proof}\n\tLet $X = \\{(\\ell,t) \\in G \\times T~|~ \\ell \\notin \\textsf{fit}(t)\\}$. Define $R = G \\times T \\setminus X$ as the set of all pairs of items where the first fits in the place of the second. \n\tLet  $Q \\subseteq \\mathbb{R}^{R}$ be the set (polytope) of  all the vectors $\\bar{\\rho} \\in \\mathbb{R}^{R}$ which satisfy the following constraints:\n\t\n\t\n\t\n\t\n\t\\begin{align}\n\t\t\t\\displaystyle \t\\bar{\\rho}_{\\ell,t} \\geq 0   ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~\t~~~~~~~~~~~\\forall (\\ell,t) \\in R~~~~~~~\t\\label{eq:rho1}\\\\\n\t\t\t\\rule{0pt}{1.8em}\n\t\t\t\\displaystyle \t\\sum_{t \\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\rho}_{\\ell,t} \\geq 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\t~~~~~~\\forall \\ell \\in  G ~~~~~~~~~~~~\t\\label{eq:rhoB}\\\\\n\t\t\t\\rule{0pt}{1.8em}\n\t\t\t\\displaystyle \t\\sum_{\\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\rho}_{\\ell,C} \\cdot s(\\ell)+\\sum_{\\ell \\in I \\setminus G} \\bar{\\lambda}_{\\ell,C} \\cdot s(\\ell) \\leq (1-s(C)) \\bar{x}_C ~~~~~~~~~~~~~\t~~~~~~\\forall C \\in {\\mathcal{C}} \\cap T ~~~~~~\t\\label{eq:rhoD}\\\\\n\t\t\t\\rule{0pt}{1.8em}\n\t\t\t\\displaystyle \t\\sum_{\\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\rho}_{\\ell,t} \\leq L_{t,G} ~~~~~~~~~~~~~~~~~~~~~~~~~~\t~~~~~~\\forall t \\in T ~~~~~~~~~~~~\t\\label{eq:rhoS}\n\t\t\\end{align}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\tNow, we define $\\bar{\\phi} \\in  \\mathbb{R}^{R}$ by $\\bar{\\phi}_{\\ell,t} = \\bar{\\lambda}_{\\ell,t}$ for all $(\\ell,t)\\in R$. As $\\bar{\\lambda}$ belongs to the $\\bar{x}$-polytope, \n\t\tit easily follows that $\\bar{\\phi} \\in Q$ since $\\bar{\\lambda}_{\\ell,t}=0$ for all $(\\ell,t) \\in (I \\times I \\cup {\\mathcal{C}}) \\setminus R$ by \\eqref{F1} (i.e., where $\\ell \\notin \\textsf{fit}(t)$). \n\t\t\n\t\t\n\t\t\n\t\t\\begin{claim}\n\t\t\t$\\bar{\\phi}$ is  a vertex of $Q$.\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}\n\t\t\tAssume towards contradiction that $\\bar{\\phi}$ is not a vertex of $Q$. Thus, there is a vector $\\bar{\\gamma} \\in \\mathbb{R}^{R}$, $\\bar{\\gamma} \\neq 0$  such that $\\bar{\\phi}+\\bar{\\gamma}, \\bar{\\phi}-\\bar{\\gamma} \\in Q$.  Thus, by \\eqref{eq:rhoB}, for all $\\ell\\in G$ we get that  \n\t\t\t\n\t\t\t$$\n\t\t  1 \\leq \t\\sum_{ t\\in T \\text{ s.t. } (\\ell,t) \\in R} (\\bar{\\phi}_{\\ell,t}+\\bar{\\gamma}_{\\ell,t}) = \t\\sum_{t\\in T\\text{ s.t. } (\\ell,t) \\in R} \\bar{\\phi}_{\\ell,t} +\t\\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} = 1+\\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} \n\t\t\t$$\n\t\t\t\n\t\t\tThe last equality is because $\\bar{\\lambda}$ satisfies constraint \\eqref{F4} of the $\\bar{x}$-polytope with equality and by the definition of $\\phi$. In addition,  \n\t\t\t$$\n\t\t1 \\leq\t\\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} (\\bar{\\phi}_{\\ell,t}-\\bar{\\gamma}_{\\ell,t}) = \\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\phi}_{\\ell,t} -\t\\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} = 1-\\sum_{t\\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} \n\t\t\t$$\n\t\t\t\tThe last equality is because $\\bar{\\lambda}$ satisfies constraint \\eqref{F4} of the $\\bar{x}$-polytope with equality and by the definition of $\\phi$. Hence, by the above it follows that \n\t\t\t\\begin{equation}\n\t\t\t\t\\sum_{t\\in T\\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t}=0.\n\t\t\t\t\\label{eq:gamzero}\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\t\n\t\n\t\t\t\n\t\t\t\n\t\t\tFurthermore, for all $t \\in T$  the following hold by \\eqref{eq:rhoS}:\n\t\t\t\n\t\t\t$$\n\t L_{t,G} \\geq \t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} (\\bar{\\phi}_{\\ell,t}+\\bar{\\gamma}_{\\ell,t})= \t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\phi}_{\\ell,t} +\t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t}\n\t\t\t$$\n\t\t\t\n\t\t\tand \n\t\t\t$$\n\tL_{t,G} \\geq \t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} (\\bar{\\phi}_{\\ell,t}-\\bar{\\gamma}_{\\ell,t}) = \t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\phi}_{\\ell,t}-\t\\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t}\n\t\t\t$$\n\t\t\t\n\t\t\t\n\t\t\tHence, by the above, because for all $(\\ell,t) \\in R$ it holds that $\\bar{\\lambda}_{\\ell,t} = \\bar{\\phi}_{\\ell,t}$, and that by \\eqref{F1} for all $(\\ell,t) \\notin R$ it holds that $\\bar{\\lambda}_{\\ell,t} = 0$,  for all $t \\in T$ it follows that:\n\t\t\t\\begin{equation}\n\t\t\t\\sum_{ \\ell \\in G} \\bar{\\lambda}_{\\ell,t}-L_{t,G}\t \\leq \\sum_{ \\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} \\leq L_{t,G}- \\sum_{ \\ell \\in G} \\bar{\\lambda}_{\\ell,t}.\n\t\t\t\t\\label{eq:TTT}\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\tDefine $\\bar{\\lambda}^+,\\bar{\\lambda}^- \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}$ by $\\bar{\\lambda}^+_{\\ell,t} = \\bar{\\lambda}^-_{\\ell,t} = \\bar{\\lambda}_{\\ell,t}$ for $(\\ell,t) \\in \\left( I \\times (I \\cup \\mathcal{C}) \\right) \\setminus R$, and $\\bar{\\lambda}^+_{\\ell,t} = \\bar{\\lambda}_{\\ell,t}+\\bar{\\gamma}_{\\ell,t},~\\bar{\\lambda}^-_{\\ell,t} = \\bar{\\lambda}_{\\ell,t}-\\bar{\\gamma}_{\\ell,t}$ for $(\\ell,t)\\in R$.  Since $\\bar{\\gamma}\\neq 0$ it follows that $\\bar{\\lambda}^+ \\neq \\bar{\\lambda}^-$. \n\t\t\t\n\t\t\tWe now show that $\\bar{\\lambda}^+, \\bar{\\lambda}^-$ are in the $\\bar{x}$-polytope. Let $\\ell \\in I , t \\in I \\cup \\mathcal{C}$ such that  $\\ell \\notin \\textsf{fit}(t)$. Therefore, $(\\ell,t) \\notin R$; thus, $\\bar{\\lambda}^+_{\\ell,t} =  \\bar{\\lambda}_{\\ell,t} = 0$ because $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope. We conclude that \\eqref{F1} is satisfied for $\\bar{\\lambda}^+$. Let $C \\in \\mathcal{C}$. Then, \n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\sum_{\\ell \\in I} \\bar{\\lambda}^+_{\\ell,C} \\cdot  s(\\ell) =  \\sum_{\\ell \\in G \\text{ s.t. } (\\ell,t) \\in R} (\\bar{\\phi}_{\\ell,C}+\\bar{\\gamma}_{\\ell,C}) \\cdot s(\\ell)+\\sum_{\\ell \\in I \\setminus G} \\bar{\\lambda}_{\\ell,C} \\cdot s(\\ell)  \\leq \\left(1-s(C)\\right) \\cdot \\bar{x}_C.\n\t\t\t\t\\label{eq:+1}\n\t\t\t\\end{equation}\n\t\t\t\n\t\tThe inequality is because $\\bar{\\phi}+\\bar{\\gamma} \\in Q$. By \\eqref{eq:+1} we conclude that \\eqref{F2} is satisfied for $\\bar{\\lambda}^+$.  Let $G' \\in \\mathcal{G}$ and $C \\in \\mathcal{C}$. Then, if $G' \\neq G$ it holds that:\n\t\t\t\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\sum_{\\ell \\in G'}  \\bar{\\lambda}^+_{\\ell,C} =   \\sum_{\\ell \\in G}  \\bar{\\lambda}_{\\ell,C} \\leq \\bar{x}_C  \\cdot (k(G)-|G \\cap C|) =  L_{C,G}. \n\t\t\t\t\\label{aaeq:+2}\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\tThe inequality is because $\\bar{\\lambda}$ is in the $\\bar{x}$-polytope and satisfies \\eqref{F5}. Otherwise, $G' = G$. \n\t\t\t\n\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\sum_{\\ell \\in G}  \\bar{\\lambda}^+_{\\ell,C} ={} & \t\\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C}+ \\sum_{\\ell \\in G} \\bar{\\gamma}_{\\ell,C} =   \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C}+ \\sum_{\\ell \\in G \\text{ s.t. } (\\ell,C) \\in R} \\bar{\\gamma}_{\\ell,C}\\\\  \\leq{} &\n\t\t\t\t\t\t\t \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C} + \\left(L_{C,G}-\\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,C} \\right) = L_{C,G}= \\bar{x}_C  \\cdot (k(G)-|G \\cap C|).\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\label{aaeq:+2a}\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\tThe inequality is by \\eqref{eq:TTT}.  By \\eqref{aaeq:+2} and \\eqref{aaeq:+2a} we conclude that \\eqref{F5} is satisfied for $\\bar{\\lambda}^+$. \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\t\n\t\t\tLet $j \\in I$. We split into two cases.\n\t\t\t\n\t\t\t\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $j \\notin G$. Then, by \\eqref{fitl}  and \\eqref{F1} for all $\\ell \\notin \\textsf{group}(j)$ it holds that $\\bar{\\lambda}^+_{\\ell,j} = 0$ and for all $\\ell' \\in I$ it holds that  $\\bar{\\lambda}^+_{\\ell',j} = \\bar{\\lambda}_{\\ell',j}$. Therefore, \t\t\\begin{equation*}\n\t\t\t\t\t\\sum_{\\ell \\in I}  \\bar{\\lambda}^+_{\\ell,j} =   \\sum_{\\ell \\in \\textsf{group}(j)}  \\bar{\\lambda}_{\\ell,C}^+ =  \\sum_{\\ell \\in \\textsf{group}(j)}  \\bar{\\lambda}_{\\ell,C} \\leq  \\sum_{C \\in {\\mathcal{C}}[j]}  \\bar{x}_C. \n\t\t\t\t\\end{equation*}\n\n\t\\item $j \\in G$. Then,  by \\eqref{fitl}  and \\eqref{F1} for all $\\ell \\notin G$ it holds that $\\bar{\\lambda}^+_{\\ell,j} = 0$. Therefore, \n\t\n\t\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\t\\sum_{\\ell \\in I}  \\bar{\\lambda}^+_{\\ell,j} ={} & \t\\sum_{\\ell \\in G}  \\bar{\\lambda}^+_{\\ell,j} =   \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,j}+ \\sum_{\\ell \\in G \\text{ s.t. } (\\ell,j) \\in R} \\bar{\\gamma}_{\\ell,j} \\\\  \\leq{} &\n\t\t \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,j}+  \\left(L_{j,G}- \\sum_{\\ell \\in G} \\bar{\\lambda}_{\\ell,j} \\right) = L_{j,G} =  \\sum_{C \\in {\\mathcal{C}}[j]} \\bar{x}_C. \n\t\t\\end{aligned}\n\t\\end{equation*} The inequality is by \\eqref{eq:TTT}. \n\t\n\n\t\n\t\t\t\\end{enumerate}\n\t\t\t\tBy the above, we conclude that \\eqref{F3} is satisfied for $\\bar{\\lambda}^+$. \n\t\t\t\n\t\t\t\n\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\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\t\n\t\t\tLet $\\ell \\in I$. We split into two cases.\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $\\ell \\notin G$. \n\t\t\t\t$$\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\lambda}^+_{\\ell,t} = \\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\lambda}_{\\ell,t} \\geq 1.$$ The last inequality is because $ \\bar{\\lambda}$ satisfies constraint \\eqref{F4} of the $\\bar{x}$-polytope. \n\t\t\t\t\n\t\t\t\t\\item $\\ell \\in G$.  $$\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\lambda}^+_{\\ell,t} \\geq \\sum_{t \\in T} \\left(\\bar{\\lambda}_{\\ell,t}+ \\bar{\\gamma}_{\\ell,t}\\right) = \\sum_{t \\in T} \\bar{\\lambda}_{\\ell,t}+\\sum_{t \\in T \\text{ s.t. } (\\ell,t) \\in R} \\bar{\\gamma}_{\\ell,t} = \\sum_{t \\in T} \\bar{\\lambda}_{\\ell,t} \\geq 1.$$ The second equality is by \\eqref{eq:gamzero}. \n\t\t\t\t\n\t\t\t\\end{enumerate} By the above,  constraint \\eqref{F4} is satisfied for $\\bar{\\lambda}^+$. Overall, we conclude that $\\bar{\\lambda}^+$ is in the $\\bar{x}$-polytope. \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\tUsing a symmetric argument it follows that $\\bar{\\lambda}^-$ is also in the $\\bar{x}$-polytope.  Since $\\bar{\\lambda}= \\frac{\\bar{\\lambda}^+ + \\bar{\\lambda}^-}{2}$, it follows that $\\bar{\\lambda}$ is not a vertex of the $\\bar{x}$-polytope. Contradiction. Thus, $\\bar{\\phi}$ is a vertex of $Q$. \n\t\t\\end{proof}\n\t\t\n\t\t\n\t\tLet $F= \\{\\ell \\in G~|~\\exists t \\in T \\text{ s.t. } \\bar{\\phi}_{\\ell,t} \\in (0,1)\\}$. By the definition of $\\bar{\\phi}$, it suffices to show that $|F| \\leq 2|T|$.\n\t\tFor any $\\ell\\in I$ define $\\delta_{\\ell} = |\\{t \\in T~|~(\\ell,t) \\in R\\}|$ to be all of the corresponding pairs of $\\ell$ in $R$. It follows that $|R| = \\sum_{\\ell\\in I} \\delta_{\\ell}$. For $\\ell \\in F$, up to $\\delta_{\\ell}-1$ of the corresponding inequalities of $\\ell$ in \\eqref{eq:rho1} and \\eqref{eq:rhoB} are tight: to satisfy \\eqref{eq:rhoB} for $\\ell$ at least two inequalities corresponding to $\\ell$ in \\eqref{eq:rho1} are not tight. In addition, for $\\ell \\in G \\setminus F$, up to $\\delta_{\\ell}$ corresponding constraints in  \\eqref{eq:rho1} and \\eqref{eq:rhoB} for $\\ell$ are tight: to satisfy \\eqref{eq:rhoB} for $\\ell$ at least one inequality corresponding to $\\ell$ in \\eqref{eq:rho1} is not tight. Moreover, there are at most $2|T|$ constraints in \\eqref{eq:rhoD} and \\eqref{eq:rhoS} that can be tight. Hence, the number of tight constraints is at most\n\t\t\n\t\t\n\t\t\\begin{equation}\n\t\t\t\\label{eq:tight}\n\t\t\t\\sum_{\\ell\\in F} (\\delta_{\\ell} -1) + \\sum_{\\ell \\in G \\setminus F} \\delta_\\ell +2|T| \\leq |R|-|F|+2|T|.\n\t\t\\end{equation}\n\t\t\n\t\t\n\t\tAs $\\bar{\\phi}$ is a vertex of $Q$, there are at least $|R|$ tight inequalities. Thus, $|F|\\leq 2|T|$. Note that for all $t \\in I \\cup {\\mathcal{C}} \\setminus T$ and $\\ell \\in I$ it holds that $\\bar{\\lambda}_{\\ell,t} = 0$ by the constraints of the $\\bar{x}$-polytope; thus, such entries cannot be fractional, and the number of fractional entries corresponding to  items in $G$ is at most $2|T|$. \n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\noindent{\\bf Proof of Lemma~\\ref{O(1)}:}\n\t\n\tNote that for all $t \\in I \\cup {\\mathcal{C}} \\setminus T$ and $\\ell \\in I$ it holds that $\\bar{\\lambda}_{\\ell,t} = 0$ by the constraints of the $\\bar{x}$-polytope; thus, such entries cannot be non-integral. In addition, by Lemma~\\ref{lem:FewFractionalGroups} and Lemma~\\ref{lem:slot-types} the number of non-integral entries is at most $2|T| \\cdot |T|$. Therefore, $$2|T| \\cdot |T| \\leq 2\\left(|\\textsf{supp}(\\bar{x})|+k\\cdot |\\textsf{supp}(\\bar{x})|\\right)^2 \\leq 2\\left(2k \\cdot |\\textsf{supp}(\\bar{x})|\\right)^2  = 8 k^2 |\\textsf{supp}(\\bar{x})|^2.$$ The first inequality is because $|I \\cap T|$ is upper bounded by the size of $\\textsf{supp}(\\bar{x})$ times the number of items in each configuration in $\\textsf{supp}(\\bar{x})$. \\qed\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nLet ${\\mathcal{A}}$ be an algorithm  that accepts as input ${\\varepsilon} > 0$. \nWe say the running time of ${\\mathcal{A}}$ is\n$\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ if there is a two-variable polynomial $p(x,y)$ such that $p(|{\\mathcal{I}}|,\\frac{1}{{\\varepsilon}})$ is an upper bound on the running time of ${\\mathcal{A}}({\\mathcal{I}}, {\\varepsilon})$. To allow a simpler presentation of the results\nwe assume throughout the paper that  the set of items $I$ is $\\{1,2,\\ldots,n\\}$, and the items are sorted in non-increasing order by sizes, i.e., $s(1)\\geq s(2)\\geq \\ldots s(n)$. \n\n\nOur scheme initially transforms a given BPP instance into one having a structure which depends on the parameter \n${\\varepsilon} >0$.  In this new instance, only a small number of groups may contain relatively large items. \nLet $K:(0,0.1) \\to \\mathbb{R}$, where\n  $K({\\varepsilon}) =  {\\varepsilon}^{-{{\\varepsilon}^{-2}}}$ for all ${\\varepsilon} \\in (0,0.1)$. We use\n  $K$ for defining a structured instance.  \n\\begin{definition}\n\t\\label{def:structuring}\n\tGiven a BPP instance ${\\mathcal{I}} = \\left(I, \\mathcal{G},s,k\\right)$ and ${\\varepsilon}\\in (0,0.1)$, we say that ${\\mathcal{I}}$\n\tis ${\\varepsilon}$-{\\em structured} if there is ${\\mathcal{B}} \\subseteq {\\mathcal{G}}$ such that  $|{\\mathcal{B}}| \\leq K({\\varepsilon})$ and for all $G \\in {\\mathcal{G}} \\setminus {\\mathcal{B}}$ and $\\ell \\in G$ it holds that~$s(\\ell) < {\\varepsilon}^2$. \n\\end{definition}\n\n\nFollowing the structuring step, our \nscheme proceeds to solve BPP on the structured instance. As a final step,\nthe packing found for the structured instance is transformed into \na packing of the original instance. This is formalized in the next result.\n\\begin{lemma}\n\t\\label{lem:reductionReconstruction}\n\tThere is a pair of algorithms, $\\textsf{Reduce}$ and $\\textsf{Reconstruct}$, which satisfy the following.\n\t\\begin{enumerate}\n\t\t\n\t\t\\item \n\t\tGiven a BPP instance ${\\mathcal{J}}$ and ${\\varepsilon} >0$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$,\n\t\talgorithm $\\textsf{Reduce}$ returns in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ an ${\\varepsilon}$-structured BPP instance ${\\mathcal{I}}$, where $\\textnormal{OPT}({\\mathcal{I}}) \\leq \\textnormal{OPT}({\\mathcal{J}})$.\\label{condition:reduction1}\n\t\t\n\t\t\\item \n\t\tGiven a BPP instance ${\\mathcal{J}}$, ${\\varepsilon} >0$ such that ${\\varepsilon}^{-1}\\in \\mathbb{N}$, and a packing $A'$ for ${\\mathcal{I}}= \\textsf{Reduce}({\\mathcal{J}}, {\\varepsilon})$ of size $m'$,\n\t\talgorithm $\\textsf{Reconstruct}$ returns in time $\\textnormal{poly} (| {\\mathcal{I}} |, \\frac{1}{{\\varepsilon}})$ a packing $A$ for the instance ${\\mathcal{J}}$ of size $m$, where $m \\leq m'+13 {\\varepsilon}\\cdot \\textnormal{OPT}({\\mathcal{J}})+1$.\\label{condition:reduction2} \n\t\\end{enumerate}\n\t\n\\end{lemma}\n\nThe structured instance ${\\mathcal{I}}$ is constructed from $\\mathcal{J}$ by reassigning items of size at least ${\\varepsilon}^2$ from all but a few groups to a new\ngroup. The reconstruction algorithm modifies  the packing of ${\\mathcal{I}}$ such that each bin in the solution is a configuration of $\\mathcal{J}$. The proof of Lemma~\\ref{lem:reductionReconstruction} (given in Section~\\ref{sec:reduction}) is \ninspired by ideas of~\\cite{DW17,Jansen_et_al:2019,DKS21}.\nBy Lemma~\\ref{lem:reductionReconstruction}, an AFTPAS for ${\\varepsilon}$-structured BPP instances  implies an AFTPAS for general BPP instances.\n\n\n\\section{Omitted Proofs of Section~\\ref{sec:reduction}}\n\\label{sec:reductionProofs}\n\nIn this section we refer to a BPP instance ${\\mathcal{J}}$ fixed in Section~\\ref{sec:reduction} and a given parameter ${\\varepsilon} \\in (0,0.1]$; in addition, we use ${\\mathcal{I}} = \\textsf{Reduce}({\\varepsilon},{\\mathcal{J}})$ as in Section~\\ref{sec:reduction}. \n\\subsection{Proofs From the Reduction}\n\n\n\\begin{lemma}\n\t\\label{lem:k_val}\n\tFor every minimal pivot $w\\in \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}$ of $\\mathcal{J}$ it holds that $s(I_w) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}(\\mathcal{J})$. \n\\end{lemma}\n\n\\begin{proof}\n\t\n\n\n\n\t\n\n\n\t\n\t\n\t$$ s(I_w) \\leq {\\varepsilon} \\cdot s\\left( \\bigcup_{i \\in \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}} I_i \\right) \\leq {\\varepsilon} \\cdot s(I) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}}).$$\n\t\n\t\n\tThe first inequality is because $I_2, \\ldots, I_{{\\varepsilon}^{-1}+1}$ are ${\\varepsilon}^{-1}$ disjoint subsets of items and by \\eqref{eq:k} it holds that $I_w$ is an interval of minimum total size. The last inequality is because each bin may contain total size at most $1$; thus, the total size of items is a lower bound to the optimum. \n\n\\end{proof}\n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:few_large_groups}:} \n\t\n\tIf $\\textnormal{OPT}({\\mathcal{J}}) = 0$ there are no items and the claim trivially follows. Otherwise, let $${\\mathcal{B}} = \\big\\{G \\in {\\mathcal{G}}~\\big|~|G \\cap  (H_w \\cup I_w)| \\geq {\\varepsilon}^{2w+4}\\big\\}$$ be the subset of groups that contain at least ${\\varepsilon}^{2w+4}$ $w$-heavy and $w$-medium items. The following holds for all $G \\in {\\mathcal{B}}$:\n\t\n\t\\begin{equation}\n\t\t\\label{eq:feww}\n\t\t\\begin{aligned}\n\t\t\ts(G) \\geq{} & s\\left(G \\cap (H_w \\cup I_w)\\right) = \\sum_{\\ell \\in G \\cap (H_w \\cup I_w)} s(\\ell) \\geq \\sum_{\\ell \\in G \\cap (H_w \\cup I_w)} {\\varepsilon}^{w+1} =  {\\varepsilon}^{w+1} \\cdot |G \\cap (H_w \\cup I_w)|  \\\\ \\geq{} &  {\\varepsilon}^{w+1} \\cdot {\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}) = {\\varepsilon}^{3w+5} \\cdot \\textnormal{OPT}({\\mathcal{J}}). \n\t\t\\end{aligned}\n\t\\end{equation} The second inequality is because the size of a $w$-medium item is at least ${\\varepsilon}^{w+1}$. The third inequality is because $G \\in {\\mathcal{B}}$. Therefore, \n\t\n\t$$|{\\mathcal{B}}| = \\sum_{G \\in {\\mathcal{B}}} 1 \\leq   \\sum_{G \\in {\\mathcal{B}}} \\frac{s(G)}{{\\varepsilon}^{3w+5} \\cdot \\textnormal{OPT}({\\mathcal{J}})} \\leq  \\frac{s(I)}{{\\varepsilon}^{3w+5} \\cdot \\textnormal{OPT}({\\mathcal{J}})} \\leq  \\frac{\\textnormal{OPT}({\\mathcal{J}})}{{\\varepsilon}^{3w+5} \\cdot \\textnormal{OPT}({\\mathcal{J}})} = {\\varepsilon}^{-3w-5}.$$ The first inequality is by \\eqref{eq:feww}. \\qed\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\tThe next two lemmas combined are essentially the proof of Condition~1 in Lemma~\\ref{lem:reductionReconstruction}. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{lemma}\n\t\t\\label{lem:shiftingNotIncreaseOPT2}\n\t\tFor ${\\mathcal{I}} = \\textsf{Reduce}({\\varepsilon},{\\mathcal{J}})$ it holds that $OPT(\\mathcal{I}) \\leq OPT({\\cal J})$.\n\t\\end{lemma}\n\t\n\t\n\t\\begin{proof}\n\t\tLet $A = (A_1, \\ldots, A_m)$ be a packing of ${\\mathcal{J}}$ and let $w \\in \\{2, \\ldots, {\\varepsilon}^{-1}+1\\}$ chosen in Step~\\ref{step:pivot} of Algorithm~\\ref{alg:reduction}. We show that $A$ is also a packing of ${\\mathcal{I}}$ which concludes the proof. By the definition of packing, we prove the following. \n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item It holds that $A$ is a partition of $I$. Follows because $A$ is a packing of ${\\mathcal{J}}$ and because that by Step~\\ref{Ir} the sets of items in ${\\mathcal{I}},{\\mathcal{J}}$ are identical. \n\t\t\t\n\t\t\t\\item For all $i \\in [m]$ it holds that $s(A_i) \\leq 1$. By Step~\\ref{Ir} the set of items and the size functions in ${\\mathcal{I}},{\\mathcal{J}}$ are identical and the proof follows. \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\item For all $G \\in {\\mathcal{G}}_R$ and $i \\in [m]$ it holds that $|G \\cap A_i| \\leq k_R(G)$. We split into three cases by Step~\\ref{kr}. \n\t\t\t\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $G \\in {\\mathcal{G}}_L(w)$. Then, $|G \\cap A_i| \\leq k(G) = k_R(G)$ where the inequality is because $A$ is a packing of ${\\mathcal{J}}$ and the equality is by Step~\\ref{kr}. \n\t\t\t\t\n\t\t\t\t\\item $G = G' \\setminus H_w \\text{ s.t. } G' \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w)$. Then, \n\t\t\t\t\n\t\t\t\t$$|G \\cap A_i| \\leq |(G \\cup (G' \\cap H_w)) \\cap A_i| = |G' \\cap A_i| \\leq k(G') = k_R(G).$$  The second inequality is because $A$ is a packing of ${\\mathcal{J}}$. The last equality is by Step~\\ref{kr}. \n\t\t\t\t\n\t\t\t\t\\item $G = \\Gamma_w$. Then, $|G \\cap A_i| = |\\Gamma_w \\cap A_i| \\leq |\\Gamma_w| < |\\Gamma_w+1| = k_R(\\Gamma_w) = k_R(G)$ where the second equality is by Step~\\ref{kr}. \n\t\t\t\\end{enumerate}\n\t\t\\end{itemize}\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{lemma}\n\t\t\\label{lem:rounding12}\n\t\t${\\mathcal{I}}$ is ${\\varepsilon}$-structured. \n\t\\end{lemma}\n\t\n\t\n\t\n\t\n\t\\begin{proof}\n\t\tLet $w \\in \\{2, \\ldots, {\\varepsilon}^{-1}+1\\}$ chosen in Step~\\ref{step:pivot} of Algorithm~\\ref{alg:reduction}. Therefore, \n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\big|\\{G{} & \\in {\\mathcal{G}}_R~|~ \\exists \\ell \\in G \\text{ s.t. } s(\\ell) \\geq {\\varepsilon}^2\\}\\big| \\leq \\big|\\{G \\in {\\mathcal{G}}_R~|~ G \\cap H_w \\neq \\emptyset\\}\\big| \\leq |{\\mathcal{G}}_L(w)|+|\\{\\Gamma_w\\}|  \\\\ \\leq{} &   {\\varepsilon}^{-3w-5}+1 \\leq {\\varepsilon}^{-3w-6} \\leq {\\varepsilon}^{-3{\\varepsilon}^{-1}-3-6} \\leq  {\\varepsilon}^{-4{\\varepsilon}^{-1}} \\leq  {\\varepsilon}^{-{\\varepsilon}^{-2}} =  K({\\varepsilon}).\n\t\t\t\\end{aligned}\n\t\t\\end{equation*} The first inequality is because for all $\\ell \\in I \\setminus H_w$ it holds that $s(\\ell) \\leq {\\varepsilon}^2$ because $w \\geq 2$ by \\eqref{eq:k}. \n\t\\end{proof}\n\t\n\t\n\t\n\t\\subsection{Proofs From the Reconstruction}\n\t\n\tFor the following, let $A = (A_1, \\ldots, A_m)$ be a packing of ${\\mathcal{I}}$ and let $F_A = (U_1, \\ldots, U_{m'})$ (see \\eqref{F_A}). In addition, for all $w \\in \\{2,\\ldots, {\\varepsilon}^{-1}+1\\}$ let $Y_w = I \\setminus (H_w \\cup I_w)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\n\t\\label{claim:tu2}\n\tFor all $ i\\in [m']$ and $G \\in {\\mathcal{G}}$ it holds that $|G \\cap Y_w\\cap U_i| \\leq k(G)$.\n\\end{claim}\n\n\\begin{proof}\nIf $i \\in [m'] \\setminus [m]$, then by \\eqref{F_A} it holds that $|G \\cap Y_w\\cap U_i| =0 \\leq k(G)$. Otherwise, we split into two cases by Step~\\ref{step:groups} of Algorithm~\\ref{alg:reduction}. \n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item $G \\in {\\mathcal{G}}_{L}(w)$. Then, \\begin{equation*}\n\t\t\t\\label{eq:FAHelp}\n\t\t\t|G \\cap Y_w\\cap U_i| = |G \\cap Y_w \\cap \\left(( A_i \\setminus (\\Gamma_w \\cup \\Omega_w)) \\cup B_i\\right)| \\leq |G \\cap Y_w \\cap A_i| \\leq |G \\cap A_i| \\leq k_R(G) = k(G).\n\t\t\\end{equation*} The first inequality is because $(\\Omega_w \\cup \\Gamma_w) \\cap Y_w = \\emptyset$ by \\eqref{Iu}, \\eqref{X:X}, and because $B_i \\subseteq \\Gamma_w$ by Lemma~\\ref{lem:Bgreedy}. The third inequality is because $A$ is a packing of ${\\mathcal{I}}$. The last equality is by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}. \n\t\t\n\t\t\\item $G  \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. Then, by \\eqref{Is} it holds that $G \\setminus H_w \\in {\\mathcal{G}}_{S}(w)$. Therefore, \\begin{equation*}\n\t\t\t|G \\cap Y_w\\cap U_i| = |G \\cap Y_w \\cap \\left(( A_i \\setminus (\\Gamma_w \\cup \\Omega_w)) \\cup B_i\\right)| \\leq |G \\cap Y_w \\cap A_i| \\leq |G \\cap A_i| \\leq k_R(G \\setminus H_w) = k(G).\n\t\t\\end{equation*}  The first inequality is because $B_i \\subseteq \\Gamma_w$ by Lemma~\\ref{lem:Bgreedy}. The third inequality is because $A$ is a packing of ${\\mathcal{I}}$. The last equality is by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}. \n\t\t\n\t\\end{enumerate}\n\t\n\t\n\\end{proof}\n\\begin{claim}\n\t\\label{claim:tu}\n\tFor all $ i\\in [m']$ and $G \\in {\\mathcal{G}}$ it holds that $|F_A(i,G)| \\leq |G \\cap U_i \\cap H_w|$. \n\\end{claim}\n\n\\begin{proof}\n\t\n\tWe split into two cases by Step~\\ref{step:groups} of Algorithm~\\ref{alg:reduction}. \n\n\n\\begin{enumerate}\n\t\\item $G \\in {\\mathcal{G}}_{L}(w)$.\n\t\n\t\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t|F_A(i,G)| =  \\max\\{|G \\cap U_i|-k(G),0\\} \\leq{} &  \\max\\{k_R(G)-k(G),0\\}  =  \\max\\{k(G)-k(G),0\\} = 0.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\t\n\tThe first equality is because for any bin $i \\in [m']$ and group $G \\in {\\mathcal{G}}$ it holds that $F_A(i,G)$ is an exclusion-minimal subset of $w$-light items from $U_i$ such that $| (U_i \\setminus F_A(i,G) )  \\cap G| \\leq k(G)$. The first inequality is because $A$ is a packing of ${\\mathcal{I}}$. The second equality is by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}.  \n\t\n\t\\item $G  \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. Then, \t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t|F_A(i,G)| = {} & \\max\\{ |G \\cap U_i|-k(G),0\\} = \\max\\{ |G \\cap U_i \\cap H_w|+|G \\cap U_i \\cap Y_w|-k(G),0\\} \\\\ \\leq{} & \\max\\{ |G \\cap U_i \\cap H_w|+k(G)-k(G),0\\} = |G \\cap U_i \\cap H_w|.\n\t\t\\end{aligned}\n\t\\end{equation*} The first equality is because for any bin $i \\in [m']$ and group $G \\in {\\mathcal{G}}$ it holds that $F_A(i,G)$ is an exclusion-minimal subset of $w$-light items from $U_i$ such that $| (U_i \\setminus F_A(i,G) )  \\cap G| \\leq k(G)$. The second equality is because $\\Omega_w \\cap U_i = \\emptyset$ by \\eqref{X:X}, \\eqref{F_A}, and because \\textsf{Fill}(A) is a partition of items in $\\Gamma_w$ by Lemma~\\ref{lem:Bgreedy}. The inequality is by Claim~\\ref{claim:tu2}.\n\t\n\\end{enumerate}\t\n\t\n\t\n\n\t\n\\end{proof}\n\n\n\\begin{claim}\n\t\\label{du:help}\n\tFor all $G \\in {\\mathcal{G}}$ it holds that $|D(A) \\cap G| \\leq {\\varepsilon}^{2w+4} \\textnormal{OPT}({\\mathcal{J}})$. \n\\end{claim}\n\n\\begin{proof}\n\tWe split into three cases by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}. \n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item $G \\in {\\mathcal{G}}_{L}(w)$. Then, we use the following inequality.\n\t\t\n\t\t\\begin{equation}\n\t\t\t\\label{DAS}\n\t\t\t|G \\cap U_i| = |G \\cap \\left(( A_i \\setminus (\\Gamma_w\\cup \\Omega_w) ) \\cup B_i\\right)| = |G \\cap A_i| \\leq k_R(G) = k(G).\n\t\t\\end{equation} \n\t\t\n\t\tThe second equality is because $(\\Gamma_w \\cup \\Omega_w) \\cap G = \\emptyset$ by \\eqref{Iu} and $B_i \\subseteq \\Gamma_w$ by Lemma~\\ref{lem:Bgreedy}. Therefore,\n\t\t\n\t\t\n\t\t$$|D(A) \\cap G| = \\sum_{i \\in [m']} |F_A(i,G)| = \\sum_{i \\in [m']}  \\max\\left\\{ |G \\cap U_i| - k(G),0 \\right\\} = 0.$$ The second equality is because for any bin $i \\in [m']$ and group $G \\in {\\mathcal{G}}$ it holds that $F_A(i,G)$ is an exclusion-minimal subset of $w$-light items from $U_i$ such that $| (U_i \\setminus F_A(i,G) )  \\cap G| \\leq k(G)$. The last equality is by \\eqref{DAS}. \n\t\t\n\t\t\n\t\t\n\t\t\\item $G  \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. Now,\n\t\t\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t|D(A) \\cap G| ={} & \\left|\\bigcup_{i \\in [m']} F_A(i,G)\\right| \\leq \\sum_{i \\in [m']} |G \\cap U_i \\cap H_w|  = |G \\cap H_w| \\leq {\\varepsilon}^{2w+4} \\textnormal{OPT}({\\mathcal{J}}).\n\t\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\t\n\t\tThe first inequality is by Claim~\\ref{claim:tu}. The second equality is by \\eqref{F_A} and because $A$ is a packing of ${\\mathcal{I}}$. The last inequality is by Lemma~\\ref{lem:few_large_groups}. \n\t\t\n\t\\end{enumerate}\n\t\n\t\n\t\n\t\n\t\n\t\n\\end{proof}\n\n\n\\begin{claim}\n\t\\label{sd:help}\n\tFor all $i \\in [m']$ and $G \\in {\\mathcal{G}}$ it holds that $s(F_A(i,G)) \\leq {\\varepsilon} \\cdot s(U_i \\cap H_w \\cap G)$. \n\\end{claim}\n\n\\begin{proof}\n\t\n\tIf $F_A(i,G) = \\emptyset$ then the claim is trivially satisfied. Otherwise, by Claim~\\ref{claim:tu} it holds that $U_i \\cap H_w \\cap G \\neq \\emptyset$. Therefore, \n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\ts(F_A(i,G)) ={} & \\sum_{\\ell \\in F_A(i,G)} s(\\ell) \\leq \\sum_{\\ell \\in F_A(i,G)} \\max_{\\ell \\in F_A(i,G)} s(\\ell) =  |F_A(i,G)|  \\cdot \\max_{\\ell \\in F_A(i,G)} s(\\ell) \\leq |U_i \\cap H_w \\cap G|  \\max_{\\ell \\in F_A(i,G)} s(\\ell) \\\\ \\leq{} &  |U_i \\cap H_w \\cap G| \\cdot  {\\varepsilon} \\cdot \\min_{\\ell \\in U_i \\cap H_w \\cap G} s(\\ell) \\leq  {\\varepsilon} \\cdot \\sum_{\\ell \\in U_i \\cap H_w \\cap G} s(\\ell) = {\\varepsilon} \\cdot s(U_i \\cap H_w \\cap G).\n\t\t\\end{aligned}\n\t\\end{equation*} The second inequality is by Claim~\\ref{claim:tu}. The third inequality is because the size of the largest item in $Y_w$ is at least ${\\varepsilon}^{-1}$ times smaller compared to the smallest item in $H_w$ by \\eqref{Interval}. \n\t\n\t\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\t\\label{lem:s(D)}\n\t$s(D(A)) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})$.\n\\end{lemma}\n\n\\begin{proof}\n\t\n\t\\comment{\n\t\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\ts(D(A)) ={} & \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']} s(F_A(i,G)) \\leq \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']} {\\varepsilon} \\cdot s(U_i \\cap H_w \\cap G) \\leq  {\\varepsilon} \\cdot \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']}  \\cdot s(U_i \\cap G) \\\\ ={} & {\\varepsilon} \\cdot \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']}  \\left( s(A_i \\cap G)-s((\\Gamma_w \\cup \\Omega_w) \\cap A_i)-s(B_i \\cap G) \\right) \\\\ \\leq{} & {\\varepsilon} \\cdot \\left( s(I)-s(\\Gamma_w)+s\\left(\\bigcup_{b \\in [m]} B_i\\right)\\right) \\leq {\\varepsilon} \\cdot s(I) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}}).\n\t\t\\end{aligned}\n\t\\end{equation*} \n}\n\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t\ts(D(A)) ={} & \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']} s(F_A(i,G)) \\leq \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']} {\\varepsilon} \\cdot s(U_i \\cap H_w \\cap G) \\leq  {\\varepsilon} \\cdot \\sum_{G \\in {\\mathcal{G}}} \\sum_{i \\in [m']}  \\cdot s(U_i \\cap G) \\\\ \\leq{} & {\\varepsilon} \\cdot s(I) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}}).\n\t\\end{aligned}\n\\end{equation*} \nThe first equality is by \\eqref{D}. The first inequality is by Claim~\\ref{sd:help}. \n\\end{proof}\n\n\n\n\n\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{thm:D}:} The proof follows by Claim~\\ref{lem:s(D)} and Claim~\\ref{du:help}. \n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\n\t\\label{lem:shiftingCanBeUsedForI2}\n\tGiven a packing $A$ for $\\mathcal{I} = \\textsf{Reduce}({\\varepsilon}, \\mathcal{J})$ of size $m$, Algorithm~\\ref{Alg:RECON} returns in time $\\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$ a packing $F$ for the instance ${\\cal J}$ of size $\\phi$ where $\\phi<m+13{\\varepsilon}\\cdot \\textnormal{OPT}({\\mathcal{J}})+1$. \n\\end{lemma}\n\n\\begin{proof}\n\t\n\tIn the proof of the lemma we use several auxiliary claims. For simplicity, by Step~\\ref{step:3}, let $F_A \\setminus E = (Z_1 \\ldots Z_{m'})$, $\\textsf{Greedy}(\\mathcal{E}) = (X_1 \\ldots X_{x})$,  $F = \\left( F_A \\setminus E \\right) \\oplus \\textsf{Greedy}(\\mathcal{E})$, and $F = (F_1, \\ldots, F_{\\phi})$. Also, for the following let $(B_1, \\ldots, B_m, R) = \\textsf{Fill}(\\mathcal{J}, A)$. Finally, recall the definition of $\\mathcal{{\\mathcal{I}}}$-partition from Section~\\ref{sec:reduction}. For $\\beta = {\\varepsilon}^{-w-2}$, we define $P_1,\\ldots, P_q$ to be the $\\mathcal{{\\mathcal{I}}}$-partition: a partition of $\\Gamma_w$ by a non decreasing order of item sizes such that for all $1\\leq i \\leq j<q$ and $\\ell \\in P_i, y \\in P_j$ it holds that $s(\\ell) \\geq s(y)$, $|P_i| = \\ceil{\\frac{|\\Gamma_w|}{\\beta}}$, and $P_{q}$ contains the remaining items from $\\Gamma_w$. It follows that $q \\leq \\beta$ and we define $q({\\mathcal{I}}) = q$.\n\t\n\t\\begin{claim}\n\t\t\\label{lem:F}\n\t\t$F$ is a packing of ${\\mathcal{J}}$.\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tWe prove the necessary conditions of packing as follows. \n\t\t\\begin{enumerate}\n\t\t\t\\item $F$ is a partition of $I$.  Let $\\ell \\in I$. By the following three complementary cases, we conclude that $F$ is a partition of $I$.\n\t\t\t\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item $\\ell \\in E$. Then, there is exactly one $j \\in [x]$  such that $\\ell \\in X_j$ by Lemma~\\ref{thm:greedy} and \\eqref{Ie}; hence,  there is exactly one $i \\in [\\phi]$ such that $\\ell \\in F_i$ because by Step~\\ref{step:3} of Algorithm~\\ref{Alg:RECON} for all $i \\in [m']$ it holds that $\\ell \\notin Z_i$. %\n\t\t\t\t\n\t\t\t\t\\item $\\ell \\in \\Gamma_w$. Then, there is exactly one $i \\in [m]$ such that $\\ell \\in A_i$ because $A$ is a packing of ${\\mathcal{I}}$. Then, by Lemma~\\ref{lem:Bgreedy}  and by \\eqref{F_A} there is exactly one $i \\in [m']$ such that $\\ell \\in Z_i$. Hence,  there is exactly one $i \\in [\\phi]$ such that $\\ell \\in F_i$ because by \\eqref{Ie} for all $j \\in [x]$ it holds that $\\ell \\notin X_j$.   \n\t\t\t\t\n\t\t\t\t\\item $\\ell \\in I \\setminus (\\Gamma_w \\cup E)$. Then, there is exactly one $i \\in [m]$ such that $\\ell \\in A_i$ because $A$ is a packing of ${\\mathcal{I}}$. Then, by \\eqref{F_A} there is exactly one $i \\in [m']$ such that $\\ell \\in Z_i$. Hence,  there is exactly one $i \\in [\\phi]$ such that $\\ell \\in F_i$ because by \\eqref{Ie} for all $j \\in [x]$ it holds that $\\ell \\notin X_j$.   \n\t\t\t\\end{itemize}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\item Let $i \\in [\\phi]$. By Step~\\ref{step:reconl} of Algorithm~\\ref{Alg:RECON} it holds that $F_i = Z_i$ or $F_i = X_j$ for $j = i-m'$. We split into two cases based on the latter distinction. \n\t\t\t\n\t\t\t\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item $F_i = Z_i$. \tThen, by \\eqref{F_A} if $U_i = \\{\\ell\\}$ for $\\ell \\in R$ it holds that $s(F_i) = s(Z_i) = s(\\ell) \\leq 1$ because the sizes of items are bounded by $1$. Otherwise, by \\eqref{F_A} it holds that $Z_i = ((A_i \\setminus (\\Gamma_w\\cup \\Omega_w)) \\cup B_i) \\setminus E$. Therefore, we use the following inequalities. \n\t\t\t\t\n\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{XW}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\ts(B_i) ={} & \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} \\sum_{\\ell \\in B_i \\cap P_j} s(\\ell) \\leq \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} \\sum_{\\ell \\in B_i \\cap P_j} \\max_{\\ell \\in P_j} s(\\ell) \\\\ \\leq{} & \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} |B_i \\cap P_j| \\cdot \\max_{\\ell \\in P_j} s(\\ell). \n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation} The first equality is because $B_i \\cap P_1 = \\emptyset$ by Lemma~\\ref{lem:Bgreedy}. In addition, \n\t\t\t\t\n\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{XW2}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~}{} & |B_i \\cap P_j| \\cdot \\max_{\\ell \\in P_j} s(\\ell) \\leq \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} |A_i \\cap P_{j-1}| \\cdot \\max_{\\ell \\in P_{j}} s(\\ell)  \\\\ \\leq{} & \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} |A_i \\cap P_{j-1}| \\cdot \\min_{\\ell \\in P_{j-1}} s(\\ell) \\\\ \\leq{} & \\sum_{j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}~} \\sum_{\\ell \\in A_i \\cap P_{j-1}} s(\\ell) \\leq s(A_i \\cap \\Gamma_w). \n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation} The first inequality is by Lemma~\\ref{lem:Bgreedy}. The third inequality is because for all $j \\in [q({\\mathcal{I}})-1]$, $\\ell \\in P_j, y \\in P_{j+1}$ it holds that $s(\\ell) \\geq s(y)$ by the definition of the ${\\mathcal{I}}$-partition.  Then, \n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\ts(F_i) ={} & s(U_i) = s\\big(((A_i \\setminus (\\Gamma_w \\cup \\Omega_w)) \\cup B_i) \\setminus E\\big) \\\\ \\leq{} & s\\big((A_i \\setminus (\\Gamma_w \\cup \\Omega_w) \\cup B_i)\\big)   \\\\ ={} & s(A_i \\setminus (\\Gamma_w \\cup \\Omega_w))+s(B_i) \\\\ \\leq{} & s(A_i \\setminus (\\Gamma_w \\cup \\Omega_w))+s(A_i \\cap \\Gamma_w) \\leq s(A_i) \\leq 1. \n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation*} The second inequality is by \\eqref{XW} and \\eqref{XW2}. The last inequality is since $A$ is a packing of~${\\mathcal{I}}$. \n\t\t\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\item $F_i =X_j$ for $j = i-m'$. Then, It holds that $s(F_i) \\leq 1$ by Lemma~\\ref{thm:greedy}.\n\t\t\t\\end{itemize}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\item Let $G \\in {\\mathcal{G}}$ and $i \\in [\\phi]$. We split into two cases by Step~\\ref{step:reconl}.  \\begin{itemize}\n\t\t\t\t\\item $F_i = X_j$ for $j = i-m'$. Then, $$|F_i \\cap G| = |X_j \\cap G| = |X_j \\cap E \\cap G| \\leq k_E(G \\cap E) = k(G).$$ The second and the last equalities are by \\eqref{Ie}. The inequality is by Lemma~\\ref{thm:greedy}. \n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\item $F_i = Z_i$. Then, we split into two complementary cases. If $G \\in {\\mathcal{G}}_{L}(w)$:\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t|F_i \\cap G| = |Z_i \\cap G| = |((A_i \\setminus (\\Gamma_w \\cup \\Omega_w)) \\cup B_i) \\cap G| = |A_i \\cap G| \\leq k_R(G) = k(G). \n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation*} The second equality is by \\eqref{F_A}. The third equality is because $G \\in {\\mathcal{G}}_{L}(w)$; hence, $G \\cap \\Gamma_w = \\emptyset$ and $G \\cap B_i \\cap \\Omega_w = \\emptyset$ by \\eqref{Iu} and \\eqref{X:X}, respectively. The inequality is because $A$ is a packing of ${\\mathcal{I}}$. The last equality is by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}. Otherwise, $G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. Then,  \t\\begin{equation}\n\t\t\t\t\t\\label{XW3}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t|F_i \\cap G| \\leq{} & |F_i \\cap G \\cap H_w|+|F_i \\cap G \\cap Y_w| \\\\ \\leq{} &  |F_i \\cap G \\cap H_w|+\\max\\{0, k(G)-|F_i \\cap G \\cap H_w|\\}  \\\\ ={} & \\max\\{ |F_i \\cap G \\cap H_w|,  k(G)\\} \\leq \\max\\{  |B_i \\cap G|,  k(G)\\} \\leq k(G).\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation}\n\t\t\t\t\n\t\t\t\tThe  first equality is because $\\Omega_w \\cap  G \\cap U_i = \\emptyset$ by \\eqref{F_A} since $G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. The second inequality is by \\eqref{D}: we discard to $F_A(i,G)$ only $w$-light items until reaching at most $k(G)$ items in $U_i$. The last equality is by \\eqref{F_A} and because $G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_L(w)$. The last inequality is by Lemma~\\ref{lem:Bgreedy}. \n\t\t\t\\end{itemize}\n\t\t\\end{enumerate}\n\t\t\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{claim:IESize}\n\t\t$s(E) \\leq 2{\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})$. \n\t\\end{claim}\n\t\n\t\n\t\\begin{proof}\n\t\t\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\ts(E) \\leq{} & s(I_w)+s(D(A)) \\leq 2 {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}}).\n\t\t\t\\end{aligned}\n\t\t\\end{equation*} The first inequality is by \\eqref{Ie}. The second inequality is by Lemma~\\ref{lem:k_val} and  Lemma~\\ref{thm:D}.\n\t\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{claim:IESize3}\n\t\tFor all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap E| \\leq2{\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})$. \n\t\\end{claim}\n\t\\begin{proof}\n\t\t\n\t\tWe split into two cases. \n\t\t\n\t\t\\begin{enumerate}\n\t\t\t\\item $G \\in {\\mathcal{G}}_{L}(w)$. Then, \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t|G \\cap E| ={} & |G \\cap D(A)| = \\left| \\bigcup_{i \\in [m']} F_A(i,G)\\right| = \\sum_{i \\in [m']} \\max\\{|G \\cap U_i|-k(G),0\\} \\\\ \\leq{} & \\sum_{i \\in [m']} \\max\\{k_R(G)-k(G),0\\}  = \\sum_{i \\in [m']} \\max\\{k(G)-k(G),0\\} = 0.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\t\t\n\t\t\tThe first equality is by \\eqref{Ie}. The first inequality is because $A$ is a packing of ${\\mathcal{I}}$ and by \\eqref{F_A}. The fourth equality is by Step~\\ref{kr} of Algorithm~\\ref{alg:reduction}.  \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\item $G \\in {\\mathcal{G}} \\setminus {\\mathcal{G}}_{L}(w)$. We use the following inequality. \n\t\t\t\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq:||d}\n\t\t\t\t|G \\cap E \\cap Y_w| = |G \\cap D(A)| \\leq {\\varepsilon} \\cdot \\textnormal{OPT} ({\\mathcal{J}}).\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\tThe equality is by \\eqref{D} and \\eqref{Ie}. The inequality is by Lemma~\\ref{thm:D}. Therefore, \n\t\t\t\n\t\t\t$$|G \\cap E| = |G \\cap E \\cap Y_w|+|G \\cap E \\cap (H_w \\cup I_w)| \\leq  2{\\varepsilon} \\cdot \\textnormal{OPT} ({\\mathcal{J}}).$$\n\t\t\t\n\t\t\t\n\t\t\tThe inequality is by Lemma~\\ref{lem:few_large_groups} and by \\eqref{eq:||d}. \n\t\t\\end{enumerate}\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{claim:IESize2}\n\t\t$V(\\mathcal{E}) \\leq  2{\\varepsilon} \\cdot \\textnormal{OPT} ({\\mathcal{J}})$. \n\t\\end{claim}\n\t\\begin{proof}\n\t\t\n\t\t$$V(\\mathcal{E}) = \\max_{G \\in {\\mathcal{G}}_E} \\ceil{\\frac{|G|}{|k_E(G)|}} \\leq  \\max_{G \\in {\\mathcal{G}}_E} |G| =  \\max_{G \\in {\\mathcal{G}}} |G \\cap E| \\leq   2{\\varepsilon} \\cdot \\textnormal{OPT} ({\\mathcal{J}}).$$ The first inequality is because for all $G \\in {\\mathcal{G}}_E$ it holds that $k_E(G) \\geq 1$. The second equality is by \\eqref{Ie}. The last inequality is by Claim~\\ref{claim:IESize3}. \n\t\t\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{claim}\n\t\t\\label{lem:sizeF}\n\t\t$\\phi<m+13{\\varepsilon}\\cdot \\textnormal{OPT}({\\mathcal{J}})+1$\n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\t\n\t\t\n\t\t\n\t\t\\begin{equation*}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\phi = {} & m'+x \\leq m+|R|+x \\leq m+{\\varepsilon}\\cdot\\textnormal{OPT}({\\mathcal{J}})+1+(1+2) \\cdot \\max\\{2s(E),s(E)+V(\\mathcal{E})\\} \\\\ \\leq{} & m+13{\\varepsilon}\\cdot\\textnormal{OPT}({\\mathcal{J}})+1.\n\t\t\t\\end{aligned}\n\t\t\\end{equation*} \n\t\t\n\t\t\n\t\t\n\t\tThe first equality is by Step~\\ref{step:reconl} of Algorithm~\\ref{Alg:RECON}. The first inequality is by \\eqref{F_A}. The second inequality is by Lemma~\\ref{lem:Bgreedy} and by Lemma~\\ref{thm:greedy}. The last inequality is by Claim~\\ref{claim:IESize} and Claim~\\ref{claim:IESize2}. \n\t\\end{proof}\n\t\n\t\\begin{claim}\n\t\t\\label{claim:reconRuningTime}\n\t\tThe running time of Algorithm~\\ref{Alg:RECON} is $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$. \n\t\\end{claim}\n\t\n\t\\begin{proof}\n\t\tBy Lemma~\\ref{lem:Bgreedy}, Step~\\ref{step:1} can be computed in time $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$. By \\eqref{Ie}, the discarded instance in Step~\\ref{step:2} can be computed in time $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$. By Lemma~\\ref{thm:greedy}, Step~\\ref{step:3} can be computed in time $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$; also, concatenating two tuples takes linear time in the size of the instance. The overall running time is therefore $\\textnormal{poly}(|{\\mathcal{J}}|,\\frac{1}{{\\varepsilon}})$. \n\t\\end{proof}\n\t\n\tThe proof of Lemma~\\ref{lem:shiftingCanBeUsedForI2} follows by Claim~\\ref{lem:F}, Claim~\\ref{lem:sizeF}, and Claim~\\ref{claim:reconRuningTime}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\vspace{0.08in}\n\\noindent{\\bf Proof of Lemma~\\ref{lem:reductionReconstruction}:}\n\nCondition~\\ref{condition:reduction1} holds By Lemma~\\ref{lem:shiftingNotIncreaseOPT2} and Lemma~\\ref{lem:rounding12}. Condition~\\ref{condition:reduction2} holds By Lemma~\\ref{lem:shiftingCanBeUsedForI2}. \\qed\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Algorithm \\textsf{Fill}}\n\\label{sec:fill}\n\nIn this section we prove Lemma~\\ref{lem:Bgreedy}. Recall the definition of $\\mathcal{{\\mathcal{I}}}$-partition from Section~\\ref{sec:reduction}. For $\\beta = {\\varepsilon}^{-w-2}$, we define $P_1,\\ldots, P_q$ to be the $\\mathcal{{\\mathcal{I}}}$-partition: a partition of $\\Gamma_w$ by a non decreasing order of item sizes such that for all $1\\leq i \\leq j<q$ and $\\ell \\in P_i, y \\in P_j$ it holds that $s(\\ell) \\geq s(y)$, $|P_i| = \\ceil{\\frac{|\\Gamma_w|}{\\beta}}$, and $P_{q}$ contains the remaining items from $\\Gamma_w$. It follows that $q \\leq \\beta$ and we define $q({\\mathcal{I}}) = q$. In Algorithm \\textsf{Fill} we start with $B_1, \\ldots, B_m$ empty sets and with $R = \\Gamma_w$. In each iteration, we try to move an item from $R$ to some $B_i, i \\in [m]$ if it is possible to do so without violating one of the conditions of the lemma. When the above cannot be done anymore, we return $(B_1, \\ldots, B_m,R)$. The pseudocode of Algorithm \\textsf{Fill} is given in Algorithm~\\ref{alg:fill}. For the proof, we use the following claims. Let $(B_1, \\ldots, B_m, R) = \\textsf{Fill}({\\mathcal{J}},A)$. \n\n\\begin{algorithm}[h]\n\t\\caption{$\\textsf{Fill}(\\mathcal{J}, A = (A_1, \\ldots, A_m))$}\n\t\\label{alg:fill}\n\t\n\t\n\t\n\tInitialize $R \\leftarrow \\Gamma_w$,  $B_i \\leftarrow \\emptyset ~:~ \\forall i \\in [m]$.\\label{fill:init}\n\t\n\t\\While{$\\exists i \\in [m], j \\in \\{2,\\ldots, q({\\mathcal{I}})\\}, \\ell \\in R \\cap P_j \\textnormal{ s.t. } |B_i \\cap P_j| < |A_i \\cap P_{j-1}| ~\\textnormal{and}~ |\\textnormal{\\textsf{group}}(\\ell) \\cap  B_i| < k(G)$ \\label{fill:while}}{\n\t\t\n\t\t\t\n\t\t\t$B_i \\leftarrow B_i \\cup \\{\\ell\\}$.\\label{fill:1}\n\t\t\t\n\t\t\t$R \\leftarrow R \\setminus \\{\\ell\\}$.\\label{fill:2}\n\t\t\t\n\t\t\n\t}\n\t\n\tReturn $(B_1, \\ldots, B_m, R)$.\\label{fill:return}\n\\end{algorithm}\n\n\n\\begin{claim}\n\t\\label{claim:fillRunningTime}\n\tThe running time of Algorithm~\\ref{alg:fill} is $\\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$.\n\\end{claim}\n\n\n\n\\begin{proof}\n\t\n\tBy Step~\\ref{fill:while}, Step~\\ref{fill:1}, and Step~\\ref{fill:2}, each iteration of the while loop of the algorithm takes linear time by trying at most all items. In addition, there are $\\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$ iterations of the while loop because by Step~\\ref{fill:while}, Step~\\ref{fill:1}, and Step~\\ref{fill:2} in each iteration an item is moved from $R$ to some $B_i, i \\in [m]$ and never moved again. Therefore, there are at most $|I| = \\textnormal{poly}(|{\\mathcal{J}}|, \\frac{1}{{\\varepsilon}})$ iterations and the claim follows. \n\t\n\\end{proof}\n\n\\begin{claim}\n\t\\label{claim:fill1}\n\tFor each $k \\in \\mathbb{N}$, after $k$ iteration of Step~\\ref{fill:while} the following hold.\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item  $(B_1, \\ldots, B_m, R)$ is a partition of $\\Gamma_w$.  \n\t\t\n\t\t\\item For all $i \\in [m]$ and $j \\in \\{2, \\ldots,q({\\mathcal{I}})\\}$ it holds that $|B_i \\cap P_j| \\leq |A_i \\cap P_{j-1}|$ and $|B_i \\cap P_1| = 0$.\n\t\t\n\t\t\\item For all $i \\in [m]$ and $G \\in {\\mathcal{G}}$ it holds that $|B_i| \\leq k(G)$.\n\t\t\n\t\t\n\t\\end{enumerate}   \n\\end{claim}\n\n\\begin{proof}\n\tWe prove the claim by induction on $k$. For the base case, let $k = 0$. Therefore, \n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item  By Step~\\ref{fill:init} it holds that $(B_1, \\ldots, B_m, R) = (\\emptyset, \\ldots, \\emptyset, \\Gamma_w)$, which is a partition of $\\Gamma_w$.  \n\t\t\n\t\t\\item By Step~\\ref{fill:init}, for all $i \\in [m]$ and $j \\in \\{2, \\ldots,q({\\mathcal{I}})\\}$ it holds that $|B_i \\cap P_j| = |\\emptyset \\cap P_j| = 0 \\leq |A_i \\cap P_{j-1}|$ and $|B_i \\cap P_1| = 0$.\n\t\t\n\t\t\\item By Step~\\ref{fill:init}, for all $i \\in [m]$ and $G \\in {\\mathcal{G}}$ it holds that $|B_i| = |\\emptyset| =  0 \\leq 1 \\leq k(G)$.\n\t\t\n\t\t\n\t\\end{enumerate}   \n\t\n\tAssume that the claim holds for some $k \\in \\mathbb{N}$. Now, for the step of the induction observe $k+1$. Let $(B^k_1, \\ldots, B^k_m,R^k)$, $(B^{k+1}_1, \\ldots, B^{k+1}_m,R^{k+1})$ be the object $(B_1, \\ldots, B_m, R)$ before and after iteration $k+1$, respectively. Let $i \\in [m], j \\in \\{2, \\ldots, q({\\mathcal{I}})\\}$ and $\\ell \\in P_j$ such that $\\ell$ is moved from $R^k$ to $B^{k+1}_i$ in iteration $k+1$ by Step~\\ref{fill:1} and Step~\\ref{fill:2}. \n\t\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item  By the assumption of the induction, it holds that $(B^k_1, \\ldots, B^k_m,R^k)$ is a partition of $\\Gamma_w$. Let $y \\in \\Gamma_w$. If $y = \\ell$, then $y \\in B^{k+1}_i$ and do not belong to any other set in the partition by Step~\\ref{fill:1} and Step~\\ref{fill:2}. Otherwise, $y \\neq \\ell$; then, there is exactly one set in  $(B^k_1, \\ldots, B^k_m,R^k)$ to which $y$ belongs by the assumption of the induction; also, in iteration $k+1$, by Step~\\ref{fill:1} and Step~\\ref{fill:2} it holds that $y$ remains in the same set since only $\\ell$ is relocated in this iteration.\n\t\t\n\t\t\\item Let  $i' \\in [m]$ and $j' \\in \\{2, \\ldots,q({\\mathcal{I}})\\}$. If $i' = i$ and $j' = j$, then  \n\t\t\n\t\t$$|B^{k+1}_i \\cap P_j| = |\\{\\ell\\}|+|B^k_i \\cap P_{j-1}| \\leq |A_i \\cap P_{j-1}|.$$\n\t\t\n\t\tThe equality is by Step~\\ref{fill:1} and Step~\\ref{fill:2}. The inequality is by Step~\\ref{fill:while}. Otherwise, it holds that $i' \\neq i$ or $j' \\neq j$. Therefore, \n\t\t\n\t\t$$|B^{k+1}_i \\cap P_j| = |B^k_i \\cap P_{j-1}| \\leq |A_i \\cap P_{j-1}|.$$\n\t\t\n\t\tThe equality is by Step~\\ref{fill:1} and Step~\\ref{fill:2}. The inequality is by the assumption of the induction.\n\t\t\n\t\t\\item Let  $i' \\in [m]$ and $G \\in {\\mathcal{G}}$. If $i' = i$ and $G = \\textsf{group}(\\ell)$, then  \n\t\t\n\t\t$$|B^{k+1}_i \\cap G| = |\\ell \\cap G|+|B^k_i \\cap G| \\leq k(G).$$\n\t\t\n\t\tThe equality is by Step~\\ref{fill:1} and Step~\\ref{fill:2}. The inequality is by Step~\\ref{fill:while}. Otherwise, it holds that $i' \\neq i$ or $G \\neq \\textsf{group}(\\ell)$. Therefore, \n\t\t\n\t\t$$|B^{k+1}_i \\cap G| = |B^k_i \\cap G| \\leq k(G).$$\n\t\t\n\t\tThe equality is by Step~\\ref{fill:1} and Step~\\ref{fill:2}. The inequality is by the assumption of the induction.\n\t\t\n\t\t\n\t\\end{enumerate}   \n\\end{proof}\n\n\n\\begin{claim}\n\t\\label{claim:fill2}\n\t$|R \\setminus P_{q({\\mathcal{I}})}| \\leq {\\varepsilon}^2 \\cdot \\textnormal{OPT}({\\mathcal{J}})$. \n\\end{claim}\n\n\\begin{proof}\n\tLet $j \\in \\{2,\\ldots, q({\\mathcal{I}})\\}$, and let $\\ell \\in R \\cap P_j$ at the end of Algorithm~\\ref{alg:fill}; in addition, let $G = \\textsf{group}(\\ell)$. Therefore, by Step~\\ref{fill:while} at least one of the following holds for all $i \\in [m]$.\n\t\n\t\\begin{itemize}\n\t\t\\item (i) $|B_i \\cap P_j| = |A_i \\cap P_{j-1}|$.\n\t\t\n\t\t\\item (ii) $|G \\cap B_i| = k(G)$. \n\t\\end{itemize}\n\t\n\t\n\tNow, define the set of indices in $[m]$ in which the second condition defined above holds. \n\t\n\t\\begin{equation}\n\t\t\\label{fill:T}\n\t\tT = \\{i \\in [m]~|~ |G \\cap B_i| = k(G)\\}.\n\t\\end{equation}\n\t\n\tWe use the following inequality \n\t\n\t\n\t\\begin{equation}\n\t\t\\label{FF1}\n\t\t\\begin{aligned}\n\t\t\t|P_{j-1}| ={} & \\sum_{i \\in [m]} |P_{j-1} \\cap A_i| = \\sum_{i \\in [m] \\setminus T} |P_{j-1} \\cap A_i|+\\sum_{i \\in T} |P_{j-1} \\cap A_i| =  \\sum_{i \\in [m] \\setminus T} |P_j \\cap B_i|+\\sum_{i \\in T} |P_{j-1} \\cap A_i| \\\\ \\leq{} & \\sum_{i \\in [m]} |P_j \\cap B_i|  +{\\varepsilon}^{-w} \\cdot |T| \\leq  \\sum_{i \\in [m]} |P_j \\cap B_i|+ {\\varepsilon}^{-w} \\cdot {\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}) \\\\ ={} &  \\sum_{i \\in [m]} |P_j \\cap B_i|+ {\\varepsilon}^{w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}).\n\t\t\\end{aligned}\n\t\\end{equation} The first equality is because $A$ is a packing of $P_j$ in particular. The third equality is by \\eqref{fill:T} and by (i).  The first inequality is because $\\Gamma_w \\subseteq H_w$; hence, the sizes of all items are at least ${\\varepsilon}^{w}$ and there can be at most ${\\varepsilon}^{-w}$ items from $\\Gamma_w$ in $A_i, \\forall i \\in [m]$ because $A$ is a packing. The second inequality is because for each $i \\in T$ it holds that $G \\cap B_i \\neq \\emptyset$ by \\eqref{fill:T}; thus, there are at least $|T|$ $w$-heavy items in $G$ because $B_i \\subseteq \\Gamma_w$. Therefore, we get by Lemma~\\ref{lem:few_large_groups} that $|T| \\leq  {\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}})$ since each $w$-small group contains at most $ {\\varepsilon}^{2w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}})$ $w$-heavy items in particular. \n\t\n\tTherefore, using the above for any $j \\in \\{2,\\ldots, q({\\mathcal{I}})\\}$:\n\t\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t|R \\setminus P_{q({\\mathcal{I}})}| ={} &\\sum_{j \\in \\{2,\\ldots,q({\\mathcal{I}})\\}} |P_{j-1}| - \\sum_{j \\in \\{2,\\ldots,q({\\mathcal{I}})-1\\}} \\sum_{i \\in [m]} |P_j \\cap B_i| \\leq  {\\varepsilon}^{w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}) \\cdot q({\\mathcal{I}}) \\\\ \\leq{} & {\\varepsilon}^{w+4} \\cdot \\textnormal{OPT}({\\mathcal{J}}) \\cdot {\\varepsilon}^{-w-2}  = {\\varepsilon}^{2} \\cdot \\textnormal{OPT}({\\mathcal{J}})\n\t\t\\end{aligned}\n\t\\end{equation*} The first equality is because $(B_1, \\ldots, B_m,R)$ is a partition of $\\Gamma_w$ by Claim~\\ref{claim:fill1}. The first inequality is by \\eqref{FF1}. \n\\end{proof}\n\n\n\n\\begin{claim}\n\t\\label{claim:fillR}\n\t$|R| \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})+1$. \n\\end{claim}\n\n\n\n\\begin{proof}\n\t\n\tWe use the following inequality.\n\t\n\t\\begin{equation}\n\t\t\\label{FF2}\n\t\t\\begin{aligned}\n\t\t\t|R \\cap P_{q({\\mathcal{I}})}| \\leq{} & |P_{q({\\mathcal{I}})}| \\leq \\ceil{\\beta^{-1} \\cdot |\\Gamma_w|} \\leq \\beta^{-1} \\cdot |\\Gamma_w|+1 = {\\varepsilon}^{w+2}  \\cdot |\\Gamma_w|+1 \\\\ \\leq{} & {\\varepsilon}^{w+2}  \\cdot {\\varepsilon}^{-w} \\cdot \\textnormal{OPT}({\\mathcal{J}}) +1 = {\\varepsilon}^2 \\cdot \\textnormal{OPT}({\\mathcal{J}})+1. \n\t\t\\end{aligned}\n\t\\end{equation} The second inequality is by the definition of ${\\mathcal{I}}$-partition. The fourth inequality is because $\\Gamma_w \\subseteq H_w$ by \\eqref{Iu}; hence, in each bin in a packing of ${\\mathcal{J}}$ there can be at most ${\\varepsilon}^{-w}$ items from $\\Gamma_w$ since each item is $w$-heavy and has a size at least ${\\varepsilon}^{-w}$; it follows that $ |\\Gamma_w|  \\leq {\\varepsilon}^{-w} \\cdot \\textnormal{OPT}({\\mathcal{J}})$. Now, \n\t\n\t$$|R| = |R \\cap P_{q({\\mathcal{I}})}|+|R \\setminus P_{q({\\mathcal{I}})}| \\leq |P_{q({\\mathcal{I}})}|+|R \\setminus P_{q({\\mathcal{I}})}| \\leq {\\varepsilon}^2 \\cdot \\textnormal{OPT}({\\mathcal{J}})+1+2{\\varepsilon}^2 \\cdot \\textnormal{OPT}({\\mathcal{J}}) \\leq {\\varepsilon} \\cdot \\textnormal{OPT}({\\mathcal{J}})+1.$$ The second inequality is by \\eqref{FF2} and by Claim~\\ref{claim:fill2}. \n\\end{proof}\n\n\n\\noindent{\\bf Proof of Lemma~\\ref{lem:Bgreedy}:} By Claim~\\ref{claim:fill1} and Claim~\\ref{claim:fillR} the output of the algorithm satisfies the required properties, since by Step~\\ref{fill:return} the returned value is the direct outcome of the while loop. In addition, the running time satisfies the properties by Claim~\\ref{claim:fillRunningTime}. \n\n\n\n\n\\section{Algorithm $\\textsf{Shift}$}\n\\label{sec:AlgPrototype}\n\n Let $\\bar{y}$ be a prototype satisfying the conditions of Lemma~\\ref{lem:Cnf}.\n Algorithm \\textsf{Shift} constructs from  $\\bar{y}$ a good \n prototype $\\bar{z}$ by reducing the size of the support of $\\bar{y}$. \n The algorithm uses {\\em classes}, i.e., partitions of groups into sets, where each set contains items of (roughly) similar size. Each class has up to ${\\varepsilon}^{-10}$\n representatives to which items in this class are mapped. In the prototype constructed by \\textsf{Shift}, each configuration $C \\in \\textsf{supp}(\\bar{y})$ is replaced by a set of\n representatives to which the items in $C$ are mapped.\n Recall that $L = \\{\\ell \\in ~|~ s(\\ell) > {\\varepsilon}^2\\}$ is the set of {\\em large} items in $I$; let $I \\setminus L$ be  the set of {\\em small} items. \n \n  We define classes for two types of groups, as explained below. Given an item $\\ell \\in I$,\n   the {\\em frequency} of $\\ell$ is given by $f_{\\bar{y}}(\\ell) = \\sum_{C \\in \\mathcal{C}[\\ell]} \\bar{y}_C$; that is, the fractional number of configurations\n containing $\\ell$  w.r.t $\\bar{y}$. \n The frequency of a subset of items $I' \\subseteq I$ is then\n$f_{\\bar{y}}(I') = \\sum_{\\ell \\in I'} f_{\\bar{y}}(\\ell)$. \nTo obtain a prototype $\\bar{z}$ with small support, only items from a small number of classes may remain in the constructed configurations. Thus, items belonging to \ngroups with low frequencies of small items, or groups with no large items, are discarded \nfrom their hosting configurations.  \n \n Specifically, for any $G \\in {\\mathcal{G}}$, let $f_{\\bar{y}}(G \\setminus L)$ be the frequency of small items in $G$. Now, let $\\eta = \\min \\{|{\\mathcal{G}}|,{\\varepsilon}^{-12}\\}$, then the {\\em significant} groups $\\mathcal{G}_{\\eta} = \\{G_1, \\ldots, G_{\\eta}\\}$ are the first $\\eta$ groups in ${\\mathcal{G}}$, where the groups are sorted in non-increasing order by the frequency of small items in each.\n   \n \n \n \\begin{lemma}\n \t\\label{lem:significantDiscard}\n \t\n \tThe following properties hold for $\\bar{y}$.\t\\begin{enumerate}\n \t\t\\item $\\sum_{\\ell \\in I \\setminus L} f_{\\bar{y}}(\\ell) \\cdot s(\\ell) \\leq {\\varepsilon} \\cdot \\|\\bar{y}\\|$.\n \t\t\n \t\t\\item  For all $G \\in \\mathcal{G} \\setminus  \\mathcal{G}_{\\eta}$ it holds that $f_{\\bar{y}}(G \\setminus L) \\leq {\\varepsilon} \\cdot \\|\\bar{y}\\|$. \n \t\\end{enumerate}\n \t\n \t\n \\end{lemma} Recall that any configuration $C \\in \\textsf{supp}(\\bar{y})$ satisfies (a) $s(C \\setminus L) \\leq {\\varepsilon}$, and (b) $|C|\\leq {\\varepsilon}^{-10}$.  The first property in Lemma~\\ref{lem:significantDiscard} follows from (a), and the second property follows from the selection of $\\eta$ and (b).  \n\nIn the prototype $\\bar{z}$ constructed by algorithm {\\sf Shift},\nthe selection of the empty configuration (i.e., $\\bar{z}_{\\emptyset}$) is increased. In the $\\bar{z}$-polytope,  this enables to fully assign the small items from non-significant groups to the empty-configuration type, as these items are discarded from the \nconfigurations selected by \n$\\bar{z}$. Using Lemma~\\ref{lem:significantDiscard}, a slight increase in  the selection of the empty-configuration suffices for the $\\bar{z}$-polytope to be non-empty. However, this can be done effectively only for small items. Hence, in addition to the significant groups, we take for the configurations of the constructed prototype also groups containing large items. Formally, Define the {\\em massive} groups as all groups containing at least one large item: \n\n\\begin{equation}\n \t\\label{Agroups}\n \t{\\mathcal{A}} = \\{G \\in {\\mathcal{G}}~|~ G \\cap L \\neq \\emptyset\\}.\n \\end{equation} Recall that ${\\mathcal{I}}$ is an ${\\varepsilon}$-structured instance; thus, $K({\\varepsilon}) = {\\varepsilon}^{-{\\varepsilon}^{-2}}$ is an upper bound on $|{\\mathcal{A}}|$. We define the {\\em important groups} as ${\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ containing all significant and massive groups. The construction of classes relies on the fractional grouping technique, which finds a partition of an important group into a small number of sets, each of roughly the same frequency according to $\\bar{y}$. Items of the same set can be treated as having the same size, as in the classic linear shifting technique \\cite{fernandez1981bin}. \n \n  We use fractional grouping separately for each \n  important group. Consider an important group $G \\in \\mathcal{G}_{\\eta} \\cup {\\mathcal{A}}$.  For the construction of  the small classes of $G$, we use the following inductive definition.\\footnote{Recall that for any two items $\\ell,j \\in I$ such that $s(\\ell) > s(j)$ it holds that $\\ell < j$.} Let $\\min (G)$ ($\\max (G)$) denote the minimal (maximal) index of an item in $G$. Define $q_0 = 1+\\max (G)$. For $i \\geq 1$, let $\\upsilon = 3{\\varepsilon}^{-2}$, and \n  \n  \\begin{equation}\n\t\\label{eq:q}\n\t\tq_i = \\max \\biggl\\{  \\ell \\in G ~\\bigg|~ f_{\\bar{y}} \\bigl(\\{j \\in G ~|~ \n\t\\ell\t  \\leq j < q_{i-1} \\}\\bigr)~ \\geq ~{\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\| \\biggr\\}.\n\\end{equation} \n\nIf the maximum in \\eqref{eq:q} is defined over a empty set then  $q_i =  \\min (G)$,\nand we set $\\tau(G) = i$ to be the number of small classes of $G$.  In words, $q_i$ is the maximal index of an item in $G$ such that the frequency of the set of items with indices \nin the range $\\{ q_i, \\ldots , q_{i-1}-1 \\}$\nis at least ${\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|$. Now, for any $i \\in [\\tau(G)]$, define $\\Delta_i(G) = \\{\\ell \\in G ~|~ q_i  \\leq \\ell <  q_{i-1} \\}$ as the $i$-th {\\em class} of $G$, which contains all items in $G$ of indices smaller than $q_{i-1}$ and at lease $q_i$. \n \n\\begin{observation}\n\t\\label{ob:FG}\n\tGiven $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$, the classes $\\Delta_1(G), \\ldots, \\Delta_{\\tau(G)}(G)$  of $G$ satisfy the following. \n\t\n\\begin{enumerate}\n\t\\item $\\tau(G)\\leq {\\varepsilon}^{-\\upsilon-10}$\n\t\n\t\\item For all $i \\in [\\tau-1]$ and $\\ell \\in \\Delta_i(G)$ it holds that $\\Delta_{i+1}(G) \\subseteq \\textnormal{\\textsf{fit}}\\left( \\ell\\right)$.\n\t\n\t\\item For all $i \\in [\\tau]$ it holds that $f_{\\bar{y}}(\\Delta_i(G)) \\leq {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|+2$.\n\t\n\t\\item For all $i \\in [\\tau-1]$ it holds that $f_{\\bar{y}}(\\Delta_i(G)) \\geq {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\|$.\n\\end{enumerate}\n\\end{observation}\n\nObservation~\\ref{ob:FG} follows from \\eqref{eq:q}. Specifically, the third propertyholds since  $f_{\\bar{y}}(\\ell) \\leq 2$ for all $\\ell \\in I$, by the properties of $\\bar{y}$. By Observation~\\ref{ob:FG}, for any $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$ and $ i \\in [\\tau(G)-1]$, each item in $\\Delta_{i+1}(G)$ fits in place of a slot $\\ell \\in \\Delta_i(G)$. This is useful for constructing a prototype $\\bar{z}$ such that the $\\bar{z}$-polytope contains a point in which items in $\\Delta_{i+1}(G)$ are assigned to at most ${\\varepsilon}^{-10}$ representatives from $\\Delta_i(G)$. Thus, given at most ${\\varepsilon}^{-10}$ representatives $\\Delta'_i(G) \\subseteq \\Delta_i(G)$, we can define $\\bar{z}$ such that $f_{\\bar{z}} \\left( \\Delta'_i(G) \\right) \\approx f_{\\bar{y}} \\left(\\Delta_{i+1}(G)\\right)$ and the size of the support of $\\bar{z}$ depends solely on a function of ${\\varepsilon}$. Finally, define the set of classes as all the classes of important groups; that is,\n\n\\begin{equation}\n\t\\label{eq:classS}\n\t\\mathcal{Q} =  \\bigcup_{G \\in {\\mathcal{G}}_{\\eta}\\cup {\\mathcal{A}}} \\{\\Delta_{1}(G), \\ldots, \\Delta_{\\tau(G)}(G)\\}. \n\\end{equation} Observe that ${\\varepsilon}^{-\\upsilon-10}\\cdot \\left( \\eta+K({\\varepsilon}) \\right)$ is an upper bound on $|\\mathcal{Q}|$: there are $\\eta$ significant groups and at most $K({\\varepsilon})$ massive groups because ${\\mathcal{I}}$ is ${\\varepsilon}$-structured; in addition, each group has at most ${\\varepsilon}^{-\\upsilon-10}$ classes, by Observation~\\ref{ob:FG}. Note that the classes in $\\mathcal{Q}$ do not intersect; moreover, except for the small items from non-important groups which are absent in these classes, each item belongs to exactly one class in $\\mathcal{Q}$. \n\n\n\n\nWe now define a mapping from configurations to representatives from a subset of the classes. Observe that for groups $G \\in {\\mathcal{G}}$ with $k(G) >1$, a configuration may contain more than  one item in $G$. Hence, given a configuration $C \\in \\textsf{supp}(\\bar{y})$ and a class $\\Phi \\in \\mathcal{Q}$, items in $C \\cap\\Phi$ are mapped to the subset of  $|C \\cap \\Phi|$  items in $\\Phi$ of minimal size; that is, the number of items which belong to $\\Phi$ in $C$ does not change by the mapping. We construct below a mapping for all items from important groups. \n\n\n\n\nGiven a subset of items $S \\subseteq I$, sorted in increasing order by item indices, let \n$\\textsf{last}_k(S)$ be the set of the last $k$ items in $S$, where $k \\in \\{0,1, \\ldots , |S|\\}$. For example, given $S = \\{9, 13, 88, 103, 1093\\}$, we have that $\\textsf{last}_3(S) = \\{88, 103, 1093\\}$. Given $\\Phi \\in \\mathcal{Q}$ and $C \\in \\textsf{supp}(\\bar{y})$, the items in $C \\cap \\Phi$ are mapped to $\\textsf{last}_{|C \\cap \\Phi|}(\\Phi)$, the last $|C \\cap \\Phi|$ items in $\\Phi$; thus, the number of items in $C \\cap \\Phi$ does not change.  Finally, given $C \\in \\textsf{supp}(\\bar{y})$, the {\\em projection} of $C$ is: \n\n\\begin{equation}\n\t\t\\label{eq:mapS,eq:mapM}\n\tP(C) = \\bigcup_{\\Phi\\in \\mathcal{Q}} \\textsf{last}_{|C \\cap\\Phi|}(\\Phi). \n\\end{equation} By the next lemma, the prototype induced by replacing each configuration $C \\in \\textsf{supp}(\\bar{y})$ by the projection of $C$ has a support of a small size.\n\n \n\n \\begin{lemma}\n\t\n\t\\label{lem:P(C)}\n\tFor all $C \\in \\textnormal{\\textsf{supp}}(\\bar{y})$ the following hold for the projection of $C$.\n\t\\begin{enumerate}\n\t\t\n\t\n\t\t\n\t\t\\item   $s\\left(P(C)\\right) \\leq s(C)$.\n\t\t\n\t\t\\item $\\textnormal{\\textsf{fit}}(C)  \\subseteq \\textnormal{\\textsf{fit}}(P(C))$. \n\t\t\n\t\t\\item $|P(C)| \\leq {\\varepsilon}^{-10}$.\n\t\t\n\t\t\\item $|\\{P(C')~|~C' \\in \\textnormal{\\textsf{supp}}(\\bar{y})\\}| \\leq Q({\\varepsilon})-3K({\\varepsilon})$. \n\t\t\n\t\t\\item For all $G \\in {\\mathcal{G}}$ it holds that $|G \\cap P(C)| \\leq |G \\cap C|$. \n\t\\end{enumerate}\n\\end{lemma}\n\nProperties 1-3 of Lemma~\\ref{lem:P(C)} hold since for any $C \\in \\textsf{supp}(\\bar{y})$ and $\\Phi \\in \\mathcal{Q}$, the items in $C \\cap \\Phi$ are mapped by $P$ to $|C \\cap \\Phi|$ items of smaller or equal size from the same group. The fourth property holds since the number of classes is bounded, and $|C \\cap \\Phi| \\leq {\\varepsilon}^{-10}$. The last bound follows \nby noting that the items of each class are mapped to the same number of items, whereas items from non-important groups are discarded from the configuration. \n\n The projection function is utilized to construct a good prototype, where the fractional selection of a configuration $C \\in \\textsf{supp}(\\bar{y})$ is added to the corresponding entry of the projection $P(C)$. To guarantee that the returned prototype $\\bar{z}$ has non-empty $\\bar{z}$-polytope, the selection of each configuration is slightly increased. In addition, the selection of the empty-configuration and of a small number of configurations of one slot is further augmented. \n \n\n\\begin{algorithm}[htb]\n\t\\caption{$\\textsf{Shift} (\\bar{y}$)}\n\t\\label{Alg:shifting}\n\t\n\nConstruct the classes $\\mathcal{Q}$.\\label{step:classesQ}\n\n\tDefine $\\bar{u} =  \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right)\\left(  \\sum_{C \\in \\textsf{supp}(\\bar{y})} \\bar{y}_C \\cdot \\mathbbm{1}^{P(C)}  \\right)$.\\label{step:u}\n\n\treturn $\\bar{z} = \\bar{u}+\\left( 4 {\\varepsilon} \\cdot \\|\\bar{y}\\| +{\\varepsilon}^{-3} \\right)\\cdot \\mathbbm{1}^{\\emptyset} +\\sum_{G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}} \\left(  {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\| + 2 \\right)\\cdot \\mathbbm{1}^{r(G)}$.\\label{step:shiftreturn}\n\\end{algorithm}\n \n Specifically, for each $G \\in {\\mathcal{G}}$, let $r(G) = \\{ \\min G \\}$ be the {\\em maximal slot} of $G$; this is a configuration containing the item of maximal size (and minimal index) in $G$. For each $C \\in {\\mathcal{C}}$ define the {\\em indicator} of $C$, a vector $\\mathbbm{1}^C \\in [0,1]^{{\\mathcal{C}}}$ such that $\\mathbbm{1}^C_C = 1$, and for all $C' \\in {\\mathcal{C}} \\setminus \\{C\\}$ define $\\mathbbm{1}^C_{C'} = 0$. Algorithm \\textsf{Shift} computes the prototype $\\bar{z}$ in two steps. First, we construct a prototype $\\bar{u}$ as  a linear combination of the indicators of all configurations $C \\in {\\mathcal{C}}$, where $C$ contributes $ \\left(1+\\frac{2{\\varepsilon}^{-\\upsilon}}{\\|\\bar{y}\\|}\\right) \\cdot \\bar{y}_C$ to $\\bar{u}_{P(C)}$. Then, given $\\bar{u}$, we construct $\\bar{z}$ by adding $4{\\varepsilon} \\|\\bar{y}\\|+{\\varepsilon}^{-3}$ to $\\bar{u}_{\\emptyset}$ and adding $ {\\varepsilon}^{\\upsilon} \\cdot \\|\\bar{y}\\| + 2$ to $r(G)$ for each $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$. It can be easily shown that $\\|\\bar{z}\\| \\leq (1+5{\\varepsilon})\\|\\bar{y}\\|+Q({\\varepsilon})$. The pseudocode of algorithm \\textsf{Shift} is given in Algorithm~\\ref{Alg:shifting}. The proof of Lemma~\\ref{lem:Cnf} is given in Appendix~\\ref{app:omittedShifting}. \n\n \n To show that the $\\bar{z}$-polytope is non-empty, we construct a point $\\bar{\\psi} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ which satisfies most of the constraints of the $\\bar{u}$-polytope. Then, based on $\\bar{\\psi}$, we construct another point $\\bar{\\lambda} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ and show that $\\bar{\\lambda}$ is in the $\\bar{z}$-polytope. For all $C \\in {\\mathcal{C}}$, let $P^{-1}(C) = \\{C' \\in {\\mathcal{C}}~|~ P(C') = C\\}$ be all configurations mapped by the projection to $C$. The construction of $\\bar{\\psi}$ relies on a point $\\bar{\\gamma} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ in the $\\bar{y}$-polytope, where for all $\\ell, j \\in I$ such that $\\ell \\neq j$ it holds that $\\bar{\\gamma}_{\\ell,j} = 0$. \n \n \n The point $\\bar{\\psi}$ is defined as follows. For all $\\ell \\in I$ and $C \\in {\\mathcal{C}}$ define $\\bar{\\psi}_{\\ell,C} = \\sum_{C' \\in P^{-1}(C)} \\bar{\\gamma}_{\\ell,C'}$. That is, $\\ell$ is assigned to $C$ as the fractional assignment of $\\ell$ by $\\bar{\\gamma}$ to configurations projected to $C$.  Additionally, for any $G \\in {\\mathcal{G}}_{\\eta} \\cup {\\mathcal{A}}$, $i \\in [\\tau(G)-1]$, $\\ell \\in \\Delta_{i+1}(G)$, and $j \\in \\Delta_{i}(G)$ define $$\\bar{\\psi}_{\\ell,j} =   \\frac{f_{\\bar{u}}(j)}{f_{\\bar{u}}(\\Delta_{i}(G))} \\cdot \\bar{\\gamma}_{\\ell,\\ell}.$$ This is where the {\\em shifting} comes into play: all items from the important group $G$ in class $\\Delta_{i+1}(G)$ are assigned to at most ${\\varepsilon}^{-10}$ slots in the consecutive class. By \\eqref{eq:mapS,eq:mapM}, Observation~\\ref{ob:FG}, and Step~\\ref{step:u} of Algorithm~\\ref{Alg:shifting}, it follows that $f_{\\bar{u}} \\left(\\Delta_{i}(G)\\right) \\approx f_{\\bar{y}} \\left(\\Delta_{i+1}(G)\\right)$ and $\\Delta_{i+1}(G) \\subseteq \\textsf{fit}(\\min \\Delta_{i}(G))$. Hence, most of the constraints in the $\\bar{u}$-polytope of items in $\\Delta_{i+1}(G)$ are satisfied. For any other entry $(\\ell',t') \\in I \\times (I \\cup {\\mathcal{C}})$ not defined above, define $\\bar{\\psi}_{\\ell',t'} = 0$. \n \n\t\\begin{claim}\n\t\\label{ob:psi}\n\t$\\bar{\\psi}$ satisfies constraints \\eqref{F1}, \\eqref{F2}, \\eqref{F5}, \\eqref{F3} of the $\\bar{u}$-polytope. \n\\end{claim} \n\nConstraint \\eqref{F1} of the $\\bar{u}$-polytope is satisfied for $\\bar{\\psi}$ by Observation~\\ref{ob:FG} and Lemma~\\ref{lem:P(C)}. Constraint \\eqref{F2} is satisfied by Lemma~\\ref{lem:P(C)}; and constraint \\eqref{F3} follows from Observation~\\ref{ob:FG}. As $\\bar{\\psi}$ may fail to satisfy constraint \\eqref{F4} of the $\\bar{u}$-polytope, we define $\\bar{\\mu} \\in \\mathbb{R}^{I}_{\\geq 0}$ such that for all $\\ell \\in I$ $\\bar{\\mu}_{\\ell} = \\max \\left\\{0,1-\\sum_{t \\in I \\cup {\\mathcal{C}}} \\bar{\\psi}_{\\ell,t}\\right\\}$ is the fractional assignment that is missing for item $\\ell$ to satisfy constraint \\eqref{F4} for $\\psi$.\n\n\t\\begin{claim}\n\t\\label{lem:mu}\n\tThe following hold for $\\bar{\\mu}$.\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\textnormal{\\textsf{supp}}(\\bar{\\mu}) \\setminus L \\subseteq \\textnormal{\\textsf{fit}}(\\emptyset)$.\n\t\t\n\t\t\t\\item For all  $\\ell \\in \\textnormal{\\textsf{supp}}(\\bar{\\mu})$ it holds that $\\ell \\in \\textnormal{\\textsf{fit}}\\left(r(\\textnormal{\\textsf{group}}(\\ell))\\right)$.\n\t\t\n\t\t\\item \tFor all $G \\in {\\mathcal{G}}$ it holds that $\\sum_{\\ell \\in G} \\bar{\\mu}_{\\ell} \\leq {\\varepsilon} \\cdot \\|\\bar{y}\\|+2$.\n\t\t\n\t\t\\item \t$\\sum_{\\ell \\in I\\setminus L} \\bar{\\mu}_{\\ell}  \\cdot s(\\ell) \\leq {\\varepsilon}\\cdot\\|\\bar{y}\\|$.\n\t\\end{enumerate}\n\t\n\\end{claim} \n\nThe proof of Properties 3 and 4 in Claim~\\ref{lem:mu} follows from Lemma~\\ref{lem:significantDiscard} and Observation~\\ref{ob:FG}, since $\\textsf{supp}(\\bar{\\mu})$ contains only (i) small items from non-important groups, and (ii) items from the first class of each important group. \n\n\nNow, we define another point $\\bar{\\rho} \\in \\mathbb{R}^{I \\times (I \\cup \\mathcal{C})}_{\\geq 0}$ based on $\\bar{\\mu}$. For all $\\ell \\in I \\setminus L$ define $\\bar{\\rho}_{\\ell,\\emptyset} = \\bar{\\mu}_{\\ell}$. Also, for all $G \\in {\\mathcal{G}}$, $\\ell \\in L \\cap G$ define $\\bar{\\rho}_{\\ell, r(G)} = \\bar{\\mu}_{\\ell}$; for any other $(\\ell,t) \\in I \\times \\left(I \\cup {\\mathcal{C}}\\right)$, define $\\bar{\\rho}_{\\ell,t} = 0$. Finally, using the definitions of $\\bar{\\rho}$ and $\\bar{\\psi}$, let $\\bar{\\lambda} = \\bar{\\rho}+\\bar{\\psi}$. By the above, all items that are not fully assigned by $\\bar{\\psi}$ are assigned to the empty-configuration or to the configuration containing a single item that is largest in the group. The constraints of the $\\bar{z}$-polytope are satisfied, since the empty-configuration and the maximal slots of important groups have increased selection.  \nThe next claim follows from Claims~\\ref{ob:psi} and~\\ref{lem:mu}. \n\n\t\\begin{claim}\n\t\\label{lem:shifting1}\n\t$\\bar{\\lambda}$ is in the $\\bar{z}$-polytope. \n\\end{claim}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFormulation of quantum mechanics avoiding operators  was given at the beginning of quantum era by Weyl--Wigner--Groenewold--Moyal (WWGM)\\cite{wwgm}\nas the $\\hbar$--deformation of classical mechanics:\nObservables are functions taking values in the classical phase space $(P_I,Q_I)$ whose combinations are constructed through the star product\n\\begin{equation}\n\\label{SCT}\n\\star   =\n\\exp \\left[\n\\frac{i\\hbar }{2} \\left(\n\\frac{\\overleftarrow{\\partial}}{\\partial Q^I}\n\\frac{\\overrightarrow{\\partial}}{\\partial P_I}\n-\\frac{\\overleftarrow{\\partial}}{\\partial P_I}\n\\frac{\\overrightarrow{\\partial}}{\\partial Q^I}\n\\right) \\right] ,\n\\end{equation}\nwhere $\\overleftarrow{\\partial}$ $(\\overrightarrow{\\partial})$ indicates\nthat the derivative is applied to the left (right). Although the WWGM\nformulation of quantum mechanics has some shortcomings, like the\nlack of positive definite probabilities, it turned out to be crucial\nto establish quantum mechanics in noncommutative coordinates: One\ntreats coordinates as commutative and imply\nthe noncommutativity by the star product given with an\nantisymmetric, constant deformation parameter $\\theta_{IJ}$  as\n\\begin{equation}\n\\label{NSP}\n\\star_\\theta   \\equiv\n\\exp \\left[\n\\frac{i}{2}\\theta_{IJ}\n\\frac{\\overleftarrow{\\partial}}{\\partial Q^I}\n\\frac{\\overrightarrow{\\partial}}{\\partial Q^J}\n \\right] .\n\\end{equation}\n$\\theta$--deformed  Hamiltonian  systems are built inserting the star product (\\ref{NSP})  between the bilinear or higher terms\nappearing in the original quantum Hamiltonians. Indeed, this is equivalent to the shift of coordinates\n\\begin{equation}\n\\label{SH}\n Q_I\\rightarrow  Q_I-\\frac{1}{2\\hbar}\\theta_{IJ}  P^{op}_J,\n\\end{equation}\nwhere $P_I^{op} =-i\\hbar\\frac{\\partial}{\\partial Q^I}$  is used. As it is obvious when the original Hamiltonian does not possess any coordinate\ndependence this procedure will not furnish any $\\theta$--deformation. Especially, it is not adequate to consider spin degrees of freedom.\n\nThere is an alternative method of defining\nquantum Hamiltonians in noncommutative spaces\\cite{oj-09}\nas far as in the starting Hamiltonian there exist terms which can be interpreted as minimally coupled gauge fields\nwhich may be either functions or matrix valued.\nIn this work we would like to discuss this alternative procedure of obtaining dynamical systems in noncommuting coordinates\nand apply it to some interesting physical systems where the Hamiltonians are functions or matrices.\n\nEmploying the star product (\\ref{NSP}) or the equivalent shift (\\ref{SH}), diverse physical systems\nlike Hall effect in noncommuting coordinates\\cite{ao} and quantum phases in noncommutative spaces\\cite{ao,glr,cpst,mz,wl,prfn} were considered.\nIn all these dynamical problems there are some external fields which can be interpreted as gauge fields\ninteracting with the particles in terms of the minimal coupling procedure.\nWithin the alternative method  we first\nshow that Hall effect in noncommutative coordinates can be formulated where resulting Hall conductivity is $\\theta$--deformed or not,\ndepending on the realization adopted. Then, we apply the new deformation procedure to obtain    velocity\nindependent formulations of\nAharonov--Bohm (AB)\\cite{ab}, Aharonov--Casher (AC)\\cite{ac}, He--McKellar--Wilkens (HMW)\\cite{hmw} and Anandan\\cite{anan,flr} phases\nin noncommutative coordinates.\nMost of the earlier formulations yielded velocity dependent quantum phases in noncommutative spaces,\nin spite of the fact that the distinguished property of the original phases is their independence  from the velocity of  scattered particles.\nWe presented  a unified  formulation of the $\\theta$--deformed quantum phases yielding velocity dependent terms.\nThe difference between them and our\nmethod is clarified. In the last section we\ndiscuss the results obtained and their consequences. In particularly we discuss how to select the suitable realization.\n\n\n\n\\section{The Alternative $\\theta$--deformation of Quantum Mechanics}\n\nGenerating classical mechanics as the $\\hbar \\rightarrow 0$ limit  of  quantum mechanics can be best perceived by the WWGM method.\nLet $(P_I , Q_I )$ denote the classical variables corresponding to\nthe quantum phase space variables $( P^{op}_I , Q^{op}_I);$\n$I =1,\\cdots ,M.$  Multiplication of observables in\nthe former  space is given by the usual operator product,\nhowever in the latter  the star product (\\ref{SCT}) is employed to carry out multiplications.\nIn the WWGM approach, to imitate quantum commutators one introduces the Moyal brackets\n$$\n[f(P,Q ) , g(P,Q ) ]_\\star \\equiv f(P,Q) \\star g(P,Q ) - g(P,Q ) \\star f(P,Q ) ,\n$$\nwhere  the observables $f(P,Q )$ and $g(P,Q )$  are some functions. Hence the\nclassical limit  of the commutators is equivalent to\n\\begin{equation}\n{\\lim_{\\hbar \\rightarrow 0}} \\frac{-i}{\\hbar } [f(P,Q ) , g(P,Q ) ]_\\star\n= \\{ f(P,Q) , g(P,Q) \\} \\equiv\n  \\frac{\\partial f}{\\partial Q^I} \\frac{\\partial g}{\\partial P_I}\n-\\frac{\\partial f}{\\partial P_I}\\frac{\\partial g}{\\partial Q^I} ,\\label{ocl}\n\\end{equation}\nwhich is the Poisson bracket.\nWhen the observables are  matrices  whose elements are  $M_{kl}(P,Q)$ and $N_{kl}(P,Q),$\nthe Moyal bracket is\n\\begin{equation}\n\\label{MM}\n\\left( [M(P,Q) , N(P,Q ) ]_\\star \\right)_{kl}  =  M_{km}(P,Q ) \\star N_{ml}(P,Q )\n- N_{km}(P,Q ) \\star M_{ml}(P,Q ) .\n\\end{equation}\nHowever, its classical limit (\\ref{ocl}),\nin addition to the Poisson brackets, yields\na singular commutator of matrices.\nHence, instead of the classical limit (\\ref{ocl}) we\ndeal with the  ``semiclassical'' limit defined by retaining the  terms\nup to  $\\hbar $ of  the Moyal bracket (\\ref{MM}):\n\\begin{equation}\n \\{M(P,Q ) , N(P,Q ) \\}_C \\equiv\n\\frac{-i}{\\hbar}[M,N]+\n\\frac{1}{2} \\{ M(P,Q ) , N(P,Q) \\}\n-\\frac{1}{2} \\{ N(P,Q ) , M(P,Q ) \\} . \\label{SCB}\n\\end{equation}\nThe first term is the\ncommutator of matrices, it should not be confused with the quantum mechanical commutator.\nObviously, the bracket (\\ref{SCB}) does not satisfy Jacobi identities. However, we consider a semiclassical approach\nin which the semiclassical limit  is taken after performing multiplication of the observables in terms of the star product.\nHence, vanishing of  the first two terms of the Moyal bracket relation\n$$\n\\frac{-i}{\\hbar}\\left(\\{M,\\{ N,L \\}_\\star \\}_\\star +\n\\{N,\\{ L,M \\}_\\star\\}_\\star  +\n\\{L,\\{ M,N \\}_\\star\\}_\\star \\right) = {\\cal O}_{-1} (\\frac{1}{\\hbar})+{\\cal O}_0 (\\hbar^0)+{\\cal O}_1 (\\hbar) +\\cdots\n$$\nshould be considered in  the semiclassical limit of the Jacobi identity.\nIndeed, one can show that the semiclassical limit of the Jacobi identity\n\\begin{eqnarray}\n{\\cal O}_{-1} (\\frac{1}{\\hbar})+{\\cal O}_0 (\\hbar^0) & = &\n-\\frac{i}{\\hbar}[M,[N,L]]+[M, \\{ N , L \\}]\n-[M , \\{ L , N\\}] \\nonumber \\\\\n&&\n+\\{M ,[N,L]\\}\n -\\{[N,L],M\\}\n + ({\\rm cyclic\\ permutations\\ of}\\  M,N,L) =0 \\nonumber\n\\end{eqnarray}\nis satisfied.\nMoreover, one can observe that the semiclassical limit of the Leibniz rule defined  as\n$$\n\\{M\\star N, L\\}_C=\\{M,L\\}_C\\star N+ M\\star \\{N,L\\}_C ,\n$$\nis also satisfied at the $\\hbar$ order.\nAlthough, these considerations are essential for a consistent definition of the semiclassical method of matrix observables, in this work we will\ndeal with a quantum phase space algebra obtained somehow utilizing the bracket (\\ref{SCB}). Obviously, adopted operator realization\nof commutators among the phase space variables should  satisfy the usual Jacobi identities as we will discuss.\n\nDynamical systems in noncommutative space can be formulated in terms of\nthe  following first order matrix Lagrangian (for the most general case see \\cite{o1}),\nby choosing the coordinates as $Q_I=(r_\\alpha ,p_\\alpha),$ where\n $\\alpha,\\beta =1,\\cdots ,{\\rm d}, $\n\\begin{equation}\n\\label{lag}\nL   = {\\dot r}^\\alpha \\left[ \\frac{p_\\alpha}{2}\\mathbb{I} +\\rho A_\\alpha (r) \\right]\n-\\frac{{\\dot p}^\\alpha}{2} \\mathbb{I}\\left[r_\\alpha  + \\frac{\\theta_{\\alpha \\beta}}{ \\hbar} p^\\beta  \\right] - H_0(r,p) .\n\\end{equation}\n$\\mathbb{I}$ denotes the unit matrix possessing the same dimension with the matrix valued gauge field ${\\mathcal A}_\\alpha .$ We denoted the coupling constant as  $\\rho$.\nAlthough (\\ref{lag}) is classical, $\\hbar$ is present to furnish the constant, antisymmetric noncommutativity parameter $\\theta_{\\alpha \\beta }$ with\nthe dimension $\\left(\\sf{length}\\right)^2$.\nThe canonical momenta  $P_I=(P_r^\\alpha =\\partial L\/ \\partial \\dot r_\\alpha ,\\  P_p^\\alpha =\\partial L \/ \\partial \\dot p_\\alpha),$\ncorresponding to the coordinates $Q_I=(r_\\alpha ,p_\\alpha),$\nyield the dynamical  constraints\n\\begin{eqnarray}\n\\psi^{1\\alpha} & \\equiv & (P_r^\\alpha -\\frac{1}{2}p^\\alpha )\\mathbb{I} -\\rho {\\mathcal A}^\\alpha ,\\nonumber \\\\\n\\psi^{2\\alpha} & \\equiv & (P_p^\\alpha +\\frac{1}{2}r^\\alpha )\\mathbb{I} +\\frac{\\theta^{\\alpha \\beta}}{ \\hbar} p_\\beta . \\nonumber\n\\end{eqnarray}\nThey satisfy the semiclassical relations\n\\begin{eqnarray}\n\\{\\psi^1_\\alpha ,\\psi^1_\\beta \\}_C & = & \\rho F_{\\alpha \\beta} ,\\nonumber \\\\\n\\{\\psi^2_\\alpha ,\\psi^2_\\beta \\}_C & = & \\frac{\\theta_{\\alpha \\beta}}{ \\hbar},\\nonumber \\\\\n\\{\\psi^1_\\alpha ,\\psi^2_\\beta \\}_C & = & -\\delta_{\\alpha \\beta} , \\nonumber\n\\end{eqnarray}\nwhere we introduced the field strength:\n\\begin{equation}\nF_{\\alpha \\beta} =\n\\frac{\\partial A_\\beta}{\\partial r^\\alpha}\n-\\frac{\\partial A_\\alpha}{\\partial r^\\beta}\n-\\frac{i\\rho }{\\hbar} [A_\\alpha ,A_\\beta ] .\\label{fst}\n\\end{equation}\nWe would like to emphasize the fact that commutators appearing in this semiclassical formulation are the ordinary matrix commutators.\nObviously, $\\psi^z_\\alpha ;\\ z=1,2,$ are second class constraints, so that one can take them into account by introducing\nthe semiclassical Dirac bracket defined as\n\\begin{equation}\n\\label{sdb}\n\\{M,N\\}_{CD} \\equiv \\{M,N\\}_C -\\{M,\\psi^z\\}_C \\ {\\cal C}^{-1}_{zz^\\prime}\\ \\{\\psi^{z^\\prime},N\\}_C  ,\n\\end{equation}\nin terms of ${\\cal C}^{-1}$ which is\nthe inverse of\n$$\n{\\cal C}^{zz^\\prime}_{\\alpha \\beta}=\n\\{\\psi^z_\\alpha , \\psi^{z^\\prime}_\\beta\\}_C .\n$$\nTherefore,\nomitting $\\mathbb{I}$ on the left hand sides, one can show that\nthe following relations are satisfied,\n\\begin{eqnarray}\n\\{r^\\alpha,r^\\beta \\}_{CD}  & = &\n \\frac{ \\theta^{\\alpha\\beta  }}{\\hbar} , \\label{rr}\\\\\n\\{p^\\alpha,p^\\beta \\}_{CD}  & = &\n\\rho  F^{\\alpha\\beta  }   -\\frac{\\rho^2}{\\hbar}(F\\theta F)^{\\alpha\\beta} ,  \\label{yy}\\\\\n\\{r^\\alpha,p^\\beta \\}_{CD}  & = &\n\\delta^{\\alpha \\beta } - \\frac{\\rho}{\\hbar} (\\theta F)^{\\alpha\\beta} ,   \\label{ry}\n\\end{eqnarray}\nkeeping the terms at the first order in $\\theta$ and  at the second\norder in $\\rho .$\nWe abbreviated $(\\theta F)^{\\alpha\\beta} \\equiv \\theta^{\\alpha\\gamma}F_{\\gamma}^{\\beta} ,\\ (F\\theta\nF)^{\\alpha\\beta} \\equiv F^{\\alpha\\gamma} \\theta_{\\gamma}^\\sigma F_{\\sigma}^{\\beta}$.\n\nThe semiclassical brackets (\\ref{SCB})  differ from the Poisson brackets up to commutators of matrices, so that for observables which are\nnot matrices (\\ref{sdb})  reduce to the ordinary Dirac brackets. Therefore, we can extend the canonical quantization rules to embrace the matrix observables\nby the substitution of the basic brackets (\\ref{rr})--(\\ref{ry})\nwith the quantum commutators as\n$\\{, \\}_{CD} \\rightarrow \\frac{1}{i\\hbar} [ ,]_q .$ Then, we are furnished with\nthe generalized algebra\n\\begin{eqnarray}\n{[\\hat r^\\alpha ,\\hat r^\\beta ]_q}  & = &\n i \\theta^{\\alpha\\beta  } ,\\label{rrq}\\\\\n{[ \\hat p^\\alpha,\\hat p^\\beta ]_q } & = &\ni\\hbar \\rho  F^{\\alpha\\beta  }   -i\\rho^2(F\\theta F)^{\\alpha\\beta} , \\label{yyq}\\\\\n{[\\hat r^\\alpha,\\hat p^\\beta ]_q } & = &\ni\\hbar \\delta^{\\alpha \\beta } - i\\rho (\\theta F)^{\\alpha\\beta} ,  \\label{ryq} \\\\\n{[\\hat p^\\alpha , \\hat r^\\beta ]_q}  & = &\n-i\\hbar \\delta^{\\alpha \\beta } + i\\rho (F \\theta )^{\\alpha\\beta}   .\\label{yrq}\n\\end{eqnarray}\nWe denoted quantum commutators by the subscript $q$ to distinguish them from matrix commutators.\nBecause of being  first order in $\\theta$, the right hand sides of\n(\\ref{rrq})--(\\ref{yrq}) may only possess ${\\hat r}_\\alpha|_{\\theta =0} =r_\\alpha $ dependence.\nHence $F_{\\alpha\\beta}$ is still as in (\\ref{fst}).\nThis is the starting point of the alternative method of establishing  quantum mechanics in noncommutative coordinates.\nA realization of the generalized algebra (\\ref{rrq})--(\\ref{yrq}) and a Hamiltonian $H_0(p,r)$ should be provided.\nLet us  deal with the operators\n\\begin{eqnarray}\n\\hat{p}_\\alpha &=& D_\\alpha\n-\\frac{\\rho}{2\\hbar}F_{\\alpha\\beta}\\theta^{\\beta\\gamma}D_\\gamma, \\label{R1}\\\\\n\\hat{r}_\\alpha &=& r_\\alpha -\\frac{1}{2\\hbar}\\theta_{\\alpha\\beta}D^\\beta \\label{R2},\n\\end{eqnarray}\nwhere the covariant derivative is\n$$\nD_\\alpha = -i\\hbar\\frac{\\partial}{\\partial r^\\alpha}-\\rho A_\\alpha .\n$$\nOne can demonstrate\nthat (\\ref{R1}) and (\\ref{R2})\nsatisfy the algebra  (\\ref{rrq})--(\\ref{yrq}) and the Jacobi identities, as far as the conditions\n\\begin{equation}\n\\label{cons}\n  -i\\hbar\\nabla_\\alpha F_{\\beta \\gamma}  -\\rho [ A_\\alpha , F_{\\beta \\gamma}] = 0,\\quad  [F_{\\alpha \\beta },F_{\\gamma \\delta }]= 0\n\\end{equation}\nare fulfilled.\nTo illustrate the method let the initial Hamiltonian be $H_0 (p)=p^2\/2m .$\nSubstituting $p$ with the quantum operator (\\ref{R1}) one obtains the $\\theta$--deformed Hamiltonian\n\\begin{equation}\n\\label{IHAM}\nH_0 (\\hat p)\\equiv {\\hat H}_{nc}= \\frac{1}{2m} \\left( D_\\alpha\n-\\frac{\\rho}{2\\hbar}F_{\\alpha\\beta}\\theta^{\\beta\\gamma}D_\\gamma \\right)^2 .\n\\end{equation}\nSetting $\\theta =0$ yields the Hamiltonian operator\n\\begin{equation}\n\\label{IH}\n\\hat H= \\frac{1}{2m} \\left(  -i\\hbar\\frac{\\partial}{\\partial\nr_\\alpha}-\\rho A_\\alpha\\right)^2 .\n\\end{equation}\nTherefore, (\\ref{IHAM}) gives the noncommutative dynamics corresponding to the Hamiltonian  (\\ref{IH}).\n\nLet us present another representation of the algebra (\\ref{rrq})--(\\ref{yrq})  which will be utilized in the subsequent sections.\nAs far as (\\ref{cons})  are valid, one can prove that\n\\begin{eqnarray}\n\\label{rea21} \\hat{p}_\\alpha &=& -i\\hbar\\nabla_\\alpha + \\frac{\\rho}{2}\nF_{\\alpha\\beta}(r^\\beta+ 2i \\theta^{\\beta\\gamma}\\nabla_\\gamma), \\\\\n\\label{rea22} \\hat{r}_\\alpha &=& r_\\alpha\n-\\frac{1}{2\\hbar}\\theta_{\\alpha\\beta}(-i\\hbar\\nabla^\\beta-\\frac{\\rho}{2}F^{\\beta\\gamma}r_\\gamma)\\end{eqnarray}\nconstitute  another realization of the algebra (\\ref{rrq})--(\\ref{yrq}). It is worth mentioning  that\nin this representation only the gauge invariant field strength $F_{\\alpha\\beta}$ appears, in contrast\nto (\\ref{R1})--(\\ref{R2}) where the gauge field $A_\\alpha$ explicitly appears.\n\n\\section{Hall Effect in Noncommutative Coordinates}\n\nWe would like to study  the Hall effect in  noncommutative\nspace in the light of the alternative method prescribed in Section 2,\nadopting different realizations.\nThis problem was addressed previously in \\cite{ao} where the\nHall conductivity was shown to acquire a $\\theta$--deformation factor.\nHowever this resulted to the cost of  an unnatural overall factor in the definition of electric current.\nWe will show that\n$\\theta$--deformation of the Hall conductivity appears naturally in some realizations. Though in \\cite{om}\na  natural $\\theta$--deformed Hall conductivity was achieved, it was within the semiclassical approach of Section 2.\n\nWe consider\nan electron moving on the two-dimensional plane $r_i=(x, y)$ in the presence  of the\nuniform external in-plane electric field $E$ and the uniform external perpendicular\nmagnetic field $ B.$ The latter is taken into account by\nthe field strength\n$F_{ij}=\\epsilon_{ij}B,$\nand fixing $\\rho=-e\/c .$\nHence the  generalized algebra (\\ref{rrq})--(\\ref{yrq}) yields\n\\begin{eqnarray}\n{[\\hat x ,\\hat y ]_q}  & = & i \\theta  ,\\label{rrqH}\\\\\n{[ \\hat p_i,\\hat p_j ]_q } & =  & -\\frac{ieB\\hbar}{c} \\left(1 - \\frac{eB\\theta}{\\hbar c}  \\right)\\epsilon_{ij}, \\label{yyqH}\\\\\n{[\\hat r_i,\\hat p_j ]_q } & = & i\\hbar \\left( 1 - \\frac{e\\theta B}{\\hbar c} \\right) \\delta_{ij}  .\\label{yrqH}\n\\end{eqnarray}\nWe choose the electric field to lie in the direction of the $x$--axis. Thus the Hamiltonian is taken  as\n\\begin{equation}\n\\label{H0}\nH = \\frac{1}{2m}\\hat{p}_i^2 + e E \\hat{x} .\n\\end{equation}\nFirst  consider  the  realization of (\\ref{rrqH})--(\\ref{yrqH})\ngiven in (\\ref{R1})-(\\ref{R2}) by choosing the symmetric gauge $A_i=\\frac{eB}{2c}\\epsilon_{ij}r_j,$\nbut  ignoring the terms at the\n$e^2\/c^2$ order:\n\\begin{eqnarray}\n\\hat{p}_i^{(1)} &=& (1-\\frac{e\\theta B}{2\\hbar c})p_i - \\frac{eB}{2c}\\epsilon_{ij}r^j ,\\label{rir1} \\\\\n\\hat{r}_i^{(1)} &=& (1-\\frac{e\\theta B}{4\\hbar c})r_i -\\frac{\\theta}{2\\hbar}\n\\epsilon_{ij}p^j . \\label{rir2}\n\\end{eqnarray}\nBy plugging  (\\ref{rir1}) and (\\ref{rir2}) into (\\ref{H0})\none obtains the Hamiltonian\n\\begin{equation}\n\\label{hr11}\nH^{(1)}=\\frac{1}{2m}\\left[ (1-2\\kappa)p_i-\\frac{eB}{2c}\\epsilon_{ij}r^j\n\\right]^2 +eE(1-\\kappa)x-\\frac{eE\\theta}{2\\hbar}p_y ,\n\\end{equation}\nwhere we defined\n$p_i\\equiv -i\\hbar\\nabla_i$ and $ \\kappa=\\frac{e\\theta B}{4\\hbar c}.$\nOn the other hand  keeping the terms at the $e^2\/c^2$ order yields  the following realization\n\\begin{eqnarray}\n\\hat{p}_i^{(2)} &=& (1-\\frac{e\\theta B}{2\\hbar c})p_i -\n\\frac{eB}{2c}(1-\\frac{e\\theta B}{2\\hbar c})\\epsilon_{ij}r^j ,\\label{rir3} \\\\\n\\hat{r}_i^{(2)} &=& (1-\\frac{e\\theta B}{4\\hbar c})r_i -\\frac{\\theta}{2\\hbar}\n\\epsilon_{ij}p^j .\\label{rir4}\n\\end{eqnarray}\nWhen these are substituted in (\\ref{H0}), one gets the $\\theta$--deformed Hamiltonian as\n\\begin{equation} \\label{hr12}\n{H}^{(2)}=\\frac{1}{2m}(1-2\\kappa)^2\\left[ p_i -\n\\frac{eB}{2c}\\epsilon_{ij}r^j \\right]^2 + eE(1-\\kappa)x -\n\\frac{eE\\theta}{2\\hbar}p_y .\n \\end{equation}\n\nAnother realization of (\\ref{rrqH})--(\\ref{yrqH}) can be read from\n(\\ref{rea21})--(\\ref{rea22}): It does not refer to the explicit form of the vector field,\n\\begin{eqnarray} \\hat{p}_i^{(3)} &=& (1-\\frac{e\\theta B}{\\hbar c})p_i\n- \\frac{eB}{2c}\\epsilon_{ij}r^j ,\\label{rir5} \\\\\n\\hat{r}_i^{(3)} &=& (1+\\frac{e\\theta B}{4\\hbar c})r_i -\\frac{\\theta}{2\\hbar}\n\\epsilon_{ij}p^j . \\label{rir6}\n\\end{eqnarray}\nPlugging (\\ref{rir5}) and (\\ref{rir6}) into (\\ref{H0})\nwill produce the deformed Hamiltonian\n\\begin{equation} \\label{hr2} {H}^{(3)} =\n\\frac{1}{2m}\\left[(1-4\\kappa)p_i-\\frac{eB}{2c}\\epsilon_{ij}r^j \\right]^2 +\neE(1+\\kappa)x - \\frac{eE\\theta}{2\\hbar}p_y .\n\\end{equation}\n\nIt is not surprising that there exist different $\\theta$-deformations of an underlying Hamiltonian. \nHowever, as we will argue in Section 5, preferring one to others is possible by adopting an\ninterpretation of the $\\theta$-deformation and then specifying the adequate $\\theta$-deformed physical quantities\nderived  from the deformed Hamiltonians.   \n\nNow we will examine these Hamiltonians in detail. In order to discuss the eigenvalue problem\nof the Hamiltonians we perform the following change of variables\n$$\n{z}= {x}+i {y}, \\qquad {p}_z = \\frac{1}{2}({p}_x-i {p}_y) .\n$$\nLet us introduce two sets of creation and annihilation operators:\n \\begin{eqnarray}\nb = 2i \\gamma {p}_z + \\frac{eB}{2c}\\beta{\\bar{z}} + \\lambda_-; &\nd = 2i\\gamma p_z - \\frac{eB}{2c}\\beta{\\bar{z}}, &\\nonumber \\\\\nb^\\dagger = -2i\\gamma {p}_{\\bar{z}} + \\frac{eB}{2c}\\beta {z} + \\lambda_-  ;&\nd^\\dagger = -2i\\gamma p_{\\bar{z}}- \\frac{eB}{2c}\\beta{z} .& \\nonumber\n\\end{eqnarray}\nThe constant coefficients $\\gamma,\\ \\beta$ and $\\lambda_-$ will be fixed for each\nHamiltonian separately. These two mutually commuting sets  of operators\nsatisfy the commutation relations\n$$\n[b, b^\\dagger] =\n2m\\hbar\\gamma\\beta\\omega, \\qquad\\qquad [d, d^\\dagger]= -2m\\hbar\\gamma\\beta\\omega ,\n$$\nwhere $\\omega=eB\/mc$ is the cyclotron frequency. Each of the  Hamiltonians\n(\\ref{hr11}), (\\ref{hr12}) and (\\ref{hr2}) can be written in terms\nof creation and annihilation operators   in the  form\n\\begin{equation}\n\\label{HG}\n{H} = \\frac{1}{4m}(b b^\\dagger + b^\\dagger b) - \\frac{\\lambda_+}{2m}(d+ d^\\dagger) - \\frac{\\lambda_-^2}{2m} ,\n\\end{equation}\nwhere the constant  $\\lambda_+$ is also\ngoing to be fixed for each Hamiltonian separately. The natural definition of the current operator\nis\n\\begin{equation}\n {{J}_i}=\n \\frac{ie\\rho_e}{\\hbar}[ H,r_i ]= \\frac{e\\rho_e\n \\gamma}{m}(\\gamma{p}_i -\\beta \\frac{eB}{2c}\\epsilon_{ij}r^j +{a}_i) ,\n\\end{equation}\nwhere $\\rho_e$ stands for electron density and   ${a}_i=(0,\n-\\frac{emE\\theta}{2\\hbar\\gamma})$. The expectation value of the current\noperator $<{{J}_i}>$ can be calculated with respect to the\neigenstates  of the Hamiltonian (\\ref{HG}) given as\\cite{ao}\n$$\n|n,\\alpha,\\theta> =\n\\frac{1}{ \\sqrt{(2m\\hbar \\gamma \\beta)^n n!} }\n\\exp\\{i(\\alpha y+\\frac{m\\gamma \\beta }{ 2\\hbar }xy)\\}  (b^{\\dagger})^n|0>,\n\\qquad n=0,1,2...,\\qquad \\alpha\\in \\mathbb{R}.\n$$\nBy definition $b|0>=0.$\nOnce the current\noperator is obtained in terms of creation and annihilation operators,\nthe calculation is straightforward. Indeed, one can easily show that expectation value of\nthe $x$-component of current vanishes:\n\\begin{equation} <{J}_x> = \\langle\nn,\\alpha,\\theta| \\frac{e\\rho_e\\gamma}{2 i m}(b-b^\\dagger)|n,\\alpha,\\theta \\rangle= 0 .\n\\end{equation}\nOn the other hand\nexpectation value of the $y$--component leads to the Hall conductivity $\\sigma_H :$\n\\begin{eqnarray} <{J}_y> &=&\n\\frac{e\\rho_e \\gamma}{m} \\langle n,\\alpha,\\theta |\\left( \\frac{b+b^\\dagger}{2}\n-\\lambda_- -\\frac{emE\\theta}{2\\hbar\\gamma } \\right)|n,\\alpha,\\theta\\rangle \\nonumber \\\\\n&=& -\\frac{e\\rho_e }{m}\\left( \\gamma \\lambda_- +\\frac{emE\\theta}{2\\hbar } \\right)=\\sigma_H  E  .\\end{eqnarray}\n\nNow we will analyze each Hamiltonian separately:\nThe coefficients related to (\\ref{hr11}), (\\ref{hr12}) and (\\ref{hr2}) are presented\nin Table 1.\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline \\hfill             & $\\gamma$    & $\\beta$   & $\\lambda_+ $                                     & $\\lambda_- $   \\\\\n\\hline $H^{(1)}$   & $  1-2\\kappa $    & $1$       & $(1-\\kappa)\\lambda + \\frac{emE\\theta}{ 4\\hbar} $ & $(1-\\kappa)\\lambda - \\frac{emE\\theta}{4\\hbar} $ \\\\\n\\hline $H^{(2)}$   & $1-2\\kappa$    & $1-2\\kappa$  & $(1+2\\kappa) \\lambda $                              & $\\lambda$ \\\\\n\\hline $H^{(3)}$   & $1-4\\kappa$    & $1$       & $(1+\\kappa)\\lambda +\\frac{emE\\theta}{4\\hbar} $   & $(1+\\kappa)\\lambda -\\frac{emE\\theta}{4\\hbar} $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Coefficients of the diverse realizations in terms of  $ \\kappa=\\frac{e\\theta B}{4\\hbar c}$ and  $\\lambda=\\frac{ m c E}{B} .$}\n\\end{table}\n\nFor the Hamiltonian (\\ref{hr12}) the\nHall conductivity does not acquire any $\\theta$--deformation:\n\\begin{equation}\n\\label{hcon2}\n \\sigma_H^{(2)}= -\\frac{\\rho_e ec}{B} .\n\\end{equation}\nConversely, one can observe that, although their coefficients differ  the\nHamiltonians (\\ref{hr11}) and (\\ref{hr2})   give\nthe same result for the Hall conductivity:\n\\begin{equation}\n\\label{hcon1}\n\\sigma_H^{(1)} (\\theta)=\\sigma_H^{(3)}(\\theta)  = -\\frac{\\rho_e e c}{B}(1-\\frac{e\\theta B}{2\\hbar c})  .\n \\end{equation}\nThis result is compatible with the one obtained in\n\\cite{ao} although the deformation factors do not coincide. However, in \\cite{ao}\n$\\theta$--deformation results due to a specific choice of overall factor in the definition of current, here\nthe current does not possess any unnatural coefficient in its definition.\n\nThere are some interesting features. In the\nrealization (\\ref{R1})-(\\ref{R2}) if one does not keep the $e^2\/c^2$ terms Hall conductivity acquires a deformation factor, in contrary to the\ncase where the $e^2\/c^2$ terms are kept. Higher order corrections in the realization sweeps out the $\\theta$ dependence of the Hall\nconductivity.  Another curious result is the fact that although their structures are different, two Hamiltonians   (\\ref{hr11}) and (\\ref{hr2})\nlead to the same deformation factor for the Hall conductivity. Consequences of obtaining various types of $\\theta$--deformations\nof the Hall conductivity will be  discussed in the last section.\n\n\n\\section{Quantum Phases in Noncommutative Space}\n\n\nThe existing formulations of the quantum phases in noncommutative\ncoordinates can mainly be distinguished by their dependence on\nmomentum eigenvalues: The formulations of \\cite{glr,cpst,mz,wl,prfn} depend on\nmomentum eigenvalues but the ones in \\cite{ao} and \\cite{o3} do not\npossess any momentum dependence as the original quantum phases.\nExcept \\cite{o3} where a (semi)classical approach was used, all of these formulations implement\nnoncommutativity in terms of the star product (\\ref{NSP}) or the equivalent coordinate shift\n(\\ref{SH}), however interpretation of the $\\theta$ dependent terms appearing in  Hamiltonians differ.  \nBecause of adopting the   interpretation of \\cite{ao}, \n the $\\theta$--deformed phases which we obtain are also  momentum independent.\nThis issue will be clarified at the end of this section. First, we would like to discuss the existing formulations.\nAlthough in  \\cite{glr,cpst,mz,wl,prfn} different phases are considered we will show that they\ncan be formulated in a unified manner. The starting Hamiltonian operator is\n\\begin{equation}\n\\label{OTH}\nH=\\frac{1}{2m} \\left(  p_\\alpha -\\rho A_\\alpha ( r )\\right)^2 ,\n\\end{equation}\nwhere $\\rho$ is a constant and the configuration is chosen such that the scalar potential term vanishes. One  implements\nnoncommutativity by the shift\n\\begin{equation}\n\\label{sfr}\nr_\\alpha \\rightarrow r_\\alpha -\n\\frac{1}{2\\hbar}\\theta_{\\alpha \\beta }p^\\beta =r_\\alpha -\n\\frac{1}{2\\hbar}\\theta_{\\alpha \\beta } \\left(\\hbar  k^\\beta +\\rho A^\\beta (r)\\right) ,\n\\end{equation}\nwhere $k_\\alpha$ is the eigenvalue of the kinetic  momentum\noperator:\n\\begin{equation}\n\\label{KMO}\n\\left( p_\\alpha -\\rho A_\\alpha (r)\\right)\\psi (r)= \\hbar k_\\alpha \\psi (r).\n\\end{equation}\nHence, (\\ref{OTH}) can be expanded at the first order in $\\theta$  as\n\\begin{equation}\n\\label{HOT}\nH=\\frac{1}{2m} \\left[  p_\\alpha -\\rho A_\\alpha ( r) + \\frac{\\rho}{2\\hbar}\\theta^{\\beta\\sigma}\n\\left(\\hbar  k_\\sigma +\\rho A_\\sigma ( r)\\right)\\partial_\\beta A_\\alpha ( r) \\right]^2.\n\\end{equation}\nIdentifying,\n\\begin{equation}\n\\label{aT}\n \\tilde{A}_\\alpha ( r, \\theta)= A_\\alpha ( r) - \\frac{1}{2\\hbar}\\theta^{\\beta\\sigma}\n \\left(\\hbar  k_\\sigma +\\rho A_\\sigma ( r)\\right)\\partial_\\beta A_\\alpha ( r)\n\\end{equation}\nas the gauge field in noncommutative space, one defines the $\\theta$--deformed quantum phase by\n\\begin{equation}\n\\label{GDO}\n\\Phi (\\theta ) =\\frac{i\\rho}{\\hbar} \\oint  \\tilde{A}_\\alpha ( r, \\theta) dr^\\alpha .\n\\end{equation}\nDifferent phases can be considered by choosing the original field $A_\\alpha$ appropriately.\nTo study the AB phase on the noncommutative plane let the nonvanishing components of the deformation parameter be\n$$\n\\theta_{ij}=\\theta\\epsilon_{ij},\n$$\nwhere $i,j=1,2.$ Moreover, choose $\\rho=-e\/c$ and\nan appropriate 3--vector potential $\\bm{A},$ whose third component\nvanishes $A_3=0.$ Hence, (\\ref{GDO}) leads to\n$$\n \\Phi_{AB}^I (\\theta ) =-\\frac{ie}{\\hbar c}\\oint  \\bm{A}( r) \\cdot d\\bm{r}\n-\\frac{i m e\\theta}{2\\hbar c} \\oint\n\\left[\\left(\\bm{v}\\times\\bm{\\nabla} A_i\\right)_3 -\\frac{e}{\\hbar m\nc} \\left( \\bm{A} \\times \\bm{\\nabla} A_i \\right)_3\\right] d r_i , \n$$\nwhere $\\bm{k}=m\\bm{v}.$  This is the deformed phase obtained in\n\\cite{glr} and \\cite{cpst}.\n\nTo formulate the\nAC, HMW and Anandan phases in noncommutative coordinates we set\n\\begin{equation}\n\\label{gan}\nc \\rho\\bm{A}=\n\\bm{\\mu} \\times \\bm{E} -\\bm{d} \\times \\bm{B}\n \\end{equation}\nwhere   $\\bm{\\mu}$ and $\\bm{d}$ are the magnetic and the electric dipole moments which are proportional to the Pauli spin matrices $\\bm \\sigma$.\nWe deal with the standard configuration where dipole moments are in  $z$--direction  and the external electric and magnetic fields are in the polar radial direction,\nso that $\\bm{\\mu}\\cdot \\bm{B}=0,\\ \\bm{d}\\cdot \\bm{E}=0.$ Moreover, let there be no change in the dipoles along\nthe external fields: $\\bm{E}\\cdot \\bm{\\nabla}\\mu=0,\n\\bm{B}\\cdot\\bm{\\nabla}d=0.$ After implying these conditions,\ninsert (\\ref{aT}) into (\\ref{GDO})  to obtain\n\\begin{equation} \\label{GDOA}\n\\Phi^A (\\theta ) =\\frac{i}{\\hbar c} \\oint \\left(\\bm{\\mu} \\times \\bm{E} -\\bm{d}\n\\times \\bm{B} \\right) \\cdot d\\bm{r} +\\frac{i}{2\\hbar^2c^2} \\theta_{ab}\\oint\n(\\bm{k} +\\bm{\\mu} \\times \\bm{E} -\\bm{d} \\times\n\\bm{B})_a\\partial_b (\\bm{\\mu} \\times \\bm{E} -\\bm{d} \\times\n\\bm{B})\\cdot d \\bm{r} ,\n \\end{equation}\nwhere $a,b=1,2,3.$ For $\\bm{d}=0$ the $\\theta$--deformation\nof the AC phase obtained in \\cite{mz} follows\n\\begin{equation}\n\\label{GDO3}\n \\Phi^{AC} (\\theta ) =\\frac{i}{\\hbar c} \\oint\n(\\bm{\\mu} \\times \\bm{E} ) \\cdot d\\bm{r} +\\frac{i}{2\\hbar^2 c^2}\n\\theta_{ab}\\oint (\\bm{k}+\\bm{\\mu} \\times \\bm{E} )_a\\partial_b\n(\\bm{\\mu} \\times \\bm{E} )\\cdot d \\bm{r} .\n\\end{equation}\nFor $\\bm{\\mu}=0,$ the HMW\nphase in noncommuting coordinates is obtained\nin accord with \\cite{wl} as\n\\begin{equation} \\label{GDO4}\n\\Phi^{HMW} (\\theta ) =-\\frac{i}{\\hbar c} \\oint (\\bm{d} \\times \\bm{B})\n\\cdot d\\bm{r} -\\frac{i}{2\\hbar^2 c^2} \\theta_{ab}\\oint (\\bm{k}-\\bm{d}\n\\times \\bm{B})_a\\partial_b (\\bm{d} \\times \\bm{B})\\cdot d \\bm{r} .\n\\end{equation}\nBy putting (\\ref{GDO3}) and (\\ref{GDO4}) together\n$$ \n\\Phi^{A1} (\\theta )=  \\Phi^{AC} (\\theta\n)+\\Phi^{HMW} (\\theta ) ,\n$$\n which means ignoring the terms behaving as\n$\\mu d$ in (\\ref{GDOA}), the deformation of \\cite{prfn}\nfollows\\footnote{There is a discrepancy of sign which seems due to a\nmisprint in Eq. (33) of \\cite{prfn}.}. Although we used\n3--dimensional vectors the formalism is effectively 2--dimensional\nbecause of the selected configurations leading to the AC and HMW phases.\n\nThe approach of \\cite{ao} differs from the above formulation. In \\cite{ao} one  considers the $\\theta$--deformed Hamiltonian\ndefined as the generalization of the one obtained in the uniform transverse magnetic field $B.$\nIn terms of the related path integral one  identifies\n$$\n\\tilde{A}_i(\\theta , r) =\\left(1- \\frac{e\\theta B}{4\\hbar c}\\right)^{-1} A_i (r).\n$$\nThen, one  employs it in (\\ref{GDO}) with $\\rho=-e\/c$ to get the AB phase in noncommutative coordinates as\n$$\n\\label{abp} \\Phi_{AB}^{II} (\\theta ) =-\\frac{ie}{\\hbar c } \\left( 1+ \\frac{e\\theta B}{4\\hbar c}\\right)\\oint  A_i ( r) dr_i .\n$$\n\nNow, let us present our approach following in part the  receipt\ngiven in \\cite{ao}. We deal with the configurations leading to\nvanishing scalar potentials, so that in general the Hamiltonian in\nnoncommutative coordinates is written in terms of $\\hat p$ which is a\nrealization of the algebra (\\ref{rrq})--(\\ref{yrq}) as\n \\begin{equation} \\label{HNCD}\n H^{nc}=\\frac{ \\hat{p}^2}{2m} .\n  \\end{equation}\nObviously, different realizations\nwill lead to different Hamiltonians. Let $(r_\\alpha,p_\\alpha)$ define the\nclassical phase space variables corresponding to the operators\n$(r^{op}_\\alpha,\\; p_\\alpha^{op}=-i\\hbar \\partial_\\alpha )  .$ The classical\nHamiltonian $H_{\\text{eff}}(r,p)$ will be obtained from the related\nHamiltonian operator in noncommutative space by substituting\n$p_\\alpha^{op} , r^{op}_\\alpha $ with the c-number variables $p,r.$ To\nkeep the discussion general let us define the classical\n$\\theta$--deformed Hamiltonian corresponding to  (\\ref{HNCD}) as \\begin{equation}\n\\label{HNC} H^{nc}_{\\text{eff}}= a_{\\alpha \\beta}(r,\\theta) p_\\alpha p_\\beta\n+ b_\\alpha (r,\\theta)p_\\alpha +c (r,\\theta), \\end{equation} without specifying the\ncoefficients $a_{\\alpha \\beta}(r,\\theta),\\  b_\\alpha (r,\\theta)$ and $c\n(r,\\theta).$ Plugging  (\\ref{HNC}) into  the  path integral \\begin{equation}\n\\label{pde} Z= N\\int d^dp\\; d^dr\\; \\exp \\left\\{ \\frac{i}{\\hbar}\\int\ndt[p^\\alpha\\dot{r}_\\alpha -H_{\\text{eff}}(p,r)] \\right\\}, \\end{equation} where $N$ is\nthe normalization factor, yields the partition function in the\n$d$--dimensional  phase space:\n$$\nZ= N\\int d^dp\\; d^dr\\; \\exp \\left\\{\\frac{i}{\\hbar}\n\\int dt\\, \\left[ p^\\alpha \\left( \\dot{{r}}_\\alpha -  b_\\alpha (r,\\alpha )\\right) - a_{\\alpha \\beta}(r,\\theta) p_\\alpha p_\\beta - c (r,\\theta) \\right]\\right\\} .\n$$\nIntegration over the momenta\ngives the partition function in configuration space with the normalization factor $N^\\prime $ as\n$$\nZ= N^\\prime \\int d^dr\\; \\exp \\left\\{\\frac{i}{\\hbar} \\int\ndt \\,  \\left[ \\frac{1}{4} a^{-1}_{\\alpha \\beta}(r,\\theta)\\left( \\dot{{r}}_\\alpha\n- b_\\alpha (r,\\theta )\\right)\\left( \\dot{{r}}_\\beta -  b_\\beta (r,\\theta\n)\\right) - c (r,\\theta) \\right]\\right\\} .\n$$\nThis  can be written as\n$$\nZ= N^\\prime \\int d^dr\\; \\exp \\left\\{\\frac{i}{\\hbar}  S\n+\\frac{i}{\\hbar} \\int d{r}_\\alpha     {\\mathcal A}^\\alpha (r, \\theta ) \\right\\} ,\n$$\nin terms of\n$$\nS = \\int \\, dt\\,\n[\\frac{1}{4}a_{\\alpha\\beta}^{-1}(r, \\theta)(\\dot{r}^\\alpha\\dot{r}^\\beta+b^\\alpha(r,\n\\theta) b^\\beta(r, \\theta))-c(r, \\theta)]\n$$\nand the $\\theta$--deformed gauge field defined as\n\\begin{equation}\n\\label{ABIZ}\n{\\mathcal A}_\\alpha (r, \\theta ) \\equiv - \\frac{1}{2}a^{-1}_{\\alpha \\beta}(r,\\theta)  b^\\beta (r,\\theta ).\n\\end{equation}\nHence, in general we can introduce the quantum phase as follows\n\\begin{equation}\n\\label{PH}\n\\Phi= \\frac{i}{\\hbar} \\oint  {\\mathcal A}^\\alpha (r, \\theta ) dr_\\alpha=\n-\\frac{i}{2\\hbar} \\oint a^{-1}_{\\alpha \\beta}(r,\\theta)  b^\\beta (r,\\theta )dr_\\alpha .\n\\end{equation}\n\nAs the first specific example we would like to discuss the\nAB phase in noncommutative space adopting some different realizations.\nHence,\nlet the  particles be\nconfined to move on the $r_i=(x,y)$ plane,\n in the presence of an infinitely long,  tiny solenoid placed along the z--axis.\nObviously we set $\\rho =-e\/c,$ moreover  the nonvanishing components of $\\theta$ and $F$ are\n$$\n\\theta_{ij}=\\epsilon_{ij}\\theta ,\\  F_{ij}=\\epsilon_{ij}F_{12} =\n\\left\\lbrace\n\\begin{array}{ll}\n\\epsilon_{ij} B & {\\rm in}, \\\\\n 0 & {\\rm out}.\n\\end{array}\n\\right\\rbrace\n$$\nExcept on the solenoid, the conditions (\\ref{cons}) are fulfilled, due to the fact that\n$F_{12}$ is constant inside the solenoid and vanishes outside the solenoid.\nThus, we are equipped with the realizations  (\\ref{R1})--(\\ref{R2}) and (\\ref{rea21})--(\\ref{rea22})\nin a consistent manner. We first deal with the realization given in (\\ref{R1}) but ignore the $e^2\/c^2$ terms, so that\nthe related coefficients are\n\\begin{equation}\n\\label{c1}\na_{ij}^{(1)}(r, \\theta)=\\frac{1}{2m} \\left(1-\\frac{eF_{12}\\theta}{2\\hbar c}\\right)^2 \\delta_{ij}, \\qquad b_i^{(1)}(r,\\theta)\n=\\frac{e}{mc}\\left(1-\\frac{eF_{12}\\theta}{2\\hbar c}\\right) A_i .\n\\end{equation}\nThe trajectory in (\\ref{PH}) is chosen to\nenclose the origin, thus it yields\n\\begin{equation}\n\\label{NAB1}\n \\Phi_{AB}^{nc(1)}=\n-\\frac{ie}{\\hbar c} \\oint \\left(1+\\frac{eF_{12}\\theta}{2\\hbar c}\\right) A_idr_i=\n-\\frac{ie}{\\hbar c} \\left(1+\\frac{e\\theta B}{2\\hbar c}\\right) \\int \\epsilon_{ij}\\nabla_iA_j ds=\n\\left(1+\\frac{e\\theta B}{2\\hbar c}\\right) \\Phi_{AB} ,\n\\end{equation}\nwhere the AB phase is given in terms of $\\Phi_0=hc\/e$ and the cross--sectional area of the solenoid  $S$ as\n$$\n\\Phi_{AB} =-2\\pi i\\frac{BS}{ \\Phi_0} .\n$$\n\nWhen we consider (\\ref{R1}) keeping the $e^2\/c^2$ terms the coefficients become\n\\begin{equation}\n\\label{NAB2}\na_{ij}^{(2)}(r,\n\\theta)=\\frac{1}{2m} \\left(1-\\frac{eF_{12}\\theta}{2\\hbar c}\\right)^2 \\delta_{ij}, \\quad b_i^{(2)}(r,\n\\theta)=\\frac{e}{mc} \\left(1-\\frac{eF_{12}\\theta}{2\\hbar c}\\right)^2 A_i.\n\\end{equation}\nObserve that (\\ref{ABIZ}) does not acquire any $\\theta$--deformation. As a\nresult of this the phase is not deformed:\n\\begin{equation}\n\\Phi_{AB}^{nc(2)}= \\Phi_{AB} \\label{abp2} .\n\\end{equation}\n\nFor the realization  (\\ref{rea21}) one can\nread the coefficients as follows\n\\begin{equation}\n\\label{NAB3}\na_{ij}^{(3)}(r, \\theta)=\n\\frac{1}{2m}\\left(1-\\frac{eF_{12}\\theta}{\\hbar c}\\right)^2 \\delta_{ij},\n\\quad b_i^{(3)}(r, \\theta)=-\\frac{eB}{2mc} \\left(1-\\frac{eF_{12}\\theta}{\\hbar c}\\right)\\epsilon_{ij}r_j .\n\\end{equation}\nHence, the $\\theta$--deformed AB\nphase is deduced as\n\\begin{equation}\n\\label{abp3}\n\\Phi_{AB}^{nc(3)}=\n\\frac{ie}{2\\hbar c} \\oint  \\left(1+\\frac{eF_{12}\\theta}{\\hbar c}\\right) F_{12} \\epsilon_{ij}r_jdr_i\n=\\left(1+\\frac{e\\theta B}{\\hbar c}\\right) \\Phi_{AB} ,\n\\end{equation}\nwhere we used  $S=\\oint \\epsilon_{ij} r_i dr_j \/2.$\n\nSimilar to the Hall conductivity (\\ref{hcon2}), the realization (\\ref{R1}) when the terms at the order of  $e^2\/c^2$  are  kept, i.e. (\\ref{NAB2}),\ndoes not procure any $\\theta$--deformation of the AB phase (\\ref{abp2}). However, the other realizations (\\ref{c1}) and (\\ref{NAB3}) led to\n (\\ref{NAB1}) and  (\\ref{abp3}) with different $\\theta$--dependent factors, in contrary to the Hall effect where they\n yielded the same factor as is given in (\\ref{hcon1}). An approach to determine which formulation should be preferred is presented\nin the last section.\n\nTo discuss the AC, HMW and Anandan phases in noncommutative coordinates we will consider the\nrealization (\\ref{rea21}) in 3 dimensions: $a,b=1,2,3.$ In general it  leads to the\n$\\theta$--deformed gauge field (\\ref{gan}), where\n$$\na_{a}^b=\\frac{1}{2m}\\left(\\delta_{a}^b-\\frac{2\\rho}{\\hbar}F_{ac}\\theta^{cb}\\right)\n$$\nand\n$$\nb_a= \\frac{\\rho}{2m}\\left(F_{ab}-\\frac{\\rho}{\\hbar}\\theta_{ac}F^{cd}F_{db}\\right)r^b.\n$$\nAs far as the conditions (\\ref{cons}) are satisfied this construction is valid also for non-Abelian gauge fields.\nThe $\\theta$--deformed\nphase factor is\n\\begin{equation}\n\\label{lp}\n\\Phi^{nc} =\n-\\frac{i\\rho}{2\\hbar} \\oint \\left( F^{ab}+\n\\frac{\\rho}{\\hbar}F^{ac}F_{cd}\\theta^{db} \\right)r_a dr_b\n\\end{equation}\nNow we specify the gauge field as in (\\ref{gan}) which is appropriate to discuss the AC, HMW and Anandan phases\nand consider  the configuration:  $\\bm{\\mu}=\\mu \\hat{z},\n\\bm{d}=d\\hat{z};$  $\\bm{\\mu}\\cdot \\bm{B}=0,\\ \\bm{d}\\cdot \\bm{E}=0$ and $\\bm{E}\\cdot \\bm{\\nabla}\\mu=0,\n\\bm{B}\\cdot\\bm{\\nabla}d=0$. Hence the problem is effectively 2--dimensional. The gauge field (\\ref{gan}) is now Abelian and the nonvanishing components\nof the field strength are\n\\begin{equation}\n\\label{fnv}\n F_{ij}=\\epsilon_{ij}(-\\mu \\bm{\\nabla}\\cdot\n\\bm{E}+d\\bm{\\nabla}\\cdot \\bm{B}) .\n\\end{equation}\nMoreover, we consider the noncommutative plane by setting\n$\\theta_{ij}=\\epsilon_{ij}\\theta .$ As usual\nthe electromagnetic fields are taken in the radial direction and\ntheir divergence vanish except in  the infinitesimal\nregions around the origin where they satisfy\\cite{o3}\n$$\n\\bm{\\nabla}\\cdot\\bm{E}=\\frac{\\lambda_e}{s^\\prime}, \\qquad\\quad\n\\bm{\\nabla}\\cdot\\bm{B}=\\frac{\\lambda_m}{s^{\\prime\\prm}}.\n$$\nWe introduced   $s^\\prime$ and $s^{\\prime\\prm}$ which are, respectively, the areas of\nthe infinitesimal regions where $\\bm{\\nabla}\\cdot\\bm{E}$ and $\\bm{\\nabla}\\cdot\\bm{B}$ are\nnonvanishing. Obviously, $s^\\prime$ and $s^{\\prime\\prm}$  do not play any role in the original definition of\nthe Anandan  phase:\n\\begin{equation}\n\\label{AO}\n\\Phi_A=\\frac{\\rho}{2}\\oint F_{ij}r^i dr^j= -\\frac{1}{\\hbar c}(\\mu \\lambda_e-d\\lambda_m) .\n\\end{equation}\nThe field strength(\\ref{fnv}) satisfies the conditions (\\ref{cons}) so that we can use the realization leading to the phase\n(\\ref{lp}).\nThe Anandan phase in noncommutative space can be calculated as\n\\begin{equation}\n\\label{nA}\n\\Phi_A^{nc}= \\Phi_A\\left[1+\\theta \\left(\\frac{\\mu\\lambda_e}{\\hbar c s^\\prime}-\\frac{d\\lambda_m}{\\hbar c\ns^{\\prime\\prm}}\\right)\\right] ,\n\\end{equation}\nwhere  (\\ref{AO}) is employed.\n\nImposing, respectively, $\\lambda_e=0$ and $\\lambda_m=0$  in (\\ref{nA}),\nthe noncommutative AC and HMW phases can be deduced as\n\\begin{eqnarray}\n\\Phi_{AC}^{nc} & = & \\frac{d\\lambda_m}{\\hbar c}\\left( 1-\\theta \\frac{d\\lambda_m}{\\hbar c s^{\\prime\\prm}}\\right) , \\nonumber \\\\\n\\Phi_{HMW}^{nc} & = & -\\frac{\\mu \\lambda_e}{\\hbar c}\n\\left(1+\\theta \\frac{\\mu\\lambda_e}{\\hbar c s\\prime}\\right) . \\nonumber\n\\end{eqnarray}\n\nLet us compare the constructions of quantum phases in noncommutative coordinates given in\n \\cite{glr}--\\cite{prfn}  and the ones obtained here. We showed that\nthe former formulations  result from the $\\theta$--deformed gauge field given in (\\ref{aT}) which is obtained from the\nHamiltonian (\\ref{HOT}) by employing the eigenvalues of the kinetic momentum (\\ref{KMO}), so that one gets rid of the momentum operators $p_\\alpha.$\nInstead of eliminating  them,\nwhat happens if  one keeps the momentum operators  $p_\\alpha$ in the definition of the $\\theta$--deformed Hamiltonian? In this case,\nthe resulting Hamiltonian would be written as\n\\begin{equation}\n\\label{HOT1}\n\\tilde{H}=\\frac{1}{2m} \\left[ \\left( \\delta_\\alpha^\\sigma -\\frac{\\rho}{2\\hbar}\\theta^{\\beta\\sigma}\n\\partial_\\beta A_\\alpha ( r) \\right) p_\\sigma -\\rho A_\\alpha ( r)  \\right]^2.\n\\end{equation}\nNow, employing  the corresponding classical Hamiltonian in the path integral (\\ref{pde}) and integrating over the momenta would have resulted\nin the $\\theta$--deformed gauge field as in (\\ref{ABIZ}) which is momentum independent.\nIn fact, the latter procedure is the one which we adopted, though our deformation procedure is different than employing the coordinate  shift\n(\\ref{sfr}). Hence, interpretation of the $\\theta$--deformed terms in Hamiltonians as a contribution to the gauge field or to the kinetic term is the main\ndifference between these approaches. In our formulation deformed phases are independent of the velocity of the scattered particles\nwhich is one of the main features of the original quantum phases.\n\n\n\n\\section{Discussions}\n\nWe discussed in detail the alternative method of establishing quantum mechanics in noncommutative coordinates which is\napplicable to dynamical systems whose  observables are either functions of\nphase space variables or matrices which may be independent of phase space coordinates.\nThe alternative procedure  itself leads to different deformed dynamical systems\ndepending on the adopted representation of the deformed algebra (\\ref{rrq})--(\\ref{yrq})  which is equivalent to identify the $\\theta$--deformed\nquantum phase space variables.\n\nWithin  the alternative procedure we discussed the Hall effect in noncommutative coordinates in Section 3 and considered the\nquantum phases in noncommutative space in Section 4. Depending on\nto the realization adopted the resulting Hall conductivity as well as AB phase\nacquire diverse deformation factors in noncommutative coordinates.\nAlthough at first sight this may seem to be a pathological fact, as  we will explain it is  an embarrassment of riches permitting us\nto choose the  realization adequate to the problem considered.\nOne of the interpretations of the noncommutativity of coordinates is\nto consider it as an effective method of introducing interactions whose dynamical origins can be complicated\\cite{ao,sc}. Once we determine which\nrealization leads to the desired effective theory we can select to work within that representation. Let us illustrate this considering\nthe integer quantum Hall effect. Demanding gauge invariance of the extended states\nyields quantization of the AB phase gained by the  electrons in Hall effect which permits  one to obtain the integer\nquantum Hall effect\\cite{LAF}.  \nIt is possible to extend this formulation to  noncommutative space employing \nnoncommutative versions of the Hall effect and the AB phase  obtained in this work.  Depending on \nthe  effective theory one desires to obtain by fixing the noncommutativity parameter $\\theta$ in the deformed\n integer quantum Hall effect,\n one can select the appropriate deformations of the Hall effect and the AB phase.\nThen, the related Hamiltonian can be taken as the as the starting point of formulating a\nfield theory which may be utilized to find out some other aspects.\nFor example, we can employ the related noncommutative field theory to\nderive Green functions and obtain the quantized Hall effect in noncommutative coordinates similar to the ordinary case\\cite{AA}.\nThese are currently under inspection.\n\nWe clarified the relation and discrepancies between our approach and\nthe existing works on defining the quantum phases in  noncommutative spaces.\nWe showed that in general quantum phases in noncommutative spaces either for Abelian or for non-Abelian gauge fields\ncan be defined independent of the velocities of the particles, as the underlying commutative phases.\nWe discussed the receipt on general grounds and in particularly applied it to the noncommutative plane, due to the fact that\nconfigurations of the observed AB and AC quantum phases are effectively two-dimensional.\nOur results can also be applied to condensed matter systems like graphen which are effectively two-dimensional and where the quantum phases play an\nimportant role. This is  an attractive problem because it may give some clues in testing the possible advantages of introducing noncommutative coordinates.\n\nObviously, we retained the first order contribution in $\\theta$ and up to some few orders in coupling constants. However,\nour method provides  a systematic receipt of deriving the higher contributions either in $\\theta$ or in coupling constants. Moreover, introducing\nanother deformation parameter similar to $\\theta$ to render the  momenta noncommutative even for vanishing $F_{\\alpha\\beta},$ is straightforward in our formulation.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet\n\\[\n\\Gamma _n = \\left\\{ {P = (p_1 ,p_2 ,...,p_n )\\left| {p_i > 0,\\sum\\limits_{i\n= 1}^n {p_i = 1} } \\right.} \\right\\},\n\\quad\nn \\ge 2,\n\\]\n\n\\noindent\nbe the set of all complete finite discrete probability distributions. For all $P,Q \\in \\Gamma _n $, let us consider two generalized symmetric divergence measures. These measures are well known in the literature on information theory and statistics.\n\n\\bigskip\nLet us consider the measure\n\\begin{equation}\n\\label{eq1}\n\\xi _s (P\\vert \\vert Q) =\n\\begin{cases}\n {L_s (P\\vert \\vert Q) = \\left[ {s(s - 1)} \\right]^{ - 1}\\left[\n{\\sum\\limits_{i = 1}^n {\\left( {\\frac{p_i^s + q_i^s }{2}} \\right)\\left(\n{\\frac{p_i + q_i }{2}} \\right)} ^{1 - s} - 1} \\right],} & {s \\ne 0,1} \\\\\n {I(P\\vert \\vert Q) = \\frac{1}{2}\\left[ {\\sum\\limits_{i = 1}^n {p_i \\ln\n\\left( {\\frac{2p_i }{p_i + q_i }} \\right) + \\sum\\limits_{i = 1}^n {q_i \\ln\n\\left( {\\frac{2q_i }{p_i + q_i }} \\right)} } } \\right],} & {s = 1} \\\\\n {T(P\\vert \\vert Q) = \\sum\\limits_{i = 1}^n {\\left( {\\frac{p_i + q_i }{2}}\n\\right)\\ln \\left( {\\frac{p_i + q_i }{2\\sqrt {p_i q_i } }} \\right)} ,} & {s =\n0} \\\\\n\\end{cases}\n\\end{equation}\n\n\\noindent\nfor all $P,Q \\in \\Gamma _n $\n\n\\bigskip\nThe measure (\\ref{eq1}) was studied for the first time by Taneja \\cite{tan2} and is called \\textit{generalized symmetric} \\textit{arithmetic and geometric mean divergence.} The measure (\\ref{eq1}) admits the following particular cases:\n\n\\bigskip\n\\noindent\n(i) $\\xi _{ - 1} (P\\vert \\vert Q) = \\frac{1}{4}\\Delta (P\\vert \\vert Q)$.\\\\\n(ii) $\\xi _1 (P\\vert \\vert Q) = I(P\\vert \\vert Q)$.\\\\\n(iii) $\\xi _{1 \/ 2} (P\\vert \\vert Q) = 4\\;d(P\\vert \\vert Q)$.\\\\\n(iv) $\\xi _0 (P\\vert \\vert Q) = T(P\\vert \\vert Q)$.\\\\\n(v) $\\xi _2 (P\\vert \\vert Q) = \\frac{1}{16}\\Psi (P\\vert \\vert Q)$.\n\n\\bigskip\n\\noindent where\n\\begin{align}\n\\Delta (P\\vert \\vert Q) & = \\sum\\limits_{i = 1}^n {\\frac{(p_i - q_i )^2}{p_i +\nq_i }} ,\\notag\\\\\n\\intertext{and}\nd(P\\vert \\vert Q) &= 1 - \\sum\\limits_{i = 1}^n {\\left( {\\frac{\\sqrt {p_i } +\n\\sqrt {q_i } }{2}} \\right)} \\left( {\\sqrt {\\frac{p_i + q_i }{2}} } \\right).\\notag\n\\end{align}\n\n\\noindent\nare the \\textit{triangular discrimination} and \\textit{d-divergence} respectively. The measures $I(P\\vert \\vert Q)$ and $T(P\\vert \\vert Q)$ are the well-known Jensen-Shannon divergence \\cite{sib, bur} and Arithmetic-Geometric mean divergence \\cite{tan2},  respectively.\n\n\\bigskip\nLet us consider now the other measure\n\\begin{equation}\n\\label{eq2}\n\\zeta _s (P\\vert \\vert Q) = \\begin{cases}\n {J_s (P\\vert \\vert Q) = \\left[ {s(s - 1)} \\right]^{ - 1}\\left[\n{\\sum\\limits_{i = 1}^n {\\left( {p_i^s q_i^{1 - s} + p_i^{1 - s} q_i^s }\n\\right) - 2} } \\right],} & {s \\ne 0,1} \\\\\n {J(P\\vert \\vert Q) = \\sum\\limits_{i = 1}^n {\\left( {p_i - q_i } \\right)\\ln\n\\left( {\\frac{p_i }{q_i }} \\right),} } & {s = 0,1} \\\\\n\\end{cases}\n\\end{equation}\n\n\\noindent\nfor all $P,Q \\in \\Gamma _n $\n\n\\bigskip\nThe measure (\\ref{eq2}) can be seen in Burbea and Rao \\cite{bur} and Taneja \\cite{tan3}. The expression (\\ref{eq2}) admits the following particular cases:\n\n\\bigskip\n\\noindent\n(i) $\\zeta _{ - 1} (P\\vert \\vert Q) = \\zeta _2 (P\\vert \\vert Q) = \\frac{1}{2}\\Psi (P\\vert \\vert Q)$,\\\\\n(ii) $\\zeta _0 (P\\vert \\vert Q) = \\zeta _1 (P\\vert \\vert Q) = J(P\\vert \\vert Q)$,\\\\\n(iii) $\\zeta _{1 \/ 2} (P\\vert \\vert Q) = 8\\,h(P\\vert \\vert Q)$,\n\\bigskip\n\n\\noindent where\n\\begin{align}\n\\Psi (P\\vert \\vert Q) & = \\chi ^2(P\\vert \\vert Q) + \\chi ^2(Q\\vert \\vert P) =\n\\sum\\limits_{i = 1}^n {\\frac{(p_i - q_i )^2(p_i + q_i )}{p_i q_i }} ,\\notag\\\\\n\\intertext{and}\nh(P\\vert \\vert Q)&  = 1 - B(P\\vert \\vert Q) = \\frac{1}{2}\\sum\\limits_{i = 1}^n\n{(\\sqrt {p_i } - \\sqrt {q_i } )^2} ,\n\\end{align}\n\n\\noindent\nare the \\textit{symmetric }$\\chi ^2$\\textit{-- divergence} and \\textit{Hellinger's discrimination} respectively. The measure $J(P\\vert \\vert Q)$ is the well-known J-divergence \\cite{jef}. For detailed study of the measures (\\ref{eq1}) and (\\ref{eq2}) refer to Taneja \\cite{tan2, tan3}.\n\n\\bigskip\nThe symmetric divergence measures (\\ref{eq1}) and (\\ref{eq2}) admit several particular cases. An inequality among these measures \\cite{tan2} is given by\n\\begin{equation}\n\\label{eq3}\n\\frac{1}{4}\\Delta (P\\vert \\vert Q) \\le I(P\\vert \\vert Q) \\le h(P\\vert \\vert\nQ) \\le 4\\,d(P\\vert \\vert Q)\n \\le \\frac{1}{8}J(P\\vert \\vert Q) \\le T(P\\vert \\vert Q) \\le \\frac{1}{16}\\Psi\n(P\\vert \\vert Q).\n\\end{equation}\n\nAn improvement over the inequalities given in (\\ref{eq3}) can be seen in Taneja \\cite{tan4, tan5}.\n\n\\bigskip\nIn this paper our aim is to prove metric space properties of the square root of the measures (\\ref{eq1}) and (\\ref{eq2}).\n\n\\section{Generalized Divergence Measures and Metric Spaces}\n\nIn this section we shall prove the metric space property of the square root of the measures given in (\\ref{eq1}) and (\\ref{eq2}).\n\n\\subsection{JS and AG -- Divergences of Type s}\n\nLet the function $\\zeta _s (p,q):{\\rm R}^ + \\times {\\rm R}^ + \\to {\\rm R}^ + $be defined as\n\\begin{equation}\n\\label{eq4}\n\\zeta _s (p,q) = \\begin{cases}\n {L_s (p,q) = \\frac{1}{s(s - 1)}\\left[ {\\left( {\\frac{p^s + q^s}{2}}\n\\right)\\left( {\\frac{p + q}{2}} \\right)^{1 - s} - \\left( {\\frac{p + q}{2}}\n\\right)} \\right],} & {s \\ne 0,1} \\\\\n {I(p,q) = \\frac{p}{2}\\ln \\left( {\\frac{2p}{p + q}} \\right) + \\frac{q}{2}\\ln\n\\left( {\\frac{2q}{p + q}} \\right),} & {s = 0} \\\\\n {T(p,q) = \\left( {\\frac{p + q}{2}} \\right)\\ln \\left( {\\frac{p + q}{2\\sqrt\n{pq} }} \\right),} & {s = 1} \\\\\n\\end{cases}\n\\end{equation}\n\n\\noindent\nIn view of (\\ref{eq4}), we can write\n\\begin{equation}\n\\label{eq5}\n\\zeta _s (P\\vert \\vert Q) = \\sum\\limits_{i = 1}^n {\\zeta _s (p_i ,q_i )} ,\n\\end{equation}\n\n\\noindent\nfor all $P,Q \\in \\Gamma _n $\n\n\\begin{theorem} The measure given by $\\sqrt {\\zeta _s (p,q)} $ is a metric over ${\\rm R}^ + $.\n\\end{theorem}\n\n\\begin{proof}  (i) Initially we shall prove the result for $s \\ne 0,1$. It is sufficient to show the triangle inequality:\n\\begin{equation}\n\\label{eq6}\n\\sqrt {L_s (p,q)} \\le \\sqrt {L_s (p,r)} + \\sqrt {L_s (r,q)} ,\n\\quad\n\\forall p,q,r \\in {\\rm R}^ + ,\n\\quad\ns \\ne 0,1\n\\end{equation}\n\n\\noindent\nLet us write\n\\[\nK_{pq} (r) = \\sqrt {L_s (p,r)} + \\sqrt {L_s (r,q)} .\n\\]\n\n\\noindent\nNow we shall prove that $K_{pq} $ has only one minimum at $r = p = q$. The derivative o f$K_{pq} $ with respect to $r$ is\n\\[\n{K}'_{pq} (r) = \\frac{{L}'_s (p,r)}{2\\sqrt {L_s (p,r)} } + \\frac{{L}'_s\n(r,q)}{2\\sqrt {L_s (r,q)} },\n\\]\n\n\\noindent where\n\\begin{align}\n{L}'_s (p,r) & = \\frac{d}{dr}L_s (p,r)\\notag\\\\\n& = \\frac{(1 - s)r^{ - s}(p + r)^s + (p^{1 - s} + r^{1 - s})s(p + r)^{s - 1}\n- 2^s}{s(s - 1)2^{s + 1}}\\notag\\\\\n& \\mathop = \\limits_{\\frac{p}{r} = t} \\frac{(1 - s)(1 + t)^s + s(t^{1 - s} +\n1)(t + 1)^{s - 1} - 2^s}{s(s - 1)2^{s + 1}}. \\notag\n\\end{align}\n\n\\noindent\nAlso, we can write\n\\[\n\\sqrt {L_s (p,r)} \\mathop = \\limits_{\\frac{p}{r} = t} \\sqrt r \\sqrt {L_s (t,1)} ,\n\\]\n\n\\noindent\nMultiply $K'_{pq} $ by $2\\sqrt r $ and define the function $h\\left( t \\right)$ by setting\n\\[\n\\frac{2\\sqrt r {L}'_s (p,r)}{\\sqrt {L_s (p,r)} }\\mathop =\n\\limits_{\\frac{p}{r} = t} h_{L_s } (t) = \\frac{n_{L_s } (t)}{d_{L_s } (t)},\n\\]\n\n\\noindent where\n\\[\nn_{L_s } (t) = \\left. {\\frac{d}{dr}L_s (p,r)} \\right|_{\\frac{p}{r} = t}\n\\]\n\n\\noindent and\n\\[\nd_{L_s } (t) = \\sqrt {L_s (t,1)} .\n\\]\n\n\\noindent\nThus the sign of $h_{L_s } (t)$ depends only on the sign of $n_{L_s } (t)$. We have\n\\[\n{n}'_{L_s } (t) = - \\frac{(1 + t)^{s - 2}}{2^{s + 1}}\\left[ {(1 + t)(1 + t^{\n- s}) + (1 + t^{1 - s})} \\right].\n\\]\n\n\\noindent\nThis give\n\\bigskip\n${n}'_{L_s } (t) < 0,\\;\\forall t > 0\\;\\mbox{and}\\;\\forall s  \\Rightarrow n_{L_s } (t)$ decreases monotonically in $\\left( {0, + \\infty }\n\\right)$.\n\n \\noindent\nAs $h_{L_s } (1) = 0$,  $n_{L_s } (t)$ changes the sign at $t = 1$, therefore $h_{L_s } (t)$ changes the sign at $t = 1$. This gives\n\\[\nh_{L_s } (t)\\begin{cases}\n { > 0,} & {t < 1} \\\\\n { < 0,} & {t > 1} \\\\\n\\end{cases}\n\\]\n\n\\noindent for any $s$.\n\n\\bigskip \\noindent\nAs $\\frac{p}{r} = t$, then $\\frac{q}{r} = \\frac{q}{p}\\frac{p}{r} = \\beta t$, where $\\beta = \\frac{q}{p}$. Therefore,\n\\[\n2\\sqrt r \\frac{dK_{L_s } }{dr} = h_{L_s } (t) + h_{L_s } (t\\beta ).\n\\]\n\n\\noindent\nNow, suppose $\\beta > 1\\mbox{ }\\left( {q > p} \\right)$, this give:\n\\begin{itemize}\n\\item for $t < \\frac{1}{\\beta }$: $h_{L_s } (t)$ and $h_{L_s } (\\beta t)$ have the same sign +\n\\item for $t > 1$: $h_{L_s } (t)$ and $h_{L_s } (\\beta t)$ have the same sign --\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow t\\beta > 1 \\Rightarrow h_{L_s } (t\\beta ) < 0$\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow h_{L_s } (t) > 0$\n\\end{itemize}\n\n\\bigskip \\noindent\nFinally, we have for $t \\in \\left( {\\frac{1}{\\beta },1} \\right)$, $h_{L_s } (t) > 0$ e $h_{L_s } (t\\beta ) < 0$. Since $\\left| {h_{L_s } \\left( t \\right)} \\right| > \\left| {h_{L_s } \\left( {t\\beta } \\right)} \\right|$\n($h_{L_s } $is monotonically decreasing), then $h_{L_s } (t) + h_{L_s } (\\beta t) > 0$. For $t > 1$, $h_{L_s } (t) < 0$ e $h_{L_s } (t\\beta ) < 0$ and $h_{L_s } (t) + h_{L_s } (\\beta t) < 0$.\n\n\\bigskip \\noindent\nTherefore, $\\frac{dK_{L_s } }{dr}$ indeed changes from positive to negative sign at $t = 1 \\quad \\left( {r = p} \\right)$ so that there is a minimum at $t = 1$. Now, we shall show that this happens only once.\n\n\\bigskip \\noindent\nSince $h_{L_s } $ is monotonically decreasing this implies that ${h}'_{L_s } < 0$ and we know that the function $h_{L_s } $changes the sign only once.  This gives\n\\[\n\\frac{d}{dt}\\left( {h_{L_s } (t) + h_{L_s } (t\\beta )} \\right) = {h}'_{L_s }\n(t) + {h}'_{L_s } (t\\beta ) < 0,\n\\]\n\n\\noindent\nThe case $q < p$ can be investigated in a similar fashion. Symmetry of $L_s $ allows us to take $t = \\frac{q}{r}$ and $\\frac{p}{r} = \\beta t$ with $\\beta = \\frac{p}{q} > 1$. From this we conclude that there is a minimum at $r = q$.\n\n\\bigskip \\noindent\nRepeating the same process by substituting $t: = \\frac{q}{r}$ and $\\frac{p}{r} = \\beta t$ with $\\beta = \\frac{p}{q}$ we conclude that the function ${K}'_{pq} $ also changes the sign at $r = q$. This proves the result (\\ref{eq6}) for $s \\ne 0,1.$\n\n\\bigskip\n(ii) Now we shall prove the result for $s = 1$. We have to show that\n\\[\n\\sqrt {T(p,q)} \\le \\sqrt {T(p,r)} + \\sqrt {T(r,q)} , \\quad \\forall p,q,r \\in {\\rm R}^ + .\n\\]\n\n\\noindent\nLet us write\n\\[\nT_{pq} (r) = \\sqrt {T(p,r)} + \\sqrt {T(r,q)} ,\n\\]\n\n\\noindent then obviously,\n\\[\n{T}'_{pq} (r) = \\frac{{T}'(p,r)}{2\\sqrt {T(p,r)} } + \\frac{{T}'(r,q)}{2\\sqrt\n{T(r,q)} }.\n\\]\n\n\\noindent\nNow, we have\n\\begin{align}\n{T}'(p,r) & = \\frac{d}{dr}T(p,r)\\notag\\\\\n& = \\frac{1}{2}\\ln \\left( {\\frac{p + r}{2\\sqrt {pr} }} \\right) + \\left(\n{\\frac{p + r}{2}} \\right)\\frac{d}{dr}\\left[ {\\ln \\left( {\\frac{p + r}{2\\sqrt\n{pr} }} \\right)} \\right]\\notag\\\\\n& = \\frac{1}{2}\\ln \\left( {\\frac{p + r}{2\\sqrt {pr} }} \\right) + \\frac{r - p}{4r}. \\notag\n\\end{align}\n\n\\noindent\nThis give\n\\[\n\\frac{{T}'(p,r)}{2\\sqrt {T(p,r)} } = \\frac{h_T \\left( t \\right)}{\\sqrt {32r}\n}\\left| {_{t = \\frac{p}{r}} } \\right.\n\\]\n\n\\noindent where\n\\[\nh_T (t): = \\frac{2\\ln \\left( {\\frac{t + 1}{2\\sqrt t }} \\right) + (1 -\nt)}{\\sqrt {(t + 1)\\ln \\left( {\\frac{(1 + t)}{2\\sqrt t }} \\right)} }.\n\\]\n\n\\noindent\nLet us take now $\\frac{q}{r} = \\beta t$ where $\\beta = \\frac{q}{p}$, then we can write\n\\[\n\\left. {\\sqrt {32r} \\,{T}'_{pq} (r)} \\right|_{t = \\frac{p}{r}} = h_T (t) +\nh_T (\\beta t).\n\\]\n\n\\noindent\nLet us study now the function $h_T (t)$. Call $n_T (x)$ and $d_T \\left( t \\right)$ the functions in the numerator and denominator of $h_T (t)$, respectively. Since we know that $d_T (t) > 0,\\;\\forall x > 0$, then the sign of $h_T (t)$ is determined by $n_T (t)$.\n\n\\bigskip \\noindent\nNow,\n\\[\n{n}'_T (t) = 2\\frac{\\sqrt t }{\\left( {t + 1} \\right)}\\frac{d}{dt}\\left(\n{\\frac{t + 1}{\\sqrt t }} \\right) - 1\n = - \\frac{\\left( {1 + t^2} \\right)}{t(t + 1)}.\n\\]\n\n\\bigskip \\noindent\nFrom this we conclude that $n_T (t)$ is decreasing $\\forall t > 0$ and ${n}'_T (t) \\ne 0$, $\\forall t \\in {\\rm R}$. Since $n_T (1) = 0$ then $n_T (t)$ changes the sign at $t = 1$. This gives that $h_T (t)$ changes the sign at $t = 1$.\n\n\\bigskip \\noindent\nThus we conclude that $h_T (t)$ is decreasing function of $x$ and\n\\[\nh_T (t)\\begin{cases}\n { > 0,} & {t < 1} \\\\\n { < 0,} & {t > 1} \\\\\n\\end{cases}\n\\]\n\n\\noindent\nSet $\\beta > 1$. In this case,\n\\begin{itemize}\n\\item for $t < \\frac{1}{\\beta }$: $h_T (t)$ and $h_T (\\beta t)$ have the same sign +\n\\item for $t > 1$: $h_T (t)$ and $h_T (\\beta t)$ have the same sign --\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow t\\beta > 1 \\Rightarrow h_T (t\\beta ) < 0$\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow h_T (t) > 0$\n\\end{itemize}\n\n\\bigskip \\noindent\nFinally, for $t \\in \\left( {\\frac{1}{\\beta },1} \\right)$, $h_T (t) > 0$ e $h_T (t\\beta ) < 0$. Thus we observe that the sign of $h_T (t) + h_T (\\beta t)$ may change in $\\left( {\\textstyle{1 \\over \\beta },1} \\right)$. Now, we shall show that this happens only once.\n\n\\bigskip \\noindent\nSince ${h}'_T $ is monotonically decreasing this implies that ${h}'_T < 0$ and we know that the function $h_T $changes the sign only once. This gives\n\\[\n\\frac{d}{dt}\\left( {h_T (t) + h_T (t\\beta )} \\right) = {h}'_T (t) + {h}'_T (t\\beta ) < 0,\n\\]\n\n\\noindent\nRepeating the same process by substituting $t: = \\frac{q}{r}$ and\n$\\frac{p}{r} = \\beta t$ with $\\beta = \\frac{p}{q}$ we conclude that the\nfunction ${T}'_{pq} $ also changes the sign at $r = q$.\n\n\\bigskip\n(iii) For $s = 0$ the result is already given in Endres and Schindelin \\cite{ens}.\n\\end{proof}\n\n\\subsection{J -- Divergences of Type s}\n\nLet the function $\\xi _s (p,q):{\\rm R}^ + \\times {\\rm R}^ + \\to {\\rm R}^ + $be defined as\n\\begin{equation}\n\\label{eq7}\n\\xi _s (p,q) = \\begin{cases}\n {J_s (p,q) = \\left[ {s(s - 1)} \\right]^{ - 1}\\left[ {p^sq^{1 - s} + p^{1 -\ns}q^s - (p + q)} \\right],} & {s \\ne 0,1} \\\\\n {J(p,q) = (p - q)\\ln \\left( {\\frac{p}{q}} \\right)} & {s = 0,1} \\\\\n\\end{cases}\n\\end{equation}\n\nIn view of (\\ref{eq7}), we can write\n\\begin{equation}\n\\label{eq8}\n\\xi _s (P\\vert \\vert Q) = \\sum\\limits_{i = 1}^n {\\xi _s (p_i ,q_i )} ,\n\\end{equation}\n\n\\noindent\nfor all $P,Q \\in \\Gamma _n $\n\n\\begin{theorem} The measure given by $\\sqrt {\\xi _s (p,q)} $ is a metric space over ${\\rm R}^ + $.\n\\end{theorem}\n\n\\begin{proof} (i) Initially we shall prove the result for $s \\ne 0,1$. It is sufficient to show the triangle inequality:\n\\begin{equation}\n\\label{eq9}\n\\sqrt {J_s (p,q)} \\le \\sqrt {J_s (p,r)} + \\sqrt {J_s (r,q)} ,\n\\quad\n\\forall p,q,r \\in {\\rm R}^ + .\n\\end{equation}\n\n\\bigskip \\noindent\nLet us write\n\\[\nF_{pq} (r) = \\sqrt {J_s (p,r)} + \\sqrt {J_s (r,q)} ,\\;p \\ne q,\n\\]\n\n\\noindent then obviously,\n\\[\n{F}'_{pq} (r) = \\frac{{J}'_s (p,r)}{2\\sqrt {J_s (p,r)} } + \\frac{{J}'_s\n(r,q)}{2\\sqrt {J_s (r,q)} }.\n\\]\n\n\\bigskip \\noindent\nNow, we have\n\\begin{align}\n{J}'_s (p,r) & = \\frac{d}{dr}J_s (p,r)\\notag\\\\\n& = \\frac{(1 - s)p^sr^{ - s} + sp^{1 - s}r^{s - 1} - 1}{s(s - 1)}\\notag\\\\\n& \\mathop = \\limits_{\\frac{p}{r} = t} \\frac{t^sr + t^{1 - s}r - tr - r}{s(s - 1)}\\notag\n\\end{align}\n\n\\bigskip \\noindent\nAlso, we can write\n\\[\n\\sqrt {J_s (p,r)} \\mathop = \\limits_{p = rt} \\sqrt r \\sqrt {J_s (t,1)} ,\n\\]\n\n\\noindent\nLet us write\n\\[\n\\sqrt r \\frac{{J}'_s (p,r)}{\\sqrt {J_s (t,1)} }\\mathop =\n\\limits^{\\frac{p}{r} = t} h_{J_s } (t) = \\frac{n_{J_s } (t)}{d_{J_s } (t)},\n\\]\n\n\\noindent where\n\\[\nn_{J_s } (t) = \\left. {\\frac{d}{dr}J_s (p,r)} \\right|_{\\frac{p}{r} = t}\n\\]\n\n\\noindent and\n\\[\nd_{J_s } (t) = \\sqrt {J_s (t,1)} .\n\\]\n\n\\noindent\nThus the sign of $h_{J_s } (t)$ depends on the sign of $n_{J_s } (t)$.\n\\[\n{n}'_{J_s } (t) = - t^{s - 1} - t^{ - s}.\n\\]\n\n\\noindent\nThus\n${n}'_{J_s } (t) < 0,\\;\\forall t > 0\\;\\mbox{and}\\;\\forall s \\in {\\rm R} - \\{0,1\\}  \\Rightarrow n_{J_s } (t)$is decreasing $\\forall t > 0$\n\n\\bigskip \\noindent\nAs $h_{J_s } (1) = 0$, $n_{J_s } (t)$ changes the sign at $t = 1$ and therefore $h_{J_s } (t)$ changes the sign at $t = 1$. Thus for any $s$, we have\n\\[\nh_{J_s } (t)\\begin{cases}\n { > 0,} & {t < 1} \\\\\n { < 0,} & {t > 1} \\\\\n\\end{cases}\n\\]\n\n\\noindent\nAs $\\frac{p}{r} = t$, then $\\frac{q}{r} = \\frac{q}{p}\\frac{p}{r} = \\beta t$, where $\\beta = \\frac{q}{p}$. Therefore,\n\\[\n\\sqrt r \\frac{dJ_s }{dr} = h_{J_s } (t) + h_{J_s } (t\\beta ).\n\\]\n\n\\noindent\nNow,\n\\begin{itemize}\n\\item for $t < \\frac{1}{\\beta }$: $h_{J_s } (t)$ and $h_{J_s } (\\beta t)$ have the same sign +\n\\item for $t > 1$: $h_{J_s } (t)$ and $h_{J_s } (\\beta t)$ have the same sign --\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow t\\beta > 1 \\Rightarrow h_{J_s } (t\\beta ) < 0$\n\\item for $t \\in \\left( {\\frac{1}{\\beta },1} \\right) \\Rightarrow h_{J_s } (t) > 0$\n\\end{itemize}\n\n\\noindent\nFinally, for $t \\in \\left( {\\frac{1}{\\beta },1} \\right)$, $h_{J_s } (t) > 0$ e $h_{J_s } (t\\beta ) < 0$. Thus we observe that the sign of $h_{J_s } (t) + h_{J_s } (\\beta t)$ may change in $\\left( {\\textstyle{1 \\over \\beta },1} \\right)$. Now, we shall show that this happens only once.\n\n\\bigskip \\noindent\nSince ${h}'_{J_s } $ is monotonically decreasing this implies that  ${h}'_{J_s } < 0$ and we know that the function $h_{J_s } $changes the sign only once. This gives\n\\[\n\\frac{d}{dt}\\left( {h_{J_s } (t) + h_{J_s } (t\\beta )} \\right) = {h}'_{J_s }\n(t) + {h}'_{J_s } (t\\beta ) < 0.\n\\]\n\n\\noindent\nRepeating the same process by substituting $t: = \\frac{q}{r}$ and $\\frac{p}{r} = \\beta t$ with $\\beta = \\frac{p}{q}$ we conclude that the function ${T}'_{pq} $ also changes the sign at $r = q$.\n\n\\bigskip \\noindent\n(ii) For $s = 0,1$, the result follows by the continuity of the function $\\xi _s (p,q)$ with respect to $s$.\n\\end{proof}\n\n\\section{Asymptotic Approximation}\n\nIn this section we shall bring asymptotic approximation of the measures given by (\\ref{eq1}) and (\\ref{eq2}). For this, first we shall  give a general result for Csisz\\'{a}r's $f$-divergence then the other cases become as particular.\n\n\\bigskip\nGiven a function$f:(0,\\infty ) \\to {\\rm R}$, the \\textit{f-divergence} measure introduced by Csisz\\'{a}r's \\cite{csi} is given by\n\\begin{equation}\n\\label{eq10}\nC_f (P\\vert \\vert Q) =\n\\sum\\limits_{i = 1}^n {q_i f\\left( {\\frac{p_i }{q_i }} \\right)} ,\n\\end{equation}\n\n\\noindent\nfor all $P,Q \\in \\Gamma _n $.\n\n\\bigskip\nThe following result is well known in the literature \\cite{csi}.\n\n\\begin{result}  If the function $f$ is convex and normalized, i.e., $f(1) = 0$, then the \\textit{f-divergence}, $C_f (P\\vert \\vert Q)$ is \\textit{nonnegative} and \\textit{convex} in the pair of probability distribution $(P,Q) \\in \\Gamma _n \\times \\Gamma _n $.\n\\end{result}\n\nBased on Result 3.1, we can prove some properties of the measures (\\ref{eq1}) and (\\ref{eq2}).\n\\begin{theorem}\nIf $f$ is twice differentiable at $x = 1$and ${f}''(1) > 0$. Also $f(1) = 0$, then\n\\begin{equation}\n\\label{eq11}\nC_f (P\\vert \\vert Q) \\approx \\frac{{f}''(1)}{2}\\chi ^2(P\\vert \\vert Q) .\n\\end{equation}\n\n\\noindent\nEquivalently,\n\\[\n\\frac{C_f (P\\vert \\vert Q)}{\\chi ^2(P\\vert \\vert Q)} \\to\n\\frac{{f}''(1)}{2}\\,\\,\\ \\mbox{as}  \\,\\,P \\to Q.\n\\]\n\\end{theorem}\n\n\\begin{proof} From Taylor's series expansion, we have\n\\[\nf(x) = {f}'(1)(x - 1) + \\frac{{f}''(1)}{2}(x - 1)^2 + k(x)(x - 1)^2,\n\\]\n\n\\noindent\nwhere $f(1) = 0$ and $k(x) \\to 0$ as $x \\to 1$. Hence\n\\[\nq_i f\\left( {\\frac{p_i }{q_i }} \\right) = {f}'(1)(p_i - q_i ) +\n\\frac{{f}''(1)}{2}\\frac{(p_i - q_i )^2}{q_i }\n + \\,k\\left( {\\frac{p_i }{q_i }} \\right)\\frac{(p_i - q_i )^2}{q_i }.\n\\]\n\nApproximating $p_i \\to q_i $ and summing over all $i = 1,2,....n$ we get the required result.\n\\end{proof}\n\n\\begin{proposition} The following results hold:\n\\begin{itemize}\n\\item[(i)] $\\zeta _s (P\\vert \\vert Q) \\approx \\frac{1}{8}\\chi ^2(P\\vert \\vert Q)$,\n$\\forall s \\in {\\rm R}$.\n\\item[(ii)] $\\xi _s (P\\vert \\vert Q) \\approx \\chi ^2(P\\vert \\vert Q)$, $\\forall s\n\\in {\\rm R}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} (i) For all $x > 0$ and $s \\in ( - \\infty ,\\infty )$, let us consider\n\\begin{equation}\n\\label{eq12}\n\\psi _s (x) = \\begin{cases}\n {\\left[ {s(s - 1)} \\right]^{ - 1}\\left[ {\\left( {\\frac{x^{1 - s} + 1}{2}}\n\\right)\\left( {\\frac{x + 1}{2}} \\right)^s - \\left( {\\frac{x + 1}{2}}\n\\right)} \\right],} & {s \\ne 0,1} \\\\\n {\\frac{x}{2}\\ln x - \\left( {\\frac{x + 1}{2}} \\right)\\ln \\left( {\\frac{x +\n1}{2}} \\right),} & {s = 0} \\\\\n {\\left( {\\frac{x + 1}{2}} \\right)\\ln \\left( {\\frac{x + 1}{2\\sqrt x }}\n\\right),} & {s = 1} \\\\\n\\end{cases},\n\\end{equation}\n\n\\noindent\nin (\\ref{eq10}), then we have $C_f (P\\vert \\vert Q) = \\zeta _s (P\\vert \\vert Q)$, where $\\zeta _s (P\\vert \\vert Q)$ is as given by (\\ref{eq1}).\n\n\\bigskip \\noindent\nWe have\n\\[\n\\psi _s ^\\prime (x) = \\begin{cases}\n {(s - 1)^{ - 1}\\left[ {\\frac{1}{s}\\left[ {\\left( {\\frac{x + 1}{2x}}\n\\right)^s - 1} \\right] - \\frac{x^{ - s} - 1}{4}\\left( {\\frac{x + 1}{2}}\n\\right)^{s - 1}} \\right],} & {s \\ne 0,1} \\\\\n { - \\frac{1}{2}\\ln \\left( {\\frac{x + 1}{2x}} \\right),} & {s = 0} \\\\\n {1 - x^{ - 1} - \\ln x - 2\\ln \\left( {\\frac{2}{x + 1}} \\right),} & {s = 1}\n\\\\\n\\end{cases}\n\\]\n\n\\noindent and\n\\begin{equation}\n\\label{eq13}\n\\psi _s ^{\\prime \\prime }(x) = \\left( {\\frac{x^{ - s - 1} + 1}{8}}\n\\right)\\left( {\\frac{x + 1}{2}} \\right)^{s - 2}.\n\\end{equation}\n\n\\noindent\nThis gives\n\\begin{equation}\n\\label{eq14}\n\\psi _s ^{\\prime \\prime }(1) = \\frac{1}{4}.\n\\end{equation}\n\n\\noindent\nExpression (\\ref{eq11}) together with (\\ref{eq13}) and (\\ref{eq14}) completes the proof of part (i).\n\n\\bigskip \\noindent\nIn particular, when $s = 0$ in (\\ref{eq12}) the result is obtained in Endres and Schindelin \\cite{ens}.\n\n\\bigskip\n(ii) For all $x > 0$ and $s \\in ( - \\infty ,\\infty )$, let us consider\n\\begin{equation}\n\\label{eq15}\n\\phi _s (x) = \\begin{cases}\n {\\left[ {s(s - 1)} \\right]^{ - 1}\\left[ {x^s + x^{1 - s} - (1 + x)}\n\\right],} & {s \\ne 0,1} \\\\\n {(x - 1)\\ln x,} & {s = 0,1} \\\\\n\\end{cases},\n\\end{equation}\n\n\\noindent\nin (\\ref{eq10}), then we have $C_f (P\\vert \\vert Q)=\\xi _s \\left( {P\\vert \\vert Q} \\right)$, where $\\xi _s \\left( {P\\vert \\vert Q} \\right)$ is given by (\\ref{eq2}).\n\n\\bigskip \\noindent\nWe have\n\\[\n\\phi _s ^\\prime (x) = \\begin{cases}\n {\\left[ {s(s - 1)} \\right]^{ - 1}\\left[ {s(x^{s - 1} + x^{ - s}) + x^{ - s}\n- 1} \\right],} & {s \\ne 0,1} \\\\\n {1 - x^{ - 1} + \\ln x,} & {s = 0,1} \\\\\n\\end{cases},\n\\]\n\n\\noindent and\n\\begin{equation}\n\\label{eq16}\n\\phi _s ^{\\prime \\prime }(x) = x^{s - 2} + x^{ - s - 1}.\n\\end{equation}\n\n\\noindent\nThis gives\n\\begin{equation}\n\\label{eq17}\n\\phi _s ^{\\prime \\prime }(1) = 2.\n\\end{equation}\n\n\\noindent\nExpression (\\ref{eq17}) together with (\\ref{eq10}) and (\\ref{eq11}) completes the proof of part (ii).\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe concept of {\\it test pooling} was apparently invented by Robert Dorfman~\\cite{Dorfman} in 1943 who suggested that it would be more effective to test WW2 would-be recruits for syphilis by mixing the blood samples of several recruits and test the pool for antigens. If the pool tests negative then all the pool members are deemed healthy; otherwise, each member of the pool is tested separately.  A simple analysis (see next section) shows that for a given  probability $p$ that a recruit is infected there is an optimum pool size $s_1(p)$ that minimizes the expected number of needed tests.  The lower the $p$, the larger the $s_1(p)$ and the lower the expected number of tests required.  Dorfman's analysis has been further refined and generalized to deal with various problems such as false negatives~\\cite{LLZA} and studied as part of the broad topic of Combinatorial Group Testing~\\cite{CGT1, CGT2}.\n\nNote that some recursive and adaptive approaches dear to computer scientists, such as binary search, often may not work for this problem: there are pragmatic limitations on (a) the size of the pool beyond which dilution results in too many false negatives; (b) the number of samples available from a given specimen; and (c) the total time required to produce an answer.  \n\nNevertheless the emergence of the COVID-19 pandemic and the cost and scarcity of tests for the underlying virus has revived an enormous interest in test pooling. For COVID-19 test pooling has been shown to be doable with pools as large as 64~\\cite{Yelin} and is already in use in several countries including Germany~\\cite{Frankfurt} and Israel~\\cite{Yelin}.\n\n\\section*{Double pooling}\n\nThe purpose of this note is to propose a simple variation on Dorfman's approach that we call {\\it double pooling}; for clarity, we refer to Dorfman's method as \\emph{single pooling}.  Double pooling works as follows:  given a probability $p$ of a positive test, pick an optimal size $s_2(p)$ for the pool size.  (The optimal $s_2(p)$ is larger than the corresponding optimal $s_1(p)$ for single pooling.)  Divide the population to be tested into non-overlapping pools of size $s_2$ (the division is assumed to be random) \\textit{twice}.  Thus, now every patient belongs to \\textit{two} pools and is tested in two parallel rounds, $A$ and $B$.   For every patient if both the pools test positive then test the patient individually. Otherwise consider that patient cleared.  If the pool tests do not ever produce false negatives the algorithm is clearly correct.  (The false positives only reduce efficiency.)\n\nIt turns out the double pooling is particularly advantageous for $p < 2\\%$ corresponding to testing a large population of asymptomatic patients but it is more efficient than single pooling even for $p=10\\%$.\n\nWe will discuss double pooling in the next section in more detail, but to see its advantages and build an intuitive understanding we start with an example.  \n\nAssume that $p=0.01112$.  It turns out $s_1=10$ is the optimal size for single pooling with this $p$ and results in an expected cost of $\\approx 0.206$ tests\/patient, a nice improvement over testing everyone.  However using double pooling the optimum $s_2(p)$ is 23 and the expected cost further declines to just $0.145$, an almost $30$\\% improvement.  (These quantities will be obvious from our analysis later.)\n\nAt first blush these gains might seem surprising, but here is a quick-and-dirty computation:  Assume that we are testing 1000 patients and remember $p\\approx0.011$ so we will posit we have exactly 11 positive patients. \n\\begin{itemize}\n  \\item For single pooling, since $s_1 = 10$, we start with $100$ tests.  Assuming all positive cases end in separate pools (an upper bound) we will need to do another 110 tests to deal with all the suspicious cases, hence $210$ total tests (which is close enough to $206$.)\n  \n  \\item For double pooling, since $s_2 = 23$, we  do twice 44 tests with 23 patients each (88 tests).  In each round  at most 11 tests  will come back positive raising suspicions about a total of  $22 \\times 11 = 242$ healthy patients.  Thus a given healthy patient has probability $0.24$ to be a suspect in Round A and the same $0.24$ probability in round B. These are quasi-independent, hence the probability of being suspected twice is only $0.24 \\times 0.24 = 0.0576$.  Thus we expect to have less than \n $57$ ($\\approx 0.0576 \\times (1000-11)$) healthy patients that were in a positive pool in both rounds and will have to be retested.  In addition the 11 truly positive patients will be retested as well.    Thus the total number of tests is $88 + 57 + 11 = 156$ (which is reasonably close to the claimed $145$ since we overestimated and also because in fact we now cover $44 \\times 23=1012$ patients).  \n\\end{itemize}\n\nIn conclusion, the ``magic\"  of double pooling comes from a paradigm that has been observed in many other situations, e.g.,~Bloom filters~\\cite{Bloom, DBLP:journals\/im\/BroderM03} and balanced allocations~\\cite{DBLP:journals\/siamcomp\/AzarBKU99, DBLP:journals\/tpds\/Mitzenmacher01}.  Although the probability of being ``unlucky\" in a given trial might be high, the probability of being unlucky in two or more independent trials decreases dramatically. \n\n\\section*{Analysis}\n\n\nConsider the expected cost attributable to  one patient in the single pooling situation, where the size of the pool is $s$:\n\\begin{itemize}\n    \\item if the patient is positive, then the cost is $1\/s + 1$ (the patient's share of the pool + their individual test);\n    \\item if the patient is negative, then the cost is $1\/s + 1 \\times \\big(1-(1-p)^{(s-1)}\\big)$  (the patient's share of the pool + their individual test \\textit{iff} not all the other patients are healthy).\n\\end{itemize}\n\n\\noindent \nSince the probability of being positive is $p$, the total expected cost per patient in this case is \n\\begin{equation} \\label{tcsp}\np \\left(1+\\frac{1}{s}\\right)+(1-p) \\left(1-(1-p)^{s-1}+\\frac{1}{s}\\right)\n= \n\\frac{1}{s} + 1- (1-p)^s.\n\\end{equation}\n\nTo determine the pool size that minimizes the total for a given $p$, we take the derivative of the cost with respect to $s$:\n\\begin{equation} \n\\frac{\\partial}{\\partial s} \\left( \\frac{1}{s} + 1- (1-p)^s \\right)\n=- (1-p)^s \\ln(1-p)- \\frac{1}{s^2},\n\\end{equation}\n and set it to $0$.\n \nThe solution of interest can be expressed in terms of the Lambert $W$ function%\n\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Lambert_W_function}} \nnamely \n\\begin{equation}\ns_1 \n= \\left. 2 W\\left( -\\frac{1}{2} \\sqrt{-\\log(1-p)} \\right) \\middle\/ \\log(1-p)\n\\right. .\n\\end{equation}\n\nLet us define $p_{10}$ as the value of $p$ for which  the optimum $s_1$ is exactly 10.  It turns out that $p_{10}\\approx0.01112$ which is the value we used in the introductory example.  More generally, Figure~\\ref{fig:s1} shows the optimum integer $s_1$ as a function of $p$. \n\n\\begin{figure}[h]\n\\centerline{\n\\includegraphics{s1.pdf}\n}\n\\caption{Optimum $s_1$ as a function of $p$.  \\label{fig:s1}}\n\\end{figure}\n\n\\bigskip\n\nLet us  turn to double pooling: now each patient will be assigned to two random pools each of size $s$.  A patient will be tested individually \\textit{iff} both their pools test positive.  Again let us look at the expected cost induced by the testing of one patient:\n\\begin{itemize}\n    \\item if the patient is positive, then the cost is $2\/s + 1$ (the patient's share of the two pools + their individual test);\n    \\item if the patient is negative, then the cost is $2\/s + 1 \\times \\bigr(1-(1-p)^{(s-1)}\\bigl)^2$  (the patient's share of the pools + their individual test \\textit{iff} \\emph{both} pools test positive).\n\\end{itemize}\nHence the total expected cost is\n\\begin{equation} \\label{tcdp}\np \\left(1+\\frac{2}{s}\\right)+q  \\left(\\bigl(1-q^{s-1}\\bigr)^2+\\frac{2}{s}\\right)\n=\n\\frac{2}{s} + p + q \\bigl(1-q^{s-1}\\bigr)^2,\n\\end{equation}\nwhere for brevity $q$ stands for $1-p$.\n\nAs before, to determine the pool size that minimizes the total cost for a given $p$, we take the partial derivative of the cost with respect to~$s$:\n\\begin{equation} \n\\frac{\\partial}{\\partial s} \n\\left(\n\\frac{2}{s} + p + q \\bigl(1-q^{s-1}\\bigr)^2\n\\right)\n=  - \\frac{2}{s^2} - \n2 q^{s} \\bigl(1-q^{s-1}\\bigr) \\ln q,\n\\end{equation}\nand set it to $0$.\n\nThis has to be solved numerically at each $p$.  Solving this at $p = p_{10} \\approx 0.01112$ we see that the optimum $s_2 = 23$.  More generally, Figure~\\ref{fig:s2} shows the optimum integer $s_2$ as a function of $p$ and Figure~\\ref{fig:c12} shows the expected cost per patient tested as a function of $p$ for both single and double pooling using optimal integer values of $s_1(p)$ and $s_2(p)$.\n\n\\begin{figure}[h]\n\\centerline{\n\\includegraphics{s2.pdf}\n}\n\\caption{Optimum $s_2$ as a function of $p$.  \\label{fig:s2}}\n\\end{figure}\n\n\\bigskip\n\n\n\n\\begin{figure}[h]\n\\centerline{\n\\includegraphics{c12.pdf}\n}\n\\caption{Expected cost using the optimum $s_1$ and $s_2$ as a function of $p$.  \\label{fig:c12}}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centerline{\n\\includegraphics{savings.pdf}\n}\n\\caption{\\% savings of double pooling over single pooling as a function of $p$.  \\label{fig:savings}}\n\\end{figure}\n\n\\bigskip\n\nIn principle we can generalize double pooling to  $k$-pooling, whereby each patient participates in $k$ independent pools in $k$ parallel rounds. The expected cost becomes\n\\begin{equation} \\label{tckp}\n\\frac{k}{s} + p + q \\bigl(1-q^{s-1}\\bigr)^k.\n\\end{equation}\nDepending on $p$ this can yield further improvements but they are probably impractical especially if $p$ gets larger. (With triple testing for $p=p_{10}$ and about 1000 samples  we would need only 128 tests with pools of size  36, and with quadruple testing only 122 tests with pools of size 47.).  Even more asymptotically efficient tests can be constructed~\\cite{MezardToninelli, PoratRothschild}, but it is unclear if they can be practical.\n\n\\section*{Conclusions}\nWe presented \\emph{double pooling}, a simple, easy-to-implement variation on test pooling, that in certain ranges for $p$, the  a priori probability of a positive tests, is significantly more efficient than the standard single pooling approach.  Figure~\\ref{fig:savings} shows the percentage of savings of double pooling over single pooling as a function of $p$.  We can see that double pooling is particularly advantageous for $p$ below $2\\%$  corresponding to large scale testing of asymptomatic patients, but is still at least $10\\%$ better than single pooling all the way up to $p=5.4\\%$. \n\n\nOur analysis assumes sampling from an infinite distribution, but in practice, double pooling can be implemented  after accumulating a fairly small collection of samples.  (There is a small efficiency penalty due  to the correlation between rounds that we will discuss in the final version.)  The main disadvantage of double pooling is that it is more sensitive to dilution-induced false negatives for two reasons: one physical:  the pools used are larger; and  one mathematical: a true positive sample will be missed if either of its two pools produces a false negative.  We will discuss this further in the final version. \n\nWe are reaching out to our colleagues in the medical field to find out whether double pooling is practically usable for COVID testing and will update our note with their feedback. \n\nPresently, there is an extraordinary flurry of activity  and independent work on group testing for COVID.  This includes an analysis of a single pooling method~\\cite{Gollier} and a proposal based on binary search~\\cite{Gossner}. It might well be the case that independent researchers have already obtained the same results presented here.  We are encouraging members of the community to send us their comments and feedback.\n\n\\subsection*{Acknowledgments}\n\nWe thank our colleagues Fernando Pereira and Tam\\'{a}s Sarl\\'{o}s for many  useful comments.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{s:introduction}\n\n    This work is a continuation of the survey search for bright\ncompact radio sources. Several major applications require an\nextended list of sources with positions known at the nanoradian\nlevel: geodetic observations including space navigation; very\nlong baseline interferometry (VLBI) phase-referencing of weak\ntargets, and differential astrometry. For satisfying the needs of\nthese applications, 878 sources were observed under various\ngeodetic and astrometric programs from 1979 thorough 2002, and\nover 80\\,\\% of them were detected. Results of these observations\nwere presented in the catalog ICRF-Ext.2\n\\citep{icrf-ext2-2004} that contains positions of 776 sources.\n   Additionally, 2952 flat-spectrum sources were observed in nineteen\n24-hour sessions from 1994 through 2005 in the Very Long Baseline Array\n(VLBA) Calibrator Survey (VCS) program. The positions of 2505 sources\nwere determined from the observations of the VCS project: VCS1\n\\citep{VCS1}, VCS2 \\citep{VCS2}, VCS3 \\citep{VCS3}, and VCS4\n\\citep{VCS4}. Since 364 sources are listed in both the ICRF-Ext.2 and\nthe VCS catalogs, the total number of sources for which positions were\ndetermined with VLBI in IVS (International VLBI Service for astrometry\nand geodesy) and VCS1 to VCS4 experiments is 2917. Among them, 2468\nsources, or 85\\,\\%, are considered acceptable calibrators: having at\nleast eight successful observations at both X band (central frequency\n8.6~GHz) and S band (central frequency 2.3~GHz), and the\nsemi-major axis of the error ellipse of their coordinates being less\nthan 25~nrad ($\\approx$~5~mas). When observations from both the geodetic\nprograms and VCS1--4 are combined, the overall catalog provides fairly\ngood sky coverage. The probability of finding a calibrator within\n4\\degr\\ of any target north of declination $-40\\degr$ is 98.1\\,\\%.\n\nIn this paper we present an extension to the VCS catalogs, called the\nVCS5 catalog. It concentrates on the brightest flat-spectrum sources\nnorth of declination $-30\\degr$ not previously observed with VLBI under\ngeodetic and absolute astrometry programs. VCS5 is different from the\nprevious campaigns since its prime goal is to collect data needed for\nastrophysical analysis of active galactic nuclei (see\n\\S\\ref{s:objectives}).\n\nSince the observations, calibration, astrometric solutions and imaging\nare similar to that of VCS1--4, most of the details are described by\n\\cite{VCS1} and \\cite{VCS3} and will not be repeated here.\nIn \\S\\ref{s:objectives} we discuss\nscientific objectives for the VCS5 survey. In \\S\\ref{s:selection} we\ndescribe the strategy for selecting the 675 candidate sources observed in\nthree 24-hour VCS5 sessions with the VLBA based on analysis of the\navailable multi-frequency non-VLBI continuum radio measurements. The\nsame strategy was successfully applied by us earlier to select one\nhundred objects with the strongest estimated flux density at 8.6~GHz in\nthe framework of the VCS4 survey. Sixty seven out of these one hundred\nVCS4 candidates showed X~band correlated flux density greater than\n0.2~Jy \\citep{VCS4}. In \\S\\ref{s:obs_anal} we briefly outline the\nobservations and data processing. We present the VCS5 catalog in\n\\S\\ref{s:catalog}, and summarize our results in \\S\\ref{s:summary}.\n\n\\section{Scientific objectives for the VCS5 survey}\n\\label{s:objectives}\n\n     There are two main scientific objectives for the VCS5 survey. We\nwould like to perform statistical analysis of physical properties of a\ndeep sample of compact AGNs on the basis of milliarcsecond scale images\nmeasured simultaneously at S~band and X~band with the VLBA. A cursory\nanalysis of the sample of the 2917 VCS\/ICRF sources observed with the\nVLBA in the S\/X mode revealed that it is nearly complete down to\n0.5~Jy but becomes incomplete at lower flux densities. As a result,\npossible usage of this largest collection of VLBI data for statistical\nanalysis of properties of active galactic nuclei at milliarcsecond\nscales is limited. The first goal of the current VCS5 project is to\nobserve the remaining bright sources with expected correlated flux\ndensities in the range 200~mJy to 600~mJy to create a statistically\ncomplete sample  of extragalactic flat-spectrum radio sources with\nintegrated flux density  at milliarcsecond scales greater than 200~mJy\nat X~band. This will make results of the VCS survey significantly more\nuseful for astrophysical applications. The uniformity of VCS data\nreduction as well as the completeness and homogeneity of the source\nsample will guarantee robust results from further statistical studies.\n\nThe second goal is to find more compact sources and to measure their\npositions precisely for use in geodetic applications including space\nnavigation, VLBI phase referencing of weak targets, and differential\nastrometry. Many applications prefer a more distant bright calibrator to\na near-by but weaker calibrator, since more time can be spent on the\ntarget. The VCS5 observations cover almost all remaining {\\em bright}\ncalibrators with correlated flux density at or greater that 200~mJy.\n\n\n\\section{Source selection}\n\\label{s:selection}\n\nOur source selection goal was to find all flat-spectrum radio sources\nbrighter than 0.2~Jy at 8.6~GHz that were missing from the VCS1 to VCS4 and\nICRF-Ext2 catalogs. We define flat radio spectrum as having a spectral\nindex $\\alpha>-0.5$ ($S\\sim\\nu^\\alpha$). To compile a list of missing\nobjects, we first selected all sources from the NVSS catalog\n\\citep{NVSS} with flux density at 1.4 GHz $S>50$~mJy, declination\n$\\delta>-30\\degr$, a Galactic latitude $|b|>1\\fdg5$, and not identified\nwith Galactic objects. Since the NVSS catalog is more than 99\\,\\%\ncomplete for flux density $S>50$~mJy, it is unlikely that sources with\nhighly inverted spectra and flux density $S>200$~mJy at 8.6~GHz will\nhave been missed.\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm]{f1a.eps}\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm]{f1b.eps}\n}\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=0cm 0cm 0cm 0cm]{f1c.eps}\n  \\includegraphics[trim=0cm 0cm 0cm 0cm]{f1d.eps}\n}\n\\end{center}\n\\caption\nIllustration of the candidates selection procedure. The data from all\navailable publications were located through the CATS database.\nThe dotted falling lines represent $\\alpha=-0.5$.\nJ1109+3738 was not selected since its integrated spectral\nindex was less than $-0.5$. \nJ0446+5802 was not selected since its flux density extrapolated to\n8.6~GHz was less than 170~mJy. J1250+0216 and J2210+2013 satisfied all\nthe selection criteria mentioned in \\S~\\ref{s:selection} and were\nselected. Their $uv$ and image data are presented in\nFigure~\\ref{f:images}.\n\\label{f:selection}\n}\n\\end{figure}\n\nWe then searched the CATS database \\citep{CATS} containing almost all\nradio catalogs to find flux density measurements at other radio\nfrequencies for the selected NVSS sources.\nThese data were supplemented by results of the 1--22 GHz\ninstantaneous broad-band spectra measurements of $\\sim$3000\nextragalactic flat-spectrum radio sources which we performed at the\ntransit mode 600~m ring radio telescope \\mbox{RATAN-600} of the Russian\nAcademy of Sciences \\citep[see, e.g.,][]{Kovalev_etal99}. The collected\ndata were then analyzed semi-automatically, and bad data points, wrong\nidentifications, multiple data points corresponding to different\ncomponents of the same extended object were flagged. We found\nthat we could compile a complete sample of sources with total flux\ndensity spectrum\nflatter than $\\alpha=-0.5$, and with estimated total flux density of\n$S>170$~mJy at 8.6~GHz. In this complete sample were 675 candidates\nnot previously observed in geodetic VLBI mode, and these are the sources\nselected for VCS5 observations.  Figure~\\ref{f:selection} presents\nexamples of plots of the total flux density spectra collected by the\nCATS database which we used for source selection.\n\nOur analysis of the multi-frequency catalogs and RATAN observations used\nfor selection indicates that we have found almost all of the sources\nwith spectral index greater than $-0.5$ and estimated total flux density\nat 8.6~GHz $S>170$~mJy. It is based on the fact that many used catalogs\nincluding NVSS \\citep{NVSS}, FIRST \\citep{FIRST}, 87GB \\citep{87GB}, GB6\n\\citep{GB6}, CLASS \\citep{Myers_etal03}, JVAS \\citep{JVAS1,JVAS2,JVAS3},\nPMN \\citep{Wright_etal94,Griffith_etal95,Wright_etal96}, and PKSCAT90\n\\citep{pkscat90}, are complete down to 150--250~mJy and below. This\nshould provide us with a sample of the same completeness\ncharacteristics. However, it is well known that flat spectrum sources\nare variable \\citep[e.g.,][]{KelPau_ARAA68}; consequently, the\nvariability corrupts at some level our estimations of spectral\nindex and total flux density. The membership of a source in the\ncompleteness sample is also changeable and depends on the observation\nepochs of the various compilation surveys. The quantitative analysis of\ncompleteness of the resulting  correlated flux density limited sample of\nthe sources from the combined ICRF-Ext.2 and VCS1 to VCS5 catalogs will have\nto take into account the frequency dependent variability properties\n\\citep[e.g.,][]{KKNB02} as well as the compactness characteristics of flat\nspectrum sources \\citep[e.g.,][]{PopovKovalev99,2cmPaperIV}. This is\nbeyond the scope of the present paper and is deferred to another\npublication. We expect the present sample to be sufficiently complete,\nrobust, and unbiased for most statistical studies of flat-spectrum radio\nsources.\n\n\\section{Observations and data processing}\n\\label{s:obs_anal}\n\n  The VCS5 observations were carried out in three 24-hour observing\nsessions with the VLBA on 2005 July 8, July 9, and July 20. Each of the\n675 target sources was observed in two scans of 120 seconds each. The\ntarget sources were observed in a sequence designed to minimize loss of\ntime from antenna slewing. In addition to these objects, 97 strong\nsources were taken from the GSFC astrometric catalog\n\\mbox{\\tt{}2004f\\_astro}\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/solutions\/astro}}.\nObservations of three or four strong sources from this list were made every\n1\\,h to 1.5\\,h, 70\\,s to 80\\,s seconds per scan. These observations were scheduled\nin such a way that at each VLBA station at least one of these sources\nwas observed at an elevation angle less than 20\\degr, and at least one\nat an elevation angle greater than 50\\degr. The purpose of these\nobservations was to provide calibration for mis-modeled atmospheric path\ndelays and to tie the VCS5 source positions to the ICRF catalog\n\\citep{icrf98}. The list of tropospheric\ncalibrators\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/vcs\/tropo\\_cal.html}}\nwas selected from the sources that, according to the 2~cm VLBA survey\nresults \\citep{2cmPaperIV}, showed the greatest compactness index, i.e.\\\nthe ratio of the correlated flux density measured at long VLBA spacings\nto the flux density integrated over the VLBA image. In total, 772\ntargets and calibrators were observed. The antennas were on-source about\n65\\,\\% of the time.\n\n  Similar to the previous VLBA Calibrator Survey observing campaigns\n\\citep[e.g.,][]{VCS4}, we used the VLBA dual-frequency geodetic mode,\nobserving simultaneously at S~band and X~band. Each band was separated into\nfour 8~MHz channels (IFs) which spanned 140 MHz around 2.3~GHz and 490\nMHz around 8.6~GHz to provide precise measurements of group\ndelays for astrometric processing. Since the a~priori coordinates of\ncandidates were expected to have errors of up to 30\\arcsec, the data\nwere correlated with an accumulation period of 1~second in 64~spectral\nchannels in order to provide an extra-wide window for fringe search.\n\n  Processing of the VLBA correlator output was done in three steps. In\nthe first step the data were calibrated using the Astronomical Image\nProcessing System (AIPS) \\citep{aips}. In the second step data were\nimported to the Caltech DIFMAP package~\\citep{difmap}, $uv$ data\nflagged, and maps were produced using an automated procedure of hybrid\nimaging developed by Greg Taylor~\\citep{difmap-script} which we adopted\nfor our needs. We were able to reach the VLBA image thermal noise level\nfor most of our CLEAN images \\citep{VLBA_summ}. Errors on our estimates\nof correlated flux density values for sources stronger than\n$\\sim$100~mJy were dominated by the accuracy of amplitude\ncalibration, which for the VLBA, according to \\citet{VLBA_summ}, is at\nthe level of 5\\,\\% at 1~GHz to 10~GHz. An additional error is introduced by\nthe fact that our frequency channels are widely spread over receiver\nbands while the VLBA S~band and X~band gain-curve parameters are measured\naround 2275~GHz and 8425~MHz respectively \\citep{VLBA_summ}, and the noise\ndiode spectrum is not ideally flat. However, this should not add more\nthan a few percent to the total resulting error. Our error estimate was\nconfirmed by comparison of the flux densities integrated over the VLBA\nimages with the single-dish flux densities which we measured with\n\\mbox{RATAN--600} in June and August 2005 for slowly varying sources\nwithout extended structure. The methods of single-dish observations and\ndata processing can be found in \\citet{Kovalev_etal99}. In the third\nstep, the data were imported to the Calc\/Solve program, group delays\nambiguities were resolved, outliers eliminated, and coordinates of new\nsources were adjusted using ionosphere-free combinations of X~band and\nS~band group delay observables of the three VCS5 sessions, 19 VCS1 to\nVCS4 experiments and 3976 24-hour International VLBI Service for\nastrometry and geodesy (IVS)\nexperiments\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/solutions\/2005c}} in\na single least square solution. Positions of 3486 sources were estimated\nincluding all detected VCS5 sources: 590 targets and 97 tropospheric\ncalibrators. Boundary conditions were imposed requiring zero\nnet-rotation of position adjustments of the 212 sources listed as\ndefining sources in the ICRF catalog with respect to their coordinates\nfrom that catalog.\n\n  In a separate solution, coordinates of the 97 well known tropospheric\ncalibrators were estimated from the VCS5 observing sessions only.\nComparison of these estimates with coordinates derived from the 3976 IVS\ngeodetic\/astrometric sessions provided us a measure of the accuracy of\nthe coordinates from the VCS5  observing campaign. The differences in\ncoordinate estimates were used for computation of parameters $a$ and\n$b(\\delta)$ of an error inflation model in the form $ \\sqrt{ (a\\,\n\\sigma)^2 + b(\\delta)^2}$, where $\\sigma$ is an uncertainty derived from\nthe fringe amplitude signal to noise ratio using the error propagation\nlaw and $\\delta$ is declination. More details about the analysis and\nimaging procedures can be found in \\cite{VCS1} and \\cite{VCS3}. The\nhistogram of source position errors is presented in\nFigure~\\ref{f:errhist}.\n\n\\begin{figure}[b]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\n\\includegraphics[trim=0cm 0cm 0cm 0cm]{f2.eps}\n}\n\\end{center}\n\\caption{Histogram of semi-major error ellipse of position errors.\nThe last bin shows errors exceeding 9~mas.\nSee explanation of different assigned source classes in\n\\S\\,\\ref{s:obs_anal}, \\S\\,\\ref{s:catalog}.\n\\label{f:errhist}\n}\n\\end{figure}\n\n\\begin{figure*}[p]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=-0.5cm 5cm 0cm 0cm]{f3a.eps}\n  \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3b.eps}\n  \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3c.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=0cm  0.0cm 0cm 0cm,angle=270]{f3d.eps}\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3e.eps}\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3f.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=-0.5cm 5cm 0cm 0cm]{f3g.eps}\n  \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3h.eps}\n  \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3i.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=0cm  0.0cm 0cm 0cm,angle=270]{f3j.eps}\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3k.eps}\n  \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3l.eps}\n}\n\\end{center}\n\\figcaption{\\footnotesize \nFrom top to bottom.\n{\\em Row~1:}\nNaturally weighted CLEAN images at S~band (2.3~GHz). The lowest contour\nis plotted at the leven given by ``clev'' in each panel title (Jy\/beam),\nthe peak brightness is given by ``max'' (Jy\/beam). The contour levels\nincrease by factors of two. The dashed contours indicate negative flux.\nThe beam is shown in the bottom left corner of the images.\n{\\em Row~2:}\nDependence of the correlated flux density at S~band on projected\nspacing. Each point represents a coherent\naverage over one 2~min observation on an individual interferometer\nbaseline. The error bars represent only the statistical errors.\n{\\em Row~3:} Naturally\nweighted CLEAN images at X~band (8.6~GHz).\n{\\em Row~4:} Dependence of the\ncorrelated flux density at X~band on projected spacing.\n\\label{f:images}\n}\n\\end{figure*}\n\n  In total, 590 out of 675 sources were detected and yielded at least\ntwo good points for position determination. This 87\\,\\% detection rate \nconfirms the validity of the applied candidate selection procedure\n(\\S\\ref{s:selection}). It should be noted that, due to an omission, the\nlist of target sources contained 21 objects previously observed and\ndetected in the VCS4 campaign.\n\nHowever, not all of these 590 sources are suitable as phase reference\ncalibrators or as targets for geodetic observations. Following\n\\citet{VCS3} we consider a source suitable as a calibrator if 1) the\nnumber of good X\/S pairs of observations is eight or greater in order to\nrule out the possibility of a group delay ambiguity resolution error,\nand 2) the position error before re-weighting is less than 5~mas\nfollowing the strategy adopted in processing VCS observations. Only 433\nsources satisfy this calibrator criteria. Other detected sources were\nsomewhat resolved and\/or below the detection limit of these observations\nof 60 mJy. Some of these may become suitable phase calibrators for\nfuture experiments with higher data rates and more sensitivity than the\nVCS surveys. Among the 157 non-calibrators, 53 sources  had less than\neight observations at X~band and less than eight observations S~band.\nPositions  of these sources we consider as unreliable, because we \ncannot rule out errors in group delay ambiguity resolution. The other 104\nsources had eight or more observations at one of the bands, so we can rule\nout the possibility of group delay ambiguity  resolution errors, and\ntherefore, we consider that our estimates of positions of these sources\nare reliable.\n\n\\section{The VCS5 catalog}\n\\label{s:catalog}\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim= 0.00cm 0cm 0cm 0cm]{f4a.eps}\n  \\includegraphics[trim= 1.5cm 0cm 0cm 0cm,clip]{f4b.eps}\n}\n\\end{center}\n\\figcaption{\nDistributions of flux density integrated over VLBA image for all\ndetected VCS5 sources (columns 10 and 12 of the Table~\\ref{t:cat}).\n\\label{f:hist_s_spi}\n}\n\\end{figure}\n\nThe VCS5 catalog of 590 detected target sources is listed in\nTable~\\ref{t:cat}. The first column gives source class: ``C'' if the\nsource can be used as a calibrator, ``N'' if it cannot but determined\npositions are reliable, ``U''---non-calibrator, unreliable positions.\nThe second and third columns give IVS source name (B1950 notation), and\nIAU name (J2000 notation). The fourth and fifth columns give measured\nsource coordinates at the J2000.0 epoch. Columns 6 and 7 give inflated\nsource position uncertainties in right ascension (without $\\cos\\delta$\nfactor) and declination in mas, and column 8 gives the correlation\ncoefficient between the errors in right ascension and declination. The\nnumber of group delays used for position determination is listed in\ncolumn 9. Columns 10 and 12 give the estimate of the flux density\nintegrated over the entire map in janskies at X~band and S~band\nrespectively. This estimate was computed as a sum of all CLEAN\ncomponents if a CLEAN image was produced. If we did not have enough\ndetections of the visibility function to produce a reliable image, the\nintegrated flux density was estimated as the median of the correlated\nflux density measured at projected spacings less than 25~M$\\lambda$ and\n7~M$\\lambda$ for X~band and S~bands respectively.  The integrated flux\ndensity means the total flux density with spatial frequencies less than\n4~M$\\lambda$ at X~band and 1~M$\\lambda$ at S~band filtered out, or in\nother words, the flux density from all components within a region about\nor less than 50~mas at X~band and 200~mas at S~band. Columns 11 and 13\ngive the flux density of unresolved components estimated as the median\nof correlated flux density values measured at projected spacings greater\nthan 170~M$\\lambda$ for X~band and greater than 45~M$\\lambda$ for\nS~band. For some sources no estimates of the integrated and\/or\nunresolved flux density are presented, because either no data were\ncollected on the baselines used in the calculations, or these data were\nunreliable. Column 14 gives the data type used for position estimation:\nX\/S stands for ionosphere-free linear combination of X and S wide-band\ngroup delays; X stands for X-band-only group delays; and S stands for\nS-band-only group delays. Some sources for which less than eight pairs\nof X~band and S~band group delay observables were available had two or\nmore observations at X~band and\/or S~band. For these sources either\nX~band or S~band only estimates of coordinates are listed in the VCS5\ncatalog.\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n  \\includegraphics[trim=0cm 0cm 0cm 0cm,angle=270]{f5.eps}\n}\n\\end{center}\n\\figcaption{\nThe probability (filled circles) of finding a calibrator in any given\ndirection within a circle of a given radius, north of declination\n$-30\\degr$. All sources from 3976 IVS geodetic\/astrometric sessions and\n22 VCS1 to VCS5 VLBA sessions that are classified as calibrators are taken\ninto account.\n\\label{f:prob}\n}\n\\end{figure}\n\n\n  In addition to this table, the html version of the catalog is posted\non the Web\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/vcs5}}. For each\nsource there are eight links: to a pair of postscript images of the\nsource at X~band and S~band; a pair of plots of correlated flux\ndensity as a function of baseline length projected to the source plane;\na pair of FITS files of CLEAN components of naturally weighted source\nimages; and to a pair of FITS files with calibrated $uv$ data. This\ndataset is also accessible from the NRAO\narchive\\footnote{\\url{http:\/\/archive.nrao.edu}} which links the files to\nthe Virtual Observatory. The positions and the plots are also accessible\nfrom the updated NRAO VLBA Calibrator Search\nweb-page\\footnote{\\url{http:\/\/www.vlba.nrao.edu\/astro\/calib}}.\n\n\\begin{deluxetable*}{c l l l r r r r r r r r r c}\n\\tablewidth{0pt}\n\\tablecaption{\\rm The VCS5 catalog \\label{t:cat}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n   \\colhead{} &\n   \\multicolumn{2}{c}{Source name}                   &\n   \\multicolumn{2}{c}{J2000.0 Coordinates}           &\n   \\multicolumn{2}{c}{Errors (mas)}                  &\n   \\colhead{}                                        &\n   \\colhead{}                                        &\n   \\multicolumn{4}{c}{Correlated flux density (Jy)}  &\n   \\colhead{}\n   \\vspace{0.5ex} \\\\\n   \\multicolumn{9}{c}{}         &\n   \\multicolumn{2}{c}{8.6 GHz}  &\n   \\multicolumn{2}{c}{2.3 GHz}  \\vspace{0.5ex} \\\\\n  \n   \\colhead{Class}    &\n   \\colhead{IVS}      &\n   \\colhead{IAU}      &\n   \\colhead{Right ascension} &\n   \\colhead{Declination}     &\n   \\colhead{$\\Delta \\alpha$} &\n   \\colhead{$\\Delta \\delta$} &\n   \\colhead{Corr}   &\n   \\colhead{\\# Obs} &\n   \\colhead{Total } &\n   \\colhead{Unres } &\n   \\colhead{Total } &\n   \\colhead{Unres } &\n   \\colhead{Band}   \n   \\vspace{0.5ex} \\\\\n   \\colhead{(1)}    &\n   \\colhead{(2)}    &\n   \\colhead{(3)}    &\n   \\colhead{(4)}    &\n   \\colhead{(5)}    &\n   \\colhead{(6)}    &\n   \\colhead{(7)}    &\n   \\colhead{(8)}    &\n   \\colhead{(9)}    &\n   \\colhead{(10)}   &\n   \\colhead{(11)}   &\n   \\colhead{(12)}   &\n   \\colhead{(13)}   &\n   \\colhead{(14)} \n   }\n\\startdata\nC  & \\objectname{0008$+$006} & \\objectname{J0011$+$0057}  & 00 11 30.403309 & $+$00 57 51.87984  &   1.02 &   2.00 &     0.114 &     25 &   0.09 &    0.07 &      \\nodata &      \\nodata &  X   \\\\\nC  & \\objectname{0009$+$467} & \\objectname{J0012$+$4704}  & 00 12 29.302900 & $+$47 04 34.73946  &   0.77 &   1.08 &  $-$0.371 &     35 &   0.13 &    0.12 &    0.10 & $<$0.06 &  X\/S \\\\\nN  & \\objectname{0013$-$240} & \\objectname{J0016$-$2343}  & 00 16 05.738818 & $-$23 43 52.18956  &  30.58 &  16.85 &  $-$0.808 &     17 &     \\nodata &      \\nodata &    0.15 &    0.08 &  S   \\\\\nC  & \\objectname{0015$-$054} & \\objectname{J0017$-$0512}  & 00 17 35.817204 & $-$05 12 41.76727  &   0.46 &   0.92 &  $-$0.278 &     54 &   0.20 &    0.12 &    0.14 &    0.09 &  X\/S \\\\\nC  & \\objectname{0015$-$280} & \\objectname{J0017$-$2748}  & 00 17 59.006128 & $-$27 48 21.57153  &   1.75 &   3.51 &     0.712 &     32 &   0.24 &    0.16 &    0.20 &    0.06 &  X\/S \\\\\nC  & \\objectname{0034$+$078} & \\objectname{J0037$+$0808}  & 00 37 32.197173 & $+$08 08 13.05750  &   0.38 &   0.50 &  $-$0.225 &     76 &   0.25 &    0.14 &    0.19 &    0.10 &  X\/S \\\\\nC  & \\objectname{0035$-$037} & \\objectname{J0038$-$0329}  & 00 38 20.794340 & $-$03 29 58.96178  &   0.32 &   0.63 &  $-$0.347 &     77 &   0.20 &    0.16 &    0.31 &    0.21 &  X\/S \\\\\nN  & \\objectname{0036$-$191} & \\objectname{J0039$-$1854}  & 00 39 16.924431 & $-$18 54 05.61863  &   4.97 &   9.83 &     0.749 &      8 &   0.12 &      \\nodata &    0.18 &    0.10 &  X\/S \\\\\nC  & \\objectname{0037$+$011} & \\objectname{J0040$+$0125}  & 00 40 13.525489 & $+$01 25 46.35014  &   0.99 &   1.94 &  $-$0.124 &     29 &   0.11 & $<$0.06 & $<$0.06 &      \\nodata &  X   \\\\\nC  & \\objectname{0041$+$677} & \\objectname{J0044$+$6803}  & 00 44 50.759589 & $+$68 03 02.68607  &   0.91 &   0.58 &  $-$0.130 &     59 &   0.28 &    0.22 &    0.14 & $<$0.06 &  X\/S \\\\\n\\enddata\n\\tablecomments{\nTable~\\ref{t:cat} is presented in its entirety in the electronic edition\nof the Astronomical Journal. A portion is shown here  for guidance\nregarding its form and contents. Assigned source class in (1) is `C' for\ncalibrator, `N' for non-calibrator with reliable coordinates,\n`U' for non-calibrator with unreliable coordinates; see\n\\S\\,\\ref{s:obs_anal}, \\S\\,\\ref{s:catalog} for details. Units of right ascension\nare hours, minutes and seconds; units of declination are degrees,\nminutes and seconds.\n}\n\\end{deluxetable*}\n\n\\begin{deluxetable}{l l c c}\n\\tablewidth{0pt}\n\\tablecaption{\\rm Sources not detected in VCS5 VLBA observations\\label{t:nondetected}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n   \\multicolumn{2}{c}{Source name}                   &\n   \\multicolumn{2}{c}{J2000.0 Coordinates}           \n   \\vspace{0.5ex} \\\\\n   \\colhead{IVS}      &\n   \\colhead{IAU}      &\n   \\colhead{Right ascension} &\n   \\colhead{Declination} \n   \\vspace{0.5ex} \\\\\n   \\colhead{(1)}    &\n   \\colhead{(2)}    &\n   \\colhead{(3)}    &\n   \\colhead{(4)}     \n   }\n\\startdata\n\\objectname{0032$-$011} & \\objectname{J0034$-$0054} & 00 34 43.93 & $-$00 54 13.0 \\\\\n\\objectname{0039$+$211} & \\objectname{J0041$+$2123} & 00 41 45.10 & $+$21 23 41.1 \\\\\n\\objectname{0104$+$650} & \\objectname{J0107$+$6521} & 01 07 51.35 & $+$65 21 20.5 \\\\\n\\objectname{0137$-$273} & \\objectname{J0139$-$2705} & 01 39 55.42 & $-$27 05 29.4 \\\\\n\\objectname{0212$-$214} & \\objectname{J0214$-$2113} & 02 14 40.73 & $-$21 13 28.2 \\\\\n\\objectname{0258$+$356} & \\objectname{J0301$+$3551} & 03 01 47.96 & $+$35 51 24.5 \\\\\n\\objectname{0426$+$351} & \\objectname{J0430$+$3516} & 04 30 14.41 & $+$35 16 23.1 \\\\\n\\objectname{0434$-$225} & \\objectname{J0436$-$2226} & 04 36 34.31 & $-$22 26 18.6 \\\\\n\\objectname{0506$-$196} & \\objectname{J0508$-$1935} & 05 08 19.03 & $-$19 35 56.4 \\\\\n\\objectname{0512$-$129} & \\objectname{J0515$-$1255} & 05 15 17.51 & $-$12 55 27.8 \\\\\n\\enddata\n\\tablecomments{\nTable~\\ref{t:nondetected} is presented in its entirety in the electronic\nedition of the Astronomical Journal. A portion is shown here  for\nguidance regarding its form and contents. Units of right ascension are\nhours, minutes and seconds; units of declination are degrees, minutes\nand seconds. The 85 sources presented in this electronic table include\nthe gravitation lens \\objectname{1422+231} which was detected, but not\nprocessed in VCS5. The J2000 source positions are taken from the NVSS\nsurvey \\citep{NVSS}, they were used for VCS5 VLBA observing and\ncorrelation.\n}\n\\end{deluxetable}\n\nTable~\\ref{t:nondetected} presents 85 sources not detected in VCS5 VLBA\nobservations. Source positions used for observations and correlation are\npresented. They are taken from \\cite{NVSS}. The correlated flux density\nfor the non-detected sources is estimated to be less than 60~mJy at 2.3~GHz\nand 8.6~GHz.\n\nFigure~\\ref{f:images} presents examples of naturally weighted contour\nCLEAN images as well as correlated flux density versus projected spacing\ndependence for three sources: the strongest VCS5 source at X~band,\n\\objectname{J1250$+$0216}, with two bright components resolved at X~band and\nnot resolved at S~band; a steep spectrum source with extended structure,\n\\objectname{J0932$+$6507}; and the source with the most inverted\nspectrum and very compact structure at the milliarcsecond scale,\n\\objectname{J2210$+$2013}.\n\nFigure~\\ref{f:hist_s_spi} presents histograms of the 2.3~GHz and 8.6~GHz\nintegrated flux density for 590 detected VCS5 sources, 132 out of which\nhave integrated flux density $S\\ge 200$~mJy at 8.6~GHz. Their addition\nto the previously observed sources will form the statistically complete\nsample north of declination $-30\\degr$. It is interesting to note that\nmany of the discovered VCS5 sources have inverted radio spectra. The\n50~mJy cutoff for the original selection of sources from the NVSS\ncatalog allowed us to add inverted-spectrum objects to the list of\ncandidates. A few tens of new compact VCS5 objects with high flux\ndensity on VLBA baselines will be useful for geodetic applications. \n\n   The sky calibrator density for different radii of a search circle for\ndeclination $\\delta>-30\\degr$ is presented in Figure~\\ref{f:prob}. As\ndiscussed in \\cite{VCS4}, the addition of these sources to the VLBA\nCalibrator list did not affect significantly the density for the search\nradius of 4\\degr, but increases it for smaller search circles, e.g.,\nthe probability of finding a calibrator within $2\\fdg5$ is now 83\\,\\%. This\nis beneficial for many applications requiring bright compact calibrator\nwithin 2\\degr to 3\\degr of a target.\n\n\\section{Summary}\n\\label{s:summary}\n\nThe VCS5 Survey has made a significant step towards constructing a\nhomogeneous statistically complete sample of flat-spectrum compact\nextragalactic radio sources north of declination $-30\\degr$ with\nintegrated VLBA flux density greater than about 200~mJy at 8~GHz. The\nVCS5 Survey has added 569 new sources, not previously observed with VLBI\nunder geodesy and absolute astrometry programs. Among them, 433 sources\nare suitable as phase referencing calibrators and as target sources for\ngeodetic applications. After processing the VCS5 experiments, the total\nnumber of sources with positions known at the  nanoradian level is 3486,\nand the number of VLBI calibrators has  grown from 2472 to 2905.  This\npool of calibrators was formed from analysis of 22 VLBA Calibrator\nSurvey and 3976 24-hour IVS astrometry and geodesy observing\nsessions.\n\nIn the present paper we do not yet provide quantitative estimates of\ncompleteness of our list of compact flat-spectrum sources. In order to\nget these estimates we are going to (i) complete the homogeneous imaging\nof all of 3486 sources and get estimates of their integrated flux\ndensities at milliarcsecond scales in the X~band and S~band, (ii) complete\nprocessing of instantaneous single-dish multi-frequency, multi-epoch flux\ndensity measurements with \\mbox{RATAN--600} for this sample, (iii)\nobserve with the VLBA a total-flux-density limited sample of all sources\nregardless of their spectral index over a relatively large portion of\nthe sky complemented with simultaneous multi-frequency single-dish\nmeasurements. The latter will allow us to assess whether conclusions\ndrawn from the VLBI flat-spectrum source samples can be extended to the\nwhole population of extragalactic objects regardless their continuum\nspectrum characteristics.\n\n\n\n\\acknowledgments\n\n   We would like to thank A.~Roy and the anonymous referee for useful\ncomments which helped to improve the manuscript.\n     \\facility[NRAO(VLBA)]{The National Radio Astronomy Observatory is a\nfacility of the National Science Foundation operated under cooperative\nagreement by Associated Universities, Inc.} We thank the staff of the\nVLBA for carrying these observations in their usual efficient manner.\n     Y.~Y.~Kovalev is a Research Fellow of the Alexander von Humboldt\nFoundation. This work was done while D.~Gordon worked for Raytheon,\nunder NASA contract NAS5--01127.  \\mbox{RATAN--600} observations were\npartly supported by the  Russian Ministry of Education and Science, the\nNASA JURRISS Program (project W--19611), and the Russian Foundation for\nBasic Research (projects 01--02--16812 and 05--02--17377). The authors\nmade use of the database CATS \\citep{CATS} of the Special Astrophysical\nObservatory. This research has made use of the NASA\/IPAC Extragalactic\nDatabase (NED) which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology, under contract with the National\nAeronautics and Space Administration.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{What is {\\sc Cinderella}?}\n\n{\\sc Cinderella} is an abbreviation of ``{\\bf C}omparison of {\\bf INDE}pendent {\\bf REL}ative {\\bf L}east-squares {\\bf A}mplitudes''. It provides a quantitative comparison between the DFT amplitude spectra of time-resolved astronomical measurements.\n\nThe {\\sc SigSpec} technique (Reegen~2005, 2007) allows to determine probabilities for coincident peaks in the DFT amplitude spectra of different datasets quantitatively and in a statistically unbiased way. The theoretical background of this procedure is introduced by Reegen et al.~(2008).\n\n{\\sc Cinderella} uses the standard output of the program {\\sc SigSpec}, which represents the results of a cascade of consecutive prewhitenings employing least-squares fits to obtain amplitudes and phases of signal components. Following the {\\sc SigSpec} terminology, {\\sc Cinderella} returns a spectral significances (hereafter abbreviated by `sig') rather than a probability. This manual uses cross-references to {\\sc SigSpec} frequently. In these cases, the reader is referred to the {\\sc SigSpec} manual (Reegen 2009).\n\n\\section{Projects}\\label{CINDERELLA_Projects}\n\n{\\sc Cinderella} is called by the command line\n\n\\begin{scriptsize}\\begin{verbatim}\nCinderella <project>\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent where {\\tt <project>} is the name (or path, if desired) of the {\\sc Cinderella} project. The project structure is strictly consistent with {\\sc SigSpec}.\n\nBefore running the program, the user has to provide\n\n\\begin{enumerate}\n\\item at least two time series input files consistent with the {\\sc SigSpec} MultiFile mode (see {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}, and ``Time series input files'', p.\\,\\pageref{CINDERELLA_Time series input files}), and\n\\item a directory {\\tt <project>} containing the {\\sc SigSpec} result files corresponding to the time series input files (see ``{\\sc SigSpec} result files'', p.\\,\\pageref{CINDERELLA_SigSpec result files}).\n\\end{enumerate}\n\nFurthermore, the user may pass a set of specifications to {\\sc Cinderella} by means of a file {\\tt <project>.cnd} (see ``The {\\tt .cnd} file'', p.\\,\\pageref{CINDERELLA_The .cnd file}). This file is expected in the same folder as the time series input files and the project directory. For specifications not given by the user, defaults are used.\n\n{\\sc Cinderella} is designed to answer two different questions, depending on the problem it is applied to. By default, the program runs through both modes simultaneously and provides both conditional (p.\\,\\pageref{CINDERELLA_Conditional Mode}) and composed sigs (p.\\,\\pageref{CINDERELLA_Composed Mode}).\n\n\\section{Input}\\label{CINDERELLA_Input}\n\n\\subsection{Time series input files}\\label{CINDERELLA_Time series input files}\n\n{\\sc Cinderella} expects at least two time series input files named according to {\\tt \\#index\\#.<project>.dat}. In this context, {\\tt \\#index\\#} is a placeholder for a six-digit index of the file. Note that only files with consecutive indices are appropriately recognised by {\\sc Cinderella}. The conventions are compatible with the {\\sc SigSpec} MultiFile mode (see {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}), whence the most convenient preparation of data for {\\sc Cinderella} is the {\\sc SigSpec} MultiFile computation.\n\nThe only restrictions to the format of the time series input files are that the number of items per row has to be constant for all rows in the file and that the columns have to be seperated by at least one whitespace character or tab. Dataset entries need not to be numeric, except for the columns specified as time, observable, and weights. The conventions for specifying these three column types are consistent with {\\sc SigSpec}. See {\\sc SigSpec} manual, pp.\\,\\pageref{SIGSPEC_Time series columns representing time and observable}, \\pageref{SIGSPEC_Time series columns containing statistical weights} for details.\n\n\\figureDSSN{f1.eps}{{\\sc SigSpec} result files used as input for the sample project {\\tt CinderellaNative}. The bottom panel refers to the file {\\tt CinderellaNative\/000000.result.dat}, which is used as the comparison dataset. Above, the files {\\tt CinderellaNative\/000001.result.dat} to {\\tt CinderellaNative\/000008.result.dat} are displayed from bottom to top. The underlying time series represent Gaussian noise plus a single sinusoidal signal at a frequency of $3.125\\,\\mathrm{d}^{-1}$ (grey line) with different amplitudes.}{CINDERELLA_normalrun.res}{!htb}{clip,angle=0,width=110mm}\n\n\\subsection{{\\sc SigSpec} result files}\\label{CINDERELLA_SigSpec result files}\n\nThe {\\sc SigSpec} result files {\\tt \\#index\\#.result.dat} are located in the project directory and contain a list of all sig maxima associated to each of the input time series {\\tt \\#index\\#.<project>.dat}. A {\\sc SigSpec} result file consists of seven columns. A full description is provided by the {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_Result files}. {\\sc Cinderella} uses only five columns:\n\n\\begin{itemize}\n\\item the frequency [inverse time units] (column 1),\n\\item the sig (column 2),\n\\item the amplitude [units of observable] (column 3),\n\\item rms scatter of the time series before prewhitening (column 5), and\n\\item point-to-point scatter of the time series before prewhitening (column 6).\n\\end{itemize}\n\nThe last line of a result file contains only two non-zero values in columns 5 and 6. These represent the rms and point-to-point scatter of the time series after the last prewhitening step, correspondingly, and are also used by {\\sc Cinderella}.\n\n\\vspace{12pt}\\noindent{\\bf Example.}\\label{CINDERELLA_EXnormalrun} {\\it The sample project {\\tt CinderellaNative} provides a run without any additional options by typing {\\tt Cinderella CinderellaNative}. The files {\\tt 000000.CinderellaNative.dat}, ..., {\\tt 000008.CinderellaNative.dat} are the same as {\\tt 000038.diffsig.dat} to {\\tt 000046.diffsig.dat} in the {\\sc SigSpec} example {\\tt diffsig} (Reegen 2009, p.\\,\\pageref{SIGSPEC_EXdiffsig}), correspondingly.\n\nThe {\\sc SigSpec} result files {\\tt 000000.result.dat} to {\\tt 000008.result.dat} are provided in the project directory {\\tt CinderellaNative}. They contain all signal components found with sig $>$ 2 and are displayed in Fig.\\,\\ref{CINDERELLA_normalrun.res}\\it .\n\nThe screen output at runtime starts with a standard header.}\n\n\\begin{scriptsize}\\begin{verbatim}\n CCCCC  ii            dd                      ll  ll\nCC   CC               dd                      ll  ll\nCC      ii n nnn   ddddd  eeee  r rrr   eeee  ll  ll   aaaa\nCC      ii nn  nn dd  dd ee  ee rr  rr ee  ee ll  ll  aa  aa\nCC      ii nn  nn dd  dd ee  ee rr     ee  ee ll  ll      aa\nCC      ii nn  nn dd  dd eeeeee rr     eeeeee ll  ll   aaaaa\nCC      ii nn  nn dd  dd ee     rr     ee     ll  ll  aa  aa\nCC   CC ii nn  nn dd  dd ee  ee rr     ee  ee ll  ll  aa  aa\n CCCCC  ii nn  nn  ddd d  eeee  rr      eeee   ll  ll  aaa a\n\n\nComparison of INDEpendent RELative Least-squares Amplitudes\nVersion 1.0\n************************************************************\nby Piet Reegen\nInstitute of Astronomy\nUniversity of Vienna\nTuerkenschanzstrasse 17\n1180 Vienna, Austria\nRelease date: April 29, 2008\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The program starts with processing the command line, checking if a present directory {\\tt CinderellaNative} is present, and searching for a configuration file {\\tt CinderellaNative.cnd} (see ``The {\\tt .cnd} file'', p.\\,\\pageref{CINDERELLA_The .cnd file}). Since there is no such file present, {\\sc Cinderella} produces a warning message.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** start **************************************************\n\nChecking availability of project directory CinderellaNative...\nproject directory CinderellaNative ok.\n\nWarning: CndFile_LoadCnd 001\n         Failed to open .cnd file.\n\\end{verbatim}\\end{scriptsize}\n\n{\\it Now {\\sc Cinderella} counts the time series input files and checks for corresponding {\\sc SigSpec} result files.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** MultiFile count ****************************************\n\nNumber of files                           9\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The next step is to count the rows in each {\\sc SigSpec} result file.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** count rows in SigSpec result files *********************\n\nCinderellaNative\/000000.result.dat:    106 rows\nCinderellaNative\/000001.result.dat:     21 rows\nCinderellaNative\/000002.result.dat:     21 rows\nCinderellaNative\/000003.result.dat:     21 rows\nCinderellaNative\/000004.result.dat:     21 rows\nCinderellaNative\/000005.result.dat:     21 rows\nCinderellaNative\/000006.result.dat:     21 rows\nCinderellaNative\/000007.result.dat:     21 rows\nCinderellaNative\/000008.result.dat:     21 rows\n\\end{verbatim}\\end{scriptsize}\n\n{\\it Before reading the input files, {\\sc Cinderella} performs the assignment of {\\tt target}, {\\tt comp} and {\\tt skip} tags to the datasets (see ``Dataset Types'', p.\\,\\pageref{CINDERELLA_Dataset Types}). Since there is no file {\\tt CinderellaNative.cnd} available, the defaults are used: the first file, {\\tt 000000.CinderellaNative.dat} is considered a comparison dataset, the eight remaining files are interpreted as targets.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** dataset type assignment ********************************\n\n\nWarning: CndFile_Cind 001\n         Failed to open .cnd file.\n         Assigning default types.\n\n000000.CinderellaNative.dat: comparison (default)\n000001.CinderellaNative.dat: target (default)\n000002.CinderellaNative.dat: target (default)\n000003.CinderellaNative.dat: target (default)\n000004.CinderellaNative.dat: target (default)\n000005.CinderellaNative.dat: target (default)\n000006.CinderellaNative.dat: target (default)\n000007.CinderellaNative.dat: target (default)\n000008.CinderellaNative.dat: target (default)\n\nnumber of target datasets                 8\nnumber of comparison datasets             1\nnumber of datasets to skip                0\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The time series and {\\sc SigSpec} result files are read, and {\\sc Cinderella} displays the frequency resolution and the mean observable for each time series.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** read input files ***************************************\n\n000000.CinderellaNative.dat: Rayleigh resolution 0.1089880382935977\n000000.CinderellaNative.dat: mean observable     0.0074422789800987\nCinderellaNative\/000000.result.dat\n000001.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000001.CinderellaNative.dat: mean observable    -0.0704869463068650\nCinderellaNative\/000001.result.dat\n000002.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000002.CinderellaNative.dat: mean observable    -0.0875484339175066\nCinderellaNative\/000002.result.dat\n000003.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000003.CinderellaNative.dat: mean observable    -0.1046099215281657\nCinderellaNative\/000003.result.dat\n000004.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000004.CinderellaNative.dat: mean observable    -0.1216714091388107\nCinderellaNative\/000004.result.dat\n000005.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000005.CinderellaNative.dat: mean observable    -0.1387328967494604\nCinderellaNative\/000005.result.dat\n000006.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000006.CinderellaNative.dat: mean observable    -0.1557943843601256\nCinderellaNative\/000006.result.dat\n000007.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000007.CinderellaNative.dat: mean observable    -0.1728558719707690\nCinderellaNative\/000007.result.dat\n000008.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000008.CinderellaNative.dat: mean observable    -0.1899173595814251\nCinderellaNative\/000008.result.dat\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The amplitudes in the {\\sc SigSpec} result file of the comparison dataset (file {\\tt CinderellaNative\/000000.result.dat} have to be transformed in order to be comparable to the target datasets (see ``Amplitude transformation'', p.\\,\\pageref{CINDERELLA_Amplitude transformation}). {\\sc Cinderella} uses the rms residual as a measure for this transformation. This is the default setting.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** amplitude transformation *******************************\n\nby rms error: file                        0\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The core of {\\sc Cinderella} consists of three different analyses. The first part of these is the computation of conditional sigs for each pair of target vs.~comparison datasets. In this case we have eight target datasets and one comparison dataset, which results in eight operations. In general, the total number of operations performed in this step is the product of the number of target datasets times the number of comparison datasets. Detailed information on the computation of pairwise conditional sigs is found in ``Single-comparison output'' (p.\\,\\pageref{CINDERELLA_Single-comparison output}).}\\pagebreak\n\n\\begin{scriptsize}\\begin{verbatim}\n*** pairwise Cinderella analysis ***************************\n\n     1 vs.      0: conditional sig\n     2 vs.      0: conditional sig\n     3 vs.      0: conditional sig\n     4 vs.      0: conditional sig\n     5 vs.      0: conditional sig\n     6 vs.      0: conditional sig\n     7 vs.      0: conditional sig\n     8 vs.      0: conditional sig\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The second part of the {\\sc Cinderella} analysis is a computation of mean conditional sigs for each target dataset over all comparison datasets (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Multi-comparison output}). Since there is only one comparison dataset available in this example, the output of this procedure is the same as of the previous pairwise analysis. Finally, {\\sc Cinderella} evaluates composed sigs for each target dataset. This calculation is applied to the ``raw'' {\\sc SigSpec} results and also to the conditional sigs obtained by the previous operation. Details on the computation of composed sigs are found in ``Composed Mode'' (p.\\,\\pageref{CINDERELLA_Composed Mode}).}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** total Cinderella analysis ******************************\n\n     1: conditional sig\n     2: conditional sig\n     3: conditional sig\n     4: conditional sig\n     5: conditional sig\n     6: conditional sig\n     7: conditional sig\n     8: conditional sig\ncomposed sig\ncomposed sig of conditional sigs\n\\end{verbatim}\\end{scriptsize}\n\n{\\it On exit, {\\sc Cinderella} displays a goodbye message.}\n\n\\begin{scriptsize}\\begin{verbatim}\nFinished.\n\n************************************************************\n\nThank you for using Cinderella!\nQuestions or comments?\nPlease contact Piet Reegen (reegen@astro.univie.ac.at)\nBye!\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The {\\sc Cinderella} output for this example is discussed in the subsequent chapters.}\n\n\\subsection{The {\\tt .cnd} file}\\label{CINDERELLA_The .cnd file}\n\nAn optional file {\\tt <project>.cnd} consists of a set of keywords and arguments defining project-specific parameters for {\\sc Cinderella}. If this file is not present in the same folder as the time series input files, {\\sc Cinderella} uses a set of default parameters.\n\n\\vspace{12pt}\n{\\bf Caution: {\\sc Cinderella} demands a carriage-return character at the end of the file {\\tt <project>.cnd}, otherwise the program hangs!}\n\\vspace{12pt}\n\nLines in the {\\tt .cnd} file starting with a {\\tt \\#} character are ignored by {\\sc Cinderella}. This provides the possibility to write comments into the file.\n\n\\section{Indexing}\\label{CINDERELLA_Indexing}\n\nBy default, {\\sc Cinderella} expects the file index of time series and {\\sc SigSpec} results to start at zero. If the user wants the program to start at a different index, the keyword {\\tt mfstart} may be given in the {\\tt .cnd} file. This keyword is followed by an integer representing the desired start index. Furthermore, the software takes into account as many files with consecutive indices as available. The number of indices to process may be restricted by means of the keyword {\\tt multifile}, followed by an integer representing the last index to take into account. The use of the keywords {\\tt mfstart} and {\\tt multifile} is strictly consistent with the {\\sc SigSpec} conventions ({\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}).\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt index} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), but both the time series input files and the {\\sc SigSpec} result files are now enumerated from {\\tt 004847} to {\\tt 004855} rather than from {\\tt 000000} to {\\tt 000008}. The file {\\tt index.cnd} consists of a single line,\n\n\\begin{scriptsize}\\begin{verbatim}\nmfstart 4847\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent Since the keyword {\\tt multifile} is not specified in the file {\\tt index.cnd}, {\\sc Cinderella} uses all input files available through consecutive indices.\n\nThe computations are entirely the same as for {\\tt CinderellaNative}, but all indices are consistently incorporated by {\\sc Cinderella}. E.\\,g., the screen output for the single-comparison computations is now:\n\n\\begin{scriptsize}\\begin{verbatim}\n*** pairwise Cinderella analysis ***************************\n\n  4848 vs.   4847: conditional sig\n  4849 vs.   4847: conditional sig\n  4850 vs.   4847: conditional sig\n  4851 vs.   4847: conditional sig\n  4852 vs.   4847: conditional sig\n  4853 vs.   4847: conditional sig\n  4854 vs.   4847: conditional sig\n  4855 vs.   4847: conditional sig\n\\end{verbatim}\\end{scriptsize}\\sf\n\n\\section{Dataset Types}\\label{CINDERELLA_Dataset Types}\n\nIn order to avoid unnessecarily high computational effort and redundant output, if comparing all possible pairs of time series input files, {\\sc Cinderella} provides the possibility to specify which pairs of target\/comparison datasets to take into account. Moreover, the user has the opportunity to identify datasets to be ignored.\n\n{\\sc Cinderella} will produce one so-called single-comparison output file (see ``Single-comparison output'', p.\\,\\pageref{CINDERELLA_Single-comparison output}) for each target-comparison pair. If there is more than one comparison dataset available, additional files are generated for each target. They contain summaries concerning all comparison datasets examined for the target (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Multi-comparison output}) and ``Output for composed mode'', p.\\,\\pageref{CINDERELLA_Output for composed mode}).\n\nContrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file.\n\n\\subsection{Target datasets}\\label{CINDERELLA_Target datasets}\n\nThe keyword {\\tt target}\\label{CINDERELLA_keyword.target} in the {\\tt .cnd} file is used for the specification of a target dataset. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Multiple declarations of {\\tt target} are supported. If no {\\tt .cnd} file is available, {\\sc Cinderella} uses the first time series input file (i.\\,e. the one with the start index) as the only target dataset. \n\n\\subsection{Comparison datasets}\\label{CINDERELLA_Comparison datasets}\n\nThe keyword {\\tt comp}\\label{CINDERELLA_keyword.comp} in the {\\tt .cnd} file is used for the specification of a comparison dataset. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Contrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file. If no {\\tt .cnd} file is available, {\\sc Cinderella} uses all time series input files -- except for the first one, which is considered target data -- as comparison datasets.\n\n\\subsection{Datasets to ignore}\\label{CINDERELLA_Datasets to ignore}\n\nThe keyword {\\tt skip}\\label{CINDERELLA_keyword.skip} in the {\\tt .cnd} file is used for the specification of a dataset not to be taken into account for computation. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Contrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file.\n\n\\subsection{Default type}\\label{CINDERELLA_Default type}\n\nTo enhance the convenience for the user, not all files need to be specified by the keywords {\\tt target}, {\\tt comp} and {\\tt skip}. The keyword {\\tt deftype}\\label{CINDERELLA_keyword.deftype} may be used to assign a default dataset type.\n\\begin{enumerate}\n\\item Use {\\tt deftype target} to assign the target attribute by default. If no {\\tt deftype} keyword is provided, this setting is activated.\n\\item Use {\\tt deftype comp} to assign the comp attribute by default.\n\\item Use {\\tt deftype skip} to assign the skip attribute by default.\n\\end{enumerate}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt types} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt types.cnd} contains the two lines\n\n\\begin{scriptsize}\\begin{verbatim}\ndeftype target\ncomp 0\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent This reproduces the default assignment of data types, and {\\sc Cinderella} performs the same calculations as for the project {\\tt CinderellaNative}. The only difference is the screen output:\n\n\\begin{scriptsize}\\begin{verbatim}\n*** dataset type assignment ********************************\n\n000000.types.dat: comparison\n000001.types.dat: target\n000002.types.dat: target\n000003.types.dat: target\n000004.types.dat: target\n000005.types.dat: target\n000006.types.dat: target\n000007.types.dat: target\n000008.types.dat: target\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent The fact that {\\tt (default)} is not attached to the file list indicates that {\\sc Cinderella} uses the specifications given in the file {\\tt types.cnd} rather than the standard settings applied to the project {\\tt CinderellaNative}.\\sf\n\n\\section{Conditional Mode}\\label{CINDERELLA_Conditional Mode}\n\nThe conditional sig is a measure of the probability that a signal component in a target star occurs, although a coincident signal component is found in a comparison star or sky background. It provides an answer to the question, ``What is the probability that a signal component with given amplitude and sig in the target data is not due to the same process that produces a coincident signal component with given amplitude and sig in the comparison data?''\n\nThe conditional {\\sc Cinderella} mode is comparable to the differential sig ({\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_Differential significance spectra}), although the numerical results are not the same.\n\n\\begin{itemize}\n\\item For the differential mode of {\\sc SigSpec}, the full spectral information is available. Thus {\\sc SigSpec} handles the DFT spectra as continuous functions. {\\sc Cinderella} accesses only a list of peaks detected by {\\sc SigSpec}. Deviations of corresponding peak frequencies in comparison and target spectra cannot be handled as accurately as in the case of differential sig computation.\n\\item The differential mode of {\\sc SigSpec} compares power integrals over the entire frequency range under consideration for the transformation of amplitudes from comparison into target data. Since {\\sc Cinderella} deals with a list of frequencies rather than the entire spectra, different strategies to transform amplitudes have to be employed. See ``Amplitude transformation'', p.\\,\\pageref{CINDERELLA_Amplitude transformation}.\n\\end{itemize}\n\n\\subsubsection{Single-comparison output}\\label{CINDERELLA_Single-comparison output}\n\nThe single-comparison output files contain three columns:\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item conditional sig,\n\\item conditional csig.\n\\end{enumerate}\n\nEach file refers to the analysis of a single target-comparison dataset pair. Correspondingly, two six-digit indices {\\tt \\#target\\#}, {\\tt \\#comparison\\#} are used to form the filenames, {\\tt \\#target\\#.cd.\\#comparison\\#.dat} (conditional mode) and {\\tt \\#target\\#.cd.\\#comparison\\#.dat} (composed mode).\n\n\\vspace{12pt}\\noindent{\\bf Example.} {\\it The output file {\\tt 000004.cd.000026.dat} contains conditional sigs for {\\tt 000004.<project>.dat} as target data and {\\tt 000026.<project>.dat} as comparison data.}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it In the sample {\\tt CinderellaNative}, there are 8 single-compari\\-son output files: {\\tt 000001.cd.000000.dat} to {\\tt 000008.cd.000000.dat}. Since there is only one comparison dataset available, these files are redundant, because {\\tt \\#target\\#.cd.dat} $=$ {\\tt \\#target\\#.cd.000000.dat}.\\sf\n\n\\subsubsection{Multi-comparison output}\\label{CINDERELLA_Multi-comparison output}\n\nFor each target dataset, a multi-comparison output file {\\tt \\#target\\#.cd.dat} is generated for each target dataset. The three columns represent\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item mean conditional sig for all comparison datasets,\n\\item mean conditional csig for all comparison datasets.\n\\end{enumerate}\n\nIf there is only one comparison dataset available, the single-comparison and multi-comparison output files are identical.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt CinderellaNative} contains 8 multi-comparison output files in the project directory: {\\tt 000001.cd.dat} to {\\tt 000008.cd.dat}.\\sf\n\n\\subsection{Candidate selection}\\label{CINDERELLA_Candidate selection}\n\nFor each target dataset, {\\sc Cinderella} scans all comparison datasets, searching for coincident signal components. A pair of signal components is considered coincident, if the frequencies match to an accuracy that may be specified by the user, who may also define what {\\sc Cinderella} shall do if a comparison dataset does not contain a match for a given target frequency.\n\nIf more than one coincident frequency associated to a given target frequency is found, {\\sc Cinderella} chooses the candidate with the lowest conditional or composed sig in order to obtain the most conservative solution.\n\n\\subsubsection{Frequency resolution}\\label{CINDERELLA_Frequency resolution}\n\nThere are mainly two interpretations of the frequency resolution. It is either calculated as the inverse time interval width (Rayleigh frequency resolution),\n\\begin{equation}\\label{CINDERELLA_EQ Rayleigh}\n\\delta f := \\frac{1}{T}\\: ,\n\\end{equation}\nor as\n\\begin{equation}\\label{CINDERELLA_EQ Kallinger}\n\\delta f := \\frac{1}{T\\sqrt{\\mathrm{sig}\\left( A\\right)}}\\: ,\n\\end{equation}\nwhere $\\mathrm{sig}\\left( A\\right)$ denotes the sig of an amplitude $A$. This definition is called Kallinger resolution (Kallinger, Reegen \\& Weiss 2008).\n\nIn order to enhance the flexibility of {\\sc Cinderella}, the frequency resolution is computed as \\begin{equation}\\label{CINDERELLA_EQ fres}\n\\delta f := \\frac{1}{T\\sqrt{\\mathrm{sig}\\left( A\\right)^{\\tau}}}\\: ,\n\\end{equation}\nwhere the floating-point number $\\tau\\in\\left[ 0,1\\right]$ may be defined using the keyword {\\tt tol}\\label{CINDERELLA_keyword.tol} in the {\\tt .cnd} file. The special value $\\tau = 0$ transforms eq.\\,\\ref{CINDERELLA_EQ fres} into eq.\\,\\ref{CINDERELLA_EQ Rayleigh}, setting $\\tau = 1$ provides eq.\\,\\ref{CINDERELLA_EQ Kallinger}.\nThe default value is $\\tau = 0$.\n\n{\\sc Cinderella} checks for frequencies in the comparison datasets are within the frequency resolution around each frequency in the target dataset.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt cand} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt cand.cnd} contains the line\n\n\\begin{scriptsize}\\begin{verbatim}\ntol 2\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent The frequency tolerance parameter is increased compared to the default value 0, which means that the intervals taken into account to search for corresponding signal components are tendentially narrower. This setting is for demonstration only; in normal applications, only frequency tolerance parameters ranging from 0 to 1 will make sense.\n\nThe effect of this modification is visible, e.\\,g., comparing the output files {\\tt 000001.cd.000000.dat} of the project {\\tt CinderellaNative} to the project {\\tt cand}. In the project {\\tt CinderellaNative}, this file contains the line\n\n\\begin{scriptsize}\\begin{verbatim}\n58.3815412948909298 -8.8063413553403507 -5.6290446018702553\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent whereas the corresponding line in the project {\\tt cand} is\n\n\\begin{scriptsize}\\begin{verbatim}\n58.3815412948909298       2.1858505500938747       0.3972034177439077\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent In the project {\\tt CinderellaNative}, the 14th component in the {\\sc SigSpec} result file {\\tt 000001.result.dat} in the project directory is related to the 47th component in the file {\\tt 000000.result.dat}. The two frequencies differ by 0.054, and the Rayleigh frequency resolution of the target dataset is $0.091$, which is sufficient for a correspondence. In the project {\\tt cand}, the target sig of 2.387 becomes relevant. Eq.\\,\\ref{CINDERELLA_EQ fres} \\it yields a frequency resolution 0.042 for this component, which is now too small for a coincidence. The corresponding line in the file {\\tt 000001.cd.000000.dat} consistently indicates that no coincident peak is found for this signal component. In this case, {\\sc Cinderella} uses a default sig threshold for the comparison data, see ``Spectral significance threshold'', below.\\sf\n\n\\subsubsection{Spectral significance threshold}\\label{CINDERELLA_Spectral significance threshold}\n\nIf no coincidence in a comparison dataset is detected for a given target frequency, i.\\,e., if {\\sc Cinderella} does not find a valid candidate for this target frequency, a default value is used for the sig in the comparison dataset. The user may specify this {\\sc Cinderella} threshold by means of the keyword {\\tt defsig}\\label{CINDERELLA_keyword.defsig} in the {\\tt .cnd} file. The same specification may be set for the default csig\\footnote{abbreviation for cumulative sig} using the keyword {\\tt defcsig}\\label{CINDERELLA_keyword.defcsig}. If one of these keywords is not provided, $\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$ is used correspondingly by default. According to Reegen~(2007), this is the expected value of the sig for white noise. The underlying assumption is that the residuals after prewhitening of all significant signal components in the comparison dataset represent pure noise, i.\\,e.~do not contain any further unresolved signal.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt cand} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt cand.cnd} contains the two lines\n\n\\begin{scriptsize}\\begin{verbatim}\ndefsig 0\ndefcsig 1\n\\end{verbatim}\\end{scriptsize}\n\nThe second row in the output file {\\tt 000001.cd.000000.dat} of the project {\\tt CinderellaNative}\n\n\\begin{scriptsize}\\begin{verbatim}\n30.7091991449662061 3.8506295783390758 3.6090130812823817\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent using the default sig and csig threshold $\\frac{\\pi}{4}\\log\\mathrm{e}$, because the comparison dataset does not contain a coincident signal component. The second row in the corresponding file of the project {\\tt cand} is\n\n\\begin{scriptsize}\\begin{verbatim}\n30.7091991449662061 4.1917236667995361 2.9501071697428420\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent Since the default sig is lower in the project {\\tt cand}, the resulting conditional sig is higher. On the other hand, the default csig is higher, which causes the resulting conditional csig to drop down.\\sf\n\n\\subsection{Amplitude transformation}\\label{CINDERELLA_Amplitude transformation}\n\nThe assumption that instrumental and environmental artifacts use to be additive in terms of intensity may create needs to transform amplitudes in mag from the comparison into the target spectra, if the conditional {\\sc Cinderella} mode is applied. The amplitude transformation is only performed to obtain conditional sigs.\n\nThree different strategies to adjust comparison amplitudes are offered, according to the specifications in the {\\tt .cnd} file.\n\n\\begin{itemize}\n\\item In photometry, the photon statistics may be employed to transform comparison into target amplitudes, if the mean magnitudes of the stars are known (Reegen et al.~2008). If the keyword {\\tt transam:mean}\\label{CINDERELLA_keyword.transam:mean} is provided in the {\\tt .cnd} file, {\\sc Cinderella} uses the mean observables $\\left< m_C\\right>$, $\\left< m_T\\right>$ of the comparison and target time series, respectively, to transform the comparison amplitude $A_C$ into a target amplitude $A_T$ according to\n\\begin{equation}\\label{CINDERELLA_EQtransam_mean}\nA_T = 2.5\\log\\left[ 1 + \\frac{10^{-0.4\\left(\\left< m_C\\right> - A_C\\right)} - 10^{-0.4\\left< m_C\\right>}}{10^{-0.4\\left< m_T\\right>}}\\right]\\: .\n\\end{equation}\n\\item If the keyword {\\tt transam:rms}\\label{CINDERELLA_keyword.transam:rms} is specified in the {\\tt .cnd} file, {\\sc Cinderella} interprets the residual rms errors $\\sigma _C$, $\\sigma _T$ of the comparison and target time series\\footnote{with all significant signal prewhitened}, respectively, as measures of the photon noise levels and evaluates the transformed amplitude according to\n\\begin{equation}\\label{CINDERELLA_EQtransam_rms}\nA_T = 2.5\\log\\left[ 1 + \\frac{\\sigma _C^2}{\\sigma _T^2}\\left( 10^{0.4\\,A_C} - 1\\right)\\right]\\: .\n\\end{equation}\n\\item The keyword {\\tt transam:ppsc}\\label{CINDERELLA_keyword.transam:ppsc} in the {\\tt .cnd} file causes {\\sc Cinderella} to use Eq.\\,\\ref{CINDERELLA_EQtransam_rms} employing residual point-to-point scatter instead of residual rms error for both $\\sigma _C$ and $\\sigma _T$.\n\\item If the keyword {\\tt transam:none}\\label{CINDERELLA_keyword.transam:none} is specified in the {\\tt .cnd} file, no amplitude transformation is performed at all, i.\\,e., {\\sc Cinderella} assumes $A_T = A_C$.\n\\end{itemize}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-mean} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The time series data are considered to represent millimag photometry. The comparison dataset is assumed to refer to a 5 mag star, whereas the target datasets shall correspond to a 15 mag star. The resulting time series input files are {\\tt 000000.transam-mean.dat} {\\tt 000001.transam-mean.dat} to {\\tt 000008.transam-mean.dat}. The keyword\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:mean\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent in the file {\\tt transam-mean.cnd} forces {\\sc Cinderella} to employ the mean magnitudes of the datasets for the amplitude transformation.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt CinderellaNative} contains an amplitude transformation based on the rms residual, which is the default method.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-ppsc} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The line\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:ppsc\n\\end{verbatim}\\end{scriptsize}\n\nin the file {\\tt transam-ppsc.cnd} forces {\\sc Cinderella} to employ the residual point-to-point scatters of the datasets for the amplitude transformation.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-none} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The line\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:none\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent in the file {\\tt transam-none.cnd} switches off the amplitude transformation.\\sf\n\n\\section{Composed Mode}\\label{CINDERELLA_Composed Mode}\n\nThe composed sig is a measure of the probability that two coincident signal components occur in two different datasets. This implements a logical `and', providing an answer to the question, ``What is the probability that two different datasets show coincident signal components with given amplitudes and sigs?''\n\nThe composed mode is useful for, e.\\,g., photometry of the same star in different filters, or if two short datasets of the same object obtained in different years are examined.\n\nNote that the composed sig in the {\\sc SigSpec} result files (see {\\sc SigSpec} manual) is consistently defined, but applies to the set of significant signal components displayed in the file, whereas {\\sc Cinderella} refers to the composed sig of signal components found in two or more different datasets.\n\nContrary to the candidate selection procedure in conditional mode (p.\\,\\pageref{CINDERELLA_Candidate selection}), the frequency interval between the lowest and the highest frequency found in all target datasets is scanned in steps defined by half the frequency resolution (p.\\,\\pageref{CINDERELLA_Frequency resolution}). For each of the frequencies under consideration, {\\sc Cinderella} computes a composed sig, basically following the introduction by Reegen et al.~(2008). Since {\\sc Cinderella}'s composed mode takes into account all signal components in all datasets, statistical weights have to be introduced that put more emphasis to signal frequencies closer to the frequency under consideration. Hence the composed sig $\\mathrm{csig}\\left( A_n\\right)$ (annotation by Reegen et al.~2008) assigned to an arbitrary frequency $f$ is evaluated according to\n\\begin{equation}\n\\log\\left[ 1-10^{\\mathrm{csig}\\left( A_n\\right)}\\right] = \\frac{1}{N}\\sum _{n=1}^{N}\\mathrm{e} ^{-\\frac{1}{2}\\left[\\frac{f-f_n}{\\min\\left(\\delta f_n\\right)}\\right]^2}\\log\\left[ 1 - 10^{-\\mathrm{sig}\\,\\left( A_n\\right)}\\right]\\: .\n\\end{equation}\nIn this context, the total number of signal components in all target datasets is denoted $N$, $f_n$ referring to the frequency of one of these signal components. The minimum frequency resolution $\\min\\left(\\delta f_n\\right)$ incorporates the definition by Eq.\\,\\ref{CINDERELLA_EQ fres}.\n\nAn interpolation loop is used to exactly identify the maxima of this composed sig, which are written to the output file.\n\n\\subsection{Output for composed mode}\\label{CINDERELLA_Output for composed mode}\n\nThe calculation of the composed sig is applied to all target datasets at once. A file {\\tt cp.dat} is generated, which contains the composed sigs of all target datasets. The three columns refer to:\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item composed sig for all target datasets,\n\\item composed csig for all target datasets.\n\\end{enumerate}\n\nThe composed sigs are also calculated for the conditional sigs, i.\\,e., the files {\\tt \\#target\\#.cd.dat} (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Single-comparison output}), and written to a file {\\tt cc.dat}. The column format is the same as for the file {\\tt cp.dat}.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it For the sample project {\\tt CinderellaNative}, the project directory contains a file {\\tt cp.dat} with the composed sigs (and csigs) for all target datasets (provided by the {\\sc SigSpec} result files {\\tt 000001.result.dat} to {\\tt 000008.result.dat}). Furthermore, a file {\\tt cc.dat} is found in the project directory. It contains the composed sigs (and csigs) applied to the conditional ones for all target datasets, i.\\,e., the composed sigs are evaluated using the multi-comparison output files generated by the conditional mode, {\\tt 000001.cd.dat} to {\\tt 000008.cd.dat}.\\sf\n\n\\section{Keywords Reference}\\label{CINDERELLA_Keywords Reference}\n\nThis section is a compilation of all keywords accepted by {\\sc Cinderella}. A brief description of arguments and default values is given. The type of argument is provided by either {\\tt <int>}, {\\tt <double>}, or {\\tt <string>}, and default values are given in parentheses, e.\\,g.~{\\tt (2)}. Empty parentheses indicate that there is no default setting.\n\n\\subsubsection{\\tt col:obs <int> (2)}\n\nobservable column index (unique), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:obs}\n\n\\subsubsection{\\tt col:time <int> (1)}\n\ntime column index (unique), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:time}\n\n\\subsubsection{\\tt col:weights <int>}\n\nweights column index (also multiple), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:weights}\n\n\\subsubsection{\\tt comp <int> (all except start index)}\n\nspecification of time series input files to be regarded as comparison datasets, p.\\,\\pageref{CINDERELLA_keyword.comp}\n\n\\subsubsection{\\tt defcsig <double> ($\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$)}\n\nthreshold to be used for the csig, if no coincidence is found in a comparison dataset, p.\\,\\pageref{CINDERELLA_keyword.defcsig}\n\n\\subsubsection{\\tt defsig <double> ($\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$)}\n\nthreshold to be used for the sig, if no coincidence is found in a comparison dataset, p.\\,\\pageref{CINDERELLA_keyword.defsig}\n\n\\subsubsection{\\tt deftype <target\/comp\/skip> (target)}\n\nspecifies the type of dataset to be assigned to a time series by default, p.\\,\\pageref{CINDERELLA_keyword.deftype}\n\n\\subsubsection{\\tt skip <int> ()}\n\nspecification of time series input files not to be taken into consideration, p.\\,\\pageref{CINDERELLA_keyword.skip}\n\n\\subsubsection{\\tt target <int> (start index)}\n\nspecification of time series input files to be regarded as target datasets, p.\\,\\pageref{CINDERELLA_keyword.target}\n\n\\subsubsection{\\tt tol <double> (0)}\n\n{\\sc Cinderella} frequency tolerance parameter, p.\\,\\pageref{CINDERELLA_keyword.tol}\n\n\\subsubsection{\\tt transam:mean}\n\namplitude transformation using the mean observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:mean}\n\n\\subsubsection{\\tt transam:none}\n\nno amplitude transformation at all, p.\\,\\pageref{CINDERELLA_keyword.transam:none}\n\n\\subsubsection{\\tt transam:ppsc}\n\namplitude transformation using the point-to-point scatter of the residual observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:ppsc}\n\n\\subsubsection{{\\tt transam:rms} (default)}\n\namplitude transformation using the rms residual observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:rms}\n\n\\section{Online availability}\n\nThe ANSI-C code is available online at {\\tt http:\/\/www.sigspec.org}. For further information, please contact P.~Reegen, {\\tt peter.reegen@univie.ac.at}.\n\n\\acknowledgments{\nPR received financial support from the Fonds zur F\\\"or\\-de\\-rung der wis\\-sen\\-schaft\\-li\\-chen Forschung (FWF, projects P14546-PHY, P17580-N2) and the BM:BWK (project COROT). Furthermore, it is a pleasure to thank M.~Gruberbauer (Univ.~of Vienna), D.\\,B.~Guenther (St.~Mary's Univ., Halifax), M.~Hareter, D.~Huber, T.~Kallinger (Univ.~of Vienna), R.~Kusch\\-nig (UBC, Vancouver), J.\\,M. Matthews (UBC, Vancouver), A.\\,F.\\,J.~Moffat (Univ.~de Montreal), D.~Punz (Univ.~of Vienna), S.\\,M. Rucinski (D.~Dunlap Obs., Toronto), D.~Sasselov (Harvard-Smithsonian Center, Cambridge, MA), L.~Schneider (Univ.~of Vienna), G.\\,A.\\,H.~Walker (UBC, Vancouver), W.\\,W. Weiss, and K.~Zwintz (Univ.~of Vienna) for valuable discussion and support with extensive software tests. I acknowledge the anonymous referee for a detailed examination of both this publication and the corresponding software, as well as for the constructive feedback that helped to improve the overall quality a lot. Finally, I address my very special thanks to J.\\,D.~Scargle for his valuable support.\n}\n\n\\References{\nKallinger, T., Reegen, P., Weiss, W.\\,W.~2008, A\\&A, 481,  571\\\\\n\nReegen, P.~2005, in {\\it The A-Star Puzzle}, Proceedings of IAUS 224, eds. J. Zverko, J. Ziznovsky, S.J. Adelman, W.W. Weiss (Cambridge: Cambridge Univ.~Press), p.~791\\\\\n\nReegen, P.~2007, A\\&A, 467, 1353\\\\\n\nReegen, P.~2009, CoAst, submitted\\\\\n\nReegen, P., Gruberbauer, M., Schneider, L., Weiss, W.\\,W.~2008, A\\&A, 484, 601\\\\\n\nZwintz, K., Marconi, M., Kallinger, T., Weiss, W.\\,W.~2004, in {\\it The A-Star Puzzle}, Proceedings of IAUS 224, eds. J.~Zverko, J.~Ziznovsky, S.\\,J.~Adelman, W.\\,W.~Weiss (Cambridge: Cambridge Univ.~Press), p.\\,353\\\\\n\nZwintz, K., Weiss, W.\\,W.~2006, A\\&A, 457, 237\\\\\n\n}\n\n\\end{document}\n\n\\section*{Abstract}\\normalsize\\sf}{}\n\\newcommand{\\References}[1]{\\begin{flushleft}{\\large References\\\\}\\vspace*{2mm}\\small #1 \\end{flushleft}}\n\n\\newcommand{\\chapterDSSN}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headDSSN}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\chapterdssn}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headdssn}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\figureDSSN}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figuredssn}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand {\\strich} {\\begin{center} \\vspace{0.5cm} \\rule{1.5cm}{0.3mm} \\end{center}}\n\\renewcommand{\\chaptername}{}\n\\newcommand{\\acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\n\\section*{Affiliations of the authors} \\smallskip\\par\\noindent\\hangindent=4.25truemm $^1$ ~\\,LESIA-Observatoire de Paris-CNRS (UMR~8109)-Univ. Paris~6-Univ. Paris~7,\\\\pl. J. Janssen, F-92195 Meudon, France\n\n\n\n\n\n\\newcommand{\\Authors}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\authors}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\Author}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\renewcommand{\\author}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\Address}[1]{\\begin{center}\\small\\sf #1 \\end{center}}\n\\newcommand{\\address}[1]{\\begin{center}\\small\\sf #1 \\end{center}}\n\n\\newcommand{\\Accepted}[1]{{\\vspace{2mm}\\small \\noindent  \\hspace*{0mm}Accepted: } \\small #1 \\normalsize}\n\n\\newcommand{\\Objects}[1]{{\\vspace{2mm}\\small \\noindent  \\hspace*{0mm}Individual Objects: } \\small #1 \\normalsize}\n\n\\newcommand{\\Multiobjects}[1]{{\\vspace{3mm}\\small \\noindent Individual Objects: }\\small\\sf \\hangindent=7truemm \\hangafter=1 #1}\t\n\n\\newcommand{\\discussion}[2]{{\\textbf{\\small#1:}}{ \\small #2 \\newline \\normalsize}}\n\n\\newcommand{\\listofparticipants}[2]\n{\\chapter[\\sf\\normalsize #\n \\footnotesize \\hspace*{10mm}\n \n\\sf\\normalsize][]\n{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]\n{\\fancyplain{}{\\sffamily \\thepage\n}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}\n{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize\n}}}\n\n\n\\newcommand{\\participant}[5]{ \\noindent\n{#1} {\\textbf{#2}}, {#3}, {#4}, {\\textit{#5}\\normalsize} \\\\ \\hangindent=0truemm}\n\n\\renewenvironment{abstract}{\\section*{Abstract}\\normalsize\\sf}{}\n\\newcommand{\\References}[1]{\\begin{flushleft}{\\large References\\\\}\\vspace*{2mm}\\small #1 \\end{flushleft}}\n\n\n\n\\newcommand{\\chapterCoAst}[2]\n{\\chapter[\\sf\\normalsize #1\\\\ \n\\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]\n{#1\\\\\n\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\n\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\n\n\n\n\\newcommand{\\sectionheading}[1]\n{\\chapter[\\sf\\large\\textbf{#1} \\footnotesize \n\\sf\\normalsize][]\n{ \\huge \\vspace{20mm} #1\\\\}\n}\n\n\n\n\n\\newcommand{\\headDSSN}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\chapterdssn}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headdssn}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\figureCoAst}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figureDSSN}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figuredssn}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand {\\strich} {\\begin{center} \\vspace{0.5cm} \\rule{1.5cm}{0.3mm} \\end{center}}\n\\renewcommand{\\chaptername}{}\n\n\\newcommand{\\acknowledgments}[1]{\\vspace*{5mm}\\noindent  \\textbf{Acknowledgments.} #1}\n\\newcommand{\\acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\n\n\\section*{Editorial statement for the following paper (entitled \\\\ {\\it MOST found evidence for solar-type oscillations in the K2 giant star HD\\,20884)} }\n{\\it \\small The main message of the following paper is to report the detection of nonradial pulsation in a K giant. This is an important and controversial topic.\n\nA key issue for this investigation is the reliability of the extracted frequencies. Your editor has spent his life with Fourier amplitude spectra of high-precision ground-based photometry. From that experience, Figure 3 of the paper\nrepresents the natural result of instrumental instabilities and atmospheric transparency changes and does not deserve a second look. However, these are satellite observations without the typical ground-based difficulties -- but there may be different problems.\n\nThe statistical analyses can only help us in a limited way. It does not matter much whether one applies my favorite criterion of amplitude signal\/noise ratio < 4, the SigSpec values, or another method. All these statistical analyses rely on assumptions of the nature of noise present in the data. Again, in ground-based photometry, we have found that the statistics are not reliable for low-frequency peaks. The obvious method to verify that certain peaks are not noise is to look at a comparison sample: the authors provide the results for another star with a simpler light curve. In my opinion, the issue is not yet solved.\\normalsize}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\bigskip\n\nFor $a,b>0$ with $a\\neq b$, the Schwab-Borchardt mean $SB(a,b)$ [1-3] of $a$ and $b$ is\ngiven by\n\\begin{equation*}\nSB(a,b)=\\begin{cases}\n\\frac{\\sqrt{b^2-a^2}}{\\cos^{-1}{(a\/b)}}, &\\quad  a<b, \\\\\n\\frac{\\sqrt{a^2-b^2}}{\\cosh^{-1}{(a\/b)}}, &\\quad a>b,\n\\end{cases}\n\\end{equation*}\nwhere $\\cos^{-1}(x)$ and $\\cosh^{-1}(x)=\\log(x+\\sqrt{x^{2}-1})$ are the inverse cosine and inverse hyperbolic cosine functions, respectively. Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for Schwab-Borchardt mean can be found in the literature [1-7]. Very recently, the Neuman mean $N(a,b)=(a+b^{2}\/SB(a,b))\/2$ derived from the Schwab-Borchardt was introduced and researched by Neuman in [8].\n\n\nLet $N_{AG}(a,b)=N(A(a,b), G(a,b))$, $N_{GA}(a,b)=N(G(a,b), A(a,b))$, $N_{QA}(a,b)=N(Q(a,b), A(a,b))$ and $N_{AQ}(a,b)=N(A(a,b), Q(a,b))$ be the Neuman means, where $G(a,b)=\\sqrt{ab}$, $A(a,b)=(a+b)\/2$ and $Q(a,b)=\\sqrt{(a^{2}+b^{2})\/2}$ are the classical geometric, arithmetic and quadratic means of $a$ and $b$, respectively. Then Neuman [8] proved that the inequalities\n$$G(a,b)<N_{AG}(a, b)<N_{GA}(a, b)<A(a,b)<N_{QA}(a, b)<N_{AQ}(a, b)<Q(a,b)$$\nhold for all $a, b>0$ with $a\\neq b$.\n\nLet $a>b>0$ and $v=(a-b)\/(a+b)\\in (0, 1)$. Then we clearly see that\n\\begin{equation}\nG(a,b)=A(a,b)\\sqrt{1-v^{2}}, \\quad Q(a,b)=A(a,b)\\sqrt{1+v^{2}},\n\\end{equation}\nand the following explicit formulas for $N_{AG}(a,b)$, $N_{GA}(a,b)$, $N_{QA}(a,b)$ and\n$N_{AQ}(a, b)$ are given in [8]\n\\begin{equation}\nN_{AG}(a,b)=\\frac{1}{2}A(a,b)\\left[1+(1-v^{2})\\frac{\\tanh^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{GA}(a,b)=\\frac{1}{2}A(a,b)\\left[\\sqrt{1-v^{2}}+\\frac{\\sin^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{QA}(a,b)=\\frac{1}{2}A(a,b)\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{AQ}(a,b)=\\frac{1}{2}A(a,b)\\left[1+(1+v^{2})\\frac{\\tan^{-1}v}{v}\\right],\n\\end{equation}\nwhere $\\tanh^{-1}(x)=\\log[(1+x)\/(1-x)]\/2$, $\\sin^{-1}(x)$, $\\sinh^{-1}(x)=\\log(x+\\sqrt{1+x^{2}})$ and $\\tan^{-1}(x)$ are the inverse hyperbolic tangent, inverse sine, inverse hyperbolic sine and inverse tangent functions, respectively.\n\nIn [8], Neuman also proved that the double inequalities\n$$\\alpha_{1} A(a,b)+(1-\\alpha_{1})G(a,b)<N_{GA}(a,b)<\\beta_{1} A(a,b)+(1-\\beta_{1})G(a,b),$$\n$$\\alpha_{2} Q(a,b)+(1-\\alpha_{2})A(a,b)<N_{AQ}(a,b)<\\beta_{2} Q(a,b)+(1-\\beta_{2})A(a,b),$$\n$$\\alpha_{3} A(a,b)+(1-\\alpha_{3})G(a,b)<N_{AG}(a,b)<\\beta_{3} A(a,b)+(1-\\beta_{3})G(a,b),$$\n$$\\alpha_{4} Q(a,b)+(1-\\alpha_{4})A(a,b)<N_{QA}(a,b)<\\beta_{4} Q(a,b)+(1-\\beta_{4})A(a,b)$$\nhold for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{1}\\leq 2\/3$, $\\beta_{1}\\geq \\pi\/4$,\n$\\alpha_{2}\\leq 2\/3$, $\\beta_{2}\\geq (\\pi-2)\/[4(\\sqrt{2}-1)]=0.689\\ldots$, $\\alpha_{3}\\leq 1\/3$, $\\beta_{3}\\geq 1\/2$,\n$\\alpha_{4}\\leq 1\/3$ and $\\beta_{4}\\geq [\\log(1+\\sqrt{2})+\\sqrt{2}-2]\/[2(\\sqrt{2}-1)]=0.356\\ldots$.\n\nIn [9], the authors presented the best possible parameters $\\alpha_{1}, \\alpha_{2}, \\beta_{1}, \\beta_{2}\\in [0, 1\/2]$ and\n$\\alpha_{3}, \\alpha_{4}, \\beta_{3}, \\beta_{4}\\in [1\/2, 1]$ such that the double inequalities\n$$G(\\alpha_{1}a+(1-\\alpha_{1})b, \\alpha_{1}b+(1-\\alpha_{1})a)<N_{AG}(a,b)<G(\\beta_{1}a+(1-\\beta_{1})b, \\beta_{1}b+(1-\\beta_{1})a),$$\n$$G(\\alpha_{2}a+(1-\\alpha_{2})b, \\alpha_{2}b+(1-\\alpha_{2})a)<N_{GA}(a,b)<G(\\beta_{2}a+(1-\\beta_{2})b, \\beta_{2}b+(1-\\beta_{2})a),$$\n$$Q(\\alpha_{3}a+(1-\\alpha_{3})b, \\alpha_{3}b+(1-\\alpha_{3})a)<N_{QA}(a,b)<Q(\\beta_{3}a+(1-\\beta_{3})b, \\beta_{3}b+(1-\\beta_{3})a),$$\n$$Q(\\alpha_{4}a+(1-\\alpha_{4})b, \\alpha_{4}b+(1-\\alpha_{4})a)<N_{AQ}(a,b)<Q(\\beta_{4}a+(1-\\beta_{4})b, \\beta_{4}b+(1-\\beta_{4})a)$$\nhold for all $a, b>0$ with $a\\neq b$.\n\nThe main purpose of this paper is to find the greatest values  $\\alpha_{1}$, $\\alpha_{2}$, $\\alpha_{3}$, $\\alpha_{4}$, $\\alpha_{5}$, $\\alpha_{6}$, $\\alpha_{7}$, $\\alpha_{8}$ and the least values $\\beta_{1}$, $\\beta_{2}$, $\\beta_{3}$, $\\beta_{4}$, $\\beta_{5}$, $\\beta_{6}$, $\\beta_{7}$, $\\beta_{8}$ such that the double inequalities\n$$A^{\\alpha_{1}}(a,b)G^{1-\\alpha_{1}}(a,b)<N_{GA}(a,b)<A^{\\beta_{1}}(a,b)G^{1-\\beta_{1}}(a,b),$$\n$$\\frac{\\alpha_{2}}{G(a,b)}+\\frac{1-\\alpha_{2}}{A(a,b)}<\\frac{1}{N_{GA}(a,b)}<\\frac{\\beta_{2}}{G(a,b)}+\\frac{1-\\beta_{2}}{A(a,b)},$$\n$$A^{\\alpha_{3}}(a,b)G^{1-\\alpha_{3}}(a,b)<N_{AG}(a,b)<A^{\\beta_{3}}(a,b)G^{1-\\beta_{3}}(a,b),$$\n$$\\frac{\\alpha_{4}}{G(a,b)}+\\frac{1-\\alpha_{4}}{A(a,b)}<\\frac{1}{N_{AG}(a,b)}<\\frac{\\beta_{4}}{G(a,b)}+\\frac{1-\\beta_{4}}{A(a,b)},$$\n$$Q^{\\alpha_{5}}(a,b)A^{1-\\alpha_{5}}(a,b)<N_{AQ}(a,b)<Q^{\\beta_{5}}(a,b)A^{1-\\beta_{5}}(a,b),$$\n$$\\frac{\\alpha_{6}}{A(a,b)}+\\frac{1-\\alpha_{6}}{Q(a,b)}<\\frac{1}{N_{AQ}(a,b)}<\\frac{\\beta_{6}}{A(a,b)}+\\frac{1-\\beta_{6}}{Q(a,b)},$$\n$$Q^{\\alpha_{7}}(a,b)A^{1-\\alpha_{7}}(a,b)<N_{QA}(a,b)<Q^{\\beta_{7}}(a,b)A^{1-\\beta_{7}}(a,b),$$\n$$\\frac{\\alpha_{8}}{A(a,b)}+\\frac{1-\\alpha_{8}}{Q(a,b)}<\\frac{1}{N_{QA}(a,b)}<\\frac{\\beta_{8}}{A(a,b)}+\\frac{1-\\beta_{8}}{Q(a,b)}$$\nhold for all $a, b>0$ with $a\\neq b$ .\n\n\n\\bigskip\n\\section {Lemmas}\n\\bigskip\n\nIn order to prove our main results we need several lemmas, which we present in this section.\n\n\\setcounter{equation}{0}\n\n\\begin{lemma} (See [10, Theorem 1.25])  Let $-\\infty<a<b<\\infty$, $f, g: [a, b]\\rightarrow (-\\infty, \\infty)$ be continuous on $[a,b]$ and differentiable on $(a, b)$, and $g^{\\prime}(x)\\neq 0$ on $(a,b)$. If $f'(x)\/g'(x)$ is increasing (decreasing) on $(a,b)$, then so are\n$$\\frac{f(x)-f(a)}{g(x)-g(a)}, \\quad\\quad\\ \\frac{f(x)-f(b)}{g(x)-g(b)}.$$\nIf $f'(x)\/g'(x)$ is strictly monotone, the the monotonicity in the conclusion is also strict.\n\\end{lemma}\n\n\\begin{lemma} (See [11, Lemma 1.1])  Suppose that the power series $f(x)=\\sum_{n=0}^{\\infty}a_{n}x^{n}$ and $g(x)=\\sum_{n=0}^{\\infty}b_{n}x^{n}$ have\nthe radius of convergence $r>0$ and $b_{n}>0$ for all $n\\geq 0$. If the sequence $\\{a_{n}\/b_{n}\\}$ is (strictly) increasing (decreasing) for all $n\\geq 0$, then the function $f(x)\/g(x)$ is also (strictly) increasing (decreasing) on $(0, r)$.\n\\end{lemma}\n\n\\begin{lemma} The function\n\\begin{equation}\nf_{1}(x)=\\frac{\\log [\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)}\n\\end{equation}\nis strictly increasing from $(0, \\pi\/2)$ onto $(2\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (3.1) that\n\\begin{equation}\nf_{1}(0)=\\frac{2}{3},\n\\end{equation}\n\\begin{equation}\nf_{1}\\left(\\frac{\\pi}{2}^{-}\\right)=1.\n\\end{equation}\n\nLet $g_{1}(x)=\\log [\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2$, $h_{1}(x)=\\log(\\cos x)$, $g_{2}(x)=\\sin(2x)-2x\\cos(2x)$ and $h_{2}(x)=[2x+\\sin(2x)]\\sin^{2}x$. Then simple computations lead to\n\\begin{equation}\ng_{1}(0^{+})=h_{1}(0)=g_{2}(0)=h_{2}(0)=0,\n\\end{equation}\n\\begin{equation}\nf_{1}(x)=\\frac{g_{1}(x)}{h_{1}(x)}, \\quad \\frac{g'_{1}(x)}{h'_{1}(x)}=\\frac{g_{2}(x)}{h_{2}(x)},\n\\end{equation}\n\\begin{equation}\n\\frac{g'_{2}(x)}{h'_{2}(x)}=\\frac{1}{\\frac{1}{2}+\\frac{\\sin(2x)}{2x}}.\n\\end{equation}\n\nIt is well known that the function $\\sin x\/x$ is strictly decreasing on $(0, \\pi)$, hence equation (2.6) leads to the conclusion that the function\n$g'_{2}(x)\/h'_{2}(x)$ is strictly increasing on $(0, \\pi\/2)$. Therefore, Lemma 2.3 follows from Lemma 2.1 and (2.2)-(2.5) together with the monotonicity of $g'_{2}(x)\/h'_{2}(x)$.\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{2}(x)=\\frac{\\log [2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}\n\\end{equation}\nis strictly increasing from $(0, \\infty)$ onto $(1\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.7) that\n\\begin{equation}\nf_{2}(0^{+})=\\frac{1}{3},\n\\end{equation}\n\\begin{equation}\n\\lim_{x\\rightarrow \\infty}f_{2}(x)=1.\n\\end{equation}\n\nLet $g_{3}(x)=\\log [2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2$ and $h_{3}(x)=\\log[\\cosh(x)]$. Then simple computations lead to\n\\begin{equation}\nf_{2}(x)=\\frac{g_{3}(x)}{h_{3}(x)}, \\quad g_{3}(0^{+})=h_{3}(0)=0,\n\\end{equation}\n\\begin{equation}\n\\frac{g'_{3}(x)}{h'_{3}(x)}=\\frac{\\sinh(4x)-4x\\cosh(2x)+2\\sinh(2x)-4x}{\\sinh(4x)+4x\\cosh(2x)-2\\sinh(2x)-4x}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sum_{n=1}^{\\infty}\\frac{\\left(2^{2n}-2n\\right)2^{2n+2}}{(2n+1)!}x^{2n+1}}\n{\\sum_{n=1}^{\\infty}\\frac{\\left(2^{2n}+2n\\right)2^{2n+2}}{(2n+1)!}x^{2n+1}}\n\\end{equation*}\n\\begin{equation*}\n=\\frac{\\sum_{n=0}^{\\infty}\\frac{\\left(2^{2n+2}-2n-2\\right)2^{2n+4}}{(2n+3)!}x^{2n}}\n{\\sum_{n=0}^{\\infty}\\frac{\\left(2^{2n+2}+2n+2\\right)2^{2n+4}}{(2n+3)!}x^{2n}}.\n\\end{equation*}\nLet\n\\begin{equation}\na_{n}=\\frac{\\left(2^{2n+2}-2n-2\\right)2^{2n+4}}{(2n+3)!}, \\quad b_{n}=\\frac{\\left(2^{2n+2}+2n+2\\right)2^{2n+4}}{(2n+3)!}.\n\\end{equation}\nThen\n\\begin{equation}\nb_{n}>0, \\quad \\frac{a_{n+1}}{b_{n+1}}-\\frac{a_{n}}{b_{n}}=\\frac{(3n+2)2^{2n+2}}{\\left(2^{2n+3}+n+2\\right)\\left(2^{2n+1}+n+1\\right)}>0\n\\end{equation}\nfor all $n\\geq 0$.\n\nIt follows from Lemma 2.2 and (2.11)-(2.13) that the function $g'_{3}(x)\/h'_{3}(x)$ is strictly increasing on $(0, \\infty)$. Therefore, Lemma 2.4 follows from Lemma 2.1 and (2.8)-(2.10) together with the monotonicity of $g'_{3}(x)\/h'_{3}(x)$.\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{3}(x)=\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}\n\\end{equation}\nis strictly increasing from $(0, \\pi\/2)$ onto $(2\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.14) that\n\\begin{equation}\nf_{3}\\left(0^{+}\\right)=\\frac{2}{3},\n\\end{equation}\n\\begin{equation}\nf_{3}\\left(\\pi\/2\\right)=1.\n\\end{equation}\n\nLet $g_{4}(x)=2x-\\sin(2x)$ and $h_{4}(x)=(1-\\cos x)[2x+\\sin(2x)]$. Then simple computations lead to\n\\begin{equation}\nf_{3}(x)=\\frac{g_{4}(x)}{h_{4}(x)}, \\quad g_{4}(0)=h_{4}(0)=0,\n\\end{equation}\n\\begin{equation*}\ng'_{4}(x)=4\\sin^{2}x,\n\\end{equation*}\n\\begin{equation*}\nh'_{4}(x)=2\\sin^{2}x\\cos x-4\\cos^{3}x+4\\cos^{2}x+2x\\sin x,\n\\end{equation*}\n\\begin{equation}\ng'_{4}(0)=h'_{4}(0)=0,\n\\end{equation}\n\\begin{equation}\n\\frac{g''_{4}(x)}{h''_{4}(x)}=\\frac{4}{9\\cos x+\\frac{x}{\\sin x}-4},\n\\end{equation}\n\\begin{equation}\n\\left(9\\cos x+\\frac{x}{\\sin x}\\right)'=-8\\sin x-\\frac{[2x-\\sin(2x)]\\cos x}{2\\sin^{2}x}<0\n\\end{equation}\nfor $x\\in (0, \\pi\/2)$.\n\nTherefore, Lemma 2.5 follows easily from Lemma 2.1 and (2.15)-(2.20).\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{4}(x)=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}\n\\end{equation}\nis strictly increasing from $(0, \\infty)$ onto $(1\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.21) that\n\\begin{equation}\nf_{4}(x)=\\frac{\\frac{1}{4}\\sinh(3x)+\\frac{1}{4}\\sinh(x)-\\sinh(2x)+x\\cosh(x)}{\\frac{1}{4}\\sinh(3x)+\\frac{1}{4}\\sinh(x)-\\frac{1}{2}\\sinh(2x)+x\\cosh(x)-x}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sum_{n=1}^{\\infty}\\frac{3^{2n+1}-2^{2n+3}+8n+5}{4[(2n+1)!]}x^{2n+1}}{\\sum_{n=1}^{\\infty}\\frac{3^{2n+1}-2^{2n+2}+8n+5}{4[(2n+1)!]}x^{2n+1}}\n\\end{equation*}\n\\begin{equation*}\n=\\frac{\\sum_{n=0}^{\\infty}\\frac{3^{2n+3}-2^{2n+5}+8n+13}{4[(2n+3)!]}x^{2n}}{\\sum_{n=0}^{\\infty}\\frac{3^{2n+3}-2^{2n+4}+8n+13}{4[(2n+3)!]}x^{2n}}.\n\\end{equation*}\nLet\n\\begin{equation}\na_{n}=\\frac{3^{2n+3}-2^{2n+5}+8n+13}{4[(2n+3)!]}, \\quad b_{n}=\\frac{3^{2n+3}-2^{2n+4}+8n+13}{4[(2n+3)!]}.\n\\end{equation}\nThen simple computations lead to\n\\begin{equation}\nb_{n}>\\frac{3^{2n+3}-2^{2n+4}}{4[(2n+3)!]}=\\frac{2^{2n+3}\\left[\\left(\\frac{3}{2}\\right)^{2n+3}-2\\right]}{4[(2n+3)!]}>0\n\\end{equation}\n\\begin{equation}\n\\frac{a_{n+1}}{b_{n+1}}-\\frac{a_{n}}{b_{n}}=\\frac{\\left(135\\times 3^{2n}-24n-31\\right)2^{2n+4}}{\\left(3^{2n+3}-2^{2n+4}+8n+13\\right)\\left(3^{2n+5}-2^{2n+6}+8n+21\\right)}>0\n\\end{equation}\nfor all $n\\geq 0$.\n\nNote that\n\\begin{equation}\nf_{4}(0^{+})=\\frac{1}{3},  \\quad \\lim_{x\\rightarrow \\infty}f_{4}(x)=\\lim_{n\\rightarrow \\infty}\\frac{a_{n}}{b_{n}}=1.\n\\end{equation}\n\nTherefore, Lemma 2.6 follows easily from Lemma 2.2 and (2.22)-(2.26).\n\n\n\\bigskip\n\\section{Main Results}\n\\bigskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nA^{\\alpha_{1}}(a,b)G^{1-\\alpha_{1}}(a,b)<N_{GA}(a,b)<A^{\\beta_{1}}(a,b)G^{1-\\beta_{1}}(a,b),\n\\end{equation}\n\\begin{equation}\n\\frac{\\alpha_{2}}{G(a,b)}+\\frac{1-\\alpha_{2}}{A(a,b)}<\\frac{1}{N_{GA}(a,b)}<\\frac{\\beta_{2}}{G(a,b)}+\\frac{1-\\beta_{2}}{A(a,b)}\n\\end{equation}\nholds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{1}\\leq 2\/3$, $\\beta_{1}\\geq 1$, $\\alpha_{2}\\leq 0$ and\n$\\beta_{2}\\geq 1\/3$.\n\\end{theorem}\n{\\em Proof.} We clearly see that inequalities (3.1) and (3.2) can be rewritten as\n\\begin{equation}\n\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\alpha_{1}}<\\frac{N_{GA}(a,b)}{G(a,b)}<\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\beta_{1}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{2}<\\frac{\\frac{1}{G(a,b)}-\\frac{1}{N_{GA}(a,b)}}{\\frac{1}{G(a,b)}-\\frac{1}{A(a,b)}}<1-\\alpha_{2},\n\\end{equation}\nrespectively.\n\nSince both the geometric mean $G(a,b)$ and arithmetic mean $A(a,b)$ are symmetric and homogeneous of degree 1, without loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.3) we know that inequalities (3.3) and (3.4) are equivalent to\n\\begin{equation}\n\\alpha_{1}<\\frac{\\log\\left[\\frac{1}{2}\\left(1+\\frac{\\sin^{-1}(v)}{v\\sqrt{1-v^{2}}}\\right)\\right]}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}<\\beta_{1}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{2}<\\frac{\\sin^{-1}v-v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})(v\\sqrt{1-v^{2}}+\\sin^{-1}v)}<1-\\alpha_{2},\n\\end{equation}\nrespectively.\n\nLet $x=\\sin^{-1}(v)$. Then $x\\in (0, \\pi\/2)$,\n\\begin{equation}\n\\frac{\\log\\left[\\frac{1}{2}\\left(1+\\frac{\\sin^{-1}(v)}{v\\sqrt{1-v^{2}}}\\right)\\right]}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}\n=\\frac{\\log[\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)},\n\\end{equation}\n\\begin{equation}\n\\frac{\\sin^{-1}v-v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})(v\\sqrt{1-v^{2}}+\\sin^{-1}v)}=\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}.\n\\end{equation}\n\n\nTherefore, inequality (3.1) holds for all $a,b>0$ with $a\\neq b$ follows from (3.5) and (3.7) together with Lemma 2.3,\nand inequality (3.2) holds for all $a,b>0$ with $a\\neq b$ follows from (3.6) and (3.8) together with Lemma 2.5.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nA^{\\alpha_{3}}(a,b)G^{1-\\alpha_{3}}(a,b)<N_{AG}(a,b)<A^{\\beta_{3}}(a,b)G^{1-\\beta_{3}}(a,b),\n\\end{equation}\n\\begin{equation}\n\\frac{\\alpha_{4}}{G(a,b)}+\\frac{1-\\alpha_{4}}{A(a,b)}<\\frac{1}{N_{AG}(a,b)}<\\frac{\\beta_{4}}{G(a,b)}+\\frac{1-\\beta_{4}}{A(a,b)}\n\\end{equation}\nhold for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{3}\\leq 1\/3$, $\\beta_{3}\\geq 1$, $\\alpha_{4}\\leq 0$ and $\\beta_{4}\\geq 2\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.9) and (3.10) can be rewritten as\n\\begin{equation}\n\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\alpha_{3}}<\\frac{N_{AG}(a,b)}{G(a,b)}<\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\beta_{3}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{4}<\\frac{\\frac{1}{G(a,b)}-\\frac{1}{N_{AG}(a,b)}}{\\frac{1}{G(a,b)}-\\frac{1}{A(a,b)}}<1-\\alpha_{4},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then it follows from (1.1) and (1.2) that inequalities (3.11) and (3.12) are equivalent to\n\\begin{equation}\n\\alpha_{3}<\\frac{\\log\\left[\\frac{1}{\\sqrt{1-v^{2}}}+\\frac{\\sqrt{1-v^{2}}}{v}\\tanh^{-1}(v)\\right]-\\log 2}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}<\\beta_{3}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{4}<\\frac{v+(1-v^{2})\\tanh^{-1}(v)-2v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})[v+(1-v^{2})\\tanh^{-1}(v)]}<1-\\beta_{4},\n\\end{equation}\nrespectively.\nLet $x=\\tanh^{-1}(v)\\in (0, \\infty)$. Then simple computations lead to\n\\begin{equation}\n\\frac{\\log\\left[\\frac{1}{\\sqrt{1-v^{2}}}+\\frac{\\sqrt{1-v^{2}}}{v}\\tanh^{-1}(v)\\right]-\\log 2}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}\n\\end{equation*}\nand\n\\begin{equation}\n\\frac{v+(1-v^{2})\\tanh^{-1}(v)-2v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})[v+(1-v^{2})\\tanh^{-1}(v)]}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}.\n\\end{equation*}\n\nTherefore, inequality (3.9) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{3}\\leq 1\/3$ and $\\beta_{3}\\geq 1$ follows from (3.13) and (3.15) together with Lemma 2.4, and inequality (3.10) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{4}\\leq 0$ and $\\beta_{4}\\geq 2\/3$ follows from (3.14) and (3.16) together with Lemma 2.6.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nQ^{\\alpha_{5}}(a,b)A^{1-\\alpha_{5}}(a,b)<N_{AQ}(a,b)<Q^{\\beta_{5}}(a,b)A^{1-\\beta_{5}}(a,b),\n\\end{equation}\n\\begin{equation}\n\\frac{\\alpha_{6}}{A(a, b)}+\\frac{1-\\alpha_{6}}{Q(a,b)}<\\frac{1}{N_{AQ}(a,b)}<\\frac{\\beta_{6}}{A(a, b)}+\\frac{1-\\beta_{6}}{Q(a,b)}\n\\end{equation}\nhold for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{5}\\leq 2\/3$, $\\beta_{5}\\geq 2\\log(\\pi+2)\/\\log 2-4=0.7244\\ldots$, $\\alpha_{6}\\leq[6+2\\sqrt{2}-(1+\\sqrt{2})\\pi]\/(\\pi+2)=0.2419\\ldots$ and $\\beta_{6}\\geq 1\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.17) and (3.18) can be rewritten as\n\\begin{equation}\n\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\alpha_{5}}<\\frac{N_{AQ}(a,b)}{A(a,b)}<\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\beta_{5}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{6}<\\frac{\\frac{1}{A(a,b)}-\\frac{1}{N_{AQ}(a,b)}}{\\frac{1}{A(a,b)}-\\frac{1}{Q(a,b)}}<1-\\alpha_{6},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.5) we clearly see that inequalities (3.19) and (3.20) are equivalent to\n\\begin{equation}\n\\alpha_{5}<\\frac{2\\log(1+\\frac{1+v^{2}}{v}\\tan^{-1}(v))-2\\log 2}{\\log(1+v^{2})}<\\beta_{5}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{6}<\\frac{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)-v\\right]\\sqrt{1+v^{2}}}{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)+v\\right](\\sqrt{1+v^{2}}-1)}<1-\\alpha_{6},\n\\end{equation}\nrespectively.\n\nLet $x=\\tan^{-1}(v)$. Then $x\\in (0, \\pi\/4)$,\n\\begin{equation}\n\\frac{2\\log(1+\\frac{1+v^{2}}{v}\\tan^{-1}(v))-2\\log 2}{\\log(1+v^{2})}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)}=f_{1}(x)\n\\end{equation*}\nand\n\\begin{equation}\n\\frac{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)-v\\right]\\sqrt{1+v^{2}}}{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)+v\\right](\\sqrt{1+v^{2}}-1)}\n\\end{equation}\n\\begin{equation*}\n\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}=f_{3}(x).\n\\end{equation*}\nNote that\n\\begin{equation}\nf_{1}\\left(\\frac{\\pi}{4}\\right)=\\frac{2\\log(\\pi+2)}{\\log 2}-4,\n\\end{equation}\n\\begin{equation}\nf_{3}\\left(\\frac{\\pi}{4}\\right)=\\frac{(2+\\sqrt{2})(\\pi-2)}{\\pi+2}=1-\\frac{6+2\\sqrt{2}-(1+\\sqrt{2})\\pi}{\\pi+2}.\n\\end{equation}\n\nTherefore, inequality (3.17) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{5}\\leq 2\/3$ and $\\beta_{5}\\geq 2\\log(\\pi+2)\/\\log 2-4$ follows from (3.21), (3.23), (3.25) and Lemma 2.3, and inequality (3.18) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{6}\\leq[6+2\\sqrt{2}-(1+\\sqrt{2})\\pi]\/(\\pi+2)$ and $\\beta_{6}\\geq 1\/3$ follows from (3.22), (3.24), (3.26) and Lemma 2.5.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nQ^{\\alpha_{7}}(a,b)A^{1-\\alpha_{7}}(a,b)<N_{QA}(a,b)<Q^{\\beta_{7}}(a,b)A^{1-\\beta_{7}}(a,b),\n\\end{equation}\n\\begin{equation}\n\\frac{\\alpha_{8}}{A(a, b)}+\\frac{1-\\alpha_{8}}{Q(a,b)}<\\frac{1}{N_{QA}(a,b)}<\\frac{\\beta_{8}}{A(a, b)}+\\frac{1-\\beta_{8}}{Q(a,b)}\n\\end{equation}\nhold for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{7}\\leq 1\/3$, $\\beta_{7}\\geq 2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]\/\\log 2-2=0.3977\\ldots$, $\\alpha_{8}\\leq[2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})]\/[\\sqrt{2}+\\log(1+\\sqrt{2})]=0.5603\\ldots$ and $\\beta_{8}\\geq 2\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.27) and (3.28) can be rewritten as\n\\begin{equation}\n\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\alpha_{7}}<\\frac{N_{QA}(a,b)}{A(a,b)}<\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\beta_{7}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{8}<\\frac{\\frac{1}{A(a,b)}-\\frac{1}{N_{QA}(a,b)}}{\\frac{1}{A(a,b)}-\\frac{1}{Q(a,b)}}<1-\\alpha_{8},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.4) we clearly see that inequalities (3.29) and (3.30) are equivalent to\n\\begin{equation}\n\\alpha_{7}<\\frac{2\\log\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}(v)}{v}\\right]-2\\log 2}{\\log\\left(1+v^{2}\\right)}<\\beta_{7}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{8}<\\frac{\\left[v\\left(1+v^{2}\\right)+\\sqrt{1+v^{2}}\\sinh^{-1}(v)\\right]-2v\\sqrt{1+v^{2}}}{(\\sqrt{1+v^{2}}-1)\\left[v\\sqrt{1+v^{2}}+\\sinh^{-1}(v)\\right]}<1-\\alpha_{8},\n\\end{equation}\nrespectively.\n\nLet $x=\\sinh^{-1}(v)$. Then $x\\in (0, \\log(1+\\sqrt{2}))$,\n\\begin{equation}\n\\frac{2\\log\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}(v)}{v}\\right]-2\\log 2}{\\log\\left(1+v^{2}\\right)}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}=f_{2}(x),\n\\end{equation*}\n\\begin{equation}\n\\frac{\\left[v\\left(1+v^{2}\\right)+\\sqrt{1+v^{2}}\\sinh^{-1}(v)\\right]-2v\\sqrt{1+v^{2}}}{(\\sqrt{1+v^{2}}-1)\\left[v\\sqrt{1+v^{2}}+\\sinh^{-1}(v)\\right]}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}=f_{4}(x).\n\\end{equation*}\nNote that\n\\begin{equation}\nf_{2}[\\log(1+\\sqrt{2})]=\\frac{2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]}{\\log 2}-2,\n\\end{equation}\n\\begin{equation}\nf_{4}[\\log(1+\\sqrt{2})]=\\frac{(2+\\sqrt{2})\\log(1+\\sqrt{2})-2}{\\sqrt{2}+\\log(1+\\sqrt{2})}\n\\end{equation}\n\\begin{equation*}\n=1-\\frac{2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})}{\\sqrt{2}+\\log(1+\\sqrt{2})}.\n\\end{equation*}\n\nTherefore, inequality (3.27) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{7}\\leq 1\/3$ and $\\beta_{7}\\geq 2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]\/\\log 2-2$ follows from (3.31), (3.33), (3.35) and Lemma 2.4, and inequality (3.28) holds\nfor all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{8}\\leq[2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})]\/[\\sqrt{2}+\\log(1+\\sqrt{2})]$ and $\\beta_{8}\\geq 2\/3$ follows from (3.32), (3.34), (3.36) and Lemma 2.6.\n\n\n\n\\medskip\n\n\\noindent{\\bf Conflict of Interests}\n\n\\noindent{The authors declare that there is no conflict of interests regarding the publication of this paper.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe two-dimensional toy models of quantum gravity are a very useful play ground for understanding the quantum nature of geometry quantitatively.   \nThis is because many of them are simple enough to be dealt with analytically and complex enough to observe non-trivial quantum effects. \nLattice regularizations in particular are known to be quite powerful tools for investigating non-perturbative quantum effects analytically. \nAmongst these, two-dimensional Euclidean dynamical triangulations ($2$d EDT) \\cite{Ambjorn:1985az,Ambjorn:1985dn,David:1984tx,Billoire:1985ur,Kazakov:1985ea,Boulatov:1986jd} \n(see a pedagogical textbook \\cite{Ambjorn:1997di}) and two-dimensional causal dynamical triangulations ($2$d CDT) \\cite{Ambjorn:1998xu} (see a detailed review \\cite{Ambjorn:2022btk}) are good practical examples. \nThe former and the latter, respectively, are Euclidean and Lorentzian lattice models \nbased on Regge's discretization of geometries \\cite{Regge:1961px}. \n\n$2$d EDT discretizes Euclidean geometries by equilateral triangles and defines a regularized quantum amplitude as a sum over distinct triangulated geometries. \nMatrix models and combinatorics can be used for calculating such a statistical sum analytically (whenever possible). \nBy virtue of analytic treatments, one can explicitly remove the regularization through the continuum limit to calculate physical observables. \nWhat is remarkable is that exactly the same value of observables can be reproduced from a genuine continuum field theory called the Liouville quantum gravity \\cite{Polyakov:1981rd, Knizhnik:1988ak, David:1988hj, Distler:1988jt}. \nThis means that $2$d EDT serves as a well-defined regularization of the Liouville quantum gravity.     \n\n \n$2$d CDT is a Lorentzian lattice model of quantum geometry which respects a global time foliation \nand prohibits the creation of so-called baby universes. \nOne can calculate the sum over such Lorentzian triangulated geometries using simple combinatorics, \nand take the continuum limit to remove the regularization. \nAll these processes can be performed analytically at least for the plain model without coupling to a matter. \nIt has been shown in Ref.~\\cite{Ambjorn:2013joa} that the resulting continuum theory is known to be in the same universality class of the projectable Ho\\v rava-Lifshitz quantum gravity (projectable HL QG) \\cite{horava} in two dimensions, which is different from the Liouville quantum gravity \\footnote{In fact, it has been shown that a direct lattice discretization of the $2$d projectable HL gravity, \nwhich has a lattice action different from that of the $2$d CDT, reproduces the same large-scale physics in the continuum limit \\cite{Glaser:2016smx}.}.      \n  \n          \n\n$2$d HL QG is a quantum field theory in two dimensions, \nwhich has a preferred foliation structure. \nThis model is invariant only under the subclass of diffeomorphisms that respects the foliation, \nknown as the foliation-preserving diffeomorphisms\\footnote{At the cost of full diffeomorphism invariance,  \nHL QG has been designed originally as a model of quantum gravity in higher dimensions such that it has a good convergence at UV in keeping with unitarity, \nand would approximately recover the diffeomorphism invariance at IR \\cite{horava}}. \n$2$d projectable HL QG is a certain version of HL QG where the time-time component of the metric called the lapse function is projectable, i.e. a function only of time. \nIn this chapter, we wish to explain in detail the relation between projectable HL QG and  CDT in two dimensions\\footnote{The relation between HL QG and CDT in four dimension has been pointed out first in Ref.~\\cite{Horava:2009if} by looking at an observable called the spectral dimension, and the resemblance of the CDT phase diagram to a Lifshitz phase diagram has been shown in Ref.~\\cite{Ambjorn:2010hu}.}.   \n \n \nIn fact one can generalize $2$d CDT in such a way that the creation and annihilation of (a finite number of) baby universes, and the formation of wormholes (handles) are allowed to occur in keeping with the foliation structure. \nThis model is called the generalized CDT (GCDT) introduced first as a continuum theory \\cite{Ambjorn:2007jm, Ambjorn:2008ta} and later defined at the discrete level \\cite{Ambjorn:2008gk, Ambjorn:2013csx}. \nThe full continuum description of GCDT is given by the so-called string field theory for CDT \\cite{Ambjorn:2008ta} in which the string means the one-dimensional closed spatial universe, \nand the baby universes and wormholes can be realized in terms of the splitting and joining interactions of strings. \n\n\nOne of remarkable facts is that focusing on a certain amplitude, i.e. loop-to-loop amplitude, one can read off a one-dimensional effective theory \nthat takes in all possible baby universe and wormhole contributions in an effective manner \\cite{Ambjorn:2009wi, Ambjorn:2009fm}: \nThe $1$d effective theory is a one-body quantum theory even though GCDT is a many-body theory that allows both creation and annihilation of strings. \nIt is known that one can correctly reproduce the $1$d effective theory if quantizing the projectable HL gravity with a certain bi-local interaction term \\cite{Ambjorn:2021wou, Ambjorn:2021ysb}. \nThis topic will be treated in this chapter. \n        \n\nFurthermore, the $1$d effective theory that includes all contributions of baby universes and wormholes is known to be reproduced \nif one assumes that the cosmological constant of the continuum limit of $2$d CDT is not really a constant but fluctuates in time \\cite{Ambjorn:2021wdm}. \nThis idea leads to a certain realization of Coleman's mechanism about the cosmological constant \\cite{Coleman:1988tj} in the context of CDT \\cite{Ambjorn:2021wdm}, \nwhich will be also explained in this chapter. \n\n\nThe rest of this chapter is organized as follows. \nIn Sec.~\\ref{sec:2dcdt}, a self-contained introduction to $2$d CDT is presented. \nWe show that taking the continuum limit the physics of $2$d CDT can be described as a one-dimensional quantum system with a Hamiltonian.    \n$2$d projectable HL QG is explained in Sec.~\\ref{sec:2dHLQG}. Through the path-integral quantization we read off the quantum Hamiltonian that is equivalent to the one obtained in the continuum limit of $2$d CDT. \nThereby one can confirm that the continuum limit of $2$d CDT is $2$d projectable HL QG.   \nIn Sec.~\\ref{sec:sumoverallgeneraingenera}, we introduce GCDT that is a generalization of $2$d CDT such that baby universes and wormholes are introduced so as to be compatible with the foliation, \nand determine the $1$d effective theory obtained through the sum over all genera. \nIn particular, we explain in detail that quantizing $2$d projectable HL gravity with a bi-local interaction yields the $1$d effective theory, and discuss Coleman's mechanism in $2$d CDT. \nSec.~\\ref{sec:summary} is devoted to summary.   \n\n \n\n    \n\n\n\\section{$2$d causal dynamical triangulations}\n\\label{sec:2dcdt}\nTwo-dimensional causal dynamical triangulations ($2$d CDT) \\cite{Ambjorn:1998xu} is a lattice model of quantum geometries based on Regge's discretization \\cite{Regge:1961px}. \nIn this section, we give an overview of $2$d CDT, and in particular explain how to obtain the quantum Hamiltonian through the continuum limit.    \n\nWe start with a two-dimensional globally hyperbolic manifold equipped with a global time foliation: \n\\[\n\\mathcal{M} = \\bigcup_{t\\in \\mathbb{R}} \\Sigma_t\\ , \n\\label{eq:m}\n\\]\nwhere each leaf $\\Sigma_t$ is a one-dimensional Cauchy ``surface'' (line). \nOne approximates the manifold with a foliation in such a way that the continuous label $t$ is discretized by integers, i.e. $t\\in \\mathbb{Z}$;  \neach leaf (line) is partitioned by vertices connected by isometric edges; \nvertices among neighboring time steps are connected by isometric edges to form a triangulation of strip (see Fig.~\\ref{fig:triangulationofstrip}). \n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.6]{triangulationofstrip.png}\n\\caption{A triangulation of a strip: Thick and thin lines are space-like and time-like edges, respectively.}\n\\label{fig:triangulationofstrip}       \n\\end{figure}\nThe edges at a given time step and those connecting vertices in different time steps are respectively space-like and time-like edges \nsince the squared edge lengths of the space-like edge $a^2_s$ and the time-like edge $a^2_t$ are given by \n\\[\na^2_s = \\varepsilon^2\\ , \\ \\ \\ a^2_t = - \\alpha \\varepsilon^2\\ ,\n\\label{eq:latticespacing}\n\\]  \nwhere $\\alpha$ is a positive number and $\\varepsilon$ is a lattice spacing that serves as a UV cutoff. \n\n\n$2$d CDT deals with a set of restricted class of Lorentzian triangulations as discussed above. \nIn particular, we consider that the topology of the one-dimensional universe (a graph consisting of vertices and edges at a given time) is either $S^1$ or $[0,1]$, \nand the topology will not change during (discrete) time propagation. \nSince the topology is fixed the curvature term plays no role in two dimensions and only the discrete analogue of the cosmological constant term, $\\Lambda_0 \\int d^2x \\sqrt{-g}$ is used as the lattice action of $2$d CDT: \n\\[\nS_T [\\lambda, \\alpha] \n= - \\frac{\\lambda}{\\varepsilon^2} \n\\left( \n\\frac{\\sqrt{4\\alpha +1}}{4}\\varepsilon^2\\ n(T)  \n\\right)\\ , \n\\label{eq:action}\n\\]      \nwhere $\\lambda\/\\varepsilon^2$ is the bare cosmological constant with the dimensionless number $\\lambda$, \n$n(T)$ the number of triangles in a triangulation $T$,  \nand the term inside the parentheses denotes the total area of the triangulation. \nIt is useful to rotate to the Euclidean signature which can be performed by changing $\\alpha \\to - \\alpha -i0$. \nAccordingly the lattice action (\\ref{eq:action}) changes as follows:\n\\[\niS_T [\\lambda, \\alpha] \n\\to \niS_T [\\lambda, -\\alpha -i0] = - \\lambda \\frac{\\sqrt{4\\alpha -1}}{4} n(T)\n\\equiv - \\lambda n(T)\n\\ , \n\\label{eq:rotation}\n\\]  \nwhere $\\alpha$ is chosen to be greater than $1\/4$ otherwise the triangle inequalities will not be satisfied after the rotation.  \nIn any case we have absorbed the parameter $\\alpha$ by the redefinition of the dimensionless cosmological constant $\\lambda$. \n\nThe amplitude of the one-dimensional universe that starts with $\\ell_1$ edges and end up with $\\ell_2$ edges after the discrete time step $t$ is \ngiven by the sum over all allowed triangulations: \n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\sum_{T\\in \\mathcal{T}^{(a)}(\\ell_1,\\ell_2,t)} e^{-\\lambda n(T)}\n= \\sum_n e^{-\\lambda n} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n)\n\\ , \n\\label{eq:amplitude}\n\\]  \nwhere $\\mathcal{T}^{(a)}$ is a set of triangulations whose topology is $[0,1] \\times [0,1]$ for $a=0$ and $S^1 \\times [0,1]$ for $a=1$, \nand \n\\[\n\\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n) \n= \\# \\left\\{ T \\in \\mathcal{T}^{(a)} \\ \\bigl{|}\\ n(T)=n \\right\\}\\ . \n\\label{eq:numberoftriangulations}\n\\]  \nWhen defining the amplitude (\\ref{eq:amplitude}), we do not allow the one-dimensional universe to vanish during the discrete time propagation. \nFor later convenience, we introduce a marked amplitude: \n\\[\nG^{(-1)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\ell_1 G^{(1)}_{\\lambda} (\\ell_1, \\ell_2; t)\\ , \n\\label{eq:markedamplitude}\n\\]\nwhere one of the edges in the initial one-dimensional universe is marked. This is because there exist $\\ell_1$ possible ways of marking the edges.   \nThe three kinds of amplitude should satisfy the composition law: \n\\[\nG^{(1)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(1)}_{\\lambda} (\\ell_1, \\ell; t_1)\\ \\ell\\ G^{(1)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \\\\\nG^{(0)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(0)}_{\\lambda} (\\ell_1, \\ell; t_1)\\ G^{(0)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \\\\\nG^{(-1)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(-1)}_{\\lambda} (\\ell_1, \\ell; t_1)\\  G^{(-1)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \n\\label{eq:compositionlaw}\n\\]\nwhere for $a=1$ one need to multiply the amputated amplitude by $\\ell$ since there exist $\\ell$ possible ways of gluing to recover the whole amplitude. \n\n\nIt is convenient to introduce the generating function of the number of triangulations. \nUsing the notation \n\\[\ng=e^{-\\lambda}\\ , \n\\label{eq:g}\n\\]\nwe define the generating function: \n\\[\n\\widetilde{G}^{(a)} (g,x,y;t) \n&= \\sum^{\\infty}_{\\ell_1=1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t)\\notag \\\\\n&= \\sum^{\\infty}_{\\ell_1=1} \\sum^{\\infty}_{\\ell_2=1} \\sum_n x^{\\ell_1} y^{\\ell_2} g^{n} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n)\\ ,\n\\label{eq:generatingfunction} \n\\]  \nwhere in the context of quantum gravity, $x$ and $y$ are related to the boundary cosmological constants, $\\lambda_1$ and $\\lambda_2$, that control the size of the boundaries: \n\\[\nx=e^{-\\lambda_1}\\ , \\ \\ \\ y=e^{-\\lambda_2}\\ . \n\\label{eq:xy}\n\\]\nOne can reconstruct the amplitude from the generating function through the following relation:\n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\oint_{\\mathcal{C}_1} \\frac{dx}{2\\pi i x^{\\ell_1 +1}} \\oint_{\\mathcal{C}_2} \\frac{dy}{2\\pi i y^{\\ell_2 +1}}\\ \\widetilde{G}^{(a)} (g,x,y;t)\\ , \n\\label{eq:amplitudefromgeneratingfunction}\n\\]\nwhere the contour $\\mathcal{C}_1$ ($\\mathcal{C}_2$) is chosen to enclose $x=0$ ($y=0$) and to ensure the convergence of $\\widetilde{G}^{(a)} (g,x,y;t)$. \nOne can derive the relation (\\ref{eq:amplitudefromgeneratingfunction}) using the identity:\n\\[\n\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z^{n +1}} = \\delta_{n,0}\\ , \\ \\ \\ (n \\in \\mathbb{Z})\\ ,\n\\]   \nwhere the contour $\\mathcal{C}$ encloses $z=0$. \nWe provide the composition law for the generating function when $a=0,-1$ in preparation for later calculations:\n\\[\n\\widetilde{G}^{(a)} (g,x,y;t_1+t_2) \n= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(a)} (g,x,z^{-1};t_1)\\  \n\\widetilde{G}^{(a)} (g,z,y;t_2)\\ ,\\ \\  (a=0,-1)\\ , \n\\label{eq:compositionlaw2}\n\\]\nwhere the contour encloses $z=0$, and for fixed $g$, $x$ and $y$ lies inside the radius of convergence \nfor $\\widetilde{G}^{(a)} (g,x,z^{-1};t_1)$ as the series in $1\/z$ and for $\\widetilde{G}^{(a)} (g,z,y;t_2)$ as the series in $z$, \nwhich is possible as we will see.   \n\nIn what follows, we will discuss the one-step amplitude $G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1)$. \nThis is because it becomes an important object when computing the whole amplitude.    \n\n\n\n\n\\subsection{Counting triangulations}\n\\label{subsec:countingtriangulations}\n\nIn this section we focus on the one-step amplitude $G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1)$ which is the sum over triangulations of a strip as shown in Fig.~\\ref{fig:triangulationofstrip}: \n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1) \n= e^{-\\lambda (\\ell_1 + \\ell_2)} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2)\\ , \n\\label{eqonesteoamplitude} \n\\]\nand count the number of triangulations $\\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2)$. \nBased on simple combinatorics, one can calculate the case of $a=1$ which has the $S^1 \\times [0,1]$ topology:\n\\[\n\\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\frac{1}{\\ell_1 + \\ell_2} \n\\begin{pmatrix}\n\\ell_1 + \\ell_2 \\\\\n\\ell_1\n\\end{pmatrix}\n= \\frac{(\\ell_1 + \\ell_2-1)!}{\\ell_1 ! \\ell_2 !}\\ .\n\\label{eq:n1}\n\\]\nBecause of the property (\\ref{eq:markedamplitude}), \none can easily compute the case of $a=-1$ that the topology is $S^1 \\times [0,1]$ and one of the edges in the initial one-dimensional universe is marked: \n\\[\n\\mathcal{N}^{(-1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\ell_1 \\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\frac{\\ell_1 (\\ell_1 + \\ell_2-1)!}{\\ell_1 ! \\ell_2 !}\\ .\n\\label{eq:n-1}\n\\] \nConcerning the case of $a=0$ whose topology is $[0,1] \\times [0,1]$, there exist several possibilities depending on the restriction on the leftmost and rightmost triangles. \nIf the rightmost triangle is the upward triangle (downward triangle) and the leftmost triangles is the downward triangle (upward triangle), then the counting of triangulations yields \n\\[\n\\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\begin{pmatrix}\n\\ell_1 + \\ell_2 -2 \\\\\n\\ell_1 - 1\n\\end{pmatrix}\n= \\frac{(\\ell_1 + \\ell_2-2)!}{(\\ell_1 -1) ! (\\ell_2 -1) !}\\ .\n\\label{eq:n0}\n\\] \nIn the following, we will use eq.~(\\ref{eq:n0}) in the case of $a=0$ for computational simplicity\\footnote{\nIf we do not impose any restriction on the leftmost and rightmost triangles, the number of triangulations becomes \n$\\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) = \\ell_1 ! ( \\ell_1 + \\ell_2 )!\/(\\ell_1! \\ell_2!)$. \n}. \n\nThe one-step generating functions can be derived inserting eqs.~(\\ref{eq:n1}), (\\ref{eq:n-1}) and (\\ref{eq:n0}) into eq.~(\\ref{eq:generatingfunction}): \n\\[\n\\widetilde{G}^{(1)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= - \\ln \\left( \n\\frac{1-gx-gy}{(1-gx)(1-gy)}\n\\right)\\ ; \n\\label{eq:generatingfunction1}\\\\\n\\widetilde{G}^{(-1)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(-1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= \\frac{g^2xy}{(1-gx)(1-gx-gy)} \\ ; \n\\label{eq:generatingfunction-1}\\\\\n\\widetilde{G}^{(0)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= \\frac{g^2xy}{1-gx-gy} \\ .\n\\label{eq:generatingfunction0}\n\\]\nIn fact, one can also obtain eq.~(\\ref{eq:generatingfunction-1}) through $\\widetilde{G}^{(-1)} (g,x,y;1) = x \\frac{\\partial}{\\partial x} \\widetilde{G}^{(1)} (g,x,y;1)$. \n\nAlternatively, it is possible to compute the one-step generating functions directly by simple combinatorics: \n\\[\n\\widetilde{G}^{(1)} (g,x,y;1) \n&= \\sum^{\\infty}_{s=1} \\frac{1}{s} \\left( \\sum^{\\infty}_{k=1} (gx)^k \\sum^{\\infty}_{l=1} (gy)^l  \\right)^s \n= - \\ln \\left( \n\\frac{1-gx-gy}{(1-gx)(1-gy)}\n\\right)\\ ; \n\\label{eq:generatingfunction1v2}\\\\\n\\widetilde{G}^{(-1)} (g,x,y;1) \n&= \\sum^{\\infty}_{k=0} \\left( gx \\sum^{\\infty}_{l=0} (gy)^l \\right)^k - \\sum^{\\infty}_{k=0} (gx)^k \n= \\frac{g^2xy}{(1-gx)(1-gx-gy)} \\ ; \n\\label{eq:generatingfunction-1v2}\\\\\n\\widetilde{G}^{(0)} (g,x,y;1) \n&= \\sum^{\\infty}_{s=1} \\frac{1}{s} \\left( \\sum^{\\infty}_{k=1} (gx)^k \\sum^{\\infty}_{l=1} (gy)^l  \\right)^s \n= \\frac{g^2xy}{1-gx-gy} \\ .\n\\label{eq:generatingfunction0v2}\n\\] \n\n\n\\subsection{Continuum limit}\n\\label{subsec:continuumlimit}\n   \nAll is now set for computing the amplitude in the continuum limit. \nIn this section, however, instead of directly computing the amplitude in the continuum limit, \nwe will derive the differential equation that the continuum amplitude satisfies. \n\nBefore going into details any further, let us explain some basic facts of the continuum limit. \nIn order to remove the cutoff $\\varepsilon$ through the continuum limit, one has to tune the bare coupling constants $(g,x,y)$ to their critical values $(g_c,x_c,y_c)$. \nAt the critical values, the generating function hits the radii of convergence and therefore becomes non-analytic. \nApproaching such a critical point, infinitely many triangles and boundary edges become important in the summation of the generating function, \ni.e. essentially the average number of triangles and boundary edges become infinity at the critical point. \nHaving this in mind, one may intuitively understand that the continuous surface would be obtained if $(g,x,y) \\to (g_c,x_c,y_c)$ and $\\varepsilon \\to 0$ in a correlated manner. \n\nIntroducing $\\lambda_c = - \\ln [g_c]$, $\\lambda_{1c} = - \\ln [x_c]$ and $\\lambda_{2c} = - \\ln [y_c]$, \none can transmute the dimension of the lattice spacing $\\varepsilon$ into the dimension of the renormalized bulk and boundary cosmological constants through the continuum limit:  \n\\[\n\\Lambda = \\lim_{\\substack{\\lambda \\to \\lambda_c \\\\ \\varepsilon \\to 0}} \\frac{\\lambda - \\lambda_c}{\\varepsilon^2}\\ , \\ \\ \\ \nX = \\lim_{\\substack{\\lambda_1 \\to \\lambda_{1c} \\\\ \\varepsilon \\to 0}} \\frac{\\lambda_1 - \\lambda_{1c}}{\\varepsilon}\\ , \\ \\ \\ \nY = \\lim_{\\substack{\\lambda_2 \\to \\lambda_{2c} \\\\ \\varepsilon \\to 0}} \\frac{\\lambda_2 - \\lambda_{2c}}{\\varepsilon}\\ ,\n\\label{eq:renormalizedcc}\n\\]         \nwhere $\\Lambda$ is the renormalized bulk cosmological constant, and $X$ and $Y$ are the renormalized boundary cosmological constants. \nTherefore, the divergent bare cosmological constants get additive renormalizations so as to obtain the finite renormalized cosmological constants that set the scale at IR.   \n\nIn the following, we discuss the continuum limit in detail with respect to each topology of spacetime. \n\n\\subsubsection{$S^1 \\times [0,1]$ topology}\n\\label{subsubsec:s1times01}\n\n\nWe consider the case of $a=-1$, i.e. $S^1 \\times [0,1]$ topology with a marked boundary.   \nUsing the composition law (\\ref{eq:compositionlaw2}) and the one-step generating function (\\ref{eq:generatingfunction-1v2}), \none obtains\n\\[\n\\widetilde{G}^{(-1)} (g,x,y;t+1) \n&= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(-1)} (g,x,z^{-1};1)\\  \n\\widetilde{G}^{(-1)} (g,z,y;t)\\notag \\\\\n&=\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i} \n\\frac{g^2x}{(1-gx)^2 (z-g\/(1-gx))} \\frac{\\widetilde{G}^{(-1)} (g,z,y;t)}{z} \\notag\\\\\n&= \\frac{gx}{1-gx}\\ \\widetilde{G}^{(-1)} \\left( g, \\frac{g}{1-gx} ,y;t \\right)\\ . \n\\label{eq:-1relation}\n\\]\nIn the last equality, we have picked up a pole at $z=g\/(1-gx)$, and there exists no pole at $z=0$ since $\\frac{\\widetilde{G}^{(-1)} (g,z,y;t)}{z}$ is regular. \nThrough iterative use of eq.~(\\ref{eq:-1relation}), one can analytically compute the generating function, and extract the information of the critical point \\cite{Ambjorn:1998xu}.       \nHowever, we do not compute the generating function directly to obtain the critical point. \nInstead, we follow the procedure shown in \\cite{Ambjorn:2022btk}: \nOne assumes the existence of the critical point, and determines the value of the critical coupling constants from the consistency.  \n\nWe assume the critical point characterized by the critical coupling constants $(g_c,x_c,y_c)$ \nand use the following parametrization:\n\\[\ng=g_c e^{-\\varepsilon^2 \\Lambda}, \\ \\ \\ \nx=x_c e^{-\\varepsilon X}\\ , \\ \\ \\ \ny=y_c e^{-\\varepsilon Y}\\ . \n\\label{eq:parametrization}\n\\]\nAssuming the scalings \n\\[\nT = \\varepsilon t\\ , \\ \\ \\ L_1 = \\varepsilon \\ell_1\\ , \\ \\ \\ L_2 = \\varepsilon \\ell_2\\ , \n\\label{eq:continuumtandl}\n\\]\nwe introduce the renormalized amplitude and the renormalized generating function at the critical point by the multiplicative renormalizations:\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n&= \\lim_{\\varepsilon \\to 0} C_{\\varepsilon}\\ G^{(-1)}_{\\lambda} (\\ell_1,\\ell_2; t)\\ , \\label{eq:continuumg}\\\\\n\\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \n&= \\lim_{\\varepsilon \\to 0} \\widetilde{C}_{\\varepsilon}\\ \\widetilde{G}^{(-1)} (g, x, y; t)\\ , \\label{eq:continuumgtilde}\n\\]\nwhere $C_{\\varepsilon}$ and $\\widetilde{C}_{\\varepsilon}$ are real functions of $\\varepsilon$ that will be fixed below. \nThe function $C_{\\varepsilon}$ can be determined in such a way that \nthe composition law (\\ref{eq:compositionlaw}) holds in the continuum limit as \n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n= \\int^{\\infty}_{0}dL\\  \nG^{(-1)}_{\\Lambda} (L_1,L; T_1) G^{(-1)}_{\\Lambda} (L,L_2; T_2)\\ , \n\\label{eq:compositionlawcontinuum} \n\\]\nwhich is possible if $C_{\\varepsilon} = \\varepsilon^{-1}$. \nThe function $\\widetilde{C}_{\\varepsilon}$ can be determined in such way that eq.~(\\ref{eq:generatingfunction}) makes sense in the continuum limit, i.e. \n \\[\n\\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \n= \\int^{\\infty}_{0} dL_1 \\int^{\\infty}_{0}dL_2\\ e^{-XL_1} e^{-YL_2}  \nG^{(-1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:continuumlaplacetr}\n\\]\nwhich is possible if $\\widetilde{C}_{\\varepsilon} = \\varepsilon\/(x_c y_c)$. \n\nUsing the scaling behavior (\\ref{eq:continuumgtilde}), eq.~(\\ref{eq:-1relation}) can yield the sensible continuum limit if the critical coupling constants satisfy \n\\[\n\\frac{g_c x_c}{1-g_c x_c} = 1\\ , \\ \\ \\ \\frac{g_c}{1-g_c x_c} = x_c \\ \\ \\ \n\\Rightarrow \\ \\ \\ \ng_c = \\frac{1}{2}\\ , \\ \\ \\ x_c = 1\\ .\n\\]  \nNow we wish to take the continuum limit of eq.~(\\ref{eq:-1relation}). \nFor notational convenience, we redefine the renormalized coupling constants as follows:\n\\[\ng= \\frac{1}{2} e^{-\\varepsilon^2 \\Lambda} \\equiv \\frac{1}{2} \\left( 1 - \\frac{1}{2}\\varepsilon^2 \\Lambda \\right), \\ \\ \\ \nx= e^{-\\varepsilon X} \\equiv 1 - \\varepsilon X \\ , \\ \\ \\ \ny= e^{-\\varepsilon Y} \\equiv 1 - \\varepsilon Y \\ . \n\\label{eq:parametrization2}\n\\]\nPlugging eqs.~(\\ref{eq:continuumtandl}), (\\ref{eq:parametrization2}) into eq.~(\\ref{eq:-1relation}), \none obtains the differential equation: \n\\[\n\\frac{\\partial}{\\partial T} \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T)\n= - \\frac{\\partial}{\\partial X} \\left[ (X^2-\\Lambda) \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \\right]\\ . \n\\label{eq:differentialeq-1}\n\\]\nDoing a little math, one can also derive the continuum description of eq.~(\\ref{eq:amplitudefromgeneratingfunction}):\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= \\int^{c+i\\infty}_{c-i\\infty} \\frac{dX}{2\\pi i} \\int^{c+i\\infty}_{c-i\\infty} \\frac{dY}{2\\pi i}\\\ne^{L_1 X} e^{L_2 Y}\\ \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T)\\ ,\n\\label{eq:inverselaplacetr} \n\\]\nwhere $c$ is a suitable real number. \nUsing the inverse Laplace transform (\\ref{eq:inverselaplacetr}) and eq.~(\\ref{eq:differentialeq-1}), \none obtains the differential equation that the continuum amplitude satisfies:\n\\[\n\\frac{\\partial}{\\partial T} G^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(-1)} (L_1)\\ G^{(-1)}_{\\Lambda} (L_1,L_2; T)\\ ,  \n\\label{eq:differentialeq-1v2}\n\\]\nwhere \n\\[\n\\hat{H}^{(-1)}(L) = - L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\\ . \n\\label{hamiltonian-1}\n\\] \nAs a result, one can interpret the continuum limit of $2$d CDT as a quantum system of the one-dimensional universe with length $L$ \nthat propagates in time $T$ following the quantum Hamiltonian (\\ref{hamiltonian-1}). \nThe quantum Hamiltonian (\\ref{hamiltonian-1}) is hermitian with respect to the inner product:\n\\[ \n\\int^{\\infty}_{0} \\frac{dL}{L} \\phi^* (L)\\ (\\hat{H}^{(-1)} \\psi) (L)\n=\n\\int^{\\infty}_{0} \\frac{dL}{L} (\\hat{H}^{(-1)} \\phi)^* (L)\\ \\psi (L)\\ . \n\\label{eq:innerproduct-1} \n\\]  \n \nThe differential equation for the un-marked amplitude can be easily read off inserting the continuum limit of eq.~(\\ref{eq:markedamplitude})\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= L_1 G^{(1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:continuummarkedamplitude} \n\\]\ninto eq.~(\\ref{eq:differentialeq-1v2}): \n\\[\n\\frac{\\partial}{\\partial T} G^{(1)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(1)} (L_1)\\ G^{(1)}_{\\Lambda} (L_1,L_2; T)\\ ,  \n\\label{eq:differentialeq1v2}\n\\]\nwhere \n\\[\n\\hat{H}^{(1)}(L) = - \\frac{\\partial^2}{\\partial L^2} L + \\Lambda L\\ . \n\\label{hamiltonian1}\n\\]   \nThe quantum Hamiltonian (\\ref{hamiltonian1}) is hermitian with respect to the inner product:\n\\[\n\\int^{\\infty}_{0} LdL \\phi^* (L)\\  (\\hat{H}^{(1)} \\psi) (L)\n=\n\\int^{\\infty}_{0} LdL (\\hat{H}^{(1)} \\phi)^* (L)\\ \\psi (L)\n\\ . \n\\label{eq:innerproduct1} \n\\]  \n\n \n \n\n\\subsubsection{$[0,1] \\times [0,1]$ topology}\n\\label{subsubsec:01times01}\n\nLet us consider the case of $a=0$, i.e. $[0,1] \\times [0,1]$ topology. \nWe basically follow the procedure shown in Sect.~\\ref{subsubsec:s1times01}.   \nUsing the composition law (\\ref{eq:compositionlaw2}) and the one-step generating function (\\ref{eq:generatingfunction0v2}), \none obtains\n\\[\n\\widetilde{G}^{(0)} (g,x,y;t+1) \n&= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(0)} (g,x,z^{-1};1)\\  \n\\widetilde{G}^{(0)} (g,z,y;t)\\notag \\\\\n&=\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i} \n\\frac{g^2x}{(1-gx)(z-g\/(1-gx))} \n\\frac{\\widetilde{G}^{(0)} (g,z,y;t)}{z} \\notag\\\\\n&= gx\\ \\widetilde{G}^{(0)} \\left( g, \\frac{g}{1-gx} ,y;t \\right)\\ . \n\\label{eq:0relation}\n\\]\nFrom eq.~(\\ref{eq:0relation}), one may obtain a sensible continuum limit \nif the critical coupling constants are the same as before, i.e. $(g_c,x_c,y_c)=(1\/2,1,1)$, and if the multiplicative renormalization is treated carefully:  \n\\[\n\\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T) \n= \\lim_{\\varepsilon \\to 0}  \\frac{\\varepsilon}{2^t}  \\widetilde{G}^{(0)} (g, x, y; t)\\ . \n\\label{eq:continuumgtilde0}\n\\]\nIn fact, this assumption yields the correct continuum limit. \nPlugging eqs.~(\\ref{eq:continuumtandl}), (\\ref{eq:parametrization2}) into eq.~(\\ref{eq:0relation}) and using eq.~(\\ref{eq:continuumgtilde0}), \none obtains the differential equation:\n\\[\n\\frac{\\partial}{\\partial T} \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T)\n= - \\left( X + (X^2 - \\Lambda) \\frac{\\partial}{\\partial X} \\right) \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T) \\ . \n\\label{eq:differentialeq0}\n\\] \nDefining the continuum amplitude in such a way that the inverse Laplace transform (\\ref{eq:inverselaplacetr}) holds in the case of $a=0$ as well, i.e. \n\\[\nG^{(0)}_{\\Lambda} (L_1,L_2; T) \n= \\int^{c+i\\infty}_{c-i\\infty} \\frac{dX}{2\\pi i} \\int^{c+i\\infty}_{c-i\\infty} \\frac{dY}{2\\pi i}\\\ne^{L_1 X} e^{L_2 Y}\\ \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T)\\ ,\n\\label{eq:inverselaplacetr0} \n\\]\nand using eq.~(\\ref{eq:inverselaplacetr0}), the differential equation (\\ref{eq:differentialeq0}) becomes \n\\[\n\\frac{\\partial}{\\partial T} G^{(0)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(0)} (L_1)\\ G^{(0)}_{\\Lambda} (L_1,L_2; T)\\ ,  \n\\label{eq:differentialeq0v2}\n\\]\nwhere $\\hat{H}^{(0)}$ is the quantum Hamiltonian obtained in Refs.~\\cite{DiFrancesco:2000nn, Durhuus:2001sp}:  \n\\[\n\\hat{H}^{(0)}(L) = - \\frac{\\partial}{\\partial L} L \\frac{\\partial}{\\partial L} + \\Lambda L\\ . \n\\label{hamiltonian0}\n\\]   \nThe quantum Hamiltonian (\\ref{hamiltonian0}) is hermitian with respect to the inner product:\n\\[\n\\int^{\\infty}_{0} dL \\phi^* (L)\\ (\\hat{H}^{(0)}\\psi) (L)\n=\n\\int^{\\infty}_{0} dL (\\hat{H}^{(0)} \\phi)^* (L)\\ \\psi (L)\n\\ . \n\\label{eq:innerproduct0} \n\\]  \n\n\n\n\\subsection{Short summary of $2$d CDT}\n\\label{subsec:shortsummaryof2dcdt}\nAs discussed in Sect.~\\ref{subsec:continuumlimit}, the continuum limit of $2$d CDT is described by the quantum mechanics of a one-dimensional universe with length $L$ that propagates in time $T$ \nbased on the Hamiltonian $\\hat{H}^{(a)}$: \n\\[\n\\hat{H}^{(-1)} = - L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(1)} = - \\frac{\\partial^2}{\\partial L^2} L + \\Lambda L\\ , \\ \\ \\\n\\hat{H}^{(0)} = - \\frac{\\partial}{\\partial L} L \\frac{\\partial}{\\partial L} + \\Lambda L\\ ,\n\\label{eq:hamiltonians}\n\\]   \nwhere the label $a$ classifies the topology of the one-dimensional universe: \n$S^1$ and $[0,1]$ for $a=1$ and $a=0$, respectively. \nWhen $a=-1$, the closed one-dimensional universe is marked. \nLet us define the eigenstates of $L$ as $|L\\rangle_{a}$ that satisfy \nthe completeness relation:  \n\\[\n1 = \\int^{\\infty}_{0}L^{a}dL\\ |L\\rangle_{a} {}_{a}\\langle L | \\ \\ \\ \n\\Leftrightarrow \n\\ \\ \\ \n{}_a \\langle L' | L \\rangle_{a} = \\frac{1}{L^a} \\delta (L - L')\\ . \n\\label{eq:completenessrelations}\n\\] \nNote that $|L \\rangle_{-1} = L | L \\rangle_1$. \nOne can then express the amplitudes as matrix elements:\n\\[\nG^{(1)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{1}\\langle L_2 | e^{-T \\hat{H}^{(1)}}  |L_1 \\rangle_{1}\\ , \\\\\nG^{(-1)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{1}\\langle L_2 | e^{-T \\hat{H}^{(-1)}}  |L_1 \\rangle_{-1}\\ , \\\\\nG^{(0)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{0}\\langle L_2 | e^{-T \\hat{H}^{(0)}}  |L_1 \\rangle_{0}\\ , \n\\label{eq:matrixelement2}\n\\] \nUsing eq.~(\\ref{eq:completenessrelations}), one can show that the composition laws hold: \nFor $a=-1,0$, \n\\[\nG^{(a)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n= \\int^{\\infty}_{0}dL\\  \nG^{(a)}_{\\Lambda} (L_1,L; T_1) G^{(a)}_{\\Lambda} (L,L_2; T_2)\\ ,\n\\label{eq:compositionlawcontinuum_a-10} \n\\]\nand for $a=1$,  \n\\[\nG^{(1)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n&= \\int^{\\infty}_{0}dL\\  \n^{(1)}_{\\Lambda} (L_1,L; T_1) L G^{(1)}_{\\Lambda} (L,L_2; T_2)\\ . \n\\label{eq:compositionlawcontinuum_a1}\n\\]\n\n\n\n\n\\section{$2$d projectable Ho\\v rava-Lifshitz quantum gravity}\n\\label{sec:2dHLQG}\n\nWe wish to introduce the classical field theory that reproduces the continuum limit of $2$d CDT once it is quantized. \nThe field theory is a certain version of the two-dimensional Ho\\v rava-Lifshitz gravity ($2$d HL gravity). \n\n\nThe starting point is the same class of manifold with a foliation (\\ref{eq:m}) where $\\Sigma_t$ is a one-dimensional space labelled by $t$: \n\\[\n\\Sigma_t = \\{ x^{\\mu} \\in \\mathcal{M}\\ |\\ f(x^{\\mu}) = t \\}\\ , \\ \\ \\ \\text{with} \\ \\ \\ \\mu = 0,1\\ . \n\\label{eq:sigma} \n\\] \nChoosing that $f(x^{\\mu}) =x^0$, the time direction can be decomposed into the two directions, i.e. the normal and the tangential to $\\Sigma_t$: \n\\[\n\\left( \\partial_t \\right)^{\\mu} \n= \\frac{\\partial x^{\\mu}}{\\partial t} \n= N n^{\\mu} + N^1 E^{\\mu}_{1}\\ , \n\\label{eq:timedirection}\n\\] \nwhere $n^{\\mu}$ and $E^{\\mu}_1$ are respectively the unit normal vector and the tangent vector defined as \n\\[\nn^{\\mu} = \n\\left(\n\\frac{1}{N}, - \\frac{N^1}{N}\n\\right)\\ , \\ \\ \\ \nE^{\\mu}_1 = \\delta^{\\mu}_1\\ . \n\\label{eq:normalandtangent}\n\\]  \nHere $N$ and $N^1$ are the Lapse function and the shift vector. \nThrough the use of eq.~(\\ref{eq:timedirection}), \none can parametrize the metric $g_{\\mu \\nu}$ on $\\mathcal{M}$ as follows:\n\\[\nds^2 \n= g_{\\mu \\nu}  dx^{\\mu} dx^{\\nu} \n= - N^2  dt^2 + h_{11}  \\left(dx +N^1 dt \\right) \\left( dx +N^1 dt \\right) \\ , \n\\label{eq:admmetric}\n\\]  \nwhere $t=x^0$ and $x=x^1$; $h_{11}$ is the spatial metric on $\\Sigma_t$ defined as $h_{11} = E^{\\mu}_{1} E^{\\nu}_1 g_{\\mu \\nu}$. \n\n$2$d HL gravity is a field theory that preserves the structure of the time foliation, or in other words, it is invariant under \nthe foliation-preserving diffeomorphisms (FPD):      \n\\[\nt \\to t + \\xi^0 (t)\\ , \\ \\ \\ x \\to x + \\xi^1 (t,x)\\ . \n\\label{eq:fpd}\n\\]\nThe fields transform under FPD as follows:\n\\[\n\\delta_{\\xi} h_{11} &= \\xi^0 \\partial_0 h_{11} + \\xi^1 \\partial_1 h_{11} + 2h_{11}\\partial_1 \\xi^1\\ , \\label{eq:deltah11}\\\\\n\\delta_{\\xi} N_1 &= \\xi^{\\mu} \\partial_{\\mu} N_1 + N_1 \\partial_{\\mu} \\xi^{\\mu} + h_{11} \\partial_0 \\xi^1\\ , \\label{eq:deltan1}\\\\\n\\delta_{\\xi} N &= \\xi^{\\mu}\\partial_{\\mu} N + N \\partial_0 \\xi^0\\ . \\label{eq:deltan}\n\\] \nwhere $N_1 = h_{11}N^1$. \nHere if a function is a constant on each foliation $\\Sigma_t$, such a function is called projectable. \nIn fact, implementing FPD the projectable Lapse function, i.e. $N=N(t)$, stays as a function only of time.  \nThe HL gravity with the projectable Lapse function is dubbed the projectable HL gravity. \nSince it is $2$d projectable HL gravity that reproduces the continuum limit of $2$d CDT once it is quantized, \nfrom now we focus on this special version of HL gravity.   \n\nThe action of $2$d projectable HL gravity is given by \n\\[\nI =\n\\int dt\\ \\mathcal{L}\n=\\frac{1}{\\kappa} \n\\int dtdx\\ \nN(t) \\sqrt{h(t,x)} \n\\left(\n(1-\\eta) K^2(t,x) - 2 \\widetilde{\\Lambda}\n\\right)\\ , \n\\label{eq:hlaction}\n\\]\nwhere $\\mathcal{L}$ is the Lagrangian; \n$\\eta$, $\\widetilde{\\Lambda}$ and $\\kappa$ are a dimensionless parameter, \nthe cosmological constant and the (dimensionless) gravitational coupling constant, respectively; \n$h$ is the determinant of the spatial metric $h_{11}$, i.e. $h=h_{11}$; \n$K$ is the trace of the extrinsic curvature $K_{11}$ given by \n\\[\nK_{11} = \\frac{1}{2N} \\left( \\partial_0 - 2\\nabla_1 N_1 \\right)\\ , \\ \\ \\ \n\\text{with}\\ \\ \\ \n\\nabla_1 N_1 = \\partial_1 N_1 - \\Gamma^1_{11} N_1\\ .\n\\label{eq:extrinsiccurvature}\n\\] \nHere $\\Gamma^1_{11}$ is the spatial Christoffel symbol: \n\\[\n\\Gamma^1_{11} \n= \\frac{1}{2} h^{11} \\partial_1 h_{11}\\ . \n\\label{eq:christoffel}\n\\]\nOne can in principle add higher spatial derivative terms to the action (\\ref{eq:hlaction}). \nHowever, such terms are not necessary because the model is renormalizable in two dimensions without introducing them, \nand therefore we omit such terms. \n\nThe continuum limit of $2$d CDT can be precisely obtained if quantizing the $2$d projectable HL gravity \nwith the following identification of parameters: \n\\[\n\\Lambda = \\frac{ \\widetilde{\\Lambda} }{2(1-\\eta)}\\ , \\ \\ \\ \n\\eta < 1\\ , \\ \\ \\ \n\\kappa = 4(1-\\eta)\\ , \n\\label{eq:parameteridentification}\n\\]\nwhere $\\Lambda$ is the renormalized cosmological constant of CDT defined by eq.~(\\ref{eq:parametrization2}).   \n\n\n\n\n\n\\subsection{Quantization}\n\\label{sec:quantization}\n\nLet us overview the quantization of $2$d projectable HL gravity shown in Ref.~\\cite{Ambjorn:2013joa} (see also Ref.~\\cite{Li:2014bla} for another article examining this issue). \n\nWe introduce the conjugate momentum of $\\sqrt{h}$ as $\\Pi$, which satisfy the Poisson bracket: \n\\[\n\\left\\{ \n\\sqrt{h(t,x)}, \\Pi (t,x')\n\\right\\}\n= \\delta (x-x')\\ . \n\\label{eq:poisson}\n\\] \nThrough the Legendre transformation of the Lagrangian (\\ref{eq:hlaction}), \none obtains the Hamiltonian of $2$d projectable HL gravity:\n\\[\nH = \\int dx\n\\left( \n\\Pi(t,x) \\partial_t \\sqrt{h (t,x)} \n\\right)\n- \\mathcal{L} \n= \nN(t) \\mathcal{C}(t) +\n\\int dx\\ \nN_1 (t,x) \\mathcal{C}^1 (t,x)\n\\ . \n\\label{eq:classicalhamiltonian}\n\\]\nSince $2$d projectable HL gravity is a singular system due to the invariance under FPD, \nthere exist two kinds of constraint:  \n\\[\n\\mathcal{C}^1 (t,x)  \n&= - \\frac{\\partial_1 \\Pi (t,x)}{\\sqrt{h (t,x)}} \\approx 0\\ , \\label{eq:momentumconst}\\\\\n\\mathcal{C} (t)\n&= \\int dx\\ \n\\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2(t,x) \\sqrt{h(t,x)} + \\frac{2}{\\kappa} \\widetilde{\\Lambda} \\sqrt{h(t,x)}\n\\right)\n\\approx 0 \n\\ , \\label{eq:classicalhamiltonianconst} \n\\]  \nwhere $\\mathcal{C}^1 (t,x) \\approx 0$ is the momentum constraint, \nand $\\mathcal{C} (t) \\approx 0$ is the Hamiltonian constraint which is global because of the projectable Lapse function\\footnote{\nThe Hamiltonian and the momentum constraints come from the consistency conditions that \nthe primary constraints, $\\Pi_N \\approx 0$ and $\\Pi_{N_1} \\approx 0$, should be preserved under the time flow \nwhere $\\Pi_N$ and $\\Pi_{N_1}$ are the conjugate momenta of $N$ and $N_1$, respectively.   \n}.   \n\nThe strategy is to solve the momentum constraint (\\ref{eq:momentumconst}) at the level of classical theory, i.e. \n\\[\n\\mathcal{C}^1 (t,x) = 0 \\ \\ \\ \n\\Rightarrow \\ \\ \\ \n\\Pi (t,x) = \\Pi (t)\\ , \n\\label{eq:solvemomentumconst} \n\\] \nmeaning that the conjugate momentum becomes a function only of time. \nApplying eq.~(\\ref{eq:solvemomentumconst}), \nthe Hamiltonian (\\ref{eq:classicalhamiltonian}) reduces to the one for the one-dimensional system: \n\\[\nH =\nN(t) \\left(\n\\frac{\\kappa}{4 (1-\\eta)} \\Pi^2(t) L(t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} L(t) \n\\right)\\ ,   \n\\ \\ \\\n\\text{with} \\ \\ \\ \nL(t) = \\int dx \\sqrt{h(t,x)}\\ , \n\\label{eq:reducedhamiltonian}\n\\]  \nwhere $L(t)$ is the invariant length of the one-dimensional universe. \nLet us discuss solutions to the Hamiltonian constraint. \nIf $(\\eta -1)\\widetilde{\\Lambda}>0$, one has a solution:\n\\[\n\\Pi^2 = \\frac{8(\\eta - 1)}{\\kappa^2}\\ \\widetilde{\\Lambda}\\ , \n\\label{eq:hamiltonianconstsol1}\n\\] \nwhich means that the extrinsic curvature is a constant. \nOn the other hand, if $(\\eta -1)\\widetilde{\\Lambda}<0$, \nthe only solution is \n\\[\nL=0\\ . \n\\label{eq:hamiltonianconstsol2}\n\\]\n\nHereafter we apply the parametrization (\\ref{eq:parameteridentification}): We choose \n$(\\eta -1)\\widetilde{\\Lambda}<0$, set the unimportant dimensionless gravitational constant as $\\kappa = 4(1-\\eta)$, and redefine the cosmological constant as $\\Lambda = \\frac{ \\widetilde{\\Lambda} }{2(1-\\eta)}$.  \nSince $\\kappa > 0$, this means that $\\eta <1$ which selects the correct sign of the kinetic term, and the positive cosmological constant $\\widetilde{\\Lambda}>0$.  \nThe dynamics of the classical $1$d system with the Hamiltonian (\\ref{eq:reducedhamiltonian}) can be alternatively described by the following action: \n\\[\nS= \\int^{1}_{0} dt \\left(\n\\frac{\\dot{L}^2(t)}{4N(t)L(t)} - \\Lambda N(t)L(t)\n\\right)\\ , \n\\label{eq:reducedaction}\n\\]  \nwhere $\\dot{L}(t) := \\frac{d}{dt} L(t)$. \nWe then introduce the proper time:\n\\[\n\\tau (s) = \\int^{s}_{0}dt\\ N(t)\\ , \\ \\ \\ s\\in [0,1]\\ .\n\\label{eq:propertime}\n\\]\nSince the proper time (\\ref{eq:propertime}) is invariant under the reparametrization of time, $t\\to t + \\xi^0 (t)$, \nif one fixes the Lapse function as $N(\\tau)=1$, the length of the one-dimensional universe $L(\\tau)$ is also invariant under the time redefinition. \nTherefore, it makes sense to discuss the amplitude such that the one-dimensional universe with the length $L_1:=L(\\tau=0)$ propagates in the proper time $\\tau$, \nand ends up with the universe whose length is given by $L_2 := L_2(\\tau = T)$. \n\n\n\nWith this understanding, we consider such an amplitude based on the path integral. \nFor convenience, we rotate $t \\to it$, which is possible thanks to the foliation and introduce the Euclidean action:\n\\[\nS_E = \n\\int^{1}_{0} dt \\left(\n\\frac{\\dot{L}^2(t)}{4N(t)L(t)} + \\Lambda N(t)L(t)\n\\right)\\ .  \n\\label{eq:euclideanaction}\n\\] \nUsing the Euclidean action (\\ref{eq:euclideanaction}), the amplitude becomes \n\\[\n\\mathcal{G}_{\\Lambda} (L_1,L_2;T)\n= \\int \\frac{\\mathcal{D}N(t)}{\\text{Diff [0,1]}} \\int^{L(1)=L_2}_{L(0)=L_1} \\mathcal{D}L(t)\\ e^{-S_E [N(t), L(t)]}\\ , \n\\label{eq:pathintegral}\n\\]\nwhere \n\\[\nT := \\int^{1}_{0} dt\\ N(t)\\ . \n\\label{eq:T}\n\\]   \nWe fix the Lapse function as $N(\\tau)=1$ introducing the corresponding Faddeev-Popov (FP) determinant. \nSince the FP determinant only gives an overall constant, we will omit it in the following. \nAfter the gauge fixing, the amplitude (\\ref{eq:pathintegral}) becomes\n\\[\n\\mathcal{G}_{\\Lambda} (L_1,L_2;T) \n= \\int^{L(T)=L_2}_{L(0)=L_1} \n \\mathcal{D}L(\\tau)\\ \n \\exp \\left[ \n - \\int^T_0 d\\tau \n \\left(\n \\frac{\\dot{L}^2(\\tau)}{4L(\\tau)} + \\Lambda L(\\tau)\n \\right)\n \\right]\\ , \n \\label{eq:pathintegral2}\n\\]     \nwhere $\\dot{L}(\\tau) := \\frac{d}{d \\tau} L(\\tau)$. \n\nSo far, we have not specified the integral measure $\\mathcal{D}L(\\tau)$. \nWe apply the three kinds of measure given by \n\\[\n\\mathcal{D}^{(a)}L(\\tau) \n= \\prod^{\\tau=T}_{\\tau=0} L^a(\\tau)\\ dL(\\tau)\\ , \\ \\ \\ (a=0,\\pm 1)\\ .\n\\label{eq:measure}\n\\] \nAccordingly, we consider the three kinds of amplitude, i.e. $\\mathcal{G}^{(a)}_{\\Lambda}(L_1,L_2;T)$, \nand rewrite them introducing the quantum Hamiltonian $\\hat{H}^{(a)}$:\n\\[\n\\mathcal{G}^{(a)}_{\\Lambda}(L_1,L_2;T) \n= {}_{a}\\langle L_2 | e^{-T \\hat{H}^{(a)}}  |L_1 \\rangle_{a}\\ ,\n\\label{eq:amplitudematrix}\n\\] \nwhere the eigenstates of $L$ satisfy the completeness relation: \n\\[\n1 = \\int^{\\infty}_{0}L^{a}dL\\ |L\\rangle_{a} {}_{a}\\langle L | \\ \\ \\ \n\\Leftrightarrow \n\\ \\ \\ \n{}_a \\langle L' | L \\rangle_{a} = \\frac{1}{L^a} \\delta (L - L')\\ . \n\\label{eq:completenessrelations2}\n\\] \nIn order to read off the quantum Hamiltonian $\\hat{H}^{(a)}$, we discretize the proper time interval in steps of $\\varepsilon$, \nand calculate the one-step matrix element $G^{(a)}_{\\Lambda}(L,L';\\varepsilon)$. \nThe normalization can be fixed so as to satisfy the following equation:\n\\[\n\\lim_{\\varepsilon \\to 0} \\int^{\\infty}_{0} L^a dL\\ \\mathcal{G}^{(a)}_{\\Lambda}(L,L';\\varepsilon) = 1\\ , \n\\label{eq:normalization} \n\\] \nwhich comes from the completeness relation (\\ref{eq:completenessrelations2}). \nThe result is \n\\[\n\\mathcal{G}^{(a)}_{\\Lambda}(L,L';\\varepsilon)  \n= \\frac{(LL')^{(1-a)\/2}}{L' \\sqrt{4\\pi \\varepsilon L'}} e^{-\\frac{(L-L')^2}{4\\varepsilon L'} - \\Lambda \\varepsilon L'}\\ . \n\\label{eq:onesteppropagator}\n\\] \nIntegrating the one-step amplitude together with a function, $\\psi_a (L) = {}_a \\langle L | \\psi \\rangle$, for $\\varepsilon \\ll 1$, \none can read off the quantum Hamiltonian:  \n\\[\n\\psi_a (L'; \\varepsilon) \n&= {}_{a}\\langle L' | e^{- \\varepsilon \\hat{H}^{(a)}}  | \\psi \\rangle \\notag \\\\ \n&= \\int^{\\infty}_{0} L^a dL\\ {}_{a}\\langle L' | e^{- \\varepsilon \\hat{H}^{(a)}}  |L \\rangle_{a} {}_a \\langle L | \\psi \\rangle \\notag \\\\  \n& \\cong \\psi_a (L') - \\varepsilon \\hat{H}^{(a)}\\psi_a (L') + \\mathcal{O} (\\varepsilon^{3\/2})\\ .  \n\\label{eq:readoffhamiltonian} \n\\]   \nUsing eq.~(\\ref{eq:onesteppropagator}), one obtains    \n\\[\n\\hat{H}^{(-1)} (L) = - L \\frac{d^2}{dL^2} + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(0)} (L) =  - \\frac{d}{dL} L \\frac{d}{dL}  + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(1)} (L) =  - \\frac{d^2}{dL^2} L + \\Lambda L\\ , \n\\label{eq:hamiltonianhorava}\n\\]\n\n\nThe quantum Hamiltonians (\\ref{eq:hamiltonianhorava}) obtained by quantizing $2$d projectable HL gravity are \nprecisely equivalent to those obtained by the continuum limit of $2$d CDT (see eqs.~(\\ref{hamiltonian-1}), (\\ref{hamiltonian1}) and (\\ref{hamiltonian0})). \nThe amplitudes are related as follows:\n\\[\n\\mathcal{G}^{(-1)}_{\\Lambda}(L,L';T) = L' G^{(-1)}_{\\Lambda} (L,L';T)\\ , \\ \\ \\  \n\\mathcal{G}^{(a)}_{\\Lambda}(L,L';T) = G^{(a)}_{\\Lambda} (L,L';T)\\ , \\ \\ \\  (a=0,1)\\ . \n\\label{eq:relationamplitudes}\n\\]\nThereby, we understand that the classical field theory that reproduces the continuum limit of $2$d CDT once it is quantized is indeed \n$2$d projectable HL gravity. The projectable Lapse function allows us to introduce the reparametrization-invariant proper time, \nand to reduce the $2$d field theory to the $1$d system.   \n\n\n\n\n\n\\section{Sum over all wormholes and baby universes}\n\\label{sec:sumoverallgeneraingenera}\n\nIn the CDT model, the spatial topology change is not allowed to occur by definition. \nOne can generalize the $2$d CDT model in such a way that spatial topology changes do occur in keeping with the foliation structure, \nand the universality class is the same as that of $2$d CDT.   \nSuch a model is called generalized CDT (GCDT). \nGCDT can be constructed as both discretized and continuum models. \nHere of course the continuum model can be obtained by the continuum limit of the discretized model, \nbut one can directly construct the continuum GCDT model promoting the one-dimensional quantum-mechanical system discussed in Sec.~\\ref{sec:2dcdt} to a $2$d field theory \nthat includes the splitting and joining interactions of the one-dimensional spatial universe. \nSuch a field theory is dubbed the string field theory for CDT, in which the string means the one-dimensional universe \\cite{Ambjorn:2008ta}. \nIn this section, we introduce the string field theory for CDT, and briefly explain the fact that one can take the sum over all wormholes (i.e. handles) and baby universes \\cite{Ambjorn:2009wi, Ambjorn:2009fm}. \nHere the baby universe is a portion of geometry that is pinched off from the ``parent universe'' and vanishes into the vacuum. \nWe also introduce an effective one-body theory that reproduces the many-body effects coming from the splitting and joining interactions.  \nWe then discuss those effects in the context of HL gravity \\cite{Ambjorn:2021wou}. \nIn the end, we show that a sort of Coleman's mechanism works when taking into account all contributions of wormholes and baby universes non-perturbatively \\cite{Ambjorn:2021wdm}.            \n\n\n\n\nWe introduce an operator that creates a marked closed string (i.e. a marked closed one-dimensional universe) with length $L$, $\\Psi^{\\dagger} (L)$, \nand an operator that annihilates a length-$L$ closed string without a mark, $\\Psi (L)$. \nThese operators satisfy the following commutators: \n\\[\n[\\Psi (L), \\Psi^{\\dagger} (L')] = \\delta (L-L')\\ , \\ \\ \\ \n[\\Psi (L), \\Psi (L')] \n= [\\Psi^{\\dagger} (L), \\Psi^{\\dagger} (L')] \n=0\\ . \n\\label{eq:commutators}\n\\]   \nThe vacuum state $| \\text{vac} \\rangle$ is defined by the equation:  \n$\\Psi (L) | \\text{vac} \\rangle  = 0$. \nThe CDT amplitude (\\ref{eq:matrixelement2}) can be expressed by sandwiching the one-body Hamiltonian: \n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2;T)\n= \\langle \\text{vac} | \\Psi (L_2)\\ \ne^{-T \\mathcal{H}^{(-1)_{\\text{free}}} (L_1)}\\\n\\Psi^{\\dagger} (L_1)\n| \\text{vac} \\rangle\\ , \n\\label{eq:2ndamplitude-1}\n\\]\nwhere  \n\\[\n\\mathcal{H}^{(-1)}_{\\text{free}} (L)\n= \\int^{\\infty}_{0} \\frac{dL}{L} \n\\Psi^{\\dagger} (L) \\left(\n-L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\n\\right) \n\\Psi (L)\\ . \n\\label{eq:2ndhamiltonian-1}\n\\]\nHereafter we omit the superscript $(-1)$ for avoiding notational complexity. \nAdding splitting and joining interactions into the free Hamiltonian (\\ref{eq:2ndhamiltonian-1}), \none obtains the full Hamiltonian of the string field theory for CDT:\n\\[\n\\mathcal{H}\n&=\\mathcal{H}_{\\text{free}} \n-g_s \\int^{\\infty}_{0}dL_1 \\int^{\\infty}_0 dL_2 \\Psi^{\\dagger} (L_1) \\Psi^{\\dagger} (L_2) (L_1 + L_2) \\Psi (L_1 + L_2) \\notag \\\\ \n&\\ \\ \\ - \\alpha g_s \\int^{\\infty}_{0}dL_1 \\int^{\\infty}_0 dL_2 \\Psi^{\\dagger} (L_1 + L_2) L_1 \\Psi (L_1) L_2 \\Psi (L_2) \\notag \\\\\n&\\ \\ \\ - \\int^{\\infty}_{0} dL\\delta (L)\\Psi (L)\\ ,\n\\label{eq:fullhamiltonian-1}\n\\]\nwhere the second, third and fourth terms respectively mean the splitting interaction with the string coupling constant $g_s$, \nthe joining interaction with the coupling constant $\\alpha g_s$, \nand the term associated with a string vanishing into the vacuum. \nHere the parameter $\\alpha$ is introduced for counting the number of handles (i.e. wormholes). \nOne can in principle calculate the amplitude for the process such that $m$ closed strings propagate in time and end up with \n$n$ closed strings: \n\\[\nA(L_1, \\cdots, L_m; L'_1, \\cdots, L'_n; T) \n= \\langle \\text{vac} | \n\\Psi (L'_1) \\cdots \\Psi (L'_n)\ne^{-T \\mathcal{H}} \n\\Psi^{\\dagger} (L_1) \\cdots \\Psi^{\\dagger} (L_m)\n| \\text{vac} \\rangle\\ , \n\\label{eq:fullamplitude}\n\\]\n\n\\subsection{Effective theory}\n\\label{sec:effectivetheory}\n\nLet us consider the full propagator $A(L_1, L_2;T)$ that includes the sum over all genera and baby universes. \nWe can set $\\alpha = 1$ without loss of generality since the parameter $\\alpha$ plays a supplementary role and we are interested in taking the sum over all genus contributions.  \nSomewhat miraculously, the full propagator defined in the many-body system with the Hamiltonian (\\ref{eq:fullhamiltonian-1}) can be effectively described by \nthe one-body system \\cite{Ambjorn:2009wi, Ambjorn:2009fm}: \n\\[\nA(L_1,L_2;T) \n= \\langle L_2 | e^{-T \\hat{H}_{\\text{eff}} (L_1)} | L_1 \\rangle\\ , \n\\label{eq:fullpropagator}\n\\] \nwhere $\\hat{H}$ is the effective Hamiltonian given by \n\\[\n\\hat{H}_{\\text{eff}} (L) = - L \\frac{d^2}{dL^2} + \\Lambda L - g_s L^2\\ . \n\\label{eq:effectivehamiltonian}\n\\] \nThis is possible because there exists a bijection called Ambj\\o rn-Budd bijection \\cite{Ambjorn:2013csx} such that \none can map each geometry generated in GCDT to a branched polymer with loops at the discrete level. \nThe last term in the Hamiltonian (\\ref{eq:effectivehamiltonian}), $-g_s L^2$, expresses all the effects originated with the baby universes and the wormholes. \nNote that the Hamiltonian (\\ref{eq:effectivehamiltonian}) is not bounded from below because of the last term, \nbut in fact this system is known to be ``classical incomplete,'' which means that the Hamiltonian has discrete energy spectra, \nand a set of square integrable eigenfunctions (see e.g. Ref.~\\cite{Ambjorn:1992ve} for a pedagogical explanation about the classical incomplete systems). \nA similar deformation has been observed in the $c=1$ non-critical string theory \\cite{Moore:1991sf, Betzios:2020nry}. \n\nThe full propagator (\\ref{eq:effectivehamiltonian}) can be also described in terms of the path-integral:\n\\[\nA(L_1,L_2;T) \n= \\int^{L(T) = L_2}_{L(0) = L_1}\n\\mathcal{D}L(\\tau)\\ \n\\exp \\left[\n- \\int^{T}_{0}d\\tau \n\\left(\n\\frac{\\dot{L}^2 (\\tau)}{4L(\\tau)} + \\Lambda L(\\tau) - g_s L^2 (\\tau)\n\\right)\n\\right]\\ ,\n\\label{eq:pathintegralfullpropagator}\n\\]\nwhere the integral measure is given by \n\\[\n\\mathcal{D}L(\\tau) \n= \\prod^{\\tau = T}_{\\tau = 0} L^{-1} (\\tau) dL (\\tau)\\ . \n\\label{eq:measure-1full}\n\\]\nIn order for the functional integral (\\ref{eq:pathintegralfullpropagator}) to be well-defined, one need to choose the boundary conditions on $L(\\tau)$ at infinity such that \nthe kinetic term counteracts the unboundedness of the potential. \nIf one generalizes the integral measure (\\ref{eq:measure-1full}) as \n\\[\n\\mathcal{D}^{(a)}L(\\tau) \n= \\prod^{\\tau = T}_{\\tau = 0} L^{a} (\\tau) dL (\\tau)\\ , \\ \\ \\ (a=0, \\pm 1)\\ ,\n\\label{eq:measureafull}\n\\]   \none can recover all possible orderings of the effective Hamiltonian (\\ref{eq:effectivehamiltonian}) following the procedure explained in Sec.~\\ref{sec:quantization}.   \n\n\nInterestingly, one can reproduce the full propagator (\\ref{eq:pathintegralfullpropagator}) \nif one considers that the cosmological constant $\\Lambda$ in eq.~(\\ref{eq:pathintegral2}) is not a constant but fluctuates independently in time around $\\Lambda$, \nfollowing the Gaussian distributions with a standard deviation $\\sigma = 2\\sqrt{g_s}$:  \n\\[\nA(L_1,L_2;T) \n= \\int \\mathcal{D}\\nu (\\tau)\\ e^{- \\frac{1}{4g_s} \\int^{T}_{0} d\\tau\\ \\nu^2(\\tau)} \\mathcal{G}_{\\Lambda+\\nu} (L_1,L_2;T)\\ ,\n\\label{eq:fluctuations}\n\\]\nwhere \n\\[\n\\mathcal{G}_{\\Lambda+\\nu} (L_1,L_2;T) \n:= \\int^{L(T)=L_2}_{L(0)=L_1} \n \\mathcal{D}L(\\tau)\\ \n \\exp \\left[ \n - \\int^T_0 d\\tau \n \\left(\n \\frac{\\dot{L}^2(\\tau)}{4L(\\tau)} + (\\Lambda + \\nu (\\tau)) L(\\tau)\n \\right)\n \\right]\\ .\n \\label{eq:pathintegralnu}\n\\]\nTherefore, all the contributions coming from the sum over all wormholes and baby universes \ncan be fully taken in if the cosmological ``constant'' in (the continuum limit of) $2$d CDT or projectable HL quantum gravity where no wormholes and baby universes exist \nis not really a constant but fluctuates in time. This would lead to a realization of Coleman's mechanism that will be discussed in Sec.~\\ref{sec:coleman}.        \n\nIn the next section, we will show that the full propagator can be also obtained if quantizing $2$d projectable HL gravity with an effective wormhole interaction term.  \n\n\n\n\n\n\\subsection{Wormhole interaction in $2$d projectable HL gravity}\n\\label{sec:wormholeinteraction}\n\nLet us consider the $2$d projectable HL gravity with a space-like wormhole interaction given by \nthe following action: \n\\[\nI_{\\text{w}}  &=\n\\frac{1}{\\kappa} \n\\int dtdx\\ \nN(t) \\sqrt{h(t,x)} \n\\left(\n(1-\\eta) K^2(t,x) - 2 \\widetilde{\\Lambda}\n\\right)\\notag \\\\\n&\\ \\ \\ + \\beta \\int dt N(t) \\int dx_1 dx_2 \\sqrt{h(t,x_1)} \\sqrt{h(t,x_2)}\n\\ , \n\\label{eq:wormholeaction}\n\\]\nwhere $\\beta$ is a dimension-full coupling constant. \nThe last bi-local term can be interpreted as an effective interaction term for a space-like wormhole connecting two distant regions at a given $t$. \nThis bi-local term is allowed to be included since it is invariant under FPD.  \n\n\n\nFollowing essentially the same procedure explained in Sec.~\\ref{sec:quantization}, \nlet us quantize the system defined by the action (\\ref{eq:wormholeaction}). \nIntroducing the conjugate momentum of the density $\\sqrt{h}$ as $\\Pi$, we introduce the Poisson bracket (\\ref{eq:poisson}). \nImplementing the Legendre transform, one obtains the corresponding Hamiltonian:   \n\\[\nH_{\\text{w}} \n= N(t)\\ \\mathcal{C}_{\\text{w}} (t) \n+ \\int dx N_1 (t,x)\\ \\mathcal{C}^1_{\\text{w}} (t,x)\\ , \n\\label{eq:hw}\n\\]\nwhere \n\\[\n\\mathcal{C}^1_{\\text{w}}  (t,x)  \n&= - \\frac{\\partial_1 \\Pi (t,x)}{\\sqrt{h (t,x)}} \\approx 0\\ , \\label{eq:momentumconst2}\\\\\n\\mathcal{C}_{\\text{w}}  (t)\n&= \\int dx\\ \n\\biggl(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2(t,x) \\sqrt{h(t,x)} + \\frac{2}{\\kappa} \\widetilde{\\Lambda} \\sqrt{h(t,x)} \\notag \\\\\n& \\ \\ \\ \n- \\beta \\sqrt{h(t,x)} \n\\int dx_2 \\sqrt{h(t,x_2)}\n\\biggl)\n\\approx 0 \n\\ . \\label{eq:classicalhamiltonianconst2} \n\\] \nThe constraints (\\ref{eq:momentumconst2}) and (\\ref{eq:classicalhamiltonianconst2}) are the momentum constraint and the Hamiltonian constraint, respectively. \nSolving the momentum constraint (\\ref{eq:momentumconst2}) at the classical level as before, \nthe Hamiltonian (\\ref{eq:hw}) reduces to the following one-dimensional one:\n\\[\nH_{\\text{w}}\n= N(t) \\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2 (t) L(t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} L(t) \n- \\beta L^2 (t)\n\\right)\\ , \n\\label{eq:hw2}\n\\] \nwhere $L(t) := \\int dx \\sqrt{h(t,x)}$. The Hamiltonian (\\ref{eq:hw2}) is subject to the Hamiltonian constraint: \n\\[\nL(t)\n\\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2 (t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} \n- \\beta L (t) \n\\right)\n\\approx 0 \n\\ .  \n\\label{eq:classicalhamiltonianconst3}\n\\]  \nHere we choose the CDT parametrization (\\ref{eq:parameteridentification}). \nA solution to the Hamiltonian constraint (\\ref{eq:classicalhamiltonianconst3}) is \n\\[\n\\Pi^2 = - \\Lambda + \\beta L \\ge 0\\ , \\ \\ \\ \\text{for}\\ \\ \\ \\sqrt{\\Lambda} L \\ge 1\/\\xi\\ , \n\\label{eq:solutiontohamiltonianconstw1}\n\\]  \nwhere $\\xi$ is a dimensionless parameter given by $\\xi = \\beta\/ \\Lambda^{3\/2}$. \nFor $\\sqrt{\\Lambda} L < 1\/\\xi$, the only allowed solution is $L=0$.  \n\n\nWhen quantizing the system, if we follow the same procedure described in Sec.~\\ref{sec:quantization}, and set \n$\\beta = g_s$, one can reproduce the path-integral of the full propagator (\\ref{eq:pathintegralfullpropagator}). \nRemember the boundary condition for the path-integral (\\ref{eq:pathintegralfullpropagator}), i.e. \nthe kinetic term should counteract the unboundedness of the potential term at $L=\\infty$. \nThis balance between the kinetic and potential terms is precisely what is reflected in the classical Hamiltonian constraint (\\ref{eq:solutiontohamiltonianconstw1}).   \n \n\n\n\n\n\\subsection{Coleman's mechanism} \n\\label{sec:coleman}\n\nIn this section, we discuss a sort of Coleman's mechanism in the context of two-dimensional gravity based on CDT briefly.  \n\nLet us define the two kinds of Wheeler-deWitt equation: \n\\[\n\\hat{H} W_0 (L) = 0\\ , \\ \\ \\ \\hat{H}_{\\text{eff}} W(L) = 0\\ , \n\\label{eq:wdw}\n\\]\nwhere $\\hat{H} := \\hat{H}^{(-1)}$ introduced in eq.~(\\ref{eq:hamiltonians}). \nThe solutions to the Wheeler-deWitt equations are the Hartle-Hawking wave functions given by  \n\\[\nW_0 (L) = e^{-\\sqrt{\\Lambda} L}\\ , \\ \\ \\ \nW(L) = \\frac{\\text{Bi} (\\xi^{-2\/3} - \\xi^{1\/3} \\sqrt{\\Lambda} L)}{ \\text{Bi} (\\xi^{-2\/3}) } \n+ c\\ \\text{Ai} ( \\xi^{-2\/3} - \\xi^{1\/3} \\sqrt{\\Lambda} L)\\ , \n\\label{eq:hartlehawking}\n\\]    \nwhere $\\text{Ai}$ and $\\text{Bi}$ are the standard Airy functions, \n$\\xi$ is a dimensionless string coupling constant measured by the cosmological constant, i.e. \n$\\xi := g_s\/ \\Lambda^{3\/2}$, \nand $c$ is an undetermined dimensionless constant. \nThe Hartle-Hawking wave function $W_0 (L)$ is the one for (the continuum theory of) $2$d CDT, \ni.e. neither baby universe nor wormhole contributions are included. \nOn the other hand, $W(L)$ is the Hartle-Hawking wave function including all possible contributions of baby universes and wormholes non-perturbatively.      \n\nWe wish to explore the behavior of the non-perturbative Hartle-Hawking wave function $W(L)$ (see Fig.~\\ref{fig:HH}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.4]{HH.pdf}\n\\caption{A plot of the Hartle-Hawking wave functions, $W_0$ (dashed line) and $W$ (solid line), for $\\xi=1\/3$ and $c=0$:  \nThe horizontal axis is $\\sqrt{\\Lambda} L$ and the vertical axis is either $W_0$ or $W$.  \n}\n\\label{fig:HH}       \n\\end{figure} \nFor $\\sqrt{\\Lambda}L \\ll 1\/\\xi$, one obtains the asymptotic expansion: \n\\[\nW(L) \\sim e^{-\\sqrt{\\Lambda}L} = W_0(L)\\ . \n\\label{eq:asymptoticexpansion}\n\\] \nTherefore, when the size of the one-dimensional universe is small enough, \nthe physics is very closed to the one without baby universes and wormholes, and it is essentially governed by the cosmological constant.  \nThe wave function in this region decreases exponentially, and this behavior does not change at any finite order of perturbation.   \n \nHowever, once the size of the universe is large enough, i.e. $\\sqrt{\\Lambda} > 1\/\\xi$, the wave function starts oscillating, and the behavior is governed by \nthe string coupling constant instead of the cosmological constant:\n\\[\nW(L) \\sim 1\/(g^{1\/3}_s L)^{1\/4}\\ . \n\\label{eq:largebehavior}\n\\] \nThis drastic change happens due to the infinitely many wormholes and baby universes. \nThe similar behavior has been observed in the context of non-critical string theory \\cite{Moore:1991sf, Betzios:2020nry}. \n\nFrom the discussion above, we observe that a sort of Coleman's mechanism works:  \nFor a large universe, the cosmological constant is not important enough to govern the physics\\footnote{\nThe Hartle-Hawking wave function $W(L)$ is not normalizable, but similar arguments are valid on the normalizable energy eigenstates \\cite{Ambjorn:2021wdm}. \n}.    \n\n\n\\section{Summary}\n\\label{sec:summary} \n\nWe have reviewed the relation between the two-dimensional causal dynamical triangulations ($2$d CDT) and the two-dimensional projectable Ho\\v rava-Lifshitz quantum gravity ($2$d projectable HL QG). \n\nIn the first part, it has been shown that the physics described by the continuum limit of $2$d CDT coincides with the one obtained quantizing $2$d projectable HL gravity. \nThis is confirmed because the quantum Hamiltonians of both models are exactly the same. \nThe system is expressed in terms of quantum mechanics of a $1$d extended object, i.e. a $1$d universe.   \nIt would be too hasty to consider that this scenario also holds for the higher dimensional cases. \nIn fact, it has been shown that in $2+1$ dimensions numerical studies of the so-called locally causal dynamical triangulations (LCDT) that relaxes the proper-time foliation of CDT and requires the local causality reproduce an intriguing specialty of CDT, \nan emergence of the de-Sitter-like geometry \\cite{Jordan:2013awa} (see e.g. Ref.~\\cite{Loll:2019rdj} for the higher-dimensional CDT). \n\nIn the second part, we have introduced the generalized CDT (GCDT) that permits baby universes and wormholes to form in keeping with the foliation, \nand in particular the construction based on the string field theory for CDT has been explained. \nHere the string means the $1$d universe, and the string field theory is constructed in such a way that the free part reproduces the CDT amplitudes, \nand the splitting and joining interactions of string are introduced to create baby universes and wormholes. \n\n\nFocusing on the loop-to-loop amplitude, we have introduced an effective $1$d theory that includes all the contributions coming from the sum over all possible baby universes and wormholes. \nFrom the point of view of HL gravity, the effective theory can be precisely reproduced if introducing a bi-local interaction term into the action of $2$d projectable HL gravity and if quantizing the system. \nIn addition, the effective theory can be also obtained considering that the cosmological constant of $2$d CDT is not a constant but it fluctuates in time. \nThis leads to Coleman's mechanism in $2$d CDT such that for a large universe the cosmological constant is not important enough to govern the physics.       \n \n \nAlthough we have not discussed issues of the coupling to matter, $2$d CDT coupled to Yang-Mills theory has been solved analytically in Ref.~\\cite{Ambjorn:2013rma}, \nand it has been shown that the quantum Hamiltonian obtained in Ref.~\\cite{Ambjorn:2013rma} can be reproduced quantizing $2$d projectable HL gravity coupled to Yang-Mills theory \\cite{Ipsen:2015ckl}. \nIn fact, we know very little about the analytical treatment of the coupling to matter compared to the situation of the $2$d dynamical triangulations and the Liouville quantum gravity. \nThis direction need to be explored in the future.  \n         \n\nWhat is remarkable is that following the standard Wilsonian renormalization group, one can take the continuum limit of the lattice model, $2$d CDT, \nand find the continuum quantum field theory, $2$d projectable HL QG, which is in the same universality class of $2$d CDT. \nA missing piece is the continuum quantum field theory of GCDT that is described by metric components and allows us to compute all the amplitudes defined by the string field theory for CDT, \nalthough we have the effective field theory, the $2$d projectable HL gravity with a bi-local interaction, that reproduces the restricted class of GCDT amplitudes once it is quantized. \nWe wish to unveil the underlying continuum quantum field theory of GCDT, through which we can understand something inherently interesting about quantum geometries for sure.    \n\n\n      \n\n\n\\begin{acknowledgement}\nYS would like to thank \nJan Ambj\\o{}rn, \nLisa Glaser, \nYuki Hiraga, \nYoshiyasu Ito \nand \nYoshiyuki Watabiki, for wonderful collaborations on the topics related to this chapter. \nGreat ideas appeared in this chapter attribute to them. However, if conceptual mistakes exist, YS is responsible for them.    \n\\end{acknowledgement}\n\n\n\n\\input{references}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{}\n\\subsection{}\n\n\\begin{theorem}[Optional addition to theorem head]\n\\end{theorem}\n\n\\begin{proof}[Optional replacement proof heading]\n\\end{proof}\n\n\\begin{figure}\n\\includegraphics{filename}\n\\caption{text of caption}\n\\label{}\n\\end{figure}\n\n\n\\begin{equation}\n\\end{equation}\n\n\\begin{equation*}\n\\end{equation*}\n\n\\begin{align}\n  &  \\\\\n  &\n\\end{align}\n\n\\section{Introduction}\nThe motivation behind this note is the following question: what are the measures $\\mu$ on the Heisenberg group ${\\mathbb{H}}^n$ which guarantee that the (correct notion of) Riesz transform is bounded from $L^2(\\mu)$ to itself? This question (or some variant of it) with $\\mathbb{R}^n$ instead of ${\\mathbb{H}}^n$, was one of the major starting point of the theory that came to be known as \\textit{quantitative rectifiability}. This area of geometric measure theory has seen an impressive development in the past thirty years, starting with the landmark works of Peter Jones \\cite{jones} and David and Semmes \\cite{david-semmes91}, \\cite{david-semmes93}, through the solution of fundamental questions in complex analysis, such as the Painlev\\'{e} problem (see \\cite{mmv}, \\cite{david98}, \\cite{tolsa03}), to more recent applications to harmonic analysis, see for example \\cite{ntv} and \\cite{ahmmt}.\n\nIn the last years, there has been an increasing interest in developing such a quantitative theory in different contexts than that of Euclidean spaces; examples of these are parabolic spaces and Heisenberg groups, or, more generally, Carnot groups. The former appear in the study of caloric measure. The latter arise naturally in the study of certain hypoelliptic operators, in the sense that the natural translations and dilations for these operators are those characterising the spaces; the Heisenberg group is the most important prototypical example, and the related operator is the so-called Kohn Laplacian; see \\cite{blu} for a comprehensive study of stratified Lie groups and the corresponding operators. \n\nWe should mention that the study of Heisenberg geometry can be approached from different perspectives and with different applications in mind; for example, see \\cite{ny} for a connection with theoretical computer science. \n\nTo be a little more specific: the starting motivation to develop a theory of quantitative rectifiability connected to our initial question, is to understand basic issues such as the removable sets for harmonic function (with respect to the relevant sub-Laplacian), or to give a characterisation of those domains where the Dirichlet problem (again, for the relevant sub-Laplacian) is well-posed. We want to underline, however, that a theory of quantitative rectifiability in the Heisenberg setting has its own, purely geometric, intrinsic appeal.\n\nIn the last couple of years, there has been some progress towards an answer to our initial question; see for example \\cite{cfo}, \\cite{fo18} and \\cite{o18}. In this note we give a necessary condition to be imposed on a Radon measure $\\mu$ on ${\\mathbb{H}}^n$ for the Riesz transform to be $L^2(\\mu)$ bounded. Here $R_\\mu$ is the singular integral operator whose kernel is the horizontal gradient of the fundamental solution of the Heisenberg sub-Laplacian, as defined in \\cite{chousionis2012singular}. See Section \\ref{preliminaries} for precise definitions.  \n\n\\begin{theorem} \\label{t:main}\nLet $\\mu$ be a Radon measure on ${\\mathbb{H}}^n$ such that $R_\\mu$ is bounded on $L^2(\\mu)$ with norm $C_1$, and such that $\\mu(F) = 0$ whenever $\\dim_H(F) \\leq 2$. Then there exists a constant $C_2$ such that for all balls $B(x,r) \\subset {\\mathbb{H}}^n$, we have\n\\begin{align}\\label{e:main}\n    \\mu(B(x,r)) \\leq C_2 r^{2n +1}.\n\\end{align}\nHere $C_2$ depends only on $n$ and $C_1$, and the ball $B(x,r)$ is defined with respect to the Kor\\'{a}nyi metric, see Section \\ref{preliminaries}.\n\\end{theorem}\n\nA corresponding statement holds in the Euclidean setting, and is a result of David, \\cite{david-wavelets}, Part III, Proposition 1.4. See  \\cite{orponen-notes}, Proposition 6.9 for a more detailed proof. Let $\\mathcal{R}^{d}_\\mu$ denote the standard $d$-dimensional Riesz transform in $\\mathbb{R}^n$.\n\\begin{theorem}\nAssume that $\\mu$ is a non-atomic Radon measure on $\\mathbb{R}^n$ such that $\\mathcal{R}^{d}_\\mu$ is bounded on $L^2(\\mu)$ with norm $C_1$. Then, for all Euclidean balls $B_{\\mathbb{R}^n}(x,r)\\subset\\mathbb{R}^n$ we have\n\\begin{align}\\label{e:growth}\n\\mu (B_{\\mathbb{R}^n}(x,r)) \\leq C_2 r^d    \n\\end{align}\nHere $C_2$ depends only on $C_1$, $n$, and $d$. \n\\end{theorem}\nA measure satisfying \\eqref{e:growth} (or \\eqref{e:main}) is said to have \\textit{polynomial growth}. \nLet us give a couple of remarks. \n\\begin{remark}\nAlthough the result itself (both in the Euclidean and Heisenberg case) is neither hard nor deep, it is nevertheless very useful. For example, most tools developed in the last two decades that take quantitative rectifiability beyond Ahlfors regular measures still need polynomial growth\\footnote{With some exceptions, see for example \\cite{azzam-schul18}, or \\cite{badger-schul15}.} (see for example the book by Tolsa \\cite{tolsa-book}). Thus, we expect that our result will be quite useful, too. \n\\end{remark}\n\\begin{remark}\nWhile the two results above look similar, there is actually a difference, in the sense that, in the Heisenberg case, there actually exist lower dimensional measures which give a bounded Riesz transform, but are not atomic.\n\nThis is \\textit{not} a byproduct of the proof, but rather a fact of the Heisenberg geometry. Indeed, the $2$-dimensional $t$-axis (or any Heisenberg translate of it) gives a bounded $(2n+1)$-dimensional Riesz transform; this is simply because on these sets the kernel vanishes identically, see \\eqref{e:K-norm}.\n\nOne can construct a more interesting example in the vertical plane of the one dimensional Heisenberg group ${\\mathbb{H}}$, say. Consider a tube of height $1$ and radius $\\varepsilon_1^2$ around the $t$-axis, and take the intersection with the vertical plane. Call the resulting rectangle $R_{1,1}$. Cut out from $R_1$ two smaller rectangles $R_{2,1}$ and $R_{2,2}$, one in the top right corner and one in the bottom left corner, both of height $\\varepsilon_2$ and width $\\varepsilon_2^2$, for some $\\varepsilon_2\\le \\varepsilon_1\/4$. We proceed in this way, so that after $k$ steps we have $2^{k-1}$ disjoint rectangles $\\{R_{k,i}\\}_i$ of height $\\varepsilon_k$ and width $\\varepsilon_k^2$. Consider the natural probability measure $\\mu$ on the Cantor-like set $C=\\bigcap_k\\bigcup_i R_{k,i}$. It is not difficult to show that, if $\\varepsilon_k\\to 0$ are small enough, the Heisenberg Riesz transform is bounded on $L^2(\\mu)$; the idea is that the set is concentrated along the $t$-axis, and thus the kernel is very small (see \\eqref{e:K-norm} below). Depending on the choice of $(\\varepsilon_k)$ we have $\\dim_H(C)\\in [0,2].$\n\\end{remark}\n\n\\subsection*{Organisation of the article} In Section \\ref{preliminaries} we briefly recall basic facts about Heisenberg groups and the Riesz transform. We also introduce a family of ``dyadic cubes'' suitable to our setting. \n\nSection \\ref{s:main lemma} is dedicated to Lemma \\ref{l:main lemma}, our main technical lemma. Roughly speaking, we show that if a measure $\\mu$ is such that $R_{\\mu}$ is bounded on $L^2(\\mu)$, and there is some cube $Q_0$ with a very high concentration of $\\mu$ (i.e. $\\mu(Q_0)\\gg \\ell(Q_0)^{2n+1}$), then we can find a family ${\\mathsf{HD}}(Q_0)$ of much smaller cubes, contained in $Q_0$, such that\n\\begin{enumerate}\n    \\item[a)] a very large portion of measure  $\\mu$ on $Q_0$ is concentrated on the cubes from ${\\mathsf{HD}}(Q_0)$,\n    \\item[b)] the family ${\\mathsf{HD}}(Q_0)$ is relatively small, in the sense that it consists of few cubes.\n\\end{enumerate}\n\nIn Section \\ref{s:iteration} we show that if the polynomial growth condition \\eqref{e:main} is not satisfied, then we can find a cube satisfying the assumptions of our main lemma. This in turn allows us to start an iteration algorithm, consisting of using the main lemma countably many times, that results in constructing a set $Z$  with $\\mu(Z)>0$ and $\\dim_H(Z)\\le 2.$ This finishes the proof of Theorem \\ref{t:main}.\n\n\n\\subsection*{Acknowledgements}\nWe thank Katrin F\\\"{a}ssler and Tuomas Orponen for introducing us to the Heisenberg group and for suggesting the problem. \n\nThe bulk of this work was done while the two authors were attending the Simons Semester in Geometric Analysis at IMPAN in the Autumn 2019. We thank Tomasz Adamowicz and the staff at the institute for their hospitality.\n\n\\section{Preliminaries}\\label{preliminaries}\nIn our estimates we will often use the notation $f\\lesssim g$ which means that there exists some absolute constant $C$ for which $f\\le Cg.$ If the constant $C$ depends on some parameter $t$, we will write $f\\lesssim_t g.$ Notation $f\\approx g$ will stand for $f\\lesssim g\\lesssim f,$ and $f\\approx_t g$ is defined analogously. For simplicity, in our estimates we will suppress the dependence on dimension $n$ and on absolute constants $\\lambda,\\ \\Lambda$ (see \\eqref{e:balls in cubes}).\n\n\\subsection{Heisenberg group}\nIn this paper we consider the $n$-th Heisenberg group with exponential coordinates (see \\cite{cdpt} or \\cite{faessler-notes} for a swift introduction to the Heisenberg group in a context close to ours). In practice, we will denote a point $p \\in {\\mathbb{H}}^n$ as $(z,t) \\in \\mathbb{R}^{2n} \\times \\mathbb{R}$, and $z=(x_1,...,x_n,y_1,...,y_n)$. In these coordinates the group law in ${\\mathbb{H}}^n$ takes the form \n\\begin{align*}\n    p \\cdot q = \\ps{z + z', t+t' + \\frac{1}{2} \\sum_{i=1}^n (x_i y_i' + y_i x_i')},\n\\end{align*}\nwhere $p=(z,t)$ and $q=(z',t')$. The identity element is the origin $(0,0)$ and the inverse is given by $p^{-1}=(-z,-t)$. We make ${\\mathbb{H}}^n$ into a metric space by setting $ d(p,q) := \\|q^{-1}\\cdot p\\|_{\\mathbb{H}}$, where \n\\begin{align}\\label{e:korany}\n    \\|p\\|_{\\mathbb{H}}^4 := |z|^4 + 16t^2,\n\\end{align}\nand $|z|$ denotes the Euclidean norm of $z\\in\\mathbb{R}^{2n}$.\n\nNote that $\\|\\cdot\\|_{\\mathbb{H}}$ is $1$-homogeneous with respect to the anisotropic dilation $p \\mapsto \\lambda p = (\\lambda z, \\lambda^2 t)$, $\\lambda >0$. The metric $d$ is sometimes called the Kor\\'{a}nyi metric.\n\nGiven $p\\in{\\mathbb{H}}^n$ and $r>0$ we set\n\\begin{equation*}\n    B(p,r)=\\ck{q\\ |\\ d(p,q)\\le r},\\quad U(p,r) = \\ck{q\\ |\\ d(p,q)< r}.\n\\end{equation*}\nFor $\\alpha>0$ we will write $\\mathcal{H}^{\\alpha}$ to denote the usual $\\alpha$-dimensional Hausdorff measure with respect to metric $d$. For $A\\subset{\\mathbb{H}}^n$ we set $\\dim_H(A)$ to be the Hausdorff dimension of $A$. \n\nIt follows easily from the definition of the Kor\\'{a}nyi metric that for all $p\\in{\\mathbb{H}}^n$ and $r>0$ we have\n\\begin{equation}\\label{e:measure of balls}\n    \\mathcal{H}^{2n+2}(B(p,r)) = \\mathcal{H}^{2n+2}(B(0,1))\\, r^{2n+2}.\n\\end{equation}\nThus, even though the topological dimension of ${\\mathbb{H}}^n$ is $2n+1$, the Hausdorff dimension of ${\\mathbb{H}}^n$ is equal to $2n+2$. For the sake of brevity we set $D:=2n+2.$ Usually one denotes the Hausdorff dimension of ${\\mathbb{H}}^n$ by $Q$, but we have decided to save that letter for cubes; hence the non-standard notation.\n\nIt is also easy to check that if $\\mathcal{L}^{2n+1}$ denotes the usual Lebesgue measure on $\\mathbb{R}^{2n+1}\\simeq {\\mathbb{H}}^n,$ then we have a constant $C>0$ such that\n\\begin{equation}\\label{e:lebesgue and hausdorff}\n    \\mathcal{L}^{2n+1}=C\\mathcal{H}^{D}.\n\\end{equation}\n\n\\subsection{Heisenberg Riesz transform}\nRecall that, for a function $u : {\\mathbb{H}}^n \\to \\mathbb{R}$, the horizontal gradient of $u$ is given by\n\\begin{align*}\n    \\nabla_{\\mathbb{H}} u := \\ps{ X_1 u,...,X_n u, Y_1 u, ..., Y_n u},\n\\end{align*}\nwhere the vector fields $X_1,\\dots,X_n,Y_1,\\dots, Y_n$ and $\\frac{\\partial}{\\partial t}$ represent the left invariant translates of the canonical basis at the identity. In particular, $X_1,\\dots,X_n,Y_1,\\dots, Y_n$ span the horizontal distribution in ${\\mathbb{H}}^n$. \n\nThe Heisenberg sublaplacian $\\Delta_{\\mathbb{H}}$ is given by $\\sum_{i=1}^n X_i^2 + Y_i^2$, and its fundamental solution is \n\\begin{align*}\n    G(p) := c_n \\|p\\|_{\\mathbb{H}}^{2-D}.\n\\end{align*}\n\nThe $(D-1)$-dimensional Riesz kernel in ${\\mathbb{H}}^n$, first considered in \\cite{chousionis2012singular}, is given by\n    $K(p)=\\nabla_{{\\mathbb{H}}} G(p).$\nThe Riesz transform is formally defined as\n\\begin{align*}\n    R_\\mu f (p) = \\int_{\\mathbb{H}^n} K(q^{-1}\\cdot p)f(q) \\, d\\mu(q).\n\\end{align*}\nSince it is not clear whether the integral above converges, one considers the truncated Riesz transform given by the formula\n\\begin{align*}\n    R_{\\mu, \\delta}f(p)= \\int_{{\\mathbb{H}}^n \\setminus B(p,\\delta)} K(q^{-1}\\cdot p)f(q)\\ d\\mu(q),\n\\end{align*}\nfor $\\delta>0$. We say that $R_{\\mu}$ is bounded on $L^2(\\mu)$ if the truncated operators $R_{\\mu, \\delta}$ are bounded on $L^2(\\mu)$ uniformly in $\\delta>0.$\n\nOne can easily check that the Riesz kernel is actually equal to\n\\begin{align*}\n    & K(z, t) \\\\\n    & = n\\, \\ps{ \\frac{ -2x_1 |z|^2 + 8y_1 t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4\n    }}, \\,\\cdots ,    \\frac{ -2x_n |z|^2 + 8y_n t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4\n    }}, \\, \\frac{ -2y_1 |z|^2 - 8x_1 t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4}}, \\cdots , \\frac{ -2y_n |z|^2 - 8x_n t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4}}}.\n\\end{align*}\nHence,\n\\begin{align} \\label{e:K-norm}\n    |K(z,t)|^2 = n^2 \\, \\frac{ 4 |z|^2 }{(|z|^4 + 16t^2)^{n+1}}.\n\\end{align}\nThis implies the curious fact that \n    $|K(z,t)| \\leq C$\nwhenever\n\\begin{align} \\label{e:paraboloid}\n    |z| \\leq  \\, 16 |t|^{n+1},\n\\end{align}\nwhich is a `paraboloidal' double cone around $t$-axis with vertex at the origin. This fact will play a key role in the subsequent analysis. \n\nChousionis and Mattila showed in \\cite[Proposition 3.11]{chousionis2012singular} that the Riesz kernel is a standard kernel. In particular, it satisfies the following continuity property: whenever  $q_1, q_2\\neq p\\in {\\mathbb{H}}^n$, we have\n\\begin{equation*}\\label{e:kernel continuous}\n    |K(p^{-1}\\cdot q_1) - K(p^{-1}\\cdot q_2)| \\lesssim \\max\\bigg\\{\\frac{d(q_1,q_2)}{d(p,q_1)^{D}},\\frac{d(q_1,q_2)}{d(p,q_2)^{D}} \\bigg\\}.\n\\end{equation*}\nTaking $p=0$ and $q_1 = \\tilde{q_1}^{-1}\\cdot \\tilde{p},\\ q_2 = \\tilde{q_2}^{-1}\\cdot \\tilde{p},$ one gets immediately that for all  $\\tilde{q_1}, \\tilde{q_2}\\neq \\tilde{p}\\in {\\mathbb{H}}^n$\n\\begin{equation}\\label{e:kernel continuous 2}\n    |K(\\tilde{q_1}^{-1}\\cdot \\tilde{p}) - K(\\tilde{q_2}^{-1}\\cdot \\tilde{p})| \\lesssim \\max\\bigg\\{\\frac{d(\\tilde{q_1},\\tilde{q_2})}{d(\\tilde{p},\\tilde{q_1})^{D}},\\frac{d(\\tilde{q_1},\\tilde{q_2})}{d(\\tilde{p},\\tilde{q_2})^{D}} \\bigg\\}.\n\\end{equation}\n\n\\subsection{Dyadic cubes}\nWe are going to use a family of decompositions of ${\\mathbb{H}}^n$ into subsets that share many properties with the standard dyadic cubes from $\\mathbb{R}^n$. The most classical constructions of this kind are due to Chirst \\cite{christ1990b} and David \\cite{David1988}, but for us it will be more convenient to use the ``cubes'' constructed in \\cite{kaenmaki2012}.\n\nFirst, note that given any ball $B(p,2r)$, one may use the $5r$-covering lemma and the property \\eqref{e:measure of balls} to conclude that there exists some absolute constant $m$ such that $B(p,2r)$ may be covered by $m$ balls $B(p_i,r)$, where $\\ck{p_i}_{i=1}^m$ are points in $B(p,2r)$. That is, ${\\mathbb{H}}^n$ is geometrically doubling. In particular, we can use {\\cite[Theorem 2.1, Remark 2.2]{kaenmaki2012}}.\n\n\\begin{lemma}[\\cite{{kaenmaki2012}}] \\label{l:cubes}\n\tFor all $k\\in\\mathbb{Z}$ there exists a family of subsets of ${\\mathbb{H}}^n$, denoted by $\\mathcal{D}_k$, such that\n\t\\begin{enumerate}\n\t\t\\item ${\\mathbb{H}}^n = \\bigcup_{Q\\in\\mathcal{D}_k}Q$,\n\t\t\\item if $k\\ge l$, and $Q\\in \\mathcal{D}_k,\\ P\\in\\mathcal{D}_l$, then either $Q\\cap P=\\varnothing$ or $Q\\subset P$,\n\t\t\\item for every $Q\\in \\mathcal{D}_k$ there exists $p_Q\\in Q$ such that\n\t\t\\begin{equation}\\label{e:balls in cubes}\n\t\tU(p_Q,\\lambda2^{-k})\\subset Q\\subset B(p_Q,\\Lambda2^{-k})\n\t\t\\end{equation}\n\t\tfor some absolute constants $\\lambda,\\Lambda>0$.\n\t\\end{enumerate}\n\\end{lemma}\n\nLet us stress once more that we will not keep track of how various parameters appearing in the proof depend on $\\lambda$ and $\\Lambda$.\n\nWe set $\\mathcal{D} = \\bigcup_k \\mathcal{D}_k$. For $Q\\in\\mathcal{D}_k$ we define the sidelength of $Q$ as $\\ell(Q)=2^{-k}$. Clearly, by \\eqref{e:measure of balls} and \\eqref{e:balls in cubes}, for $Q\\in\\mathcal{D}$ we have\n\\begin{equation*}\n    \\mathcal{H}^{{D}}(Q) \\approx \\ell(Q)^{{D}}.\n\\end{equation*}\nIt follows that if $Q\\in\\mathcal{D}$, then for $k\\ge 0$\n\\begin{equation}\\label{e:dyadic property}\n\\#\\ck{P\\in\\mathcal{D}\\ |\\ P\\subset Q,\\ \\ell(P)=2^{-k}\\ell(Q) }\\approx 2^{kD}.\n\\end{equation}\n\nGiven a Radon measure $\\mu$ and $Q\\in\\mathcal{D}$ we will denote the $(D-1)$-dimensional density of $\\mu$ in $Q$ by\n\\begin{equation*}\n    \\Theta_{\\mu}(Q) = \\frac{\\mu(Q)}{\\ell(Q)^{D-1}}.\n\\end{equation*}\nFor simplicity, we will suppress the dependence on $\\mu$ and simply write $\\Theta(Q)$.\n\n\\section{Main lemma}\\label{s:main lemma}\n   Our main tool in the proof of Theorem \\ref{t:main} is the following lemma.\n\t\\begin{lemma}\\label{l:main lemma}\n\t\tLet $\\mu$ be a Radon measure on ${\\mathbb{H}}^n$ such that $R_{\\mu}$ is bounded on $L^2(\\mu)$ with norm $C_1$. There exist constants $A=A(n)>1,\\ s=s(A,n)\\in (0,1\/2)$ and $M=M(C_1,n)>100$ such that the following holds.\n\t\t\n\t\tSuppose that $Q_0\\in\\mathcal{D}$ satisfies $\\Theta(Q_0)\\ge M$. Set $N = \\floor{A^{-2} \\log(\\Theta(Q_0))}$. Then, the family of high density cubes\n\t\t\\begin{equation*}\n\t\t{\\mathsf{HD}}(Q_0) = \\ck{Q\\in \\mathcal{D}\\ |\\ Q\\subset Q_0,\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)>2\\,\\Theta(Q_0)}\n\t\t\\end{equation*}\n\t\tsatisfies\n\t\t\\begin{equation}\\label{e:HD are thicc}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) \\ge (1-\\Theta(Q_0)^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\tMoreover, we have\n\t\t\\begin{equation}\\label{e:HD packing}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\ell(Q)^2\\le C_p\\,  \\ell(Q_0)^2\n\t\t\\end{equation}\n\t\tfor some dimensional constant $C_p$ (``$p$'' stands for ``packing'').\n\t\\end{lemma}\n\t\n\tThe rest of this section is dedicated to proving the lemma above. For brevity of notation, we set $\\Theta_0=\\Theta(Q_0)$. Observe that the integer $N$ was chosen in such a way that\n\t\\begin{equation}\\label{e:choice of N}\n\t2^{A^2N}\\approx \\Theta_0\\ge M.\n\t\\end{equation}\n\tIn particular, we have $N\\ge N_0$ for some very big $N_0$ depending on $M$ and $A$.\n\t%\n\n\n\n\n\t\n\tWe split the proof of Lemma \\ref{l:main lemma} into several steps.\n\t\n\tFirst, note that by the pigeonhole principle and \\eqref{e:dyadic property}, we can find a cube $Q_1 \\in \\mathcal{D}$ with sidelength $\\ell(Q_1) = 2^{-AN} \\ell(Q_0)$ such that \n\t\\begin{align}\\label{e:assumption Q1}\n\t\\mu(Q_1) \\gtrsim \\frac{\\mu(Q_0)}{2^{AND}}.    \n\t\\end{align}\n\tWithout loss of generality, by applying the appropriate translation, we can assume that $Q_1$ is centred at the origin, i.e. $p_{Q_1}=0$.\n\tSet\n\t\\begin{align*}\n\tT:=\\ck{ (z,t) \\in Q_0 \\, |\\, |z| \\leq 2^{-N} \\ell(Q_0) }\n\t\\end{align*}\n\tand for any $\\kappa>0$ set \n\t\\begin{equation*}\n\t{ T_{\\kappa}}:=\\ck{ (z,t) \\in Q_0 \\, |\\, |z| \\leq \\kappa\\, 2^{-N} \\ell(Q_0) }.\n\t\\end{equation*}\n\tObserve that $Q_1\\subset T.$ In a sense, $T$ can be seen as a tube with vertical axis passing through $p_{Q_1}=0$. Note also that for any cube $Q\\subset Q_0 \\setminus T$ we have $\\dist(Q,Q_1)\\gtrsim 2^{-N}\\ell(Q_0).$\n\t\n\tWe start by proving a few preliminary results.\n\t\\begin{lemma}\\label{c:number-cubes}\n\tThere are at most $C(\\kappa)\\, 2^{2N}$ cubes of sidelength $2^{-N}\\ell(Q_0)$ contained in $T_\\kappa$.\n\t\\end{lemma}\n\t\\begin{proof}\n\tObserve that since $0\\in Q_0$, and by \\eqref{e:balls in cubes} $Q_0\\subset B(p_{Q_0},\\Lambda\\ell(Q_0))$, we have $Q_0\\subset B(0,2\\Lambda\\ell(Q_0)).$ Hence,\n\t\\begin{align*}\n\t    T_{\\kappa} &\\subset \\ck{ (z,t)\\in B(0,2\\Lambda\\ell(Q_0))\\ |\\ |z|\\le \\kappa\\, 2^{-N} \\ell(Q_0) }\\\\\n\t    &\\subset \\ck{ (z,t)\\in {\\mathbb{H}}^n\\ |\\ |z|\\le \\kappa\\, 2^{-N} \\ell(Q_0),\\ 16|t|^2\\le (2\\Lambda \\ell(Q_0))^4 }=: \\widetilde{T}_{\\kappa}.\n\t\\end{align*}\n\tBy \\eqref{e:lebesgue and hausdorff},\n\t\\begin{equation*}\n\t    \\mathcal{H}^{D}(\\widetilde{T}_{\\kappa}) = C\\mathcal{L}^{2n+1}(\\widetilde{T}_{\\kappa}) \\approx (\\kappa 2^{-N} \\ell(Q_0))^{2n}(2\\Lambda \\ell(Q_0))^2 \\approx_{\\kappa} 2^{-2nN}\\ell(Q_0)^{D}.\n\t\\end{equation*}\n\tIt follows that $\\mathcal{H}^{D}(T_{\\kappa})\\lesssim_{\\kappa}2^{-2nN}\\ell(Q_0)^{D}.$ On the other hand, recall that for any cube $Q$ with sidelength $\\ell(Q)=2^{-N}\\ell(Q_0)$ we have $\\mathcal{H}^{D}(Q)\\approx 2^{-ND}\\ell(Q_0)^{D}$. Since all such cubes are pairwise disjoint, we get\n\t\\begin{equation*}\n\t    \\#\\ck{Q\\in\\mathcal{D}\\ |\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ Q\\subset T_{\\kappa} } \\lesssim\t\\fr{\\mathcal{H}^{D}(T_{\\kappa})}{2^{-ND}\\ell(Q_0)^{D}}\\lesssim_{\\kappa} \\fr{2^{-2nN}\\ell(Q_0)^{D}}{2^{-N(2n+2)}\\ell(Q_0)^{D}}=2^{2N}.\n\t  \\end{equation*}\n\t\\end{proof}\n\t\\begin{lemma}\\label{c:dense-tube}\n\t\tLet $Q\\in\\mathcal{D}$ satisfy $Q \\subset Q_0 \\setminus T$ and $\\ell(Q) = \\ell(Q_1)=2^{-AN}\\ell(Q_0)$. Then\n\t\t\\begin{equation*}\n\t\t    \\mu(Q) \\le \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}}.\n\t\t\\end{equation*}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tSuppose the claim above is false. Then we can find a cube $Q_2 \\subset Q_0 \\setminus T$ with $\\ell(Q_2)=2^{-AN}\\ell(Q_0)$ such that\n\t\t\\begin{equation}\\label{e:assumption Q2}\n\t\t\\mu(Q_2) \\geq \\frac{\\mu(Q_0)}{\\Theta_0 \\,2^{AND}}.\n\t\t\\end{equation}\n\t\tLet $0< \\delta < \\dist(Q_1, Q_2)$, let $p\\in Q_2$ be arbitrary, and consider \n\t\t\\begin{equation*}\n\t\tR_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p) = \\int_{Q_1} K(q^{-1} \\cdot p) \\, d\\mu(q).   \n\t\t\\end{equation*}\n\t\tBy triangle inequality,\n\t\t\\begin{equation}\\label{e:split}\n\t\t|R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)| \\geq \\av{ \\int_{Q_1} K(p) \\, d\\mu(q) } - \\av{ \\int_{Q_1} K(q^{-1}\\cdot p) - K(p) \\, d\\mu(q) }.\n\t\t\\end{equation}\n\t\tWe estimate the first term as follows. Note that, since $p \\in Q_2$ and $Q_2$ lies outside $T$, then, writing $p=(z, t)$ and using \\eqref{e:K-norm}, we have\n\t\t\\begin{align*}\n\t\t|K(p)|^2 \\approx \\frac{|z|^2}{(|z|^4 + 16t^2)^{n+1}} \\gtrsim \\frac{|z|^2}{\\ell(Q_0)^{4(n+1)}} \\geq 2^{-2N}\\ell(Q_0)^{-4n-2} = 2^{-2N}\\ell(Q_0)^{-2D+2}.\n\t\t\\end{align*}\n\t\tAnd thus we also have \n\t\t\\begin{equation}\\label{e:first term estimate}\n\t\t\\av{ \\int_{Q_1} K(p)\\, d\\mu(q) } = \\av{K(p)}\\mu(Q_1) \\gtrsim 2^{-N} \\, \\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{equation}\n\t\t\n\t\tFor the second term in \\eqref{e:split} we use the continuity of the kernel $K$ \\eqref{e:kernel continuous 2} and the fact that $d(p,q) \\approx \\|p\\|_{{\\mathbb{H}}}\\ge 2^{-N}\\ell(Q_0)$ (because $p\\in Q_2\\subset Q_0\\setminus T$):\n\t\t\\begin{align}\n\t\t|K(q^{-1} \\cdot p) - K(p)| \\lesssim \\fr{\\|q\\|_{{\\mathbb{H}}}}{\\min(\\|p\\|_{{\\mathbb{H}}}, d(p,q))^{D}}\\lesssim \\fr{2^{-AN}\\ell(Q_0)}{(2^{-N}\\ell(Q_0))^{D}} = \\fr{2^{-AN+DN}}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{align}\n\t\tTaking $A\\ge 2D$ we get\n\t\t\\begin{equation*}\n\t\t\\av{ \\int_{Q_1} K(q^{-1} \\cdot p) - K(p) \\, d\\mu(q) }\\lesssim 2^{-AN\/2}\\fr{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{equation*}\n\t\tTogether with \\eqref{e:first term estimate} and \\eqref{e:split}, assuming $N_0$ bigger than some absolute constant (recall that $N\\ge N_0$), this gives\n\t\t\\begin{equation*}\n\t\t|R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)|\\gtrsim 2^{-N} \\, \\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}\n\t\t\\end{equation*}\n\t\tfor all $p\\in Q_2$.\n\t\t\n\t\tNow, we use the estimate above and the $L^2(\\mu)$ boundedness of $R_{\\mu}$ to get\n\t\t\\begin{equation*}\n\t\t2^{-N}\\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}\\mu(Q_2)^{\\frac{1}{2}} \\lesssim \\ps{\\int |R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)|^2 \\, d\\mu(p)}^{\\frac{1}{2}} \\leq C_1 \\mu(Q_1)^{\\frac{1}{2}}.\n\t\t\\end{equation*}\n\t\tOur assumptions on $Q_1$ \\eqref{e:assumption Q1} and $Q_2$ \\eqref{e:assumption Q2} yield\n\t\t\\begin{align*}\\label{e:est-cauchy}\n\t\tC_1 & \\gtrsim 2^{-N}\\frac{\\mu(Q_1)^{\\frac{1}{2}}\\mu(Q_2)^{\\frac{1}{2}}}{\\ell(Q_0)^{D-1}}\\gtrsim 2^{-N}\\fr{\\mu(Q_0)}{2^{AND}\\ell(Q_0)^{D-1}}\\Theta_0^{-1\/2} = 2^{-AND-N}\\Theta_0^{1\/2} \\notag\\\\\n\t& \\overset{\\eqref{e:choice of N}}{\\approx} 2^{-AND-N}\\, 2^{A^2 N\/2}.\n\t\t\\end{align*}\n\t\tTaking $A\\ge 5D$ we can bound the last term from below in the following way:\n\t\t\\begin{equation*}\n\t\t    2^{-AND-N+A^2 N\/2}\\ge  2^{A^2N\/4}\\overset{\\eqref{e:choice of N}}{\\gtrsim} M^{1\/4}.\n\t\t\\end{equation*}\n\t\tPutting together the estimates above gives $C_1\\gtrsim M^{1\/4}$, which is a contradiction for $M=M(C_1,n)$ big enough.\n\t\\end{proof}\n\t\n\tWe immediately get the following corollary.\n\t\\begin{corollary}\n\tWe have\n\t\\begin{equation}\\label{e:T2 thicc}\n\t\\mu({T_2})\\ge (1-\\Theta_0^{-1})\\mu(Q_0).\n\t\\end{equation}\n\t\\end{corollary}\n\t\\begin{proof}\n\tObserve that if $Q\\in\\mathcal{D}$ satisfies $\\ell(Q) = \\ell(Q_1) = 2^{-AN}\\ell(Q_0)$ and $Q\\not\\subset T_2$, then we have $Q\\cap T=\\varnothing$ (assuming $A$ large enough with respect to $\\Lambda$). It follows that $Q$ satisfies the assumptions of Lemma \\ref{c:dense-tube}, and so\n\t\\begin{equation*}\n\t    \\mu(Q)\\le2^{-AND}\\Theta_0^{-1}\\mu(Q_0).\n\t\\end{equation*}\n\tSumming over all such $Q$ and using \\eqref{e:dyadic property} yields\n\t\\begin{equation*}\n\t    \\mu(Q_0\\setminus T_2) \\le \\Theta_0^{-1}\\mu(Q_0).\n\t\\end{equation*}\n\t\\end{proof}\n\tRecall that\n\t\\begin{equation*}\n\t{\\mathsf{HD}}(Q_0) = \\ck{Q\\in \\mathcal{D}\\ |\\ Q\\subset Q_0,\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)>2\\Theta_0  },\n\t\\end{equation*}\n\tand that $\\Lambda$ is the absolute constant such that $Q\\subset B(p_Q,\\Lambda\\ell(Q)).$ Without loss of generality, we may assume $\\Lambda>2$.\n\t\n\n\tWe are ready to prove the first part of Lemma \\ref{l:main lemma}, the estimate \\eqref{e:HD are thicc}.\n\t\\begin{lemma}\\label{l:measureHD}\n\t\tThere exists $s=s(A,n)\\in (0,1\/2)$ such that\n\t\t\\begin{equation}\\label{e:HD are thicc 2}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) \\ge (1-\\Theta_0^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tWe will prove \\eqref{e:HD are thicc 2} by contradiction. Suppose that\n\t\t\\begin{equation}\\label{e:HD not thicc :(}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) < (1-\\Theta_0^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\tSet\n\t\t\\begin{equation*}\n\t\t{\\mathsf{LD}}(Q_0) = \\ck{Q\\in\\mathcal{D}\\ |\\ Q\\subset T_{2\\Lambda},\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)\\le 2\\Theta_0  }.\n\t\t\\end{equation*}\n\t\tIt is easy to see that the cubes from ${\\mathsf{HD}}(Q_0)\\cup {\\mathsf{LD}}(Q_0)$ cover $T_{2}$. If we assume $\\Theta_0\\ge M>100$, and $s<1\/2$, then $\\Theta_0^{-s}\/2\\ge \\Theta_0^{-1}$, and so by \\eqref{e:T2 thicc} and \\eqref{e:HD not thicc :(} we get\n\t\t\\begin{equation}\\label{e:LD has plenty mass}\n\t\t\\sum_{Q\\in{\\mathsf{LD}}(Q_0)}\\mu(Q) \\ge \\fr{\\Theta_0^{-s}}{2}\\mu(Q_0).\n\t\t\\end{equation}\n\t\t\n\t\tOn the other hand, recall from Lemma \\ref{c:number-cubes} that there are at most $C2^{2N}$ cubes of sidelength $2^{-N}\\ell(Q_0)$ contained in $T_{2\\Lambda}$, where $C=C(\\Lambda,n)$. Moreover, for any $Q\\in {\\mathsf{LD}}(Q_0)$ we have\n\t\t\\begin{equation*}\n\t\t\\mu(Q)\\le 2 \\Theta_0 \\ell(Q)^{D-1}= 2\\, \\mu(Q_0)\\fr{\\ell(Q)^{D-1}}{\\ell(Q_0)^{D-1}} =2^{-N(D-1)+1}\\mu(Q_0).\n\t\t\\end{equation*}\n\t\tIn consequence,\n\t\t\\begin{equation*}\n\t\t\\sum_{Q\\in{\\mathsf{LD}}(Q_0)}\\mu(Q) \\le C 2^{2N} 2^{-N(D-1)+1}\\mu(Q_0).\n\t\t\\end{equation*}\n\t\tThis contradicts \\eqref{e:LD has plenty mass} because\n\t\t\\begin{equation*}\n\t\tC\\, 2^{-ND+3N+1} = 2\\, C\\,  (2^{-A^2N})^{(-D+3)A^{-2}} \\overset{\\eqref{e:choice of N}}{\\le}\\widetilde{C}(n)\\Theta_0^{(-D+3)A^{-2}}\\le \\fr{\\Theta_0^{-s}}{2},\n\t\t\\end{equation*}\n\t\tchoosing $s=s(A,n)$ small enough.\n\t\\end{proof}\n\tWe move on to the second part of Lemma \\ref{l:main lemma}, i.e. the packing estimate \\eqref{e:HD packing}.\n\t\\begin{lemma}\\label{l:HD-in-tube}\n\tWe have\n\t\\begin{equation}\\label{e:HD-in-tube}\n\t    \\bigcup_{Q\\in{\\mathsf{HD}}(Q_0)}Q\\subset T_{2\\Lambda}.\n\t\\end{equation}\n\tIn consequence, \n\t\\begin{equation}\\label{e:HD packing 2}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\ell(Q)^2\\lesssim \\ell(Q_0)^2.\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\tWe will prove that for $Q\\in{\\mathsf{HD}}(Q_0)$ we have $Q\\cap T_2\\neq\\varnothing.$ Then, since $\\ell(Q)=2^{-N}\\ell(Q_0)$, it follows easily from \\eqref{e:balls in cubes} that indeed $Q\\subset T_{\\Lambda+2}(Q_0)\\subset T_{2\\Lambda}(Q_0)$.\n\t\n\tWe argue by contradiction. Suppose that $Q\\in {\\mathsf{HD}}(Q_0)$ and $Q\\cap T_2=\\varnothing$. Consider the cubes $\\{P_i\\}_{i\\in I}$ with $\\ell(P_i)=2^{-AN}\\ell(Q_0) = 2^{-(A-1)N}\\ell(Q)$ and $P_i\\subset Q$. Then, $Q=\\bigcup_i P_i,$ for all $i\\in I$ we have $P_i\\cap T_2=\\varnothing,$ and $\\# I\\approx 2^{(A-1)ND}$ by \\eqref{e:dyadic property}. \n\t\n\tWe use Lemma \\ref{c:dense-tube} to conclude that for all $i\\in I$\n\t\\begin{equation*}\n\t    \\mu(P_i) \\le \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}}.\n\t\\end{equation*}\n\tSumming over $i\\in I$ yields\n\t\\begin{equation*}\n\t    \\mu(Q) = \\sum_{i\\in I}\\mu(P_i)\\le \\# I\\cdot \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}} \\approx 2^{(A-1)ND} \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}} = \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{ND}},\n\t\\end{equation*}\n\tso that\n\t\\begin{equation*}\n\t    \\Theta(Q) = \\fr{\\mu(Q)}{(2^{-N}\\ell(Q_0))^{D-1}} \\lesssim \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{ND}}\\cdot \\frac{1}{2^{-N(D-1)}\\ell(Q_0)^{D-1}} = \\frac{\\Theta_0}{\\Theta_0 \\, 2^{N}}=2^{-N}\\le 1.\n\t\\end{equation*}\n\tBut this contradicts the assumption $Q\\in {\\mathsf{HD}}(Q_0)$:\n\t\\begin{equation*}\n\t    \\Theta(Q)\\ge 2\\Theta_0\\ge 2M>1,\n\t\\end{equation*}\n\tand so the proof of \\eqref{e:HD-in-tube} is finished.\n\t\n\tConcerning \\eqref{e:HD packing 2}, note that by \\eqref{e:HD-in-tube} and Lemma \\ref{c:number-cubes} we have\n\t\\begin{equation} \\label{e:HD-card}\n\t    \\# {\\mathsf{HD}}(Q_0) \\lesssim 2^{2N}.\n\t\\end{equation}\n\tHence,\n\t\\begin{align*}\n\t    \\sum_{Q \\in {\\mathsf{HD}}(Q_0)} \\ell(Q)^2 = \\ell(Q_0)^2\\,  2^{-2N}  \\sum_{Q \\in {\\mathsf{HD}}(Q_0)} 1  \\lesssim \\ell(Q_0)^2. \n\t\\end{align*}\n\t\\end{proof}\n\\section{Iteration argument}\\label{s:iteration}\nTo complete the proof of Theorem \\ref{t:main}, we assume that the measure $\\mu$ does not satisfy the polynomial growth condition \\eqref{e:main}. Then we will use Lemma \\ref{l:main lemma} countably many times to construct a set $Z$ with positive $\\mu$-measure and with Hausdorff dimension at most $2$. \n\nSuppose that there exists a ball $B(x,r)$ with $\\mu(B(x,r)) \\geq C_2 r^{2n+1}$; if $C_2$ is big enough, we can find a cube $Q_0 \\in \\mathcal{D},\\ Q\\subset B(x,r)$ such that \n\t\\begin{align*}\n\t\\Theta(Q_0) \\ge M, \n\t\\end{align*}\n\twhere $M$ is the constant from Lemma \\ref{l:main lemma}.\n\nLet $A>1$ be as in Lemma \\ref{l:main lemma}. Following the notation of Lemma \\ref{l:main lemma}, for an arbitrary cube $Q\\in\\mathcal{D}$ with $\\Theta(Q)\\ge M$, set\n\\begin{equation*}\n    N(Q) := \\floor{A^{-2} \\log(\\Theta(Q))}\n\\end{equation*}\nand\n\\begin{equation*}\n    {\\mathsf{HD}}(Q):=\\left\\{ P \\in \\mathcal{D} \\, |\\, P\\subset Q,\\, \\ell(P) =2^{-N(Q)}\\ell(Q), \\, \\Theta(P) > 2 \\Theta(Q) \\right\\}.\n\\end{equation*}\nPut $Z_{0}:=Q_0,\\ {\\mathsf{HD}}_0:=\\{Q_0\\},\\ {\\mathsf{HD}}_1:={\\mathsf{HD}}(Q_0),$ and $Z_1:=\\bigcup_{Q\\in{\\mathsf{HD}}_1} Q$. Proceeding inductively, for all $j\\ge 2$ we define\n\\begin{gather*}\n    {\\mathsf{HD}}_{j} := \\bigcup_{Q \\in {\\mathsf{HD}}_{j-1}} {\\mathsf{HD}}(Q),\\\\\n    Z_{j} := \\bigcup_{Q\\in{\\mathsf{HD}}_j} Q.\n\\end{gather*}\nNote that for each $j$ the cubes in ${\\mathsf{HD}}_j$ form a disjoint family. Moreover, $\\{Z_j\\}_{j\\ge 0}$ form a decreasing sequence of sets, that is $Z_{j+1}\\subset Z_j.$ Define\n\\begin{equation*}\n   Z := \\bigcap_{j \\geq 0} Z_j.\n\\end{equation*}\n\\begin{claim}\\label{c:positive}\nWe have\n\\begin{equation*}\n    \\mu(Z)\\gtrsim_{M,s}\\mu(Q_0).\n\\end{equation*}\n\\end{claim}\n\\begin{proof}\nObserve that for $Q\\in{\\mathsf{HD}}_j$ we have\n\\begin{equation}\\label{e:estimate densities}\n    \\Theta(Q)\\ge 2^{j}\\Theta(Q_0)\\ge 2^{j}M.\n\\end{equation}\nIn particular, $\\Theta(Q)\\ge M$ and so we may apply Lemma \\ref{l:main lemma} to $Q$. It follows that for any $j \\geq 0$ we have\n\\begin{multline*}\n    \\mu(Z_{j+1})=\\sum_{Q \\in {\\mathsf{HD}}_{j+1}} \\mu(Q) = \\sum_{Q \\in {\\mathsf{HD}}_{j}} \\sum_{P \\in {\\mathsf{HD}}(Q)} \\mu(P) \\stackrel{\\eqref{e:HD are thicc}}{\\geq} \\sum_{Q \\in {\\mathsf{HD}}_{j}} (1-\\Theta(Q)^{-s})\\mu(Q)\\\\\n    \\overset{\\eqref{e:estimate densities}}{\\ge} \\sum_{Q \\in {\\mathsf{HD}}_{j}} (1-2^{-js}M^{-s})\\mu(Q) = (1-2^{-js}M^{-s}) \\mu(Z_{j}).\n\\end{multline*}\nUsing this estimate $(j+1)$ times we arrive at\n\\begin{equation}\\label{e:HD_jbig}\n    \\mu(Z_{j+1}) \\geq \\prod_{i=0}^{j}(1-2^{-is}M^{-s}) \\mu(Q_0).\n\\end{equation}\nSince $Z_j$ form a sequence of decreasing sets, we get by the continuity of measure \n\\begin{equation*}\n    \\mu(Z) = \\lim_{j\\to\\infty} \\mu(Z_j)\\ge \\prod_{i=0}^{\\infty}(1-2^{-is}M^{-s}) \\mu(Q_0) = C(s,M)\\mu(Q_0),\n\\end{equation*}\nwhere $C(s,M)$ is positive and finite because $\\sum_{i=0}^{\\infty} 2^{-is}<\\infty$.\n\\end{proof}\n\\begin{claim}\\label{c:dim} We have\n\\begin{equation*}\n    \\textstyle{\\dim_H(Z)} \\leq 2.\n\\end{equation*}\n\\end{claim}\n\\begin{proof}\nRecall that $N(Q) = \\floor{A^{-2} \\log(\\Theta(Q))}$. It follows from \\eqref{e:estimate densities} that for $Q\\in{\\mathsf{HD}}_j$ we have $N(Q)\\ge C_3jA^{-2}$ for some absolute constant $C_3>0$. Thus, for $Q\\in{\\mathsf{HD}}_j$ and $P\\in{\\mathsf{HD}}(Q)$\n\\begin{equation*}\n    \\ell(P)=2^{-N(Q)}\\ell(Q)\\le 2^{-C_3jA^{-2}}\\ell(Q).\n\\end{equation*}\nUsing this observation $j$ times we get that for $P\\in{\\mathsf{HD}}_{j+1}$\n\\begin{equation*}\n    \\ell(P)\\le 2^{-C_4 j(j+1)A^{-2}}\\ell(Q_0),\n\\end{equation*}\nwhere $C_4=C_3\/2.$\nHence, the cubes from ${\\mathsf{HD}}_j$ form coverings of $Z$ with decreasing diameters, well suited for estimating the Hausdorff measure of $Z$. \n\nLet $0<\\varepsilon<1,\\ 0<\\delta<1$ be small. Let $j\\ge 0$ be so big that for $Q\\in{\\mathsf{HD}}_{j}$ we have $\\mathrm{\\, diam}(Q)\\le\\Lambda\\ell(Q)\\le\\delta$. Then,\n\\begin{equation}\\label{e:Hausdorff estimate}\n   \\mathcal{H}_{\\delta}^{2+\\ve}(Z) \\le \\Lambda^{2+\\varepsilon}\\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2+\\varepsilon}\\le \\Lambda^{2+\\varepsilon}(2^{-C_4j(j-1)A^{-2}}\\ell(Q_0))^{\\varepsilon}\\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2}.\n\\end{equation}\nIt follows by \\eqref{e:HD packing} that\n\\begin{equation*}\n    \\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2} = \\sum_{P\\in{\\mathsf{HD}}_{j-1}}\\sum_{Q\\in{\\mathsf{HD}}(P)}\\ell(Q)^{2}\\le C_p\\sum_{P\\in{\\mathsf{HD}}_{j-1}}\\ell(P)^{2}.\n\\end{equation*}\nUsing the estimate above $j$ times, and putting it together with \\eqref{e:Hausdorff estimate} we arrive at\n\\begin{equation*}\n    \\mathcal{H}_{\\delta}^{2+\\ve}(Z) \\le \\Lambda^{2+\\varepsilon} (C_p)^j\\, \\,2^{-\\varepsilon C_4 j(j-1)A^{-2}}\\,  \\ell(Q_0)^{2+\\varepsilon}.\n\\end{equation*}\nThe right hand side above converges to 0 as $j\\to\\infty$ (just note that the exponent at $C_p$ is linear in $j$ while the exponent at $2$ is quadratic in $j$). Hence, $\\mathcal{H}_{\\delta}^{2+\\ve}(Z)=0.$ Letting $\\delta\\to 0$ we get $\\mathcal{H}^{2+\\ve}(Z)=0.$ Since this is true for arbitrarily small $\\ve>0$, it follows that \n\\begin{equation*}\n    \\textstyle{\\dim_H(Z)}  = \\inf\\{t\\ge 0\\ :\\ \\mathcal{H}^t(Z)=0\\} \\le 2.\n\\end{equation*}\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{t:main}]\nWe have found a set $Z \\subset \\mathbb{H}^{n}$ of dimension smaller than or equal to $2$ (Claim \\ref{c:dim}) but which nevertheless has positive $\\mu$-measure (Claim \\ref{c:positive}). This contradicts the assumptions of Theorem \\ref{t:main}. Thus, there exists $C_2=C_2(n,C_1)$ such that $\\mu(B(x,r)) \\leq C_2 r^{2n+1}$ for all $x\\in{\\mathbb{H}}^n$ and $r>0$.\n\\end{proof}\n\n\\subsection*{Funding}\nD. D\\k{a}browski was supported by Spanish Ministry of Economy and Competitiveness, through the Mar\\'ia de Maeztu Programme for Units of Excellence in R\\&D (grant MDM-2014-0445), and also partially supported by the Catalan Agency for Management of University and Research Grants (grant 2017-SGR-0395), and by the Spanish Ministry of Science, Innovation and Universities (grant MTM-2016-77635-P).\n\nM. Villa was supported by The Maxwell Institute Graduate School in Analysis and its\nApplications, a Centre for Doctoral Training funded by the UK Engineering and Physical\nSciences Research Council (grant EP\/L016508\/01), the Scottish Funding Council, Heriot-Watt\nUniversity and the University of Edinburgh.\n\nBoth authors were partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nObservations and numerical (radiation-)MHD models of the formation of massive protostars along with their accretion disks and collimated jets have made clear progress of the last decade. Even though there are several observational examples of massive protostellar jets \\citep[see e.g.][]{Purser2016}, direct observations and estimations of the radii of their associated accretion disks are rarer. The source for the HH 80-81 radio jet, for example, has an accretion disk with an estimated radius of $950\\text{--}1300\\unit{au}$, and an inner radius of $\\lesssim 170\\unit{au}$, with the mass of the protostar in the range of $4\\text{--}18\\unit{M_\\odot}$ \\citep{Girart2017, Carrasco-Gonzalez2010}.\n\n\n\n\nRecently, a new generation of multi-scale observations of the star-forming region IRAS 21078+5211 \\citep{Moscadelli2021multi, Moscadelli2022} have revealed the existence of a disk-jet system around a protostar of $5.6\\pm 2 \\unit{M_\\odot}$. Emission peaks from molecular rotational transitions of $\\mathrm{CH_3CN}$ and $\\mathrm{HC_3N}$ trace the kinematical footprint of a Keplerian-like accretion disk of around $400\\unit{au}$ in diameter surrounding the protostar. For the first time, the direct study of the launching mechanism of the jet has been possible, thanks to water maser observations performed with Very Long Base Interferometry (VLBI) \\citep{Moscadelli2022}, which traced individual streamlines in the jet and reached a resolution of $0.05\\unit{au}$.\n\nIn \\cite{Moscadelli2022}, we modeled the disk-jet system in IRAS 21078+5211 with a resistive-radiation-gravito-magnetohydrodynamical simulation of a forming massive star starting from the collapse of a cloud core. We followed the evolution of the protostar, the formation of the accretion disk, and the launching, acceleration, collimation, propagation and termination of the magnetically-driven outflows, which self-consistently form within the computational domain. Thanks to the use of an axisymmetrical grid, we were able to reach unprecedented resolutions in the regions surrounding the forming massive star (up to $0.03\\unit{au}$), allowing for the comparison with the VLBI data (which has a resolution of $0.05\\unit{au}$). We found good agreement between the simulated and observed disk size (given the protostellar mass) as well as the main features of the jet. The simulation was part of a larger study that examines the effects of the natal environment on the formation of a massive star. We present the full series of simulations in this article, with special focus on the dynamics of the accretion disk, while the dynamics of the magnetically-driven outflows are the focus of an upcoming article (hereafter Paper II).\n\n\n\n\nPreviously, the numerical simulations performed by \\cite{Anders2018} were able to show the self-consistent magneto-centrifugal launching of the jet and a tower flow driven by magnetic pressure in the context of massive star formation. The present study builds on their work by improving the treatment of the thermodynamics of the gas and dust with the inclusion of radiation transport with the gray flux-limited diffusion approximation \\citep{Kuiper2020}, and increasing the resolution of the grid. Those changes enable us to examine the vertical structure of the accretion disk. Previous numerical studies of the formation of a massive star in an environment with magnetic fields did not considered magnetic diffusion (e.g. \\citealt{Myers2013} and \\citealt{Seifried2011}) or approximated the thermodynamics of the system with the use of a barotropic \\citep[e.g.][]{Machida2020} or isothermal \\citep{Anders2018} equation of state instead of solving for radiation transport. Recent studies by \\cite{Mignon-Risse2021} and \\cite{Commercon2022}, which include ambipolar diffusion as the non-ideal MHD effect and do include radiation transport, find indications of a magneto-centrifugally launched jet (although their grid does not properly resolve the launching region), and smaller disks with stronger magnetic fields. The question of the size of the disk is explored in more detail in the present article, as we find an opposite trend in our results. For further literature review, we refer the reader to Sect. \\ref{S: previous}.\n\n\nBecause of computational costs, previous efforts have focused on only a handful of natal environmental conditions for the massive (proto)star. Our setup, in turn, allows us to explore a wide range of natal environmental conditions and their impact on the outcome of the formed massive star and its disk-jet system.  In section \\ref{S: model method} we describe the model, physical effects considered, and the parameter space of the simulations. We present an overview of the evolution of the system based on the fiducial case of the parameter space in \\ref{S: evolution}. In section \\ref{S: dynamics}, we focus on the dynamical processes observed in the accretion disk, i.e., the effect of the magnetic field, Ohmic dissipation and radiation transport. In section \\ref{s:parameter-study} we explore the dependence of the disk size and evolution with the initial conditions for the gravitational collapse, and finally in section \\ref{S: previous} we offer a comparison of our results with previous numerical studies.\n\n\n\n\n\\section{Model and method} \\label{S: model method}\n\\subsection{Description of the model}\nWe model axisymmetrically the gravitational collapse of a cloud core of mass $M_C$ and a radius of $R_C=0.1\\unit{pc}$ (cf. Fig. \\ref{overview_evol}), which has an initial density profile of the form\n\\begin{equation}\n\\rho(r,t=0) = \\rho_0 \\left(\\frac{r}{r_0}\\right)^{\\beta_\\rho}.\n\\end{equation}\nThe constant $r_0$ is chosen to be $1\\unit{au}$ and $\\rho_0$ is determined by computing the mass of the cloud with the integral of the density profile over $0 < r < R_C$. The cloud is in slow initial rotation, given by an initial angular velocity profile of the form\n\\begin{equation}\n\\Omega(R,t=0) = \\Omega_0 \\left( \\frac{R}{R_0} \\right)^{\\beta_\\Omega},\n\\end{equation}\nwhere $R$ is the cylindrical radius and $R_0$ is chosen to be $10\\unit{au}$. The constant $\\Omega_0$ is fixed by computing the ratio of rotational to gravitational energy $\\zeta \\equiv E_r\/E_g$ from the initial density and rotation profiles; $\\zeta$ is then a parameter of the simulation.\n\nThe cloud core is threaded by an initially uniform magnetic field directed parallel to the rotation axis. Its magnitude $B_0$ is determined by the normalized mass-to-flux ratio $\\bar \\mu$,\n\\begin{equation}\n\\bar \\mu \\equiv \\frac{M_C\/\\Phi_B}{\\left(M_C\/\\Phi_B\\right)_\\mathrm{crit}}.\n\\end{equation}\nThe denominator of the previous expression represents the critical value of the mass-to-flux ratio, i.e., the value for which the gravitational collapse is halted by the magnetic field under idealized conditions. We take the value from \\cite{MouschoviasSpitzer1976}. The magnetic flux $\\Phi_B$ is simply computed as the flux across the midplane, $B_0 \\cdot \\pi R_C^2$, which in magnitude is the same flux that enters or leaves the spherical cloud. \n\n\\subsection{Physics}\n\nThe simulations solve for the equations of magnetohydrodynamics, with the addition of Ohmic resistivity as a non-ideal effect, self-gravity, the gravitational force from the forming massive star, and radiation transport for the thermal emission from the gas and dust, on an axisymmetric grid. Next, we describe the set of equations solved.\n\nThe dynamics of the gas and dust are followed by numerically solving the following system of conservation laws, using the Pluto code \\citep{Mignone2007}:\n\\begin{equation}\n\\partial_t \\rho + \\vec \\nabla \\cdot (\\rho \\vec v) = 0,\n\\end{equation}\n\\begin{equation}\n{\\partial_t (\\rho \\vec v) } + \\vec \\nabla \\cdot \\left[  \\rho \\vec v \\otimes \\vec v -  \\tfrac{1}{4\\pi}{\\vec B \\otimes \\vec B} + P_t  \\mathsf{I} \\right] = \\rho \\vec{a}_\\text{ext},\n\\end{equation}\n\\begin{equation}\n{\\partial_t \\vec B} + \\vec \\nabla \\times (c \\vec{\\mathcal E}) = 0,\n\\end{equation}\n\\[ {\\partial_t (E^K+E^\\text{th}+E^B)} \\]\n\\begin{equation} \\label{e:eneq}\n\\quad\\quad + \\vec \\nabla \\cdot \\left[ (E^K+E^\\text{th} +  P)\\vec v + c \\vec{\\mathcal E} \\times \\vec B \\right] = \\rho \\vec v \\cdot \\vec a_\\text{ext},\n\\end{equation}\nwhich correspond to the continuity equation, momentum equation, induction equation and energy conservation equation, respectively. The gas is weakly ionized and has density $\\rho$, velocity $\\vec v$ and magnetic field $\\vec B$. The magnetic field must also satisfy the solenoidality condition $\\vec \\nabla \\cdot \\vec B = 0$. The total pressure $P_t = P + \\tfrac{1}{8\\pi}{B^2}$ is composed of the thermal and magnetic pressures. The electric field $\\vec{\\mathcal E}$ can be directly substituted by the expression\n\\begin{equation}\nc\\vec{\\mathcal E} = -\\vec v \\times \\vec B + \\eta \\vec \\nabla \\times \\vec B,\n\\end{equation}\nwhere $\\eta(\\rho,T)$ is the Ohmic resistivity. The energy density of Eq. \\ref{e:eneq} is separated into its components: kinetic ($E^K$), thermal or internal ($E^\\mathrm{th}$) and magnetic ($E^B$). The acceleration source term $\\vec a_\\text{ext}$ is a sum of the gravity of the forming star $\\vec a_{g\\star} = -GM_\\star\/r^2\\, \\vec e_r$, the self-gravity of the gas $\\vec a_\\text{sg}$ and the viscosity source term $\\vec a_\\nu$.\n\n\nThe self-gravity acceleration source term is given by $\\vec a_\\text{sg} = -\\nabla \\Phi_\\text{sg}$. The self-gravity potential $\\Phi_\\text{sg}$ is found by solving Poisson's equation $\\nabla^2 \\Phi_\\text{sg} = 4\\pi G\\rho$ with a diffusion Ansatz, according to \\cite{Kuiper2010circ}. Self-gravity is inherently a three-dimensional effect, and a cloud with the characteristics we consider here is expected to produce an accretion disk with spiral arms. Since we perform the simulations on a two-dimensional axisymmetric grid, we mimic the missing angular momentum transport by the gravitational torques with the addition of a shear viscosity source term $\\vec a_\\nu = \\vec \\nabla \\Pi\/\\rho$, where the viscosity tensor $\\Pi$ is fixed with the $\\alpha$-parametrization of \\cite{ShakuraSunyaev1973} and no bulk viscosity is considered. This approach was extensively studied in \\cite{Kuiper2011}, and we refer the reader to that article for more details on the computation of the viscosity tensor, as well as comparisons of the efficacy of angular momentum transport using this approach in comparison to non-viscous 3D models.\n\n\nWe treat radiation transport with the gray flux-limited diffusion approximation, by using the module Makemake described in ample detail in \\cite{Kuiper2020}. Specifically, we point  the reader to sections 2.3.2 and 2.3.4 in that reference. In a nutshell, two equations are solved simultaneously for treating radiation transport. The first one is the zeroth-moment of the radiation transport equation within a given volume, which essentially says that the net emitted energy density (emission minus absorption) that is not lost through the boundary as a radiative flux, has to be stored as a local radiative energy density. This equation is simplified by assuming  that the radiative flux can be written as a diffusion term in terms of the radiation energy density (hence flux-limited diffusion). The second equation describes the changes to $E^\\text{th}$: the net absorbed energy density (absorption minus emission) has to be stored as internal energy in the gas. Both energy fields are solved without previous assumption of equilibrium between them; this is the so-called two-temperature approach \\cite{Commercon2011rad}. In all but one of the simulations (see Sect. \\ref{S: disk var eta} for details), a constant value of the opacity for the dust and gas of $1\\unit{cm^2\\, g^{-1}}$ was used. The initial dust-to-gas mass ratio is set to $1\\%$. The irradiation from the forming massive star is not considered in this study.\n\n\nWe use the resistivity model by \\cite{Machida2007} (which is based on a numerical study by \\citealt{Nakano2002})\n\\begin{equation}\n\\eta = \\frac{740}{X_e} \\left({\\frac{T}{10\\unit{K}}}\\right)^{1\/2} \\left[  1 - \\tanh \\left( \\frac{n_H}{10^{15} \\unit{cm^{-3}}} \\right) \\right] \\unit{cm^2\\, s^{-1}},\n\\end{equation}\nwhere we use for simplicity $n_H=\\rho\/\\mu_H$ as the number density of hydrogen nuclei ($\\mu_H$ being the molecular weight of hydrogen) and the ionization fraction $X_e$ is given by\n\\begin{equation}\nX_e = 5.7\\cdot 10^{-4} \\left(\\frac{n_H}{\\unit{cm^{-3}}}\\right)^{-1}.\n\\end{equation}\n\n\nAt all moments in time, the properties of the formed massive star are computed using the evolutionary tracks for high-mass stars calculated by \\cite{HosokawaOmukai2009evol} and the instantaneous values of the stellar mass and accretion rate.\n\n\\subsection{Boundary conditions}\nThe computational domain assumes symmetry with respect to the rotation axis and equatorial symmetry with respect to the midplane. This means that scalar quantities and the velocity are simply reflected across the $z$-axis and the midplane. In the case of the magnetic field, it is reflected across the $z$-axis, but across the midplane, the parallel component switches sign instead of the normal component. The same symmetries imply that the boundaries across the azimuthal direction are periodic. Both the inner and outer boundaries (i.e., the sink cell and the outermost part of the cloud) impose a zero gradient condition for the magnetic field and the azimuthal and polar components of the velocity; for the radial component of the velocity and the density, only outflow but no inflow is allowed. Therefore, all matter that goes inside of the sink cell is considered as accreted.\n\n\\subsection{Initial conditions: parameter space and numerical configuration}\n\n\\begin{table}\n\\caption{Grid resolutions}\n\\label{t:grid}\n\\centering\n\\begin{tabular}{c c c c c}\n\\hline\\hline\nGrid & $N_r$ & $N_\\theta$ & $\\Delta x_\\text{min}\\ [\\unit{au}]$ & $\\Delta x_{1000}\\  [\\unit{au}]$ \\\\\n\\hline\nx1 & 56 & 10 & 0.51 & 160 \\\\\nx2 & 112 & 20 & 0.25 & 80 \\\\\nx4 & 224 & 40 & 0.12 & 40 \\\\\nx8 & 448 & 80 & 0.06 & 20\\\\\nx16 & 896 & 160 & 0.03 & 10\\\\\n\\hline \n\\end{tabular}\n\\end{table}\n\n\nWe present an extensive parameter study of the system, with the express aim of aiding in the understanding of the physical processes that intervene in the formation and evolution of the disk and the magnetically-driven outflows. In total, we present results from 31 different runs. Instead of describing all the parameters in multiple tables, we show a summary of the parameter space as a diagram in the right panel of Fig. \\ref{overview_evol}.\n\n\nWe used five different axisymmetrical grids with increasing resolution, that we name x1, x2, x4, x8 and x16. The grid for the radial coordinate $r$ increases logarithmically with distance, starting from the radius of the inner boundary ($3\\unit{au}$ in the fiducial case), to the radius of the cloud core ($0.1\\unit{pc}$). We often refer to the inner boundary as the sink cell. Using the assumed symmetries of the system, the colatitude $\\theta$ extends from $0$ to $\\pi\/2$ radians, and it is divided linearly into cells such that they have approximately the same dimensions in both the radial and polar directions for each distance to the center of the cloud. Table \\ref{t:grid} details the number of cells in each direction, the minimum cell size and a reference cell size at $1000\\unit{au}$ for all the grids used in this study, and assuming the sink cell size of the fiducial case.\n\n\nThe parameters with direct relation to physical quantities we considered are: the mass of the cloud core $M_C$, the resistivity model used, the normalized mass-to-flux ratio $\\bar \\mu$, the initial density profile exponent $\\beta_\\rho$, the initial rotation profile exponent $\\beta_\\Omega$, and the ratio of the rotational-to-gravitational energy $\\zeta$. The values for each parameter in the fiducial case are highlighted in blue in Fig. \\ref{overview_evol}. The fiducial case was run on all the grids. Modifications of the fiducial case are then investigated while keeping the rest of the parameters constant. Because of this, we refer in the paper to each simulation by the modified parameter followed by the grid it was run on; for example, $M_C = 150\\unit{M_\\odot}\\text{ [x8]}$ refers to a cloud of $150\\unit{M_\\odot}$, with the rest of the parameters as in the fiducial case for grid x8. The full set of values with direct relation to physical quantities was run on the base grid x4, and some selected values, which are underlined in Fig. \\ref{overview_evol}, were run on grid x8 as well. During the preparation of this study, we also ran a part of this physical parameter space on the lower resolution grids, however, we do not include the results here because they are redundant. Details of the choice of the parameters, as well as the comparison of results obtained, are presented in Sect. \\ref{s:parameter-study}.\n\n\nThe numerical convergence of the results was studied by changing the values of the sink cell radius, the strength of the $\\alpha$ shear viscosity and the Alfv\u00e9n limiter using the grid x2 as a base. A more detailed explanation of these latter  parameters, as well as the results of the convergence study can be found in Appendix \\ref{s:convergence}.\n\n\n\n\n\\section{Overview of the evolution of the system}\\label{S: evolution}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{imgraw\/overview_evol}\n\t\\caption{Initial conditions, overview of the evolution of the system, and summary of the parameter space presented in this study.}\n\t\\label{overview_evol}\n\\end{figure*}\n\nFigure \\ref{overview_evol} shows a schematic overview of the features observed during the evolution of the system, based on the results of the fiducial case but generally present in most of the simulations of the study. We describe these characteristic features and processes in the following subsections case-by-case.\n\n\\subsection{Gravitational infall epoch}\nAs soon as the simulation begins, the cloud starts collapsing under its own gravity. The flow quickly becomes super-Alfv\u00e9nic (the kinetic flow velocity is larger than the Alfv\u00e9n speed), and the magnetic field lines are dragged by the infall accordingly. The result of this collapse is that the configuration of magnetic field lines adopts an ``hourglass'' shape, as described in \\cite{Galli2006}, and that has been observed in \\cite{Beltran2019}.\n\n\\subsection{Protostar}\nA massive (proto)star is formed at the center of the cloud, which reaches a mass of $\\sim 20\\unit{M_\\odot}$ after $30\\unit{kyr}$. The luminosity of the (proto)star is dominated by the conversion of gravitational energy into radiation during the first $20\\unit{kyr}$, and only after that time the stellar luminosity becomes comparable to the accretion luminosity. We point out that both the stellar and accretion luminosities are not taken into account in this study, but we will report on these radiative feedback effects in future work.\n\n\\subsection{Magneto-centrifugal epoch}\n\nDue to the conservation of angular momentum, gas coming from large scales rotates faster as it reaches the center of the cloud. At $t\\sim 5\\unit{kyr}$, enough angular momentum is transported onto the center of the cloud to form a Keplerian-like accretion disk as a result. The disk grows in time due to the continued infall of material from large scales.\n\n\nThe accretion disk can be morphologically divided into a thin and a thick layer. The thin layer of the disk  is a region of the midplane consisting of material in rotation with speeds close to the Keplerian speed. The density of the thin layer increases towards the center of the cloud $10^{-16}$ to $10^{-11}\\unit{g\\, cm^{-3}}$. Enclosing the thin layer, there is a less dense thick layer (typical densities: $10^{-14}\\unit{g\\,cm^{-3}}$) which is also rotating.\n\n\nRoughly at the same time as the disk forms, magnetically-driven outflows are launched, which we classify according to their speeds and driving mechanisms. The jet is launched by the magneto-centrifugal mechanism (in a way similar to the model of \\citealt{BlandfordPayne1982}) and it reaches typical speeds higher than $100\\unit{km\\,s^{-1}}$. Rotation drags the magnetic field lines, as it has been observed in \\cite{Beuther2020} and \\cite{Girart2013}. This creates a magnetic pressure gradient that drives a slower and broader tower flow (cf. the model of \\citealt{Lynden-Bell2003}). Its typical speeds are of the order of $10\\unit{km\\,s^{-1}}$, and it broadens with time. A discussion of the dynamical processes present in the launching of outflows is offered in Paper~II.\n\n\nOn the interface between inflow along the thick layer of the disk and outflow along the cavity, an additional structure is identified, which we refer as the \\emph{cavity wall}. Densities at the cavity wall are higher than those in the cavity, but also higher than the surrounding infalling envelope and the thick layer of the disk. In terms of its dynamics, material from the cavity wall contributes episodically to both the inflow and the outflow.\n\n\\subsection{Magnetic braking epoch}\n\nAs magnetic field lines are dragged by rotation, the resulting magnetic tension exerts a torque on the gas that brakes it, infalling as a consequence of the angular momentum loss. The inclusion of Ohmic dissipation makes magnetic braking negligible at early times in the simulation, but at around $t\\sim 15 \\unit{kyr}$, magnetic braking starts to affect the innermost region of the disk and the cavity wall ($r\\lesssim 30\\unit{au}$). As a result, a magnetically-braked region is formed, where the material is mostly infalling but where the disk densities are kept because of constant replenishment through material from the cavity wall.\n\n\nThe extraction of angular momentum from the inner region of the cloud by magnetic braking modifies or even interrupts the magneto-centrifugal mechanism that drives the jet, but not the tower flow, which is mostly present at large scales. The tower flow broadens over time because the continuous dragging of magnetic field lines by rotation increases the magnetic pressure gradient on top of the disk, which is itself growing in time as well. This outflow broadening mechanism offers an explanation for the earliest outflow broadening observed and discussed in \\cite{Beuther2005}. An analysis and discussion of the effects of magnetic braking in outflows, as well as their propagation over time into the large scales of the cloud is offered in Paper~II, which focuses on the outflow physics of the same simulation data analyzed here.\n\n\n\\section{Dynamics of the disk} \\label{S: dynamics}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{imgraw\/thindisk_keplerianity.pdf}\n\t\\caption{Keplerianity on the midplane, computed with the fiducial simulation on grid x8.}\n\t\\label{thindisk_keplerianity}\n\\end{figure}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{imgraw\/thindisk_energies.pdf}\n\t\\caption{Contributions to the specific energy in the thin layer of the disk, calculated with the fiducial simulation on grid x8.}\n\t\\label{thindisk_energies}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{imgraw\/stratification_torus_1}\n\t\\caption{Dynamics of the thick layer of the disk. The panels of the figure are grouped in columns such that they show: the density structure of the thick layer (\\emph{a} and \\emph{e}), the offset from Keplerianity at three different heights in the thick layer (\\emph{b} and \\emph{f}), the contributions to the specific energy (\\emph{c} and \\emph{g}), and the vertical balance of forces (\\emph{d} and \\emph{h}). Data from the fiducial simulation on grid x8 was used for this figure.}\n\t\\label{thickdisk_dynamics}\n\\end{figure*}\n\nAfter giving an overview on the general evolution of the system, we aim at understanding the dynamical processes present in the disk in detail by taking the fiducial case and examining selected terms of the equations of motion. With the identification of these processes, we are able to perform a more effective and comprehensive comparison of results for the full parameter space, which we present in Sect. \\ref{s:parameter-study}.\n\n\\subsection{Radial dynamics in the disk}\n\nThe forming massive star accretes mass through the accretion disk, which implies that the velocity in the disk must have a component in the radially-inwards direction. The torque necessary for the angular momentum transport in the disk is provided by gravitational torques of forming spiral arms, which is modeled here through the use of the $\\alpha$ shear viscosity model. Nevertheless, $v_\\phi \\gg v_R$ in the disk, and so, it is useful to examine the effects of the different forces that act in the disk under the assumption of radial equilibrium.\n\n\nIn cylindrical coordinates $(R,\\phi,z)$, the radial component of the MHD momentum equation, setting $\\theta=\\pi\/2$ (midplane), assuming axisymmetry, is given by\n\\begin{equation}\\label{e:mhd-R}\n\\frac{\\partial (\\rho v_R)}{\\partial t} + \\nabla \\cdot \\left( \\rho v_R \\vec v - \\frac{B_R \\vec B}{4\\pi}  \\right) + \\frac{\\partial P_t}{\\partial R} = \\rho g_R + \\frac{\\rho v_\\phi^2 - B_\\phi^2\/(4\\pi)}{R},\n\\end{equation}\nwhere the divergence operator $\\nabla \\cdot$ acts according to\n\\begin{equation}\\label{e:nablaop}\n\\nabla \\cdot \\vec f(R,z) = \\frac{1}{R}\\frac{\\partial (R f_R)}{\\partial R} + \\frac{\\partial f_z}{\\partial z}\n\\end{equation}\nonto the vector fields $\\vec f(R,z) \\in \\{ \\rho v_R \\vec v, B_R\\vec B\\}$; $\\vec g$ is the total gravitational force from the star and self-gravity, and $P_t$ is the sum of the thermal and magnetic pressures.\n\n\n\nIn order to examine the forces that define radial equilibrium in the disk, we consider a parcel of gas moving in a circular orbit, for which we set $v_R = 0$ in eq. \\ref{e:mhd-R} and obtain\n\n\\begin{equation}\\label{e:mhd-R2}\n \\frac{\\rho v_\\phi^2 - B_\\phi^2\/(4\\pi)}{R} = -\\rho g_R + \\frac{\\partial P}{\\partial R} +  \\frac{\\partial}{\\partial R}\\left(\\frac{B^2}{8\\pi}\\right) -\\nabla \\cdot  (B_R \\vec B).\n\\end{equation}\n\nFor $\\vec B = \\vec 0$, $P = 0$, and considering the midplane, one recovers the well-known condition for gravito-centrifugal equilibrium (on the co-rotating frame), which defines the Keplerian velocity, $v_K = \\sqrt{GM(r)\/R}$; $M(r)$ being the enclosed mass (dominated by the mass of the protostar for small $r$). The use of the enclosed mass is an approximation to the use of the full potential given by the Poisson equation.\n\n\\subsubsection{Keplerianity}\nThe degree of gravito-centrifugal equilibrium in the whole disk can be measured by computing the ratio of the centrifugal force density to the gravitational force density:\n\\begin{equation}\n\\frac{a^\\mathrm{c}_R}{\\rho g_R} = \\frac{\\rho v_\\phi^2}{r\\sin\\theta} \\cdot \\frac{r^2}{\\rho G M(r) \\sin\\theta} = \\frac{v_\\phi^2}{v_K^2\\sin^3\\theta} .\n\\end{equation}\nThis motivates us to define the offset from Keplerianity as\n\\begin{equation}\n\\text{offset from Keplerianity} = \\left|\\frac{v_\\phi}{v_K\\sin^{3\/2}\\theta}\\right|-1,\n\\end{equation}\nwhich in the midplane corresponds to the fractional difference of the azimuthal velocity with respect to the Keplerian value.\n\n\nFigs. \\ref{thindisk_keplerianity} and \\ref{thickdisk_dynamics} show the offset from Keplerianity presented as a percentage, in such a way that the negative values indicate sub-Keplerianity, and the positive values, super-Keplerianity. In the midplane, there is a Keplerian region within $\\pm 25 \\%$ for all times, which corresponds to the thin accretion disk. The Keplerianity drops outside of the disk, which corresponds to the infalling envelope. At the exterior boundary of the disk, the mostly-rotating material encounters the mostly-infalling material from the region of the envelope, creating a centrifugal barrier that is seen as a slight super-Keplerianity in Fig. \\ref{thindisk_keplerianity}. Even though we ignore the effect of viscosity in the analysis of this section, we note that, as it transports angular momentum outwards, its effect is to increase positively the deviation from Keplerianity. The rest of the deviations from Keplerianity in the disk are discussed in the following sections.\n\nPanels \\emph{b} and \\emph{f} of Fig. \\ref{thickdisk_dynamics} contrast the values of the offset from Keplerianity in the midplane with values in planes parallel to it, which slice through the thick layer of the disk and reveal its dynamical structure. In general, the thick layer also shows Keplerianity within $\\pm 25\\%$, but it decreases with altitude because the cylindrical-radial component of gravity also decreases with altitude. The regions close to the cavity become increasingly super-Keplerian, while the regions close to the infalling envelope become increasingly sub-Keplerian.\n\n\\subsubsection{Specific energies}\nWe study the dynamics of the thin and thick layers of the disk by computing the terms of the Bernoulli equation, that is, the contributions to the specific energy of the material. This allows us to isolate the role of each term in the equations of magnetohydrodynamics and compare its relative importance. The results of the energy calculations are available in Fig. \\ref{thindisk_energies} for the midplane, and Fig. \\ref{thickdisk_dynamics} for the plane $z=z_1$. The contributions to the specific energy we considered are:\n\\begin{itemize}\n\\item the gravitational specific energy \\begin{equation}e^\\text{grav} = \\frac{GM(r)}{r},\\end{equation}\n\\item the thermal specific energy \\begin{equation} \\label{e:energies-thermal}\ne^\\text{th} = \\frac{P}{\\rho (\\Gamma - 1)},\n\\end{equation}\n\\item the contribution to the specific kinetic energy by the azimuthal component of velocity \\begin{equation}\ne^{K}_\\phi = \\frac{v_\\phi^2}{2},\n\\end{equation}\n\\item the contribution to the specific kinetic energy by the spherical-radial component of velocity\n\\begin{equation}\ne^K_r = \\frac{v_r^2}{2},\n\\end{equation}\n\\item the contribution to the specific magnetic energy by the toroidal component of the magnetic field\n\\begin{equation}\ne^B_\\text{tor} = \\frac{B_\\phi^2}{8\\pi \\rho},\n\\end{equation}\n\\item the contribution to the specific magnetic energy by the poloidal component of the magnetic field\n\\begin{equation}\ne^B_\\text{pol} = \\frac{B_r^2 + B_\\theta^2}{8\\pi \\rho}.\n\\end{equation}\n\\end{itemize}\n\n\n\nInitially, the specific gravitational energy increases as a function of $r$ because it is given by the enclosed mass (see Fig. \\ref{thindisk_energies}a). As the collapse progresses (Fig. \\ref{thindisk_energies}b), the gravity of the forming massive star becomes more important, which means that the curve for $e^\\text{grav}$ has a point-source-like behavior close to the center of the cloud. At large scales, the enclosed mass of the envelope becomes important. As expected, the infalling envelope has $e^K_r > e^K_\\phi$, but the advection of angular momentum from large to small scales causes $e^K_\\phi$ to increase until $e^K_\\phi > e^K_r$, forming the disk. When the magnetic field lines are dragged by rotation in the disk, the toroidal contribution to the magnetic energy increases.\n\n\\subsubsection{Thermal and magnetic pressure} \\label{S: dynamics - PB}\n\nThe following two sections examine the contributions of the remaining terms of eq. \\ref{e:mhd-R2} to the equilibrium of cylindrical radial forces in the disk and their impact to the values of Keplerianity shown in Fig. \\ref{thindisk_keplerianity}.\n\n\n\nFirst, we only consider the additional radial support that the thermal pressure gradient $\\partial P\/\\partial R$ gives to the material in the disk, which be inferred by using eq. \\ref{e:energies-thermal} and Figs. \\ref{thindisk_energies} and \\ref{thindisk_denstemp}. In agreement with the theory, we find that the thermal pressure gradient provides an additional small support against gravity and therefore makes material in radial equilibrium to appear to be slightly sub-Keplerian.\n\n\n\nIn Gaussian units, the magnetic pressure and magnetic energy density correspond to the same quantity, which means that we can use the specific energy density to study the magnetic pressure support in the disk. In the infalling envelope, the poloidal component of the magnetic field contributes the most to magnetic pressure (see, e.g., Fig. \\ref{thindisk_energies}b), given that the initial magnetic field distribution is parallel to the rotation axis. When the disk is formed, the toroidal component of the magnetic field increases in general because the magnetic field lines are dragged by rotation. The toroidal magnetic field becomes dominant over the poloidal component, which additionally allows us to neglect the magnetic tension term $\\nabla \\cdot (B_R \\vec B)$ from eq. \\ref{e:mhd-R2} in most parts of the disk.\n\n\nThe magnetic pressure $P_B = \\rho (e^B_\\text{tor} + e^B_\\text{pol})$ decreases with distance in the infalling envelope (compare Figs. \\ref{thindisk_energies}b and  \\ref{thindisk_denstemp}a), and overall in the disk as well, although the more complex structure of the magnetic field in the thin and thick layers of the disk means that there are regions of local increase with distance. Similarly to our argument with the thermal pressure gradient, $v_\\phi < v_K$ for a negative magnetic pressure gradient. Therefore, the radius of the disk should be slightly larger compared to the non-magnetic case, given that $v_\\phi$ decreases with distance and the angular momentum contained at larger $R$ becomes sufficient to support a circular orbit. This argument is revisited in Sect. \\ref{S: disk var B} after considering the effects of magnetic braking (Sect. \\ref{S: disk MB}) and magnetic diffusion (Sect. \\ref{S: disk m diff}).\n\n\nAdditionally to magnetic pressure, the second term of the left hand side of eq. \\ref{e:mhd-R2} reveals that the azimuthal component of the magnetic field reduces the effectiveness of the centripetal force required to keep a circular orbit. This can be measured by comparing the azimuthal component of the kinetic and magnetic energies:\n\\begin{equation}\n\\frac{B_\\phi^2}{4\\pi R} \\cdot \\frac{R}{\\rho v_\\phi^2}  = \\frac{e^B_\\text{tor}}{e^K_\\phi}.\n\\end{equation}\nIn the thin layer of the disk, the contribution of this term is negligible, but in the thick layer, they are comparable. In terms of Keplerianity, this term can cause the material to be slightly super-Keplerian; this can be observed when comparing panels \\emph{f} and \\emph{g} of Fig.  \\ref{thickdisk_dynamics}.\n\n\n\n\\subsection{Ohmic dissipation} \\label{S: disk m diff}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{imgraw\/Rem}\n\\caption{Magnetic Reynolds number, computed with the fiducial simulation on grid x8. For $t=10\\unit{kyr}$, the radius of the disk is around $80\\unit{au}$, while for $t=22.75\\unit{kyr}$, the exterior radius is $\\approx 700\\unit{au}$ and the interior radius is $\\approx 20\\unit{au}$.}\n\\label{Rem}\n\\end{figure}\n\n\n\nAccording to the model of Ohmic resistivity by \\cite{Machida2007}, which we use, the resistivity increases progressively with density for the number densities we obtain here ($n_H \\lesssim 10^{13}\\unit{cm^{-3}}$). We computed the ratio of the magnitude of the induction and diffusion terms in the induction equation, which is analogous to the magnetic Reynolds number, as\n\\begin{equation}\\label{e:Rem}\n\\mathrm{Re}_m = \\frac{|\\vec \\nabla \\times (\\vec v \\times \\vec B)|}{|\\eta \\vec \\nabla \\times  (\\vec \\nabla \\times \\vec B)|},\n\\end{equation}\nand show the results in Fig. \\ref{Rem}, which are computed for two instants of time from the fiducial case. At $t=10\\unit{kyr}$, the plasma is fully diffusive in most regions of the thin layer of the disk; the thick layer is dominated by advection but it experiences magnetic diffusion in around one part per hundred; and the cavity wall is also partially diffusive. Later in time, at $t=22.75\\unit{kyr}$, only the highest density regions close to the massive protostar are dominated by diffusion, the thin layer experiences magnetic diffusion of around one part per hundred, and the thick layer is almost completely dominated by advection.\n\n\nOhmic resistivity acts mostly on the toroidal component of the magnetic field, as it can be seen by comparing the energy curves in Figs. \\ref{thindisk_energies} and \\ref{thickdisk_dynamics}. In the thick layer of the disk at late times (Fig. \\ref{thickdisk_dynamics}g) $e_\\text{tor}^B$ roughly follows $e_\\phi^K$. This sets the picture for the case where magnetic diffusion is low. On the other hand, in the thin layer of the disk, at early times (Fig. \\ref{thindisk_energies}b), $e_\\text{tor}^B$ and $e_\\phi^K$ are clearly decoupled: the toroidal magnetic field component becomes low inside of the disk, corresponding to the region where diffusion dominates over advection. In the outer disk, where the density is lower, the resistivity is lower as well, and the toroidal component of the magnetic field is allowed to increase. \n\n\n\\subsection{Magnetic braking} \\label{S: disk MB}\nThe momentum equation in the azimuthal direction, considering axisymmetry and circular motion, reduces to\n\\begin{equation}\n\\frac{\\partial (\\rho v_\\phi)}{\\partial t}  =  \\frac{1}{R^2}\\frac{\\partial}{\\partial R}(R^2 B_\\phi B_R) + \\frac{\\partial}{\\partial z}(B_\\phi B_z).\n\\end{equation}\nThe right hand side of this equation is equal to zero if there are no magnetic fields, and it leads to the conservation of angular momentum. The terms on the right hand side constitute the azimuthal component of the magnetic tension force, which exerts a torque that reduces angular momentum locally. This process is known as magnetic braking, and it can be understood as a consequence of the dragging of magnetic field lines by rotation \\citep{Galli2006}.\n\nEven though Ohmic resistivity reduces the toroidal component of the magnetic field (cf. $e^B_\\text{tor}$ in Fig. \\ref{thindisk_energies}b), it does not completely suppress it. Over time, the magnetic field lines are wound enough so that magnetic braking happens in the innermost parts of the disk. In Fig. \\ref{thindisk_energies}c, which corresponds to $t=22.75\\unit{kyr}$,  $e^B_\\text{tor}$ is higher in the disk than when it is measured at $t=10\\unit{kyr}$, while $e^K_\\phi$ decreases over the same interval of time in the inner $20\\unit{au}$; those quantities show the reduction of angular momentum due to magnetic tension. In the face of the reduction of angular momentum, the material falls toward the protostar, as evidenced by the increase in $e^K_r$; this creates the magnetically-braked region. The cavity wall and the thick layer of the disk are also affected by magnetic braking; material from several directions is then delivered to the magnetically-braked region.\n\n\n\\subsection{Vertical support}\n\nIn the vertical direction (i.e., parallel to the rotation axis), the momentum equation reads\n\\begin{equation}\\label{e:mhd-z}\n\\frac{\\partial (\\rho v_z)}{\\partial t} + \\nabla \\cdot \\left(\\rho v_z \\vec v  - \\frac{B_z \\vec B}{4\\pi}\\right) + \\frac{\\partial P_t}{\\partial z} = \\rho g_z\n\\end{equation}\nwith the same definition of the divergence as given in eq. \\ref{e:nablaop}. Considering a parcel of gas moving in a circular orbit ($v_z=0$), and given that in the disk region $B_z \\ll B_\\phi$, eq. \\ref{e:mhd-z} becomes\n\\begin{equation}\n\\frac{\\partial P}{\\partial z} + \\frac{\\partial}{\\partial z} \\left( \\frac{B^2}{8\\pi} \\right) = \\rho g_z.\n\\end{equation}\n\n\nThe ratio of the  thermal to magnetic pressure is equivalent to the ratio of the specific thermal energy to the total magnetic energy. In the thin layer of the disk (Fig. \\ref{thindisk_energies}), thermal pressure dominates over magnetic pressure, from which we conclude that the thin layer is supported mainly by thermal pressure (at late times, magnetic pressure becomes comparable but not much higher than thermal pressure). On the contrary, the thick layer of the disk is supported by magnetic pressure, as evidenced by Figs. \\ref{thickdisk_dynamics}d and \\ref{thickdisk_dynamics}h, where we present a direct comparison of both pressure gradients to gravity in a vertical slice that crosses the disk. As the vertical component of gravity vanishes at the midplane, both curves increase towards $z=0$. In the thick layer of the disk, thermal pressure decreases, while the magnetic pressure gradient compensates gravity. At the cavity wall, thermal pressure increases because it has a higher density than the thick layer of the disk. Finally, in the cavity, magnetic pressure dominates over gravity in the vertical direction, which is expected in the case of a magnetically-driven outflow.\n\n\n\\subsection{Effects of other forms of magnetic diffusion} \\label{S: other magn diff}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{imgraw\/brhox16-700.png}\n\\caption{Distribution of the magnetic field strength as a function of density. This histogram was computed with the fiducial simulation run on grid x16, for time $t=7\\unit{kyr}$. The color scale indicates the polar angle: the dark purple colors correspond to the midplane (with the disk corresponding to $\\rho \\gtrsim 10^{-15} \\unit{g\\,cm^{-3}}$) and the light blue colors correspond to the rotation axis (where the jet is located).}\n\\label{brho}\n\\end{figure}\n\nIn this study, the effects of ambipolar diffusion and the Hall effect are not considered, and this formally constitutes a caveat of our approach which we plan to address in a future study. In this section, we discuss the expected effects that other forms of magnetic diffusivity might have on the results reported in the previous sections, with a focus on the fiducial case. We examine every region of interest for the discussion: the accretion disk, the infalling envelope and the outflow cavity. The simulation catalog also includes an investigation of the effects of the resistivity model; the results can be used to further assess the missing magnetic diffusivity and are presented in Sect. \\ref{S: disk var eta}.\n\nFirst, we consider only the accretion disk. The curves for ambipolar diffusivity and Ohmic resistivity given in e.g. \\cite{Marchand2022,Marchand2016} and \\cite{Tsukamoto2020} reveal that, at least for the range of number densities $10^8 \\lesssim n_H \\lesssim 10^{13}\\unit{cm^{-3}}$, which are found in the thin layer of the accretion disk, the values of the diffusivities are similar within the uncertainties of the dust models considered for their derivation, differing by about a factor of ten. The distribution of the magnetic field strength with density is presented in Fig. \\ref{brho}, where we use a color scale that highlights with darker colors the midplane. For the densities of the accretion disk ($\\rho > 10^{-14}\\unit{g\\,cm^{-3}}$), the magnetic field strength increases with density, until it reaches between $0.1$ and $1\\unit{G}$. When comparing the magnetic field distribution to the results of a recent study by \\cite{Commercon2022}, who considered the effects of ambipolar diffusion but not Ohmic dissipation, we find consistent results for the thin layer of the disk. In the thick layer, however, the addition of higher, more realistic magnetic diffusivities might reduce the vertical magnetic pressure support. This means that thinner accretion disks than those reported here would be expected in a more realistic setting.\n\nAs for the effects of magnetic braking, we expect that ambipolar diffusion and the Hall effect reduce the magnetic Reynolds number, that is, the material would become more diffusive. Contrary to Ohmic dissipation, ambipolar and Hall diffusivities depend on magnetic field, and so the reduction could be of particular importance at later times, when considerable magnetic flux has been transferred to the central regions of the cloud core. This would cause a delay in the formation of the magnetically-braked region in the inner disk (see also the discussion in Sect. \\ref{S: disk var eta}). On the other hand, magnetic braking could play an important role in determining the total angular momentum transferred to the forming massive star, as well as the growth of HII regions by terminating the magnetically-driven outflows (Martini et al., in prep.). An investigation of the role of all forms of magnetic diffusivity in the long-term evolution of the accretion disk and the jet is necessary. However, at least for early times ($t\\lesssim 20\\unit{kyr}$) in the formation of the massive star, our results fit observational constraints (see Sect. \\ref{S: obs}).\n\n\nAs a second region of interest, we consider the infalling envelope. According to the curves by \\cite{Marchand2022}, ambipolar diffusivities tend to be higher at lower densities ($n_H < 10^{8} \\unit{cm^{-3}}$). This suggests that we are missing key magnetic diffusion during the early phases of gravitational collapse, where densities are low. However, there are theoretical and observational results that suggest that the collapse phase is still reasonably modeled in our simulations. In this simulation series, we consider the case of weak magnetic fields (see the discussion on this assumption in Sect. \\ref{S: disk var B}). Because ambipolar diffusivities increase with magnetic field strength, we expect magnetic diffusion to play a stronger role in cases when the magnetic fields are initially strong. In the studies by \\cite{Commercon2022} and \\cite{Mignon-Risse2021}, who considered $\\bar \\mu = 5$ and ambipolar diffusion but no Ohmic dissipation, the magnetic field distribution for low densities coincides well with what is depicted in Fig. \\ref{brho}. Based on this, we would not expect a strong qualitative divergence of our results for the gravitational collapse epoch ($t \\lesssim 5 \\unit{kyr}$) upon the addition of ambipolar diffusion to our models. From the observational point of view, the role of ambipolar diffusion as a necessary enabler for gravitational collapse in a magnetized cloud core was recently questioned in \\cite{2022Natur.601...49C}. These latter authors found a supercritical low-mass prestellar core, which is not yet self-gravitating. This means that the magnetic field was diffused before the gravitational collapse takes place, that is, earlier than predicted by the classical picture of ambipolar diffusion turning a subcritical core supercritical and in a consistent way with our simplification.\n\n\nLastly, we take a look at the magnetic diffusivities in the outflow cavity, which is the focus of Paper II. Observations of protostellar jets have found evidence of ionized material \\citep[see e.g.,][]{Moscadelli2021multi, Rodriguezkamenetzky2017, Carrasco-Gonzalez2021, Guzman2010}, possibly from shock ionization. As discussed in \\cite{Marchand2016,Marchand2022}, ionization leads to a general decrease in the magnetic diffusivities. So far, the models of ambipolar diffusion and the Hall effect done in e.g., \\cite{Marchand2016,Marchand2022} and \\cite{Tsukamoto2020} do not consider the shock ionization in the outflow cavity. Material in the outflow cavity and close to the massive protostar is very low density and is in presence of strong magnetic fields, which means that neglecting shock ionization would overestimate magnetic diffusivities coming from the ambipolar diffusion and Hall effect terms, when comparing to the curves by \\cite{Marchand2022}. By only considering Ohmic dissipation (for which the resistivity is low for low densities and independent of magnetic field strength), the material in the cavity behaves closer to ideal MHD theory, in agreement to what has been found from observational evidence (see Sect. \\ref{S: obs} and \\cite{Moscadelli2022}). This can be seen in Fig. \\ref{brho}, where the magnetic field strengths in the cavity (light-colored dots) are able to increase while keeping the values constrained in the disk (dark-colored dots). A realistic treatment of the problem, however, would require the consideration of all forms of magnetic diffusivity (which would probably impact the disk dynamics) and the effect that thermal ionization and shock ionization have on the magnetic diffusivities corresponding to the outflow cavity.\n\n\n\n\\subsection{Density and temperature profiles}\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{imgraw\/x8-profiles}\n\t\\caption{(a) Density and (b) temperature profiles on the midplane, for the fiducial simulation on grid x8.}\n\t\\label{thindisk_denstemp}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{imgraw\/thickdisk_denstemp.pdf}\n\t\\caption{(a) Density and (b) temperature for two different altitudes in the thick layers of the disk (blue lines), and the corresponding profile for the thin layer (black). The data corresponds to the fiducial simulation on grid x8.}\n\t\\label{thickdisk_denstemp}\n\\end{figure}\n\n\nFigure \\ref{thindisk_denstemp} presents the density and temperature profiles for the midplane. The thin layer has densities over $10^{-16} \\unit{g\\,cm^{-3}}$, and they reach over $10^{-11}\\unit{g\\,cm^{-3}}$ at the inner boundary. For $t\\lesssim 15\\unit{kyr}$, the density profile of the thin layer follows an approximate power law. Later, the magnetically-braked region is formed, corresponding to the high density feature observed for $r \\lesssim 30 \\unit{au}$ in Fig. \\ref{thindisk_denstemp}a. Some of the material of the magnetically-braked region is drawn from the thin layer of the disk: the density profile of the thin layer reveals the existence of a plateau close to its magnetic braking radius, which coincides as well with the decrease in the azimuthal contribution to the specific kinetic energy due to magnetic braking (Fig. \\ref{thindisk_energies}c). However, another part of the material in the magnetically-braked region corresponds to the inflows from the cavity wall and thick layer of the disk, both of which also experience magnetic braking and continuously replenish it.\n\n\nThe temperature profile of the thin layer follows roughly a power law of the form $T\\propto r^{-1\/2}$, and it experiences a general increase over time. The transition between the disk and the envelope is accompanied by a change in the temperature gradient; the transition between the magnetically-braked region and the thin layer of the disk is more subtle. We remind the reader that the present simulations do not include the effect of irradiation from the star, for which we anticipate two additional features not present in Fig. \\ref{thindisk_denstemp}b, namely, an increase in temperature due to the flux from the forming massive star, and the existence of the dust evaporation front \\citep[see][]{Kuiper2010circ}. Those features are only relevant for $t\\gtrsim 20 \\unit{kyr}$.\n\n\n\nFig. \\ref{thickdisk_denstemp}a presents the density profile at two different altitudes crossing the thick layer of the disk at the time $t=10\\unit{kyr}$ (when $M_\\star = 3 \\unit{M_\\odot}$), \n accompanied by a comparison with the density in the midplane. Differently than the thin layer of the disk, the thick layer has a nearly uniform density of $\\sim 10^{-14}\\unit{g\\, cm^{-3}}$. The density drop for $r\\lesssim 10 \\unit{au}$ corresponds to the outflow cavity. The mean density in the thick layer decreases slightly with time, however, it roughly remains within the same order of magnitude throughout the simulation.\n\n\nBecause the density is nearly uniform, the temperature of the thick layer of the disk remains also relatively uniform, at around $200\\unit{K}$ in comparison to the power-law temperature of the thin layer. This means that it should be possible in principle to observe the vertical stratification of the disk by distinguishing the line emission from each layer.\n\n\n\\section{Dependence of the resulting disk properties on initial cloud properties} \\label{s:parameter-study}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{comp-img\/rdisk-norm}\n\t\\caption{Radius of the disk ($R_\\text{disk}$) and its magnetic braking radius ($R_\\text{mb}$) for different initial values of the mass of the cloud, magnetic field, density profile, rotational profile, rotational energy and resistivity models. The transparent lines in the panels that show $R_\\text{mb}$ indicate the full data, while the solid lines show the moving average. The colored dots in panel A4 indicate $t=30\\unit{kyr}$ in each simulation. For all the panels in rows B and C, $M_C=100\\unit{M_\\odot}$ was used.}\n\t\\label{rdisk}\n\\end{figure*}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{comp-img\/mstar-norm}\n\t\\caption{Mass of the protostar, corresponding to the mass in the sink cell, for several values of the (\\emph{a}--\\emph{b}) mass of the cloud, (\\emph{c}) resistivity model, (\\emph{d}) mass-to-flux ratio, (\\emph{e}) density profile, and (\\emph{f}) rotation profile and rotational energy. Panel \\emph{b} shows the same results as panel \\emph{a}, but normalized in such a way that scalability can be readily seen. For panels (c)--(f), $M_C = 100\\unit{M_\\odot}$ and $t_\\text{ff} = 53.73\\unit{kyr}$.}\n\t\\label{mstar}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{comp-img\/disk_morph}\n\t\\caption{Density structure (column 1) and specific energies (column 2) of the disk for selected values of the normalized values of the mass-to-flux ratio (rows B and C), and rotation profile (column D). All the snapshots were taken at $t=15\\unit{kyr}$ of evolution. All cases consider a rotational-to-gravitational energy of 4\\%.}\n\t\\label{disk_morph}\n\\end{figure}\n\nWe perform a series of simulations to explore the large parameter space of initial conditions. We vary the fiducial model model described above in terms of the initial mass of the cloud core, its density and rotation profiles, its rotational-to-gravitational energy ratio, its initial magnetic field strength and probe the effect of different models of Ohmic resistivity. In this section, we discuss how those changes affect the processes discussed in Sect. \\ref{S: dynamics}. As a probe, we use the radial extent of the Keplerian-like disk (between the magnetic braking radius and the [outer] radius), defined as the region where the material is within $\\pm 25\\%$ Keplerianity, and the results of which are presented in Fig. \\ref{rdisk} for the whole parameter space. Similarly, the mass of the forming massive star is presented in Fig. \\ref{mstar} for the whole parameter space. We also examine differences in the density structure of the disk for selected values of the parameters, and relate them to the dynamical information revealed by the specific energy; the results are shown in Fig. \\ref{disk_morph}.\n\n\\subsection{Dependence on cloud core mass} \\label{S: disk var M}\n\nCloud cores of masses of $50, 100, 150$ and $200 \\unit{M_\\odot}$ were considered, which form stars ranging masses from $6$ to $75\\unit{M_\\odot}$ after $30\\unit{kyr}$ of evolution (see Fig. \\ref{mstar}a). Further evolution and additional physical effects which are not considered here would be necessary to determine the final mass of the star. When changing the initial core mass of the fiducial model, we also change the normalization of the rotation profile in order to keep the rotational-to-gravitational energy ratio as in the fiducial model. The mass and size of the cloud core determines its free-fall timescale\n\\begin{equation}\n\tt_\\text{ff} = \\left(\\frac{\\pi^2 R_C^3}{8GM_C}\\right)^{1\/2},\n\\end{equation}\nwhich is $52.4\\unit{kyr}$ for the fiducial case of $M_C = 100\\unit{M_\\odot}$ and $R_C = 0.1\\unit{pc}$. Panels A1 and A2 of Fig. \\ref{rdisk} show that the disk and the magnetically-braked region are smaller for the lower-mass cases when plotted against time, however, when investigated as a function of the stellar mass as a fraction of the initial mass of the cloud (panels A3 and A4), both the size of the disk and the magnetic braking radius coincide well. We find that resulting curves roughly follow the empirical linear relation\n\\begin{equation}\n R_\\text{disk}(m) \\approx R_\\text{onset} + R_\\infty \\left( m - m_\\text{onset} \\right) ,\n\\end{equation}\nwhere we define\n\\begin{equation}\nm \\equiv \\frac{M_\\star}{M_C}.\n\\end{equation}\n$R_\\text{onset} \\equiv 3\\unit{au}$ is the minimum radius of the disk we can detect in the simulation data (the size of the sink cell), $m_\\text{onset}=0.016$ is the corresponding normalized mass of the protostar for $R_\\text{onset}$, and $R_\\infty=6380\\unit{au}$ can be interpreted as the approximate value of the radius of the disk at the hypothetical end of the collapse where $m\\sim 1$. This latter value will never be reached in reality because of stellar feedback: \\cite{KuiperHosokawa2018}, e.g., found that the disk lifetime is expected to be of only a few $10^5\\unit{yr}$ (see also  \\citealt{Kee2019} for the conditions for limiting accretion through the disk). A similar empirical disk size formula was previously offered in \\cite{Kuiper2011}.\n\nWe also verified that the mass of the protostar, expressed as a fraction of the mass of the cloud, scales with time presented as fraction of the free-fall timescale (cf. panels \\emph{a} and \\emph{b} of Fig. \\ref{mstar}). This result indicates that despite the differences in the scale-dependent microphysics, namely, the radiative transfer, resistivity and self-gravity, the disk scales well with the free-fall timescale, which allows our results to be used with a more ample range of cloud masses than what is presented here. Defining $\\tau \\equiv t\/t_\\text{ff}$, we also find an approximate fit for the mass of the protostar with a power law:\n\\begin{equation}\n  m(\\tau) \\approx k_1\\, \\tau^{k_2},\n\\end{equation}\nwhere $k_1 \\approx 0.56$ and $k_2 \\approx 1.8$.\n\n\\subsection{Ohmic resistivity} \\label{S: disk var eta}\n\nThe Ohmic resistivity model controls the diffusivity of the plasma, and therefore it has an impact on magnetic braking. We studied the effects of using the resistivity model of \\cite{Machida2007} in two versions: the full expression dependent on both density and temperature, and the same expression but with a fixed temperature of $\u00ba   T=10\\unit{K}$, as it was used in the isothermal study of \\cite{Anders2018}. We denote the first one as $\\eta(\\rho,T)$ and the second one as $\\eta (\\rho)$ given the near linear behavior of the resistivity in the range of densities for which we are interested. We estimate that the version that considers the temperature dependence differs at most by a factor of $100$ in regions close to the protostar with respect to the version with the fixed temperature. We compare our results as well with the ideal MHD case ($\\eta = 0$) and the non-magnetic (hydrodynamical) case ($\\eta\\to \\infty$). In order to mimic the magnetic diffusivity expected in the disk region if ambipolar diffusion were taking into account, we also include a simulation with an artificially high Ohmic resistivity (ten times the temperature-independent value, denoted as $10\\eta(\\rho)$). This factor is motivated by the difference in the curve for ambipolar diffusivity and Ohmic resistivity in \\cite{Marchand2022}, for disk densities. There is an additional difference between the simulations of this series: while most simulations of this study consider a constant dust and gas opacity of $1\\unit{cm^2\\, g^{-1}}$, the simulation with $\\eta(\\rho,T)$ uses the dust opacity table from \\cite{LaorDraine1993} and a constant gas opacity of $10^{-2}\\unit{cm^2\\, g^{-1}}$.\n\n\n\nFor the rest of Sect. \\ref{s:parameter-study}, we present the size of the disk as a function of the stellar mass as a fraction of the cloud mass, instead of time, because the calculation of the offset from Keplerianity (which defines the disk) is dependent on the gravity of the (proto)star, and by fixing it we are able to compare and assess the effects of the other terms that influence the radial equilibrium of the disk, namely, the angular momentum and pressure support. Additionally, our results can be scaled as a result of the discussion in Sect. \\ref{S: disk var M}.\n\n\nIn the panel B1 of Fig. \\ref{rdisk}, we observe that the magnetically-braked region develops earlier (when the mass of the star is lower, cf. Fig. \\ref{mstar}c) in the ideal MHD case, as compared to the corresponding curves for the simulations that consider resistivity. The case with $\\eta(\\rho)$ requires a more massive protostar (more time evolution) for the magnetic braking radius to appear, a trend that is confirmed with the simulations that consider higher resistivity: the temperature-dependent resistivity delays the appearance of magnetic braking, while the simulation with the artificial factor of ten in resistivity delays it even further. As expected, the disk in the case with no magnetic fields does not develop a magnetic braking radius. A disk is able to form at all in the ideal MHD case because the relatively weak magnetic field considered for this parameter scan.\n\n\nFrom this evidence, we infer that the presence of resistivity delays the action of magnetic braking compared to the results only considering ideal MHD. Higher resistivities delay magnetic braking more, however, as discussed in Sect. \\ref{S: disk MB}, because magnetic diffusion does not completely suppress the toroidal magnetic field, the Lorentz force continues to grow until enough magnetic tension force is built to decelerate the material. As a result, the innermost parts of the disk lose their centrifugal support.\n\n\nThe disk radius (Fig. \\ref{rdisk}B2) is in general larger in the magnetic cases in comparison to the non-magnetic case. Moreover, the disk radius of the run with no resistivity (ideal MHD) is marginally larger than the ones with resistivity. Magnetic braking removes angular momentum from the disk because of the torque produced by magnetic tension; therefore, a smaller disk would be expectable. However, magnetic tension is highest in the inner disk, where the magnetic field lines are wound the most, and not in the outer disk. As discussed in Sect. \\ref{S: dynamics - PB}, magnetic pressure can provide an additional radial support for the disk. This means that in the ideal MHD case, where the toroidal magnetic field is high in the disk, the additional magnetic pressure supports a larger disk. Magnetic diffusion lowers the toroidal magnetic field, and therefore the disk does not have radial magnetic pressure support: this can be seen at early times (where $M_\\star < 5\\unit{M_\\odot}$ or $m = 0.05$) where the simulations with $\\eta(\\rho,T)$ and $10\\eta(\\rho)$ yield a disk radius that almost matches the one obtained in the non-magnetic case. As the toroidal magnetic field increases over time due to the winding of magnetic field lines, the magnetic pressure support also increases and the disk becomes larger than the non-magnetic case.\n\n\nGiven that magnetic braking delivers mass onto the protostar, we investigate the role of resistivity in the mass of the forming massive star. The corresponding curves for the mass of the protostar as a function of time are shown in Fig. \\ref{mstar}c. The results for the resistive and non-resistive cases are very similar. The stellar mass, however, increases more in the non-magnetic case in comparison to the magnetic cases. This happens because the protostar in the non-magnetic case accretes material not only through the disk, but also through the infalling envelope along the bipolar directions. On the contrary, in the magnetic cases, there are magnetic outflows that impede accretion through the axis, resulting in a slightly lower mass.\n\n\n\\subsection{Initial magnetic field strength} \\label{S: disk var B}\n\n\nWe study values of the normalized mass-to-flux ratio ranging from $\\bar \\mu = 2$ (strongest magnetic field) to $\\bar\\mu = 40$ (weakest magnetic field).%\n\n\nThe results of the radial extent of the centrifugally-supported disk are shown in the panels B1 and B2 of Fig. \\ref{rdisk}. The curve for $\\bar \\mu = 2$ goes quickly to zero, which means that the structure formed is strongly sub-Keplerian and it is not classified as a Keplerian-like disk according to our criterion. A more detailed study of the specific energies revealed that the structure has a comparable radial and azimuthal velocities. Our result of no disk for $\\bar \\mu = 2$ should be taken with caution, because we do not include ambipolar diffusion in our calculations. As ambipolar diffusivities increase with magnetic field (contrary to Ohmic dissipation), the cases with strong magnetic fields are missing magnetic diffusivity that may allow the disk to form (see the discussions in Sects. \\ref{S: other magn diff} and \\ref{S: previous amb diff}).\n\n\nThe other values of the mass-to-flux ratio confirm that magnetic braking is responsible for the formation of the magnetically-braked region because higher initial magnetic fields build enough magnetic tension earlier. Additionally, the magnetically-braked region is larger for stronger magnetic fields. The radius of the disk, on the other hand, is larger for stronger magnetic fields, which supports the hypothesis that higher magnetic pressure provides additional support against gravity and allows material from larger radii (with lower angular momentum) to be part of the disk. As for the accretion onto the protostar, Fig. \\ref{mstar}d reveals that higher magnetic fields do tend to deliver more mass onto the protostar, due to stronger magnetic braking at late stages.\n\n\nThe rows A--C of Fig. \\ref{disk_morph} present a comparison of the morphological differences of the disk when it has evolved for $t=15\\unit{kyr}$ under different values of the mass-to-flux ratio. For $\\bar \\mu = 40$, the weaker magnetic field produces a disk with a thick layer that is smaller than the thin layer. Both regions of the disk are of the same size for $\\bar \\mu = 20$, which confirms magnetic pressure as the nature of the vertical support in the thick disk. However, the part of the thin layer that is not enveloped by the thick layer of the disk becomes inflated by magnetic pressure, as the curve for the toroidal contribution to the specific magnetic energy reveals in Fig. \\ref{disk_morph}B2: in the outer disk ($r\\sim 100\\unit{au}$), $e^B_\\text{tor}$ becomes higher than $e^\\text{th}$. Nevertheless, a similar analysis of the specific energies at later times reveals that the thick layer of the disk grows over time until it completely envelops the thin layer (which happens at around $t\\sim 19\\unit{kyr}$). At $t\\approx 25.5\\unit{kyr}$, the thin and thick disk layers of the disk of the case with $\\bar \\mu = 40$ become morphologically and dynamically similar to the results shown in panels A1 and A2 corresponding to the $\\bar \\mu = 20$ at $15\\unit{kyr}$; the difference being the size of the disk and the mass of the protostar. For a stronger initial magnetic field ($\\bar \\mu = 5$, row C of Fig. \\ref{disk_morph}), the thick layer becomes thicker because of the increased magnetic pressure.\n\n\\subsection{Initial density profile}\nA steeper initial density profile accelerates the accretion onto the protostar at early times because of the increased concentration of mass (and therefore gravity) in the center of the cloud. Apart from a density profile with $\\beta_\\rho = -1.5$, we also tried a profile with $\\beta_\\rho = -2$. The mass of the protostar (Fig. \\ref{mstar}e) increases rapidly at early times with $\\beta_\\rho = -2$, but the accretion rate decelerates over time, which is the opposite behavior observed for $\\beta_\\rho = -1.5$. This behavior can be readily seen from the power-law fits to those curves: the normalization constant is similar in both cases, but while the fiducial case is described by $m \\propto \\tau^{1.80}$, the steeper initial density profile exhibits $m \\propto \\tau^{0.94}$. The  radius of the disk (Fig. \\ref{rdisk}C2) is larger in the case of the shallower density profile when seen as a function of the stellar mass and its behavior is longer linear. However, given that the stellar mass increases more rapidly in the case of the steeper density profile, in this latter case, a disk of a given size is observed earlier in time. A similar comparison of stellar masses and time reveals that magnetic braking happens at roughly the same time for both simulations, because the magnetic field is wound by rotation in a similar way in both cases. While the magnetically-braked region appears at $M \\approx 3\\unit{M_\\odot}$ when $\\beta_\\rho = -1.5$, it appears at $M\\approx 17\\unit{M_\\odot}$ for $\\beta_\\rho = -2$; however, the protostar reaches both masses at the same time, $t \\approx 0.2 t_\\text{ff} \\approx 10.7\\unit{kyr}$. As a result, we conclude that a disk that does not exhibit a magnetically-braked region at a given time can be obtained for a range of protostellar masses by adjusting the initial density profile.\n\n\\subsection{Initial rotation profile} \\label{S: disk var rotprof}\n\n\nWe investigated the effect of two scenarios: initial solid-body rotation and a rotation profile that scales with the cylindrical radius at $\\Omega \\propto R^{-\\beta_\\Omega}$, with $\\beta_\\Omega = \\beta_\\rho\/2 = -0.75$. This choice of $\\beta_\\Omega$ keeps the initial ratio of rotational to gravitational energy independent of the radius of the cloud. Additionally, the particular value of $-0.75$ produces a cloud for which the offset from Keplerianity is uniformly negative. We complemented this parameter scan with an investigation of the effects of using a lower initial rotational-to-gravitational energy ratio $\\zeta$ of 1\\% instead of 4\\% as in the fiducial case.\n\n\nAn inspection between panels D1 and A1 of Fig. \\ref{disk_morph} reveals that the thick layer of the disk is flatter in the cloud with the steep rotation profile. The higher angular velocity at the center of the cloud allows for the formation of a larger accretion disk earlier in the simulation, in comparison with the cloud initially rotating as a solid-body. Panel C4 of Fig. \\ref{rdisk} confirms this: the curves for the disk radius for $\\beta_\\Omega = -0.75,\\,\\zeta = \\{0.01,0.04\\}$ are in general higher than the corresponding curves for $\\beta_\\Omega = 0$, and resemble a power law rather than a linear function as it is the case for solid-body rotation. For a fixed value of $\\beta_\\Omega$, the disk becomes larger with more content of initial rotational energy. Additionally, both curves for $\\beta_\\Omega = -0.75$ show that the disk forms at very early times during the simulation compared to $\\beta_\\Omega = 0$: for the solid-body case, angular momentum conservation during the gravitational collapse provides the necessary increase of angular momentum in the center of the cloud in order to build up a disk, while the steep rotation profile already provides some of this angular momentum from the start. Interestingly, the cases $\\beta_\\Omega = -0.75,\\,\\zeta = 0.01$ and $\\beta_\\Omega = 0,\\,\\zeta = 0.04$ produce a similar disk radius. The mass of the formed protostar (Fig. \\ref{mstar}f) is lower in the cases with a steep rotation profile and higher initial rotational energy. This is consequence of the earlier formation of a disk that is also larger (and therefore more massive), at it reduces the efficiency of accretion onto the protostar (in \\citealt{Oliva2020}, we utilized this fact to study disk fragmentation in the absence of magnetic fields). The magnetic braking radius of the disk is very similar for all cases when plotted as a function of the protostellar mass, however, given that the masses grow different in time, magnetic braking appears later in the case with $\\beta_\\Omega = 0,\\,\\zeta = 0.04$.\n\n\nFrom the dynamical point of view, a comparison between panels D2 and A2 of Fig. \\ref{disk_morph} reveals that in general, the different contributions to the specific energy have a more uniform radial gradient. Given that the offset from Keplerianity in the midplane can also be computed as $1-2e^{K}_\\phi\/e^\\text{grav}$, the disk formed with the steep rotation profile is more uniformly Keplerian. This is also manifested when comparing the curves for $e^{K}_r$ inside of the disk in both cases.\n\n\n\n\n\n\\section{Comparison to observations} \\label{S: obs}\n\n\\subsection{Natal environment of massive star formation}\nThe parameter space in our simulation set was motivated by observational typical values found in regions of massive star formation. Surveys of cloud cores in the early stages of massive star formation have found power-law density profiles with exponents ranging in $1.5\\lesssim -\\beta_\\rho \\lesssim 2.6$ \\citep[see, for example,][]{2022AA...657A...3G,Beuther2018, 2002ApJ...566..945B, 2002ApJS..143..469M, 2003A&A...409..589H}, with fragmenting cores having typical values of around $\\beta_\\rho \\approx -1.5$. We take values in the extremes of this interval.\n\nFor the initial angular momentum, we start from the results of the survey by \\citep{Goodman1993}, who found that the the ratio of rotational to gravitational energy of cloud cores ($\\zeta$) is of a few per cent, with typical values of around 2\\%. Those authors fitted linear gradients to the cloud cores that had signs of rotation, and found solid-body profiles. Our second choice for the initial rotation profile, $\\beta_\\Omega = \\beta_\\rho\/2$, is motivated by keeping $\\zeta$ independent of the radius of the cloud core, as explained in Sect. \\ref{S: disk var rotprof}.\n\nThe values of the mass-to-flux ratio are motivated by the supercriticality in star-forming cloud cores \\citep[e.g.][]{Beuther2020, Girart2013} and the following analysis. In low-mass prestellar cores, the magnitudes of the magnetic field have been found to be of the order of a few to tens of micro Gauss \\citep{2022Natur.601...49C, 2008ApJ...680..457T}. As the pre-collapse conditions in the massive star formation case are currently not known, we consider two scenarios (see also the discussion in \\citealt{Machida2020}):\n\\begin{enumerate}\n\\item That the normalized mass-to-flux ratio is independent of mass, which means that magnetic field strengths of 100 to $1000\\,\\mu\\mathrm{G}$ are required in the cloud prior to the gravitational collapse.\n\\item That the magnetic field strength of the high-mass star formation case is similar to the low-mass counterpart, which implies that the mass-to-flux ratio must be high.\n\\end{enumerate}\nThe first case is considered in the low mass-to-flux ratios in our parameter space ($\\bar \\mu \\sim 5$), while the second case corresponds to high mass-to-flux ratios, including our fiducial case ($\\bar \\mu = 20$).\n\n\\subsection{Size of the disk}\nThe analysis in Sect. \\ref{s:parameter-study} has shown that the radius of the disk is more strongly determined by the initial density and rotation (angular momentum) profiles of the large scale initial condition, as compared to the magnetic field and resistivity models, within the ranges of those values expected from observations. Panel B4 of Fig. \\ref{rdisk} shows that when $M_\\star\/M_C \\sim 0.15$, the average $R_\\text{disk}$ for the values of $\\bar \\mu$ we considered is around $850\\unit{au}$, with a variation of the order of $\\pm 200\\unit{au}$. However, the radius of the disk and its growth rate vary much more with the density profile and the rotation profile (panels C2 and C4 of Fig. \\ref{rdisk}). Smaller disks are produced with a lower and flatter initial distribution of angular momentum, and with shallower initial density profiles. This conclusion is crucial for the comparison of our results with previous studies, which is done in Sect. \\ref{S: previous}.\n\nWith regards to the available observational evidence of disk-jet systems around forming massive stars, we see that in principle it is possible to obtain a variety of disk sizes that are not only dependent on the age of the system but also on the initial distribution of matter and angular momentum, and not so strongly determined by magnetic braking in the outer regions of the disk. For example, assuming that the molecular cloud has the typical values of $M_C = 100\\unit{M_\\odot}$ in a sphere of radius $0.1\\unit{pc}$ of our fiducial case, we obtain a disk of radius $\\sim 1200 \\unit{au}$ when the mass of the protostar is $M_\\star \\sim 10 \\unit{M_\\odot}$, if we assume a steep and fast initial rotation profile ($\\beta_\\Omega = -0.75$, $\\zeta = 0.04$), which would provide a possible initial configuration for the observations of HH 80-81 \\citep{Girart2017}. \n\n\nIn \\cite{Moscadelli2022}, we utilized the fiducial simulation run on grid x16 to model the accretion disk and jet of IRAS 21078+5211. For that particular system, observations of molecular rotational transitions had revealed the existence of a Keplerian-like accretion disk of around $200 \\unit{au}$ in size \\citep{Moscadelli2021multi}. In order to find a model of our catalog that was consistent with the observations, in \\cite{Moscadelli2022} we used the mass of the protostar, the radius of the disk, and the morphology of the ejected material to constrain the parameter space and the time elapsed. In that study, we found strong agreement between the velocity field of the observed water masers and the velocity streamlines predicted by the chosen model from our catalog. This good agreement in turn constitutes evidence that the results for the size of the disk as a function of the pre-collapse conditions --which we present in Sect. \\ref{s:parameter-study}-- are in line with what is expected from observations. We note, however, that the wide streamline traced by the maser points (Fig. 2 of \\citealt{Moscadelli2022}) is slightly closer to the assumed disk plane compared to the reference wide streamline from the simulations. This can be caused by a number of factors, for example, a slightly different assumed perspective between the simulations and the observations, but a possible explanation is that the disk in IRAS 21078+5211 is thinner than what is predicted in the simulations.\n\n\n\n\n\\section{Comparison with previous numerical studies} \\label{S: previous}\n\nThe present study is a continuation of the work started by \\cite{Anders2018}. Their simulations correspond to our fiducial case but with a grid equivalent to our x2 grid, and used an isothermal equation of state. An isothermal equation of state yields a much smaller pressure scale height of the disk or thinner disk, respectively. In combination with the rather coarse grid resolution, the thickness of the disk could not be resolved on the numerical grid, making an in-depth convergence study problematic. In that sense, the addition of radiation transport and higher resolutions has enabled us to resolve the thin layer of the disk, and clearly differentiate it from the surrounding region that we call the thick layer of the disk. Also, they do not report about the existence of the magnetically-braked region, however, it is visible upon close examination of their Fig. 21 as a region where the vertical velocity is negative close to the center of the disk. This was originally thought to be a numerical effect caused by the additional mass created by the Alfv\u00e9n limiter, however, we performed a parameter scan with several values of the Alfv\u00e9n limiter, the highest values of which produce negligible artificial mass, and found a magnetically-braked region in all of them.\n\n\\subsection{Studies with ideal MHD}\n\\cite{Banerjee2007}, \\cite{Seifried2011}, \\cite{Myers2013} and \\cite{Rosen2020} conducted simulations of the formation of massive stars under the assumption of ideal MHD. Although \\cite{Banerjee2007} observed magnetically-driven outflows, their disk is always sub-Keplerian. \\cite{Myers2013} started from a cloud of $300 M_\\odot$, an initial velocity profile with supersonic turbulence and no rotation, and $\\bar \\mu = 2$ with $B_z \\propto R^{-1\/2}$. They included grey radiative transfer and radiative protostellar feedback, and used a 3D AMR grid with a maximum resolution of $10 \\unit{au}$, as well as an additional isothermal high resolution run ($\\Delta x = 1.25 \\unit{au}$). They only find a disk in their high resolution run; it has a radius of $40\\unit{au}$ when $t\\sim 0.2 t_\\mathrm{ff}$, for which $M_\\star \\sim 3.5 M_\\odot$. It is not trivial to compare their results to ours because of the difference in initial velocity fields, and the use of an isothermal equation of state. Using the scalability of our results with the mass of the cloud, we find the radius of the disk to be $128\\unit{au}$ for $t = 0.2 t_\\mathrm{ff}$ in the simulation for $\\bar \\mu = 2$. However, if we take $M_\\star\/M_C$ and compute the corresponding fraction of the free-fall timescale, we get $t\/t_\\mathrm{ff} \\approx 0.104$, for which our disk has a radius of $\\sim 30 \\unit{au}$, which is in line with the fact that supersonic turbulence delays the formation of an accretion disk as enough angular momentum has to assemble close to the forming star. \\cite{Rosen2020} also considered a cloud core with supersonic turbulence and radiation transport, however their coarse grid (minimum cell size of $20\\unit{au}$) only allows them to partially resolve a disk-like structure and therefore a direct comparison to our results is not possible.\n\n\\cite{Seifried2011} considered a setup similar to ours in terms of the mass and radius of the cloud, as well as its initial density and rotation profiles, but used a cooling function instead of radiation transport, and only considered the ideal MHD approximation. The code FLASH with an AMR grid of maximum resolution of $4.7\\unit{au}$ was used. The authors explored several values of $\\bar \\mu$ ranging from 2.6 to 26, but with a uniform plasma $\\beta$, and obtained a Keplerian-like accretion disk for $\\bar \\mu = 26$. They observe a drop in Keplerianity for the inner parts of the disk in the case of $\\bar \\mu = 10.4$, and credit it to magnetic braking, in agreement with our results; however, they do not observe a disk for $\\bar \\mu < 5.2$ because the high density of the thin layer of the disk decouples the gas from the magnetic field im resistive MHD runs.\n\n\\subsection{Studies including Ohmic resistivity}\n\\cite{Matsushita2017} and \\cite{Machida2020} use the same Ohmic resistivity model as we do \\citep{Machida2007}, but both consider a barotropic equation of state instead of solving for radiation transport. In \\cite{Matsushita2017}, a disk was formed, however, their analyses are strongly focused on the magnetic outflows. \\cite{Machida2020} consider cloud cores of radius $0.2\\unit{pc}$ with masses ranging from $11$ to $545$ solar masses, an enhanced Bonnor-Ebert density profile, and slow initial solid-body rotation. If we expand our cloud to $0.2 \\unit{pc}$, keeping the same density profile as in the fiducial case, the mass of the cloud becomes $283 \\unit{M_\\odot}$, which means that our setup is similar to their simulation series E. The authors also make a parameter scan with the  normalized mass-to-flux ratio with values ranging from 2 to 20, and use a three-dimensional AMR grid with a maximum resolution of $0.62\\unit{au}$. After $10\\unit{kyr}$ of evolution, they find disk radii of a few hundred astronomical units, which develops spiral arms and fragments for some configurations.\n\n\n\\subsection{Studies including ambipolar diffusion} \\label{S: previous amb diff}\n\n\\cite{Mignon-Risse2021} and \\cite{Commercon2022} studied the formation of massive stars from $100 \\unit{M_\\odot}$ cloud cores of radius $0.2\\unit{pc}$, with a centrally-condensed initial density profile; which roughly correspond to our configuration $M_C = 50\\unit{M_\\odot}$. The cloud cores are initially in solid body rotation and are threaded by a uniform magnetic field determined from the normalized mass-to-flux ratio $\\bar \\mu = 5 \\text{ and } 2$. They treated radiation transport with the flux-limited diffusion approach and gray stellar irradiation; for magnetic diffusivity, they considered ambipolar diffusion using the model from \\cite{Marchand2016} but no Ohmic resistivity, and ran their simulations on a three-dimensional AMR grid with the code RAMSES down to a maximum resolution of $5\\unit{au}$. In a nutshell, those studies constitute the ones which include the most similar physical ingredients but to our setup but with ambipolar diffusion instead of Ohmic dissipation as the resistive effect and utilizing a different grid method.\n\nBoth studies report on thin accretion disks which are vertically supported by thermal pressure, in agreement with our results; however, they do not observe a surrounding thick layer of the disk supported by magnetic pressure. %\nOpposite to our findings, the authors find smaller disks with stronger magnetic fields, however, they do find a bigger disk for their ideal MHD run compared to the runs with ambipolar diffusion. In that case, the disk is not only bigger, but also inflated and therefore less dense, in agreement with our results. They attribute the finding of smaller disks to higher ionization in the outer disk, and therefore more magnetic braking. In turn, our results indicate that magnetic braking is highest in the inner disk, where the fluid and the magnetic field lines are dragged faster by rotation, although resistivity causes the dragging to be partial and delayed because of the dominance of magnetic diffusion. Moreover, in our results for the ideal MHD case we observe simultaneously the largest disk and the strongest magnetic braking, because it mostly affects the inner disk.\n\nA comparison of the radius of the disk that we report here and the disk size reported in \\cite{Commercon2022} can only be made qualitatively and with several considerations in mind, apart from the difference in the non-ideal MHD effect considered. First, the density profiles in both studies are different because of the presence of an initial density plateau in the inner $\\sim 4125\\unit{au}$ in \\cite{Commercon2022} and \\cite{Mignon-Risse2021}, in contrast to our continuous power-law slope down to the smallest scales. This density plateau affects the gravitational collapse and therefore, the elapsed time and mass of the protostar cannot be directly compared. Second, the initial rotational to gravitational energy ratio of the cloud is not the same in the fiducial cases of both studies, but we can roughly compare our $\\zeta = 0.04$ case to the ``fast'' case studied in \\cite{Commercon2022}, which has $\\zeta = 0.05$. This is of special importance because of what is shown in Fig. \\ref{rdisk}C4: the radius of the disk is strongly influenced by the initial rotation of the cloud, even more than the initial magnetic field (cf. Fig. \\ref{rdisk}B4) or resistivity (Fig. \\ref{rdisk}B2). Finally, the disk identification criterion is not the same in both studies: while \\cite{Commercon2022} use a set of criteria that involves the radial and vertical dynamics of the disk as well as a density threshold, we consider a purely dynamical criterion (radial equilibrium between gravity and centrifugal force) as it is more usual in observational studies. With this in mind, we note that in our simulation $M_C = 50\\, \\unit{M_\\odot}\\ \\text{[x4]}$, the  radius of the disk is around $470\\unit{au}$ after $27\\unit{kyr}$, while their disk is around the same value after $51\\unit{kyr}$. However, the mass of the protostar at that time is lower in our case ($4.6\\unit{M_\\odot}$) than theirs ($9.1\\unit{M_\\odot}$); both the difference in time and mass can be reasonably attributed to the initial density profile. Additionally, while we find that the disk grows in size with time, most of the nonideal MHD simulations by \\cite{Commercon2022} find disks that stop growing at around $100\\unit{au}$.\n\n\n\\cite{Commercon2022} and \\cite{Mignon-Risse2021} do not report the existence of a magnetically-braked region, for which we offer the following explanations, apart from the different criteria for disk identification we already discussed. First, ambipolar diffusion adds magnetic diffusivity, which may have the effect of delaying the formation of the magnetically-braked region. In more detail: we report on the fact that higher Ohmic resistivities delay the onset of magnetic braking and with it, the formation of the magnetically-braked region. Ambipolar diffusion also decouples the magnetic field from the flow (even though it may act in a different direction than Ohmic resistivity), which leads us to assume that a similar delaying effect on magnetic braking could be obtained upon the inclusion of ambipolar diffusion in our calculations. We note, however, that if this delay in the onset of magnetic braking is longer than the timescale of gravitational collapse of the cloud, the massive star and the disk may form without ever developing a magnetically-braked region. %\nSecond, spatial resolution: their 3D AMR grid has a minimum cell size (maximum spatial resolution) of $5\\unit{au}$, while our grid has minimum cell sizes of the order of $10^{-1}\\unit{au}$ close to the sink cell for the simulation series x1, x2 and x4, and $10^{-2}\\unit{au}$ for the simulation series x8 and x16. Third, their sink particle algorithm requires the definition of an accretion radius, set to four times the minimum cell size, i.e., $20\\unit{au}$, which is the size of the magnetically-braked region for most of the time in our fiducial case. As a consequence, gas orbiting the massive protostar within the accretion radius or in the forming magnetically-braked region cannot be resolved on their finest AMR level while it is resolved on our spherical grid.\n\nFinally, we mention the study by \\cite{Masson2016}, who carried out simulations of low-mass star formation considering both ideal MHD and ambipolar diffusion. Although we cannot compare our results quantitatively with theirs, there are several qualitative similarities. The authors observe an accretion disk vertically supported by thermal pressure, with magnetic pressure dominating above the disk. Their figure 14 shows a structure with relatively high angular momentum enveloping the disk for both the diffusive and ideal cases, which is reminiscent of the thick layer of the disk we observe, however, due to the different definitions of the disk used in both studies, we cannot establish a direct correspondence. The same figure also shows that in the ideal MHD case, the inner parts of the disk are destroyed by magnetic braking at late times. Nevertheless, their corresponding run that considers ambipolar diffusion also exhibits for late times a region of low angular momentum (probably due to magnetic braking) in a conical region around the inner disk and that could deliver material to it, in a way that is reminiscent of the early stages of magnetic braking we observe in our simulations. Those results seem to support the idea that increasing the diffusivity leads to a delay but not complete suppression of magnetic braking in a magnetized disk embedded in a collapsing cloud.\n\n\\section{Summary and conclusions}\n\nWe have modeled the formation of a massive star with a series of 30 magnetohydrodynamical simulations including Ohmic resistivity, radiation transport from thermal emission of the dust and gas, and self gravity. The series of simulations covered a wide range of cloud masses, magnetic field strengths, density profiles, rotation profiles, and ratios of rotational to gravitational energy, in line with currently estimated values from observations. We also performed a convergence study to test the robustness of our results.\n\nAfter analyzing the fiducial case of our parameter study in depth, we found the following general features of the system:\n\n\\begin{itemize}\n\t\\item After the initial gravitational collapse, a Keplerian-like accretion disk is formed.\n\t\\item The accretion disk is divided into two layers: a thin layer supported vertically by thermal pressure, and a surrounding thick layer supported by magnetic pressure. The thin layer of the disk appears only in simulations with sufficiently high resolution.\n\t\\item At early times magnetic diffusion due to Ohmic resistivity is strong in the inner parts of the disk and it greatly reduces magnetic braking there during the magneto-centrifugal epoch ($5 \\unit{kyr} \\lesssim t \\lesssim 15\\unit{kyr}$). As time progresses, and the magnetic field lines are continuously dragged by rotation, magnetic braking is observed in the innermost $\\sim 50\\unit{au}$ of the disk for the fiducial case in our parameter space.\n\t\\item Magnetic pressure can increase the size of the accretion disk.\n\\end{itemize}\n\nWhen examining the full parameter space of initial conditions for the onset of gravitational collapse, we find that:\n\\begin{itemize}\n\t\\item Our results for the size of the disk and the mass gain of the protostar scale with the initial mass of the cloud, despite the non-scalability of self-gravity and the thermodynamics considered.\n\t\\item The thickness of the thick layer of the disk is controlled by the initial magnetic field strength.\n\t\\item For a cloud with an initial density profile $\\rho \\propto r^{-1.5}$ and in solid-body rotation, the disk grows roughly linearly in size as $R_\\text{disk} \\approx [ 6380 M_\\star\/M_C - 98 ]\\unit{au} $. The stellar mass grows approximately like $M_\\star \\propto (t\/t_\\text{ff})^{1.5..1.9}$.\n\t\\item The size of the disk is more strongly determined by the initial distribution of density and rotational energy in the cloud than by the strength of the magnetic field.\n\t\\item Multiple initial configurations of the cloud can produce a given disk size and (proto)stellar mass. This means that observations of disk-jet systems constrain (as opposed to determine) the possible conditions for the onset of gravitational collapse, and more measurements (distribution and strength of magnetic fields, for example) are needed to break the degeneracies.\n\\end{itemize}\n\nIn a follow-up article in preparation, we will perform a dynamical analysis of the magnetically-driven outflows of the same dataset we present here.\n\n\\begin{acknowledgements}\nWe thank Richard Nies for his contributions to the analysis of part of the dataset at the early stages of the project. GAO acknowledges financial support from the Deutscher Akademischer Austauschdienst (DAAD), under the program Research Grants - Doctoral Projects in Germany, and complementary financial support for the completion of the Doctoral degree by the University of Costa Rica, as part of their scholarship program for postgraduate studies in foreign institutions. RK acknowledges financial support via the Emmy Noether and Heisenberg Research Grants funded by the German Research Foundation (DFG) under grant no.~KU 2849\/3 and 2849\/9.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCoupling molecules to the quantized radiation field inside an optical cavity creates a set of new photon-matter hybrid states, which are commonly referred to as polaritons,\\cite{Ebbesen2016ACR,kowalewski_manipulating_2017,Flick2017pnas,Ribeiro2018,Feist2018,Mandal2019JPCL} which have been shown to facilitate new chemical reactivities.\\cite{Hutchison2012ACIE,Ebbesen2016ACR,Thomas2019S,Mandal2019JPCL,Mandal2020JPCB} Theoretical investigations play a crucial role in understanding the fundamental limit and basic principles in this emerging field,\\cite{Mandal2019JPCL,Feist2018,Luk2017jctc,Groenhof2018jpcl,Groenhof2019jpcl,Groenhof2019jpcl,Groenhof2021JCP} as these polariton chemical reactions often involve a rich dynamical interplay among the electronic, nuclear, and photonic degrees of freedom (DOFs). Accurately simulating polaritonic quantum dynamics remains a challenging task and is beyond the scope of photochemistry or quantum optics.\\cite{kowalewski_manipulating_2017}\n\nThe trajectory-based non-adiabatic dynamics approaches\\cite{Tully2012jcp,BarbattiSH,Mai2015} play an important role in simulating the non-adiabatic dynamics of the coupled electronic-nuclear DOFs. Two of the most commonly used mixed quantum-classical (MQC) methods are the Ehrenfest and fewest switches surface hopping (FSSH) approaches.\\cite{Tully,tully94jcp} Both approaches describe the electronic subsystem quantum mechanically, and treat the nuclear DOFs classically. It is thus a natural idea for the theoretical chemistry community to extend these two approaches to investigate polariton chemistry by treating the electronic-photonic DOFs (or so-called polariton subsystem) quantum mechanically and the nuclear DOFs classically. Incorporating the description of the photon field into the MQC methods has become a basic strategy to simulate polariton chemistry.\\cite{Luk2017jctc,Groenhof2018jpcl,Groenhof2019jpcl,Groenhof2021JCP,Fregoni2018,Fregoni2020,fregoni_strong_2020,Zhang2019jcp} The key ingredient in the MQC simulations of polariton dynamics is the expression of the nuclear gradient. Recently, we derived a rigorous expression of the nuclear gradient using the quantum electrodynamics (QED) Hamiltonian without making the usual approximations,\\cite{Zhou2022} such as the rotating wave approximation. These gradient expressions, together with the corresponding MQC approaches (Ehrenfest and FSSH approaches), is valid for any any number of electronic states or Fock states at any light-matter coupling strength. However, the inherent semi-classical approximation in these approaches can lead to the break-down of detailed balance\\cite{ParandekarJCTC2006} (incorrect long time population) in Ehrenfest dynamics and the creation of artificial electronic coherence\\cite{subotnik2016arpc} or incorrect\nchemical kinetics\\cite{subotnik2016arpc} for the FSSH dynamics without invoking \\textit{ad hoc} decoherence corrections.\n\nIn response to these theoretical challenges, a wide range of non-adiabatic dynamics approaches have been developed in the diabatic representation. Many of them belong to the family of non-adiabatic mapping dynamics which are based on the Meyer-Miller-Stock-Thoss (MMST) mapping formalism.\\cite{MeyerJCP1979,StockPRL1997,ThossPRA1999} These methods include partial linearized density matrix\\cite{HuoJCP2011,HuoMP2012} (PLDM), symmetrical quasi-classical\\cite{CottonJCP2013,CottonJPCA2013} (SQC), the quantum-classical Liouville equation (QCLE) dynamics.\\cite{HsiehJCP2012,HsiehJCP2013} In particular, the recently developed $\\gamma$-SQC has been shown\\cite{CottonJCP2019_2} to provide impressively accurate non-adiabatic photo-dissociation quantum dynamics with coupled Morse potentials through the adjusted zero-point energy (ZPE) parameter of the mapping variables, thus appearing to be a promising method to simulate on-the-fly quantum dynamics of complex molecular systems. In addition, the spin-mapping Linearized Semi-Classical approach (spin-LSC)\\cite{richardson2019,richardson2020,Duncan2022JCP}, which uses generalized spin mapping representation\\cite{richardson2020} for the electronic DOF as well as the Linearization approximation\\cite{MillerJCP98,shi2004} for the nuclear DOF, has also shown a significant improvement of the population dynamics in the system-bath model problems (such as in spin-boson systems\\cite{richardson2019} and many-state exciton Hamiltonians of light-harvesting complexes.\\cite{richardson2020}) The $\\gamma$-SQC approach has already demonstrated\\cite{WeightJCP2021} its ability to outperform MQC approaches (Ehrenfest and FSSH) in describing the electronic non-adiabatic dynamics for {\\it ab initio} on-the-fly simulations. These new mapping approaches should, in principle, also outperform the MQC methods in simulating the polaritonic non-adiabatic dynamics that happens in the electron-photon subspace coupling to the motion of the nuclei. Unfortunately, to the best of our knowledge, there are only limited studies of using mapping dynamics to investigate polariton chemistry for model systems with strict diabatic states.\\cite{Mandal2019JPCL,Chowdhury2021jcp,Mandal2020JPCB}\n\nRecently, we have developed the quasi-diabatic (QD) propagation scheme\\cite{MandalJCTC2018,MandalJCP2018,MandalJPCA2019,SandovalJCP2018,ZhouJCPL2019, WeightJCP2021} as a general framework to seamlessly combine a diabatic quantum dynamics approach, such as the mapping based methods,\\cite{CottonJCP2019_2,richardson2020} with the adiabatic outputs of an electronic structure method. The QD propagation scheme uses the adiabatic states at a reference nuclear geometry (the so-called ``crude adiabatic\" states) as the {\\it locally well-defined} diabatic states during a short-time propagation and then dynamically updates the QD basis at each consecutive nuclear propagation step. In this propagation scheme, one does not construct a {\\it global diabatic representation} but instead, uses a sequence of locally diabatic representations (one for each short-time segment) to propagate the dynamics. We have both analytically\\cite{MandalJCTC2018} and numerically\\cite{MandalJPCA2019,SandovalJCP2018} demonstrated that the QD scheme provides exactly the same results compared to the direct diabatic quantum dynamics at the single trajectory level. \n\nIn this work, we generalize the QD propagation scheme to simulate polariton non-adiabatic dynamics in a molecule-cavity hybrid system. In particular, we use the adiabatic-Fock state at a reference nuclear geometry as the locally well-defined diabatic basis to propagate the polariton dynamics, and dynamically update the definition of these local diabatic states between two consecutive propagation steps. These adiabatic-Fock states are tensor products of the electronic adiabatic states for the molecular system and the Fock states of the photon field inside an optical cavity. We use the Shin-Metiu (SM) model\\cite{Shin1995jcp,Hoffmann2020} as the ``\\textit{ab initio}\" model molecular system to investigate strong and ultra-strong light-matter interactions between a molecule and an optical cavity. Through numerical simulations, we demonstrate the accuracy of using both $\\gamma$-SQC\\cite{CottonJCP2019_2} and spin-LSC\\cite{richardson2019,richardson2020} to obtain non-adiabatic polariton dynamics, which outperforms widely used MQC approaches.\n\n\\section{Theory and Methods} \\label{methods}\n\\subsection{The Pauli-Fierz QED Hamiltonian}\nThe Pauli-Fierz (PF) QED Hamiltonian for one molecule coupled to quantized radiation field inside an optical cavity can be written as \n\\begin{equation}\\label{total_PF_H}\n\\hat{H}=\\hat{T}_\\mathrm{n}+\\hat{H}_\\mathrm{en}+\\hat{H}_\\mathrm{p}+\\hat{H}_\\mathrm{enp}+\\hat{H}_\\mathrm{d},\n\\end{equation}\nwhere $\\hat{T}_\\mathrm{n}$ represents the nuclear kinetic energy operator, $\\hat{H}_\\mathrm{en}$ is the electronic Hamiltonian that describes electron-nucleus interactions. Further, $\\hat{H}_\\mathrm{p}$, $\\hat{H}_\\mathrm{enp}$, and $\\hat{H}_\\mathrm{d}$ represent the photonic Hamiltonian, electronic-nuclear-photonic interactions, and the dipole self-energy (DSE) term, respectively. A full derivation of this Hamiltonian, as well as its connection with the various atomic cavity QED models can be found in the Appendix of Ref.~\\citenum{Chowdhury2021jcp}.\n\nThe electronic-nuclear potential $\\hat{H}_\\mathrm{en}$, which describes the common molecular Hamiltonian excluding the nuclear kinetic energy is described as follows\n\\begin{equation}\\label{eqn:Hen}\n\\hat{H}_\\mathrm{en} = \\hat{T}_\\mathrm{e} + \\hat{V}_\\mathrm{ee} + \\hat{V}_\\mathrm{en} + \\hat{V}_\\mathrm{nn}.\n\\end{equation}\nThe above expression includes electronic kinetic energy, electron-electron interaction, electron-nucleus interaction and nucleus-nucleus interaction. The expressions of these four terms can be found in previous work.\\cite{Doltsinis2002jctc,Schafer2018pra,Marxbook} Modern electronic structure theory have been developed around solving the eigenvalue problem of $\\hat{H}_\\mathrm{en}$, providing the following electronically adiabatic energy and its corresponding state\n\\begin{equation}\\label{HenPsi}\n\\hat{H}_\\mathrm{en}|\\phi_\\alpha(\\mathbf{R}) \\rangle= E_{\\alpha}({\\mathbf R})|\\phi_\\alpha(\\mathbf{R}) \\rangle.\n\\end{equation}\nHere, $|\\phi_\\alpha(\\mathbf{R}) \\rangle$ represents the $\\alpha_\\mathrm{th}$ many-electron adiabatic state for a given molecular system, with the adiabatic energy $E_{\\alpha}({\\mathbf R})$.\n\nFor clarity, we restrict our discussions to the cavity with only one photonic mode, and all the formulas presented here can be easily generalized into a more realistic, many-mode cavity. The photonic Hamiltonian is written as \n\\begin{equation}\\label{eqn:Hp}\n\\hat{H}_\\mathrm{p}=\\frac12 \\left(\\hat{p}_\\mathrm{c}^2+\\omega_\\mathrm{c}^2\\hat{q}_\\mathrm{c}^2\\right)=\\hbar\\omega_\\mathrm{c} \\Big(\\hat{a}^{\\dagger}\\hat{a}+\\frac12 \\Big),\n\\end{equation}\nwhere $\\hat{q}_\\mathrm{c}=\\sqrt{\\hbar\/2 \\omega_\\mathrm{c}}(\\hat{a}^{\\dagger}+\\hat{a})$ and $\\hat{p}_\\mathrm{c}=i\\sqrt{\\hbar\\omega_\\mathrm{c}\/2}(\\hat{a}^{\\dagger}-\\hat{a})$ are photon field operators, $\\hat{a}^{\\dagger}$ and $\\hat{a}$ are the photonic creation and annihilation operators, respectively and $\\omega_\\mathrm{c}$ is the photon frequency.\n\nThe light-matter coupling term (electronic-nuclear-photonic interactions) under the dipole gauge is expressed as \n\\begin{equation}\\label{eq:enp}\n\\hat{H}_\\mathrm{enp}=\\omega_\\mathrm{c}\\hat{q}_\\mathrm{c} ( \\boldsymbol{\\lambda} \\cdot {\\hat{\\boldsymbol \\mu}})=g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot  {\\hat{\\boldsymbol \\mu}} (\\hat{a}^{\\dagger} + \\hat{a}).\n\\end{equation}\nwhere $\\boldsymbol{\\lambda}=\\lambda \\cdot{\\boldsymbol\\epsilon}$ characterizes the cavity photon field strength, ${\\boldsymbol\\epsilon}$ is the direction of the field polarization. The photon field strength is determined by the volume of the cavity as $\\lambda=\\sqrt{1\/\\epsilon_{0}\\mathcal{V}_{0}}$, where $\\epsilon_{0}$ is the permittivity inside the cavity and $\\mathcal{V}_{0}$ is the effective quantization volume inside the cavity. Another way to characterize the light-matter coupling strength is using $g_\\mathrm{c}=\\sqrt{\\hbar\\omega_\\mathrm{c}\/2}\\lambda$. Note that the common notation used in the literature,\\cite{Kowalewski2016JPCL,Mandal2019JPCL} the definition of $g_\\mathrm{c}$ also includes $\\boldsymbol{\\lambda} \\cdot {\\hat{\\boldsymbol \\mu}}$. Further, the total dipole operator of both electrons and nuclei is defined as\n\\begin{equation}\\label{dipole}\n{\\hat{\\boldsymbol \\mu}} = -\\sum_ie \\hat{{\\bf r}}_i +\\sum_{j} Z_{j} e \\hat{\\bf{R}}_{j}, \n\\end{equation}\nwhere $-e$ is the charge of the electron and $Z_{j} e $ is the charge of the $j_\\mathrm{th}$ nucleus.\n\nFinally, the DSE term is expressed as\n\\begin{equation}\\label{eqn:Hd}\n\\hat{H}_\\mathrm{d}=\\frac12 (\\boldsymbol{\\lambda} \\cdot \\hat{\\boldsymbol \\mu})^2=\\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega_\\mathrm{c}} (\\boldsymbol{\\epsilon} \\cdot \\hat{\\boldsymbol \\mu})^2.\n\\end{equation}\nThis is a necessary term in the PF Hamiltonian and ensures both gauge invariance of the Hamiltonian\\cite{Taylor2020PRL,Mandal2020JPCB} and a bounded ground state.\\cite{Rokaj2018JPB,Mandal2020JPCB,Schaefer2020AP} In this work, we do not consider the cavity loss. The cavity loss can be effectively incorporated by using Lindblad dynamics approaches with the MQC simulations.\\cite{Koessler2022}\n\nFor the molecule-cavity hybrid system, a convenient basis for quantum dynamics simulations could be the photon-dressed electronic adiabatic states \n\\begin{equation}\\label{adia-fock}\n|\\psi_i({\\mathbf R}) \\rangle=|\\phi_\\alpha({\\mathbf R}) \\rangle \\otimes |n\\rangle \\equiv|\\phi_{\\alpha}({\\bf R}),n\\rangle,\n\\end{equation}\nwhere quantum number $i\\equiv\\{\\alpha,n\\}$ indicates both the adiabatic electronic state of the molecule and the Fock state. Note that we have introduced a shorthand notation in Eq.~\\ref{adia-fock}, which will be used throughout the rest of this paper. This is one of the most straightforward choices of basis for the hybrid system because of the readily available adiabatic electronic information (\\textit{e.g.,} wavefunctions, energies, and the dipole matrix) from electronic structure calculations that we need to construct the elements of Hamiltonian. \n\nIn the MQC simulation, such as the Ehrenfest or FSSH approach, or the recently developed mapping non-adiabatic approaches, the total molecular Hamiltonian is expressed as\n\\begin{equation}\\label{totalH}\n\\hat{H}=\\hat{T}_\\mathrm{n}+\\hat{V},\n\\end{equation}\nwhere $\\hat{T}_\\mathrm{n}$ represents the nuclear kinetic energy operator, and $\\hat{V}$ represents the rest of the Hamiltonian. For a bare molecular system, $\\hat{V}=\\hat{H}_\\mathrm{en}$ expressed in Eq.~\\ref{eqn:Hen}.\nFor a molecule-cavity hybrid system, \n\\begin{equation}\\label{eqn:Henp}\n\\hat{V}=\\hat{H}_\\mathrm{en}+\\hat{H}_\\mathrm{p}+\\hat{H}_\\mathrm{enp}+\\hat{H}_\\mathrm{d}\\equiv \\hat{H}_\\mathrm{pl},\n\\end{equation}\nwhich is commonly referred to as the polariton Hamiltonian,\\cite{Flick2017jctc,Flick2017pnas} also denoted as $\\hat{H}_\\mathrm{pl}$. In a similar way that electronic adiabatic states are defined in Eq.~\\ref{HenPsi}, one can further define the polaritonic state\\cite{Flick2017jctc,Flick2017pnas} as the eigenstate of $\\hat{V}=\\hat{H}_\\mathrm{pl}$ (see definition in Eq.~\\ref{eqn:Henp}) through the following eigenequation\n\\begin{equation}\\label{polariton}\n\\hat{H}_\\mathrm{pl}|\\mathcal{E}_{J}({\\bf R})\\rangle=\\mathcal{E}_{J}({\\bf R})|\\mathcal{E}_{J}({\\bf R})\\rangle,\n\\end{equation}\nwhere $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is the polariton state with polariton energy $\\mathcal{E}_{J}({\\bf R})$. The polariton eigenstate can be expressed as \n\\begin{equation}\\label{pol-exp}\n|\\mathcal{E}_{J}({\\bf R})\\rangle=\\sum_{\\alpha,n}c^{J}_{\\alpha, n}({\\mathbf R})|\\phi_{\\alpha}({\\bf R}),n\\rangle,\n\\end{equation}\nwhere $c^{J}_{\\alpha, n}({\\mathbf R})=\\langle \\phi_{\\alpha}({\\bf R}),n|\\mathcal{E}_{J}({\\bf R})\\rangle$ and $\\mathcal{E}_{J}({\\bf R})$ can be obtained by diagonalizing the matrix of $\\hat{V}=\\hat{H}_\\mathrm{pl}$ (constructed from the adiabatic-Fock state basis in Eq.~\\ref{adia-fock}) as\n\\begin{equation}\\label{eqn:diag}\n\\mathbf{U}^{\\dagger}[V({\\mathbf R})]\\mathbf{U} = [\\mathcal{E}({\\mathbf R})],\n\\end{equation}\nwhere\n\\begin{equation}\n[V({\\mathbf R})]_{ij} = \\langle\\psi_i({\\mathbf R})| \\hat{V} |\\psi_j({\\mathbf R}) \\rangle.\n\\end{equation}\nNote that the ${\\bf R}$-dependence of $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is entirely coming from the ${\\bf R}$-dependence of the adiabatic states $|\\phi_\\alpha({\\mathbf R}) \\rangle$, and the Fock state $|n\\rangle$ is completely ${\\bf R}$-independent. Meanwhile, the $|\\mathcal{E}_{J}({\\bf R})\\rangle$ is the eigenstate of $\\hat{V}$, whereas the adiabatic state $|\\phi_\\alpha({\\mathbf R}) \\rangle$ is only the eigenstate of $\\hat{H}_\\mathrm{en}$, and not for $\\hat{V}$.\n\n\\subsection{Quasi-Diabatic Propagation Scheme for Molecular Cavity QED}\\label{sec:QD}\nThe QD propagation scheme explicitly addresses the discrepancy between accurate quantum dynamics methods in the diabatic representation and the electronic structure methods in the adiabatic representation. The essential idea of the QD scheme is to use the electronic adiabatic states associated with a reference geometry as the local diabatic states during a short-time quantum propagation and dynamically updates the definition of the QD states along the time-dependent nuclear trajectory.\\cite{MandalJCTC2018,MandalJCP2018,MandalJPCA2019,SandovalJCP2018,ZhouJCPL2019, WeightJCP2021}\n\nIn this work, we apply the QD propagation scheme to the case of molecular cavity QED. This requires the use of a convenient basis with a reference nuclear geometry as the $\\it locally$ well-defined diabatic basis, in the sense that its character is fixed (which is automatically guaranteed because of the fixed reference geometry by construction) as well as it is a complete basis (which is only true when the geometry is close to this reference geometry). The potential candidate for this basis is the adiabatic-Fock state $|\\psi_i({\\mathbf R}) \\rangle=|\\phi_\\alpha({\\mathbf R}),n\\rangle$ (Eq.~\\ref{adia-fock}), which is not the same as the polariton states $|\\mathcal{E}_{J}({\\bf R})\\rangle$ (Eq.~\\ref{pol-exp}) except for the zero-coupling limit. In this work, we use the adiabatic-Fock state as the convenient choice due to its simplicity in terms of the polariton coupling and nuclear gradient expressions in the QD propagation framework.\n\nConsider a short-time propagation of the nuclear DOFs during $t\\in[t_0, t_1]$, where the nuclear positions evolve from ${\\bf R}(t_0)$ to ${\\bf R}(t_1)$, and the corresponding adiabatic-Fock basis (defined in Eq.~\\ref{adia-fock}) are $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\}$ and $\\{|\\psi_{j}{\\bf R}(t_1))\\rangle\\}$. We uses the basis $\\{|\\psi_{i}({\\bf R}_0)\\rangle\\equiv|\\phi_\\alpha({\\mathbf R_0}),n\\rangle\\}$ at the reference nuclear geometry ${\\bf R}(t_0)$ as the {\\it diabatic} basis during this short-time propagation, such that\n\\begin{equation}\\label{eqn:qdidea}\n|\\psi_{i}({\\bf R}_{0})\\rangle\\equiv|\\psi_{i}({\\bf R}(t_0))\\rangle,~~\\mathrm{for}~t\\in[t_0,t_1].\n\\end{equation}\nWith the above QD basis defined independently of ${\\bf R}(t)$ within each propagation segment, the electronic derivative couplings vanish while $\\hat{V}({\\bf R}(t))$ in the QD basis becomes off-diagonal. With this local diabatic basis, all of the necessary diabatic quantities can be evaluated and used to propagate quantum dynamics during $t\\in[t_0,t_1]$. \n\nDuring this propagation step, the matrix element of $\\hat{V}$ in the QD basis is evaluated as\n\\begin{equation}\\label{eqn:vijt} \nV_{\\alpha\\beta, m n}({\\bf R}(t))  = \\langle \\phi_\\alpha ({\\bf R}_0), m| \\hat V ({\\bf R}(t))| \\phi_\\beta ({\\bf R}_0), n\\rangle.\n\\end{equation}\nFor on-the-fly simulations, this quantity is obtained from a linear interpolation\\cite{Rossky-Webster} between $V_{{\\alpha}{\\beta},m n} ({\\bf R}_{0})$ and $V_{\\alpha\\beta,m n}({\\bf R}(t_1))$ as follows\n\\begin{align}\\label{eqn:interpolation}\n&V_{\\alpha\\beta,m n}({\\bf R}(t))= V_{\\alpha\\beta,m n}({\\bf R}_{0}) \\\\\n&~~~~~+ \\frac {(t - t_{0})}{(t_{1} - t_{0})}\\big[V_{\\alpha\\beta,mn}({\\bf R}(t_{1})) - V_{\\alpha\\beta,mn}({\\bf R}_{0})\\big].\\nonumber\n\\end{align} \nThe above linear interpolation scheme can be further improved in future work and one potential choice is the recently developed norm-preserving interpolation scheme.\\cite{meek2014evaluation,jain2016efficient}\n\nIt is straightforward to evaluate $V_{{\\alpha}{\\beta},m n} ({\\bf R}_{0})$ and $V_{\\alpha\\beta,m n}({\\bf R}(t_1))$ separately for the molecule-cavity hybrid system, as discussed below. Using electronic {\\it ab initio} calculation, as well as the properties of $\\hat{a}^{\\dagger}$ and $\\hat{a}$ for the photonic DOF, we can explicitly evaluate each term of $V_{\\alpha\\beta,m n}({\\bf R}_{0})$ (see Eq.~\\ref{eqn:Henp}) as follows\n\\begin{subequations}\n\\begin{align}\n&H^\\mathrm{en}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{en} ({\\bf R}_{0})|\\phi_\\beta({\\bf R}_{0}), n\\rangle  \\nonumber \\\\ \n&~~~= E_{\\alpha}({\\bf R}_{0}) \\delta_{{\\alpha},{\\beta}} \\delta_{m,n} \\\\\n&H^\\mathrm{p}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{p} | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber \\\\ \n&~~~= \\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\\\\n&H^\\mathrm{enp}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{enp} ({\\bf R}_{0}) | \\phi_\\beta({\\bf R_{0}}), n \\rangle \\nonumber \\\\\n&~~~= g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\beta}({\\bf R}_{0}) \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1} \\big) \\\\\n&H^\\mathrm{d}_{\\alpha\\beta,m n}({\\bf R}_{0}) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{d} ({\\bf R}_{0}) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\\n&~~~= \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega_\\mathrm{c}} \\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\gamma} ({\\bf R}_{0})) (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\gamma\\beta} ({\\bf R}_{0})) \\delta_{m,n} \\nonumber\\\\ &\n~~~\\equiv D^2_{\\alpha\\beta}({\\bf R}_{0}) \\delta_{m,n},\n\\end{align}\n\\end{subequations}\nwhere $\\hat{H}_\\mathrm{p}$ (see its definition in Eq.~\\ref{eqn:Hp}) is an ${\\bf R}$-independent operator, the sum $\\sum_{\\gamma}$ in the matrix element of $\\hat{H}_\\mathrm{d}$ runs over the diabatic states, and $D^{2}_{\\alpha\\beta}$ denotes the elements of DSE. Further, the matrix element of the dipole operator under the diabatic representation is expressed as\n\\begin{equation}\\label{dipole-mat}\n{\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}_{0})\\equiv\\langle \\phi_\\alpha ({\\bf R}_{0}) |\\hat{\\boldsymbol\\mu} ({\\bf R}_{0})|\\phi_\\beta ({\\bf R}_{0}) \\rangle.\n\\end{equation}\nSimilarly, at time $t_1$, the matrix element $V_{\\alpha\\beta, m n} ({\\bf R}(t_1))=\\langle \\phi_\\alpha ({\\bf R}_{0}), m| \\hat V ({\\bf R}(t_1))| \\phi_\\beta ({\\bf R}_{0}), n\\rangle$ can also be written explicitly, with each term expressed as follows\n\\begin{subequations}\n\\begin{align}\n&H^\\mathrm{en}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle\\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{en} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\\n&~~~= H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1)) \\delta_{m,n} \\\\\n&H^\\mathrm{p}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{p} | \\phi_\\beta({\\bf R}_{0}),n \\rangle \\nonumber\\\\ \n&~~~= \\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\\\\n&H^\\mathrm{enp}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{enp} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0}), n \\rangle \\nonumber\\\\ \n&~~~= g_\\mathrm{c} \\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\beta}({\\bf R}(t_1)) \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1} \\big)\\\\\n&H^\\mathrm{d}_{\\alpha\\beta,mn}({\\bf R}(t_{1})) = \\langle \\phi_\\alpha({\\bf R}_{0}), m | \\hat{H}_\\mathrm{d} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R_{0}}), n \\rangle \\nonumber\\\\ \n&~~~=\\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\alpha\\gamma} ({\\bf R}(t_1))) (\\boldsymbol{\\epsilon} \\cdot {\\boldsymbol\\mu}_{\\gamma\\beta} ({\\bf R}(t_1))) \\delta_{m,n} \\nonumber\\\\ \n&~~~\\equiv D^2_{\\alpha\\beta}({\\bf R}(t_1)) \\delta_{m,n}, \n\\end{align}\n\\end{subequations}\nwhere $H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1))\\equiv\\langle\\phi_\\alpha({\\bf R}_{0})| \\hat{H}_\\mathrm{en} ({\\bf R}(t_1)) | \\phi_\\beta({\\bf R}_{0})\\rangle$, and ${\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1))\\equiv\\langle \\phi_\\alpha ({\\bf R}_{0}) |\\hat{\\boldsymbol\\mu} ({\\bf R}(t_1))|\\phi_\\beta ({\\bf R}_{0}) \\rangle$.\\\\\n\nTo conveniently calculate $H^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1))$ and ${\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1))$, we use the following relations\n\\begin{subequations}\n\\begin{align}\nH^{\\mathrm{en}}_{\\alpha\\beta}({\\bf R}(t_1)) & =\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\tilde{H}^{\\mathrm{en}}_{\\lambda\\nu}({\\bf R}(t_1)) S^{\\dagger}_{\\beta\\nu} \\label{eqn:elect2} \\\\\n{\\boldsymbol\\mu}_{\\alpha\\beta} ({\\bf R}(t_1)) & =\\sum_{\\lambda\\nu}S_{\\alpha\\lambda}\\tilde{{\\boldsymbol\\mu}}_{\\lambda\\nu} ({\\bf R}(t_1)) S^{\\dagger}_{\\beta\\nu}, \\label{eqn:mu2}\n\\end{align}\n\\end{subequations}\nwhere the matrix elements at ${\\bf R}(t_1)$ are expressed as\n\\begin{subequations}\n\\begin{align}\n\\tilde{H}^{\\mathrm{en}}_{\\lambda\\nu}({\\bf R}(t_1)) & =\\langle \\phi_{\\lambda}({\\bf R}(t_{1}))| \\hat{H}_\\mathrm{en} ({\\bf R}(t_1))|\\phi_{\\nu}({\\bf R}(t_{1})) \\rangle \\\\\n&=E_{\\lambda}({\\bf R}(t_1))\\delta_{\\lambda\\nu} \\nonumber \\\\\n\\tilde{\\boldsymbol\\mu}_{\\lambda\\nu} ({\\bf R}(t_1)) & =\\langle \\phi_{\\lambda}({\\bf R}(t_{1}))| \\hat{\\boldsymbol\\mu} ({\\bf R}(t_1)) |\\phi_{\\nu}({\\bf R}(t_{1})) \\rangle,\n\\end{align}\n\\end{subequations}\nand the overlap matrix between two electronic adiabatic states (with two different nuclear geometries) are  \n\\begin{subequations}\n\\begin{align}\nS_{\\alpha\\lambda}&= \\langle \\phi_{\\alpha}({\\bf R}_0)|\\phi_{\\lambda}({\\bf R}(t_{1}))\\rangle\\label{eqn:wf_overlap}\\\\\nS^{\\dagger}_{\\beta\\nu}&= \\langle \\phi_{\\nu}({\\bf R}(t_{1}))|\\phi_{\\beta}({\\bf R}_0)\\rangle.\\label{eqn:wf_overlap_diag}\n\\end{align}\n\\end{subequations}\nUsing the above information, as well as Eq.~\\ref{eqn:interpolation}, we can obtain each tern of $V_{\\alpha\\beta,m n}({\\bf R}(t))$ for propagating the dynamics of the quantum subsystem that contains both electronic and photonic DOFs.\n\nNext, we need to evaluate the nuclear gradients to propagate the dynamics of the classical subsystem, which contains the nuclear DOFs. In particular, we need to evaluate the nuclear gradients on each term of $\\nabla V_{\\alpha\\beta,mn}({\\bf R}(t_1))$. First, let us focus on the gradient term from the electron-nuclear coupling term\\cite{ZhouJCPL2019} as follows\n\\begin{align}\\label{dHen}\n&{\\nabla} H^\\mathrm{en}_{\\alpha\\beta,m n}({\\bf R}(t_1))\\\\\n&= {\\nabla} \\langle \\phi_{\\alpha} ({\\bf R}(t_0)),m|\\hat{H}_\\mathrm{en}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0)),n\\rangle \\nonumber\\\\\n&=\\langle \\phi_{\\alpha} ({\\bf R}(t_0))|{\\nabla}\\hat{H}_\\mathrm{en}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0))\\rangle\\cdot\\langle m|n\\rangle \\nonumber\\\\\n&=\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\langle \\phi_\\lambda({{\\bf R}({t_1})}) |\\nabla{\\hat{H}}_{\\mathrm{en}} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle S^{\\dagger}_{\\beta\\nu}\\cdot \\delta_{mn}. \\nonumber\n\\end{align}\nHere, from the first to the second line, we have used the fact that neither $\\langle\\phi_{\\alpha}({\\bf R}_0)|$ nor $\\langle m|$ are ${\\bf R}$-dependent, which allows ${\\nabla}$ to bypass both and directly act on $\\hat{V}({\\bf R}(t_{1}))$. We have also used the fact that $\\hat{H}_\\mathrm{en}$ does not contain any photonic operators. The gradient term $\\langle \\phi_\\lambda({{\\bf R}({t_1})}) |\\nabla{\\hat{H}}_{\\mathrm{en}} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle$ can be evaluated using the following well-known equality\\cite{tully94jcp}\n\\begin{equation}\\label{eqn:HF}\n\\langle \\phi_{\\lambda} ({\\bf R}) |\\nabla \\hat{H}_\\mathrm{en}({\\bf R}) | \\phi_{\\nu} ({\\bf R}) \\rangle = \n\\begin{cases}\n     \\nabla E_{\\lambda} \\quad & (\\lambda=\\nu) \\\\\n      {\\bf d}_{\\lambda\\nu} \\left( E_{\\nu} - E_{\\lambda} \\right) \\quad & (\\lambda \\neq \\nu), \n\\end{cases}\n\\end{equation}\nwhere the non-adiabatic coupling (NAC) vector (or so-called  derivative coupling) is \n\\begin{equation}\\label{elec-nac}\n{\\bf d}_{\\lambda\\nu}=\\langle \\phi_{\\lambda} ({\\bf R}) |\\nabla|\\phi_{\\nu} ({\\bf R}) \\rangle.\n\\end{equation}\n\nFor the gradient on the matrix $H^\\mathrm{p}_{\\alpha\\beta,m n}$, because there is no nuclear DOF in $\\hat{H}_\\mathrm{p}$, thus \n\\begin{equation}\n\\nabla H^\\mathrm{p}_{\\alpha\\beta,m n}({\\bf R}({t_1}))= \\nabla \\big[\\hbar\\omega_\\mathrm{c} (n+\\frac12)\\delta_{\\alpha,\\beta} \\delta_{m,n}\\big] = 0.\n\\end{equation}\n\nFor the gradient on the light-matter interaction term $H^\\mathrm{enp}_{\\alpha\\beta,m n}$, we have\n\n\\begin{widetext}\n\\begin{align}\\label{eqn:gradient_en}\n&\\nabla H^\\mathrm{enp}_{\\alpha\\beta,m n}({\\bf R}(t_1))= {\\nabla}\\langle \\phi_{\\alpha} ({\\bf R}(t_0)),m|\\hat{H}_\\mathrm{enp}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0)),n\\rangle \\\\\n& = \\langle \\phi_{\\alpha} ({\\bf R}(t_0))|{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}(t_0))\\rangle\\cdot \\boldsymbol{\\epsilon}\\cdot  g_\\mathrm{c} \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1}\\big)\\nonumber\\\\\n&=\\sum_{\\lambda\\nu}S_{\\alpha\\lambda} \\langle \\phi_\\lambda({{\\bf R}({t_1})})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R}(t_1))| \\phi_\\nu({{\\bf R}(t_1)}) \\rangle S^{\\dagger}_{\\beta\\nu}\\cdot \\boldsymbol{\\epsilon}\\cdot g_\\mathrm{c} \\big(\\sqrt{n}\\delta_{m,n-1} + \\sqrt{n+1}\\delta_{m,n+1}\\big),\\nonumber\n\\end{align}\n\\end{widetext}\n\nwhere $S_{\\alpha\\lambda}$ and $S^{\\dagger}_{\\beta\\nu}$ are defined in Eq.~\\ref{eqn:wf_overlap} and Eq.~\\ref{eqn:wf_overlap_diag}, respectively. To evaluate the term $\\langle\\phi_\\lambda({\\bf R})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R})| \\phi_\\nu({\\bf R)}\\rangle$ that appears in Eq.~\\ref{eqn:gradient_en}, we use a simple relation based on the chain rule as follows\n\\begin{widetext}\n\\begin{align}\\label{eq:muderive}\n&\\langle\\phi_\\lambda({\\bf R})|\\nabla\\hat{\\boldsymbol\\mu} ({\\bf R})| \\phi_\\nu({\\bf R})\\rangle=\\nabla\\langle\\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle - \\langle\\nabla \\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle-\\langle \\phi_{\\lambda}({\\bf R})|\\hat{\\boldsymbol\\mu} ({\\bf R})|\\nabla\\phi_{\\nu}({\\bf R})\\rangle \\nonumber \\\\\n&=\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R}) -\\sum_{\\gamma}\\langle\\nabla \\phi_{\\lambda}({\\bf R})|\\phi_{\\gamma}({\\bf R})\\rangle {\\boldsymbol\\mu}_{\\gamma \\nu}({\\bf R})-\\sum_{\\gamma}{\\boldsymbol\\mu}_{\\lambda\\gamma} ({\\bf R})\\langle\\phi_{\\gamma} ({\\bf R}) |\\nabla\\phi_{\\nu}({\\bf R})\\rangle\\nonumber\\\\\n&=\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})+ \\sum_{\\gamma}\\big[{\\bf d}_{\\lambda \\gamma}({\\bf R}) {\\boldsymbol\\mu}_{\\gamma \\nu}({\\bf R}) -{\\boldsymbol\\mu}_{\\lambda\\gamma}({\\bf R}){\\bf d}_{\\gamma\\nu}({\\bf R})\\big], \n\\end{align}\n\\end{widetext}\nwhere ${\\bf d}_{\\lambda\\gamma}({\\bf R})=\\langle\\phi_{\\lambda} ({\\bf R})|\\nabla\\phi_{\\gamma}({\\bf R})\\rangle=-\\langle\\nabla\\phi_{\\lambda}({\\bf R})|\\phi_{\\gamma}({\\bf R})\\rangle$ and ${\\bf d}_{\\gamma\\nu}({\\bf R})=\\langle\\phi_{\\gamma} ({\\bf R})|\\nabla\\phi_{\\nu}({\\bf R})\\rangle=-\\langle\\nabla\\phi_{\\gamma}({\\bf R})|\\phi_{\\nu}({\\bf R})\\rangle$ are the electronic derivative couplings (defined in Eq.~\\ref{elec-nac}). Note that from the first line to the second line, we have inserted $\\hat{\\mathcal P}=\\sum_{\\gamma}|\\phi_{\\gamma}({\\bf R})\\rangle\\langle\\phi_{\\gamma}({\\bf R})|$, which is the resolution of identity in the electronic subspace. As one can clearly see, this term requires the evaluation of derivative coupling ${\\bf d}_{\\lambda\\gamma}({\\bf R})$ and ${\\bf d}_{\\gamma\\nu}({\\bf R})$, and the derivative on dipole matrix element $\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})$.\nFor most of the electronic structure methods, the derivatives on dipole matrix elements $\\nabla {\\boldsymbol\\mu}_{\\lambda\\nu}({\\bf R})$ are not implemented. Nevertheless, recent theoretical development has made these quantities available.\\cite{Zhang2019jcp}\n\nFinally, the gradient from the DSE term is expressed as follows\n\\begin{widetext}\n\\begin{align} \\label{eqn:gradient_d}\n&\\nabla H^\\mathrm{d}_{\\alpha\\beta,m n}({\\bf R}(t_1))= {\\nabla} \\langle \\phi_{\\alpha} ({\\bf R}_{0}),m|\\hat{H}_\\mathrm{d}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0}), n\\rangle \\\\\n=& {\\nabla} \\Big[\\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\Big] \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\delta_{m,n} \\nonumber \\\\\n=& \\Big[\\sum_{\\gamma} (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |\\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\nonumber  \\\\\n& + (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\alpha} ({\\bf R}_{0}) | \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}_{0})\\rangle) (\\boldsymbol{\\epsilon} \\cdot \\langle \\phi_{\\gamma} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle) \\Big] \\frac{g_{\\mathrm{c}}^2}{\\hbar\\omega} \\delta_{m,n},\\nonumber\n\\end{align}\n\\end{widetext}\nwhere the term $\\langle \\phi_{\\alpha} ({\\bf R}(t_0)) | \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\gamma}({\\bf R}(t_0))\\rangle$ can be evaluated using Eq.~\\ref{eqn:mu2}, and the $\\langle \\phi_{\\alpha} ({\\bf R}_{0}) |{\\nabla} \\hat{\\boldsymbol \\mu}({\\bf R}(t_1))|\\phi_{\\beta}({\\bf R}_{0})\\rangle$ type of derivative can be computed in the same fashion as elaborated in Eq.~\\ref{eqn:gradient_en}.\n\nUsing the matrix elements $V_{\\alpha\\beta,m n}({\\bf R}(t))$ (Eq.~\\ref{eqn:interpolation}) and the nuclear gradient $\\nabla V_{\\alpha\\beta,mn}({\\bf R})$ (as outlined in Eqs.~\\ref{dHen} -~\\ref{eqn:gradient_d}), one can in principle use any trajectory-based approaches or wavepacket approaches with guiding trajectories\\cite{IzmaylovJCP2018,Ben-Nun1998jcp,mcEhrenfest} to propagate the quantum dynamics in the time step of $t\\in[t_0,t_1]$. During the next short-time propagation segment $t\\in[t_1,t_2]$, the QD scheme adopts a new reference geometry ${{\\bf R}'_{0}}\\equiv {\\bf R}(t_1)$ and new {\\it diabatic} basis $|\\psi_{j}({{\\bf R}'_{0}})\\rangle\\equiv|\\psi_{j}({\\bf R}(t_1))\\rangle=|\\phi_{\\beta}({\\bf R}(t_{1})),m\\rangle$. Between the $t\\in [t_0,t_1]$ propagation and the $t \\in [t_1,t_2]$ propagation segments, all of the necessary quantities will be transformed from $\\{|\\psi_{i}({\\bf R}_{0})\\rangle\\}$ to the $\\{|\\psi_{j}({\\bf R}'_{0})\\rangle\\}$ basis, using the relation \n\\begin{equation}\\label{eqn:basis}\n    |\\psi_{j}({\\bf R}(t_{1}))\\rangle=\\sum_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle|\\psi_{i}({\\bf R}(t_{0}))\\rangle.\n\\end{equation}\n\nFor the model calculations in this work, these overlap integrals are evaluated as \n\\begin{equation}\\label{overlap}\n\\langle\\psi_{i}({\\bf R}(t_{0}))|\\psi_{j}({\\bf R}(t_{1}))\\rangle=\\langle\\phi_{\\alpha}({\\bf R}(t_{0}))|\\phi_{\\beta}({\\bf R}(t_{1}))\\rangle\\cdot\\langle n|m\\rangle,\n\\end{equation}\nwhere the electronic adiabatic state overlaps $\\langle\\phi_{\\alpha}({\\bf R}(t_{0}))|\\phi_{\\beta}({\\bf R}(t_{1}))\\rangle$ are directly calculated using the discrete variable representation (DVR) basis, and the Fock states are orthonormal to each other $\\langle n|m\\rangle=\\delta_{n,m}$. \nWhen performing the transformation in Eq.~\\ref{eqn:basis} (as well as in  Eq.~\\ref{map-trans2} for the non-adiabatic mapping methods), the eigenvectors maintain their mutual orthogonality subject to a very small error when they are expressed in terms of the previous basis due to the incompleteness of the basis.\\cite{GranucciJCP2001,PlasserJCP2012} Nevertheless, the orthogonality remains to be well satisfied among  $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\}$ or $\\{|\\psi_{j}({\\bf R}(t_1))\\rangle\\}$. This small numerical error generated from each step can, however, accumulate over many steps and cause a significant error at longer times, leading to non-unitary dynamics.\\cite{GranucciJCP2001,PlasserJCP2012} This potential issue can be easily resolved by using orthonormalization procedure among the vectors of the overlap matrix composed by $\\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle$, as been done in our previous work\\cite{MandalJCP2018} for simulating photo-induced charge transfer dynamics. Here, we perform the L\\\"owdin orthogonalization procedure\\cite{LowdinJCP1950} as commonly used in the local diabatization approach\\cite{GranucciJCP2001} to ensure unitary propagation.\n\nAs the nuclear geometry closely follows the reference geometry throughout the propagation, the QD basis forms a convenient and compact basis. Note that, in principle, one needs an infinite set of crude adiabatic states $\\{|\\psi_{i}({\\bf R}_0)\\rangle\\}$ to represent the time-dependent electronic wavefunction because the electronic wavefunction could change rapidly with the motion of the nuclei, and the crude adiabatic basis is only convenient when the reference geometry ${\\bf R}_{0}$ is close to the nuclear geometry ${\\bf R}$. By dynamically updating the basis in the QD scheme, the time-dependent electronic wavefunction is expanded with the ``moving crude adiabatic basis\"\\cite{IzmaylovJCP2018} that explores the most relevant and important parts of the Hilbert space, thus requiring only a few states for quantum dynamics propagation. \n\n\\subsection{Non-adiabatic Mapping Dynamics Methods}\\label{subsec:mapping_method}\nThe Meyer-Miller-Stock-Thoss (MMST) formalism\\cite{MeyerJCP1979,StockPRL1997,ThossPRA1999} maps the discrete electronic DOFs onto continuous phase space variables. In the strict diabatic basis $\\{|i\\rangle\\}$ (in the sense that $\\langle i|\\nabla|j\\rangle=0$ for all $|i\\rangle$ and $|j\\rangle$), the total Hamiltonian in Eq.~\\ref{totalH} is expressed as\n\\begin{equation}\\label{Diabatic-Hamiltonian}\n\\hat{H}={\\frac {{\\bf P}^2} {2{\\bf M}}}+\\sum_{i}V_{ii}(\\hat{\\bf R})|i\\rangle\\langle i|+\\sum_{i\\neq j}V_{ij}(\\hat{\\bf R})|i\\rangle\\langle j|,\n\\end{equation}\nwhere $V_{ij}(\\hat{\\bf R})=\\langle i|\\hat{V}(\\hat{\\bf r},\\hat {\\bf R})|j\\rangle$ are the matrix elements of the electronic Hamiltonian. Note that here $|i\\rangle$ is used to represent the strict diabatic basis, and not to be confused with the adiabatic-Fock state $|\\psi_i({\\mathbf R}) \\rangle=|\\phi_{\\alpha}({\\bf R}),n\\rangle$ introduced in Eq.~\\ref{adia-fock}. Nevertheless, based on the QD scheme, these adiabatic-Fock states with a reference geometry ${\\bf R}_{0}$ will be used as the diabatic state in the neighborhood of the reference geometries, as indicated in Eq.~\\ref{eqn:qdidea}.\n \nIn the non-adiabatic mapping approach, the Hamiltonian operator in Eq.~\\ref{Diabatic-Hamiltonian} is transformed into the following MMST Hamiltonian\n\\begin{equation} \\label{eq:mapham} \n\\mathcal{H}_\\mathrm{m}={\\frac {{\\bf P}^2} {2{\\bf M}}}+{\\frac{1}{2}}\\sum_{ij}V_{ij}({\\bf R})\\left(p_{i}p_{j}+q_{i}q_{j}-2\\gamma_{j}\\delta_{ij}\\right),\n\\end{equation}\nwhere $2\\gamma_{j}$ is viewed as a parameter\\cite{MillerFD2016} which specifies the ZPE of the mapping oscillators.\\cite{UweJCP1999,MillerFD2016,richardson2019,richardson2020} In principle, $2\\gamma_{j}$ is state-specific and trajectory-specific.\\cite{CottonJCP2019_2} The MMST mapping Hamiltonian has been historically justified by Stock and Thoss using harmonic oscillator's raising and lowering operators as the mapping operator.\\cite{StockPRL1997,ThossPRA1999} Recently, it has been derived using the $SU(N)$ Lie group theory or so-called generalized spin mapping approach.\\cite{richardson2020}\n\nClassical trajectories are generated based on Hamilton's equations of motion (EOM)\n\\begin{subequations}\\label{eq:mapeqn}\n\\begin{eqnarray}  \n    \\dot q_{j} &=& \\partial \\mathcal{H}_\\mathrm{m}\/ \\partial p_{j};~~\\dot p_{i} = -\\partial \\mathcal{H}_\\mathrm{m} \/ \\partial q_{i}\\\\\n    \\dot {\\bf R} &=& \\partial \\mathcal{H}_\\mathrm{m}\/ \\partial {\\bf P};~~ \\dot {\\bf P}=-\\partial \\mathcal{H}_\\mathrm{m} \/ \\partial {\\bf R}= {\\bf F}, \n\\end{eqnarray}\n\\end{subequations}\nwith the nuclear force expressed as\n\\begin{equation}\n\\label{eq:force}\n{\\bf F}=-{\\frac {1} {2}}\\sum_{ij}\\nabla V_{ij}({\\bf R})\\big(p_{i}p_{j}+q_{i}q_{j}-2\\gamma_{j}\\delta_{ij}\\big).\n\\end{equation}\nOverall, the MMST mapping provides a consistent classical footing for both electronic and nuclear DOFs, and the non-adiabatic transitions between electronic states are captured through the classical motion of the fictitious harmonic oscillators. The non-adiabatic dynamics obtained from this formalism have shown good performance in the {\\it ab initio} on-the-fly dynamics.\\cite{ZhouJCPL2019,HuJCTC2021,WeightJCP2021}\n\nTo sample the initial electronic condition and estimate the population, it is also convenient to use the action-angle variables, $\\{\\varepsilon_{j}, \\theta_{j}\\}$, which are related to the canonical mapping variables $\\{p_{j}, q_{j}\\}$ through \n\\begin{equation}\\label{action-angle}\n    \\varepsilon_j = \\frac{1}{2}\\left(p_j^2 + q_j^2 \\right);~~~\\theta_j =-\\tan^{-1}\\left( \\frac{p_j}{q_j}\\right),\n\\end{equation}\nand the inverse relations\n\\begin{equation}\\label{eqn:action_to_mapping}\n    q_j = \\sqrt{2 \\varepsilon_j }\\cos(\\theta_j);~~p_j =-\\sqrt{2 \\varepsilon_j}\\sin(\\theta_j),\n\\end{equation}\nwhere $\\varepsilon_j$ is a positive-definite action variable that is directly proportional to the mapping variables' radius in action-space.\\cite{CottonJCP2019_2} \n\nThe SQC approach calculates the population of electronic state $|j\\rangle$, which is to be evaluated as\\cite{MillerFD2016}\n\\begin{align}\\label{eqn:wignersqc}\n\\rho_{jj}(t)&=\\mathrm{Tr}_{\\bf R}\\left[\\hat{\\rho}(0)e^{i\\hat{H}t\/\\hbar}|j\\rangle\\langle j|e^{-i\\hat{H}t\/\\hbar}\\right]\\\\\n&\\approx\\int d\\boldsymbol{\\tau} \\rho_\\mathrm{W}({\\bf P},{\\bf R})W_i({\\boldsymbol \\varepsilon}(0))W_j({\\boldsymbol \\varepsilon}(t)),\\nonumber\n\\end{align}\nwhere $\\hat{\\rho}(0)=\\hat{\\rho}_{\\bf R} \\otimes |i\\rangle \\langle i|$ is the initial density operator, $\\rho_\\mathrm{W}({\\bf P},{\\bf R})$ is the Wigner transform of $\\hat{\\rho}_{\\bf R}$ operator for the nuclear DOFs, ${\\boldsymbol \\varepsilon} = \\{\\varepsilon_1,\\varepsilon_2,...,\\varepsilon_\\mathcal{N}\\}$ is the positive-definite action variable vector for $\\mathcal{N}$ electronic states,\\cite{CottonJCP2019_2} $W_i({\\boldsymbol \\varepsilon})= \\delta (\\varepsilon_i - (1+\\gamma_{i}))\\prod_{i \\neq j} \\delta(\\varepsilon_j-\\gamma_{j})$ is the Wigner transformed action variables,\\cite{CottonJCP2016} and $d\\boldsymbol{\\tau}\\equiv d{\\bf P}\\cdot d{\\bf R}\\cdot d{\\boldsymbol \\varepsilon} \\cdot d\\boldsymbol{\\theta}$. For practical reasons, the above delta functions in $W_i({\\boldsymbol \\varepsilon})$ are broadened using a distribution function (so-called window function) that can be used to bin the resulting electronic action variables in action-space.\\cite{MillerFD2016} Further, we use the $\\gamma$-SQC approach,\\cite{CottonJCP2019_2} which uses a {\\it state-specific} and {\\it trajectory-specific} $\\gamma_{j}$ parameter in Eq.~\\ref{eq:mapham} to correct the initial force according to the initially populated state. This method has been proven to provide very accurate non-adiabatic dynamics in model photo-dissociation problems (coupled Morse potential), as well as outperform FSSH (with decoherence correction) in {\\it ab initio} on-the-fly simulations.\\cite{HuJCTC2021, WeightJCP2021} The details of $\\gamma$-SQC are provided in Appendix~\\ref{apsec:details_of_sqc}.\n\nFor the spin-LSC approach,\\cite{richardson2019,richardson2020} one chooses a universal ZPE parameter $2\\gamma_{j}=\\Gamma$ for all states and trajectories. The spin-LSC population dynamics is calculated as\n\\begin{align}\\label{eqn:wignerlsc}\n\\rho_{jj}(t)&=\\mathrm{Tr}_{\\bf R}\\left[\\hat{\\rho}_{R}\\otimes|i\\rangle\\langle i|e^{i\\hat{H}t\/\\hbar}|j\\rangle\\langle j|e^{-i\\hat{H}t\/\\hbar}\\right]\\\\\n&\\approx\\int d\\boldsymbol{\\tau} \\rho_\\mathrm{W}({\\bf P},{\\bf R}) [|i\\rangle\\langle i|]_\\mathrm{s}(0)\\cdot[|j\\rangle\\langle j|]_\\mathrm{\\bar{s}}(t),\\nonumber\n\\end{align}\nwhere the population estimators are obtained from the Stratonovich-Weyl transformed electronic projection operators, with the expressions as follows\\cite{richardson2020}\n\\begin{subequations}\\label{sw-estimator}\n\\begin{align}\n&\\big[|i\\rangle\\langle i|\\big]_\\mathrm{s}=\\frac{1}{2}(q^2_i+p^2_i-\\Gamma)\\\\\n&\\big[|j\\rangle\\langle j|\\big]_\\mathrm{\\bar s}\n=\\frac{\\mathcal{N}+1}{2(1+\\frac{\\mathcal{N}\\Gamma}{2})^2}\\cdot (q^2_{j}+p^2_{j})-\\frac{1-\\frac{\\Gamma}{2}}{1+\\frac{\\mathcal{N}\\Gamma}{2}}.\n\\end{align}\n\\end{subequations}\nThe parameter $\\Gamma$ is related to the radius of the generalized Bloch sphere $r_\\mathrm{s}$ through $\\Gamma=\\frac{2}{\\mathcal N}(r_\\mathrm{s}-1)$, where $\\mathrm{s}$ and $\\mathrm{\\bar s}$ are complementary indices in the Stratonovich-Weyl transform. Among the vast parameter space, one of the best performing choices\\cite{richardson2019,richardson2020} is when $r_\\mathrm{s}=r_\\mathrm{\\bar s}=\\sqrt{\\mathcal{N}+1}$, which is referred to as $\\mathrm{s}=\\mathrm{W}$, leading to a ZPE parameter\n\\begin{equation}\\label{Gamma}\n\\Gamma=\\frac{2}{\\mathcal{N}}(\\sqrt{\\mathcal{N}+1}-1),\n\\end{equation}\nas well as the identical expression of $[|i\\rangle\\langle i|]_\\mathrm{s}$ and $[|j\\rangle\\langle j|]_\\mathrm{\\bar s}$ in Eq.~\\ref{sw-estimator}. We further use the focused initial condition\\cite{richardson2019,richardson2020} that replaces the sampling of the mapping variables in the $d\\boldsymbol{\\tau}$ integral of Eq.~\\ref{eqn:wignerlsc} with specific values of the mapping variables, such that $\\frac{1}{2}(q^2_i+p^2_i-\\Gamma)=1$ for initially occupied state $|i\\rangle$ and $\\frac{1}{2}(q^2_j+p^2_j-\\Gamma)=0$ for the initially unoccupied states $|j\\rangle$. The angle variables $\\{\\theta_{j}\\}$ (Eq.~\\ref{action-angle}) are randomly sampled\\cite{richardson2020} in the range of $[0,2\\pi)$.\n\nUsing the QD propagation scheme, one can directly perform non-adiabatic using both $\\gamma$-SQC and spin-LSC in their original diabatic formalism, with the information from the ``ab initio\" polaritonic calculations of the molecule-cavity hybrid system. Using the schemes outlined in Sec.~\\ref{sec:QD}, one can obtain the polariton coupling $\\langle\\psi_{i}({\\bf R}_{0})|\\hat{V}({\\bf R})|\\psi_{j}({\\bf R}_{0})\\rangle$ (see Eq.~\\ref{eqn:vijt} and Eq.~\\ref{eqn:interpolation}) and nuclear gradient $\\nabla\\langle\\psi_{i}({\\bf R}_{0})|\\hat{V}({\\bf R})|\\psi_{j}({\\bf R}_{0})\\rangle$ (see Eq.~\\ref{dHen}-Eq.~\\ref{eqn:gradient_d}), which are the necessary ingredients to solve the MMST mapping EOMs in Eq.~\\ref{eq:mapeqn} and Eq.~\\ref{eq:force}. Between two propagation steps, the QD basis is transformed from $\\{|\\psi_{i}({\\bf R}(t_0))\\rangle\\equiv|\\phi_\\alpha({\\mathbf R}(t_0)),n\\rangle\\}$ to $\\{|\\psi_{j}({\\bf R}(t_1))\\rangle\\equiv|\\phi_\\beta({\\mathbf R}(t_1)),m\\rangle\\}$. This leads to the corresponding transform of mapping variables between the two consecutive QD bases as follows\\cite{MandalJCTC2018,ZhouJCPL2019}\n\\begin{subequations}\\label{map-trans2}\n\\begin{align}\n    &\\sum_{i} q_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))|\\psi_{j}({\\bf R}(t_{1}))\\rangle \\rightarrow q_{j}\\\\\n    &\\sum_{i} p_{i} \\langle \\psi_{i}({\\bf R}(t_{0}))| \\psi_{j}({\\bf R}(t_{1}))\\rangle \\rightarrow p_{j},\n\\end{align}\n\\end{subequations}\nwhere the overlaps between the two steps are evaluated using Eq.~\\ref{overlap} (see discussion under that equation). More computational details for the $\\gamma$-SQC and spin-LSC are provided in section \\ref{sec:comp-detail}.\n\n\\section{Computational Details}\n\\subsection{The Model System}\nIn this work, we use the asymmetrical Shin-Metiu model\\cite{Shin1995jcp,Hoffmann2020} as the ``\\textit{ab initio}\" model molecular system to investigate strong light-matter interactions between a molecule and an optical cavity. The model contains a transferring proton (nucleus) and an electron, as well as two fixed ions labeled as donor (D) and acceptor (A), as shown in Fig.~\\ref{fig:adiabatic_PES}a. This model is usually used to describe the proton-coupled electron transfer (PCET) reaction and has been studied recently using the exact factorization approach to investigate how cavity can influence chemical reactivities.\\cite{Maitra2019,Hoffmann2020,Maitra2021} The electron-nuclear interaction potential operator $\\hat{H}_\\mathrm{en}$ (c.f. Eq.~\\ref{eqn:Hen}) is expressed as\n\\begin{equation}\\label{Hen}\n\\hat{H}_\\mathrm{en}=\\sum_{\\sigma=\\pm 1} \\left( \\frac{1}{|R+\\frac{\\sigma L}{2}|}-\\frac{\\mathrm{erf}\\Big( \\frac{|r+\\frac{\\sigma L}{2}|}{a_{\\sigma}} \\Big)}{|r+\\frac{\\sigma L}{2}|} \\right) - \\frac{\\mathrm{erf}\\Big(\\frac{|R-r|}{a_f}\\Big)}{|R-r|},\n\\end{equation}\nwhere the first term represents the potential of the transferring proton, the second term represents the potential of the transferring electron, and the third term represents the electron-proton coupling. We choose the same parameters used in Ref.~\\citenum{Hoffmann2020}, which is $L=19 $ a.u., $a_+=3.1$ a.u., $a_{-}=4.0$ a.u., $a_f=5.0$ a.u. and the proton mass is $M=1836$ a.u. To calculate the electronic properties of the SM model, we use the Sinc discrete variable representation (DVR) basis\\cite{Colbert1992} to represent the electronic adiabatic states. These adiabatic states $|\\phi_{\\alpha}(R)\\rangle$ are computed on-the-fly for a given nuclear configuration $R$ by solving Eq.~\\ref{HenPsi}. The details are provided in Appendix~\\ref{Exact}.\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_1.pdf}\n  \\end{minipage}%\n   \\caption{(a) The schematic illustration of the asymmetrical SM model, where one proton and one electron can transfer between two fixed ions (donor and acceptor). The distance between the donor and acceptor is 19 a.u. Here $V_p$ and $V_e$ are the potential of the transferring proton and electron, respectively. (b) The first two adiabatic electronic states, (c) NAC between these two electronic states and (d) their permanent and transition dipole moments of the SM model. The PESs of the polaritonic states inside the cavity are obtained with the light-matter coupling strength (e) $g_\\mathrm{c}=0.001$ and (f) $g_\\mathrm{c}=0.005$. The color used in (e) and (f) is coded according to $\\langle \\hat{a}^{\\dagger}\\hat{a}\\rangle$, as shown in the upper position of panel (e).}\n\\label{fig:adiabatic_PES}\n\\end{figure}\n\nFig.~\\ref{fig:adiabatic_PES}a also depicts the model potential in Eq.~\\ref{Hen}, with the black curve depicting the proton potential (the first term in Eq.~\\ref{Hen}) and the green curve depicting the electron potential (the second term in Eq.~\\ref{Hen}). Fig.~\\ref{fig:adiabatic_PES}b presents the two lowest adiabatic electronic states of the SM model (red and blue curves). There is an avoided crossing between the ground and the first excited state potential energy surfaces (PESs) near $R=2.0$ a.u.. Fig.~\\ref{fig:adiabatic_PES}c presents the NAC between them (the green curve), which shows a strong coupling near the avoided crossing region. The matrix elements of the dipole operator under the adiabatic representation (Eq.~\\ref{dipole-mat}) of the SM model are presented in Fig.~\\ref{fig:adiabatic_PES}d. \n\nWhen coupling the SM molecular model with the cavity, the photon frequency of the cavity mode is chosen as $\\hbar \\omega_\\mathrm{c}=2.721$ eV ($\\approx \\, 0.1 $ a.u.). Further, we assume that the cavity field polarization direction ${\\boldsymbol{\\epsilon}}$ is always aligned with the direction of the dipole operator $\\hat{\\boldsymbol\\mu}$, such that ${\\boldsymbol{\\epsilon}}\\cdot{\\boldsymbol\\mu}_{\\gamma\\nu}={\\mu}_{\\gamma\\nu}$ (for $\\{\\nu,\\gamma\\}=\\{e,g\\}$) where ${\\mu}_{\\gamma\\nu}$ is the magnitude of $\\hat{\\boldsymbol\\mu}$. Explicitly considering the angle between ${\\boldsymbol{\\epsilon}}$ and $\\hat{\\boldsymbol\\mu}$ will generate a polariton induced conical intersection (even for a diatomic molecule), which will induce geometric phase effects.\\cite{Farag2021PCCP} We consider two different light-matter coupling strengths $g_\\mathrm{c}=0.001$ a.u. and $g_\\mathrm{c}=0.005$ a.u. in this work. The normalized coupling strength is often defined as\\cite{Kockum2019rev} $\\eta\\equiv g_\\mathrm{c}\\cdot|\\boldsymbol{\\epsilon} \\cdot  {{\\boldsymbol \\mu}}_{eg}|\/\\omega_\\mathrm{c}$ where $|\\boldsymbol{\\epsilon} \\cdot {{\\boldsymbol \\mu}}_{eg}|$ is the typical magnitude of the transition dipole projected along the field polarization direction. For the coupling strength considered above (and taking $\\hbar\\omega_\\mathrm{c}\\approx 2.721$ eV for the model calculation), the normalized coupling strength is $\\eta = 0.06$ (for $g_\\mathrm{c}=0.001$ a.u.) and $\\eta = 0.3$ (for $g_\\mathrm{c}=0.005$ a.u.). When $0.1<\\eta<1$, the light and matter interaction achieves the ultra-strong coupling regime,\\cite{Kockum2019rev,Nori2019natphys} which is difficult to achieve but still within the reach of the current experimental setup.\\cite{Shegai_NC_2020,Haran2016} Thus, besides the pure theoretical value to derive the exact nuclear gradient expression, our computational results are also within the reach of the near future experimental setup. \n\n\nThe polaritonic PESs $\\mathcal{E}_{J}(R)$ associated with polariton states $|\\mathcal{E}_{J}(R)\\rangle$ (see their definition in Eq.~\\ref{polariton}) are presented in Fig.~\\ref{fig:adiabatic_PES}e with the light-matter coupling strength $g_\\mathrm{c}=0.001$ a.u. and in Fig.~\\ref{fig:adiabatic_PES}f with the light-matter coupling strength $g_\\mathrm{c}=0.005$. These polariton potentials are color coded (as shown in the inset of panel (e)) based on the expectation value of $\\langle \\hat{a}^{\\dagger}\\hat{a}\\rangle$ indicated on top of this panel. Note that this should not be viewed as a ``photon number\" operator under the dipole gauge used in the PF Hamiltonian\\cite{Mandal2020JPCL,Schaefer2020AP} because the rigorous photon number operator should be obtained by applying the Power-Zienau-Woolley (PZW) Gauge transformation\\cite{PZW,Cohen-Tannoudji,Taylor2020PRL} on the photon number operator $\\hat{a}^{\\dagger}\\hat{a}$. Nevertheless, it can be viewed as an approximate estimation of the photon number when the light-matter couplings are not in the ultra-strong coupling regime.\n\\cite{Forn-Diaz2019rmp}\n\nThe initial state (for $t=0$) of the molecule-cavity hybrid system is\n\\begin{equation}\\label{eq:initial-state}\n|\\Phi(t=0)\\rangle=|e,0\\rangle\\otimes|\\chi\\rangle,\n\\end{equation}\nwhich corresponds to a Franck-Condon excitation of the hybrid system to the $|e,0\\rangle$ state, with $|\\chi\\rangle$ as the initial nuclear wavefunction. For the SM model in this work, we use $\\chi(R)=\\langle R|\\chi\\rangle\\sim \\exp[-M\\omega_{0}(R-R_{0})^2\/2\\hbar]$, where $M$ is the mass of the proton (nucleus in the SM model), $R_0$ is the position with a minimum potential energy of the ground electronic state. Here, $\\chi(R)$ is the vibrational ground state wavefunction on the ground electronic states, centered at $R_0$ under the harmonic approximation, with the harmonic oscillation frequency being $\\omega_{0}$. We use the parameters in the original reference\\cite{Hoffmann2020} for $R_{0}=-4$ and $\\omega_{0}=0.000382$ a.u. To solve the exact quantum dynamics, we use the DVR basis for the nuclear DOF and the adiabatic-Fock state for the electronic-photonic subsystem. The details of the exact quantum dynamics are provided in Appendix~\\ref{Exact}. \n\n\\subsection{Details of $\\gamma$-SQC and spin-LSC Dynamics}\\label{sec:comp-detail}\n\nTo perform the $\\gamma$-SQC dynamics, we need to sample the initial condition for the quantum subsystem. In this work, we first sample the action-angle variables $\\{\\varepsilon_{j}, \\theta_{j}\\}$ then transform them to the mapping variables$\\{p_{j},q_{j}\\}$ using Eq.~\\ref{eqn:action_to_mapping}. Among them, the action variables $\\{\\varepsilon_{j}\\}$ are sampled according to the window function in Eq.~\\ref{EQ:TriangleWindow}, and the angle variables $\\{\\theta_{j}\\}$ are randomly sampled from $[0,2\\pi)$. The triangle window is used in this work, although the square window generates similar results. \n\nFor the spin-LSC dynamics, we use the focused initial conditions\\cite{richardson2020} as described in section \\ref{subsec:mapping_method}, where the action variable $\\varepsilon_{i}$ is set to be $1+\\Gamma\/2$ for the initially occupied state and $\\Gamma\/2$ for the initially unoccupied state, with $\\Gamma$ expressed in Eq.~\\ref{Gamma}. The angle variables $\\{\\theta_{j}\\}$ are randomly generated between $[0,2\\pi)$ as in the $\\gamma$-SQC method. The canonical mapping variables are obtained from Eq.~\\ref{eqn:action_to_mapping}.\n\nThe initial nuclear distribution of all trajectory-based simulations (Ehrenfest, FSSH, $\\gamma$-SQC and spin-LSC) are generated by sampling the Wigner density \n\\begin{equation}\\label{wigner}\n[\\langle R|\\chi\\rangle]_\\mathrm{w}=\\frac{1}{\\hbar\\pi}e^{-M(P^2+\\omega^2(R-R_{0})^2)\/\\omega\\hbar},\n\\end{equation}\nwhich is the Wigner transformation of the nuclear wavefunction $\\chi(R)=\\langle R|\\chi\\rangle$ in the initial state (see Eq.~\\ref{eq:initial-state}). Here, $R$ and $P$ are the nuclear coordinate and momentum, respectively. The initial state for the electronic-photonic subsystem is set to $|e0\\rangle$. The nuclear time step used in the QD-$\\gamma$-SQC and QD-spin-LSC is $dt=0.1$ fs, with $100$ equally spaced electronic time steps for the mapping variables' integration during each nuclear time step. The equation of motion in Eq.~\\ref{eq:mapeqn}-Eq.~\\ref{eq:force} are integrated using a second-order symplectic integrator for the MMST variables\\cite{kelly2012mapping,church2018} for a given nuclear time step, and these mapping variables are transformed based on Eq.~\\ref{map-trans2} between two adjacent nuclear time steps due to the change of the QD basis. The population dynamics using all MQC and mapping methods were averaged over $5000$ trajectories, although $3000$ trajectories were enough to produce the basic trend of the polariton dynamics, see Fig. S3 in the Supplementary Material. The light-matter coupling strength $g_\\mathrm{c}$ was chosen to be $0.001$ and $0.005$, according to our previous work.\\cite{Zhou2022}\n\nWe also benchmark the results of non-adiabatic mapping dynamics approaches with commonly used MQC approaches, including the Ehrenfest dynamics and the FSSH method. The details of these two MQC approaches are provided in Appendix~\\ref{apsec:Ehrenfest_FSSH_Details}. In particular, the Ehrenfest dynamics is equivalent to choosing $\\gamma=0$ in the mapping theory (see Eq.~\\ref{eq:mapham}) and an initial action-angle variables condition (see Eq.~\\ref{action-angle} and Eq.~\\ref{eqn:action_to_mapping}) of $\\varepsilon_j=\\delta_{ij}$ (for the initially occupied state $|i\\rangle$) and $\\theta_{j}=0$ (for all state $|j\\rangle$). One can thus use the same QD scheme and the mapping equation to obtain the results of the Ehrenfest dynamics.\\cite{Subotnik2016JCP} \n\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_2.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states in Shin-Metiu-cavity model obtained from (a) Ehrenfest dynamics, (b) FSSH approach, (c) $\\gamma$-SQC method, and (d) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Two electronic states and two Fock states are considered in the simulation, and the light-matter coupling strength $g_\\mathrm{c}=0.001$ a.u.}\n\\label{fig:gc001_results}\n\\end{figure}\n\n\\section{Results}\nFig.~\\ref{fig:gc001_results} presents the population dynamics of the adiabatic-Fock states simulated using the approximate methods (open circles), including the MQC approaches (Ehrenfest and FSSH) and the mapping dynamics methods ($\\gamma$-SQC and spin-LSC), compared to the numerically exact results (solid lines). The light-matter coupling strength is chosen to be $g_\\mathrm{c}=0.001$ a.u. The system is initially prepared in the $|e0\\rangle$ state and decays quickly into the $|g1\\rangle$ state during the first $\\sim12$ fs due to the large light-matter coupling strength from the large transition dipole moment ($\\boldsymbol{\\mu}_{eg}$) between adiabatic electronic state $|g\\rangle$ and $|e\\rangle$ (as shown in Fig.~\\ref{fig:adiabatic_PES}d). Then, the system starts to oscillate between $|e0\\rangle$ and $|g1\\rangle$ until about $20$ fs. All of the MQC and mapping dynamics methods can describe the above process reasonably well compared to the exact results. After that, all the dynamics results (including the exact one) show a fast population increase of the $|g0\\rangle$ state, which is due to the electronic NAC $d_{eg}$ that directly couples the $|e0\\rangle$ state to $|g0\\rangle$ state (gold lines). All of the approximate methods can qualitatively describe such a trend, but the MQC methods (panels a-b) are less accurate compared to the mapping-based methods (panels c-d), in terms of the rising of the $|g0\\rangle$ population as well as its long time plateau. Moreover, both Ehrenfest and FSSH dynamics predict a significant population transfer from $|g1\\rangle$ to $|e1\\rangle$ state (panels a-b) as an {\\it artifact} that is not shown in the exact dynamics results. In contrast, the $\\gamma$-SQC and spin-LSC methods perform much better, where the population transfer process from $|g1\\rangle$ to $|e1\\rangle$ state is largely suppressed (panels c-d). Overall, the mapping methods outperform the MQC methods in this small light-matter coupling case. It is worth mentioning that the population dynamics results obtained with the FSSH method can be significantly improved if one uses the proper estimator.\\cite{Landry2013JCP} We have provided details of this approach and numerical results in Appendix~\\ref{apsec:Ehrenfest_FSSH_Details}. Even so, the FSSH method is still facing many challenges from the improper treatment of the quantum coherence and frustrated hop problems, which have been widely discussed.\\cite{StockPRL1997,subotnik2016arpc,PrezdoSH,Granucci2007} \n\n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_3.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states in Shin-Metiu-cavity model obtained from (a) Ehrenfest dynamics, (b) FSSH approach, (c) $\\gamma$-SQC method, and (d) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Two electronic states and two Fock states are considered in the simulation, and the light-matter coupling strength $g_\\mathrm{c}=0.005$ a.u.}\n\\label{fig:gc005_results}\n\\end{figure}\n\nFig.~\\ref{fig:gc005_results} presents the polariton population dynamics with the coupling strength $g_\\mathrm{c}=0.005$ a.u. The oscillation between $|e0\\rangle$ and $|g1\\rangle$ state population appears much earlier and faster compared to the $g_\\mathrm{c}=0.001$ results due to the larger light-matter coupling between $|e0\\rangle$ and $|g1\\rangle$ states (see Eq.~\\ref{eq:enp}). Further, the $|g0\\rangle$ and $|e1\\rangle$ states are also getting populated at an earlier time, due to the permanent dipole $\\mu_{gg}$ and $\\mu_{ee}$ that couples $|g1\\rangle$ state to $|g0\\rangle$ state and $|e0\\rangle$ state to $|e1\\rangle$ state, respectively. Similar to the $g_\\mathrm{c}=0.001$ case, all the MQC and mapping dynamics provide a reasonable accuracy for the population dynamics at a short time, while the mapping methods perform much better than the MQC methods at a longer time. In addition, the spin-LSC method outperforms $\\gamma$-SQC method in the description of $|g0\\rangle$ and $|e0\\rangle$ state population after $t=20$ fs, as shown in Fig.~\\ref{fig:gc005_results}c-d.\n\nUntil now, all of our simulations are restricted in the Hilbert subspace formed by two electronic states ($|g\\rangle$ and $|e\\rangle$) and two photonic Fock states ($|0\\rangle$ and $|1\\rangle$). The system could explore a larger Hilbert space due to the increasing light-matter coupling strength. Thus, we systematically check the polariton dynamics using the exact wavepacket dynamics method with a larger number of electronic adiabatic states and Fock states, as shown in Fig. S1 and S2 in the Supplementary Material. The results show that, for the small light-matter coupling strength case ($g_\\mathrm{c}=0.001$ a.u.), truncation to the Hilbert subspace formed by two electronic states and two Fock states is enough to give an accurate description of the population dynamics for the SM model studied in this work. However, for the larger coupling strength case ($g_\\mathrm{c}=0.005$ a.u.), the polariton dynamics will converge when including four adiabatic electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$ with energies in ascending order) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$ with photon number in ascending order). \n\\begin{figure}[ht!]\n \\centering\n  \\begin{minipage}[t]{1.0\\linewidth}\n     \\centering\n     \\includegraphics[width=\\linewidth]{Figure_4.pdf}\n  \\end{minipage}%\n   \\caption{The population dynamics of the adiabatic-Fock states with (a,b) Ehrenfest dynamics, (c,d) FSSH approach, (e,f) $\\gamma$-SQC method, and (g,h) spin-LSC dynamics. The population dynamics are obtained with the approximate methods (open circles) and exact quantum propagation (solid lines). Four adiabatic electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$ with energies in ascending order) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$ with photon number in ascending order) are considered in the simulations and the light-matter coupling strength is  $g_\\mathrm{c}=0.005$ a.u. Only the adiabatic-Fock states with observable populations of more than $0.01$ are plotted.}\n\\label{fig:more-states}\n\\end{figure}\n\nTo further test the performance of the mapping methods ($\\gamma$-SQC and spin-LSC) as well as the MQC methods (Ehrenfest and FSSH) in such a large Hilbert subspace, which includes sixteen states formed by the tensor product of four electronic states ($|g\\rangle$, $|e\\rangle$, $|f\\rangle$, $|h\\rangle$) and four Fock states ($|0\\rangle$, $|1\\rangle$, $|2\\rangle$, $|3\\rangle$). Fig.~\\ref{fig:more-states} presents the results of using Ehrenfest dynamics (a-b), FSSH (c-d), $\\gamma$-SQC (e-f) and spin-LSC (g-h). Besides the adiabatic-Fock states already appear in the four-states subspace ($|g0\\rangle$, $|e0\\rangle$, $|g1\\rangle$ and $|e1\\rangle$), we can also see some other states ($|h0\\rangle$, $|f0\\rangle$, $|g2\\rangle$ and $|f1\\rangle$) are populated due to the increasing light-matter coupling strength. All of the MQC (Ehrenfest, FSSH) and mapping ($\\gamma$-SQC and spin-LSC) dynamics results provide accurate agreement with the exact one in the short time ($< 20$ fs). After that, all of the methods start to generate less accurate results (especially for the $|g0\\rangle$ population). Note that both $\\gamma$-SQC and spin-LSC perform less accurately compared to the situation in a smaller Hilbert subspace (Fig.~\\ref{fig:gc005_results}). This is because both $\\gamma$-SQC and spin-LSC are sensitive to the number of states of the system. For $\\gamma$-SQC, more states means less trajectory landed in the population action window.\\cite{Cotton_JCP_2019_many_states} For spin-LSC, the ZPE correction $\\Gamma$ (Eq.~\\ref{Gamma}) explicitly depends on the number of states $\\mathcal{N}$. This suggests a need for the future development of more accurate dynamics approaches. The current work, nevertheless, paves the way for those future methods to be directly used for simulating on-the-fly polariton quantum dynamics.\n\n\\section{Conclusions}\nIn this work, we generalize the quasi-diabatic (QD) propagation scheme\\cite{MandalJCP2018,ZhouJCPL2019,MandalJPCA2019} to simulate the non-adiabatic polariton dynamics in molecule-cavity hybrid systems. The adiabatic-Fock states, which are the tensor product states of the adiabatic electronic states of the molecule and photon Fock states, are used as the {\\it locally} well-defined {\\it diabatic} states for the dynamics propagation.\\cite{ZhouJCPL2019,MandalJPCA2019} These locally well-defined diabatic states allow using any diabatic quantum dynamics methods for dynamics propagation, and the definition of these states will be updated at every nuclear time step. The benefit of using such adiabatic-Fock states is that one can conveniently obtain the electronic adiabatic states energies, the nuclear gradient, the dipole moments and NACs between these states, which are necessary ingredients in molecular cavity QED simulations. \n\nWe use the recently developed non-adiabatic mapping dynamics approaches, $\\gamma$-SQC\\cite{CottonJCP2019_2} and spin-LSC\\cite{richardson2020} to investigate polariton dynamics of a Shin-Metiu model coupled to an optical cavity.\\cite{Hoffmann2020,Zhou2022} To benchmark the results of the obtained polariton dynamics, we performed simulations using the Ehrenfest dynamics and the FSSH approaches, as well as the numerically exact polariton wavepacket propagation. The results show that the mapping methods can accurately describe the population dynamics of the molecule-cavity system both at a short time and a longer time when compared to exact dynamics results. In addition, the mapping methods outperform the Ehrenfest and FSSH approaches at a long time dynamics. The numerical results also demonstrate that the performance of the mapping methods ($\\gamma$-SQC and spin-LSC) becomes less accurate with an increased number of states in the simulation, indicating the need for future theoretical development.\n\nWe envision that the theoretical development in this work will provide the emerging polariton chemistry field with a general theoretical tool that enables direct {\\it ab initio} on-the-fly simulations of polariton photochemical processes. We also anticipate that the theoretical developments in this work will enable many recently developed diabatic quantum dynamics approaches to directly simulate polariton quantum dynamics.\n\n\\section*{Acknowledgments}\nThis work was supported by the National Science Foundation CAREER Award under Grant No. CHE-1845747 and by a Cottrell Scholar award (a program by Research Corporation for Science Advancement). Computing resources were provided by the Center for Integrated Research Computing (CIRC) at the University of Rochester. D.H. appreciates valuable discussions with Wanghuai Zhou. We also appreciate valuable suggestions by Prof. Yihan Shao.\n\n\\section*{Conflict of Interest}\nThe authors have no conflicts to disclose.\n\n\\section*{Supplementary Material}\nSee Supplementary Material for additional results of the convergence test for the number of trajectories and the number of adiabatic electronic states and Fock states.\n\n\\section*{Availability of Data}\t\nThe data that support the findings of this study are available from the corresponding author upon a reasonable request.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\subsection{Background}\n\\label{subsec:background} Let $P = \\{p_1,\\ldots,p_n\\}$ be a set of\n$n$ points in $\\mathbb{R}^3$. The \\emph{2-center problem} for $P$ is to\nfind two congruent balls of minimum radius whose union covers $P$.\nThis is a special case of the general $p$-center problem in\n$\\mathbb{R}^d$, which calls for covering a set $P$ of $n$ points in\n$\\mathbb{R}^d$ by $p$ congruent balls of minimum radius. If $p$ is\npart of the input, the problem is known to be NP-complete~\\cite{MK}\neven for $d=2$, so the complexity of algorithms for solving the\n$p$-center problem, for any fixed $p$, is expected to increase more\nthan polynomially in $p$. Agarwal and Procopiuc showed that the\n$p$-center problem in $\\mathbb{R}^d$ can be solved in $n^{O(p^{1-1\/d})}$\ntime~\\cite{AP}, improving upon a naive $n^{O(p)}$-solution. At the\nother extreme end, the 1-center problem (also known as the\n\\emph{smallest enclosing ball} problem) is known to be an LP-Type\nproblem, and can thus be solved in $O(n)$ randomized expected time\nin any fixed dimension, and also in deterministic linear time\n~\\cite{CM, NML, NMLT}. Faster approximate solutions to the general\n$p$-center problem have also been proposed~\\cite{AP, BHI, BE}.\n\nIf $d$ is not fixed, the 2-center problem in $\\mathbb{R}^d$ is\nNP-Complete~\\cite{MK2}. The 2-center problem in $\\mathbb{R}^2$ has a\nrelatively rich history, mostly in the past two decades. Hershberger\nand Suri~\\cite{HS} showed that the decision problem of determining\nwhether $P$ can be covered by two disks of a given radius $r$ can be\nsolved in $O(n^2 \\log n)$ time. This has led to several\nnearly-quadratic algorithms~\\cite{ASP, DE, JK} that solve the\noptimization problem, the best of which, due to Jaromczyk and\nKowaluk~\\cite{JK}, runs in $O(n^2 \\log n)$ deterministic time.\nSharir~\\cite{MS} considerably improved these bounds and obtained a\ndeterministic algorithm with $O(n \\log^9 n)$ running time. His\nalgorithm combines several geometric techniques, including\nparametric searching, searching in monotone matrices, and dynamic\nmaintenance of planar configurations. Chan~\\cite{TC} (following an\nimprovement by Eppstein~\\cite{DEF}) improved the running time to\n$O(n \\log^2 n \\log^2 \\log n)$.\n\nThe only earlier work on the 2-center problem in $\\mathbb{R}^3$ we are aware of is by Agarwal~\\textsl{et~al.}~\\cite{AES},\nwhich presents an algorithm with $O(n^{3+\\varepsilon})$ running time, for any $\\varepsilon > 0$. It uses\na rather complicated data structure for dynamically maintaining upper and lower envelopes of bivariate functions.\n\n\\subsection{Our results}\n\\label{subsec:results} We present two randomized algorithms for the\n2-center problem in $\\mathbb{R}^3$. We first present an algorithm whose\nexpected running time is $O(n^3 \\log^5 n)$. It is conceptually a\nnatural generalization of the earlier algorithms for the planar\n2-center problem~\\cite{ASP, DE, JK}; its implementation however is\nconsiderably more involved. The second algorithm runs in $O((n^2\n\\log^5 n) \/(1-r^*\/r_0)^3)$ expected time, where $r^*$ is the common\nradius of the 2-center balls and $r_0$ is the radius of the smallest\nenclosing ball of $P$.  This is based on some of the ideas in\nSharir's planar algorithm~\\cite{MS}, but requires several new\ntechniques. As in the previous algorithms, we first present\nalgorithms for the decision problem: given $r > 0$, determine\nwhether $P$ can be covered by two balls of radius $r$. We then\ncombine it with an adaptation of Chan's randomized optimization\ntechnique~\\cite{TCG} to obtain a solution for the optimization\nproblem. In both cases, the asymptotic expected running time of the\noptimization algorithm is the same as that of the decision procedure\n(which itself is deterministic).\n\nThe paper is organized as follows. Section~\\ref{sec:sketches}\nbriefly sketches our two solutions. Section~\\ref{sec:cubic_alg}\npresents the near-cubic algorithm, and\nSection~\\ref{sec:improved_alg} presents the improved algorithm. A\nkey ingredient of both algorithms is a dynamic procedure for testing\nwhether the intersection of a collection of balls in $\\mathbb{R}^3$\nis nonempty. We present the somewhat technical details of this\nprocedure in Section~\\ref{sec:spherical_polytopes}, and conclude in\nSection~\\ref{sec:discussion} with a few open problems.\n\n\\section{Sketches of the Solutions}\n\\label{sec:sketches}\n\n\\subsection{The near-cubic algorithm}\n\\label{subsec:n^3_sketch} To solve the decision problem, in the less\nefficient but conceptually simpler manner, we use a standard\npoint-plane duality, and replace each point $p \\in P$ by a dual\nplane $p^*$, and each plane $h$ by a dual point $h^*$, such that the\nabove-below relations between points and planes are preserved. We\nnote that if $P$ can be covered by two balls $B_1, B_2$ (not\nnecessarily congruent), then there exists a plane $h$ (containing\nthe circle ${\\partial}{B_1} \\cap {\\partial}{B_2}$, if they intersect at all, or\nseparating $B_1$ and $B_2$ otherwise) separating $P$ into two\nsubsets $P_1, P_2$, such that $P_1 \\subset B_1$ and $P_2 \\subset\nB_2$. We therefore construct the arrangement ${\\cal A}$ of the set $\\{p^*\n\\mid p \\in P\\}$ of dual planes. It has $O(n^3)$ cells, and each cell\n$\\tau$ has the property that, for any point $w \\in \\tau$, its primal\nplane $w^*$ separates $P$ into two subsets of points, $P_\\tau^+$ and\n$P_\\tau^-$, which are the same for every $w \\in \\tau$, and depend\nonly on $\\tau$. We thus perform a traversal of ${\\cal A}$, which proceeds\nfrom each visited cell to a neighbor cell. When we visit a cell\n$\\tau$, we check whether the subsets $P_\\tau^+$ and $P_\\tau^-$ can\nbe covered by two balls of radius $r$, respectively. To do so, we\nmaintain dynamically the intersection of the sets $\\{B_r(p)\\mid p\n\\in P_\\tau^+\\}$, $\\{B_r(p)\\mid p \\in P_\\tau^-\\}$, where $B_r(p)$ is\nthe ball of radius $r$ centered at $p$, and observe that (a) any\npoint in the first (resp., second) intersection can serve as the\ncenter of a ball of radius $r$ which contains $P_\\tau^+$ (resp.,\n$P_\\tau^-$), and (b) no ball of radius $r$ can cover $P_\\tau^+$\n(resp., $P_\\tau^-$) if the corresponding intersection is empty.\nMoreover, when we cross from a cell $\\tau$ to a neighbor cell\n$\\tau'$, $P_\\tau^+$ changes by the insertion or deletion of a single\npoint, and $P_\\tau^-$ undergoes the opposite change, so each of the\nsets of balls $\\{B_r(p)\\mid p \\in P_\\tau^+\\}$, $\\{B_r(p)\\mid p \\in\nP_\\tau^-\\}$ changes by the deletion or insertion of a single ball.\nAs we know the sequence of updates in advance, maintaining\ndynamically the intersection of either of these sets of balls can be\ndone in an offline manner. Still, the actual implementation is\nfairly complicated. It is performed using a variant of the\nmulti-dimensional parametric searching technique of\nMatou\\v{s}ek~\\cite{JM} (see also~\\cite{TCA, CMS, NPT}). The same\nprocedure is also used by the second improved algorithm. For the\nsake of readability, we describe this procedure towards the end of\nthe paper, in Section~\\ref{sec:spherical_polytopes}.\n\n\nThe main algorithm uses a segment tree to represent the sets\n$P_\\tau^+$ (and another segment tree for the sets $P_\\tau^-$).\nRoughly, viewing the traversal of ${\\cal A}$ as a sequence $\\Sigma$ of\ncells, each ball $B_r(p)$ has a \\emph{life-span} (in $P_\\tau^+$),\nwhich is a union of contiguous maximal subsequences of cells $\\tau$,\nin which $p \\in P_\\tau^+$, and a complementary life-span in\n$P_\\tau^-$. We store these (connected portions of the) life-spans as\nsegments in the segment tree. Each leaf of the tree represents a\ncell $\\tau$ of ${\\cal A}$, and the balls stored at the nodes on the path\nto the leaf from the root are exactly those whose centers belong to\nthe set $P_\\tau^+$ (or $P_\\tau^-$). By precomputing the intersection\nof the balls stored at each node of the tree, we can express each of\nthe intersections $\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^+\\}$ and\n$\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^-\\}$, for each cell $\\tau$, as\nthe intersection of a logarithmic number of precomputed\nintersections (see also~\\cite{DE}). We show that such an\nintersection can be tested for emptiness in $O(\\log^5 n)$ time. This\nin turn allows us to execute the decision procedure with a total\ncost of $O(n^3 \\log^5 n)$. We then return to the original\noptimization problem and apply a variant of Chan's randomization\ntechnique~\\cite{TCG} to solve the optimization problem by a small\nnumber of calls to the decision problem, obtaining an overall\nalgorithm with $O(n^3 \\log^5 n)$ \\emph{expected} running\ntime.\\footnote{\\small The earlier algorithm in~\\cite{AES} follows\nthe same general approach, but uses an even more complicated, and\nslightly less efficient machinery for dynamic emptiness testing of\nthe intersection of congruent balls.}\n\n\\subsection{The improved solution}\n\\label{subsec:n^2_sketch} The above algorithm runs in nearly cubic\ntime because it has to traverse the entire arrangement ${\\cal A}$, whose\ncomplexity is $O(n^3)$. In Section~\\ref{sec:improved_alg} we improve\nthis bound by traversing only portions of ${\\cal A}$, adapting some of the\nideas in Sharir's improved solution for the planar\nproblem~\\cite{MS}. Specifically, Sharir's algorithm solves the\ndecision problem (for a given radius $r$) in three steps, treating\nseparately three subcases, in which the centers $c_1, c_2$ of the\ntwo covering balls are, respectively, far apart ($|c_1 c_2| > 3r$),\nat medium distance apart ($r < |c_1 c_2| \\leq 3r$) and near each\nother ($|c_1 c_2| \\leq r$). We base our solution on the techniques\nused in the first two cases, which, for simplicity, we merge into a\nsingle case (as done in~\\cite{DEF} for the planar case), and extend\nit so that we only need to assume that $|c_1c_2| \\geq \\beta r$, for\nany fixed $\\beta > 0$. In more detail, letting $B_r(p)$ denote the\ndisk of radius $r$ centered at a point $p$, Sharir's algorithm\nguesses a constant number of lines $l$, one of which separates the\ncenters $c_1, c_2$ of the respective solution disks $D_1,D_2$, so\nthat the set $P_L$ of the points to the left of $l$ is contained in\n$D_1$. We then compute the intersection $K(P_L) = \\bigcap_{p \\in\nP_L} B_r(p)$, and intersect each ${\\partial}{B_r(p)}$, for $p \\in P_R = P\n\\setminus P_L$ (the subset of points to the right of $l$), with\n${\\partial}{K(P_L)}$. It is easily seen that ${\\partial}{K(P_L)}$ has linear\ncomplexity and that each circle ${\\partial}{B_r(p)}$, for $p \\in P_R$,\nintersects it at two points (at most). This produces $O(n)$ critical\npoints (vertices and intersection points) on ${\\partial}{K(P_L)}$ and\n$O(n)$ arcs in between. As argued in~\\cite{MS}, it suffices to\nsearch these points and arcs for possible locations of the center of\n$D_1$ (and dynamically test whether the balls centered at the\nuncovered points have nonempty intersection).\n\nGeneralizing this approach to $\\mathbb{R}^3$, we need to guess a separating plane $\\lambda$, to retrieve the subset $P_L \\subseteq P$ of points to the left of $\\lambda$, to compute ${\\partial}{K(P_L)}$ (which, fortunately, still has only linear complexity), to intersect ${\\partial}{B_r(p)}$, for each $p \\in P_R$, with ${\\partial}{K(P_L)}$, and to form the arrangement of the resulting intersection curves. Each cell of this arrangement is a candidate for the location of the center of the left covering ball $B_1$, and for each placement in $\\tau$, $B_1$ contains the same fixed subset of $P$ (which depends only on $\\tau$).\n\nHowever, the complexity of the resulting arrangement $M_K$ on\n${\\partial}{K(P_L)}$ might potentially be cubic. We therefore compute only\na portion $M$ of $M_K$, which suffices for our purposes, and prove\nthat its complexity is only $O(n^2)$. This is the main geometric\ninsight in the improved algorithm, and is highlighted in\nLemma~\\ref{lemma:quadratic_K_P_L}. We show that if there is a\nsolution then $O(1\/\\beta^3)$ guesses suffice to find a separating\nplane. This implies that the running time of the improved decision\nprocedure is $O((1\/\\beta^3) n^2\\log^5 n)$. Thus, it is nearly\nquadratic for any fixed value of $\\beta$. We show that one can take\n$\\beta = 2(r_0\/r -1)$, where $r_0$ is the radius of the smallest\nenclosing ball of $P$.\n\nTo solve the optimization problem, we conduct a search on the\noptimal radius $r^*$, using our decision procedure, starting from\nsmall values of $r$ and going up, halving the gap between $r$ and\n$r_0$ at each step\\footnote{\\small We have to act in this manner to\nmake sure that we do not call the decision procedure with values of\n$r$ which are too close to $r_0$, thereby losing control over the\nrunning time.}, until the first time we reach a value $r > r^*$.\nThen we use a variant of Chan's technique~\\cite{TCG}, combined with\nour decision procedure, to find the exact value of $r^*$. The way\nthe search is conducted guarantees that its cost does not exceed the\nbound $O((1\/\\beta^3) n^2 \\log^5 n)$, for the separation parameter\n$\\beta = 2(r_0\/r^* -1)$ for $r^*$. Hence, we obtain a randomized\nalgorithm that solves the 2-center problem for any positive\nseparation of $c_1$ and $c_2$, and runs in $O((n^2 \\log^5 n)\n\/(1-r^*\/r_0)^3)$ expected time.\n\n\n\n\\section{A Nearly Cubic Algorithm}\n\\label{sec:cubic_alg}\n\n\\subsection{The decision procedure}\n\\label{subsec:decision_procedure} In this section we give details of\nthe implementation of our less efficient solution, some of which are\nalso applicable for the improved solution. Recall from the\ndescription in Section~\\ref{sec:sketches} that the decision\nprocedure, on a given radius $r$, constructs two segment trees $T^+,\nT^-$, on the life-spans of the balls $B_r(p)$, for $p \\in P$ (with\nrespect to the tour of the dual plane arrangement ${\\cal A}$). Each leaf\nis a cell $\\tau$ of ${\\cal A}$, and the balls, whose centers belong to\n$P_\\tau^+$ (resp., $P_\\tau^-$), are those stored at nodes on the\npath from the root to $\\tau$ in $T^+$ (resp., $T^-$).\n\nFor each node $u$ of $T^+$, let $S_u$ denote the intersection of all\nthe balls (of radius $r$) stored at $u$. We refer to each $S_u$ as a\n\\emph{spherical polytope}; see~\\cite{BCT, BLNP, BN} for (unrelated)\nstudies of spherical polytopes. We compute each $S_u$ in $O(|S_u|\n\\log |S_u|)$ deterministic time, using the algorithm by\nBr\\\"{o}nnimann et al.~\\cite{BCM} (see also~\\cite{CS, ER} for\nalternative algorithms). Since the arrangement ${\\cal A}$ consists of\n$O(n^3)$ cells, standard properties of segment trees imply that the\ntwo trees require $O(n^3 \\log n)$ storage and $O(n^3 \\log^2 n)$\npreproccessing time.\n\nClearly, the intersection $K(P_\\tau^+)$ (resp., $K(P_\\tau^-)$) of\nthe balls whose centers belong to $P_\\tau^+$ (resp., $P_\\tau^-$) is\nthe intersection of all the spherical polytopes $S_u$, over the\nnodes $u$ on the path from the root to $\\tau$ in $T^+$ (resp.,\n$T^-$).\n\n\\paragraph{Intersection of spherical polytopes.}\nLet ${\\cal S} = \\{S_1,\\ldots,S_t\\}$ be the set of $t = O(\\log n)$\nspherical polytopes stored at the nodes of a path from the root to a\nleaf of $T^+$ or of $T^-$, where, as above, a spherical polytope is\nthe intersection of a finite set of balls, all having the common\nradius $r$. Each $S_i$ is the intersection of some $n_i$ balls, and\n$\\sum_{i=1}^t n_i \\leq n$. Our current goal is to determine, in\npolylogarithmic time, whether the intersection $K$ of the spherical\npolytopes in ${\\cal S}$ is nonempty. If this is the case for at least one\npath of $T^+$ and for the same path in $T^-$ then $r^* \\leq r$, and\notherwise $r^* >r$. Moreover, if there exist a pair of such paths\nfor which both intersections have nonempty interior, then $r^* < r$\n(because we can then slightly shrink the balls and still get a\nnonempty intersection). If no such pair of paths have this property,\nbut there exist pairs with nonempty intersections (with at least one\nof them being degenerate) then $r^* = r$.\n\nThe algorithm for testing emptiness of $K$ is technical and fairly\ninvolved. For the sake of readability, we delegate its description\nto Section~\\ref{sec:spherical_polytopes}. It uses a variant of\nmultidimensional parametric searching which somewhat resembles\nsimilar techniques used in ~\\cite{TCA, CMS, JM, NPT}. It is\nessentially independent of the rest of the algorithm (with some\nexceptions, noted later). We summarize it in the following\nproposition.\n\n\\begin{proposition}\n\\label{prop:intersection_test} Let ${\\cal S}$ be a collection of spherical\npolytopes, each defined as the intersection of at most $n$ balls of\na fixed radius $r$. Let $N$ denote the sum, over the polytopes of\n${\\cal S}$, of the number of balls defining each polytope. After a\npreprocessing stage, which takes $O(N \\log n)$ time and uses $O(N)$\nstorage, we can test whether any $t \\leq \\log n$ polytopes of ${\\cal S}$\nhave a nonempty intersection in $O(\\log^5 n)$ time, and also\ndetermine whether the intersection has nonempty interior.\n\\end{proposition}\n\nHence, we check, for each cell $\\tau$, whether each of $K(P_\\tau^+)$\nand $K(P_\\tau^-)$ are nonempty and non-degenerate. To this end, we\ngo over each path of $T^+$, and over the same path of $T^-$, and\ncheck, using the procedure described in\nProposition~\\ref{prop:intersection_test}, whether the spherical\npolytopes along the tested paths (of $T^+$ and of $T^-$) have a\nnonempty intersection (and whether these intersections have nonempty\ninteriors). We stop when a solution for which both $K(P_\\tau^+)$ and\n$K(P_\\tau^-)$ are nonempty and non-degenerate is obtained, and\nreport that $r^* < r$. Otherwise, we continue to test all cells\n$\\tau$. If at least one degenerate solution is found (i.e., a\nsolution where both $K(P_\\tau^+), K(P_\\tau^-)$ are nonempty, and at\nleast one of them has nonempty interior), we report that $r^* = r$,\nand otherwise $r^* > r$.\n\nBy proposition~\\ref{prop:intersection_test}, the cost of this\nprocedure is $O(n^3 \\log^5 n)$. This subsumes the cost of all the\nother steps, such as constructing the arrangement ${\\cal A}$ and the\nsegment trees $T^+, T^-$. We therefore get a decision procedure\nwhich runs in $O(n^3 \\log^5 n)$ (deterministic) time.\n\n\\subsection{Solving the optimization problem}\n\\label{subsec:optimization_problem}\nWe now combine our decision procedure with the randomized\noptimization technique of Chan~\\cite{TCG}, to obtain an algorithm\nfor the optimization problem, which runs in $O(n^3 \\log^5 n)$\n\\emph{expected} time. Our application of Chan's technique, described\nnext, is somewhat non-standard, because each recursive step has also\nto handle global data, which it inherits from its ancestors.\n\nChan's technique, in its ``purely recursive'' form, takes an optimization problem that has to compute an optimum value $w(P)$ on an input set $P$. The technique replaces $P$ by several subsets $P_1,\\ldots,P_s$, such that $w(P) = \\min\\{w(P_1),\\ldots,w(P_s)\\}$, and $|P_i| \\leq \\alpha|P|$ for each $i$ (here $\\alpha < 1$ and $s$ are constants). It then processes the subproblems $P_i$ in a \\emph{random} order, and computes $\\displaystyle\\min_i w(P_i)$ by comparing each $w(P_i)$ to the minimum $w$ collected so far, and by replacing $w$ by $w(P_i)$ if the latter is smaller.\\footnote{\\small So the value of $w$ keeps shrinking.} Comparisons are performed by the decision procedure, and updates of $w$ are computed recursively. The crux of this technique is that the expected number of recursive calls (in a single recursive step) is only $O(\\log s)$, and this (combined with some additional enhancements, which we omit here) suffices to make the expected cost of the whole procedure asymptotically the same as the cost of the decision procedure, for \\emph{any} values of $s$ and $\\alpha$. Technically, if the cost $D(n)$ of the decision procedure is\n$\\Omega(n^\\gamma)$, where $\\gamma$ is some fixed positive constant, the expected running time is $O(D(n))$ provided that\n\\begin{equation}\n\\label{equation:lnr}\n(\\ln s + 1) \\alpha^\\gamma < 1.\n\\end{equation}\nHowever, even when (\\ref{equation:lnr}) does not hold ``as\nis'', Chan's technique enforces it by compressing $l$ levels of the\nrecursion into a single level, for $l$ sufficiently large, so its expected cost is still $O(D(n))$. See~\\cite{TCG} for details.\n\n\nTo apply Chan's technique to our decision procedure,\nwe pass to the dual space, where each point $p \\in P$ is mapped to a\nplane $p^*$, as done in the decision procedure. We obtain the set $P^* =\n\\{p^* \\mid p \\in P\\}$ of dual planes, and we consider its arrangement\n${\\cal A} = {\\cal A}(P^*)$, where each cell $\\tau$ in ${\\cal A}$ represents an\nequivalence class of planes in the original space, which separate $P$\ninto the same two subsets of points $P_\\tau^+, P_\\tau^-$.\n\nTo decompose the optimization problem into subproblems, as required\nby Chan's technique, we construct a $(1\/\\varrho)$-\\emph{cutting} of\nthe dual space. We recall that, given a collection $\\it{H}$ of $n$\nhyperplanes in $\\mathbb{R}^d$ and a parameter $1 \\leq \\varrho \\leq\nn$, a $(1\/\\varrho)$-\\emph{cutting} of ${\\cal A}(\\it{H})$ of size $q$ is a\npartition of space into $q$ (possibly unbounded) openly disjoint\n$d$-dimensional simplices $\\Delta_1,\\ldots,\\Delta_q$, such that the\ninterior of each simplex $\\Delta_i$ is intersected by at most\n$n\/\\varrho$ of the hyperplanes of $\\it{H}$. See~\\cite{JMC} for more\ndetails.  We use the following well known result~\\cite{BC, CF}:\n\n\\begin{lemma}\n\\label{lemma:cutting}\nGiven a set $\\it{H}$ of $n$ hyperplanes in $\\mathbb{R}^d$, a\n$(1\/\\varrho)$-cutting of ${\\cal A}(\\it{H})$ of size $O(\\varrho^d)$ can be constructed\nin time $O(n\\varrho^{d-1})$, for any $\\varrho \\leq n$.\n\\end{lemma}\n\nReturning to our setup, we construct a $(1\/\\varrho)$-cutting for\n${\\cal A}(P^*)$, for a specific constant value of $\\varrho$, that we will\nfix later, and obtain $O(\\varrho^3)$ simplices, such that the\ninterior of each of them is intersected by at most $n\/\\varrho$\nplanes of $P^*$. Each simplex $\\Delta_i$ corresponds to one\nsubproblem and contains some (possibly only portions of) cells\n$\\tau_1,\\ldots,\\tau_k$ of the arrangement ${\\cal A}$. We recall that each\ncell $\\tau_j$ represents an equivalence class of planes which\nseparate $P$ into two subsets of points $P_{\\tau_j}^+$ and\n$P_{\\tau_j}^-$. Hence, $\\Delta_i$ represents a collection of such\nequivalence classes. All these subproblems have in common the sets\n$(P^*)_{\\Delta_i}^+$, $(P^*)_{\\Delta_i}^-$, consisting,\nrespectively, of all the planes that pass fully above $\\Delta_i$ and\nthose that pass fully below $\\Delta_i$. (These sets are dual to\nrespective subsets $P_{\\Delta_i}^+, P_{\\Delta_i}^-$ of $P$, where\n$P_{\\Delta_i}^+$ is contained in all the sets $P_{\\tau_j}^+$, for\nthe cells $\\tau_j$, that meet $\\Delta_i$, and symmetrically for\n$P_{\\Delta_i}^-$.) Note that most of the dual planes belong to\n$(P^*)_{\\Delta_i}^+ \\cup (P^*)_{\\Delta_i}^-$; the ``undecided''\nplanes are those that cross the interior of $\\Delta_i$, and their\nnumber is at most $n\/\\varrho$. We denote the set of these planes as\n$(P^*)_{\\Delta_i}^0$ (and the set of their primal points as\n$P_{\\Delta_i}^0$).\n\n\nTo apply Chan's technique, we construct two segment trees on the\narrangement of $(P^*)_{\\Delta_i}^0$, as described in Section~\\ref{subsec:decision_procedure}. Consider one of these segment trees, $T^+$,\nthat maintains the set of balls ${\\cal B}^+ = \\{B_r(p) \\mid p \\in\nP_{\\tau_j}^+\\}$. Each cell $\\tau_j$ in $\\Delta_i$ is represented by a\nleaf of $T^+$. Each ball is represented as a collection of disjoint\nlife-spans, with respect to a fixed tour of the cells of ${\\cal A}((P^*)_{\\Delta_i}^0)$,\nwhich are stored as segments in $T^+$, as described earlier.\nIn addition, we compute the intersection of the balls centered\nat the points of $P_{\\Delta_i}^+$, in $O(n \\log n)$ time, and store it at\nthe root of $T^+$. Note that, as we go down the recursion, we keep\nadding planes to $(P^*)_{\\Delta_i}^+$, that is, points to $P_{\\Delta_i}^+$, and the\nactual set $P_{\\Delta_i}^+$ of points dual to the planes above the\ncurrent $\\Delta_i$ is the union of logarithmically many subsets, each\nobtained at one of the ancestor levels of the recursion, including the current step.\nHowever, we cannot inherit the precomputed intersections of the balls\nin these subsets of $P_{\\Delta_i}^+$ from the previous levels, since, as we go down the\nrecursion, Chan's technique keeps `shrinking' the radius of the balls. Hence, each\ntime we have to solve a decision subproblem, we compute the\nintersection of the balls centered at the points of $P_{\\Delta_i}^+$\n(collected over all the higher levels of the recursion)\nfrom scratch. (See below for details on the additional cost incurred\nby this step.) We build a second segment tree $T^-$\nthat maintains the balls of ${\\cal B}^- = \\{B_r(p) \\mid p \\in\nP_{\\tau_j}^-\\}$, in a fully analogous manner. The running time so far (of the decision procedure)\nis $O(n \\log n + m^3 \\log^2 m)$, where $m$ is the\nnumber of planes in $(P^*)_{\\Delta_i}^0$ and $n$ is the size of the\ninitial input set $P$.\n\nTo solve the decision procedure for a given subproblem associated\nwith a simplex $\\Delta_i$, we test, by going over all the\nroot-to-leaf paths in $T^+$ and $T^-$, whether there exists a cell\n$\\tau$ (overlapping $\\Delta_i$), for which the intersections of the\nspherical polytopes on the two respective paths in $T^+$ and $T^-$\nare nonempty (and, if nonempty, whether they both have nonempty\ninteriors). The overall cost of this step, iterating over the\n$O(m^3)$ cells of ${\\cal A}((P^*)_{\\Delta_i}^0)$ and applying the\nprocedure from Section~\\ref{subsec:decision_procedure} for\nintersecting spherical polytopes, is $O(m^3 \\log^5 n)$.\n\nWhen the recursion bottoms out, we have two subsets\n$P_{\\Delta_i}^+$, and $P_{\\Delta_i}^-$ of $O(n)$ points, and a\nconstant number of points in $P_{\\Delta_i}^0$. Hence, we try the\nconstant number of possible separations of $P_{\\Delta_i}^0$ into an\nordered pair of subsets $P_1$ and $P_2$, and, for each of these separations, we compute the two\nsmallest enclosing balls of the sets $P_{\\Delta_i}^+ \\cup P_1$ and\n$P_{\\Delta_i}^- \\cup P_2$ in linear time. If both $P_{\\Delta_i}^+\n\\cup P_1$ and $P_{\\Delta_i}^- \\cup P_2$ can be covered by balls of\nradius $r$, for at least one of the possible separations of\n$P_{\\Delta_i}^0$ into two subsets, then we have found a solution for\nthe 2-center problem. (Discriminating between $r^* = r$ or $r^* < r$\nis done as in Section~\\ref{subsec:decision_procedure}.)\n\nWe now apply Chan's technique to this decision procedure. Note that\nthis application is not standard because the recursive subproblems are\nnot ``pure'', as they also involve the ``global'' parameter $n$. We\ntherefore need to exercise some care in the analysis of the expected\nperformance of the technique.\n\nSpecifically, denote by $T(m,n)$ an upper bound on the expected\nrunning time of the algorithm, for preprocessing\na recursive subproblem involving $m$ points, where the initial input\nconsists of $n$ points. Then $T(m,n)$ satisfies the following recurrence.\n\n\\begin{equation}\n\\label{eqn:T(m,n)}\nT(m,n) \\leq \\left\\{ \\begin{array}{ll}\n\\ln(c\\varrho^3)T(m\/\\varrho,n) + O(m^3 \\log^5 n + n\\log n),   & \\mbox{for $m \\geq \\varrho$,}\\\\\nO(n),                                        & \\mbox{for $m <\n\\varrho$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $c$ is an appropriate absolute constant (so that $c\\varrho^3$\nbounds the number of cells of the cutting), and $\\varrho$ is chosen\nto be a sufficiently large constant so that (\\ref{equation:lnr})\nholds (with $s = c\\varrho^3, \\alpha = 1\/\\varrho$, and $\\gamma = 3$).\nIt is fairly routine (and we omit the details) to show that the\nrecurrence (\\ref{eqn:T(m,n)}) yields the overall bound $O(n^3 \\log^5\nn)$ on the expected cost of the initial problem; i.e., $T(n,n) =\nO(n^3 \\log^5 n)$. We thus obtain the following intermediate result.\n\n\\begin{theorem}\nLet $P$ be a set of $n$ points in $\\mathbb{R}^3$. A 2-center for $P$\ncan be computed in $O(n^3 \\log^5 n)$ randomized expected time.\n\\end{theorem}\n\n\\section{An Improved Algorithm}\n\\label{sec:improved_alg}\n\n\\subsection{An improved decision procedure $\\Gamma$}\n\\label{subsec:improved_decision_procedure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{beta_a_s.pstex_t}\n\\caption{\\small \\sf The points $q_1, q_1', q_2, q_2'$ prevent $|c_1\nc_2|$ from getting smaller.} \\label{figure:beta}\n\\end{center}\n\\end{figure}\n\nConsider the decision problem, where we are given a radius $r$ and a parameter $\\beta > 0$, and\nhave to determine whether $P$ can be covered by two balls of radius\n$r$, such that the distance between their centers $c_1, c_2$ is at\nleast $\\beta r$. (Details about supplying a good lower bound for $\\beta$ will be given in Section~\\ref{sec:separated_centers_optimization}.)\nBy this we mean that there is no placement of two balls of radius $r$, which cover $P$, such that the distance between their centers is smaller than $\\beta r$; see Figure~\\ref{figure:beta}.\n\nThis assumption is easily seen to imply the following property: Let\n$C_{12}$ denote the intersection circle of ${\\partial}{B_1}$ and ${\\partial}{B_2}$\n(assuming that $B_1 \\cap B_2 \\neq \\emptyset$). Then any hemisphere\n$\\nu$ of ${\\partial}{B_1}$, such that (a) the plane $\\pi$ through $c_1$\ndelimiting $\\nu$ is disjoint from $C_{12}$, and (b) $\\nu$ and\n$C_{12}$ lie on different sides of $\\pi$, must contain a point $q$\nof $P$, for otherwise we could have brought $B_1$ and $B_2$ closer\ntogether by moving $c_1$ in the normal direction of $\\pi$, into the\nhalfspace containing $c_2$ (and $C_{12}$). See\nFigure~\\ref{figure:case2_assumption}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{case2_assumption_s.pstex_t}\n\\caption{\\small \\sf The plane $\\pi$ passes through $c_1$ and is\ndisjoint from $C_{12}$. The hemisphere $\\nu$ delimited by $\\pi$,\nwhich lies on the side of $\\pi$ not containing $C_{12}$, must\ncontain a point $q$ of $P$.} \\label{figure:case2_assumption}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{new_case2_a_s.pstex_t}\n\\caption{\\small \\sf $v_1$ is the leftmost point of the intersection\ncircle $C_{12}$.} \\label{figure:new_case2_1}\n\\end{center}\n\\end{figure}\n\n\\smallskip\n\\noindent{\\bf Guessing orientations and separating planes.} We\nchoose a set $D$ of canonical orientations, so that the maximum\nangular deviation of any direction $u$ from its closest direction in\n$D$ is an appropriate multiple $\\alpha$ of $\\beta$. The connection\nbetween $\\alpha$ and $\\beta$ is given by the following reasoning.\nFix a direction $v \\in D$ so that the angle between the orientation\nof $c_1 c_2$ and $v$ is at most $\\alpha$. Rotate the coordinate\nframe so that $v$ becomes the $x$-axis. As above, let $C_{12}$\ndenote the intersection circle of ${\\partial}{B_1}$ and ${\\partial}{B_2}$\n(assuming that the balls intersect). Let $v_1$ be the leftmost point\nof $C_{12}$ (in the $x$-direction); see\nFigure~\\ref{figure:new_case2_1}. If $B_1$ and $B_2$ are disjoint\n(which only happens when $|c_1 c_2| > 2r$) we define $v_1$ to be the\nleftmost point of $B_2$. To determine the value of $\\alpha$, we note\nthat (in complete analogy with Sharir's algorithm in the\nplane~\\cite{MS}) our procedure will try to find a $yz$-parallel\nplane, which separates $c_1$ from $v_1$. For this, we want to ensure\nthat $x(v_1) - x(c_1) > \\beta r\/4$, say, to leave enough room for\nguessing such a separating plane. Let $\\theta$ denote the angle\n$\\varangle v_1c_1c_2$ (see Figure~\\ref{figure:new_case2_2}). Using\nthe triangle inequality on angles, the angle between\n$\\overrightarrow{c_1 v_1}$ and the $x$-axis is at most $\\theta +\n\\alpha$, so $x(v_1) - x(c_1) \\geq r \\cos(\\theta + \\alpha)$. Hence,\nto ensure the above separation, we need to choose $\\alpha$, such\nthat $\\cos(\\theta + \\alpha) > \\beta\/4$. Since $|c_1 c_2| \\geq \\beta\nr$, we have $\\cos \\theta \\geq \\beta\/2$. Hence, it suffices to choose\n$\\alpha$, such that\n$$\\alpha \\leq \\cos^{-1} \\frac{\\beta}{4} - \\cos^{-1} \\frac{\\beta}{2} =\n    \\sin^{-1} \\frac{\\beta}{2} - \\sin^{-1} \\frac{\\beta}{4}\n    = \\Theta(\\beta).$$\nWith this constraint on $\\alpha$, the size of $D$ is\n$\\Theta\\left(1\/\\alpha^2\\right) = \\Theta\\left(1\/\\beta^2\\right)$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{choosing_alpha_s2.pstex_t}\n\\caption{\\small \\sf $x(v_1) - x(c_1) \\geq r \\cos(\\theta + \\alpha)$.}\n\\label{figure:new_case2_2}\n\\end{center}\n\\end{figure}\n\nWe draw $O(1\/\\beta)$ $yz$-parallel planes, with horizontal\nseparation of $\\beta r\/4$, starting at the leftmost point of $P$\n(with respect to the guessed orientation). One of these planes will\nseparate $v_1$ from $c_1$. Thus, the total number of guesses that we\nmake (an orientation in $D$ and a separating plane) is\n$O(1\/\\beta^3)$. The following description pertains to a correct\nguess, in which the properties that we require are satisfied. (If\nall guesses fail, the decision procedure has a negative answer.)\n\n\\smallskip\n\\noindent{\\bf Reducing to a 2-dimensional search.}\nBy the property noted above, the left hemisphere $\\nu_{\\lambda_0}$ of ${\\partial}{B_1}$, delimited by\nthe $yz$-parallel plane $\\lambda_0$ through $c_1$,\nmust pass through at least one point $q$ of $P$ (see Figure~\\ref{figure:new_case2_left}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{leftsemiball_s2.pstex_t}\n\\caption{\\small \\sf The separating plane $\\lambda$ and its parallel\ncopy $\\lambda_0$ through $c_1$. The hemisphere $\\nu_{\\lambda_0}$ of\n${\\partial}{B_1}$ to the left of $\\lambda_0$ must contain a point $q$ of\n$P$.} \\label{figure:new_case2_left}\n\\end{center}\n\\end{figure}\n\n\nLet $P_L$ denote the subset\nof points of $P$ lying to the left of $\\lambda$. Then $P_L$\nmust be fully\ncontained in $B_1$ and contain $q$.\nWe compute the intersection $K(P_L) = \\bigcap \\{B_r(p) \\mid p \\in P_L\\}$\nin $O(n \\log n)$ time~\\cite{BCM}. If $K(P_L)$ is empty, then $P_L$ cannot be covered\nby a ball of radius $r$ and we determine that the currently assumed\nconfiguration does not yield a positive solution for the decision problem.\nOtherwise, since $P_L \\subseteq B_1$, $c_1$\nmust lie in\n$K(P_L)$. Moreover, since $q \\in P_L$ lies on the \\emph{left} portion\nof ${\\partial}{B_1}$, $c_1$ must\nlie on the \\emph{right} portion of the boundary of $K(P_L)$. Finally, since $c_1$ lies to the\nleft of $\\lambda$, only the portion $\\sigma_L$ of the right part of ${\\partial}{K(P_L)}$ to the left of $\\lambda$ has to be considered.\nIf $K(P_L)$ is disjoint from $\\lambda$ then $\\sigma_L$ is just the right\nportion of ${\\partial}{K(P_L)}$. Otherwise, $\\sigma_L$ has a ``hole'', bounded by ${\\partial}{K(P_L)} \\cap \\lambda$, which is a convex piecewise-circular curve, being the boundary of the intersection of the disks $B_r(p) \\cap \\lambda$, for $p \\in P_L$.\n\nWe partition $\\sigma_L$ into \\emph{quadratically many} cells, such\nthat if we place the center $c_1$ of the left solution ball $B_1$ in\na cell $\\tau$, then, no matter where we place it within $\\tau$,\n$B_1$ will cover the same subset of points from $P$. To construct\nthis partition, we intersect, for each $p \\in P_R = P \\setminus\nP_L$, the sphere ${\\partial}{B_r(p)}$ with $\\sigma_L$ and obtain a curve\n$\\gamma_p$ on $\\sigma_L$; this curve bounds the portion of the\nunique face of ${\\partial}{K(P_L \\cup \\{p\\})}$ within $\\sigma_L$. Hence,\nwithin $K(P_L)$, it is a closed connected curve (it may be\ndisconnected within $\\sigma_L$, though). Let $M$ denote the\narrangement formed on $\\sigma_L$ by the curves $\\gamma_p$, for $p\n\\in P_R$, and by the arcs of $\\sigma_L$. Apriori, $M$ might have\ncubic complexity, if many of the $O(n^2)$ pairs of curves $\\gamma_a,\n\\gamma_b$, for $a,b \\in P_R$, traverse a linear number of common\nfaces of $\\sigma_L$, and intersect each other on many of these\nfaces, in an overall linear number of points. Equivalently, the\n``danger'' is that the intersection circle $C_{ab}$ of a\ncorresponding pair of spheres ${\\partial}{B_r(a)}, {\\partial}{B_r(b)}$, for $a,b\n\\in P_R$, could intersect a linear number of faces of $\\sigma_L$\n(and each of these intersections is also an intersection point of\n$\\gamma_a$ and $\\gamma_b$). See Figure~\\ref{figure:new_case2_cubic}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\n\\input{new_case2_cubic.pstex_t}\n\n\\caption{\\small \\sf In a general setup (different than ours), an intersection circle of two balls (the dotted circle) may\nintersect a linear number of faces of ${\\partial}{K(P_L)}$.}\n\\label{figure:new_case2_cubic}\n\\end{center}\n\\end{figure}\n\n\\smallskip\n\\noindent{\\bf Complexity of $M$.} Fortunately, in the assumed\nconfiguration, this cubic behavior is impossible --- $C_{ab}$ can\nmeet only a constant number of faces of $\\sigma_L$. Consequently,\nthe overall complexity of $M$ is only quadratic. This crucial claim\nfollows from the observation that, for $C_{ab}$ to intersect many\nfaces of $\\sigma_L$, it must have many short arcs, each delimited by\ntwo points on $\\sigma_L$ and lying outside $K(P_L)$. The main\ngeometric insight, which rules out this possibility, and leads to\nour improved algorithm, is given in the following lemma.\n\n\\begin{lemma}\n\\label{lemma:quadratic_K_P_L}\nLet $\\lambda$ be a $yz$-parallel plane, which separates $v_1$ from $c_1$.\nLet $P_L \\subseteq P$ be the subset of points of $P$ to the left of $\\lambda$, and let $P_R = P \\setminus P_L$.\nLet $C_{ab}$ denote the intersection circle of ${\\partial}{B_r(a)}, {\\partial}{B_r(b)}$, for some pair of points $a, b \\in P_R$, and let $q \\in P_L$. If the arc $\\omega = C_{ab} \\setminus B_r(q)$ is smaller than a semicircle of $C_{ab}$, then at least one of its endpoints must lie to the right of $\\lambda$.\n\\end{lemma}\n\n\\begin{proof}The situation and its analysis are depicted in Figure~\\ref{figure:lemma_setup}. To slightly simplify the analysis, and without loss of generality, assume that $r = 1$. Let $h$ be the plane passing through $a$, $b$ and $q$. Let $c_{ab}$\ndenote the midpoint of $ab$, and let $w$ denote the center of the\ncircumscribing circle $Q$ of $\\triangle qab$. Denote the distance $|ab|$\nby $2x$, and the radius of $Q$ by $y$ (so $|wp_1|=|wp_2|=|wq|=y$).\nNote that $c_{ab}$ and $w$ lie in $h$ and that $y \\ge x$.\nObserve that $c_{ab}$ is the center of the intersection circle $C_{ab}$ of\n${\\partial}{B_r(a)}$ and ${\\partial}{B_r(b)}$. See Figure~\\ref{figure:lemma_setup}(a).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular*}{0.8\\textwidth}{c c c}\n            \\hspace{-35pt} {\\input{h_setup_s2.pstex_t} } &  & {\\input{C_ab_s2.pstex_t} } \\\\\n    \\small (a) & \\hspace{15pt} & \\small (b) \\\\\n    \\hspace{45pt} & {\\input{abq_s2.pstex_t} } & \\hspace{45pt}\\\\\n     \\hspace{-35pt} & \\small (c) &\n          \n          \n           \\end{tabular*}\n\n       \\vspace{20pt}\n\\caption{\\small \\sf The setup in Lemma~\\ref{lemma:quadratic_K_P_L}:\n           (a) the setup within the plane $h$; (b) the setup within\n    $C_{ab}$; (c) $ww'$ lies on the bisector of $ab$ in\n           the direction that gets away from $q$.}\n\n\\label{figure:lemma_setup}\n\\end{center}\n\\end{figure}\n\nThe intersection points $z,z'$ of $C_{ab}$ and ${\\partial}{B_r(q)}$ are the\nintersection points of the three spheres\n${\\partial}{B_r(a)}$, ${\\partial}{B_r(b)}$, and ${\\partial}{B_r(q)}$.\nThey lie on the line $\\ell$ passing through $w$ and orthogonal\nto $h$, at equal distances $\\sqrt{1-y^2}$ from $w$. See Figure~\\ref{figure:lemma_setup}(b).\n(If $y>1$ then $z$ and $z'$ do not exist, in which case $C_{ab}$ does not intersect ${\\partial}{B_r(q)}$; in what follows we assume\nthat $y\\le 1$.)\nHence, within $C_{ab}$, $zz'$ is a chord of length $2\\sqrt{1-y^2}$.\nIn the assumed setup, $z$ and $z'$\ndelimit a short arc $\\omega$ of $C_{ab}$, which lies outside $B_r(q)$,\nso points on the arc are (equally) closer to $a$ and $b$ than to\n$q$.\n\n\nHence, the projection of the arc $\\omega$ onto $h$ is a small\ninterval $ww'$, which lies on the bisector of $ab$ in the\ndirection that gets away from $q$; that is, it lies on the Voronoi\nedge of $ab$ in the diagram ${\\rm Vor}(\\{a,b,q\\})$ within $h$. See Figure~\\ref{figure:lemma_setup}(c).\nMoreover, $c_{ab}$ also lies on the bisector, but it has to lie on the\nother side of $w$, or else the smaller arc $\\omega$ would have to lie\ninside $B_r(q)$. That is, $c_{ab}$ has to be closer to $q$ than to $a$\nand $b$. Since $\\lambda$ separates $a$ and $b$ from $q$, it also separates\n $c_{ab}$ from $q$. Moreover, the preceding arguments are easily seen to imply that $wq$ crosses $ab$ (as in Figure~\\ref{figure:lemma_setup}(a)), which implies that $\\lambda$ also separates $q$ and $w$, so\n$w$ has to lie to the right of $\\lambda$. Since $z$ and $z'$ lie on\ntwo sides of $w$ on the line $\\ell$, at least one of them has to lie on\nthe same side of $\\lambda$ as $w$ (i.e., to the right of\n$\\lambda$). This completes the proof.\n\\end{proof}\n\nLet $a,b \\in P_R$ and consider those arcs of $C_{ab}$ which lie outside $K(P_L)$ but their endpoints lie on $\\sigma_L$. Clearly, all these arcs are pairwise disjoint. At most one such arc can be larger than a semicircle. Let $\\omega$ be an arc of this kind which is smaller than a semicircle, and let $q \\in P_L$ be such that one endpoint of $\\omega$ lies on ${\\partial} B_r(q)$. Then $\\omega' = C_{ab} \\setminus B_r(q)$ is contained in $\\omega$ and therefore is also smaller than a semicircle. By Lemma~\\ref{lemma:quadratic_K_P_L}, exactly one endpoint of $\\omega'$ lies to the right of $\\lambda$ (the other endpoint lies on $\\sigma_L$).\nNote that $C_{ab}$ cannot have more than two such short arcs lying outside $K(P_L)$, since, due to the convexity of $C_{ab}$, only two arcs of $C_{ab}$ can have their two endpoints lying on opposite sides of $\\lambda$. Hence the number of arcs of $C_{ab}$ under consideration is at most 3, implying that $\\gamma_a$ and $\\gamma_b$ intersect at most three times, and thus the complexity of $M$ is $O(n^2)$, as asserted.\n\n\\smallskip\n\\noindent{\\bf Constructing and searching $M$.} The next step of the\nalgorithm is to compute $M$. We have already constructed\n${\\partial}{K(P_L)}$, in $O(n \\log n)$ time, and, in additional linear\ntime, we can compute its portion $\\sigma_L$ to the left of $\\lambda$\n(we omit the straightforward details). We compute the intersection\ncurve $\\gamma_p$ of $B_r(p)$ and $\\sigma_L$, for each $p \\in P_R$,\nin $O(n \\log n)$ time, by computing the intersection $K(P_L \\cup\n\\{p\\})$, and obtaining the curve which bounds the portion of the\nunique face of ${\\partial}{K(P_L \\cup \\{p\\})}$ within $\\sigma_L$. If\nnecessary, we also split $\\gamma_p$ into portions, such that each\nportion is contained in a different face of $\\sigma_L$. The total\ncost of computing all curves $\\{\\gamma_p \\mid p \\in P_R\\}$, and\nspreading them along the faces of $\\sigma_L$, is $O(n^2 \\log n)$.\nThen, for each face $f$ of $\\sigma_L$, we consider the portions of\nall the arcs $\\gamma_p$, for $p \\in P_R$, within $f$, and compute\ntheir arrangement (which is the portion of $M$ which lies in $f$).\nTo this end, we use standard line-sweeping~\\cite{book}, to report\nall the intersections of $n$ curves in the plane in $O((n+k) \\log\nn)$ time, where $k = k_f$ is the complexity of the resulting\narrangement on $f$. Hence, the total cost of computing the portion\nof $M$ on all the faces of $\\sigma_L$ is $\\sum_{f \\in \\sigma_L}\nO((n+k_f) \\log n) = O(n^2 \\log n) + O(\\log n) \\cdot \\sum_{f \\in\n\\sigma_L} k_f = O(n^2 \\log n)$, since the complexity of $M$ is\n$O(n^2)$.\n\nWe next perform a traversal of the cells of $M$ in a manner similar to the one used in Section~\\ref{sec:cubic_alg}, via a tour, which proceeds from each visited cell to an adjacent one. For each cell $\\tau$ that we visit, we place\nthe center $c_1$ of $B_1$ in $\\tau$, and maintain dynamically the subset\n$P_\\tau^+$ of points of $P$ not covered by $B_1$. (Here, unlike the algorithm of Section~\\ref{sec:cubic_alg}, the complementary set  $P_\\tau^-$ is automatically covered by $B_1$ and there is no need to test it.) As before,\nwhen we move\nfrom one cell $\\tau$ to an adjacent cell $\\tau_1$, $P_{\\tau_1}^+$ gains\none point or loses one point. This implies that this tour generates only $O(n^2)$ connected life-spans of the points of $P$, where a life-span of a point $p$ is a maximal connected interval of the tour, in which $p$ belongs to $P_\\tau^+$. We can thus use a segment tree $T_M$ to store these life-spans, as before. Each\nleaf $u$ of $T_M$ represents a cell $\\tau$ of $M$, and the balls not containing $\\tau$ are those with life-spans that are stored at the nodes on the path from the root to $u$. Since $M$ has a quadratic number of\ncells, $T_M$ has a total\nof $O(n^2)$ leaves. Arguing exactly as in Section~\\ref{subsec:decision_procedure}, we can compute $T_M$ in overall $O(n^2 \\log^2 n)$ time, and the total storage used by $T_M$ is $O(n^2 \\log n)$.\n\nAs in Section~\\ref{subsec:decision_procedure}, we next test, for\neach leaf $u$ of $T_M$, whether the spherical polytopes along the\npath from the root to $u$ have non-empty intersection. We do this\nusing the parametric search technique described in\nProposition~\\ref{prop:intersection_test}, which takes $O(\\log^5 n)$\ntime for each path, for a total of $O(n^2 \\log^5 n)$. More\nprecisely, as above, we also need to distinguish between $r = r^*$\nand $r > r^*$. We therefore stop only when both the intersection\nalong the path and the cell of $\\sigma_L$ corresponding to $u$ are\nnon-degenerate, and then report that $r^* < r$. Otherwise, we\ncontinue running the above procedure over all paths of $T_M$, and\nrepeat it for each of the $O(1\/\\beta^3)$ combinations of an\norientation $v$ and a separating plane $\\lambda$. If we find at\nleast one (degenerate\\footnote{Note that $\\bigcap\\{B_r(p) \\mid p \\in\nP_\\tau^-\\}$ is non-degenerate if $\\tau$ is a 2-face or an edge. If\n$\\tau$ is a vertex we test for degeneracy as in the procedure in\nSection~\\ref{subsec:decision_procedure}. Determining whether\n$\\bigcap\\{B_r(p) \\mid p \\in P_\\tau^+\\}$ is degenerate is also\nperformed using that procedure.}) solution, we report that $r^* =\nr$, and otherwise conclude that $r^*\n> r$. Hence, the cost of handling Case~2, and thus also the overall\ncost of the decision procedure, is $O((1\/\\beta^3) n^2 \\log^5 n)$.\n\n\\subsection{Solving the optimization problem}\n\\label{sec:separated_centers_optimization}\n\nWe now combine the decision procedure $\\Gamma$ described in Section~\\ref{subsec:improved_decision_procedure} with the randomized optimization\ntechnique of Chan~\\cite{TCG} (as briefly described in Section~\\ref{subsec:optimization_problem}),\nto obtain a solution for the optimization problem.\n\nThe decision procedure $\\Gamma$, on a specified radius $r$, relies on an apriori knowledge of a lower bound $\\beta$ for the separation ratio\n$|c_1c_2|\/r$. To supply such a $\\beta$, let $r_0$ denote the radius of the smallest\nenclosing ball of $P$, and observe that if there exist two balls $B_1, B_2$ of radius $r$ covering $P$ then the smallest ball $B^*$ enclosing $B_1 \\cup B_2$ must be at least as large as the smallest enclosing ball of $P$, so its radius must be at least $r_0$. Since this radius is\n$(1+\\beta\/2)r$ (see Figure~\\ref{figure:new_case2_SEB}), we have\n$(1+ \\beta\/2)r \\geq r_0$ or $\\beta \\geq 2(r_0\/r -1)$.\nIt follows that the running time of the decision procedure $\\Gamma$ is\n$$O\\left(\\frac{1}{\\beta^3} n^2 \\log^5 n\\right) =\nO\\left(\\frac{1}{\\left(1-r\/r_0 \\right)^3} n^2 \\log^5 n\\right).$$\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{B_s2.pstex_t}\n\\caption{\\small \\sf The smallest enclosing ball $B^*$ of $B_1 \\cup\nB_2$.} \\label{figure:new_case2_SEB}\n\\end{center}\n\\end{figure}\n\nChan's technique starts with a very big $r$ (for all practical\npurposes we can start with $r = r_0$) and shrinks it as it iterates\nover the subproblems. Therefore, running Chan's technique in a\nstraightforward manner, starting with $r = r_0$, will make it\npotentially very inefficient, because the initial executions of\n$\\Gamma$, when $r$ is still close to $r_0$, may be too expensive due\nto the large constant of proportionality (not to mention the run at\n$r_0$ itself, which the algorithm cannot handle at all). We need to\nfine-tune Chan's technique, to ensure that we do not consider values\nof $r$ which are too close to $r_0$. To do so, we consider the\ninterval $(0, r_0)$ which contains $r^*$, and run an ``exponential\nsearch'' through it, calling $\\Gamma$ with the values $r_i = r_0\n\\left(1-1\/2^i\\right)$, for $i = 1,2, \\ldots$, in order, until the\nfirst time we reach a value $r' = r_i \\geq r^*$. Note that $1-\nr'\/r_0 = 1\/2^i$ and $1\/2^i < 1- r^*\/r_0 < 1\/2^{i-1}$, so our lower\nbound estimates for the separation ratio $\\beta$ at $r'$ and at\n$r^*$ differ by at most a factor of $2$, so the cost of running\n$\\Gamma$ at $r'$ is asymptotically the same as at $r^*$. Moreover,\nsince the (constants of proportionality in the) running time bounds\non the executions of $\\Gamma$ at $r_1,\\ldots,r_i$ form a geometric\nsequence, the overall cost of the exponential search is also\nasymptotically the same as the cost of running $\\Gamma$ at $r^*$. We\nthen run Chan's technique, with $r'$ as the initial minimum radius\nobtained so far. Hence, from now on, each call to $\\Gamma$ made by\nChan's technique will cost asymptotically no more than the cost of\ncalling $\\Gamma$ with $r'$ (which is asymptotically the same as\ncalling $\\Gamma$ with $r^*$).\n\n\\paragraph{Combining Chan's technique with the decision procedure $\\Gamma$.}\nTo apply Chan's technique with our decision procedure, we use the\nsame cutting-based decomposition as in\nSection~\\ref{subsec:optimization_problem}. That is, we replace each\npoint $p \\in P$ by its dual plane $p^* \\in P^*$, and construct a\n$(1\/\\varrho)$-cutting of ${\\cal A}(P^*)$, for some sufficiently large\nconstant parameter $\\varrho>0$. We then apply Chan's technique to\nthe resulting subproblems (where each subproblem corresponds to a\nsimplex $\\Delta_i$ of the cutting), using the improved decision\nprocedure $\\Gamma$ on each of them, and recursing into some of them,\nas required by the technique. As in Section~\\ref{sec:cubic_alg}, the\nrecursion and the application of the decision procedure are not\n``pure'', because they need to consider also those planes that miss\nthe current simplex. (Note that in the problem decomposition we use,\nfor simplicity, the full 3-dimensional arrangement ${\\cal A}(P^*)$, of\ncubic size. This, however, does not affect the asymptotic running\ntime, because we have only a constant number of subproblems, and\nChan's technique recurses into only an expected logarithmic number\nof them.) Given a radius $r$, we compute the lower bound $\\beta =\n2\\left(\\frac{r_0}{r} -1\\right)$ for the separation ratio\n$\\frac{|c_1c_2|}{r}$, where $c_1, c_2$ are the centers of the two\ncovering balls, as above. Consider the application of $\\Gamma$ to a\nsubproblem represented by a simplex $\\Delta_i$ of the cutting. The\npresence of ``global'' points (those dual to planes passing above or\nbelow $\\Delta_i$) forces us, as in\nSection~\\ref{subsec:optimization_problem}, to modify the ``pure''\nversion of $\\Gamma$ described above. We use the same notations as in\nSection~\\ref{sec:cubic_alg}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{h_lambda_s2.pstex_t}\n\n\\caption{\\small \\sf $h_\\lambda$ does not contain any point of\n$P_{\\Delta_i}^+$.} \\label{figure:h_lambda}\n\\end{center}\n\\end{figure}\n\nWe again rotate the coordinate axes, in $O\\left(1\/\\beta^2\\right)$\nways (in the same manner as in the ``pure'' decision procedure), and\ndraw $O(1\/\\beta)$ $yz$-parallel planes, such that, at the correct\norientation, one of these planes, $\\lambda$, separates $c_1$ from\n$v_1$ (if there is a solution for $r$). As in the pure case, we may\nassume that the $x$-span of $P$ is at most $5r$; a larger span is\nhandled earlier. We assume, without loss of generality, that\n$P_{\\Delta_i}^- \\subseteq B_1$, and that $P_{\\Delta_i}^+ \\subseteq\nB_2$. Recall also that the points in the left halfspace $h_\\lambda$\nbounded by $\\lambda$ are all contained in $B_1$. Moreover, the plane\n$\\pi$ containing the intersection circle $C_{12}$ is dual to a point\n$\\pi^*$, which has to separate $(P^*)_{\\Delta_i}^+$ from\n$(P^*)_{\\Delta_i}^-$. Hence, all the points of $P_{\\Delta_i}^+$ have\nto lie on the other side of $\\pi$, and in $B_2$, which is easily\nseen to imply that none of them can lie in $h_\\lambda$. See\nFigure~\\ref{figure:h_lambda}. We thus verify that $P_{\\Delta_i}^+\n\\cap h_\\lambda = \\emptyset$, aborting otherwise the guess of\n$\\lambda$. (Note that, in contrast, points of $P_{\\Delta_i}^-$ can\nalso lie to the right of $\\lambda$.)\n\nWe now have a subset $P_L \\subseteq P_{\\Delta_i}^0$ of $O(m)$ points\nto the left of $\\lambda$, which are assumed, together with the\npoints of $P_{\\Delta_i}^-$, to be contained in $B_1$. Note however\nthat, for Lemma~\\ref{lemma:quadratic_K_P_L} to hold, we have to\ndefine $\\sigma_L$ only in terms of the points to the left of\n$\\lambda$. Therefore, we compute the surface $\\sigma_L' = {\\partial}{K(P_L\n\\cup (P_{\\Delta_i}^- \\cap h_\\lambda)) \\cap h_\\lambda}$ and search on\nit for a placement of the center $c_1$ of $B_1$. However, since the\nremaining points of $P_{\\Delta_i}^-$ are also assumed to belong to\n$B_1$, we need to consider only the portion of $\\sigma_L'$ inside\n$\\bigcap\\{B_r(p) \\mid p \\in P_{\\Delta_i}^- \\setminus h_\\lambda\\}$.\nLet $\\sigma_L''$ denote this portion. It is easy to compute\n$\\sigma_L''$ in $O(n \\log n)$ time. It is easily checked that $c_1$\nmust lie on $\\sigma_L''$ (if there is a solution for the current\nsituation). So far, the cost of the decision procedure also depends\n(cheaply --- see below) on the initial input size $n$, but the\nsaving in this setup comes from the fact that it suffices to\nintersect the $O(m)$ spheres ${\\partial}{B_r(p)}$, for $p \\in\nP_{\\Delta_i}^0 \\setminus h_\\lambda$, with $\\sigma_L''$ to obtain the\nmap $M$, since only the points of $P_{\\Delta_i}^0$ are\n``undecided''. (The points of $P_{\\Delta_i}^+$ are always placed in\n$B_2$ as already discussed.)\n\nNote that $\\sigma_L''$ need not to be connected, so it may seem\nimpossible to visit all the cells of $M$ in a single connected tour.\nNevertheless, we will be able to do it, in a manner detailed below.\nWe thus build a segment tree $T_M$ to maintain the subset $P'(c_1)$\nof points of $P$ not covered by $B_1$. We build and query $T_M$ as\nis done in Section~\\ref{subsec:decision_procedure}, except for the\nfollowing modifications. First, note that the points of\n$P_{\\Delta_i}^+$ are assumed to be contained in $B_2$. Thus, the\npoints of $P_{\\Delta_i}^+$, that in the decision procedure were\nconsidered in building $M$, do not need to be considered as part of\n$M$ now, rather it is enough to build the spherical polytope\n$\\bigcap \\{B_r(p) \\mid p \\in P_{\\Delta_i}^+ \\}$ and place it at the\nroot of $T_M$. Second, we claim that $M$ is of complexity $O(m n)$.\nTo see this, let ${\\cal C}^0$ denote the set of curves $\\{{\\partial}{B_r(p)} \\cap\n\\sigma_L'' \\mid p \\in P_{\\Delta_i}^0\\}$. Each pair of curves of\n${\\cal C}^0$ can intersect each other in only a constant number of points,\nas proved in Section~\\ref{subsec:improved_decision_procedure}.\nHence, the complexity of the arrangement of the $O(m)$ curves in\n${\\cal C}^0$, formed on $\\sigma_L''$, is $O(m^2)$. However, $\\sigma_L''$\nitself is of complexity $O(n)$, and each edge of $\\sigma_L''$ may\nintersect the curves of ${\\cal C}^0$ at $O(m)$ points. Hence, the\ncomplexity of the map $M$ is $O(m n)$, but the number of its\nvertices that lie in the interior of the faces of $M$ is only\n$O(m^2)$.\n\nTo overcome the possible disconnectedness of $\\sigma_L''$, we\nproceed as follows. We consider the (connected) network of the\n$O(n)$ edges of $\\sigma_L'$, and intersect each of these edges with\nthe $m$ balls $B_r(p)$, for $p \\in P_{\\Delta_i}^0$. We construct a\ntour of this network, which visits $O(mn)$ arcs along the edges of\n$\\sigma_L'$, and append to this ``master tour'' separate tours of\neach face of $\\sigma_L''$. We get in this way a single grand tour of\nthe cells of $M$ (which also traverses some superfluous arcs of\n$\\sigma_L' \\setminus \\sigma_L''$), of length $O(mn)$, which has the\nincremental property that we need: Moving from any cell or arc of\nthe tour to a neighbor cell or arc incurs an insertion or a deletion\nof a single point into\/from $P'(c_1)$.\n\n\n\\paragraph{Running time.}\nFor each cell of $M$ we run the procedure described in\nProposition~\\ref{prop:intersection_test} for determining whether the\nintersection of the corresponding spherical polytopes is nonempty\n(and whether it has nonempty interior). Therefore, solving each\nsubproblem requires $O(m n \\log^5 n)$ time. The $O(m n \\log n)$ time\nrequired to build $M$, and the $O(n \\log n)$ time required to\nconstruct the intersection of the balls in $\\{B_r(p) \\mid p \\in\nP_{\\Delta_i}^+\\}$, are all subsumed in that cost. Repeating this for\neach of the $O(1\/\\beta^3)$ guesses of an orientation and a\nseparating plane, results in $O\\left((1\/\\beta^3) mn \\log^5 n\\right)$\nrnning time. When the recursion bottoms out, we handle it the same\nway as in Section~\\ref{subsec:optimization_problem}.\n\nArguing similarly to the less efficient solution, we obtain the\nfollowing recurrence for the maximum expected cost $T(m, n)$ of\nsolving a recursive subproblem involving $m$ ``local'' points, where\n$n$ is the number of initial input points in $P$.\n\n\\begin{equation}\n\\label{eqn:T(m,n)_2} T(m,n) \\leq \\left\\{ \\begin{array}{ll}\n\\ln(c\\varrho^3)T(m\/\\varrho,n) + O\\left((1\/\\beta^3) m n \\log^5 n\\right),   & \\mbox{for $m \\geq \\varrho$,}\\\\\nO(n),                                        & \\mbox{for $m <\n\\varrho$,}\n\\end{array}\\right.\n\\end{equation}\nwhere $c$ is an appropriate absolute constant (as in\nSection~\\ref{subsec:optimization_problem}), $\\varrho$ is the\nparameter of the cutting, chosen to be a sufficiently large constant\n(to satisfy (\\ref{equation:lnr}), as above, with $\\gamma = 2$), and\n$\\beta = 2\\left(r_0\/r'-1\\right)$, where $r'$ is the value of $r$ at\nwhich the initial exponential search is terminated.\n\nIt can be shown rather easily (and we omit the details, as we did in\nthe preceding section), that the recurrence~(\\ref{eqn:T(m,n)_2})\nyields the overall bound $O\\left((1\/\\beta^3) n^2 \\log^5 n\\right)$ on\nthe expected cost of the initial problem; i.e.,\n$$T(n,n) = O\\left((1\/\\beta^3) n^2 \\log^5 n\\right).$$ We thus finally obtain\nour main result:\n\\begin{theorem}\nLet $P$ be a set of $n$ points in $\\mathbb{R}^3$. A 2-center for $P$\ncan be computed in $O((n^2 \\log^5 n)\/(1-r^*\/r_0)^3 )$ randomized\nexpected time, where $r^*$ is the radius of the balls of the\n2-center for $P$ and $r_0$ is the radius of the smallest enclosing\nball of $P$.\n\\end{theorem}\n\n\\section{Efficient Emptiness Detection of Intersection of Spherical\nPolytopes} \\label{sec:spherical_polytopes} In this section we\ndescribe an efficient procedure for testing emptiness (and\nnon-degeneracy) of the intersection of spherical polytopes, as\nprescribed in Proposition~\\ref{prop:intersection_test}. Let ${\\cal S}$ be\na collection of spherical polytopes, each defined as the\nintersection of at most $n$ balls of a fixed radius $r$. Fix a\nspherical polytope $S \\in {\\cal S}$. To simplify the forthcoming analysis,\nwe assume that the centers of the balls involved in the polytopes of\n${\\cal S}$ are in general position, meaning that no five of them are\nco-spherical, and that there exists at most one quadruple of centers\nlying on a common sphere of radius $r$. As is well known, each ball\n$b$ participating in the intersection $S$ contributes at most one\n(connected) face to ${\\partial}{S}$ (see~\\cite{ER}). The vertices and edges\nof $S$ are the intersections of two or three bounding spheres,\nrespectively (at most one vertex might be incident to four spheres).\nHence ${\\partial}{S}$ is a planar (or, rather, spherical) map with at most\n$|S|$ faces, which implies that the complexity of ${\\partial}{S}$ is\n$O(|S|)$.\n\nWe preprocess $S$ into a point-location structure. We first\npartition ${\\partial}{S}$ into its upper portion ${\\partial}{S}^+$ and lower\nportion ${\\partial}{S}^-$. We project vertically each of ${\\partial}{S}^+$ and\n${\\partial}{S}^-$ onto the $xy$-plane and obtain two respective planar maps\n$M^+$ and $M^-$ (see Figure~\\ref{figure:projection}). For each face\n$\\zeta$ of each map we store the ball $b$ that created it; that is,\n$\\zeta$ is the projection of the (unique) face of ${\\partial}{S}$ that lies\non ${\\partial}{b}$. The $xy$-projection $S^*$ of $S$ is equal to both\nprojections of ${\\partial}{S^+}$, ${\\partial}{S^-}$, and is bounded by a convex\ncurve $E^*$ that is the concatenation of the $xy$-projections of\ncertain edges of $S$ and of portions of horizontal equators of some\nof its balls.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{projection_log8.pstex_t}\n\\caption{\\small \\sf Projecting ${\\partial}{S_i}^-$ vertically onto the\n$xy$-plane (left), and the point location structure for the\nresulting map $M_i^-$ (right).} \\label{figure:projection}\n\\end{center}\n\\end{figure}\n\nWe apply the standard point-location algorithm of Sarnak and\nTarjan~\\cite{ST} to each of the maps $M^+, M^-$. That is, we divide\neach planar map into slabs by parallel lines (to the $y$-axis)\nthrough each of the endpoints (and locally $x$-extremal points) of\nthe arcs obtained by projecting the edges of ${\\partial}{S}$, including the\nnew equatorial arcs. Using the persistent search structure\nof~\\cite{ST}, the total storage is linear in $|S|$ and the\npreprocessing cost is $O(|S| \\log |S|)$, where $|S|$ is the number\nof balls forming $S$. To locate a point $q_0$ in $M^+$ (or in\n$M^-$), we first find the slab in the $x$-structure that contains\n$q_0$, and then find the two curves between which $q_0$ lies in the\n$y$-structure.\\footnote{\\small All these standard details are\npresented to make more precise the infrastructure used by the\nhigher-dimensional routines $\\Pi_1$ and $\\Pi_2$.}\n\nTo determine whether $q \\in S^*$, we locate the face $\\zeta^+$\n(resp., $\\zeta^-$) of the map $M^+$ (resp., $M^-$) that contains\n$q$, as just described. Each of these faces can be a 2-face, an edge\nor a vertex. We therefore retrieve a set ${\\cal B}^+$ (resp., ${\\cal B}^-$) of\nthe one, two, or three or four balls associated (respectively) with\nthe 2-face, edge or vertex containing $q$. (We omit here the easy\nconstruction of witness balls when the faces $\\zeta^+$ and $\\zeta^-$\nare not associated with any ball, that is, $q \\notin S^*$.)\n\nLet ${\\cal B}$ denote the set ${\\cal B}^+ \\cup {\\cal B}^-$. We observe that $q \\in\nS^*$ if and only if the $z$-vertical line $\\lambda_q$ through $q$\nintersects $S$. Moreover, we have, by construction, $\\lambda_q \\cap\nS = \\lambda_q \\cap (\\bigcap {\\cal B})$. Hence $q \\in S^*$ if and only if\n$s \\coloneqq \\lambda_q \\cap (\\bigcap {\\cal B}) \\neq \\emptyset$. Clearly,\nif we put $N = \\sum_{S \\in {\\cal S}} |S|$, then the preprocessing stage\ntakes a total of $O(N \\log n)$ time and requires $O(N)$ storage.\n\nNext, let $S_1,\\ldots,S_t$ be $t \\leq \\log n$ spherical polytopes of\n${\\cal S}$, for which we want to determine whether $K = \\bigcap_{i=1}^t\nS_i$ is nonempty (and, if so, whether it has nonempty interior). We\nsolve this problem by employing a technique similar to the\nmulti-dimensional parametric searching technique of\nMatou\\v{s}ek~\\cite{JM} (see also~\\cite{AES, TCA, CMS, NPT}). We\nsolve in succession the following three subproblems, $\\Pi_0(q)$,\nwhere $q$ is a point in the $xy$-plane, $\\Pi_1(l)$, where $l$ is a\n$y$-parallel line in the $xy$-plane, and $\\Pi_2$, over the entire\n$xy$-plane. In the latter problem we wish to to determine whether\nthe $xy$-projection $K^*$ of $K$ is nonempty. During the execution\nof the algorithm for solving $\\Pi_2$, we call recursively the\nalgorithm for solving $\\Pi_1(l)$, for certain $y$-parallel lines $l\n\\subset \\mathbb{R}^2$, and we wish to determine whether $K^*$ meets\n$l$. If so, then $\\Pi_2$ is solved directly (with a positive\nanswer). Otherwise, we wish to determine which side of $l$, within\n$\\mathbb{R}^2$, can meet $K^*$ (since $K^*$ is convex, there can\nexist at most one such side). The recursion bottoms out at certain\npoints $q \\in l$, on which we run $\\Pi_0(q)$ to determine whether\n$K^*$ contains $q$. If so, then $\\Pi_1(l)$ is solved directly (with\na positive answer). Otherwise, we determine which side of $q$,\nwithin $l$, can meet $K^*$, and continue the search accordingly.\n\nOur solutions to the subproblems $\\Pi_k$, $0\\leq k\\leq 2$, are based\non generic simulations of the standard point-location machinery of\nSarnak and Tarjan~\\cite{ST} mentioned above. In each of the\nsubproblems, if we find a point in $f \\cap K^*$, for the respective\npoint, line, or the entire $xy$-plane $f$, we know that $K \\neq\n\\emptyset$ and stop right away. If $f \\cap K^* = \\emptyset$, we want\nto ``prove'' it, by returning a small set of \\emph{witness balls}\n$b_1,\\ldots,b_y$, where, for each $j$, $b_j$ is one of the balls\nthat participates in some spherical polytope $S_i$ (so $b_j\n\\supseteq S_i$), so that their intersection $K_0 = \\bigcap_{j=1}^y\nb_j$ satisfies $f \\cap K_0^* = \\emptyset$ (where, as above, $K_0^*$\nis the $xy$-projection of $K_0$). If $K_0 = \\emptyset$ then $K =\n\\emptyset$ too and we stop. Otherwise (when $f$ is a line or a\npoint), $K_0$ determines the side of $f$ (within $\\mathbb{R}^2$ if\n$f$ is a line, or within the containing line $l$ if $f$ is a point)\nthat might meet $K^*$; the opposite side is asserted at this point\nto be disjoint from $K^*$. We use this information to perform binary\nsearch (or, more precisely, parametric search) to locate $K^*$\nwithin the flat, from which we have recursed into $f$.  The\nexecution of the algorithm for solving $\\Pi_2$ will therefore either\nfind a point in $K$ or determine that $K = \\emptyset$, because it\nhas collected a small (as we will show, polylogarithmic) number of\nwitness balls, whose intersection, which has to contain $K$, is\nfound to be empty.\n\n\\paragraph{Solving $\\Pi _0(q)$ for a point $q$.}\n\\label{subsec:Pi_0}\nHere we have a point $q \\in \\mathbb{R}^2$ and we wish to determine\nwhether $q \\in K^*$. To do so, we locate $q$ in each of the maps\n$M_i^+$ (the $xy$-projection of ${\\partial} S_i^+$) and $M_i^-$ (the\n$xy$-projection of ${\\partial} S_i^-$), for each $i=1,\\ldots,t$. If $q$\nlies outside the projection of at least one polytope $S_i$ then $q\n\\notin K^*$, and we return the witness balls that prove that $q\n\\notin S_i^*$. Otherwise, as explained above, each point location\nreturns a set ${\\cal B}_i$ of $O(1)$ witness balls for $S_i$. We compute\nthe $t$ line segments $s_i = \\lambda_q \\cap (\\bigcap{\\cal B}_i)$, for each\n$i = 1,\\ldots,t$, where $\\lambda_q$ is, as above, the $z$-vertical\nline through $q$. We then have $K_0 \\coloneqq \\lambda_q \\cap K =\n\\bigcap_{i=1}^t s_i$, so it suffices to compute this intersection\n(in $O(t)$ time) and test whether it is nonempty. If $K_0$ is\nnonempty, then we have found a point $q'$ in $K$. Otherwise, we\nreturn the set ${\\cal B}_0 = \\bigcup \\{{\\cal B}_i \\mid 1 \\leq i \\leq t\\}$ of up\nto $5 \\log n$ balls as witness balls for the higher-dimensional step\n(involving the $y$-parallel line containing $q$).\n\nThe time complexity for solving $\\Pi_0(q)$ is $O(\\log^2 n)$, since\nit takes $O(\\log n)$ time to compute, for each of the $O(\\log n)$\nspherical polytopes $S_i$, the intersection $\\lambda_q \\cap S_i$.\n\n\\paragraph{\\bf Solving $\\Pi _1(l)$ for a line $l$.}\n\\label{subsec:Pi_1}\nHere we have a $y$-parallel line $l \\subset \\mathbb{R}^2$ and we\nwish to determine whether $K^*$ meets $l$. We first locate $l$ in\neach of the planar maps $M_i^+$ and $M_i^-$ of each $S_i$, and find\nthe slabs $\\psi_i^+$ and $\\psi_i^-$, which contain $l$ (in some\ncases $l$ is the common bounding line of two adjacent slabs\n$\\psi_i'$ and $\\psi_i''$ of $M_i^+$ or of $M_i^-$, so we retrieve\nboth slabs). We then run a binary search through the $y$-structure\nof each of the obtained slabs to find a point in $K^* \\cap l$, if\none exist. In each step of the search, within some fixed slab\n$\\psi_0$, we consider an arc $\\gamma$ of the $y$-structure, and\ndetermine whether $K^*$ meets $l$ above or below $\\gamma$ (within\n$\\mathbb{R}^2$), assuming $K^* \\cap l \\neq \\emptyset$. To this end,\nwe find the intersection point $q_0 = l \\cap \\gamma$, and run the\nalgorithm for solving $\\Pi_0(q_0)$ (see Figure~\\ref{figure:pi_1}).\nIf $q_0 \\in K^*$, then we have found a point $q'$ in $K$, and we\nimmediately stop. Otherwise, we have a set ${\\cal B}_0$ of up to $5 \\log\nn$ balls returned by the algorithm for solving $\\Pi_0(q_0)$. We test\nwhether the $xy$-projection $K_0^*$ of $\\bigcap {\\cal B}_0$ intersects\n$l$. If $K_0^* \\cap l = \\emptyset$, then (due to the convexity of\n$K$) we know which side of $l$ (within $\\mathbb{R}^2$) meets $K^*$,\nand we return ${\\cal B}_0$ as a set of witness balls for the\nhigher-dimensional (planar) step. Otherwise (again due to the\nconvexity of $K$), we know which side of $\\gamma$, within $l$, meets\n$K^*$, and we continue the search through the $y$-structure of\n$\\psi_0$ on this side. We continue the search in this manner, until,\nfor each $S_i$, we obtain an interval $\\xi_i$ of $l$ between two\nconsecutive arcs of the $y$-structure of $\\psi_0$, which meets $K^*$\n(assuming $K^* \\cap l \\neq \\emptyset$). Let $\\Xi$ denote the\ncollection of all these intervals. Clearly, $K^* \\cap l \\subseteq\n\\bigcap \\Xi$. We find the lowest endpoint $E^-$ among the top\nendpoints of the intervals in $\\Xi$ and the highest endpoint $E^+$\namong the bottom endpoints of the intervals in $\\Xi$, and test\nwhether $E^-$ is above $E^+$. If so, we consider the set ${\\cal B}_1$ of\nup to $10 \\log n$ witness balls returned by the algorithms for\nsolving $\\Pi_0(E^-)$ and $\\Pi_0(E^+)$. If the $xy$-projection\n$K_1^*$ of $\\bigcap {\\cal B}_1$ intersects $l$, then $K^*$ meets $l$ and\nwe stop immediately, for we have found that $K$ is nonempty.\nOtherwise, we know which side of $l$ (within $\\mathbb{R}^2$) can\nmeet $K^*$, and we return ${\\cal B}_1$ as a set of witness balls for the\nhigher (planar) recursive level. If $E^-$ is not above $E^+$, then\n$K^* \\cap l = \\emptyset$ and we return ${\\cal B}_1$ as a set of witness\nballs for the higher (planar) recursive level as well.\\footnote{With\nsome care, the number of witness balls can be significantly reduced.\nWe do not go into this improvement, because handling the witness\nballs is an inexpensive step, whose cost is subsumed by the cost of\nthe other steps of the algorithm.}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{pi_1_log8.pstex_t}\n\\caption{\\small \\sf The line $l$ on which we run $\\Pi_1(l)$. The\npoint $q_0$ on which we run $\\Pi_0(q_0)$ is the intersection point\nof $l$ with some arc $\\gamma$.} \\label{figure:pi_1}\n\\end{center}\n\\end{figure}\n\nA naive implementation of the above procedure takes $O(\\log^4 n)$\ntime, since for each of the $O(\\log n)$ spherical polytopes $S_i$ we\nrun a binary search through the $y$-structure of at most two slabs\nof each of the maps $M_i^+$ and $M_i^-$, and in each of the binary\nsearch steps, we run the algorithm for solving $\\Pi_0(q_0)$ for some\npoint $q_0$. The other substeps take less time. However, we can\nimprove the running time by implementing it in a parallel manner and\nsimulating the parallel version sequentially with a smaller number\nof calls to $\\Pi_0$.\n\nWe only parallelize the binary searches through the $y$-structure of\neach $M_i^+$ and $M_i^-$, since the other substeps take less time.\nTo this end, we use $O(\\log n)$ processors, one for each of the\nplanar maps $M_i^+$ and $M_i^-$, and we run in parallel the binary\nsearch through the $y$-structure of each planar map using $O(\\log\nn)$ parallel steps. In each parallel step we need to ``compare''\n$O(\\log n)$ arcs with $K^*$ (one arc for each of the planar maps\n$M_i^+$, $M_i^-$). We therefore intersect each such arc with $l$ and\nobtain a set $Q$ of $O(\\log n)$ intersection points. We then run a\nbinary search through the points of $Q$ (to locate $K^*$) using\n$\\Pi_0$. This determines the outcome of the comparisons of each of\nthe arcs with $K^*$, and the parallel execution can proceed to the\nnext step. Applying this approach to each of the $O(\\log n)$\nparallel steps results in an $O(\\log^3 n \\log \\log n)$-time\nalgorithm for solving $\\Pi_1(l)$. However, we can slightly improve\nthis bound further using a simple variant of Cole's\ntechnique~\\cite{RC}. More precisely, in each parallel step we have a\ncollection $Q$ of $O(\\log n)$ weighted points, one for each map,\nwhich we need to compare with $K^*$. We select the (weighted) median\npoint $q_0$ of $Q$ and run $\\Pi_0(q_0)$. This determines the\noutcomes of the comparisons between $K^*$ and each of the points in\n$Q$ which lie to the opposite side of $q_0$ to the side containing\n$K^*$. Points in $Q$ which lie in the same side of $q_0$ as $K^*$,\nin level $j$ of the parallel implementation, are given weight\n$1\/4^{j-1}$ and we try to resolve their comparison to $K^*$ in the\nnext step. An easy calculation (simpler than the one used by Cole)\nshows that this method adds only $O(\\log n)$ steps to the $O(\\log\nn)$ parallel steps of the searches, and now in each parallel step we\nperform only one call to $\\Pi_0$ (see~\\cite{RC} for more details).\nTherefore, the total running time of $\\Pi_1(l)$ is $O(\\log^3 n)$.\n\n\\paragraph{\\bf Solving $\\Pi _2$.}\n\\label{subsec:Pi_2}\nWe next consider the main problem $\\Pi _2$, where we want to\ndetermine whether $K^* \\neq \\emptyset$ (i.e., whether $K \\neq\n\\emptyset$). We use parametric searching, in which we run the point\nlocation algorithm that we used for solving $\\Pi_0$, in the\nfollowing generic manner.\n\nIn the first stage of the generic point location, we run a binary\nsearch through the slabs of each of the planar maps $M_i^+$ and\n$M_i^-$, for $i=1,\\ldots,t$. In each step of the search through any\nof the maps, we take a line $l_0$ delimiting two consecutive slabs\nof the map, and run the algorithm for solving $\\Pi _1(l_0)$, thereby\ndeciding on which side of $l_0$ to continue the search. At the end\nof this stage, unless we have already found a point in $K$ or\ndetermined that $K$ is empty, we obtain a single slab in each map\nthat contains $K^*$.  Let $\\psi$ denote the intersection of these\nslabs, which must therefore contain $K^*$ (unless $K$ is empty). The\ncost of this part of the procedure is $O(\\log^5 n)$.\n\nIn the next stage of the generic point location, we consider each\nmap $M_i^+$ or $M_i^-$ (for simplicity we refer to it just as $M_i$)\nseparately, and run a binary search through the $y$-structure of its\nslab $\\psi_i$ that contains $\\psi$. In each step of the search we\nconsider an arc $\\gamma$ of the $y$-structure, and determine which\nside of $\\gamma$ (within the slab $\\psi$), can meet $K^*$, assuming\nthat $\\psi \\cap K^* \\neq \\emptyset$; if $\\gamma \\cap K^* \\neq\n\\emptyset$ we will detect it and stop right away. Before describing\nin detail how to resolve each comparison with an arc $\\gamma$, we\nnote that this results in $O(\\log n)$ comparisons of arcs $\\gamma$\nto $K^*$ for each of the $O(\\log n)$ planar maps $M_i^+$ and\n$M_i^-$. However, we can reduce the number of comparisons to $O(\\log\nn)$ in total, by simulating (sequentially) a parallel implementation\nof this step, as follows. There are $O(\\log n)$ parallel steps, and\nin each step we execute a single step of the binary search in each\nof the maps $M_i^+, M_i^-$. In each parallel step we need to compare\n$K^*$ to a set $G$ of $O(\\log n)$ arcs, one of each of the planar\nmaps $M_i^+, M_i^-$. Consider the portion ${\\cal A}'(G)$ of the\narrangement ${\\cal A}(G)$ of the arcs in $G$ which lies in $\\psi$. Let\n$L(G)$ denote the set of $O(\\log^2 n)$ $y$-parallel lines which pass\nthrough the vertices of ${\\cal A}'(G)$. We run a binary search through the\nlines of $L(G)$, using calls to the algorithm for $\\Pi_1$ to guide\nthe search, to locate $K^*$ amid these lines, in a total of\n$O(\\log^3 n \\log \\log n)$ running time. This step (if it did not\nfind a line crossing $K^*$) may trim $\\psi$ to a narrower slab\n$\\psi'$ in which $K^*$ must lie if $K^* \\neq \\emptyset$. Put $G' =\n\\{\\gamma \\cap \\psi' \\mid \\gamma \\in G\\}$, and observe that the arcs\nof $G'$ are pairwise disjoint and form a sorted sequence in the\n$y$-direction. We then perform a binary search through the arcs in\n$G'$, using $O(\\log \\log n)$ comparisons to $K^*$. Each comparison\nis carried out in $O(\\log^4 n)$ time, in a manner detailed below.\nOnce the binary search is terminated, we can determine the outcomes\nof the comparisons between $K^*$ and each of the arcs in $G'$ and\nproceed to the next parallel step. Applying this approach to each of\nthe $O(\\log n)$ parallel steps results in an $O(\\log^5 n \\log \\log\nn)$-algorithm for solving $\\Pi_2$. We again use an appropriate\nvariant of Cole's technique to improve the running time by a $\\log\n\\log n$ factor, in a manner similar to the one described in the\nsolution of $\\Pi_1$.\n\nTo carry out a comparison between an arc $\\gamma \\in G'$ and $K^*$,\nwe act under the assumption that $\\gamma \\cap K^* \\neq \\emptyset$,\nand try to locate a point of $\\gamma \\cap K^*$ in each of the other\nmaps. Suppose, to simplify the description, that we managed to\nlocate the entire $\\gamma$ in a single face of each of the other\nmaps $M_j^+$, $M_j^-$. This yields a set ${\\cal B}$ of $O(t)$ balls, so\nthat a point $v \\in \\gamma$ lies in $K^*$ if and only if it lies in\nthe $xy$-projection $K_0^*$ of $\\bigcap {\\cal B}$. We then test whether\n$\\gamma$ intersects $K_0^*$. If so, we have found a point in $K$ and\nstop right away. Suppose then that $K_0^* \\cap \\gamma = \\emptyset$.\nIf $K_0^* \\cap \\psi' = \\emptyset$ then $K$ must be empty, because we\nalready know that $K^* \\subset \\psi'$. If $K_0^* \\cap \\psi' \\neq\n\\emptyset$, then we know on which side of $\\gamma$ to continue the\nbinary search in (the portion within $\\psi'$ of) $\\psi_i$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\input{pi_2_log8.pstex_t}\n\\caption{\\small \\sf Comparing $\\gamma \\cap K^*$ with $\\delta$. The\noutcome of $\\Pi_1(l_0)$ determines (a) the side of $\\delta$ in which\nthe search in $\\psi_j$ should continue, and (b) the portion of\n$\\gamma$ which can still meet $K^*$. The subslab $\\psi'$ is drawn\nshaded.} \\label{figure:pi_2}\n\\end{center}\n\\end{figure}\n\n\n\n\nIn general, though, $\\gamma$ might split between several cells of a\nmap $M_j$, where $M_j$ denotes, as above, one of the maps $M_j^+$ or\n$M_j^-$. This forces us to narrow the search to a subarc of\n$\\gamma$, in the following manner. We run a binary search through\nthe $y$-structure of the corresponding slab $\\psi_j$ of $M_j$, which\ncontains $\\psi'$, and repeat it for each of the maps $M_j$. In each\nstep of the search, we need to compare $\\gamma$ (or, more precisely,\nsome point in $\\gamma \\cap K^*$) with some arc $\\delta$ of $\\psi_j$,\nwhich we do as follows. If $\\gamma$ lies, within $\\psi'$, completely\non one side of $\\delta$, we continue the binary search in $\\psi_j$\non that side of $\\delta$. If $\\gamma$ intersects $\\delta$, we pick\nan intersection point $v$ of $\\gamma$ and $\\delta$, pass a\n$y$-parallel line $l_0 \\subset \\mathbb{R}^2$ through $v$, and run\nthe non-generic version of the algorithm to solve $\\Pi_1(l_0)$. (See\nFigure~\\ref{figure:pi_2}.) As before, if $l_0 \\cap K^* \\neq\n\\emptyset$ we detect this and stop. Otherwise, we know which of the\ntwo portions of $\\gamma$, delimited by $v$, can intersect $K^*$. We\nrepeat this step for each of the at most four intersection points of\n$\\gamma$ and $\\delta$ (observing that these are elliptic arcs), and\nobtain a connected portion $\\gamma'$ of $\\gamma$, delimited by two\nconsecutive intersection points, whose relative interior lies\ncompletely above or below $\\delta$, so that $\\gamma \\cap K^*$, if\nnonempty, lies in $\\gamma'$. This allows us to resolve the generic\ncomparison with $\\delta$, and continue the binary search through\n$\\psi_j$. (On the fly, each comparison with a line $l_0$ narrows\n$\\psi'$ still further.)\n\nTo make this procedure more efficient, we perform the binary\nsearches through the slabs $\\psi_j$ in parallel, as follows. As\nbefore, we run in parallel the binary searches through each of the\nslabs $\\psi_j$ using $O(\\log n)$ parallel steps. In each parallel\nstep we need to compare a set $D$ of $O(\\log n)$ arcs to $\\gamma$,\none arc $\\delta$ from each planar map $M_j$. We intersect each of\nthe arcs in $D$ with $\\gamma$ and obtain a set $Z$ of $O(\\log n)$\nintersection points. Let $L_Z$ denote the set of the $O(\\log n)$\n$y$-parallel lines which pass through the points of $Z$. We run a\nbinary search through the lines of $L_Z$, using calls to the\nalgorithm for $\\Pi_1$ to guide the search, in a total of $O(\\log^3 n\n\\log \\log n)$ running time. We obtain a connected portion $\\gamma'$\nof $\\gamma$, delimited by two consecutive intersection points of\n$Z$, whose relative interior lies completely above or below each\n$\\delta \\in D$, so that $\\gamma \\cap K^*$, if nonempty, lies in\n$\\gamma'$. This allows us to resolve each comparison between $K^*$\nand an arc $\\delta \\in D$, assuming that $\\gamma \\cap K^* \\neq\n\\emptyset$, and we continue the binary search through each $M_j$ in\nthe same manner.\n\nWe again use a variant of Cole's technique~\\cite{RC} to slightly\nimprove this bound further. In each parallel step we have a\ncollection $Z$ of $O(\\log n)$ weighted points, each of which is an\nintersection point of $\\gamma$ with some arc $\\delta$ from one of\nthe planar maps $M_j$, and we need to compare each of the points of\n$Z$ with $K^*$. Let $D$ denote the set of these active arcs.\n\nNote that each arc $\\delta$ participating in this step\ncontributes (at most) four points to $Z$, for a total of at most\n$4|D|$ points. We perform three steps of a (weighted) binary search on\nthe points of $Z$, where each step takes the weighted median $z_0$\nof an appropriate portion of $Z$, and calls $\\Pi_1(l_0)$, where $l_0$\nis the vertical line through $z_0$. These $\\Pi_1$-steps resolve the\ncomparisons with $K^*$ of all but $1\/8$ of the points of $Z$, that is,\nat most $(1\/8)\\cdot 4|D| = |D|\/2$ points of $Z$ are still unresolved.\n\nIn other words, after the three calls\nto the algorithm for solving $\\Pi_1$ (in the first parallel\nstep of the execution), we can determine the outcomes of the\ncomparisons of at least half of the arcs in $D$ with $K^*$. We can\nthen proceed in this manner and apply Cole's technique (as before),\nby using only a constant number of calls to $\\Pi_1$ in each of the\n$O(\\log n)$ parallel steps of searching in all the maps.\nThis reduces a $\\log \\log n$ factor from the bound of the\nrunning time, so it is only $O(\\log^5n)$ time.\n\nWhen these searches terminate, we end up with a 2-face in each\n$M_j$, in which $\\gamma \\cap K^*$ lies (if nonempty), and we reach\nthe scenario described in a preceding paragraph. As explained there,\nwe can now either determine that $K \\neq \\emptyset$, or that $K =\n\\emptyset$, or else we know which side of $\\gamma$, within $\\psi_i$\n(or, rather, within $\\psi'$) can contain $K^*$, and we continue the\nbinary search through $\\psi_i$ on that side.\n\nWhen the binary search through $\\psi_i$ terminates, we have a 2-face\n$\\zeta_i$ of $M_i$, where $K^*$ must lie, and we retrieve the ball\n$b_i$ corresponding to $\\zeta_i$. We repeat this step to each of the\nmaps $M_i^+$ and $M_i^-$ of each of the $t$ spherical polytopes\n$S_i$, and obtain a set ${\\cal B}_1$ of $2t$ balls. In addition, the\nsearches through the maps $M_i^+$ and $M_i^-$ may have trimmed\n$\\psi'$ to a narrower strip $\\psi''$, and have produced a set ${\\cal B}_2$\nof witness balls, so that the $xy$-projection of their intersection\nlies inside $\\psi''$. ${\\cal B}_2$ may consist of a total of $O(t^3 \\log^2\nn)$ witness balls, as is easy to verify. In addition, the\nsecond-level searches produce an additional collection ${\\cal B}_2'$,\nconsisting of balls corresponding to faces of the maps $M_j^+$ and\n$M_j^-$, in which the second-level searches have ended; their\noverall number is $O(t^2 \\log n)$. Put $K_2 = \\bigcap ({\\cal B}_1 \\cup\n{\\cal B}_2 \\cup {\\cal B}_2')$. Hence $K \\neq \\emptyset$ if and only if $K_2 \\neq\n\\emptyset$.\n\nAs already noted, the overall running time of the emptiness\ndetection is $O(\\log^5 n)$.\n\nSo far, we have only determined whether $K$ is empty or not.\nHowever, to enable the decision procedure to discriminate between\nthe cases $r^* = r$ and $r^* < r$ we need to refine the algorithm,\nso that it can also determine whether $K$ has nonempty interior (we\nrefer to an intersection $K$ with this property as\n\\emph{non-degenerate}).  To do so, we make the following\nmodifications to the algorithm described above. Each step in the\nemptiness testing procedure which detects that $K \\neq \\emptyset$\nobtains a specific point $w$ that belongs to $K$. Moreover, $w$\nbelongs to the intersection $K_1$ of polylogarithmically many\nwitness balls, and does not lie on the boundary of any other ball.\nThis is because each of the procedures $\\Pi_0, \\Pi_1$, or $\\Pi_2$\nlocates the $xy$-projection $w^*$ of $w$ (which, for $\\Pi_1$ and\n$\\Pi_2$ is a generic, unknown point in $K$) in each of the maps\n$M_i^+, M_i^-, i = 1,\\ldots,t$, and the collection of the witness\nballs gathered during the various steps of the searches contains all\nthe balls that participate in the corresponding spherical polytopes\n$S_i$ on whose boundary $w$ can lie. Thus, when we terminate with a\npoint $w \\in K$, we find, among the polylogarithmically many witness\nballs, the at most four balls whose boundaries contain $w$ (recall\nour general position assumption), and test whether their\nintersection is the singleton $\\{w\\}$. It is easily checked that\nthis is equivalent to the condition that $K$ is degenerate.\n\nThis completes the description of the algorithm, and concludes the\nproof of Proposition~\\ref{prop:intersection_test}.\n\n\n\\section{Discussion and Open Problems.}\n\\label{sec:discussion} In this paper we presented two algorithms for\ncomputing the 2-center of a set of points in $\\mathbb{R}^3$. The first\nalgorithm takes near-cubic time, and the second one takes\nnear-quadratic time provided that the two centers are not too close\nto each other. Note that our second algorithm may be slightly\nrevised, so that it receives, in addition to $P$, a parameter\n$\\epsilon > 0$ as input, and returns a solution for the 2-center\nproblem for $P$, if $\\epsilon \\leq 1-\\frac{r^*}{r_0}$. To this end,\nwe run the exponential search until we reach a value of $r$ with\n$1-\\frac{r}{r_0} \\leq \\epsilon$. If along the search we have found a\nvalue of $r$ such that $r \\geq r^*$, we stop the search and run\nChan's technique with the constraint that $r^* \\leq r$, as above.\nOtherwise, we have $r^* > r_0(1-\\epsilon)$ and we may return the\nsmallest enclosing ball of $P$ as an $\\epsilon$-approximate solution\nfor the 2-center problem. This way, we ensure that the running time\nof our algorithm is $O(\\epsilon^{-3} n^2\\log^5 n)$.\n\nAn obvious open problem is to design an algorithm for the 2-center\nproblem that runs in near-quadratic time on all point sets in\n$\\mathbb{R}^3$.  Another interesting question is whether the 2-center\nproblem in $\\mathbb{R}^3$ is \\emph{{\\sc 3sum}-hard} (see~\\cite{GO}\nfor details), which would suggest that a near-quadratic algorithm is\n(almost) the best possible for this problem.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:Introduction}}\n\n\nThe organic-inorganic interfaces formed between peptides and surfaces \nare of interest to diverse fields like medicine, optoelectronics and energy storage \\cite{WangGiovanni2013, GuoCahen2016, Khatayevich, Mannoor, Guy, Zhao, Sarikaya:2003-biomim, Costa:2015:IMPORTANT, Heinz:2016:Review, Walsh:2017ci}.\nAmino acids and their oligomers -- i.e. peptides -- are particularly interesting materials because they are naturally biocompatible and offer a rich functional space already at the amino acid level, that can be extended by the combinatorial increase of molecular motifs available through the peptide bond formation. In these setups, the inorganic component offers a platform to immobilize and template the bioorganic counterpart, as well as to record electronic signals from interactions or reactions.\nHowever, emergent behavior makes it impossible to base the description of such interfaces solely on the study of its independent components. \n\nOn the experimental front, progress on the fundamental understanding of these interfaces has been achieved with recent soft landing techniques \\cite{Rauschenbach2017} that allow amino acids and peptides of a few tens of building blocks to be deposited on solid surfaces and subsequently imaged in ultra-high vacuum conditions employing scanning tunneling microscopy (STM) \\cite{Rauschenbach2016}. \nThese experiments have shown that small sequence modifications, changes in the protonation of a peptide, and charge transfer at the interface can yield drastic changes in the two-dimensional structure and self-assembly~\\cite{Rauschenbach2017, Abb2016, Humblot2016}.  \n\nExplaining the physical mechanisms that govern adsorption and pattern formation calls for systematic theoretical investigations with atomistic resolution of such interfaces. \nHowever, such studies are challenging as they require (i) accurate energetics for a system build-up of elements across the periodic table and where considerable charge rearrangement and chemical reactions can occur (ii) sampling and representing a large conformational space, and (iii) dealing with structure motifs that can only be represented by unit-cells containing hundreds of atoms. Pioneering work that used density-functional theory (DFT) to study amino-acid at inorganic substrates has focused on small or rigid amino acids, and a limited portion of their conformational space \\cite{DiFelice2003, DiFelice2004, Ghiringhelli2006, Arrouvel2007, Iori2008}. \nOne of the first studies that was dedicated to larger amino acids was performed by Hong and coworkers \\cite{Hong2009} and illustrates the challenge of properly sampling the large structure space of flexible biomolecules. These studies have clarified that an accurate energy function is only one of the ingredients needed to properly predict the structure of peptides at surfaces, with the sampling of structure space being just as important.\n\n\nWe propose to analyze how the structure space of a complex building block with multiple functional groups is affected by changes in its (non-biological) environment. To achieve this goal, we analyze a database of the arginine (Arg) amino-acid and its protonated counterpart (Arg-H$^+$) in the gas-phase and interfaced with Cu(111), Ag(111) and Au(111). Arginine is a good test bed because it is a relatively small molecule, but yet contains a very flexible side-chain and allows for different stable protonation states. We show that it is possible to perform an exhaustive conformational space exploration while still treating the potential-energy surface (PES) with DFT, thus capturing the charge rearrangements at the interface.\nThe outcome of such a computational structure-search is a large number of stationary-states on the respective PES, where not only the global minimum is of interest. Also the relative positioning of other local minima with respect to the structural degrees of freedom, as well as with respect to the energy scale of the PES, are relevant because they can reveal different basins and structures of interest under different conditions~\\cite{ropo2016trends, Rossi_2013, Rossi_2014, Schubert_2015, Baldauf_2015}.\nIn order to handle such high-dimensional data, we make use of recent advances in machine-learning methods that can help to visualize conformational preferences of adsorbed molecules~\\cite{Ceriotti2011, Tribello2012, Ceriotti2013, De2016, Bartok2017, De2017}.\n\nIn the following, we discuss the impact of the protonation state and the presence or absence of the surface on the accessible conformational space of arginine. We start by describing the procedure we followed to build the database of Arg and Arg-H$^+$ on Cu(111), Ag(111), and Au(111), based on thousands of first-principles structure optimizations which we make available to the community~\\cite{DOI_of_the_data_set_at_NOMAD}.\nWe then analyze this database and show how different patterns of bonding and charge transfer induce fundamental changes in the accessible conformational space. We also provide an analysis of property trends across the different metallic surfaces, including protonation-dependent stereoselective binding to the surfaces and deprotonation propensities.\n\n\n\\section{Computational methods \\label{sec:CompMethods}}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{F1_Protonation_states.jpg}\n\\caption{a) Pictorial representation of the arginine amino acid, including labels of chemical groups and atoms. b) Protomers of Arg that are addressed in this work. c) Protomers of Arg-H$^+$ that are addressed in this work. }\n\\label{fig:protonationsketch}\n\\end{figure}\n\nArginine is an amino acid with a flexible side-chain of three (aliphatic) CH$_2$ groups and a guanidino group. \nA depiction of the Arg molecule including the labeling of the different chemical groups and specific atoms we will refer to in the manuscript is shown in Fig. \\ref{fig:protonationsketch}(a). \nIn the context of this publication we use the term \\textit{protonation state} to distinguish between Arg and its singly-protonated form Arg-H$^+$. We use the word \\textit{protomers} to distinguish different arrangements of protons within molecules of the same sum formula, for example the protomers \\textbf{P1} to \\textbf{P5} of Arg or the protomers \\textbf{P6} and \\textbf{P7} of Arg-H$^+$, shown in Fig. \\ref{fig:protonationsketch}(b) and (c). In this section, we describe the computational setup, convergence tests, and sampling strategies we employed in order to build the database of Arg and Arg-H$^+$ adsorbed on different metal surfaces.\n\n\nAll the electronic structure calculations were carried out using the numeric atom-centered basis set all-electron code FHI-aims \\cite{Blum2009, Havu2009}. We used the standard \\textit{light} settings of FHI-aims for all species, except when stated otherwise. For modeling the adsorbed molecules, a $5\\times6$ surface unit cell with $4\\times4\\times1$ $k$-point sampling was employed. The slab contains 4 layers and we added a 50 \\AA~ vacuum in the $z$ direction in order to separate periodic images of the system. \nA surface unit cell of this size does not completely isolate neighboring molecules on the surface plane.\nIn order to estimate the magnitude of this spurious interaction, we calculated binding energies for three  Arg and three Arg-H$^+$ structures adsorbed on Cu(111) using different surface unit cell sizes. As shown in the SI (Tables S3 and S4), the relative binding energies change by no more than 50 meV when reaching a $10\\times12$ cell. \n\nAn accurate description of the dispersion interactions that play a role in weakly-bonded adsorbates on metallic surfaces can be achieved with newly-proposed methods that take into account electronic screening and the the many-body nature of the dispersion term \\cite{Hermann2019}; however, such methods are not yet computationally feasible for large-scale studies.\nConsidering dispersion interactions is, however, crucial to model such interfaces \\cite{Ruiz2016, Ruiz-thesis, Liu2013, Liu2012, Al-Saidi2012, VanRuitenbeek2012, Wagner2012, Carrasco2014}. We thus employ the PBE exchange-correlation functional augmented by the TS-vdW$^{\\text{surf}}$ \\cite{Ruiz2016} dispersion correction between pairs of atoms that involve the adsorbed molecule (molecule-molecule and molecule-surface dispersion). \nAdding pairwise vdW corrections for pairs of metal atoms did not result in a systematic improvement of the lattice constants due to the complex many-body nature of the electronic screening within the metallic bulk.  \nBecause the PBE lattice constants for Cu, Ag, and Au are in good agreement with experimental data (see Table S1 of the SI) and the electronic structure itself is not changed by the inclusion of these types of vdW interactions, we chose to use the simplest setup. \nWe optimized all structures \nuntil all forces in the system were below 0.01\\,eV\/\\AA\\,. \nWe also fixed the two bottom layers of the slabs in all optimizations. \nA dipole correction was applied in $z$ direction to compensate for the dipole formed by the asymmetric surface configurations. \n\n\\subsection{Database Generation \\label{sec:ModelGeenration}}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{F2_Relative_Energies.jpg}\n\\caption{(a-d) Correlation plots of relative energies of Arg or Arg-H$^+$ conformers on Cu, Ag, and Au (111) surfaces. Each dot corresponds to the same conformer optimized on the two surfaces addressed in each panel, color coded with respect to the RMSD (heavy atoms only) between the superimposed optimized structures without taking surface atoms into consideration. (e) Binding energies of Arg and Arg-H$^+$ on Cu(111), Ag(111) and Au(111) surfaces.}\n\\label{fig:relrel}\n\\end{figure}\n\n\nThe sampling of the structure space of the amino acid in two protonation states on metallic surfaces was performed by starting from a previously published data-set comprising stationary points of isolated amino acids and dipeptides \\cite{Ropo2016,ropo2016trends}. \nFor Arg, 1206 structures were present in the database.\nIn order to reduce the number of possibilities, but keeping a representative share of the structures, we considered the 300 lowest energy conformers, the 27 highest energy conformers, and 125 conformers uniformly spanning the energy range in between. \nFor the Arg-H$^+$ amino acid, all 215 structures present in the gas-phase data set were used in this study.\n\nWe distinguish \\textit{upstanding} positions of the molecules where the largest principal axis of the molecule is approximately perpendicular to the surface plane, from \\textit{flat lying} positions with an arrangement parallel to the surface.\nFor Arg, 3 lying configurations per structure were generated by randomly placing the molecule flat on the Cu(111) surface and then rotating it by $120^{\\circ}$ around the principal axis. \nTwo upstanding configurations were generated for the 25 lowest-energy gas-phase structures by first placing the molecule in a random upright orientation and then flipping it. \nFor Arg-H$^+$ a similar procedure was adopted: flat lying positions were created by $90^{\\circ}$ rotations around the principal axis and upstanding configurations were created for 27 structures that were uniformly selected from the whole energy range. \nIn summary, we considered a total of 1156 conformers of Arg@Cu(111) and 914 conformers of Arg-H$^+$@Cu(111).\n\nAll optimized structures that fell within a range of 0.5 eV from the global minimum on Cu(111) were transferred to Ag(111) and Au(111) and further optimized. In addition, we randomly picked 105 Arg-H$^+$ structures representing the higher energy range on Cu(111) to be further optimized on Ag(111) and Au(111). Moreover, for Arg, 180 randomly picked structures representing the higher energy range were considered on Ag(111) and 61 on Au(111). The total amount of calculated structures for each case is summarized in Table \\ref{tbl:totalnmbrofstructs}. \n\n\\begin{table}\n\\center\n \\caption{Number of calculated Arg and Arg-H structures in isolation and adsorbed on Cu(111), Ag(111) and Au(111).}\n \\label{tbl:totalnmbrofstructs}\n\\begin{tabular}{|r|c|c|c|c|}\n\\hline\n     & Gas phase & Cu(111) & Ag(111) & Au(111) \\\\\n\\hline\nArg  & 1206      & 1156    & 327     & 209  \\\\\n\\hline\nArg-H$^+$ & 215       & 914      & 718       & 721         \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe checked that this strategy ensured a sufficient sampling of the low-energy range of both Arg and Arg-H$^+$ on Ag(111) and Au(111) by analyzing the alterations in relative energy hierarchies on the different surfaces. In Fig. \\ref{fig:relrel}(a-d) each dot corresponds to a conformer that was optimized first on the Cu(111) surface and then post-relaxed on Ag(111) or Au(111). Within the lowest 0.5 eV range, we do not observe any significant rearrangement of the energy hierarchy with respect to the Cu(111) surface. The energy hierarchies of both Arg and Arg-H$^+$ on the Ag(111) and Au(111) surfaces are almost identical. The most pronounced outliers in all plots correlate with a higher root mean square displacement (RMSD) of the molecular atoms (i.e. disregarding the surface-adsorption site), thus pointing to a structural rearrangement of the molecule.\n\n\\subsection{Structure space representation and analysis \\label{sec:soap-map}}\n\nWe analyse the structure space of all systems considered, as well as its alterations by employing a dimensionality reduction procedure that makes it more intuitive to understand the high-dimensional space. Following Ref. \\cite{De2017}, we represent the local atom-centered environments of the structures through the smooth overlap of atomic positions (SOAP)\\cite{Bartok2017} descriptors. We then obtain the similarity matrix between different conformers with the regularized entropy match kernel (REMatch) \\cite{De2016}. We used SOAP descriptors with a cutoff of 5.0 \\AA, a Gaussian broadening of $\\sigma=0.5$ \\AA~ and an intermediate regularization parameter $\\gamma$=0.01. SOAP kernels were calculated only for heavy atoms in the molecule (disregarding metal and hydrogen atoms) and were obtained using the GLOSIM package \\cite{De2016, GLOSIM}. \n\nFor the dimensionality-reduced representation, we here chose to use the multi-dimensional scaling (MDS) algorithm as implemented in the \\texttt{scikit-learn} package\\cite{scikit-learn}. This algorithm is similar to the Sketchmap algorithm previously employed in Ref. \\cite{De2017}, but we found it more suitable for the data at hand, which is composed of decorrelated local stationary-points, instead of structures generated from molecular dynamics trajectories. In short, the low-dimensional map was obtained considering all calculated structures of Arg in the gas-phase and through minimizing the stress function\n\\begin{equation}\n\\sigma = \\sum_{i\\neq j} (D_{ij}-d_{ij})^2,  \n\\end{equation}\nwhere $D_{ij}$ is the similarity between structures $i$ and $j$ in the high-dimensional space and $d_{ij}$ the Cartesian distance in the low-dimensional (2D) space. \nWe then projected structures in different environments onto the pre-computed map of gas-phase Arg by fixing the parameters of the map and finding the low-dimensional coordinates $x$ that minimize  \n\\begin{equation}\n\\sigma_P(x)= \\sum_{i=1}^{N_{\\text{Arg}}} \\left(|X-X_i|-|x-x_i|\\right)^2 \n\\label{eq:proj}\n\\end{equation}\nwhere the sum runs over all structures in the optimized gas-phase Arg map and $X$ represents the high-dimensional representation of the new structure to be projected. In order to classify structural patterns, we employ the following notations:\n\\begin{itemize}\n    \\item We represent the protomers by the labels shown in Fig. \\ref{fig:protonationsketch}(b) and (c). \n    \\item We identify the presence of strong intramolecular hydrogen bonds (H-bonds) whenever the distances between the hydrogen connecting donor and acceptor are below 2.5 \\AA. We label the H-bond pattern between two atoms in the molecules according to the nomenclature shown in Fig. S3 in the SI.\n    \\item We further classify the structures according to the longest distance between two heavy atoms in the molecule.\n\\end{itemize}\n\n\\subsection{Binding energies and charge rearrangement \\label{sec:BindingEnergies}}\n\nThe binding energies for each structure were calculated as follows. For neutral Arg we computed \n$\nE_{\\textsf{b}} =  E_{\\textsf{mol@surf}} - E_{\\textsf{surf}} - E_{\\textsf{mol}}, \n$\nwhere $E_{\\textsf{mol@surf}}$ corresponds to the total energy of the interface, $E_{\\textsf{surf}}$ is the total energy of the pristine metallic slab and $E_{\\textsf{mol}}$ the total energy of the lowest energy gas-phase conformer.\n\nFor charged Arg-H$^+$, we considered the binding energy of a two-step reaction. \\textit{First}, the interface is formed between the charged molecule and the clean surface:\n$\nE_{\\textsf{b1}} = E_{\\textsf{mol}^+\\textsf{@surf}} - E_{\\textsf{surf}} - E_{\\textsf{mol}^+},\n$\nwhere $E_{\\textsf{mol}^+}$ the total energy of the most stable gas-phase conformer of the isolated charged molecule. \\textit{Second}, an electron from the metal neutralizes the unit cell where the adsorbed molecule is located, yielding\n$\nE_{\\textsf{b2}} = E_{\\textsf{mol@surf}} - E_{\\textsf{mol}^+\\textsf{@surf}} - E_{f},\n$\nwhere $E_f$ corresponds to the Fermi energy of the metallic surface. The final binding energy is thus considered to be\n\\begin{equation}\nE_{\\textsf{b}}^+ = E_{\\textsf{b1}} + E_{\\textsf{b2}} = E_{\\textsf{mol@surf}} - E_{\\textsf{surf}} - E_{f}- E_{\\textsf{mol}^+}.\n\\end{equation}\nFor reference, we report the values we used for $E_f$ at each surface in Table S5. These binding energies are shown in Fig. \\ref{fig:relrel}d, and will be discussed in detail in Section \\ref{sec:Results}.\n\n\nIn addition,  for conformers within 0.1 eV of the respective global minimum of each system, we have calculated free energies at finite temperatures within the harmonic approximation \\cite{McQuarrie, Fultz2010}, in order to address question about thermal stability of such structures. We have calculated\n$\nF_{\\textsf{harm}}(T) = E_{\\textsf{PES}} + F_{\\textsf{vib}}(T),\n$\nwhere $E_{\\textsf{PES}}$ is the total energy obtained from DFT, and we have used textbook expressions for the harmonic vibrational Helmholtz free energy $F_{\\textsf{vib}}(T)$. We have calculated the Hessian matrix only taking into account displacements of the adsorbate. The surface was only considered as an external field, which is a good approximation for physisorbed systems. \n\nTo address charge rearrangements after adsorption on the surface, we calculated electron density differences for selected  structures on each surface by computing\n$\n\\Delta\\rho = \\rho_{\\textsf{mol@surf}} - \\rho_{\\textsf{surf}} - \\rho_{\\textsf{mol}},  \n$\nand in the case of Arg and \n$\n\\Delta\\rho^{(+)} = \\rho_{\\textsf{mol@surf}} - \\rho_{\\textsf{surf}} - \\rho_{\\textsf{mol}^{(+)}},\n$\nin the case of Arg-H$^+$. In these expressions, $\\rho_{\\textsf{mol@surf}}$ is the total electron density of the interface, $\\rho_{\\textsf{surf}}$ is the electron density of the slab without molecule, and $\\rho_{\\textsf{mol}}$ and $\\rho_{\\textsf{mol}^+}$ are electron densities of neutral Arg and charged Arg-H$^+$ molecules with the same geometries as in interface. The $+$ sign denotes that the final density difference integrates to +1 electron in the case of Arg-H$^+$.\n\n\\section{Results and discussion \\label{sec:Results}}\n\n\\subsection{The unconstrained structure space: Arg in isolation \\label{sec:Isolation}}\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=0.87\\linewidth]{F3_Arg_isolated.jpg}\n\\caption{Low-dimensional map of Arg stationary points on the PES. Only points linked to structures with a relative energy of 0.5 eV or lower are colored. Representative structures of all conformer families are visualized as well as their H-bond distances (in turquoise) and longest distance between two heavy atoms (in red) of the molecule. The maps are colored with respect to a) relative energy, b) longest distance, and c) H-bond pattern. The size of the dots also reflect their relative energy, with larger dots corresponding to lower energy structures.}\n\\label{fig:Arg_Sketchmap}\n\\end{figure}\n\nWe start by analysing the unconstrained conformational space of Arg in isolation, which is formed by more than 1200 local stationary states \\cite{Ropo2016,ropo2016trends}. In order to rationalize different structural arrangements in this space, we utilize the dimensionality-reduction algorithm discussed in Section \\ref{sec:soap-map} and build a two-dimensional map. On this map, shown in Figure \\ref{fig:Arg_Sketchmap}(a), each dot represents one structure. A close proximity of the dots implies similarities of the heavy-atom arrangement between the conformations. This is the low-dimensional map that is taken as a reference for comparison throughout this manuscript. \n\nWe proceed to color-code the dots on the map according to different properties. In Fig. \\ref{fig:Arg_Sketchmap}(a) we show the map colored by the relative energy of each structure with respect to the global minimum $\\Delta E_{\\text{rel}}$. We only color structures with $\\Delta E_{\\text{rel}} < 0.5$ eV. The region with $\\Delta E_{\\text{rel}} < 0.1$ eV is colored red and is represented by 32 different structures that occupy different parts of the map. The dominant protomer (98.6\\%) among these conformers is the one labeled \\textbf{P1} in Fig. \\ref{fig:protonationsketch}, i.e. non-zwitterionic. \nHowever, the lowest energy structure, labeled \\textit{a} in panel (a) of Fig. \\ref{fig:Arg_Sketchmap}, is protomer \\textbf{P3}, with a shared proton between the carboxylic and the guanidino group. \nThis structure is compact, with the longest distance within the molecule of only 5.01 \\AA~ and presenting two strong intramolecular H-bonds.\nZwitterionic protomers, denoted as \\textbf{P4} and \\textbf{P5} in Fig. \\ref{fig:protonationsketch}, do not appear in the gas-phase.\n\nInspecting the map in Fig. \\ref{fig:Arg_Sketchmap}(a), it is clear that low-energy conformers are almost exclusively present in the upper hemisphere of the plot. \nThis can be rationalized in terms of the structural motifs that occupy these two halves of conformational space:\nIn Fig. \\ref{fig:Arg_Sketchmap}(b), we color-code the dots in terms of the longest extension of the conformers. \nWhile the upper hemisphere features compact structures, the lower hemisphere of the map is populated by extended conformers (with longest extensions between 7.5 \\AA~ and 10.0 \\AA).\nMany of them do not contain any H-bond or contain only one H-bond between the carboxyl and amino group.\nExtended conformers of Arg are energetically unfavoured in the gas-phase as the formation of strong H-bonds is crucial for the stabilization of Arg in isolation. \nComparing the different plots in Figure \\ref{fig:Arg_Sketchmap}, we see that all low-energy structures with $\\Delta E_{\\text{rel}} < 0.1$ eV are indeed compact with one or two H-bonds. \n\nIn Fig. \\ref{fig:Arg_Sketchmap}(c), we identify in total 13 different families with respect to the number and character of H-bonds in the molecule, with $\\Delta E_{\\text{rel}} < 0.5$ eV. Representative structures of all families are shown in panel (a). \nThis family classification  helps us understand why in Fig. \\ref{fig:Arg_Sketchmap}(a) there are structures of higher energies at similar regions as structures with lower energies. Even though these structures are typically in the same protomeric state and have a similar arrangement of the heavy-atoms, the carboxyl group can rotate, giving rise to different H-bond patterns.\nThese different arrangements can give rise to energy differences of up to 0.2 eV, as exemplified in Fig. S4 in the SI. Including hydrogens in the SOAP descriptors used to build the 2D map could provide a better energy separation, but it would prevent us from comparing different protonation states, as shown in the next section.\n\n\\subsection{Adding a proton: Arg-H$^+$ in isolation \\label{sec:Proton}}\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=0.9\\linewidth]{F4_ArgH_isolated.jpg}\n\\caption{Representative conformers of the populated structure families within 0.5 eV of the global minimum of isolated Arg-H$^+$ and low-dimensional projections of all populated conformers onto the Arg map. Grey dots represent all structures from the original map of isolated Arg in Figure 3, and serve as a guide to the eye. The maps are colored with respect to a) relative energy, b) longest distance within the molecule, and c) H-bond pattern.}\n\\label{fig:arghmaps}\n\\end{figure}\n\nArg-H$^+$ is of particular interest as it is the most abundant form of this amino acid under physiological pH conditions \\cite{BolgerBook1995}, and we thus investigate changes of the conformational space introduced by the addition of a proton to the Arg amino-acid. \nTo that end, we plot a projection of all stationary points of the Arg-H$^+$ PES with $\\Delta E_{\\text{rel}} < 0.5$ eV (referenced to its own global minimum) onto the map that was previously created for Arg. \nIn Fig. \\ref{fig:arghmaps}(a), we color the dots in the map according to $\\Delta E_{\\text{rel}}$, in Fig. \\ref{fig:arghmaps}(b) according to the longest distance between heavy atoms in the molecule, and in Fig. \\ref{fig:arghmaps}(c) according to the H-bond pattern. The grey dots in the maps represent all points in the Arg map of Figure \\ref{fig:Arg_Sketchmap} and are shown for ease of comparison.\n\nThe unique conformation types of Arg-H$^+$ can be grouped into 8 different families in this energy range, which are represented in Fig. \\ref{fig:arghmaps}(a). Most families only have one H-bond and there are no zwitterionic protomers. \nThis means that in isolation only the protomer \\textbf{P6} is populated.\nIt is worth noting that under physiological conditions (in solution), the zwitterionic protomer \\textbf{P7} is preferred. \n\n\nThere are only two very similar structures (same family) with $\\Delta E_{\\text{rel}} < 0.1$ eV in this case. The global minimum, labeled \\textit{a} in Fig. \\ref{fig:arghmaps}(a), contains two H-bonds within the molecule, between atoms N-N$\\mathrm{\\varepsilon}$ and O1-N$\\eta$ (see Fig. \\ref{fig:protonationsketch}). \nThis particular structure resembles the lowest-energy structure of Arg with a proton added to the carboxyl group.\nThis protonation results in an extension of the molecule by around 1 \\AA. \nThat correlates with the location of the lowest-energy structure being slightly shifted on the map towards the region containing more extended structures.\n\nWe note that the structure space of Arg-H$^+$ is contained within the conformational space of Arg and also drastically reduced in numbers if compared to Arg: \nThere are only 108 structures with $\\Delta E_{\\text{rel}} < 0.5$ eV, compared to 1179 structures in the Arg case. \nIn this energy range, regions of the map with very compact and very extended structures are not populated in this protonation state. \nThis can be traced to the constraint imposed by the addition of the proton, that make extended structures less stable due to the strong driving force to neutralize the charge imbalance created by the proton on the guanidino group.\nTo rationalize why the most compact conformers are also less populated, we show in Fig. \\ref{fig:ArghArg_electrondensitydifferencce} the electron-density differences between the lowest energy Arg-H$^+$ conformer and an Arg conformer created by fixing the same Arg-H$^+$ structure, but neutralizing the charge and removing the hydrogen connected to the carboxyl group. This modification yields the same covalent connectivity observed in the global minimum of Arg. \nWe show isosurfaces corresponding to electron accumulation in Arg-H$^+$ in red and electron depletion in Arg-H$^+$ (accumulation in Arg) in blue. \nWe observe a density surplus between the O1 and N$\\eta$ atoms in Arg, favoring the formation of a stronger H-bond leading to a more compact structure.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{diffelectron.jpg}\n\\caption{Electron density difference between Arg-H$^+$ and Arg calculated by neutralizing the charge and removing the hydrogen connected to the carboxyl group (marked in green) from the lowest energy structure of Arg-H$^+$. The isosurfaces of electron density with value $\\pm0.005$ e\/Bohr$^3$ corresponding to the a) regions of electron accumulation on Arg-H$^+$ and b) where the electron depletion on Arg-H$^+$, both compared to Arg.}\n\\label{fig:ArghArg_electrondensitydifferencce}\n\\end{figure}\n\n\\subsection{Adsorption of Arg on Cu, Ag, Au (111) surfaces}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{F6_Arg_surfaces.jpg}\n\\caption{\nLow-dimensional projections of conformers of Arg adsorbed on a) Cu(111), b) Ag(111), and c) Au(111), onto the gas-phase Arg map of Fig. 3. Only conformers within 0.5 eV from the respective global minimum are colored. Grey dots represent all structures from the original map of gas-phase Arg, and serve as a guide to the eye.\nIn each panel, representative structures are shown from two perspectives: a side view where molecule and surface are shown (bottom), and the corresponding top view (top) where only the molecule is shown. The longest distance within each visualized conformer is reported in red and H-bond lengths are reported in turquoise.}\n\\label{fig:argsurf_sketchmap}\n\\end{figure}\n\nWe now turn to the analysis of the conformational space of Arg when in contact with metal surfaces, namely Cu(111), Ag(111), and Au(111). \nIn Figure \\ref{fig:argsurf_sketchmap}, we show map-projections of the stationary points with $\\Delta E_{\\text{rel}} < 0.5$ eV (referenced to each respective global minimum) of Arg adsorbed on the three surfaces.\nThe conformational space of Arg upon adsorption is reduced and the conformers occupy similar regions of the map as the ones from Arg-H$^+$. \nWe will learn in the following that this is mainly due to the formation of strong bonds with the surface that result in steric constraints of the space and also partially due to electron donation from the molecule to the metallic surfaces.\n\nThe lowest energy structure lies on the same part of the map on all surfaces, which is different from the area where the gas-phase global minimum of Arg was located. These conformers, labeled \\textit{a} in Figure \\ref{fig:argsurf_sketchmap}(a), (b) and (c) form a strong H-bond between atoms O1 and N$\\epsilon$. The longest distance within molecule lies between 7.20-7.35 \\AA. This structure binds most strongly to all three surfaces through its amino and carboxyl groups. \n\nOther low-energy structures in all surfaces form strong bonds to the surfaces only through the carboxyl group, as exemplified by the structure labeled \\textit{b} in all panels of Fig. \\ref{fig:argsurf_sketchmap}. \nThese bonds are formed most favorably on \\textit{top} positions, i.e. vertically on top of a surface metal atom.\nIn particular for Cu(111), the atomic spacing of the Cu atoms on the surface favors both oxygens to bind on \\textit{top} positions simultaneously.\n\nThe favorable formation of these bonds is connected with the fact that all conformers with $\\Delta E_{\\text{rel}} < 0.2$ eV are in the protomeric state \\textbf{P3}, in which the carboxyl group is deprotonated.\nThe bonds to the surface and a favorable vdW attraction effectively flatten the molecular conformation, thus energetically favoring more elongated structures. \nProtomers of type \\textbf{P1}, which was dominant in the gas-phase, only appear  with $\\Delta E_{\\text{rel}} > 0.3$ eV on Cu and Ag, and with $\\Delta E_{\\text{rel}} > 0.2$ eV on Au.\nProtomers \\textbf{P4} and \\textbf{P5} are again not observed. \nRegarding the intramolecular H-bond patterns, within 0.5 eV from the global minimum\nwe can identify 7 different families on Cu(111), \nand 6 families on both Ag(111) and Au(111). \nThese families contain H-bonds where the carboxyl group predominantly participates. All families are represented in Table S4 in the SI.\n\n\n\n\\subsection{Adsorption of Arg-H$^+$ on Cu, Ag, Au (111) surfaces}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.83\\linewidth]{F7_ArgH_surfaces.jpg}\n\\caption{Low-dimensional projections of conformers of Arg-H$^+$ adsorbed on a) Cu(111), b) Ag(111), and c) Au(111), onto the gas-phase Arg map of Fig. 3. Only conformers within 0.5 eV from the respective global minimum are colored. Grey dots represent all structures from the original map of gas-phase Arg, and serve as a guide to the eye.\nIn each panel, representative structures are shown from two perspectives: a side view where molecule and surface are shown (bottom), and the corresponding top view (top) where only the molecule is shown. The longest distance within each visualized conformer is reported in red and H-bond lengths are reported in turquoise.}\n\\label{fig:arghsurf_map}\n\\end{figure}\n\nFinally, we characterize the conformational-space changes arising from the simultaneous addition of a proton and the adsorption onto metallic surfaces.\nIn Figure \\ref{fig:arghsurf_map}, we show the projection of Arg-H$^+$ conformers adsorbed on Cu, Ag, and Au(111) onto the map of isolated Arg. \nThese projections, in particular the comparison of the plots in Figs. \\ref{fig:argsurf_sketchmap} and \\ref{fig:arghsurf_map}, reveal that \nthe conformational space of adsorbed Arg-H$^+$ is larger than the one of adsorbed Arg. \nWhile Arg-H$^+$ features more than 500 conformers within $ \\Delta E_{\\text{rel}} < 0.5$ eV, Arg only counts about 150 conformers in the same energy range.\nInterestingly, the adsorption of Arg-H$^+$ to a metal surface also results in an increase of the occupied structure space in comparison to isolated Arg-H$^+$ (108 structures with $ \\Delta E_{\\text{rel}} < 0.5$ eV), shown in Fig. \\ref{fig:arghmaps}. \nIn fact, the structures occupy similar regions of the map as the ones occupied by Arg-H$^+$, with the addition of extended structures that are located in the bottom of the map.\n\n\nWe identify 4 different families on Cu(111) and 3 on Ag(111) and Au(111) with $ \\Delta E_{\\text{rel}} < 0.1$ eV. \nRepresentative conformers of these families are shown in Fig. \\ref{fig:arghsurf_map}. \nThe lowest energy conformer, labeled \\textit{a} in Fig. \\ref{fig:arghsurf_map}(a)-(c), appears on all surfaces at the same region of the map as for adsorbed Arg.  The largest distance within the molecule lies around 7 \\AA~ and it also has a strong H-bond linking the carboxyl-O and the N$\\varepsilon$ atoms. \nThe structure, however, does not present the same orientation to the surface as compared to the lowest energy conformer of Arg, and does not form strong bonds with the surface.\nWith the exception of the extended structure on Cu(111), labeled \\textit{d} in  Fig. \\ref{fig:arghsurf_map}(a), all conformers with $ \\Delta E_{\\text{rel}} < 0.1$ eV on all surfaces contain one intramolecular H-bond involving either \ncarboxyl-O and N$\\varepsilon$ atom (labeled a), \nbackbone N and N$\\varepsilon$ atoms (labeled b) or \ncarboxyl O and a N$\\eta$ atom (labeled c). \nCompared to adsorbed Arg, adsorbed Arg-H$^+$ structures become on average 1.0 \\AA~ more extended (see SI). The guanidino and carboxyl groups often lie parallel to the surface. \nThe protomer \\textbf{P6}, the only one present in the gas-phase, is dominantly populated also on the surfaces. \nHowever, we do observe very few conformers in the zwitterionic \\textbf{P7} state. These structures are at least 0.2 eV higher in energy than than the global minimum. \n\nWith respect to the number of bonds that Arg-H$^+$ forms with the surface, the picture is very different from adsorbed Arg. \nWithin the lower 0.15 eV, we do not observe short (strong) bonds of O or N atoms to the surfaces. \nThis lack of constraint by the surface contributes to the increased structure space of adsorbed Arg-H$^+$. \nIn addition, the molecule accepts electrons from the surface, becoming less positively charged, as we discuss in detail in the next section. \nWe conclude that Arg-H$^+$ interacts with the metallic surfaces mostly through van der Waals and electrostatic interactions.\n\n\\subsection{Electronic structure and trends across surfaces}\n\n\nIn the previous sections we focused on structural aspects of the adsorbed molecules and the most prominent bonds the molecules make with the metallic surfaces. In the following, we will discuss different aspects of the molecule-surface interactions with the goal of identifying trends across these systems.\n\nThe binding energies for all surfaces were calculated as discussed in Section \\ref{sec:BindingEnergies}. \nThe larger negative values in Fig. \\ref{fig:relrel}(d) correspond to stronger binding of the molecule to the surface. \nIn the case of adsorbed Arg, many conformers bind to Cu more strongly than to Ag and Au.\nAs mentioned previously, Arg forms strong bonds to the surfaces, but the binding of the deprotonated carboxyl group of Arg to the Cu(111) surface is geometrically favored as discussed above.\nIn the case of adsorbed Arg-H$^+$, there is no pronounced difference in binding strengths to the surfaces and the values are comparable to binding energies obtained for Arg adsorbed on Cu(111). \nThis correlates with the observation that the interaction of Arg-H$^+$ to the surfaces happens mostly through dispersion and electrostatic interactions. Despite the strong binding to the surface, it is also visible in Fig. \\ref{fig:relrel}(a) that the interaction of Arg-H$^+$ with the surface does not strongly template the conformations of this molecule, implying a low corrugation (i.e. homogeneity) of the molecule-surface interaction and allowing for a larger variety of conformers with similiar energies. This is in contrast to the molecule-surface interation of Arg, that is more inhomogeneous due to the formation of bonds through specific chemical groups.\nAdditionally, we have estimated harmonic vibrational free energies for all conformers with $\\Delta E_{\\text{rel}} < 0.1$ eV in each surface. In contrast to what has been reported for longer helical peptides \\cite{Rossi_2013,Schubert_2015}, the global minimum remains the same in all cases, as reported in Fig. S6 in the SI. For Arg-H$^+$ we observe relative energy rearrangements of up to 50 meV at 300 K, which changes the relative energy hierarchy of conformers less stable than the global minimum.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Closest_sketch_com_v1.jpg}\n\\caption{Low dimensional projections of adsorbed Arg and Arg-H$^+$ on Cu(111), Ag(111) and Au(111) color-coded with respect to the distance of the center of mass of the molecule with respect to the surface. Grey dots represent all structures from the original map of isolated Arg where the projection was made, and serve as a guide to the eye. }\n\\label{fig:Closest_map}\n\\end{figure}\n\nWe then focus on the distance between the molecule and the surfaces. We define this quantity by measuring the distance of the center of mass (COM) of the molecule with respect to the  \nsurface plane defined by the top layer of surface atoms.\nThese distances are collected in Fig. \\ref{fig:Closest_map}. \nThe COM is closer to Cu(111) than to Ag(111) and Au(111) for both Arg and Arg-H$^+$. \nIn addition, in all surfaces, Arg lies closer than Arg-H$^+$, in agreement with the observation that Arg forms stronger bonds to the surface. \nThe extended structures of Arg-H$^+$, at the bottom of the maps, tend to be closer to the surface than those that have H-bonds within the molecule, likely due to the stronger vdW attraction to the surface by extended conformations. \n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{F10_UpDown_representation.jpg}\n\\caption{Orientation of the C$_\\alpha$H group in a) \\textit{up} orientation (hydrogen pointing towards vacuum) and b) \\textit{down} orientation (hydrogen pointing towards the surfaces). c) The amount of structures with \\textit{up} and \\textit{down} orientation within 0.1\/0.5 eV from the global minimum of each surface.}\n\\label{fig:UpDown}\n\\end{figure}\n\n\nThe differences in COM distance to the surfaces between Arg and Arg-H$^+$ are apparently related to the preferred orientation of the chiral center of the molecule to the surface. \nThe chiral C$_\\alpha$ carbon can point its bonded hydrogen towards the surface (labeled \\textit{down} in the following), or towards the vacuum region (labeled \\textit{up} in the following).\nExamples of different molecular orientation are shown in Fig. \\ref{fig:UpDown}(a).\nThe dominant orientation with respect to the surface is different in the cases of Arg and Arg-H$^+$, as evidenced by the numbers presented in Fig. \\ref{fig:UpDown}(b). \nThe lower energy structures are mostly in the \\textit{up} orientation for Arg and mostly in the \\textit{down} orientation for Arg-H$^+$ (see also map in Fig. S7).\nAs discussed in the previous sections, despite showing different orientations of their C$_\\alpha$H groups, the lowest energy structures for both molecules adsorbed on each surface have very similar conformations. \nSince the addition or removal of a proton can apparently alter the preference of the chiral-center orientation, we propose that it could template different chiralities of  self-assembled super-structures on the surface \\cite{LingenfelderThesis}.\n\n\\begin{figure}[hb!]\n\\includegraphics[width=\\textwidth]{F13_El_difference_Z.jpg}\n\\caption{Electronic-density difference averaged over the directions parallel to the surface for the lowest energy conformers of Arg adsorbed on Cu(111) (a), Ag(111) (b), and Au(111) (c), as well as of Arg-H$^+$ adsorbed on Cu(111) (d), Ag(111) (e), and Au(111) (f). Positive values (red) correspond to electron density accumulation and negative values (blue) correspond to electron density depletion.\nIn each panel, we also show a side and top view of the 3D electronic density rearrangement.  Blue isosurfaces correspond to an electron density of +0.05 e\/Bohr$^3$ and red isosurfaces to -0.05 e\/Bohr$^3$. \\label{fig:charge-re}}\n\\end{figure}\n\nWe then investigated the rearrangement of the electronic density upon binding of the molecules to the different surfaces. \nIn Fig. \\ref{fig:charge-re} we show the electronic density rearrangement created by the lowest energy conformer at each surface, integrated over the axis parallel to the surface, overlaid on the side-view of the 3D density rearrangement. In addition, we show a top view of the density rearrangement in each case. Examples of further conformers are summarized in the SI, Figs. S8-S13. The data shows that Arg donates electrons to the surface, while Arg-H$^+$ accepts electrons from the surface. We have checked this propensity for selected conformers by integration of the electronic density rearrangement around the molecule and by calculating the Hirshfeld charge remaining on the molecule for the full database (see SI, Table S6). When comparing Hirshfeld charges on the molecule and those obtained from the electronic density rearrangement, we observe that Hirshfeld charges are always 0.3-0.5 e underestimated. \nIn addition, we observe that the depletion and accumulation of charge is not uniform through the lateral extension of the molecule. \nThis behavior is consistent with the level alignment predicted by the PBE Kohn-Sham energy levels, as shown in Fig. S14 in the SI. However, we note that quantitative values of charge transfer are often inaccurate at this level of theory, as characterized in Refs. \\cite{Egger_2015, Liu_2017}. Optimally tuned range-separated hybrid functionals would yield more accurate values, but their computational cost is prohibitive for the use in this whole database. Nevertheless, hybrid-functional calculations of selected conformers (see SI, Fig. S15) confirm the qualitative trend. Therefore, we conclude that the protonation state again critically impacts these systems, in this case by qualitatively changing the redistribution of electronic charge.   \n\n\n\\begin{figure}[ht!]\n \\centering\n \\includegraphics[width=0.65\\textwidth]{Figure_DehydroEnergies.jpg}\n \\caption{Energy differences upon hydrogen dissociation for selected conformers (see text and SI) of Arg and Arg-H$^+$ on all metallic surfaces. $\\Delta E = E_{\\textbf{dep}}-E$, where $E_{\\textbf{dep}}$ is the total energy of the dissociated structure after optimization (including the adsorbed hydrogen) and $E$ the energy of the optimized intact structure. A negative $\\Delta E$ indicates that deprotonation is favored.}\n \\label{fig:deprotonation}\n \\end{figure}\n\n\nIt was observed experimentally that amino acids can undergo deprotonation on reactive surfaces \\cite{BARLOW1998322, BarlowRaval2004, METHIVIER201588, MATEOMARTI2002191, MARTI200485, EralpTrolle2010}. \nHere we also investigated whether deprotonation of Arg and Arg-H$^+$  was favorable on any of the surfaces studied here, the details of the procedure can be found in the SI. The results are summarized in Fig. \\ref{fig:deprotonation}. They show that deprotonation of Arg-H$^+$ is favorable in Cu(111), such that Arg-H$^+$ would be predominantely deprotonated. The population of deprotonated Arg on Cu(111) could reach about 0.1\\% of the molecules at room temperature, from a simple Arrhenius estimation. Given that we have not observed any spontaneous dissociation upon optimization of Arg-H$^+$ on Cu(111), we conclude that, although favorable, this dissociation of H does not occur without a barrier. In all other surfaces, the barrier for dissociation would be rather high for both molecules.\n\n\n\\section{Conclusions}\n\nIn this paper, we have characterized the conformational space of the arginine amino acid in its neutral and protonated form in different non-biological environments, i.e. in isolation and in contact with metallic surfaces. In particular, we have analyzed how and why different parts of the conformational space become accessible or excluded depending on the protonation state and the environment, showing the importance of bond formation and charge rearrangement in these systems. \n\nThis study included the construction of a database based on thousands of structures optimized by density-functional theory including dispersion interactions. The construction of this database is a result in itself. We found, for example, that it is advantageous to start from a comprehensive sampling of the conformational space of the least-constrained molecular form, which in our case was the neutral Arg amino acid in the gas-phase. This is evidenced by the fact that in our low-dimensional projections, all low-energy conformers we observe on the surfaces, for both Arg and Arg-H$^+$ lie among structural conformations that were already present in the gas-phase sampling of Arg, albeit often with high relative energies. In addition, we find that within Cu, Ag, and Au surfaces, relative energies between different conformers are largely preserved when changing the substrate.\n\nIn recent years much effort has been invested in developing and improving classical force fields (FF) for simulation of organic-inorganic interfaces, including peptides on surfaces \\cite{Iori2009, Wright2013, Heinz2013}. However, as we discuss in the SI, the existing force fields are not accurate for these scenarios where amino acids are in contact with metal surfaces and under vacuum conditions. Nevertheless, these situations are extremely relevant for scanning tunneling microscopy experiments \\cite{Rauschenbach2017, Rauschenbach2016, Abb2016, KoslowskiThesis} and other technological applications \\cite{WangGiovanni2013, GuoCahen2016}. As we illustrate in the SI, even though these force-fields can sample the relevant areas of conformational space, they are not able to capture consistent energy hierarchies. Additionally, the molecular chemical groups show a preference to adsorb on different surface sites, which could have considerable impact on self-assembly studies. Databases such as this one will serve as an important source of data for further parametrization and improvement of these potentials. \n\nRegarding the structural space of Arg and Arg-H$^+$ adsorbed on (111) surfaces of Cu, Ag and Au, we have learned the following:\nThe adsorption of Arg leads to the formation of strong bonds to the surface that involve mostly the carboxyl and amino groups. This stabilizes the protomer that we label \\textbf{P3} in this work, where the carboxyl group is deprotonated and the side chain is protonated. This is different to the dominant protomer in the gas phase \\textbf{P1}. The bonds to the surface sterically constrain the conformations of this molecule, thus decreasing the amount of structures\nwith respect to the numbers observed in the gas phase. \nThe molecule also donates electrons to the surface, becoming slightly positively charged. \nWe do not observe fully extended structures lying on the surface and most conformers exhibit intramolecular H-bonds. The majority of conformers of Arg in the low-energy region adsorbs with the C$_{\\alpha}$H chiral center pointing the hydrogen atom away from the surfaces.\n\nArginine in its protonated form, i.e. Arg-H$^+$ is the most abundant form of this amino-acid in biological environments, where it typically adopts the zwitterionic protomer \\textbf{P7}. \nIn the gas-phase, we observe that the non-zwitterionic state \\textbf{P6} is dominant and that the addition of the proton decreases the number of allowed conformations with respect to isolated Arg due to the added electrostatic interactions, and the passivation of the carboxyl group that would otherwise be involved in intramolecular H-bonds. \nUpon adsorption to the metallic surfaces, we observe that the protomer \\textbf{P6} is still dominant and that there are no strong bonds formed to the surface. In addition, this molecule receives electrons from the surface, thus becoming less positively charged. Both effects conspire to yield a homogeneous (flat) molecule-surface interaction and a relatively high population of different structures in the low-energy range. Contrary to Arg, most low-energy conformers of Arg-H$^+$ adsorb with the C$_{\\alpha}$-H chiral center pointing the hydrogen atom towards to the surfaces. Finally, through the calculation of dissociation energies, we also conclude that the deprotonation of Arg-H$^+$ is energetically favorable only on Cu(111). \n\nOur observations regarding the preferred protomers and deprotonation propensities discussed above are consistent with the observations in the literature that the adsorption of amino acids in their anionic and deprotonated form is common on reactive metals like Cu(111) \\cite{Costa:2015:IMPORTANT}. \nOne pronounced difference that we find among surfaces is the average adsorption height of the molecules: They follow the trend Cu(111) $<$ Ag(111) $<$ Au(111), and Arg is always closer than Arg-H$^+$ to the same respective surface.\n\nThe set of electronic-structure calculations presented here show that a flexible amino-acid like Arginine presents a rich conformational space involving different protomeric states and molecule orientations with respect to the surface, allied to a complex charge rearrangement. Going forward, it is clear that the likes of this study based solely on DFT cannot become a routine method due to the elevated computational cost. Addressing the whole breadth of amino acids as well as self assembly of these structures on surfaces will profit from this study as a benchmark and a means to develop models, possibly based on different machine-learning techniques \\cite{Todorovic2019, Oganov2009, Dieterich2010, fafoom, genarris, HORMANN2019, Olsson8265}, that can bypass the cost of thousands of DFT structure optimizations.\n\n\n\\section*{Availability of the data}\nThe data presented here is has been uploaded to the NOMAD repository \\cite{DOI_of_the_data_set_at_NOMAD}.\n\n\n\\section*{Acknowledgements}\n This work was supported by the Max Planck-EPFL Center for Molecular Nanoscience and Technology. We acknowledge Michele Ceriotti for insightful discussions and critical comments to the manuscript. We also acknowledge fruitful discussions with Yair Litman, Stephan Rauschenbach and Sabine Abb. MR also acknowledges support from the Max Planck Research Network on Big-Data-Driven Science (BigMAX).\n\n\n\n\\section{Details of the calculations}\n\nFor Cu, Ag and Au, the bulk lattice constants were determined by optimizing the fcc unit cell. The convergence criteria were set to 0.001 eV\/\\AA~ for the final forces, 10$^{-4}$ e\/Bohr$^{3}$ for the charge density, and 10$^{-5}$ eV for the total energy of the system. A 30$\\times$30$\\times$30 k-grid mesh  was used for the sampling of the Brillouin zone. Relativistic effects were considered by the zeroth order regular approximation (ZORA) \\cite{VanLenthe2000, VanWullen1998}. The values obtained with the PBE functional\\cite{Perdew1997} are in good agreement with previous works \\cite{Liu2013, Haas2009} and are shown in Table \\ref{tbl:Bulk_constants}. In that Table, we compare these values with lattice constants obtained when including pairwise van der Waals dispersion from the original Tkatchenko-Scheffler scheme (+vdW)\\cite{Tkatchenko2009} and from the one that includes an effective electronic screening optimized for metallic surfaces (+vdW$^{\\text{surf}}$)\\cite{Ruiz2016}\\footnote{We here used the original parameters published in Ref. \\cite{Ruiz2016}.}. \n\n\\begin{table}[h!]\n\\center\n \\caption{Lattice constants (in \\AA) of bulk metals determined with the PBE, PBE+vdW and PBE+vdW$^\\text{surf}$ functionals (\\textit{light} settings). }\n \\label{tbl:Bulk_constants}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nMethod     & Cu & Ag & Au \\\\\n\\hline\nPBE                   & 3.633     & 4.156   & 4.157   \\\\\n\\hline\nPBE+vdW               & 3.545     & 4.077     & 4.114               \\\\\n\\hline\nPBE+vdW$^{\\text{surf}}$ & 3.604     & 4.022   & 4.173    \\\\\n\\hline\nExp              & 3.603      & 4.069     & 4.065      \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\center\n\\caption{Fermi energies calculated with the PBE functional for the 4-layer slabs with (111) surface orientation used in our calculations of the binding energies of Arg-H$^+$ to the different surfaces. All values in eV.}\n\\label{tbl:fermi}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n& Cu & Ag  & Au \\\\ \\hline\nSlab $E_f$ & -4.73     & -4.30     & -5.02   \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\center\n\\caption{Relative binding energies (in eV) of relaxed Arg@Cu for different surface unit cell sizes with a 8$\\times$8$\\times$1 k-grid for the cell sizes less than 10$\\times$12 and 4$\\times$4$\\times$1 for the 10$\\times$12 unit cell. All numbers are reported with respect to the binding energy for the structure A modelled with a $5\\times6$ surface unit cell.}\n\\label{tbl:surf_unit_cell_ArgCu}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nsize     & A & B & C \\\\\n\\hline\n5$\\times$6                   & 0.000     &  0.011   & 0.216  \\\\\n\\hline\n6$\\times$6                   & -0.011     &   -0.013    & 0.190              \\\\\n\\hline\n6$\\times$7                   & -0.021     &  -0.030     & 0.174    \\\\\n\\hline\n10$\\times$12                 &   -0.048     &  -0.053  &       0.151  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{Conv_unit_cell_ArgCu.jpg}\n\\caption{Structures that were used for the surface unit cell size convergence test of Arg@Cu. Image unit cell size is $5 \\times 6$.}\n\\label{fig:surfaceunitcellArgCu}\n\\end{figure}\n\n\\begin{table}[h!]\n\\center\n \\caption{Relative binding energies (in eV) of relaxed Arg-H$^+$@Cu for different surface unit cell sizes with a 8$\\times$8$\\times$1 k-grid for the cell sizes less than 10$\\times$12 and 4$\\times$4$\\times$1 for the 10$\\times$12 unit cell. All numbers are reported with respect to the binding energy for the structure A modelled with a $5\\times6$ surface unit cell.}\n\\label{tbl:surf_unit_cell_ArgHCu}\n\\begin{tabular}{|r|c|c|c|}\n\\hline\nsize     & A & B & C \\\\\n\\hline\n5$\\times$6                   & 0.000     &  0.080   & 0.035  \\\\\n\\hline\n6$\\times$6                   & -0.050     &   0.041    & -0.017              \\\\\n\\hline\n6$\\times$7                   & -0.055     &  0.029     & -0.033    \\\\\n\\hline\n10$\\times$12                 & -0.044      & -0.007  &   -0.057     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{Conv_unit_cell_ArgHCu.jpg}\n\\caption{Structures that were used for surface unit cell size convergence test for Arg-H$^+$@Cu. Image unit cell size is $5 \\times 6$.}\n\\label{fig:surfaceunitcellArgHCu}\n\\end{figure}\n\\clearpage\n\n\\section{Family Classification According to Hydrogen Bond Patterns}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{F4_Hbond_Notations.jpg}\n\\caption{Labeling of all H-bond patterns considered in this manuscript.}\n\\label{fig:hbonds}\n\\end{figure}\n\\clearpage\n\\section{Structure space representation}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.99\\linewidth]{Many_conf_similar.jpg}\n\\caption{Representative conformers with similar backbone structure but different H-bonds within the molecule. The different H-bond pattern can cause energy differences of up to 0.2 eV for similar structures, as discussed in the main text.}\n\\label{fig:hbonds}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.99\\linewidth]{Families_on_surfaces.jpg}\n\\caption{Projection of Arg and Arg-H$^+$ conformers adsorbed on the different metalic surfaces on the low-dimensional map of gas-phase Arg, colored according to the H-bond pattern.}\n\\label{fig:hbonds}\n\\end{figure}\n\n\\begin{table}[]\n\\caption{Number of different families within 0.1\/0.5 eV energy range for different systems.}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n           & Arg    & Arg-H$^+$ & \\multicolumn{3}{c|}{Arg} & \\multicolumn{3}{c|}{Arg-H$^+$} \\\\ \\hline\nAtomnames    &        &      & Cu     & Ag     & Au     & Cu      & Ag     & Au     \\\\ \\hline\nNO         & 0 \/599 & 0\/2  & 0\/2    & 0\/0    & 0\/0    & 1 \/162  & 0\/124  & 0\/98   \\\\ \\hline\nN-N$\\varepsilon$        & 0 \/70  & 0\/11 & 5\/10   & 0\/9    & 1\/9    & 4 \/80   & 3\/79   & 3\/77   \\\\ \\hline\nN-N$\\eta1$        & 7 \/87  & 0\/20 & 0\/11   & 0\/12   & 0\/13   & 0 \/78   & 0\/78   & 0\/73   \\\\ \\hline\nN-N$\\varepsilon$, O1-N$\\eta1$   & 2 \/11  & 2\/4  & 1\/2    & 2\/6    & 4\/6    & 0 \/4    & 0\/4    & 0\/5    \\\\ \\hline\nO1-N$\\varepsilon$        & 2 \/115 & 0\/31 & 6\/56   & 8\/62   & 9\/56   & 11\/152  & 6\/146  & 6\/140  \\\\ \\hline\nO1-N$\\eta1$        & 16\/237 & 0\/37 & 0\/66   & 0\/70   & 0\/71   & 5 \/135  & 4\/115  & 5\/109  \\\\ \\hline\nO1-N$\\varepsilon$, O1-N$\\eta1$   & 5 \/27  & 0\/2  & 0\/1    & 0\/1    & 0\/2    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\varepsilon$   & 0 \/5   & 0\/1  & 0\/0    & 0\/0    & 0\/0    & 0 \/1    & 0\/1    & 0\/1    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\eta1$   & 0 \/8   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\varepsilon$, N-N$\\eta1$   & 0 \/8   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\varepsilon$, O1-N$\\varepsilon$   & 0 \/7   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nN-N$\\eta1$, O1-N$\\eta1$ & 0 \/2   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\nO1-N$\\eta1$, O2-N$\\eta2$   & 0 \/3   & 0\/0  & 0\/0    & 0\/0    & 0\/0    & 0 \/0    & 0\/0    & 0\/0    \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\\section{Harmonic free energies}\nFree energies were calculated in the harmonic approximation \\cite{McQuarrie, Fultz2010} for selected molecules adsorbed on surfaces within 0.1 eV range.\n\n$$\nF(\\textrm{T}) = E_{\\textrm{PES}} + F_\\textrm{vib}(\\textrm{T}) + F_\\textrm{rot}(\\textrm{T}),\n$$\n\nwhere $E_{\\textrm{PES}}$ is the total energy obtained with DFT (PBE+vdW$^{\\textrm{surf}}$ functional), and\n\n$$\nF_{\\mathrm{vib}}(\\mathrm{T})=\\sum_{i}^{3 N-6}\\left[\\frac{\\hbar \\omega_{i}}{2}+k_{\\mathrm{B}} T \\ln \\left(1-e^{-\\beta \\hbar \\omega_{i}}\\right)\\right],\n$$\n\nwhere N is the total number of atoms in the molecule (metal atoms were not displaced and were taken into account in external field), $k_\\textrm{B}$ is Boltzmann constant, $T$ is the temperature, $\\omega_i$ are vibrational frequencies obtained by diagonalization of Hessian matrix with use of developing version of phonopy-FHI-aims \\cite{phonopy, phonopy-fidaynyan} and\n \n$$\nF_\\textrm{rot}(\\textrm{T}) =-k_\\textrm{B} T \\textrm{ln} \\left[ \\frac{\\sqrt{\\pi}}{\\sigma} \\left(  \\frac{8\\pi^2 I k_\\textrm{B} T}{h^2}  \\right) \\right],\n$$\nwhere $I$ is the moment of inertia of the molecule obtained after diagonalization of the inertia tensor of the molecule. For the adsorbed conformers, rotational contributions are completely neglected since rotation around all principal axes of the molecule become internal vibrational modes of the system.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.9\\textwidth]{Free_energies_300.jpg}\n\\caption{Harmonic free energies calculated for adsorbed structures within the lowest 0.1 eV total-energy range. E$_{\\textbf{PES}}$ corresponds to the total energy of the system obtained at DFT level and F$_{\\textbf{harm}}$ corresponds to the free energy of the system at 300 K calculated as described above.}\n\\label{sup:Harmonic_free_energies_300}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{F11_UpDown_sketch.jpg}\n\\caption{Low dimensional maps of Arg and Arg-H$^+$ adsorbed on Cu(111), Ag(111) and Au(111) color-coded with respect to the orientation of the C$_\\alpha$H group. Blue correspond to \\textit{up} orientation and red correspond to \\textit{down} orientation of the C$_\\alpha$H group.}\n\\label{fig:UpDown_map}\n\\end{figure}\n\n\\clearpage\n\\section{Charge rearrangement while on the surface}\n\n\n\\begin{table}[ht]\n\\caption{Calculated charge on the molecule with use of Hirshfeld partial charge analysis and by integration of the electron density difference in the molecular region. Values are in electrons. Conformations are depicted in the following Figures S8-13.}\n\\center\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\nConformer & Hirshfeld & Integral & Conformer & Hirshfeld & Integral \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Cu}     & \\multicolumn{3}{c|}{Arg-H$^+$@Cu}     \\\\ \\hline\na     & 0.11      & 0.19     & a     & 0.29      & 0.85     \\\\ \\hline\nb     & 0.03      & 0.30     & b     & 0.30      & 0.85     \\\\ \\hline\nc     & 0.04      & 0.31     & c     & 0.31      & 0.84     \\\\ \\hline\nd     & 0.08      & 0.26     & d     & 0.43      & 0.88     \\\\ \\hline\ne     & 0.01      & 0.24     & e     & 0.46      & 0.85     \\\\ \\hline\nf     & 0.11      & 0.30     & f     & 0.38      & 0.82     \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Ag}     & \\multicolumn{3}{c|}{Arg-H$^+$@Ag}     \\\\ \\hline\na     & 0.04      & 0.15     & a     & 0.28      & 0.83     \\\\ \\hline\nb     & -0.08     & 0.23     & b     & 0.30      & 0.83     \\\\ \\hline\nc     & -0.03     & 0.24     & c     & 0.31      & 0.82     \\\\ \\hline\nd     & -0.06     & 0.21     & d     & 0.43      & 0.86     \\\\ \\hline\ne     & -0.13     & 0.16     & e     & 0.46      & 0.85     \\\\ \\hline\nf     & 0.05      & 0.14     & f     & 0.36      & 0.86     \\\\ \\hline\n\\multicolumn{3}{|c|}{Arg@Au}     & \\multicolumn{3}{c|}{Arg-H$^+$@Au}     \\\\ \\hline\na     & 0.06      & 0.05     & a     & 0.32      & 0.86     \\\\ \\hline\nb     & -0.01     & 0.29     & b     & 0.29      & 0.86     \\\\ \\hline\nc     & 0.00      & 0.30     & c     & 0.34      & 0.85     \\\\ \\hline\nd     & -0.10     & 0.25     & d     & 0.48      & 0.91     \\\\ \\hline\ne     & 0.01      & 0.23     & e     & 0.49      & 0.90     \\\\ \\hline\nf     & 0.06      & 0.31     & f     & 0.43      & 0.92     \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgCu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Cu(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgCu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgAg.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Ag(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgAg}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgAu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg on Au(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgAu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHCu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Cu(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHCu}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHAg.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Ag(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHAg}\n\\end{figure}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{Electron_density_diff_ArgHAu.jpg}\n\\caption{Side and top views of the adsorbed structures of Arg-H$^+$ on Au(111). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3.}\n\\label{sup:Charge_rearrangement_ArgHAu}\n\\end{figure}\n\nIn order to take a look in electronic level alignments of interface system after adsorption the projected, angular-momentum resolved partial density of states (pDOS) averaged over all atoms of each species  were calculated and normalized per molecule and per surface respectively. For corresponding isolated molecular geometry HOMO and LUMO levels were calculated and plotted together with interface pDOS. These calculations were performed with higher number of k-grid points: 6x6x1. Gaussian broadening was chosen to be 0.05. \n\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=0.8\\textwidth]{DOS_PDOS.jpg}\n\\caption{Projected densities of states of the lowest energy structures on each surface. Filled area corresponds to the occupied states below highest occupied state (VBM) of the whole system. HOMO (black solid line) and LUMO (black dashed line) are the states of the corresponding gas-phase molecular conformer calculated with the same geometry as it adopts when adsorbed. The Fermi energy of the pristine slab is depicted with blue dashed line.} \n\\label{sup:DOS_PDOS}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=1.0\\textwidth]{PBE0.jpg}\n\\caption{Side and top views of the adsorbed structures of a) Arg on Cu(111) (conformer c in Fig. S8) and b) Arg-H$^+$ on Cu(111) (conformer a in Fig. S11). Dashed black lines correspond to: average z position of the atoms in the lowest layer of the surface (left), average z position of  atoms in the highest layer of the surface (middle), centre of the mass of the molecule (right). Red\/blue solid lines (and also red\/blue regions) correspond to the electron density accumulation\/depletion, calculated as discussed in the manuscript in the section 2.3 with PBE0 functional.} \n\\label{sup:DOS_PDOS}\n\\end{figure}\n\n\\clearpage\n\\section{Deprotonation on the surfaces}\n\nIn Arg, we found it most favorable to detach the proton from the guanidino group, while for Arg-H$^+$, it was most favorable to detach the proton from the carboxyl group. In both cases we note that the final adsorbed species is a hydrogen, i.e., it does not carry a positive charge. We chose three representative conformers  at each surface: the lowest energy structure and other two with different H-bonds within the molecule. We placed the detached proton at a distance of at least 2.5 \\AA~ from the molecule and fully optimized the dissociated structures. Comparing the energy difference between the final and initial states gives a lower limit for the dissociation barrier.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.9\\textwidth]{Deprotonation.jpg}\n\\caption{Representative structures that were analyzed for the calculation of the deprotonation energies. Colored structures represent the deprotonated relaxed structure. The green translucent structures represent the initial structure from which hydrogen was removed and placed on surface. The hydrogen that was removed is highlighted in bright green. $\\Delta E$ (see main text) is also reported in each panel.}\n\\label{sup:Deprotonation}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\includegraphics[width=0.9\\textwidth]{Deprotonation_ver_0.jpg}\n\\caption{All structures that were analyzed for the calculation of the deprotonation energies. $\\Delta E$ (see main text) is also reported in each panel.}\n\\label{supfig:Deprotonation_v0}\n\\end{figure}\n\n\\clearpage\n\n\\section{Comparison of DFT with INTERFACE-FF}\n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=1.05\\textwidth]{Energies_dft_vs_FF.jpg}\n  \\caption{Comparison of the relative energies obtained from DFT optimized structures and the same structures after post-relaxation in with the INTERFACE force field. Dots on the diagonal line represent an optimal correlation. The red area marks structures that lie in the lower 0.5 eV energy range in DFT but above the 0.5 eV energy range in INTERFACE-FF. The green area marks the structures that are in the lower 0.5 eV energy range regardless of the level of theory. The grey area marks the structures that are above the 0.5 eV energy range in DFT but below the 0.5 eV energy range in INTERFACE-FF.}\n  \\label{sup:RelEn_ArgHCu_DFTvsFF}\n\\end{figure}\n\n\n\nSelected local minima obtained at DFT level of theory were optimized with the  INTERFACE-FF\\cite{Heinz:2016:Review} \nusing the NAMD package \\cite{Phillips2005}. Calculations were performed with periodic boundary conditions with the same cell size and number of Cu atoms as in the DFT calculations. \nWe obtained parameters for certain protonation states as described in the following.\nFor the calculation of Arg, two protomers \\textbf{P1} and \\textbf{P3} (as denoted in the main text) had to be prepared.\nThey are called ``ARN'' (P1)  and  ``ARZ'' (P3) in the topology file that is provided. \n\nThe parametrization of\n``ARZ'' proceeded by taking the C-terminus in the deprotonated form (PRES CTER) and the N-terminus in the neutral form (PRES NNEU) from \\texttt{top-all36-prot.rtf}, such that the protonation is COO-NH2 with total charge 0. The partial charge of the guanidino group is +1. Other parameters were taken from ARG (\\texttt{top-all22-prot-metals.inp}) which is in the protonated form, by default, in CHARMM force field.\n\n\\begin{figure}[htbp]\n  \\centering\n    \\includegraphics[width=0.95\\textwidth]{FF_DFT_sketchmaps.jpg}\n  \\caption{Low-dimensional map of the conformational space of the Arg and Arg-H$^+$ molecules adsorbed on the Cu(111) surface. The map was optimized considering all DFT and INTERFACE-FF structures. Green dots represent conformations obtained at DFT level of theory and red dots represent conformations obtained after geometry optimization with INTERFACE-FF. Close proximity of the dots reflects their structural similarity.}\n  \\label{sup:ArgCu_moleculeDFTvsFF_Sketchmap}\n\\end{figure}\n\nThe parametrization of ``ARN'' proceeded by taking the parameters for neutral C-terminus and N-terminus (PRES CNEU and PRES NNEU from \\texttt{top-all36-prot.rtf}) and the deprotonated methyl-guanidinium group (RESI MGU2) from \\texttt{par-all36-cgenff.rtf}. Atom types and parameters for MGU2 were copied from \\texttt{par-all36-cgenff.rtf}. The missing parameters related to joining MGU2 with the rest of Arg were manually added to the topology file. They were obtained from the corresponding values appearing in the protonated Arg, assuming that CG331 == CT2, NG311 == NC2, HGPAM1 == HC, NG2D1 == NC2, CG2N1 == C, where needed. The partial charge of CD atom was manually adjusted (decreased by 0.1) in order to have a neutral molecule with neutral guanidino group. Parameters for the neutral C-terminus and N-terminus were taken from the  \\texttt{top-all36-prot.rtf} file (PRES CNEU and PRES NNEU) and manually added to the customized file of topology. \n\n\n\nArgH named as ``ARX'' has total charge +1 (partial charge of guanidino group is also +1) and COOH-NH2 termini which is neutral. All the other parameters were directly taken from the INTERFACE-FF \\texttt{all22-prot-metals} topology and parameter files. \n\nWe conclude from Fig. \\ref{sup:RelEn_ArgHCu_DFTvsFF} that DFT (PBE+vdW$^{\\text{surf}}$) and the INTERFACE-FF yield very different energy hierarchies. However, from Fig. \\ref{sup:ArgCu_moleculeDFTvsFF_Sketchmap}, we conclude that both levels of theory represent a similar conformational space. However, Table \\ref{sup:table-ff-dft-adsorption-sites} shows that DFT and the FF yield different adsorption site preferences for the amino and carboxyl groups. In particular, DFT predicts that O will adsorb almost exclusively on top sites, consistent with the accepted adsorption site preference of CO groups on the pristine Cu(111) surface. The FF predicts a larger population of other adsorption sites, in particular hollow sites, compared to DFT. \n\n\\begin{table}[htbp]\n\\center\n\\caption{Surface site adsorption preferences of chosen chemical groups in Arg and Arg-H$^+$. All numbers are reported as a percentage of the total number of conformers optimized with DFT (PBE+vdW$^{\\text{surf}}$) and the INTERFACE force field. \\label{sup:table-ff-dft-adsorption-sites}}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n\\hline\n                & \\multicolumn{4}{c|}{Arg@Cu(111)}               & \\multicolumn{4}{c|}{Arg-H$^+$@Cu(111)}                     \\\\ \\hline\n                & \\multicolumn{2}{c|}{Amino} &  \\multicolumn{2}{c|}{Carboxyl}    & \\multicolumn{2}{c|}{Amino} & \\multicolumn{2}{c|}{Carboxyl} \\\\ \\hline\nAdsorption site & DFT          & FF          & DFT        & FF   & DFT          & FF          & DFT            & FF           \\\\ \\hline\nTop             & 80           & 53          & 76         & 48   & 59           & 50          & 70             & 45           \\\\ \\hline\nBridge          & 9            & 18          & 14         & 18   & 18           & 20          & 15             & 22           \\\\ \\hline\nHollow-FCC      & 5            & 13          & 4          & 17   & 13           & 15          & 7              & 16           \\\\ \\hline\nHollow-HCP      & 6            & 16          & 5          & 17   & 10           & 15          & 9              & 18           \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect1}\n\nIn recent years there has been a revived interest in 2D shell models because of unconventional materials and extremely small aspect to thickness ratio, such as for instance thin polymeric films or biological membranes. For classical engineering materials and for non-extreme aspect to thickness ratios, available 3D FEM-Codes may readily be used such that the need for a truly 2D shell model does not arise anymore. However, for ultrathin specimens the application of a 3D  constitutive law is not  clear at all.  In these extreme cases one is led to employ a 2D shell model.\nThis paper is concerned with one such model, the geometrically non-linear resultant theory of shells. We consider the 6-parameter model of shells which involves two independent kinematic fields: the translation vector field and the rotation tensor field (in total 6 independent scalar kinematic variables). This theory of shells is one of the most general, and it is also very effective in the treatment of complex shell problems, as can be seen from the works \\cite{Pietraszkiewicz10,Eremeyev11,Pietraszkiewicz11}, among others. The resultant  6-parameter theory of shells was originally proposed by Reissner \\cite{Reissner74} and it has been considerably developed subsequently. An account of these developments and main achievements have been presented in the books of Libai and Simmonds \\cite{Libai98} and    Chr\\'o\\'scielewski, Makowski and  Pietraszkiewicz \\cite{Pietraszkiewicz-book04}.\nIn this approach,\nthe 2D equilibrium equations and static boundary conditions of the shell are derived exactly\nby direct through-the-thickness integration of the stresses in the 3D balance laws of linear and angular momentum. The kinematic fields are then constructed on the 2D level using the integral identity of the virtual work principle. Following this procedure, the 2D model is expressed in terms of stress resultants and work--averaged deformation fields defined on the shell base surface. It is interesting  that the kinematical structure of 6-parameter shells (involving the translation vector and rotation tensor) is identical to the kinematical structure of Cosserat shells (defined as material surfaces endowed with a triad of rigid directors describing the orientation of points). From this point of view, the 6-parameter theory of shells is related to the shell model proposed initially by the Cosserat brothers \\cite{Cosserat09neu} and developed by many authors, such as  Zhilin \\cite{Zhilin76}, Zubov \\cite{Zubov97}, Altenbach and Zhilin \\cite{Altenbach04}, Eremeyev and Zubov \\cite{Eremeyev08}, B\\^{\\i}rsan and Altenbach \\cite{Birsan10}. Using the so--called derivation approach, Neff \\cite{Neff_plate04_cmt,Neff_plate07_m3as} has established independently a Cosserat--type model for initially planar shells (plates) which is very similar to the  6-parameter resultant shell model. A   comparison between these two models has been presented in the paper \\cite{Birsan-Neff-Plates}, in the case of plates.\n\nOn the other hand, we should mention that the kinematic structure of the 6-parameter shell model is different from the kinematic structure of the so--called \\emph{Cosserat surfaces}, which are defined as material surfaces with one or more deformable directors attached to every point, see \\cite{Naghdi72,Antman95,Rubin00,Rubin04,Altenbach-Erem12,Altenbach-Erem-Review}. For instance, the kinematics of Cosserat surfaces with one deformable director  is characterized also by 6 degrees of freedom (3 for the position of material points and 3 for the orientation and stretch of the material line element through the thickness), which differ essentially from the 6 degrees of freedom in the 6-parameter resultant shell model.\n\nThe topic of existence of solutions for the 2D equations of linear and non-linear elastic shells has been treated in many works.   The results that can be found in the literature refer to various types of shell models and they employ different techniques, see e.g. \\cite{Koiter60,Simo89.1,Simo92,Steigmann08,Steigmann12,Sansour92,Aganovic06,Aganovic07,Tiba02,Davini75,Birsan08,Wisniewski10,Wisniewski12,Ibrahim94,Badur-Pietrasz86}. The method of formal asymptotic expansions is one method of investigation which allows   for the derivation and justification of plate and shell models. The existence theory for linear or nonlinear shells is presented in details in the books of Ciarlet \\cite{Ciarlet97,Ciarlet98,Ciarlet00}, together with many historical remarks and bibliographic references. Another fruitful approach to the existence theory of 2D plate and shell models (obtained as limit cases of 3D models) is the $\\Gamma$-convergence analysis of thin structures, see e.g. \\cite{Neff_Chelminski_ifb07,Neff_Danzig05,Neff_Hong_Reissner08,Paroni06,Paroni06b}. Concerning the geometrically  non-linear 6-parameter theory of elastic shells, there is no existence theorem published in the literature yet, as far as we are aware of. In the case of \\emph{linear} micropolar shells, the existence of weak solutions has been recently proved in \\cite{EremeyevLebedev11}. Existence results for the related (very similar) Cosserat--type model of initially planar shells have been established by Neff \\cite{Neff_plate04_cmt,Neff_plate07_m3as}. In \\cite{Neff_Chelminski_ifb07,Neff_Danzig05,Neff_Hong_Reissner08} the linearized version of this model has been analyzed and compared with the\nclassical membrane and bending plate models given by the Reissner--Mindlin or Kirchhoff--Love theories.\n\nIn the present work, we prove the existence of minimizers for the minimization problem of the total potential energy associated to the deformation of geometrically non-linear 6-parameter elastic shells. We emphasize that our work is not concerned with the derivation of the 2D shell model, but it presents existence results for the well-established 2D theory of 6-parameter elastic shells. It should be mentioned from the beginning that this model refers to shells made of a simple (classical) elastic material, not a generalized (Cosserat or micropolar) continuum. However, the rotation tensor field appears naturally in this theory, in the course of the exact through-the-thickness reduction of the 3D formulation of the problem to the 2D one \\cite{Libai98,Pietraszkiewicz-book04,Eremeyev06}. Thus, in spite of the above mentioned similarity with the kinematics of Cosserat shells, the material of the shell in the resultant 6-parameter model is described as a simple continuum (without any specific microstructure or material length scale). On the other hand, in the case of dimensional reduction of the 3D equations of micropolar shell-like bodies one can obtain the same 6-parameter theory with modified 2D constitutive equations, see e.g. \\cite{Altenbach-Erem09} for the linear case and \\cite{Neff_plate04_cmt,Neff_plate07_m3as,Zubov09} for the nonlinear case, or one can obtain more complex theories as in \\cite{Eringen67}.\n\nFor the proof of existence, we employ the direct methods of the calculus of variations and extend the techniques presented in  \\cite{Neff_plate04_cmt,Neff_plate07_m3as} to the case of general shells (with non-vanishing curvature in the reference configuration). In Section 2 we present briefly the kinematics of general  6-parameter shells and the equations of equilibrium. In Section 3 we give some alternative formulas for the strain tensor and  curvature tensor, which are written in the direct tensorial notation as well as in the component (matrix) notation. These expressions are needed subsequently in the proof of our main result.\nIn Section 4 we formulate the two-field minimization problem for general elastic shells, corresponding to mixed--type boundary conditions. Under the assumptions of convexity and coercivity of the quadratic strain energy function (physically linear material response), we prove the existence of minimizers over a large  set of admissible pairs. Thus, the minimizing solution pair is of class $\\boldsymbol{H}^1(\\omega, \\mathbb{R}^3)$ for the translation vector and   $\\boldsymbol{H}^1(\\omega, SO(3))$ for the rotation tensor. The existence result is valid for general anisotropic elastic shells having arbitrary geometry of the reference configuration.\nSection 5 includes some applications of the existence theorem and discussions of special cases. We present a convenient way to choose the initial directors and the parametrization of the reference surface. Then, we consider separately the cases of isotropic shells, orthotropic shells, and composite layered shells and we present   the respective forms of the strain energy densities. Applying the theorem stated previously, we establish the conditions on the constitutive coefficients that ensure the existence of minimizers in each situation. This analysis shows the usefulness of our theoretical result in the treatment of practical  problems for elastic shells.\n\n\n\n\n\\section{General 6-parameter resultant shells}\\label{sect2}\n\n\n\nConsider a general 6-parameter shell and denote with $S^0$ the base surface of the shell in the reference (initial) configuration and with $S$ the base surface in the deformed configuration. Let $O$ be a fixed point in the Euclidean space and $\\{\\boldsymbol{e}_1,\\boldsymbol{e}_2,\\boldsymbol{e}_3\\}$ the fixed orthonormal vector basis. The reference configuration is represented by the position vector $\\boldsymbol{y}^0$ (relative to the point $O$) of the base surface $S^0$ \\emph{plus} the structure tensor $\\boldsymbol{Q}^{0}$. The structure tensor is a second order proper orthogonal tensor which can be described by an orthonormal triad of directors  $\\{\\boldsymbol{d}^0_1,\\boldsymbol{d}^0_2,\\boldsymbol{d}^0_3\\}$  attached to every point \\cite{Pietraszkiewicz-book04,Eremeyev06}. Thus the reference (initial) configuration is characterized by the functions\n\\begin{equation}\\label{1}\n\\begin{array}{l}\n    \\,\\boldsymbol{y}^0:\\omega\\subset \\mathbb{R}^2\\rightarrow\\mathbb{R}^3,\\qquad\\qquad\\,\\,\\boldsymbol{y}^0=\\boldsymbol{y}^0(x_1,x_2) ,\\\\\n    \\boldsymbol{Q}^{0}:\\omega\\subset \\mathbb{R}^2\\rightarrow SO(3), \\qquad\\,\\,\\, \\boldsymbol{Q}^{0} = \\boldsymbol{d}_i^0 (x_1,x_2)\\otimes \\boldsymbol{e}_i\\,,\n    \\end{array}\n\\end{equation}\nwhere thus $(x_1,x_2)$ are material curvilinear coordinates on the surface $S^0\\,$. Throughout the paper Latin indexes $i,j,...$ take the values $\\{1,2,3\\}$, while Greek indexes $\\alpha,\\beta,...$ the values $\\{1,2\\}$. The usual Einstein summation convention over repeated indexes is employed. We assume that the curvilinear coordinates $(x_1,x_2)\\in\\omega$  range over a bounded open domain $\\omega$ (with Lipschitz boundary $\\partial\\omega$) of the $Ox_1x_2$ plane, see Figure \\ref{Fig1}.\nLet us denote the partial derivative with respect to $x_\\alpha$ by $\\partial_\\alpha f=\\frac{\\partial f}{\\partial x_\\alpha}\\,\\,$, for any function $f$. We designate by $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2 \\}$ the (covariant) base vectors in the tangent plane of $S^0$ and by $\\boldsymbol{n}^0$ the unit normal to $S^0$ given by\n\\begin{equation}\\label{2}\n   \\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0 = \\dfrac{ \\partial\\boldsymbol{y}^0}{\\partial x_\\alpha}\\,,\\qquad \\boldsymbol{n}^0=  \\dfrac{ \\boldsymbol{a}_1\\times\\boldsymbol{a}_2}{ \\| \\boldsymbol{a}_1\\times\\boldsymbol{a}_2\\|}\\,\\,.\n\\end{equation}\nThe reciprocal (contravariant) basis $\\{\\boldsymbol{a}^1,\\boldsymbol{a}^2 \\}$ of the tangent plane is defined by\n$ \\boldsymbol{a}^\\alpha\\cdot\\boldsymbol{a}_\\beta =\\delta^\\alpha_\\beta$ (the Kronecker symbol). We also use the notations\n$$\\boldsymbol{a}_3=\\boldsymbol{a}^3=\\boldsymbol{n}^0, \\quad a_{\\alpha\\beta}=\\boldsymbol{a}_\\alpha\\cdot \\boldsymbol{a}_\\beta\\,,\\quad a^{\\alpha\\beta}=\\boldsymbol{a}^\\alpha\\cdot\\boldsymbol{a}^\\beta,\\quad a=\\sqrt{\\det(a_{\\alpha\\beta})_{2\\times 2}}>0.$$\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{Fig1}\n\\put(-398,43){\\small{$O$}} \\put(-411,20){\\small{$\\boldsymbol{e}_1$}} \\put(-378,45){\\small{$\\boldsymbol{e}_2$}}\n\\put(-432,2){$x_1$} \\put(-316,31){$x_2$}\n    \\put(-415,64){\\small{$\\boldsymbol{e}_3$}} \\put(-372,14){$\\omega$}\n    \\put(-357,88){$\\boldsymbol{y}^0(x_1,x_2)$} \\put(-228,82){$\\boldsymbol{R}(x_1,x_2)$}\n    \\put(-195,60){$\\boldsymbol{y}(x_1,x_2)$} \\put(-410,105){$\\boldsymbol{Q}^{0}(x_1,x_2)$}\n    \\put(-293,130){${S}^0$} \\put(-264.5,145){\\small{$\\boldsymbol{d}^0_1$}} \\put(-264,173){\\small{$\\boldsymbol{d}^0_2$}} \\put(-311,191){\\small{$\\boldsymbol{d}^0_3$}}\n    \\put(-67.5,173.5){\\small{$\\boldsymbol{y} $}} \\put(-33,167){$\\boldsymbol{d}_1$} \\put(-39.5,195){\\small{$\\boldsymbol{d}_2$}} \\put(-81,202){\\small{$\\boldsymbol{d}_3$}}\n\\put(-220,204){$\\boldsymbol{\\chi}( \\boldsymbol{y}^0)$}\n\\put(-220,225){$\\boldsymbol{Q }^e( \\boldsymbol{y}^0)$} \\put(-307,150){\\small{$\\boldsymbol{y}^0$}}\n \\put(-72,120){$ {S}$}\n\\caption{The base surface $S^0$ of the shell in the initial configuration, the base surface $S$ in the deformed configuration, and the fictitious planar reference configuration $\\omega$. The orthonormal triads of vectors $\\{\\boldsymbol{e}_i\\}\\,$, $\\{\\boldsymbol{d}_i^0 \\}$ and $\\{\\boldsymbol{d}_i\\}$ are related through the relations $ \\boldsymbol{d}_i=\\boldsymbol{Q }^e\\boldsymbol{d}_i^0=\\boldsymbol{R}\\boldsymbol{e}_i \\,$ and $\\boldsymbol{d}_i^0=\\boldsymbol{Q}^{0}\\boldsymbol{e}_i\\,$, where $\\boldsymbol{Q}^e  $ is the elastic rotation field, $\\boldsymbol{Q}^{0}$ is the  initial rotation, and $\\boldsymbol{R}$ is the total rotation field.}\n\\label{Fig1}      \n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{Fig2}\n     \\put(-354,14){$\\omega\\times (\\!-\\frac{h}{2}\\,,\\frac{h}{2}) $}\n    \\put(-271,115){$ \\boldsymbol{\\Theta}\\big( \\omega\\times(\\!-\\frac{h}{2}\\,,\\frac{h}{2})\\big) $}\n \\put(-94,105){$ \\boldsymbol{\\varphi}\\big( \\omega\\times(\\!-\\frac{h}{2}\\,,\\frac{h}{2})\\big) $}\n \\put(-294,63){$\\boldsymbol{F}^0\\!=\\!\\boldsymbol{P} $}\n    \\put(-172,16){$ \\bar{\\boldsymbol{F}}$}\n \\put(-150,125){$ \\boldsymbol{F}^e$}\n\\caption{\\small{Multiplicative decomposition of the total deformation gradient $\\,\\bar{F}\\,=\\,F^eF^0\\,$ into the elastic shell deformation gradient $F^e$ and the initial deformation gradient $F^0=P$. Interpretation in terms of reconstructed three-dimensional quantities. The elastic response is governed by $F^e$. The curved initial configuration corresponds to the intermediate stress free configuration in multiplicative plasticity.}}\n\\label{Fig2}      \n\\end{center}\n\\end{figure}\n\n\nFor the deformed configuration of the shell, let $\\boldsymbol{y}(x_1,x_2)$ denote the position vector (relative to $O$) and $\\{ \\boldsymbol{d}_i(x_1,x_2)\\}$ the orthonormal triad of directors attached to the point with initial curvilinear coordinates $(x_1,x_2)$. The deformed configuration is completely characterized by the functions\n\\begin{equation}\\label{3}\n    \\boldsymbol{y}=\\boldsymbol{\\chi}(\\boldsymbol{y}^0),\\qquad \\boldsymbol{Q}^e= \\boldsymbol{d}_i\\otimes \\boldsymbol{d}_i^0\\in SO(3),\n\\end{equation}\nwhere $\\boldsymbol{\\chi}:S^0\\rightarrow\\mathbb{R}^3$ represents the deformation of the base surface and the proper orthogonal tensor field $\\boldsymbol{Q}^e$ is the (effective)  elastic rotation. The displacement vector is defined as usual by $\\boldsymbol{u}= \\boldsymbol{y}-\\boldsymbol{y}^0\\,$.\n\nWe mention that the role of the triads of directors $\\{\\boldsymbol{d}_i^0 \\}$ and $\\{\\boldsymbol{d}_i\\}$ is to determine the structure tensor $\\boldsymbol{Q}^{0}$ and the rotation tensor $\\boldsymbol{Q}^{e}$ of the shell, respectively. Thus, the directors do not describe here any microstructure of the material. According to the derivation procedure of the 6-parameter shell model, the kinematical fields $\\boldsymbol{y}$ and $\\boldsymbol{Q}^e$ are uniquely defined as the work--conjugate averages of 3D deformation distribution over the shell thickness, whose virtual values enter the virtual work principle of the shell (see \\cite{Pietraszkiewicz-book04,Pietraszkiewicz-book02}).\n\nIn view of \\eqref{1} and \\eqref{3}, the deformed configuration can alternatively be characterized by the functions\n$$\\boldsymbol{y}=\\boldsymbol{y}(x_1,x_2)= \\boldsymbol{\\chi}\\big( \\boldsymbol{y}^0(x_1,x_2)\\big),\\qquad\n\\boldsymbol{R}(x_1,x_2)=\\boldsymbol{Q}^e\\boldsymbol{Q}^{0}= \\boldsymbol{d}_i(x_1,x_2)\\otimes \\boldsymbol{e}_i\\in SO(3),$$\nwhere the vector $\\boldsymbol{y}$ and the orthogonal tensor $\\boldsymbol{R}$ are fields defined over $\\omega$.  The  orthogonal tensor field   $\\boldsymbol{Q}^e$ represents the \\emph{elastic rotation} tensor between the reference and deformed configurations \\cite{Pimenta04,Pimenta09}. The tensor $\\boldsymbol{Q}^{0}$ is the     \\emph{initial rotation}  field, while $\\boldsymbol{R}= \\boldsymbol{Q}^e\\boldsymbol{Q}^{0} $ describes the\n\\emph{total rotation} from the fictitious planar reference configuration $\\omega$ (endowed with the triad $\\{\\boldsymbol{e}_i\\}$) to the deformed configuration $S$. We mention that the tensors $\\boldsymbol{Q}^{0}$   and $\\boldsymbol{R}$ are also called the \\emph{structure tensors} of the reference and deformed configurations, respectively \\cite{Pietraszkiewicz-book04,Eremeyev06}.\nThe following relations hold\n\\begin{equation}\\label{4}\n    \\boldsymbol{Q}^e=\\boldsymbol{R}\\boldsymbol{Q}^{0,T}\\,,\\qquad \\boldsymbol{d}_i^0=\\boldsymbol{Q}^{0} \\boldsymbol{e}_i\\,,\\qquad \\boldsymbol{d}_i=\\boldsymbol{Q}^e\\boldsymbol{d}_i^0= \\boldsymbol{R}  \\boldsymbol{e}_i\\,.\n\\end{equation}\nUsually, the initial directors $\\,\\boldsymbol{d}_i^0\\,$ are chosen such that $\\,\\boldsymbol{d}_3^0 = \\boldsymbol{n}^0$ and $ \\boldsymbol{d}_\\alpha^0$ belong to the tangent plane of $S^0$ (see Remark \\ref{rem6}). This assumption is not necessary in general and we do not use it in the proof of our existence result.\n\nLet $\\boldsymbol{F}=\\text{Grad}_s\\boldsymbol{y}=\\partial_\\alpha \\boldsymbol{y}\\otimes\\boldsymbol{a}^\\alpha$ denote the (total) shell deformation gradient tensor. The strong form of the equations of equilibrium for 6-parameter shells can be written in the form \\cite{Eremeyev06}\n\\begin{equation}\\label{5}\n    \\mathrm{Div}_s\\, \\boldsymbol{N}+\\boldsymbol{f}=\\boldsymbol{0},\\qquad \\mathrm{Div}_s\\, \\boldsymbol{M} + \\mathrm{axl}(\\boldsymbol{N}\\boldsymbol{F}^T-\\boldsymbol{F}\\boldsymbol{N}^T)\n    +\\boldsymbol{c}=\\boldsymbol{0},\n\\end{equation}\nwhere $\\boldsymbol{N}$ and $\\boldsymbol{M}$ are the internal surface stress resultant and stress couple tensors of the 1$^{st}$ Piola--Kirchhoff type, while $\\boldsymbol{f}$ and $\\boldsymbol{c}$ are the external surface resultant force and couple vectors applied to points of $S$, but measured per unit area of $S^0\\,$. The operators Grad$_s$ and  Div$_s$ are the surface gradient and surface divergence, respectively, defined intrinsically in \\cite{Gurtin-Murdoch-75,Murdoch-90}. The superscript $(\\cdot)^T$ denotes the transpose and axl$(\\,\\cdot)$ represents the axial vector of a skew--symmetric tensor.\n\nLet $\\boldsymbol{\\nu}$ be the external unit normal vector to the boundary curve $\\partial S^0$ lying in the tangent plane. We consider boundary conditions of the type \\cite{Pietraszkiewicz04,Pietraszkiewicz11}\n\\begin{equation}\\label{6}\n\\begin{array}{l}\n\\boldsymbol{N}\\boldsymbol{\\nu}=\\boldsymbol{n}^*,\\qquad \\boldsymbol{M}\\boldsymbol{\\nu}=\\boldsymbol{m}^*\\qquad\\mathrm{along}\\,\\,\\,\\partial S^0_f\\,,\\\\\n    \\quad\\boldsymbol{y}=\\boldsymbol{y}^* ,\\qquad\\quad\\,\\, \\boldsymbol{R}=\\boldsymbol{R}^* \\qquad\\mathrm{along}\\,\\,\\,\\partial S^0_d\\,,\n    \\end{array}\n\\end{equation}\nwhere $\\partial S^0=\\partial S^0_f\\cup \\partial S^0_d $ is a disjoint partition  of $S^0\\,$ ($\\partial S^0_f\\cap \\partial S^0_d=\\emptyset$) with length($\\partial S^0_d )>0$. Here, $\\boldsymbol{n}^*$ and $\\boldsymbol{m}^*$ are the external boundary resultant force and couple vectors respectively, applied along the deformed boundary $\\partial S$, but measured per unit length of $\\partial S^0_f\\subset\\partial S^0\\,$. We denote by $\\partial \\omega_f$ and $\\partial \\omega_d$ the subsets of the boundary curve $\\partial \\omega$ which correspond to $\\partial S^0_f$ and $\\partial S^0_d$ respectively, through the mapping $\\boldsymbol{y}^0\\,$.\n\nThe weak form associated to these local balance equations for shells has been presented in \\cite{Libai98,Pietraszkiewicz-book04,Pietraszkiewicz04}.\n\n\n\n\n\n\n\\section{Elastic shell strain and curvature measures}\\label{sect3}\n\n\n\nAccording to \\cite{Eremeyev06,Pietraszkiewicz-book04}, the elastic shell strain tensor $\\boldsymbol{E}^e$ in the material representation is given by\n\\begin{equation}\\label{7}\n    \\boldsymbol{E}^e=\\boldsymbol{Q}^{e,T}\\text{Grad}_s\\,\\boldsymbol{y}- \\text{Grad}_s\\,\\boldsymbol{y}^0=\\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}- \\partial_\\alpha \\boldsymbol{y}^0 \\big) \\otimes \\boldsymbol{a}^\\alpha\\,,\n\\end{equation}\nsince $\\text{Grad}_s\\boldsymbol{y}=\\partial_\\alpha \\boldsymbol{y} \\otimes\\boldsymbol{a}^\\alpha\\,$.\nIt is useful to write  the strain tensor $\\boldsymbol{E}^e$ in the alternative form\n\\begin{equation*}\n\\begin{array}{l}\n\\boldsymbol{E}^e=\\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y} - \\boldsymbol{a}_\\alpha\\big) \\otimes \\boldsymbol{a}^\\alpha=  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{a}^\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{a}^3\\big)- \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{a}^i\\big) \\\\\n\\quad\\,\\,\\, =  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big) - {1\\!\\!\\!\\:1 } _3\n\\end{array}\n\\end{equation*}\nor equivalently, since $\\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big)= \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1}$,\n\\begin{equation}\\label{13}\n\\begin{array}{l}\n    \\boldsymbol{E}^e  =  \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1} - {1\\!\\!\\!\\:1 } _3 \\,\\,\\,=\\,\\, \\, \\overline{\\boldsymbol{U}}^{\\,e} - {1\\!\\!\\!\\:1 } _3\\,\\,,\\vspace{4pt}\\\\\n    \\quad\\text{with}\\qquad \\overline{\\boldsymbol{U}}^{\\,e}= \\big(\\boldsymbol{Q}^{e,T}\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1},\n    \\end{array}\n\\end{equation}\nwhere $ {1\\!\\!\\!\\:1 } _3=\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_i$ is the identity tensor and  $\\overline{\\boldsymbol{U}}^{\\,e}$ represents the non-symmetric \\emph{elastic shell stretch tensor}, which can be seen as the 2D analog of the 3D non-symmetric Biot--type stretch tensor \\cite{Neff_Biot07} or the first Cosserat deformation tensor \\cite[page 123, eq. (43)]{Cosserat09neu} for the shell.\nLet us denote by  $\\boldsymbol{P}$  the tensor defined by\n\\begin{equation}\\label{14}\n    \\boldsymbol{P}  =  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i = \\partial_\\alpha \\boldsymbol{y}^0\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\,.\n\\end{equation}\nThen, from \\eqref{13} and \\eqref{14} we get\n\\begin{equation}\\label{14,1}\n\\begin{array}{l}\n    \\boldsymbol{E}^e =\\, \\overline{\\boldsymbol{U}}^{\\,e} - {1\\!\\!\\!\\:1 } _3\\, = \\,\\boldsymbol{Q}^{e,T} \\big( \\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^{e}\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\boldsymbol{P}^{-1}- {1\\!\\!\\!\\:1 } _3\\,,\\vspace{4pt}\\\\\n    \\qquad\\,\\,\\,\\,\\overline{\\boldsymbol{U}}^{\\,e}= \\,\\boldsymbol{Q}^{e,T} \\big( \\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^{e}\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\big)  \\boldsymbol{P}^{-1}.\n    \\end{array}\n\\end{equation}\nIn the sequel, it is useful to write the elastic shell strain tensors   in component form, relative to the fixed tensor basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$. Let $E^e=\\big(E^e_{ij}\\big)_{3\\times 3}$   be the matrix of components for the tensor $\\boldsymbol{E}^e=E^e_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\,$. In general, we decompose any second order tensor $\\boldsymbol{T}$ in the form $\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j$ and denote by  $T=\\big(T_{ij}\\big)_{3\\times 3}$ the matrix of components. Also, for any vector\n$\\boldsymbol{v}=v_{i }\\boldsymbol{e}_i$ we designate by $v=\\big(v_{i }\\big)_{3\\times 1}$ the column matrix of components.\n\\begin{remark}\\label{rem1}\nThe matrix of components $P=\\big(P_{ij}\\big)_{3\\times 3}$ for the tensor defined in \\eqref{14} can be specified in terms of its 3 column vectors as follows\n\\begin{equation}\\label{14bis}\n    P=\\Big(\\,\\partial_1 y^{0}\\,\\Big|\\, \\partial_2 y^{0}\\,\\Big|\\, n^0\\,\\Big)_{3\\times 3}=\\Big(\\,\\nabla y^{0}\\,\\Big|\\, n^0\\,\\Big)_{3\\times 3}= \\Big(\\,a_{1}\\,\\Big|\\, a_{2}\\,\\Big|\\, n^0\\, \\Big)_{3\\times 3}\\,\\,\\,.\n\\end{equation}\nWe mention that the tensor\n$\\boldsymbol{P}$ introduced in \\eqref{14} can be seen as a three-dimensional  (deformation) gradient\n\\begin{equation}\\label{14tris}\n\\begin{array}{c}\n\\boldsymbol{P}= \\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,x_3)_{\\big|x_3=0}\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\Theta}(x_1,x_2,x_3):=\\boldsymbol{y}^0(x_1,x_2)+x_3\\,\\boldsymbol{n}^0(x_1,x_2),\n\\end{array}\n\\end{equation}\nand it satisfies  $\\det\\boldsymbol{P} =  \\sqrt{\\det a_{\\alpha\\beta}}  =a>0$, so that the inverse $\\boldsymbol{P}^{-1}$ exists.\nThe mapping $\\,\\boldsymbol{\\Theta}:\\,\\omega\\times\\big(\\!-\\frac{h}{2}\\,,\\frac{h}{2}\\,\\big)\\rightarrow\\mathbb{R}^3$  has been introduced previously in \\cite{Ciarlet00,Ciarlet05,Neff_plate04_cmt} and employed for the geometrical description of 3D shells ($h$ denotes the thickness of the shell).\n\\end{remark}\nBy virtue of \\eqref{13} and \\eqref{14}, we obtain the following  matrix form for the strain tensor $\\boldsymbol{E}^e$\n\\begin{equation}\\label{15}\n    E^e= \\Big(\\,Q^{e,T}\\partial_1 y\\,\\Big|\\, Q^{e,T}\\partial_2 y\\,\\Big|\\, n^0\\,\\Big) \\,P^{-1} -  {1\\!\\!\\!\\:1 } _3\\,,\n\\end{equation}\nwhere $ {1\\!\\!\\!\\:1 } _3=\\big(\\delta_{ij}\\big)_{3\\times 3}$ is the unit matrix. The matrix $E^e$ can be written equivalently as\n\\begin{equation}\\label{15,1}\n\\begin{array}{l}\n    E^e \\,=\\, Q^{e,T}\\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} -  {1\\!\\!\\!\\:1 } _3\\,,\\qquad\\text{or}\\vspace{4pt}\\\\\n    E^e\\,=\\,\\, \\overline{U}^{\\,e}- {1\\!\\!\\!\\:1 } _3 \\,\\,= \\,\\, Q^{e,T}\\,F^e -  {1\\!\\!\\!\\:1 } _3 \\,\\, = \\,\\, Q^{e,T}\\bar{F}  \\,P^{-1}-   {1\\!\\!\\!\\:1 } _3\\,,\n    \\end{array}\n\\end{equation}\nwith\n\\begin{equation}\\label{15,2}\n\\begin{array}{l}\n    \\overline{U}^{\\,e}= Q^{e,T}\\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} = Q^{e,T}\\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1}\\,,\\vspace{4pt}\\\\\n    F^e:= \\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\,P^{-1} = \\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) \\big(\\nabla \\Theta(x_1,x_2,0)\\big)^{-1}\\,,  \\vspace{4pt}\\\\\n    \\bar{F}\\,\\,:= \\Big(\\,\\partial_1 y\\,\\Big|\\,  \\partial_2 y\\,\\Big|\\, Q^{e} n^0\\,\\Big) = \\Big(\\,\\nabla y\\,\\Big|\\, Q^{e} n^0\\,\\Big) ,\\vspace{4pt}\\\\\n    F^0:=P=\\Big(\\,\\partial_1 y^{0}\\,\\Big|\\, \\partial_2 y^{0}\\,\\Big|\\, n^0\\,\\Big)= \\Big(\\,\\nabla y^0\\,\\Big|\\,   n^0\\,\\Big), \\vspace{4pt}\\\\\n    \\qquad\\qquad\\qquad\\,\\,\\,\\qquad\\bar{F}\\,\\,\\,=\\,\\,F^e\\,\\,F^0\\,.\n    \\end{array}\n\\end{equation}\nIn order to see a parallel with the classical  multiplicative decomposition  into elastic and plastic parts from finite elasto-plasticity \\cite{Neff_Cosserat_plasticity05,HutterSFB02}, we may interpret $F^e$ as an elastic shell mid-surface deformation gradient and $F^0=P$ as an initial deformation gradient. Both are gradients of suitably defined mappings, see Remark \\ref{rem2} and  Figure \\ref{Fig2}, in contrast to the case of elasto-plasticity. In our context, the elastic material response is defined in terms of the elastic part of the deformation, e.g. $\\,E^e\\,=\\, Q^{e,T}\\,F^e -  {1\\!\\!\\!\\:1 } _3 \\,$, cf. \\eqref{15,1}.\n\n\n\n\n\n\\begin{remark}\\label{rem2}\nAlthough the resultant shell model is truly a 2D theory, we may always consider artificially reconstructed three-dimensional quantities. In this sense, similar to the  context of Remark \\ref{rem1}, the tensor $\\,\\bar{\\boldsymbol{F}}=\\partial_\\alpha \\boldsymbol{y}\\otimes \\boldsymbol{e}_\\alpha+  \\boldsymbol{Q}^e\\boldsymbol{n}^0  \\otimes \\boldsymbol{e}_3\\,$, which has the components matrix  $\\,\\bar{F}\\,$ is a three-dimensional deformation gradient\n\\begin{equation}\\label{15,3}\n\\begin{array}{c}\n\\bar{\\boldsymbol{F}}= \\nabla \\, \\boldsymbol{\\varphi}(x_1,x_2,x_3)_{\\big|x_3=0}\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\varphi}(x_1,x_2,x_3):=\\boldsymbol{y}(x_1,x_2)+x_3\\,\\boldsymbol{Q}^e(x_1,x_2) \\boldsymbol{n}^0(x_1,x_2)\\\\\n\\qquad\\qquad\\qquad\\qquad\\,\\,\\, =   \\boldsymbol{y}(x_1,x_2)+x_3\\,\\boldsymbol{Q}^e(x_1,x_2)\\,      \\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,0)\\boldsymbol{e}_3.\n\\end{array}\n\\end{equation}\nHere, the mapping $\\,\\boldsymbol{\\varphi}:\\,\\omega\\times\\big(\\!-\\frac{h}{2}\\,,\\frac{h}{2}\\,\\big)\\rightarrow\\mathbb{R}^3$ is a 3D deformation of the body, in terms of the given 2D quantities $\\boldsymbol{y}(x_1,x_2)$ and $\\boldsymbol{Q}^e(x_1,x_2)$. Similarly,\n\\begin{equation*}\n\\begin{array}{c}\n\\boldsymbol{F}^e:= \\nabla \\, \\boldsymbol{\\varphi}^e\\Big(\\boldsymbol{\\Theta}(x_1,x_2,x_3)_{\\big|x_3=0}\\Big)\\,\\,,\\qquad\\mathrm{with}\\vspace{4pt}\\\\\n \\boldsymbol{\\varphi}^e\\big(\\boldsymbol{\\Theta}(x_1,x_2,x_3)\\big):=\n \\boldsymbol{\\varphi}(x_1,x_2,x_3).\n\\end{array}\n\\end{equation*}\nHowever, we note that $\\bar{\\boldsymbol{F}}$ cannot be interpreted as the  3D deformation\ngradient of the real 3D shell, because in general the initial normals become\narbitrarily curved after deformation.\n\\end{remark}\n\n\nIn terms of the total rotation $\\boldsymbol{R}$ and the initial rotation $\\boldsymbol{Q}^{0}$, the elastic shell strain tensor is expressed by\n\\begin{equation}\\label{8}\n    \\boldsymbol{E}^e=\\boldsymbol{Q}^{0}\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} - \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0 \\big) \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nThen, we have\n$$\\boldsymbol{E}^e=\\boldsymbol{Q}^{0}\\big[\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} \\!\\!-\\! \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big)\\! \\otimes\\! \\boldsymbol{e}_\\alpha\\big] \\big(  \\boldsymbol{e}_i\\otimes \\boldsymbol{a}^i\\big)= \\boldsymbol{Q}^{0}\\big[\\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y}\\!-\\! \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big) \\!\\otimes\\! \\boldsymbol{e}_\\alpha\\big]\n \\boldsymbol{P}^{-1}$$\nwhich can be written in matrix form as follows\n\\begin{equation}\\label{16}\n    E^e= Q^0\\,H \\,P^{-1} \\quad\\text{with}\\quad H:=\\Big(\\,R^T\\partial_1 y- Q^{0,T}\\partial_1 y^{0}\\,\\Big|\\, R^T\\partial_2 y- Q^{0,T}\\partial_2 y^{0}\\,\\Big|\\,\\, 0\\,\\,\\Big)_{3\\times 3}\\, .\n\\end{equation}\\smallskip\n\nOn the other hand, the elastic shell curvature  tensor $\\boldsymbol{K}^e$ in the material description is defined by \\cite{Eremeyev06,Pietraszkiewicz-book04}\n\\begin{equation}\\label{9}\n     \\boldsymbol{K}^e=\\big[\\boldsymbol{Q}^{e,T}\\text{axl}(\\partial_\\alpha \\boldsymbol{R}  \\boldsymbol{R}^T)- \\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^{0}\\boldsymbol{Q}^{0,T})\\big] \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nIn order to write $\\boldsymbol{K}^e$ in a  form more convenient to us, we use relations of the type\n\\begin{equation}\\label{10}\n    \\tilde{\\boldsymbol{Q}}^{ T}\\text{axl}(\\partial_\\alpha\\tilde{ \\boldsymbol{Q}} \\, \\tilde{\\boldsymbol{Q}}^{ T})=  \\text{axl}(\\tilde{\\boldsymbol{Q}}^{ T} \\partial_\\alpha \\tilde{ \\boldsymbol{Q}}  ),\\qquad \\text{axl}(\\tilde{\\boldsymbol{Q}} \\boldsymbol{A}\\tilde{\\boldsymbol{Q}}^{ T}) = \\tilde{\\boldsymbol{Q}}  \\,\\text{axl}(\\boldsymbol{A}),\n\\end{equation}\nwhich hold true for any   orthogonal tensor $\\tilde{\\boldsymbol{Q}} \\in SO(3)$ and any skew--symmetric tensor $\\boldsymbol{A}\\in \\frak{so}(3)$ (see e.g., \\cite{Birsan-Neff-AnnRom12}). Using \\eqref{10} in \\eqref{9} we can write the elastic  curvature tensor $\\boldsymbol{K}^e$ in the equivalent forms\n\\begin{equation}\\label{11}\n    \\boldsymbol{K}^e= \\boldsymbol{Q}^{e,T}\\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^e  \\boldsymbol{Q}^{e,T}) \\otimes \\boldsymbol{a}^\\alpha= \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e ) \\otimes \\boldsymbol{a}^\\alpha ,\n\\end{equation}\nor,\n$$\\boldsymbol{K}^e= \\big[\\, \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e) \\otimes \\boldsymbol{e}_\\alpha\\,\\big]\n \\big(  \\boldsymbol{a}_i\\otimes \\boldsymbol{e}_i\\big)^{-1}  = \\big[\\, \\text{axl}(\\boldsymbol{Q}^{e,T} \\partial_\\alpha \\boldsymbol{Q}^e) \\otimes \\boldsymbol{e}_\\alpha\\,\\big]\n \\boldsymbol{P}^{-1}.\n$$\nThen, the matrix of components $K^e=\\big(K^e_{ij}\\big)_{3\\times 3}$ is given by\n\\begin{equation}\\label{17}\n    K^e= \\Big(\\,\\,\\text{axl}(Q^{e,T}\\partial_1 Q^e) \\,\\,\\Big|\\,\\, \\text{axl}(Q^{e,T}\\partial_2 Q^e) \\,\\,\\Big|\\, \\,0\\,\\,\\Big) \\,P^{-1}.\n\\end{equation}\n\nIf we express $\\boldsymbol{K}^e$ in terms of the total rotation $\\boldsymbol{R}$ and the initial rotation $\\boldsymbol{Q}^{0}$, we get\n\\begin{equation}\\label{12}\n    \\boldsymbol{K}^e= \\boldsymbol{Q}^{0} \\big[ \\text{axl}(\\boldsymbol{R}^T \\partial_\\alpha \\boldsymbol{R}  )- \\text{axl}(\\boldsymbol{Q}^{0,T}\\partial_\\alpha \\boldsymbol{Q}^{0})\\big] \\otimes \\boldsymbol{a}^\\alpha\\,.\n\\end{equation}\nThis relation can be written as\n\\begin{equation}\\label{12,1}\n\\begin{array}{c}\n    \\boldsymbol{K}^e=\\boldsymbol{K} - \\boldsymbol{K}^0,\\qquad \\text{with} \\qquad \\boldsymbol{K}:= \\boldsymbol{Q}^{0}   \\text{axl}(\\boldsymbol{R}^T \\partial_\\alpha \\boldsymbol{R}  ) \\otimes \\boldsymbol{a}^\\alpha\\,,\\vspace{4pt} \\\\\n      \\boldsymbol{K}^0:= \\boldsymbol{Q}^{0}  \\text{axl}(\\boldsymbol{Q}^{0,T}\\partial_\\alpha \\boldsymbol{Q}^{0})  \\otimes \\boldsymbol{a}^\\alpha\\,\\,=\\,\\,\\text{axl}(\\partial_\\alpha \\boldsymbol{Q}^{0}\\,\\boldsymbol{Q}^{0,T}) \\otimes \\boldsymbol{a}^\\alpha\\,,\n    \\end{array}\n\\end{equation}\nwhere the tensor $\\boldsymbol{K}$ is the total  curvature tensor, while $\\boldsymbol{K}^0$ is the initial  curvature (or structure curvature tensor of $S^0$). In view of  \\eqref{12} and  \\eqref{12,1},\nthe matrix $K^e=\\big(K^e_{ij}\\big) $ is given by\n\\begin{equation}\\label{18}\n\\begin{array}{c}\n    K^e\\,\\,=\\,\\,Q^0\\,L\\,P^{-1}\\,\\,=\\,\\,K-K^0\\qquad\\text{with}  \\vspace{4pt}\\\\ L:= \\Big(\\,\\,\\text{axl}(R^T\\partial_1 R)-\\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\Big|\\,\\, \\text{axl}(R^T\\partial_2 R) -\\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\Big|\\, \\,0\\,\\,\\Big)_{3\\times 3}\\, ,\\vspace{4pt}\\\\\n    \\,\\, K= Q^0\\Big(\\,\\,\\text{axl}(R^T\\partial_1 R)  \\,\\,\\Big|\\,\\, \\text{axl}(R^T\\partial_2 R) \\,\\,\\Big|\\, \\,0\\,\\,\\Big)P^{-1}, \\vspace{4pt}\\\\\n    \\qquad K^0= Q^0\\Big(\\,\\, \\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\Big|\\,\\,  \\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\Big|\\, \\,0\\,\\,\\Big)P^{-1}.\n    \\end{array}\n\\end{equation}\nIn what follows, we shall use the expressions \\eqref{16} and \\eqref{18} of the elastic shell strain measures $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ written with tensor components in the basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$.\n\\begin{remark}\nAs expected, the case of zero strain and bending measures corresponds to a rigid body mode of the shell. Indeed, if $\\boldsymbol{E}^e=\\boldsymbol{0}$ and $\\boldsymbol{K}^e=\\boldsymbol{0}$, then from \\eqref{7} and \\eqref{11} we obtain\n$$\\partial_\\alpha\\boldsymbol{y}=\\boldsymbol{Q}^e\\,\\partial_\\alpha\\boldsymbol{y}^0\\qquad \\mathrm{and}\\qquad \\partial_\\alpha \\boldsymbol{Q}^e=\\boldsymbol{0}.$$\nHence, it follows that $\\boldsymbol{Q}^e$ is constant and\n$$\\boldsymbol{y}=\\boldsymbol{Q}^e \\boldsymbol{y}^0+ \\boldsymbol{c}\\qquad (\\boldsymbol{c}=\\mathrm{constant}),$$\nwhich means that the shell undergoes a rigid body motion with constant translation $\\,\\boldsymbol{c}\\,$ and constant rotation $\\,\\boldsymbol{Q}^e$.\n\\end{remark}\n\\begin{remark}\nIn the case when the base surface $S^0$ of the initial configuration of the shell is planar we may assume that $S^0$ coincides with $\\,\\omega$. In this situation we have $\\,\\boldsymbol{a}_i=\\boldsymbol{e}_i\\,$, $\\boldsymbol{P}=  {1\\!\\!\\!\\:1 } _3\\,$, and\nthe above strain and curvature measures coincide with those defined for the Cosserat model of planar--shells introduced in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\\begin{remark}\nIn view of \\eqref{11} or \\eqref{17}, the elastic shell  curvature tensor $\\,\\boldsymbol{K}^e$ is an analog of the second Cosserat deformation tensor in the 3D theory, see the original Cosserats book \\cite[page 123, eq. (44)]{Cosserat09neu}.\n\\end{remark}\n\n\n\n\n\n\n\\section{Variational formulation for elastic shells}\\label{sect4}\n\n\n\n\nLet us denote the strain energy density of the elastic shell by $W=W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$. According to the hyperelasticity assumption, the    internal surface stress resultant $\\boldsymbol{N}$ and stress couple tensor $\\boldsymbol{M}$ are expressed by the constitutive equations in the form\n\\begin{equation}\\label{19}\n    \\boldsymbol{N}=\\boldsymbol{Q}^e\\,\\dfrac{\\partial\\, W}{\\partial \\boldsymbol{E}^e}\\,\\,,\\qquad \\boldsymbol{M}=\\boldsymbol{Q}^e\\,\\dfrac{\\partial\\, W}{\\partial \\boldsymbol{K}^e}\\,\\,.\n\\end{equation}\nIn this paper we assume that the strain energy density $W$ is a quadratic function of its arguments $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$. Thus, the considered model  is physically linear and geometrically non-linear. The explicit form of the strain energy function $W$ is presented in \\cite{Libai98,Eremeyev06} for isotropic, hemitropic or orthotropic elastic shells.\nIn general, the coefficients of the strain energy function $W$ depend on the structure curvature tensor $\\,\\boldsymbol{K}^0$, see \\cite{Eremeyev06}.\nIn \\cite{Chroscielewski11}, the case of composite (layered) shells is investigated and the expression of the energy density is established. These special cases will be discussed in Section \\ref{sect5}.\n\n\nConsider the usual Lebesgue spaces $\\big(L^p(\\omega),\\|\\cdot\\|_{L^p(\\omega)}\\big)$, $p\\geq 1$, and Sobolev space $\\big(H^1(\\omega),\\|\\cdot\\|_{H^1(\\omega)}\\big)$. We denote by $\\boldsymbol{L}^p(\\omega,\\mathbb{R}^3)$ (respectively $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$) the space of all vector fields $\\boldsymbol{v}=v_i\\boldsymbol{e}_i$ such that $v_i\\in L^p(\\omega)$ (respectively $v_i\\in H^1(\\omega)$). Similarly, we denote the sets\n$    \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})=\\{\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j \\,|\\, T_{ij}\\in H^1(\\omega)\\}$ ,   $\\boldsymbol{H}^1(\\omega,SO(3))= \\{\\boldsymbol{T}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3}) \\,|\\,\\boldsymbol{T}\\in SO(3)\\}$ , $ \\boldsymbol{L}^p(\\omega,\\mathbb{R}^{3\\times 3})=\\{\\boldsymbol{T}=T_{ij}\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j \\,|\\, T_{ij}\\in L^p(\\omega)\\}$ , $ \\boldsymbol{L}^p(\\omega,SO(3))= \\{\\boldsymbol{T}\\in \\boldsymbol{L}^p(\\omega,\\mathbb{R}^{3\\times 3}) \\,|\\,\\boldsymbol{T}\\in SO(3)\\}$.\nThe norm of a tensor $\\boldsymbol{T}$ is defined by $\\|\\boldsymbol{T}\\|^2=\\text{tr}(\\boldsymbol{T} \\boldsymbol{T}^T)=T_{ij}T_{ij}\\,$.\n\n\nConcerning the boundary-value problem \\eqref{5}, \\eqref{6}, we assume the existence of a function $\\Lambda(\\boldsymbol{y},\\boldsymbol{R})$ representing the potential of external surface loads $\\boldsymbol{f}$, $\\boldsymbol{c}$, and boundary loads $\\boldsymbol{n}^*$, $\\boldsymbol{m}^*$ (cf. \\cite{Pietraszkiewicz04}).\n\nWe consider the following two--field minimization problem associated to the deformation of elastic shells: find the pair $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ in the admissible set $\\mathcal{A}$ which realizes the minimum of the functional\n\\begin{equation}\\label{20}\nI(\\boldsymbol{y},\\boldsymbol{R})=\\int_{S^0} W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,\\mathrm{d}S - \\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\qquad\\mathrm{for}\\qquad (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A},\n\\end{equation}\nwhere d$S$ is the area element of the surface $S^0\\,$. The admissible set $\\mathcal{A}$ is defined by\n\\begin{equation}\\label{21}\n    \\mathcal{A}=\\big\\{(\\boldsymbol{y},\\boldsymbol{R})\\in\\boldsymbol{H}^1(\\omega, \\mathbb{R}^3)\\times\\boldsymbol{H}^1(\\omega, SO(3))\\,\\,\\big|\\,\\,\\,  \\boldsymbol{y}_{\\big| \\partial S^0_d}=\\boldsymbol{y}^*, \\,\\,\\boldsymbol{R}_{\\big| \\partial S^0_d}=\\boldsymbol{R}^* \\big\\},\n\\end{equation}\nwhere the boundary conditions are to be understood in the sense of traces.\nThe tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ are expressed in terms of $(\\boldsymbol{y},\\boldsymbol{R})$ through the relations \\eqref{8} and \\eqref{12}. If we write $W=W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)=\\tilde{W}(\\nabla\\boldsymbol{y},\\boldsymbol{R},\\nabla\\boldsymbol{R})$, then referring the integral to the (fictitious reference) domain $\\omega$, the change of variable formula clearly  gives\n\\begin{equation}\\label{22}\n\\begin{array}{l}\n    \\displaystyle{\\int_{S^0}} W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,\\mathrm{d}S=\\displaystyle{\\int_\\omega } W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)\\,a(x_1,x_2)\\, \\mathrm{d}x_1\\mathrm{d}x_2 \\\\\n    \\qquad\\,\\,\\, = \\displaystyle{\\int_\\omega } \\tilde{W}\\big(\\nabla\\boldsymbol{y}(x_1,x_2),\\boldsymbol{R}(x_1,x_2),\\nabla\\boldsymbol{R}(x_1,x_2)\\big)\\, \\det\\big(\\nabla \\boldsymbol{\\Theta} (x_1,x_2,0)\\big)\\, \\mathrm{d}x_1\\mathrm{d}x_2 ,\n    \\end{array}\n\\end{equation}\nwhere $a=\\sqrt{\\det(a_{\\alpha\\beta})}$ is the notation introduced previously. The variational principle for the total energy of elastic shells with respect to the functional \\eqref{20} has been presented in \\cite{Pietraszkiewicz04}, Sect.2. We decompose the loading potential $\\Lambda(\\boldsymbol{y},\\boldsymbol{R})$ additively as follows\n\\begin{equation}\\label{23}\n\\begin{array}{c}\n    \\Lambda(\\boldsymbol{y},\\boldsymbol{R})=\\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R}) + \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R}),\\\\\n    \\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})= \\displaystyle{\\int_{S^0}}\\! \\boldsymbol{f}\\!\\cdot\\! \\boldsymbol{u}\\, \\mathrm{d}S + \\Pi_{S^0}(\\boldsymbol{R}), \\quad\n    \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R})= \\displaystyle{\\int_{\\partial S^0_f}}\\! \\boldsymbol{n}^*\\!\\cdot\\! \\boldsymbol{u}\\, \\mathrm{d}l + \\Pi_{\\partial S^0_f}(\\boldsymbol{R}).\n    \\end{array}\n\\end{equation}\nwhere $\\boldsymbol{u}=\\boldsymbol{y}-\\boldsymbol{y}^0$ is the displacement vector and d$l$ is the element of length along the boundary curve $\\partial S^0_f\\,$. In \\eqref{23}, $\\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})$ is the potential of the external surface loads $\\boldsymbol{f}, \\boldsymbol{c}$, while $\\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{Q}^e)$ is the potential of the external boundary loads $\\boldsymbol{n}^*, \\boldsymbol{m}^*$. The expression of the load potential functions $\\,\\,\\Pi_{S^0}\\,,  \\,\\Pi_{\\partial S^0_f}:\\boldsymbol{L}^2( \\omega ,SO(3))\\rightarrow\\mathbb{R}$ are not given explicitly, but they are assumed to be continuous and bounded operators. Of course, the integrals over $S^0$ and $\\partial S^0_f$ appearing in \\eqref{23} can be transformed like in \\eqref{22} into integrals over $\\omega$ and $\\partial \\omega_f\\,$, respectively.\n\nWe mention that one can consider more general cases of external loads in the definition of the loading potential \\eqref{23}, such as for example tracking loads.\n\n\n\\subsection{Main result: Existence of minimizers}\n\n\n\nThis theorem states the existence of minimizers to the minimization problem \\eqref{20}--\\eqref{23}.\n\\begin{theorem}\\label{th1}\nAssume that the external loads satisfy the conditions\n\\begin{equation}\\label{24}\n    \\boldsymbol{f}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^3),\\qquad  \\boldsymbol{n}^*\\in \\boldsymbol{L}^2(\\partial\\omega_f,\\mathbb{R}^3),\n\\end{equation}\nand the boundary data satisfy the conditions\n\\begin{equation}\\label{25}\n    \\boldsymbol{y}^*\\in\\boldsymbol{H}^1(\\omega ,\\mathbb{R}^3),\\qquad \\boldsymbol{R}^*\\in\\boldsymbol{H}^1(\\omega, SO(3)).\n\\end{equation}\nAssume that the following conditions concerning the initial configuration are fulfilled: $\\,\\boldsymbol{y}^0:\\omega\\subset \\mathbb{R}^2\\rightarrow\\mathbb{R}^3$ is a continuous injective mapping and\n\\begin{equation}\\label{26}\n    \\begin{array}{c}\n   \\boldsymbol{y}^0\\in\\boldsymbol{H}^1(\\omega ,\\mathbb{R}^3),\\qquad \\boldsymbol{Q}^{0}\\in\\boldsymbol{H}^1(\\omega, SO(3)),\n   \\end{array}\n\\end{equation}\n\\begin{equation}\\label{26,1}\n    \\begin{array}{c}\n   \\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0\\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^3)\\quad \\big(\\,\\mathrm{i.e.}\\,\\,\\, \\nabla\\boldsymbol{y}^0\\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^{3\\times 2})\\big), \\vspace{6pt}\\\\\n    \\det\\big(a_{\\alpha\\beta}(x_1,x_2)\\big)\\geq a_0^2 >0\\,\\,,\n    \\end{array}\n\\end{equation}\nwhere $a_0$ is a constant. The strain energy density $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ is assumed to be a quadratic convex function of $(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ and $W$ is coercive, in the sense that  there exists a constant $C_0>0$ with\n\\begin{equation}\\label{26bis}\n    W( {\\boldsymbol{E}^e}, {\\boldsymbol{K}^e})\\,\\geq\\, C_0\\,\\big( \\,   \\|\\boldsymbol{E}^e\\|^2 +  \\|\\boldsymbol{K}^e\\|^2\\,\\big).\n\\end{equation}\nThen, the minimization problem \\eqref{20}--\\eqref{23} admits at least one minimizing solution pair\n$(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in  \\mathcal{A}$.\n\\end{theorem}\n\\begin{remark}\nThe hypotheses \\eqref{26,1} can be written equivalently in terms of the tensor $\\boldsymbol{P}=\\nabla \\, \\boldsymbol{\\Theta}(x_1,x_2,0)$ as\n\\begin{equation}\\label{26,2}\n    \\begin{array}{c}\n   \\boldsymbol{P} \\in \\boldsymbol{L}^\\infty(\\omega ,\\mathbb{R}^{3\\times 3}),\\qquad \\det\\boldsymbol{P} \\geq a_0 >0\\,\\,,\n    \\end{array}\n\\end{equation}\nin view of the relations \\eqref{14} and \\eqref{14bis}. Since $\\boldsymbol{y}^0$ represents the position vector of the reference base surface $S^0$ (which is bounded), the conditions \\eqref{26}$_1$ and \\eqref{26,1}$_1$ can be written together in the form $\\boldsymbol{y}^0 \\in\\boldsymbol{W}^{1,\\infty}(\\omega ,\\mathbb{R}^3)$.\n\\end{remark}\n\\begin{proof}\nWe employ the direct methods of the calculus of variations. We show first that there exists a constant $C>0$ such that\n\\begin{equation}\\label{27}\n    |\\,\\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\,|\\,\\leq\\, \\,\\,C\\,\\big(\\,\\|\\boldsymbol{y}\\|_{H^1(\\omega)}+1\\big),\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in\\mathcal{A}.\n\\end{equation}\nIndeed, since $\\boldsymbol{a}_\\alpha\\in\\boldsymbol{L}^\\infty(\\omega,\\mathbb{R}^3)$ it follows that $a=\\sqrt{\\det(a_{\\alpha\\beta})}\\in{L}^\\infty(\\omega)$.\nWe also have $\\|\\boldsymbol{R}\\|^2=\\text{tr}(\\boldsymbol{R}\\boldsymbol{R}^T)=3$, $\\,\\forall\\boldsymbol{R}\\in SO(3)$. Taking into account the hypotheses \\eqref{24} and the boundedness of $\\,\\,\\Pi_{S^0}\\,$ and $ \\,\\Pi_{\\partial S^0_f}\\,$, we deduce from \\eqref{23} that\n\\begin{equation*}\n \\begin{array}{l}\n     |\\,\\Lambda(\\boldsymbol{y},\\boldsymbol{R})\\,|\\,\\leq\\, |\\, \\Lambda_{S^0}(\\boldsymbol{y},\\boldsymbol{R})\\, | + |\\,\n    \\Lambda_{\\partial S^0_f}(\\boldsymbol{y},\\boldsymbol{R}) \\,|\\,\\leq   \\,\n     C_1\\,\\|\\boldsymbol{y}-\\boldsymbol{y}^0\\|_{L^2(\\omega)} +C_2\\, \\|\\boldsymbol{y}-\\boldsymbol{y}^0\\|_{L^2(\\partial\\omega_f)} \\\\\n     \\qquad +\\,|\\, \\Pi_{S^0}(\\boldsymbol{R})\\,|+ | \\,\\Pi_{\\partial S^0_f}(\\boldsymbol{R})\\,| \\,\\leq \\, C_3 \\|\\boldsymbol{y}\\|_{L^2(\\omega)}+C_4\\|\\boldsymbol{y}\\|_{H^1(\\omega)} +C_5\\,,\n     \\end{array}\n\\end{equation*}\nfor some positive constants $C_k>0$. Then, the inequality \\eqref{27} holds.\n\nIn what follows, we employ the component form of the elastic  strain tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$, written as matrices $E^e$ and $K^e$ in \\eqref{16} and \\eqref{18}, respectively. Let us show next that there exists a positive constant $\\lambda_0>0$ such that\n\\begin{equation}\\label{28}\n    \\|\\,\\boldsymbol{E}^e\\,\\|=\\|\\,E^e\\,\\|\\,\\geq\\, \\lambda_0\\,\\|\\,H\\,\\|\\,,\\qquad  \\|\\,\\boldsymbol{K}^e\\,\\|=\\|\\,K^e\\,\\|\\,\\geq\\, \\lambda_0\\,\\|\\,L\\,\\|\\,,\n\\end{equation}\nwhere the matrices $H=\\big(H_{ij}\\big)_{3\\times 3}$ and $L=\\big(L_{ij}\\big)_{3\\times 3}$ are introduced in \\eqref{16} and \\eqref{18}. Indeed, since $E^e=Q^0HP^{-1}$ and $Q^0\\in SO(3)$ we have\n\\begin{equation}\\label{29}\n    \\|\\,E^e\\,\\|^2=\\|\\,Q^0HP^{-1}\\,\\|^2=\\|\\,HP^{-1}\\,\\|^2=\\text{tr}\\big[ HP^{-1} (HP^{-1})^T \\big]=\\text{tr}\\big[ H(P^{T}\\!P)^{-1} H^T \\big].\n\\end{equation}\nFrom \\eqref{14bis} we deduce that\n\\begin{equation}\\label{30}\n    P^{T}P =\\begin{bmatrix} a_{11} & a_{12} &  0 \\\\\n                                 a_{12} & a_{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\qquad\\text{and therefore}\\qquad\n    (P^{T}P)^{-1}=\\begin{bmatrix} a^{11} & a^{12} &  0 \\\\\n                                 a^{12} & a^{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\,\\,.\n\\end{equation}\nInserting \\eqref{30} into \\eqref{29} we obtain\n\\begin{equation}\\label{31}\n     \\|\\,E^e\\,\\|^2=a^{\\alpha\\beta}H_{i\\alpha}H_{i\\beta} = a^{\\alpha\\beta}H_{1\\alpha}H_{1\\beta}+ a^{\\alpha\\beta}H_{2\\alpha}H_{2\\beta}+ a^{\\alpha\\beta}H_{3\\alpha}H_{3\\beta}\\,,\\,\\,\\,\\text{with}\\,\\,\\, H=\\big(H_{ij}\\big)_{3\\times 3}\\,,\n\\end{equation}\nsince $H_{i3}=0$ according to \\eqref{16}.\nIn virtue of \\eqref{26,1}, it follows that the matrix $\\big(a_{\\alpha\\beta}\\big)_{2\\times 2}$ and its inverse matrix\n$\\big(a^{\\alpha\\beta}\\big)_{2\\times 2}=\\big(a_{\\alpha\\beta}\\big)^{-1}$  satisfy\n$$\\big(a_{\\alpha\\beta}\\big)\n\\in\\boldsymbol{L}^\\infty(\\omega, \\mathbb{R}^{2\\times 2})\\quad\\text{and}\\quad \\big(a^{\\alpha\\beta}\\big)\n\\in\\boldsymbol{L}^\\infty(\\omega, \\mathbb{R}^{2\\times 2}).$$\nThen, the smallest eigenvalue of the positive definite symmetric matrix $\\big(a^{\\alpha\\beta}(x_1,x_2)\\big)_{2\\times 2}$ is greater than a positive constant $\\lambda_0^2>0$ and consequently\n\\begin{equation}\\label{32}\n    a^{\\alpha\\beta}\\!(x_1,x_2)\\,\\,v_\\alpha\\, v_\\beta\\,\\geq\\, \\lambda_0^2\\,v_\\gamma\\, v_\\gamma\\,,\\qquad \\forall\\,(x_1,x_2)\\in\\omega,\\,\\,\\, \\forall\\,v_1,v_2\\in\\mathbb{R}.\n\\end{equation}\nUsing  inequality \\eqref{32} for each individual sum in the right-hand side of \\eqref{31} we deduce that $\\|E^e\\|^2\\geq \\lambda_0^2\\,H_{i\\alpha}H_{i\\alpha}=\\lambda_0^2 \\,\\|H\\|^2$, i.e. the inequality \\eqref{28}$_1$ is proved. The proof of the inequality \\eqref{28}$_2$ is identical. In view of \\eqref{28}$_1$ and \\eqref{16} we have\n\\begin{equation*}\n\\begin{array}{l}\n\\|\\,E^e\\,\\|^2\\geq \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2} \\|\\,R^T\\partial_\\alpha y\\!-\\! Q^{0,T}\\partial_\\alpha y^{0}\\,\\|^2   \\\\\n \\qquad\\qquad=\n \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2}\\big( \\|\\,R^T\\partial_\\alpha y\\|^2\\!-2\\langle\\,   R^T\\partial_\\alpha y\\, , Q^{0,T}\\partial_\\alpha y^{0}\\,\\rangle +\\|\\,Q^{0,T}\\partial_\\alpha y^{0} \\|^2\\big)\\\\\n \\qquad\\qquad = \\lambda_0^2\\displaystyle{\\sum_{\\alpha=1}^2}\\big( \\|\\,\\partial_\\alpha y\\|^2-2\\langle\\,   R^T\\partial_\\alpha y\\, , Q^{0,T}\\partial_\\alpha y^{0}\\,\\rangle +\\|\\,\\partial_\\alpha y^{0} \\|^2\\big)\n ,\n \\end{array}\n\\end{equation*}\nwhere $\\langle\\,S,T\\,\\rangle=\\text{tr}[ST^T]$ is the scalar product of  two matrices $S,T$. Integrating over $\\omega$ and using the Cauchy--Schwarz inequality we obtain\n$$\\|\\,E^e\\,\\|^2_{L^2(\\omega)} \\geq\n \\lambda_0^2\\sum_{\\alpha=1}^2\\big( \\|\\,\\partial_\\alpha y\\|^2_{L^2(\\omega)}-2\\|\\, \\partial_\\alpha y\\|_{L^2(\\omega)} \\|\\,\\partial_\\alpha y^{0}\\,\\|_{L^2(\\omega)} +\\|\\,\\partial_\\alpha y^{0 } \\|^2_{L^2(\\omega)}\\big),$$\nor\n\\begin{equation}\\label{33}\n    \\|\\,\\boldsymbol{E}^e\\,\\|^2_{L^2(\\omega)} \\geq\n \\lambda_0^2\\,\\big( \\|\\,\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\big) - \\bar{C}_1\\|\\, \\boldsymbol{y}\\,\\|_{H^1(\\omega)}+\\bar{C}_2\\,,\n\\end{equation}\nfor some positive constants $\\bar{C}_1>0$, $\\bar{C}_2>0$. Let us show that the functional $I(\\boldsymbol{y},\\boldsymbol{R})$ is bounded from below over the admissible set $\\mathcal{A}$. By virtue of \\eqref{22}, \\eqref{26,1}$_2$ and \\eqref{27} we can write\n$$I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0 \\!\\!\\int_\\omega \\|\\,\\boldsymbol{E}^e\\,\\|^2\\,a\\, \\mathrm{d}x_1\\mathrm{d}x_2 - \\Lambda(\\boldsymbol{y},\\boldsymbol{R}) \\geq C_0\\,a_0 \\|\\,\\boldsymbol{E}^e\\,\\|^2_{L^2(\\omega)}  - C\\,\\big(\\,\\|\\boldsymbol{y}\\|_{H^1(\\omega)}+1\\big)$$\nand using \\eqref{33} we deduce that there exist the constants $\\bar{C}_3>0$ and $\\bar{C}_4$ such that\n\\begin{equation}\\label{34}\n    I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0\\,a_0 \\lambda_0^2\\,\\big( \\|\\,\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\big) - \\bar{C}_3\\|\\, \\boldsymbol{y}\\,\\|_{H^1(\\omega)}-\\bar{C}_4\\,,\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A},\n\\end{equation}\nwith $\\,a_0\\,$ specified by \\eqref{26,1}.\nWe observe that the vector field $\\boldsymbol{y}-\\boldsymbol{y}^*\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ satisfies $\\boldsymbol{y}-\\boldsymbol{y}^*=\\boldsymbol{0}$ on $\\partial\\omega_d\\,$. Applying the Poincar\\'e--inequality we infer the existence of a constant $c_p>0$ such that\n\\begin{equation}\\label{35}\n     \\| \\partial_1 (\\boldsymbol{y}-\\boldsymbol{y}^*)\\|^2_{L^2(\\omega)} +\\| \\partial_2 (\\boldsymbol{y}-\\boldsymbol{y}^*) \\|^2_{L^2(\\omega)}  \\,\\geq\\, c_p\\,\\|\\, \\boldsymbol{y}-\\boldsymbol{y}^*\\,\\|_{H^1(\\omega)}^2\\,\\,.\n\\end{equation}\nUsing inequalities of the type $\\|\\partial_\\alpha \\boldsymbol{y}\\|_{L^2(\\omega)}^2\\geq \\big(\\|\\partial_\\alpha (\\boldsymbol{y}-\\boldsymbol{y}^*)\\|_{L^2(\\omega)}- \\| \\partial_\\alpha \\boldsymbol{y}^*\\|_{L^2(\\omega)}\\big)^2$ and \\eqref{35} we find that\n\\begin{equation*}\n\\begin{array}{l}\n\\|\\partial_1 \\boldsymbol{y}\\|^2_{L^2(\\omega)} +\\|\\partial_2 \\boldsymbol{y} \\|^2_{L^2(\\omega)}\\geq  c_p\\,\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}^2 \\\\\n\\qquad\\quad\\,\\,\\,\\, -2\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}\\big( \\|\\partial_1 \\boldsymbol{y}^*\\|_{L^2(\\omega)} +\\|\\partial_2 \\boldsymbol{y}^* \\|_{L^2(\\omega)}\\big)  +\n\\big(\\|\\,\\partial_1 \\boldsymbol{y}^*\\|^2_{L^2(\\omega)} +\\|\\,\\partial_2 \\boldsymbol{y}^* \\|^2_{L^2(\\omega)}\\big).\n\\end{array}\n\\end{equation*}\nFrom the last inequality and \\eqref{34} follows that there exist some constants $\\bar{C}_5>0$ and $\\bar{C}_6$ with\n\\begin{equation}\\label{36}\n    I(\\boldsymbol{y},\\boldsymbol{R})\\geq C_0\\,a_0 \\lambda_0^2\\,c_p\\,\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)}^2-\\bar{C}_5\\| \\boldsymbol{y}-\\boldsymbol{y}^*\\|_{H^1(\\omega)} +\\bar{C}_6\\,,\\quad\\forall\\,(\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}.\n\\end{equation}\nSince the constant $C_0\\,a_0 \\lambda_0^2\\,c_p>0$, the function $I(\\boldsymbol{y},\\boldsymbol{R})$ is bounded from below over   $\\mathcal{A}$.  Hence, there exists an infimizing sequence $\\big\\{(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty \\subset\\mathcal{A}$ such that\n\\begin{equation}\\label{37}\n    \\lim_{n\\rightarrow \\infty} I(\\boldsymbol{y}_n,\\boldsymbol{R}_n) = \\,\\inf\\, \\big\\{I(\\boldsymbol{y},\\boldsymbol{R})\\, \\big|\\,  (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}\\big\\}.\n\\end{equation}\nIn view of the conditions \\eqref{25} we have $I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)<\\infty$. The infimizing sequence $\\big\\{(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty$ can be chosen such that\n\\begin{equation}\\label{38}\n    I(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\,\\leq \\,I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)\\,< \\infty\\,, \\qquad \\forall\\,n\\geq 1.\n\\end{equation}\nTaking into account \\eqref{36} and \\eqref{38} we see that the sequence $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$. Then, we can extract a subsequence of $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$  (not relabeled) which converges weakly in  $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ and moreover, according to Rellich's selection principle, it converges strongly in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$, i.e. there exists an element $\\hat{\\boldsymbol{y}}\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$ such that\n\\begin{equation}\\label{39}\n    \\boldsymbol{y}_n  \\rightharpoonup \\hat{ \\boldsymbol{y}} \\quad\\mathrm{in}\\quad \\boldsymbol{H}^1(\\omega, \\mathbb{R}^3),\\qquad \\mathrm{and}\\qquad \\boldsymbol{y}_n \\rightarrow\\hat{ \\boldsymbol{y}} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^3).\n\\end{equation}\nFor any $n\\in\\mathbb{N}$, let us denote by $\\boldsymbol{E}^e_n$ and $\\boldsymbol{K}^e_n$ the strain measures corresponding to the fields $(\\boldsymbol{y}_n,\\boldsymbol{R}_n)$, defined by the relations \\eqref{8} and \\eqref{12}. We have $\\boldsymbol{E}^e_n, \\boldsymbol{K}^e_n\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ and let $E_n^e,K_n^e$ be the matrices of components in the basis $\\{\\boldsymbol{e}_i\\otimes\\boldsymbol{e}_j\\}$, given by \\eqref{16} and \\eqref{18} in terms $\\{y_n,R_n\\}$. From \\eqref{20}, \\eqref{26,1}$_2$ , \\eqref{26bis}, \\eqref{27} and \\eqref{38} we get\n$$ C_0a_0\\,\\|\\,\\boldsymbol{K}^e_n\\,\\|^2_{L^2(\\omega)} \\leq\n\\int_\\omega W(\\boldsymbol{E}^e_n,\\boldsymbol{K}^e_n)\\,a(x_1,x_2)\\, \\mathrm{d}x_1\\mathrm{d}x_2 \\leq I(\\boldsymbol{y}^*,\\boldsymbol{R}^*)+ C\\,\\big(\\,\\|\\boldsymbol{y}_n\\|_{H^1(\\omega)}+1\\big).\n$$\nSince $\\big\\{ \\boldsymbol{y}_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3)$, it follows from the last inequalities that $\\big\\{ \\boldsymbol{K}^e_n \\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. In view of \\eqref{28}$_2\\,$, we deduce that $\\big\\{\\text{axl}(\\boldsymbol{R}_n^{T}\\partial_\\alpha \\boldsymbol{R}_n)\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$, or equivalently $ \\big\\{ \\partial_\\alpha \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, for $\\alpha=1,2$. Since $\\boldsymbol{R}_n\\in SO(3)$ we have $\\|\\boldsymbol{R}_n\\|^2=3$ and thus we can infer that the sequence $ \\big\\{ \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$\nis bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})$. Hence, there exists a subsequence of $ \\big\\{ \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ (not relabeled) and an element $\\hat{\\boldsymbol{R}}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3\\times 3})$ with\n\\begin{equation}\\label{40}\n    \\boldsymbol{R}_n \\rightharpoonup       \\hat{\\boldsymbol{R}}    \\quad\\mathrm{in}\\quad \\boldsymbol{H}^1(\\omega, \\mathbb{R}^{3\\times3}) , \\qquad\\mathrm{and}\\qquad          \\boldsymbol{R}_n  \\rightarrow    \\hat{\\boldsymbol{R}} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}).\n\\end{equation}\nWe can show for the limit that $\\hat{\\boldsymbol{R}}\\in SO(3)$. Indeed, since $\\boldsymbol{R}_n\\in SO(3)$ we have\n$$\\|\\,\\boldsymbol{R}_n \\hat{\\boldsymbol{R}}{}^T - {1\\!\\!\\!\\:1 } _3\\|_{L^2(\\omega)}= \\|\\,\\boldsymbol{R}_n ( \\hat{\\boldsymbol{R}}{}^T -\\boldsymbol{R}_n^T)\\|_{L^2(\\omega)}=  \\|\\,  \\hat{\\boldsymbol{R}}  -\\boldsymbol{R}_{n }\\|_{L^2(\\omega)} \\longrightarrow 0,\n$$\ni.e. $\\boldsymbol{R}_n\\hat{\\boldsymbol{R}}{}^T\\rightarrow {1\\!\\!\\!\\:1 } _3$ in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. On the other hand, we can write\n$$\\|\\,\\boldsymbol{R}_n \\hat{\\boldsymbol{R}}{}^T -\\hat{\\boldsymbol{R}} \\hat{\\boldsymbol{R}}{}^T\\|_{L^1(\\omega)}= \\|\\, ( \\boldsymbol{R}_{n }  -\\hat{\\boldsymbol{R}})\\hat{\\boldsymbol{R}}{}^T\\|_{L^1(\\omega)} \\leq 3  \\|\\,   \\boldsymbol{R}_{n }  -\\hat{\\boldsymbol{R}} \\|_{L^2(\\omega)}\\,  \\|\\,  \\hat{\\boldsymbol{R}} \\|_{L^2(\\omega)} \\longrightarrow 0,\n$$\nwhich means that $\\boldsymbol{R}_n\\hat{\\boldsymbol{R}}{}^T\\rightarrow\\hat{\\boldsymbol{R}}\\hat{\\boldsymbol{R}}{}^T$ in $\\boldsymbol{L}^1(\\omega,\\mathbb{R}^{3\\times 3})$. Consequently, we find $\\hat{\\boldsymbol{R}}\\hat{\\boldsymbol{R}}{}^T= {1\\!\\!\\!\\:1 } _3$ so that $\\hat{\\boldsymbol{R}}\\in\\boldsymbol{H}^1(\\omega,SO(3))$.\n\nBy virtue of the relations  $(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\in \\mathcal{A}$ and \\eqref{39}, \\eqref{40},   we derive that $\\hat{\\boldsymbol{y}}=\\boldsymbol{y}^*$ on $\\partial S^0_d$ and $\\hat{\\boldsymbol{R}}=\\boldsymbol{R}^*$ on $\\partial S^0_d\\,$ in the sense of traces. Hence, we obtain that the limit pair satisfies $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in \\mathcal{A}$.\n\nLet us construct the elements $\\hat{\\boldsymbol{E}^e}, \\hat{\\boldsymbol{K}^e} \\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ defined in terms of the fields $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ by the relations \\eqref{8} and \\eqref{12}. Then, the matrices of components $\\hat{E}^e,\\hat{K}^e$ are expressed in terms of the components $(\\hat{y},\\hat{R})$ by \\eqref{16} and \\eqref{18}, i.e.\n\\begin{equation}\\label{41}\n    \\begin{array}{l}\n     \\hat{E}^e= Q^0\\big(\\,\\hat{R}{}^T\\partial_1 \\hat{y}- Q^{0,T}\\partial_1 y^{0}\\,\\big|\\, \\hat{R}{}^T\\partial_2 \\hat{y}- Q^{0,T}\\partial_2 y^{0}\\,\\big|\\,\\, 0\\,\\,\\big) \\,P^{-1} ,\\vspace{4pt}\\\\\n     \\hat{K}^e=Q^0\\big(\\,\\text{axl}(\\hat{R}{}^T\\partial_1 \\hat{R})\\!-\\!\\text{axl}(Q^{0,T}\\partial_1 Q^{0}) \\,\\,\\big|\\,\\, \\text{axl}(\\hat{R}{}^T\\partial_2 \\hat{R})\\! -\\!\\text{axl}(Q^{0,T}\\partial_2 Q^{0})\\,\\,\\big|\\, \\,0\\,\\big)P^{-1} \\!.\n     \\end{array}\n\\end{equation}\n\nNext, we want to show that there exist some subsequences (not relabeled) of $\\{\\boldsymbol{E}^e_n\\}$ and $\\{\\boldsymbol{K}^e_n\\}$ such that\n\\begin{equation}\\label{42}\n    \\boldsymbol{E}^e_n  \\rightharpoonup  \\hat{\\boldsymbol{E}^e} \\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}),\\quad\\text{and} \\qquad  \\boldsymbol{K}^e_n \\rightharpoonup \\hat{\\boldsymbol{K}^e}\\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega, \\mathbb{R}^{3\\times3}).\n\\end{equation}\nAs shown above, the sequence $ \\big\\{ \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3 })$. It follows that $ \\big\\{ \\partial_\\alpha \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3 })$ and then the sequence $ \\big\\{ \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{y}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3 })$, since $ \\boldsymbol{R}_n\\in SO(3)$. Consequently, there exists a subsequence (not relabeled) and an element $\\boldsymbol{\\xi}_\\alpha\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$ such that\n\\begin{equation}\\label{43}\n    \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{y}_n \\,\\,\\rightharpoonup \\,\\, \\boldsymbol{\\xi}_\\alpha\\qquad\\text{in} \\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3}).\n\\end{equation}\nOn the other hand, let $\\boldsymbol{\\phi}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3})$ be an arbitrary test function. Then, using the properties of the scalar product  we deduce\n\\begin{equation*}\n    \\begin{array}{l}\n    \\displaystyle{\\int_\\omega}\\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n - \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n    \\qquad\\quad=\n    \\displaystyle{\\int_\\omega}  \\hat{\\boldsymbol{R}}{}^T \\big( \\partial_\\alpha \\boldsymbol{y}_n - \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n        +\\displaystyle{\\int_\\omega}  \\big( \\boldsymbol{R}_n^T- \\hat{\\boldsymbol{R}}{}^T\\big) \\partial_\\alpha \\boldsymbol{y}_n  \\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n     \\qquad\\quad   =\\displaystyle{\\int_\\omega}  \\big( \\partial_\\alpha \\boldsymbol{y}_n - \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\cdot\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi} \\, \\mathrm{d}x_1\\mathrm{d}x_2  +\n        \\displaystyle{\\int_\\omega} \\!\\!\\big\\langle \\boldsymbol{R}_{n}\\!\\!-\\! \\hat{\\boldsymbol{R}}\\,, \\partial_\\alpha \\boldsymbol{y}_n \\!\\otimes\\!\\boldsymbol{\\phi}\\rangle \\mathrm{d}x_1\\mathrm{d}x_2\n    \\vspace{4pt}\\\\\n    \\qquad\\qquad\\,\\, \\leq\\|\\boldsymbol{R}_{n}\\!-\\! \\hat{\\boldsymbol{R}}\\|_{L^2(\\omega)}\\|\\partial_\\alpha \\boldsymbol{y}_n \\!\\otimes\\!\\boldsymbol{\\phi}\\|_{L^2(\\omega)} \\!+\\!\\! \\displaystyle{\\int_\\omega}  \\big( \\partial_\\alpha \\boldsymbol{y}_n \\!- \\!\\partial_\\alpha \\hat{\\boldsymbol{y}}\\big) \\!\\cdot\\!\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi} \\, \\mathrm{d}x_1\\mathrm{d}x_2 \\,,\n\\end{array}\n\\end{equation*}\nsince the relations \\eqref{39}, \\eqref{40} and $\\hat{\\boldsymbol{R}}\\boldsymbol{\\phi}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3})$ hold, and $\\|\\partial_\\alpha \\boldsymbol{y}_n\\otimes\\boldsymbol{\\phi}\\|$ is bounded. Thus, we get\n\\begin{equation}\\label{44}\n    \\displaystyle{\\int_\\omega}\\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n \\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\longrightarrow\n    \\displaystyle{\\int_\\omega}\\big( \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}\\big)\\cdot\\boldsymbol{\\phi} \\,\\mathrm{d}x_1\\mathrm{d}x_2,\\quad\\forall\\, \\boldsymbol{\\phi}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^3).\n\\end{equation}\nBy comparison of \\eqref{43} and \\eqref{44} we find $\\boldsymbol{\\ell}_\\alpha= \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{y}}\\,$, which means that\n$\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n   \\rightharpoonup\n     \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}}$ in $ \\boldsymbol{L}^2(\\omega,\\mathbb{R}^3)$ ,\nor equivalently\n\\begin{equation}\\label{45}\n    \\big(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{y}_n  - \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0 \\big)\\quad \\rightharpoonup\\quad\n    \\big( \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{y}} -  {\\boldsymbol{R}}{}^T_0 \\partial_\\alpha \\boldsymbol{y}^{0}\\big)\\quad\\mathrm{in}\\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^3).\n\\end{equation}\nTaking into account \\eqref{16}, \\eqref{41}$_1$ and the hypotheses \\eqref{26}, \\eqref{26,1}, we obtain from \\eqref{45} that $E^e_n\\rightharpoonup\\hat{E}^e$ in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, i.e. the relation \\eqref{42}$_1$ holds.\n\nTo prove \\eqref{42}$_2$ we start from the fact that the sequence $ \\big\\{ \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{R}_n\\big\\}_{n=1}^\\infty$ is bounded in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3 })$, as we proved previously. Then, there exists a subsequence (not relabeled) and an element $\\boldsymbol{ \\zeta}_\\alpha\\in \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$ such that\n\\begin{equation}\\label{46}\n    \\boldsymbol{R}_n^T\\partial_\\alpha \\boldsymbol{R}_n \\rightharpoonup \\boldsymbol{ \\zeta}_\\alpha\\qquad\\text{in} \\quad \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3}).\n\\end{equation}\nOn the other hand, for any test function $\\boldsymbol{\\mathit{\\Phi}}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3\\times 3})$ we can write\n\\begin{equation*}\n\\begin{array}{l}\n    \\displaystyle{\\int_\\omega}\\big\\langle \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_{n} - \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{R}}  \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2=\n    \\displaystyle{\\int_\\omega}\\big\\langle   \\hat{\\boldsymbol{R}}{}^T \\big(\\partial_\\alpha  \\boldsymbol{R}_{n} - \\partial_\\alpha\\hat{\\boldsymbol{R}}\\big) \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\vspace{2pt}\\\\\n     \\qquad+\n    \\displaystyle{\\int_\\omega}\\big\\langle \\big( \\boldsymbol{R}_n^T- \\hat{\\boldsymbol{R}}{}^T\\big) \\partial_\\alpha \\boldsymbol{R}_{n}  \\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2\n    \\leq\n     \\displaystyle{\\int_\\omega}\n     \\big\\langle \\partial_\\alpha \\boldsymbol{R}_{n} - \\partial_\\alpha \\hat{\\boldsymbol{R}}  \\,,\\, \\hat{\\boldsymbol{R}}\\boldsymbol{\\mathit{\\Phi}} \\,\\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2\n     \\vspace{2pt}\\\\\n     \\qquad+\n     \\|\\boldsymbol{R}_{n}- \\hat{\\boldsymbol{R}}\\|_{L^2(\\omega)}\\,\\| \\partial_\\alpha \\boldsymbol{R}_n \\boldsymbol{\\mathit{\\Phi}}^T \\, \\|_{L^2(\\omega)} \\longrightarrow 0,\n\\end{array}\n\\end{equation*}\nsince $\\hat{\\boldsymbol{R}}\\boldsymbol{\\mathit{\\Phi}}\\in\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$, $\\|\\partial_\\alpha \\boldsymbol{R}_n \\boldsymbol{\\mathit{\\Phi}}^T\\|$ is bounded, and relations \\eqref{40} hold. Consequently, we have\n$$\\displaystyle{\\int_\\omega}\\big\\langle \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_{n}\\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2 \\longrightarrow\n    \\displaystyle{\\int_\\omega}\\big\\langle  \\hat{\\boldsymbol{R}}{}^T \\partial_\\alpha \\hat{\\boldsymbol{R}}\\,,\\, \\boldsymbol{\\mathit{\\Phi}} \\big\\rangle \\,\\mathrm{d}x_1\\mathrm{d}x_2,\\quad\\forall \\,\\boldsymbol{\\mathit{\\Phi}}\\in  \\boldsymbol{C}_0^\\infty(\\omega,\\mathbb{R}^{3\\times3}), $$\nand by comparison with \\eqref{46} we deduce that $\\boldsymbol{ \\zeta}_\\alpha= \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}}\\,$, i.e. the convergence $ \\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_n\\rightharpoonup \\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}}$ holds in $\\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3})$. It follows that\n$$\\big[\\text{axl}(\\boldsymbol{R}_n^T \\partial_\\alpha \\boldsymbol{R}_n) - \\text{axl}(\\boldsymbol{R}^{T}_0 \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]\n \\,\\,\\,\\rightharpoonup \\,\\,\\,\n\\big[\\text{axl}(\\hat{\\boldsymbol{R}}{}^{T}\\partial_\\alpha \\hat{\\boldsymbol{R}})- \\text{axl}(\\boldsymbol{R}^{T}_0 \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]\n\\quad\\text{in}\\,\\, \\boldsymbol{L}^2(\\omega,\\mathbb{R}^{3\\times 3}),$$\nand from \\eqref{18}, \\eqref{26}, \\eqref{26,1} and \\eqref{41}$_2$ we derive that the convergence \\eqref{42}$_2$ holds true. \\medskip\n\nIn the last step of the proof we use the convexity of the strain energy density $W$. In view of \\eqref{42}, we have\n\\begin{equation}\\label{47}\n     \\int_\\omega W(\\hat{\\boldsymbol{E}^e},\\hat{\\boldsymbol{K}^e})\\,a(x_1,x_2)\\,\\mathrm{d}x_1\\mathrm{d}x_2\\,\\leq \\, \\liminf_{n\\to\\infty}  \\int_\\omega W( {\\boldsymbol{E}^e_n}, {\\boldsymbol{K}^e_n})\\,a(x_1,x_2)\\,\\mathrm{d}x_1\\mathrm{d}x_2.\n\\end{equation}\nsince $W$ is convex in $(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$. Taking into account the hypotheses \\eqref{24}, the continuity of the load potential functions      $\\,\\,\\Pi_{S^0}\\,$, $ \\,\\Pi_{\\partial S^0_f}\\,$,    and the convergence relations \\eqref{39}$_2$ and \\eqref{40}$_2\\,$, we deduce\n\\begin{equation}\\label{48}\n  \\Lambda(\\hat{\\boldsymbol{y}}, \\hat{\\boldsymbol{R}})=  \\lim_{n\\to\\infty} \\Lambda(\\boldsymbol{y}_n, \\boldsymbol{R}_n).\n\\end{equation}\nFrom \\eqref{20}, \\eqref{22}, \\eqref{47} and \\eqref{48} we get\n\\begin{equation}\\label{49}\n    I(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\,\\leq\\,  \\liminf_{n\\to\\infty} \\, I(\\boldsymbol{y}_n,\\boldsymbol{R}_n)\\,.\n\\end{equation}\nFinally, the relations \\eqref{37} and \\eqref{49} show that\n$$ I(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\,=\\,\n\\,\\inf\\, \\big\\{I(\\boldsymbol{y},\\boldsymbol{R})\\, \\big|\\,  (\\boldsymbol{y},\\boldsymbol{R})\\in \\mathcal{A}\\big\\}.\n$$\nSince $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})\\in\\mathcal{A}$, we conclude that $(\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{R}})$ is a minimizing solution pair of our minimization problem. The proof is complete.\n\\hfill\\end{proof}\n\\begin{remark}\nThe solution fields satisfy the following regularity conditions\n$$\\hat{\\boldsymbol{y}}\\in \\boldsymbol{H}^1(\\omega,\\mathbb{R}^{3})\n,\\qquad\n\\hat{\\boldsymbol{R}}\\in\\boldsymbol{L}^\\infty(\\omega,SO(3))\n\\cap \\boldsymbol{H}^1(\\omega,SO(3)) .\n$$\nThus, the position vector $ \\,\\hat{\\boldsymbol{y}}\\,$ and the total rotation field $\\,\\hat{\\boldsymbol{R}}\\,$ may fail to be continuous, according to the limit case of Sobolev embedding.\n\\end{remark}\n\\begin{remark}\nWe observe that the boundary conditions imposed on the orthogonal tensor $\\boldsymbol{R}$ can be relaxed in the definition of the admissible set $\\mathcal{A}$. Thus, one can prove the existence of minimizers for the minimization problem \\eqref{20} over the following larger admissible set\n\\begin{equation*}\n    \\tilde{\\mathcal{A}} =\\big\\{(\\boldsymbol{y},\\boldsymbol{R})\\in\\boldsymbol{H}^1(\\omega,\\mathbb{R}^3) \\times\\boldsymbol{H}^1(\\omega, SO(3))\\,\\,\\, \\big|\\,\\, \\,\\, \\boldsymbol{y}_{\\big| \\partial\\omega_d}=\\boldsymbol{y}^*  \\big\\}.\n\\end{equation*}\nThis assertion can be proved in the same way as the Theorem \\ref{th1}. For a discussion of   possible alternative boundary conditions for the   field $\\boldsymbol{R}$ on $\\partial\\omega_d$ we refer to the works \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Applications of the theorem and discussions}\\label{sect5}\n\n\n\n\nIn this section we present some important special cases for the choice of the energy density $W$ where the Theorem \\ref{th1} can be successfully applied to show the existence of minimizers.\n\nLet us discuss first on the choice of the 3 initial directors $\\{\\boldsymbol{d}_i^0\\}$ in the reference configuration, i.e. the specification of the proper orthogonal tensor $\\boldsymbol{Q}^{0}=\\boldsymbol{d}_i^0 \\otimes \\boldsymbol{e}_i\\,$. One judicious choice for the tensor $\\boldsymbol{Q}^{0}$ is the following\n\\begin{equation}\\label{49,1}\n    \\boldsymbol{Q}^{0}=\\text{polar}(\\boldsymbol{P})=\\text{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big),\n\\end{equation}\nwhere $\\boldsymbol{P}=\\boldsymbol{a}_{i}\\otimes\\boldsymbol{e}_i=\\partial_\\alpha \\boldsymbol{y}^0\\otimes\\boldsymbol{e}_\\alpha +\\boldsymbol{n}^0\\otimes \\boldsymbol{e}_3$ has been introduced previously in \\eqref{14} and $\\text{polar}(\\boldsymbol{T})$ denotes the orthogonal tensor given by the polar decomposition of any tensor $\\boldsymbol{T}$.\n\\begin{remark}\\label{rem6}\nIf the tensor $\\boldsymbol{Q}^{0}$ is given by \\eqref{49,1}, then the (initial) directors $\\boldsymbol{d}_\\alpha^0$ belong to the tangent plane at any point of $S^0$ and $\\boldsymbol{d}_3^0= \\boldsymbol{n}^0$. Indeed, let $\\boldsymbol{P}=\\boldsymbol{Q}^{0}\\boldsymbol{U}^0$ be the polar decomposition of $\\boldsymbol{P}$.\nUsing the matrices of components in the $\\{\\boldsymbol{e}_{i}\\otimes\\boldsymbol{e}_j\\}$ tensor basis, we write this relation as $P=Q^0U^0$, and from \\eqref{30} we derive consecutively\n\\begin{equation*}\n    U^{0,T} U^{0}=P^{T}P =\\begin{bmatrix} a_{11} & a_{12} &  0 \\\\\n                                 a_{12} & a_{22} &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\\quad\n    U^0=\n    \\begin{bmatrix} u_{11}^0 & u_{12}^0 &  0 \\\\\n                                 u_{12}^0 & u_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\\quad\n    \\big(U^0\\big)^{-1}=\n    \\begin{bmatrix} \\bar{u}_{11}^0 & \\bar{u}_{12}^0 &  0 \\\\\n                                 \\bar{u}_{12}^0 & \\bar{u}_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix},\n\\end{equation*}\nwhere $u_{\\alpha\\beta}^0$ and $\\bar{u}_{\\alpha\\beta}^0$ are some given real functions of $(x_1,x_2)$. In view of \\eqref{14bis}, it follows\n\\begin{equation}\\label{49,2}\n   Q^0= P\\,\\big(U^0\\big)^{-1}= \\Big(\\,a_{1}\\,\\Big|\\, a_{2}\\,\\Big|\\, n^0\\, \\Big)_{3\\times 3}\n   \\begin{bmatrix} \\bar{u}_{11}^0 & \\bar{u}_{12}^0 &  0 \\\\\n                                 \\bar{u}_{12}^0 & \\bar{u}_{22}^0 &  0 \\\\\n                                 0 & 0 & 1\n    \\end{bmatrix}\\qquad\\Rightarrow\\qquad Q^0e_3=n^0,\n\\end{equation}\nfrom which we can see that the third column of the matrix $\\,Q^0$ is equal to $\\,n^0$.\nOn the other hand, by the definition \\eqref{1}$_{2\\,}$, the initial rotation field $ \\boldsymbol{Q}^{0}$ is given by $ \\boldsymbol{Q}^{0}= \\boldsymbol{d}_i^{0}\\otimes  \\boldsymbol{e}_i\\,$ and the matrix $Q^0$ can be written in column form as\n\\begin{equation}\\label{49,3}\n   Q^0 = \\Big(\\,d_{1}^0\\,\\Big|\\, d_{2}^0\\,\\Big|\\, d_3^0\\, \\Big)_{3\\times 3}\\,\\,\\,.\n\\end{equation}\nIf we compare \\eqref{49,2} and  \\eqref{49,3} we find that $d_3^0=n^0$. Thus, we have\n$\\boldsymbol{d}_3^{0}= \\boldsymbol{n}^{0}$ and  $\\{\\boldsymbol{d}_1^0,\\boldsymbol{d}_2^0\\}$ is an orthonormal basis in the tangent plane, at any point of $S^0$.\n\nIf we choose the tensor $\\boldsymbol{Q}^{0}$ as in \\eqref{49,1}, then in order to satisfy \\eqref{26}$_2$ we need to consider an additional regularity assumption on the initial configuration, namely\n$$\\mathrm{polar}(\\boldsymbol{P})=\\mathrm{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big)\\in \\boldsymbol{H}^1(\\omega,SO(3)),$$\nwhich is equivalent to $\\mathrm{Curl}\\big[\\mathrm{polar}\\big( \\nabla \\boldsymbol{\\Theta}(x_1,x_2,0)\\big)\\big]\\in \\boldsymbol{L}^2(\\omega,SO(3))\\,$, cf. \\cite{Neff_curl08}. A stronger sufficient condition is $\\boldsymbol{\\Theta}\\in \\boldsymbol{W}^{1,\\infty}(\\omega,\\mathbb{R}^3)\\cap  \\boldsymbol{H}^2(\\omega,\\mathbb{R}^3)$.\n\\end{remark}\n\nIt is possible to simplify the form of the equations in the case of an orthogonal parametrization of the initial surface $S^0\\,$. If we   assume that the curvilinear coordinates $(x_1,x_2)$ are   such that the basis $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal, then the initial surface $S^0$ is formally parametrized by orthogonal arc-length coordinates \\cite{Chroscielewski11} and we have\n\\begin{equation}\\label{50}\n    \\boldsymbol{a}_\\alpha=\\boldsymbol{a}^\\alpha,\\qquad a_{\\alpha\\beta}= a^{\\alpha\\beta}=\\delta_{\\alpha\\beta}\\,.\n\\end{equation}\n\\begin{remark}\nThe \\emph{Theorema Egregium} (Gauss) can be put into the following form: the Gaussian curvature $K$ can be found given the full knowledge of the first fundamental form of the surface and expressed via the first fundamental form and its partial derivatives of first and second order (the Brioschi formula). Therefore, the Gaussian curvature of an embedded smooth surface in $\\mathbb{R}^3$ is invariant under local isometries, i.e. if the parametrization $\\boldsymbol{y}^0:\\omega\\subset\\mathbb{R}^2\\rightarrow \\mathbb{R}^3$ of the surface from a flat reference configuration $\\omega$ is given such that  $\\big(\\nabla \\boldsymbol{y}^0\\big)^T \\nabla \\boldsymbol{y}^0= {1\\!\\!\\!\\:1 } _2$ (the basis $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal), then the curvature $K$ of the surface $\\boldsymbol{y}^0(\\omega)$ is necessarily zero. This is only the case for developable surfaces.\n\n\nFor general surfaces it is therefore impossible to determine, even locally, an orthonormal parametrization. However, in FEM approaches one may think in a discrete pointwise manner as in \\cite{Chroscielewski11}.\n\nFor example, let $S^0$ be a cylindrical surface (which is a developable surface) with generators parallel to $\\boldsymbol{e}_3\\,$. The position vector $\\boldsymbol{y}^0$ is given by\n$$\\boldsymbol{y}^0=\\boldsymbol{y}^0(\\theta,z)=r_0\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\nr_0\\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2+ z \\, \\boldsymbol{e}_3\\qquad (r_0>0\\,\\,\\mathrm{constant}).\n$$\nChoosing the curvilinear coordinates $x_1=\\theta, x_2=z$, we have\n$$\\boldsymbol{a}_1= \\partial_1 \\boldsymbol{y}^0=- \\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\n\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2,\\quad\n\\boldsymbol{a}_2=\\partial_2 \\boldsymbol{y}^0=\\boldsymbol{e}_3\n,\\quad\n\\boldsymbol{n}^0=\\cos\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_1+\n\\sin\\dfrac{\\theta}{r_0}\\, \\boldsymbol{e}_2\\,,$$\nso that $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ is orthonormal.\n\\end{remark}\\smallskip\n\nIn view of \\eqref{49,1}--\\eqref{50}, we obtain in this case that $\\boldsymbol{Q}^{0}=\\boldsymbol{P}$ (since $\\boldsymbol{U}^0= {1\\!\\!\\!\\:1 } _3$ and $\\text{polar}(\\boldsymbol{P})=  \\boldsymbol{P}\\in SO(3)$) and the directors $\\{\\boldsymbol{d}_i^0\\}$ in the reference configuration coincide with $\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}$ in each point of $S^0\\,$, i.e.\n\\begin{equation}\\label{51}\n    \\boldsymbol{d}_\\alpha^0=\\boldsymbol{a}_\\alpha= \\partial_\\alpha \\boldsymbol{y}^0\\,,\\qquad\n    \\boldsymbol{d}_3^0= \\boldsymbol{a}_3=\\boldsymbol{n}^0,\\qquad \\boldsymbol{a}_i= \\boldsymbol{Q}^{0}\\boldsymbol{e}_i\\,,\\qquad \\boldsymbol{d}_i=\\boldsymbol{R} \\boldsymbol{Q}^{0,T}\\boldsymbol{a}_i\\,.\n\\end{equation}\nThe expressions of the elastic strain measures $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ may be simplified  in this situation. By virtue of \\eqref{50} and \\eqref{51} we get\n\\begin{equation}\\label{52}\n\\begin{array}{l}\n    \\boldsymbol{Q}^{0} \\big(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} -  \\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{y}^0\\big) = \\boldsymbol{Q}^{0}  \\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} -  \\boldsymbol{a}_{ \\alpha} \\vspace{3pt}\\\\\n    \\qquad\\qquad\\quad\\qquad\\qquad\\qquad\n     = \\big( \\boldsymbol{a}_i\\cdot\\boldsymbol{Q}^{0}  \\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i} \\big) \\boldsymbol{a}_{ i}\n     = \\big( \\boldsymbol{d}_i\\cdot\\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i} \\big) \\boldsymbol{a}_{ i}\\,.\n     \\end{array}\n\\end{equation}\nWe can write\n$\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}=(\\boldsymbol{d}_{i}\\otimes\\boldsymbol{e}_i)^T (\\partial_\\alpha \\boldsymbol{d}_{j}\\otimes\\boldsymbol{e}_j)= (\\boldsymbol{d}_{i}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} )\n\\boldsymbol{e}_i\\otimes \\boldsymbol{e}_j\\,$,\nso that we find\n$$\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R})= \\dfrac{1}{2}\\,e_{ijk}(\\boldsymbol{d}_{k}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} )\\boldsymbol{e}_i\\,,\\qquad \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})= \\dfrac{1}{2}\\,e_{ijk}(\\boldsymbol{a}_{k}\\cdot \\partial_\\alpha \\boldsymbol{a}_{j} )\\boldsymbol{e}_i\\,,\n$$\nwhere $e_{ijk}$ is the permutation symbol. The last relations and $\\boldsymbol{Q}^{0,T} \\boldsymbol{a}_i =\\boldsymbol{e}_i$ yield\n\\begin{equation}\\label{53}\n\\begin{array}{c}\n    \\boldsymbol{Q}^{0} \\big[\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}) - \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})\\big]= \\big[\\big(\\text{axl}(\\boldsymbol{R}^T\\partial_\\alpha \\boldsymbol{R}) - \\text{axl}(\\boldsymbol{Q}^{0,T} \\partial_\\alpha \\boldsymbol{Q}^{0})\\big)\\!\\cdot\\! \\boldsymbol{e}_i\\big]\\boldsymbol{a}_i \\\\\n    =\n    \\dfrac{1}{2}\\,e_{ijk}\\big[ (\\boldsymbol{d}_{k}\\cdot \\partial_\\alpha \\boldsymbol{d}_{j} ) - (\\boldsymbol{a}_{k}\\cdot \\partial_\\alpha \\boldsymbol{a}_{j} )\\big]\\boldsymbol{a}_i\\,.\n    \\end{array}\n\\end{equation}\nUsing the relations \\eqref{8}, \\eqref{12}, \\eqref{52} and \\eqref{53} we decompose the strain tensor $\\boldsymbol{E}^e$ and the curvature tensor $\\boldsymbol{K}^e$ in the basis $\\{\\boldsymbol{a}_i\\otimes \\boldsymbol{a}_\\alpha\\}$ as follows\n\\begin{equation}\\label{54}\n\\begin{array}{l}\n    \\boldsymbol{E}^e=\\tilde{E}^e_{i\\alpha}\\boldsymbol{a}_i\\otimes\\boldsymbol{a}_\\alpha\\,,\\qquad\n    \\tilde{E}^e_{i\\alpha}= \\boldsymbol{d}_i  \\cdot   \\partial_\\alpha \\boldsymbol{y} - \\delta_{\\alpha i}\\,,\\qquad\n        \\boldsymbol{K}^e=\\tilde{K}^e_{i\\alpha}\\boldsymbol{a}_i\\otimes\\boldsymbol{a}_\\alpha\\,,\\\\\n        \\tilde{K}^e_{1\\alpha}= \\boldsymbol{d}_3\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{2} \\! -\\! \\boldsymbol{a}_3\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{2}, \\quad\n    \\tilde{K}^e_{2\\alpha}= \\boldsymbol{d}_1\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{3} \\!- \\! \\boldsymbol{a}_1\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{3}, \\quad\\!\n    \\tilde{K}^e_{3\\alpha}= \\boldsymbol{d}_2\\!\\cdot\\!\\partial_\\alpha \\boldsymbol{d}_{1} \\!-\\! \\boldsymbol{a}_2 \\!\\cdot\\!\\partial_\\alpha \\boldsymbol{a}_{1} .\n\\end{array}\n\\end{equation}\nFor later reference, we introduce the notations\n\\begin{equation}\\label{54,1}\n\\boldsymbol{E}^e_{\\parallel}=\\boldsymbol{E}^e- (\\boldsymbol{n}^0\\otimes\\boldsymbol{n}^0) \\boldsymbol{E}^e,\\qquad\\boldsymbol{K}^e_{\\parallel}=\\boldsymbol{K}^e- (\\boldsymbol{n}^0\\otimes\\boldsymbol{n}^0) \\boldsymbol{K}^e\\,.\n\\end{equation}\nThen, from \\eqref{54} we get\n\\begin{equation}\\label{55}\n\\begin{array}{l}\n    \\boldsymbol{E}^e_{\\parallel}=\\tilde{E}^e_{\\alpha\\beta}\\boldsymbol{a}_\\alpha\\otimes\\boldsymbol{a}_\\beta\\,,\n    \\qquad\n     \\boldsymbol{n}^0\\boldsymbol{E}^e=\\boldsymbol{E}^{e,T}\\boldsymbol{n}^0 =\\tilde{E}^e_{3\\alpha}\\boldsymbol{a}_\\alpha\\,, \\\\\n     \\boldsymbol{K}^e_{\\parallel}=\\tilde{K}^e_{\\alpha\\beta}\\boldsymbol{a}_\\alpha\\otimes\\boldsymbol{a}_\\beta\\,,\n    \\qquad\n     \\boldsymbol{n}^0\\boldsymbol{K}^e=\\boldsymbol{K}^{e,T}\\boldsymbol{n}^0 =\\tilde{K}^e_{3\\alpha }\\boldsymbol{a}_\\alpha\\,.\n     \\end{array}\n\\end{equation}\nIf we denote the matrices by $\\tilde{E}^e=\\big(\\tilde{E}^e_{ij}\\big)_{3\\times 3}\\,$, $\\tilde{K}^e=\\big(\\tilde{K}^e_{ij}\\big)_{3\\times 3}\\,$, and also\n\\begin{equation}\\label{55bis}\n\\begin{array}{l}\n    \\tilde{E}^e_{\\parallel}=   \\begin{bmatrix} \\tilde{E}^e_{11} & \\tilde{E}^e_{12}  \\\\\n                                 \\tilde{E}^e_{21} & \\tilde{E}^e_{22}    \\end{bmatrix}=\n    \\begin{bmatrix} \\boldsymbol{d}_1  \\cdot    \\partial_1 \\boldsymbol{y} \\!-\\!1 \\,\\, & \\,\\,\n    \\boldsymbol{d}_1  \\cdot   \\partial_2 \\boldsymbol{y} \\\\\n    \\boldsymbol{d}_2  \\cdot     \\partial_1 \\boldsymbol{y} \\,\\, & \\,\\, \\boldsymbol{d}_2  \\cdot   \\partial_2 \\boldsymbol{y}\\!-\\!1\n    \\end{bmatrix}, \\vspace{4pt}\\\\\n    \\big(\\tilde{E}^{e,T}n^0\\big)= \\begin{bmatrix}       \\tilde{E}^e_{31} & \\tilde{E}^e_{32}    \\end{bmatrix}=\n     \\begin{bmatrix}    \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{y} \\,\\, & \\,\\,\n     \\boldsymbol{d}_3 \\! \\cdot   \\partial_2 \\boldsymbol{y}     \\end{bmatrix},\n    \\vspace{4pt}\\\\\n     \\tilde{K}^e_{\\parallel}= \\begin{bmatrix} \\tilde{K}^e_{11} & \\tilde{K}^e_{12}  \\\\\n                                 \\tilde{K}^e_{21} & \\tilde{K}^e_{22}    \\end{bmatrix} =\n    \\begin{bmatrix} \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{d}_2 \\!-\\! \\boldsymbol{a}_3 \\! \\cdot    \\partial_1 \\boldsymbol{a}_2 \\,\\, &\n    \\,\\, \\boldsymbol{d}_3 \\! \\cdot    \\partial_2 \\boldsymbol{d}_2 \\!-\\! \\boldsymbol{a}_3 \\! \\cdot    \\partial_2 \\boldsymbol{a}_2   \\\\\n    \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{d}_3 \\!-\\! \\boldsymbol{a}_1 \\! \\cdot    \\partial_1 \\boldsymbol{a}_3 \\,\\,   & \\,\\, \\boldsymbol{d}_1 \\! \\cdot    \\partial_2 \\boldsymbol{d}_3  \\!-\\! \\boldsymbol{a}_1 \\! \\cdot    \\partial_2 \\boldsymbol{a}_3   \\end{bmatrix},\n                                \\vspace{4pt} \\\\\n    \\big(\\tilde{K}^{e,T}n^0\\big)= \\begin{bmatrix}       \\tilde{K}^e_{31} & \\tilde{K}^e_{32}    \\end{bmatrix}=\n    \\begin{bmatrix}   \\boldsymbol{d}_2  \\cdot    \\partial_1 \\boldsymbol{d}_1 \\!-\\! \\boldsymbol{a}_2  \\cdot    \\partial_1 \\boldsymbol{a}_1  \\,\\, & \\,\\, \\boldsymbol{d}_2  \\cdot    \\partial_2 \\boldsymbol{d}_1 \\!-\\! \\boldsymbol{a}_2  \\cdot    \\partial_2 \\boldsymbol{a}_1  \\end{bmatrix},\n     \\end{array}\n\\end{equation}\n then the relations \\eqref{54} and \\eqref{55} can be written in matrix form\n\\begin{equation}\\label{55,1}\n\\begin{array}{l}\n    \\tilde{E}^e=\\begin{bmatrix} \\tilde{E}^e_{11} & \\tilde{E}^e_{12} &  0 \\\\\n                                 \\tilde{E}^e_{21} & \\tilde{E}^e_{22} &  0 \\\\\n                                 \\tilde{E}^e_{31} & \\tilde{E}^e_{32} & 0\n    \\end{bmatrix}\\! = \\!\\! \\begin{bmatrix} \\big(\\,\\tilde{E}^e_{\\parallel}\\,\\big)_{2\\times 2} & 0_{2\\times 1}  \\vspace{10pt} \\\\\n                                 \\big(\\tilde{E}^{e,T}\\!n^0\\big)_{1\\times 2} & 0\n    \\end{bmatrix}_{3\\times 3}\\!\\! \\!\\!=\\!\\! \\begin{bmatrix} \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{y} \\!-\\!1 & \\boldsymbol{d}_1 \\! \\cdot   \\partial_2 \\boldsymbol{y} &  \\!\\!0 \\\\\n                                \\boldsymbol{d}_2 \\! \\cdot     \\partial_1 \\boldsymbol{y}  & \\boldsymbol{d}_2 \\! \\cdot   \\partial_2 \\boldsymbol{y}\\!-\\!1 & \\!\\! 0 \\\\\n                                 \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{y}  & \\boldsymbol{d}_3 \\! \\cdot   \\partial_2 \\boldsymbol{y} &  \\!\\!0\n    \\end{bmatrix}\\!,\\vspace{4pt}\\\\\n    \\tilde{K}^e=\\begin{bmatrix} \\tilde{K}^e_{11} & \\tilde{K}^e_{12} &  0 \\\\\n                                 \\tilde{K}^e_{21} & \\tilde{K}^e_{22} &  0 \\\\\n                                 \\tilde{K}^e_{31} & \\tilde{K}^e_{32} & 0\n    \\end{bmatrix}  =  \\begin{bmatrix} \\big(\\,\\tilde{K}^e_{\\parallel}\\,\\big)_{2\\times 2} & 0_{2\\times 1}  \\vspace{10pt} \\\\\n                                 \\big(\\tilde{K}^{e,T}\\!n^0\\big)_{1\\times 2} & 0\n    \\end{bmatrix}_{3\\times 3} \\vspace{4pt}\\\\\n    \\qquad\\qquad = \\tilde{K}- \\tilde{K}^0 = \\begin{bmatrix} \\boldsymbol{d}_3 \\! \\cdot    \\partial_1 \\boldsymbol{d}_2  & \\boldsymbol{d}_3 \\! \\cdot    \\partial_2 \\boldsymbol{d}_2  &  0 \\\\\n                                \\boldsymbol{d}_1 \\! \\cdot    \\partial_1 \\boldsymbol{d}_3   & \\boldsymbol{d}_1 \\! \\cdot    \\partial_2 \\boldsymbol{d}_3  &  0 \\\\\n                                \\boldsymbol{d}_2 \\! \\cdot    \\partial_1 \\boldsymbol{d}_1   & \\boldsymbol{d}_2 \\! \\cdot    \\partial_2 \\boldsymbol{d}_1  & 0\n    \\end{bmatrix} -\n    \\begin{bmatrix} \\boldsymbol{a}_3 \\! \\cdot    \\partial_1 \\boldsymbol{a}_2  & \\boldsymbol{a}_3 \\! \\cdot    \\partial_2 \\boldsymbol{a}_2  &  0 \\\\\n                                \\boldsymbol{a}_1 \\! \\cdot    \\partial_1 \\boldsymbol{a}_3   & \\boldsymbol{a}_1 \\! \\cdot    \\partial_2 \\boldsymbol{a}_3  &  0 \\\\\n                                \\boldsymbol{a}_2 \\! \\cdot    \\partial_1 \\boldsymbol{a}_1   & \\boldsymbol{a}_2 \\! \\cdot    \\partial_2 \\boldsymbol{a}_1  & 0\n    \\end{bmatrix}\n         \\end{array}\n\\end{equation}\nThese expressions are completely similar to the strain measures for planar--shells introduced in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\nLet us discuss next some important classes of elastic shells.\n\n\n\\subsection{Isotropic shells}\n\n\nIn the resultant 6-parameter theory of shells, the strain energy density for isotropic shells has been presented in various forms. The simplest expression of $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ has been proposed in the papers \\cite{Pietraszkiewicz-book04,Pietraszkiewicz10} in the form\n\\begin{equation}\\label{56}\n   \\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= \\,\\,\\,C\\big[\\,\\nu \\,(\\mathrm{tr} \\boldsymbol{E}^e_{\\parallel})^2 +(1-\\nu)\\, \\mathrm{tr}(\\boldsymbol{E}^{e,T}_{\\parallel}  \\boldsymbol{E}^e_{\\parallel} )\\big]  + \\alpha_{s\\,}C(1-\\nu) \\, \\boldsymbol{n}^0 \\boldsymbol{E}^e \\boldsymbol{E}^{e,T}  \\boldsymbol{n}^0 \\\\\n    \\qquad\\qquad\\qquad\\,\\, +\\,D \\big[\\,\\nu\\,(\\mathrm{tr} \\boldsymbol{K}^e_{\\parallel})^2 + (1-\\nu)\\, \\mathrm{tr}(\\boldsymbol{K}^{e,T}_{\\parallel}  \\boldsymbol{K}^e_{\\parallel} )\\big]  + \\alpha_{t\\,}D(1-\\nu) \\,    \\boldsymbol{n}^0 \\boldsymbol{K}^e \\boldsymbol{K}^{e,T}  \\boldsymbol{n}^0,\n\\end{array}\n\\end{equation}\nwhere $C=\\frac{E\\,h}{1-\\nu^2}\\,$ is the stretching (in-plane) stiffness of the shell, $D=\\frac{E\\,h^3}{12(1-\\nu^2)}\\,$ is the bending stiffness, $h$ is the thickness of the shell, and $\\alpha_s\\,$, $\\alpha_t$ are two shear correction factors. Also,    $E $ and $ \\nu$ denote the Young modulus and  Poisson ratio of the isotropic and homogeneous material. By the numerical treatment of non-linear shell problems, the values of the shear correction factors have been set to  $\\alpha_s=5\/6$, $\\alpha_t=7\/10$ in \\cite{Pietraszkiewicz10}. The value $\\alpha_s=5\/6$ is a classical suggestion, which has been previously deduced analytically by Reissner in the case of plates \\cite{Reissner45,Naghdi72}.\nAlso, the value $\\,\\alpha_t=7\/10\\,$ was proposed earlier in \\cite[see p.78]{Pietraszkiewicz-book79} and has been suggested in the work \\cite{Pietraszkiewicz79}.\nHowever, the discussion concerning the possible values of  shear correction factors for shells is long and controversial in the literature \\cite{Naghdi72,Naghdi-Rubin95}.\n\nWith the help of the matrices \\eqref{55bis}, we can express the strain energy density \\eqref{56}   in the alternative form\n\\begin{equation}\\label{57}\n\\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= \\\\\n    =C(1\\!-\\!\\nu)\\big(\\|\\text{dev}_2\\,\\text{sym}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\!+ \\|\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\big)\\!+\n    C \\dfrac{1\\!+\\!\\nu}{2}\\big(\\text{tr}\\, \\tilde{E}^e_{\\parallel}\\big)^2\n         + \\alpha_sC(1\\!-\\!\\nu)\\,\\|\\tilde{E}^{e,T}n^0  \\|^2 \\vspace{2pt}\\\\\n    +D(1\\!-\\!\\nu)\\big(\\|\\text{dev}_2\\,\\text{sym}\\, \\tilde{K}^e_{\\parallel}\\,\\|^2\\!+ \\|\\text{skew}\\, \\tilde{K}^e_{\\parallel}\\,\\|^2\\big)\\!\n         + D \\dfrac{1\\!+\\!\\nu}{2}\\big(\\text{tr}\\, \\tilde{K}^e_{\\parallel}\\big)^2 + \\alpha_tD(1\\!-\\!\\nu)\\,\\|\\tilde{K}^{e,T}n^0   \\|^2\\!,\n\\end{array}\n\\end{equation}\nwhere $\\,\\text{sym}\\, X= \\frac{1}{2}\\big(X+X^T\\big)$ is the symmetric part, $\\,\\text{skew}\\, X= \\frac{1}{2}\\big(X-X^T\\big)$ is the skew-symmetric part,\nand $\\,\\,\\text{dev}_2\\, X= X-\\frac{1}{2}\\big(\\text{tr}\\,X\\big) {1\\!\\!\\!\\:1 } _2\\,\\,$ is the deviatoric part of any $2\\times 2$ matrix  $X $. The coefficients in \\eqref{57} are expressed in terms of the Lam\\'e constants of the material $\\lambda$ and $\\mu$  by the relations\n\\begin{equation*\n    C \\dfrac{1\\!+\\!\\nu}{2}\\,=h\\,\\dfrac{\\mu(2\\mu\\!+\\!3\\lambda)}{2\\mu+\\lambda}\\,,\\quad\\! C(1\\!-\\!\\nu)=2\\mu h,\\quad\\!\n    D \\dfrac{1\\!+\\!\\nu}{2}\\,=\\dfrac{h^3}{12}\\,\\,\\dfrac{\\mu(2\\mu\\!+\\!3\\lambda)}{2\\mu+\\lambda}\\,,\\quad\\!\n    D(1\\!-\\!\\nu)=\\dfrac{\\mu h^3}{6}\\,.\n\\end{equation*}\nThen, we obtain  that the given quadratic form \\eqref{57} is positive definite if and only if the coefficients $E$ and $\\nu$ satisfy the inequalities\n\\begin{equation}\\label{57,1}\n    E>0,\\qquad -1<\\nu<\\dfrac{1}{2}\\,\\,.\n\\end{equation}\nIn terms of  the  Lam\\'e moduli of the material,   the  inequalities \\eqref{57,1} are equivalent to\n$$\\mu>0,\\qquad 2\\mu+3\\lambda>0.$$\nThese conditions are guaranteed by the positive definiteness of the 3D quadratic elastic strain energy for isotropic materials.\nThus, we find that the strain energy $W$ is convex and satisfies the coercivity condition \\eqref{26bis}, so that the hypotheses of Theorem \\ref{th1} are fulfilled. Applying Theorem \\ref{th1} we obtain (under suitable assumptions on the given load and boundary data, and the reference configuration $( \\boldsymbol{y}^0,\\boldsymbol{Q}^{0})$) the existence of minimizers for isotropic shells with strain energy density in the form \\eqref{56}.\\smallskip\n\nIn \\cite{Eremeyev06}, Eremeyev and Pietraszkiewicz have proposed a more general form of the strain energy density, namely\n\\begin{equation}\\label{58}\n    \\begin{array}{l}\n   2 W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)=  \\alpha_1\\big(\\mathrm{tr}  \\boldsymbol{E}^e_{\\parallel}\\big)^2 +\\alpha_2  \\mathrm{tr} \\big(\\boldsymbol{E}^e_{\\parallel}\\big)^2    + \\alpha_3 \\mathrm{tr}\\big(\\boldsymbol{E}^{e,T}_{\\parallel}  \\boldsymbol{E}^e_{\\parallel} \\big)  + \\alpha_4     \\boldsymbol{n}^0 \\boldsymbol{E}^e \\boldsymbol{E}^{e,T}  \\boldsymbol{n}^0 \\\\\n    \\qquad\\qquad\\qquad + \\beta_1\\big(\\mathrm{tr}  \\boldsymbol{K}^e_{\\parallel}\\big)^2 +\\beta_2  \\mathrm{tr} \\big(\\boldsymbol{K}^e_{\\parallel}\\big)^2    + \\beta_3 \\mathrm{tr}\\big(\\boldsymbol{K}^{e,T}_{\\parallel} \\boldsymbol{K}^e_{\\parallel} \\big)  + \\beta_4     \\boldsymbol{n}^0 \\boldsymbol{K}^e \\boldsymbol{K}^{e,T}  \\boldsymbol{n}^0.\n\\end{array}\n\\end{equation}\nThe eight coefficients $\\alpha_k\\,$, $\\beta_k$ ($k=1,2,3,4$) can depend in general on the structure curvature tensor  $ \\boldsymbol{K}^0=\\text{axl}\\big(\\partial_\\alpha \\boldsymbol{Q}^{0}\\boldsymbol{Q}^{0,T}\\big) \\otimes \\boldsymbol{a}^\\alpha$ of the reference configuration. For the sake of simplicity, we assume in our discussion that the coefficients $\\alpha_k$ and $\\beta_k$ are constant.\nWe can decompose the strain energy density  \\eqref{58}  in the  in-plane part  $W_{\\text{plane}}(\\boldsymbol{E}^e)$ and the curvature part $W_{\\text{curv}}(\\boldsymbol{K}^e)$ and write their expressions using the matrices of components \\eqref{55bis} in  form\n\\begin{equation}\\label{59}\n    \\begin{array}{c}\n    W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= W_{\\text{plane}}(\\boldsymbol{E}^e\\big)+ W_{\\text{curv}}(\\boldsymbol{K}^e\\big)\\,,\n\\end{array}\n\\end{equation}\n\\begin{equation*}\n    \\begin{array}{c}\n    2W_{\\text{plane}}(\\boldsymbol{E}^e)= (\\alpha_2\\!+\\!\\alpha_3) \\| \\text{sym}\\, \\tilde{E}^e_{\\parallel}\\|^2\\!+ (\\alpha_3\\!-\\!\\alpha_2)\\|\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\|^2\\!+ \\alpha_1\\big(\\mathrm{tr}  \\tilde{E}^e_{\\parallel}\\big)^2 +  \\alpha_4\\|\\tilde{E}^{e,T}n^0  \\|^2, \\vspace{4pt}\\\\\n    2 W_{\\text{curv}}(\\boldsymbol{K}^e)= (\\beta_2\\!+\\!\\beta_3) \\| \\text{sym}\\, \\tilde{K}^e_{\\parallel}\\|^2\\!+ (\\beta_3\\!-\\!\\beta_2)\\|\\text{skew}\\, \\tilde{K}^e_{\\parallel}\\|^2\\!+ \\beta_1\\big(\\mathrm{tr}  \\tilde{K}^e_{\\parallel}\\big)^2\n    + \\beta_4\\|\\tilde{K}^{e,T}n^0  \\|^2.\n    \\end{array}\n\\end{equation*}\nThe in-plane part of the energy density \\eqref{59} can be written equivalently as\n\\begin{equation}\\label{59,1}\n \\begin{array}{l}\n    2W_{\\text{plane}}(\\boldsymbol{E}^e)= \\underbrace{(\\alpha_2+\\alpha_3)\\, \\|\\,\\text{dev}_2 \\,\\text{sym}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2 }_{\\text{in-plane shear--stretch energy}}  \\,+\\, \\underbrace{(\\alpha_3-\\alpha_2)\\,\\|\\,\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\,}_{\\text{in--plane drill rotation energy}}\\vspace{4pt}\\\\\n    \\qquad\\qquad\\quad\\,\\,\\, +\\,\\underbrace{\\Big(\\alpha_1+\\dfrac{\\alpha_2 + \\alpha_3}{2}\\Big)\\big(\\mathrm{tr} \\, \\tilde{E}^e_{\\parallel}\\big)^2}_{\\text{in-plane elongational stretch energy}}  \\,+\\,\\underbrace{\\, \\alpha_4\\,\\|\\,\\tilde{E}^{e,T}n^0 \\, \\|^2}_{\\text{transverse shear energy}}.\n    \\end{array}\n\\end{equation}\nThe above forms of the strain energy $W$ are expressed in terms of the components of the tensors $\\boldsymbol{E}^e$ and $\\boldsymbol{K}^e$ in the basis $\\{\\boldsymbol{a}_i\\otimes \\boldsymbol{a}_\\alpha\\}\\,$, i.e. in terms of the elements of the matrices \\eqref{55bis}.\nDenoting with $\\,\\,\\mu_c^{\\text{drill}}\\,$ the coefficient $\\,(\\alpha_3-\\alpha_2)\\,$  in   \\eqref{59,1}, we remark that the   term\n\\begin{equation}\\label{59,2}\n    \\mu_c^{\\text{drill}}\\,\\|\\,\\text{skew}\\, \\tilde{E}^e_{\\parallel}\\,\\|^2\\,,\\qquad \\text{with}\\qquad \\mu_c^{\\text{drill}}\\,:=\\,\\alpha_3-\\alpha_2\\,,\n\\end{equation}\ndescribes the quadratic in-plane drill rotation energy of the shell. We call the coefficient $\\,\\,\\mu_c^{\\text{drill}}\\,$  the \\emph{linear in-plane rotational couple modulus}, in analogy to the Cosserat couple modulus in the three-dimensional Cosserat theory \\cite{Neff_zamm06}.\n\\begin{remark}\nThe planar isotropic Cosserat shells have been investigated also in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}, using a model derived directly from the 3D equations of Cosserat elasticity. We mention that the expressions \\eqref{59}, \\eqref{59,1} of the strain energy density are essentially the same as the strain energy of the Cosserat model for planar shells \\cite{Neff_plate04_cmt}. By comparing these two approaches (6-parameter resultant shells and Cosserat model) we deduce the following identification of the constitutive coefficients $\\alpha_1\\,,...,\\alpha_4$\n\\begin{equation}\\label{59,3}\n    \\alpha_1=h\\,\\dfrac{2\\mu\\lambda}{2\\mu+\\lambda}\\,,\\quad \\alpha_2=h(\\mu-\\mu_c),\\quad \\alpha_3=h(\\mu+\\mu_c),\\quad \\alpha_4=\\kappa\\, h (\\mu+\\mu_c),\n\\end{equation}\nwhere $\\,\\mu_c\\,$ is the Cosserat couple modulus of the 3D continuum, and $\\,\\kappa\\,$ is a formal shear correction factor. From \\eqref{59,2}, \\eqref{59,3} we observe that\n\\begin{equation}\\label{59,4}\n    \\mu_c^{\\mathrm{drill}}\\,=\\,\\alpha_3-\\alpha_2\\,=\\,2h\\,\\mu_c\\,,\n\\end{equation}\nwhich means that the in-plane rotational couple modulus $\\,\\mu_c^{\\mathrm{drill}}\\,$ of the Cosserat shell model is determined by the Cosserat couple modulus $\\,\\mu_c\\,$ of the 3D Cosserat material.\n\nThe relations \\eqref{59,3} are similar to the corresponding relations in the linear theory of micropolar plates, see \\cite[Eqs.(45)]{Altenbach-Erem09}. From a mathematical viewpoint, the difference between the two sets of relations consists in the notations used and the value of the shear correction factor.\n\\end{remark}\n\n\nLooking at \\eqref{59} and \\eqref{59,1} we observe that  the quadratic form $ W(\\boldsymbol{E}^e,\\boldsymbol{K}^e) $ is positive definite  if and only if the coefficients verify the conditions\n\\begin{equation}\\label{60}\n    \\begin{array}{l}\n    2\\alpha_1+\\alpha_2+\\alpha_3>0,\\quad \\alpha_2+\\alpha_3>0,\\quad \\alpha_3-\\alpha_2>0, \\quad\\alpha_4>0,\\\\\n    2\\beta_1+\\beta_2+\\beta_3>0,\\quad \\beta_2+\\beta_3>0,\\quad \\beta_3-\\beta_2>0,\\quad  \\beta_4>0,\n\\end{array}\n\\end{equation}\nProvided that the conditions \\eqref{60} are satisfied, the strain energy function $W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)$ is convex and coercive in the sense of \\eqref{26bis}. By virtue of Theorem \\ref{th1}, in this case the minimization problem associated to the deformation of isotropic elastic shells admits at least one solution.\n\n\\begin{remark}\nThe same conditions \\eqref{60} have been imposed in \\cite{EremeyevLebedev11} to establish existence results in the \\emph{linearized} theory of micropolar (6-parameter) shells.\n\\end{remark}\n\\begin{remark}\nThe case $\\,\\mu_c^{\\mathrm{drill}}=0\\,$ (i.e., $\\alpha_3-\\alpha_2=0$) is not uniformly positive definite. However, with a slight change of the resultant shell model, one can prove the existence of minimizers using similar methods as in \\cite{Neff_plate07_m3as}. A linearization of such a model  leads exactly to the Reissner kinematics with 5 degrees of freedom \\cite{Neff_plate07_m3as}, where the in-plane drill rotation is absent. The physical meaning of the in-plane rotational stiffness $\\,\\mu_c^{\\mathrm{drill}}=\\alpha_3-\\alpha_2\\,$ in the resultant shell model is not entirely clear to us.\n\nSince only two independent rotations are required to orient a unit director field, a distinctive feature of classical plate and shell theories is a rotation field defined in terms of only \\emph{two} independent degrees of freedom. Rotations about the director itself -- the so-called drill rotation -- are irrelevant and for that matter undefined in classical shell theory.\n\\end{remark}\n\n\n\n\\subsection{Orthotropic shells}\n\n\n\nThe constitutive equations for orthotropic shells have been presented in \\cite{Eremeyev06} within the 6-parameter resultant shell  theory. The expression of the strain energy density in terms of the tensor components defined in \\eqref{54} is given by\n\\begin{equation}\\label{61}\n   2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= C_{\\alpha\\beta\\gamma\\delta}^E\\, \\tilde{E}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta} +  D_{\\alpha\\beta}^E\\, \\tilde{E}^e_{3\\alpha } \\tilde{E}^e_{3\\beta} +\n      C_{\\alpha\\beta\\gamma\\delta}^K\\,   \\tilde{K}^e_{\\alpha\\beta} \\tilde{K}^e_{\\gamma\\delta}+      D_{\\alpha\\beta}^K\\, \\tilde{K}^e_{3\\alpha } \\tilde{K}^e_{3\\beta}\n\\end{equation}\nwhere $C_{\\alpha\\beta\\gamma\\delta}^E\\,$, $C_{\\alpha\\beta\\gamma\\delta}^K\\,$, $D_{\\alpha\\beta}^E\\,$ and $D_{\\alpha\\beta}^K\\,$ are material constants which satisfy the following symmetry relations\n$$ C_{\\alpha\\beta\\gamma\\delta}^E=  C_{\\gamma\\delta\\alpha\\beta}^E\\,,\\quad D_{\\alpha\\beta}^E= D_{\\beta\\alpha}^E\\,,\\quad\n      C_{\\alpha\\beta\\gamma\\delta}^K=  C_{\\gamma\\delta\\alpha\\beta}^K\\,,\\quad D_{\\alpha\\beta}^K= D_{\\beta\\alpha}^K\\,.$$\nWe observe that the quadratic function \\eqref{61} is coercive if and only if the following symmetric matrices are positive definite\n\\begin{equation}\\label{62}\n    \\begin{bmatrix} C_{1111}^E & C_{1122}^E  &  C_{1112}^E  &   C_{1121}^E   \\\\\n                    C_{1122}^E & C_{2222}^E  &   C_{2212}^E  &   C_{2221}^E   \\\\\n                    C_{1112}^E  & C_{2212}^E   &  C_{1212}^E &   C_{1221}^E  \\\\\n                    C_{1121}^E  &  C_{2221}^E   &  C_{1221}^E   &  C_{2121}^E   \\end{bmatrix},\n    \\begin{bmatrix} C_{1111}^K & C_{1122}^K  &  C_{1112}^K  &   C_{1121}^K   \\\\\n                    C_{1122}^K & C_{2222}^K  &   C_{2212}^K  &   C_{2221}^K   \\\\\n                    C_{1112}^K  & C_{2212}^K   &  C_{1212}^K &   C_{1221}^K  \\\\\n                    C_{1121}^K  &  C_{2221}^K   &  C_{1221}^K   &  C_{2121}^K   \\end{bmatrix},\n    \\begin{bmatrix} D_{11 }^E & D_{12}^E   \\\\\n                    D_{12}^E & D_{22}^E    \\end{bmatrix},\n    \\begin{bmatrix} D_{11 }^K & D_{12}^K   \\\\\n                    D_{12}^K & D_{22}^K    \\end{bmatrix}\\!.\n\\end{equation}\nIn the situation when the matrices \\eqref{62} are positive definite, then the strain energy $W$ given by \\eqref{61} satisfies the hypotheses of Theorem \\ref{th1}. Then, we can use our theoretical results to derive the existence of minimizers for orthotropic shells.\n\n\n\n\\subsection{Composite layered shells}\n\n\n\nLet us analyze the case of composite shells made of a finite number of individually homogeneous layers. According to \\cite{Chroscielewski11}, the strain energy density of such type of shells can be written by means of the tensor components \\eqref{54} in the form\n\\begin{equation}\\label{63}\n\\begin{array}{l}\n    2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= A_{\\alpha\\beta\\gamma\\delta} \\, \\tilde{E}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta} +  D_{\\alpha\\beta\\gamma\\delta} \\, \\tilde{K}^e_{\\alpha \\beta} \\tilde{K}^e_{\\gamma\\delta} +\n      B_{\\alpha\\beta\\gamma\\delta} ( \\tilde{E}^e_{\\alpha\\beta} \\tilde{K}^e_{\\gamma\\delta} + \\tilde{K}^e_{\\alpha\\beta} \\tilde{E}^e_{\\gamma\\delta})\\vspace{4pt}\\\\\n      \\qquad\\qquad\\qquad\\quad\n      +      S_{\\alpha\\beta} \\, \\tilde{E}^e_{3\\alpha } \\tilde{E}^e_{3\\beta}\n        +      G_{\\alpha\\beta} \\,        \\tilde{K}^e_{3\\alpha } \\tilde{K}^e_{3\\beta}\\,,\n        \\end{array}\n\\end{equation}\nwhere $A_{\\alpha\\beta\\gamma\\delta}\\,$, $B_{\\alpha\\beta\\gamma\\delta}\\,$, $D_{\\alpha\\beta\\gamma\\delta}\\,$, $S_{\\alpha\\beta}\\,$ and $G_{\\alpha\\beta}\\,$ are the constitutive coefficients of composite elastic shells, which have been determined in \\cite{Chroscielewski11} in terms of the material\/geometrical parameters of the layers. They satisfy the symmetry conditions\n$$A_{\\alpha\\beta\\gamma\\delta}=  A_{\\gamma\\delta\\alpha\\beta}\\,,\\quad D_{\\alpha\\beta\\gamma\\delta} = D_{\\gamma\\delta\\alpha\\beta}\\,,\\quad\n      S_{\\alpha\\beta}=  S_{\\beta\\alpha}\\,,\\quad G_{\\alpha\\beta} = G_{\\beta\\alpha}\\,.$$\nIn the constitutive relation \\eqref{63} one can observe a multiplicative coupling of the strain tensor $\\boldsymbol{E}^e$ with the curvature tensor $\\boldsymbol{K}^e$ for composite shells. Let us denote by $A$, $D$ and $B$ the $4\\times 4$ matrices of material constants\n\\begin{equation*}\n\\begin{array}{c}\n    A=\\begin{bmatrix} A_{1111}  & A_{1122}   &  A_{1112}   &   A_{1121}    \\\\\n                    A_{1122}  & A_{2222}   &   A_{2212}   &   A_{2221}    \\\\\n                    A_{1112}   & A_{2212}    &  A_{1212}  &   A_{1221}   \\\\\n                    A_{1121}   &  A_{2221}    &  A_{1221}    &  A_{2121}    \\end{bmatrix},\\qquad\n    D=\\begin{bmatrix} D_{1111}  & D_{1122}   &  D_{1112}   &   D_{1121}    \\\\\n                    D_{1122}  & D_{2222}   &   D_{2212}   &   D_{2221}    \\\\\n                    D_{1112}   & D_{2212}    &  D_{1212}  &   D_{1221}   \\\\\n                    D_{1121}   &  D_{2221}    &  D_{1221}    &  D_{2121}    \\end{bmatrix},\\\\\n    B=\\begin{bmatrix} B_{1111}  & B_{1122}   &  B_{1112}   &   B_{1121}    \\\\\n                    B_{2211}  & B_{2222}   &   B_{2212}   &   B_{2221}    \\\\\n                    B_{1211}   & B_{1222}    &  B_{1212}  &   B_{1221}   \\\\\n                    B_{2111}   &  B_{2122}    &  B_{2112}    &  B_{2121}    \\end{bmatrix}.\n\\end{array}\n\\end{equation*}\nOne can show that the necessary and sufficient condition for the coercivity of the strain energy function \\eqref{63} is that the following matrices are positive definite\n\\begin{equation*}\n    \\mathbb{C}=\\begin{bmatrix} A_{4\\times 4 }  & B_{4\\times 4 }   \\\\\n                    B_{4\\times 4 } & D_{4\\times 4 }    \\end{bmatrix}_{8\\times 8}\\,,\\quad\n                    \\mathbb{S}= \\begin{bmatrix} S_{11 }  & S_{12}    \\\\\n                    S_{12}  & S_{22}     \\end{bmatrix}_{2\\times 2}\\,,\\quad\n                    \\mathbb{G}= \\begin{bmatrix} G_{11 }  & G_{12}    \\\\\n                    G_{12}  & G_{22}     \\end{bmatrix}_{2\\times 2}\\,.\n\\end{equation*}\nWith these notations, one may write the strain energy density \\eqref{63} in the matrix form\n\\begin{equation*}\n\\begin{array}{l}\n   2W(\\boldsymbol{E}^e,\\boldsymbol{K}^e)= V\\,\\mathbb{C}\\,V^T+ \\big(\\,\\tilde{E}^e_{31}\\,,\\,\\tilde{E}^e_{32}\\big)\\, \\mathbb{S}\\, \\big(\\,\\tilde{E}^e_{31}\\,,\\,\\tilde{E}^e_{32}\\big)^T + \\big(\\,\\tilde{K}^e_{31}\\,,\\,\\tilde{K}^e_{32}\\big)\\, \\mathbb{G}\\, \\big(\\,\\tilde{K}^e_{31}\\,,\\,\\tilde{K}^e_{32}\\big)^T, \\vspace{4pt}\\\\\n   \\qquad\\quad\\text{with}\\qquad V= \\big(\\,\\tilde{E}^e_{11}\\,,\\,\\tilde{E}^e_{22}\\,,\\, \\tilde{E}^e_{12}\\,,\\,\\tilde{E}^e_{21}\\,,\\,\\tilde{K}^e_{11}\\,,\\,\\tilde{K}^e_{22}\\,,\\, \\tilde{K}^e_{12}\\,,\\,\\tilde{K}^e_{21}\\big)_{1\\times 8}\\,\\,.\n\\end{array}\n\\end{equation*}\nIn conclusion, if the matrices $\\mathbb{C}$, $\\mathbb{S}$ and $\\mathbb{G}$ are positive definite, then we can apply  Theorem \\ref{th1} for the strain energy density given by \\eqref{63} and prove the existence of minimizers for composite layered shells.\n\n\n\\begin{remark}\nThe results and conclusions presented above are obviously valid also in the case of \\emph{plates}, i.e. when the reference base surface $S^0$ is \\emph{planar}. However, many of the formulas  for general shells can be significantly simplified in the case of plates, since   the 3 orthonormal bases\n$\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\boldsymbol{n}^0\\}\\,$, $\\{\\boldsymbol{d}^0_1,\\boldsymbol{d}^0_2,\\boldsymbol{d}^0_3\\}$ and $\\{\\boldsymbol{e}_1,\\boldsymbol{e}_2,\\boldsymbol{e}_3\\}$ can be considered identical.\n\nThe corresponding existence results for 6-parameter geometrically non-linear plates (planar shells) has been presented in \\cite{Birsan-Neff-Plates} for isotropic or anisotropic materials, and in \\cite{Birsan-Neff-AnnRom12} for composite planar--shells. We mention that, in the case of isotropic plates, the existence theorem can be obtained from the more general results concerning Cosserat planar--shells presented by the second author in \\cite{Neff_plate04_cmt,Neff_plate07_m3as}.\n\\end{remark}\n\n\nIn a forthcoming contribution we will extend our existence results to the 6-parameter resultant shell model with physically non-linear behavior and show the invertibility of the reconstructed deformation gradient $\\bar{F}$.\n\n\n\\bigskip\\bigskip\n\\small{\\textbf{Acknowledgements.}\nThe first author (M.B.) is supported by the german state  grant: ``Programm des Bundes und der L\\\"ander f\\\"ur bessere Studienbedingungen und mehr Qualit\\\"at in der Lehre''. We thank our many friends who have made substantial comments on a preliminary version of the paper.\n\n\n\n\n\n\\bibliographystyle{plain}\n{\\footnotesize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA set $\\{a_1, a_2, \\dots, a_m\\}$ of $m$ distinct nonzero rationals\nis called {\\it a rational Diophantine $m$-tuple} if $a_i a_j+1$ is a perfect square\nfor all $1\\leq i < j\\leq m$.\nDiophantus discovered a rational Diophantine quadruple\n$\\{\\frac{1}{16}, \\frac{33}{16}, \\frac{17}{4}, \\frac{105}{16} \\}$.\nThe first example of a Diophantine quadruple in integers, the set $\\{1, 3, 8, 120\\}$,\nwas found by Fermat. In 1969, Baker and Davenport \\cite{B-D} proved that Fermat's set\ncannot be extended to a Diophantine quintuple in integers.\nRecently, He, Togb\\'e and Ziegler proved that there are no Diophantine quintuples\nin integers \\cite{HTZ} (the nonexistence of Diophantine sextuples in integers was proved in \\cite{D-crelle}).\nEuler proved that there are infinitely many rational Diophantine quintuples.\nThe first example of a rational Diophantine sextuple, the\nset $\\{11\/192, 35\/192, 155\/27, 512\/27, 1235\/48, 180873\/16\\}$, was found by Gibbs \\cite{Gibbs1},\nwhile Dujella, Kazalicki, Miki\\'c and Szikszai \\cite{DKMS} recently proved that there are infinitely\nmany rational Diophantine sextuples (see also \\cite{Duje-Matija,DKP,DKP-split}). It is not known whether there\nexists any rational Diophantine septuple.\nFor an overview of results on\nDiophantine $m$-tuples and its generalizations see \\cite{Duje-Notices}.\n\nThe problem of extendibility and existence of Diophantine $m$-tuples\nis closely connected with the properties of the corresponding elliptic curves.\nLet $\\{a,b,c\\}$ be a rational Diophantine triple. Then there exist nonnegative rationals\n$r,s,t$ such that $ab+1=r^2$, $ac+1=s^2$ and $bc+1=t^2$.\nIn order to extend the triple $\\{a,b,c\\}$ to a quadruple,\nwe have to solve the system of equations\n\\begin{equation} \\label{eq2}\nax+1= \\square, \\quad bx+1=\\square, \\quad cx+1=\\square.\n\\end{equation}\nWe assign the following elliptic curve to the system \\eqref{eq2}:\n\\begin{equation} \\label{e1}\nE:\\qquad  y^2=(ax+1)(bx+1)(cx+1).\n\\end{equation}\nWe say that the elliptic curve $E$ is induced by the rational Diophantine triple $\\{a,b,c\\}$.\n\nSince the curve $E$ contains three $2$-torsion points\n$$ A=\\Big( -\\frac{1}{a}, 0 \\Big), \\quad B=\\Big( -\\frac{1}{b}, 0 \\Big), \\quad\nC=\\Big( -\\frac{1}{c}, 0 \\Big), $$\nby Mazur's theorem \\cite{Mazur}, there are at most four possibilities for the torsion group\nover $\\mathbb{Q}$ for such curves: $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/2\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$\nand $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nIn \\cite{D-glasnik}, it was shown that all these torsion groups actually appear. Moreover, it was shown that every elliptic curve with torsion\ngroup $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$ is induced by a Diophantine triple\n(see also \\cite{CG}). Questions about the ranks of elliptic curves induced by Diophantine triples were\nstudied in several papers\n(\\cite{ADP, D-rocky, D-bordo,D-glasnik, D-JB-S, D-Peral-LMSJCM, D-Peral-RACSAM, D-Peral-JGEA,D-Peral-highrank}).\nIn particular, such curves were used\nfor finding elliptic curves with the largest known rank\nover $\\mathbb{Q}$ and $\\mathbb{Q}(t)$ with torsion groups\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$\n(\\cite{D-Peral-LMSJCM,D-Peral-JGEA}) and\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ (\\cite{D-Peral-RACSAM}).\n\nIn this paper, we consider the question how small can be the rank of\nelliptic curves induced by rational Diophantine triples.\nWe will see that it is easy to find rational Diophantine triples with elements with mixed signs\nwhich induce elliptic curves with rank $0$ and that there exist such curves\nwith torsion groups $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ and\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nHowever, the problem of finding\nsuch examples of rational Diophantine triples with positive elements is much harder, and they exist\nonly for torsion $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nWe describe the method for finding candidates for such curves\nand by using {\\tt magma} we are able to find one explicit example.\n\n\n\\section{Conditions for point $S$ to be of finite order}\n\nApart from three $2$-torsion points $A$, $B$ and $C$, the curve $E$\ncontains also the following two obvious rational points:\n$$ P = (0, 1), \\quad S = \\Big( \\frac{1}{abc}, \\frac{rst}{abc} \\Big). $$\nIt is not so obvious, but it is easy to verify that $S = 2R$, where\n$$ R = \\Big( \\frac{rs + rt + st + 1}{abc}, \\frac{(r + s)(r + t)(s + t)}{abc} \\Big). $$\nThus, a necessary condition for $E$ to have the rank equal to $0$ is that\nthe points $P$ and $S$ have finite order.\nThe triple $\\{a,b,c\\}$ is regular, i.e. $c=a+b\\pm 2r$ if and only if $S=\\mp 2P$\n(see \\cite{D-bordo}).\n\nBy Mazur's theorem and the fact that $S\\in 2E(\\mathbb{Q})$, we have the following possibilities:\n\\begin{itemize}\n\\item $mP = \\mathcal{O}$, $m=3,4,6,8$;\n\\item $mS = \\mathcal{O}$, $m=2,3,4$.\n\\end{itemize}\n\nIn particular, since the point $P$ cannot be of order $2$, is it not possible to have simultaneously\nrank equal to $0$ and torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/2\\mathbb{Z}$.\n\nBy the coordinate transformation $x\\mapsto \\frac{x}{abc}$, $y\\mapsto \\frac{y}{abc}$, applied\nto the curve $E$, we obtain the equivalent curve\n\\begin{equation} \\label{e2}\nE':\\qquad  y^2=(x+ab)(x+ac)(x+bc),\n\\end{equation}\nand the points $A$, $B$, $C$, $P$ and $S$ correspond to\n$A'=(-bc,0)$, $B'=(-ac,0)$, $C'=(-ab,0)$, $P'=(0,abc)$ and $S'=(1,rst)$, respectively.\nIn the next lemma, we will investigate all possibilities for point $S$ to be of finite order.\n\n\\begin{lemma} \\label{l:mS}\n\\mbox{}\n\\begin{itemize}\n\\item[(i)] The condition $2S=\\mathcal{O}$ is equivalent to\n$$ (ab+1)(ac+1)(bc+1)=0. $$\n\\item[(ii)] The condition $3S=\\mathcal{O}$ is equivalent to\n$$ 3+4(ab+ac+bc)+6abc(a+b+c)+12(abc)^{2}-(abc)^{2}(a^2 + b^2 + c^2 - 2ab - 2ac - 2bc)=0. $$\n\\item[(iii)] The point $S$ is of order $4$ if and only if\n$$ ((ab+1)^2 - ab(c-a)(c-b))((ac+1)^2 - ab(c-a)(c-b))((bc+1)^2 - ab(c-a)(c-b)) = 0. $$\n\\end{itemize}\n\\end{lemma}\n\n\\proof\n\\mbox{}\n\\begin{itemize}\n\\item[(i)] The condition $2S'= \\mathcal{O}$ implies $rst=-rst$, i.e. $rst=0$, and\n$$ (ab+1)(ac+1)(bc+1)=0. $$\n\n\\item[(ii)]\nFrom $3S'=\\mathcal{O}$, i.e. $x(2S')=x(-S')=x(S')$,\nthe formulas for doubling points of elliptic curves give\n\\begin{align*}\n&3+(ab+ac+bc)\\\\\n&=\\dfrac{9+4(ab+ac+bc)^{2}+(abc(a+b+c))^{2}+12(ab+ac+bc)}{4r^{2}s^{2}t^{2}}\\\\\n&\\ \\ \\ \\ +\\dfrac{6abc(a+b+c)+4abc(ab+ac+bc)(a+b+c)}{4r^{2}s^{2}t^{2}}.\n\\end{align*}\nThus we get\n\\begin{align*}\n&4\\left((abc)^{2}+abc(a+b+c)+(ab+ac+bc)+1\\right)(3+ab+ac+bc)\\\\\n&=9+12(ab+ac+bc)+\\left(6abc(a+b+c)+4(ab+ac+bc)^{2}\\right)\\\\\n&\\ \\ \\ \\ \\ \\ +4abc(ab+ac+bc)(a+b+c)+\\left(abc(a+b+c)\\right)^{2},\n\\end{align*}\nwhich is equivalent to\n\\begin{align*}\n&3+4(ab+ac+bc)+6abc(a+b+c)+12(abc)^{2}\\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ - (abc)^{2}(a^{2} + b^{2} + c^{2} - 2ab - 2ac - 2bc)=0.\n\\end{align*}\n\n\\item[(iii)]\nThe condition that the point $S'$ is of order $4$ is equivalent to $2S'\\in \\{A',B',C'\\}$.\nLet us assume that $2S'=C'$ (other two cases are completely analogous).\nFrom the formulas for doubling points of elliptic curves, we get\n\\begin{align*}\n&2+(bc+ac)\\\\\n&=\\dfrac{9+4(ab+ac+bc)^{2}+\\left(abc(a+b+c)\\right)^{2}+12(ab+ac+bc)}{4r^{2}s^{2}t^{2}}\\\\\n&\\ \\ \\ \\ +\\dfrac{6abc(a+b+c)+4abc(ab+ac+bc)(a+b+c)}{4r^{2}s^{2}t^{2}},\n\\end{align*}\nwhich is equivalent to\n$$\\left(1+2ab-abc(c-a-b)\\right)^{2}=0, $$\nor\n$$(ab+1)^{2}=ab (c-a)(c-b) .$$\n\\end{itemize}\n\\qed\n\n\n\\section{Rank zero curves for triples with mixed signs}\n\nLet us now consider three possibilities for $mS = \\mathcal{O}$.\n\nAssume first that $2S= \\mathcal{O}$. By Lemma \\ref{l:mS}(i), we have $(ab+1)(ac+1)(bc+1)=0$,\nso we conclude that $a,b,c$ cannot have the same sign.\nIf we allow the mixed signs, then in this case we may assume that $b=-1\/a$.\nIn \\cite{D-glasnik}, the following parametrization of rational Diophantine triples of the form\n$\\{a,-1\/a,c\\}$ is given:\n$$ a=\\frac{ut+1}{t-u}, \\quad b=\\frac{u-t}{ut+1}, \\quad c=\\frac{4ut}{(ut+1)(t-u)}. $$\nTo find examples with rank $0$, let us assume that the triple $\\{a,-1\/a,c\\}$ is regular.\nThis condition leads to $(u^2-1)(t^2-1)=0$, so we may take $u=1$.\nIf we take e.g. $t=2$, we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{3, -\\frac{1}{3}, \\frac{8}{3} \\Big\\}. $$\n\n\\medskip\n\nAssume now that $3S= \\mathcal{O}$.\nIf we also have $3P= \\mathcal{O}$, then $P=\\pm S$, a contradiction.\nHence, if the point $P$ has finite order, the only possibility that $P$ is of order $6$.\nThis implies $2P=\\pm S$ and $c=a+b\\mp 2r$.\nBy inserting $b=(r^2-1)\/a$ and $c=a+b+2r$ in the condition from Lemma \\ref{l:mS}(ii), we get\n$$ (2ar-1+2r^2) (-a+2ar^2-2r+2r^3) (2a^2r-a-2r+4ar^2+2r^3)=0. $$\nThus,\n$$ a= \\frac{-2r(r^2-1)}{-1+2r^2}, \\quad \\mbox{or} \\quad\n\\frac{-(-1+2r^2)}{2r}, \\quad \\mbox{or} \\quad\n\\frac{1-4r^2 \\pm \\sqrt{1+8r^2}}{4r}. $$\nTake\n\\begin{equation} \\label{eq:z2z6a}\n (a,b,c)=\\left(\\frac{-2r(r-1)(r+1)}{-1+2r^2}, \\frac{-(-1+2r^2)}{2r}, \\frac{(-1+2r)(2r+1)}{2(-1+2r^2)r}\\right).\n\\end{equation}\nThen the condition $ab>0$ is equivalent to $r>1$ or $r<-1$, while the condition\n$bc>0$ is equivalent to $-1\/2 < r < 1\/2$. Hence, $a,b,c$ cannot have the same sign.\n\nThe case\n$$ (a,b,c)=\\left(\\frac{-(-1+2r^2)}{2r}, \\frac{-2r(r-1)(r+1)}{-1+2r^2}, \\frac{(-1+2r)(2r+1)}{2(-1+2r^2)r}\\right) $$\nis the same as the previous case, just $a$ and $b$ are exchanged.\n\nFinally, let $8r^2+1 = (2rt+1)^2$, to get rid of a square root in the third case. It gives $r = \\frac{-t}{-2+t^2}$. Then\n$$ (a,b,c)=\\left(\\frac{-t(t-2)(t+2)}{2(-2+t^2)}, \\frac{2(t-1)(t+1)}{(-2+t^2)t}, \\frac{-(-2+t^2)}{2t}\\right) $$\n(or $a$ and $b$ exchanged).\nThe condition $ac>0$ is equivalent to $t>2$ or $t<-2$, while the condition\n$bc>0$ is equivalent to  $-1 < t < 1$. Hence, is this case also $a,b,c$ cannot have the same sign.\n\nIf we allow the mixed signs, then we can obtain examples with rank $0$, e.g. from triples\nof the form (\\ref{eq:z2z6a}). E.g. for $t=4$ we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{-\\frac{12}{7}, \\frac{15}{28}, -\\frac{7}{4} \\Big\\}. $$\n\n\\medskip\n\nIt remains the case when the point $S$ is of order $4$.\nThen the point $R$, such that $2R=S$ is of order $8$ and therefore\nthe torsion group of $E$ is $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nAs we already mentioned in the introduction, it is shown in \\cite{D-glasnik} that\nevery elliptic curve over $\\mathbb{Q}$ with this torsion group\nis induced by a rational Diophantine triple.\nMore precisely, any such curve is induced by a Diophantine triple of the form\n\\begin{equation}\\label{z2z8T}\n\\left\\{ \\frac{2T}{T^2-1}, \\,\\, \\frac{1-T^2}{2T}, \\,\\, \\frac{6T^2-T^4-1}{2T(T^2-1)} \\right\\}.\n\\end{equation}\nIt is clear that the elements of (\\ref{z2z8T}) have mixed signs.\nBy taking $T=2$ we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{ \\frac{4}{3}, -\\frac{3}{4}, \\frac{7}{12} \\Big\\}. $$\n\n\n\n\\section{An example of rank zero curve for triple with positive elements}\n\nIn a previous section, we showed that for rational Diophantine triples with all positive elements\nwe cannot have rank $0$ and torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$\nor $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$. So the only remaining possibility\nis the torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nSince the elements of (\\ref{z2z8T}) clearly have mixed signs ($ab=-1$),\nand all curves with torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$\nare induced by (\\ref{z2z8T}), on the first sight we might think that\ntriples with all positive elements are not possible for this torsion group.\nHowever, it is shown in \\cite{D-Peral-RACSAM} that this is not true.\nNamely, we may have a triple with positive elements which induce the same curve as\n(\\ref{z2z8T}) for certain rational number $T$.\n\nBut in \\cite{D-Peral-RACSAM} it remained open whether it is possible to\nobtain simultaneously torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$\nand rank $0$ for triples with positive elements, although some candidates for such triples\nare mentioned.\n\nAs in the previous section, we assume that the point $S$ is of order $4$,\nand we take $b=(r^2-1)\/a$, $c=a+b+2r$. By inserting this in the first factor\n$$ (ab+1)^2 - ab(c-a)(c-b) $$\nin Lemma \\ref{l:mS}(iii), we get the quadratic equation in $a$:\n$$ (2r^3-2r)a^2+(4r^4-6r^2+1)a+2r^5+2r-4r^3 = 0.$$\nIts discriminant,\n$$ 1-4r^4+4r^2 $$\nshould be a perfect square.\nThe quartic curve defined by this equation is birationally equivalent to the elliptic curve\n$$ E_1: \\quad Y^2 = X^3+X^2+X+1 $$\nwith rank $1$ and a generator $P_1=(0, 1)$.\nThus, by computing multiples of the point $P_1$ on the curve $E_1$\n(adding the $2$-torsion point $T_1=(-1,0)$ has a same effect as changing $r$ to $-r$),\nand transferring them back to the quartic, we obtain candidates for the solution of our problem.\nHowever, we have to satisfy the condition that all elements of the corresponding triple are positive\n(it is enough that all elements have the same sign, since by multiplying all elements from\na rational Diophantine triple by $-1$ we obtain again a rational Diophantine triple).\nThe first two multiples of $P$ producing the\ntriples with positive elements are $6P$ and $11P$.\n\nThe point $6P$ gives\n$r=-\\frac{3855558}{3603685}$ and the triple\n$$ (a,b,c) = \\left(\n\\frac{1884586446094351}{25415891646864180},\n\\frac{14442883687791636}{7402559392524605},\n\\frac{60340495895762708555}{14487505263205637124} \\right). $$\nWe were not able to determine that the rank of the correspoding curve.\nNamely, both {\\tt magma} and {\\tt mwrank}\ngive that $0 \\leq \\mbox{rank} \\leq 2$. Assuming the Parity conjecture the rank should be equal to $0$ or $2$.\n\nThe point $11P$ gives\n$r=\\frac{35569516882766685106979}{32383819387240952672281}$ and the triple $(a,b,c)$, where\n\\begin{align*}\na&= \\frac{69705492951192675600645567228019184577147632882703132983}{132014843349912467692901303836561266921302184459536763120}, \\\\ b&= \\frac{47826829880079829075801189563942620732062701095548790400}{122336669420709509303637442647966391336596694969835459327}, \\\\ c&=\n \\frac{47982111146649404421749331709393501777791774558546217987550257759801}{15400090753918257364093484910580652390786084055043677020804056653840}.\n\\end{align*}\n(By comparing $j$-invariants, we get that the same curve is induced by (\\ref{z2z8T}) for\n$T=\\frac{18451786408106133183649}{41916048174422594852689}$.)\nFor the corresponding curve, both {\\tt mwrank} and {\\tt magma} function {\\tt MordellWeilShaInformation}\ngive that $0 \\leq \\mbox{rank} \\leq 4$.\nHowever, {\\tt magma} (version  {\\tt V2.24-7}) function {\\tt TwoPowerIsogenyDescentRankBound},\nwhich implements the algorithm by Fisher from \\cite{Fisher}, gives that the rank is equal to $0$\n(at step $5$, just beyond $4$-descent, but not yet $8$-descent).\nHence, we found an example of a rational Diophantine triple with positive elements for\nwhich the induced elliptic curve has the rank equal to $0$. Let us mention that the same {\\tt magma} function applied to the\ncurve mentioned above corresponding to the point $6P$ gives only $\\mbox{rank} \\leq 2$.\nThis construction certainly gives\ninfinitely many multiples of $P$ which produce triples with positive elements\n(the set $E_1(\\mathbb{Q})$ is dense in $E_1(\\mathbb{R})$, see e.g. \\cite[p.78]{Skolem}).\nHowever, it is hard to predict distribution of ranks in such families of elliptic curve,\nso we may just speculate that there might be infinitely many curves in this family with rank $0$.\n\n\\bigskip\n\n{\\bf Acknowledgements.}\nA.D. was supported by the Croatian Science Foundation under the project no.~IP-2018-01-1313.\nHe also acknowledges support from the QuantiXLie Center of Excellence, a project\nco-financed by the Croatian Government and European Union through the\nEuropean Regional Development Fund - the Competitiveness and Cohesion\nOperational Programme (Grant KK.01.1.1.01.0004).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\\noindent\nA crucial requirement for understanding AGN\nevolution and demographics\nis the ability to select a sample for study \nin a complete and unbiased way.\nX-ray emission has been frequently used for AGN selection,  \nand can distinguish the high energy emission \nassociated with mass accretion by a black hole (BH)\nfrom inactive galaxies and stars.\nCombining wide-area X-ray surveys \nwith the ability to classify large numbers of objects spectroscopically via \nthe Sloan Digital Sky Survey \n\\citep[SDSS,][]{2000AJ....120.1579Y,2006AJ....131.2332G}\nprovides a powerful tool for the study of AGN.\n\nSPIDERS (SPectroscopic IDentification of \\textit{eROSITA} \nSources; PIs Merloni and Nandra) is an \nSDSS-IV \\citep{2017AJ....154...28B} \neBOSS \\citep{2016AJ....151...44D} subprogramme\nthat is currently conducting optical spectroscopy of extragalactic X-ray detections \nin wide-area \\textit{ROSAT} and \\textit{XMM-Newton} surveys\n\\citep{2017MNRAS.469.1065D}.\nLying at the bright end of the X-ray source population, \nthese sources will also be detected by \\textit{eROSITA}\n\\citep{2012arXiv1209.3114M, 2016SPIE.9905E..1KP}.\nThe current SDSS DR14 \\citep{2018ApJS..235...42A} SPIDERS sample\nis a powerful resource for the multiwavelength analysis of AGN.\nThis work aims to capitalise on the wealth of information already available\nby providing detailed optical spectral measurements,\nas well as estimates of BH masses and Eddington ratios.\n\nAn accurate measurement of the central supermassive black hole (SMBH) mass\nis necessary for the study of AGN \nand their coevolution with their host galaxies.\nBH mass has been found to scale \nwith a number of host galaxy spheroid properties;\nstellar velocity dispersion\n\\citep[the $\\rm M_{BH}-\\sigma$ relation, e.g.][]{2000ApJ...539L..13G, 2001ApJ...547..140M, 2002ApJ...574..740T},\nstellar mass \\citep[e.g.][]{1998AJ....115.2285M},\nand luminosity \\citep[e.g.][]{1995ARA&A..33..581K}.\nThese correlations suggest a symbiotic evolution \nof SMBHs and their host galaxies.\n\nReverberation mapping (RM) has been used to measure the \napproximate radius of the broad-line region (BLR) in AGN\n\\citep[e.g.][]{1972ApJ...171..467B, 1982ApJ...261...35C, 1982ApJ...255..419B, 1993PASP..105..247P, 2015PASP..127...67B, 2015ApJS..216....4S}.\nThis technique involves measuring the time delay between \nvariations in the continuum emission,\nwhich is expected to arise from the accretion disk,\nand the induced variations in the broad emission lines.\nIt was found that different emission lines have different time delays,\nwhich is expected if the BLR is stratified, \nwith lines of lower ionisation being emitted \nfurther from the central ionising source \\citep[e.g.][]{1986ApJ...305..175G}.\nFor example, the high ionisation line CIV$\\rm \\lambda$1549 \nhas a shorter time delay than H$\\rm \\beta$ \\citep{2000ApJ...540L..13P}.\n\nThe RM effort has also revealed a tight relationship between the \ncontinuum luminosity and the radius of the BLR \\citep{2000ApJ...533..631K, 2006ApJ...644..133B, 2009ApJ...697..160B}.\nTherefore, by using the measured luminosity as a proxy for the BLR radius,\nand measuring the BLR line-of-sight velocity\nfrom the width of the broad emission lines,\nBH masses can be estimated from a single spectrum\n\\citep{2002ApJ...571..733V, 2002MNRAS.337..109M, 2006ApJ...641..689V, 2011ApJ...742...93A, 2012ApJ...753..125S, 2013BASI...41...61S}.\nThis approach is known as the single-epoch, or photoionisation, method.\n\nSince H$\\beta$ is the most widely studied RM emission line\nit is therefore considered to be the most reliable line to use for single-epoch mass estimation. \nIn addition, AGN H$\\beta$ emission lines typically exhibit \na clear inflection point between the broad and narrow line components,\nmaking the virial full width at half maximum (FWHM) measurement \nrelatively straightforward (see section~\\ref{bl_decomp}).\nThe MgII line width correlates well with that of H$\\beta$ \n(see section~\\ref{compare_mass}),\nand therefore MgII has also been used for single-epoch mass estimation\n\\citep[e.g.][]{2002MNRAS.337..109M}.\nFor SDSS spectra, either H$\\beta$ or MgII \nis visible in the redshift range 0\\,$\\leq$\\,z\\,$\\lesssim$\\,2.5. \n\nAt higher redshifts, \nthe broad, high-ionisation line CIV$\\lambda1549$ is available.\nThe CIV line width does not correlate strongly with that of low ionisation lines\n\\citep[e.g.][]{2005MNRAS.356.1029B, 2012MNRAS.427.3081T}\nand this, along with the presence of a large blueshifted component\n\\citep[e.g.][]{2002AJ....124....1R}\nmakes it difficult to employ CIV for mass estimation.\nA number of calibrations have been developed \nwhich aim to improve the mass estimates derived from CIV\n\\citep[][]{2012ApJ...759...44D, 2013MNRAS.434..848R, 2013ApJ...770...87P, 2017MNRAS.465.2120C},\nhowever, whether CIV can provide reliable mass estimates when compared with \nlow ionisation lines is still a subject of debate\n\\citep[see][]{2018MNRAS.478.1929M}.\n\nThis paper is organised as follows:\nthe selection of a reliable subsample of sources \nto be used for optical spectral fitting \nis discussed in section~\\ref{data}.\nSection~\\ref{spec_fit} describes the method used \nto fit the H$\\beta$ and MgII emission line regions.\nThe methods for estimating BH mass and bolometric luminosity \nare discussed in section~\\ref{estimate_mass}.\nThe X-ray flux measurements used in this work are discussed in section~\\ref{xray_flux}.\nSection~\\ref{access} describes where the catalogue containing the results of this work can be accessed.\nA comparison between the UV and optical spectral fitting results \nis given in section~\\ref{compare_uv_opt}.\nSection~\\ref{prop} provides a discussion of the sample properties, \nand finally, section~\\ref{limits} includes a discussion of the reliability and limitations of\nthe fitting procedure.\n In order to facilitate a direct comparison with previous studies based on X-ray surveys,\na concordance flat $\\rm \\Lambda CDM$ cosmology was adopted\nwhere $\\Omega_{\\rm M}$=0.3, $\\Omega_{\\Lambda}$=1-$\\Omega_{\\rm M}$,\nand H$_{0}$=70\\,km\\,s$^{-1}$\\,Mpc$^{-1}$.\n\n\n\n\\section{Preparing the Input Catalogue}\\label{data}\n\n\\subsection{X-ray Data}\n\n\\noindent\nThe \\textit{ROSAT} sample used in this work \nis part of the second \\textit{ROSAT} all-sky survey (2RXS) \ncatalogue \\citep{2016A&A...588A.103B},\nwhich has a limiting flux of $\\rm \\sim10^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$,\nwhich corresponds to a luminosity of $\\rm \\sim10^{43}\\,erg\\,s^{-1}$ at z = 0.5.\nCompared to the first \\textit{ROSAT} data release \\citep{1999A&A...349..389V},\nthe 2RXS catalogue is the result of an improved detection algorithm,\nwhich uses a more detailed background determination \nrelative to the original \\textit{ROSAT} pipeline.\nA full visual inspection of the 2RXS catalogue has been performed,\nwhich provides a reliable estimate of its spurious source content\n\\citep[see][]{2016A&A...588A.103B}.\nThe first \\textit{XMM-Newton} slew survey catalogue\nrelease 1.6\n\\citep[XMMSL1;][]{2008A&A...480..611S} was also used in this work.\nThis catalogue includes observations made by \nthe European Photon Imaging Camera (EPIC) pn detectors \nwhile slewing between targets,\nand has a limiting flux of $\\rm 6\\times10^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$\nin the soft band,\nwhich corresponds to a luminosity of $\\rm 5.8\\times10^{44}\\,erg\\,s^{-1}$ at z=0.5.\n\n\n\\begin{figure*}[h]\n\t\\centering\n                \\includegraphics[width=0.65\\textwidth]{z_Lx_4.png}\n\\caption{\\footnotesize\nSoft X-ray luminosity versus spectroscopic redshift\nfor the samples presented in this work\nand the following previously published X-ray selected samples;\nXMM-XXL \\citep{2016MNRAS.457..110M, 2016MNRAS.459.1602L},\nCDFS \\citep{2017ApJS..228....2L},\nSTRIPE82X \\citep{2016ApJ...817..172L},\nCOSMOS \\citep{2016ApJ...817...34M, 2016ApJ...830..100M, 2016ApJ...819...62C},\nAEGIS-X \\citep{2015ApJS..220...10N}, and the \nLockman Hole deep field (LHDF) \\citep{2008A&A...479..283B, 2012ApJS..198....1F}.\nFor each sample, the 0.5-2 keV luminosities are shown,\nexcept for the 2RXS sample, where the 0.1-2.4 keV luminosities are shown,\nand the XMMSL1 sample, \nwhere the 0.2-2 keV luminosities from \\citet{2008A&A...480..611S} are shown.\nThe detection limit for the 2RXS and XMMSL1 samples \nare shown by the solid and dashed grey lines respectively.\nThe X-ray luminosities for the 2RXS sample are derived from the classical \nflux estimates described in section~\\ref{2RXS_flux},\nhowever it is noted here that some low count rate \n2RXS sources do not have flux estimates.\nFor sources that were detected in both 2RXS and XMMSL1,\nonly the XMMSL1 luminosities are shown.\nSources classified as stars have not been included in this figure.}\n\\label{compare_samples}\n\\end{figure*}\n\n\n\\subsection{The SPIDERS Programme}\\label{spiders_prog}\n\nThe SPIDERS programme has been providing SDSS spectroscopic observations \nof 2RXS and XMMSL1 \nsources\\footnote{The SPIDERS programme targets both point-like and extended X-ray sources. This work focuses on the counterparts to point-like X-ray detections, \nwhich are predominantly AGN,\nand therefore, the samples discussed in the subsequent paragraphs \nare derived from the SPIDERS-AGN programme.}\nin the eBOSS footprint. \nBefore the start of the eBOSS survey in 2014, \nthe SPIDERS team compiled a sample of X-ray selected spectroscopic targets\nand submitted this sample for spectroscopic follow-up using the BOSS spectrograph \nas part of the eBOSS\/SPIDERS subprogramme\n\\citep[see][for further details on the SPIDERS programme]{2017MNRAS.469.1065D}.\nAs of the end of eBOSS in February 2019,\nthe eBOSS\/SPIDERS survey has covered a sky area of $\\rm 5321 \\, deg^{2}$.\nThe SDSS DR14 SPIDERS sample presented in this work\ncovers an area of $\\rm \\sim2200 \\, deg^{2}$ \n($\\rm \\sim 40\\%$ of the final eBOSS\/SPIDERS area).\n\nThe spectroscopic completeness achieved by the SPIDERS survey \nas of SDSS DR14 in the eBOSS area is \n$\\sim 53\\%$ for the sample as a whole, \n$\\sim 63 \\%$ considering only high-confidence X-ray detections \n(see section~\\ref{clean_sample}),\nand $\\sim 87 \\%$ considering sources with high-confidence X-ray detections and \noptical counterparts with magnitudes in the nominal survey limits \n($\\rm 17\\leq m_{\\rm Fiber2,i}\\leq22.5$).\nOutside the eBOSS area, the spectroscopic completeness of this sample is lower:\n$\\sim 28\\%$ for the sample as a whole, \n$\\sim 39 \\%$ considering only high-confidence X-ray detections,\nand $\\sim 57 \\%$ considering sources with high-confidence X-ray detections and \noptical counterparts with magnitudes in the nominal survey limits.\nThe spectroscopic completeness of the SDSS DR16 SPIDERS sample \ninside and outside the eBOSS area is expected to be similar \nto that of the sample presented here.\n\nIn addition to those targeted during eBOSS\/SPIDERS, \na large number of 2RXS and XMMSL1 sources \nreceived spectra during the SDSS-I\/II\n\\citep[2000-2008;][]{2000AJ....120.1579Y}\nand the SDSS-III \\citep{2011AJ....142...72E}\nBOSS \\citep[2009-2014;][]{2013AJ....145...10D} surveys.\nThis paper includes spectra obtained \nby eBOSS\/SPIDERS up to DR14 (2014-2016) \nas well as spectra from SDSS-I\/II\/III.\n\n\n\n\\subsection{Identifying IR Counterparts}\\label{nway_match}\n\nTo identify SPIDERS spectroscopic targets,\nthe Bayesian cross-matching algorithm ``NWAY'' \\citep{2018MNRAS.473.4937S}\nwas used to select AllWISE \\citep{2013yCat.2328....0C} \ninfrared (IR) counterparts for the 2RXS and XMMSL1 \nX-ray selected sources in the BOSS footprint.\nThe AllWISE catalogue consists of data obtained \nduring the two main survey phases of the \nWide-field Infrared Survey Explorer mission\n\\citep[WISE;][]{2010AJ....140.1868W}\nwhich conducted an all-sky survey \nin the 3.4, 4.6, 12, and 22 $\\rm \\mu m$ bands \n(magnitudes in these bands are denoted [W1], [W2], [W3], and [W4] respectively).\nThe matching process used the colour-magnitude priors \n[W2] and [W2-W1] \\citep[see][]{2017MNRAS.469.1065D}\nwhich, at the depth of the 2RXS and XMMSL1 surveys, \ncan distinguish between the correct counterparts and chance associations.\nThese colours would not be efficient if the 2RXS survey was much deeper\n\\citep[see][for a complete discussion]{2018MNRAS.473.4937S}.\nThe resulting 2RXS and XMMSL1 catalogues\nwith AllWISE counterparts contained 53455 and 4431 sources respectively.\nAllWISE positions were then matched to \nphotometric counterparts, where available, in SDSS.\n\n\n\n\\subsection{Comparison with Previous X-ray Surveys}\n\nFigure~\\ref{compare_samples} displays the \nsources in the 2RXS and XMMSL1 samples\nwhich have spectroscopic redshifts and measurements of the soft X-ray flux.\nFor comparison, a series of previously published X-ray selected samples \nthat have optical spectroscopic redshifts are also shown.\nThe large number of sources present in the 2RXS and XMMSL1 samples \nmotivated the optical spectroscopic analysis discussed in the following sections.\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{tree_2RXS.png}\n\\includegraphics[width=0.49\\textwidth]{tree_XMMSL.png}\n\\caption{\\footnotesize \nSequence of quality cuts applied to the 2RXS and XMMSL1 samples \nto produce the subsample used for spectral analysis.\nThe starting points (2RXS-AllWISE and XMMSL1-AllWISE)\nare the full samples of 2RXS and XMMSL1 selected sources \nwith AllWISE IR counterparts in the BOSS footprint\n(see section~\\ref{nway_match}).\nThe steps in grey are those that have been discussed \nin \\citet{2017MNRAS.469.1065D}.}\n\\label{cuts}\n\\end{figure*}\n\n\n\\subsection{Selecting a Reliable Subsample}\\label{clean_sample}\n\n\\noindent\nThe selection of SPIDERS spectroscopic targets \nwas discussed in detail by \\citet{2017MNRAS.469.1065D}.\nThis section summarises the selection steps \ndiscussed in detail by \\citet{2017MNRAS.469.1065D}\nand describes the additional cuts made in this work to \nselect a sample for spectral analysis.\nThe sequence of selection criteria used and the resulting sample size \nare shown in figure~\\ref{cuts}.\n\n2RXS sources with an X-ray detection likelihood (EXI\\_ML)\n$\\leq \\,$10 were excluded since these detections are considered \nhighly uncertain with a spurious fraction \n$\\geq20\\%$ \\citep[see][]{2016A&A...588A.103B}. \nXMMSL1 sources with an X-ray detection likelihood \n(XMMSL\\_DET\\_ML\\_B0) $\\leq \\,$10 were also excluded.\n\\citet[][figure 1]{2018MNRAS.473.4937S} show the \ndistribution of flux with detection likelihood for both samples.\nThese cuts returned 23245\/53455 2RXS and 3803\/4431 XMMSL1 sources.\n\nThe following cuts, which were described in detail in \\citet{2017MNRAS.469.1065D},\nhave also been applied to the sample:\n\n\\begin{itemize}\n\\item For each X-ray source, \\citet{2018MNRAS.473.4937S} give \nthe probability, p\\_any, that a reliable counterpart exists among the \npossible AllWISE associations. Sources with p\\_any < 0.01 were removed.\nThis returned 23046\/23245 2RXS and 3558\/3803 XMMSL1 sources. \\\\\n\n\\item For each X-ray source, the most probable AllWISE counterpart was chosen\nby selecting sources with match\\_flag=1.\nThis returned 20585\/23046 2RXS and 3321\/3558 XMMSL1 sources. \\\\\n\n\\item For each AllWISE counterpart, the brightest SDSS-DR13 photometric source within the AllWISE matching radius was selected using FLAG\\_SDSSv5b\\_best=1.\nThis returned 19385\/20585 2RXS and 3063\/3321 XMMSL1 sources. \\\\\n\n\\item Cases where the AllWISE-SDSS separation \nexceeded 3\\,arcsec were removed.\nThis returned 18575\/19385 2RXS and 2893\/3063 XMMSL1 sources. \\\\\n\\end{itemize}\n\n\\noindent\nThe results of these constraints are displayed in grey in figure~\\ref{cuts}.\nAs shown above, sources with match\\_flag=1 were targeted;\nhowever, for 14\\% of the 2RXS sample and 10\\% of the XMMSL1 sample,\nmore than one counterpart was highly likely.\nThis implies that either the counterpart association was not reliable, \nor that the X-ray detection was the result of emission from multiple sources.\nThese sources were not included in the discussion of \noptical spectral properties as a function of X-ray properties in section~\\ref{aox_section}.\nAfter selecting the brightest SDSS-DR13 \\citep{2017ApJS..233...25A} \nphotometric source, there were 15 cases where two unique 2RXS sources \nwere matched to the same AllWISE\/SDSS counterpart \nand 3 cases where two unique XMMSL1 sources \nwere matched to the same AllWISE\/SDSS counterpart. \nThese sources were also removed.\n\nOf these samples with reliable SDSS photometric counterparts, \n8777 2RXS and 1315 XMMSL1 sources have received spectra during SDSS-I\/II\/III \nwhile 1122 2RXS and 221 XMMSL1 sources have received spectra \nduring the SPIDERS programme (including SEQUELS), resulting in a sample of \n9899 2RXS and 1536 XMMSL1 sources with spectra as of DR14.\nThe distribution of SDSS i band fiber2 magnitudes for this sample \n(showing the different spectroscopic programmes)\nis presented in figure~\\ref{origin_spec}.\nDue to targeting constraints (as discussed in section~\\ref{spiders_prog}), \nthe sample completeness is much lower \noutside of the nominal magnitude limits for the survey \n($\\rm 17\\leq m_{\\rm Fiber2,i}\\leq22.5$ for eBOSS).\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{mag_2RXS.png}\n\\includegraphics[width=0.49\\textwidth]{mag_XMMSL.png}\n\\caption{\\footnotesize \nDistribution of i-band fiber magnitudes (fiber2Mag).\nThe coloured curves represent all of the sources with spectra,\nand the survey from which the spectra were taken.\nThe grey histogram displays the X-ray sources with a reliable SDSS photometric counterpart, including stars \nwhich cannot be targets for spectroscopy due to their brightness.}\n\\label{origin_spec}\n\\end{figure*}\n\n\n\\subsection{Source Classification}\\label{source_classification}\n\n\\noindent\nVisual inspection results for each object in this sample \nare available from a combination of literature sources \n\\citep{2007AJ....133..313A, 2010AJ....139..390P, 2010AJ....139.2360S, 2017A&A...597A..79P} and the SPIDERS group.\nThe SPIDERS visual inspection \n\\citep[see][for further details]{2017MNRAS.469.1065D}\nprovides a visual confirmation of the SDSS pipeline redshift and object classification.\nThe results of this inspection include \na flag indicating the confidence of the redshift, ``CONF\\_BEST'',\nwhich can take the values 3 (highly secure), \n2 (uncertain), 1 (poor\/unusable), 0 (insufficient data).\nA confirmation of the source classification \nwas also added during the visual inspection, \nwhich uses the categories QSO, broad absorption line QSO (BALQSO), \nblazar, galaxy, star, and none.\n\\citet{2007AJ....133..313A} provide the broad line AGN (BLAGN) and narrow line AGN (NLAGN) classifications, which are defined based on the presence or absence of \nbroad (FWHM\\,>\\,1000$\\rm \\,km\\,s^{-1}$) permitted emission lines.\n\nThe main goal of this work is to analyse the type 1 AGN in the SPIDERS sample,\nand therefore only sources that have been classified via their optical spectra \nas either ``BLAGN'' or ``QSO'' were selected for spectroscopic analysis.  \nThis returned 7805\/9899 2RXS and 1192\/1536 XMMSL1 sources.\nSince the categories ``BLAGN'' and ``QSO'' are based on different classification criteria,\nthere will be some overlap between the two sets of sources.\nTherefore, no distinction will be made between the two categories;\ninstead, both sets of objects will be considered type 1 AGN in this work.\n\n\n\n\\subsection{Contamination from Starburst Galaxies}\n\n\\noindent\nAlthough our sample probes luminosity ranges\ntypically associated with AGN emission,\nstarburst galaxies are also powerful X-ray sources\nand may be present as contaminants in our AGN sample.\nThe X-ray emission from starburst galaxies is expected to \noriginate from a number of energetic phenomena \nincluding supernova explosions and X-ray binaries\n\\citep[e.g.][]{2002A&A...382..843P}.\nTherefore, the X-ray emission from starburst galaxies \ncan be expected to be correlated with the star formation rate (SFR).\nUsing their sample of luminous infrared galaxies, \nand a sample of nearby galaxies from \\citet{2003A&A...399...39R},\n\\citet{2011A&A...535A..93P} found that the total SFR\nis related to the soft X-ray luminosity as follows:\n\n\\begin{equation}\\label{SFR}\n\\rm SFR_{IR+UV}\\,(M_{\\odot}\\,yr^{-1}) = 3.4 \\times 10^{-40}\\,L_{0.5-2keV} \\, (erg \\, s^{-1})\n\\end{equation}\n\n\\noindent\n\\citet{2015A&A...579A...2I}, figure 3, show the specific \nSFR for the COSMOS \\citep{2007ApJS..172....1S} \nand GOODS \\citep{2004ApJ...600L..93G} surveys \nfor a series of redshift bins in the range $\\rm 0.2 <  z < 1.4$.\nThe peak of the redshift distribution of the 2RXS\/XMMSL1 samples \npresented in this work is $\\sim0.25$.\nTherefore, assuming that the COSMOS\/GOODS sample in the redshift bin 0.2-0.4\nis a good representative of the 2RXS\/XMMSL1 samples, \nthe upper limit on the SFR that can be expected \nfor galaxies in our sample is $\\rm \\sim50\\,M_{\\odot}\\,yr^{-1}$.\nAccording to equation~\\ref{SFR}, this corresponds to a soft X-ray luminosity of \n$\\rm \\sim10^{41}\\,erg\\,s^{-1 }$,\nwhich is below the lower range probed by our samples \n$\\rm (\\sim10^{42} \\, erg \\, s^{-1}$, see figure~\\ref{compare_samples}).\n\n\n\n\\subsection{Redshift Constraints}\\label{selection}\n\n\\noindent\nUsing the ``CONF\\_BEST'' flag, sources with uncertain redshift or spectral classification \n(identified during the visual inspection of the sample) were also removed.\nThis process resulted in a sample of 7795\/7805 2RXS sources \nand 1190\/1192 XMMSL1 sources.\nIn the spectral fitting procedure (described in section~\\ref{spec_fit}), \nthe H$\\beta$ and MgII lines were fit independently.\nSources with H$\\beta$ and MgII present in their optical spectrum \nwere selected using the following logic:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm H_{\\beta}: \\rm (SN\\_MEDIAN\\_ALL > 5) \\,\\,\\&\\&\\,\\, \\\\\n\\rm (( (INSTRUMENT == SDSS) \\,\\,\\&\\&\\,\\, (0 < Z\\_BEST < 0.81) )\\,\\,||\\,\\, \\\\\n\\rm ( (INSTRUMENT == BOSS) \\,\\,\\&\\&\\,\\, (0 < Z\\_BEST < 1.05)))\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm MgII: \\rm (SN\\_MEDIAN\\_ALL > 5) \\,\\,\\&\\&\\,\\, \\\\\n\\rm (( (INSTRUMENT == SDSS) \\,\\,\\&\\&\\,\\, (0.45 < Z\\_BEST < 2.1) )\\,\\,||\\,\\, \\\\\n\\rm ( (INSTRUMENT == BOSS) \\,\\,\\&\\&\\,\\, (0.38 < Z\\_BEST < 2.5)) )\n\\end{split}\n\\end{equation}\n\n\\noindent\nDifferent redshift ranges have been used because the \nBOSS spectrograph has a larger wavelength coverage than the SDSS spectrograph.\nIn some cases, parts of the fitting region will have been redshifted \nout of the SDSS\/BOSS spectrograph range \\citep{2013AJ....146...32S}, \nand therefore will not be fit.\nHowever, the redshift limits where chosen\nso that both samples contain the broad lines used for estimating BH mass.\nSources with a median signal-to-noise ratio (S\/N) less than or equal to five \nper resolution element were excluded from the spectral analysis \nsince for these sources the broad line decomposition \nand resulting BH mass estimates may be unreliable\n\\citep[see][]{2009ApJ...692..246D,2011ApJS..194...45S}.\n\nTable~\\ref{spec_sample} lists the numbers of sources \nwith spectral coverage of either H$\\beta$ or MgII, \nwhile figure~\\ref{clean_subsample} shows the redshift distribution of these sources.\nThere are 711 cases where the same optical counterpart \nwas detected by both 2RXS and XMMSL1.\nThe final combined (2RXS and XMMSL1) sample for spectral analysis  \ncontains 7790 unique type 1 sources.\n\n\n\\begin{table}\n\\caption{\\footnotesize \nThe coverage of the H$\\beta$ and MgII emission lines \nin the two samples used in this work.\nThere are 711 sources which were detected in both \nthe 2RXS and XMMSL1 surveys. \nThe ``total'' row lists the total number of unique sources \nobtained from combining the 2RXS and XMMSL1 samples.}\n\\centering  \n\\begin{tabular}{l c c c c c}      \n\\toprule \n\\midrule\n\n\\multicolumn{1}{c}{}\n& \\multicolumn{1}{c}{MgII}\n& \\multicolumn{1}{c}{H$\\beta$}\n& \\multicolumn{1}{c}{H$\\beta$ and MgII}\n& \\multicolumn{1}{c}{H$\\beta$ or MgII}\n\\\\\n\\midrule\n2RXS & 3310 & 6268 & 2234 & 7344 \\\\ \nXMMSL1 & 314 & 1070 & 227 & 1157 \\\\\nTotal & 3473 & 6654 & 2337 & 7790 \\\\\n\\midrule\n\\bottomrule \\\\\n\\end{tabular}\n\\label{spec_sample}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{z_hist_spec.png}\n\\caption{\\footnotesize\nRedshift distribution of the sample of type 1 AGN with coverage of $\\rm H\\beta$ \nand\/or MgII.}\n\\label{clean_subsample}\n\\end{figure}\n\n\n\n\\section{Spectral Analysis}\\label{spec_fit}\n\n\\noindent\nA series of scripts have been written to perform spectral fits\nusing the MPFIT least-squares curve fitting routine \n\\citep{2009ASPC..411..251M}.\nEach spectrum was corrected for Milky Way extinction \nusing the extinction curve from \\citet{1989ApJ...345..245C},\nand the dust map from \\citet{1998ApJ...500..525S},\nwith an R$\\rm_{V}$=3.1.\nNo attempt has been made to estimate and correct for \nthe intrinsic (host) extinction of each source\\footnote{\nAlso note that extinction laws (e.g. Calzetti and Prevot) \nare based on samples of nearby SB and irregular type galaxies. \nDue to the lack of nearby passive galaxies,\nan extinction law for these galaxy types is not yet available.}.\nMeasured line widths were corrected for the resolution of the \\\nSDSS\/BOSS spectrographs.\nThe H$\\beta$ and MgII emission line regions \nwere fit independently using similar\nmethods described in the following \nsections\\footnote{For each model parameter,\nthe 1-sigma uncertainties from MPFIT were adopted.}.\n\n\n\n\\subsection{Iron Emission Template}\\label{fe}\n\n\\noindent\nAGN typically exhibit FeII emission consisting of \na large number of individual lines \nacross the optical and UV regions of the spectrum.\nThese lines appear to be blended, \nprobably due to the motion of the gas from which they are emitted,\nand the magnitude of this broadening varies significantly from source to source.\nThe presence of FeII emission in the optical and UV portions of the spectrum can be\na significant complication when attempting to accurately measure line profiles.\nTherefore it is crucial that the model used to derive line widths for BH mass measurements also accounts for the nearby FeII emission.\n\nFigure~\\ref{fe_template} shows the two FeII templates used in this work;\nthe \\citet{2001ApJS..134....1V} and \\citet{1992ApJS...80..109B}\ntemplates used for the UV and optical regions of the spectrum, respectively. \nBoth of these templates have been derived from \nthe narrow line Seyfert 1 galaxy I Zwicky 1\nwhich, due to its bright FeII emission and narrow emission lines, \nis an ideal candidate for generating the FeII template.\nIn order to model the observed blending of the FeII emission,\nthe templates were convolved with a Gaussian \nwhose width was included as a free parameter in the fitting procedure.\n\n\n\\begin{figure}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{iron_template_uv.png}\\\\\n                \\includegraphics[width=0.49\\textwidth]{iron_template_opt.png}\n\\caption{\\footnotesize\nUpper panel: The \\citet{2001ApJS..134....1V} FeII template \nwhich was used when fitting the MgII emission line region.\nLower panel: The \\citet{1992ApJS...80..109B} FeII template\nwhich was used when fitting the H$\\beta$ emission line region. \nIn both cases, the original template is shown (blue) \nalong with the same template convolved with a Gaussian with FWHM\\,=\\,4000\\,km\\,s$^{-1}$ (red).\nThe vertical dashed lines correspond to the position of MgII and H$\\beta$ \nat 2800$\\AA$ and 4861$\\AA$.}\n\\label{fe_template}\n\\end{figure}\n\n\n\\begin{figure*}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{spec-1159-52669-0470.png}\n                \\includegraphics[width=0.49\\textwidth]{spec-0423-51821-0250.png}\n\\caption{\\footnotesize\nExamples of model fits to the H$\\beta$ \n(left panel; plate=1159, MJD=52669, fiber=470) \nand MgII (right panel; plate=423, MJD=51821, fiber=250) spectral regions.\nThe model components are colour-coded as follows;\npower law (orange), iron emission (violet), broad lines (blue), \nnarrow lines (yellow), [OIII] shifted wings (red), and the total model (red).\nThe panels beneath the spectra show the data\/model ratio.}\n\\label{examples}\n\\end{figure*}\n\n\n\\subsection{H$\\beta$}\\label{HB_fit_method}\n\n\\noindent\nThe region from 4420-5500\\,$\\AA$ was fit for each spectrum. \nThe continuum model consisted of a power law,\na galaxy template, and the \\citet{1992ApJS...80..109B} FeII emission template.\nThe FeII template was convolved with a Gaussian while fitting, \nand the width of this Gaussian, along with the normalisation of the template \nwere included as free parameters in the fit (see section~\\ref{fe}).\nPrevious spectral analyses of AGN spectra have assumed \nan early-type galaxy component in the model \\citep{2017MNRAS.472.4051C}.\nFollowing this method, we use an early-type SDSS galaxy\ntemplate\\footnote{template number 24 on \\\\ http:\/\/classic.sdss.org\/dr5\/algorithms\/spectemplates\/}  in the fit, and the normalisation of this template as well as the \nnormalisation and slope of the power law were also free parameters.\nThe use of a single, early-type galaxy template is an approximation, \nhowever, it is considered to be justified since \nAGN are typically found to reside in bulge dominated galaxies\n\\citep[e.g.][]{2005ApJ...627L..97G, 2007ApJ...660L..19P},\nand the spectroscopic fiber collects emission mostly from \nthe bulge (which is characterised by an old stellar population)\nand the active nucleus.\n\nThe [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007 narrow lines\nwere each fit with two Gaussians, one used to fit the narrow core, \nand an additional Gaussian to account for the presence of \nblue-shifted wings which are often detected \\citep{2005AJ....130..381B}. \nA single Gaussian was used to fit the HeII$\\lambda$4686 emission line.\nTo avoid overfitting the H$\\beta$ line, the fitting process was run four times, \nwith one, two, three, and four\\footnote{Three broad Gaussians are used in addition to a single narrow component to account for the three distinct broad components \nthat are expected to be present\n(see section~\\ref{VBC_section})\nin at least some sources \\citep{2010MNRAS.409.1033M}.} \nGaussian components used to fit the H$\\beta$ line.\nFor each fit, the velocity width and peak wavelength of one of the Gaussian components was fixed to that of [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007 \nin order to aid the identification of the narrow H$\\beta$ component.\nThe normalisation ratio of the [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007\nlines was fixed to the expected value of 1:3 \\citep[e.g.][]{2000MNRAS.312..813S}.\nThe best-fit model was then selected using the Bayesian information criterion \n\\citep[BIC,][]{1978AnSta...6..461S},\nwhich can be written as \n\n\\begin{equation}\\nonumber\n\\rm BIC = ln(n)k+\\chi^{2} \n\\end{equation}\n\n\\noindent\nwhere n is the number of data points, k is the number of model parameters,\nand $\\rm \\chi^{2}$ is the chi-square of the fit.\nThe preferred model is that with the lowest BIC.\nAn example of a fit to the H$\\beta$ spectral region \nis shown in the left panel of figure~\\ref{examples}.\n\n\n\n\\subsubsection{Broad Line Decomposition}\\label{bl_decomp}\n\n\\noindent\nThe narrow H$\\beta$ and [OIII] components \nare required to have widths $\\rm \\leq800\\,km\\,s^{-1}$.\nAny of the additional Gaussians used to fit MgII and H$\\beta$\nwith FWHM\\,$\\rm > 800\\,km\\,s^{-1}$ are considered ``broad''.\nThis threshold of 800$\\rm \\, km\\, s^{-1}$ is taken from the approximate division between broad and narrow FWHM distributions in the lower panels of figure~\\ref{line_width}.\nThe virial FWHM used for BH mass estimation \nis the FWHM of the line profile defined by the \nsum of these broad Gaussian components\n(see figure~\\ref{line_decomp}).\nA major challenge with using the single-epoch method for estimating BH mass \nis decomposing the broad and narrow components of the line\nin order to measure the virial FWHM.\nFigure~\\ref{line_decomp} presents an example \nof the decomposition of a broad H$\\beta$ line.\nIn this case, the narrow H$\\beta$ core can be easily distinguished \nand removed before measuring the virial FWHM.\nHowever, there are many cases \nwhere the broad and narrow components are blended,\nmaking it difficult to successfully identify the appropriate virial FWHM.\nThere are also cases where there is a clear distinction between\ntwo broad line components that are shifted in wavelength \nrelative to each other (known as ``double-peaked emitters''). \nHow one should interpret \nthe single-epoch BH mass estimates \nfor these unusual objects is uncertain \n(also see section~\\ref{reliability}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{line.png}\n\\caption{\\footnotesize \nAn example of the decomposition of a typical AGN H$\\beta$ line \n(plate=7276, MJD=57061, fiber=470).\nThe horizontal dashed line represents the FWHM used for BH mass estimation.\nThe vertical dashed line indicates the rest-frame wavelength of H$\\beta$.\nSee section~\\ref{bl_decomp} for further details.}\n\\label{line_decomp}\n\\end{figure}\n\n\n\n\\subsection{MgII}\\label{mgII_fit}\n\n\\noindent\nThe region from 2450-3050$\\AA$ was fit for each spectrum. \nAs in the case of the H$\\beta$ fits, a power law,\nan early-type galaxy template \n\\citep[5\\,Gyr old elliptical galaxy;][]{1998ApJ...509..103S, 2007ApJ...663...81P},\nand the \\citet{2001ApJS..134....1V} FeII emission template\nwere used to fit the continuum.\nAgain, the FeII template normalisation, and width of the \nGaussian smoothing applied to the template, \nwere included as free parameters in the fit.\nThe MgII line is a doublet; however, due to the close spacing \nand virial broadening of the lines,\nit usually appears as a single broad component in AGN spectra.\nThe narrow MgII line cores are usually not observed in AGN spectra,\ntherefore the MgII profile was fit using three broad Gaussians.\nAn example of a fit to the MgII spectral region \nis presented in the right panel of figure~\\ref{examples}.\n\n\n\\begin{table*}[h]\n\\caption{\\footnotesize \nBH mass calibrations used in this work.\nA, B, and C are the calibration constants for single-epoch mass estimation\n(see equation~\\ref{mass_est}).}\n\\centering  \n\\begin{tabular}{l c c c c c}      \n\\toprule \n\\midrule\n\n\\multicolumn{1}{c}{}\n& \\multicolumn{1}{c}{A}\n& \\multicolumn{1}{c}{B}\n& \\multicolumn{1}{c}{C}\n& \\multicolumn{1}{c}{Reference}\n\\\\\n\\midrule\nMgII, L$_{3000\\AA}$ & 1.816 & 0.584 & 1.712 & \\citet{2012ApJ...753..125S} \\\\\nH$_{\\beta}$, L$_{5100\\AA}$ & 0.91 & 0.5 & 2 & \\citet{2006ApJ...641..689V} \\\\ \nH$_{\\beta}$, L$_{5100\\AA}$ & 0.895 & 0.52 & 2 & \\citet{2011ApJ...742...93A} \\\\\n\\midrule\n\\bottomrule \\\\\n\\end{tabular}\n\\label{calib}\n\\end{table*}\n\n\n\\begin{figure*}[h!]\n\\begin{center}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_1.png}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_2.png}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_3.png}\n\\caption{\\footnotesize \\label{mass_hist}\nDifferences between the BH mass calibrations used in this work\n(see section~\\ref{estimate_mass} for further details).\nThe vertical blue lines indicate the mean value of each distribution.\nThe standard deviation, $\\rm \\sigma$, of each distribution is also shown.} \n\\end{center}\n\\end{figure*}\n\n\n\\section{Bolometric Luminosity and BH Mass Estimation}\\label{estimate_mass}\n\n\\noindent\nBolometric luminosities were estimated from \nthe monochromatic luminosities \nusing the bolometric corrections from \n\\citet{2006ApJS..166..470R, 2011ApJS..194...45S}:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm L_{Bol} = 5.15 \\, L_{3000\\AA} \\\\\n\\rm L_{Bol} = 9.26 \\, L_{5100\\AA}\n\\end{split}\n\\end{equation}\n\n\\noindent\nThese bolometric corrections have been derived using mean AGN SEDs;\nhowever, \\citet{2006ApJS..166..470R} note that \nusing a bolometric correction resulting from a single mean SED \ncan result in bolometric luminosities with inaccuracies up to 50$\\%$.\n\nUnder the assumption that the BLR gas is virialised, \nthe single-epoch method can be used to estimate BH mass as follows:\n\n\\begin{equation}\\label{mass_est}\n\\rm log\\left(\\frac{M_{BH}}{M_{\\odot}}\\right) = A + B\\,log\\left(\\frac{\\lambda L_{\\lambda}}{10^{44}\\,erg\\,s^{-1}}\\right)+C\\,log\\left(\\frac{FWHM}{km\\,s^{-1}}\\right)\n\\end{equation}\n\n\\noindent\nwhere $\\rm L_{\\lambda}$ is the monochromatic luminosity\nat wavelength $\\lambda$,\nand FWHM is the full width at half maximum of the broad component of the emission line.\nA, B, and C are constants that are calibrated using RM results\nand vary depending on which line is used. \n\nOver the years, many groups have provided \ncalibrations of equation~\\ref{mass_est} for MgII and H$\\beta$.\nIn this work, the calibrations from \n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A} are used for H$\\beta$.\n\\citet{2006ApJ...641..689V} based their work on \nan updated study of the $\\rm R_{BLR} - L$ relationship \n\\citep{2005ApJ...629...61K, 2006ApJ...644..133B} and \na reanalysis of the RM mass estimates \\citep{2004ApJ...613..682P}\nand therefore presented an improved mass calibration relative to previous studies.\n\\citet{2011ApJ...742...93A} provide a mass calibration that is based on the \n$\\rm R_{BLR} - L$ relationship from \\citet{2009ApJ...705..199B}.\n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A} \nboth provide similar calibrations for single-epoch H$\\beta$ mass estimation,\nas can be seen from the left panel of figure~\\ref{mass_hist}.\n\nThe \\citet{2012ApJ...753..125S} calibration is used in this work for MgII.\nThis calibration is based on a sample of \n60 high-luminosity ($\\rm L_{5100\\AA} > 10^{45.4} erg \\, s^{-1}$)\nquasars in the redshift range 1.5 - 2.2.\n\\citet{2012ApJ...753..125S} use the \\citet{2006ApJ...641..689V} \nmass estimates as a reference when determining their MgII calibration.\nThe centre and right panels of figure~\\ref{mass_hist}\nshow the comparison between the \\citet{2012ApJ...753..125S} MgII calibration\nand the $\\rm H \\beta$ calibrations from \n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A}.\nThese calibrations agree reasonably well,\nwith the standard deviation $\\rm \\sigma \\simeq 0.3$ in both cases,\nwhich is likely due to the fact that these BH mass estimates were derived \nusing two different emission lines.\nA list of the three BH mass calibrations used in this work \nis given in Table~\\ref{calib}.\n\nBH masses were computed for each of these calibrations \nand are included in the catalogue (see section~\\ref{columns}). \nBH masses were only estimated for sources with a detected \nbroad line component (see section~\\ref{bl_decomp}).\nThese BH mass estimates were then used to estimate \nthe Eddington luminosity and the Eddington ratio\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  L_{Edd}=4\\pi c G M_{BH} m_{p}\/\\sigma_{T} \\\\\n\\rm \\lambda_{E} = L_{Bol}\/L_{Edd}\n\\end{split}\n\\end{equation}\n\n\\noindent\nwhere c is the speed of light, \nG is the gravitational constant,\nM$\\rm_{BH}$ is the BH mass,\nm$\\rm_{p}$ is the proton mass,\nand $\\rm \\sigma_{T}$ is the Thomson scattering cross-section.\n\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{flux_comparison_2.png}\n\\caption{\\label{bay_class_flux} \\footnotesize \nComparison between the classical and Bayesian methods \nfor estimating 2RXS fluxes. The deviation from a ratio of one at fainter fluxes \nresults from the attempt to correct for the effect of the Eddington bias\n(see section~\\ref{2RXS_flux} for further details).} \n\\end{center}\n\\end{figure}\n\n\n\\begin{figure*}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{uv_opt_lum.png}\n                \\includegraphics[width=0.49\\textwidth]{compare_lbol.png}\n                \\includegraphics[width=0.49\\textwidth]{virial_fwhm.png}\n                \\includegraphics[width=0.49\\textwidth]{compare_mass.png}\n\\caption{\\footnotesize \nComparison between the results obtained from fitting \nthe H$\\beta$ and MgII regions of the spectrum.\nUpper left: 5100$\\AA$ and 2500$\\AA$ monochromatic luminosities.\nUpper right: Bolometric luminosities estimated from the\n5100$\\AA$ and 3000$\\AA$ monochromatic luminosities.\nLower left: Virial FWHM measured using H$\\beta$ and MgII.\nLower right: BH mass estimates derived from H$\\beta$ \n(using the \\citet{2011ApJ...742...93A} calibration) \nand MgII (using the \\citet{2012ApJ...753..125S} calibration).\nIn each panel, the solid black line is the unity line.}\n\\label{compare}\n\\end{figure*}\n\n\n\\section{X-ray Flux Estimates}\\label{xray_flux}\n\n\\noindent\nSince X-ray detections are available for all objects in this sample,\nX-ray flux estimates have also been included in the catalogue.\nXMMSL1 fluxes in the 0.2-12\\,keV range \nfrom \\citet{2008A&A...480..611S} are included.\n\\citet{2008A&A...480..611S} convert the XMMSL1 count rates \nto fluxes using a spectral model consisting of \nan absorbed power law with a photon index of 1.7 \nand N$\\rm_{H}=3\\times10^{20}cm^{-2}$.\nThe 2RXS fluxes were estimated using the method outlined below.\n\n\\subsection{Estimating 2RXS X-Ray Fluxes}\\label{2RXS_flux}\n\nMany of the sources in the 2RXS sample \nhave flux measurements close to\nthe \\textit{ROSAT} flux limit ($\\sim$10$\\rm ^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$).\nTherefore, when estimating fluxes for this sample,\nit was necessary to correct for the Eddington bias.\nThis was done by adopting a Bayesian method \nto derive a probability distribution of fluxes based on the known \ndistribution of AGN as a function of flux.\nFollowing \\citet{1991ApJ...374..344K}, \\citet{2009ApJS..180..102L},\nand \\citet{2011MNRAS.414..992G},\nthe probability of a source having flux f$\\rm_{X}$, \ngiven an observed number of counts C, is \n\n\\begin{equation}\\label{prob_flux}\n\\rm P(f_{X}, C) = \\frac{T^{C}e^{-T}}{C!}\\pi(f_{X})\n\\end{equation}\n\n\\noindent where C is the total number of observed source and background counts,\nT is the mean expected total counts in the detection cell for a given flux,\nand $\\rm \\pi(f_{X})$ is the prior, which is the \ndistribution of AGN per unit X-ray flux interval.\nThe exact expression for the prior was taken from\n\\citet{2008MNRAS.388.1205G}, equation 1.\n\nSource and background counts were taken from \nthe 2RXS catalogue \\citep{2016A&A...588A.103B}.\nA flux-count rate conversion factor,\nwhich was required to estimate T in equation~\\ref{prob_flux},\nwas derived using \\texttt{XSPEC} \\citep{1996ASPC..101...17A}\nassuming a model consisting of \na power law (with $\\rm \\Gamma=2.4$ following \\citet{2017MNRAS.469.1065D})\nabsorbed by the Milky Way column density.\nThis method was used to estimate \nthe flux in the full ROSAT band (0.1-2.4\\,keV)\nas well the monochromatic flux at 2\\,keV.\n\nThe fluxes resulting from the method described above \nwith and without applying the prior (termed ``Bayesian'' and ``classical'', respectively)\nare compared in Figure~\\ref{bay_class_flux}.\nThe disagreement between the two flux estimates increases with decreasing flux,\nwhich is expected since, without the prior, \nthe classical method fails to account for the Eddington bias. \nLow count rate sources in this sample would be assigned \nunrealistically low Bayesian fluxes. \nTo avoid this, the flux was left as undetermined\nwhen the Bayesian flux estimate was more than a factor of ten smaller \nthan the classical flux estimate.\n\n\n\n\\section{Accessing the Data}\\label{access}\n\n\\noindent\nThe results from the spectral analysis discussed above,  \nalong with X-ray flux measurements and visual inspection results,\nhave been made available in an SDSS DR14 value added catalogue\nwhich is available at \nhttp:\/\/www.sdss.org\/dr14\/data\\_access\/value-added-catalogs\/.\nAdditionally, an extended version of the catalogue will be maintained at \nhttp:\/\/www.mpe.mpg.de\/XraySurveys\/SPIDERS\/SPIDERS\\_AGN\/\nThe column description for the catalogue is given in appendix~\\ref{columns}.\n\n\n\\begin{figure*}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{lbol_z_shen.png}\n                \\includegraphics[width=0.49\\textwidth]{mass_lbol_shen.png}\n\\caption{\\footnotesize \nBolometric luminosity versus redshift (left panel) and \nbolometric luminosity versus BH mass (right panel)\nfor the sample presented in this work\nand the \\citet{2011ApJS..194...45S} sample.\nSources with $\\rm H_{\\beta}$-derived BH masses are shown in blue,\nand sources with MgII-derived BH masses are displayed in green.}\n\\label{properties}\n\\end{figure*}\n\n\n\\section{Comparing the UV and Optical Fitting Results}\\label{compare_uv_opt}\n\n\\noindent\nA subsample of sources have spectral measurements available from \nboth the MgII and H$\\beta$ spectral regions. \nIn order to test the consistency of the independent fits to these two regions,\nproperties measured from both were compared.\n\n\\subsection{$L_{2500\\AA}-L_{5100\\AA}$ Relation}\n\n\\noindent\nA  subsample of AGN whose spectra cover the rest-frame wavelengths \n2500$\\AA$ and 5100$\\AA$\nwas selected using the following criteria:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  ( (INSTRUMENT = SDSS) \\,\\,\\&\\&\\,\\, (0.52 < redshift < 0.8) ) \\,\\, || \\\\  \n\\rm ( (INSTRUMENT = BOSS) \\,\\,\\&\\&\\,\\, (0.46 < redshift <1.04) ) \n\\end{split}\n\\end{equation}\n\n\\noindent\nOf this sample, 1718 sources had reliable measurements of both \n$\\rm L_{2500\\AA}$ and $\\rm L_{5100\\AA}$.\nThe $\\rm L_{2500\\AA}-L_{5100\\AA}$ relation was fit \nusing the LINMIX \\citep{2007ApJ...665.1489K} package.\nLINMIX is a Bayesian linear regression algorithm \nthat accounts for uncertainties in both dependent and independent variables,\nas well as non-detections.\nThe upper left panel of figure~\\ref{compare}\nshows the $\\rm L_{2500\\AA}-L_{5100\\AA}$ distribution and\nthe best-fit relation\n\n\\begin{equation}\\label{flux_conversion}\n\\begin{split}\n\\rm Log_{10}(L_{5100\\AA}) = (0.841\\pm 0.007) Log_{10}(L_{2500\\AA}) \\\\\n\\rm -(6.0 \\pm 0.3)\n\\end{split}\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.0151.\nThe comparison between the estimated bolometric luminosities \nderived from the 3000$\\AA$ and 5100$\\AA$ monochromatic fluxes\nis shown in the upper right panel of figure~\\ref{compare}.\nEquation~\\ref{flux_conversion} can be used to estimate L$\\rm_{2500\\AA}$\nfrom L$\\rm_{5100\\AA}$, which allows low redshift sources to be included \nin the $\\rm \\alpha_{OX}$ analysis discussed in section~\\ref{aox_section}.\n\n\n\n\\subsection{Comparing MgII and H$\\beta$ FWHM Measurements}\\label{compare_mass}\n\nA  subsample of AGN whose spectra cover the broad\n$\\rm H\\beta$ and MgII emission lines \nwas selected using the following criteria\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  ( (INSTRUMENT = SDSS) \\,\\,\\&\\&\\,\\, (0.45 < redshift < 0.81) ) \\,\\, || \\\\  \n\\rm ( (INSTRUMENT = BOSS) \\,\\,\\&\\&\\,\\, (0.38 < redshift <1.05) ) \n\\end{split}\n\\end{equation}\n\n\\noindent\nOf this sample, 2323 sources had FWHM measurements\nfor both H$\\beta$ and MgII.\nThe lower left panel of figure~\\ref{compare}\ndisplays the virial FWHM measurements from H$\\beta$ and MgII.\nThe resulting best-fit relation, fit using LINMIX, is\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm Log_{10}(FWHM_{MgII}) = (0.65\\pm0.01)\\, Log_{10}(FWHM_{H\\beta}) \\\\\n\\rm +(1.21\\pm0.04)\n\\end{split}\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.005.\nThis deviation from the one-to-one relation has also been observed by \n\\citet{2009ApJ...707.1334W}, who reported a slope of $0.81\\pm0.02$,\nand \\citet{2012ApJ...753..125S}, who found a slope of $0.57\\pm0.09$.\nThe single-epoch BH mass relations (equation~\\ref{mass_est})\naccount for the $\\rm FWHM_{MgII}-FWHM_{H\\beta}$ slope;\nwhen the correct BH mass calibration is used,\nthe H$\\beta$ and MgII virial FWHM measurements \nyield BH masses that are in close agreement\n(see the lower right panel of figure~\\ref{compare}).\n\n\n\n\\section{Sample Properties}\\label{prop}\n\n\\noindent\nFigure~\\ref{properties} presents the comparison between\nthis sample and the full sample of SDSS DR7 AGN \nwith optical spectral properties measured by \\citet{2011ApJS..194...45S}\nin the bolometric luminosity-redshift and bolometric luminosity-BH mass planes.\nAs discussed in section~\\ref{intro}, $\\rm H_{\\beta}$-derived BH masses \nare used where available (shown in blue), \nwhile MgII-derived masses are used \nfor the remaining higher-redshift sources (shown in green).\nThe left panel of figure~\\ref{properties} \nshows that this sample populates \nthe low-redshift, high-luminosity region of the parameter space,\nwhich is partially due to the high flux threshold of the X-ray selection.\nFrom the right panel of figure~\\ref{properties} \nit can be seen that the sample presented in this work \nappears to be well bounded by the Eddington limit \nat least up to $\\rm M_{BH}\\simeq 10^{9.5}\\,M_{\\odot}$.\n\n\n\n\\subsection{H$\\beta$ Line Components}\\label{VBC_section}\n\n\\noindent\nSection~\\ref{HB_fit_method} described how the H$\\beta$ line profile\nwas fit with either one, two, three, or four Gaussian components.\nFigure~\\ref{line_width} displays the resulting distribution of \n$\\rm H\\beta$ FWHM measurements\n(the panels are split based on the \nnumber of Gaussian components required to fit the line).\nThere is a clear peak in the distribution at low FWHM\nassociated with the narrow H$\\beta$ core\ntypically measured at a few hundred $\\rm km\\,s^{-1}$.\nAbove $\\rm \\sim 1000\\,km\\,s^{-1}$ the distribution is bimodal\n(in the two lower panels) with a large number of sources showing evidence for the \n``very broad component'' (VBC) of $\\rm H\\beta$\nat FWHM$\\rm \\,\\geq 10000\\,km\\,s^{-1}$\nalso discussed in \\citet{2010MNRAS.409.1033M}.\n\nIt has been suggested that the VBC\nis emitted from a distinct physical region, \nand is possibly the result of line emission from the accretion disk \n\\citep[e.g.][]{2009MNRAS.400..924B}.\nIf the VBC represents emission from the accretion disk,\nthen a strong VBC may result in a bias towards  \na higher BH mass estimate, since the \nsingle-epoch method assumes a calibration that is based on \nthe luminosity-BLR radius relation.\nHowever, since the kinematics and physical origin of the VBC remains uncertain,\ndetected VBCs have not been excluded from the broad line profiles \nused to measure the virial FWHM in this analysis \n(as discussed in section~\\ref{bl_decomp}).\n\n\n\\begin{figure}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{component_hist.png}\n\\caption{\\footnotesize \nDistribution of H$\\beta$ Gaussian FWHM values.\nThe panels are split based on the number of Gaussians required to fit the line.\nThe coloured histograms each represent one of up to four \npossible Gaussians used to fit the H$\\rm_{\\beta}$ line. \nThe grey histograms represent the sum of the individual coloured histograms.}\n\\label{line_width}\n\\end{figure}\n\n\n\\begin{figure*}\n                \\includegraphics[width=0.49\\textwidth]{r_fwhm.png}\n                \\includegraphics[width=0.49\\textwidth]{r_fwhm_eddrat.png}\n\\caption{\\footnotesize \nFWHM of the broad component of H$\\beta$ versus R$\\rm _{FeII}$.\nThe left panel displays the sample described in section~\\ref{4DE1_section} (grey), \nand sources with a median spectral S\/N$\\,\\geq\\,$20 (blue).\nThe right panel presents the same subsample of high S\/N sources, \ncolour coded to indicate the trend in Eddington ratio across the distribution.\nThe grey dashed line marks the division between \npopulation A ($\\rm FWHM\\,H_{\\beta}^{BC}\\leq4000\\,km\\,s^{-1}$) and \npopulation B ($\\rm FWHM\\,H_{\\beta}^{BC}\\geq4000\\,km\\,s^{-1}$) sources.\nThe size of the typical uncertainties, multiplied by a factor of five,\nin both variables for the high-S\/N subsample is also shown.}\n\\label{4DE1}\n\\end{figure*}\n\n\n\\subsection{This Sample in the 4D Eigenvector 1 Context}\\label{4DE1_section}\n\n\\noindent\nThe 4D Eigenvector 1 (4DE1) system \n\\citep{1992ApJS...80..109B, 2000ApJ...536L...5S, 2011BaltA..20..427S}\naims to define a set of parameters that uniquely account for AGN diversity.\nTwo main 4DE1 parameters are\nthe FWHM of the broad component of H$\\beta$ ($\\rm FWHM \\, H_{\\beta}^{BC}$)\nand the strength of the FeII emission relative to that of H$\\beta$, defined as \n\n\\begin{equation}\\nonumber\n\\rm R_{FeII} = F_{FeII}\/F_{H\\beta}\n\\end{equation}\n\n\\noindent\nwhere $\\rm F_{FeII}$ and $\\rm F_{H\\beta}$\nare the fluxes of the \nFeII emission in the 4434-4684$\\rm \\AA$ range\nand broad $\\rm H\\beta$ line, respectively.\nA sample of 2098 sources with measurements of these parameters \nand reliable spectral fits ($\\rm 0 \\le \\chi^{2}_{\\nu,H\\beta} \\le 1.2$)\nwas selected. The left panel of figure~\\ref{4DE1} shows \nthe distribution of this sample \nin the 4DE1 parameter space (grey).\nIt is expected that a reliable measurement of the FeII component \nwill be difficult for many of the lower S\/N sources\n\\citep[see][]{2003ApJS..145..199M}.\nFor this reason, the subset of sources in figure~\\ref{4DE1} \nwith a median S\/N greater than or equal to 20 per resolution element is also shown (blue).\nThe right panel of figure~\\ref{4DE1} presents the higher S\/N sources, \ncolour-coded as a function of Eddington ratio.\nThe expected trend of increasing Eddington ratio \ntowards smaller $\\rm FWHM\\,H_{\\beta}^{BC}$ and larger R$\\rm_{FeII}$ \nis observed for this sample of high-S\/N sources.\nTypically, sources with both high $\\rm R_{FeII}$\nand high $\\rm FWHM \\, H_{\\beta}^{BC}$ are not observed. \nIf these sources exist, they may be difficult to detect \nsince strong FeII emission might conceal a faint $\\rm H_{\\beta}$ broad component.\nThe potential bias in the 4DE1 plane source distribution \ndue to model limitations and spectral S\/N \nis discussed in sections~\\ref{sim} and \\ref{sim_bias}.\n\nThe grey dashed line in the right panel of figure~\\ref{4DE1}\nindicates the division between \npopulation A ($\\rm FWHM\\,H_{\\beta}^{BC}\\leq4000\\,km\\,s^{-1}$) \nand population B ($\\rm FWHM\\,H_{\\beta}^{BC}\\geq4000\\,km\\,s^{-1}$)\nsources in the 4DE1 context \\citep[see][]{2011BaltA..20..427S}.\nPopulation A sources often possess Lorentzian broad line profiles,\nand it has been suggested that Gaussian fits to population A broad lines \nwill result in an underestimation of the BH mass \\citep[see][]{2014AdSpR..54.1406S}.\n\n\n\n\\subsection{Relationship Between AGN X-ray and Optical Emission}\\label{aox_section}\n\n\\noindent\nQuasars exhibit a non-linear relationship between their X-ray and UV emission,\nusually represented by the $\\rm \\alpha_{OX}$ parameter\n\n\\begin{equation}\\nonumber\n\\rm \\alpha_{OX} = \\frac{Log(L_{2\\,keV}\/L_{2500\\,\\AA})}{Log(\\nu_{2\\,keV}\/\\nu_{2500\\,\\AA})}\n\\end{equation}\n\n\\noindent\nwhere $\\rm L_{2\\,keV},\\, L_{2500\\,\\AA},\\, \\nu_{2\\,keV},\\, and \\, \\nu_{2500\\,\\AA}$\nare the monochromatic luminosities and frequencies at \n2\\,keV and $\\rm 2500\\,\\AA$, respectively\n\\citep{2003AJ....125.2876V, 2005AJ....130..387S, 2006AJ....131.2826S, 2007ApJ...665.1004J, 2008ApJS..176..355K, 2009ApJ...690..644G, 2009ApJS..183...17Y, 2010A&A...512A..34L}.\nThe $\\rm \\alpha_{OX}$ parameter is considered to be \na proxy for the relative contribution of\nthe UV accretion disk emission and the \nX-ray emission from the surrounding corona\nto the total luminosity.\nIn order to study this relationship, \na sample of sources with measurements of the \n2keV,  2500\\AA, and 5100\\AA \\, luminosities was selected.\nFor lower redshift sources without spectral coverage of 2500\\AA,\nequation~\\ref{flux_conversion} was used to estimate the 2500\\AA\\, luminosity\nfrom the 5100\\AA\\, luminosity.\nExtended sources were removed in order to prevent additional scatter in the relationship due to the contribution of the host galaxy.\nThis was done by requiring that the SDSS g band ``stellarity''\\footnote{\nFor a description of how cModelMag\\_g and psfMag\\_g are measured see\nhttps:\/\/www.sdss.org\/dr12\/algorithms\/magnitudes\/} \n(defined as S(g) = cModelMag\\_g- psfMag\\_g) \nlies between $\\pm$0.1. \nThis sample does not contain X-ray sources with \nmore than one potential AllWISE counterpart and therefore avoids cases where \nthe X-ray detection includes emission from more than one object.\nThis selection process resulted in a sample of 4777 sources.\nFigure~\\ref{aox} shows the $\\rm \\alpha_{OX}$ parameter versus \nthe monochromatic luminosity at 2500$\\AA$.\nThe $\\rm \\alpha_{OX}-L_{2500\\AA}$ relation\nwas fit using LINMIX, which gave the following best-fit result\n\n\\begin{equation}\\nonumber\n\\rm \\alpha_{OX} = 2.39\\pm0.16 - (0.124\\pm0.005) Log(L_{2500\\AA})\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.0034.\nThis slope is consistent with previous results from the literature\n\\citep[e.g.][]{2008ApJS..176..355K}.\n\n\n\\begin{figure}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{aox_L2500.png}\n\\caption{\\footnotesize\n$\\rm \\alpha_{OX}$ versus UV luminosity.\nThe dashed line is the best linear fit to the distribution.\nThe size of the typical uncertainties in both variables is also shown.}\n\\label{aox}\n\\end{figure}\n\n\n\n\\section{Interpreting the Data and Limitations}\\label{limits}\n\n\\noindent\nIn this section, the reliability and limitations of the sample will be discussed.\n\n\\subsection{Measuring the FeII Emission}\n\n\\noindent\nDistinguishing the FeII component from the continuum emission \nbecomes more difficult when using low S\/N spectra.\nIn addition, for a given S\/N, it may also be more difficult to detect FeII emission \nif the intrinsic broadening of the FeII lines is large, \nsince broader, blended FeII emission lines are more likely to be fit \nby the model as continuum emission \\citep[see][]{2003ApJS..145..199M}.\nUsing simulated AGN spectra, \\citet{2003ApJS..145..199M} \nestimate the minimum detectable optical FeII emission \nas a function of H$\\beta$ width for different bins of S\/N.\n\nA poor fit to the FeII emission may affect the accuracy of the BH mass estimates,\nsince FeII emission can influence measurements of both the broad line width (see section~\\ref{feII_FWHM}) and the continuum luminosity.\nFeII emission may also conceal a broad $\\rm H \\beta$ component\nthus biasing a source's position in the 4DE1 plane (figure~\\ref{4DE1}).\nThese potential issues are tested in the following three sections.\n\n\n\n\\subsubsection{Accuracy of the Broad Emission Line FWHM Measurements \nfor Sources with FeII Continuum Emission}\\label{feII_FWHM}\n\nA poor fit to the FeII emission may affect the measurement of \nthe broad emission line width.\nTo quantify the magnitude of this effect, the $\\rm H\\beta$ fitting script \n(using four Gaussians to fit $\\rm H\\beta$) was run with and without the FeII template\non a sample of $\\rm \\sim 400$ randomly selected sources.\nThe fit without the FeII template represents the most extreme case \nwhere the FeII emission is completely ignored by the model.\nTherefore, the change in line widths measured by these two models \nshould be the upper limit on what can be expected \nfor cases where the FeII fit is inadequate.\nFigure~\\ref{feII_test_plot} shows that the line width dispersion induced by \nignoring the presence of FeII emission is $\\rm \\simeq 640 \\,km\\,s^{-1}$.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{width_hist.png}\n\\caption{\\footnotesize\nComparison of the $\\rm H \\beta$ FWHM measurements\nderived using a model with and without an FeII template.\nThe vertical dashed line shows the mean value of the distribution.}\n\\label{feII_test_plot}\n\\end{figure}\n\n\n\\subsubsection{Model Limitations in Detecting Sources in the 4DE1 Plane}\\label{sim}\n\n\\noindent\nSources with both large $\\rm R_{FeII}$ and large $\\rm FWHM\\,H_{\\beta}^{BC}$\nare typically not observed, however,\nthis absence may be due to model limitations;\nat high $\\rm R_{FeII}$ the broad $\\rm H\\beta$ component \nmay be concealed beneath the FeII emission, \nand therefore may not be detected.\nThe experiment outlined in this section was carried out in order to determine \nwhether the spectral fitting code used in this work\nwould return accurate measurements \nfor sources with high $\\rm R_{FeII}$ and $\\rm FWHM\\,H_{\\beta}^{BC}$ values.\n\nA parameter space defined by \n$\\rm 0.1 \\leq R_{FeII} < 5$ and\n$\\rm 1000\\,\\,km\\,s^{-1} \\leq FWHM\\,H_{\\beta}^{BC} < 15000 \\,\\,km\\,s^{-1}$\nwas divided into a 12$\\times$12 grid.\n10 S\/N bins between 5 and 50 \n(a representative range for the samples presented in this work)\nwere selected for each point on the grid,\nand 10 spectra were simulated for each \n$\\rm R_{FeII}-\\rm FWHM \\, H_{\\beta}^{BC}-S\/N$ combination, \nresulting in 14400 simulated spectra.\nFor the parameters that were fixed in this experiment, \nthe interquartile mean of the best-fit values for the type 1 AGN in this sample were used.\nThe $\\rm H\\beta$ line profile was modelled with one narrow and one broad Gaussian.\nThe wavelength range was set to $\\rm 4420 - 5500 \\AA$ (as in section~\\ref{HB_fit_method})\nand the logarithmic wavelength \nspacing\\footnote{https:\/\/www.sdss.org\/dr12\/spectro\/spectro\\_basics\/} \nwas set to be equal to that of SDSS spectra;\n\n\\begin{equation}\\nonumber\n\\rm Log_{10}\\lambda_{i+1} - Log_{10}\\lambda_{i} = 0.0001\n\\end{equation}\n\nThese spectra were fit using a version of the $\\rm H\\beta$ fitting script \nwhich used one narrow and one broad Gaussian component to fit $\\rm H\\beta$.\nThe minimum S\/N required for the fitting script to return the correct \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations\nwas then determined.\nIn order to consider an $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\ncombination detectable at a given S\/N,\nat least  7\/10 spectra were required to have best fit \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values \nthat agreed with the input values.\n\nFigure~\\ref{4DE1_sim_1} shows the detectable \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\ncombinations for each point on the grid, along with the minimum S\/N required to \ndetect that combination. \nIt is clear from figure~\\ref{4DE1_sim_1} that \nat the S\/N levels available in this sample,\na large region of the 4DE1 parameter space would not be detected.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{rfeII_fwhm_sn_test3.png}\n\\caption{\\footnotesize\nMinimum S\/N required to detect a range of \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations.\nThe blue grid indicates the range of the parameter space covered\nin the simulation described in section~\\ref{sim}.\nPoints on the grid that do not have a minimum S\/N indicator represent \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations\nthat are not detectable even at the highest S\/N used in this experiment.\nSources detected at these $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ \ncombinations are likely to be spurious (see figure~\\ref{sn_slice}).}\n\\label{4DE1_sim_1}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{rfeII_fwhm_sn_5.png}\n\\includegraphics[width=0.49\\textwidth]{rfeII_fwhm_sn_455.png}\n\\caption{\\footnotesize\nComparison between the measured and simulated \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values\nfor the highest and lowest S\/N bins used in section~\\ref{sim}.\nFor clarity, each figure displays only one of the ten sets of spectra\nfor that S\/N.\nThe values used to simulate the spectra (grey points)\nare connected to the corresponding best fit measurements\n(except for cases where either the broad $\\rm H\\beta$\nor FeII components were not detected).\nFor clarity, the figures do not show a small number of \nunphysically high $\\rm R_{FeII}$ measurements.}\n\\label{sn_slice}\n\\end{figure*}\n\n\n\\subsubsection{Bias in the $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\nDistribution due to Model Limitations}\\label{sim_bias}\n\n\\noindent\nFigure~\\ref{sn_slice} shows the comparison between the simulated and measured\n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values \nfor the highest and lowest S\/N bins used in section~\\ref{sim}.\nThe left panel of figure~\\ref{sn_slice} shows that at low S\/N\nthe results are clearly biased against high  \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values.\nAt higher S\/N (figure~\\ref{sn_slice}, right panel), \nthe accuracy of the lower left quadrant measurements is significantly improved.\nHowever, even at S\/N=45.5 \n(which is approximately the upper end of the S\/N distribution \nof the samples presented in this work)\nthe high $\\rm R_{FeII}$ - $\\rm FWHM \\, H_{\\beta}^{BC}$ measurements \ndeviate significantly from the corresponding ``true'' values.\nThis may suggest that the L-shaped distribution of sources\nin the 4DE1 plane (e.g. figure~\\ref{4DE1})\nis at least in part due to model limitations.\n\n\n\n\\subsection{Reliability of the Single-Epoch Method for Mass Estimation}\\label{reliability}\n\n\\noindent\nAssuming that AGN broad emission lines are produced by gas \nwhose motion is dominated by the gravitational potential of the central SMBH,\nthe single-epoch method is expected to produce \nreliable mass estimates when compared to RM \\citep[see][]{2006ApJ...641..689V},\nwith a systematic uncertainty of 0.3-0.4 dex.\nHowever, it is not clear how to measure \nthe virial FWHM of lines that deviate from this norm.\n\nThe spectrum shown in the left panel of figure~\\ref{unusual_fig}\nis an example of a source which exhibits \na double-peaked H$\\beta$ line profile,\nwhere a clear inflection point is visible \nbetween two velocity-shifted broad line components.\nDouble-peaked broad line profiles in AGN \nare expected to be the result of emission from the accretion disk\n\\citep{1988MNRAS.230..353P, 1989ApJ...339..742C, 1994ApJS...90....1E, 2003AJ....126.1720S, 2003ApJ...599..886E}.\n\\citet{2007MNRAS.376.1335Z} have found that single-epoch BH mass estimates \nobtained from double-peaked line profiles \nare significantly larger than BH mass estimates \nderived from stellar velocity dispersion measurements.\n\\citet{2007MNRAS.376.1335Z} suggest that this discrepancy \nis the result of an overestimation of the BLR radius\nby the single-epoch mass calibrations for these objects.\nTherefore, the BH mass estimates provided in this work for sources \nwhich exhibit double-peaked broad emission lines \nshould be treated with caution.\n\nThe right panel of figure~\\ref{unusual_fig} shows\nan example of narrow absorption in the UV portion of the spectrum\ncaused by intervening absorbing material along the line of sight to the AGN.\nSources identified during the visual inspection \nas having narrow absorption lines \nhave been flagged in the catalogue (column 189; flag\\_abs).\nThese sources were fit using the model described in section~\\ref{mgII_fit} \nwith the absorption line regions masked.\nHowever, in many cases, the absorption features distort the broad MgII line, \nand therefore the resulting BH mass estimates may not be reliable.\n\n\n\\begin{figure*}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{spec-7512-56777-0321.png}\n                \\includegraphics[width=0.49\\textwidth]{spec-8188-57348-0946}\n\\caption{\\footnotesize\nLeft panel: Example of a source with a double-peaked H$\\rm \\beta$ line profile\n(plate=7512, MJD=56777, fiber=321).\nRight panel: Example of a source showing narrow UV absorption features\nwhich have been masked when fitting the model\n(plate=8188, MJD=57348, fiber=946). \nSee section~\\ref{reliability} for details.}\n\\label{unusual_fig}\n\\end{figure*}\n\n\n\n\\section{Conclusions}\n\n\\noindent\nThis work presents a catalogue of spectral properties for\nall SPIDERS type 1 AGN up to SDSS DR14.\nVisual inspection results were used to select a reliable subsample for spectral analysis,\nand the spectral regions around H$\\beta$ and MgII were fit with a multicomponent model.\nUsing the single-epoch method, BH masses, bolometric luminosities, \nEddington ratios, along with additional spectral parameters were measured. \nA catalogue containing these results has been made available\nas part of a set of SDSS DR14 value added catalogues.\nThis catalogue also includes the results of a visual inspection of the sample,\nand is available at \nhttp:\/\/www.mpe.mpg.de\/XraySurveys\/SPIDERS\/SPIDERS\\_AGN\/.\n\n\n\n\\section{Acknowledgements}\nDC has participated in the International Max Planck Research School \non Astrophysics at the Ludwig Maximilians University Munich.\nDC also acknowledges financial support from the Max Planck Society.\nDC would also like to thank Riccardo Arcodia and Julien Wolf\nfor many helpful discussions.\nThe authors would like to thank Josephine Reisinger \nfor her contribution to the visual inspection of sources in this sample.\nFinally, the authors would like to thank \nthe anonymous referee for providing a thorough critique of the paper \nwhich greatly improved its content.\n\nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, \nthe U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges\nsupport and resources from the Center for High-Performance Computing at\nthe University of Utah. The SDSS web site is www.sdss.org.\n\nSDSS-IV is managed by the Astrophysical Research Consortium for the \nParticipating Institutions of the SDSS Collaboration including the \nBrazilian Participation Group, the Carnegie Institution for Science, \nCarnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, \nInstituto de Astrof\\'isica de Canarias, The Johns Hopkins University, \nKavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ \nUniversity of Tokyo, Lawrence Berkeley National Laboratory, \nLeibniz Institut f\\\"ur Astrophysik Potsdam (AIP),  \nMax-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), \nMax-Planck-Institut f\\\"ur Astrophysik (MPA Garching), \nMax-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), \nNational Astronomical Observatories of China, New Mexico State University, \nNew York University, University of Notre Dame, \nObservat\\'ario Nacional \/ MCTI, The Ohio State University, \nPennsylvania State University, Shanghai Astronomical Observatory, \nUnited Kingdom Participation Group,\nUniversidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, \nUniversity of Colorado Boulder, University of Oxford, University of Portsmouth, \nUniversity of Utah, University of Virginia, University of Washington, University of Wisconsin, \nVanderbilt University, and Yale University.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation,\nthe Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. \nThe SDSS-III web site is http:\/\/www.sdss3.org\/.\n\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including \nthe University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, \nthe French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrof\\'isica de Canarias, \nthe Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, \nMax Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, \nNew York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, \nthe Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, \nUniversity of Virginia, University of Washington, and Yale University.\n\nFunding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, \nthe Participating Institutions, the National Science Foundation, the U.S. Department of Energy, \nthe National Aeronautics and Space Administration, the Japanese Monbukagakusho, \nthe Max Planck Society, and the Higher Education Funding Council for England. \nThe SDSS Web Site is http:\/\/www.sdss.org\/.\n\nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. \nThe Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, \nUniversity of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, \nthe Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, \nthe Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), \nLos Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), \nNew Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, \nPrinceton University, the United States Naval Observatory, and the University of Washington.\n\nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory\/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration.\n\nPlot colours were in part based on www.ColorBrewer.org, \nby Cynthia A. Brewer, Penn State.\nThe TOPCAT tool \\citep{2005ASPC..347...29T} was used during this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe description of perturbative quantum field theory in terms of Hopf algebras \\cite{Kre98,CK99,CK00} (cf. \\cite{CM04c,CM04b,CM06}) has led to a better understanding of the combinatorial structure of renormalization. At the same time, it provided a beautiful interaction of quantum field theory with several parts of mathematics. \n\nAlthough the Hopf algebra was modelled on a scalar quantum field theory, it can be generalized to any quantum field theory. However, subtleties appear in the case of quantum gauge theories. Gauge theories are field theories that are invariant under a certain group, known as the gauge group. In the process of renormalization, one has to deal with the `overcounting' due to the gauge degrees of freedom. Furthermore, one has to understand how the gauge symmetries manifest themselves in the quantum field theory. Important here are the so-called Ward-Takahashi (or Slavnov-Taylor) identities.\n\nThe first example of a gauge theory is quantum electrodynamics (QED), which has the abelian group $U(1)$ as a gauge group. The Hopf algebraic structure of Feynman graphs in QED was desrcibed in \\cite{BK99,KD99} and more recently in \\cite{VP04}. A slightly different approach is taken in \\cite{BF01,BF03}, where Hopf algebras on planar binary trees were considered. \nThe anatomy of a gauge theory with gauge group $SU(3)$ (known as quantum chromodynamics) has been discussed in the Hopf algebra setting in \\cite{Kre05}, with a central role played by the Dyson-Schwinger equations (see also \\cite{KY06}). However, a full understanding of renormalization of general gauge theories is still incomplete. This paper is a modest attempt towards understanding this by working out explicitly the Hopf algebraic structure underlying renormalization of quantum electrodynamics, thereby implementing the Ward-Takahashi identities in a compatible way.\n\n\\bigskip\n\nWe start in Section \\ref{sect:hopf} by defining the commutative Hopf algebra $H$ of Feynman graphs for quantum electrodynamics. The approach we take differs from the one in \\cite{BF03} where a noncommutative Hopf algebra was considered. This was motivated by the fact that in QED, Feynman amplitudes are (noncommuting) matrices. Instead, we work with a commutative Hopf algebra and encode the matricial form of the Feynman amplitudes into the distributions giving the external structure of the graph. Our approach has the advantage of still being able to use the duality between commutative Hopf algebras and affine group schemes in the formulation of the BPHZ-procedure.\n\nIn Section \\ref{sect:birkhoff}, we associate a Feynman amplitude to a Feynman graph, as dictated by the Feynman rules. The BPHZ-formula of renormalization is obtained as a special case of the Birkhoff decomposition for affine group schemes \\cite{CK99}. \n\nThen in Section \\ref{sect:wt}, we incorporate the Ward-Takahashi (WT) identities as relations between graphs, much as was done by 't Hooft and Veltman in \\cite{HV73}. We show that these relations are compatible with the coproduct in the Hopf algebra $H$, so that they define a Hopf ideal. This reflects the physical fact that WT-identities are compatible with renormalization (in the dimensional regularization and minimal subtraction scheme). \n\nThe Feynman amplitudes can now be defined on the quotient Hopf algebra of $H$ by this ideal, and we conclude that the counterterms as well as the renormalized Feynman amplitudes satisfy the WT-identities in the physical sense ({i.e.}, between Feynman amplitudes). In particular, we arrive at the well-known identity $Z_1=Z_2$ as derived by Ward in \\cite{War50}. \n\n\n\n\\section{Hopf algebra structure of Feynman graphs in QED}\n\\label{sect:hopf}\nQuantum electrodynamics in 4 dimensions is given by the following classical Lagrangian,\n\\begin{align*}\n\\mathcal{L}=- \\frac{1}{4} F_{\\mu\\nu} F^{\\mu\\nu} - \\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2 +\\overline\\psi \\left( \\gamma^\\mu(\\partial_\\mu+ e A_\\mu)-m\\right) \\psi,\n\\end{align*}\nwhere $F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$ is the {\\it field strength} of the {\\it gauge potential} $A_\\mu$, $\\psi$ is the spinor describing the electron with mass $m$ and electric charge $e$ and $\\gamma^\\mu, \\mu=1,\\ldots,4$ are the $4 \\times 4$ Dirac matrices. Finally, the real parameter $\\xi$ is the {\\it gauge-fixing parameter}. \n\nThe Lagrangian $\\mathcal{L}$ describes the dynamics and the interaction as well as the gauge fixing: we can write $\\mathcal{L}=\\mathcal{L}_0 + \\mathcal{L}_\\mathrm{int}+\\mathcal{L}_\\mathrm{gf}$, where the free, interaction and gauge fixing parts are\n\\begin{align*}\n\\mathcal{L}_0&=A_\\mu D_{\\mu\\nu} A_\\nu + \\overline\\psi (\\gamma^\\mu \\partial_\\mu -m) \\psi,\\\\\n\\mathcal{L}_\\mathrm{int}&= e \\overline\\psi A_\\mu \\psi.\\\\\n\\mathcal{L}_\\mathrm{gf} &= -\\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2.\n\\end{align*}\nThe adjective ``quantum'' for electrodynamics is justified when this Lagrangian is used in computing probability amplitudes. In perturbation theory, one expands such amplitudes in terms of Feynman diagrams which are the graphs constructed from the following vertices, as dictated by the form of the Lagrangian:\n\\begin{center}\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{photon,label=$A_\\mu$}{l,v}\n      \\fmf{photon,label=$A_\\nu$}{v,r}\n    \\end{fmfgraph*}\\qquad\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain,label=$\\overline\\psi$}{l,v}\n      \\fmf{plain,label=$\\psi$}{v,r}\n    \\end{fmfgraph*}\\qquad\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon,label=$A_\\mu$}{l,v}\n      \\fmf{plain,label=$\\overline\\psi$}{r1,v}\n      \\fmf{plain,label=$\\psi$}{v,r2}\n    \\end{fmfgraph*}\n\\end{center}\n\\begin{rem}\n\\label{rem:el-mass}\nIn fact, we will also need the 2-point vertex \n\\begin{fmfgraph*}(40,10)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfdot{v}\n    \\end{fmfgraph*}\n, associated to the electron mass term, similar to \\cite{BF01}. \n\\end{rem}\nWe will focus on the so-called {\\it one-particle irreducible} (1PI) {\\it graphs}, which are graphs that are not trees, and that cannot be disconnected by cutting a single internal edge. In QED, there are three types of 1PI graphs that are of interest in renormalization theory: the vacuum polarization, the  electron self-energy and the full vertex graphs:\n\\begin{center}\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmflabel{$\\Gamma(q,\\mu,\\nu)=q$}{l}\n      \\fmflabel{$q$}{r}\n      \\fmf{photon,label=$\\mu$}{l,v}\n      \\fmf{photon,label=$\\nu$}{v,r}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\\\\\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmflabel{$\\Gamma(p)=p$}{l}\n      \\fmflabel{$p$}{r}\n      \\fmf{fermion}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\\\\\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$\\Gamma(q,\\mu;p)=q$}{l}\n      \\fmflabel{$p$}{r1}\n      \\fmflabel{$p+q$}{r2}\n      \\fmf{photon,label=$\\mu$}{l,v}\n      \\fmf{fermion}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\n\\end{center}\n\\bigskip\nHere the blob stands for any 1PI graph of the type dictated by the external lines.\n\n\\bigskip\n\nWe now describe the {\\it external structure} of a Feynman graph $\\Gamma$, assigning (among other data) momenta to the external legs. In physics, the Feynman amplitude of $\\Gamma$ (dictated by the Feynman rules, see below) is evaluated with respect to this external structure (see below). Mathematically, the external structure is a distribution on the space of Feynman amplitudes, understood as test functions on a suitable space. \n\nMore explicitly, in the case of QED the (Euclidean) Feynman amplitudes $U(\\Gamma)$ are functions in $C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$ where\n\\begin{align*}\nE_\\Gamma=\\left\\{ (q_1,\\ldots,q_n,\\mu_1,\\ldots,\\mu_n,p_{n+1}, \\ldots, p_N) : \\sum p_i+q_j=0,  \\right\\}\n\\end{align*}\nwith $\\{1,\\ldots,n\\}$ the set of indices labelling the external photon lines of $\\Gamma$ with spatial index $\\{\\mu_1,\\ldots,\\mu_n\\}$ respectively, and $\\{n+1,\\ldots,N\\}$ is the set of its external electron lines. The factor $M_4(\\C)$ can be understood from the fact that in QED, Feynman amplitudes are matrix-valued functions on $E_\\Gamma$. \n\nFor QED, the external structure is thus given by an element in the space of distributions on $C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$; we denote this space by $\\left(C^\\infty(E_\\Gamma) \\otimes M_4(\\C)\\right)'$. For example, for the above full vertex graph $\\Gamma$, $E_\\Gamma=\\left\\{ (q,\\mu,p,p'): \\sum q+ p+p'=0 \\right\\}$ and a typical distribution is given by $\\sigma^{kl}=\\delta_{q,\\mu,p,-p-q} \\otimes e^{kl}$ in terms of a Dirac mass and the (standard) dual basis $\\{ e^{kl}\\}$ of $M_4(\\C)'$. Evaluation on the Feynman amplitude $U(\\Gamma) \\in C^\\infty(E_\\Gamma)\\otimes M_4(\\C)$ is then given by the pairing\n\\begin{align*}\n\\langle \\sigma^{kl}, U(\\Gamma) \\rangle = U(\\Gamma)(q,\\mu;p,-p-q)_{kl}\n\\end{align*}\nNote that $p'=-p-q$ is translated in the diagram by a reversal of the corresponding arrow with associated momentum $p+q$. \n\n\\bigskip\n\nWe will be interested in the following special external structures for the different graphs introduced above. A full vertex graph $\\Gamma(q,\\mu;p)$ at zero momentum transfer (meaning $q=0$) can be written in terms of the two form factors $F_1, F_2$ (cf. \\cite[Section 6.2]{PS95})\n\\begin{align}\nU(\\Gamma)(q=0,\\mu;p)=\\gamma_\\mu F_1(p^2) + \\frac{\\not{p} \\gamma_\\mu \\not{p}}{p^2} F_2(p^2).\n\\end{align}\nwhere $\\not{p}=\\sum_\\mu \\gamma^\\mu p_\\mu$. Only the first form factor $F_1$ requires renormalization; therefore, we define the following external structure:\n\\begin{align}\n\\label{ext1}\n\\langle \\sigma_\\ext{1},f \\rangle &= \\frac{1}{16}\\sum_\\mu \\mathrm{tr~} \\gamma_\\mu f(q=0,\\mu;p=0),\\qquad \\forall f \\in C^\\infty(E_\\Gamma) \\otimes M_4(\\C),\n\\end{align}\nwhere the normalization comes from $\\mathrm{tr~} \\gamma^\\mu \\gamma_\\mu = 16$. For the Ward-Takahashi identities, we will also need the following external structure,\n\\begin{align}\n\\langle \\sigma_\\ext{0,\\mu, p}^{kl}, f \\rangle = f(q=0, \\mu; p)_{kl},\n\\end{align}\nwhich puts finite momentum $p$ on the ingoing and outgoing electron lines, but zero momentum on the external photon line (zero-momentum transfer). This external structure allows us to treat Ward-Takahashi identities in massless QED as well. \n\n\\begin{rem}\nThe form factors $F_1$ and $F_2$ can also be recovered separately by using the following two distributions \\cite{KD99},\n\\begin{align}\n\\langle \\sigma',f \\rangle &= \\mathrm{tr~} \\sum_\\mu \\gamma^\\mu f(q=0, \\mu;p),  \\\\\n\\langle \\sigma'',f \\rangle &= \\mathrm{tr~} \\sum_\\mu \\not{p} p_\\mu f(q=0, \\mu;p).\n\\end{align}\n\\end{rem}\n\n\\bigskip\n\nFor an electron self-energy graph $\\Gamma(p)$, we have $E_\\Gamma = \\{ (p)\\}$ and the corresponding Feynman amplitude is written in terms of two form factors:\n\\begin{align}\nU(\\Gamma)(p)=\\not{p} A(p^2) + m B(p^2).\n\\end{align}\nCorrespondingly, there are two distributions, which we choose of the form,\n\\begin{align}\n\\label{ext0}\n\\langle \\sigma_\\ext{0},f \\rangle &= \\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial p_\\mu} f(p) \\big|_{p=0} +  m^{-1} ~\\mathrm{tr~} f(p=0),\\\\\n\\label{ext2}\n\\langle \\sigma_\\ext{2},f \\rangle &= \\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial p_\\mu} f(p) \\big|_{p=0},\n\\end{align}\nfor all $f \\in C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$. Also, there is an external structure which puts finite momentum on the ingoing and outgoing electron lines,\n\\begin{align}\n\\langle \\sigma_\\ext{p_\\mu} , f \\rangle = \\frac{\\partial}{\\partial p_\\mu} f(p)\n\\end{align}\n\nFinally, for a vacuum polarization graph $\\Gamma$, $E_\\Gamma = \\{ (q, q', \\mu,\\nu) : q+q'=0 \\} \\equiv \\{ (q, \\mu,\\nu )\\}$, we let $\\sigma_\\ext{3}$ be the external structure that satisfies\n\\begin{align}\n\\label{ext3}\n\\langle \\sigma_\\ext{3}, \\left(- \\delta_{\\mu\\nu}q^2+ q_\\mu q_\\nu\\right)\\otimes M \\rangle = \\mathrm{tr~} M,\n\\end{align}\nfor all $M \\in M_4(\\C)$. \n\n\\begin{rem}\nThe labelling of the external structures by $\\ext{0}, \\ext{1}, \\ext{2}$ and $\\ext{3}$ differs from the one in \\cite{CK00}, where they were labelled by $\\ext{0}$ and $\\ext{1}$. Here we follow \\cite{BF01}, so that the labels correspond to the renormalization constants $m_0, Z_1, Z_2, Z_3$, as they appear in the Lagrangian,\n\\begin{align}\n\\label{eq:L-ren}\n\\mathcal{L} =  - \\frac{1}{4} Z_3 A_\\mu D_{\\mu\\nu} A_\\nu - \\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2 + Z_2 \\overline\\psi \\left( \\gamma^\\mu \\partial_\\mu - m_0\\right) \\psi  + e Z_1 \\overline\\psi \\gamma^\\mu A_\\mu \\psi .\n\\end{align}\nThe precise correspondence will be given in equation \\eqref{eq:ren-const} below.\n\\end{rem}\n\n\\bigskip\n\nLet $H$ be the free commutative algebra generated by pairs $(\\Gamma, \\sigma)$ with $\\Gamma$ a 1PI graph and $\\sigma \\in \\left(C^{\\infty} (E_\\Gamma) \\otimes M_4(\\C) \\right)'$ a distribution encoding its external structure. The product in $H$ can be understood as the union of graphs, with induced external structure. More precisely, for two 1PI graphs $\\Gamma_\\sigma$ and $\\Gamma'_{\\sigma'}$, the product $\\Gamma_\\sigma \\cdot \\Gamma'_{\\sigma'}$ is the graph $\\Gamma \\cup \\Gamma'$ with external structure given by $\\sigma \\otimes \\sigma' \\in \\left(C^\\infty(E_\\Gamma \\times E_{\\Gamma'}) \\otimes M_4(\\C)\\right)'$. The algebra $H$ is a (positively) graded algebra $H=\\oplus_{n\\in \\N} H^n$ , with the grading given by the loop number of the graph. \n\nIn the following, we will also write $\\Gamma_\\sigma$ for the pair $(\\Gamma,\\sigma)$, and sometimes even $\\Gamma$. We will shorten the notation for graphs equipped with the special external structures defined above by writing $\\Gamma_\\ext{k} = (\\Gamma, \\sigma_\\ext{k})$.\n\nWe define a coproduct $\\Delta :H\\to H \\otimes H$ as follows; if $\\Gamma$ is a 1PI graph, then one sets\n\\begin{align}\n\\label{coproduct}\n\\Delta \\Gamma = \\Gamma \\otimes 1 + 1 \\otimes \\Gamma + \\sum_{\\gamma \\subset \\Gamma} \\gamma_{(k)} \\otimes \\Gamma\/\\gamma_{(k)}.\n\\end{align}\nHere $\\gamma$ is a proper subset of the graph $\\widetilde\\Gamma$ formed by the internal edges of $\\Gamma$ ({i.e.} $0 \\subsetneq \\gamma \\subsetneq \\widetilde \\Gamma$). The connected components $\\gamma'$ of $\\gamma$ are 1PI graphs with the property that the set of edges of $\\Gamma$ that meet $\\gamma'$ have two or three elements. The sum runs over all multi-indices $k$, one index for each 1PI connected component of $\\gamma$, and one denotes by $\\gamma'_{(k)}$ the 1PI graph that has external structure as defined in equation \\eqref{ext1}-\\eqref{ext3}, with $k=3$ in the case of a vacuum polarization, $k=1$ for a full vertex graph and $k=0$ or $2$ in the case of an electron self-energy. In equation \\eqref{coproduct}, $\\gamma_{(k)}$ denotes the disjoint union of all graphs $\\gamma'_{(k)}$ associated to the connected components of $\\gamma$ and the graph $\\Gamma\/\\gamma_{(k)}$ is the graph $\\Gamma$ (with the same external structure) with each $\\gamma'$ reduced to a vertex of type $(k)$. More explicitly, if $\\gamma'=$\n\\begin{fmfgraph*}(30,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}, then for $k=2$ one replaces this graph in $\\Gamma$ by the line     \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{2}$}}{v}\n\\end{fmfgraph*}\n$ \\equiv$\n      \\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n\\end{fmfgraph*}\n, and for $k=0$ by \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v}\n\\end{fmfgraph*}\n$ \\equiv $\n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfdot{v}\n\\end{fmfgraph*} (cf. Remark \\ref{rem:el-mass}).\nIn the case of the vacuum polarization and the full vertex graph, one replaces $\\gamma'$ by the corresponding vertex. \n\nBy complete analogy with \\cite{CK99}, we find that with this coproduct, $H$ becomes a connected graded Hopf algebra, {i.e.} $H=\\oplus_{n \\in \\N} H^n$, $H^0=\\C$ and\n\\begin{align*} \n\\Delta(H^n) = \\sum_{k=0}^n H^k \\otimes H^{n-k}.\n\\end{align*}\nIndeed, $H^0$ consists of complex multiples of the empty graph, which is the unit in $H$, so that $H^0=\\C 1$. Moreover, from general results on graded Hopf algebras, we obtain the antipode inductively, \n\\begin{align}\n\\label{antipode}\nS(X)=-X - S(X')X'',\n\\end{align}\nwhere $\\Delta(X) = X\\otimes 1 + 1\\otimes X+ \\sum X' \\otimes X''$.\n\nWe give a few examples in order to clarify the coproduct. \n\\begin{align*}\n\\Delta\\bigg(\n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n\\bigg) &= \n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n \\otimes 1 + 1 \\otimes \n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v2,v3,r}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n_\\ext{2}\n\\otimes\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v2,v3,r}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n_\\ext{0}\n\\otimes\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v3,v4,r}\n      \\fmfdot{v3}\n      \\fmf{photon,left,tension=0}{v1,v4}\n\\end{fmfgraph*}}\n\\end{align*}\n\\begin{multline*}\n\\Delta\\bigg(\n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n\\bigg) = \\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}} \\otimes 1\n+ \n1 \\otimes \\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n_\\ext{2}\n\\otimes \n\\parbox{50pt}{\n\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n\\end{fmfgraph*}}\n\\\\[5mm]\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n_\\ext{0}\n\\otimes \n\\parbox{50pt}{\n\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmfdot{v3}\n\\end{fmfgraph*}}\n+ \n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n    \\fmfforce{(.33w,0h)}{b}\n    \\fmfleft{l}\n    \\fmfright{r}\n    \\fmf{plain}{l,v1,v3,v5,v6,v7,r}\n    \\fmffreeze\n    \\fmf{photon}{b,v3}\n    \\fmf{photon,left,tension=0}{v5,v6}\n    \\fmf{photon,left,tension=0}{v1,v7}\n    \\end{fmfgraph*}}\n_\\ext{1}\n\\otimes \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n    \\fmfforce{(.66w,0h)}{b}\n    \\fmfleft{l}\n    \\fmfright{r}\n    \\fmf{plain}{l,v1,v2,v3,v6,v7,r}\n    \\fmffreeze\n    \\fmf{photon}{b,v6}\n    \\fmf{photon,left,tension=0}{v2,v3}\n    \\fmf{photon,left,tension=0}{v1,v7}\n    \\end{fmfgraph*}}\n_\\ext{1}\n\\otimes \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n\\end{multline*}\n\n\n\\section{Birkhoff decomposition and renormalization}\n\\label{sect:birkhoff}\nIn \\cite{CK99}, Connes and Kreimer understood renormalization of perturbative quantum field theory in terms of a Birkhoff decomposition. In particular, using the dimensional regularization (dim-reg) and minimal subtraction scheme, they have proved that the BPHZ-formula is a special case of the Birkhoff decomposition of a loop on a small circle in the complex plane, centered at the dimension of the theory, and taking values in a pro-unipotent Lie group. Before applying this to the case of QED, we briefly recall the Birkhoff decomposition, while referring to \\cite{CK99} and \\cite{CM04b} for more details.\n\nThe Birkhoff decomposition provides a procedure to extract a finite value from a singular expression. More precisely, let $C \\in \\mathbb{P}^1(\\C)$ be a smooth simple curve and let $C_\\pm$ denote its two complements with $\\infty \\in C_-$. The Birkhoff decomposition of a loop $\\gamma :C \\to G$, taking values in a complex Lie group $G$, is a factorization of the form\n\\begin{align*}\n\\gamma(z) = \\gamma_-(z)^{-1} \\gamma_+(z) ,\\qquad z \\in \\C\n\\end{align*}\nwhere $\\gamma_\\pm$ are boundary values of holomorphic maps (denoted by the same symbol)\n\\begin{align*}\n\\gamma_\\pm : C_\\pm \\to G.\n\\end{align*}\nThe normalization $\\gamma_-(\\infty)=1$ ensures uniqueness of the decomposition (if it exists). \n\nThe evaluation $\\gamma \\to \\gamma_+(z_0)\\in G$ is a natural principle to extract a finite value from the (possibly) singular expression $\\gamma(z_0)$. This gives a multiplicative removal of the pole part for a meromorphic loop $\\gamma$, when we let $C$ be an infinitesimal circle centered at $z_0$. \n\nExistence of the Birkhoff decomposition has been established for prounipotent complex Lie groups in \\cite{CK99}, and, more generally, in \\cite{CM04b} in the setting of affine group schemes. Recall that affine group schemes are dual to commutative Hopf algebras in the following sense. For a commutative Hopf algebra $H$, we understand an affine group scheme as a functor $G:A \\to G(A)$ from the category of commutative algebras to the category of groups in the following way. For a given algebra $A$, one defines the group $G(A)$ to be the set of all homomorphisms from $H$ to $A$, with product, inverse and unit given as the duals with respect to the coproduct, antipode and counit $\\epsilon$ respectively, {i.e.}\n\\begin{align*}\n\\phi_1 * \\phi_2(X) &= \\langle \\phi_1 \\otimes \\phi_2, \\Delta(X) \\rangle,\\\\\n\\phi^{-1}(X)&=\\phi(S(X)),\\\\\ne(X)&=\\epsilon (X).\n\\end{align*}\nThe following result is the algebraic translation of the Birkhoff decomposition to (pro-unipotent) affine group schemes, expressed in terms of the commutative (connected graded) Hopf algebra $H$ underlying this affine group scheme. Its proof can be found in \\cite{CK99} (see also \\cite{CM04b}). We assume that $C$ is an infinitesimal circle centered at $0 \\in \\C$ and denote by $K$ the field of convergent Laurent series, with arbitrary radius of convergence and by $\\mathcal{O}$ the ring of convergent power series. Furthermore, we set $\\mathcal{Q}=z^{-1} \\C([z^{-1}])$. \n\\begin{thm}[Connes-Kreimer]\n\\label{birkhoff}\nLet $\\phi:H \\to K$ be an algebra homomorphism from a commutative connected graded Hopf algebra $H$ to the field $K$ defined above. The Birkhoff decomposition of the corresponding loop is obtained recursively from the equalities,\n\\begin{align*}\n\\phi_-(X)&=-T \\left( \\phi(X)+\\sum \\phi_-(X') \\phi(X'') \\right),\n\\end{align*}\nwhere we have written $\\Delta(X) = X \\otimes 1 + 1 \\otimes X + \\sum X' \\otimes X''$ for $X \\in H$ and $T$ is the projection on the pole part in $z$, and $\\phi_+= \\phi_- * \\phi$, or explicitly, \n\\begin{align*}\n\\phi_+(X)&=\\phi(X) + \\phi_-(X) + \\sum \\phi_-(X') \\phi(X'').\n\\end{align*}\nThe maps $\\phi_-$ and $\\phi_+$ are homomorphisms from $H$ to $\\mathcal{Q}$ and $\\mathcal{O}$ respectively.\n\\end{thm}\nThe loop $\\gamma : C \\to G$ defined in terms of $\\phi: H \\to K$ by $\\gamma(z)(X):=\\phi(X)(z)$ therefore factorizes as $\\gamma=\\gamma_-^{-1} \\gamma_+$ where $\\gamma_\\pm:C_\\pm \\to G$ are defined in like manner in terms of $\\phi_\\pm$. The fact that they are holomorphic maps on $C_+$ and $C_-$, respectively, follows from the properties that $\\phi_+$ maps to $\\mathcal{O}$ whereas $\\phi_-$ maps to $\\mathcal{Q}$. \n\n\\bigskip\n\nLet us see how this applies in the case of quantum electrodynamics and in particular, how it gives the BPHZ procedure of renormalization of it. Let us start by giving the Feynman rules for the graphs in our Hopf algebra, allowing us to associate a Feynman amplitude to a graph $\\Gamma$. We will use dim-reg (see \\cite[Ch.4]{Col84}) and obtain the regularized (bare) Feynman amplitudes as integrals\n\\begin{align}\n\\label{eq:feynm-ampl-pre}\nU_\\Gamma (q_1,\\ldots , q_n,\\mu_1,\\ldots,\\mu_n,p_{n+1},\\ldots,p_N)(z)\n=\\int d^{4-z}k_1 \\cdots d^{4-z} k_L ~I_\\Gamma(q,\\mu,p,k_1,\\ldots,k_L)\n\\end{align}\nIf we work in the Euclidean setting, the integrand is given by the following Feynman rules:\n\\begin{enumerate}\n\\item Assign a factor $\\left( -\\frac{\\delta_{\\mu\\nu}}{p^2+\\mathrm{i} \\epsilon} + \\frac{p_\\mu p_\\nu}{(p^2+\\mathrm{i}\\epsilon)^2} (1-\\xi) \\right)$ to each internal photon line. \n\\item Assign a factor $\\frac{1}{\\gamma^\\mu p_\\mu + m}$ to each internal electron line.\n\\item Assign a factor $e \\gamma^\\mu$ to each 3-point vertex. The apparent dependence on the index $\\mu$ is resolved by summing over $\\mu$ in combination with the attached photon line (having also an index $\\mu$). \n\\item Assign a factor $m$ to the 2-point vertex. \n\\item Assign a momentum conservation rule to each vertex.\n\\end{enumerate}\nIn rules {\\it 1.} and {\\it 2.}, we have introduced the IR-regulators $\\mathrm{i} \\epsilon$ in order to resolve divergences at small momenta.\n\\begin{ex}\nConsider the following electron self-energy graph\\\\\n\\begin{center}\n\\begin{fmfgraph*}(150,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v2}\n      \\fmf{fermion,label=$p-k$}{v2,v3}\n      \\fmf{fermion}{v3,r}\n      \\fmf{photon,left,tension=0,label=$k$}{v2,v3}\n\\end{fmfgraph*}\n\\end{center}\nAccording to the Feynman rules, the integrand for this graph is \n\\begin{align*}\nI_\\Gamma(p,k)=(e\\gamma^\\mu) \\frac{1}{\\gamma^\\kappa (p_\\kappa + k_\\kappa) + m} (e\\gamma^\\nu) \\left( -\\frac{\\delta_{\\mu\\nu}}{k^2+\\mathrm{i} \\epsilon} + \\frac{k_\\mu k_\\nu}{(k^2+\\mathrm{i} \\epsilon)^2} (1-\\xi) \\right) \n\\end{align*}\nwith summation over repeated indices understood. \n\\end{ex}\nAs can be seen from equation \\eqref{eq:feynm-ampl-pre}, a Feynman amplitude $U_\\Gamma$ corresponding to a 1PI graph $\\Gamma$ is an element in $C^\\infty(E_\\Gamma) \\otimes M_4(\\C) \\otimes K$, and a map $U:H \\to K$ can be defined by setting\n\\begin{align} \n\\label{eq:feynm-ampl}\nU (\\Gamma_\\sigma)(z)=e^{2-N} \\langle \\sigma, U_\\Gamma \\rangle.\n\\end{align} \nThe factor $e^{2-N}$ is introduced in order to keep track of the loop number of the graph in terms of powers of $e^2$. Indeed, with $N$ the number of external edges of $\\Gamma$, the power of $e^2$ appearing in the above expression is exactly the loop number $L$ of $\\Gamma$. \n\nThe counterterm $C(\\Gamma)$ for a graph $\\Gamma$ is given as the negative part of the Birkhoff decomposition of $U$ applied to $\\Gamma$ and the renormalized value $R(\\Gamma)$ as the positive part. Indeed, in this case, the above formul{\\ae} give precisely the recursive BPHZ-procedure of subtracting divergences, dealing with possible subdivergences recursively, so that $C=U_-$ and $R=U_+$. More explicitly,\n\\begin{align*}\nC(\\Gamma) &= -T \\left( \nU(\\Gamma) + \\sum_{\\gamma \\subset \\Gamma} C(\\gamma_{(k)}) U(\\Gamma\/\\gamma_{(k)}) \\right),\\\\\nR(\\Gamma) &=  \nU(\\Gamma) + C(\\Gamma) + \\sum_{\\gamma \\subset \\Gamma} C(\\gamma_{(k)}) U(\\Gamma\/\\gamma_{(k)}).\n\\end{align*}\n\n\n\\begin{rem}\nThe relation between the renormalization constants in the Lagrangian (see equation \\eqref{eq:L-ren}) and the counterterms, was given by Dyson in \\cite{Dys49}. In terms of an expansion of Feynman graphs of the specified type, they read (compare with equation (73)-(76) in \\cite{BF01})\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:ren-const}\nm_0 &= Z_2 m + \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} \n    C(\\Gamma_\\ext{0}),\\\\\nZ_1 &= 1 + \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{1}),\\\\\nZ_2 &= 1 - \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{2}),\\\\\nZ_3 &=1 - \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{photon}{l,v}\n      \\fmf{photon}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{3}).\n\\end{aligned}\n\\end{equation}\n\\end{rem}\n\n\n\\section{Ward-Takahashi identities}\n\\label{sect:wt}\nIn quantum electrodynamics, the Ward-Takahashi identities give relations between Feynman amplitudes for full vertex graphs and electron self-energy graphs. Our goal is to translate these identities into relations on the Hopf algebra $H$, thus giving relations between Feynman graphs. More precisely, we consider a quotient of the Hopf algebra $H$ by a Hopf ideal generated by so-called {\\it Ward-Takahashi (WT) elements}, thereby establishing the well-known fact (see for example \\cite{Col84, PS95}) that the WT-identities are compatible with renormalization. \n\nFirst, we introduce the following notation. Let $\\Gamma$ be an electron self-energy graph, and number the internal electron lines that are not part of an electron loop by $i$. We define $\\Gamma(i)$ to be the graph $\\Gamma$ with insertion of a photon line on the $i$'th electron line.\n\n\\begin{defn}\nFor every electron self-energy graph $\\Gamma$, we define the {\\rm Ward-Takahashi element} $W(\\Gamma) \\in H$ associated to $\\Gamma$ by\n\\begin{align*}\nW(\\Gamma)(p,q)_{kl} := \\sum_i \\sum_\\mu q_\\mu \\Gamma(i)(q,\\mu;p)_{kl} + \\Gamma(p+q)_{kl} - \\Gamma(p)_{kl}.\n\\end{align*}\nThe first term is understood as the pair $(\\Gamma(i),\\sigma) \\in H$, with external structure given by the distribution $\\sigma^{kl}=\\sum q^\\mu \\delta_{q,\\mu,p,-p-q} \\otimes e^{kl}$. \nMoreover, we set $W'_\\mu(\\Gamma)(p) := \\frac{\\partial}{\\partial q_\\mu} W(\\Gamma)(p,q) \\big|_{q=0}$ and $W''(\\Gamma):=\\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial q_\\mu} W(\\Gamma)(p,q) \\big|_{p=q=0}$; in other words\n\\begin{align*}\nW'_\\mu(\\Gamma)(p) &= \\sum_i \\Gamma(i)_\\ext{0,\\mu,p} + \\Gamma_\\ext{p_\\mu},\\\\\nW''(\\Gamma) &= \\sum_i \\Gamma(i)_\\ext{1} + \\Gamma_\\ext{2}.\n\\end{align*}\n\\end{defn}\n\\noindent In what follows, we will suppress the matrix indices $k,l$ in $W(\\Gamma)$ for notational convenience.\n\nThe following notation due to 't Hooft and Veltman of double headed photon lines provides a compact way to denote the above external structure in terms of diagrams:\n\\begin{align*}\n\\parbox{40pt}{\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmf{doublehead,label=$q$}{l,v}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n= \\sum q^\\mu ~\\parbox{40pt}{\n\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{photon,label=$(\\noexpand q,,\\noexpand \\mu)$}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\\\\[3mm]\n\\end{align*}\nIn diagrams, this leads to the familiar expressions for the WT-identities (see for example \\cite[Ch. 7]{PS95}),\n\\begin{align*}\nW(\\Gamma)(p,q)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}\n} \n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{doublehead, label=$q$}{l,v}\n      \\fmf{fermion,label=$p$}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n+\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p+q$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n-\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\\\[5mm]\nW_\\mu'(\\Gamma)(p)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}} \n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{photon,label=$(\\noexpand 0,,\\noexpand \\mu)$}{l,v}\n      \\fmf{fermion}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n+\\frac{\\partial}{\\partial p_\\mu}~\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\\\[5mm]\nW''(\\Gamma)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}} \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}_\\ext{1}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,30)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}_\\ext{2}\n\\end{align*}\nfor $\\Gamma=$ \\begin{fmfgraph*}(40,11)\n      \\fmfforce{(0w,.25h)}{l}\n      \\fmfforce{(1w,.25h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}\nany electron self-energy graph. \n\n\\begin{thm}\n\\label{thm:WT}\nThe ideal generated (in $H$) by $W(\\Gamma)$, $W_\\mu'(\\Gamma)$ and $W''(\\Gamma)$ for all 1PI electron self-energy graphs $\\Gamma$ is a Hopf ideal.\n\\end{thm}\nBefore stating the proof of this, we remark that it is thus possible to define the quotient Hopf algebra $\\widetilde H$ of $H$, with the induced coproduct, counit and antipode.\n\\begin{proof}\nRecall that an ideal $I$ in a Hopf algebra $H$ is called a Hopf ideal if\n$$\n\\Delta(I) \\subseteq I \\otimes H + H\\otimes I, \\qquad \\epsilon(I)=0, \\qquad S(I) \\subseteq I.\n$$\nIn our case of a connected graded Hopf algebra $H$, the last property follows from the first. Indeed, since $S$ is given inductively by equation \\eqref{antipode}, we find that $S(I)\\subseteq I$.\n\nWe set $\\Delta'(X) := \\Delta(X) - X \\otimes 1 + 1 \\otimes X= \\sum X' \\otimes X''$ and denote the ideal generated by the WT-identities by $I$ . Clearly, it is enough to establish $\\Delta'(I) \\subseteq I \\otimes H + H \\otimes I$. Moreover, since $\\Delta$ is an algebra map by definition, it is enough to establish $\\Delta' (W(\\Gamma)) \\in I \\otimes H + H \\otimes I$ and the analogous expressions for $W'_\\mu(\\Gamma)$ and $W''(\\Gamma)$ for any electron self-energy graph $\\Gamma$. \n\n\\bigskip\n\nLet us start by illustrating the general argument in the following special case of the self-energy graph $\\Gamma$ displayed in Figure \\ref{figure-blocks}.\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{fmfgraph*}(90,60)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v1,e12,v2,v3,e34,v4,v5,e56,v6,e67,v7,v8,e89,v9,v10,r}\n      \\fmffreeze\n      \\fmf{photon,left,tension=0}{v1,v9}\n      \\fmf{photon,left,tension=0}{v2,v3}\n      \\fmf{photon,left,tension=0}{v4,v5}\n      \\fmf{photon,left,tension=0}{v6,v10}\n      \\fmf{photon,left,tension=0}{v7,v8}\n      \\fmffreeze\n      \\fmf{dots,right,tension=0,label=$\\alpha_1$}{e12,e34}\n      \\fmf{dots,right,tension=0}{e34,e12}\n      \\fmf{dots,right,tension=0,label=$\\alpha_2$}{e34,e56}\n      \\fmf{dots,right,tension=0}{e56,e34}\n      \\fmf{dots,right,tension=0,label=$\\alpha'$}{e67,e89}\n      \\fmf{dots,right,tension=0}{e89,e67}\n\\end{fmfgraph*}\n\\end{center}\n\\caption{An electron self-energy graph with blocks $\\alpha=\\alpha_1\\cdot \\alpha_2$ and $\\alpha'$.}\n\\label{figure-blocks}\n\\end{figure}\nWe label the internal electron lines from left to right, from 1 to 9. \nLet us consider the first term in $W(\\Gamma)$, and split its image under $\\Delta'$ in a term in which $\\gamma$ contains the electron line $i$ and a term in which it does not:\n$$\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)= \\sum_{\\gamma \\subset\\Gamma} C(\\gamma) + D(\\gamma),\n$$\nwith $C(\\gamma):=  \\sum_{i \\in \\gamma} \\sum_k \\gamma(i)_\\ext{k} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k}$ and $D(\\gamma):=\\sum_{i\\notin \\gamma} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}$. \n\nFor example, if $\\gamma=\\alpha_1 \\cdot \\alpha_2$ forms the `block' of concatenated electron self-energy graphs inside $\\Gamma$, we find,\n\\begin{align*}\nC(\\gamma) &= \\sum_{i=2,4} \\sum_{k_1,k_2} \\gamma(i)_\\ext{k_1,k_2} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k_1,k_2}\\\\\n&=\\sum_k\n(\\widetilde\\alpha_1 \\cdot \\alpha_2)_\\ext{1,k} \\otimes \\Gamma\/\\gamma_\\ext{1,k}(e_1)\n + (\\alpha_1\\cdot\\widetilde\\alpha_2)_\\ext{k,1} \\otimes \\Gamma\/\\gamma_\\ext{k,1}(e_2),\n\\end{align*}\nwhere for $m=1,2$, $\\widetilde \\alpha_m$ denotes the electron self-energy graph $\\alpha_m$ with an external photon line attached to its internal electron line and $e_m$ is the electron line that corresponds to $\\alpha_m$ in the quotient $\\Gamma\/\\gamma$. On the other hand,\n\\begin{align*}\nD(\\gamma) &=\\sum_{i=1,3,5,\\ldots,9} \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\Gamma(i)\/\\gamma_\\ext{k_1,k_2} \\\\\n&= \\sum_k (\\alpha_1 \\cdot \\alpha_2)_\\ext{2,k} \\otimes \\Gamma \/ \\gamma_\\ext{2,k}(1)\n+ (\\alpha_1 \\cdot\\alpha_2)_\\ext{k,2} \\otimes \\Gamma \/ \\gamma_\\ext{k,2}(3) \\\\\n&+\\sum_k (\\alpha_1 \\cdot \\alpha_2)_\\ext{0,k} \\otimes \\Gamma \/ \\gamma_\\ext{0,k}(1)\n+ (\\alpha_1 \\cdot \\alpha_2)_\\ext{k,0} \\otimes \\Gamma \/ \\gamma_\\ext{k,0}(3) \\\\\n& +  \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\sum_{i=5,\\ldots,9} \\Gamma\/\\gamma_\\ext{k_1,k_2}(i).\n\\end{align*}\nHere we wrote explicitly the external structure $k=0,2$ for the electron self-energy graph in $\\gamma$ for which an external photon line is attached to its incoming electron line (i.e. for $\\alpha_1$ if $i=1$ and for $\\alpha_2$ if $i=3$). Also, in the last line we used a labelling of the internal electron lines of the quotient $\\Gamma\/\\gamma$ in terms of the electron lines of $\\Gamma$ when $i \\notin \\gamma$.\n\nThe first two terms of $D(\\gamma)$ combine with $C(\\gamma)$ forming the WT-elements $W''(\\alpha_m)$, and also the last three terms of $D(\\gamma)$ combine to give,\n\\begin{align*}\nC(\\gamma)+D(\\gamma)=&\\sum_k W''(\\alpha_1) (\\alpha_2)_\\ext{k} \\otimes \\Gamma\/\\gamma_\\ext{2,k} (e_1)\n+(\\alpha_1)_\\ext{k} W''(\\alpha_2) \\otimes \\Gamma\/\\gamma_\\ext{k,2} (e_2)\\\\\n&+ \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\sum_{\\parbox{45pt}{\\centering \\tiny{$j$ insertion \\\\points in $\\Gamma\/\\gamma$}}} \\Gamma\/\\gamma_\\ext{k_1,k_2}(j),\n\\end{align*}\nusing for the last term the fact that the quotient $\\Gamma\/(\\alpha_m)_\\ext{0}$ gives rise to a 2-point vertex, and thus to a new internal electron line. It is easy to see that the first two terms in the above equation are in $I \\otimes H$.\n\nThe two other terms in $\\Delta'(W(\\Gamma))$ are of the form $\\sum_\\gamma \\gamma_{k_1,k_2} \\otimes \\Gamma\/\\gamma_\\ext{k_1,k_2}$ and combine, for fixed $\\gamma=\\alpha_1 \\cdot \\alpha_2$ as above, with the last term in the previous equation to give precisely $\\gamma_\\ext{k_1,k_2} \\otimes W(\\Gamma\/\\gamma_\\ext{k_1,k_2}) \\in H \\otimes I$. We conclude that the term in $\\Delta'(W(\\Gamma))$ arising from the subgraph $\\gamma=\\alpha_1\\cdot \\alpha_2$ is an element in $I \\otimes H + H \\otimes I$.\nSimilar arguments show that such relations hold for all subgraphs of $\\Gamma$.\n\n\n\n\n\n\n\n\\bigskip\n\nLet us now turn to the proof of the claim that $\\Delta' (W(\\Gamma)) \\in I \\otimes H + H \\otimes I$ for any electron self-energy graph $\\Gamma$. For the first term in $W(\\Gamma)$ we have\n\\begin{align*}\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)=\n\\Delta'\\left( \\sum_i\n\\parbox{40pt}{\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmf{doublehead}{l,v}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\right) = \n\\sum_i \\sum_{\\gamma \\subset \\Gamma(i)}  \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k},\n\\end{align*}\nwhere $k$ is a multi-index labelling the external structure on the connected components of $\\gamma$ and $i$ runs over all internal electron lines of $\\Gamma$ that are not part of a loop. Also, we suppressed the external structure of $\\Gamma(i)$ as it appears in $W(\\Gamma)(p,q)$.\n\nIn general, $\\gamma$ can be written as a union $\\gamma=\\gamma_E\\cdot \\gamma_P \\cdot \\gamma_V \\cdot \\gamma_E(i)$, where $\\gamma_E$ consists of electron self-energy graphs, $\\gamma_V$ of full vertex graphs, $\\gamma_P$ of vacuum polarization graphs and $\\gamma_E(i)$ of electron self-energy graphs with a photon line inserted at the $i$'th electron line (with this labelling inherited from $\\Gamma$). Graphs of the type $\\gamma_P(i)$ and $\\gamma_V(i)$ do not appear in the sum, because there are no corresponding vertices of valence 3 and 4 in QED. \nNote that the multi-index $k$ only affects the subgraphs in $\\gamma_E$, for each of which it can take the values $0$ and $2$. We split the above sum in a term in which $\\gamma_\\ext{k}$ contains the electron line $i$ as an internal line and those in which it does not:\n\\begin{align}\n\\label{cop-i:2}\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)= \\sum_{\\gamma \\subset \\Gamma}\n\\left(\n\\sum_{i\\in \\gamma_E} \\gamma(i)_\\ext{k} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k}\n+\\sum_{i\\notin \\gamma_E} \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}\n\\right),\n\\end{align}\nwhere $\\gamma=\\gamma_E \\cdot \\gamma_V \\cdot \\gamma_P$. We will write $\\Delta'\\left( \\sum_i \\Gamma(i) \\right)=\\sum_\\gamma C(\\gamma) + D(\\gamma)$ for the two terms appearing in equation \\eqref{cop-i:2}.\n\nBefore examining the terms $C$ and $D$, we note that for each $\\gamma$, $\\gamma_E$ can be split into `blocks' each of which appear in the graph $\\Gamma$ by concatenation of electron self-energy graphs (as in Figure \\ref{figure-blocks}). We will denote such blocks by $\\alpha \\subset \\gamma_E$ and write $C(\\gamma)$ as\n\\begin{align*}\nC(\\gamma)= \n\\sum_{\\parbox{60pt}{\\centering \\tiny{$\\alpha \\subset \\gamma_E$\\\\ $\\alpha=\\alpha_1 \\cdots \\alpha_l$}}} \\sum_{m=1}^l \\sum_{i \\in \\alpha_m} \\sum_k \\alpha_m(i)_\\ext{1} ~(\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma \/ \\gamma_\\ext{k} \\left( e_m \\right)\n\\end{align*}\nwhere $e_m$ is the electron line corresponding to $\\alpha_m$ in the quotient $\\Gamma\/\\gamma$. In words, we separated the sum over the internal electron lines $i$ of $\\gamma_E$ into a sum over all blocks $\\alpha$ in $\\gamma_E$, together with a sum over its connected components $\\alpha_m$ and a sum over the internal electron lines that are part of $\\alpha_m$. Then, with $i \\in \\alpha_m$, the quotient $\\Gamma(i)\/\\gamma(i)_\\ext{k}$ is given by the replacement of $\\alpha_m$ by a 3-vertex, followed by the quotient by the other graphs that constitute $\\gamma$, and with external structure inherited from $\\Gamma(i)$.\nNote that since $\\alpha_m(i)$ is a full vertex graph, there is only the external structure $\\alpha_m(i)_\\ext{1}$. This fixes $k_m$ inside the multi-index $k$ to be $2$, and by a slight abuse of notation, we let $k$ also denote the external structures of the elements in $(\\gamma-\\alpha_m)$. \n\nThe term $D(\\gamma)$ can be split in a sum over $i$ of electron lines that are external edges for $\\gamma_E$ and those that are not,\n\\begin{align*}\nD(\\gamma)= \\sum_{i \\in \\partial \\gamma_E} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k} + \\sum_{i \\notin \\gamma \\cup \\partial \\gamma_E} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}.\n\\end{align*}\nAgain, we will write this in terms of the blocks $\\alpha$ forming $\\gamma_E$:\n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{k_m} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_m)$}}{v3}\n    \\end{fmfgraph*}}\n\\right)\n\/ \n (\\gamma-\\alpha_m)_\\ext{k}\n\\\\\n&+\\sum_\\alpha \\sum_k \n(\\alpha_l)_\\ext{k_l} (\\gamma-\\alpha_l)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_l \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_l)$}}{v1}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_l)_\\ext{k} \n+ \\sum_{i \\notin \\gamma \\cup \\partial \\gamma_E}\\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k},\n\\end{aligned}\n\\end{equation*}\nwhere the first term arises from $i$ being the ingoing line of $\\alpha_m$ for $m=1,\\cdots l$ and the second term from $i$ being the outgoing line for $\\alpha_l$, thus equal to the sum of all external edges in $\\alpha$. \n\n\\medskip\n\nWriting explicitly the two terms for $k_m=0,2$ one obtains, \n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\bigg(\\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{2} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n\\\\\n&+\n\\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{0} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v3}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_m)_\\ext{k} \n\\\\\n&+\\sum_k \n(\\alpha_l)_\\ext{k_l} (\\gamma-\\alpha_l)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_l \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_l)$}}{v1}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_l)_\\ext{k} \n\\bigg)\n+ \\sum_{i \\notin \\gamma \\cap \\partial \\gamma_E}\\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}.\n\\end{aligned}\n\\end{equation*}\nwhere in the first line, we have used the fact that $\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(2)$}}{v3}\n    \\end{fmfgraph*}}\n\\right) = \\Gamma\/\\alpha_m (e_m)$, in the above notation.\n\n\\bigskip\n\n\\noindent Since each vertex \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v}\n\\end{fmfgraph*}\nadds an electron line to $\\Gamma$, we can combine the last three terms to obtain\n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{2} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n+ \\sum_k \\gamma_\\ext{k} \\otimes \\sum_{\\parbox{45pt}{\\centering \\tiny{$j$ insertion \\\\points in $\\Gamma\/\\gamma$}}} (\\Gamma\/\\gamma_\\ext{k}) (j).\n\\end{aligned}\n\\end{equation*}\nThus, we obtain for the sum of $C$ and $D$,\n\\begin{equation}\n\\begin{aligned}\n\\label{cop-i:CD}\nC(\\gamma) + D(\\gamma) &=  \\sum_{\\alpha,m,k}\n\\left( \\sum_{i \\in \\alpha_m}(\\alpha_m)(i)_\\ext{2} +(\\alpha_m)_\\ext{2} \\right)(\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n\\\\\n&+ \\sum_k \\gamma_\\ext{k} \\otimes \\sum_j \\left( \\Gamma\/\\gamma_\\ext{k} \\right) (j),\n\\end{aligned}\n\\end{equation}\nwhere we recognize the WT-element $W''(\\alpha_m)$ for each 1PI self-energy graph $\\alpha_m\\subset \\Gamma$,\n\\begin{align*}\n  W''(\\alpha_m)=\\sum_{i \\in \\alpha_m}(\\alpha_m)(i)_\\ext{1} +(\\alpha_m)_\\ext{2}.\n\\end{align*}\nCombining the second term in equation \\eqref{cop-i:CD} with the two other terms in $\\Delta'(W(\\Gamma))$ (which involve the coproduct $\\Delta'(\\Gamma)=\\sum_\\gamma \\gamma_\\ext{k} \\otimes \\Gamma\/\\gamma_\\ext{k}$, it follows that \n$$\n\\Delta'(W(\\Gamma)) = \\sum_{\\gamma \\subset \\Gamma}   \\sum_{\\alpha,m,k} W''(\\alpha_m) (\\gamma-\\alpha_m)_\\ext{k} \\otimes \\Gamma \/ \\gamma_\\ext{k} (e_m) + \\sum_{\\gamma \\subset \\Gamma} \\gamma_\\ext{k} \\otimes W\\left( \\Gamma\/\\gamma_\\ext{k} \\right),\n$$\nso that $\\Delta'(W(\\Gamma))\\in I \\otimes H + H \\otimes I$. \n\nThe other inclusions $\\Delta'(W_\\mu'(\\Gamma)) \\in I \\otimes H + H \\otimes I$ and $\\Delta'(W''(\\Gamma)) \\in I \\otimes H + H \\otimes I$ follow from the latter equation by the definitions of $W_\\mu'(\\Gamma)$ and $W''(\\Gamma)$ in terms of $W(\\Gamma)$. \n\\end{proof}\n\n\n\\bigskip\n\n\\section{Conclusions}\n\\label{sect:concl}\nWe have shown that the Ward-Takahashi identities can be implemented on the Hopf algebra of Feynman graphs of QED as relations that define a Hopf ideal. The quotient Hopf algebra $\\widetilde H$ by this Hopf ideal is still commutative but has the WT-identities `built in'. The Feynman amplitudes $U$ on $\\widetilde H$ defined by equation \\eqref{eq:feynm-ampl} therefore automatically satisfy the WT-identities in the physical sense ({i.e.} between Feynman amplitudes). Moreover, Theorem \\ref{birkhoff} applies and shows that the counterterms and the renormalized Feynman amplitudes are given in terms of the algebra maps $C=U_-: \\widetilde H \\to \\mathcal{Q}$ and $R=U_+ : \\widetilde H \\to \\mathcal{O}$, respectively. In particular, the counterterms as well as the renormalized Feynman amplitudes satisfy the WT-identities. The well-known relation $Z_1=Z_2$ between the renormalization constants as derived by Ward in \\cite{War50} is now an easy consequence of their definition in \\eqref{eq:ren-const},\n\\begin{align*}\nZ_1 - Z_2 & = \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} \n\\sum_{\\parbox{45pt}{\\centering \\tiny{$i$ insertion \\\\points in $\\Gamma$}}} \nC(\\Gamma(i)_\\ext{1}) + C(\\Gamma_\\ext{2})\\\\\n&= \\sum_{\\Gamma} C\\left( W''(\\Gamma) \\right) = 0 .\n\\end{align*}\nIn the first line we have used the fact that the contribution $C(\\Gamma(i)_\\ext{1})$ vanishes whenever $i$ is part of an electron loop, as follows by explicit computation using the Feynman rules (cf. \\cite[p.240-241]{PS95}).\n\n\n\n\\section*{Acknowledgements}\nI would like to thank Matilde Marcolli for discussion and remarks, and {\\\"O}zg{\\\"u}r Ceyhan for several comments.\n\n\n\n\\end{fmffile}\n\n\\pagebreak\n\\newcommand{\\noopsort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe interplay between physical concepts and geometric structures is at the heart of fundamental theoretical physics. The prime example is the role of Riemannian geometry in the formulation and understanding of general relativity. It is therefore natural to ask what novel geometric structures will be needed to encompass the ideas behind a theory of quantum gravity.\n\nA promising candidate for such a theory is string theory. Due to the extended nature of the fundamental string and the presence of dualities, it has proven useful to work with a \\emph{doubled geometry}. This is formed by extending spacetime to include extra coordinates corresponding to the winding modes of the string (whereas usual coordinates correspond to the momentum modes). This doubled geometry can be viewed as the target space of the fundamental string which is manifestly invariant under T-duality transformations exchanging winding and momentum modes. An effective action for this setup is provided by double field theory (DFT) \\cite{Siegel:1993xq,Siegel:1993th,Hull:2009mi}. The doubled space can also be interpreted as the phase space of the fundamental string which then leads to the formulation of metastring theory \\cite{Freidel:2014qna,Freidel:2015pka}.\n\nThe geometric structure of this doubled space is that of a \\emph{para-Hermitian manifold} as was first demonstrated by Vaisman \\cite{Vaisman:2012ke,Vaisman:2012px}. Such geometry provides the {\\it kinematical structure} for the theory in the sense that it augments the doubled space with a differentiable structure, naturally adapted to the T-duality covariant setting. The dynamical data is then given by a generalized metric and generalized fluxes on the doubled geometry. Such a setup has been dubbed \\emph{Born geometry} \\cite{Freidel:2013zga}.\n\nIn this contribution we give an executive summary of para-Hermitian geometry in the context of the doubled geometry of string theory and how it is related to generalized geometry based on \\cite{Freidel:2017yuv,Svoboda:2018rci,Freidel:2018tkj} (see \\cite{Chatzistavrakidis:2018ztm} for related developments). We give the basic ingredients of Born geometry and present the unique torsion-free compatible connection for Born geometry. This can be seen as the analogy to the Levi-Civita connection for Riemannian geometry. We will also explain how the fluxes naturally arise in this setting. \\textcolor{white}{\\speaker{F.~J.~Rudolph, D.~Svoboda} }\n\n\n\n\n\\section{Generalized Kinematics}\nIt has been observed that the vector fields on the extended spacetime obey a new algebra relation which is different than the usual Lie algebra of vector fields given by the Lie bracket. One is therefore led to replace the Lie bracket with a new bracket operation called the {\\it D-bracket} which leads to a redefinition of the notion of {\\it differentiable structure} on the manifold, i.e. the Cartan calculus. All usual objects whose definitions use the Lie bracket, such as the torsion and Nijenhuis tensors, need to be replaced by their counterparts using the D-bracket. We call this new differentiable structure on the manifold the {\\it generalized kinematical structure}. In this section we describe that such structure on an extended spacetime is naturally given by the data of a para-Hermitian manifold.\n\n\\subsection*{Para-Hermitian Geometry}\nWe now briefly introduce all important concepts of para-Hermitian geometry. For more details consult \\cite{Cruceanu} and references therein.\n\n\\begin{Def}\nLet $\\mathcal{P}$ be a $2n$-dimensional manifold. An almost para-Hermitian structure on $\\mathcal{P}$ is a pair $(\\eta,K)$, where $\\eta$ is a metric with signature $(n,n)$ and $K \\in \\text{End}(T\\mathcal{P})$ with $K^2=\\mathbbm{1}$. These two are compatible in the sense that $K$ is an anti-isometry of $\\eta$: $\\eta(K\\cdot,K\\cdot)=-\\eta(\\cdot,\\cdot)$.\n\\end{Def}\n\nA consequence of the definition is that $K$ has two eigenbundles of rank $n$ corresponding to the eigenvalues $\\pm 1$, which we will in the following denote by $L$ and ${\\tilde{L}}$. The fact that $L$ and ${\\tilde{L}}$ have the same rank makes $K$ an almost {\\it para-complex} structure.  Another consequence of the definition is that the tensor $\\omega \\coloneqq \\eta\\circ K$ is a non-degenerate two-form, i.e. an {\\it almost symplectic structure}, sometimes called the fundamental form. If $\\mathrm{d} \\omega=0$, we call $(\\eta,K)$ an (almost) {\\it para-K\\\"ahler} structure.\n\nIt is useful to realize the analogy of para-Hermitian geometry with its complex counterpart, Hermitian geometry. Indeed, the only difference here is that the para-complex structure $K$ squares to $+\\mathbbm{1}$ as opposed to $-\\mathbbm{1}$ in the complex case. The isometry condition of a complex structure then also comes with the opposite sign. \n\nJust as in the complex case, we can invoke the notion of {\\it integrability} which is governed by the {\\it Nijenhuis tensor}:\n\\begin{align}\\label{eq:nijenhuis}\n\\begin{aligned}\nN_K(X,Y)&\\coloneqq\\frac{1}{4}\\Big( [X,Y]+[KX,KY]-K([KX,Y]+[X,KY])\\Big)\\\\\n&=P[{\\tilde{P}} X,{\\tilde{P}} Y]+{\\tilde{P}}[P X,P Y],\n\\end{aligned}\n\\end{align}\nwhere we introduced the projections onto $L$ and ${\\tilde{L}}$:\n\\begin{align}\\label{eq:K-projections}\nP=\\frac{1}{2}(\\mathbbm{1}+K) \\quad \\mathrm{and} \\quad {\\tilde{P}}=\\frac{1}{2}(\\mathbbm{1}-K).\n\\end{align}\nFrom the last line of \\eqref{eq:nijenhuis}, it is obvious that $N_K$ vanishes if and only if both bundles $L$ and ${\\tilde{L}}$ are Frobenius integrable, i.e. closed under the Lie bracket. When this is the case, one gets local charts $X^I=(x^i,\\tilde{x}^i)$, $i=1\\cdots n$, such that $L=\\text{span}\\{\\partial_i=\\frac{\\partial}{\\partial x^i}\\}$ and ${\\tilde{L}}=\\text{span}\\{\\tilde{\\partial}^i=\\frac{\\partial}{\\partial \\tilde{x}_i}\\}$ and the manifold looks locally like a product manifold.\n\nWe can also notice one of the most striking features of para-complex geometry which sets it apart from complex geometry -- the integrability of the eigenbundles $L$ and ${\\tilde{L}}$ is independent on each other, meaning that $L$ can be integrable while ${\\tilde{L}}$ is not and vice versa. This gives rise to the notion of {\\it half-integrability}. In the subsequent discussion we will not reference integrability when it is not relevant and omit the ``almost'' prefix, while explicitly calling the para-Hermitian structure (half-)integrable whenever relevant.\n\n\n\\subsection*{The D-bracket}\nWe now explain how the para-Hermitian data $(\\eta,K)$ naturally defines a D-bracket on the underlying manifold. The central theorem for this discussion is the following:\n\n\\begin{Thm}\nLet $(\\mathcal{P},\\eta,K)$ be an almost para-Hermitian manifold. There exists a unique bracket operation $[\\![\\ , \\ ]\\!]$: $\\Gamma(T\\mathcal{P})\\times \\Gamma(T\\mathcal{P}) \\rightarrow \\Gamma(T\\mathcal{P})$ satisfying the following properties:\n\\begin{enumerate}\n\\item $[\\![ X,fY]\\!] = f[\\![ X,Y]\\!]+X[f]Y,\\quad$ \\hfill Leibniz property\n\\item $X[\\eta(Y,Z)] = \\eta([\\![ X,Y]\\!],Z)+\\eta(Y,[\\![ X,Z]\\!])$ \\hfill  Compatibility with $\\eta$ \\\\\n$\\eta(Y,[\\![ X,X]\\!] ) = \\eta([\\![ Y,X]\\!],X),\\quad$ \n\\item $\\mathcal{N}_K(X,Y)=[\\![ X,Y]\\!] +[\\![ K X,K Y]\\!]  - K\\big([\\![ K X,Y]\\!]  + [\\![ X,K Y]\\!]\\big)=0,\\quad$ \\hfill Compatibility with $K$\n\\item $\n[\\![ PX,PY ]\\!] = P([PX,PY]),\\quad[\\![ {\\tilde{P}} X,{\\tilde{P}} Y ]\\!] ={\\tilde{P}}([ {\\tilde{P}} X,{\\tilde{P}} Y]),$ \\hfill Relationship with the Lie bracket.\n\\end{enumerate}\nfor any vector fields $X,Y\\in\\Gamma(T\\mathcal{P})$ and any function $f\\in C^\\infty(\\mathcal{P})$. \n\\end{Thm}\nIn the above statement we introduced the {\\it generalized Nijenhuis tensor} of $K$, $\\mathcal{N}_K$. One can glean that this tensorial quantity is nothing else than the usual Nijenhuis tensor, except the Lie bracket is replaced with the D-bracket.\n\nThe following statement shows that the D-bracket not only exists, but can be easily written out in a convenient form:\n\n\\begin{Prop}\nThe D-bracket on a para-Hermitian manifold $(\\mathcal{P},\\eta,K)$ is given by \n\\begin{align}\\label{eq:D-bracket_canonical}\n\\eta([\\![ X,Y]\\!],Z)=\\eta(\\nc_XY-\\nc_YX,Z)+\\eta(\\nc_ZX,Y),\n\\end{align}\nwhere $\\nc$ is the {\\it canonical connection}, which can be defined via the Levi-Civita connection $\\lc$ of $\\eta$:\n\\begin{align*}\n\\nc_XY=P\\lc_X(PY)+{\\tilde{P}}\\lc_X({\\tilde{P}} Y).\n\\end{align*}\n\\end{Prop}\nIn the expression \\eqref{eq:D-bracket_canonical} we can notice that the first two terms, which are skew-symmetric under the exchange of $X$ and $Y$, are reminiscent of how one can define a Lie bracket using a torsionless connection. Even though $\\nc$ is not torsionless, in the limit when $(\\eta,K)$ is para-K\\\"ahler, it is easy to check that $\\nc=\\lc$ and therefore $\\nc_XY-\\nc_YX=[X,Y]$.\n\nWe should also point out that the D-bracket in particular coincides with the D-bracket appearing in the physics literature, where this operation is considered usually on flat manifolds, $\\mathbb{R}^{2n}$ or tori $T^{2n}$, both of which are simple examples of flat para-Hermitian manifolds.\n\\begin{Cor}\nOn a flat para-Hermitian manifold, the D-bracket takes the form \n\\begin{align*}\n[\\![ X,Y ]\\!]=\\left(X^I\\partial_IY^J-Y^I\\partial_IX^J+\\eta_{IL}\\eta^{KJ}Y^I\\partial_KX^L\\right)\\partial_J,\n\\end{align*}\nwhere $\\partial_I=(\\partial_i,\\tilde{\\partial}^i)$ are the partial derivatives along the coordinates $(x^i,\\tilde{x}_i)$.\n\\end{Cor}\n\nWe emphasize here that even though the D-bracket is unique, the choice of connection to express it via the formula \\eqref{eq:D-bracket_canonical} is not unique and $\\n^c$ is only one choice of such connection. In \\cite{Svoboda:2018rci}, the set of connections that yield the D-bracket are called {\\it adapted}.\n\n\\subsection*{Generalized Torsion}\nThe usual torsion of a linear connection $\\n$ can be understood as a tensorial quantity measuring how much the skew combination $\\n_XY-\\n_YX$ deviates from the Lie bracket, $[X,Y]$. The same is true for a so-called generalized torsion. In the light of the equation \\eqref{eq:D-bracket_canonical}, we define\n\n\\begin{Def}\nLet $(\\mathcal{P},\\eta,K)$ be a para-Hermitian manifold and $\\n$ a linear connection. The {\\it generalized torsion} of $\\n$ is the tensorial quantity defined by\n\\begin{align*}\n\\mathcal{T}^\\n(X,Y,Z)\\coloneqq \\eta(\\n_XY-\\n_YX,Z)+\\eta(\\n_ZX,Y)-\\eta([\\![ X,Y]\\!],Z)\n\\end{align*}\n\\end{Def}\n\nFrom this definition we see that it precisely captures how much does the cyclic combination of $\\n$, $\\eta(\\n_XY-\\n_YX,Z)+\\eta(\\n_ZX,Y)$ deviate from the D-bracket. If $\\mathcal{T}^\\n=0$, $\\n$ via this formula defines  a D-bracket.\n\n\n\\subsection*{Courant Algebroids and the Relationship to the D-bracket}\nIn this section we explore how the D-bracket relates to the Dorfman bracket. We explain that when $K$ is integrable, the D-bracket can be seen as a sum of two Dorfman brackets corresponding to certain Courant algebroids. In particular, this equips the tangent bundle of the para-Hermitian manifold with two natural Courant algebroid structures defined purely in terms of the para-Hermitian data.\n\nWe start off with a definition of a Courant algebroid.\n\\begin{Def}\nLet $E\\rightarrow M$ be a vector bundle. A Courant algebroid is a quadruple $(E,a,\\langle\\ , \\ \\rangle_E,[\\![\\ ,\\ ]\\!])$, where $a$ is bundle map $E\\rightarrow TM$ called the anchor, $\\langle\\ , \\ \\rangle_E:\\Gamma(E)\\times\\Gamma(E)\\rightarrow C^\\infty(M)$ is a non-degenerate symmetric pairing and $[\\![\\ ,\\ ]\\!]:\\Gamma(E)\\times\\Gamma(E)\\rightarrow \\Gamma(E)$ is a bracket operation called the Dorfman bracket\\footnote{The Dorfman bracket is sometimes called the Dorfman derivative or generalized Lie derivative.}, such that\n\\begin{enumerate}\n\\item $a(X)\\langle Y,Z\\rangle_E=\\langle[\\![ X,Y]\\!],Z\\rangle_E+\\langle Y,[\\![ X,Z]\\!]\\rangle_E$\n\\item $\\langle[\\![ X,X]\\!],Y\\rangle_E=\\frac{1}{2}a(Y)\\langle X,X\\rangle_E$\n\\item $[\\![ X, [\\![ Y,Z]\\!]\\br=[\\![\\bl X,Y]\\!], Z]\\!]+[\\![ Y,[\\![ X,Z]\\!]\\br$,\n\\end{enumerate}\n\\end{Def}\nA canonical example is given by the {\\it standard Courant algebroid}, where $E=(T\\oplus T^*)M$, $a:(T\\oplus T^*)M\\rightarrow TM$ is the natural projection, the pairing is given by $\\langle x+\\alpha ,y+\\beta \\rangle=\\alpha(y)+\\beta(x)$ and the bracket operation is the Dorfman bracket:\n\\begin{align*}\n[\\![ x+\\alpha,y+\\beta ]\\!] = [x,y]+\\mathcal L_x\\beta+\\imath_y\\mathrm{d} \\alpha.\n\\end{align*}\nwhere $x,y\\in\\Gamma(TM)$, $\\alpha,\\beta\\in\\Gamma(T^*M)$ and $\\mathcal L_x$ is the Lie derivative. We further introduce the following notation:\n\\begin{Def}\nLet $[\\![\\ ,\\ ]\\!]$ be the D-bracket on a para-Hermitian manifold. Then the {\\it projected brackets} $[\\![\\ ,\\ ]\\!]_P$ and $[\\![\\ ,\\ ]\\!]_{\\tilde{P}}$ are defined as follows\n\\begin{align*}\n\\eta([\\![ X,Y]\\!]_P,Z)&=\\eta(\\nc_{PX}Y-\\nc_{PY}X,Z)+\\eta(\\nc_{PZ}X,Y)\\\\\n\\eta([\\![ X,Y]\\!]_{\\tilde{P}},Z)&=\\eta(\\nc_{{\\tilde{P}} X}Y-\\nc_{{\\tilde{P}} Y}X,Z)+\\eta(\\nc_{{\\tilde{P}} Z}X,Y).\n\\end{align*}\nThe brackets are related by $[\\![ X,Y]\\!]=[\\![ X,Y]\\!]_P+[\\![ X,Y]\\!]_{\\tilde{P}}$.\n\\end{Def}\n\nWe continue with the following simple observation. Since $L$ and ${\\tilde{L}}$ are isotropic with respect to $\\eta$, whenever $\\tilde{x} \\in \\Gamma({\\tilde{L}})$, $\\eta(\\tilde{x},\\cdot)$ is an element of $\\Gamma(L^*)$ because it contracts only with vectors in $L$. In fact, any section of $L^*$ is of this form by non-degeneracy of $\\eta$. We therefore have the following vector bundle isomorphism:\n\\begin{align*}\n\\begin{aligned}\n\\rho:\\ T\\mathcal{P} &= L\\oplus {\\tilde{L}}\\rightarrow L\\oplus L^*\\\\\nX &= x+\\tilde{x} \\mapsto x+\\eta(\\tilde{x}).\n\\end{aligned}\n\\end{align*}\nand similarly we can define $\\tilde{\\rho}:L\\oplus {\\tilde{L}} \\rightarrow {\\tilde{L}} \\oplus {\\tilde{L}}^*$. Moreover, when $L$ is an integrable distribution (i.e. whenever ${\\tilde{P}}[PX,PY]=0\\ \\forall X,Y \\in \\Gamma(T\\mathcal{P})$), the manifold $\\mathcal{P}$ is {\\it foliated} by half-dimensional manifolds $\\ell_i$ called leaves. This means that there is a partition of the $2n$-dimensional manifold $\\mathcal{P}$ into $n$-dimensional manifolds as $\\mathcal{P}=\\bigcup_i \\ell_i$. We can therefore view $\\mathcal{P}$ as an $n$-dimensional manifold (given by the union of the leaves $\\ell_i$ of the foliation) and to avoid confusion we denote this $n$-dimensional manifold as $\\mathcal{F}$. Then, by definition, $L=T\\mathcal{F}$ and $T\\mathcal{P}=L\\oplus L^*=(T\\oplus T^*)\\mathcal{F}$ in a natural way. It turns out that the projected bracket $[\\![\\ ,\\ ]\\!]_P$ is nothing else than the Dorfman bracket of the standard Courant algebroid $(T\\oplus T^*)\\mathcal{F}$ mapped under the map $\\rho$:\n\n\n\\begin{Thm}\nLet $(\\mathcal{P},\\eta,K)$ be an almost para-Hermitian manifold and $[\\![\\ ,\\ ]\\!]$, $[\\![\\ ,\\ ]\\!]_P,[\\![\\ ,\\ ]\\!]_{\\tilde{P}}$ the associated D-bracket and projected brackets, respectively. Whenever $L$ is integrable, $T\\mathcal{P}$ acquires the structure of a Courant algebroid $(T\\mathcal{P},P,\\eta,[\\![\\ ,\\ ]\\!]_P)$. The map $\\rho$ is then an isomorphism of Courant algebroids:\n\\begin{align*}\n(T\\mathcal{P},P,\\eta,[\\![\\ ,\\ ]\\!]_P) \\overset{\\rho_+}{\\longrightarrow}((T\\oplus T^*)\\mathcal{F},a,\\langle\\ ,\\ \\rangle,[\\![\\ ,\\ ]\\!]_{\\mathcal{F}}),\n\\end{align*}\nwhere $((T\\oplus T^*)\\mathcal{F},a,\\langle\\ ,\\ \\rangle,[\\![\\ ,\\ ]\\!]_{\\mathcal{F}})$ is the standard Courant algebroid of $\\mathcal{F}$. This means that\n\\begin{align*}\n\\rho [\\![ X,Y]\\!]_P=[\\![ \\rho X,\\rho Y]\\!]_\\mathcal{F}, \\quad \\eta(X,Y)=\\langle \\rho X,\\rho Y\\rangle, \\quad \\rho\\circ P=a\\circ \\rho.\n\\end{align*}\nAn analogous statement holds for ${\\tilde{L}}$ where the Courant algebroid structure is then given by \\\\ $(T\\mathcal{P}, {\\tilde{P}}, \\eta, [\\![\\ ,\\ ]\\!]_{\\tilde{P}})$.\n\\end{Thm}\n\n\\section{Born Geometry}\n\\label{sec:borngeo}\n\nWe have seen that para-Hermitian geometry is a para-complex geometry $(\\mathcal{P},K)$ with a compatible neutral metric $\\eta$, i.e. one of signature $(n,n)$. We have also seen how if $L$ is integrable we can recover an $n$-dimensional manifold $\\mathcal{F}$ with $L=T\\mathcal{F}$ and and a standard Courant algebroid structure which can be identified with the physical spacetime. In order to make connection with a string background, we also need to recover the physical background fields on $\\mathcal{F}$, i.e. the spacetime metric $g$ and the Kalb-Ramond field $B$.\n\nTherefore the next step is to add another compatible metric $\\mathcal{H}$ of signature $(2n,0)$ to the picture, giving rise to what has been named Born geometry \\cite{Freidel:2013zga}. The additional Riemannian structure $\\mathcal{H}$ defines two more endomorphisms of the tangent bundle which -- along with the para-complex structure $K$ already in place -- form a para-quaternionic structure \\cite{Freidel:2015pka,Freidel:2013zga,Ivanov:2003ze}.\n\n\\begin{Def}\nLet $(\\mathcal{P},\\eta,\\omega)$ be a para-Hermitian manifold and let $\\mathcal{H}$ be a Riemannian metric satisfying\n\\begin{align}\n\\eta^{-1}\\mathcal{H}=\\mathcal{H}^{-1}\\eta,\\quad \\omega^{-1}\\mathcal{H}=-\\mathcal{H}^{-1}\\omega .\n\\end{align}\nThen we call the triple $(\\eta,\\omega,\\mathcal{H})$ a Born structure on $\\mathcal{P}$ where $\\mathcal{P}$ is called a Born manifold and $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$ a Born geometry.\n\\end{Def}\nNote that in this definition we view $(\\eta,\\omega,\\mathcal{H})$ as maps $T\\mathcal{P}\\rightarrow T^*\\mathcal{P}$. One can show \\cite{Freidel:2018tkj} that there always exists a choice of frame on $T\\mathcal{P}=L\\oplus{\\tilde{L}}$ where the generalized metric takes the form \n\\begin{equation}\n\\mathcal{H} = \\begin{pmatrix}\ng & 0 \\\\ 0 & g^{-1}\n\\end{pmatrix}\n\\label{eq:genmetric}\n\\end{equation}\nwhere $g$ is a Riemannian metric on $L=T\\mathcal{F}$. The appearance of the B-field will be discussed below in the context of twisting the para-Hermitian structure. \n\nIn order to understand the nature of a Born structure we review its three fundamental structures. First, as we have seen, it contains an almost {\\it para-Hermitian} structure $(\\omega, K)$ with compatibility\n\\begin{equation}\nK^2=\\mathbbm{1},\\qquad \\omega(KX,KY)=-\\omega(X,Y). \n\\end{equation}\nNext, the compatibility between $\\eta$ and $\\mathcal{H}$ implies that $J=\\eta^{-1}\\mathcal{H} \\in \\mathrm{End}\\, T\\mathcal{P}$ defines what we refer to as a  {\\it  chiral} structure\\footnote{The projectors $P_\\pm=\\frac12(\\mathbbm{1}\\pm J)$ define, similarly to projectors $P$, ${\\tilde{P}}$, a splitting of the tangent bundle $T\\mathcal{P} = C_+ \\oplus C_-$ which mirrors the right\/left chiral splitting of string theory. In other words,\ngiven a worldsheet $X:\\Sigma \\rightarrow \\mathcal{P}$ with $\\Sigma$ a Riemann surface, we have that $\\partial_z X$ takes values in $C_+$ and $\\partial_{\\bar{z}} X$ in $C_-$. Hence $J$ is called a chiral structure.} $(\\eta,J)$ on $\\mathcal{P}$\n\\begin{align}\nJ^2=\\mathbbm{1},\\qquad \n\\eta(JX,JY)=\\eta(X,Y).\n\\end{align}\nFinally, the compatibility between  $\\mathcal{H}$ and $\\omega$ defines an almost {\\it Hermitian} structure $(\\mathcal{H},I)$ on $\\mathcal{P}$\n\\begin{equation}\nI^2=-\\mathbbm{1},\\qquad \\mathcal{H}(IX,IY)= \\mathcal{H}(X,Y).\n\\end{equation}\nThese three structures, para-Hermitian, chiral and Hermitian satisfy in turn an extra compatibility equation\n\\begin{equation}\nKJI=\\mathbbm{1},\n\\end{equation}\nwhich follows directly from their definition $K=\\omega^{-1}\\eta$, $J=\\eta^{-1} \\mathcal{H}$, $I=\\mathcal{H}^{-1}\\omega$. This means that the triple $(I,J,K)$  form an almost para-quaternionic structure \n\\begin{align}\n-I^2=J^2=K^2=\\mathbbm{1}, \\qquad \n\\{I,J\\}=\\{J,K\\}=\\{K,I\\}=0,\\qquad \nKJI=\\mathbbm{1},\n\\end{align}\nwhere $\\{\\ , \\ \\}$ is the anti-commutator. \n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\centering\n\\resizebox{0.9\\textwidth}{!}{%\n\\begin{tabular}{@{}ccc@{}}\n\\toprule\n$I={\\mathcal{H}}^{-1}{\\omega}=-\\omega^{-1}\\mathcal{H}$ & $J={\\eta}^{-1}{\\mathcal{H}}={\\mathcal{H}}^{-1}{\\eta}$ & $K={\\eta}^{-1}{\\omega}={\\omega}^{-1}{\\eta}$  \\vspace{8pt} \\\\\n$-I^2=J^2=K^2=\\mathbbm{1}$                     & $\\{I,J\\}=\\{J,K\\}=\\{K,I\\}=0$           & $IJK=-\\mathbbm{1}$     \\vspace{8pt} \\\\\n$\\mathcal{H}(IX,IY)=\\mathcal{H}(X,Y)$              & $\\eta(IX,IY)= -\\eta(X,Y)$         & $\\omega(IX,IY)=\\omega(X,Y)$             \\\\\n$\\mathcal{H}(JX,JY)=\\mathcal{H}(X,Y)$              & $\\eta(JX,JY)= \\eta(X,Y)$          & $\\omega(JX,JY)=-\\omega(X,Y)$            \\\\\n$\\mathcal{H}(KX,KY)=\\mathcal{H}(X,Y)$              & $\\eta(KX,KY)= -\\eta(X,Y)$         & $\\omega(KX,KY)=-\\omega(X,Y)$            \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\caption{Summary of structures in Born geometry. Here $\\{\\ , \\ \\}$ is the anti-commutator.}\n\\label{tab:BornGeo}\n\\end{table}\n\n\n\\section{Born Connection}\nWe have now all ingredients in place to examine the Born connection. It is the unique connection compatible with the Born geometry, i.e. with the three defining structures $(\\eta,\\omega,\\mathcal{H})$, which has no generalized torsion. The Born connection can be seen as the analogue of the Levi-Civita connection which is the unique, torsionfree, metric-compatible connection for Riemannian geometry.\n\n\\begin{Thm}\\label{th:BornConnection}\nLet $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$ be a Born geometry with $K=\\eta^{-1}\\omega$ the corresponding almost para-Hermitian structure. Then there exists a unique connection called the Born connection denoted by $\\n^{\\mathrm{B}}$ which\n\\begin{itemize}\n  \\item is compatible with the Born geometry $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$,\n  \\item has a vanishing generalized torsion ${\\mathcal{T}}=0$.\n\\end{itemize}\nThis connection can be explicitly expressed in terms of the three defining structures $(\\eta,\\omega,\\mathcal{H})$ of the Born geometry and the canonical D-bracket. It can be concisely written in terms of the para-Hermitian structure $K$ and the chiral projections $X_\\pm=\\frac{1}{2}(\\mathbbm{1}\\pm J)X$ as\n\\begin{equation}\n\\n^{\\mathrm{B}}_XY = [\\![ X_-,Y_+ ]\\!]_+ + [\\![ X_+,Y_-]\\!]_- + (K[\\![ X_+,KY_+ ]\\!])_+ + (K[\\![ X_-,KY_-]\\!])_- .\n\\label{eq:BornBracket}\n\\end{equation}\n\\end{Thm}\n\nThe theorem is proven in \\cite{Freidel:2018tkj}.\n\n\\section{Fluxes}\nWe have seen in previous sections that any para-Hermitian structure $(\\eta, K)$ gives rise to its unique associated D-bracket. In this section we will explain how the {\\it twisted} D-bracket appears in this context. We will see that the fluxes twisting the bracket appear when we choose a different para-Hermitian structure $K'$ compatible with the same $\\eta$, and express its associated D-bracket in the splitting given by the original $K$. The appearance of fluxes can be therefore intuitively understood as a choice of a splitting not compatible with the D-bracket at hand and the fluxes can be seen as certain obstructions of an integrability of this splitting.\n\nWe now define a transformation of the para-Hermitian structure analogous to a B-field transformation in generalized geometry.\n\n\\begin{Def}\nLet $(\\eta,K)$ be a para-Hermitian structure on $\\mathcal{P}$. A B-transformation of $K$ is given by\n\\begin{align*}\nK\\overset{e^{B}}{\\longmapsto} K_{B}=e^{B} K e^{-{B}},\\quad\ne^{B}\\coloneqq\n\\begin{pmatrix}\n\\mathbbm{1} & 0 \\\\\nB & \\mathbbm{1}\n\\end{pmatrix} \\in \\text{End}(T\\mathcal{P})\n\\end{align*}\nwhere the map $e^B$ is expressed in the matrix representation corresponding to the splitting $L\\oplus L$ and $B:L\\rightarrow {\\tilde{L}}$ is a skew map such that $\\eta(BX,Y)=-\\eta(X,BY)$.\n\\end{Def}\nIt is easy to see that ${B}$ can be given by either a two-form $b$ or a bivector $\\beta$,\n\\begin{align}\\label{eq:b-beta}\n\\eta(BX,Y)=b(X,Y)=\\beta(\\eta (X),\\eta (Y)),\n\\end{align}\nwhere $b$ is of type $(+2,-0)$ and $\\beta$ is of type $(+0,-2)$, so we can write $b(X,Y)=b(\\tilde{x},\\tilde{y})$. In coordinates, we have\n\\begin{align*}\nb=b_{ij}\\mathrm{d} x^i\\wedge \\mathrm{d} x^j,\\quad \\beta=\\beta_{ij} \\tilde{\\partial}^i\\wedge \\tilde{\\partial}^j.\n\\end{align*}\n\nIt can be checked that $K_B$, the B-transformation of $K$, is a new (almost) para-Hermitian structure. $K$ and $K_B$ share the same $-1$ eigenbundle, but the $+1$ eigenbundle is sheared by the map $B$: $L\\mapsto L_B=L+B(L)$. The B-transformation therefore in general does not preserve integrability of the para-Hermitian structure.\n\nWhen the starting para-Hermitian structure is in fact an integrable para-K\\\"ahler structure, the D-bracket associated to $K_B$ carries with respect to $K$ well-known fluxes from physical literature:\n\n\\begin{Prop}\nLet $K_B$ be a B-transformation of a para-K\\\"ahler structure $(\\mathcal{P},\\eta,K)$. Then the D-bracket associated to $K_B$ is given by\n\\begin{align}\\label{twistedDbrac}\n\\eta([\\![ X,Y]\\!]^{B},Z)=\\eta([\\![ X,Y]\\!]^D ,Z)-(\\mathrm{d} b)(X,Y,Z).\n\\end{align}\nwhere $[\\![\\ ,\\ ]\\!]$ denotes the D-bracket of $K$. The different components of $\\mathrm{d} b$ with respect to the splitting of $K_B$ yield \n\\begin{align*}\n\\mathrm{d} b^{(3,0)_B}&=\\hat{H}=H+\\tilde{R}\\\\\n\\mathrm{d} b^{(3,0)_B}&=\\tilde{Q},\n\\end{align*}\nwhere $\\hat{H}$ is the Covariantized H-flux, $H$ is the usual H-flux and $\\tilde{R}$ and $\\tilde{Q}$ are the dual R- and Q-fluxes.\n\\end{Prop}\n\nWe see the appearance of the {\\it dual} fluxes, meaning $\\tilde{R}$ is a three-vector on $\\tilde{L}$, while the usual R-flux is usually understood as a three-vector on $L$, i.e. $R$ and $\\tilde{R}$ have an opposite index structure and similarly with $Q$ and $\\tilde{Q}$. If we wanted to achieve the usual $Q$ and $R$, we would have to perform the dual B-transformation, shearing ${\\tilde{L}}\\mapsto {\\tilde{L}}+B({\\tilde{L}})$. Indeed, this has been done in \\cite{Marotta:2018myj}, where it was additionally also shown that the simultaneous B-transformation shearing both in $L$ and ${\\tilde{L}}$ directions yields the whole hierarchy of fluxes appearing in DFT. \n\nLet us conclude this section by giving the effect of a B-transformation on the generalized metric $\\mathcal{H}$ as given in \\eqref{eq:genmetric}. The transformed generalized metric takes the form\n\\begin{equation}\n\\mathcal{H} \\overset{e^{B}}{\\longmapsto} e^{-B}\\mathcal{H} e^{-B} = \n\\begin{pmatrix}\ng - b g^{-1} b& bg^{-1} \\\\ -g^{-1}b & g^{-1}\n\\end{pmatrix}\n\\end{equation}\nwhich is familiar from generalized geometry and DFT.\n\n\\section{Conclusion}\nThe construction of a para-Hermitian manifold with a corresponding foliation (when one of the subdistributions is integrable) to recover the physical spacetime of a string theory background has been summarized here. This puts the idea of a doubled target space for the fundamental string on a firm mathematical base and spells out the precise relation to generalized geometry and double field theory. \n\nOn top of this generalized kinematical structure one can include the dynamical fields via a generalized metric which corresponds to a choice of Riemannian metric on the subspace, giving rise to a Born geometry. The generalized fluxes appear via a twisting of the para-Hermitian structure and hence the D-bracket. This allows for an inclusion of the B-field and subsequently the other fluxes of string theory.\n\nConcrete examples of para-Hermitian and Born geometries such as $\\mathcal{P} = TM$, i.e. where the doubled space is the tangent bundle of some manifold, can be found in \\cite{Marotta:2018myj}. Group manifolds such as Drinfel'd doubles can also be viewed from a para-Hermitian perspective and provide another class of examples \\cite{Marotta:2018myj,Mori:2019slw,inprep}.\n\nIn future work, we aim to develop this formalism further and show how it applies in various well-known settings where T-duality arises, such as topological T-duality \\cite{tduality1,tduality2,tduality3}.\n\n\\section*{Acknowledgments}\nThe authors would like to thank the organizers of the ``Corfu Summer Institute 2018'' for the invitation. F.J.R. is supported by the Max-Planck-Society. The work of D.S. is supported by NSERC Discovery Grant 378721.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{he} drilling machines are commonly used in factories to drill holes in materials. In this paper, a failure detection system for Valmet AB's drilling machines is investigated using Valmet AB's drilling sounds. A drilling machine operated by Valmet AB includes many drill bits used for drilling thousands of small holes in metal surfaces. Drill bits that break can cause serious damage to a product since they lead to a lack of holes in the metal surface. As a result, a technician manually punches holes into metal sheets that are missing holes. It is a very time-consuming and expensive process. Hence, technicians usually check the drilling machine every 10 minutes so that if any drill bit breaks they can replace it before the drilling machine continues. Furthermore, shutting the machine off and on every 10 minutes is labor-intensive and time-consuming. Hence, the factory would benefit from an automated system that can classify drilling sounds and notify technicians when drill bits break. Our machine failure detection proposal builds on the fact that skilled technicians can detect a drill that isn't working properly by listening to its sound when cutting metal.\n\nTypically, large, balanced datasets were used to build a machine failure detection system. Data anomalies related to the damaged machine in Valmet AB are probably only a small proportion of the total data set. As a result of the imbalance of real-world datasets, CNN tends to be biased; for instance, CNN will struggle to predict minority classes if they have fewer data. The collection and labeling of sounds can be a time-consuming and expensive process for a company. These operational costs can be reduced by using data augmentation techniques to transform datasets. Additionally, in order to create variations that are representative of what a deep learning model might see in reality, data augmentation techniques enable deep learning models to be more robust.\n\nSound applications use standard augmentation methods for extending small datasets. There are classic and advanced techniques in sound augmentation. The conventional approach of sound augmentation methods includes time stretching, pitch shifting, volume control, adding noise, time-shifting, etc \\cite{noauthor_augment_nodate}. Research in \\cite{tran2021detecting} showed that some augmentation methods are not appropriate to apply in our short sounds (20.83 ms or 41.67 ms). The advanced approach for data augmentation is using Synthetic Minority Over-sampling Technique (SMOTE) \\cite{chawla_smote:_2002} or  Modified Synthetic Minority Over-sampling Technique (MSMOTE)\\cite{5403368}, etc. Overall, data augmentation helps to increase the number of sounds in the dataset. This improves the prediction accuracy of deep learning models and reduces data overfitting. Additionally, it increases the generalization of deep learning models by creating variable data. \n\nRecently, deep learning augmentation methods such as generative adversarial network (GAN) ~\\cite{8698632,9006268,9411996,9390834,8615782,8985892,8717183, 8962219, 8857905} have been widely used as data augmentation methods in computer vision. However, GANs require large amounts of training data to generate effective augmented data.\nVariational Autoencoder (VAE) was proposed in 2014 by Kingma and Welling \\cite{kingma2014autoencoding}. VAE has been used in domain adaption in face recognition frameworks~\\cite{9432318}, network representation learning~\\cite{9449483}, detect anomaly in log file systems~\\cite{8944863}, statistically extract latent space hidden in sampled data to control the robot~\\cite{9143393}, to generate more data for software fault prediction~\\cite{8672311}, traffic anomaly detection in videos~\\cite{9531567}, for time series anomaly detection~\\cite{9053558}, etc.\nVAE and different improved versions of VAE have been used to generate synthetic data in remote sensing images~\\cite{9393473}, for phrase-level melody representation learning to generate music~\\cite{9054554}, to generate sensor data for chiller fault diagnosis~\\cite{9264218}, to generate conditional handwritten characters~\\cite{9511404, 8803129}, etc.\nThe purpose of this study is to generate new drilling sounds using VAE. To the best of our knowledge, this is the first time VAE was investigated from a drilling sound augmentation standpoint for the machine failure detection system. \n\nThe classification of sound signals generated by machines has been investigated in recent years. In conventional machine learning techniques, statistical features from acoustic signals are extracted and classified~\\cite{Lee2016},~\\cite{Kemalkar2017},~\\cite{Zhang2019a}. In machine failure analysis, machine learning algorithms are commonly used because of their robustness and adaptability on small datasets~\\cite{9252126}. These conventional machines learning methods  include Support Vector Machine (SVM)~\\cite{fengqi_compound_2006,Li2011TheAO,deng_novel_2019}, k\u2011Nearest Neighbors~\\cite{7360161, WANG2016201, GLOWACZ201965}, decision tree~\\cite{saravanan_fault_2009}, ANN and radial basis function\n(RBF)~\\cite{1265797,10.1016\/j.neucom.2011.03.043, doi:10.1177\/1077546313518816}, etc.\n\nThe use of deep learning to detect mechanical faults by analyzing acoustic signals has been successfully applied~\\cite{polat2020fault,zhang2020deep,islam2018motor,verstraete2017deep,chen2017vibration,Long2020a,Paul2019,Luo2018,Ince2016}. According to recent studies, deep learning architectures can be trained to identify sound signals by using image representation, such as Mel frequency cepstral coefficients (MFCCs)~\\cite{Zhang2017,8635051}, spectrogram~\\cite{Boddapati2017}, Mel spectrogram~\\cite{Mushtaq2021,9252126}, log-Mel spectrogram~\\cite{tran2021detecting}. \n\nWhen detecting failures on rotating machines, a conventional approach of handcrafted feature extraction and selection is often used. However, extracting and selecting the right features can be challenging in order to use in a classifier or to accurately detect faults in an industrial setting. It is possible for both sound and noise to occur simultaneously in a realistic soundscape. Additionally, the drill sound waveform is complex and short, making it difficult to detect (around 20.83~ms and 41.67~ms). It is therefore not guaranteed that the extracted features of raw audio signals are sufficient for classification. \n\nThe use of vibration sensors is common when detecting machine faults. On the contrary, our methods of detecting broken drills were based on sound signals for these reasons. Drill bits for each drilling machine in Valmet AB are available in quantities of 90 or 120. A vibration sensor is required for each drill bit in order to detect fractures. Mounting 90 to 120 vibration sensors on a drilling machine simultaneously and classifying these vibrations is complicated and expensive. In addition to being cautious about smooth and accurate holes when cutting metal, you must also avoid metal sticking to the drill bit. Dust and sediment are therefore removed from metal surfaces with water. Using vibration sensors in a very wet environment is problematic. For all of the reasons outlined above, we chose sound over vibration to detect broken drills in the drilling machine.\n\n\n\nThe rest of the paper is organized as follows. Section II introduces Valmet AB's dataset. Section III presents the method for data augmentation using VAE and our proposed machine failure detection system. Section IV presents the experiment result of training VAE to synthesize new drilling sounds, the classification results on the augmented dataset, and the comparison study. Section V is the conclusion. \n\n\n\\section{Dataset}\nFour microphones with AudioBox iTwo Studio were used to capture the sounds with a sampling frequency of 96 kHz from Valmet AB's drill machines in Sundsvall, Sweden. The drilling machines at Valmet AB consist of two types of drill bits, one contains 90 drill bits, the other 120 drill bits. The sample duration for sounds in the Valmet AB dataset is very short, around 20.83~ms and 41.67~ms corresponding to 2000 sample points. The original Valmet dataset includes two categories: the \"Anomaly\" class (67 anomalous sounds) and the \"Normal\" class (67 normal sounds). \"Anomaly\" class includes all sounds recorded when the drill bit was broken. The \"Normal\" class includes sounds recorded when the drilling machine was operating properly. \n\n\\section{Methodology}\n\\subsection{Variational Autoencoder}\nA variational autoencoder (VAE)~\\cite{kingma2014autoencoding},~\\cite{rezende2014stochastic} is the architecture that belongs to the field of probabilistic graphical and variational Bayesian. A VAE is composed of an encoder and a decoder (Figure \\ref{fig:Figure1}). The purpose of a VAE model isn't to replicate input sounds but to generate random variations on this input from a continuous space. The most important part of VAE is the continuous latent space that makes interpolation easier. VAE first takes an input sound and creates a two-dimensional vector (mean and variance) from a random variable (the encoding). A sampled encoding is obtained from this vector and passed to the decoder.  The decoder decodes this 1D vector and recreates the original sound. The decoder is capable of learning from all nearby points on the same latent space because encodings are generated from distributions with the same mean and variance as inputs.\n\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=0.95\\linewidth]{Images\/Figure1.pdf}\n  \\caption{Autoencoder.}\n  \\label{fig:Figure1}\n\\end{figure}\n\nThe divergence between probability distributions was controlled by Kullback\u2013Leibler divergence. Kullback\u2013Leibler divergence is minimized by optimizing mean and variance to closely resemble that of the target mean and variance. \n\n\\subsection{Using VAE for drilling sound synthesis}\nWe created a VAE to generate new drilling sounds from the sounds in the original dataset. VAE differs from traditional autoencoders in that it does not reconstruct the input using the encoding-decoding process. VAE instead imposes a probability distribution on the latent space and learns it so that the decoder produces outputs. The distribution of these outputs matches the observed data. New data are generated by sampling from this distribution. In this paper, we train a VAE on sounds in our dataset and generate new sounds that closely resemble the original sounds. Figure \\ref{fig:EncoderDecoder} shows the architecture of our VAE. \n\n\n\\subsubsection{Encoder}\nThe encoder includes four one-dimensional convolutional (Conv1D) layers interspersed with four ResnetEncoder blocks, as shown in the left side of Figure \\ref{fig:EncoderDecoder}. The inputs of the encoder are 1999-length vectors corresponding to 1999 sample points of each sound in the dataset. The architecture of ResNetEncoder block is shown on the right side of Figure \\ref{fig:EncoderDecoder}. ResNetEncoder block includes two one-dimensional convolutional layers and two instance normalization layers. A channel's features are normalized by instance normalization. When the learning rate is adjusted linearly with batch size, it has a more stable accuracy than the batch normalization in a wide range of small-batch sizes.\n\n\\subsubsection{Decoder}\nThe decoder includes four ResnetDecoder blocks interspersed with four one-dimensional convolutional transpose (Conv1DTranpose) layers, as shown in Figure \\ref{fig:EncoderDecoder}. The input\\_shape of the decoder is the latent dimension vector from the encoder. ResNetDecoder block includes two one-demensional convolutional layers and two batch normalization layers. A transposed convolution layer has the opposite transformation as a normal convolution layer. This is achieved by transforming the latent space that has the shape of the output of a convolution into something that has the shape of its input while maintaining a connectivity pattern that is compatible with the convolution. \n\n\n\n\\subsubsection{Reparameterization trick}\nThe output of the encoder is the latent distribution. The next step is the sampling step which samples the variance and the mean vectors to pass to the decoder. However, this sampling step creates a bottleneck because the back propagation can not run through a random node, the parameterization trick is usually used to address it. As a result, we approximate \\(Z\\) using the decoder parameters and another parameter as follows:\n\\begin{equation}\n    Z = \\mu + \\sigma*\\epsilon,\n\\end{equation}\nwhere $\\mu$ represent the mean and $\\sigma$ represent the standard deviation of a Gaussian distribution. $\\epsilon$ is an auxiliary variable derived from a standard normal distribution that preserves the stochasticity of \\(Z\\). The parameterization trick is despicted in Figure~\\ref{fig:Figure1}.\n\n\\subsubsection{Loss function}\nThe loss function in VAE is the sum of the reconstruction loss and Kullback\u2013Leibler loss (KL loss) \\cite{MAL-056}. The reconstruction loss measures the difference between our original input sound and the decoder output sound by using the binary cross-entropy loss. \n\nThe KL\\_loss ($KL\\_loss$) is calculated by optimizing the single sample Monte Carlo estimation as below: \n\n\\begin{equation}\nKL\\_loss = \\log p(x|Z)+\\log p(Z)-\\log q(Z|x)\n\\end{equation}\n\n\\begin{figure*}[!ht]\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{Images\/EncoderDecoder.pdf}\n  \\caption{The variational autoencoder.}\n  \\label{fig:EncoderDecoder}\n\\end{figure*}\nwhere $x$ represents the input sound, $Z$ represents the latent variable, $p(x)$ is the probability distribution of the sound, $p(Z)$ is the probability distribution of the latent variable, and $p(x|Z)$ is the distribution of generating data given latent variable, The probability distribution $p(Z|x)$ describes how our data is projected into latent space. $q(Z|x)$ is the inference model with \\(q(Z|x) \\approx p(Z|x)\\)~\\cite{MAL-056}.  \n\n\\subsection{The proposed machine failure detection system}\nIn this study, we attempted to categorize the drill sounds for the purpose of detecting drill bit failure in Valmet AB, a manufacturing in Sundsvall, Sweden. A fairly small balanced dataset for the two classes was a challenge for us when applying deep learning model. Therefore, we proposed employing VAE (a deep learning model) to generate more synthetic drilling sounds from the modest original drilling sounds as a data augmentation strategy. The original sounds were combined with the generated sounds from VAE to form the enhanced dataset. Then, a low-pass filter with a passband of 22\\kern 0.16667em000 Hz and a high-pass filter with a passband of 1000 Hz were applied to this dataset as part of the preprocessing procedure. After preprocessing, these sounds were converted into Mel spectrogram and classified using the pre-trained network AlexNet. Our proposed procedure is shown in Figure~\\ref{fig:systemArch}.\n\n\\begin{figure*}[!ht]\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{Images\/SystemArchitecture.pdf}\n  \\caption{The proposed system architecture.}\n  \\label{fig:systemArch}\n\\end{figure*}\n\n\\section{Experimental results}\nThe experiment for this paper was conducted on both python and Matlab. \n\\subsection{Training VAE for drilling sound synthesis}\n\n\\subsubsection{Data preparation and training VAE}\nTraining VAE for sound synthesis is implemented using tensorflow, numpy, pandas and librosa. The number of epoch is 20 and batch size is 8. The dimensionality of the latent space is set to 2 so the latent space is visualized as a plane. The sample rate is 96\\kern0.16667em000. The training lists included 34 anomalous sounds and 34 normal drill sounds. The testing lists included 33 anomalous drill sounds and 33 normal drill sounds. The sounds are shuffled and batched using \\textit{tf.data}.\n\n\nThe training process was iterated over the training lists. For each iteration, the sound was passed through the encoder to obtain a set of mean and log-variance for the $q(Z|x)$. This approximate posterior is then sampled using the reparameterization trick. The parameterized samples were then passed through the decoder to get the distribution $p(x|Z)$.\n\nAfter training VAE on the training sounds, we generated 48 drilling sounds for the \"Anomaly\" class and 48 normal drilling sounds for the \"Normal\" class. \n\n\n\\subsubsection{Measuring sound similarities between the original sound and the synthesized sound}\nWe hypothesis that VAE maybe reduce the noise in the synthesized data. However, by listening to the synthesized drilling sounds, the newly generated drilling sounds closely follow the pitch of the original sounds. Hence, we assessed the similarity between a sound in a testing set and a synthesized drilling sound from VAE using the cross-correlation and the spectral coherence between two sounds using Matlab.\n\n\\textbf{Cross-correlation between two sounds:}\n\nWe measured the similarities between an original sound in the anomaly class and the synthesized sound from VAE. These two sounds have the same sampling rate of 96\\kern 0.16667em000 Hz but have different lengths. The original sound has a length of 41.67 ms whereas the synthesized sound has a length of around 20.83 ms. The calculation of the difference between two sounds is difficult because of the different lengths so we had to extract the common part of the two sound signals using the cross-correlation. Figure~\\ref{fig:cross-correlation} shows the cross-correlation between the original sound and the synthesized sound. The value of cross-correlation coefficient is from -1.0 to 1.0. If the cross-correlation value is closer to 1, the two sounds are more similar. The high peak in the third subplot shows that the original sound correlated to the synthesized sound.\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/cross_correlation.eps}\n  \\caption{The cross-correlation between the original sound and the synthesized sound.}\n  \\label{fig:cross-correlation}\n\\end{figure}\n\n\\textbf{The spectral coherence between two sounds:}\n\nPower spectrum shows how much power is present per frequency. Correlation between signals in the frequency domain is referred to as spectral coherence. The power spectrum of Figure~\\ref{fig:spectral_coherence1} shows that the original sound and synthesized sound have correlated components in the range of 0 Hz and 6890 Hz. \n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/spectral_coherence.pdf}\n  \\caption{The power spectrum.}\n  \\label{fig:spectral_coherence1}\n\\end{figure}\n\nWe show the coherence estimate in the first subplot of Figure~\\ref{fig:spectral_coherence2}. Coherence values range from 0 to 1. For example, we marked three highest peak of the coherence values as 0.494153, 0.55872, and 0.377155, as shown in Figure~\\ref{fig:spectral_coherence2}. The higher the coherence, the more correlated the corresponding frequency components are. There are approximately 45 degrees of phase lag between the 1125 Hz components and the 4312.5 components. The phase lag between the 9187.5 Hz components is approximately 38 degrees.\n\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/spectral_coherence2.pdf}\n  \\caption{The spectral coherence.}\n  \\label{fig:spectral_coherence2}\n\\end{figure}\n\n\\subsection{Classification result using original and synthesized sounds}\nWe mixed the orginal dataset that contained 67 sounds for each class with the generated dataset that contained 48 synthesized sounds for each class. We obtained the augmented dataset that included two classes, \"Normal\" class (115 sounds) and \"Anomaly\" class (115 sounds).\n\\subsubsection{Sound pre-processing}\nThe sample rate of sounds in our augmented dataset is 96\\kern 0.16667em000, but the sound of a drill typically ranges from roughly 1000 to 22\\kern 0.16667em000 Hz. The low-pass filter are used to eliminate frequencies higher than 22\\kern 0.16667em000 Hz. The frequency below 1000 Hz was then removed using a high pass filter.\n\n\\subsubsection{Convert sounds to Mel spectrograms}\nWe converted sounds into Mel spectrograms using the Mel spectrums of 1999-point periodic Hann windows with 512-point overlap. 1999-point FFT was used to convert to the frequency domain and pass this through 32 half-overlapped triangular bandpass filters. \nFigure~\\ref{fig:MelSpec}a shows the Mel spectrogram of a sound without applying low-pass filter and high pass filter. Figure~\\ref{fig:MelSpec}b shows the Mel spectrogram after applying low-pass filter and high pass filter. \n\n\n\n\\begin{figure}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.49\\linewidth]{Images\/Break_61_MelSpect.jpg}} \n    \\subfigure[]{\\includegraphics[width=0.49\\linewidth]{Images\/Break_61_lowpass_highpass_MelSpect.jpg}} \n    \\caption{The Mel spectrogram of an original sound in the \"Anomaly\" class: (a) without applying low-pass filter and high-pass filter  (b) apply low-pass filter and high-pass filter.}\n    \\label{fig:MelSpec}\n\\end{figure}\n\n\n\\subsubsection{Classification result}\nTo demonstrate the effectiveness of using VAE for generating new sounds, we proceed to classify two classes in the augmented dataset (mix between original sounds and synthesized sounds) using several pre-trained networks as in Table~\\ref{tablecompaAccuracy}. This experiment is conducted on Matlab 2021a. In overall, pre-trained AlexNet shows the best accuracy (94.12\\%) on our dataset. AlexNet is a simple CNN with only eight layers that was trained to classify 1000 objects from the ImageNet dataset. \n\n\\begin{table}[h]\n\\centering\n\\caption{The accuracy of different pre-trained models on the augmented dataset.}\n\\label{tablecompaAccuracy}\n\\begin{tabular}{l|c}\n\\hline\n\\textbf{Pre-trained model}                  & \\textbf{Accuracy (\\%)} \\\\ \n\\hline\nGoogLeNet               & 89.71                       \\\\ \\hline\n\\textbf{AlexNet}         & \\textbf{94.12}                       \\\\ \\hline\nVGG16   & 88.24                     \\\\ \\hline\nVGG19         & 85.29                       \\\\ \\hline\nEfficientnetb0         & 86.76                       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nBecause after training VAE to generate new drilling sounds, our augmented dataset is extended but still small. Hence, we fine-tuned a common AlexNet with transfer learning instead of training a new CNN from scratch. The learned features from pre-trained AlexNet can easily and quickly transfer to a new task of classification the \"Normal\" class and \"Anomaly\" class in the augmented dataset. Sounds in the augmented dataset were converted into Mel spectrogram, as described above.  \n\nWe utilized 70\\% of Mel spectrograms (162 sounds) for training and 30\\% for validation (68 sounds). The input size of AlexNet is 227-by-227-by-3, where 3 represents three R, G, B channels.  The fully connected layer and classification layer of AlexNet were used to classify 1000 classes. To fine-tuned AlexNet to classify only two classes in our augmented dataset, the fully connected layer was replaced with a new fully connected layer that has only two outputs. the classification layer was replaced with a new classification layer that has two output classes. \n\nWe augmented the training set (162 sounds) by flipping the Mel spectrograms along the x-axis, translating them via the x-axis and the y-axis randomly in the range of (-30, 30) pixels, and scaling them via the x-axis and the y-axis randomly in the range of (0.9, 1.1).  \n\nFor the training option, mini-batch size was 10, the validate frequency (\\(valFreq\\)) is calculated as below:\n\\begin{equation}\nvalFreq = training\\_file\/ mini\\_batch\\_size\n\\end{equation}\nwhere training\\_file is the number of the training file. Max epochs were 160, the initial leaning rate is 3e-4, shuffle every epoch.\n\nThe overall accuracy when fine-tuning AlexNet on our augmented dataset reaches 94.12\\%. The confusion matrix is shown in Figure \\ref{fig:ConfusionMatrix_ImgAugmentation} whereas the accuracy when detect \"Anomaly\" class reaches 97.06\\%.\n\n\n\\begin{figure*}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.35\\linewidth]{Images\/AlexNet_ConfusionMatrix.pdf}} \n    \\subfigure[]{\\includegraphics[width=0.35\\linewidth]{Images\/AlexNet_ConfusionMatrixPercent.pdf}} \n    \\caption{The confusion matrix when fine-tuning AlexNet to classify \"Normal\" and \"Anomaly\" classes on the augmented dataset: (a) in number (b) in percentage.}\n    \\label{fig:ConfusionMatrix_ImgAugmentation}\n\\end{figure*}\n\n\\subsection{Analyzing the original dataset with pre-trained Alexnet}\nThis section compares the accuracy of the pre-trained network AlexNet on the original dataset and on the augmented dataset. We trained pre-trained AlexNet with the same training option on the original drilling dataset, the accuracy reached 87.5\\%, as shown in Table ~\\ref{tablecompare1}. Meanwhile, we found that when we combined the original drilling sounds and synthesized sounds derived from VAE (the augmented dataset), the classification accuracy was 94.12\\%, which is 10.15\\% higher than when we used the small original dataset. The confusion matrix for the original dataset is shown in Figure~\\ref{fig:ConfusionMatrix_original}.\n\n\n\\begin{table}[h]\n\\centering\n\\caption{The comparison of AlexNet on the original dataset and the augmented dataset. No. of sounds indicates the number of sounds.}\n\\label{tablecompare1}\n\\begin{tabular}{l|l|c}\n\\hline\n\\textbf{Dataset}              & \\textbf{No. of sounds} & \\textbf{Accuracy (\\%)} \\\\ \\hline\nThe original drilling dataset & 134                   & 87.5                      \\\\ \\hline\nThe augmented dataset         & 230                   & 94.12                  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure*}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Images\/AlexNet_ConfusionMatrix_OriginalDataset.pdf}} \n    \\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Images\/AlexNet_ConfusionMatrix_Percent_OriginalDataset.pdf}} \n    \\caption{The confusion matrix when fine-tuning AlexNet to classify \"Normal\" and \"Anomaly\" classes on the original dataset.: (a) The confusion matrix in number  (b) The confusion matrix in percentage (\\%).}\n    \\label{fig:ConfusionMatrix_original}\n\\end{figure*}\n\n\\begin{comment}\n\\subsubsection{A comparison of the classification accuracy between pre-processing and without pre-processing sounds}\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{Images\/Number_confusion_matrix_WITHOUT_LowPass_HighPass.jpg}\n     \\caption{The confusion matrix in number.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{Images\/Percent_confusion_matrix_WITHOUT_LowPass_HighPass.jpg}\n    \\caption{The confusion matrix in percentage (\\%).}\n  \\end{subfigure}\n  \\caption{The confusion matrix when classify \"Normal\" and \"Anomaly\" classes of the augmented dataset without using low-pass filter and high-pass filter.}\n  \\label{fig:ConfusionMatrix_without_low_high_pass_filters}\n\\end{figure*}\n\nIn this section, we examined the accuracy of the pre-trained AlexNet on the augmented dataset with and without using low-pass filter and high-pass filter for pre-processing sounds. Sounds on the augmented dataset were converted into Mel spectrograms without applying low-pass filter and high-pass filter. The pre-trained AlexNet was trained on these Mel spectrograms with the same training option. As shown in Table~\\ref{tablecompare2}, the classification accuracy of the pre-trained AlexNet reached 88.24\\% that is lower than pre-processing sound by high-pass filter and low-pass filter before converting to Mel spectrogram (92.65\\%). Figure~\\ref{fig:ConfusionMatrix_without_low_high_pass_filters} despicts the confusion matrix. \n\n\\begin{table}[!ht]\n\\centering\n\\caption{A comparison of the classification accuracy between pre-processing and without pre-processing sounds.}\n\\label{tablecompare2}\n\\begin{tabular}{l|c}\n\\hline\n\\textbf{}                  & \\textbf{Accuracy (\\%)} \\\\ \n\\hline\nLow-pass filter (22\\kern 0.16667em000 Hz) and \\\\high-pass filter (1000 Hz)   & 92.65                       \\\\ \\hline\nWithout pre-processing sounds   & 88.24                       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\end{comment}\n\n\\section{Conclusion}\nDrill failure detection is critical for the industry to reduce drilling machine downtime. The unavailability of drilling sounds in the dataset, on the other hand, made it impossible to adequately train the failure detection model. As a result, employing VAE to synthesize\u00a0new drilling sounds from existing drilling sounds can help to supplement the small sound dataset. A case study is presented here that uses 134 drill sounds classified into two categories (normal and anomalous) in a Valmet AB dataset. We used a VAE to synthesize 96 new drilling sounds (48 normal sounds and 48 anomaly sounds) from the original drilling sounds. VAE may, although, minimize the noise in synthetic data. However, listening to the synthetic drilling sounds reveals that the newly generated drilling sounds roughly match the pitch of the original sounds. The newly synthesized sounds and original sounds were combined to create the augmented sound dataset to train a discrimination model. We fine-tuned AlexNet to classify Mel spectrograms of sounds in the augmented dataset. The result is also compared to fine-tuned AlexNet on the original dataset. The overall classification accuracy reached 94.12\\%. The result is promising to build a real-time machine fault detection system in the industry.\n\n\\section*{Acknowledgment}\nThis research was supported by the EU Regional Fund, the MiLo Project (No. 20201888), and the Acoustic sensor set for AI monitoring systems (AISound ) project. The authors would like to thank Valmet AB for providing the drill sound dataset. \n\nThe computations\/data handling was enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at the SNIC center is partially funded by the Swedish Research Council through grant agreement no. 2018-05973.\n\n\\Urlmuskip=0mu plus 1mu\\relax\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\\section{Introduction}\n\n\\citet{Einstein1936} showed that the light curve of a source microlensed\nby a single foreground compact object is given by an extremely simple\nsymmetric bell-shape, described analytically by a very compact\nformula, now known as Paczy\\'nski curve \\citep{Pacz1986}. It is somewhat\nfrustrating that by adding just another lens the complexity of\nmicrolensing explodes so dramatically that after almost 30 years of\nactive theoretical and observational research a complete\nclassification of all possible light curve morphologies is still\nmissing even in the simplest static case! The lack of a complete\nknowledge of the light curve zoology represents a considerable\nhandicap in the modelling of real microlensing events. In fact, in\norder to set initial conditions for fitting, one may follow two\nroutes: either blindly set-up a dense grid in the parameter space or\nidentify good initial seeds with light curve morphologies close to the\none we wish to model. The first approach is more systematic but can be\ntime consuming and redundant; furthermore, it does not guarantee the\ncompleteness of the exploration of all possible corners, which may\nremain hidden in the space between consecutive points in the grid. The\nsecond approach promises to be more efficient in terms of computing\ntime but needs to be supported by a robust and rigorous theoretical\nframework in order to be safely pursued.\n\nHistorically, it is not uncommon for modellers to explore specific\nmorphological traits of light curves to narrow down the parameter\nspace to be searched, as has been done by authors such as\n\\citet{Mao1995,Dominik1996, DiStefano1997, Albrow1999, Dominik1999a,\n  Han2008e}, but literature that systematically covers the whole range\nof possible morphologies is more scarce. The modelling of observed\nmultiple-lens microlensing light curves requires extensive computation\nof the magnification curves. Much effort has been invested into\nspeeding up the modelling process, by improving the parametrisation\n\\citep{An2002, Cassan2008b, Bennett2010, Bennett2012, Penny2014}, by\nemploying neural networks to map light curve features to model light\ncurves \\citep{Vermaak2007}. Of course this development happened\nalongside of substantial advances in the code implementation of\nexisting algorithms.\n\n\\citet{Mao1995} discussed a new method for modelling binary\nmicrolensing events: the positions and amplitudes of binary light\ncurve extrema are compared to those stored in a pre-compiled\n(unblended, point-source) light curve library to find promising\ncandidate events, which in turn provide initial parameter sets for a\nmore conventional fitting procedure. This approach works well for\nmulti-peak events, where the source trajectory passes over or close to\nthe binary caustics.\n\n\\citet{DiStefano1997} developed the library approach further by\ndescribing any binary-lens light curve by the set of coefficients of\nChebyshev basis polynomials. They note that the Chebyshev expansion\nwill never exactly match the microlensing light curve, because there\nwill be extra extrema and inflection points, but an arbitrarily\nprecise agreement can be achieved (limited by computational power) by\nfurther expansion. In this way, a model search can be refined until\nthe photometric precision of the data points is matched. They find\nmodel parameter solutions to \\emph{smooth} and \\emph{caustic-crossing}\nlight curves by comparing the rough characteristics of the light curve\n(positions of extrema and inflection points and the magnification\nvalues at these points) with a pre-computed light curve library and\nthen searching the nearby environment in the physical parameter space\nwith an increased sampling density until they find a match (or\nmultiple matches) that satisfies the desired precision. In principle,\nthis method is quite good at finding degenerate solutions and\nhigher-order or even non-microlensing parameters can easily be\nintegrated, but again it remains unclear whether all relevant\nparameter-space regions have corresponding entries in the library.\nThe optimistic assertion that the ``morphological features change in a\nway that is gradual and consistent as the physical parameters are\nchanged'' \\citep{DiStefano1997} is most likely true for smooth light\ncurves, but for caustic-crossing light curves, we know that very small\nchanges in the source trajectory can have dramatic implications for\nthe number of extrema and their relative positions.\n\n\\citet{Night2008} make a broad distinction between \\emph{smooth} light\ncurves and \\emph{caustic-crossing} light curves, but the\nclassification is not based on the light curve itself, but on the\nsource trajectory and its closeness to the caustics, i.e. the known\nsimulation parameters, not the observable data. They come to the\nconclusion that the ratio of smoothly-perturbed to caustic-crossing\nbinary-lens light curves is rather low in survey detections, which can\npartly be explained by the fact that caustic-crossing peaks stand out\nunambiguously, whereas smooth perturbations often can have a range of\ncompeting explanations (such as binary sources, parallax\neffects, orbital motion).\n\nIn \\citet{Bozza2012}, a\ndetailed morphological assessment is used for the modelling of\nOGLE-2008-BLG-510 and furthermore the groundwork is laid for a\nreal-time binary event modelling code (further based on\n\\citet{Bozza2001} and \\citet{Bozza2010}). The code relies on a\nwide choice of starting conditions (``seeds'') from where a search\nfor \\emph{local} $\\chi^2$ minima is carried out. The choice of seeds\nis based on the morphology of the binary caustics, with the assumption\nthat binary-lens light curves sampled from a given region of the\nparameter space lie on a smooth slope of the $\\chi^2$ landscape\n\\emph{as long as the morphology of the light curves does not\n  change}. The \\emph{morphology} is understood, in this case, as a\ngiven peak sequence of caustic crossings and grazings, with any\nnewly created or destroyed peak leading to a change in morphology.\n\nOur intention, with the present work, is to take the move from the\npath traced by \\citet{Mao1995} and \\citet{DiStefano1997} and achieve\nthe first complete classification of light curves in the binary\nmicrolensing problem. By studying peak-number plots, we can separate\ngroupings of light curves in the binary-lens parameter space. We are\nnot concerned with directly establishing light curve models, but we\nwant to ensure that we classify all possible light curves. We then\nwant to improve our understanding of the relations between the\nparameter space and the light curves.\n\nThe variety of microlensing light curves can seem overwhelming, but\nthe trained eye recognises familiar patterns and translates them back\nto the parameter space. In fact, the shape of a microlensing light\ncurve does follow certain rules: not any arbitrary curve can be\ninterpreted as a microlensing light curve. Specifically, the limited\ntopologies of the binary lens magnification maps allow only for a\nlimited range of light curve morphologies.\n\nConsequently, the fundamental idea of this work is to identify the\nbuilding blocks of microlensing light curves and develop a\nclassification scheme that can be directly applied to observed light\ncurves and that allows for a significant narrowing of the modelling\nparameter space, while, unlike any other approach, guaranteeing\ncompleteness. We want to gain a good understanding of the range of\npossible light curves and how the identified morphological classes\nrelate to subspaces of the modelling parameter space. As a first step,\nwe focus on an in-depth study of the equal-mass binary lens, while the\nframework developed applies to the general case. On reviewing the\nproperties of this special case in Sec.~\\ref{sec:binarylensing}, we\nuse the opportunity to introduce a convenient notation for caustic\nelements. Sec.~\\ref{sec:classes} introduces our morphology\nclassification scheme, which is based on the four fundamental peak\ntypes that occur in microlensing; we also discuss the practicalities,\nsuch as the light curve simulation, the peak counting and the\nidentification of iso-maxima regions with light curve morphologies. In\nSec.~\\ref{sec:results}, we summarise and discuss the current results\nof this study, we leave some further considerations to\nSec.~\\ref{sec:considerations}, and stress its future potential in\nSec.~\\ref{sec:discussion}, while the bulk of the content is shown in\ntabular and graphical form in Table~\\ref{tab:morph_class} and in\nFigures~\\ref{fig:morphclass_expl_lcs_0} to\n~\\ref{fig:morphclass_expl_lcs_7}.\n\n\n\\section{Microlensing of equal-mass binary systems}\n\\label{sec:binarylensing}\n\n\\subsection{Parametrisation}\n\\label{sec:parametrisation}\n\nGravitational microlensing is characterised by the angular Einstein radius\n\\begin{equation}\n\\theta_\\mathrm{E} = \\sqrt{\\frac{4GM}{c^2}\\,\\frac{D_\\mathrm{S} -\n    D_\\mathrm{L}}{D_\\mathrm{L}D_\\mathrm{S}}}\\,,\n\\end{equation}\n\nwhere $M$ is the total mass of the (foreground) lens object, while\n$D_\\mathrm{L}$ and $D_\\mathrm{S}$ denote the lens and source distances\nfrom the observer. In the course of a microlensing event, the\nseparation between each pair of images is of the order of\n$\\theta_\\mathrm{E}$, which is less than a milliarcsecond for typical\nobserved events with the source in the bulge ($D_\\mathrm{S} \\sim 8$\nkpc), and the lens being a main sequence star half-way to the source\n($D_\\mathrm{L} \\sim 4$ kpc, $M\\sim 0.3 M_\\odot$)\n\nIf one assumes uniform, rectilinear relative proper motion $\\mu$\nbetween the lens and source, the magnification due to a single point\nlens is described by only three parameters: $u_0$, the closest angular\nimpact of the source to the centre of mass expressed in units of\n$\\theta_\\mathrm{E}$, the Einstein radius crossing time $t_{\\text{E}}\n\\equiv \\theta_{\\text{E}}\/\\mu$, and the time of closest approach $t_0$\nof the source to the centre of mass of the lens system, which is\ntypically used to fix the epoch of observations, but is irrelevant for\nthe light curve shape. Beyond the single lens parameters, we need the\nbinary mass ratio $q = m_2\/m_1$, where $m_1$ is the primary mass of\nthe binary lens system and $m_2$ the secondary mass in units of the\ntotal mass of the system $M$, the angular separation of the binary\ncomponents $s$ in units of $\\theta_\\mathrm{E}$, the angular source\nstar radius $\\rho$ still in units of $\\theta_\\mathrm{E}$, and the\nangle $\\alpha$, between the direction vector from the primary to the\nsecondary and the direction of the source relative to the lens, see\nalso Fig.~\\ref{fig:alpha_impact_definition}.  We assume uniform,\nrectilinear relative proper motion between source and lens for the\nsimulations and ignore higher-order effects.  The observed light curve\nis the sum of the source flux $F_S$, amplified by the microlensing\neffect $A(t;u_0,t_E,t_0,q,s,\\rho,\\alpha)$, and the blend flux $F_B$\ncontributed by unresolved sources\n\\begin{equation}\nF(t)=F_S A(t)+F_B.\n\\end{equation}\nFor the purposes of this morphological study, $F_S$ and $F_B$ have no\nimpact as they just represent a multiplicative and an additive factor\nrespectively.\n\n\\begin{figure\n  \\centering\n  \\resizebox{0.6\\columnwidth}{!}{\\input{figures\/u_0-alpha-latex.pdf_t}}\n  \\hspace{1cm}\n  \\caption{Our definition of $u_0$ and $\\alpha$. The impact parameter\n    $u_0$ is positive, when source and lens (centre of mass) pass each\n    other on the right-hand side as projected on the plane of\n    sky. $\\alpha$ is the angle between the binary axis (pointing from\n    primary to secondary mass) and the source trajectory.}\n  \\label{fig:alpha_impact_definition}\n\\end{figure}\n\nOur parametrisation is equivalent to the convention detailed in\n\\citet[Appendix A]{Skowron2011}, except that we regard the source\nrather than the lens system as moving, resulting in a difference of\n$\\alpha^\\text{here}= \\alpha_{0}^\\text{Skowron} - \\pi$.  A change by\n$\\pi$ just means the source is travelling in the opposite direction on\nthe same trajectory which does not affect the morphology of the light\ncurve, in other words it is a time reversal of the light curve.  More\non the parameter space symmetries in Sec.~\\ref{sec:isopeak}.\n\n\n\\subsection{Caustics}\n\\label{sec:caustics}\n\nIn the theoretical treatment of multiple lens systems, caustics are\nsingular lines where the flux of a point source is infinitely\nmagnified. \\citet{Schneider1986} have shown that there are exactly\nthree distinct caustic topologies for the case of an equal-mass binary\nlens. \\citet{Erdl1993} confirmed this to be true for arbitrary mass\nratios. They also noted the transition points in the binary lens\nseparation where the caustic topology changes depending on the lens\nmass ratio $q$ (also cf. \\citet{Dominik1999d}). A caustic enters the\n\\emph{close-separation} topology domain when $s < s_c$,\n\\begin{align}\n  m_1 m_2 &= \\frac{1}{s_c^8} \\left( \\frac{1-s_c^4}{3},\n  \\right)^3\\label{eq:closetop}\n\\end{align}\nand will show the \\emph{wide-separation} topology when $s_w < s$,\n\\begin{align}\n  s_w^2 &= \\left( \\sqrt[3\\phantom{3}]{m_1} + \\sqrt[3\\phantom{3}]{m_2}.\n  \\right)^3,\\label{eq:widetop}\n\\end{align}\n\nThe three topologies (close, intermediate, wide) are shown in\nFigs. \\ref{fig:closenot}-\\ref{fig:widenot} for representative choices\nof the separation parameter. These figures also contain the labels of\nthe notation to be introduced and discussed in Section\n\\ref{sec:notation}. An isolated pair of lenses close to each other\n(i.e. $s < \\sqrt{2}\/2$ for $q = 1$) result in three caustics\n(Fig.~\\ref{fig:closenot}): one diamond shaped at the centre of mass,\nand two small, triangular, secondary caustics set off from the binary\naxis. If the angular separation of the two lenses is of the order of\none Einstein radius, there will be only one central, relatively large,\nsix-cusped caustic, see Fig.~\\ref{fig:internot}. For the equal-mass\nbinary lens ``of the order of'' means the exact range $\\sqrt2\/2 < s <\n2$. If the two lenses are far from each other ($s > 2$), two diamond\nshaped caustics close to the true position of the lenses result. We\nrecollect that caustic lines are always concave in the coplanar binary\nlens case relevant for Galactic microlensing applications.\n\\citet{Petters2001} go into more mathematical detail in describing\ncaustics through singularity theory of differentiable maps.\n\nMoving from point to extended sources, the singularities of the lens\nmap are regularised by an integration over the finite angular disk of\nthe source. As \\citet[][Fig. 9]{Schneider1986} have shown, an increase\nin angular source size leads to decreased peak magnification, a\nbroadening of the peak width and a displacement of the peak, which\nmeans that the maximum will occur later when a larger source enters a\ncaustic (and earlier at the exit). Magnification maps of extended\nsources feature closed high-magnification lines that can be easily\nrecognised as originating from the smoothing-out of caustics. These\nhigh magnification lines asymptotically approach the mathematical\ncaustics as the source size shrinks to zero.\n\nAs an aside, introducing a third lens can lead to exceedingly more\ncomplicated caustic structures \\citep{Rhie2002}. \\citet{Danek2015a,\n  Danek2015b} have set out to explore the full range of triple-lens\ncaustic topologies. To quote just one very specific example, the case\nof three masses positioned at the tips of an equilateral triangle with\ntwo equal masses at $(1-\\mu)\/2$ and a third mass at $\\mu$, boasts 10\ndifferent caustic topologies. Many of those can be found in other\ntriple-lens scenarios, but the list of ten is nowhere close to\ncovering the whole range possible.\n\n\n\\begin{figure\n  \\centering\n  \\includegraphics[width=0.8\\columnwidth]{figures\/close_notation_.pdf}\n  \\caption{Caustic feature notation of the close-separation binary lens.}\n  \\label{fig:closenot}\n\\end{figure}\n\\begin{figure\n  \\centering\n  \\includegraphics[width=0.65\\columnwidth]{figures\/intermediate_notation_.pdf}\n  \\caption{Caustic feature notation of the intermediate-separation binary lens.}\n  \\label{fig:internot}\n\\end{figure}\n\\begin{figure\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figures\/wide_notation_.pdf}\n  \\caption{Caustic feature notation of the wide-separation binary lens.}\n  \\label{fig:widenot}\n\\end{figure}\n\n\n\\subsection{Notation for caustic elements}\n\\label{sec:notation}\n\nAll extrema of a binary-lens light curve can be traced back to\nfeatures of the caustic of the lens system. We have developed a\n``shorthand'' notation for these features, sketched in\nFigures~\\ref{fig:closenot}, \\ref{fig:internot} and~\\ref{fig:widenot}\nand listed in Table~\\ref{tab:notation}. In this study, we use and\ndepict this shorthand only for the equal-mass binary lens, but we\npoint out its universal applicability to binary-lens caustics of any\nmass ratio.\n\nWe denote folds of a caustic with a lower-case letter and cusps with\nan upper-case letter. Local maxima arise either when the source\ntrajectory approaches or crosses a fold or a cusp.  We discuss the\npeak types in detail in Sec.~\\ref{sec:typology}.\n\nWe recall that the magnification of a point source diverges as\n$f_{c,on}\/\\delta y$ if one hits the cusp along its axis and as\n$f_{c,off}\/\\delta y^{2\/3}$ if one hits it off-axis\n\\citep{Schneider1992}. For fold crossing, the magnification diverges\nas $f_{f}\/\\delta y^{1\/2}$ when approaching the singular line from the\ninside. Following these basic analytic formulae, all cusps are\nstrongly magnifying compared to their immediate surroundings. The\nstrength of magnification varies considerably between one point on a\nfold line and another depending on the factor $f_f$, which becomes\nweaker as we move off the binary lens axis.\n\nIt is the cusps closest to the binary axis that are\nthe strongest in comparison. In the equal-mass binary case, regardless\nof the specific topology, the points of maximum magnification are the\ntwo ``A''-cusps on the binary axis, followed by those parts of the\n``a''-folds closest to the axis.\n\nThe four off-axis cusps (``B'') in the intermediate case,\ncf. Fig.~\\ref{fig:internot}, can be traced across different\nseparations. When the two lenses are moved closer together, the\na-folds will eventually merge and split the single caustic line into\nthree separate caustics. The newly created cusps are denoted by\n``C''. A similar metamorphosis takes place, when the two lenses are\nset further apart, except that in this case the ``b''-folds will merge\nto form the new ``D''-cusps.\n\nIn the close topology, the closer the two lenses are positioned, the\nfurther the two triangular, secondary caustics will move out from the\naxis and they will continually decrease in size and strength, whereas\nthe central caustic only decreases in size but gains in strength,\nuntil the binary lens becomes indistinguishable from a single lens for\n$s \\to\\, 0$.\n\nConversely, in the wide topology, the two arrow-shaped caustics become\nmore and more symmetric towards a diamond shape and decrease in size,\nuntil for $s \\to \\infty$ the B-cusps point perpendicular to the\naxis and the D-cusps become more equal in strength to the\nA-cusps. Ultimately the two caustics shrink to two points, at which\nstage two independent single lenses will be observed rather than one\nbinary system.\n\nAll peaks arising from features closer to or facing the lens on the\nleft side are furnished with an index ``$_1$'', whereas those nearer\nthe right side lens are indexed ``$_2$''. We also want to distinguish\nthe symmetric caustic features, which are mirrored across the binary\naxis. Quite arbitrarily, we denote them with ``$_\\text{t}$'' or top,\nif they are on the left-hand side of the binary axis (looking from\nprimary to secondary) and ``$_\\text{b}$'' or bottom, if they lie on\nthe right-hand side. Figures~\\ref{fig:closenot}, \\ref{fig:internot}\nand \\ref{fig:widenot} better illustrate the ``logic'' behind this\nchoice.\n\\begin{table}\n  \\centering\n  \\begin{tabular}[c]{ l l }\n    \\toprule\n    Notation & Meaning\\\\\n    \\midrule\n    a, b & fold\\\\\n    A, B, C, D & cusp\\\\\n    $_\\text{1}$ & nearer binary mass 1 \\\\\n    $_\\text{2}$ & nearer binary mass 2 \\\\\n    $_\\text{t}$ & ``above'' binary axis (i.e. left of binary vector)\\\\\n    $_\\text{b}$ & ``below'' binary axis (i.e. right of binary vector)\\\\\n    $_\\text{p}$ & primary caustic (in close-separation case)\\\\\n    $_\\text{s}$ & secondary caustic (in close-separation case)\\\\\n    $\\left[\\right.$a\\dots; $\\left[ \\right.$A\\dots & caustic entry (via fold; via cusp)\\\\\n    \\dots a$\\left.\\right]$; \\dots A$\\left.\\right]$  & caustic exit (via fold; via cusp)\\\\\n    $\\left[\\right.$\\dots a \\dots$\\left.\\right]$ & fold grazing (always inside (or \\emph{on}) caustic for binary case) \\\\\n    \\dots A \\dots &  cusp grazing (always outside (or \\emph{on}) caustic for binary case) \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Caustic feature notation, also illustrated in\n    the sketches in Figures~\\ref{fig:closenot},~\\ref{fig:internot}\n    and~\\ref{fig:widenot}.}\n  \\label{tab:notation}\n\\end{table}\n\nIn the special case of an equal-mass binary under examination, we have\na second symmetry axis through the centre of mass, i.e. through the\nmidpoint between the two lenses and perpendicular to the binary\naxis. This does not affect the choice of notation.  The chosen caustic\nfeature notation scheme covers \\emph{all} scenarios with two point\nlenses, including mass ratios very different from unity.  The notation\nscheme is summarised in Table~\\ref{tab:notation}.\n\n\n\\section{Classification scheme and methodology}\n\\label{sec:classes}\n\nHaving revisited the basic structure of equal-mass binary lenses and\nhaving established an alphanumeric notation to identify every fold and\ncusp in each of the three topologies, we now move to the\nclassification of microlensing light curves. First, we define a light\ncurve morphology based solely on observable features of light curves\n(Section \\ref{sec:typology}). By spanning the whole parameter\nspace of binary microlensing, we simulate light curves (Section\n\\ref{sec:simulation}) and assign them to the corresponding morphology\nclass. In this way, we can identify every region in the parameter\nspace in which the same morphology arises as the result of the\nencounter of a determinate sequence of caustic features by the source\nalong its trajectory (Section \\ref{sec:isopeak}).\n\n\\subsection{The four peak types in microlensing}\n\\label{sec:typology}\n\nGiven that the most obvious characteristic of microlensing light\ncurves is the sequence and shape of their local extrema, this sequence\nprovides a natural taxonomic key for our light curve classification\nscheme. We propose that a class of light curves can be identified by\nthe common sequence of peak types. We then recognise that any\nmicrolensing light-curve maximum is created by one of four basic\nmechanisms.  We discuss the four peak types in detail below, but in\nshort summary they are:\n\\begin{enumerate}\n  \\setlength{\\itemsep}{1pt}\n  \\setlength{\\parskip}{0pt}\n  \\setlength{\\parsep}{0pt}\n\\item \\label{item:=C} a cusp grazing (\\textoverbar{C}),\n\\item \\label{item:F} a caustic fold entry (F-) or exit (-F),\n\\item \\label{item:C} a cusp entry (C-) or exit (-C),\n\\item \\label{item:=F} a fold grazing (-\\textoverbar{F}-).\n\\end{enumerate}\nNow, in detail:~\\ref{item:=C} the \\emph{cusp grazing},\n\\textoverbar{C}: The peak that arises when the source passes outside\nthe caustic but close enough to one of the cusps to\npass over the lobe of increased magnification, is a ``cusp\ngrazing''. We unambiguously call a light curve ``cusp-grazing'', if\nthe source trajectory is outside the caustic pre and post-peak and\nonly a single peak results. The Paczy\\'nski curve can be understood as\na grazing of the point caustic (or infinite-order cuspoid) of the\nsingle lens. The name \\emph{Paczy\\'nski curve} should be reserved for\nsingle lens light curves only, but in the limits where a binary lens\nresembles a single lens, when the source does not pass close to the\ncaustics or when the caustics are very small relative to the solid\nangle of the source, a single-peaked light curve will result. We do\nnot register any morphological difference to the cusp grazing in the\nnarrow sense.\n\n\\ref{item:F} the \\emph{fold entry\/exit}, F-\/-F: When the source\nenters on a caustic fold, this creates a very distinctly shaped curve\n(cf. \\citet{Schneider1992, Gaudi2002c}), with a steep, almost vertical\nrise followed by a more parabolic fall, which does not descend as low\nas the caustic-exterior magnification. The morphology is mirrored in\nthe fold exit. A pair of fold entry and exit peaks give rise to the\nfamiliar \\emph{double caustic crossing} signature.\n\n\\ref{item:C} the \\emph{cusp entry\/exit}, C-\/-C: If the caustic is\nentered or exited along a cusp, the peak will have a more symmetric\nshape, because the lobe outside the caustic and the close proximity of\nthe fold lines on the inside of the caustic attenuate the gradient of\nthe passage on both sides. The fact that the magnification in the\ncaustic interior is increased can help to distinguish it from a\ncusp-grazing\\footnote{\\citet{Mao2013} have recently shown that this\nis not necessarily the case for a multi-planar lens distribution.}.\n\n\\ref{item:=F} the ``interior fold approach'' or \\emph{fold grazing},\n\\mbox{-\\textoverbar{F}-:} This type of peak occurs inside the caustic,\nwhile the source trajectory passes close to a caustic fold. Due to the\nconcavity of the caustic lines, the fold-grazing peak will only be\nobserved if it is an \\emph{interior} approach. A special case is the\npeak that occurs when two or more caustic lines are close enough or\nstrong enough to raise the magnification of an extended area between\nthem, giving rise to a peak that cannot be directly attributed to one\nsingle fold.\n\nThese ``building blocks'' of microlensing light curves can be\nsequenced, subject to a few rules:\n\\begin{itemize}\n  \\setlength{\\itemsep}{1pt}\n  \\setlength{\\parskip}{0pt}\n  \\setlength{\\parsep}{0pt}\n\\item a caustic entry must be followed by a caustic exit\\footnote{For\n    $n>3$ lenses the number of entries and exits may be unequal as\n    caustic lines can be intersecting and nesting. For $n=2$, one\n    caustic entry must be followed by one caustic exit, before\n    another caustic entry can occur.}\n\\item a caustic exit cannot occur, if the caustic has not been entered\n  before\n\\item a fold grazing can only take place inside a\n  causti\n\\item a cusp grazing can only take place outside a\n  causti\n\\item due to the concave curvature, a straight caustic-crossing\n  trajectory must exit by a fold (or cusp) different from the one\n  through which it entered \\citep{Cassan2010}\n\\end{itemize}\n\nAll binary microlensing light curves (in the parameter space\nconsidered in this study) adhere to these rules, but just conforming\nto these rules does not guarantee a microlensing light curve since the\npossible caustic topologies are limited \\citep{Erdl1993}.\n\nIt is well known that similar light curve morphologies may arise in\ncompletely different situations, with source trajectories interacting\nwith different cusps or folds in different topologies. Such\ndisconnected regions can be identified by specifying the folds and\ncusps involved using the notation introduced in\nSec. \\ref{sec:notation}. Then the symbols identifying a sequence of\npeaks conforming to a specific morphology class (e.g. F-F\n\\textoverbar{C}), can be replaced by the corresponding caustic\nelements involved (e.g. $[a_{t_1} b_t]B_{t_2}$). Since the folds and\ncusp symbols already carry subscripts, in order to generate more\nreader-friendly sequences, we indicate the caustic entry and exits by\nsquare brackets and suppress the bar for the grazings. So a fold entry\nis ``[a...'', a cusp entry ``[A...'', with the exit being ``...]''.  A\nfold grazing is ``[...a...]'' and a cusp grazing is given by\n``A''. These notations detailing the caustic features involved in the\nlight curve morphology sequence are also summarised in Table\n\\ref{tab:notation}. We will use the synthetic notation (e.g.  F-F\n\\textoverbar{C}) for identifying a light curve morphology class\nirrespective of its possible interpretations in terms of source\ntrajectories and caustics involved, and the detailed notation\n($[a_{t_1} b_t]B_{t_2}$ in this example) to identify the iso-maxima\nregion(s) in the parameter space giving rise to that specific\nmorphology.\n\nTo see an example of a light curve classification ``at work'',\nconsider the light curve in Fig.~\\ref{fig:morphclass_expl_lcs_0}(h)\nwhere we see a (symmetric) cusp entry (C-) paired with an (asymmetric)\nfold exit (-F) and a post-caustic grazing of the cusp lobe\n(\\textoverbar{C}),\n\\begin{align*}\n    \\underbrace{\\text{C-F}}_{\\substack{\\text{caustic}\\\\\\text{traversal}}}\n  \\underbrace{\\text{\\textoverbar{C}}}_{\\substack{\\text{cusp}\\\\\n      \\text{lobe}}}.\n\\end{align*}\nThe detailed sequence specifying the folds and cusps involved in this\nlight curve is $[A_1 a_{t2}] A_2$.\n\nFig.~\\ref{fig:morphclass_expl_lcs_1}(d) gives a nice example with a\nclear-cut fold entry \\mbox{(F-)}, followed by a second peak still inside the\ncaustic, which can only be an inner fold approach (-\\textoverbar{F}-),\na fold exit (-F) and followed by a final cusp lobe grazing\n(\\textoverbar{C}), so we classify it as\n\\begin{align*}\n  \\underbrace{\\text{F-\\textoverbar{F}-F}}_{\\substack{\\text{caustic}\\\\\\text{traversal}}}\n  \\underbrace{\\text{\\textoverbar{C}}}_{\\substack{\\text{cusp}\\\\\n      \\text{lobe}}}.\n\\end{align*}\nThe detailed sequence specifying the folds and cusps involved in this\nlight curve is $[a_{b1} a_{t1} a_{t2}] B_{t2}$.\n\nKeeping the caustic geometry fixed and displacing the source\ntrajectory, we can appreciate the changes in the light curve\nmorphology, with peaks merging or disappearing while other peaks\nappear or separate in two. These transition morphologies need some\nfurther attention in order to be assigned to specific classes without\nambiguities.\n\nIn this respect, consider the case of Fig.~\\ref{fig:cg_morphs},\nrepresenting the morphing from two fold crossings F-F to a cusp\ngrazing \\textoverbar{C}. When the extended source trajectory cuts a\ncusp nearly perpendicularly to its axis, the light curve features a\ntransition morphology with a single peak preceded and followed by\nderivative discontinuities, typical of fold crossings (trajectories\nST2 and ST3 in Fig.~\\ref{fig:cg_morphs}). Introducing a new\nintermediate ``cusp cutting'' class would not be very useful, since\nthe detection of the two discontinuities at the base of the peak could\nnever be unambiguously assessed in real observations. Only a very\ndetailed analysis of the light curve would distinguish a cusp-cutting\nfrom a cusp-grazing trajectory. Keeping in mind that the purpose of\nour study is to identify regions in the parameter space that may give\nrise to independent seeds for model searches, we assign these\ncusp-cutting peaks to the broader cusp grazing class, extending its\ndefinition by including all trajectories for which the cusp cutting\ndoes not give rise to two fold-crossing peaks with a dip\nin between. In some sense, this statement is already contained in the\nabove definition, in which we required that the source is outside the\ncaustic pre- and post-peak and only one peak occurs. This specific\nexample should help avoiding any confusion.\n\n\n\\begin{figure}\n  \\centering\n\\parbox{\\columnwidth}{\n\\includegraphics[width=\\columnwidth]{figures\/mp_st_ss_x.png}\\\\[1ex]\n\\includegraphics[width=\\columnwidth]{figures\/cg_morph_lcs.png}\n}\n{\\small\n\\begin{picture}(0,0)\n  \\put(-110,130){ST 1}\n  \\put(10,130){ST 2}\n  \\put(-110,60){ST 3}\n  \\put(10,60){ST 4}\n\\end{picture}\n}\n\\caption{Comparison of \\emph{fold-crossing} and \\emph{cusp-grazing}\n  source trajectories (ST 1 to 4) and resulting light curves. The\n  angular source size is indicated by the white circles. From left to\n  right, the light curve morphology evolves from a double fold\n  crossing F-F for ST 1 to a cusp grazing \\textoverbar{C} (ST 2, 3 and\n  4). Where exactly this transition occurs depends on the angular\n  source size; with a smaller source, ST 2 would also lead to a\n  double-peaked fold crossing.}\n\\label{fig:cg_morphs}\n\\end{figure}\nIt follows that the peak classification does not just depend on the\nsource trajectory relative to the lens positions, but equally on the\nangular source size relative to the caustic size. I.e. a given source\ntrajectory (e.g. ST 2 in Fig.~\\ref{fig:cg_morphs}) can yield an F-F\nmorphology for a smaller source and a \\textoverbar{C} morphology for a\nlarger source, whereas for a given source size ST 1 can result in an\nF-F pair, but ST 2 will only show a single peak and be classified as\ncusp-grazing \\textoverbar{C}.\n\nAnother situation almost complementary to the previous one occurs when\na fold grazing morphs into two fold crossings as the source trajectory\nchanges from fully internal to tangent and then secant to the\nfold. Adopting the same convention as before, we extend the ``fold\ngrazing'' class to include the transition peak occurring when the\nsource moves tangentially to the fold, as long as the peak remains\nsingle.\n\nTransition morphologies can be more complicated than these two cases\nillustrated above and may also involve changes in the caustic\ntopologies. In Fig.~\\ref{fig:beak_to_beak}, we have a fold-grazing\nsource trajectory C-\\=F-C, across a caustic that is close to the\nlimits of the intermediate-to-wide transition, which morphs into a\ncusp exit\/entry pair, C-C C-C, with an increased source size.\n\nIn summary, all sorts of transitions can be safely treated by adopting\nthe extended definitions of cusp grazing and fold grazing classes just\ndescribed, including the transition peaks before they split into\ntwo. Now we are ready to apply our classification scheme to\narbitrarily complicated light curves without facing any more\nambiguities.\n\n\n\n\n\\begin{figure}\n  \\centering\n\\parbox{\\columnwidth}{\n\\includegraphics[width=\\columnwidth]{figures\/beak_to_beak_mp_st_ss_x.png}\\\\[1ex]\n\\includegraphics[width=\\columnwidth]{figures\/beak_to_beak_lcs.png}\n}\n{\\small\n\\begin{picture}(0,0)\n  \\put(-100,132){$\\rho = 0.01$}\n  \\put(20,132){$\\rho = 0.03$}\n  \\put(-100,60){$\\rho = 0.05$}\n  \\put(20,60){$\\rho = 0.07$}\n\\end{picture}\n}\n\\caption{Classification in the case of a beak-to-beak\n  metamorphosis. The magnification curves result from the same source\n  trajectory, but with different source sizes (as indicated by the\n  white circles). The smallest source produces an unambiguous\n  \\emph{fold-grazing}, as the central peak occurs inside the caustic\n  (C-\\=F-C). Interestingly, the larger sources create a central\n  \\emph{pair} of peaks instead, thus leading us to classify the light\n  curves as C-C C-C. This might seem counterintuitive, before one\n  considers the convoluted magnification pattern, where it becomes\n  clear that a larger source shifts the position of the beak-to-beak\n  fold merger -- thereby causing the caustic topology change to occur\n  at a smaller separation compared to the smaller source.}\n\\label{fig:beak_to_beak}\n\\end{figure}\n\n\n\\subsection{Light curve simulation and processing}\n\\label{sec:simulation}\n\nIn order to achieve a complete classification of binary lens light\ncurve morphologies, we process simulated light curves. We then\nconsider light curves grouped in the parameter space by their number\nof maxima. The parameter space we want to cover is the equal-mass lens\n($q=1$), the separation $s$ across all topologies and the source\ntrajectory parameters $0 \\leq \\alpha < 2\\pi$ and $u_0$ as far as new\nmorphologies can be expected to occur. We use an extended source with\nangular radius $0.002\\,\\theta_{\\text{E}}$. For each light curve we\nrecord the number of peaks and visualise the results in peak-number\nplots (over $\\alpha$ and $u_0$). The resulting iso-maxima regions are\nexamined with regard to the contained light curve\nmorphologies. Broadly speaking, an iso-maxima region, covering a\n``bundle'' of neighbouring source trajectories, corresponds to a\nspecific sequence of caustic features. One step up in the\nclassification hierarchy, different iso-maxima regions are collected\nin morphology classes (as introduced in Sec.~\\ref{sec:classes}). The\nfixed source size used in our investigation is small enough to probe\nthe caustics of an equal-mass binary lens in detail but large enough\nto let cusp crossings occur in finite regions of the parameter\nspace. Different choices will cause a slight shift of the boundaries\nof the iso-maxima regions (cf. Figs.~\\ref{fig:cg_morphs}\nand \\ref{fig:beak_to_beak}). This point is further discussed in\nSec.~\\ref{sec:considerations}.\n\nIn our examination of the equal-mass binary lens case, we simulate\nmicrolensing light curves for all (relevant) volumes of the\n($s$,~$\\alpha$,~$u_0$) parameter space. We simulate the light curves\nwith inverse ray shooting, using a software library written in 2010 by\nMarnach\\footnote{Published at\n  \\url{https:\/\/github.com\/smarnach\/luckylensing}.}. Assuming static\nlenses, this means we can compute magnification maps for every\n($q$,~$s$) set, fold them with the source star profile with a radius\n$\\rho$ and then extract a large number of light curves differing in\n$\\alpha$ and $u_0$ at virtually no computational cost.  During the\npeak counting, numerical noise can create artificial peaks and\ntroughs, especially for source trajectories that run at a small angle\nto fold lines.  To avoid these, we require a minimal difference\nbetween the maximum and the minima on either side of 5\\% of the\nnearest local minimum value, before a trough-peak-trough occurrence is\ncounted as a peak. Because of this threshold, sometimes true peaks\nwill be disregarded in the maxima counting algorithm. But this is\nunlikely to make us miss a whole iso-maxima region, as generally the\nregion boundary (where the formerly disregarded peak becomes\nsignificant) will only be slightly shifted in the\n($u_0$,~$\\alpha$) plane.\n\n\n\\subsection{Iso-maxima regions}\n\\label{sec:isopeak}\n\nPer examined separation, we plot the number of local maxima per light\ncurve over $\\alpha$, $u_0$ of its source trajectory, see\nFig.~\\ref{fig:regions100}. In the resulting plot, we can identify and\nisolate regions of a uniform peak number, which we call\n\\emph{iso-maxima regions}.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/q1-00_s1-00_sr0-002_3000_1600_labelled_newcolour.png}\n  \\caption{Plot of the number of maxima per light curve in the first\n    quadrant of the ($u_0$,~$\\alpha$) parameter space for the\n    equal-mass binary lens at separation $s = 1.0$ (intermediate\n    caustic topology): \\emph{white} means the light curve has a single\n    peak, \\emph{dark blue} means six peaks, \\emph{higher values} are\n    numerical noise in this instance. Each labelled region represents\n    a grouping of source trajectories and corresponding light curves\n    that follow a specific caustic feature sequence, see\n    Table~\\ref{tab:peaks100}. Rarely are two regions with the same\n    number of peaks directly connected (cf. III b and III g above).}\n  \\label{fig:regions100}\n\\end{figure*}\n\nDue to the inherent symmetries, we can restrict ourselves to the first\nquadrant, $0 < u_0$, $0 < \\alpha < \\pi\/2$.  Beyond the trivial\nperiodicity of $\\alpha$ with period $2\\pi$, there are several\nsymmetries in the two-dimensional $(u_0, \\alpha)$ space. Generally,\nfor a binary lens,\n\\begin{align}\n  (u_0, \\alpha) \\Leftrightarrow  (- u_0, - \\alpha)\n\\end{align}\nis an exact degeneracy, which is caused by the intrinsic symmetry of\nthe binary lens across the binary\naxis. \\citet[Appendix~A]{Skowron2009} argues (and this has been common\npractice for a while, see e.g. \\citet{Albrow1999b}) that models for\nstatic binaries should be expressed in the range $u \\geq 0$ and $0\n\\leq \\alpha < 2\\pi$, with the exception of cases that display parallax\neffect where the apparent source position can appear on both sides of\nthe lens. We generally subscribe to this view, nonetheless it is\ninstructive to, at least once, visualise the ``full'' parameter space,\nsee Sec.~\\ref{sec:results}. %\n\nSince we are interested in the morphology only,\n\\begin{align}\n  (u_0, \\alpha) \\Leftrightarrow (- u_0, \\alpha + \\pi),\n\\end{align}\ngives the symmetry of a time reversal (where the sign of $u_0$ has to\nchange according to the convention, because the source now passes the\nlens on the other side).\nWe can also combine the two,\n\\begin{align}\n  (u_0, \\alpha) \\Leftrightarrow (u_0, \\pi - \\alpha).\n\\end{align}\nFor the special case of the equal-mass binary, we also have a perfect\ndegeneracy\n\\begin{align}\n  u_0 \\Leftrightarrow -u_0,\n\\end{align}\ni.e. the plot is axis-symmetric in $u_0$.\n\nWe note that whenever one moves from one iso-maxima region to a\nneighbouring one, the morphology of the light curve peaks changes --\nnaturally, because the border will be overstepped whenever a peak is\ncreated or destroyed. We record the caustic feature sequence for the\nlight curves of each region, see Table~\\ref{tab:peaks100}, and realise\nthat in a given quadrant, there are no two iso-maxima regions with the\nsame number of peaks that contain the same sequence of caustic\nfeatures.\n\nWe then map the caustic features to the broader peak typology, thereby\nreducing the complexity of the light curve description and enabling us\nto collate different regions in more general morphology classes.\n\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}[c]{ l >{ $\\rm} l <{ $} l }\n    \\toprule\n    Region label &  \\text{Caustic feature sequence}  &  Morphology class  \\\\\n    \\midrule\n   \n    I a   &  A_1\\text{ or }B_{t1}     &  \\=C   \\\\\n    \\midrule\n    II a  &   [A_1 A_2]        &  C-C \\\\\n    II b  &   [a_{t1} a_{t2}]  &  F-F   \\\\\n    II c  &   B_{t1} B_{t2}    &  C-C \\\\\n    II d  &   [a_{b1} a_{t2}]  &  F-F  \\\\\n    II e  &   [a_{t1} b_t]     &  F-F \\\\\n    II f  &   [a_{b1} b_t]     &  F-F \\\\\n    II g  &    A_1 B_{t1}      &  C-C \\\\\n    II h  &   [b_b b_t]        &  F-F \\\\\n    II i  &    [b_b b_t]       &  F-F \\\\\n    \\midrule\n    III a &   [A_1 a_{t2}] A_2   & C-F \\=C  \\\\\n    III b &   A_1 [a_{t1} a_{t2}]    &  \\=C F-F  \\\\\n    III c &   [a_{t1} b_t a_{t2}]    &  F-\\=F-F \\\\\n    III d &   A_1 [a_{b1} a_{t2}]   &  \\=C F-F \\\\\n    III e &   [a_{t1} b_t] B_{t2}   &  F-F \\=C \\\\\n    III f &   [a_{b1} a_{t1} a_{t2}]   &  F-\\=F-F \\\\\n    III g &   A_1 [a_{t1} B_{t2}]   &  \\=C F-C \\\\\n    III h &   A_1 [a_{t1} b_t]   &   \\=C F-F \\\\\n    III i &   [a_{b1} a_{t2}] B_{t2}   &  F-F \\=C \\\\\n    III j &   [a_{b1} a_{t1} B_{t2}]   &  F-\\=F-C \\\\\n    III k &   [a_{b1} b_t] B_{t2}       &  F-F \\=C \\\\\n    III l &   [a_{b1} a_{t1} b_t]   &  F-\\=F-F \\\\\n    III m &   B_{b1} [a_{b1} b_t]   &  \\=C F-F \\\\\n    III n &   [b_b a_{b1} b_t]   &  F-\\=F-F \\\\\n    III o &   [a_{b1} a_{t1}] B_{t1}   &  F-F \\=C \\\\\n    III p &   B_{b1} A_1 B_1   &  \\=C \\=C \\=C   \\\\\n    \\midrule\n    IV a  &   A_1 [a_{t1} a_{t2}] A_2       & \\=C F-F \\=C \\\\\n    IV b  &   [a_{t1} b_t] [b_t a_{t2}]     & F-F F-F     \\\\\n    IV c  &   A_1 [a_{b1} a_{t2}] A_2       & \\=C F-F \\=C \\\\\n    IV d  &   A_1 [a_{t1} b_t] B_{t2}       & \\=C F-F \\=C \\\\\n    IV e  &   [a_{b1} a_{t1}] [a_{t1} a_{t2}] & F-F F-F   \\\\\n    IV f  &   A_1 [a_{t1} a_{t2}] B_{t2}    & \\=C F-F \\=C \\\\\n    IV g  &   [a_{b1} a_{t1} a_{t2}] B_{t2} & F-\\=F-F \\=C   \\\\\n    IV h  &   [a_{b1} a_{t1}] [a_{t1} B_{t2}] & F-F F-C   \\\\\n    IV i  &   [a_{b1} a_{t1} b_t] B_{t2}    & F-\\=F-F \\=C \\\\\n    IV j  &   B_{b1} [a_{b1} a_{t2}] B_{t2} & \\=C F-F \\=C \\\\\n    IV k  &   [a_{b1} a_{t1}] [a_{t1} b_t]  & F-F F-F     \\\\\n    IV l  &   [b_b a_{b1}] [a_{b1} b_t]     & F-F F-F     \\\\\n    IV m  &   B_{b1} [a_{b1} a_{t1} b_t]    & \\=C F-\\=F-F   \\\\\n    IV n  &   [b_b a_{b1} a_{t1} b_t]       & F-\\=F-\\=F-F     \\\\\n    IV o  &   B_{b1} [a_{b1} a_{t1}] B_{t1} & \\=C F-\\=F-F   \\\\\n    \\midrule\n    V a   &   [a_{b1} a_{t1}] [a_{t1} a_{t2}] B_{t2}  &  F-F F-F \\=C \\\\\n    V b   &   [a_{b1} a_{t1}] [a_{t1} b_{t}] B_{t2}   &  F-F F-F \\=C \\\\\n    V c   &   B_{b1} [a_{b1} a_{t1}] [a_{t1} b_t]     &  \\=C F-F F-F \\\\\n    V d   &   [b_b a_{b1}] [a_{b1} a_{t1} b_t]        &  F-F F-\\=F-F   \\\\\n    V e   &   B_{b1} [a_{b1} a_{t2}] [a_{t2} b_t]     &  \\=C F-F F-F \\\\\n    \\midrule\n    VI a  &   [b_b a_{b1}] [a_{b1} a_{t1}] [a_{t1} b_t]   &  F-F F-F F-F \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Caustic feature sequences for the\n    iso-maxima regions in Fig.~\\ref{fig:regions100} ($q=1.0$,\n    $s=1.0$). Each sequence is unique to its region, while the\n    morphology classes generally span several independent regions.}\n  \\label{tab:peaks100}\n\\end{table}\n\n\n\\section{Results}\n\\label{sec:results}\n\nFocussing on the equal-mass binary lens, we analysed peak-number plots\nspanning all three caustic topologies and the two transitioning cases:\nclose ($s = 0.5$, 0.65), close-to-intermediate ($s = 0.7$),\nintermediate ($s = 0.85$, 1.0, 1.5), intermediate-to-wide ($s = 2.05$)\nand wide ($s = 2.5$).  As discussed in Sec.~\\ref{sec:simulation}, we\nwere motivated to use an extended source with an angular radius, $\\rho\n= 0.002$ (in units of $\\theta_\\mathrm{E}$) and work with a peak threshold\nof 5\\% above the nearest minima. \\begin{comment}The detailed\n  iso-maxima region results are printed in Table~\\ref{tab:morph_class}\n \n; we provide a summary and\nan overview here.\\end{comment}\n\nWithin the peak-number plots, we know the light curve composition in\neach (substantial) iso-maxima region, i.e. we know which sequence of\ncaustic features produces the observable peaks of all light curves in\nthat region. We note that it is mostly a bijective mapping, with only\nvery few regions containing more than one kind of caustic feature\nsequences. In no case, do two unconnected regions share the same\ncaustic feature sequence.\n\nThe light curves (and iso-maxima regions) are collected in morphology\nclasses, where each peak is morphologically classified as one of the\nfollowing: cusp-grazing, cusp-crossing, fold-crossing or\nfold-grazing. A substantial subset of morphology classes can be found\nin all examined separation settings. Other classes only appear when a\nhigher or lower separation leads to multi-caustic topologies, whereas\nthe specific example of a double fold grazing is necessarily limited\nto the intermediate caustic cases.\n\nThe extreme variety of binary microlensing phenomenology can be\nappreciated by summarising the results of our search in a few\nnumbers. We have found 73 different light curve morphologies according\nto our classification based on the sequence of peaks. These\nmorphologies arise in 232 independent regions of the parameter\nspace. The simplest morphologies can be obtained in many different\nways. For example, the simplest caustic crossing light curve class,\nF-F, can be found in 7 disconnected volumes of the parameter space. If\nwe add shoulders to this caustic crossing, with the classical sequence\n\\overbar{C} F-F \\overbar{C}, we find 9 different volumes. We emphasise\nthe fact that thanks to our thorough investigation we are able to\nquantify the exact number of independent physical models that can\nqualitatively reproduce an observable light curve for the first time!\nMore complicated morphologies with multiple caustic crossings are\nrarer and appear in fewer regions of the parameter space. A\nmicrolensing light curve for an equal-mass binary lens can have up to\n10 peaks, if the source moves near the vertical axis of a close\nconfiguration.\n\n\n\\section{Further considerations}\n\\label{sec:considerations}\n\n\n\\paragraph*{Source size}\nA hypothetical point source is often useful in theoretical studies of\nthe behaviour of gravitational lenses, but because we want to examine\nthe range of real, observable light curve morphologies, we use an\nextended source size of $0.002\\,\\ensuremath{\\theta_{\\textit{E}}}$ for our simulations. The\nsource size does influence the shape of a light curve, as discussed in\nSec.~\\ref{sec:classes}. A pair of fold crossings can be merged\ninto a single peak, a whole caustic can be crossed and appear as a\nsingle peak, but as long as the solid angle of the source area is\nsmall relative to the caustic extent, the absolute size will not\nchange the number of distinct morphologies that can be studied. For\nthe studied mass ratio $q=1$, we can afford to use a moderately large\nsource that reduces the numerical noise in our samples. Meaningful\nstudies of planetary mass ratios $q\\lesssim 10^{-3}$, require a\nsmaller source size. We point out that not all of the peak types of\nSec.~\\ref{sec:typology} can be simulated with a point source:\nthe cusp crossing can only occur, if the point source enters the\ncaustic \\emph{exactly} over the infinitesimal cusp point. The\nprobability for this occurrence is therefore zero.\n\nTwo peaks will generally merge into one, if their angular separation\nis smaller than the angular source diameter (disregarding\nlimb-darkening effects). In our simulations the source has a diameter\nof $4\\times 10^{-3}\\, \\theta_{\\text{E}}$, i.e. peaks within $4\\times\n10^{-4}\\, t_{\\text{E}}$ of each other would be missed. We work with\nthe assumption that a larger source size can only lead to a smaller\nnumber of identified morphologies. This has been visually demonstrated\nfor $q=1.0$, $s=1.0$ in Fig.~\\ref{fig:pnp_rho_comp}.\n\\citet{Liebig2014} also documents the entirety of morphological\nclasses for a source radius of $10^{-2}\\, \\theta_{\\text{E}}$ and they\nare a subset of the morphology presented here.\n\n\\begin{figure}\n  \\centering\n  \\vspace{5mm}\n  \\includegraphics[width=.325\\textwidth]{figures\/q-1_0-s-1_00-sr-0_005-x.png}\n  \\includegraphics[width=.325\\textwidth]{figures\/q-1_0-s-1_00-sr-0_010-x.png}\n  \\includegraphics[width=.325\\textwidth]{figures\/q-1_0-s-1_00-sr-0_020-x.png}\n  \\caption{Comparison of peak-number plots resulting from different\n    source sizes (from top to bottom: $\\rho = 0.005$, 0.01,\n    0.02$\\,\\theta_\\mathrm{E}$), scale and ranges as in\n    Fig.~\\ref{fig:regions100}: \\emph{x-axis:} $0 \\leq u_0 \\leq 1.0$,\n    \\emph{y-axis:} $0.0 \\leq \\alpha \\leq \\pi\/2$. The change in\n    iso-maxima regions is subtle, but noticeable. The smallest source\n    not only leads to more iso-maxima regions, but also to more\n    numerical artefacts. Also compare Fig.~\\ref{fig:regions100}, where\n    $\\rho = 0.002\\,\\theta_\\mathrm{E}$.}\n  \\label{fig:pnp_rho_comp}\n\\end{figure}\n\n\n\n\\paragraph*{Error margin}\nWhile we aim for completeness, due to the numerical nature of our\nstudy we have to ignore very small sub-regions of the studied\nparameter space and therefore might have missed out on a particular\nlight curve morphology.  Within this space we have examined all\niso-maxima regions larger than 10 by 10 pixel, i.e. $10^2 \\times\n1\/(u_0\\text{-sampling}) \\times\n\\frac{\\pi}{2}\/(\\alpha\\text{-sampling})$, meaning that within a given\nEinstein radius and with our sampling of 1600, the probability to\nobserve that particular light curve morphology is smaller than\n$\\lesssim 1\/16000$.\n\n\\paragraph*{Caustic feature regions}\nIt is highly instructive to consider the complete parameter space\nvolume that corresponds to a peak created by a particular caustic\nfeature. This \\emph{caustic feature region} will cover several\niso-maxima regions, where the light curves show one or two peaks due\nto this particular caustic element, see an example in\nFig.~\\ref{fig:isomax_full}. If this information content could be\nproperly harnessed, it would provide an immediate key for the mapping of\nthe lens system to the light curve and from the light curve morphology\nto the caustic.\n\nFor caustic-crossing point sources, this problem often reduces to\nregistering the intersection points of the straight source trajectory\nwith the caustic, which is a mathematically well-defined\nproblem. However, for non-caustic crossing peaks (i.e. cusp and fold\napproaches), caustic lines do not provide sufficient information. It\nwould be necessary to study the magnification map around caustics in\norder to pin down the position of the maximum magnification along the\nsource trajectory and then assign this position to a nearby caustic\nfold or cusp for classification. This approach appears too complex to\nimplement in practice.\n\n\n\\begin{figure\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figures\/eq-mass-bin-10-800-800_full_newcolour_a_1.png}\n  \\caption{Extended plot of the iso-maxima regions for $s = 1.0$ to\n    illustrate existing symmetries and the seamless continuation of\n    iso-maxima regions beyond the first quadrant. %\n    We have marked a caustic feature region: the green outline frames\n    the area where the ${\\rm a_{t1}}$ fold gives rise to a light curve\n    peak, more specifically the top and bottom regions contain the\n    fold entry ${\\rm [a_{t1} \\ldots]}$ whereas the middle region\n    contains the fold exit ${\\rm [\\ldots a_{t1}]}$. The green shade\n    marks areas where the ${\\rm a_{t1}}$ fold is crossed twice\n    (requiring ${\\rm [\\ldots a_{t1}]\\,[a_{t1} \\ldots]}$ to be part of\n    the light curve). Moving to a slightly smaller $u_0$ from the\n    shaded area, the light curves will display the fold grazing ${\\rm\n      [\\ldots a_{t1} \\ldots]}$.}\n  \\label{fig:isomax_full}\n\\end{figure}\n\n\\section{Conclusion and future prospects}\n\\label{sec:discussion}\n\nWe have compiled an unprecedented catalogue of microlensing light\ncurve morphologies for the equal-mass binary lens. We realised that\nall peaks in microlensing light curves can be classified in just four\ncategories: cusp-grazing, cusp-crossing, fold-crossing or\nfold-grazing. In order to achieve this complete classification, we\nhave developed a general notation scheme for the features of\nbinary-lens caustics. Our tool, plots of peak number over $u_0$ and\n$\\alpha$, serves to provide insight into the the microlensing\nparameter space.\n\nBefore this work, statements of the diversity of binary microlensing\nlight curves were only made on reasonable but vague arguments. With\nour detailed study we are able to assign numbers to all specific cases\nand open the way to more quantitative studies of binary microlensing.\n\nApart from the pure taxonomical aspects, which are very interesting\nfrom the theoretical point of view, Table~\\ref{tab:morph_class} stands\nout as a very powerful tool for modellers to relate an observed light\ncurve to all possible regions of the parameter space in which this\nlight curve can be found. This capability would help the construction\nof more fail-safe algorithms that will guarantee a full exploration of\nthe microlensing parameter space. In practice, seeds for fitting\nalgorithms can be placed in the middle of each iso-maxima region so as\nto ensure a full exploration of all possible cases. Among the\ncurrently running platforms for modelling using this principle for\nsetting initial conditions, we mention RTModel\n\\citep{Bozza2010,Bozza2012}. The inclusion of our catalogue in the\ntemplate library consulted by RTModel would further diminish the\nchances of missing any particular region in the parameter space.\n\nAnother interesting aspect that can be further investigated is the\nprobability of the occurrence of a given morphology. Having traced the\niso-maxima regions in the parameter space, it should not be difficult\nto translate the volumes of the iso-maxima regions in probabilities\nnormalised by a physically motivated measure. In this way, we would be\nable to quantify how ``rare'' or common a morphology is. The current\niso-maxima plots would already suffice for probabilities at fixed lens\nseparations. However, for a more complete and useful study, one should\nmove through different lens separations with a much smaller sampling\nstep, so as to characterise the shapes and the volumes of the regions\nin a more accurate way. Furthermore, the final result would depend on\nthe assumed prior distribution function for the separations of binary\nsystems. Summing up, the study of the relative frequencies of the\ndifferent morphology classes is certainly one of the most interesting\ndirections opened up by our work, which deserves the greatest\nattention and an adequate space in dedicated future works.\n\nWe have only very briefly mentioned the existence of caustic-feature\nregions as ``meta regions'' to the iso-maxima regions, i.e. the\ncombination of all iso-maxima regions containing one specific,\ncaustic-related peak. Unfortunately, we have not yet found a good way\nto extract and preserve the information about these meta regions, but\nin fact they can provide a more fundamental understanding of the\nparameter space, since iso-maxima regions are basically just\n``stacks'' of caustic-feature regions. In contrast to iso-maxima\nregions, caustic-feature regions are smooth structures and, like the\ncaustics they are derived from, they change continuously over the\nparameter space. If their boundaries could be analytically derived\nfrom the caustic lines, an elegant automatic classification could be\nachieved.\n\nFinally, we must remember that our work is limited to the equal-mass\nstatic lens case. Higher order effect such as parallax and orbital\nmotion would dramatically increase the complexity of the\nclassification, adding very few new morphologies (at least in\nreasonable physical cases) and would mainly distort existing iso-maxima\nregions. The only really relevant and humanly achievable upgrade of\nour catalogue should include a variable mass ratio.\n\nWhile previous literature has shown only up to three pairs of caustic\ncrossings for a single microlensing caustic\n\\citep[e.g.][]{Cassan2010}, it is possible to draw a microlensing\ntrajectory experiencing five pairs of caustic crossings for a binary\nlens with a mass ratio slightly smaller than one, see\nFig.~\\ref{fig:fiveDCC}.\n\\begin{figure}\n  \\centering\n  \\input{figures\/five_dcc}\n  \\caption{Five pairs of double caustic crossings in one ten-maxima\n    light curve. This ``quintuple F-F'' morphology cannot be created\n    with an equal-mass binary.  The light curve is plotted as\n    magnification on a logarithmic scale in the range from 1 to 50\n    (dotted line at magnification 10), the time axis covers four\n    Einstein times. The small plot shows the corresponding caustic\n    topology and source trajectory (4.2 by 4.2 Einstein angles). The\n    caustic is computed with Caustic Finder by Schmidt, published 2008\n    at \\url{causticfinder.sourceforge.net}.}\n  \\label{fig:fiveDCC}\n\\end{figure}\nThis is a morphology that is not present in this equal-mass\ncatalogue. Indeed, most of the new morphologies appearing in\nunequal-mass binary microlensing would come from the close topology.\nOur plots of the iso-peak regions in the ($u_0,\\alpha$) space are done\nat fixed separation $s$. We can follow the evolution of iso-maxima\nregions with a variation of $s$ in different plots and it is easy to\nrefine the sampling in $s$ in order to catch all possible regions\narising only in limited ranges of $s$. The addition of a new parameter\nwould make the search much more complicated, as we should trace the\nevolution of iso-maxima regions in a two-dimensional space with the\ndanger that tiny intersections may escape a search with a too coarse\ngrid. New tricks are needed in order to carry out this search\nefficiently and without omissions. The equal-mass catalogue represents\na good basis for this exploration and an already powerful map for the\ncomprehension of microlensing of binary systems.\n\n\n\\section*{Acknowledgements}\n\\label{sec:ack}\nThis publication was supported by NPRP grant NPRP-09-476-1-78 from the\nQatar National Research Fund (a member of Qatar Foundation).\n\n\\bibliographystyle{mn2e_fix}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\ \\ \\  A classical problem in theory of minimal surfaces of  \n$\\mathbb{R}^{3}$ is to determine the number of the missing points on the \nimage of the Gauss map, in other words, if $M$ is a complete minimal surface \nand $G: M\\longrightarrow \\mathbb{S}^{2}$ is the Gauss map of $M$, how many \npoints  in $\\mathbb{S}^{2} \\backslash G(M)$ are there? Fujimoto showed that if \n$M$ is not the plane, then $\\sharp (\\mathbb{S}^{2}\\backslash G(M)) \\leqslant 4$ \nand is exactly $4$ for the Scherk's surface. Osserman, studied this problem \nunder the hypothesis that the total curvature is finite. In this case, he \nshowed that $\\sharp (\\mathbb{S}^{2}\\backslash G(M)) \\leqslant 3$.\n Recently, Jorge and Mercuri \\cite{kn:J-M} improved Osserman's estimate in the  \nfollowing theorem\n\\begin{thm}[Jorge, Mercuri]\\label{thm:thJM}\n If the Gauss map of a complete minimal surface of $\\mathbb{R}^3$ with finite \ntotal curvature misses $3$ or more points of the sphere then it is a plane.\n\\end{thm}\nThis estimative is sharp since catenoid's Gauss map omits exactly 2 points. \nSo the problem is fully answered for this class of surfaces.\nTheorem 1.1 follows from analysis of immersions with Gauss map missing three \npoints.\n\n A natural question arises: Which other class of surfaces have the same \nconformal type of the complex plane and Gauss map as meromorphic functions?\n \n\n \n \n \n As first choice we have the Bryant's Surfaces. Those surfaces share many \nproperties with minimal surfaces of $\\mathbb{R}^3$ and act like an analogue \nof minimal surfaces in hyperbolic 3-space. Indeed, Collin, Hauswirth and \nRosenberg \\cite{kn:C-H-R-1} proved an analogue of Osserman's result for Bryant's \nsurfaces, \nthat is, the hyperbolic Gauss map of a complete Bryant's surface with finite \ntotal curvature misses at most 3 points.\n \n Kawakami \\cite{kn:K} proved partial results for other classes of surfaces. He \nstudied \nthe hyperbolic Gauss map of algebraic Bryant Surfaces and CMC-1 faces in de \nSitter 3-space. Algebraic Bryant Surfaces are Bryant Surfaces with finite total \ncurvature dual (dual in the Umehara and Yamada's sense). CMC-1 faces are \nspacelike CMC-1 surfaces in de Sitter space with a specific kind of singularities. \nKawakami showed that hyperbolic Gauss map of a complete algebraic Bryant's \nSurfaces and of a complete CMC-1 face can miss 3 points at most, otherwise it is \nconstant.    \n \nIn this paper, we use Jorge and Mercuri's ideas to prove the following \ntheorem:\n\\begin{claim}\\label{th}\nLet $M$ be one of the following:\n\\begin{enumerate}\n \\item[(1)] a  complete Bryant surface with finite total \ncurvature or,\n\\item[(2)] a complete algebraic Bryant surface,\n\\item[(3)]  a complete CMC-1 face of finite type with elliptic ends.\n\\end{enumerate}\n  Then the hyperbolic Gauss map $G$ of $M$ \nomit at most 2 points, otherwise $G$ is constant. Further, in case of constant \nhyperbolic Gauss map $M$ is a  horosphere for items (1) and (2) and a \nhorosphere space like type surface for item (3).  \n\\end{claim}\nAll results in this theorem are sharp. \n \nA surface is {\\em parabolic} if positive harmonic functions are constant.  \n By \\cite{kn:J-Meeks} the volume growth of geodesics ball $B_r(x_o)$ of a \ncomplete \nminimal surface into $\\R^3$ with finite total curvature is of type $r^2$. In \n\\S5 we study the decay of curvature of those surfaces of theorem 1 and \nconcluded that all of them have $\\text{vol}_M(B_r)\\leq cr^2,\\ r\\geq r_0$ form \nsome constant $c_0$ and fixed $r_0>0.$ This implies next result.  \n\n\n\\begin{claim}\\label{th2}\n Let $M$ be one complete surface of type:\n\\begin{enumerate}\n \\item[(1)] minimal surface with finite total curvature  into $\\R^3$;\n \\item[(2)] Algebraic Bryant surface with finite total curvature;\n \\item[(3)] Algebraic Bryant surface endowed with the dual metric;\n \\item[(4)] CMC-1 Face surface of finite type, elliptic ends endowed with the lift metric.\n\\end{enumerate}\nThen $M$ is parabolic. If $M$ has a K\\\"ahler structure then all bounded \nholomorphic map $g\\colon M\\to\\pc$ is constant. \n\\end{claim}\n\\begin{rem}\n All surfaces in items (1),\\ (2),\\ (3),\\ (4), are isometric to a complete \nminimal surface $M$ into $\\R^3$ with finite total curvature. Then they have a \nnatural K\\\"ahler structure. If the Gauss map $G$ of $M$ miss at least one point \n$y\\in\\sph^2$  we can rotated $M$ inside $\\R^3$ to making $y=(0,\\ 0,\\ 1).$ If \n$g\\colon M\\to\\pc$ is the stereographic projection of $G$ then $g$ is \nholomorphic. Hence all surfaces described in the theorem \\ref{th2} are K\\\"ahler. \n Unfortunately, if $g(M)=\\pc\\setminus\\{a,\\ b\\}$ we can not do a lifting of $g$ \nto the unit disk $\\D$ by a modular function ot get a easy proof of theorem 1.1 \nunless $M$ is simply connected.      \n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\\section{Hyperbolic Space}\n\n \n\\ \\ \\  Let $\\mathbb{L}^{4}$ be Lorentz-Minkowski 4-space with Lorentzian metric\n \n\\begin{equation}\\label{HS1}\n  \\langle (x_{0}, x_{1}, x_{2}, x_{3}), (y_{0}, y_{1}, y_{2}, y_{3}) \\rangle = \n-x_{0}y_{0} + x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}\n\\end{equation}\n \n Define the hyperbolic $3$-space \n \n\\begin{eqnarray*}\n \\mathbb{H}^{3} = \\{ v \\in \\mathbb{L}^{4} | \\langle v, v\\rangle = -1, x_{0}(v) \n> 0  \\} \n\\end{eqnarray*}\n  \nwith the induced metric from $\\mathbb{L}^{4}$. Give to $\\mathbb{H}^{3}$ the \norientation which $v_{1}, v_{2},v_{3}$ forms a oriented base of \n$T_{v}\\mathbb{H}^{3}$ if, and only if, $v, v_{1}, v_{2},v_{3}$ forms a base of \n$\\mathbb{L}^{4}$. We have that $ \\mathbb{H}^{3}$ is a Riemannian manifold \n3-dimensional simply connected with constant mean curvature $-1$. \n \n The space $\\mathbb{H}^{3}$ is not compact, but can be compacted by adding a \n\"sphere at infinity\" $\\mathbb{S}_{\\infty}^{2}$ in such way that the rigid \nmotions of $\\mathbb{H}^{3}$ extend to homeomorphisms of \n$\\overline{\\mathbb{H}^{3}} = \\mathbb{S}_{\\infty}^{2} \\cup \\mathbb{H}^{3}$. \n \n\n Identify $ \\mathbb{L}^{4}$ with the set $Herm(2) = \\{X^{*} = X \\}$ of $2 \n\\times 2$ hermitian matrices in the follow way\n \n \\begin{equation}\\label{HS2}\n  (x_{0}, x_{1}, x_{2}, x_{3}) \\mapsto \\left[\\begin{array}{c c }\n                                            x_{0} + x_{3}&x_{1} + ix_{2}\\\\      \n                                     \n                                            x_{1} - ix_{2}&x_{0} - x_{3}\n                                            \\end{array}\\right]\n \\end{equation}\n\n where $i = \\sqrt{-1}$.\n \n With this identification, we can see $ \\mathbb{H}^{3}$ as $$ \\mathbb{H}^{3} = \n\\{ XX^{*} | X \\in SL(2, \\ \\mathbb{C}) \\} \\ \\ \\ \\ \\ X^{*} = \\overline{X}^{t}$$ \nwith the metric \n $$\\langle X, Y \\rangle = - \\frac{1}{2}tr(X,\\tilde{Y}), \\ \\langle X, X\\rangle = \n-det(X)$$ where $\\tilde{Y}$ is the cofactor of $Y$. \n $PSL(2, \\mathbb{C}) = SL(2, \\mathbb{C})\/\\{\\pm id\\}$ acts isometrically on \n$\\mathbb{H}^{3}$  by \n \n \\begin{equation}\\label{HS3}\n  X \\mapsto YXY^{*} , \\ X \\in \\mathbb{H}^{3} \\, \\ Y \\in PSL(2, \\mathbb{C})\n \\end{equation}\n \n Note that the map $PSL(2, \\mathbb{C}) \\longrightarrow Herm(2)$ such that $F \n\\mapsto FF^{*}$ take values in $\\mathbb{H}^{3}$.    \n\n\\subsection{Bryant Surfaces}\n\n\\ \\ \\ Surfaces of constant mean curvature (CMC) are objects of great interest \nbecause they are critic points of functional area with respect to volume \npreserving variations that fix their boundaries. Minimal surfaces are a special \nclass of CMC surfaces that are critical for all variations, not just volume \npreserving ones. Beside that, minimal surfaces can be described by a pair of \nholomorphic functions, called Weierstrass data. Many properties of minimal \nsurfaces can be obtained using this Weierstrass data.\n\nIn 1970, Lawson described a correspondence between minimal surfaces and CMC-1 \nin $\\mathbb{H}^{3}$, the \"Lawson's correspondence\". So, as the minimal surfaces \nare described by a pair of holomorphic function, the Weierstrass data, one may \nask if CMC-1 surfaces in $\\mathbb{H}^{3}$ can be described in a similar way. \nSuch question was answered by Robert Bryant \\cite{kn:B} in 1987. He showed that this \nclass of surfaces can be described by a pair of holomorphic functions, known as \nthe \\textit{Bryant data}. \n \n\\begin{thm}[Bryant Representation]\\label{thBriantRep}\n Let $M$ be a Riemann's surface and $F: M \\longrightarrow SL(2, \\mathbb{C})$ \nconformal immersion such that$det(F^{-1}dF) = 0$. Let $\\phi: M \\longrightarrow \n\\mathbb{H}^{3}$ be such that $\\phi = FF^{*}$. So $\\phi$ is a immersion from $M$ \nto $\\mathbb{H}^{3}$ with constant mean curvature 1. Conversely, if $\\phi: M \n\\longrightarrow \\mathbb{H}^{3}$ is a immersion with constant mean curvature 1, \nthere is a holomorphic lift $\\phi$ to universal recovery $\\tilde{F}: \\tilde{M} \n\\longrightarrow SL(2, \\mathbb{C})$, such that $det(F^{-1}dF) = 0$\n and $\\phi = \\tilde{F}\\tilde{F}^{*}$.\n \n\\end{thm}\n\nBecause of this work, the CMC-1 surfaces in hyperbolic 3-space became known as \n\\textit{Bryant Surfaces}.\n\n\\begin{rem}\nA holomorphic map $F: M \\longrightarrow SL(2, \\mathbb{C})$, such that \n$det(F^{-1}dF) = 0$ is called \\textit{holomorphic null immersion}.\n\\end{rem}\n\n\\begin{rem}\nLet $F: M \\longrightarrow SL(2, \\mathbb{C})$ be a map such that  $det(F^{-1}dF) \n= 0$. Locally, we can write:\n\n  \\begin{equation}\\label{BR1}\n  F^{-1}dF =  \\left[\\begin{array}{c c }\n                     g & -g^{2}\\\\                                           \n                     1 & -g\n                     \\end{array}\\right] \\omega\n \\end{equation}\n \nWhere $g$ is a meromorphic function and $\\omega$ is a holomorphic 1-form. \nIndeed, write\n \n   \\begin{equation}\\label{BR2}\n  F(z) =  \\left[\\begin{array}{c c }\n                     A(z) & B(z)\\\\                                           \n                     C(z) & D(z)\n                     \\end{array}\\right] \\in SL(2, \\mathbb{C})\n \\end{equation}\n \n Where $z$ is conformal local coordinate and $A, B, C, D$ are holomorphic. So, \ntake $$g = - \\frac{dB}{dA}, \\ \\omega = AdC - CdA$$\n  \n\\end{rem}\n\n As $F$ is holomorphic, the forms $\\omega$ e $g^{2}\\omega$ are also \nholomorphic, and  $F^{-1}dF$ never vanish, by the fact that $F$ is a immersion. \nTherefore poles of $g$ are zeros of $\\omega$, and a pole of order $k$ of $g$is a \nzero of order $2k$ of $\\omega$. The pair $(g, \\omega)$ is the Bryant data \nassociated to $F$.\n \n\\begin{rem}\n We can write the holomorphic 1-form $\\omega$, in local coordinates, as $\\omega \n= f(z)dz$. So, we can write the Bryant data as $(g, f)$.\n\\end{rem} \n \n Let $F$ be  a immersion of $M$ in $SL(2, \\mathbb{C})$ such that \n \n\\begin{equation}\\label{BR3}\n F^{-1}dF =  \\left[\\begin{array}{c c }\n                     g & -g^{2}\\\\                                           \n                     1 & g\n                     \\end{array}\\right] \\omega \n\\end{equation}\n\nBy Bryant's representation theorem and Lawson's correspondence, if $(g, f)$ is \na Bryant data of a Bryant surface\nso the metric and Gaussian curvature can be described using $g$ and $F$ by. \n\n$$ds^{2} = (1 + |g|^{2})^{2}|f|^{2}dz^{2}$$\n\n$$\\kappa(z) = \\frac{-4|g'|^2}{|f|^2(1+|g|^2)^4} $$\n\nNote that these are the same expressions for the Weierstrass data and minimal \nsurfaces. \n\n\\begin{rem}\n Note That the Weierstrass representation can be obtained as the limit of \nBryant's theorem by collapsing the Lie group $SL(2, \\mathbb{C})$ in the \nabelian group $\\mathbb{C}^{3}$. \n\\end{rem}\n\n\\subsection{Hyperbolic Gauss Map}\n\nIn 1986, Epstein introduced the hyperbolic Gauss map while he studied immersed \nsurfaces in the hyperbolic 3-space in Poincar\u00e9 model. Later, Bryant \\cite{kn:B} \nreintroduced the concept in the hyperboloid of Minkowski space model, in the \nfollowing way:\n\n\\begin{defn}\\label{HGMdef}\n Let $M$ be a immersed surface in the hyperbolic 3-space. Define the \n\\textit{hyperbolic Gauss Map} \n $G:M \\longrightarrow \\mathbb{S}_{\\infty}^{2}$ in the following way:\nLet $z \\in M$, take $\\gamma$ normal oriented geodesic starting from $z$. Define \n$G(z)$ as the point where $\\gamma$\nintercepts the ideal boundary $\\mathbb{S}_{\\infty}^{2}$ of $\\mathbb{H}^{3}$. In \nother words $$G(z) = \\lim\\limits_{t \\rightarrow \\infty} \\gamma (t)$$\n\\end{defn}\n\nFor Bryant surfaces we have the follow characterization. Let $M$ be a Bryant \nsurface and $F:M \\longrightarrow SL(2, \\mathbb{C})$ the conformal immersion \ngiven by theorem $2.1$. Denote \n\n\\begin{eqnarray}\\label{HGM1}\n F(z) = \\left[\\begin{array}{c c}\n                     A(z) & B(z)\\\\                                           \n                     B(z) & D(z)\n                     \\end{array}\\right]\n\\end{eqnarray}\n\nthen,\n\n\\begin{eqnarray}\n G(z) = \\frac{dA}{dC} = \\frac{dB}{dD}\\label{HGM2}\n\\end{eqnarray}\n\nFollowing the terminology introduced by Umehara and Yamada,  $g(z) =  - \n\\frac{dB}{dA}$ is called \\textit{Secondary Gauss Map}. Note that the secondary \nGauss map is the map $g$ from Bryant data $(g, \\omega)$.\nthe motivation for the name comes from the fact that in a minimal surface \nimmersed in $\\mathbb{R}^{3}$ with  Weierstrass data $(\\tilde{g}, \\tilde{f})$ \nthe map $\\tilde{g}$ can be seen as the composition of Gauss map with \nstereographic projection. So, the map $\\tilde{g}$ has 2 fundamentals roles: \n\n\\begin{enumerate}\n \\item[(i)] Describe the metric. \n \\item[(ii)] Describe the stereographic projection of unit normal vector. \n\\end{enumerate}\n\nBut in the case of Bryant data $(g, f)$, the roles $(i)$ e $(ii)$ are played by \n2 different maps, $g$ plays the role $(i)$ and $G$ plays the role $(ii)$. \nTherefore, we can think that the Bryant representation has 2 Gauss Maps, the \nHyperbolic Gauss Map and the Secondary Gauss Map.\n\nUmehara e Yamada showed the following relationship between the Hyperbolic \nGauss Map and Secondary Gauss Map.\n\n\\begin{eqnarray*}\n S(g) - S(G) = 2Q\n\\end{eqnarray*}\n\nwhere\n\n\\begin{eqnarray*}\n S(g) &=& S_{z}(g)dz \\\\\n S_{z}(g) &=& \\left( \\frac{g''}{g'} \\right) ' - \\frac{1}{2} \\left( \n\\frac{g''}{g'} \\right) ^{2} \\\\\n Q &=& \\omega dg\n\\end{eqnarray*}\n\n$S_{z}(g)$ is the \\textit{Schwarzian derivate} and $Q$ is the \\textit{Hopf's \ndifferential}.\n\n\n\\subsection{Finite Total Curvature}\n\n\\ The basic tools in the modern theory of minimal immersions $\\phi: M^{2} \n\\longrightarrow \\mathbb{R}^{3}$ was establish by Osserman \\cite{kn:O}. In this \npaper he shows  the following  results about complete minimal surfaces $M$ of \n$\\mathbb{R}^3$ with finite total curvature:\n\\begin{enumerate}\n \\item There is a compact Riemann surface $\\overline{M}$ of finite genus $\\mu$ \nand  a finite set $E=\\{p_1,\\cdots,p_k\\}\\subset\\overline{M}$ such that $M$ is \nconformal to $\\overline{M}\\setminus E.$\n\\item The Gauss map $G\\colon M\\to\\mathbb{S}^2$ extends to a conformal branched \ncovering map $G\\colon \\overline{M}\\to\\mathbb{S}^2$.\n\\item $\\sharp\\big(\\mathbb{S}^2\\setminus G(M)\\big)\\leq 3$ unless $M$ is a flat \nplane.\n\\end{enumerate}\nThe points $p_{i} \\in E$ or some times one neighborhood of them are known as \n\\textit{ends of $M$.} In the hyperbolic context those items have similar \nresults with fill exception. \n\n\\begin{thm}[Bryant, \\cite{kn:B}]\\label{thBryant}\n Let $M$ a complete Bryant surface with finite total curvature immersed in \n$\\mathbb{H}^{3}$. Then there is a compact Riemann surface $\\overline{M}$ and a \nfinite set $E = \\{p_{1}, \\ ... \\ , p_{n} \\} \\subset \\overline{M}$ such that $M$ \nis conformal to \n $\\overline{M} \\setminus E$.\\end{thm}\n\\begin{thm}[Collin, Hauswirth, Rosenberg, \\cite{kn:C-H-R-1}]\\label{thCHR1}\n Let $M$ be a immersed complete Bryant surface with finite total curvature in \n$\\mathbb{H}^{3}$. Then the hyperbolic Gauss map $G$ of $M$ omit at most \n$3$ points unless $G$ is constant and $M$ is a horosphere.\n\\end{thm} \n \n \n \n \n\n The hyperbolic Gauss map is holomorphic but does not necessarily has \nmeromorphic extension to the ends. In fact some ends could be an essential \nsingularity of the Gauss map and motivated the next definition. \n\n\\begin{defn}\nLet $M$ be a complete Bryant surface with finite total curvature and $E = \n\\{p_{1}, \\ ... \\ , p_{n} \\}$ the ends of $M$. We say that an end $p_{i}$ is \n\\textit{regular} if the hyperbolic Gauss map does extend meromorphically to \n$p_{i}$. Otherwise the end $p_{i}$  is called \\textit{irregular}.\n\\end{defn}\n\nThe existence of irregular ends is the first big difference between minimal \nsurfaces and Bryant surfaces.\n\n\n\n\\subsection{Dual Surfaces}\n\n\\ \\ \\ It is well known that a minimal surface $M$ with finite total \ncurvature immersed in $\\mathbb{R}^{3}$ satisfies the Osserman's inequality\n\n\\begin{eqnarray}\n \\frac{1}{2 \\pi} \\int _{M} KdA \\leq (\\chi (M) - n)\\label{SD1}\n\\end{eqnarray}\n\nwhere $K$ is the Gaussian curvature of $M$ and $n$ is the number of ends of the \nsurface. For Bryant surfaces with finite total curvature, there is no analogue  \nrelation. In fact, Umehara and Yamada \\cite{kn:U-Y-2} showed that Bryant surfaces satisfy \nonly the Cohn-Vossen inequality\n\n\\begin{eqnarray}\\label{SD2}\n  \\frac{1}{2 \\pi} \\int _{M} KdA < \\chi (M)\n\\end{eqnarray}\n\n\\ \\ \\ Motivated by this fact, Umehara and Yamada \\cite{kn:U-Y-2} introduced the concept of \ndual surface, in a way that if $\\phi : M^{2} \\longrightarrow \\mathbb{H}^{3}$ is \na CMC-1 immersion and $\\phi^{\\sharp} : \\tilde{M}^{2} \\longrightarrow \n\\mathbb{H}^{3}$ is the dual immersion, then $M$ satisfies an analogue of \nOsserman's inequality, but in terms of dual surface. \n\n\\ \\ \\ Let $\\phi: M \\longrightarrow \\mathbb{H}^{3}$ be a complete CMC-1 \nimmersion with finite total curvature. Let $G$ be the hyperbolic Gauss map, \n$(g,\\omega)$ its Bryant data and $Q = \\omega dg$ the Hopf differential of \n$\\phi$. \n\n\\begin{defn}\\label{SDdef}\n Define the dual CMC-1 immersion $\\phi^{\\sharp}: \\tilde{M} \\longrightarrow \n\\mathbb{H}^{3}$ associated to Bryant data $(g, \\omega)$ of $\\phi$ by\n \n \\begin{eqnarray*}\n\\phi^{\\sharp} = (F^{\\sharp})(F^{\\sharp})^{*}  \n \\end{eqnarray*}\n\nwhere $F$ satisfies $(4)$, $F^{\\sharp}$ is the inverse of the matrix $F$ and \n$\\tilde{M}$ is the universal cover of $M$.\n\\end{defn}\n\n\\begin{rem}\nObserve that $\\phi^{\\sharp}$ is not necessarily \\textit{single-valued} in $M$. \nBut is easy see that $\\phi^{\\sharp}$ is \\textit{single-valued} in $M$ if, and \nonly if, $g$ is \\textit{single-valued} in $M$.\n\\end{rem}\n\nLet $(g^{\\sharp}, \\omega^{\\sharp})$ be the pair defined by\n\n\\begin{equation}\\label{SD3}\n  (F^{\\sharp})^{-1}dF^{\\sharp} =  \\left[\\begin{array}{c c }\n                     g^{\\sharp} & -(g^{\\sharp})^{2}\\\\                           \n                \n                     1 & -g^{\\sharp}\n                     \\end{array}\\right] \\omega^{\\sharp}\n \\end{equation}\n\nthus, taking $(\\phi^{\\sharp})^{\\sharp}$, we get a map \n$(F^{\\sharp})^{\\sharp}: \\tilde{\\tilde{M}} \\longrightarrow \\mathbb{H}^{3}$ such \nthat, $(F^{\\sharp})^{\\sharp} = (F^{-1})^{-1} = F$, this shows that \n$(\\phi^{\\sharp})^{\\sharp} = \\phi$.\n\nNow we show a important result due to Umehara and Yamada \\cite{kn:U-Y-2} that shows the \nrelationship between the Bryant data and its dual data.\n\n\\begin{prop}[Umehara, Yamada]\\label{thUY}\n Let $\\phi^{\\sharp}$ be the dual immersion of the CMC-1 immersion $\\phi$ of a \nsurface $M$ in a hyperbolic 3-space. Let $(g, \\omega)$ be the Bryant data of \n$\\phi$. So, $\\phi^{\\sharp}$ is a CMC-1 immersion of $M$ in the hyperbolic \n3-space, with hyperbolic Gauss map $G^{\\sharp}$, Bryant data $(g^{\\sharp}, \n\\omega^{\\sharp})$ and Hopf differential $Q^{\\sharp}$  $\\phi^{\\sharp}$ given \nby:\n \n \\begin{eqnarray}\n  G^{\\sharp} = g \\ , \\ g^{\\sharp} = G \\ , \\ \\omega^{\\sharp} = - \\frac{Q}{dG} \\ \n, \\ Q^{\\sharp} = - Q\n \\end{eqnarray}\n\n\\end{prop}\n\n\\begin{rem}\n Observe that the dual surface changes the hyperbolic Gauss map with the \nsecondary Gauss map of $M$.\n\\end{rem}\n\nIf $\\phi$ is a CMC-1 immersion in hyperbolic 3-space and $\\phi^{\\sharp}$ its \ndual immersion. By the above theorem, $\\phi^{\\sharp}$ is a CMC-1 immersion \nalso, so it admits a Bryant data $(g^{\\sharp}, \\omega^{\\sharp})$. Thus the \nmetric of the dual immersion is given by\n \n\\begin{eqnarray}\n ds^{\\sharp 2} &=& (1 + |g^{\\sharp}|^{2})^{2}\\omega^{\\sharp} \n\\overline{\\omega^{\\sharp}} \\\\\n               &=& (1 + |G|^{2})^{2}\\frac{Q}{dG} \\overline{(\\frac{Q}{dG})}  \n\\end{eqnarray}\n\nSuch metric is called \\textit{dual metric}.\n\n\\begin{rem}\n Observe that the metric $ds^{\\sharp 2}$ is well defined and \\textit{single \nvalued} in $M$. So, make sense take $M$ with the metric $ds^{\\sharp 2}$, such \nsurface is a Bryant surface with Bryant data $(G, \\frac{-Q}{dG})$.\n\\end{rem}\n\nThere is a relation between the metric $ds^{2}$ and the dual metric $ds^{\\sharp \n2}$ \\cite{kn:Z}\n\n\\begin{prop}[Yu]\\label{thYu}\nThe dual metric $ds^{\\sharp 2}$ is complete (resp. non degenerated) if, and \nonly if, the metric $ds^{2}$ is complete (resp. non degenerated).\n\\end{prop}\n\n\n\n\\subsubsection{Algebraic Bryant Surfaces}\n\n\nAs the dual metric is well defined in $M$, we have the following definition:\n\n\\begin{defn}\\label{SDdefDualTC}\nLet $M$ be a Bryant surface. Define the \\textit{dual total curvature} of $M$, \nas  \n\n\\begin{eqnarray}\nK_{T}^{\\sharp} = \\int _{M} - K^{\\sharp}dA^{\\sharp} .\\label{SD4}\n\\end{eqnarray}\n\nWhere $K^{\\sharp}$ and $dA^{\\sharp}$ are the Gaussian curvature and the area \nelement of the surface $M$ with the dual metric.\n\n\\end{defn}\n\nObserve that the dual total curvature is the area of $M$ with respect to the \n(singular) metric induced by Fubini-Study metric in $\\mathbb{C}\\mathbb{P}^{1}$.\n\n\\begin{defn}\\label{SDdefFTC}\n Let $M$ be a Bryant surface. We say that $M$ is \\textit{algebraic} if $M$ has \nfinite dual total curvature.\n\\end{defn}\n\nThe algebraic Bryant surfaces satisfy the following theorem\n\n\\begin{thm}[Bryant, Huber, Z. Yu]\\label{thBHYu}\nLet $M be$ a algebraic Bryant surface. So:\n \n \\begin{enumerate}\n  \\item[(i)] $M$ is biholomorphic to $\\overline{M_{\\gamma}} \\setminus E$, where \n$\\overline{M_{\\gamma}}$ is a  closed surface of genus $\\gamma$ and  $E \\subset \n\\overline{M_{\\gamma}}$ is a finite set $E = \\{p_{1} , ... , p_{m} \\}$.\n  \\item[(ii)] the dual Bryant data $(G, \\omega^{\\sharp})$ extends \nmeromorphically to $\\overline{M_{\\gamma}}$.\n \\end{enumerate}\n\nThe points of set $E$ are the \\textit{ends} of $M$. \n\\end{thm}\n\nUsing this, Umehara and Yamada \\cite{kn:U-Y-2} deduced an analogue to Osserman's \ninequality to dual surfaces. Explicitly they proved that\n\n\\begin{thm}[Umehara, Yamada]\n Let $M$ be a Riemann surface and $\\phi:M \\longrightarrow \\mathbb{H}^{3}$ a \nCMC-1 complete conformal immersion with finite dual total curvature. Let \n$\\phi^{\\sharp} : {M}^{2} \\longrightarrow \\mathbb{H}^{3}$ be the dual immersion. \nthen,\n\n\\begin{eqnarray}\n \\frac{1}{2 \\pi} \\int _{M} K^{\\sharp}dA^{\\sharp} \\leq (\\chi (M) - n) \\label{SD5}\n\\end{eqnarray}\n\nWhere $\\tilde{K}$ and $dA^{\\sharp}$ the dual Gaussian curvature and the dual \narea element, and $n$ is the number of ends of the original surface $M$. \n \n\\end{thm}\n\nDue o this properties, one may ask if there is a Picard Type theorem for the \nhyperbolic Gauss map of algebraic Bryant surfaces too. Indeed, some partial \nresults were obtained by Kawakami \\cite{kn:K}.\n\n\n\\section{De Sitter 3-Space}\n\n\\ \\ \\ In this section we study the \\textit{CMC-1 faces}. This CMC-1 faces are \nspacelike CMC-1 surfaces in de Sitter 3-space with some kind of singularities. \nSuch surfaces share a many properties with Bryant surfaces, in particular, they \nhave an analogue for the Bryant representation. For CMC-1 faces, is possible to \ndefine a hyperbolic Gauss map in a similar way we did in surfaces immersed in \nhyperbolic 3-space, so we can estimate the number of omitted points in the \nimage of this map. In this work we show a sharp estimative for this number.\n\n\nLets give a brief description of \\textit{de Sitter} space.\n\nLet $\\mathbb{L}^{4}$ be the Lorentz-Minkowski 4-space with Lorentz metric\n \n \\begin{equation}\\label{DeS1}\n  \\langle (x_{0}, x_{1}, x_{2}, x_{3}), (y_{0}, y_{1}, y_{2}, y_{3}) \\rangle = \n-x_{0}y_{0} + x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}\n \\end{equation}\n \nDefine the \\textit{de Sitter} 3-space $\\mathbb{S}_{1}^{3}$ as\n\n \\begin{eqnarray*}\n \\mathbb{S}_{1}^{3} = \\{ v \\in \\mathbb{L}^{4} | \\langle v, v\\rangle = 1\\} \n \\end{eqnarray*}\n\nwith the induced metric $\\mathbb{L}^{4}$. $\\mathbb{S}_{1}^{3}$ is a \n3-dimensional Lorentz manifold  simply connected with constant sectional \ncurvature 1.\n\nWe identify $\\mathbb{L}^{4}$ with the set of hermitian matrices $2 \\times 2$  \n$Herm(2) = \\{X^{*} = X \\}$ by\n\n \\begin{equation}\\label{DeS2}\n  (x_{0}, x_{1}, x_{2}, x_{3}) \\mapsto \\left[\\begin{array}{c c }\n                                            x_{0} + x_{3}&x_{1} + ix_{2}\\\\      \n                                     \n                                            x_{1} - ix_{2}&x_{0} - x_{3}\n                                            \\end{array}\\right]\n \\end{equation}\n\n where $i = \\sqrt{-1}$.\n \nWith this identification, we can define\n \n \\begin{equation}\\label{DeS3}\n  e_{0} = \\left[\\begin{array}{c c }\n                       1 & 0 \\\\                                           \n                       0 & 1\n                \\end{array}\\right] \\ , \\ e_{1} = \\left[\\begin{array}{c c }\n                                                        0 & 1 \\\\                \n                           \n                                                        1 & 0\n                                                       \\end{array}\\right]  \\ , \n\\ e_{2} = \\left[\\begin{array}{c c }\n                                                                                \n                      0 & i \\\\                                           \n                                                                                \n                      -i & 0\n                                                                                \n                \\end{array}\\right]   \\ , \\ e_{3} = \\left[\\begin{array}{c c }\n                                                                                \n                                                                  1 & 0 \\\\      \n \n                                    \n                                                                                \n                                                                  0 & -1\n                                                                                \n                                                           \\end{array}\\right]\n \\end{equation}\n \n Thus, \n \n \\begin{eqnarray*}\n  \\mathbb{S}_{1}^{3} & = & \\{X | X = X^{*} \\ , \\ det(X) = -1 \\} \\\\\n                     & = & \\{Fe_{3}F^{*} | F \\in SL(2, \\mathbb{C}) \\}\n \\end{eqnarray*}\n\nwith the metric $$\\langle X, Y \\rangle = - \\frac{1}{2} tr(Xe_{2}(Y^{t})e_{2}).$$\n \nWhere $X^{*} = \\overline{X^{t}}$.\n \n\\begin{rem}\nObserve that with above definition $\\langle X, X \\rangle = - detX$.\n\\end{rem}\n \n\\subsection{CMC-1 Face}\nIn this section, we define CMC-1 faces, enumerate some important results and at \nlast we prove a Picard type theorem for this surfaces. \n \n\\begin{defn}\\label{CMC1Facedef}\nA immersion $f: M^{2} \\longrightarrow \\mathbb{S}_{1}^{3}$ of a surface $M$ is \ncalled \\textit{spacelike} if the induced metric in $M$ is positive definite.\n\\end{defn}\n\nAs an analogue for the Bryant surfaces, we have the following theorem \\cite{kn:A-A}:\n \n\\begin{thm}[Aiyama-Akutagawa]\\label{thAiAk}\nLet $D$ be a simply connected domain in $\\mathbb{C}$ and $z_{0} \\in D$ a base \npoint. Let \n  \n  $$g:D \\longrightarrow (\\mathbb{C} \\cup \\{\\infty \\}) \\setminus \\{z \\in \n\\mathbb{C} | |z| \\leq 1 \\} $$\n \nbe a meromorphic function and $\\omega$ a holomorphic 1-form in $D$ such that\n \n $$d\\hat{s}^{2} = (1 + |g|^{2})^{2}\\omega \\overline{\\omega}$$\n \nis a Riemannian metric in $D$.\n \nTake $F = (F_{ij}) : D \\longrightarrow SL(2, \\mathbb{C})$ holomorphic immersion \nsuch that $F(z_{0}) = e_{0}$ and\n \n\\begin{equation}\\label{DeS4}\n  F^{-1}dF =  \\left[\\begin{array}{c c }\n                     g & -g^{2}\\\\                                           \n                     1 & -g\n                     \\end{array}\\right] \\omega\n\\end{equation}\n\nthen $f: D \\longrightarrow \\mathbb{S}_{1}^{3}$ defined by\n$$f = Fe_{3}F^{*}$$\n\nis a conformal spacelike immersion, with constant mean curvature 1.\n\nThe induced metric $ds^{2}$ in $D$, satisfies\n\n$$ds^{2} = (1 - |g|^{2})^{2}\\omega \\overline{\\omega}$$\n \nConversely, every CMC-1 immersion of a simply connected surface has this form.\n \\end{thm}\n\n\\begin{rem}\n The pair $(g, \\omega)$ is called \\textit{Aiyama data}.\n\\end{rem}\n\nFollowing Umehara and Yamada definitions for CMC-1 immersions in \n$\\mathbb{H}^{3}$, we define\n\n\\begin{defn}\nLet $f:D \\longrightarrow \\mathbb{S}_{1}^{3}$ and $F = (F_{ij})$ as above. \nDefine the \\textit{hyperbolic Gauss Map} $G$ of $f$ as\n \n $$ G = \\frac{dF_{11}}{dF_{21}} = \\frac{dF_{21}}{dF_{22}}. $$\n \nWe define also the \\textit{Hopf differential} $Q$ of immersion by \n \n $$Q = \\omega dg.$$\n \n\\end{defn}\n\n\\begin{rem}\nThe hyperbolic Gauss map has the following meaning. Let \n$\\mathbb{S}_{\\infty}^{2}$ be the pointing future ideal boundary ideal of \n$\\mathbb{S}_{1}^{3}$. $\\mathbb{S}_{\\infty}^{2}$ can be identified with \n$\\mathbb{C} \\cup \\{ \\infty \\}$. So, given $z \\in D$, take $\\gamma$ geodesic in \n$\\mathbb{S}_{1}^{3}$ with initial velocity  as the unity normal vector of \n$f(D)$ in $f(z)$. Then $G(z)$ is the point where $\\gamma$ intercepts the ideal \nboundary $\\mathbb{S}_{\\infty}^{2}$\n\\end{rem}\n\n\\begin{rem}\nObserve that given $f: D \\longrightarrow \\mathbb{S}_{1}^{3}$ with $f = \nFe_{3}F^{*}$, then $\\hat{f}: D \\longrightarrow \\mathbb{H}^{3}$ given by \n$\\hat{f} = FF^{*}$ is a conformal CMC-1 immersion, with metric $d\\hat{s}^{2}$ \nand the same hyperbolic Gauss map $G$ and Hopf differential $Q$ of $f$.\n\\end{rem}\n\n\\begin{defn}\nLet $M$ be a oriented surface. A smooth map $f: M \\longrightarrow \n\\mathbb{S}_{1}^{3}$ is called CMC-1 map if there is an open dense set $W \n\\subset M$ such that $f|_{W}$ is a spacelike immersion. A point $p \\in M$ is \ncalled a singular point of $f$ if the indexed metric $ds^{2}$ is degenerated in \n$p$.\n\\end{defn}\n\n\\begin{defn}\nLet $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a CMC-1 map and $W \\subset M$ \nan open dense set such that $f|_{W}$ is a CMC-1 immersion. A point $p \\in M \n\\setminus W$ is an \\textit{admissible singular point} if:\n \n \\begin{enumerate}\n  \\item[(1)] There is a map $\\beta : U \\cap W \\longrightarrow \\mathbb{R}^{+}$ \nof class $C^{1}$, where $U$ is a neighborhood of $p$, such that $\\beta ds^{2}$ \nextends to a Riemannian metric in $U$ of class $C^{1}$.\n  \\item[(2)] $df(p) \\neq 0$, that is, $df$ has rank 1 in $p$.\n \\end{enumerate}\n\nWe say that a CMC-1 map $f$ is a CMC-1 face if all its singular points are \nadmissible.\n\\end{defn}\n\n\\begin{prop}\\label{DeS-prop1}\nLet $M$ be a oriented surface and $f: M \\longrightarrow \\mathbb{S}_{1}^{3}$ 0a \nCMC-1 face where $W \\subset M$ is the open dense such that $f|_{W}$ is a CMC-1 \nimmersion. Then there is a only one complex structure $J$ on $M$ such that\n \n\\begin{enumerate}\n \\item[(1)] $f|_{W}$ is conformal with respect to $J$.\n \\item[(2)] There is a immersion $F: \\tilde{M} \\longrightarrow SL(2, \n\\mathbb{C})$ which is holomorphic with respect to $J$, such that \n \n $$det(dF) = 0 \\ e \\ f o \\varrho = Fe_{3}F^{*}$$\n \nWhere $\\varrho: \\tilde{M} \\longrightarrow M$ is the universal cover of $M$.\n\\end{enumerate}\n\n \n$F$ is called \\textit{holomorphic null lift}.\n \n\\end{prop}\n\n\\begin{rem}\nThe holomorphic null lift is unique except of right multiplication for a \nconstant matrix in $SU(1, 1)$.\n\\end{rem}\n\nBy the above proposition, given a CMC-1 face $f:M \\longrightarrow \n\\mathbb{S}_{1}^{3}$ , always exists a complex structure $J$ in $M$. \nHenceforward, $M$ will be treated as a Riemann surface with this complex \nstructure.\n\n\\begin{prop}\\label{DeS-prop2}\nLet $M$ be a Riemann surface and $F: M \\longrightarrow SL(2, \\mathbb{C})$ a \nholomorphic null immersion. Assume that the symmetric $(0, 2)$-tensor \n$det[d(Fe_{3}F^{*})]$ is not identically zero. Then \n \n $$f = Fe_{3}F^{*} : M \\longrightarrow \\mathbb{S}_{1}^{3}$$\n \nis a CMC-1 face, and a point $p \\in M$ is a singular point of $M$ if, and only \nif, $det[d(Fe_{3}F^{*})]_{p} = 0$. Beside that, $ - det[d(FF^{*})]$ is positive \ndefinite on $M$.\n\\end{prop}\n\n\\begin{rem}\nThe hypothesis of $det[d(Fe_{3}F^{*})] \\neq 0$ is essential. In fact, take $F: \nM \\longrightarrow \\mathbb{S}_{1}^{3}$ holomorphic null immersion such that \n$$F(z) = \\left[\\begin{array}{c c }\n                     z + 1 & - z\\\\                                           \n                      z    & - z + 1\n                     \\end{array}\\right] $$\n                     \nSo, $f = Fe_{3}F^{*}$ degenerates in every point of $\\mathbb{C}$, therefore \ndoes not provides a immersion. It comes from the fact that \n$det[d(Fe_{3}F^{*})]$ \nis identically zero.\n\\end{rem}\n\nUsing the above theorems, is possible to extend Aiyama-Akutagawa representation \nto CMC-1 faces which domain is not simply connected.\n\n\\begin{rem}\nLet $F$ be a holomorphic null lift a CMC-1 face $f$ with Aiyama data $(g, \n\\omega)$. Let $B \\in SU(1, 1)$ be a constant matrix, that is\n \n $$ B =  \\left[\\begin{array}{c c }\n                \\overline{p} & - q\\\\                                           \n               - \\overline{} & p\n               \\end{array}\\right] \\in SU(1, 1) \\ , \\ p\\overline{p} - q \n\\overline{q} = 1 $$\n               \nThus, $FB$ is a holomorphic null lift of $f$. The Aiyama data $(\\hat{g}, \n\\hat{\\omega})$ corresponding to $(FB)^{-1}d(FB)$ is given by\n\n$$\\hat{g} = \\frac{pg + q}{\\overline{q}g + \\overline{p}} \\ e \\ \\hat{\\omega} = \n(\\overline{q}g + \\overline{p})^{2} \\omega. $$\n\n2 Aiyama datas $(g, \\omega)$ and $(\\hat{g}, \\hat{\\omega})$ are called \n\\textit{equivalents} if satisfy the above equality for some $B \\in SU(1, 1)$. \nWe call the equivalence class of Aiyama data $(g, \\omega)$ as Aiyama data \nassociated to $f$.\n\\end{rem}\n\n\\begin{rem}\n\nObserve that, if $(g, \\omega)$ e $(\\hat{g}, \\hat{\\omega})$ are 2 equivalents \nAiyama data, and if $G$ and $\\hat{G}$ are the hyperbolic Gauss maps associated \nwith this data, then\n\n$$ G = \\frac{dF_{11}}{dF_{21}} \\ \\ \\ e \\ \\ \\ \\hat{G} = \n\\frac{\\overline{p}dF_{11} + \\overline{q}dF_{12}}{\\overline{p}dF_{21} + \n\\overline{q}dF_{22}} $$\n\nTherefore, $G = \\hat{G}$, that is, the hyperbolic Gauss map does not depend of \nthe choice of $F$ in the  class. \n\\end{rem}\n\n\\subsection{CMC-1 Faces With Elliptics Ends}\n\n\\begin{defn}\\label{CMC1Face-defEE}\n Let $M$ be a Riemann surface, and $f: M \\longrightarrow \\mathbb{S}_{1}^{3}$ a \nCMC-1 face. Let \n $ds^{2} = f^{*}(ds_{\\mathbb{S}_{1}^{3}}^{2})$. $f$ is  \\textit{complete} \n(resp, of \\textit{finite type}) if there is a compact set $C$ and a symmetric \n$(0, 2)$-tensor $T$ in $M$ such that \n $T$ vanishes in $M \\setminus C$ and $ds^{2} + T$ is a complete Riemannian \nmetric (resp. has finite total curvature).\n\\end{defn}\n\n\\begin{rem}\n For CMC-1 immersions in $\\mathbb{S}_{1}^{2}$, the Gaussian curvature $K$ is \nnot negative, So the total curvature is the same as the absolute total \ncurvature. However, for CMC-1 faces with singular points, the total curvature \nnever is finite. \n\\end{rem}\n\n\\begin{rem}\nThe universal cover of a  complete CMC-1 face (resp finite type) is not \nnecessarily complete (resp finite type), Because the singular set may not be \ncompact the universal cover.\n\\end{rem}\n\nLet $f: M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of \nfinite type. So $(M, ds^{2} + T)$ is a  complete Riemannian surface with finite \ntotal curvature. Therefore, $M$ has finite topological type.\n\nLet $\\phi: \\tilde{M} \\longrightarrow M$ be the universal cover of $M$, and $F: \n\\tilde{M} \\longrightarrow SL(2, \\mathbb{C})$ a holomorphic null lift of a CMC-1 \nface $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$. Fix a point $z_{0} \\in M$. Let \n$\\gamma: [0, 1] \\longrightarrow M$ a loop such that $\\gamma (0) = \\gamma (1) = \nz_{0}$. So there is a unique deck transformation $\\tau$ of $\\tilde{M}$ \nassociated to the homotopy class of $\\gamma$. Define the \\textit{monodromy \nrepresentation} $\\Phi_{\\gamma}$ of $F$ by\n\n$$Fo\\tau = F\\Phi_{\\gamma}.$$\n\nAs $f$ is well defined in $M$, $\\Phi_{\\gamma} \\in SU(1, 1)$ for every loop \n$\\gamma$. Therefore, $\\Phi_{\\gamma}$ is conjugated to one of the following \nmatrices\n\n$$ E_{1} =  \\left[\\begin{array}{c c }\n                e^{i\\theta} & 0\\\\                                           \n                          0 & e^{i\\theta}\n               \\end{array}\\right] \\ ou \\ E_{2} = \\pm \\left[\\begin{array}{c c }\n                                                           cosh s & senh s\\\\    \n                                       \n                                                           senh s & cosh s\n                                                        \\end{array}\\right] \\ ou \n\\ E_{3} = \\pm \\left[\\begin{array}{c c }\n                                                                                \n                        1 + i & 1\\\\                                           \n                                                                                \n                            1 & 1 - i\n                                                                                \n                     \\end{array}\\right] $$\n                                                                                \n                     De Sitter 3-Space\nfor $\\theta \\in [0, 2 \\pi)$, $s \\in \\mathbb{R} \\setminus \\{0\\}$.\n\n\\begin{defn}\n Let $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of \nfinite type with holomorphic null lift $F$. A end of $f$ is called \n\\textit{elliptic, hyperbolic or parabolic} if tis monodromy representation is \nconjugated to $E_{1}$, $E_{2}$ or $E_{3}$ in \n $SU(1, 1)$ respectively.\n\\end{defn}\n\n\\begin{rem}\n As every matrix in  $SU(2, \\mathbb{C})$ is conjugated to $E_{1}$ in $SU(2, \n\\mathbb{C})$, CMC-1 immersions in \n $\\mathbb{H}^{3}$ have similar properties to CMC-1 faces with elliptic ends in \n$\\mathbb{S}_{1}^{3}$.\n\\end{rem}\n\n\n\\begin{prop}\\label{CMC1Face-prop1}\nLet $V$ be a neighborhood of a end of $f$ and $f|_{V}$ a spacelike CMC-1 \nimmersion of finite total curvature, which is complete in the end. Suppose that \nthe end is elliptic. So there is a holomorphic null lift $F: \\tilde{V} \n\\longrightarrow SL(2, \\mathbb{C})$ of $f$ with associated Aiyama data $(g, \n\\omega)$ such that $$d\\hat{s}^{2}|_{V} = (1 + |g|^{2})^{2}|\\omega|^{2}$$ is \nsingle valued in $V$. Besides, $d\\hat{s}^{2}$ has complete total curvature and \nis complete at the end. \n\\end{prop}\n\n\\begin{prop}\\label{CMC1Face-prop2}\nLet $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of finite \ntype with elliptic ends. So, there is a compact Riemann surface $\\overline{M}$ \nand a finite number of points $p_{1}, ... , p_{n} \\in \\overline{M}$, such that \n$M$ \u00e9 biholomorphic to $\\overline{M} \\setminus \\{p_{1}, ... , p_{n} \\}$. \nBesides, the Hopf differential $Q$ of $f$ extends meromorphically to \n$\\overline{M}$.\n\\end{prop}\n\nSimilarly to immersed Bryant surfaces in hyperbolic 3-space, the hyperbolic \nGauss map does not extends necessarily to the ends. Because of this, we have \nthe following definition\n\n\\begin{defn}\\label{CMC1Face-defreg}\nLet $f: M \\longrightarrow \\mathbb{S}_{1}^{3}$ a CMC-1 face. A end $p_{j}$ of \n$M$ is \\textit{regular} if\n the hyperbolic Gauss map does extend meromorphically to $p_{j}$, otherwise is \ncalled \\textit{irregular}.\n\\end{defn}\n\nLet $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a CMC-1 face of finite type \nwith elliptic ends. Thus $M = \\overline{M} \\setminus \\{p_{1}, ... , p_{n} \\}$. \nLet $G$ and $Q$ the hyperbolic Gauss map and the Hopf differential of $f$ \nrespectively.\n\n\\begin{defn}\\label{CMC1Face-defmetric}\n We call the metric \n\n $$d\\hat{s}^{\\sharp 2} = (1 + \n|G|^{2})^{2}\\left(\\frac{Q}{dG}\\right)\\overline{\\left(\\frac{Q}{dG}\\right)}$$\n\n of \\textit{lift metric} of CMC-1 face $f$. Besides\n \n $$-(K_{d\\hat{s}^{\\sharp 2}})d\\hat{s}^{\\sharp 2}|_{V}  =  \n\\frac{4dGd\\overline{G}}{(1 + |G|^{2})^{2}} $$\n \n \\end{defn}\n\n\\begin{rem}\n Observe that $d\\hat{s}^{\\sharp 2}$ and $-(K_{d\\hat{s}^{\\sharp \n2}})d\\hat{s}^{\\sharp 2}$ are given in terms of $G$ and $Q$, and such functions \nare defined in $M$, thus $d\\hat{s}^{\\sharp 2}$ and $-(K_{d\\hat{s}^{\\sharp \n2}})d\\hat{s}^{\\sharp 2}$ are defined in $M$ also.\n\\end{rem}\n\n Fujimori \\cite{kn:F} showed that\n\n\\begin{prop}\\label{CMC1Face-thFuj}\n Let $f:M \\longrightarrow \\mathbb{S}_{1}^{3}$ be a CMC-1 face. Assume that each \nend of $f$ elliptic and regular. Thus, if $f$ is complete and of finite type, \nthen the lift metric $d\\hat{s}^{\\sharp 2}$ is complete and of finite total \ncurvature in $M$.\n\\end{prop}\n\n\n\n\n\n\n\n\n\n\\section{The proof of theorem 1}\n\n\n\\ \\ \\ \n\n\n\\begin{proof}\n Suppose that $\\phi:M \\longrightarrow \\mathbb{H}^{3}$ is a complete algebraic \nCMC-1 immersion with hyperbolic \n Gauss map $G$ and Bryant data $(g, \\omega)$. Then, take  $\\phi^{\\sharp}: M \\longrightarrow \\mathbb{H}^{3}$\nits dual immersion with dual data $(G, \\omega ^{\\sharp})$ and let $\\psi: M \\longrightarrow \\mathbb{R}^{3}$ be \nthe minimal immersion cousin of the $\\phi^{\\sharp}$, with Weierstrass data $(G, \\omega ^{\\sharp})$. Note that\n $\\psi$ is a complete minimal surface with finite total curvature, since $\\phi^{\\sharp}$ is complete\n  and the total curvature of $\\psi$ coincides with total dual curvature of \n$\\psi^{\\sharp}$, witch is finite since\n  $\\phi$ is algebraic. Besides, the Gauss map of $\\psi$ coincides with the hyperbolic Gauss map $G$ of $\\phi$.\n Then by theorem \\ref{thm:thJM}, $G$ omits 2 points at most.\n\n Suppose now that $M$ is a complete Bryant surface with finite total curvature. \nThen $M$ is conformal \nto $\\overline{M} \\setminus E$, where $\\overline{M}$ is a compact surface and $E$ is a finite set, the ends\nof the surface. Suppose that $M$ has at least one irregular end $p \\in E$. \nThen $p$ is a essential \nsingularity of $G$, so by Big Picard's theorem, $G$ omits 2 points at most. \nTherefore we can suppose \nthat all ends of $M$ are regular. If all ends are regular, then $M$ is a algebraic Bryant surface,\nand therefore, $G$ can omits 2 points at most.\n\n Suppose now that $M$ is a a complete CMC-1 face of finite type with elliptic \nends. Let $(g, \\omega)$\nbe its Aiyama data. Define $\\phi: M \\longrightarrow \\mathbb{H}^{3}$ the immersion wich has\n$(g, \\omega)$ as its Bryant data. Then, $\\phi: M \\longrightarrow \\mathbb{H}^{3}$ is a complete\nCMC-1 immersion. Note that the lift metric of the CMC-1 face $M$ coincides with the\ndual metric of $\\phi$. Then $\\phi(M)$ is a algebraic Bryant surface, because $M$ has finite type.\nObserve that the hyperbolic Gauss map $G$ of $M$ coincides with the hyperbolic Gauss map of \n$\\phi: M \\longrightarrow \\mathbb{H}^{3}$. Therefore, $G$ can miss 2 points at most.\n   \n\\end{proof}\n\n\\begin{rem}\nThis estimative is sharp since:\n\n\\begin{enumerate}\n\\item[(i)] the catenoid cousin is a complete Bryant surface with finite total curvature\nwitch hyperbolic Gauss map that omits exactly 2 points.\n\\item[(ii)] $\\mathbb{C} \\setminus \\lbrace 0, 1 \\rbrace$ is a complete algebraic Bryant \nsurface witch hyperbolic Gauss map that omits exactly 2 points.\n\\item[(iii)]  the elliptical catenoid is a complete CMC-1 face of finite type with elliptic\n ends with hyperbolic Gauss map that omits exactly 2 points.\n\\end{enumerate}  \n\\end{rem}\n\n\n\\begin{cor}\n Let $M$ be a properly embedded Bryant surface of finite topological \ntype. Then, the hyperbolic Gauss map is constant and $M$ is a horosphere or the \nimage of hyperbolic Gauss map omits 2 points at most.\n\\end{cor}\n\n\\begin{proof}\n Collin, Hauswirth and Rosenberg showed in \\cite{kn:C-H-R-2} that a\n properly embedded \nBryant surface of finite topological type has finite total curvature, thus the \nresult follows from the previous theorem.\n\\end{proof}\n\n\\section{Curvature estimate and volume growth.}\n\nWe begin with the following theorem\n\n\\begin{thm}\\label{thm5.1}\nLet $M$ be one complete surface of type:\n\\begin{enumerate}\n\\item[(1)] minimal surface with finite total curvature  into $\\R^3$;\n \\item[(2)] Algebraic Bryant surface with finite total curvature;\n \\item[(3)] Algebraic Bryant surface endowed with the dual metric;\n \\item[(4)] CMC-1 Face surface of finite type, elliptic ends endowed with the lift metric.\n\\end{enumerate}\nIf $\\kappa\\colon M\\to\\R$ is the curvature of $M$, there is a function $k: [0, \n\\infty) \n\\longrightarrow [0, \\infty)$, at least $C^{2}$ such that,\n \n \\begin{enumerate}\n  \\item[(i)] $0 \\geq \\kappa(p) \\geq -k(r(p)),\\quad \\forall\\ p\\in M,\\ \nr(p)=\\text{dist}^M(p,p_0),\\ p_0\\ \\text{fixed},$\n  \\item[(ii)] $\\int_{0}^{\\infty} sk(s)ds<\\infty.$\n \\end{enumerate}\n\n\\end{thm}\n\n\\begin{proof}\nThe Bryant surface with finite total curvature and regular ends is isometric to \nits cousin, that is, to one complete minimal surface into $\\R^3$ with finite \ntotal curvature. \nThe CMC-1 Face dual surface of finite type and elliptic ends is isometric to \nthe dual of one\n algebraic Bryant surface (with finite total dual curvature) and both are \nisometric to one complete minimal surface into $\\R^3$ with finite total \ncurvature. \n\n\n We know that there is a compact Riemann surface $\\overline{M}$ \nand a finite set $E = \\{w_{1}, \\ ... \\ , w_{m} \\}$ such that $M$ is conformal \nto $\\overline{M} \\smallsetminus E$, where $w_{i}$ are the ends of the surface. \nThe Gauss map, for a minimal surface, and the hyperbolic Gauss map, for the\nBryant surface, $G$ of $M$ does extend meromorphically to the ends, and \n$G:\\overline{M} \\longrightarrow \\mathbb{S}_{\\infty}^{2}$ is a branched \ncovering. \nTake $R_{0} > 0$ such that $\\overline{M} \\setminus B_{R_{0}}^{M}(o)$ is the \nfinite \nunion of the subends $E_{1}, ... , E_{m}$ . Bryant Proposition 4 \\cite{kn:B} \nshowed \nthat\nis possible parametrize each subend $E_{i}$ using the Bryant data $(g, f)$, \nin$\\{z \\in \\mathbb{C} \/ 0 < \n|z| < r_{j} \\ , \\ \\ r_{j} > 0\\}$  by\n \n \\begin{equation}\n\t\t\\left\\{ \n\t\t\\begin{array}{l l}\n\t\tg(z) = z^{1 + \\beta (w_{i})} \\hat{g}(z) \\\\\n\t\tf(z) = \\frac{1}{z^{1 + I(w_{i})}} \\hat{f}(z)\n                \\end{array} \\right.\n\\end{equation}\n\n Where $\\hat{g}$ and $\\hat{f}$ are analytic, never vanish and extend \nmeromorphically to $z=0$. Is well known that is possible to get a similar \nparametrization for minimal surfaces with finite total curvature.\n \n Let $\\gamma_j=\\partial E_{j}.$ For each $|z|<r_{j}$ let $\\alpha_{j}$ be the \npath $re^{i \\theta}$, $|z| \\leqslant r \\leqslant r_{j}$. We have \n \n\\begin{eqnarray*}\n \\text{dist}^M(\\gamma_{j},z) &=& inf_{\\alpha} \\{ l(\\alpha) \\} \\\\\n                           & \\leq & l(\\alpha) \\\\\n                           &=& \\int_{\\alpha} |ds| \\\\\n                           & \\leq & \\int_{r_j}^{|z|}|f|(1+|g|^2)|dz| \\\\\n                           &=&  \\int_{r_{j}}^{|z|} \\frac{1}{|z|^{1 + \nI(w_{j})}}|\\hat{f}(z)|(1 + |z|^{2 + 2 \\beta(w_{j})}|\\hat{g}(z)|^{2})|dz| \\\\\n                           &=&  \\int_{r_{j}}^{|z|} \\frac{1}{|z|^{1 + \nI(w_{j})}}|\\hat{f}(z)|(1 + |z|^{2 + 2 \\beta(w_{j})}|\\hat{g}(z)|^{2})|dz| \\\\\n\\end{eqnarray*}\n\nObserve that $\\hat{f}$ e $\\hat{g}$ are bounded in $ 0 < |z| < r_{j}$. So, in \nthis set, we have\n\n\\begin{eqnarray*}\ndist_{M}(\\gamma_{j},z) & \\leq & \\int_{r_{j}}^{|z|} \\frac{1}{|z|^{1 + \nI(w_{j})}}|{\\hat{f}}(z)|(1 + |z|^{2 + 2 \\beta(w_{j})}|{\\hat{g}}(z)|^{2})|dz| \\\\\n                       & \\leq & \\int_{r_{j}}^{|z|} \\frac{1}{|z|^{1 + I(w_{j})}} \nc_{1}(1 + |r_{j}|^{2 + + 2 \\beta(w_{j})}c_{2}^{2})|dz| \\\\\n                       & = & \\int_{r_{j}}^{|z|} \\frac{1}{|z|^{1 + \nI(w_{j})}}c_{3} |dz| \\\\\n\\end{eqnarray*}\n\nTherefore\n\n\\begin{eqnarray}\n \\text{dist}^M(\\gamma_{j},z) \\leq \\frac{c_{3}}{|z|^{I(w_{j})}}.\n\\end{eqnarray}\n\nBeside that, the Gaussian curvature $\\kappa$ satisfies\n\n\\begin{eqnarray*}\n |\\kappa(z)| &=& \\frac{4|g'|^2}{|f|^2(1+|g|^2)^4} \\\\\n             & \\leq & \\frac{4|z^{\\beta(w_{j})}\\hat{g}(z) + z^{1 + \n\\beta(w_{j})}\\hat{g}'(z)|^2}{|\\frac{1}{z^{1 + I(w_{j})}}\\hat{f}(z)|^2(1+|z^{1 + \n\\beta(w_{j})}g_{1}(z)|^2)^4} \n\\end{eqnarray*}\n\nObserve that $1 \\leq |1 + |g(z)|^{2}|$, therefore $\\frac{1}{|1 + |g(z)|^{2}|} \n\\leq 1$. Besides, $\\hat{f}$, $\\hat{g}$ and $\\hat{g}'$ are bounded in $ 0 < |z| \n< r_{j}$. So,\n\n\\begin{eqnarray*}\n |\\kappa(z)| & \\leq & \\frac{4|z^{\\beta(w_{j})}\\hat{g}(z) + z^{1 + \n\\beta(w_{j})}\\hat{g}'(z)|^2}{|\\frac{1}{z^{1 + I(w_{j})}}\\hat{f}(z)|^2(1+|z^{1 + \n\\beta(w_{j})}\\hat{g}(z)|^2)^4} \\\\\n             & \\leq & \\frac{4(|z|^{\\beta(w_{j})}|\\hat{g}(z)| + |z|^{1 + \n\\beta(w_{j})}|\\hat{g}'(z)|)^2}{\\frac{|\\hat{f}(z)|^2}{|z|^{2 + \n2I(w_{j})}}(1+|z^{1 + \\beta(w_{j})}\\hat{g}(z)|^2)^4} \\\\\n             & = & 4 \\frac{|z|^{2 + 2I(w_{j})}(|z|^{\\beta(w_{j})}|\\hat{g}(z)| + \n|z|^{1 + \\beta(w_{j})}|\\hat{g}'(z)|)^2}{|\\hat{f}(z)|^2(1+|z^{1 + \n\\beta(w_{j})}\\hat{g}(z)|^2)^4} \\\\\n             & \\leq & 4 \\frac{|z|^{2 + 2I(w_{j})}(|z|^{\\beta(w_{j})}\\hat{c}_{1} \n+ |z|^{1 + \\beta(w_{j}}\\hat{c}_{2})}{\\hat{c}_{3}} \\\\\n             & = & \\hat{c}_{4}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})}(1 + |z|)^{2} \n\\\\\n             & = & \\hat{c}_{4}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})}(1 + \nr_{j})^{2} \\\\\n             & = & \\hat{c}_{5}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})}\n\\end{eqnarray*}\n\nWhere $|\\hat{g}(z)| \\leq \\hat{c}_{1}$, $|\\hat{g}'(z)| \\leq \\hat{c}_{2}$ and \n$|\\hat{3}(z)| \\geq \\hat{c}_{3}$ for $|z| \\leq r_{j}$, and\n$\\hat{c}_{4} = 4.max\\{\\hat{c}_{1}, \\hat{c}_{2} \\}$.\n\nNow, let $\\rho(z)$ be the geodesic distance from $o \\in M$ to $z$. Note that\n $\\rho(z) \\leq \\frac{c_{3}}{|z|^{I(w_{j}}}$, so\n\n\\begin{eqnarray*}\n |z|^{I(w_{j})} & \\leq & \\frac{c_{3}}{\\rho} \n\\end{eqnarray*}\n\ntherefore,\n\n\\begin{eqnarray*}\n |z| & \\leq & \\frac{c_{3}}{\\rho^{\\frac{1}{I(w_{j})}}} \n\\end{eqnarray*}\n\nReplacing this in the Gaussian curvature inequality above\n\n\\begin{eqnarray*}\n |\\kappa(z)| & \\leq & c_{4}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})} \\\\\n - \\kappa(z) & \\leq & c_{4}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})} \\\\\n   \\kappa(z) & \\geq & - c_{4}|z|^{2 + 2I(w_{j}) + 2\\beta(w_{j})} \\\\\n             & \\geq & - c_{4} \\left( \\frac{c_{5}}{\\rho^{\\frac{1}{I(w_{j})}}} \n\\right)^{2 + 2I(w_{j}) + 2\\beta(w_{j})} \\\\\n             & = & - \\frac{c_{6}}{\\rho^{2 + 2\\frac{1 + \\beta(w_{j})}{I(w_{j})}}}\n\\end{eqnarray*}\n\nThus, we have\n\n\\begin{eqnarray}\n 0 \\geq \\kappa(z) \\geq - \\frac{c_{6}}{\\rho^{2 + \\frac{1 + \n\\beta(w_{j})}{I(w_{j})}}} \\ , \\ para \\ |z| < r_{j}\n\\end{eqnarray}\n\n\nTake $c_{0} \\geq max_{j} \\{ c_{6} \\}$, and define\n\n\\begin{eqnarray*}\n \\epsilon = inf \\{2 \\frac{1 + \\beta(w_{j}}{I(w_{j})} | 1 \\leq j \\leq m \\}\n\\end{eqnarray*}\n\nand\n\n\\begin{eqnarray*}\n \\kappa_{o}  = sup_{\\rho(p)} |\\kappa(z)|.\n\\end{eqnarray*}\n\nTake a \\textit{cut off function} $k: [0, \\infty) \\longrightarrow [0, \\infty)$ \nof class $C^{\\infty}$ such that $k(s) \\geq |\\kappa(z)|$ for all\n$\\rho(p) \\leq R_{o}$ and $k(s) = \\frac{c_{o}}{s^{2 + \\epsilon}}$ , for all $s \n\\geq R_{o}$.\n\nSo we have\n\n\\begin{eqnarray*}\n \\int_{0}^{\\infty} sk(s)ds & = & \\int_{0}^{R_{o}} sk(s)ds + \n\\int_{R_{o}}^{\\infty} sk(s)ds \\\\\n                           & \\leq & \\int_{0}^{R_{o}} s\\kappa_{o} ds + \n\\int_{R_{o}}^{\\infty} s \\frac{c_{o}}{s^{2 + \\epsilon}} ds \\\\\n                           & = & \\kappa_{o}R_{o}^{2} + \\int_{R_{o}}^{\\infty} \n\\frac{c_{o}}{s^{1 + \\epsilon}} ds \\\\\n                           & = & C_{7} + c_{o} \\lim\\limits_{y \\rightarrow \n\\infty} \\int_{R_o}^{y} \\frac{1}{s^{1 + \\epsilon}} ds \\\\\n                           & = & C_{7} -\\frac{c_{o}}{\\epsilon}  \\lim\\limits_{y \n\\rightarrow \n\\infty} \\left( \\frac{1}{y^{\\epsilon}} - \\frac{1}{R_{o}^{\\epsilon}} \\right) \\\\\n                           & < & \\infty.\n\\end{eqnarray*}\n\\end{proof}\n\\begin{lem}\n Let $M$ be a complete surface with curvature $\\kappa,\\ 0\\geq \\kappa\\geq k,$ \nwhere $k(r)$ is the function of last theorem. If $B_r$ is the geodesic ball of \n$M$ with fixed center then there are constant $c$ and $r_0>0$ such that \n$\\text{vol}(B_r)\\leq cr^2$ for all $r\\geq r_0.$\n\\end{lem}\n\\begin{proof}\nLet $M_o$ be a model with metric $dr^2+h^2dt^2$ where $h''\/h=k$  the function \n$k(s)$ defined in theorem \\ref{thm5.1} item (ii). By lemma 4.5 of \\cite{kn:G-W}\nthere is a constant $c_0>1$ such that $r\\leq h(r)\\leq c_0r$. \n\nSet $\\exp=\\exp_{p_0}\\colon\\R^2\\to M,\\ \\R^2=T_{p_0}M.$ Let $F$ be the set \n$M\\setminus \\mathcal{C}(p_0)$ where $\\mathcal{C}(p_0)$ is the cut locus of \n$p_0.$ Let $U$ be the closure of the connected component of $\\exp^{-1}(F)$ \npassing at the origin. The metric into $U$ endowed by $\\exp$ has the expression \n$dr^2+g^2dt^2.$  If $B_r$ is the geodesic ball with \ncenter $p_0$ then \n\\[\n \\text{vol}_M(B_r)=\\int_{U\\cap\\{|z|\\leq r\\}}dM,\n\\]\nBy the Rauch comparison theorem we have $g(r)\\leq h(r)\\leq c_0r$ for all $ \nr_0\\leq r\\leq r^\\star,$ where $[r_0,\\ r^\\star]$ is the interval where the \nradial geodesic $\\gamma(r)$ is into $U$. Hence \n\\begin{equation}\n \\text{vol}_M(B_r)\\leq  \\text{vol}_M(B_{r_0})+c_1\\int_{r_0}^rsds\\leq \nc_2r^2\\quad r\\geq r_0.\n\\end{equation}\n\n\n\\end{proof}\n\n\\section{Proof of theorem \\ref{th2}}\n It is \nwell know that $\\text{vol}(B_r)\\leq c_0r^2$ of geodesic balls imply those \nsurfaces are parabolic. For example \n\\[\n \\int_{r_0}^\\infty\\frac{rdr}{\\text{vol}(B_r)}\\geq \n\\text{const.}\\int_{r_0}^r\\frac{ds}{s}=\\text{const.}\\ln(r\/r_0),\\quad r>r_0,\n\\]\nand by \n\\cite{kn:Grig} theorem 11.14 $M$ is parabolic. Then all surfaces of theorem 2 \nare parabolic. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nThe tiny scale structures on scales of tens of astronomical units\nwere observed in the interstellar medium (ISM).\nThese structures have been detected by 21 cm absorption lines against \nquasars with VLBI techniques (Dieter, Welch, \\& Romney 1976),\nagainst pulsars with time variability (Frail et al. 1994) and\nagainst close binary stars in optical interstellar lines\n(Meyer \\& Blades 1996).\nThe tiny scale structure is seen in all directions for which it has been \nsearched, which suggests that it is ubiquitous,\nnot associated with large extinction (Heiles 1997).\nAU-sized structures have been seen not only in atomic gas, but also\nin molecular gas.\nLanger et al. (1995) observed several clumps ranging in size from \n0.007 $-$ 0.021 pc \nin the Taurus molecular cloud 1 (TMC1) with interferometer techniques.\nThe mass of these smallest fragments is of the order of $0.01 M_{\\sun}$.\nThese small-scale structures appear to be gravitationally unbound \nby a large factor.\nThe presence of these tiny scale atomic and molecular\nstructures in variety of the interstellar medium\nsuggests that their formation mechanisms have some similarities.\n\nModels of ISM have been studied by many authors.\nField, Goldsmith, \\& Habing (1969) first showed that \ncold ($T\\sim 50$ K) neutral medium (CNM) and \nwarm ($T\\sim 8000$ K) neutral medium (WNM)\ncan be in pressure equilibrium.\nCox \\& Smith (1974) showed that supernova explosions can greatly modify \nthe general aspect of the ISM.\nMcKee \\& Ostriker (1977) developed a theory of \na supernova-dominated ISM.\nThe late phase supernova remnant (SNR) expands \nuntil the internal pressure drops to the\nambient interstellar pressure ($n$ = 0.1 cm$^{-3}$, \\, $T$ = 10$^4$ K).\nThis maximum radius $R_{\\rm max}$ = 10$^{2.1}$ pc is reached at a time\n$t_{\\rm max}$ = 10$^{6.3}$ yr.\nThe total volume $SR^3_{\\rm max}t_{\\rm max}$ which all SNRs occupy in the\nmaximum expansion time $t_{\\rm max}$\nexceeds the volume of the galactic disk,\nwhere $S \\sim$ 10$^{-2}$ yr$^{-1}$ is the supernova rate in our galaxy.\nThis means that the \nSNRs overlap before they are dissipated.\nThus, the study of shock propagation into ISM is important.\n\n\nIn this paper, we study the basic processes in the evolution of\nISM with a strong shock wave\nby means of one-dimensional hydrodynamic calculations \nwith non-equilibrium thermal and chemical processes. \nThe processes we include in our calculation are \nphotoelectric heating from small grains and PAHs, \nheating and ionization by cosmic rays and X-rays,\nheating by H$_2$ formation and destruction,\natomic line cooling from hydrogen Ly$\\alpha$, \n\\ion{C}{2}, \\ion{O}{1}, \\ion{Fe}{2}, and \\ion{Si}{2},\nrovibrational line cooling from H$_2$ and CO, \nand atomic and molecular collisions with grains.\nEffects of self-gravity and magnetic field are not treated in this paper.\nIn the next section, we explain the details of our numerical methods.\nResults are presented in section 3.\nIn section 4, we discuss the formation of small molecular cloudlets.\nSection 5 is for summary.\n\n\\section{NUMERICAL METHODS}\n\\subsection{Hydrodynamic Equations}\n\nTo study the thermal and dynamical evolution of shock-compressed layer,\nwe use a one-dimensional plane parallel numerical method.\nThe hydrodynamics module of our scheme is based on the second-order \nGodunov method. \nWe use the result of Riemann problem at each cell interface to calculate \nnumerical fluxes, \nwithout introducing any artificial viscosity,  \nwhich characterizes the Godunov method.  \nHigher-order Godunov method has been the ``state-of-the-art'' \nnumerical method in last two decades (see e.g., van Leer 1997), \nand widely used in astrophysical context \n(see e.g., Truelove et al. 1998) .  \nThe hydrodynamic equations for the system are written as follows:\n\\begin{equation}\n\\frac{d\\rho}{dt}+\\rho\\frac{\\partial v}{\\partial X}=0,\n\\end{equation}\n\\begin{equation}\n\\rho\\frac{dv}{dt}=-\\frac{\\partial P}{\\partial X},\n\\end{equation}\n\\begin{equation}\nP=\\left(1.1+x_{\\rm e}-\\frac{x_2}{2}\\right)n k_{\\rm B}T,\n\\end{equation}\n\\begin{equation}\n\\rho=1.4m_{{\\rm H}}n,\n\\end{equation}\n\\begin{equation}\n\\frac{1}{\\gamma-1}\\frac{dP}{dt}\n-\\frac{\\gamma}{\\gamma-1}\\frac{P}{\\rho}\\frac{d\\rho}{dt}\n=n\\Gamma-n\\Lambda\n+\\frac{\\partial}{\\partial X}K\\frac{\\partial T}{\\partial X}.\n\\end{equation}\nIn the above equations, $n$ is the number density of hydrogen nuclei,\n$\\rho$ is the mass density of the gas,\n$v$ is the velocity of fluid elements, $P$ is the gas pressure,\nand $K$ is the coefficient of thermal conductivity.\nWe have assumed an abundance of He atom is $0.1 n$.\nFor the range of temperatures and ionized fractions considered,\nthe dominant contribution to thermal conductivity is that \ndue to neutral atoms, for which\n$K = 2.5 \\times 10^3 T^{1\/2}\\,{\\rm cm^{-1} K^{-1} s^{-1}}$ (Parker 1953).\nWe use the ratio of the specific heats $\\gamma=5\/3$ for simplicity. \n$\\Gamma$ and $\\Lambda$ are heating and cooling rate per hydrogen nucleus,\nrespectively.\nThese rates depend on number density $n$, temperature $T$, \nelectron fraction $x_{\\rm e}=n({\\rm e^-})\/n$, \nH$_2$ fraction $x_2=2n({{\\rm H}_2})\/n$, \nand CO fraction $x_{{\\rm CO}}=n({{\\rm CO}})\/n$.\nThe gas behind the shock front is chemical non-equilibrium state.\nTherefore we solve three chemical equations for \n$x_{\\rm e}$, $x_2$, and $x_{{\\rm CO}}$ \nand hydrodynamic equations simultaneously.\n\n\\subsection{Heating and Cooling Processes}\nWolfire et al. (1995, hereafter W95) calculated the thermal equilibrium state \nin the neutral atomic phase.\nWe follow their calculation for $n<10^3 \\, {\\rm cm^{-3}}$.\nWe also calculate molecular processes for \n$10^3 \\, {\\rm cm^{-3}}<n<10^6 \\, {\\rm cm^{-3}}$\n(see Figure \\ref{fig:equilibrium}).\nContributions to the heating of an interstellar cloud are from \nthe photoelectric emission from small grains and PAHs, \nionization by cosmic rays and soft X-rays,\nand the formation and photodissociation of H$_2$.\nFollowing W95, the local FUV field is set to 1.7 times Habing's estimate \n(i.e., $G_0=1.7$).\nThe cooling function is dominated by line emission from H, \nC, O, Si, and Fe, \nby rovibrational lines from H$_2$ and CO, \nand by atomic and molecular collisions with dust grains.\nTo solve these thermal processes in non-equilibrium, \nwe solve a set of \nthree\ntime-dependent equations for ionization and recombination \nof hydrogen, and formation and dissociation of H$_2$ and CO.\nSelf-shielding effects are also included in the calculation of \nH$_2$ photodissociation.\nThe details of these processes are presented in Appendix \\ref{THERMAL}.  \nFigure \\ref{fig:equilibrium} shows the equilibrium temperature, pressure,\nand chemical fractions as functions of number density.\nImportant physical timescales are also shown in Figure \\ref{fig:equilibrium}. \n\n\n\\subsection{Initial and Boundary Conditions}\n\nThe strong shock wave is characterized by Mach number. \nTo analyze the evolution of ISM swept up by a strong shock wave,\nwe set up a plane-parallel cloud which collides against a rigid wall.\nFigure \\ref{fig:configure} shows the schematic configuration of\nour calculation.\nThis configuration also corresponds to a face-on collision between \ntwo identical clouds.\n\nWe have generated a grid of calculations for\nvarying initial densities and collision velocities.\nThe typical models are summarized in table \\ref{tbl-1}. \nInitial temperatures and chemical fractions are taken to be\nthe value of thermal and chemical equilibrium state\n(see Figure \\ref{fig:equilibrium}).\nThe velocity difference, \n$V_{\\rm d}$, defined by upstream velocity minus downstream \nvelocity corresponds to the range of Mach number $M = 2 - 12$.\nThis range of Mach number in the WNM ($T \\sim 8000$ K) corresponds to\nthe sweeping-up speed of momentum-driven SNRs.\nExternal FUV and X-ray radiation penetrate from both sides of  \ntwo identical clouds.\nThe extinction of the radiation is evaluated by \nthe cumulative column density of each grid.\nThe largest total column density we considered is that of the\nstandard \\ion{H}{1} cloud ($10^{19 - 20}\\,{\\rm cm^{-2}}$).\n\n\\section{RESULTS}\\label{section:RESULTS}\n\n\\subsection{Shock Propagation into WNM}\\label{subsection:WNM}\n\nIn this section, we analyze the model W6 where a shock wave propagates\ninto WNM.\nFigure \\ref{fig:WNM Layer} shows the result of calculation.\nSnapshots at \n$t=8.0\\times 10^4$, $1.3\\times 10^5$, $2.5\\times 10^5$, and $2.9\\times 10^6$ \nyr are shown.\nThe horizontal axis in logarithmic scale denotes distance from the rigid wall.\nFigure \\ref{fig:WNM Layer}a shows the evolution of pressure.\nThe shock-compressed layer is almost isobaric.\nThe shock front propagates from left to right.\nFigure \\ref{fig:WNM Layer}b shows the evolution of temperature.\nLyman $\\alpha$ cooling is so efficient that the temperature of\npost-shock gas quickly decreases to the pre-shock value.\nIn the next subsection, we analyze the shock front in detail.\nBehind the shock front, cooling dominates heating because density is larger\nand temperature decreases monotonically.\nAround $t \\simeq 10^5$ yr, \ntemperature starts to decline rapidly because of thermal instability.\nThis instability is mainly driven by \\ion{O}{1} (63 \\micron) and \\ion{C}{2} \n(157 \\micron) line cooling. \nEventually temperature attains thermal equilibrium value \n$T \\simeq 19$ K at $n \\simeq 2.5\\times 10^3 \\, {\\rm cm^{-3}}$. \nMain coolant of collapsed layer is \\ion{C}{2} (see \nFigure \\ref{fig:equilibrium}c). \nFigure \\ref{fig:WNM Layer}c shows the evolution of number density.\nA thin layer is formed \nby thermal instability in shock-compressed layer.\nThe width of thermally-collapsed layer is much shorter than the\nshock-compressed layer.\nHowever column density of the thermally-collapsed layer becomes\ncomparable after $t=2.6\\times 10^5$ yr (dotted line on Figure\n\\ref{fig:WNM Layer}c).\nFigure \\ref{fig:WNM Layer}d shows the evolution of velocity.\nFigure \\ref{fig:WNM Layer}e shows the evolution of electron fraction.\nHigh shock temperature raises the ionization degree in the post-shock gas.\nThe timescale of recombination of electron in post-shock layer \nis larger than the timescale of gas cooling. \nFigure \\ref{fig:WNM Layer}f shows \nthe evolution of H$_2$ number fraction.\nBecause the timescale of H$_2$ formation is much\nlarger than cooling timescale (see Figure \\ref{fig:equilibrium}d),\nthis thermal instability is not driven by H$_2$ cooling.\nIn the highest density region at final time step (dot-dash),\n1.3\\% of the hydrogen is in H$_2$ and \n0.0002\\% of the carbon is in CO.\nNote that the \npre-shock H$_2$ number fraction at final time step (dot-dash) is larger\nthan the initial fraction,\nbecause H$_2$ photodissociation radiation from the left hand\nside is shielded by large H$_2$\ncolumn density in the collapsed layer in this one-dimensional\nplane-parallel model.\n\n\\subsection{Structure Across The Shock Front}\n\nIn this subsection, we analyze the detailed structure across\nshock front which propagates into WNM.\nThe same calculation as model W6 at $t=1\\times 10^4$ are shown in\nFigure \\ref{fig:shock front}.\nDashed lines show the calculation \nwhich used the same initial grid spacing as model W6.\nWe also calculated the model with 320 times higher spatial resolution\n(solid lines) to resolve detailed structure of the shock front.\nDotted lines denote the same calculation without cooling and heating \nprocesses.\nThe width of the shock front is on the order of collision mean free path,\n$\\ell=1\/n\\sigma_{col}\\approx 0.0003\/n$ pc.\nFor a strong shock without cooling, post-shock temperature\nis determined by Rankine-Hugoniot relation as \n$T_2=(\\gamma-1)mV^2_{\\rm d}\/2k_{\\rm B}$.\nOn the other hands, \nshock temperature with cooling is estimated by the following equation: \n\\begin{equation}\n\\frac{k_{\\rm B}T_{\\rm s}}{\\gamma-1}+E_{\\rm loss}=\\frac{1}{2}mV^2_{\\rm d},\n\\end{equation}\nwhere \n$E_{\\rm loss}=\\int n(\\Lambda-\\Gamma)\\, dt\n\\approx n\\Lambda_{\\rm Ly\\alpha}(T_{\\rm s}) \\ell\/V_{\\rm d}$.\nThis analytic estimation provides $\\log T_{\\rm s} \\ {\\rm [K]}=4.8$ in\nthis model. \nEven the lowest resolution model can provide the same values of \nphysical quantities in the post-shock region.  \n\nRadiative precursor which is not treated in our models can change \npre-shock temperature and ionization.\nShull \\& McKee (1979) have calculated the ionization of the pre-shock\ngas which is caused by the diffuse radiation from the post-shock\ngas in their models of interstellar shock waves with $v_s=$ 40 -- 130 km\/s. \nThey have concluded that sufficiently fast shocks ($v_{\\rm s}>110$ km\/s)\ncompletely ionize the pre-shock gas.\nTo study this effect,\nwe also calculated the model with fully ionized pre-shock gas (model I12),\nand found no significant difference between the model I12 and W12.\nTherefore we neglect the effect of radiative precursor in the rest of\nthis paper.\n\n\\subsection{Thermal Instability of Isobarically Cooling WNM}\n\nFigure \\ref{fig:n-Pw,n-Tw} shows the evolution of the fluid element \nwhich has the maximum density at each time step.\nDensity, temperature and pressure evolve on the dashed lines \nfrom left to right.\nSolid and thick lines correspond to the thermally stable equilibrium. \nDotted lines correspond to the thermally unstable equilibrium. \nThe pre-shock gas is a\nthermally stable WNM ($n=0.1 \\, {\\rm cm^{-3}}, \\ T=8000$ K). \nIn Appendix \\ref{LINEAR}, \nwe investigate the stability of isobarically contracting gas\nby linear perturbation theory to analyze the evolution on the dashed lines.\nWe have used the post-shock values of hydrodynamic calculation as the\nchemical compositions in the linear analysis\n($x_{\\rm e}=0.1$, $x_2=10^{-6.3}$, $x_{\\rm CO}=10^{-23}$).\nBecause the chemical reaction timescale is longer than the cooling timescale.\nShaded area in Figure \\ref{fig:n-Pw,n-Tw}b denotes \nthermally unstable region determined by the linear analysis.\nLarger electron fraction makes cooling rate larger,\nso that the unstable region is wider than the unstable region of\nequilibrium gas.\nThis shaded region shows that thermal instability is inevitable\nwhen WNM cools into CNM.\n\nNext, we discuss about the characteristic length and timescale of \nthe thermal instability.\nThe linear growth rate of the thermal instability \nas a function of perturbation wavelength is calculated with \na pair of given temperature and density.\nThe growth time of the instability is shown in Figure \\ref{fig:growth43}.\nIn this figure, the vertical axis denotes the unperturbed temperature.\nThe value of the constant pressure, \n$P_{\\rm c}\/k_{\\rm B}=5 \\times 10^4 \\,{\\rm K\/cm^3}$,\nis adopted from the result of our non-linear calculation.\nThe density can be deduced from the relation, $n=P_{\\rm c}\/k_{\\rm B}T$.\nThe horizontal axis denotes a quarter of perturbation wavelength.\nContours of growth time are depicted in this wavelength-temperature plane.\nIn Figure \\ref{fig:growth43}b,\nthe temperature evolution of our non-linear calculation \n(Figure \\ref{fig:WNM Layer}b) is superposed\nupon the growth rate contours (Figure \\ref{fig:growth43}a). \nThermal instability occurs at temperature less than 7300 K.\nThe instability becomes drastic at temperature below 1000 K \nwhich corresponds to the growth time less than $10^4$ yr.\nIn this way we can understand the collapse of the layer in terms of\nthermal instability.\nThe shortest wavelength of unstable perturbation \nis $\\lambda_{\\rm min} \\sim 8\\times 10^{-5} {\\rm pc}\\approx$ 16 AU.  \nThe thickness of the thermally-collapsed layer increases with time \nthrough the accretion of gas.\n\n\\subsection{Shock Propagation into CNM}\\label{subsection:CNM}\n\nIn this section, we analyze the model C10 where shock wave propagates\ninto CNM.\nAlthough Smith (1980) has done similar calculation,\nhis calculation does not have sufficient dynamic range to resolve \nthermally collapsing layer which is of our interest.\nFigure \\ref{fig:CNM Layer} shows the results of our calculation\nwhere Mach number $M=10$ \nwhich corresponds to the velocity difference of 10 km\/s.\nSnapshots at $t=3.1\\times 10^3$, $5.2\\times 10^4$, $5.7\\times 10^4$,\nand $8.8\\times 10^4$ yr are shown. \nThe shock front is almost adiabatic.\nThe post-shock pressure can be estimated by Rankine-Hugoniot relation as\nfollows:\n\\begin{equation}\n\\frac{P_2}{P_1}=1+\\frac{\\gamma(\\gamma+1)}{4}M^2+\\frac{\\gamma}{4}M\n\\sqrt{(\\gamma+1)^2M^2+16},\n\\end{equation}\nwhere M is the Mach number in the center of mass frame.\nWhen M equals to 10, the post-shock pressure is 224 times higher than the\nambient pressure.\nDetails of post-shock structure is more complicated than \nthat in the WNM case.\nThe post-shock gas, of which temperature and density are determined by\nadiabatic shock condition, is thermally unstable \n(see Figure \\ref{fig:n-Pc,n-Tc}b).\nThe shock front decelerates after post-shock layer collapsed.\nAfter thermally collapsed layer is formed, \naccretion shock is formed inside the shock-compressed layer.\nAccretion shock propagates into compressed layer.\nIn the highest density region at final time step,\n4.9\\% of the hydrogen is in H$_2$ and \n0.02\\% of the carbon is in CO.\nNote that the abundances of these molecules are still increasing\nbecause the \nH$_2$\nformation timescale is much longer than collapse timescale\n$t\\simeq 10^5$ yr (see Figure \\ref{fig:equilibrium}d).\n\n\n\n\\subsection{Thermal Instability of Isobarically Cooling CNM}\n\nFigure \\ref{fig:n-Pc,n-Tc} shows the evolution of the fluid element\nwhich has the maximum density at each time step.\nShaded area of Figure \\ref{fig:n-Pc,n-Tc}b denotes thermally unstable region.\nIn this case of linear analysis (Appendix \\ref{LINEAR}), \nchemical compositions in thermal and chemical\nequilibrium states of each densities are used. \nThis Figure shows how thermally stable CNM becomes unstable.\nThe growth time of the instability is shown in Figure \\ref{fig:growth46}.\nIn Figure \\ref{fig:growth46}b,\nwe superpose the temperature evolution of our non-linear calculation \n(Figure \\ref{fig:CNM Layer}b)       \nupon the growth rate contours (Figure \\ref{fig:growth46}a). \nThe shortest wavelength of unstable perturbation is \n$\\lambda_{\\rm min} \\sim 2.5\\times 10^{-5} {\\rm pc}\\approx$ 5 AU.  \n\n\\subsection{Results from Various Initial Parameters}\n\nCalculations are done \nat nine different initial densities ($10^{-2}<n<10^2$),\nand eleven velocity differences ($2<$ Mach number $<12$). \nThe column density is $10^{20} \\ {\\rm cm^{-2}}$. \nWe follow the calculation until $t=10^5$ -- $10^6$ yr.\nThe number fraction of H$_2$ in the highest density region\nis shown in Figure \\ref{fig:chemical abundance}. \nThe horizontal axis denotes the initial number density of hydrogen nuclei.\nThe vertical axis denotes the velocity difference.\nSolid lines denote the 8 \\% boundary of H$_2$ number fraction.  \nHigher velocity produces more abundance of H$_2$ \nin the thermally-collapsed layer.  \nNote that the fragmentation of the thermally-collapsed layer will change\nthe column density of the fragments, which change the efficiency of \nH$_2$ self-shielding.\nTherefore, the amount of H$_2$ in the fragments\nwould be different from the value obtained in this paper.\nTo determine the realistic abundance of H$_2$,\ntwo or three-dimensional hydrodynamic calculation is required. \n\n\n\n\\section{DISCUSSION}\n\nOur hydrodynamic calculation shows that thermal instability makes the\ncollapsed layer with considerable amount of H$_2$.\nIn this section, we discuss the consequence of this instability.\n\n\\subsection{Fragmentation of Thermally-Collapsed Layer}\n\nThe shock propagation into ISM can make the thermally-collapsing gas\ninside the shock-compressed layer.\nThe initial thickness of the thermally-collapsed layer is \ndozens of AU and agrees with the prediction by the linear analysis. \nThe thickness of the thermally-collapsed layer increases with time\nthrough the accretion of gas.\nThe thermally collapsing layer is dynamically unstable\nalso in the Y- and Z-direction, so that perturbations \nin the layer will grow and lead to fragmentation of the layer.\nWe expect that this layer will break up into very small cloudlets\nwhich have different translational velocities.\nThis velocity dispersion of cloudlets produced by the passage of SN shocks\nmay make an origin for the observed interstellar ``turbulent'' \nvelocities.\nExtremely high pressure in the tiny-scale structure observed in\n\\ion{H}{1} clouds (Heiles 1997) is consistent with our shock propagation\nmodel.\n\n\\subsection{Molecular Clouds in The Galaxy}\n\nIt seems difficult to detect these cloudlets in emission lines \nof CO molecules (see e.g., Liszt \\& Wilson 1993, Liszt 1994). \nMany shell-like or filamentary structures in the \\ion{H}{1} 21-cm \nobservation maps \n(Hartmann \\& Burton 1997)\nare reminiscent of overlapping shells of old SNRs. \nIf these structures in \\ion{H}{1} maps are really the \nresults of SNR shocks,  \nmany tiny molecular cloudlets should be hidden in the shell \nand should be ubiquitous in the Galaxy. \nMcKee \\& Ostriker (1977) showed that the recycling SNRs in the \ngalaxy would overlap before they are dissipated.\nThe overlapping filamentary region may produce larger clouds. \n\n\n\\section{SUMMARY}\n\nWe have done one-dimensional hydrodynamic calculations\nfor the propagation of a strong shock wave into WNM and CNM\nincluding detailed thermal and chemical processes. \nOur results show that the shock propagation into WNM can make thin and\ndense H$_2$ layer by the thermal instability inside the\nshock-compressed layer.\nThe shock propagation into CNM can also make the thermally-collapsed layer.\nWe predict that this thermally-collapsed layer will fragment\ninto small molecular cloudlets.\nOur subsequent work which includes two or three-dimensional\ncalculation is aimed to explore \nhow these cloudlets have velocity dispersion, and form larger clouds.\n\n\\acknowledgments\nWe are grateful to the anonymous referee for helpful comments, \nwhich have improved the manuscript.\nWe would like to thank Shoken M. Miyama for discussions and \ncontinuous encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPolarizabilities represent fundamental properties of hadrons, encoding\ntheir linear response to externally applied fields. Experimentally, they\nmanifest themselves, e.g., in the non-Born part of the low-energy Compton\nscattering amplitude. While the leading low-energy response is controlled\nby the static polarizabilities found in the presence of constant external\nfields, at subsequent orders of a derivative expansion, the effective hadron\nHamiltonian becomes sensitive to temporal and spatial structures in the\napplied fields. The present study focuses, specifically, on weak, spatially\nconstant external electric fields $\\vec{E} $ applied to a neutron. When the\nfields are sufficiently weak, nonlinear effects, i.e., terms in the\neffective neutron Hamiltonian of higher order than quadratic in the\nexternal field, can be neglected, and, to zeroth order in spatial\nderivatives as well as second order in temporal derivatives of $\\vec{E} $,\nthe neutron mass shift can be expanded as\n\\begin{equation}\n\\Delta m = m(\\vec{E} )-m(0) = -\\frac{1}{2} \\alpha_{E} \\vec{E}^2 \\\n-\\frac{1}{2} \\gamma_{E1} \\vec{\\sigma } \\cdot\n(\\vec{E} \\times \\dot{\\vec{E} } ) \\\n-\\frac{1}{2} \\alpha_{E\\nu } \\dot{\\vec{E} }^2 \\ + \\ldots\n\\label{heff}\n\\end{equation}\nThe static electric polarizability $\\alpha_{E} $ has been investigated\nin a number of lattice studies, cf., e.g.,\n\\cite{wilepap,shintani,elpol,polprog,alexan,detlat}.\nHere, the goal is to perform a first evaluation of the electric spin\npolarizability $\\gamma_{E1} $ within lattice QCD. An effort in this direction\nis particularly timely in view of an ongoing experimental program at\nHI$\\gamma $S to measure this polarizability in isolation for the first\ntime, albeit in a proton; previously, only the forward and backward spin\npolarizabilities $\\gamma_{0} $, $\\gamma_{\\pi } $ of the proton\n\\cite{spexp1,spexp2}, which contain $\\gamma_{E1} $ in combination with\nother polarizabilities, had been accessed experimentally.\n\n\\section{External field and four-point function scheme}\nConsider a spatially constant external electromagnetic gauge field $A_i $\ngenerating an electric field $E_i = \\partial_0 A_i $. It enters the\nlattice link variables $U_i $ as an additional phase,\n$U_i \\rightarrow \\exp (iaq A_i ) \\cdot U_i $, where $a$ denotes the\nlattice spacing and $q$ the quark electric charge matrix. To evaluate only\ncontributions to the neutron mass shift quadratic in $\\vec{E}$, it is\nsufficient to expand the relevant neutron two-point function in powers\nof $\\vec{E}$ from the outset. Thus, one can perform a fully dynamical\ncalculation using existing unperturbed gauge ensembles:\nExpanding $\\exp (iaq A_i ) = 1+iaq A_i -(a^2 q^2 \/2) A_i^2 +\\ldots $ and\ninserting into the lattice action decomposes the latter into an\nunperturbed part and an external field-dependent part, $S=S_0 + S_{ext} $,\nwhere $S_{ext} $ is essentially a standard $j\\cdot A$ coupling of the\nexternal field to the vector current\\footnote{Additional\ncontact terms proportional to $A_i^2 $, cf.~\\cite{elpol,polprog},\ncan be excluded in the present context, as seen below.}.\nAs a consequence, the neutron two-point function expands as\n\\begin{equation}\n\\left\\langle N_{\\beta } (y) \\bar{N}_{\\alpha } (x) \\right\\rangle =\n\\begin{array}{c}\n\\int [DU] [D\\psi ] [D\\bar{\\psi } ] \\ \\exp (-S_0 ) \\\n\\left( 1 - S_{ext} + S_{ext}^{2} \/2 + \\ldots \\right)\nN_{\\beta } (y) \\bar{N}_{\\alpha } (x) \\\\\n\\hline\n\\int [DU] [D\\psi ] [D\\bar{\\psi } ] \\ \\exp\n(-S_0 ) \\ \\left( 1 - S_{ext} + S_{ext}^{2} \/2 + \\ldots \\right)\n\\end{array}\n\\end{equation}\nPerforming the quark integrations\nyields the diagrammatic representation for the contributions to the\nneutron two-point function quadratic in $\\vec{A} $ depicted in\nFig.~\\ref{diagrams}.\n\\begin{figure}[h]\n\\vspace{-0.5cm}\n\\hspace{1.1cm} \\epsfig{file=diagrams2.ps,width=13cm}\n\\vspace{-0.9cm}\n\\caption{Diagrams contributing to the neutron mass shift; crosses denote\n$j\\cdot A$ insertions.}\n\\label{diagrams}\n\\vspace{-0.15cm}\n\\end{figure}\nIn the present study, only the connected contributions in Fig.~\\ref{diagrams}\nwere evaluated. The four-point function scheme at which one has thus arrived\nis particularly suited to isolate the spin polarizability $\\gamma_{E1} $\nfrom the other contributions appearing in (\\ref{heff}): As will be specified\nin more detail below, each vertex in Fig.~\\ref{diagrams} is associated with\nan external field insertion $\\vec{A} = A_1 \\vec{e}_{1} + A_2 \\vec{e}_{2} $;\nhowever, in view of (\\ref{heff}), only contributions $\\sim A_1 A_2 $\n\\linebreak are of \\pagebreak\ninterest, which can be isolated by populating one vertex with only $A_1 $\nand the other vertex with only $A_2 $ in each diagram (and, thus, any\ncontact terms proportional to $A_i^2 $ can also be disregarded, as noted\nabove).  This not only discards the signals from the contributions\ncontrolled by $\\alpha_{E} $ and $\\alpha_{E\\nu } $ in (\\ref{heff}), but also\ntheir statistical noise\\footnote{Note that evaluating the difference\nbetween the mass shifts obtained with neutrons polarized in two\nopposing directions does not achieve this; that\neliminates the signal, but not the noise associated with the $\\alpha_{E} $\nand $\\alpha_{E\\nu } $ contributions.}. The very specific control over the\ndifferent contributions afforded by the present scheme\nthus aids in reducing the numerical uncertainties.\n\nChoosing the neutron state to be polarized in the 3-direction,\na suitable external gauge field is\n\\begin{equation}\n\\vec{A} = ( a_1^0 + a_1^1 t ,\\ a_2^0 + a_2^1 t + a_2^2 t^2 ,\\ 0) \\ ,\n\\label{gena}\n\\end{equation}\nwhich generates a constant $E_1 $ and an $E_2 $ varying linearly with time,\nas called for by (\\ref{heff}). An ambiguity remains in the choice of the\nconstants $a_1^0 $, $a_2^0 $ and $a_2^1 $; fixing $E_1 $ and $\\dot{E}_2 $\nonly determines $a_1^1 $ and $a_2^2 $. In particular, the finite spatial\nlattice torus is distinct from an infinite spatial domain in that the\nconstant potentials $a_i^0 $ cannot be wholly eliminated\nusing gauge transformations. Due to the torus boundary\nconditions, only the residual discrete invariance\n$A_i \\longrightarrow A_i +2\\pi \/q_{min} L$ remains, where $L$ is the\nspatial extent of the torus and $q_{min}$ the smallest unit of electrical\ncharge in the theory. Fractionally different $a_i^0 $ correspond to\ndifferent physics; namely, they represent flavor-dependent Bloch momenta\nin view of the minimal substitution $p_i \\rightarrow (p_i -q a_i^0 )$.\nThus, despite the dependences on the constants $a_1^0 $, $a_2^0 $\nand $a_2^1 $ being partially related to one another via space-time symmetries,\nthe neutron mass shift is influenced by additional physics distinct from\nthe spin polarizability effect,\n\\begin{equation}\n\\Delta m = c_1 \\vec{\\sigma } \\cdot (\\vec{A} \\times \\vec{E} ) \\\n-\\frac{1}{2} \\gamma_{E1} \\vec{\\sigma } \\cdot\n(\\vec{E} \\times \\dot{\\vec{E} } )\n\\label{haeff}\n\\end{equation}\neven after having restricted the calculation to contributions $\\sim A_1 A_2 $.\nThe additional Bloch momentum dependence\\footnote{\\label{foot3}Note that a\nseparate dependence of the form $\\vec{A} \\times \\dot{\\vec{E} } $ can be\nexcluded as follows: Working in Euclidean space, each time derivative acquires\na factor $i$ compared to Minkowski space. The terms in (\\ref{haeff}) both\ncontain an odd number of time derivatives, and $\\Delta m $ will\ncorrespondingly be extracted from the imaginary part of the neutron two-point\nfunction. By contrast, any putative physical $\\vec{A} \\times \\dot{\\vec{E} } $\ndependence cannot (and will explicitly be seen not to) arise in the imaginary\npart.} must be disentangled from the polarizability effects in order to\nextract $\\gamma_{E1} $. To this end, several choices of external\nfields of the type (\\ref{gena}) will be considered and contrasted below.\n\n\\section{Extracting the neutron mass shift in the adiabatic approximation}\nIn the presence of external fields of the form (\\ref{gena}), the Hamiltonian\nof the system under consideration is time-dependent. One facet of this\nwhich deserves to be noted is the connection to the Bloch momenta discussed\nabove. One can, e.g., write $A_1 $ in (\\ref{gena}) in two equivalent ways:\n$A_1 = a_1^0 + a_1^1 t \\equiv a_1^1 (t-t_0 )$, i.e., a translation of the\nsystem by the time $t_0 $ corresponds to the introduction of a Bloch momentum\n$a_1^0 = -a_1^1 t_0 $. This, of course, simply expresses the fact that the\nelectric field $E_1 $ accelerates or decelerates the Bloch currents.\nCorrespondingly, the Bloch momentum dependence of the neutron mass shift\n\\pagebreak\nwill manifest itself in a characteristic time dependence of the neutron\ntwo-point function, as will be seen below.\n\nIn general, the time evolution operator for a time-dependent Hamiltonian\nacquires an intricate time dependence. Here, an adiabatic approximation\nwill be adopted to interpret the measured behavior of the neutron two-point\nfunction: Since the external field can be taken to be arbitrarily weak, it\nseems plausible to consider the case that the strong dynamics equilibrate\nthe system rapidly on the scale of the temporal variation of the external\nfield. In this situation, the behavior of the two-point function (now\nprojected onto zero momentum states polarized in the 3-direction) is, at\ntimes $t$ sufficiently large for excited states to have decayed,\n\\begin{equation}\nG_{\\uparrow} (p=0,t) =\n\\sum_{\\vec{y} } \\ \\mbox{Tr}\n\\left( \\frac{1+\\gamma_{4} }{2} (1-i\\gamma_{3} \\gamma_{5} )\n\\left\\langle N (y) \\bar{N} (x) \\right\\rangle \\right)\n\\ \\ = \\ \\\nW \\ \\exp \\left( -\\int dt^{\\prime } \\, m(t^{\\prime } ) \\right)\n\\end{equation}\nInserting expansions of the normalization and the mass\\footnote{Note\nthat the omission of a linear term $m^{(1)} $ in the ansatz\n(\\ref{massexp}) implies that the absence of a permanent net electric current\nor electric dipole moment in an unperturbed neutron has already been put in.}\nin the external fields (where the superscript specifies the order in the\nexternal field),\n\\begin{eqnarray}\nW &=& W_0 + W^{(1)} [\\vec{A} (t)] + W^{(2)} [\\vec{A} (t)] +\\ldots \\\\\nm &=& m_0 + m^{(2)} [\\vec{A} (t)] +\\ldots\n\\label{massexp}\n\\end{eqnarray}\none obtains specifically for the contribution quadratic in the external\nfield\n\\begin{equation}\nG^{(2)}_{\\uparrow} (p=0,t) \\ \\ = \\ \\\nW_0 \\ \\exp (-m_0 \\, t) \\left( \\frac{W^{(2)} [\\vec{A} (t)]}{W_0 }\n-\\int dt^{\\prime } \\, m^{(2)} [\\vec{A} (t^{\\prime } )] \\right)\n\\label{corr2}\n\\end{equation}\nIn view of this, the mass shift $m^{(2)} [\\vec{A} (t)]$ can be obtained,\nafter dividing out the unperturbed correlator\n$G^{(0)} (p=0,t) = W_0 \\ \\exp (-m_0 \\, t)$,\nfrom the temporal slope of the ratio {\\em only at a stationary point} of\nthe time evolution, where the time dependences of $W^{(2)} [\\vec{A} (t)]$\nand $m^{(2)} [\\vec{A} (t)]$ are relegated $\\mbox{\\hspace{1.5cm} } $\n\\vspace{-0.2cm}\n\\begin{table}[h]\n\\hspace{0.05cm}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$am_l $ & $am_s $ & $ m_{\\pi } $ & \\# configs \\\\\n\\hline \\hline\n0.01 & 0.05 & 357 MeV & 448 \\\\\n\\hline\n0.05 & 0.05 & 759 MeV & 425 \\\\\n\\hline\n\\end{tabular}\n\\caption{$N_f =2+1$, $20^3 \\times 64 $ MILC asqtad\n\\hspace{8.6cm} $\\mbox{\\ \\ } $\nensembles with $a=0.124\\, \\mbox{fm} $ used in the\n\\hspace{8.43cm} $\\mbox{\\ \\ } $\npresent investigation.}\n\\label{tabmilc}\n\\end{table}\n\\vspace{-4cm}\n\n\\hspace{6.3cm} \\parbox{8.02cm}{to quadratic order in $t$.\nPhysically, this occurs when the quark Bloch currents cease to flow and turn\naround into the opposite direction due to the forcing by the external\nelectric field. Only at such a stationary point do the strong dynamics have\nthe opportunity to form a bona fide neutron, the polarizability of which\ncan thus be extracted only in the vicinity \\linebreak of that particular\npoint in time.}\n\n\\section{Numerical results}\nThe neutron two-point function was evaluated using the MILC ensembles listed\nin Table~\\ref{tabmilc}.\nEach configuration was HYP-smeared and chopped into two $20^3 \\times 32 $\nsublattices with temporal Dirichlet boundary conditions. Domain wall valence\nquarks were employed. To improve statistics, averages over neutron spin in\nthe positive and negative 3-directions were taken, as well as over the\nexternal field (\\ref{gena}) and its rotation by $\\pi \/4$ around the\n3-axis. Figs.~\\ref{fig759ab}-\\ref{fig357c} display results for the ratio\n$R_2 (t) = G_{\\uparrow }^{(2)} (p=0,t) \/ G^{(0)} (p=0,t)$ for several cases\nof external field.\n\nConsider first Figs.~\\ref{fig759ab} and \\ref{fig759c}, which pertain to the\ncase of pion mass $m_{\\pi } = 759$ MeV. To exhibit the different effects at\nplay, Fig.~\\ref{fig759ab}~(left) shows the result of merely using an\n\\pagebreak\nexternal field of the form $\\vec{A} = (a_1 ,\\ e_2 (t-t_0 ) ,\\ 0)$, which is\nexpected to yield a time-independent effective dynamics isolating\nthe $\\vec{\\sigma } \\cdot (\\vec{A} \\times \\vec{E} )$ dependence in\n(\\ref{haeff}). Indeed, the correlator ratio $R_2 (t)$ exhibits linear\nbehavior, corresponding to a constant neutron mass\nshift determined by the slope of $R_2 (t)$, cf.~(\\ref{corr2}).\n\\begin{figure}[t]\n\\vspace{-0.1cm}\n\\epsfig{file=data05.01.ps,width=4.9cm,angle=-90} \\hspace{0.1cm}\n\\epsfig{file=data05.02.ps,width=4.9cm,angle=-90}\n\\caption{Neutron two-point function ratio $R_2 $ obtained in the\n$m_{\\pi } =759\\, \\mbox{MeV} $ ensemble, in the presence of the external\nfields $\\vec{A} = (a_1 ,\\ e_2 (t-t_0 ) ,\\ 0)$ (left) and\n$\\vec{A} = (a_1 ,\\ \\dot{e}_2 (t-t_0 )^2 \/2 ,\\ 0)$ (right), where\n$t_0 =6a$. The neutron source is located at $t=0$, electromagnetic\nfields are in Gaussian units. Linear and quadratic fits, respectively,\nwere performed using the fit range $t\/a \\in [4,8]$.}\n\\label{fig759ab}\n\\end{figure}\nContinuing with the more complex external field\n$\\vec{A} = (a_1 ,\\ \\dot{e}_2 (t-t_0 )^2 \/2 ,\\ 0)$,\nthe expected behavior of the correlator ratio $R_2 (t)$ is parabolic\nin the vicinity of the stationary point $t=t_0 $, since now the electric\nfield entering the $\\vec{\\sigma } \\cdot (\\vec{A} \\times \\vec{E} )$\ndependence of (\\ref{haeff}) is linear in time, crossing zero at $t=t_0 $.\nOn the other hand, an additional mass shift proportional to\n$\\vec{\\sigma } \\cdot (\\vec{A} \\times \\dot{\\vec{E} } )$ would manifest\nitself as an additional linear term in $R_2 (t)$, with the effect of\nshifting the parabola away from $t=t_0 $. Fig.~\\ref{fig759ab}~(right),\ndepicting the corresponding numerical result, corroborates the expected\nbehavior; there is no shift of the parabolic time dependence away from\n$t=t_0 $, and therefore no additional dependence on the combination\n$\\vec{\\sigma } \\cdot (\\vec{A} \\times \\dot{\\vec{E} } )$, as expected on\ngeneral grounds, cf.~footnote~\\ref{foot3} further above. One can moreover\nverify that the quadratic coefficient of the parabola in\nFig.~\\ref{fig759ab}~(right) is compatible with the slope in\nFig.~\\ref{fig759ab}~(left) within numerical error, consistent with\nthe above interpretation.\n\\begin{figure}[h]\n\\vspace{0.15cm}\n\\hspace{0.05cm}\n\\epsfig{file=data05.12.ps,width=4.9cm,angle=-90}\n\\caption{Neutron two-point function ratio $R_2 $ ob-\n\\hspace{7.53cm} $\\mbox{\\ \\ } $\ntained in the $m_{\\pi } =759\\, \\mbox{MeV} $ ensemble, given the ex-\n\\hspace{7.62cm} $\\mbox{\\ \\ } $\nternal field $\\vec{A} = (e_1 (t-t_0 ) ,\\ \\dot{e}_2 (t-t_0 )^2 \/2 ,\\ 0)$,\nwhere\n\\hspace{7.58cm} $\\mbox{\\ \\ } $\n$t_0 =6a$. The neutron source is located at $t=0$, and\n\\hspace{7.58cm} $\\mbox{\\ \\ } $\nelectromagnetic fields are in Gaussian units. Cubic\n\\hspace{7.54cm} $\\mbox{\\ \\ } $\nfit was performed using the fit range $t\/a \\in [4,8]$.}\n\\label{fig759c}\n\\vspace{-8.67cm}\n\\end{figure}\n\n\\hspace{7.3cm} \\parbox{7.02cm}{Finally, employing the full external field\n$\\vec{A} = (e_1 (t-t_0 ) ,\\ \\dot{e}_2 (t-t_0 )^2 \/2 ,\\ 0)$,\none would expect to see cubic behavior of $R_2 (t)$ around the stationary\npoint $t=t_0 $, since now the time dependences of $A$ and\n$E$ in $\\vec{\\sigma } \\cdot (\\vec{A} \\times \\vec{E} )$\ncombine to render the latter proportional to $(t-t_0 )^2 $.\nIf this were the only contribution to the neutron mass shift, the slope\nof $R_2 (t)$ at the inflection point $t=t_0 $ would vanish; on the other\nhand, an additional contribution to the mass shift proportional to\n$\\vec{\\sigma } \\cdot (\\vec{E} \\times \\dot{\\vec{E} } )$ would manifest\nitself as an additional linear term in $R_2 (t)$, the slope of which\ncould therefore be read off at the inflection point of the overall cubic\nbehavior. This slope, of course, determines the electric \\hspace{-0.039cm}\nspin \\hspace{-0.039cm} polarizability \\hspace{-0.039cm} $\\gamma_{E1} $\n\\hspace{-0.039cm} of \\hspace{-0.039cm} the \\hspace{-0.039cm} neutron}\nwhich is the ultimate objective of the analysis. Indeed, the corresponding\nnumerical result for $R_2 (t)$ shown in Fig.~\\ref{fig759c} exhibits the\nexpected behavior, including a nonvanishing slope at the stationary point\n$t=t_0 $. From this slope, one extracts the value\n$\\gamma_{E1} = 0.0057(8) \\cdot 10^{-4} \\, \\mbox{fm}^{4} $, from connected\ncontributions only, at the pion mass $m_{\\pi } = 759$ MeV.\n\n\\begin{figure}[t]\n\\vspace{-0.2cm}\n\\epsfig{file=data01.01.ps,width=4.9cm,angle=-90} \\hspace{0.1cm}\n\\epsfig{file=data01.02.ps,width=4.9cm,angle=-90}\n\\caption{Correlator ratios $R_2 $ analogous to Fig.~2,\nexcept in the $m_{\\pi } =357\\, \\mbox{MeV} $ ensemble, with fits\nperformed using the fit range $t\/a \\in [3,5]$.}\n\\label{fig357ab}\n\\vspace{-0.1cm}\n\\end{figure}\n\nTurning to Figs.~\\ref{fig357ab} and \\ref{fig357c}, which present analogous\nresults for the lighter pion mass $m_{\\pi } = 357$ MeV, the principal\ndifference lies in the substantially larger statistical uncertainties, which\npreclude a clear identification of the functional form of the neutron\ncorrelator beyond the time $t=5a$ in the plots. As a result, fits in a\nsymmetric time range around the stationary point $t=6a$, as employed in\nthe $m_{\\pi } = 759$ MeV case, are not viable at the present level of\nstatistics. Instead, pending improvement of the numerical accuracy, a fit\nin the time range $t\/a\\in [3,5]$ was used to identify the preliminary value\n$\\gamma_{E1} = 0.031(21) \\cdot 10^{-4} \\, \\mbox{fm}^{4} $, from connected\ncontributions only, at the pion mass $m_{\\pi } = 357$ MeV. Nevertheless,\nthe data depicted in Figs.~\\ref{fig357ab} and \\ref{fig357c} display behavior\nentirely analogous to the one seen in the heavier pion mass case for small\ntimes, where the numerical fluctuations are under control.\n\nFinally, Fig.~\\ref{figeft} relates the two data points for the spin\npolarizability $\\gamma_{E1} $ extracted above to partially\nquenched chiral perturbation theory \\cite{detcsb}, adjusted to match the\npresent connected diagram calculation. The corresponding expression\ndepends on a number of low energy constants, all but one of which are\ncomparatively well determined from either experiment or previous lattice\ncalcula- $\\mbox{\\hspace{0.5cm}}$\n\\begin{figure}[h]\n\\vspace{-0.54cm}\n\\hspace{0.05cm}\n\\epsfig{file=data01.12.ps,width=4.9cm,angle=-90}\n\\caption{Correlator ratio $R_2 $ analogous to Fig.~3,\n\\hspace{7.78cm} $\\mbox{\\ \\ } $\nexcept\\, in\\, the\\, $m_{\\pi } =357\\, \\mbox{MeV} $\\, ensemble,\\, with\\, fit\n\\hspace{7.8cm} $\\mbox{\\ \\ } $\nperformed using the fit range $t\/a \\in [3,5]$.}\n\\label{fig357c}\n\\vspace{-6.94cm}\n\\end{figure}\n\n\\hspace{7.3cm} \\parbox{7.02cm}{tions on the same ensembles. The lone poorly\ndetermined parameter, $g_1 $, has been adjusted such as to generate the two\ncurves in Fig.~\\ref{figeft} which pass through the upper and lower ends of\nthe error bar of the $\\gamma_{E1} $ measurement at $m_{\\pi } = 357$ MeV. This\nyields a rather stringent estimate of\n$g_1 = -0.15\\,\\,^{+0.01+0.04}_{-0.01-0.10}\\,$,\nwhere the first uncertainty quantifies the spread depicted in\nFig.~\\ref{figeft}, and the second one was obtained by varying the other\nlow energy constants within reasonable bounds. Note that errors due to\nthe truncation of the chiral expansion were not estimated.}\n\n\\newpage\n\n\\section{Summary}\nThe present investigation provides first lattice QCD results for the\nelectric spin polarizability $\\gamma_{E1} $ of the neutron, albeit\nincluding connected diagrams only. The extraction of this quantity was\nfacilitated by the adopted four-point function approach, which allows\nfor the exact elimination of other contributions to the neutron mass\nshift in the external electric field considered, namely, ones associated\nwith the static polarizability $\\alpha_{E} $ and the dispersion\npolarizability $\\alpha_{E\\nu} $. In addition, the effects of constant\nexternal gauge fields, corresponding to quark Bloch momenta on a finite\nspatial torus, had to be carefully disentangled from the spin\npolarizability effect itself. A clear signal for $\\mbox{\\hspace{0.5cm}}$\n\\begin{figure}[h]\n\\vspace{-0.18cm}\n\\hspace{0.05cm}\n\\epsfig{file=new_eftdata_short.ps,width=4.9cm,angle=-90}\n\\caption{Measured lattice data in relation to par-\n\\hspace{7.7cm} $\\mbox{\\ \\ } $\ntially quenched chiral perturbation theory, cf.~text.}\n\\label{figeft}\n\\vspace{-6.67cm}\n\\end{figure}\n\n\\hspace{7.3cm} \\parbox{7.02cm}{\nthe connected contribution to $\\gamma_{E1} $ was obtained at the pion mass\n$m_{\\pi } = 759\\, \\mbox{MeV} $; also a preliminary extraction at\n$m_{\\pi } = 357\\, \\mbox{MeV} $ proved possible, although a refinement of\nthe statistical accuracy is desirable. The most striking feature of the\nconnected contribution to $\\gamma_{E1} $ obtained here is its smallness\ncompared to the result expected for full QCD from chiral perturbation\ntheory \\cite{detcsb}, namely, a negative $\\gamma_{E1} $ larger in modulus\nby roughly two orders of magnitude. This suggests that the electric spin\npolarizability $\\gamma_{E1} $ of the neutron is dominated by the\ndisconnected contributions.}\n\n\\vspace{0.099cm}\n\n\\section*{Acknowledgments}\nDiscussions with F.~X.~Lee, presenting related work at this conference,\nand W.~Detmold are gratefully acknowledged. Computations were performed using\nChroma \\cite{chroma} on U.S.~DOE\/USQCD resources at Fermilab.\nThis work was supported by U.S.~DOE grant DE-FG02-96ER40965.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of discrete dynamical systems is concerned with the the behavior of the iterates of a continuous map on a (usually compact) metric space. The most interesting and studied examples include maps that, in some sense, `mix' the space. For a nice survey on the subject see the article by Kolyada and Snoha \\cite{Kolyada97}.\n\nFormally, let $X$ be a topological space, and let $f$ be a continuous map from $X$ to $X$, write\n$$f^n(x)=\\underbrace{f\\circ f \\circ \\cdots \\circ f}_{n-fold}$$\nto denote the $n^{th}$ iteration of the map $f$.\n\nWe say $f$ is topologically transitive if given any two non-empty open sets $U$ and $V$ of $X$ there exists a natural number $n$ such that $f^n(U) \\cap V \\neq \\emptyset.$\n\nA continuous map $f:X\\to X$ is said to be topologically mixing (or just mixing) if given any two non-empty open sets $U$ and $V$ of $X$ there exists a natural number $N$ such that $f^n(U) \\cap V \\neq \\emptyset$, whenever $n>N$. Not surprisingly, mixing is a stronger condition than topological transitivity. The irrational rotation of the circle is a topologically transitive map that is not mixing.  \n\n\nIt is important to note that the general theory of discrete dynamical systems is usually not concerned with any underlying algebraic structure of the space $X$. Operator theorists, on the other hand, are usually interested in the dynamics of maps preserving the linear structure of the underlying space. In this vein, we will study the dynamics of continuous endomorphisms on topological groups.\n\nIn the setting of linear operators acting on a Fr\\'echet space, a very celebrated result is the set of sufficient conditions for an operator to be mixing known as the Hypercyclicity Criterion. The result first appeared in Kitai \\cite{KITAI82} and was later independently rediscovered by Gethner and Shapiro \\cite{GETHNER87}. Chan \\cite{CHAN01} and Moothatu \\cite{MOO09} independently proved that there is an analogous set of sufficient conditions for a continuous epimorphism to be mixing in the setting of topological groups. These conditions will be discussed in the following section.\n\n\\section{Main Results}\n\nIn what follows $G$ will denote a  metric, complete, separable topological semigroup with identity: a polish semigroup for short.\n\nRecall that for a semigroup $G$ with the semigroup operation written as multiplication, an endomorphism is a map $\\varphi:G\\to G$ such that $\\varphi(gh)=\\varphi(g)\\varphi(h)$ for all $g$ and $h$ in $G$. When $G$ in endowed with a topology compatible with the semigroup structure, we can study the behavior under iteration of such maps.\n\nWe introduce the following definition:\n\n\\begin {definition}[Semigroup Mixing Criterion]\nLet $G$ be a polish semigroup with identity $e$.  We say that a continuous endomorphism $\\varphi:G \\to G$ satisfies the Mixing Criterion if there exists dense sets $F$ and $H$ of $G$, and maps $\\psi_n:F\\to F$ such that, for any $f \\in F$ and $h \\in H$:\n\n(i) $\\varphi^{n}(h)\\to e$ as $n\\to \\infty$.\n\n(ii) $\\psi_n (f) \\to e$ as $n \\to \\infty$.\n\n(iii) $\\varphi^{n}\\psi_n (f)\\to f$ as $n \\to \\infty$.\n\n\\end {definition}\n\nChan \\cite{CHAN01} and Moothatu \\cite{MOO09} independently showed that if $\\varphi$ is a continuous endomorphism on a polish semigroup satisfying the Mixing Criterion, then $\\varphi$ is topologically mixing. Formally, we have.\n\n\\begin{theorem}[Chan \\cite{CHAN01} and Moothatu \\cite{MOO09}]\nLet  $G$ be a polish semigroup with identity $e$ and $f:G \\to G$ a continuous endomorphism.  If $\\varphi$ satisfies the Semigroup Mixing Criterion then $\\varphi$ is mixing.\n\\end{theorem}\n\nWe now apply Theorem 1 to a particular family of maps on the complex unit circle. We will denote by $\\T$ the set of complex numbers of modulus one endowed with the topology induce by the arclength metric, i.e, the distance between two points in $\\T$ is given by the length of the shortest arc joining them. Complex multiplication is then compatible with this topology and we have that $\\T$ is a compact polish group.  We will show that the homomorphisms of the form $f(z)=z^n$, where $n \\geq 2$, are weakly mixing.  Indeed, we have:\n\n\\begin{proposition}\nFix $n \\in \\N$ such that $n \\geq 2$ and let $f:\\T \\to \\T$ be given by $f(z)=z^n$, then f is mixing.\n\\end{proposition}\n\\begin{proof}\nNotice that $f(z)=z^n$ is clearly a continuous endomorphim on $\\T$; we will show that it is mixing by showing it satisfies the hypothesis of the Semigroup Mixing Criterion.  First recall that every element $ z\\in \\T$ can be written uniquely as $z=e^{i\\theta}$ with $0\\leq\\theta<2\\pi$.  Now, consider the set\n\\[\nF_k^n=\\left\\{z \\in \\T ; z^{n^k}=1\\right\\}\n\\]\nWe claim that $F^n=\\cup_{i=1}^{\\infty}F_i^n$ is dense in $\\T$.\n\nNote that $F_i^n$ is a set of $n^i$ evenly spaced points on the unit circle. Thus, $F_i^n$ partitions the unit circle into $n^i$ arcs of the same length. Also note that $F_{i+1}^n$ contains the points in $F_i$ and further subdivides each arc into $n$ arcs of the same length. Consider some fixed $z \\in \\T$ and an arbitrary $\\epsilon>0$. We can find $i$ large enough so that the smallest distance between two points in $F_i^n$ is less than $\\epsilon$ and thus the minimal distance between any element of $\\T$ and some element of $F_i^n$ is less than $\\epsilon$. Thus at least one element of $F_i \\subseteq F$ must be in the $\\epsilon$-ball around $z$.\n\nNow, if we iterate the map $f$, we get that\n\n\\begin{align*}\n& f(z)=z^n, \\\\\n& f^2(z)=f(f(z))=(z^n)^n=z^{n^2}, \\\\\n& f^3(z)=f(f^2(z))=(z^{n^2})^n=z^{n^3}\\\\\n& \\vdots\\\\\n& f^r(z)=z^{n^r}.\\\\\n\\end{align*}\nIf $z \\in F^n$ (thus $z \\in F_k^n$ for some $k$), then for large $N$, $f^N(z)=f^{N-n^k}(f^{n^k}(z))=f^{N-n^k}(1)=1$ so $ f^{r}(z) \\to 1$ as $r \\rightarrow \\infty$.\n\nNow, for each $r \\in \\N$, define $\\psi_r: \\T \\to \\T$ by $\\psi_r(e^{i\\theta})=e^{i{\\frac{\\theta}{n^r}}}$. Observe that $\\psi_r(z) \\to 1$ as $r \\to \\infty$ for all $ z \\in \\T$ and that $f^{r}(\\psi_{r}(z))=z$ for all $z \\in \\T$.  Hence, $f$ is mixing.\n\\end{proof}\n\nIt is a well known result \\cite{RepTheory} that $f(z) = z^n$ where $n \\geq 2$ are the only continuous endomorphisms of $\\T$. Thus, we have the following corollary.\n\n\\begin{corollary}\nEvery continuous epimorphism of $\\T$, except for the identity, is mixing.\n\\end{corollary}\n\nWe now consider the continuous epimorphisms of $\\prod_{i=1}^{k} \\T \\equiv \\T^k$. Since Hom(G,H) distributes over direct sums, we know that all continuous epimorphisms  $f: \\T^n \\rightarrow \\T^n$ are of the form $$f(z_1, z_2, \\cdots, z_n)= (z_1^{m_{1,1}} z_2^{m_{1,2}} \\cdots z_n^{m_{1,n}}, z_1^{m_{2,1}} z_2^{m_{2,2}} \\cdots z_n^{m_{2,n}}, \\cdots, z_1^{m_{n,1}} z_2^{m_{n,2}} \\cdots z_n^{m_{n,n}})$$ where $m_{i,j}$ is a nonnegative integer.  We can show that a large class of such maps is mixing. We have:\n\n\n\\begin{proposition}\nLet $\\sigma$ be a permutation on the set $\\{1, 2, \\cdots, n\\}$. Then the map $f: \\T^k \\rightarrow \\T^k$ defined by $f(z_1, z_2, \\cdots, z_k) = (z_{\\sigma^{-1}(1)}^{m_1}, z_{\\sigma^{-1}(2)}^{m_2}, \\cdots, z_{\\sigma^{-1}(k)}^{m_k})$  where $m_i\\in \\N$ and $m_i \\geq 2$, for all $i$, is mixing if $\\gcd\\{m_j | j \\in \\widetilde{\\sigma}(i)\\} > 1$ for all $i$ (where $\\widetilde{\\sigma}(i)$ is the orbit of i) or if $\\sigma$ is the identity permutation.\n\\end{proposition}\n\n\\begin{proof}\nIf $f(z_1, z_2. \\cdots, x_k)=(z_1^ {m_1}, z_2^ {m_2}, \\cdots, z_k^ {m_k})$ then $f$ is the product of topologically mixing maps and hence, topologically mixing. \\cite{LinearChaos}.\n\nNow, suppose $\\gcd\\{m_j | j \\in \\widetilde{\\sigma}(i)\\} > 1$ for all $i$.\n\nNotice that $f$ is a continuous endomorphism since it can be viewed as a product of continuous endomorphism composed with a permutation of coordinates. We will show $f$ is mixing by showing it satisfies the Semigroup Mixing Criterion.\n\nLet $F^s = \\cup_{i=1}^{\\infty}F_r^s$ where $F_r^s$ is defined as it was in the proof of Proposition 1. Let $s_i = \\gcd\\{m_j | j \\in \\widetilde{\\sigma}(i)\\}$ and let $F = F^{s_1} \\times F^{s_2} \\times \\cdots \\times F^{s_k}$. Note that $F$ is a dense subset of $\\T^n$.\n\nWe will now show that for $z=(z_1, z_2, ..., z_k) \\in F$, $f^n(z_1, z_2, \\cdots, z_k) \\rightarrow 1$. Observe that\n\\[\n\\begin{aligned}\n f^n(z_1, z_2, \\cdots, z_k) &= \\big(\\big(\\big(\\big(z_{\\sigma^{-n}(1)}^{m_{\\sigma^{-(n-1)}(1)}}\\big)^{\\cdots}\\big)^{m_{\\sigma^{-1}(1)}}\\big)^{m_1}, \\cdots,  \\big(\\big(\\big(z_{\\sigma^{-n}(k)}^{m_{\\sigma^{-(n-1)}(k)}}\\big)^{\\cdots}\\big)^{m_{\\sigma^{-1}(k)}}\\big)^{m_k}\\big)\\\\\n&= \\big(z_{\\sigma^{-n}(1)}^{{m_{\\sigma^{-(n-1)}(1)}}\\cdots{m_{\\sigma^{-1}(1)}}{m_1}}, \\cdots, z_{\\sigma^{-n}(k)}^{{m_{\\sigma^{-(n-1)}(k)}}\\cdots{m_{\\sigma^{-1}(k)}}{m_k}}\\big).\n\\end{aligned}\n\\]\nNote that the $i^{th}$ coordinate of the image of\n\n$f^n(z_1, z_2, \\cdots, z_k)$ is $z_{\\sigma^{-n}(i)}^{{m_{\\sigma^{-(n-1)}(i)}}\\cdots{m_{\\sigma^{-1}(i)}}{m_i}}$. For instance if we let $k=5$ and let $\\sigma = (1,3,5,2,4)$. Then $f(z_1, z_2, \\cdots, z_k) = (z_4^{m_1}, z_5^{m_2}, z_1^{m_3}, z_2^{m_4}, z_3^{m_5})$ and the $2^{nd}$ coordinate of $f^3(z_1, z_2, \\cdots, z_k)$ is $\\big(\\big(z_{\\sigma^{-3}(2)}^{m_{\\sigma^{-2}(2)}}\\big)^{m_{\\sigma^{-1}(2)}}\\big)^{m_2} = {{{z_1}^{m_3}}^{m_5}}^{m_2}$\n\nNote that $s_{\\sigma^{-(n)}(i)}^n \\bigm| ({{m_{\\sigma^{-(n-1)}(i)}}\\cdots{m_{\\sigma^{-1}(i)}}{m_i}})$. Since $z_{\\sigma^{-N}(i)} \\in F^{s_{\\sigma^{-N}(i)}}$,  for large $N$ the $i^{th}$ coordinate of $f^N$ is\n$$z_{\\sigma^{-N}(i)}^{{{m_{\\sigma^{-(N-1)}(i)}}\\cdots{m_{\\sigma^{-1}(i)}}{m_i}}} = \\big(z_{\\sigma^{-N}(i)}^{s_{\\sigma^{-(N)}(i)}^N}\\big)^w = 1^w = 1$$ for all $i$, where $w$ is some positive integer. Thus $f^n \\rightarrow 1$ as $n \\rightarrow \\infty$.\n\nNow for each $n \\in \\N$ define $\\psi_n : \\T^k \\rightarrow \\T^k$ by $$\\psi_n(z_1, \\cdots, z_k) = z_{\\sigma^{n}(1)}^{(-m_{\\sigma^n(1)}){\\cdots}(-m_{\\sigma(1)})}, \\cdots, z_{\\sigma^{n}(k)}^{(-m_{\\sigma^n(k)}){\\cdots}(-m_{\\sigma(k)})}.$$\n\nFor instance if we let $k=5$ and let $\\sigma = (1,3,5,2,4)$. Then the $1^{st}$ coordinate of $\\psi_3(z_1, z_2, \\cdots, z_k)$ is $z_{\\sigma^{3}(1)}^{(-m_{\\sigma^3(1)})(-m_{\\sigma^2(1)})(-m_{\\sigma(1)})} = {{{z_2}^{-m_2}}^{-m_5}}^{-m_3}$\nObserve that since $m_i>2$ for all $i$, $\\psi_n(z_1, \\cdots, z_k) \\rightarrow 1$ as $n \\rightarrow \\infty$ and that $f^n(\\psi_n(z)) = z$ for all $z \\in \\T^k$. Hence $f$ is mixing.\n\n\n\n\\end{proof}\n\nNotice that not all continuous endomorphisms of this form are mixing. For instance, when $m_{i,j} = k_j$ for all $i$ , we get $$f(z_1, z_2, \\cdots, z_n) =\n(z_1^{k_1} z_2^{k_2} \\cdots z_n^{k_n}, z_1^{k_1} z_2^{k_2} \\cdots z_n^{k_n}, \\cdots, z_1^{k_1} z_2^{k_2} \\cdots z_n^{k_n})$$\nThe  range of $f$ is a subset of the diagonal of $\\T^k$ (elements of the form $(z,z, \\cdots, z)$). Since the diagonal of $\\T^k$ is a closed subset, its complement is and open set which does not intersect the orbit of any element in the diagonal. Hence $T$ cannot be mixing.\n\nA natural question is whether any polish group supports a mixing endomorphism.  We show that if the group in question can be written as a countable infinite product of isomorphic copies of one of its closed subgroups, then the answer is affirmative.  More precisely, we have the following theorem\n\n\\begin{theorem}\\label{InfiniteThm}\n\\label{ext}\nLet $G$ be a metrizable separable topological group and let $\\varphi:G \\to G$ be a continuous endomorphism. There exists a continuous endomorphism $\\Phi: \\prod_{i=0}^{\\infty} G \\to \\prod_{i=0}^{\\infty} G$ such that\n\\item[(i)] $\\Phi$ is mixing, and\n\\item[(ii)]$\\pi_0 \\circ \\Phi=\\pi \\circ \\varphi$ where $\\pi_0: \\prod_{i=0}^{\\infty} G \\to G$ is the natural projection to the $0^{th}$ coordinate.\n\\end{theorem}\n\n\\begin{proof}\nWe will construct the endomorphism $\\Phi$ and check it is mixing by verifying it satisfies the Semigroup Mixing Criterion.\n\nNotice if we consider a metric $d$ on $G$ which is bounded by 1, we  can define a metric $\\rho$ on $\\prod_{i=0}^{\\infty} G$ via\n \\[\n \\rho(g,h)=\\sum_{i=0}^{\\infty}\\frac{d(g_i,h_i)}{2^i}\n \\]\n\nFor each $g\\in \\prod_{i=0}^{\\infty} G$ write\n\\[\ng=(g_0,g_1, g_2,\\cdots),\\text{   } g_i \\in G,\n\\]\nand define the maps\n\\[\n\\Phi(g)=\\left(\\varphi(g_0)g_1, g_2, g_3, \\cdots \\right)\n\\] and\n\n\\[\n\\Psi(g)=\\left(e,g_0,g_1,g_2, \\cdots \\right)\n\\]\nwhere $e$ is the identity in $G$.\n\nIt is clear that $\\Phi$ is a continuous endomorphism on $\\prod_{i=0}^{\\infty} G$ with $\\pi_0 \\circ \\Phi=\\pi_0 \\circ \\varphi$. Also, for all $g\\in \\prod_{i=0}^{\\infty} G$, $\\Phi(\\Psi(g))=g$ and $\\Psi^ng=\\tilde{e}$  as $n\\to \\infty$ where the identity element $\\tilde{e}$ of $\\prod_{i=0}^{\\infty} G$ is  $(e,e,e,\\cdots)$. We will now verify that $\\Phi$ is mixing. Let $H$ be a dense set in $G$ and consider the subgroup $\\tilde{D}$ in $\\prod_{i=0}^{\\infty} G$ whose elements are of the form $\\left(h_0, h_1, h_2, \\cdots,h_k, e, e, e,\\dots \\right)$ for some natural $k$ and $h_i \\in H$.  This is clearly dense in $\\prod_{i=0}^{\\infty} G$ and for any element $h=\\left(h_0, h_1, h_2, \\cdots,h_k, e, e, e,\\dots \\right) \\in \\tilde{D}$, we have\n\\[\n\\Phi^k(h)=\\left(\\phi^k(h_0)\\varphi^{k-1}(h_1)\\cdots\\varphi(h_{k-1})h_k, e, e, \\cdots \\right)\n\\]and\n\n\\[\n\\Psi^k(\\Phi^k(h))=(\\underbrace{e,e, \\cdots, e}_\\text{ k  positions},\\phi^k(h_0)\\phi^{k-1}(h_1)\\cdots\\phi(h_{k-1})d_k, e, e, \\cdots).\n\\]\nand\n\nNotice that\n \\[\n \\begin{aligned}\n \\rho(\\Psi^k(\\Phi^k(h)), \\tilde{e}) &=\\frac{d(\\phi^k(h_0)\\phi^{k-1}(h_1)\\cdots\\phi(h_{k-1})h_k,e)}{2^k} \\\\\n &\\leq\\frac{1}{2^k} \\to 0 \\text{ as } k \\to \\infty.\n \\end{aligned}.\n \\]\n\n Now, consider the set\n \\[ \\tilde{C}=\\{g(\\Psi^n(\\Phi^n(g))^{-1},g \\in \\tilde{D}, n \\in \\mathbb{N}\\}.\n \\] Notice that\n \\[\n \\begin{aligned}\n \\lim_{n\\to \\infty}g(\\Psi^n(\\Phi^n(g))^{-1}&=g\\lim_{n\\to \\infty}(\\Psi^n(\\Phi^n(g))^{-1}\\\\\n  &=g(\\lim_{n\\to \\infty}(\\Psi^n(\\Phi^n(g))^{-1}\\\\\n &=ge \\\\\n &=g\n \\end{aligned}\n \\] which shows that $\\tilde{C}$ is dense is $\\tilde{G}$.\n\nAlso, let $c \\in \\tilde{C}$, so $c=g(\\Psi^n(\\Phi^n(g))^{-1}$ for some $g \\in \\tilde{G}$ and we have\n\\[\n\\begin{aligned}\n\\Phi^n(c) &=\\Phi^n(g(\\Psi^n(\\Phi^n(g))^{-1})\\\\\n&=\\Phi^n(g)(\\Phi^n(\\Psi^n(\\Phi^n(g))))^{-1}\\\\\n&=\\Phi^n(g)\\tilde{e}(\\Phi^n(g))^{-1}\\\\\n&=\\tilde{e}.\n\\end{aligned}\n\\] Hence, $\\Phi^n \\to \\tilde{e}$ on $\\tilde{C}$ and by Theorem, we conclude $\\Phi$ is mixing.\n \\end{proof}\n\nTheorem \\ref{InfiniteThm} in particular shows that every continuous endomorphism of $\\T$ (most of which are mixing) has an extension to a mixing endomorphism of the countable infinite toroidal group, $\\T^{\\infty}$. In fact, every continuous endomorphism of $\\T^k$ (some of which are not mixing) has an extension to a mixing endomorphism of $\\T^{\\infty}$ since the $\\T^{\\infty}$ is the countable infinite product of any $\\T^k$. Lastly, the theorem shows that every countable infinite product of a polish semigroup admits a topologically mixing endomorphism, since the identity map or the trivial map on the semigroup can be extended to mixing endomorphism on the product.\n\n\\section{Future Work}\n\nA natural extension of our result would be to characterize all the mixing maps on the finite and infinite toroidal groups.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\bf{Introduction}}\n\nQED in $(1+1)$ dimension, e.g., Schwinger model \\cite{SCH} is a\nvery interesting field theoretical model. It has been widely\nstudied over the years by several authors in connection with the\nmass generation, confinement aspect of fermion (quark), charge\nshielding etc. \\cite{LO, COL, CAS, AG}. The description of this\nmodel in noncommutative space time has also been found to give\ninteresting result \\cite{APR, APR1}. Effect of non commutativity\nin space time has showed up as an interacting background with the\nmassive boson of the usual vector Schwinger model. In the\nSchwinger model massless fermion interact with the Abelian gauge\nfield. Photon acquires mass via a kind of dynamical symmetry\nbreaking and the quarks disappear from the physical spectra. The\nexactly solvable nature of the model leads to express this model\nin terms of canonical boson field. It is a remarkable feature of\n(1+1) dimensional exactly solvable interacting fermionic field\ntheory. Gauge invariance was there in the fermionic version of the\nmodel. So initially the model in the bosonised version too studied\nmaintaining gauge invariance. However gauge non invariant\nregularized version of this model also found to be an important\nfield theoretical model \\cite{AR}. Here we find that a one\nparameter class of regularization commonly used to study the\nchiral Schwinger model has been introduced in the vector Schwinger\nmodel. For a specific choice,\n i.e., for the vanishing value of the parameter the model reduces to\n the usual vector Schwinger model but for\n the other admissible value of this parameter the phase space structure\n as well as  the the physical spectra\ngets altered remarkably. This new regularization leads to a change\nin the confinement scenario of the quark too. In fact, the quarks\ngets liberated as it was happened in the Chiral Schwinger model\n\\cite{JR, RABIN, GIR, GIR1, GIR2}.\n\nRecently, we find that the Schwinger model is studied adding\nmasslike term for gauge field with the lagrangian at the classical\nlevel \\cite{USHA1, USHA2}. This model is structurally equivalent\nto the model studied in \\cite{AR}. The masslike term for gauge\ngauge field are introduced in the two models with different\nperspective. In \\cite{AR}, masslike term occurred as a one loop\ncorrection in order to remove the divergence of the fermionic\ndeterminant appeared during bosonization whereas in \\cite{USHA1,\nUSHA2}, the author studied the Schwinger model with the masslike\nterm for gauge field at the classical level. In \\cite{AR}, the\nauthors had reasonably fair  motivation to introduce the masslike\nterm since it is known that in QED, a regularization gets involved\nwhen one calculates the effective action by integrating the\nfermions out. The ambiguity in the regularization has been\nexploited by different authors in different times in (1+1)\ndimensional QED and Chiral QED and different interesting scenarios\nhave been resulted in \\cite{JR, RABIN, GIR, GIR1, GIR2, PM, MG,\nKH, ABD}. The most remarkable one is the chiral Schwinger model\nstudied by Jackiw and Rajaraman \\cite{JR}. They saved the long\nsuffering of the chiral generation of the Schwinger model due to\nHagen \\cite{HAG} from the non-unitary problem introducing a one\nparameter class of regularization. In \\cite{CASA}, the authors\nstudied the Schwinger model introducing masslike term for the\ngauge field at the classical level and showed that for a\nparticular value of the ambiguity parameter the lost gauge\ninvariance of the so called nonconfining (anomalous) Schwinger\nmodel \\cite{AR} gets restored. In \\cite{USHA1, USHA2}, however the\nauthors presented a surprising and untrustworthy result adding the\nsame masslike term at the classical level. There we find that the\nmass generated for the boson is $m=\\sqrt{2}e$ and it does not\ncontain the parameter involved within the masslike term of the\ngauge field!\nIf we look into the the work \\cite{AR}, a structurally equivalent\nmodel to \\cite {USHA1}, we find that the theoretical spectrum\ncontains a massive and a massless boson and the mass of the\nmassive boson acquires a generalized expression with the ambiguity\nparameter. However a massless boson is there in \\cite{USHA1,\nUSHA2}. A parameter free mass term may appear if the added term\nworks as gauge fixing. However the added term was not a legitimate\ngauge fixing term in \\cite{USHA1, USHA2}. A question therefore,\nautomatically comes how did the authors get such untrustworthy\nresult in \\cite{USHA1, USHA2}? The model is thus reinvestigated in\nthis note.\n\nThe vector Schwinger model is described by the following\ngenerating functional.\n\\begin{equation} Z[A] = \\int\nd\\psi d\\bar\\psi e^{\\int d^2x{\\cal L}_f}\n\\end{equation}\\b with\n\\begin{equation}\n{\\cal L}_f = \\bar\\psi\\gamma^\\mu[i\\partial_\\mu + e\\sqrt\\pi\nA_\\mu]\\psi \\nonumber\n\\end{equation}\nThe effective bosonized lagrangian density obtained by integrating\nout the  the fermion one by is\n\\begin{equation}\n{\\cal L}_B = \\frac{1}{2}\\partial_\\mu\\phi \\partial^\\mu\\phi -\ne\\epsilon_{\\mu\\nu}\\partial^\\nu\\phi A^\\mu. \\label{BLD}\n\\end{equation}\nIf electromagnetic background is introduced with masslike term for\nthe gauge field the model reads\n\\begin{equation}\n{\\cal L}_B = \\frac{1}{2}\\partial_\\mu\\phi \\partial^\\mu\\phi -\ne\\epsilon_{\\mu\\nu}\\partial^\\nu\\phi A^\\mu -\n\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} + \\frac{1}{2}m_0 A_{\\mu}A^{\\mu}.\n\\label{BLM}\n\\end{equation}\nHere Lorentz indices runs over the two values $0$ and $1$\ncorresponding to the two space time dimension and the rest of the\nnotation are standard. The antisymmetric tensor is defined with\nthe convention $\\epsilon_{01}=+1$. The coupling constant $e$ has\none mass dimension in this situation. The parameter $m_0$ is\nintroduced to represent the masslike term for the gauge field at\nthe classical level in the same way as it was done in the\nThirring-Wess model \\cite{THIR}. In \\cite{USHA1, USHA2} the\nauthors used $m_0=ae^2$. Here we  would like to mention that the\nauthors in \\cite{USHA1} used a regularization parameter $M$ in the\ngeneralized bosonized lagrangian and set it to zero along with few\nothers parameter to get vector Schwinger mode. Then again the\nauthor added a mass like term for the gauge field ${\\cal L}_m =\n\\frac{1}{2}ae^2A_\\mu A^\\mu$ and termed this $a$ again as standard\nregularization parameter. It is highly confusing at this level.\n\nLet us now proceed to study the phase space structure of the\n model. To this end it is necessary to calculate the\nthe momenta corresponding to the field $A_0$, $A_1$ and $\\phi$.\nFrom the standard definition of the momentum we obtain\n\\begin{equation}\n\\pi_0=0 \\label{M1},\n\\end{equation}\n\\begin{equation}\n\\pi_1 = F_{01}\\label{M2},\n\\end{equation}\n\\begin{equation}\n\\pi_\\phi = \\dot\\phi - eA_1\\label{M3},\n\\end{equation}\nwhere $\\pi_0$, $\\pi_1$ and $\\pi_\\phi$ are the momenta\ncorresponding to the field $A_0$, $A_1$ and $\\phi$. Using the\nequations (\\ref{M1}), (\\ref{M2}) and (\\ref{M3}), the Hamiltonian\ndensity are calculated:\n\\begin{equation}\n{\\cal H} = \\frac{1}{ 2}(\\pi_\\phi +eA_1)^2 + \\frac{1}{2}\\pi_1^2 +\n\\frac{1}{2}\\phi'^2 + \\pi_1A_0' - eA_0\\phi' - {1\\over 2}m_0(A_0^2 -\nA_1^2).\\end{equation} Note that $\\omega = \\pi_0 \\approx 0$, is the\nfamiliar primary constraint of the theory. The preservation of the\n constraint $\\omega$\nrequires $[\\omega(x), H(y)] = 0$, which leads to the Gauss' law as\na secondary constraint:\n\\begin{equation}\n\\tilde\\omega = \\pi_1' + e\\phi' + m_0A_1 \\approx 0. \\label{SCO}\n\\end{equation}\nThe constraints (\\ref{M1}) and (\\ref{SCO}) form a second class\nset. Treating (\\ref{M1}) and (\\ref{SCO}) as strong condition one\ncan eliminate $A_0$ and obtain the reduced Hamiltonian density as\nfollows.\n\\begin{equation}\n{\\cal H}_r = \\frac{1}{2}(\\pi_\\phi + eA_1)^2 +\n\\frac{1}{2m_0}(\\pi'_1 + e\\phi')^2 + \\frac{1}{2}({\\pi_1}^2 +\n\\phi'^2)^2 + \\frac{1}{2}m_0A_1^2. \\label{RHAM}\n\\end{equation}\nAccording to the Dirac's prescription \\cite{DIR} of quantizing a\ntheory with second class constraint the Poisson brackets becomes\ninadequate for this situation. This type of systems however remain\nconsistent with the Dirac brackets \\cite{DIR}. It is\nstraightforward to show that the Dirac brackets between the fields\ndescribing the reduced hamiltonian (\\ref{RHAM}) remain canonical.\nUsing the canonical Dirac brackets the following first order\nequations of motion  are found out from the reduced Hamiltonian\ndensity (\\ref{RHAM}).\n\\begin{equation}\n\\dot A_1= \\pi_1 -\\frac{1}{m_0}(\\pi_1'' + e\\phi''), \\label{EQM1}\n\\end{equation}\n\\begin{equation}\n\\dot\\phi = \\pi_\\phi + eA_1,\\label{EQM2}\\end{equation}\n\\begin{equation}\n\\dot \\pi_\\phi = (1+ \\frac{e^2}{m_0})\\phi'' + \\frac{e}{m_0}\\pi_1''\n, \\label{EQM3}\n\\end{equation}\n\\begin{equation}\n\\dot\\pi_1 = -e\\pi_\\phi - (m_0+e^2)A_1. \\label{EQM4}\n\\end{equation}\nNote that in \\cite{USHA1},  all the equations of motion except the\nequation  of motion corresponding to equation (\\ref{EQM3}) were\nidentical. In \\cite{USHA1}, the calculational mistake started from\nthat erroneous equation of motion. A little algebra converts the\nabove first order equations (\\ref{EQM1}), (\\ref{EQM2}),\n(\\ref{EQM3}) and (\\ref{EQM4}) into the following second order\nequations:\n\\begin{equation}\n(\\Box + (m_0+ e^2)\\pi_1 = 0, \\label{SP1}\n\\end{equation}\n\\begin{equation}\n\\Box[\\pi_1 + \\frac{m_0+e^2}{e}\\phi] = 0. \\label{SP2}\n\\end{equation}\nEquation (\\ref{SP1}) describes a massive boson field with mass $m\n=\\sqrt{m_0+ e^2}$ whereas equation (\\ref{SP2}) describes a\nmassless scalar field. The result clearly shows that the mass\nacquires a generalized expression with the parameter involved in\nthe masslike term at the classical level as it can be expected\nfrom the result of the paper \\cite{AR}. This boson can be\nidentified with the photon that acquired mass via a dynamical\nsymmetry breaking. The massless boson (\\ref{SP2}) is equivalent to\nthe massless fermion in (1+1) dimension. So fermion here does not\nconfine. It remains free as it has been found in chiral Schwinger\nmodel \\cite{JR, RABIN, GIR, GIR1, GIR2} and the so called\nnonconfining Schwinger model \\cite{AR, AR1}. In this context we\nshould mention that there is some confusion in the literature\nregarding the conclusion concerning confinement and de-confinement\nscenario of fermion but the result in this context is considered\nto be more or less standard now \\cite{LO, AR, GIR2, PM, MG, AR1}.\nThe nature of the theoretical spectrum becomes more transparent if\nwe calculate the fermionic propagator to which we now tern.\n\nTo calculate fermion propagator one needs to work with the\noriginal fermionic model. The calculation is analogous to the so\ncalled nonconfining Schwinger model \\cite{AR}. The same\ncalculation for chiral Schwinger model is available in \\cite{GIR,\nGIR1}. The effective action obtained by integrating out $\\phi$\nfrom the bosonized action (\\ref{BLM}) is\n\\begin{equation}\nS_{eff}=\\int d^2x\\frac{1}{2}[A_\\mu(x)\nM^{\\mu\\nu}A_\\nu(x)],\\end{equation} where,\n\\begin{equation}\nM^{\\mu\\nu}= m_0 g^{\\mu\\nu} - \\frac{\\Box\n+e^2}{\\Box}\\tilde\\partial^\\mu\\tilde\\partial^\\nu.\\end{equation}\nHere we have used the standard notation\n$\\tilde\\partial^\\mu=\\epsilon^{\\mu\\nu}\\partial_\\nu.$\n The gauge field propagator is just\nthe inverse of $M^{\\mu\\nu}$ and it is found to be\n\\begin{equation}\n\\Delta_{\\mu\\nu}(x-y)=\\frac{1}{m_0}[g_{\\mu\\nu}+ \\frac{\\Box +e^2}{\n\\Box(\\Box + (m_0+e^2))} \\tilde\\partial_\\mu\\tilde\\partial_\\nu\n]\\delta(x-y).\n\\end{equation}\nNote that the two poles of propagator are found at the expected\npositions. One at zero and another at $m_0 + e^2$ indicating a\nmassive and a massless excitations.\n\nSetting an  Ansatz, for the Green function of the Dirac operator\n$(i\\partial\\!\\!\\!\\!\/ -gA\\!\\!\\!\\!\/)$, enable us to construct the\npropagator of the original fermion $\\psi$. The conventional\nconstruction of the Ansatz is\n\\begin{equation}\nG(x,y;A)=e^{ie(\\Phi(x)-\\Phi(y))}S_F(x-y),\\label{ANST}\n\\end{equation} where $S_F$  is  the  free, massless fermion\npropagator and $\\Phi$ is determined when the Ansatz (\\ref{ANST})\nis plugged into the equation for the Green function. From the\nstandard construction the Green function can be written down as\n\\begin{equation}\nG(x,y;A)=e^{ie\\int d^2zA^\\mu(z)J_\\mu(z)}S_F(x-y),\\end{equation}\nwhere the {\\it current} $J_\\mu$ has the following expression.\n\\begin{equation}\nJ_\\mu=(\\partial^z_\\mu  +  \\gamma_5\\tilde\\partial^z_\\mu)(D_F(z- x)\n-D_F(z-y)).\\end{equation} Here $D_F$ is the propagator of a\nmassless  free  scalar  field. Such propagators have to be\ninfra-red regularized in two dimensions \\cite{LO}:\n\\begin{equation}\nD_F(x)=-{i\\over 4\\pi}ln(-\\mu^2x^2+i0),\\end{equation} where $\\mu$\nis the infra-red regulator mass.\n\nFinally we obtain the  fermion  propagator by functionally\nintegrating $G(x,y;A)$ over the gauge field:\n\\begin{eqnarray}\nS'_F&=&\\int {\\cal D}A e^{\\frac{i}{2}\\int d^2z~(A_\\mu(z)\nM^{\\mu\\nu}A_\\nu(z) + 2eA_\\mu   J^\\mu)}S_F(x-y)\\nonumber\\\\&=&{\\cal\nN} \\exp[\\frac{D_F}{\\frac{m_0^2 + m_0e^2}{e^4}}+\n\\frac{\\Delta_F(m^2=m_0 +\ne^2)}{\\frac{m_0+e^2}{e^2}}]~~S_F.\\label{PROP} \\end{eqnarray} Here\n$\\Delta_F$ is the propagator of a massive free scalar field and\n${\\cal N}$ is a wave function renormalization factor.\n\nBoth the results therefore confirms strongly that the mass\nacquires a generalized expression with bare coupling constant and\nthe parameter involved in the masslike term for the gauge field.\nIt is known that in vector Schwinger model this type of parameter\nfree mass generates. In fact, in the bosonized vector Schwinger\nmodel no such parameter exists if gauge fixing is not introduced\nat the lagrangian level. In this context, note that setting $a=0$,\nin the bosonized lagrangian of the so called nonconfining\nSchwinger model \\cite{AR} one can obtain vector Schwinger model\n\\cite{SCH}.\n\nSo from our result it can be concluded that it is the\ncalculational error that led the authors of \\cite{USHA1, USHA2} to\nland in to this type of untrustworthy result. We have pointed out\nin the body of this paper where the the actual error started.\nActually, the use of that erroneous equation corresponding to the\nequation  (\\ref{EQM3}) led them to reach to the wrong expression\nof the mass term for photon. The fermionic propagator presented in\n\\cite{USHA1, USHA2} also carried the same error. In\nequation(\\ref{PROP}) of this note the correct expression fermionic\noperator is calculated.\n\nAs a concluding remark we would like to mention that the term\n$\\frac{1}{2}ae^2A^\\mu A_\\mu$ is not a legitimate gauge fixing\nterm. With this term the vector Schwinger model turns into the so\ncalled nonconfining Schwinger model \\cite{AR, AR1}. Introduction\nof proper gauge fixing term like $\\frac{\\alpha}{2e^2}(\\partial_\\mu\nA^\\mu)^2$ in the starting lagrangian of vector Schwinger  model\nhowever leads to a result that corresponds to parameter free mass\nterm for the photon though there exists a parameter $\\alpha$ in\nthe starting lagrangian.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we develop an algorithm to invert the $j$--function in quasilinear time, and give an application to testing whether an elliptic curve has complex multiplication.\n\nIt is known that the inverse of $j$ is equal to a ratio of two Gaussian hypergeometric functions, though we do not believe any analysis has been made of the running time or required precision to invert $j$ by this method.\nOur method is similar to that of \\cite{fasttheta}, which makes use of addition formulae between Jacobi's theta functions in order to reduce the argument to a fixed compact set -- here we repeatedly make use of the modular polynomial \n\\begin{align*}\n \\Phi_2(X,Y)=X^3&+Y^3-X^2Y^2+1488X^2Y+1488XY^2-162000X^2-162000Y^2\\\\+&40773375XY+8748000000X+8748000000Y-157464000000000,\n\\end{align*}\nwhich has the property that the roots of $\\Phi_2(j(\\tau),z)$ are $j(2\\tau)$, $j\\left(\\frac{\\tau}{2}\\right)$, and $j\\left(\\frac{\\tau+1}{2}\\right)$,\n to either compute $j(2^k\\tau)$, the logarithm of which is a close approximation to $-2^{k+1}\\pi\\tau$, or to manipulate the argument of $j$ into a compact set to which Newton's method may be applied.\n\n We define the \\emph{regulated precision} of an approximation $\\tilde{\\alpha}$ to $\\alpha$ to be\n\\begin{equation*}\n \\frac{\\lvert\\alpha-\\tilde{\\alpha}\\rvert}{\\max\\{1,\\lvert\\alpha\\rvert\\}},\n\\end{equation*}\nand denote by $M(P)$ the computational complexity of multiplication of two $P$--bit integers, which by a recent result of Harvey and Hoeven \\cite{fastmult} may be taken to be $O(P\\log P)$. We obtain the following,\n\\begin{theorem}\n Suppose that $\\tilde{j}$ is an approximation to $j(\\tau)$, $\\tau\\in\\mathcal{F}$, of regulated precision $2^{-P}$, with $P\\geq400$. Let $Q=P\/6$ if $\\lvert \\tau-i\\rvert\\leq2^{-30}$ or $\\left\\lvert \\tau-\\frac{\\pm1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$, and $Q=P-\\max\\{11\\log P,100\\}$ otherwise. Then we may obtain an approximation to $\\tau$ of relative precision $2^{-Q}$ in time\n \\begin{equation*}\n  O(M(P)\\log (P)^2).\n \\end{equation*}\n\\end{theorem}\nThe $j$--function has two ramification points in its fundamental domain, which entails the loss of precision in its inversion when $j$ is close to $0$ or $1728$.\nWe apply this algorithm to test for complex multiplication of elliptic curves -- given an approximation to the $j$--invariant of an elliptic curve and a bound upon its height and degree, we may invert it and determine if the inverse is a quadratic irrational, determining also the discriminant,\n\\begin{theorem}\n Suppose that $j(\\tau)$ is the $j$--invariant of an elliptic curve $E$, with $j(\\tau)$ of degree bounded by $d$ and height bounded by $H\\geq e^e$. Then it may be determined from $d$, $H$, and an approximation to $j$ of regulated precision $2^{-300d^2\\log H(\\log d+\\log\\log H)^2-200}$ whether $E$ has complex multiplication, and if so the associated discriminant, in time, letting $T=d^2\\log H(\\log d+\\log\\log H)^2$, \n\\begin{equation*}\n O(M(T)(\\log T)^2).\n\\end{equation*}\n\\end{theorem}\nPrevious methods include that of \\cite{achter}, based on reduction of elliptic curves at primes, which has an unconditional running time of $O(H^{cd})$, and assuming the Generalized Riemann Hypothesis a running time of $O(d^2(\\log H)^2)$, and two tests of \\cite{xavier}, comprising a deterministic algorithm based on Galois representations associated to torsion points, with running time $O(d^{c_1}(\\log H)^{c_2})$, with an ineffective implicit constant, and a probabilistic algorithm, also of polynomial running time.\n\nWe note that one may also apply our algorithm for the inversion of $j$ to detecting isogenies between two elliptic curves of running time $O(N\\log N\\log\\log N)$, where the degree of the isogeny is bounded by $N$, though our implicit constant is ineffective as explicit bounds on the coefficients of modular polynomials $\\Phi_N(X,Y)$ for composite indices are not currently available.\n\\section{Preliminaries}\nWe will denote by $\\mathcal{F}$ the usual fundamental domain of $j(z)$, $\\left\\{z|-\\frac{1}{2}<\\text{\\normalfont Re}(z)\\leq\\frac{1}{2},\\lvert z\\rvert>1\\right\\}\\cup\\{z|\\lvert z\\rvert=1,\\text{\\normalfont Re}(z)\\geq0\\}$.\nThroughout we will make use of the following results,\n\\begin{lemma}[Lemma 1 of \\cite{j2079}]\\label{j_q}\nIf $\\tau\\in\\mathcal{F}$\n \\begin{equation*}\n  \\left\\lvert j(\\tau)-e^{-2\\pi i\\tau}\\right\\rvert\\leq2079.\n \\end{equation*}\n\\end{lemma}\n\n\\begin{theorem}[Kantorovich, \\cite{kantorovich}]\\label{kantorovich}\n Let $F:S(x_0,R)\\subset X\\to Y$ have a continuous Fr\\'{e}chet derivative in $\\overline{S(x_0,r)}$. Moreover, let \\emph{(i)} the linear operation $\\Gamma_0=[F'(x_0)]^{-1}$ exist; \\emph{(ii)} $\\lVert\\Gamma_0 F(x_0)\\rVert\\leq\\eta$; \\emph{(iii)} $\\lVert\\Gamma_0F''(x)\\rVert\\leq K$ $(x\\in\\overline{S(x_0,r)})$. Now, if\n \\begin{equation*}\n  h=K\\eta\\leq\\frac{1}{2}\n \\end{equation*}\nand \n\\begin{equation*}\n r\\geq\\frac{1-\\sqrt{1-2h}}{h}\\eta,\n\\end{equation*}\nthen $F(x)=0$ will have a solution $x^*$ to which the Newton method is convergent. Here,\n\\begin{equation*}\n \\lVert x^*-x_0\\rVert\\leq r_0.\n\\end{equation*}\nFurthermore, if for $h<\\frac{1}{2}$,\n\\begin{equation*}\n r<r_1=\\frac{1+\\sqrt{1-2h}}{h}\\eta,\n\\end{equation*}\nor for $h=\\frac{1}{2}$\n\\begin{equation*}\n r\\leq r_1,\n\\end{equation*}\nthe solution $x^*$ will be unique in the sphere $\\overline{S(x_0,r)}$. The speed of convergence is characterized by the inequality\n\\begin{equation*}\n \\lVert x^*-x_k\\rVert\\leq\\frac{1}{2^k}(2h)^{2^k}\\frac{\\eta}{h}\n\\end{equation*}\nfor $k=0,1,2,\\ldots$.\n\\end{theorem}\nWe note that the condition \n\\begin{equation*}\n r\\geq\\frac{1-\\sqrt{1-2h}}{h}\\eta\n\\end{equation*}\nmay be replaced with\n\\begin{equation*}\n r\\geq 2\\eta.\n\\end{equation*}\n\n\n\\section{Inversion of $j(z)$}\nFirstly, if $\\lvert j \\rvert\\leq 2^{-P\/2}$ or $\\lvert j-1728\\rvert\\leq 2^{-P\/3}$, we return $\\tau = \\frac{1+i\\sqrt{3}}{2}$ or $\\tau = i$ respectively, and otherwise, we split the fundamental domain of $j$ into $4$ sections -- $\\text{\\normalfont Im}(\\tau)\\geq 3$, $\\lvert\\tau-i\\rvert\\leq 2^{-{31}}$, $\\left\\lvert\\tau-\\frac{1+\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, $\\left\\lvert\\tau-\\frac{-1+\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$ and the remaining compact subset of the fundamental domain. We may determine $\\tau$ with sufficient precision by taking a low precision inverse via the following expression for $j^{-1}$ in terms of Gaussian hypergeometric functions -- with $\\alpha$ a solution to \n\\begin{equation*}\n j(\\tau)=\\frac{1728}{4\\alpha(1-\\alpha)},\n\\end{equation*}\n$\\tau$ is equal to either\n\\begin{align*}\n i\\frac{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,1-\\alpha\\right)}{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\alpha\\right)}.\n\\end{align*}\nor the negative of its inverse.\nIf $j$ is sufficiently large, or close to $0$ or $1728$, then we do not need to evaluate this in order to determine which section of the fundamental domain $\\tau$ lies in, so we need only compute the above to a fixed precision in a compact set, at points which it is easily shown are bounded away from zeros of $_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,z\\right)$, so this takes only constant time.\n\\iffalse\nFirstly, we note that in the case of real $j$, one may take the elliptic curve\n\\begin{equation*}\n E:y^2=x^3-\\frac{27j}{j-1728}x-\\frac{27j}{j-1728},\n\\end{equation*}\nwhich has $j$--invariant $j$, and use the algorithm of (???, p.391) to compute a basis $\\omega_1,\\omega_2$ of the lattice of $E$, taking $\\tau = \\frac{\\omega_2}{\\omega_1}$, and then transforming this to the fundamental domain to obtain the inverse of $j$, which would give a faster algorithm, at least for $j$ away from $0$ and $1728$.\n\\fi\n\\subsection{Large $j$}\n\nThroughout this section we assume $\\text{\\normalfont Im}(\\tau)\\geq 3$, and will repeatedly make use of the consequent fact that $\\lvert j(\\tau)\\rvert\\geq 10^8$. We will make use of the modular polynomial $\\Phi_2(X,Y)$ to obtain an approximation to $j(2\\tau)$, and repeat the process until we have an approximation to $j(2^k\\tau)$, where $k$ is sufficiently large, at which point taking the logarithm of $j(2^k\\tau)$ gives a good approximation to $-2^{k+1}\\pi\\tau$, owing to the $q$--series of $j$.\n\n\\begin{proposition}\\label{phi2discrepancy}\n Let $j(\\tau)=j$, and suppose that $\\text{\\normalfont Im}(\\tau)\\geq3$ and $\\tilde{j}$ is an approximation to $j$ of relative precision at least $2^{-P}$, with $P\\geq300$. Then the largest root, in absolute value, of $\\Phi_2(\\tilde{j},z)$ is an approximation to $j(2\\tau)$ of relative precision at least $2^{-P+2}$.\n\\end{proposition}\n\\begin{proof}\n Let $f(z) = \\Phi_2(j,z)$ and $g(z) =\\Phi_2(\\tilde{j},z)$. We first bound the coefficients of $f(z)-g(z)$. For the coefficient of $z^2$, we have, with $\\tilde{j}=j+\\delta$, \n \\begin{align*}\n  \\left\\lvert -j^2+1488j-162000-(-(j+\\delta)^2+1488(j+\\delta)-162000)\\right\\rvert &= \\lvert 2\\delta j+\\delta^2+1488\\delta\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert,\n \\end{align*}\nas $\\lvert \\delta\\rvert\\leq2^{-300}\\lvert j\\rvert$ and $\\lvert j\\rvert\\geq 10^8$.\nFor the coefficient of $z$, we have\n\\begin{align*}\n  &\\left\\lvert 1488j^2 +40773375j+8748000000-(1488(j+\\delta)^2+40773375(j+\\delta)+8748000000)\\right\\rvert\\\\\n  &\\quad\\quad\\quad= \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  &\\quad\\quad\\quad\\leq 3210\\lvert\\delta\\rvert\\lvert j\\rvert.\n \\end{align*}\n For the constant coefficient, we have\n \\begin{align*}\n  &\\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\\\\n  &\\quad\\quad\\quad\\leq 3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{align*}\n We now bound the values of $g(j(\\tau'))$ for $\\tau'\\in\\left\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\right\\}$.\n For $j(2\\tau)$, as $\\lvert j(2\\tau)\\rvert\\leq 1.02\\lvert j\\rvert^2$,\n \\begin{align*}\n  \\lvert g(j(2\\tau))\\rvert &= \\lvert g(j(2\\tau))-f(j(2\\tau))\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert(1.02\\lvert j\\rvert^2)^2+3210\\lvert\\delta\\rvert\\lvert j\\rvert\\cdot1.02\\lvert j\\rvert^2+3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n  &\\leq 2.2\\lvert\\delta\\rvert\\lvert j\\rvert^5.\n \\end{align*}\n For $g(j(\\tau'))$, $\\tau'\\in\\left\\{\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\right\\}$,  firstly, the absolute values of $j\\left(\\frac{\\tau}{2}\\right)$ and $j\\left(\\frac{\\tau+1}{2}\\right)$ are bounded above and below by $e^{2\\pi\\text{\\normalfont Im}(\\tau)\/2}\\pm2079$, and the absolute value of $j$ is bounded above and below by $e^{2\\pi\\text{\\normalfont Im}(\\tau)}\\pm2079$. As $\\text{\\normalfont Im}(\\tau)\\geq 3$, these yield \n\\begin{equation}\n0.83\\lvert j\\rvert^{1\/2}\\leq\\left\\lvert j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert, \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\leq 1.17\\lvert j\\rvert^{1\/2},\n\\end{equation}\nso \n\\begin{align*}\n  \\lvert g(j(\\tau'))\\rvert &= \\lvert g(j(\\tau'))-f(j(\\tau'))\\rvert\\\\\n  &\\leq 2.1\\lvert\\delta\\rvert\\lvert j\\rvert(1.17\\lvert j\\rvert^{1\/2})^2+3210\\lvert\\delta\\rvert\\lvert j\\rvert(1.17\\lvert j\\rvert^{1\/2})+3.4\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n  &\\leq 20\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{align*}\nLet $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g(z)$. Then for $\\beta_i$ the closest root of $g$ to $j(2\\tau)$,\n\\begin{equation*}\n \\lvert j(2\\tau)-\\beta_i\\rvert \\leq \\left( 2.2\\lvert\\delta\\rvert\\lvert j\\rvert^5\\right)^{1\/3}\\leq \\left(2.2\\cdot2^{-100}\\lvert j\\rvert^6\\right)^{1\/3}\\leq10^{-9}\\lvert j\\rvert^2,\n\\end{equation*}\nand we let $\\beta_0$ be the closest root of $g$ to $j(2\\tau)$. As $0.98\\lvert j\\rvert^2\\leq j(2\\tau)\\leq1.02\\lvert j\\rvert^2$, $0.97\\lvert j\\rvert^2\\leq \\lvert\\beta_0\\rvert\\leq1.03\\lvert j\\rvert^2$. \nFor $\\tau'=\\frac{\\tau}{2}$, with $\\beta_i$ the closest root of $g$ to $j(\\tau')$,\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_i\\rvert \\leq \\left( 20\\lvert\\delta\\rvert\\lvert j\\rvert^2\\right)^{1\/3}\\leq \\left(20\\cdot2^{-300}\\lvert j\\rvert^3\\right)^{1\/3}\\leq10^{-29}\\lvert j\\rvert,\n\\end{equation*}\nand in particular, \n\\begin{equation}\n \\lvert\\beta_i\\rvert\\leq 10^{-29}\\lvert j\\rvert+1.17\\lvert j\\rvert^{1\/2}<0.97\\lvert j\\rvert^2\\leq\\lvert\\beta_0\\rvert,\n\\end{equation}\nso $\\beta_i\\neq \\beta_0$. Let this root of $g$ be $\\beta_1$. Now we may improve the bound on $\\lvert j(\\tau')-\\beta_1\\rvert$,\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_1\\rvert^2\\leq \\frac{20\\lvert\\delta\\rvert\\lvert j\\rvert^2}{\\lvert j(\\tau')-\\beta_0\\rvert}\\leq\\frac{20\\lvert\\delta\\rvert\\lvert j\\rvert^2}{0.98\\lvert j\\rvert^2-1.17\\lvert j\\rvert^{1\/2}}\\leq30\\cdot2^{-300}\\lvert j\\rvert,\n\\end{equation*}\nso\n\\begin{equation*}\n \\lvert j(\\tau')-\\beta_1\\rvert\\leq 10^{-88}\\lvert j\\rvert^{1\/2},\n\\end{equation*}\nand consequently \n\\begin{equation*}\n 0.82\\lvert j\\rvert^{1\/2}\\leq \\lvert\\beta_1\\rvert\\leq1.18\\lvert j\\rvert^{1\/2}.\n\\end{equation*}\nWe now bound $\\beta_2$. By our bound on the difference between the constant coefficients of $f$ and $g$, the constant coefficient $-\\beta_0\\beta_1\\beta_2$ of $g$ is bounded in absolute value by\n\\begin{equation*}\n \\left\\lvert j(2\\tau)j\\left(\\frac{\\tau}{2}\\right)j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert + 3.4\\delta\\lvert j\\rvert^2\\leq 1.4\\lvert j\\rvert^3+0.01\\lvert j\\rvert^3\\leq 1.5\\lvert j\\rvert^3,\n\\end{equation*}\nso that\n\\begin{equation*}\n \\lvert\\beta_2\\rvert\\leq \\frac{1.5\\lvert j\\rvert^{3}}{0.97\\lvert j\\rvert^2\\cdot0.82\\lvert j\\rvert^{1\/2}}\\leq 2\\lvert j\\rvert^{1\/2}.\n\\end{equation*}\nReturning to $g(j(2\\tau))$, we now bound below the terms $\\lvert j(2\\tau)-\\beta_i\\rvert$ for $i=1,2$ in order to improve our inequality for $\\lvert j(2\\tau)-\\beta_0\\rvert$. We now have, for $i=1,2$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_{i}\\rvert &\\geq \\lvert j(2\\tau)\\rvert - 2\\lvert j\\rvert^{1\/2}\\\\\n &\\geq0.97\\lvert j\\rvert^2.\n\\end{align*}\nThis now improves our bound on the difference of $\\beta_0$ to $j(2\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_0\\rvert &\\leq \\frac{2.2\\delta\\lvert j\\rvert^5}{\\lvert j(2\\tau)-\\beta_1\\rvert\\lvert j(2\\tau)-\\beta_2\\rvert} \\\\\n &\\leq \\frac{2.2\\delta\\lvert j\\rvert^5}{(0.97\\lvert j\\rvert^2)^2}\\\\\n &\\leq 2.4\\delta\\lvert j\\rvert.\n\\end{align*}\nSo the relative precision of $\\beta_0$ as an approximation to $j(2\\tau)$ is bounded by\n\\begin{equation*}\n \\frac{\\lvert j(2\\tau)-\\beta_0\\rvert}{\\lvert j(2\\tau)\\rvert}\\leq \\frac{2.4\\lvert \\delta\\rvert\\lvert j\\rvert}{0.97\\lvert j\\rvert^2}\\leq2.5\\frac{\\lvert\\delta\\rvert}{\\lvert j\\rvert}\\leq 2^{-P+2}.\n\\end{equation*}\n\\end{proof}\n\n\\begin{proposition}\n If $\\text{\\normalfont Im}(\\tau)\\geq 3$ and $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision $2^{-P}$, where $P$ is at least $300$, applying Newton's method to $\\Phi_2(\\tilde{j},z)$, with starting point\n \\begin{equation*}\n  \\tilde{j}^2-2\\cdot744\\tilde{j}-2\\cdot196884+744^2+744,\n \\end{equation*}\nwill obtain an approximation to $j(2\\tau)$ of relative precision $2^{-P+3}$ after at most $[2\\log P+\\log\\log\\lvert j\\rvert]$ steps of Newton iteration.\n\\end{proposition}\n\\begin{proof}\nWe first give a rough approximation to $\\beta_0$. Writing\n\\begin{equation*}\n j(\\tau) = e^{-2\\pi i\\tau}+744+196884e^{2\\pi i\\tau} + f(\\tau),\n\\end{equation*}\nwe have\n\\begin{align*}\n j(\\tau)^2-2\\cdot744j(\\tau)&+2\\cdot19688+2\\cdot744^2+744 \\\\\n &= e^{-4\\pi i\\tau}+744+196884^2e^{4\\pi i\\tau}+2(196884e^{2\\pi i\\tau}+e^{-2\\pi i\\tau})f(\\tau)+f(\\tau)^2\n\\end{align*}\nso that, as $\\text{\\normalfont Im}(\\tau)\\geq 3$, $f$ is maximized when $\\text{\\normalfont Re}(\\tau)=0$, and $f(\\text{\\normalfont Im}(\\tau))$ is decreasing in $\\text{\\normalfont Im}(\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau) &- (j(\\tau)^2-744j(\\tau)+19688-744^2+744)\\rvert\\\\\n &\\leq (196884+196884^2)e^{-4\\pi\\text{\\normalfont Im}(\\tau)}+\\lvert f(\\text{\\normalfont Im}(2\\tau))\\rvert\\\\\n &\\quad\\quad+2(196884e^{2\\pi \\text{\\normalfont Im}(\\tau)}+e^{2\\pi \\text{\\normalfont Im}(\\tau)})\\lvert f(\\text{\\normalfont Im}(\\tau))\\rvert+\\lvert f(\\text{\\normalfont Im}(\\tau))\\rvert^2\\\\\n &\\leq 10^{-7}+\\lvert f(6)\\rvert+10^6\\lvert f(3)\\rvert+\\lvert f(3)\\rvert^2\\\\\n &\\leq 0.4\n\\end{align*}\nWe now let \n\\begin{equation*}\n \\gamma_0 = \\tilde{j}^2-744\\tilde{j}+19688-744^2+744.\n\\end{equation*}\nThis will be our starting point for Newton iteration to find $\\beta_0$. We now bound the terms appearing in Kantorovich's theorem. Firstly,\n\\begin{align*}\n\\Phi_2(\\tilde{j},z)=z^3&+(-\\tilde{j}^2+1488\\tilde{j}-162000)z^2\\\\\n  &+(1488\\tilde{j}^2+40773375\\tilde{j}+8748000000)z\\\\\n  &+\\tilde{j}^3-162000\\tilde{j}^2+8748000000\\tilde{j}-157464000000000.\n \\end{align*}\nNow it is clear, as $\\lvert j-\\tilde{j}\\rvert\\leq2^{-300}\\lvert j\\rvert$, that $\\lvert \\gamma_0-j(2\\tau)\\rvert\\leq 0.4+\\frac{\\lvert j\\rvert^2}{2^{90}}$, and so as $0.98\\lvert j\\rvert^2\\leq\\lvert j(2\\tau)\\rvert\\leq1.02\\lvert j\\rvert^2$, we have $0.979\\lvert j\\rvert^2\\leq\\lvert \\gamma_0\\rvert\\leq1.021\\lvert j\\rvert^2$. we will take the $r$ of Kantorovich's theorem to be $0.009\\lvert j\\rvert^2$, and give bounds for the disc $\\lvert\\gamma-\\gamma_0\\rvert\\leq0.009\\lvert j\\rvert^2$. For $\\gamma$ in this disc, $0.97\\lvert j\\rvert^2\\leq\\lvert \\gamma\\rvert\\leq1.03\\lvert j\\rvert^2$ and in addition $\\lvert\\tilde{j}\\rvert\\leq1.01\\lvert j\\rvert$, so for an upper bound on the first derivative, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert&\\leq 3\\lvert\\gamma\\rvert^2+2(\\lvert\\tilde{j}\\rvert^2+1488\\lvert\\tilde{j}\\rvert+162000)\\lvert\\gamma_0\\rvert\\\\\n  &\\quad+(1488\\lvert\\tilde{j}\\rvert^2+40773375\\lvert\\tilde{j}\\rvert+8748000000)\\\\\n  &\\leq 3.2\\lvert j\\rvert^4+2.2\\lvert j\\rvert^4+10^{-7}\\lvert j\\rvert^4\\\\\n  &\\leq 5.5\\lvert j\\rvert^4,\n\\end{align*}\nand for a lower bound, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert&\\geq 3\\lvert\\gamma\\rvert^2-2(\\lvert\\tilde{j}\\rvert^2+1488\\lvert\\tilde{j}\\rvert+162000)\\lvert\\gamma_0\\rvert\\\\\n  &\\quad-(1488\\lvert\\tilde{j}\\rvert^2+40773375\\lvert\\tilde{j}\\rvert+8748000000)\\\\\n  &\\geq 2.82\\lvert j\\rvert^4-2.1\\lvert j\\rvert^4-10^{-7}\\lvert j\\rvert^4\\\\\n  &\\geq 0.71\\lvert j\\rvert^4.\n\\end{align*}\nFor our upper bound on the function at $\\gamma_0$, as\n $\\lvert\\beta_1\\rvert, \\lvert\\beta_2\\rvert\\leq2\\lvert j\\rvert^{1\/2}$, and as $\\lvert\\beta_0-j(2\\tau)\\rvert\\leq 2^{-298}\\lvert j\\rvert^2$ we have $\\lvert\\gamma_0-\\beta_0\\rvert\\leq0.4+2^{-298}\\lvert j\\rvert^2$, so\n\\begin{align*}\n \\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert&=\\lvert\\gamma_0-\\beta_0\\rvert\\lvert\\gamma_0-\\beta_1\\rvert\\lvert\\gamma_0-\\beta_2\\rvert\\\\\n &\\leq(0.4+2^{-289}\\lvert j\\rvert^2)(1.03\\lvert j\\rvert^2+2\\lvert j\\rvert^{1\/2})^2\\\\\n &\\leq 2^{-54}\\lvert j\\rvert^6.\n\\end{align*}\nFor the second derivative, we have for an upper bound\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},\\gamma)\\rvert&\\leq6\\lvert\\gamma\\rvert+2\\lvert \\tilde{j}\\rvert^2+1488\\lvert \\tilde{j}\\rvert+162000\\\\\n &\\leq 8.3\\lvert j\\rvert^2,\n\\end{align*}\nand for a lower bound,\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},\\gamma_0)\\rvert&\\geq6\\lvert\\gamma\\rvert-2\\lvert \\tilde{j}\\rvert^2-1488\\lvert \\tilde{j}\\rvert-162000\\\\\n &\\geq 3.8\\lvert j\\rvert^2.\n\\end{align*}\nNow, by Theorem \\ref{kantorovich}, as\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert\\lvert\\Phi_2''(\\tilde{j},\\gamma)\\rvert}{\\lvert\\Phi_2'(\\tilde{j},\\gamma)\\rvert^2}\\leq\\frac{2^{-54}\\lvert j\\rvert^6\\cdot8.3\\lvert j\\rvert^2}{(0.71\\lvert j\\rvert^4)^2}\\leq2^{-49}<\\frac{1}{2},\n\\end{equation*}\nand\n\\begin{align*}\n2\\eta &= 2\\frac{\\lvert\\Phi_2(\\tilde{j},\\gamma_0)\\rvert}{\\lvert\\Phi_2'(\\tilde{j},\\gamma_0)\\rvert}\\leq \\frac{2^{-54}\\lvert j\\rvert^6}{0.71\\lvert j\\rvert^4}\\leq0.001\\lvert j\\rvert^2,\\\\\n r&=0.009\\lvert j\\rvert^2,\n\\end{align*}\nNewton's method will converge to the root $\\beta_0$, with a rate of convergence\n\\begin{equation*}\n \\lvert \\gamma_k - \\beta_0\\rvert\\leq\\frac{1}{2^k}2^{-48\\cdot2^{k}}\\frac{\\lvert\\Phi_2'(\\tilde{j},\\gamma_0)\\rvert}{\\lvert\\Phi_2''(\\tilde{j},\\gamma_0)\\rvert}\\leq\\frac{1}{2^k}2^{-48\\cdot2^k}\\frac{5.5\\lvert j\\rvert^4}{3.8\\lvert j\\rvert^2}\\leq2^{-48\\cdot2^k}\\lvert j\\rvert^2\n\\end{equation*}\nfor $k\\geq 1$. In particular, when $k\\geq [2\\log P+2\\log\\log\\lvert j\\rvert]$, $\\lvert \\gamma_k-\\beta_0\\rvert\\leq 2^{-P}$. Now as, by Proposition \\ref{phi2discrepancy}, $\\beta_0$ is an approximation to $j(2\\tau)$ of relative precision $2^{-P+2}$, it may be easily shown that after $[2\\log P+2\\log\\log\\lvert j\\rvert]$ steps, $\\gamma_k$ will be an approximation to $j(2\\tau)$ of relative precision $2^{-P+3}$.\n\\end{proof}\n\n\\begin{lemma}\n Suppose that $\\lvert j(\\tau)\\rvert\\geq 2^{P+12}$, and $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision at least $2^{-P}$. Then\n \\begin{equation*}\n  -\\frac{\\log\\tilde{j}}{2\\pi}\n \\end{equation*}\nis an approximation to $\\tau$ of absolute precision $2^{-P}$.\n\\end{lemma}\n\\begin{proof}\n Let $j(\\tau)+\\delta = \\tilde{j}$. As\n\\begin{equation*}\n \\lvert j(\\tau)-e^{-2\\pi\\tau}\\rvert \\leq 2079,\n\\end{equation*}\nwe have\n\\begin{equation*}\n \\log(\\tilde{j}) = -2\\pi\\tau+ \\log\\left(1+\\frac{\\delta+2079\\theta}{j(\\tau)}\\right),\n\\end{equation*}\nwhere $\\lvert\\theta\\rvert\\leq1$. So as $\\lvert\\log(1+z)\\rvert\\leq1.1\\lvert z\\rvert$ when $\\lvert z\\rvert\\leq 0.05$,\n\\begin{equation*}\n \\lvert\\log\\tilde{j}+2\\pi \\tau\\rvert \\leq 1.1\\cdot2^{-P}+0.6\\cdot2^{-P}\\leq 1.7\\cdot2^{-P}.\n\\end{equation*}\nSo that \n\\begin{equation*}\n \\left\\lvert\\tau-\\left(-\\frac{\\log\\tilde{j}}{2\\pi}\\right)\\right\\rvert\\leq \\frac{1.7}{2\\pi}2^{-P}\\leq 2^{-P}.\n\\end{equation*}\n\\end{proof}\n\nNow the algorithm to invert $j$ when $\\text{\\normalfont Im}(\\tau)\\geq 3$ proceeds as follows -- if $\\lvert j\\rvert\\leq 2^{P+12}$, first iteratively compute approximations to $j(2^k\\tau)$ by Newton's method applied to $\\Phi_2(\\tilde{j},z)$, up to $k = \\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)\\right]+1$. Then calculate the logarithm of $j(2^k\\tau)$ to relative precision $2^{-P-2}$ (note that $\\lvert \\tau\\rvert\\geq 1$), and divide by $-2\\pi$, where we have calculated $2\\pi$ to relative precision $2^{-P-2}$. This process entails a loss of precision of at most $5\\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)\\right]+2$, which is bounded by $11\\log P$ when $P\\geq 400$, and the precision at all applications of Newton's method is at least $2^{-300}$, so our assumptions on the precision of our approximations in the propositions of this section are satisfied at each application. As the computational complexity of Newton's method is $O(M(P)\\log P)$, and of the complex logarithm and computing $\\pi$ are $O(M(P)(\\log P)^2)$, if $\\lvert j\\rvert\\leq2^{P+12}$, the algorithm has time complexity\n\\begin{equation*}\n O\\left(M(P)\\left[2\\log\\left(\\frac{P+12}{\\text{\\normalfont Im}(\\tau)}\\right)+1\\right][2\\log P+2\\log\\log\\lvert j\\rvert]\\right) = O\\left(M(P)(\\log P)^2\\right),\n\\end{equation*}\nand if $\\lvert j\\rvert\\geq 2^{P+12}$, time complexity\n\\begin{equation*}\n O\\left(M(P)(\\log P)^2\\right),\n\\end{equation*}\nwhere the implicit constants are not too large and could be made effective. \n\n\\subsection{Near 1728 and 0}\nWhen $\\tau$ is close to either $i$ or $\\frac{\\pm1+i\\sqrt{3}}{2}$, we will make use of the modular polynomial $\\Phi_2(X,Y)$ to obtain an approximation to one of $j\\left(2\\tau\\right)$, $j\\left( \\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$, for which the SL$_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to either $2\\tau$, $\\frac{\\tau}{2}$, or $\\frac{\\tau+1}{2}$ will lie in the aforementioned compact set, to which we may apply Newton's method. We carry out the analysis only for $i$ and $\\frac{1+i\\sqrt{3}}{2}$, as when $\\left\\lvert\\tau-\\frac{-1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, the SL$_2(\\mathbb{Z})$--equivalent $\\tau+1$ satisfies $\\left\\lvert(\\tau+1)-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$.\n\n\n\\begin{lemma}\\label{taylor_series_at_i}\n If $\\lvert\\delta\\rvert\\leq 2^{-30}$ then \n  \\begin{equation*}\n\\left\\lvert j(i+\\delta)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert \\leq 0.07\\lvert \\delta\\rvert^2,\n\\end{equation*}\nand\n\\begin{equation*}\n\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}+\\delta\\right)-\\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{3!}\\delta^3\\right\\rvert \\leq 0.07\\lvert \\delta\\rvert^3.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe first bound the coefficients of the Taylor series of $j$ at $z=i$ and $z=\\frac{1+i\\sqrt{3}}{2}$. Firstly, by Theorem 1 of \\cite{jcoefficients}, the coefficient of $e^{2\\pi in\\tau}$ in the $q$--expansion of $j$ is bounded by $4e^{4\\pi\\sqrt{n}}$, so the corresponding coefficient of the $k$'th derivative is bounded by $(2\\pi n)^ke^{4\\pi\\sqrt{n}}$. Further, for $n\\geq 1$, $e^{4\\pi\\sqrt{n}-\\sqrt{3}\\pi n}\\leq 100e^{-\\pi n}$, so we have the following bound of the derivatives at these two points,\n\\begin{align*}\n \\lvert j^{(k)}(i)\\rvert, \\left\\lvert j^{(k)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert&\\leq (2\\pi)^ke^{2\\pi}+744+400\\sum_{n=1}^\\infty(2\\pi n)^ke^{-\\pi n}.\n\\end{align*}\nWe now bound the sum in this expression. Firstly, we have the identity\n\\begin{equation*}\n \\sum_{n=1}^\\infty e^{-\\pi nx} = \\frac{1}{e^{\\pi x}-1},\n\\end{equation*}\nand so consider the derivatives of this function, expressing the derivative as sums of the form\n\\begin{equation*}\n \\sum \\alpha_i\\frac{e^{a\\pi x}}{(e^{\\pi x}-1)^b},\n\\end{equation*}\nwhere each term occurring in the expression for the $k$'th derivative is derived from taking the derivative of either the numerator or denominator of a term occurring in the $k-1$'th derivative, i.e. there is no collection of terms with $(a,b)$ equal. It is clear that there are at most $2^k$ terms, and, passing from one derivative to the next, $\\lvert\\alpha_i\\rvert$ may increase by at most $\\pi\\max\\{a,b\\}$, and that $a\\leq b\\leq k+1$, and $a\\leq k$. So $\\lvert\\alpha_i\\rvert\\leq \\pi^k(k+1)!$, and a bound for the whole expression evaluated at $1$ is therefore\n\\begin{equation*}\n 2^k\\pi^k(k+1)!\\frac{e^{k\\pi}}{(e^{\\pi}-1)^k}\\leq 14^kk!.\n\\end{equation*}\nNow we have the bounds\n\\begin{align*}\n \\lvert j^{(k)}(i)\\rvert, \\left\\lvert j^{(k)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert&\\leq (2\\pi)^ke^{2\\pi}+744+400\\cdot14^kk!\\\\\n &\\leq1700\\cdot14^kk!.\n\\end{align*}\nAt $z=i$ and $z=\\frac{1+i\\sqrt{3}}{2}$, the Taylor series expansions for $j(z)$ are\n\\begin{align*}\n j(z) = 1728 + \\sum_{n=2}^{\\infty}c_n(z-i)^n,\\\\\n j(z) = \\sum_{n=3}^{\\infty}c'_n\\left(z-\\frac{1+i\\sqrt{3}}{2}\\right)^n.\n\\end{align*}\nwhere $c_n$ and $c_n'$ are bounded in absolute value by $1400\\cdot14^k$.\n We now bound the tails of the sums of the Taylor series,\n \\begin{equation*}\n  \\sum_{n=3}^\\infty \\lvert c_n\\rvert\\lvert\\delta\\rvert^n\\leq \\lvert\\delta\\rvert^3\\sum_{n=0}^\\infty1700\\cdot14^{n+3}\\lvert\\delta\\rvert^n,\n \\end{equation*}\nand \n\\begin{equation*}\n  \\sum_{n=4}^\\infty \\lvert c_n'\\rvert\\lvert\\delta\\rvert^n\\leq \\lvert\\delta\\rvert^4\\sum_{n=0}^\\infty1700\\cdot14^{n+4}\\lvert\\delta\\rvert^n.\n \\end{equation*}\n Now as $\\lvert\\delta\\rvert\\leq 2^{-30}$,\n \\begin{equation*}\n  \\sum_{n=0}^\\infty1700\\cdot14^{n+4}\\lvert\\delta\\rvert^n=\\frac{1700\\cdot14^4}{1-14\\lvert\\delta\\rvert}\\leq 7\\cdot10^7,\n \\end{equation*}\nso that\n \\begin{align*}\n\\left\\lvert j(i+\\delta)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert &\\leq 2^{-30}\\cdot7\\cdot10^7\\lvert\\delta\\rvert^2\\leq 0.07\\lvert \\delta\\rvert^2,\\\\\n\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}+\\delta\\right)-\\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{3!}\\delta^3\\right\\rvert &\\leq 2^{-30}\\cdot7\\cdot10^7\\lvert\\delta\\rvert^3\\leq 0.07\\lvert \\delta\\rvert^3.\n \\end{align*}\n\\end{proof}\nFurthermore, we note that\n$49700\\geq\\lvert j^{(2)}(i)\\rvert\\geq 49600$, and $275000\\geq\\left\\lvert j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert\\geq 274000$, so if $\\lvert\\tau - i\\rvert\\leq 2^{-30}$, by the previous lemma, we have\n\\begin{equation*}\n2.4\\cdot10^4\\lvert\\delta\\rvert^2\\leq \\lvert j-1728\\rvert\\leq2.5\\cdot10^4\\lvert\\delta\\rvert,\n\\end{equation*}\nand if $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq 2^{-30}$,\n\\begin{equation*}\n 4.5\\cdot10^4\\lvert\\delta\\rvert^3\\leq\\lvert j\\rvert\\leq4.6\\cdot10^4\\lvert\\delta\\rvert^3,\n\\end{equation*}\nfrom which we deduce the following,\n\\begin{lemma}\\label{j_to_tau_small}\nIf $\\lvert\\tau-i\\rvert\\leq 2^{-30}$ and $P\\geq 300$, then:\n\\begin{enumerate}\n \\item If $\\lvert j(\\tau) - 1728\\rvert\\leq 2^{-P}$, then $\\lvert\\tau - i\\rvert\\leq 2^{-P\/2-7}$.\n \\item If $\\lvert j(\\tau) - 1728\\rvert\\geq 2^{-P}$, then $\\lvert\\tau - i\\rvert\\geq 2^{-P\/2-8}$.\n \\end{enumerate}\n If $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$ and $P\\geq 300$, then:\n \\begin{enumerate}\n \\item If $\\lvert j\\rvert\\leq 2^{-P}$, then $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-P\/3-5}$.\n \\item If $\\lvert j\\rvert\\geq 2^{-P}$, then $\\left\\lvert\\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq2^{-P\/3-6}$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{taylor_series_at_2i}\n Suppose that $\\lvert\\delta\\rvert\\leq 2^{-28}$. Then\n \\begin{equation*}\n\\left\\lvert j\\left(2i+\\delta\\right)-j\\left(2i\\right)-j'\\left(2i\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert,\n\\end{equation*}\nand \n\\begin{equation*}\n\\left\\lvert j\\left(i\\sqrt{3}+\\delta\\right)-j\\left(i\\sqrt{3}\\right)-j'\\left(i\\sqrt{3}\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe proceed similarly to the previous lemma -- as $17e^{-2\\pi n}\\geq e^{4\\pi n-2\\sqrt{3}\\pi n}$, \n\\begin{equation*}\n \\lvert j^{(k)}(2i)\\rvert,\\lvert j^{(k)}(\\sqrt{3}i)\\rvert \\leq (2\\pi)^ke^{4\\pi}+744+68\\sum_{n=1}^{\\infty}(2\\pi n)^ke^{-2\\pi n},\n\\end{equation*}\nand a similar analysis to the previous lemma bounds the sum in this inequality by\n\\begin{equation}\n (2\\pi)^ke^{4\\pi}+744+68\\cdot 13^kk! \\leq 300000\\cdot13^kk!.\n\\end{equation}\n So when $\\lvert\\delta\\rvert\\leq 2^{-28}$,\n \\begin{align*}\n\\left\\lvert j\\left(2i+\\delta\\right)-j(2i)-j'\\left(2i\\right)\\delta\\right\\rvert &\\leq \\sum_{n=2}^{\\infty}300000\\cdot13^n\\delta^n\\\\\n&\\leq \\sum_{n=2}^{\\infty}300000\\cdot13^n\\delta^n\\\\\n&\\leq 2^{-28}\\cdot300000\\cdot13^2\\lvert\\delta\\rvert\\sum_{n=0}^{\\infty}13^n2^{-28n}\\\\\n&\\leq0.2\\lvert \\delta\\rvert.\n\\end{align*}\n and as our bound for the terms of the Taylor series applies to both expansions, we similarly have\n \\begin{align*}\n\\left\\lvert j\\left(i\\sqrt{3}+\\delta\\right)-j\\left(i\\sqrt{3}\\right)-j'\\left(i\\sqrt{3}\\right)\\delta\\right\\rvert \\leq 0.2\\lvert \\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\nWe now give two lemmas on the separation and closeness of the three preimages of roots of the modular polynomial $\\Phi_2(j(\\tau),z)$, for $\\tau$ near $i$ and $\\frac{1+i\\sqrt{3}}{2}$.\n\\begin{lemma}\\label{preimage_closeness}\n Suppose that $\\tau\\in\\mathcal{F}$. Then, if $\\tau = i+\\delta$,\n \\begin{align*}\n  \\lvert 2\\tau - 2i\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert -\\frac{2}{\\tau}- 2i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert,\n \\end{align*}\nand if $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$,\n\\begin{align*}\n  \\lvert \\left(2\\tau-1\\right) - \\sqrt{3}i\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert \\left(1-\\frac{2}{\\tau}\\right)- \\sqrt{3}i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert,\\\\\n  \\left\\lvert \\left(-1-\\frac{2}{\\tau-1}\\right)- \\sqrt{3}i\\right\\rvert&\\leq 2\\lvert\\delta\\rvert.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\nFirst note that as $\\tau\\in\\mathcal{F}$, $\\lvert\\tau\\rvert\\geq 1$. Let $2\\tau = 2i+\\delta_1$, and $-\\frac{2}{\\tau}=2i+\\delta_2$. For $\\delta_1$, $\\lvert2\\tau-2i\\rvert = 2\\lvert \\delta\\rvert$, and for $\\delta_2$,\n\\begin{align*}\n \\lvert\\delta_2\\rvert=\\left\\lvert -\\frac{2}{\\tau}-2i\\right\\rvert &= \\left\\lvert \\frac{-2-2i\\tau}{\\tau}\\right\\rvert\\\\\n &=\\frac{\\lvert2i\\delta\\rvert}{\\lvert\\tau\\rvert}\\\\\n &\\leq 2\\lvert\\delta\\rvert\n\\end{align*}\nIf $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, let $2\\tau-1 = \\sqrt{3}i+ \\delta_1$, $\\left(1-\\frac{2}{\\tau}\\right) = \\sqrt{3}i+\\delta_2$, and $\\left(1-\\frac{2}{\\tau-1}\\right)=\\sqrt{3}i+\\delta_3$. For $\\delta_1$,\n\\begin{align*}\n \\lvert\\delta_1\\rvert = \\left\\lvert 2\\tau-1-\\sqrt{3}i\\right\\rvert\\leq 2\\lvert\\delta\\rvert,\n\\end{align*}\nfor $\\delta_2$,\n\\begin{align*}\n \\lvert\\delta_2\\rvert = \\left\\lvert \\left(1-\\frac{2}{\\tau}\\right)- \\sqrt{3}i\\right\\rvert& = \\left\\lvert\\frac{\\tau-2-\\sqrt{3}i\\tau}{\\tau}\\right\\rvert\\\\\n &=\\left\\lvert\\frac{(1-\\sqrt{3}i)\\delta}{\\tau}\\right\\rvert\\\\\n &\\leq2\\lvert\\delta\\rvert,\n\\end{align*}\nand for $\\delta_3$,\n\\begin{align*}\n \\lvert\\delta_3\\rvert = \\left\\lvert \\left(-1-\\frac{2}{\\tau-1}\\right)- \\sqrt{3}i\\right\\rvert &= \\left\\lvert\\frac{-\\tau+1-2-\\sqrt{3}i\\tau+\\sqrt{3}i}{\\tau}\\right\\rvert\\\\\n &=\\frac{\\lvert(1+\\sqrt{3}i)\\delta\\rvert}{\\lvert\\tau-1\\rvert}\\\\\n &\\leq 2\\lvert\\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\\label{preimage_separation}\n If $\\tau = i+\\delta$, $\\tau \\in \\mathcal{F}$, with $\\lvert \\delta\\rvert \\leq 2^{-30}$, then the distance between $2\\tau$ and $-\\frac{2}{\\tau}$ is at least $3.99\\lvert\\delta\\rvert$, and the distance between either of these and the point in $\\mathcal{F}$ which is $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $\\frac{\\tau+1}{2}$ is at least $0.99$ in magnitude. If $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, $\\tau\\in\\mathcal{F}$, then the distance between any pair of the three points lying in $\\mathcal{F}$ which are $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $2\\tau$, $\\frac{\\tau}{2}$, and $\\frac{\\tau+1}{2}$ is at least $3.46\\lvert\\delta\\rvert$.\n\\end{lemma}\n\\begin{proof}\n Firstly, for the distance between $\\tau$ and $-\\frac{2}{\\tau}$, we have\n  \\begin{align*}\n   2\\tau +\\frac{2}{\\tau} &= \\frac{2(\\tau^2+1)}{\\tau}\\\\\n   &=\\frac{4i\\delta+2\\delta^2}{\\tau}.\n  \\end{align*}\n  As $\\lvert\\delta\\rvert\\leq 2^{-30}$, $\\lvert\\delta\\rvert^2\\leq2^{-30}\\lvert\\delta\\rvert$, and $\\lvert i+\\delta\\rvert\\leq1+2^{-30}$, so\n  \\begin{equation*}\n   \\left\\lvert2\\tau-\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\geq3.99\\lvert\\delta\\rvert.\n  \\end{equation*}\n  Now for $-\\frac{2}{\\tau+1}+1$, we have\n  \\begin{align*}\n   -\\frac{2}{\\tau+1}+1-i &= \\frac{-2+\\tau+1-i\\tau-i}{\\tau+1}\\\\\n   &=\\frac{\\delta-i\\delta}{1+i+\\delta}\n  \\end{align*}\n  so that \n\\begin{equation}\n \\left\\lvert\\left(-\\frac{2}{\\tau+1}+1\\right)-i\\right\\rvert \\leq 2\\lvert\\delta\\rvert\n\\end{equation}\nand so either $-\\frac{2}{\\tau+1}+1$ or $-\\left(-\\frac{2}{\\tau+1}+1\\right)^{-1}$ is $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $\\frac{1+\\tau}{2}$ and in $\\mathcal{F}$, and its distance from $i$ is bounded by $\\left\\lvert 1+\\frac{2\\lvert\\delta\\rvert}{1-2\\lvert\\delta\\rvert}\\right\\rvert\\leq 2^{-28}$. By Lemma \\ref{preimage_closeness}, the distance of either $2\\tau$ or $-\\frac{2}{\\tau}$ from $2i$ is at most $2\\lvert\\delta\\rvert\\leq2\\cdot2^{-30}$, so both of these are separated from either $-\\frac{2}{\\tau+1}+1$ or $-\\left(-\\frac{2}{\\tau+1}+1\\right)^{-1}$ by a distance of at least $0.99$.\n\nNow for $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent points to $2\\tau$, $\\frac{\\tau}{2}$, and $\\frac{\\tau+1}{2}$ are $2\\tau-1$, $1-\\frac{2}{\\tau}$, and $-1-\\frac{2}{\\tau-1}$ respectively. For their differences, we have,\n \\begin{align*}\n   2\\tau-1 -\\left(1-\\frac{2}{\\tau}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau}.\n  \\end{align*}\n  So as $\\lvert \\tau\\rvert\\leq1+2^{-30}$,\n  \\begin{equation*}\n   \\left\\lvert 2\\tau-1 -\\left(1-\\frac{2}{\\tau}\\right)\\right\\rvert\\geq 3.46\\lvert \\delta\\rvert\n  \\end{equation*}\n  For the first and third points,\n  \\begin{align*}\n   2\\tau-1 -\\left(-1-\\frac{2}{\\tau-1}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau-1}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau-1}.\n  \\end{align*}\nAs $\\lvert\\tau-1\\rvert\\leq1+2^{-30}$, we have the lower bound\n\\begin{equation*}\n  \\left\\lvert2\\tau-1 -\\left(-1-\\frac{2}{\\tau-1}\\right)\\right\\rvert\\geq3.46\\lvert \\delta\\rvert.\n\\end{equation*}\nFor the second and third points,\n\\begin{align*}\n   \\left(1-\\frac{2}{\\tau}\\right) -\\left(-1-\\frac{2}{\\tau-1}\\right) &= \\frac{2(\\tau^2-\\tau+1)}{\\tau(\\tau-1)}\\\\\n   &=\\frac{i2\\sqrt{3}\\delta+2\\delta^2}{\\tau(\\tau-1)}.\n\\end{align*}\nSo we obtain the lower bound\n\\begin{equation*}\n  \\left\\lvert\\left(1-\\frac{2}{\\tau}\\right) -\\left(-1-\\frac{2}{\\tau-1}\\right)\\right\\rvert\\geq3.46\\lvert \\delta\\rvert.\n\\end{equation*}\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{bounds_for_images}\n If $\\tau = i+\\delta$, with $\\lvert\\delta\\rvert\\leq2^{-30}$, then \n \\begin{equation*}\n  7.18\\cdot10^{6}\\lvert\\delta\\rvert\\leq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\leq7.25\\cdot10^6\\lvert\\delta\\rvert ,\n \\end{equation*}\n and \n \\begin{equation*}\n   2.4\\cdot10^5\\leq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert,  \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\leq3.1\\cdot10^5,\n \\end{equation*}\n and if $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$, with $\\lvert\\delta\\rvert\\leq 2^{-30}$, then for $\\tau_1,\\tau_2\\in\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\}$, $\\tau_1\\neq\\tau_2$,\n \\begin{equation*}\n  1.15\\cdot10^6\\lvert\\delta\\rvert\\leq\\lvert j(\\tau_1)-j(\\tau_2)\\rvert\\leq 1.34\\cdot10^6\\lvert\\delta\\rvert.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n We first use the Taylor series expansion of $j(z)$ at $2i$. By Lemma \\ref{preimage_closeness}, we may apply Lemma \\ref{taylor_series_at_2i}, and using also Lemma \\ref{preimage_separation}, we obtain the lower bound\n \\begin{align*}\n  \\left\\lvert j(2\\tau)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert&\\geq \\left\\lvert j(2i)+j'(2i)\\delta_1-j(2i)-j'(2i)\\delta_2\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &=\\left\\lvert j'(2i)\\right\\rvert\\left\\lvert \\delta_1- \\delta_2\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &=\\left\\lvert j'(2i)\\right\\rvert\\left\\lvert 2\\tau+\\frac{2}{\\tau}\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n  &\\geq 1.8\\cdot10^6\\cdot3.99\\rvert\\delta\\lvert - 0.4\\lvert\\delta\\rvert\\\\\n  &\\geq 7.18\\cdot10^{6}\\lvert\\delta\\rvert.\n \\end{align*}\n Similarly we have an upper bound of \n \\begin{equation*}\n  \\left\\lvert j(2\\tau)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\leq1.81\\cdot10^6\\cdot4\\rvert\\delta\\lvert+ 194\\lvert\\delta\\rvert\\leq7.25\\cdot10^6\\lvert\\delta\\rvert.\n \\end{equation*}\n By Lemma \\ref{preimage_separation}, the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent point in $\\mathcal{F}$ to $\\frac{\\tau+1}{2}$ is at most $2\\lvert\\delta\\rvert\\leq2^{-29}$ from $i$, so \n \\begin{equation*}\n  \\left\\lvert j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\leq e^{2.01\\pi}+2079\\leq e^{7.9},\n \\end{equation*}\n and by Lemma \\ref{preimage_closeness}, $2\\tau,-\\frac{\\tau}{2}$ are at most $2^{-29}$ from $2i$,\n \\begin{equation*}\n  \\left\\lvert j\\left(2\\tau\\right)\\right\\rvert,\\left\\lvert j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\geq e^{3.99\\pi}-2079\\geq e^{12.5},\n \\end{equation*}\n so that \n \\begin{align*}\n  3.1\\cdot10^5\\geq\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert,  \\left\\lvert j\\left(-\\frac{\\tau}{2}\\right)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\geq 2.4\\cdot10^5.\n \\end{align*}\nNow using the Taylor series of $j(z)$ at $i\\sqrt{3}$, by Lemma \\ref{preimage_closeness}, we may apply Lemma \\ref{taylor_series_at_2i}, and using also Lemma \\ref{preimage_separation}, with $\\tau_1$, $\\tau_2$ any distinct pair of $2\\tau$ ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $2\\tau-1$), $\\frac{\\tau}{2}$, ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $1-\\frac{2}{\\tau}$), and $\\frac{\\tau+1}{2}$ ($\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent to $-1-\\frac{1}{\\tau-1}$), with $\\tau_j = i\\sqrt{3}+\\delta_j$,\n\\begin{align*}\n \\left\\lvert j(\\tau_1)-j(\\tau_2)\\right\\rvert &\\geq \\left\\lvert j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_1-(j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_2)\\right\\rvert - 0.4\\lvert\\delta\\rvert\\\\\n &=\\lvert j'(\\sqrt{3}i)\\rvert\\lvert\\tau_1-\\tau_2\\rvert-0.4\\lvert\\delta\\rvert\\\\\n &\\geq 334500\\cdot3.46\\lvert\\delta\\rvert-0.4\\lvert\\delta\\rvert\\\\\n &\\geq1.15\\cdot10^6\\lvert\\delta\\rvert,\n\\end{align*}\nand similarly for an upper bound we have\n\\begin{align*}\n \\left\\lvert j(\\tau_1)-j(\\tau_2)\\right\\rvert &\\leq \\left\\lvert j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_1-(j(\\sqrt{3}i)+j'(\\sqrt{3}i)\\delta_2)\\right\\rvert + 0.4\\lvert\\delta\\rvert\\\\\n &=\\lvert j'(\\sqrt{3}i)\\rvert\\lvert\\tau_1-\\tau_2\\rvert+0.4\\lvert\\delta\\rvert\\\\\n &\\leq 334600\\cdot4\\lvert\\delta\\rvert+0.4\\lvert\\delta\\rvert\\\\\n &\\leq1.34\\cdot10^6\\lvert\\delta\\rvert.\n\\end{align*}\n\\end{proof}\n\n\\subsubsection{$j$ near $1728$}\nWe now bound the discrepancy between the roots of $\\Phi_2(j,z)$ and $\\Phi_2(\\tilde{j},z)$ when $\\lvert\\tau-i\\rvert\\leq2^{-20}$ and $\\lvert \\tilde{j}-1728\\rvert\\geq 2^{-P\/3}$.\n\\begin{lemma}\\label{phi2discrepancy_points_1728}\n Suppose that $\\tilde{j}$ is an approximation to $j(\\tau)$ of relative precision $2^{-P}$, with $P\\geq 300$, $\\lvert\\tau-i\\rvert\\leq2^{-30}$, and $\\left\\lvert \\tilde{j}-1728\\right\\rvert\\geq 2^{-P\/3}$. Then the relative precision of any root of $\\Phi_2(\\tilde{j},z)$ to the closest root of $\\Phi_2(j,z)$ is at least $2^{-P\/3+10}$\n\\end{lemma}\n\\begin{proof}\nFirstly, as in the proof of Lemma \\ref{phi2discrepancy}, with $f(z) = \\Phi_2(j,z)$ and $g(z) = \\Phi_2(\\tilde{j},z)$, and $\\tilde{j}=j+\\delta$, we will bound the size of the coefficients of $f-g$. We note that $\\lvert j\\rvert\\geq1700$. For $z^2$, we have\n\\begin{equation*}\n \\lvert 2j\\delta+\\delta^2+1448\\delta\\rvert\\leq 3\\lvert\\delta\\rvert\\lvert j\\rvert,\n\\end{equation*}\nfor $z$, we have\n\\begin{equation*}\n \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  \\leq 30000\\lvert\\delta\\rvert\\lvert j\\rvert,\n\\end{equation*}\nand for the constant term\n\\begin{equation*}\n  \\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\leq 3100\\lvert\\delta\\rvert\\lvert j\\rvert^2.\n \\end{equation*}\nNow evaluating $g$ at $j(2\\tau)$, we have, noting that $\\lvert j(2\\tau)\\rvert\\leq 0.1\\lvert j\\rvert^2$,\n\\begin{align*}\n \\lvert g(j(2\\tau))\\rvert &= \\lvert f(j(2\\tau))-g(j(2\\tau))\\rvert\\\\\n &\\leq 3\\lvert\\delta\\rvert\\lvert j\\rvert(0.1\\lvert j\\rvert^2)^2+30000\\lvert\\delta\\rvert\\lvert j\\rvert(0.1\\lvert j\\rvert^2)+3100\\lvert\\delta\\rvert\\lvert j\\rvert^2\\\\\n &\\leq 0.04\\lvert j\\rvert^5\n\\end{align*}\nLetting $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g$, where $\\beta_0$ is the closest root of $g(z)$ to $j(2\\tau)$,\n\\begin{align*}\n \\lvert j(2\\tau)-\\beta_0\\rvert\\lvert j(2\\tau)-\\beta_1\\rvert\\lvert j(2\\tau)-\\beta_2\\rvert&\\leq0.04\\lvert\\delta\\rvert\\lvert j\\rvert^5\\\\\n \\lvert j(2\\tau)-\\beta_0\\rvert&\\leq10^5\\lvert\\delta\\rvert^{1\/3}\\\\\n &\\leq 1.2\\cdot10^6\\cdot2^{-P\/3},\n\\end{align*}\nand similarly for the nearest root of $g(z)$ to each of the roots of $f(z)$. As $\\lvert\\tilde{j}-1728\\rvert\\geq 2^{-P\/3}$, $\\lvert j-1728\\rvert\\geq 2^{-P\/3}-1800\\cdot2^{-P}\\geq 2^{-P\/3-1}$, and by Lemma \\ref{j_to_tau_small}, $\\lvert \\tau - i\\rvert\\geq 2^{-P\/6-9}$, so that by Lemma \\ref{bounds_for_images}, \n\\begin{equation*}\n \\left\\lvert j(2\\tau) - j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert\\geq7.18\\cdot10^7\\cdot2^{- P\/6-9}.\n\\end{equation*}\nNow letting $\\beta_i$ and $\\beta_j$ be the nearest roots to $j(2\\tau)$ and $j\\left(\\frac{\\tau}{2}\\right)$ respectively,\n\\begin{align*}\n \\lvert\\beta_i-\\beta_j\\rvert &= \\left\\lvert (\\beta_i-j(2\\tau))-j(2\\tau)-\\left(\\left(\\beta_j-j\\left(\\frac{\\tau}{2}\\right)\\right)-j\\left(\\frac{\\tau}{2}\\right)\\right)\\right\\rvert\\\\\n &\\geq \\left\\lvert j(2\\tau) - j\\left(\\frac{\\tau}{2}\\right)\\right\\rvert - \\lvert j(2\\tau)-\\beta_i\\rvert - \\left\\lvert j\\left(\\frac{\\tau}{2}\\right)-\\beta_j\\right\\rvert\\\\\n &\\geq 7.18\\cdot10^7\\cdot 2^{-P\/6-9} - 1.2\\cdot10^6\\cdot2^{-P\/3}\\\\\n &\\geq 7.18\\cdot10^7\\cdot 2^{-P\/6-9} - 0.01\\cdot2^{-P\/6}\\\\\n &\\geq 1.4\\cdot10^5\\cdot2^{-P\/6}.\n\\end{align*}\nIn particular, $\\beta_i$ and $\\beta_j$ are distinct, and given that, by Lemma \\ref{bounds_for_images}, $\\left\\lvert j(2\\tau)-j\\left(\\frac{\\tau+1}{2}\\right)\\right\\rvert\\geq2.4\\cdot10^5\\geq7.18\\cdot10^7\\cdot2^{-P\/6-9}$, one may similarly deduce a separation of $2\\cdot10^{5}$ between the closest root to $j\\left(\\frac{\\tau+1}{2}\\right)$ and either of the other two. So each root $j_i$ of $\\Phi_2(j,z)$ has a unique associated closest root $\\beta_i$ of $\\Phi_2(\\tilde{j},z)$, which satisfies\n\\begin{equation*}\n \\lvert j_i - \\beta_i\\rvert\\leq 1.2\\cdot10^6\\cdot2^{-P\/3}.\n\\end{equation*}\nThe relative precision of $\\beta_i$ to its closest root is then bounded by \n\\begin{equation*}\n \\frac{1.2\\cdot10^6\\cdot2^{-P\/3}}{\\lvert j(2\\tau)\\rvert}, \\frac{1.2\\cdot10^6\\cdot2^{-P\/3}}{\\lvert j\\left(\\frac{\\tau}{2}\\right)\\rvert}\\leq 2^{-P\/3 + 3}\n\\end{equation*}\nif $\\beta_i$ is the root close to $j(2\\tau)$ or $j\\left(\\frac{\\tau}{2}\\right)$, and \n\\begin{equation*}\n \\frac{1.2\\cdot10^{6}\\cdot2^{-P\/3}}{\\lvert j(\\frac{\\tau+1}{2})\\rvert}\\leq 2^{-P\/3+10}\n\\end{equation*}\nfor the root close to $ j\\left(\\frac{\\tau+1}{2}\\right)$.\n\\end{proof}\nFinally we show that Newton's method may be used to obtain an approximation to $j(2\\tau)$.\n\\begin{proposition}\nLet $j=j(\\tau)$, where $\\lvert\\tau-i\\rvert\\leq2^{-30}$, and $\\tilde{j}$ be an approximation to $j$ of relative precision $2^{-P}$, $P\\geq 300$, such that $\\lvert\\tilde{j}-1728\\rvert\\geq2^{-P\/3}$. Let\n\\begin{equation*}\n \\tau_0 = i+\\sqrt{\\frac{\\tilde{j}-1728}{j^{(2)}(i)}},\n\\end{equation*}\nwhere the sign of the square root is chosen arbitrarily.\nThen Newton's method applied to $\\Phi_2(\\tilde{j},z)$, with starting point $j(2\\tau_0)$ will converge to either the root of $\\Phi_2(\\tilde{j},z)$ closest to $j(2\\tau)$, or the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau}{2}\\right)$, and after $[2\\log P]$ steps will produce an approximation either $j(2\\tau)$ or $j\\left(\\frac{\\tau}{2}\\right)$ of relative precision $2^{-P\/3+11}$.\n\\end{proposition}\n\\begin{proof}\n Firstly, we bound the difference between $\\tau$ and $\\tau_0$. We let $\\tau = i+\\delta$. By Lemma \\ref{taylor_series_at_i},\n \\begin{equation*}\n  \\left\\lvert j(\\tau)-1728-\\frac{j^{(2)}(i)}{2}\\delta^2\\right\\rvert\\leq0.07\\lvert\\delta\\rvert^2,\n \\end{equation*}\nso that\n\\begin{equation*}\n \\left\\lvert\\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}-\\delta\\right\\rvert\\left\\lvert\\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}+\\delta\\right\\rvert\\leq \\frac{388}{j^{(2)}(i)}\\lvert\\delta\\rvert^2+\\frac{2^{-P+1}\\lvert j\\rvert}{j^{(2)}(i)}\\leq 3\\cdot10^{-6}\\lvert\\delta\\rvert^2.\n\\end{equation*}\nNow we will show that, taking some branch of the square root above, we will obtain a good starting point for Newton's method. Let $\\epsilon = \\sqrt{\\frac{2(\\tilde{j}-1728)}{j^{(2)}(i)}}$, where the branch of the square root is arbitrary. Firstly, one of $\\left\\lvert\\epsilon-\\delta\\right\\rvert$ or $\\left\\lvert\\epsilon+\\delta\\right\\rvert$ is $\\geq\\lvert\\delta\\rvert$, and so the other is $\\leq 3\\cdot10^{-6}\\lvert\\delta\\rvert$. In the first case, $\\lvert\\tau_0-\\tau\\rvert\\leq3\\cdot10^{-6}\\lvert\\delta\\rvert$, and in the second case,\n\\begin{equation*}\n \\left\\lvert\\tau_0-\\left(-\\frac{1}{\\tau}\\right)\\right\\rvert=\\left\\lvert\\frac{-1+i\\delta+i\\epsilon+1+\\delta\\epsilon}{\\tau}\\right\\rvert\\leq\\lvert\\delta+\\epsilon\\rvert+\\lvert\\delta\\epsilon\\rvert\\leq3.1\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nNow as $\\lvert\\delta\\rvert\\leq2^{-30}$, $j'(z)$ is bounded in absolute value by $1.81\\cdot10^6$ between $2\\tau_0$ and either of $2\\tau$ or $-\\frac{2}{\\tau}$, and the distance from $2\\tau_0$ to the closest of these is bounded by $6.2\\cdot10^{-6}\\lvert\\delta\\rvert$, so either\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j(2\\tau)\\right\\rvert\\leq 12\\lvert\\delta\\rvert\n\\end{equation*}\nor\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j\\left(-\\frac{2}{\\tau}\\right)\\right\\rvert\\leq 12\\lvert\\delta\\rvert.\n\\end{equation*}\nNow we bound the terms occurring in Kantorovich's criterion. Let $\\beta_0,\\beta_1,\\beta_2$ be the roots of $\\Phi_2(\\tilde{j},z)$, with $\\beta_0$ the closest root to $j(2\\tau_0)$, and $\\beta_1$ the other root near $j(2i)$. Firstly, as $\\lvert j\\rvert\\geq2^{-P\/2}$, by Lemma \\ref{bounds_for_images}, the above, and Lemma \\ref{phi2discrepancy_points_1728}, we have, for $2\\tau_0$,  \n\\begin{align*}\n  \\left\\lvert j(2\\tau_0)-\\beta_0\\right\\rvert&\\leq 12\\lvert\\delta\\rvert + 2^{-P\/3+10}\\leq12.1\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_1\\right\\rvert&\\leq 7.25\\cdot10^6\\lvert\\delta\\rvert+2^{-P\/3+10}\\leq 7.26\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_2\\right\\rvert&\\leq3.1\\cdot10^5+2^{-P\/3+10}\\leq3.11\\cdot10^5\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_1\\right\\rvert&\\geq7.18\\cdot10^6\\lvert\\delta\\rvert-2^{-P\/3+10}\\geq 7.17\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0)-\\beta_2\\right\\rvert&\\geq2.4\\cdot10^5-2^{-P\/3+10}\\geq2.39\\cdot10^5,\n \\end{align*}\n and similarly, for any $\\tau'$ satisfying $\\lvert\\tau'-2\\tau\\rvert\\leq35\\lvert\\delta\\rvert$ (which we take to ensure the condition on $r$ is satisfied), we have, as $\\lvert j'(z)\\rvert\\leq1.81\\cdot10^{6}$ between $2\\tau_0$ and $\\tau'$, and as $\\lvert\\delta\\rvert\\leq2^{-30}$,\n \\begin{align*}\n  \\left\\lvert j(\\tau')-\\beta_0\\right\\rvert&\\leq 6.4\\cdot10^{7}\\lvert\\delta\\rvert+12\\lvert\\delta\\rvert\\leq0.06,\\\\\n \\left\\lvert j(\\tau')-\\beta_1\\right\\rvert&\\leq 6.4\\cdot10^{7}\\lvert\\delta\\rvert+7.26\\cdot10^6\\lvert\\delta\\rvert\\leq0.06,\\\\\n \\left\\lvert j(\\tau')-\\beta_2\\right\\rvert&\\leq6.4\\cdot10^{7}\\lvert\\delta\\rvert+3.11\\cdot10^5\\leq3.2\\cdot10^5.\n \\end{align*}\nFor our bound on $\\Phi_2(\\tilde{j},z)$ evaluated at $j(\\tau')$, we have\n\\begin{align*}\n \\left\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\right\\rvert &= \\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_1\\rvert\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\leq2.8\\cdot10^{13}\\lvert\\delta\\rvert^2.\n\\end{align*}\nFor the first derivative, we have\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert &= ( j(\\tau')-\\beta_0)( j(\\tau')-\\beta_1)+( j(\\tau')-\\beta_0)( j(\\tau')-\\beta_2)+( j(\\tau')-\\beta_1)( j(\\tau')-\\beta_2)\\\\\n &\\geq \\lvert j(\\tau')-\\beta_1\\rvert\\lvert j(\\tau')-\\beta_2\\rvert-\\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_1\\rvert-\\lvert j(\\tau')-\\beta_0\\rvert\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\geq 7.17\\cdot10^6\\cdot2.39\\cdot10^5\\lvert\\delta\\rvert - 12.1\\cdot7.26\\cdot10^6\\lvert\\delta\\rvert^2- 12\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert\\\\\n &\\geq 1.7\\cdot10^{12}\\lvert\\delta\\rvert,\n\\end{align*}\nand for the second derivative,\n\\begin{align*}\n \\left\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\right\\rvert &\\leq 2\\lvert j(\\tau')-\\beta_0\\rvert+2\\lvert j(\\tau')-\\beta_1\\rvert+2\\lvert j(\\tau')-\\beta_2\\rvert\\\\\n &\\leq 6.5\\cdot10^5.\n\\end{align*}\nThese now give\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert^2}\\leq\\frac{2.8\\cdot10^{13}\\lvert\\delta\\rvert^2\\cdot6.5\\cdot10^5}{(1.7\\cdot10^{12}\\lvert\\delta\\rvert)^2}\\leq 2^{-17}<\\frac{1}{2},\n\\end{equation*}\nand for the condition on $r$, we have\n\\begin{align*}\n r&=35\\lvert\\delta\\rvert,\\\\\n 2\\eta&=2\\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}\\leq2\\frac{2.8\\cdot10^{13}\\lvert\\delta\\rvert^2}{1.7\\cdot10^{12}\\lvert\\delta\\rvert}\\leq34\\lvert\\delta\\rvert,\n\\end{align*}\nwhich ensures convergence. \nFor the rate of convergence, we need a lower bound on the second derivative and an upper bound on the first derivative, which we have as follows,\n\\begin{align*}\n \\left\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\right\\rvert &\\geq 2\\lvert j(\\tau')-\\beta_2\\rvert-2\\lvert j(\\tau')-\\beta_0\\rvert-2\\lvert j(\\tau')-\\beta_1\\rvert\\\\\n &\\geq 4.7\\cdot10^5,\n\\end{align*}\nand\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert &\\leq \\lvert j(2\\tau_0)-\\beta_1\\rvert\\lvert j(2\\tau_0)-\\beta_2\\rvert+\\lvert j(2\\tau_0)-\\beta_0\\rvert\\lvert j(2\\tau_0)-\\beta_1\\rvert\\\\\n &\\quad\\quad\\quad+\\lvert j(2\\tau_0)-\\beta_0\\rvert\\lvert j(2\\tau_0)-\\beta_2\\rvert\\\\\n &\\leq 7.25\\cdot10^6\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert + 12.1\\cdot7.25\\cdot10^6\\lvert\\delta\\rvert^2+ 12.1\\cdot3.11\\cdot10^5\\lvert\\delta\\rvert\\\\\n &\\leq 2.3\\cdot10^{12}\\lvert\\delta\\rvert,\n\\end{align*}\nwhich gives a bound on the convergence, for $k\\geq 1$, of\n\\begin{equation*}\n \\frac{1}{2^k}2^{-17\\cdot2^k}\\frac{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}\\leq2^{-17\\cdot2^k}.\n\\end{equation*}\nSo in order to obtain an absolute precision of $2^{-P}$, $[2\\log P]$ steps will suffice. By Lemma \\ref{phi2discrepancy_points_1728}, this approximation to $\\beta_0$ will then be an approximation to either $j(2\\tau)$ or $j\\left(-\\frac{2}{\\tau}\\right)$ of relative precision $2^{-P\/3+11}$.\n\\end{proof}\n\n\\subsubsection{$j$ near $0$}\n\n\nWe now bound the discrepancy between the roots of $\\Phi_2(j,z)$ and $\\Phi_2(\\tilde{j},z)$ when $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$ and $\\lvert \\tilde{j}\\rvert\\geq 2^{-P\/2}$.\n\\begin{lemma}\\label{phi2discrepancy_points_0}\n Let $j=j(\\tau)$, where $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-30}$, and suppose that $\\tilde{j}$ is an approximation to $j$ of absolute precision $2^{-P}$, with $P\\geq 300$, and $\\left\\lvert \\tilde{j}\\right\\rvert\\geq 2^{-P\/2}$. Then relative precision of any root of $\\Phi_2(\\tilde{j},z)$ to its closest root of $\\Phi_2(j,z)$ is at most $2^{-P\/3 + 2}$, and the roots of $\\Phi_2(\\tilde{j},z)$ are separated by at least $2^{-P\/6+8}$.\n\\end{lemma}\n\\begin{proof}\nFirstly, as in the proof of the Lemma \\ref{phi2discrepancy_points_1728}, with $f(z) = \\Phi_2(j,z)$ and $g(z) = \\Phi_2(\\tilde{j},z)$, we will bound the size of the coefficients of $f-g$. Let $\\tilde{j}=j+\\delta$. For the coefficient of $z^2$, we have\n\\begin{equation*}\n \\lvert 2j\\delta+\\delta^2+1448\\delta\\rvert\\leq 1500\\lvert\\delta\\rvert,\n\\end{equation*}\nfor the coefficient of $z$,\n\\begin{equation*}\n \\lvert 2976\\delta j+1488\\delta^2+40773375\\delta\\rvert\\\\\n  \\leq 4.1\\cdot10^7\\lvert\\delta\\rvert,\n\\end{equation*}\nand for the constant term\n\\begin{equation*}\n  \\left\\lvert 3\\delta j^2+3\\delta^2j+\\delta^3-16200\\delta j-16200\\delta^2+8748000000\\delta\\right\\rvert\\leq 8.8\\cdot10^9\\lvert\\delta\\rvert.\n \\end{equation*}\n Now evaluating $g(z)$ at $j(\\tau')$, for $\\tau'\\in\\{2\\tau,\\frac{\\tau}{2},\\frac{\\tau+1}{2}\\}$, as $\\lvert j(\\tau')\\rvert\\leq60000$, we have\n \\begin{align*}\n  \\lvert g(j(2\\tau))\\rvert&\\leq60000^2\\cdot1500\\lvert\\delta\\rvert + 60000\\cdot4.1\\cdot10^7\\lvert\\delta\\rvert + 60000\\cdot8.8\\cdot10^9\\lvert\\delta\\rvert\\\\\n  &\\leq 6\\cdot10^{14}\\lvert\\delta\\rvert.\n \\end{align*}\nLetting $\\beta_0,\\beta_1,\\beta_2$ be the roots of $g(z)$, we have, as $\\lvert\\tilde{j}\\rvert\\geq2^{-P\/2}$, which impels $\\lvert j\\rvert\\geq 2^{-P\/2-1}$,\n\\begin{align*}\n \\lvert\\beta_0-j(\\tau_i)\\rvert\\lvert\\beta_1-j(\\tau_i)\\rvert\\lvert\\beta_2-j(\\tau_i)\\rvert&\\leq6\\cdot10^{14}\\lvert\\delta\\rvert\\\\\n &\\leq2^{-P+50}\\\\\n\\end{align*}\nNow for any $\\beta_i$, letting $j(\\tau_j)$ be the nearest root of $f(z)$, we have\n\\begin{equation*}\n \\lvert\\beta_i-j(\\tau_j)\\rvert\\leq2^{-P\/3+17}.\n\\end{equation*}\nBy Lemma \\ref{j_to_tau_small}, as $\\lvert j\\rvert\\geq2^{-P\/2}$,  \n\\begin{equation*}\n \\left\\lvert \\tau - \\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq 2^{-P\/6-9 },\n\\end{equation*}\nwhich yields, by Lemma \\ref{bounds_for_images}, for $i\\neq j$,\n\\begin{equation*}\n \\lvert j(\\tau_i)-j(\\tau_j)\\rvert\\geq 2^{-P\/6+9}.\n\\end{equation*}\nSimilarly to the previous lemma, we now observe that, with $\\beta_i$ the closest root to $j(\\tau_i)$,\n\\begin{align*}\n \\lvert\\beta_i-\\beta_j\\rvert &\\geq \\lvert j(\\tau_i) - j(\\tau_j)\\rvert - 2\\cdot2^{-P\/3+17}\\\\\n &\\geq 2^{-P\/6+9} - 2^{-P\/3+17}\\\\\n &\\geq 2^{-P\/6+8}\n\\end{align*}\nand so $\\beta_i$, $\\beta_j$ are distinct. So each root has a unique closest associated root, with relative precision\n\\begin{equation*}\n \\frac{2^{-P\/3+17}}{\\lvert j(\\tau_j)\\rvert} \\leq 2^{-P\/3+2}.\n\\end{equation*}\n\\end{proof}\n\n\n\n\n\\begin{proposition}\nLet $j=j(\\tau)$, where $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\leq2^{-31}$, and $\\tilde{j}$ be an approximation to $j$ of relative precision $2^{-P}$ such that $\\lvert\\tilde{j}\\rvert\\geq2^{-P\/2}$. Let\n\\begin{equation*}\n \\tau_0 = \\frac{1+i\\sqrt{3}}{2}+\\sqrt[3]{\\frac{6\\tilde{j}}{j^{(3)}(i)}},\n\\end{equation*}\nwhere the branch of the cube root is chosen arbitrarily.\nThen Newton's method applied to $\\Phi_2(\\tilde{j},z)$, with starting point $j(2\\tau_0)$ will converge to either the root of $\\Phi_2(\\tilde{j},z)$ closest to $j(2\\tau)$, the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau}{2}\\right)$, or the root of $\\Phi_2(\\tilde{j},z)$ closest to $j\\left(\\frac{\\tau+1}{2}\\right)$, and after $[2\\log P]$ steps will produce an approximation to either $j(2\\tau)$, $j\\left(\\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$ of relative precision $2^{-P\/3+3}$.\n\\end{proposition}\n\\begin{proof}\n Let $\\tau = \\frac{1+i\\sqrt{3}}{2}+\\delta$. By Lemma \\ref{taylor_series_at_i},\n \\begin{equation*}\n  \\left\\lvert j(\\tau) - \\frac{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}{6}\\delta^3\\right\\rvert\\leq 0.07\\lvert\\delta\\rvert^3,\n \\end{equation*}\nso that\n\\begin{equation*}\n  \\left\\lvert \\frac{6\\tilde{j}}{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)} -\\delta^3\\right\\rvert\\leq 1.6\\cdot10^{-6}\\lvert\\delta\\rvert^3.\n\\end{equation*}\nWe let $\\epsilon = \\sqrt[3]{\\frac{6\\tilde{j}}{j^{(3)}\\left(\\frac{1+i\\sqrt{3}}{2}\\right)}}$. Considering the geometry of the cube roots of $\\delta^3$, the product of the two furthest from $\\epsilon$ is at least $\\lvert\\delta\\rvert^2$, so letting the closest be $\\delta_0$, we have\n\\begin{equation*}\n\\lvert\\epsilon - \\delta_0\\rvert\\leq 1.6\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nWe now show that any branch of the cube root taken for $\\epsilon$ will result $2\\tau_0-1$ being very close to one of the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to one of $2\\tau,\\frac{\\tau}{2}, \\frac{\\tau+1}{2}$. We have\n\\begin{align*}\n \\left\\lvert\\frac{1+i\\sqrt{3}}{2}+\\epsilon- \\left(-\\frac{1}{\\tau}+1\\right)\\right\\rvert&=\\frac{\\left\\lvert \\tau+i\\sqrt{3}\\tau+2+2\\epsilon\\tau-2\\tau\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert -\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}-\\delta+i\\frac{\\sqrt{3}}{2} - \\frac{3}{2}+i\\sqrt{3}\\delta+2+(1+i\\sqrt{3})\\epsilon+2\\epsilon\\delta\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert(1+i\\sqrt{3})\\epsilon-(1-i\\sqrt{3})\\delta+2\\epsilon\\delta\\right\\rvert}{\\left\\lvert2\\tau\\right\\rvert}\\\\\n &\\leq\\lvert\\epsilon - e^{4i\\pi\/3}\\delta\\rvert+\\frac{\\lvert\\delta\\epsilon\\rvert}{2},\n\\end{align*}\nand \n\\begin{align*}\n \\left\\lvert\\frac{1+i\\sqrt{3}}{2}+\\epsilon- \\left(-\\frac{1}{\\tau-1}\\right)\\right\\rvert&=\\frac{\\left\\lvert \\tau-1+i\\sqrt{3}(\\tau-1)+2+2\\epsilon(\\tau-1)\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert -\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}+(1+i\\sqrt{3})\\delta-i\\frac{\\sqrt{3}}{2}-\\frac{3}{2}+2+(-1+i\\sqrt{3})\\epsilon +2\\epsilon\\delta\\right\\rvert}{2\\lvert\\tau\\rvert}\\\\\n &=\\frac{\\left\\lvert(1-i\\sqrt{3})\\epsilon-(1+i\\sqrt{3})\\delta-2\\epsilon\\delta\\right\\rvert}{\\left\\lvert2\\tau\\right\\rvert}\\\\\n &\\leq\\lvert\\epsilon - e^{2i\\pi\/3}\\delta\\rvert+\\frac{\\lvert\\delta\\epsilon\\rvert}{2}.\n\\end{align*}\nIn particular, the difference between $2\\tau_0-1$ and the closest of the $\\text{\\normalfont SL}_2(\\mathbb{Z})$--equivalent elements of $\\mathcal{F}$ to $2\\tau,\\frac{\\tau}{2}, \\frac{\\tau+1}{2}$, which we let be $\\tau_1$, and the others $\\tau_2,\\tau_3$, is bounded by\n\\begin{equation*}\n 2\\left\\lvert\\epsilon-\\delta_0\\right\\rvert+\\lvert\\delta\\epsilon\\rvert\\leq3.3\\cdot10^{-6}\\lvert\\delta\\rvert.\n\\end{equation*}\nNow as the absolute value of the derivative of $j(z)$ is bounded in absolute value by $3.4\\cdot10^5$ between $\\tau_0$ and $\\tau_1$, we have\n\\begin{equation*}\n \\left\\lvert j(2\\tau_0)-j(\\tau_1)\\right\\rvert\\leq 1.2\\lvert\\delta\\rvert.\n\\end{equation*}\nNow by the bound of Lemma \\ref{phi2discrepancy_points_0} on the discrepancy of the roots of $\\Phi_2(\\tilde{j},z)$ and $\\Phi_2(j,z)$, and the bounds on the distances between the distinct roots of Lemma \\ref{bounds_for_images}, letting $\\beta_0$ be the closest root of $\\Phi_2(\\tilde{j},z)$ to $j(2\\tau_0)$, and $\\beta_1,\\beta_2$ the other two roots, we collect the relevant bounds for Kantorovich's criterion,\n\\begin{align*}\n \\left\\lvert j(2\\tau_0) - \\beta_0\\right\\rvert&\\leq 1.2\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0) - \\beta_1\\right\\rvert, \\left\\lvert j(2\\tau_0) - \\beta_2\\right\\rvert&\\leq 1.35\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(2\\tau_0) - \\beta_1\\right\\rvert, \\left\\lvert j(2\\tau_0) - \\beta_2\\right\\rvert&\\geq 1.14\\cdot10^6\\lvert\\delta\\rvert,\n\\end{align*}\nand with $r=4\\lvert\\delta\\rvert$, for any $\\tau'$ such that $\\lvert\\tau'-2\\tau_0\\rvert\\leq4\\lvert\\delta\\rvert$, we have, as the derivative of $j$ is bounded in absolute value by $3.4\\cdot10^{5}$ here,\n\\begin{align*}\n \\left\\lvert j(\\tau') - \\beta_0\\right\\rvert&\\leq 1.37\\cdot10^6\\lvert\\delta\\rvert\\\\\n \\left\\lvert j(\\tau') - \\beta_1\\right\\rvert, \\left\\lvert j(\\tau') - \\beta_2\\right\\rvert&\\leq 2.71\\cdot10^6\\lvert\\delta\\rvert\\\\\n\\end{align*}\nNow we bound the various terms of Kantorovich's criterion. Firstly,\n\\begin{align*}\n \\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert&\\leq 1.2\\lvert\\delta\\rvert\\cdot(1.35\\cdot10^6\\lvert\\delta\\rvert)^2\\\\\n &\\leq 2.2\\cdot10^{12}\\lvert\\delta\\rvert^3,\n\\end{align*}\nfor the first derivative, we have\n\\begin{align*}\n \\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert&\\geq (1.14\\cdot10^6\\lvert\\delta\\rvert)^2 - 2\\cdot1.2\\lvert\\delta\\rvert\\cdot1.35\\cdot10^6\\lvert\\delta\\rvert\\\\\n &\\geq 1.2\\cdot10^{12}\\lvert\\delta\\rvert^2,\n\\end{align*}\nand for the second derivative,\n\\begin{align*}\n \\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert&\\leq 2\\cdot1.37\\cdot10^6\\lvert\\delta\\rvert+4\\cdot2.71\\cdot10^6\\lvert\\delta\\rvert\\\\\n &\\leq1.36\\cdot10^7\\lvert\\delta\\rvert,\n\\end{align*}\nso that we have\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert\\lvert\\Phi_2''(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert^2}\\leq\\frac{2.2\\cdot10^{12}\\lvert\\delta\\rvert^3\\cdot1.36\\cdot10^7\\lvert\\delta\\rvert}{(1.2\\cdot10^{12}\\lvert\\delta\\rvert^2)^2}\\leq 2^{-15}<\\frac{1}{2},\n\\end{equation*}\nand for the condition on $r$ we have\n\\begin{align*}\n r &=4\\lvert\\delta\\rvert\\\\\n 2\\eta&=2\\frac{\\lvert\\Phi_2(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}\\leq3.8\\lvert\\delta\\rvert,\n\\end{align*}\nensuring convergence. For the rate of convergence, we have the upper bound on the first derivative as follows,\n\\begin{align*}\n \\left\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\right\\rvert \\leq 1.4\\cdot10^{12}\\lvert\\delta\\rvert^2.\n\\end{align*}\nFor the lower bound on the second derivative, we have that\n\\begin{equation*}\n \\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert = \\lvert6 j(\\tau') - 2(\\beta_0+\\beta_1+\\beta_2)\\rvert.\n\\end{equation*}\nNow the second term in the absolute value is the coefficient of $z^2$ of $\\Phi_2(\\tilde{j},z)$, which is equal to \n\\begin{equation*}\n -\\tilde{j}^2+1488\\tilde{j}-162000.\n\\end{equation*}\nBy Lemma \\ref{taylor_series_at_i}, $\n\\lvert j\\rvert\\leq\\left(\\frac{\\left\\lvert j\\left(\\frac{1+i\\sqrt{3}}{2}\\right)\\right\\rvert}{6}+0.07\\right)\\lvert\\delta\\rvert^3$, and as $\\lvert\\tilde{j}-j\\rvert\\leq2^{-P}\\leq\\lvert j\\rvert^2$, $\\lvert\\tilde{j}\\rvert\\leq1.01\\lvert j\\rvert$, so\n\\begin{equation*}\n \\lvert-\\tilde{j}^2+1488\\tilde{j}\\rvert\\leq4.13\\cdot10^8\\lvert\\delta\\rvert^3\\leq0.1\\lvert\\delta\\rvert.\n\\end{equation*}\nIn particular, $-(\\beta_1+\\beta_2+\\beta_3)=162000+\\theta_1$, where $\\lvert\\theta_1\\rvert\\leq 0.1\\lvert\\delta\\rvert$.\nAs $\\lvert\\delta\\rvert\\leq2^{-31}$, $\\lvert\\tau'-2\\tau_0\\rvert\\leq4\\lvert\\delta\\rvert$, and $\\lvert2\\tau_0-1-i\\sqrt{3}\\rvert\\leq(2+3.3\\cdot10^{-6})\\lvert\\delta\\rvert$, \n\\begin{equation*}\n \\lvert\\tau'-1-i\\sqrt{3}\\rvert\\leq 6.1\\lvert\\delta\\rvert\\leq2^{-28},\n\\end{equation*}\nand hence by Lemma \\ref{taylor_series_at_2i},\n\\begin{equation*}\n \\lvert j(\\tau') - j(i\\sqrt{3})\\rvert\\geq \\lvert  j'(i\\sqrt{3})\\delta\\rvert-1.3\\lvert\\delta\\rvert\\geq 334000\\lvert\\delta\\rvert.\n\\end{equation*}\nIn particular, as $j(i\\sqrt{3})=54000$,  $j(2\\tau_0) = 54000+\\theta_2$, where $\\lvert\\theta_2\\rvert\\geq334000\\lvert\\delta\\rvert$.\nNow combining these bounds, we have\n\\begin{align*}\n  \\lvert 6 j(\\tau') - 2(\\beta_0+\\beta_1+\\beta_2)\\rvert &=\\lvert 324000+6\\theta_2-324000+2\\theta_1\\rvert\\\\\n  &\\geq 6\\lvert\\theta_2\\rvert - 2 \\lvert\\theta_1\\rvert\\\\\n  &\\geq6\\cdot334000\\lvert\\delta\\rvert - 0.2\\lvert\\delta\\rvert\\\\\n  &\\geq2\\cdot10^6\\lvert\\delta\\rvert.\n\\end{align*}\nReturning to the convergence, we have\n\\begin{equation*}\n \\frac{\\lvert\\Phi_2'(\\tilde{j},j(2\\tau_0))\\rvert}{\\lvert\\Phi_2''(\\tilde{j},j(\\tau'))\\rvert}\\leq\\frac{1.4\\cdot10^{12}\\lvert\\delta\\rvert^2}{2\\cdot10^6\\lvert\\delta\\rvert}\\leq2^{-10}, \n\\end{equation*}\nso a bound for the rate of convergence is\n\\begin{equation*}\n 2^{-15\\cdot2^k}.\n\\end{equation*}\nIn particular, to obtain an absolute precision of $2^{-P}$, $[2\\log P]$ steps will suffice.  The approximant obtained will then, by Lemma \\ref{phi2discrepancy_points_0}, be an approximation of one of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right)$, or $j\\left(\\frac{\\tau+1}{2}\\right)$, of relative precision at least $2^{-P\/3+3}$.\n\\end{proof}\n\n\\subsubsection{Running time}\nNow we have shown the time required to obtain an approximation to one of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right),$ or $j\\left(\\frac{\\tau+1}{2}\\right)$ is $O(M(P)\\log P)$, and the obtained value may then be used as an input to Newton's method on the compact set described in the subsequent section. In order to determine which of $j(2\\tau),j\\left(\\frac{\\tau}{2}\\right),j\\left(\\frac{\\tau+1}{2}\\right)$ was computed above, after obtaining an inverse $\\tau_0$, we compute $j(2\\tau_0)$ and $j\\left(\\frac{\\tau_0}{2}\\right)$, and return the argument corresponding to whichever was closest to our initial approximation $\\tilde{j}$ to $j$, after applying elements of $\\text{\\normalfont SL}_2(\\mathbb{Z})$ to move it to the fundamental domain (which takes $O(P)$ time). In particular we will obtain an approximation of precision at least $2^{-P\/3+12}$.\n\n\\subsection{Newton iteration on the compact set}\n\nFor $\\tau$ such that $\\tau \\in \\mathcal{F}$, $\\text{\\normalfont Im}(\\tau)\\leq 3.1$, $\\lvert \\tau - i\\rvert\\geq2^{-32}$, $\\left\\lvert\\tau-\\frac{1+i\\sqrt{3}}{2}\\right\\rvert\\geq 2^{-32}$, we make use of an algorithm due to Dupont, \\cite{fastj} for the quasilinear evaluation of $j(\\tau)$ to relative precision, in order to invert $j(z)$ by the secant method. As we are considering a compact set, there is some fixed precision our starting points for the secant method may be in order to obtain convergence for any $\\tau$.\nWe compute the low precision inverses of $j$ by the formula\n\\begin{align*}\n \\tau_0 &= i\\frac{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\frac{1}{2}+\\frac{1}{2}\\sqrt{1-\\frac{1728}{\\tilde{j}}}\\right)}{_2F_1\\left(\\frac{1}{6},\\frac{5}{6},1,\\frac{1}{2}-\\frac{1}{2}\\sqrt{1-\\frac{1728}{\\tilde{j}}}\\right)},\\\\\n \\tau_1 &= \\tau_0 - \\frac{j(\\tau_0)-j}{j'(\\tau_0)},\n\\end{align*}\nand we note that as $j$ is bounded away from $1728$, the arguments of the Gaussian hypergeometric functions are bounded away from their singularities at $0$ and $1$, and $j'$ is bounded away from $0$, so computation of the two points to a fixed precision takes a fixed amount of time. Upon a failure of convergence (or slow convergence), we may simply increase the precision of our starting points and repeat the process -- as there is a uniform bound on the required precision, this requires only constant time. As the secant method converges quadratically,  and the evaluation of $j(z)$ via the algorithm of $\\cite{fastj}$ takes $O(M(P)\\log P)$ time, the computational complexity of obtaining an approximation of precision $2^{-P+1}$ requires time $O(M(P)(\\log P)^2)$. \n\nA similar analysis to that of Lemma \\ref{taylor_series_at_i} gives a bound of \\begin{equation*}\n\\left\\lvert j^{k}(z_0)\\right\\rvert\\leq3\\cdot10^8\\cdot14^kk!                                                                                  \\end{equation*}\nfor $z_0$ in our compact set, so that if $z_1=z_0+\\lvert\\delta\\rvert$, with $\\lvert\\delta\\rvert\\leq2^{-150}$, \n\\begin{align*}\n \\lvert j(z_0)+j'(z_0)\\delta-j(z_1)\\rvert&\\leq3\\cdot10^8\\sum_{n=2}^\\infty\\lvert\\delta\\rvert^n14^n\\\\\n &\\leq3\\cdot10^8\\cdot14^2\\cdot2^{-150}\\cdot\\lvert\\delta\\rvert\\sum_{n=0}^\\infty2^{-150n}14^n\\\\\n &\\leq5\\cdot10^{-35}\\lvert\\delta\\rvert.\n\\end{align*}\nIt may be verified that $\\lvert j'(z)\\rvert\\geq10^{-19}$ in our compact set, so\n\\begin{equation*}\n \\lvert j(z_0)-j(z_1)\\rvert\\geq \\lvert j'(z_0)\\rvert\\lvert\\delta\\rvert-5\\cdot10^{-35}\\lvert\\delta\\rvert\\geq 10^{-20}\\lvert\\delta\\rvert,\n\\end{equation*}\nand in particular once we have computed by the secant method $\\tau^*$ such that $\\lvert \\tilde{j}-j(\\tau^*)\\rvert\\leq2^{-P}$, $\\lvert j-j(\\tau^*)\\rvert\\leq3\\cdot10^8\\cdot2^{-P}$, and so, with $j(\\tau)=j$, and $\\tau^*=\\tau+\\delta$, we have \n\\begin{equation*}\n \\lvert\\delta\\rvert\\leq 3\\cdot10^8\\cdot10^{20}\\cdot2^{-P}\\leq2^{-P+100}.\n\\end{equation*}\n\\subsection{$j$ very close to $0$ or $1728$}\nNow if $\\lvert\\tilde{j}\\rvert\\leq 2^{-P\/2}$, or $\\lvert\\tilde{j}-1728\\rvert\\leq2^{-P\/3}$, we simply return $\\frac{1+i\\sqrt{3}}{2}$ or $i$ respectively, and this will, by Lemma \\ref{j_to_tau_small}, be an approximation to the inverse of $j$ of absolute, and as $\\lvert\\tau\\rvert\\geq 1$, relative precision $2^{-P\/6}$.\n\n\\iffalse\nWe do not perform an analysis of the required precision for the secant method, but note that an initial precision of $2^{-90}$ would suffice for Newton's method in this situation, so the constant time required does not seem excessive.\n\\fi\n\n\\iffalse\n The following bounds hold,\n\\begin{align*}\n \\left\\lvert j(\\tau)\\right\\rvert &\\leq 6\\cdot10^5\\\\\n \\left\\lvert j'(\\tau)\\right\\rvert &\\leq 4\\cdot10^6\\\\\n \\left\\lvert j''(\\tau)\\right\\rvert &\\leq 3\\cdot10^7\\\\\n \\left\\lvert j'(\\tau)\\right\\rvert &\\geq 3\\cdot10^{-5}\\\\\n \\left\\lvert j''(\\tau)\\right\\rvert &\\geq 4\n\\end{align*}\n\\fi\n\\iffalse\nso that by  (KATOROVICH), if we start with a low--precision approximation $\\tau_0$ to $\\tau$ of absolute precision $2^{-90}$, then Newton's method applied to the function $j(z)-\\tilde{j}$ will converge to the pre--image of $\\tilde{j}$ under $j(z)$, as\n\\begin{equation}\n \\frac{(j(\\tau_0)-\\tilde{j})j''(\\tau_0)}{j'(\\tau_0)^2}\\leq \n\\end{equation}\n\\fi\n\n\\iffalse\n\\begin{equation*}\n j(z) = 32\\frac{\\theta_{00}(0;z)^8+\\theta_{01}(0;z)^8+\\theta_{10}(0;z)^8}{\\theta_{00}(0;z)^8\\theta_{01}(0;z)^8\\theta_{10}(0;z)^8},\n\\end{equation*}\n\\fi\n\n\\section{Testing CM}\n\\iffalse\nHere we consider the problem of determining whether an elliptic curve has complex multiplication. We take as input an elliptic curve in the form\n\\begin{equation*}\n y^2=4x^3+\\alpha x+\\beta\n\\end{equation*}\nwith the following data,\n\\begin{itemize}\n \\item Approximations to $\\alpha$ and $\\beta$ of relative precision ,\n \\item The Mahler measures $M_\\alpha$, $M_\\beta$ and degrees $d_\\alpha$, $d_\\beta$ of $\\alpha$ and $\\beta$.\n\\end{itemize}\n\nWe first compute the $j$-invariant of the elliptic curve,\n\\begin{equation*}\n j = 1728\\frac{\\alpha^3}{\\beta^3-27\\alpha^3}.\n\\end{equation*}\nAs $\\beta^3\\neq27\\alpha^3$, by Liouville's inequality, we have\n\\begin{equation*}\n \\lvert\\beta^3-27\\alpha^3\\rvert\\geq28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}.\n\\end{equation*}\nFirst note that as $\\lvert\\alpha\\rvert\\leq M_\\alpha$, and $\\lvert\\beta\\rvert\\leq M_\\beta$, $\\lvert\\delta_1\\rvert,\\lvert\\delta_2\\rvert\\leq1$. Let our approximations to $\\alpha$, $\\beta$ be $\\tilde{\\alpha}=\\alpha+\\delta_1$, and $\\tilde{\\beta}=\\alpha+\\delta_2$, with $\\lvert\\delta_1\\rvert,\\lvert\\delta_2\\rvert\\leq\\delta$. The discrepancy between our approximation of $\\beta^3-27\\alpha^3$ and its true value is bounded as follows,\n\\begin{align*}\n \\lvert(\\beta+\\delta_1)^3-27(\\alpha+\\delta_2)^3 - \\beta^3-27\\alpha^3\\rvert&\\leq 3\\delta_2\\lvert\\beta\\rvert^2+3\\delta_1^2\\lvert\\beta\\lvert+\\delta_1^3+81\\delta_2\\lvert\\alpha\\rvert+81\\lvert\\delta_1\\rvert^2\\lvert\\alpha\\rvert+27\\lvert\\delta_2\\rvert^3\\\\\n &\\leq250\\delta\\max\\{M_\\alpha,M_\\beta\\},\n\\end{align*}\nand so on inverting this, if the difference of our approximation \n\nSo as our approximations to $\\beta$ and $\\alpha$ have precision at least\n\\begin{equation*}\n \\left(250^{-1}28^{-3d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta-3}M_\\beta^{-3d_\\alpha-3}\\right)^4 =: \\epsilon^4,\n\\end{equation*}\nour approximation of the inverse of $\\beta^3-27\\alpha^3$ is\n\\begin{align*}\n \\left\\lvert\\frac{1}{\\beta^3-27\\alpha^3}-\\frac{1}{\\tilde{\\beta}^3-27\\tilde{\\alpha}^3}\\right\\rvert &\\leq \\frac{250\\delta\\max\\{M_\\alpha,M_\\beta\\}}{28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}\\left(28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}-250\\delta\\max\\{M_\\alpha,M_\\beta\\}\\right)}\\\\\n &\\leq 2\\cdot250^{-2}M_\\alpha^{-5}M_\\beta^{-5}\\epsilon^2,\n\\end{align*}\nand we let this approximation be $\\gamma_1$.\nOur approximation to $\\alpha^3$ satisfies\n\\begin{equation*}\n \\lvert(\\alpha+\\delta_2)^3-\\alpha^3\\rvert\\leq7\\delta\\lvert\\alpha\\rvert^2,\n\\end{equation*}\nand we let this be $\\gamma_2$. So on multiplying this with the inverse of $\\tilde{\\beta}^3-27\\tilde{\\alpha}^3$, we obatin an approximation $\\tilde{j}$ to $j$ with error bounded as follows,\n\\begin{equation*}\n \\left\\lvert1728\\gamma_1\\gamma_2-1728\\frac{\\alpha^3}{\\beta^3-27\\alpha^3}\\right\\rvert\\leq 1728(250^{-2}M_\\alpha^{-5}M_\\beta^{-5}\\epsilon^2\\lvert\\gamma_2\\rvert+7\\delta M_\\alpha\\lvert\\gamma_1\\rvert+250^{-1}7\\delta M_\\alpha\\epsilon).\n\\end{equation*}\nas \n\\begin{align*}\n \\lvert\\gamma_1\\rvert&\\leq 2\\cdot28^{d_\\alpha d_\\beta}M_\\alpha^{3d_\\beta}M_\\beta^{3d_\\alpha},\\\\\n \\lvert\\gamma_2\\rvert&\\leq 7M_\\alpha^3,\n\\end{align*}\nour approximation to $j$ has absolute precision at least $\\epsilon^2$. Furthermore,\n\\begin{equation*}\n \\lvert j\\rvert \\geq 28^{-d_\\alpha d_\\beta}M_\\alpha^{-3d_\\beta}M_\\beta^{-3d_\\alpha}\\cdot M_\\alpha^{-3}\\geq \\epsilon,\n\\end{equation*}\nso\n\\begin{equation*}\n \\frac{\\left\\lvert\\tilde{j}-j\\right\\rvert}{\\lvert j\\rvert}\\leq \\frac{\\epsilon^2}{\\epsilon}\\leq \\epsilon,\n\\end{equation*}\nso our approximation has relative precision at least $\\epsilon$. As $\\lvert j\\rvert\\leq \\frac{1}{\\epsilon}$, \n\\begin{equation*}\n \\lvert j-e^{2\\pi i\\tau}\\rvert\\leq 2079,\n\\end{equation*}\nand $\\epsilon \\leq 250^{-4}$, $\\lvert \\tau\\rvert\\leq \\frac{-\\log\\epsilon}{2\\pi}+2$.\nApplying our algorithm to calculate the inverse of $j$, we obtain an approximation to its inverse $\\tau$ of relative precision $\\epsilon^{1\/4}\\log\\frac{1}{\\epsilon}$, and so an approximation of absolute precision $\\epsilon^{1\/4}$.\n \\fi\n \n Firstly, given the input of the $j$--invariant of an elliptic curve $E$, its degree $d$, and a bound on its height $H\\geq e^e$, we may bound the maximum discriminant from which $j$ may arise, if $E$ were to have complex multiplication. We first consider $\\lvert D\\rvert\\geq16$. The principal form of a discriminant has an associated $\\tau$ of imaginary part $\\frac{i \\sqrt{\\lvert D\\rvert}}{2}$, so as $\\lvert D\\rvert\\geq 16$, $M(j(\\tau))\\geq e^{\\pi\\sqrt{\\lvert D\\rvert}}-2079 \\geq e^{3.13\\sqrt{\\lvert D\\rvert}}$. In particular, as $M(j)\\leq H^d$, \n \\begin{equation*}\n  \\lvert D\\rvert\\leq \\frac{d^2(\\log H)^2}{9.7}.\n \\end{equation*}\n Now the $\\tau$ associated to a binary quadratic form of discriminant of absolute value $\\leq N$ has real part bounded in height by $2N$, and the square of the imaginary part bounded in height by $4N^2$, so we will determine if the preimage of $j$ is a quadratic irrational satisfying these conditions. \n \n \n \n We first bound the degree of  $j(z)$ at a quadratic irrational of discriminant $D$. For a fundamental discriminant $D$, we have the bound (Proposition 2.2, \\cite{paulin}),\n \\begin{equation*}\n  h(D)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert),\n \\end{equation*}\n and in the case of non--fundamental discriminants, by Theorem 7.4 of \\cite{buell}, for an odd prime $p$,\n \\begin{equation*}\n  h(p^2D)\\leq (p+1)h(D)\\leq\\frac{p+1}{p\\pi}\\sqrt{\\lvert p^2D\\rvert}(2+\\log\\lvert D\\rvert)\\leq\\frac{4}{3\\pi}\\sqrt{\\lvert p^2D\\rvert}(2+\\log\\lvert p^2D\\rvert)\n \\end{equation*}\n and for $2$, if $D\\not\\equiv0(8)$,\n \\begin{equation*}\n h(4D)\\leq 3h(D)\\leq \\frac{3}{2\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert D\\rvert)\\leq \\frac{3}{2\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert 4D\\rvert)\n \\end{equation*}\nand if $D\\equiv 0(4)$,\n\\begin{equation*}\n h(4D)\\leq 2h(D)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert D\\rvert)\\leq\\frac{1}{\\pi}\\sqrt{\\lvert 4D\\rvert}(2+\\log\\lvert 4D\\rvert).\n\\end{equation*}\nIn particular, we have the bound\n\\begin{equation*}\n h(D)\\leq\\frac{3}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert).\n\\end{equation*}\n For the Mahler measure of $j(\\tau)$, as $\\lvert j(\\tau)\\rvert\\leq e^{2\\pi\\text{\\normalfont Im}(\\tau)}+2079\\leq9.1e^{2\\pi\\text{\\normalfont Im}(\\tau)}$, we have the upper bound in terms of the reduced binary quadratic forms $ax^2+bxy+cy^2$ of discriminant $D$,\n \\begin{equation*}\n  M(j(\\tau))\\leq9.1^{h(D)}\\exp\\left(2\\pi\\sqrt{\\lvert D\\rvert}{\\sum_{\\substack{(a,b,c)\\\\ \\text{ reduced}}}\\frac{1}{a}}\\right).\n \\end{equation*}\n The number of times each $a$ may occur is bounded by the number of solutions of $b^2\\equiv d(2a)$, which is bounded by twice the number of distinct divisors $r(a)$ of $a$. By \\cite{divisor_problem}, we have the bound\n \\begin{equation*}\n  A(x) = 2\\sum_{1\\leq a\\leq x}r(A)\\leq 2x\\log x+0.4x+2x^{1\/2},\n \\end{equation*}\n so that by Abel's summation formula,\n \\begin{align*}\n  \\sum_{1\\leq a\\leq h(D)}\\frac{r(a)}{a} &= \\frac{A(h(D))}{h(D)}+\\int_{1}^{h(D)}\\frac{A(y)}{y^2}dy\\\\\n  &\\leq \\frac{(\\log h(D))^2}{2}+1.2\\log(h(D))+2.2\\\\\n  &\\leq 1.3\\log\\lvert D\\rvert^2,\n \\end{align*}\n yielding a bound of\n \\begin{equation*}\n  M(j(\\tau))\\leq9.1^{h(D)}e^{5.2\\pi\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2}\\leq e^{5.9\\pi\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2}.\n \\end{equation*}\n In particular, by Liouville's inequality, if $j\\neq j(\\tau)$, where $\\tau$ is of discriminant $D$,\n \\begin{align*}\n  \\lvert j - j(\\tau)\\rvert &\\geq 2^{-\\frac{3d}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert)} H^{-\\frac{3d}{2\\pi}\\sqrt{\\lvert D\\rvert}(2+\\log\\lvert D\\rvert)}e^{5.9\\pi d\\sqrt{\\lvert D\\rvert}(\\log\\lvert D\\rvert)^2},\n \\end{align*}\n and substituting our bound on $\\lvert D\\rvert$, we obtain the lower bound\n\\begin{equation*}\n \\exp\\left(-30d^2\\log H(\\log d+\\log\\log H)^2\\right).\n\\end{equation*}\n  Now as $\\lvert j\\rvert\\leq H^d$, letting $j=j(z_0)$, where $z_0\\in\\mathcal{F}$, as $\\lvert j(z_0)\\rvert\\geq e^{2\\pi\\text{\\normalfont Im}(z_0)}-2079$, we have $\\text{\\normalfont Im}(z_0)\\leq \\frac{d\\log H}{2\\pi}+2$. By differentiating the $q$--expansion of $j$, and taking an upper bound, (note that the sum of the absolute values of the terms of the tail is decreasing), we have, as $d\\log H\\geq0$,\n  \\begin{equation*}\n   \\lVert j'\\rVert_{B(z_0,1)}\\leq 2\\pi e^{d\\log H+6\\pi}+800\\leq e^{d\\log H+21},\n  \\end{equation*}\nso that if $\\lvert z_0 - \\tau\\rvert\\leq1$, \n\\begin{equation*}\n \\lvert j - j(\\tau)\\rvert\\leq\\lvert z_0 - \\tau\\rvert e^{d\\log H+21}.\n\\end{equation*}\nCombining this with our separation result, noting that $\\log d+\\log\\log H\\geq1$, if\n\\begin{equation*}\n \\lvert z_0 - \\tau\\rvert\\leq \\exp\\left(-31d^2\\log H(\\log d+\\log\\log H)^2-21\\right),\n\\end{equation*}\nthen $j=j(\\tau)$, and so $j$ is a singular modulus. Now as our method of inverting $j$ from an input of regulated precision $2^{-P}$ obtains its inverse with relative precision at least $2^{-P\/6}$, to obtain a result of absolute precision $2^{-Q}$, as $\\lvert \\tau\\rvert\\leq\\frac{d\\log H}{2\\pi}+2$, we will need an input of relative precision $2^{-6Q-2\\log(d\\log H+2)}$. In particular, with an input of relative precision\n\\begin{equation*}\n 2^{-300d^2\\log H(\\log d+\\log\\log H)^2-200},\n\\end{equation*}\nour computed approximation $\\tilde{z_0}$ to $z_0$ will have sufficient precision to determine whether\n\\begin{equation*}\n \\lvert z_0 - \\tau\\rvert\\leq\\lvert z_0 - \\tilde{z_0}\\rvert+\\lvert\\tau-\\tilde{z_0}\\rvert\n\\end{equation*}\nis sufficiently small.\nIn order to obtain our candidate $\\tau$, we develop the continued fractions of the real part and the square of the imaginary part of our computed inverse $\\tilde{\\tau}$ -- as the discriminants under consideration are bounded by $\\lvert D\\rvert$, the distance between the real parts of any two preimages of the singular moduli is at least \n\\begin{equation*}\n 19^{-1}d^{-4}(\\log H)^{-4},\n\\end{equation*}\nand of the squares imaginary parts is at least\n\\begin{equation*}\n 19^{-2}d^{-8}(\\log H)^{-8}.\n\\end{equation*}\nSo we develop the continued fractions of the real part of $\\tilde{z_0}$ and the square of the imaginary part of $\\tilde{z_0}$ until their difference from the convergents is bounded by  \n\\begin{equation*}\n 19^{-3}d^{-8}(\\log H)^{-8},\n\\end{equation*}\nfor once this holds of the convergents, they will form the only possible quadratic irrational inverse of $j$.\nIf the height of the real convergent $c_r$ is greater than $\\frac{d^2(\\log H)^2}{9.7}$, or the height of the square of the imaginary convergent $c_i$ is greater than $\\frac{d^4(\\log H)^4}{90}$, or the square root of $c_i$ is a rational number, then we may conclude that $j$ is not a singular modulus -- otherwise, letting $\\tau = c_r+i\\sqrt{c_i}$, we may test if $\\lvert \\tilde{z_0}-\\tau\\rvert$ satisfies our condition for $j=j(\\tau)$. If so, then $j$ is a singular modulus, and otherwise $j$ is not a singular modulus for $\\tau$ of discriminant $\\lvert D\\rvert\\geq16$. It is clear that the computational complexity of the calculation of the convergents and other operations in this algorithm is dominated by that of inverting $j$, so we obtain a running time, with $T=d^2\\log H(\\log d+\\log\\log H)^2$ of \n\\begin{equation*}\n O(M(T)(\\log T)^2).\n\\end{equation*}\nFor testing discriminants with $\\lvert D\\rvert\\leq16$, we may simply take the list of all $\\tau$ of discriminant $\\leq16$, and test whether $j$ is equal to them. The degree of $j(\\tau)$ for such $\\tau$ is bounded by $2$, and its Mahler measure is bounded by $3\\cdot10^6$, so if \n\\begin{equation*}\n \\lvert j-j(\\tau)\\rvert\\leq2^{-2d}H^{-2d}\\left(3\\cdot10^6\\right)^{-d}\\leq\\exp(2d(33+\\log H)),\n\\end{equation*}\nthen $j=j(\\tau)$. So with an approximation $\\tilde{j(\\tau)}$ to $j(\\tau)$ of absolute precision $2^{-4d(33+\\log H)+2}$, then we will be able to establish whether $j=j(\\tau)$. By the algorithm to compute $j$ of \\cite{fastj}, which requires time $O(M(P)\\log P)$ for relative precision of $P$, as $\\lvert j(\\tau)\\rvert\\leq 3\\cdot10^{6}$, computing $j$ to the required absolute precision possible in time\n\\begin{equation*}\n O(M(d\\log H)(\\log d+\\log\\log H)),\n\\end{equation*}\nwhich again is $O\\left(M(T)(\\log T)^2\\right)$.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Key notions}\n\nA decay of a particle $A$ is a process of the type\n $A \\rightarrow B_1 + B_2 + \\ldots B_n$, where  the particles  $B_i$ are  the decay products. Decays are ubiquitous in physics. Examples include the emission of photons from excited atoms or nuclei, alpha and beta emission, decays of composite subatomic particles (for example, neutrons or  pions) and decays of elementary particles (for example, a muon decaying to one electron and two neutrinos).\n\nMost decays follow an exponential law.  The probability that a decay takes place within the time-interval\n  $[t, t+ \\delta t]$, for $t > 0 $,  equals  $p(t) \\delta t$, where the probability density $p(t)$ is given by\n\\begin{eqnarray}\np(t) = \\Gamma e^{- \\Gamma t}. \\label{probdec}\n\\end{eqnarray}\nThe {\\em decay constant} $\\Gamma$ is positive and has dimensions of inverse time.\n\nWe note that for any two instants of time $t_1$ and $t_2$,\n\\begin{eqnarray}\np(t_2) = p(t_1) e^{-\\Gamma(t_2-t_1)},\n\\end{eqnarray}\nHence, the exponential law remains invariant under a shift of the     initial moment of time $t = 0$. The decay of an ensemble of $A$ particles after a moment of time $t$ carries no memory of any properties prion to $t$. In classical  probability theory, exponential decays correspond to {\\em Markovian processes}.\n\nExperiments typically involve a large number of decaying particles with identical preparation and the detection of  some of  the decay products. Recording the number of detection events within given time intervals $[t, t+\\delta t]$, we can reconstruct the probability density $p(t)$ associated to the decay. Hence,  $p(t)$ is a {\\em directly observable quantity}.\n\nThe focus of this paper is the derivation of the probability density $p(t)$ from the rules of quantum mechanics. We present different approaches to the problem, we apply them to specific physical problems and we analyze their underlying assumptions and their limitations.\n\n\n\n\n\n\n\n\n\n\\subsection{The quantum description of decays}\n\nThe aim of the quantum mechanical description of a decay process is to construct the probability density $p(t)$ from first principles. This probability density is different from the ones usually considered in quantum theory, because  the random variable $t$ is   temporal.  We cannot use Born's rule, because there is no self-adjoint operator for time in quantum theory \\cite{Pauli}. Indeed, the construction of quantum probabilities in which time is a random variable is an old problem---for reviews, see,  Ref.   \\cite{ToAbooks}. There are several different approaches that lead to different results, even for elementary problems, for example, constructing probabilities for the time of arrival \\cite{ML} or   specifying the time it takes a particle to tunnel through a potential barrier \\cite{tunt}.\n\n \\medskip\n {\\em The persistence amplitude method.}\nMost studies of decays avoid  a direct construction of $p(t)$.  Instead they focus on a slightly different issue,  namely, on finding the probability that the quantum system persists in its initial  configuration. In quantum theory, the notion of a configuration refers to the set of all quantum states compatible with a specific property. Mathematically, it corresponds  to a subspace of the system's Hilbert space, and it is represented by the associated projection operator. For example, if the defining property of the initial configuration is that a particle is confined by a potential well in a spatial region $U$, the relevant subspace corresponds to the projection operator $\\hat{P}_U = \\int_U dx |x\\rangle \\langle x|$ that describes the property $\"x \\in U\"$ for the particle position $x$.\n\n In some unstable systems, the subspace of the initial configuration is one-dimensional, and it coincides with the quantum state  $|\\psi\\rangle$ at which the system has been initially prepared. For such systems,\nit is convenient to employ the\n   {\\em persistence amplitude} (or {\\em survival amplitude})\n\n\\begin{eqnarray}\n{\\cal A}_{\\psi}(t) = \\langle \\psi|e^{-i\\hat{H}t}|\\psi\\rangle. \\label{persiss}\n\\end{eqnarray}\nwhere $\\hat{H}$ is the Hamiltonian of the system---for the mathematical properties of the persistence amplitude, see Refs. \\cite{GhiFo, Peres80}.\n\n\nThe modulus square of\n  ${\\cal A}_{\\psi}(t)$ is the {\\em persistence probability} (or {\\em survival probability}), i.e., the probability  a system prepared at the state $|\\psi\\rangle$ at $t = 0$ will still  be found at $|\\psi\\rangle$ by a measurement at a later time $t$. The persistence probability is identical with the {\\em fidelity} between the initial state and its time-evolution.\n\n\n   It is then suggested that the decay probability density be defined as\n\\begin{eqnarray}\np(t) = - \\frac{d}{dt} |{\\cal A}_{\\psi}(t)|^2, \\label{decprob2}\n\\end{eqnarray}\ni.e., the decay is assumed to happen when the system leaves  $|\\psi\\rangle$.\n\nEq. (\\ref{decprob2}) is a reasonable candidate probability density provided that (i) all states  normal to $|\\psi\\rangle$ that can be reached via Hamiltonian evolution correspond to decay products, and (ii) the probability of the reverse  process to the decay is negligible.\n\nIf condition (i) is not satisfied, then the propositions A = \"the system left $|\\psi\\rangle$\" and B = \"the decay happened\" are not identical: B implies A, but A does not imply B. Hence,  Eq. (\\ref{decprob2}) cannot be identified with the decay probability. For example,  if\n $|\\psi\\rangle$ evolves to a different state $|\\phi\\rangle$ that describes the initial particle $A$,  a part of the persistence amplitude will describe  Rabi-type oscillations between $|\\psi\\rangle $ and $|\\phi \\rangle$. The persistence probability may not be a decreasing function of $t$, and, hence, $p(t)$ may take negative values. It will not be a genuine probability density: Eq. (\\ref{decprob2}) will not predict the number of particles detected at each  moment of time.\n\nCondition (ii) is satisfied if there are many more states available to the decay products than to the initial particle. Furthermore, the configuration of the experiment must be such as to allow the decay products to `explore' their available states. This means that the decay products  leave the locus of their production and they are measured far away. Consider for example an excited atom contained in a cavity. For some cavity geometries,  the emitted photon does not exit the cavity immediately, and it may be reabsorbed by the atom at a later time. The persistence amplitude of the excited atomic state may has an oscillating component. In this case, the candidate probability density (\\ref{decprob2}) does not correlate with  photodetection records.\n\n\nFor systems that satisfy conditions (i) and (ii), Eq. (\\ref{decprob2}) provides a  reasonable  construction of the decay probability density. It works best in model systems that incorporate the conditions (i) and (ii)  above into their definition. On such example is   the Lee model \\cite{Lee} that is presented in   Sec. 3. However, even in such models the candidate probability density (\\ref{persiss}) may become negative outside the exponential decay regime---see, Sec. 3.1.3.\n\n\nWe note that the interpretation of the persistence probability as the fidelity of the initial state makes it a useful tool also for the study of a broader class of physical phenomena than decays \\cite{BPSZ}. Hence, the quantity (\\ref{decprob2}) may be of physical interest, even if it takes negative values and cannot be identified with the probability density for decay.\n \\medskip\n\n {\\em Probability currents.}\n An alternative elementary description of decays is available, whenever we can  associate a probability-current operator $\\hat{\\pmb J}({\\pmb x}, t)$ to one of the decay products. For example, for non-relativistic particles of mass $m$ satisfying Schr\\\"odinger's equation with Hamiltonian $\\hat{H}$, the current operator is\n\\begin{eqnarray}\n \\hat{\\pmb J}({\\pmb x}, t) = \\frac{1}{2m}e^{i\\hat{H} t}\\left[ \\hat{\\pmb p} \\delta^3(\\hat{\\pmb x} - {\\pmb x}) + \\delta^3(\\hat{ \\pmb x} - {\\pmb x}) \\hat{\\pmb p} \\right] e^{-i\\hat{H}t},\n\\end{eqnarray}\nwhere $\\hat{\\pmb x}$ and $\\hat{\\pmb p}$ are the standard position and momentum operators, respectively.\nThe expectation value of  $\\hat{\\pmb J}({\\pmb x}, t)$ on a   state $|\\psi\\rangle $ reproduces the standard expression for the probability current $\\frac{1}{m}[\\mbox{Im} \\psi^*({\\pmb x}, t) {\\pmb \\nabla} \\psi({\\pmb x}, t)]$  associated to the solution $\\psi({\\pmb x}, t) := \\langle {\\pmb x}|e^{-i \\hat{H}t}|\\psi\\rangle$ of Schr\\\"odinger's equation.\n\nGiven a current operator  $\\hat{\\pmb J}({\\pmb x}, t)$ for one of the decay products, we can evaluate the flux  $\\Phi_C(t) = \\int_C d^2{\\pmb \\sigma} \\cdot \\langle \\psi| \\hat{\\pmb J}({\\pmb x}, t)|\\psi\\rangle$ through any surface $C$. In many set-ups, the probability flux  coincides with the particle flux through this surface, which is  a directly measurable quantity. Then, the flux $\\Phi_S(t)$ over a two-sphere surrounding the unstable system is expected to be proportional to the decay probability density $p(t)$.\n\n The main limitation of this method is that it cannot be consistently applied to relativistic systems, because of difficulties in defining probability currents that are both  Lorentz covariant and causal \\cite{RoHo}. For this reason, it has mainly been used only in non-relativistic settings, in particular, for decays that can be described in terms of tunneling \\cite{LL, Perel}.\n\n Another problem of the probability current method is the possibility of back-flow: the current operator has  negative eigenvalues even for states with strictly positive momentum \\cite{Brack}. Hence, in some cases the probability flux may turn out to be negative for some times $t$--especially when the current is evaluated near the locus of the decay \\cite{Winter}. In such cases the flux is not a reliable measure of detected outgoing particles.\n\n \\medskip\n\n {\\em Detector models.} A more rigorous description of quantum decays requires the incorporation of  the measurement process  \\cite{EkSi, GhiFo}. In fact, the idea that quantum decays cannot be explained solely in terms of unitary time evolution was  crucial to the early interpretational discussions of quantum theory \\cite{Heismem}. We have to take into account   the irreversibility of the quantum measurements on the decay products  in order to avoid theoretical discrepancies. This is the case, for example, in systems characterized by competing decay channels. If one ignores the effects of the apparatus, the persistence amplitude  exhibits large oscillations, thereby contradicting the standard classical description of sequential decays \\cite{FoGhi}.\n\nIn quantum measurement theory, the consistent treatment of the measuring apparatus allows of the description of quantum observables in terms of Positive Operator Valued Measures (POVM), i.e., a set of positive operators $\\hat{\\Pi}(a)$, where $a$ are the values of the physical magnitude recorded by the apparatus. Then, given an initial state $\\hat{\\rho}$ of the system, the probability that the result $a$ is obtained is given by $Tr\\left[\\hat{\\rho}_0 \\hat{\\Pi}(a)\\right]$.\n\n In recent years, a new class of model for particle detection has been developed, allowing    for  the construction of quantum observables for the time $t$ of a detection event \\cite{AnSav}---see, also \\cite{An08} for an early application to the decay problem. In this paper, we present a study of decays, using one such\nobservable $\\hat{\\Pi}(t)$ for detection time  that can be obtained by  elementary arguments similar to the ones of standard photodetection theory \\cite{Glauber}.\n\n\n\\subsection{This paper}\nThe aim of this paper is to provide the\n reader with tools for addressing a large class of decay problems. We present the most common approximation schemes, as well as criteria for checking when they fail. We prefer to work with models that admit simple solutions and controlled approximations, but we introduce more complex models for the explicit demonstration of important physical issues.\n\nA key motivation for this paper is to provide explicit arguments that the traditional methods for describing decays (persistence amplitude and probability current) are inapplicable for many problems of current physical interest.  These methods work excellently for exponential decays, but they lead to negative values of the decay probability $p(t)$ outside the exponential regime. The problem is not a failure of some approximation,  the  very definition of such methods cannot guarantee positivity. We believe that a first-principles construction of decay probabilities is essential for describing decays in novel regimes, for example, in entangled systems \\cite{AnHu2, CVS17, C18}, attosecond tunneling ionization \\cite{atto} or decay oscillations in nuclear physics \\cite{GSI}.\n\n\n\nThe paper is organized as follows.\n  Sec. 2 presents the simplest method for the study of decays, namely, the evaluation of the persistence amplitude as a line integral on the complex energy space. This method is particularly suitable for perturbative decays, i.e., systems in which the decay originates from a small term in the Hamiltonian of the system. It also applies to relativistic systems described by quantum field theory.\n\n  We show that   exponential decay is common in perturbative decays: it originates  from the separation of energy scales in the persistence amplitude. Nonetheless, it is neither exact nor generic. Exponential decay fails at very early and very late times, and also if the energy of the initial state is close to a resonance of the system.\n\n In  Sec. 3, we present the Lee model that describes the decaying particle as a  two-level system. This model can easily be adapted to problems in different branches of physics, including high energy physics, nuclear physics, atom optics, and condensed matter. Here, we present its applications to photo-emission and beta decay. We also employ the model in order to demonstrate the breakdown of the persistence amplitude method outside the exponential decay regime.\n\n Sec. 4 describes the decays of an atom in a cavity, providing an explicit example of decays that are non-exponential at all times because of resonance.\n\nIn Sec. 5, we study non-perturbative decays due to quantum tunneling, using the probability current method. We show that exponential decays originate from a decoherence condition, namely the lack of interferences between different attempts of the particle to tunnel through the barrier.\n\nIn Sec. 6, we revisit the Lee  model using probabilities constructed from a simple temporal observable, analogous to the one used in photodetection theory. We show that the detection probability reproduces the results of the persistence amplitude method, but can also be used in regimes where the latter fails.\n\nThe material in this article is mostly at the level of a graduate course in quantum mechanics. Familiarity with quantum many-particle systems is presupposed.\nSecs. 2, 3 and  5 provide a minimal   introduction to quantum decays, explaining the emergence of the exponential decay law and its limitations,  developing methods that can be applied to different branches of physics, and presenting some important    physical examples.\nThe Appendix contains a sketch of  additional results that can be used as exercises.\n\n\\section{Perturbative evaluation of the persistence amplitude}\n\nIn this section, we study decays in which the persistence amplitude can be evaluated perturbatively. All physical properties of the decays are encoded into a function defined on the complex energy plane, the {\\em self-energy function}. We identify the conditions under which the exponential law emerges, and we identify regimes for non-exponential decays.\n\n\\subsection{ Preliminaries}\nBy definition, the initial state\n $|\\psi\\rangle$ of an unstable system is not an eigenstate of the Hamiltonian $\\hat{H}$. However, in many problems of physical interest, $|\\psi\\rangle$ is close to an eigenstate of $\\hat{H}$, in the sense that  $\\hat{H}$  is of the form\n\\begin{eqnarray}\n\\hat{H} = \\hat{H}_0 + \\hat{V}, \\label{Hamper}\n\\end{eqnarray}\nwhere $|\\psi\\rangle$ is an eigenstate of $\\hat{H}_0$ and  $\\hat{V}$ is a small perturbation. For example, $\\hat{H}_0$ may be the Hamiltonian of an atom, and $\\hat{V}$ the interaction Hamiltonian of the atom with the quantum electromagnetic field.\nIn this case, we evaluate the persistence amplitude   perturbatively.\n\nLet us denote by $|b\\rangle$ the eigenstates of $\\hat{H}_0$ and by $E_b$ the corresponding eigenvalues. Then,\n\\begin{eqnarray}\n\\hat{H}_0 |b\\rangle = E_b|b\\rangle.\n\\end{eqnarray}\nWe will take the initial state $|\\psi\\rangle$ to be one of the eigenstates, say $|a\\rangle$, and we will write  $V_{ab} := \\langle a|\\hat{V}|b\\rangle$.\n\nWithout loss of generality, we     assume that  $\\langle a|\\hat{V}|a\\rangle =0$. If $\\langle a|\\hat{V}|a\\rangle \\neq 0$,\n  we can always  redefine\n \\begin{eqnarray}\n \\hat{H}_0' = \\hat{H}_0 + \\sum_a \\langle a|\\hat{V}|a\\rangle |a\\rangle \\langle a| \\hspace{1cm}  \\hat{V}' = \\hat{V} - \\sum_a \\langle a|\\hat{V}|a\\rangle |a\\rangle \\langle a|,\n  \\end{eqnarray}\nso that  $|a\\rangle$ is an eigenstate of $\\hat{H}_0'$ and $\\langle a|\\hat{V}'|a\\rangle = 0$.\n\n\\medskip\n\n\\noindent {\\em The resolvent.} The perturbative calculation of the persistence amplitude is simplified by using the {\\em resolvent} associated to the Hamiltonian operator:\n  $(z - \\hat{H})^{-1}$, for $z \\in {\\pmb C}$. We will often write the resolvent  as $\\frac{1}{z - \\hat{H}}$.\n\n\n\nThe resolvent is related to the evolution operator $e^{-i\\hat{H}t}$ by\n\n\n\\begin{eqnarray}\ne^{-i\\hat{H}t} = \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi  } \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\hat{H}},  \\label{propagator2}\n\\end{eqnarray}\nwhere $\\epsilon > 0$  and $t > 0$.\n\nEq. (\\ref{propagator2}) follows from  the identity\n\\begin{eqnarray}\ne^{- i \\omega t} =  \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi  } \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\omega}. \\label{idtycmplx}\n\\end{eqnarray}\n To prove Eq. (\\ref{idtycmplx}) we evaluate the line integral\n\n \\begin{eqnarray}\nI(t) = \\oint_{C_+} dz \\frac{e^{-izt}}{z - \\omega}, \\label{cont}\n\\end{eqnarray}\nalong the negative-oriented contour $C_+$ of Fig. 1. At the lower half of the imaginary plane $Im z  =  - y$, for $y >0$. Therefore, the integrand along the semicircle is proportional to $e^{-yt}$ and vanishes as the radius of the semi-circle tends to infinity. Hence,\n\\begin{eqnarray}\nI(t) = \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E+ i \\epsilon - \\omega}.\n\\end{eqnarray}\nThe contour $C_+$ includes the single pole of the integrand (\\ref{cont}), at $z = \\omega$. Using Cauchy's residue theorem, we arrive at Eq. (\\ref{idtycmplx}).\n\nBy integrating along the contour $C_-$, we can similarly prove that\n\\begin{eqnarray}\n\\lim_{\\epsilon \\rightarrow 0}   \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E -  i \\epsilon - \\hat{H}} = 0 ,  \\label{propagator2b}\n\\end{eqnarray}\nfor $\\epsilon > 0$  and $t > 0$. The   integral vanishes because the contour $C_-$ does not enclose any pole.\n\n\n\\begin{figure}[]\n\\includegraphics[height=6cm]{complexcurve} \\caption{ \\small Integration contours $C_{\\pm}$ for calculating the integrals  (\\ref{idtycmplx}) and (\\ref{propagator2b}).  The integrals are evaluated at the limit where the radius $r$ of the semi-circle goes to infinity. Both curves are traversed clock-wise and therefore have negative orientation.}\n\\end{figure}\n\n\nA crucial property of the resolvent is that it can be expanded in a perturbative series. For a Hamiltonian $\\hat{H}$ of the form (\\ref{Hamper}),\n \\begin{eqnarray}\n  (z - \\hat{H})^{-1}\n = (z - \\hat{H}_0 - \\hat{V})^{-1} = [(z-\\hat{H}_0) (1 - (z-\\hat{H}_0)^{-1}\\hat{V})]^{-1}\\nonumber \\\\\n  = (1 - (z-\\hat{H}_0)^{-1}\\hat{V})^{-1}(z-\\hat{H}_0)^{-1}. \\nonumber\n \\end{eqnarray}\n Using the geometric series formula $(1 - \\hat{A})^{-1} = \\sum_{n=0}^{\\infty} \\hat{A}^n$, we obtain\n\\begin{eqnarray}\n\\frac{1}{z - \\hat{H}} = \\frac{1}{z - \\hat{H}_0} + \\frac{1}{z - \\hat{H}_0}  \\hat{V} \\frac{1}{z - \\hat{H}_0}  + \\frac{1}{z - \\hat{H}_0}  \\hat{V}\\frac{1}{z - \\hat{H}_0}  \\hat{V} \\frac{1}{z - \\hat{H}_0}  + \\ldots \\label{seriesprop}\n\\end{eqnarray}\n\n\\subsection{The random phase approximation}\nWe construct the persistence amplitude  ${\\cal A}_a(t)$ for an initial state $|a\\rangle$ that is an eigenstate of $\\hat{H}_0$. By  Eq. (\\ref{propagator2}),\n \\begin{eqnarray}\n {\\cal A}_a(t) = \\lim_{\\epsilon \\rightarrow 0} \\frac{i}{2\\pi } \\int_{-\\infty }^{\\infty } dE e^{-iEt}  G_a(E+i\\epsilon) , \\label{ampll}\n \\end{eqnarray}\n where\n  \\begin{eqnarray}\n  G_a(z) =  \\langle a|(z - \\hat{H})^{-1}|a\\rangle.\n \\end{eqnarray}\n\n We evaluate $G_a(z)$ using the perturbative series (\\ref{seriesprop}). We assume that $\\hat{V}$ is of first order to  some  dimensionless parameter $\\lambda << 1$.  The zeroth-order contribution to $G_a(z)$ is\n  $ \\frac{1}{z - E_a}$. The first-order contribution is\n $ \\frac{1}{(z - E_a)^2} \\langle a|\\hat{V}|a\\rangle$ = 0.\n\n The second-order term is  $ \\frac{1}{(z - E_a)^2} \\langle a|\\hat{V} \\frac{1}{z - \\hat{H}_0} \\hat{V}|a\\rangle$. Writing $ \\frac{1}{z - \\hat{H}_0} =  \\sum_b \\frac{1}{z - E_b} |b\\rangle \\langle b|$, this term becomes $ \\frac{1}{(z - E_a)^2} \\Sigma_a(z) $, where\n\\begin{eqnarray}\n\\Sigma_a(z) = \\sum_b \\frac{|V_{ab}|^2}{z - E_b}, \\label{sigmaa}\n\\end{eqnarray}\nis the {\\em self-energy} function of the state $|a\\rangle$.\n The third-order term is\n \\begin{eqnarray}\n\\sum_{bc} \\frac{1}{(z-E_a)^2} \\frac{ V_{ab} V_{bc} V_{ca}}{(z-E_b)(z-E_c)}, \\nonumber\n\\end{eqnarray}\nand the fourth-order term is\n\\begin{eqnarray}\n\\sum_{bcd} \\frac{1}{(z-E_a)^2} \\frac{ V_{ab} V_{bc} V_{cd}V_{da}}{(z-E_b)(z-E_c)(z-E_d)}. \\nonumber\n\\end{eqnarray}\n The procedure can be continued ad infinitum.\n\nWe assume that the Hamiltonian $\\hat{H}_0$ has continuous spectrum for $E > \\mu$, for some parameter $\\mu$. The continuous spectrum corresponds to the kinetic energy of the decay products. With this assumption, we invoke the   {\\em Random Phase Approximation} (RPA), according to which\n\n\\begin{eqnarray}\n\\sum_c \\frac{V_{ac}V_{cb}}{z-E_c} \\simeq \\delta_{ab} \\Sigma_a(z). \\label{rpa}\n\\end{eqnarray}\nThe reasoning for Eq. (\\ref{rpa}) is the following. Suppose that the system is contained in a box of volume $V$ with periodic boundary conditions and that it contains a large number $N$ of degrees of freedom. The RPA is the assumption that the phases of the matrix elements $\\langle a| \\hat{V}|b\\rangle$, for $b \\neq a$ are {\\em randomized} in the continuous limit, i.e., in the limit where $N$ and $V$ goes to infinity, with $N\/V$ constant. By `randomized', we mean that the phases of $\\langle a| \\hat{V}|b\\rangle$ do not exhibit any periodicity, or quasi-periodicity as $b$ varies. Hence, the summation over $b$ is a sum of many random phases, and is expected to be much smaller than the term for $a = b$ that involves no such phases. Eq. (\\ref{rpa}) then follows.\n\n  The RPA was initially postulated in systems with  a large but finite number of degrees for freedom, in condensed matter \\cite{BohmPi, GeBr} and\nin nuclear physics \\cite{Bloch, Nami}. However, it  also applies to systems with an infinite number of degrees of freedom, i.e., quantum fields.  Indeed, the name `self-energy'  for the function (\\ref{sigmaa}) originates from quantum field theory.\n\n\n\n\n\nBy the RPA,     odd-order terms in the expansion of $G_a(z)$  vanish. Furthermore, the terms of order $2n$ for $n$ integer equal $\\Sigma_a(z)^n\/(z-E_a)^{n+1}$. Hence, $G_a(z)$ is give by  a geometric series,\n\\begin{eqnarray}\nG_a(z) = \\frac{1}{z-E_a} \\sum_{n=0}^{\\infty} \\frac{\\Sigma_a(z)^n}{(z-E_a)^n} = \\frac{1}{z-E_a} \\frac{1}{1 - \\frac{\\Sigma_a(z)}{z-E_a}} = \\frac{1}{z - E_a - \\Sigma_a(z)}. \\label{asas}\n\\end{eqnarray}\n\n Eq. (\\ref{asas}) is accurate to order $\\lambda^2$. Hence, it can be obtained without the RPA, solely by a perturbative analysis. Most treatments of RPA assume a weaker condition that Eq. (\\ref{rpa}), so that the resulting expression for  $G_a(z)$ involves a  self-energy function that coincides with Eq. (\\ref{sigmaa}) only to second order in $\\lambda$---for details, see,   Ref. \\cite{NNP96}. In the present treatment, the RPA gives the same results with a second-order perturbative expansion. In general, it has a larger domain of validity.\n\n\n\\subsection{Structure of the self-energy function}\nEqs. (\\ref{asas}) and (\\ref{ampll}) imply that\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi  } \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E + i \\epsilon- E_a - \\Sigma_a(E+i\\epsilon)}. \\label{ampldec}\n\\end{eqnarray}\nIf the self-energy function $\\Sigma_a(z)$ were analytic, the integral (\\ref{ampldec}) could be evaluated by integrating     along the contour of Fig. 1, and using Cauchy's theorem. However, this is not the case, the self-energy function is discontinuous and it may contain poles or branch points.\n\nTo see this, consider the definition     (\\ref{sigmaa}) of  $\\Sigma_a(z)$. Since the spectrum of $\\hat{H}_0$ is continuous for energies larger than $\\mu$, $\\Sigma_a(z)$ is\n     divergent along the half-line $D = \\{ z \\in {\\pmb C}|\\; \\mbox{Re} z > \\mu, \\mbox{Im}z = 0\\}$. Shifting the Hamiltonian by a constant term, we can always choose    $\\mu = 0$, so that $D = {\\pmb R}^+$.\n\nTo analyze the behavior of  $\\Sigma_a(z)$ near $D$, we define\n\\begin{eqnarray}\n\\Sigma_a(E^{\\pm}) = \\lim_{\\eta \\rightarrow 0} \\Sigma_a(E\\pm i\\eta),\n \\end{eqnarray}\nfor $\\eta > 0$. By definition,\n\\begin{eqnarray}\n\\mbox{Re} \\Sigma_a(E\\pm  i \\eta) = \\sum_b \\frac{|V_{ab}|^2(E-E_b)}{(E  - E_b)^2 +  \\eta^2}.\n\\end{eqnarray}\nIt follows that $\\mbox{Re} \\Sigma_a(E^+) = \\mbox{Re} \\Sigma_a(E^-)$, leading to the definition of the {\\em level-shift function}\n\n\\begin{eqnarray}\nF_a(E) := \\mbox{Re}  \\Sigma_a(E^{\\pm})  \\label{faE}\n\\end{eqnarray}\nOn the other hand, the imaginary part of $\\Sigma_a(z)$\n\\begin{eqnarray}\n\\mbox{Im} \\Sigma_a(E\\pm  i \\eta) = \\mbox{Im} \\sum_b \\frac{|V_{ab}|^2}{E  - E_b + i \\eta} = \\mp \\eta \\sum_b \\frac{|V_{ab}|^2}{(E  - E_b)^2 +  \\eta^2}, \\label{imsa}\n\\end{eqnarray}\n is discontinuous as the half-line $D$ is crossed.  Eq. (\\ref{imsa}) implies that   $\\mbox{Im} \\Sigma(E^-) = - \\mbox{Im} \\Sigma(E^+)\\geq 0$.\n\nWe define the {\\em decay function}\n\\begin{eqnarray}\n\\Gamma_a(E) := 2 \\mbox{Im}  \\Sigma(E^-) > 0, \\label{gammaE}\n\\end{eqnarray}\n so that\n \\begin{eqnarray}\n \\Sigma_a(E^{\\pm}) = F_a(E) \\mp \\frac{i}{2} \\Gamma_a(E).\n \\end{eqnarray}\n  The discontinuity of $\\Sigma_a$  across $D$  is\n$\\Delta \\Sigma_a(E) := \\Sigma_a(E^+) - \\Sigma_a(E^-) = - i \\Gamma_a(E)$.\nObviously,  $\\Gamma_a(E) = 0$ for $E < 0$.\n\n  Eq. (\\ref{ampldec}) becomes\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi  }   \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)}. \\label{ampldecb}\n\\end{eqnarray}\nSince $\\Gamma(E) = 0$ for $E < 0$,\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi }  \\left( \\int_{-\\infty }^{0} \\frac{dE e^{-iEt}}{E  - E_a - F_a(E)}+ \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)} \\right). \\label{ampldecc}\n\\end{eqnarray}\n\nOn the other hand, Eq. (\\ref{propagator2b}) implies that\n\\begin{eqnarray}\n \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dE e^{-iEt}}{E - i \\epsilon- E_a - \\Sigma_a(E^-)} = 0,\n\\end{eqnarray}\nhence,\n\\begin{eqnarray}\n  \\int_{-\\infty }^{0 } \\frac{dE e^{-iEt}}{E -   E_a - F_a(E)}  = -  \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{E -   E_a - F_a(E) - \\frac{i}{2}  \\Gamma_a(E)}.\n\\end{eqnarray}\nSubstituting into Eq. (\\ref{ampldecb}), we obtain\n\\begin{eqnarray}\n{\\cal A}_a(t) &=& \\frac{i}{2 \\pi  }  \\int_{0}^{\\infty } dE e^{-iEt} \\left[ \\frac{1}{E  - E_a - F_a(E) + \\frac{i}{2} \\Gamma_a(E)} - \\frac{1}{E  - E_a - F_a(E) - \\frac{i}{2} \\Gamma_a(E)}\\right] \\nonumber \\\\\n&=&  \\frac{1}{2 \\pi }  \\int_{0 }^{\\infty } \\frac{dE \\Gamma_a(E) e^{-iEt}}{ [E - E_a - F_a(E)]^2 + \\frac{1}{4}[\\Gamma_a(E)]^2}. \\label{mainampl}\n\\end{eqnarray}\n\n\n\n\nEq. (\\ref{mainampl}) is the main result of this section, an explicit formula relating the persistence amplitude to the components of the self-energy function.\n\n\n\\subsection{The Wigner-Weisskopf approximation}\n\nThe integral  (\\ref{mainampl}) involves the functions $F_a(E)$ and $\\Gamma_a(E)$ in the denominator. These function are second-order with respect  the perturbation parameter $\\lambda << 1$. If  $|F_a(E)| << E_a$ and $\\Gamma(E) << E_a$ for $E$ in the vicinity of $E_a$, we can evaluate the persistence amplitude using the {\\em Wigner-Weisskopf Approximation} (WWA) \\cite{WWA}.\n\nThe WWA essentially postulates the substitution of the Lorentzian-like function of $E$ in Eq. (\\ref{mainampl}) with an actual Lorentzian. The justification is the following.  The integral (\\ref{mainampl}) is dominated by values of $E$  within distance of order $\\lambda^2$ from  $E \\simeq E_a$. For these values, the integrand is of order $\\lambda^{-2}$, otherwise it is of order $\\lambda^0$.   Hence, with an error of order $\\lambda^2$,\n we   can substitute the energy-shift function $F_a(E)$ with the constant\n\\begin{eqnarray}\n\\delta E := F_a(E_a)\n\\end{eqnarray}\n and the decay function $\\Gamma_a(E)$ with the constant\n \\begin{eqnarray}\n \\Gamma  := \\Gamma_a(E_a).\n \\end{eqnarray}\n Within an error of the same order of magnitude,\n we extend the range of integration to $(-\\infty, \\infty)$. Thus, we  obtain an elementary integral\n\\begin{eqnarray}\n{\\cal A}_a(t) =  \\frac{\\Gamma_a}{2 \\pi }  \\int_{-\\infty}^{\\infty } \\frac{dE   e^{-iEt}}{ (E - E_a -  \\delta E )^2 + \\frac{1}{4}\\Gamma^2} =  e^{ -i (E_a +\\delta E ) t - \\frac{\\Gamma }{2}t},  \\label{ampl3}\n\\end{eqnarray}\n\n\n Substituting Eq.  (\\ref{ampl3}) in Eq. (\\ref{decprob2}), we conclude\n\\begin{eqnarray}\np(t) = \\Gamma  e^{-\\Gamma t}.\n\\end{eqnarray}\n\n Hence, the WWA leads to an exponential decay law with a decay constant $\\Gamma $ that is determined by the imaginary part of the self-energy function. The real part of the self-energy function leads to a shift\n $\\delta E $ of the energy level $E_a$, usually referred to as the {\\em Lamb shift}.\n\n\n\n\n The same expression for the decay constant $\\Gamma $ is  given by {\\em Fermi's golden rule}. To see this, we use\nEq. (\\ref{imsa}),\n\\begin{eqnarray}\n\\Gamma  =  \\lim_{\\eta \\rightarrow 0} \\sum_b \\frac{2\\eta |V_{ab}|^2}{(E_a  - E_b)^2 +  \\eta^2}.\n\\end{eqnarray}\nSince $\\lim_{\\eta \\rightarrow 0} \\frac{\\eta}{x^2 +\\eta^2} = \\pi \\delta(x)$, we obtain Fermi's decay rate\n\\begin{eqnarray}\n\\Gamma  = 2 \\pi \\sum_{b, E_b = E_a} |V_{ab}|^2. \\label{fgg}\n\\end{eqnarray}\n\n\n\n\n\n\nA more rigorous derivation of Eq. (\\ref{ampl3}) from Eq. (\\ref{ampldecb}) employs the notion of the  {\\em van Hove limit} \\cite{vanHo}.  This limit is obtained as follows. First, we change the time variable to $\\tilde{t} = \\lambda^2 t$, and we define $x = (E-E_a)\/\\lambda^2$. Then, Eq. (\\ref{ampldecb}) becomes\n\\begin{eqnarray}\ne^{iE_at} {\\cal A}_a(t) =  \\frac{i}{2 \\pi  } \\lim_{\\bar{\\epsilon} \\rightarrow 0} \\int_{-\\infty }^{\\infty } \\frac{dx e^{-ix \\tilde{t}}}{x + i \\bar{\\epsilon}-  \\tilde{F}_a(E_a+\\lambda^2x ) + \\frac{i}{2} \\tilde{\\Gamma}_a(E_a+\\lambda^2x)}, \\label{aaavH}\n\\end{eqnarray}\nwhere $\\tilde{\\Gamma} (E) = \\lambda^{-2}\\Gamma(E)$ and $\\tilde{F} (E) = \\lambda^{-2}F (E)$ are of order $\\lambda^0$; $\\bar{\\epsilon} = \\epsilon\/\\lambda^2$ can still be chosen arbitrarily small. The van-Hove limit consists of taking the limit $\\lambda \\rightarrow 0$ in the r.h.s. of Eq. (\\ref{aaavH}), while keeping $\\tilde{t}$ constant. Then,\n\\begin{eqnarray}\ne^{iE_at} {\\cal A}_a(t) =  \\frac{i}{2 \\pi  }  \\int_{-\\infty }^{\\infty } \\frac{dx e^{-ix \\tilde{t}}}{x   - \\tilde{F}_a(E_a) + \\frac{i}{2} \\tilde{\\Gamma}_a(E_a)}, \\label{aaavH2}\n\\end{eqnarray}\nEq. (\\ref{aaavH2}) can be straightforwardly evaluated using the contour integral of Fig. 1. It leads to  Eq. (\\ref{ampl3}). Hence, the WWA  is equivalent to the imposition of the van Hove limit on the decay amplitude.\n\nWhile the van Hove limit of the persistence amplitude is always well defined, we have to keep in mind that in physical systems $\\lambda^2$ is always finite and non-zero. In specific systems, the van Hove limit may misrepresent the form of the persistence amplitude. This is the case if $\\Gamma_a(E)$ strongly varies with $E$ within a distance of order $\\lambda^2$ from $E_a$.  Hence, taking the van Hove limit is justified only if the self-energy function is sufficiently `flat' in the vicinity of $E_a$ \\cite{BaRa}.\n\n\n\n\n\n\\subsection{Beyond   exponential decay }\nWe derived  exponential decay as a consequence of two approximations, the RPA and WWA. Since the RPA is redundant for a second-order approximation to the self-energy function,  WWA is the only approximation that needs to be considered in the weak coupling regime.\n\n\\medskip\n\n\\noindent {\\em Very early times.} First, we note that exponential decay cannot be valid at very early times. This is a general statement that originates from the definition (\\ref{decprob2}).\nWe Taylor-expand the persistence amplitude around $t = 0$, to obtain    ${\\cal A}_{\\psi} = 1 - i t \\langle \\hat{H} \\rangle - \\frac{t^2}{2} \\langle \\hat{H}^2 \\rangle + \\ldots$. Keeping terms up to order    $t^2$,\nthe probability density becomes\n\\begin{eqnarray}\n|{\\cal A}_{\\psi} |^2 = 1 - (\\Delta H)^2 t^2 + \\ldots.\n\\end{eqnarray}\nEq.  (\\ref{decprob2}) implies that\n\\begin{eqnarray}\np(t) =  (\\Delta H)^2 t.\n\\end{eqnarray}\n It follows that $p(0) =0$, while in   exponential decays,  $p(0) = \\Gamma$.  This violation of exponential decay at early times is a special case of the  so called {\\em quantum Zeno effect} \\cite{MiSu, FP08}, and it has been verified experimentally  \\cite{Zeno}.\n\nThe early time behavior of a decaying system can also be identified by an uncertainty relation for the persistence probability, first derived by Mandelstam and Tamm \\cite{MaTa45}, see, also \\cite{Bhatta, Dodo}. For a system in a state $|\\psi\\rangle$ and Hamiltonian $\\hat{H}$, the Kennard-Robertson uncertainty relation gives,\n \\begin{eqnarray}\n \\Delta H \\Delta A \\geq \\frac{1}{2} |\\langle \\psi| [\\hat{H}, \\hat{A}]|\\psi\\rangle|, \\label{kerob}\n \\end{eqnarray}\n for any observable $\\hat{A}$. For the Heisenberg evolved observable $\\hat{A}(t) = e^{i\\hat{H}t} \\hat{A} e^{-i\\hat{H}t}$,  Eq. (\\ref{kerob}) implies the so called Mandelstam-Tamm inequality\n\\begin{eqnarray}\n\\Delta H \\Delta A(t) \\geq \\frac{1}{2} \\large|\\langle \\psi| \\frac{\\partial \\hat{A}(t)}{\\partial t} |\\psi\\rangle \\large| \\label{mata1}\n\\end{eqnarray}\nChoosing   $\\hat{A} = |\\psi\\rangle \\langle \\psi|$, the expectation $ \\langle \\psi|\\hat{A}(t)|\\psi\\rangle$ coincides with the survival probability of $|\\psi\\rangle$, which we will denote by $a(t)$.  Hence, Eq.\n (\\ref{mata1}) becomes\n  \\begin{eqnarray}\n  \\frac{|\\dot{a}|}{ \\sqrt{a-a^2}} \\leq 2\\Delta H. \\label{mata2}\n  \\end{eqnarray}\n\n  Eq. (\\ref{mata2}) holds for all quantum states and Hamiltonians. It is easy to verify that it is  not compatible with the exponential decay law at early times. Setting $a(t) = e^{-\\Gamma t}$, we find that Eq. (\\ref{mata2} is satisfied only if $t > \\Gamma^{-1} \\ln \\left[ 1 + \\frac{\\Gamma^2}{4 (\\Delta H)^2}\\right]$.\n\n  Assuming that $\\dot{a} \\leq 0$, we can integrate inequality   (\\ref{mata2}), to obtain\n\\begin{eqnarray}\n\\cos^{-1} \\sqrt{a(t)} \\leq  \\Delta H t  \\label{mata3}\n\\end{eqnarray}\nEq. (\\ref{mata3})  is satisfied trivially for\n $t > \\frac{\\pi}{2\\Delta H}$; for $t < \\frac{\\pi}{2\\Delta H}$, it implies that  $a(t) \\geq    \\cos^2(\\Delta H t)$. The half-life $\\tau_h$  of the state is defined by the requirement\n  $a(\\tau_h) = \\frac{1}{2}$. Hence, Eq.  (\\ref{mata3}) leads to an uncertainty relation between energy spread and half-life\n\\begin{eqnarray}\n\\Delta H \\tau_h \\geq \\frac{\\pi}{4},\n\\end{eqnarray}\n that applies even in regimes where exponential decay fails\\footnote{For  $\\tau$ such that $a(\\tau) = 0$, Eq.  (\\ref{mata3}) leads to the uncertainty relation $\\Delta H \\tau \\geq \\frac{\\pi}{4}$. The time $\\tau$ is interpreted as the minimum time required for the system to arrive to a state orthogonal to $|\\psi\\rangle$, and the uncertainty relation is said to define a limit to the `speed of quantum evolution', and, consequently, to the speed of quantum computation. The inequality has been improved \\cite{MaLe, LeTo} in order to also incorporate the case of $\\Delta H \\rightarrow \\infty$. It has then been broadly generalized, for example, to open-system dynamics \\cite{MLOS} and classical systems \\cite{Shan}---for a review, see, Ref. \\cite{DeCa}.  }.\n\n\\medskip\n\n\\noindent {\\em The persistence amplitude in terms of a contour integral. } In order to establish the  range of validity of the WWA, we must evaluate the survival amplitude (\\ref{ampldec}) without approximations. To this end, we analytically continue  $\\Gamma_a(E)$ to the lower imaginary plane, and we define   $\\Sigma^-_a(z) := \\Sigma_a(z)$ and $ \\Sigma^+_a(z) := \\Sigma_a(z) - i \\Gamma_a(z)$.   The functions $\\Sigma^{\\pm}_a(z)$ can be viewed as components of a multi-valued complex function: $\\Sigma^+$ corresponds to  the first Riemann sheet, and $\\Sigma^-$ corresponds to the second Riemann sheet \\cite{colth}.\n\nWe define a line integral over the contour $C_-$ of Fig. 1.  The contribution from the  circle at infinity vanishes, hence, taking the limit $\\epsilon \\rightarrow 0$, we obtain\n\n \\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{i}{2 \\pi } \\oint_{C_-} dz e^{-izt} \\left(\\frac{1}{z-E_a - \\Sigma^+_a(z)} -  \\frac{1}{z-E_a - \\Sigma^-_a(z)}  \\right)+  I_a(t),  \\label{amplnew}\n\\end{eqnarray}\nwhere $I_a(t)$ is the contribution of the negative real axis\\footnote{We can use a different integration contour, consisting of the positive real axis, an arc of the circle at infinity, and a half line $N$ that starts from the latter and ends at the origin---for an example, see, Fig. 5. As long as the contour encloses the physically relevant poles near the positive real axis, the analysis remains unchanged. Then, $I_a(t)$ is a line integral along $N$.\nThe choice   of the negative imaginary axis for $N$ is particularly useful for calculating the long time limit of Eq. (\\ref{amplnew}). In fact, we use it implicitly in the derivation of Eq. (\\ref{iat0}).}.\n\\begin{eqnarray}\nI_a(t) = \\frac{1}{2\\pi }\\int_0^{\\infty} \\frac{dx e^{ixt} \\Gamma_a(-x)}{[x+  E_a  +F_a(-x)]^2 + \\frac{1}{4}\\Gamma_a(-x)^2}. \\label{remainder}\n\\end{eqnarray}\n\n\n If $\\Sigma^{\\pm}_a(z)$ are meromorphic functions in the region of the complex plane enclosed by the contour $C_-$, the line integral in Eq. (\\ref{amplnew}) is evaluated by finding the poles of the integrand inside $C_-$. To this end, we must  solve the equation\n\\begin{eqnarray}\nz-E_a - \\Sigma_a^{\\pm}(z) = 0. \\label{poleq}\n\\end{eqnarray}\nWe will refer to this contribution to the survival amplitude as the {\\em pole term}; we will refer to $I_a(t)$ as the {\\em remainder term}.\n\n\n\\medskip\n\n\\noindent {\\em The pole term.}   Unless $E_a$ is very close to a point of divergence of $\\Sigma_a^{\\pm} $, we expect that $|\\Sigma_a(E_a^+)|\/ E_a$  is  much smaller than unity. Hence, there  exist a solution to Eq. (\\ref{poleq}) within a distance of order $\\lambda^2$ from the point $z = E_a$.  Setting $z = E + \\lambda^2 x$, we find that $ \\lambda ^2 x = \\Sigma^{\\pm}_a(E_x + \\lambda^2x) = \\Sigma_a^{\\pm} (E_a) + O(\\lambda^4)$. Hence, Eq. (\\ref{poleq}) is satisfied for\n\n\\begin{eqnarray}\nz = E_a + F_a(E_a) \\mp \\frac{i}{2} \\Gamma_a(E_a) + O(\\lambda^4). \\label{BBpole}\n\\end{eqnarray}\nThe pole with the plus sign is outside $C_-$, hence, it does not contribute to the contour integral. The pole with the minus sign reproduces the result of the WWA\\footnote{Hence, the WWA justifies a common statement of scattering theory, that unstable particles correspond to  poles of the $S$-matrix on the second Riemann sheet \\cite{colth, RNewt}. Indeed, the poles of the S-matrix provide a simple way for identifying the basic properties of an unstable state. However, the $S$-matrix formalism  gives a coarse description of a decay process. It  enforces the constancy of transition rates \\cite{Weinberg} by averaging over a long time interval, and, for this reason, it cannot discern transient phenomena, including deviation from exponential decay.}. The associated residue is $[1 - F_a'(E_a) + \\frac{i}{2}\\Gamma'(E_a)]^{-1}$.\n\nLet all other solutions to Eq.   (\\ref{poleq}) inside $C_-$ be  $z = \\alpha_i$ and $R_i$ the associated residues. Then, by Cauchy's theorem,\n\n\\begin{eqnarray}\n{\\cal A}_a(t)  = \\frac{e^{ -i (E_a +\\delta E ) t - \\frac{\\Gamma }{2}t}}{1- F_a'(E_a) + \\frac{i}{2}\\Gamma_a'(E_a)}  + K_a(t) + I_a(t), \\label{accuracy}\n \\end{eqnarray}\n where\n $K_a(t) = \\sum_i R_i e^{-i\\alpha_i t}$. Hence, the WWA is valid if both terms $K_a(t)$ and $I_a(t)$ are negligible.\n\n  If the self-energy function is analytic in the region enclosed by the contour $C_-$, then, typically,  other roots $z = \\alpha_i$ to Eq. (\\ref{poleq}) are further away from the real axis than the perturbative root (\\ref{BBpole}). This means that   $|Im \\alpha_i| >> \\Gamma$. Hence, $K_a(t)$  vanishes much faster than the WWA term, and can be neglected after an initial transient period.\n\n   If\n $\\Sigma^{\\pm}_a(z)$ has divergences near or on the real axis, then there may exist other roots, as close to the real axis as the perturbative root  (\\ref{BBpole}).  There is no general rule here, and, in principle, a case-by-case study is required.\n   Of particular importance is the possibility of {\\em resonance}. If $E_a$ is sufficiently close to   a pole of $\\Sigma_a(z)$, then the condition $|\\Sigma_a(E_a^+)|\/ E_a <<1$ will not hold. Then, there is no perturbative pole, the WWA fails, and decays are likely to be non-exponential.\n\n\\medskip\n\n \\noindent{\\em Late times.} The contribution of the pole term to the persistence amplitude drops exponentially.  The remainder term $I_a(t)$ drops as an inverse power law, hence, it dominates at sufficiently late times \\cite{Helund, Nami2}.\n\nTo see this, we change  variables to $y = x t$, and write Eq. (\\ref{remainder}) as\n\n\\begin{eqnarray}\nI_a(t) = \\frac{1}{2\\pi t}\\int_0^{\\infty} \\frac{e^{iy} \\Gamma_a(-y\/t)}{[y\/t + E_a +F_a(-y\/t)]^2 + \\frac{1}{4}\\Gamma_a(-y\/t)^2  }, \\label{eqyb}\n\\end{eqnarray}\n As $t \\rightarrow \\infty $, the denominator becomes $[E_a+F_a(0)]^2 + \\frac{1}{4}\\Gamma_a(0)^2 = E_a^2 +O(\\lambda^2)$.\n  Hence,\n\n\\begin{eqnarray}\nI_a(t) \\simeq    \\frac{1}{2\\pi E_a^2 t} \\int_0^\\infty dy e^{i y} \\Gamma_a(-y\/t) =  \\frac{1 }{2\\pi E_a^2  }  \\int_0^\\infty dx e^{i xt} \\Gamma_a(-x). \\label{iat3}\n\\end{eqnarray}\n\n\nThe long -time limit of $I_a(t)  $ is determined by  the behaviour of $\\Gamma_a(z)$ near zero.\n  Let $\\Gamma_a(z) \\simeq A z^{n}$ as $x \\rightarrow 0$, for some positive constant $A$ and integer $n$. We evaluate the integral $\\int_0^\\infty dx e^{i xt} (-x)^n$ for imaginary time $t = i \\tau$, with $\\tau >0$, to obtain $(-1)^n n!\\tau^{1+n}$. We analytically continue back to $t$, to obtain\n\n\\begin{eqnarray}\n{\\cal A}_a(t)=  - \\frac{A n! }{2\\pi E_a^2 i^{n+1} } \\frac{1}{t^{n+1}}. \\label{iat0}\n\\end{eqnarray}\n\n\nThe persistence amplitude  drops as an inverse power of   $t$. Hence, in the long-time limit, decays are characterized by an inverse-power law and not an exponential one. In most systems, the inverse-power behavior appears is time-scales too large to be measurable. Nonetheless, it has been experimentally confirmed in luminescence decays of dissolved organic\nmaterial \\cite{longtime}, with the exponent depending on the material.\n\n\n\\bigskip\n\n\\noindent We summarize the results of our analysis.\n\\\\\n\\noindent 1. The exponential law always fails at very short and very long times.\n\\\\\n\\noindent 2. If the self-energy function has no divergences near or on the real axis,  the exponential law is very accurate at all intermediate times.\n\\\\\n\\noindent 3. If the self-energy function has divergences near or on the real axis,  there may be corrections to exponential decay from other poles.\n\\\\\n\\noindent 4. Exponential decay likely fails on resonance, i.e., for energies near a divergence point of the self-energy function.\n\n\n\n\n\n\\section{Lee's model}\nIn this section, we present a general model for decays that can be applied to many different physical situations. This model originates from the work of T.D. Lee \\cite{Lee}. We shall consider two versions of the model, one where the decay is accompanied by the emission of a bosonic particle and one where the decay is accompanied by the emission of two fermions. The former describes  spontaneous emission of photons, the latter describes beta decay.\n\\subsection{Bosonic emission}\n\\subsubsection{Definitions and properties}\nWe consider decays of the form $A' \\rightarrow A + B$, in which the emitted bosonic particle $B$ is much lighter than the particles $A'$ and $A$. Examples of this type of decay are the following.\n\\begin{itemize}\n\\item  $A'$ is the excited state of a nucleus, or of an atom, or of a molecule, $A$ is the corresponding ground state and   $B$ is a photon.\n\\item  $A'$ is a heavy nucleus that decays to  the nucleus $A$ and an alpha particle.\n\\item  $A'$ and $A$ are baryons and  $B$ is a light meson.\n\\end{itemize}\nThe key idea in Lee's model is to ignore all degrees of freedom pertaining to the motion of the  heavy particles. The heavy particles are then treated as a two-level system (2LS).The ground state $|g\\rangle$ corresponds to the particle $A$ and the excited state $|e\\rangle$ to the particle   $A'$. The $B$ particle is described by a bosonic Fock space ${\\cal F}_B$. We   denote the vacuum of ${\\cal F}_B$ by\n  $|0\\rangle$ and the creation and annihilation operators by $\\hat{a}_r$ and $\\hat{a}^{\\dagger}_r$. The latter satisfy the canonical commutation relations\n  \\begin{eqnarray}\n  [\\hat{a}_r, \\hat{a}_s] = 0, \\hspace{0,3cm}   [\\hat{a}^{\\dagger}_r, \\hat{a}^{\\dagger}_s] = 0, \\hspace{0,3cm}    [\\hat{a}_r, \\hat{a}^{\\dagger}_s] = \\delta_{rs}.\n  \\end{eqnarray}\n  The indices $r, s, \\ldots$ denote the possible states of a single $B$ particle, i.e., they label a (generalized) basis on the associated single-particle  Hilbert space.\n\nThe Hilbert space of the total system is  ${\\pmb C}^2 \\otimes {\\cal F}_B$. The Hamiltonian\n  $\\hat{H}_L$ of Lee's model consists of three terms\n\\begin{eqnarray}\n\\hat{H}_L = \\hat{H}_A + \\hat{H}_B + \\hat{V} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\hat{H}_A = \\frac{1}{2} \\Omega (\\hat{1} + \\hat{\\sigma}_3) , \\label{vlee1}\n\\end{eqnarray}\nis the 2LS Hamiltonian, and $\\Omega$ stands for the energy difference of the two levels;\n\\begin{eqnarray}\n\\hat{H}_B =  \\sum_r \\omega_r \\hat{a}^{\\dagger}_r \\hat{a}_r, \\label{vlee2}\n\\end{eqnarray}\nis the   Hamiltonian for non-interacting particles $B$ particles;\n\\begin{eqnarray}\n\\hat{V} = \\sum_r \\left(g_r \\hat{\\sigma}_+   \\hat{a}_r + g^*_r  \\hat{\\sigma}_-   \\hat{a}_r^{\\dagger} \\right), \\label{vlee3}\n\\end{eqnarray}\n  is the interaction term. It describes the excitation of the 2LS accompanied by the absorption of a B particle, and the decay of the 2LS accompanied by the emission of a $B$ particle.\n   The coefficients  $g_r$ depend upon the physical system under consideration.\n\nThe initial state for Lee's model is\n\\begin{eqnarray}\n|A'\\rangle = |e\\rangle \\otimes |0\\rangle,\n\\end{eqnarray}\ni.e., the 2LS is excited and no $B$-particle is present.\n\n We evaluate the survival amplitude\n  ${\\cal A}(t) := \\langle A' |e^{-i\\hat{H}_L t}|A'\\rangle$ using the series (\\ref{seriesprop}) for $\\hat{H}_0 = \\hat{H}_A + \\hat{H}_B$.  We find that\n\\begin{eqnarray}\n(z-\\hat{H}_0)^{-1}\\hat{V}(z-\\hat{H}_0)^{-1}|A'\\rangle =  \\frac{1}{z - \\Omega} |g\\rangle \\otimes \\sum_r \\frac{g_r}{z- \\omega_r} \\hat{a}^{\\dagger}_r|0\\rangle,\n\\end{eqnarray}\nhence,  the  matrix elements\n $\\langle \\psi|\\hat{V}|A'\\rangle$ are non-zero only for $|\\psi\\rangle$\nare of the form $|g\\rangle \\otimes \\hat{a}^{\\dagger}_r|0\\rangle$.  In particular, $\\langle A'|\\hat{V}|A'\\rangle = 0$.\n\nThe next term in the perturbative series is\n\\begin{eqnarray}\n(z-\\hat{H}_0)^{-1}\\hat{V} (z - \\hat{H}_0)^{-1} \\hat{V}(z - \\hat{H}_0)^{-1}|A'\\rangle = \\frac{\\Sigma(z)}{(z- \\Omega)^2}  |A'\\rangle, \\label{leetwo}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Sigma(z) = \\sum_r \\frac{|g_r|^2}{z-\\omega_r}, \\label{selee}\n\\end{eqnarray}\nis the self-energy function for the initial state  $|A'\\rangle$.\n\nSince the second-order term is proportional to the zero-th order one, the above expressions are reproduced to all orders of perturbation. We obtain\n\\begin{eqnarray}\n[(z - \\hat{H}_0)^{-1}\\hat{V}]^{2n}(z - \\hat{H}_0)^{-1} |A'\\rangle = \\frac{1}{z- \\Omega} \\left(\\frac{ \\Sigma(z)}{z- \\Omega}\\right)^n|A'\\rangle \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n[(z - \\hat{H}_0)^{-1}\\hat{V}]^{2n+1} (z - \\hat{H}_0)^{-1} |A'\\rangle = \\frac{1}{z- \\Omega} \\left(\\frac{ \\Sigma(z)}{z- \\Omega}\\right)^n|g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r } |0\\rangle. \\nonumber\n\\end{eqnarray}\nHence,\n\\begin{eqnarray}\n\\frac{1}{z -\\hat{H}}|A'\\rangle =  \\frac{1}{z-\\Omega} \\sum_{n=0}^{\\infty} \\frac{\\Sigma(z)^n}{(z-\\Omega)^n} \\left(|A'\\rangle + |g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r} |0\\rangle\\right)\\nonumber \\\\\n = \\frac{1}{z - \\Omega - \\Sigma(z)}\\left(|A'\\rangle + |g\\rangle \\otimes \\sum_r \\frac{g_r \\hat{a}^{\\dagger}_r}{z- \\omega_r} |0\\rangle\\right). \\label{decay555}\n\\end{eqnarray}\n\nWe conclude that\n\\begin{eqnarray}\nG(z)   =  \\frac{1}{z - \\Omega - \\Sigma(z)}. \\label {asas2}\n\\end{eqnarray}\n\nWe obtained Eq.  (\\ref{asas}) by resumming the full perturbative series (\\ref{seriesprop}), without any approximation. This means that\n Lee's model incorporates the RPA in its definition.\n\n\n\\subsubsection{Spontaneous emission from atoms}\nWe will evaluate the self-energy function in a simple model where the emitting particles have zero spin and zero rest mass, i.e., scalar photons \\cite{AnHu}. This model ignores the effects of polarization, but otherwise it describes well the emission of photons by excited atoms. The inclusion of polarization changes $\\Sigma(z)$  only by a multiplicative factor that  can be absorbed in a redefinition of the coupling constant.\n\nIn this model, the basis $r$ corresponds to photon momenta ${\\pmb k}$, $\\omega_r$ corresponds to $\\omega_{\\pmb k } = |{\\pmb k}|$, and the summation over $r$ corresponds to integration with   measure $\\frac{d^3k}{(2\\pi)^3}$. If  the size $a_0$ of the emitting atom is much smaller than the wavelengths of the emitted radiation, we can describe the atom-radiation interaction in the dipole approximation  \\cite{QO}. Then, the coupling coefficients are  $g_{\\pmb k} = \\frac{\\lambda }{\\sqrt{\\omega_{\\pmb k}}}e^{i {\\pmb k}\\cdot{\\pmb r}}$, where $\\lambda $ is a dimensionless constant and ${\\pmb r}$ is the position vector of the atom\\footnote{The coupling originates from an interaction Hamiltonian of the form $\\lambda[\\hat{\\sigma}_-\\hat{\\phi}^{(-)}({\\pmb r}) + \\hat{\\sigma}_+\\hat{\\phi}^{(+)}({\\pmb r})] $, where the scalar field components $\\hat{\\phi}^{(\\pm)}  ({\\pmb r}) $ are given by Eqs. (\\ref{scalar}, \\ref{chik}).}.\n\n We substitute into Eq. (\\ref{selee}) for the self-energy function, to obtain\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2}{2 \\pi^2} \\int_0^{\\infty} \\frac{k dk}{z - k}. \\label{seel4}\n\\end{eqnarray}\n\nThe integral in Eq. (\\ref{seel4}) diverges as $k \\rightarrow \\infty$.  However, photon energies much larger than $a_0^{-1}$, where $a_0$ is the size of the atom, are not physically relevant. Hence, we regularize the integral (\\ref{seel4}) by introducing a high-frequency cut-off $\\Lambda >> \\Omega$,\n\n  \\begin{eqnarray}\n   \\Sigma (z) = \\frac{\\lambda^2}{2 \\pi^2}  \\int_0^{\\Lambda} \\frac{k  dk}{z-k} = - \\frac{\\lambda^2}{2 \\pi^2} \\left[ \\Lambda + z (\\ln(\\Lambda - z) - \\ln (-z)) \\right]. \\label{seel5}\n  \\end{eqnarray}\n\n  There are many other ways to regularize the integral (\\ref{seel4}), for example, by inserting an exponential cut-off function  $e^{-k\/\\Lambda}$. The choice of regularization does not affect the form of $\\Sigma(z)$ for the physical range of values of $z$, i.e., $|z|<< \\Lambda$. However, it introduces an arbitrariness in the behaviour of $\\Sigma(z)$ for $z$ of the order of $\\Lambda$. In particular,   the apparent branch-point at $z = \\Lambda$ in Eq. (\\ref{seel5}) is a artefact of the regularization. As far as the physically relevant values of $z$ are  concerned,  there is no error in substituting\n   $\\ln(\\Lambda - z) \\simeq \\ln \\Lambda$ in Eq. (\\ref{seel5}). Furthermore, it is convenient to absorb the constant term in $\\Sigma(z)$ into a redefinition of the frequency  $\\tilde{\\Omega} = \\Omega -  \\frac{\\lambda^2\\Lambda}{2 \\pi^2}$.\n   With these modifications, Eq. (\\ref{seel5}) becomes\n\n      \\begin{eqnarray}\n   \\Sigma (z) =   \\frac{\\lambda^2  }{2 \\pi^2} \\left[ -  (\\ln\\Lambda) z  + z \\ln (-z) \\right]. \\label{seel6}\n  \\end{eqnarray}\n\n   The logarithm in Eq. (\\ref{seel6}) is defined in the principal branch,  i.e., its argument lies in  $(-\\pi, \\pi]$. When evaluating $\\Sigma(E^-)$, we substitute $z = E - i \\eta$, for $\\eta > 0$. Hence, $\\ln(-z) = \\ln E + \\ln[-(1-i\\eta\/E)]$. As $\\eta \\rightarrow 0$,  $1-i\\eta\/E \\simeq e^{-i\\eta\/E}$. We have two options for the $-1$ term in the logarithm, we can express it either as $e^{i \\pi}$ or as $e^{-i \\pi}$. The first choice gives $\\ln (e^{i (\\pi-\\eta\/E)})$, hence, the argument lies in the principal branch. The second choice gives $\\ln (e^{i (-\\pi-\\eta\/E)})$, and the argument lies outside the principal branch. Only the first choice is acceptable. Hence, $\\ln[-(1-i\\eta\/E)] = i(\\pi-\\eta\/E)$, and  $\\lim_{\\eta \\rightarrow 0} \\ln[-(E-i\\eta)] = \\ln E + i \\pi$. It follows that\n\n\\begin{eqnarray}\n\\Sigma(E^-) = - \\frac{\\lambda^2}{2 \\pi^2} \\left[  E \\ln(\\Lambda\/E) - i \\pi E ) \\right].\n\\end{eqnarray}\nBy Eqs. (\\ref{faE}) and (\\ref{gammaE}),\n\\begin{eqnarray}\nF(E) &=& - \\frac{\\lambda^2}{2 \\pi^2}     E \\ln(\\Lambda\/E)   \\\\\n\\Gamma(E) &=& \\frac{\\lambda^2}{ \\pi}E.  \\label{gammaE2}\n\\end{eqnarray}\n\n\nHence, the decay constant in the WWA is\n\\begin{eqnarray}\n\\Gamma = \\Gamma(\\tilde{\\Omega}) = \\frac{\\lambda^2}{ \\pi}\\tilde{\\Omega}.\n\\end{eqnarray}\nThe Lamb shift depends on the cut-off parameter $\\Lambda$ and it induces a renormalization of the frequency $\\tilde{\\Omega}$.\n\n\n\\bigskip\n\n\\noindent{\\em Validity of exponential decay.}\nWe examine the domain of validity of the WWA. First, we consider the contribution from other poles. To this end, we look for solutions to equation\n \\begin{eqnarray}\n z - \\tilde{\\Omega} - \\frac{\\lambda^2}{2 \\pi^2} z \\ln(1 - \\Lambda\/z) = 0.  \\label{sol3}\n \\end{eqnarray}\n Eq. (\\ref{sol3}) admits one solution  for $z$ near  $\\Lambda$. To see this, we write $z = \\Lambda ( 1 +x)$ for $|x| << 1$. We obtain\n$x   \\simeq  e^{-\\frac{2\\pi^2}{\\lambda^2}} << 1$ to leading order in $\\lambda^2$.  This solution is possibly an artefact of the regularization, but in any case it corresponds to oscillations much more rapid than any physically relevant time scale. It averages to zero in any measurement with  temporal resolution of order $\\sigma_T >> \\Lambda^{-1}$. We conclude that the contribution of other poles to the decay probability is negligible.\n\nNext, we evaluate the remainder term, Eq. (\\ref{iat0}). By Eq. (\\ref{gammaE2}), $A = \\frac{\\lambda^2}{\\pi}$ and $n = 1$. Hence,\n\\begin{eqnarray}\nI(t) = \\frac{\\Gamma}{2 \\pi \\tilde{\\Omega}^3 t^2}. \\label{asyem}\n\\end{eqnarray}\nFor sufficiently large $t$ the remainder term dominates over the exponential term $e^{- \\Gamma t\/2}$ in the persistence amplitude. The relevant time-scale $\\tau$ is found from the solution of the equation $|I(\\tau) |= e^{- \\Gamma \\tau\/2}$ at large $\\tau$. We set $\\Gamma \\tau\/ 2 =x$ and $\\alpha = \\frac{1}{8\\pi} (\\Gamma\/\\tilde{\\Omega})^3$, to obtain an equation for $x$,\n\\begin{eqnarray}\n2 \\ln x - x =  \\ln \\alpha. \\label{eqqq}\n\\end{eqnarray}\n Solutions to Eq. (\\ref{eqqq}) for different values of $\\Gamma\/\\tilde{\\Omega}$ are given in the following able.\n\n \\bigskip\n\n \\begin{tabular}{|c||c|c|c|c|c|c|}\n   \\hline\n  \n   $\\Gamma\/\\tilde{\\Omega}$ & $10^{-2}$ & $10^{-3}$ & $10^{-4}$ & $10^{-5}$ & $10^{-6}$ & $10^{-7}$ \\\\\n   \\hline\n   x & 23.3 & 30.8 & 38.1 & 45.4 & 52.5 & 59.8 \\\\\n   \\hline\n  $ e^{-\\Gamma\\tau}$ & $5\\cdot 10^{-21}$ & $2 \\cdot 10^{-27}$ & $8 \\cdot 10^{-34}$ & $4 \\cdot 10^{-40}$  & $2 \\cdot 10^{-46}$  &$ 10^{-52}$ \\\\\n   \\hline\n \\end{tabular}\n\n \\bigskip\n\nEven for values of $\\Gamma\/\\tilde{\\Omega}$ as large as $0.01$,    the exponential decay law breaks down at a time where less than $1:10^{20}$ of the initial   atoms remains in the excited state. We conclude that any deviations from exponential decay in photo-emission are negligible outside the quantum Zeno regime.\n\n\\subsubsection{Emission of a massive boson}\nWe adapt the model of Sec. 3.1.2 to describe the emission of a   bosonic particle of mass $\\mu$.  The model is identical to that of Sec. 3.1.2, except for the form of the energy function $\\omega_{\\pmb k}$. We assume non-relativistic energies for the product particle, hence,  $\\omega_{\\pmb k}= \\mu + \\frac{{\\pmb k}^2}{2 \\mu}$. This model is a variation of the one in Refs. \\cite{Ghiold, AA} that describes  the decay of a heavy baryon to a lighter one   with the emission of a pion. It allows for an analytic calculation of the persistence amplitude at all times. Hence,  we can witness the transition between the exponential and the power-law regimes without worrying about the validity of specific approximation schemes.\n\nThe self-energy function is\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2}{2 \\pi^2} \\int_0^{\\infty} \\frac{k^2 dk}{\\left(\\mu + \\frac{k^2}{2\\mu}\\right) \\left(z - \\mu - \\frac{k^2}{2\\mu}\\right)} = -  \\frac{\\sqrt{2}\\lambda^2\\mu}{\\pi^2}\\int_0^{\\infty} \\frac{dxx^2}{(1+x^2)(x^2 - \\zeta)},\n\\end{eqnarray}\nwhere we set $\\zeta = z\/\\mu - 1$ and $x = k\/(\\sqrt{2}\\mu)$.  The integral over $x$ is evaluated to $\\frac{\\pi}{2}(1 +\\sqrt{-\\zeta})^{-1}$. Hence,\n\\begin{eqnarray}\n\\Sigma(z) =  -  \\frac{\\lambda^2\\mu}{\\sqrt{2}\\pi} \\frac{1}{1 + \\sqrt{ - \\frac{z - \\mu}{\\mu}}}. \\label{sigmazint}\n\\end{eqnarray}\n\nIn the non-relativistic regime, the relevant values of $z$ satisfy $|z - \\mu| << \\mu$, so we can approximate $(1 + \\sqrt{ - \\frac{z - \\mu}{\\mu}})^{-1}$ with $1 -  \\sqrt{ -\\frac{z - \\mu}{\\mu}}$. We shift the  energy of  the ground state by $\\mu$, so that the energy of the  product  particle starts at  $z = 0$ rather than at $z = \\mu$. We also absorb the constant $\\Sigma(0)$ into $\\Omega$. Then, the  energy of the 2LS is $\\tilde{\\Omega} = \\Omega - \\mu  -  \\frac{\\lambda^2\\mu}{\\sqrt{2}\\pi}$, and the self-energy function becomes\n\\begin{eqnarray}\n\\Sigma(z) =  -  \\frac{\\lambda^2}{\\pi} \\sqrt{-\\frac{\\mu z}{2}} . \\label{sigmazin}\n\\end{eqnarray}\nFor $E > 0$, the level-shift function $F(E)$ vanishes because $\\Sigma(E^{\\pm})$ is purely imaginary. The decay function is\n\\begin{eqnarray}\n\\Gamma(E) = \\frac{\\lambda^2}{\\pi} \\sqrt{2m E},\n\\end{eqnarray}\nand the decay constant  $\\Gamma = \\frac{\\lambda^2}{\\pi} \\sqrt{2m \\tilde{\\Omega}}$.\n\nThe persistence amplitude (\\ref{mainampl}) becomes\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{\\Gamma}{2\\pi} \\int_0^{\\infty} \\frac{dE e^{-iEt} \\sqrt{E\/\\tilde{\\Omega}}}{(E - \\tilde{\\Omega})^2 + \\frac{\\Gamma^2E}{4\\tilde{\\Omega}}},\n\\end{eqnarray}\nwhere we wrote $\\Gamma(E) = \\Gamma \\sqrt{E\/\\tilde{\\Omega}}$. We change variables to $x = E\/\\tilde{\\Omega}$, to obtain\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{\\sqrt{2}\\gamma}{\\pi} \\int_0^{\\infty} \\frac{dx e^{-ix (\\tilde{\\Omega t})} \\sqrt{x}}{(x-1)^2+ 2 \\gamma^2 x},\n\\end{eqnarray}\nwhere $\\gamma = \\frac{\\Gamma}{2\\sqrt{2}\\tilde{\\Omega}} << 1$. The denominator is a binomial with two roots, at $x = x_{\\pm}:= 1- \\gamma^2 \\pm i \\gamma\\sqrt{2-\\gamma^2}$. We use the identity\n\\begin{eqnarray}\n\\frac{x_+-x_-}{(x-x_+)(x-x_-)} = \\frac{1}{x-x_+} - \\frac{1}{x-x_-},\n\\end{eqnarray}\nand change variables to $y = x \\tilde{\\Omega}t$, to obtain\n\n\\begin{eqnarray}\n{\\cal A}(t) = \\frac{R\\left(  (1- \\gamma^2 + i \\gamma\\sqrt{2-\\gamma^2})\\Omega t\\right) - R\\left(  (1- \\gamma^2 - i \\gamma\\sqrt{2-\\gamma^2})\\Omega t\\right)}{2\\pi i\\sqrt{1 - \\frac{1}{2} \\gamma^2}\\sqrt{\\tilde{\\Omega t}}}.\n\\end{eqnarray}\nWe defined the function\n\\begin{eqnarray}\nR(a) := \\int_0^{\\infty} \\frac{dy e^{-iy}\\sqrt{y}} {y - a}. \\label{fai}\n\\end{eqnarray}\nFor $\\mbox{Re} \\; a > 0$, the integral (\\ref{fai})  can be expressed analytically in terms of the Fresnel integrals $C(x) = \\int_0^x ds \\cos(s^2)$ and $S(x) =  \\int_0^x ds \\cos(s^2)$. We find \\cite{ASt},\n\\begin{eqnarray}\nR(a) = (1-i) \\sqrt{\\frac{\\pi }{2}}-\\pi  e^{-i a} \\sqrt{-a}-(1+i)e^{-i a}\\pi \\sqrt{a}\\left[C\\left(\\sqrt{\\frac{2}{\\pi }} \\sqrt{a}\\right)+i S\\left(\\sqrt{\\frac{2}{\\pi }} \\sqrt{a}\\right)\\right]. \\label{closedat}\n\\end{eqnarray}\nEq. (\\ref{closedat}) is a closed expression for the persistence amplitude that is valid for all times. The logarithm of the persistence probability $|{\\cal A}(t)|^2$ as a function of time is plotted in Fig. 2. The agreement with exponential decay is excellent until $t \\simeq 15 \\Gamma^{-1}$. The transition to power-law decay is accompanied by increasingly stronger oscillations that originate from the asymptotic behavior of the Fresnel integrals.\n\nIt is important to emphasize that the oscillations of Fig. 2 signify the breakdown not only of exponential decay, but of the  persistence probability method. They imply that the probability density (\\ref{decprob2}) takes negative values. Hence, the   method does not correlate with the experimental records, namely, the number of product particles recorded at each moment of time. The result strongly suggests that the persistence probability method is not reliable outside the regime of exponential decay.\n\n\\begin{figure}\n\\includegraphics[height=6cm]{per4}\n\\caption{The persistence probability as a function of $\\Gamma t$ for $\\gamma = 0.05$. }\n\\end{figure}\n\n\\subsection{Fermionic emission}\n\\subsubsection{The Hamiltonian}\nNext, we consider decays of the form $A' \\rightarrow A + B_1 + B_2$, in which   $B_1$ and $B_2$ are fermionic particles, much lighter than $A'$ and $A$. The most important example of this type is beta decay, where $A'$ and $A$ are nuclei, $B_1$ is an electron (or positron) and $B_2$ is an anti-neutrino (or a neutrino). Again the nucleus is described as a 2LS, with a ground state $|g \\rangle$ and an excited state $|e\\rangle$.\n\nA fermionic Fock space ${\\cal F}_1$ is associated to the  particle $B_1$ and a fermionic Fock space ${\\cal F}_2$ is associated to the particle $B_2$. The corresponding ground states are $|0\\rangle_1$ and $|0 \\rangle_2$, respectively. The creation and annihilation operators on ${\\cal F}_1$ will be denoted as $\\hat{c}_{r}$ and $\\hat{c}_{r}^{\\dagger}$, and the creation and annihilation operators  on ${\\cal F}_2$ as $\\hat{d}_{l}$ and $\\hat{d}_{l}^{\\dagger}$. They satisfy the canonical anti-commutation relations\n\\begin{eqnarray}\n[\\hat{c}_r, \\hat{c}_s]_+ = [\\hat{c}_r^{\\dagger}, \\hat{c}^{\\dagger}_s]_+ = 0, \\hspace{0.4cm} [\\hat{c}_r, \\hat{c}^{\\dagger}_s]_+ = \\delta_{rs}\n\\end{eqnarray}\n\\begin{eqnarray}\n[\\hat{d}_l, \\hat{d}_m]_+ = [\\hat{d}_l^{\\dagger}, \\hat{d}^{\\dagger}_m]_+ = 0, \\hspace{0.4cm} [\\hat{d}_l, \\hat{d}^{\\dagger}_m]_+ = \\delta_{lm}.\n\\end{eqnarray}\nIn the above, $r$ and $s$ are labels of a basis on the Hilbert space of a single $B_1$ particle;  $l$ and $m$   are labels of a basis on the Hilbert space of a single $B_2$ particle. The Hilbert space of the total system is  ${\\pmb C}^2 \\otimes {\\cal F}_1 \\otimes {\\cal F}_2$\n\nThe Hamiltonian for this system again consists of three parts $\\hat{H}_L = \\hat{H}_A + \\hat{H}_B + \\hat{V}$, where\n\\begin{eqnarray}\n\\hat{H}_A &=& \\frac{1}{2} \\Omega (\\hat{1} + \\hat{\\sigma}_3)  , \\label{vleeb1}\\\\\n\\hat{H}_B &=&   \\sum_r \\omega_r \\hat{c}^{\\dagger}_r \\hat{c}_r   +     \\sum_r \\tilde{\\omega}_l \\hat{d}^{\\dagger}_l \\hat{d}_l  , \\label{vleeb2}\\\\\n\\hat{V} &=& \\sum_{r,l} \\left(g_{r,l} \\hat{\\sigma}_+   \\hat{c}_r   \\hat{d}_l + g_{r,l}^*  \\hat{\\sigma}_-   \\hat{c}_r^{\\dagger}  \\hat{d}_l^{\\dagger} \\right). \\label{vleeb3}\n\\end{eqnarray}\nIn Eq. (\\ref{vleeb2}), $\\omega_r$ and $\\tilde{\\omega}_l$ are energy eigenvalues of a single $B_1$ and $B_2$ particle, respectively. The coefficients $g_{r, l}$ are model-dependent.\n\nThe self-energy function  for an initial state $|A'\\rangle = |e\\rangle \\otimes |0\\rangle_1 \\otimes |0\\rangle_2$ is\n\\begin{eqnarray}\n\\Sigma(z) = \\sum_{r, l} \\frac{|g_{r,l}|^2}{z - \\omega_r - \\tilde{\\omega}_l}. \\label{sig2f}\n\\end{eqnarray}\n\n\\subsubsection{Beta decay}\nWe consider a simplified model for beta decay, in which both emitted particles have zero spin and mass. This model is similar to the original Fermi theory of weak interactions \\cite{Fermi, Wilson}.\n The zero-mass approximation is reasonable, if the energy $\\Omega$ is much larger than the masses of the emitted particles, as is often the case in beta decay. The zero spin approximation is bad; spin is important in the weak interactions.\n\n\n\nIn this model, the basis $r$ corresponds to momenta ${\\pmb p}$, the basis $l$ to momenta ${\\pmb q}$, $\\omega_r$\n  corresponds to $\\omega_{\\pmb k } = |{\\pmb p}|$, and $\\tilde{\\omega}_l$\n  corresponds to $\\tilde{\\omega}_{\\pmb q } = |{\\pmb q}|$. The summation over $r$ corresponds to integration over  $\\frac{d^3p}{(2\\pi)^3}$ and the summation over $s$ corresponds to integration over  $\\frac{d^3q}{(2\\pi)^3}$. The appropriate constants $g_{r,l}$ for beta decay are\n \\begin{eqnarray}\n  g_{{\\pmb p}, {\\pmb q}} = \\frac{1}{\\mu^2}e^{i({\\pmb p}+{\\pmb q})\\cdot{\\pmb r}}\n  \\end{eqnarray}\n   where $\\mu$ has dimensions of mass\\footnote{In terms of the standard parameters of the theory of weak interactions, $\\mu^{-2} = G_F V_{ud} {\\cal M}_n$, where $G_F$ is Fermi's constant, $V_{ud}$ is an element of the Kobayashi-Maskawa matrix, and ${\\cal M}_n$ the amplitude for the nuclear transition \\cite{nuclear}. The dimensionality of $\\mu$ comes from the inverse mass squared dimension of Fermi's constant.   } and   ${\\pmb r}$ is the position vector of the atom.\n\n   We substitute to  Eq. (\\ref{sig2f}), to obtain\n  \\begin{eqnarray}\n  \\Sigma(z) = \\frac{1}{64 \\pi^6 \\mu^4} \\int \\frac{d^3p d^3q}{z - |\\pmb p| - |{\\pmb q}|} = \\frac{1}{16 \\pi^4 \\mu^4} \\int_0^{\\infty}p^2dp \\int_0^{\\infty} q^2 dq  \\frac{1}{z-p-q}.\n  \\end{eqnarray}\nWe change the integration  variables to $y = p+q, \\xi = p-q$. Then,\n  \\begin{eqnarray}\n  \\Sigma(z) =   \\frac{1}{256 \\pi^4 \\mu^4} \\int_0^{\\infty}\\frac{dy}{z-y} \\int_0^y d\\xi (y^2-\\xi^2)^2 = \\frac{1}{480\\pi^4 \\mu^4}\\int_0^{\\infty} \\frac{dy y^5}{z-y}, \\label{sz2f}\n  \\end{eqnarray}\nThe integral in Eq. (\\ref{sz2f}) diverges, so we introduce a high-frequency cut-off $\\Lambda << \\mu$ and restrict the integration over $y$ to the interval $[0, \\Lambda]$. Thus, we obtain\n\\begin{eqnarray}\n\\Sigma(z) = - \\frac{1}{240 \\pi^4 \\mu^4} \\left[ \\Lambda^5\\left(\\frac{1}{5} + \\frac{z}{4 \\Lambda}  + \\frac{z^2}{3\\Lambda^2} + \\frac{z^3}{2\\Lambda^3} + \\frac{z^4}{\\Lambda^4}\\right)+z^5 \\ln \\Lambda -z^5 \\ln(-z)   \\right].\n\\end{eqnarray}\nwhere we approximated  $\\ln(\\Lambda- z) \\simeq \\ln \\Lambda$.\n\nAs in Sec. 3.1.2, the branch point $z = 0$ is logarithmic. Following the same procedure, we evaluate the level-shift and decay functions\n\n\n\\begin{eqnarray}\nF(E) &=& - \\frac{\\Lambda^5}{1200\\pi^4 \\mu^4} \\left(1+ \\frac{5E}{4 \\Lambda} + \\frac{5E^2}{3\\Lambda^2} + \\frac{5E^3}{2\\Lambda^3} + \\frac{5E^4}{\\Lambda^4}\\right) - \\frac{E^5}{240 \\pi^4 \\mu^4}  \\ln\\frac{\\Lambda}{E}\n\\\\\n\\Gamma(E) &=& \\frac{E^5}{120 \\pi^3 \\mu^4}.  \\label{gamma2ef}\n\\end{eqnarray}\nThe beta decay rate is\n\\begin{eqnarray}\n\\Gamma = \\Gamma(\\Omega) =  \\frac{\\Omega^5}{120 \\pi^3 \\mu^4}\n\\end{eqnarray}\n\n\n\\section{Resonant decays}\nIn this section, we consider decays in presence of resonance. To this end, we study  a  variation of the spontaneous emission model of Sec. 3.1.2, in which the 2LS is  within a cavity, consisting of two (infinite) parallel metal plates at distance $L$.  The cavity is perfect, so the field satisfies Dirichlet boundary conditions on the plates.   The model has been studied in Refs. \\cite{AnHu, Cum}, but most of the results presented here are new.\n\n\\subsection{The self-energy function}\n\nIn the directions parallel to the cavity the photon momenta take continuous values. The momentum in the perpendicular direction is an  integer multiple  of the fundamental frequency\n\\begin{eqnarray}\n\\omega_0 = \\frac{\\pi}{L}.\n\\end{eqnarray}\n  Thus, the index $r$ of the bosonic Lee model corresponds to the pair $({\\pmb k}, n)$, where ${\\pmb k}$ is a two-dimensional vector parallel to the plates and $n = 0, 1, 2, \\ldots$. The energies are $\\omega_{{\\pmb k},n} = \\sqrt{{\\pmb k}^2 +n^2 \\omega_0^2}$ and the mode summation corresponds to $\\sum_{n=0}^{\\infty} \\int \\frac{d^2k}{(2\\pi)^2}$.\n\n The coupling constants have the same dependence on energy as in Sec. 3.1.2, but we express them as\n  \\begin{eqnarray}\n  g_{{\\pmb k},n} = \\lambda \\sqrt{\\frac{\\omega_0}{\\omega_{{\\pmb k},n}} },\n  \\end{eqnarray}\n  in terms of the dimensionless constant $\\lambda$.\n\nThen, the self-energy function takes the form\n\\begin{eqnarray}\n\\Sigma(z) = \\frac{\\lambda^2 \\omega_0}{2\\pi} \\sum_{n=0}^{\\infty}\\int_0^{\\infty} \\frac{kdk}{\\omega_{{\\pmb k},n}(z- \\omega_{{\\pmb k},n})}  = - \\frac{\\lambda^2 \\omega_0}{2\\pi}  \\sum_{n=0}^{\\infty}  \\int_{n \\omega_0 -z}^{\\infty} \\frac{dx}{x}, \\label{szcav0}\n\\end{eqnarray}\n where in the last step we set $x = \\omega_{{\\pmb k},n} - z$.\n\n The integral for $\\Sigma(z)$ in Eq. (\\ref{szcav0}) diverges at high energies. We regularize by introducing a high-energy cut-off $\\Lambda$ in the integral over $x$, and a maximum integer $N = \\Lambda\/\\omega_0$ in the summation over $n$.  Then,\n \\begin{eqnarray}\n \\Sigma(z) &=& -\\frac{\\lambda^2 \\omega_0}{2\\pi} \\left[ N \\ln \\frac{\\Lambda}{\\omega_0} - \\sum_{n=0}^{N} \\ln(n - z\/\\omega_0)\\right]   \\nonumber \\\\\n &=& -\\frac{\\lambda^2 \\omega_0}{2\\pi} \\left[\\frac{\\Lambda}{\\omega_0}  \\ln \\frac{\\Lambda}{\\omega_0} - \\ln\\Gamma(N - z\/\\omega_0) +  \\ln\\Gamma( -z\/\\omega_0) \\right],\n \\end{eqnarray}\n where $\\ln\\Gamma(z)$ is the logarithmic gamma function. Since the physical values of $z$ are much smaller than $\\Lambda$,  we use the asymptotic form of the logarithmic gamma function (Stirling's formula)\n \\begin{eqnarray}\n   \\ln\\Gamma(z) \\simeq z \\ln z -z, \\label{stirling}\n \\end{eqnarray}\nto obtain\n\\begin{eqnarray}\n\\Sigma(z) =  - \\frac{\\lambda^2\\Lambda }{2\\pi} -    \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z-  \\frac{\\lambda^2 \\omega_0}{2\\pi} \\ln\\Gamma( -z\/\\omega_0). \\label{szcav}\n\\end{eqnarray}\n As in Sec. 3.1.2, we  incorporate the constant part of $\\Sigma(z)$ into a frequency redefinition: $\\tilde{\\Omega} =  \\Omega - \\frac{\\lambda^2\\Lambda }{2\\pi}$, so that\n\n\\begin{eqnarray}\n\\Sigma(z) =  - \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z  - \\frac{\\lambda^2 \\omega_0}{2\\pi} \\ln\\Gamma( -z\/\\omega_0). \\label{szcav2}\n\\end{eqnarray}\n\n\nThe logarithmic gamma function has no poles, but it has infinitely many branch points at all negative integers. The identity $\\Gamma(z+1) = z \\Gamma(z)$ implies that\n\\begin{eqnarray}\n\\ln \\Gamma(-x) = - \\ln(-x) - \\ln(-x+1) - \\ldots - \\ln(-x + [x]) +\\ln\\Gamma[-x+[x]+1], \\label{lngaid}\n\\end{eqnarray}\nfor $x > 0$ and where $[x]$ is the integer part of $x$.\n\nEq. (\\ref{lngaid}) implies that\n $\\Sigma(E^-)$   involves the sum of  $[E\/\\omega_0]+1$ logarithms with negative argument. Each logarithm contributes a term $ \\pi$ to the imaginary part of $\\Sigma(E^-)$, hence,\n\\begin{eqnarray}\n  \\Gamma(E)  =  \\lambda^2 \\omega_0 ([E\/\\omega_0]+1).\n\\end{eqnarray}\n\n The level-shift function $F(E)$ involves a term $ - \\ln(E\/\\omega_0) -   \\ln(E\/\\omega_0 - 1) - \\ln(E\/\\omega_0 - [E\/\\omega_0]) + \\ln \\Gamma( [E\/\\omega_0] + 1 -  E\/\\omega_0)$, which can be written as\n$ \\ln \\left( \\frac{\\Gamma(1+ [E\/\\omega_0] - E\/\\omega_0)   \\Gamma(E\/\\omega_0 - [E\/\\omega_0]) }{ \\Gamma(E\/\\omega_0) }\\right)$. Hence,\n  \\begin{eqnarray}\n  F(E) =   - \\frac{\\lambda^2\\log (\\Lambda\/\\omega_0) }{2\\pi}z + \\frac{\\lambda^2\\omega_0 }{2\\pi} \\ln \\left( \\frac{ \\Gamma(E\/\\omega_0) }{ \\Gamma(E\/\\omega_0 - [E\/\\omega_0])\\Gamma(1+ [E\/\\omega_0] - E\/\\omega_0)  }\\right).\n  \\end{eqnarray}\n  Both $F(E)$ and $\\Gamma(E)$ have finite  discontinuities across the resonances $E = n \\omega_0$, for $n = 1, 2, \\ldots$.\n\n\n\n\\medskip\n\n\\noindent {\\em Large cavity.} For a large cavity ($L \\Omega >>1$), $\\omega_0$ is very small, so we expand the logarithmic gamma function in Eq. (\\ref{szcav2}) using  Stirling's formula. Then,\n\\begin{eqnarray}\n\\Sigma(z) =   \\frac{\\lambda^2  }{2\\pi}\\left[ - ( \\ln \\Lambda - 1)z + z \\ln(-z) \\right] . \\label{szcav2b}\n\\end{eqnarray}\nEq. (\\ref{szcav2b}) has the same functional dependence on $z$ with the self-energy function of Eq. (\\ref{seel6})   that is obtained in absence of a cavity. If the coupling constant and the cut-off parameters are properly redefined, the two expressions coincide.\n\n\\medskip\n\nWe evaluate the persistence amplitude using with Eq. (\\ref{mainampl}). We define $n_c = [\\tilde{\\Omega}\/\\omega_0] $ and $x_0 = \\tilde{\\Omega}\/\\omega_0 - n_c $. Then, we express the integral $\\int_0^{\\infty} dE$   as $\\sum_{n=0}^{\\infty} \\int_{n \\omega_0}^{(n+1)\\omega_0}dE$, where $n = [E\/\\omega_0]$.\n\\begin{eqnarray}\n{\\cal A}_a(t) = \\frac{\\lambda^2  }{2 \\pi} \\sum_{n=0}^{\\infty}(n+1)e^{-i n \\omega_0 t} \\int_0^1 dx \\frac{e^{-i(\\omega_0t)x}}{[x + n - n_c - x_0 - \\frac{\\lambda^2 }{2\\pi} f_n(x)]^2 + \\frac{\\lambda^4 }{4}(n+1)^2 } \\label{offres}\n\\end{eqnarray}\n where we substituted $ E = \\omega_0(n+x)$ and we wrote\n\\begin{eqnarray}\nf_n(x) =   - \\ln(\\Lambda\/\\omega_0) (n +x)  + \\ln \\frac{ \\Gamma(n+x)}{\\Gamma(1-x)\\Gamma(x)}.\n\\end{eqnarray}\nThe function $f_n(x)$ diverges near $x = 0$ and near $x = 1$. Using the expansion,  $\\Gamma(x) \\simeq \\frac{1}{x}  + \\ldots$ near $x = 0$, we obtain\n\\begin{eqnarray}\nf_n(x) &=&   \\ln x  + \\ln (n-1)! -   \\ln(\\Lambda\/\\omega_0) n \\; \\; (n > 0), \\label{fn0}\n\\\\ f_n(1-x) &=& \\ln x + \\ln n! -   \\ln(\\Lambda\/\\omega_0) (n+1), \\label{fn1}\n\\end{eqnarray}\nfor $x << 1$.\n\n\n The integrals in Eq. (\\ref{offres}) are dominated by  values of $x$ for which the denominator is of order $\\lambda^2$.\n Their behavior  depends crucially on whether the system is near resonance or not.\n\n\n\n\n\n\n\\subsection{ Off resonance}\n\nFirst, we assume that the atomic frequency is far from the cavity's resonances. Hence, $x_0$ and $1 - x_0$ are much larger than $O(\\lambda^2)$. The integrals are dominated by their values near solutions to the equation\n\\begin{eqnarray}\nx + n - n_c - x_0 - \\frac{\\lambda^2 }{2\\pi} f_n(x) = 0, \\label{eqoll}\n\\end{eqnarray}\nfor $x\\in [0, 1]$. For $n = n_c$, Eq. (\\ref{eqoll}) admits a perturbative solution at $x = x_0 + \\frac{\\lambda^2 }{2\\pi} f_{n_c}(x_0) + O(\\lambda^4)$.\n\nThere are   other solutions to Eq. (\\ref{eqoll}), near the boundaries  $x = 0$ and $x = 1$ where $f_n(x)$ diverges.\n It is easy to show that these solutions lie at distance  smaller than $e^{-\\frac{2 \\pi x_0}{\\lambda^2}}$ or $e^{-\\frac{2 \\pi (1-x_0)}{\\lambda^2}}$ from the boundaries,  and that their contribution to the integral is negligible.\n\nTherefore, the dominant term to the persistence amplitude corresponds to $n = n_c$. Since the integral is dominated by values near the perturbative solution with a width of order $\\lambda^2$, we can extend the range of integration to $(-\\infty, \\infty)$. But then, we recover the integral (\\ref{ampl3}) of the WWA, with\n\n\\begin{eqnarray}\n\\delta E &=&  - \\frac{\\lambda^2 \\omega_0\\log (\\Lambda\/\\omega_0) }{2\\pi} (n_c  + x_0)  +  \\frac{\\lambda^2\\omega_0 }{2\\pi} \\ln \\left( \\frac{\\Gamma(n_c + x_0) }{\\Gamma(1- x_0 )  \\Gamma(x_0)  }\\right) \\\\\n\\Gamma &=&  \\lambda^2 \\omega_0 (n_c + 1).\n\\end{eqnarray}\n\nAs expected, the persistence amplitude off-resonance does not differ significantly from the persistence amplitude of an atom outside a cavity.\n\n\n\\subsection{Resonance}\nThe condition for resonance is that either $x_0$ or $1 - x_0$ is of order $\\lambda^2$. In the former case,\n$\\Omega_R$ is just above the resonance frequency $n_R \\omega_0$, for $n_R = n_c$; we define the detuning parameter $\\delta = x_0$.\nIn the latter  case, $\\Omega_R$ is just below the resonance frequency $n_R \\omega_0$, for $n_R = n_c + 1$; we define the detuning parameter as $\\delta =   x_0 - 1$.\n\nTwo terms in the series (\\ref{offres}) dominate. The first corresponds to $n = n_R$. In this term, the denominator of the integral is of order $\\lambda^2$ near $x = 0$. Therefore, we can use the approximation (\\ref{fn0}) for $f_n(x)$. We change variables to $y = \\frac{\\lambda^2}{2\\pi}x$ and we extend the limit of integration for $y$ form $\\frac{2\\pi}{\\lambda^2}>> 1$ to $\\infty$. Then, this term equals $e^{-in_R\\omega_0 t} G_-(\\frac{\\Gamma_0t}{\\pi}, -d, (n_R+1)\\pi)$, where\n\n\\begin{eqnarray}\nG_{\\pm}(s, a, b) :=  \\frac{b}{\\pi}  \\int_0^{\\infty} \\frac{dy e^{- i    s y}}{(y \\pm \\ln y - a)^2  + b^2 },  \\label{gsab}\n\\end{eqnarray}\n\\begin{eqnarray}\n d := \\ln \\frac{2 \\pi}{\\lambda^2} - \\frac{2\\pi\\delta}{\\lambda^2} - \\ln (n_R-1)!  +   \\ln(\\Lambda\/\\omega_0) n_R,\n  \\end{eqnarray}\n  and $\\Gamma_0 =   \\lambda^2 \\omega_0$.\n\n The second term corresponds to $n = n_R - 1 $. In this term, the denominator is of order $\\lambda^2$ near $x = 1$. Again, we can use the approximation (\\ref{fn1}) for $f_n(x)$. We change variables to $y = \\frac{2\\pi}{\\lambda^2}(1-x)$ and take the limit of integration for $y$ to $\\infty$. Then, this term equals $e^{-in_R\\omega_0 t} G_+^*(\\frac{\\Gamma_0t}{\\pi}, d, \\frac{1}{2}n_R\\pi)$. Hence,\n \\begin{eqnarray}\n{\\cal A} (t) =  e^{-in_R \\omega_0 t} \\left[ G_-[\\frac{\\Gamma_0t}{\\pi}, -d,  (n_R+1)\\pi ] + G^*_+[\\frac{\\Gamma_0t}{\\pi}, d,  n_R \\pi ]   \\right], \\label{atres}\n\\end{eqnarray}\n\n\n\n\nFor general values of $a$ and $b$ the functions $G_{\\pm}(s, a,b)$ can only evaluated numerically. In general, they decay with increasing $s$, with some oscillations for positive $a$.  Plots of $G_{\\pm}(s, a, b)$ as a function of $s$ for different values of $a$ and $b$ are given in Fig. 3.\n\n\\begin{figure}\n\\includegraphics[height=10cm]{functionG}\n\\caption{The real and the imaginary part of   $G_{\\pm}(s, a, b)$ are plotted as  functions of $s$ for different choices of $a$ and   $b$. }\n\\end{figure}\nFor  $a >>b $  , the contribution of the logarithm to the integral (\\ref{gsab}) is negligible, and the integral is little affected if the range is extended to $(-\\infty, \\infty)$. Then,\n\\begin{eqnarray}\nG_{\\pm}(s, a, b) = \\frac{b}{\\pi}   \\int_{-\\infty}^{\\infty} \\frac{dy e^{-i    s y}}{(y- a )^2  + b^2 } = e^{-b s-ias}\n\\end{eqnarray}\nIn contrast, if $|a|>> b$ with $a <0$, $G_{\\pm}(s, a, b)$ is of order $(|a|\/b)^2$. We readily  verify  that Eq. (\\ref{atres}) recovers the exponential decay form for $ |d| >> \\pi n_R$.\n\nIn all other regimes, the decays are non exponential. This can be seen in  Fig. 4, where the logarithm of the persistence probability   is plotted as a function of $\\Gamma_0t\/\\pi$ for different values of $d$. The graph becomes a straight line for larger values of $d$, signaling exponential decay. Deviations  appear when a tiny fraction of the initial  2LS remains excited. For several values of $d$, the persistence probability is not a decreasing function of $t$. Again, this signifies a failure of the definition (\\ref{decprob2}) for the decay probability.\n\n\\begin{figure}\n\\includegraphics[height=7cm]{persistence}\n\\caption{The logarithm of the persistence probability $\\ln |{\\cal A}(t)|^2$ as a function of $\\Gamma_0t\/\\pi$ for different values of $d$ and for $n_R = 1$.   }\n\\end{figure}\n\nThe behavior of the persistence probability derived here is typical  for decaying systems with energies close to a branch point of the self-energy function. \n In quantum field theory, such branch points appear at energy thresholds, i.e., for energies near  the activation energy $E_0$ of a new decay channel. For example, if the energy of a photon becomes $2m_e$, where $m_e$ is the electrons's mass, the photon decay to a electron-positron pair is possible. In such cases, the persistence amplitude receives two distinct contributions, one from energies slightly beneath and one from energies slightly above the threshold.  For discussions of non-exponential decays due to threshold effects, see, Refs. \\cite{LZMM, RZ93, JMSST, DJN09}.\n\n\n\n\\section{Decay through barrier tunneling}\nThe methodology developed in Sec. 2 applies to decays that originate from a small perturbation in the Hamiltonian. In this section, we consider non-perturbative decays that can be understood in terms of tunneling. Examples of such decays are the alpha emission of nuclei, tunneling ionization of atoms due to an external field, and leakage of particles from a trap.\nWe will consider a simple model of a particle in one dimension that mimics the classic treatment of alpha decay by Gamow and by Gurney and Condon \\cite{Gamow, GuCo}.\n\n \\subsection{Set-up}\n\n\\noindent {\\em Dynamics.} We consider a particle in the half-line ${\\pmb R}^+ = [0, \\infty)$ in presence of  a potential $V(x)$. The potential vanishes outside $ [a, b]$, where $a$ and $b$ are microscopic lengths.\n\nIt is convenient to express the potential in terms of the transmission and reflection coefficients  of the  Schr\\\"odinger operator  $\\hat{H} = \\frac{\\hat{p}^2}{2m}+ V(\\hat{x})$ over the full real line.  $\\hat{H}$    has two generalized eigenstates $f_{k\\pm}(x)$ for each value of energy $E = \\frac{k^2}{2m}$.\n\\begin{eqnarray}\nf_{k+}(x) &=& \\left\\{ \\begin{array}{cc} \\frac{1}{\\sqrt{2\\pi}} (e^{ikx} + R_k e^{-ikx}), & x < a\\\\\n \\frac{1}{\\sqrt{2\\pi}} T_k e^{ikx}& x > b\n \\end{array}\\right. \\nonumber \\\\\n f_{k-}(x) &=& \\left\\{ \\begin{array}{cc} \\frac{1}{\\sqrt{2\\pi}}  T_k e^{-ikx}, & x < a\\\\\n\\frac{1}{\\sqrt{2\\pi}} (e^{ikx} + \\tilde{R}_k e^{ikx})& x > b\n \\end{array}\\right. \\label{f+-}\n\\end{eqnarray}\nwhere   the complex amplitudes $T_k$, $R_k$ and $\\tilde{R}_k$ satisfy\n\\begin{eqnarray}\n|T_k|^2+|R_k|^2 = 1, \\hspace{0.75cm}  |R_k| = |\\tilde{R}_k|,\n  \\hspace{0.75cm}  T^*_k R_k +T_k \\tilde{R}_k^* = 0. \\label{idtyscat}\n\\end{eqnarray}\n$T_k$ is the transmission amplitude, $R_k$ is the reflection amplitude for a right-moving particle and $\\tilde{R}_k$ is the reflection amplitude for a left-moving particle.\n\nWhen the range of $x$ is restricted into the half-line, the generalized eigenfunction $g_k$ of the Schr\\\"odinger operator   is the linear combination of $f_{k+}$ and $f_{k-}$ that satisfies  $g_E(x) = 0$, i.e.,\n\\begin{eqnarray}\ng_k =  \\left( -\\frac{T_k}{1+R_k} f_{k+} + f_{k-}\\right).\n\\end{eqnarray}\nThis implies that\n\\begin{eqnarray}\ng_{k}(x) = \\left\\{ \\begin{array}{cc} -\\frac{2i}{\\sqrt{2\\pi}} \\frac{T_k}{1+R_k} \\sin kx, & x < a\\\\\n \\frac{1}{\\sqrt{2\\pi}} \\left[ e^{- ikx} - e^{iS_k} e^{ikx} \\right] & x > b\n \\end{array}\\right. , \\label{gkx}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\ne^{i S_k} = \\frac{T_k^2}{1+R_k} - \\tilde{R}_k =  \\frac{1+R^*_k}{1+R_k} \\left(\\frac{T_k^2}{|T_k|^2}\\right), \\label{eithe}\n\\end{eqnarray}\nis the reflection amplitude of a left-moving particle. The absolute value of the reflection amplitude is unity, because there is no possibility of transmission to $x < 0$.\n\nThe eigenfunctions $g_k(x)$ are normalized so that $\\int_0^{\\infty} g_k(x)^* g_{k'}(x) = \\delta(k-k')$. We will represent them by kets $|k\\rangle_D$.\n\n\\medskip\n\n\\noindent {\\em Initial state.} We consider an initial state $\\psi_0(x)$ with the following four properties. First, it vanishes  outside $[0, a]$. Second, it belongs to the Hilbert space of square-integrable harmonic functions on ${\\pmb R}^+$ subject to Dirichlet boundary conditions. Hence, it can be expressed as\n\\begin{eqnarray}\n\\psi_0(x) = \\sqrt{\\frac{2}{\\pi}} \\int_0^{\\infty} \\sin(kx) \\tilde{\\psi}_0(k). \\label{Dirich}\n\\end{eqnarray}\nThe function $\\tilde{\\psi}_0(k)$ is defined for $k > 0$. Extending to $k <0$ by $\\tilde{\\psi}_0(-k) = -\\tilde{\\psi}(k)$, we can write Eq. (\\ref{Dirich}) as $\\psi_0(x) = - \\frac{i}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} dk e^{ikx} \\tilde{\\psi}_0(k)$. By Eq. (\\ref{gkx}),\n\\begin{eqnarray}\n{}_D\\langle k|\\psi_0\\rangle = i \\frac{T^*_k}{1+R^*_k}   \\tilde{\\psi}_0(k).\n\\end{eqnarray}\nThird, we assume that $\\psi_0(x)$ is real-valued. For example, $\\psi_0$ may be an eigenstate of a  Schr\\\"odinger operator $\\hat{H}'$ with a different potential $U(x)$.  The physical interpretation of this condition is that we prepare the system in an eigenstate of $\\hat{H}'$ and at time $t = 0$ we change the potential to $V(x)$, for example, by switching on an external electric field. Fourth,  we assume that   $\\psi_0$  has a sharp energy distribution with respect to the Hamiltonian $\\hat{H}$: the energy spread for $\\psi_0$ is much smaller than the mean energy.\n\n\n\n \\medskip\n Given an initial state $\\psi_0$, we find the state at time $t$\n \\begin{eqnarray}\n \\psi_t(x) = \\int_0^{\\infty} dk {}_D\\langle k|\\psi_0\\rangle  g_k(x) e^{ - i\\frac{k^2}{2m}t}. \\label{evolvt}\n \\end{eqnarray}\n The  definition  (\\ref{decprob2}) of the decay probability in terms of the persistence probability  is not appropriate for this problem, because the initial configuration of the system does not correspond to an one-dimensional subspace, i.e., the particle may remain confined by the potential without remaining in its initial state. A better choice is perhaps to define the decays probability as $p(t) = - \\frac{d}{dt} W(t)$, where\n$W(t) = \\int_0^a dx|\\psi_t(x)|^2$ is the  {\\em non-escape probability}, i.e., the probability that the particle is found in $[0, a]$ at time $t$---see, Ref. \\cite{CMM95} for the relation between persistence and non-escape probability. We note that the definition of the non-escape probability  is   arbitrary:\n we could have equally well  taken the integration range to $[0, b]$.\n\n The main drawback of both the non-escape and the persistence probability  is that they  lack a clear operational interpretation.\n We do not carry out measurements of particles at  microscopic scales inside the confining potential, rather we measure the number of particles that have tunneled from the barrier, as recorded by a detector  far from the tunneling region. Furthermore, the persistence probability and the non-escape probability are not always  increasing functions of $t$ \\cite{Peshk}. Hence, they may lead to negative values for $p(t)$.\n\nHere, we will proceed  by calculating the probability flux far from the potential region, i.e., we evaluate\n\\begin{eqnarray}\nJ(x, t) = \\frac{1}{m}\\mbox{Im} \\left[ \\psi^*_t(x) \\partial_x \\psi_t( x)\\right],\n\\end{eqnarray}\nfor $x >> b$.\n\n Eqs. (\\ref{gkx}) and (\\ref{evolvt}) imply that\n\\begin{eqnarray}\n\\psi_t(x)  = -\\frac{i}{\\sqrt{2\\pi}} \\int _0^{\\infty} dk  \\frac{T^*_k}{1+R^*_k}  \\left[ e^{iS_k +ikx}-  e^{-ikx}\\right] e^{-i\\frac{k^2}{2m}t}  \\tilde{\\psi}_0(k). \\label{psitxx}\n\\end{eqnarray}\n\nFor $ x>> b$, there is no stationary phase in the integral involving $e^{-ikx -i\\frac{k^2}{2m}t}$. Hence, its contribution to $\\psi_t(x)$ is much smaller than that of  the integral involving $e^{ikx -i\\frac{k^2}{2m}t}$, which has a stationary phase. By Eq.  (\\ref{eithe}),\n\\begin{eqnarray}\n\\psi_t(x) = -\\frac{i}{\\sqrt{2\\pi}} \\int _0^{\\infty} dk \\frac{T_k}{1+R_k}  e^{ikx -i\\frac{k^2}{2m}t } \\tilde{\\psi}_0(k). \\label{psitx}\n\\end{eqnarray}\n  The integral (\\ref{psitx}) contains only out-coming waves, hence, there is  no backflow in the probability current. Note that, in general, $J(x, t)$ can take negative values close to the barrier \\cite{Winter}, and hence, it is not reliable for the description of near-field experiments, i.e., when a particle detector is placed at a microscopic distance from the tunneling region.\n\nWe expand $(1+R_k)^{-1} = \\sum_{n=0}^{\\infty}(-R_k)^n$, to write Eq. (\\ref{psitx}) as\n\\begin{eqnarray}\n\\psi_t(x) = -\\frac{i}{\\sqrt{2\\pi}}\\sum_{n=0}^{\\infty}\\int _0^{\\infty} dk T_k(-R_k)^n  e^{ikx -i\\frac{k^2}{2m}t } \\tilde{\\psi}_0(k). \\label{psitx1b}\n\\end{eqnarray}\n\n\\subsection{Exponential decay }\nEq. (\\ref{psitx}) is accurate for the asymptotic behavior of the wave-function at $x >> b$. To proceed further, we exploit the fact that $\\psi_0(k)$ is strongly peaked about a specific value $k_0$, and we evaluate Eq. (\\ref{psitx}) in the saddle-point approximation. To this end, we write\n  $R_k = -|R_k|e^{i\\phi_k}$ and $T_k = |T_k|e^{i\\chi_k}$, so that Eq. (\\ref{psitx1b}) becomes\n\\begin{eqnarray}\n\\psi_t(x) =  - \\frac{i}{\\sqrt{2\\pi}}  \\sum_{n=0}^{\\infty}  \\int _0^{\\infty}dk |T_kR_k^n|  e^{ikx -i\\frac{k^2}{2m}t + i \\chi_k + i n \\phi_k} \\tilde{\\psi}_0(k). \\label{psitx2}\n\\end{eqnarray}\nThen, we extend the range of integration of $k$ to $(-\\infty, \\infty)$ setting $\\tilde{\\psi}_0(-k) =  - \\tilde{\\psi}_0(k)$ for negative $k$. The integral is not affected, because the additional terms involve a term $e^{-i|k|x -i\\frac{k^2}{2m}t}$, with no stationary phase. Then, we  approximate $|T_k| \\simeq |T_{k_0}|, |R_k| \\simeq |R_{k_0}|$, $k^2 \\simeq k_0^2 + 2 k_0 (k-k_0)$, $\\phi_k \\simeq \\phi_{k_0}+ \\phi'_{k_0} (k-k_0)$, $\\chi_k \\simeq \\chi_{k_0}+ \\chi'_{k_0} (k-k_0)$. The resulting integral is simply the inverse Fourier transform of $\\tilde{\\psi}_0(k)$. Hence,\n\\begin{eqnarray}\n\\psi_t(x) =  T_{k_0}  e^{ik_0x -i\\frac{k_0^2}{2m}t } \\sum_{n=0}^{\\infty} (-R_{k_0})^n \\psi_0(x - \\frac{k_0t}{m} + \\chi'_{k_0} + n \\phi'_{k_0}) \\label{psitx3}\n\\end{eqnarray}\n\n\nEq. (\\ref{psitx3}) has a natural interpretation in terms of classical concepts. The particle makes successive attempts to cross the barrier at $x = a$. On failure, it is reflected back,  it is reflected again at $x = 0$, and then it makes a new attempt. The $n$-th term in the sum of Eq. (\\ref{psitx3}) is the amplitude associated to a particle that succeeded in crossing the barrier at its\n $(n+1)$-th attempt: it is proportional to $T_{p_0}$ (one success) and to $R^n_{p_0}$ (after $n$ failures).\n\nSince $\\psi_0(x)$ has support only on $[0, a]$, $\\psi_t(x)$ vanishes for $t < t_0 := m (x -a + \\chi_{k_0}')\/p_0$. The time-scale $t_0$ has an obvious classical interpretation: it is the time it takes a particle inside the barrier region to traverse the distance to point $x$. The term $\\chi_{k_0}$ corresponds to the Wigner-Bohm time delay due to the particle crossing the\nclassically forbidden region \\cite{BohmWig}. We rewrite Eq. (\\ref{psitx3}) as\n\\begin{eqnarray}\n\\psi_t(x) =  T_{k_0}  e^{ik_0x -i\\frac{k_0^2}{2m}t } \\theta(t-t_0)\\sum_{n=0}^{\\infty} (-R_{k_0})^n \\psi_0\\left(a - \\frac{k_0(t - t_0 )}{m} + n\\Delta x \\right)  \\label{psitx4}\n\\end{eqnarray}\nwhere we defined $\\Delta x = \\phi'_{k_0}$ the position-shift between successive terms in the series (\\ref{psitx4}). If $|\\Delta x| > a$, the partial amplitudes at different $n$ do not overlap. Hence, there is no quantum interference between different attempts of the particle to cross the barrier. For $|\\Delta x| < a$, there is  quantum interference between $M = [a\/|\\Delta x|]$ successive attempts to cross the barrier.\n\nSince we assumed the initial state $\\psi_0$ to be almost monochromatic at energy $\\frac{k_0^2}{2m}$, the dominant contribution to the current is $ \\frac{k_0}{m} |\\psi_t(x)|^2$. Hence,\n\\begin{eqnarray}\nJ(t,x) &=& \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\sum_{n=0}^{\\infty} \\sum_{\\ell=0}^{\\infty}(-R_{k_0})^n(-R_{ k_0}^{*})^{\\ell}\\nonumber \\\\\n&\\times& \\psi_0(a- \\frac{k_0(t - t_0 )}{m} + \\ell \\Delta x )\\psi_0(a- \\frac{k_0(t - t_0 )}{m} + n \\Delta x ) \\label{jxt1}\n\\end{eqnarray}\nTerms in the summation with $|n- \\ell| > M$ vanish because the corresponding wave functions do not overlap. Then, we write\n\\begin{eqnarray}\nJ(t,x) = \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\sum_{N=0}^{\\infty}|R_{k_0}|^N \\rho\\left(\\frac{k_0(t - t_0 )}{m} - \\frac{1}{2} N\\Delta x  \\right), \\label{jtx3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\rho(x) = \\sum_{j=-M}^M \\psi_0(a - x - \\frac{j}{2} \\Delta x) \\psi_0(a - x +\\frac{j}{2} \\Delta x) e^{i j\\phi_{k_0}}.\n\\end{eqnarray}\nThe function $\\rho(x)$ is localized within a width of order $a$.\n\nIn order to connect Eq. (\\ref{jtx3}) with experiments, we have to treat both $x$ and $t$ as macroscopic variables. This means that they can be measured with an accuracy of order $\\sigma_X$ and $\\sigma_T$, respectively, that is much larger than the microscopic scales that characterize the system. Hence, $\\sigma_X >> a$ and $\\sigma_T >> ma\/k_0$. At such scales, the width of $\\rho(x)$ is negligible, and we can substitute it with a delta function,\n\\begin{eqnarray}\n\\rho(x) \\simeq \\alpha \\delta(x), \\label{Markovtunel}\n\\end{eqnarray}\nwhere $\\alpha = \\int_{-\\infty}^{\\infty}dx \\rho(x)$ is a number close to unity. In particular,  $\\alpha =1 $ for $M = 0$. In this regime, we can also approximate the sum over $N$ with an integral, so that\n\\begin{eqnarray}\nJ(t,x) &=& \\alpha \\frac{k_0}{m} |T_{k_0}|^2  \\theta(t-t_0) \\int_0^{\\infty}dN |R_{k_0}|^N \\delta\\left(\\frac{k_0(t - t_0 )}{m} - \\frac{1}{2} N\\Delta x  \\right)\\nonumber \\\\  &=& \\alpha \\frac{|T_{k_0}|^2}{\\Delta t}  \\theta(t-t_0) e^{\\log |R_{p_0}|^2 \\frac{(t-t_0)}{\\Delta t}} \\label{jx5}\n\\end{eqnarray}\nwhere  $\\Delta t = m \\Delta x\/k_0$ has the classical interpretation as the time between two successive attempts of the particle to cross the barrier. For $|T_{k_0}| << 1$, $\\log|R_{p_0}|^2 \\simeq -|T_{p_0}|^2$. Then, Eq. (\\ref{jx5}) describes exponential decay,\n\n\\begin{eqnarray}\nJ(t,x) =  \\alpha \\Gamma  e^{-\\Gamma  (t-t_0) } \\theta (t-t_0), \\label{expdec}\n\\end{eqnarray}\n with a decay constant\n\\begin{eqnarray}\n\\Gamma = \\frac{ |T_{k_0}|^2}{\\Delta t}. \\label{Gammat}\n\\end{eqnarray}\nthat does not depend on the detailed properties of the initial state.\n\n\nNote that  exponential decay fails at early times; the derivation of  Eq. (\\ref{expdec}) requires that $|t - t_0 |>> \\Delta t$.\n\n\nIn deriving the exponential decay law, we employed the saddle point approximation. This it is reasonably accurate for   $\\cap$-shaped potentials. In general, it does not\napply  to potentials with multiple transmission and reflection points, like the double well potential \\cite{Matsu}. Such potentials may trap the particle in an intermediate region, and they require an analysis of the escape from this region. The  escape satisfies an exponential decay law, except for energies near resonance \\cite{AnSav13}.\n\nThe exponential decay law also fails at very long times, when wave-function dispersion becomes important. To see this, we change variables to  $y = \\frac{k^2t}{2m}$ in Eq. (\\ref{psitx}) for $\\psi_t(x)$. The dominant term at $t \\rightarrow \\infty$ is\n\\begin{eqnarray}\n\\psi_t(x) =  - i \\sqrt{\\frac{m}{4\\pi t}} A_0  \\int_0^{\\infty} \\frac{dy}{y}  e^{-iy} \\tilde{\\psi}_0(\\sqrt{2my\/t}). \\label{asymptunel}\n\\end{eqnarray}\nwhere $A_0 $ stands for $\\lim_{k\\rightarrow \\infty} T_k\/(1+R_k)$. In general, $A_0 \\neq 0$, as can be readily checked in elementary systems. Hence, the asymptotic behavior of $\\psi_t(x)$ depends on  the infrared behavior of $\\tilde{\\psi}_0$. For  a power law dependence, $\\psi_0 \\sim k^n$ near $k = 0$, Eq. (\\ref{asymptunel}) gives $\\psi_t(x) \\sim t^{-\\frac{n+1}{2}}$. It follows that $J(t, x) \\sim t^{-(n+1)}$, i.e., the flux decays  with an inverse power law. We note here that the asymptotic regime captures some of the information of the initial state \\cite{MDCG}.\n\n\nThe key property in deriving the exponential decay law is the lack of interference between  different attempts of the particle to cross the barrier. Let us assume that the maximum number of successive attempts that interfere in the probability amplitude is $M$. The decay  time scale $\\Gamma^{-1}$ corresponds to $|T_{k_0}|^{-2}$  attempts to cross the barrier. As long as\n\\begin{eqnarray}\n|T_{k_0}|^{-2} >> M, \\label{tkm}\n\\end{eqnarray}\nthe effects of interference are negligible. This also implies that the particle has very short `memory' about its past attempts to cross the barrier. Hence, the memory time-scale  is much shorter than the decay time-scale. This feature is known as the {\\em Markov property}.\n\nIn absence of quantum interferences and memory effects,  decays due to tunneling are indistinguishable from classical probabilistic processes that can be described using elementary arguments.  Consider a classical particle that attempts to cross a barrier with probability $w << 1$ of success\\footnote{The non-zero probability to cross the barrier needs not be quantum mechanical in origin. The particle may interact with a stochastic environment, such as a thermal bath. Then, the crossing of the  barrier may be due to a random force.}. After $N$ attempts, the survival probability is $(1-w)^N \\simeq e^{-Nw}$. If every attempt takes time $\\Delta t$, then for $N >> 1$, the system is described by an exponential decay law with constant $\\Gamma = w \/\\Delta t$, in full agreement with Eq. (\\ref{Gammat}).\n\n\n\n\n\\subsection{Alternative description of tunneling decays}\nWe can understand the emergence of exponential decay in tunneling using a different argument that does not rely on the saddle-point approximation. Instead, we use the analyticity properties of the time-evolution operator. For the full development of such methods, see, Ref. \\cite{RNewt}, and for applications to tunneling decays, see, Refs. \\cite{CP75, CMM95, CCM09}.\n\nAssume that we can analytically extend $\\tilde{\\psi}_0$ to the fourth quadrant of the complex plane. Then, we can write $\\psi_t(x) = K(t, x) + I_N(t,x)$, where\n\n\\begin{eqnarray}\nK(t, x) = \\frac{i}{\\sqrt{2\\pi}}  \\oint_C dz  \\frac{T_z}{1+R_z}  e^{izx -i\\frac{z^2}{2m}t } \\tilde{\\psi}_0(z), \\label{ktn}\n\\end{eqnarray}\nis a line integral along the contour $C$ of Fig. 5, and $I_N(t)$ is the integral across the line segment $N$ of $C$.  By Cauchy's theorem, we can evaluate $K(t,x)$  in terms of the poles of the integrand in the interior of $C$.\n\n\\begin{figure}\n\\includegraphics[height=7cm]{curven}\n\\caption{The integration contour of the line-integral (\\ref{ktn}). The contour is traversed clockwise. }\n\\end{figure}\nLet us denote by $z_n = q_n -i \\gamma_n$ the poles of $\\frac{T_z}{1+R_z} $ in  the interior of $C$. The integer $n$ labels the poles, in the interior of $C$, $q_n$ and $\\gamma_n$ are positive. Then,\n\\begin{eqnarray}\n\\psi_t(x) = \\sum_n c_n \\tilde{\\psi}_0(q_n -i \\gamma_n) e^{iq_n x - i\\frac{q_n^2-\\gamma_n^2}{2m}t - \\frac{q_n\\gamma_n}{m}(t - \\frac{mx}{q_n})} + I_N(t),\n\\end{eqnarray}\nfor some complex constants $c_n$. Each term in the sum, is suppressed by an exponential factor $\\exp[\\frac{q_n\\gamma_n}{m}(t - \\frac{x}{q_n})]$, for $t > \\frac{x}{q_n}$. For an almost monochromatic state $\\tilde{\\psi}_0$ with energy $k_0$, only a small number of poles with $q_n$ near $k_0$ contribute. Furthermore, the contribution   $I_N$ to $\\psi_t(x)$ drops exponentially for $t > mx\/k_0$, i.e., after the earliest possible time of detection.\n\nFor simplicity, let us assume that the contribution of only one pole at $n = n_0$ is significant, and that $q_{n_0} \\simeq k_0$. Then, for $t > mx\/k_0$,\n \\begin{eqnarray}\n \\psi_t(x) \\sim   e^{ik_0 x - i\\frac{k_0^2-\\gamma_{n_0}^2}{2m}t - \\frac{k_0\\gamma_{n_0}}{m}|t - \\frac{mx}{k_0}|}.\n \\end{eqnarray}\n  Therefore, the flux $J(t, x)$ is proportional to $e^{-2 \\frac{k_0\\gamma_{n_0}}{m} |t - \\frac{mx}{k_0}|}$. The decay constant is\n  \\begin{eqnarray}\n  \\Gamma = \\frac{2k_0\\gamma_{n_0}}{m}, \\label{decaytb}\n  \\end{eqnarray}\n   and it is determined {\\em solely} by the pole of $\\frac{T_z}{1+R_z} $ near $z = k_0$. No information about the initial state other than its energy is required.\n\nThe above analysis also provides a criterion for the breakdown of exponential decay. If the initial state allows for the contribution of different poles $z_n$, such that there is a significant variation in the values of $\\gamma_n$, then corrections to exponential decay, or even its breakdown are possible.\n\nFor example, consider an initial state $ \\psi_0 = a_1 \\psi_1 + a_2 \\psi_2$ that is a superposition of  two almost monochromatic states $\\psi_1$ and $\\psi_2$ with energies $\\frac{k_1^2}{2m}$ and $\\frac{k_2^2}{2m}$. Furthermore, assume that $k_1$ is close to   one pole of $T_z\/(1+R_z)$ at $n = n_1$, and $k_2$ close to  another pole of $T_z\/(1+R_z)$ at $n = n_2$, and that there is no overlap. Then, for $t > \\max \\{ mx\/k_1, mx\/k_2 \\}$,\n\\begin{eqnarray}\n \\psi_t(x) \\sim c_1   e^{ik_1 x - i\\frac{k_1^2-\\gamma_{n_1}^2}{2m}t - \\frac{k_0\\gamma_{n_1}}{m}|t - \\frac{x}{k_1}|} +\n  c_2   e^{ik_2 x - i\\frac{k_2^2-\\gamma_{n_2}^2}{2m}t - \\frac{k_0\\gamma_{n_2}}{m}|t - \\frac{x}{k_2}|},\n\\end{eqnarray}\nfor some constants $c_1$ and $c_2$.\n\n\nThe dominant contribution to the current is\n\\begin{eqnarray}\nJ(t, x)  &=& |c_1|^2 k_1 e^{- \\Gamma_1 |t - \\frac{mx}{k_1}|} + |c_2|^2 k_2 e^{- \\Gamma_2 |t - \\frac{mx}{k_2}|}\n\\nonumber \\\\\n&+& (k_1 + k_2) e^{- \\frac{1}{2} \\Gamma_1 |t - \\frac{mx}{k_1}| - \\frac{1}{2} \\Gamma_2 |t - \\frac{mx}{k_2}| } \\mbox{Re} \\left[ c_1c_2^* e^{i \\theta(t, x)}\\right],\n\\end{eqnarray}\nwhere   $\\Gamma_i = \\frac{2k_i\\gamma_{n_i}}{m}$, and the interference phase is\n\\begin{eqnarray}\n\\theta(t, x) = (k_1 - k_2)x  - \\frac{k_1^2 -k_2^2}{2m} t + \\frac{\\gamma_{n_1}^2 - \\gamma_{n_2}^2}{2m}t.\n\\end{eqnarray}\nThe flux is characterized by an exponential decay with a periodic modulation due to the energy difference between the interfering states. This is the well-known phenomenon of {\\em quantum beats}.\n\n\n\\section{Detection probabilities}\nIn the previous sections, we employed two methods for constructing the decay probability, namely, persistence probabilities and probabilities currents. Both methods work fine for exponential decays, where the decay probability is determined by a single parameter $\\Gamma$. Outside exponential decay they have a restricted domain of validity. The key problem is that they are not guaranteed to define positive-definite probabilities.  This is due to the fact that they do not express probabilities in terms of measurement outcomes for concrete observables, These probabilities are  guaranteed to be positive by the rules of quantum theory.\n\nA rigorous  description of decays requires a consideration of the explicit measurement scheme through which the decay products are detected, and the construction of appropriate measurement observables. The latter correspond to\n  positive operators $\\hat{\\Pi}(t)$, in which the detection time $t$ appears as a random variable. Then, given an initial state $\\hat{\\rho}_0$, the detection probability $p(t)$ is determined by $Tr\\left[\\hat{\\rho}_0 \\hat{\\Pi}(t)\\right]$.  A scheme for constructing temporal observables of this type has been developed in \\cite{AnSav}. Here, we will present an elementary example of such observables that generalizes the well-established photodetection model by Glauber \\cite{Glauber}.\n\n\nSuppose that one of the decay products is a particle that is described by  quantum field  operators $\\hat{\\phi}({\\pmb x})$ and Hamiltonian $\\hat{H}$. The field operators are split into a positive frequency part $\\hat{\\phi}^{(+)}({\\pmb x})$ that contains annihilation operators and a negative frequency part $\\hat{\\phi}^{(-)}({\\pmb x})$ that contains creation operators. Consider an elementary apparatus located at a point ${\\pmb x}$ that gives a detection signal at time $t$ after having absorbed the incoming particle. The amplitude associated to this process is then proportional to $\\hat{\\phi}^{(+)}({\\pmb x})|\\psi_t\\rangle$, where  $|\\psi_t \\rangle$  is the state of the quantum field at time $t$. The probability of detection is the determined by the modulus square of this amplitude. It is given by\n  Glauber's formula\n\\begin{eqnarray}\nP(t, {\\pmb x})  = C  \\langle \\psi_t| \\hat{\\phi}^{(-)}({\\pmb x})\\hat{\\phi}^{(+)}({\\pmb x})|\\psi_t\\rangle, \\label{Glauber}\n\\end{eqnarray}\nwhere   $C$ is a normalization constant.  We do not obtain normalized probabilities, because there is a non-zero probability that the particle will not be detected and this probability depends on the field-state.\n\nEq. (\\ref{Glauber}) was first proposed by Glauber for photodetection. In Glauber's theory,  the role of $\\hat{\\phi}$ is played by the electric field, and the absorption interaction corresponds to the dipole coupling between the electromagnetic field and a macroscopic detectors.    Glauber's formula is a special case of a larger class of particle detection observables that can be defined in quantum fields \\cite{AnSav}.\n\n\nWe will apply Glauber's formula to the bosonic Lee model. The field operators associated to the bosonic creation and annihilation operators are\n\\begin{eqnarray}\n\\hat{\\phi}^{(+)}({\\pmb x}) = \\sum_r \\hat{a}_r \\chi_r({\\pmb x}) \\hspace{1cm}  \\hat{\\phi}^{(-)}({\\pmb x}) = \\sum_r \\hat{a}_r \\chi^*_r({\\pmb x}) \\label{scalar}\n\\end{eqnarray}\nwhere $\\chi_r(x)$ are eigenfunctions of the single-particle Hamiltonian. For particles in three dimensions,  $r$ corresponds to the three-momentum ${\\pmb k}$, and\n\\begin{eqnarray}\n\\chi_{\\pmb k}({\\pmb x}) = \\frac{1}{\\sqrt{2 \\omega_{\\pmb k}} }e^{i{\\pmb k} \\cdot {\\pmb x}}. \\label{chik}\n\\end{eqnarray}\n\nBy Eq. (\\ref{decay555}),\n\\begin{eqnarray}\n\\hat{\\phi}^{(+)}({\\pmb x}) \\frac{1}{z - \\hat{H}} |A'\\rangle = \\frac{V(z; {\\pmb x})}{z - \\Omega - \\Sigma(z)} |g\\rangle \\otimes  |0\\rangle,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nV_ {\\pmb x}(z)  = \\sum_r \\frac{g_r \\chi_r({\\pmb x})  }{z- \\omega_r}. \\label{Vz}\n\\end{eqnarray}\nThen, Eq. (\\ref{Glauber}) gives\n\\begin{eqnarray}\nP(t, {\\pmb x})  = C  |{\\cal B}(t, {\\pmb x})|^2, \\label{Glauber2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\cal B}(t, {\\pmb x}) =   \\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty+i \\epsilon }^{\\infty+i \\epsilon } \\frac{dE V_ {\\pmb x}(E)  e^{-iEt}}{[E - \\Omega - \\Sigma_a(E) }.\n\\end{eqnarray}\nFollowing the same steps that lead to  Eq. (\\ref{mainampl}), we find that\n\\begin{eqnarray}\n{\\cal B}(t,  {\\pmb x}) =\n   \\int_{0 }^{\\infty } \\frac{dE e^{-iEt}}{2\\pi}  \\frac{ \\frac{1}{2}\\Gamma(E) [V_{\\pmb x}^+(E) +V_{\\pmb x}^-(E) ] + i [E - \\Omega - F(E)][V_{\\pmb x}^+(E) - V_{\\pmb x}^-(E) ]  }{ [E - \\Omega - F(E)]^2 + \\frac{1}{4}[\\Gamma(E)]^2} \\label{mainampl44}\n\\end{eqnarray}\n where\n \\begin{eqnarray}\n V_{\\pmb x}^{\\pm}(E, {\\pmb x}) :=   \\lim_{\\eta \\rightarrow 0 } V_{\\pmb x}(E \\pm i \\eta).\n \\end{eqnarray}\n\nWe evaluate the amplitude (\\ref{mainampl44})  in the WWA, in which the dominant contribution to the integral comes\n from values of $E$ near $\\Omega$. Hence,\n\\begin{eqnarray}\n{\\cal B}(t,  {\\pmb x}) &\\simeq&\n  \\frac{1}{2 \\pi }  \\int_{-\\infty }^{\\infty } dE e^{-iEt}  \\frac{ \\Gamma  [V_{\\pmb x}^+(\\Omega) +V_{\\pmb x}^-(\\Omega) ]  + i [E - \\Omega - \\delta E][V_{\\pmb x}^+(\\Omega)  - V_{\\pmb x}^-(\\Omega) ]   }{ [E - \\Omega - \\delta E]^2 + \\frac{1}{4}\\Gamma^2} \\nonumber \\\\\n&=& \\frac{1}{2} e^{-i (\\Omega t + \\delta E)t - \\frac{1}{2}\\Gamma t}   V_+(\\Omega; {\\pmb x})   , \\label{mainampl44b}\n\\end{eqnarray}\nwhere we set $\\Gamma = \\Gamma(\\Omega)$ and $\\delta E = F(\\Omega)$. Hence, we obtain\n\\begin{eqnarray}\nP(t, {\\pmb x})  = \\frac{1}{4} C|V_{\\pmb x}^+(\\Omega) |^2 e^{-\\Gamma t}. \\label{decayGl}\n\\end{eqnarray}\nThe constant $C$ can be determined by normalizing over all particle detection events, i.e., by the requirement that\n\\begin{eqnarray}\n\\int_0^{\\infty} dt \\oint_{S} d^2n  P(t, {\\pmb x}) =1  , \\label{normalk}\n\\end{eqnarray}\nwhere $S$ is a two-sphere at distance $r$ from the location of the 2LS, ${\\pmb n}$ is a unit vector such that ${\\pmb x} = r {\\pmb n}$ on $S$.\n\nWe conclude  that the WWA guarantees exponential decay with constant $\\Gamma$, {\\em irrespective of the method used}. However, outside the exponential decay regime the probability density (\\ref{Glauber2}) differs significantly from Eq. (\\ref{decprob2}). In particular, it is guaranteed to be always positive.\n\n\n\n\\medskip\n\nAs an example, we revisit the photoemission model of Sec. 3.1.2. We employ Eqs. (\\ref{chik}) and (\\ref{Vz}), to obtain\n\\begin{eqnarray}\nV_{\\pmb x}(z) = \\frac{\\lambda}{8\\sqrt{2} \\pi^3} \\int \\frac{d^3k}{|{\\pmb k}|(z - |{\\pmb k}|)} e^{i {\\pmb k} \\cdot {\\pmb x}}.\n\\end{eqnarray}\nWe introduce spherical coordinates $(k, \\theta, \\phi)$ for ${\\pmb k}$, so that\n\\begin{eqnarray}\nV_{\\pmb x}(z) = \\frac{\\lambda}{4\\sqrt{2} \\pi^2} \\int_0^{\\infty} \\frac{kdk}{z- k} \\int_0^{\\pi} d\\theta \\sin \\theta e^{ikr \\cos \\theta} =\n \\frac{\\lambda}{2\\sqrt{2} \\pi^2 r} \\int_0^{\\infty} \\frac{dk \\sin(kr)}{z-k}, \\label{vzr}\n\\end{eqnarray}\ni.e., $V_{\\pmb x}(z)$ depends only on the radial coordinate $r = |{\\pmb x}|$.\n\nWe evaluate the integral (\\ref{vzr}) to\n\\begin{eqnarray}\nV_r(z) =  \\frac{\\lambda}{2\\sqrt{2} \\pi^2r}  \\left[ [\\gamma + \\ln(-rz) + \\mbox{Cin}(rz)] \\sin rz - \\mbox{si}(rz)\\cos rz\\right],\n \\end{eqnarray}\nwhere the functions\n $\\mbox{Cin}$ and $\\mbox{si}$ are    defined as\n\\begin{eqnarray}\n\\mbox{Cin}(z) = \\int_0^z dt \\frac{1-\\cos t}{t} \\hspace{1cm} \\mbox{si}(z) = \\int_z^{\\infty} dt \\frac{\\sin t}{t}, \\label{CiSi}\n\\end{eqnarray}\nand $\\gamma$ is the Euler-Mascheroni constant \\cite{ASt}.\n\nWe straightforwardly evaluate\n\\begin{eqnarray}\nV_r^{\\pm}(E;r) = \\frac{\\lambda}{2\\sqrt{2} \\pi^2r}  \\left[ [\\gamma + \\ln(rE) + \\mbox{Cin}(rE)] \\sin rE - \\mbox{si}(rE)\\cos rE \\mp i \\pi \\sin(rE)\\right].\n\\end{eqnarray}\n\nWe assume that the detectors are located at macroscopic distance from the decaying atom, so that $\\tilde{\\Omega}r >> 1$. Then, the terms involving the trigonometric integrals vanish, and the imaginary part of $V_+(\\Omega, r)$  dominates. Eq. (\\ref{decayGl}) gives\n\\begin{eqnarray}\nP(t, r) =  \\frac{\\lambda^2 \\sin^2(\\tilde{ \\Omega} r)}{32 \\pi^2 r^2} C e^{-\\Gamma t}.\n\\end{eqnarray}\n The sinusoidal dependence  on $r$ disappears if we average $P(t, r)$ over a thin shell of width $d >> \\tilde{\\Omega}^{-1}$ at distance $r$, since $\\langle\\sin^2(\\tilde{\\Omega}r)\\rangle = \\frac{1}{2}$. Then, the normalization condition (\\ref{normalk}) is satisfied for $C = (16 \\tilde{\\Omega})^{-1} $.\n\n  Note that outside the exponential decay regime, the probability density (\\ref{Glauber2}) leads to different predictions from the persistence probability method. The latter predicts an asymptotic probability density decaying with $t^{-5}$. Eq. (\\ref{mainampl44}) leads to an asymptotic decay of ${\\cal B}$ with $t^{-2}$, hence,  the probability density (\\ref{Glauber2})\ndecays as $t^{-4}$.\n\n\n\\section{Conclusions}\nWe presented an overview of the quantum description of decay processes. We showed that the emergence of the exponential decay law is explained in terms of a scale separation. In perturbative decays, the scale separation refers to energy: exponential decays emerge when the released energy associated to the decay  is much larger the energy of the interaction, as described by the self-energy function. In non-perturbative decays, the scale separation refers to time: exponential decay emerges when the decay time-scale is much larger than the time-scale of coherence between different attempts of the particle to cross the barrier.\n\nExponential decay may be extremely common, but it is not universal. It is not valid at very early and very late times, and in specific systems, it is not relevant at all. There is good experimental evidence for non-exponential decays, some of which pose persistent theoretical puzzles \\cite{GSI}. Our increasing access and control of multi-partite\/multi-particle systems is expected to uncover further unconventional types of decay---for example,  memory effects due to interaction with an environment \\cite{nonMark, BKEC},  effects due to the entanglement of the initial state \\cite{AnHu2, CVS17, C18}, or effects from particle statistics in many-particle systems \\cite{ADCM, TS11, CL11, DC11, MG11}. Furthermore, studies of    decay dynamics in many-particle quantum systems have demonstrated the need of  a genuinely many-particle characterization \\cite{ PSC12, HZHB13, DC16}, i.e., going beyond\ndescription in terms of single-particle observables.  We believe that a significant upgrade of traditional methods for quantum decays will be needed, in order to address such challenges.\n\n\\section*{Acknowledgements}\nResearch was supported by Grant No. E611 from the Research Committee of the University of Patras via the \"K. Karatheodoris\" program.\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}{0.15\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic1.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:A}\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.17\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:B}\t\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.16\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:C}\n\t\\end{subfigure}\n\t\\quad \\quad \\quad \\quad\n\t\\begin{subfigure}{0.15\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic4.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:motivation:D}\n\t\\end{subfigure}\n\t\\\\ \n\t\\begin{subfigure}{0.19\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic5.pdf}\n\t\t\\caption{Tied filtering}\n\t\t\\label{fig:motivation:E}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.23\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic6.pdf}\n\t\t\\caption{Ranking filtering}\n\t\t\\label{fig:motivation:F}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.20\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic7.pdf}\n\t\t\\caption{Gaussian-induced filtering}\n\t\t\\label{fig:motivation:G}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t\\begin{subfigure}{0.17\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\textwidth]{pic8.pdf}\n\t\t\\caption{$3\\times 3$ regular filtering}\n\t\t\\label{fig:motivation:H}\n\t\\end{subfigure}\n\t\\caption{Different filters operations on graph vertices. Examples of one-hop subgraphs are given in (a)-(d), where $v_0$ is the reference vertex and each vertex is assigned to a signal. The tied filtering (e) summarizes all neighbor vertices, and generates the same responses to (a) and (b) under the filter $f$, {i.e.}} \\def\\etal{{et.al}, $f(\\sum \\widetilde{w_{0i}}x_i)=f(1.9)$, although the two graphs are completely different in structures. The ranking filtering (f) sorts\/prunes neighbor vertices and then performs different filtering on them. It might result into the same responses $f_1(1)+f_2(3)+f_3(1)+f_4(4)$ to different graphs such as (b) and (c), where the digits in red boxes denote the ranked indices and the vertex of dashed box in (b) is pruned. Moreover, the vertex ranking is uncertain\/non-unique for equal connections in (d).To address these problems, we derive edge-induced GMM to coordinate subgraphs as shown in (g). Each of Gaussian model can be viewed as one variation component (or direction) of subgraph. Like the standard convolution (h), the Gaussian encoding is sensitive to different subgraphs, \\eg, (a)-(d) will have different responses. Note $f, f_i$ are linear filters, and the non-linear activation functions are put on their responses.\n\t}\n\t\\label{fig:motivation}\n\\end{figure*}\n\nAs witnessed by the widespread applications, graph is one of the most successful models to conduct structured and semi-structured data, ranging from text~\\cite{defferrard2016convolutional}, bioinformatics~\\cite{yanardag2015deep,niepert2016learning,song2018eeg} and social network~\\cite{gomez2017dynamics,orsini2017shift} to images\/videos~\\cite{marino2016more,cui2018context,cui2017spectral}. Among these applications, learning robust representations from structured graphs becomes the main topic. To this end, various methods have come forth in recent years. Graph kernels~\\cite{yanardag2015deep} and recurrent neural networks (RNNs)~\\cite{scarselli2009graph} are the most representative ones. Graph kernels usually take the classic R-convolution strategy~\\cite{haussler1999convolution} to recursively decompose graphs into atomic sub-structures and then define local similarities between them. RNNs based methods sequentially traverse neighbors with tied parameters in depth. With the increase of graph size, graph kernels would suffer diagonal dominance of kernels~\\cite{scholkopf2002kernel} while RNNs would have the explosive number of combinatorial paths in the recursive stage.\n\nRecently convolutional neural networks (CNNs)~\\cite{lecun2015deep} have achieved breakthrough progresses on representing grid-shaped image\/video data. In contrast, graphs are with irregular structures and fully coordinate-free on vertices and edges. The vertices\/edges are not strictly ordered, and can not be explicitly matched between two graphs. To generalize the idea of CNNs onto graphs, we need to solve this problem therein that the same responses should be produced for those homomorphic graphs\/subgraphs when performing convolutional filtering. To this end, recent graph convolution methods~\\cite{defferrard2016convolutional,atwood2016diffusion,hamilton2017inductive} attempted to aggregate neighbor vertices as shown in Fig.~\\ref{fig:motivation:E}. This kind of methods actually employ a fuzzy filtering ({i.e.}} \\def\\etal{{et.al}, a tied\/shared filter) on neighbor vertices because only first-order statistics (mean) is used. Two examples are shown in Fig.~\\ref{fig:motivation:A} and Fig.~\\ref{fig:motivation:B}. Although they have different structures, the responses on them are fully equal. Oppositely, Niepert \\etal~\\cite{niepert2016learning} ranked neighbor vertices according to weights of edges, and then used different filters on these sorted vertices, as shown in Fig.~\\ref{fig:motivation:F}. However, this rigid ranking method will suffer some limitations: i) probably consistent responses to different structures (\\eg, Fig.~\\ref{fig:motivation:B} and Fig.~\\ref{fig:motivation:C}) because weights of edges are out of consideration after ranking; ii) information loss of node pruning for a fixed-size receptive field as shown in Fig.~\\ref{fig:motivation:B}; and iii) ranking ambiguity for equal connections as shown in Fig.~\\ref{fig:motivation:D}; and iv) ranking sensitivity to (slightly) changes of edge weights\/connections.\n\nIn this paper we propose a Gaussian-induced graph convolution framework to learn graph representation. For a coordinate-free subgraph region, we design an \\textit{edge-induced} Gaussian mixture model (EI-GMM) to implicitly coordinate the vertices therein. Specifically, the edges are used to regularize Gaussian models such that variations of subgraph can be well-encoded. In analogy to the standard convolutional kernel as shown in Fig.~\\ref{fig:motivation:H}, EI-GMM can be viewed as a coordinate normalization by projecting variations of subgraph into several Gaussian components. For example, the four subgraphs w.r.t. Fig.~\\ref{fig:motivation:A}$\\sim$\\ref{fig:motivation:D} will have different representations\\footnote{Suppose three Gaussian models are $\\mcN(0,1), \\mcN(0,2)$ and $\\mcN(0,3)$, then we can compute the responses on (a)-(d) respectively as $f_1([0.49, -0.93])+f_2([0.17, -0.65])+f_3([0.07, -0.44])$, $f_1([0.35, -0.73])+f_2([0.15, -0.58])+f_3([0.10, -0.64]$, $f_1([0.35, -0.71])+f_2([0.15, -0.39])+f_3([0.10, -0.43])$, $f_1([0.46, -0.99])+f_2([0.18, -0.62])+f_3([0.08, -0.42])$. Please refer to incoming section.} through our Gaussian encoding in Fig.~\\ref{fig:motivation:G}. To make the network inference forward, we transform Gaussian components of each subgraph into the gradient space of multivariate Gaussian parameters, instead of employing the sophisticated EM algorithm. Then the filters (or transform functions) are performed on different Gaussian components like latticed kernels on different directions in Fig.~\\ref{fig:motivation:H}. Further, we derive a \\textit{vertex-induced} Gaussian mixture model (VI-GMM) to favor dynamic coarsening of graph. We also theoretically analyze the approximate equivalency of VI-GMM to weighted graph cut~\\cite{dhillon2007weighted}. Finally, EI-GMM and VI-GMM can be alternately stacked into an end-to-end optimization network.\n\nIn summary, our main contributions are four folds: i) propose an end-to-end Gaussian-induced convolutional neural network for graph representation; ii) propose edge-induced GMM to encode variations of different subgraphs; iii) derive vertex-induced GMM to perform dynamic coarsening of graphs, which is an approximation to the weighted graph cut; iv) verify the effectiveness of our method and report state-of-the-art results on several graph datasets.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.83\\textwidth]{pic9.pdf}\n\t\\caption{The GIC network architecture. The GIC main contains two module: convolution layer (EI-GMM) and coarsening layer (VI-GMM). The GIC stacks several convolution and coarsening layers alternatively and iteratively. More details can be found in incoming section.}\n\t\\label{fig:network:A}\n\\end{figure*}\n\n\n\\section{Related Work}\n\nGraph CNNs mainly fall into two categories: spectral and spatial methods. Spectral methods~\\cite{bruna2013spectral,scarselli2009graph,henaff2015deep,such2017robust,li2018action,li2018spatio} construct a series of spectral filters by decomposing graph Laplacian, which often suffers high-computational burden. To address this problem, the fast local spectral filtering method~\\cite{defferrard2016convolutional} parameterizes the frequency responses as a Chebyshev polynomial approximation. However, as shown in Fig.~\\ref{fig:motivation:B}, after summarizing all nodes, this method will discard topology structures of a local receptive field. This kind of methods usually require equal sizes of graphs like the same sizes of images for CNNs~\\cite{kipf2016semi}. Spatial methods attempt to define spatial structures of adjacent vertices and then perform filtering on structured graphs. Diffusion CNNs~\\cite{atwood2016diffusion} scans a diffusion process across each node. PATCHY-SAN~\\cite{niepert2016learning} linearizes neighbors by sorting weights of edges and deriving convolutional filtering on graphs, as shown in Fig.~\\ref{fig:motivation:C}. As an alternative, random walks based approach is also used to define the neighborhoods~\\cite{perozzi2014deepwalk}. For the linearized neighbors, RNNs~\\cite{li2015gated} could be used to model the structured sequences. Similarly, NgramCNN~\\cite{luo2017deep} serializes each graph by introducing the concept of $n$-gram block. \nGAT~\\cite{velickovic2017graph} attempts to weight edges through the attention mechanism. WSC~\\cite{jiang2018walk} attempts to aggregate walk fields defined by random walks into Gaussian mixture models. Zhao~\\cite{zhao2018work} attempts to define a standard network with different graph convolutions. Besides, some variants~\\cite{hamilton2017inductive,duran2017learning,zhang2018tensor} employ the aggregation or propagation of local neighbor nodes. Different from these tied filtering or ranking filtering methods, we use Gaussian models to encode local variations of graph. Also different from the recent mixture models~\\cite{monti2017geometric}, which uses GMM to only learn the importance of adjacent nodes, our method uses weighted GMM to encode the distributions of local graph structures.\n\n\n\\section{The GIC Network}\n\n\\subsection{Attribute Graph}\n\nHere we consider an undirected attribute graph $\\mcG=(\\mcV,\\A,\\X)$ of $m$ vertices (or nodes), where $\\mcV=\\{v_i\\}_{i=1}^{m}$ is the set of vertices, $\\A$ is a (weighted) adjacency matrix, and $\\X$ is a matrix of graph attributes (or signals). The adjacency matrix $\\A\\in\\mbR^{m\\times m}$ records the connections between vertices. If $v_i, v_j$ are not connected, then $A(v_i,v_j)=0$, otherwise $A(v_i,v_j)\\neq 0$. We sometimes abbreviate $A(v_i,v_j)$ as $A_{ij}$. The attribute matrix $\\X\\in\\mbR^{m\\times d}$ is associated with the vertex set $\\mcV$, whose $i$-th row $\\X_{i}$ (or $\\X_{v_i}$) denotes a $d$-dimension attribute of the $i$-th node ({i.e.}} \\def\\etal{{et.al}, $v_i$).\n\nThe graph Laplacian matrix $\\L$ is defined as $\\L = \\D-\\A$, where $\\D\\in\\mbR^{m\\times m}$ is the diagonal degree matrix with $D_{ii}=\\sum_{j}A_{ij}$. The normalized version is written as $\\L^{norm} = \\D^{-1\/2}\\L\\D^{-1\/2}= \\I-\\D^{-1\/2}\\A\\D^{-1\/2}$.\nwhere $\\I$ is the identity matrix. Unless otherwise specified, we use the latter. We give the definition of subgraph used in the following.\n\\begin{defn} \\label{def:graph}\n\tGiven an attribute graph $\\mcG=(\\mcV,\\A,\\X)$, the attribute graph $\\mcG'=(\\mcV',\\A',\\X')$ is a subgraph of $\\mcG$, denoted $\\mcG'\\subseteq\\mcG$, if (i) $\\mcV'\\subseteq\\mcV$, (ii) $\\A'$ is the submatrix of $\\A$ w.r.t. the subset $\\mcV'$, and (iii) $\\X'=\\X_{\\mcV'}$.\n\\end{defn}\n\n\\subsection{Overview}\n\nThe GIC network architecture is shown in Fig.~\\ref{fig:network:A}. Given an attribute graph $\\mcG^{(0)}=(\\mcV^{(0)},\\A^{(0)},\\X^{(0)})$, where the superscript denotes the layer number, we construct multi-scale receptive fields for each vertex based on the adjacency matrix $\\A^{(0)}$. Each receptive field records $k$-hop neighborhood relationships around the reference vertex, and forms a local centralized subgraph. To encode the centralized subgraph, we project it into edge-induced Gaussian models, each of which defines one variation ``direction\" of the subgraph. We perform different filtering operations on different Gaussian components and aggregate all responses as the convolutional output. After the convolutional filtering, the input graph $\\mcG^{(0)}$ is transformed into a new graph $\\mcG^{(1)}=(\\mcV^{(1)},\\A^{(1)},\\X^{(1)})$, where $\\mcV^{(1)}=\\mcV^{(0)}$ and $\\A^{(1)}=\\A^{(0)}$. To further abstract graphs, we next stack a coarsening layer on the graph $\\mcG^{(1)}$. The proposed vertex-induced GMM is used to downsample the graph $\\mcG^{(1)}$ into the low-resolution graph $\\mcG^{(2)}=(\\mcV^{(2)},\\A^{(2)},\\X^{(2)})$. Taking the convolution and coarsening modules, we may alternately stack them into a multi-layer GIC network, With the increase of layers, the receptive field size of filters will become larger, so the higher layer can extract more global graph information. In the supervised case of graph classification, we finally concatenate with a fully connected layer followed by a softmax loss layer.\n\n\\subsection{Multi-Scale Receptive Fields} \n\nIn the standard CNN, receptive fields may be conveniently defined as latticed spatial regions. Thus convolution kernels on grid-shaped structures are accessible. However, the construction of convolutional kernels on graphs are intractable due to coordinate-free graphs, \\eg, unordered vertices, unfixed number of adjacent edges\/vertices. To address this problem, we resort to the adjacent matrix $\\A$, which expresses connections between vertices. Since $\\A^k$ exactly records the $k$-step reachable vertices, we may construct a $k$-neighbor receptive field by using the $k$-order polynomial of $\\A$, denoted as $\\psi_k(\\A)$. Taking the simplest case, $\\psi_k(\\A)=\\A^k$ reflects the $k$-hop neighborhood relationships. In order to remove the scale effect, we may normalize $\\psi_k(\\A)$ as $\\psi_k(\\A) \\text{diag} (\\psi_k(\\A)\\1)^{-1}$, which describes the reachable possibility in a $k$-hop walking. Formally, we define the $k$-th scale receptive field as a subgraph.\n\\begin{defn}\\label{def:receptive}\n\tThe $k$-th scale receptive field around a reference vertex $v_i$ is a subgraph $\\mcG_{v_i}^k=(\\mcV',\\A',\\X')$ of the k-order graph $(\\mcV, \\tbA=\\psi_k(\\A), \\X)$, where $\\mcV'=\\{v_j|\\wtA_{ij}\\neq 0\\}\\cup\\{v_i\\}$, $\\A'$ is the submatrix of $\\tbA$ w.r.t. $\\mcV'$, and $\\X'=\\X_{\\mcV'}$.\n\\end{defn}\n\n\\subsection{Convolution: Edge-Induced GMM}\n\nGiven a reference vertex $v_i$, we can construct the centralized subgraph $\\mcG_{v_i}^k$ of the $k$-th scale.\nTo coordinate the subgraph, we introduce Gaussian mixture models (GMMs), each of which may be understood as one principal direction of its variations. To encode the variations accurately, we jointly formulate attributes of vertices and connections of edges into Gaussian models. The edge weight $A'(v_i,v_j)$ indicates the relevance of $v_j$ to the central vertex $v_i$. The higher weight is, the stronger impact on $v_i$ is. So the weights can be incorporated into a Gaussian model by observing $A'(v_i,v_j)$ times. As the likelihood function, it is equivalent to raise the power $A'(v_i,v_j)$ on Gaussian function, which is proportional to $\\mcN(\\X'_{v_j}, \\bmu, \\frac{1}{A'(v_i,v_j)}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta})$. Formally,  we estimate the probability density of the subgraph $\\mcG_{v_i}^k$ from the $C_1$-component GMM,\n\\begin{align}\np_{v_i}(\\X'_{v_j};\\bTheta_1,A'_{ij}) &= \\sum_{c=1}^{C_1}\\pi_c\\mcN(\\X'_{v_j}; \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c), \\nonumber \\\\\n&\\st \\pi_c>0, \\sum_{c=1}^{C_1}\\pi_c=1, \\label{eqn:GMM_edge}\n\\end{align}\nwhere $\\bTheta_1=\\{\\pi_1,\\cdots,\\pi_{C_1}, \\bmu_1,\\cdots,\\bmu_{C_1},\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_1,\\cdots,\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_{C_1}\\}$ are the mixture parameters, $\\{\\pi_c\\}$ are the mixture coefficients, $\\{\\bmu_c, \\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c\\}$ are the parameters of the $k$-th component, and $A'_{ij}>0$ \\footnote{In practice, we normalize $\\A'$ into a non-negative matrix.}. Intuitively, edge weight $A'_{ij}$ is, the stronger impact of the node $v_j$ w.r.t. the reference vertex $v_i$ is. We will refer to the model in Eqn.~(\\ref{eqn:GMM_edge}) as the \\textit{edge-induced Gaussian mixture model} (EI-GMM).\n\nIn what follows, we assume all attributes of nodes are independent on each other, which is often used in signal processing. That means, the covariance matrix $\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c$ is diagonal, so we denote it as $\\text{diag}(\\mathbf{\\sigma}_c^2)$. To avoid the explicit constraints for $\\pi_c$ in Eqn.~(\\ref{eqn:GMM_edge}), we adopt the soft-max normalization with the re-parameterization variable $\\alpha_c$, {i.e.}} \\def\\etal{{et.al}, $\\pi_c = {\\exp(\\alpha_c)}\/{\\sum_{k=1}^{C_1}\\exp(\\alpha_k)}$. Thus, the entire subgraph log-likelihood can be written as\n\\begin{align}\n\\zeta(\\mcG_{v_i}^k) &= \\sum_{j=1}^m\\ln p_{v_i}(\\X'_{v_j};\\bTheta_1,\\A') \\nonumber \\\\\n&= \\sum_{j=1}^m\\ln\\sum_{c=1}^{C_1}\\pi_c\\mcN(\\X'_{v_j}; \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c),\\label{eqn:GMM_log}\n\\end{align}\nTo infer forward, instead of the expectation-maximization (EM) algorithm, we use the gradients of the subgraph with regard to the parameters of the EI-GMM model $\\bTheta_1$, motivated by the recent Fisher vector work~\\cite{sanchez2013image}, which has been proven to be effective in representation.\n\nFor a convenient calculation, we simplify the notations, $\\mcN_{jc} = \\mcN(\\X'_{v_j}, \\bmu_c, \\frac{1}{A'_{ij}}\\mathbf{\\sigma}_c^2)$ and $ Q_{jc}=\\frac{\\pi_c\\mcN_{jc} }{\\sum_{k=1}^{C_1}\\pi_k\\mcN_{jk}}$, then we can derive the gradients of model parameters from Eqn.~(\\ref{eqn:GMM_log}) as follows\n\\begin{align}\n&\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\bmu_c}  \\!\\!=\\!\\! \\sum_{j=1}^m\\frac{A'_{ij}Q_{jc}(\\X'_{v_j}-\\bmu_c)}{\\mathbf{\\sigma}_c^2}, \\quad \\nonumber \\\\\n&\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\mathbf{\\sigma}_c}  \\!\\!=\\!\\! \\sum_{j=1}^m \\frac{Q_{jc}(A'_{ij}(\\X'_{v_j}-\\bmu_c)^2-\\mathbf{\\sigma}_c^2)}{\\mathbf{\\sigma}_c^3},\n\\end{align} \nwhere the division of vectors means a term-by-term operation. Note we do not use $\\partial \\zeta(\\mcG_{v_i}^k) \/ \\partial\\alpha_c$ due to no improvement in our experience. The gradients describe the contribution of the corresponding parameters to the generative process. The subgraph variations are adaptively allocated to $C_1$ Gaussian models. Finally, we ensemble all gradients w.r.t. Gaussian model ({i.e.}} \\def\\etal{{et.al}, directions of graph) to analogize the collection of local square receptive field on image. Formally, for the $k$-scale receptive field $\\mcG_{v_i}^k$ around the vertex $v_i$, the attributes produced from Gaussian models are filtered respectively and then concatenated,\n\\begin{align}\nF(\\mcG_{v_i}^k,\\bTheta_1,f) &= \\text{ReLU}(\\sum_{c=1}^{C_1} f_i(\\text{Cat}[\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\bmu_c},\\frac{\\partial \\zeta(\\mcG_{v_i}^k)}{\\partial\\mathbf{\\sigma}_c}]), \\label{eqn:gau_fea}\n\\end{align}\nwhere $\\text{Cat}[\\cdot, \\cdot]$ is a concatenation operator, $f_i$ is a linear filtering function ({i.e.}} \\def\\etal{{et.al}, a convolution function) and ReLU is the rectified linear unit. Therefore we can produce the feature vectors that have same dimensionality depending on the number of Gaussian models for different subgraphs. If the soft assignment distribution $ Q_{jc} $ is sharply peaked on a single value of one certain Gaussian for the vertex $v_{j}$, the vertex will be only projected onto one Gaussian direction.\n\n\\subsection{Coarsening: Vertex-Induced GMM}\n\nLike the standard pooling in CNNs, we need to downsample graphs so as to abstract them as well as reduce the computational cost. However, the pooling on images are tailored for latticed structures, and cannot be used for irregular graphs. One solution is to use some clustering algorithms to partition vertices to several clusters, and then produce a new vertex from each cluster. However, we expect that two vertices should not fall into the same cluster with a larger possibility if there is a high transfer difficulty between them. To this end, we derive vertex-induced Gaussian mixture models (VI-GMM) to weight each vertex. To utilize the edge information,  we construct a latent observation $\\phi(v_i)$ w.r.t. each vertex $v_i$ from the graph Laplacian (or adjacent matrix if semi-positive definite), {i.e.}} \\def\\etal{{et.al}, the kernel calculation $\\langle\\phi(v_i),\\phi(v_j)\\rangle=L_{ij}$. Moreover, for each vertex $v_i$, we define an influence factor $w_i$ for Gaussian models. Formally, given $C_2$ Gaussian models, VI-GMM is written as\n\\begin{align}\np(\\phi(v_i);\\bTheta_2,w_i) &= \\sum_{c=1}^{C_2}\\pi_c\\mcN(\\phi(v_i); \\bmu_c, \\frac{1}{w_i}\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}_c), \\nonumber \\\\\n&\\st  w_i=h(\\X_{v_i})>0,\n\\end{align}\nwhere $h$ is a mapping function to be learnt. To reduce the computation cost of matrix inverse on $\\mathbf{\\Sigma}} \\def\\bmu{\\mathbf{\\mu}}  \\def\\btheta{\\mathbf{\\theta}} \\def\\bTheta{\\mathbf{\\Theta}$, we specify it as an identity matrix. Then we have\n\\begin{align}\np(\\phi(v_i);\\bTheta_2,w_i) =  \\sum_{c=1}^{C_2}\\frac{\\pi_c}{(\\frac{2\\pi}{w_i})^{d\/2}}\\exp^{-\\frac{w_i}{2}\\|\\phi(v_i)-\\bmu_c\\|^2}, \n\\end{align}\nGiven a graph with $m$ vertices, the objective is to maximize the following log-likelihood:\n\\begin{align}\n\\argmax_{\\bTheta_2} \\zeta(\\bTheta_2) = \\sum_{i=1}^m \\ln\\sum_{c=1}^{C_2}\\pi_c\\mcN(\\phi(v_i); \\bmu_c, \\frac{1}{w_i}\\I)). \\label{eqn:likelihood}\n\\end{align}\n\nTo solve above model in Eqn.~(\\ref{eqn:likelihood}), we use the iterative expectation maximization algorithm, which has closed-form solution at each step. Meanwhile, the algorithm may automatically conduct the required constraints. The graphical clustering process is summarized as follows:\n\n(1) {E-Step}: the posteriors, {i.e.}} \\def\\etal{{et.al}, the $i$-th vertex for the $c$-th cluster, are updated with\n$p_{ic} = \\frac{\\pi_c p(\\phi(v_i);\\btheta_c, w_i)}{\\sum_{k=1}^C \\pi_k p(\\phi(v_i);\\btheta_k, w_i)}$,\nwhere $\\btheta_c$ is the $c$-th Gaussian parameters, and $\\bTheta_2=\\{\\btheta_1,\\cdots,\\btheta_{C_2}\\}$.\n\n(2) {M-Step}: we optimize Gaussian parameters $\\pi, \\bmu$. The parameter estimatation is given by $\n\\pi_c = \\frac{1}{m}\\sum_{i=1}^m r_{ic},\n\\bmu_c =\\frac{\\sum_{v_i\\in \\mcG_c}w_i\\phi(v_i)}{\\sum_{v_i\\in \\mcG_c}w_i}$.\n$\\pi_c$ indicates the energy summation of all vertices assigned to the cluster $c$, and $\\bmu_c$ may be understood as a doubly weighted ($w_i, r_{ic}$) average on the cluster $c$.\n\nAfter several iterations of the two steps, we perform hard quantification. The $i$-th vertex is assigned as the class with the maximum possibility, formally, $r_{ic} = 1$ if $c=\\argmax_{k} p_{ik}$, otherwise 0. Thus we can obtain the cluster matrix $\\P\\in\\{0,1\\}^{m\\times C_2}$, where $P_{ic}=1$ if the $i$-th vertex falls into the cluster $c$. During coarsening, we take maximal responses of each cluster as the attributes of new vertex, and derive a new adjacency matrix by using $\\P^\\intercal} \\def\\st{\\text{s.t.~}\\A\\P$.\n\nIt is worth noting that we need not compute the concrete $\\phi$ during the clustering process. The main calculation $\\|\\phi(v_i)-\\bmu_c\\|^2$ in EM can be reduced to the kernel version:\n$K_{ii}-\\frac{2\\sum_{v_j\\in\\mcG_c}w_jK_{ij}}{\\sum_{v_j\\in\\mcG_c}w_j}\n+\\frac{\\sum_{v_j,v_k\\in\\mcG_c}w_jw_k K_{jk}}{(\\sum_{v_j\\in\\mcG_c}w_j)^2}$,\nwhere $K_{ij} = \\langle\\phi(v_i),\\phi(v_j)\\rangle$.\nIn practice, we can use the graph Laplacian $\\L$ as the kernel. In this case, we can easily reach the following proposition, which is relevant to graph cut~\\cite{dhillon2007weighted}.\n\\begin{prop}\n\tIn EM, if the kernel matrix takes the weight-regularized graph Laplacian, {i.e.}} \\def\\etal{{et.al}, $\\mcK= \\text{diag}(\\w)\\L \\text{diag}(\\w)$, then VI-GMM is equal to an approximate optimization of graph cut, {i.e.}} \\def\\etal{{et.al}, $\n\t\\min \\sum_{c=1}^C\\frac{\\text{links}(\\mcV_c, \\mcV\\backslash \\mcV_c)}{w(\\mcV_c)}$,\n\twhere $\\text{links}(\\mcA, \\mcB)=\\sum_{v_i\\in\\mcA,v_j\\in\\mcB} A_{ij}$, and $w(\\mcV_c)=\\sum_{j\\in\\mcV_c}w_j$.\n\\end{prop}\n\n\n\\section{Experiments}\n\n\\subsection{Graph Classification}\n\nFor graph classification, each graph is annotated with one label. We use two types of datasets: Bioinformatics and Network datasets. The former contains MUTAG~\\cite{debnath1991structure}, PTC~\\cite{toivonen2003statistical}, NCI1 and NCI109~\\cite{wale2008comparison}, ENZYMES~\\cite{borgwardt2005protein} and PROTEINS~\\cite{borgwardt2005protein}. The latter has COLLAB~\\cite{leskovec2005graphs}, REDDIT-BINARY, REDDIT-MULTI-5K, REDDIT-MULTI-12K, IMDB-BINARY and IMDB-MULTI.\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Comparisons with state-of-the-art methods.}\n\t\\begin{sc}\n\t\\scalebox{0.85}{\n\t\t\\begin{tabular}{|l| c c c| c c | c c c |c |c |c |c|}\n\t\t\t\\toprule\n\t\t\tDataset \n\t\t\t&PSCN  &DCNN  &NgramCNN   &FB  &DyF   &WL   &GK  &DGK  &RW   &SAEN  & GIC \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{MUTAG}\n\t\t\t&92.63   &66.98  &\\textbf{94.99}  &84.66   &88.00    &78.3   &81.66     &82.66      &83.72     &84.99 &94.44   \\\\\t\t\n\t\t\t& $\\pm$4.21  &--    &\\textbf{$\\pm$5.63} &$\\pm$2.01 & $\\pm$2.37  & $\\pm$1.9 & $\\pm$2.11  & $\\pm$1.45  & $\\pm$1.50   & $\\pm$1.82  \t&$\\pm$4.30      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{PTC}\n\t\t\t&60.00   &56.60  &68.57  &55.58   &57.15     &--     &57.26     &57.32   &57.85    &57.04  &\\textbf{77.64}   \\\\\t\t\t\n\t\t\t& $\\pm$4.82  &--   &$\\pm$1.72  &2.30   & $\\pm$1.47  & --  &  $\\pm$1.41  &  $\\pm$1.13  & $\\pm$1.30\n\t\t\t&  $\\pm$ 1.30  & \\textbf{$\\pm$ 6.98}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{NCI1}\n\t\t\t&78.59   &62.61     &--  &62.90   &68.27  &83.1   &62.28   &62.48   &48.15    &77.80  &\\textbf{84.08}    \\\\\t\t\t\n\t\t\t& $\\pm$1.89  &--   &--  &$\\pm$0.96  & $\\pm$0.34  & $\\pm$0.2  & $\\pm$0.29  &  $\\pm$0.25  &  $\\pm$0.50  & $\\pm$ 0.42  & \\textbf{$\\pm$1.77}     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{NCI109}\n\t\t\t& --   &62.86    &--  &62.43  & 66.72   & \\textbf{85.2}    & 62.60   & 62.69   & 49.75 \t & --  & 82.86    \\\\\t\t\t\n\t\t\t& --   &--   &--  &$\\pm$1.13    & $\\pm$ 0.20  & \\textbf{$\\pm$ 0.2}  & $\\pm$ 0.19  &  $\\pm$ 0.23  &  $\\pm$ 0.60\t& --  & $\\pm$ 2.37     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{ENZYMES}\n\t\t\t& --    &18.10    &--  &29.00   & 33.21   & 53.4   & 26.61   & 27.08   & 24.16   & --   & \\textbf{62.50}  \\\\\t\t\t\n\t\t\t& --    &--    &--  &$\\pm$1.16  & $\\pm$ 1.20  & $\\pm$ 1.4   &  $\\pm$ 0.99  & $\\pm$ 0.79  &  $\\pm$ 1.64\n\t\t\t&--  & \\textbf{$\\pm$ 5.12}     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{PROTEINS}\n\t\t\t& 75.89  &--   &75.96  &69.97 & 75.04   & 73.7  & 71.67   & 71.68  & 74.22    & 75.31  & \\textbf{77.65}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.76  &--   &$\\pm$2.98  &$\\pm$1.34  & $\\pm$ 0.65  &  $\\pm$ 0.5  & $\\pm$ 0.55  &  $\\pm$ 0.50  & $\\pm$ 0.42\t&  $\\pm$ 0.70 & \\textbf{$\\pm$ 3.21}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{COLLAB}\n\t\t\t& 72.60  &--    &--  &76.35 & 80.61 & -- & 72.84 & 73.09 & 69.01 & 75.63 & \\textbf{81.24}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.15  &--   &--  &1.64  & $\\pm$ 1.60  & --  & $\\pm$ 0.28  & $\\pm$ 0.25  & $\\pm$ 0.09\n\t\t\t& $\\pm$ 0.31  & \\textbf{$\\pm$ 1.44}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-B}\n\t\t\t& 86.30  &--     &--  &88.98 &\\textbf{89.51}  &75.3 & 77.34 & 78.04 & 67.63  & 86.08 & 88.45   \\\\\t\t\t\n\t\t\t& $\\pm$ 1.58  &--   &--  &$\\pm$2.26  & \\textbf{$\\pm$ 1.96}  & $\\pm$ 0.3  & $\\pm$ 0.18  & $\\pm$ 0.39  & $\\pm$ 1.01\n\t\t\t& $\\pm$ 0.53  & $\\pm$ 1.60     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-5K}\n\t\t\t& 49.10  &--      &--  &50.83   & 50.31 & --  & 41.01 & 41.27 & --  &\\textbf{52.24} &51.58 \\\\\t\t\n\t\t\t& $\\pm$ 0.70  &--    &--  &1.83  &$\\pm$ 1.92  & --  & $\\pm$ 0.17  & $\\pm$ 0.18  & --\n\t\t\t& \\textbf{$\\pm$ 0.38}  & $\\pm$ 1.68   \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{REDDIT-12K}\n\t\t\t& 41.32  &--      &--  &42.37  & 40.30 & --  & 31.82 & 32.22 & --  & \\textbf{46.72} & 42.98 \\\\\t\t\t\n\t\t\t& $\\pm$ 0.42  &--    &--  &1.27   & $\\pm$ 1.41  & -- & $\\pm$ 0.08  & $\\pm$ 0.10  & --\n\t\t\t& \\textbf{$\\pm$ 0.23} & $\\pm$ 0.87     \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{IMDB-B}\n\t\t\t& 71.00   &--    &71.66  &72.02 & 72.87 & 72.4 & 65.87 & 66.96 & 64.54 \t& 71.26 & \\textbf{76.70}     \\\\\t\t\t\n\t\t\t& $\\pm$ 2.29  &--    &$\\pm$2.71  &$\\pm$4.71  & $\\pm$ 4.05  & $\\pm$ 0.5  & $\\pm$ 0.98  & $\\pm$ 0.56  & $\\pm$ 1.22\n\t\t\t& $\\pm$ 0.74  & \\textbf{$\\pm$ 3.25}      \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{IMDB-M}\n\t\t\t& 45.23 &--     &50.66  &47.34  & 48.12 & -- & 43.89 & 44.55 & 34.54 & 49.11 & \\textbf{51.66}      \\\\\t\t\t\n\t\t\t& $\\pm$ 2.84  &--  &$\\pm$4.10  &3.56  & $\\pm$ 3.56  & -- & $\\pm$ 0.38  & $\\pm$ 0.52  & $\\pm$ 0.76\n\t\t\t& $\\pm$ 0.64  & \\textbf{$\\pm$ 3.40}      \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\end{sc}\n\t\\label{table:state-of-the-art}\n\\end{table*}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Node label prediction on Reddit and PPI data (micro-averaged F1 score).}\n\t\\label{table:multi-label}\n\t\\begin{sc}\t\n\t\t\\scalebox{0.9}{\n\t\t\t\\begin{tabular}{l c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset      & Reddit  & PPI   \\\\\n\t\t\t\t\\midrule\t\n\t\t\t\tRandom       & 0.042   & 0.396                     \\\\\n\t\t\t\tRaw features & 0.585   & 0.422                       \\\\\n\t\t\t\tDeep walk    & 0.324   & --                       \\\\\n\t\t\t\tDeep walk + features  & 0.691  & --               \\\\\n\t\t\t\tNode2Vec + regression  & 0.934 & -- \\\\\n\t\t\t\tGraphSAGE-GCN  & 0.930   & 0.500                       \\\\\t\t\t\n\t\t\t\tGraphSAGE-mean  & 0.950   & 0.598                       \\\\\n\t\t\t\tGraphSAGE-LSTM  & \\textbf{0.954}   & 0.612                       \\\\\n\t\t\t\tGIC    & 0.952   & \\textbf{0.661}                     \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\n\n\\subsubsection{Experiment Settings} \n\nWe verify our GIC on the above bioinformatics and social network datasets. In default, GIC mainly consists of three graph convolution layers, each of which is followed by a graph coarsening layer, and one fully connected layer with a final softmax layer as shown in Fig~\\ref{fig:network:A}. Its configuration can simply be set as C(64)-P(0.25)-C(128)-P(0.25)-C(256)-P-FC(256), where C, P and FC denote the convolution, coarsening and fully connected layers respectively. The choices of hyperparameters are mainly inspired from the classic VGG net. For example, the coarsening factor is 0.25 (w.r.t. 0.5$\\times$0.5 in VGG), the attribute dimensions at three conv. layers are 64-128-256 (w.r.t. the channel numbers of conv1-3 in VGG). The scale of respective field and the number of Gaussian components are both set to 7. We train GIC network with stochastic gradient descent for roughly 300 epochs with a batch size of 100, where the learning rate is 0.1 and the momentum is 0.95. \n\nIn the bioinformatics datasets, we exploit labels and degrees of the vertices to generate initial attributes of each vertex. In the social network datasets, we use degrees of vertices. We closely follow the experimental setup in PSCN~\\cite{niepert2016learning}. We perform 10-fold cross-validation, 9-fold for training and 1-fold for testing. The experiments are repeated 10 times and the average accuracies are reported.\n\n\n\\subsubsection{Comparisons with the State-of-the-arts}\n\nWe compare our GIC with several state-of-the-arts, which contain graph convolution networks (PSCN~\\cite{niepert2016learning}, DCNN~\\cite{atwood2016diffusion}, NgramCNN~\\cite{luo2017deep}), neural networks (SAEN~\\cite{orsini2017shift}), feature based algorithms (DyF~\\cite{gomez2017dynamics}, FB~\\cite{bruna2013spectral}), random walks based methods (RW~\\cite{gartner2003graph}), graph kernel approaches (GK~\\cite{shervashidze2009efficient}, DGK~\\cite{yanardag2015deep}, WL~\\cite{morris2017glocalized}). We present the comparisons with the state-of-the-arts, as shown in Table~\\ref{table:state-of-the-art}. All results come from the related literatures. We have the following observations.\n\nDeep learning based methods on graphs (including DCNN, PSCN, NgramCNN, SAEN and ours) are superior to those conventional methods in most cases. The conventional kernel methods usually require the calculation on graph kernels with high-computational complexity. In contrast, these graph neural networks attempt to learn more abstract high-level features by performing inference-forward, which need relatively low computation cost.\n\nCompared with recent graph convolution methods, ours can achieve better performance on most datasets, such as PTC, NCI1, NCI109, ENZYMES and PROTEINS. The main reason should be that local variations of subgraphs are accurately described with Gaussian component analysis.\n\nThe proposed GIC achieves state-of-the-art results on most datasets. The best performance is gained in some bioinformatics datasets and some social network datasets including PTC, NCI1, ENZYMES, PROTEINS, COLLAB, IMDB-BINARY and IMDB-MULTI. Although NgramCNN, DyF, WL and SEAN approaches have obtained the best performance on MUTAG, REDDIT-BINARY, NCI109, REDDIT-MULTI-5K and REDDIT-MULTI-12K respectively, our method is fully comparable to them.\n\n\n\\subsection{Node Classification}\n\nFor node classification, one node is assigned one\/multiple labels. It is challenging if the label set is large. During training, we only use a fraction of nodes and their labels. The task is to predict the labels for the remaining nodes. Following the setting in~\\cite{hamilton2017inductive}, we conduct the experiments on Reddit data and PPI data. For a fair comparison to graphSAGE~\\cite{hamilton2017inductive}, we use the same initial graph data, mini-batch iterators, supervised loss function and neighborhood sample. The other network parameters are similar to graph classification except removing the coarsening layer.\n\nTabel~\\ref{table:multi-label} summarizes the comparison results. Our GIC can obtain the best performance 0.661 on PPI data and a comparable result 0.952 on Reddit data. The raw features provide an important initial information for node multi-label classification. Based on the raw features, deep walk~\\cite{perozzi2014deepwalk} improves about 0.36 (micro-F1 scores) on Reddit data. Meanwhile, we conduct an experiment of node2vec and use regression model to classification. Our method gains better performance than node2vec~\\cite{grover2016node2vec}. Comparing different aggregation methods like GCN~\\cite{kipf2016semi}, mean and LSTM, our GIC has a significant improvement about 0.16 on PPI data and gains a competitive performance on Reddit data. The results demonstrate our approach is robust to infer unknown labels of partial graphs.\n\n\n\\subsection{Model Analysis}\n\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{The verification of our convolution and coarsening.}\n\t\\label{table:convandpooling}\n\t\\begin{sc}\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\t\\multirow{2}{*}{Dataset}        & ChebNet   & GCN   & GIC  &\\multirow{2}{*}{GIC} \\\\\n\t\t\t\t& w\/ VI-GMM   & w\/ VI-GMM & w\/o VI-GMM & \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG          & 89.44 $\\pm$ 6.30      & 92.22 $\\pm$ 5.66   & 93.33 $\\pm$ 4.84  & \\textbf{94.44 $\\pm$ 4.30} \\\\\n\t\t\t\tPTC            & 68.23 $\\pm$ 6.28      & 71.47 $\\pm$ 4.75   & 68.23 $\\pm$ 4.11  & \\textbf{77.64 $\\pm$ 6.98} \\\\\n\t\t\t\tNCI1           & 73.96 $\\pm$ 1.87      & 76.39 $\\pm$ 1.08   & 79.17 $\\pm$ 1.63  & \\textbf{84.08 $\\pm$ 1.77}\t\\\\\t\t\n\t\t\t\tNCI109         & 72.88 $\\pm$ 1.85      & 74.92 $\\pm$ 1.70   & 77.81 $\\pm$ 1.88  & \\textbf{82.86 $\\pm$ 2.37} \\\\\n\t\t\t\tENZYMES        & 52.83 $\\pm$ 7.34      & 51.50 $\\pm$ 5.50   & 52.00 $\\pm$ 4.76  & \\textbf{62.50 $\\pm$ 5.12} \\\\\n\t\t\t\tPROTEINS       & 78.10 $\\pm$ 3.37      & \\textbf{80.09 $\\pm$ 3.20}    & 78.19 $\\pm$ 2.04 & 77.65 $\\pm$ 3.21 \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Comparisons on $K$ and $C_1$.}\n\t\\label{table:GMM}\n\t\\begin{sc}\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset        & $K,C_1=1$                & $K,C_1=3$                  & $K,C_1=5$                  & $K,C_1=7$       \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG         & 67.77 $\\pm$ 11.05        & 83.88 $\\pm$ 5.80           & 90.55 $\\pm$ 6.11           & \\textbf{94.44 $\\pm$ 4.30}    \\\\\n\t\t\t\tPTC           & 72.05 $\\pm$ 8.02         & 77.05 $\\pm$ 4.11           & 76.47 $\\pm$ 5.58           & \\textbf{77.64 $\\pm$ 6.98}    \\\\\n\t\t\t\tNCI1         & 71.21 $\\pm$ 1.94         & 83.26 $\\pm$ 1.17        & \\textbf{84.47 $\\pm$ 1.64}  & 84.08 $\\pm$1.77\\\\\t\t\t\t\n\t\t\t\tNCI109         & 70.02 $\\pm$ 1.57         & 81.74 $\\pm$ 1.56           & \\textbf{83.39 $\\pm$ 1.65}  & 82.86 $\\pm$ 2.37            \\\\\n\t\t\t\tENZYMES        & 33.83 $\\pm$ 4.21         & \\textbf{64.00 $\\pm$ 4.42}  & 63.66 $\\pm$ 3.85           & 62.50 $\\pm$ 5.12            \\\\\n\t\t\t\tPROTEINS       & 75.49 $\\pm$ 4.00         & 77.47 $\\pm$ 3.37           & \\textbf{78.10 $\\pm$ 2.96}  & 77.65 $\\pm$ 3.21            \\\\\n\t\t\t\t\\bottomrule\t\t\t\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Comparisons on the layer number.}\n\t\\label{table:layer}\n\t\\begin{sc}\t\n\t\t\\scalebox{0.68}{\n\t\t\t\\begin{tabular}{l c c c c}\n\t\t\t\t\\toprule\n\t\t\t\tDataset       & $N=2$                  & $N=4$                        & $N=6$                      & $N=8$          \\\\\n\t\t\t\t\\midrule\n\t\t\t\tMUTAG         & 86.66 $\\pm$ 8.31       & 91.11 $\\pm$ 5.09             & 93.88 $\\pm$ 5.80           & \\textbf{94.44 $\\pm$ 4.30}   \\\\\n\t\t\t\tPTC           & 64.11 $\\pm$ 6.55       & 74.41 $\\pm$ 6.45             & 75.29 $\\pm$ 6.05           & \\textbf{77.64 $\\pm$ 6.98}   \\\\\n\t\t\t\tNCI1          & 71.82 $\\pm$ 1.85       & 81.36 $\\pm$ 1.07             & 83.01 $\\pm$ 1.54           & \\textbf{84.08 $\\pm$ 1.77}   \\\\\n\t\t\t\tNCI109        & 71.09 $\\pm$ 2.41       & 80.02 $\\pm$ 1.67             & 81.60 $\\pm$ 1.83           & \\textbf{82.86 $\\pm$ 2.37}    \\\\\n\t\t\t\tENZYMES       & 42.33 $\\pm$ 4.22       & 61.83 $\\pm$ 5.55             & \\textbf{64.83 $\\pm$ 6.43}  & 62.50 $\\pm$ 5.12          \\\\\n\t\t\t\tPROTEINS      & 77.38 $\\pm$ 2.97       & \\textbf{79.81 $\\pm$ 3.84}    & 78.37 $\\pm$ 4.00           & 77.65 $\\pm$ 3.21          \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{sc}\n\\end{table}\n\n\\textbf{EI-GMM and VI-GMM}: To directly analyze convolution filtering with EI-GMM, we compare our method with ChebNet~\\cite{defferrard2016convolutional} and GCN~\\cite{kipf2016semi} approaches by using the same coarsening mechanism VI-GMM.  As shown in Table~\\ref{table:convandpooling}, under the same coarsening operation, our GIC is superior to ChebNet+VI-GMM and GCN+VI-GMM. It indicates EI-GMM can indeed encode the variations of subgraphs more effectively. On the other hand, we remove the coarsening layer from our GIC. For different size graphs, we pad new zero vertices into a fixed size and then concatenate attributes of all vertices for classification. As shown in this table, the performance of GIC still outperforms GIC without VI-GMM coarsening, which verifies the effectiveness of the coarsening layer VI-GMM.\n\n\\textbf{$K$ and $C_1$}: The kernel size $K$ and the number of Gaussian components $C_1$ are the most crucial parameters. Generally, the $C_1$ is proportional to the $K$. The reason is that the larger receptive field usually contains more vertices ({i.e.}} \\def\\etal{{et.al}, a relative large subgraph). Thus we simply take the equal values for them, $K=C_1 = \\{1,3,5,7\\}$. The experimental results are shown in Table~\\ref{table:GMM}. With the increase of $K, C_1$, the performance improves at most cases. The reasons are two folds: i) with increasing receptive field size, the convolution will cover the farther hopping neighbors; ii) with the increase of $C_1$, the variations of subgraphs are encoded more accurately. But for the larger values of $K$ and $C_1$ will increase the computational burden. Moreover, the overfitting phenomenon might occur with the increase of model complexity. Take the example of NCI109, in the first convolution layer, the encoded attributes (in Eqn.~(\\ref{eqn:gau_fea})) will be $2\\times39\\times7=546$ for each scale of receptive field, where $39$ is the dimension of attributes (w.r.t the number of node labels) and $7$ is the number of Gaussian components. Thus, for 7 scales of receptive field, the final encoded attributes will be $546\\times7=3822$ dimensions, which will be mapping to 64 dimensions by the function $f=[f_1,\\cdots,f_{C_1}]$ in Eqn.~(\\ref{eqn:gau_fea}). Thus the model parameter is $3822\\times64=244608$ in the first layer. Similarly, if the number of node label is 2, the model parameter will sharply decrease into $18816$. Besides, the parameter complexity is related to the number of classes and nodes. The comparison results in Table~\\ref{table:GMM} demonstrate the trend of the parameters $K$ and $C_1$ in our GIC framework.\n\n\\textbf{Number of stacked layers}: Here we test on the number of stacked network layers with $N=2,4,6,8$. When $N=2$, only one fully connected layer and one softmax layer are preserved. When $N=4$, we add two layers: the convolution layer and the coarsening layer. When continuing to stack both, the depth of network will be 6 and 8. The results are shown in Table~\\ref{table:layer}. Deeper networks can gain better performance in most cases, because the larger receptive field is observed and more abstract structures will be extracted in the topper layer. Of course, there is an extra risk of overfitting due to the increase of model complexity.\n\n\\textbf{An analysis of computation complexity}: In the convolution layer, the computational costs of receptive fields and Gaussian encoding are about $O(Km^2)$ and $O(C_1d^2)$ respectively, where $m, d$ are number of nodes and the feature dimensionality. Generally, $K=C_1\\ll d<m$. In the coarsening layer, the time complexity is about $O(pm^2+md)$, where $p$ is iteration number of the EM algorithm. In all, suppose the whole GIC alternatively stacks $n$ convolution and coarsening layers, the entire time complexity is $O(n(K+p)m^2+nC_1d^2+nmd)$.\n\n\n\\section{Conclusion}\nIn this paper, we proposed a novel Gaussian-induced convolution network to handle with general irregular graph data. Considering the previous spectral and spatial methods do not well characterize local variations of graph, we derived edge-induced GMM to adaptively encode subgraph structures by projecting them into several Gaussian components and then performing different filtering operations on each Gaussian direction like the standard CNN filters on images. Meanwhile, we formulated graph coarsening into vertex-induced GMM to dynamically partition a graph, which was also proven to be equal to graph cut. Extensive experiments in two graphic tasks (i.e. graph and node classification) demonstrated the effectiveness and superiority of our GIC compared with those baselines and state-of-the-art methods. In the future, we would like to extend our method into more applications to irregular data.\n\n\n\\section{Acknowledgments}\nThe authors would like to thank the Chairs and the anonymous reviewers for their critical and constructive comments and suggestions. This work was supported by the National Science Fund of China under Grant Nos. 61602244, 61772276, U1713208 and 61472187 and Program for Changjiang Scholars.\n \n\n\\small\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary Material}\n\\hrulefill\n\n\\vspace{4 cm}\n\n\\twocolumngrid\n\n\\section{Minimization of the perpendicular field component}\nAs evidenced in Fig.~\\ref{fig:fr_Qi_perp} (a) and (b), $B_{\\perp}$ shifts strongly the \nresonance frequency. We use this susceptibility to minimize the spurious perpendicular field \ncomponent during in-plane sweeps. For each value of $B_{\\parallel}$, we trace the phase \nresponse of the resonator at a fixed frequency (close to $f_{\\mathrm{r}}$) in a narrow field range of $B_{\\perp}$ (cf. Fig.~\\ref{fig:field_alignment}). The maximum of the phase response, corresponding to the optimal compensating $B_{\\perp}$, is determined from quadratic fit. We observe a compensation field linearly-dependent on the in-plane magnetic field, hence, a minor chip tilt with respect to Helmholtz coils is the most probable origin for the unwanted perpendicular field component. A typical compensation field of $\\approx~\\SI{1.94}{\\milli\\tesla}$ required for $B_{\\parallel}~=~\\SI{200}{\\milli\\tesla}$ (cf. Fig.~\\ref{fig:field_alignment}) corresponds to a misalignment angle below \\SI{1}{\\degree}.\n\n\\begin{figure}[htb]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {comp_field}}\n  \\caption{Typical measurement used to calibrate the compensation field $B_{\\perp}$. The phase response of the resonator at a fixed frequency close to $f_{\\mathrm{r}}$ is shown in red and the quadratic fit in blue. We choose the value of the compensation field which maximizes the phase response. For the example shown here, res. A and $B_{\\parallel}~=~\\SI{0.2}{\\tesla}$, the compensation field is $B_{\\perp}~\\approx~\\SI{1.94}{\\milli\\tesla}$.}\n  \\label{fig:field_alignment}\n\\end{figure} \n\n\\section{In-plane field sweeps along orthogonal axes}\nA comparison between the resonance frequency shift for two different in-plane magnetic field directions is shown for res. B in Fig.~\\ref{fig:B_para_comp}. Magnetic field sweeps along the resonator's axis (red) and perpendicular to the resonator's axis (gray) overlap closely, which indicates that in our case the orientation of the in-plane field does not significantly influence the superconducting properties of the film. Since the effective areas of the resonator parallel and perpendicular to the its axis are 60 $\\times$ different, the suppression of the superconductor's gap appears to be the main mechanism responsible for the measured change of kinetic inductance under in-plane magnetic field (cf. Eq.~\\ref{eq:delta_fr} in the main text).  \n\n\\begin{figure}[htb]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {B_para_comp}}\n  \\caption{Resonance frequency shift versus in-plane magnetic field for res. B (measured in the cylindrical waveguide), applied along two orthogonal directions: parallel (red) and perpendicular (gray) to the resonator's axis. The fact that the measurements overlap, for different effective side areas of the resonator exposed to in-plane field, indicates that screening currents do not play a significant role. Active compensation of the spurious perpendicular field component is performed for both sweeps, according to the procedure shown in Fig.~\\ref{fig:field_alignment}.}\n  \\label{fig:B_para_comp}\n\\end{figure} \n\n\\section{Self-Kerr effect vs. in-plane field}\n\n\\begin{figure}[hb!]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {self_Kerr}}\n  \\caption{Self-Kerr coefficient of res. A under in-plane field. (a) Resonance frequency shift vs. the average photon number, $\\overline{n}$, for $B_{||}~=~\\SI{0}{\\tesla}, \\SI{0.3}{\\tesla}, \\SI{0.5}{\\tesla}, \\SI{0.65}{\\tesla}$. The self-Kerr coefficient is extracted from the data above $\\overline{n}~\\approx~1~\\cdot~10^{5}$ (indicated by vertical black dashed line) using a linear fit, $\\Delta f_{\\mathrm{r}}~=~-K_{11}\\overline{n}$. This resonator bifurcates at $\\approx~4~\\cdot~10^{5}$ photons. (b) Fitted self-Kerr coefficient vs. $B_{||}$. }\n  \\label{fig:self_Kerr}\n\\end{figure} \n\nThe resonance frequency shift as a function of the number of circulating photons in the resonator, known as the self-Kerr effect, is measured under in-plane magnetic field for res. A (cf. Fig.~\\ref{fig:self_Kerr}~(a)).\nNote that for $B_{||}~>~\\SI{0.5}{\\tesla}$ the resonance frequency shift is no longer linear in the low photon number region. Furthermore at \\SI{0.65}{\\tesla}, $\\Delta~f_{\\mathrm{r}}$ first increases and then decreases. The exact cause of this behaviour is unclear. It can be attributed to induced superconducting fluxons, due to imperfect compensation of the perpendicular field in the end regions of the resonator, at the current nodes. These fluxons can act as quasiparticle traps\\cite{Nsanzineza2014}, which become more efficient at higher circulating power in the resonator when the mobility of the quasiparticles is higher. Trapping quasiparticles into these areas could slightly reduce the effective quasiparticle density, $n_{\\mathrm{qp}}$, and as $f_{\\mathrm{r}}~\\propto~1\/\\sqrt{L_{\\mathrm{k}}}~\\propto~1\/\\sqrt{n_{\\mathrm{qp}}}$, result in an increased $f_{\\mathrm{r}}$.\n\nThe extracted self-Kerr coefficient under in-plane magnetic field up to \\SI{650}{\\milli\\tesla} is shown in Fig.~\\ref{fig:self_Kerr}~(b). As discussed in Ref.~\\cite{Maleeva2018}, the self-Kerr coefficient of grAl is $K_{11}~\\propto~f_{\\mathrm{r}}^2\/j_{\\mathrm{c}}$, where $f_{\\mathrm{r}}$ and $j_{\\mathrm{c}}$ are the resonant frequency and the critical current density. Because both $f_{\\mathrm{r}}^{2}$ and $j_{\\mathrm{c}}$ are $\\propto~1\/L_{\\mathrm{k}}$, $K_{11}$ is expected to be field-independent\\cite{Maleeva2018}.\n\n\\section{Onset of the plastic regime under perpendicular magnetic field}\n\nThe transition of resonator A to the plastic regime (cf. Fig.~\\ref{fig:fr_Qi_perp}) is visible during three consecutive sweeps with $B_{\\perp}^{\\mathrm{max}} = \\SI{0.7}{\\milli\\tesla}, \\SI{0.8}{\\milli\\tesla}~\\mathrm{and}~\\SI{0.9}{\\milli\\tesla}$ (cf. Fig.~\\ref{fig:onset_plastic}). When $B_{\\perp}$ is ramped up to \n\\SI{0.8}{\\milli\\tesla}, the onset of the plastic regime is evidenced by a sharp drop in \nthe $Q_{\\mathrm{i}}$ for $B_{\\perp}~>~\\SI{0.7}{\\milli\\tesla}$, which is explained by fluxons permeating the film (cf. Fig.~\\ref{fig:onset_plastic} (c)).\n\n\\begin{figure}[hb!]\n  \\center{\\includegraphics[width=\\columnwidth]\n        {onset_plastic}}\n  \\caption{Measurements of $Q_{\\mathrm{i}}$ versus perpendicular magnetic field for res. A, evidencing the onset of the plastic regime during the sweeps to $B_{\\perp}^{\\mathrm{max}} = \\SI{0.7}{\\milli\\tesla}$ (a - b), $B_{\\perp}^{\\mathrm{max}} = \\SI{0.8}{\\milli\\tesla}$ (c - d) and $B_{\\perp}^{\\mathrm{max}} = \\SI{0.9}{\\milli\\tesla}$ (e - f). Horizontal blue and red arrows indicate the directions of the magnetic field sweep. \n}\n  \\label{fig:onset_plastic}\n\\end{figure}\n\n\\end{document}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nThe core task of Collaborative Filtering (CF) in recommendation is learning the representations of users and items from historical interactions, such that the cosine or Euclidean distance between two representations reflects their semantic similarity. \nHowever, one of the main challenges in learning high-quality representations in CF is the popularity bias issues in data \\cite{unfairness,Crowd,MostPop,Beyond_Parity}.\nTypically, the training data distribution is usually long-tailed, \\emph{e.g., } a few head items occupy most of the interactions, whereas the majority of tail items are unpopular and receive relatively little attention.\nDue to this imbalanced property of the training data, the resultant CF model will inherit the popularity bias and, in many cases, amplify the bias by over-recommending the head items and under-recommending the tail items. \nAs a result, the popularity bias is causing the biased representations with poor generalization ability, leading the recommendations to deviate from users' actual interests.\n\n\nMotivated by the existence of such biases, popularity debiasing methods that aim to lift the tail performance have been studied extensively. \nUnfortunately, most prevalent debiasing techniques focus on the trade-off between the head and tail evaluations, \\emph{e.g., } post-processing re-ranking \\cite{Calibration,RankALS,rescale,FPC,PC_Reg}, balanced training loss \\cite{ESAM,ALS+Reg,Regularized_Optimization,PC_Reg}, sample re-weighting \\cite{ips,IPS-C,IPS-CN,Propensity_SVM-Rank,YangCXWBE18,UBPR}, and head bias removal by causal inference \\cite{CausE,PDA,DecRS,MACR}. \nWorse still, many of them hold the assumptions that are infeasible in reality, \\emph{e.g., } the balanced test distribution is known in advance to guide the hyperparameters adjustment \\cite{DICE, MACR}, and a small unbiased subset data is existent for training unbiased model \\cite{KDCRec,CausE}. \nConsequently, these methods encounter the difficulty that tail performance rise sacrifices the head performance and leads to a severe overall performance drop.\nWe argue that the head and tail trade-off results in low-quality representations, which hardly improve the generalization ability of debiasing models.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/intro.pdf}\n    \\vspace{-5pt}\n    \\caption{Illustration of separative and discriminative representations. Dots represent item representations, while star refers to the user representation. Positive and negative samples are in red and blue, respectively.}\n    \\label{fig:intro}\n    \\vspace{-15pt}\n\\end{figure}\n\nWe conjecture that a debiased model truly improves long-tailed performance in CF by learning better user\/item representations, but not by playing a head and tail performance trade-off. \nFor the recommendation task, the user\/item is required to be mapped into not only separable but also discriminative representation space,\nsince it is impractical to pre-collect all the users' positive responses.\nTaking an implicit-feedback recommender as an example, it only leverages positive user-item interactions, such as clicks, to learn personalized user preferences, and the lack of such clicks does not indicate a negative response from the users.\nSuch a positive-unlabeled problem asks the learned representations to be discriminative and generalized enough to predict users' preferences without true labels.\nThe two critical properties for discriminative representations are high compactness for positive samples and large dispersion for negative samples, as shown in Figure \\ref{fig:intro}. \nHowever, the softmax loss \\cite{softmax} only encourage the separability to preserve maximal information \\cite{alignment_uniformity}.\n\nThe purpose of this paper, therefore, is to modify the softmax loss to a more general and highly efficient BC loss, short for Bias-margin Contrastive loss, to obtain discriminative representations in the recommendation task.\nSpecifically, we impose an interaction-wise bias-dependent angular margin between every positive user\/item pair and negative samples.\nThe size of the angular margin can be quantitatively controlled by a bias degree extractor.\nIf the user-item pair's popularity bias degree is low, which means it is a hard example, we further add a larger margin to strongly squeeze the tightness of those under-represented samples (usually inactivated users with tail items) to improve generalization. \nIn contrast, when the value of bias degree extractor is high, \\emph{e.g., } user-item interactions can be correctly predicted using only popularity information of user-item pair, we adjust the softmax loss by no margin or smaller margin to reduce the importance of these biased examples (in most cases, some head user-item pairs).\nBy optimizing BC loss, the user\/item representations become more discriminative by pulling positive representations together and pushing negative representations apart.\nAs a result, BC loss can significantly improve both head and tail performance.\n\nThe BC loss has several desirable advantages.\nFirst, it has a clear and intuitive geometric interpretation, as elaborated in Figure \\ref{fig:intro}.\nFrom a geometric view, we impose a popularity bias-related margin on a hypersphere manifold to enlarge dissimilar representations' distance.\nSecond, the interaction-wise adaptive margin forces the optimization to concentrate on the hard examples, a simple but efficient hard example mining strategy. \nThe implicit effect of the hard example mining strategy is to encourage the model to shift its attention to more informative hard samples instead of relying on biased samples.\nThird, theoretical analysis in Section 2 reveals that BC loss tends to learn a low-entropy cluster for positive samples (\\emph{e.g., } high similar item\/user compactness) and a high-entropy feature space (\\emph{e.g., } large dissimilar item\/user dispersion). \nIn this case, BC loss benefits alleviate the popularity bias as well as the general recommendation task.\n\n\nOur main contributions can be summarized as follows.\n\\begin{itemize}[leftmargin=*]\n\\item To the best of our knowledge, we are among the first to reduce the popularity bias in CF by enhancing the discriminative power of the loss function.\n\\item We devise a generalized softmax loss - BC loss, equipped with an intuitive geometric interpretation, an efficient hard example mining strategy, and a clear theoretical analysis.\n\\item Promising results demonstrate that BC loss not only alleviates popularity bias amplification effectively but also improves both head and tail performance in both imbalanced and balanced evaluations.\n\\end{itemize}\nBC loss opens up the possibility of constructing a highly discriminative loss function for the traditional recommendation.\nConsidering the theoretical guarantee and empirical effectiveness of BC loss, we recommend using BC loss not only as a debiasing technique but also as a standard loss in CF.\n\n\n\n\n\n\n\n\n\\section{Methodology}\n\n\nThe personalized item recommendation is usually based on implicit feedback \\cite{softmax} whose goal is to retrieve a subset of interesting items for a given user from a large catalog of items.\nWe consider the standard formulation of implicit feedback recommendation as a top-n ranking problem.\nGiven a historical interaction dataset $\\Set{O}^{+}=\\{(u,i)|y_{ui}=1\\}$ with $u \\in \\Space{U}$ a user, $i \\in \\Space{I}$ an item, and $y_{ui}$ indicating the user $u$ has adopted item $i$ before, a scoring function $\\hat{y}: \\Space{U}\\times \\Space{I} \\rightarrow \\Space{R}$ is optimized to reflect the preference of the user for the item. \n\nFor a single user-item pair, recommender utilizes an item encoder $\\phi: \\Space{I} \\rightarrow \\Space{R}^d$, a user encoder $\\psi: \\Space{U} \\rightarrow \\Space{R}^d$, and a similarity function $s: \\Space{R}^d \\times \\Space{R}^d \\rightarrow \\Space{R}$. \nThese functions are composed as follows: \n\\begin{equation*}\n    \\hat{y}(u,i) = s(\\psi(u), \\phi(i))\n\\end{equation*}\nThe choice of item\/user encoder, the backbone of the recommender, can be roughly categorized into ID-based (\\emph{e.g., } MF \\cite{BPR,SVD++}, NMF \\cite{NMF}, CMN \\cite{cmn}), history-based (\\emph{e.g., } SVD++ \\cite{SVD++}, FISM \\cite{FISM}, MultVAE \\cite{MultVAE}), and graph-based models (\\emph{e.g., } NGCF \\cite{NGCF}, PinSage \\cite{PinSage}, LightGCN \\cite{LightGCN}). \nMoreover, the design of similarity function $s(\\dot)$ can be dot product, cosine similarity, or more complex multilayer perceptron (MLP) \\cite{NMF}.\n\nTo the end, we conduct our experiments in the representative of the conventional backbone MF and the SOTA backbone LightGCN. \nFurthermore, a cosine similarity is used as the similarity function, suggested in \\cite{RendleKZA20}, for better interpretation, which is denoted as\n\\begin{equation*}\n    s(\\psi(u), \\phi(i))=\\frac{<\\psi(u), \\phi(i)>}{\\norm{\\psi(u)} \\cdot \\norm{\\phi(i)}} \t\\doteq \\cos{(\\hat{\\theta}_{ui})}\n\\end{equation*}\nin which $\\hat{\\theta}_{ui}$ is the angle between the representation of a user $\\psi(u)$ and an item $\\phi(i)$.\n\n\\textbf{Classical Learning Strategy.}\nThe classical learning strategy of collaborative filtering (CF) model can be classified into three groups: point-wise loss (\\emph{e.g., } binary cross-entropy \\cite{CrossEntropy}, mean square error \\cite{SVD++}), pair-wise loss (\\emph{e.g., } BPR \\cite{BPR}, WARP \\cite{WARP}) and softmax loss \\cite{softmax}. \nPoint-wise and pair-wise loss are long-standing and most-used objective functions in CF. \nHowever, large-scale studies theoretically and empirically confirm that both point-wise and pair-wise losses will inevitably propagate more information towards head user-item pairs, which causes gradient shifts and amplifies popularity bias \\cite{PC_Reg,unfairness,tang2020investigating}. \nOn the other hand, recent years have witnessed a growing academic interest in using softmax loss in recommender systems \\cite{sgl,CLRec,S3-Rec}, due to its wide success in self-supervised contrastive learning. \n\nHence, in this paper, we set sampled softmax loss as the learning objective and minimize it to train the parameters over a dataset of size $|\\Set{O}^{+}|$:\n\\begin{gather*}\n    \\min\\mathbf{\\mathop{\\mathcal{L}}}_0 = -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}}\n\\end{gather*}\nwhere $(u,i)\\in\\Set{O}^{+}$ is one observed interaction of user $u$, while $\\Set{N}_{u}=\\{j|y_{uj}=0\\}$ is the sampled unobserved items that $u$ did not interact with before; $\\tau$ is the hyper-parameter known as the temperature in softmax \\cite{tau}. \n\nAlthough, softmax loss has been outstandingly successful in various fields due to its inherent property of alignment, which assigns similar features to cohere together \\cite{alignment,WuXYL18}. \nSoftmax loss, as a widespread self-supervised representation learning loss, encourages the learned features uniformly distributed to preserve maximal information for helping downstream tasks \\cite{alignment_uniformity,SaunshiPAKK19}. \nUnfortunately, the modification of softmax loss to enhance discriminative representation as well as reduce the popularity bias has seldomly been studied. \nTherefore, the goal of this paper is to conduct a more generic and broadly applicable variant of softmax loss in CF tasks, which can fundamentally improve the long-tail performance in CF.  \n\n\\subsection{Bias-Margin Contrastive Loss}\n\n\\textbf{Popularity bias extractor.} \nHow do we know if an example exhibits popularity bias? \nHow could we quantify the degree of popularity bias for a single user-item pair?\nAlthough it is impractical to directly measure the distribution shift without accessing the test data, CF models are unlikely to predict well, given only biased information.\nSpecifically, the biased information could be user popularity statistics $p_u \\in \\Space{P}$, the number of historical items that user $u$ interacted with before, and the item popularity statistics $p_i \\in \\Space{P}$, the number of observed interactions that item $i$ is involved in. \nIf the CF models could still correctly predict the user's preference using only $p_u$ and $p_i$, it is likely due to CF models learning the popularity bias.\nMoreover, the extent of the prediction recovery will capture the degree of popularity bias.\n\nSimilarly as a standard CF model, the popularity bias extractor can be formalized as a function $\\hat{y}_b: \\Space{P}\\times \\Space{P} \\rightarrow \\Space{R}$ and composed of an item popularity encoder $\\phi_b:\\Space{P} \\rightarrow \\Space{R}_d$, a user popularity encoder $\\psi_b: \\Space{P} \\rightarrow \\Space{R}^d$, and a cosine similarity function $s: \\Space{R}^d \\times \\Space{R}^d \\rightarrow \\Space{R}$, which is defined as\n\\begin{equation*}\n    \\hat{y}_b(p_u,p_i) = s(\\psi_b(p_u), \\phi_b(p_i)) \\doteq cos(\\hat{\\xi}_{ui})\n\\end{equation*}\nin which $\\hat{\\xi}_{ui}$ is the angle between the user popularity embedding $\\psi_b(p_u)$ and item popularity embedding $\\phi_b(p_i)$.\n\nThe popularity bias extractor $\\hat{y}_b$ can be trained using sampled softmax loss $\\mathbf{\\mathop{\\mathcal{L}}}_b$:\n\\begin{gather} \\label{equ:bias_extractor}\n    \\min\\mathbf{\\mathop{\\mathcal{L}}}_b = -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\xi}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\xi}_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\xi}_{uj})\/\\tau)}}\n\\end{gather}\n\nThis optimization forces the popularity bias extractor  $\\hat{y}_b$  to reconstruct the historical interaction and assess the popularity degree from an interaction perspective.\nThose interactions that can not be predicted well using only popularity statistics are hard examples, which are more informative for representation learning.\nIn other words, the popularity bias extractor is helpful to identify the hard examples in the training set and carefully measures how hard it is for each user-item pair.\n\n\\textbf{Bias-margin Contrastive Loss (BC loss).} \nInstead of designing a weighted combination of softmax loss and a popularity bias penalty, we propose a more elegant way to learn discriminative representations by adding popularity bias-related angular margin.\nOur Bias-margin Contrastive Loss (BC loss) is formulated as:\n\\begin{gather}\\label{equ:bc_loss}\n    \\mathbf{\\mathop{\\mathcal{L}}}_{BC} = -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}}\n\\end{gather}\n\nWhere $M_{ui}$ is the interaction-wise additive angular margin corresponding to user $u$ and item $i$. \n\nWith larger $M_{ui}$, the additive angular margin becomes larger and the learning objective becomes harder.\nIntuitive, we prefer larger angular margin for the hard example, which helps the CF model concentrates on the hard examples to compensate biases, more detailed illustrations in Section \\ref{sec:three_advantages}.\n\nIn this study, we constrain the margin in a popularity bias view:\n\\begin{gather}\n    M_{ui} = \\min \\{\\hat{\\xi}_{ui}, \\pi - \\hat{\\theta}_{ui}\\}\n\\end{gather}\nWhere the purpose of the upper bound $\\pi - \\hat{\\theta}_{ui}$ is to restrict the $\\cos(\\hat{\\theta}_{ui} + \\cdot)$ to be a monotonically decreasing function corresponding to the angular margin. \n\nThe choice of bias-margin $M_{ui}$ is various. \nA series of margin selections, such as $M_{ui} = \\min\\{\\lambda \\cdot \\hat{\\xi}_{ui}, \\pi - \\hat{\\theta}_{ui}\\}$, is also reasonable, where $\\lambda$ controls the strength of the bias-margin.\nMeanwhile, by carefully constructing a monotonically decreasing similarity function, we can get ride of the restriction: $\\pi - \\hat{\\theta}_{ui}$.\nHowever, Such modifications are heuristic-based, and come at the price of increased complexity or additional hyper-parameters.\nWe encourage the reader to further explore, but will not list and compare them in this article.\n\nBC loss is extremely easy to implement in recommendation task, which only needs revise several lines of code. \nMoreover, BC loss only adds negligible computational complexity during training compared with softmax loss, but achieves better representations as clearly shown in Section \\ref{sec:three_advantages} and \\ref{sec:toy_example}. \nHence, we recommend using BC loss not only as a debiasing technique but also as a standard loss in recommender system.\n\n\n\\subsection{Three Advantages of BC Loss} \\label{sec:three_advantages}\nOur proposed BC loss has three advantages: an intuitive geometric interpretation, a novel hard example mining strategy, and clear theoretical properties.\n\n\\textbf{Geometric Interpretation.}\nWe firstly revisit the softmax loss by looking its ranking criteria. To simplify the geometric interpretation, we analyze one user $u$ with one positive item $i$ and only two negative items $j_1, j_2$. Then the posterior probabilities obtained by softmax loss are:\n\\begin{gather*}\n    \\frac{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj_1})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj_2})\/\\tau)}}\n\\end{gather*}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subcaptionbox{Softmax Loss (2D)\\label{fig:2d_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.38\\linewidth]{figures\/2d-softmax.pdf}}\n\t\\subcaptionbox{BC Loss (2D)\\label{fig:2d_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.38\\linewidth]{figures\/2d-bc.pdf}}\n\t\\subcaptionbox{Softmax Loss (3D)\\label{fig:3d_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.38\\linewidth]{figures\/3d-softmax.pdf}}\n\t\\subcaptionbox{BC Loss (3D)\\label{fig:3d_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.38\\linewidth]{figures\/3d-bc.pdf}}\n\t\\vspace{-10pt}\n\t\\caption{Geometric Interpretation of softmax loss and BC loss. The first row is the 2D hypersphere manifold for user representation constraint, and the second row is the 3D constraint. The dark red region indicates the discriminative user constraint for each loss, while the light red region is for comparison.}\n\t\\label{fig:geometric_interpretation}\n\t\\vspace{-13pt}\n\\end{figure}\n\n\nIn the training stage, the softmax loss requires $\\hat{\\theta}_{ui} > \\hat{\\theta}_{uj_1}, \\hat{\\theta}_{ui} > \\hat{\\theta}_{uj_2}$ to model the basic assumption in recommendation task.\nConcretely, if an item $i$ has been viewed by user $u$, then we assume that the user $u$ prefers the item $i$ over all other non-observed items $j_1,j_2$. \nWe will skip the analysis for BPR loss since the constraint and geometric interpretation can be easily obtained by considering one negative item.\n\nWhat if we add an angular margin to making the criteria more stringent? Intuitively, we instead requires $\\hat{\\theta}_{ui} + M_{ui} > \\hat{\\theta}_{uj_1}, \\hat{\\theta}_{ui} + M_{ui} > \\hat{\\theta}_{uj_2}$ where $M_{ui}$ is a bias-related value. By directly formulating this idea into the modified softmax loss, we have the posterior probabilities for BC loss, similar in Equation \\eqref{equ:bc_loss}:\n\n\\begin{gather*}\n    \\frac{\\exp{(\\cos(\\hat{\\theta}_{ui}+ M_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj_1})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj_2})\/\\tau)}}\n\\end{gather*}\nWe can see that BC loss is more rigor about the ranking assumption. More detailed explanations of assumption are given in Section \\ref{sec:hard_example}. \n\nThe comparison of geometric interpretation for softmax loss and BC loss is depicted in Figure \\ref{fig:geometric_interpretation}. \nAssume the learned items $i,j_1,j_2$ are given, and softmax and BC losses are optimized to the same value.\nThe constraint boundaries for correctly ranking the user's preference in Softmax are $\\hat{\\theta}_{ui} = \\hat{\\theta}_{uj_1}, \\hat{\\theta}_{ui} = \\hat{\\theta}_{uj_2}$, while in BC loss, the constraint boundaries are $\\hat{\\theta}_{ui} + M_{ui} = \\hat{\\theta}_{uj_1}, \\hat{\\theta}_{ui} + M_{ui} = \\hat{\\theta}_{uj_2}$.\nGeometrically, from softmax loss to BC loss, it is a more stringent circle-like region on the unit sphere in the 3D case, as illustrated in Figure \\ref{fig:3d_bc}.\nNote that a larger bias-related margin $M_{ui}$ leads to a smaller hypercycle-like region, which is an explicit discriminative constraint on a manifold.\nLimited constraint regions squeeze the tightness of similar items, at the same time, enlarge the dissimilar item's distance.\nAs the dimension of representations increases, BC loss has more difficult learning objective requirements, which exponentially decreases the area of constraint regions for correct ranking.\nConsequently, BC loss is gradually powerful to learn discriminative representations in recommendation tasks.\n\n\n\n\\textbf{Hard Example Mining Strategy.} \\label{sec:hard_example}\nWe argue that BC loss naturally inherits the idea of hard example mining to improve training efficiency and effectiveness. \n\nThe key for hard example mining strategy is to mine the most valuable samples.\nThe popularity bias extractor $\\hat{y}_b$, defined in Equation \\eqref{equ:bias_extractor}, can be viewed as a hard example detector.\nIf the value of popularity bias is high, which means the interaction could be predicted by using only popularity information, then the corresponding user-item pair is a biased sample.\nIn most cases, those biased samples are activated users with famous items.\nTake a news recommender system as an example; the reader is possible to click the news only because it is a top story, not because its topic is a long-term interest for that reader. \nHere, the bias-related margin $M_{ui}$ is close to zero.\nIn other words, we still make the basic assumption that the reader favors the popular news over all other unobserved news.\n\nIn contrast, the popularity bias extractor identifies the hard examples when the popularity statistics can not successfully predict the interactions.\nThose user-item pairs, in general, are inactivated users with unpopular items.\nRecall the news recommendation example; the reader clicking a piece of niche news is more likely to be attracted by interesting keywords or specific news topics.  \nThose interactions are of importance in helping the recommendation model understand user behavior.\nSpecifically, the bias margin $M_{ui}$ is significant in this situation, and we assume a more stringent assumption that the reader prefers this unpopular news by a large margin to other unclicked news.\nThe large margin makes the objective function more challenging to learn. \nAs a result, leading the model shifts its attention to hard examples.\n\n\n\\textbf{Theoretical Properties of BC Loss.}\nWe show that BC loss has compactness part and dispersion part on representation learning based on the following theorem.\n\n\\begin{theorem}\nAssuming the representations of users and items are normalized, then minimizing the BC loss is equivalent to simultaneously minimizing a compactness part and a dispersion part:\n\\begin{align*}\n    \\mathbf{\\mathop{\\mathcal{L}}}_{BC} &  \\propto \\underbrace{\\sum_{u=1}^{|\\Space{U}|}\\norm{\\Mat{v}_u-\\Mat{c}_u}^2 + \\sum_{i=1}^{|\\Space{I}|}\\norm{\\Mat{v}_i-\\Mat{c}_i}^2}_{\\text{Compactness part}}\\underbrace{-\\sum_{u=1}^{|\\Space{U}|}\\sum_{j\\in\\Set{N}_{u}}\\norm{\\Mat{v}_u-\\Mat{v}_j}}_{\\text{Dispersion part}} \\\\\n    & \\propto \\underbrace{\\mathcal{H}(\\Mat{V}|Y)}_{\\text{Compactness}}\\:\\: \\underbrace{- \\:\\: \\mathcal{H}(\\Mat{V})}_{\\text{Dispersion}}\\: = \\:- \\mathcal{I}(\\Mat{V};Y)\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof}\nThroughout this section, for simplicity, $\\Mat{v}_u \\doteq \\psi(u)$ and $\\Mat{v}_i \\doteq \\phi(i)$. \nWe use the upper-case letter $\\Mat{V} \\in \\Space{V}$ to indicate the random vector of representation, where $\\Space{V} \\subseteq \\Space{R}^d$ is the representation space.\nWe also use the normalization assumption for representations to connect cosine and Euclidean distances, \\emph{i.e., } if $\\norm{\\Mat{v}_u}=1, \\forall u$ and $\\norm{\\Mat{v}_i}=1, \\forall i$, then $\\Mat{v}_u^T\\Mat{v}_i = 1 - \\frac{1}{2} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2$. \n\nIt is clear that we can find a upper bound $m$, \\emph{s.t. } $-1 < \\cos(\\hat{\\theta}_{ui}+M_{ui}) \\leq \\Mat{v}_u^T\\Mat{v}_i - m <  1 $. \nLet $\\Set{P}_u = \\{i|y_{ui} =1\\}$ denotes the positive items set for user $u$, $\\Set{P}_i = \\{u|y_{ui} =1\\}$ refers to the positive user set for item $i$, and $\\Set{N}_{u}=\\{i|y_{ui}=0\\}$ represents the negative items set for user $u$.\nLet us start by splitting the BC loss into two terms:\n\\begin{align} \n    \\mathbf{\\mathop{\\mathcal{L}}}_{BC} & \\geq  -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{\\{(\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau}\\}}{\\exp{\\{(\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau\\}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\{(\\Mat{v}_u^T\\Mat{v}_j)\/\\tau\\}}} \\nonumber\\\\\n    & = -\\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} \\nonumber\\\\\n    & + \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\log \\{\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}\\}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\}}\\} \\label{eq:split_bc}\n\\end{align}\n\nLet $\\Mat{c}_{u} = \\frac{1}{|\\Set{P}_u|}\\sum_{i\\in \\Set{P}_{u}}\\Mat{v}_i$ denotes the hard mean of all items in the positive items set for user $u$. \nSimilarly, let $\\Mat{c}_{i} = \\frac{1}{\\Set{P}_{i}} \\sum_{u\\in\\Set{P}_{i}} \\Mat{v}_{u}$. \nLet the symbol $\\eqc$ indicates equality up to a multiplicative and\/or additive constant. \nWe first analyze the first term in Equation \\eqref{eq:split_bc} by illustrating it to a compactness term of the center loss.\n\\begin{align}\n    -\\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} \n    & =  \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\frac{\\norm{\\Mat{v}_u-\\Mat{v}_i}^2}{2\\tau} +\\frac{m-1}{\\tau} \\nonumber\\\\\n    & \\eqc \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2 \\nonumber\\\\\n    & = \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} (\\norm{\\Mat{v}_u}^2 + \\norm{\\Mat{v}_i}^2 - 2\\Mat{v}_u^T\\Mat{v}_i)\\nonumber\\\\\n    & = \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} (\\norm{\\Mat{v}_u}^2 - \\Mat{v}_u^T\\Mat{v}_i )\n    + \\sum_{i=1}^{|\\Space{I}|}\\sum_{u\\in\\Set{P}_i} (\\norm{\\Mat{v}_i}^2 - \\Mat{v}_u^T\\Mat{v}_i )\\nonumber \\\\\n    & \\eqc \\sum_{u=1}^{|\\Space{U}|}\\norm{\\Mat{v}_u-\\Mat{c}_u}^2 + \\sum_{i=1}^{|\\Space{I}|}\\norm{\\Mat{v}_i-\\Mat{c}_i}^2 \\label{eq:conpactness}\n\\end{align}\nwhere we use the properties of normalization in the first line and the definitions of $\\Mat{c}_i$ and $\\Mat{c}_u$ in the last line. \nNote that, we technically link the K-means to the first term of our BC loss in Equation \\eqref{eq:split_bc}. \n\nCombining the two term in Equation \\eqref{eq:conpactness}, we obtain:\n\\begin{align*}\n    -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} \\eqc \\sum_{\\Mat{v}\\in \\Space{V}} \\norm{\\Mat{v} - \\Mat{c}_{v}}^2\n\\end{align*}\n\nFollowing \\cite{BoudiafRZGPPA20}, we can then interpret the first term in Equation \\eqref{eq:split_bc} as a conditional cross-entropy between $\\Mat{V}$ and another random variable $\\bar{\\Mat{V}}$, whose conditional distribution given $Y$ is a standard Gaussian $\\bar{\\Mat{V}}|Y \\sim \\mathcal{N}(\\Mat{c}_{\\Mat{V}},I)$:\n\\begin{align}\n    -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} & \\eqc\n    \\mathcal{H}(\\Mat{V};\\bar{\\Mat{V}}|Y)=\\mathcal{H}(\\Mat{V}|Y)+\\mathcal{D}_{KL}(\\Mat{V}||\\bar{\\Mat{V}|Y}) \\nonumber\\\\\n    & \\propto \\mathcal{H}(\\Mat{V}|Y) \\label{eq:first-term}\n\\end{align}\nAs a consequence, minimizing the first term in Equation \\eqref{eq:split_bc} is equivalent to minimizing $\\mathcal{H}(\\Mat{V}|Y)$.\nThis concludes the proof for the first compactness part of BC loss.\n\nWe then analyze the second term in Equation \\eqref{eq:split_bc} to demonstrate its dispersion property:\n\\begin{align}\n & \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\log \\{\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}\\}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\{(\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\}}\\}\\nonumber\\\\\n & \\geq \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\log{(\n \\sum_{j\\in\\Set{N}_{u}}\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\}})} \\nonumber\\\\\n & \\geq \\sum_{u=1}^{|\\Space{U}|}\\sum_{j\\in\\Set{N}_{u}}\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\}}\\nonumber\\\\\n & \\eqc - \\sum_{u=1}^{|\\Space{U}|}\\sum_{j\\in\\Set{N}_{u}} \\norm{\\Mat{v}_u - \\Mat{v}_j}^2 \\label{eq:dispersion}\n\\end{align}\nWhere we drop the redundant term with the compactness objective in the second line, and use the Jensen's inequality in the third line. Following \\cite{WangS11}, the negative of term in Equation \\eqref{eq:dispersion} is the entropy estimator $\\hat{\\mathcal{H}}(\\Mat{V})$. Then, we know that minimizing the second term in Equation \\eqref{eq:split_bc} is equivalent to maximizing $\\mathcal{H}(\\Mat{V})$:\n\\begin{align} \\label{eq:second-term}\n         \\sum_{u=1}^{|\\Space{U}|}\\sum_{i \\in \\Set{P}_u} \\log \\{\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}\\}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\}}\\} \\propto -\\mathcal{H}(\\Mat{V})\n\\end{align}\nCombining Equation \\eqref{eq:first-term} and \\eqref{eq:second-term}, we conclude the proof.\n\\end{proof}\n\n\\textbf{Discussion.}\nThe BC loss breaks down into a compactness part and a dispersion part.\nFor the compactness part, $\\Mat{c}_{u}$ is the mean of all items that user $u$ has interaction with, which is a simple representation for the interest of user $u$.\nMoreover, $\\Mat{c}_{i}$ denotes the hard mean of all users in the positive users set for item $i$, which is the most straightforward portrait of all users who love item $i$.\nIn other words, the compactness part of BC loss encourages each user to be close to his interests while each item is tight to the portrait of all users who interact with the specific item.\nFrom the entropy view, the compactness part of BC loss tends to learn a low-entropy cluster for positive samples, \\emph{i.e., } high similar item\/user compactness.\nOn the other hand, the dispersion part of BC loss maximizes the pair-wise euclidean distance for negative samples, which is equivalent to forcing negative samples to stand far apart from one another.\nFrom the entropy view, the dispersion part enforces the spread of representations to learn a high-entropy representation space \\emph{i.e., } large dissimilar user\/item separation degree. \n\n\n\n\n\n\n\\subsection{Visualization of A Toy Experiment} \\label{sec:toy_example}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\subcaptionbox{BPR loss for (head)\\label{fig:head_bpr}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/head_bpr.pdf}}\n\t\\subcaptionbox{Softmax loss (head)\\label{fig:head_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/head_info.pdf}}\n\t\\subcaptionbox{BC loss (head)\\label{fig:head_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/head_bc.pdf}}\n\t\\subcaptionbox{IPS-CN\\cite{IPS-CN} (head) \\label{fig:head_ips}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/head_ips.pdf}}\n\t\\subcaptionbox{BPR loss (tail)\\label{fig:tail_bpr}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/tail_bpr.pdf}}\n\t\\subcaptionbox{Softmax loss (tail)\\label{fig:tail_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/tail_info.pdf}}\n\t\\subcaptionbox{BC loss (tail)\\label{fig:tail_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/tail_bc.pdf}}\n\t\\subcaptionbox{IPS-CN\\cite{IPS-CN} (tail)\\label{fig:tail_ips}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.24\\linewidth]{figures\/tail_ips.pdf}}\n\t\\vspace{-5pt}\n\t\\caption{Visualization of item representations learned with different loss functions on Yelp2018. The identical head\/tail user and the selected 500 items are drawn in (a-d)\/(e-h). The first row of each subfigure shows the 3D item representations projected on the unit sphere. The second row of each subfigure shows the angle distribution of both positive and negative items for the specific head\/tail user. \n\tBC loss encourages the majority positive items to fall into the group closest to the user and further achieves the smallest mean positive angle.\n\tSome item representations are almost clustered at one point by learning BC loss.\n\tMoreover, only BC loss shows a 0.5-radian margin for the tail user between positive and negative samples.\n\tThese phenomena verify that, compared to training with other loss functions, the BC loss simultaneously encourages similar user\/item compactness and enlarges dissimilar user\/item dispersion, strongly enhancing discriminative power for a better recommendation.}\n\t\\label{fig:toy-example}\n\t\\vspace{-8pt}\n\\end{figure*}\n\nTo intuitively visualize the effect of BC loss in recommendation task, a toy example on Yelp2018 \\cite{LightGCN} dataset is presented.\nWe modify the two-layer LightGCN by reducing the output number of the last hidden layer to three, which means the dimension of the representations is three. \nSo we can directly normalize the obtained 3-dimensional representations and plot them on a 3D unit sphere for visualization in Figure \\ref{fig:toy-example}.\nWe train the recommendation models with identical lightGCN backbone over the training dataset by minimizing different loss functions.\nFor the same head or tail user (green star), we plot 500 items' representations in the unit sphere containing all positive items (red dots) and randomly selected negative items (blue dots) from both training and testing sets. \nThe angle distribution (the second row of each subfigure) of positive and negative samples for a specific user quantitatively shows the discriminative power of each losses. \nWe observe that:\n\n\\begin{itemize}[leftmargin = *]\n    \\item \\textbf{In both head and tail user cases, BC loss learns more discriminative representations.} \n    As Figures \\ref{fig:head_bc} and \\ref{fig:tail_bc} show, for head and tail users, BC loss encourages around 40\\% and 50\\% of positive items to fall into the group closest to user representations, respectively. \n    In other words, the representations of those samples are almost clustered at one point.\n    BC loss also achieves the smallest distance \\emph{w.r.t.}~ mean positive angle.\n    This verifies that BC loss tends to learn a high similar item\/user compactness.\n    Moreover, a 0.5-radian margin is clearly shown in Figure \\ref{fig:tail_bc} between positive items and negative items, reflecting a highly discriminative power.\n    We conjecture that, compared with softmax loss performance in Figures \\ref{fig:head_softmax} and \\ref{fig:tail_softmax}, the compactness and dispersion properties of BC loss come from the incorporating of interaction-wise popularity bias-related margin.\n    \n    \\item \\textbf{The representations learned by BPR and Softmax losses are not discriminative enough.}\n    Under the supervision of BPR and Softmax losses, the item representations are separably allocated in a wide range of the unit sphere, where blue and red points occupy almost the same space area, demonstrated in Figures \\ref{fig:head_bpr} and \\ref{fig:tail_softmax}.\n    Furthermore, it seems only a negligible overlapping for angle distribution of positive and negative samples in Figure \\ref{fig:tail_bpr}.\n    However, because the number of positive and negative items is likely to be more than ten times different for the tail user, a small overlap will cause a large number of negative items to rank in front of positive items, significantly destroying the accuracy of recommendation.\n    Consequently, it is not satisfactory to directly use BPR and softmax losses for the task of personalized recommendation.\n    \n    \\item \\textbf{The IPS-CN, a well-known popularity debiasing method in CF, is prone to lift the tail performance by sacrificing the representation learning for the head.}\n    Compared with BPR loss in Figure \\ref{fig:tail_bpr}, IPS-CN learns better item representations for the tail user, which only a slight part of distribution overlap for positive and negative samples, illustrated in Figure \\ref{fig:tail_ips}.\n    However, for the head user in Figure \\ref{fig:head_ips}, the positive and negative item representations are mixed in a large space and cannot distinguish between each other.\n    This results in a huge performance drop for head evaluations; more results can be found in Experiments.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nAt the core of leading collaborative filtering (CF) models is \\wx{the learning of} high-quality representations of users and items from historical interactions.\nHowever, most CF models easily suffer from the popularity bias issue in the interaction data \\cite{unfairness,Crowd,MostPop,Beyond_Parity}.\nSpecifically, the training data distribution is typically long-tailed, \\emph{e.g., } a few head items occupy most of the interactions, whereas the majority of tail items are unpopular and receive little attention.\nThe CF models built upon the imbalanced data are prone to learn the popularity bias and even amplify it by over-recommending head items and under-recommending tail items.\nAs a result, the popularity bias causes the biased representations with poor generalization ability, making recommendations deviate from users' actual preferences.\n\n\nMotivated by concerns of popularity bias, studies on debiasing have been conducted to lift the tail performance.\nUnfortunately, most prevalent debiasing strategies focus on the trade-off between head and tail evaluations (see Table \\ref{tab:subgroups}), including post-processing re-ranking \\cite{Calibration,RankALS,rescale,FPC,PC_Reg}, balanced training loss \\cite{ESAM,ALS+Reg,Regularized_Optimization,PC_Reg}, sample re-weighting \\cite{ips,IPS-C,IPS-CN,Propensity_SVM-Rank,YangCXWBE18,UBPR}, and head bias removal by causal inference \\cite{CausE,PDA,DecRS,MACR}.\nWorse still, many of them hold some assumptions that are infeasible in practice, such as the balanced test distribution is known in advance to guide the hyperparameters' adjustment \\cite{DICE, MACR}, or a small unbiased data is present to train the unbiased model \\cite{KDCRec,CausE}.\nConsequently, \\wx{they pursue improvements on tail items but exacerbate the performance sacrifice of head items, leading to a severe overall performance drop.}\nThe trade-off between \\wx{the} head and tail evaluations \\wx{results in} suboptimal representations, which derails the generalization ability.\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\subcaptionbox{BPR loss (head)\\label{fig:head_bpr}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/head_bpr.pdf}}\n\t\\subcaptionbox{Softmax loss (head)\\label{fig:head_softmax}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/head_info.pdf}}\n    \\subcaptionbox{IPS-CN \\cite{IPS-CN} (head) \\label{fig:head_ips}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/head_ips.pdf}}\n\t\\subcaptionbox{BC loss (head)\\label{fig:head_bc}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/head_bc.pdf}}\n\n\t\\subcaptionbox{BPR loss (tail)\\label{fig:tail_bpr}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/tail_bpr.pdf}}\n\t\\subcaptionbox{Softmax loss (tail)\\label{fig:tail_softmax}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/tail_info.pdf}}\n    \\subcaptionbox{IPS-CN \\cite{IPS-CN} (tail)\\label{fig:tail_ips}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/tail_ips.pdf}}\n\t\\subcaptionbox{BC loss (tail)\\label{fig:tail_bc}}{\n\t    \\vspace{-6pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/tail_bc.pdf}}\n\n\t\\caption{Visualizations of item representations learned by LightGCN \\cite{LightGCN} on Yelp2018~\\cite{LightGCN}, where subfigures (a-d)\/(e-h) depict the identical head\/tail user as a green star, while the red and blue points denote positive and negative items, respectively. In each subfigure, the first row presents the 3D item representations projected on the unit sphere, while the second row shows the angle distribution of items \\emph{w.r.t.}~ the specific user and the statistics of mean angles.\n    Compared to other losses, BC loss learns better head representations (\\emph{cf. } with the smallest mean positive angle, the vast majority of positive items fall into the group closest to the user) and tail representations (\\emph{cf. } a clear margin exists between positive and negative items for the tail user). \\za{BC loss learns a more reasonable representation distribution that is locally clustered and globally separated.} See more details in Appendix \\ref{sec:toy_example}.}\n\t\\label{fig:toy-example}\n\t\\vspace{-10pt}\n\\end{figure*}\n\n\n\n\\wxx{\nIn this paper, we conjecture that an ideal debiasing strategy should learn high-quality head and tail representations with powerful discrimination and generalization abilities, rather than playing a trade-off game between the head and tail performance.\nHere we follow the prior studies \\cite{IPS-CN,DICE,ips,IPS-C} to focus on one key ingredient in representation learning: the loss function.\nFigure \\ref{fig:toy-example} depicts the item representations, which is optimized via two non-debiasing losses (BPR \\cite{BPR} and Softmax \\cite{SampledSoftmaxLoss}) and one debiasing loss (IPS-CN \\cite{IPS-CN}).\nWherein, representation discrimination is reflected in how well the positive items of a user are apart from the negatives.\nOur insights are:\n(1) For a user, the non-debiasing losses are inadequate to discriminate his\/her positive and negative items well, since their representations are largely overlapped as Figures \\ref{fig:head_bpr} and \\ref{fig:head_softmax} show;\n(2) Although IPS-CN achieves better discrimination power in the tail group than BPR (\\emph{cf. } positive items get smaller angles to the ego user in Figure \\ref{fig:tail_ips}, as compared to Figure \\ref{fig:tail_bpr}), it gets worse discrimination ability in the head (\\emph{cf. } positive items hold larger angles to the ego user in Figure \\ref{fig:head_ips}, as compared to Figure \\ref{fig:head_bpr}).\n}\n\n\n\n\n\n\n\nTowards this end, we incorporate \\underline{B}ias-aware margins into \\underline{C}ontrastive Loss and devise a simple yet effective \\textbf{BC Loss} to guide the head and tail representation learning of CF models.\nSpecifically, we first \\wx{employ} a bias degree extractor to \\wx{quantify the influence of interaction-wise popularity} --- that is, how well an interaction is predicted, when only popularity information of the target user and item is used.\nInteractions involving inactive users and unpopular items often \\wx{align with} lower bias degrees, indicating that popularity fails to reflect user preference faithfully.\nIn contrast, \\wx{interactions with active users and popular items are spurred by the popularity information, thus easily inclining to high bias degrees.}\nWe then move on to \\wx{train} the CF model by converting the bias degrees into the angular margins between user and item representations.\nIf the bias degree is low, we impose a larger margin to strongly squeeze the tightness of representations.\nIn contrast, if the bias degree is large, we \\wx{exert} a small or vanishing margin to reduce the influences of biased representations.\n\\wx{Through this way, for each ego user's representation, BC quantitatively controls its bias-aware margins with item representations --- adaptively intensifying the representation similarity among positive items, while diluting that among negative items.}\nBenefiting from stringent and discriminative representations, BC loss significantly improves both head and tail performance.\n\n\n\n\n\n\n\n\n\nFurthermore, BC loss has \\wx{three} desirable advantages.\nFirst, it has a clear geometric interpretation, as \\wx{illustrated} in Figure \\ref{fig:geometric_interpretation}.\nSecond, it brings forth a simple but effective mechanism of hard example mining (See Appendix \\ref{sec:hard_example}).\nThird, we theoretically reveal that BC loss tends to learn a low-entropy cluster for positive pairs (\\emph{e.g., } \\wx{compactness of matched users and items}) and a high-entropy space for negative pairs (\\emph{e.g., } \\wx{dispersion of unmatched users and items})  (See Theorem \\ref{theorem}).\nConsidering the theoretical guarantee and empirical effectiveness, \\wx{we argue that} BC loss is not only promising to alleviate popularity bias, but also suitable as \\wx{a} standard learning strategy in CF.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminary of Collaborative Filtering (CF)}\n\n\\textbf{Task Formulation.} \\label{sec:task-formulation}\nPersonalized recommendation is retrieving a subset of items from a large catalog to match user preference.\nHere we consider a \\wx{typical} scenario, collaborative filtering (CF) with implicit feedback \\cite{softmax}, which can be framed as a top-$N$ recommendation problem.\nLet $\\Set{O}^{+}=\\{(u,i)|y_{ui}=1\\}$ be the historical interactions between users $\\Set{U}$ and items $\\Set{I}$, where $y_{ui}=1$ indicates that user $u\\in\\Set{U}$ has adopted item $i\\in\\Set{I}$ before.\nOur goal is to optimize a CF model $\\hat{y}: \\Space{U}\\times\\Space{I}\\rightarrow\\Space{R}$ that latches on user preference towards items.\n\n\n\n\\textbf{Modeling Scheme.}\nScrutinizing leading CF models \\cite{BPR,LightGCN,SVD++,MultVAE}, we systematize the common paradigm as a combination of three modules: user encoder $\\psi(\\cdot)$, item encoder $\\phi(\\cdot)$, and similarity function $s(\\cdot)$.\nFormally, we depict one CF model as $\\hat{y}(u,i) = s(\\psi(u), \\phi(i))$,\nwhere $\\psi:\\Space{U}\\rightarrow\\Space{R}^d$ and $\\phi:\\Space{I}\\rightarrow\\Space{R}^d$ encode the identity (ID) information of user $u$ and item $i$ into $d$-dimensional representations, respectively;\n$s:\\Space{R}^d \\times\\Space{R}^d\\rightarrow\\Space{R}$ measures the similarity between user and item representations.\nIn literature, there are various choices of encoders and similarity functions:\n\\begin{itemize}[leftmargin=*]\n    \\item Common encoders roughly fall into three groups: ID-based (\\emph{e.g., } MF \\cite{BPR,SVD++}, NMF \\cite{NMF}, CMN \\cite{cmn}), history-based (\\emph{e.g., } SVD++ \\cite{SVD++}, FISM \\cite{FISM}, MultVAE \\cite{MultVAE}), and graph-based (\\emph{e.g., } GCMC \\cite{GCMC}, PinSage \\cite{PinSage}, LightGCN \\cite{LightGCN}) fashions.\n    Here we select two high-performing encoders, MF and LightGCN, as the backbone models being optimized.\n\n    \\item The widely-used similarity functions include dot product \\cite{BPR}, cosine similarity \\cite{RendleKZA20}, and neural networks \\cite{NMF}.\n    As suggested in the recent study \\cite{RendleKZA20}, cosine similarity is a simple yet effective and efficient similarity function in CF models, having achieved strong performance.\n    For better interpretation, we take a geometric view and denote it by:\n    \\begin{gather}\\label{eq:cosine-similarity}\n        s(\\psi(u), \\phi(i))=\\frac{\\Trans{\\psi(u)}\\phi(i)}{\\norm{\\psi(u)} \\cdot \\norm{\\phi(i)}}\\doteq \\cos{(\\hat{\\theta}_{ui})},\n    \\end{gather}\n    in which $\\hat{\\theta}_{ui}$ is the angle between the user representation $\\psi(u)$ and item representation $\\phi(i)$.\n\\end{itemize}\n\n\n\n\n\n\\textbf{Learning Strategy.}\nTo optimize the model parameters, CF models mostly frame the top-$N$ recommendation problem into a supervised learning task, and resort to one of three classical learning strategies:\npointwise loss (\\emph{e.g., } binary cross-entropy \\cite{CrossEntropy}, mean square error \\cite{SVD++}),\npairwise loss (\\emph{e.g., } BPR \\cite{BPR}, WARP \\cite{WARP}), and softmax loss \\cite{softmax}.\nAmong them, pointwise and pairwise losses are long-standing and widely-adopted objective functions in CF.\nHowever, extensive studies \\cite{PC_Reg,unfairness,tang2020investigating} have analytically and empirically confirmed that using pointwise or pairwise loss is prune to propagate more information towards the head user-item pairs, which\namplifies popularity bias.\n\n\n\nSoftmax loss is much less explored in CF than its application in other domains like CV \\cite{ArcFace,SphereFace}.\nRecent studies \\cite{RendleKZA20,CLRec,S3-Rec,DBLP:conf\/www\/Lian0C20,DBLP:conf\/recsys\/YiYHCHKZWC19} find that it inherently conducts \\wx{hard example mining} over multiple negatives and aligns well with the ranking metric, thus \\wx{attracting a surge of interest} in recommendation.\nHence, we cast the minimization of softmax loss \\cite{SampledSoftmaxLoss} as the representative learning strategy:\n\\begin{gather}\n    \\mathbf{\\mathop{\\mathcal{L}}}_0 = -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}},\n\\end{gather}\nwhere $(u,i)\\in\\Set{O}^{+}$ is one observed interaction of user $u$, while $\\Set{N}_{u}=\\{j|y_{uj}=0\\}$ is the set of sampled unobserved items that $u$ did not interact with before; $\\tau$ is the hyper-parameter known as the temperature in softmax \\cite{tau}.\nNonetheless, modifying softmax loss to enhance the discriminative power of representations and alleviate the popularity bias remains largely unexplored.\nTherefore, \\wx{our work aims to} devise a more generic and broadly-applicable variant of softmax loss for CF tasks, which can improve the long-tail performance fundamentally.\n\\section{Methodology of BC Loss}\nOn the basis of softmax loss, we devise our BC loss and present its desirable characteristics.\n\n\n\\subsection{{Popularity Bias Extractor}}\nBefore mitigating popularity bias, we need to quantify the influence of popularity bias on a single user-item pair.\nOne straightforward solution is to compare the performance difference between the biased and unbiased evaluations.\n\\wx{However, this is not feasible as} the unbiased data is usually unavailable in practice.\nStatistical metrics of popularity could be a reasonable proxy of the biased information, such as user popularity statistics $p_{u}\\in\\Space{P}$ (\\emph{i.e., } the number of historical items that user $u$ has interacted with before) and item popularity statistics $p_{i}\\in\\Space{P}$ (\\emph{i.e., } the number of observed interactions that item $i$ is involved in).\nIf \\wx{the impact of} the interaction between $u$ and $i$ can be captured well based solely on such statistics, the model is susceptible to exploiting popularity bias for prediction.\nHence, we argue that the popularity-only prediction will delineate the influence of bias.\n\n\n\nTowards this end, we first train an additional module, termed popularity bias extractor, which only takes the popularity statistics as input to make prediction.\n\\wx{Similar} to the modeling of CF (\\emph{cf. } Section \\ref{sec:task-formulation}), the bias extractor is formulated as a function $\\hat{y}_b:\\Space{P}\\times\\Space{P}\\rightarrow\\Space{R}$:\n\\begin{gather}\\label{eq:bias-prediction}\n    \\hat{y}_b(p_u,p_i) = s(\\psi_b(p_u), \\phi_b(p_i)) \\doteq cos(\\hat{\\xi}_{ui}),\n\\end{gather}\nwhere the user popularity encoder $\\psi_b:\\Space{P}\\rightarrow\\Space{R}^d$ and the item popularity encoder $\\phi_b:\\Space{P}\\rightarrow\\Space{R}^d$ map the popularity statistics of user $u$ and item $i$ into $d$-dimensional popularity embeddings $\\psi_b(p_u)$ and $\\phi_b(p_i)$, respectively;\n$s:\\Space{R}^d\\times\\Space{R}^d\\rightarrow\\Space{R}$ is \\wx{the cosine similarity function} between popularity embeddings (\\emph{cf. } Equation \\eqref{eq:cosine-similarity}).\n$\\hat{\\xi}_{ui}$ is the angle between $\\psi_b(p_u)$ and $\\phi_b(p_i)$.\n\nWe then minimize the following softmax loss to optimize the popularity bias extractor:\n\\begin{gather} \\label{eq:bias-extractor}\n    \\mathbf{\\mathop{\\mathcal{L}}}_b = -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\xi}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\xi}_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\xi}_{uj})\/\\tau)}}.\n\\end{gather}\nThis optimization enforces the extractor to reconstruct the historical interactions \\za{using only biased information (\\emph{i.e., } popularity statistics)} and makes the reconstruction reflect the interaction-wise bias degree.\n\\za{As shown in Appendix \\ref{sec:bias_extractor}, interactions with active users and popular items are inclining to learn well via Equation \\eqref{eq:bias-extractor}.}\nFurthermore, we can distinguish hard interactions based on the bias degree, \\wx{\\emph{i.e., } the interactions that can be hardly predicted by popularity statistics} ought to be more informative for representation learning in the target CF model.\nIn a nutshell, the popularity bias extractor underscores the bias degree of each user-item interaction, which substantively reflects how hard it is to be predicted.\n\n\n\\subsection{BC Loss}\nWe move on to \\wx{devise} a new BC loss for the target CF model.\nOur BC loss stems from softmax loss but converts the interaction-bias degrees into the bias-aware angular margins among the representations to enhance the discriminative power of representations.\nOur BC loss is:\n\\begin{align}\\label{equ:bc_loss}\n    \\mathbf{\\mathop{\\mathcal{L}}}_{\\text{BC}} =\n    -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}},\n\\end{align}\nwhere $M_{ui}$ is the bias-aware angular margin for the interaction $(u,i)$ defined as:\n\\begin{gather}\n    M_{ui} = \\min \\{\\hat{\\xi}_{ui}, \\pi - \\hat{\\theta}_{ui}\\}\n\\end{gather}\nwhere $\\hat{\\xi}_{ui}$ is derived from the popularity bias extractor (\\emph{cf. } Equation \\eqref{eq:bias-prediction}), and $\\pi - \\hat{\\theta}_{ui}$ is the upper bound to restrict $\\cos(\\cdot + M_{ui})$ to be a monotonically decreasing function.\nIntuitively, if a user-item pair $(u,i)$ is the hard interaction \\wx{that can hardly be} reconstructed by its popularity statistics, it holds a high value of $\\hat{\\xi}_{ui}$ and leads to a high value of $M_{ui}$; henceforward, BC loss imposes the large angular margin $M_{ui}$ between the negative item $j$ and positive item $i$ and optimizes the representations of user $u$ and item $i$ to lower $\\hat{\\xi}_{ui}$.\nSee more details and analyses in Section \\ref{sec:three_advantages}.\n\n\\wx{It is noted that} BC loss is extremely easy to implement in recommendation tasks, which only needs to revise several lines of code.\nMoreover, compared with softmax loss, BC loss only adds negligible computational complexity during training \n\\za{(\\emph{cf. } Table \\ref{tab:elapse_time})}\nbut achieves more discriminative representations. \nHence, we recommend \\wx{to use} BC loss not only as a debiasing strategy to alleviate the popularity bias, but also as a standard loss in recommender models to enhance the discriminative power.\nNote that the modeling of $M_{ui}$ is worth exploring, such as the more complex version $M_{ui} = \\min\\{\\lambda \\cdot \\hat{\\xi}_{ui}, \\pi - \\hat{\\theta}_{ui}\\}$ where $\\lambda$ controls the strength of the bias-margin.\nMeanwhile, carefully designing a monotonically decreasing function helps to get rid of the upper bound restriction.\nWe will leave the exploration of bias-margin in future work.\n\n\n\\section{Analyses of BC Loss}\\label{sec:three_advantages}\n\nWe analyze desirable characteristics of BC loss.\nSpecifically, we start by presenting its geometric interpretation, and then show its theoretical properties \\emph{w.r.t.}~ compactness and dispersion of representations. The hard mining mechanism of BC loss is discussed in Appendix \\ref{sec:hard_example}. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\subcaptionbox{Softmax Loss (2D)\\label{fig:2d_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/2d-softmax.pdf}}\n\t\\subcaptionbox{BC Loss (2D)\\label{fig:2d_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/2d-bc.pdf}}\n\t\\subcaptionbox{Softmax Loss (3D)\\label{fig:3d_softmax}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/3d-softmax.pdf}}\n\t\\subcaptionbox{BC Loss (3D)\\label{fig:3d_bc}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.23\\linewidth]{figures\/3d-bc.pdf}}\n\n\t\\caption{Geometric Interpretation of softmax loss and BC loss in 2D and 3D hypersphere. \n\n\tThe dark red region indicates the discriminative user constraint, while the light red region is for comparison.}\n\t\\label{fig:geometric_interpretation}\n\t\\vspace{-10pt}\n\\end{figure}\n\n\\subsection{\\textbf{Geometric Interpretation}}\nHere we probe into the ranking criteria of softmax loss and BC loss, from the geometric perspective.\nTo simplify the geometric interpretation, we analyze one user $u$ with one observed item $i$ and only two unobserved items $j$ and $k$.\nThen the posterior probabilities obtained by softmax loss are: $\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uk})\/\\tau)}}$.\nDuring training, softmax loss encourages the ranking criteria $\\hat{\\theta}_{ui} < \\hat{\\theta}_{uj}$ and $\\hat{\\theta}_{ui} < \\hat{\\theta}_{uk}$ to model the basic assumption that the observed interaction $(u,i)$ indicates more positive cues of user preference than the unobserved interactions $(u,j)$ and $(u,k)$.\n\nIntuitively, to make the ranking criteria more stringent, we can impose an angular margin $M_{ui}$ on it and establish a new criteria $\\hat{\\theta}_{ui} + M_{ui} < \\hat{\\theta}_{uj}$ and $\\hat{\\theta}_{ui} + M_{ui} < \\hat{\\theta}_{uk}$.\nDirectly formulating this idea arrives at the posterior probabilities of BC loss: $\\frac{\\exp{(\\cos(\\hat{\\theta}_{ui}+ M_{ui})\/\\tau)}}{\\exp{(\\cos(\\hat{\\theta}_{ui}+M_{ui})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uj})\/\\tau)}+\\exp{(\\cos(\\hat{\\theta}_{uk})\/\\tau)}}$.\nObviously, BC loss is more rigorous about the ranking assumption compared with softmax loss. See Appendix \\ref{sec:hard_example} for more detailed explanations.\n\n\nWe then depict the geometric interpretation and comparison of softmax loss and BC loss in Figure \\ref{fig:geometric_interpretation}.\nAssume the learned representations of $i$, $j$, and $k$ are given, and softmax and BC losses are optimized to the same value.\nIn softmax loss, the constraint boundaries for correctly ranking user $u$'s preference are $\\hat{\\theta}_{ui} = \\hat{\\theta}_{uj}$ and $\\hat{\\theta}_{ui} = \\hat{\\theta}_{uk}$; whereas, in BC loss, the constraint boundaries are $\\hat{\\theta}_{ui} + M_{ui} = \\hat{\\theta}_{uj}$ and $\\hat{\\theta}_{ui} + M_{ui} = \\hat{\\theta}_{uk}$.\nGeometrically, from softmax loss (\\emph{cf. } Figure \\ref{fig:3d_softmax}) to BC loss (\\emph{cf. } Figure \\ref{fig:3d_bc}), it is a more stringent circle-like region on the unit sphere in the 3D case.\nFurther enlarging the margin $M_{ui}$ will lead to a smaller hyperspherical cap, which is an explicit discriminative constraint on a manifold.\nAs a result, limited constraint regions squeeze the tightness of similar items \\wx{and encourages the separation of dissimilar items.}\nMoreover, with the increase of representation dimension, BC loss has more restricted learning requirements, exponentially decreasing the area of constraint regions for correct ranking, and becomes progressively powerful to learn discriminative representations.\n\n\n\n\n\n\n\n\\subsection{\\textbf{Theoretical Properties}}\nBC loss improves head and tail representation learning by enforcing the compactness of matched users and items, while imposing the dispersion of unmatched users and items. See detailed proof in Appendix \\ref{sec:proof}.\n\n\n\n\n\n\\begin{restatable}{theorem}{bclosstheorem} \\label{theorem}\nLet $\\Mat{v}_u \\doteq \\psi(u)$, $\\Mat{v}_i \\doteq \\phi(i)$, and $\\Mat{c}_{u} = \\frac{1}{|\\Set{P}_u|}\\sum_{i\\in \\Set{P}_{u}}\\Mat{v}_i$, $\\Mat{c}_{i} = \\frac{1}{|\\Set{P}_{i}|} \\sum_{u\\in\\Set{P}_{i}} \\Mat{v}_{u}$, where $\\Set{P}_u = \\{i|y_{ui} =1\\}$ and $\\Set{N}_{u}=\\{i|y_{ui}=0\\}$ are the sets of user $u$'s positive and negative items, respectively; $\\Set{P}_i = \\{u|y_{ui} =1\\}$ is the set of item $i$'s positive users.\nAssuming the representations of users and items are normalized, the minimization of BC loss is equivalent to minimizing a compactness part and a dispersion part simultaneously:\n\\begin{align}\n    \\mathbf{\\mathop{\\mathcal{L}}}_{BC} \\geq \\underbrace{\\sum_{u\\in\\Set{U}}\\norm{\\Mat{v}_u-\\Mat{c}_u}^2 + \\sum_{i\\in\\Set{I}}\\norm{\\Mat{v}_i-\\Mat{c}_i}^2}_{\\text{Compactness part}}\\underbrace{-\\sum_{u\\in\\Set{U}}\\sum_{j\\in\\Set{N}_{u}}\\norm{\\Mat{v}_u-\\Mat{v}_j}^2}_{\\text{Dispersion part}}\n    \\propto \\underbrace{H(\\Mat{V}|Y)}_{\\text{Compactness}}\\:\\: \\underbrace{-H(\\Mat{V})}_{\\text{Dispersion}}.\n\\end{align}\n\\end{restatable}\n\n\n\\noindent\\textbf{Discussion.}\n$\\Mat{c}_{u}$ is the averaged representations of all items that $u$ has interacted with, which describes $u$'s interest; similarly, $\\Mat{c}_{i}$ profiles item $i$'s user group.\nFor the compactness part, BC loss forces the user's positive items to be user-centric and vice versa.\n\\wx{From} the entropy perspective, compactness part tends to learn a low-entropy cluster for positive interactions, \\emph{i.e., } high compactness for similar users and items.\nFor the dispersion part, for users and items from unobserved interactions, BC loss maximizes the pairwise euclidean distance between their representations and encourages them to be distant from each other;\n\\wx{Hence, from} the entropy viewpoint, dispersion part levers the spread of representations to learn a high-entropy representation space, \\emph{i.e., } large separation degree for dissimilar users and items.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\nWe aim to answer the following research questions:\n\\begin{itemize}[leftmargin=*]\n   \\item \\textbf{RQ1:} How does BC Loss perform compared with debiasing strategies in various evaluations?\n   \\wx{\\item \\textbf{RQ2:} Does BC loss cause the trade-off between head and tail performance?}\n   \\item \\textbf{RQ3:} What are the impacts of the components (\\emph{e.g., } temperature, margin) on BC Loss?\n\\end{itemize}\n\n\\noindent\\textbf{Baselines \\& Datasets.} \nSOTA debiasing strategies in various research lines are compared: sample re-weighting (IPS-CN \\cite{IPS-CN}), bias removal by causal inference (MACR \\cite{MACR}, CausE \\cite{CausE}), and regularization-based framework (sam+reg \\cite{Regularized_Optimization}).\nExtensive experiments are conducted on \\za{eight} real-world benchmark datasets: Tencent \\cite{Tencent}, Amazon-Book \\cite{Amazon-Book}, Alibaba-iFashion \\cite{Alibaba-ifashion}, Yelp2018 \\cite{LightGCN}, Douban Movie \\cite{Douban}, \\za{Yahoo!R3 \\cite{Yahoo}, Coat \\cite{ips}} and KuaiRec \\cite{KuaiRec}.\nFor comprehensive comparisons, almost all standard test distributions in CF are covered in the experiments: balanced test set \\cite{MACR,DICE,KDCRec}, randomly selected imbalanced test set \\cite{ESAM,ZhuWC20}, temporal split test set \\cite{PDA,DecRS,Regularized_Optimization}, and unbiased test set \\cite{ips,KuaiRec,Yahoo}.\nSee more experiments on KuaiRec, \\za{Yahoo!R3, and Coat} for unbiased test evaluation in Appendix \\ref{sec:KuaiRec} and \\za{more comparison results between BC loss and other standard losses (most widely used BPR \\cite{BPR}, newest proposed CCL \\cite{CCL} and SSM \\cite{SSM}) in Appendix \\ref{sec:standard_loss}.}\n\n\\begin{table*}[t]\n\\caption{Overall debiasing performance comparison in balanced and imbalanced test sets. \n}\n\\vspace{-5pt}\n\\label{tab:overall}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|cc|cc|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{1}{c|}{} & \\multicolumn{4}{c|}{Tencent}       & \\multicolumn{4}{c|}{Amazon-book}   & \\multicolumn{4}{c}{Alibaba-iFashion} \\\\\n\\multicolumn{1}{c|}{} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c|}{Imbalanced} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c|}{Imbalanced} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c}{Imbalanced} \\\\\n \n\\multicolumn{1}{c|}{} & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall  & NDCG    & Recall  & NDCG   \\\\\n\\midrule\nMostPop   &  0.0002     &   0.0002 &  0.0384     &    0.0208    &     0.0001   &      0.0001  & 0.0102    &    0.0063    &      0.0003      &  0.0001   & 0.0212     &   0.0084      \\\\\\midrule\nMF                   & 0.0052 & 0.0040 & \\underline{0.0982} & \\underline{0.0643} & 0.0109 & 0.0103 & \\underline{0.0856} & \\underline{0.0638} & 0.0056  & 0.0028  & \\underline{0.0843}  & \\underline{0.0411} \\\\\n+ IPS-CN             & \\underline{0.0075} & \\underline{0.0058} & 0.0686 & 0.0421 & 0.0132 & 0.0123 & 0.0765 & 0.0554 & 0.0050  & 0.0027  & 0.0551  & 0.0255 \\\\\n+ CausE              & 0.0056 & 0.0043 & 0.0687 & 0.0468 & 0.0115 & 0.0105 & 0.0720 & 0.0551 & 0.0005  & 0.0003  & 0.0185  & 0.0086 \\\\\n+ sam+reg &  0.0070 &  0.0054  & 0.0406 & 0.0266 &  0.0141  & 0.0132  & 0.0599  &   0.0443  &  0.0067    &  0.0032       &      0.0305   &  0.0146      \\\\\n+ MACR               & 0.0067 & 0.0046 & 0.0326 & 0.0241 & \\underline{0.0181} & \\underline{0.0146} & 0.0292 & 0.0229 & \\underline{0.0086}  & \\underline{0.0041}  & 0.0650  & 0.0331 \\\\\n+ BC Loss            & \\textbf{0.0087}* & \\textbf{0.0068}* & \\textbf{0.1298}* & \\textbf{0.0904}* & \\textbf{0.0221}* & \\textbf{0.0202}* & \\textbf{0.1198}* & \\textbf{0.0948}* & \\textbf{0.0095}*  & \\textbf{0.0048}*  & \\textbf{0.0967}*  & \\textbf{0.0487}* \\\\\\midrule\nImp. \\%              &  16.0\\%  &  17.2\\%      &    32.2\\%  &  40.1\\%      &   22.1\\%     &    38.4\\%    &    40.0\\%    &  49.6\\%      &    10.5\\%     &    17.1\\%     &    14.7\\%     & 18.5\\%        \\\\\\midrule\\midrule\nLightGCN             & 0.0055 & 0.0042 & \\underline{0.1065} & \\underline{0.0712} &0.0123 & 0.0116 & \\underline{0.0941} & \\underline{0.0724} & 0.0036  & 0.0017  & \\underline{0.0660}  & \\underline{0.0322} \\\\\n+ IPS-CN             & 0.0072 & 0.0054 & 0.0900 & 0.0599 & 0.0148 & 0.0136 & 0.0836 & 0.0639 & 0.0038  & 0.0017  & 0.0658  & 0.0317 \\\\\n+ CausE              & 0.0055 & 0.0040 & 0.0966 & 0.0665 & 0.0134 & 0.0121 & 0.0926 & 0.0717 & 0.0029  & 0.0013  & 0.0449  & 0.0221 \\\\\n+ sam+reg & \\underline{0.0076}  & \\underline{0.0056}  & 0.0653  & 0.0436  &   0.0157 & 0.0149  &  0.0773  &  0.0600  &    \\underline{0.0056} & \\underline{0.0027}    & 0.0502  &  0.0252 \\\\\n+ MACR               & 0.0075 & 0.0050 & 0.0731 & 0.0532 & \\underline{0.0183} & \\underline{0.0153} & 0.0767 & 0.0600 & 0.0033  & 0.0015  & 0.0475  & 0.0238 \\\\\n+ BC Loss            & \\textbf{0.0095}* & \\textbf{0.0073}* & \\textbf{0.1194}* & \\textbf{0.0832}* & \\textbf{0.0257}* & \\textbf{0.0227}* & \\textbf{0.1123}* & \\textbf{0.0903}* & \\textbf{0.0077}*  & \\textbf{0.0037}*  & \\textbf{0.0992}*  & \\textbf{0.0510}* \\\\\\midrule\nImp. \\%              &   25.0\\%     &  30.1\\%      &   12.1\\%     &    16.9\\%    &  40.4\\%      &    48.4\\%    &   19.3\\%     &   24.7\\%     &   37.5\\%      &   37.0\\%      &   50.3\\%      &   58.4\\%    \\\\\n\\bottomrule\n\\end{tabular}}\n\\vspace{-15pt}\n\\end{table*}\n\n\n\n\\subsection{Performance Comparison (RQ1)}\n\n\\subsubsection{\\textbf{Evaluations on Imbalanced and Balanced Test Sets}}\n\\noindent\\textbf{Motivation.}\nMany prevalent debiasing methods assume that test distribution is known in advance \\cite{MACR,DICE,ESAM}, \\emph{i.e., } the validation set has similar distribution with the test set. \nMoreover, only an imbalanced or balanced test set is evaluated.\nHowever, in real-world applications, the test distributions are usually unavailable and can even reverse the prior in the training distribution.\nWe conjecture that a good debiasing recommender is required to perform well on both imbalanced and balanced test distributions. \nIn our settings, no information about the balanced test is provided in advance.\n\n\\noindent\\textbf{Data Splits.}\nThe models are identical across both imbalanced and balanced evaluations.\nThe test distribution in the balanced evaluation is uniform, \\emph{i.e., } randomly sample $15\\%$ of interactions with equal probability \\emph{w.r.t.}~ items.\nBesides, the test splits for the imbalanced test are similarly long-tailed like the train and validation sets, \\emph{i.e., } randomly split the remaining interactions into training, validation, and imbalanced test sets ($60\\%:10\\%:15\\%$).\n\n\\noindent\\textbf{Results.} Table \\ref{tab:overall} reports the comparison of performance in imbalanced and balanced test evaluations.\nThe best performing methods are bold and starred, while the strongest baselines are underlined; Imp.\\% measures the relative improvements of BC loss over the strongest baselines.\nWe observe that:\n\\begin{itemize} [leftmargin = *]\n\\item \\textbf{BC loss significantly outperforms the state-of-the-art baselines \nin both balanced and imbalanced evaluations across all datasets.}\nIn particular, it achieves consistent improvements over the best debiasing baselines and original CF models by $12.1\\%\\sim 58.4\\%$.\nThis clearly demonstrates that BC loss not only effectively alleviates the amplification of popularity bias but also improves the discriminative power of representations.\n\\wx{Moreover, Table \\ref{tab:elapse_time} shows the computational costs of all methods. Compared to the backbone models, BC loss only adds negligible time complexity.}\n\n\\item \\textbf{Debiasing baselines sacrifice the imbalanced performance and perform inconsistently across datasets.}\nDebiasing strategies generally achieve higher balanced performance at the expense of a large imbalanced performance drop. \nSpecifically, the strongest baselines over all imbalanced test sets are \\wx{the} original CF models.\nWorse still, as the degree of data sparsity increases, some debiasing methods fail to quantify the popularity bias and limit their bias removal ability.  \nFor example, in the sparest Alibaba-iFashion dataset, the results of MF+IPS-CN, MF+CausE, LightGCN+MACR, and LightGCN+CausE on the balanced evaluation are lower than original CF models (MF or LightGCN).\nIn contrast, benefiting from popularity bias-aware margin, BC loss can learn discriminative representations that accomplish more profound user and item understanding, leading to higher head and tail recommendation quality.\n\\end{itemize}\n\n\n\\subsubsection{\\textbf{Evaluations on Temporal Split Test Set}}\n\\noindent\\textbf{Motivation.}\n\\begin{wraptable}{r}{0.6\\columnwidth}\n  \\centering\n  \\vspace{-25pt}\n  \\caption{The performance comparison on Douban dataset. }\n  \\label{tab:time_split}\n  \\vspace{-5pt}\n  \\resizebox{0.6\\columnwidth}{!}{\n  \\begin{tabular}{l|ccc|ccc}\n  \\toprule\n   & \\multicolumn{3}{c|}{MF} & \\multicolumn{3}{c}{LightGCN} \\\\\n  \\multicolumn{1}{c|}{} & HR & Recall & NDCG & HR & Recall & NDCG \\\\\\midrule\n  Backbone & \\underline{0.2924} & \\underline{0.0294} & \\underline{0.0472} & \\underline{0.3543} & \\underline{0.0313} & \\underline{0.0602} \\\\\n  + IPS-CN & 0.2514 & 0.0174 & 0.0324 & 0.3212 & 0.0261 & 0.0502 \\\\\n  + CausE & 0.2725 & 0.0203 & 0.0376 & 0.3403 & 0.0275 & 0.0514 \\\\\n  + sam+reg & 0.2826 & 0.0191 & 0.0390 & 0.2944 & 0.0252 &  0.0488 \\\\\n  + MACR & 0.1084 & 0.0087 & 0.0163 & 0.3127 & 0.0271 & 0.0519 \\\\\n  + BC loss & \\textbf{0.3742}* & \\textbf{0.0324}* & \\textbf{0.0601}* & \\textbf{0.3562}* & \\textbf{0.0346}* & \\textbf{0.0652}* \\\\\\midrule\n  Imp. \\% & 28.0\\% & 10.2\\% & 27.3\\% & 0.5\\% & 10.4\\% & 8.3\\% \\\\\\bottomrule\n  \\end{tabular}}\n  \\vspace{-15pt}\n\\end{wraptable}\nIn real applications, popularity bias dynamically changes over time.\nHere we consider temporal split test evaluation on Douban Movie where the historical interactions are sliced into the training, validation, and test sets (7:1:2) according to the timestamps.\n\n\n\\noindent\\textbf{Results.}\nAs Table \\ref{tab:time_split} shows, BC loss is steadily superior to all baselines \\emph{w.r.t.}~ all metrics on Douban Movie.\nFor instance, it achieves significant improvements over the MF and LightGCN backbones \\emph{w.r.t.}~ Recall@20 by 10.2\\% and 10.4\\%, respectively.\nThis validates that BC loss endows the backbone models with better robustness against the popularity distribution shift and alleviates the negative influence of popularity bias.\nSurprisingly, none of the debiasing baselines could maintain a comparable performance to the backbones.\nWe ascribe the failure to their preconceived idea of tail items, which possibly change over time.\n\n\n\n\n\n\n\\begin{table*}[t]\n  \\caption{The performance evaluations of head, mid, and tail on Tencent dataset. \n \n  }\n  \\label{tab:subgroups}\n  \\vspace{-5pt}\n  \\resizebox{\\textwidth}{!}{\\begin{tabular}{l|llll|llll}\n  \\toprule\n  \\multicolumn{1}{c|}{} & \\multicolumn{4}{c|}{Balanced NDCG@20} & \\multicolumn{4}{c}{Imbalanced NDCG@20} \\\\\n  \\multicolumn{1}{c|}{} & \\multicolumn{1}{c}{Tail} & \\multicolumn{1}{c}{Mid} & \\multicolumn{1}{c}{Head} & \\multicolumn{1}{c|}{Overall} & \\multicolumn{1}{c}{Tail} & \\multicolumn{1}{c}{Mid} & \\multicolumn{1}{c}{Head} & \\multicolumn{1}{c}{Overall} \\\\\\midrule\n  MF & 0.00004 & 0.00097 & 0.01250 & 0.00402 & 0.00021 & 0.00197 & 0.06837 & 0.06431 \\\\\n  + IPS-CN & $0.00009 ^{\\color{+}+125 \\%}$ & $ 0.00212^{\\color{+}+ 119\\%}$ & $ 0.01684^{\\color{+}+ 35\\%}$ & $ 0.00575^{\\color{+}+43 \\%}$ & $0.00056 ^{\\color{+}+167 \\%}$ & $0.00401 ^{\\color{+}+ 104\\%}$ & $ 0.04439^{\\color{-}-35\\%}$ & $ 0.04205^{\\color{-}-35\\%}$ \\\\\n  + CausE & $0.00008 ^{\\color{+}+ 100\\%}$ & $ 0.00149^{\\color{+}+54 \\%}$ & $0.01168 ^{\\color{-}-7\\%}$ & $ 0.00430^{\\color{+}+ 7\\%}$ & $ 0.00038^{\\color{+}+81 \\%}$ & $0.00253 ^{\\color{+}+ 28\\%}$ & $ 0.04876^{\\color{-}- 29\\%}$ & $ 0.04680^{\\color{-}- 27\\%}$ \\\\\n  + sam-reg & $0.00006 ^{\\color{+}+50 \\%}$ & $0.00135 ^{\\color{+}+39 \\%}$ &\n  $ 0.01573^{\\color{+}+ 26\\%}$ & $ 0.00535^{\\color{+}+ 33\\%}$ & $0.00011 ^{\\color{-}-48 \\%}$ &$0.00281 ^{\\color{+}+43 \\%}$ & $0.02850 ^{\\color{-}-58 \\%}$ & $0.02661 ^{\\color{-}-59 \\%}$\\\\\n  + MACR & $\\mathbf{0.00188} ^{\\color{+}+ 4600\\%}$ & $\\mathbf{0.00521} ^{\\color{+}+437 \\%}$ & $0.00555 ^{\\color{-}-56 \\%}$ & $0.00456 ^{\\color{+}+13 \\%}$ & $\\mathbf{ 0.00370}^{\\color{+}+ 1662\\%}$ & $ 0.00615^{\\color{+}+ 212\\%}$ & $ 0.02748^{\\color{-}- 60\\%}$ & $ 0.02413^{\\color{-}- 62\\%}$ \\\\\n  + BC loss & $ 0.00024^{\\color{+}+ 500\\%}$ & $ 0.00355^{\\color{+}+ 266\\%}$ & $ \\mathbf{0.01831}^{\\color{+}+46 \\%}$ & $ \\mathbf{0.00680} ^{\\color{+}+ 69\\%}$ & $0.00142 ^{\\color{+}+ 576\\%}$ & $ \\mathbf{0.00712}^{\\color{+}+261 \\%}$ & $\\mathbf{0.09552} ^{\\color{+}+ 40\\%}$ & $\\mathbf{0.09040} ^{\\color{+}+ 41\\%}$ \\\\\\midrule\\midrule\n  LightGCN & 0.00025 & 0.00193 & 0.01136 & 0.00417 & 0.00094 & 0.00391 & 0.07561 & 0.07121 \\\\\n  + IPS-CN & $0.00140 ^{\\color{+}+460 \\%}$ & $0.00241 ^{\\color{+}+25 \\%}$ & $0.01560 ^{\\color{+}+ 37\\%}$ & $0.00544 ^{\\color{+}+ 30\\%}$ & $0.00109 ^{\\color{+}+ 16\\%}$ & $0.00522 ^{\\color{+}+34 \\%}$ & $ 0.06333^{\\color{-}- 16\\%}$ & $ 0.05993^{\\color{-}- 16\\%}$ \\\\\n  + CausE & $ 0.00006^{\\color{-}- 76\\%}$ & $ 0.00138^{\\color{-}- 29\\%}$ & $ 0.01177^{\\color{+}+ 4\\%}$ & $0.00403 ^{\\color{-}- 3\\%}$ & $ 0.00040^{\\color{-}-57 \\%}$ & $0.00279 ^{\\color{-}- 29\\%}$ & $0.06996 ^{\\color{-}-7 \\%}$ & $0.06650 ^{\\color{-}-7 \\%}$ \\\\\n  + sam-reg & $0.00006 ^{\\color{-}-76 \\%}$ & $0.00120 ^{\\color{-}-38 \\%}$ & $ 0.01727^{\\color{+}+52 \\%}$ & $0.00560 ^{\\color{+}+ 34\\%}$ & $0.00024 ^{\\color{-}-74 \\%}$ & $0.00253 ^{\\color{-}- 35\\%}$ & $0.04647 ^{\\color{-}-39 \\%}$ & $0.04355 ^{\\color{-}- 39\\%}$ \\\\\n  + MACR & $\\mathbf{0.00287} ^{\\color{+}+ 1048\\%}$ & $\\mathbf{0.00461} ^{\\color{+}+139 \\%}$ & $0.00454 ^{\\color{-}- 60\\%}$ & $0.00501 ^{\\color{+}+20 \\%}$ & $ \\mathbf{0.00389}^{\\color{+}+ 313\\%}$ & $ \\mathbf{0.00635}^{\\color{+}+ 62\\%}$ & $0.04058 ^{\\color{-}- 46\\%}$ & $0.05323 ^{\\color{-}-25 \\%}$ \\\\\n  + BC loss & $0.00057 ^{\\color{+}+ 128\\%}$ & $0.00321 ^{\\color{+}+ 66\\%}$ & $\\mathbf{0.01943} ^{\\color{+}+ 71\\%}$ & $\\mathbf{0.00730} ^{\\color{+}+ 75\\%}$ & $ 0.00125^{\\color{+}+33 \\%}$ & $0.00516 ^{\\color{+}+ 32\\%}$ & $ \\mathbf{0.08823}^{\\color{+}+ 17\\%}$ & $ \\mathbf{0.08320}^{\\color{+}+17 \\%}$\\\\\\bottomrule\n  \\end{tabular}}\n  \\vspace{-10pt}\n  \\end{table*}\n\n\\subsection{Head, Mid, \\& Tail Performance (RQ2)}\n\n\\noindent\\textbf{Motivation.}\nTo further evaluate whether BC loss lifts the tail performance by inevitably sacrificing the head performance, we divide the test set of Tencent into three subgroups, according to the interaction number of each item: head (popular items that are in the top third), mid (normal items in the middle), and tail (unpopular items in the bottom third).\nMost previous studies focus on average NDCG@20 for evaluation, especially balanced test evaluations \\cite{MACR,DICE}. \nHowever, average metrics could be insufficient to reflect the performance of each subgroup.\nA trivial solution to achieve high performance is promoting the rankings of low-popularity items in the recommendations.\nIn this case, only the average metrics are not reliable on the balanced test.\nTherefore, we report the performance of individual subgroups on both balanced and imbalanced test sets for a more comprehensive comparison.\n\n\n\n\n\\noindent\\textbf{Results.}\nTable \\ref{tab:subgroups} shows the evaluations of the head, mid, and tail subgroups.\nThe red and blue numbers in percentage separately refer to the improvement and decline of each method relative to the original CF model (MF or LightGCN).\nWe find that:\n\\begin{itemize}[leftmargin = *]\n    \\item \\textbf{BC loss is the only method that consistently yields remarkable improvements in every subgroup.}\n    With a closer look at the head evaluation, BC loss shows its ability to learn more discriminative representations for popular items across imbalanced and balanced settings.\n    In particular, it achieves significant improvements over MF and LightGCN \\emph{w.r.t.}~ head NDCG by 40\\% and 17\\% in the imbalanced test evaluation, respectively.\n    We attribute improvements to the usage of bias-aware margin, which boosts the recommendation quality for the tail and head items.\n    \n    \\item \\textbf{As the performance comparison among subgraphs in the imbalanced scenario shows, the baselines enhance the tail performance but sacrifice the head performance.}\n   \n    Specifically, these baselines hardly maintain the head performance and show a clear trade-off trend between the head and tail performance.\n    Taking MACR as an example, although the great improvement ($+1662\\%$) over MF is achieved in the tail subgraph, it brings in the dramatic drop ($-62\\%$) in the head subgraph, which lowers the overall performance by a big drop ($-60\\%$). \n    Here we ascribe the trade-off to blindly promoting the rankings of tail items for matched and unmatched, rather than improving the discriminative power of representations.\n\\end{itemize}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\subcaptionbox{Varying margins.\\label{fig:margin}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.43\\linewidth]{figures\/margin_tencent_mf.pdf}}\n\t\\subcaptionbox{MF+BC loss\\label{fig:tau_mf}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.26\\linewidth]{figures\/effect_tau_mf.pdf}}\n\t\\subcaptionbox{LightGCN+BC loss\\label{fig:tau_Lightgcn}}{\n\t    \\vspace{-5pt}\n\t\t\\includegraphics[width=0.26\\linewidth]{figures\/effect_tau_lgn.pdf}}\n\t\\vspace{-5pt}\n\t\\caption{(a) Comparisons with a varying margin; (b-c) Temperature $\\tau$ sensitivity analysis on Tencent.}\n\t\\label{fig:study-bc-loss}\n\t\\vspace{-10pt}\n\\end{figure*}\n\n\n\\subsection{Study on BC Loss (RQ3)}\n\\label{sec:ablation}\n\n\\textbf{Effect of Bias-aware Margin.}\nFigure \\ref{fig:margin} displays the performance on balanced and imbalanced test sets on Tencent among softmax loss, BC loss with constant margin M \\cite{ArcFace}, and BC loss with adaptive bias-aware margin.\nBC loss achieves the best performance, illustrating that bias-aware margin indeed is effective at reducing popularity bias and learning high-quality representations.\n\n\\textbf{Effect of Temperature $\\tau$.}\nBC loss has one hyperparameter to tune --- temperature $\\tau$ in Equation \\eqref{equ:bc_loss}.\nIn Figure \\ref{fig:tau_mf} and \\ref{fig:tau_Lightgcn}, both balanced and imbalanced evaluations exhibit the concave unimodal functions of $\\tau$, where the curves reach the peak almost synchronously in a small range of $\\tau$. For example, MF+BC loss gets the best performance when $\\tau=0.05$ and $\\tau=0.07$ in balanced and imbalanced settings, respectively;\nWe observe similar trends on other datasets and skip them due to the space limit.\nThis justifies that BC loss does not suffer from the trade-off between the balanced and imbalanced evaluations and improves the generalization without sacrificing the head performance.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\nPrevalent popularity debiasing strategies in CF roughly fall into four research lines.\n\n\\textbf{Post-processing re-ranking methods} \\cite{Calibration,RankALS,rescale,FPC,PC_Reg} are applied to the output of the recommender system without changing the representations of users and items. \nThe purposes of modifying the ranking of models can be various: Calibration \\cite{Calibration} ensures that the past interests proportions of users are expected to maintain at the same level; RankALS \\cite{RankALS} aims to increase the diversification of recommendation; \nFPC \\cite{FPC} investigates the popularity bias in the dynamic recommendation by rescaling the predicted scores.\n\n\\textbf{Regularization-based frameworks} \\cite{ESAM,ALS+Reg,Regularized_Optimization,PC_Reg} explore the use of regularization that provides a tunable mechanism for controlling the trade-off between recommendation accuracy and coverage.\nThe difference among these methods is the design of penalty terms: \nALS+Reg \\cite{ALS+Reg} defines intra-list distance as the penalty to achieve the fair recommendation; ESAM \\cite{ESAM} introduces the attribute correlation alignment, center-clustering, and self-training regularization to learn good feature representations; sam-reg \\cite{Regularized_Optimization} regularizes the biased correlation between user-item relevance and item popularity; Reg \\cite{PC_Reg} decouples the item popularity with the model preference predictions.\n\n\\textbf{Sample re-weighting methods} \\cite{ips,IPS-C,IPS-CN,Propensity_SVM-Rank,YangCXWBE18,UBPR}, also known as Inverse Propensity Score (IPS), view the item popularity in the training set as the propensity score and exploit its inverse to re-weight loss of each instance.\nTo address the high variance of re-weighted loss, many of them \\cite{IPS-CN,IPS-C} further employ normalization or smoothing penalty to attain a more stable output.\nHowever, the unreliability of methods is due to their measurement of the propensity score, leveraging the item frequency but failing to consider interaction-wise popularity bias.\n\n\\textbf{Bias removal by causal inference methods} \\cite{CausE,KDCRec, DICE, PDA, DecRS, MACR}, getting inspiration from the recent success of counterfactual inference, specify the role of popularity bias in assumed causal graphs and mitigate the bias effect on the prediction. \nHowever, the causal structure is heuristically assumed based on the author's understanding, without any theoretical guarantee.\n\nBC loss opens up a possibility of conventional debiasing methods in CF that mitigate the popularity bias by enhancing the discriminative power. \nRecent studies, boosting the discriminative feature spaces by modified softmax loss are mainly discussed in face recognition, where a constant margin is added \\cite{ArcFace} to better classify.\nWe transfer it in CF and compare it with BC loss in Figure \\ref{fig:margin}.\n\n\n\\section{Conclusion} \\label{sec:conclusion}\nDespite the great success in collaborative filtering, today's popularity debiasing methods are still far from \\wx{being able to improve} the recommendation quality. \nIn this work, we proposed a simple yet effective BC loss, utilizing popularity bias-aware margin to eliminate the popularity bias.\nGrounded by theoretical proof, clear geometric interpretation and real-world visualization study, BC loss boosts the head and tail performance by learning a more discriminative representation space.\nExtensive experiments verify that the remarkable improvement in head and tail evaluations on various test sets indeed comes from \\wx{the} better representation rather than \\wx{simply} catering to the tail.\n\nThe limitations of BC loss are in three respects, which will be addressed in future work: 1) the modeling of bias-aware margin is worth exploring, which could significantly influence the performance of BC loss, 2) multiple important biases, such as exposure and selection bias, are not considered, and 3) more experiments comparing BC loss to standard CF losses (\\emph{e.g., } cross-entropy, WARP) are needed to \\wx{further} demonstrate the power of BC loss in regular recommendation tasks (\\za{See comparison to BPR, CCL and SSM in Appendix \\ref{sec:standard_loss}}). \nWe believe \\wx{that} this work provides a potential research direction to diagnose the debiasing of long-tail ranking and will inspire more works.\n\\section*{Checklist}\n\n\n\n\n\\begin{enumerate}\n\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes{}\n  \\item Did you describe the limitations of your work?\n    \\answerYes{See Section~\\ref{sec:conclusion}.}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNo{This paper proposes a novel debiasing algorithm for recommendation system, which does not have any negative societal impacts.}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes{}\n\\end{enumerate}\n\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results?\n    \\answerYes{}\n        \\item Did you include complete proofs of all theoretical results?\n    \\answerYes{See Appendix~\\ref{sec:proof}.}\n\\end{enumerate}\n\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{We include code by URL in abstract.}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{See Appendix~\\ref{sec:implementation}.}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerYes{}\n        \\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes{See Appendix~\\ref{sec:implementation}.}\n\\end{enumerate}\n\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n    \\answerYes{}\n  \\item Did you mention the license of the assets?\n    \\answerYes{}\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n    \\answerYes{}\n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n    \\answerNA{}\n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNA{}\n\\end{enumerate}\n\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerNA{}\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA{}\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA{}\n\\end{enumerate}\n\n\n\\end{enumerate}\n\n\n\n\\section{In-depth Analysis of BC loss}\n\\subsection{Visualization of A Toy Experiment} \\label{sec:toy_example}\n\n\n\nHere we visualize a toy example on the Yelp2018 \\cite{LightGCN} dataset to showcase the effect of BC loss.\nSpecifically, we train a two-layer LightGCN whose embedding size is three, and illustrate the 3-dimensional normalized representations on a 3D unit sphere in Figure \\ref{fig:toy-example} (See the magnified view in Figure \\ref{fig:magnified-toy-example}).\nWe train the identical LightGCN backbone with different loss functions: BPR loss, softmax loss, BC loss, and IPS-CN \\cite{IPS-CN}.\nFor the same head\/tail user (\\emph{i.e., } green stars), we plot 500 items in the unit sphere covering all positive items (\\emph{i.e., } red dots) and randomly-selected negative items (\\emph{i.e., } blue dots) from both the training and testing sets.\nMoreover, the angle distribution (the second row of each subfigure) of positive and negative items for a certain user quantitatively shows the discriminative power of each loss.\nWe observe that:\n\\begin{itemize}[leftmargin = *]\n    \\item \\textbf{BC loss learns more discriminative representations in both head and tail user cases. Moreover, BC loss learns a more reasonable representation distribution that is locally clustered and globally separated.} \n    As Figures \\ref{fig:head_bc} and \\ref{fig:tail_bc} show, for head and tail users, BC loss encourages around 40\\% and 55\\% of positive items to fall into the group closest to user representations, respectively.\n    In other words, these item representations are almost clustered to a small region.\n    BC loss also achieves the smallest distance \\emph{w.r.t.}~ mean positive angle.\n    This verifies that BC loss tends to learn a high similar item\/user compactness.\n    Moreover, Figure \\ref{fig:tail_bc} presents a clear margin between positive and negative items, reflecting a highly-discriminative power.\n    Compared to softmax loss in Figures \\ref{fig:head_softmax} and \\ref{fig:tail_softmax}, the compactness and dispersion properties of BC loss come from the incorporation of interaction-wise bias-aware margin.\n    \n    \\item \\textbf{The representations learned by standard CF losses - BPR loss and softmax loss - are not discriminative enough.}\n    Under the supervision of BPR and softmax losses, item representations are separably allocated in a wide range of the unit sphere, where blue and red points occupy almost the same space area, as Figures \\ref{fig:head_bpr} and \\ref{fig:head_softmax} demonstrate.\n    Furthermore, Figure \\ref{fig:tail_bpr} shows only a negligible overlap between positive and negative items' angle distributions.\n    However, as the negative items are much more than the positive items for the tail user, a small overlap will make many irrelevant items rank higher than the relevant items, thus significantly hindering the recommendation accuracy.\n    Hence, directly optimizing BPR or softmax loss might be suboptimal for the personalized recommendation tasks. \n    \n    \\item \\textbf{IPS-CN, a well-known popularity debiasing method in CF, is prone to lift the tail performance by sacrificing the representation learning for the head.}\n    Compared with BPR loss in Figure \\ref{fig:tail_bpr}, IPS-CN learns better item representations for the tail user, which achieves smaller mean positive angle as illustrated in Figure \\ref{fig:tail_ips}.\n    However, for the head user in Figure \\ref{fig:head_ips}, the positive and negative item representations are mixed and cannot be easily distinguished. Worse still, the representations learned by IPS-CN has a larger mean positive angle for head user compared to BPR loss. \n    This results in a dramatic performance drop for head evaluations.\n\\end{itemize}\n\n\n\n\n\\subsection{Hard Example Mining Mechanism - one desirable property of BC loss:}\\label{sec:hard_example}\n\nWe argue that the mechanism of adaptively mining hard interactions is inherent in BC loss, which improves the efficiency and effectiveness of training.\nDistinct from softmax loss that relies on the predictive scores to mine hard negative samples \\cite{sgl} and leaves the popularity bias untouched, our BC loss considers the interaction-wise biases and adaptively locates hard informative interactions.\n\nSpecifically, the popularity bias extractor $\\hat{y}_b$ in Equation \\eqref{eq:bias-prediction} can be viewed as a hard sample detector.\nConsidering an interaction $(u,i)$ with a high bias degree $cos(\\hat{\\xi}_{ui})$, we can only use its popularity information to predict user preference and attain a vanishing bias-aware angular margin $M_{ui}$.\nHence, interaction $(u,i)$ \\wx{will} plausibly serve as the biased and easy \\wx{sample, if it involves the} active users and popular items.\nIts close-to-zero margin makes $(u,i)$'s BC loss approach softmax loss, thus downgrading the ranking criteria to match the basic assumption of softmax loss.\n\nIn contrast, if the popularity statistics are deficient in recovering user preference via the popularity bias extractor, the interaction $(u,i)$ garners the low bias degree $cos(\\hat{\\xi}_{ui})$ and exerts the significant margin $M_{ui}$ on its BC loss.\nHence, it could work as the hard \\wx{sample}, which typically covers the tail users and items, and yields a more stringent assumption that user $u$ prefers the tail item over \\wx{the} other popular items by a large margin.\nSuch a significant margin makes the losses more challenging to learn.\n\nIn a nutshell, BC loss adaptively prioritizes the interaction \\wx{samples} based on their bias degree and leads the CF model to shift its attention to hard \\wx{samples}, thus improving both head and tail performance, compared with softmax loss (\\emph{cf. } Section \\ref{sec:ablation}).\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{theorem}}\n\\label{sec:proof}\n\n\\bclosstheorem*\n\n\\begin{proof}\nLet the upper-case letter $\\Mat{V} \\in \\Space{V}$ be the random vector of representation and $\\Space{V} \\subseteq \\Space{R}^d$ be the representation space.\nWe use the normalization assumption of representations to connect cosine and Euclidean distances, \\emph{i.e., } if $\\norm{\\Mat{v}_u}=1$ and $\\norm{\\Mat{v}_i}=1$, $\\Mat{v}_u^T\\Mat{v}_i = 1 - \\frac{1}{2} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2$, $\\forall u,i$. \n    \n    \nLet $\\Set{P}_u = \\{i|y_{ui} =1\\}$ be the set of user $u$'s positive items , $\\Set{P}_i = \\{u|y_{ui} =1\\}$ to be the set of item $i$'s positive users, and $\\Set{N}_{u}=\\{i|y_{ui}=0\\}$ be the set of user $u$'s negative items.\nClearly, there exists an upper bound $m$, \\emph{s.t. } $-1 < \\cos(\\hat{\\theta}_{ui}+M_{ui}) \\leq \\Mat{v}_u^T\\Mat{v}_i - m <  1 $.\nTherefore, we can analyze BC loss, which has the following relationships:\n   \n   \n   \n   \n    \\begin{align} \n        \\mathbf{\\mathop{\\mathcal{L}}}_{BC} \\geq &  -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{((\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau})}{\\exp{((\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{((\\Mat{v}_u^T\\Mat{v}_j)\/\\tau)}} \\nonumber\\\\\n        &= -\\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}  + \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}}). \\label{eq:split_bc}\n    \\end{align}\n    We now probe into the first term in Equation \\eqref{eq:split_bc}:\n\\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}\n        =&  \\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\norm{\\Mat{v}_u-\\Mat{v}_i}^2}{2\\tau} +\\frac{m-1}{\\tau} \\nonumber\\\\\n        \\eqc& \\sum_{(u,i)\\in\\Set{O}^{+}} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2 \\nonumber\\\\\n       \n        =& \\sum_{u\\in\\Set{U}}\\sum_{i \\in \\Set{P}_u} (\\norm{\\Mat{v}_u}^2 - \\Mat{v}_u^T\\Mat{v}_i )\n        + \\sum_{i\\in\\Set{I}}\\sum_{u\\in\\Set{P}_i} (\\norm{\\Mat{v}_i}^2 - \\Mat{v}_u^T\\Mat{v}_i )\\nonumber \\\\\n        \\eqc& \\sum_{u\\in\\Set{U}}\\norm{\\Mat{v}_u-\\Mat{c}_u}^2 + \\sum_{i\\in\\Set{I}}\\norm{\\Mat{v}_i-\\Mat{c}_i}^2, \\label{eq:conpactness}\n    \\end{align}\n    where the symbol $\\eqc$ indicates equality up to a multiplicative and\/or additive constant; $\\Mat{c}_{u} = \\frac{1}{|\\Set{P}_u|}\\sum_{i\\in \\Set{P}_{u}}\\Mat{v}_i$ is the averaged representation of all items that $u$ has interacted with, which describes $u$'s interest; $\\Mat{c}_{i} = \\frac{1}{|\\Set{P}_{i}|} \\sum_{u\\in\\Set{P}_{i}} \\Mat{v}_{u}$ is the averaged representation of all users who have adopted item $i$, which profiles its user group.\n   \n    We further analyze Equation \\eqref{eq:conpactness} from the entropy view by conflating the first two terms:\n    \\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} \\eqc \\sum_{\\Mat{v}} \\norm{\\Mat{v} - \\Mat{c}_{v}}^2,\n    \\end{align}\n    where $\\Mat{v}\\in\\{\\Mat{v}_{u}|u\\in\\Set{U}\\}\\cup\\{\\Mat{v}_{i}|i\\in\\Set{I}\\}$ summarizes the representations of users and items, with the mean of $\\Mat{c}_{v}$.\n    Following \\cite{BoudiafRZGPPA20}, we further interpret this term as a conditional cross-entropy between $\\Mat{V}$ and another random variable $\\bar{\\Mat{V}}$ whose conditional distribution given $Y$ is a standard Gaussian $\\bar{\\Mat{V}}|Y \\sim \\mathcal{N}(\\Mat{c}_{\\Mat{V}},I)$:\n    \\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} & \\eqc\n        H(\\Mat{V};\\bar{\\Mat{V}}|Y)=H(\\Mat{V}|Y)+D_{KL}(\\Mat{V}||\\bar{\\Mat{V}}|Y) \\nonumber\\\\\n        & \\propto H(\\Mat{V}|Y),\\label{eq:first-term}\n    \\end{align}\n    where $H(\\cdot)$ denotes the cross-entropy, and $D_{KL}(\\cdot)$ denotes the $KL$-divergence.\n    As a consequence, the first term in Equation \\eqref{eq:split_bc} is positive proportional to $H(\\Mat{V}|Y)$.\n   \n    This concludes the proof for the first compactness part of BC loss.\n    \n    We then inspect the second term in Equation \\eqref{eq:split_bc} to demonstrate its dispersion property:\n    \\begin{align}\n     & \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}})\\nonumber\\\\\n     \\geq& \\sum_{(u,i)\\in\\Set{O}^{+}} \\log{(\n     \\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}})} \\nonumber\\\\\n     \\geq& \\sum_{u\\in\\Set{U}}\\sum_{j\\in\\Set{N}_{u}}\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\nonumber\\\\\n     \\eqc& - \\sum_{u\\in\\Set{U}}\\sum_{j\\in\\Set{N}_{u}} \\norm{\\Mat{v}_u - \\Mat{v}_j}^2, \\label{eq:dispersion}\n    \\end{align}\n    where we drop the redundant terms aligned with the compactness objective in the second line, and adopt Jensen's inequality in the third line.\n    As shown in prior studies \\cite{WangS11}, minimizing this term is equivalent to maximizing entropy $H(\\Mat{V})$:\n    \\begin{align} \\label{eq:second-term}\n        \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}}) \\propto -\\mathcal{H}(\\Mat{V}).\n    \\end{align}\n    As a result, the second term in Equation \\eqref{eq:split_bc} works as the dispersion part in BC loss.\n\\end{proof}\n\n\n\\begin{landscape}\n    \\begin{figure}[t]\n        \\centering\n        \\subcaptionbox{BPR loss (head)\\label{fig:head_bpr}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_bpr.pdf}}\n        \\subcaptionbox{Softmax loss (head)\\label{fig:head_softmax}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_info.pdf}}\n        \\subcaptionbox{IPS-CN \\cite{IPS-CN} (head) \\label{fig:head_ips}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_ips.pdf}}\n        \\subcaptionbox{BC loss (head)\\label{fig:head_bc}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_bc.pdf}}\n    \n        \\subcaptionbox{BPR loss (tail)\\label{fig:tail_bpr}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_bpr.pdf}}\n        \\subcaptionbox{Softmax loss (tail)\\label{fig:tail_softmax}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_info.pdf}}\n        \\subcaptionbox{IPS-CN \\cite{IPS-CN} (tail)\\label{fig:tail_ips}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_ips.pdf}}\n        \\subcaptionbox{BC loss (tail)\\label{fig:tail_bc}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_bc.pdf}}\n       \n        \\caption{Magnified View of Figure \\ref{fig:toy-example}.}\n        \\label{fig:magnified-toy-example}\n        \\vspace{-20pt}\n    \\end{figure}\n    \\end{landscape}\n\n\n\\section{Experiments} \\label{sec:app_exp}\n\n\n\\begin{table}[b]\n    \\centering\n    \\vspace{-10pt}\n    \\caption{Dataset statistics.}\n   \n    \\label{tab:dataset-statistics}\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lrrrrrrr}\n    \\toprule\n     & KuaiRec & Douban Movie & Tencent & Amazon-Book & Alibaba-iFashion & Yahoo!R3 & Coat\\\\ \\midrule\n    \\#Users & 7175 & 36,644 & 95,709 & 52,643 & 300,000 & 14382 & 290 \\\\\n    \\#Items & 10611 & 22,226 & 41,602 & 91,599 & 81,614 & 1000 & 295 \\\\\n    \\#Interactions & 1062969 & 5,397,926 & 2,937,228 & 2,984,108 & 1,607,813 & 129,748 & 2,776 \\\\\n    Sparsity & 0.01396 & 0.00663 & 0.00074 & 0.00062 & 0.00007 \n    & 0.00902 &  0.03245\\\\ \\midrule\n    $D_{KL}$-Train & 1.075 & 1.471 & 1.425 & 0.572 & 1.678 & 0.854 & 0.356 \\\\\n    $D_{KL}$-Validation & 1.006 & 1.642 & 1.423 & 0.572 & 1.705 & 0.822 & 0.350\\\\\n    $D_{KL}$-Balanced & - & - & 0.003 & 0.000 & 0.323 & - & -  \\\\\n    $D_{KL}$-Imbalanced & - & - & 1.424 & 0.571 & 1.703 & - & - \\\\ \n    $D_{KL}$-Temporal & - & 1.428 & - & - & -  & - & - \\\\\n    $D_{KL}$-Unbiased & 1.666 & - & - & - & -& 0.100 & 0.109 \\\\ \\bottomrule\n    \\end{tabular}}\n\\end{table}\n\n\\subsection{Experimental Settings} \\label{sec:implementation}\n\\textbf{Datasets.} \nWe conduct experiments on five real-world benchmark datasets: Tencent \\cite{Tencent}, Amazon-Book \\cite{Amazon-Book}, Alibaba-iFashion \\cite{Alibaba-ifashion}, Douban Movie \\cite{Douban}, and KuaiRec \\cite{KuaiRec}.\nAll datasets are public and vary in terms of size, domain, and sparsity.\nTable \\ref{tab:dataset-statistics} summarizes the dataset statistics, where the long-tail degree is monitored by KL-divergence between the item popularity distribution and the uniform distribution, \\emph{i.e., } $D_{KL}(\\hat{P}_{data}|| \\text{Uniform})$.\nA larger KL-divergence value indicates that the heavier portion of interactions concentrates on the head of distribution. \nIn the stage of pre-processing data, we follow the standard 10-core setting \\cite{PDA,KGAT} to filter out the items and users with less than ten interactions.\n\n\\noindent\\textbf{Data Splits.}\nFor comprehensive comparisons, almost all standard test distributions in CF are covered in the experiments: balanced test set \\cite{MACR,DICE,KDCRec}, randomly selected imbalanced test set \\cite{ESAM,ZhuWC20}, temporal split test set \\cite{PDA,DecRS,Regularized_Optimization}, and unbiased test set \\cite{ips,KuaiRec,Yahoo}. \nThree datasets (\\emph{i.e., } Tencent, Amazon-Book, Alibaba-iFashion) are partitioned into both balanced and randomly selected imbalanced evaluations. \nAs an intervention test, the balanced evaluation (\\emph{i.e., } uniform distribution) is frequently employed in recent debiasing CF approaches \\cite{MACR,DICE,KDCRec,Causal_inference}.\nDouban is split based on the temporal splitting strategy \\cite{MengMMO20}. \nKuaiRec is an unbiased, fully-observed dataset in which the feedback of the test set's interaction is explicitly collected.\n\n\n\\vspace{5pt}\n\\noindent\\textbf{Evaluation Metrics.}\nWe adopt the all-ranking strategy \\cite{KricheneR20}, \\emph{i.e., } for each user, all items are ranked by the recommender model, except the positive ones in the training set. To evaluate the quality of recommendation, three widely-used metrics are used: Hit Ratio (HR@$K$), Recall@$K$, Normalized Discounted Cumulative Gain (NDCG@$K$), where $K$ is set as $20$ by default.\n\n\\vspace{5pt}\n\\noindent\\textbf{Baselines.}\nWe validate our BC loss on two widely-used CF models, MF \\cite{BPR} and LightGCN \\cite{LightGCN}, which are representatives of the conventional and state-of-the-art CF models.\nWe compare BC loss with the popular debiasing strategies in various research lines: sample re-weighting (IPS-CN \\cite{IPS-CN}), bias removal by causal inference (MACR \\cite{MACR}, CausE \\cite{CausE}), and regularization-based framework (sam+reg \\cite{Regularized_Optimization}).\n\\za{We also compare BC loss to other standard losses used in collaborative filtering including most commonly used loss (BPR loss \\cite{BPR}) and newest proposed softmax losses (CCL \\cite{CCL} and SSM \\cite{SSM})}.\n\n\\vspace{5pt}\n\\noindent\\textbf{Parameter Settings.}\nWe conduct experiments using a Nivida-V100\nGPU (32 GB memory) on a server with a\n40-core Intel CPU (Intel(R) Xeon(R) CPU E5-2698 v4).\nWe implement our BC loss in PyTorch.\n\\wx{Our codes, datasets, and hyperparameter settings are available at \\url{https:\/\/anonymous.4open.science\/r\/BC-Loss-8764\/model.py} to guarantee reproducibility.}\nFor a fair comparison, all methods are optimized by Adam \\cite{Adam} optimizer with the batch size as 2048, embedding size as 64, learning rate as 1e-3, and the coefficient of regularization as 1e-5 in all experiments.\nFollowing the default setting in \\cite{LightGCN}, the number of embedding layers for LightGCN is set to 2.\nWe adopt the early stop strategy that stops training if Recall@20 on the validation set does not increase for 10 successive epochs.\nA grid search is conducted to tune the critical hyperparameters of each strategy to choose the best models \\emph{w.r.t.}~ Recall@20 on the validation set.\n\\za{For softmax, SSM, and BC loss, we search $\\tau$ in [0.06, 0.14] with a step size of 0.02.}\nFor CausE, 10\\% of training data with balanced distribution are used as intervened set, and $cf\\_pen$ is tuned in $[0.01,0.1]$ with a step size of 0.02.\nFor MACR, we follow the original settings to set weights for user branch $\\alpha=1e-3$ and item branch $\\beta=1e-3$, respectively. We further tune hyperparameter $c=[0, 50]$ with a step size of 5.\n\\za{For CCL loss, we search $w$ in $\\{1,2,5,10,50,100,200\\}$, $m$ in the range $[0.2,1]$ with a step size of 0.2. When it come to the number of negative samples, the softmax, SSM, CCL, and BC loss set 128 for MF backbone and in-batch negative sampling for LightGCN models.}\n\n\\subsection{Training Cost} \n\\begin{table}[t]\n    \\centering\n    \\caption{Training cost on Tencent (seconds per epoch\/in total). }\n   \n    \\label{tab:elapse_time}\n    \\resizebox{0.9\\linewidth}{!}{\n    \\begin{tabular}{l|rrrrrr}\n    \\toprule\n     & Backbone & +IPS-CN & +CausE & +sam+reg & +MACR & +BC loss\\\\ \\midrule\n    MF & 15.5 \/ 17887 & 17.8 \/ 10662 & 16.6 \/ 1859 & 18.2 \/ 3458 & 160 \/ 17600 &  36.1 \/ 12815\\\\\n    LightGCN & 78.6 \/ 4147 & 108 \/ 23652 & 47.2 \/ 3376 & 49.8 \/ 10458 & 135 \/ 20250 & 283 \/ 7075\\\\ \\bottomrule\n    \\end{tabular}}\n\\end{table}\n\nIn terms of time complexity, as shown in Table \\ref{tab:elapse_time}, we report the time cost per epoch and in total of each baselines on Tencent.\nCompared with backbone methods (\\emph{i.e., } MF and LightGCN), BC loss adds very little computing complexity to the training process. \n\n\n\n\\subsection{Evaluations on Unbiased Test Set} \\label{sec:KuaiRec}\n\n\\noindent\\textbf{Motivation.}\nBecause of the missing-not-at-random condition in a real recommender system, offline evaluation on collaborative filtering and recommender system is commonly acknowledged as a challenge.\nTo close the gap, Yahoo!R3 \\cite{Yahoo} and Coat \\cite{ips} are widely used, which offer unbiased test sets that are collected using the missing-complete-at-random (MCAR) concept.\nAdditionaly, newly proposed KuaiRec \\cite{KuaiRec} also provides a fully-observed unbiased test set with 1,411 users over 3,327 videos.\nWe conduct experiments on all these datasets for comprehensive comparison, and KuaiRec is also included as one of our unbiased evaluations for two key reasons:\n1) It is significantly larger than existing MCAR datasets (\\emph{e.g., } Yahoo! and Coat); \n2) It overcomes the missing values problem, making it as effective as an online A\/B test.\n\n\\begin{table}[t]\n  \\centering\n  \\caption{The performance comparison on KuaiRec dataset. }\n  \\label{tab:KuaiRec}\n  \\resizebox{0.7\\columnwidth}{!}{\n  \\begin{tabular}{l|ccc|ccc}\n  \\toprule\n   & \\multicolumn{3}{c|}{Validation} & \\multicolumn{3}{c}{Unbiased Test} \\\\\n  \\multicolumn{1}{c|}{} & HR & Recall & NDCG & HR & Recall & NDCG \\\\\\midrule\n  LightGCN & 0.299 & 0.069 & 0.051 & 0.104 & 0.0038  & 0.0064  \\\\\n  + IPS-CN & 0.255 & 0.056 & 0.042 & \\underline{0.109} & \\underline{0.0073} & \\underline{0.0083}  \\\\\n  + CausE & 0.292 & 0.067 & 0.050 & 0.101 & 0.0056 & 0.0077 \\\\\n  + sam+reg & 0.274 & 0.060 &  0.047 & 0.107 & 0.0069 & 0.0080 \\\\\n  + BC loss & 0.343 & 0.076 & 0.062 & \\textbf{0.139}* & \\textbf{0.0077}* & \\textbf{0.0115}* \\\\\\midrule\n  Imp.\\% & - & - & - & 27.5\\% & 4.05\\% & 38.6\\% \\\\\\bottomrule\n  \\end{tabular}}\n  \\vspace{-15pt}\n\\end{table}\n\n\\noindent\\textbf{Parameter Settings.} For BC loss on Yahoo!R3 and Coat, we search $\\tau_1$ in [0.05, 0.21] with a step size of 0.01, and $\\tau_2$ in [0.1, 0.6] with a step size of 0.1, and search the number of negative samples in [16, 32, 64, 128]. We adopt the batch size of 1024 and learning rate of 5e-4.\n\n\\noindent\\textbf{Results.}\nTable \\ref{tab:KuaiRec} and \\ref{tab:Yahoo&Coat} illustrate the unbiased evaluations on KuaiRec, Yahoo!R3, and Coat dataset using LightGCN and MF as backbone models, respectively.\nThe best performing methods are bold and starred, while the strongest baselines are underlined;\nImp.\\% measures the relative improvements of BC loss over the strongest baselines.\nBC loss is consistently superior to all baselines \\emph{w.r.t.}~ all metrics. \nIt indicates that BC loss truly improves the generalization ability of recommender.\n\n\n\n\\subsection{Performance Comparison with Standard Loss Functions in CF} \\label{sec:standard_loss}\n\n\\noindent\\textbf{Motivation.}\nTo verify the effectiveness of BC loss as a standard learning strategy in collaborative filtering, we further conduct the experiments over various datasets between BC loss, BPR loss, CCL loss \\cite{CCL}, and SSM loss \\cite{SSM}. We choose these three losses as baselines for two main reasons: 1) BPR loss is the most commonly applied in recommender system; 2) SSM and CCL are most recent proposed losses, where SSM is also a softmax loss and CCL employs a global margin. \n\n\\noindent\\textbf{Results.}\nTable \\ref{tab:loss_compare} reports the performance on both balanced and imbalanced test sets on various datasets among different losses. \nWe have two main observations: (1) Clearly, our BC loss consistently outperforms CCL and SSM; (2) CCL and SSM achieve comparable performance to BC loss in the imbalanced evaluation settings, while performing much worse than BC loss in the balanced evaluation settings. This indicates the superiority of BC loss in alleviating the popularity bias, and further justifies the effectiveness of the bias-aware margins.\n\nTable \\ref{tab:Yahoo&Coat} shows the unbiased evaluations on Yahoo!R3 and Coat dataset. \nWith regard to all criteria, BC loss constantly outperforms all other losses. \nIt verifies the effectiveness of instance-wise bias margin of BC loss.\n\n\\begin{table*}[t]\n\\caption{The performance comparison on Tecent, Amazon-book and Alibaba-iFashion datasets. \n}\n\\label{tab:loss_compare}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|cc|cc|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{1}{c|}{} & \\multicolumn{4}{c|}{Tencent}       & \\multicolumn{4}{c|}{Amazon-book}   & \\multicolumn{4}{c}{Alibaba-iFashion} \\\\\n\\multicolumn{1}{c|}{} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c|}{Imbalanced} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c|}{Imbalanced} &\n  \\multicolumn{2}{c}{Balanced} &\n  \\multicolumn{2}{c}{Imbalanced} \\\\\n \n\\multicolumn{1}{c|}{} & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall  & NDCG    & Recall  & NDCG   \\\\\n\\midrule\n\nBPR    & 0.0052 & 0.0040 & 0.0982 & 0.0643 & 0.0109 & 0.0103 & 0.0850 & 0.0638 & 0.0056  & 0.0028  & 0.0843 & 0.0411 \\\\\nSSM            & 0.0055 & 0.0045 & \\underline{0.1297} & \n\\underline{0.0872} & 0.0156 & 0.0157 & 0.1125 & \\underline{0.0873} & \n\\underline{0.0079}  & \\underline{0.0040}  & \\underline{0.0963}  & \\underline{0.0436} \\\\\nCCL           & \\underline{0.0057} & \\underline{0.0047} & 0.1216 & 0.0818 & \\underline{0.0175} & \\underline{0.0167} & \\underline{0.1162} & \\underline{0.0927} & 0.0075  & 0.0038  & 0.0954  & 0.0428 \\\\\nBC Loss            & \\textbf{0.0087}* & \\textbf{0.0068}* & \\textbf{0.1298}* & \\textbf{0.0904}* & \\textbf{0.0221}* & \\textbf{0.0202}* & \\textbf{0.1198}* & \\textbf{0.0948}* & \\textbf{0.0095}*  & \\textbf{0.0048}*  & \\textbf{0.0967}*  & \\textbf{0.0487}* \\\\\\midrule\nImp. \\%    &  52.6\\%  &  44.7\\%      &    0.1\\%  &  3.7\\%      &   26.3\\%     &    21.0\\%    &    3.1\\%    &  2.3\\%      &    20.3\\%     &    20.0\\%     &    0.4\\%     & 11.7\\%        \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{The performance comparison on Yahoo!R3 and Coat dataset. }\n  \\label{tab:Yahoo&Coat}\n  \\resizebox{0.6\\columnwidth}{!}{\n  \\begin{tabular}{l|cc|cc}\n  \\toprule\n   & \\multicolumn{2}{c|}{Yahoo!R3} & \\multicolumn{2}{c}{Coat} \\\\\n  \\multicolumn{1}{c|}{} & Recall & NDCG & Recall & NDCG \\\\\\midrule\n  \n  IPS-CN &  0.1081 & 0.0487 & 0.1700 & 0.1377 \\\\\n  CausE  & 0.1252 & 0.0537 & \\underline{0.2329} & 0.1635  \\\\\n  sam+reg & 0.1198 & 0.0548 & 0.2303 & 0.1869 \\\\\n  MACR & 0.1243 & 0.0539 & 0.0798 & 0.0358 \\\\\\midrule\n  BPR & 0.1063 & 0.0476 & 0.0741 & 0.0361\\\\\n  SSM & \\underline{0.1470} & \n  \\underline{0.0688}  & 0.2022 & 0.1832\\\\\n  CCL & 0.1428 & 0.0676 & 0.2150 & \\underline{0.1885} \\\\\n  BC loss  & \\textbf{0.1487}* & \\textbf{0.0706}* & \\textbf{0.2385}* & \\textbf{0.1969}* \\\\\\midrule\n  Imp.\\% & 1.2\\% & 2.6\\% & 2.4\\% & 4.5\\% \\\\\\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\subsection{Study on BC Loss (RQ3) - Effect of Popularity Bias Extractor} \\label{sec:bias_extractor}\n\n\\noindent\\textbf{Motivation.} To check the effectiveness of popularity bias extractor, we need to answer two main questions: 1) what kinds of interactions will be learned well by the bias extractor? What does the learned bias-angle distribution look like? 2) can BC loss benefit from the bias margin extracted according to the popularity bias extractor in various groups of interactions? We devise two distinct experiments to tackle these two problems.\n\n\\noindent\\textbf{Experiment Setting A.}\nUsers can be divided into three parts: head, mid, and tail, based on their popularity scores. Analogously, items can be partitioned into the head, mid, and tail parts. As such, we can categorize all user-item interactions into nine subgroups. \nWe have visualized the learned angles for various types of interactions in the following table \\ref{fig:angles}. \n\n\\noindent\\textbf{Results A.} The table \\ref{fig:angles} shows the learned angles over all subgroups. We find that interactions between head users and head items tend to hold small angles. Moreover, as evidenced by the high standard deviation, the interactions stemming from the same subgroup types are prone to receive a wide range of angular values. This demonstrates the variability and validity of instance-wise angular margins.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{figures\/angle.png}\n    \\caption{Visualization of popularity bias angle for different types of interactions on Tencent.}\n    \\label{fig:angles}\n\\end{figure*}\n\n\n\\noindent\\textbf{Experiment Setting B.}\nWe partition interactions into disjoint subgroups on Tencent dataset, based on the bias degree estimated by our popularity bias extractor. These subgraphs are composed of (1) hard interactions with low bias degrees (40\\% of total interactions, Popularity rank > 1000), and (2) easy interactions with high bias degrees (15\\% of total interactions, Popularity rank < 100), respectively. Then, we use four losses (BPR, CCL, SSM, and BC losses) to train the same MF backbone.\n\n\\noindent\\textbf{Results B.}\nFigure \\ref{eq:bias-extractor} depicts the performance of each loss in the hard-interaction subgroup, easy-interaction subgraph, and all-interaction group, during the training phase. With the increase of training epochs, the performance of BPR drops dramatically in the hard-interaction subgroup, while increasing in the all-interaction group. Possible reasons are BPR is easily influenced by the interaction-wise bias: BPR focuses largely on the majority subgroups of easy interactions, while sacrificing the performance of the minority subgroups of hard interactions. In contrast, the performance of BC loss consistently increases over every subgroup during training. This indicates that BC loss is able to mitigate the negative influences of popularity bias, thus further justifying that the popularity bias extract captures the interaction-wise bias well.\n\n\n\n\\begin{figure*}[b]\n\t\\centering\n\t\\subcaptionbox{Hard interactions}{\n\t\t\\includegraphics[width=0.32\\linewidth]{figures\/easy.png}}\n\t\\subcaptionbox{Easy interactions}{\n\t\t\\includegraphics[width=0.32\\linewidth]{figures\/hard.png}}\n\t\\subcaptionbox{All interactions}{\n\t\t\\includegraphics[width=0.32\\linewidth]{figures\/average.png}}\n\t\\caption{(a) The performance of BPR loss on hard interactions consistently drops; (b)- (c). BC loss shows the promise to mitigate the popularity bias by utilizing the popularity bias extractor.}\n\t\\label{fig:pop_extractor}\n\\end{figure*}\n\n\n\\newpage\n\\za{\n\\subsection{Visualization of interaction bias degree}\nFigures \\ref{fig:item_pop} and \\ref{fig:user_pop} illustrate the relations between popularity scores and bias degree extracted by popularity bias extractor \\emph{w.r.t.}~ user and item sides, respectively. Specifically, positive trends are shown, where their relations are also quantitatively supported by Pearson correlation coefficients. (0.7703 and 0.662 for item and user sides, respectively). It verifies the power of popularity embeddings to predict the popularity scores --- that is, user popularity embeddings derived from the popularity bias extractor are strongly correlated and sufficiently predictive to user popularity scores; analogously to the item side.}\n\\begin{figure*}[t]\n\t\\centering\n\t\\subcaptionbox{Bias degree \\emph{w.r.t.}~ Item popularity bias \\label{fig:item_pop}}{\n\t  \n\t\t\\includegraphics[width=0.43\\linewidth]{figures\/item_pop.png}}\n\t\\subcaptionbox{Bias degree \\emph{w.r.t.}~ User popularity bias\\label{fig:user_pop}}{\n\t  \n\t\t\\includegraphics[width=0.43\\linewidth]{figures\/user_pop.png}}\n\t\\caption{Visualizations of relationships between interaction bias degree estimated by our popularity bias extractor and item\/user popularity statistics. Pearson correlation coefficients are provided.}\n\\end{figure*}\n\n\n\n\n\\begin{table}[b]\n  \\centering\n  \\caption{Model architectures and hyper-parameters}\n  \\label{tab:parameter}\n  \\resizebox{0.6\\columnwidth}{!}{\n  \\begin{tabular}{l|c|c|c|c|c|c}\n  \\toprule\n   & \\multicolumn{5}{c}{BC loss hyper-parameters} \\\\\\midrule\n   \n  \\multicolumn{1}{c|}{} & $\\tau_1$ & $\\tau_2$ & lr & batch size & No. negative samples \\\\\\midrule\n  \\textbf{MF} \\\\\\midrule\n  Tencent &  0.06 &  0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n  iFashion & 0.08 & 0.1 & 1e-3 & 2048 & 128  \\\\\\midrule\n  Amazon & 0.08 & 0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n  Douban  & 0.08 & 0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n  Yahoo!R3 & 0.15 & 0.2 & 5e-4 & 1024 & 128 \\\\\\midrule\n  Coat & 0.09 & 0.4  & 5e-4 & 1024 & 64\\\\\\midrule\n  \\textbf{LightGCN} \\\\\\midrule\n    Tencent &  0.12 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n    iFashion & 0.14 & 0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n  Amazon & 0.08 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n  Douban  & 0.14 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n  \\end{tabular}}\n\\end{table}\n\\section{In-depth Analysis of BC loss}\n\\subsection{Visualization of A Toy Experiment} \\label{sec:toy_example}\n\n\n\nHere we visualize a toy example on the Yelp2018 \\cite{LightGCN} dataset to showcase the effect of BC loss.\nSpecifically, we train a two-layer LightGCN whose embedding size is three, and illustrate the 3-dimensional normalized representations on a 3D unit sphere in Figure \\ref{fig:toy-example} (See the magnified view in Figure \\ref{fig:magnified-toy-example}).\nWe train the identical LightGCN backbone with different loss functions: BPR loss, softmax loss, BC loss, and IPS-CN \\cite{IPS-CN}.\nFor the same head\/tail user (\\emph{i.e., } green stars), we plot 500 items in the unit sphere covering all positive items (\\emph{i.e., } red dots) and randomly-selected negative items (\\emph{i.e., } blue dots) from both the training and testing sets.\nMoreover, the angle distribution (the second row of each subfigure) of positive and negative items for a certain user quantitatively shows the discriminative power of each loss.\nWe observe that:\n\\begin{itemize}[leftmargin = *]\n    \\item \\textbf{BC loss learns more discriminative representations in both head and tail user cases. Moreover, BC loss learns a more reasonable representation distribution that is locally clustered and globally separated.} \n    As Figures \\ref{fig:head_bc} and \\ref{fig:tail_bc} show, for head and tail users, BC loss encourages around 40\\% and 55\\% of positive items to fall into the group closest to user representations, respectively.\n    In other words, these item representations are almost clustered to a small region.\n    BC loss also achieves the smallest distance \\emph{w.r.t.}~ mean positive angle.\n    This verifies that BC loss tends to learn a high similar item\/user compactness.\n    Moreover, Figure \\ref{fig:tail_bc} presents a clear margin between positive and negative items, reflecting a highly-discriminative power.\n    Compared to softmax loss in Figures \\ref{fig:head_softmax} and \\ref{fig:tail_softmax}, the compactness and dispersion properties of BC loss come from the incorporation of interaction-wise bias-aware margin.\n    \n    \\item \\textbf{The representations learned by standard CF losses - BPR loss and softmax loss - are not discriminative enough.}\n    Under the supervision of BPR and softmax losses, item representations are separably allocated in a wide range of the unit sphere, where blue and red points occupy almost the same space area, as Figures \\ref{fig:head_bpr} and \\ref{fig:head_softmax} demonstrate.\n    Furthermore, Figure \\ref{fig:tail_bpr} shows only a negligible overlap between positive and negative items' angle distributions.\n    However, as the negative items are much more than the positive items for the tail user, a small overlap will make many irrelevant items rank higher than the relevant items, thus significantly hindering the recommendation accuracy.\n    Hence, directly optimizing BPR or softmax loss might be suboptimal for the personalized recommendation tasks. \n    \n    \\item \\textbf{IPS-CN, a well-known popularity debiasing method in CF, is prone to lift the tail performance by sacrificing the representation learning for the head.}\n    Compared with BPR loss in Figure \\ref{fig:tail_bpr}, IPS-CN learns better item representations for the tail user, which achieves smaller mean positive angle as illustrated in Figure \\ref{fig:tail_ips}.\n    However, for the head user in Figure \\ref{fig:head_ips}, the positive and negative item representations are mixed and cannot be easily distinguished. Worse still, the representations learned by IPS-CN has a larger mean positive angle for head user compared to BPR loss. \n    This results in a dramatic performance drop for head evaluations.\n\\end{itemize}\n\n\n\n\\subsection{Hard Example Mining Mechanism - one desirable property of BC loss:}\\label{sec:hard_example}\n\nWe argue that the mechanism of adaptively mining hard interactions is inherent in BC loss, which improves the efficiency and effectiveness of training.\nDistinct from softmax loss that relies on the predictive scores to mine hard negative samples \\cite{sgl} and leaves the popularity bias untouched, our BC loss considers the interaction-wise biases and adaptively locates hard informative interactions.\n\nSpecifically, the popularity bias extractor $\\hat{y}_b$ in Equation \\eqref{eq:bias-prediction} can be viewed as a hard sample detector.\nConsidering an interaction $(u,i)$ with a high bias degree $cos(\\hat{\\xi}_{ui})$, we can only use its popularity information to predict user preference and attain a vanishing bias-aware angular margin $M_{ui}$.\nHence, interaction $(u,i)$ \\wx{will} plausibly serve as the biased and easy \\wx{sample, if it involves the} active users and popular items.\nIts close-to-zero margin makes $(u,i)$'s BC loss approach softmax loss, thus downgrading the ranking criteria to match the basic assumption of softmax loss.\n\nIn contrast, if the popularity statistics are deficient in recovering user preference via the popularity bias extractor, the interaction $(u,i)$ garners the low bias degree $cos(\\hat{\\xi}_{ui})$ and exerts the significant margin $M_{ui}$ on its BC loss.\nHence, it could work as the hard \\wx{sample}, which typically covers the tail users and items, and yields a more stringent assumption that user $u$ prefers the tail item over \\wx{the} other popular items by a large margin.\nSuch a significant margin makes the losses more challenging to learn.\n\nIn a nutshell, BC loss adaptively prioritizes the interaction \\wx{samples} based on their bias degree and leads the CF model to shift its attention to hard \\wx{samples}, thus improving both head and tail performance, compared with softmax loss (\\emph{cf. } Section \\ref{sec:ablation}).\n\n\n\n\n\\subsection{Proof of Theorem \\ref{theorem}}\n\\label{sec:proof}\n\n\\bclosstheorem*\n\n\\begin{proof}\nLet the upper-case letter $\\Mat{V} \\in \\Space{V}$ be the random vector of representation and $\\Space{V} \\subseteq \\Space{R}^d$ be the representation space.\nWe use the normalization assumption of representations to connect cosine and Euclidean distances, \\emph{i.e., } if $\\norm{\\Mat{v}_u}=1$ and $\\norm{\\Mat{v}_i}=1$, $\\Mat{v}_u^T\\Mat{v}_i = 1 - \\frac{1}{2} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2$, $\\forall u,i$. \n    \n    \nLet $\\Set{P}_u = \\{i|y_{ui} =1\\}$ be the set of user $u$'s positive items , $\\Set{P}_i = \\{u|y_{ui} =1\\}$ to be the set of item $i$'s positive users, and $\\Set{N}_{u}=\\{i|y_{ui}=0\\}$ be the set of user $u$'s negative items.\nClearly, there exists an upper bound $m$, \\emph{s.t. } $-1 < \\cos(\\hat{\\theta}_{ui}+M_{ui}) \\leq \\Mat{v}_u^T\\Mat{v}_i - m <  1 $.\nTherefore, we can analyze BC loss, which has the following relationships:\n   \n   \n   \n   \n    \\begin{align} \n        \\mathbf{\\mathop{\\mathcal{L}}}_{BC} \\geq &  -\\sum_{(u,i)\\in\\Set{O}^{+}}\\log\\frac{\\exp{((\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau})}{\\exp{((\\Mat{v}_u^T\\Mat{v}_i - m)\/\\tau)}+\\sum_{j\\in\\Set{N}_{u}}\\exp{((\\Mat{v}_u^T\\Mat{v}_j)\/\\tau)}} \\nonumber\\\\\n        &= -\\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}  + \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}}). \\label{eq:split_bc}\n    \\end{align}\n    We now probe into the first term in Equation \\eqref{eq:split_bc}:\n\\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}\n        =&  \\sum_{(u,i)\\in\\Set{O}^{+}} \\frac{\\norm{\\Mat{v}_u-\\Mat{v}_i}^2}{2\\tau} +\\frac{m-1}{\\tau} \\nonumber\\\\\n        \\eqc& \\sum_{(u,i)\\in\\Set{O}^{+}} \\norm{\\Mat{v}_u-\\Mat{v}_i}^2 \\nonumber\\\\\n       \n        =& \\sum_{u\\in\\Set{U}}\\sum_{i \\in \\Set{P}_u} (\\norm{\\Mat{v}_u}^2 - \\Mat{v}_u^T\\Mat{v}_i )\n        + \\sum_{i\\in\\Set{I}}\\sum_{u\\in\\Set{P}_i} (\\norm{\\Mat{v}_i}^2 - \\Mat{v}_u^T\\Mat{v}_i )\\nonumber \\\\\n        \\eqc& \\sum_{u\\in\\Set{U}}\\norm{\\Mat{v}_u-\\Mat{c}_u}^2 + \\sum_{i\\in\\Set{I}}\\norm{\\Mat{v}_i-\\Mat{c}_i}^2, \\label{eq:conpactness}\n    \\end{align}\n    where the symbol $\\eqc$ indicates equality up to a multiplicative and\/or additive constant; $\\Mat{c}_{u} = \\frac{1}{|\\Set{P}_u|}\\sum_{i\\in \\Set{P}_{u}}\\Mat{v}_i$ is the averaged representation of all items that $u$ has interacted with, which describes $u$'s interest; $\\Mat{c}_{i} = \\frac{1}{|\\Set{P}_{i}|} \\sum_{u\\in\\Set{P}_{i}} \\Mat{v}_{u}$ is the averaged representation of all users who have adopted item $i$, which profiles its user group.\n   \n    We further analyze Equation \\eqref{eq:conpactness} from the entropy view by conflating the first two terms:\n    \\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} \\eqc \\sum_{\\Mat{v}} \\norm{\\Mat{v} - \\Mat{c}_{v}}^2,\n    \\end{align}\n    where $\\Mat{v}\\in\\{\\Mat{v}_{u}|u\\in\\Set{U}\\}\\cup\\{\\Mat{v}_{i}|i\\in\\Set{I}\\}$ summarizes the representations of users and items, with the mean of $\\Mat{c}_{v}$.\n    Following \\cite{BoudiafRZGPPA20}, we further interpret this term as a conditional cross-entropy between $\\Mat{V}$ and another random variable $\\bar{\\Mat{V}}$ whose conditional distribution given $Y$ is a standard Gaussian $\\bar{\\Mat{V}}|Y \\sim \\mathcal{N}(\\Mat{c}_{\\Mat{V}},I)$:\n    \\begin{align}\n        -\\sum_{(u,i)\\in\\Set{O}^{+}}\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau} & \\eqc\n        H(\\Mat{V};\\bar{\\Mat{V}}|Y)=H(\\Mat{V}|Y)+D_{KL}(\\Mat{V}||\\bar{\\Mat{V}}|Y) \\nonumber\\\\\n        & \\propto H(\\Mat{V}|Y),\\label{eq:first-term}\n    \\end{align}\n    where $H(\\cdot)$ denotes the cross-entropy, and $D_{KL}(\\cdot)$ denotes the $KL$-divergence.\n    As a consequence, the first term in Equation \\eqref{eq:split_bc} is positive proportional to $H(\\Mat{V}|Y)$.\n   \n    This concludes the proof for the first compactness part of BC loss.\n    \n    We then inspect the second term in Equation \\eqref{eq:split_bc} to demonstrate its dispersion property:\n    \\begin{align}\n     & \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}})\\nonumber\\\\\n     \\geq& \\sum_{(u,i)\\in\\Set{O}^{+}} \\log{(\n     \\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}})} \\nonumber\\\\\n     \\geq& \\sum_{u\\in\\Set{U}}\\sum_{j\\in\\Set{N}_{u}}\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}\\nonumber\\\\\n     \\eqc& - \\sum_{u\\in\\Set{U}}\\sum_{j\\in\\Set{N}_{u}} \\norm{\\Mat{v}_u - \\Mat{v}_j}^2, \\label{eq:dispersion}\n    \\end{align}\n    where we drop the redundant terms aligned with the compactness objective in the second line, and adopt Jensen's inequality in the third line.\n    As shown in prior studies \\cite{WangS11}, minimizing this term is equivalent to maximizing entropy $H(\\Mat{V})$:\n    \\begin{align} \\label{eq:second-term}\n        \\sum_{(u,i)\\in\\Set{O}^{+}} \\log (\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_i - m}{\\tau}}+\\sum_{j\\in\\Set{N}_{u}}\\exp{\\frac{\\Mat{v}_u^T\\Mat{v}_j}{\\tau}}) \\propto -\\mathcal{H}(\\Mat{V}).\n    \\end{align}\n    As a result, the second term in Equation \\eqref{eq:split_bc} works as the dispersion part in BC loss.\n\\end{proof}\n\n\n\n\\begin{landscape}\n    \\begin{figure}[t]\n        \\centering\n        \\subcaptionbox{BPR loss (head)\\label{fig:head_bpr}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_bpr.pdf}}\n        \\subcaptionbox{Softmax loss (head)\\label{fig:head_softmax}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_info.pdf}}\n        \\subcaptionbox{IPS-CN \\cite{IPS-CN} (head) \\label{fig:head_ips}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_ips.pdf}}\n        \\subcaptionbox{BC loss (head)\\label{fig:head_bc}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/head_bc.pdf}}\n    \n        \\subcaptionbox{BPR loss (tail)\\label{fig:tail_bpr}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_bpr.pdf}}\n        \\subcaptionbox{Softmax loss (tail)\\label{fig:tail_softmax}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_info.pdf}}\n        \\subcaptionbox{IPS-CN \\cite{IPS-CN} (tail)\\label{fig:tail_ips}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_ips.pdf}}\n        \\subcaptionbox{BC loss (tail)\\label{fig:tail_bc}}{\n            \\vspace{-6pt}\n            \\includegraphics[width=0.23\\linewidth]{figures\/tail_bc.pdf}}\n       \n        \\caption{Magnified View of Figure \\ref{fig:toy-example}.}\n        \\label{fig:magnified-toy-example}\n        \\vspace{-20pt}\n    \\end{figure}\n    \\end{landscape}\n\n\n\\section{Experiments} \\label{sec:app_exp}\n\n\n\\begin{table}[b]\n    \\centering\n    \\vspace{-10pt}\n    \\caption{Dataset statistics.}\n   \n    \\label{tab:dataset-statistics}\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lrrrrrrr}\n    \\toprule\n     & KuaiRec & Douban Movie & Tencent & Amazon-Book & Alibaba-iFashion & Yahoo!R3 & Coat\\\\ \\midrule\n    \\#Users & 7175 & 36,644 & 95,709 & 52,643 & 300,000 & 14382 & 290 \\\\\n    \\#Items & 10611 & 22,226 & 41,602 & 91,599 & 81,614 & 1000 & 295 \\\\\n    \\#Interactions & 1062969 & 5,397,926 & 2,937,228 & 2,984,108 & 1,607,813 & 129,748 & 2,776 \\\\\n    Sparsity & 0.01396 & 0.00663 & 0.00074 & 0.00062 & 0.00007 \n    & 0.00902 &  0.03245\\\\ \\midrule\n    $D_{KL}$-Train & 1.075 & 1.471 & 1.425 & 0.572 & 1.678 & 0.854 & 0.356 \\\\\n    $D_{KL}$-Validation & 1.006 & 1.642 & 1.423 & 0.572 & 1.705 & 0.822 & 0.350\\\\\n    $D_{KL}$-Balanced & - & - & 0.003 & 0.000 & 0.323 & - & -  \\\\\n    $D_{KL}$-Imbalanced & - & - & 1.424 & 0.571 & 1.703 & - & - \\\\ \n    $D_{KL}$-Temporal & - & 1.428 & - & - & -  & - & - \\\\\n    $D_{KL}$-Unbiased & 1.666 & - & - & - & -& 0.100 & 0.109 \\\\ \\bottomrule\n    \\end{tabular}}\n\\end{table}\n\n\n\n\n\\subsection{Experimental Settings} \\label{sec:implementation}\n\\textbf{Datasets.} \nWe conduct experiments on eight real-world benchmark datasets: Tencent \\cite{Tencent}, Amazon-Book \\cite{Amazon-Book}, Alibaba-iFashion \\cite{Alibaba-ifashion}, Yelp2018 \\cite{LightGCN}, Douban Movie \\cite{Douban}, \\za{Yahoo!R3 \\cite{Yahoo}, Coat \\cite{ips}} and KuaiRec \\cite{KuaiRec}.\nAll datasets are public and vary in terms of size, domain, and sparsity.\nTable \\ref{tab:dataset-statistics} summarizes the dataset statistics, where the long-tail degree is monitored by KL-divergence between the item popularity distribution and the uniform distribution, \\emph{i.e., } $D_{KL}(\\hat{P}_{data}|| \\text{Uniform})$.\nA larger KL-divergence value indicates that the heavier portion of interactions concentrates on the head of distribution. \nIn the stage of pre-processing data, we follow the standard 10-core setting \\cite{PDA,KGAT} to filter out the items and users with less than ten interactions.\n\n\\noindent\\textbf{Data Splits.}\nFor comprehensive comparisons, almost all standard test distributions in CF are covered in the experiments: balanced test set \\cite{MACR,DICE,KDCRec}, randomly selected imbalanced test set \\cite{ESAM,ZhuWC20}, temporal split test set \\cite{PDA,DecRS,Regularized_Optimization}, and unbiased test set \\cite{ips,KuaiRec,Yahoo}. \nThree datasets (\\emph{i.e., } Tencent, Amazon-Book, Alibaba-iFashion) are partitioned into both balanced and randomly selected imbalanced evaluations. \nAs an intervention test, the balanced evaluation (\\emph{i.e., } uniform distribution) is frequently employed in recent debiasing CF approaches \\cite{MACR,DICE,KDCRec,Causal_inference}.\nDouban is split based on the temporal splitting strategy \\cite{MengMMO20}. \nKuaiRec is an unbiased, fully-observed dataset in which the feedback of the test set's interaction is explicitly collected.\n\n\n\\vspace{5pt}\n\\noindent\\textbf{Evaluation Metrics.}\nWe adopt the all-ranking strategy \\cite{KricheneR20}, \\emph{i.e., } for each user, all items are ranked by the recommender model, except the positive ones in the training set. To evaluate the quality of recommendation, three widely-used metrics are used: Hit Ratio (HR@$K$), Recall@$K$, Normalized Discounted Cumulative Gain (NDCG@$K$), where $K$ is set as $20$ by default.\n\n\\vspace{5pt}\n\\noindent\\textbf{Baselines.}\nWe validate our BC loss on two widely-used CF models, MF \\cite{BPR} and LightGCN \\cite{LightGCN}, which are representatives of the conventional and state-of-the-art CF models.\nWe compare BC loss with the popular debiasing strategies in various research lines: sample re-weighting (IPS-CN \\cite{IPS-CN}), bias removal by causal inference (MACR \\cite{MACR}, CausE \\cite{CausE}), and regularization-based framework (sam+reg \\cite{Regularized_Optimization}).\n\\za{We also compare BC loss to other standard losses used in collaborative filtering including most commonly used loss (BPR loss \\cite{BPR}) and newest proposed softmax losses (CCL \\cite{CCL} and SSM \\cite{SSM})}.\n\n\\vspace{5pt}\n\\noindent\\textbf{Parameter Settings.}\nWe conduct experiments using a Nivida-V100\nGPU (32 GB memory) on a server with a\n40-core Intel CPU (Intel(R) Xeon(R) CPU E5-2698 v4).\nWe implement our BC loss in PyTorch.\n\\wx{Our codes, datasets, and hyperparameter settings are available at \\url{https:\/\/github.com\/anzhang314\/BC-Loss} to guarantee reproducibility.}\nFor a fair comparison, all methods are optimized by Adam \\cite{Adam} optimizer with the batch size as 2048, embedding size as 64, learning rate as 1e-3, and the coefficient of regularization as 1e-5 in all experiments.\nFollowing the default setting in \\cite{LightGCN}, the number of embedding layers for LightGCN is set to 2.\nWe adopt the early stop strategy that stops training if Recall@20 on the validation set does not increase for 10 successive epochs.\nA grid search is conducted to tune the critical hyperparameters of each strategy to choose the best models \\emph{w.r.t.}~ Recall@20 on the validation set.\n\\za{For softmax, SSM, and BC loss, we search $\\tau$ in [0.06, 0.14] with a step size of 0.02.}\nFor CausE, 10\\% of training data with balanced distribution are used as intervened set, and $cf\\_pen$ is tuned in $[0.01,0.1]$ with a step size of 0.02.\nFor MACR, we follow the original settings to set weights for user branch $\\alpha=1e-3$ and item branch $\\beta=1e-3$, respectively. We further tune hyperparameter $c=[0, 50]$ with a step size of 5.\n\\za{For CCL loss, we search $w$ in $\\{1,2,5,10,50,100,200\\}$, $m$ in the range $[0.2,1]$ with a step size of 0.2. When it come to the number of negative samples, the softmax, SSM, CCL, and BC loss set 128 for MF backbone and in-batch negative sampling for LightGCN models.}\n\n\n\n\\subsection{Training Cost} \n\\begin{table}[t]\n    \\centering\n    \\caption{Training cost on Tencent (seconds per epoch\/in total). }\n   \n    \\label{tab:elapse_time}\n    \\resizebox{0.9\\linewidth}{!}{\n    \\begin{tabular}{l|rrrrrr}\n    \\toprule\n     & Backbone & +IPS-CN & +CausE & +sam+reg & +MACR & +BC loss\\\\ \\midrule\n    MF & 15.5 \/ 17887 & 17.8 \/ 10662 & 16.6 \/ 1859 & 18.2 \/ 3458 & 160 \/ 17600 &  36.1 \/ 12815\\\\\n    LightGCN & 78.6 \/ 4147 & 108 \/ 23652 & 47.2 \/ 3376 & 49.8 \/ 10458 & 135 \/ 20250 & 283 \/ 7075\\\\ \\bottomrule\n    \\end{tabular}}\n\\end{table}\n\n\nIn terms of time complexity, as shown in Table \\ref{tab:elapse_time}, we report the time cost per epoch and in total of each baselines on Tencent.\nCompared with backbone methods (\\emph{i.e., } MF and LightGCN), BC loss adds very little computing complexity to the training process. \n\n\n\n\n\\subsection{Evaluations on Unbiased Test Set} \\label{sec:KuaiRec}\n\n\\noindent\\textbf{Motivation.}\nBecause of the missing-not-at-random condition in a real recommender system, offline evaluation on collaborative filtering and recommender system is commonly acknowledged as a challenge.\nTo close the gap, Yahoo!R3 \\cite{Yahoo} and Coat \\cite{ips} are widely used, which offer unbiased test sets that are collected using the missing-complete-at-random (MCAR) concept.\nAdditionaly, newly proposed KuaiRec \\cite{KuaiRec} also provides a fully-observed unbiased test set with 1,411 users over 3,327 videos.\nWe conduct experiments on all these datasets for comprehensive comparison, and KuaiRec is also included as one of our unbiased evaluations for two key reasons:\n1) It is significantly larger than existing MCAR datasets (\\emph{e.g., } Yahoo! and Coat); \n2) It overcomes the missing values problem, making it as effective as an online A\/B test.\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Performance comparison on KuaiRec dataset. }\n  \\label{tab:KuaiRec}\n  \\resizebox{0.7\\columnwidth}{!}{\n  \\begin{tabular}{l|ccc|ccc}\n  \\toprule\n   & \\multicolumn{3}{c|}{Validation} & \\multicolumn{3}{c}{Unbiased Test} \\\\\n  \\multicolumn{1}{c|}{} & HR & Recall & NDCG & HR & Recall & NDCG \\\\\\midrule\n  LightGCN & 0.299 & 0.069 & 0.051 & 0.104 & 0.0038  & 0.0064  \\\\\n  + IPS-CN & 0.255 & 0.056 & 0.042 & \\underline{0.109} & \\underline{0.0073} & \\underline{0.0083}  \\\\\n  + CausE & 0.292 & 0.067 & 0.050 & 0.101 & 0.0056 & 0.0077 \\\\\n  + sam+reg & 0.274 & 0.060 &  0.047 & 0.107 & 0.0069 & 0.0080 \\\\\n  + BC loss & 0.343 & 0.076 & 0.062 & \\textbf{0.139}* & \\textbf{0.0077}* & \\textbf{0.0115}* \\\\\\midrule\n  Imp.\\% & - & - & - & 27.5\\% & 4.05\\% & 38.6\\% \\\\\\bottomrule\n  \\end{tabular}}\n  \\vspace{-15pt}\n\\end{table}\n\n\\noindent\\textbf{Parameter Settings.} For BC loss on Yahoo!R3 and Coat, we search $\\tau_1$ in [0.05, 0.21] with a step size of 0.01, and $\\tau_2$ in [0.1, 0.6] with a step size of 0.1, and search the number of negative samples in [16, 32, 64, 128]. We adopt the batch size of 1024 and learning rate of 5e-4.\n\n\\noindent\\textbf{Results.}\nTable \\ref{tab:KuaiRec} and \\ref{tab:Yahoo&Coat} illustrate the unbiased evaluations on KuaiRec, Yahoo!R3, and Coat dataset using LightGCN and MF as backbone models, respectively.\nThe best performing methods are bold and starred, while the strongest baselines are underlined;\nImp.\\% measures the relative improvements of BC loss over the strongest baselines.\nBC loss is consistently superior to all baselines \\emph{w.r.t.}~ all metrics. \nIt indicates that BC loss truly improves the generalization ability of recommender.\n\n\n\n\n\n\n\\subsection{Performance Comparison with Standard Loss Functions in CF} \\label{sec:standard_loss}\n\n\\noindent\\textbf{Motivation.}\nTo verify the effectiveness of BC loss as a standard learning strategy in collaborative filtering, we further conduct the experiments over various datasets between BC loss, BPR loss, CCL loss \\cite{CCL}, and SSM loss \\cite{SSM}. We choose these three losses as baselines for two main reasons: 1) BPR loss is the most commonly applied in recommender system; 2) SSM and CCL are most recent proposed losses, where SSM is also a softmax loss and CCL employs a global margin. \n\n\\noindent\\textbf{Results.}\nTable \\ref{tab:loss_compare} reports the performance on both balanced and imbalanced test sets on various datasets among different losses. \nWe have two main observations: (1) Clearly, our BC loss consistently outperforms CCL and SSM; (2) CCL and SSM achieve comparable performance to BC loss in the imbalanced evaluation settings, while performing much worse than BC loss in the balanced evaluation settings. This indicates the superiority of BC loss in alleviating the popularity bias, and further justifies the effectiveness of the bias-aware margins.\n\nTable \\ref{tab:Yahoo&Coat} shows the unbiased evaluations on Yahoo!R3 and Coat dataset. \nWith regard to all criteria, BC loss constantly outperforms all other losses. \nIt verifies the effectiveness of instance-wise bias margin of BC loss.\n\n\\begin{table*}[t]\n\\caption{Performance comparison on Tecent, Amazon-book and Alibaba-iFashion datasets.}\n\\label{tab:loss_compare}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|cc|cc|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{1}{c|}{} & \\multicolumn{4}{c|}{Tencent}       & \\multicolumn{4}{c|}{Amazon-book}   & \\multicolumn{4}{c}{Alibaba-iFashion} \\\\\n\\multicolumn{1}{c|}{} &\n    \\multicolumn{2}{c}{Balanced} &\n    \\multicolumn{2}{c|}{Imbalanced} &\n    \\multicolumn{2}{c}{Balanced} &\n    \\multicolumn{2}{c|}{Imbalanced} &\n    \\multicolumn{2}{c}{Balanced} &\n    \\multicolumn{2}{c}{Imbalanced} \\\\\n   \n\\multicolumn{1}{c|}{} & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall & NDCG   & Recall  & NDCG    & Recall  & NDCG   \\\\\n\\midrule\nBPR    & 0.0052 & 0.0040 & 0.0982 & 0.0643 & 0.0109 & 0.0103 & 0.0850 & 0.0638 & 0.0056  & 0.0028  & 0.0843 & 0.0411 \\\\\nSSM            & 0.0055 & 0.0045 & \\underline{0.1297} & \n\\underline{0.0872} & 0.0156 & 0.0157 & 0.1125 & \\underline{0.0873} & \n\\underline{0.0079}  & \\underline{0.0040}  & \\underline{0.0963}  & \\underline{0.0436} \\\\\nCCL           & \\underline{0.0057} & \\underline{0.0047} & 0.1216 & 0.0818 & \\underline{0.0175} & \\underline{0.0167} & \\underline{0.1162} & \\underline{0.0927} & 0.0075  & 0.0038  & 0.0954  & 0.0428 \\\\\nBC Loss            & \\textbf{0.0087}* & \\textbf{0.0068}* & \\textbf{0.1298}* & \\textbf{0.0904}* & \\textbf{0.0221}* & \\textbf{0.0202}* & \\textbf{0.1198}* & \\textbf{0.0948}* & \\textbf{0.0095}*  & \\textbf{0.0048}*  & \\textbf{0.0967}*  & \\textbf{0.0487}* \\\\\\midrule\nImp. \\%    &  52.6\\%  &  44.7\\%      &    0.1\\%  &  3.7\\%      &   26.3\\%     &    21.0\\%    &    3.1\\%    &  2.3\\%      &    20.3\\%     &    20.0\\%     &    0.4\\%     & 11.7\\%        \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Performance comparison on Yahoo!R3 and Coat dataset. }\n    \\label{tab:Yahoo&Coat}\n    \\resizebox{0.4\\columnwidth}{!}{\n    \\begin{tabular}{l|cc|cc}\n    \\toprule\n     & \\multicolumn{2}{c|}{Yahoo!R3} & \\multicolumn{2}{c}{Coat} \\\\\n    \\multicolumn{1}{c|}{} & Recall & NDCG & Recall & NDCG \\\\\\midrule\n    \n    IPS-CN &  0.1081 & 0.0487 & 0.1700 & 0.1377 \\\\\n    CausE  & 0.1252 & 0.0537 & \\underline{0.2329} & 0.1635  \\\\\n    sam+reg & 0.1198 & 0.0548 & 0.2303 & 0.1869 \\\\\n    MACR & 0.1243 & 0.0539 & 0.0798 & 0.0358 \\\\\\midrule\n    BPR & 0.1063 & 0.0476 & 0.0741 & 0.0361\\\\\n    SSM & \\underline{0.1470} & \n    \\underline{0.0688}  & 0.2022 & 0.1832\\\\\n    CCL & 0.1428 & 0.0676 & 0.2150 & \\underline{0.1885} \\\\\n    BC loss  & \\textbf{0.1487}* & \\textbf{0.0706}* & \\textbf{0.2385}* & \\textbf{0.1969}* \\\\\\midrule\n    Imp.\\% & 1.2\\% & 2.6\\% & 2.4\\% & 4.5\\% \\\\\\bottomrule\n    \\end{tabular}}\n  \\end{table}\n  \n \n  \\subsection{Study on BC Loss (RQ3) - Effect of Popularity Bias Extractor} \\label{sec:bias_extractor}\n  \n  \\noindent\\textbf{Motivation.} To check the effectiveness of popularity bias extractor, we need to answer two main questions: 1) what kinds of interactions will be learned well by the bias extractor? What does the learned bias-angle distribution look like? 2) can BC loss benefit from the bias margin extracted according to the popularity bias extractor in various groups of interactions? \n  We devise the following experiment to tackle the aforementioned problems.\n  \n  \\noindent\\textbf{Experiment Setting.}\n  Users can be divided into three parts: head, mid, and tail, based on their popularity scores. Analogously, items can be partitioned into the head, mid, and tail parts. As such, we can categorize all user-item interactions into nine subgroups. \n  We have visualized the learned angles for various types of interactions in the following table \\ref{fig:angles}. \n  \n  \\noindent\\textbf{Results.} The table \\ref{fig:angles} shows the learned angles over all subgroups. We find that interactions between head users and head items tend to hold small angles. Moreover, as evidenced by the high standard deviation, the interactions stemming from the same subgroup types are prone to receive a wide range of angular values. This demonstrates the variability and validity of instance-wise angular margins.\n  \n  \\begin{figure*}\n      \\centering\n      \\includegraphics[width=0.5\\linewidth]{figures\/angle.png}\n      \\caption{Visualization of popularity bias angle for different types of interactions on Tencent.}\n      \\label{fig:angles}\n  \\end{figure*}\n  \n  \n  \n  \n  \n  \n  \n \n  \n\n\\subsection{Visualization of interaction bias degree}\nFigures \\ref{fig:item_pop} and \\ref{fig:user_pop} illustrate the relations between popularity scores and bias degree extracted by popularity bias extractor \\emph{w.r.t.}~ user and item sides, respectively. Specifically, positive trends are shown, where their relations are also quantitatively supported by Pearson correlation coefficients. (0.7703 and 0.662 for item and user sides, respectively). It verifies the power of popularity embeddings to predict the popularity scores --- that is, user popularity embeddings derived from the popularity bias extractor are strongly correlated and sufficiently predictive to user popularity scores; analogously to the item side.\n  \\begin{figure*}[t]\n      \\centering\n      \\subcaptionbox{Bias degree \\emph{w.r.t.}~ Item popularity bias \\label{fig:item_pop}}{\n        \n          \\includegraphics[width=0.43\\linewidth]{figures\/item_pop.png}}\n      \\subcaptionbox{Bias degree \\emph{w.r.t.}~ User popularity bias\\label{fig:user_pop}}{\n        \n          \\includegraphics[width=0.43\\linewidth]{figures\/user_pop.png}}\n      \\caption{Visualizations of relationships between interaction bias degree estimated by our popularity bias extractor and item\/user popularity statistics. Pearson correlation coefficients are provided.}\n  \\end{figure*}\n  \n  \n  \n  \n  \\begin{table}\n    \\centering\n    \\caption{Model architectures and hyper-parameters}\n    \\label{tab:parameter}\n    \\resizebox{0.6\\columnwidth}{!}{\n    \\begin{tabular}{l|c|c|c|c|c|c}\n    \\toprule\n     & \\multicolumn{5}{c}{BC loss hyper-parameters} \\\\\\midrule\n     \n    \\multicolumn{1}{c|}{} & $\\tau_1$ & $\\tau_2$ & lr & batch size & No. negative samples \\\\\\midrule\n    \\textbf{MF} \\\\\\midrule\n    Tencent &  0.06 &  0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n    iFashion & 0.08 & 0.1 & 1e-3 & 2048 & 128  \\\\\\midrule\n    Amazon & 0.08 & 0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n    Douban  & 0.08 & 0.1 & 1e-3 & 2048 & 128 \\\\\\midrule\n    Yahoo!R3 & 0.15 & 0.2 & 5e-4 & 1024 & 128 \\\\\\midrule\n    Coat & 0.09 & 0.4  & 5e-4 & 1024 & 64\\\\\\midrule\n    \\textbf{LightGCN} \\\\\\midrule\n      Tencent &  0.12 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n      iFashion & 0.14 & 0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n    Amazon & 0.08 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n    Douban  & 0.14 &  0.1 & 1e-3 & 2048 & in-batch\\\\\\midrule\n    \\end{tabular}}\n  \\end{table}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent X-ray observations of galaxy clusters reveal various phenomena\nin the gaseous haloes of clusters, such as mergers, cavities, shocks\nand cold fronts (CFs).  CFs are thought to be contact discontinuities,\nwhere the density and temperature jump, while the pressure\nremains continuous (up to projection\/resolution effects).  They are\ncommon in clusters \\citep{markevitch07} and vary in morphology (they\nare often arcs but some linear or filamentary CFs exist), contrast\n(the density jump is scattered around a value of $\\sim 2$), quiescence\n(some are in highly disturbed merging regions, and some in smooth,\nquiet locations), and orientation (some are arcs around the cluster\ncentres, and some are radial, or spiral in nature).  CFs have been postulated to\noriginate from cold material stripped during mergers \\citep[][ for\n  example]{markevitch00} or sloshing of the intergalactic medium\n\\citep[IGM; ][]{markevitch01,ascasibar06}.\nIn some cases, a metallicity gradient is observed across the CF,\nindicating that it results from stripped gas, or radial motions of gas\n\\citep[see ][and references therein]{markevitch07}.  Shear is often\nfound along CFs in relaxed cluster cores, implying nearly sonic bulk flow\nbeneath (towards the cluster's core) the CF \\citep{keshet10}.  Whether\nCFs are present at very large radii ($\\gtrsim 500~{\\rm kpc}$) is\ncurrently unknown because of observational limitations.\n\nCold fronts are not rare. \\citet{markevitch03} find CFs in more than\nhalf of their cool core clusters, suggesting that most, if not all\nsuch clusters harbor CFs \\citep{markevitch07}. \\citet{ghizzardi10}\nsurvey a sample of 42 clusters using XMM-Newton \nand find that  19 out of 32 nearby ($z<0.075$) clusters host at least one CF. Based on \nobservational limitations, and orientations of cold fronts, they\nconclude that this is a lower limit. While many CFs are directly\nattributed to mergers, out of $23$  clusters in their\nsample that seem relaxed, $10$ exhibit\ncold fronts. These 10 have systematically lower \ncentral entropy values. Another useful compilation of clusters for\nwhich cold fronts have been observed can be found in\n\\citet{owers09}. Out of the 9 high-quality Chandra observations of clusters\nwith observed cold fronts, 3 of them, RXJ1720.1+2638, MS1455.0+2232\nand Abel 2142 have non-disturbed morphologies.\n\nIn what follows, we propose that some CFs are produced by merging of\nshocks.  When two shocks propagating in the same direction merge, a CF is always\nexpected to form.  Its parameters can be calculated by solving the\nshock conditions and corresponding Riemann problem.  Most of the\nscenarios in which shocks are produced (quasars, AGN jets, mergers)\npredict that shocks will be created at, or near, a cluster centre, and\nexpand outwards (albeit not necessarily isotropically).  When a\nsecondary shock trails a primary shock, it always propagates faster (supersonically with\nrespect to the subsonic downstream flow of the primary shock), so\ncollisions between outgoing shocks in a cluster are inevitable if they\nare generated within a sufficiently short time interval. We note that\ncontact discontinuities can also form from head on collisions between\nshocks, and various other geometrical configurations. We restrict\nourselves in this paper to merging of two shock propagating in roughly\nthe same direction.\n\nIn \\S~\\ref{sec:riemann} we solve for the parameters of CFs that are\ncaused by merging two arbitrary planar shocks. We show results of\nshock-tube hydrodynamic simulation that agree with the calculated\nanalytical prediction. In \\S~\\ref{sec:spherical}, we describe our spherical\nhydrodynamic simulations and realistic cluster profiles that are later\nused to investigate\nhow these\nshocks may be produced, and discuss various ways to produce shocks in\ncluster settings. We identify two general mechanisms that\ncreate shocks in clusters: energetic explosions near the centre, or\nchanges and recoils in the potential well of the cluster that\ncorrespond to various merger events.\nThese mechanisms are simulated in \\S~\\ref{sec:explosions} and \\S~\\ref{sec:mergers} \nand also serve to demonstrate the robustness of the shock\ninduced cold fronts (SICFs) that form.\nIn \\S~\\ref{sec:virial} we follow cluster growth and accretion over a\nHubble time by using self-consistent 1D hydrodynamic simulations of gas and\ndark-matter. The virial shock that naturally forms serves as a primary\nshock, and we demonstrate that SICFs form when secondary shocks\npropagating from the centre merge with it. We show that oscillations of this\nshock around its steady state position can produce a series of SICFs. \nIn \\S~\\ref{sec:stabil_observe} we discuss aspects regarding the stability and\nsustainability of cold fronts in general, and derive some\nobservational  predictions that could serve to distinguish between\nCFs produced via this and other mechanisms. In \\S~\\ref{sec:summary}\nwe discuss our findings and conclude. Compact expressions for SICF\nparameters can be found in appendix \\S~\\ref{sec:gamma53}.\n\n\\section{Merging of Two Trailing Shocks}\n\\label{sec:riemann}\n\nIn this section we examine two planar shocks propagating in the same\ndirection through a homogeneous medium and calculate the parameters of\nthe contact\ndiscontinuity that forms when the \nsecond shock (secondary) overtakes the leading (primary) one.  This problem is fully\ncharacterized (up to normalization) by the Mach numbers of the primary\nand secondary shocks, $M_0$ and $M_1$.  At the instant of collision, a\ndiscontinuity in velocity, density and pressure develops,\ncorresponding to a Riemann problem \\citep[second case\nin][\\S93]{landau59}.  The discontinuity evolves into a (stronger)\nshock propagating in the initial direction and a reflected rarefaction\nwave, separated by a contact discontinuity which we shall refer to as\na shock induced CF (SICF).  The density is higher (and the entropy\nlower) on the SICF side closer to the origin of the shocks.  In\nspherical gravitational systems, this\nyields a Rayleigh-Taylor stable configuration if the shocks are\nexpanding outwards.  Now, we derive the discontinuity parameters.\n\n\\subsection{The Discontinuity Contrast}\n\n\\begin{figure}\n\\includegraphics[width=3.5in, trim =190pt 82pt 200pt 30pt,clip=true ]{mach_scheme}\n\\caption{Schematics of shock merging. Pressure p is shown vs. an\n  arbitrary spatial coordinate x, which is not fixed between the\n  panels. {\\it Top:} two shocks, propagating according to the arrows,\n  induce transitions between zones $0\\rightarrow 1$ and between $1\\rightarrow 2$. {\\it Middle\n  panel}, the instance of collisions between the shocks. The transition\n  $0\\rightarrow 2$ does not correspond to consistent shock jump\n  conditions. {\\it Bottom:} at the Lagrangian location of the\n  collision an isobaric discontinuity between $3_{o}$ and $3_{i}$\n  forms, separating between a forward moving shock \n  ($0\\rightarrow 3_{o}$) and a reflected rarefaction (smooth transition\n  between $2$ and $3_{i}$).\\label{fig:scheme}}\n\\end{figure}\n\nWe consider an ideal gas with an adiabatic index $\\gamma.$ Before\nthe shocks collide, denote the unshocked region as zone $0$, the\nregion between the two shocks as zone $1$, and the doubly shocked\nregion as zone $2$ (these definitions are demonstrated schematically\nin fig. \\ref{fig:scheme}).  After the merging, regions $0$ and $2$ remain\nintact, but zone $1$ vanishes and is replaced by two regions, zone\n$3_{o}$ (the outer region for outgoing\nspherical shocks; adjacent to zone $0$) and zone $3_{i}$ (inner; adjacent to $2$),\nseparated by the SICF.  We refer to the plasma density, pressure,\nvelocity and speed of sound respectively as $\\rho$, $p$, $u$ and $c$.\nVelocities of shocks at the boundaries between zones are denoted by\n$v_i$, with $i$ the upstream zone.  We rescale all parameters by the\nunshocked parameters $\\rho_0$ and $p_0$.\n\nThe velocity of the leading shock is\n\\begin{eqnarray}\nv_0&=&u_0+M_0~c_0,\\label{eq:v0}\n\\label{eq:v}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nc_i&=&\\left(\\frac{\\gamma p_i}{\\rho_i}\\right)^{1\/2}\n\\end{eqnarray}\nfor each zone $i$.  Without loss of generality, we measure velocities\nwith respect to zone $0$, implying $u_0=0$.\n\nThe state of zone $1$ is related to zone $0$ by the Rankine-Hugoniot conditions,\n\\begin{eqnarray}\np_1&=&p_0\\frac{2\\gamma M_0^2-\\gamma+1}{\\gamma+1}, \\label{eq:purho1}\\\\\nu_1&=& u_0 + \\frac{p_1-p_0}{\\rho_0(v_0-u_0)}, \\label{eq:purho2}\\\\\n\\rho_1&=&\\rho_0\\frac{u_0-v_0}{u_1-v_0}. \\label{eq:purho3}\n\\label{eq:purho}\n\\end{eqnarray}\nThe equations are ordered such that each equations uses only known\nvariables from previous equations.\nThe state of the doubly shocked region (zone $2$) is related to zone\n$1$ by reapplying equations \\ref{eq:v}-~\\ref{eq:purho} with the\nsubscripts $0,1$ replaced respectively by $1,2$, and $M_0$ replaced by\n$M_1$.\n\nAcross the contact\ndiscontinuity that forms as the shocks collide (the SICF), the\npressure and velocity are continuous but\nthe density, temperature and entropy are not; the CF contrast is\ndefined as ${q}\\equiv\\rho_{3{i}}\/\\rho_{3{o}}$.  Regions $0$ and\n$3_{o}$ are related by the Rankine-Hugoniot jump conditions across\nthe newly formed shock,\n\\begin{eqnarray}\np_3&=&p_0\\frac{(\\gamma+1)\\rho_{3{o}}-(\\gamma-1)\\rho_0}{(\\gamma+1)\\rho_0-(\\gamma-1)\\rho_{3{o}}}, \\label{eq:u2u0a}\\\\\nu_3-u_0&=&\\left[(p_3-p_0)\\left(\\frac{1}{\\rho_0}-\\frac{1}{\\rho_{3{o}}}\\right)\\right]^{1\/2} \\,. \\label{eq:u2u0b}\n\\end{eqnarray}\nThe adiabatic rarefaction from pressure $p_2$ down to $p_3=p_{3{i}}=p_{3{o}}$ is determined by\n\\begin{eqnarray}\nu_2-u_3&=&-\\frac{2c_2}{\\gamma-1}\\left[1-\\left(\\frac{p_3}{p_2}\\right)^{(\\gamma-1)\/2\\gamma}\\right] . \\label{eq:u2u0c}\n\\end{eqnarray}\nThe system can be solved by noting that the sum of\neqs. (\\ref{eq:u2u0b}) and (\\ref{eq:u2u0c}) equals $u_2-u_0$, fixing\n$\\rho_{3{o}}$ as all other parameters are known from\neqs. (\\ref{eq:v0}-\\ref{eq:u2u0a}).\n\nFinally, the Mach number of the new shock and the rarefacted density are given by\n\\begin{eqnarray}\nM_f^2 & = & \\frac{2\\rho_{3{o}}\/\\rho_0}{(\\gamma+1)-(\\gamma-1)\\rho_{3{o}}\/\\rho_0} \\,, \\\\\n\\rho_{3{i}}&=&\\rho_2(p_3\/p_2)^{1\/\\gamma} . \\label{eq:rho3}\n\\end{eqnarray}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{m0m1}\n\\caption{Contrast ${q}$ (labeled contours) of a CF generated by a\n  collision between two trailing \n  shocks with Mach numbers $M_0$ and $M_1,$ for $\\gamma=5\/3$ .\n  The color map shows the final Mach number $M_f$ normalized by $M_0~M_1.$\\label{fig:m0m1}}\n\\end{figure}\nClosed-form expressions for ${q}$ and $M_f$ are presented in \\S~\\ref{sec:gamma53}.\nFigure \\ref{fig:m0m1} shows the discontinuity contrast $q$ for various\nvalues of $M_0$ and $M_1,$ for $\\gamma=5\/3.$  Typically ${q}\\sim 2,$\nranging from $1.45$ for $M_0=M_1=2$, for example, to\n${q}_{max}=2.653$. The figure colorscale shows the\ndimensionless factor $f\\equiv M_f\/(M_0 M_1)$, ranging between\n$0.75$ and $1$ for $1<\\{M_0,M_1\\}<100$.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{figRmax}\n\\caption{ The maximal SICF contrast (solid, left axes) as a function of $\\gamma$.\nIt occurs in a collision between a very strong shock $M_0\\to \\infty$ and a trailing shock of finite Mach number $M_1$ (dashed, right axes).\nFor $\\gamma= 5\/3$, ${q}_{max} = 2.65$ with $M_1 = 6.65$.\\label{fig:qmax}\n}\n\\end{figure}\nThe maximal contrast ${q}_{max}$ depends only on the\nadiabatic index, as shown in fig. \\ref{fig:qmax}.\nIt occurs for a very strong primary shock $M_0\\to\\infty$ and a finite secondary $M_1$;\nsee \\S~\\ref{sec:gamma53} for details.\nFor $\\gamma=5\/3$ we find ${q}_{max}=2.653$, achieved\nfor $M_0\\gg 1$ and $M_1\\simeq 6.65$.\n\n\\subsection{Planar Hydrodynamic Example of SICF Formation}\n\\label{sec:planar}\n\\begin{figure}\n\\includegraphics[width=3.6in, trim=35 0 0 65, clip=true]{planar}\n\\caption{Shock tube simulation of a contact discontinuity (SICF)\n  formed by merging of two shocks. The piston propagates from $x=0$ in the positive\n  direction at a supersonic speed creating the shock between\n  $0\\rightarrow 1$. At t=.04 the piston's velocity\n  abruptly increases causing a second shock to propagate between\n  $1\\rightarrow 2$. At the position and time where the shocks merge a contact\n  discontinuity forms between regions $3_{o}$ and $3_{i}$. {\\it Top:}\n  density (normalized so that $\\rho_0=1$). Red lines mark the\n  trajectories of some Lagrangian cell boundaries. The zones here are\n  analougous to those in \\S~\\ref{sec:riemann} and\n  fig.~\\ref{fig:scheme}. {\\it Bottom:} entropy\n  (normalized so that $S_0\\equiv T_0\/\\rho_0^{2\/3}=1$). Note that there\n  is no entropy change between zone $2$ and $3_{i}$ because the\n  gas rarefies adiabatically between them. The contrast of the SICF,\n  ${q}$, is the density jump between zones $3_{o}$ and\n  $3_{i}$. \\label{fig:planar}\n}\n\\end{figure}\nTo demonstrate the formation of an SICF we present in\nfig.~\\ref{fig:planar} results from a 1D planar hydrodynamic code. The\nnumerical scheme of the hydrodynamic calculation is Lagrangian, with\nthe locations and velocities\ndefined at edges of cells and thermodynamic variables defined at the centres of\ncells.  The density $\\rho$ and internal energy $e$, are the free thermodynamic\nvariables, setting the pressure $p$, temperature $T$ and the\nentropy $S$ according to an ideal gas equation of state. Throughout\nthe\npaper we consider monoatomic gas, with an adiabatic constant of\n$\\gamma=5\/3$ unless stated otherwise. The time propagation is\ncalculated by a leap-frog explicit scheme. The timesteps here are\nadaptive, and set by a standard Courant Friedrichs Levy\ncondition. Discontinuities are integrated over by employing\nfirst and second order Von Neumann artificial viscosity.\n\nThe simulation presented in fig.~\\ref{fig:planar} was run with 500\ncells. Without loss of\ngenerality, we can assume $\\rho_0=e_0=T_0=1$. The definition of\ntemperature here implies an arbitrary\nheat capacity that does not enter into the equations of motions. We define\nthe entropy as $S\\equiv T\/\\rho^{\\gamma-1}$ (the exponent of the usual\nformal definition) so $S_0=1$ as well.\nWe use external boundary conditions of a piston propagating into the\nmaterial from $X=0$ in the positive direction at Mach number\n$M_1=2.35$. At time $t=0.04$ the piston's velocity is increased to\ncreate a second shock propagating in the same direction, with\n$M_2=1.76$ with respect to the previously shocked gas. The contact\ndiscontinuity that forms has a density ratio\\footnote{The SICF\n  strength is calculated in the simulation according to\n  ${q}=(S_{3{i}}\/S_{3{o}})^{-1\/\\gamma}$ because in the rarefied zone, $3_{i}$ the\n  density is not constant, and so harder to measure directly, while\n  the entropy is constant throughout the zone.} ${q}\\equiv\n\\rho_{3{i}}\/\\rho_{3{o}}=1.466$ while the theoretical prediction\naccording \nto equations \\ref{eq:v}-\\ref{eq:rho3} is ${q}=1.459$ - in satisfactory\nagreement for the resolution used, and for the accuracy needed in\nthe discussion presented here. The\nsame technique has also been performed for larger Mach numbers and\nyielded results with comparable accuracy.\n\n\\section{Numerical Investigation of Spherical Cold Fronts }\n\\label{sec:spherical}\n\\subsection{Driving Shocks}\nWe explore different ways to drive central shocks through clusters of\ngalaxies. The first, most intuitive method is external energy\ninjection at the centre (i.e. explosions), which would correspond, for\nexample, to mechanical energy released in \nAGN bursts. Observations in X-ray and radio indicate a wide variety of\nshocks and sound waves \\citep{fabian03b,forman07} emitted repeatedly,\nperhaps even periodically from a central AGN.\nMany clusters also include hot and diffuse bubbles, that are most likely inflated\nby AGN jets that are decollimated by the ambient cluster gas\n\\citep[see review by ][and reference\ntherein]{mcnamara07}.\nIf these bubbles are a plausible mechanism to counteract the\novercooling problem, the energies associated with them must be\ncomparable to the cooling rate of the clusters \\citep[$\\gtrsim 10^{44}{\\rm\n  erg~sec}^{-1}$; ][]{edge90,david93,markevitch98} over some fraction\nof the Hubble time, indicating that the total energy injection is\n$\\gtrsim 10^{61}{\\rm erg}$.\n The process of inflation of these bubbles sends\nshocks through the ICM, but at the same time the extremely hot,\nunderdense bubbles alter the radial\nstructure of the ICM in a non trivial manner. In \\S~\\ref{sec:explosions} we\ndemonstrate (noting the limitations of the spherically symmetric\nanalysis in that case) that mergers between shocks driven by such central\nexplosions can produce SICFs.\n\nShocks can also be driven gravitationally. Mergers of galaxies and subhaloes\nchange the structure and depth of the potential well causing sound\nwaves and shocks to propagate through the haloes. These shocks\ntypically do not considerably disturb the ICM structure, leaving a\nstatic  gaseous halo after they propagate. We show in\n\\S~\\ref{sec:mergers} that abrupt changes to the gravitational\npotential of the halo directly produce shocks. In particular, in the\nfirst passage stage of a merger, two outward propagating shocks are\nproduced: one when the halo contracts due to the increase of the\ngravitational force, and another when the halo abruptly rarefies once\nthe merging subhalo flies out. The numerical investigation of explosion and merger\ndriven SICF is performed using a spherical one\ndimensional hydrodynamic code within a static potential well\n\\citep[a version of the code ``Hydra'' documented in ][that is stripped from the\ndynamic dark matter evolution and self-gravitating gas]{bd03} that,\nalong with the initial conditions, is described in\n\\S~\\ref{sec:hydra}.\n\nIn \\S~\\ref{sec:virial} we investigate the interaction of a secondary\nshock with the virial shock that is always expected at edges of\nclusters. To produce this primary shock, we simulate the\ncosmological evolution of a cluster by using a hydrodynamic\nscheme that includes a self-gravitating gas that flows within self-gravitating,\ndynamic dark matter shells, and starts from spherical cosmological\noverdensities at a redshift $z=100$. We show that perturbations can either be\ninvoked manually, or occur naturally when infalling dark matter has\nsufficiently radial orbits so it interacts with the central core\ndirectly. In this context, we also show that low amplitude\nperturbations in the core send sound waves that can steepen into a\nshock that interacts with the virial shock.\n\n\\subsection{The ``Hydra'' Hydrodynamical Code}\n\\label{sec:hydra}\nThe cosmological implications of shock interactions in clusters are tested\nusing 1D, spherical hydrodynamic simulations performed with\n``Hydra'' \\citep{bd03}.\nThe code is run here in two modes: cosmological and static DM (dark\nmatter) modes. In \\S~\\ref{sec:staticdm} we\nuse the code in the static DM mode, a simplified\nconfiguration in which the baryons do not self-gravitate, and there\nis no evolution of the dark matter. Rather, baryons start\nin hydrostatic equilibrium within a fixed cluster-like potential well. In\n\\S~\\ref{sec:virial} we use the code in its cosmological mode,\nutilizing the full\ncapabilities of the code to evolve baryons and dark matter\nself-consistently from a cosmological initial perturbation at high\nredshift to $z=0$. Next, we\nsummarize the physical processes implemented in both modes\nof ``Hydra'' and then highlight the difference in\nboundary conditions and initial conditions corresponding\nto each modus operandi.\n\nSimilar to the planar code described in\n\\S~\\ref{sec:planar}, the code uses a Lagrangian scheme. The\npositions of baryonic spherical shells are defined by their boundaries (with the\ninnermost boundary at $r=0$), and the thermodynamic properties are\ndefined at centres of shells. A summary of the fluid equations that\nare solved,\nand the numerical difference scheme is available in \\citet{bd03}.\nThe code (in its cosmological mode) also evolves thin discrete dark\nmatter shells, that propagate\nthrough the baryons, and interact with them gravitationally. The full\ndiscrete force equation for the ``i''th baryonic shell is thus:\n\\begin{equation}\n\\ddot{r_i}=-4\\pi r^2\\frac{\\Delta P_i}{\\Delta\n  M_i}-\\frac{GM_i}{(r_i+\\epsilon)^2}+\\frac{j_i^2}{r_i^3},\n\\label{eq:force}\n\\end{equation}\nwith $r_i,M_i,j_i$ the radius of boundary $i$, the mass (baryonic +\nDM) enclosed\nwithin that radius, and the specific angular momentum prescribed to\nthis shell respectively, and $\\Delta P_i,\\Delta M_i$ the pressure\ndifference and baryonic mass difference between\nthe centres of shells $i+1$ and\n$i$. $\\epsilon$ is the gravitational smoothing length ($50~{\\rm pc}$ and\n $500~{\\rm pc}$ in the static DM and cosmological modes respectively).\n\nThe dark matter and baryonic shells are assigned\nangular momentum that acts as an outward centrifugal force designed\nto repel shells that are close to the singularity at the centre, in\nanalogy to the angular momentum of a single star, or of\nthe gaseous or stellar disk in realistic systems, keeping them from falling into the centre\nof the potential well in the absence of pressure support. In\nthe cosmological mode the shells are initially expanding with a\nmodified Hubble expansion, and the angular momentum is added as each shell\nturns around, corresponding to the physical generation of angular\nmomentum via tidal torquing around the maximal expansion radius. In\nthe static DM mode, the angular momentum is present from the beginning\nof the simulation, and is taken into account when the hydrostatic\ninitial profile is calculated. The angular momentum prescription of\nthe baryons is determined by the requirement that as gas passes\nthrough the virial radius, its angular momentum will be some fraction\n\\citep[$\\lambda=0.05$,][]{bullock01_j} of the value needed to support\nit on circular motions. As\nexpected, the gas only becomes angular momentum supported near the\ncentre, and when gas cooling is turned on (in the simulations\npresented below, the angular momentum of the baryons is never important, and\nis presented here only for completeness). The dark matter is\nprescribed angular momentum according to the eccentricity of a shell's\ntrajectory\nas it falls in through the virial radius: $j=v_{\\rm vir}r_{\\rm vir}\\sqrt{2(1-\\beta)},$\n\\citep[corresponding to: $v_\\theta^2=v_\\phi^2=(1-\\beta)v_r^2$;][\n\\S~4.2]{bt87}. $v_{\\rm vir},r_{\\rm vir}$ are the virial velocity and\nradius, and $v_r,v_\\theta,v_\\phi$ are the radial and two tangential\nvelocity components of an infalling trajectory. \nLow values\nof $\\beta$ indicate large angular momentum, and $\\beta=1$ corresponds\nto purely radial orbits. $\\beta$ is constant for all the dark matter shells in the\nsimulation. Gas cooling can be turned on, by interpolating from a\ntabulated version of \\citet{sutherland93}, with a prescribed metallicity.\n\nRather than use a leap-frog integration scheme, which is sufficient for most explicit\nhydrodynamic schemes, we use here 4th order Runge-Kutta integration\nover the coupled dark matter and baryonic timestep. This allows us to\nderive a timestep criterion for the dark matter shells, by comparing\nchanges in the\nvelocities and radii of the dark matter and baryons between\nthe 1st and 4th order propagation and\nlimit this difference to some fraction of the velocity and radius. The\ntimestep is determined by the combined Courant Friedrichs Levy\ncondition and this requirement. If the criterion is not met, the\ncode reverts to the beginning of the timestep and recalculates the\nnext step with a reduced timestep.\n\nThe numerical integration scheme,\nangular momentum and cooling is described in \\citet{bd03}. In\naddition, we report here on two new ingredients to the calculation:\nadaptive changes to the grid, and 1D convective model.\nBaryonic shells are now split and merged if they become too thick or\nthin respectively. A shell is split if its width is larger\nthan some fraction (typically $0.2$) of its radius, or if it is larger\nthan some fraction (typically $4$) of the shell directly below it and\nabove it, provided that it is larger than a\nminimal value (typically $2~{\\rm kpc}$), and that the shell is not in\nthe angular momentum\nsupported region near the centre of the halo. Shells are merged when the\ncumulative width of two adjacent shells is smaller than some\nvalue (typically $0.1~{\\rm kpc}$). Shell merging is helpful in cases\nwhere two shells are pressed\ntogether, decreasing the timestep to unreasonable values (reasonable\ntimesteps for cosmological simulations are above $10^{-7}~{\\rm Gyr}$).\nAn additional component included in ``Hydra'' for the first time is a 1D mixing\nlength convection model, that acts to transfer energy outwards in regions where\nthe entropy profile is non-monotonic \\citep{spiegel63}. This model\nis described in detail in \\citet{bd10}, and is used here only in the final\ncase described in \\S~\\ref{sec:virial} in its maximal form: bubbles can\nrise until they reach the local speed of sound. In this work, the\ninclusion of the maximally efficient mixing length theory convection ensures\nthat the entropy is always monotonically increasing, in an energetically\nself-consistent way.\n\nThe initial conditions for a cosmological run are derived by requiring an\naverage mass evolution history for a halo with some final mass \\citep{neistein06}. The\noverdensity of each shell at the initial time is calculated so that the shell would pass\nthrough its virial ($r_{180}$) radius at the required time \\citep[the\nprocedure is\ndescribed in detail in][]{bdn07}. This procedure has an additional\nbenefit of setting the initial grid in a way that would ensure that\nshells will pass through the virial radius at constant time intervals.\nIn this procedure, and in the\ncosmological simulation, a force corresponding to the cosmological\nconstant has been added to\neq.~\\ref{eq:force}, making the simulation completely consistent with\n$\\Lambda$CDM \\citep{bdn07} (the cosmological parameters used\nthroughout the paper are:\n$\\Omega_m=0.3,~\\Omega_\\Lambda=0.7,~h=0.7$). This has little overall\neffect, and essentially no\nimpact once the shells are collapsed, and the overdensities become non-linear.\n\nIn the stripped down, static DM mode, there are no dark matter shells,\nand the gas does not self-gravitate. Instead, the value $M_i$ of\neq. \\ref{eq:force} is interpolated from a predefined lookup table.\nIn this mode, a rigid external boundary condition is applied, in\ncontrast to a vacuum boundary condition in the cosmological mode. The\ninner boundary condition in both cases is trivially satisfied because\nall terms in eq. \\ref{eq:force} are individually zero there.\n\n\\begin{figure}\n\\includegraphics[width=3.6in, trim=30 0 10 0, clip=true]{profile}\n\\caption{Radial profiles of cluster initial conditions. {\\it Top:}\n  density (red, left axis) and total enclosed mass (green, right axis) of\n  the cluster. {\\it Bottom:} Temperature (red, left axis) and entropy\n  (green, right axis), as a function of radius (upper x axis) and\n  virial fraction (lower x axis). The total mass profile, and the temperature\n  profile are prescribed, as well as a pivot point in the entropy\n  (value of $10~{\\rm keV~cm^2}$ at $r=0.01R_{\\rm vir}$). The density\n  and entropy profiles are calculated so that the gas is in\n  hydrostatic equilibrium. The\n  temperature profile is set to exhibit a decrease by a factor\n  $\\approx 1.5$ between $0.1$ to $0.01R_{\\rm vir}$.\\label{fig:profile}\n}\n\\end{figure}\nFor the static DM simulations we wish to define initial conditions\nwhich are typical of those in clusters. The total mass profile is\ndefined by an NFW profile \\citep{nfw97}, with\n$M_{\\rm vir}=3\\times 10^{14}M_\\odot,$ and concentration set to $c=5.4$ according\nto \\citet{bullock01_c}. The virial radius is $1730~{\\rm kpc}.$ A central galaxy (BCG) is\nmodeled by a Hernquist profile \\citep{hernquist90} superimposed on the\nNFW profile, with a sharp cutoff at $10~{\\rm kpc}$ and concentration\n$a=6~{\\rm kpc}$ normalized to $M_{BCG}=3\\times 10^{11}M_\\odot.$ We define an\ninward decreasing temperature profile \\citep{leccardi08} that  peaks\nat $r_T=0.1~R_{\\rm vir}$ at $T(r_T)=T_{\\rm\n  vir}=2.7\\times 10^7K$ and drops inwards\naccording to $T(r)=T_{\\rm vir}(r\/r_T)^{0.2}$. Outwards, it is linearly\ndecreasing from $T(r_T)=T_{\\rm vir}$ to $T(R_{\\rm vir})=0.7~T_{\\rm\n  vir}.$ Once the\nentropy of a single point is defined, the\ndensity profile can be constructed by the hydrostatic requirement\nwithout further degeneracy.\nWe set $S(0.01R_{\\rm vir})=10~{\\rm keV~cm^2}$ as our pivot point \\citep{cavagnolo09}. The\nfinal baryonic mass of the resulting profile in the ICM is $2.5\\times 10^{13}M_\\odot,$\nwhich is $\\approx 8\\%$ of the total mass. The\ndensity, total mass, temperature and entropy profiles are plotted in\nfig.~\\ref{fig:profile}. We note that while the constructed initial\nprofile reasonably\nresembles that of a realistic cluster, the mechanism that is studied\nhere, of SICFs, is not sensitive to these details; any merging between\nshocks, in any profile, should produce them.\n\n\\section{Static DM simulations}\n\\label{sec:staticdm}\n\\subsection{Cluster Explosions}\n\\label{sec:explosions}\n\\begin{figure}\n\\includegraphics[width=3.6in, trim=35 0 0 65, clip=true]{explode_b}\n\\caption{Time evolution of an initially hydrostatic halo showing \n  explosion driven shocks from the centre propagating outwards. The\n  energy injection parameters are described in the text. {\\it Top:}\n  entropy color map on the radius-time plane. The red thin lines mark\n  the position of every 50th Lagrangian shell boundary. The black\n  dots mark \n  shocks, defined as artificial viscosity pressure exceeding\n  $10^{-11}erg~cm^{-3}$. These shocks are driven by two explosions, at $t=0.1~{\\rm Gyr}$ and\n  $0.13~{\\rm Gyr}$. The entropy of the shocked gas is saturated\n  in this colormap, but the\n  entropy jump of the SICF is visible at the Lagrangian location of collision\n  between the shocks. {\\it middle:} temperature colormap of the same\n  simulation. The temperature increase induced by the shocks, and the\n  temperature jump at the SICF (by a factor of $1.3$) is visible. {\\it Bottom:} Pressure\n  colormap. The pressure across the SICF is smooth, as expected for\n  cold fronts. \\label{fig:explode}\n}\n\\end{figure}\nThe first mechanism we investigate to produce shocks is that of cosmic\nexplosions. Many processes can cause abrupt energy injection at\ncentres of clusters, among which are AGNs or starbursts at the BCG and\ngalaxy-galaxy mergers. The discussion in this section is confined to\ninjection to the baryonic component itself (changes to the\ngravitational potential well will be discussed next). As shocks sweep\nthrough matter,\nthey constantly loose energy to heating the gas, and when the shocks\nexpand spherically, they also weaken due to the geometrical increase\nin the surface of the shock \\footnote{\\citet{waxman93,kushnir05}\n  showed that for power law density profiles and strong shocks, as\n  long as the mass diverges with radius, or, $\\rho_{\\rm gas}\\propto\n  r^{-w},$ with $w\\leq 3$, the shock will weaken}. We \nseek here to create shocks that will survive to some considerable \nfraction of the halo\nradius, requiring that the amount of energy that is injected will be\ncomparable to the total thermal energy of the gas in the cluster. This\nenergy scale is similar to that required to solve the overcooling\nproblem, indicating that any solution to the cluster overcooling\nproblem via strong shocks will produce shocks that will expand to a\nsignificant fraction of the halo radius.\n\nFig.~\\ref{fig:explode} describes the time dependent\nevolution of an initially hydrostatic cluster atmosphere, as two\nexplosions are driven through it. The simulation has $1,000$\nlogarithmically spaced shells, between $0.0005~R_{\\rm vir}$ and\n$R_{\\rm  vir}.$ Here, and in the following examples, the cooling is\nturned off, which is justified in the absence of any feedback\nmechanism in the simulation that would counteract the\novercooling. The explosions are driven by setting a homologic\n($v=v_{\\rm exp}r\/r_{\\rm exp}$) velocity profile with\nhighly a supersonic velocity ($v_{\\rm exp}$), sharply cutting off beyond some\nradius $r_{\\rm exp}$. For\nthe first explosion, these parameters are $v_{\\rm exp}=10,000~{\\rm\n  km~sec}^{-1}$ and $r_{exp}=20~{\\rm kpc},$ injecting\n$1.7\\times 10^{61}~{\\rm erg}$ into the core of the cluster. The second\n  explosion, $3\\times 10^7~{\\rm yr}$ later, has a velocity of $v_{\\rm exp}=20,000~{\\rm km~sec}^{-1}$ at $r_{exp}=40~{\\rm kpc},$ and\ninjects $4.7\\times 10^{61}~{\\rm erg}.$ These large values are not a-typical for\nthe energy estimated in radio bubbles \\citep{birzan04,mcnamara07}. While no buoyancy is\npossible within a spherical symmetric framework, the explosions do\ncreate extremely hot and dilute regions, and drive strong shocks.\nThe two shocks can be traced in fig.~\\ref{fig:explode} by the breaks in the plotted\nLagrangian lines (velocity jump), by\nthe temperature jump, and by the regions in the simulation with large\nartificial viscosity (black dots). At the location of merging\nbetween the shocks (at $t=0.15~{\\rm Gyr}$ and radius $\\approx 150~{\\rm\n  kpc}$), an SICF (contact discontinuity) forms (visible in\nthe entropy colormap) that propagates with the matter. We have\nfollowed the simulation for $1~{\\rm Gyr}$ and, as expected in the\nabsence of any diffusive mechanisms, the SICF retains its strength.\n\nThe energies and time interval between the shocks have been chosen arbitrarily to\ncreate a shock collision at $\\approx 150~{\\rm kpc}.$ A series of\nsharp, frequent and strong central shocks is expected in some\nself-consistent radiative and\nmechanical AGN feedback models \\citep{ciotti07,ciotti09}, and is\nindicated in some observations \\citep[for example, ][for Perseus and\nM87 respectively]{fabian03b,forman05}. Non-spherical,\nfrequent collisions may occur between the shocks produced by\nthe two bubbles of a radio active AGN, at a position roughly above one\nof the bubbles \\citep[the generation of two shocks corresponding to observed\nbubbles, as well as\nradial soundwaves that steepen into a shock is discussed\nin][]{fabian03b}. The shock from the near-by bubble (primary) arrives to that position\n first, and\nthe shock from the opposite bubble (secondary) arrives later. Given our 1D\nframework, we do not attempt to\nmap the parameter\nspace of possible explosions. Since shocks weaken as they propagate\noutwards, weaker shocks which occur within a shorter time interval\nwill produce smaller radii SICF with the same strength as stronger\nshocks driven within a larger time interval.\n\n\\subsection{Gravitational Perturbations: Merger Induced Shocks}\n\\label{sec:mergers}\nIf the core of an otherwise hydrostatic cluster is gravitationally\nperturbed, the reaction of the gas to the perturbation often results\nin shocks. While explosions, discussed in \\S~\\ref{sec:explosions},\ntypically considerably heat the shocked gas\nnear the centre, gravitational perturbations make a more subtle  change to\nthe gas, while forming strong outward expanding shocks. The core is perturbed\nwhen gas or dark matter\nis accreted to the centre of the cluster, either smoothly (creating\nweak perturbations) or by mergers, (causing a more violent and abrupt\nshock). In this section we describe the shocks that occur as a\nresult of a large change to the core potential depth of a cluster due\nto mergers. In \\S~\\ref{sec:virial} we show that weak oscillations in\nthe core send sound waves through the gas, that steepen into shocks.\n\n\\begin{figure}\n\\includegraphics[width=3.6in, trim=35 0 0 65, clip=true]{flyby}\n\\caption{Same as fig.~\\ref{fig:explode}, but now the\n  potential well is modified by artificially adding a central object\n  of mass  $3\\times 10^{13}M_\\odot$, which is then\n  removed $2\\times 10^7{\\rm yr}$ later (corresponding to a first passage of\n  a 1:10 ratio subhalo). Note that the scales are different from fig.~\\ref{fig:explode}. \\label{fig:flyby}\n}\n\\end{figure}\nFig.~\\ref{fig:flyby} shows a merger event of a cluster with the same initial conditions\ndescribed in \\S~\\ref{sec:hydra} and fig.~\\ref{fig:profile} with a\ndense substructure with total mass of \n$3\\times 10^{13}M_\\odot$ (a $10\\!\\!:\\!\\!1$\nmerger ratio). The\nsimulation describes, in a simplified form, how the\nmain halo will react to the first passage of the substructure near its\ncore. During its first pericenter passage, the subcluster will first\ncause the core to contract, and then, as it flies out, allow the core\nto relax again. Here, we model this in a simplistic way by adding, and\nthen removing (after $2\\times 10^7~{\\rm yr}$), a massive object at the centre of the simulated\ncluster. The addition of the mass causes all the cluster's atmosphere to contract,\naffecting the inner region more than the outer. A coherent\nflow inwards begins, that stops as an outward expanding shock passes\nthrough the cluster, readjusting it to a new hydrostatic equilibrium\n(note in fig.~\\ref{fig:flyby} that after the passage of the primary shock the post-shock gas\nis almost at rest). Once the central object is removed, the halo jolts out from the\ninside out (again, because the addition of the central force affects\nthe inner core the most), causing another shock to\noccur. The energy for these two shocks is taken from\nthe orbital energy of the merging subcluster, and would cause it to\nspiral in. In the figure both shocks are clearly visible in the\ntemperature maps, and through the black dots corresponding to strong\nartificial viscosity. At the instant of merging between the shocks (at $t=0.044~{\\rm Gyr}$ and radius $\\approx 25~{\\rm\n  kpc}$),\nan SICF forms which is seen in the entropy and temperature\ncolormaps. The time interval for the merger interaction is chosen manually, but\ndoes not seem improbable noting that an object approaching the centre\nat $1000{~\\rm km~sec}^{-1}$, roughly the virial velocity, passes\n$20~{\\rm kpc}$ during that period. Shocks associated with sequential\ncore-passages of an inspiraling subhalo could also induce large scale\nSICFs. While shocks produced by merger\nevents have been seen in simulations \\citep{mccarthy07},\nthe formation of SICF in these simulations hasn't\nbeen tested yet, and is left for future work.\n\n\\section{Virial Shocks and SICFs}\n\\label{sec:virial}\nAt the edges of galaxy clusters there are virial shocks with typical\nMach numbers $\\sim 30-100,$ heating the gas from $\\sim 10^4~{\\rm K}$\nto $\\sim 10^7~{\\rm K}$ by converting kinetic energy into thermal\nenergy. Virial shocks are probably observed by Suzaku \\citep{george09,hoshino10}.\nThe rate of expansion of a virial shock is set on average by the\nmass flux and velocity of the infalling material\n\\citep[see for example][]{bertschinger85}. Secondary shocks that originate from\nthe centre of clusters, due, for example, to the mechanisms discussed in\n\\S~\\ref{sec:spherical}\nwill collide and merge with the virial shock, creating SICFs at the\nlocations of the virial shock at the corresponding times. Since\nshock mergers with virial shocks require only one additional shock to form\n(rather than two in the general cases described in\n\\S~\\ref{sec:staticdm}), and no\nparticular timing is required, such SICFs are perhaps even more\nprobable. When the Mach number of the primary (virial) shock is $M_0\\gtrsim 3$,\nwhich is always the case,\nthe strength of the SICF becomes almost independent of the secondary\nshock, allowing for a rather narrow range of strengths around\n${q}\\approx 2$ as long as the secondary shock is sufficiently strong (fig.~\\ref{fig:m0m1}).\n\nWe simulate the merging of a virial shock with a secondary by evolving self consistently the dark matter and\nbaryons from an initial perturbation in the cosmological mode described in \\S~\\ref{sec:hydra}.\nThese initial conditions provide a reasonable laboratory for gas dynamics\nin the ICM, producing correct temperatures and\ndensities, and tracking the\nvirialization process of the gas in a self consistent way. In order to\ncapture the production of a secondary shock and its interaction with\nthe virial shock in 1D simulation, we examine various artificial\nschemes which mimic the full 3D dynamics, as discussed below. A more\ndetailed simulation of cluster evolution would require 3D simulations\nthat include AGN feedback and merger and smooth accretion\nfor baryons and dark matter.\nThe cosmological simulations here have been performed with $2,000$\nbaryonic shells, and $10,000$ dark matter shells, sufficient according\nto our convergence tests.\n\nFor the average mass accretion histories used here\n(\\S~\\ref{sec:hydra}), the\naccretion rate grows with halo mass, even after taking into account\nthe late time decline in accretion rates \\citep{dekel09}, so\nshells that fall later are more massive. When late\n massive dark matter shells fall into the inner core, their\nmass becomes comparable to that of the gas enclosed within that shell, causing\nit to vibrate stochastically. These core vibrations drive sound\nwaves that are emitted from the centre\nand steepen to create shocks (the energy for these\nshock is taken from the orbits of the dark matter shells). We can\ncontrol the level of stochastic noise by increasing or decreasing the\nangular\nmomentum prescribed to the dark matter. In the first two ``synthetic''\nexamples shown here, we use $\\beta=0.7$ (see \\S~\\ref{sec:hydra}) to reduce noise in the\nevolution, and then manually add perturbations at time $-7~{\\rm Gyr}$\n($z=0.84$). In the last example, we decrease the dark matter angular\nmomentum ($\\beta=0.991$), and use the numerical stochastic noise\nas a proxy for the perturbations that halo cores undergo during their\nformation and accretion. We demonstrate that this noise naturally give rise\nto shocks that interact with the virial shock. This interaction seems\nto be quasi periodic, and we suggest a mechanism for driving this behavior. In\nthis final example, the entropy profile is sometimes non-monotonic,\nimplying that convection might be important. We\n test the effect of convection using our 1D mixing length theory and\nshow that it does not significantly reduce the strength of the SICFs\nthat are produced, and does not change the overall evolution significantly.\n\n\\subsection{Manual Initiation of Secondary Shocks}\n\\label{sec:manual_shocks}\n\\begin{figure*}\n\\includegraphics[width=7.in, trim=40 0 50 0, clip=true]{cosmo_madd}\n\\caption{Cosmological mode simulation of the evolution of a\n  $M(z=0)=3\\times 10^{14}M_\\odot$ cluster. The gray lines mark the evolution of\n  every 50th Lagrangian shell,\n  and the colormaps are entropy ({\\it top}) and temperature ({\\it\n    bottom}). The initial Hubble expansion, turnaround and the virial\n  shock are visible.\nInitially, at $z=100$, shells expand due to the\n  near-Hubble flow. At time $-7~{\\rm Gyr}$ a central object with mass\n  $3\\times 10^{13}M_\\odot$ is permanently added, causing a shock to\n  propagate outwards. The collision between this shock and the virial\n  shock creates an SICF visible in the entropy colormap which\n  persists to $z=0$.\\label{fig:cosmo_madd}\n}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=7.in, trim=40 0 50 0, clip=true]{cosmo_msin}\n\\caption{A cosmological mode simulation with the same initial\n  conditions as in fig.~\\ref{fig:cosmo_madd}, but with the core\n  perturbed briefly between $-7$ and $-6.75~{\\rm Gyr}$ (see\n  \\S~\\ref{sec:manual_shocks} for details). The perturbation\n  is a sinusoid in time, with an amplitude of $3\\times 10^{13}M_\\odot$, and\n  a period of $5\\times 10^7~{\\rm Gyr},$ following by a constant left-over\n  mass of  $5\\times 10^{12}M_\\odot.$ Weak shocks and sound waves are sent\n  outwards, steepening into a single shock that collides with the\n  virial shock, producing an SICF.\\label{fig:cosmo_msin}\n}\n\\end{figure*}\nFig.~\\ref{fig:cosmo_madd} describes a typical evolution of a cluster\nthat ends up with mass\n$3\\times 10^{14}M_\\odot$ at $z=0$, and subsequent interaction of the\nvirial shock with a\nsecondary merger induced shock. The merger here is $10\\!\\!:\\!\\!1$, with a\ncentral mass of $3\\times 10^{13}M_\\odot$ instantaneously added at  $-7~{\\rm\n  Gyr}$ (at that time the cluster's virial mass\nis $\\approx 10^{14}M_\\odot$). Unlike the procedure described in\n\\S~\\ref{sec:mergers}, we do \nnot remove the merged mass, creating only one shock. The Hubble\nexpansion, turnaround and virial shocks are seen,\nas well as the SICF formation at time $\\approx -5.5~{\\rm Gyr}$ and at\nradius $\\approx 700~{\\rm kpc}$. After the deepening of the gravitational\npotential well, the secondary shock mediates the re-establishment of\nhydrostatic equilibrium. Thus, the post-shock gas, and the SICF, are roughly at\nrest in the Eulerian laboratory frame of reference.\nThe post-shock temperatures, particularly below $100~{\\rm kpc}$ are\nunrealistically high, corresponding, perhaps, to disturbed morphologies seen in\ncluster merger events \\citep[the bullet cluster; ][]{tucker98}, although no entropy inversion occurs there. This\nis remedied somewhat when the shock is initiated by a series of weaker\nshocks and sound waves that steepen into a strong shock further from\nthe cluster's centre (fig.~\\ref{fig:cosmo_msin}), and is completely\nalleviated when the shocks are due to persistent stochastic noise near\nthe core (\\S~\\ref{sec:natural_shocks}), generating profiles which are\nalso consistent with quiescent cool core clusters.\n\nWe produce a weaker, more local perturbation by engineering a\nperturber that sends weak shocks and sound waves that steepen into a\nshock.\nFig.~\\ref{fig:cosmo_msin} shows an evolution of the same cluster as in\nfig.~\\ref{fig:cosmo_madd}, but with a central perturber who's mass is\nsinusoidally changing between $-7<t<-6.75~{\\rm Gyr}$ with an amplitude\nof  $3\\times 10^{13}M_\\odot$  and a period of of $5\\times 10^7~{\\rm\n  Gyr}.$ After that, a constant mass of $5\\times 10^{12}M_\\odot$ is\nleft. The steepening of the shocks is visible, and the temperature\nincrease of the gas by the secondary is much smaller, because,\ncontrary to the permanent  mass addition in fig.~\\ref{fig:cosmo_madd},\nno net contraction is caused. This example serves to show how shocks\ncan form from ``noise'' generated near the core, but still heat the\ncentre from the inside out, in a way that would correspond to a merger\nevent but not to a quiescent cool core cluster. The entropy\nprofile here becomes non-monotonic near the centre.\n\n\\subsection{Self Consistent Initiation of Secondary Shocks}\n\\label{sec:natural_shocks}\nWe allow numerical noise to perturb our hydrodynamic simulation by\nreducing the angular momentum of dark matter so that late infalling,\nmassive dark matter shells penetrate the cluster core. The resulting\ntime evolution and final profile are shown in figures\n~\\ref{fig:cosmo_ad}~-~\\ref{fig:prof}. As soon as the cluster progenitor\nforms, at high redshift, sound waves and weak shocks begin propagating through the\ncluster, as the gas constantly adjusts to the varying potential well\nnear the centre. The weak shocks steepen and accumulate to create\nstrong shocks that propagate outwards and merge with the virial shock,\ncreating an SICF on each merger instance. From fig.~\\ref{fig:cosmo_ad}\nit is evident that the shocks are weak, and that they do not alter the\ninner profile considerably.\n\nThe shocks that are invoked by the stochastic noise seem to be\nperiodic, with a period corresponding to about twice the sound (or\nshock) crossing time through the cluster. Lacking a formal analysis\nfor this oscillatory mode, we suggest, based on these simulations, the\nfollowing explanation. The location of the virial shock is regulated\nsuch that the\npressure below (towards the cluster's centre) will support the shock\nand the ram pressure of the infalling gas\n\\citep{bertschinger85}. While this process would\nimply a smooth expansion of the shock, the virial shock can\noscillate around its steady state trajectory in\nresponse to perturbations in the halo. When the post-shock gas is\nover-pressurized, the virial shock adjusts itself by expanding\nfaster, relaxing this overpressure. If the relaxation overshoots, the\npressure would drop below its steady state value, causing the shock to\ndecelerate. The halo thus oscillates between these two \nphases depending on the virial \nshock's velocity: in one it is overly slow,\nproducing pressurization of the\npost-shock gas, and in the other it is overly fast, causing\nrarefaction. These modes are interlaced. A\nrarefaction cycle begins with the secondary hitting the virial\nshock sending\nrarefaction waves inwards that are reflected at the centre and\npropagate outwards, ultimately reaching the virial shock\nmaking it\ndecelerate. A compression, or secondary shock cycle\ncollects compression waves that are transmitted inwards when the\nvirial shock is\ntoo slow, and are reflected through the centre, ultimately steepening\ninto what would become the secondary shock. Circumstantial evidence\nfor this mode can be seen in fig.~\\ref{fig:time} where the\nshocks are automatically traced, and the trajectory of one rarefaction\nwave reflected from a secondary-virial shock merger is highlighted by\nhand (dashed curve, based on a high resolution time sequence). Note\nthat this periodicity is also seen (albeit much \nmore weakly) in the two\nmanual excitations of a secondary shock described in figures\n\\ref{fig:cosmo_madd} and \\ref{fig:cosmo_msin}. The periodic SICFs that\nare produced can be seen in the entropy profiles in the upper panel of\nfig.~\\ref{fig:cosmo_ad}, in the lines marking the gradient of the\nentropy in fig.~\\ref{fig:time}, and in the final profile of this\nsimulation shown in fig.~\\ref{fig:prof}.\nThe shocks and rarefactions propagating though the halo, and the\nalternating speed of the virial shock\\footnote{A movie of the evolution of radial\nprofiles in time is available at\n\\url{http:\/\/www.cfa.harvard.edu\/~ybirnboi\/SICF\/sicf.html} } are observed in the 3D galactic halo simulations of \\citet{kh09}\\footnote{Du{\\v s}an Kere{\\v s},\n(private communication). Propagation of merger induced shocks and the\nsubsequent bounce of the virial shock as these secondaries merge with\nit are seen in\n\\url{http:\/\/www.cfa.harvard.edu\/~dkeres\/movies\/B1hr_10n128_large_gas.mp4}. There,\nleft panel is gas density, right panel temperature, and a particularly\nnotable shock forms at $z\\approx 1.7$.}.\nNote that this qualitative argument is similar\n  to the one described in \\S~\\ref{sec:riemann}. The overpressure\nthere is caused by a secondary shock that perturbs the equilibrium\nsolution of a single shock: the primary accelerates, and a rarefaction is\nsent back, relaxing the gas.\n\n\\begin{figure*}\n\\includegraphics[width=7.in, trim=40 0 50 0, clip=true]{cosmo_ad}\n\\caption{A cosmological mode adiabatic simulation of cluster\n  evolution with initial conditions as described in the\n  \\S~\\ref{sec:natural_shocks}. The \n  angular momentum of the dark matter is sufficiently small to allow\n  massive, late accretion of dark matter shells to propagate and reach\n  the centre, increasing the amount of stochastic ``noise'' with\n  respect to the levels seen in figures \\ref{fig:cosmo_madd} and\n  \\ref{fig:cosmo_msin}. A series of secondary shocks merge with the\n  virial shock, creating a series of SICFs.\\label{fig:cosmo_ad}\n}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=3.3in]{time_ent}\n\\caption{Evolution of a galaxy cluster from cosmological initial\n  perturbations. The final mass of the cluster at $z=0$ is $3\\times\n  10^{14}M_\\odot.$ The radius and time of Lagrangian shells (every\n  $25^{th}$ shell) are  plotted in red thin lines. Shocks are traced by large\n  Lagrangian derivatives of the entropy ($d{\\rm ln}S\/dt>.1~{\\rm Gyr}^{-1}$,\n  blue dots), and CFs are traced by their large entropy gradients\n  ($\\partial{\\rm ln} S\/\\partial {\\rm ln}r>0.5$, green dots).\n  Rarefaction waves can be seen as small motions of the Lagrangian\n  shells (illustrated by the dashed magenta curve, manually added\n  based on a time series analysis). Their trajectory is approximately\n  the reflection of the preceding outgoing compression, time inverted\n  about the last secondary-virial shock collision.\n  \\label{fig:time}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in, trim =40pt 0pt 50pt 0pt,clip=true ]{final_profiles}\n\\caption{Thermodynamic profile of the simulated cluster shown in\n  figures \\ref{fig:cosmo_ad}-\\ref{fig:time} at $z=0$ (solid lines) and of the same\n  simulation with maximal convection (dashed lines). {\\it Top:}\n  entropy (red, left axis) and pressure (green,right\n  axis). The entropy of the convective simulation has been raised by\n  $0.5~{\\rm dec}$ to distinguish between the models. {\\it Bottom:} density (red, left axis) and temperature\n  (green, right axis). The positions of CFs in the adiabatic, non-convective\n  simulations are marked with blue dashed\n  vertical lines.\\label{fig:prof}\n  }\n\\end{figure}\n\n The calculations presented in Figures\n\\ref{fig:cosmo_ad}-\\ref{fig:prof} are adiabatic.  When cooling is\nturned on, and\nin the absence of feedback, this simulation suffers from overcooling,\ncreating an overmassive BCG of $2\\times 10^{12}M_\\odot$ and BCG\naccretion rates (corresponding to star\nformation rates) of $\\gtrsim 100~M_\\odot{\\rm yr}^{-1}$, and the luminosity\nexceeds the $L_x-T$ relation \\citep{edge90,david93,markevitch98}. The excitation of the basic mode, and\nperiodic CFs, are observed in all these simulations. Since the virial shock's\nstrength is non-monotonic, it is not surprising that between SICF\nformation, the post-shock entropy is non-monotonic. This introduces a\nproblem in 1D simulations where convection due to entropy inversion\ncannot naturally occur. We test this by rerunning the adiabatic\nsimulation with 1D convection according to mixing length theory\n\\citep{spiegel63}, with the mixing length coefficient set such that\nthe convection is limited only by the requirement that bubbles never\nexceed the sonic speed. This acts to redistribute the energy and\nentropy between SICFs, but does not change the overall dynamics. The\nfinal profile in this case is plotted in the dashed lines of\nfig.~\\ref{fig:prof}. The strength of the SICF in this case is reduced,\nbut it is still clearly visible, with typical values of ${q}\\approx\n1.5$. It is highly improbable that convection operates in its maximal physical\nefficiency, particularly in the presence of ICM magnetic\nfields. \\citet{parrish08b,parrish09} show that in the presence of weak\nmagnetic fields the effective convection is smaller by many orders of\nmagnitude than the heat fluxes carried by conduction, implying that it\nis highly suppressed from its mixing length theoretical value (Ian\nParrish, private communication).\n\nWe note a qualitative similarity between reverberations of the virial accretion shocks\ndiscussed here, and another prominent case of accretion shocks in\nastrophysics - of type II core collapse supernovae. \\citet{burrows07} find that\nstalled accretion shocks around type II cores are unstable to 2D\n(g-mode) reverberation that,\nafter enough cycles, accumulate\nsufficient energy and amplitude to cause\nan outbreak of the stalled shock. While the\nstanding accretion shock instability (SASI) is predominantly 2D, and\nacts on different scales than discussed here, it does draw its\nenergy from the accreted gas, and is perturbed by sound waves\nthat are emitted from the vibrating core, that accumulate to a single\nunstable mode.\n\n\\section{Stability and Observability of Shock Induced Cold Fronts}\n\\label{sec:stabil_observe}\n\\subsection{Stability}\n\\label{sec:stability}\nThe stability of CFs limits the time duration over which they are\ndetectable, and so is important when comparing CF formation models\nwith observations.  Various processes can cause a gradual breakup or\nsmearing of the CF.  They act in other CF formation models as well.\n\nThermal conduction and diffusion of particles across the CF smears the\ndiscontinuity on a timescale that is set by the thermal velocity and\nthe mean free path of the protons. The Spitzer m.f.p. $\\lambda$ in an\nunmagnetized plasma with typical cluster densities is a few kpc, and\ndepends on the thermal conditions on both sides of the discontinuity\n\\citep{markevitch07}. Taking $\\lambda \\sim 10~{\\rm kpc}$ and a thermal\nvelocity $\\bar{v}\\sim 1000~{\\rm km \\, sec}^{-1},$ and assuming that a\nCF is visible if it is sharper than $L_{\\rm obs}\\sim 10~{\\rm kpc},$ we\nget a characteristic timescale for CF dissipation,\n\\begin{equation}\nt\\sim \\frac{L_{\\rm obs}^2}{D}\\sim\\frac{3L_{\\rm\n    obs}^2}{\\bar{v}\\lambda}\\sim 10^7{\\rm yr}.\n\\end{equation}\nThis result indicates that in order for shock induced CFs to be\nobservable, either (i) they are formed frequently \\citep[e.g., by a\nseries of AGN bursts; see][]{ciotti07,ciotti09}; or (ii) magnetic\nfields reduce the m.f.p. considerably, as some evidence suggests\n\\citep[see the discussion in][]{lazarian06}.\n\nHeat flux driven buoyancy instability \\citep[HBI; ][]{parrish08,quataert08,parrish09} tends to\npreferentially align magnetic fields perpendicular to the heat flow in\nregions where the temperature decreases in the direction of gravity.\nThis effect has been argued to reduce radial diffusion within the inward cooling\nregions in the cores of cool core clusters.\nAs suggested in \\citet{parrish08}, the temperature gradient across a\nCF is opposite to the direction of\ngravity, and HBI instability is expected to quickly form there (over $8~{\\rm\n  Myr}$ in their example), aligning the magnetic fields parallel to\nthe discontinuity and\ndiminishing radial diffusion. This possibility needs to be addressed\nfurther by detailed numerical magneto-hydrodynamic simulations.\nAlso, the timescale for achieving local thermal equilibrium between\nthe electrons and ions below the virial shocks are long. It is unclear\nwhat the proper diffusion \ncoefficients are in that case, and a particle transport analysis needs\nto be applied. These important issues are left for future work.\n\n\\begin{figure}\n\\includegraphics[width=3.3in]{mr}\n\\caption{Fractional decline (contours) in initial CF contrast ${q}$\n  induced by a collision with a shock of Mach number $M_1$ arriving from the dense CF side, for $\\gamma=5\/3.$\n  The vertical dashed line corresponds to the maximal SICF contrast, ${q}_{max}$.\n  \\label{fig:mr}}\n\\end{figure}\nCFs  could also be degraded by subsequent shocks that sweep outwards across\nthe CF. This is true regardless of their formation mechanisms.\nHowever, the CF contrast is only slightly diminished by this\neffect. This is shown analytically by recalculating the Riemann\nproblem described in \\S~\\ref{sec:riemann} but starting from a general\ncontact discontinuity with strength \n${q}$ instead of a primary shock. For ${q}=2$ this decrement is between $4\\%$ for $M_1<2$, to\n$20\\%$ for $M_1\\gtrsim 10$, as\nshown in fig.~\\ref{fig:mr}. Such passing shocks seed\nRichtmyer-Meshkov instabilities which could cause the CF to break\ndown.  Although less efficient than Rayleigh-Taylor instabilities,\nthese instabilities operate regardless of the alignment with gravity.\nThe outcome depends on $M_1$ and on any initial small perturbations in\nthe CF surface.  However, the instability is suppressed if the CF\nbecomes sufficiently smoothed.\n\nIt is worth noting that KHI, which could break down CFs formed through\nram pressure stripping and sloshing, does not play an important role\nin SICFs because there is little or no shear velocity across them.\nHowever, the stabilizing alignment of magnetic fields caused by\nsuch shear \\citep{markevitch07,keshet10} is also not expected here.\n\n\\subsection{Observability}\n\\label{sec:observe}\n\\citet{owers09} present high quality Chandra observations of\n$3$ relaxed CFs. They all seem concentric with respect to the cluster\ncentre, and spherical in appearance. They interpret these as\nevidence for sloshing, and find spiral characteristics in $2$ of\nthem. In the cold front sample of \\citet{ghizzardi10}, $10$ of the\nrelaxed morphology clusters also host cold fronts. We argue here that\nsome CFs of this type could originate from \nshock mergers.  In the SICF model, CFs are spherical, unlike the\ntruncated CFs formed by ram pressure stripping or sloshing, so no\nspecial projection orientation is required.  On the other hand, SICFs\ndo not involve metallicity discontinuities, observed in some of these\nCFs.  The observed contrasts of these three CFs in \\citet{owers09} are quite uniform, with best\nfit values in the range $2.0-2.1$ (with $\\sim 20\\%$ uncertainty) in\nall three, as expected for SICFs\\footnote{However, shocks sweeping\n  over CFs could diminish their contrast towards ${q}\\sim 2$,\nregardless of CF origin.  A larger, more complete sample of CFs is\nneeded to determine the origin\/s of the relaxed population.}.\n\nThe following characteristics are specific to SICFs, and could thus be\nused to differentiate between SICFs and CFs formed by other\nmechanisms. These are also predictions for SICFs that are expected to\nform at radii that are not observable today.\n\n{\\bf Morphology.} SICFs are quasi-spherical around the source of the\nshocks. In contrast, for example, a galactic merger defines an orbit plane, and the\ncorresponding perturbation (stripped material or sloshing and centre displacement)\nwill create CFs\nparallel to\nthis plane.  In some sloshing\nscenarios \\citep{ascasibar06} the CFs extend considerably above the\nplane, but would never cover all\nviewing angles; an observed\nclosed CF ``ring'' would strongly point towards an SICF.  The statistical\nproperties of a large CF sample could thus be used to distinguish\nbetween the different scenarios. \\citet{ghizzardi10} identify 23\napparently relaxed clusters, 10 of which exhibit cold fronts. They find a\n{\\it perfect} correlation between central entropy levels and cold fronts, with\nthe 10 lowest central entropy clusters exhibiting cold fronts. This\ncorrelation is also consistent with convection of low entropy gas from the\ncentre \\citep{markevitch07,keshet10}, but would require that all $10$\nCFs are viewed from a favorable viewing angle. We offer here an\nalternative interpretation, in which the low\nentropy and short cooling times imply relatively more active\nAGNs and consequently the formation of shocks on shorter periods\n\\citep{ciotti09}. The SICFs  naturally cover $4\\pi$ of the sky,\nalleviating the viewing angle statistical requirement.\n\n{\\bf Amplitude.} SICFs have distinct entropy and density contrasts\nthat depend weakly on shock parameters; $q$ is typically larger than\n$\\sim 1.4$ (assuming $M_1\\geq 2$) and is always smaller than\n$q_{max}=2.65$. This is consistent with observations.\n\n{\\bf Extent.} If cluster oscillations, as described in\n\\S~\\ref{sec:natural_shocks} are  excited, SICF radii\nshould be approximately logarithmically spaced.\nAny shock that expands and collides with the virial shock will create\nan SICF at the location of the virial shock, far beyond the core.\nThe external SICFs are\nyounger, and would appear sharper owing to less degradation due to the\nprocesses discussed in \\S~\\ref{sec:stability}. This concentric CF\ndistribution thus resembles tree rings. Deep observations capable of\ndetecting CFs at $r\\sim 1~{\\rm Mpc}$  are predicted to find SICFs\n(fig.~\\ref{fig:prof}). Such distant CFs occur naturally only in the\nSICF model \\citep[observations by Suzaku may be able\nto probe this range with sufficient quality in the near future;\n][]{hoshino10}.\n\n{\\bf Plasma diagnostics.}  Shocks are known to modify plasma\nproperties in a non-linear manner, for example by accelerating\nparticles to high energies and amplifying\/generating magnetic\nfields. The plasmas on each side of an SICF may thus differ, being\nprocessed either by two shocks or by one, stronger shock.  This may\nallow indirect detection of the CF, in particular if the two shocks\nwere strong before the collision.  For example, enhanced magnetic\nfields below the CF may be observable as excess synchrotron emission\nfrom radio relics that extend across the CF, in nearby clusters, using\nfuture high-resolution radio telescopes (MWA,LOFAR, SKA).\n\n\\section{Summary and Discussion}\n\\label{sec:summary}\n\nOur study consists of two parts. The first is a general, analytic discussion\nabout cold fronts that form as a result of a merging between a primary and\nsecondary shock propagating in the same direction. This is a novel\nmechanism to create cold fronts discussed here for the first time.\nThe second part\ndescribes the possible relevance of this SICF mechanism in cluster\nenvironments using 1D spherical hydrodynamical simulations.\n\nWe have shown in \\S~\\ref{sec:riemann} that when shocks moving in the\nsame direction merge they generate a CF. The density contrast across\nthe CF is calculated as\na function of the Mach numbers of the two shocks. It is typically\nlarger than $1.4$ (if $M\\gtrsim2$), and is always smaller than\n${q}_{max}=2.65$. We support the analytical calculation by a detailed\ninvestigation of a shock tube planar hydrodynamical simulation.\n\nIn \\S~\\ref{sec:staticdm} and \\S~\\ref{sec:virial}, using a 1D spherical hydrodynamic code, we demonstrate that SICFs in\nclusters are a natural consequence of shocks that are generated at\ncentres of clusters. We entertain two ways to invoke shocks: by\ninjecting large amounts of energy near the centre (corresponding to AGN\noutbursts), and by abruptly changing potential the well of the cluster\n(corresponding to merger events).\n Finally, by simulating cluster evolution from initial cosmological\n perturbations over a Hubble time we show that outgoing shocks that\n merge with the virial shock create very distinctive CFs. These shocks\n can be caused by a critical event (merger or explosion) but can also\n be invoked by stochastic oscillations of the cluster's core, caused\n by  accretion of low angular momentum substructure that perturbs the\n core. We show that a reverberation\nmode exists in the haloes of galaxies and clusters that causes\nperiodic merging between the virial shocks and secondary shocks,\nproducing an SICF every cycle.\n The simulated SICF contrast is\nconsistent with the theoretical predictions of \\S~\\ref{sec:riemann}.\nA more thorough investigation of this potentially important mode,\nincluding its stability in 3D is left for future works.\n\n\nWe then discuss in \\S~\\ref{sec:stability} the\nsurvivability and degradation in time of CFs after they are formed. The\nCF discontinuity is smeared over time by diffusion,\nat a rate that depends on the unknown nature and amplitude of magnetic\nfields.\nCFs are susceptible to heat-flux-driven buoyancy instability\n\\citep[HBI; ]{parrish08},\nwhich could align the magnetic field tangent to the CF and potentially\nmoderate further diffusion. SICFs, like all other CFs, are subject to\nRichtmyer-Meshkov instabilities from subsequent shocks passing through\nthe cluster. Such collisions also reduce the CF contrast until it reaches\n$q\\sim 2$.  Unlike most other CF models, an SICF is not expected to\nsuffer from KHI.\n\nThe predicted properties of SICFs are presented in\n\\S\\ref{sec:observe}, and reproduce some of the CF features discussed\nin \\citet{owers09} and \\citet{ghizzardi10}.  In particular, we suggest that CFs in relaxed\nclusters, with no evidence of mergers, shear, or chemical\ndiscontinuities, may have formed by shock collisions.  We list the\nproperties of SICFs that could distinguish them from CFs formed by other\nmechanisms.\nThe SICF model predicts quasi-spherical CFs which are concentric about\nthe cluster centre, with contrast ${q} \\sim 2,$\nand possibly extending as far out as the virial shock.\nAn observed closed (circular\/oval) CF could only be an SICF.\nIn the specific case of cluster reverberation,\na distinct spacing pattern between CFs is expected.\nIt may be possible to detect them indirectly, for example as\ndiscontinuities superimposed on peripheral radio emission.\n\nShocks originating from the cluster centre naturally occur in feedback\nmodels that are invoked to solve the overcooling problem.  They are\nalso formed by mergers of substructures with the BCG.  Thus, SICFs\nshould be a natural phenomenon in clusters.  Further work is needed to\nassess how common SICFs are with respect to other types of CFs, and to\ncharacterize inner SICFs that could result, for example, from\nmergers between offset AGN shocks.  The properties of SICFs\nin 3D will be pursued in future work.\n\n\\section*{Acknowledgements}\nWe thank I. Parrish, Du{\\v s}an Kere{\\v s} and M. Markevitch for\nuseful discussions and the referee, Trevor Ponman, for helpful\nsuggestions.  YB acknowledges the \nsupport of an ITC fellowship from the Harvard College Observatory.  UK\nacknowledges support by NASA through Einstein Postdoctoral Fellowship\ngrant number PF8-90059 awarded by the Chandra X-ray Centre, which is\noperated by the Smithsonian Astrophysical Observatory for NASA under\ncontract NAS8-03060.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\nA non-well-founded proof is usually defined as a possibly infinite tree of formulas (sequents) that is constructed according to inference rules of a proof system and, in addition, that satisfies a particular condition on infinite branches. A cyclic, or circular, proof can be defined as a finite pointed graph of formulas (sequents) which unraveling yields a non-well-founded proof. These proofs turn out to be an interesting alternative to traditional proofs for logics with inductive and co-inductive definitions, fixed-point operators and similar features. For example, proof systems allowing cyclic proofs can be defined for the modal $\\mu$-calculus \\cite{AfLe}, the Lambek calculus with iteration \\cite{Kuz} and for Peano arithmetic \\cite{Sim}. In the last case, these proofs can be understood as a formalization of the concept of proof by infinite descend.\n\nStructural proof theory of deductive systems allowing cyclic and non-well-founded proofs seems to be underdeveloped. In \\cite{ForSan}, J.~Fortier and L.~Santocanale considered the case of the $\\mu$-calculus with additive connectives. They present a procedure eliminating all applications of the cut-rule from a cyclic proof and resulting an infinite tree of sequents. Though they don't show that this tree satisfies a guard condition on infinite branches, which is necessary for non-well-founded proofs in the $\\mu$-calculus. Unfortunately, we don't know other syntactic cut-elimination results for systems with non-well-founded proofs.    \nHere we present a sequent calculus for the Grzegorczyk modal logic allowing non-well-founded proofs and obtain the cut-elimination theorem for it. This article is an extended version of the conference paper \\cite{SavSham}.\n\nThe Grzegorczyk modal logic $\\mathsf{Grz}$ is a well-known modal logic \\cite{Maks}, which can be characterized by reflexive partially ordered Kripke frames without infinite ascending chains. This logic is complete w.r.t. the arithmetical semantics, where the modal connective $\\Box$ corresponds to the strong provability operator \\textit{\"... is true and provable\"} in Peano arithmetic. There is a translation from $\\mathsf{Grz}$ into the G\\\"{o}del-L\\\"{o}b provability logic $\\mathsf{GL}$ such that \n$$\\mathsf{Grz} \\vdash A \\Longleftrightarrow \\mathsf{GL} \\vdash A^\\ast,$$ where $A^\\ast$ is obtained from $A$ by replacing all subformulas of the form $\\Box B$ by $B \\wedge \\Box B$.\n\nRecently a new proof-theoretic presentation for the G\\\"{o}del-L\\\"{o}b provability logic $\\mathsf{GL}$ in the form of a sequent calculus  allowing non-well-founded proofs was given in \\cite{Sham, Iemhoff}. \nWe wonder whether cyclic and, more generally, non-well-founded proofs can be fruitfully considered in the case of $\\mathsf{Grz}$.\nWe consider a sequent calculus allowing non-well-founded proofs for the Grzegorczyk modal logic and present the cut-elimination theorem for the given system.\nIn order to avoid nested co-inductive and inductive reasoning, we adopt an approach from denotational semantics of computer languages, where program types are interpreted as ultrametric spaces and fixed-point combinators are encoded using the Banach fixed-point theorem (see \\cite{BaMa}, \\cite{Esc}, \\cite{Breu}). We consider the set of non-well-founded proofs of $\\mathsf{Grz}$ and various sets of operations acting on theses proofs as ultrametric spaces and define our cut-elimination operator using the Prie\\ss-Crampe fixed-point theorem (see \\cite{PrCr}), which is a strengthening of the Banach's theorem. \nIn this paper, we also establish that in our sequent system it is sufficient to consider only unravellings of cyclic proofs instead of arbitrary non-well-founded proofs in order to obtain all provable sequents. As an application of cut-elimination, we obtain the Lyndon interpolation property for the Grzegorczyk modal logic $\\mathsf{Grz}$ proof-theoretically. \n \n\n\n\nRecall that the Craig interpolation property for a logic $\\mathsf{L}$ says that if $A$ implies $B$, then there is an interpolant, that is, a formula $I$ containing only common variables of $A$ and $B$ such that $A$ implies $I$ and $I$ implies $B$. The Lyndon interpolation property is a strengthening of the Craig one that also takes into consideration negative and positive occurrences of the shared propositional variables; that is, the variables occurring in $I$ positively (negatively) must also occur both in $A$ and $B$ positively (negatively).\n\nThough the Grzegorczyk logic has the Lyndon interpolation property \\cite{Maks2}, there were seemingly no syntactic proofs for this result.  \nIt is unclear how Lyndon interpolation can be obtained from previously introduced sequent systems for $\\mathsf{Grz}$ \\cite{Avron, BorGen, DyNe} by direct proof-theoretic arguments because these systems contain inference rules in which a polarity\nchange occurs under the passage from the principal formula in the conclusion to its immediate ancestors in the premise. In this article, we give a syntactic proof of Lyndon interpolation for the Grzegorczyk modal logic as an application of our cut-elimination theorem.\n  \nThe paper is organised as follows. In Section \\ref{SecPrel} we recall a standart sequent calculus for $\\mathsf{Grz}$. In Section \\ref{SecNWF} we introduce the proof system $\\mathsf{Grz_{\\infty}}$ that allows non-well-founded proofs and prove its equivalence to the standard one. In Section \\ref{SecUlt} we recall basic notions of the theory of ultrametric spaces and consider several relevant examples. In Section \\ref{SecAdm} we state admissability of several rules for our system that will be used later. In Section \\ref{SecCut} we establish the cut elimination result for the system $\\mathsf{Grz}_\\infty$ syntactically. In \\ref{SecLyn} we prove the Lyndon interpolation property for the logic $\\mathsf{Grz}$. Finally, in Section \\ref{SecCyc} we establish that every provable sequent of $\\mathsf{Grz}_\\infty$ has a cyclic proof.\n\n\n\n\n\\section{Preliminaries}\n\\label{SecPrel}\nIn this section we recall the Grzegorczyk modal logic $\\mathsf{Grz}$ and define an ordinary sequent calculus for it.\n\n\\textit{Formulas} of $\\mathsf{Grz}$, denoted by $A$, $B$, $C$, are built up as follows:\n$$ A ::= \\bot \\,\\,|\\,\\, p \\,\\,|\\,\\, (A \\to A) \\,\\,|\\,\\, \\Box A \\;, $$\nwhere $p$ stands for atomic propositions. \nWe treat other boolean connectives and the modal operator $\\Diamond$ as abbreviations:\n\\begin{gather*}\n\\neg A := A\\to \\bot,\\qquad\\top := \\neg \\bot,\\qquad A\\wedge B := \\neg (A\\to \\neg B),\n\\\\\nA\\vee B := (\\neg A\\to B),\\qquad\\Diamond A := \\neg\\Box \\neg A.\n\\end{gather*}\n\n\n\nThe Hilbert-style axiomatization of $\\mathsf{Grz}$ is given by the following axioms and inference rules:\n\n\\textit{Axioms:}\n\\begin{itemize}\n\\item[(i)] Boolean tautologies;\n\\item[(ii)] $\\Box (A \\rightarrow B) \\rightarrow (\\Box A \\rightarrow \\Box B)$;\n\\item[(iii)] $\\Box A \\rightarrow \\Box \\Box A$;\n\\item[(iv)] $\\Box A \\rightarrow A$;\n\\item[(v)] $\\Box(\\Box(A \\rightarrow \\Box A) \\rightarrow A) \\rightarrow \\Box A$.\n\\end{itemize}\n\n\\textit{Rules:} modus ponens, $A \/ \\Box A$. \\\\\n\n\nNow we define an ordinary sequent calculus for $\\mathsf{Grz}$. A \\textit{sequent} is an expression of the form $\\Gamma \\Rightarrow \\Delta$, where $\\Gamma$ and~$\\Delta$ are finite multisets of formulas. For a multiset of formulas $\\Gamma = A_1,\\dotsc, A_n$, we define $\\Box \\Gamma$ to be $\\Box A_1,\\dotsc, \\Box A_n$.\n\n\n\nThe system $\\mathsf{Grz_{Seq}}$, which is a variant of the sequent calculus from \\cite{Avron}, is defined by the following initial sequents and inference rules: \n\\begin{gather*}\n\\AXC{ $\\Gamma, A \\Rightarrow A, \\Delta $ ,}\n\\DisplayProof \\qquad\n\\AXC{ $\\Gamma , \\bot \\Rightarrow \\Delta $ ,}\n\\DisplayProof\\\\\\\\\n\\AXC{$\\Gamma , B \\Rightarrow \\Delta $}\n\\AXC{$\\Gamma \\Rightarrow A,\\Delta $}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Gamma , A \\to B \\Rightarrow \\Delta$}\n\\DisplayProof\\;,\\qquad\n\\AXC{$\\Gamma , A \\Rightarrow B, \\Delta $}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Gamma \\Rightarrow A \\to B, \\Delta$}\n\\DisplayProof\\;,\\\\\\\\\n\\AXC{$\\Gamma, B, \\Box B \\Rightarrow \\Delta $}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Gamma , \\Box B \\Rightarrow \\Delta$}\n\\DisplayProof\\;,\\qquad\n\\AXC{$ \\Box\\Pi, \\Box(A\\to\\Box A) \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box_{Grz}}$}\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A ,\\Delta $}\n\\DisplayProof \\;.\n\\end{gather*}\n\\begin{center}\n\\textbf{Fig. 1.} The system $\\mathsf{Grz_{Seq}}$\n\\end{center}\n\nThe cut rule has the form\n\\begin{gather*}\n\\AXC{$\\Gamma\\Rightarrow A,\\Delta$}\n\\AXC{$\\Gamma,A\\Rightarrow\\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma\\Rightarrow\\Delta$}\n\\DisplayProof\n\\end{gather*}\n\nwhere $A$ is called the \\emph{cut formula} of the given inference. \n\n\\begin{lem} \\label{prop}\n$\\mathsf{Grz_{Seq}} + \\mathsf{cut}\\vdash \\Gamma\\Rightarrow\\Delta$ if and only if $\\mathsf{Grz} \\vdash \\bigwedge\\Gamma\\to\\bigvee\\Delta $. \n\\end{lem}\nThis lemma is completely standard, so we omit the proof.\n\n\n\\begin{thm}\\label{cutelimgrz}\nIf $\\mathsf{Grz_{Seq}} + \\mathsf{cut}\\vdash \\Gamma\\Rightarrow\\Delta$, then $\\mathsf{Grz_{Seq}} \\vdash \\Gamma\\Rightarrow\\Delta$. \n\n\\end{thm}\n\n\nA syntactic cut-elimination for $\\mathsf{Grz}$ was obtained by M.~Borga and P.~Gentilini in \\cite{BorGen}. In this paper, we will give another proof of this cut-elimination theorem in the next sections. \n\n \n\n\n\n\n\n\n\\section{Non-well-founded proofs}\n\\label{SecNWF}\nIn this section we introduce a sequent calculus for $\\mathsf{Grz}$ allowing non-well-founded proofs and define two translations that connect ordinary and non-well-founded sequent systems.\n\nInference rules and initial sequents of the sequent calculus $\\mathsf{Grz_\\infty}$ have the following form:\n\\begin{gather*}\n\\AXC{ $\\Gamma, p \\Rightarrow p, \\Delta $ ,}\n\\DisplayProof\\qquad\n\\AXC{ $\\Gamma , \\bot \\Rightarrow  \\Delta$ ,}\n\\DisplayProof \\\\\\\\\n\\AXC{$\\Gamma , B \\Rightarrow \\Delta $}\n\\AXC{$\\Gamma \\Rightarrow A,\\Delta $}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Gamma , A \\to B \\Rightarrow \\Delta$}\n\\DisplayProof\\;,\\qquad\n\\AXC{$\\Gamma , A \\Rightarrow B, \\Delta $}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Gamma \\Rightarrow A \\to B, \\Delta$}\n\\DisplayProof\\;,\\\\\\\\\n\\AXC{$\\Gamma, B, \\Box B \\Rightarrow \\Delta $}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Gamma , \\Box B \\Rightarrow \\Delta$}\n\\DisplayProof\\;,\\qquad\n\\AXC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \\;.\n\\end{gather*}\n\\begin{center}\n\\textbf{Fig. 2.} The system $\\mathsf{Grz}_\\infty$\n\\end{center}\n\nThe system $\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$ is defined by adding the rule ($\\mathsf{cut}$) to the system $\\mathsf{Grz_\\infty}$.\nAn \\emph{$\\infty$--proof} in $\\mathsf{Grz}_\\infty$ ($\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$) is a (possibly infinite) tree whose nodes are marked by\nsequents and whose leaves are marked by initial sequents and that is constructed according to the rules of the sequent calculus. In addition, every infinite branch in an $\\infty$--proof must pass through a right premise of the rule ($\\Box$) infinitely many times. A sequent $\\Gamma \\Rightarrow \\Delta$ is \\emph{provable} in $\\mathsf{Grz}_\\infty$ ($\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$) if there is an $\\infty$--proof in $\\mathsf{Grz}_\\infty$ ($\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$) with the root marked by $\\Gamma \\Rightarrow \\Delta$.\n\n\nThe \\emph{$n$-fragment} of an $\\infty$--proof is a finite tree obtained from the $\\infty$--proof by cutting every branch at the $n$th from the root right premise of the rule ($\\Box$). The $1$-fragment of an $\\infty$--proof is also called its \\emph{main fragment}. We define the \\emph{local height $\\lvert \\pi \\rvert$ of an $\\infty$--proof $\\pi$} as the length of the longest branch in its main fragment. An $\\infty$--proof only consisting of an initial sequent has height 0.\n\n\n\nFor instance, consider an $\\infty$--proof of the sequent $\\Box(\\Box(p \\rightarrow \\Box p) \\rightarrow p) \\Rightarrow p$: \n\n\\begin{gather*}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F, p\\Rightarrow p$}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F,p\\Rightarrow \\Box p,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$ F \\Rightarrow p\\to\\Box p,p$}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$p, F \\Rightarrow p$}\n\\AXC{$\\vdots$}\n\\noLine\n\\UIC{$ F \\Rightarrow p$} \n\\LeftLabel{$\\mathsf{\\Box}$} \n\\BIC{$p, F \\Rightarrow \\Box p$}\n\\LeftLabel{$\\mathsf{}\\to_R$}\n\\UIC{$ F \\Rightarrow p\\to\\Box p$}\n\\LeftLabel{$\\mathsf{\\Box}$} \n\\BIC{$ F \\Rightarrow \\Box(p\\to \\Box p),p$} \n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Box(p \\rightarrow \\Box p) \\rightarrow p, F \\Rightarrow p$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,} \n\\UIC{$F \\Rightarrow p $}\n\\DisplayProof\n\\end{gather*}\nwhere $F=\\Box(\\Box(p \\rightarrow \\Box p) \\rightarrow p) $. \nThe local height of this $\\infty$--proof equals to 4 and its main fragment has the form\n\\begin{gather*}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F, p\\Rightarrow p$}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F,p\\Rightarrow \\Box p,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$ F \\Rightarrow p\\to\\Box p,p$}\n\\AXC{\\qquad \\qquad \\qquad \\qquad}\n\\LeftLabel{$\\mathsf{\\Box}$} \n\\BIC{$ F \\Rightarrow \\Box(p\\to \\Box p),p$} \n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Box(p \\rightarrow \\Box p) \\rightarrow p, F \\Rightarrow p$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ .} \n\\UIC{$F \\Rightarrow p $}\n\\DisplayProof\n\\end{gather*}\nWe denote the set of all $\\infty$-proofs in the system $\\mathsf{Grz}_{\\infty} +\\mathsf{cut} $ by $\\mathcal P$. \nFor $\\pi, \\tau\\in\\mathcal P$, we write $\\pi \\sim_n \\tau$ if $n$-fragments of these $\\infty$-proofs are coincide. For any $\\pi, \\tau\\in\\mathcal P$, we also set $\\pi \\sim_0 \\tau$.\n\nNow we define two translations that connect ordinary and non-well-founded sequent calculi for $\\mathsf{Grz}$. \n\n\\begin{lem}\\label{AtoA}\nWe have  $\\mathsf{Grz}_{\\infty} \\vdash \\Gamma,A\\Rightarrow A,\\Delta$ for any sequent $\\Gamma \\Rightarrow \\Delta$ and any formula $A$.\n\\end{lem}\n\\begin{proof}\nStandard induction on the structure of $A$.\n\\end{proof}\n\n\\begin{lem}\\label{Grz-schema}\nWe have $\\mathsf{Grz}_{\\infty}\\vdash\\Box(\\Box(A \\rightarrow \\Box A) \\rightarrow A) \\Rightarrow A$ for any formula $A$.\n\\end{lem}\n\\begin{proof}\nConsider an example of $\\infty$--proof for the sequent $\\Box(\\Box(p \\rightarrow \\Box p) \\rightarrow p) \\Rightarrow p$ given above. We transform this example into an $\\infty$--proof for $\\Box(\\Box(A \\rightarrow \\Box A) \\rightarrow A) \\Rightarrow A$ by replacing $p$ with $A$ and adding required $\\infty$--proofs instead of initial sequents using Lemma \\ref{AtoA}.  \n\\end{proof}\n\nRecall that an inference rule is called admissible (in a given proof system) if, for any instance of the rule, the conclusion is provable whenever all premises are provable. \n\\begin{lem}\\label{weakening}\nThe rule\n\\begin{gather*}\n\\AXC{$\\Gamma\\Rightarrow\\Delta$}\n\\LeftLabel{$\\mathsf{weak}$}\n\\UIC{$\\Pi,\\Gamma\\Rightarrow\\Delta,\\Sigma$}\n\\DisplayProof\n\\end{gather*}\nis admissible in the systems $\\mathsf{Grz_{Seq}} $ and $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$.\n\\end{lem}\n\\begin{proof}\nStandard induction on the structure (local height) of a proof of $\\Gamma\\Rightarrow\\Delta$.\n\\end{proof}\n\n\\begin{thm}\\label{seqtoinfcut}\nIf $\\mathsf{Grz_{Seq}}+\\mathsf{cut}\\vdash\\Gamma\\Rightarrow\\Delta$, then $\\mathsf{Grz}_{\\infty}+\\mathsf{cut}\\vdash\\Gamma\\Rightarrow\\Delta$.\n\\end{thm}\n\\begin{proof}\nAssume $\\pi$ is a proof of $\\Gamma\\Rightarrow\\Delta$ in $\\mathsf{Grz_{Seq}}+\\mathsf{cut}$. By induction on the size of $\\pi$ we prove $\\mathsf{Grz}_{\\infty}+\\mathsf{cut}\\vdash\\Gamma\\Rightarrow\\Delta$. \n\nIf $\\Gamma \\Rightarrow \\Delta $ is an initial sequent of $\\mathsf{Grz_{Seq}}+\\mathsf{cut}$, then it is provable in $\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$ by Lemma \\ref{AtoA}.\nOtherwise, consider the last application of an inference rule in $\\pi$. \n\nThe only non-trivial case is when the proof $\\pi$ has the form \n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Box \\Pi,\\Box(A\\to\\Box A)\\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box_{Grz}}$}\n\\RightLabel{ ,}\n\\UIC{$\\Sigma,\\Box\\Pi\\Rightarrow \\Box A, \\Lambda$}\n\\DisplayProof\n\\end{gather*} \nwhere $\\Sigma,\\Box\\Pi = \\Gamma$ and $\\Box A, \\Lambda = \\Delta$. By the induction hypothesis there is an $\\infty$--proof $\\xi$ of $\\Box \\Pi,\\Box(A\\to\\Box A)\\Rightarrow A$ in $\\mathsf{Grz}_{\\infty}+\\mathsf{cut}$.\n\nWe have the following $\\infty$--proof $\\lambda$ of $\\Box \\Pi\\Rightarrow A$ in $\\mathsf{Grz}_{\\infty}+\\mathsf{cut} $:\n\\begin{gather*}\n\\AXC{$\\xi^{\\prime}$}\n\\noLine\n\\UIC{$\\Box \\Pi,\\Box(A\\to\\Box A)\\Rightarrow A,A$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Box \\Pi\\Rightarrow G,A$}\n\\AXC{$\\xi$}\n\\noLine\n\\UIC{$\\Box \\Pi,\\Box(A\\to\\Box A)\\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Box \\Pi\\Rightarrow G$}\n\\LeftLabel{$\\Box$}\n\\BIC{$\\Box \\Pi\\Rightarrow\\Box G,A$}\n\\AXC{$\\theta$}\n\\noLine\n\\UIC{$\\Box\\Pi,\\Box G \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ ,}\n\\BIC{$\\Box\\Pi\\Rightarrow A$}\n\\DisplayProof\n\\end{gather*}\nwhere $G= \\Box(A \\rightarrow \\Box A) \\rightarrow A$, $\\xi^{\\prime}$ is an $\\infty$--proof of $\\Box \\Pi,\\Box(A\\to\\Box A)\\Rightarrow A,A$ obtained from $\\xi$ by Lemma \\ref{weakening} and $\\theta$ is an $\\infty$--proof of $\\Box\\Pi,\\Box G \\Rightarrow A$, which exists by Lemma \\ref{Grz-schema} and Lemma \\ref{weakening}.\n\nThe required $\\infty$--proof for $\\Sigma,\\Box\\Pi\\Rightarrow \\Box A, \\Delta$ has the form\n\\begin{gather*}\n\\AXC{$\\lambda'$}\n\\noLine\n\\UIC{$\\Sigma,\\Box\\Pi\\Rightarrow A,\\Lambda$}\n\\AXC{$\\lambda$}\n\\noLine\n\\UIC{$\\Box\\Pi\\Rightarrow A$}\n\\LeftLabel{$\\Box$}\n\\RightLabel{ ,}\n\\BIC{$\\Sigma,\\Box\\Pi\\Rightarrow \\Box A, \\Lambda$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\lambda'$ is an $\\infty$-proof of the sequent $\\Sigma,\\Box\\Pi\\Rightarrow A,\\Lambda$ obtained from the $\\infty$-proof $\\lambda$ by Lemma \\ref{weakening}.\n\nThe cases of other inference rules being last in $\\pi$ are straightforward, so we omit them.\n\n\\end{proof}\n\n\n\nFor a sequent $\\Gamma\\Rightarrow\\Delta$, let $Sub(\\Gamma\\Rightarrow\\Delta)$ be the set of all subformulas of the formulas from $\\Gamma \\cup\\Delta$.\nFor a finite set of formulas $\\Lambda$, let $\\Lambda^\\ast$ be the set $\\{\\Box(A\\to\\Box A)\\mid A\\in\\Lambda\\}$.\n\n\\begin{lem} \\label{translation\nIf $\\mathsf{Grz_\\infty}\\vdash \\Gamma\\Rightarrow\\Delta$, then $\\mathsf{Grz_{Seq}} \\vdash \\Lambda^\\ast,\\Gamma\\Rightarrow\\Delta$ for any finite set of formulas $\\Lambda$.\n\\end{lem}\n\\begin{proof}\nAssume $\\pi$ is an $\\infty$--proof of the sequent $\\Gamma\\Rightarrow\\Delta$ in $\\mathsf{Grz}_\\infty$ and $\\Lambda$ is a finite set of formulas.\nBy induction on the number of elements in the finite set $Sub(\\Gamma\\Rightarrow\\Delta)\\backslash \\Lambda$ with a subinduction on $\\lvert \\pi \\rvert$, we prove $\\mathsf{Grz_{Seq}} \\vdash \\Lambda^\\ast,\\Gamma\\Rightarrow\\Delta$. \n\n\nIf $\\lvert \\pi \\rvert=0$, then $\\Gamma\\Rightarrow\\Delta$ is an initial sequent. We see that the sequent $\\Lambda^\\ast,\\Gamma\\Rightarrow\\Delta$ is an initial sequent and it is provable in $\\mathsf{Grz_{Seq}}$.\nOtherwise, consider the last application of an inference rule in $\\pi$. \n\nCase 1. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Gamma,A\\Rightarrow B,\\Sigma$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma\\Rightarrow A\\to B,\\Sigma$}\n\\DisplayProof\n\\end{gather*}\nwhere $A\\to B,\\Sigma = \\Delta$.\nNotice that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. By the induction hypothesis for $\\pi^\\prime$ and $\\Lambda$, the sequent $\\Lambda^\\ast,\\Gamma,A\\Rightarrow B,\\Sigma$ is provable in  $\\mathsf{Grz_{Seq}}$. \nApplying the rule ($\\mathsf{\\to_R}$) to it, we obtain that the sequent $\\Lambda^\\ast,\\Gamma\\Rightarrow\\Delta$ is provable in $\\mathsf{Grz_{Seq}}$.\n\nCase 2. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Sigma, B\\Rightarrow \\Delta$}\n\\AXC{$\\pi^{\\prime\\prime}$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow A,\\Delta$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Sigma, A\\to B\\Rightarrow \\Delta$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Sigma, A\\to B = \\Gamma$. We see that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. By the induction hypothesis for $\\pi^\\prime$ and $\\Lambda$, the sequent $\\Lambda^\\ast,\\Sigma, B\\Rightarrow \\Delta$ is provable in  $\\mathsf{Grz_{Seq}}$. Analogously, we have $\\mathsf{Grz_{Seq}} \\vdash \\Lambda^\\ast,\\Sigma \\Rightarrow A,\\Delta$. Applying the rule ($\\mathsf{\\to_L}$), we obtain that the sequent $\\Lambda^\\ast,\\Sigma, A\\to B \\Rightarrow\\Delta$ is provable in $\\mathsf{Grz_{Seq}}$.\n\nCase 3. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Sigma,A,\\Box A\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$\\Sigma,\\Box A\\Rightarrow \\Delta$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Sigma, \\Box A = \\Gamma$. We see that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. By the induction hypothesis for $\\pi^\\prime$ and $\\Lambda$, the sequent $\\Lambda^\\ast,\\Sigma, A, \\Box A\\Rightarrow \\Delta$ is provable in  $\\mathsf{Grz_{Seq}}$. Applying the rule ($\\mathsf{refl}$), we obtain $\\mathsf{Grz_{Seq}} \\vdash \\Lambda^\\ast,\\Sigma, \\Box A\\Rightarrow\\Delta$. \n\nCase 4. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow A, \\Sigma$}\n\\AXC{$\\pi^{\\prime\\prime}$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Phi, \\Box \\Pi = \\Gamma$ and $\\Box A, \\Sigma =\\Delta$.\n\nSubcase 4.1: the formula $A$ belongs to $\\Lambda$. We see that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. By the induction hypothesis for $\\pi^\\prime$ and $\\Lambda$, the sequent $\\Lambda^\\ast,\\Phi, \\Box \\Pi \\Rightarrow A, \\Sigma$ is provable in  $\\mathsf{Grz_{Seq}}$. \nThen we see\n\\begin{gather*}\n\\AXC{$\\mathsf{Ax}$}\n\\noLine\n\\UIC{$\\Lambda^\\ast,\\Box A,\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\AXC{$\\Lambda^\\ast,\\Phi, \\Box \\Pi \\Rightarrow A,  \\Sigma$}\n\\LeftLabel{$\\mathsf{weak}$}\n\\UIC{$\\Lambda^\\ast,\\Phi, \\Box \\Pi \\Rightarrow A,  \\Box A,\\Sigma$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$(\\Lambda\\backslash\\{A\\})^\\ast,A\\to\\Box A,\\Box(A\\to\\Box A),\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$(\\Lambda\\backslash\\{A\\})^\\ast,\\Box(A\\to\\Box A),\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\DisplayProof\n\\end{gather*}\nwhere the rule ($\\mathsf{weak}$) is admissible by Lemma \\ref{weakening}.\n\nSubcase 4.2: the formula $A$ doesn't belong to $\\Lambda$. We have that the number of elements in $Sub(\\Box\\Pi\\Rightarrow A)\\backslash(\\Lambda\\cup \\{A\\})$ is strictly less than the number of elements in $Sub(\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma)\\backslash\\Lambda$. Therefore, by the induction hypothesis for $\\pi^{\\prime\\prime}$ and $\\Lambda\\cup \\{A\\}$, the sequent $\\Lambda^\\ast,\\Box(A\\to\\Box A),\\Box \\Pi \\Rightarrow A$ is provable in  $\\mathsf{Grz_{Seq}}$. Then we have\n\\begin{gather*}\n\\AXC{$\\Lambda^\\ast,\\Box(A\\to\\Box A),\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box_{Grz}}$}\n\\RightLabel{ .}\n\\UIC{$\\Lambda^\\ast,\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\DisplayProof\n\\end{gather*}\n\\end{proof}\nFrom Lemma \\ref{translation} we immediately obtain the following theorem.\n\\begin{thm}\\label{inftoseq}\nIf $\\mathsf{Grz_\\infty}\\vdash \\Gamma\\Rightarrow\\Delta$, then $\\mathsf{Grz_{Seq}} \\vdash \\Gamma\\Rightarrow\\Delta$.\n\\end{thm}\n\n\n\n\\section{Ultrametric spaces}\n\\label{SecUlt}\nIn this section we recall basic notions of the theory of ultrametric spaces (cf. \\cite{Shor}) and consider several examples concerning $\\infty$-proofs\n\nAn \\emph{ultrametric space} $(M,d)$ is a metric space that satisfies a stronger version of the triangle inequality: for any $x,y,z\\in M$\n$$d(x,z) \\leqslant \\max \\{d(x,y), d(y,z)\\}.$$ \n\nFor $x\\in M$ and $r\\in [0, + \\infty)$, the set $B_r(x)=\\{ y\\in M \\mid d(x,y) \\leqslant r\\}$ is called the \\emph{closed ball} with \\emph{center $x$} and \\emph{radius $r$}. Recall that a metric space $(M,d)$ is \\emph{complete} if any descending sequence of closed balls, with radii tending to $0$, has a common point. An ultametric space $(M,d)$ is called \\emph{spherically complete} if an arbitrary descending sequence of closed balls has a common point.\n\nFor example, consider the set $\\mathcal P$ of all $\\infty$-proofs of the system $\\mathsf{Grz}_{\\infty} +\\mathsf{cut} $. We can define an ultrametric $d_{\\mathcal P} \\colon {\\mathcal P} \\times {\\mathcal P} \\to [0,1]$ on $\\mathcal P$ by putting\n\\[d_{\\mathcal P}(\\pi,\\tau) = \n\\inf\\{\\frac{1}{2^n} \\mid  \\pi \\sim_n \\tau\\}.\n\\]\nWe see that $d_{\\mathcal P}(\\pi , \\tau) \\leqslant 2^{-n}$ if and only if $\\pi \\sim_n \\tau$. Thus, the ultrametric $d_{\\mathcal P}$ can be considered as a measure of similarity between $\\infty$-proofs.\n\n\\begin{prop}\\label{ComplP}\n$(\\mathcal{P},d_{\\mathcal P})$ is a (spherically) complete ultrametric space.\n\\end{prop} \n\n \nConsider the following characterization of spherically complete ultrametric spaces. Let us write $x \\equiv_r y$ if $d(x,y)\\leqslant r$.\nTrivially, the relation $\\equiv_r$ is an equivalence relation for any ultrametric space and any number $r\\geqslant 0$.\n\n\n\n\\begin{prop}\\label{sphcomp}\nAn ultametric space $(M,d)$ is spherically complete if and only if for any sequence $(x_i)_{i\\in \\mathbb N}$ of elements of $M$, where $x_i\\equiv_{r_i}x_{i+1}$ and  \n$r_i \\geqslant r_{i+1}$ for all $i\\in \\mathbb N$, there is a point $x$ of $M$ such that $x \\equiv_{r_i} x_i$ for any $i\\in\\mathbb N$.\n\\end{prop}\n\\begin{proof}\n($\\Rightarrow$) Assume $(M,d)$ is a spherically complete ultrametric space. Consider a sequence $(x_i)_{i\\in \\mathbb N}$ of elements of $M$ such that $x_i\\equiv_{r_i}x_{i+1}$ and $r_i \\geqslant r_{i+1}$ for all $i\\in \\mathbb N$.  Then the sequence $(B_{r_i}(x_i))$ is a descending sequence of closed balls, and therefore by spherical completeness has a common point $x$. Trivially, the point $x$ satisfies the desired conditions.\n\n($\\Leftarrow$) Assume there is a descending sequence of closed balls $(B_{r_i}(x_i))$. We have that $x_0 \\equiv_{r_0} x_1 \\equiv_{r_1} \\dots$ and $r_i \\geqslant r_{i+1}$ for all $i\\in \\mathbb N$. So there is an element $x\\in M$ such that $x \\equiv_{r_i} x_i$, meaning it lies in all the balls.\n\\end{proof}\n\nIn an ultrametric space $(M,d)$, a function $f \\colon M\\to M$ is called \\emph{non-expansive} if $d(f(x),f(y)) \\leqslant d(x,y)$ for all $x,y\\in M$.\nFor ultrametric spaces $(M, d_M)$ and $(N, d_N)$, the Cartesian product $M \\times N$ can be also considered as an ultrametric space with the metric $$d_{M \\times N} ((x_1,y_1),(x_2,y_2)) = \\max \\{d_M(x_1,x_2), d_N(y_1,y_2) \\}.$$ \n\n\nLet us consider another example. For $m\\in \\mathbb N$, let $\\mathcal F_m$ denote the set of all non-expansive functions from $\\mathcal P^m$ to $\\mathcal P$. Note that any function $\\mathsf u\\colon \\mathcal P^m\\to\\mathcal P$ is non-expansive if and only if for any tuples $\\vec\\pi$ and $\\vec\\pi'$, and any $n\\in\\mathbb N$ we have \n$$\\pi_1\\sim_n\\pi'_1,\\dotsc,\\pi_m\\sim_n\\pi'_m\\Rightarrow \\mathsf u(\\vec\\pi)\\sim_n\\mathsf u(\\vec\\pi').$$\nNow we introduce an ultrametric for $\\mathcal F_m$. For $\\mathsf{a,b}\\in \\mathcal F_m$, we write \n$\\mathsf a\\sim_{n,k}\\mathsf b$ if $\\mathsf a(\\vec{\\pi})\\sim_n\\mathsf b(\\vec{\\pi})$ for any $\\vec{\\pi}\\in\\mathcal P^m$ and, in addition, $\\mathsf a(\\vec{\\pi}) \\sim_{n+1}\\mathsf b(\\vec{\\pi})$ whenever $\\Sigma_{i=1}^m\\lvert\\pi_i\\rvert< k$.\\footnote{This definition is inspired by \\cite[Subsection 2.1]{GiMi}.}\nAn ultrametric $l_m$ on $\\mathcal F_m$ is defined by \n\\[l_m(\\mathsf a,\\mathsf b)=\\frac{1}{2}\\inf\\{\\frac{1}{2^n}+\\frac{1}{2^{n+k}}\\mid \\mathsf a\\sim_{n,k}\\mathsf b\\}.\\]\nWe see that $l_m(\\mathsf a,\\mathsf b) \\leqslant 2^{-n-1}+2^{-n-k-1}$ if and only if $\\mathsf a\\sim_{n,k}\\mathsf b$.\n\n\\begin{prop}\\label{SphCom}\n$(\\mathcal F_m, l_m)$ is a spherically complete ultrametric space.\n\\end{prop}\n\\begin{proof}\nAssume we have a series\n$\\mathsf a_0\\sim_{n_0,k_0}\\mathsf a_1\\sim_{n_1,k_1}\\dotso  $,\nwhere the sequence $r_i=2^{-n_i}+2^{-n_i-k_i-1}$ is non-increasing. From Proposition \\ref{sphcomp}, it is sufficient to find a function $\\mathsf a\\in\\mathcal F_m$ such that $\\mathsf a\\sim_{n_i,k_i}\\mathsf a_i$ for all $i\\in \\mathbb N$.\n\n\nSuppose $\\lim_{i\\to\\infty}r_i=0$. Consider a tuple $\\vec{\\pi}\\in\\mathcal P^m $. We have that $\\lim_{i\\to\\infty}n_i=+ \\infty$ and $\\mathsf a_0 (\\vec{\\pi})\\sim_{n_0}\\mathsf a_1 (\\vec{\\pi})\\sim_{n_1}\\dotso$ .\nBy Proposition \\ref{ComplP}, there is an $\\infty$-proof $\\tau$ such that $\\tau \\sim_{n_i} \\mathsf a_i (\\vec{\\pi})$ for all $i\\in \\mathbb N$. We define $\\mathsf a (\\vec{\\pi}) = \\tau$. We need to check that the mapping $\\mathsf a$ is non-expansive. If for tuples $\\vec\\pi$ and $\\vec\\pi'$ we have $\\pi_1\\sim_n\\pi'_1,\\ldots,\\pi_m\\sim_n\\pi'_m$ for some $n\\in\\mathbb N$, then we can choose $i$ such that $n_i>n$. We have $\\mathsf a(\\vec\\pi)\\sim_n \\mathsf a_i(\\vec\\pi)\\sim_n\\mathsf a_i(\\vec\\pi')\\sim_n\\mathsf a(\\vec\\pi')$. Therefore $\\mathsf a(\\vec\\pi)\\sim_n \\mathsf a(\\vec\\pi')$ and the mapping $\\mathsf a$ is non-expansive.\n\nIf $\\lim_{i\\to\\infty}r_i>0$, then $\\lim_{i\\to\\infty}n_i= n$ for some $n \\in \\mathbb{N}$. We have two cases: either $\\lim_{i\\to\\infty}k_i= k$ for a number $k \\in \\mathbb{N}$, or $\\lim_{i\\to\\infty}k_i= +\\infty$. In the first case, there is $j\\in\\mathbb N$ such that $(n_i,k_i)=(n_j,k_j)$ for all $i>j$, and we can take $\\mathsf a_j$ as $\\mathsf a$. Here the mapping $\\mathsf a$ is obviously non-expansive.\nIn the second case, for a tuple $\\vec{\\pi}$ we define $\\mathsf a (\\vec{\\pi})$ to be $\\mathsf{a}_j (\\vec{\\pi})$, where $j= \\min \\{ i \\in \\mathbb{N} \\mid n_i= n \\text{ and } \\sum_{s=1}^m\\lvert\\pi_s\\rvert < k_i \\}$. For all tuples $\\vec\\pi$ and $\\vec\\pi'$ we have $\\mathsf a(\\vec\\pi)\\sim_0 \\mathsf a(\\vec\\pi')$. If for tuples $\\vec\\pi$ and $\\vec\\pi'$ and some $n\\geqslant 1$ we have $\\pi_1\\sim_n\\pi'_1,\\ldots,\\pi_m\\sim_n\\pi'_m$, then $\\sum_{s=1}^m\\lvert\\pi_s\\rvert=\\sum_{s=1}^m\\lvert\\pi_s'\\rvert=t$ and $\\mathsf a(\\vec\\pi)=\\mathsf a_j(\\vec\\pi)\\sim_n \\mathsf a_j(\\vec\\pi')=\\mathsf a(\\vec\\pi')$, where $j=\\min \\{ i \\in \\mathbb{N} \\mid n_i= n \\text{ and } t < k_i \\}$. Therefore $\\mathsf a(\\vec\\pi)\\sim_n \\mathsf a(\\vec\\pi')$ and the mapping $\\mathsf a$ is non-expansive.\n\n\n\n\n\n\n\n\\end{proof}\n\n\nIn an ultrametric space $(M,d)$, a function $f\\colon M \\to M$ is called \\emph{(strictly) contractive} if $d(f(x),f(y)) < d(x,y)$ when $x \\neq y$. \n\n\nNotice that any operator $\\mathsf U\\colon\\mathcal F_m\\to \\mathcal F_m$ is strictly contractive if and only if for any $\\mathsf a,\\mathsf b\\in\\mathcal F_m$, and any $n,k\\in\\mathbb N$ we have \n$$\\mathsf a\\sim_{n,k}\\mathsf b\\Rightarrow \\mathsf U(\\mathsf a)\\sim_{n,k+1}\\mathsf U(\\mathsf b).$$\n\n\nNow we state a generalization of the Banach's fixed-point theorem for ultrametric spaces that will be used in the next sections.\n\n\n\\begin{thm}[Prie\\ss-Crampe \\cite{PrCr}]\\label{fixpoint}\nLet $(M,d)$ be a non-empty spherically complete ultrametric space. Then every strictly contractive mapping $f\\colon M\\to M$ has a unique fixed-point. \n\\end{thm}\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Admissible rules}\\label{SecAdm}\n\n\n\n\nIn this section, for the system $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$, we state admissibility of auxiliary inference rules, which will be used in the proof of the  cut-elimination theorem. \n\nRecall that the set $\\mathcal P$ of all $\\infty$-proofs of the system $\\mathsf{Grz}_{\\infty} +\\mathsf{cut} $ can be considered as an ultrametric space with the metric $d_{\\mathcal P}$.\n\nBy $\\mathcal{P}_n$ we denote the set of all $\\infty$-proofs that do not contain applications of the cut rule in their $n$-fragments. We also set $\\mathcal{P}_0= \\mathcal{P}$.\n\nA mapping $\\mathsf u:\\mathcal P^m\\to \\mathcal P$ is called \\emph{adequate} if for any $n\\in\\mathbb N$ we have $\\mathsf u(\\pi_1,\\ldots,\\pi_n)\\in\\mathcal P_n$, whenever $\\pi_i\\in\\mathcal P_n$ for all $i\\leqslant n$. \n\n\nIn $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$, we call a single-premise inference rule \\emph{strongly admissible} if there is a non-expansive adequate mapping $\\mathsf{u}\\colon\\mathcal{P} \\to \\mathcal{P}$ that maps any $\\infty$-proof of the premise of the rule to an $\\infty$-proof of the conclusion. The mapping $\\mathsf{u}$ must also satisfy one additional condition: $\\lvert \\mathsf{u}(\\pi)\\rvert \\leqslant \\lvert \\pi \\rvert$ for any $\\pi \\in \\mathcal{P}$.\n\nIn the following lemmata, non-expansive mappings are defined in a standard way by induction on the local heights of $\\infty$-proofs for the premises. So we omit further details.\n\n\\begin{lem}\\label{strongweakening}\nFor any finite multisets of formulas $\\Pi$ and $\\Sigma$, the inference rule\n\\begin{gather*}\n\\AXC{$\\Gamma\\Rightarrow\\Delta$}\n\\LeftLabel{$\\mathsf{wk}_{\\Pi, \\Sigma}$}\n\\UIC{$\\Pi,\\Gamma\\Rightarrow\\Delta,\\Sigma$}\n\\DisplayProof\n\\end{gather*}\nis strongly admissible in $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$.\n\\end{lem}\n\n\n\\begin{lem}\\label{inversion}\nFor any formulas $A$ and $B$, the rules\n\\begin{gather*}\n\\AXC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\LeftLabel{$\\mathsf{li}_{A \\to B}$}\n\\UIC{$\\Gamma ,B \\Rightarrow  \\Delta$}\n\\DisplayProof\\qquad\n\\AXC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\LeftLabel{$\\mathsf{ri}_{A \\to B}$}\n\\UIC{$\\Gamma  \\Rightarrow  A, \\Delta$}\n\\DisplayProof\\\\\\\\\n\\AXC{$\\Gamma  \\Rightarrow   A \\rightarrow B, \\Delta$}\n\\LeftLabel{$\\mathsf{i}_{A \\to B}$}\n\\UIC{$\\Gamma ,A \\Rightarrow B, \\Delta$}\n\\DisplayProof\\qquad\n\\AXC{$\\Gamma  \\Rightarrow   \\bot, \\Delta$}\n\\LeftLabel{$\\mathsf{i}_{\\bot}$}\n\\UIC{$\\Gamma  \\Rightarrow \\Delta$}\n\\DisplayProof\\qquad \n\\AXC{$\\Gamma \\Rightarrow \\Box A, \\Delta$}\n\\LeftLabel{$\\mathsf{li}_{\\:\\Box A}$}\n\\UIC{$\\Gamma \\Rightarrow A, \\Delta$}\n\\DisplayProof\n\\end{gather*}\nare strongly admissible in $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$.\n\\end{lem}\n\n\\begin{lem}\\label{weakcontraction}\nFor any atomic proposition $p$, the rules\n\\begin{gather*}\n\\AXC{$\\Gamma , p,p \\Rightarrow  \\Delta$}\n\\LeftLabel{$\\mathsf{acl}_{p}$}\n\\UIC{$\\Gamma ,p \\Rightarrow  \\Delta$}\n\\DisplayProof\\qquad\n\\AXC{$\\Gamma \\Rightarrow p,p, \\Delta$}\n\\LeftLabel{$\\mathsf{acr}_{p}$}\n\\UIC{$\\Gamma \\Rightarrow p, \\Delta$}\n\\DisplayProof\n\\end{gather*}\nare strongly admissible in $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$.\n\\end{lem}\n\n\n\n\n\n\\section{Cut elimination}\n\\label{SecCut}\n\n\nIn this section, we construct a continuous function from $\\mathcal{P}$ to $\\mathcal{P}$, which maps any $\\infty$-proof of the system $\\mathsf{Grz}_{\\infty} +\\mathsf{cut}$ to a cut-free $\\infty$-proof of the same sequent. \n\nLet us call a pair of $\\infty$-proofs $(\\pi,\\tau)$ a \\emph{cut pair} if $\\pi$ is an $\\infty$-proof of the sequent $\\Gamma\\Rightarrow\\Delta,A$ and $\\tau$ is an $\\infty$-proof of the sequent $A, \\Gamma\\Rightarrow \\Delta$ for some $\\Gamma, \\Delta$, and $A$. For a cut pair $(\\pi,\\tau)$, we call the sequent $\\Gamma\\Rightarrow \\Delta$ its \\emph{cut result} and the formula $A$ its cut formula.\n\nFor a modal formula $A$, a non-expansive mapping $\\mathsf{u}$ from $\\mathcal P \\times \\mathcal P$ to $\\mathcal P$ is called \\emph{$A$-removing} if it maps every cut pair $(\\pi,\\tau)$ with the cut formula $A$ to an $\\infty$-proof of its cut result.\nBy $\\mathcal R_A$, let us denote the set of all $A$-removing mappings.\n\n\\begin{lem}\\label{Comp R_A}\nFor each formula $A$, the pair $(\\mathcal R_A,l_2)$ is a non-empty spherically complete ultrametric space.\n\n\\end{lem}\n\\begin{proof}\nThe proof of spherical completeness of the space $(\\mathcal R_A,l_2)$ is analoguos to the proof of Proposition \\ref{SphCom}.\n\nWe only need to check that the set $\\mathcal R_A$ is non-empty. Consider the mapping $\\mathsf u_{cut}:\\mathcal P^2\\to\\mathcal P$ that is defined as follows. For a cut pair $(\\pi,\\tau)$ with the cut formula $A$, it joins the $\\infty$-proofs $\\pi$ and $\\tau$ with an appropriate instance of the rule $(\\mathsf{cut})$ . For all other pairs, the mapping $\\mathsf u_{cut}$ returns the first argument.\n\nClearly, $\\mathsf u_{cut}$ is non-expansive and therefore lies in $\\mathcal R_A$.\n\\end{proof}\n\nIn what follows, we use nonexpansive adequate mappings $ \\mathsf{wk}_{\\Pi, \\Sigma}$, $\\mathsf{li}_{A\\to B}$, $\\mathsf{ri}_{A\\to B}$, $\\mathsf{i}_{A\\to B}$, $\\mathsf{i}_\\bot$, $\\mathsf{li}_{\\Box A}$, $\\mathsf{acl}_{p}$, $\\mathsf{acr}_{p}$ from Lemma \\ref{strongweakening}, Lemma \\ref{inversion}, and Lemma \\ref{weakcontraction}.\n\n\n\\begin{lem}\\label{repadeq}\nFor any atomic proposition $p$, there exists an adequate $p$-removing mapping $\\mathsf {re}_p$.\n\\end{lem}\n\\begin{proof}\n\nAssume we have two $\\infty$-proofs $\\pi$ and $\\tau$. If the pair $(\\pi,\\tau)$ is not a cut pair or is a cut pair with the cut formula being not $p$, then we put $\\mathsf{re}_{p}(\\pi,\\tau)=\\pi$. \nOtherwise, we define $\\mathsf{re}_{p}(\\pi,\\tau)$ by induction on $\\lvert \\pi\\rvert$. Let the cut result of  the pair $(\\pi,\\tau)$ be $\\Gamma\\Rightarrow \\Delta$.\n\nIf $\\lvert \\pi\\rvert=0$, then $\\Gamma\\Rightarrow \\Delta, p$ is an initial sequent. Suppose that $\\Gamma\\Rightarrow \\Delta$ is also an initial sequent. Then $\\mathsf{re}_{p}(\\pi,\\tau)$ is defined as the $\\infty$-proof consisting only of the sequent $\\Gamma\\Rightarrow \\Delta$. If $\\Gamma\\Rightarrow \\Delta$ is not an initial sequent, then $\\Gamma$ has the form $p,\\Phi$, and $\\tau$ is an $\\infty$-proof of the sequent $p,p,\\Phi \\Rightarrow \\Delta$. Applying the non-expansive adequate mapping $\\mathsf{acl}_p$ from Lemma \\ref{weakcontraction}, we put $\\mathsf{re}_{p}(\\pi,\\tau) := \\mathsf{acl}_p (\\tau)$.   \n\nNow suppose that $\\lvert \\pi \\rvert >0$. We define $\\mathsf{re}_{p}(\\pi,\\tau)$ according to the last application of an inference rule in $\\pi$:  \n\\begin{align*}\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma,B\\Rightarrow C,\\Sigma,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Gamma\\Rightarrow B\\to C,\\Sigma,p$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{re}_p(\\pi_0, \\mathsf{i}_{B \\to C}(\\tau))$}\n\\noLine\n\\UIC{$\\Gamma,B\\Rightarrow C,\\Sigma$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma\\Rightarrow B\\to C,\\Sigma$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma, C\\Rightarrow \\Delta, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow B,\\Delta, p$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Sigma, B\\to C\\Rightarrow \\Delta, p$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{re}_p (\\pi_0, \\mathsf{li}_{B\\to C} (\\tau))$}\n\\noLine\n\\UIC{$\\Sigma, C\\Rightarrow \\Delta$}\n\\AXC{$\\mathsf{re}_p (\\pi_1, \\mathsf{ri}_{B\\to C} (\\tau))$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow B,\\Delta$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Sigma, B\\to C\\Rightarrow \\Delta$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma,B,\\Box B\\Rightarrow \\Delta,p$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Sigma,\\Box B\\Rightarrow \\Delta,p$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{re}_p(\\pi_0, \\mathsf{wk}_{ B, \\emptyset} (\\tau)$}\n\\noLine\n\\UIC{$\\Sigma,B,\\Box B\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$\\Sigma,\\Box B\\Rightarrow \\Delta$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma\\Rightarrow B,\\Delta, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma,B \\Rightarrow \\Delta, p$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\BIC{$\\Gamma\\Rightarrow \\Delta, p$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{re}_p (\\pi_0, \\mathsf{wk}_{\\emptyset,B} (\\tau))$}\n\\noLine\n\\UIC{$\\Gamma\\Rightarrow B,\\Delta$}\n\\AXC{$\\mathsf{re}_p (\\pi_1, \\mathsf{wk}_{B,\\emptyset} (\\tau))$}\n\\noLine\n\\UIC{$\\Gamma, B\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma\\Rightarrow \\Delta$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma, p$}\n\\DisplayProof\n,\\tau\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{re}_p(\\pi_0, \\mathsf{li}_{\\: \\Box B}(\\tau))$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma$}\n\\DisplayProof\n\\end{align*}\n\nThe mapping $\\mathsf{re}_p$ is well defined. \n\nNow we claim that $\\mathsf{re}_p$ is non-expansive. It sufficient to check that for any pairs $(\\pi,\\tau)$ and $(\\pi^\\prime,\\tau^\\prime)$, and any $n\\in\\mathbb N$ we have \n\\[\\pi\\sim_n\\pi',\\; \\tau\\sim_n\\tau'\\Rightarrow  \\mathsf{re}_p(\\pi,\\tau)\\sim_n \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime).\\]\n\nAssume we have two pairs of $\\infty$-proofs $(\\pi,\\tau)$ and $(\\pi^\\prime,\\tau^\\prime)$ such that $\\pi\\sim_n\\pi'$, $\\tau\\sim_n\\tau'$.\nIf $n=0$, then trivially $ \\mathsf{re}_p(\\pi,\\tau)\\sim_n \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime)$. If $n>0$, then main fragments of $\\pi$ and $\\pi^\\prime$ ($\\tau$ and $\\tau^\\prime$) are identical.\nSuppose that $(\\pi,\\tau)$ is not a cut pair or is a cut pair with the cut formula being not $p$. Then the same condition holds for the pair $(\\pi^\\prime,\\tau^\\prime)$. In this case, we have \n\\[ \\mathsf{re}_p(\\pi,\\tau) = \\pi \\sim_n \\pi^\\prime =  \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime).\\] \n\nOtherwise, $(\\pi,\\tau)$ and $(\\pi^\\prime,\\tau^\\prime)$ \nare cut pairs with the cut formula $p$.\nNow the statement $\\mathsf{re}_p(\\pi,\\tau)\\sim_n \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime)$ is proved by induction on $\\lvert \\pi\\rvert=  \\lvert \\pi^\\prime\\rvert$ with the case analysis according to the definition of $\\mathsf{re}_p$. \n\nIf the last inference in $\\pi$ is an application of the rule $(\\mathsf{\\to_R})$, then $\\pi$ and $\\pi^\\prime$ have the following forms\n\\[\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma,B\\Rightarrow C,\\Sigma,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Gamma\\Rightarrow B\\to C,\\Sigma,p$}\n\\DisplayProof\\qquad\n\\AXC{$\\pi^\\prime_0$}\n\\noLine\n\\UIC{$\\Gamma,B\\Rightarrow C,\\Sigma,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma\\Rightarrow B\\to C,\\Sigma,p$}\n\\DisplayProof\n\\]\nwhere $\\pi_0 \\sim_n \\pi^\\prime_0$. Since $\\mathsf{i}_{B \\to C}$ is non-expansive, we have $\\mathsf{i}_{B \\to C}(\\tau) \\sim_n \\mathsf{i}_{B \\to C}(\\tau^\\prime)$. Consequently, by the induction hypothesis for pairs $(\\pi_0, \\mathsf{i}_{B \\to C}(\\tau))$ and $(\\pi^\\prime_0, \\mathsf{i}_{B \\to C}(\\tau^\\prime))$, we have $\\mathsf{re}_p(\\pi_0, \\mathsf{i}_{B \\to C}(\\tau)) \\sim_n \\mathsf{re}_p(\\pi^\\prime_0, \\mathsf{i}_{B \\to C}(\\tau))$. Thus we obtain $\\mathsf{re}_p(\\pi,\\tau)\\sim_n \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime)$.\n\nConsider the case when the last inference in $\\pi$ is an application of the rule $(\\mathsf{\\Box})$. We have that $\\pi$ and $\\pi^\\prime$ have the following forms\n\\[\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma, p$}\n\\DisplayProof\\qquad\n\\AXC{$\\pi^\\prime_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma, p$}\n\\AXC{$\\pi^\\prime_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma, p$}\n\\DisplayProof\n\\] \nwhere $\\pi_0 \\sim_n \\pi^\\prime_0$ and $\\pi_1 \\sim_{n-1} \\pi^\\prime_1$. Since $\\mathsf{li}_{\\: \\Box B}$ is non-expansive, we have $\\mathsf{li}_{\\: \\Box B}(\\tau) \\sim_n \\mathsf{li}_{\\: \\Box B}(\\tau^\\prime)$. Consequently, by the induction hypothesis for pairs $(\\pi_0, \\mathsf{li}_{\\: \\Box B}(\\tau))$ and $(\\pi^\\prime_0, \\mathsf{li}_{\\: \\Box B}(\\tau^\\prime))$, we have $\\mathsf{re}_p(\\pi_0, \\mathsf{li}_{\\: \\Box B}(\\tau)) \\sim_n \\mathsf{re}_p(\\pi^\\prime_0, \\mathsf{li}_{\\: \\Box B}(\\tau))$. Thus we obtain $\\mathsf{re}_p(\\pi,\\tau)\\sim_n \\mathsf{re}_p(\\pi^\\prime,\\tau^\\prime)$.\n\nAll the other cases are treated similarly, so we omit them. We have that the mapping $\\mathsf{re}_p$ is non-expansive.\nIt remains to check that the mapping $\\mathsf{re}_p$ is adequate.\n \nAssume we have a pair of $\\infty$-proofs $(\\pi,\\tau)$ such that $\\pi \\in \\mathcal{P}_n$ and $\\tau\\in\\mathcal{P}_n $. We claim that $\\mathsf{re}_p(\\pi,\\tau) \\in \\mathcal{P}_n$. If $n=0$, then trivially $ \\mathsf{re}_p(\\pi,\\tau) \\in \\mathcal{P}_0=\\mathcal{P}$. If the pair $(\\pi,\\tau)$ is not a cut pair or is a cut pair with the cut formula being not $p$, then we have $ \\mathsf{re}_p(\\pi,\\tau) = \\pi \\in \\mathcal{P}_n$. \n\nNow suppose $(\\pi,\\tau)$ is a cut pair with the cut formula $p$ and $n>0$. We proceed by induction on $\\lvert \\pi\\rvert$ with the case analysis according to the definition of $\\mathsf{re}_p$. Let us consider the cases of inference rules $(\\mathsf{\\to_L})$ and $(\\mathsf{\\Box})$. \n\nIf the last inference in $\\pi$ is an application of the rule $(\\mathsf{\\to_L})$, then $\\pi$ has the following form\n\\[\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma, C\\Rightarrow \\Delta, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow B,\\Delta, p$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Sigma, B\\to C\\Rightarrow \\Delta, p$}\n\\DisplayProof\n\\]\nwhere $\\pi_0, \\pi_1 \\in \\mathcal{P}_n$. Since $\\mathsf{li}_{B \\to C}$ and $\\mathsf{ri}_{B \\to C}$ are  adequate, we have $\\mathsf{li}_{B \\to C}(\\tau)\\in \\mathcal{P}_n$ and $\\mathsf{ri}_{B \\to C}(\\tau) \\in \\mathcal{P}_n$. Consequently, by the induction hypothesis for the pairs $(\\pi_0, \\mathsf{li}_{B \\to C}(\\tau))$ and $(\\pi_1, \\mathsf{ri}_{B \\to C}(\\tau))$, we have $\\mathsf{re}_p(\\pi_0, \\mathsf{li}_{B \\to C}(\\tau))\\in \\mathcal{P}_n$ and $\\mathsf{re}_p(\\pi_1, \\mathsf{ri}_{B \\to C}(\\tau))\\in \\mathcal{P}_n$. Thus we obtain $\\mathsf{re}_p(\\pi,\\tau)\\in \\mathcal{P}_n$.\n\nConsider the case when the last inference in $\\pi$ is an application of the rule $(\\mathsf{\\Box})$.\nThen $\\pi$ has the form\n\\[\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma, p$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma, p$}\n\\DisplayProof\n\\] \nwhere $\\pi_0 \\in \\mathcal{P}_n$ and $\\pi_1 \\in \\mathcal{P}_{n-1}$. Since $\\mathsf{li}_{\\: \\Box B}$ is adequate, we have $\\mathsf{li}_{\\: \\Box B}(\\tau) \\in \\mathcal{P}_n$. Consequently, by the induction hypothesis for the pair $(\\pi_0, \\mathsf{li}_{\\: \\Box B}(\\tau))$, we have $\\mathsf{re}_p(\\pi_0, \\mathsf{li}_{\\: \\Box B}(\\tau)) \\in \\mathcal{P}_n$. Thus we obtain $\\mathsf{re}_p(\\pi,\\tau)\\in \\mathcal{P}_n$.\n\nNotice that the last inference in $\\pi$ differs from an application of the rule $(\\mathsf{cut})$. The remaining cases of inference rules $(\\to_\\mathsf{R})$ and $(\\mathsf{refl})$ are treated in a similar way to the case of $(\\to_\\mathsf{L})$, so we omit them.\n\n\\end{proof}\n\n\\begin{lem}\\label{reboxadeq}\nGiven an adequate $B$-removing mapping $\\mathsf{re}_B$, there exists an adequate $\\Box B$-removing mapping $\\mathsf{re}_{\\Box B}$.\n\\end{lem}\n\\begin{proof}\nAssume we have an adequate $B$-removing mapping $\\mathsf{re}_B$.\nThe required $\\Box B$-removing mapping $\\mathsf{re}_{\\Box B}$ is obtained as the fixed-point of a contractive operator $\\mathsf G_{\\Box B} \\colon \\mathcal R_{\\Box B} \\to \\mathcal R_{\\Box B}$. \n\nFor a mapping $\\mathsf u\\in \\mathcal R_{\\Box B}$ and a pair of $\\infty$-proofs $(\\pi,\\tau)$, the $\\infty$-proof $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)$ is defined as follows. \nIf $(\\pi,\\tau)$ is not a cut pair or a cut pair with \nthe cut formula being not $\\Box B$, then we put $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)=\\pi$. \n\n\n\n\nNow let $(\\pi,\\tau)$ be a cut pair with \nthe cut formula $\\Box B$ and the cut result $\\Gamma\\Rightarrow \\Delta$.\nIf $\\lvert \\pi\\rvert=0$ or $\\lvert \\tau \\rvert=0$, then $\\Gamma\\Rightarrow \\Delta$ is an initial sequent. In this case, we define $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)$ as the $\\infty$-proof consisting only of the sequent $\\Gamma\\Rightarrow \\Delta$. \n\nSuppose that $\\lvert \\pi\\rvert>0$ and $\\lvert \\tau \\rvert>0$. We define $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)$ according to the last application of an inference rule in $\\pi$:\\\\\n\\begin{align*}\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma,C\\Rightarrow D,\\Sigma,\\Box B$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$\\Gamma\\Rightarrow C\\to D,\\Sigma,\\Box B$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{u}(\\pi_0, \\mathsf{i}_{C \\to D}(\\tau))$}\n\\noLine\n\\UIC{$\\Gamma,C\\Rightarrow D,\\Sigma$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma\\Rightarrow C\\to D,\\Sigma$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma, D\\Rightarrow \\Delta, \\Box B$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow C,\\Delta, \\Box B$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Sigma, C\\to D\\Rightarrow \\Delta, \\Box B$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{u} (\\pi_0, \\mathsf{li}_{C\\to D} (\\tau))$}\n\\noLine\n\\UIC{$\\Sigma, D\\Rightarrow \\Delta$}\n\\AXC{$\\mathsf{u} (\\pi_1, \\mathsf{ri}_{C\\to D} (\\tau))$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow C,\\Delta$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Sigma, C\\to D\\Rightarrow \\Delta$}\n\\DisplayProof\n\\end{align*}\n\\begin{align*}\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma,C,\\Box C\\Rightarrow \\Delta,\\Box B$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Sigma,\\Box C\\Rightarrow \\Delta,\\Box B$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{u}(\\pi_0, \\mathsf{wk}_{ C, \\emptyset} (\\tau)$}\n\\noLine\n\\UIC{$\\Sigma,C,\\Box C\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$\\Sigma,\\Box C\\Rightarrow \\Delta$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma\\Rightarrow C,\\Delta, \\Box B$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma,C \\Rightarrow \\Delta, \\Box B$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\BIC{$\\Gamma\\Rightarrow \\Delta, \\Box B$}\n\\DisplayProof\n,\\tau\n\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{u} (\\pi_0, \\mathsf{wk}_{\\emptyset,C} (\\tau))$}\n\\noLine\n\\UIC{$\\Gamma\\Rightarrow C,\\Delta$}\n\\AXC{$\\mathsf{u} (\\pi_1, \\mathsf{wk}_{C,\\emptyset} (\\tau))$}\n\\noLine\n\\UIC{$\\Gamma, C\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma\\Rightarrow \\Delta$}\n\\DisplayProof\n\\\\\\\\\n\\left(\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow C, \\Sigma, \\Box B$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box C, \\Sigma, \\Box B$}\n\\DisplayProof\n,\\tau\\right)\n&\\longmapsto\n\\AXC{$\\mathsf{u}(\\pi_0, \\mathsf{li}_{\\: \\Box C}(\\tau))$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow C, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$} \n\\RightLabel{ .}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box C, \\Sigma$}\n\\DisplayProof\n\\end{align*}\n\nConsider the case when $\\pi$ has the form \n\\[\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma$}\n\\DisplayProof\n\\]\nWe define $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)$ according to the last application of an inference rule in $\\tau$. \nIf the last inference is an application of the rule $\\mathsf{(refl)}$ with the principal formula being not $\\Box B$ or is an application of the rule $\\mathsf{(\\Box)}$ without the formula $\\Box B$ in the right premise, then $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)$ is defined similarly to the previous cases of $\\mathsf{(refl)}$ and $(\\mathsf{\\Box})$. Cases of inference rules $\\mathsf{(\\to_L)}$, $\\mathsf{(\\to_R)}$, and $\\mathsf{(cut)}$ are also completely similar to the previous cases of $\\mathsf{(\\to_L)}$, $\\mathsf{(\\to_R)}$, and $\\mathsf{(cut)}$.\n\n\n\nIf the last application of an inference rule in $\\tau$ is an application of the rule $\\mathsf{(refl)}$ with the principal formula $\\Box B$, then we put\n\\[\\mathsf G_{\\Box B}(\\mathsf u) \\colon\n\\left(\\pi,\n\\AXC{$\\tau_0$}\n\\noLine\n\\UIC{$\\Gamma,B,\\Box B\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UnaryInfC{$\\Gamma,\\Box B\\Rightarrow \\Delta$}\n\\DisplayProof\n\\right)\n\\longmapsto\\mathsf{re}_{B}(\\pi_0,\\mathsf{u}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)).\n\\]\n\nIt remains to consider the case when $\\tau$ has the form\n\\[\n\\AxiomC{$\\tau_0$}\n\\noLine\n\\UnaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow C, \\Sigma^\\prime$}\n\\AxiomC{$\\tau_1$}\n\\noLine\n\\UnaryInfC{$\\Box B,\\Box \\Pi^\\prime \\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{.}\n\\BinaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow \\Box C, \\Sigma^\\prime$}\n\\DisplayProof\n\\]\nWe define $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\n\\tau)$ as\n\\[\n\\AxiomC{$\\mathsf{u}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)$}\n\\noLine\n\\UnaryInfC{$\\Phi^\\prime, \\Box \\Pi^\\prime \\Rightarrow C, \\Sigma^\\prime$}\n\\AxiomC{$\n\\mathsf{u}\n\\left(\\pi',\n\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\n\\right)$}\n\\noLine\n\\UnaryInfC{$\\Box\\Pi\\backslash \\Box\\Pi^\\prime, \\Box\\Pi^\\prime\\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BinaryInfC{$\\Phi^\\prime, \\Box \\Pi^\\prime \\Rightarrow \\Box C, \\Sigma^\\prime$}\n\\DisplayProof\n\\]\nwhere $\\pi'$ equals to\n\\[\n\\AxiomC{$\\mathsf{wk}_{\\Box\\Pi^\\prime\\backslash\\Box\\Pi,C}(\\pi_1)$}\n\\noLine\n\\UnaryInfC{$\\Box\\Pi^\\prime\\backslash \\Box\\Pi, \\Box\\Pi\\Rightarrow B,C$}\n\\AxiomC{$\\pi_1$}\n\\noLine\n\\UnaryInfC{$\\Box\\Pi\\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BinaryInfC{$\\Box\\Pi^\\prime\\backslash \\Box\\Pi,\\Box\\Pi\\Rightarrow\\Box B, C$}\n\\DisplayProof\n\\]\nNotice that $\\Box\\Pi^\\prime\\backslash \\Box\\Pi,\\Box\\Pi=\\Box\\Pi\\backslash \\Box\\Pi^\\prime,\\Box\\Pi^\\prime$.\n\nNow the operator $\\mathsf {G_{\\Box B}}$ is well-defined.\nBy the case analysis according to the definition of $\\mathsf {G_{\\Box B}}$, we see that $\\mathsf {G_{\\Box B}} (\\mathsf u) $ is non-expansive and belongs to $\\mathcal R_{\\Box B}$ whenever $\\mathsf u \\in \\mathcal R_{\\Box B}$.\n\nWe claim that $\\mathsf {G_{\\Box B}}$ is contractive. It sufficient to check that for any $\\mathsf u, \\mathsf v\\in \\mathcal R_{\\Box B}$ and any $n,k\\in\\mathbb N$ we have \n\\[\\mathsf u\\sim_{n,k}\\mathsf v \\Rightarrow \\mathsf {G_{\\Box B}(u)}\\sim_{n,k+1}\\mathsf{G_{\\Box B}(v)}.\\]\n\n\nAssume there are two $\\Box B$-removing mappings $\\mathsf u$ and $\\mathsf v$ such that $\\mathsf u\\sim_{n,k}\\mathsf v$. Consider an arbitrary pair of $\\infty$-proofs $(\\pi,\\tau)$. By the case analysis according to the definition of $\\mathsf {G_{\\Box B}}$, we prove that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. In addition, we check that if $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$, then $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n+1}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. \n\nIf the pair $(\\pi,\\tau)$ is not a cut pair or is a cut pair with the cut formula being not $\\Box B$, then we have $ \\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau) = \\pi =\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. \nOtherwise, consider last inferences in $\\pi$ and $\\tau$.  \n\nSuppose that the $\\infty$-proof $\\pi$ has the form\n\\[\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Sigma,C,\\Box C\\Rightarrow \\Delta,\\Box B$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ .}\n\\UIC{$\\Sigma,\\Box C\\Rightarrow \\Delta,\\Box B$}\n\\DisplayProof\n\\]\nThen we have $\\mathsf{u}(\\pi_0, \\mathsf{wk}_{ C, \\emptyset} (\\tau))\\sim_n \\mathsf{v}(\\pi_0, \\mathsf{wk}_{ C, \\emptyset} (\\tau))$. Hence, $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. \n\nIn addition, if $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$, then $\\lvert\\pi_0\\rvert+\\lvert\\mathsf{wk}_{ C, \\emptyset} (\\tau)\\lvert<k$. Thus, $\\mathsf{u}(\\pi_0, \\mathsf{wk}_{ C, \\emptyset} (\\tau))\\sim_{n+1} \\mathsf{v}(\\pi_0, \\mathsf{wk}_{ C, \\emptyset} (\\tau))$. We obtain that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n+1}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. \n\nNow let us consider the case when $\\pi$ and $\\tau$ have the forms\n\\[\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma$}\n\\DisplayProof\\quad\n\\AXC{$\\tau_0$}\n\\noLine\n\\UIC{$\\Gamma,B,\\Box B\\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ .}\n\\UnaryInfC{$\\Gamma,\\Box B\\Rightarrow \\Delta$}\n\\DisplayProof\n\\]\nWe see that $\\mathsf{u}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0) \\sim_n \\mathsf{v}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)$. Since the mapping $\\mathsf{re}_{B}$ is non-expansive, we have \n\\[\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau) = \\mathsf{re}_{B}(\\pi_0,\\mathsf{u}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)) \\sim_n \\mathsf{re}_{B}(\\pi_0,\\mathsf{v}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)) = \\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau).\\]\n\nIn addition, if $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$, then $\\lvert\\mathsf{wk}_{B,\\emptyset}(\\pi)\\rvert+\\lvert\\tau_0\\lvert<k$. Consequently, $\\mathsf{u}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0) \\sim_{n+1} \\mathsf{v}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)$.\nThus, we have\n\\[\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau) = \\mathsf{re}_{B}(\\pi_0,\\mathsf{u}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)) \\sim_{n+1} \\mathsf{re}_{B}(\\pi_0,\\mathsf{v}(\\mathsf{wk}_{B,\\emptyset}(\\pi),\\tau_0)) = \\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau).\\]\n\nLet us consider the main case when $\\pi$ and $\\tau$ have the forms \n\\[\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma$}\n\\DisplayProof\\quad\n\\AxiomC{$\\tau_0$}\n\\noLine\n\\UnaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow C, \\Sigma^\\prime$}\n\\AxiomC{$\\tau_1$}\n\\noLine\n\\UnaryInfC{$\\Box B,\\Box \\Pi^\\prime \\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BinaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow \\Box C, \\Sigma^\\prime$}\n\\DisplayProof\n\\]\nWe have $\\mathsf{u}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)\\sim_n\\mathsf{v}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)$, and $\\mathsf{u}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)\\sim_{n}\\mathsf{v}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)$.\nHence, $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$. \n\nIf $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$, then\n$\\mathsf{u}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)\\sim_n\\mathsf{v}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)$ and $\\mathsf{u}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)\\sim_{n+1}\\mathsf{v}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)$, because $\\lvert \\mathsf{li}_{\\Box C}(\\pi)\\rvert +\\lvert \\tau_0\\rvert< k$. We obtain $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\sim_{n+1}\\mathsf G_{\\Box B}(\\mathsf v)(\\pi,\\tau)$.\n\n\n\n\nAll the other cases of lowermost inferences in $\\pi$ and $\\tau$ are treated similarly, so we omit them. We have that the operator $\\mathsf {G_{\\Box B}}$ is contractive. \n\nNow we define the required $\\Box B$-removing mapping $\\mathsf{re}_{\\Box B}$ as the fixed-point of the the operator $\\mathsf G_{\\Box B} \\colon \\mathcal R_{\\Box B} \\to \\mathcal R_{\\Box B}$, which exists by Lemma \\ref{Comp R_A} and Theorem \\ref{fixpoint}.\n\n\nIt remains to check that the mapping $\\mathsf{re}_{\\Box B}$ is adequate. For some $n,k\\in\\mathbb N$, let us call a mapping $\\mathsf u\\in \\mathcal R_{\\Box B}$ $(n,k)$-adequate if it satisfies the following two conditions: $\\mathsf u(\\pi,\\tau)\\in\\mathcal P_i$ for any $i\\leqslant n$ and any $\\pi, \\tau \\in \\mathcal P_i$; $\\mathsf u(\\pi,\\tau)\\in\\mathcal P_{n+1}$ whenever $\\pi, \\tau \\in \\mathcal P_{n+1}$ and $\\lvert \\pi\\rvert + \\lvert \\tau \\rvert <k$. \n\nWe claim that $\\mathsf G_{\\Box B}(\\mathsf u)$ is $(n,k+1)$-adequate whenever $\\mathsf{u}$ is a $(n,k)$-adequate mapping from $\\mathcal R_{\\Box B}$. Consider an arbitrary pair of $\\infty$-proofs $(\\pi,\\tau)$ and an $(n,k)$-adequate mapping $\\mathsf u\\in\\mathcal R_{\\Box B}$. By the case analysis according to the definition of $\\mathsf {G_{\\Box B}}$, we prove that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal{P}_i$ if $i\\leqslant n$ and $\\pi, \\tau \\in \\mathcal P_i$. In addition, we check that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal P_{n+1}$ if $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$ and $\\pi, \\tau \\in \\mathcal P_{n+1}$. \n\nIf the pair $(\\pi,\\tau)$ is not a cut pair or is a cut pair with the cut formula being not $\\Box B$, then we have $ \\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau) = \\pi $. In this case, the required statements trivially hold.\nOtherwise, consider last inferences in $\\pi$ and $\\tau$.  \n\nSuppose that the $\\infty$-proof $\\pi$ has the form\n\\[\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma,C\\Rightarrow D,\\Sigma,\\Box B$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ .}\n\\UIC{$\\Gamma\\Rightarrow C\\to D,\\Sigma,\\Box B$}\n\\DisplayProof\n\\]\nSince the mapping $\\mathsf{i}_{ C\\to D}$ is adequate, we have $\\mathsf{u}(\\pi_0, \\mathsf{i}_{ C\\to D} (\\tau))\\in \\mathcal{P}_i$ whenever $i\\leqslant n$ and $\\pi, \\tau \\in \\mathcal P_i$. In this case, we see that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in  \\mathcal P_{i}$.\n\nIf $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$, then $\\lvert\\pi_0\\rvert+\\lvert\\mathsf{i}_{ C\\to D}(\\tau)\\lvert<k$. Thus, $\\mathsf{u}(\\pi_0, \\mathsf{i}_{ C\\to D} (\\tau))\\in  \\mathcal P_{n+1}$ whenever $\\pi, \\tau \\in \\mathcal P_{n+1}$. We obtain that $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal P_{n+1}$. \n\nSuppose that $\\pi$ and $\\tau$ have the forms \n\\[\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow B, \\Sigma$}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow B$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box B, \\Sigma$}\n\\DisplayProof\\quad\n\\AxiomC{$\\tau_0$}\n\\noLine\n\\UnaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow C, \\Sigma^\\prime$}\n\\AxiomC{$\\tau_1$}\n\\noLine\n\\UnaryInfC{$\\Box B,\\Box \\Pi^\\prime \\Rightarrow C$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BinaryInfC{$\\Phi^\\prime, \\Box B, \\Box \\Pi^\\prime \\Rightarrow \\Box C, \\Sigma^\\prime$}\n\\DisplayProof\n\\]\nTrivially, $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal P_{0} = \\mathcal{P}$. Thus consider the case when $\\pi, \\tau \\in \\mathcal{P}_i$ for some $0<i \\leqslant n$. Then we see that $\\pi_0, \\tau_0 \\in \\mathcal{P}_i$ and $\\pi_1, \\tau_1 \\in \\mathcal{P}_{i-1}$.\nSince the mapping $\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}$ and $\\mathsf{li}_{\\Box C}$ are adequate, we have $\\mathsf{u}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)\\in \\mathcal{P}_i$ and $\\mathsf{u}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)\\in \\mathcal{P}_{i-1}$.\nHence, $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal{P}_i$. \n\nIf $\\lvert\\pi\\rvert+\\lvert\\tau\\lvert<k+1$ and $\\pi,\\tau \\in \\mathcal{P}_{n+1}$, then\n$\\lvert \\mathsf{li}_{\\Box C}(\\pi) \\rvert + \\lvert \\tau_0 \\rvert <k$ and $\\mathsf{u}(\\mathsf{li}_{\\Box C}(\\pi),\\tau_0)\\in \\mathcal{P}_{n+1}$. Also, we see that $\\pi'\\in  \\mathcal{P}_n$ and $\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1) \\in \\mathcal{P}_n$. Hence, $\\mathsf{u}\\left(\\pi',\\mathsf{wk}_{\\Box\\Pi\\backslash\\Box\\Pi^\\prime,\\varnothing}(\\tau_1)\\right)\\in \\mathcal{P}_{n}$.\nWe obtain $\\mathsf G_{\\Box B}(\\mathsf u)(\\pi,\\tau)\\in \\mathcal{P}_{n+1}$.\n\nWe omit other cases of lowermost inferences in $\\pi$ and $\\tau$, because they are treated similarly.\n\n\nWe established that $\\mathsf G_{\\Box B}(\\mathsf u)$ is $(n,k+1)$-adequate for any $(n,k)$-adequate $\\mathsf u\\in\\mathcal R_{\\Box B}$. Notice that if a mapping $\\mathsf{u} $ is $(n,k)$-adequate for all $k\\in\\mathbb N$, then it is also $(n+1,0)$-adequate. Now by induction on $n$ with a subinduction on $k$, we immediately obtain that the mapping $\\mathsf {re}_{\\Box B}$, which is a fixed-point of $\\mathsf G_{\\Box B}$, is $(n,k)$-adequate for all $n,k\\in\\mathbb N$. Therefore the mapping $\\mathsf {re}_{\\Box B}$ is adequate. \n\n\n\n\\end{proof}\n\n\n\n\\begin{lem}\\label{reabadeq}\nFor any formula $A$, there exists an adequate $A$-removing mapping $\\mathsf{re}_A$.\n\\end{lem}\n\\begin{proof}\n\nWe define $\\mathsf{re}_A$ by induction on the structure of the formula $A $.\n\n\n\nCase 1: $A$ has the form $p$. In this case, $\\mathsf{re}_{p} $ is defined by Lemma \\ref{repadeq}.\n \nCase 2: $A$ has the form $\\bot$. Then we put $\\mathsf{re}_{\\bot}(\\pi,\\tau):= \\mathsf{i}_\\bot (\\pi)$, where $\\mathsf{i}_\\bot$ is a non-expansive adequate mapping from Lemma \\ref{inversion}.\n\nCase 3: $A$ has the form $B\\to C$. Then we put    \n\\[\\mathsf{re}_{B \\to C}(\\pi,\\tau):=\\mathsf{re}_C(\\mathsf{re}_B(\\mathsf{wk}_{\\emptyset, C}(\\mathsf{ri}_{B\\to C}(\\tau)),\\mathsf{i}_{B\\to C}(\\pi)),\\mathsf{li}_{B\\to C}(\\tau))\\,\\]\nwhere $\\mathsf{ri}_{B \\to C}$, $\\mathsf{i}_{B \\to C}$, $\\mathsf{li}_{B \\to C}$ are non-expansive adeqate mappings from Lemma \\ref{inversion} and $\\mathsf{wk}_{\\emptyset, C}$ is a non-expansive adequate mapping from Lemma \\ref{strongweakening}.\n\nCase 4: $A$ has the form $\\Box B$. By the induction hypothesis, there is an adequate $B$-removing mapping $\\mathsf{re}_B$. The required $\\Box B$-removing mapping $\\mathsf{re}_{\\Box B}$ exists by Lemma \\ref{reboxadeq}.\n\\end{proof}\n\n\n\n\n\n\n\n\nA mapping $\\mathsf{u}\\colon \\mathcal P \\to\\mathcal P$ is called \\emph{root-preserving} if it maps $\\infty$-proofs to $\\infty$-proofs of the same sequents. Let $\\mathcal{T}$ denote the set of all root-preserving non-expansive mappings from $\\mathcal P$ to $\\mathcal P$. \n\n\\begin{lem}\\label{Comp T}\nThe pair $(\\mathcal T, l_1)$ is a non-empty spherically complete ultrametric space.\n\\end{lem}\n\\begin{proof}\nThe proof of spherical completeness of the space $(\\mathcal T, l_1)$ is analogous to the proof of Proposition \\ref{SphCom}.\nThe space is obviously non-empty, since the identity function lies in $\\mathcal T$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{thm}[cut-elimination]\n\\label{infcuttoinf}\nIf $\\mathsf{Grz_\\infty} +\\mathsf{cut}\\vdash\\Gamma\\Rightarrow\\Delta$, then $\\mathsf{Grz_\\infty}\\vdash\\Gamma\\Rightarrow\\Delta$.\n\\end{thm}\n\\begin{proof}\nWe obtain the required cut-elimination mapping $\\mathsf{ce}$ as the fixed-point of a contractive operator $\\mathsf F \\colon \\mathcal T \\to T$. \n\nFor a mapping $\\mathsf u\\in \\mathcal T$ and an $\\infty$-proof $\\pi$, the $\\infty$-proof $\\mathsf F(\\mathsf u)(\\pi)$ is defined as follows. If $\\lvert \\pi\\rvert=0$, then we define $\\mathsf F(\\mathsf u)(\\pi)$ to be $\\pi$. \n\n\n\n\nOtherwise, we define $\\mathsf F(\\mathsf u)(\\pi)$ according to the last application of an inference rule in $\\pi$:\n\\begin{gather*}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma , B \\Rightarrow  \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow  A, \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_L}$}\n\\BIC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_1)$}\n\\noLine\n\\UIC{$\\Gamma , B \\Rightarrow  \\Delta$}\n\\AXC{$\\mathsf u(\\pi_2)$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow  A, \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\\begin{gather*}\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow  B , \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_R}$}\n\\UIC{$\\Gamma \\Rightarrow  A \\rightarrow B , \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_0)$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow  B , \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma \\Rightarrow  A \\rightarrow B , \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\\begin{gather*}\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma, A, \\Box A \\Rightarrow   \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Gamma , \\Box A\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_0)$}\n\\noLine\n\\UIC{$\\Gamma, A, \\Box A \\Rightarrow   \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma , \\Box A\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\\begin{align*}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n&\\longmapsto\n\\AXC{$\\mathsf u(\\pi_1)$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\mathsf{u}(\\pi_2)$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof\\\\\\\\\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\BIC{$\\Gamma\\Rightarrow  \\Delta$}\n\\DisplayProof \n&\\longmapsto\n\\mathsf {re}_A(\\mathsf u(\\pi_0),\\mathsf u(\\pi_1)).\n\\end{align*}\nNow the operator $\\mathsf F$ is well-defined.\nBy the case analysis according to the definition of $\\mathsf F$, we see that $\\mathsf F (\\mathsf u) $ is non-expansive and belongs to $\\mathcal T$ whenever $\\mathsf u \\in \\mathcal T$.\n\nWe claim that $\\mathsf {F}$ is contractive. It sufficient to check that for any $\\mathsf u, \\mathsf v\\in \\mathcal T$ and any $n,k\\in\\mathbb N$ we have \n\\[\\mathsf u\\sim_{n,k}\\mathsf v \\Rightarrow \\mathsf {F(u)}\\sim_{n,k+1}\\mathsf{F(v)}.\\]\n\nAssume there are mappings $\\mathsf u$ and $\\mathsf v$ from $\\mathcal{T}$ such that $\\mathsf u\\sim_{n,k}\\mathsf v$. Consider an arbitrary $\\infty$-proof $\\pi$. By the case analysis according to the definition of $\\mathsf {F}$, we prove that $\\mathsf F(\\mathsf u)(\\pi)\\sim_{n}\\mathsf F(\\mathsf v)(\\pi)$. In addition, we check that if $\\lvert\\pi\\rvert<k+1$, then $\\mathsf F(\\mathsf u)(\\pi)\\sim_{n+1}\\mathsf F(\\mathsf v)(\\pi)$. \n\nLet us consider the case when $\\pi$ has the form\n\\[\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n\\]\nWe have $\\mathsf u(\\pi_0) \\sim_n \\mathsf v(\\pi_0)$ and  $\\mathsf u(\\pi_1) \\sim_n \\mathsf v(\\pi_1)$. Thus, $F(\\mathsf u)(\\pi) \\sim_n F(\\mathsf v)(\\pi)$.\n\nIf $\\lvert\\pi\\rvert<k+1$, then $\\lvert\\pi_0\\rvert<k$.\nWe have $\\mathsf u(\\pi_0) \\sim_{n+1} \\mathsf v(\\pi_0)$ and  $\\mathsf u(\\pi_1) \\sim_n \\mathsf v(\\pi_1)$.\nHence, $F(\\mathsf u)(\\pi) \\sim_{n+1} F(\\mathsf v)(\\pi)$.\n\nNow consider the case when $\\pi$ has the form\n\\[\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\] \nWe have $\\mathsf u(\\pi_0) \\sim_n \\mathsf v(\\pi_0)$ and  $\\mathsf u(\\pi_1) \\sim_n \\mathsf v(\\pi_1)$.\nSince the mapping $\\mathsf {re}_A$ is non-expansive, we see that\n\\[F(\\mathsf u)(\\pi) = \\mathsf {re}_A(\\mathsf u(\\pi_0),\\mathsf u(\\pi_1))\n\\sim_n \\mathsf {re}_A(\\mathsf v(\\pi_0),\\mathsf v(\\pi_1))= F(\\mathsf v)(\\pi).\\]\n\nMoreover, if $\\lvert\\pi\\rvert<k+1$, then $\\lvert\\pi_0\\rvert<k$ and $\\lvert\\pi_1\\rvert<k$. We obtain $\\mathsf u(\\pi_0) \\sim_{n+1} \\mathsf v(\\pi_0)$ and  $\\mathsf u(\\pi_1) \\sim_{n+1} \\mathsf v(\\pi_1)$. Hence,\n\\[F(\\mathsf u)(\\pi) = \\mathsf {re}_A(\\mathsf u(\\pi_0),\\mathsf u(\\pi_1))\n\\sim_{n+1} \\mathsf {re}_A(\\mathsf v(\\pi_0),\\mathsf v(\\pi_1))= F(\\mathsf v)(\\pi).\\]\n\nOther cases are straightforward, so we omit them. We have that the operator $\\mathsf {F}$ is contractive. \n\nNow we define the required cut-elimination mapping $\\mathsf{ce}$ as the fixed-point of the the operator $\\mathsf F \\colon \\mathcal T \\to \\mathcal T$, which exists by Lemma \\ref{Comp T} and Theorem \\ref{fixpoint}.\n\nFor some $n,k\\in\\mathbb N$, let us call a mapping $\\mathsf u\\in \\mathcal T$ $(n,k)$-free if it satisfies the following two conditions: $\\mathsf u(\\pi)\\in\\mathcal P_n$ for any $\\pi \\in \\mathcal P$; $\\mathsf u(\\pi)\\in\\mathcal P_{n+1}$ whenever $\\lvert \\pi\\rvert<k$. \n\n\nWe claim that $F(\\mathsf u)$ is $(n,k+1)$-free whenever $\\mathsf{u}$ is a $(n,k)$-free mapping from $\\mathcal T$. Consider an arbitrary $\\infty$-proof $\\pi$ and an $(n,k)$-free mapping $\\mathsf u\\in\\mathcal T$. By the case analysis according to the definition of $\\mathsf {F}$, we prove that $F(\\mathsf u)(\\pi)\\in \\mathcal{P}_n$ for any $\\pi \\in \\mathcal P$. In addition, we check that $\\mathsf F(\\mathsf u)(\\pi)\\in \\mathcal P_{n+1}$ if $\\lvert\\pi\\rvert<k+1$. \n\nLet us consider the case when $\\pi$ has the form\n\\[\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n\\]\nWe have $\\mathsf u(\\pi_0) ,\\mathsf u(\\pi_1) \\in \\mathcal{P}_n$. Hence, $F(\\mathsf u)(\\pi) \\in \\mathcal{P}_n$.\n\nIf $\\lvert\\pi\\rvert<k+1$, then $\\lvert\\pi_0\\rvert<k$. We have $\\mathsf u(\\pi_0) \\in \\mathcal{P}_{n+1}$ and $\\mathsf u(\\pi_1) \\in \\mathcal{P}_n$. Hence, $F(\\mathsf u)(\\pi) \\in \\mathcal{P}_{n+1}$.\n\nNow we consider the case when $\\pi$ has the form\n\\[\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow \\Delta$}\n\\LeftLabel{$\\mathsf{cut}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\] \nWe have $\\mathsf u(\\pi_0) ,\\mathsf u(\\pi_1) \\in \\mathcal{P}_n$. Since the mapping $\\mathsf {re}_A$ is adequate, we see that $F(\\mathsf u)(\\pi) = \\mathsf {re}_A(\\mathsf u(\\pi_0),\\mathsf u(\\pi_1)) \\in \\mathcal{P}_n$.\n\nIf $\\lvert\\pi\\rvert<k+1$, then $\\lvert\\pi_0\\rvert<k$ and $\\lvert\\pi_1\\rvert<k$. We obtain $\\mathsf u(\\pi_0), \\mathsf{u}(\\pi_1) \\in \\mathcal{P}_{n+1}$. Hence, $F(\\mathsf u)(\\pi) = \\mathsf {re}_A(\\mathsf u(\\pi_0),\\mathsf u(\\pi_1))\\in \\mathcal{P}_{n+1}$.\n\nWe omit other cases of lowermost inferences in $\\pi$, because they are trivial.\n\n\nWe established that $\\mathsf F(\\mathsf u)$ is $(n,k+1)$-free for any $(n,k)$-free $\\mathsf u\\in\\mathcal T$. Notice that if a mapping $\\mathsf{u} $ is $(n,k)$-free for all $k\\in\\mathbb N$, then it is also $(n+1,0)$-free. Now by induction on $n$ with a subinduction on $k$, we immediately obtain that the mapping $\\mathsf {ce}$, which is a fixed-point of $\\mathsf F$, is $(n,k)$-free for all $n,k\\in\\mathbb N$. Therefore, for any $\\infty$-proof $\\pi$, the $\\infty$-proof $\\mathsf {ce}(\\pi)$ does not contain instances of the rule $\\mathsf {(cut)}$. \n\n\n\n\n\nNow assume $\\mathsf{Grz_\\infty} +\\mathsf{cut}\\vdash\\Gamma\\Rightarrow\\Delta$. Take an $\\infty$-proof of the sequent $\\Gamma\\Rightarrow\\Delta$ in the system $\\mathsf{Grz_\\infty} +\\mathsf{cut}$ and apply the mapping $\\mathsf{ce}$ to it. We obtain an $\\infty$-proof of the same sequent in the system $\\mathsf{Grz_\\infty}$.\n\\end{proof}\nTheorem \\ref{cutelimgrz} is now established as a direct consequence of Theorem \\ref{seqtoinfcut}, Theorem \\ref{infcuttoinf}, and Theorem \\ref{inftoseq}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Lyndon interpolation}\n\\label{SecLyn}\nThe Lyndon interpolation property for the logic was established in \\cite{Maks} on the basis of Kripke semantics. Here we present a proof-theoretic argument for the same result. \n\nFor a formula or a set of formulas $X$, let $\\mathsf{pos}(X)$ be the set of atomic propositions that have positive occurrences in $X$ and let $\\mathsf{neg}(X)$ be the set of atomic propositions with negative occurrences in $X$. For a sequent $\\Gamma\\Rightarrow\\Delta$, let $\\mathit{pos}(\\Gamma\\Rightarrow\\Delta)=\\mathit{pos}(\\Delta)\\cup \\mathit{neg}(\\Gamma)$ and $\\mathit{neg}(\\Gamma\\Rightarrow\\Delta)=\\mathit{neg}(\\Delta)\\cup \\mathit{pos}(\\Gamma)$.\n\nRecall that for a finite set of formulas $\\Lambda$, the set $\\Lambda^\\ast$ is $\\{\\Box(A\\to\\Box A)\\mid A\\in\\Lambda\\}$.\n\n\n\n\\begin{lem}\nFor any finite sets of formulas $\\Lambda_1$, $\\Lambda_2$, \nif $\\mathsf{Grz_\\infty}\\vdash \\Gamma_1,\\Gamma_2\\Rightarrow\\Delta_1,\\Delta_2$, then there exists a formula $I$ called an interpolant of this sequent, such that $\\mathit{pos}(I)\\subset \\mathit{neg}(\\Gamma_1\\Rightarrow\\Delta_1)\\cap \\mathit{pos}(\\Gamma_2\\Rightarrow\\Delta_2)$, $\\mathit{neg}(I)\\subset \\mathit{pos}(\\Gamma_1\\Rightarrow\\Delta_1)\\cap \\mathit{neg}(\\Gamma_2\\Rightarrow\\Delta_2)$, and\n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Gamma_1\\rightarrow\\bigvee\\Delta_1 \\cup \\{ I\\} \\quad \\text{and} \\quad\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast \\cup\\{ I\\} \\cup  \\Gamma_2\\rightarrow\\bigvee\\Delta_2\\:.$$\n\\end{lem}\n\\begin{proof}\nAssume $\\pi$ is an $\\infty$--proof of the sequent $\\Gamma_1,\\Gamma_2\\Rightarrow\\Delta_1,\\Delta_2$ in $\\mathsf{Grz}_\\infty$ and $\\Lambda_1$, $\\Lambda_2$ are finite sets of formulas.\nWe prove the statement of the lemma by induction on the sum of cardinalities $\\mathit{card}(Sub(\\Gamma_1\\Rightarrow\\Delta_1)\\backslash \\Lambda_1 )+\\mathit{card}(  Sub(\\Gamma_2\\Rightarrow\\Delta_2)\\backslash  \\Lambda_2)$ with a subinduction on $\\lvert \\pi \\rvert$.\n\n\n\nSuppose $\\lvert \\pi \\rvert=0$. Then $\\Gamma_1,\\Gamma_2\\Rightarrow\\Delta_1,\\Delta_2$ is an initial sequent. \nIf $\\Gamma_1\\Rightarrow\\Delta_1$ ($\\Gamma_2\\Rightarrow\\Delta_2$) is an initial sequent, then we put $I:= \\bot$ ($I:= \\top$). Otherwise, we have an atomic proposition $p$ such that $p \\in \\Gamma_1$ and $p\\in \\Delta_2$ ($p \\in \\Gamma_2$ and $p\\in \\Delta_1$). In this case, we put $I:= p$ ($I:= \\neg p$).\n\nNow consider the last application of an inference rule in $\\pi$. \n\nCase 1. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Gamma_1,\\Gamma_2,A\\Rightarrow B,\\Sigma$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma_1,\\Gamma_2\\Rightarrow A\\to B,\\Sigma$}\n\\DisplayProof\n\\end{gather*}\nwhere $A\\to B,\\Sigma = \\Delta_1, \\Delta_2$.\nNotice that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. \nIf $A \\to B \\in \\Delta_1$ ($A \\to B \\in \\Delta_2$), then we put $\\Gamma^\\prime_1:= \\Gamma_1 \\cup \\{A\\}$, $\\Delta^\\prime_1:= (\\Delta_1\\backslash \\{A\\to B\\})\\cup \\{B\\}$, $\\Gamma^\\prime_2:=\\Gamma_2$, $\\Delta^\\prime_2:=\\Delta_2$ ($\\Gamma^\\prime_2:= \\Gamma_2 \\cup\\{A\\}$, $\\Delta^\\prime_2:= (\\Delta_2\\backslash \\{A\\to B\\})\\cup\\{B\\}$, $\\Gamma^\\prime_1:=\\Gamma_1$, $\\Delta^\\prime_1:=\\Delta_1$). We see that $\\pi^\\prime$ is an $\\infty$-proof of the sequent $\\Gamma^\\prime_1,\\Gamma^\\prime_2\\Rightarrow\\Delta^\\prime_1,\\Delta^\\prime_2$ in $\\mathsf{Grz}_\\infty$. By the induction hypothesis for $\\pi^\\prime$, $\\Lambda_1$ and $\\Lambda_2$, there is the corresponding interpolant $I^\\prime$. We set $I:=I^\\prime$.\n\nCase 2. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Sigma, B\\Rightarrow \\Delta_1,\\Delta_2$}\n\\AXC{$\\pi^{\\prime\\prime}$}\n\\noLine\n\\UIC{$\\Sigma \\Rightarrow A,\\Delta_1,\\Delta_2$}\n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Sigma, A\\to B\\Rightarrow \\Delta_1,\\Delta_2$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Sigma, A\\to B = \\Gamma_1,\\Gamma_2$. We see that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. \nIf $A \\to B \\in \\Gamma_1$ ($A \\to B \\in \\Gamma_2$), then we put $\\Gamma^\\prime_1:= (\\Gamma_1  \\backslash \\{A\\to B\\})\\cup \\{B\\}$, $\\Gamma^{\\prime\\prime}_1:= \\Gamma_1  \\backslash \\{A\\to B\\}$, \n$\\Delta^\\prime_1:= \\Delta_1  $, $\\Delta^{\\prime\\prime}_1:= \\Delta_1    \\cup \\{A\\}$, $\\Gamma^\\prime_2:=\\Gamma_2$, $\\Gamma^{\\prime\\prime}_2:=\\Gamma_2$,\n$\\Delta^\\prime_2:=\\Delta_2$, $\\Delta^{\\prime\\prime}_2:=\\Delta_2$\n($\\Gamma^\\prime_2:= (\\Gamma_2  \\backslash \\{A\\to B\\})\\cup \\{B\\}$, $\\Gamma^{\\prime\\prime}_2:= \\Gamma_2  \\backslash \\{A\\to B\\}$, \n$\\Delta^\\prime_2:= \\Delta_2  $, $\\Delta^{\\prime\\prime}_2:= \\Delta_2    \\cup \\{A\\}$, \n$\\Gamma^\\prime_1:=\\Gamma_1$, $\\Gamma^{\\prime\\prime}_1:=\\Gamma_1$,\n$\\Delta^\\prime_1:=\\Delta_1$, $\\Delta^{\\prime\\prime}_1:=\\Delta_1$). We see that $\\pi^\\prime$ and $\\pi^{\\prime\\prime}$ are $\\infty$-proofs of $\\Gamma^\\prime_1,\\Gamma^\\prime_2\\Rightarrow\\Delta^\\prime_1,\\Delta^\\prime_2$ and $\\Gamma^{\\prime\\prime}_1,\\Gamma^{\\prime\\prime}_2\\Rightarrow\\Delta^{\\prime\\prime}_1,\\Delta^{\\prime\\prime}_2$ in $\\mathsf{Grz}_\\infty$. By the induction hypothesis for $\\pi^\\prime$, $\\Lambda_1$ and $\\Lambda_2$, there is a interpolant $I^\\prime$. Analogously, by the induction hypothesis for $\\pi^{\\prime\\prime}$, $\\Lambda_1$ and $\\Lambda_2$, there is a interpolant \n$I^{\\prime\\prime}$. If $A \\to B \\in \\Gamma_1$, then we set $I:=I^\\prime \\vee I^{\\prime\\prime}$. Otherwise, when $A \\to B \\in \\Gamma_2$, we set $I:=I^\\prime \\wedge I^{\\prime\\prime}$.\n\nCase 3. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Sigma,A,\\Box A\\Rightarrow \\Delta_1,\\Delta_2$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,}\n\\UIC{$\\Sigma,\\Box A\\Rightarrow \\Delta_1,\\Delta_2$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Sigma, \\Box A = \\Gamma_1,\\Gamma_2$. If $\\Box A \\in \\Gamma_1$ ($\\Box A \\in \\Gamma_2$), then we put $\\Gamma^\\prime_1:= \\Gamma_1 \\cup \\{A\\}$, $\\Delta^\\prime_1:= \\Delta_1$, $\\Gamma^\\prime_2:=\\Gamma_2$, $\\Delta^\\prime_2:=\\Delta_2$ ($\\Gamma^\\prime_2:= \\Gamma_2 \\cup\\{A\\}$, $\\Delta^\\prime_2:= \\Delta_2$, $\\Gamma^\\prime_1:=\\Gamma_1$, $\\Delta^\\prime_1:=\\Delta_1$). We see that $\\pi^\\prime$ is an $\\infty$-proof of the sequent $\\Gamma^\\prime_1,\\Gamma^\\prime_2\\Rightarrow\\Delta^\\prime_1,\\Delta^\\prime_2$ in $\\mathsf{Grz}_\\infty$. By the induction hypothesis for $\\pi^\\prime$, $\\Lambda_1$ and $\\Lambda_2$, there is the corresponding interpolant $I^\\prime$. We set $I:=I^\\prime$.\n \nCase 4. Suppose that $\\pi$ has the form\n\\begin{gather*}\n\\AXC{$\\pi^\\prime$}\n\\noLine\n\\UIC{$\\Phi, \\Box \\Pi \\Rightarrow A, \\Sigma$}\n\\AXC{$\\pi^{\\prime\\prime}$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Phi, \\Box \\Pi \\Rightarrow \\Box A, \\Sigma$}\n\\DisplayProof\n\\end{gather*}\nwhere $\\Phi, \\Box \\Pi = \\Gamma_1, \\Gamma_2$ and $\\Box A, \\Sigma =\\Delta_1, \\Delta_2$.\n\nSubcase 4.1: the formula $\\Box A \\in \\Delta_1$ and $A \\in \\Lambda_1 $ ($\\Box A \\in \\Delta_2$ and $A \\in \\Lambda_2$). We see that $\\lvert \\pi^\\prime \\rvert < \\lvert \\pi \\rvert $. \nWe put $\\Gamma^\\prime_1:= \\Gamma_1 $, $\\Delta^\\prime_1:= (\\Delta_1 \\backslash \\{\\Box A\\}) \\cup \\{A\\} $, $\\Gamma^\\prime_2:=\\Gamma_2$, $\\Delta^\\prime_2:=\\Delta_2$ ($\\Gamma^\\prime_2:= \\Gamma_2 $, $\\Delta^\\prime_2:= (\\Delta_2 \\backslash \\{\\Box A\\}) \\cup \\{A\\} $, $\\Gamma^\\prime_1:=\\Gamma_1$, $\\Delta^\\prime_1:=\\Delta_1$). We see that $\\pi^\\prime$ is an $\\infty$-proof of the sequent $\\Gamma^\\prime_1,\\Gamma^\\prime_2\\Rightarrow\\Delta^\\prime_1,\\Delta^\\prime_2$ in $\\mathsf{Grz}_\\infty$. By the induction hypothesis for $\\pi^\\prime$, $\\Lambda_1$ and $\\Lambda_2$, there is the corresponding interpolant $I^\\prime$ such that\n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Gamma^\\prime_1\\rightarrow\\bigvee\\Delta^\\prime_1 \\cup \\{ I^\\prime\\} \\quad \\text{and} \\quad\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast \\cup\\{ I^\\prime\\} \\cup  \\Gamma^\\prime_2\\rightarrow\\bigvee\\Delta^\\prime_2\\:.$$\nIf $\\Box A \\in \\Delta_1$ and $A \\in \\Lambda_1 $, then $\\Box (A \\to \\Box A) \\in \\Lambda_1^\\ast$. Thus we have $\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\rightarrow (A \\to \\Box A)$. It follows that we can replace the formula $A$ in $\\Delta^\\prime_1$ by $\\Box A$ and obtain \n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Gamma^\\prime_1\\rightarrow\\bigvee\\Delta_1 \\cup \\{ I^\\prime \\} \\:.$$\nNote also that $\\Gamma^\\prime_1:= \\Gamma_1 $, $\\Gamma^\\prime_2:=\\Gamma_2$, $\\Delta^\\prime_2:=\\Delta_2$. Hence, \n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Gamma_1\\rightarrow\\bigvee\\Delta_1 \\cup \\{ I^\\prime \\}\\quad \\text{and} \\quad\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast \\cup\\{ I^\\prime\\} \\cup  \\Gamma_2\\rightarrow\\bigvee\\Delta_2\\:.$$\nWe set $I:=I^\\prime$. The case  $\\Box A \\in \\Delta_2$ and $A \\in \\Lambda_2 $ is analogous. \n\nSubcase 4.2: the formula $\\Box A \\in \\Delta_1$ and $A \\nin \\Lambda_1 $ ($\\Box A \\in \\Delta_2$ and $A \\nin \\Lambda_2$). We have that $\\Gamma_1, \\Gamma_2 = \\Phi, \\Box \\Pi$ and $\\Delta_1, \\Delta_2 = \\Box A, \\Sigma$. We split $\\Box \\Pi$ into two multisets $\\Box \\Pi_1 \\subset \\Gamma_1$ and $\\Box \\Pi_1 \\subset \\Gamma_2$ such that $\\Box \\Pi= \\Box \\Pi_1, \\Box \\Pi_2$. Put $\\Gamma^{\\prime\\prime}_1:= \\Box \\Pi_1 $, $\\Delta^{\\prime\\prime}_1:=  \\{A\\} $, $\\Gamma^{\\prime\\prime}_2:=\\Box \\Pi_2$, $\\Delta^{\\prime\\prime}_2:=\\emptyset$ ($\\Gamma^{\\prime\\prime}_1:= \\Box \\Pi_1 $, $\\Delta^{\\prime\\prime}_1:=  \\emptyset $, $\\Gamma^{\\prime\\prime}_2:=\\Box \\Pi_2$, $\\Delta^{\\prime\\prime}_2:=\\{A\\}$).\nWe see that $\\pi^{\\prime\\prime}$ is an $\\infty$-proof of the sequent $\\Gamma^{\\prime\\prime}_1,\\Gamma^{\\prime\\prime}_2\\Rightarrow\\Delta^{\\prime\\prime}_1,\\Delta^{\\prime\\prime}_2$ in $\\mathsf{Grz}_\\infty$.\n\nIf $\\Box A \\in \\Delta_2$ and $A \\nin \\Lambda_2 $, then \n$\\mathit{card}(Sub(\\Box\\Pi_1\\Rightarrow)\\backslash \\Lambda_1 )+\\mathit{card}(  Sub(\\Box\\Pi_2\\Rightarrow A)\\backslash  (\\Lambda_2\\cup \\{A\\})$ is strictly less than\n$\\mathit{card}(Sub(\\Gamma_1\\Rightarrow\\Delta_1)\\backslash \\Lambda_1 )+\\mathit{card}(  Sub(\\Gamma_2\\Rightarrow\\Delta_2)\\backslash  \\Lambda_2)$. Hence by the induction hypothesis for $\\pi^{\\prime\\prime}$, $\\Lambda_1$ and $\\Lambda_2\\cup \\{A\\}$ \nthere exists an interpolant $I^{\\prime\\prime}$ such that\n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Box \\Pi_1\\rightarrow  I^{\\prime\\prime} \\quad \\text{and} \\quad\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast \\cup \\{\\Box (A \\to \\Box A)\\}\\cup\\{ I^{\\prime\\prime}\\} \\cup  \\Box \\Pi_2\\rightarrow A\\:.$$\n\nFrom the left condition we immediately obtain $$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Box \\Pi_1\\rightarrow  \\Box I^{\\prime\\prime} \\quad \\text{and} \\quad \\mathsf{Grz} \\vdash \\bigwedge \\Lambda_1^\\ast \\cup\\Gamma_1\\rightarrow  \\bigvee\\Delta_1 \\cup\\{\\Box  I^{\\prime\\prime} \\}\\:.$$\n\nFrom the right condition we see $$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast  \\cup\\{ I^{\\prime\\prime}\\} \\cup  \\Box \\Pi_2\\rightarrow (\\Box (A \\to \\Box A) \\to A)\\:.$$\nThus, \n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast  \\cup\\{ \\Box I^{\\prime\\prime}\\} \\cup  \\Box \\Pi_2\\rightarrow \\Box (\\Box (A \\to \\Box A) \\to A)\\:.$$\nApplying the axiom (v) of $\\mathsf{Grz} $ we see that \n$$\\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast  \\cup\\{ \\Box I^{\\prime\\prime}\\} \\cup  \\Box \\Pi_2\\rightarrow \\Box A\\quad \\text{and} \\quad \\mathsf{Grz} \\vdash \\bigwedge \\Lambda_2^\\ast  \\cup\\{ \\Box I^{\\prime\\prime}\\} \\cup  \\Gamma_2\\rightarrow \\bigvee\\Delta_2 \\:.$$\nNow we can set $I:=\\Box I^{\\prime\\prime}$.\nIn the case $\\Box A \\in \\Delta_2$, $A \\nin \\Lambda_2$ we can analogously set $I:=\\Diamond I^{\\prime\\prime}$.\n\n\\end{proof}\n\n\\begin{thm}[Lyndon interpolation] \nIf $\\mathsf{Grz} \\vdash A \\rightarrow B$, then there exists a formula $C$, called an interpolant of $A \\rightarrow B$, such that $\\mathit{pos} (C) \\subset \\mathit{pos} (A) \\cap \\mathit{pos} (B)$, $\\mathit{neg} (C) \\subset \\mathit{neg} (A) \\cap \\mathit{neg} (B)$, and \\[\\mathsf{Grz }\\vdash A \\rightarrow C , \\qquad \\mathsf{Grz} \\vdash C \\rightarrow B.\\] \n\\end{thm} \n\\begin{proof}\nAssume $\\mathsf{Grz} \\vdash A \\rightarrow B$. By Lemma \\ref{prop}, we have $\\mathsf{Grz_{Seq}} + \\mathsf{cut} \\vdash A \\Rightarrow B$. Applying Theorem \\ref{seqtoinfcut} and Theorem \\ref{infcuttoinf} we obtain $\\mathsf{Grz}_\\infty \\vdash A \\Rightarrow B$. Applying the previous lemma with $\\Lambda_1=\\Lambda_2=\\emptyset$, $\\Gamma_1= \\{ A\\}$, $\\Delta_2=\\{ B\\}$, $\\Gamma_2=\\Delta_1=\\emptyset $, we find an interpolant for $A \\to B$.    \n\\end{proof}\n\n\\section{Cyclic proofs}\n\\label{SecCyc}\n\nThere exists a simple class of $\\infty$-proofs that is sufficient to derive all theorems of $\\mathsf{Grz}_\\infty$. An $\\infty$-proof is called \\emph{regular} if it contains only finitely many nonisomorphic subtrees.\n\nRegular $\\infty$-proofs have useful finite representations called cyclic (circular) proofs.\nA \\emph{cyclic proof} of a sequent $\\Gamma \\Rightarrow \\Delta$ is a pair $(\\kappa, d)$,\nwhere $\\kappa$ is a finite tree of sequents constructed according to the rules of $\\mathsf{Grz}_\\infty$ with\nthe root marked by $\\Gamma \\Rightarrow \\Delta$, and $d$ is a function with the following properties: the\nfunction $d$ is defined on the set of all leaves of $\\kappa$ that are not marked by initial\nsequents; the image $d(a)$ of a leaf $a$ lies on the path from the root of $\\kappa$ to the leaf $a$; there is a right premise of the rule $(\\Box)$ between $a$ and $d(a)$; $a$ and $d(a)$ are marked by the same sequents. If the function $d$\nis defined at a leaf $a$, then we say that the nodes $a$ and $d(a)$ are joined by a back-link.\nHere is an example of a cyclic proof for the sequent $\\Box(\\Box(p \\rightarrow \\Box p) \\rightarrow p) \\Rightarrow p$: \n\n\\begin{gather*}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F, p\\Rightarrow p$}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$ F,p\\Rightarrow \\Box p,p$}\n\\LeftLabel{$\\mathsf{\\to_R}$}\n\\UIC{$ F \\Rightarrow p\\to\\Box p,p$}\n\\AXC{\\textsf{Ax}}\n\\noLine\n\\UIC{$p, F \\Rightarrow p$}\n\\AXC{$ F \\Rightarrow p $ \\tikzmark{a}}\n\\LeftLabel{$\\mathsf{\\Box}$} \n\\BIC{$p, F \\Rightarrow \\Box p$}\n\\LeftLabel{$\\mathsf{}\\to_R$}\n\\UIC{$ F \\Rightarrow p\\to\\Box p$}\n\\LeftLabel{$\\mathsf{\\Box}$} \n\\BIC{$ F \\Rightarrow \\Box(p\\to \\Box p),p$} \n\\LeftLabel{$\\mathsf{\\to_L}$}\n\\BIC{$\\Box(p \\rightarrow \\Box p) \\rightarrow p, F \\Rightarrow p$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ ,} \n\\UIC{$F \\Rightarrow p $ \\tikzmark{b}}\n\\DisplayProof\n\\begin{tikzpicture}[overlay,remember picture, >=latex,distance=4cm]\n    \\draw[->, thick] ({pic cs:a}) to [out=0,in=-20]({pic cs:b});\n \\end{tikzpicture}\n\\end{gather*}\nwhere $F=\\Box(\\Box(p \\rightarrow \\Box p) \\rightarrow p) $. \n\nThe notion of cyclic proof determines the same provability\nrelation as the notion of regular $\\infty$-proof. Obviously, each cyclic proof can be\nunravelled into a regular $\\infty$-proof. The converse is also true.\n\\begin{prop}[cf. \\cite{Sham}, Proposition 3.1]\nAny regular $\\infty$-proof of $\\mathsf{Grz}_\\infty$ can be obtained by unraveling a cyclic proof.\n\\end{prop}\n\n\n\n\nIn the rest of the section we establish that any sequent provable in $\\mathsf{Grz}_\\infty$ has a cyclic proof.\n\n\n\n\\begin{lem}\\label{contraction}\nFor any formula $A$, the rules\n\\begin{gather*}\n\\AXC{$\\Gamma , A,A \\Rightarrow  \\Delta$}\n\\LeftLabel{$\\mathsf{cl}_{A}$}\n\\UIC{$\\Gamma ,A \\Rightarrow  \\Delta$}\n\\DisplayProof\\qquad\n\\AXC{$\\Gamma \\Rightarrow A,A, \\Delta$}\n\\LeftLabel{$\\mathsf{cr}_{A}$}\n\\UIC{$\\Gamma \\Rightarrow A, \\Delta$}\n\\DisplayProof\n\\end{gather*}\nare admissible in $\\mathsf{Grz}_{\\infty}$.\n\\end{lem}\n\\begin{proof}\nAssume $\\pi$ and $\\tau$ are $\\infty$-proofs of the sequents $\\Gamma , A,A \\Rightarrow  \\Delta$ and $\\Gamma \\Rightarrow A,A, \\Delta$ in the system $\\mathsf{Grz}_{\\infty}$.\nLet $\\xi$ be an $\\infty$-proof of the sequent $\\Gamma , A \\Rightarrow  A,\\Delta$ in $\\mathsf{Grz}_{\\infty}$, which exists by Lemma \\ref{AtoA}.\n\nThe required $\\infty$-proofs of the sequents $\\Gamma , A \\Rightarrow  \\Delta$ and $\\Gamma  \\Rightarrow  A,\\Delta$ are defined by setting $\n\\mathsf{cl}_{A}(\\pi)=\\mathsf{re}_A(\\xi, \\pi)$ and \n$\n\\mathsf{cr}_{A}(\\tau) =\\mathsf{re}_A(\\tau,\\xi)$, where $\\mathsf{re}_A$ is an $A$-removing mapping from Lemma \\ref{reabadeq}. Since the mapping $\\mathsf{re}_A$ is adequate, the $\\infty$-proofs $\n\\mathsf{cl}_{A}(\\pi)$ and $\\mathsf{cr}_{A}(\\tau)$ do not contain applications the rule $(\\mathsf{cut})$.\n \n\\end{proof}\n\nLet $\\mathcal{T}^\\ast$ denote the set of all root-preserving mappings from the set of $\\infty$-proofs of $\\mathsf{Grz}_\\infty$ to itself.\n\n\\begin{lem}\\label{Comp T*}\nThe pair $(\\mathcal T^\\ast, l_1)$ is a non-empty spherically complete ultrametric space.\n\\end{lem}\n\\begin{proof}\nThe proof of spherical completeness of the space $(\\mathcal T^\\ast, l_1)$ is analogous to the proof of Proposition \\ref{SphCom}.\nThe space is obviously non-empty, since the identity function lies in $\\mathcal T^\\ast$.\n\\end{proof}\n\nAn application of the modal rule $(\\Box)$ \n\\[\\AXC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ .}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n\\]\nis called \\emph{slim} if the multiset $\\Pi$ here is a set.  \nAn $\\infty$-proof is called \\emph{slim} if every application of the rule $(\\Box)$ in it is slim.\n\n\\begin{lem}\nIf $\\mathsf{Grz}_\\infty\\vdash \\Gamma \\Rightarrow \\Delta$, then the sequent $\\Gamma \\Rightarrow \\Delta$ has a slim $\\infty$-proof in $\\mathsf{Grz}_\\infty$.\n\\end{lem} \n\\begin{proof}\n\n\nWe construct a mapping $\\mathsf {slim}\\in \\mathsf T^\\ast$ that maps an $\\infty$-proof to a slim $\\infty$-proof of the same sequent. This mapping is defined as the fixed-point of a contractive operator $\\mathsf H\\colon \\mathcal T^\\ast \\to \\mathcal T^\\ast $.\n\nFor a mapping $\\mathsf u\\in \\mathcal T^\\ast$ and an $\\infty$-proof of $\\mathsf{Grz}_\\infty$, the $\\infty$-proof $\\mathsf H(\\mathsf u)(\\pi)$ is defined as follows.\nIf $\\lvert \\pi\\rvert=0$, then we put $\\mathsf H(\\mathsf u)(\\pi)=\\pi$. \n\nOtherwise, we define $\\mathsf H(\\mathsf u)(\\pi)$ according to the last application of an inference rule in $\\pi$:\n\\begin{gather*}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma , B \\Rightarrow  \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow  A, \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_L}$}\n\\BIC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_1)$}\n\\noLine\n\\UIC{$\\Gamma , B \\Rightarrow  \\Delta$}\n\\AXC{$\\mathsf u(\\pi_2)$}\n\\noLine\n\\UIC{$\\Gamma \\Rightarrow  A, \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_L}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma , A \\rightarrow B \\Rightarrow  \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\\begin{gather*}\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow  B , \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_R}$}\n\\UIC{$\\Gamma \\Rightarrow  A \\rightarrow B , \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_0)$}\n\\noLine\n\\UIC{$\\Gamma, A \\Rightarrow  B , \\Delta$}\n\\LeftLabel{$\\mathsf{\\rightarrow_R}$}\n\\RightLabel{ ,}\n\\UIC{$\\Gamma \\Rightarrow  A \\rightarrow B , \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\\begin{gather*}\n\\AXC{$\\pi_0$}\n\\noLine\n\\UIC{$\\Gamma, A, \\Box A \\Rightarrow   \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\UIC{$\\Gamma , \\Box A\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_0)$}\n\\noLine\n\\UIC{$\\Gamma, A, \\Box A \\Rightarrow   \\Delta$}\n\\LeftLabel{$\\mathsf{refl}$}\n\\RightLabel{ .}\n\\UIC{$\\Gamma , \\Box A\\Rightarrow  \\Delta$}\n\\DisplayProof \n\\end{gather*}\n\nFor a multiset $\\Pi$, we denote its underlying set by $\\Pi^S$. The case of the modal rule is as follows:\n\\begin{gather*}\n\\AXC{$\\pi_1$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\pi_2$}\n\\noLine\n\\UIC{$\\Box \\Pi \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n\\longmapsto\n\\AXC{$\\mathsf u(\\pi_1)$}\n\\noLine\n\\UIC{$\\Gamma, \\Box \\Pi \\Rightarrow A, \\Delta$}\n\\AXC{$\\mathsf {u}(\\pi^\\prime_2)$}\n\\noLine\n\\UIC{$\\Box \\Pi^S \\Rightarrow A$}\n\\LeftLabel{$\\mathsf{\\Box}$}\n\\RightLabel{ ,}\n\\BIC{$\\Gamma, \\Box \\Pi \\Rightarrow \\Box A, \\Delta$}\n\\DisplayProof \n\\end{gather*}\nwhere $\\pi^\\prime_2$ is an $\\infty$-proof of the sequent $\\Box \\Pi^S \\Rightarrow A$ in $\\mathsf{Grz}_\\infty$. This $\\infty$-proof exists by the\nadmissibility of contraction rules obtained in Lemma \\ref{contraction}.\n \nNow it can be easily shown, that if $\\mathsf u\\sim_{n,k} \\mathsf v$ for some $\\mathsf u, \\mathsf v\\in \\mathcal T^\\ast$ and $n,k\\in\\mathbb N$, then $\\mathsf {H(u)}\\sim_{n,k+1}\\mathsf{H(v)}$. Therefore the mapping $\\mathsf H$ is a contractive operator on a spherically complete ultrametric space. Thus, it has a fixed-point, which we denote by $\\mathsf {slim}$.\n\nIn analogous way to the proofs of Lemma \\ref{reboxadeq} and Theorem \\ref{infcuttoinf}, we see that $\\mathsf {slim}(\\pi)$ is a slim $\\infty$-proof for any $\\pi$.\n\\end{proof}\n\n\n\\begin{thm}\nIf $\\mathsf{Grz}_\\infty\\vdash \\Gamma \\Rightarrow \\Delta$, then there is a cyclic proof for the given sequent. \n\\end{thm}\n\\begin{proof}\nLet $\\pi$ be a slim $\\infty$-proof of the sequent $\\Gamma \\Rightarrow \\Delta$ obtained in the previous lemma. Note that all formulas from $\\pi$ are subformulas of the formulas from $\\Gamma \\cup \\Delta$. Consequently, the $\\infty$-proof $\\pi$ contains only finitely many different sequents that occur as right premises of the rule $(\\Box)$ in it. By $k$, we denote the number of these sequents. \n\nLet $\\xi$ be the $(k+2)$-fragment of the $\\infty$-proof $\\pi$. Consider any branch $a_0, a_1, \\dotsc, a_n$ in $\\xi$ connecting the root with a leaf $a_n$ that is not marked by an initial sequent. This branch containes a pair of different nodes $a$ and $b$ determining coinciding right premises of the rule $(\\Box)$. Assuming that $b$ is further from the root of $\\xi$ than $a$, we cut the branch under consideration at the node $b$ and connect $b$, which has become a leaf, with $a$ by a back-link. Applying\na similar operation to the remaining branches of $\\xi$, we turn the $(k+2)$-fragment of $\\pi$ into a cyclic proof of the sequent $\\Gamma \\Rightarrow \\Delta$.  \n\n\n\\end{proof}\n\n\n\\paragraph*{Funding.} The article was prepared within the framework of the Basic\nResearch Program at the National Research University \nHigher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project \n'5-100'. This work is also supported by the Russian Foundation for Basic Research, grant 15-01-09218a.\n\n\\paragraph*{Acknowledgements.} The second author heartily thanks his beloved wife Maria Shamkanova for her warm and constant support. He is also indebted to the Lord God Almighty. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nInflation of a cylindrical rubber tube usually generates a prototypical localization instability, known as localized bulging, and such an interesting phenomenon was first recorded in detail and primarily analyzed by \\cite{mallock}. With applications to bulge initiation and propagation in pressure vessels \\citep{jam1984,ijms1988,book2007} and aneurysm formation in human arteries \\citep{new2011,sa2018}, inflation of a cylindrical hyperelastic tube offers a pertinent paradigm to study the mechanism of localized instabilities \\citep{ky1990,ky1991,prsa2021}. In addition, recent advances on inflated tubes and localized bulging can shed light on bifurcation in arteries affected by Marfan's syndrome \\citep{mrc2009,mrc2010}, bulge prevention in energy harvesting devices \\citep{re2013}, inflation-driven soft robots \\citep{jin2021}, inflation of nematic elastomers \\citep{he2020}, and bulge formation subject to additional effects of electric actuation \\citep{lu2015}, swelling \\citep{dm2017},  magnetic field \\citep{ijss2018}, and plasticity \\citep{takla2}. \n\nIn general, the evolution of localized bulging is mainly composed of three typical periods, including bulge initiation, growth, and propagation \\citep{wang2019}. Specifically, experimental evidence suggests that the diameter of the bulge will reach a maximum in the propagation stage, and a two-phase deformation can be observed \\citep{ky1990}. On the theoretical side, the pioneering works on bifurcation analysis of inflated membrane tubes and thick tubes were conducted by \\cite{ho79,ho791}. Later, \\cite{fu2008} found that a zero mode, which is usually viewed as another uniform deformation, precisely generates localized bulging, and a bifurcation condition was derived for a membrane tube based on the dynamical systems theory. Since then, multiple fundamental problems were investigated, such as solution stability \\citep{pearce2010,fuxie2010,ams2012}, influence of material constitution \\citep{pearce2012}, imperfection sensitivity \\citep{fuxie2012}, and dynamic inflation \\citep{dynamic}. Within the framework of nonlinear elasticity, an explicit bifurcation condition for thick tubes was derived by \\cite{fu2016}, and the accuracy of membrane assumption was examined. This bifurcation condition paves a convenient way to study localized bugling in rotating cylinders \\citep{wang2017}, fiber-reinforced tubes \\citep{wangfu2018}, and layered tubes \\citep{liu2019,ye2019}. Furthermore, a comprehensive understanding of localized bulging can provide useful insight into localized necking \\citep{eml2018}. We mention that the evolution of a bulge can only be handled in the nonlinear regime. To perform post-bifurcation analysis for inflated thick-walled cylinders, \\cite{fead2013} presented a numerical procedure by virtue of nonlinear finite element models. Within the framework of nonlinear elasticity,  \\cite{ye2020} carried out a weakly nonlinear analysis to deduce the amplitude equation for a bulge and discovered an interesting parametric domain where necking occurs instead of bulging. Recently, the analysis of localizations was further extended to situations where surface tension is involved \\cite{wang2020,emery2021,emery20211,fu2021}. Recent papers have shown a growing interest in the community for such a paradigmatic localization problem.\n\nIn addition to the investigations mentioned above, some effort was dedicated to establishing a reduced model applicable to inspect bulge formation and propagation in cylindrical hyperelastic tubes. \\cite{audoly2018} derived a one-dimensional model for the analysis of localized bulging in long tubes making use of nonlinear membrane theory and asymptotic expansion. A fundamental assumption is that localization can be viewed as a long-wavelength periodic mode. Subsequently, \\cite{biggins2020} proposed an energy method in virtue of asymptotic expansions to track the nonlinear evolution of an inhomogeneous solution and further to identify phase separation of typical localization such as beading, bulging, and necking.\n\nAs mentioned earlier, localized bulging in inflated tubes can be viewed as an analogue to aneurysm formation in arteries \\citep{sh2001,fu2012}. Accordingly, theoretical or numerical analysis was performed to supply mechanical insight into possible pathogenesis behind aneurysms, see \\cite{ijes2014,fu2015,vn2017,vn2018} and the references therein. In particular, our previous analysis reveals a fact that an aneurysm is more prone to appear as the innermost layer of an artery stiffens \\citep{ye2019}. This qualitatively interprets why the risk of aneurysm formation for a person becomes higher with aging \\citep{fg2015}.    \n\nGenerally speaking, two loading conditions, i.e. either the resultant axial force or the axial length is specified, are adopted in experiments. The former was adopted in \\cite{ky1990,ky1991,wang2019,prsa2021} while the latter was used in \\cite{ijms2006,ijms2008,guo2016,wang2019,prsa2021}. On the one hand, the resultant axial force can be fixed by suspending a dead weight at one end while air comes into the tube from the other end. In this scenario, the curve of pressure versus volume ratio possesses an $N$-shape. Therefore, localized bulging is induced by a limit-point instability \\citep{ijes1971,ijnm2007,wt2018} and the propagation pressure can be predicted by Maxwell's equal-area rule \\citep{jam1984}. On the other hand, the fixed axial length can be acquired by imposing a pre-stretch on the tube and then fixing the total length. In this situation, the pressure-volume ratio curve may be monotonic. The effect of pre-stretch on pressurized thin tubes was examined by \\cite{mao2014,hnijms}. Furthermore, human arteries usually suffer a fixed pre-stretch \\textit{in vivo} \\citep{bmm2014}. In this study, we also employ these two loading approaches.\n\nIt is pointed out that most existing literature on localized bulging is mainly concerned with a homogeneous or piecewise homogeneous material. Yet functionally graded materials (abbreviated by FGMs) with continuously varied physical or mechanical properties have unfolded many unusual functions such as high-temperature resistance and diminishing stress or strain concentration. This new type of material first emerged in the middle of the 1980s by mixing metal and ceramic phases in a manageable way. In doing so, it will acquire a desirable material property that distributes spatially. Afterwards, the mechanics of FGM has rapidly taken a center stage in solid mechanics \\citep{ke2008,zhong2012,jha2013}. Especially, the popularity of 3D printing techniques substantially facilitates the fabrication of FGMs. Besides, human arteries chiefly consist of intima, media, and adventitia \\citep{je2000,hgo}, which can be viewed as a sandwich structure and further a special graded structure. In most practical applications, we intend to prevent localized bulging or aneurysm formation, such as an Anaconda wave-energy extraction device \\citep{re2013}. Naturally, improving the stiffness of a homogeneous tube will raise the critical pressure inducing localized bulging since pressure is proportional to the elastic modulus. However, an extremely stiff structure will lose the ability to deform subjected to a similar load and hence will degrade certain structural or physiological performance, particularly for arteries. Naturally, a compromised way is to partially stiffen. Then a well-imposed question arises.  Where is the optimal position to be strengthened, the innermost surface, the outermost surface, or the middle part? In fact, the question has been perfectly solved by nature. It is the intermediate layer that is hardest, and this is also the optimization result of artery evolution \\citep{je2000,jctr2012}. But why? Previous studies have shown the potential effect of modulus gradient form on the stress distribution in functionally graded rubberlike cylinders and spheres \\citep{mms2011} or pattern transition in growing graded tubular tissues \\citep{liu2020}. It is therefore of fundamental significance to clarify the influence of material inhomogeneity, different modulus gradients as well as the maximum modulus mismatch on bulge initiation, growth, and propagation. This motivates the current study. Although deformation or bifurcation analysis was carried by \\cite{bb2009} for inflated and everted circular cylinders comprised of a graded Mooney-Rivlin material and by \\cite{chenwq2017} for pressurized graded cylindrical tubes, respectively, localization instability was still not addressed. In this study, we aim at furnishing a thorough analysis of localized bulging in graded hyperelastic tubes under the combined action of internal pressure and axial stretching and unveiling the mechanism behind structure optimization of arteries.\n\nThis paper is organized as follows. In Section 2, we characterize the primary deformation generated by inflation for a general material constitution and a generic modulus gradient and establish all required formulas used in the bifurcation condition within the framework of nonlinear elasticity. Section 3 demonstrates the numerical strategy for identifying bulge initiation and presents a detailed parametric study for the onset of localized bulging. A finite element model is constructed and validated in Section 4. A nonlinear analysis based on finite element analysis is performed in Section 5 and the bulge propagation is investigated by Maxwell's equal-area rule. Finally, some conclusions are given in Section 6.\n\n\\section{Primary deformation and bifurcation condition}\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.8]{fig1}\\caption{(Color online) A graded tube with inner radius A, outer radius B and length $L$ in its stress-free state.}\\label{fig1}\n\\end{figure}\nAs shown in Figure \\ref{fig1}, a cylindrical tube with inner radius $A$ and outer radius $B$ in its initial state is considered in this study. Since we focus on a graded soft tube, it is assumed that the elastic modulus of the tube is no longer uniformly distributed in the radial direction but is homogeneous in the axial direction. When such a graded tube is subjected to an internal pressure $P$ and an axial extension, a primary deformation is first generated, and the inner and outer radii become $a$ and $b$, respectively. Then we define the initial state as the reference configuration while the primary state the current configuration. Assuming that the tube is composed of incompressible hyperelastic material, we can use  $W(\\mathbf{F})$ to represent the strain-energy function where $\\mathbf{F}$ is the deformation gradient and is subjected to a constraint $\\operatorname{det} \\mathbf{F}=1$, which is the so-called incompressibility condition.\n\nFor the current problem, it is convenient to adopt cylindrical polar coordinates in both the reference and current configurations. By determining the centroid as the origin, the position coordinates of the same material point in the reference and current states are denoted by $(R, \\Theta, Z)$ and $(r, \\theta, z)$, respectively, and the tube occupies $A\\leqslant R\\leqslant B$ and $-L\/2\\leqslant Z\\leqslant L\/2$ in the reference state. Bearing in mind that the primary deformation is axisymmetric, we can write the deformation gradient as\n\\begin{equation}\n\\mathbf{F}=\\lambda_1\\bm e_r\\otimes\\bm e_r+\\lambda_2\\bm e_\\theta\\otimes\\bm e_\\theta+\\lambda_3\\bm e_z\\otimes\\bm e_z,\\label{eq2_1}\n\\end{equation} \nwhere $\\lambda_i$ $(i=1,2,3)$ is the principal stretch in $i$-direction and the subscripts 1, 2 and 3 denote the $r$-, $\\theta$- and $z$-directions. Furthermore, the common orthonormal basis $\\{\\bm e_r, \\bm e_\\theta, \\bm e_z\\}$ is used in both states. In particular, the first two principal stretches are equal to\n\\begin{equation}\n\\lambda_1=\\dfrac{\\textrm{d} r}{\\textrm{d} R},~~\\lambda_2=\\dfrac{r}{R},\\label{eq2_2}\n\\end{equation}\nand the third one $\\lambda_3$ is a constant since the primary deformation is homogeneous in the $z$-direction. \n\nThe strain-energy function can be written as a function of these principal stretches $W(\\lambda_1,\\lambda_2,\\lambda_3)$. Accordingly, the non-zero components of Cauchy Stress tensor are given by \n\\begin{align}\n\\sigma_{ii}=\\lambda_iW_{,i}-p,~~\\textrm{no summation}, \\label{eq2_3}\n\\end{align}\nwhere $p$ depicts the Lagrange multiplier enforcing the incompressibility condition. Meanwhile, we have defined that a comma means derivative with respect to the corresponding variable, for instance, $W_{,2}=\\partial W\/\\partial \\lambda_2$. \n\nIt can be readily checked that the primary deformation is governed by the sole equilibrium equation: \n\\begin{align}\n\\dfrac{\\textrm{d} \\sigma_{rr}}{\\textrm{d} r}+\\dfrac{\\sigma_{rr}-\\sigma_{\\theta\\theta}}{r}=0.\\label{eq2_4}\n\\end{align}\n\nIn order to facilitate subsequent analysis, we define two parameters according to the incompressibility condition \n\\begin{align}\n\\lambda=\\lambda_2=\\frac{r}{R},~\\lambda_z=\\lambda_3.\\label{eq2_5}\n\\end{align}\nCorrespondingly, the first principal stretch is determined by $\\lambda_1=1\/(\\lambda \\lambda_z)$. In addition, the following relations can be derived :\n\\begin{align}\n&r^2=\\lambda_z^{-1}(R^2-A^2)+a^2,~~\\theta=\\Theta,~~z=\\lambda_zZ.\\label{eq2_6}\n\\end{align}\n\nNext, we replace $\\lambda_1$ in $W$ and employ a reduced strain-energy function $w$ which is determined by\n\\begin{align}\nw(\\lambda,\\lambda_z)\\equiv W(1\/(\\lambda \\lambda_z),\\lambda,\\lambda_z).\\label{eq2_7}\n\\end{align}\nSubsequently, we find that\n\\begin{align}\nw_{,1}=W_{,2}-\\frac{1}{\\lambda^2\\lambda_z}W_{,1},~~w_{,2}=W_{,3}-\\frac{1}{\\lambda\\lambda_z^2}W_{,1}.\\label{eq2_8}\n\\end{align}\n\nSubstituting the stress components in (\\ref{eq2_3}) into the governing equation (\\ref{eq2_4}) and utilizing (\\ref{eq2_8}) yield\n\\begin{align}\n\\sigma_{rr}=\\int_{r_0}^{r}\\frac{\\lambda w_{,1}}{r}\\textrm{d}r, \\label{eq2_9}\n\\end{align}\nwhere $r_0$ is a constant. Actually $w$ is dependent on $R$ for a graded material. It can be deduced from $(\\ref{eq2_5})_1$ and $(\\ref{eq2_6})_1$ that $R=A\\sqrt{\\lambda_z\\lambda_a^2-1}\/\\sqrt{\\lambda_z\\lambda^2-1}$. Then we apply a variable exchange $\\lambda=r\/R$ and a new identity of the Cauchy stress $\\sigma_{rr}$ is given by\n\\begin{equation}\n\\sigma_{rr}=\\int_{\\lambda_0}^\\lambda\\frac{w_{,1}}{1-\\lambda^2\\lambda_z}\\textrm{d}\\lambda,\\label{eq2_10}\n\\end{equation}\nwhere $\\lambda_0$ is another constant to be determined later and we have replaced $R$ by $A\\sqrt{\\lambda_z\\lambda_a^2-1}\/\\sqrt{\\lambda_z\\lambda^2-1}$ such that the integrand in (\\ref{eq2_10}) is only a function of $\\lambda$ (see also equation (\\ref{eq2_15})). \n\nBefore we proceed further, it is appropriate to introduce two hoop stretches at the inner surface and outer surface as follows\n\\begin{equation}\n\\lambda_a=\\frac{a}{A},~~\\lambda_b=\\frac{b}{B},\\label{eq2_11}\n\\end{equation}\nand their relation can be obtained from (\\ref{eq2_6})\n\\begin{equation}\n\\lambda_b=\\sqrt{\\frac{B^2-A^2+A^2\\lambda_a^2\\lambda_z}{B^2\\lambda_z}}.\\label{eq2_12}\n\\end{equation}\n\nIt is assumed that the outer surface of the tube is traction-free and the inner surface suffers a pressure $P$. Therefore, we obtain\n\\begin{equation}\n\\sigma_{rr}|_{\\lambda=\\lambda_b}=0,~~\\sigma_{rr}|_{\\lambda=\\lambda_a}=-P.\\label{eq2_13}\n\\end{equation}\nIt then follows from the boundary condition $(\\ref{eq2_13})_{1}$ that the constant of integration $\\lambda_0$ is identical to $\\lambda_b$. In addition, the other one $(\\ref{eq2_13})_{2}$ furnishes\n\\begin{equation}\nP=\\int_{\\lambda_b}^{\\lambda_a}\\frac{w_{,1}}{1-\\lambda^2\\lambda_z}\\textrm{d}\\lambda.\\label{eq2_14}\n\\end{equation}\nActually, the pressure $P$ can be regarded as a function of $\\lambda_a$ and $\\lambda_z$. Similarly, the resultant axial force $N$ in any cross-section is also dependent on  $(\\lambda_a,\\lambda_z)$ and holds the following form\n\\begin{align}\nN=2\\pi\\int_a^b\\sigma_{zz}r\\textrm{d}r-P\\pi a^2=\\pi A^2(\\lambda_a^2\\lambda_z-1)\\int_{\\lambda_b}^{\\lambda_a}\\frac{2\\lambda_z w_{,2}-\\lambda w_{,1}}{(\\lambda^2\\lambda_z-1)^2}\\lambda \\textrm{d}\\lambda.\\label{eq2_15}\n\\end{align}\n\nIt should be mentioned that (\\ref{eq2_14}) and (\\ref{eq2_15}) are valid for both homogeneous and graded materials. For graded tubes, the reduced strain-energy function in these two expressions contains radius-dependent elastic modulus. Moreover, (\\ref{eq2_14}) can be seen as a special case in \\cite{chenwq2017} where both the inner and outer surfaces suffer pressures.\n\nCurrently, the primary deformation induced by inflation has been fully characterized by the above two equations for a graded hyperelastic tube of finite thickness. In practice, there are usually two different end conditions, i.e. either the axial force $N$ or the axial length $\\lambda_zL$ is fixed. The former case can be attained by imposing an object of dead weight at one end and leaving the other end free. Correspondingly, (\\ref{eq2_14}) and (\\ref{eq2_15}) provide two algebraic equations for the two unknowns $\\lambda_a$ and $\\lambda_z$ and solving them for a given pressure and a given axial force yields the solution that characterizes the primary deformation. This kind of loading type has been used in the experimental investigations carried out by \\cite{ky1990,ky1991,guo2016,wang2019, prsa2021}. In the other loading condition, the tube is first uniformly stretched and then the two ends are fastened.  As a result, the axial length or the axial stretch $\\lambda_z$ is fixed. Furthermore, the axial force $N$ is not controlled and may increase, decrease or remain constant during the inflation process. We emphasize that this scenario corresponds to the $in$ $vivo$ situation for human arteries \\citep{bmm2014}. Also, such an experimental setup was employed by \\cite{ijms2006,ijms2008,wang2019,prsa2021}. Since $\\lambda_z$ is prescribed, the internal pressure $P$ is only a function of $\\lambda_a$, and the primary deformation is fully determined by equation (\\ref{eq2_14}).\n\nIn an earlier study by \\cite{chenwq2017}, various bifurcation behaviors were investigated based on the incremental theory for functionally graded tubes subjected to internal pressure, external pressure, and end compression, and periodic modes in the axial and hoop directions were analyzed. In this study, we consider that the graded tube suffers a combined action of internal pressure and an axial extension. Therefore, zero-mode may become preferred \\citep{fu2008}, resulting in localized bulging in soft cylindrical tubes. On the one hand, if either the resultant axial force $N$ or the prescribed axial stretch $\\lambda_z$ is greater enough, localized bulging can disappear \\citep{liu2019}.   On the other hand, if the axial stretch $\\lambda_z$ is less than a threshold value, global buckling can replace localized bulging \\citep{lin2020,prsa2021}. In this paper, we intend to investigate localized bulging (aneurysm formation) in graded tubes and further to elucidate the influence of modulus gradient on bulge formation, growth, and propagation. To this end, we merely concentrate on the situation that localized bulging is preferred as a result of the primary bifurcation.\n\nBearing in mind that we have denoted the internal pressure $P$ and the resultant axial force $N$ as functions of the stretches $\\lambda_a$ and $\\lambda_z$, it is suitable to resort to an explicit bifurcation condition that the Jacobian of $P$ and $F$ in terms of variables $\\lambda_a$ and $\\lambda_z$ vanishes. This concise bifurcation condition was first proposed by \\cite{fu2016} in the framework of finite elasticity. Later, its applicability to localized bulging in bilayer tubes was verified numerically using FE analysis in commercial software Abaqus \\citep{liu2019,ye2019}. Recently, \\cite{yu2021} offered analytical proof of this bifurcation condition with two relatively weak assumptions. The first one is that the strain-energy function is isotropic while the second one is that the deformations before bulge initiation and in the propagation stage are both axisymmetric. So, it also supports the validity of this bifurcation condition when applied to graded tubes. Writing explicitly, we obtain the bifurcation condition for localized bulging as follows\n\\begin{equation}\nJ(P,N)=\\frac{\\partial P}{\\partial\\lambda_a}\\frac{\\partial N}{\\partial\\lambda_z}-\\frac{\\partial P}{\\partial \\lambda_z}\\frac{\\partial N}{\\partial\\lambda_a}=0.\\label{eq2_16}\n\\end{equation}\n\nWith the aid of (\\ref{eq2_16}), the first bifurcation points for fixed axial force and fixed axial length can be identified, respectively, by the following equations\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\textrm{fixed axial force (loading type I):}~~~\nJ(P,N)=0,~~\nN(\\lambda_a,\\lambda_z)=N_0,\\\\&\n\\textrm{fixed axial length (loading type II):}~~~\nJ(P,N)=0,~~\n\\lambda_z=\\lambda_{z0},\n\\end{aligned}\n\\right.\\label{eq2_17}\n\\end{equation}\nwhere $N_0$ and $\\lambda_{z0}$ are given constants. For convenience, we abbreviate loading type I (II) as LTI (LTII) from now on. Indeed, equation $(\\ref{eq2_17})_1$ furnishes two critical stretch values $\\lambda_a^c$ and $\\lambda_z^c$ for LTI where localized bulging initiates, and equation $(\\ref{eq2_17})_2$ determines one critical stretch value $\\lambda_a^c$ for LTII. Subsequently, the critical pressures $P_{cr}$ for both loading types can be calculated from equation (\\ref{eq2_14}).\n\nNext, we suppose that the tube is composed of the incompressible Gent material in all illustrative examples, and the strain-energy function is given by\n\\begin{equation}\nW=-\\frac{\\mu(R)}{2}J_m\\textrm{ln}\\bigg(1-\\frac{\\lambda_1^2+\\lambda_2^2+\\lambda_3^2-3}{J_m}\\bigg),\\label{eq2_18} \n\\end{equation}\nwhere $\\mu(R)$ stands for the radius-dependent shear modulus for a graded tube and $J_m$ is a material parameter representing the maximum extensibility. In particular, we set $J_m=97.2$ which is typical for rubber \\citep{gent}. Note that this model can well capture the bulge initiation, bulge growth and bulge propagation in finite element simulations \\citep{liu2019,ye2019}, and these three distinctive stages were commonly seen in experiments \\citep{wang2019}. \n\nAs mentioned earlier, we are also concerned with how different modulus gradients affect the onset of localized bulging and bulge evolution. We emphasize that an accurate modulus distribution in a graded structure is difficult to describe. When fabricated polydimethylsiloxane (PDMS) material is exposed to UV radiation, Young's modulus may have an exponential distribution, resulting in a graded PDMS structure \\citep{chen2018}. Generally speaking, the modulus distribution may be varied in different situations. A power law was adopted in \\cite{bb2009} and a linear function was employed in \\cite{chenwq2017}. In our previous study for growth-induced surface instabilities, it was found that different modulus distributions can alter pattern transition \\citep{liu2020}. Moreover, it was shown in \\cite{ye2019} that a stiffer inner layer makes the structure less stable. If a sandwich tube has a stiffer or softer core and the two faces share the same shear modulus, there must exist a competition among these three layers and may cause some interesting results. This intuitively motivates the investigation on a non-monotonic modulus distribution, which can further be used to model human arteries. As a result, we employ three modulus functions as follows\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\textrm{linear distribution:}~\n\\mu(R)=\\mu_B\\left[(\\beta_1-1)\\dfrac{R-B}{A-B}+1\\right],\\\\&\n\\textrm{exponential distribution:}~\\mu(R)=\\mu_B\\left[(\\beta_1-1)\\textrm{exp}\\left(\\zeta\\dfrac{A-R}{B-A}\\right)+1\\right],\\\\&\n\\textrm{sinusoidal distribution:}~\\mu(R)=\\mu_B\\left[(\\beta_2-1)\\sin\\left(\\dfrac{R-A}{B-A}\\pi\\right)+1\\right],\n\\end{aligned}\n\\right.\\label{eq2_19}\n\\end{equation}\nwhere $\\beta_1=\\mu(A)\/\\mu_B$ signifies the ratio of the shear modulus of the inner surface to that of the outer surface and $\\mu_B=\\mu(B)$. The $\\beta>1$ means that the inner surface is stiffer than the outer surface. In the sinusoidal distribution, we define $H=(B+A)\/2$ and let $\\beta_2=\\mu(H)\/\\mu_B$. Moreover, the parameter $\\zeta$ in $(\\ref{eq2_19})_2$ refers to as the decay rate. In fact, letting $R=B$ in $(\\ref{eq2_19})_1$ yields $\\mu(B)=\\mu_B\\left((\\beta_1-1)\\textrm{exp}(-\\zeta)+1\\right)$. In order to satisfies $\\mu(B)=\\mu_B$, we must have $\\zeta=+\\infty$. In our illustrative examples, we specify $\\zeta=30$ as an approximation such that exp$(-30)\\approx 9.358\\times10^{-14}$. Meanwhile, the effect of $\\zeta$ is out of the scope of the current study. We then plot these three modulus distributions in Figure \\ref{fig2} to depict the details.\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=1.2]{fig2}\\caption{Three typical distributions of the shear modulus $\\mu$ in the radial direction. The parameters are given by $\\beta_1=\\beta_2=20$, $A\/B=0.8$, $\\zeta=30$.}\\label{fig2}\n\\end{figure}\n\nIn this section, we have derived the expressions of the pressure $P$ and the resultant axial force $N$ for the primary deformation of a graded soft tube. Meanwhile, a theoretical framework for determining the bulge initiation is established for a general material model and a general modulus gradient. In the next section, a detailed parametric analysis will be carried out based on the bifurcation condition (\\ref{eq2_16}).\n\n\\section{Theoretical analysis of bulge initiation}\nWe have indicated that this study only focuses on the situation where localized mode is preferred. Accordingly, the prescribed axial force $N_{0}$ and the prescribed axial stretch $\\lambda_{z0}$ in (\\ref{eq2_17}) should lie in a proper domain. In the succeeding calculations, we specify $N_0=0$ and $\\lambda_{z0}=1.5$ and suppose that the analysis can also shed light on qualitative effects of different parameters and modulus gradients for other $N_0$ and $\\lambda_{z0}$. To facilitate further analysis, we introduce some dimensionless quantities as follows\n\\begin{align}\nA^*=\\dfrac{A}{B},~~N^*=\\dfrac{N}{\\mu_B B^2},~~\\mu^*=\\dfrac{\\mu}{\\mu_B},~~P^*=\\dfrac{P}{\\mu_B},~~w^*=\\dfrac{w}{\\mu_B}.\n\\end{align}\nCorrespondingly, the bifurcation condition for localized bulging is rewritten as $J(P^*, N^*)=0$.\n\nIn practice, analytical expressions of the dimensionless pressure $P^*$ and the dimensionless force $N^*$ can be derived for many homogeneous material models such as the neo-Hookean model, Mooney-Rivlin model, Ogden model, and Gent model. Accordingly, equation (\\ref{eq2_17}) contains two sets of nonlinear algebraic equations, and solving them numerically can lead to the critical stretch $\\lambda_a^c$. However, when the shear modulus becomes position-dependent in the Gent model, explicit formulas for $P^*$ and $N^*$ may be difficult to acquire for a complex modulus distribution, for instance, an exponential one. Therefore, we need to employ a numerical integration scheme. To construct a general computation framework for identifying the first bifurcation point, we will introduce our solution procedure first and then illustrate the bifurcation results.\n\n\\subsection{Solution strategy}\nWithout loss of generality, we apply the exponential modulus to exhibit the solution procedure, and all symbolic calculations are carried out in $Mathematica$ \\citep{math2019}. This selection gives rise to that exponential functions are involved in the integrands of (\\ref{eq2_14}) and (\\ref{eq2_15}). So explicit primitive functions of these two integrations become difficult to obtain. We shall resort to the built-in command $NIntegrate$ in $Mathematica$. In the case of $N_0=0$, the two unknowns $\\lambda_a$ and $\\lambda_z$ can be identified simultaneously according to $(\\ref{eq2_17})_1$. In order to generate effective numerical scheme, we define some necessary quantities as follows\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&P_1(\\lambda,\\lambda_a,\\lambda_z)=\\dfrac{w^*_{,1}}{1-\\lambda^2\\lambda_z},\\\\&\nN_1(\\lambda_a,\\lambda_z)=\\pi(A^*)^2(\\lambda^2_a\\lambda_z-1),~~\nN_2(\\lambda,\\lambda_a,\\lambda_z)=\\lambda\\dfrac{2\\lambda_zw^*_{,2}-\\lambda w^*_{,1}}{(\\lambda^2\\lambda_z-1)^2}.\n\\end{aligned}\n\\right.\\label{eq3_2}\n\\end{equation}\nBy use of the above new notations, we rewrite $P^*$ and $N^*$ as\n\\begin{align}\nP^*=\\int_{\\lambda_b}^{\\lambda_a}P_1(\\lambda,\\lambda_a,\\lambda_z)\\textrm{d}\\lambda,~~N^*=N_1(\\lambda_a,\\lambda_z)\\int_{\\lambda_b}^{\\lambda_a}N_2(\\lambda,\\lambda_a,\\lambda_z)\\textrm{d}\\lambda.\\label{eq3_3}\n\\end{align}\n\nIt is known that the bifurcation condition $J(P^*,N^*)=0$ actually describes a curve in the $(\\lambda_a,\\lambda_z)$-plane. Similarly, the loading condition $N^*=0$ provides another curve in the same plane. So, the intersection point corresponds to the first bifurcation point where localized bulging occurs. It is found from equation (\\ref{eq2_12}) that $\\lambda_b$ is a function of $\\lambda_a$ and $\\lambda_z$. According to the derivative of integral with variable bounds, we arrive at \n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\dfrac{\\partial P^*}{\\partial \\lambda_a}=\\int_{\\lambda_b}^{\\lambda_a}\\dfrac{\\partial P_1}{\\partial \\lambda_a}\\textrm{d}\\lambda+P_1(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_a}-P_1(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_b}\\dfrac{\\partial \\lambda_b}{\\partial \\lambda_a}, \\\\&\n\\dfrac{\\partial P^*}{\\partial \\lambda_z}=\\int_{\\lambda_b}^{\\lambda_a}\\dfrac{\\partial P_1}{\\partial \\lambda_z}\\textrm{d}\\lambda-P_1(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_b}\\dfrac{\\partial \\lambda_b}{\\partial \\lambda_z},\\\\&\n\\dfrac{\\partial N^*}{\\partial \\lambda_a}=\\dfrac{\\partial N_1}{\\partial \\lambda_a}\\int_{\\lambda_b}^{\\lambda_a}N_2\\textrm{d}\\lambda+N_1\\left(\\int_{\\lambda_b}^{\\lambda_a}\\dfrac{\\partial N_2}{\\partial \\lambda_a}\\textrm{d}\\lambda+N_2(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_a}-N_2(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_b}\\dfrac{\\partial \\lambda_b}{\\partial \\lambda_a}\\right), \\\\&\n\\dfrac{\\partial N^*}{\\partial \\lambda_z}=\\dfrac{\\partial N_1}{\\partial \\lambda_z}\\int_{\\lambda_b}^{\\lambda_a}N_2\\textrm{d}\\lambda+N_1\\left(\\int_{\\lambda_b}^{\\lambda_a}\\dfrac{\\partial N_2}{\\partial \\lambda_z}\\textrm{d}\\lambda-N_2(\\lambda,\\lambda_a,\\lambda_z)\\Big |_{\\lambda=\\lambda_b}\\dfrac{\\partial \\lambda_b}{\\partial \\lambda_z}\\right).\n\\end{aligned}\n\\right.\\label{eq3_4}\n\\end{equation}\nAlthough all integrals in (\\ref{eq3_4}) can not be solved analytically, we emphasize that it still formulates the theoretical foundation of our numerical scheme. If all geometric and material parameters are specified, we could numerically determine the implicit function of $\\lambda_z(\\lambda_a)$ according to $J(P^*, N^*)=0$. \n\nNext, we briefly summarize the solution procedure by specifying $A^*=0.4$, $\\beta_1=30$, $\\zeta=30$ and $J_m=97.2$. This parametric setting denotes a thick grade tube where the inner surface is stiffer. We first give an initial value $\\lambda_z=1.001$ and seek a solution to $J(P^*, N^*)=0$ using Newton's iteration method. In doing so, a pair of solution $\\{\\lambda_z=1.001,\\lambda_a=2.026\\}$ can be obtained. We then adopt a loop algorithm starting from $\\lambda_z=1.001$ with a small incremental step. It is well-known that the classical Newton's method for solving nonlinear algebraic equations is sensitive to the initial guess. Especially for complex problems, a proper initial guess can make the solution approach more efficient. Under the assumption that the curve of $J(P^*, N^*)=0$ is smooth, a nice initial guess can be found around the neighborhood of the solution of the previous step. By doing so, the relation between $\\lambda_a$ and $\\lambda_z$ can be found. We emphasize that this solution procedure can be applied to another nonlinear algebraic equation $N^*=0$, which offers the second implicit function of $\\lambda_z$ and $\\lambda_z$. For simplicity, extra computation details are omitted. In the case of fixed axial length, the value $\\lambda_z=1.5$ supplies a horizontal line in the $(\\lambda_a,\\lambda_z)$-plane. We supply the graphical interpretation of the solution strategy for the bulge initiation. Currently, determination of the first bifurcation point is equivalent to identifying the intersection of $N^*=0$ and $J(P^*,N^*)=0$ or the counterpart of $\\lambda_z=1.5$ and $J(P^*,N^*)=0$, as shown in Figure \\ref{fig3}. For the prescribed parameters given earlier, we find that $\\lambda_z^c=1.182$ and $\\lambda_a^c=1.932$ if $N^*=0$ while $\\lambda_a^c=1.852$ for $\\lambda_z=1.5$.\n\nSo far, we have narrated the solution strategy for identifying the bulge initiation for graded soft tubes of arbitrary thickness. Especially, it is a numerical-scheme-based framework. Although we only employ the Gent model, this framework also supplies a calculation platform for a general material model and a generic modulus distribution. In the following analysis, we shall apply the solution procedure to the three different modulus functions in (\\ref{eq2_19}) and aim to elucidate the influence of modulus gradient as well as the position where maximum modulus attains on the bulge initiation.\n \n\\begin{figure}[!htbp]\n\\centering\n\\subfigure[The resultant axial force is fixed.]{\\includegraphics[scale=0.78]{fig3a}{\\label{fig3a}}}\\hspace{5mm}\n\\subfigure[The axial length is fixed.]{\\includegraphics[scale=0.78]{fig3b}{\\label{fig3b}}}\n\\caption{Contour plots of the bifurcation condition and the corresponding loading curves ($N^*=0$ or $\\lambda_z=1.5$) when the exponential modulus gradient $(\\ref{eq2_19})_2$ is used, and the parameters are given by $A^*=0.4$, $\\beta_1=30$, $\\zeta=30$ and $J_m=97.2$. The intersections indicate the first bifurcation point resulting in localized bulging.}\\label{fig3}\n\\end{figure}\n\n\\subsection{LTI: Fixed axial force}\nFirst of all, we consider the case of fixed axial force. The aim is to unravel the effect of different geometrical and material parameters and various modulus gradients on the bulge initiation. It is emphasized that there are at most four free parameters in the graded Gent model, videlicet, the dimensionless inner radius $A^*$, the ratio of shear modulus $\\beta_1$ or $\\beta_2$, the $J_m$ and the decay rate $\\zeta$ in $(\\ref{eq2_19})_2$. As introduced before, the last two parameters will be fixed by $J_m=97.2$ and $\\zeta=30$ in all illustrative examples, and their influence is beyond the scope of this study. As a result, there are two independent parameters $A^*$ and $\\beta_1$ (or $\\beta_2$). The former measures the thickness of the graded tube while the latter displays the maximum modulus variation. In addition to these parameters, another important factor is the modulus distribution. Therefore, we shall assign a representative value of $\\beta_1$ or $\\beta_2$ and exhibit the results for different inner radii and repeat this procedure when $A^*$ turns into the fixed parameter.\n\nApplying the solution strategy outlined in the preceding subsection, the dependences of the critical stretch $\\lambda_a^c$ on the dimensionless inner radius $A^*$ are shown in Figure \\ref{fig4}. The modulus ratios are set to be $5$  in Figure \\ref{fig4a} and $30$ in Figure \\ref{fig4b}, respectively. It can be seen that the critical stretch $\\lambda_a^c$ always decreases monotonically with increased $A^*$. Note that varying $A^*$ is equivalent to altering the thickness of the tube. This implies the fact that localized bulging is invariably easier to take place in a thinner tube, regardless of the modulus distribution. The curves for the sinusoidal function are always higher, leading to a greater critical stretch for a given $A^*$ and further a more stable structure compared to the other two modulus distributions. We observe that the curves for the sinusoidal function in both figures are almost identical. It is therefore concluded that the modulus ratio $\\beta_2$ has a weak influence on the critical bifurcation stretch $\\lambda_a^c$. Furthermore, the dashed line is higher than the solid one in \\ref{fig4a} while becomes lower in \\ref{fig4b}. This indicates that a graded tube with the shear modulus decaying exponentially from the inner surface is more stable than the counterpart where the shear modulus decays linearly when $\\beta_1=5$. However, if $\\beta_1=30$, a reverse inference can be obtained. Ultimately, as $A^*$ goes to unity, the deviations among all curves almost vanish, so modulus gradient can be ignored in extremely thin tubes.\n\n\\begin{figure}[!htbp]\n\\centering\n\\subfigure[$\\beta_1=\\beta_2=5$.]{\\includegraphics[scale=0.78]{fig4a}{\\label{fig4a}}}\\hspace{5mm}\n\\subfigure[$\\beta_1=\\beta_2=30$.]{\\includegraphics[scale=0.78]{fig4b}{\\label{fig4b}}}\n\\caption{The critical stretch $\\lambda_a^c$ versus the dimensionless inner radius $A^*$ when the three modulus functions in (\\ref{eq2_19}) are adopted. The resultant axial force is given by $N^*=0$, and the other parameters are given by $\\zeta=30$ and $J_m=97,2$.}\\label{fig4}\n\\end{figure}\n\nThe above analysis primarily reveals the complexity of the influence of modulus gradient on the bifurcation threshold. In the following study, the modulus ratios $\\beta_1$ and $\\beta_2$ are taken into account to explore the impact of the modulus ratio as well as the modulus distribution. Figure \\ref{fig5} displays the relations between $\\lambda_a^c$ and the modulus ratio for three modulus gradients by specifying $A^*=0.7$. The right figure highlights the detail of the left one when the modulus ratio is around unity. Seen from Figure \\ref{fig5}, we find that the critical stretch approaches a limit value for all modulus functions as $\\beta_1$ or $\\beta_2$ increases. If the shear modulus has a sinusoidal distribution, the bulge initiation is nearly independent of the modulus ratio $\\beta_2$, especially when $\\beta_2>1$. Yet the critical stretch is always a decreasing function for the other two modulus distributions. Specifically, if the shear modulus grows exponentially from the inner surface ($\\beta_1<1$), the bifurcation threshold only varies in a very small range compared to the linearly increasing counterpart, and a linear distribution always delays the occurrence of localized bulging. Nevertheless, when $\\beta_1>1$, the bifurcation threshold decreases fast and starts to be impervious to the variation of $\\beta_1$ as $\\beta_1$ approaches practically 30 when the linear function $(\\ref{eq2_19})_1$ is selected.  On the one hand, it is observed from figure \\ref{fig5} that these three curves connect at $\\beta_1=\\beta_2=1$ where a graded tube is reduced to a homogeneous one. On the other hand, the solid line and dashed line intersect at another point and we define the corresponding horizontal coordinate as $\\beta_c$. It can be solved that $\\beta_c=12.78$ for $A^*=0.7$. Accordingly, the exponential distribution retards the bifurcation for $1<\\beta_1<\\beta_c$ while diminishes the bulge initiation for $\\beta>\\beta_c$. In other words, compared to a linearly decayed shear modulus from the inner surface, an exponential counterpart gives rise to a more stable structure as $\\beta$ is lower than $\\beta_c$ while resulting in a less stable structure otherwise. \n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.78]{fig5}\\caption{The critical stretch $\\lambda_a^c$ versus the ratio of modulus when the three modulus functions in (\\ref{eq2_19}) are adopted. (a) The modulus ratio ranges from 0.01 to 101 and (b) the modulus ratio ranges from 0.01 to 1. The right figure is a blowup of the right figure when the modulus ratio is small. The resultant axial force is given by $N^*=0$, and the other parameters are given by $A^*=0.7$, $\\zeta=30$ and $J_m=97,2$.}\\label{fig5}\n\\end{figure}\n\nTo verify that the main characteristic in Figure \\ref{fig5} is irrelevant to the special choice $A^*=0.7$, we further present the dependence of $\\lambda^c_a$ on the modulus ratio for three modulus functions in Figure \\ref{fig6} by considering a thicker tube where $A^*=0.5$. Likewise, the right part exhibits a blowup when the modulus ratio is around unity. It can be seen that Figure \\ref{fig6} differs from Figure \\ref{fig5} in the range of the vertical axis. In addition to this small divergence, these two figures are qualitatively the same. So similar conclusion can be drawn and we identify $\\beta_c=11.22$ for $A^*=0.5$. \n\nIn summary, if other conditions remain identical and only the thickness can be varied, a higher thickness always delays bulge formation in a pressurized graded tube. However, the effect of the modulus gradient is quite distinctive. Although we only select three prototypical modulus functions as shown in (\\ref{eq2_19}), the bifurcation threshold versus the modulus experiences various tendencies. When the shear modulus either grows monotonically or declines monotonically from the inner surface, the critical stretch $\\lambda_a^c$ is constantly a decreasing function as $\\beta_1$ increases and will attain a constant value for large enough $\\beta_1$. This implies that the bifurcation threshold remains practically unaffected when the modulus mismatch between the inner and outer surfaces is fairly high. Furthermore, these two curves intersect at $\\beta_1=1$ and $\\beta_1=\\beta_c>1$, leading to a transition zone $1<\\beta_1<\\beta_c$ where a graded tube with shear modulus distributing exponentially is more stable. Yet a non-monotonically sinusoidal distribution of the shear modulus leads to another situation. The curve of $\\lambda_a^c$ versus $\\beta_c$ is slowly increasing if $\\beta_2<1$ and becomes approximately a horizontal line once $\\beta_2$ passes the value of unity. Specifically, the critical stretch $\\lambda_a^c$ is practically identical to that of the homogeneous counterpart. As a result, this special graded tube has similar deformation behavior to its homogeneous analogue. We mention that a graded tube with the shear modulus distributed sinusoidally can be viewed as an effectively continuous simulacrum of a sandwich tube. Thus, we note that a sandwich tube with a hard internal core can resist the internal pressure inducing aneurysm formation without influence on the critical stretch. From the viewpoint of a structure optimization, such a composition reduces material cost but increases the critical pressure giving rise to aneurysm formation. Bearing in mind that human arteries are composed of a sandwich structure where the core (intermediate layer) has the largest elastic modulus \\citep{je2000,jctr2012}, our theoretical prediction agrees well with the natural evolution and optimization of human arteries.\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.78]{fig6}\\caption{The critical stretch $\\lambda_a^c$ versus the ratio of modulus when the three modulus functions in (\\ref{eq2_19}) are adopted. (a) The modulus ratio ranges from 0.01 to 101 and (b) the modulus ratio ranges from 0.01 to 1. The right figure is a blowup of the right figure when the modulus ratio is small. The resultant axial force is given by $N^*=0$, and the other parameters are given by $A^*=0.5$, $\\zeta=30$ and $J_m=97,2$.}\\label{fig6}\n\\end{figure}\n\n\n\\subsection{LTII: Fixed axial length}\nSecondly, we address the other commonly adopted loading condition where the axial length is prescribed by stretching a rubber tube and then fixing two ends. A noteworthy feature in LTII is that the curve of pressure against stretch $\\lambda_a$ or volume ratio $\\lambda_a^2\\lambda_z$ may lose $N$-shape and further may be a monotonically increasing function. Consequently, the limit-point theory fails to identify the bifurcation point, and we resort to the bifurcation condition derived by \\cite{fu2016}. \n\nPrevious studies of localized bulging in inflated bilayer tubes have indicated that the effect of material and geometrical parameters on the bifurcation threshold directing to localized bulging has no essential distinction \\citep{liu2019,ye2019}. Therefore, we anticipate that similar bifurcation results as that in LTI will be observed here. To this end, we specify the modulus ratio and plot the dependence of $\\lambda_a^c$ on the dimensionless inner radius $A^*$ in Figure \\ref{fig7} while Figure \\ref{fig8} displays $\\lambda_a^c$ against the modulus ratio $\\beta_1$ or $\\beta_2$. As explained earlier, the axial stretch is given by $\\lambda_z=1.5$ in all illustrative examples. For comparison, Figure \\ref{fig7a} shows the outcomes for three modulus functions when $\\beta_1=\\beta_2=5$ while Figure \\ref{fig7b} exhibits the counterparts when $\\beta_1=\\beta_2=30$. As expected, the curve for the sinusoidal function is the highest. In addition, the curve for a linear function is lower than the one for exponential function as $\\beta_1=5$ but becomes higher when $\\beta_1$ is transferred to 30. These features are consistent with those in Figure \\ref{fig4}, so we omit more detailed descriptions.\n\nThen we explore the influence of modulus ratio depicted in Figure \\ref{fig8}. It can be seen that the tendencies and shapes of all curves are consistent with those in Figures \\ref{fig5} and \\ref{fig6}. Similarly, we determine the second intersect of the solid line and the dashed line as $\\beta_c=13.01$. \n\\begin{figure}[!htbp]\n\\centering\n\\subfigure[$\\beta_1=\\beta_2=5$.]{\\includegraphics[scale=0.78]{fig7a}{\\label{fig7a}}}\\hspace{5mm}\n\\subfigure[$\\beta_1=\\beta_2=30$.]{\\includegraphics[scale=0.78]{fig7b}{\\label{fig7b}}}\n\\caption{The critical stretch $\\lambda_a^c$ versus the dimensionless inner radius $A^*$ when the three modulus functions in (\\ref{eq2_19}) are adopted. The axial stretch is given by $\\lambda_z=1.5$, and the other parameters are given by $\\zeta=30$ and $J_m=97,2$.}\\label{fig7}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.78]{fig8}\\caption{The critical stretch $\\lambda_a^c$ versus the ratio of modulus when the three modulus functions in (\\ref{eq2_19}) are adopted. (a) The modulus ratio ranges from 0.01 to 101 and (b) the modulus ratio ranges from 0.01 to 1. The right figure is a blowup of the right figure when the modulus ratio is small. The axial stretch is given by $\\lambda_z=1.5$, and the other parameters are given by $A^*=0.7$, $\\zeta=30$ and $J_m=97,2$.}\\label{fig8}\n\\end{figure}\n\nAt present, the influence of the tube thickness, the shear modulus ratio, and the modulus distribution on bulge initiation is revealed by the use of the bifurcation condition $J(P^*, N^*)=0$ and based on the solution procedure established at the beginning of this section. We emphasize that in practical applications not only the bulge formation but also the bulge evolution is of great interest because material rapture often occurs in the growth or propagation stage. However, theoretically characterizing the solution path for bulge growth is extremely difficult. Only weakly nonlinear analysis can be carried out for thick tubes in the framework of finite elasticity \\citep{ye2020}. Unfortunately, the information of weakly nonlinear behavior has limited guidance in the bulge growth stage since the bifurcation nature of localized bulging is subcritical, and experimentally observed amplitude is always finite. To acquire more knowledge of bulge growth in inflated graded tubes, we shall establish a finite element model in commercial software Abaqus in the next section.\n\n\\section{Finite element model}\nIn this section, we pursue constructing a finite element (FE) model in Abaqus to conduct a post-buckling analysis of localized bulging in a pressurized graded tube. Although the incompressible Gent model has not been built in the software, one could write UHYPER subroutine codes following the user guideline \\citep{abaqus}. In our previous studies \\citep{liu2019,ye2019}, homogeneous Gent model was supplemented into Abaqus employing UHYPER subroutine coding. Another important issue is to model modulus gradient. In practice, we can discretize the tube into many sub-tubes in the thickness direction and each sub-tube is composed of a homogeneous Gent material where the shear modulus can be determined according to the given modulus function. This is equivalent to approximating a continuous modulus function by a staircase one. As the mesh size tends to zero, the deformation behavior of a layered tube would perfectly coincide with that of the original graded one. Based on this idea, pattern transition induced by volumetric growth in graded structures was investigated using FE analysis based on this idea \\cite{liu2020}. Moreover, another more convenient way is to make use of the linearly temperature-dependent elastic modulus option in Abaqus \\citep{prsa2014,chen2018}. In this alternative approach, the temperature distribution will exactly conform to the modulus distribution and the thermal expansion coefficient must be set to be zero. \n\nIn principle, both approaches can be applied to simulate the deformation and instability of a graded tube under the combined action of internal pressure and axial extension in Abaqus. Notwithstanding, it should be pointed out that the former method may need more grids to guarantee the quality of meshes and may further require additional computation costs. So we employ the latter method and write the UHYPER subroutine codes in Fortran for the graded Gent model by creating a connection between the shear modulus and the temperature field. Ultimately, a graded Gent model with arbitrary modulus function is well established in Abaqus. In doing so, we can carry out FE simulations for the localized bulging.\n\nBefore proceeding further, we shall check the validity of the FE model. For that purpose, we calculate the bifurcation threshold using FE analysis and compare the results with the corresponding theoretical predictions. To eliminate the end effects of a grade tube with finite length, we construct all FE models by specifying the length to overall diameter by 30, or equivalently $L\/B=60$. Furthermore, the built-in command ``Buckle\" can not be applied to solve the eigenvalue problem when the critical mode is zero, and we utilize the Module ``Static, Riks\" to perform a fully nonlinear analysis. \n\nTo trigger localized bulging, certain structures or physical imperfections need to be involved in the FE model. In the case of fixed axial force, we refer to the practical end conditions in an inflation experiment where both the two ends are fastened and the radial movement is confined \\citep{wang2019}. So in the FE model, the radial displacements on the two ends are restricted and this can be regarded as a geometrical imperfection. For fixed axial length, the shear modulus at the center $Z=0$ is assigned to $\\mu(R)-0.001$ if $\\beta_1$ (or $\\beta_2$) is less than 30 or otherwise a greater magnitude $\\mu(R)-0.01$ shall be used. In addition, A  quadratic 3D hybrid element with reduced integration (C3D20RH) is employed in all subsequent FE simulations.\n\nFor convenience, the FE model is validated by taking the linear modulus function $(\\ref{eq2_19})_1$ as an example, and all comparisons are summarized in Figure \\ref{fig9}. The theoretical solutions are denoted by solid lines while the FE results are represented by red dots. The resultant axial force is specified by $N^*=0$ in Figures \\ref{fig9a} and \\ref{fig9b} and the axial stretch is fixed by $\\lambda_z=1.5$ in Figures \\ref{fig9c} and \\ref{fig9d}, respectively. It is found that the critical stretch based on FEA agrees well with that from the theoretical model. This implies the credibility and robustness of the established FE model and the UHYPER subroutine codes for the graded Gent model.\n\n\\begin{figure}[!htbp]\n\\centering\n\\subfigure[The modulus ratio is given by $\\beta_1=30$.]{\\includegraphics[scale=0.78]{fig9a}{\\label{fig9a}}}\\hspace{5mm}\n\\subfigure[The inner radius is given by $A^*=0.7$.]{\\includegraphics[scale=0.78]{fig9b}{\\label{fig9b}}}\n\\subfigure[The modulus ratio is given by $\\beta_1=30$.]{\\includegraphics[scale=0.78]{fig9c}{\\label{fig9c}}}\\hspace{5mm}\n\\subfigure[The inner radius is given by $A^*=0.7$.]{\\includegraphics[scale=0.78]{fig9d}{\\label{fig9d}}}\n\\caption{(Color online) Comparisons of the critical stretch $\\lambda_a^c$ between the theoretical predictions and the FE results when the shear modulus varies linearly. The resultant axial force is fixed by $N^*=0$ in top subfigures while the axial stretch is given by $\\lambda_z=1.5$ in the bottom subfigures. The red dots denote the FE solutions and the solid lines correspond to the theoretical ones.}\\label{fig9}\n\\end{figure}\n\nNow, a robust FE model for simulating localized bulging in pressurized graded tubes has been formulated. In particular, the modulus gradient in the FE model can be arbitrary. So the FE model paves a convenient way to implement a fully nonlinear analysis of bulge evolution. In the next section, both bulge growth and bulge propagation will be investigated for a graded tube.\n\n\\section{Bulge growth and propagation}\nIt is known that a complete process of localized bulging in pressurized tubes contains three typical stages, consisting of bulge initiation, bulge growth, and bulge propagation. When the internal pressure passes a critical value, a bulge profile appears. With increased pressure, the diameter of the bulge grows until it reaches a maximum. Then we define a new parameter $\\lambda_a^m$ that depicts the largest hoop stretch at the inner surface. As a result, the $\\lambda_a^mA^*$ indicates the maximum inner radius for the bulge. Afterwards, the growing bulge ceases to expand in the radial direction and only dilates in the axial direction at constant pressure, and the instability enters into the propagation stage. We emphasize that an ideal propagation stage only exists in an infinitely long tube, so the maximum radius cannot be exactly reached in a tube of finite length.\n\nIn many actual applications, such as soft robots and flexible actuation, bulge profile is harmful to the normal function of these devices, so the parametric analysis for the onset of localized bulging can supply useful insight into bulge suppression. If localized bulging has emerged, material rupture may appear either in the growth stage or during the period of bulge propagation. In practice, the rupture risk is possibly relevant to the size of a bulge or the magnitude of circumferential stretch. For clinicians, the diameter of an aneurysm can be used to evaluate the danger of aneurysm fracture \\citep{jb2012}. It is therefore of great significance to estimate how large a bulge can finally attain. As noted earlier, human arteries are composed of three layers where the intermediate layer has the highest elastic modulus, and we can approximately take a non-monotonic modulus distribution to qualitatively elucidate the mechanism behind artery evolution. Hence, the section aims to reveal the effect of modulus ratio and material gradient on the maximum magnitude of the bulge and to provide further insight into how a normal artery is constructed to resist aneurysm formation while keeping proper elasticity.\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=1]{fig10}\\caption{(Color online) The dependence of pressure $P^*$ on volume ratio $v$ for the primary deformation subjected to inflation of a graded tube where the sinusoidal distribution $(\\ref{eq2_19})_3$ is applied. The resultant axial force is fixed by $N^*=0$ and the parameters are give by $A^*=0.7$ and $\\beta_2=30$. }\\label{fig10}\n\\end{figure}\n\nIt is pointed out that the propagation stage is a two-phase deformation and is composed of two uniform states joined by a transition zone. When the resultant axial force is fixed, the curve of pressure versus $v$ or $\\lambda_a$ presents an $N$-shape where the first stationary point corresponds to the onset of localized bulging. The $v=\\lambda_a^2\\lambda_z$ denotes the ratio of the volume enclosed in the undeformed tube to that in the inflated state. In this scenario, the pressure under which a bulge starts to propagate can be determined according to Maxwell's equal-area rule \\citep{jam1984}. To clearly illustrate this methodology, we consider a graded tube with sinusoidally varying shear modulus subjected to inflation, and the resultant axial force is fixed by $N^*=0$. Figure \\ref{fig10} sketches the relation between the dimensionless pressure $P^*$ and the volume ratio $v$ when the dimensionless inner radius $A^*$ is set to be 0.7. The marked pressure $P_k^*$ depicts the origination of bulge propagation. Maxwell's equal-area rule yields\n\\begin{align}\nP^*_k(v_1-v_0)=\\int_{v_0}^{v_1}P^*(v)\\textrm{d}v, \\label{eq5_1}\n\\end{align}\nwhere $v_0$ corresponds to a state before localized bulging and $v_1$ the counterpart after bulge has propagated two both ends. Referring to the solution strategy outlined in Section 3.1, we can also program in $Mathematica$ to identify the exact values of $P_k^*$, $v_0$ and $v_1$, and the technicality is neglected for brevity. For instance, for the parameters used in plotting Figure \\ref{fig10}, we acquire that $P_k^*=3.318$, $v_0=1.575$ and $v_1=192.87$. However, when the axial length is specified, the pressure versus $v$ or $\\lambda_a$ may become monotonic, so Maxwell's equal-area rule is no longer valid. Notwithstanding, it is naturally expected that the principal conclusion for fixed axial force can offer qualitatively guidance for other loading methods as usual.\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=1]{fig11}\\caption{Illustration of dependences of $\\lambda_a^m$ on $\\beta_1$ when the linear and exponential functions are employed or $\\beta_2$ when the sinusoidal function is applied. The resultant axial force is given by $N^*=0$ and the dimensionless inner radius $A^*$ is prescribed by $0.7$.}\\label{fig11}\n\\end{figure}\n \nIn the subsequent analysis, we shall illuminate the influence of the modulus ratio as well as the material gradient on the maximum circumferential stretch $\\lambda_a^m$ based on equation (\\ref{eq5_1}). Similar to the parametric analysis for the onset of localized bulging, we also plot in Figure \\ref{fig11} the curves for three modulus functions in (\\ref{eq2_19}) when the modulus ratio $\\beta_1$ or $\\beta_2$ ranges from 1 to 101 and the dimensionless inner radius is given by $A^*=0.7$. This implies that the shear modulus of the inner (or middle) surface is equal to or larger than that of the outer surface. Moreover, it is a wonder that the trend of each curve is identical to that in Figures \\ref{fig5} and \\ref{fig6}. Similarly, the sinusoidal distribution is highest among these curves, indicating that a localized bugle can attain a greater radius in this case compared to a linearly decayed or an exponentially decayed modulus distribution if the peak value of shear modulus and the thickness remain the same. Meanwhile, the curve of sinusoidal distribution increases slightly from $\\beta_2=1$ and soon becomes virtually flat as $\\beta_2$ is around 10. Consequently, such a special non-monotonic modulus gradient brings a negligible change in the final deformed state. Note that these curves intersect at $\\beta_1=\\beta_2=1$ where a homogeneous counterpart is obtained. Furthermore, we only focus on the linear and exponential distributions since the unique difference is the modulus gradient. It can be seen that both curves are monotonically decreasing as $\\beta_1>1$. Especially, within the domain $1<\\beta_1<\\beta_s$ where $\\beta_s=8.67$, a pressurized graded tube with the shear modulus distributed exponentially can finally undergo a larger bulge compared to the linear counterpart if the axial force is specified. On the other hand, with increased $\\beta_1$, the dependence of the final magnitude of a bulge on the modulus ratio becomes weaker and weaker until the modulus ratio has no influence.\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=1]{fig12}\\caption{(Color online) Graphical interpretation of the scaled amplitude $\\lambda_a(0)-\\lambda_\\infty$. The white horizontal line marks the axis of the tube.}\\label{fig12}\n\\end{figure}\n\nAs of now, we have performed a theoretical analysis on the limit magnitude of the bulge in a graded tube subject to combined fixed axial force and internal pressure in the light of Maxwell's equal area. An interesting discovery is that a sinusoidal modulus function does practically not alter the final size of a bulge compared to its homogeneous counterpart. Bearing in mind that a similar situation takes place for the bulge initiation, we are left to speculate what will happen in the bulge growth stage for a homogeneous tube and a graded tube where the shear modulus distributes sinusoidally. To this end, we apply the established FE model in the previous section and carry out a fully nonlinear analysis. At first, we define the amplitude of localized bulging. As illustrated in Figure \\ref{fig12}, the hoop stretch is dependent not only on the radial coordinate $R$ but also on the axial coordinate $Z$ after bulge initiation. In a perfect system, namely, the tube is infinitely long, bulge solution holds translation invariance. So we can assume that localized bulging, associated with the greatest diameter of the bulge, always appears at the center of the tube $Z=0$. Then the dimensionless radius at the inner surface and $Z=0$ writes $\\lambda_a(0)A^*$. Furthermore, we denote the corresponding radius for the uniform inflation region by $\\lambda_\\infty A^*$, where $\\lambda_\\infty$ stands for the hoop stretch at the infinity. In a uniform inflation process, the $\\lambda_a$ is independent of $Z$ hence we get $\\lambda_a=\\lambda_\\infty$. In an illustrative FE example, we identify $\\lambda_\\infty$ in the non-bulging region, as shown in Figure \\ref{fig12}. The scaled amplitude of a bulge is given by $\\lambda_a(0)-\\lambda_\\infty$. \n\nOn specifying $A^*=0.7$, we show the bifurcation diagrams for a homogeneous tube and a graded one in Figure \\ref{fig13} using FE analysis. The modulus ratio for the graded tube is given by $\\beta_2=30$, and the $\\beta_2$ is reduced to unity in a homogeneous tube. In particular, Figure \\ref{fig13a} displays the results when the resultant axial force is prescribed while Figure \\ref{fig13b} plots the counterparts when the axial length is fixed. From the bifurcation diagram, it is observed that the amplitude of bulge is trivial in the uniform inflation where $\\lambda_\\infty$ is reduced to the circumferential stretch in the primary deformation $\\lambda_a$. As the inflation continues, especially when $\\lambda_\\infty$ passes the critical value $\\lambda_a^c$, a non-trivial solution resulting in localized bulging takes place and the scaled amplitude $\\lambda_a(0)-\\lambda_\\infty$ gets a positive value. Afterwards, the amplitude increases with reduced $\\lambda_\\infty$. This implies the fact that the existing bulge dilates in both the radial and the axial directions while the non-bulging part shrinks. Remarkably, we again find an extremely good coincidence for the deformation processes between the homogeneous tube and the graded one no matter it is the resultant axial force $N^*$ or the axial stretch $\\lambda_z$ that is fixed. Combining the analysis for the bulge initiation and the maximum size of a bulge, it is therefore summarized that a graded tube with sinusoidally distributed shear modulus can undergo a higher internal pressure and remain the deformation feature at the same time. For instance, the deformation paths in Figure \\ref{fig13a} are identical while the associated critical pressures for a graded tube and a homogeneous one are $P_{cr}^*=5.248$ and $P_{cr}^*=0.271$, respectively. \n\n\\begin{figure}[!htbp]\n\\centering\n\\subfigure[The resultant axial force is fixed by $N^*=0$.]{\\includegraphics[scale=0.9]{fig13a}{\\label{fig13a}}}\\hspace{5mm}\n\\subfigure[The axial length is fixed by $\\lambda_z=1.5$.]{\\includegraphics[scale=0.9]{fig13b}{\\label{fig13b}}}\n\\caption{(Color online) Comparisons of the bifurcation diagrams between a graded tube where the shear modulus function is given by $(\\ref{eq2_19})_3$ and a homogeneous counterpart. The parameters are given by $A^*=0.7$ and $\\beta_2=30$. The black solid lines represent the results for the homogeneous tube while the red dashed lines the counterparts for the graded one. The profiles of these blue dots highlighted in the figure are displayed in Figures \\ref{fig14} and \\ref{fig15}, respectively.}\\label{fig13}\n\\end{figure}\n\nWe also highlight three pairs of $\\{\\lambda_\\infty,\\lambda_a(0)-\\lambda_\\infty\\}$ by blue dots for the graded tube in Figures \\ref{fig13a} and \\ref{fig13b}, respectively. In particular, we select a state before instability and two states after bulge initiation. The corresponding coordinates $\\{\\lambda_\\infty,\\lambda_a(0)-\\lambda_\\infty\\}$ read $\\{1.4,0\\}$, $\\{1.267, 4.004\\}$ and $\\{1.244, 5.373\\}$ from the bottom to the top in Figure \\ref{fig13a} while in Figure \\ref{fig13b} the counterparts are given by $\\{1.4,0\\}$, $\\{1.209, 3.253\\}$ and $\\{1.176, 4.771\\}$. The deformed profiles for the three highlighted points are exhibited in Figure \\ref{fig13a}. In the right part, we further plot the cross-sections enclosed in the dashed rectangular to offer a clear view of the stress distribution on the inner surface. The aspect ratio of length to diameter is 30. As introduced in Section 4, the radial displacement of two ends are restricted and this setup serves as a geometric imperfection that can be used to trigger aneurysm formation at a critical pressure. It can be seen that the end effect decays very fast and has no influence on the deformation around the center where localized bulging is expected to appear. As the stretch passes the critical stretch $\\lambda_a^c$, a bulge occurs at the center of the tube and starts to expand in the radial and axial orientations. At the same time, the inner or outer diameter at the non-bulging area (outside the dashed rectangular in Figure \\ref{fig14}) suffers a small but continuous drop, which explains why the curve in the amplitude diagram turns back when $\\lambda_\\infty$ exceeds the critical stretch $\\lambda_a^c$. From the sectional view, it is also found that the stress and strain are both concentrated on the center of the bulge, which may cause potential rapture. \n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.8]{fig14}\\caption{(Color online) Deformed profiles of the three marked points in Figure \\ref{fig13a} for the graded tube when the axial force is fixed by $N^*=0$. (a) to (c) show the entire deformations at different periods while (d) to (e) illustrate the sectional views. The values of $\\lambda_\\infty$ and $\\lambda_a(0)-\\lambda_\\infty$ are listed below the corresponding subfigures. The parameters are given by $A^*=0.7$ and $\\beta_2=30$.}\\label{fig14}\n\\end{figure}\n\nFinally, we plot in Figure \\ref{fig15} the deformed configurations of all blue points in Figure \\ref{fig13b}. The layout is similar to that in Figure \\ref{fig14}. We emphasize that an alternative physical defect that the shear modulus at the middle cross-section $Z=0$ is reduced to $\\mu(R)-0.01$ is employed in the case of fixed axial length. In doing so, the center of the tube is slightly softer than other positions. Then localized bulging occurs at the center if the internal pressure attains a critical value. Additionally, the cross-sections in the dashed rectangular are shown to depict evidently the stress distribution on the inner surface.  \n\n\n\\begin{figure}[!htbp]\n\\centering\\includegraphics[scale=0.8]{fig15}\\caption{(Color online) Deformed profiles of the three marked points in Figure \\ref{fig13b} for the graded tube when the axial stretch is fixed by $\\lambda_z=1.5$. (a) to (c) show the entire deformations at different periods while (d) to (e) illustrate the sectional views. The values of $\\lambda_\\infty$ and $\\lambda_a(0)-\\lambda_\\infty$ are listed below the corresponding subfigures. The parameters are given by $A^*=0.7$ and $\\beta_2=30$.}\\label{fig15}\n\\end{figure}\n\n\n\\section{Concluding remarks}\nLocalized bulging or aneurysm formation in inflated graded cylindrical tubes of arbitrary thickness was investigated in detail within the framework of finite elasticity. The primary deformation where a tube dilates uniformly was analytically determined to formulate the expressions of the internal pressure and the resultant axial force. Meanwhile, a concise bifurcation condition in terms of the internal pressure and the resultant axial force has been applied and a semi-analytical solution procedure was proposed to determine the onset of localized bulging. Two typical loading conditions, defined by fixed axial force and fixed axial length, were adopted throughout the paper. In particular, the case of fixed axial force has potential applications in soft pneumatic actuators \\citep{he2020,li2022} while the other end condition corresponds to the \\textit{in vivo} status of human arteries \\citep{bmm2014}. For all illustrative examples, the incompressible Gent material was engaged and three archetypal modulus gradients were used, containing a linear, an exponential, and a sinusoidal function. Then a thorough theoretical analysis was conducted for the critical stretch $\\lambda_a^c$ at the inner surface. Note that the structure is more stable with a higher $\\lambda_a^c$ in a volume control problem. In this sense, a thicker tube is always less susceptible to suffering localized bulging regardless of a concrete form of the material gradient. Remarkably, it turns out that the critical stretch is sensitive to the specific modulus distribution applied. In particular, a sinusoidal function almost does not influence $\\lambda_a^c$, no matter what values of the ratio of the shear modulus on the middle position to that on the outer surface (denoted by $\\beta_2$). Furthermore, a linear or an exponential one will reduce $\\lambda_a^c$ as the ratio of the shear modulus on the inner surface to that on the outer surface (denoted by $\\beta_1$) increases. However, a larger enough $\\beta_1$ ceases to affect the critical stretch. It is emphasized that a qualitatively similar conclusion can be obtained for both end conditions. The current analysis on bulge initiation implies that not only the modulus mismatch but also the exact position where maximum modulus attains is vital to the initiation of localized bulging. Bearing in mind that an accurate measurement of elastic modulus of human arteries is extremely difficult \\citep{prsa2019} and we only have a general understanding that the media (intermediate layer) of an artery occupies a larger elastic modulus \\cite{je2000,jctr2012}, the current analysis concerning sinusoidally distributed shear modulus implies that the deformation is almost insensitive to the practical value of the shear modulus of the middle surface. \n\nTo track the evolution of a bulge, a finite element model incorporating material inhomogeneity is established using Abaqus UHYPER subroutine coding. This model can be used to perform nonlinear analysis for localized bulging in inflated graded tubes for an arbitrary modulus gradient. The robustness of the established FE model was validated by comparing the bulge initiations using finite element analysis to the theoretical predictions when a linear modulus gradient is applied. Ultimately, bulge propagation for the case of fixed axial force was analytically studied according to Maxwell's equal-area rule, and the influence of material gradient, as well as the modulus ratio on the maximum magnitude of a bulge, were clarified. Combing the analysis for the onset of localized bulging, it is found that either the material gradient or the modulus ratio has the same effect on the deformed profiles at the critical state and at the propagation stage. Furthermore, we compared the bifurcation diagrams between a graded tube where the shear modulus decays sinusoidally from the middle surface to both lateral boundaries and its homogeneous counterpart based on finite element simulations. Interestingly, these two bifurcation diagrams are nearly identical for both loading types. Therefore, this work provides an answer for one fundamental problem arising from structure optimization, i.e. stiffening the intermediate surface enhances the ability of a structure to resist internal pressure, and neither the critical stretch nor the deformation path varies compared with the homogeneous counterpart. This conclusion is perfectly consistent with the actual situation of human arteries. It is expected that the current analysis would supply useful insight into localization instabilities in graded structure as well as into aneurysm prevention in functional soft devices.\n\n\n\\section*{Acknowledgments}\nThe work was supported by the National Natural Science Foundation of China (Grant Nos 12072227, 12072225, and 12121002). The Abaqus simulations were carried out on TianHe-1 (A) at the National Supercomputer Center in Tianjin, China. We thank Prof. Yibin Fu at Keele University for valuable advice and discussion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum coherence arises due to the superposition principle and is an essential property of quantum systems.\nBasic features of coherence were well studied through \\cite{glauber1963coherent}, where they were investigated\nin the context of quantum optics. But these studies have focused only on the detection of quantum coherence\nwhile a method to estimate coherence was not available. A procedure to measure coherence using a quantum\ninformation theoretic framework was introduced by Baumgratz et. al. in Ref.~\\cite{baumgratz2014quantifying}.\nThis method is used to estimate the coherence of finite dimensional systems.  This led to some fundamental results in the field of resource\ntheory of quantum coherence \\cite{winter2016operational,brandao2015reversible,streltsov2017colloquium,chitambar2019quantum,chitambar2015relating}.\nSubsequently quantum coherence has been studied in  the contexts, where the system is either in a\nnon-inertial frame of reference \\cite{wang2016irreversible,he2018multipartite} or where the system is in contact\nwith an external environment \\cite{Pati2018Banerjee,chandra2019time,radhakrishnan2019dynamics,cao2020fragility}.\nBut most of these investigations have been carried out on quantum systems with finite number of degrees\nof freedom like qubits.\n\nQuantum communication requires a quantum description of the interaction and propagation of electromagnetic waves.  An infinite number of\ndegrees of freedom with continuous spectrum is needed to describe the electromagnetic waves. Hence from both theoretical and experimental\nperspective, continuous variable (CV) states \\cite{braunstein2005quantum,adesso2014continuous,weedbrook2012gaussian,wang2007quantum}\nof infinite dimensional systems are very important. Continuous variable systems constitute an extremely powerful resource to quantum\ninformation processing. One particular class of continuous variable states is the Gaussian states\n\\cite{weedbrook2012gaussian,olivares2012quantum,wang2007quantum} for which the quantum state has a representation in terms\nof Gaussian functions. To a first approximation, the ground states of the thermal states of a quantum system is a\nGaussian state. Similarly there are dynamical operations in which quantum\nstates are transformed to another Gaussian states and these are referred to as Gaussian operations. The Gaussian operations are linear in\nnature and any nonlinear operation can be approximated to a Gaussian operation to a satisfactory level of accuracy. In light of these reasons,\nthe Gaussian states occupy a special place in the study of continuous variable systems. The first quantum resource to be studied extensively\nin a Gaussian state is its entanglement. Since a covariance matrix along with the displacement vectors completely characterizes a Gaussian\nstate,  the entanglement can be quantified using a covariance matrix based approach. To measure the quantum coherence of Gaussian states\na measure based on relative entropy, using covariance matrix and displacement vectors was introduced in\nRef.~\\cite{xu2016quantifying}. The fundamental postulates of {\\it (i)} positivity {\\it (ii)} monotonicity and {\\it (iii)} convexity\nwere verified for this relative entropy measure.\n\nIn most of the investigations on discrete and continuous variable systems, we do not consider the effect of the\nexternal environment on the system. But in reality the quantum systems are exposed to an environment, which acts as a bath\nand affects the characteristics of the system by constantly interacting with it. To understand the effects of the environment and\nincorporate them in the characterization of the quantum resources, we take an open quantum system approach. Here we model\nthe environment as a quantum many-body object which is coupled to the system. Based on the strength of the coupling and the\nother characteristics of the system and the environment, there is a dynamical change in the quantum resource. Depending on\nthe bath spectrum and whether the system is weakly or strongly coupled to the environment, the system can exhibit a Markovian or\nnon-Markovian dynamics \\cite{breuer2016colloquium,de2017dynamics}. A quantum resource can completely disappear and then\nreappear a phenomenon known as revival. These features are dependent not only on the initial properties of the bath and the system\nbut also on how they are coupled to each other. Understanding this dynamical change is essential to the fabrication of quantum devices\n\\cite{shnirman2003noise,astafiev2006temperature,yoshihara2006decoherence,tong2006signatures,kakuyanagi2007dephasing,\nseoanez2007dissipation,ribeiro2015non,vyas2020master}.\nIt  provides us a clear idea of the operational time of the quantum devices, i.e., the time within which the quantum operations have to\nbe completed before the quantum resources vanish.  The time evolution of the entanglement of finite dimensional systems has been\ninvestigated in a very detailed manner and some unique features like sudden death of entanglement\n\\cite{yu2004finite, yu2006sudden,yu2009sudden} and entanglement revival\n\\cite{bellomo2007non,deng2021sudden,xu2010experimental,mazzola2009sudden}\ndue to environmental back action have been observed. For continuous variable systems, investigations on the dynamics of\nentanglement have been done  \\cite{Liu2007Goan,Zhang2007hong,Paz2008dynamics}. An open quantum system study of coherence\nhas been carried out using an atom-field interaction model \\cite{chandra2019time} and a central spin model\n\\cite{radhakrishnan2019dynamics}. The dynamics of coherence and its distribution was extensively  investigated for the different\nbipartite and tripartite states. But both these investigations were carried out on finite dimensional system. For the continuous variable\nsystems, the quantum coherence dynamics has not been investigated so far. In this work we investigate the dynamics of coherence\nof a single mode Gaussian states {\\it viz} the coherent state, the squeezed state and the displaced squeezed state.\n\nThe plan of the manuscript is as follows:  In Sec.~\\ref{states and coherence} we give a brief introduction to the continuous variable\nquantum state and explain the Gaussian states in detail.  Also we describe the covariance matrix based coherence measure introduced\nin Ref.~\\cite{xu2016quantifying}. The description of the Gaussian state in contact with a non-Markovian environment is\ndescribed in Sec.~{\\ref{singlemodeNME}.\nFor the single mode coherent states the dynamics of quantum coherence is discussed in Sec.~\\ref{coherentstates}. Next we\nstudy the effect of non-Markovian environment on squeezed states in Sec.~\\ref{squeezedstates}.  An analysis of coherence in\nthe single mode displaced squeezed state is shown in Sec.~\\ref{displacedsqueezedstate}.  Finally in Sec.~\\ref{crossovertransition},\nwe investigate the crossover from Markovian to non-Markovian dynamics.  We present our conclusions in Sec. \\ref{conclusions}.\n\n\n\\section{Gaussian States and Quantification of coherence}\n\\label{states and coherence}\n\n\\subsection{Gaussian states}\nA  continuous variable quantum system \\cite{braunstein2005quantum,adesso2014continuous} has an infinite dimensional\nHilbert space in which observables have a continuous eigenspectrum.\nAn example of a continuous-variable system is the electromagnetic field in which the quantized radiation  modes are represented by the\nbosonic modes. The tensor product Hilbert space corresponding to these modes is $\\otimes_{k=1}^{n} H_k$,  where `$n$' is the\nnumber of bosonic modes.  For a given mode `$k$' of the bosonic field, the operators  $a_k^{\\dagger}$ and $a_k$ are the corresponding\ncreation and annihilation operators.  Apart from the bosonic field operators, the continuous-variable system can also be described using\nthe quadrature operators $\\{x_k,p_k\\}$.  The quadrature field operators can be arranged in the form of a $2n$-dimensional vector as\n${\\bm \\xi} = \\left(x_1, p_1,...,x_n, p_n\\right)$.  Here the $2n$-dimensional vectors ${\\bm \\xi}$ contains the quadrature pairs of all the $n$ modes.\nIn terms of the bosonic field operators, the quadrature operators $\\{x_k,p_k\\}$ are expressed as\n\\begin{align*}\nx_k = (a_k + a_k^{\\dagger}),  \\qquad  p_k = -i (a_k - a_k^{\\dagger}),\n\\end{align*}\nand they are canonically conjugate to each other. These quadrature field operators act similar to the canonically conjugate position and\nmomentum operators of a harmonic oscillator.\n\nIn the present work we study a single-mode continuous variable system.  The two quadrature operators corresponding to the single mode\ncontinuous variable system are $\\xi_1=x=(a + a^{\\dagger})$  and $\\xi_2=p=-i (a - a^{\\dagger})$ and the 2D-vector\n${\\bm \\xi} = \\{\\xi_{1}, \\xi_{2} \\}$. The quadrature operators satisfy the canonical commutation relations\n$\\left[ \\xi_i, \\xi_j\\right]=2i \\Omega_{ij}$, where $\\Omega_{ij}$ are the elements of the matrix\n\\begin{align}\n\\bm{\\Omega} = \\left(\\begin{array}{cc}\n 0  & 1\\\\\n-1 & 0\\\\\n\\end{array}\n\\right).\n\\end{align}\nIn particular, we study the coherence dynamics of the Gaussian states. The Gaussian states \\cite{weedbrook2012gaussian,olivares2012quantum}\nare defined as the states with Gaussian Wigner function. For a Gaussian state, the Wigner quasiprobability distribution is\n\\begin{align}\n\\nonumber\nW({\\bm \\xi}) = \\frac{1}{2\\pi \\sqrt{\\rm{det}~\\bm{V}}} \\exp \\left\\{ - \\frac{1}{2} ({\\bm  \\xi - \\bm{\\overline \\xi} }).\n\\bm{V}^{-1} ({\\bm  \\xi - \\bm{\\overline \\xi} })^{T}  \\right\\}\n\\label{wigner}\n\\end{align}\nA Gaussian state is completely characterized by the first and the second moments of the quadrature field operators.  Here the\nvector of the first moments ${\\overline{\\bm \\xi}} = (\\langle \\xi_1 \\rangle, \\langle \\xi_2 \\rangle)$ and the covariance matrix\n${\\bm V}$\n\\begin{align}\nV_{ij} = \\langle \\{ \\Delta \\xi_i, \\Delta \\xi_j \\} \\rangle = {\\rm Tr} \\left( \\{ \\Delta \\xi_i, \\Delta \\xi_j \\} \\rho \\right),\n\\end{align}\nwhere $\\Delta \\xi_i=\\xi_i - \\langle \\xi_i \\rangle$ is the fluctuation operator and\n$\\{ \\Delta \\xi_i, \\Delta \\xi_j \\} = (\\Delta \\xi_i \\Delta \\xi_j + \\Delta \\xi_j \\Delta \\xi_i )\/2$.  For the single mode quantum system under\nconsideration, the matrix elements of the covariance matrix are\n\\begin{eqnarray}\nV_{ii} &=& \\langle \\xi_i^2 \\rangle - \\langle \\xi_i \\rangle^2, \\\\\nV_{ij} &=& \\frac{1}{2} \\langle \\xi_i \\xi_j + \\xi_j \\xi_i \\rangle - \\langle \\xi_i \\rangle \\langle \\xi_j \\rangle.\n\\label{covele}\n\\end{eqnarray}\nTo reflect the positivity of the density matrix, the covariance matrix has to satisfy the uncertainty relation \\cite{simon1994quantum}\n$\\bm{V} + i \\bm{\\Omega} \\ge 0$.\n\n\\subsection{Measurement of Coherence}\n\nThe quantum coherence of a qubit system is measured as follows:  First we define the set of incoherent states $\\mathcal{I}$,\ni.e., the states with zero coherence. Next we use a suitable distance measure to find the distance between the\narbitrary density matrix and the closest incoherent states. In Ref.~\\cite{baumgratz2014quantifying}, the authors used the\nrelative entropy measure to quantify the amount of coherence in the system. In general the relative entropy distance\nbetween two density matrices is\n\\begin{align}\n\\mathcal{D}(\\rho) =  \\min_{\\sigma} S(\\rho \\| \\sigma) = \\min_{\\sigma} {\\rm Tr} (\\rho \\log_{2} \\rho - \\rho \\log_{2} \\sigma),\n\\label{coher1}\n\\end{align}\nwhere $\\rho$ is the given quantum state and $\\sigma$ is the reference state.  Using this formulation we can find the\ncoherence in the system by assuming the reference state to be the incoherent state.  It has been proved that for the relative\nentropy of coherence, the closest incoherent state is $\\rho_d=\\sum_n \\rho _{nn} \\vert n \\rangle \\langle n \\vert$, which is the\ndiagonal state of the density matrix. Hence it is not necessary to perform the minimization to determine the quantum coherence.\nThe relative entropy measure reads:\n\\begin{eqnarray}\nC(\\rho) =  S(\\rho \\| \\rho_d) = S(\\rho_{d}) - S(\\rho),\n\\label{coher2}\n\\end{eqnarray}\nwhere $S(\\rho) = - {\\rm tr} (\\rho \\log_{2} \\rho)$ is the vonNeumann entropy of the system. So far, the relative entropy of coherence is the\nmost widely used measure. The quantum coherence of continuous variable system was first investigated in\n\\cite{zhang2016quantifying}. Here the authors used a relative entropy\nbased quantifier of coherence by considering the density matrix $\\rho$ in the Fock basis. While this method is a valid process\nto quantify coherence, it does not give a closed form expression for general Gaussian states. Alternatively in\nRef.~\\cite{xu2016quantifying}, the coherence measure for a one-mode Gaussian state $\\rho$ was defined as\n\\begin{align}\nC(\\rho) = \\inf_{\\delta}  S(\\rho \\| \\delta).\n\\end{align}\nHere $S(\\rho \\| \\delta)$ is the relative entropy measure with $\\delta$ being an incoherent Gaussian state\nand the infimum runs over all incoherent Gaussian states. Also, it was proved that\na one mode Gaussian state is incoherent if and only if it is a thermal state. Hence for a Gaussian state, the minimization\nis achieved for thermal states of the form\n\\begin{eqnarray}\n\\rho_d = {\\overline \\rho} = \\sum_{n=0}^{\\infty} \\frac{\\overline{n}^n}{(1+\\overline{n})^{n+1}} \\vert n \\rangle \\langle n \\vert,\n\\label{thermal}\n\\end{eqnarray}\nwhere $\\overline{n}={\\rm Tr} [a^\\dagger a {\\overline \\rho}]$ is the mean number.  For the Gaussian states, the entropy of a\ngiven state $\\rho$  can be written  as\n\\begin{align}\nS(\\rho) = \\frac{\\nu+1}{2} \\log_{2}  \\frac{\\nu+1}{2} - \\frac{\\nu-1}{2} \\log_{2}  \\frac{\\nu-1}{2},\n\\end{align}\nwhere $\\nu = \\sqrt{{\\rm det} {\\bm V}}$.  Based on these expressions, the relative entropy based coherence measure for the\nGaussian states is\n\\begin{eqnarray}\nC(\\rho) &=& S(\\rho \\| {\\overline \\rho}) ={ \\rm Tr} (\\rho \\log_2 \\rho - \\rho \\log_2 {\\overline \\rho}) \\\\\n&=& \\frac{\\nu-1}{2} \\log_2 \\frac{\\nu-1}{2} - \\frac{\\nu+1}{2} \\log_2 \\frac{\\nu+1}{2} \\nonumber \\\\\n&+& (\\overline{n}+1) \\log_2 (\\overline{n}+1) - \\overline{n} \\log_2 \\overline{n}.\n\\label{coher3}\n\\end{eqnarray}\nHence the quantum coherence of a Gaussian state is completely determined by  the determinant of the covariance matrix\nand the mean number $\\overline{n}$ associated with the Gaussian state.  But this measure can be used to compute\ncoherence of only Gaussian states. It is important to note that when the system-environment interaction Hamiltonian\nis bilinear in the creation and annihilation operators, the Gaussian nature of the quantum states is preserved\nduring the time evolution \\cite{schumaker1986quantum,olivares2012quantum}.\n\n\n\\section{Single mode Gaussian states in a Non-Markovian environment}\n\\label{singlemodeNME}\nA study of the dynamics of quantum coherence of qubit systems have been investigated both theoretically\n\\cite{Pati2018Banerjee,chandra2019time,radhakrishnan2019dynamics} and experimentally \\cite{cao2020fragility,ding2021experimental}.\nOn the theoretical side, the dynamics was characterized in atomic systems and spin systems for both Markovian\nand non-Markovian environments. Experimentally in \\cite{cao2020fragility,ding2021experimental}, the dynamics of\ncoherence, entanglement and mutual information were compared in qubit system. In the present work we consider a\ncontinuous variable system consisting  of a single bosonic mode of frequency $\\omega_{0}$ coupled to a general\nnon-Markovian environment \\cite{xiong2010exact,zhang2012general,ali2014exact,ali2015non,xiong2015non}\nat finite temperature. The single bosonic mode can be either a quantum optical field, a superconducting resonator\nor a nano-mechanical oscillator. A structured bosonic\nreservoir with a collection of infinite modes of varying frequencies can describe a general non-Markovian environment. The entire\nsystem comprising of the single bosonic mode and the reservoir can be described by the Fano-Anderson Hamiltonian\n\\cite{anderson1961localized,fano1961effects}. This Hamiltonian was introduced in the context of atomic  \\cite{fano1961effects}\nand condensed mater physics \\cite{anderson1961localized} and has been used to study several different models.\nThe Fano-Anderson Hamiltonian of the system and bath combination is:\n\\begin{align}\nH = \\hbar \\omega_0 a^{\\dagger}a + \\hbar \\sum_k \\omega_k b_k^{\\dagger}b_k\n      + \\hbar \\sum_k  \\left(\\mathcal{V}_k a^{\\dagger} b_k  + \\mathcal{V}_k^{\\ast} b_k^{\\dagger}a \\right),\n\\label{Tot}\n\\end{align}\nwhere $\\omega_{0}$ is the system frequency and $a^{\\dag}$ ($a$) is the creation (annihilation) operator\ncorresponding to the bosonic mode (system) and $b^{\\dag}_{k}$ ($b_{k}$) is the creation (annihilation) operator of\nthe $k^{th}$ mode of the bosonic reservoir with frequency $\\omega_{k}$.  The factor $\\mathcal{V}_{k}$ represents\nthe coupling strength between the system and the environment.\n\nThe dynamics of the single mode system and the environment is solved using the Heisenberg equation of motion\napproach.  The time evolution of the bosonic field operator and the  environment operators reads:\n\\begin{align}\na(t) = e^{\\frac{i H t}{\\hbar}} a e^{-\\frac{i H t}{\\hbar}},  \\quad \\quad  b_k(t) = e^{\\frac{i H t}{\\hbar}} b_k e^{-\\frac{i H t}{\\hbar}}.\n\\end{align}\nIn the Heisenberg picture, these operators satisfy the following equations of motion,\n\\begin{eqnarray}\n&  & \\frac{d}{dt} a(t) =  -\\frac{i}{\\hbar}  \\left[ a(t), H \\right] = -i \\omega_0 a(t) - i \\sum_k \\mathcal{V}_k b_k(t),  \\quad\n\\label{EOM1} \\\\\n&  & \\frac{d}{dt} b_k(t) = - \\frac{i}{\\hbar} \\left[ b_k(t), H \\right] = -i \\omega_k b_k(t) - i \\mathcal{V}_k^{\\ast} a(t).   \\quad\n\\label{EOM2}\n\\end{eqnarray}\nSolving for Eqn. (\\ref{EOM2}) for $b_{k}$ we get the solution\n\\begin{align}\nb_k(t) = b_k(0) e^{-i\\omega_k t} - i \\mathcal{V}_k^{\\ast} \\int_0^t  d\\tau~a(\\tau)~e^{-i\\omega_k(t-\\tau)}.\n\\label{EOM3}\n\\end{align}\nSubstituting Eq. (\\ref{EOM3}) in (\\ref{EOM1}) we arrive at the following quantum Langevin equation \\cite{ali2017nonequilibrium}\n\\begin{align}\n{\\dot a}(t) + i \\omega_0 a(t)  + \\int_0^t  d\\tau g(t,\\tau) a(\\tau) = - i \\sum_k \\mathcal{V}_k b_k(0) e^{-i\\omega_k t}.\n\\label{langevinequation}\n\\end{align}\nHere the integral kernel $g(t,\\tau)  = \\sum_k |\\mathcal{V}_k|^2 e^{-i\\omega_k (t-\\tau)}$ characterizes the\nnon-Markovian memory effects between the system and the environment.  For the continuous environment\nspectrum $g(t,\\tau) = \\int_0^{\\infty} d\\omega J(\\omega) e^{-i\\omega(t-\\tau)}$, where\n$J(\\omega)=\\varrho(\\omega) |\\mathcal{V}(\\omega)|^2$ is the spectral density which characterizes all the\nnon-Markovian memory of the environment on the system. Here $\\varrho(\\omega)$\nbeing the density of states of the environment and $\\omega$ is the continuously varying frequency of the bath.\nDue to the linearity of Eq. (\\ref{langevinequation}) the time evolved bosonic operator $a(t)$ can be expressed\n\\cite{de2000continuous} in terms of the initial field operators $a(0)$ and $b_{k}(0)$ of the system and the environment as\n\\begin{align}\na(t) = u(t) a(0) + f(t).\n\\label{fieldoperators}\n\\end{align}\nThe time-dependent coefficient $u(t)$ and noise operator $f(t)$ are determined by the quantum Langevin equation,\nand they satisfy the two integrodifferential equations given below:\n\\begin{eqnarray}\n\\frac{d}{dt} u(t) &=&   - i \\omega_0 u(t) - \\int_0^t d\\tau g(t,\\tau) u(\\tau),\n\\label{utime} \\\\\n\\frac{d}{dt} f(t)  &=&   -i \\omega_0 f(t)  -  \\int_0^t d\\tau g(t,\\tau) f(\\tau) \\nonumber  \\\\\n                            &   & - i \\sum_k \\mathcal{V}_k b_k(0) e^{-i\\omega_k t}.\n\\label{ftime}\n\\end{eqnarray}\nTo determine $u(t)$ we solve Eqn. (\\ref{utime}) numerically with the initial condition $u(0) =1$.\nHere the solution for the noise operator obtained using the initial condition $f(0) =0$ is\n\\begin{align}\nf(t) = - i \\sum_k \\mathcal{V}_k b_k(0) \\int_0^t d\\tau e^{-i\\omega_k \\tau} u(t,\\tau).\n\\label{ft2}\n\\end{align}\nThe nonequilibrium thermal fluctuation can be evaluated using the quantity\n\\begin{align}\n\\langle f^{\\dagger}(t) f(t)\\rangle = v(t) = \\! \\! \\int_{0}^{t}  \\! \\! \\! d\\tau_1   \\int_{0}^{t^\\prime} \\! \\! \\!   d\\tau_2 u(t,  \\tau_1)\n{\\widetilde g}(\\tau_1,\\tau_2) u^{\\ast}(t^{\\prime},  \\tau_2),\n\\label{fdf}\n\\end{align}\nwhere we consider the initial state of the total system to be $\\rho_{tot}(0)=\\rho_S(0) \\otimes \\rho_E(0)$.\nHere the initial environment state  $\\rho_E(0) = \\exp(-\\beta H_E)\/{\\rm Tr}[\\exp(-\\beta H_E)]$ is the thermal state\nfor the Hamiltonian $H_E=\\sum_k \\hbar \\omega_k b_k^{\\dagger}b_k$, where $\\beta = 1\/k_{B} T$ is the inverse\ntemperature and $k_B$ is the Boltzmann constant.\n\nWhen the environment has a continuous spectrum, the time correlation function reads:\n\\begin{align}\n{\\widetilde g}(\\tau_1,\\tau_2)  = \\int_0^{\\infty} d\\omega  J(\\omega) {\\bar n}(\\omega) e^{-i\\omega(\\tau_1-\\tau_2)},\n\\end{align}\nwhere ${\\bar n}(\\omega)=1\/(e^{\\hbar \\omega \/ k_{B} T}-1)$ is the initial particle number distribution of the\nbosonic environment.  From the Eqs. (\\ref{fieldoperators}) to (\\ref{fdf}), one can obtain the time-dependent\naverage values of the system operators as follows:\n\\begin{align}\n\\label{avg1}\n&\\langle a(t) \\rangle = u(t) \\langle a(0) \\rangle, ~~\\langle a^{\\dagger}(t) \\rangle = u^{\\ast}(t) \\langle a^{\\dagger}(0) \\rangle,  \\\\\n\\label{avg2}\n&\\langle a(t) a(t) \\rangle = (u(t))^{2}  \\langle a(0) a(0) \\rangle, \\\\\n\\label{avg3}\n&\\langle a^{\\dagger}(t) a^{\\dagger}(t)\\rangle = (u^{\\ast}(t))^{2}  \\langle a^{\\dagger}(0) a^{\\dagger}(0) \\rangle, \\\\\n\\label{avg4}\n&\\langle a^{\\dagger}(t) a(t) \\rangle =  |u(t)|^{2}  \\langle a^{\\dagger}(0) a(0) \\rangle + v(t).\n\\end{align}\nHere we consider $\\langle f(t)\\rangle=\\langle f^{\\dagger}(t)\\rangle=0$ and also\n$\\langle f(t) f(t) \\rangle=\\langle f^{\\dagger}(t) f^{\\dagger}(t)\\rangle=0$, since the reservoir is initially in a thermal state\nuncorrelated to the system.  Using the time-dependent average values in Eqs (\\ref{avg1}) -(\\ref{avg4}), we can\nevaluate the time evolved first and second moments of the quadrature operators namely,\n$\\langle \\xi_1(t) \\rangle$, $\\langle \\xi_2(t) \\rangle$, $\\langle \\xi_1^2(t) \\rangle$,\n$\\langle \\xi_2^2(t) \\rangle$, $\\langle \\xi_1(t) \\xi_2(t) \\rangle$, and $\\langle \\xi_2(t) \\xi_1(t) \\rangle$.\nThe elements of the time evolved covariance matrix are\n\\begin{eqnarray}\nV_{11} &=& 1+ 2 v(t) + 2  |u(t)|^{2} \\,  {\\rm Cov}(a^{\\dagger}(0), a(0))  \\nonumber  \\\\\n                   &   &  + (u(t))^{2} \\, {\\rm Var}(a(0)) + (u^{\\ast}(t))^{2} \\, {\\rm Var}(a^{\\dagger}(0)), \\qquad\n\\label{V11}\n\\end{eqnarray}\n\\begin{eqnarray}\nV_{22} &=& 1+ 2 v(t) + 2  |u(t)|^{2} \\,  {\\rm Cov}(a^{\\dagger}(0), a(0))  \\nonumber  \\\\\n                   &   &  - (u(t))^{2} \\, {\\rm Var}(a(0)) - (u^{\\ast}(t))^{2} \\, {\\rm Var}(a^{\\dagger}(0)), \\qquad\n\\label{V22}\n\\end{eqnarray}\n\\begin{equation}\nV_{12}  =  i  (u^{\\ast}(t))^{2} \\; {\\rm Var}(a^{\\dag} (0))  - i ((u(t))^{2}  {\\rm Var}(a(0)).\n\\end{equation}\nwhere ${\\rm Cov}(a,b) = \\langle a b \\rangle - \\langle a \\rangle \\langle b \\rangle$ and ${\\rm Var}(a) = {\\rm Cov}(a,a)$ and\n$V_{12} = V_{21}$ due to the symmetry of the covariance matrix.\n\n\nOnce the initial state and the bath parameters are known, the time evolved covariance matrix elements are completely\ndetermined using the nonequilibrium Green's function $u(t)$ and $v(t)$. To calculate the Green's function we need to\nspecify the spectral density $J(\\omega)$ of the environment. In our work we consider a Ohmic-type spectral density which\ncan simulate a large class of thermal baths \\cite{leggett1987dynamics}\n\\begin{align}\nJ(\\omega) = \\eta ~\\omega \\left( \\frac{\\omega}{\\omega_c} \\right)^{s-1} ~e^{-\\omega\/\\omega_c},\n\\label{sd}\n\\end{align}\nwhere $\\eta$ is the coupling strength between the system and the environment and $\\omega_{c}$ is the frequency cut-off\nof the environmental spectra. A localized mode is generated when the system-environment coupling approaches a\ncritical value $\\eta_{c} = \\omega_{0} \/(\\omega_{c} \\Gamma(s))$ where $\\Gamma(s)$ is the Gamma function.\nDepending on the value of $s$, the environment is classified as Ohmic for $s=1$,\nsub-Ohmic for $s<1$ and super-Ohmic for $s>1$. Throughout our work we use a scaled temperature\n$T_{s} = k_{B} T\/ \\hbar \\omega_{0}$, where $\\omega_{0}$ is the system frequency. The coherence dynamics for\ndifferent environmental conditions, and various system-environment couplings is studied for pure Gaussian\nstates through our work.  Since the Hamiltonian under consideration (\\ref{Tot}) is bilinear in the creation and annihilation operators,\nduring evolution, the Gaussian states preserve their form and remain Gaussian.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure1.pdf}\n\\caption{The time evolution of quantum coherence of a coherent state with parameter $\\alpha$ in contact with a Ohmic bath is shown for\nweakly coupled systems ($\\eta = 0.01 \\; \\eta_{c}$) in (a) at low temperature $T_{s} =1$, (b) at high temperature $T_{s} =20$ and\nfor the strongly coupled systems ($\\eta = 2.0 \\; \\eta_{c}$) in (c) at low temperature  $T_{s} =1$, (d) at high temperature $T_{s} =20$.\nWe use the cut-off frequency $\\omega_{c} = 5.0 \\; \\omega_{0}$. }\n\\label{fig1}\n\\end{figure}\n\n\n\\section{Quantum Coherence dynamics of coherent states in a Non-Markovian environment}\n\\label{coherentstates}\nGlauber \\cite{glauber1963coherent} defined a coherent state $| \\alpha \\rangle$ as the eigenstate of the annihilation operator $a$ with eigenvalue\n$\\alpha \\in \\mathbb{C}$.  We can generate the coherent state from the ground state $|0 \\rangle$ of an oscillator through the action\nof the displacement operator $ D(\\alpha)  = \\exp( \\alpha a^{\\dag}  - \\alpha^{\\ast} a)$, where $a^{\\dag}$ and $a$ are the\nannihilation and creation operators of the standard harmonic oscillator. In the number basis, the coherent states\ncan be expressed as\n\\begin{align}\n|\\alpha \\rangle = \\exp\\left( - \\frac{1}{2} |\\alpha|^{2} \\right)  \\sum_{n=0}^{\\infty}  \\frac{\\alpha^{n}}{\\sqrt{n !}}  |n \\rangle.\n\\end{align}\nwhere $|n \\rangle$ is the oscillator number basis. The coherent states are Gaussian wavepackets\nwhich does not spread and has minimal uncertainty.  Thus for a free classical mode, the closest quantum analogue is the\ncoherent state.  Hence the coherent state is very useful in quantum optics especially to study the state of the quantized\nelectromagnetic field.  The time evolution of the quantum coherence of a coherent state subjected to a spectral density of the\nform given in Eq. (\\ref{sd}) is reported in this section. Depending on the value of the parameter $s$, the environments are\nclassified as Ohmic $s=1$,  sub-Ohmic $s<1$ and super-Ohmic $s>1$.\n\n\\noindent{\\it Ohmic bath  (s=1):} \\\\\nThe time dynamics of quantum coherence of a coherent state for a Ohmic bath with spectral density\n$J(\\omega) = \\eta \\omega \\exp(- \\omega \/ \\omega_{c})$ is shown in Fig. \\ref{fig1}.  From the plots Figs \\ref{fig1} (a) - (d) we find\nthat higher the value of the coherent state parameter `$\\alpha$' higher the initial amount of coherence and also the\nvalue at a given time.   In Fig  \\ref{fig1}(a) and \\ref{fig1}(b) the time variation of coherence, when the system is weakly coupled to the\nenvironment i.e., $\\eta =0.01 \\, \\eta_{c} $ is shown.  Particularly in Fig. \\ref{fig1}(a) we look at the low temperature limit and in Fig. \\ref{fig1}(b),\nthe high temperature regime of the coherence dynamics.  From both these plots we find that quantum coherence decreases\nmonotonically with time.  But it falls faster at higher temperature because apart from the environmental effects, the\nthermal effects also contribute to the system decoherence.  The dynamics of coherence when the system is strongly coupled\nto the environment i.e., for $\\eta=2.0 \\,  \\eta_{c}$ is shown in Fig. \\ref{fig1}(c) and \\ref{fig1}(d). Here we can see that the coherence initially decreases\nmonotonically and then there is an oscillatory phase.  This oscillatory  phase is because of the environmental back action\ndue to the non-Markovian nature of the bath. For lower tempertures as shown in Fig. \\ref{fig1}(c), the coherence saturates at a\nhigher value compared to the high temperature limit given in Fig. \\ref{fig1}(d).\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure2.pdf}\n\\caption{The coherence dynamics of a coherent mode with parameter $r$ in contact with sub-Ohmic bath is studied for\nweakly coupled systems ($\\eta = 0.01 \\; \\eta_{c}$) in (a) low temperature $T_{s} =1$, (b) high temperature $T_{s} =20$.\nFor the strongly coupled systems ($\\eta = 2.0 \\; \\eta_{c}$)  the plots are (c) at low temperature $T_{s}=1$ and (d) at high temperature $T_{s}=20$.\nWe use the cut-off frequency $\\omega_{c} = 5.0 \\; \\omega_{0}$. }\n\\label{fig2}\n\\end{figure}\n\n\\noindent{\\it Sub-Ohmic bath (s=1\/2):}   \\\\\nTo investigate the coherence dynamics in the sub-Ohmic region we consider $s=1\/2$ in Eq. (\\ref{sd}).\nThe corresponding spectral density reads $J(\\omega) = \\eta \\sqrt{\\omega \\, \\omega_{c}} \\exp(- \\omega\/ \\omega_{c})$.\nThe results corresponding to this analysis is presented in Fig. \\ref{fig2}.   The time variation of\ncoherence for the weak coupling limit  with $\\eta =0.01 \\, \\eta_{c} $ is given in Fig. \\ref{fig2} (a) and (b).  In Fig. \\ref{fig2} (a) the\nlow temperature  limit is analyzed and we find that the coherence decreases monotonically with time.  At higher\ntemperatures also, coherence decreases monotonically but the fall is much faster as seen in Fig. \\ref{fig2} (b).\nThis faster fall is because at high temperatures apart from the dissipation, the thermal fluctuations also cause\na decoherence in the system. The coherence dynamics when the system is strongly coupled to the environment $\\eta=2.0 \\,  \\eta_{c}$ is\npresented in Fig. (\\ref{fig2}) (c) and (d) corresponding to the low and high temperature cases respectively.  In the\nlow temperature case illustrated in Fig. \\ref{fig2} (c), we find a revival of coherence due to the non-Markovian effects\nof the bath.  Such a coherence revival is not observed in the high temperature limit even in the strong coupling case.\nThis is because the temperature affects the environmental back action on the system, a feature which has also\nbeen observed in Ref. [].\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure3.pdf}\n\\caption{A coherent mode in contact with a super-Ohmic bath is studied for weakly interacting systems ($\\eta = 0.01 \\; \\eta_{c}$) for (a) $T_{s}=1$\nand (b) $T_{s} = 20$ and also for strongly interacting systems ($\\eta = 2.0 \\; \\eta_{c}$) with (c) $T_{s} =1$ and (d) $T_{s}=20$.  The cut-off\nfrequency used is $\\omega_{c} = 5.0 \\; \\omega_{0}$.}\n\\label{fig3}\n\\end{figure}\n\n\\noindent{\\it Super-Ohmic bath (s=3):}   \\\\\nFor the super-Ohmic bath we consider $s=3$ to investigate the dynamics of coherence. The spectral density in this case\nreads $J(\\omega) = \\eta \\, \\omega_{c} (\\omega \/ \\omega_{c})^{3} \\exp(- \\omega\/ \\omega_{c})$. The coherence evolution\nin this case is described through the plots in Fig. \\ref{fig3} (a) to (d) considering different coupling strengths and different\ntemperatures.  In  Fig. \\ref{fig3} (a) and (b) we analyze the coherence variation in the weak coupling limit when $\\eta =0.01 \\, \\eta_{c} $.\nHere we can see that the coherence decreases monotonically both in the low and high temperature limits.  Due to temperature\ndependent decoherence effects, the rate of decrease is higher in Fig \\ref{fig3} (b).   The nature of coherence variation in\nthe strong coupling limit ($\\eta=2.0 \\,  \\eta_{c}$) is examined in Fig. \\ref{fig3} (c) and (d) respectively.  In both these cases we see that\nthe coherence decreases initially and then saturates at a finite value a phenomenon known as coherence freezing.  But the\nvalue at which coherence freezes is different in Fig. \\ref{fig3} (c) and Fig. \\ref{fig3} (d) and is also dependent on the parameter\n$\\alpha$.  For a higher $\\alpha$, the coherence freezes at a higher value. The general decrease in coherence\nis due to the environmental effects.  For the same environment at different temperatures, the system experiences\ntemperature dependent decoherence effects.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure4.pdf}\n\\caption{A squeezed mode in contact with a Ohmic bath is studied for weakly interacting systems ($\\eta = 0.01 \\; \\eta_{c}$) for (a) $T_{s}=1$\nand $T_{s} =20$ and also for strongly coupled systems ($\\eta = 2.0 \\; \\eta_{c}$) for  (a) $T_{s}=1$  and $T_{s} =20$.  The cut-off frequency used is\n$\\omega_{c} = 5.0 \\, \\omega_{0}$. }\n\\label{fig4}\n\\end{figure}\n\n\\section{Non-Markovian dynamics of Squeezed states}\n\\label{squeezedstates}\nA squeezed state has a minimum value for the product of the dispersion of the position and momentum operators.\nFor a squeezed state \\cite{walls1983squeezed}, the Hamiltonian consists of terms quadratic in the creation and the\nannihilation operator. The Gaussian unitary corresponding to the one mode squeezing operator is\n\\begin{align*}\nS(r) = \\exp \\left[ r \\left(a^{2} -( a^{\\dag})^{2} \\right) \\right].\n\\end{align*}\nHere we assume $r \\in \\mathbb{R}$ for the sake of convenience.  The Bogoliubov transformation of the annihilation and creation\noperator based on the squeezing operator reads:\n\\begin{align}\nS^{\\dag} (r) a S(r) = a \\cosh r - a^{\\dag} \\sinh r,, \\nonumber \\\\\nS^{\\dag}(r) a^{\\dag} S(r) = a^{\\dag} \\cosh r  - a \\sinh r, \\nonumber\n\\end{align}\nWhen this squeezing operator is applied to a vaccum state we can generate a squeezed vaccum state\n\\begin{align*}\n|0,r \\rangle = S(r) |0 \\rangle =  \\frac{1}{\\sqrt{\\cosh r}}  \\sum_{n=0}^{\\infty}  \\frac{\\sqrt{(2n)!}}{2^{n} n!}  \\tanh^{n} r  |2n \\rangle.\n\\end{align*}\nIn this state, the variance of one of the quadrature operator is below the quantum shot noise and hence it is called a\nsqueezed state.  To compensate the squeezing in one quadrature, there is an antisqueezing in the other quadrature.\nThe dynamics of quantum coherence of the single mode squeezed state is presented in this section, for the spectral\ndensity Eq. (\\ref{sd}), considering the Ohmic, sub-Ohmic and the super-Ohmic conditions.\n\n\\noindent{\\it Ohmic bath  (s=1) :} \\\\\nThe Ohmic bath has a spectral density $J(\\omega) = \\eta \\omega \\exp(-\\omega\/\\omega_{c})$ and the dynamics of quantum\ncoherence of a single mode squeezed state is given through the plots in Fig. (\\ref{fig4}).  When the squeezed single mode\nsystem is weakly coupled ($\\eta =0.01 \\, \\eta_{c} $) to the environment, the coherence evolution is shown through Fig. \\ref{fig4}(a) and\nFig. \\ref{fig4}(b) for the high and the low temperature limits respectively.  We do not observe non-Markovian effects in both the limits.\nBut the coherence falls faster in the high temperature limit due to temperature dependent decoherence effects.  Under\nthe conditions when the system is strongly coupled to the environment ($\\eta=2.0 \\,  \\eta_{c}$), the coherence evolution is shown in\nFig. \\ref{fig4} (c) and Fig. \\ref{fig4} (d) respectively.  We observe a very small non-Markovian effects in the low temperature\nlimit.  It is absent in the high temperature limit due to the thermal decoherence effects.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure5.pdf}\n\\caption{The transient dynamics of coherence for a single squeezed mode in contact with a sub-Ohmic bath is studied for weakly\ninteracting systems ($\\eta = 0.01 \\; \\eta_{c}$) for (a) $T_{s} =1$ and (b) $T_{s} =20$ and strongly interacting systems ($\\eta = 2.0 \\; \\eta_{c}$)\n(c) $T_{s} =1$ and (d) $T_{s}=20$ and using a cut-off frequency of $\\omega_{c} = 5.0 \\; \\omega_{0}$. }\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure6.pdf}\n\\caption{For a squeezed mode in contact with a super-Ohmic bath, the coherence dynamics is studied for weakly interacting systems\n($\\eta = 0.01 \\; \\eta_{c}$) for (a) $T_{s} =1$ and (b) $T_{s} =20$ and strongly coupled systems ($\\eta = 2.0 \\; \\eta_{c}$)\n(c) $T_{s}=1$ and (d) $T_{s} =20$.  The cut-off frequency used is $\\omega_{c} = 5.0 \\; \\omega_{0}$.  }\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{figure7.pdf}\n\\caption{The temporal evolution of the quantum coherence $(C)$ as a function of time ($\\omega_{0} t$) and coupling strength\n($\\eta\/\\eta_{c}$) is given for the displaced squeezed state in contact with an Ohmic bath.  The plot is divided into four columns\nas (a) $\\alpha =0.1, r =0.1$, (b) $\\alpha =0.1, r=2.0$ (c) $\\alpha = 4.0, r=0.1$ and (d) $\\alpha =4.0, r = 2.0$.  In each column\nthere are two plots one for $T_{s} = 1$ and $T_s =20$.  The cut-off frequency used is $\\omega_{c} = 5.0 \\; \\omega_{0}$.  }\n\\label{fig7}\n\\end{figure*}\n\n\n\\noindent{\\it Sub-Ohmic bath (s=1\/2):}   \\\\\nFor a sub-Ohmic bath with spectral density, $J(\\omega) = \\eta \\sqrt{\\omega \\, \\omega_{c}} \\exp(- \\omega\/ \\omega_{c})$,\nthe coherence dynamics of the single mode squeezed state is shown via the plots in Fig. \\ref{fig5} (a) - (d).  In Fig.\n\\ref{fig5} (a) and \\ref{fig5}(b), the system is analyzed when it is weakly coupled to the bath ($\\eta=0.01 \\, \\eta_{c} $).  Here we\nsee that at high temperature the coherence falls faster due to thermal decoherence. The strong coupling effects\n($\\eta=2.0 \\,  \\eta_{c}$) are analyzed in the plots \\ref{fig5}(c) and \\ref{fig5}(d) respectively.  In the low temperature limit, there is\na small non-Markovian effect, but in the high temperature limit, there is no environmental back action. Overall,\nwe find that coherence falls faster with increase in temperature.\n\n\\noindent{\\it Super-Ohmic bath (s=3):}   \\\\\nTo study the effects of a super-Ohmic environment, we consider a bath with a spectral density\n$J(\\omega) = \\eta \\, \\omega_{c} (\\omega \/ \\omega_{c})^{3} \\exp(- \\omega\/ \\omega_{c})$.  The plots in Fig. \\ref{fig6}\n(a) - (d) show the dynamics of the quantum coherence of the squeezed single mode state.  The change of coherence\nwith time, when the system is weakly coupled to the bath ($\\eta =0.01 \\, \\eta_{c} $) is shown in Fig. \\ref{fig6} (a) and (b).  The\ncoherence falls monotonically in the weak coupling limit and the rate of fall of coherence increases with increase in\nthe temperature.  In the strong coupling limit i.e., when ($\\eta=2.0 \\,  \\eta_{c}$), the coherence initially decreases but saturates at a\nfinite value, a phenomenon known as coherence freezing.  The rate of fall of coherence and the saturation value of coherence\nis dependent on the temperature and for higher temperature the coherence saturates at a lower value.\nIn general we find that the amount of coherence is directly proportional to the squeezing parameter $r$, under all the\nenvironmental conditions.\n\n\n\\section{Quantum coherence evolution of displaced squeezed states}\n\\label{displacedsqueezedstate}\n\nA  displaced squeezed \\cite{kim1989properties} state has a general form as given below:\n\\begin{align*}\n|\\alpha, r \\rangle = D(\\alpha) S(r) |0\\rangle,\n\\end{align*}\nwhere $|0 \\rangle$ is the vaccum state and $D(\\alpha)$ and $S(r)$ are the displacement and squeezing operators\ngiven below:\n\\begin{align*}\nD(\\alpha) = \\exp(\\alpha a^{\\dag} - \\alpha^{\\ast} a);   \\quad \\quad  S(r) = \\exp(r a^{2} - r (a^{\\dag})^{2}).\n\\end{align*}\nHere $\\alpha \\in \\mathbb{C}$ and $r \\in \\mathbb{R}$ are the relevant parameters.  In this state, we investigate the transient dynamics\nof quantum coherence for different values of the displacement parameter $\\alpha$ and the squeezing parameter $r$.\nThe single mode displaced squeezed state is examined for the spectral density Eq. (\\ref{sd}), considering the Ohmic,\nsub-Ohmic and the super-Ohmic conditions.\n\n\\noindent{\\it Ohmic bath  (s=1) :} \\\\\nOhmic bath has a spectral density  $J(\\omega) = \\eta \\omega \\exp(-\\omega\/\\omega_{c})$ and the coherence dynamics\nof the single mode displaced squeezed state for this spectrum is given through the $3D$ plots in Fig. \\ref{fig7}.\nThe coherence $C(\\rho)$ being the vertical axis and the parameters $\\omega_{0}t$ and $\\eta\/\\eta_{c}$ along the orthogonal\nhorizontal axes.  The figure is split  in to four columns labelled from (a) to (d) corresponding to the four different states of the displaced\nsqueezed states. The first row gives the plots in the low temperature limit ($T_{s} = 1$) and the second row the\nhigh temperature regime ($T_{s} = 20$).\n\nIn the first column i.e., Fig. \\ref{fig7} (a), we present the dynamics for the parameter values of ($\\alpha =0.1, r =0.1$).\nBoth in the high and low temperature limit, the coherence falls monotonically to zero in a very fast manner which can\nbe referred to as coherence sudden death in analogy  with the sudden death of entanglement.  In the low temperature\nlimit the coherence revives and saturates at a finite value, which does not happen when the temperature is high.  After\nrevival in the low temperature limit, the coherence attains saturation and here it shows a non-Markovian feature that survives\nfor a long time.  When the squeezing is increased to high values ($\\alpha=0.1, r=2.0$), the coherence behavior is shown in Fig. \\ref{fig7} (b).\nHere we again find a coherence sudden death in both the low and high temperature limits.  When the displacement\nvalues are higher, the coherence is shown in Fig. \\ref{fig7} (c) and \\ref{fig7} (d) for the values ($\\alpha=4.0, r=0.1$)\nand ($\\alpha=4.0, r=2.0$) respectively.  The system showns coherence sudden death after which there is a coherence\nrevival and saturation in this limit.  The system exhibits non-Markovian behavior for the parameters after the revival.\nFrom all the plots Fig. \\ref{fig7} (a) to (d) we can see that the saturation value is higher when ($\\alpha=4.0, r=0.1$)\nand on increasing the squeezing parameter it decreases as can be seen from Fig \\ref{fig7} (d) corresponding to the parameter values\n($\\alpha=4.0, r=2.0$).\n\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{figure8.pdf}\n\\caption{The coherence dynamics of a displaced squeezed mode is shown in the figure above as a function of time\n$\\omega_{0}t$,  and $\\eta\/\\eta_{c}$ when it is in contact with a sub-Ohmic bath.  The plot is divided into four columns\n(a) $\\alpha =0.1, r =0.1$, (b) $\\alpha =0.1, r=2.0$, (c) $\\alpha = 4.0, r=0.1$, and (d) $\\alpha =4.0, r = 2.0$. In each\ncolumn there are two plots for $T_{s} =1$ and $T_{s} =20$. The cut-off frequency used is $\\omega_{c} = 5.0 \\; \\omega_{0}$.}\n\\label{fig8}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[scale=0.5]{figure9.pdf}\n\\caption{The time evolution of coherence of a displaced squeezed mode in contact with a super-Ohmic bath is shown in\nthe figure above as a funciton of $\\omega_{0}t$,  and $\\eta\/\\eta_{c}$. There are four columns in the plot\n(a) $\\alpha =0.1, r =0.1$, (b) $\\alpha =0.1, r=2.0$, (c) $\\alpha = 4.0, r=0.1$, and (d) $\\alpha =4.0, r = 2.0$.\nIn each column there are two plots for $T_{s} =1$ and $T_{s} =20$. The cut-off frequency used is\n$\\omega_{c} = 5.0 \\; \\omega_{0}$.}\n\\label{fig9}\n\\end{figure*}\n\n\\noindent{\\it Sub-Ohmic bath (s=1\/2):}   \\\\\nThe single displaced squeezed mode on being affected by a sub-Ohmic bath is given through the plots in Fig. \\ref{fig8}.\nThe corresponding spectral density is $J(\\omega) = \\eta \\sqrt{\\omega \\, \\omega_{c}} \\exp(- \\omega\/ \\omega_{c})$.\nThe $3D$ plots describing the variation of coherence given as a set of plots, where the first row represents the\nlow temperature regime ($kT=1$) and the second row representing the high temperature regime ($kT=20$).\nWe label the columns from (a) to (d) in the plots.\n\nFor the parameter values of ($\\alpha =0.1$,  $r=0.1$) and ($\\alpha =0.1$,  $r=2.0$), the coherence dynamics\nis given in Fig. \\ref{fig8} (a) and (b) respectively.  Here in the low temperature limit we observe both coherence\nsudden death and coherence revival and also a non-Markovian effect in the coherence obtained after the revival.\nMeanwhile in the high temperature limit, we observe only coherence sudden death and there is no revival.\nIn the regimes ($\\alpha =4.0$,  $r=0.1$) and ($\\alpha =4.0$,  $r=2.0$) we  see the coherence again decaying\nwithin a short interval of time as shown through the plots in Fig. \\ref{fig8} (c) and (d) respectively.  The coherence sudden\ndeath and the coherence revival is observed both in the low and high temperature limit.  Also the non-Markovian\neffects are clearly seen in these plots.  In the sub-Ohmic limit, we see coherence revivals only in the high $\\alpha$\nregion.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure10.pdf}\n\\caption{The $3$-D plots $\\frac{{\\rm d} C}{{\\rm d} \\eta_{s}}$ Vs $\\eta_{s}$ Vs $\\omega_{0} t$ for the squeezed coherent\nstate with parameters $\\alpha = 4.0, r = 0.1$ is given above.  In Fig. (a) and (c) we study the low temperature ($T_{s} =1$)\nand the high temperature limit ($T_{s} = 20$) when the state is in contact with an Ohmic bath. The low temperature ($T_{s} =1$)\nand high temperature limit ($T_{s} = 20$) is shown in Figs. (b) and (d) respectively, when the state is exposed to a  sub-Ohmic\nbath. The cut-off frequency used is $\\omega_{c} = 5.0 \\; \\omega_{0}$.}\n\\label{fig10}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figure11a.pdf}\n\\includegraphics[width=8.0 cm]{figure11b.pdf}\n\\caption{The dissipation coefficient $\\gamma(t)$ is calculated from the exact solution of $u(t)$.\nWe present here the contour plot of the dissipation coefficient $\\gamma(t)$ and its time derivative\n${\\rm d}\\gamma\/{\\rm d}t$ with varying time and coupling strength. In Fig. (a) and (b) we show\nthe transient behavior of $\\gamma(t)$ for Ohmic ($s=1$) and sub-Ohmic ($s=1\/2$) bath spectrum respectively.\nWe next plot the time derivative ${\\rm d}\\gamma\/{\\rm d}t$ with varying time and coupling strength\nwhen the system is in contact with an Ohmic bath (c) and sub-Ohmic bath (d) respectively. The cut-off frequency\n$\\omega_c = 5.0 \\omega_0$.}\n\\label{fig11}\n\\end{figure}\n\n\\noindent{\\it Super-Ohmic bath (s=3):}   \\\\\nThe quantum coherence dynamics of the single displaced squeezed mode, when it is exposed to a super-Ohmic\nenvironment is given here.  The spectral density used for this computation is\n$J(\\omega) = \\eta \\, \\omega_{c} (\\omega \/ \\omega_{c})^{3} \\exp(- \\omega\/ \\omega_{c})$.  The\nresults pertaining to this study is given as $3D$ plots in Fig. \\ref{fig9},  with the coherence $C(\\rho)$ being along\nthe vertical axis and the parameters $\\omega_{0} t$ and $\\eta\/\\eta_{c}$ along the orthogonal horizontal axes.\nThe first row represents the low temperature regime ($T_{s}=1$) and the second row the high temperature regime\n($T_{s}=20$) and the columns in the figure are labelled from (a) to (d).\n\n\\begin{figure}\n\\includegraphics[width=9.5cm]{figure12.pdf}\n\\caption{The fluctuation coefficient $\\widetilde{\\gamma}(t)$ is calculated from\nthe exact solution of $u(t)$ and $v(t)$. We present here the contour plot of the\nfluctuation coefficient $\\widetilde{\\gamma}(t)$ for Ohmic ($s=1$) spectrum in the\nlow temperature (a) $T_{s}=1$ and in the higher temperature (c) $T_{s}=20$\nrespectively. We next show the contour plot of fluctuation coefficient $\\widetilde{\\gamma}(t)$\nfor sub-Ohmic ($s=1\/2$) spectrum in the low temperature (b) $T_{s}=1$ and in the high\ntemperature (d) $T_{s}=20$. The cut-off frequency in all the cases $\\omega_c = 5.0 \\omega_0$.}\n\\label{fig12}\n\\end{figure}\n\nIn Fig. {\\ref{fig9}} (a) and (b) the coherence dynamics is given for parameter values of  ($\\alpha =0.1$,  $r=0.1$)\nand  ($\\alpha =0.1$,  $r=2.0$) respectively.  For the parameter values ($\\alpha =4.0$,  $r=0.1$) and\n ($\\alpha =4.0$,  $r=2.0$), the dynamics is give through the plots in Fig. {\\ref{fig9}} (c) and (d).  From all the\nplots we observer that the coherence decays monotonically and then attains a saturation value at which the\ncoherence freezes for the rest of the evolution.  But the fall of coherence and the value at which the coherence\nfreezes is dependent on the temperature and higher the temperature, lower the saturation value.  This is\nbecause apart from the loss of coherence due to dissipation, some of the coherence is also\nlost due to the thermal decoherence.  From Fig. \\ref{fig9} (a) we can see that initial coherence is low\nwhen both the $\\alpha$ and $r$ values are low.  On increasing either the displacement parameter $\\alpha$\nor the squeezing parameter $r$, we find that the system has a higher amount of initial coherence i.e.,\ncoherence at $t=0$ as shown in Fig. \\ref{fig9} (b) and (c).  Further, from these two cases, we also observe\nthat the saturation value is higher when $\\alpha$ is higher.  Finally in \\ref{fig9} (d) we examine the situation\nwhere both the displacement parameter ($\\alpha$) and the squeezing parameter ($r$) are higher.\nIn this case we find that the initial amount of coherence is comparable to the cases in Fig. \\ref{fig9} (b) and (c),\nbut the fall of coherence and the saturation value more closely resembles the case in  Fig. \\ref{fig9}  (c),\nwhere  ($\\alpha =4.0$,  $r=0.1$).  An increase in either the displacement parameter or the squeezing parameter\nincreases the quantum coherence in the system.\n\n\n\\section{Markovian to non-Markovian crossover}\n\\label{crossovertransition}\nThe present work considers a Gaussian state in contact with an external non-Markovian environment.  When we\nconsider a Ohmic or sub-Ohmic bath coupled to the system, we initially observe a Markovian evolution for the coherent\nstate, squeezed state and displaced squeezed state. Depending on whether the coupling is weak or strong, the\ndynamics continues to remain either Markovian or switches over to non-Markovian behavior.  In this current section\nwe examine the crossover behavior in the dynamics of a displaced squeezed state.\n\nLet us consider the evolution of a displaced squeezed state with the parameter values of $\\alpha = 4.0 $ and $r = 0.1 $.\nWhen the coupling between the bath and the system is weak, we observe that the dynamics of the system is\ncompletely Markovian  where the system experiences either a death of coherence or coherence freezing.  In\nthis case, the correlation time of the bath is much shorter than the system evolution time. The irreversible loss\nof information occurs, because the bath states are restored very quickly and so any information received from\nthe state is lost. For systems which have a stronger coupling with the bath, the initial coherence decay is Markovian\nand then there is a sudden switch where the dynamics becomes non-Markovian.  In the non-Markovian phase there\nis a back flow of information from the environment to the system. To check this behaviour, we look at Fig. \\ref{fig10},\nthe $3D$ plots of $\\frac{{\\rm d} C}{{\\rm d} \\eta_{s}}$ Vs $\\eta_{s}$ Vs $\\omega_{0} t$ in both the low temperature\n($T_{s} = 1$) and the high temperature ($T_{s} = 20$) limits.  In Fig. \\ref{fig10} (a), the plot shows ${\\rm d} C \/ {\\rm d} \\eta_{s}$\nfor the Ohmic spectrum in the low temperature limit. Here we find that initially, the slope is $-ve$ and monotonic implying a Markovian nature.\nIncreasing the coupling strength, we observe that the slope changes from $-ve$ to $+ve$ indicating a change from\nMarkovian to non-Markovian behavior. For the Ohmic bath in the low temperature regime this crossover in the behavior\nof dynamics is very abrupt and this sudden change is reminiscent of a phase transition.  In the initial phase we observe\n$\\tau_{b} \\ll \\tau_{s}$, where $\\tau_{b}$ is the relaxation time of the bath and $\\tau_{s}$ is the evolution time of the system.\nAfter the crossover (i.e., transition) the time scales are related as $\\tau_{b} \\approx \\tau_{s}$ indicative of a new phase.\nIn the low temperature limit $(T_{s} = 1)$, the sub-Ohmic regime is described in Fig. \\ref{fig10} (b).  Here there is a region\nwhere the coherence remains constant before the transition to the non-Markovian evolution.  The high temperature\n$(T_{s} = 20)$ case of the $3$D plots are shown in Fig. \\ref{fig10} (c) and (d) for the Ohmic and sub-Ohmic baths respectively.\nFrom the plots we can observe that transition region between the Markovian and non-Markovian regimes is more pronounced.\nAlso the non-Markovian effect is decreased indicating a lesser amount of environmental back flow of coherence. This\nreduced amount of back flow is due to the decoherence caused by thermal effects.\n\nThe dynamical behavior of the continuous variable system can also be studied using a quantum master equation.\nThe crossover between Markovian and non-Markovian can be perceived from the changing behavior of the\ndecay rates in the master equation. The total density matrix $\\rho_{tot}$ describing the continuous variable system and\nenvironment has a dynamics governed by the quantum evolution operator\n$\\rho_{tot}(t) = \\exp\\left( - \\frac{i}{\\hbar} H t \\right)  \\rho_{tot} (0) \\exp\\left( \\frac{i}{\\hbar} H t \\right)$.\nThe initial state of the system is considered to be $\\rho_{tot}(0) = \\rho_{s} (0) \\otimes \\rho_{E} (0) $,\nwhere $\\rho_{E}(0) = \\exp( - \\beta H_{E}) \/ ({\\rm Tr} \\exp(- \\beta H_{E}) )$  as proposed in\n\\cite{feynman1963Vernon,caldeira1983path}. Tracing over the environmental degrees of freedom we can get the\nreduced density matrix of the system. The master equation for the reduced density matrix reads:\n\\begin{eqnarray}\n\\frac{{\\rm d} \\rho(t)}{{\\rm d} t} &=& - i \\omega_{0}^{\\prime} (t) [ a^{\\dag} a, \\rho(t) ]  \\nonumber  \\\\\n                                                    &   &   + \\gamma(t) [ 2 a \\rho(t) a^{\\dag}  - a^{\\dag} a \\rho(t) - \\rho(t) a^{\\dag} a ]  \\nonumber \\\\\n                                                    &   &   + \\widetilde{\\gamma} (t) [ a \\rho(t) a^{\\dag} + a^{\\dag} \\rho(t) a - a^{\\dag} a \\rho(t)\n                                                                         - \\rho(t) a a^{\\dag} ]  \\qquad\n\\end{eqnarray}\nwhere the coefficients are\n\\begin{align*}\n \\omega_{0}^{\\prime} (t)  = {\\rm Im} \\left[ \\frac{\\dot{u}(t)}{u(t)} \\right],   \\qquad   \\gamma(t) = - {\\rm Re}  \\left[ \\frac{\\dot{u}(t)}{u(t)} \\right] \\\\\n \\widetilde{\\gamma}(t)  =  \\dot{v}(t) - 2 v(t) {\\rm Re}  \\left[ \\frac{\\dot{u}(t)}{u(t)} \\right].\n\\end{align*}\nHere $\\omega_{0}^{\\prime} (t)$ is the renormalized frequency of the single mode system and $\\gamma(t)$ and $\\widetilde{\\gamma}(t)$ are the\ndissipation and fluctuation coefficients. In Fig.~\\ref{fig11} (a) and (b), we present a contour plot of the dissipation coefficient\n$\\gamma(t)$ varying with time and coupling strength. The Ohmic case is described through the plot\nin \\ref{fig11}(a), where in the weak coupling limit $\\eta_{s} \\leq 1$  we find that the dissipation is almost a constant\nthroughout the entire evolution. For the strong coupling regime $\\eta_{s} \\geq 1$, the dissipation rate $\\gamma(t)$ increases\ninitially and then decreases. This change in the direction of the decay rate with time, indicates a backflow of information from\nthe environment to the system.  A similar behavior of the dissipation constant $\\gamma(t)$ is observed in the sub-Ohmic case\nas shown in \\ref{fig11}(b).  The reverse flow of information is more transparent from the contour plot of the slope of $\\gamma(t)$.\nWe show contour plot of the time derivative of the decay rate (${\\rm d}\\gamma\/{\\rm d}t$) for the Ohmic case in Fig.~\\ref{fig11} (c)\nand sub-Ohmic case in Fig.~\\ref{fig11} (d).  From the plot we observe that the slope of the dissipation coefficient $\\gamma(t)$,\ntransitions from being positive to negative.  This is indicative of a crossover from the Markovian to a non-Markovian regime.\n\nNext, we study the variation of the fluctuation coefficient namely $\\widetilde{\\gamma}(t)$ in Fig. \\ref{fig12}.\nWe show the contour plot of fluctuation coefficient for Ohmic spectrum in the low temperature limit ($T_{s}=1$)\nin Fig.~\\ref{fig12} (a) and for the high temperature limit ($T_{s}=20$) in Fig. \\ref{fig12} (c).  In the strong coupling\nregime  ($\\eta_{s} \\geq 1$), the fluctuation coefficient $\\widetilde{\\gamma}(t)$ rises initially and then starts decreasing\nwith time.  This changing behavior of $\\widetilde{\\gamma}(t)$ indicates the environmental back action in the strong coupling\nregime.  The contour plot of fluctuation coefficient for sub-Ohmic spectrum in the low temperature ($T_{s} = 1$)\nis shown in Fig.~\\ref{fig12} (b).  For the high temperature limit ($T_{s} = 20$) is given in Fig. \\ref{fig12} (d).  We observe a\n similar dynamical crossover of $\\widetilde{\\gamma}(t)$ for the sub-Ohmic spectral density  as well.  Hence the\n environmental back flow of information \\cite{breuer2009measure,liu2011experimental,chruscinski2014degree}\n i.e., the non-Markovian behavior can be obtained from the dynamical behavior of the dissipation and the fluctuation\n coefficients.  The dynamical crossover observed in the coherence dynamics of a single mode state,\n stands verified by the analysis carried out from the dynamics of the fluctuation and dissipation coefficients of the\n master equation of the state.\n\n\n\n\\section{Conclusion}\n\\label{conclusions}\nThe transient dynamics of quantum coherence of a single mode Gaussian state is analyzed in  the open quantum\nsystem formalism.  For this we consider a non-Markovian environment, which is a collection of infinite bosonic modes\nof varying frequencies.  The entire analysis is carried out in the finite temperature limit with arbitrary system-bath coupling\nstrength.  To measure the quantum coherence we use the relative entropy measure which estimates the distance to the\nthermal state.  Hence the quantum coherence can be completely characterized by the determinant of the covariance\nmatrix and the average of the number operator of the Gaussian states.  We solve the quantum Langevin equation for the\nbosonic mode operators to obtain the time evolved covariance matrix elements.  The time evolution of the field operators are\ndetermined by the two basic nonequilibrium Green's functions namely $u(t)$ and $v(t)$.\n\nThe time dynamics of quantum coherence is studied for three distinct experimentally  realizable Gaussian states.  The\ninvestigations have been carried out under three different environmental spectral densitites {\\it viz} Ohmic, sub-Ohmic\nand super-Ohmic spectral densitites.  When the system interacts weakly with the environment, the quantum coherence\ndecreases monotonically with time.  The coherence decay rate is relatively lower for the super-Ohmic bath when compared\nwith the Ohmic and the sub-Ohmic baths.  Also the rate of decay increases with the temperature.  In general we observe\na coherence death for the Ohmic and the sub-Ohmic baths in the short time limit.  For the super-Ohmic bath we observe coherence freezing\nwhich is a saturation of coherence at a finite value.  In the strong coupling regime, the coherence decreases initially\nand then starts increasing resulting in a revival of coherence for the Ohmic and the sub-Ohmic baths.  This is followed by\na stabilization of coherence with an oscillatory phase.  This oscillatory behavior is due to the environmental backaction\nwhich is a feature of non-Markovian dynamics in the system.  Hence in our study we observe a transition or crossover\nfrom the Markovian dynamics to the non-Markovian dynamics.  The non-Markovian memory effect helps us to restore\nquantum resources in the strong coupling limit. The rate at which the crossover in the dynamics occurs\ndepends on the parameters of the Gaussian state as well as on the environmental parameters.  To verify the existence of the crossover\nwe study the dynamical behavior of the quantum system using a master equation approach.  Here we compute the dissipation and\nfluctuation coefficients in the master equation of the reduced density matrix. Throughout the evolution, the\ndissipation rate is almost a constant in the weak coupling limit.  In the strong coupling limit, the dissipation rate\nincreases initially and then decreases, which implies a Markovian to non-Markovian transition.  A  plot of the time\nderivative of the decay rate shows that the slope of the dissipation coefficient changes from being positive to negative.\nThis is a clear indication of the crossover from the Markovian to the non-Markovian dynamics.  This crossover\nfeature is also verified from the dynamics of the fluctuation coefficient.  Thus we find that the quantum system has both\nMarkovian and non-Markovian behavior at different times and there is a dynamical crossover from the Markovian to\nthe non-Markovian regime.   This change from Markovian to non-Markovian regime is not a gradual change.\nInstead it is either a sudden change or there is a time period between the Markovian and non-Markovian regimes,\nwhere there is no dynamical change. This raises an important question as to whether this dynamical change can\nbe considered as some kind of phase transition between two different regimes with totally different relaxation time\nscales.  While the results seems to point towards such a conclusion, a more detailed study from the point of view\nof quantum phase transitions is needed to confirm and validate this, and such a study will form the scope of our future works.\nA similar observation on an abrupt change from Markovian to non-Markovian dynamics has also been made in\nRef. \\cite{pang2018abrupt}.  Here the authors study a single qubit system in contact with a single qubit environment.\nIn our work we consider a single mode Gaussian state in contact with a structured bosonic reservoir with a collection of\ninfinite modes to describe a general non-Markovian environment.  The investigations carried out in the present work can be\nexperimentally verified by measuring the quantum coherence of Gaussian systems\n\\cite{bowen2004experimental,furusawa1998unconditional,yonezawa2004demonstration,diguglielmo2007experimental}.\n\n\n\\section*{Acknowledgements}\nMd.~Manirul Ali was supported in part by the Centre for Quantum Science and Technology, Chennai Institute of Technology, India,\nvide funding number  CIT\/CQST\/2021\/RD-007.\nChandrashekar Radhakrishnan was supported in part by a seed grant from IIT Madras to the Centre for\nQuantum Information, Communication and Computing.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:Intro}\nTremendous advancements in sensor technology and data-driven machine learning have enabled exciting applications such as automatic health monitoring and autonomous cars. In many cases, the lack of data in certain regions of the domain reveals important structure. For instance, the sensors on a car driving through a parking lot might have dense observation points in 3D except inside pillars. Such \\emph{voids} in the data are ubiquitous across applications whether it is a subspace of unattainable configurations for a robot~\\cite{farber2018configuration}, regions without network coverage~\\cite{Ghrist2005} or missing measurements in an experimentally-determined chemical structure~\\cite{townsend2020representation}, to name a few~\\cite{aktas2019persistence}.  \n\nA standard approach to extract structural information from data proceeds by first encoding pairwise relationships in a problem via a graph and then analysing its properties.  Recent advances in Graph Neural Networks have enabled practical learning in the domain of graphs and have provided approximate solutions to difficult graph  problems~\\cite{hamilton2017representation}. Despite the wealth of techniques underpinned by solid graph theory, this approach is fundamentally  misaligned with problems where the relationships involve multiple points, and topological \\& geometric structure must be encoded beyond pairwise interactions.\n\nFortunately, higher dimensional combinatorial structures come to the rescue in the form of simplicial complexes, the powerhorse of topological data analysis~\\cite{chazal2017introduction}. Interfacing between combinatorics and geometry, simplicial complexes capture multi-scale relationships and facilitate the passage from local structure to global invariant features. These features occur in the form of homology groups, intuitively perceived as {\\em holes}, or {\\em voids}, in any desired dimension. Alas, this expressive power comes with a burden, that of high computational complexity, and difficulty in localization of said voids~\\cite{chen2011hardness}.\n\nFor every hard computational problem there seems to exist a neural network approximation~\\cite{xu2018powerful}. Nevertheless, homology and simplicial complexes have only recently started to follow suit~\\cite{bodnar2021weisfeilerMPNNS, ebli2020simplicial,bunch2020simplicial}, with inference {\\em of} homological information still lacking.\n\nThe key insight in this paper is to guide learning on simplicial complexes by flipping the conventional view of approximation. We propose a GNN model for localizing homological information in the form of a distance function of each point of a complex to its the nearest homology generating features. \nInstead of using the most-significant eigenvectors of the relevant Laplacian operator we focus on the subspace spanned by the eigenvectors corresponding to the lowest eigenvalues. The justification is that homology-related information is contained in its null space. We implement this idea by calculating the most-significant subspace of an inverted version of the operator (see Sec.~\\ref{subsec:Ltp}). Figure~\\ref{fig:diff} shows the result of twelve diffusion iterations performed using the conventional view and compares it with our inverted operator. Although diffusion is insufficient to localise homology, it highlights the tendency of the inverted operator to localize cycles. \n\nThe main contributions in the paper are:\n\\begin{enumerate}\n    \\item A novel way to represent simplicial complexes as computational graphs suitable for inference via GNNs, the {\\em Hodge Laplacian graphs}, focusing on the dimension of interest (see Section~\\ref{subsec:SCGNN}), and\n    \\item A new homology-aware graph convolution framework operating on the proposed Hodge Laplacian graphs, taking advantage of the spectral properties of a shifted-inverted version of the Hodge Laplacian (see Section~\\ref{subsec:SFGNN}, Eq.~\\eqref{eq:SFGC}).\n\\end{enumerate}\n\nThe rest of the paper is structured as follows: in Section~\\ref{sec:Prelims} all the necessary theoretical background is presented, followed by relevant work, in Section~\\ref{sec:Related}. We describe our proposed model in detail in Section~\\ref{sec:Model}, followed by thorough evaluation and discussion, in Section~\\ref{sec:Experiments}.\n\n\\newcommand{0.11}{0.11}\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/2D_ones_init\/double_torus_2D_L1_12_steps.png}\n         \\caption{$\\hL_1$}\n         \\label{fig:hL1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/2D_ones_init\/double_torus_2D_L1-k3_12_steps.png}\n         \\caption{$\\hL_1$ (rank-3)}\n         \\label{fig:hL1_k3}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/2D_ones_init\/double_torus_2D_shiftedPinv_12_steps.png}\n         \\caption{$\\Ltpd{1}$}\n         \\label{fig:Ltpd1}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/2D_ones_init\/double_torus_2D_shiftedPinv-k3_12_steps.png}\n         \\caption{$\\Ltpd{1}$(rank-3)}\n         \\label{fig:Ltpd1_k3}\n     \\end{subfigure}\n    \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/3D_ones_init\/double_torus_3D_L1_12_steps.png}\n         \\caption{$\\hL_1$}\n         \\label{fig:hL1_3D}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/3D_ones_init\/double_torus_3D_L1-k5_12_steps.png}\n         \\caption{$\\hL_1$ (rank-5)}\n         \\label{fig:hL1_3D_k5}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/3D_ones_init\/double_torus_3D_shiftedPinv_12_steps.png}\n         \\caption{$\\Ltpd{1}$}\n         \\label{fig:Ltpd1_3D}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/3D_ones_init\/double_torus_3D_shiftedPinv-k5_12_steps.png}\n         \\caption{$\\Ltpd{1}$(rank-5)}\n         \\label{fig:Ltpd1_3D_k5}\n     \\end{subfigure}\n        \\caption{\\label{fig:diff}Twelve diffusion iterations based on the 1-dimensional Hodge Laplacian $\\hL_{1}$, and its shifted pseudoinverse $\\Ltpd{1}$ of a double annulus in 2D (four leftmost) and a double torus in 3D (four rightmost). \\subref{fig:hL1_k3}, \\subref{fig:Ltpd1_k3}, \\subref{fig:hL1_3D_k5}, and \\subref{fig:Ltpd1_3D_k5} are low-rank approximations of the respective Laplacians based on the top 3 and 5 eigenpairs corresponding to the largest magnitude eigenvalues. \n       \n        }\n        \\label{fig:laplacian_diffusion}\n\\end{figure*}\n\n\\section{Preliminaries}\\label{sec:Prelims}\n\\subsection{The Simplicial Laplacian Operators}\\label{subsec:homology}\n\nAn {\\em abstract simplicial complex} is a collection $K$ of subsets of a finite set $S$ satisfying two axioms: first, for each $v$ in $S$ the singleton set $\\{v\\}$ lies in $K$, and second, whenever some $\\sigma \\subset S$ lies in $K$, every subset of $\\sigma$ must also lie in $K$. The constituent subsets $\\sigma \\subset S$ which lie in $K$ are called {\\em simplices}, and the dimension of each such $\\sigma$ is one less than its cardinality, i.e., $\\dim \\sigma = |\\sigma|-1$. By far the most familiar examples of simplicial complexes are (undirected, simple) {\\em graphs}; each graph $G = (V,E)$ forms a simplicial complex whose $0$-dimensional simplices are given by the vertex set $V$ and $1$-dimensional simplices constitute the edge set $E$. The passage from graphs to simplicial complexes is motivated by the compelling desire to model phenomena beyond pairwise interactions using higher-dimensional simplices.\n\n\\subsubsection{Homology Groups}\n\nTo each directed graph $G=(V,E)$ one can associate an {\\em incidence matrix}, which is best viewed as a linear map $A:\\mathbb{R}[E] \\to \\mathbb{R}[V]$ from a real vector space spanned by edges to the vector space spanned by the vertices. The entry of $A$ in the column corresponding to a directed edge $e:v \\to v'$ and the row corresponding to a vertex $u$ is prescribed by\n\\[\nA_{u,e} = \\begin{cases} -1  & \\text{if } u = v, \\\\\n                        1 & \\text{ if } u = v', \\text{ and} \\\\\n                        0 & \\text{ otherwise.}\n        \\end{cases}.\n\\]\nWriting $r$ for the rank of $A$, the number of connected components and loops in $G$ equals $|V|-r$ and $|E|-r$, respectively. Thus, one can learn the geometry of $G$ from the linear algebraic data given by its adjacency matrix.\n\nThis linear algebraic success story admits a remarkable simplicial sequel. Fix a simplicial complex $K$ and write $K_d$ to indicate the set of all $d$-simplices in $K$. We seek linear maps $\\partial_d:\\mathbb{R}[K_d] \\to \\mathbb{R}[K_{d-1}]$ to play the role of the $d$-dimensional incidence matrices. To build these {\\em boundary operators}, one first orders the vertices in $K_0$ so that each $d$-simplex $\\sigma \\in K$ can be uniquely expressed as a list $\\sigma = [v_0,\\ldots,v_d]$ of vertices in increasing order. The desired matrix $\\partial_d$ is completely prescribed by the following action on each such $\\sigma$:\n\\begin{align}\\label{eq:sbound}\n\\partial_d(\\sigma) = \\sum_{i=0}^d (-1)^i \\cdot  \\sigma_{-d}\n\\end{align}\nwhere $\\sigma_{-d} := [v_0,\\dots,\\hat{v}_i,\\ldots,v_d]$ is the $(d-1)$-simplex obtained by removing the $i$-th vertex $v_i$ from $\\sigma$.\n\nThese higher incidence operators assemble into a sequence of vector spaces and linear maps:\n\\begin{align}\\label{eq:chcomp}\n\\xymatrix{\n\\cdots\n\\ar@{->}[r]^--{\\partial_{d+1}} \n& \\mathbb{R}[K_{d}] \\ar@{->}[r]^{\\partial_{d}} & \\mathbb{R}[K_{d-1}] \\ar@{->}[r]^--{\\partial_{d-1}} & \\cdots.\n}\n\\end{align}\nIt follows from \\eqref{eq:sbound} that for each $d > 0$ the composite $\\partial_d \\circ \\partial_{d+1}$ is the zero map, so the kernel of $\\partial_d$ contains the image of $\\partial_{d+1}$ as a subspace, $\\im\\partial_{d+1} \\subseteq \\ker \\partial_{d}$.\n\nFor each $d \\geq 0$, the $d$-th {\\bf homology group} of $K$ is the quotient vector space $\\mathcal{H}_d(K) := {\\ker \\partial_d}\/{\\im \\partial_{d+1}}$ of $k$-cycles $\\mathcal{Z}_k=\\ker \\partial_d$ by $(k+1)$-boundaries  $\\mathcal{B}_k=\\im \\partial_{d+1}$. The basis of $\\mathcal{H}_d(K)$ contains equivalence classes of $d$-dimensional {\\em voids} or {\\em loops} $[g_i]$, i.e. $\\mathcal{H}_d(K)=\\text{span}\\{[g_1], \\dots, [g_k]\\}$, each $[g_i]$ describing a family of loops that cannot be contracted to a point, and cannot be continuously deformed into another family $[g_j]$, $i\\neq j$.  Consequently, the dimension of $\\mathcal{H}_d(K)$ provides us with a topological invariant, namely, the $k$-th \\textbf{betti number} $\\beta_d=\\text{rank}(\\mathcal{H}_d(K))$, which counts the number of $d$-dimensional voids in $K$.\n\nEach $d$-cycle $g \\in \\mathcal{Z}_d$ is a formal sum of $d$-simplices satisfying $\\partial_d (g)=0$. By assigning weights $w:K_d \\rightarrow \\mathbb{R}_+$ to these simplices, one can thus define the length of $g$ by adding together weights of its constituent simplices, i.e., $\\text{len}(g)=\\sum_{\\sigma \\in g} w(\\sigma)$. An {\\em optimal} homology basis is the one whose generators have minimum length among all possible bases. \n\nReplacing each matrix $\\partial_d$ in \\eqref{eq:chcomp} by its transpose $\\partial_d^T$, one similarly obtains the $d$-th {\\bf cohomology group} of $K$, denoted $\\mathcal{H}^d(K;\\mathbb{R})$. It is a straightforward consequence of the rank-nullity theorem that there are isomorphisms $\\mathcal{H}_d(K;\\mathbb{R}) \\cong \\mathcal{H}^d(K;\\mathbb{R})$ between homology and cohomology groups.  \n\n\n\n\\subsubsection{Hodge Laplacians}\n\n{\\em Hodge Laplacians} \\cite{horak2013spectra} are to graph Laplacians what simplicial boundary operators are to adjacency matrices.  Given a simplicial complex $K$ and the corresponding sequence \\eqref{eq:chcomp}, both composites $\\mathcal{L}_d^\\text{up} := \\partial_{d+1} \\partial_{d+1}^T$ and $\\mathcal{L}_d^\\text{down} := \\partial_d^T \\partial_d$ furnish linear maps $\\mathbb{R}[K_d] \\to \\mathbb{R}[K_d]$. The $d$-th Hodge Laplacian is their sum:\n\\begin{align}\\label{eq:hodgelapl}\n\\mathcal{L}_d:=\\mathcal{L}_d^\\text{up}+\\mathcal{L}_d^\\text{down}.\n\\end{align}\n\nAn immediate consequence of this definition is that the standard graph Laplacian agrees with the 0-th Hodge Laplacian. The nullity of the graph Laplacian  equals the number of connected components of the underlying graph. Similarly, the kernel of the $d$-th Hodge Laplacian of a simplicial complex $K$ is isomorphic the corresponding $d$-th homology group~\\cite{eckmann1944harmonische}:\n\\begin{align}\n    \\ker \\mathcal{L}_d(K)\\cong \\mathcal{H}_d(K;\\mathbb{R}).\n\\end{align}\n\n\n\\subsection{Graph Neural Networks}\n\n {\\em Graph Neural Networks} (GNNs) provide a general framework for {\\em Geometric Deep Learning}~\\cite{bronstein2021geometric}, where the input domain, represented by a {\\em graph} $G=(V,E)$, is allowed to vary together with the signals that are defined on it. More concretely, the {\\em Message Passing Graph Neural Network} (MPGNN) framework generalizes the convolution operation on the edges of a graph $G$ by employing a simple message passing scheme between features of nodes $h_u, u\\in V$, and their neighbors $v\\in \\mathcal{N}_u$. \n \nThe output of each layer $\\ell$ for each node $u$ can be broadly formulated as:\n\\begin{align}\\label{eq:gnn}\nh_u^{\\ell+1}=\\phi \\left( h_u^\\ell, \\bigoplus_{v\\in \\mathcal{N}_u} w_{u,v} \\cdot \\psi \\left(h_u^\\ell,h_v^\\ell\\right) \\right),\n\\end{align}\nwith $\\bigoplus$ being a {\\em permutation invariant} aggregation, $\\phi$ and $\\psi$ learnable functions,  and $w_{u,v}$ the weight of edge $(u,v)\\in E$. Under this formulation, learnable parameters of $\\phi$ and $\\psi$ are shared across all nodes in the graph network. \n\nEach message passing layer can be described more compactly using matrix notation:\n\\begin{align}\n    H^{\\ell+1}=\\phi \\left( \\tilde{L}\\psi(H^\\ell) \\right),\n\\end{align}\nwhere $\\tilde{L}=AWA^T$ is the weighted graph Laplacian matrix, and $H^0$ the $|V| \\times F$ matrix of initial node features. This formulation highlights the similarities of GNNs with Laplacian diffusion operations, a fact that we will largely exploit.\n\nThe output of a number of message passing iterations results to latent {\\em node embeddings}, largely based on the local graph topology at each node. Such embeddings can be subsequently used for node regression, node classification, or, via feature aggregation of all nodes, for graph classification and aggregation tasks.\n\n\n\\section{Related work}\\label{sec:Related}\n\n\\subsubsection{Homology Localization} \n\n\nThe {\\em minimum basis problem} in computational topology involves extracting optimal homology generators, with optimality usually expressed in terms of norm or length minimization of cycles. In dimensions exceeding one, this is an NP-hard problem~\\cite{chambers2009minimum,chen2011hardness}, whereas the 1-dimensional case succumbs to a polynomial time algorithm~\\cite{dey2010approximating}. This latter fact that spawned a significant body of work examining special cases and computational improvements~\\cite{borradaile2016minimum, chen2010measuring, dey2011optimal,erickson2005greedy, Busaryev2011,chen2021decomposition}. \n\nWhile the aforementioned methods generally output sets of simplices that form optimal homology generators in their respective class, the rest of the simplices in the complex remain largely oblivious to the location of such optimal cycles in relation to themselves. In our work we attempt to characterize each simplex in the complex with respect to its nearest homology generator, while gaining in efficiency (once the model is sufficiently trained). More similar to our line of work,~\\cite{Ebli2019harmonicclustering} implements a homology-aware clustering method for point data.\n\n\n\\subsubsection{Topological methods in ML}\n\n\nWith the marriage of homology and ML~\\cite{Hensel2021TopMLsurvey, love2021topDL, montufar2020can, Hofer2019LearningBarcodes}, it did not take long for GNNs to meet their higher dimensional counterparts in the form of simplicial~\\cite{bodnar2021weisfeilerMPNNS, ebli2020simplicial,bunch2020simplicial}, cell~\\cite{hajij2021cell, bodnar2021weisfeilerCWNs}, hypergraph~\\cite{feng2019hypergraph}, and sheaf~\\cite{hansen2020sheaf} neural networks. Most higher dimensional extensions of GNNs aim to operate on the full complex, and redefine the convolution operation in terms of the corresponding Laplacian operator. Contrary to such generalizations, we still operate on a graph. The key difference is that our graph is derived from adjacency and Hodge Laplacian information of the complex at the dimension of interest.\n\n\n\\subsubsection{Pseudoinverse \\& Hodge Laplacians in GNNs}\n\nThe pseudoinverse and shifted versions of the Laplacian operator are not new in the context of GNNs~\\cite{klicpera2019diffusion,wu2019simplifying,alfke2021pseudoinverse}. Nevertheless, they only consider spectral manipulations of the ``classic\" graph Laplacian, whose kernel is usually of no practical interest, as long as the graph is connected.\n\nMore closely to our work,~\\cite{roddenberry2019hodgenet, schaub2018flow} consider edge-flows for signal denoising, interpolation, and source localization based on the {\\em linegraph Laplacian}, and the 1-dimensional down Hodge Laplacian $\\hL_1^\\text{down}$, based on a proxy graph resulting from interchanging edges and nodes. Nevertheless, their analysis remains restricted on graph structures, disregarding any homological features. \n\n\n\n\n\\section{Dist2Cycle}\\label{sec:Model}\n\n\nHere we present a model for learning homology-aware distance functions on simplicial complexes.\n\n\\subsection{Shifted Inverted Hodge Laplacians} \\label{subsec:Ltp}\n\nThe spectral properties of the Hodge Laplacian matrices provide salient information regarding the geometry and topology of a simplicial complex, as hinted in Section~\\ref{sec:Prelims}. Furthermore, Laplacian flow dynamical systems on simplicial complexes tend to stabilize towards specific spectral regions of the Laplacian~\\cite{muhammad2006control}. Nevertheless, the choice of the Laplacian operator with which diffusion is performed impacts greatly the energy distribution on the simplices of interest. \n\nIn Figures~\\ref{fig:laplacian_diffusion}\\subref{fig:hL1},\\subref{fig:hL1_k3},\\subref{fig:hL1_3D},\\subref{fig:hL1_3D_k5} we perform 12 diffusion steps according to the Hodge Laplacian $\\hL_1$ (and its low-rank approximation using the top 3 and 5 eigenpairs, respectively), namely,\n\\begin{align} \\label{eq:diffusion}\n\\bm{x}_{i+1} =\\bm{x}_i + \\hL_1 \\bm{x}_i,\n\\end{align} \nwith $\\bm{x}_0=[1\\ 1\\ \\dots\\ 1]^T$ as the initial signal on the 1-simplices of the complex. We show the absolute value of the resulting flow vector at each simplex, on two basic examples. Energy tends to concentrate at well connected simplices, while ignoring the homological features that we are interested in.\n\nThe Laplacian diffusion of~\\eqref{eq:diffusion} can be seen as a simplified version of a graph convolution that takes place in GNNs~\\eqref{eq:gnn}, with all nonlinearities and learnable parameters pruned, and trivial initial features. Thus, in order to focus our attention on optimal homology generators, we must invert the spectrum of the Hodge Laplacian $\\hL_1$, while making sure that its kernel will replace the part of the spectrum corresponding to its top eigenvalues. For this purpose we employ a {\\em shifted inverted} version of the Hodge Laplacian\n\\begin{align*}\n\\Ltpd{d}=(\\mathbb{1}+\\hL_{d})^{+},\n\\end{align*}\nwhich makes the $\\ker \\hL_d$ the prominent part of the spectrum of $\\Ltpd{d}$ with eigenvalue 1, onto which the diffusion asymptotically converges. The effect this modified Laplacian matrix has on diffusion is shown in \\ref{fig:laplacian_diffusion}\\subref{fig:Ltpd1},\\subref{fig:Ltpd1_k3},\\subref{fig:Ltpd1_3D},\\subref{fig:Ltpd1_3D_k5}.\n\n\n\n\\begin{figure*}[htbp]\n    \\centering\n    \\begin{tabular}{@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}}\n       \n        \\multicolumn{8}{c}{{Ours}} \\\\\n         \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/cropped\/BEST_pts_140_filtV_1_3883159230772615_cid_6_cmplx_ours.png} &          \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/pts_200_filtV_0_014619974523328298_cid_27_cmplx_ours.png} &          \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/pts_180_filtV_0_07363109339738495_cid_3_cmplx_ours.png} &          \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/cropped\/WORST_pts_70_filtV_0_0023980842642273175_cid_0_cmplx_ours.png} &             \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/BEST_pts_120_filtV_0_018349590565120224_cid_20_cmplx_ours.png} &          \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/pts_130_filtV_0_0770273102594478_cid_49_cmplx_ours.png} &          \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/pts_120_filtV_0_5535622674081626_cid_1_cmplx_ours.png} &          \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/WORST_pts_120_filtV_0_014305282858018277_cid_7_cmplx_ours.png}\\\\\n       \n       \n         \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/cropped\/BEST_pts_140_filtV_1_3883159230772615_cid_6_cmplx_ref.png} &         \\includegraphics[,width=0.11\\textwidth]{images\/RESOUT\/2D\/pts_200_filtV_0_014619974523328298_cid_27_cmplx_ref.png} &         \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/pts_180_filtV_0_07363109339738495_cid_3_cmplx_ref.png}  &         \\includegraphics[width=0.11\\textwidth]{images\/RESOUT\/2D\/cropped\/WORST_pts_70_filtV_0_0023980842642273175_cid_0_cmplx_ref.png} &\n       \n         \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/BEST_pts_120_filtV_0_018349590565120224_cid_20_cmplx_ref.png} &         \\includegraphics[,width=0.15\\textwidth]{images\/RESOUT\/3D\/pts_130_filtV_0_0770273102594478_cid_49_cmplx_ref.png} &         \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/pts_120_filtV_0_5535622674081626_cid_1_cmplx_ref.png}  &         \\includegraphics[width=0.15\\textwidth]{images\/RESOUT\/3D\/WORST_pts_120_filtV_0_014305282858018277_cid_7_cmplx_ref.png}\\\\\n             \\multicolumn{8}{c}{{Reference}} \n       \n    \\end{tabular}\n\\caption{Qualitative comparisons of selected complexes from the 2D \\texttt{TORI} (four left) and 3D (four right) test set. The 1-simplices of the complexes are color-coded according to their distance from the nearest homology cycle, with blue indicating close proximity to an optimal homology cycle, and red indicating large distance from a homology cycle. }    \\label{fig:complviz}\n\\end{figure*}\n \n \n\\subsection{From Simplicial Complexes to Graph Neural Networks}\\label{subsec:SCGNN}\n\\newcommand{0.11}{0.11}\n\\begin{figure}[h!]\n     \\centering\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/Laplacian_graph\/complex.png}\n         \\caption{Example complex $K$.}\n         \\label{fig:complex}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/Laplacian_graph\/complex_Ldown.png}\n         \\caption{Laplacian graph $G_{\\hL_1^\\text{down}}$.}\n         \\label{fig:complex_Ldown}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/Laplacian_graph\/complex_Lup.png}\n         \\caption{Laplacian graph $G_{\\hL_1^\\text{up}}$.}\n         \\label{fig:complex_Lup}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.11\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/Laplacian_graph\/complex_L.png}\n         \\caption{Laplacian graph $G_{\\hL_1}$.}\n         \\label{fig:complex_L}\n     \\end{subfigure}\n     \\caption{Laplacian graph constructions on example complex $K$ for 1-simplices (square nodes with red edges overlaid on top of the original complex).}\n     \\label{fig:laplacian_graphs}\n\\end{figure}\n\nIn order to employ the GNN framework for inference on the complex, we need to express the space of $k$-simplices accordingly. Furthermore, we desire the resulting graph structure to retain the spectral properties of the Laplacian operator of interest. \n\nWe interpret the $|K_d|\\times |K_d|$ Hodge Laplacian operator $\\hL_d$ (or $\\Ltpd{d}$) as the weighted graph Laplacian $\\tilde{L}_\\text{GNN}=AWA^T$ of a computational graph $G_\\text{GNN}=(V_\\text{GNN}, E_\\text{GNN})$. Under this lens, each $d$-simplex $\\sigma$ of the original complex $K$ becomes a node of $G_\\text{GNN}$, i.e. $\\sigma \\in V_\\text{GNN}$, and weighted edges are drawn according to the adjacency information encoded in $\\hL_d$ ($\\Ltpd{d})$. Namely, a weighted edge $(\\sigma, \\tau)\\in E_\\text{GNN}$ is drawn between nodes corresponding to $k$-simplices $\\sigma$ and $\\tau$, whenever the (potentially normalized) Laplacian operator $\\hL_d$ ($\\Ltpd{d}$) contains a nonzero entry in the respective position. This entry is also used to weight the corresponding edge $w_{\\sigma, \\tau}=\\hL_{\\sigma,\\tau}$, allowing self-loops. Figure~\\ref{fig:laplacian_graphs} provides an example of the resulting graph, what we call the {\\em Hodge Laplacian graph}, when using $\\hL_1^\\text{down}$, $\\hL_1^\\text{up}$, and $\\hL_1$ for extracting adjacency relations on 1-simplices (sans self-loops, for easier vizualization). A somewhat similar approach is followed in~\\cite{roddenberry2019hodgenet}, with their mapping akin to the graph in Figure~\\ref{fig:complex_Ldown} minus the 2-simplices, as they are only dealing with graphs.\n\n\nAs mentioned in Section~\\ref{subsec:Ltp}, we are interested in capturing the spectrum, and thus the connectivity information of $\\Ltp$, which is in general a dense matrix and hence computationally prohibitive to work with directly.\nTo overcome this issue, we impose the sparsity structure of $\\hL$ to $\\Ltpd{}$, masking all entries of $\\Ltpd{}$ that are zero in the original, sparse, Hodge Laplacian $\\hL$. If we denote by $A$ the adjacency matrix encoding the connectivity of $\\hL$, with\n\\[\nA_{u,v} = \\begin{cases} 1 & \\text{ if } \\hL_{u,v}\\neq 0, \\text{ and} \\\\\n                        0 & \\text{ otherwise}\n        \\end{cases},\n\\]\nthis can be achieved with the Hadamard product $A \\odot \\Ltp$. The resulting graph is called the {\\bf Hodge Laplacian graph} throughout this paper.\n\n\nWhile more sophisticated methods for spectral sparsification exist~\\cite{spielman2011graph}, imposing the connectivity dictated by $\\hL$ or $\\hL^\\text{down}$ seems to preserve all important adjacency information required for the task at hand, while not annihilating important spectral information. Furthermore, in the context of learning, this approach is reminiscent to inference with missing values, which GNNs are known to handle well~\\cite{Jiaxuan2020missing}\n\n\n\\begin{figure*}[htbp]\n    \\centering\n    \\begin{tabular}{@{}c@{}@{}c@{}@{}c@{}@{}c@{}@{}}\n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/2D\/cropped\/BEST_pts_140_filtV_1_3883159230772615_cid_6_dist_inset.png} &  \n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/2D\/cropped\/pts_200_filtV_0_014619974523328298_cid_27_dist_inset.png}&\n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/2D\/cropped\/pts_180_filtV_0_07363109339738495_cid_3_dist_inset.png}&\n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/2D\/cropped\/WORST_pts_70_filtV_0_0023980842642273175_cid_0_dist_inset.png}\n        \\\\\n             \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/3D\/cropped\/BEST_pts_120_filtV_0_018349590565120224_cid_20_dist_inset.png} &  \n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/3D\/cropped\/pts_130_filtV_0_0770273102594478_cid_49_dist_inset.png}&\n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/3D\/cropped\/pts_120_filtV_0_5535622674081626_cid_1_dist_inset.png}&\n        \\includegraphics[width=0.245\\textwidth]{images\/RESOUT\/3D\/cropped\/WORST_pts_120_filtV_0_014305282858018277_cid_7_dist_inset.png}\n    \\end{tabular}\n    \\caption{Plots of predicted distances (blue) of the simplices sorted in increasing ground truth distance value (orange) for the Tori dataset in 2D (top) and 3D (bottom). The plots show four cases (columns) ranging from the best (left) to the worst (right). }\n    \\label{fig:complplots}\n\\end{figure*}\n \n \n\\subsection{Shifted Inverted Laplacian GNNs for Homology Localization} \\label{subsec:SFGNN}\n\nWe are now ready to propose a {\\em Simplicial Neural Network} model for homology localization. By following the construction described in Section~\\ref{subsec:SCGNN} we obtain a weighted computational graph $G_\\text{GNN}=(V_\\text{GNN}, E_\\text{GNN})$, with weights according to $\\Ltp$ and adjacency dictated by $\\hL$ (or $\\hL^\\text{down}$). Thus, graph convolution (message passing) on the $G_\\text{GNN}$ can be summarized as:\n\\begin{align} \\label{eq:SFGC}\n    H^{\\ell+1}=\\phi \\left( A \\odot \\Ltpd{d} H^\\ell W^\\ell \\right),\n\\end{align}\nwhere $A$ is the adjacency matrix describing the selected sparsification regime according to $\\hL$ (or $\\hL^\\text{down}$), $\\Ltpd{d}$ the {\\em shifted inverted Hodge Laplacian} in dimension $d$ (Section~\\ref{subsec:Ltp}), and $\\odot$ denoting the Hadamard product. The learnable weights of the model at layer $\\ell$ are denoted as $W^\\ell$, and $\\phi$ can be any activation function, such as ReLU, Sigmoid, etc. Finally, $H^\\ell$ is the $|V_\\text{GNN}|\\times F$ feature matrix having the $F$-dimensional features of each node (i.e. $d$-simplex) as rows.  \n\nTo aid the task of homology localization, we encode both local and global information at each node. Locality is incorporated by computing betti numbers $[\\beta_0, \\dots, \\beta_{d+1}]$ of the {\\em link} at each $d$-simplex $\\sigma$  --- this is the subcomplex consisting of all simplices $\\tau$ for which $\\sigma \\cap \\tau$ is empty and $\\sigma \\cup \\tau$ is a simplex in $K$. Global features manifest in the form of spectral embeddings of the $d$-simplices in the space spanned by singular vectors corresponding to the largest $k$ singular values of $\\Ltpd{d}$. Denoting the appropriately permuted singular value decomposition (SVD) of $\\Ltpd{d}$ as $\\Ltpd{d}=U\\Sigma V^T$ with $\\Sigma$ containing in its diagonal the singular values of $\\Ltp$ in descending order, the rows of the matrix $U_{1:k}$ formed by the first $k$ singular vectors constitute coordinates of the simplices in the eigenspace of $\\Ltpd{d}$. Due to the shift-invert operation of $\\Ltp$, this scheme effectively embeds the $d$-simplices in the spectral subspace corresponding to the kernel of $\\hL_{d}$, i.e. the space encoding homological information. \n\n\n\\begin{figure*}[htbp]\n     \\centering\n     \\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/2D\/2D-BEST_binned_MSEerrors_log.png}\n         \\caption{}\n         \\label{fig:2DMSE}\n     \\end{subfigure} &\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/2D\/2D-BEST_simplices_error.png}\n         \\caption{}\n         \\label{fig:2Dsimplices}\n     \\end{subfigure}&\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/2D\/2D-BEST_betti_error.png}\n         \\caption{}\n         \\label{fig:2Dbetti}\n     \\end{subfigure} &\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/2D\/2D-BEST_maxCycle_error.png}\n         \\caption{}\n         \\label{fig:2DmaxCycle}\n     \\end{subfigure} \\\\\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/3D\/3D-BEST_binned_MSEerrors_log.png}\n         \\caption{}\n         \\label{fig:3DMSE}\n     \\end{subfigure} &\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/3D\/3D-BEST_simplices_error.png}\n         \\caption{}\n         \\label{fig:3Dsimplices}\n     \\end{subfigure} &\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/3D\/3D-BEST_betti_error.png}\n         \\caption{}\n         \\label{fig:3Dbetti}\n     \\end{subfigure} &\n     \\hfill\n     \\begin{subfigure}[b]{0.248\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/RESOUT\/3D\/3D-BEST_maxCycle_error.png}\n         \\caption{}\n         \\label{fig:3DmaxCycle}\n     \\end{subfigure}\n     \\end{tabular}\n     \n     \n     \\caption{MSE error plots for the \\texttt{TORI} dataset in 2D (top) and 3D (bottom). The first column shows MSE error aggregates against the stratified distance values (x-axis). Green dashed line: mean, red line: median, box limits: quartiles, whiskers: error range. Columns 2, 3 and 4 are plots of MSE against the number of simplices, the homology rank count $\\beta_1$, and the maximum cycle length, respectively. Shaded regions are standard deviation, and insets provide histograms for the respective parameters in the test set.}\n     \\label{fig:2D3Derrors}\n\\end{figure*}\n\n\\section{Evaluation}\\label{sec:Experiments}\nIn this section we describe the function learned by our model, the dataset we developed to train and evaluate our model (Section~\\ref{subsec:dataset}), the experiments conducted (Section~\\ref{subsec:setting}) and an analysis of the results (Section~\\ref{subsec:results}). \n\n\\subsection{Nearest optimal homology generator}\n\nOur model approximates a function that depends on the optimal homology generators. To validate our model, we only consider  1-simplices since efficient algorithms for ground truth data only exist for $d=1$. We  denote by $\\mathcal{Q}_1$ the set of optimal generators of $\\mathcal{H}_1$. For each simplex $\\sigma \\in K_1$ we seek to learn its distance from the nearest optimal $g\\in \\mathcal{Q}_1$,\n\\begin{align}\\label{eq:distTocycle}\n    f(\\sigma)=\\min_{g\\in \\mathcal{Q}_1} \\hat{d}(\\sigma,g).\n\\end{align}\nAs distance $d(\\sigma, g)$ between a $k$-simplex $\\sigma$ and a $k$-dimensional homology generator we consider the minimum number of $k$-simplices required to reach any $k$-simplex $\\rho \\in g$ participating in a $k$-cycle $g$. To keep the function complex-independent, we then normalize the distance in the range $[0,1]$, obtaining $\\hat{d}(\\cdot)$, with simplices near an optimal homology generator attaining values close to zero.\n\n \n\\subsection{\\texttt{TORI} Dataset}\\label{subsec:dataset}\nOur \\texttt{TORI} datasets consist of {\\em Alpha} complexes~\\cite{edelsbrunner2010alpha} that originate from considering ``snapshots\" of filtrations~\\cite{edelsbrunner2010computational} on points sampled from tori manifolds, in 2 and 3 dimensions.We seek to capture richness of homological information,  controlability in terms of scalability in the number of simplices and homology cycles, as well as ease of visualization.\n\nWe first sampled 400 point clouds from randomly generated configurations of tori and pinched tori, with number of ``holes\" ranging from 1 to 5, to which Gaussian noise is added. We then constructed Alpha filtrations on the collection of point clouds, i.e. sequences of simplicial complexes dictated by a monotonically increasing distance parameter $\\alpha$. Tracking homological changes in the sequence of complexes results in a {\\em barcode} representation, with one bar per homology feature, that spans a range of $\\alpha$ values. The longer the bar, the more persistent, and possibly ``significant\", a homological feature is. From these barcodes we considered the 5 most persistent features, expressed by the longest bars. The birth and death values of these features were deemed as appropriate points to capture ``snapshots\" of the complexes, guaranteed to contain interesting large and small scale homological information. This pipeline yields a collection of 2000 complexes for each dataset. The generality of the datasets stems from the spurious homological features occuring while considering filtrations of noisy point clouds, as confirmed by the histogram insets describing the test sets in Figure~\\ref{fig:2D3Derrors}. The number of simplices ranges from tens to thousands, and betti numbers, i.e. number of homology cycles, from 0 up to 66. Furthermore, homology generators present in the dataset can contain from 3 up to 60 1-simplices.\n\n\nWe calculate the reference function using optimal homology generators of $\\mathcal{H}_1$ via {\\em Shortloop}~\\cite{dey2010approximating} by calculating the normalized ``hop\" distance of each simplex to its nearest optimal generator(see Eq.\\eqref{eq:distTocycle}).\n\n\n\\subsection{Experimental settings}\\label{subsec:setting}\n\nWe used a GNN with 12 graph convolutional layers (for 2D as well as 3D), as described by Eq.~\\eqref{eq:SFGC}, and 128 hidden units. We chose LeakyReLU activations ($\\phi$ in Eq.~\\eqref{eq:gnn}) with negative slope $r=0.02$ for the layers, and a hyperbolic tangent Tanh for the output. Neighbor activations are aggregated via a summation ($\\bigoplus$ in Eq.~\\eqref{eq:gnn}). Learnable weights undergo Kaiming uniform initialization~\\cite{he2015delving}. Finally, node features are the result of concatenating the betti numbers describing the homology of the link at each simplex, with its 5-dimensional spectral embedding.\n\nThe dataset is split into training (80\\%) and testing (20\\%) sets and the models were trained for 1000 epochs, with a mini-batch size of 5 complexes using an Intel Xeon E5-2630 v.4 processor, a TITAN-X 64GB GPU and 64GB of RAM, using CUDA 10.1. The GNN model was implemented using the dgl library~\\cite{wang2019dgl} with the Torch backend~\\cite{paszke2017automatic}. All simplicial and homology computations were handled by the Gudhi library~\\cite{gudhi:urm}. \n\nWe apply a Laplacian smoothing post-processing step. Let $\\bm{x}$ the output of the model, i.e. the inferred distances for each 1-simplex, and $\\hat{L}=D^{-1\/2}(D-A)D^{-1\/2}$ the normalized graph Laplacian of the 1-skeleton of the complex $K$, i.e. the underlying graph spanned by the 0 and 1-simplices of $K$. The signal at the simplices are smoothed using \n$\\bm{x}'=\\bm{x}-\\hat{L}\\bm{x}$.\n\n \n\\subsection{Results}\\label{subsec:results}\n\nThe main quantitative results can be found in Figure~\\ref{fig:complplots} and Figure~\\ref{fig:2D3Derrors} with qualitative examples in Figure~\\ref{fig:complviz}. We report mean squared error (MSE) between the predicted and reference relative distances. Since the range of $\\hat{d}(\\cdot)$ is within $[0,1]$, MSE can never exceed $1$, i.e. $100\\%$ error.\n\nIn 2D as well as 3D  our model learns the homology-parametrized distance function by achieving an MSE of $3.81\\% \\ (0.0381)$ (2D), and $4.47\\% \\ (0.0447)$ (3D), respectively. Figures~\\ref{fig:2DMSE} and~\\ref{fig:3DMSE} provide insights into the distribution of error at different distances (x-axis) from optimal generators. In 2D the error is monotonically increasing with the distance from the homology generators,  whereas in 3D the model performs slightly better at relative distances of about a third from the optimal cycles. In both settings, areas far away from the homology cycles attain the maximum mean MSE but never in excess of 15\\%.\n\n\\Cref{fig:2Dsimplices,fig:2Dbetti,fig:2DmaxCycle}, and \\labelcref{fig:3Dsimplices,fig:3Dbetti,fig:3DmaxCycle} investigate the scalability of the model as the number of simplices, homology features ($\\beta_1$) and maximum cycle lengths, increase. The insets provide histograms of the respective parameter counts in the test sets to shed light into the standard deviation (shaded). The parameter values are non-uniformly represented in the test sets, partly explaining the larger variance towards the lower ends of the value ranges. Our model scales well in all three parameters and we observe that the error  decreases for larger numbers of simplices. The maximum MSE across all parameters never exceeded 15\\%.  \n\nQualitative assessment of the model's performance is provided in Figure~\\ref{fig:complviz} for examples from the test sets. The comparisons are arranged from lowest error (left) to maximum error (right) within our dataset.  The top row of figures visualize the predicted distances projected onto the complex while the bottom shows the ground truth. Cool areas indicate close proximity to an optimal homology generator. \n\nFigure~\\ref{fig:complplots} plots reference distance values (orange)  and our model's output (blue) for each simplex against the rank of the simplex's distance from an optimal generator (X axis). Ideally, the blue curve should be monotonically increasing and should closely match the orange curve.\n\nBoth in 2D and 3D the model can handle multiple homology cycles of various lengths, even greater than the number of convolutional layers that usually dictate the receptive field of each simplex. Problematic appear to be the cases where components of the complexes have trivial homology, evident from the rightmost column in Figure~\\ref{fig:complplots}. In such cases the model attempts to detect homological structure, where none exists.\n\n\\subsection{Limitations and Conclusion}\n\nOur model entifies simplices that are distant to homology cycles, aided by the shifted-inverted Hodge Laplacian based graph convolution and the simplified simplex adjacency construction. We experimented with a traditional approach of using a Hasse graph analogue of the complex but that model failed to learn. Another advantage of our model is that both large and small scale homology cycles are consistently localized. Although we restricted our analysis to $\\mathcal{H}_1$ for ease of evaluation, the generality of our model allows direct extension to higher dimensional simplices.  \n\nOne drawback of our model is that it performs poorly when components of complexes  contain no higher dimensional homological information whatsoever. In such cases, it hallucinates homology cycles while they do not exist. \n\nAnother limitation is the variance of the inferred function on the complex as shown in Figure~\\ref{fig:complplots}. This variance stems from two sources: First, the target function is piecewise constant; and second, the adjacency structure that we use to sparsify an otherwise complete graph (for creating the GNN computational graph) alters the spectrum of the kernel. \n\nThe choice of a spectral sparsification method with theoretical guarantees is an interesting avenue of future work. Needless to say, our results are promising even with a basic GNN architecture. We foresee exciting opportunities to improvements in the architecture. \n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:intro} \nMaterial differentiation and quantification using a standard single-energy computed tomography (SECT) is extremely challenging because different materials may have the same CT value\\cite{ref1}. To tackle this challenge, dual-energy CT (DECT) takes full advantage of the energy dependence of the linear attenuation coefficient by scanning the patients using two different energy spectra\\cite{ref2, ref3, ref4, ref5, ref6,lee2017feasibility,petrongolo2018single,xue2019accurate}. This enables DECT imaging providing energy- and material-selective images, and having been very widely used in clinical practice for many applications, such as virtual monochromatic imaging\\cite{ref7, ref8}, differentiating intracerebral hemorrhage from iodinated contrast\\cite{ref9}, automated bone removal in CT angiography\\cite{ref10, ref11, ref12, ref13}, virtual noncontrast-enhanced imaging\\cite{ref14, ref15, ref16, ref17, ref18, ref19} and urinary stone characterization\\cite{ref20, ref21, ref22, ref25}. However, it is still an open and challenging task for clinical DECT imaging due to complex practical implementations, proprietary patents for major CT vendors, and less popularity for DECT scanners compared to the standard SECT scanners.\n\nSince the low- and high-energy CT images acquired from the DECT scanners have the same anatomical structures, there is substantial redundant anatomical information between the DECT images. For the scanned patients using the same DECT imaging protocols, the low- and high-energy CT images are also correlated in the energy domain, resulting in information redundancies in the energy-domain\\cite{ref26, ref27}. Meanwhile, both DECT images are reconstructed using fully-sampled projection data which have to meet the classical Shannon-Nyquist theorem in angular-data sampling to reconstruct artifacts-free images. By fully exploiting the anatomical consistency and energy domain correlation between the DECT images, it is possible to provide high-quality artifacts-free DECT images using conventional SECT images together with sparse sampling projection data at different energy levels.\n\n\\begin{figure*}[t\n    \\centering\n        \\includegraphics[width=0.8\\textwidth]{workflow.png}\n        \\caption{The workflow of the proposed fully-sampled low-energy and single-view high-energy DECT imaging approach. During the training phase, the denoising DECT images together with the single-view dual-energy projections are used to train the projection domain convolutional neural network (CNN) and the MD-CNN. In the testing phase, the trained networks use the input low-energy images and the single-view dual-energy projections to infer the corresponding high-energy images.}\n    \\label{fig:1}\n\\end{figure*}\n\nCT imaging with the full use of as low as reasonably achievable (ALARA) principle has been commonly accepted in routine practice and further reducing radiation dose from CT scanning is clinically favorable and has been extensively studied for almost two decades. Deep learning (DL) has recently been proved to be a powerful tool for mapping complex relationships and incorporating existing knowledge into an inference model through feature extraction and representation learning\\cite{ref28, ref29, ref30, ref31, ref32, ref33, ref34, ref35, ref36, ref37}. It has also been applied in low-dose CT\\cite{ref38, ref39, ref40, ref41} and DECT imaging\\cite{ref42, ref43, ref44, ref45, ref46}.\n\nTo further significantly reduce the radiation dose of DECT imaging, in this study, we synergically exploit the energy-domain correlation and anatomical consistency between DECT images by leveraging the deep learning approach and the seamless integration of the correlation and consistency in a data-driven DECT imaging process, and eventually push the sparse sampling to the limit of a single projection view and demonstrate the feasibility of high-performance DECT imaging using a deep learning approach termed fully low-energy and single high-energy DECT imaging (FLESH-DECT).\n\n\n\n\\section{Method}\n\\label{sect:method}\nThe flowchart of proposed FLESH-DECT strategy is shown in Fig.~\\ref{fig:1}.\nThe input to the model is a single-view high-energy projection together with the low-energy image \\(I_{low}\\) which is reconstructed using fully-sampled low-energy projection data. For the low-energy image, it is firstly denoised with a denoising network to mitigate the impact of image noise. Instead of training a network directly mapping high-energy images from the low-energy images, we use a convolutional neural network (CNN) to perform material-decomposition-type operations and it is termed material decomposition CNN (MD-CNN). The input of the MD-CNN is the denoised low-energy image \\(I_{low}^{de}\\) while the output is a \"material component\" matrix \\(A\\). The matrix \\(A\\) has the same image size as \\(I_{low}^{de}\\) but with multiple channels each of which corresponds to a pseudo material-specific image. The values are the percentages of corresponding \"basis material\" on these pixels. We ensure that the sum of the percentages equals to one (mass conversation) for each unique pixel. Furthermore, another CNN is used to pre-process the differences between the given high-energy projections and its corresponding low-energy projections. This projection-domain network\nis used to fill the gap between the denoised images and the non-denoised projection. Since CT forward projection can be regarded as linear summations of pixel values, we can use the least squares method to solve the corresponding CT values of each \"material\" \\(b_{dif}\\) according to the matrix \\(A\\) and the pre-processed projection difference. The estimated high-energy image is calculated as the summation of low-energy images, and the inner product of matrix \\(A\\) and vector \\(b_{dif}\\).\nDetailed formula derivation is described in the following subsections.\n\n\\subsection{DECT Imaging}\nIn CT imaging, the attenuation coefficient at each position can be represented as a linear combination of basis materials' attenuation coefficient\\cite{ref50}.\n\\begin{equation}\n    \\mu=\\alpha_1\\mu_1+\\alpha_2\\mu_2+\\dots+\\alpha_m\\mu_m\n    \\label{eq:1}\n\\end{equation}\nwhere \\(m\\) is the number of basis materials, \\(\\alpha_i\\) is the percentage of the \\(i\\)-th basis material and \\(\\mu_i\\) is the attenuation coefficient of the \\(i\\)-th basis material. Since CT values in Hounsfield Unit (HU) can be represented as the linear transformation of attenuation coefficient \\(\\mu\\) with the following equation:\n\\begin{equation}\n    HU=1000\\times\\frac{\\mu-\\mu_{water}}{\\mu_{water}}\n    \\label{eq:2}\n\\end{equation}\nEq.\\eqref{eq:1} can also be written as:\n\\begin{equation}\n    HU=\\alpha_{1}HU_{1}+\\alpha_{2}HU_{2}+\\dots+\\alpha_{m}HU_{m}\n    \\label{eq:3}\n\\end{equation}\nwhere \\(HU_{i}\\) stands for the Hounsfield Unit CT value for the \\(i\\)-th basis material.\nConsidering there are \\(n_{pix}=W{\\times}H\\) pixels in an image slice, we can write Eq.\\eqref{eq:3} into the following matrix multiplication form\n\\begin{equation}\n    I=A\\cdot{b}^T\n    \\label{eq:4}\n\\end{equation}\nwhere \\(I\\) is the image vector sized \\(n_{pix}{\\times}1\\) containing  CT values at each pixel, \\(A={[\\alpha_{ij}]}_{n_{pix}{\\times}m}\\) is the material component matrix, \\(\\alpha_{ij}\\) stands for the percentage of material \\(j\\) at pixel \\(i\\), \\(b={[HU_i]}_{m{\\times}1}\\) consists of HU values for each material.\n\nIn DECT, there are two different images \\(I_{low}\\) and \\(I_{high}\\). The material component matrix \\(A\\) remains the same for both images because pixel compositions do not change between low- and high-energy scans. Therefore, we have the following equations\n\\begin{equation}\n    \\begin{cases}\n        &I_{low}=A\\cdot{b}_{low}^T\\\\\n        &I_{high}=A\\cdot{b}_{high}^T\n    \\end{cases}\n    \\label{eq:5}\n\\end{equation}\nBy subtracting the high-energy equation from the low-energy equation, we get\n\\begin{equation}\n    I_{dif}=A\\cdot{b}_{dif}^T\n    \\label{eq:6}\n\\end{equation}\nwhere \\(I_{dif}\\) is the difference image between \\(I_{low}\\) and \\(I_{high}\\), \\(b_{dif}\\) is the difference between \\(b_{low}\\) and \\(b_{high}\\). Let \\(P_{high}\\) and \\(P_{low}\\) be the given high- and low- energy projection measurements, and \\(R\\) be the projection matrix sized \\(n_{ray}{\\times}n_{pix}\\) corresponding to the high-energy view. We have the following equation\n\\begin{equation}\n\\begin{split}\n    R{\\cdot}A\\cdot{b}_{dif}^T & = R{\\cdot}I_{dif} \\\\\n                               & = R{\\cdot}I_{high}-R{\\cdot}I_{low} \\\\\n                               & = P_{high}-P_{low}\n    \\label{eq:7}\n\\end{split}\n\\end{equation}\nFor the fully-sampled low-energy and single-view high-energy CT imaging task, the unknows in Eq.\\eqref{eq:7} are the material component matrix \\(A\\) and the corresponding difference values \\(b_{dif}\\). Assuming that we have the material component matrix \\(A\\), let \\(M=R{\\cdot}A\\in\\Re^{m{\\times}n_{ray}}\\), \\(P_{dif}=P_{high}-P_{low}\\in\\Re^{1{\\times}n_{ray}}\\), the difference values \\(b_{dif}\\) can be calculated by solving the equation \\(M{\\cdot}b^{T}=P_{dif}\\). In regular CT imaging, we have \\(n_{ray}>>m\\), the best \\(b_{dif}\\) can therefore be found using least-squares method which can be computed with Cholesky decomposition, i.e.,\n\\begin{equation}\n\\begin{split}\n    b_{dif} & =\\operatornamewithlimits{argmin}\\limits_{\\tilde{b}}{\\|M{\\cdot}\\tilde{b}^T-P_{dif}\\|}_2^2 \\\\\n            & =[(M^TM)^{-1}M^TP_{dif}]^T\n    \\label{eq:8}\n\\end{split}\n\\end{equation}\nThe only task now is to estimate the material component matrix \\(A\\) from the low energy image \\(I_{low}\\).\n\n\\subsection{Material decomposition-based dual-energy CT mapping}\nDue to its ability to learn complex relationships and incorporate existing knowledge into a nonlinear mapping model, a dedicated CNN model (termed MD-CNN) is used to estimate the material component matrix \\(A\\). When designing the MD-CNN model, a major challenge is the lack of training labels due to unknown materials in the images and their percentages. To tackle this challenge, we train the MD-CNN indirectly. We firstly denoise the dual-energy image pairs, and the denoised low-energy images \\(I_{low}^{de}\\) are inputted into the MD-CNN to acquire material component matrices \\(A_{DL}\\). Meanwhile, we put the projection differences into another 1-D projection domain CNN to preprocess the projection data. Then, we compute \\(b_{dif}\\) using Eq.\\eqref{eq:8}. The estimated denoised high-energy images \\(I_{high}^{dl}\\) can therefore be calculated as\n\\begin{equation}\n    I_{high}^{dl}=I_{low}^{de} + A_{DL}{\\cdot}b_{dif}^T\n    \\label{eq:10}\n\\end{equation}\nAn image similarity loss is calculated between the denoised high-energy image \\(I_{high}^{de}\\) and the DL-estimated image \\(I_{DL}^{de}\\), and the mean squared error (MSE) loss is used for the task:\n\\begin{equation}\n    \\mathcal{L}_{high}=\\frac{1}{n}{\\|I_{high}^{de}-I_{DL}^{de}\\|}_2^2\n    \\label{eq:11}\n\\end{equation}\nInstead of getting material component label, we focus on the target high-energy image and it is not necessary to specify each channel in \\(A_{DL}\\) to represent real material or linear combination of different materials. Meanwhile, since matrix \\(A_{DL}\\) is supposed to be the material component matrix, it should be able to recover the input denoised low-energy image as well, resulting in the follow loss function:\n\\begin{equation}\n    \\mathcal{L}_{low}=\\frac{1}{n}{\\|I_{low}^{de}-A_{DL}{\\cdot}b_{low}^T\\|}_2^2\n    \\label{eq:12}\n\\end{equation}\nThe same strategy in Eq.\\eqref{eq:8} is used to calculate the HU values \\(b_{low}\\) for each \"material\" under low energy,\n\\begin{equation}\n    b_{low}=\\operatornamewithlimits{argmin}\\limits_{\\tilde{b}}{\\|A_{DL}{\\cdot}\\tilde{b}^T-I_{low}^{de}\\|}_2^2\n    \\label{eq:13}\n\\end{equation}\nThe final loss function is computed as the summation of \\(\\mathcal{L}_{low}\\) and \\(\\mathcal{L}_{high}\\), i.e.,\n\\begin{equation}\n    \\mathcal{L}=\\mathcal{L}_{low}+\\mathcal{L}_{high}\n    \\label{eq:14}\n\\end{equation}\n\nDuring the inference phase, the low-energy images are also firstly denoised and then inputted into the trained MD-CNN for \\(A_{DL}\\). The projection domain CNN preprocesses the projection difference \\(P_{dif}\\). The estimated difference images are calculated according to Eq.\\eqref{eq:8}. The difference between the training and inference is the estimated high-energy image is calculated as the summation of the original low-energy image \\(I_{low}\\) and the difference image \\(I_{dif}\\) in inference phase.\n\n\\subsection{Network details}\n\\subsubsection{Denoising CNN}\n\nWe employ the denoising network in our previous work\\cite{ref46} to reduce the DECT image noise. The network uses a plain structure which encompasses 13 convolution layers to learn the residual between the input image and the denoised image. The first 12 layers are convolution layers with kernel size \\(3\\times3\\) and each layer is followed by a batch normalization layer (BN) and a rectified linear unit (ReLU) activation. The last layer is a convolution layer with kernel size \\(1\\times1\\) fusing the result. Fig.~\\ref{fig:2} shows the detailed structure of the denoising CNN. The denoised image is computed as the summation of the input image and the output from the last layer.\n\n\\begin{figure}[t\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{denoise_cnn.png}\n        \\caption{Architecture of the fully convolutional network for image denoising. A plain structure which encompasses 13 convolution layers is applied to learn the residual between the input image and the denoised image. }\n    \\label{fig:2}\n\\end{figure}\n\n\\subsubsection{MD-CNN}\n\nFor the material decomposition network, we employ a U-Net-type structure\\cite{ref51} which has a large receptive field and is quite suitable for many medical image processing tasks. There are 10 normal \\(3\\times3\\) convolution layers in the proposed network (Fig.~\\ref{fig:3}). Each convolution layer is followed by a BN layer and a ReLU activation layer. There are 3 resolution levels in total. For down-sampling, we use convolution layer with kernel size\nof \\(2\\times2\\) and stride equal to 2. Each strided convolution layer is also followed by a BN layer and a ReLU activation layer. For up-sampling, we use bilinear interpolation to double both image width and height. At the end of the network, a convolution layer with kernel size\nof \\(1\\times1\\) is added to fuse the channels. Since the values on each output channel are supposed to be the percentages of corresponding \"basis material\", we apply softmax after the convolution layer with kernel size\nof \\(1\\times1\\) to make sure that the sum of materials' proportion equals to 1 at each pixel.\n\n\\begin{figure}[t\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{img_cnn2.png}\n        \\caption{Structure of the MD-CNN. A much simplified UNet-like structure with 14 convolution layers is used here to estimate the percentages of each corresponding \"basis material\". Numbers under each block show\nthe number of channels of the multichannel feature maps.}\n    \\label{fig:3}\n\\end{figure}\n\n\\subsubsection{Projection-domain CNN}\n\nTo enhance the robustness of the lease-square problem in Eq.\\eqref{eq:8}, the projection-domain network (Fig.~\\ref{fig:4}) is applied to slightly refine the inputting projection difference and to make its noise level to be consistent with that of the denoised DECT difference image, and we employ a concise 4-layer network for the task. A residual learning structure is used here for an easier startup at the beginning iterations. All basic convolution layers have a kernel with size of \\(1\\times5\\) except for the last one which has a kernel with size of \\(1\\times1\\). The first three convolution layers are followed by a BN layer and a ReLU activation layer.\n\n\\begin{figure}[t\n    \\centering\n        \\includegraphics[width=0.35\\textwidth]{proj_cnn2.png}\n        \\caption{Structure of the projection-domain CNN. A concise 4-layer network is employed to slightly refine the inputting projection difference. Numbers under each block show the number of channels of the multichannel feature maps.}\n    \\label{fig:4}\n\\end{figure}\n\n\\subsubsection{Network training}\nThe denoising CNN was implemented in MATLAB with MatConvNet framework\\cite{ref47}. Denoising was performed as a data preprocessing step for all DECT images. The material decomposition network and projection-domain network are implemented using Python with Tensorflow framework\\cite{ref48}. Those two networks were trained together in an end-to-end fashion. The parameters in the networks were optimized using ADAM algorithm\\cite{ref49} with \\(\\beta_1=0.9\\) and \\(\\beta_2=0.999\\). Learning rate was set to \\(10^{-3}\\) during the training. The training set was randomly split into small batches in each epoch with batch size of 8. The proposed network was trained for 200 epochs in total. We perform validation after each training epoch and the model with best validation loss was selected as the final model for testing. The number of materials \\(m\\) was set to 10 in our experiments. It has to note that the model was able to achieve reasonable results even with \\(m=3\\). The networks were trained and tested on a workstation with configurations as follows: CPU is Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz; GPU is NVIDIA RTX 2080 Ti with 12G memory.\n\n\\subsection{Projectors for different CT geometries}\nIn order to calculate matrix \\(M\\) in Eq.\\eqref{eq:8}, projector corresponding to the CT geometry is indispensable in our algorithm. There are mainly two types of 2D CT geometries, fan-beam and parallel-beam. The fan-beam geometry can be further divided into two sub-types, equiangular and equispacing. For each above-mentioned geometry, we developed a projector and trained a new model for evaluation.\n\\subsubsection{Equiangular fan-beam}\nEquiangular geometry is mainly implemented with arc detectors which keep the angles between two adjacent detector pixels and the source-detector-distance the same. We test our model in the anterior-posterior (AP) direction, but it can be easily extended to any other projection view.\nTo acquire projection data using the 2D equiangular fan-beam geometry, we first calculate the intersection of each projection ray with each image row. The values at each intersection points are computed using linear interpolation. Suppose the image size is \\(W{\\times}H\\), and a matrix \\(I'\\) with size of \\(n_{ray}{\\times}H\\) can be obtained using the following rebinning equation:\n\\begin{equation}\n    I'(\\phi, y)=I((D-y)\\tan(\\phi), y)\n    \\label{eq:15}\n\\end{equation}\nwhere \\(\\phi\\) is the angle between the projection ray and the central ray (as shown in Fig.~\\ref{fig:8}), and \\(y\\) is the image pixel position along y-axis in the Cartesian coordinate system centered at image center \\(O\\). \\(D\\) is the distance between projection source and the rotation center which is overlapped with the image center \\(O\\). After the linear interpolation, we calculate the summation of each column in matrix \\(I'\\) to obtain raysum $S$ which is weighted by the distance to yield the final projection $P$:\n\\begin{equation}\n    P(\\phi)=\\frac{dy}{\\cos{\\phi}}S(\\phi)\n    \\label{eq:16}\n\\end{equation}\nwhere \\(dy\\) is the image spacing in y-axis.\nWe set \\(D=600mm\\), \\(dy=0.5mm\\), the number of detector channels \\(n_{ray}=800\\) and the angle between adjacent channels \\(ds=\\frac{7}{11000}rad\\) for all images.\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{arc_beam.png}\n        \\caption{Illustration of equiangular fan-beam geometry. }\n    \\label{fig:8}\n\\end{figure}\n\n\\subsubsection{Equispacing fan-beam}\nFlat detectors with equal spacing between the adjacent channels are commonly used to implement the equispacing geometry. We use similar strategy to implement the equispacing fan-beam projection in the AP direction.\nIn this case, the rebinning procedure is computed with the following equation:\n\\begin{equation}\n    I'(s, y)=I(\\frac{s(D-y)}{L}, y)\n    \\label{eq:17}\n\\end{equation}\nwhere \\(s\\) is the distance between current channel and the detector center (direction included), and \\(L\\) is the source to detector distance, as shown in Fig.~\\ref{fig:9}. The rest part of the implementation is the same as equiangular projection. For the parameters, we have \\(D=600mm\\), \\(L=1100mm\\), \\(dy=0.5mm\\), \\(n_{ray}=800\\) and the spacing between nearby channels \\(ds=0.78mm\\).\n\n\\begin{figure}[tb]\n    \\centering\n        \\includegraphics[width=0.48\\textwidth]{flat_beam.png}\n        \\caption{Illustration of equispacing fan-beam geometry. }\n    \\label{fig:9}\n\\end{figure}\n\n\\subsubsection{Parallel-beam}\nFor the parallel-beam scenario, We assume that the detector channel has the same spacing as the image pixels and each X-ray projects exactly through an image column in the AP direction. Therefore, the projection in the AP direction can be computed as the summation of each image column.\n\n\\section{Data Specification}\n\\subsection{Training data for the denoising network}\nThe AAPM Low-Dose CT Grand Challenge data are used to train the denoising network. This dataset consists of routine dose CT and the corresponding simulated low-dose CT data from 10 patients. The routine dose scanning voltage is 100 kV or 120 kV and the X-ray tube current varies from 200mA to 500mA. The detector has \\(736\\times64\\) elements, and each element has a size of \\(1.2856\\times1.0947mm^2\\). The source-to-axial distance is \\(59.5cm\\) and the source-to-detector distance was \\(108.56cm\\). All the images were reconstructed to slice thickness of \\(1.0 mm\\) and \\(512\\times512\\) pixel. The pixel size varies from \\(0.66\\times0.66mm^2\\) to \\(0.78\\times0.78mm^2\\). To simulate the low-dose CT data, Poisson noise was introduced into the routine dose to mimic a noise level that corresponded to \\(25\\%\\) of the routine dose, and noisy projection are reconstructed to yield the low-dose image. \n\n\\subsection{DECT imaging dataset}\nClinical DECT images of 22 patients who underwent iodine contrast-enhanced DECT exams were collected for the study. All the exams were performed in Nanjing General PLA Hospital, China, with the approval by the institutional review board and patient consent forms. The DECT images (5753 slices in total) were acquired using a SOMATOM Definition Flash DECT scanner (Siemens Healthineers, Forchheim, Germany) after administering iodine contrast agent. The low- and high-energy of the DECT scans were 100 kV and 140 kV, respectively. All CT images were reconstructed using the filtered back-projection (FBP) algorithm provided by the commercial CT vendor. The dataset were split into training set, validation set and testing set randomly with 16, 3 and 3 patients included respectively.\n\n\\begin{figure}[t\n    \\centering\n        \\includegraphics[width=\\linewidth]{fig2_2.png}\n        \\caption{Example results on a testing slice and the difference with respect to the corresponding real high-energy image. The first, third and fifth row display the images on axial, sagittal and coronal view. From left to right are real 100 kV image, real 140 kV image and result from proposed method, respectively. The second, fourth and sixth row are the corresponding differences with real 140 kV images. The CT images are displayed with a window width=300 HU and center=50 HU while the difference images are displayed under window width=300 HU and center=0 HU.}\n    \\label{fig:5}\n\\end{figure}\n\n\\section{Results}\n\nWe first focus on the results of equiangular geometry, and comparison results using different geometries are presented at the end this section.\nFig.~\\ref{fig:5} shows original DECT images and the 140 kV images predicted using the proposed method for a testing patient. The first, second and third columns show the original 100 kV images, the original 140 kV images, and the predicted results, respectively. The first, third, and fifth rows show CT images in transverse, sagittal, and coronal planes, respectively. The second, fourth, and sixth rows show difference images with respect to the corresponding real high-energy CT images in transverse, sagittal, and coronal planes, respectively.\n\\begin{table}[t\n    \\begin{center}\n        \\caption{Quantitative comparisons between the predicted and the real high-energy CT images for the testing patients. }\n        \\begin{tabular}{ c | c c c c c }\n            \\hline\\hline\n             & MSE & PSNR & SSIM & HU error & Time(s) \\\\\n            \\hline\n            Patient1 & 875.06 & 36.80 & 0.8702 & 1.65\\(\\pm\\)1.03 & 2.65 \\\\\n            Patient2 & 887.60 & 36.99 & 0.8789 & 2.09\\(\\pm\\)1.48 & 2.31 \\\\\n            Patient3 & 927.47 & 36.89 & 0.8745 & 1.50\\(\\pm\\)0.82 & 3.34 \\\\\n            \\hline\\hline\n        \\end{tabular}\n        \\label{tab:1}\n       \n    \\end{center}\n\\end{table}\n\\begin{figure}[t\n    \\centering\n        \\includegraphics[width=\\linewidth]{fig6_2.png}\n        \\caption{VNC images and Iodine maps reconstructed using original DECT, DL-DECT, and FLESH-DECT images. The first, third and fifth row are the VNC images on axial, sagittal and coronal view, respectively, while the second, fourth and sixth row are the corresponding Iodine maps. }\n    \\label{fig:6}\n\\end{figure}\nAs can be seen, the proposed DL-derived high-energy images are highly consistent with the original high-energy images. There are some differences at sharp boundaries which also appear in difference images between original high- and low-images. Those differences may be motion introduced difference between original DECT images because there is approximately 90 degrees out of phase for the low- and high-energy data acquired using a dual-source DECT scanner. When inputting the low-energy image into the model, the model performed prediction based on the anatomical structure of the low image and can not reflect the change with respect to the original high-energy image.\n\n\\begin{table}[b\n    \\begin{center}\n        \\caption{Quantitative comparisons of the VNC image and the iodine maps reconstructed using original DECT and FLESH-DECT images.}\n        \\begin{tabular}{ c c | c c | c c }\n            \\hline\\hline\n            \\multicolumn{2}{c}{\\multirow{2}{*}{}} & \\multicolumn{2}{|c|}{VNC} & \\multicolumn{2}{c}{Iodine Map} \\\\\n            & & MSE & PSNR & MSE & PSNR \\\\\n            \\hline\n            \\multirow{2}{*}{Patient1} & DL-DECT & 3882.34 & 29.03 & 0.0313 & 23.65 \\\\\n            & FLESH-DECT & \\textbf{3808.97} & \\textbf{29.11} & \\textbf{0.0308} & \\textbf{23.73} \\\\\n            \\hline\n            \\multirow{2}{*}{Patient2} & DL-DECT & 4216.36 & 28.95 & 0.0342 & 23.78 \\\\\n            & FLESH-DECT & \\textbf{4128.01} & \\textbf{29.04} & \\textbf{0.0335} & \\textbf{23.87} \\\\\n            \\hline\n            \\multirow{2}{*}{Patient3} & DL-DECT & 3508.35 & 29.71 & 0.0313 & 24.85 \\\\\n            & FLESH-DECT & \\textbf{3431.13} & \\textbf{29.81} & \\textbf{0.0306} & \\textbf{24.95} \\\\\n            \\hline\\hline\n        \\end{tabular}\n        \\label{tab:2}\n    \\end{center}\n\\end{table}\n\n\nQuantitative metrics were calculated to evaluate the accuracy of the predicted high-energy CT images. We use the well-established metrics MSE, PSNR and SSIM to assess the image similarity between the real and the predicted high-energy images. Additionally, more than 100 region-of-interests (ROIs) were randomly selected for each testing volume on homogeneous areas (e.g. liver and stomach). We calculated the mean HU value differences on those ROIs and the results show that the averaged HU error between the predicted and the original high-energy image is smaller than 2.09 HU. For the computation time, it takes around 2.5 seconds for the proposed method to process 300 slices. All quantitative results are shown in Table \\ref{tab:1}.\nSince most of the differences come from noise, we also compared the denoised DL-predicted images with the denoised high-energy images. In this case, the DL-predicted images are calculated by adding the denoised low-energy image to the DL-estimated difference image. For those denoised images, the proposed method achieves average MSE of 171.09, PSNR of 44.33 and SSIM of 0.9848.\n\nIn our previous work \\cite{ref46}, we propose the DL-DECT method to estimate \\(I_{dif}\\) directly from \\(I_{low}\\). In this work, with the introduction of the additional high-energy single-view projection, we can further enhance the accuracy of the predicted high-energy image and eventually the accuracy of the material- and energy-specific images. To show the benefit of the additional high-energy projection, we quantitatively compared the proposed FLESH-DECT method with the DL-DECT method.\nVirtual noncontrast (VNC) images and iodine maps are derived to demonstrate the clinical utility of the proposed method. Fig. \\ref{fig:6} depicts the VNC images and iodine maps reconstructed using different methods on the transverse, sagittal and coronal planes. Both the DL-DECT and FLESH-DECT algorithms provide high-quality VNC and iodine images that are consistent with the images generated by original DECT images. Quantitative metrics on VNC and iodine images are shown in Table \\ref{tab:2}. The results demonstrate FLESH-DECT can provide high-quality material-specific images and it outperforms the DL-DECT method.\n\n\\begin{table}[t\n    \\begin{center}\n        \\caption{Quantitative Noise Level Comparison on VNCs and Iodine Maps.}\n        \\begin{tabular}{ c c | c c }\n            \\hline\\hline\n            \\multicolumn{2}{c}{Standard Deviation on ROIs} & VNC & Iodine Map \\\\\n            \\hline\n            \\multirow{2}{*}{Patient1} & Real & 81.58 & 0.3363 \\\\\n            & DL-DECT & 29.14 & 0.1618 \\\\\n            & FLESH-DECT & \\textbf{28.09} & \\textbf{0.1499} \\\\\n            \\hline\n            \\multirow{3}{*}{Patient2} & Real & 74.69 & 0.2837 \\\\\n            & DL-DECT & 24.81 & 0.1017 \\\\\n            & FLESH-DECT & \\textbf{24.12} & \\textbf{0.0998} \\\\\n            \\hline\n            \\multirow{3}{*}{Patient3} & Real & 76.95 & 0.2875 \\\\\n            & DL-DECT & 25.97 & 0.0946 \\\\\n            & FLESH-DECT & \\textbf{25.67} & \\textbf{0.0891}\\\\\n            \\hline\\hline\n        \\end{tabular}\n        \\label{tab:3}\n       \n    \\end{center}\n\\end{table}\n\nFrom the VNC images and Iodine maps, we find that the DL-derived VNC images and iodine maps show a remarkably reduced noise level compared with those generated from original DECT images. Here we also compare the noise level by calculating the standard deviation in ROIs. More than 500 ROIs were selected randomly in homogeneous areas. The mean standard deviations in ROIs on each testing patient are provided and compared in Table \\ref{tab:3}. The mean standard deviations of the images derived using FLESH-DECT are close to that of the DL-DECT method which is much lower than those from the original DECT images. This result shows that the FLESH-DECT method maintains the denoising feature.\n\n\\begin{figure*}[tb]\n    \\centering\n        \\includegraphics[width=0.98\\textwidth]{result.png}\n        \\caption{Results on three testing slices for different geometry models. From left to right are original 100 kV images, original 140 kV images, parallel-beam results, equispacing fan-beam results and equiangular fan-beam results, respectively. }\n    \\label{fig:7}\n\\end{figure*}\n\nWe tested the proposed method with several 2D CT geometries. Example results on a testing volume using different geometries are displayed in Fig. \\ref{fig:7}. All CT images are displayed with window width=300HU and center=50HU while difference images are displayed with window width=300HU and center=0HU. As can be seen, all models are able to provide competitive results which are highly consistent with the original high energy image. For better comparison, we also calculate quantitative metrics on those results (Table \\ref{tab:4}). The differences between the results obtained using the proposed method with different geometries are marginal and all of them are superior to our previous DL-DECT method which does not utilize the additional single-view projection. Overall, the proposed method reduces mean-squared error (averaged for all testing cases) from 1858.32 to 898.35 while increasing PSNR from 33.83 to 36.89 and SSIM from 0.8641 to 0.8744.\n\n\\begin{table}[b\n    \\begin{center}\n        \\caption{Quantitative Results for Different Geometries.}\n        \\begin{tabular}{ c c | c c c }\n            \\hline\\hline\n            \\multicolumn{2}{c}{} & MSE & PSNR & SSIM \\\\\n            \\hline\n            \\multirow{4}{*}{Patient1} & DL-DECT & 2466.07 & 32.37 & 0.8558 \\\\\n            & Parallel & \\textbf{866.48} & \\textbf{36.84} & \\textbf{0.8707} \\\\\n            & Equiangular & 875.06 & 36.80 & 0.8702 \\\\\n            & Equispacing & 871.65 & 36.82 & 0.8703 \\\\\n            \\hline\n            \\multirow{4}{*}{Patient2} & DL-DECT & 1530.75 & 34.61 & 0.8723 \\\\\n            & Parallel & \\textbf{876.68} & \\textbf{37.03} & \\textbf{0.8793} \\\\\n            & Equiangular & 887.60 & 36.99 & 0.8789 \\\\\n            & Equispacing & 882.76 & 37.01 & 0.8789 \\\\\n            \\hline\n            \\multirow{4}{*}{Patient3} & DL-DECT & 1578.13 & 34.61 & 0.8660 \\\\\n            & Parallel & 927.47 & 36.89 & 0.8745 \\\\\n            & Equiangular & 932.40 & 36.88 & 0.8741 \\\\\n            & Equispacing & \\textbf{927.15} & \\textbf{36.90} & \\textbf{0.8747} \\\\\n            \\hline\\hline\n        \\end{tabular}\n        \\label{tab:4}\n    \\end{center}\n\\end{table}\n\n\n\\begin{table}[t\n    \\begin{center}\n        \\caption{Computation Time for Different Methods in Seconds.}\n        \\begin{tabular}{ c c c c }\n            \\hline\\hline\n            & Patient1 & Patient2 & Patient3 \\\\\n            \\hline\n            Slices & 308 & 265 & 387 \\\\\n            \\hline\n            DL-DECT & 5.43 & 4.68 & 6.86 \\\\\n            Parallel & \\textbf{2.33} & \\textbf{2.04} & \\textbf{2.94} \\\\\n            Equiangular & 2.65 & 2.31 & 3.34 \\\\\n            Equispacing & 2.53 & 2.18 & 3.23 \\\\\n            \\hline\\hline\n        \\end{tabular}\n        \\label{tab:5}\n        \\vspace{-5mm}\n    \\end{center}\n\\end{table}\n\nThe computation time for different methods are displayed and compared in Table \\ref{tab:5}. Note that computation time for the denoising CNN is not included in the results. The denoising network takes about 0.01 seconds for each slice and the denoising time is similar to all methods. Compared to the DL-DECT method, the proposed method speeds-up the computation time by 2-fold which can be attributed to the reduced number of weights and the simple network structure. There are some differences among the time using different geometries which means the computational cost of the proposed method depends on the projector.\n\n\n\\section{Discussion}\nThere are substantial redundant information and correlation in both anatomical structure and energy-domain between the low- and high-energy DECT images. By incorporating the redundancy and the correlation into a deep learning model, it's possible to provide material- and energy-specific images using standard SECT scanners, which has the potential to alleviate the need for premium DECT scanners. In addition, compared to the standard fully low- and high-energy sampling DECT mechanism, the use of sparse sampling at the second energy level can significantly reduce the radiation dose of DECT imaging.\n\nOur results show both the DL-DECT and FLESH-DECT methods achieve high-performance DECT imaging by using the input low-energy CT data, and quantitative analysis shows FLESH-DECT outperforms DL-DECT in terms of HU accuracy and calculation speed. The superior performance of the FLESH-DECT can be attributed to the additional single-view high-energy projection. Different from the DL-DECT method which directly infers a high-energy image using the incorporated prior knowledge, the proposed FLESH-DECT method uses the learned knowledge to fit the measured high-energy projection. Namely, the high-energy projection introduces a penalty to constrain the projection generated by the predicted high-energy images to be consistent with the measurement, which in turn enhances the accuracy of the predicted images. The single-view high-energy projection can be obtained shortly before or after the standard SECT low-energy data acquisition. For example, one may use a prescanning scout-view image (with the corresponding high-energy kV setting) as the single-view high-energy projection, and existing SECT systems are able to implement these scanning protocols without modifying the hardware.\n\nFLESH-DECT is suitable for different geometries. In this study, we have tested the method using 2D geometries (fan-beam and parallel beam). However, the method can be applied straightforwardly to 3D geometries by extending the networks and the projection operators into 3D scenarios. Since the reconstructed size is smaller than the projection view size in the rotation axis for 3D case, projections generated using the reconstructed image volume cannot match the measured projection. To solve this issue, one can use the middle part of the measured projection which does not propagate through the region beyond the image volume.\n\n\\begin{figure*}[tb]\n    \\centering\n        \\includegraphics[width=0.98\\textwidth]{mat_decomp.png}\n        \\caption{An example of the output of the MD-CNN. Each image represents a channel in the resulting matrix \\(A_{DL}\\). All images are displayed under the same window with center=0.5 and width=1.0.}\n    \\label{fig:10}\n\\end{figure*}\n\nThe proposed method employs the MD-CNN to generate \"material-decomposition\" maps $A$ from 100 kV images. Since basis materials in the image do not change under X-ray at different energy levels, accurate material-decomposition maps are able to provide CT images under any spectrum if combined with the projection view acquired using the corresponding spectrum (e.g. 120 kV image from 120 kV projection, 150 kV image from 150 kV projection). In our experiments, the models were trained and tested using DECT images under 100 kV\/Sn 140 kV scanning protocol. Therefore, the generated \"material-decomposition\" maps are likely to be optimized for this specific protocol and may not be applied to images acquired using a different protocol (such as 80 kV\/Sn 140 kV). An example is shown in Fig. \\ref{fig:10}. However, if the model was trained using images acquired from several different spectra, the \"material-decomposition\" maps would be much closer to the real ones and the model would have the potential to generate images under different spectra without re-training the MD-CNN. Also, regularizers may further be introduced and applied to matrix $A$ during network training to enhance the robustness of the model.\n\nThere is a denoising-CNN included in the flowchart of FLESH-DECT. We use this network to reduce the impact of image noise and the results show its effectiveness. However, the denoising network is not mandatory and it can be replaced with other image denoising techniques~\\cite{ma2011low}, such as non-local mean (NLM)~\\cite{zhang2013iterative}, block-matching and 3D filtering (BM3D)~\\cite{salehjahromi2017spectral}, without seeing severe degradation in performance. The denoising step may also be removed when the inputting extremely-high quality images.\n\nDespite all the advantages and potentials mentioned above, the proposed method has limitations. FLESH-DECT relies highly on the deep neural network to perform the material-decomposition-like operation. Since the domain knowledge learned by the network highly depends on training data, it is unlikely to provide reasonable results when there is a huge difference between training and testing data. For example,\nmodels trained under 100 kV\/Sn 140 kV protocol may not generate correct results when inputting low-energy images scanned under 70 kV protocol. However, these limitations can be solved by training different models for different protocols.\n\n\n\\section{Conclusion}\nIn this paper, we proposed a deep learning approach to perform DECT imaging using a low-energy image and a single-view high-energy projection. Compared to the standard DECT imaging, the approach can provide superior material-specific images with significantly reduced noise. It also has the potential to simplify the system design and reduce the radiation dose, and allows us to perform high-quality DECT imaging without the conventional hardware-based DECT solutions. The approach may significantly extend the usage of the widespread standard SECT scanners by providing advanced DECT clinical applications, such as urinary stone characterization and as differentiating intracerebral hemorrhage from iodinated contrast, and thus lead to a new paradigm of SECT imaging.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\nWe demonstrate that the absence of stable quasiparticle excitations on parts of the Fermi surface, known as the ``nodal-antinodal dichotomy\" in underdoped cuprate superconductors, can be reproduced in models of strongly correlated electrons defined via a holographic dual. We show analytically that the anisotropy of the quantum critical continuum, which is a feature of these models, may lead to washing out the quasiparticle peak in one direction while leaving it intact in the perpendicular one. The effect relies on the qualitatively different scaling of the self-energy in different directions. Using the explicit example of the anisotropic Q-lattice model, we  demonstrate  how this effect emerges due to specific features of the near horizon geometry of the black hole in the dual description. \n}\n\\setcounter{tocdepth}{1}\n\\tableofcontents\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nSystems of strongly correlated electrons are not always well-described in terms of stable quasiparticle excitations. An experimental example is the phenomenon of the nodal-antinodal dichotomy in underdoped cuprate superconductors~\\cite{shen2005nodal}, where sharp quasiparticle peaks are observed only on some parts of the Fermi surface. Non-quasiparticle regimes have been demonstrated to exist in theoretical models of strongly correlated electrons constructed using holography, also known as the AdS\/CFT correspondence \\cite{Cubrovic:2009ye,Faulkner:2009wj,Faulkner:2011tm}. In these models the features of the near horizon geometry of the black hole in the dual description have been shown to govern the self-energy part of the fermionic two-point function in such a way that the inverse quasiparticle lifetime would grow faster than its energy, rendering the stable quasiparticle concept practically useless. \n\nOne may describe this holographic effect as the coupling of the would-be quasiparticles to a ``quantum critical continuum'' bath which mediates their decay, even at small frequencies where the usual elastic decay channels are kinematically forbidden~\\cite{Faulkner:2010tq}. In systems without rotational symmetry, e.g. due to the underlying crystal lattice, this quantum critical continuum may be present only in some directions in momentum space, and hence it is possible for the quasiparticle excitations to be stable only on some parts of the Fermi surface.\n\nWe sketch such a scenario in figure~\\ref{fig:cartoon}. In figure~\\ref{fig:cartoon_continuum} we show the two contributions that we may heuristically think of as combining to give the full fermion spectral function. The plot shows these contributions as functions of momentum in two directions in momentum space, which we label as \\(k_n\\) and \\(k_a\\), where the subscripts stand for (anti)nodal. The blue curve shows the anisotropic quantum critical continuum, which is only present for momentum along one direction (\\(k_n\\)). The dashed orange curves show sharp resonance peaks at the Fermi momentum. In figure~\\ref{fig:cartoon_peaks} we show the resulting spectral function. Where the peak overlaps with the quantum critical continuum, as happens along the \\(k_a\\) cut in the figure, interactions between the two broaden the peak, destroying the quasiparticle interpretation. On the other hand, where a peak does not overlap with the continuum it remains sharp and the quasiparticle interpretation is preserved. We refer to these two situations as antinodal and nodal respectively.\n\n\\begin{figure}\n    \\begin{subfigure}{0.5\\textwidth}\n        \\includegraphics[width=\\textwidth]{figs\/cartoon_continuum.pdf}\n        \\caption{Cuts of contributions to spectral function}\n        \\label{fig:cartoon_continuum}\n    \\end{subfigure}\\begin{subfigure}{0.5\\textwidth}\n        \\includegraphics[width=\\textwidth]{figs\/cartoon_peaks.pdf}\n        \\caption{Cuts of the full spectral function}\n        \\label{fig:cartoon_peaks}\n    \\end{subfigure}\n    \\caption{Cartoons of the fermion spectral function in the anisotropic strongly correlated systems considered in this work, as a function of momentum in two different directions in momentum space that we label as \\(k_n\\) and \\(k_a\\). \\textbf{(a):} The spectral function receives contributions from two sources. A quantum critical continuum (solid blue) which exists only for momentum in one direction, which we take to be \\(k_n\\), and resonance peaks (dashed orange). \\textbf{(b):} When these two contributions are combined any peaks that overlap the continuum are broadened, and thus lose their interpretation as stable quasiparticle excitations.\n    }\n    \\label{fig:cartoon}\n\\end{figure}\n\nIn this work we provide a comprehensive illustration of how to engineer the above mechanism in holography, showing that it may be realised by an appropriate choice of anisotropic infrared (IR) scaling symmetry. For simplicity we will only focus on cases that preserve a two-fold rotational symmetry, such that the nodes and antinodes occur along orthogonal directions in momentum space. We comment about a possible generalisation to the experimentally relevant case of four-fold rotational symmetry in section~\\ref{sec:Discussion}.\n\nWe start by considering analytically the general case of a holographic model with an anisotropic IR scaling geometry at zero temperature in section~\\ref{sec:anis}. Then we specialise to the particular model of the holographic Q-lattice. This model describes a (2+1)-dimensional quantum system with a scalar operator that has a linear profile in one direction \\(x\\). This breaks the translational symmetry in the $x$ direction as well as rotational symmetry. What is important for us is that this model has regimes in which the deep infrared geometry has the aforementioned anisotropic IR scaling behaviour. The self-energy of the quasiparticle excitation then demonstrates nodal behaviour in the \\(y\\) direction and antinodal in the \\(x\\) direction.\n\nIn section~\\ref{sec:zeroT} we consider such Q-lattices at zero temperature and demonstrate the finite width of the quasiparticle peak already at the Fermi surface in the $x$-direction, while in the perpendicular $y$-direction the peak remains sharp and the ``quantum critical'' self-energy is exponentially suppressed. We also evaluate the fermionic spectral function at the Fermi surface at finite temperature and demonstrate the absence of quasiparticle peaks in certain directions and the behaviour of the corresponding width as a function of temperature. The three appendices are devoted to the details of our analytical treatment (appendix~\\ref{app:matching}) and two different numerical methods: shooting at zero temperature, appendix~\\ref{app:zeroT_shooting} and pseudospectral relaxation at finite temperature, appendix~\\ref{app:finiteT_relaxation}. Finally, in appendix~\\ref{app:extra_plots} we show some additional plots of numerical results.  \n\n\n\\section{\\label{sec:anis}Fermionic Green's functions and anisotropic quantum critical continua}\n\\input{tex\/sec_anis.tex}\n\n\n\n\\section{\\label{sec:zeroT}\\label{sec:finiteT} Explicit example: anisotropic Q-lattice}\n\\input{tex\/sec_zeroT.tex}\n\n\n\n\n\n\n\\section{\\label{sec:Discussion}Discussion}\n\nIn this work we have studied the effects of anisotropy of the quantum critical continuum in systems of strongly correlated and strongly entangled quantum matter defined via their holographic duals. In the dual gravitational description, the features of quantum criticality are encoded in the scaling behaviour of the deep IR geometry. We focus on holographic systems with anisotropic deep IR geometries, with scaling behaviour with exponents obeying the conditions~\\eqref{eq:conditions_for_dichotomy}, in particular, with a negative dynamical exponent in one direction. We show analytically that in such systems the fermion spectral function behaves at small frequencies qualitatively differently in the two different directions in momentum space, in such a way as to yield an apparent nodal-antinodal dichotomy on the Fermi surface. We check this result by an explicit numerical calculation in a concrete holographic model.\n\nThe crucial part of our finding is that the behaviour we observe is solely dictated by the features of the continuum and the associated scaling exponents (modelled by the IR geometry), and is largely independent of the features of the probe (such as the mass or charge of the probe fermion). In this way it is quite general and would emerge for almost any probe used to describe phenomenological data. We thus might hope that this mechanism for producing a nodal-antinodal dichotomy would work for any type of quantum criticality, whether it can be described by holography or not. Our main statement is therefore that the angular behaviour of the fermionic self-energy, as extracted from e.g. ARPES data in experiments on underdoped cuprates, may reveal important information about the anisotropy of the quantum criticality underlying the system and doesn't have to be explained in terms of fermiology, for example by resonance scattering between the different Fermi surfaces.\n\nOur probe-agnostic treatment goes even further actually. It is easy to see that the deep IR geometry affects bosonic two-point correlation functions in a similar way to fermionic ones, and so one doesn't have to restrict oneself to fermionic probes only. Our mechanism predicts that the anisotropic features of a bosonic probe, which can resolve the angular structure of the finite momentum response, will closely follow the nodal-antinodal dichotomy observed in the fermionic probes, since both are dictated by the same quantum criticality. We suggest that this prediction can be directly checked at available momentum resolved EELS experimental facilities~\\cite{vig2017measurement,kogar2017signatures} provided the angular dependence is under control. We leave a full holographic exploration of the behaviour of bosonic probes in models of the type considered here to future work.\n\nThere are a number of other interesting directions for further research. Although our mechanism for producing a nodal-antinodal dichotomy is probe agnostic in the sense that it is independent of the mass and charge of the probe fermion, the dichotomy may be destroyed by other interactions, depending on how these interactions scale in the IR. To see why, recall that the key to obtaining a quantum critical continuum in the \\(x\\) direction is that, apart from the terms proportional to frequency, the zero-derivative terms in equation~\\eqref{eq:fermion_ir_eom_main_text} decay at least as fast as \\(\\zeta^{-1}\\) in the deep IR \\(\\zeta \\to\\infty\\). If we add non-minimal interaction terms to the Dirac equation that decay slower than this, they will destroy the quantum critical continuum.\n\nOne interesting and well-motivated non-minimal interaction is a dipole coupling, proportional to \\(\\slashed{F} \\Xi\\), which can arise in top-down constructions~\\cite{Bah:2010yt,Bah:2010cu} and has been used in holographic models of Mott insulators~\\cite{Edalati:2010ww,Edalati:2010ge}. It is straightforward to show that including such an interaction adds terms to equation~\\eqref{eq:fermion_ir_eom_main_text} that decay in the deep IR as \\(\\zeta^{-(1+\\nu_A)}\\). Given the constraint \\(\\nu_A \\geq 0\\) from equation~\\eqref{eq:exponent_conditions}, this interaction decays rapidly enough as \\(\\zeta\\to\\infty\\) that it will not destroy the quantum critical continuum.\n\nOn the other hand, suppose the gravitational background contains some scalar field \\(\\Phi\\). One could add to the Dirac equation~\\eqref{eq:Dirac_equation} a Yukawa-like coupling proportional to \\(\\Phi^n \\Xi\\) \\cite{Wu:2019orq}, where \\(n\\) is some power that we assume is positive in order to avoid a singularity at \\(\\Phi=0\\). If the scalar field scales as \\(\\Phi \\propto r^{\\a_\\Phi}\\) in the IR then this adds terms to equation~\\eqref{eq:fermion_ir_eom_main_text} proportional to \\(\\zeta^{-1+\\gamma}\\) with \\(\\gamma =\\nu_\\theta + n \\a_\\Phi (2\\nu_\\theta-1)\\). If \\(\\a_\\Phi<0\\), i.e. if the scalar field diverges as a power law in the IR, then for sufficiently large \\(n\\) we will find \\(\\gamma>0\\) even when \\(\\nu_\\theta<0\\), thus destroying the quantum critical continuum. For example, if we choose \\(\\Phi = e^{\\phi}\\) in the Q-lattice model studied in section~\\ref{sec:zeroT} then we have \\(\\a_\\Phi = -1\/4\\) and \\(\\nu_\\theta = -1\/6\\), and consequently \\(\\gamma > 0\\) for \\(n>1\/2\\). It would be valuable to perform a complete analysis of possible couplings and their effects on the continuum.\n\nNote also that while in our case the anisotropy of the IR geometry was caused by the explicit Q-lattice, this is not the only way to create such geometries. In particular, holographic models with spontaneous formation of spatially inhomogeneous order would naturally affect the IR geometry in a similar way. This is relevant for various models with spontaneous charge density waves~\\cite{Baggioli:2022pyb} including the holographic model of doped Mott insulators~\\cite{Andrade:2017ghg} and its homogeneous toy-model version~\\cite{Andrade:2018gqk}.\n\nIn another direction, it should be noted that our simple model produces a nodal-antinodal dichotomy with two-fold rotational symmetry, where the ``nodes'' and ``anti-nodes'' have an angular separation of $\\pi\/2$. This is in contrast to known examples of the dichotomy in real materials, where the rotational symmetry is four-fold~\\cite{shen2005nodal}: the nodes are separated from the anti-nodes by $\\pi\/4$ (and are also strictly aligned with the crystal lattice). In order to reproduce this phenomenology in a holographic approach one could introduce a periodic 2-dimensional crystal lattice, which will break rotations down to a discrete group and imprint the four-fold anisotropy onto the deep IR geometry. Such periodic lattice constructions have been realised for example in refs.~\\cite{Donos:2014yya,Donos:2020viz,Balm:2022bju}. The behaviour of the low frequency, finite momentum two-point functions should then be dictated by the same principles as outlined in section~\\ref{sec:anis}. It would be interesting to find such a periodic lattice model with four-fold anisotropic scaling behaviour analogous to the conditions~\\eqref{eq:conditions_for_dichotomy},which should then display the desired four-fold nodal-antinodal dichotomy, or its softer version, as described in the end of section~\\ref{sec:anis}. The advantage of our simplified model with two-fold rotational symmetry is the possibility for more complete analytic control, drastically improving our understanding of the underlying mechanism for the dichotomy.\n\n\n\\section*{Acknowledgements}\nWe thank Koenraad Schalm, Jan Zaanen, Erik van Heumen and Elias Kiritsis for useful communication and insightful comments.  A.K. acknowledges the hospitality of the Lorentz Institute for Theoretical Physics, where the preliminary results of this work have been discussed.  The work of A.K is supported by VR Starting Grant 2018-04542 of Swedish Research Council. The numerical computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC), partially funded by the Swedish Research Council through grant agreement no. 2018-05973, at SNIC Science Cloud and PDC Center for High Performance Computing, KTH Royal Institute of Technology. Nordita is supported in part by Nordforsk.\n\n\n\n\n\n\n\n\\subsection[kx term dominant]{\\(\\bar{k}_x\\) term dominant}\n\\label{sec:matching_kx_dominant}\n\nWe first consider the case in which for \\(\\zeta \\gg 1\\) and \\(\\bar{k}_x \\sim \\mathcal{O}(1)\\):\n\\begin{itemize}\n    \\item \\(\\bar{k}_x \\zeta^{\\nu_x} \\gg \\bar{m} \\zeta^{\\nu_\\theta}\\), either because \\(\\bar{m} = 0\\) or because \\(\\nu_x > \\nu_\\theta\\); and\n    \\item \\(\\bar{k}_x \\zeta^{\\nu_x} \\gg \\bar{q} \\zeta^{2\\nu_\\theta-\\nu_A}\\), either because \\(\\bar{q} = 0\\) or because \\(\\nu_x > 2\\nu_\\theta-\\nu_A. \\)\n\\end{itemize}\nThen at sufficiently large \\(\\zeta\\), the IR equations~\\eqref{eq:fermion_eom_ir} are well approximated by\n\\begin{equation}\n    \\psi_\\pm' + i \\left(\\bar{\\omega} \\mp \\frac{\\bar{k}_x}{\\zeta^{1- \\nu_x}} \\right) \\chi_\\pm = 0,\n    \\qquad\n    \\chi_\\pm'  + i \\left(\\bar{\\omega} \\pm \\frac{\\bar{k}_x}{\\zeta^{1 - \\nu_x}} \\right) \\psi_\\pm = 0,\n    \\label{eq:fermion_eom_ir_kx}\n\\end{equation}\nIt will be convenient to rewrite these equations by defining a rescaled radial coordinate \\(s = \\zeta \\left(\\bar{\\omega}\/\\bar{k}_x \\right)^{1\/(1-\\nu_x)}\\), in terms of which they read\n\\begin{equation}\n    \\psi_\\pm'(s) + i \\k \\left(1 \\mp \\frac{1}{s^{1-\\nu_x}}\\right) \\chi_\\pm(s) = 0,\n    \\qquad\n    \\chi_\\pm'(s) + i \\k \\left(1 \\pm \\frac{1}{s^{1-\\nu_x}}\\right) \\psi_\\pm(s) = 0,\n    \\label{eq:fermion_eom_ir_kx_s}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:kappa_definition}\n    \\k \\equiv \\left(\\frac{\\bar{k}_x}{\\bar{\\omega}^{\\nu_x}}\\right)^{1\/(1-\\nu_x)}\n\\end{equation}\nWe will be interested in approximate solutions of equation~\\eqref{eq:fermion_eom_ir_kx_s} when \\(\\bar{k}_x \\sim \\mathcal{O}(1)\\) and \\(\\bar{\\omega} \\ll 1\\). We will be interested in cases where \\(\\nu_x < 0\\), and thus from equation~\\eqref{eq:kappa_definition} \\(\\k \\ll 1\\).\n\nWe solve equation~\\eqref{eq:fermion_eom_ir_kx_s} at small \\(\\k\\) by the following matching procedure. For \\(s \\gg 1\\) we can neglect the terms propotional to \\(s^{-1\/(1-\\nu_x)}\\), and thus the solution obeying ingoing boundary conditions is that given in equation~\\eqref{eq:fermion_solution_deep_ir}. In terms of \\(\\k\\) and \\(s\\) this solution reads\n\\begin{equation}\n    \\psi_{\\pm,R}(s) = e^{i \\k s},\n    \\qquad\n    \\chi_{\\pm,R}(s) = - e^{i \\k s},\n    \\label{eq:approximate_solution_R_s}\n\\end{equation}\nwhere we have added the subscripts \\(R\\) as a reminder that this solution applies to the right of \\(s=1\\). Near \\(s \\sim \\mathcal{O}(1)\\) we expect the neglected terms in equation~\\eqref{eq:fermion_eom_ir_kx_s} proportional to \\(s^{-1\/(1-\\nu_x)}\\) to become important. However, for \\(\\k \\ll 1\\) they only have a small effect on the solution. To see this, we write the full solution as \\(\\psi_\\pm(s) = \\psi_{\\pm,R}(s) + \\d \\psi_\\pm(s)\\), where \\(\\d \\psi_{\\pm}(s) \\to 0\\) as \\(s \\to \\infty\\), and similar for \\(\\chi_\\pm\\). Substituting into equation~\\eqref{eq:fermion_eom_ir_kx_s}, we then find\n\\begin{align}\n    \\d\\psi_\\pm'(s) + i \\left(1 \\mp \\frac{1}{s^{1-\\nu_x}} \\right) \\k\\, \\d\\chi_\\pm(s) \\pm i \\k \\frac{e^{i\\k s}}{s^{1-\\nu_x}} &= 0,\n    \\nonumber \\\\\n    \\d \\chi_\\pm'(s)+ i \\left(1 \\pm \\frac{1}{s^{1-\\nu_x}} \\right) \\k\\,\\d\\psi_\\pm(s) \\pm i \\k \\frac{e^{i\\k s}}{s^{1-\\nu_x}} &= 0.\n    \\label{eq:fermion_eom_ir_kx_deviation_1}\n\\end{align}\nWhen \\(\\d\\psi_\\pm\\) and \\(\\d\\chi_\\pm\\) are small, then the terms proportional to \\(\\k \\, \\d\\psi_\\pm\\) and \\(\\k \\, \\d\\chi_\\pm\\) are doubly small in the small-\\(\\k\\) limit, and may be neglected to leading order in this limit. The approximate solution to equation~\\eqref{eq:fermion_eom_ir_kx_deviation_1} subject to the boundary conditions \\(\\d\\psi_\\pm, \\d\\chi_\\pm \\to 0\\) as \\(s \\to \\infty\\) is then\n\\begin{equation}\n    \\d\\psi_\\pm(s) \\approx \\d\\chi_\\pm(s) \\approx \\pm i \\k (-i \\k )^{- \\nu_x} \\Gamma(\\nu_x, - i \\k s) = \\mp \\frac{i \\k s^{\\nu_x}}{\\nu_x} + \\mathcal{O}(\\k^{1+|\\nu_x|}),\n\\end{equation}\nwhere \\(\\Gamma\\) is the incomplete Gamma function, and on the right-hand side we have expanded for fixed \\(s\\) and small \\(\\k\\). The right-hand side vanishes in the limit \\(\\k \\to 0\\), and thus \\(\\psi_{\\pm,R}\\) and \\(\\chi_{\\pm,R}\\) indeed provide good approximate around \\(s=1\\) at low frequencies. This also justifies a posteriori our neglect of the terms in equation~\\eqref{eq:fermion_eom_ir_kx_deviation_1} proportional to \\(\\k \\, \\d \\psi_\\pm\\) and \\(\\k \\, \\d \\chi_\\pm\\). On the other hand, this approximation breaks down at \\(s \\ll 1\\), where the incomplete gamma functions grow very large.\n\nFor \\(s \\ll 1\\), the terms in equation~\\eqref{eq:fermion_eom_ir_kx_s} proportional to \\(s^{-(1-\\nu_x)}\\) become dominant, and we obtain the approximate solution\n\\begin{align}\n    \\psi_{\\pm,L}(s) &\\approx a_\\pm \\exp\\left(\\frac{\\k}{\\nu_x}s^{\\nu_x}\\right) +  b_\\pm \\exp\\left(-\\frac{\\k}{\\nu_x}s^{\\nu_x}\\right),\n    \\nonumber\\\\\n    \\chi_{\\pm,L}(s) &\\approx \\mp i a_\\pm \\exp\\left(\\frac{\\k}{\\nu_x}s^{\\nu_x}\\right) \\pm i b_\\pm \\exp\\left(-\\frac{\\k}{\\nu_x}s^{\\nu_x}\\right),\n    \\label{eq:approximate_solution_L_s}\n\\end{align}\nwhere we have added the subscripts \\(L\\) as a reminder that this solution applies to the left of \\(s=1\\). An argument almost identical to the one we made for \\((\\psi_{\\pm,R},\\chi_{\\pm,R})\\) shows that \\((\\psi_{\\pm,L},\\chi_{\\pm,L})\\) are good approximations to the full solution of equation~\\eqref{eq:fermion_eom_ir_kx_s} near \\(s \\sim \\mathcal{O}(1)\\) when \\(\\k \\ll 1\\).\n\nWe fix the integration constants \\(a_\\pm\\) and \\(b_\\pm\\) in equation~\\eqref{eq:approximate_solution_L_s} by matching to equation~\\eqref{eq:approximate_solution_R_s} at some \\(s_m~\\mathcal{O}(1)\\), near which both solutions provide good approximations. Setting \\(\\psi_{\\pm,L}(s_m) = \\psi_{\\pm,R}(s_m)\\) and \\(\\chi_{\\pm,L}(s_m) = \\chi_{\\pm,R}(s_m)\\), we can solve for the integration constants to find\n\\begin{align}\n    a_\\pm  &= \\frac{1}{\\sqrt{2}} \\exp\\left(i \\k s_m - \\frac{\\k }{\\nu_x} s_m^{\\nu_x}\\mp \\frac{i \\pi}{4}\\right) = \\frac{e^{\\mp i \\pi\/4}}{\\sqrt{2}} + \\mathcal{O}(\\k),\n    \\nonumber \\\\\n    b_\\pm &= \\frac{1}{\\sqrt{2}} \\exp\\left(i \\k s_m + \\frac{\\k }{\\nu_x} s_m^{\\nu_x} \\pm i \\frac{\\pi}{4}\\right) = \\frac{e^{\\pm i \\pi\/4}}{\\sqrt{2}} + \\mathcal{O}(\\k).\n    \\label{eq:fermion_greens_function_matching_coefficients}\n\\end{align}\nAs a check of our matching procedure we plot the relative errors \\(|\\psi_{+,m} - \\psi_+|\/|\\psi_+|\\) and \\(|\\chi_{+,m} - \\chi_+|\/|\\chi_+|\\) for \\(\\nu_x=-1\/3\\) and \\(s_m=1\\), and sample values of \\(\\k\\) in figure~\\ref{fig:example_matching}, where \\((\\psi_+,\\chi_+)\\) are a numerical solution of equation~\\eqref{eq:fermion_eom_ir_kx_s} with ingoing boundary conditions, while \\((\\psi_{+,m},\\chi_{+,m})\\) are matched approximations, given by equation~\\eqref{eq:approximate_solution_R_s} for \\(s > 1\\) and equation~\\eqref{eq:approximate_solution_L_s} for \\(s<1\\), with coefficients~\\eqref{eq:fermion_greens_function_matching_coefficients}. The relative error indeed becomes very small at small \\(\\k\\).\n\\begin{figure}\n    \\begin{center}\n    \\includegraphics{figs\/example_matching_psi.pdf}\n    \\includegraphics{figs\/example_matching_chi.pdf}\n    \\end{center}\n    \\caption{\n        Demonstration of the validity of the matching procedure at small \\(\\k\\). We plot the error in \\(\\psi_+\\) and \\(\\chi_+\\) obtained from the matching calculation relative to a numerical solution of equation~\\eqref{eq:fermion_eom_ir_kx_s} with ingoing boundary conditions \\(\\psi_+ \\approx e^{i\\k s}\\) and \\(\\chi_+ \\approx e^{-i\\k s}\\) at large \\(s\\). In other words, what is plotted is the absolute value of the difference between the matching and numerical solutions, divided by the numerical solution. For concreteness we take \\(\\nu_x = -1\/3\\) (the same value used in section~\\ref{sec:zeroT}) and \\(s_m=1\\). The relative error clearly becomes very small at small \\(\\k\\).\n    }\n    \\label{fig:example_matching}\n\\end{figure}\n\nReturning to the radial coordinate \\(\\zeta = s (\\bar{k}_x\/\\bar{\\omega})^{1\/(1-\\nu_x)}\\), we have found the following approximate solution to equation~\\eqref{eq:fermion_eom_ir_kx}, obeying ingoing boundary conditions at \\(\\zeta \\to \\infty\\) and valid at small \\(\\bar{\\omega}\\) and fixed \\(\\bar{k}_x\\),\n\\begin{align}\n    \\psi_\\pm(\\zeta) &= \\begin{cases}\n        e^{i \\bar{\\omega} \\zeta}, & \\zeta > \\zeta_m,\\\\\n        \\sqrt{2} \\cosh \\left(\\frac{\\bar{k}_x}{\\nu_x} \\zeta^{\\nu_x} - \\frac{i\\pi}{4} \\right),\\phantom{\\mp} & \\zeta < \\zeta_m.\n    \\end{cases}\n    \\nonumber \\\\\n    \\chi_\\pm(\\zeta) &= \\begin{cases}\n        - e^{i \\bar{\\omega} \\zeta}, & \\zeta > \\zeta_m,\\\\\n        \\mp i \\sqrt{2} \\sinh \\left(\\frac{\\bar{k}_x}{\\nu_x} \\zeta^{\\nu_x} - \\frac{i\\pi}{4} \\right), & \\zeta < \\zeta_m.\n    \\end{cases}\n    \\label{eq:kx_dominant_matching_solution}\n\\end{align}\nwhere \\(\\zeta_m = s_m(\\bar{k}_x\/\\bar{\\omega})^{1\/(1-\\nu_x)}\\).\n\nAs \\(\\zeta\\) is decreased from infinity, a point will eventually be reached where equation~\\eqref{eq:fermion_eom_ir_kx} ceases to provide a good approximation to the full equations of motion, and thus the matching solution~\\eqref{eq:kx_dominant_matching_solution} will cease to provide a good approximate solution. This is both because of the terms we have neglected from equation~\\eqref{eq:fermion_eom_ir} and because the metric functions and gauge field will depart from their scaling forms~\\eqref{equ:scaling}. At low frequencies the approximate location at which this occurs will be independent of frequency, being determined by the relative size of the neglected terms and the terms proportional to \\(\\bar{k}_x\\) in equation~\\eqref{eq:fermion_eom_ir_kx}.\n\nWe can thus choose to evaluate the IR Green's function \\(\\mathcal{G}_{\\pm} = - i \\chi_\\pm(\\zeta_0)\/\\psi_\\pm(\\zeta_0)\\) at some frequency-independent \\(\\zeta_0\\) satisfying \\(1 \\ll \\zeta_0 \\ll \\zeta_m\\) (note that \\(\\zeta_m\\to\\infty\\) as \\(\\bar{\\omega}\\to0\\)), where we choose \\(\\zeta_0\\) to lie within the range of validity of the matching solution. We can then use equation~\\eqref{eq:kx_dominant_matching_solution} to evaluate the IR Green's function, obtaining\n\\begin{equation}\n    \\mathcal{G}_{\\pm}(\\omega,k_x,0) \\approx i, \n    \\qquad\n    \\bar{\\omega} \\ll 1.\n\\end{equation}\nRecalling that the low frequency scaling of \\(\\Im \\mathcal{G}_\\pm\\) determines the low frequency scaling of the imaginary part of the full Green's function, we find that the spectral function tends to a non-zero constant at small frequency.\n\n\n\\subsection[q term dominant]{\\(\\bar{q}\\) term dominant}\n\\label{sec:matching_q_dominant}\n\nNow we consider the case in which the first subleading terms in equation~\\eqref{eq:fermion_eom_ir} at large \\(\\zeta\\) are the ones proportional to \\(\\bar{q}\\). This will occur if \\(\\nu_x < 2\\nu_\\theta-\\nu_A\\) and \\(\\bar{m}=0\\).\\footnote{Notice that for non-zero mass and \\(\\nu_\\theta \\leq 0\\), the mass term always dominates over the charge term, \\(\\bar{m} \\zeta^{\\nu_\\theta} \\gg \\bar{q} \\zeta^{2\\nu_\\theta-\\nu_A}\\) at large \\(\\zeta\\), due to the requirement \\(\\nu_A \\geq 0\\) in equation~\\eqref{eq:exponent_conditions}.} In this case, at sufficiently large \\(\\zeta\\) the IR equations~\\eqref{eq:fermion_eom_ir} will be well approximated by\n\\begin{equation}\n    \\psi_\\pm'   + i \\left(\\bar{\\omega} + \\frac{\\bar{q}}{\\zeta^{1-2\\nu_\\theta+\\nu_A}}  \\right) \\chi_\\pm = 0,\n    \\qquad\n    \\chi_\\pm'  + i \\left(\\bar{\\omega} + \\frac{\\bar{q}}{\\zeta^{1-2\\nu_\\theta+\\nu_A}} \\right) \\psi_\\pm = 0.\n    \\label{eq:fermion_eom_ir_q}\n\\end{equation}\nThese can be solved exactly. With ingoing boundary conditions we have\n\\begin{equation}\n    \\psi_\\pm(\\zeta) = \\exp\\left(i \\bar{\\omega} \\zeta - i \\frac{\\bar{q}\\zeta^{-(\\nu_A - 2 \\nu_\\theta)} }{\\nu_A-2\\nu_\\theta} \\right),\n    \\qquad\n    \\chi_\\pm(\\zeta) = - \\exp\\left(i \\bar{\\omega} \\zeta - i \\frac{\\bar{q}\\zeta^{-(\\nu_A - 2 \\nu_\\theta)} }{\\nu_A-2\\nu_\\theta} \\right),\n    \\label{eq:q_dominant_solution}\n\\end{equation}\nvalid over the region for which equation~\\eqref{eq:fermion_eom_ir_q} provides a good approximation to the full equations of motion. The eigenvalues of the IR Green's function are \\(\\mathcal{G}_\\pm = - i \\chi_\\pm(\\zeta_0)\/\\psi_\\pm(\\zeta_0)\\), where \\(\\zeta_0\\) is some large value of \\(\\zeta\\) within this region. From equation~\\eqref{eq:q_dominant_solution} we find\n\\begin{equation}\n    \\mathcal{G}_\\pm(\\omega,k_x,0) \\approx i, \\qquad \\bar{\\omega} \\ll 1.\n\\end{equation}\nAgain, the fermion spectral function tends to a non-zero value at zero frequency.\n\n\\subsection[m term dominant]{\\(\\bar{m}\\) term dominant}\n\\label{sec:matching_m_dominant}\n\nFinally, we consider the case in which for \\(\\zeta \\gg 1\\) and \\(\\bar{k}_x \\sim \\mathcal{O}(1)\\) that \\(\\bar{m} \\zeta^{\\nu_\\theta} \\gg \\bar{k}_x \\zeta^{\\nu_x} \\). This requires \\(\\nu_\\theta > \\nu_x\\).\nThe calculation is very similar to that of section~\\ref{sec:matching_kx_dominant}, so we will be brief with the details. At sufficiently large \\(\\zeta\\), the IR equations~\\eqref{eq:fermion_eom_ir} are well approximated by\n\\begin{equation}\n    \\psi_\\pm' + \\frac{\\bar{m}}{\\zeta^{1 - \\nu_\\theta}} \\psi_\\pm + i\\bar{\\omega}  \\chi_\\pm = 0,\n    \\qquad\n    \\chi_\\pm' - \\frac{\\bar{m}}{\\zeta^{1 - \\nu_\\theta}} \\chi_\\pm  + i \\bar{\\omega} \\psi_\\pm = 0,\n    \\label{eq:fermion_eom_ir_m}\n\\end{equation}\nIt will be convenient to rewrite these equations by defining a rescaled radial coordinate \\(s = \\zeta \\left(\\bar{\\omega}\/\\bar{m} \\right)^{1\/(1-\\nu_\\theta)}\\) (note that this is a different \\(s\\) from section~\\ref{sec:matching_kx_dominant}), in terms of which they read\n\\begin{equation}\n    \\psi_\\pm'(s) + \\frac{\\mathfrak{m}}{s^{1-\\nu_\\theta}} \\psi(s)+ i \\mathfrak{m} \\chi(s) = 0,\n    \\qquad\n    \\chi_\\pm'(s) - \\frac{\\mathfrak{m}}{s^{1-\\nu_\\theta}} \\chi(s) + i \\mathfrak{m} \\chi(s) = 0,\n    \\label{eq:fermion_eom_ir_m_s}\n\\end{equation}\nwhere\n\\begin{equation}\n    \\mathfrak{m} = \\left(\\frac{m}{\\omega^{\\nu_\\theta}}\\right)^{1\/(1-\\nu_\\theta)} \\ll 1.\n\\end{equation}\n\nAt large \\(s\\) we neglect the terms in equation~\\eqref{eq:fermion_eom_ir_m_s} proportional to \\(s^{-1\/(1-\\nu_\\theta)}\\), obtaining the ingoing solution\n\\begin{equation}\n    \\psi_{\\pm,R}(s) = e^{i \\mathfrak{m} s},\n    \\qquad\n    \\chi_{\\pm,R}(s) = - e^{i \\mathfrak{m} s}.\n\\end{equation}\nOn the other hand, at small \\(s\\) the terms in equation~\\eqref{eq:fermion_eom_ir_m_s} proportional to \\(s^{-1\/(1-\\nu_\\theta)}\\) are dominant, and we obtain the approximate solution\n\\begin{equation}\n    \\psi_{\\pm,L}(s) = a_\\pm e^{-\\mathfrak{m} s^{\\nu_\\theta}\/\\nu_\\theta},\n    \\qquad\n    \\chi_{\\pm,L}(s) = b_\\pm e^{\\mathfrak{m} s^{\\nu_\\theta}\/\\nu_\\theta},\n\\end{equation}\nwhere \\(a_\\pm\\) and \\(b_\\pm\\) are integration constants.\n\nAn argument similar to the one made in section~\\ref{sec:matching_kx_dominant} shows that both \\((\\psi_{\\pm,L},\\chi_{\\pm,L})\\) and \\((\\psi_{\\pm,R}, \\chi_{\\pm,R})\\) are good approximate solutions at \\(s \\sim \\mathcal{O}(1)\\) when \\(\\mathfrak{m} \\ll 1\\). We can thus fix the integration constants by setting \\(\\psi_{\\pm,L}(s_m) = \\psi_{\\pm,R}(s_m)\\) and \\(\\chi_{\\pm,L}(s_m) = \\chi_{\\pm,R}(s_m)\\). This yields \\(a_\\pm = 1\\) and \\(b_\\pm = -1\\) to leading order at small \\(\\mathfrak{m}\\) (and thus at small \\(\\bar{\\omega}\\)). In terms of the radial coordinate \\(\\zeta\\), the approximate matching solution is then\n\\begin{equation}\n    \\psi_\\pm(\\zeta) = \\begin{cases}\n        e^{i\\bar{\\omega} \\zeta}, & \\zeta > \\zeta_m,\n        \\\\\n        e^{\\bar{m} \\zeta^{\\nu_\\theta}\/\\nu_\\theta}, & \\zeta < \\zeta_m,\n    \\end{cases}\n    \\qquad\n    \\chi_\\pm(\\zeta) = \\begin{cases}\n        e^{i\\bar{\\omega} \\zeta}, & \\zeta > \\zeta_m,\n        \\\\\n        -e^{\\bar{m} \\zeta^{\\nu_\\theta}\/\\nu_\\theta}, & \\zeta < \\zeta_m,\n    \\end{cases}\n    \\label{eq:m_dominant_matching_solution}\n\\end{equation}\nwhere \\(\\zeta_m = s_m (\\bar{m}\/\\bar{\\omega})^{1\/(1-\\nu_\\theta)}\\). The IR Green's function is \\(\\mathcal{G}_{\\pm} = - i \\chi_\\pm(\\zeta_0)\/\\psi_\\pm(\\zeta_0)\\) for some \\(\\zeta_0\\) satisfying \\(1 \\ll \\zeta_0 \\ll \\zeta_m\\). Evaluating this using equation~\\eqref{eq:m_dominant_matching_solution}, we find that at leading order at small frequencies\n\\begin{equation}\n    \\mathcal{G}_{\\pm}(\\omega,k_x,0) \\approx i, \\qquad \n    \\bar{\\omega} \\ll 1.\n\\end{equation}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpoken dialogue systems (SDS) have been deployed in many gadgets such as smartphone assistants and smart speakers in the past ten years.\nMillions of users now use them daily. In these systems, dialogue is regarded as a means to complete some tasks by the machine.\nThe typical tasks include command and control of the machine or application software (e.g., music player and alarm) and simple information retrieval (e.g., weather and routing).\nThe task goals are objective and shared by the system and users.\nThus, the dialogue should be completed as soon as possible.\nThere is a big gap from the human-human dialogue, in which task goals are not definite and the duration is flexible.\n\nConversational robots are another application of the spoken dialogue systems, but they should be designed in a different principle from ``smart'' devices mentioned above.\nWe do not need a robot to ask for weather information and alarm.\nInstead, the conversational robots are expected to simply talk to or listen to humans, as humans do.\nAn ultimate goal is to realize human-level dialogue, which is engaging and pleasant.\nApart from the quality of dialogue, there is a significant difference in the style of the dialogue.\nIn the current human-machine interface, a user is assumed to utter one sentence per one turn, which corresponds to a command or a query, to which the system will respond.\nOn the other hand, in natural human dialogue, a participant utters many sentences per one turn, during which the counterpart makes backchannels.\nThe difference can be analogous to half-duplex versus full-duplex in the communication channel.\nIt is weird to nod or backchannel to a machine, but a human-looking android would help the realization of human-level dialogue.\n\nWith these backgrounds, we are conducting a project to develop an autonomous android ERICA~\\cite{dylan2016roman,inoue2016sigdial,kawahara2018iwsds}, who behaves and interacts just like a human, including facial look and expression, gaze and gesture, and spoken dialogue.\nAn ultimate criterion is to pass a Total Turing Test, that is to convince people that ERICA's interaction is comparable to a human, or it is indistinguishable from a remote-operated android.\nWe hope that through this project we can make clear what is missing or critical in natural interaction.\nERICA is expected to replace some social roles currently done by a human, or used for conversation skill training in a realistic setting.\n\nIn this article, the major challenges and key technologies in this autonomous android are described.\nIn particular, the systems developed for attentive listening and job interview are explained with evaluations.\n\n\n\\section{Android ERICA}\n\nFigure~\\ref{fig:erica} shows a snapshot of ERICA.\nShe does not move around, but generates elaborate facial expressions and movements of heads, eyes, and body attitudes while sitting on a chair, so that she can behave and interacts exactly like a human.\nWe have also developed ASR (automatic speech recognition) and TTS (text-to-speech) systems as well as the human tracking system dedicated to ERICA.\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[width=90mm]{erica-crop.pdf}\n  \\end{center}\n  \\caption{Android ERICA}\n  \\label{fig:erica}\n\\end{figure}\n\n\\subsection{Social interaction tasks}\n\nWe have explored a number of social roles suited to ERICA that can take advantage of human-like presence and would realize human-like dialogue.\nA majority of tasks conducted by the current SDSs such as information services are not adequate; they are better suited to smartphones and smart speakers.\nWhile most conventional robots are engaged in physical tasks such as moving objects or delivering goods, recently many kinds of communicative robots are designed and introduced in public spaces.\nThey serve as an attendant~\\cite{fujie2009conversation} or a receptionist~\\cite{bohus2016models}.\nThey are effective for attracting people, but the resulting interaction is usually very shallow and short such as a predefined script and simple guidance.\nIn recent years, chatting systems are also developed intensively, but the dialogue is also shallow and not so engaging. In contrast, interaction with ERICA should leverage physical presence and involve face-to-face communication.\nTherefore, we design ``social interaction'' tasks in which human-like presence matters and long deep interaction is exchanged. Here, dialogue itself is a task, and the goal of dialogue may be mutual understanding or appealing. We assign a realistic social role to ERICA, so matched users are seriously engaged beyond chatting. Specifically, we set up the following four tasks.\nThey are compared in Table~\\ref{table:task_for_erica}.\n\n\\subsubsection{Attentive listening}\n\nIn this task, ERICA mostly listens to senior people talking about topics such as memorable travels and recent activities~\\cite{lala2017}.\nAttentive listening is being recognized as effective for maintaining the communication ability of senior people, and many communicative robots are designed for this task.\nThe role of ERICA is to encourage users to speak for long.\nIn this sense, attentive listening is similar to counseling~\\cite{devault2014simsensei}.\n\n\\subsubsection{Job interview (practice)}\n\nWhile dialogue systems have been investigated for casual interviews~\\cite{kobori2016small}, a job interview is very important for both applicants, typically students, and companies hiring them.\nEach side makes a lot of preparations including rehearsal.\nIn this setting, ERICA plays the role of interviewer by asking questions.\nShe provides a realistic simulation, and is expected to replace a human interviewer in the future.\n\n\\subsubsection{Speed dating (practice)}\n\nSpeed dating is widely held for giving an opportunity for people to find a partner.\nIn this setting, two persons meet for the first time and talk freely to introduce themselves, and see if the counterpart can be a good match.\nThere was a study~\\cite{ranganath2009s} that analyzed a corpus of speed dating.\nIn our setting, ERICA plays the role of the female participant by talking about topics such as hobbies and favorite foods.\nShe provides a realistic simulation and gives proper feedbacks according to the dialogue.\n\n\\subsubsection{Lab guide}\n\nA robot is often used for introducing laboratories and museums.\nIt is expected to attract people, and needs to keep their engagement by providing proper interaction.\n\n\\begin{table}[t]\n  \\caption{Dialogue tasks designed for ERICA}\n  \\label{table:task_for_erica}\n  \\begin{center}\n  \\begin{tabular}{lcccc}\n  \\hline\n  & attentive listening & job interview & speed dating & lab guide \\\\\n  \\hline\n  role of system & listen & ask & all & talk \\\\\n  dialogue initiative & user & system & mixed & system \\\\\n  main speaker & user & user & both & system \\\\\n  main listener & system & system & both & user \\\\\n  turn-taking & few & explicit & complicate & explicit \\\\\n  \\hline\n  \\end{tabular}\n  \\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[width=150mm]{system-crop.pdf}\n  \\end{center}\n  \\caption{System configuration for android ERICA}\n  \\label{fig:system}\n\\end{figure}\n\n\\subsection{System configuration} \\label{sec:system}\n\nFigure~\\ref{fig:system} depicts the whole system configuration for android ERICA.\nThe input devices are a 16-channel microphone array and a depth camera (Kinect v2).\nThese sensors are placed within the interaction environment but not physically within or on the android robot.\nThe sensors do not require any human contact or pre-dialogue calibration, so users can start and engage in dialogue naturally without awareness of any sensors (as opposed to using hand microphones, for example).\nWe use the sensors for sound source localization, voice activity detection, and speech enhancement to obtain the segmented speech of the target user~\\cite{carlos2016iros}.\nThe enhanced speech is then fed into the ASR (automatic speech recognition) module which uses an acoustic-to-subword end-to-end neural network model.\nSimultaneously, prosodic information such as fundamental frequency (F0) and power is extracted from the enhanced speech~\\cite{carlos2016iros}.\nThe ASR result is then fed into the dialogue manager to generate a system response which will be then played by a TTS (text-to-speech) engine designed for ERICA\\footnote{\\url{https:\/\/voicetext.jp\/news\/product\/151023\/}}.\nThe TTS engine is also enabled to play non-linguistic utterances such as backchannels, fillers, and laughing.\nLip and head movements are controlled based on the generated TTS speech~\\cite{carlos2012iros,sakai2015roman}.\n\n\n\n\\section{Technologies for human-level conversations}\n\nFor human-level conversation, linguistic processing and understanding is necessary but non-linguistic behaviors are similarly important. \nWe briefly explain several of these components for ERICA.\n\n\\subsection{Backchannel generation}\n\nBackchannels are short utterances uttered by listeners such as ``{\\it yeah}'' in English and ``{\\it un}'' in Japanese.\nListeners utter backchannels toward speakers to stimulate further talk and also to express the listener's attention and interest in the conversation.\nIn order to generate backchannels, systems need to predict the timing of backchannels, which has been tackled by many works using prosodic features~\\cite{ward2000prosodic,truong2010rule}.\nWhile Existing backchannel generation systems make a prediction of backchanneling after the end of user utterances segmented by IPUs (inter-pausal units), we implement frame-wise continuous prediction of backchannel timing with a logistic regression that predicts if the system should utter a backchannel within the next 500 milliseconds~\\cite{lala2017}.\nWe also proposed a prediction model of backchannel form (type) based on both prosodic and linguistic features~\\cite{kawahara2016prediction}.\n\n\\subsection{Flexible turn-taking using TRP}\n\nFlexible and robust turn-taking is a required function for natural communication between the system and a user.\nEnd-of-turn prediction has been widely studied using linguistic and prosodic features extracted from the user's utterance~\\cite{skantze2017sigdial,masumura2017interspeech}.\nExisting models were trained with actual turn-taking behaviors in dialogue corpora, but the turn-taking decision is sometimes arbitrary so model training can become unstable.\nTherefore, we take into account the concept of TRPs (transition-relevance places)~\\cite{sacks1974TRP} where the current turn could be completed, and are independent of the actual end-of-turn instances.\nWe manually annotated these TRP instances in our human-robot dialogue corpus~\\cite{hara2019interspeech}.\nWe then proposed a two-step turn-taking prediction model where the system first detects the TRP, and if detected, predicts if the system actually takes the turn.\nBy decomposing complex turn-taking prediction into TRP detection and subsequent turn prediction, the accuracy of turn-taking was improved.\n\nFrom the perspective of live spoken dialogue systems, another required function for the turn-taking module is to predict how long the system waits until it actually takes the turn.\nA simple approach is to set a fixed silence threshold time, but it may cause false cut-in (interruption) or unnaturally long silences.\nOur system integrates the above-mentioned turn-taking prediction model using TRP labels with an FSTTM (Finite-State Turn-Taking Machine)~\\cite{Raux2009,Lala2018} to determine the silence time the system will wait for.\nIt can quickly take the turn with a high probability of being end-of-turn, while it can also wait longer if the user says utterances such as fillers or hesitations.\n\n\\subsection{Engagement recognition}\n\nEngagement is defined as how much a user is interested in the current dialogue, and keeping users engaged is an important factor that leads to ``long and deep'' dialogue~\\cite{oertel2020engagement}.\nTo this end, engagement recognition has been widely studied by investigating multi-modal user behaviors~\\cite{dhall2019emotiw}.\nWe proposed an engagement recognition model based on listener behaviors such as backchannels, laughs, head nods, and eye contact~\\cite{inoue2018engagement}.\nWe automatically detected these listener behaviors to implement a real-time engagement recognition system.\nThis system was applied to the laboratory guide dialogue task with ERICA to control the android robot's behaviors according to the level of user engagement~\\cite{inoue2019iwsds}.\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[width=160mm]{response-list-ver2-crop.pdf}\n  \\end{center}\n  \\caption{List of listener responses in our attentive listening system (Examples of generated responses are shown in right side in this figure. Underline means the focus focus word.)}\n  \\label{fig:response_list}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[width=80mm]{recording-crop.pdf}\n  \\end{center}\n  \\caption{Snapshot of dialogue experiment for our attentive listening system with ERICA}\n  \\label{fig:experiment}\n\\end{figure}\n\n\\section{Attentive Listening System}\n\nWe implemented an attentive listening system with ERICA~\\cite{lala2017,inoue2020sigdial}.\nThe aforementioned backchannel generation model is a core component in this system.\nBesides backchannels, the system generates a variety of listener responses: partial repeats, elaborating questions, assessments, generic sentimental responses, and generic responses, as depicted in Figure~\\ref{fig:response_list}.\nPartial repeats and elaborating questions are generated based on the {\\it focus word} of user utterances.\nAssessments and generic sentimental responses are based on the sentiment (positive or negative) of user utterances.\n\nWe have conducted dialogue experiments with a total of 40 senior people and confirmed that they could engage with ERICA for 5-7 minutes without a conversation breakdown.\nFigure~\\ref{fig:experiment} shows the snapshot of this experiment.\nWe also manually evaluated each system response and found that about 60\\% of the system responses were acknowledged as appropriate, which means the other 40\\% responses can be improved.\nThe novelty of our experiment is to compare our system with a human listener implemented in a WOZ (Wizard-of-OZ) setting with a hidden human operator tele-operating ERICA~\\cite{inoue2020sigdial}.\nFrom the subjective evaluation, our system achieved comparable scores against the WOZ setting in basic skills of attentive listening such as {\\it encouragement to talk}, {\\it focused on the talk}, and {\\it actively listening}.\nOn the other hand, there is still a gap between our system and human listeners for more sophisticated skills such as {\\it dialogue understanding}, {\\it showing interest}, and {\\it empathy towards the user}.\nFrom this experiment we could identify the type of skills that differed between our system and human listeners.\nIn future work, we will improve the response generation modules to increase the ratio of appropriate responses and close the gap with human listeners.\n\n\n\\section{Job Interview System}\n\nWe also implemented a job interview system where ERICA plays the role of an interviewer~\\cite{inoue2020icmi}.\nExisting job interview systems give the same pre-defined questions to any interviewees, so interviews tend to be tedious and far from real human-human interviews.\nTo make interviews more realistic, our system generates follow-up questions dynamically based on initial responses from the interviewee with two different approaches.\nThe first approach is based on assessing the quality of the interviewee's utterance using a checklist of items that ``{\\it should be mentioned}'' in the job interview.\nFor example, when the interviewee could not mention what he or she wants to contribute to the company, the system generates a follow-up question such as ``{\\it Which part of our company do you want to contribute to?}''. \nThe second approach is based on keyword extraction from the interviewee's response.\nFor example, when the interviewee said ``{\\it I have an experience on machine learning.}'', then the system extracts a keyword as ``{\\it machine learning}'' and then generates a follow-up question such as ``{\\it Could you explain more about machine learning?}''\n\nWe conducted a dialogue experiment with university students to compare our system with a baseline system that asked only pre-defined basic questions.\nWe found that our system was significantly better than the baseline system in the quality of the questions.\nIt was also found that the perceived presence of the android interviewer was enhanced by the follow-up questions~\\cite{inoue2020icmi}.\nAnother research question in this job interview scenario is how much the interviewer's appearance affects the above-reported experimental result, so we also conducted the same experiment with a virtual agent, as shown in Figure~\\ref{fig:mmd}.\nAs a result, we confirmed a similar result for the follow-up questions even with the virtual agent interviewer, but the presence of the virtual agent interviewer was not enhanced.\nThe presence of the interviewer is an important factor in creating an interview system with a realistically tense atmosphere.\nTherefore, it is more effective to implement the follow-up questions with android ERICA as the job interviewer than a virtual agent.\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[width=120mm]{mmd-crop.pdf}\n  \\end{center}\n  \\caption{Difference of appearance between ERICA and virtual agent in our job interview dialogue experiment}\n  \\label{fig:mmd}\n\\end{figure}\n\n\\section{Conclusions}\n\nBecause of the pervasiveness of SNS as a communication channels and then COVID-19, importance of face-to-face communication is questioned or emphasized.\nThe android ERICA provides a testbed for investigating what are important factors for the quality communications.\nThe android is expected to replace some social roles under and after the COVID-19 era.\n\n\\section*{Acknowledgments}\nThis work was supported by JST ERATO Grant number JPMJER1401 and JSPS KAKENHI Grant number JP19H05691.\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{sec:intro}\nIn recent years, higher-order perturbative calculations of the quark and gluon form factors in massless Quantum Chromodynamics (QCD) have generated a great deal of interest. \nThis is primarily due to their relevance to two of the most important Large Hadron Collider processes, the production of a Drell-Yan lepton pair \\cite{Drell:1970wh}\nand the production of a Higgs boson via gluon fusion \\cite{Georgi:1977gs}. \nAt the most basic level, a calculation of the quark and gluon form factors at some fixed order provides, respectively, the purely virtual QCD corrections to Drell-Yan lepton production \nand gluon-fusion Higgs boson production in the infinite top-quark-mass limit~\\cite{Wilczek:1977zn,Shifman:1978zn,Ellis:1979jy,Inami:1982xt}.\nFurthermore, the cusp and collinear anomalous dimensions can be extracted from $\\epsilon^{-2}$ and $\\epsilon^{-1}$ poles of the bare form factors.\nThey determine the structure of the quark and gluon jet functions, objects which play an important role in the theory of the infrared divergences of multi-leg, massless scattering amplitudes\nand in the theory of soft-gluon resummation (see {\\it e.g.} reference \\cite{Becher:2014oda} for a recent review).\n\nEven if all real radiation is very soft relative to the scale of the hard scattering, the purely virtual corrections to a process are only one part of the story.\nUsing the eikonal approximation to treat all real radiation allows one to give an approximate cross section for a process at higher orders in QCD perturbation theory.\nIn reference \\cite{Harlander:2002wh}, this so-called soft-virtual approximation was shown\nto capture a sizable fraction of the total gluon-fusion Higgs boson production cross section at next-to-next-to-leading order.\nAlthough three-loop calculations of both form factors through the finite terms \\cite{Baikov:2009bg,Lee:2010cga,Gehrmann:2010ue} were key ingredients for recent\nthird-order, soft-virtual calculations of the cross sections for Drell-Yan lepton production and gluon-fusion Higgs boson production, much more work is required to obtain the final results. \nAs usual, it is also necessary to consider the radiation of additional partons and then combine all relevant real radiative\ncorrections with the purely virtual contributions. Only then are infrared-finite results obtained which can be convoluted with the appropriate parton distribution functions to produce meaningful numbers.\nIn a series of papers \\cite{Anastasiou:2013srw,Li:2013lsa,Duhr:2013msa,Anastasiou:2014vaa,Li:2014bfa,Ahmed:2014cla,Li:2014afw}, results were obtained which,\nin principle, could have determined both the Drell-Yan lepton production cross section and the gluon-fusion Higgs boson production cross section to levels of precision sufficient for the purposes of LHC physics.\nAll of this work culminated in a new milestone, namely two independent calculations of the parton-level cross sections for both processes in the soft-virtual\nnext-to-next-to-next-to-leading order (N$^3$LO) approximation \\cite{Anastasiou:2014vaa,Ahmed:2014cla,Li:2014afw}. \n\nA systematic inclusion of power-corrections in the soft-virtual Higgs-production\nanalysis~\\cite{Anastasiou:2014lda} shows significant contributions beyond the\nleading term. While the small numerical impact at high expansion order suggests that this approximate N$^3$LO prediction is sufficiently precise for the purposes of Large Hadron Collider phenomenology,\nit is still interesting to carry out the exact N$^3$LO calculation to put the analysis on more rigorous grounds.\nIt has long been known that the accuracy of predictions for the above-mentioned production processes can be improved significantly if appropriate towers of logarithms \ncoming from soft radiation near the production threshold are resummed~\\cite{Kramer:1996iq}.\nTo improve upon the current state-of-the-art, one could carry out a soft parton resummation of the next-to-next-to-next-to-leading Sudakov logarithms (N$^3$LL order) \nand then match this onto the exact fixed order N$^3$LO prediction once the result becomes available.\nAt this stage, it would also be of interest \nto include a resummation of additional terms in the virtual matrix elements which appear due to the fact that $s$-channel processes have a time-like momentum transfer \\cite{Ahrens:2008qu,Ahrens:2008nc}.\n\nIn fact, some partial progress towards this more ambitious goal has already been realized.\nGiven the impressive number of recently-obtained \nresults \\cite{Hoschele:2012xc,Anastasiou:2013mca,Kilgore:2013gba,Hoschele:2014qsa,Duhr:2014nda,Dulat:2014mda,Anastasiou:2015ema,Anastasiou:2015yha,Bonvini:2014joa,Catani:2014uta,deFlorian:2014vta,Anzai:2015wma}, \nit seems clear that the exact parton-level N$^3$LO results lie within reach and that,\nfurthermore, matching them with N$^3$LL Sudakov resummations will not be an issue once all of the required ingredients\nbecome available. Actually, almost all of the quantities required for a complete N$^3$LL resummation have already been available in the literature for quite some time, both for Drell-Yan and for Higgs. However, in order to properly carry out a \nN$^3$LL resummation, one needs the four-loop cusp anomalous dimensions. Although the three-loop cusp anomalous dimensions were calculated long ago \\cite{Moch:2004pa},\nsurprisingly little has been reported at one order higher; several interesting techniques have been \ndeveloped \\cite{Panzer:2013cha,Panzer:2014gra,Panzer:2014caa,Panzer:2015ida,vonManteuffel:2014qoa,vonManteuffel:2014ixa,Ablinger:2012ph,Ablinger:2015tua,Ruijl:2014hha,Lee:2012te,Henn:2013nsa}\nand a number of relevant master integrals (mainly propagators\\footnote{One four-loop three-point integral with off-shell external momenta was presented in \\cite[example~3.6]{Panzer:2014gra}.}) computed, \nsee \\cite{Baikov:2010hf,Smirnov:2010hd} and \\cite{Lee:2011jf,Lee:2011jt}\\footnote{Recently, it was reported that these computations can now be reproduced using the public software package\n{\\tt SummerTime} \\cite{Lee:2015eva}.}, but it is arguably the case that more progress has been made on the analogous problem\nin maximally supersymmetric gauge theory \\cite{Boels:2012ew,Boels:2015yna}.\n\nBesides the phenomenological motivation given above, a calculation of the four-loop cusp anomalous dimensions also has the potential to answer a long-standing question about the infrared structure of massless QCD. \nThrough three-loop order, it has been observed that the gluon cusp anomalous dimension can be derived from the quark cusp anomalous dimension by simply rescaling it. \nThe calculation of both the quark and the gluon cusp anomalous dimensions at four loops is therefore essential to see if this pattern continues to hold. \nThese four-loop calculations will provide the first non-trivial test of the putative Casimir scaling property of QCD:\nthe proposal that, to all loop orders in massless QCD, the gluon cusp anomalous dimension is the same as the quark cusp anomalous dimension up to an overall factor of $C_A\/C_F$, where $C_A = N_c$ and $C_F = (N_c^2-1)\/(2 N_c)$\nare respectively the quadratic Casimir invariants of the adjoint and the fundamental representations of the QCD gauge group.\\footnote{We assume a $SU(N_c)$ gauge group throughout this article.} \nA breakdown of Casimir scaling at four loop order would have profound implications. For example, it would mean that the four-loop infrared divergences in massless gauge theory scattering amplitudes at subleading color\ncannot straightforwardly be described in an abstract way which treats quarks and gluons on an equal footing \\cite{Becher:2009qa,Gardi:2009qi}.\n\nIn this paper we demonstrate the advantages of our approach \\cite{vonManteuffel:2014qoa} in a complete rederivation of the well-studied one-, two-, and three-loop form factors in perturbative QCD. \nWe found it particularly elegant to use a suitable basis of finite master integrals and made three main observations:\n\\begin{itemize}\n\t\\item\nThe $\\epsilon$ pole structure of the unexpanded form factors becomes absolutely explicit and has the striking feature that the most complicated integrals do not contribute to the cusp anomalous dimensions.\n\n\t\\item\nFinite integrals can be computed exactly and automatically in terms of multiple polylogarithms. In fact, our work constitutes the first complete analytical check of the weight eight, \n$\\mathcal{O}\\left(\\epsilon^2\\right)$ three-loop results published in \\cite{Gehrmann:2010tu}.\n\n\t\\item\nOn the numerical side, we obtained more than an order of magnitude improvement in both run time and precision for a fixed number of Monte Carlo integrand evaluations by employing a basis of finite integrals.\nRemarkably, we have since observed that, quite generally, one can substantially increase the reach and reliability of publicly available sector decomposition programs \\cite{Binoth:2000ps,Bogner:2007cr,Smirnov:2013eza,Borowka:2015mxa} \nif one first rotates to a basis of finite integrals. We will explore this in detail in a separate publication.\n\\end{itemize}\nTowards the end of this work, we take a first look at the computation of the four-loop cusp anomalous dimensions and explain how our method will allow for a significant reduction of the problem. \nWe illustrate our point at four loops by computing a non-trivial integral in an irreducible top-level sector. \nDue to the fact that the leading term in its $\\epsilon$ expansion consists solely of transcendental numbers of weight seven, it is not expected to contribute to the cusp anomalous dimensions.\n\nThis article is organized as follows. In Section~\\ref{sec:notation}, we define our $L$-loop bare form factors in massless QCD, introduce notation that we use,\nand state some useful facts about the structure of the form factors at one, two, and three loops. \nIn Section~\\ref{sec:method}, we describe our method of computation, focusing on its non-standard features.\nIn Sections \\ref{sec:1Lffs}, \\ref{sec:2Lffs}, and \\ref{sec:3Lffs}, we present our results for the one-, two- and three-loop bare form factors written as linear combinations of finite master integrals. \nWhen written in this way, the results have the remarkable feature that many of the most complicated master integrals do not contribute to the $\\epsilon^{-2}$ pole terms. \nWe explore this further in Section~\\ref{sec:cusps} and give an example of this phenomenon at the four-loop level,\nbased on the exact computation of a four-loop form factor with {\\texttt{HyperInt}}, a package for the evaluation of convergent Feynman integrals \\cite{Panzer:2014caa}.\nIn Appendix~\\ref{sec:finite}, we extend theoretical arguments given in reference \\cite{vonManteuffel:2014qoa}, showing that,\nfor scattering and decay processes which admit a Euclidean region respecting the kinematical constraints, one can always find a basis of finite loop integrals.\n\nAs a crucial supplement, ancillary files are available on arXiv.org which contain the quark and gluon form factors in terms of finite master integrals at one, two, and three loops,\nas well as all of our finite master integrals and final results $\\epsilon$-expanded to weight eight.\nWe also provide our highly-automated and parallelized setup for the computation of the $\\epsilon$-expansions using {\\texttt{HyperInt}} as described in Appendix~\\ref{sec:hyperint}.\nGiven sufficient computer resources, all of the Feynman integrals discussed in this paper may be straightforwardly reproduced by following the instructions given in the ancillary files.\n\n\\section{Notation and Conventions}\n\\label{sec:notation}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{oneloopffFeynmandiags}\n\\caption{The one-loop Feynman diagrams for the quark (upper panel) and gluon (lower panel) form factors in massless QCD. \nAt each order in the bare strong coupling constant, the effective coupling of the Higgs boson to gluons can be obtained by matching full QCD with a massive top quark onto an effective field theory in which the massive top quark is integrated out.}\n\\label{fig:samplediagrams}\n\\end{figure}\n\nIn this section, we give our definitions of the unrenormalized, massless quark and gluon form factors, $\\mathcal{F}_{\\rm bare}^q\\left(\\alpha_s^{\\rm bare}, (p_1 + p_2)^2, \\mu_\\epsilon^2, \\epsilon\\right)$ \nand $\\mathcal{F}_{\\rm bare}^g\\left(\\alpha_s^{\\rm bare}, (p_1 + p_2)^2, \\mu_\\epsilon^2, \\epsilon\\right)$. \nWe also establish some notation and state some facts about the general structure of our bare results at one, two, and three loops.\nThe $L$-loop contribution to the quark form factor is defined by the interference of the tree-level amplitude for $\\gamma^*(p_1+p_2) \\to q(p_1) \\bar{q}(p_2)$ with the $L$-loop corrections to this amplitude in massless QCD, \nnormalized to the tree-level amplitude squared. For all interferences, the sum over spin and color degrees of freedom is implied.\nSimilarly, the $L$-loop contribution to the gluon form factor is defined by the interference of the tree-level diagram for $h(p_1+p_2) \\to g(p_1) g(p_2)$ in the infinite top quark mass limit \nwith the $L$-loop corrections to this process in massless QCD, normalized to the tree-level amplitude squared (see Figure \\ref{fig:samplediagrams} for a visualization at the one-loop level). \n\nOur bare form factors have a perturbative expansion of the form\n\\begin{align}\n\\label{eq:expbareq}\n\\mathcal{F}_{\\rm bare}^q\\left(\\alpha_s^{\\rm bare}, (p_1 + p_2)^2, \\mu_\\epsilon^2, \\epsilon\\right) &= 1 + \\sum_{L = 1}^\\infty \\left(\\frac{\\alpha_s^{\\rm bare}}{4\\pi}\\right)^L \n\\left(\\frac{4\\pi \\mu_\\epsilon^2}{-(p_1+p_2)^2}\\right)^{L \\epsilon} \\frac{\\mathcal{F}_L^q(\\epsilon)}{\\Gamma^L(1-\\epsilon)}\n\\\\\n\\label{eq:expbareg}\n\\mathcal{F}_{\\rm bare}^g\\left(\\alpha_s^{\\rm bare}, (p_1 + p_2)^2, \\mu_\\epsilon^2, \\epsilon\\right) &= 1 + \\sum_{L = 1}^\\infty \\left(\\frac{\\alpha_s^{\\rm bare}}{4\\pi}\\right)^L \n\\left(\\frac{4\\pi \\mu_\\epsilon^2}{-(p_1+p_2)^2}\\right)^{L \\epsilon} \\frac{\\mathcal{F}_L^g(\\epsilon)}{\\Gamma^L(1-\\epsilon)},\n\\end{align}\nwhere $\\alpha_s^{\\rm bare}$ is the bare strong coupling constant, $(p_1 + p_2)^2$ is the momentum transfer squared, $\\mu_\\epsilon$ is the 't Hooft scale, and\n$\\epsilon$ is the parameter of dimensional regularization \\cite{'tHooft:1972fi}.\nThe dependence on the scale $(p_1+p_2)^2$ is trivial and may be reconstructed at any time by power counting. We therefore set\n\\begin{equation}\n(p_1+p_2)^2 = -1\n\\end{equation}\nto simplify our notation.\n\nThe master integrals for the form factors have traditionally been calculated in $d=4-2\\epsilon$ dimensions.\nIn the present paper, however, we make extensive use of dimensionally-shifted basis integrals.\nWe employ Minkowskian propagators and choose an absolute normalization of\n\\begin{equation}\n\\left(\\frac{\\Gamma(d\/2 - 1)}{i \\pi^{d\/2}}\\right)^{L}\n\\end{equation}\nfor our $L$-loop Feynman integrals defined in $d$ dimensions in order to prevent the appearance of spurious constants in our results.\nWe also consider propagators of higher multiplicity\nin our scalar Feynman integrals.\nThese we visualize by placing dots on the edges of the associated graphical representations,\nwhere a propagator power $\\nu + 1$ is represented by an edge with $\\nu$ dots on it.\nAs a concrete example, let us consider the one-loop Feynman integral which we actually use in Section \\ref{sec:1Lffs} below. It is the one-loop bubble integral in $d=6 - 2\\epsilon$ with both propagators squared:\n\\begin{align}\\label{eq:miff1a3}\n\t\\dimgraph{ff1a_2_3}{6} \n\t&= \n\t\\left. \\frac{\\Gamma(2-\\epsilon)}{i \\pi^{3-\\epsilon}}\\int\\!\\!\\frac{\\mathrm{d}^{6-2\\epsilon}k_1}{((k_1+p_1)^2)^2((k_1-p_2)^2)^2} ~\\right|_{(p_1+p_2)^2 = -1}\n\t\\nonumber\\\\\n\t&= \\frac{\\Gamma(2-\\epsilon)\\Gamma^2(1-\\epsilon)\\Gamma(1+\\epsilon)}{\\Gamma(2-2\\epsilon)}\n\t\\nonumber\\\\\n\t&= 1 + \\epsilon + 2 \\epsilon^2 + \\mathcal{O}\\left(\\epsilon^3\\right).\n\\end{align}\n\nIn Tables \\ref{tab:ff1fams}, \\ref{tab:ff2fams}, and \\ref{tab:ff3fams} below, we present the integral families that we used to uniquely parametrize all of our Feynman integrals. \nIn these tables, the loop momenta are denoted by $k_i$, $i=1,2,3$, and each momentum $q$ in the lists represents a scalar propagator $1\/q^2$.\nAt one-loop, the form factors are extremely simple and the three non-zero diagrams of Figure \\ref{fig:samplediagrams} can be covered with just one integral family, called $\\mathrm{A_1}$ in this work, see Table \\ref{tab:ff1fams}.\nThe color structure at one-loop is also extremely simple; $\\mathcal{F}_1^q(\\epsilon)$\nis directly proportional to $C_F$ and $\\mathcal{F}_1^g(\\epsilon)$ is directly proportional to $C_A$. \nThe two-loop form factors, on the other hand, are more complicated and have an interesting history.\nIt took three attempts to correctly calculate the two-loop quark form factor through the finite terms \\cite{Gonsalves:1983nq,Kramer:1986sg,Matsuura:1987wt}, \nand the analogous calculation for the two-loop gluon form factor was first published more than a decade later in reference \\cite{Harlander:2000mg}. \nIn reference \\cite{Gehrmann:2005pd}, the two-loop form factors were finally computed to all orders in $\\epsilon$ in terms of Gamma functions and generalized hypergeometric functions. One must use two integral families\nto cover the two-loop Feynman diagrams, chosen as $\\mathrm{A_2}$ and $\\mathrm{B_2}$ in this paper, see Table \\ref{tab:ff2fams}. Both $\\mathcal{F}_2^q(\\epsilon)$ and $\\mathcal{F}_2^g(\\epsilon)$ have three color structures,\nsome of which depend on the number of massless quarks, $N_f$. The color structures $C_F^2$, $C_F C_A$, $C_F N_f$ appear in the expression for $\\mathcal{F}_2^q(\\epsilon)$\nand the color structures $C_A^2$, $C_A N_f$, and $C_F N_f$ appear in the expression for $\\mathcal{F}_2^g(\\epsilon)$.\n\nNeedless to say, the calculation of the three-loop form factors is harder still. To obtain even approximate results at $\\mathcal{O}\\left(\\epsilon^0\\right)$ \ntook many years of work by a large number of researchers \\cite{Moch:2005tm,Gehrmann:2006wg,Heinrich:2007at,Heinrich:2009be,Baikov:2009bg}. Analytical\nresults for the three-loop form factors through the finite terms were obtained a short time later \\cite{Lee:2010cga,Gehrmann:2010ue}. \nThe three-loop form factor master integrals were first computed numerically to $\\mathcal{O}\\left(\\epsilon^2\\right)$ in reference \\cite{Lee:2010ik} using dimensional recurrence relations \\cite{Lee:2009dh} but, \nwith the help of the celebrated PSLQ algorithm \\cite{PSLQ}, the authors were able to recover the analytical solutions.\\footnote{At the time, it was conjectured that zeta and multiple zeta values of, at most, \nweight eight would appear in the higher-order results required to expand the three-loop form factors through to $\\mathcal{O}\\left(\\epsilon^2\\right)$.} \nThe explicit higher-order results for the three-loop masters are, regrettably, scattered over three different articles \\cite{Gehrmann:2005pd,Lee:2010ug,Lee:2010ik}. \nNevertheless, it should be stressed that reference \\cite{Lee:2010ik} provided the bulk of the unknown higher-order terms in the $\\epsilon$ expansions of the master integrals\nrequired for a calculation of the three-loop form factors up to and including contributions of $\\mathcal{O}\\left(\\epsilon^2\\right)$ \\cite{Gehrmann:2010tu}. \nSome time later, a subset of the higher-order results for the three-loop masters were confirmed by first solving an auxiliary system of differential equations for analogous integrals with two off-shell legs \nand then performing an asymptotic analysis on the results obtained \\cite{Henn:2013nsa}. Three integral families are needed to cover all three-loop Feynman diagrams which contribute. \nThe integral families we use coincide with those of reference \\cite{Gehrmann:2010ue} and are labeled $\\mathrm{A_3}$, $\\mathrm{B_3}$, $\\mathrm{C_3}$ in Table \\ref{tab:ff3fams}. \nThe color structures $C_F^3$, $C_F^2 C_A$, $C_F C_A^2$, $C_F^2 N_f$, $C_F C_A N_f$, $C_F N_f^2$, and $(d_{abc}d_{abc}\/N_c) N_{q\\gamma}$ appear in the expression for $\\mathcal{F}_3^q(\\epsilon)$ \nand the color structures $C_A^3$, $C_A^2 N_f$, $C_A C_F N_f$, $C_F^2 N_f$, $C_A N_f^2$, and $C_F N_f^2$ appear \nin the expression for $\\mathcal{F}_3^g(\\epsilon)$. Here, $N_{q\\gamma} =  (1\/e_q){\\sum_{q^\\prime} e_{q^\\prime}}$ is the charge-weighted sum of the $N_f$ quark flavors normalized to the charge of the primary quark $q$\nand $d_{abc}d_{abc} = (N_c^2 - 1)(N_c^2 - 4)\/N_c$ for $SU(N_c)$.\n\n\\begin{table}\n\\centering\n\\begin{tabular}[h!]{l}\nFamily $\\mathrm{A_1}$\\\\[1mm]\n\\hlinewd{2pt}\n\\rule{0pt}{2ex}~~~$k_1+p_1$\\\\\n\\rule{0pt}{2ex}~~~$k_1-p_2$\\\\\n\\rule{0pt}{2ex}~~~$k_1$\n\\end{tabular}\n\\caption{One integral family which covers all one-loop form factor diagrams.}\n\\label{tab:ff1fams}\n\\end{table}\n\\begin{table}\n\\centering\n\\begin{tabular}[h!]{ll}\nFamily $\\mathrm{A_2}$\\hspace{5mm} & Family $\\mathrm{B_2}$\\hspace{5mm}\\\\[1mm]\n\\hlinewd{2pt}\n\\rule{0pt}{2ex}~$k_1+p_1$ & $k_1+p_1$\\\\\n\\rule{0pt}{2ex}~$k_2+p_1$ & $k_2+p_1$\\\\\n\\rule{0pt}{2ex}~$k_1-p_2$ & $k_1-p_2$\\\\\n\\rule{0pt}{2ex}~$k_2-p_2$ & $k_1-k_2-p_2$\\\\\n\\rule{0pt}{2ex}~$k_1-k_2$ & $k_1-k_2$\\\\\n\\rule{0pt}{2ex}~$k_1$     & $k_2$\\\\\n\\rule{0pt}{2ex}~$k_2$     & $k_1$\n\\end{tabular}\n\\caption{Two integral families which cover all two-loop form factor diagrams.}\n\\label{tab:ff2fams}\n\\end{table}\n\\begin{table}\n\\centering\n\\begin{tabular}[h!]{lll}\nFamily $\\mathrm{A_3}$\\hspace{5mm} & Family $\\mathrm{B_3}$\\hspace{5mm} & Family $\\mathrm{C_3}$\\hspace{5mm}\\\\[1mm]\n\\hlinewd{2pt}\n\\rule{0pt}{2ex}~$k_1$         & $k_1$         & $k_1$\\\\\n\\rule{0pt}{2ex}~$k_2$         & $k_2$         & $k_2$\\\\\n\\rule{0pt}{2ex}~$k_3$         & $k_3$         & $k_3$\\\\\n\\rule{0pt}{2ex}~$k_1-k_2$     & $k_1-k_2$     & $k_1-k_2$\\\\\n\\rule{0pt}{2ex}~$k_1-k_3$     & $k_1-k_3$     & $k_1-k_3$\\\\\n\\rule{0pt}{2ex}~$k_2-k_3$     & $k_1-k_2-k_3$ & $k_2-k_3$\\\\\n\\rule{0pt}{2ex}~$k_1-p_1$     & $k_1-p_1$     & $k_1-k_3-p_2$\\\\\n\\rule{0pt}{2ex}~$k_1-p_1-p_2$ & $k_1-p_1-p_2$ & $k_1-p_1-p_2$\\\\\n\\rule{0pt}{2ex}~$k_2-p_1$     & $k_2-p_1$     & $k_2-p_1$\\\\\n\\rule{0pt}{2ex}~$k_2-p_1-p_2$ & $k_2-p_1-p_2$ & $k_1-k_2-p_2$\\\\\n\\rule{0pt}{2ex}~$k_3-p_1$     & $k_3-p_1$     & $k_3-p_1$\\\\\n\\rule{0pt}{2ex}~$k_3-p_1-p_2$ & $k_3-p_1-p_2$ & $k_3-p_1-p_2$\n\\end{tabular}\n\\caption{Three integral families which cover all three-loop form factor diagrams.}\n\\label{tab:ff3fams}\n\\end{table}\n\n\\section{Computational Method}\n\\label{sec:method}\n\\begin{figure}\n\\centering\n\\begin{align*}\n&\\figgraph{.35}{ff1a_2_3}{6}\n\\\\\n&\\figgraph{.35}{ff2a_4_15}{6} \\quad \\figgraph{.35}{ff2a_3_22}{8} \\quad \\figgraph{.35}{ff2a_4_58}{10} \\quad \\figgraph{.35}{ff2b_6_63}{8}\n\\\\\n&\\figgraph{.35}{ff3a_4_172}{10} \\quad \\figgraph{.35}{ff3a_5_662}{8} \\quad \\figgraph{.35}{ff3a_5_158}{6} \\quad \\figgraph{.35}{ff3a_5_412}{10}\n\\\\\n&\\figgraph{.35}{ff3a_5_433}{8} \\quad \\figgraph{.35}{ff3a_6_2695}{6} \\quad \\figgraph{.35}{ff3a_6_1683}{10} \\quad \\figgraph{.35}{ff3a_6_691}{8}\n\\\\\n&\\figgraph{.35}{ff3a_6_1433}{10} \\quad \\figgraph{.35}{ff3a_6_429}{6} \\quad \\figgraph{.35}{ff3a_6_444}{6} \\quad \\figgraph{.35}{ff3b_7_1770}{6}\n\\\\\n&\\figgraph{.35}{ff3b_7_1780}{6} \\quad \\figgraph{.35}{ff3b_7_1766}{8} \\quad \\figgraph{.35}{ff3a_7_758}{6} \\quad \\figgraph{.35}{ff3b_7_1722}{4}\n\\\\\n&\\figgraph{.35}{ff3c_8_2959}{8} \\quad \\figgraph{.35}{ff3b_8_2750}{4} \\quad \\figgraph{.35}{ff3b_8_1662}{6} \\quad \\figgraph{.35}{ff3a_9_1790}{6}\n\\\\\n&\\figgraph{.35}{ff3b_9_1790}{6} \\quad \\figgraph{.35}{ff3c_9_1015}{6}\n\\end{align*}\n\\caption{The one-, two-, and three-loop finite form factor master integrals used in Sections \\ref{sec:1Lffs}, \\ref{sec:2Lffs}, and \\ref{sec:3Lffs}\nfor $\\mathcal{F}_1^q(\\epsilon)$, $\\mathcal{F}_1^g(\\epsilon)$, $\\mathcal{F}_2^q(\\epsilon)$, $\\mathcal{F}_2^g(\\epsilon)$, $\\mathcal{F}_3^q(\\epsilon)$, and $\\mathcal{F}_3^g(\\epsilon)$.}\n\\label{fig:basis}\n\\end{figure}\nOur calculation of the massless form factors is based on a variant of the method of dimension-shifts and dots recently introduced by us\nin reference~\\cite{vonManteuffel:2014qoa}. Its salient features can be summarized as follows.\nUsing Feynman diagrams, we express all relevant loop amplitudes in terms of scalar Feynman integrals which are then reduced to a set of basis integrals using integration-by-parts reductions.\nFor our basis integrals, we select Feynman integrals which are \\emph{finite} in the $\\epsilon \\to 0$ limit.\nStarting from the Feynman parameter representations of these integrals, we Taylor expand the integrands about $\\epsilon = 0$. Finally, we use\nmodern analytical integration techniques to integrate all expansion coefficients in terms of zeta and multiple zeta values. \nWe now proceed to describe how our calculations were carried out at a more technical level of detail.\n\nAs a first step, we generate all of the Feynman diagrams using {\\tt Qgraf} \\cite{Nogueira:1991ex} and compute the interferences required to extract the one-, two-, and three-loop form factors.\nFor this task, we use the QCD interference calculator built into the {\\tt Reduze 2} program \\cite{vonManteuffel:2012np,Studerus:2009ye,Bauer:2000cp,fermat}. \nThe result of this computation is a linear combination of scalar loop integrals whose integrands have numerator insertions up to a certain maximal rank,\nwhich are then mapped onto inverse propagators. We find that, in 't Hooft-Feynman gauge, there are up to two inverse propagators at one loop, \nup to four inverse propagators at two loops, and up to five inverse propagators at three loops.\nNext, the loop integrals are systematically reduced to a minimal set of basis integrals, the so-called master integrals.\nThese reductions are obtained from integration by parts identities in $d$ spacetime\ndimensions \\cite{Tkachov:1981wb,Chetyrkin:1981qh} with a variant of Laporta's algorithm \\cite{Laporta:2001dd}, \nas implemented in the {\\tt Reduze 2} program.\nAt this point, we employ the well-known standard integral basis of corner\nintegrals in $4-2\\epsilon$ dimensions:\none one-loop integral, four two-loop integrals, and twenty-two three-loop integrals.\n\nIn order to compute the master integrals using their Feynman parameter representations, we switch to a basis of \\emph{finite master integrals}.\\footnote{In this work, we chose to make use of scalar Feynman integrals without numerator insertions. \nHowever, it might be beneficial to take integrals with irreducible numerators into account, as this provides more finite integrals in a given number of dimensions.}\nIn previous work \\cite{vonManteuffel:2014qoa}, we showed that one can make use of so-called quasi-finite master integrals, master integrals which have convergent Feynman parameter integrations but, \npotentially, a trivial overall $1\/\\epsilon$ divergence.\nHere, we find it more convenient to work with truly finite integrals in order to make it manifestly visible which master integrals contribute to which $\\epsilon$ poles of the bare form factors.\nAs explained in reference \\cite{vonManteuffel:2014qoa} and Appendix~\\ref{sec:finite}, such a finite basis can always be constructed in dimensional regularization for any multi-leg, \nmulti-loop process, provided that there exists a Euclidean region which respects all kinematical constraints.\n\nFor each of the irreducible topologies, we enumerate finite integrals as described in Appendix~\\ref{sec:finite}, using an automated job in the development version of {\\tt Reduze 2}.\nOut of many possible choices, we find it convenient\nto keep the dotted graphical representations of our master integrals as symmetric as possible while aiming for masters possessing high maximal weights at leading order in $\\epsilon$.\nWe discard sets of master integrals for which the reduced interferences contain spurious poles worse than $\\epsilon^{-2 L}$ at $L$-loop order.\nIt is essential to avoid such spurious poles if one wishes to be able to read off by eye which master integrals contribute to which $\\epsilon$ poles.\nAll other things being equal, we also find it natural to pick candidates which inhabit smaller spacetime dimensions and have fewer dots.\nA systematic analysis of the first 100 finite integrals produced by our algorithm shows\nthat, for each sector, it is sufficient to use the lowest possible spacetime dimension ({\\it i.e.} the spacetime dimension at which finite integrals first appear) to determine the highest possible maximal weight at leading order in $\\epsilon$\nconsistent with our spurious pole veto.\\footnote{Although we have no proof that one cannot do better, we were unable to find a candidate to cover \nthe unique non-factorizable, three-point, eight-line integral topology which both had the highest maximal weight possible at leading order in $\\epsilon$ (five) and resulted in reduced three-loop integrands without $\\epsilon^{-7}$ poles.} \nIn a few cases we picked non-minimal spacetime dimensions in order to allow for symmetric dottings.\nThe results of our analysis are depicted in Figure \\ref{fig:basis}.\n\nThe next step is to relate the traditional integral basis to the basis of finite integrals using dimension shifts\nand integration by parts relations, see Section 2.3 of reference~\\cite{vonManteuffel:2014qoa} for details.\nFor this purpose, we must solve reduction identities for integrals besides the ones needed for the interferences.\nDespite the fact that some of our finite integrals carry a rather large number\nof dots, these reductions are moderate in computational complexity when compared\nto the reductions required for the form factor Feynman diagrams. For the convenience of the reader, our arXiv submission includes the relevant lists of replacement rules required to pass from the standard Reduze 2\nbases of corner integrals to the finite integral bases employed in this work (see \\file{BasisToFinite.m}).\n\nAt this stage, we have reexpressed the one-, two-, and three-loop form factors as linear combinations of master integrals whose Feynman parameter integral representations converge in the $\\epsilon \\to 0$ limit.\nIf our basis integrals satisfy certain technical requirements, we can apply modern, highly-automated, analytical integration algorithms to compute the Taylor expansion about $\\epsilon = 0$ for our master integrals.\nThe {\\texttt{HyperInt}} package \\cite{Panzer:2014caa}, written in {\\texttt{Maple}}, provides powerful routines for the evaluation of Euclidean linearly reducible Feynman integrals\\footnote{Roughly speaking, a Feynman integral \nis linearly reducible if its Feynman parametrization can be integrated out parameter-by-parameter in terms of multiple polylogarithms with rational arguments.} which possess convergent Feynman parameter representations.\nThe package implements the algorithms presented in \\cite{Brown:2008um,Brown:2009ta} and all underlying principles are discussed in \\cite{Panzer:2015ida}. In \\cite{Panzer:2014gra}, \nit was pointed out that all three-loop form factor integrals happen to be linearly reducible. This non-obvious fact implies that one can use the {\\texttt{HyperInt}} approach to compute the $\\epsilon$-expansion of the three-loop form factors. \nEven though the lower-loop results have been known to all orders in $\\epsilon$ for some time, we use {\\texttt{HyperInt}} to evaluate the $\\epsilon$-expansion of the form factors to weight eight in all cases. \nAt one loop, this requires an $\\epsilon$ expansion of the form factors up to $\\mathcal{O}\\left(\\epsilon^6\\right)$,\n at two loops, this requires an $\\epsilon$ expansion of the form factors up to $\\mathcal{O}\\left(\\epsilon^4\\right)$, \n and, at three loops, this requires an $\\epsilon$ expansion of the form factors up to $\\mathcal{O}\\left(\\epsilon^2\\right)$. In the following three sections, we summarize our results.\n\n\\section{One-Loop Form Factors}\n\\label{sec:1Lffs}\nFor the bare one-loop form factors we find\n\\begin{align}\n\\label{eq:1Lffq}\n\\mathcal{F}_1^q(\\epsilon) &=\\casimir{C_F} \\left\\{\\pole{\\frac{1}{\\epsilon^2}}\\left[a_1 \\dimgraph{ff1a_2_3}{6}\\right]\\right\\}\n\\end{align}\n\\begin{align}\n\\label{eq:1Lffg}\n\\mathcal{F}_1^g(\\epsilon) &=\\casimir{C_A} \\left\\{\\pole{\\frac{1}{\\epsilon^2}}\\left[b_1 \\dimgraph{ff1a_2_3}{6}\\right]\\right\\},\n\\end{align}\nwhere we have abbreviated the rational functions of $\\epsilon$ as\n\\begin{align}\n\\label{eq:1Lffqcoeff}\na_1 &= \\frac{-2 + \\epsilon-2 \\epsilon^2}{1-\\epsilon}\\\\\n\\label{eq:1Lffgcoeff}\nb_1 &= \\frac{-2 (1-3 \\epsilon+2 \\epsilon^2+\\epsilon^3)}{(1-\\epsilon)^2}.\n\\end{align}\nHere and in the following sections, the notation is such that both the graphically represented master integrals and the abbreviated rational\ncoefficient functions are finite at $\\epsilon = 0$ and possess a Taylor series expansion starting at $\\mathcal{O}\\left(\\epsilon^0\\right)$, recall the form of \\ Eq.\\ \\eqref{eq:miff1a3}.\nThis notation makes all divergences completely explicit and shows which integral topologies contribute to specific $\\epsilon$ poles of the form factors.\n\nAt the one-loop level, the Casimir scaling property is reflected in the fact that \n$a_1$ is equal to $b_1$ in the $\\epsilon \\to 0$ limit.\nExpanding Eqs.\\ \\eqref{eq:1Lffq} and \\eqref{eq:1Lffg} to $\\mathcal{O}\\left(\\epsilon^6\\right)$, \nwe find complete agreement with the results of references \\cite{Gehrmann:2010ue} and \\cite{Gehrmann:2010tu}. Further details of our analysis of the one-loop quark and gluon form factors are available online at arXiv.org \nin the ancillary files \\file{Fq.m} and \\file{Fg.m}.\n\n\\section{Two-Loop Form Factors}\n\\label{sec:2Lffs}\nFor the bare two-loop form factors we find\n\\begin{align}\n\\label{eq:2Lffq}\n\\mathcal{F}_2^q(\\epsilon) &=\\casimir{C_F^2} \\left\\{\\pole{\\frac{1}{\\epsilon^4}} \\left[c_1 \\dimgraph{ff2a_4_15}{6} + c_2 \\dimgraph{ff2a_3_22}{8}\\right]\n+\\pole{\\frac{1}{\\epsilon^3}} \\left[c_3 \\dimgraph{ff2a_4_58}{10}\\right]+\\pole{\\frac{1}{\\epsilon}} \\left[c_4 \\dimgraph{ff2b_6_63}{8}\\right]\\right\\}\n\\nonumber\\\\[-1pt] &\n+ \\casimir{C_F C_A} \\left\\{\\pole{\\frac{1}{\\epsilon^4}} \\left[c_5 \\dimgraph{ff2a_3_22}{8}+c_6 \\dimgraph{ff2a_4_58}{10}\\right]+\\pole{\\frac{1}{\\epsilon}} \\left[c_7 \\dimgraph{ff2b_6_63}{8}\\right]\\right\\}\n\\nonumber\\\\[-1pt] &\n+ \\casimir{C_F N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^3}} \\left[c_8 \\dimgraph{ff2a_4_58}{10}\\right]\\right\\}\n\\end{align}\n\\begin{align}\n\\label{eq:2Lffg}\n\\mathcal{F}_2^g(\\epsilon) &=\\casimir{C_A^2} \\left\\{\\pole{\\frac{1}{\\epsilon^4}} \\left[d_1 \\dimgraph{ff2a_4_15}{6}+d_2 \\dimgraph{ff2a_3_22}{8}+d_3 \\dimgraph{ff2a_4_58}{10}\\right]\n+\\pole{\\frac{1}{\\epsilon}} \\left[d_4 \\dimgraph{ff2b_6_63}{8}\\right]\\right\\}\n\\nonumber\\\\[-1pt] &\n+ \\casimir{C_A N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^3}} \\left[d_5 \\dimgraph{ff2a_3_22}{8}+d_6 \\dimgraph{ff2a_4_58}{10}\\right]+d_7 \\dimgraph{ff2b_6_63}{8}\\right\\}\n\\nonumber\\\\[-1pt] &\n+ \\casimir{C_F N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^2}} \\left[d_8 \\dimgraph{ff2a_3_22}{8}+d_9 \\dimgraph{ff2a_4_58}{10}\\right]+d_{10} \\dimgraph{ff2b_6_63}{8}\\right\\}.\n\\end{align}\nAt this order in QCD perturbation theory, the Casimir scaling property can no longer be seen by eye; for example, the leading infrared divergences in the quark form factor exponentiate \\cite{Frenkel:1976bj}\nand the $C_F^2$ color structure can therefore not contribute to the two-loop quark anomalous dimension at all. What can be deduced from Eqs.\\ \\eqref{eq:2Lffq} and \\eqref{eq:2Lffg}, \nhowever, is that the finite two-loop non-planar form factor integral in the above\ncannot contribute to the two-loop quark and gluon cusp anomalous dimensions; it cannot contribute to the $\\epsilon^{-2}$ pole terms of either form factor because\nit is finite and has prefactors which, at worst, diverge like $\\epsilon^{-1}$ in the $\\epsilon \\to 0$ limit. Expanding Eqs.\\ \\eqref{eq:2Lffq} and \\eqref{eq:2Lffg} to $\\mathcal{O}\\left(\\epsilon^4\\right)$, \nwe find complete agreement with the results of references \\cite{Gehrmann:2010ue} and \\cite{Gehrmann:2010tu}. Further details of our analysis of the two-loop quark and gluon form factors are available online at arXiv.org \nin the ancillary files \\file{Fq.m} and \\file{Fg.m}.\n\n\\section{Three-Loop Form Factors}\n\\label{sec:3Lffs}\nFor the bare three-loop form factors we find\n\\begin{align}\n\\label{eq:3Lffq}\n\\mathcal{F}_3^q(\\epsilon) &=\n\\casimir{C_F^3} \\left\\{\\pole{\\frac{1}{\\epsilon^6}} \\left[e_1 \\dimgraph{ff3a_4_172}{10}+e_2 \\dimgraph{ff3a_5_662}{8}+e_3 \\dimgraph{ff3a_5_158}{6}+e_4 \\dimgraph{ff3a_6_2695}{6}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_5 \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^5}} \\left[e_6 \\dimgraph{ff3a_5_412}{10}+e_7 \\dimgraph{ff3a_5_433}{8}+e_8 \\dimgraph{ff3a_6_1683}{10}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^4}} \\left[e_9 \\dimgraph{ff3a_6_429}{6}\\right]+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{10} \\dimgraph{ff3a_6_691}{8}+e_{11} \\dimgraph{ff3a_6_444}{6}+e_{12} \\dimgraph{ff3b_7_1770}{6}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{13} \\dimgraph{ff3b_7_1780}{6}+e_{14} \\dimgraph{ff3b_7_1766}{8}+e_{15} \\dimgraph{ff3c_8_2959}{8}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{16} \\dimgraph{ff3b_8_1662}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}} \\left[e_{17} \\dimgraph{ff3a_7_758}{6}+e_{18} \\dimgraph{ff3b_7_1722}{4}+e_{19} \\dimgraph{ff3b_8_2750}{4}+e_{20} \\dimgraph{ff3a_9_1790}{6}+e_{21} \\dimgraph{ff3b_9_1790}{6}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{22} \\dimgraph{ff3c_9_1015}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F^2 C_A} \\left\\{\\pole{\\frac{1}{\\epsilon^6}} \\left[e_{23} \\dimgraph{ff3a_4_172}{10}+e_{24} \\dimgraph{ff3a_5_662}{8}+e_{25} \\dimgraph{ff3a_5_158}{6}+e_{26} \\dimgraph{ff3a_5_412}{10}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{27} \\dimgraph{ff3a_6_1683}{10}+e_{28} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^5}} \\left[e_{29} \\dimgraph{ff3a_5_433}{8}\\right]+\\pole{\\frac{1}{\\epsilon^4}} \\left[e_{30} \\dimgraph{ff3a_6_429}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{31} \\dimgraph{ff3a_6_691}{8}+e_{32} \\dimgraph{ff3a_6_444}{6}+e_{33} \\dimgraph{ff3b_7_1770}{6}+e_{34} \\dimgraph{ff3b_7_1780}{6}+e_{35} \\dimgraph{ff3b_7_1766}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{36} \\dimgraph{ff3c_8_2959}{8}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{37} \\dimgraph{ff3b_8_1662}{6}\\right]+\\pole{\\frac{1}{\\epsilon}} \\left[e_{38} \\dimgraph{ff3a_7_758}{6}+e_{39} \\dimgraph{ff3b_7_1722}{4}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{40} \\dimgraph{ff3b_8_2750}{4}+e_{41} \\dimgraph{ff3a_9_1790}{6}+e_{42} \\dimgraph{ff3b_9_1790}{6}+e_{43} \\dimgraph{ff3c_9_1015}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F C_A^2} \\left\\{\\pole{\\frac{1}{\\epsilon^6}} \\left[e_{44} \\dimgraph{ff3a_4_172}{10}+e_{45} \\dimgraph{ff3a_5_662}{8}+e_{46} \\dimgraph{ff3a_5_158}{6}+e_{47} \\dimgraph{ff3a_5_412}{10}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{48} \\dimgraph{ff3a_5_433}{8}+e_{49} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^4}} \\left[e_{50} \\dimgraph{ff3a_6_429}{6}\\right]+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{51} \\dimgraph{ff3a_6_444}{6}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+e_{52} \\dimgraph{ff3b_7_1770}{6}+e_{53} \\dimgraph{ff3b_7_1780}{6}+e_{54} \\dimgraph{ff3b_7_1766}{8}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{55} \\dimgraph{ff3b_8_1662}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}} \\left[e_{56} \\dimgraph{ff3a_7_758}{6}+e_{57} \\dimgraph{ff3b_7_1722}{4}+e_{58} \\dimgraph{ff3a_9_1790}{6}+e_{59} \\dimgraph{ff3b_9_1790}{6}+e_{60} \\dimgraph{ff3c_9_1015}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F^2 N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^5}} \\left[e_{61} \\dimgraph{ff3a_5_412}{10}+e_{62} \\dimgraph{ff3a_5_433}{8}+e_{63} \\dimgraph{ff3a_6_1683}{10}+e_{64} \\dimgraph{ff3a_6_1433}{10}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^4}} \\left[e_{65} \\dimgraph{ff3a_4_172}{10}\\right]+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{66} \\dimgraph{ff3a_6_429}{6}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{67} \\dimgraph{ff3a_6_691}{8}\n+e_{68} \\dimgraph{ff3b_7_1770}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F C_A N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^5}} \\left[e_{69} \\dimgraph{ff3a_4_172}{10}+e_{70} \\dimgraph{ff3a_5_412}{10}+e_{71} \\dimgraph{ff3a_5_433}{8}+e_{72} \\dimgraph{ff3a_6_1433}{10}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{73} \\dimgraph{ff3a_6_429}{6}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{74} \\dimgraph{ff3a_6_444}{6}+e_{75} \\dimgraph{ff3b_7_1770}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F N_f^2} \\left\\{\\pole{\\frac{1}{\\epsilon^4}}\\left[e_{76} \\dimgraph{ff3a_6_1433}{10}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{\\frac{d_{abc}d_{abc}}{N_c} N_{q\\gamma}} \\left\\{\\pole{\\frac{1}{\\epsilon^5}} \\left[e_{77} \\dimgraph{ff3a_4_172}{10}+e_{78} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^4}}  \\left[e_{79} \\dimgraph{ff3a_5_433}{8}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^3}} \\left[e_{80} \\dimgraph{ff3a_5_662}{8}+e_{81} \\dimgraph{ff3a_5_158}{6}+e_{82} \\dimgraph{ff3a_5_412}{10}+e_{83} \\dimgraph{ff3a_6_429}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^2}} \\left[e_{84} \\dimgraph{ff3a_6_444}{6}+e_{85} \\dimgraph{ff3b_7_1770}{6}+e_{86} \\dimgraph{ff3b_7_1780}{6}+e_{87} \\dimgraph{ff3b_7_1766}{8}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}} \\left[e_{88} \\dimgraph{ff3b_8_1662}{6}\\right]+\\left[e_{89} \\dimgraph{ff3b_7_1722}{4}+e_{90} \\dimgraph{ff3a_9_1790}{6}+e_{91} \\dimgraph{ff3b_9_1790}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\epsilon} \\left[e_{92} \\dimgraph{ff3a_7_758}{6}\\right]\\right\\}\n\\end{align}\n\\begin{align}\n\\label{eq:3Lffg}\n\\mathcal{F}_3^g(\\epsilon) &=\n\\casimir{C_A^3} \\left\\{\\pole{\\frac{1}{\\epsilon^6}} \\left[f_1 \\dimgraph{ff3a_4_172}{10}+f_2 \\dimgraph{ff3a_5_662}{8}+f_3 \\dimgraph{ff3a_5_158}{6}+f_4 \\dimgraph{ff3a_5_412}{10}+f_5 \\dimgraph{ff3a_5_433}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_6 \\dimgraph{ff3a_6_2695}{6}+f_7 \\dimgraph{ff3a_6_1683}{10}+f_8 \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^4}} \\left[f_9 \\dimgraph{ff3a_6_429}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^3}} \\left[f_{10} \\dimgraph{ff3a_6_691}{8}+f_{11} \\dimgraph{ff3a_6_444}{6}+f_{12} \\dimgraph{ff3b_7_1770}{6}+f_{13} \\dimgraph{ff3b_7_1780}{6}+f_{14} \\dimgraph{ff3b_7_1766}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{15} \\dimgraph{ff3c_8_2959}{8}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[f_{16} \\dimgraph{ff3b_8_1662}{6}\\right]+\\pole{\\frac{1}{\\epsilon}} \\left[f_{17} \\dimgraph{ff3a_7_758}{6}+f_{18} \\dimgraph{ff3b_7_1722}{4}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{19} \\dimgraph{ff3b_8_2750}{4}+f_{20}\\dimgraph{ff3a_9_1790}{6}+f_{21} \\dimgraph{ff3b_9_1790}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_A^2 N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^5}} \\left[f_{22} \\dimgraph{ff3a_4_172}{10}+f_{23} \\dimgraph{ff3a_5_662}{8}+f_{24} \\dimgraph{ff3a_5_158}{6}+f_{25} \\dimgraph{ff3a_5_412}{10}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{26} \\dimgraph{ff3a_5_433}{8}+f_{27} \\dimgraph{ff3a_6_1683}{10}+f_{28} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^3}} \\left[f_{29} \\dimgraph{ff3a_6_429}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon^2}} \\left[f_{30} \\dimgraph{ff3a_6_691}{8}+f_{31} \\dimgraph{ff3a_6_444}{6}+f_{32} \\dimgraph{ff3b_7_1770}{6}+f_{33} \\dimgraph{ff3b_7_1780}{6}+f_{34} \\dimgraph{ff3b_7_1766}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{35} \\dimgraph{ff3c_8_2959}{8}\\right]+\\pole{\\frac{1}{\\epsilon}} \\left[f_{36} \\dimgraph{ff3b_8_1662}{6}\\right]+\\left[f_{37} \\dimgraph{ff3a_7_758}{6}+f_{38} \\dimgraph{ff3b_7_1722}{4}\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.+f_{39} \\dimgraph{ff3b_8_2750}{4}+f_{40} \\dimgraph{ff3a_9_1790}{6}+f_{41} \\dimgraph{ff3b_9_1790}{6}+f_{42} \\dimgraph{ff3c_9_1015}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_A C_F N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^5}} \\left[f_{43} \\dimgraph{ff3a_4_172}{10}+f_{44} \\dimgraph{ff3a_5_662}{8}+f_{45} \\dimgraph{ff3a_5_158}{6}+f_{46} \\dimgraph{ff3a_5_433}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{47} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^4}} \\left[f_{48} \\dimgraph{ff3a_5_412}{10}+f_{49} \\dimgraph{ff3a_6_1683}{10}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[f_{50} \\dimgraph{ff3a_6_691}{8}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{51} \\dimgraph{ff3a_6_444}{6}+f_{52} \\dimgraph{ff3b_7_1770}{6}+f_{53} \\dimgraph{ff3b_7_1780}{6}+f_{54} \\dimgraph{ff3b_7_1766}{8}+f_{55} \\dimgraph{ff3c_8_2959}{8}\n\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}} \\left[f_{56} \\dimgraph{ff3a_6_429}{6}+f_{57} \\dimgraph{ff3b_8_1662}{6}\\right]+\\left[f_{58} \\dimgraph{ff3a_7_758}{6}+f_{59} \\dimgraph{ff3b_7_1722}{4}+f_{60} \\dimgraph{ff3b_8_2750}{4}\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left. +f_{61} \\dimgraph{ff3a_9_1790}{6}+f_{62} \\dimgraph{ff3b_9_1790}{6}+f_{63} \\dimgraph{ff3c_9_1015}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+\\casimir{C_F^2 N_f} \\left\\{\\pole{\\frac{1}{\\epsilon^4}} \\left[f_{64} \\dimgraph{ff3a_5_433}{8}+f_{65} \\dimgraph{ff3a_6_1433}{10}\\right]+\\pole{\\frac{1}{\\epsilon^3}} \\left[f_{66} \\dimgraph{ff3a_4_172}{10}+f_{67} \\dimgraph{ff3a_5_158}{6}\n\\right.\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n\\left.\n+f_{68} \\dimgraph{ff3a_5_412}{10}\\right]+\\pole{\\frac{1}{\\epsilon^2}} \\left[f_{69} \\dimgraph{ff3a_5_662}{8}+f_{70} \\dimgraph{ff3a_6_444}{6}+f_{71} \\dimgraph{ff3b_7_1780}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}} \\left[f_{72} \\dimgraph{ff3b_7_1770}{6}+f_{73} \\dimgraph{ff3b_7_1766}{8}\\right] + \\left[f_{74} \\dimgraph{ff3a_6_429}{6}+f_{75} \\dimgraph{ff3a_9_1790}{6}+f_{76} \\dimgraph{ff3c_9_1015}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\epsilon \\left[f_{77} \\dimgraph{ff3a_7_758}{6}+f_{78} \\dimgraph{ff3b_7_1722}{4}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_A N_f^2} \\left\\{\\pole{\\frac{1}{\\epsilon^4}}\\left[f_{79} \\dimgraph{ff3a_4_172}{10}+f_{80} \\dimgraph{ff3a_5_158}{6}+f_{81} \\dimgraph{ff3a_5_412}{10}\\right]+\\pole{\\frac{1}{\\epsilon^2}}\\left[f_{82} \\dimgraph{ff3a_6_1433}{10}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}}\\left[f_{83} \\dimgraph{ff3b_7_1780}{6}\\right]\\right\\}\n\\nonumber\\\\[5pt] & \\hspace{-6.2ex}\n+ \\casimir{C_F N_f^2} \\left\\{\\pole{\\frac{1}{\\epsilon^3}}\\left[f_{84} \\dimgraph{ff3a_4_172}{10}+f_{85} \\dimgraph{ff3a_5_412}{10}\\right]+\\pole{\\frac{1}{\\epsilon^2}}\\left[f_{86} \\dimgraph{ff3a_5_158}{6}\\right]\n\\right.\n\\nonumber\\\\[-1pt] &\n\\left.\n+\\pole{\\frac{1}{\\epsilon}}\\left[f_{87} \\dimgraph{ff3b_7_1780}{6}\\right]\n\\right\\}.\n\\end{align}\nIn total, six of the twenty-two finite three-loop form factor master integrals in Eqs.\\ \\eqref{eq:3Lffq} and \\eqref{eq:3Lffg} above turn out not to contribute to the three-loop cusp anomalous dimensions (see Figure \\ref{fig:noncuspintegrals}). \nThese include, in particular, all of the most complicated, nine-line, finite three-loop masters. Expanding Eqs.\\ \\eqref{eq:3Lffq} and \\eqref{eq:3Lffg} to $\\mathcal{O}\\left(\\epsilon^2\\right)$, \nwe find complete agreement with the results of references \\cite{Gehrmann:2010ue} and \\cite{Gehrmann:2010tu}. Further details of our analysis of the three-loop quark and gluon form factors are available online at arXiv.org \nin the ancillary files \\file{Fq.m} and \\file{Fg.m}.\n\\begin{figure}\n\\centering\n\\begin{align*}\n&\\figgraph{.35}{ff3a_7_758}{6} \\qquad \\figgraph{.35}{ff3b_7_1722}{4} \\qquad \\figgraph{.35}{ff3b_8_2750}{4}\n\\\\\n&\\figgraph{.35}{ff3a_9_1790}{6} \\qquad \\figgraph{.35}{ff3b_9_1790}{6} \\qquad \\figgraph{.35}{ff3c_9_1015}{6}\n\\end{align*}\n\\caption{The finite three-loop form factor master integrals in $\\mathcal{F}_3^q\\left(\\epsilon\\right)$ and $\\mathcal{F}_3^g\\left(\\epsilon\\right)$ (Eqs.\\ \\eqref{eq:3Lffq} and \\eqref{eq:3Lffg})\nwhich do not contribute to the three-loop cusp anomalous dimensions.}\n\\label{fig:noncuspintegrals}\n\\end{figure}\n\n\\section{Towards the Four-Loop Cusp Anomalous Dimensions}\n\\label{sec:cusps}\nAs has long been known, a convenient way to calculate the cusp anomalous dimensions to four loops is to perform the four-loop quark and gluon form factor calculations and then extract the four-loop cusp anomalous dimensions\nfrom the first non-trivial poles in the parameter of dimensional regularization. \nThe $\\epsilon^{-8} - \\epsilon^{-3}$ poles at four loops are predicted by the renormalization group equations satisfied by the form factors, in terms of known lower-loop results,\nand the $\\epsilon^{-2}$ poles may be used to extract the four-loop cusp anomalous dimensions (see {\\it e.g.} reference~\\cite{Moch:2005tm} for details).\\footnote{In reference \\cite{Gehrmann:2010ue},\nthe one-, two-, and three-loop form factors are given to sufficiently high orders in the $\\epsilon$ expansion for the purposes of this analysis.}\n\nNormally, for dimensionally-regulated, $L$-loop bare amplitudes in quantum field theory, which\ncan be expressed in terms of multiple zeta values or, more generally, multiple polylogarithms, one expects\ncontributions of, at most, weight $2 L + n$ in the Laurent expansion coefficient of order $n$.\\footnote{We are not aware of any proof that this will always be the case. \nHowever, to the best of our knowledge, this weight bound turns out to hold in every explicit higher-order calculation performed to date.}\nIn fact, given our experience at lower loop orders, it is natural to hope\nthat the four-loop cusp anomalous dimensions are rational linear combinations of the numbers $1$, $\\zeta_2$, $\\zeta_3$, $\\zeta_2^2$, $\\zeta_2 \\zeta_3$, $\\zeta_5$, $\\zeta_2^3$, and $\\zeta_3^2$.\nIf we further assume that there are no cancellations of spurious constants between different master integrals, we\nconclude that integrals which contain other transcendental constants at leading order in $\\epsilon$ should not contribute to the four-loop cusp anomalous dimensions.\nFor illustration, we have chosen the finite, non-planar twelve-line integral\\footnote{Here we use the convention $\\displaystyle \\zeta_{5,3}=\\sum_{0<n<m} n^{-3} m^{-5}$.}\n\\begin{align}\\label{eq:4L}\n\t\\dimgraph{ff4c_12_163253_pure}{6} \n\t& =\n\t\\tfrac{18}{5} \\zeta^2_2 \\zeta_3 - 5 \\zeta_2\\zeta_5\n\t+\\Big(\n\t\t24 \\zeta_2 \\zeta_3\n\t\t+20 \\zeta_5\n\t\t-\\tfrac{188}{105} \\zeta_2^3\n\t\t-17 \\zeta_3^2\n\t\t+9 \\zeta_2^2 \\zeta_3\n\t\\nonumber\\\\&\n\t\t-47 \\zeta_2 \\zeta_5\n\t\t-21 \\zeta_7\n\t\t+\\tfrac{6883}{2100} \\zeta_2^4\n\t\t+\\tfrac{49}{2} \\zeta_2 \\zeta_3^2\n\t\t+\\tfrac{1}{2} \\zeta_3 \\zeta_5\n\t\t-9 \\zeta_{5,3}\n\t\\Big)\\epsilon\n\t+\\mathcal{O}\\left(\\epsilon^2\\right),\n\\end{align}\nwhich is of pure weight seven at leading order in $\\epsilon$ and should thus not contribute to the four-loop cusp anomalous dimensions in our approach.\nWe computed \\eqref{eq:4L} analytically with the programs in the ancillary files as described in appendix~\\ref{sec:hyperint} and checked our result to four significant digits numerically using {\\tt FIESTA 3} \\cite{Smirnov:2013eza},\na program which provides a highly-automated framework for the numerical evaluation of Feynman integrals via sector decomposition \\cite{Binoth:2000ps}.\n\nIn fact, we see at four loops indications of a pattern analogous to what was observed at two and three loops. The simplest integral topologies ({\\it e.g.} those with many bubble insertions) give rise to finite Feynman integrals\nwhich are likely to contribute to all or almost all poles in the form factors, due to the fact that they have relatively small maximal weights at leading order in the $\\epsilon$ expansion. \nOn the other hand, already at nine lines we found several examples of integral topologies which are not likely to be relevant to the calculation of the $\\epsilon^{-2}$ poles in the form factors; in these sectors, we have identified\nfinite integrals which have maximal weights of seven or even eight at leading order in $\\epsilon$.\n\nOut of all four-loop form factor integral topologies, only fifteen fail {\\texttt{HyperInt}}'s test for linear reducibility out of the box and ten of these are top-level topologies. Altogether, this means that roughly 90\\% of all relevant\nFeynman integrals can be accessed using the methods implemented in {\\texttt{HyperInt}}. Given our experience so far, \nit seems possible that at least the set of finite Feynman integrals which contribute to the four-loop cusp anomalous dimensions could be computable analytically using the technology that we have developed.\nLet us point out, however, that the integration by parts reductions required for the four-loop form factors in QCD are challenging to calculate.\nIn order to overcome the limitations of available programs, an implementation of the new\nintegration by parts reduction technique described in reference \\cite{vonManteuffel:2014ixa} is under development.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper, we studied the bare quark and gluon form factors in massless QCD using a recently-developed approach to the computation of virtual corrections in quantum field theory. Using integration by parts reduction,\nwe found compact expressions for the one-, two-, and three-loop form factors in terms of linearly reducible, finite master integrals. \nLinear reducibility and finiteness together guarantee that, at least in principle, the deterministic integration algorithms implemented in {\\texttt{HyperInt}}\ncan immediately be applied to analytically compute the $\\epsilon$ expansion of these master integrals to the order required.\nWe evaluated all master integrals to sufficiently high order and then checked all of the literature on the subject.\nAt the three-loop level, our analysis constitutes the first complete independent analytical check on the\npublished higher-order results.\n\nThe extraction of the cusp anomalous dimensions at $L$ loop order requires knowledge of the $L$-loop form factors expanded through to $\\mathcal{O}\\left(\\epsilon^{-2}\\right)$.\nWith our choice of finite master integrals, only a subset of the full set of master integrals contributes to these poles in $\\epsilon$.\nAt the three-loop level, we have shown explicitly that none of the most complicated (nine-line) form factor integrals contribute to the cusp anomalous dimensions.\nAlthough our analysis of the four-loop form factor integrals is still ongoing,\nwe have found that the finite integrals in many sectors seem to follow an analogous pattern and are not expected to contribute to the four-loop cusp anomalous dimensions.\nOnce the necessary interferences and integration by parts reductions become available,\nit will be possible to algebraically check all four-loop integral topologies to see whether or not they contribute to the $\\epsilon^{-2}$ pole terms.\n\nWe encountered fifteen irreducible four-loop form factor topologies for which a straightforward test of linear reducibility failed.\nThese integrals are interesting in their own right, even if it turns out that they do not contribute to the four-loop cusp anomalous dimensions.\nInvestigating them is worthwhile because they will likely be relevant to the finite parts of the four-loop form factors \nand studying them could quite possibly lead to the development of even more powerful methods for the analytical evaluation of Feynman integrals.\nFor example, it might be that these integrals become linearly reducible once one makes a suitable change of variables. \nNon-trivial examples of this phenomenon were reported recently in \\cite[section~4]{Panzer:2014gra} and \\cite[section~2.1]{Panzer:2014fla},\nbut it is not yet completely clear how to determine whether or not a given topology which fails to be linearly reducible can be treated in this way.\n\nIt may be that, at least initially, some of the four-loop form factor integrals defeat all known analytical methods. \nWhether or not an exact determination of all basis integrals is possible to sufficiently high order in the $\\epsilon$ expansion, our method of dimension shifts and dots allows one to make use of public software packages for sector decomposition\n(principally {\\tt sector\\_decomposition} \\cite{Bogner:2007cr}, {\\tt FIESTA 3} \\cite{Smirnov:2013eza}, and {\\tt SecDec 3} \\cite{Borowka:2015mxa}) \nin situations where it would almost certainly not be possible to proceed in the conventional approach utilized in {\\it e.g.} reference \\cite{Boels:2015yna}.\nThe analysis upon which this assertion is based will be discussed at length in a forthcoming publication on numerical applications of our method.\n\n\\section*{Acknowledgments}\nWe gratefully acknowledge Bas Tausk for bringing reference \\cite{Tkachov:1996wh} and follow-up work to our attention and Cedric Studerus for discussions.\nWe are grateful to Johannes Schlenk for pointing out a misprint in the diagram\nin \\eqref{eq:4L} in the original version of this paper to us.\nThe work of RMS was supported by the European Research Council (ERC)\nthrough the Advanced Grant EFT4LHC and Grant 647356 (CutLoops).\nEP was supported by ERC Grant 257638 via the CNRS at the IHES and the Humboldt-Universit\\\"{a}t zu Berlin. Our figures were generated using {\\tt Jaxodraw} \\cite{Binosi:2003yf}, based on {\\tt AxoDraw} \\cite{Vermaseren:1994je}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\\section{Introduction}\nIn many real-world pattern recognition tasks, obtaining labeled examples to train classification algorithms is much more expensive than obtaining unlabeled examples. These tasks include document and image classification \\cite{Nigam2000} where unlabeled objects can easily be downloaded from the web, part of speech tagging \\cite{Elworthy1994}, protein function prediction \\cite{Weston2005} and many others. Using unlabeled data to improve the training of a classification procedure, however, requires semi-supervised variants of supervised classifiers to make use of this additional unlabeled data. Research into semi-supervised learning has therefore seen an increasing amount of interest in the last decade \\cite{Chapelle2006}.\n\nIn supervised learning adding additional labeled training data improves performance for most classification routines. This does not generally hold for semi-supervised learning \\cite{Cozman2006}. Adding additional unlabeled data may actually deteriorate classification performance. This can happen when the underlying assumptions of the model do not hold. In effect, disregarding the unlabeled data can lead to a better solution.\n\nIn this work we consider linear discriminant analysis (LDA) applied to classification. Several semi-supervised adaptations of this supervised procedure have been proposed. These approaches may suffer from the problem that additional unlabeled data degrade performance. To counter this problem, \\cite{Loog2014a} introduced moment constrained LDA, which offers a more robust type of semi-supervised LDA. The recently introduced idea of implicitly constrained estimation \\cite{Krijthe2013}, is another method that relies on constraints given by the unlabeled data. We compare these two approaches to other semi-supervised methods, in particular, expectation maximization and self-learning, and empirically study in what sense we can expect improvement by employing any of these semi-supervised methods.\n\nThe contributions of this work are the following:\n\n\\begin{itemize}\n  \\item Introduce a new, principled approach to semi-supervised LDA: implicitly constrained LDA\n  \\item Offer a comparison of semi-supervised versions of linear discriminant analysis\n  \\item Explore ways in which we can expect these semi-supervised methods to offer improvements over the supervised variant, in particular in terms of the log likelihood\n\\end{itemize}\n\nThe rest of this paper is organized as follows. After discussing related work, we introduce several approaches to semi-supervised linear discriminant analysis. These methods are then compared on an illustrative toy problem and in an empirical study using several benchmark datasets. We end with a discussion of the results and conclude.\n\n\\section{Related Work}\nSome of the earliest work on semi-supervised learning was done by \\cite{McLachlan1975,Taylor1977} who studied the self-learning approach applied to linear discriminant analysis. This has later also been referred to as Yarowsky's algorithm \\cite{Yarowsky1995}. This approach is closely related to Expectation Maximization \\cite{Abney2004}, where, in a generative model, the unknown labels are integrated out of the likelihood function and the resulting marginal likelihood is maximized \\cite{Dempster1977}. More recent work on discriminative semi-supervised learning has focussed on introducing assumptions that relate unlabeled data to the labeled objects \\cite{Chapelle2006}. These assumptions usually take the form of either a manifold assumption \\cite{Zhu2003}, encoding that labels change smoothly in a low-dimensional manifold, or a low-density class separation assumption used in, for instance, transductive support vector machines \\cite{Bennett1998,Joachims1999} and entropy regularization \\cite{Grandvalet2005}. \n\nWork on semi-supervised LDA has tried to incorporate unlabeled data by leveraging the increase in accuracy of estimators of quantities that do not rely on labels. An approach relying on the more accurate estimate of the total covariance matrix of both labeled and unlabeled objects is taken for dimensionality reduction in Normalized LDA, proposed by \\cite{Fan2008} and similar work by \\cite{Cai2007}. In addition to this covariance matrix, \\cite{Loog2014a} also include the more accurate estimate of the overal mean of the data and propose two solutions to solve a subsequent optimization problem. Building on these results, in \\cite{Krijthe2013} we introduced implicitly constrained least squares classification, a semi-supervised adaptation of least squares classification. Since this procedure proved both theoretically and practically successful for a discriminative classifier, here we consider whether the idea of implicitly constrained semi-supervised learning can be extended to generative classifiers such as LDA.\n\n\\section{Methods}\nWe will first introduce linear discriminant analysis as a supervised classification algorithm and discuss different semi-supervised procedures. We will consider 2-class classification problems, where we are given an $N_l \\times d$ design matrix $\\mathbf{X}$, where $N_l$ is the number of labeled objects and $d$ is the number of features. For these observations we are given a label vector $\\mathbf{y}=\\{0,1\\}^{N_l}$. Additionally, in the semi-supervised setting we have an $N_u \\times d$ design matrix $\\mathbf{X}_u$ without a corresponding $\\mathbf{y}_u$ for the unlabeled observations.\n\n\\subsection{Supervised LDA}\nIn supervised linear discriminant analysis, we model the 2 classes as having multivariate normal distributions with the same covariance matrix $\\boldsymbol{\\Sigma}$ and differing means $\\boldsymbol{\\mu}_1$ and $\\boldsymbol{\\mu}_2$. To estimate the parameters of this model, we maximize the likelihood, or, equivalently, the log likelihood function:\n\\begin{align}\n\\label{eq:lda}\nL(\\theta|\\mathbf{X},\\mathbf{y})= \\sum_{i=1}^{N_l} & y_i \\log(\\pi_1 \\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu}_1,\\mathbf{\\Sigma})) \\nonumber \\\\\n& +(1-y_i) \\log(\\pi_2 \\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma}))\n\\end{align}\nwhere $\\theta=\\left( \\pi_1,\\pi_2, \\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma} \\right)$, $\\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu},\\mathbf{\\Sigma})$ denotes the density of a multivariate normal distribution with mean $\\boldsymbol{\\mu}$ and covariance $\\mathbf{\\Sigma}$ evaluated at $\\mathbf{x}_i$ and $\\pi_c$ denotes the prior probability for class $c$. The closed form solution to this maximization is given by the estimators:\n\\begin{align}\n\\label{eq:ldasolution}\n\\hat{\\pi_1} &= \\frac{\\sum_{i=1}^{N_l} y_i}{N_l}, \\hspace{20px} \\hat{\\pi_2} = \\frac{\\sum_{i=1}^{N_l} (1-y_i)}{N_l}  \\nonumber \\\\\n\\hat{\\boldsymbol{\\mu}}_1 &= \\frac{\\sum_{i=1}^{N_l} y_i \\mathbf{x}_i}{\\sum_{i=1}^{N_l} y_i}, \\hspace{20px} \\hat{\\boldsymbol{\\mu}}_2 = \\frac{\\sum_{i=1}^{N_l} (1-y_i) \\mathbf{x}_i}{\\sum_{i=1}^{N_l} (1-y_i)}  \\nonumber  \\\\\n\\hat{\\mathbf{\\Sigma}} &= \\frac{1}{N_l} \\sum_{i=1}^{N_l} y_i (\\mathbf{x}_i-\\boldsymbol{\\mu}_1) (\\mathbf{x}_i-\\boldsymbol{\\mu}_1)^T  \\nonumber \\\\\n& \\hspace{20px} + (1-y_i) (\\mathbf{x}_i-\\boldsymbol{\\mu}_2) (\\mathbf{x}_i-\\boldsymbol{\\mu}_2)^T \n\\end{align}\nWhere the maximum likelihood estimator $\\hat{\\mathbf{\\Sigma}}$ is a biased estimator for the covariance matrix. Given a set of labeled objects $(\\mathbf{X},\\mathbf{y})$, we can estimate these parameters and find the posterior for a new object $\\mathbf{x}$ using:\n\\begin{equation}\np(c=1|\\mathbf{x})=\\frac{\\pi_1 \\mathcal{N}(\\mathbf{x}|\\hat{\\boldsymbol{\\mu}}_1,\\hat{\\mathbf{\\Sigma}})}{\\sum_{c=1}^{2} \\pi_c \\mathcal{N}(\\mathbf{x}|\\hat{\\boldsymbol{\\mu}}_c,\\hat{\\mathbf{\\Sigma}})}\n\\end{equation}\nThis posterior distribution can be employed for classification by assigning objects to the class for which its posterior is highest. We now consider several adaptations of this classification procedure to the  semi-supervised setting.\n\n\\subsection{Self-Learning LDA (SLLDA)}\nA common and straightforward adaptation of any supervised learning algorithm to the semi-supervised setting is self-learning, also known as bootstrapping or Yarowsky's algorithm \\cite{McLachlan1975,Yarowsky1995}. Starting out with a classifier trained on the labeled data only, labels are  predicted for the unlabeled objects. These objects, with their imputed labels are then used in the next iteration to retrain the classifier. This new classifier is now used to relabel the unlabeled objects. This is done until the predicted labels on the unlabeled objects converge. \\cite{Abney2004} studies the underlying loss that this procedure minimizes and proves its convergence.\n\n\\subsection{Expectation Maximization LDA (EMLDA)}\nAssuming the mixture model of Equation \\eqref{eq:lda} and treating the unobserved labels $\\mathbf{y}_u$ as latent variables, a possible adaptation of this model is to add a term for the unlabeled data to the objective function and to integrate out the unknown labels, $\\mathbf{y}_u$, to find the marginal likelihood:\n\\begin{align}\n\\l(\\theta|\\mathbf{X},\\mathbf{y},\\mathbf{X}_u)=\\prod_{i=1}^{N_l}\\left(\\pi_1 \\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu}_1,\\mathbf{\\Sigma})\\right)^{y_i} \\left(\\pi_2 \\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma})\\right)^{1-y_i}  \\nonumber \\\\ \n\\times \\prod_{i=1}^{N_u} \\sum_{c=1}^2 \\pi_c \\mathcal{N}(\\mathbf{x}_i|\\boldsymbol{\\mu}_c,\\Sigma)\n\\end{align}\nMaximizing this marginal likelihood, or equivalently, the log of this function is harder then the supervised objective in Equation \\eqref{eq:lda}, since the expression contains a log over a sum. However, we can solve this optimization problem using the well-known expectation maximization (EM) algorithm \\cite{Dempster1977, Nigam2000}. In EM, the log over the sum is bounded from below through Jensen's inequality. In the M step of the algorithm, we maximize this bound by updating the parameters using the imputed labels obtained in the E step. In practice, the M step consists of the same update as in Equation \\eqref{eq:ldasolution}, where the sum is no longer over the labeled objects but also the unlabeled objects using the imputed posteriors, or responsibilities, from the E step. In the E step the lower bound is made tight by updating the imputed labels using the posterior under the new parameter estimates. This is done until convergence. \nIn effect this procedure is very similar to self-learning, where instead of hard labels, a probability over labelings is used. Both self-learning and EM suffer from the problem of wrongly imputed labels that can reinforce their wrongly imputed values because the parameters are updated as if they were the true labels.\n\n\\subsection{Moment Constrained LDA (MCLDA)}\nAn alternative to the EM-like approaches like EMLDA and SLLDA was proposed by \\cite{Loog2010} in the form of moment constrained parameter estimation. The main idea is that there are certain constraints that link parameters that are calculated using feature values alone, with parameters which require the labels. In the case of LDA \\cite{Loog2014a}, for instance, the overal mean of the data is linked to the means of the two classes through:\n\\begin{equation}\n\\label{eq:constraintmean}\n\\boldsymbol{\\mu}_t=\\pi_1 \\boldsymbol{\\mu}_1 + \\pi_2 \\boldsymbol{\\mu}_2\n\\end{equation}\nWere $\\boldsymbol{\\mu}_t$ is the overal mean on all the data and therefore does not depend on the labels.\nThe total covariance matrix $\\mathbf{\\Sigma}_t$ is linked to the within-class covariance matrix $\\mathbf{\\Sigma}$ and between-class covariance matrix $\\mathbf{\\Sigma}_b$, the covariance matrix of the means. Only the latter two rely on the labels:\n\\begin{equation}\n\\label{eq:constraintcovariance}\n\\mathbf{\\Sigma}_t=\\mathbf{\\Sigma} + \\mathbf{\\Sigma}_b\n\\end{equation}\nRecognizing that the unlabeled data allow us to more accurately estimate the parameters in these constraints that do not rely on the labels, \\cite{Loog2014a} points out that this more accurate estimate will generally violate the constraints, meaning the other label-dependent estimates should be updated accordingly.\n\nAn ad hoc way to update the parameters based on these more accurate estimates \\cite{Loog2014a} leads to the following updated moment constrained estimators:\n\\begin{align}\n\\hat{\\boldsymbol{\\mu}}_c^{MC} & =\\hat{\\boldsymbol{\\mu}}_c - \\sum_{j=1}^{2} \\hat{\\pi}_j \\hat{\\boldsymbol{\\mu}}_j-\\hat{\\boldsymbol{\\mu}}_t \\\\\n\\hat{\\mathbf{\\Sigma}}^{MC} &= \\hat{\\mathbf{\\Theta}}^{\\frac{1}{2}} \\hat{\\mathbf{\\Sigma}}_t^{\\frac{1}{2}} \\hat{\\mathbf{\\Sigma}} \\hat{\\mathbf{\\Sigma}}_t^{\\frac{1}{2}} \\hat{\\mathbf{\\Theta}}^{\\frac{1}{2}}\n\\end{align}\nwhere $\\hat{\\boldsymbol{\\mu}}_t$ and $\\hat{\\mathbf{\\Theta}}$ are the overal mean and overal covariance estimated on all labeled and unlabeled data, while $\\hat{\\mathbf{\\Sigma}}_t$ is the overal covariance estimated on the labeled data alone.\n\nAlternatively and slightly more formally, \\cite{Loog2012b} forces the constraints to be satisfied by maximizing the likelihood on the labeled objects under the constraints in Equations \\eqref{eq:constraintmean} and \\eqref{eq:constraintcovariance}. This leads to a non-convex objective function that can be solved numerically. In this work we use the simpler ad hoc constraints.\n\n\\subsection{Implicitly Constrained LDA (ICLDA)}\nThe former approach requires the identification of specific constraints. Ideally, we would like these constraints to emerge implicitly from a choice of supervised learner and a given set of unlabeled objects. Implicitly constrained semi-supervised learning attempts to do just that. The underlying intuition is that if we could enumerate all possible $2^{N_u}$ labelings, and train the corresponding classifiers, the classifier based on the true but unknown labels is in this set. This classifier would generally outperform the supervised classifier. Two problems arise:\n\n\\begin{enumerate}\n\\item How do we find a classifier in this set that is close to the one based on the true but unknown labels?\n\\item How do we efficiently traverse this enormous set of possible labelings without having to enumerate them all?\n\\end{enumerate}\nAs for the first problem: a safe way to know how well a solution performs in terms of our supervised objective is to estimate its performance using the labeled objects. We therefore propose the following objective:\n\n\\begin{equation}\n\\label{eq:iclda}\n\\operatorname*{arg\\,max}_{\\left( \\pi_1,\\pi_2, \\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma}\\right) \\in \\mathcal{C}_\\theta} L(\\pi_1,\\pi_2, \\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma}|\\mathbf{X},\\mathbf{y})\n\\end{equation}\nwhere\n\\begin{align}\n\\mathcal{C}_{\\boldsymbol{\\theta}} = \\left\\{ \\operatorname*{arg\\,max} L(\\pi_1,\\pi_2, \\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma}| \\mathbf{X}_e, \\mathbf{y}_e) : \\mathbf{y}_u \\in [0,1]^{N_u} \\right\\} \\nonumber\n\\end{align}\nand $\\mathbf{X}_e=[\\mathbf{X}^T \\mathbf{X}_u]^T, \\mathbf{y}_e=[\\mathbf{y}^T \\mathbf{y}_u^T]^T$ are the design matrix and class vector extended with the unlabeled data. This can be interpreted as optimizing the same objective function as supervised LDA, with the additional constraint that the solution has to attainable by a particular assignment of responsibilities (partial assignments to classes) for the unlabeled objects.\n\nAs for the second problem: since, for a given imputed labeling, we have a closed form solution for the parameters, the gradient of the supervised loss \\eqref{eq:iclda} with respect to the responsibilities $\\mathbf{y}_u$ can be found using\n\\begin{equation}\n\\frac{\\partial L(\\theta|\\mathbf{X},\\mathbf{y})}{\\partial \\mathbf{y_u}} = \\frac{\\partial L(\\theta|\\mathbf{X},\\mathbf{y})}{\\partial \\theta} \\frac{\\partial \\phi(\\mathbf{y}_u)}{\\partial \\mathbf{y}_u} \n\\end{equation}\nwhere $\\phi(\\mathbf{y}_u)=\\theta$ is the function that has as input a particular labeling of the points, and outputs the parameters $\\theta=\\left(\\pi_1,\\pi_2, \\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\mathbf{\\Sigma}\\right)$, similar to Equation \\eqref{eq:ldasolution}.\n\nThis can be used to efficiently move through the set of solutions using a simple gradient ascent procedure that takes into account the $[0,1]$ bounds on the responsibilities.\n\n\\section{Experimental setup and results}\nWe present simulations on an illustrative toy dataset and a set of benchmark datasets. Other than the classifiers covered in the previous section, we also include the LDA classifier trained using all labels of the unlabeled data (LDAoracle) as an upper bound on the performance of any semi-supervised procedure. The experiments can be reproduced using code from the authors' website. \n\n\\subsection{Toy problems}\nTo illustrate the behaviour of ICLDA when compared to EMLDA we consider two toy datasets. In both cases we have two multivariate normal distributions centered at respectively $\\boldsymbol{\\mu}_1=[1,1]^T$ and $\\boldsymbol{\\mu}_2=[-1,-1]^T$ and equal covariance $\\mathbf{\\Sigma}=0.6 \\mathbf{1}$, with $\\mathbf{1}$ the $2 \\times 2$ identity matrix. An example is given in Figure \\ref{fig:toyplots}. In the bottom row, these two gaussians correspond to the different classes. In the top row, we consider the case where the decision boundary is actually perpendicular to the boundary in the other setting. This means that the bottom row corresponds exactly to the assumptions of EM, while this is not the case in the top row. Figure \\ref{fig:toyplots} illustrates what happens in a particular sample from this problem were we draw 10 labeled and 990 unlabeled objects. When the assumption does not hold, EMLDA forces the decision boundary to fall between the two gaussian clusters leading to a much worse solution then the supervised LDA based on only a few labeled examples. The ICLDA solution does not deviate from the correct boundary by much. When the assumptions do hold, EMLDA finds the correct boundary, as expected, while ICLDA only does minor adjustments in this case. \n\nWhile one could claim that ICLDA is more robust, one could also expect ICLDA to never lead to any improvements. Figure \\ref{fig:learningcurvegauss} shows the results when resampling from the data distribution in the second example and shows that ICLDA does lead to improvement on average in the second dataset, while not making the mistake in the first dataset where the LDA assumptions do not hold.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.45]{toyplots-crop.pdf}\n\\caption{Behaviour on the two-class two dimensional gaussian datasets, with 10 labeled objects and 990 unlabeled objects. The first row shows the scatterplot and the trained responsibilities for respectively ICLDA and EMLDA on a dataset where the decision boundary does not adhere to the assumptions of EMLDA. The second row shows the results when the decision boundary is in between the two Gaussian classes. The black line indicates the decision boundary of a supervised learner trained using only the labeled data. Note that in the first row, the responsibilities of EM are very different from the true labels, while IC is not as sensitive to this problem.}\n\\label{fig:toyplots}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.40]{learningcurvegaussian-crop.pdf}\n\\caption{Semi-supervised learning curve on the Gaussian data set using 500 repeats. The shaded regions indicate one standard error around the mean. Since their assumptions hold exactly, SLLDA and EMLDA work very well. ICLDA also outperforms the supervised LDA.}\n\\label{fig:learningcurvegauss}\n\\end{figure}\n\n\\subsection{Simulations on Benchmark datasets}\nWe test the behaviour of the considered procedures using datasets from the UCI machine learning repository \\cite{Bache2013}, as well as from \\cite{Chapelle2006}. Their characteristics can be found in Table \\ref{table:datasets}.\n\n\\begin{table}[!t]\n\\caption{Description of the datasets used in the experiments}\n\\label{table:datasets}\n\\centering\n\\begin{tabular}{|rrrr|}\n  \\hline\n Name & \\# Objects & \\#Features & Source \\\\ \n  \\hline\n  Haberman & 305 &   4 & \\cite{Bache2013} \\\\ \n  Ionosphere & 351 &  33 & \\cite{Bache2013} \\\\ \n  Parkinsons & 195 &  20 & \\cite{Bache2013} \\\\ \n  Pima & 768 &   9 & \\cite{Bache2013} \\\\ \n  Sonar & 208 &  61 & \\cite{Bache2013} \\\\ \n  SPECT & 265 &  23 & \\cite{Bache2013} \\\\ \n  SPECTF & 265 &  45 & \\cite{Bache2013} \\\\ \n  Transfusion & 748 &   4 & \\cite{Bache2013} \\\\ \n  WDBC & 568 &  30 & \\cite{Bache2013} \\\\\n  BCI & 400 & 118 & \\cite{Chapelle2006} \\\\ \n   \\hline\n\\end{tabular}\n\\end{table}\n\nA cross-validation experiment was carried out as follows. Each of the datasets were split into $10$ folds. Every fold was used as a validation set once, while the other nine folds were used for training. The data in these nine folds was randomly split into a labeled and an unlabeled part, where the labeled part had $\\max(2d,10)$ objects, while the rest was used as unlabeled objects. This procedure was repeated $20$ times and the average error and average negative log likelihood (the loss function of LDA) on the test set was determined. The results can be found in Tables \\ref{table:cvresults-error} and \\ref{table:cvresults-loss}.\n\nTo study the behaviour of these classifiers for differing amount of unlabeled data, we estimated semi-supervised learning curves by randomly drawing $\\max(2d,10)$ labeled objects from the datasets, and using an increasing randomly chosen set as unlabeled data. The remaining objects formed the test set. This procedure was repeated 500 times and the average and standard error of the classification error and negative log likelihood were determined. The learning curves for 3 datasets can be found in Figure \\ref{fig:errorcurves}.\n\n\\input{cvtable-error.tex}\n\\input{cvtable-loss.tex}\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.5]{errorcurves-crop.pdf}\n\\caption{Learning curves for increasing amounts of unlabeled data for the error rate as well as the loss (negative log likelihood) for three datasets using 500 repeats. The shaded regions indicate one standard error around the mean.}\n\\label{fig:errorcurves}\n\\end{figure*}\n\nWe find that overal in terms of error rates (Table \\ref{table:cvresults-error}), MCLDA seems to perform best, being both more robust than the EM approaches as well as effective in using the unlabeled information in improving error rates. While ICLDA is robust and has the best performance on 2 of the datasets, it is conservative in that it does not show improvements in terms of the classification error for many datasets where the other classifiers do offer improvements.\n\nThe picture is markedly different when we consider the log likelihood criterion that supervised LDA optimizes, evaluated on the test set (Table \\ref{table:cvresults-loss}). Here ICLDA outperforms all other methods on the majority of the datasets. \n\n\n\n\n\n\n\n\n\\section{Discussion}\nThe results show that ICLDA provides the safest option among the semi-supervised approaches to LDA that we considered. At least in terms of the log-likelihood, it provides the best performance by far. A particularly interesting observation is that the implicit constraints, in many a case, seem to add a lot to the constraints that MCLDA enforces, as also the latter classifier is consistently outperformed by ICLDA in terms of the log likelihoods achieved on the test set. As yet, we have no full understanding in what way the implicit constraints add restrictions beyond the moment constraints, apart from the fact that the former are stricter than the latter.  A deeper insight into this issue might cast further light on the workings of ICLDA.\n\nWhile it is the safest version, ICLDA may be \\emph{too} safe, in that it does not attain performance improvement in terms of classification error in many cases where MCLDA or the EM approaches do offer improvements. In terms of the loss on the test set, however, ICLDA is the best performing method. Since this is the objective minimized by supervised LDA as well, perhaps this is the best we could hope for in a true semi-supervised adaptation of LDA. We found similar empirical and theoretical performance results in terms of improvements in the loss on the test set when applying the implicitly constrained framework to the least squares classifier \\cite{Krijthe2013}. How then, this improvement in ``surrogate'' loss relates to the eventual goal of classification error, is unclear, especially for a non-margin based loss function such as the negative log likelihood \\cite{Bartlett2006}. However, since ICLDA does offer the best behaviour of supervised LDA's loss on the test set, ICLDA could be considered a step towards a principled semi-supervised version of LDA.\n\nAn open question regarding the objective function associated with ICLDA is to what extent it is convex. The solution in terms of the responsibility vector $\\mathbf{y}_u$ is non-unique: different labelings of the points can lead to the same parameters. In terms of the parameters, however, the optimization seems to converge to a unique global optimum. While we do not have a formal proof of this, as in the case of implicitly constrained least squares classification, we conjecture that the objective function is convex in the parameters, at least in the case in which we choose to parameterize LDA by means of its canonical parameters \\cite{Lehmann1998}.  In this case, LDA does lead to a convex optimization problem.\n\nWe find that the behaviour of EMLDA is more erratic than that of SLLDA. The hard label assignments could have a regularizing effect on the semi-supervised solutions, making self-learning a good and fast alternative to self-learning. Note that safer versions of SLLDA and EMLDA could be obtained by introducing a weight parameter to control the influence of the unlabeled data \\cite{McLachlan1975}. In the limited labeled data setting, it is hard to correctly set this parameter. While this may help when dealing with larger sample sizes, the constraint approaches bring us closer to methods that always perform at least as well as their supervised counterpart. \n\n\\section{Conclusion}\nICLDA is a principled and robust adaptation of LDA to the semi-supervised setting. In terms of error rates, it may be overly conservative. When measured in terms of the loss on the test set, however, it outperforms other semi-supervised methods. It therefore seems that there are opportunities for robust semi-supervised learners, although the performance criterion that we should consider may not be the error rate, but rather the loss that the supervised learner minimizes \\cite{Loog2014b}.\n\n\n\n\\section*{Acknowledgment}\nThis work was partly funded by project P24 of the Dutch public\/private research network COMMIT.\n\n\n\n\\bibliographystyle{IEEEtranS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiant Molecular Clouds (GMCs) are the nursery of massive stars. They are of prime importance to the evolution of their host galaxies, \nbut there are still uncertainties on how to best estimate their masses. At their low temperatures ($\\sim10-50$~K), the molecular masses cannot be measured through the emission of the H$_2$ molecule because its first rotational transition arises from an energy level at a temperature of 510~K. \nGenerally in galaxies, low-J transitions from the second most abundant molecule, CO, are the tracer of the molecular mass. \nDust emission is an alternative tracer of the gas mass, which has been used to estimate dust masses \\citep{draine07b,galametz11} and to calibrate the $X_{\\rm CO}$ conversion factor from CO luminosity to H$_2$ mass  \\citep{israel97,leroy11}. In the Milky Way and nearby galaxies, where GMCs are spatially resolved, the $X_{\\rm CO}$-factor has been also estimated using the relationships between cloud properties assuming virial equilibrium \\citep{solomon87,bolatto08}. \n\nCO emission does not have a one-to-one correspondence with H$_2$. Observations in the Galaxy \\citep{grenier05,abdo10, planck11a19} and models of photodissociation regions (PDRs) \\citep[e.g.][]{wolfire10} show that CO does not trace the molecular gas over the full extent of clouds. CO-dark H$_2$ gas was also observed in the Large Magellanic Cloud (LMC) with dust observations from Spitzer and Herschel \\citep{bernard08, romanduval10,galliano11}.  In the Small Magellanic Cloud (SMC), \\citet{rubio04} and later \\citet{bot07,bot10} reported differences up to a factor $\\sim$10 between GMCs masses derived from the dust emission (based on 870~$\\mu$m and 1.2~mm emission) and the virial theorem. Additional evidence for H$_2$ gas without CO emission is provided by observations of the [\\ion{C}{ii}]$\\lambda$158$\\mu$m line emission in Magellanic irregular galaxies \\citep{madden97, israel11} and the Milky Way \\citep{velusamy10}.\nPDR models show that the column density threshold for CO to become the main carbon species has a much stronger dependence on the metallicity than that for the gas to become molecular \\citep[e.g.][]{wolfire10}. This is because CO is shielded from photodissociation by dust while H$_2$ is self-shielded. Whereas in the Galaxy there is a contribution to CO emission from diffuse molecular gas \\citep{liszt10}, in low metallicity galaxies the CO may predominately come from dense, high column density clumps embedded in a diffuse molecular envelop with weak or no CO emission \\citep{israel88,lequeux94}. \n \nThe motivation of this paper is to compare estimates of the molecular cloud masses in a low metallicity environment, the LMC. The LMC is at 50 kpc \\citep{persson04}. It presents a metallicity, measured for \\ion{H}{ii} regions, of 12+log(O\/H)=8.37 \\citep{keller06}, that corresponds to $Z$$\\simeq$$0.5~Z_{\\sun}$. Extensive CO(1--0) surveys have been done in the Magellanic Clouds. In the LMC, the first one done by \\citet{cohen88} and subsequently the SEST \\citep{israel93}, NANTEN \\citep{fukui08} and MAGMA \\citep{hughes10,wong11} surveys gave a large set of GMCs. Also, several detailed studies have been done in the LMC (e.g. \\citealt{garay02} for molecular complex No. 37; \\citealt{israel03} for N11; \\citealt{johansson98}, \\citealt{kutner97}, \\citealt{pineda09} for 30Dor).  All of these studies report a CO-to-H$_2$ conversion factor estimated from the comparison between the virial and CO luminosity masses of the molecular clouds larger than in the Galaxy.\n \nThis paper is based on CO(J$=$1--0 and J$=$2--1) and 1.2~mm continuum observations of the N11 star-forming region in the LMC. We present the observations and compare the CO luminosity of molecular clouds with their masses estimated with the virial theorem and the millimeter dust emission. Unlike what has been done with far-IR observations from Spitzer and what is now being done with Herschel \\citep{bernard08, galliano11, leroy11}, the computation of molecular masses from the millimeter emission of the dust is not very sensitive to the dust temperature because it is on the Rayleigh-Jeans spectral side of the energy distribution. Further, the comparison between dust and virial masses has seldom been performed because the molecular clouds have to be resolved to estimate virial masses. With single dish telescopes and space IR observations (Spitzer and Herschel) this is possible only in the Milky Way and the Magellanic Clouds. This paper extends the earlier studies on the SMC done by \\citet{rubio04} and \\citet{bot07,bot10} to intermediate metallicities in the LMC. Soon with ALMA, this can be extended to more distant galaxies in the Local group. \n \n\nThe paper is organized as follows. In Section 2, we present observations of CO in the J$=$2--1 transition line, new data in the transition J$=$1--0, and a continuum map at 1.2~mm. We identify the N11 molecular clouds using the CO(2--1) data and derive their physical properties (CO luminosity, size, line-width and virial mass) in Section 3. In Section 4, we derive millimeter dust fluxes from the continuum observations. In Section 5, we present the three estimations of molecular masses, and the comparison between them. We show for which values of the $X_{\\rm CO}$-factor and the millimeter dust emissivity per hydrogen, as a function of the virial parameter of the clouds, the masses are identical. Finally, Section 6 presents the conclusions.\n\n\n\n\\section{Observations in the N11 region}\n\n\\subsection{The N11 star-forming region}\nN11 \\citep{henize56} is the second largest and brightest \\ion{H}{ii} region in the LMC, and hosts several stellar clusters (see Fig.~\\ref{fig:opt}). Its main feature is a cavity with an inner diameter of $\\sim$700\\arcsec\\ ($\\sim$170 pc). In its center there is the OB association LH9 \\citep{lucke70} while in its northeast part there is the N11B nebula which is associated with another stellar cluster, LH10. The clusters are 7 and 3~Myr old, respectively \\citep{mokiem07}.  \\citet{parker92} show that most of the ionizing flux in N11 comes from these two clusters. IR observations \\citep{barba03} reveal that the star formation is on-going in N11B. \\citet{israel03} presented CO(1--0) observations of N11 with a resolution of FWHM$=$45\\arcsec. They identified molecular clouds and computed their physical properties. Recently, \\citet{israel11} showed that even though the [\\ion{C}{ii}]$\\lambda$158$\\mu$m and CO emission lines present similar spatial distributions, the [\\ion{C}{ii}] emission extends beyond that of CO. Also, it is observed that generally [\\ion{C}{ii}] does not peak where CO peaks but it is offset in the direction of the ionizing stars. \n\n   \\begin{figure}[ht]\n   \\centering\n   \\includegraphics[width=8cm]{fig1_n11_b.ps}\n      \\caption{Image obtained from the combination of hydrogen, sulfur and oxygen bandpasses (Magellanic Clouds Emission-Line Survey MCELS, C. Aguilera, C. Smith and S. Points\/NOAO\/AURA\/NSF).  Contours correspond to the CO(2--1) line emission. We indicate the position of the OB associations in N11, LH9 in the central cavity of N11, LH10 in the nebula N11B, LH13 in N11C and LH14 in N11E.} \\label{fig:opt}\n   \\end{figure}\n  \n\n\\subsection{SEST Observations: CO emission lines}\\label{sec:sest}\n\nThe observations of the CO emission lines were taken with the SEST (Swedish-ESO Submillimeter Telescope) at La Silla Observatory, Chile, during January and September in 2001. SEST was a single dish antenna of 15 meter of diameter operating in the range of frequencies 70$-$365~GHz. It was decommissioned in 2003 with the beginning of operations at the APEX (Atacama Pathfinder Experiment) telescope.\n\nWe mapped the central part of the N11 star-forming region in three different regions, which are listed in Table~\\ref{tab:sest}. We observed simultaneously the CO~J$=$1--0 and CO~J$=$2--1 lines at the frequencies 115~GHz and 230~GHz with a full width beam at half maximum (FWHM) of 45\\arcsec(11~pc linear size) and 23\\arcsec(6~pc linear size), respectively. The spatial sampling of the telescope pointings for both frequencies is 24\\arcsec. We note that our CO(2--1) observations are not fully-sampled. However, this does not have a significant impact for the goal of this paper (see \\S~ \\ref{ss:molclouds}). The system temperature varied between 360 and 713~K at 115~GHz and between 235 and 467~K at 230~GHz, during the observations.\n\nThe observations were done in position switch mode with a fixed reference off position free of CO emission. A narrow band AOS high-resolution (HRS) spectrometer with 2000 channels, 80 MHz bandwidth and 41.7 kHz channel separation (corresponding to 0.055 km~s$^{-1}$ for the $^{12}$CO~J$=2-1$ line) was used as back end. Intensity calibration was done using the standard chopper-wheel technique. The pointing accuracy, checked during the observations on RDor, was better than 2\\arcsec.\n\n   \\begin{figure*}\\label{fig:co21em}\n   \\centering\n   \\includegraphics[width=13.5cm]{fig2_n11_d.ps}\n         \\caption{Integrated CO(2--1) emission line of the N11 star forming region, observed with the SEST. Contour corresponds to 2.5 K~km~s$^{-1}$. The dots represent the observed positions. Red crosses mark the position of the molecular clouds listed in Table~\\ref{tab:nub}. Offset positions are relative to $\\alpha:$$04^{h}56^{m}42^{s}.47$, $\\delta:$$-66^{\\circ}28^{\\prime}42\\farcs0$ J2000.0.} \\label{fig:co21}\n   \\end{figure*}\n   \n\n\\begin{table}[h]\n\\begin{minipage}[t]{\\columnwidth}\n\\caption{Characteristics of the SEST observations in the N11 star-forming region.}\n\\label{tab:sest}\n\\centering\n\\renewcommand{\\footnoterule}{} \n\\begin{tabular}{ccccc}        \n\\hline\\hline                 \n\\multirow{2}{*}{Zone} & \\multicolumn{2}{c}{Coordinates} & $\\sigma_{\\rm CO(2-1)}^a$  & $\\sigma_{\\rm CO(1-0)}^b$ \\\\  \n& R.A. & DEC.& \\multicolumn{2}{c}{K}   \\\\  \n\\hline                       \n  Map 1 & $04^{h}57^{m}25^{s}.00$ & $-66^{\\circ}24^{\\prime}00\\farcs0$ & 0.25 & 0.30 \\\\\n  Map 2 & $04^{h}55^{m}36^{s}.00$ & $-66^{\\circ}30^{\\prime}00\\farcs0$  & 0.22 & 0.33 \\\\\n  Map 3 & $04^{h}56^{m}25^{s}.00$ & $-66^{\\circ}36^{\\prime}00\\farcs0$ & 0.20 & 0.27 \\\\\n\\hline                                   \n\\end{tabular}\n\\end{minipage}\n$^a$ computed within the 0.055~${\\rm km~s^{-1}}$ channel width. \\\\\n$^b$ computed within the 0.11~${\\rm km~s^{-1}}$ channel width.\n\\end{table}  \n\n\nThe data reduction was done with CLASS (Continuum and Line Analysis Single-dish Software\\footnote{http:\/\/www.iram.fr\/IRAMFR\/GILDAS}). The complete set of spectra were visually inspected one-by-one, we did not find any anomalies in the spectra. We fitted and subtracted a baseline of second order to each spectrum to correct for residual atmospheric and instrumental emission. The observed  velocity resolution is 0.11 and 0.055~km~s$^{-1}$, for CO(1--0) and CO(2--1), respectively. Finally, to obtain the radiation temperature of the sources, we corrected for the antenna efficiency $\\eta$$=$0.7 and 0.5 for CO(1--0) and CO(2--1), respectively \\citep{johansson98}. The flux measurements have a photometric calibration uncertainty of 20\\%. Table~\\ref{tab:sest} lists the 1$\\sigma$ noise in a 0.11 and 0.055~km~s$^{-1}$ channel band, for CO(1--0) and CO(2--1) respectively, in each line emission and mapped zone. These values were estimated as the average value of the standard deviation computed in 20 line-free channels at velocities of about 259~km~s$^{-1}$.\n\nFigure~\\ref{fig:co21} shows the integrated CO(2--1) line emission $I_{\\rm CO}$, in units of K~km~s$^{-1}$. This image was obtained by integrating the CO emission in a velocity window defined in the total spectrum, from 265 to 289~km~s$^{-1}$. Figure~\\ref{fig:chan} presents the CO(2--1) channel maps integrated in bins of 2~km~s$^{-1}$. From these channel maps we can distinguish several components at different velocities.\n\n\\begin{figure*} \\centering\n\\includegraphics[width=16cm,trim = 22mm 0mm 50mm 0mm]{channel_maps_clouds.ps}\n\\vspace{1cm}\n\\caption{Channel maps for the N11 region. Contours illustrate the CO emission in the transition J$=$2--1 smoothed over 4 pixels. The first contour is 0.75 K km s$^{-1}$, with a spacing of 0.8 K km s$^{-1}$. From the top-left to the bottom-right image, we present channel maps starting from 266 km s$^{-1}$ to 290 km s$^{-1}$, with a velocity width of 2 km s$^{-1}$. Offset positions are the same as in Fig.~\\ref{fig:co21em}. We mark in red the molecular clouds identified with CPROPS.} \\label{fig:chan}\n\\end{figure*}\n\nFor each CO transition line, we computed the total emission by adding all the spectra in the field and integrating the emission in the velocity range  265$-$289~km~s$^{-1}$. The total emission corresponds to 1416 K~km~s$^{-1}$ and 1630 K~km~s$^{-1}$ for CO(1--0) and CO(2--1), respectively.\n   \n\\subsection{SIMBA observations: millimeter continuum emission at 1.2~mm}\\label{sec:12mm}\n\n\nTo measure the dust emission in N11, we observed the continuum emission at 1.2~mm. This observation was obtained with the SEST IMaging Bolometer Array (SIMBA) at La Silla Observatory in October 2002. SIMBA was a 37 channel hexagonal array operating at 250~GHz. The bandwidth in each channel was about 90~GHz. The Half Power Beamwidth (HPBW) of each element was 24\\arcsec.\n\nThe observations covered the central and northern parts of the N11 complex. The individual maps were produced using the fast scanning mode. The elimination of the correlated sky noise, the co-addition of individual maps, and the photometry was done using the MOPSI package\\footnote{MOPSI is a data reduction software package for IR and radio data developed by R. Zylka, IRAM, Grenoble, France}. The $\\tau$ zenith opacity was determined performing skydips every 2 hours during the day, and every 3 hours during the night. The flux determination was done for each individual observing run by observing Uranus. The flux measurements in the final coadded map have a photometric calibration uncertainty of 20\\%. The sensitivity of the final image is 4~mJy~beam$^{-1}$ (1$\\sigma$). Figure~\\ref{fig:12mm} shows the observed emission of N11 at 1.2~mm, with CO(2--1) contours overlaid.  \n\n   \\begin{figure}[!h]\n   \\centering\n\\includegraphics[width=11cm]{graficos_n11_newab.ps}\n      \\caption{Millimeter continuum emission at 1.2~mm of the N11 star forming region, observed with the SIMBA. Black contour represents the emission at 3$\\sigma$$=$12~mJy~beam$^{-1}$. Red contour is the CO(2--1) emission at 2.5~K~km~s$^{-1}$.}\\label{fig:12mm}\n   \\end{figure}\n   \nThis image was used to measure the millimeter continuum flux of the molecular clouds in N11, identified with the CO observations. We measure the emission in an elliptical aperture defined by the molecular cloud extent at the position of the central coordinates of the cloud. We subtracted the nearby sky emission measured next to the source on an empty image area.  The flux errors correspond to the noise level (standard deviation) in units of Jy~beam$^{-1}$, measured locally on an OFF area next to the cloud, multiplied by the area of the beam and by the root square of the number of beams within the aperture size. The final flux errors were obtained using error propagation of the flux uncertainties and the 20\\% photometric calibration errors. \n\n\n\\subsection{Ancillary data}\n\nWe complement our data set with two published fully reduced images at 8.6 and 4.8 GHz obtained with the ATCA (Australia Telescope Compact Array) in February 2003 \\citep{dickel05}. These observations are part of two surveys of the LMC. The angular resolution is 22\\arcsec and 35\\arcsec (FWHM beam width) at 8.6 and 4.8 GHz, respectively. The flux uncertainties for the observations are 5\\%. \nTo measure the fluxes of these radio emission images, we did aperture photometry in the same way as for the emission at 1.2~mm.\n\n\n\n\\section{Molecular clouds in N11}  \n\nIn this section, we present the observational results obtained from the SEST observations in N11. \n\n\\subsection{Identification of molecular clouds}\\label{ss:molclouds}\n\nThe identification of the molecular clouds in N11 was done using the CO(2--1) line emission observations, which has an angular resolution twice that of the CO(1--0) line emission. \nWe use the CPROPS algorithm \\citep{rosolowsky06} to identify the individual molecular clouds. This program has been used in several other studies of molecular clouds, including the LMC \\citep[e.g. ][]{fukui08,hughes10,wong11}. CPROPS is an IDL procedure which identifies clouds from a data-cube using the spectral and spatial structure of the line emission. The physical parameters of the identified molecular clouds are measured as moments of the emission along the spectral and spatial axis. The CPROPS algorithm works as follows. It defines a mask from a $\\sigma_{\\rm rms}$ value of the data. Within this mask, pixels with emission higher than a threshold value of 5$\\sigma_{\\rm rms}$, over at least two consecutive velocity channels, are identified as cores of the mask. \nThe mask includes pixels with brightness larger than 2$\\sigma_{\\rm rms}$ emission which meet the two following conditions. They are spatially and spectrally connected to the core pixels and they are not spectrally isolated (i.e. at least two consecutive channels are above 2$\\sigma_{\\rm rms}$). Every core is decomposed into candidates molecular clouds. The parameters used for this decomposition were: minimum area of 4 beams, line-widths larger than 1.5~km~s$^{-1}$, and a minimum ratio between the flux of the cloud at the peak and at its edge of 3$\\sigma_{\\rm rms}$. The algorithm defines elliptical molecular clouds, which are extrapolated to the 0~K isosurface to estimate what we would measure with no noise. For a detailed explanation of CPROPS, we direct the reader to \\citet{rosolowsky06}. \n\n\nTo increase the SNR in our data, we smoothed the data in the velocity axis to a spectral resolution of 0.5~km~s$^{-1}$. CPROPS identified 22 individual molecular clouds. We discarded three clouds with low signal-to-noise ratio ($<$4) and which areas are close to our 4 beams threshold. Our CO(2--1) data is not fully-sampled. It was observed at about twice the Nyquist frequency (\\S~\\ref{sec:sest}). \n However, the undersampling of the observations do not impact the conclusions of this paper. We quantify the impact of the undersampling on the parameters of the clouds identified by CPROPS with a simple test. We run CPROPS for our fully sampled CO(1--0) (angular spacing of 24\\arcsec\\ and beam size of 45\\arcsec) data cube with a minimum cloud area of one CO(1--0) beam. The results of this run can be directly compared with the CPROPS run on the CO(2--1), for which we have used a minimum cloud area of four CO(2--1) beams. \n In Figure~\\ref{fig:under}, the cloud radii and velocity dispersions for both runs on the CO(1--0) and CO(2--1) data-sets are compared. The error bars are 1-sigma errors on the parameters computed by CPROPS. Although the two cloud parameters do not match exactly one-to-one, the two sets of points are statistically close to each other. There is no systematic difference in radii, and only a slight difference by a factor of 1.2  between the mean velocity dispersion for the CO(1--0) and CO(2--1) lines. We conclude that the undersampling of the CO(2--1) observations has a minor impact on the paper conclusions which are all of statistical nature.  The difference in line widths implies a factor 1.5 systematic on our estimates of the virial masses. We do not correct our 2--1 line widths for this factor but consider it as a systematic uncertainty in our scientific analysis.\n\n\n\\begin{figure}[ht]  \n\\centering\n\\includegraphics[width=9cm]{Undersampling_CO21.ps}\n\\caption{Comparison between the radius and velocity dispersion, for molecular clouds obtained from CPROPS runs the CO(1--0) (red\nsquares) and CO(2--1) (black triangles) data set. The error bars correspond to the 1-sigma errors computed by CPROPS.}\\label{fig:under}\n\\end{figure}\n\nFigure~\\ref{fig:co21} presents the positions of the molecular clouds in the N11 region. Channel maps in Figure~\\ref{fig:chan} show the boundaries of the clouds. Table~\\ref{tab:nub} lists the final set of individual molecular clouds identified with CPROPS and their properties. The radii correspond to the geometrical mean of the ellipse semi-axis. CPROPS gives velocity dispersions which we convert to line-widths at half maximum with the relation $\\Delta v = \\sigma_{\\rm v}\\,2\\,\\sqrt{2{\\rm\\, ln(2)}}$. Uncertainties on the radii, line-widths and CO luminosities come from the fitting procedure in CPROPS \\citep{rosolowsky06}. Error bars on the CO luminosity also include the 20\\% uncertainty on the calibration. Clouds \\#1 and \\#5 present different velocity components and might be separated into 2 clouds each. The mean local standard of rest velocity $V_\\mathrm{LSR}$ is 278~km~s$^{-1}$. \n\nThe total intensity of the identified molecular clouds corresponds to 61\\% and 50\\% of the total emission (see \\S~\\ref{sec:sest}) for the CO(1$-$0) and CO(2--1) emission lines, respectively.\n\n\n\\input{tabla_nube.tex}\n\n\\subsection{Comparison with previous data sets}\n\nThe N11 star-forming region was previously observed in CO(1--0) as part of the ESO-SEST Key Programme \\citep{israel03} with an angular resolution of 45\\arcsec. They identified 14 molecular clouds in the same area that we cover. We compare our list with their results. All of the clouds previously identified are in our list of clouds. Our CPROPS run could not resolve two of their clouds (\\#16 and \\#17, Table~\\ref{tab:i03}), due to the undersampling of the observations. These clouds are small ($\\sim$30\\arcsec, i.e. 7~pc, of radius) and close to each other (40\\arcsec, i.e. 10~pc, less than twice the sampling of our observations). Thanks to the higher J$=$2--1 CO resolution, we find six additional clouds. Table~\\ref{tab:i03} lists the correspondence between both studies, comparing radii, line-widths and virial masses. Even though there are differences in the data-sets and methodologies, global parameters of the molecular clouds are similar.  Statistically, our line-widths are larger than those of \\citet{israel03}, which yields larger virial masses. We checked with the data that this difference mainly comes from the way the line-widths are measured. Those reported by \\citet{israel03} are measured on the spectrum at the emission peak of the molecular clouds, while our values are computed on the spectrum integrated over the entire molecular clouds. The higher sensitivity and resolution of our data could also contribute to this difference. We may have included weaker velocity components located in the same line-of-sight of the clouds, which were not detected before.\n\n\nWe also compare our results with those of the MAGMA survey in the whole LMC (CO(1--0) observations, FWHM$=$45\\arcsec) \\citep{wong11}. Like us, they identified molecular clouds using CPROPS. The parameters used in their decomposition to identify molecular clouds are a minimum ratio between the flux of the cloud at the peak and at its edge of 2$\\sigma$.\n In the same area that we observed, they identified 18 clouds. 15 clouds in their and our list coincide. Two of the remaining three clouds are part of a single cloud in our list, while the third one is small ($<$5~pc) and was not detected by our CPROPS run.  Statistically, our CO(2--1) molecular clouds present line-width slightly broader, by a median factor of 1.3.  If we take into account the 1.2 factor which comes from the undersampling of the CO(2--1) observations -- offset value between the mean velocity dispersions of the CO(1--0) and CO(2--1) clouds presented in Section~\\ref{ss:molclouds} --, this factor increases slightly to 1.5. \n A comparison of our molecular clouds with those identified across the entire LMC \\citep{wong11} is displayed in Figures~\\ref{fig:corr1} and \\ref{fig:corr2}, which show that the N11 star-forming region is a singular region in the LMC.\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{p{0.36cm}ccccccc}\n\\hline \\hline\n\\noalign{\\smallskip}\n\\multicolumn{4}{c}{This work} & \\multicolumn{4}{c}{\\citet{israel03}} \\\\\n\\hline\n\\noalign{\\smallskip}\nRef.  & $R$ & $\\Delta v$ & $M_{\\rm vir}$ & Ref. & $R$ &$\\Delta v$ & $M_\\mathrm{vir}$ \\\\\n  \\#    &    pc   & {\\small km~s$^{-1}$}  &    {\\small 10$^3$$M_{\\odot}$}   & \\#  &  pc & {\\small km~s$^{-1}$}  &{\\small 10$^3$$M_{\\odot}$} \\\\\n\\hline\n\\noalign{\\smallskip}\n\t1\t& 13.2\t&\t7.1\t&\t125.2\t&\t16\t\t& 7.7\t\t&\t5.8\t&\t49\t\t\\\\\n\t\t& \t\t&\t\t&\t\t\t&\t17\t\t& 8.6\t\t&\t1.7\t&\t4.5\t\t\\\\\n\t2\t& $<$5.6\t&\t3.9\t&\t$<$17 \t&\t$-$      \t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t3\t& 9.4\t\t&\t4.2\t&\t31.0  \t&\t$-$\t\t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t4\t& 8.9\t\t&\t4.1\t&\t27.6  \t&\t15\t\t& 10.7\t&\t3.8\t&\t29\t\t\\\\\n\t5\t& 12.8\t&\t5.5\t&\t71.6  \t&\t11 \t\t& 7.7\t\t&\t4.5\t&\t30\t\t\\\\\n\t6\t& $<$5.6\t&\t4.7\t&\t$<$25\t&\t$-$ \t\t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t7\t& 3.2\t\t&\t3.6\t&\t7.7\t\t&\t13 \t\t& $<$5\t&\t3.1\t&\t$<$9\t\t\\\\\n\t8\t& 4.8\t\t&\t6.8\t&\t41.7  \t&\t10 \t\t& 7.4\t\t&\t6.1\t&\t49\t\t\\\\\n\t9\t& 9.2\t\t&\t5.3\t&\t49.1  \t&\t$-$ \t\t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t10\t& 13.1\t&\t4.5\t&\t50.2 \t\t&\t14 \t\t& $<$5\t&\t4.0\t&\t$<$15\t\\\\\n\t11\t& 5.0\t\t&\t2.8\t&\t7.1 \t\t&\t$-$\t\t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t12\t& 6.2\t\t&\t3.5\t&\t14.1  \t&\t 2\t\t& 7.3\t\t&\t2.7\t&\t10\t\t\\\\\n\t13\t& 16.0\t&\t5.9\t&\t104.2  \t&\t1\t\t& 10.6\t&\t2.7\t&\t15\t\t\\\\\n\t14\t& 10.4\t&\t5.9\t&\t67.7 \t\t&\t4\t\t& 11.2\t&\t5.7\t&\t69\t\t\\\\\t\t\n\t15 \t& 7.5\t\t&\t4.2\t&\t25.2  \t&\t3\t\t& 11.3\t&\t2.9\t&\t18\t\t\\\\\n\t16\t& 6.8\t\t&\t4.5\t&\t26.2 \t\t&\t$-$\t\t& $-$\t&\t$-$\t&\t$-$\t\t\\\\\n\t17\t& 6.1\t\t&\t4.5\t&\t22.8  \t&\t 9\t\t& 7.0\t\t&\t2.6\t&\t9\t\t\\\\\n\t18\t& 14.4\t&\t2.8\t&\t20.7  \t&\t6\t\t& 19.9\t&\t2.5\t&\t24\t\t\\\\\n\t19\t& 7.7\t\t&\t2.4\t&\t8.4\t\t&\t 7\t\t& $<$5\t&\t1.9\t&\t$<$4\t\t\\\\\n\\hline\n\\end{tabular}\n\\caption{Comparison between molecular clouds detected in CO(2$-$1) and previous clouds detected in CO(1$-$0) by \\citet{israel03}. To compare the masses, we scale the $M_{\\rm vir}$ values from \\citet{israel03} to use the formula for spherical clouds with a density profile $\\propto r^{-1}$. }\\label{tab:i03}\n\\end{center}\n\\end{table}\n\n\\subsection{Line ratio $R_{2-1\/1-0}=I_\\mathrm{CO(2-1)}\/I_\\mathrm{CO(1-0)}$}\n\n\nAs described in \\S 2, we mapped the central part of N11 simultaneously in CO(1--0) and CO(2--1) line emission. From these lines, we compute the ratio of line intensities in K~km~s$^{-1}$, $R_{2-1\/1-0}$. To compare these lines, we need to have the data at the same spatial resolution. Thus, we convolve the CO(2--1) data set (FWHM$=$23\\arcsec) with a gaussian kernel to the 45\\arcsec\\ resolution of the CO(1--0) data. Maps of integrated emission for CO(1--0) and CO(2--1) are obtained similarly as described in \\S~\\ref{sec:sest}, by integrating the emission within a spectral window in the velocity range 265-289~km~s$^{-1}$. From these two maps, we derive a $R_{2-1\/1-0}$ ratio map for the entire region. We estimate the average $R_{2-1\/1-0}$ ratio for each molecular cloud by measuring the emission within an aperture defined by the size of the molecular clouds as identified by CPROPS. Uncertainties on this ratio include the 20\\% photometric uncertainties. Table~\\ref{tab:nub} lists these values. \nFigure~\\ref{fig:rat} shows the CO(1--0) and CO(2--1)  intensity relation for each pixel in our observations of the N11 region and for each molecular cloud. The median value for the $R_{2-1\/1-0}$ ratio is 1.0. This is close to the median value of 1.2 reported by \\citet{israel03b} for another sample of molecular clouds in the LMC. In Figure~\\ref{fig:rat} we observe some dispersion of the pixels around the line. There are a few outliers.\nFour molecular clouds, 2, 6, 8 and 10, show large $R_{2-1\/1-0}$ values. Clouds 6 and 8 are associated with the embedded IR stellar cluster in N11 identified by \\citet{barba03}. \n\n\n\\begin{figure}[ht]  \\centering\n   \\includegraphics[width=9cm]{Razones_lineas_scatter_pixpix.ps}\n      \\caption{$I_{\\rm CO(1-0)}$ vs. $I_{\\rm CO(2-1)}$ line emission for each pixel in N11 (black diamonds) and for each molecular cloud in Table~\\ref{tab:nub} (red circles). The cloud fluxes were divided by 7. The line corresponds to the median value of the ratio $R_{2-1\/1-0}$$=$1.0.}\\label{fig:rat}\n      \\end{figure}\n\n\\subsection{Correlations between physical parameters}\n     \n\\begin{figure}[ht]\n\\centering\n      \\includegraphics[width=20cm]{graficos_fitexy_05kms_a.ps}  \n      \\caption{Correlation graphics. a) log($L_{\\rm CO}$) vs. $\\sigma_{\\rm v}$, b) log($L_{\\rm CO}$) vs. $R$ and c) $\\sigma_{\\rm v}$ vs. $R$.  Straight red lines: best fits to our data, using the Galactic exponents. The shadow area represent the uncertainties for individual data points. Black dot-dashed lines: Galactic relations measured by \\citet{solomon87}. Green dashed lines: correlations for GMCs in the LMC from the MAGMA survey by \\citet{hughes10}. Blue dot-dashed lines: correlation for dwarf galaxies from \\citet{bolatto08}. Green crosses represent the molecular clouds identified in the MAGMA survey of the LMC by \\citet{wong11}.}\\label{fig:corr1}\n\\end{figure}\n\nIn this section we compare the physical properties of the N11 clouds with empirical relations derived from wider surveys in the Galaxy and the LMC.\nWe determine the correlations between the physical properties of the molecular clouds $L_{\\rm CO(1-0)}$, $R$, $M_\\mathrm{vir}$  and $\\sigma_{\\rm v}$ presented in Table~\\ref{tab:nub}. $L_{\\rm CO(1-0)}$ values are estimated as the $L_{\\rm CO(2-1)}$ measures scaled by the $R_{2-1\/1-0}$ estimates. Since the range of the molecular clouds luminosities in N11 is limited, we cannot estimate reliable values for the exponents of the power law correlations between the different parameters. Thus, we fit the N11 relations with the exponents found by \\citet{solomon87} for GMCs in the Galactic molecular ring. The multiplicative factor of the following equations result from an error-weighted fit of all data points and the error bars are the dispersion of the individual points,\n\\begin{eqnarray}  \\label{eq:rel1}\nL_{\\rm CO}&=&6.9_{-4.0}^{+9.0}~R^{2.5}, \\\\  \\label{eq:rel2}\nL_{\\rm CO}&=&42_{-28}^{+81}~\\sigma_{\\rm v}^{5},\\\\ \\label{eq:rel3}\nM_{\\rm vir}&=&91_{-42}^{+79} ~L_{\\rm CO}^{0.81}, \\\\ \n\\sigma_{\\rm v}&=&0.72_{-0.15}^{+0.19}~R^{0.5}. \\label{eq:rel4}\n\\end{eqnarray}\n\nFigure~\\ref{fig:corr1} shows the relations $L_{\\rm CO}$ vs. $\\sigma_{v}$, $L_{\\rm CO}$ vs. $R$ and $\\sigma_{v}$ vs. $R$. Figure~\\ref{fig:corr2} presents the relation between the virial mass and the CO luminosity. In these figures, we compare the N11 molecular clouds with LMC clouds identified by \\citet{wong11} as part of the MAGMA survey (green crosses). The best fits for our clouds, using the Galactic exponents, correspond to the solid lines and the shadow areas to the 1$\\sigma$-error in the multiplicative factor. We also include, in segmented lines, the relations obtained by \\citet{solomon87} for the Galactic molecular clouds, \\citet{bolatto08} for GMCs from dwarf galaxies and \\citet{hughes10} for GMCs in the LMC. \n\n\\section{Millimeter dust emission at 1.2~mm}\\label{sec:mm}\n\n In this section, we estimate the emission of the dust in N11 at 1.2~mm combining the SIMBA (FWHM$=$24\\arcsec), SEST (FWHM$=$23\\arcsec) and ATCA 8.6~GHz (FWHM$=$22\\arcsec) observations. To do that, we convolved the SEST and ATCA images to the spatial resolution of the SIMBA observations.\n\nIn a molecular cloud, the continuum emission at 1.2~mm, $S^{\\rm total}_{\\rm1.2mm}$, is the sum of free-free and dust radiation. Our observations obtained with the SIMBA bolometer (see \\S~\\ref{sec:12mm}) provide us with the continuum emission at 1.2~mm. The CO(2--1) emission line falls in the SIMBA bandwidth. Thus, to measure the dust emission at 1.2~mm for the N11 molecular clouds,  the free-free and CO line contribution have to be subtracted.  \n\nWe use centimeter-wave observations (at 8.6~GHz) to estimate the contribution from free-free to the 1.2~mm emission. Continuum emission at centimeter wavelengths is composed by two contributions, thermal emission from \\ion{H}{ii} gas (free-free emission), and synchrotron emission. Since the spectral index of the thermal emission is flatter than that of the synchrotron emission and the synchrotron emission is not correlated with GMCs, we assume that the emission at 8.6~GHz is mainly tracing the free free emission. We scale the radio continuum emission at 8.6~GHz to 250~GHz (1.2~mm). For an optically thin medium, as the emission at sub-mm\/mm wavelengths, the free-free emission depends on the frequency as $S_{\\nu}\\propto \\nu^{-0.1}$. From this relation, we estimate the free-free emission at 1.2~mm as:\n\\begin{equation}\\label{eq:ff}\nS^\\mathrm{ff}_\\mathrm{1.2mm} = \\left( \\frac{250~\\mathrm{GHz}}{8.64~\\mathrm{GHz}}\\right) ^{-0.1}~S_\\mathrm{8.64\\,GHz}.\n\\end{equation}\nUncertainties on $S^\\mathrm{ff}_{\\rm 1.2mm}$ are computed as the quadratic sum of the errors, including the flux calibration uncertainty (5\\% of the flux in the convolved and scaled image), and the uncertainty on the free-free contribution. The latter was estimated comparing, for each cloud, the emission at 4.8 and 8.6~GHz smoothed to a common resolution of 35\\arcsec. The error is the absolute value of the difference between the 1.2~mm free-free flux estimated from the 8.6~GHz observation and that computed from the geometric mean value of the 4.8 and 8.6~GHz fluxes.\n\nWe estimate the contribution of the CO(2--1) emission line to the SIMBA bolometer as follows. This contribution can be written as $S_\\mathrm{CO(2-1)}$$=$$2\\mathrm{k}\\nu^{3}\\mathrm{c}^{-3} \\Omega I_\\mathrm{CO}\/\\Delta v$ where $\\nu$$=$230~GHz is the frequency of the CO(2--1) line, $I_{\\rm CO}$ the emission line intensity, $\\Omega$ the solid angle of the Gaussian beam of SIMBA in steradian, and $\\Delta v$$=$90~GHz the frequency width of the bolometer. We obtain an expression that we use to compute the contribution of the CO(2--1) line and its error bar:\n\n\\begin{equation}\\label{eq:co}\n\\left[ \\frac{S^\\mathrm{CO}_\\mathrm{1.2mm}}{\\mathrm{mJy~beam^{-1}}}\\right] \\simeq 0.20~\\left[ \\frac{I_\\mathrm{CO}}{\\mathrm{K~km~s^{-1}}}\\right].\n\\end{equation}\n\nTo obtain the dust emission, we subtract the free-free and CO contributions from the total continuum emission at 1.2~mm. The error on the dust emission is the quadratic sum of three errors: the error on the 1.2~mm flux including the 20\\% uncertainty on the photometric calibration, the error on the free-free subtraction and the error on the CO line contribution. Figure~\\ref{fig:12mmcont} shows the different contributions to the emission at 1.2~mm in the N11 region. From the left to the right images: the total emission at 1.2~mm, the free-free emission at 1.2~mm estimated from the ATCA emission at 8.6~GHz, the CO(2--1) line contribution to the total 1.2~mm emission observed with SIMBA, and the resulting dust emission. \nTable~\\ref{tab:mm} lists the fluxes contributions to the total 1.2~mm emission for each molecular cloud, measured in the images (convolved to the same spatial resolution) by aperture photometry using the molecular cloud sizes computed by CPROPS and subtracting the nearby background within an aperture of similar size.\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=12.5cm]{graficos_fitexy_05kms_b.ps}\n\t\\caption{Correlation between $M_{\\rm vir}$ and $L_{\\rm CO}$ for the molecular clouds of the N11 region. Lines are the same as in Fig.~\\ref{fig:corr1}.}\\label{fig:corr2}\n\\end{figure*}\n\n   \n  \\begin{figure*}\n   \\centering\n   \\includegraphics[width=6.2cm,angle=90,viewport=290 0 520 680,clip]{mm_co_ff_dust_emission_n11.ps}\n      \\caption{Different contributions to the emission at 1.2~mm. a) Total emission observed with SIMBA. b) Free-free emission at 1.2~mm estimated from the ATCA image at 8.6~GHz, as showed in Eq.~\\ref{eq:ff}. c) CO(2--1) contribution to the 1.2~mm emission as observed through the SIMBA bolometer array (see Eq.~\\ref{eq:co}). The red contour corresponds to the CO(2--1) emission at 0.5~mJy~beam$^{-1}$. d) N11 dust emission at 1.2~mm obtained after subtraction of the free-free and CO contributions to the total 1.2~mm emission.\nAll the images are in units of mJy beam$^{-1}$. Intensities of the images b) and c) were amplified by a factor 6. In all the images, the black contour correspond to 3$\\sigma$ (12~mJy~beam$^{-1}$) of the total 1.2~mm emission. }\n         \\label{fig:12mmcont}\n   \\end{figure*} \n   \n\n\n\\section{Molecular masses}\\label{sec:mass}\n\nIn this section, we discuss and compare three methods to estimate the masses of the molecular clouds based on the CO luminosity, the virial theorem and the millimeter dust emission.\n\n\\subsection{Masses from CO luminosity, $M_{\\rm CO}$}\\label{sec:mco}\nIn the Galaxy, the mass of a molecular cloud is traditionally estimated from its CO emission. \nThe ratio between the CO(1$-$0) integrated intensity $W_{\\rm CO}$ and the H$_{2}$\\ column density $N(\\mathrm{H}_{2})$, is the $X_{\\rm CO}$-factor ($N({\\rm H}_{2})=X_{\\rm CO}~W_{\\rm CO}$). This conversion factor is uncertain because it depends on the cloud properties. In the Milky Way, it has been calibrated with $\\gamma$$-$ray emission \\citep[i.e.][]{bloemen86}, the assumption of virial equilibrium \\citep[i.e.][]{solomon87}, and dust extinction \\citep[i.e.][]{lombardi06}. These empirical studies yield values in the range 2$-$4$\\tten{20}$ cm$^{-2}$~(K~km~s$^{-1}$)$^{-1}$.   Most recent values from $\\gamma-$ray in the Gould Belt are lower \\citep{abdo10}, but in average in the Milky Way the studies agree with the previous range of values \\citep{bolatto13}.\n\nSeveral studies \\citep[e.g., ][]{cohen88,israel97,fukui08,hughes10} have shown that the Galactic conversion factor does not apply to molecular clouds in the LMC, where the estimated values are between 2 and 4 times higher than the Galactic one. \nIn this work, we use the $X_{\\rm CO}$-factor computed from the molecular clouds observed and identified by the MAGMA survey in the LMC \\citep{hughes10} which corresponds to $4.7\\times 10^{20} {\\rm cm^{-2}~(K~km~s^{-1})^{-1}}$. To estimate the masses based on the CO luminosity ($M_{\\rm CO}$) by using the $X_{\\rm CO(1-0)}$ factor estimated from the MAGMA survey, we scale the $L_{\\rm CO(2-1)}$ luminosities by the $R_{2-1\/1-0}$ values, both listed in Table~\\ref{tab:nub}. The inferred CO luminosity masses are also listed in Table~\\ref{tab:nub} and they range from 2 to 80~$\\times 10^3~M_{\\odot}$.\n\n\\subsection{Masses from the virial theorem, $M_\\mathrm{vir}$}\\label{sec:mvir}\n\nThe virial masses are computed from the virial equilibrium between gravitational and turbulent kinetic energies. This implies that the molecular clouds are gravitationally bound, ignoring external pressure and magnetic energy. For a spherical cloud with  a profile density of $\\rho \\propto r^{-1}$, the virial mass is $M_\\mathrm{vir} =9\/2~R \\sigma_{\\rm v}^{2}\/G$, where $R$ is the radius, $\\sigma_{v}$ the velocity dispersion of the gas in 1 dimension, and $G$ the gravitational constant \\citep{maclaren88}:\n\n\\begin{equation}\n\\frac{M_\\mathrm{vir}}{[M_\\mathrm{\\sun}]} = 190 \\left( \\frac{R}{[\\mathrm{pc}]}\\right ) \\left(\\frac{\\Delta v}{[\\mathrm{km\\,s^{-1}}]}\\right)^{2}.\n\\end{equation}\n\nTo compute $M_{\\rm vir}$ we use the line-width and the size estimated from the CO(2--1) emission, both listed in Table~\\ref{tab:nub}. The resulting $M_{\\rm vir}$ values are listed in the same table. These masses vary between 7 and 130 $\\tten{3}~M_{\\sun}$. The 1$\\sigma$ uncertainty corresponds to the quadratic sum of the errors on the line-width and the radius. In Table~\\ref{tab:nub}, we also list the ratio between $M_{\\rm vir}$ and $M_{\\rm CO}$. The median value for this ratio is 1.9 with a median absolute deviation of 0.9. This ratio increases to 2.8 if we include the offset in the velocity dispersion due to the undersampling (\\S~\\ref{ss:molclouds}). The median value of the ratio between the virial mass and $L_{\\rm CO(1-0)}$ is 21 $M_{\\odot}~{\\rm (K~km~s^{-1}~pc^2)^{-1}}$ (31 $M_{\\odot}~{\\rm (K~km~s^{-1}~pc^2)^{-1}}$ including the effect of the undersampling on the velocity dispersion). The ratios for all clouds, but clouds \\#14, 18 and 19, are within 3$\\sigma$ of this median ratio.\n\n\\subsection{Masses from the millimeter dust emission, $M_\\mathrm{mm}$}\\label{sec:mmm}\nThe masses of the molecular clouds can also be estimated from the millimeter emission of the dust. In this method, we assume that the dust properties in N11 are the same as those in the Galaxy. There is still an open debate about this hypothesis, since the far-IR dust properties in the Galaxy and in the LMC may differ \\citep{galliano11,planck11a17}.\n\nWe approximate the spectral dependence of the dust emission with a grey body. Assuming that the emission is optically thin, the dust optical depth $\\tau_{d}(\\nu)$ can be written in terms of the dust column density. Then, the gas mass estimated from the dust millimeter emission is:\n\\begin{equation}\\label{eq:mmm}\nM_\\mathrm{mm} = \\frac{S_{\\nu}\\,D^{2}}{B_{\\nu}(T_\\mathrm{d})\\kappa_\\mathrm{d}(\\nu)\\,x_{\\rm d}},\n\\end{equation}\nwith $S_{\\nu}$ the millimeter cold dust emission, $D$ the distance to the LMC, $B_{\\nu}$ the black body emission at a dust temperature $T_\\mathrm{d}$, $\\kappa_\\mathrm{d}$ the absorption coefficient per unit dust mass and $x_\\mathrm{d}$ the dust-to-gas mass ratio. We can estimate $\\kappa_\\mathrm{d}$ from the emissivity per hydrogen $\\epsilon_{\\rm H}(\\nu)$ as $\\kappa_{\\rm d}(\\nu)$$=$$\\epsilon_{\\rm H}(\\nu) (x_\\mathrm{d}\\mu m_\\mathrm{H})^{-1}$, where $\\mu$ is the mass per hydrogen atom of 1.36. Our masses derived from dust emission, like the masses derived from CO luminosity (in section~\\ref{sec:mco}) and the virial masses (in section~\\ref{sec:mvir}), include the contribution from helium $\\mu=1.36$.  We estimate $\\epsilon_{\\rm H}({\\rm 1.2mm})$ using the recent results obtained with the Planck satellite. By combining Planck and IRAS observations in the Galaxy, \\citet{planck11a24} fitted the spectral energy distribution of the local ISM with a grey body function, finding $T_{\\rm dust}$$=$17.9 K and $\\beta$$=$1.8. They derived a dust emissivity at 250~$\\mu$m which agrees with that obtained from COBE observations \\citep{boulanger96}. From these Planck results, we compute the dust emissivity per hydrogen atom at 1.2~mm as $\\epsilon_{\\rm H}$(1.2mm)$=$6.1$\\tten{-27}$~cm$^{2}$. Furthermore, \\citet{planck11a25} showed that the emissivity $\\tau_{250 \\mu m}\/N_{\\rm H}$ increases by a factor of two from the atomic to the molecular gas in the Taurus molecular cloud. Thus, for the dust-to-gas mass ratio in the solar neighborhood of $x_{\\rm d}=0.007$  \\citep{draine07}, the absorption coefficient for dust in molecular clouds is $\\kappa_{\\rm d}$(1.2mm)$=$0.77~cm$^{2}$~g$^{-1}$. This value is similar to that used by \\citet{bot07} to compute the millimeter masses of the molecular clouds in the SMC. \n \nTo compute the millimeter masses, we use a single value of 20~K  for the dust temperatures. This mean value is derived from the temperature map of \\citet[][their Figure~7]{planck11a17} at the position of the N11 region. The same figure shows that, around N11, the dust temperature ranges between 15 and 25~K. We will use this range as the maximum uncertainty on the temperatures.\n\nWe assume that the dust-to-gas mass ratio scales with metallicity \\citep{issa90,james02}. Thus, $x_\\mathrm{d}$(LMC)$=$$x_{\\rm d}(\\odot)~Z_{\\rm LMC}$$=$3.5$\\tten{-3}$ for $Z_{\\rm LMC}$$=$$0.5~Z_{\\odot}$. We obtain an expression for the gas mass from the millimeter dust emission at 1.2~mm and for a dust temperature $T_{\\rm d}$$=$20~K:\n\\begin{equation}\n\\left[\\frac{M_\\mathrm{mm}}{M_{\\odot}} \\right] = 158 \\left[\\frac{S_{\\rm 1.2mm}^{\\rm dust}}{\\mathrm{mJy}} \\right],\n\\end{equation}\nwhere $S_{\\rm 1.2mm}^{\\rm dust}$ is the cold dust millimeter continuum emission in mJy at 1.2~mm. The multiplicative factor varies from 236 for $T_{\\rm d}$$=$15~K to 119 for $T_{\\rm d}$$=$25~K.  This variation in temperature gives a maximum uncertainty on the millimeter masses of a factor of 1.5.\n\nThe resulting millimeter masses are listed in Table~\\ref{tab:mm}. All clouds but one, \\#12, are within 2$\\sigma$ of the median value $M_{\\rm vir}\/M_{\\rm mm}=1.9\\pm0.4$, where the error is the median absolute deviation. Cloud \\#12 is an outlier; it is compact, barely resolved by the observations and corresponds to a peak in the millimeter emission.  We discard five molecular clouds because their estimated dust emission, after subtraction of the free-free emission, has a SNR lower than 2$\\sigma$. These clouds are in the north-east of N11, where the free-free emission is high (see Figure~\\ref{fig:12mmcont}). Their $M_{\\rm vir}\/M_{\\rm mm}$ ratio does not agree with the median value.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{cccccccc}\n\\hline \\hline\n\\noalign{\\smallskip}\n\\multirow{2}{*}{Cloud} &  $S_{\\rm 1.2mm}$ & $S_{\\rm ff}$ & $S_{\\rm CO}$ & $S_{\\rm dust}$ & $M_{\\rm mm}$ & $M_{\\rm vir}$ &  \\multirow{2}{*}{$M_{\\rm vir}\/M_{\\rm mm}$} \\\\\n\\cline{2-7}\n\\noalign{\\smallskip}\n &  \\multicolumn{4}{c}{mJy} &  \\multicolumn{2}{c}{10$^3~M_{\\odot}$}  & \\\\\n\\hline\n\\noalign{\\smallskip}\n 1\t&  74$\\pm$21\t & 13$\\pm$6\t & 23$\\pm$5\t &   27$\\pm$23\t &     $-$\t\t& 125$\\pm$11\t&  $-$ \\\\\n 3\t&  46$\\pm$13\t & 8$\\pm$4\t & 12$\\pm$6\t &   23$\\pm$15\t &    $-$\t\t&  31$\\pm$18\t&  $-$\\\\\n 4\t& 166$\\pm$47 \t & 21$\\pm$10\t & 19$\\pm$6\t &  123$\\pm$49 &   20$\\pm$8\t&  28$\\pm$11\t&   1.4$\\pm$0.8 \\\\\n 5\t& 106$\\pm$30\t & 36$\\pm$17\t & 14$\\pm$3\t &   53$\\pm$35\t &    $-$\t\t&  72$\\pm$15\t&  $-$\\\\\n 7\t&  24$\\pm$7\t & 0.2$\\pm$0.1\t &  4$\\pm$1\t &   19$\\pm$7\t &     3$\\pm$1\t&   8$\\pm$3\t&   2.5$\\pm$1.3 \\\\\n 8\t& 212$\\pm$60\t & 36$\\pm$17\t & 15$\\pm$3\t &  159$\\pm$62 &  25$\\pm$10\t&  42$\\pm$14\t&   1.7$\\pm$0.9 \\\\\n 9\t& 144$\\pm$41\t & 64$\\pm$30\t &  7$\\pm$2\t &   68$\\pm$51\t &     $-$\t\t&  49$\\pm$20\t&  $-$\\\\\n10\t& 401$\\pm$114 &125$\\pm$59\t & 12$\\pm$3\t & 261$\\pm$123&  41$\\pm$20 \t&  50$\\pm$8\t&   1.2$\\pm$0.6 \\\\\n11\t&  11$\\pm$4\t & 2.3$\\pm$ 1.1 &  3$\\pm$1\t &    5$\\pm$4\t &    $-$\t\t&   7$\\pm$3\t&  $-$ \\\\\n12\t& 142$\\pm$40\t &    6$\\pm$3\t &  6$\\pm$2\t &  129$\\pm$41 &   20$\\pm$6\t&  14$\\pm$5\t&   0.7$\\pm$0.3 \\\\\n13\t& 324$\\pm$92\t &   10$\\pm$5\t & 22$\\pm$5\t &  296$\\pm$93 &   47$\\pm$15\t& 104$\\pm$30\t&   2.2$\\pm$0.9 \\\\\n14\t& 284$\\pm$81\t &   7$\\pm$3\t & 40$\\pm$8\t &  231$\\pm$81 &   37$\\pm$13\t&  68$\\pm$17\t&   1.9$\\pm$0.7 \\\\\n15\t&  93$\\pm$27\t & 2.5$\\pm$1.2\t &  5$\\pm$2\t &   83$\\pm$27\t &    13$\\pm$4\t&  25$\\pm$14\t&   1.9$\\pm$1.3 \\\\\n16\t&  54$\\pm$16\t & 1.3$\\pm$0.6\t &  5$\\pm$1\t &   45$\\pm$16\t &    7$\\pm$3\t&  26$\\pm$5\t&   3.6$\\pm$1.6 \\\\\n17\t&  24$\\pm$8\t & 5.3$\\pm$2.5\t &  5$\\pm$1\t &   24$\\pm$8\t &     4$\\pm$1\t&  23$\\pm$7\t&   6.0$\\pm$2.9 \\\\\n18\t&  84$\\pm$26\t &           $-$ \t & 22$\\pm$5\t &   73$\\pm$27\t &     12$\\pm$4\t&  21$\\pm$8\t&   1.8$\\pm$1.1 \\\\\n19\t&  38$\\pm$12\t & 0.2$\\pm$0.1\t &  7$\\pm$2\t &   37$\\pm$12\t &     6$\\pm$2\t&   8$\\pm$4\t&   1.4$\\pm$1.0 \\\\\n\\hline\n\\end{tabular}\n\\caption{Flux contribution to the total 1.2~mm emission for N11 molecular clouds. In the second row we list dust temperatures, obtained by a SED fitting (see section~\\ref{sec:mmm}). We separate this emission into 3 different contributions: free-free, CO(2--1) and cold dust emission. Millimeter masses are computed from the cold dust emission (see section~\\ref{sec:mmm}). The error bar on dust masses ($M_{\\rm mm}$) is derived from the error on $S_{\\rm dust}$. It does not include any uncertainty on the dust temperatures. The last column shows the comparison between virial and millimeter masses.}\\label{tab:mm}\n\\end{center}\n\\end{table*}\n\n\n\\subsection{Comparing the three mass estimates}\n\nThe observations provide three independent estimates of the masses of the molecular clouds in the N11 region. Several previous studies have compared the masses derived from CO luminosity with the virial masses for LMC molecular clouds. For the first time, in this paper, we add a third mass estimate based on the millimeter dust emission. Each mass estimate has a large systematic uncertainty, which we can constrain by quantifying what is required for the different methods to yield identical gas masses.\n\nThe main uncertainty on the CO masses comes from the $X_{\\rm CO}$-factor. The $X_{\\rm CO}$-factor is the ratio between the true molecular cloud mass and the CO luminosity,\n\\begin{equation}\\label{eq:xco}\nX_{\\rm CO} = \\frac{M_{\\rm true}}{L_{\\rm CO (1-0)}}.\n\\end{equation}\n\nAs we pointed out in the introduction, in low metallicity environments the CO molecule is dissociated by the UV radiation from the nearby massive stars. Thus, the LMC $X_{\\rm CO}$-factor must be larger than that measured in the Galaxy. However, in the LMC there is a wide range of values reported in the literature, ranging from a factor 2 to 6 higher than in the Galaxy \\citep[e.g.][]{cohen88,israel97,fukui08,dobashi08,hughes10}. \n\nFor the virial mass, we assume energy balance between the gravitational binding energy and turbulent kinetic energy. In doing this, we ignore external pressures as well as the magnetic energy. These contributions are difficult to quantify. The importance of these energy terms can vary among the N11 clouds. Some of the molecular clouds could be influenced by the radiation pressure and winds from the near young massive stars. Other clouds could have a high external pressure because they are embedded within a bigger complex. Moreover, the energy terms are computed within the simplifying assumption of spherical geometry. To parametrize the difference between the true mass and the virial mass as estimated in section~\\ref{sec:mvir}, we use the $\\alpha_{\\rm vir}$ parameter  that is the ratio between the kinetic $K$ and gravity $W$ energy terms in the virial equations \\citep{mckee92}. For a molecular cloud with a density profile depending on the radius as $\\rho\\propto r^{-1}$, the $\\alpha_{\\rm vir}$ parameter is defined as\n\\begin{equation}\\label{eq:alpha}\n\\alpha_{\\rm vir} \\equiv \\frac{2~K}{|W|} =\\frac{9}{2}\\frac{R \\sigma_{\\rm v}^2}{G M_{\\rm true}}= \\frac{M_{\\rm vir}}{M_{\\rm true}}.\n\\end{equation}\n\nThe mass estimate based on the dust emission involves uncertainties on the dust temperature and the dust emissivity per hydrogen atom $\\epsilon_{\\rm H}(\\nu)$, which depends on the absorption coefficient of the dust $\\kappa_{\\rm d}(\\nu)$ and the dust-to-gas mass ratio $x_{\\rm d}$ (see section~\\ref{sec:mmm}). We have computed the masses using a single dust temperature for each cloud but, on scales of a GMC, the dust has a range of temperatures. In Section~\\ref{sec:mmm} we estimated the uncertainty on the dust mass associated with the choice of a single temperature to be less than 1.5. The dust emissivity per hydrogen atom depends on two assumptions which are difficult to validate. We assume that the dust-to-gas mass ratio scales as the metallicity and that the millimeter absorption coefficient $\\kappa_{\\rm d}$(1.2mm) does not change from the Milky Way to the LMC. This is a debated topic \\citep{planck11a17,galliano11,draine12} and it is difficult to quantify uncertainties. To parametrize the uncertainties we introduce the factor $\\mathcal{K}_{\\rm dust} = \\kappa_{\\rm d}({\\rm 1.2mm}) x_{\\rm d}\/[2.7\\times10^{-3}\\,{\\rm cm}^2{\\rm g^{-1}}]$ where the reference value is the value used to estimate the millimeter masses in section 5.3. This is related to the true molecular mass as follows, \n\\begin{equation}\\label{eq:edust}\nM_{\\rm true} = \\frac{\\mathcal{L}_{\\rm 1.2mm}({\\rm dust})}{B_{\\rm 1.2mm}(T_{\\rm dust})~\\kappa_{\\rm d}({\\rm 1.2mm}) x_{\\rm d}}=\\frac{M_{\\rm mm}}{\\mathcal{K}_{\\rm dust}},\n\\end{equation}\nwhere $\\mathcal{L}_{\\rm 1.2mm}({\\rm dust})$  is the dust luminosity at 1.2~mm per unit frequency, and $M_{\\rm mm}$ is the millimeter mass computed in section 5.3.\n\nThe measured ratios between $L_{\\rm CO}\/M_{\\rm vir}$ and $M_{\\rm mm}\/M_{\\rm vir}$ constrain the three parameters introduced in formulae~\\ref{eq:xco} to \\ref{eq:edust} if the three masses are to be identical. In the following equations we use the median values of these two ratios given in section~\\ref{sec:mvir} and \\ref{sec:mmm}, and the error is the median absolute deviation of these values. Note that these median values do not apply to all clouds. For a small number of clouds, the measured ratios are significantly different from the median values. \n\nFirst, the virial mass and the CO mass are equal if\n\\begin{equation}\nX_{\\rm CO} = 8.8\\pm3.5\\, \\times10^{20}  {\\rm cm^{-2}~(K~km~s^{-1})^{-1}} \\, \\alpha_{\\rm vir}^{-1}, \\label{eq:xco_vir}\n\\end{equation}\nand second, the virial mass and the millimeter mass are equal if\n\\begin{equation}\n\\frac{\\mathcal{K}_{\\rm dust}}{\\alpha_{\\rm vir}} = 0.6\\pm0.2, \\label{eq:kdu_vir}\n\\end{equation}\nwhere the error bar does not include the uncertainty on the dust temperature. We assume that the mean temperature of 20~K applies in average to the sample.\n\n From the previous equation, we estimate the product between the millimeter absorption coefficient and the dust-to-gas mass ratio,\n\\begin{equation}\n\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d} =  1.5\\pm 0.5~\\times 10^{-3} \\,{\\rm cm^2\\,g}^{-1}\\,\\alpha_{\\rm vir}.\\label{eq:kx_vir}\n\\end{equation}\n\n\n\\subsection{Comparison with Gould's Belt clouds in the Milky Way} \n\n\n\\begin{figure}[!ht]\n   \\centering\n  \\includegraphics[width=6.3cm,angle=90]{eH_alphavir_Fig9.ps}\n      \\caption{The ratio between the product of the millimeter absorption coefficient $\\kappa_{\\rm d}$(1.2mm) and the dust-to-gas mass ratio $x_{\\rm d}$ with the alpha parameter $\\alpha_{\\rm vir}$ versus $x_{\\rm d}$. This ratio is linearly related to the ratio between the millimeter dust luminosity and the virial mass. The data represent the median values of the $\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d}\/\\alpha_{\\rm vir}$ ratio for molecular clouds located in three different environments with different dust-to-gas mass ratio, the Gould's Belt in the Milky Way, several regions in the SMC, and the N11 region in the LMC. The error bar is the standard deviation of each population.  The values for the molecular clouds in the Gould's Belt and the SMC were extracted from \\citet{bot07}.  The solid line is the Galactic measurement scaled to different $x_{\\rm d}$ values. The shaded area represents the uncertainty on the dust temperature  (from 15 to 25~K).}\n         \\label{fig:masses}\n   \\end{figure}\n   \nIn Fig.~\\ref{fig:masses} we compare our results for the N11 clouds with earlier measurements done by \\citet{bot07} for clouds with comparable masses in the Gould's Belt (i.e. in the solar neighborhood) and the SMC. The figure shows the median ratio $\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d}\/\\alpha_{\\rm vir}$ versus the dust-to-gas mass ratio $x_{\\rm d}$, for the molecular clouds in the three different environments.  For a given dust temperature, the data points in Fig.~\\ref{fig:masses} scale with the ratio  between the dust luminosity at 1.2~mm and the virial masses (Eq.~\\ref{eq:edust}). For the Galactic and SMC clouds, these values were computed from the ratio between the millimeter and virial masses listed in Tables~3 and 4 of \\citet{bot07}. Error bars are the standard deviation of the individual values. The solid line shows the Galactic value of $\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d}\/\\alpha_{\\rm vir}$ scaled to different gas-to-dust mass ratios. The pink region represents the uncertainty on the dust temperature, assuming a typical value of $T_{\\rm d}=20$~K which varies between 15 and 25~K. The median value for the N11 clouds agree with the scaled Galactic value. This indicates that the observed difference in the millimeter dust luminosity to virial mass ratio ratio between the Galactic and N11 clouds can be accounted for by the difference in the gas metallicities. This statement dos not apply to the SMC clouds as reported by \\citet{rubio04} and \\citet{bot07,bot10}. The millimeter dust opacity $\\kappa_{\\rm d}$ could be higher in the SMC than in the Milky Way. This interpretation is supported by the difference in the sub-mm\/mm dust spectral energy distributions \\citep{israel10, planck11a17}. \\citet{draine12} have proposed that the enhanced millimeter dust emission observed in the SMC could be accounted for by dipolar magnetic emission from ferromagnetic particles.\n\n \\subsection{Discussion}\n\nOur observational results constrain the product between the $X_{\\rm CO}$-factor and the alpha virial parameter (Eq.~\\ref{eq:xco_vir}), and the ratio between the millimeter dust absorption coefficient and the virial parameter (Eq.~\\ref{eq:kx_vir}) for molecular clouds in the N11 region. In this section, we discuss the most likely values for these three parameters.\n\nIf the measured virial masses of the N11 molecular clouds are their true cloud masses, our $X_{\\rm CO}$-factor is about 3 times larger than the canonical value estimated for Galactic clouds \\citep{solomon87}.  {\\bf The effect of the undersampling of the CO(2--1) observations will increase by a factor 1.5 the difference between the averaged $X_{\\rm CO}$-factor in the Milky Way and our estimate. } Within uncertainties, it agrees with that found with the low-resolution (2\\farcm6 half-power beamwidth) NANTEN survey in the LMC by \\citet{fukui08}, estimated from the virial and CO luminosity masses. \nBut, it is about twice the value estimated by the MAGMA survey in the LMC (see Section~\\ref{sec:mco}) and that estimated by most of the studies of extragalactic GMCs, including GMCs from galaxies in the local group \\citep{blitz07, bolatto08}, as observed in Figure 7 in \\citet{leroy11}, where our $X_{\\rm CO}$-factor estimate corresponds to $\\alpha_{\\rm CO}=19~M_{\\odot}~{\\rm pc^{-2}~({\\rm K~km~s^{-1}})^{-1}}$. The $X_{\\rm CO}$-factor for the N11 clouds will agree with these previous estimates in the LMC if the $\\alpha_{\\rm vir}$ parameter is a factor $\\sim$~2 (see Eq.~\\ref{eq:xco_vir}). \n\nFor molecular clouds in the LMC, \\citet{wong11} compared the $\\alpha_{\\rm vir}$ parameter\\footnote{This value depends on the assumed $X_{\\rm CO}$-factor value.} with the CO luminous masses of the clouds (see the right panel of their Fig.~15). Using the canonical value of $X_{\\rm CO}$, most of the LMC clouds have $\\alpha_{\\rm vir}>1$ and half of them $\\alpha_{\\rm vir}>2$. Using the $X_{\\rm CO}$-factor estimated by \\citet{hughes10}, half of the clouds have $\\alpha_{\\rm vir}>1$ and only a few $\\alpha_{\\rm vir}>2$. \nFor molecular clouds in the outer Galaxy, \\citet{heyer01} found $\\alpha_{\\rm vir}$ to be larger than 2 for clouds with CO masses smaller than $3\\times 10^3~M_{\\odot}$. Most of the N11 clouds are more massive than this limit (see Table~\\ref{tab:nub}).  However, in a recent study of galactic GMCs, defined previously by \\citet{solomon87} and \\citet{heyer09}, \\citet{dobbs11} found that Galactic clouds, in the N11 mass range, have $\\alpha_{\\rm vir}>1$ and about half of them have a $\\alpha_{\\rm vir}>2$ (see their Fig.~1). This finding is in agreement with numerical simulations of the turbulent ISM in the Milky Way presented in their paper. \\citet{dobbs11} claim that there is no need for molecular clouds to be entirely bound to reproduce the observed star formation rate. Star formation will occur in the densest part of the unbound clouds, where gravity wins over turbulence. \n\nFor $\\alpha_{\\rm vir}=2$, Equation~\\ref{eq:kx_vir} yields a dust absorption coefficient for the N11 clouds of $\\kappa_{\\rm d}=0.86\\,{\\rm cm^2~g}^{-1}$. This value is similar to the Galactic value presented in Section~\\ref{sec:mmm}.  Therefore, we favor these values for the $\\alpha_{\\rm vir}$ and $\\kappa_{\\rm d}$ parameters because they do not require the dust emissivity, nor the dynamical state of the clouds, to differ between the LMC and the Milky Way. For $\\alpha_{\\rm vir}=2$, we also get a $X_{\\rm CO}$-factor akin to that estimated by the MAGMA survey in the LMC. The ratio between the virial parameter and the dust far-IR properties is plotted in Figure~\\ref{fig:masses}.  The data does not allow us to separate the two parameters $\\alpha_{\\rm vir}$ and $\\kappa_{\\rm d}$. For N11 with respect to the Galaxy we favor a variation of $\\kappa_{\\rm d}$ associated with metallicity and no variation in  $\\alpha_{\\rm vir}$. However, this interpretation does not apply to SMC clouds. The figure shows that the observed $\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d}\/\\alpha_{\\rm vir}$ values for the GMCs in the SMC are above than that estimated for the Galaxy and N11, by a factor of 3 to 4. \n\n\n\\section{Summary}\n\n\nWe have presented high sensitivity and spatial resolution observations of the CO(2--1) line emission observed with SEST, for the N11 star-forming region in the LMC. Simultaneously with this CO transition, we observed the J$=$1--0 transition, which allowed us to measure the ratio between the CO(2--1) and CO(1--0) integrated emission for the entire map. This ratio has a median value of unity for the entire region. We identified 19 molecular clouds using the CPROPS algorithm which yields their CO luminosity, sizes, line-widths and virial masses. We also presented 1.2~mm continuum observations which we use to estimate millimeter dust fluxes. In this paper, and for the first time in the LMC,  we compute molecular masses estimated from the millimeter dust luminosity ($\\mathcal{L}_{\\rm 1.2mm}({\\rm dust})$) and compare them with the masses obtained from the CO luminosity ($L_{\\rm CO}$) and virial theorem ($M_{\\rm vir}$). The main results of this study are the following ones.\n\n\\begin{itemize}\n\\item The correlations between  CO luminosity, line-widths, sizes, and virial masses in the N11 clouds are in agreement with those found in earlier CO surveys of the LMC, sampling a larger set of clouds and a larger range of cloud masses \\citep{fukui08, wong11}. \n\n\\item  The measured ratios $L_{\\rm CO}\/M_{\\rm vir}$ and $\\mathcal{L}_{\\rm 1.2mm}({\\rm dust})\/M_{\\rm vir}$ constrain the $X_{\\rm CO}$-factor and the dust emissivity per hydrogen atom at 1.2~mm as a function of the virial parameter $\\alpha_{\\rm vir}$ ($M_{\\rm vir}\/M_{\\rm true}$, the ratio between the kinetic and gravity energy terms in the virial equations).\n\n\\item The comparison between the CO and virial masses yields a conversion factor of $X_{\\rm CO}=8.8\\pm3.5~\\times10^{20}  {\\rm cm^{-2}~(K~km~s^{-1})^{-1}} \\, \\alpha_{\\rm vir}^{-1}$.  The comparison between the virial and millimeter dust luminosity yields a product between the gas-to-dust mass ratio $x_{\\rm d}$ and the millimeter absorption coefficient per unit of dust mass $\\kappa_{\\rm d}$(1.2mm) of $1.5\\pm0.5~\\times 10^{-3}~{\\rm cm^2\\,g^{-1}\\,}\\alpha_{\\rm vir}$.\n\n\\item The comparison with previous studies of molecular clouds in the LMC suggests that N11 clouds are unbound, as observed in several Galactic molecular clouds within the same range of masses \\citep{dobbs11}. The median values for the $\\alpha_{\\rm vir}$ and $\\kappa_{\\rm d}$ parameters are 2 and $0.86\\,{\\rm cm^2~g}^{-1}$, respectively. These values do not require peculiar dust properties nor dynamical clouds properties different from those observed in the Milky Way. \n\n\\item We do not find in N11 a large discrepancy between the dust millimeter and virial masses, as reported for GMCs in the SMC by \\citet{rubio04} and \\citet{bot07,bot10}. The ratio $\\kappa_{\\rm d}({\\rm 1.2mm})x_{\\rm d}\/\\alpha_{\\rm vir}$ in N11 is half of that measured for Gould's Belt molecular clouds, which can be accounted for by a factor two lower gas-to-dust mass ratio. This difference is the same as that observed between the gas metallicities. If the two samples have similar $\\alpha_{\\rm vir}$ median values, their dust far-IR properties should be also similar. \n\n\\end{itemize}\n\n\n\n\\begin{acknowledgements}\nWe are grateful to A. Leroy for helping us to compute the properties of the molecular clouds using CPROPS. \nM.R. and C.H. acknowledge support from FONDECYT grant No.1080335. M.R. and F.B. acknowledge the support from the ECOS research grant C08U03. M.R. is supported by the Chilean Center for Astrophysics FONDAP No.15010003. A.D.B. wishes to acknowledge partial support from a CAREER grant NSF-AST0955836,and from a Research Corporation for Science Advancement Cottrell Scholar award.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTwinning has become increasingly significant in modern metallurgy \\cite{Lu2009}. Deformation or mechanical twinning occurs when a region of the crystal is sheared, with each atom moving less than one lattice spacing, such that the twinned region forms a mirror image of the parent crystal about the twinning or habit plane \\cite{Cottrell1995}. Many modern materials such as TWIP steels \\cite{Rahman2014,Rahman2015a}, medium Mn steels \\cite{Kwok2022e}, TWIP Ti \\cite{Zhao2020a,Gao2018}, Mg alloys and Zr alloys \\cite{Abdolvand2016} all capitalise on twinning to enable high strain hardening rates and improve elongation. However, as Qin and Bhadeshia \\cite{Qin2008} point out, the maximum contribution of twinning to total elongation (true strain) in a random polycrystal is $<0.3$. This is compared to total elongation of 0.5--0.6 true strain in TWIP steels \\cite{DeCooman2018,DeCooman2011}. It is therefore the interaction between twins and dislocations which are of importance. In TWIP steels, long and thin twins are formed during deformation, acting as barriers to dislocation motion. The volume fraction of twins increase with increasing strain and continuously subdivides the grain, resulting in a high strain hardening rate and also elongation. This mechanism has been termed the dynamic Hall-Petch effect \\cite{Allain2004a,Bouaziz2001}.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Fig1}\n\t\\caption{Schematic ball and stick model of twinning in FCC materials. (a) Perfect FCC lattice with ABCABC stacking. (b) Passage of one, (c) two and (d) three Shockley partial dislocations on successive (111) planes building up the twin thickness. Grey regions indicate the growing twin. (e) Schematic of the interaction between dislocation and twin boundaries described by Idrissi and Schryvers \\cite{Idrissi2012}, drawn after De Cooman \\textit{et al.} \\cite{DeCooman2018}.}\n\t\\label{fig:atom-twins}\n\\end{figure*}\n\n\nIn the case of Face Centred Cubic (FCC) crystals, the origin of twinning is still a matter of academic debate with many different mechanisms being proposed although the three layer twin nucleus model by Mahajan and Chin \\cite{Mahajan1973} is increasingly being used in computational models \\cite{Steinmetz2013,DeCooman2018}. Nevertheless, the fundamentals of FCC twinning are well understood and first involves the dissociation of a full $\\frac{a}{2} \\{111\\} \\langle 110\\rangle$ into two $\\frac{a}{6} \\{111\\} \\langle112\\rangle$ Shockley partial dislocations, where $a$ is the FCC lattice spacing. In TWIP steels, the formation of partial dislocations is favoured due to the low Stacking Fault Energy (SFE). An FCC twin is then created, as shown in Figure \\ref{fig:atom-twins}, when partial dislocations propagate on successive $\\{111\\}$ planes and increasing the twin thickness. \n\nIn TWIP steels, the deformation twins are often between 20--30 nm in thickness, corresponding to the glide of approximately 100 partial dislocations \\cite{DeCooman2018}. Interestingly, these twins generally do not continue to thicken with strain once this thickness is reached, rather choosing to form new twins instead. This is compared to a larger twin thickness of $\\leq100$ nm in Cu-Al alloys \\cite{Zhang2009} and 1--2 \\textmu m in $\\alpha$-brass \\cite{Roy2014}. Therefore, while there is a general mechanism for twin thickening, there also appears to be mechanism for twin thickness saturation based on some material property such as SFE \\cite{Zhang2009} and grain size \\cite{Rahman2015,Ghaderi2011}. Unfortunately, the reasons for twin thickness saturation are not very well studied, even though the ability of TWIP steels to form a high density of very fine twins is paramount to a high degree of grain fragmentation, underpinning the dynamic Hall-Petch effect and overall tensile properties.\n\nNevertheless, Idrissi and Schryvers \\cite{Idrissi2012} studied the interaction between coherent twin boundaries and dislocations and proposed a \\enquote{pole $+$ deviation} mechanism that might help to control twin thickness. Illustrated in Figure \\ref{fig:atom-twins}e, the first step involves two Shockley partials (D$\\gamma$ and $\\gamma$A) gliding on the K\\textsubscript{1} primary slip plane which constrict and recombine under applied stress to form a full dislocation (DA). In the second step, the full dislocation meets the coherent twin boundary and dissociates again under stress into a sessile Frank partial dislocation (D$\\delta$) and a Shockley partial dislocation ($\\delta$A) according to the reaction:\n\n\\begin{equation}\n\t\\frac{a}{2}\\langle110\\rangle \\rightarrow \\frac{a}{3}\\langle111\\rangle_{sessile} + \\frac{a}{6}\\langle11\\bar{2}\\rangle\n\\end{equation}\n\n\\noindent\nwhere the sessile Frank partial dislocation (D$\\delta$) remains along the twin boundary and the Shockley partial dislocation ($\\delta$A) glides off on the K\\textsubscript{2} conjugate plane, increasing the twin thickness by one layer. Idrissi and Schryvers \\cite{Idrissi2012} then propose that the high density of sessile Frank partials on the twin boundary will affect the mobility of Shockley partials on the twinning plane (K\\textsubscript{1}), limiting the thickening of the twin. Recent computational work by Grilli \\textit{et al.} \\cite{Grilli2020} confirmed the observations by Idrissi and Schryvers \\cite{Idrissi2012} that residual dislocations on the twin boundary can completely stop twin thickening and add that materials with a larger SFE produces larger twins.\n\nTo further investigate the reasons behind the early twin thickness saturation in TWIP steels, Four Dimensional Scanning Transmission Electron Microscopy (4D-STEM) can used to map the strain state in and around a mechanical twin. 4D-STEM is a relatively new technique where an electron diffraction pattern is acquired at every point in a scan grid where sub-nm spatial resolution can be achieved \\cite{Ophus2019}. The term \\enquote{4D} refers to the collection of 2D diffraction patterns over a 2D grid. A comprehensive review of 4D-STEM and its applications in strain mapping, imaging and ptychography is available in a review by Ophus \\cite{Ophus2019}.\n\n\n\nIn this study, a 400 g ingot of steel with nominal composition of Fe-16.4Mn-0.9C-0.5Si-0.05Nb-0.05V (SFE $\\approx$ 23 mJ m\\textsuperscript{-2} \\cite{Sun2018}) based on the composition by Scott \\textit{et al.} \\cite{Scott2011} was produced by arc melting pure elements in a vacuum arc melter. The steel was cast into a copper crucible and homogenised at 1300 \\degree C for 24 h in Ar. The steel ingot was then hot rolled from $10 \\rightarrow 5$ mm (50\\% reduction) in 2 passes at 1000 \\degree C, cold rolled from $5 \\rightarrow 1.5$ mm (67\\% reduction) and finally annealed at 1000 \\degree C for 5 min. Sub-sized tensile samples with gauge dimensions of 19 $\\times$ 1.5 $\\times$ 1.5 mm were obtained from the rolled strip \\textit{via} electrical discharge machining. Tensile samples were then tested to 6\\% engineering strain (approximately 5\\% plastic strain) and to failure, both at a nominal strain rate of $10^{-3}$ s\\textsuperscript{-1}. \n\n\n\nElectron Backscattered Diffraction (EBSD) was conducted using a Sigma 300 FEG-SEM on the as-annealed and 6\\% strained conditions. A strain of 6\\% was chosen to produce a sufficient twin density in order to locate the twins using EBSD while keeping background dislocation density to a minimum. In the 6\\% strained sample, a grain with a $\\langle111\\rangle$ crystallographic direction in the vertical axis of the microscope field of view was selected and a thin foil was extracted using the Focussed Ion Beam (FIB) lift-out technique, producing a foil with its $\\langle111\\rangle$ direction parallel to the foil normal.\n\n\n\n\n\n\n\n4D-STEM was performed on the foil using the probe corrected JEOL ARM200CF Transmission Electron Microscope (TEM) at ePSIC (Oxford, United Kingdom). Diffraction patterns were collected with a Merlin (MediPix) direct electron detector. An accelerating voltage of 200 kV and a camera length of 40 cm were employed with a 10 \\textmu m CL1 aperture. A $68.2 \\times 83.1$ nm area of interest was scanned in $188 \\times 229$ real space pixels and with a $256 \\times 256$ pixel diffraction pattern captured at each of these scan positions with a 1 ms dwell time per pattern.\n\nThe in-plane elastic strain was calculated from the electron diffraction pattern at each scan location. Bragg peak identification, dataset calibration including elliptical distortion correction and diffraction shift correction and elastic strain calculation were performed with the open source py4DSTEM analysis package \\cite{Savitzky2019}. For locating Bragg peaks, a correlation power of 1 was used, corresponding to cross-correlation \\cite{Pekin2017}, and achieve subpixel precision with local Fourier up-sampling by a factor of 16 \\cite{Guizar-Sicairos2008}. For the rest of the paper, an orthogonal frame of reference with directions \\textit{1, 2, 3} will be adopted, where \\textit{3} refers to the $[111]$ direction perpendicular to the surface of the foil, \\textit{1} as $[1\\bar{2}1]$ and \\textit{2} as $[10\\bar1]$.\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Fig2}\n\t\\caption{EBSD IPF-X (right) and image quality maps of (a) as-annealed sample where black lines indicate high angle grain boundaries and white lines indicate austenite $\\Sigma3$ boundaries (rolling direction parallel to the horizontal axis and transverse direction pointing out of the page), (b) sample deformed to 6\\% strain, demonstrating fine deformation twins. (c) Engineering tensile curves of the investigated steel when tested to failure and to 6\\% strain. }\n\t\\label{fig:tensile-ebsd}\n\\end{figure}\n\n\n\nThe microstructure of the as-annealed TWIP steel is shown in Figure \\ref{fig:tensile-ebsd}a with an area weighted average grain size of 12 \\textmu m. An EBSD map of the 6\\% strained sample is shown in Figure \\ref{fig:tensile-ebsd}b. Unfortunately, due to the fine twin thickness, it is not easy to index deformation twins using EBSD. However, they can be identified as darker lines in the Image Quality (IQ) map. The tensile behaviour of the investigated steel is shown in Figure \\ref{fig:tensile-ebsd}c and has a yield strength of 360 MPa, tensile strength of 1050 MPa and total elongation of 69\\%.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics{Fig3}\n\t\\caption{(a,b) Virtual bright and (c,d) dark field images of the steel nanotwin using different virtual apertures indicated by the open red circles. (e) Average deconvolution maps, sums over real space of the identified Bragg peak locations weighted by intensity.}\n\t\\label{fig:vbf-vdf}\n\\end{figure*}\n\nFigure \\ref{fig:vbf-vdf}a--d shows the Virtual Bright and Dark Field (VBF and VDF respectively) images reconstructed from the intensity collected from the highlighted digital apertures. VBF images in Figure \\ref{fig:vbf-vdf}a--b clearly distinguish the twin from the matrix. Given the axis system we have adopted with [111] out of the plane and the diffraction vectors indexed in Figure \\ref{fig:vbf-vdf}e, it is inferred that the twins have a habit plane of $(11\\bar{1})$.\n\nIn conventional TEM, the direct beam contains a large amount of structure-dependent information. Conventional bright field imaging when used with an objective aperture to isolate the direct beam, uses electron wave phase information as well as intensity to re-interfere and reconstruct an image \\cite{Williams2009}. In VBF, only access to electron intensity in the diffraction plane is available. Therefore it is likely that the observed contrast between twin and matrix is derived from local strain, lattice rotation or dynamical effects which will alter the ratio of diffracted to direct intensity. The high angle VDF image in Figure \\ref{fig:vbf-vdf}c is similar to convention High-Angle Annular Dark Field (HAADF) imaging which detects electrons which are scattered to high angles and also suggests that there was no detectable variation in local chemistry between the twin and the matrix. The twin is explicitly highlighted in Figure \\ref{fig:vbf-vdf}d by reconstructing the spatial image from the $00\\bar{2}$ reciprocal lattice point for the twinned region only, similar to a conventional TEM dark field image. The action of twinning rotates the the diffraction pattern about the direct beam, such that the twinned and untwinned $02\\bar{2}$ peaks are separated. In Figure \\ref{fig:vbf-vdf}e, a Bragg vector map is presented (after Savitzky \\textit{et al.} \\cite{Savitzky2019}), showing the distribution of all measured Bragg peak locations for the untwinned region. It is acknowledged that a small amount of intensity is observed at the twin reciprocal lattice points even for the untwinned material. This is possibly due to the geometry of the sample, \\textit{i.e.} twinning habit plane not perfectly parallel to beam direction, and through thickness sampling of both untwinned and twinned material near the interface. \n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Fig4}\n\t\\caption{Elastic strain maps resolved in the (a) 11, (b) 22, (c) 12 and (d) 33 directions. The average 1-direction profile, perpendicular to the twin's length, was calculated by integrating all points along the \\textit{2}-direction within the highlighted region.}\n\t\\label{fig:twinstrain}\n\\end{figure*}\n\nThe strain components $\\varepsilon_{11}$, $\\varepsilon_{22}$, $\\varepsilon_{12}$ were calculated from the relative movements of the diffraction spots. This operation was performed independently for twinned and untwinned regions. A set of reference reciprocal lattice vectors were obtained by averaging the untwinned regions's reciprocal lattice basis vectors. The magnitude of the twin basis vectors were normalised to this unstrained length. As such, elastic strains are given with reference to this \\enquote{unstrained} state. The measurement could alternatively be considered as the elastic strain variation across the area of interest. \n\n\nMaps of the measured (11, 22, 12) and inferred strain components across the area of interest are shown in Figure \\ref{fig:twinstrain}. An average line profile in the 1-direction was obtained by integrating points along the 2-direction within the highlighted area. From the average profiles in Figure \\ref{fig:twinstrain}, it is striking that the average elastic strains in the 11 and 22 directions reach values as high as 6\\% at locations along the twin boundaries. Hot spots of up to 12\\% were also observed at certain points along the twin boundaries. These values far exceed other elastic strain measurements observed in the literature (Table \\ref{tab:straincompare}), even between other 4D-STEM experiments with the same methodology, \\textit{e.g.} $\\sim4\\%$ observed by Pekin \\textit{et al.} along twin boundaries in stainless steel 304 \\cite{Pekin2017}. However, it should be noted that the averaged strains have been disproportionately raised above the baseline of approximately 4\\% by the presence of hot spots. \n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Comparison of strain, $\\varepsilon$, in \\% obtained with different strain mapping methods on various materials. *SS--Stainless Steel, CP--Commercially Pure, N-PED--Nano-Precession Electron Diffraction.}\n\t\\begin{adjustbox}{width=\\columnwidth,center}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\toprule\n\t\t\tMaterial & Method & Feature & $\\varepsilon$ & Reference \\\\\n\t\t\t\\midrule\n\t\t\t\n\t\t\tAISI 321 (SS)\t&\t4D-STEM\t&\tPlanar dislocations\t&\t0.5\t& \\cite{Pekin2018} \\\\\n\t\t\tBoron Carbide & N-PED & Grain boundary &  0.75     & \\cite{Rottmann2018} \\\\\n\t\t\tPd    & HR-TEM & Twins & 2.5   & \\cite{Rosner2010} \\\\\n\t\t\tSS 304 & 4D-STEM  & Twins & 4     & \\cite{Pekin2017} \\\\\n\t\t\tCP Mg & 4D-STEM  & Twins & 4     & \\cite{Chen2020b} \\\\\n\t\t\tTWIP steel & 4D-STEM & Twins & 6     & Current \\\\\n\n\t\t\t\\bottomrule\n\t\t\\end{tabular}%\n\t\\end{adjustbox}\n\t\\label{tab:straincompare}%\n\\end{table}%\n\nThe presence of hot spots was also observed by R\\\"{o}sner \\textit{et al.} \\cite{Rosner2010} in a cold rolled Pd foil. These hot spots of approximately $\\leq25\\%$ strain were attributed to dislocation debris when mapping the strain of dislocation cores of stacking faults or partial dislocations in the foil. It should therefore also follow, that the hot spots observed at the twin boundaries in Figure \\ref{fig:twinstrain} were the result of a highly dislocated structure. Following the \\enquote{pole $+$ deviation} mechanism proposed by Idrissi and Schryvers \\cite{Idrissi2012}, the hot spots are likely to be the locations where one slip system impinges onto the twin boundary, generating a high density of sessile Frank partial dislocations at the twin boundary and also generating a very large stress field. Faulted twin boundaries as a result of impingement from another slip system were also shown by Dao \\textit{et al.} \\cite{Dao2006} in nanotwinned Cu using HR-TEM. Furthermore, since the sample was only deformed to 5\\% strain, it is likely that only a few slip systems would have been activated, hence the localisation of strain only at certain points along the entire twin boundary. \n\nIn the study by Grilli \\textit{et al.} \\cite{Grilli2020}, it was found that an additional shear stress of 400 MPa to the critical resolved shear stress for twinning in $\\alpha$-uranium (orthorhombic) was sufficient to completely retard twinning and favour slip instead. Using the analytical model from M\\\"{u}llner \\textit{et al.} \\cite{Mullner1994,Mullner1997,Mullner1994a}, it was estimated that the strain hot spots at the twin boundaries in the investigated steel resulted in a shear stress of approximately 1--1.5 GPa (shear modulus of austenite being 65 GPa \\cite{Allain2004a}), significantly larger than the shear stress needed to retard twinning in $\\alpha$-uranium. This large shear stress may help to explain why deformation twins in TWIP steels remain very thin. \n\nHowever, it should be noted that the twins imaged in the current sample were only 4 nm thick, whereas the \\enquote{saturation} twin thickness in TWIP steels is approximately 30 nm \\cite{DeCooman2018}, suggesting that these twins will continue to thicken with increasing strain. Based on the conclusions by Grilli \\textit{et al.} \\cite{Grilli2020} that twin growth is stopped when a critical density of residual dislocations on the twin boundary is reached, it is speculated that by the time the twin has grown to approximately 30 nm, the strain field along the twin boundary would have become more uniform, \\textit{i.e.} no more hot spots. The high density of dislocations along the twin boundary would then be able to completely stop the motion of Shockley partial dislocations along the twin boundary and the twin would then reach its maximum thickness. \n\n\n\n\nThe existence of hot spots may also help to explain the variability of twin thickness. De Cooman \\textit{et al.} \\cite{DeCooman2018} showed that while twins are commonly considered to be plate-like in morphology, TEM observations have shown twin thicknesses to be highly variable and twin boundaries to be highly faulted \\cite{Dao2006}. The formation of faulted twin boundaries appear to be caused by the disorderly overlapping of Shockley partial dislocations \\cite{DeCooman2018}. Therefore, it is speculated that while the hot spots may not have been sufficient to completely stop twin thickening at early strains, they will certainly disrupt the propagation of Shockley partials along the twin boundary leading to twin growth with irregular thicknesses. \n\n\n\n\n\n\n\n\n\n\n\nIn summary, the 4D-STEM technique was used to experimentally observe the strain fields in and around several deformation twins in a TWIP steel with sub-nm resolution. The results show several strain hot spots of up to 12\\% along the twin boundaries which very likely correspond to sessile Frank dislocations along the twin boundaries according to the \\enquote{pole $+$ deviation} mechanism proposed by Idrissi and Schryvers \\cite{Idrissi2012}. The observed average strain of 6\\% corresponds to a large shear stress of 1--1.5 GPa along the twin boundaries compared to other materials and is likely to be responsible for early retardation of twin growth and thickness saturation. This mechanism for twin thickness saturation helps to keep deformation twins in TWIP steels extremely fine, resulting in increased grain fragmentation with applied strain which is central to the dynamic Hall-Petch effect and underpins the impressive tensile properties of TWIP steels.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{\\baselineskip}\n\\noindent\n\\textbf{Acknowledgements}\n\\vspace{\\baselineskip}\n\nTPM and DD would like to acknowledge support from the Rolls-Royce plc - EPSRC Strategic Partnership in Structural Metallic Systems for Gas Turbines (EP\/M005607\/1), and the Centre for Doctoral Training in Advanced Characterisation of Materials (EP\/L015277\/1) at Imperial College London. TWJK is grateful for a studentship from A*STAR. AA acknowledges EPSRC grant IAA EP\/R511547\/1. We thank Diamond Light source for access and support in use of the ePSIC instrument E02 and proposal no. EM18770 that contributed to the results presented here. BHS and CO acknowledge funding from the Toyota Research Institute, and from by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract no. DE-AC02-05CH11231.  We are grateful to TB Britton for insight into electron diffraction and micromechanics and B Poole for guidance with basis rotation. We also thank VA Vorontsov and AJ Knowles for helpful conversations when planning the project. \n\n\\vspace{\\baselineskip}\n\\noindent\n\\textbf{References}\n\\vspace{\\baselineskip}\n\n\\bibliographystyle{thesis_thomas5}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGalaxy clusters host several potential accelerators of cosmic ray (CR) \nelectrons and protons, from ordinary galaxies to active galaxies (AGN) \nand cosmological shock waves, driven in the intracluster medium (ICM) \nduring the process of hierarchical cluster formation (see Blasi et al. \n2007 for a review).\n\nThe long lifetime of CR protons (or nuclei) in the ICM and the large \ngeometrical size of the magnetized region of galaxy clusters make them \nefficient storage rooms for the hadronic component of CRs produced within \ntheir volume (V\\\"olk et al. 1996, Berezinsky et al. 1997, \nEnsslin et al. 1997). The accumulation of CRs inside clusters over \ncosmological times leads to the assumption \nthat an appreciable amount of energy \nmay be stored in the ICM in the form of non-thermal particles. \nIf this energy is sufficiently high, the flux of $\\gamma$ radiation induced \nby the production and decay of neutral pions may reach potentially \ndetectable levels, thereby providing us with a powerful diagnostic tool of \nthe CR energy content of clusters (Colafrancesco \\& Blasi 1998, \nBlasi \\& Colafrancesco 1999, V\\\"olk \\& Atoyan 1999, Miniati 2003, \nPfrommer \\& En\\ss lin 2004, Wolfe et al. 2008).\n\nSo far only upper limits to the $\\gamma$-ray emission from galaxy clusters \nhave been obtained (Reimer et al.~2003; Perkins et al. 2006, \nAharonian et al. 2009a,b; Aleksic et al. 2010; Ackermann et al. 2010)\n\\footnote{See however Han et al. 2011 for Virgo}.\nThese upper limits, together with several constraints from complementary \napproaches based on radio observations lead us to conclude that \nCR protons contribute less than a few percent\nof the energy of the ICM, at least in the central Mpc--size region\n(Reimer et al. 2004, Brunetti et al. 2007, 2008, Brown et al. 2011,\nAleksic et al. 2011).\n\nCR electrons are very well traced in the ICM through their radio emission \nwhich appears in the form of diffuse (Mpc scale) \nsynchrotron {\\it giant radio halos} from the cluster X-ray emitting regions, \nand {\\it relics}, typically in the clusters' peripheral regions \n(see Ferrari et al. 2008, Venturi 2011, for recent reviews on observations).\n\n\\noindent\nGiant radio halos are the most spectacular and best studied non-thermal\nlarge scale phenomena in the universe. They appear in about $1\/3$ of the \nmost massive galaxy clusters (e.g. Giovannini et al. 1999, \nKempner \\& Sarazin 2001, Cassano et al. 2008), in a rather clear connection \nwith dynamically disturbed systems, while ``off-state'' clusters \n(those with no evidence of diffuse emission) are generally more \nrelaxed (Cassano et al. 2010a and references therein). \nThe connection between cluster mergers and radio halos suggests that \nsuch emission traces the hierarchical cluster assembly and probes the \ndissipation of gravitational energy during the dark matter-driven \nmergers that lead to the formation of clusters. \nThe physical mechanisms responsible for the generation and evolution of \nradio halos are still a matter of debate, but two main\nlines of thought have been \ndeveloped throughout the years.\n\n\\noindent\nOne is based on the idea that seed electrons \nmay be re-accelerated by turbulence produced during merger \nevents (Brunetti et al. 2001, Petrosian 2001). \nIn this class of models the $\\gamma$-ray emission is predicted to\nbe rather low, \nthough it may be substantial under the hypothesis that the electron seeds\nare secondaries produced in inelastic collisions of a subdominant \nhadronic CR component in the ICM (Brunetti \\& Blasi 2005; \nBrunetti \\& Lazarian 2011). \n\n\\noindent\nThe second line of thought is based on the idea of clusters as storage \nrooms of CR protons: the radio halos may be generated as a result of \nsynchrotron emission of secondary electrons and positrons from pp \ncollisions (Dennison 1980,\nBlasi \\& Colafrancesco 1999, Pfrommer \\& En\\ss lin 2004). \nOn one hand this idea serves as a solution to the problem of the large \nspatial dimensions of the radio emitting region, larger than the typical \nloss length of electrons: secondary electrons and positrons are \nproduced {\\it in situ} in inelastic CR collisions and radiation \nis produced near the production region. \nOn the other hand, secondary models do not explain in a {\\it natural} way \nthe observed association between cluster mergers and giant \nradio halos, in that CRs accumulate inside the ICM on cosmological \ntime scales and not in direct connection with acceleration events \nsuch as those associated with mergers\\footnote{see however Ensslin et al.\n2011, where diffusion\/transport of CRs is studied under peculiar conditions}. \nOne proposal is that \nduring merger events the cluster magnetic field becomes larger than \nin the quiescent state, thereby turning the halo on (Kushnir et al. 2009, \nKeshet \\& Loeb 2010), although studies based on Rotation Measures\nof clusters' and background radio sources disfavour this scenario\n(see Bonafede et al.~2011a and ref. therein).\nIn fact, some pieces of observations put tension on a {\\it pure} hadronic \norigin of radio halos. These include the steepening in the spectrum \n(or the very steep spectrum) of several halos (Schlickeiser et al. 1987,\nThierbach et al. 2003, Reimer et al. 2004, \nBrunetti et al. 2008, Dallacasa et al. 2009,\nGiovannini et al. 2009, Macario et al. 2010, van Veeren et al. 2011) and the \nvery large spatial extent of several halos (or their flat radio-brightness \ndistribution) (Brunetti 2004,\nMurgia et al. 2009, Donnert et al. 2010a, Brown \\& Rudnick 2011); \nin all cases observations would imply that the energy budget of CR protons \nis uncomfortably large, at least assuming that clusters are magnetised \nat $\\sim \\mu$G level, consistent with the results of \nRM (see Bonafede et al.~2010 and references therein).\n\n\\noindent\nThe most distinct prediction of models of radio halos that are based on\nsecondary particles is the \nproduction of $\\gamma$--rays (e.g., Blasi \\& Colafrancesco 1999, \nSarazin 2004, Pfrommer \\& En\\ss lin 2004, Brunetti 2009,\nBrunetti \\& Lazarian 2011, En\\ss lin et al. 2011). \nNowadays there is agreement on the fact that the abundance of secondaries\nrequired to fit the spectrum of (at least some) radio halos should produce\n$\\gamma$--ray emission detectable with the sensitivity of the Fermi-LAT,\nassuming low\/medium level of cluster magnetic field (eg. Marchegiani\net al 2007, Pfrommer 2008, Brunetti 2009, Jeltema \\& Profumo 2011).\nThus \nthe combination of the available information on the spectrum and morphology \nof radio halos in conjunction with \nthe limits on their $\\gamma$-ray emission provides \na powerful tool to gain insights into the origin of the halo emission.\n\n\\noindent\nFollowing this pathway, in \nthis paper we concentrate on the case of the Coma cluster, which \nrepresents a prototypical example of giant radio halos, with a wealth \nof data available on its spectrum and morphology. \nThe incoming upper limits on the Coma $\\gamma$-ray emission with the \nFermi-LAT telescope are invaluable in imposing stringent limits on \nthe amount of cosmic rays that can be stored in the intracluster medium \nand serve as sources of secondary electrons and positrons. \nIn particular in this paper we combine for the first time \nall available information about the\nspectral shape and morphology of the radio halo of the Coma cluster\nwith the recent $\\gamma$-ray upper limits and magnetic field\nstrengths derived from Faraday rotation measures.\nWe show that the requirement of reproducing\nthe properties of the Coma radio halo in the context of a pure\nsecondary electron model leads to a large CR energetics and fluxes \nof $\\gamma$-rays which results in a tension with the existing upper limits \nfrom the Fermi-LAT (Ackermann et al.~2010).\nThis situation is readily alleviated in the case of a\nlarge cluster magnetic field,\nhowever the magnetic fields required by the model are appreciably larger \nthan those inferred from recent Faraday rotation measures. \n\n\\noindent\nThis tension disappears when including the effect of turbulent\nreacceleration in combination with the process of injection of\nsecondary particles.\nWe adopt a physically motivated \npicture in which secondary products of cosmic ray interactions are\nreaccelerated by MHD turbulence during clusters mergers.\nIn this sense we value the physical insight behind the concept of cosmic \nray confinement and the production of secondary electrons, but we do not \nassume that these particles are ``directly'' responsible for the formation of \nthe radio halo. \nWe find that even reacceleration models of this type require a broad \nspatial distribution of the parent cosmic rays, but the expected \n$\\gamma$-ray \nfluxes are well consistent with the Fermi-LAT upper limits.\n\nThe paper is organized as follows: we discuss the hadronic model of \nradio emission in \\S \\ref{sec:hadro} and the reacceleration model \nin \\S \\ref{sec:reacc}. A critical discussion of our results \nis provided in \\S \\ref{sec:disc}.\nA $\\Lambda$CDM\ncosmology ($H_{o}=70\\,\\rm km\\,\\rm s^{-1}\\,\\rm Mpc^{-1}$,\n$\\Omega_{m}=0.3$, $\\Omega_{\\Lambda}=0.7$) is adopted throughout the paper.\n\n\n\\section{Pure hadronic models}\n\\label{sec:hadro}\n\n\\subsection{Formalism : radio and $\\gamma$--ray emission}\n\nThe decay chain responsible for the injection of secondary particles \nin the ICM due to p-p collisions is (e.g., Blasi \\& Colafrancesco 1999):\n\n$$p+p \\to \\pi^0 + \\pi^+ + \\pi^- + \\rm{anything}$$\n$$\\pi^0 \\to \\gamma \\gamma$$\n$$\\pi^\\pm \\to \\mu^{\\pm} + \\nu_\\mu ~~,~~ \\mu^\\pm\\to e^\\pm \\nu_\\mu \\nu_e \\, .$$\n\nThe initial inelastic scattering reaction occurs at threshold, \nnamely it requires protons with kinetic energy larger \nthan $T_p \\approx 300$ MeV. The injection rate of pions is \n\\begin{equation}\nQ_{\\pi}^{\\pm,o}(E_{\\pi^{\\pm,o}},t)= n_{th} c \n\\int_{p_{*}} dp N_p(p,t) \\beta_p {{ F_{\\pi}(E_{\\pi},E_p) \n\\sigma^{\\pm,o}(p)}\\over\n{\\sqrt{1 + (m_pc\/p)^2} }},\n\\label{q_pi}\n\\end{equation}\nwhere $n_{th}$ is the number density of thermal (target) protons,\n$N_p$ is the spectrum of cosmic ray protons, \nand $F_{\\pi}$ is the spectrum of pions from the collision between \na CR proton of energy $E_p$ and thermal protons \n(taken from Brunetti \\& Blasi 2005). \nThe inclusive cross--section, $\\sigma(p)$, is taken from the fitting \nformulae of Dermer (1986b), which allows us to describe separately \nthe rates of generation of $\\pi^{-}$, $\\pi^{+}$, and $\\pi^{o}$, \nand $p_* = \\max \\{ p_{_{tr} },\\, p_{\\pi} \\}$, where $p_{tr}$ is the \nthreshold momentum of protons for pion production (different for \ncharged and neutral pions). \n\nCharged pions decay into muons and then\nsecondary pairs (electrons and positrons). \nIf secondaries are not accelerated by other mechanisms and the physical \nconditions in the ICM do not change on time-scales shorter than \nthe electrons' radiative time \\footnote{see Keshet (2010) for a discussion \non effects due to variations of magnetic fields on shorter time-scales}, \ntheir spectrum approaches a stationary distribution because of the \ncompetition between injection and energy losses (e.g.,\nDolag \\& Ensslin 2000), and one can write:\n\n\\begin{equation}\nN_e^{\\pm}(p)=\n{1 \\over\n{\\Big| \\left( {{dp}\\over{dt}} \\right)_{\\rm loss} \\Big| }}\n\\int_{p}^{p_{\\rm max}}\nQ_e^{\\pm}(p) dp \\, ,\n\\label{sec_stat}\n\\end{equation}\n\n\\noindent\nwhere $Q_e^{\\pm}$ is the injection rate of secondaries \n(Blasi \\& Colafrancesco 1999; Moskalenko \\& Strong 1998), and \nradiative losses, which are dominant for $\\gamma > 10^3$ electrons \nin the ICM, can be written as (Sarazin 1999):\n\n\\begin{equation}\n\\Big| \\left( {{ d p }\\over{d t}}\\right)_{\\rm loss} \\Big|\n\\simeq 3.3 \\times 10^{-32}\n\\big( {{p\/m_e c}\\over{300}} \\big)^2 \\left[ \\left( {{ B_{\\mu G} }\\over{\n3.2}} \\right)^2 + (1+z)^4 \\right] \\, .\n\\label{loss}\n\\end{equation}\n\nAssuming a power law (momentum) distribution of CR protons, \n$N_p(p) = K_p p^{-s}$, the spectrum of secondaries \nat high energies, $\\gamma > 10^3$, \nis $N_e(p) \\propto p^{-(s+1)} {\\cal F}(p)$, \nwhere ${\\cal F}$ accounts for the log--scaling of the p-p cross--section \nat high energies and causes the spectral shape to be slightly flatter \nthan $p^{-(s+1)}$ (Brunetti \\& Blasi 2005). \nThe synchrotron spectrum from secondary e$^{\\pm}$ \nis (Rybicky \\& Lightman 1979):\n\n\\begin{equation}\nJ_{syn}(\\nu) = \\sqrt{3} {{e^3}\\over{m_e c^2}} B\n\\int_0^{\\pi\/2} d\\theta sin^2\\theta \\int dp N_e(p)\nF\\big( {{\\nu}\\over{\\nu_c}} \\big)\n\\simeq C_{syn}(\\alpha, T) X n_{th}^2\n{{B^{1+\\alpha} }\\over{B^2 + B_{cmb}^2}} \\nu^{-\\alpha} \\, ,\n\\label{jsyn}\n\\end{equation}\nwhere $C_{syn}$ is a constant, $X=\\epsilon_{CR}\/\\epsilon_{ICM}$ \nis the ratio of the energy densities of CR protons to thermal protons, \n$F$ is the synchrotron kernel, $\\nu_c$ is the critical frequency, \nand $\\alpha \\simeq s\/2 -\\Delta$ is the synchrotron spectral index, \nwhere $\\Delta \\sim 0.1$ (in the conditions\nof interest for our paper) is due to \nthe log-scaling of the cross--section.\n\nThe spectrum of $\\gamma$-rays produced by secondary particles is dominated \nby the decay of the secondary $\\pi^0$ (Dermer 1986ab; \nBlasi \\& Colafrancesco 1999, Blasi 2001) :\n\n\\begin{equation}\nQ_{\\gamma}(E_{\\gamma})\n= 2 \\int_{E_{min}}^{E_p^{max}}\n{{ Q_{\\pi^0}(E_{\\pi^0}) }\\over{\n\\sqrt{E_{\\pi}^2 - m_{\\pi}^2 c^4} }}\nd E_{\\pi} \\, ,\n\\label{qgamma}\n\\end{equation}\n\n\\noindent\nwhere $E_{min} = E_{\\gamma} + m_{\\pi}^2 c^4 \/ (4 E_{\\gamma})$. \nUnder the same assumptions as Eq.\\ref{jsyn}, the $\\gamma$-ray emissivity \nfor $h\\nu_{\\gamma} >$ GeV is:\n\n\\begin{equation}\nJ_{\\gamma}(\\nu) \n\\simeq C_{\\gamma}(s, T) X n_{th}^2 \\nu^{-\\tilde{s}} \\, ,\n\\label{jgamma}\n\\end{equation}\n\n\\noindent\nwhere $C_{\\gamma}$ is a constant and $\\tilde{s} \\sim s -1$. \nThis gives a minimum estimate of the cluster $\\gamma$--ray emission \nsince IC and non-thermal bremsstrahlung emission from secondary and \nprimary electrons provide additional \ncontributions (e.g. Sarazin 1999, Blasi 2001, Miniati 2003).\n\n\\subsection{Testing hadronic models for radio halos}\n\nThe ratio of the synchrotron and $\\gamma$--ray cluster luminosities depends \non the magnetic field in the emitting volume (from Eqs.~\\ref{jsyn} \nand \\ref{jgamma}). By considering reasonable assumptions on the magnetic \nfield in galaxy clusters several authors suggested that observations\nby the Fermi-LAT \nmay have the chance to detect $\\gamma$-rays from nearby clusters hosting \nradio halos (Pfrommer \\& En\\ss lin 2004, Marchegiani et al. 2007,\nPfrommer 2008, Wolfe et al. 2008, Brunetti 2009).\n\nMore recently, Jeltema \\& Profumo (2011) analysed the impact of \nlimits on the $\\gamma$-ray emission from clusters of galaxies (Ackermann\net al.~2010) on \nhadronic models for the origin of cluster radio halos, deriving lower \nlimits on the average cluster magnetic field. In some cases of nearby \nradio halos they find that these lower limits are close to (or larger than) \nthe magnetic field values inferred from Faraday rotation measures, \nthereby placing tension on the hadronic origin of radio halos. \nIn their analysis \nJeltema \\& Profumo used the total radio luminosity and the\n$\\gamma$-ray upper limits.\n\nHere we discuss the possibility that even more stringent limits may \nbe obtained by using the information about the morphology of the \nradio emission {\\it in addition} to the total luminosity. \nGiant radio halos have flat brightness distributions, in several \ncases flatter than the thermal X-ray emission of the hosting clusters\n(e.g. Govoni et al. 2001, Feretti et al. 2001, Brown \\& Rudnick 2011), \nimplying that most of the synchrotron emission is produced in the external \nregions where the magnetic field is smaller.\nThus, we anticipate a larger energy budget of CR protons \ncompared with that needed to explain only the total luminosity \nof radio halos (Brunetti 2004, Donnert et al. 2010a,b). \nThe consequence is also a larger $\\gamma$-ray luminosity from the hosting \nclusters (Donnert et al. 2010a).\n\n\\noindent\nGiven these premises, in Sect.~2.3 we attempt to analyse the impact \nof the Fermi-LAT upper limits from the Coma cluster. We summarize our \nassumptions as follows:\n\n\\begin{itemize}\n\n\\item[{\\it i)}]\nThe spectrum of cosmic ray protons is assumed to be a power \nlaw in momentum, as results from most acceleration mechanisms, \n$N(p) = K_p p^{-s}$;\n\n\\item[{\\it ii)}]\nThe spatial distribution of the gas in the ICM is taken to follow \nthe {\\it observed} $\\beta$--model that is parameterised according\nto thermal density at the cluster center, $n_{th}(0)$, the \ncore radius, $r_c$ kpc, and $\\beta$;\n\n\\item[{\\it iii)}]\nThe spatial distribution of the energy density of CR protons in the \ncluster volume is parametrized \nas $\\epsilon_{CR} \\propto \\epsilon_{ICM}^{1+f}$;\n\n\\item[{\\it iv)}]\nThe magnetic field strength is assumed to scale with location $r$ \nin the ICM according with \n$B(r)=B_0 \\left[\\epsilon_{ICM}(r)\/ \\epsilon_{ICM} (0)\\right]^{\\eta}$,\n$\\eta=0.5$ implies that the magnetic field energy density scales with \nthe thermal one.\n\n\\end{itemize}\n\n\\noindent\nWithin these assumptions the synchrotron emissivity from secondary \nparticles (from Eq.~\\ref{jsyn}) is:\n\n\\begin{equation}\nJ_{syn}(\\nu) \\propto\nn_{th}^{2+f} T^{1+f}\n{{ ( n_{th} T)^{\\eta(1+\\alpha)}}\\over{\nC_B^2 ( n_{th} T)^{2 \\eta} + B_{cmb}^2 }},\n\\label{JsynS}\n\\end{equation}\nwhich implies a synchrotron brightness (assuming $T$ constant \non $\\sim 1$ Mpc scale to derive simple scalings) \n$I_{syn} \\propto (1 + x^2)^{-3 \\tilde{\\beta} +1\/2}$, \nwhere $x$ is the projected distance in units of the core radius, \n$C_B$ is a constant ($B = C_B ( n_{th} T)^{\\eta}$), \nand $\\tilde{\\beta} = \\beta (2 +f +(\\alpha -1) \\eta)\/2$ \nand $\\tilde{\\beta} =\\beta (2 +f +(\\alpha +1) \\eta)\/2$ in the \ncase $B^2(x) \\gg B_{cmb}^2$ and $B^2(x) \\ll B_{cmb}^2$, respectively.\n\nFor a given model of magnetic field distribution, $(B_0,\\eta)$, \nthe three key parameters characterizing the radio emissivity and then\nthe predictions for the $\\gamma$-ray emission are thus :\n\n\\begin{itemize}\n\n\\item[1.]\n$s$: the logaritmic slope of the momentum distribution of CR protons, \ndetermined\nfrom measurements of the radio spectral index $\\alpha$.\nTypically $s \\sim 2.3-3$ (for some halos with ultra-steep \nspectrum $s > 3$);\n\n\\item[2.]\n$f$: the non-linear scaling of the energy density of CR protons \nwith the thermal\nICM energy density, derived from the shape of the\nsynchrotron brightness profile of the radio halo;\n\n\\item[3.]\n$K_p$: the normalization of the momentum distribution of CR protons,\ndetermined from the radio luminosity of the halo.\n\n\\end{itemize}\n\n\\noindent\nThe outcome of this procedure is the expected $\\gamma$-ray \nemission from $\\pi^0$--decays.\n\n\\subsection{Application to the Coma radio halo}\n\\label{sec:morph}\n\nThe radio halo in the Coma cluster (Coma C) is the prototypical cluster \nradio halo (e.g., Willson 1970, Giovannini et al. 1993). It is unique in \nthat its spectrum is measured over a wide frequency \nrange (Fig. \\ref{fig:hadroComa}, left panel). The spectrum shows \na significant steepening at high frequencies: a power-law that fits the \ndata at lower frequencies overestimates the flux measured at 2.7 and \n5 GHz by a factor 2 and 4 respectively (Schlickeiser et al. 1987, \nThierbach et al. 2003). In secondary electron models the electron \nspectrum extends in principle to very high energies, therefore no \nintrinsic cutoff is expected in the radio spectrum. \nThe observed spectral steepening is therefore inconsistent with \nsecondary electron models (Schlickeiser et al. 1987, Brunetti et al. 2001, \nPetrosian 2001, Blasi 2001, Reimer et al 2004) unless unnatural assumptions \nare made on the CR proton spectrum. \nEn\\ss lin (2002) suggested that the steepening at higher frequencies \nmay be due to the thermal SZ-decrement (suppression of the CMB spectrum \nseen in the direction of the cluster from the up-scattering \nof the CMB photons due to Compton scattering with \nthe thermal electrons in the ICM). Calculations carried out \nby Brunetti (2004) and Reimer et al. (2004) \nshowed that this is not the case when the SZ estimate is limited \nto the region covered by the halo emission. \nMore recent investigations, based \non numerical simulations confirm these calculations showing \nthat the effect of the SZ decrement on the Coma spectrum at 2.7 and 5 GHz \nis not sufficient to explain the observed \nspectral break (Donnert et al. 2010a).\n\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.4\\textwidth]{comalike_radio_secS_aB_R2p5_SZn_DATI.ps}\n\\includegraphics[width=0.4\\textwidth]{syn_brigh_SB_S1p6_NEWNEWDEISSShea.ps}}\n\\end{center}\n\\caption{{\\bf Left}: Spectrum of the Coma halo from hadronic models \nassuming a slope $s=$2.6 (solid), 2.85 (dotted), 3, 3.25 (dashed)\n(compilation of data points taken from Pizzo 2010). \nModels are normalised at 350 MHz. The steepening at higher frequencies \nin the models is due to the thermal SZ--decrement (see text). \n{\\bf Right}: The measured brightness (azimuthal averaged) profile of the \nComa halo (in arbitrary units) at 350 MHz, using observations in\nBrown \\& Rudnick (2011), is shown as a function of\ndistance in units of core radius, $r_c = 270$ kpc (red points).\nRed dashed lines mark the upper and lower envelope of the measured\nprofile taken along different directions (from Brown \\& Rudnick).\nThe new measurements are compared with previous data at 350 MHz\nfrom Govoni et al.(2001) (solid, blue and crosses) and \nwith data at 1.4 GHz from Deiss et al.(1997) (cyan points).\nThe brightness profile from hadronic models is shown assuming \n$B_0 \\sim 5 \\mu$G, $\\eta =0.5$ and $f=-1.6$ for $s=$2.6 \n(same line--type as in the left panel). The inset shows the \ncumulative flux profile calculated using the Brown \\& Rudnick azimuthal \naveraged profile, the vertical line shows the transition between \nthe regime of synchrotron--dominance (smaller distances) \nand IC--dominance (larger distances from cluster center). \nFinally the (magenta) dotted--lines show the expected shape of the \nbrightness profile assuming hadronic models with $f=-1.0$, with \nsynchrotron--dominance and IC--dominance (scalings derived\nin the paragraph below Eq.7); for display purposes the\nmagenta profiles are not normalised to the data.}\n\\label{fig:hadroComa}\n\\end{figure}\n\nThe halo shows a fairly regular morphology on a  \nscale $\\sim$1.5 Mpc with a radio brightness profile $I_{syn}(r)$ that is \nflatter than the X-ray (thermal) profile $I_X$ \n(Govoni et al. 2001, Brown \\& Rudnick 2011). \nThe global point-to-point power-law correlation between the 0.3 GHz \nradio brightness and the X-ray brightness, $I_{syn} \\propto I_X^{0.64}$, \nimplies that most of the radio emission is produced outside the \ncluster core, $r_c \\sim 270$ kpc. The brightness profile of radio\nhalos is an important observable quantity that provides \nconstraints on the spatial distribution of CRs and\nmagnetic fields in the cluster (Brunetti 2004, Pfrommert \\& En\\ss lin 2004,\nColafrancesco et al. 2005, Donnert et al 2010a).\nWe measure the radial profile of the Coma halo using the new WSRT observations\nat 350 MHz by Brown \\& Rudnick (2011) and obtain unprecedented constraints \non the brightness distribution of the halo up to 3-3.5 $r_c$ \ndistance from the cluster center.\nThe radial (azimuthally averaged) brightness profile is \nshown in Fig. \\ref{fig:hadroComa} (right\npanel) and is compared with other profiles available in the literature,\nGovoni et al. (2001) at 350 MHz, and Deiss et al (1997) at\n1.4 GHz. The new brightness profile provides constraints that are\nsignificantly better than those from previous observations and, most\nimportant, they extend to larger spatial scales. The brightness profile \nis much flatter than the one found by Deiss et al., that has been widely \nadopted in the literature  \n(Pfrommer \\& En\\ss lin 2004, Colafrancesco et al 2005, Donnert et al 2010a),\nimplying that the amount of energy in the form of \nCRs and magnetic field at distances $\\sim 2-3.5 \\, r_c$ from the\ncluster center is much larger than previously thought.\n\n\\noindent\nThe inset in the right panel of  Fig. \\ref{fig:hadroComa} shows the\ncumulative flux profile of the Coma halo (based on Brown \\& Rudnick 2011).\nAssuming a central value of the magnetic field $B_0 \\sim 5 \\mu$G \nand $\\eta=0.5$ (Bonafede et al. 2010), about 70-80\\% of the observed \nradio luminosity is produced in regions where $B^2 \\ll B_{cmb}^2$, \nwhile synchrotron dominance is expected only inside the cluster's core. \nIn the right panel of Fig. \\ref{fig:hadroComa} the (magenta) dotted-lines \nshow the case of a hadronic model of the radio \nhalo, with the shape of the expected synchrotron spectrum chosen \nto match the observed synchrotron spectral index, \n$\\alpha \\sim 1.3$ (Deiss et al. 1997), $\\beta = 0.75$, and a flat spatial \ndistribution of CR protons ($f=-1$). \nIn the relevant case $B^2 \\ll B_{cmb}^2$, the expected brightness \nis steeper than the observed one, implying that the energy density \nof CRs must increase with distance (at least up to \nabout $r \\sim 2.5-3 \\, r_c$) in orderd to fit the observed radio\nprofile. This has consequences on the total \nenergy budget of CRs in the external regions of the cluster and \non the expected $\\gamma$-ray emission.\n\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{brunetti_f2.ps}}\n\\end{center}\n\\caption{{\\bf Left}: $\\gamma$--ray spectrum due to $\\pi^0$--decay\nfrom hadronic models in Fig.~1.\nThe blue solid and dashed arrows mark the Ackermann et al.2010 \n1--10 GeV limit and that extrapolated by assuming no detection \nafter 3 years of FERMI observations, respectively.\nTo highlight the effect due to constraints from the radio brightness \nprofile on the predicted $\\gamma$--ray emission we also show\nexpectations for $\\delta=2.6$ assuming the old brightness profile\nfrom Deiss et al.~(1997) (cyan, long-dashed line) and assuming \n$f=0$ (red, long-dashed line).\n{\\bf Right}: The $\\gamma$--ray brightness profile from hadronic models\nin Fig.~1 compared with the Gaussians with 68\\% surface\nintensity containment radius $=$ 0.4 and 0.6 deg that are chosen \nin Ackermann et al. to derive the value of the Fermi-LAT upper limit\nused in our paper.}\n\\label{fig:gamma}\n\\end{figure}\n\nWe adopt a general modeling of the non-thermal CR proton distribution \nin the Coma radio halo, $\\epsilon_{CR} \\propto \\epsilon_{ICM}^{1+f}$ \nfor $r \\leq R_H$ ($R_H \\sim 3 \\, r_c$ being the halo radius) and \n$\\epsilon_{CR} \\propto \\epsilon_{ICM}$ for $r > R_H$. \nThis provides us with a conservative approach that minimizes the energy \nbudget of CR protons in the very external regions ($r > 3-3.5 \\, r_c$), \nwhere present data do \nnot provide reliable constraints. Under these conservative assumptions the \n$\\gamma$--ray brightness profile becomes steeper at projected distance \n$d \\geq R_H$ where the brightness becomes proportional to that of the \nthermal X--ray emission; from Eq. \\ref{jgamma} it is  \n$I_{\\gamma}(d)d^2 \\propto d^{-6 \\beta +3}$, implying that the \nmajority of the $\\gamma$--rays are produced within $\\sim R_H$.\n\nThe observed radio luminosity at 350 MHz and the brightness profile \nallow us to constrain $K_p$ (see Sect.~2) and consequently to estimate \nthe $\\gamma$--ray luminosity for a given model of the cluster magnetic field. \nIn order to break the degeneracy with the magnetic field properties, \nas a {\\it reference point} we start by assuming the magnetic field strength \nand its spatial distribution as derived from the analysis \nof RM (Bonafede et al. 2010). Bonafede et al. analyzed polarization \ndata for seven radio sources in the Coma cluster field observed with \nthe VLA at 3.6, 6 and 20 cm. They derived Faraday rotation measures \nwith kpc scale resolution and compared observations with simulations \nof random three-dimensional magnetic field models where the magnetic \nfield was scaled with the thermal cluster density in the \nform $B(r) \\propto B_0 n_{th}(r)^{\\eta}$. \nThey derive constraints for the magnetic field strength and \nprofile $(B_0,\\eta)$ with best values $B_0=4.7 \\, \\mu$G and $\\eta=0.5$, \nrespectively; $\\eta \\sim 0.5$ is also consistent with results \nfrom recent numerical MHD simulations of a variety of\nclusters (Bonafede et al. 2011b).\n\n\\label{sec:hydro}\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{giggia_BONAFEDE.ps}}\n\\end{center}\n\\caption{{\\bf Left}: A comparison between the allowed \nregion $(B_0, \\eta)$ from the analysis of RM by Bonafede \net al. 2010 (contours are reported at 1, 2, 3 $\\sigma$) and \nthat allowed, assuming a hadronic origin of the radio halo \nwith a slope of CR protons \n$s=2.6$, by the 2--$\\sigma$ upper limit from Ackermann \net al. (dashed), assuming no detection from 3 years \nof Fermi-LAT observations (extrapolated from Ackermann, solid line),\nand using the 2--$\\sigma$ upper limit from Han et al.(2012) (dotted).\nThe vertical dotted line marks the relevant \ncase $\\eta =0.5$, in this case a lower limit to $B_0$ obtained \nby assuming a reacceleration model (Sect.3) is reported (cyan).\n{\\bf Right}: The same as in Left panel but assuming $s=2.85$; \nfor simplicity here we do not report limits obtained by using \nHan et al. (2012).}\n\\label{fig:Bfield}\n\\end{figure}\n\nThe $\\gamma$--ray emission from $\\pi^0$ decay as calculated for \nthe same choice of parameters as in Fig. \\ref{fig:hadroComa} is \nshown in Fig. \\ref{fig:gamma} (left panel) together with the Fermi-LAT \nupper limits from the first 18 month of observations \n(Ackermann et al. 2010). We are using the Ackermann et al. limits \nderived by chosing a Gaussian (with 68\\% surface intensity containment radius\n$=$ 0.4 and 0.6 deg) spatial distributions of $\\gamma$-rays;\nthis is consistent with the spatial distribution expected from the model we \nadopted here (Fig. \\ref{fig:gamma}, right panel). \nThe comparison between the predictions of the model and the existing \nupper limits is therefore to be considered fair. \nIn Fig. \\ref{fig:gamma} (left panel, dashed) we also show the level of\nthe upper limit assuming no detection after 3 years of\nobservations;\\footnote{Ando \\& Nagai (2011)\nand Han et al. (2012) presented a analysis of 3 years of Fermi-LAT\nobservations of nearby galaxy clusters.\nThey both report a 2$\\sigma$ upper limit for the Coma cluster.}\nthat is obtained by simply scaling the Ackermann's upper limit\naccording to the square root of the exposure time.\nFor sake of clarity, \nin Fig.2 we also show the expected $\\gamma$-rays in the case we assume\na constant ratio of the CR protons and thermal energy densities in the cluster\n($f=0$, red long-dashed line), i.e. without considering the constraints \nfrom the radio brightness distribution of the Coma halo (see also\nSect.~4 and Fig.~4).\n\n\\noindent\nOur results show that a pure hadronic origin of the halo is inconsistent \nwith the Fermi-LAT upper limit, if the magnetic field properties are \nparametrized to satisfy the best fit obtained by Bonafede et al. (2010). \nRemarkably, the radio spectrum plotted in the left panel \nof Fig. \\ref{fig:hadroComa} shows that the spectrum of the halo is \nbest fitted assuming steeper CR spectra. \nThis happens because the effect of the\nnegative flux from the thermal SZ-decrement becomes more relevant  \nwhen the radio flux of the halo at high frequencies is smaller (see \nalso Pfrommer \\& En\\ss lin 2004). However, steeper CR spectra produce \nmore $\\gamma$-rays at low energy and a hadronic origin of the halo \nis readily ruled out by using the Fermi-LAT upper limits.\n\nThe choice of the value of $\\eta$ has a significant \nimpact on the expected $\\gamma$--ray emission from the cluster because\nit determines the amount of CR protons that must be assumed \nat different distances from the cluster center to match the observed \nsynchrotron profile.\nThere are some observational and theoretical arguments that suggest that\nthe ratio of turbulent and thermal pressure in clusters may increase\nin the cluster external regions (e.g. Vazza et al 2009, Iapichino\net al 2011, Churazov et al 2012) which may eventually imply that the \nratio of magnetic field and thermal energy densities \ncould increase with cluster distance (assuming that magnetic field\nand turbulent energy correlate).\nFor this reason we carry out more general calculations\nassuming $\\eta$ in the range 0--1.\nThis is shown in Fig. \\ref{fig:Bfield} \nusing present (Ackermann et al. 2010) Fermi-LAT upper limits (dashed line),\nand assuming no detection of $\\gamma$--rays from the Coma cluster \nafter 3 years of observations (solid line, obtained by simply scaling \nthe Ackermann et al. limit according to the square root of \nthe exposure time); lower limits obtained by using very recent \nFermi-LAT upper limits by Ando \\& Nagai (2011) and Han et al. (2012) are \nalso shown for the sake of completeness (dotted line).\nFor large values of $\\eta$ ($\\eta > 0.5$) the constraints are more stringent \nand only implausibly large values of $B_0$ are allowed. On the other hand \nfor small values of $\\eta$ fewer CR protons are required to match the radio \nluminosity and the brightness profile of the halo, leading to less stringent \nconstraints on the magnetic field strength.\nIn particular the minimum value of $B_0$ increases with $\\eta$,\nfrom about $B_0 \\geq 3 \\mu$G for $\\eta=0$ ($B_0 \\geq 5.5 \\mu$G for\n$\\delta =2.85$) to $B_0 \\geq 8.5 \\mu$G for $\\eta=0.5$\n($B_0 \\geq 11 \\mu$G for $\\delta =2.85$).\n\n\\noindent\nIn Fig. \\ref{fig:Bfield} we also show constraints on the\nvalues of $B_0 - \\eta$ as obtained by Bonafede et al.\nWe find disagreement between the region constrained from the\nanalysis of RM and that allowed by the lower limits on \n$B_0$ as obtained from the Fermi-LAT limits. \nIn the relevant case \n$\\epsilon_B \\propto \\epsilon_{ICM}$ ($\\eta =0.5$), \nwe find that the minimum\nmagnetic field energy density allowed by Fermi-LAT limits \nis 5-10 times larger than that of the {\\it reference} magnetic field\nenergy density, estimated from the analysis of RM, while\nfor $\\eta=0.2$ (still consistent with RM within 3$\\sigma$) the\ndiscrepancy is reduced to a factor $\\sim 2$; the discrepancy\nbecomes severe for $\\eta > 0.5$.\nIn general it is worth stressing here that our constraints using\nupper limits from 3 years of Fermi-LAT observations select  \nhigh magnetic fields, in which case even a moderate improvement\nof the $\\gamma$--ray limits implies a substantial increase of the \nminimum value of the magnetic field that is allowed (for\nexample, in the case $\\eta \\sim 0$ and $B_0 > B_{cmb}$\nit is $B_{\\rm min} \\propto\n{J_{\\gamma}}_{(ul)}^{1\/(1-\\alpha) \\sim -3}$, from Eqs.5--6).\nConsequently deeper upper limits from Fermi-LAT in the next years,\nor even a detection of the Coma cluster at a level 2-3 times below\nAckermann et al.~ limits will considerably increase the difficulties\nin the context of a {\\it pure} hadronic model for the radio halo.\n\n\\label{sec:hydro}\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{brunetti_f4_i.ps}}\n\\end{center}\n\\caption{The $\\gamma$-ray luminosity of the Coma cluster assuming hadronic \nmodels for the radio halo (assuming $B_0=4.7 \\mu$G, $\\eta=0.5$ and $s=2.6$) \nis shown as a function of $f$ and normalised to \nthat expected for $f=0$ (namely assuming a scaling \n$\\epsilon_{CR} \\propto \\epsilon_{ICM}$ as in \nJeltema \\& Profumo 2011). The inset shows the comparison between \nthe observed brightness profile of the Coma halo and expectations based \non hadronic models with $f=$-1.6, -1.0, 0 (from flatter to steeper). \nThe horizontal error-bar gives the values of $f$ as constrained \n(3--$\\sigma$) from the observed brightness profile (we consider the \nslope of the profile in the distance range 1--2.5 $r_c$). \nBlue arrows give the Fermi-LAT \nupper limits (solid$=$ Ackermann et al.(2010), \ndotted$=$ extrapolation of Ackermann's \nlimit at 3 years observation) and their implications for the parameter $f$.}\n\\label{fig:gammaprofile}\n\\end{figure}\n\nUsing the Ackermann et al.~ upper limits, in the relevant case \n$\\eta = 0.5$ Jeltema \\& Profumo (2011) \ninfer a value of the minimum magnetic field in the \ncluster center, $B_0 > 3.9 \\mu$G, a result that is \nstill consistent with the values \nconstrained from RM. The difference between our work and \nJeltema \\& Profumo (2011) essentially stems from the adopted \nspatial distribution of CRs in the cluster. \nJeltema \\& Profumo (2011) assumed $\\epsilon_{CRp} \\propto \\epsilon_{ICM}$  \nwhile in our calculations we used the spatial distribution of CR protons \nneeded to reproduce \nthe observed synchrotron brightness profile of the radio \nhalo. In order to highlight this point, in Fig. \\ref{fig:gammaprofile} \nwe show the ratio of the $\\gamma$-ray luminosity that is obtained assuming \na scaling $\\epsilon_{CRp} \\propto \\epsilon_{ICM}^{1+f}$ and that assuming \na linear scaling ($f=0$) between the energy densities of CRs and ICM. \nIn Fig. \\ref{fig:gammaprofile} we also report the constraints on $f$ \nderived from the synchrotron brightness profile of the halo and the limits \nimplied by the Fermi-LAT upper limits after 18 months \nof observations (Ackermann et al. 2010) and by assuming no \ndetection of the Coma cluster after $\\sim 3$ years of observations.\nWe confirm that assuming $f=0$ (as in Jeltema \\& Profumo) the value of \nthe central magnetic field constrained from the Ackermann \net al. $\\gamma$-ray upper limits is still consistent with the value \ninferred from RM ($B_0 = 4.7 \\mu$G, with $\\eta =0.5$, \nBonafede et al. 2010), however in this case the synchrotron brightness \nprofile of the halo is predicted to be considerably steeper, \ninconsistent with the observed one (Fig. \\ref{fig:gammaprofile}, inset).\n\nFinally we stress here that \nthe discussion above considers a {\\it reference} value for \nthe magnetic field in the Coma cluster based on an analysis of RMs.\nThese results are obtained by assuming that the RM originates entirely in \nthe ICM (Bonafede et al. 2010).\nIf a contribution local to the radio galaxies is present, as found \nin other cases (e.g. \nRudnick \\& Blundell 2003, Guidetti et al. 2011), the values of the magnetic \nfield in the ICM, as inferred from the observed RM, would be biased to higher\nvalues (see Rudnick \\& Blundell 2003) and the {\\it reference} value \nshould be considered as an upper limit to the actual cluster field.\nThis case would make our conclusions even stronger.\n\n\n\\section{Turbulent acceleration of secondary electrons \nand positrons}\n\\label{sec:reacc}\n\nModels based on turbulent reacceleration \nof seed electrons (Schlickeiser et al. 1987,\nBrunetti et al. 2001, Petrosian 2001, Fujita et al. 2003, Cassano\n\\& Brunetti 2005) rely upon the observed low efficiency of particle \nacceleration \nin the ICM, as suggested for instance by the \n$\\geq 1$ GHz cutoff in the radio spectrum of \nComa (Schlickeiser et al. 1987, Thierbach et al. 2003), and the fact\nthat radio halos are not common and are correlated with clusters' dynamics \n(see discussion in Brunetti et al. 2009). \nMore recently, (indirect) \nevidence in favor of these models has been \nprovided by the observation of radio halos with very steep \nspectra (Brunetti et al. 2008), interpreted as halos in which \nthe radio spectrum starts steepening at frequencies smaller than \nthe observing frequency, thereby favoring poorly efficient \nacceleration mechanisms.\n\nAcceleration of electrons from the thermal pool to relativistic \nenergies by MHD turbulence in the ICM faces serious \nenergetic problems (e.g. Petrosian \\& East 2008) and usually leads \nto large heating of the ICM (Blasi 2000, Petrosian \\& East 2008). \nConsequently, turbulent acceleration models start from the assumption \nof a pre-existing population of relativistic particles that provides \nthe seeds to ``reaccelerate'' during mergers. \nThe combination of the ``unknown'' distribution of the seed particles \nin the ICM with the poorly known properties of turbulence in galaxy \nclusters reduces the predictive power of this scenario (see \nhowever Cassano et al.~2010b and references therein for predictions \non the population properties of radio halos).\n\n\\noindent\nIn the context of these models an elegant possibility is based on \nclusters of galaxies as storage rooms of CRs  \n(V\\\"olk et al. 1996, Berezinsky et al. 1997 \nand Ensslin et al. 1997) and on the fact that \nthe secondary electrons and positrons produced via inelastic collisions \nbetween CR protons and thermal protons in the ICM may be reaccelerated by\nMHD turbulence during cluster mergers (Brunetti \\& Blasi 2005, \nBrunetti \\& Lazarian 2011). \nIn this sense, this model is of a hybrid nature, but its requirements \nin terms of CR energy density in clusters are very meek. \nIn this scenario, {\\it off-state} (purely hadronic) radio halos in \nalmost all galaxy clusters should have a radio luminosity \n$\\sim$10 times smaller than that of classical {\\it on-state} (turbulent) \ngiant radio halos (Brunetti \\& Lazarian 2011). \nThe recent detection of diffuse emission in quiescent clusters from \nstacked analysis of the SUMSS survey may provide the first, possible \nsupport for this scenario (Brown et al. 2011).\n\n\\noindent\nIn these hybrid models $\\gamma$-rays are produced as a result of \nthe generation \nand decay of neutral pions and of IC emission from high \nenergy ($\\sim$ TeV) secondary electrons. This realization allows \nus to gather complementary tests for this scenario based on \n$\\gamma$--ray observations. \nThe level of $\\gamma$-ray emission expected in this scenario is \nhowever significantly smaller than that in the pure hadronic case,\nsince the interaction between CR protons and their secondaries \nwith the MHD turbulence enhances the ratio of radio (synchrotron) \nand $\\gamma$-ray emission in connection with cluster \nmergers (Brunetti \\& Lazarian 2011). Therefore it is easier to \nachieve consistency between the Fermi-LAT upper limits, radio observations \nof the spectrum, morphology and Faraday RM (see Brunetti 2011 for \na recent review). \n\nIn turbulent acceleration models the spectra and morphologies of \nradio halos depend on the combination of several physical quantities. \nTo provide a simple estimate of \nthe emissivity from a population \nof reaccelerated electrons we note that the turbulent \nenergy flux that is damped by the coupling with relativistic \nelectrons is eventually converted to synchrotron \nand IC radiation (Cassano et al.~2007): the bolometric synchrotron \nemissivity can be written as:\n\n\\begin{equation}\nJ_{syn} \\propto {{F_{\\epsilon_t} \\Gamma_{e^{\\pm}}\/\\Gamma_{th}}\\over\n{1 + (B_{cmb}\/B)^2}},\n\\label{JsynT1}\n\\end{equation}\nwhere $F_{\\epsilon_t}$ is the energy injection rate of turbulence \n(per unit volume) and \n$\\Gamma_{e^{\\pm}}\/\\Gamma_{th} \n\\propto (\\epsilon_{e^{\\pm}}\/\\epsilon_{ICM}) \\sqrt{T}$ is the ratio of \nthe damping rate of the turbulence by relativistic electrons \nand by thermal ICM (the second channel of \ndamping goes into thermal heating). \nIf the emitting electrons are secondary particles reaccelerated \nby the MHD turbulence and assuming that the \nreacceleration time, $\\tau_{acc}$, is smaller than the cooling time, \n$\\tau_{loss}$, of electrons with energy $E < E_{\\nu}$ ($E_{\\nu}$ being \nthe energy of the electrons emitting at the observing frequency), \nwe have $\\epsilon_{e^{\\pm}} \\propto\n\\epsilon_{CRp} n_{th}$ and Eq. \\ref{JsynT1} reads :\n\n\\begin{equation}\nJ_{syn}(\\nu) \\propto\nn_{th}^{2+f} {{ B^2 }\\over\n{B^2 + B_{cmb}^2}} T^{3\/2+f} \\Theta (r,\\alpha),\n\\label{JsynT2}\n\\end{equation}\nwhere we made the simplified assumption \n$F_{\\epsilon_t} \\propto n_{th} T$ (see Cassano \net al.~2007 for details). $\\Theta$ is a fudge function to \naccount for several effects. One effect is that in \nthe internal, denser, region of the cluster Coulomb losses can play \na role making the reacceleration of secondaries with initial energies \n$E \\sim 100-300$ MeV more difficult. Another effect is that the damping \nrate of turbulence also depends on the ``spectral shape'' of \nrelativistic (reaccelerated) particles (Brunetti \\& Lazarian 2007) that \nis sensitive to the combination of physical parameters in the ICM. \nBased on previous calculations (Brunetti \\& Blasi 2005, \nBrunetti \\& Lazarian 2011) we expect that all these effects combine to\nmake the brightness profile flatter.\n\n\\noindent\nEq.~\\ref{JsynT2} essentially implies that the brightness profile \nfrom reacceleration of secondary particles is very similar to that \nbased on pure secondary  models (with $\\alpha \\sim 1$, Eq.~\\ref{JsynS}), \nand that two regimes exist (considering $\\Theta \\sim$ constant, to\nevaluate approximate scalings): \n(i) $I_{syn} \\propto (1 + x^2)^{-3 \\beta (2+f)\/2 +1\/2}$, \nfor $B^2(x) \\gg B_{cmb}^2$, and \n(ii) $I_{syn} \\propto (1 + x^2)^{-3 \\beta (2(1+\\eta)+f)\/2 +1\/2}$, \nfor $B^2(x) \\ll B_{cmb}^2$.\n\nPredictions from reacceleration of secondary particles differ radically\nfrom those of pure \nhadronic models at high observing frequencies, where \n$\\tau_{acc} \\sim \\tau_{loss}$ (or $\\tau_{acc} > \\tau_{loss}$). \nUnder these conditions the synchrotron spectrum steepens at the \nfrequency (Brunetti et al. 2001):\n\n\\begin{equation}\n\\nu_s \\propto {{B \\tau_{acc}^{-2}}\\over{(B^2 + B_{cmb}^2)^2}},\n\\label{nus}\n\\end{equation}\nIn the IC-dominated regime, $B < B_{cmb}$, \n$\\nu_s \\propto B \\tau_{acc}^{-2}$. Under the simple assumption that \nthe magnetic field strength depends only on distance from the cluster \ncenter ($B(r)$, {\\it homogeneus models}), the synchrotron spectrum steepens \nwith distance (due to the fact that the emission is produced \nin a magnetic field that decreases with distance, and assuming an\napproximately constant acceleration time--scale) and consequently radio \nhalos become smaller at higher observing frequencies. \nWe note indeed that observational claims exist for a decrease of \nthe halo size with frequency and a steepening of the halo spectrum \nwith projected distance (Giovannini et al. 1993, Deiss et al. 1997, see\nalso Fig.~1), \nalthough future observations are highly desirable to confirm these findings.\n\nIn this section we check that the predictions of the model based \non turbulent reacceleration of secondary electrons is consistent \nwith all existing observations and we investigate the implications \nof the model for the detectability of clusters in $\\gamma$-rays. \nThe turbulence and particle acceleration process are \nmodelled following  \nBrunetti \\& Lazarian (2011)\\footnote{We refer the readers to Brunetti \\&\nLazarian 2011 for the formalism on turbulence and stochastic particle\nre-acceleration. It allows us to model self-consistently the effect of\nturbulent\nre-acceleration on the spectra of CR protons, secondaries and \nof the non-thermal emission,\ngoing beyond the simplified scalings in Eqs.\\ref{JsynT1}--\\ref{nus}}, \nwhere compressible turbulence is injected \nat large scales, $\\sim$100-300 kpc, and decays onto smaller scales \nwhere it behaves as MHD turbulence. \nThe cascading of fast modes is \ndissipated in the collisionless regime mainly due to \ntransit--time--damping (TTD) with thermal and relativistic particles \nin the ICM. This process channels a fraction of the turbulent energy \nflux into the reacceleration of CR protons and secondary electrons. \nWe carry out a homogeneous modeling of turbulence and turbulent \nreacceleration in the Coma halo, namely we assume a constant ratio \n$\\epsilon_{tur}\/\\epsilon_{ICM}$, $\\epsilon_{tur}$ being the turbulent \nenergy density, and assume the same scalings between thermal and \n(the initial)\nnon-thermal properties as in Sect.~2.2. As a reference value we assume \nthat turbulent motions on scales $l \\leq 30$ kpc contribute to about \n5\\% of the ICM energy (implying a total turbulent budget on the largest \nscales $\\epsilon_{tur}\/\\epsilon_{ICM} \\sim 0.2$)\\footnote{This reference value\nis needed\nto reproduce radio halo properties, see Brunetti \\& Lazarian 2007, 2011}.\n\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{brunetti_f5.ps}}\n\\end{center}\n\\caption{{\\bf Left}: Brightness (azimuthal averaged) profile of the Coma\nhalo (as in Fig.~1) compared with expectations based on reacceleration models.\nAll models assume $\\epsilon_{tur}\/\\epsilon_{ICM}\\sim 0.05$ on scales \n$l < 30$ kpc and $s=$2.6. Calculations are carried out assuming\n(i) $B_0 =5 \\mu$G and $\\eta=0.5$  (solid lines) with $f=-1.15$\nand $\\Delta \\tau_r = 0.75$ Gyr (red lines) and $f=-1.3$ and\n$\\Delta \\tau_r = 0.5$ Gyr (blue lines) ($\\Delta \\tau_r$ is the period\nof reacceleration), and (ii) $B_0 =2 \\mu$G and $\\eta=0.5$ (dashed lines)\nwith $f=-1.6$ and $\\Delta \\tau_r = 0.75$ Gyr (red lines) and $f=-1.75$\nand $\\Delta \\tau_r = 0.5$ Gyr (blue lines).\n{\\bf Right}: Spectrum of the Coma halo from reacceleration models as\nin Left panel. All models are normalised at the radio halo luminosity\nat 350 MHz.}\n\\label{fig:reacc}\n\\end{figure}\n\nIn Figure \\ref{fig:reacc} we show the expected brightness profile \n(left panel) and synchrotron spectrum (right panel) of the Coma halo \nin the reference case $\\eta =0.5$ and $B_0 \\sim 5\\mu$G. \nThe spatial distribution of CR protons must be fairly flat, \n$f \\sim -1.1$ to $-1.35$ (up to $r \\sim 3 r_c$), so as to reproduce \nthe observed radio profile.\nThis is similar to the case of pure hadronic models, although less\nextreme because in the case \nof turbulent reacceleration the synchrotron profiles are slightly \nflatter (a spatially constant distribution of the energy density \nof CR protons, $f \\sim -1$, is still consistent with observations) \nfor the reasons discussed above\n(see Eq.~\\ref{JsynT2} and the paragraph below it).\nContrary to the case of pure\nhadronic models, the steepening of the synchrotron \nspectrum at higher frequencies is reproduced very well\nin the case of turbulent reacceleration \nmodels allowing a good representation of the data over almost 2 decades \nin frequency range; more (less) turbulence in \nthe ICM would produce a steepening at higher (lower)\nfrequencies. \n\nDuring a re-acceleration phase \nthe total energy budget of CR protons within the halo emitting \nvolume ($r \\sim 3 \\,r_c$) is $\\approx$4 \\% of that of the ICM. \nThis is almost 1 order of magnitude smaller than that in the case \nof pure hadronic models and implies a smaller $\\gamma$--ray emission, \nwell consistent with Fermi-LAT limits, as illustrated \nin Fig. \\ref{fig:reaccgamma}.\n\nMore $\\gamma$--rays are produced if the magnetic field in the halo \nvolume is smaller. This allows us to constrain the minimum value\nof the magnetic field that is required to have a $\\gamma$--ray flux\nconsistent with Fermi-LAT limits, under the assumption that the halo\nis generated by reacceleration of secondary particles.\nIn the relevant case $\\eta=0.5$ we derive a minimum value \n$B_0 \\geq 2 \\mu$G (Fig.~ \\ref{fig:reaccgamma}) that would imply a magnetic \nfield energy density in the ICM $\\sim 5$ times smaller than the best fit \nobtained from the analysis of Faraday RM by Bonafede et al.(2010).\nFor the sake of completeness, in Fig. \\ref{fig:reacc} we show \nthe expected synchrotron spectrum and the brightness profile of the \nComa halo obtained by assuming $B_0 = 2 \\mu$G and $\\eta =0.5$. \nIn this case the synchrotron emission is always produced under \nconditions $B_{IC}^2 \\gg B^2$ and a significantly inverted \nspatial distribution of CR protons, $f \\sim -1.5$ to $-1.8$\nis needed to match the observed brightness profile, with the\nconsequence of a large energy budget in the form of CR protons.\n\n\\label{sec:hydro}\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{brunetti_f6.ps}}\n\\end{center}\n\\caption{$\\gamma$--ray spectrum due to $\\pi^0$--decay as expected \nfrom reacceleration (hybrid) models. \nLines are labelled as in Fig.~\\ref{fig:reacc}.}\n\\label{fig:reaccgamma}\n\\end{figure}\n\n\\section{Discussion and conclusions}\n\\label{sec:disc}\n\nWe presented a combined analysis of the spectrum and morphology of \nthe giant radio halo in the Coma cluster, the available measurements \nof cluster magnetic fields from RM and the upper limits on the \n$\\gamma$-ray emission from this cluster as recently obtained by the Fermi-LAT \ntelescope (Ackermann et al. 2010)\\footnote{see also \nAndo \\& Nagai (2012) and Han et al. (2012)}, in the context of \nmodels based on secondary electrons: we concentrated on a pure secondary \nelectron model and on a scenario where secondaries are reaccelerated by MHD\nturbulence during mergers.\n\n\\subsection{Hadronic models}\n\nThe pure secondary electron model is based on the concept of CR effective \nconfinement in the ICM (V\\\"olk et al. 1996, \nBerezinsky et al. 1997) and consists of explaining the giant radio halo \nemission as the result of synchrotron emission of secondary \nelectrons (and positrons) from inelastic collisions of cosmic ray \nprotons with gas in the ICM. These collisions result in the production \nand decay of charged and neutral pions. The former lead to production \nof secondary electrons, while the latter provide a channel of continuous \nproduction of gamma radiation. \nThis model has been widely implemented also in cosmological simulations.\nThese simulations, that include, to some extent, CR physics\nand the acceleration of CRs at cosmological shocks provided a picture\nof the radio to $\\gamma$--ray properties of galaxy clusters,\nunder the assumption that the emitting particles (electrons) are\ngenerated through pp collisions in the ICM or accelerated\nat shocks (e.g., Pfrommer et al. 2008 and references therein). \nThe first simulations of this kind\npredicted that clusters would be potentially\ndetectable in $\\gamma$-rays with present\nday $\\gamma$--ray telescopes (Miniati 2003, Pfrommer 2008).\nThe most important assumption in these simulations is in the\nefficiency of particle acceleration at weak shocks that is poorly known\n(see Gabici \\& Blasi 2003, 2004 and Kang et al.~2007 for a critical view).\nMore recently, numerical\nsimulations of large-scale structure formation made an attempt to reconcile\ntheir results with the lack of detection of galaxy clusters\nin the $\\gamma$-ray band (Pinzke \\& Pfrommer 2010,\nsee also Aleksic et al. 2010,11). Although these simulations provide\nexpectations (still) consistent with the Fermi-LAT upper limits,\nit is worth mentioning that\nthey do not allow to reproduce the observed properties of radio halos if we\nassume that halos originate via secondary emission,\nin particular their (very broad) spatial extent (Donnert et al. 2010a,b;\nBrown \\& Rudnick 2011).\nIn this respect,\nfor the sake of completeness, in Fig. \\ref{fig:numerical} we show a comparison\nbetween the brightness profile of the Coma radio halo and expectations\nbased on CRs distributions from\nPfrommer et al. (2008) and from Pinzke \\& Pfrommer (2010) simulations.\nThis can be made by using the semi-analytical prescription of\nthe spatial distribution of CR in galaxy clusters,\nas derived from high resolution numerical simulations\nof clusters (Pinzke \\& Pfrommer 2010), and using parameters\nof the Coma cluster to calculate the production rate of secondaries and the\nsynchrotron emissivity. \nAccording to these simulations the ratio of\nthe CR and thermal pressure in typical non-CC clusters is quasi constant\nup to a distance $R\/R_{vir} \\sim 0.2$ and increases by a factor 2-3 up\nto the virial radius (see Fig.14 in Pinzke \\& Pfrommer 2010); such\nan increase is however\nmainly driven by the temperature decrement in the ICM at large distances.\n\n\\noindent\nIn reality, the microphysics of the ICM and of CRs in galaxy clusters\nis very complicated and unfortunately\nbeyond the capabilities of present simulations.\nFor this reason in our paper we have carried out a analysis that\ndoes not depend on the way CR protons are generated\nin the ICM and on the complex processes of CR transport\nand reacceleration that can take place in the cluster volume.\nRather than modeling these complex processes we indeed derived\nconstraints on the spectral and spatial distributions of CR protons \ndirectly from the observed spectrum and morphology of the Coma halo,\nusing parameters of the ICM from the X-ray observations (\\S 2.2).\n\n\\noindent\nWe derived the azimuthally averaged brightness profile of the Coma halo\nand used it in order to obtain constraints on the spatial profile of \nthe magnetic field \nstrength and of CRs density as functions of the distance from the \ncluster center.\nOur profile is obtained using the deep WSRT observations from Brown \\&\nRudnick (2011) and allows us to obtain unprecedented constraints\non the non-thermal cluster properties on 3--3.5 $r_c$ scales.\nWe find that fitting the volume averaged radio spectrum of the Coma halo \nand its azimuthal brightness distribution requires a rather steep spectrum \nof CR protons, $s\\sim 2.6$, in the ICM with a \nspatial distribution much flatter than the spatial distribution \nof the thermal gas. This leads \nto an exceedingly large energy content in the form of CRs confined \nin the ICM and correspondingly large $\\gamma$-ray emission, \nexceeding the Fermi-LAT upper limits. Moreover the spectral steepening \nobserved at high frequency can only marginally be explained for \nthe cases with steeper slope ($s \\geq 3$) and the combined effect of \nthe SZ decrement. This case is however the one that violates the \nFermi-LAT upper limits more clearly. \n\n\\noindent\nIf we neglect the spectral steepening in our analysis, the model\ncan potentially explain radio observations. However the Fermi-LAT \n$\\gamma$--ray limits set a lower limit to the strength of the\ncluster magnetic field that is in disagreement with a recent \nanalysis of Faraday\nrotation measures (Bonafede et al. 2010);\nthe more so for steeper spectra of the parent CR protons.\n\n\\noindent\n{\\it Pure} secondary electron models of the Coma halo\nare thus disfavoured when all available observational data\nare considered.\nIn this respect any further improvement of upper limits over the next\nseveral years, or even a $\\gamma$-ray detection of the Coma cluster \n(i.e. with flux 2-3 times smaller the Ackermann et al. limits)\nwill conclusively establish the incompatibility of a hadronic \norigin of the Coma radio halo with the radio (including RM) and $\\gamma$-ray \nobservations.\n\n\\begin{figure}\n\\begin{center}\n{\n\\includegraphics[width=0.79\\textwidth]{syn_brigh_SB_S1p6_NEWNEW_PP.ps}}\n\\end{center}\n\\caption{Azimuthal averaged brightness profile of the Coma halo \n(as in Fig.~1) compared \nwith expectations based on numerical simulations that include the \nacceleration of CR protons and the generation of secondary electrons \nin the ICM. Points show the expected synchrotron profile from secondary \nelectrons in the massive cluster gs72 from Pfrommer et al. 2008 (circles \nmark the case of radiative simulations). The solid line (black) show \nthe expectations based on the semi-analytic\nmodel of CR protons in Coma based on numerical \nsimulations (Pinzke \\& Pfrommer 2010).\nA magnetic field profile $B(r)^2 \\propto \\epsilon_{ICM}$ and\n$B_0 = 5 \\mu$G are assumed.}\n\\label{fig:numerical} \n\\end{figure}\n\n\\subsection{A comment on the most relevant assumptions}\n\n\\noindent\nA discussion of the assumptions adopted in our analysis is in order. \nConclusions derived above are based on canonical assumptions used to\nmodel non-thermal emission in galaxy clusters.\nThe most notable assumption is that the CR spectrum in the cluster volume\nis a power law in momentum, \n$N(p) \\propto p^{-s}$, and that the spectrum of secondary particles\nemitting in the radio band can be calculated under stationary\nconditions. We constrained the value of the slope $s$ from \nthe spectrum of the radio halo in the frequency \nrange $\\sim$0.1-1 GHz (Fig. \\ref{fig:hadroComa}) and the $\\gamma$-ray \nemission of the cluster is calculated by using such values of $s$. \nIn principle, the shape of the spectrum of CR protons \nmight also change with location,\nmoving away from cluster core toward the periphery, although there is \nno strong \nindication that this may be happening from the total radio spectrum of the\nComa halo\\footnote{Indeed a mixture of constributions from different power-law \nspectra results in a ``concave'' shape of the total synchrotron \nspectrum.}.\nIn this case the expected $\\gamma$--ray emission is reduced\nif the spectrum of CR protons is flatter in the regions where the majority of \n$\\gamma$--rays are produced, at distances 2--3.5 $r_c$ from the center. \nThis however would result in a halo spectrum that is flatter in the\nexternal regions, in contrast with present \nobservations that suggest a ``radial spectral steepening''\n(Giovannini et al 1993, Deiss et al 1997, this is also evident from\nthe comparison between the synchrotron brightness distributions of\nthe halo at 350 and 1400 MHz, Fig. 1).\n\n\\noindent\nThe situation may change if the spectrum of CRs becomes harder at low enough \nenergies. In this case one may expect that the flux of low energy \n$\\gamma$-rays \n(from decays of neutral pions) is also reduced allowing for some room\nfor pure hadronic models to accomodate current observational constraints.\nPresent radio data limit the \npossibility of a flattening in the radio spectrum to frequencies \n$\\nu \\leq 100$ MHz, leading to a limit to the energy where a possible \nbreak occurs in the spectrum of the primary CR protons, \n$E_b \\leq 10-20$ GeV. Under these conditions we note that\na substantial reduction of the $\\gamma$-ray luminosity due to $\\pi^0$--decay \nin the 1-10 GeV band, necessary to make hadronic models still consistent \nwith --at least-- the limits derived from the first 18 month of FERMI-LAT data\n(Ackermann et al. 2010) (Fig. \\ref{fig:gamma}), would require \na prominent CR spectral break, by $\\Delta s \\ge 0.5$.\n\nThe two points that disfavour pure hadronic models for the radio halo\nare the steepening of the halo spectrum observed at higher frequencies and \nthe inconsistency between the properties of the cluster magnetic field \nconstrained by Faraday RM and by Fermi-LAT limits under the assumption \nof a hadronic origin of the halo.\nThe latter point however might simply indicate that RM and synchrotron\nemission trace different magnetic fields. \nAs already mentioned in Sect. 2.3 the simplest way to have RM and \nsynchrotron emission sample different fields is to assume that some of \nthe RM come from regions adjacent to and influenced by the radio sources, \nin this case however the magnetic field in the ICM as inferred from the \nanalysis of RM would likely be biased to higher values of the magnetic \nfield (Rudnick \\& Blundell 2003) thus making the inconsistency between \nmagnetic fields even stronger. \nOn the other hand, \nif one assumes that RM originates entirely in the ICM, RM and synchrotron \nemission may sample different fields in the case of a highly inhomogeneous \nfields, because the synchrotron emissivity depends non-linearly on the \nmagnetic field and its fluctuations.\nPositive fluctuations however increase also the radiative losses of particles\n(assuming they vary on time scale sufficiently long,\n$> 10^7-10^8$yrs\\footnote{The minimum RM scale of about 2 kpc for Coma\nradio galaxies derived by Bonafede et al 2010, and the typical fractional\npolarization of about 10\\%, imply that any ICM fluctuation on smaller\nscales, that could be very intermittent, must be weak.}) and this is\nexpected to partially quench the expected boosting of the synchrotron\nemissivity of electrons generated in regions with medium\/high fields.\nUnder (at least quasi--) stationary conditions the emissivity reads:\n\n\\begin{equation}\n<J_{syn}>_{Volume} = \\propto\n{\\hat B}^{1+\\alpha} {{ ( 1 + (\\delta B\/{\\hat B})^2)^{(1+\\alpha)\/2} }\n\\over{ {\\hat B}^2 + \\delta B^2 + B^2_{cmb} }} \\, ,\n\\label{jsyn2}\n\\end{equation}\n\n\\noindent\nwhere we introduced a magnetic field made of a large scale \ncomponent ${\\hat B}$ and a turbulent component \n$\\delta B$, such that $<\\bf{ {\\hat B}} + \\bf{\\delta B}>_{Volume}=\n{\\bf {\\hat B}}$ and \n$<\\bf{ {\\hat B}} + \\bf{\\delta B}>^2_{Volume}= {\\hat B}^2 + \\delta B^2$.\nUnder these assumptions we estimate that the ratio of $\\gamma$--ray and \nradio cluster luminosities decreases by (only) a factor \n$\\sim 1.7$, compared to results using the formalism in Sect.~2.1, if we \nassume $\\delta B^2 \\sim {\\hat B}^2$, \nwith $\\delta B^2$ anchored to the best fit value from RM studies\n(normalisation and spatial profile). \n\nOn the other hand, RMs \nsuggest that $\\delta B\\gg B$, in which case Eq.\\ref{jsyn2} becomes\nequivalent to Eq.\\ref{jsyn} in Sect.~2.1\\footnote{\nIn the analysis by Bonafede et al. the magnetic field is \nparameterised with a power spectrum $B^2 = 8\\pi \\int P_B(k) dk$\nbetween a maximum and minimum scale and its properties are constrained \nby comparing \nobservations with simulated Faraday Rotation maps derived for different \nparameters}.\nIn this sense the question of whether Faraday RM and synchrotron emission \nmeasure the same magnetic field only lies in the capability of RM studies \nto provide a good description of the ICM magnetic field.\nFuture radio telescopes, including LOFAR and SKA, will greatly improve the\nsensitivity to RMs allowing to detect and use many tens of background \nradio sources per clusters to sample magnetic fields along \nmany lines of sight, thus removing potential biases in present studies.\n\n\\noindent\nFinally, for the sake of completeness we mention that in principle, spatial \ndiffusion of particles in inhomogeneous fields may also affect the value \nof the ratio synchrotron\/$\\gamma$-ray luminosity in hadronic models, \nif the diffusion \ntime necessary to cover the spatial scales on which field inhomogeneities \noccur is smaller \nthan both the life-time of particles and the time-scale of magnetic \nfield (local) variations. This however depends on the details of the \nmagnetic field and diffusion model.\n\n\n\\subsection{Beyond the pure hadronic model: turbulent reacceleration \nof secondaries}\n\nIn \\S \\ref{sec:reacc} we went beyond the pure hadronic model and\ndiscussed the case of the reacceleration model were secondary particles\nare reaccelerated by MHD turbulence.\nWe find that this model allows to obtain a good\ndescription of the radio spectrum and morphology of the Coma \ncluster and at the same time they may easily be compatible with \nthe Fermi-LAT upper limits on the $\\gamma$-ray emission from this cluster \nand with the measured magnetic field strength as inferred from RMs. \n\n\\noindent\nIn the simplified assumption that both the ratios of turbulent and thermal \nenergy density, $\\epsilon_{tur}\/\\epsilon_{ICM}$, and of magnetic field\nand thermal energy density , $\\epsilon_{B}\/\\epsilon_{ICM}$, are \nconstant in the radio emitting volume, the brightness profile of the \nComa halo leads us to infer \nthat the spatial distribution of CRs in the cluster should be very broad, \nflat (or slightly increasing in the external regions) on the halo size-scale.\nThis is because the particular reacceleration model adopted in this \npaper faces drawbacks that are in part similar to those of a pure hadronic\nmodel for the origin of the halo. \nThis is simply because the seed electrons \nfor reacceleration are generated by proton-proton collisions in the ICM.\nA flat spatial distribution of CRs is a fairly strong requirement, although\npossible support for a rather flat distribution of CRs,\nsignificantly broader than that of the thermal ICM,\ncomes from very recent numerical cosmological simulations\nthat include CRs accelerated at shocks (Vazza et al.~2012).\nPresent radio data do not constrain the energy density of CR protons\non scales larger than the halo size-scale. However if we speculate that \nthe flat spatial distribution of CR protons extends on larger scales the \nresulting energy budget of CRs in the cluster would be significantly\nlarger than that required by our modeling.\n\nIn general several, effects may contribute to mitigate this \nsituation. \nFirst, numerical simulations show that the turbulent \nenergy density (and the ratio $\\epsilon_{tur}\/\\epsilon_{ICM}$) increases \noutside the cores of simulated clusters (Vazza et al. 2009, 11; \nIapichino \\& Niemeyer 2008, Iapichino et al. 2011), implying a synchrotron \nbrightness profile potentially flatter than that calculated under \nthe assumption of a constant ratio $\\epsilon_{tur}\/\\epsilon_{ICM}$. \nSecond, we limit our analysis to the case $B^2 \\propto \\epsilon_{ICM}$\n($\\eta =0.5$), that gives the best fit to Faraday RM. \nOn the other hand for smaller values of $\\eta$ the spatial distribution \nof CR protons that is required by the model to match the observed synchrotron \nbrightness profile of the halo is significantly steeper with radius, \nalthough still flatter than the distribution of the thermal energy density\nof the cluster.\nFinally we note that the situation is greatly mitigated as soon as\none would relax the assumption of having only secondary electrons\nin the ICM.\nIndeed primary electrons accelerated at shocks, or during the \nactivity of cluster galaxies and AGN, can survive a substantial fraction \nof the Hubble time in the cluster outskirts (e.g. Sarazin 1999). \nThese primaries provide a natural population of seed electrons to \nreaccelerate in a turbulent ICM (Brunetti et al 2001, Petrosian 2001)\nand may significantly contribute to \nthe synchroton emission in the external regions of giant radio halos. \nIf a substantial contribution to the radio halo emission comes from \nthe reacceleration of primary electrons, the expected $\\gamma$--ray \nemission from the Coma cluster becomes even smaller than that \ncalculated in \\S \\ref{sec:reacc}. \n\n\\noindent\nOngoing and future observations in the deep non-thermal (high and very-high\nenergy) regime of the Coma cluster will therefore provide \nprecious information on its non-thermal content.\n\n\n\\section*{Acknowledgements}\nAuthors acknowledge useful comments from the anonymous referee and \ndiscussions with J. Donnert and T. Jeltema.\nGB acknowledges partial supported by INAF under grant PRIN-INAF 2009.\nLR acknowledges support, in part, by U.S. National Science Foundation grant\nAST-0908688 to the University of Minnesota.\nAB acknowledges partial supported by the DFG Research Unit 1254\n``Magnetization of interstellar and intergalactic media: the \nprospect of low frequency radio observations''.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTraffic classification (TC) is at the core of any network traffic monitoring system, and a pillar for traffic management, cyber security, quality-of-experience monitoring, and other strategic activities for network operators. It is also a very mature research topic with many surveys on the subject~\\cite{tcsurvey08comst,tcsurvey15survey,tcsurvey18comst}.\n\nFrom a chronological standpoint, we can categorize TC literature into two ``waves''.\nThe first wave ignited in the early 2000's, and centered around the use of Machine Learning (ML) methods using per-packet (e.g., packet size, packets inter-arrival time) or per-flow (e.g., total bytes, packets, ports) features as input targeting the classification of a handful of applications. Several works demonstrated that even when just a few packets of a flow were observed, the classification was accurate~\\cite{bernaille06ccr,crotti07ccr}, and could be sustained at line-rate speed~\\cite{santiago12imc}---``early'' TC was born.\n\nInspired by the success of image processing in computer vision, in the last years Deep Learning (DL) techniques re-ignited the interest towards TC, with several DL-based classifiers being proposed using as input either raw payload bytes or the same traffic features discovered during the first wave~\\cite{wang2015blackhat,wang2017isi,aceto2018tma,lopez2017access,aceto2019mirage,lotfollahi2020deep}.\nThis renewed interest stems also from the growth in adoption of traffic encryption, the extreme dynamicity of Internet traffic, and the heterogeneity of devices connecting to the Internet, especially when considering mobile ones (having an ecosystems of tools that eases the installation of new apps and their updates)~\\cite{aceto2019mirage}.\n\nAccordingly, to closely track the network traffic landscape, an effective TC system should support \\emph{continuous} model updates, as sketched in Fig.~\\ref{fig:dataflywheel}. \n\\emph{Incremental Learning} (IL), also known as \\emph{continuous} or \\emph{online} learning \\cite{Van2019}, is a discipline studying \nhow to update models to accommodate the new knowledge required to perform well the target task (e.g., a new class needs to be added to a classifier).\nThis fits well TC needs, and a few works indeed consider incremental TC. However, those works resort to datasets with only a few classes (typically less than $10$), they use per-flow features (which only enable post-mortem classification~\\cite{hat-TELECOMSYST16,isvm-MNA18}), and they do not consider scenarios where new applications are progressively added to models (needed to perform network management on new services or apps). In other words, TC literature focuses only on the problem of creating the most accurate classifier given a dataset where both ($a$) the number of classes and ($b$) the data for each class are immutable. \nIn turn, TC systems based on literature approaches use \\emph{amend and retrain} policies: to add new applications or new traffic behavior to a model one needs to ($i$) create a new training set (or expand the existing one), and ($ii$) train a new model from scratch---model updates \\emph{are not incremental}.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.65\\columnwidth]{figures\/sketch_data-flywheel_v2.pdf}\n    \\caption{Models development infinite loop.}\n    \\label{fig:dataflywheel}\n\\end{figure}\n\nDifferently, in this paper we investigate the use of IL techniques~\\cite{Van2019} to expand the knowledge of an existing DL-based traffic classifier without requiring a full retraining, namely \\emph{Class Incremental Learning} (CIL), which we explore to support real-world operations by reducing the requirement of full models retraining. \nWe apply $\\mathtt{iCarl}$\\xspace~\\cite{icarl-CVPR17}, a CIL state-of-the-art approach, to a standard 1-dimensional convolutional neural network TC model, using the publicly available $\\mathtt{MIRAGE}$-2019\\xspace dataset comprising traffic of $40$ popular Android applications. Our contributions are two-fold: ($i$) we dissect $\\mathtt{iCarl}$\\xspace's design and propose alternative options, including a revised version of the original method we named $\\mathtt{iCarl+}$\\xspace, and ($ii$) a thorough evaluation highlighting relevant trends, pitfalls, and further directions of improvement.\n\nThe rest of the paper is organized as follows:\nin Sec.~\\ref{sec:background} we review IL considering both ML and DL methodologies, and the relevant TC literature; in\nSec.~\\ref{sec:methodology} we drill down into $\\mathtt{iCarl}$\\xspace's design, and set our research questions which we evaluate in Sec.~\\ref{sec:results}; finally, in\nSec.~\\ref{sec:conclusions} we conclude pointing to future avenues of research.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/sketch_ml.pdf}\n    \\caption{Incremental learning in tree-based classifiers.}\n    \\label{fig:sketch-ml}\n\\end{figure}\n\n\\section{Background\\label{sec:background}}\n\nIncremental learning studies how to ($i$) integrate new classes and ($ii$) refresh the knowledge of already known classes without retraining a model from scratch. Models update procedures are designed to limit \\emph{catastrophic forgetting}, i.e. the model ``forgets'' the knowledge already acquired in favor of the new one~\\cite{catastrophic-PSM89,catastrophic-ICLR14}, and they rely also on a \\emph{memory} accumulating input samples in between updates. Herein, first we review how ML- and DL-based IL techniques (typically) incorporate these two principles, and then we introduce our research goals.\n\n\\subsection{Machine learning approaches}\n\nIncremental learning in ML is defined in the context of processing continuous streams of data, i.e., scenarios where is unfeasible to adopt large memories, and operate multiple scans of the data for an update.\nIn these cases, tree-based data structures offer the best performance, apt to integrate knowledge carried in either individual or batch of data~\\cite{batchvsincremental-IDA12} with well established algorithms. \n\nFor instance, in a Hoeffding decision tree~\\cite{hat-IDA09} each node buffers statistics observed in a window of data; ADWIN~\\cite{adwin-SIAM07} then detects knowledge drift in the buffered data, and an update takes place for example by branching out new nodes (Fig.~\\ref{fig:sketch-ml}-A). In a trees ensemble~\\cite{Parker2015} instead, each tree is weighted by its accuracy and, using algorithms such as AWE~\\cite{awe-KDD03}, trees with low weights are pruned from the ensemble (Fig.~\\ref{fig:sketch-ml}-B). \n\nWe underline that these algorithms, and other ML alternative techniques, update only the knowledge of already known classes, and do not incrementally add new ones. This justifies the scarcity of TC literature exploring them, since the typical dataset is a ``static snapshot''. For instance, Hoeffding Adaptive Trees are contrasted against decision trees in~\\cite{hat-TELECOMSYST16} to identify 10 classes of traffic evolving over 12 years of MAWI traces. We did not find any work adopting incremental trees ensembles, while a few works use incremental support vectors~\\cite{isvm-MNA18} and $k$-means~\\cite{inckmeans-ACISC16} although in a more constrained scenario than~\\cite{hat-TELECOMSYST16}.  \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/sketch_dl_train-transferlearning.pdf}\n    \\caption{Deep learning model internals (A) and transfer learning application (B).}\n    \\label{fig:sketch-dl-base-transferlearning}\n\\end{figure}\n\n\\subsection{Deep learning approaches\\label{sec:background-dl}}\n\nDL models are harder to evolve than trees. Yet, their ``modularity'' enables a better reuse of knowledge extraction than trees, and opens new venues for IL.\n \nTo better understand this, Fig.~\\ref{fig:sketch-dl-base-transferlearning}-A sketches the internals of a DL model: a composition of \\emph{feature extraction} and \\emph{classification}, each corresponding to a separate data transformation.\nFirst, $\\varphi : X$$\\rightarrow$$V$ transforms an input sample $x$ into a feature vector $v$. Then, $\\psi : V$$\\rightarrow$$Y$ splits the feature vectors $v$ based on their true label $y$, obtaining the final classification $\\hat{y}$. The function $\\varphi(\\cdot)$ and its parameters $\\theta_b$ are known as the \\emph{model's backbone}; $\\psi(\\cdot)$ and its parameters $\\theta_h$ are the \\emph{model's head}. The rationale of this composition is that feature extraction $\\varphi(\\cdot)$ aims to increase the input space dimensionality so to bring closer points sharing the same label while distancing points of different labels, and to ease the formulation of the classification task $\\psi(\\cdot)$. \nAt training, feature extraction and classification are performed together to discover the best combinations of parameters $(\\theta_b, \\theta_h)$ minimizing a loss function $L_c(y,\\hat{y})$ which compares the true label $y$ and the inferred one $\\hat{y}$. The $(\\theta_b, \\theta_h)$ pair embodies the model knowledge, so adding new knowledge requires changing those parameters and\/or the model architecture.\n\nGiven this model composition, the IL literature broadly falls into \\emph{three} categories: \\emph{model-growth} approaches enlarge models architecture and layers size to accommodate for new knowledge~\\cite{mallya2018, aljundi2016};  \\emph{fixed-representation} approaches rely on a rich backbone~\\cite{kemker2018, hayes2020} while altering the model's head; and \\emph{fine-tuning} adapt gradually model's backbone and head~\\cite{lwf-TPAMI18, icarl-CVPR17}.\nAll those methods try to address the limited capability of neural networks to support incremental changes, thus being significantly more sensible to catastrophic forgetting than trees.\nTo address this issue, incremental learning methods typically rely on  \\emph{Knowledge distillation}~\\cite{distillation-NIPS15}, a technique that decomposes the training loss function $L$ into two terms: a \\emph{distillation} term $L_o$ quantifying the (old) knowledge acquired, and a \\emph{classification term} $L_n$ quantifying the (new) knowledge to add.\nThe two terms can be balanced in different ways~\\cite{lwf-TPAMI18}. \n\nMoreover, differently from ML techniques, memory plays a key role since\n($i$) training requires multiple passes on the input data and\n($ii$) \\emph{rehearsal mechanisms}, i.e., re-proposing samples from a previous training set when updating the model, are important to overcome catastrophic forgetting~\\cite{Van2019,belouadah2020} and avoid the extra cost of data augmentation techniques like generative replay~\\cite{Shin2017}.\n\n\\noindent\\textbf{Transfer learning.}\n\nA model's backbone learns hidden patterns in the data, and the larger the training set (and the lower the layer in the model's architecture), the more the extracted knowledge is expected to abstract from the specific classification task at hand.\nThis observation is key in \\emph{transfer learning} where the backbone of a model $M_a$ is reused for a model $M_b$ by detaching $M_a$ head $\\theta_o$ and replacing it with a new one $\\theta_n$ tailored to the new classification task (Fig.~\\ref{fig:sketch-dl-base-transferlearning}-B). Then, $M_b$ is trained via fixed-representation or fine-tuning.\nConsidering TC, the authors of~\\cite{multitasklearn-ICCCN20} show that one can reuse the backbone of a model trained to predict flows duration and bandwidth, to train a classifier to distinguish 5 traffic classes. \n\nIn its basic formulation, transfer learning derives a new model from an existing one, hence does not perform an update. It is however also possible to incrementally add new tasks to an existing model~\\cite{neverending-COMM18}---this is known as \\emph{multi-task incremental learning}. In this case, a new head is attached to the backbone without pruning the existing ones. \nConsidering traffic classification, in~\\cite{fewshot-IEEEAccess19} authors study how to add a traffic classifier task to a model trained to classify both flow duration and flow bandwidth.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/sketch_icarl.pdf}\n    \\caption{Sketch of $\\mathtt{iCarl}$\\xspace~\\cite{icarl-CVPR17} internals.}\n    \\label{fig:sketch-icarl}\n\\end{figure}\n\n\\noindent\\textbf{Class incremental learning (CIL).}\n\nIf we do not stick to rigid taxonomy boundaries, transfer learning can also be used for CIL. To be more specific, a CIL state-of-the-art method is \\emph{Incremental Classifier and Representation Learning} ($\\mathtt{iCarl}$\\xspace)~\\cite{icarl-CVPR17}.\n$\\mathtt{iCarl}$\\xspace is a fine-tuning technique using ($i$) known classes exemplars rehearsal, ($ii$) a pre-allocated output layer, ($iii$) knowledge distillation, and ($iv$) a Nearest Mean Classifier (NMC), as sketched in Fig.~\\ref{fig:sketch-icarl}.\n\nLet us consider a base model $(\\theta_b, \\theta_h)$ trained on a $(X_o, Y_o)$ dataset and having $C$ classes. At the input, $\\mathtt{iCarl}$\\xspace selects exemplars of known classes $P$$\\subseteq$$X_o$ which are rehearsed at training.\nA herding mechanism~\\cite{herding-ICML09} selects exemplars based on their distance with respect to the class average centroid in the latent space, and keeps them in a memory of fixed size.\n\nAt the output, the model's head is a single layer (with neurons using the sigmoid activation function) of size $K$ with $K$$\\gg$$C$---the head pre-allocates \\mbox{$K$-$C$} units for future classes to be added.\nWe denote with $\\hat{g}_x$ the response vector associated to $K$ sigmoids, whereas with ${\\hat{g}_{x}^{0}}$ (resp. ${\\hat{g}_{x}^{\\mathrm{f}}}$) the subvector associated to the base $C$ (resp. future \\mbox{$K$-$C$}) classes.\nThe classification is however operated computing distance from classes' average centroid $\\mu_c$ in the latent space \\mbox{$\\hat{y} = \\ARGMIN_i \\norm{ \\varphi(x) - \\mu_c }$}---this is known as \\emph{Nearest Mean Classifier} (NMC). \nIn computer vision, this simple mechanism has been found superior to more typical parametric classification (i.e., leveraging the model head) when performing incremental learning~\\cite{nmc-TPAM13,nmc-CVPR14}.\n\nWhen performing an update, $\\mathtt{iCarl}$\\xspace first applies the current model to \\emph{both} exemplars in memory and new classes data.\nThe obtained ($C$-dimensional) soft-output $\\tilde{g}_{x}^{0}$ is used for knowledge distillation $L_{o}(\\tilde{g}_{x}^{0},\\hat{g}_{x}^{0})$.\nAdditionally, some of the $K$-$C$ extra units pre-allocated in the model's head (i.e. $\\hat{g}_{x}^{\\mathrm{f}}$), denoted with $\\hat{g}_{x}^{1}$, are ``consumed'' for the new classes.\nThese units are trained via the classification loss, whereas labels of new (resp. base) classes are one-hot (resp. zero-vector) encoded, namely $L_n(enc(y_n), \\hat{g}_{x}^{1})$.\n\n\\subsection{Our research goal}\n\nWe found scarce literature for IL applied to TC. \nIndeed, only \\cite{transflearn-INFOCOM20} presents a closed-loop TC system,\nbut authors ($i$) consider a small use-case with $10$ classes, ($ii$) adopt simple fine-tuning to introduce new classes, and ($iii$) focus on a single model update. Broadly speaking, the closely related literature, no matter the adopted methodology, share common limitations: old datasets (\\cite{isvm-MNA18,inckmeans-ACISC16} use data from $2005$, whereas \\cite{hat-TELECOMSYST16} use 2001-2014 data); very few (typically less than $10$), and very generic classes (HTTP, DNS, SSH, etc.); adopt per-flow features, thus not performing early traffic classification.\nOn the one hand, the validation of CIL techniques against outdated datasets allow to grasp insights on models generalization and conjecture on their future longevity. \nOn the other hand, some datasets are $10+$ years old, and offer just a handful of classes, which makes them unfit for our study.\nIndeed, the limited availability of public labeled datasets for TC is a well known problem,\nquite far from the abundance of datasets for computer vision- and natural language processing-related tasks.\n\nMethodology-wise, we aim to a fresh look at CIL for traffic classification.\nWe discard ML-based IL techniques since they can only expand knowledge of already known classes, and rely on handcrafted features.\nWe also discard multi-task transfer learning techniques since they lead to multi headed models which complicate systems operation---given one input sample, a multi headed model would return one label for each head, raising the problem of how to reconcile multiple output labels into one.\nWe focus instead on fine-tuning, and specifically $\\mathtt{iCarl}$\\xspace given its accuracy, limited complexity, and high scalability~\\cite{icarl-CVPR17}.  \nTo the best of our knowledge, $\\mathtt{iCarl}$\\xspace has only been applied to image classification, hence we aim to understand if its qualities remain the same when used for (mobile) TC.\nTo do so, we contribute an in-depth analysis of $\\mathtt{iCarl}$\\xspace internal design, challenging the original choices (e.g., the adoption of NMC), and thoroughly evaluating alternative choices using the publicly available $\\mathtt{MIRAGE}$-2019\\xspace which, to the best of our knowledge, is the most recent dataset on mobile traffic, and offer a large variety of classes ($40$ Android apps), thus fitting well the aim of our study.\n\n\\section{Methodology}\\label{sec:methodology}\n\nTo better understand the design space, in this section we start reviewing $\\mathtt{iCarl}$\\xspace internal components in more details, and set our research questions.\nWe then introduce the dataset, and the DL-based model we use in our evaluation.\n\n\\subsection{Dissecting $\\mathtt{iCarl}$\\xspace design}\n\n$\\mathtt{iCarl}$\\xspace is a CIL state-of-the-art approach for image classification \nwhich trades off complexity and performance with respect to alternative solutions like~\\cite{castro2018end, wu2019}, thus is the perfect candidate technique for our initial exploration of CIL for TC.\n\n$\\mathtt{iCarl}$\\xspace's design centers around two components: a \\emph{memory} at the input, and an \\emph{NMC} at the output (Fig.~\\ref{fig:sketch-icarl}), both operating on the feature vectors generated by a model's backbone. \nThe memory is fixed in size, and contains a uniform amount of exemplars for each class.\nIt follows that, at each update, each known class drops the same amount of exemplars to make room for the new classes exemplars. Such filtering is controlled by a herding mechanism, a procedure aiming to select a subset of exemplars so that the class average centroid in the latent space created from the filtered exemplars is as close as possible to the one obtained without filtering---it selects a ``herd'' of points capturing ``the body'' of the class (for more details, please refer to~\\cite{icarl-CVPR17}). \nBy design, an NMC tights to the same mechanics of centroids and point-to-centroid distances computation in the latent space. According to~\\cite{icarl-CVPR17}, these design choices aim to \\emph{decouple the non-linear feature extraction from the linear classifier}: while the NMC approximates the class feature vectors mean, the herding selection process tries to keep this mean unchanged by carefully selecting exemplars---this is the \\emph{representation learning} component in the $\\mathtt{iCarl}$\\xspace acronym, which stresses the need of a robust feature extraction. \n\nAt a closer look, we spot \\emph{two} other subtle aspects, at the border between design choices and implementation details. First, $\\mathtt{iCarl}$\\xspace adopts a sigmoid rather than a softmax activation function at the output. We find this peculiar, and while authors briefly state that both functions are equivalent, they do not evaluate the impact of softmax.\nSecondly, as introduced in Sec.~\\ref{sec:background-dl}, the output layer is pre-allocated, i.e., its size is larger than the number of classes required, and the extra units are progressively used when adding new classes.\n$\\mathtt{iCarl}$\\xspace authors do not mention this in their paper, but it is evident inspecting $\\mathtt{iCarl}$\\xspace source code.\\footnote{https:\/\/github.com\/srebuffi\/iCaRL} While at first sight this sounds like a ``smart trick'', it can cause unexpected effects when intertwined with the use of sigmoid activations (see Sec.~\\ref{sec:results-upperbound}).\n\nConsidering the overall design choices,\nin this work we target the following questions:\n\\emph{\n\\begin{itemize}\n    \\item Is it better to use an NMC, or to classify via the output layer? \n    \\item Is it better to adopt a sigmoid, or a softmax activation function at the output layer?\n    \\item Is it better to pre-allocate the output layer size, or progressively expand it?\n    \\item How sensitive is the herding selection to memory size? Does a larger memory provide better performance?\n\\end{itemize}\n}\n\nThese are all general questions of practical relevance, and to the best of our knowledge they are not fully investigated in the literature.\nYet, it is beyond the scope of this work to provide a final answer to them. Rather, we explore the questions in the context of TC using only one dataset.\nIn particular, differently from observed in~\\cite{icarl-CVPR17}, we find that ($i$) output layer classification is superior to NMC, ($ii$) softmax is superior to sigmoid, ($iii$) pre-allocation of the output layer may have a detrimental effect, and ($iv$) memory has indeed an impact in balancing catastrophic forgetting as reported in literature. We contribute our observations into a revised version of $\\mathtt{iCarl}$\\xspace, namely $\\mathtt{iCarl+}$\\xspace.\nAs we shall see in Sec.~\\ref{sec:results-il-strategies}, neither $\\mathtt{iCarl}$\\xspace nor $\\mathtt{iCarl+}$\\xspace match the performance observed in computer vision when tested on a TC dataset. Yet, $\\mathtt{iCarl+}$\\xspace significantly reduces training time, another aspect not quantified in previous literature (Sec.~\\ref{sec:results-training-time}). \n\n\\subsection{Dataset}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_class_samples.pdf}\n    \\caption{$\\mathtt{MIRAGE}$-2019\\xspace dataset composition.}\n    \\label{fig:dataset}\n\\end{figure}\n\n\\begin{figure}[!t]\n    \\includegraphics[width=\\columnwidth]{figures\/heatmap_pktsize_sortpadding_noack.pdf}\n    \\includegraphics[width=\\columnwidth]{figures\/heatmap_iat_sortpadding_noack.pdf}\n    \\includegraphics[width=\\columnwidth]{figures\/heatmap_dir_sortpadding_noack.pdf}\n    \\includegraphics[width=\\columnwidth]{figures\/heatmap_paddingratio_sortpadding_noack.pdf}\n    \\caption{$\\mathtt{MIRAGE}$-2019\\xspace dataset biflows time series properties.}\n    \\label{fig:heatmaps}\n\\end{figure}\n\nIn this work we use the $\\mathtt{MIRAGE}$-2019\\xspace~\\cite{aceto2019mirage} dataset.\nIt contains traffic related to $40$ Android apps belonging to $16$ different categories.\\footnote{\\url{http:\/\/traffic.comics.unina.it\/mirage\/app_list.html}}\n$\\mathtt{MIRAGE}$-2019\\xspace was collected at the ARCLAB laboratory of the University of Napoli ``Federico II'' with multiple measurement campaigns running between May 2017 and May 2019.\nIn each experiment a volunteer (out of $\\approx$300 students) was asked to use one of the $40$ apps while in background raw pcap files and \\texttt{strace} log-files were collected on the phones. Then, pcap were converted into JSON files where each record reports on the network activity of a different bidirectional flows (biflows) with aggregate level metrics (total bytes, packets, etc.), and per-packet level metrics (time series of packet size, direction, TCP flags, etc.).\nThe \\texttt{strace} logs instead exposed the socket-to-appID mapping as observed on the phone, so they were used as ground-truth to label the per-flow logs obtained from raw pcaps. For more details, please refer to~\\cite{aceto2019mirage}.\n\nThe $\\mathtt{MIRAGE}$-2019\\xspace dataset contains $\\approx$100k biflows distributed across apps as shown in Fig.~\\ref{fig:dataset}.\nAlthough the number of experiments for each app was comparable, biflows' distribution is heavy-tailed with the top-10 (top-20) apps accounting for $54.2$\\% ($78.9$\\%) of all biflows.\n\nIn this work, we focus on three biflow-related per-packet properties, namely packet (L4) payload size (PS), inter arrival time (IAT), and packet direction (DIR).\nThose have been successfully used in previous literature for early TC~\\cite{bernaille06ccr}.\nDiscarding packets with zero payload, we find that $86\\%$ of biflows have size $\\leq 100$ packets, while\nthe average size of biflows across the entire dataset is $180$ packets. Thus, for each flow, we extract time series of length $100$ for PS, IAT, and DIR, that we use an input for our DL classifier.\nIn case the time series are shorter than 100 packets, we apply zero-padding. Notice that we encode DIR values as $+1$ (for upstream) or -1 (for downstream), while the minimum IAT supported by the $\\mathtt{MIRAGE}$-2019\\xspace capture is $1\\mu$s, so the zero-padding does not clash with valid time series values.\n\nIn Fig.~\\ref{fig:heatmaps} we render the obtained time series as heatmaps:\neach row maps to a different app, and columns map to a different time series positions, with values  averaged across all biflows.\nThe bottom heatmap further shows the probability of zero-padding, which we also use to sort heatmaps' rows by putting more ``chatty'' apps first.\n\nConsidering packets size, all apps have a burst within the first $5$ packets, more sporadic communications within the first $25$ packets, after which they are mostly silent. Both zero-padding and packet direction heatmaps well capture this effect too.\nThe IAT's heatmap further captures the interleaving period between first and second communications. This interleave is possibly related to the nature of mobile apps which, based on HTTPS, render content in stages~\\cite{mcpa-TMA19}.\nThe right side of the IAT's heatmap also shows bursts of activity \nbut, especially for the apps at the bottom of the heatmaps, this is likely an artifact of the reduced number of packets (notice the higher zero-padding for those apps).\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/model_architecture.pdf}\n    \\caption{Select DL model architecture.}\n    \\label{fig:architecture}\n\\end{figure}\n\n\\subsection{Model architecture}\\label{sec:model-arch}\n\nIn theory, $\\mathtt{iCarl}$\\xspace can be used without changing a model's architecture, with the caveat that the original design requires to adopt a sigmoid activation as output, and the model head is expected to be a single layer. \nIn other words, $\\mathtt{iCarl}$\\xspace puts an accent on the importance of the model's backbone. $\\mathtt{iCarl}$\\xspace is originally evaluated using a convolutional neural network (CNN), which well suits TC needs too~\\cite{aceto2018tma}. Thus, we rely on the 1d-CNN architecture sketched in Fig.~\\ref{fig:architecture}.\nThe input is a $3$ channels $100$-elements time series where each channel is associated to a different packet property. \nThe considered input configuration was observed experimentally to provide higher performance with respect to configurations with ($i$) only (PS,DIR) pairs, and ($ii$) PS*DIR (i.e. signed payload sizes), with $+1\\%$ and $+6\\%$ F1 score improvement, respectively.\nWe also experimented with time series with less than 100 packets, and obtained only marginal differences in model accuracy.\nThe input layer feeds a stack of $2$ convolutional layers, with $16$ and $32$ filters of $1\\times3$ size, respectively, ReLU activations and max-pooling layers (stride $2$), followed by one fully-connected layer of size $256$ for a total depth of $3$ layers before the output layer.\nThis corresponds to $\\approx200$k parameters overall. \n\nWe underline that we do not claim any novelty on the model architecture, rather we point out that similar designs have been found accurate in previous literature~\\cite{aceto2018tma,wang2017isi}, so we consider it a good reference choice and a state-of-the-art network for TC.\nIn selecting the architecture we also intentionally avoided including payload bytes or long short term memory (LSTM) layers which could have possibly increased the model performance. Rather, we opted for a ``conservative'' base line.\nStill, we highlight that the considered IL methodology virtually applies to other DL-based traffic classifier proposals.\n\n\\section{Evaluation\\label{sec:results}}\n\nWe compared three strategies for CIL, namely $\\mathtt{iCarl}$\\xspace, fixed representation, and $\\mathtt{iCarl+}$\\xspace, with the last two derived from $\\mathtt{iCarl}$\\xspace's source code. \nWe evaluated all strategies updating \\emph{upperbound} models\ni.e., models trained from scratch using the architecture defined in Sec.~\\ref{sec:model-arch}. Upperbound models are the performance baseline for models incrementally trained using $\\mathtt{iCarl}$\\xspace.\nWe leveraged the apps variety in the $\\mathtt{MIRAGE}$-2019\\xspace to construct different scenarios varying the number of base classes, performing individual or multiple updates in sequence, and using two or more classes for each update.\nWe carried out $10$ runs for each scenario by randomizing the set of classes. Unless differently reported, all runs used a memory of $1\\mathrm{k}$ exemplars.\nWe measured TC effectiveness via (macro average) F1 score, which we split between \\emph{base classes} and \\emph{new classes} to quantify catastrophic forgetting.\n\nWe adopted the same parameters used by $\\mathtt{iCarl}$\\xspace: all models (both upperbound and updates) were trained for $200$ epochs with a learning rate of $10^{-2}$ halved every $50$ epochs, and a momentum of $0.9$; for model updates only, the loss function had an extra regularization term with $10^{-5}$ weight.\n\nIn the remainder, we introduce the selection process of the upperbound, and evaluate the design choices at the output. We then contrast the three IL strategies, and further zoom into $\\mathtt{iCarl+}$\\xspace to assess the impact of both multiple updates and memory size. We conclude reporting on model update time.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_upperbound.pdf}\n    \\caption{Classification methods accuracy in upperbound models.}\n    \\label{fig:upperbound}\n\\end{figure}\n\n\\subsection{Design choices at the output\\label{sec:results-upperbound}}\n\nWe investigated three design choices at the output: ($i$) the type of activation function, ($ii$) the type of classifier, and ($iii$) the method to expand the output layer.\nWithout loss of generality, we quantified all of them directly using upperbound models. \n\nFocusing on choices ($i$)-($ii$), Fig.~\\ref{fig:upperbound} compares three configurations: upperbound models classifying by means of the output layer via either softmax or sigmoid, and the original $\\mathtt{iCarl}$\\xspace design, i.e., a sigmoid activation coupled with an NMC. Lines correspond to averages across runs, while shaded areas capture standard deviation.\n$\\mathtt{iCarl}$\\xspace's design offers the worst performance, even worse than classifying via a model's head using a sigmoid activation.\nThis differs from results in~\\cite{icarl-CVPR17}, where $\\mathtt{iCarl}$\\xspace's authors find an NMC to be superior than classifying via the output layer trained with sigmoid.\nThe best strategy is also the most popular in literature: to classify via the model's head and adopt a softmax activation (not evaluated in~\\cite{icarl-CVPR17}). We conjecture that the NMC poor performances are possibly due to:\n($a$) our models being smaller, both in terms of architecture and training set size, than what commonly used in computer vision, hence the feature representation obtained from the model's backbone might not be robust enough to integrate well new classes;\n($b$) the lack of an effective decoupling between feature extraction and classification might be biasing NMC class centroids, since they are expected to rely on class-conditional distributions following the multivariate Gaussian distribution~\\cite{lee2018}.\n\nConsidering model's head expansion, $\\mathtt{iCarl}$\\xspace \\emph{pre-allocates} the output layer: given an upperbound model, the output layer is fixed to match a larger number of classes with respect to the one expected to add at the next update. Then, a ``fake update'' is performed to push neurons of the not-yet-seen classes to zero by replaying all available known class samples, i.e. currently available classes' labels are mapped on an extended one-hot encoding.\nIn Fig.~\\ref{fig:output-size} we take an upperbound model of 10 classes, and expand its output layer by adding $10$, $20$, and $30$ units. The first group of histograms on the left shows the base models performance; the remaining histograms show the performance after the expansion. While softmax is immune to the change, both sigmoid and NMC suffer from the expansion. We believe this is due to the intrinsic nature of the sigmoid activation function being more sensible to perturbations, which instead are smoothed out by adopting a softmax activation (thanks to its normalization).\n\n\\noindent \\emph{Takeaway: We find that the best strategy is to classify via the output layer trained with a softmax activation. This further allows to expand the output layer with no extra penalty. In Fig.~\\ref{fig:output-size} we show expansions at multiples of 10 units, but the same is true for any value. Thus, in $\\mathtt{iCarl+}$\\xspace we opt for expanding the output layer dynamically at each update to fit the precise number of classes required, and we classify with a model's head using a softmax activation.}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_output_size.pdf}\n    \\caption{Impact of output layer expansion in upperbound models (base models with 10 classes).}\n    \\label{fig:output-size}\n\\end{figure}\n\n\\subsection{Comparing incremental learning strategies\\label{sec:results-il-strategies}}\nHaving defined the upperbound, we then compared $\\mathtt{iCarl}$\\xspace, $\\mathtt{iCarl+}$\\xspace, and fixed-representation performance.\nWe considered scenarios with base models with $2$, $10$, $20$, and $30$ classes, and performed a single update adding $2$ classes. \nFig.~\\ref{fig:increment-strategies} shows the average F1 score deviation from the upperbound, separating the evaluation for known classes (top histograms) from new classes (bottom histograms). Results corresponds to the difference between the average upperbound performance and the average performance of the incremental learning strategy across $10$ runs, i.e., the lower the value, the better the performance.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_histo_f1-drop-across-policies_corrected.pdf}\n    \\caption{Comparing absolute F1 score reduction from upperbound across incremental learning methods (updates with 2 classes).\n    \\label{fig:increment-strategies}}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\newcommand{\\w}[0]{0.24\\columnwidth}\n    \\centering\n    \\subfloat[][Upperbound.]{\\includegraphics[width=\\w, trim=145 85 80 25, clip]{figures\/cm_ub_metrics_linear_cm_e1.pdf}}\n    \\subfloat[][$\\mathtt{iCarl+}$\\xspace.]{\\includegraphics[width=\\w, trim=145 85 80 25, clip]{figures\/cm_proposal_metrics_linear_cm_e1.pdf}}\n    \\subfloat[][$\\mathtt{iCarl}$\\xspace.]{\\includegraphics[width=\\w, trim=145 85 80 25, clip]{figures\/cm_vanilla_metrics_linear_cm_e1.pdf}}\n    \\subfloat[][Fixed-repr.]{\\includegraphics[width=\\w, trim=145 85 80 25, clip]{figures\/cm_lb_metrics_linear_cm_e1.pdf}}\n    \\caption{Confusion matrices for one run with a base model with 10 classes, updated with $5$ new classes.\n    }\n    \\label{fig:confusion_matrices}\n\\end{figure}\n\n$\\mathtt{iCarl}$\\xspace suffers from more catastrophic forgetting than $\\mathtt{iCarl+}$\\xspace, especially for base models with more than $10$ classes, i.e., the practical real life scenarios. Fixed-representation instead is significantly better at preserving previous knowledge (as the model backbone is frozen), but just modifying the head does not suffice to introduce the new classes. This hints to lack of generalization of the model backbone. Overall, $\\mathtt{iCarl+}$\\xspace offers the best tradeoff, but notice how a single update with 2 classes leads to $\\sim$10\\% F1 accuracy drop on average for the known classes; the drop is about 3$\\times$ larger for new classes. This signals an imbalance between distillation and classification loss. Fig.~\\ref{fig:confusion_matrices} further renders the catastrophic forgetting across the strategies showing the confusion matrix for a single run of a base model with $10$ classes updated with $5$ classes (we use $5$ instead of $2$ just for visualization purposes). Reference lines (dashed blue) separate new and old classes. Notice that in this specific instance $\\mathtt{iCarl+}$\\xspace performance drop on new classes seems mostly due to one of the 5 new classes.\n\n\\noindent \\emph{Takeaway: Performance are below expectation with respect to what reported in computer vision (e.g., average test accuracy $94.57$\\%~\\cite{Van2019}). We believe that integrating other mechanisms (e.g., alternative formulation of loss~\\cite{lwf-TPAMI18}, re-weighting the classifier~\\cite{belouadah2020-scail}, sampling strategy~\\cite{borsos2020}) we could further improve $\\mathtt{iCarl+}$\\xspace, but we leave this as future work.}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_histo_f1-drop-across-increments.pdf}\n    \\caption{Absolute F1 score reduction from upperbound when performing multiple increments in sequence of 2 classes using $\\mathtt{iCarl+}$\\xspace.}\n    \\label{fig:multi-episodes-size-2}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_histo_f1-drop-across-increments-size.pdf}\n    \\caption{Absolute F1 score reduction from upperbound when performing increments of different sizes with $\\mathtt{iCarl+}$\\xspace. \n    \\label{fig:multi-episodes-different-sizes}\n    }    \n\\end{figure}\n\n\\subsection{Performing multiple episodes}\n\nFocusing on $\\mathtt{iCarl+}$\\xspace, we now discuss how performance are affected when operating multiple updates. In Fig.~\\ref{fig:multi-episodes-size-2} we show the performance drop when executing up to $4$ episodes in sequence, each adding 2 new classes each, using base models with $2$, $10$, $20$, and $30$ classes. As before, results report the average drop from the upperbound across $10$ runs. While for new classes the F1 accuracy drop is confined in the same range, catastrophic forgetting accumulates proportionally with the number of updates.\n\nTo complement the analysis, in Fig.~\\ref{fig:multi-episodes-different-sizes} we compare the performance across multiple updates of different sizes. Specifically, given a model with $10$, $20$, and $30$ classes, we performed multiple updates to introduce $10$ new classes using either $5$ updates of $2$ classes each, $2$ updates of $5$ classes each, or $1$ update of $10$ classes. Two trends overlap: the larger the number of base classes, the more severe the F1 accuracy drop; the larger the number of classes in the update, the better the performance. \n\n\\noindent \\emph{Takeaway: From a practical standpoint, we expect to operate small updates, which in our analysis do not offer the best performance.\nThis suggests that IL should be paired with model ``checkpointing'', i.e., after a few updates a model is retrained from scratch to level its accuracy.}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_memory_wo_icarl.pdf}\n    \\caption{\n    Absolute F1 score  reduction from upperbound at different memory size (base model with 10 classes, single update of 2 classes).\n    }\n    \\label{fig:memory}\n\\end{figure}\n\n\\subsection{Herding selection and memory size}\n\nWe found both the herding selection and the memory size to have a key role for $\\mathtt{iCarl+}$\\xspace performance. \nTo investigate this aspect, we considered a scenario with a base model of $10$ classes, and performed a single update with $2$ classes using a memory storing $100$, $1$k, $10$k, $50$k, or all exemplars. We considered also and extra reference that we named ``upperbound-herded'', i.e., given an upperbound model, we applied herding selection to identify exemplars, which we used to retrain a new upperbound from scratch---the defined upperbound-herded is a balanced downsampling strategy that we use just as an extra reference baseline.\nAs before, Fig.~\\ref{fig:memory} shows the results of F1 accuracy drop with respect to the true upperbound, averaging results across 10 runs.\n\nTwo trends emerge. First, as expected, downsampling affects the upperbound but the herding selection is effective in selecting meaningful exemplars. Recall that the memory holds a uniform number of exemplars for each known class. In $\\mathtt{MIRAGE}$-2019\\xspace a class has an average training set size of 1,800 samples, but a memory of 10k (i.e., an average 44\\% reduction of training set) leads the upperbound-herded performance very close to true upperbound; it takes 10$\\times$ the number of samples to match the true upperbound performance. Second, for $\\mathtt{iCarl+}$\\xspace varying the memory size trades off the distillation and classification losses, as increasing the memory size significantly reduces catastrophic forgetting (top histograms) at the cost of F1 accuracy drop for new classes (bottom histograms).\n\n\\noindent\\emph{Takeaway: The memory is a key component to constrain catastrophic forgetting, but over-sizing it can be detrimental.}\n\n\\subsection{Update execution time\\label{sec:results-training-time}}\n\nThe killer advantage of IL is its major reduction in training time with respect to training and upperbound model. We exemplify this in Fig.~\\ref{fig:training-time}: training a model with $40$ classes corresponds to $23.04$ min, against $2.23$ min to add $2$ classes to a model with $38$.\n\n\\noindent\\emph{Takeaway: From the performance analysis, IL techniques seem still in their infancy. Yet, their light computational cost enables fast prototyping which is the key to unlock the model development infinite loop (Fig.~\\ref{fig:dataflywheel}).}\n\n\\section{Conclusions and Future Directions}\\label{sec:conclusions}\n\nIn this work we reviewed class incremental learning in DL-based traffic classification.\nWe dissected $\\mathtt{iCarl}$\\xspace's design, a state-of-the-art CIL method,\npinpointed its components, discussed alternatives design choices, and thoroughly evaluated it with the $\\mathtt{MIRAGE}$-2019\\xspace dataset. We demonstrated some limitations of $\\mathtt{iCarl}$\\xspace's design, and we proposed a revised version $\\mathtt{iCarl+}$\\xspace based on the use of ($i$) a softmax activation function, rather than an NMC classifier, and ($ii$) a dynamic output layer expansion fitting the number of new classes, rather than a pre-sized output layer with more units that what strictly required. Although we cannot generalize our observations, no previous literature provides a similar analysis. Our analysis highlights the infancy of $\\mathtt{iCarl+}$\\xspace and similar methods given their \nlower performance on traffic classification tasks if compared to what observed for computer vision. However, it is not trivial to understand the reason behind such differences. Considering their relatively small architecture training set size, we conjecture that our models' backbone do not generalize enough, so integrating new classes still require large changes which disrupt the previously acquired knowledge. To better dissect those mechanisms, we need to contrast our analysis to more datasets, and other methods CIL methods such as~\\cite{castro2018end, wu2019}.\nLikewise, more robust incremental learning methodologies includes research into alternative formulation of loss functions~\\cite{lwf-TPAMI18}, re-weighting the classifier and distillation loss~\\cite{belouadah2020-scail}, sampling strategy~\\cite{borsos2020}), and other design choices.\n\nAt a deeper level, incorporating computer vision datasets in the analysis might shed light onto \\emph{domain adaptation} issues, i.e., why mechanisms proven valuable for a domain might not work as well when applied to another domain. We believe such analysis would be useful beyond the specificity of (class) incremental learning.\n\nOverall, despite their infancy,\nthanks to their massive reduction of training computational cost, we foresee a more mature set of CIL techniques as the key to materialize the vision of automation of DL-based traffic analysis systems, and closed-loop networks operation in general. However, the roadmap to such vision includes methodologies beyond CIL, such as few-shot learning and coreset identification (model classes with a very small number of samples), open-set recognition (identify new classes), etc., which so far have received limited attention by the network traffic analysis research community.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/plot_timing.pdf}\n    \\caption{Model update completion time.}\n    \\label{fig:training-time}\n\\end{figure}\n\n\\balance\n\\footnotesize\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nGiven a bounded, connected domain $\\Omega \\subset \\mathbb R^n$ with Lipschitz boundary $\\partial \\Omega$, the classical Poincar\\' e inequality with a sharp constant $C(p,\\Omega)$ states that\n\\begin{equation}\\label{eqPoincareInequality}\n \\int_\\Omega |u|^p dx \\leqslant C(p,\\Omega) \\int_\\Omega |\\nabla u|^p dx\n\\end{equation}\nfor ``suitable\" function $u$ (usually in the Sobolev space $W^{1,p}(\\Omega)$) with vanishing mean value on $\\Omega$. Without assuming the vanishing mean value on $\\Omega$, the classical Poincar\\'e inequality reads as\n\\begin{equation}\\label{eqPoincareInequalityWithAverage}\n \\int_\\Omega |u - \\overline u|^p dx \\leqslant C(p,\\Omega) \\int_\\Omega |\\nabla u|^p dx\n\\end{equation}\nwhere $\\overline u = (1\/|\\Omega|) \\int_\\Omega u dx$ denotes the mean value (or average) of $u$ over $\\Omega$. Inequality \\eqref{eqPoincareInequality} usually holds for $1 \\leqslant p < +\\infty$ under very general assumptions on $\\Omega$, for example, it holds for domains satisfying the so-called ``segment property\" or ``cone property\"; see \\cite{Agmon, liebloss2001}. An interesting question is that how the constant $C(p,\\Omega)$ depends on the domain $\\Omega$?\n\nFor $p=2$ and $n=3$, Steklov \\cite{Ste96} showed that the constant $C(2,\\Omega)$, when $\\partial \\Omega$ is piecewise smooth, must equal $1\/\\lambda_1$ where $\\lambda_1$ is the first, non-zero eigenvalue of the following Neumann boundary condition problem\n\\[\n\\begin{cases}\n-\\Delta u = \\lambda u & \\text{ in } \\Omega,\\\\\n\\partial_{\\vec n} u = 0 & \\text{ on }\\partial \\Omega.\n\\end{cases}\n\\]\nHere $\\vec n$ is the exterior unit normal. A similar result was also obtained by Steklov \\cite{Ste97} for the Dirichlet boundary condition problem\n\\[\n\\begin{cases}\n-\\Delta u = \\lambda u & \\text{ in } \\Omega,\\\\\nu = 0 & \\text{ on }\\partial \\Omega.\n\\end{cases}\n\\]\nBased on these fundamental results, a few results for the sharp constant $C(2,\\Omega)$ are known; for example, the sharp constant $C(2, B(0,1))$ for the unit ball in $\\mathbb R^3$ is $1\/j_{1,1}$ where $j_{1,1}$ is the first positive zero of the Bessel function $J_1$; see \\cite[Subsection 2.2]{KN} and \\cite{NR}. For convex domains $\\Omega \\subset \\mathbb R^n$ with diameter $d$, in a beautiful work by Payne and Weinberger \\cite{PW}, the authors showed that \\eqref{eqPoincareInequality} for $p=2$ can be obtained from weighted Poincar\\'e inequalities in one dimension. As a consequence of this, they proved that $C(2,\\Omega) = d\/\\pi$. A similar argument applied to the case $p=1$ gives $C(1,\\Omega)=d\/2$; see \\cite{AD}. \n\nPoincar\\'e inequalities for punctured domains was also studied in \\cite{LSY}. For a general domain $\\Omega$ and arbitrary $p$, determining the Poincar\\'e constant $C(p,\\Omega)$ is a hard task since the value $C(p,\\Omega)$ depends on $p$ and the geometry of the domain $\\Omega$.\n\nIn this note, we consider \\eqref{eqPoincareInequality} for the hyperbolic space $\\mathbb H^n$ with $n \\geqslant 2$. The motivation of writing this note goes back to a recent higher order Poincar\\'e-type inequality on $\\mathbb H^n$ established by Karmakar and Sandeep in \\cite{KS2016} and subsequently by a few works such that \\cite{BG15, BGG15}; for interested readers, we refer to \\cite{MS08, Tatatu01} for further details and related issues. To go further, let us briefly recall the definition of the space $\\mathbb H^n$. \n\nThe hyperbolic space $\\mathbb H^n$ with $n\\geqslant 2$ is a complete and simply connected Riemannian manifold having constant sectional curvature equal to $-1$. There is a number of models for $\\mathbb H^n$, however, the most important models are the half-space model, the ball model, and the hyperboloid (or Lorentz) model. In this note, we are interested in the ball model since this model is especially useful for questions involving rotational symmetry. \n\nGiven $n \\geqslant 2$, we denote by $B_n$ the open unit ball in $\\mathbb R^n$. Clearly, $B_n$ can be endowed with the following Riemannian metric\n\\[\ng(x) =\\Big(\\frac 2{1-|x|^2}\\Big)^2 dx \\otimes dx,\n\\]\nwhich is then called the ball model of the hyperbolic space $\\mathbb H^n$. In local coordinates, we have $g_{ij} = (2\/(1-|x|^2))^2 \\delta_{ij}$ and $g^{ij} = ((1-|x|^2)\/2)^2 \\delta^{ij}$. Clearly, one can think that $g$ is conformal to $dx^2$ with the conformal factor $\\ln (2\/(1-|x|^2))$. Then, it is well-known that volume element of $\\mathbb H^n$ is given by \n\\[\ndV_g(x) = \\Big(\\frac2{1-|x|^2}\\Big)^n dx,\n\\]\nwhere $dx$ denotes the Lebesgue measure in $\\mathbb R^n$. Let $d(0,x)$ denote the hyperbolic distance between the origin and the point $x$. In the ball model, it is well-known that $d(0,x) = \\ln \\big( (1+|x|)\/(1-|x|) \\big)$ for arbitrary $x\\in B_n$. In this new context, we still use $\\nabla$ and $\\Delta$ to denote the Euclidean gradient and Laplacian as well as $\\la\\cdot ,\\cdot\\ra$ to denote the standard inner product in $\\mathbb R^n$. Then, in terms of $\\nabla$, $\\Delta$, and $\\la\\cdot ,\\cdot\\ra$, with respect to the hyperbolic metric $g$,  the hyperbolic gradient $\\nabla_g$, whose local coordinates is $g^{ij}\\partial_j$, and the Laplacian-Beltrami operator $\\Delta_g$, defined to be $\\text{div}_g (\\nabla \\, \\cdot )$, are given by\n\\[\n\\nabla_g = \\Big(\\frac{1-|x|^2}2\\Big)^2 \\nabla,\\quad \\Delta_g = \\Big(\\frac{1-|x|^2}2\\Big)^2 \\Delta + (n-2) \\Big( \\frac{1-|x|^2}2\\Big)^2 \\la x,\\nabla\\ra.\n\\]\nFor higher order derivatives, we shall adopt the following convention\n\\begin{equation*}\n\\nabla_g^m \\cdot =\n\\begin{cases}\n\\Delta_g^{m\/2} \\cdot &\\mbox{if $m$ is even,}\\\\\n\\nabla_g ( \\Delta_g^{(m-1)\/2} \\cdot \\,) &\\mbox{if $m$ is odd.}\n\\end{cases}\n\\end{equation*}\nThen the norm of $\\nabla_g^m$ being calculated with respect to the metric $g$ is understood as follows\n\\begin{equation*}\n|\\nabla_g^m \\cdot |_g=\n\\begin{cases}\n|\\nabla_g^m \\cdot | &\\mbox{if $m$ is even,}\\\\\n\\langle \\nabla_g^m \\, \\cdot  \\, , \\nabla_g^m \\, \\cdot \\, \\rangle_g^{1\/2} &\\mbox{if $m$ is odd.}\n\\end{cases}\n\\end{equation*}\nFor simplicity, we write $|\\nabla_g^m \\cdot |$ instead of $|\\nabla_g^m \\cdot |_g$ if no confusion occurs.\n \n\nGiven a function $f$ on $\\mathbb H^n$, we denote $\\|f\\|_p = \\lt(\\int_{\\mathbb H^n} |f|^p dV_g\\rt)^{1\/p}$ and $\\|\\nabla_g^m f\\|_p = \\| |\\nabla_g^m f|_g\\|_p$, for each $1 \\leqslant p < \\infty$ and integer $m\\geqslant 1$. We use $W^{m,p}(\\mathbb H^n)$ to denote the Sobolev space of order $m$ in $\\mathbb H^n$. In \\cite{KS2016}, the authors prove the following higher order Poincar\\'e inequality\n\\begin{equation}\\label{eq:HighOrderPoincare}\n \\|\\nabla^l_g u\\|_2 \\leqslant  \\big( \\frac 2{n-1}\\big)^{m-l} \\|\\nabla^m_g u\\|_2\n\\end{equation}\nfor all $u \\in W^{m,2}(\\mathbb H^n)$. In view of \\eqref{eq:HighOrderPoincare}, one can ask: \\textit{Whether the constant $(2\/(n-1))^{m-l}$ is sharp and do we have a similar inequality for $L^p$-norm?} We notice that it was claimed in \\cite{BG15} that the constant $(2\/(n-1))^{m-l}$ in \\eqref{eq:HighOrderPoincare} is sharp; however, we have not found any proof of this yet. In this note, we seek for an answer for the above question.\n\nIn order to state our results, for each number $1 < p < +\\infty$, let us first denote the following constant\n\\begin{equation}\\label{eqConstantC}\nC(n,m,p) =\n\\begin{cases}\n\\lt( p  p' \/(n-1)^2 \\rt)^{m\/2}&\\mbox{if $m$ is even},\\\\\n(p\/(n-1))\\lt( p  p' \/(n-1)^2\\rt)^{(m-1)\/2} &\\mbox{if $m$ is odd},\n\\end{cases}\n\\end{equation}\nwith $p' = p\/(p-1)$. Clearly when $p=2$ and hence $p'=2$, we obtain $C(n,m,2)= \\big(2\/(n-1)\\big)^m$. In this note, our first result is the following.\n\n\\begin{theorem}\\label{thmMAIN}\nGiven $p > 1$, then the following inequality holds\n\\begin{equation}\\label{eq:sharpPoincare}\n\\|u\\|_{p} \\leqslant C(n,m,p) \\|\\nabla^m_g u\\|_{p}\n\\end{equation}\nfor $u \\in W^{m,p}(\\mathbb H^n)$. Moreover, the constant $C(n,m,p)$ is sharp and is never achieved in $W^{m,p}(\\mathbb H^n)$.\n\\end{theorem}\n\nAs a consequence of Theorem \\ref{thmMAIN}, we know that the sharp constant $C(3,1,2)$ is $1\/2$ which is not $1\/j_{1,1}$ as in the Euclidean case. Let us now go back to \\eqref{eq:HighOrderPoincare}. By making use of Theorem \\ref{thmMAIN} above, we obtain the following corollary, which generalizes \\eqref{eq:HighOrderPoincare}.\n\n\\begin{corollary}\\label{corMAIN}\nGiven $p > 1$, then the following inequality holds\n\\begin{equation}\\label{eq:sharpPoincareLM}\n\\|\\nabla^l_g u\\|_{p} \\leqslant C(n,m-l,p) \\|\\nabla^m_g u\\|_{p}\n\\end{equation}\nfor $u \\in W^{m,p}(\\mathbb H^n)$. Moreover, the constant $C(n,m-l,p)$ is sharp and is never achieved in $W^{m,p}(\\mathbb H^n)$.\n\\end{corollary}\n\nAs a special case of Corollary \\eqref{corMAIN}, we conclude that the constant $(2\/(n-1))^{m-l}$ in \\eqref{eq:HighOrderPoincare} is sharp. In view of the results in \\cite{BG15}, it would be nice, since the sharp constant is never achieved, if there is an analogue of \\eqref{eq:sharpPoincare} with reminders. We leave this topic for interested readers.\n\n\n\\section{Proofs}\n\nIn this section, we prove Theorem \\ref{thmMAIN}. Our proof basically consists of two main parts. In the first part, we prove \\eqref{eq:sharpPoincare}. Then in the second part, we show that the constant $C(n,m,p)$ is sharp. We start with the first part. \n\n\\subsection{Proof of (\\ref{eq:sharpPoincare})}\n\nIt is now known that the symmetrization argument works well in the setting of hyperbolic spaces. It is not only the key tool in the proof of several important inequalities such as Adams-Moser--Trudinger in $\\mathbb H^n$ but also a key tool in the present proof. Let us now recall some facts about the rearrangement in the hyperbolic spaces. Let the function $f: \\mathbb H^n\\to \\mathbb R$ be such that \n\\[\n\\big |\\{x\\in \\mathbb H^n\\, :\\, |f(x)| > t\\} \\big | = \\int_{\\{x\\in \\mathbb H^n\\,:\\, |f(x)|>t\\}} dV_g < +\\infty\n\\]\nfor every $t >0$. Its \\textit{distribution function} is defined by\n\\[\n\\mu_f(t) = \\big |\\{x\\in \\mathbb H^n\\, :\\, |f(x)| > t\\} \\big |.\n\\]\nThen its \\textit{decreasing rearrangement} $f^*$ is defined by\n\\[\nf^*(t) = \\sup\\{s > 0\\, :\\, \\mu_f(s) > t\\}.\n\\]\nSince $f^*$ is non-increasing, the maximal function $f^{**}$ of $f^*$ is defined by\n\\[\nf^{**}(s) = \\frac 1s\\int_0^s f^*(t) dt.\n\\]\nIt is well-known for any $p \\in (1,\\infty)$ that\n\\begin{equation}\\label{eq:Hardyinequality}\n\\left(\\int_0^\\infty f^{**}(s)^p ds\\right)^{1\/p} \\leqslant p' \\left(\\int_0^\\infty f^*(s)^p ds\\right)^{1\/p}.\n\\end{equation}\nNow, we define $f^\\sharp: \\mathbb H^n \\to \\mathbb R$ by \n\\[\nf^\\sharp(x) = f^*(|B(0,d(0,x))|),\n\\]\nwhere $B(0,d(0,x))$ and $|B(0,d(0,x))|$ denote the ball centered at the origin $0$ with radius $d(0,x)$ in the hyperbolic space and its hyperbolic volume, respectively. Then for any continuous increasing function $\\Phi: [0, +\\infty) \\to [0, +\\infty)$ we have\n\\begin{equation}\\label{eqIDENTITY}\n\\int_{\\mathbb H^n} \\Phi(|f|) dV_g = \\int_{\\mathbb H^n} \\Phi(f^\\sharp) dV_g.\n\\end{equation}\nMoreover, the Polya--Szeg\\\"o principle conclude that\n\\[\n\\int_{\\mathbb H^n} |\\nabla_g f^\\sharp|^p dV_g \\leqslant \\int_{\\mathbb H^n} | \\nabla_g f |^p dV_g\n\\]\nfor any function $f:\\mathbb H^n \\to \\mathbb R$. Now we define a function on $[0,+\\infty)$ as follows\n\\[\n\\Phi(s) =  n \\om_n \\int_0^s (\\sinh r)^{n-1} dr, \\quad s\\geqslant 0.\n\\]\nClearly, $\\Phi$ is a continuous and strictly increasing function from $[0,+\\infty)$ to $[0,+\\infty)$. Let $F$ denote the inverse function of $\\Phi$, it is not hard to verify that $F$ is a continuous, strictly increasing function and satisfies\n\\begin{equation}\\label{eq:definitionofF}\ns = n \\om_n \\int_0^{F(s)} (\\sinh r)^{n-1} dr\n\\end{equation}\nfor any $s\\geqslant 0$. Depending on $m$ and for clarity, we divide this part into several small steps as follows.\n\n\\subsubsection{The case $m=1$} \n\nLet $u \\in W^{1,p}(\\mathbb H^n)$ arbitrary. Upon normalization, if necessary, we can assume that $\\|\\nabla_g u\\|_p =1$. Then by the Polya--Szeg\\\"o principle we know that $\\|\\nabla_g u^\\sharp\\|_p \\leqslant 1$. Recall, by definition, that $u^\\sharp(x) = u^*(|B(0, d(0,x))|)$. Let $\\mu_u$ denote the distribution function of $u$. For $t >0$, let $\\rho(t)$ denote the radius of the ball having hyperbolic volume $\\mu_u(t)$. Then, we have\n\\[\n\\mu_u(t) = \\int_{B(0,\\rho(t))} dV_g = n\\om_n \\int_0^{\\rho(t)} (\\sinh s)^{n-1} ds.\n\\]   \nFrom this and the definition of the function $F$, it is easy to check that\n\\[\n\\rho(t) = F(\\mu_u(t)).\n\\]  \nWe now define\n\\begin{equation}\\label{eq:vphi}\n\\varphi (s) = (n \\om_n)^{-p\/(p-1)} \\int_s^{+\\infty} (\\sinh F(t))^{-p(n-1)\/(p-1)} dt\n\\end{equation}\nand choose\n\\begin{equation}\\label{eq:g}\ng( \\varphi (s)) = u^*(s).\n\\end{equation}\nClearly the function $\\varphi$ is decreasing with\n\\[\n-\\varphi '(s) = \\big( n \\om_n (\\sinh F(s))^{n-1} \\big)^{-p\/(p-1)}.\n\\]\nConcerning to the function $g$, it is increasing and\n\\[\n\\int_0^{+\\infty} (g'(s))^p ds = \\int_{\\mathbb H^n} |\\nabla_g u^\\sharp|^p dV_g \\leqslant 1.\n\\]\nDenote $\\underline g = (g')^*$ the decreasing rearrangement of $g'$ on $(0,\\infty)$ and set\n\\begin{equation*}\nf(s) = \\int_0^{\\varphi (s)} \\underline g(t) dt.\n\\end{equation*}\nWe have $f(s) \\geqslant u^*(s)$ and \n\\[\n\\int_0^{+\\infty} \\underline g(s)^p ds =\\int_0^{+\\infty} (g'(s))^p ds \\leqslant 1.\n\\] \nVia integration by parts, we have for any $0< a < b < +\\infty$\n\\begin{equation}\\label{eq:IBPconstant}\n\\begin{split}\n\\int_a^b f(s)^p ds = & -p\\int_a^b s \\varphi '(s) \\underline g(\\varphi (s)) f(s)^{p-1} ds \\\\\n&+ b\\Big(\\int_0^{\\varphi (b)} \\underline g(s) ds\\Big)^p -a\\Big(\\int_0^{\\varphi (a)} \\underline g(s) ds\\Big)^p.\n\\end{split}\n\\end{equation}\nNext we show that \n\\begin{equation}\\label{eq:limit0}\n\\lim_{a\\to 0} a\\Big(\\int_0^{\\varphi (a)} \\underline g(s) ds\\Big)^p = \\lim_{b\\to +\\infty} b\\Big(\\int_0^{\\varphi (b)} \\underline g(s) ds\\Big)^p =0.\n\\end{equation}\nIndeed, for any $\\varepsilon > 0$, there is $R > 0$ such that $\\int_R^{+\\infty} \\underline g(s)^p ds < \\varepsilon^p$, take $s_0$ such that $\\varphi (s_0) =R$, for $0 < a < s_0$ we have\n\\begin{align*}\n\\int_0^{\\varphi (a)} \\underline g(s) ds &= \\int_0^{\\varphi (s_0)} \\underline g(s) ds + \\int_{\\varphi (s_0)}^{\\varphi (a)} \\underline g(s) ds\\\\\n&\\leqslant \\int_0^{\\varphi (s_0)} \\underline g(s) ds +\\Big(\\int_{\\varphi (s_0)}^{\\varphi (a)} \\underline g(s)^p ds\\Big)^{1\/p} \\big(\\varphi (a) -\\varphi (s_0) \\big)^{(p-1)\/p}\\\\\n&\\leqslant \\int_0^{\\varphi (s_0)} \\underline g(s) ds +\\varepsilon \\big( \\varphi (a) -\\varphi (s_0) \\big)^{(p-1)\/p}.\n\\end{align*}\nSince $n\\om_n (\\sinh F(s))^{n-1} \\geqslant (n-1) s$ for all $s > 0$, we conclude that\n\\[\\begin{split}\n\\varphi (a) -\\varphi (s_0) \\leqslant &\\int_a^{s_0} \\big( (n-1)s \\big)^{-p\/(p-1)}ds \\\\\n= &(n-1)^{-p\/(p-1)} (p-1) \\big(a^{-1\/(p-1)} -s_0^{-1\/(p-1)} \\big).\n\\end{split}\\]\nTherefore we get\n\\begin{align*}\n\\limsup_{a\\to 0} a\\Big(\\int_0^{\\varphi (a)} \\underline g(s) ds\\Big)^p \\leqslant& \\limsup_{a\\to 0} a\\Big(\\int_0^{\\varphi (s_0)} \\underline g(s) ds +\\varepsilon \\big(\\varphi (a) -\\varphi (s_0) \\big)^{(p-1)\/p}\\Big)^p\\\\\n=&\\limsup_{a\\to 0} \\Big[ a \\varepsilon^p (\\varphi (a) -\\varphi (s_0))^{p-1} \\Big]\\\\\n\\leqslant &(n-1)^{-p} (p-1)^{p-1} \\varepsilon^p \\\\\n& \\times \\limsup_{a\\to 0} a \\big(a^{-1\/(p-1)} -s_0^{-1\/(p-1)} \\big)^{p-1}\\\\\n=&(n-1)^{-p}(p-1)^{p-1} \\varepsilon^p.\n\\end{align*}\nSince $\\varepsilon >0$ is arbitrary, we get\n\\[\n\\limsup_{a\\to 0} a\\Big(\\int_0^{\\varphi (a)} \\underline g(s) ds\\Big)^p =0.\n\\]\nThe second limit in \\eqref{eq:limit0} follows from the H\\\"older inequality. Indeed, first we have\n\\[\n\\Big(\\int_0^{\\varphi (b)} \\underline g(s) ds\\Big)^p \\leqslant \\varphi (b)^{p-1}\\int_0^{\\varphi (b)} \\underline g(s)^p ds .\n\\]\nObserve that\n\\begin{equation}\\label{eq:boundofvphi}\n\\begin{split}\n\\varphi (b) \\leqslant &(n-1)^{-p\/(p-1)}\\int_b^{+\\infty} s^{-p\/(p-1)}ds \\\\\n \\leqslant &(n-1)^{-p\/(p-1)}(p-1) b^{- 1\/(p-1)},\n\\end{split}\n\\end{equation}\nwhich helps us to obtain\n\\[\nb\\Big(\\int_0^{\\varphi (b)} \\underline g(s) ds\\Big)^p \\leqslant (n-1)^{-p} (p-1)^{p-1} \\int_0^{\\varphi (b)} \\underline g(s)^p ds.\n\\]\nFrom this the conclusion follows by the fact $\\lim_{b\\to 0} \\int_0^{\\varphi (b)} \\underline g(s)^p ds =0$ since $\\varphi (b)$ tends to $0$ as $b$ tends to $0$. Thanks to $\\varphi ' \\leqslant 0$, we can denote \n\\[\nh(s) = \\underline g(\\varphi (s)) (-\\varphi '(s))^{1\/p}.\n\\]\nClearly, $\\int_0^{+\\infty} h(s)^p ds \\leqslant 1$. Making use of the H\\\"older inequality and  \\eqref{eq:IBPconstant}, we can estimate $\\int_a^b f(s)^p ds$ as follows\n\\begin{align*}\n\\int_a^b f(s)^p ds \\leqslant  & p \\left(\\int_a^b \\big( -\\varphi '(s) s \\underline g(\\varphi (s))\\big)^p ds\\right)^{1\/p} \\left(\\int_a^b f(s)^p ds\\right)^{(p-1)\/p} \\\\\n&\\qquad\\qquad + b\\Big(\\int_0^{\\varphi (b)} \\underline g(s) ds\\Big)^p -a\\Big(\\int_0^{\\varphi (a)} \\underline g(s) ds\\Big)^p.\n\\end{align*}\nDividing both sides by $\\big(\\int_a^b f(s)^p ds\\big)^{(p-1)\/p}$, then letting $a \\searrow 0$ and $b \\nearrow +\\infty$ and using \\eqref{eq:limit0}, we obtain\n\\begin{equation}\\label{eq:case1}\n\\Big(\\int_0^{+\\infty} f(s)^p ds\\Big)^{1\/p} \\leqslant p \\left(\\int_a^b \\big[ -\\varphi '(s) s \\underline g(\\varphi (s))\\big]^p ds\\right)^{1\/p}.\n\\end{equation}\nNote that the inequality $n\\om_n (\\sinh F(s))^{n-1} > (n-1) s$ for any $s>0$ and the definition of $\\varphi$ imply that\n\\[\n\\big(-\\varphi '(s) \\big)^{(p-1)\/p} s < (n-1)^{-1}, \\quad\\forall s >0.\n\\]\nCombining the latter inequality and \\eqref{eq:case1}, we obtain\n\\[\n\\Big(\\int_0^{+\\infty} f(s)^p ds\\Big)^{1\/p} < \\frac p{n-1} \\left(\\int_0^\\infty h(s)^p ds\\right)^{1\/p} \\leqslant \\frac p{n-1},\n\\]\nSince $u^* \\leqslant f$, we have\n\\[\n\\left(\\int_{\\mathbb H^n} |u|^p dV_g\\right)^{1\/p} = \\left(\\int_0^{+\\infty} (u^*(s))^p ds\\right)^{1\/p} \\leqslant \\left(\\int_0^{+\\infty} f(s)^p ds\\right)^{1\/p} < \\frac p{n-1},\n\\]\nfor any function $u\\in W^{1,p}(\\mathbb H^n)$ with $\\|\\nabla_g u\\|_p =1$. This proves \\eqref{eq:sharpPoincare} for the case $m=1$ and also shows that the constant $C(n,1,p)$ is not achieved.\n\n\\subsubsection{The case $m=2$} \n\nFor any function $u\\in W^{2,p}(\\mathbb H^n)$ such that $\\|\\Delta_g u\\|_p = 1$, denote $f = -\\Delta_g u$. It was proved in \\cite{NgoNguyen16} that\n\\[\nu^*(s) \\leqslant \\int_s^{+\\infty} \\frac{t f^{**}(t)}{[n \\omega_n (\\sinh F(t))^{n-1}]^2}dt =: h(s), \\quad\\forall s>0.\n\\]\nAs in the case $m=1$, we can easily prove that\n\\[\n\\lim_{s\\to 0^+} s h(s)^p = \\lim_{s\\to +\\infty} s h(s)^p = 0.\n\\]\nFor any $b> a> 0$, using integration by parts and the H\\\"older inequality imply that\n\\begin{align*}\n\\int_a^b h(s)^p ds &= p \\int_a^b h(s)^{p-1} \\frac{s^2 f^{**}(s)}{[n \\om_n (\\sinh F(s))^{n-1}]^2} ds + bh(b)^p -a h(a)^p\\\\\n&\\leqslant p \\left(\\int_a^b h(s)^p ds\\right)^{(p-1)\/p} \\left(\\int_a^b \\left[\\frac{s^2 f^{**}(s)}{[n \\om_n (\\sinh F(s))^{n-1}]^2}\\right]^p ds\\right)^{1\/p}\\\\\n&\\qquad\\qquad +b h(b)^p -a h(a)^p.\n\\end{align*}\nDividing both sides by $\\big(\\int_a^b h(s)^p ds\\big)^{1\/p}$ and letting $a \\searrow 0$ and $b \\nearrow +\\infty$, we obtain\n\\begin{equation}\\label{eq:case2}\n\\left(\\int_0^{+\\infty} h(s)^p ds\\right)^{1\/p} \\leqslant p \\left(\\int_0^{+\\infty} \\left[\\frac{s^2 f^{**}(s)}{[n \\om_n (\\sinh F(s))^{n-1}]^2}\\right]^p ds\\right)^{1\/p}.\n\\end{equation}\nUsing the inequalities $n \\omega_n (\\sinh F(s))^{n-1} > (n-1) s$ for any $s>0$, \\eqref{eq:Hardyinequality} and \\eqref{eq:case2}, we have\n\\[\n\\left(\\int_0^{+\\infty} h(s)^p ds\\right)^{1\/p} < \\frac{pp'}{(n-1)^2} \\left(\\int_0^{+\\infty} f^*(s)^p ds\\right)^{1\/p} \\leqslant \\frac{pp'}{(n-1)^2}.\n\\]\nSince $u^* \\leqslant h$, we then obtain\n\\[\n\\left(\\int_{\\mathbb H^n} |u|^p dV_g\\right)^{1\/p} = \\left(\\int_0^{+\\infty} (u^*(s))^p ds\\right)^{1\/p} \\leqslant \\left(\\int_0^{+\\infty} h(s)^p ds\\right)^{1\/p} < \\frac {pp'}{(n-1)^2},\n\\]\nfor any function $u\\in W^{2,p}(\\mathbb H^n)$ with $\\|\\Delta_g u\\|_p =1$. This proves \\eqref{eq:sharpPoincare} for the case $m=2$ and also shows that the constant $C(n,2,p)$ is not achieved.\n\n\\subsubsection{The case $m >2$}\n\nIn this scenario, we have two possible cases:\n\n\\noindent\\textbf{Case 1}. Suppose that $m=2k$ is even. Clearly, this case follows from the case $m=2$ by repeating $k$ times as follows\n\\[\\begin{split}\n\\|u\\|_p \\leqslant  \\frac{p  p'}{(n-1)^2} \\|\\Delta_g u\\|_p \\leqslant & \\Big(\\frac{p  p'}{(n-1)^2} \\Big)^2 \\|\\Delta_g^2 u\\|_p  \\\\\n\\leqslant & \\cdots \\leqslant \\Big(\\frac{p  p'}{(n-1)^2} \\Big)^k \\|\\Delta_g^k u\\|_p.\n\\end{split}\\] \n\n\\noindent\\textbf{Case 2}. Suppose that $m=2k+1$ is odd. This case can also be derived from the cases $m=1$ and $m=2$ as the following\n\\[\\begin{split}\n\\|u\\|_p \\leqslant \\frac p{n-1} \\|\\nabla_g u\\|_p \\leqslant & \\frac p{n-1} \\frac{p  p'}{(n-1)^2}\\|\\nabla_g (\\Delta_g u)\\|_p \\\\\n \\leqslant & \\cdots \\leqslant  \\frac p{n-1} \\Big(\\frac{p  p'}{(n-1)^2} \\Big)^k  \\|\\nabla_g (\\Delta_g^k u)\\|_p.\n\\end{split}\\] \n\nWe now move to the second part of the proof. We shall prove the sharpness of $C(n,m,p)$ given in \\eqref{eqConstantC} in the next subsection.\n\n\\subsection{The sharpness of $C(n,m,p)$}\n\nIt remains to check the sharpness of the constant $C(n,m,p)$. To do this, we will construct a function $u$ in such a way that $\\|\\nabla^m_g u\\|_p\/ \\|u\\|_p$ approximates $C(n,m,p)^{-1}$. Observe from \\eqref{eq:definitionofF} that \n$$n\\om_n (\\sinh F(s))^{n-1}\\geqslant (n-1) s$$ \nfor any $s\\geqslant 0$ and\n\\[\n\\lim_{s\\to \\infty} \\frac{n\\om_n (\\sinh F(s))^{n-1}}{(n-1) s} = 1.\n\\]\nHence, for any $\\varepsilon >0$, there is $s_0$ such that\n\\[\n(n-1) s \\leqslant n\\om_n (\\sinh F(s))^{n-1} \\leqslant (1+\\varepsilon) (n-1) s\n\\]\nfor all $s \\geqslant s_0$. For any $R > s_0$, let us construct a positive, continuous, non-increasing function $f_R$ on $[0,\\infty)$ given by\n\\begin{equation}\\label{eq:FunctionF_R}\nf_R(s) = \n\\begin{cases}\ns_0^{-1\/p}&\\mbox{if $s\\in (0,s_0)$}\\\\\ns^{-1\/p} &\\mbox{if $s\\in [s_0,R)$}\\\\\nR^{-1\/p} \\max\\{2- s\/R, 0\\} &\\mbox{if $s\\geqslant R$}.\n\\end{cases}\n\\end{equation}\nThen we define two sequences of functions $\\{v_{R,i}\\}_{i \\geqslant 0}$, $\\{g_{R,i}\\}_{i \\geqslant 1}$ as follows: first we set $v_{R,0} = f_R$, then we define $g_{R,i+1}$ as the maximal function of $v_{R,i}$, that is\n\\[\ng_{R,i+1}(s) = \\frac1s \\int_0^s v_{R,i}(t) dt,\n\\]\nand then\n\\[\nv_{R,i+1}(s) = \\int_s^{+\\infty}\\frac{t g_{R,i+1}(t)}{(n\\om_n (\\sinh F(t))^{n-1})^2} dt,\n\\]\nfor $i=0, 1,2,...$ Note that $v_{R,i}$ and $g_{R,i}$ are non-increasing functions. We can explicitly compute the function $g_{R,1}$ as follows: When $s < R$ we have\n\\[\ng_{R,1}(s) =\n\\begin{cases}\ns_0^{-1\/p}&\\mbox{if $s\\in (0,s_0)$}\\\\\np's^{-1\/p} -s_0^{1-1\/p}\/((p-1)s) &\\mbox{if $s\\in [s_0,R)$,}\n\\end{cases}\n\\]\nwhile for $s\\in [R,2R)$ we have\n\\[\ng_{R,1}(s) = \\bigg(\\Big(p'-\\frac 32\\Big) R^{1-1\/p} - \\frac{s_0^{1-1\/p}}{p-1}\\bigg)\\frac1s +2R^{-1\/p} -\\frac{R^{-1-1\/p} s}2, \\quad\\forall\\, s\\in [R,2R),\n\\]\nand finally when $s \\geqslant 2R$ we have\n\\[\ng_R(s) = \\bigg(p' R^{1-1\/p} - \\frac{s_0^{1-1\/p}}{p-1}\\bigg)\\frac1s + \\frac{R^{1-1\/p}}{2s},\\quad\\forall\\, s\\geqslant 2R.\n\\]\nNote that\n\\[\n\\int_R^{+\\infty} g_R(s)^p ds \\leqslant C\n\\]\nfor some constant $C >0$ independent of $R$\n\nIn the sequel, we use $C$ to denote various constants which are independent of $R$ and whose values can change from line to line and even in one line if no confusion occurs. We will need the following result.\n\n\\begin{proposition}\\label{decomposefunction}\nFor any $i\\geqslant 1$, there exist functions $h_{R,i}$ and $w_{R,i}$ such that $v_{R,i} = h_{R,i} + w_{R,i}$, $\\int_0^{+\\infty} |w_{R,i}|^p ds \\leqslant C$ and \n\\[\n\\frac1{(1+\\varepsilon)^{2i}} \\Big(\\frac{pp'}{(n-1)^2}\\Big)^i f_R \\leqslant h_{R,i} \\leqslant \\Big(\\frac{pp'}{(n-1)^2}\\Big)^i f_R.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nLet us define the operator $T$ acting on functions $v$ on $[0,+\\infty)$ by\n\\[\nTv(s) = \\int_s^{+\\infty} \\frac{r}{(n\\om_n (\\sinh F(r))^{n-1})^2} \\Big(\\frac1r \\int_0^r v(t) dt\\Big) dr.\n\\]\nFor simplicity, for each function $v$ on $[0,+\\infty)$ we define an associated function $\\overline{v}$ on $\\mathbb H^n$ by \n\\[\n\\overline{v}(x) = v(|B(0,d(0,x))|).\n\\]\nWith these notation, it is not hard to see that\n\\[\n\\|\\o{w_{R,i}}\\|_p = \\Big(\\int_0^{+\\infty} |w_{R,i}(s)|^p ds\\Big)^{1\/p}\n\\]\nfor any $i\\geqslant 1$ and \n$$-\\Delta_g \\o{Tw_{R,i}}(x) = \\o{w_{R,i}}(x)$$ \nfor any $x\\in \\mathbb H^n$. Hence, by the Poincar\\'e inequality, we have\n\\[\n\\int_0^{+\\infty} |Tw_{R,i}(s)|^p ds = \\|\\o{Tw_{R,i}}\\|_p^p \\leqslant C \\|\\o{w_{R,i}}\\|_p^p = C \\Big(\\int_0^{+\\infty} |w_{R,i}(s)|^p ds\\Big)^{1\/p}.\n\\]\nThus, using an induction argument, it is enough to prove this proposition for $i=1$. We will perform several explicit estimation for the function $v_{R,1}$. Note that for $s\\geqslant s_0$ we have\n\\[\n(n-1) s \\leqslant n\\om_n (\\sinh F(s))^{n-1} \\leqslant (1+\\varepsilon) (n-1) s.\n\\]\n\\noindent\\textbf{Estimate of $v_{R,1}$ when $s \\geqslant 2R$}. Clearly for $s \\geqslant 2R$, we have\n\\begin{align*}\nv_{R,1}(s)& \\leqslant \\frac1{(n-1)^2}\\int_s^{+\\infty} \\frac{(p'+1\/2)R^{1-1\/p} -s_0^{1-1\/p}\/(p-1)}{t^2} dt\\\\\n&=\\frac1{(n-1)^2}\\frac{(p'+1\/2)R^{1-1\/p} -s_0^{1-1\/p}\/(p-1)}{s},\n\\end{align*}\nand similarly we have\n\\[\nv_{R,1}(s) \\geqslant \\frac1{(1+\\varepsilon)^2(n-1)^2}\\frac{(p'+1\/2)R^{1-1\/p} -s_0^{1-1\/p}\/(p-1)}{s}.\n\\]\nThus an easy calculation shows that\n\\begin{equation}\\label{eqEstimateVOut2R}\n\\int_{2R}^{+\\infty} v_{R,1}(s)^p ds \\leqslant C.\n\\end{equation}\n \n\\noindent\\textbf{Estimate of $v_{R,1}$ when $R \\leqslant s < 2R$}. For $s\\in [R,2R)$, we first write\n\\[\nv_{R,1}(s) = v_{R,1}(2R) + \\int_s^{2R} \\frac{t g_{R,1}(t)}{(n\\om_n (\\sinh F(t))^{n-1})^2}dt.\n\\]\nThen we can estimate\n\\[\\begin{split}\nv_{R,1}(2R) +& \\frac1{(1+\\varepsilon)^2(n-1)^2}\\int_s^{2R} \\frac{g_{R,1}(t)}t dt \\\\\n&\\leqslant v_{R,1}(s) \\leqslant v_{R,1}(2R)+ \\frac1{(n-1)^2}\\int_s^{2R} \\frac{g_{R,1}(t)}t dt.\n\\end{split}\\]\nNote that $v_{R,1}(2R)$ is equivalent to $R^{-1\/p}$ and\n\\begin{align*}\n\\int_s^{2R} \\frac{g_{R,1}(t)}t dt &=\\lt(\\Big(p'-\\frac32\\Big) R^{1-1\/p} - \\frac{s_0^{1-1\/p}}{p-1}\\rt) \\Big(\\frac1s-\\frac1{2R}\\Big)\\\\\n&\\qquad\\qquad\\qquad +2R^{-1\/p} \\ln \\frac{2R}s -\\frac{R^{-1-1\/p} (2R-s)}2.\n\\end{align*}\nThis shows that\n\\begin{equation}\\label{eqEstimateVInR2R}\n\\int_R^{2R} v_{R,1}(s)^p ds \\leqslant C,\n\\end{equation}\nand that $v_{R,1}(R)$ is equivalent to $R^{-1\/p}$. Combining the estimates \\eqref{eqEstimateVOut2R} and \\eqref{eqEstimateVInR2R} gives $\\int_R^{+\\infty} v_{R,1}(s)^p ds \\leqslant C$. \n\n\\noindent\\textbf{Estimate of $v_{R,1}$ when $s_0 \\leqslant s < R$}. For $s\\in [s_0,R)$, we also write\n\\[\nv_{R,1}(s) = v_{R,1}(R) + \\int_s^{R} \\frac{t g_{R,1}(t)}{(n\\om_n (\\sinh F(t))^{n-1})^2}dt.\n\\]\nThus\n\\[\nv_{R,1}(R)+ \\frac1{(1+\\varepsilon)^2(n-1)^2}\\int_s^{R} \\frac{g_{R,1}(t)}t dt\\leqslant v_{R,1}(s) \\leqslant v_{R,1}(R)+ \\frac1{(n-1)^2}\\int_s^{R} \\frac{g_{R,1}(t)}t dt.\n\\]\nA simple computation gives\n\\[\n\\int_s^{R} \\frac{g_{R,1}(t)}t dt = pp'(s^{-1\/p} -R^{-1\/p}) -\\frac{s_0^{1-1\/p}}{p-1} \\Big(\\frac1s -\\frac1R\\Big),\n\\]\nwhich implies that\n\\[\n\\int_{s_0}^R \\lt|\\int_s^{R} \\frac{g_{R,1}(t)}t dt - \\frac{pp'}{s^{1\/p}}\\rt|^p ds \\leqslant C.\n\\]\n\n\\noindent\\textbf{Estimate of $v_{R,1}$ when $s <s_0$}. For $s \\in (0,s_0)$ we write\n\\[\nv_{R,1}(s) = v_{R,1}(s_0) + \\int_s^{s_0} \\frac{t s_0^{-1\/p}}{(n\\om_n (\\sinh F(t))^{n-1})^2}dt,\n\\]\ntherefore\n\\[\n|v_{R,1}(s)|\\leqslant C(R^{-1\/p} + s_0^{2\/n -1\/p}).\n\\]\nConsequently, we can write $v_{R,1} = h_{R,1} + w_{R,1}$ with $\\int_0^{+\\infty} |w_{R,1}|^p ds \\leqslant C$ for some constant $C$ independent of $R$ and \n\\[\n\\frac1{(1+\\varepsilon)^2} \\frac{pp'}{(n-1)^2} f_R \\leqslant h_{R,1} \\leqslant \\frac{pp'}{(n-1)^2} f_R.\n\\]\n(The way to see this is as follows: Since $\\int_R^{+\\infty} v_{R,1}(s)^p ds \\leqslant C$, we can choose $h_{R,1} = pp' (n-1)^{-2} f_R$ when $r \\geqslant R$. When $r < s_0$, we choose the same function for $h_{R,1}$. When $s_0 \\leqslant r < R$, we choose $h_{R,1} (s) = pp'(n-1)^{-2}\/s^{1\/p}$ with a remark that $f_R (s) = s^{-1\/p}$ in this scenario.) This finishes our proof of the proposition.\n\\end{proof}\n\nWe are now in a position to confirm the sharpness of $C(n,m,p)$. For clarity, we split our proof into several small steps.\n\n\\subsubsection{The sharpness of $C(n,1,p)$}\n\nWe set $u_R(x) =f_R(|B(0,d(0,x))|)$. It is not hard to see that $u_R \\in W^{1,p}(\\mathbb H^n)$. We also consider the function $k_R$ defined by\n\\[\nk_R(\\varphi (s)) \\varphi '(s) = f_R'(s).\n\\]\nTo finish our proof, we shall test the inequality with the function $u_R$. First, we use \\eqref{eqIDENTITY} to get\n\\[\n\\int_{\\mathbb H^n} u_R(x)^p dV_g = \\int_0^{+\\infty} f_R(s)^p ds = 1 + \\ln R -\\ln s_0 + \\int_0^1 (1-s)^p ds.\n\\]\nFor the gradient term, we observe that\n\\begin{align*}\n\\int_{\\mathbb H^n} |\\nabla_g u_R(x)|^p dV_g &= \\int_0^{+\\infty} k_R(s)^p ds\\\\\n&=-\\int_0^{+\\infty} k_R(\\varphi (s))^p \\varphi '(s) ds\\\\\n&=\\int_0^{+\\infty} (f_R'(s))^p (-\\varphi '(s))^{1-p} ds\\\\\n&\\leqslant \\frac{(n-1)^p}{p^p}(1+\\varepsilon)^p \\int_{s_0}^R s^{-1} ds +(n-1)^p (1+\\varepsilon)^p R^{-p-1}\\int_R^{2R} s^p ds \\\\\n&=\\frac{(n-1)^p}{p^p}(1+\\varepsilon)^p (\\ln R -\\ln s_0) + (n-1)^p (1+\\varepsilon)^p \\int_0^1 (1+s)^p ds.\n\\end{align*}\nHence\n\\[\n\\inf_{u\\in W_0^{1,p}(\\mathbb H^n) \\backslash \\{0\\}} \\frac{\\int_{\\mathbb H^n} |\\nabla_g u|^p dV_g}{\\int_{\\mathbb H^n} |u|^p dV_g} \\leqslant \\liminf_{R\\to +\\infty} \\frac{\\int_{\\mathbb H^n} |\\nabla_g u_R|^p dV_g}{\\int_{\\mathbb H^n} |u_R|^p dV_g} \\leqslant \\frac{(n-1)^p}{p^p}(1+\\varepsilon)^p.\n\\]\nSince $\\varepsilon >0$ is arbitrary, we obtain \n\\[\n\\inf_{u\\in W_0^{1,p}(\\mathbb H^n)\\backslash \\{0\\}} \\frac{\\int_{\\mathbb H^n} |\\nabla_g u|^p dV_g}{\\int_{\\mathbb H^n} |u|^p dV_g} \\leqslant \\Big(\\frac{n-1}{p}\\Big)^p.\n\\]\nHence the preceding inequality becomes equality. This proves the sharpness of $C(n,1,p)$. Next, we move to a proof for the sharpness of $C(n,2,p)$.\n\n\\subsubsection{The sharpness of $C(n,2,p)$}\n\nIn this case, we set $u_R(x) = v_{R,1}(|B(0,d(0,x))|)$, then we have \n\\[\n-\\Delta_g u_R(x) = f_R(|B(0,d(0,x))|).\n\\]\nUsing this fact and \\eqref{eqIDENTITY}, we easily obtain\n\\begin{equation}\\label{eqIntegralDeltaU}\n\\int_{\\mathbb H^n} |\\Delta_g u_R|^p dV_g = \\int_0^{+\\infty} f_R(s)^p ds = 1+ \\ln (R\/s_0) + \\int_0^1 (1-s)^pds.\n\\end{equation}\nBy Proposition \\ref{decomposefunction}, we have\n\\begin{align*}\n\\|u_R\\|_{p} &= \\Big(\\int_0^{+\\infty} v_{R,1}(s)^p ds\\Big)^{1\/p} \\\\\n&\\geqslant \\Big(\\int_0^{+\\infty} h_{R,1}(s)^p ds\\Big)^{1\/p} - \\Big(\\int_0^{+\\infty} |w_{R,1}|^p ds\\Big)^{1\/p}\\\\\n&\\geqslant \\frac1{(1+\\varepsilon)^2} \\frac{pp'}{(n-1)^2}\\Big(\\int_0^{+\\infty} f_R(s)^p ds\\Big)^{1\/p} -C\\\\\n&= \\frac1{(1+\\varepsilon)^2} \\frac{pp'}{(n-1)^2} \\Big(1 + \\ln(R\/s_0) + \\int_0^1(1-t)^p dt\\Big)^{1\/p} -C.\n\\end{align*}\nCombing this estimate and \\eqref{eqIntegralDeltaU} gives\n\\[\nC(n,2,p) \\geqslant \\liminf_{R\\to +\\infty} \\frac{\\|u_R\\|_p}{\\|\\Delta_g u_R\\|_p} \\geqslant \\frac1{(1+\\varepsilon)^2}\\frac{pp'}{(n-1)^2}.\n\\]\nSince $\\varepsilon >0$ is arbitrary, we conclude that\n\\[\nC(n,2,p) \\geqslant \\frac{pp'}{(n-1)^2}\n\\]\nand this finishes our proof for the case $m=2$.\n\n\\subsubsection{The sharpness of $C(n,2k,p)$ with $k\\geqslant 2$} \n\nIn this case, we set \n$$u_R(x) = v_{R,k}(|B(0,d(0,x))|),$$\nthen it is clear to see that \n$$(-\\Delta_g)^k u_R(x) = f_R (|B(0,d(0,x))|).$$ \nBy Proposition \\ref{decomposefunction}, we can write $v_{R,k} = h_{R,k} + w_{R,k}$ with $\\int_0^{+\\infty} |w_{R,k}|^p ds \\leqslant C$ and \n\\[\n\\frac1{(1+\\varepsilon)^{2k}} \\Big(\\frac{pp'}{(n-1)^2}\\Big)^k f_R \\leqslant h_{R,k} \\leqslant \\Big(\\frac{pp'}{(n-1)^2}\\Big)^k f_R.\n\\]\nUsing again the argument in proving the sharpness of $C(n,2,p)$, we obtain the sharpness of $C(n,2k,p)$.\n\n\\subsubsection{The sharpness of $C(n,2k+1,p)$ with $k\\geqslant 1$} \n\nIn the previous argument, we can find a function $u_R$ on $\\mathbb H^n$ such that \n$$(-\\Delta_g)^k u_R(x) = f_R(|B(0,d(0,x))|)$$\nand that\n\\[\n\\|u_R\\|_p \\geqslant \\frac1{(1+\\varepsilon)^{2k}} \\Big(\\frac{pp'}{(n-1)^2}\\Big)^k \\Big(\\int_0^{+\\infty} f_R(s)^p ds\\Big)^{1\/p}- C.\n\\]\nFrom the proof of the sharpness of $C(n,1,p)$, we know that\n\\[\n\\int_{\\mathbb H^n} |\\nabla_g (\\Delta_g^k u_R) |^p dV_g\\leqslant \\Big( \\frac{ n-1 }{p} \\Big)^p (1+\\varepsilon)^p \\ln \\frac R{s_0} + (n-1)^p (1+\\varepsilon)^p \\int_0^1(1-t)^p dt.\n\\]\nCombining these two estimate implies the sharpness of $C(n,2k+1,p)$.\n\n\n\n\n\\section*{Acknowledgments}\n\nV.H.N would like to acknowledge the support of the CIMI postdoctoral research fellowship. The research of Q.A.N is funded by the VNU University of Science under project number TN.16.01.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ALP-GMM}\n\\label{app-alp-gmm}\nALP-GMM \\cite{portelas2019} (MIT license) relies on an empirical per-task computation of Absolute Learning Progress (ALP), allowing to fit a GMM on a concatenated space composed of tasks' parameters and respective ALP. Given a task $a_{new} \\in \\mathcal{A}$ whose parameter is $p_{new} \\in \\mathcal{P}$ and on which the student's policy collected the episodic reward $r_{new} \\in \\mathbb{R}$, Its ALP is computed using the closest previous tasks $a_{old}$ (Euclidean distance) with associated episodic reward $r_{old}$:\n\\begin{equation}\n    \\label{eq:2}\n    alp_{new} = |r_{new} - r_{old}|\n\\end{equation}\nAll previously encountered task's parameters and their associated ALP, parameter-ALP for short, recorded in a history database $H$, are used for this computation. Contrastingly, the fitting of the GMM is performed every $N$ episodes on a window $\\mathcal{W}$ containing the $N$ most recent parameter-ALP. The resulting mean ALP dimension of each Gaussian of the GMM is used for proportional sampling. To adapt the number of components of the GMM online, a batch of GMMs having from 2 to $k_{max}$ components is fitted on $\\mathcal{W}$, and the best one, according to Akaike's Information Criterion \\cite{aic}, is kept as the new GMM. In all of our experiments we use the same hyperparameters as in \\cite{portelas2019} ($N=250$, $k_{max}=10$), except for the percentage of random task sampling $\\rho_{rnd}$ which we set to $10\\%$ (we found it to perform better than $20\\%$) when running ALP-GMM. See algorithm \\ref{algo:ALP-GMM} for pseudo-code and figure \\ref{ALP-GMM-pipeline} for a schematic pipeline. Note that in the main body of this paper we refer to ALP as LP for simplicity (ie. $LP_{ti}$ in $\\mathcal{C}$ from eq. \\ref{eq:craw} is equivalent to the mean ALP of Gaussians in ALP-GMM).\n\n\n\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Absolute Learning Progress Gaussian Mixture Model (ALP-GMM)}\n\t\\label{algo:ALP-GMM}\n\t\\begin{algorithmic}\n\t\n\t\\REQUIRE Student policy $s_\\theta$, parametric procedural environment generator $E$, bounded parameter space $\\mathcal{P}$, probability of random sampling $\\rho_{rnd}$, fitting rate $N$, max number of Gaussians $k_{max}$\n\t\\vspace{0.2cm}\n\t\\STATE Initialize $s_\\theta$\n\t\\STATE Initialize parameter-ALP First-in-First-Out window $\\mathcal{W}$, set max size to $N$\n\t\\STATE Initialize parameter-reward history database $H$\n\t\\LOOP[{$N$ times ~~~~{\\color{gray}~~~\\#  Bootstrap phase}}]\n\t    \\STATE Sample random $p \\in \\mathcal{P}$, send $E(a \\sim \\mathcal{A}(p))$ to $s_\\theta$, observe episodic reward $r_p$\n\t    \\STATE Compute ALP of $p$ based on $r_p$ and $H$ (see equation \\ref{eq:2})\n\t    \\STATE Store $(p,r_p)$ pair in $H$, store $(p,ALP_p)$ pair in $\\mathcal{W}$\n\t\\ENDLOOP\n\t\\LOOP[{{\\color{gray}~~~\\#  Stop after $K$ inner loops}}]\n\t\\STATE Fit a set of GMM having 2 to $k_{max}$ kernels on $\\mathcal{W}$\n\t\\STATE Select the GMM with best Akaike Information Criterion\n\t\\LOOP[{$N$ times}]\n\t    \\STATE $\\rho_{rnd} \\%$ of the time, sample a random parameter $p \\in \\mathcal{P}$\n\t    \\STATE Else, sample $p$ from a Gaussian chosen proportionally to its mean ALP value \n\t\t\\STATE Send $E(a \\sim \\mathcal{A}(p))$ to student $s_\\theta$ and observe episodic reward $r_p$\n\t    \\STATE Compute ALP of $p$ based on $r_p$ and $H$\n\t    \\STATE Store $(p,r_p)$ pair in $H$, store $(p,ALP_p)$ pair in $\\mathcal{W}$\n\t\\ENDLOOP\n\t\\ENDLOOP\n\t\\STATE \\textbf{Return} $s_\\theta$\n\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[width=0.80\\textwidth]{graphics\/alp-gmm_pipeline_example_final.pdf}\n\\caption{\\footnotesize{Schematic view of an ALP-GMM teacher's workflow from \\cite{portelas2019}}}\n\\label{ALP-GMM-pipeline}\n\\end{figure*}\n\n\\section{AGAIN}\n\\label{app-again}\n\n\n\\paragraph{IN variants.} In order to filter the list $\\mathcal{C}_{raw}$ (see eq. \\ref{eq:craw}) of GMMs extracted from a training trajectory $\\tau_{s}$ selected in training trajectory history $\\mathcal{H}$ into $\\mathcal{C}$ and use it as an expert curriculum, we remove any Gaussian with a $LP_{ti}$ below $\\delta_{LP}=0.2$ (the LP dimension is normalized between $0$ and $1$, which requires to choose an approximate potential reward range, set to $[-150,350]$ for all experiments on Box2D locomotion environments (sec. \\ref{sec:exp:walker-climber} and sec. \\ref{exp:trying-again}). When all Gaussians of a GMM are discarded, the GMM is removed from $\\mathcal{C}$. In practice, it allows  to 1) remove non-informative GMMs corresponding to the initial exploration phase of ALP-GMM, when the learner has not made any progress (hence no LP detected by the teacher), and 2) remove an entire training trajectory $\\tau_{s}$ if ALP-GMM never detected high-LP Gaussians, i.e. it failed to train student $s$. $\\mathcal{C}$ is then iterated over to generate a curricula with either of the Time-based (see algo. \\ref{int-algo}), Pool-based (see algo \\ref{inp-algo}) or Reward-based (The one used in our main experiments, see algo \\ref{inr-algo}) IN. The IN-P approach does not require additional hyperparameters. The IN-T requires an update rate $N$ to iterate over $\\mathcal{C}$, which we set to $250$ (same as the fitting rate of ALP-GMM). The IN-R approach requires to extract additional data from the first run, in the form of a list $\\mathcal{R}_{raw}$:\n\\begin{equation}\n\\label{eq:rraw}\n    \\mathcal{R}_{raw} = \\{\\mu_r^1, ...,\\mu_r^t, \\mu_r^T\\} ~~s.t~~~ |\\mathcal{R}_{raw}| = |\\mathcal{C}_{raw}|,\n\\end{equation}\nwith T the total number of GMMs in the first run (same as in $\\mathcal{C}_{raw}$), and $\\mu_r^t$ the mean episodic reward obtained by the first DRL agent during the last $50$ tasks sampled from the $t^{th}$ GMM. $\\mathcal{R}$ is simply obtained by removing any $\\mu_r^t$ that corresponds to a GMM discarded while extracting $\\mathcal{C}$ from $\\mathcal{C}_{raw}$. The remaining rewards are then used as thresholds in IN-R to decide when to switch to the next GMM in $\\mathcal{C}$.\n\n\\paragraph{AGAIN} In AGAIN (see algo. \\ref{again-algo}), the idea is to use both IN (R,T or P) and ALP-GMM (without the random bootstrapping period) for curriculum generation. Our main experiments use IN-R as it is the highest performing variant (see app. \\ref{ann:toy-exp}). This means that in the main sections of this paper, AGAIN $=$ AGAIN-R and IN $=$ IN-R. We combine the changing GMM of IN and ALP-GMM over time, simply by building a GMM $G$ containing Gaussians from the current GMM of IN and ALP-GMM. By selecting the Gaussian in $G$ from which to sample a new task using their respective LP, this approach allows to adaptively modulate the task sampling between both, shifting the sampling towards IN when ALP-GMM does not detect high-LP subspaces and towards ALP-GMM when the current GMM of IN have lower-LP Gaussians. While combining ALP-GMM to IN, we reduce the residual random sampling of ALP-GMM from $\\rho_{high}=10\\%$, used for the pretrain phase, to either $\\rho_{low}=2\\%$ for experiments presented in sec. \\ref{sec:exp:toy-env} and sec. \\ref{exp:trying-again}, or $\\rho_{low}=0\\%$  for experiments done in the Parkour environment in sec. \\ref{sec:exp:walker-climber} (here we found $\\rho_{low}=0\\%$ to be beneficial in terms of performances w.r.t. $\\rho_{low}=2\\%$, which means that the task-exploration induced by the periodic GMM fit of ALP-GMM was sufficient for exploration). In AGAIN-R and AGAIN-T, when the last GMM $p(T)$ of the IN curriculum is reached, we switch the fixed $LP_{Ti}$ values of all IN Gaussians to periodically updated LP estimates, i.e. we allow AGAIN to modulate the importance of $p(T)$ for task sampling depending on its current student's performance.\n\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Pretrain phase (helper function)}\n\t\\label{pret-algo}\n\t\\begin{algorithmic}[1]\n\t\\REQUIRE Student policy $s_\\theta$, teacher training history $\\mathcal{H}$, task-encoding parameter space $\\mathcal{P}$, LP threshold $\\delta_{LP}$, experimental pre-train budget $K_{pre}$, pre-test set size $m$, number of neighbors for student selection $k$, random sampling ratio $\\rho_{high}$, parametric procedural environment generator $E$\n\t\\vspace{0.2cm}\n\t\\STATE Init $s_{\\theta}$, train it for $K_{pre}$ env. steps with ALP-GMM($\\rho_{high}, \\mathcal{P}$) \n\t\\STATE Pre-test $s_{\\theta}$ with $m$ tasks selected uniformly over $\\mathcal{P}$ and get $KC_s^{pre}$ \\COMMENT{{\\color{gray}~~~\\# Pre-test phase}}\n\t\\STATE Apply knn algorithm in KC space of $\\mathcal{H}$, get $k$ students closest to $KC_s^{pre}$ \\STATE Among those $k$, keep the one with highest summed post training $KC^{post}$, extract its $\\mathcal{C}_{raw}$\n\t\\STATE Get $\\mathcal{C}$ from $\\mathcal{C}_{raw}$ by removing any Gaussian with $LP_{ti} < \\delta_{LP}$.\n\t\\STATE \\textbf{Return} $\\mathcal{C}$\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Inferred progress Niches - Time-based (IN-T)}\n\t\\label{int-algo}\n\t\\begin{algorithmic}[1]\n\t\n\t\\REQUIRE Student policy $s_\\theta$, teacher training history $\\mathcal{H}$, task-encoding parameter space $\\mathcal{P}$, LP threshold $\\delta_{LP}$, update rate $N$, experimental budget $K$, experimental pre-train budget $K_{pre}$, pre-test set size $m$, number of neighbors for student selection $k$, random sampling ratio $\\rho_{high}$, parametric procedural environment generator $E$\n\t\\vspace{0.2cm}\n\t\\STATE Launch Pretrain phase and get expert GMM list $\\mathcal{C}$ \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{pret-algo}}}\n\t\\STATE Initialize expert curriculum index $i_{c}$ to $0$\n\t\\LOOP[{~Stop after $K - K_{pre}$ environment steps}]\n\t    \\STATE Set $i_{c}$ to $min(i_{c}+1, len(\\mathcal{C}))$\n\t    \\STATE Set current GMM $G_{IN}$ to $i_{c}^{th}$ GMM in $\\mathcal{C}$\n\t    \\LOOP[{$N$ times}]\n\t    \\STATE Sample $p$ from a Gaussian in $G_{IN}$ chosen proportionally to its $LP_{ti}$\n\t\t\\STATE Send $E(a \\sim \\mathcal{A}(p))$ to student $s_{\\theta}$\n\t\\ENDLOOP\n\t\\ENDLOOP\n\t\\STATE Add student's training trajectory to $\\mathcal{H}$\n\t\\STATE \\textbf{Return} $s_\\theta$\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Inferred progress Niches - Pool-based (IN-P)}\n\t\\label{inp-algo}\n\t\\begin{algorithmic}[1]\n\t\n\t\\REQUIRE Student policy $s_\\theta$, teacher training history $\\mathcal{H}$, task-encoding parameter space $\\mathcal{P}$, LP threshold $\\delta_{LP}$, update rate $N$, experimental budget $K$, experimental pre-train budget $K_{pre}$, pre-test set size $m$, number of neighbors for student selection $k$, random sampling ratio $\\rho_{high}$, parametric procedural environment generator $E$\n\t\\vspace{0.2cm}\n\t\\STATE Launch Pretrain phase and get expert GMM list $\\mathcal{C}$ \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{pret-algo}}} \n\t\\STATE Initialize pool GMM $G_{IN}$, containing all Gaussians from $\\mathcal{C}$\n\t\\LOOP[{~Stop after $K - K_{pre}$ environment steps}]\n\t    \\STATE Sample $p$ from a Gaussian in $G_{IN}$ chosen proportionally to its $LP_{ti}$\n\t\t\\STATE Send $E(a \\sim \\mathcal{A}(p))$ to student $s_\\theta$\n\t\\ENDLOOP\n\t\\STATE Add student's training trajectory to $\\mathcal{H}$\n\t\\STATE \\textbf{Return} $s_\\theta$\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Inferred progress Niches - Reward-based (IN-R)}\n\t\\label{inr-algo}\n\t\\begin{algorithmic}[1]\n\t\n\t\\REQUIRE Student policy $s_\\theta$, teacher training history $\\mathcal{H}$, task-encoding parameter space $\\mathcal{P}$, LP threshold $\\delta_{LP}$, update rate $N$, experimental budget $K$, experimental pre-train budget $K_{pre}$, pre-test set size $m$, number of neighbors for student selection $k$, random sampling ratio $\\rho_{high}$, parametric procedural environment generator $E$\n\t\\vspace{0.2cm}\n\t\\STATE Launch Pretrain phase and get expert GMM list $\\mathcal{C}$ \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{pret-algo}}} \n\t\\STATE Initialize reward First-in-First-Out window $\\mathcal{W}$, set max size to $N$\n\t\\STATE Initialize expert curriculum index $i_{c}$ to $0$\n\t\\LOOP[{~Stop after $K - K_{pre}$ environment steps}]\n\t    \\STATE If $\\mathcal{W}$ is full, compute mean reward $\\mu_w$ from $\\mathcal{W}$\n\t    \\STATE ~~~~If $\\mu_w$ superior to $i_{c}^{th}$ reward threshold in $\\mathcal{R}$, set $i_{c}$ to $min(i_{c}+1, len(\\mathcal{C}))$\n\t    \\STATE Set current GMM $G_{IN}$ to $i_{c}^{th}$ GMM in $\\mathcal{C}$\n\t    \\STATE Sample $p$ from a Gaussian in $G_{IN}$ chosen proportionally to its $LP_{ti}$\n\t\t\\STATE Send $E(a \\sim \\mathcal{A}(p))$ to student $s_\\theta$ and add episodic reward $r_p$ to $\\mathcal{W}$\n\t\\ENDLOOP\n\t\\STATE Add student's training trajectory to $\\mathcal{H}$\n\t\\STATE \\textbf{Return} $s_\\theta$\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{algorithm*}[htb!]\n\t\\caption{~ Alp-Gmm And Inferred progress Niches (AGAIN)}\n\t\\label{again-algo}\n\t\\begin{algorithmic}[1]\n\t\n\t\\REQUIRE Student policy $s_\\theta$, teacher training history $\\mathcal{H}$, task-encoding parameter space $\\mathcal{P}$, LP threshold $\\delta_{LP}$, update rate $N$, experimental budget $K$, experimental pre-train budget $K_{pre}$, pre-test set size $m$, number of neighbors for student selection $k$, random sampling ratio $\\rho_{low}$ and $\\rho_{high}$, parametric procedural environment generator $E$\n\t\\vspace{0.2cm}\n\t\\STATE Launch Pretrain phase and get expert GMM list $\\mathcal{C}$ \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{pret-algo}}}\n\t\\STATE Setup new ALP-GMM($\\rho_{rnd}=0, \\mathcal{P}$) \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{algo:ALP-GMM}}} \n\t\\STATE Setup either IN-T, IN-P or IN-R \\COMMENT{{\\color{gray}~~~\\# See algo. \\ref{int-algo}, \\ref{inp-algo} and \\ref{inr-algo}}} \n\t\\LOOP[{~Stop after $K - K_{pre}$ environment steps}]\n\t    \\STATE Get composite GMM $G$ from the current GMM of both ALP-GMM and IN\n\t    \\STATE $\\rho_{low} \\%$ of the time, sample a random parameter $p \\in \\mathcal{P}$\n\t    \\STATE Else, sample $p$ from a Gaussian chosen proportionally to its $LP$ \n\t\t\\STATE Send $E(a \\sim \\mathcal{A}(p))$ to student $s_\\theta$ and observe episodic reward $r_p$\n\t    \\STATE Send $(p,r_p)$ pair to both ALP-GMM and IN\n\t\\ENDLOOP\n\t\\STATE Add student's training trajectory to $\\mathcal{H}$\n\t\\STATE \\textbf{Return} $s_\\theta$\n\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\n\\section{Considered ACL and Meta-ACL teachers}\n\\label{an:details}\n\\paragraph{Meta-ACL variants} Our proposed approach, AGAIN, is based on the combination of an inferred expert curriculum with ALP-GMM, an exploratory ACL approach. In section \\ref{sec:methods} and appendix \\ref{app-again}, we present $3$ approaches to use such an expert curriculum, giving the AGAIN-R, AGAIN-P and AGAIN-T algorithms. In our experiments, we also consider ablations were we only use the expert curriculum, giving the IN-R, IN-P and IN-T variants. We also consider two additional AGAIN variants that do not use our proposed KC-based student selection method:\n\\begin{itemize}\n    \\item AGAIN with Random selection (AGAIN\\_RND), a lower-baseline ablation were we select the training trajectory $\\tau$ from which to extract the expert curriculum randomly in history $\\mathcal{H}$.\n    \\item AGAIN with Ground Truth selection (AGAIN\\_GT), an upper-baseline using privileged information. Instead of performing the knn algorithm in the KC space, this approach directly uses the true student distribution. For instance, in the Parkour environment, given a new student $s$, AGAIN\\_GT selects the $k$ previously trained students from $\\mathcal{H}$ that are morphologically closest to $s$ (i.e. same embodiment type and closest limb sizes), and uses the training trajectory of the student with highest score $j_s$ (see sec. \\ref{sec:methods}).\n    \n   \n\\end{itemize}\nNote that both for AGAIN\\_RND and AGAIN\\_GT, there is no need to pre-test the student, which means we can use the IN expert curriculum directly at the beginning of training rather than after a pre-training phase.\n\n\n\\paragraph{ACL conditions} A first natural ACL approach to compare our AGAIN variants to is ALP-GMM, the underlying ACL algorithm in AGAIN. We also add as a lower-baseline a random curriculum teacher (Random), which samples tasks' parameters randomly over the task space.\n\nIn both the toy environment (sec. \\ref{sec:exp:toy-env}, toy env. for short) and the Parkour environment (sec. \\ref{sec:exp:walker-climber}), we additionally compare to Adaptive Domain Randomization (ADR), an ACL algorithm proposed in \\cite{OpenAI2019SolvingRC}, which is based on inflating a task distribution sampling from a predefined initially feasible task $p_{easy}$ (w.r.t a given student). Each lower and upper boundaries of each dimension of the sampling distribution are modified independently with step size $\\Delta_{step}$ whenever a predefined mean reward threshold $r_{thr}$ is surpassed over a window (of size $q$) of tasks occasionally sampled (with probability $\\rho_b$) at the sampling dimension boundary. More details can be found in \\cite{OpenAI2019SolvingRC}. In our experiments, as we do not assume access to expert knowledge over students sampled within the student distribution, we randomize the setting of $p_{easy}$ uniformly over the task space in Parkour experiments and uniformly over the $4$ possible student starting subspaces in toy env. experiments. Based on the hyperparameters proposed in \\cite{OpenAI2019SolvingRC} and on informal hyperparameter search, we use $[\\rho_b=0.5, r_{thr}=1, \\Delta_{step}=0.05, q=10]$ in toy env. experiments and $[\\rho_b=0.5, r_{thr}=230, \\Delta_{step}=0.1, q=20]$ in Parkour experiments.\n\nIn experiments described in sec \\ref{exp:trying-again}, we compare our approaches to an oracle condition (Oracle), which is a hand-made curriculum that is very similar to IN-R, except that the list $\\mathcal{C}$ is built using expert knowledge before training starts (i.e. no pre-train and pre-test phases), and all reward thresholds $\\mu_r^i$ in $\\mathcal{R}$ (see eq. \\ref{eq:rraw}) are set to $230$, which is an episodic reward value often used in the literature as characterizing a default walker having a \"reasonably efficient\" walking gate in environments derived from the Box2D gym environment BipedalWalker \\cite{poet,portelas2019}.In practice, Oracle starts proposing tasks from a Gaussian (with std of $0.05$) located at the simplest subspace of the task space (ie. low stump height and high stump spacing) and then gradually moves the Gaussian towards the hardest subspaces (high stump height and low stump spacing) by small increments ($50$ steps overall) happening whenever the mean episodic reward of the DRL agent over the last $50$ proposed tasks is superior to $230$.\n\n\n\\newpage\\section{Analysing Meta-ACL in a toy environment}\n\\label{ann:toy-exp}\n\nIn this section we report the full comparative experiments done in the toy environment, which includes comparisons with AGAIN-T and AGAIN-P to AGAIN-R, shown in table \\ref{toy-env-results-table}. We also provide visualizations of the KC-based curriculum priors selection process (see fig. \\ref{toy-exps-stud-selec-vizu}) happening after the pretraining phase in AGAIN along with a visualization of the fixed set of $96$ randomly drawn students used to perform the varying classroom experiments reported in sec. \\ref{sec:exp:toy-env} (see fig. \\ref{classroom-toyenv-vizu}).\n\n\\paragraph{Additional comparative analysis} Table \\ref{toy-env-results-table} summarizes the post-training performances obtained by our considered Meta-ACL conditions and ACL baselines on the toy environment with only $4$ possible students on a fixed set of $48$ randomly drawn students. Meta-ACL conditions are given a training trajectory $\\mathcal{H}$ created by training an initial classroom of $128$ students. Using a Reward-based iterating scheme over the inferred expert curriculum (AGAIN-R and IN-R) outperforms the Time-based and Pool-based variants ($p<.001$). This result was expected as both these last two variants do not have flexible mechanisms to adapt to the student being trained. The pool based variants (AGAIN-P and IN-P), which discard the temporal ordering of the expert curriculum are the worst performing variants, statistically significantly inferior to both Reward-based and Time-based conditions ($p<.001$).\n\n\\begin{table}[htb!]\n\\caption{\\footnotesize{\\textbf{Experiments on the toy environment.} The average performance with standard deviation after 200k episodes is reported (48 seeds per conditions). For Meta-ACL variants we report results with column 1) the regular KC-based curriculum prior selection performed after $20$k pre-training episodes, column 2) An ablation that performs the selection at random before training, and column 3) An oracle condition selecting before training the curriculum prior using student ground truth type. \\textit{*} Denotes stat. significant advantage w.r.t. ALP-GMM (Welch's t-test at $200k$ ep. with $p<0.05$).}}\n\\vspace{0.3cm}\n    \\centering\n    \\footnotesize\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nCondition & Regular        & Random  & Ground Truth  \\\\ \\midrule\nAGAIN-R   & 98.8 +- 4.8* & 55.4 +- 32.2     & 99.8 +- 0.9*         \\\\ \nIN-R      & 91.4 +- 3.4* & 26.3 +- 41.1     & 92.5 +- 3.0*         \\\\ \nAGAIN-T   & 84.3 +- 3.8    & 38.6 +- 34.1     & 89.0 +- 1.7*         \\\\ \nIN-T      & 79.0 +- 12.0   & 30.3 +- 37.3     & 88.9 +- 1.7*         \\\\ \nAGAIN-P   & 38.2 +- 7.5    & 9.3 +- 9.2      & 14.8 +- 1.2            \\\\ \nIN-P      & 40.6 +- 6.4    & 9.2 +- 9.0       & 15.1 +- 1.2            \\\\ \\midrule\nALP-GMM   & 84.6 +- 3.4    &                  &                        \\\\ \nADR       & 14.9 +- 27.4   &                  &                        \\\\ \nRandom    & 10.0 +- 0.8    &                  &                        \\\\  \\bottomrule\n\\end{tabular}\n\n\n    \\label{toy-env-results-table}\n\\end{table}\n\n\\begin{figure*}[htb!]\n    \\centering\n    \\subfloat{\\includegraphics[width=0.3\\textwidth]{graphics\/sample_cp10_varying_classroom_size_expe.pdf}}\n    \\subfloat{\\includegraphics[width=0.3\\textwidth]{graphics\/sample_for_varying_classroom_size_expe.pdf}}\n    \\caption{\\footnotesize{ Additional visualizations for the varying classroom size experiments (see sec. \\ref{sec:exp:toy-env}).\n    \\textbf{Left:} Visualization of the starting cells of students from a $10$\\% sample of a classroom of $400$ students (one per student type) trained with ALP-GMM and used to populate the training trajectory $\\mathcal{H}$. Each blue circle marks the starting cell of each student (i.e. its type) within the $2$D parameter space $\\mathcal{P}$, which is an initial learning subspace that needs to be detected by the teacher for successful training.  \\textbf{Right:}Visualization of the fixed set of $96$ randomly drawn students that have to be trained by Meta-ACL variants given $\\mathcal{H}$. As not all student types are represented in $\\mathcal{H}$, Meta-ACL approaches have to generalize their curriculum generation to these new students.}}\n    \\label{classroom-toyenv-vizu}\n\\end{figure*}\n\n\\begin{figure*}[htb!]\n\\centering\n\\subfloat[with type 0 new student]{\\includegraphics[width=0.35\\textwidth]{graphics\/metacl_prior_selection_10-06_toy_env_teacher_CEGT_expert_type_R_use_alpgmm_pt_4_sR_106.png}}\n\\subfloat[with type 1 new student]{\\includegraphics[width=0.35\\textwidth]{graphics\/metacl_prior_selection_10-06_toy_env_teacher_CEGT_expert_type_R_use_alpgmm_pt_4_sR_130.png}}\n\n\\subfloat[with type 2 new student]{\\includegraphics[width=0.35\\textwidth]{graphics\/metacl_prior_selection_10-06_toy_env_teacher_CEGT_expert_type_R_use_alpgmm_pt_4_sR_110.png}}\n\\subfloat[with type 3 new student]{\\includegraphics[width=0.35\\textwidth]{graphics\/metacl_prior_selection_10-06_toy_env_teacher_CEGT_expert_type_R_use_alpgmm_pt_4_sR_131.png}}\n\n\\caption{\\footnotesize{\\textbf{Examples of student selection process in $4$-student type toy environment.} In all figures, we plot the $2$D PCA visualization of the $KC^{pre}$ vectors (after pre-training) of the initial classroom ($128$ students) trained with ALP-GMM and used to populate the training trajectory $\\mathcal{H}$ used by AGAIN variants in our $4$-student type toy env experiments (see sec. \\ref{sec:exp:toy-env}). We then use these $4$ figures to showcase the selection process happening in $4$ different AGAIN-R runs (one per student type). Each triangle represents a student, whose ground truth type (i.e. its initial learning cell) is denoted by the orientation of the triangle. Given a new student to train, AGAIN pretrains the student, constructs its KC vector (purple border triangle), infers the k closest previously trained students from $\\mathcal{H}$ (red and golden border triangles), and use the one with highest end of training performance (i.e. highest score $s$, see sec. \\ref{sec:methods}), denoted by a golden border triangle, to infer curriculum priors for the new student.}}\n\\label{toy-exps-stud-selec-vizu}\n\\end{figure*}\n\n\\begin{figure*}[b]\n\\centering\n\\subfloat{\\includegraphics[width=0.35\\textwidth]{graphics\/concatenated_parkour_agent_images.jpg}}\\hspace{0.2cm}\\subfloat{\\includegraphics[width=0.35\\textwidth]{graphics\/concatenated_parkour_tracks_images.jpg}}\n\\caption{\\footnotesize{Visualizations of the student space and the task space of the Parkour environment.\\textbf{Left:} Examples of possible agent embodiments (randomly set for a given DRL learner before training starts). \\textbf{Right:} Examples of randomly sampled parkour tracks.}}\n\\label{parkour-env-vizu}\n\\end{figure*}\n\n\n\\vspace{10cm}\\section{Meta-ACL for DRL students in the Parkour environment}\n\\label{ann:parkour}\nIn this section we give additional details on the Parkour environment presented in section \\ref{sec:exp:walker-climber}, and we provide additional details and visualizations on the experiments that were performed on it.\n\n\\paragraph{Details on the Parkour environment.} In our experiments, we bound the wall spacing dimension of the task space to $\\Delta_{w}=[0,6]$, and the gate y position to $\\mu_{gate}=[2.5,7.5]$. In practice, given a single parameter tuple $(\\mu_{gate},\\Delta_{w})$, we actually encode a distribution of tasks, since for each new wall along the track we add an independent Gaussian noise to each wall's gate y position $\\mu_{gate}$. Examples of parkour tasks randomly sampled within these bounds are available in figure \\ref{parkour-env-vizu} (right). At the beginning of training a given DRL policy, the agent is embodied in either a bipedal walker morphology with two joints per legs or a two-armed climber morphology with 3-joints per arms ended by a grasping \"hand\". Both morphologies are controlled by torque. Climbers have an additional action dimension $g \\in [-1,1]$ used to grasp: if $g \\in [-1,0[$, the climber closes its gripper, and if $g \\in ]0,1]$ it keeps it open. To avoid falling (which aborts the episode with a $-100$ penalty) while moving forward to collect rewards, climber agents must learn to swing themselves forward by successive grasp-and-release action sequences. To increase the diversity of the student distribution, we also randomize limb sizes. See figure \\ref{parkour-env-vizu} (left) for examples of randomly sampled embodiments.\n\n\\paragraph{Soft Actor-Critic students} In our experiments, we use an implementation of Soft Actor-Critic provided by OpenAI\\footnote{https:\/\/github.com\/openai\/spinningup} (MIT license). We use a $2$ layered ($400$,$300$) network for V, Q1, Q2 and the policy. Gradient steps are performed each $10$ environment steps, with a learning rate of $0.001$ and a batch size of $1000$. The entropy coefficient is set to $0.005$.\n\n\\paragraph{Evaluation procedure} To report the performance of our students on the Parkour environment, we use two separate test sets, one per embodiment type. For walkers we use a $100$-tasks test set, uniformly sampled over a subspace of the task space with $\\Delta_w \\in [0,6]$ and $\\mu_{gate} \\in [2.5,3.6]$, which we chose based on 1) what we initially believed to be morphologically feasible for walkers, and 2) based on previously designed test sets built in recent work \\cite{portelas2019} on comparable bipedal walker experiments). For climbers, because there is no similar experiments in the literature and since it is hard to infer beforehand what will be achievable by such a morphology, we simply use a uniform test set of $225$ tasks sampled over the full task space. Importantly, the customized test set used for walkers is solely used for visualization purposes. In our AGAIN approaches, we pre-test all students with the expert-knowledge-free set of $225$ tasks uniformly sampled over the task space. \n\n\\paragraph{Compute resources} Each of the 576 seeds required to reproduce our experiments (128 seeds for the classroom and 7*64 seeds for our 7 conditions) takes 36 hours on a single cpu. This amounts to around 21 000 cpu hours. Each run requires less than 1GB of RAM.\n\n\n\n\\paragraph{Visualizing student diversity.} To assess whether our proposed multi-modal distribution of possible students in the Parkour environment do have diverse competence profiles (which is desirable as it creates a challenging Meta-ACL scenario), we plot the 2D PCA of the post training KC vector for each students of the initial classroom trained with ALP-GMM (used to populate $\\mathcal{H}$). The result, visible in figure \\ref{vizu-parkour-classroom} (top), shows that climber-students and walker-students are located in two independant clusters, i.e. they do have clearly different competence profiles. The spread of each clusters also demonstrates that variations in initial policy parameters and limb sizes also creates students with diverse learning potentials. The competence differences between walkers and climbers can also be seen in Figure \\ref{vizu-parkour-classroom} (left and right), which shows the episodic reward obtained for each of the $225$ tasks of the KC vector after training by a representative walker student (left) and climber student (right). \n\n\n\n\\begin{figure*}[htb!]\n\\centering\n\\subfloat{\\includegraphics[width=0.6\\textwidth]{graphics\/pca_KC_post_parkour.pdf}}\n\n\n\\subfloat{\\includegraphics[width=0.45\\textwidth]{graphics\/classroom_KC_0_11.pdf}}\\hspace{0.3cm}\\subfloat{\\includegraphics[width=0.45\\textwidth]{graphics\/classroom_KC_1_3.pdf}}\n\\caption{\\footnotesize{\\textbf{top:} PCA of classroom's KC vector (128 students) after being trained for 10M student steps with ALP-GMM. \\textbf{left and right:} Episodic reward obtained for each task that compose the KC vector by a walker-student (left) and a climber-student (right) of this classroom. Stars are added for all tasks for which the agent obtained more than $r=230$ (which corresponds to an efficient locomotion policy). Walkers only manage to learn tasks with very low gate positions while climbers learn only tasks with medium to high gate positions.}}\n\\label{vizu-parkour-classroom}\n\\end{figure*}\n\n\\clearpage\\newpage\\section{Applying Meta-ACL to a single student: Trying AGAIN instead of trying longer}\n\\label{ann:tryingagain}\n\nIn the following section we report all experiments on applying AGAIN variants to train a single DRL student (i.e. no history $\\mathcal{H}$), which is briefly presented in sec. \\ref{exp:trying-again}.\n\n\n\\paragraph{Parametric BipedalWalker env.} We test our modified AGAIN variants along with baselines on an existing parametric BipedalWalker environment proposed in \\cite{portelas2019}, which generates walking tracks paved with stumps whose height and spacing are defined by a $2$D parameter vector used for the procedural generation of tasks. We keep the original bounds of this task space, i.e. we bound the stump-height dimension to $\\mu_h \\in [0,3]$ and the stump-spacing dimension to $\\delta_s \\in [0,6]$. As in their work, we also test our teachers when the learning agent is embodied in a modified short-legged walker, which constitutes an even more challenging scenario (as the task space is unchanged, i.e. more unfeasible tasks). The agent is rewarded for keeping its head straight and going forward and is penalized for torque usage. The episode is terminated after 1) reaching the end of the track, 2) reaching a maximal number of $2000$ steps, or 3) head collision (for which the agent receives a strong penalty). See figure \\ref{pbw-demo} for visualizations.\n\n\n\\begin{figure}[htb!]\n\n\\centering\\includegraphics[width=0.75\\columnwidth]{graphics\/pbw_demo.pdf}\n\\caption{\\footnotesize{Parameterized BipedalWalker environment. \\textbf{Left:} Examples of generated tracks. \\textbf{Right:} The two walker morphologies tested on the environment.} One parameter tuple ($\\mu_h, \\delta_s$) actually encodes a \\textit{distribution} of tasks as the height of each stump along the track is drawn from $\\mathcal{N}(\\mu_h,0.1)$. }\n\\label{pbw-demo}\n\\end{figure}\n\n\\paragraph{Results} To perform our experiments, we ran each condition for either $10$Millions (IN and AGAIN variants) or $20$Millions (others)  environment steps ($30$ repeats). The preliminary ALP-GMM runs used in IN and AGAIN variants correspond to the first $10$ Million steps of the ALP-GMM condition (whose end-performance after $20$ Million steps is reported in table \\ref{results-table}. All teacher variants are tested when paired with a Soft-Actor Critic \\cite{sac} student, with same hyperparameters as in the Parkour experiments (see app. \\ref{ann:parkour}). Performance is measured by tracking the percentage of mastered tasks (i.e. $r>230$) from a fixed test set of $100$ tasks sampled uniformly over the task space. We thereafter report results for $2$ independent experiments done with either default walkers or short walkers.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.5\\columnwidth]{graphics\/compact_demo.pdf}\n\\caption{\\footnotesize{Given a single DRL student to train, AGAIN outperforms ALP-GMM in a parametric BipedalWalker environment. sem plotted, 30 seeds.}}\\label{trying-again-compact-app}\n\\end{figure} \n\\textit{Is re-training from scratch beneficial? - } The end performances of all tested conditions are summarized in table \\ref{results-table}. Interestingly, retraining the DRL agent from scratch in the second run gave superior end performances than fine-tuning using the weights of the first run \\textit{in all tested variants}. This showcases the brittleness of gradient-based training and the difficulty of transfer learning. Despite this, even fine-tuned variants reached superior end-performances than classical ALP-GMM, meaning that the change in curriculum strategy in itself is already beneficial.\n\n\\textit{Is it useful to re-use ALP-GMM in the second run? - } In the default walker experiments, AGAIN-R, T and P conditions mixing ALP-GMM and IN in the second run reached lower mean performances than their respective IN variants. However, the exact opposite is observed for IN-R and IN-T variants in the short walker experiments. This can be explained by the difficulty of short walker experiments for ACL approaches, leading to $16\/30$ preliminary 10M steps long ALP-GMM runs to have a mean end-performance of $0$, compared to $0\/30$ in the default walker experiments. All these run failures led to many GMMs lists $\\mathcal{C}$ used in IN to be of very low-quality, which illustrates the advantage of AGAIN that is able to emancipate from IN using ALP-GMM.\n\n\\textit{Highest-performing variants. - } Consistently with the precedent analysis, mixing ALP-GMM with IN in the second run is not essential in default walker experiments, as the best performing ACL approach is IN-P. This most likely suggests that the improved adaptability of the curriculum when using AGAIN is outbalanced by the added noise (due to the low task-exploration). However in the more complex short walker experiments, mixing ALP-GMM with IN is essential, especially for AGAIN-R, which substantially outperforms ALP-GMM and other AGAIN and IN variants (see fig. \\ref{trying-again-compact}), reaching a mean end performance of $19.0$. The difference in end-performance between AGAIN-R and Oracle, our hand-made expert using privileged information who obtained $20.1$, is not statistically significant ($p=0.6$).\n\n\\begin{table*}[]\n\\caption{\\footnotesize{\\textbf{Experiments on Parametric BipedalWalker} The avg. perf. with std. deviation after 10 Millions steps (IN and AGAIN variants) or 20 Million steps (others) is reported (30 seeds). For IN and AGAIN we also test variants that do not retrain the weights of the policy used in the second run \\textit{from scratch} but rather \\textit{fine-tune} them from the preliminary run.$\\mathbf{^{*\/-}}$ Indicates whether perf. difference with ALP-GMM is statistically significant ie. $p<0.05$ in a post-training Welch's student t-test ($\\mathbf{^{*}}$ for performance advantage w.r.t ALP-GMM and $\\mathbf{^{-}}$ for perf. disadvantage).}}\n\\vspace{0.3cm}\n    \\footnotesize\n    \\centering\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nCondition            & Short walker & Default walker \\\\ \\midrule\nAGAIN-R               & $19.0\\pm12.0^*$               & $41.6\\pm6.3^*$           \\\\\nAGAIN-R(fine-tune)    & $11.4\\pm12.9$                 & $39.9\\pm4.6$           \\\\\nIN-R                  & $13.4\\pm14.4$                 & $43.5\\pm9.6^*$           \\\\\nIN-R(fine-tune)       & $11.2\\pm12.3$                 & $40.8\\pm5.6$           \\\\\nAGAIN-T               & $15.1\\pm11.9$                 & $40.6\\pm11.5$           \\\\\nAGAIN-T(fine-tune)    & $11.4\\pm11.8$                 & $40.6\\pm3.8^*$           \\\\\nIN-T                  & $13.5\\pm13.3$                 & $43.5\\pm6.1^*$           \\\\\nIN-T(fine-tune)       & $10.7\\pm12.3$                 & $40.3\\pm7.6$           \\\\\nAGAIN-P               & $13.6\\pm12.5$                 & $41.9\\pm5.1^*$           \\\\\nAGAIN-P(fine-tune)    & $11.1\\pm12.0$                 & $41.5\\pm3.9^*$           \\\\\nIN-P                  & $14.5\\pm12.6$                 & $\\mathbf{44.3}\\pm3.5^*$  \\\\\nIN-P(fine-tune)       & $12.2\\pm12.5$                 & $41.1\\pm3.8^*$           \\\\\nALP-GMM               & $10.2\\pm11.5$                 & $38.6\\pm3.5$           \\\\\nOracle                & $\\mathbf{20.1}\\pm3.4^*$         & $27.2\\pm15.2^-$           \\\\\nRandom                & $2.5\\pm5.9^-$                   & $20.9\\pm11.0^-$           \\\\ \\bottomrule\n\\end{tabular}\n    \\label{results-table}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[htb!]\n\\centering\n\\subfloat[\\textbf{Pool-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_default_P_.pdf}}\n\\subfloat[\\textbf{Time-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_default_T_.pdf}}\n\\subfloat[\\textbf{Reward-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_default_R_.pdf}}\n\\caption{\\footnotesize{\\textbf{Evolution of performance across 20M environment steps of each condition with default bipedal walker.} Each point in each curve corresponds to the mean performance (30 seeds), defined as the percentage of mastered tracks (ie. $r>230$) on a fixed test set. Shaded areas represent the standard error of the mean. Consistently with \\cite{portelas2019}, which implements a similar approach, Oracle is prone to forgetting with default walkers due to the strong shift in task subspace (which is why it is not the best performing condition for default walker experiments.}}\n\\label{default-exps-curves}\n\\end{figure*}\n\n\\begin{figure*}[htb!]\n\\centering\n\\subfloat[\\textbf{Pool-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_short_P_.pdf}}\n\\subfloat[\\textbf{Time-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_short_T_.pdf}}\n\\subfloat[\\textbf{Reward-based} IN]{\\includegraphics[width=0.31\\textwidth]{graphics\/means_short_R_.pdf}}\n\\caption{\\footnotesize{\\textbf{Evolution of performance across 20M environment steps of each condition with short bipedal walker.} Each point in each curve corresponds to the mean performance (30 seeds), defined as the percentage of mastered tracks (ie. $r>230$) on a fixed test set. Shaded areas represent the standard error of the mean.}}\n\\label{short-exps-curves}\n\\end{figure*}\n\\section{Introduction}\n\nThe idea of organizing the learning sequence of a machine is an old concept that stems from multiple works in reinforcement learning \\citep{selfridge,Schmid}, developmental robotics \\citep{oudeyer2007intrinsic} and supervised learning \\citep{elman, bengiocl}, from which the Deep RL community borrowed the term \\textit{Curriculum Learning}. Automatic CL \\citep{portelas2020-acl-drl} refers to \\textit{teacher} algorithms able to autonomously adapt their task sampling distribution to their evolving \\textit{student}. In DRL, ACL has been leveraged to scaffold learners in a variety of multi-task control problems, including video-games \\citep{icm,rnd,montezuma-single-demo}, multi-goal robotic arm manipulation \\citep{her,curious,cideron2019self,fournier-accuracy-acl} and navigation in sets of environments \\citep{tscl,portelas2019,ADRmila,goalgan,selfpaceddrl}. Concurrently, multiple authors demonstrated the benefits of Procedural Content Generation (PCG) as a tool to create rich task spaces to train generalist agents \\citep{risiPCG,illuminating,quantif-coinrun}. The current limit of ACL is that, when applied to such large continuous task spaces, that often have few learnable subspaces, it either relies on 1) human expert knowledge that is hard\/costly to provide (and which undermines how automatic the ACL approach is), or 2) it loses a lot of time finding tasks of appropriate difficulty through \\textit{task exploration}.\n\nGiven the aforementioned impressive results on training DRL learners with ACL to generalize over tasks (which extended the classical single-task scenarios \\citep{dqn,trpo,ddpg} to multi-tasks), we propose to go further and work on training (unknown) distributions of students on continuous task spaces, thereafter referred to as \\textit{Classroom Teaching} (CT). CT defines a family of problems in which a teacher algorithm is tasked to sequentially generate multiple curricula tailored for each of its students, all having potentially varying abilities. CT differs from the problems studied in population-based developmental robotics \\citep{imgep} and evolutionary algorithms \\citep{poet} as in CT there is no direct control over the characteristics of learners, and the objective is to foster maximal learning progress over all learners rather than iteratively populating a pool of high-performing task-expert policies. Studying CT scenarios brings DRL closer to assisted education research problems and might stimulate the design of methods that alleviate the expensive use of expert knowledge in current state of the art methods \\citep{zpdes, Koedinger13}. CT can also be transposed to (multi-task) robotic training scenarios, e.g. when performing iterative design improvements on a robot, which requires to train a sequence of morphologically related (yet different) robots.\n\n\nGiven multiple students to train, no expert knowledge, and assuming at least partial similarities between each students' optimal curriculum, current \\textit{tabula-rasa} exploratory-ACL approaches that do not reuse knowledge between different students do not seem like the optimal choice. This motivates the research of what we propose to call Meta Automatic Curriculum Learning mechanisms, that is algorithms learning to generalize ACL over multiple students.\nIn this work we formalize this novel setup and propose a first Meta-ACL baseline algorithm (based on an existing ACL method \\citep{portelas2019}). Given a new student to train, our approach is centered on the extraction of adapted curriculum priors from a history of previously trained students. The prior selection is performed by matching competence vectors that are built for each student through pre-testing. We show that this simple method can bring significant performance improvements over classical ACL in both a toy environment without DRL students and on Box2D parkour environments with DRL learners.\n\n\n\\paragraph{Related Work.} To approach the problem of curriculum generation for DRL agents, recent works proposed multiple ACL algorithms based on the optimization of surrogate objectives such as learning progress \\citep{portelas2019,tscl,tscllike,curious}, diversity \\citep{diayn,metarl-carml,countbased} or intermediate difficulty \\citep{goalgan,settersolver,OpenAI2019SolvingRC,reverse-cur,ADRmila}. All these works tackled student training through independent ACL runs, while we propose to investigate how one can share information accross multiple trainings.\nWithin DRL, \\textit{Policy Distillation} \\citep{distralTeh2017,pol-dil-review} consists in leveraging one or several previously trained policies to perform \\textit{behavior cloning} on a new policy (e.g. to speed up training and\/or to leverage task-experts to train a multi-task policy). Our work can be seen as proposing a complementary toolbox aiming to perform \\textit{Curriculum Distillation} on a continuous space of tasks.\n\nSimilar ideas were developed for supervised learning by \\citep{hacohen19a-scoring-pacing, ban, banlike}. In \\citep{hacohen19a-scoring-pacing}, authors propose an approach to infer a curriculum from past training for an image classification task: they train their network once without curriculum and use its predictive confidence for each image as a difficulty measure exploited to derive an appropriate curriculum to re-train the network. Although we are mainly interested in training a classroom of diverse students, section \\ref{exp:trying-again} presents similar experiments in a DRL scenario, showing that our Meta-ACL procedure can be beneficial for a single learner that we train once and re-train using curriculum priors inferred from the first run.\n\nParallel to this work, Turcheta et. al. \\citep{safe-rl-curr-induction} studied how to infer safety constraints (i.e. curriculum) over multiple DRL students in a data-driven way to better perform on a given single task. In their work, students are only varying by their network's initialization and their teacher assumes the existence of a pre-defined discrete set of safety constraints to choose from. By contrast, we consider the problem of training generalist students with varying morphologies (and networks initializations), with a teacher algorithm choosing tasks from a continuous task space.\n\n\n\n\n\\paragraph{Main Contributions.}\n\\begin{itemize}\n    \\item Introduction to the concept of Meta-ACL, i.e. algorithms that generalize curriculum generation in Classroom Teaching scenarios, and an approach to study these algorithms. Formalization of the interaction flows between Meta-ACL algorithms and (unknown) Deep RL student distributions.\n    \\item Introduction of AGAIN, a first Meta-ACL baseline algorithm which learns curriculum priors to fasten the identification of learning progress niches for new Deep RL students.\n    \\item Design of a toy-environment and of a parametric Box2D Parkour environment featuring a multi-modal distribution of possible agent embodiments well suited to study Meta-ACL.\n    \\item Analysis of AGAIN on these environments, demonstrating the performance advantages of this approach over classical ACL, including (and surprisingly) when applied to a single student.\n  \n\\end{itemize}\n\n\n\n\\section{Meta Automatic Curriculum Learning Framework}\n\\label{sec:framework}\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=\\textwidth]{graphics\/meta-acl-pipeline_final.pdf}\n\\caption{\\footnotesize{In \\textit{Meta Automatic Curriculum Learning }(Meta-ACL), the objective is to leverage previous teaching experience to improve the curriculum generation of a new agent, whose embodiment and learning mechanisms have potentially never been seen before: the teacher has to \\textit{generalize} over students.}}\n\\label{meta-acl-fig}\n\\end{figure}\n\n\\paragraph{Black-box students} The Meta-ACL framework assumes the existence of policy learners, a.k.a students, capable of interacting in episodic control tasks. These students are assumed non-resettable, as in classical ACL scenarios. Their optimization objective is the maximization of some performance measure $P$ w.r.t the task (e.g. episodic reward, exploration). To make the framework problem-independant, we do not assume expert knowledge over the task space w.r.t the student distribution, e.g. task subspaces could be trivial for some students and unfeasible for others. The objective of Meta-ACL is precisely to autonomously infer such prior knowledge from experience in scenarios where human expert knowledge is either hard or impossible to use. Similarly, we consider a black-box teaching scenario, i.e. we do not assume the knowledge of which learning mechanisms are used by students (e.g. DRL agents, evolutionary algorithms, ...).\n\n\n\n\\paragraph{Automatic Curriculum Learning} \\hspace{-0.2cm} Given such a black-box student $s$ to train on a continuous \\textit{task space} $\\mathcal{A}$, the purpose of an ACL algorithm is to sequentially sample (parameterized) tasks for $s$, such that the following evaluation metric, used by the experimenter, is maximized:\n\\begin{equation}\n    \\label{eq:acl-obj}\n    \\max \\int_{a\\sim \\mathcal{A}} \\! P_{s,a}^E\\, \\mathrm{d}a,\n\\end{equation}\nwith $E$ the episode budget, and $P_{s,a}^E$ the end performance of student $s$ on task $a$ (e.g. exploration score, cumulative reward). Since direct optimization of such a post-training performance is difficult, ACL is often approached using proxy objectives (e.g. intermediate difficulty). Given one such proxy objective, an ACL algorithm usually relies on observing episodic behavioral features of its student w.r.t to proposed tasks (e.g. episodic rewards), allowing to infer a competence status $o \\in \\mathcal{O}$ of its learner (e.g. progress regions in $\\mathcal{A}$) which conditions task sampling. This sequential task selection unrolled by ACL methods along a student's training can be transposed into a policy search problem on a high-level non-episodic POMDP. While interacting within this POMDP, an ACL policy $\\Pi(o,s) \\rightarrow \\rho(\\mathcal{A})$ proposes task distributions $\\rho(\\mathcal{A}) \\subset \\mathcal{A}$ to its student, observes the resulting behavioral features to update its competence status $o$, and collects objective-dependant rewards (e.g. learning progress).\n\n\n\n\n\n\nIn practice, approaching this task-level control problem with classical DRL algorithms is challenging because of sample efficiency: an ACL policy has to be learned and exploited along interaction windows typically around a few tens of thousands of steps. This has to be compared to the tens of millions or sometimes billions of interaction steps necessary to train a DRL policy for robotic control tasks. For this reason, most recent ACL research has focused on reducing the teaching problem into a Multi Armed Bandit setup, which ignores the sequential dependency over student states implied in POMDP settings \\citep{tscl,tscllike,curious,portelas2019}. Although out of the scope of this paper, the use of classical DRL approaches as Meta-ACL algorithms is worth investigating in future work.\n\n\\paragraph{Meta-ACL for Classroom Teaching.} We now present the concept of Meta-ACL applied to a Classroom Teaching scenario, i.e. there is no longer a single student $s$ to be trained, but a set of students with varying abilities (e.g. due to morphology and\/or learning mechanisms) sequentially drawn from an unknown distribution $\\mathcal{S}$. The notion of meta-learning refers to \\textit{any type of learning guided by prior experience with other\ntasks} \\citep{surveymetarl}. In meta-RL, agents are \\textit{learning to learn} to act \\citep{wang2016}, i.e. their objective is to maximize performance on previously unseen test tasks after $0$ or a few learning updates. In other words, it is about leveraging knowledge from previously encountered tasks to generalize on new tasks. We propose to extend this concept into Meta-ACL, that is, algorithms that are \\textit{learning to learn} to teach, i.e. they leverage knowledge from curricula built for previous students to improve the curriculum generation for new ones.  More precisely, a Meta-ACL algorithm can be formulated as a function:\n\\begin{equation} \\label{eq:meta-acl}\n\\begin{gathered}\nf(\\Pi, \\mathcal{H}, s^{K}) \\rightarrow \\Pi^{'} ~~s.t.~~\\mathcal{H}=[\\tau_{s^0}, \\tau_{s^1}, ...,\n\\tau_{s^{K-1}}] \\\\\n \\tau_{s^{i}} = [(\\rho^0(\\mathcal{A}), o^0), (\\rho^1(\\mathcal{A}), o^1), ..., (\\rho^T(\\mathcal{A}), o^{T})],\n\\end{gathered}\n\\end{equation}\nwith $\\mathcal{H}$ the history of past $K$ training trajectories $\\tau_s$ resulting from the scaffolding of $K$ previous students with an ACL (or Meta-ACL) policy $\\Pi$, and $s^K$ the current student. Given our formalization of ACL (see eq. \\ref{eq:acl-obj}), the experimenter's evaluation objective for Meta-ACL can be expressed as follows:\n\\begin{equation}\n    \\label{eq:meta-acl-obj}\n    \\max_{f} \\int_{s\\sim \\mathcal{S}}\\int_{a\\sim \\mathcal{A}} \\! P_{s,a}^E\\, \\mathrm{d}a~\\mathrm{d}s.\n\\end{equation}\n\nAs in the case of the ACL evaluation objective expressed in eq. \\ref{eq:acl-obj}, direct optimization of eq. \\ref{eq:meta-acl-obj} is difficult as it implies the joint maximization of multiple students' performance. In our experiments, we reduce the Meta-ACL problem to the sequential independent training of a set of new students by leveraging priors from previous student trainings (with the hope to maximize performance over the entire set). Figure \\ref{meta-acl-fig} provides a visual transcription of the workflow of a Meta-ACL algorithm. While in our experiments we use a fixed-size history $\\mathcal{H}$ of ACL-trained students (to make our experiments computationally tractable), $\\mathcal{H}$ could be grown incrementally by collecting training trajectories online from Meta-ACL trainings.\n\n\n\n\\section{A first Meta-ACL baseline: Alp-Gmm And Inferred Progress Niches}\n\\label{sec:methods}\n\n\\begin{figure*}[b]\n\\centering\n\\includegraphics[width=\\textwidth]{graphics\/AGAIN-pipeline.pdf}\n\\caption{\\footnotesize{Schematic pipeline of Alp-Gmm And Inferred progress Niches (AGAIN), our proposed approach, which first leverages a preliminary run with a high-exploration ALP-GMM curriculum generator, and uses student pre-testing to infer an expert curriculum combined with a low-exploration ALP-GMM.}}\n\\label{again-fig}\n\\end{figure*}\n\nIn this section, we present AGAIN (Alp-Gmm And Inferred Progress Niches), our proposed Meta-ACL algorithm, and connect it to the formalism described in section \\ref{sec:framework}. We first give a broad overview of the approach and then provide detailed explanations of key components.\n\n\\paragraph{Overview} Figure \\ref{again-fig} provides a schematic pipeline of our Meta-ACL approach. Given a history $\\mathcal{H}$ of previously trained students and a new student $s^K \\sim \\mathcal{S}$ to train, AGAIN starts by (1) pre-training $s^K$ using ALP-GMM, an existing ACL algorithm from \\cite{portelas2019} (chosen for its simplicity and its non-reliance on expert knowledge). After pre-training, it (2) challenges the student with a set of test tasks to construct a meaningful competence profile of the student, which we thereafter refer to as a \\textit{Knowledge Component (KC) vector}. This KC vector is then (3) used to select a previously trained student $s^i$ similar to $s^K$ from which the \\textit{training trajectory} $\\tau_{s^{i}}$ is recovered. Based on $\\tau_{s^{i}}$, (4) AGAIN infers a set of curriculum priors $\\mathcal{C}$ (i.e. promising task sub-spaces). Finally, (5) the training of $s^K$ can resume using a composite curriculum generator using both an expert curriculum derived from $\\mathcal{C}$ (for exploitation), and ALP-GMM (for exploration).\n\n\n\\paragraph{1 - ALP-GMM} ALP-GMM \\citep{portelas2019} is a Learning Progress (LP) based ACL technique for continuous task spaces that does not assume prior knowledge. ALP-GMM frames the task sampling problem into a non-stationary Multi-Armed bandit setup \\citep{auer2002nonstochastic} in which arms are Gaussians spanning over the task space. The utility of each Gaussian is defined with a local LP measure derived from episodic reward comparisons. The essence of ALP-GMM is to periodically fit a Gaussian Mixture Model (GMM) on recently sampled tasks' parameters \\textit{concatenated with their respective LP}. This periodically updated GMM used for task sampling can be seen as the evolving competence status $o$ described in section \\ref{sec:framework}. The Gaussian from which to sample a new task is chosen proportionally to its mean LP dimension. Task exploration happens initially through a bootstrapping period of random task sampling and during training through residual random task sampling.\n\n\\paragraph{2,3 - KC-based curriculum priors selection.} For a new student $s^K$, given its capabilities on the considered task space, how to selected the most relevant previously trained student from which to extract curriculum priors? This problem is closely related to knowledge assessment in Intelligent Tutoring Systems setups studied in the educational data mining literature \\citep{vie-mooc-test-assembly,vie-thesis}. Inspired by these works, we use pre-tests to derive a Knowledge Component vectors $KC^{pre} \\in \\mathbb{R}^{m}$ for all trained students. Each dimensions of $KC^{pre}$ contains the episodic return of the student on the corresponding pre-test task. Given that we do not assume access to expert knowledge, we build this pre-test task set by selecting $m$ tasks uniformly over the task space. We use the same task set to build a post-training KC vector $KC^{post} \\in \\mathbb{R}^{m}$ whose dimensions are summed up to get a score $j_s \\in \\mathbb{R}$, used to evaluate the end performance of students in $\\mathcal{H}$. After the initial pre-training of $s^K$ with ALP-GMM, curriculum priors can be obtained in 3 steps: 1) pre-test $s^K$ to get its KC vector $KC^{pre}_{s^K}$, 2) infer the $k$ most similar previously trained students in KC space (using a k-nearest neighbor algorithm), and 3) use the training trajectory $\\tau_s$ of the student with maximal post-training score $j_s$ among those $k$. In essence, this method is about re-using curriculum data from a similarly-skilled and successfully-trained student.\n\n\n\\paragraph{4 - Inferred progress Niches (IN).} Given that the KC-based student selection identified the training trajectory $\\tau_{s^{i}}$ as the most promising for $s^K$, and assuming ALP-GMM as the underlying ACL teacher used for $s^i$, we can derive an expert curriculum from $\\tau_{s^{i}}$ by first considering the ordered sequence of GMMs $\\mathcal{C}_{raw}$ that were periodically fitted along training:\n\n\\begin{equation}  \\label{eq:craw}\n  \\begin{gathered}\n \\mathcal{C}_{raw} = \\{p(1), ..., p(T)\\} \\\\\n s.t.~~~p(t)= \\sum_{i=1} LP_{ti}\\mathcal{N}(\\bm{\\mu_{ti}},\\bm{\\Sigma_{ti}}),\n\\end{gathered}  \n\\end{equation}\n\nwith $T$ the total number of GMMs in the list and $LP_{ti}$ the Learning Progress of the $i^{th}$ Gaussian from the $t^{th}$ GMM. By keeping only Gaussians with $LP_{ti}$ above a predefined threshold $\\delta_{LP}$, we can get a curated list $\\mathcal{C}$ containing only Gaussians located on task subspaces on which $s^i$ experienced learning progress (i.e. curriculum priors). Given a GMM of $\\mathcal{C}$, a task is selected by 1) sampling a Gaussian proportionally to its $LP_{ti}$ value, and 2) sampling the Gaussian to obtain parameters mapping to a task. But how to decide which GMMs to use along the training of the new student $s^K$ ?\n\nWhile the simplest way to obtain such a curriculum would be to start sampling tasks from the first GMM and step to the next GMM at the same rate than the initial ALP-GMM run, we propose a more flexible \\textit{reward-based} method. This method requires to record the mean episodic reward obtained by the previously trained student $s^i$ for each GMM of $\\mathcal{C}$ (which can be done without additional assumptions or computational overhead). Given this, to select which GMM from $\\mathcal{C}$ is used to sample tasks over time along the training of $s^K$, we start with the first GMM and only iterate over $\\mathcal{C}$ once the mean episodic reward over tasks recently sampled from the current GMM matches or surpasses the mean episodic reward recorded during the initial ALP-GMM run. In app. \\ref{ann:toy-exp}, we show that this \\textit{reward-based} variant outperforms other potential methods. See app. \\ref{app-again} for algorithmic details. We name the resulting meta-learned expert curriculum approach Infered progress Niches (IN)\n\n\\paragraph{5 - Our proposed approach: AGAIN.} Simply using IN directly for $s^K$ lacks adaptive mechanisms towards the characteristics of the new student (e.g. new embodiment, different initial parameters, ...), which could lead to failure cases where the expert curriculum misses important aspects of training (e.g. detecting task subspaces that are being forgotten). Additionally, the meta-learned ACL algorithm must have the capacity to emancipate from the expert curriculum once the trajectory is completed (i.e. go beyond $\\mathcal{C}$). This motivates why our approach combines IN with an ALP-GMM teacher after the initial pre-training. The resulting Alp-Gmm And Inferred progress Niches approach (AGAIN) samples tasks from a GMM that is composed of the current mixture of both ALP-GMM and IN. See appendix \\ref{app-alp-gmm} \\& \\ref{app-again} for implementation details and pseudo-code algorithms.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments and Results}\n\\label{sec:expes}\n\nWe organize the analysis of our proposed Meta-ACL algorithm around $3$ experimental questions:\n\\begin{itemize}\n    \\item What are the properties and important components of AGAIN? In this section we will leverage a toy environment without DRL students to conduct systematic experiments.\n    \\item Does AGAIN scale well to Meta-ACL scenarios with DRL students? Here we will present a new Parkour environment that will be used to conduct our experiments.\n    \\item Can AGAIN be used for single learners? Here we will show that it can be useful to derive curriculum priors even for a single student (i.e. without any student History $\\mathcal{H}$).\n\\end{itemize}\n\n\n\\paragraph{Considered baselines and AGAIN variants.} In the following experiments we compare AGAIN to variants where 1) we directly use the expert curriculum instead of combining it with ALP-GMM (\\textit{IN} condition), and 2) where we select a training trajectory at random (\\textit{AGAIN\\_RND}) or using the ground truth student distribution (\\textit{AGAIN\\_GT}). We compare these Meta-ACL variants to ACL approaches such as random curriculum generation (\\textit{Random}), \\textit{ALP-GMM} and either Adaptive Domain Randomization (\\textit{ADR}) \\citep{OpenAI2019SolvingRC} or an expert-made \\textit{Oracle} curriculum. See appendix \\ref{an:details} for details.\n\n\\subsection{Analysing Meta-ACL in a toy environment}\n\\label{sec:exp:toy-env}\n\nTo provide in-depth experiments on AGAIN, we first emancipate from DRL students through the use of a modified version of the toy testbed presented in \\citep{portelas2019}. The objective of this environment is to simulate the learning of a student within a $2$D parameter space $\\mathcal{P} = [0,1]^2$. The parameter space is uniformly divided in $400$ square cells $C \\subset  \\mathcal{P}$, and each parameter $p \\in \\mathcal{P}$ sampled by the teacher is directly mapped to an episodic reward $r_p$ based on sampling history and whether $C$ is considered \"locked\" or \"unlocked\". Three rules enforce reward collection in $\\mathcal{P}$:~1) Every cell $C$ starts \"locked\", except a randomly chosen one that is \"unlocked\". 2) If $C$ is \"unlocked\" and $p \\in C$, then $r_p = min(|C|,100)$, with $|C|$ the cumulative number of parameters sampled within $C$ while being \"unlocked\" (if $C$ is \"locked\", then $r_p = 0$). Finally, 3) If $|C| >= 75$, adjacent cells become \"unlocked\". Given these rules, one can model students with different curriculum needs by assigning them different initially unlocked cells, which itself models what is \"easy to learn\" initially for a given student, and from where it can expand.\n\\begin{figure*}[htb!]\n\\centering\n\\subfloat{\\includegraphics[width=0.45\\textwidth]{graphics\/nips21_reg_toy_env_vizu.pdf}}\n\\subfloat{\\includegraphics[width=0.45\\textwidth]{graphics\/nips21_varying_classroom_size_expe.pdf}}\n\\caption{\\footnotesize{\\textbf{left:} By leveraging meta-learned curriculum priors w.r.t to its students, AGAIN outperforms regular ACL approaches. Avg. perfs. with \\textit{sem} (standard error of the mean) plotted, 48 seeds. The vertical dashed black line indicates when pre-training ends for Meta-ACL conditions. \\textbf{right:} Impact of classroom size and sparsity on Meta-ACL performances. Post-training ($200$k ep.) avg perfs. plotted, 96 seeds.}}\n\\label{vizu-toy-env-classroom}\n\\end{figure*}\n\n\\paragraph{Results} Instead of performing a pre-test to construct the KC vector of a student, we directly compute it by concatenating $|C|$ for all cells, giving a $400$-dimensional KC vector. This vector is computed after $20$k training episodes out of $200$k. To study AGAIN, we first populate our training trajectory history $\\mathcal{H}$ by training with ALP-GMM an initial classroom of 128 students drawn randomly from 4 fixed possible student types (i.e. 4 possible initially unlocked cell positions), and then test it on a new fixed set of $48$ random students.\n\n\\textit{Comparative analysis - } Figure \\ref{vizu-toy-env-classroom} (left) showcases performance across training for our considered Meta-ACL conditions and ACL baselines. Both AGAIN and IN significantly outperform ALP-GMM ($p<.001$ for both, using Welch's t-test at $200$k episodes). The initial performance advantage of IN w.r.t AGAIN is due to the greedy nature of IN, which only exploits the expert curriculum while AGAIN complements it with ALP-GMM for exploration. By the end of training, AGAIN outperforms IN ($p<.001$) thanks to its ability to emancipate from the curriculum priors it initially leverages. The regular KC-based curriculum priors selection used in AGAIN outperformed the random selection used in AGAIN\\_RND ($p<.001$ at $200$k episodes), while being not significantly inferior to the Ground Truth variant AGAIN\\_GT ($p=0.16$). Because we assume no expert knowledge over the set of students to train, i.e. their respective initial learning subspace is unknown, ADR -- which relies on being given an initial easy task -- fails to train most students when given randomly selected starting subspace (among the $4$ possible ones). By contrast, this showcases the ability of AGAIN to autonomously and efficiently infer such expert knowledge.\n\n\\textit{Varying classroom size experiment - } An important property that must be met by a meta-learning procedure is to have a monotonic increase of performance as the database of information being leveraged increases. Another important expected aspect of Meta-ACL is whether the approach is able to generalize to students that were never seen before. To assess whether these properties hold on AGAIN, we consider the full student distribution of the toy environment, i.e. $400$ possible student types. We populate a new history $\\mathcal{H}$ by training (with ALP-GMM) a $400$-students classroom (one per student type). We then analyse the end performance of AGAIN and IN on a fixed test set of 96 random students when given increasingly smaller subsets of $\\mathcal{H}$. The smaller the subset, the harder it becomes to generalize over new students. Results, shown in fig. \\ref{vizu-toy-env-classroom} (right), demonstrate that both AGAIN and IN do have monotonic performance increasements as the classroom grows. With as little as $10$\\% of possible students in the classroom, AGAIN statistically significantly ($p<.001$) outperforms ALP-GMM on the new student set, i.e. it generalizes to never seen before students.\n\n\n \\subsection{Meta-ACL for DRL students in the Parkour environment}\n \\label{sec:exp:walker-climber}\n \n \\begin{wrapfigure}[14]{r}{6.3cm}\n\\vspace{-0.4cm}\n\\includegraphics[width=5.7cm]{graphics\/parkour_vizu.PNG}\n\\caption{\\footnotesize{Our proposed parametric Parkour env. to study Meta-ACL with DRL students.}}\n\\label{parkour-env}\n\\end{wrapfigure}\n \nTo study Meta-ACL with DRL students, we present a Box2D Parkour environment with a $2$D parametric PCG that encodes a large space of tasks (see fig. \\ref{parkour-env}). The first parameter controls the spacing between walls that are positioned along the track, while the second parameter sets the y-position of a gate that is added to each wall. Positive rewards are collected by going forward. To simulate a multi-modal distribution of students well suited to study Meta-ACL, we randomize the student's morphology for each new training (i.e. each seed): It can be embodied in either a bipedal walker, which will be prone to learn tasks with near-ground gate positions, or a two-armed climber, for which tasks with near-roof gate positions are easiest. We also randomize the student's limb sizes which can vary from the length visible in fig. \\ref{parkour-env} to 50\\% shorter.\n\n\n\n\n\\paragraph{Results} In the following experiments our Meta-ACL variants leverage a history $\\mathcal{H}$ built from a classroom of $128$ randomly drawn Soft-Actor-Critic \\citep{sac} students (i.e. varying embodiments and initial policy weights) trained with ALP-GMM. We then compare ACL and Meta-ACL variants on a fixed set of 64 new students and report the mean percentage of mastered environments (i.e. $r>230$) from 2 fixed expert test sets (one per embodiment type) across training. The KC vector is built using a uniform pre-test set of $m=225$ tasks, performed after $2$ millions agent steps out of $10$. See appendix \\ref{ann:parkour} for additional experimental details.\n\n\\textit{Qualitative view - } Figure \\ref{wc-env-vizu} (left) showcases the evolution of task sampling when using AGAIN to train a new student. Three distinct phases emerge: 1) A pre-training exploratory phase used to gather information about the student's capabilities, 2) After building the KC vector and inferring the most appropriate curriculum priors from $\\mathcal{H}$, AGAIN paces through the resulting IN curriculum while mixing it to ALP-GMM, and 3) AGAIN emancipates from IN after completing it.\n\n\\textit{Comparative analysis - } As shown in figure \\ref{wc-env-vizu} (right), through its use of curriculum priors, AGAIN outperforms ALP-GMM on Parkour, mastering an average of $41\\%$ of the test set at $10$M steps, compared to $31\\%$ for ALP-GMM ($p<.001$) after $10.5$M steps ($0.5$M training steps added to account for AGAIN additional pre-test time). AGAIN performs better than its AGAIN\\_RND random prior selection variant, and is not statistically different ($p=0.8$) from ground truth sampling (AGAIN\\_GT), although only by the end of training. While AGAIN and IN initially have comparable performances, after $7$ Millions training steps, -- a point at which most students trained with IN or AGAIN reached the last IN GMM --, AGAIN outperforms IN by the end of training ($p<0.02$). This showcases the advantage of emancipating from the expert curriculum once completed. As in the toy environment experiments, when given randomly selected starting subspaces (since we assume no expert knowledge), ADR fails to train most students.\n\n \\begin{figure*}[htb!]\n\\centering\n\\subfloat{\\includegraphics[width=0.49\\textwidth]{graphics\/vizu_wc_AGAIN-R15.pdf}}\n\\subfloat{\\includegraphics[width=0.48\\textwidth]{graphics\/nips21_env_rndls_only_combiperf.pdf}}\n\\caption{\\footnotesize{\\textbf{left:} Example of evolution of task sampling when using AGAIN in the Parkour env. (1 seed). \\textbf{right:} Average performances of AGAIN with variants and baselines in the Parkour env.. 64 seeds, sem plotted. The vertical dashed black line indicates when pre-training ends for Meta-ACL conditions.}}\n\\label{wc-env-vizu}\n\\end{figure*}\n\n\n \\subsection{Applying Meta-ACL to a single student: Trying AGAIN instead of trying longer}\n \\label{exp:trying-again}\n\n \nGiven a single DRL student to train (i.e. no history $\\mathcal{H}$), and if we do not assume access to expert knowledge, current ACL approaches leverage task-exploration (as in ALP-GMM). We hypothesize that these additional tasks presented to the DRL learner have a cluttering effect on the gathered training data, which adds noise in its already brittle gradient-based optimization and leads to sub-optimal performances. We propose to address this problem by modifying AGAIN to fit this no-history setup and by allowing to restart the student along training. More precisely, instead of pre-testing the student to find appropriate curriculum priors in $\\mathcal{H}$, we split the training of the target student into a two stage approach where 1) the DRL student is first trained with ALP-GMM (with high-exploration), and then 2) we extract curriculum priors from the training trajectory of the first run and use them to re-train the same agent \\textit{from scratch}.\n \n\n\n \\begin{wrapfigure}[18]{r}{6.5cm}\n \\vspace{-0.2cm}\n\\includegraphics[width=6.5cm]{graphics\/nips21_short_compact_demo.pdf}\n\\caption{\\footnotesize{Given a single DRL student to train, AGAIN outperforms ALP-GMM in a parametric BipedalWalker environment. sem plotted, 32 seeds.}}\\label{trying-again-compact}\n\\end{wrapfigure} \n\\paragraph{Results.} We test our modified AGAIN along with variants and baselines on a parametric version of BipedalWalker proposed in \\citep{portelas2019},  which generates walking tracks paved with stumps whose height and spacing are defined by a PCG-encoding $2$-D parameter vector. As in their work, we test our approaches with both the default walker and a modified short-legged walker, which constitutes an even more challenging scenario (as the task space is unchanged). Performance is measured by tracking the percentage of mastered tasks from a fixed test set. See App. \\ref{ann:tryingagain} for a complete analysis. \n\nFigure \\ref{trying-again-compact} showcases our proposed approach on the short walker setup (with a SAC student \\citep{sac}). On this short walker scenario, mixing ALP-GMM with IN is essential: while IN end performances are not statistically significantly superior to ALP-GMM, AGAIN clearly outperforms ALP-GMM $(p<0.01)$, reaching a mean end performance of $19.0$. The difference in end-performance between AGAIN and Oracle, our hand-made curriculum using privileged information who obtained $20.1$, is not significant ($p=0.6$).\n\n\\section{Conclusion and Discussion}\nIn this work we attempted to motivate and formalize the study of Classroom Teaching problems, in which a set of diverse students have to be trained optimally, and we proposed to attain this goal through the use of Meta-ACL algorithms. We then presented AGAIN, a first Meta-ACL baseline, and demonstrated its advantages over classical ACL and variants for CT problems in both a toy environment and in a new parametric Parkour environment with DRL learners. We also showed how AGAIN can bring performance gains over ACL in classical single student ACL scenarios.\n\n\\paragraph{Future work}In future work, AGAIN could be improved by using adaptive approaches to build compact pre-test sets, e.g. using decision tree based test pruning methods, or by combining curriculum priors from multiple previously trained learners. While AGAIN is built on top of an existing ACL algorithm, developing an end-to-end Meta-ACL algorithm that generates curricula using a DRL teacher policy trained across multiple students is also a promising line of work to follow. \nAdditionally, this work opens-up exciting new perspectives in transferring Meta-ACL methods to educational data-mining, e.g. in MOOC scenarios, given a previously trained pilot classroom, one could use Meta-ACL to infer adaptive curricula for new students.\n\n\\paragraph{Potential negative societal impact} Meta Automatic Curriculum Learning exploits previous curriculum data of DRL students to better train new ones. As such, if previously trained students acquired unwanted biases, they could potentially be transferred and further amplified from the Meta-ACL training of new DRL students. This can have serious negative societal impacts if considering applications in socially impactful domains (e.g. deciding whether to give an insurance to someone). Further work will need to study how one\ncan safely learn curriculum priors that are not harmful in sensitive application areas.\n\n\\section*{Acknowledgments}\nThis work was supported by Microsoft Research through its PhD Scholarship Programme. All presented experiments were carried out using 1) the computing facilities MCIA (M\u00e9socentre de Calcul Intensif Aquitain) of the Universit\u00e9 de Bordeaux and of the Universit\u00e9 de Pau et des Pays de l'Adour, and 2) the HPC resources of IDRIS under the allocation 2020-[A0091011996] made by GENCI.\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nHuman-robotic fusions controlled through brain computer interfaces (BCI) have tremendous potential to impact human health and capabilities through applications like intelligent motor and cognitive prostheses~\\cite{guger1999prosthetic, muller2007control, mcfarland2008brain, fifer2013simultaneous, hotson2016individual, Zhao2017, Akinola2017}. BCI-based control approaches are currently limited by the number of independent degrees of freedom that can be reliability controlled directly.  The number of degrees of freedom on motor prostheses for example can be several dozen~\\cite{pasquina2015recent}.  This is about an order of magnitude more degrees of freedom than can be reliability generated by current non-invasive BCI systems.  The dilemma is how to control high-degree of freedom complex machines with only a few control inputs. \n\nOne approach to increase the impact of limited control inputs is through modularity and hierarchical control mechanisms~\\cite{Zhao2017, Akinola2017}.  The idea is to use the limited number of inputs to select a primitive control policy, from a library of primitive behaviors, and potentially a target. Complex tasks are performed by chaining primitive behaviors.\n\nAs an example of this scenario, consider a Universal Robots UR5~\\cite{UR} manipulator mounted on a Clearpath Husky platform~\\cite{Husky} as shown in Fig.~\\ref{fig:husky}.  The UR5 is used to demonstrate reaching, grabbing, and lifting a block on a table.  Other tasks may require performing these actions in another order, so it may be useful to learn and maintain a collection of these primitive behaviors for later use. While the underlying behavior primitives are well defined for the reach-and-grasp scenario, other example scenarios may not have as well defined or labeled primitives.  In this work, we assume that the underlying label of the behaviors shown in the task demonstrations is unknown.  \n\\begin{figure}[ht!]\n\\centering\n  \\includegraphics[width=.8\\linewidth]{img\/husky.png}\n  \\caption{Husky-UR5 Reach and Grasp Environment }\n  \\label{fig:husky}\n\\end{figure} \n\nThe questions we investigate are how might we learn and maintain the primitive library from unlabeled demonstrations and, assuming the behavior primitive library exists, how would one know when to use, adapt, or create a new primitive behavior.  We propose that the behavior library should be actively maintained to minimize redundancy and maximize the ability to reconstruct complex tasks through chains of the primitive behaviors.  In this work, we explore techniques to directly optimize for these criteria by building on methods that learn from demonstration.\n\n\nWe explore maintaining a behavior primitive library in an online learning scenario.  Given a potentially non-empty behavior primitive library and a new set of unlabeled task demonstrations, we seek to update the behavior primitive library to maximally accommodate the new demonstrations while maintaining the ability to reconstruct previously demonstrated trajectories.\n\nOur contribution is an approach called $PICO$\\xspace that simultaneously learns subtask decomposition from unlabeled task demonstrations, trains behavior primitives, and learns a hierarchical control mechanism that allows blending of primitive behaviors to create even greater behavioral diversity, an overview is shown in Fig.~\\ref{fig:overview}. Our approach directly optimizes the contents of the primitive library to maximize the ability to reconstruct unlabeled task demonstrations from sequences of primitive behaviors.\n\\begin{figure}[ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{img\/overview.png}\n  \\caption{An overview of $PICO$\\xspace. The approach takes as input unlabeled demonstrations and a library of primitive behaviors.  The goal is to predict the primitive behavior label associated with each time point in all demonstrations. Additional behavior primitive models can be trained to fill gaps that are not well represented by existing behavior primitives. }\n  \\label{fig:overview}\n\\end{figure} \n\n\\section{PRELIMINARIES}\n\n\n\n\nLearning from demonstration (LfD) and imitation learning allow agents to execute a task by observing the task being performed \\cite{Hussein:2017:ILS:3071073.3054912}. In the robotics domain, a goal of imitation learning is to produce a mapping, $\\pi$, from states to actions, known as a control \\emph{policy} \\cite{ARGALL2009469, schaal2010learning}, that has the maximum likelihood of producing the demonstration dataset $\\mathcal{D} = \\{\\rho_1,\\rho_2,\\dots,\\rho_n\\}$, where each $\\rho = ((s_1,a_1),(s_2,a_2),\\dots,(s_T,a_T)$ is a demonstration trajectory of of state, action pairs. The demonstrations can be created by another control policy \\cite{distillation}, by a human expert \\cite{konidaris2012}, or in a simulated environment \\cite{TACO18, compile2019}. Let $\\pi_\\theta$ parameterized by $\\theta$. The goal is then to optimize Equation \\ref{bc} by varying $\\theta$.\n\n\\begin{equation}\n    \\max \\mathbb{E}_\\rho[\\sum_{t=1}^T\\log\\pi_\\theta(a_t|s_t)]\\label{bc}\n\\end{equation}\n\nFollowing optimization, covariate drift can cause errors in the control process that can place the robot in a previously unobserved state. Control policies will have higher action prediction errors in parts of the state space that it has not observed, leading to poor action predictions and compounding errors with increased iterations of the policy.  One approach that has been introduced to decrease the impact of covariate shift is to introduce noise into the demonstrations used for learning \\cite{DART}.  This approach increases the amount of state space covered by the policy and improves action predictions around the demonstrations, leading to better generalization and error tolerance.  \n\n\\subsection{Model Agnostic Meta-Learning}\nIn meta-learning a model is trained on a variety of learning tasks and the parameters of the method are fine-tuned for generalization. The idea of meta-learning is to combine a set of learner models to improve performance on a task more quickly than one without pretrained models.  This is a common strategy for one-shot~\\cite{santoro2016oneshot} or few shot scenarios, where a model must be trained using one or a few examples. Some approaches for meta-learning come from the reinforcement learning~\\cite{Finn2017}, which typically differ in how they update individual learners.  Some meta-learning methods update models using gradient information~\\cite{Finn2017} and others learn how to update learners from data~\\cite{learning_to_learn,Bengio2002}.\n\n\\section{RELATED WORK}\n\n\nImitation learning alone does not provide a mechanism to generalize demonstrations to new tasks.  One mechanism to address this challenge is task decomposition, which has the goal of identifying subtasks from demonstration.  Subtasks can be made into sub-policies through imitation learning, including methods methods that combine subtask discovery with imitation learning~\\cite{TACO18,NTP18}. By decomposing demonstrations into subtasks, it becomes possible to permute the sequence of sub-policies to achieve greater task diversity and generalizability. However, decomposing demonstrations into subtasks that are maximally useful for recombination is a challenge in task decomposition~\\cite{TACO18}. \n\n\nOnce sub-task policies are established, a hierarchical control policy can be learned that identifies the sequence of policies needed to achieve a specified goal. Given a sufficiently diverse set of demonstrations the reasoning layer can be learned from a set of demonstrations~\\cite{NTP18}. Several approaches for learning hierarchical architectures for control policies from limited demonstrations have been proposed~\\cite{TACO18, NTP18, duan2017}.  We were inspired by the work on mixtures-of-experts\\cite{mixture_of_experts,experts}\nwhich includes a similar hierarchical representation.\n\n\nSome approaches assume that the behavior primitive library is fully trained in advance~\\cite{NTP18}.  In the reinforcement learning domain, the options framework~\\cite{drl_options, andreas2017, kulkarni2016} and hierarchical reinforcement learning~\\cite{HRL} are common approaches for organising hierarchies of policies.  The techniques in reinforcement learning are often predicated on being able to interact with an environment and collect a lot of data.  In this work, we focus on learning hierarchical task decomposition strategies from a limited set of demonstrations.\n\n\\subsection{Task Sketch for Sub-policy Discovery}\nSome related approaches~\\cite{andreas2017, mu2019plots} perform demonstration decomposition by combining both demonstrations and task sketches.  The literature refers to these approaches as \\textit{weakly-supervised} because the order of tasks is given and the exact transition points within a demonstration must be inferred. \n\nLet $\\mathcal{D}$ be our dataset containing\ntrajectories $\\rho = ((s_0,a_0),(s_1,a_1),\\ldots,((s_T,a_T))$ of length $T$ containing state-action tuples $(s,a)$ for state $s$ and action $a$.\nGiven a library of sub-tasks policies $\\mathcal{B}=(\\pi_1,\\pi_2,\\ldots,\\pi_K)$,  A task sketch $\\tau =(\\tau_1,\\tau_2,\\ldots,\\tau_L$) is a sequence of sub-tasks labels where $L$ is the length of the sketch.  A path is a sequence of sub-task labels $\\zeta = (\\zeta_1,\\zeta_2,\\ldots,\\zeta_T)$ where $T$ is the length of a demonstration.  We assume that $L<<T$.  We say that a path $\\zeta$ matches a task sketch $\\tau$ if $\\tau = \\zeta $ after  removing all adjacent duplicate sub-tasks in $\\zeta$.  For example, the path $(\\pi_2,\\pi_2,\\pi_2,\\pi_3,\\pi_3,\\pi_1,\\pi_1,\\pi_1,\\pi_1)$ matches the task sketch $(\\pi_2,\\pi_3,\\pi_1)$.  \n\n\n\\section{METHODS}\n\nIn this section we describe the approaches most closely aligned with our work referred to as CTC~\\cite{CTC06} and TACO~\\cite{TACO18}.  Then, we introduce our approach called Primitive Imitation for COntrol ($PICO$\\xspace).\n\n\\subsection{Connectionist Temporal Classification}\nGiven a dataset $\\mathcal{D}$ and task sketch $\\tau$, one approach to obtain a set of generalizable sub-tasks $\\mathcal{B}$ is to separately learn alignment of trajectories to the task sketch then learn the control policies for sub-tasks with behavior cloning.\nConnectionist Temporal Classification (CTC)~\\cite{CTC06} addresses the problem of aligning sequences of dissimilar lengths. There are potentially multiple ways in which a path could be aligned to a task sketch. Let $\\mathbb{Z}_{(T,\\tau)}$ be the set of all paths of length $T$ that match the task sketch $\\tau$.  The CTC objective maximises the\nprobability of the task sketch $\\tau$ given the input trajectory $\\rho$:\n\n\\begin{equation}\n   \\theta^* =  \\argmaxA_\\theta \\mathbb{E}_{(\\rho,\\tau)}[p_\\theta(\\tau | \\rho)]\n\\label{eqn:ctc1} \n\\end{equation}\n\n\\begin{equation}\n    \\theta^* =  \\argmaxA_\\theta \\mathbb{E}_{(\\rho,\\tau)}[\\sum_{\\zeta \\in \\mathbb{Z}_{(T,\\tau)}} \\prod_{t=1}^T p_\\theta(\\zeta_t,|\\rho_t)]\n    \\label{eqn:ctc2}\n\\end{equation}\n\n$p_\\theta(\\zeta_t,|\\rho)$ is commonly represented as a neural network with parameters $\\theta$ that outputs the probability of each sub-task policy in $\\mathcal{B}$.  The objective is solved efficiently using dynamic programming. Inference using the neural network model is used to find a maximum likelihood path $\\zeta$ for a trajectory $\\rho$. The labels in $\\zeta$ provide an association between state-action tuples $(s_t,a_t)$ and subtask policies $\\pi \\in \\mathcal{B}$.  The state-action policies associated with a single sub-task are used to create a sub-task policy using behavior cloning.\n\n\\subsection{Temporal Alignment for Control} Given a demonstration $\\rho$ and a task sketch $\\tau$, Temporal Alignment for Control (TACO)~\\cite{TACO18} will learn where each subtask begins and ends in the trajectory and simultaneously trains a library of sub-tasks policies $\\mathcal{B}$.  TACO maximizes the joint log likelihood of the task sequence $\\tau$ and the actions from sub-task policies contained in $\\mathcal{B}$ conditioned on the states. Let $\\textbf{a}_\\rho$ and $\\textbf{s}_\\rho$ be the set of actions and states respectively in trajectory $\\rho$.\n\n\n\\begin{equation}\n    p(\\tau,\\textbf{a}_\\rho|\\textbf{s}_\\rho) = \\sum_{\\zeta \\in \\mathbb{Z}_{(T,\\tau)}} p(\\zeta|\\textbf{s}_\\rho) \\prod_{t=1}^T \\pi_{\\zeta_t}(a_t|s_t)\n    \\label{eqn:taco}\n\\end{equation}\n\nwhere $p(\\zeta|\\textbf{s}_\\rho) $ is the product of action probabilities associated with any given path $\\zeta$. The path $\\zeta$ determines which data within $\\rho$ corresponds to each sub-task policy $\\pi$ and  $\\prod_{t=1}^T \\pi_{\\zeta_t}(a_t|s_t)$ is the behavior cloning objective from Equation \\ref{bc}.\n\n\n\\subsection{Primitive Imitation for Control (PICO)}\nIn this work, we introduce Primitive Imitation for Control. ($PICO$\\xspace). The approach differs from previous work in a few important ways. $PICO$\\xspace  similarly decomposes behavior primitives from demonstration. It optimizes the action conditioned on state and does not require a task sketch, and unlike CTC\\cite{CTC06}, our approach simultaneously learns to segment demonstrations and trains underlying behavior primitive models. \n\nWe aim to reconstruct the given trajectories as well as possible using the existing sub-task policy library.   As shown in Equation \\ref{mse}, we seek to minimize the sum of squared error between the observed action and the predicted action for all actions over all timepoints $T$ and all trajectories $\\rho \\in \\mathcal{D}$.   We refer to this objective as minimizing reconstruction error. Let  $(s_{\\rho_t},a_{\\rho_t})$ be the state-action tuple corresponding to $\\rho_t$ timepoint $t$ in trajectory $\\rho$.  The action prediction, equation \\ref{action_prediction}, is the product of the probability $p(\\pi|s_{\\rho_t})$ of a sub-task policy $\\pi$ conditioned on the state $s_{\\rho_t}$ and the action predicted by policy $\\pi(s_{\\rho_t})$ for the state $s_{\\rho_t}$.  Substituting equation \\ref{action_prediction} into \\ref{mse} results in Equation \\ref{mse2} which is the optimization problem for $PICO$\\xspace.\n\n\n\n\\begin{equation}\n\\min \\sum_{\\rho \\in \\mathcal{D}} \\sum_{t=0}^T (a_{\\rho_t}-\\hat{a}_{\\rho_t})^2\n\\label{mse}\n\\end{equation}\n\n\\begin{equation}\n    \\hat{a}_{\\rho_t} = \\sum_{\\pi \\in \\mathcal{B}} p(\\pi|s_{\\rho_t}) \\pi(s_{\\rho_t} )\n    \\label{action_prediction}\n\\end{equation}\n\n\\begin{equation}\n\\min \\sum_{\\rho \\in \\mathcal{D}} \\sum_{t=0}^T (a_{\\rho_t}-\n\\sum_{\\pi \\in \\mathcal{B}} p(\\pi|s_{\\rho_t}) \\pi(s_{\\rho_t} \n))^2\n\\label{mse2}\n\\end{equation}\n\n\n\n\n\n\\subsection{Neural Network Architecture}\n\nEstimates of both $p(\\pi|s_{\\rho_t})$ and $\\pi(s_{\\rho_t})$ are given by a recurrent neural network architecture. Figure \\ref{fig:model} gives an overview of the recurrent and hierarchical network architecture. \nWe solve for the objective in Equation \\ref{mse2} directly by back propagation through a recurrent neural network with equation \\ref{mse} as the loss function.  The model architecture is composed of two branches that are recombined to compute the action prediction at each timepoint.  \n\nTo more easily compare with other approaches that do not blend sub-task policies, we estimate the maximum likelihood sub-task policy label at each timepoint. We refer to sub-task policies as behavior primitives.  The behavior primitive label prediction is given by the maximum likelihood estimate of $\\pi$ shown in Equation \\ref{eqn:label} for time $t$ in trajectory $\\rho$. \n\n\\begin{equation}\n     \\argmaxA_{\\pi \\in \\mathcal{B}} p(\\pi|\\rho_t)\n    \\label{eqn:label}\n\\end{equation}\n\nFigure \\ref{fig:rnn} illustrates how we compute the predicted action $\\hat{a}_t$ at time $t$.  In the figure, the probability of $\\pi$ given state $s_{\\rho_t}$ is $\\lambda_\\pi = p(\\pi|s_{\\rho_t})$ for $\\pi \\in \\mathcal{B}$.  The latent representation $h_t$ at the current timepoint $t$ is a function of both the value of the latent representation of the previous state $h_{t-1}$ and the current state $s_t$\n\n\n\n\n\\begin{figure}[ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{img\/network_arch.png}\n  \\caption{Hierarchical recurrent deep network architecture for task decomposition, novel behavior primitive discovery, and behavior blending.  }\n  \\label{fig:rnn}\n\\end{figure} \n\n\n\n\n\\begin{figure}[ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{img\/pico_model.png}\n  \\caption{Neural network architecture for $PICO$\\xspace.  Given a set of input trajectories and a behavior primitive library, the core architecture follows two branches, the left most branch estimates a distribution over the behavior primitives.  The right hand branch estimates the action prediction from each primitive behavior sub-model.  We compute the predicted action as a linear combination between the behavior primitive distribution and the set of predicted actions from all behavior primitives.  }\n  \\label{fig:model}\n\\end{figure} \n\nFigure \\ref{fig:model} details the architecture used for $PICO$\\xspace based on the Husky+UR5 dataset example.  Unless otherwise specified, the fully connected (FC) layers have ReLU activations, except for the output layers from behavior primitive models.  The last layer of behavior primitive models have linear activations to support diverse action predictions.  While not shown in Figure \\ref{fig:model}, the network architecture also returns the predicted latent embedding and behavior primitive distribution for additional visualization and analysis.  \n\n\\subsection{Discovering and Training New Behavior Primitives}\n\n\nAn important aspect of our approach is the ability to discover and create new behavior primitives from a set of trajectories and a partial behavior primitive library.  $PICO$\\xspace detects and trains new behavior primitive models simultaneously.  As shown in figure \\ref{fig:rnn}, $PICO$\\xspace supports building new behavior primitive models by adding additional randomly initialized behavior models to the library prior to training.\n For our experiments, we assume that we know the correct number of missing primitives.\n\nWe define a \\emph{gap} in a trajectory as region within a demonstration where actions are not predicted with high probability using the existing behavior primitive models.\nA gap in a trajectory implies that the current library of behavior primitives is insufficient to describe a set of state-action tuples $\\rho$ in some part of the given trajectory.  This also implies that the probability $p(\\pi|\\rho_t)$ that the data $\\rho_t$ for time point $t$ was generated by the current library of behavior primitive models is low  for all $\\pi \\in \\mathcal{B}$.  These low probabilities increase the likelihood that an additional randomly initialized behavior primitive policy $\\pi_{new}$ might have a higher probability $p(\\pi_{new}|\\rho_t)>p(\\pi|\\rho_t)$ for $\\pi \\in \\mathcal{B}$. The data $\\rho_t$ is then used to train $\\pi_{new}$. For nearby data in the same gap region $\\rho_{t+1}$, it is now more likely that $p(\\pi_{new}|\\rho_{t+1})>p(\\pi|\\rho_{t+1})$ for $\\pi \\in \\mathcal{B}$.\nThis mechanism allows $\\pi_{new}$ to develop in to a new behavior primitive that is not well covered by existing primitives.\n\n\n\n\\subsection{Training Details}\n$PICO$\\xspace is trained end-to-end by back propagation. This is possible because all functions in the model are differentiable with the exception the \\texttt{argmax} function.  For experiments making use of pretrained behavior primitive models, the contents of the behavior primitive library are trained using the DART~\\cite{DART} technique for imitation learning.\n\nAs shown in Equation \\ref{mse}, the loss used to train the model is mean squared error between the predicted and observed actions over all timepoints and all demonstrations.  There is no loss term for label prediction accuracy, because we assume that the demonstrations are unlabeled.\n\n\\subsection{Metrics}\n\nTwo metrics are computed to estimate performance.  First, we evaluate mean squared error (MSE) as shown in Equation \\ref{mse} between the predicted and given action. \nSecond, we compute behavior primitive label accuracy which is a comparison between the predicted and given behavior primitive label. Label accuracy is computed as the number of matching labels divided by the total number of comparisons. Both metrics are computed over all timepoints and over all demonstrations in the test set.\n\n\n\\subsection{Baseline Implementations}\nShiarli et al. \\cite{TACO18} developed TACO, which aligned subtasks to demonstrations given a library of primitives and a \\emph{task sketch}, where a task sketch describes the sequence in which subtasks will appear. In addition, in their recent work \\cite{TACO18}, they extended the connectionist temporal classification (CTC) algorithm \\cite{CTC06}, commonly used to align sequences for speech recognition, for use with identifying subtasks. For this work, we use TACO and the extended version of CTC as baseline comparisons for our algorithm, using an open source implementation~\\footnote{https:\/\/github.com\/KyriacosShiarli\/taco}. Both were tested using MLP and RNN architectures. \n\n\n\n\\section{EXPERIMENTS AND DISCUSSION}\nWe evaluate $PICO$\\xspace using a reach-grab-lift task using a Husky+UR5 environment.  The dataset consists of 100 demonstrations of a Clearpath Husky robot with a UR5 manipulator performing a variety of reach, grasp, and lift tasks, see Figure~\\ref{fig:husky}. The number of time steps in the demonstrations varied from 1000 to 1800, but each used all three primitives: reach, grasp, and lift. \n\nThe first experiment quantifies the ability of $PICO$\\xspace to identify primitive task labels from demonstration independently from learning behavior primitives.  The second experiment evaluates the ability of $PICO$\\xspace to identify parts of demonstrations that are not represented by existing behavior primitives and rebuild the missing behavior primitive.\n\n\\subsection{Reconstruction from existing primitives}\n\n\nOur initial experiment is an ablation study that separately evaluates the estimate of the primitive behavior probability distribution and the action predictions from learning behavior primitives.  We train and freeze behavior primitive models for \\emph{reach}, \\emph{grasp}, and \\emph{lift} using the ground truth labeled data from trajectories.  We evaluated $PICO$\\xspace, TACO \\cite{TACO18}, and CTC based on label classification accuracy.  For Taco and CTC we additionally compared the methods using MLP and RNN based underlying network models.  We evaluated all methods based on an 80\/20 split of demonstrations into training and test sets.  The average of five independent runs were obtained for each approach.  In Table \\ref{table:husky}, we show the results of the comparison. \n\n\n\\begin{figure}[ht]\n\\centering\n  \\subfloat[Sample trajectory label accuracy]{\n  \\includegraphics[width=.75\\linewidth]{img\/reconstruction.png}\n   }\\break\n   \\subfloat[Missing primitive label accuracy]{\n   \\includegraphics[width=.75\\linewidth]{img\/discovery.png}\n   }\n  \\caption{Example behavior primitive label accuracy for a single test demonstration.  We compared the label predictions given by $PICO$\\xspace (red) to the ground truth (blue) (a) A sample reconstruction for a single trajectory with an existing behavior primitive library.  Timepoints are on the x-axis. and behavior primitive label is on they y-axis.  The labels 0,1, and 2 correspond to reach, grasp, and lift respectively. (b) Reconstruction of an example trajectory and discovery of a missing behavior primitive (grasp).  }\n  \\label{fig:reconstruction}\n\\end{figure}\n\n\nFigure \\ref{fig:reconstruction}(a), shows a comparisons between between the predicted label based on Equation \\ref{eqn:label} and the ground truth label.  Over all trajectories in the test set, the average label classification accuracy was 96\\% compared to the ground truth label.  The summary of results are shown in Table \\ref{table:husky}.\n\n\\subsection{Behavior Primitive Discovery}\n\n\n\nIn our next experiment, we evaluate the ability of $PICO$\\xspace to recognize and build a missing behavior primitive model.  We ran a leave-one-behavior-out experiment where one of the three primitives (i.e. reach, grasp, lift) was replaced with a randomly-initialized behavior primitive.  This experiment used the same 100 trajectories on the Husky+UR5 dataset discussed in the previous section and a 80\/20 split between training and validation sets. Again, five trials were run with the training and validation sets randomly chosen. The label accuracy and action prediction MSE are shown in \\ref{fig:labelling}. The leftmost bar shows the results with all primitives pre-trained with behavior cloning. The remaining bars show the accuracy when reach, grasp and lift, respectively, were replaced with the gap primitive. Note, the gap primitive was updated throughout the training with back-propagation such that the final primitive ideally would perform as well as the original pre-trained, behavior-cloned version; this comparison is shown with the action prediction MSE. The error bars show the standard deviation across the five trials. While the label accuracy across all three replaced primitives is approximately the same, the action prediction for the lift primitive is significantly worse. We believe this is due to the larger variance in lift trajectories. Unlike the reach and grasp which have restrictions placed on their final target position (it needs to be near the block), the final position of lift is randomly placed above the block's starting position. \n\n\n\nAs shown in the sample trajectory in Figure \\ref{fig:reconstruction}(b), the label prediction of the trained model closely aligns with the ground truth label from the example trajectory.  Over all of the test trajectories, the average label classification accuracy was 96\\%.\n\n\\begin{figure}[ht]\n\\centering\n  \\subfloat[behavior label accuracy]{\n  \\includegraphics[width=.75\\linewidth]{img\/label_accuracy.png}\n   }\n   \\break\n   \\subfloat[action prediction MSE]{\n   \\includegraphics[width=.75\\linewidth]{img\/action_mse_accuracy.png}\n   }\n  \\caption{Accuracy of $PICO$\\xspace to correctly identify a primitive's label on the validation set (twenty randomly selected trajectories). (a) The leftmost bar shows performance when all primitives are in the library, successive bars denote accuracy when the \\emph{reach}, \\emph{grasp}, and \\emph{lift} primitives are dropped out and learned from a randomly generated ``gap\" primitive. Error bars represent the standard deviation across five validation trials. (b) Mean squared error between the ground truth action and the learned model's estimate averaged across twenty randomly selected test trajectories five times.  }\n  \\label{fig:labelling}\n\\end{figure}\n\n\n\n\\subsection{Visualizing the Learned Latent Space}\n\nTo better understand the role of the embedding space for predicting the primitive probability distribution, we visualized the embedding of all states vectors from the test set in the recurrent hidden layer.  We would expect that a useful latent embedding would naturally cluster states that correspond to different primitives into distinct locations in the embedding space.\n\\begin{figure}[ht]\n\\centering\n  \\includegraphics[width=.75\\linewidth]{img\/latent2.png}\n  \\caption{The organization of the learned latent space associated with the Husky-UR5 dataset for reach, grasp, and lift (red, green, and purple respectively).    }\n  \\label{fig:latent}\n\\end{figure}\nFigure \\ref{fig:latent} shows layout of the latent space in two dimensions.  Each point corresponds to a state vector from the test dataset.  The points are colored by the ground truth label.\n\n\\subsection{Jaco Dial Domain Dataset}\n\\begin{figure}[ht]\n\\centering\n  \\includegraphics[width=.75\\linewidth]{img\/pinpad.png}\n  \\caption{The joint-domain dial scenario. A Jaco manipulator modeled in Mujoco presses a sequences of 4 keys on a dialpad.  The positions of the keys are randomly shuffled for each demonstration.  The positions of the joints and positions of the keys are given as state information. }\n  \\label{fig:pinpad}\n\\end{figure}\n\nWe also make use of the Jaco dial domain dataset\\cite{TACO18} illustrated in Figure \\ref{fig:pinpad}.  The dial dataset is composed of demonstrations from a Jaco manipulator pressing 4 keys in sequence (e.g. 3,5,4,7).  The positions of the keys are randomly shuffled for each demonstration, but the position of each key is given in the state vector.  The intention is to treat pressing an individual digit as a behavior primitive. For this dataset, label prediction accuracy is a challenging metric without a task sketch because the starting position of the jaco may not provide clues about which button will be pressed.  As the jaco gets closer to a button, it becomes more clear which button will be pressed.  The dataset of dialpad demonstrations were generated using default parameters and code from TACO\\cite{TACO18}.\n\n\n\n\n\n\n\n\\subsection{Dial Domain Comparison}\nThe goal of this comparison is to evaluate the label prediction accuracy of the metacontroller in $PICO$\\xspace.  To isolate the label predictions of the metacontroller, the behavior primitive library is pretrained on the training dataset  including 1200 demonstrations and frozen.  Label classification and action prediction accuracy is then evaluated on the test set including 280 demonstrations.\n\nThe average results of 5 runs are shown for TACO and CTC.  We evaluate each approach using the same label accuracy and action prediction metrics.  The summary of results are shown in Table \\ref{table:pinpad}.  We found that our approach achieves the highest label accuracy at 65\\%.  The overall label accuracy of $PICO$\\xspace on the dial dataset is lower than the Husky+UR5 dataset.  Additional analysis revealed that many of the mislabeling occurred at the beginning of a new key press where context about where the Jaco is moving next is weakest.  The dataset is also more challenging than than the Husky dataset because the number of unique behavior primitives has increased from 3 to 10.  \n\nAlso of note, we compare our results to TACO which is a weakly supervised approach.  TACO is given the ordering of tasks.  For task sequences of length 4, this means that a random baseline would be expected to achieve an accuracy of 25\\%.  For an unlabeled approach like $PICO$\\xspace, any of the 10 behavior primitives could be selected at each timepoint.  This means that unlabeled demonstrations the expected accuracy of a random baseline would be 10\\%.\n\n\n\\section{CONCLUSION}\n\n\n\nIn this paper, we describe $PICO$\\xspace, an approach to learn behavior primitives from unlabeled demonstrations and a partial set of behavior primitives.  We optimize a metric that directly minimizes reconstruction error for a set of demonstrations using sequences of behavior primitives. We directly compare our results to similar approaches using demonstrations generated from simulations of two different robotic platforms and achieve both better label accuracy and reconstruction accuracy as measured by action prediction mean squared error. While we have demonstrated success in these tasks, there are limitations to our approach.  The number additional primitives to add to the library must be decided prior to training. In spite of these limitations, we believe that $PICO$\\xspace is a useful contribution to the community that may be relevant in a number of different domains.\n\n\n\\begin{table}[]\n\\caption{Method comparisons using the Husky UR5 Reach and Grasp dataset.}\n\\label{table:husky}\n\\begin{tabular}{|c|c|c|}\n\\hline\nHusky UR5  & Label Accuracy & MSE Action Prediction  \\\\ \\hline\n$PICO$\\xspace    & \\textbf{96\\%}    &  \\textbf{0.053} \\\\ \\hline\nTACO (MLP)   & 74\\% &  3.59 \\\\ \\hline\nTACO (RNN)   & 73\\% &  3.75 \\\\ \\hline\nCTC (MLP) & 25\\% & 4.20 \\\\ \\hline\nCTC (RNN) & 33\\% & 2.68 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\caption{Method comparisons using the Jaco Pinpad dataset. *TACO (RNN) resulted in NaN loss after repeated attempts.}\n\\label{table:pinpad}\n\\begin{tabular}{|c|c|c|}\n\\hline\nJaco Pinpad & Label Accuracy & MSE Action Prediction  \\\\ \\hline\n$PICO$\\xspace  & \\textbf{65\\%}    &  \\textbf{0.0061} \\\\ \\hline\nTACO (MLP)   & 47\\% &  0.55 \\\\ \\hline\nTACO (RNN)   & * &  * \\\\ \\hline\nCTC (MLP) & 31\\% & 0.57 \\\\ \\hline\nCTC (RNN) & 29\\% & 0.58 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n   \n\n\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $n$ be a positive integer, \nand let $\\mathbb{F}$ be any field.\nThe \\emph{subspace lattice} $\\mathcal{P}_n(\\mathbb{F})$\n(also known as the \\emph{finite projective space})\nis the poset of linear subspaces of $\\mathbb{F}^n$.\nThe general linear group $\\mathrm{GL}_n(\\mathbb{F})$ \nacts on $\\mathcal{P}_n(\\mathbb{F})$.\nThe natural grading structure\nof $\\mathcal{P}_n(\\mathbb{F})$\nis given by $\\mathrm{GL}_n(\\mathbb{F})$-action,\nand\neach fiber (i.e., orbit) is called a \\emph{Grassmannian}.\nIn other words,\nthe Grassmannian $\\mathrm{Gr}(m,n)$ is the set of \n$m$-dimensional subspaces in $\\mathbb{F}^n$, where $0 \\le m \\le n$.\nLet $\\mathcal{B}$ denote the Borel subgroup of $\\mathrm{GL}_n(\\mathbb{F})$.\nThen, the $\\mathcal{B}$-action defines \na finer ``hyper-cubic'' grading structure\nof $\\mathcal{P}_n(\\mathbb{F})$,\nand\neach fiber contained in $\\mathrm{Gr}(m,n)$\nis called a \\emph{Schubert cell of $\\mathrm{Gr}(m,n)$}.\nSee \\cite{MR2474907} for details.\nThe author showed in \\cite{W}\nthat\nthe algebra defined from the ``hyper-cubic'' grading structure of \n$\\mathcal{P}_n(\\mathbb{F})$ together with its incidence structure\nhas a close relation to the quantum affine algebra\n$U_q(\\widehat{\\mathfrak{sl}}_2)$, if $\\mathbb{F}$ is a finite field of $q^2$ elements.\nIn this paper,\nwe study the Schubert cells of a Grassmannian \nfrom the combinatorial point of view of \\emph{association schemes}.\nMore precisely,\nwe show that the association scheme\ndefined by the $\\mathcal{B}$-action\non each Schubert cell is a \n\\emph{generalized wreath product} of \none-class association schemes\nwith the base set $\\mathbb{F}$.\nThe concept of \na generalized wreath product\nof association schemes\nwas introduced by\nR.~A.~Bailey~\\cite{MR2206477} in 2006.\nThe (usual) wreath product\nof association schemes has\nbeen actively studied (see e.g.,\n\\cite{MR2712017, MR2610292, MR2002219, MR2964723, MR3612420, MR2747797, MR3047011, MR2802177}),\nand we may view the result of this paper as demonstrating \nthe fundamental importance of Bailey's generalization as well.\n\nBefore the main discussion,\nwe briefly recall the notion of the generalized wreath product of association schemes.\nFor the definition of association schemes,\nsee \\cite{MR882540, MR2535398, MR2184345},\nand\nfor the theory of posets,\nsee \\cite{MR2868112}.\nLet $(X,\\le)$ be a nonempty finite poset.\nA subset $Y$ in $X$ is called an \\emph{anti-chain}\nif any two elements in $Y$ is incomparable.\nFor an anti-chain $Y$ in $X$,\ndefine the \\emph{down-set} (also known as the \\emph{order ideal}) by \n\\[\n\\mathrm{Down}(Y) = \\lbrace x \\in X \\mid \\text{$x < y$ for some $y \\in Y$}\\rbrace.\n\\]\nNote that this definition follows \\cite{MR2206477}\nand it is different from \\cite{MR2868112},\nwhere \n$\\mathrm{Down}(Y) \\cup Y$ is called \nthe down-set of $Y$.\nFor each $x \\in X$,\nlet $\\mathcal{Q}_x$ denote an $r_x$-class association scheme\non a set $\\Omega_x$.\nWe do not assume either $\\mathcal{Q}_x$ is symmetric\nor $\\Omega_x$ is finite.\nLet $R_{x,i}$ denote the $i$-th associate class\nfor $i \\in \\lbrace 0,1,\\ldots,r_x\\rbrace$.\nBy convention,\nwe choose the index \nso that $R_{x,0} = \\lbrace (\\omega, \\omega) \\mid \\omega \\in \\Omega_x\\rbrace$.\nWe set $\\Omega = \\prod_{x \\in X} \\Omega_x$.\nFor each anti-chain $Y$ in $X$\nand for each $(i_x)_{x \\in Y} \\in \\prod_{x \\in Y} \\lbrace 1,2,\\ldots,r_x\\rbrace$,\nlet\n$R(Y,(i_x)_{x \\in Y})$ denote the set of\n$((\\alpha_x)_{x \\in X}, (\\beta_x)_{x \\in X}) \\in \\Omega \\times \\Omega$\nsatisfying\n(i) $\\alpha_x = \\beta_x$ if $x \\in X \\setminus (Y \\cup \\mathrm{Down}(Y))$,\nand\n(ii) $(\\alpha_x, \\beta_x) \\in R_{x,i_x}$ if $x \\in Y$.\nLet $\\mathcal{R}$ denote the set of \n$R(Y,(i_x)_{x \\in Y})$ for all anti-chains $Y$ in $X$ and \n$(i_x)_{x \\in Y} \\in \\prod_{x \\in Y} \\lbrace 1,2,\\ldots,r_x\\rbrace$.\n\n\\begin{thm}[cf. {\\cite[Theorem 3]{MR2206477}}]\\label{Bailey}\nThe pair $(\\Omega, \\mathcal{R})$ is an association scheme.\n\\end{thm}\n\nThe association scheme $(\\Omega, \\mathcal{R})$ in Theorem \\ref{Bailey} is called the \n\\emph{generalized wreath product of $\\mathcal{Q}_x$\nover the poset $X$}.\nWe remark that Bailey~\\cite[Theorem 3]{MR2206477}\nassumes that each base set $\\Omega_x$ is finite\nand each association scheme $\\mathcal{Q}_x$ is symmetric.\nHowever, the theorem is still true if \nwe drop both of these assumptions.\n\n\\section{Subspace lattices}\n\nThroughout this paper, we fix a positive integer $n$ and a field $\\mathbb{F}$.\nLet $\\mathbb{F}^n$ denote the $n$-dimensional column vector space over $\\mathbb{F}$.\nBy the \\emph{subspace lattice}, denoted by \n$\\mathcal{P}_n(\\mathbb{F})$,\nwe mean the poset consisting of all subspaces\nin $\\mathbb{F}^n$ with partial order given by inclusion.\nWe fix \nthe sequence $\\lbrace V_i \\rbrace_{i = 0}^n$ in $\\mathcal{P}_n(\\mathbb{F})$ such that\neach $V_i$ consists of vectors whose\nbottom $n-i$ entries are zero.\nWe remark that \n$\\lbrace V_i \\rbrace_{i = 0}^n$ is a \\emph{(complete) flag} (i.e., a maximal chain) in $\\mathcal{P}_n(\\mathbb{F})$.\n\n\nLet $\\mathrm{Mat}_n(\\mathbb{F})$\ndenote the set of $n \\times n$ matrices with entries in $\\mathbb{F}$.\nLet $\\mathrm{GL}_n(\\mathbb{F})$ denote the set of \ninvertible matrices in $\\mathrm{Mat}_n(\\mathbb{F})$.\nObserve that \n$\\mathrm{GL}_n(\\mathbb{F})$ acts on $\\mathbb{F}^n$\nby left multiplication,\nand hence it acts on $\\mathcal{P}_n(\\mathbb{F})$.\nLet $\\mathcal{B}$ denote the (Borel) subgroup in $\\mathrm{GL}_n(\\mathbb{F})$ stabilizing\n$\\lbrace V_i \\rbrace_{i = 0}^n$.\nIn other words,\n$\\mathcal{B}$ consists of all \nupper triangular invertible matrices in $\\mathrm{Mat}_n(\\mathbb{F})$.\nIn this paper,\nwe consider the $\\mathcal{B}$-action on $\\mathcal{P}_n(\\mathbb{F})$.\n\nA matrix in $\\mathrm{Mat}_n(\\mathbb{F})$\nis said to be in \\emph{reverse column echelon form}\nif the following two conditions are met:\n\\begin{enumerate}\n\\item[(CE1)] Any zero columns are right of all nonzero columns.\n\\item[(CE2)] The last nonzero entry of a nonzero column is always\nstrictly below of the last nonzero entry of its right column.\n\\end{enumerate}\nA matrix in $\\mathrm{Mat}_n(\\mathbb{F})$\nis said to be in \\emph{reduced reverse column echelon form}\nif it is in reverse column echelon form and \nthe following third condition is also met:\n\\begin{enumerate}\n\\item[(CE3)] Every last nonzero entry of a nonzero column is $1$ and \nis the only nonzero entry in its row.\n\\end{enumerate}\n\nBy elementary linear algebra, we have the following.\n\\begin{prop}\\label{prop:bij}\nThere exists a bijection between the following two sets:\n\\begin{enumerate}\n\\item the set of all matrices in $\\mathrm{Mat}_n(\\mathbb{F})$ \nin reduced reverse column echelon form,\n\\item the set $\\mathcal{P}_n(\\mathbb{F})$ of all subspaces in $\\mathbb{F}^n$,\n\\end{enumerate}\nthat sends a matrix in $\\mathrm{Mat}_n(\\mathbb{F})$\nto its column space.\n\\end{prop}\n\\begin{proof}\nSee for instance \\cite{MR3013937}.\n\\end{proof}\n\n\\begin{ex}[$n=7$]\\label{ex1}\nLet $e_1, e_2, \\ldots, e_7$ denote the standard basis for $\\mathbb{F}^7$.\nSuppose $U$ is the $4$-dimensional subspace\nin $\\mathbb{F}^7$ given by\n$U = \\mathrm{Span}\\lbrace\n8e_1+6e_3+4e_6+2e_7,\n8e_1+9e_3+e_4+e_5,\n4e_1+e_2+5e_3+e_4,\n3e_1+e_2\\rbrace$.\nThen the following is the matrix corresponding to $U$\nby the bijection in Proposition \\ref{prop:bij}:\n\\[\nM = \n\\begin{pmatrix}\n4 & 7 & 1 & 3 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n3 & 4 & 5 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n2 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix}.\n\\]\n\\end{ex}\nObserve that\nfor \n$M, N \\in \\mathrm{Mat}_n(\\mathbb{F})$\nin reduced reverse column echelon form\nand for $G \\in \\mathcal{B}$,\nthe column space of $N$ moves to \nthat of $M$ by the $G$-action\nif and only if \n$M$ and $GN$ are column equivalent.\nSince $GN$ is in reverse column echelon form\n(but not necessarily reduced),\nthese conditions are equivalent to\n$M = GNH^{T}$ for some $H \\in \\mathcal{B}$.\nFor notational convenience, \nwe write $M \\sim N$\nif there exist $G, H \\in \\mathcal{B}$ such that $M = GNH^T$.\nObserve that $\\sim$ is an equivalence relation on $\\mathrm{Mat}_n(\\mathbb{F})$.\n\nFor the rest of this paper,\nwe will identify $\\mathcal{P}_n(\\mathbb{F})$ with the set of \nall matrices in $\\mathrm{Mat}_n(\\mathbb{F})$ in reduced reverse column echelon form\nby the bijection in Proposition \\ref{prop:bij}.\n\n\\section{The $\\mathcal{B}$-action on $\\mathcal{P}_n(\\mathbb{F})$}\n\nFor a positive integer $m$, we write \n$[m] = \\lbrace 1,2,\\ldots, m\\rbrace$.\nWe define a partial order in\nthe index set $[n] \\times [n]$ of matrices in $\\mathrm{Mat}_n(\\mathbb{F})$ by \n$(i,j) \\le (k,l)$ if $i \\le k$ and $j \\le l$.\nThis is known as the \\emph{direct product order}\nin \\cite[Section~3.2]{MR2868112}.\nFor $M \\in \\mathrm{Mat}_n(\\mathbb{F})$,\nby the \\emph{support} of $M$, denoted by $\\mathrm{Supp}(M)$, we mean \nthe subposet of $[n] \\times [n]$\nconsisting of all indices $(i,j) \\in [n] \\times [n]$ with $M_{i,j} \\neq 0$.\nThe \\emph{pivot-set} of $M$,\ndenoted by $\\mathrm{Piv}(M)$,\nis the set of all maximal elements in $\\mathrm{Supp}(M)$.\nEach element in the pivot-set is called a \\emph{pivot}.\nObserve that\n$(i,j) \\in \\mathrm{Piv}(M)$\nif and only if\n$M_{i,j} \\neq 0$ and $M_{k,l} = 0$ if $(k,l) > (i,j)$.\nWe remark that\nevery entry indexed by a pivot of a matrix in $\\mathcal{P}_n(\\mathbb{F})$ \nmust be $1$ by the condition (CE3).\n\n\\begin{lem}\\label{lem:wequiv}\nFor $M, N \\in \\mathcal{P}_n(\\mathbb{F})$,\nthe following are equivalent:\n\\begin{enumerate}\n\\item $M \\sim N$,\n\\item $\\mathrm{Piv}(M) = \\mathrm{Piv}(N)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n(i) $\\Rightarrow$ (ii)\nSuppose $M \\sim N$.\nThere exist\n$G, H \\in \\mathcal{B}$ such that $M = GNH^T$.\nIt suffices to show $\\mathrm{Piv}(N) \\subseteq \\mathrm{Piv}(M)$.\nFor $(i,j) \\in \\mathrm{Piv}(N)$,\nwe have\n$N_{i,j} = 1$ and $N_{k,l} = 0$ if $(k,l) > (i,j)$.\nSince\n$G, H$ are upper triangular,\nwe have\n\\[\nM_{k,l} = \\sum_{s = k}^n\\sum_{t = l}^n G_{k,s}N_{s,t}H_{l,t}\n=\n\\begin{cases}\nG_{i,i}H_{j,j} & \\text{if $(k,l) = (i,j)$},\\\\\n0 & \\text{if $(k,l) > (i,j)$}.\n\\end{cases}\n\\]\nSince $G, H$ are invertible,\n$G_{i,i}H_{j,j} \\neq 0$.\nThese imply $(i,j) \\in \\mathrm{Piv}(M)$\nand hence $\\mathrm{Piv}(N) \\subseteq \\mathrm{Piv}(M)$.\n\n(ii) $\\Rightarrow$ (i)\nSuppose $\\mathrm{Piv}(M) = \\mathrm{Piv}(N)$.\nTake $X \\in \\mathcal{P}_n(\\mathbb{F})$ with\n$X_{i,j} = 1$ if $(i,j) \\in \\mathrm{Piv}(M)$ and $X_{i,j} = 0$ otherwise.\nObserve that\nfor each $j \\in [n]$,\nthere exists at most one $k$ such that $(j,k) \\in \\mathrm{Piv}(M)$\nand then we\ndefine $G \\in \\mathrm{Mat}_n(\\mathbb{F})$ by\n\\[\nG_{i,j} =\n\\begin{cases}\nM_{i,k} & \\text{if $(j,k) \\in \\mathrm{Piv}(M)$ for some $k$},\\\\\n\\delta_{i,j} & \\text{if there is no $k$ such that  $(j,k) \\in \\mathrm{Piv}(M)$}\n\\end{cases}\n\\]\nfor $i,j \\in [n]$.\nThen we have $G_{i,i} = 1$ for $i \\in [n]$\nand $G_{i,j} = 0$ if $j < i$ for $i, j \\in [n]$.\nThus \n$G \\in \\mathcal{B}$.\nBy the direct calculation, we have $M = GX$\nand \nhence, $M \\sim X$.\nSimilarly we have $N \\sim X$ and so $M \\sim N$.\n\\end{proof}\n\nLet $1 \\le m \\le n-1$ and $M \\in \\mathcal{P}_n(\\mathbb{F})$\nwith $\\rank M = m$.\nNote that \nwe avoid the trivial cases $m = 0$ and $m = n$.\nSince the pivots of $M$ lie in the first $m$ columns,\n$\\mathrm{Piv}(M)$\nis an anti-chain in $[n] \\times [m]$ of size $m$.\nFor $1 \\le m \\le n-1$\nand for an anti-chain $\\alpha$ in $[n] \\times [m]$\n of size $m$,\nwe set\n\\begin{equation}\n\\mathcal{O}_{\\alpha} = \\lbrace M \\in \\mathcal{P}_n(\\mathbb{F}) \\mid\n\\mathrm{Piv}(M) = \\alpha\n\\rbrace.\n\\label{OA}\n\\end{equation}\nFor each $1 \\le m \\le n-1$ and\neach anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$,\nconsider\n$M \\in \\mathcal{P}_n(\\mathbb{F})$ with\n$M_{i,j} = 1$ if $(i,j) \\in \\alpha$ and $M_{i,j} = 0$ otherwise.\nThen we have $M \\in \\mathcal{O}_\\alpha$\nand in particular $\\mathcal{O}_\\alpha \\neq \\emptyset$.\n\n\n\\begin{prop}\\label{prop:OA}\nThe rank of any matrix in \\eqref{OA} \nis $m = |\\alpha|$.\nMoreover,\neach subset \\eqref{OA}\nis an orbit of the $\\mathcal{B}$-action on $\\mathcal{P}_n(\\mathbb{F})$.\n\\end{prop}\n\\begin{proof}\nImmediate from the construction and Lemma \\ref{lem:wequiv}.\n\\end{proof}\n\nRecall the Grassmannian $\\mathrm{Gr}(m,n)$\nand we identify $\\mathrm{Gr}(m,n)$ with a set of matrices\nby the bijection in Proposition \\ref{prop:bij}.\nIn other words,\n\\[\n\\mathrm{Gr}(m,n) = \\lbrace M \\in \\mathcal{P}_n(\\mathbb{F}) \\mid\n\\rank M = m\n\\rbrace.\n\\]\nBy Proposition \\ref{prop:OA},\neach $\\mathcal{O}_\\alpha$ in \\eqref{OA} is\na $\\mathcal{B}$-orbit in $\\mathrm{Gr}(m,n)$,\nwhere $m = |\\alpha|$.\nThus,\nit is called a \\emph{Schubert cell of a Grassmannian} \\cite{MR2474907}.\n\n\\begin{ex}[$n=7$, $m = 4$]\\label{ex2}\nTake $M \\in \\mathcal{P}_7(\\mathbb{F})$ as in Example \\ref{ex1}.\nThen we have\n$\\mathrm{Piv}(M) = \\lbrace\n(2,4), (4,3), (5,2), (7,1)\n\\rbrace$.\nMoreover,\n$\\mathcal{O}_{\\mathrm{Piv}(M)}$ is\nthe set of matrices of the form\n\\begin{equation}\n\\begin{pmatrix}\n\\ast & \\ast & \\ast & \\ast & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n\\ast & \\ast & \\ast & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n\\ast & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{pmatrix},\n\\label{ex:Oalpha}\n\\end{equation}\nwhere the symbol $\\ast$ denotes an arbitrary element\nin $\\mathbb{F}$.\n\\end{ex}\n\n\\begin{lem}\\label{lem:diag}\nLet $1 \\le m \\le n-1$\nand let $\\alpha$ denote an anti-chain in $[n] \\times [m]$ of size $m$.\nFor $M, N, M', N' \\in \\mathcal{O}_\\alpha$,\nthe following are equivalent:\n\\begin{enumerate}\n\\item $(M,N)$ moves to $(M',N')$ by the diagonal $\\mathcal{B}$-action,\n\\item $\\mathrm{Piv}(M-N) = \\mathrm{Piv}(M'-N')$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n(i) $\\Rightarrow$ (ii)\nSuppose there exist $G, H, K \\in \\mathcal{B}$\nsuch that $M' = GMH^T$ and $N' = GNK^T$.\nThen we have\n\\begin{equation}\n\\mathrm{Piv}(GM) = \\mathrm{Piv}(GN) = \\alpha,\n\\label{*}\n\\end{equation}\n\\begin{equation}\n\\mathrm{Piv}(GM-GN) = \\mathrm{Piv}(M-N)\n\\label{**}\n\\end{equation}\nsince $G \\in \\mathcal{B}$ (cf. Lemma \\ref{lem:wequiv}).\nWe write $\\alpha = \\lbrace(k_r,r) \\mid r \\in [m] \\rbrace$\nand observe that $k_1 > k_2 > \\cdots > k_m$\nand that $\\mathrm{Piv}(M-N) \\subseteq \\mathrm{Down}(\\alpha)$.\n\nTake $(i,j) \\in \\mathrm{Piv}(M-N)$.\nObserve that $j \\in [m]$ and $k_j > i$\nand hence \nthere exists $m' = \\max\\lbrace r \\in [m] \\mid k_r > i\\rbrace$.\nFor $r, l \\in [m]$ with $j \\le r \\le m'$ and $j \\le l \\le m'$,\nsince $H, K$ are upper triangular,\nwe have\n\\begin{align*}\nM'_{k_r,l} &= \\sum_{t = l}^n (GM)_{k_r,t}H_{l,t}, \\\\\nN'_{k_r,l} &= \\sum_{t = l}^n (GN)_{k_r,t}K_{l,t}.\n\\end{align*}\nIn the above equations, we have the following:\nFor each $r,l$, we have $M'_{k_r,l} = N'_{k_r,l} = \\delta_{r,l}$ by (CE3);\nFor each $r,t$, we have $(GM)_{k_r,t} = (GN)_{k_r,t}$\nsince $(k_r,t) > (i,j)$ and by \\eqref{**};\nFor each $r, t$ with $r < t$, we have $(GM)_{k_r,r} = (GN)_{k_r,r} = G_{k_r,k_r} \\neq 0$ \nand $(GM)_{k_r,t} = (GN)_{k_r,t} = 0$ \nsince $(k_r,r) \\in \\alpha$ and by \\eqref{*}.\nBy these comments,\nfor each $l \\in [m]$ with $j \\le l \\le m'$,\nboth $(H_{l,l}, H_{l,l+1}, \\ldots, H_{l,m'})$\nand $(K_{l,l}, K_{l,l+1}, \\ldots, K_{l,m'})$\nare solutions to the same system of $m'-j+1$ independent linear equations.\nHence, $H_{l,t} = K_{l,t}$ for $j,t \\in [m]$ with \n$j \\le l \\le t \\le m'$.\n\nFor $(k,l) \\in [n] \\times [n]$ with $(k,l) > (i,j)$,\nwe have\n$(GM)_{k,t} = (GN)_{k,t}$ if $t \\ge l$ since $(k,t) > (i,j)$ and by \\eqref{**},\nand we also have \n$(GM)_{k,t} = (GN)_{k,t} = 0$ if $t > m'$ by the definition of $m'$ and by \\eqref{*}.\nRecall that we have shown $H_{l,t} = K_{l,t}$ if $j \\le l \\le t \\le m'$.\nBy these comments, we have\n\\begin{align*}\nM'_{k,l} = \\sum_{t = l}^{n} (GM)_{k,t}H_{l,t}\n= \\sum_{t = l}^{n} (GN)_{k,t}K_{l,t} = N'_{k,l}.\n\\end{align*}\nSimilarly,\nwe have\n\\begin{align*}\nM'_{i,j} - N'_{i,j} = \\sum_{t = j}^{n} (GM)_{i,t}H_{j,t}\n- \\sum_{t = j}^{n} (GN)_{i,t}K_{j,t} \n= ((GM)_{i,j} - (GN)_{i,j})H_{j,j}.\n\\end{align*}\nIn the above equations,\nwe have $(GM)_{i,j} \\neq (GN)_{i,j}$ by \\eqref{**},\nand we also have $H_{j,j} \\neq 0$ since $H \\in \\mathcal{B}$.\nTherefore we obtain $M'_{i,j} \\neq N'_{i,j}$.\nThese imply $(i,j) \\in \\mathrm{Piv}(M'-N')$\nand hence $\\mathrm{Piv}(M-N) \\subseteq \\mathrm{Piv}(M'-N')$.\nSince $G, H, K$ are invertible,\nwe also have \n$\\mathrm{Piv}(M'-N') \\subseteq \\mathrm{Piv}(M-N)$.\nConsequently, we have\n$\\mathrm{Piv}(M-N) = \\mathrm{Piv}(M'-N')$.\n\n(ii) $\\Rightarrow$ (i)\nLet\n$M, N, M', N' \\in \\mathcal{O}_\\alpha$\nwith $\\mathrm{Piv}(M-N) = \\mathrm{Piv}(M'-N')$.\nTake $X \\in \\mathcal{O}_\\alpha$ with\n$X_{i,j} = 1$ if $(i,j) \\in \\alpha \\cup \\mathrm{Piv}(M-N)$ and $X_{i,j} = 0$ otherwise,\nand $Y \\in \\mathcal{O}_\\alpha$ with\n$Y_{i,j} = 1$ if $(i,j) \\in \\alpha$ and $Y_{i,j} = 0$ otherwise.\nDefine $G \\in \\mathrm{Mat}_n(\\mathbb{F})$ by\n\\[\nG_{i,j} = \\begin{cases}\nN_{i,k} & \\text{if $(j,k) \\in \\alpha$ for some $k$},\\\\\nM_{i,k} - N_{i,k} & \\text{if $(j,k) \\in \\mathrm{Piv}(M-N)$ for some $k$}, \\\\\n\\delta_{i,j} & \\text{if there is no $k$ such that $(j,k) \\in \\alpha \\cup \\mathrm{Piv}(M-N)$}\n\\end{cases}\n\\]\nfor $i,j \\in [n]$.\nThen we have $G \\in \\mathcal{B}$\nand $M = GX$ and $N = GY$.\nSimilarly there exists $G' \\in \\mathcal{B}$\nsuch that\n$M' = G'X$ and $N' = G'Y$.\nTherefore\n$(M,N)$ moves to $(M',N')$ by the diagonal action of \n$G'G^{-1}$.\n\\end{proof}\n\nLet $1 \\le m \\le n-1$\nand let $\\alpha$ denote an anti-chain in $[n] \\times [m]$ of size $m$.\nLet $M, N \\in \\mathcal{O}_\\alpha$.\nObserve that\n$\\mathrm{Piv}(M-N)$\nis an anti-chain in \n\\begin{equation}\n\\mathcal{D}(\\alpha)\n=\n\\lbrace (i,j) \\in \\mathrm{Down}(\\alpha) \\mid\n\\text{there is no $k$ such that $(i,k) \\in \\alpha$}\\rbrace.\n\\label{Dalpha}\n\\end{equation}\nFor $1 \\le m \\le n-1$\nand for an anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$\nand for an anti-chain $\\beta$ in $\\mathcal{D}(\\alpha)$,\nwe set\n\\begin{equation}\n\\mathcal{R}_{\\alpha,\\beta} = \\lbrace (M,N) \\in \\mathcal{O}_\\alpha \\times \\mathcal{O}_\\alpha \\mid \n\\mathrm{Piv}(M-N) = \\beta\n\\rbrace.\n\\label{Rab}\n\\end{equation}\nFor each $1 \\le m \\le n-1$ and\neach anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$\nand each anti-chain $\\beta$ in $\\mathcal{D}(\\alpha)$,\nconsider\n$M \\in \\mathcal{P}_n(\\mathbb{F})$ with\n$M_{i,j} = 1$ if $(i,j) \\in \\alpha \\cup \\beta$ and $M_{i,j} = 0$ otherwise,\nand $N \\in \\mathcal{P}_n(\\mathbb{F})$ with\n$N_{i,j} = 1$ if $(i,j) \\in \\alpha$ and $N_{i,j} = 0$ otherwise.\nThen we have $(M, N) \\in \\mathcal{R}_{\\alpha,\\beta}$\nand in particular $\\mathcal{R}_{\\alpha,\\beta} \\neq \\emptyset$.\n\n\\begin{prop}\\label{prop:Rab}\nLet $1 \\le m \\le n-1$\nand let $\\alpha$ denote an anti-chain in $[n] \\times [m]$ of size $m$.\nEach subset \\eqref{Rab}\nis an orbital of the $\\mathcal{B}$-action on $\\mathcal{O}_\\alpha$.\n\\end{prop}\n\\begin{proof}\nImmediate from Lemma \\ref{lem:diag}.\n\\end{proof}\n\nLet $1 \\le m \\le n-1$.\nFor an anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$,\nconsider\n\\begin{equation}\n\\mathcal{D}_1(\\alpha) = \\lbrace i \\mid \\text{there is no $k$ such that $(i,k) \\in \\alpha$}\\rbrace.\n\\label{D1alpha}\n\\end{equation}\nThen we have $|\\mathcal{D}_1(\\alpha)| = n-m$ since \n$|\\alpha| = m$.  \nFor $1 \\le i \\le n-m$,\nwe define\n$\\lambda_i = |\\lbrace j \\mid (d_i,j) \\in \\mathcal{D}(\\alpha)\\rbrace|$,\nwhere $d_i$ denotes the $i$-th smallest element in $\\mathcal{D}_1(\\alpha)$.\nThen $\\lambda = (\\lambda_1, \\lambda_2, \\ldots, \\lambda_{n-m}) \\in \\mathbb{N}^{n-m}$ is \nan integer partition (i.e., a non-increasing sequence) with largest part at most $m$,\nwhere \n\\[\n\\mathbb{N} = \\lbrace 0,1,\\ldots\\rbrace.\n\\]\nConsider the map $\\varphi_m$ which sends $\\alpha$\nto $\\lambda$.\n\nFor an integer partition $\\lambda = (\\lambda_1, \\lambda_2, \\ldots, \\lambda_l)$,\nthe \\emph{Ferrers board} of shape $\\lambda$\nis defined by\n\\[\n\\lbrace (i,j) \\in \\mathbb{N} \\times \\mathbb{N} \\mid 1 \\le i \\le l, 1 \\le j \\le \\lambda_i \\rbrace.\n\\]\nWe endow the Ferrers board\nwith direct product order in $\\mathbb{N} \\times \\mathbb{N}$.\n\n\\begin{lem}\\label{al}\nFor $1 \\le m \\le n-1$,\nthe map $\\varphi_m$ is a bijection between the following two sets:\n\\begin{enumerate}\n\\item the anti-chains in $[n] \\times [m]$ of size $m$;\n\\item the integer partitions in $\\mathbb{N}^{n-m}$ with largest part at most $m$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nLet $1 \\le m \\le n-1$.\nIt is clear that $\\varphi_m$ is a map from (i) to (ii).\nWe define the map $\\varphi_m'$ from (ii) to (i) as follows.\nFor a given integer partition $\\lambda$ in $\\mathbb{N}^{n-m}$ with largest part at most $m$,\nwe define\n$\\alpha$ as the set of maximal elements \nin the Ferrers board of shape $\\mu = \\lambda \\cup (m, m-1,\\ldots,1)$,\nwhich is the integer partition obtained by rearranging \nparts of both $\\lambda$ and $(m, m-1,\\ldots,1)$ in non-increasing order.\nSince $\\mu$ is in $\\mathbb{N}^{n}$ and its largest part is $m$,\n$\\alpha$ is an anti-chain in $[n] \\times [m]$ of size $m$.\nThe map $\\varphi_m'$ is defined to send $\\lambda$\nto $\\alpha$.\nBy construction,\n$\\varphi_m$ and $\\varphi_m'$ are inverses and hence bijections.\n\\end{proof}\n\n\n\\begin{lem}\\label{bmu}\nLet $1 \\le m \\le n-1$\nand\nlet $\\alpha$ denote an anti-chain in $[n] \\times [m]$ of size $m$.\nThe poset\n$\\mathcal{D}(\\alpha)$ in \\eqref{Dalpha} is isomorphic to\nthe Ferrers board of shape $\\varphi_m(\\alpha)$.\nMoreover,\nthere is a one-to-one correspondence\nbetween\nthe anti-chains in $\\mathcal{D}(\\alpha)$ \nand the subpartitions of $\\varphi_m(\\alpha)$.\n\\end{lem}\n\\begin{proof}\nRecall the set $\\mathcal{D}_1(\\alpha)$ in \\eqref{D1alpha}.\nThen define the map $\\psi$\nfrom\nthe Ferrers board of shape $\\varphi_m(\\alpha)$ \nto $\\mathcal{D}(\\alpha)$ by\n$\\psi(i,j) = (d_i,j)$,\nwhere $d_i$ denotes the $i$-th smallest element in $\\mathcal{D}_1(\\alpha)$.\nIt is obvious that $\\psi$ is an order-preserving bijection.\nThe first assertion follows.\nTo show the second assertion,\nwe define the map $\\rho$ from \nthe subpartitions of $\\varphi_m(\\alpha)$\nto\nthe anti-chains in $\\mathcal{D}(\\alpha)$\nby \n$\\rho(\\mu) = \\psi(\\max(\\mu))$,\nwhere $\\max(\\mu)$ is the set of all maximal elements in \nthe Ferrers board of shape $\\mu$.\nFrom the construction, \nthe map $\\rho$ is also a bijection.\nThe second assertion follows.\n\\end{proof}\n\n\\begin{ex}[$n=7$, $m=4$]\nTake the anti-chain $\\alpha = \\lbrace\n(2,4), (4,3), (5,2), (7,1)\n\\rbrace$ as in Example \\ref{ex2}.\nRecall that $\\mathcal{O}_\\alpha$ is the set of \nmatrices of the form \\eqref{ex:Oalpha}.\nThen $\\varphi_4(\\alpha) = (4,3,1)$.\nWe remark that\neach number in $(4,3,1)$\nequals the number of $\\ast$'s in each row without a pivot.\n\\end{ex}\n\n\\section{The association scheme on each Schubert cell}\n\nFor $1 \\le m \\le n-1$\nand for an anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$,\nby Propositions \\ref{prop:OA} and \\ref{prop:Rab},\nthe pair\n\\begin{equation}\n\\mathfrak{X}_\\alpha = (\\mathcal{O}_\\alpha, \\lbrace\\mathcal{R}_{\\alpha, \\beta}\\rbrace_\\beta)\n\\label{Xalpha}\n\\end{equation}\nbecomes an association scheme,\nwhere $\\beta$ runs over all anti-chains in $\\mathcal{D}(\\alpha)$ in \\eqref{Dalpha}.\nSee \\cite[Preface]{MR2184345}.\nWe remark that by Lemma \\ref{al},\nthe family of association schemes $\\lbrace \\mathfrak{X}_\\alpha \\rbrace_\\alpha$\ncan be indexed by integer partitions $\\lambda \\in \\mathbb{N}^{n-m}$ with largest part at most $m$.\nIn this case,\nthe associate classes of $\\mathfrak{X}_\\alpha = \\mathfrak{X}_\\lambda$ are indexed by \nthe subpartitions of $\\lambda$\nby Lemma \\ref{bmu}.\n\\begin{thm}\nLet $1 \\le m \\le n-1$\nand let $\\alpha$ denote an anti-chain in $[n] \\times [m]$ of size $m$.\nThe association scheme $\\mathfrak{X}_\\alpha$ in \\eqref{Xalpha} is symmetric.\n\\end{thm}\n\\begin{proof}\nImmediate from the definition of $\\mathcal{R}_{\\alpha, \\beta}$.\n\\end{proof}\n\n\\begin{thm}\nFor $1 \\le m \\le n-1$\nand for an anti-chain $\\alpha$ in $[n] \\times [m]$ of size $m$,\nthe association scheme $\\mathfrak{X}_\\alpha$ in \\eqref{Xalpha} is \nthe generalized wreath product of the one-class association schemes \nwith the base set $\\mathbb{F}$ over the poset $\\mathcal{D}(\\alpha)$ in \\eqref{Dalpha}.\n\\end{thm}\n\\begin{proof}\nFor \n$M, N \\in \\mathcal{O}_\\alpha$\nand for\nan anti-chain $\\beta$ in $\\mathcal{D}(\\alpha)$,\nwe have\n$(M,N) \\in \\mathcal{R}_{\\alpha, \\beta}$\nif and only if\n$M_{i,j} = N_{i,j}$ if $(i,j) \\not\\in \\beta \\cup \\mathrm{Down}(\\beta)$, $M_{i,j} \\neq N_{i,j}$ if $(i,j) \\in \\beta$.\nTherefore,\nthis associate relation\nis the same as that of \nthe generalized wreath product of the one-class association schemes \nwith the base set $\\mathbb{F}$ over the poset $\\mathcal{D}(\\alpha)$.\nSo the result follows.\n\\end{proof}\n\n\\section{Concluding remarks}\n\nThis paper focuses on the Schubert cells\nof a Grassmannian.\nIt would be an interesting problem to find similar results on Schubert cells for other types of BN-pairs.\n\nThe Terwilliger algebra, introduced by P.~Terwilliger \\cite{MR1203683}, \nof the wreath product of one-class association schemes\nis discussed in several papers \\cite{MR2610292, MR2747797, MR2802177}.\nWe will consider\nthe Terwilliger algebra of the generalized wreath product of one-class association schemes\nin a future paper.\n\n\\section*{Acknowledgments}\nThe author gratefully acknowledges the many helpful suggestions of his advisor, Hajime Tanaka\nduring the preparation of the paper.\nThe author thanks Paul Terwilliger\nfor giving valuable comments\nand Motohiro Ishii for drawing the author's attention to the theory of Schubert cells.\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{MR2206477}{article}{\n   author={Bailey, R. A.},\n   title={Generalized wreath products of association schemes},\n   journal={European J. Combin.},\n   volume={27},\n   date={2006},\n   number={3},\n   pages={428--435},\n   issn={0195-6698},\n}\n\n\\bib{MR882540}{book}{\n   author={Bannai, Eiichi},\n   author={Ito, Tatsuro},\n   title={Algebraic combinatorics. I},\n   subtitle={Association schemes},\n   publisher={The Benjamin\/Cummings Publishing Co., Inc., Menlo Park, CA},\n   date={1984},\n   pages={xxiv+425},\n   isbn={0-8053-0490-8},\n}\n\n\\bib{MR2712017}{thesis}{\n   author={Bhattacharyya, Gargi},\n   title={Terwilliger algebras of wreath products of association schemes},\n   type={Thesis (Ph.D.)--Iowa State University},\n   publisher={ProQuest LLC, Ann Arbor, MI},\n   date={2008},\n   pages={84},\n   isbn={978-0549-68830-3},\n}\n\n\\bib{MR2610292}{article}{\n   author={Bhattacharyya, Gargi},\n   author={Song, Sung Y.},\n   author={Tanaka, Rie},\n   title={Terwilliger algebras of wreath products of one-class association\n   schemes},\n   journal={J. Algebraic Combin.},\n   volume={31},\n   date={2010},\n   number={3},\n   pages={455--466},\n   issn={0925-9899},\n}\n\n\\bib{MR2002219}{article}{\n   author={Hanaki, Akihide},\n   author={Hirotsuka, Kaoru},\n   title={Irreducible representations of wreath products of association\n   schemes},\n   journal={J. Algebraic Combin.},\n   volume={18},\n   date={2003},\n   number={1},\n   pages={47--52},\n   issn={0925-9899},\n}\n\n\\bib{MR3013937}{collection}{\n   title={Handbook of linear algebra},\n   series={Discrete Mathematics and its Applications (Boca Raton)},\n   editor={Hogben, Leslie},\n   edition={2},\n   publisher={CRC Press, Boca Raton, FL},\n   date={2014},\n   pages={xxx+1874},\n   isbn={978-1-4665-0728-9},\n}\n\n\\bib{MR2964723}{article}{\n   author={Kim, Kijung},\n   title={Terwilliger algebras of wreath products by quasi-thin schemes},\n   journal={Linear Algebra Appl.},\n   volume={437},\n   date={2012},\n   number={11},\n   pages={2773--2780},\n   issn={0024-3795},\n}\n\n\\bib{MR2474907}{book}{\n   author={Lakshmibai, V.},\n   author={Brown, Justin},\n   title={Flag varieties},\n   series={Texts and Readings in Mathematics},\n   volume={53},\n   subtitle={An interplay of geometry, combinatorics, and representation\n   theory},\n   publisher={Hindustan Book Agency, New Delhi},\n   date={2009},\n   pages={xiv+272},\n   isbn={978-81-85931-92-0},\n}\n\n\\bib{MR2535398}{article}{\n   author={Martin, William J.},\n   author={Tanaka, Hajime},\n   title={Commutative association schemes},\n   journal={European J. Combin.},\n   volume={30},\n   date={2009},\n   number={6},\n   pages={1497--1525},\n   issn={0195-6698},\n}\n\n\\bib{MR3612420}{article}{\n   author={Song, Sung Y.},\n   author={Xu, Bangteng},\n   author={Zhou, Shenglin},\n   title={Combinatorial extensions of Terwilliger algebras and wreath\n   products of association schemes},\n   journal={Discrete Math.},\n   volume={340},\n   date={2017},\n   number={5},\n   pages={892--905},\n   issn={0012-365X},\n}\n\n\\bib{MR2868112}{book}{\n   author={Stanley, Richard P.},\n   title={Enumerative combinatorics. Volume 1},\n   series={Cambridge Studies in Advanced Mathematics},\n   volume={49},\n   edition={2},\n   publisher={Cambridge University Press, Cambridge},\n   date={2012},\n   pages={xiv+626},\n   isbn={978-1-107-60262-5},\n}\n\n\\bib{MR2747797}{article}{\n   author={Tanaka, Rie},\n   title={Classification of commutative association schemes with almost\n   commutative Terwilliger algebras},\n   journal={J. Algebraic Combin.},\n   volume={33},\n   date={2011},\n   number={1},\n   pages={1--10},\n   issn={0925-9899},\n}\n\n\\bib{MR3047011}{article}{\n   author={Tanaka, Rie},\n   author={Zieschang, Paul-Hermann},\n   title={On a class of wreath products of hypergroups and association\n   schemes},\n   journal={J. Algebraic Combin.},\n   volume={37},\n   date={2013},\n   number={4},\n   pages={601--619},\n   issn={0925-9899},\n}\n\n\\bib{MR1203683}{article}{\n   author={Terwilliger, Paul},\n   title={The subconstituent algebra of an association scheme. I},\n   journal={J. Algebraic Combin.},\n   volume={1},\n   date={1992},\n   number={4},\n   pages={363--388},\n   issn={0925-9899},\n}\n\n\\bib{W}{article}{\n   author={Watanabe, Yuta},\n   title={An algebra associated with a flag in a subspace lattice over a finite field and the quantum affine algebra $U_q(\\widehat{\\mathfrak{sl}}_2)$},\n   eprint={arXiv:1709.06329 [math.CO]},\n   date={2017},\n}\n\n\\bib{MR2802177}{article}{\n   author={Xu, Bangteng},\n   title={Characterizations of wreath products of one-class association\n   schemes},\n   journal={J. Combin. Theory Ser. A},\n   volume={118},\n   date={2011},\n   number={7},\n   pages={1907--1914},\n   issn={0097-3165},\n}\n\n\\bib{MR2184345}{book}{\n   author={Zieschang, Paul-Hermann},\n   title={Theory of association schemes},\n   series={Springer Monographs in Mathematics},\n   publisher={Springer-Verlag, Berlin},\n   date={2005},\n   pages={xvi+283},\n   isbn={978-3-540-26136-0},\n   isbn={3-540-26136-2},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\bigskip\n\n\\noindent\nYuta Watanabe \\\\\nGraduate School of Information Sciences \\\\\nTohoku University \\\\\nSendai, 980-8579 Japan\\\\\nemail: \\texttt{watanabe@ims.is.tohoku.ac.jp}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s1}\nReaction-diffusion equations arise naturally when modeling certain phenomena in biological, chemical and physical systems. In this paper we study reaction-diffusion equations for infinity Laplacian, which despite being too degenerate to realistically represent a physical diffusion process, has been studied in the framework of optimization and free boundary problems (see, for example, \\cite{ATU19}, \\cite{RST17}, \\cite{RT12,RTU15,TU17}, just to cite a few). More precisely, we establish regularity and geometric properties of solutions of the problem\n\\begin{equation}\\label{1.1}\n\\Delta_\\infty u=f(u) \\quad \\text{in} \\quad \\Omega,\n\\end{equation}\nwhere $\\Omega\\subset\\mathbb{R}^n$, $f\\in C(\\mathbb{R}_+)$ and\n\\begin{equation}\\label{1.2}\n0\\le f(\\delta t)\\leq M\\delta^\\gamma f(t),\n\\end{equation}\nwith $M>0$, $\\gamma\\in[0,3)$, $t>0$ bounded, and $\\delta>0$ small enough. Additionally, we assume that \n\\begin{equation}\\label{comparison}\nf \\text{ is non-decreasing }.\n\\end{equation}\nHere, $\\mathbb{R}_+$ is the set of non-negative numbers, and the infinity Laplacian is defined as follows:\n$$\n\\Delta_\\infty u(x):=\\sum_{i,j=1}^nu_{x_i}u_{x_j}u_{x_ix_j},\n$$\nwith $u_{x_i}=\\partial u\/\\partial x_i$. Note that the continuity of $f$ provides that $f(u)$ is bounded once $u$ is bounded. Note also that \\eqref{1.2} is quite general in the sense that it needs to hold only for $\\delta$ close to zero. For example, it holds for functions that are homogeneous of degree $\\gamma$. Condition \\eqref{comparison} is needed to guarantee the comparison principle. Solutions of \\eqref{1.1} are understood in the viscosity sense according to the following definition:\n\\begin{definition}\\label{d1.1}\n\tA function $u\\in C(\\Omega)$ is called a viscosity super-solution (resp. sub-solution) of \\eqref{1.1}, and written as $\\Delta_\\infty u\\le f(u)$ (resp. $\\ge$), if for every $\\phi\\in C^2(\\Omega)$ such that $u-\\phi$ has a local minimum at $x_0\\in\\Omega$, with $\\phi(x_0)=u(x_0)$, we have\n\t$$\n\t\\Delta_\\infty\\phi(x_0)\\leq f(\\phi(x_0)).\\quad \\textrm{(resp. $\\geq$)}\n\t$$\n\tA function $u$ is called a viscosity solution if it is both a viscosity super-solution and a viscosity sub-solution.\n\\end{definition}\n\nThe infinity Laplace operator is related to the absolutely minimizing Lipschitz extension problem: for a given Lipschitz function on the boundary of a bounded domain, find its extension inside the domain in a way that has the minimal Lipschitz constant, \\cite{A67}. It is known (see \\cite{J93}) that such function $u$ has to be an infinity harmonic one, i.e. $\\Delta_\\infty u=0$ (in the viscosity sense). The regularity issue of infinity harmonic functions received extensive attention over the years. As was shown in \\cite{ES08}, the infinity harmonic functions in the plane are $C^{1,\\alpha}$, for a small $\\alpha$ (it is conjectured that the optimal regularity is $C^{1,\\frac{1}{3}}$). In higher dimensions infinity harmonic functions are known to be everywhere differentiable (see \\cite{ES11}). \n\nAs for the inhomogeneous case of $\\Delta_\\infty u=f$, it is known that the Dirichlet problem has a unique viscosity solution, provided $f$ does not change sign (see \\cite{LW08}). Moreover, as was shown in \\cite{L14}, for bounded right hand side, the Lipschitz estimate and everywhere differentiability of solutions remain true. The case of $f$ not being bounded away from zero, mainly, when $f=u_+^\\gamma$, where $u_+:=\\max\\left(u,0\\right)$ and $\\gamma\\in[0,3)$ is a constant, was studied in \\cite{ALT16} (dead-core problem). The authors show that for such right hand side (strong absorbtion) across the free boundary $\\partial\\{u>0\\}$ non-negative viscosity solutions are of class $C^{\\frac{4}{3-\\gamma}}$. The denominator $3-\\gamma$ is related to the degree of homogeneity of the operator, which is three, i.e., $\\Delta_\\infty(Cu)=C^3\\Delta_\\infty u$, for any constant $C$. Note that for $\\gamma\\in(0,3)$ this regularity is more than the conjectured $C^{1,\\frac{1}{3}}$, i.e., we obtain higher regularity across the free boundary. This result allows to establish Hausdorff dimension estimate for the free boundary $\\partial\\{u>0\\}$ and conclude that it has Lebesgue measure zero. \n\nWe extend these results for the source term $f$ satisfying \\eqref{1.2}. In particular, it includes equations with the right hand side \n$$\nf(t)=e^t-1\\,\\,\\,\\textrm{ and }\\,\\,\\,f(t)=\\log(t^2+1)\n$$\namong others (see Section \\ref{s7} for more examples). In fact, our results are true in a broader context, when allowing ``coefficients'' in the right hand side, that is, when in \\eqref{1.1} one has $f=f(x,u)$, as long as $f(x,u)$ satisfies \\eqref{1.2} as a function of $u$ and is continuous (and bounded) as a function of $x$ (see Section \\ref{s7}). For simplicity, we restrict ourselves to the case of $f(x,u)=f(u)$.\n\nOur strategy is the following: by means of a flattening argument, we show that across the free boundary $\\partial\\{u>0\\}\\cap\\Omega$ non-negative viscosity solutions of \\eqref{1.1} are of class $C^{\\frac{4}{3-\\gamma}}$, when \\eqref{1.2} holds. When the source term is comparable to a homogeneous function of degree $\\gamma$, this result is sharp in the sense that across the free boundary non-negative viscosity solutions grow exactly as $r^{\\frac{4}{3-\\gamma}}$ in the ball of radius $r$. We also analyze the borderline (critical) case, that is, when $\\gamma=3$ (which is also the degree of the homogeneity of the infinity Laplacian). Unlike \\cite{ALT16}, $f$ is not given explicitly, which makes it harder to construct a barrier function - needed for our analysis. Nevertheless, we are able to show that in this case \\eqref{1.1} has a viscosity sub-solution whose gradient has modulus separated from zero. We use this function to build up a suitable barrier to conclude that if a viscosity solution vanishes at a point, it has to vanish everywhere. Our results remain true when the right hand side has some ``bounded coefficients'' (see Remark \\ref{r7.1}). For simplicity we restrict ourselves with the right hand side ``without coefficients''.\n\nThe paper is organized as follows: in Section \\ref{s2}, we prove an auxiliary result (flattening solutions) (Lemma \\ref{l2.2}), which we use in Section \\ref{s3} to derive the main regularity result (Theorem \\ref{t3.1}), and as a consequence, in Section \\ref{Liouville}, we obtain Liouville type theorems (Theorem \\ref{t3.2} and Theorem \\ref{t3.3}). In Section \\ref{s4}, we prove several geometric measure estimates (Theorem \\ref{t4.1} (non-degeneracy) and Corollary \\ref{c4.1} (porosity)), and conclude that the free boundary has Lebesgue measure zero (Corollary \\ref{c4.2}). In Section \\ref{s5}, when $\\gamma=3$, we show that the only non-negative viscosity solution that has zero, is the function that is identically zero (Theorem \\ref{t5.1}). Finally, in Section \\ref{s7} we bring some examples of source terms for which our results are true.\n\n\\section{Preliminaries}\\label{s2}\nIn this section we list some preliminaries, as well as prove an auxiliary lemma for future reference. We start by the comparison principle, the proof of which can be found in \\cite{CIL92,LW08}.\n\\begin{lemma}\\label{l2.1}\n\tLet $u$, $v\\in C(\\overline{\\Omega})$ be such that\n\t$$\n    \\Delta_\\infty u-f(u)\\le0, \\,\\,\\,\\Delta_\\infty v-f(v)\\ge0\\,\\,\\textrm{ in }\\,\\,\\Omega\n\t$$\n\tin the viscosity sense, and $f$ satisfy \\eqref{comparison} or $\\inf f>0$. If $u\\geq v$ on $\\partial\\Omega$, then $u\\geq v$ in $\\Omega$.\n\\end{lemma}\nThe comparison principle, together with Perron's method leads to the following result (for the proof we refer the reader to \\cite{CIL92}, for example). In fact, existence of solutions can be shown even without directly applying the comparison principle, as it was done, for example, in \\cite[Theorem 3.1]{RTU19}.\n\\begin{theorem}\\label{t2.1}\n\tIf $\\Omega\\subset\\mathbb{R}^n$ is bounded and $\\varphi\\in C(\\partial\\Omega)$ is a non-negative function, then there is a unique and non-negative function $u$ that solves the Dirichlet problem\n\t\\begin{equation}\\label{2.1}\n\t\\left\\{\n\t\\begin{aligned}\n\t\t\\Delta_\\infty u&=f(u) \\text { in } \\Omega, \\\\\n\t\tu&=\\varphi\\text { on } \\partial\\Omega\n\t\\end{aligned}\\right.\n\t\\end{equation}\nin the viscosity sense.\t\n\\end{theorem}\nThe following auxiliary lemma is a variant of the flatness improvement technique introduced in \\cite{ALT16,T13,T16} to study the regularity properties of solutions of dead-core problems.\n\\begin{lemma}\\label{l2.2} Let $g\\in L^\\infty(B_1)\\cap C(B_1)$ be a non-negative function such that \n\t$$\n\t\\|g\\|_\\infty\\le\\max\\{1,M\\}\\sup_{[0,\\|u\\|_\\infty]}f,\n\t$$\n\twhere $f$ and $M$ are as in \\eqref{1.2}. For any given $\\mu >0 $ there exists a constant $\\kappa _{\\mu}=\\kappa(\\mu,n)>0$ such that if in $B_1$ a continuous functions $v$, which vanishes at the origin and $v\\in[0,1]$, satisfies, in viscosity sense,\n\t$$\\Delta_\\infty v  - \\kappa_\\mu^4g(v)   = 0$$\\\\\n\tfor $0<\\kappa\\leq \\kappa_{\\mu}$, then \\\\\n\t$$\\sup_{B_{1\/2}} v \\leq \\mu.$$\n\\end{lemma}\n\\begin{proof}\nWe argue by contradiction assuming that there exist $\\mu^* >0$, $\\{v_i\\}_{i\\in \\mathbb{N}}$ and $\\{\\kappa_i\\}_{i\\in \\mathbb{N}}$ with $v_i(0)=0$, $0\\leq v_i\\leq1$, in $B_1$ satisfying in viscosity sense to\n\t$$\\Delta_\\infty v_i-\\kappa_i^4 g(v_i)= 0$$\\\\\n\twhere $\\kappa_i = \\text{o}(1)$, while\n\\begin{equation}\\label{2.2}\t\t\t\t\t\n\\sup_{B_{1\/2}} v_i > \\mu^*.\n\\end{equation}\\\\\nBy local Lipschitz regularity (see \\cite[Corollary 2]{L14}, for example), the sequence $\\{v_i\\}_{i\\in \\mathbb{N}}$ is pre-compact in the $C^{0,1}(B_{3\/4})$. Hence, by Arzel\\`{a}-Ascoli theorem, $v_i$ converges (up to a subsequence) to a function $v_\\infty$ locally uniformly in $B_{2\/3}$. Moreover, $v_\\infty(0)=0, \\; 0\\leq v_\\infty \\leq 1  $ and $\\Delta_\\infty v_\\infty = 0.$ The maximum principle for the infinity harmonic functions then yields $v\\equiv  0$, which contradicts to \\eqref{2.2} once $i$ is big enough.\n\\end{proof}\nThe following definition is for future reference.\n\\begin{definition}\\label{d1.2}\n\tA function $u$ is called an entire solution, if it is a viscosity solution of \\eqref{1.1} in $\\mathbb{R}^n$.\n\\end{definition}\nWe close this section by reminding the notion of porosity.\n\\begin{definition}\\label{porosity}\n\tThe set $E\\subset\\mathbb{R}^n$ is called porous with porosity $\\sigma$, if there is $R>0$ such that $\\forall x\\in E$ and $\\forall r\\in (0,R)$ there exists $y\\in\\mathbb{R}^n$ such that\n\t$$\n\tB_{\\sigma r}(y)\\subset\tB_{r}(x)\\setminus E.\n\t$$\n\\end{definition}\n\tA porous set of porosity $\\sigma$ has Hausdorff dimension not exceeding\n$n-c\\sigma^n$, where $c>0$ is a constant depending only on dimension. In particular, a porous set has Lebesgue measure zero (see \\cite{Z88}, for instance).\n\n\\section{Regularity across the free boundary}\\label{s3}\nIn this section we make use of Lemma \\ref{l2.2} and derive regularity result for viscosity solutions of \\eqref{1.1} across the free boundary $\\partial\\{u>0\\}$.\n\\begin{theorem}\\label{t3.1}\n\tIf $u$ is a non-negative viscosity solution of \\eqref{1.1}, where $f$ satisfies \\eqref{1.2}, and $x_0\\in\\partial \\{u>0\\}\\cap\\Omega$, then there exists a constant $C>0$, depending only on $\\gamma$, $\\|u\\|_\\infty$ and $\\mathrm{dist} (x_0 , \\partial \\Omega)$, such that\n\t$$u(x) \\leq C |x - x_0|^{\\frac{4}{3-\\gamma}}$$\n\tfor $x \\in \\{u>0\\} $ near $x_0$.\n\\end{theorem}\n\\begin{proof}\n\tThe idea is to use an iteration argument and carefully choose sequence of functions that allows to make use of the Lemma \\ref{l2.2}.  Observe that without loss of generality, we may assume that $x_0=0$ and $B_1\\subset\\Omega$.\n\t\n\tFor $\\mu =2^{-\\frac{4}{3-\\gamma}}$, let now $\\kappa_{\\mu} >0$ be as in Lemma \\ref{l2.2}. We then construct the first member of the sequence by setting\t\n\t$$\n\tw_0(x):= \\tau u(\\rho x) \\quad \\text{in} \\quad B_1,\n\t$$\t\n\twhere\n\t$$\n\t\\tau:= \\min \\left\\{1, \\|u\\|_\\infty^{-1} \\right\\}\\,\\,\\,\\textrm{ and }\\,\\,\\,\\rho :=\\kappa_\\mu   \\tau^{-\\frac{3-\\gamma}{4}}.\n\t$$\n\tNote that $\\tau^3\\rho^4=\\kappa_\\mu^4\\tau^\\gamma$, $w_0(0)=0$ and in $w_0\\in[0,1]$. Since $u$ is a viscosity solution of \\eqref{1.1}, then\n\t$$\n\t\\Delta_\\infty w_0(x) - \\tau^3\\rho^4 f(\\tau^{-1}w_0(x))=0\n\t$$\n\tor, equivalently,\n\t\\begin{equation}\\label{3.1}\n\t\\Delta_\\infty w_0(x)-\\kappa_\\mu^4\\tau^{ \\gamma} f(\\tau^{-1}w_0(x)) = 0.\n\t\\end{equation}\n\tSince $\\tau\\le1$, then \t\n\t$g(w_0):=\\tau^\\gamma f(\\tau^{-1}w_0)\\le f(u(\\rho x))\\le\\displaystyle\\sup_{[0,\\|u\\|_\\infty]}f$. From Lemma \\ref{l2.2}, we obtain\t\n\t$$\n\t\\sup_{B_{1\/2}} w_0 \\leq 2^{-\\frac{4}{3-\\gamma}}.\n\t$$\n\tFor $i\\in \\mathbb{N} $, we then define\n\t\\[ w_i(x) := 2^{ -\\frac{4}{3-\\gamma}}w_{i-1}(2^{-1}x). \\]\n\tand observe that $w_i(0)=0$, $w_i\\in[0,1]$ and $w_i$ satisfies \n\t$$\n\t\\Delta_\\infty w_i(x)=\\kappa_\\mu^4 2^{\\frac{4\\gamma}{3-\\gamma}i}\\tau^\\gamma f\\left(\\tau^{-1}2^{-\\frac{4}{3-\\gamma}i}w_i(x)\\right).\n\t$$\n\tUsing \\eqref{1.2}, for $i$ big we estimate\n\t$$\n\t2^{\\frac{4\\gamma}{3-\\gamma}i}\\tau^\\gamma f\\left(\\tau^{-1}2^{-\\frac{4}{3-\\gamma}i}w_i(x)\\right)\\le M\\tau^\\gamma f\\left(\\tau^{-1}w_i(x)\\right)\\le M\\displaystyle\\sup_{[0,\\|u\\|_\\infty]}f.\n\t$$\n\tOnce again applying Lemma \\ref{l2.2}, one gets\n\t\\[  \\sup_{B_{1\/2}} w_i \\leq 2^{-\\frac{4}{3-\\gamma}},  \\]\n\tor in other terms,\n\t\\[  \\sup_{B_{1\/4}} w_{i-1} \\leq 2^{-2 \\frac{4}{3-\\gamma}}. \\]\n\tContinuing this way, for $w_0$ we obtain\n\t\\begin{equation}\\label{3.2}\n\t\\sup_{B_{2^{-i}}} w_0 \\leq 2^{-i\\frac{4}{3-\\gamma}}.\n\t\\end{equation}\n\tNext, for a fixed $0<r\\leq \\frac{\\rho}{2}$, by choosing $i\\in \\mathbb{N}$ such that\n\t$$\n\t2^{-(i+1)} < \\frac{r}{\\rho}  \\leq 2^{-i},\n\t$$\n\tand using \\eqref{3.2}, we estimate\n\t\\begin{equation*}\n\t\\begin{split}\n    \\sup_{B_{r}} u &\\leq \\sup_{B_{\\rho2^{-i}}} u = \\tau^{-1}\\sup_{B_{\\rho2^{-i}}} w_0\\\\\n    &\\leq\\tau^{-1} 2^{-i\\frac{4}{3-\\gamma}} = 2^{\\frac{4}{3-\\gamma}}\\tau^{-1}2^{-(i+1)\\frac{4}{3-\\gamma}}\\\\\n    & \\leq \\left(\\tau^{-1}2\\rho^{-1}\\right)^{\\frac{4}{3-\\gamma}}r^{\\frac{4}{3-\\gamma}}\\\\\n    &=Cr^{\\frac{4}{3-\\gamma}}.\n\t\\end{split}\n\t\\end{equation*}\n\\end{proof}\nGeometrically Theorem \\ref{t3.1} means that no matter how ``bad'' the function $u$ is in $\\{u>0\\}$, it touches the free boundary $\\partial\\{u>0\\}$ smoothly. In other words, a non-negative viscosity solution of \\eqref{1.1} may have cusp singularities in its positivity set, and yet it is smooth near its free boundary. \n\n\\section{Liouville type results}\\label{Liouville}\n\nDespite the regularity information being available only across the free boundary, it is enough to derive the following Liouville type theorem.\n\\begin{theorem}\\label{t3.2}\n\tIf $u$ is an entire solution, \\eqref{1.2} holds and $u(x_0)=0$ for a $x_0\\in\\mathbb{R}^n$ with\n    \\begin{equation}\\label{3.3}\n    \tu(x) = o\\left(|x|^{\\frac{4}{3-\\gamma}}\\right),\\,\\,\\,\\textrm{ as }\\,\\,\\,|x| \\rightarrow \\infty,\n    \\end{equation}\n    then $u \\equiv0$.\n\\end{theorem}\n\\begin{proof}\n\tWithout loss of generality we may assume that $x_0=0$. For $k\\in\\mathbb{N}$, set\n\t\\[ u_k(x):= k^{\\frac{-4}{3-\\gamma}}u(kx),\\quad x\\in B_1,\\]\n\twhere $B_1$ is the ball of radius one centered at the origin. Note that $ u_k(0)=0 $. Since $u$ is an entire solution, for $x\\in B_1$ one has\n\t\\begin{equation*}\n\t\\Delta_\\infty u_k(x)=k^{\\frac{-4 \\gamma}{3-\\gamma}} f\\left(k^{\\frac{4}{3-\\gamma}} u_{k}(x)\\right).\n\t\\end{equation*}\t\n\tNote that the right hand side of the last equation satisfies \\eqref{1.2}. From Theorem \\ref{t3.1}, we then deduce that if $x_k\\in\\overline{B}_r$ is such that\n    $$\n    u_k(x_k)=\\sup_{\\overline{B}_r} u_k,\n    $$\n    where $r>0$ is small, then in $B_r$ one has\n    \\begin{equation}\\label{3.4}\n    \\|u_k \\|_\\infty\\to0,\\,\\,\\,\\textrm{ as }\\,\\,\\,k\\to\\infty.\n    \\end{equation}\n    In fact, if $|kx_k|$ remains bounded as $k\\to\\infty$, then applying Theorem \\ref{t3.1} to $u_k$ we obtain\n\t\\begin{equation}\\label{3.5}\n\tu_k (x_k) \\le C_k|x_k|^{\\frac{4}{3-\\gamma}},\n\t\\end{equation}\n\twhere $C_k>0$ and $C_k\\to0$. This implies that $u(kx_k)$ remains bounded as $k\\to\\infty$, and therefore $u_k(x_k)\\to0$, as $k\\to\\infty$, and \\eqref{3.4} is true. It remains true also in the case when $|kx_k|\\to\\infty$, as $k\\to\\infty$, since then from \\eqref{3.3} we get\n\t\\begin{equation*}\n\tu_k(x_k) \\leq |kx_k|^{-\\frac{4}{3-\\gamma}}k^{-\\frac{4}{3-\\gamma}}\\to0,\\,\\,\\,\\textrm{ as }\\,\\,\\,k\\to\\infty.\n\t\\end{equation*}\n\tNow, if there exists $y\\in\\mathbb{R}^n$ such that $u(y)>0$, by choosing $k\\in\\mathbb{N}$ large enough so $y\\in B_{kr}$ and using \\eqref{3.4} and \\eqref{3.5}, we estimate\n\t\\[\\dfrac{u(y)}{|y|^{\\frac{4}{3-\\gamma}}} \\leq  \\sup_{B_{kr}} \\dfrac{u(x)}{|x|^{\\frac{4}{3-\\gamma}}}=\\sup_{B_r}  \\dfrac{u_k(x)}{|x|^{\\frac{4}{3-\\gamma}}}\\leq \\dfrac{u(y)}{2|y|^{\\frac{4}{3-\\gamma}}},\\]\n\twhich is a contradiction.\n\\end{proof}\nIn fact, once the comparison principle holds, the condition \\eqref{3.3} can be weakened in the following sense (Theorem \\ref{t3.3} below). Let $x_0\\in\\mathbb{R}^n$ and $r>0$ be fixed, and let $u\\ge0$ be the unique solution of \\eqref{2.1} in $B_r(x_0)$ with $\\varphi\\equiv\\alpha_r>0$ constant, guaranteed by Theorem \\ref{t2.1}. Note that $u$ is a viscosity sub-solution of\n\\begin{equation}\\label{3.7}\n\\left\\{\n\\begin{aligned}\n\\Delta_\\infty v &= \\lambda v_+^{\\gamma} &&\\textrm{ in } B_r(x_0),\\\\\nv&=\\alpha_r &&\\textrm{ on } \\partial B_r(x_0),\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere\n\\begin{equation}\\label{3.6}\n\\lambda:=M^{-1}\\beta^{-\\gamma}f(\\beta),\n\\end{equation}\nand $\\beta>\\|u\\|_\\infty$ is a constant big enough so \\eqref{1.2} holds. Then the condition \\eqref{3.3} can be weakened and substituted by\n\\begin{equation}\\label{3.8}\n\t\\limsup_{|x| \\rightarrow \\infty} \\dfrac{u(x)}{|x-x_0|^{\\frac{4}{3-\\gamma}}} < \\left( \\lambda \\frac{(3-\\gamma)^4}{64(1+\\gamma)} \\right)^{\\frac{1}{3-\\gamma}},\n\\end{equation}\nwhere $\\lambda$ is defined by \\eqref{3.6}, and Theorem \\ref{t3.2} can be improved to the following variant (see Theorem \\ref{t3.3} below). The choice of the right hand side of \\eqref{3.8} comes from the explicit structure of the unique solution of \\eqref{3.7}, which, as observed in \\cite{ALT16}, is given by\n\\begin{equation}\\label{3.9}\nv(x):=\\Upsilon\\left(|x-x_0|-r + \\left(\\frac{\\alpha_r}{\\Upsilon}\\right)^{\\frac{3-\\gamma}{4}} \\right)^{\\frac{4}{3-\\gamma}}_+,\n\\end{equation}\nwhere\n\\begin{equation}\\label{3.10}\n\t\\Upsilon:= \\left(\\lambda \\frac{(3-\\gamma)^4}{64(1+\\gamma)}\\right)^{\\frac{1}{3-\\gamma}}.\n\\end{equation}\n\\begin{theorem}\\label{t3.3}\n\tLet \\eqref{1.2}, \\eqref{comparison} hold. If $u$ is an entire solution and satisfies \\eqref{3.8}, then $u\\equiv0$.\n\\end{theorem}\n\\begin{proof}\n\tOnce $r>0$ is large enough, then \\eqref{3.8} guarantees, with $\\Upsilon>0$ defined by \\eqref{3.10},\n\t\\[ \\sup_{\\partial B_r} \\dfrac{u(x)}{r^{\\frac{4}{3-\\gamma}}} \\leq \\theta\\Upsilon,\\]\n\tfor some $\\theta < 1$. On the other hand, using \\eqref{1.2}, one has that the unique solution of \\eqref{3.7}, with $\\alpha_r=\\displaystyle\\sup_{\\partial B_r(x_0)}u$, given by \\eqref{3.9}, is a viscosity sub-solution of \\eqref{1.1}. The comparison principle, Lemma \\ref{l2.1}, then implies that $u\\le v$ in $B_r(x_0)$. Letting $r\\to\\infty$, we conclude that $u\\equiv0$.\t\n\\end{proof}\n\\begin{remark}\\label{r3.1}\n\tAs can be seen from \\eqref{3.9}, the plateau of $v$, i.e., the set $\\{v=0\\}$, is the ball $\\overline{B}_{R}(x_0) $, where\n\t$$\n\t0<R:=r - \\left(\\frac{\\alpha_r}{\\Upsilon}\\right)^{\\frac{3-\\gamma}{4}}.\n\t$$\n    Since $0\\le u\\le v$, the plateau of $u$ contains the $\\overline{B}_R(x_0)$.\n\\end{remark}\n\n\\begin{remark}\\label{r3.2}\n\tNote that the inequality \\eqref{3.8} has to be strict. For example, if\n\t$$\n\tw(x):=\\Upsilon|x-x_0|^{\\frac{4}{3-\\gamma}}\n\t$$\n\tthen\n\t$$\n\t\\limsup_{|x| \\rightarrow \\infty} \\dfrac{w(x)}{|x-x_0|^{\\frac{4}{3-\\gamma}}}=\\Upsilon,\n\t$$\n\tbut $w$ is not identically zero.\n\\end{remark}\n\n\n\\section{Non-degeneracy and porosity}\\label{s4}\n\nIn this section we show that once\n\\begin{equation}\\label{5.1}\n\tf(\\delta t)\\ge N\\delta^\\gamma f(t)\\ge0,\n\\end{equation}\nwith $N>0$, $\\gamma\\in[0,3)$, $t>0$ bounded, and $\\delta>0$ small enough, then across the free boundary non-negative viscosity solutions of \\eqref{1.1} grow exactly as $r^\\frac{4}{3-\\gamma}$ in the ball $B_r$, for $r>0$ small enough. As a consequence, we conclude that the touching ground surface is a porous set, which implies that it has Hausdorff dimension less than $n$, and so its Lebesgue measure is zero (see \\cite{Z88}). We start by the following non-degeneracy theorem.\n\\begin{theorem}\\label{t4.1} \n\tLet \\eqref{5.1} hold. Let also $f$ satisfy \\eqref{comparison} or $\\inf f>0$. If $u$ is a non-negative viscosity solution of \\eqref{1.1}, then there exists a universal constant $c>0$, depending only on dimension and $\\gamma$, such that\n\t\\[ \\sup_{B_{r}(x_0)} u \\ge cr^{\\frac{4}{3-\\gamma}},  \\]\nwhere $x_0\\in\\overline{\\{u>0\\}}\\cap\\Omega$ and $0<r<\\operatorname{dist}(x_0, \\partial\\Omega)$.\n\\end{theorem}\n\\begin{proof}\n\tSince $u$ is continuous, it is enough to prove the theorem for points $x_0\\in\\{u>0\\}\\cap\\Omega$. Set\n\t$$\n\tv(x):=c|x-x_0|^{\\frac{4}{3-\\gamma}},\n\t$$\n\twith a constant $c\\in(0,\\Upsilon)$, where $\\Upsilon>0$ is defined by \\eqref{3.10}. Using \\eqref{5.1}, direct computation reveals that the choice of $c$ makes $v$ a viscosity super-solution of \\eqref{1.1} in $B_r(x_0)$, where $r>0$ is such that $B_r(x_0)\\subset\\Omega$. If $v\\ge u$ on $\\partial B_r(x_0)$, then the comparison principle, Lemma \\ref{l2.1}, would imply $v\\ge u$ in $B_r(x_0)$, contradicting to the fact that $0=v(x_0)<u(x_0)$. Hence, there is a point $y\\in\\partial B_r(x_0)$ such that $v(y)<u(y)$. We then estimate\n\t\\[ \\sup_{B_r(x_0)} u \\geq u(y) \\ge v(y) =cr^{\\frac{4}{3-\\gamma}}.\\]\t\n\\end{proof}\nAs a consequence, we obtain that the free boundary is a porous set, therefore it has Hausdorff dimension strictly less than $n$, hence its Lebesgue measure is zero.\n\nNote that from \\eqref{1.2} one has $f(0)=0$, so to use the comparison principle, Lemma \\ref{l2.1}, it is enough to assume that $f$ is non-decreasing, that is, \\eqref{comparison} holds.\n\\begin{corollary}\\label{c4.1}\n\tLet \\eqref{1.2}, \\eqref{comparison} and \\eqref{5.1} hold. If $u$ is a bounded non-negative viscosity solution of \\eqref{1.1}, then $\\partial\\{u>0\\}$ is a porous set.\n\\end{corollary}\n\\begin{proof}\n\tLet $x\\in\\partial\\{u>0\\}$ and $y\\in\\overline{B}_r(x)$ be such that\n\t$$\n\tu(y)=\\sup_{B_{r}(x)}u.\n\t$$\n\tBy Theorem \\ref{t4.1}, $u(y)\\ge cr^{\\frac{4}{3-\\gamma}}$. On the other hand, Theorem \\ref{t3.1} provides\n\t$$\n\tu(y)\\le C\\left[d(y)\\right]^{\\frac{4}{3-\\gamma}},\n\t$$\n\twhere $d(y):=\\text{dist}\\left({y,\\partial\\{u>0\\}}\\right)$. Therefore,\n\t$$\n\t\\left(\\frac{c}{C}\\right)^{\\frac{3-\\gamma}{4}}r\\le d(y).\n\t$$\n\tHence, if $\\sigma:=\\frac{1}{2}\\left(\\frac{c}{C}\\right)^{\\frac{3-\\gamma}{4}}$, one has $$\n\tB_{2\\sigma r}(y)\\subset\\{u>0\\}.\n\t$$\n\tWe now choose $\\xi\\in(0,1)$ such that for the point $z:=\\xi y+(1-\\xi)x$ we have $|y-z|=\\sigma r$. Then\n\t$$\n\tB_{\\sigma r}(z)\\subset B_{{2\\sigma}r}(y)\\cap B_r(x).\n\t$$\n\tMoreover, we have\n\t$$\n\tB_{2\\sigma r}(y)\\cap B_r(x)\\subset \\{u>0\\},\n\t$$\t\n\twhich together with the previous inclusion implies\n\t$$\n\tB_{\\sigma r}(z)\\subset B_{2\\sigma r}(y)\\cap B_r(x)\\subset B_r(x)\\setminus\\partial\\{u>0\\},\n\t$$\n\tthat is, the set $\\partial\\{u>0\\}$ is porous with porosity $\\sigma$.\n\\end{proof}\n\\begin{corollary}\\label{c4.2}\n\tIf \\eqref{1.2}, \\eqref{comparison}, \\eqref{5.1} hold, and $u$ is a viscosity solution of \\eqref{1.1}, then Lebesgue measure of the set $\\partial\\{u>0\\}$ is zero.\n\\end{corollary}\n\\section{The borderline case}\\label{s5}\nAlthough, in general, one cannot expect more than $C^{1,\\alpha}$ regularity for viscosity solutions of \\eqref{1.1}, Theorem \\ref{t3.1} provides higher and higher regularity across the free boundary, as $\\gamma\\in[0,3)$ gets closer to 3. In this section we analyze the limit case of $\\gamma=3$. The scaling property of the operator plays an essential role here, as $\\gamma=3$ is also the degree of homogeneity of the infinity Laplacian, meaning that $\\Delta_\\infty(Cu)=C^3\\Delta_\\infty u$, for any constant $C$. Observe that Theorem \\ref{t3.1} cannot be applied directly, since the estimates deteriorate as $\\gamma\\to3$. Thus, in this section \\eqref{1.2} is substituted by\n\\begin{equation}\\label{6.1}\n0\\le f(\\delta t)\\le M\\delta^3f(t),\n\\end{equation}\nwith $M>0$, $t>0$ bounded and $\\delta>0$ small. Our first observation states as follows.\n\\begin{lemma}\\label{l5.1}\n\tIf $u$ is a non-negative viscosity solution of \\eqref{1.1}, where $f$ satisfies \\eqref{6.1}, then its every zero is of infinite order.\n\\end{lemma}\n\\begin{proof}\n\tThis is a consequence of Theorem \\ref{t3.1}. To see that it is enough to rewrite \\eqref{1.2}, for $\\gamma=3$, as\n\t$$\n\tf(\\delta t)\\leq M_\\delta\\delta^{3-\\beta}f(t),\n\t$$\n\twhere $M_\\delta:=M\\delta^\\beta$ and $\\beta>0$. An application of Theorem \\ref{t3.1} with $M=M_\\delta$ leads to the conclusion that if $u(z)=0$ for $z\\in\\Omega$, then $D^nu(z)=0$, $\\forall n\\in\\mathbb{N}$.\n\\end{proof}\nFurthermore, we show that if a non-negative viscosity solution of \\eqref{1.1} vanishes at a point, then it must vanish everywhere. For $f\\equiv0$ this follows from the Harnack inequality. The particular case, when $f$ is homogeneous of degree three, that is, $f(t)=M t^3$, was studied in \\cite{ALT16}, where by means of a suitable barrier function, was concluded that if non-negative viscosity solution vanishes in an inner point, then it has to vanish everywhere. Unlike \\cite{ALT16}, our function $f$ is not given explicitly, which makes the construction of a suitable barrier function more complicated. Observe that once \\eqref{6.1} holds, then $f(0)=0$, hence $\\inf f=0$, so to use the comparison principle, one needs to assume that $f$ is non-decreasing.\n\\begin{theorem}\\label{t5.1} Let $u$ be a non-negative viscosity solution of \\eqref{1.1}, where $f$ satisfies \\eqref{comparison} and \\eqref{6.1}. If $\\{u=0\\}\\cap\\Omega\\neq\\emptyset$, then $u\\equiv0$.\n\\end{theorem}\n\\begin{proof}\n\tWe argue by contradiction, assuming that there is $x\\in\\Omega$ such that $u(x)=0$, but $u(y)>0$ for a point $y\\in\\Omega$. Without loss of generality we may assume that\n\t$$\n\tr:=\\mathrm{dist}\\left(y,\\{u=0\\}\\right)<\\frac{1}{10}\\mathrm{dist}\\left(y,\\partial\\Omega\\right).\n\t$$\n\tWe aim to construct a sub-solution of \\eqref{1.1} which stays below $u$ on $\\partial B_r(y)$.\n\t\n\tLet $w$ be an infinity sub-harmonic function in $B_r(y)$ such that $|\\nabla w|\\ge\\eta$ for $\\eta\\ge0$ constant to be chosen later. Such function can be built up as a limit, as $p\\to\\infty$, of $p$-super-harmonic functions with modulus of gradient separated from zero by $\\eta$. We refer the reader for details to \\cite{J93}.\n\n    Now if $g$ is a smooth function and $v=g(w)$, direct computation reveals that\n    $$\n    \\Delta_{\\infty} v=\\left[g'(w)\\right]^3\\Delta_{\\infty} w+\\left[g^{\\prime}(w)\\right]^{2} g^{\\prime \\prime}(w)|\\nabla w|^{4}.\n    $$\n    Thus, for $g(t)=e^t+t$,\n    \\begin{equation}\\label{5.2}\n    \\Delta_{\\infty} v\\ge\\left[g^{\\prime}(w)\\right]^{2} g^{\\prime \\prime}(w)|\\nabla w|^{4},\n    \\end{equation}\n    since $g^\\prime\\ge1$ and $\\Delta_\\infty w\\ge0$. Also $g^{\\prime\\prime}\\ge e^{-\\|w\\|_\\infty}>0$, and \\eqref{5.2} yields (recall that $|\\nabla w|\\ge\\eta$)\n    \\begin{equation}\\label{5.3}\n    \t\\Delta_\\infty v\\ge\\mu\\eta,\n    \\end{equation}\n    where $\\mu:=e^{-\\|w\\|_\\infty}>0$. Choosing\n    $$\n    \\eta>\\frac{M}{\\mu}\\max_{[0,\\|v\\|_\\infty]}f,\n    $$\n    from \\eqref{5.3} we obtain\n    $$\n    \\Delta_\\infty v-Mf(v)\\ge\\Delta_\\infty v-\\mu\\eta\\ge0,\n    $$\n    i.e., $v$ is a sub-solution of \\eqref{1.1}. The latter together with \\eqref{1.2} gives, for any small constant $\\delta>0$,\n\t$$\n\t\\Delta_\\infty\\left(\\delta v\\right)-f\\left(\\delta v\\right)\\ge\\delta^3\\left(\\Delta_\\infty v-Mf(v)\\right)\\ge0,\n\t$$\t\n\tthat is, the function $\\delta v$ is also a sub-solution of \\eqref{1.1}. We choose $\\delta>0$ small enough to guarantee\n\t$$\n\t\t\\delta v(x)\\le u(x),\\,\\,\\,x\\in\\partial B_r(y),\n\t$$\n\tand by the comparison principle, Lemma \\ref{l2.1},\n\t\\begin{equation}\\label{5.4}\n\t\t\\delta v(x)\\le u(x),\\,\\,\\,x\\in B_r(y).\n\t\\end{equation}\n    Observe, that writing \\eqref{1.2} as\n    $$\n    f(\\delta t)\\le M\\delta\\delta^2f(t),\n    $$\n    and applying Theorem \\ref{t3.1} with $\\widetilde{M}=M\\delta$, we arrive at\n    \\begin{equation}\\label{5.5}\n    \t\\sup_{B_d(z)}u\\le Cd^4,\n    \\end{equation}\n    where $z\\in\\partial B_r(y)\\cap\\partial\\{u>0\\}$, and $d>0$ is small. In fact, we choose $0<d<\\left(\\frac{\\delta\\eta}{4C}\\right)^\\frac{1}{3}$. Using the fact that $|\\nabla v|=g^\\prime|\\nabla w|\\ge\\eta$, recalling \\eqref{5.4} and \\eqref{5.5} and the choice of $d$, we estimate\n    $$\n    \\delta\\eta d\\le\\sup_{B_d(z)}\\delta|v(x)-v(z)|\\le\\sup_{B_d(z)}\\delta v\\le\\sup_{B_d(z)}u\\le Cd^4\\le\\frac{1}{4}\\delta\\eta d,\n    $$\n    which is a contradiction.\n\\end{proof}\n\n\\section{Examples and beyond}\\label{s7}\nWe close the paper with some examples of the source term, for which our results are valid. We start with the following remark.\n\n\\begin{remark}\\label{r7.1}\n\tThe results in this paper remain true, without changes in the proofs, when in \\eqref{1.1} the right hand side has continuous (bounded) coefficients, i.e., \n\t$$\n\t\\Delta_\\infty u=f(x,u),\n\t$$ \n\tas long as $f(x,u)$ satisfies \\eqref{1.2} (and \\eqref{comparison}, \\eqref{6.1} when needed) as a function of $u$.\n\\end{remark}\n\nNote that $f\\equiv0$ satisfies \\eqref{1.2}, \\eqref{comparison}, \\eqref{5.1}, therefore, our results resemble those for the non-negative infinity harmonic functions. Also, the results are true when in \\eqref{1.1} one has $f(t)=t^\\gamma$, $t\\ge0$, $\\gamma\\in[0,3)$ (studied in \\cite{ALT16}). In view of Remark \\ref{r7.1}, we can allow some coefficients too, as long as they remain bounded, that is, functions of the type $f(x,t)=g(x)t^\\gamma$, where $g$ is a non-negative continuous, bounded function. When in the last example $\\gamma=0$, i.e., the source term depends only on $x$, $f(x,t)=g(x)$, across the free boundary one has $C^{\\frac{4}{3}}$ regularity, once $g(x)\\ge0$ is continuous and bounded. Furthermore, the non-degeneracy result is true, and hence the free boundary is a porous set and has zero Lebesgue measure.\n\nAn example of the source term, not constructed via power functions is $f(t):=e^t-1$, which satisfies \\eqref{1.2} with $M=1$ and $\\gamma=0$, since $e^{\\delta t}<e^t$, for $\\delta>0$ small and $t>0$. Hence, applying Theorem \\ref{t3.1} to\n$$\n\\Delta_\\infty u=e^u-1,\n$$\nwe conclude that non-negative viscosity solutions of the above equation are $C^\\frac{4}{3}$ near the free boundary $\\partial\\{u>0\\}$. Moreover, if $u$ is an entire solution of the last equation, which vanishes at a point and\n$$\nu(x)=o\\left(|x|^\\frac{4}{3}\\right),\\,\\,\\,\\textrm{ as }\\,\\,\\,|x|\\rightarrow\\infty,\n$$\nthen Theorem \\ref{t3.2} implies that it has to be identically zero.\n\nSame conclusion can be made when in \\eqref{1.1} the source term is $f(t):=\\log (t^2+1)$. Of course, Theorem \\ref{t3.1} can be applied also for any linear combination (with continuous, bounded coefficients) of the above source terms. In fact, Theorem \\ref{t3.1} is true for any non-negative continuous function, which is non-decreasing around zero and vanishes at the origin. We point out that \\eqref{1.2} (as well as \\eqref{5.1} and \\eqref{6.1}) is required to hold only around zero, so Theorem \\ref{t3.1} is true also for source terms that are any of the above examples around zero and can be anything outside, while remaining non-negative and continuous (and in case of Theorem \\ref{t4.1} and its consequences, also non-decreasing).\n\nWe finish with an application of Theorem \\ref{t5.1}. Let $u$ be a non-negative viscosity solution of\n\\begin{equation}\\label{7.1}\n\t\\Delta_\\infty u=\\log\\left(1+u^3\\right).\n\\end{equation}\nSince $f(t)=\\log(1+t^3)$ satisfies \\eqref{1.2} with $M=1$ and $\\gamma=0$, Theorem \\ref{t3.2} implies $C^{\\frac{4}{3}}$ regularity of $u$ near the touching ground. On the other hand, the function $f(t)$ can be written as \n$$\n\\log\\left(1+t^3\\right)=t^3\\frac{\\log\\left(1+t^3\\right)}{t^3}:=t^3g(t).\n$$ \nSet $g(0)=1$. Then $g$ is continuous, bounded ($0\\le g\\le1$) function, and the non-decreasing function $f(t)=g(t)t^3$ satisfies \\eqref{6.1} with $M=1$. Applying Theorem \\ref{t5.1}, we conclude that if $u$ is a non-negative viscosity solution of \\eqref{7.1}, which is zero at a point, then it must be identically zero.\n\n\\bigskip\n\n\\noindent{\\bf Acknowledgments.} NMLD was partially supported by Instituto Federal de Educa\\c{c}\\~ao, Ci\\^encia e Tecnologia do Rio Grande do Sul. NMLD thanks the Analysis group at Centre for Mathematics of the University of Coimbra (CMUC) for fostering a pleasant and productive scientific atmosphere during his postdoctoral program. RT was partially supported by FCT -- Funda\\c c\\~ao para a Ci\\^encia e a Tecnologia, I.P., through projects PTDC\/MAT-PUR\/28686\/2017 and UTAP-EXPL\/MAT\/0017\/2017, and by CMUC -- UID\/MAT\/00324\/2013, funded by the Portuguese government through FCT and co-funded by the European Regional Development Fund through Partnership Agreement PT2020.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{intro}\n\nGiant extended emission-line regions (EELRs), with typical radii of 10--30 kpc, are found around roughly half of the low-redshift steep-radio-spectrum quasars with strong nuclear narrow-line emission \\citep{Sto87,Fu09}. Although the most luminous EELRs at low redshifts are exclusively associated with steep-spectrum radio-loud quasars, the distribution of the ionized gas typically shows no obvious connection with the radio jets or lobes, or with host galaxy morphologies. These EELRs most likely comprise gas that has been driven out of the host galaxies by superwinds \\citep{Sto02,Fu06,Fu07b,Fu08,Fu09}. But are these superwinds mostly produced by starbursts or by the quasars themselves? We have found some evidence in favor of quasar-driven outflows from momentum considerations \\citep{Fu06} and from the lack of any optical spectroscopic evidence for a very recent starburst in the radio galaxy 3C\\,79, which shows both scattered light from a hidden quasar and a strong EELR \\citep{Fu08}. However, to deal with this question more systematically, we need to work with a global star-formation indicator that we can apply to a larger sample of objects.  In particular, for the present program, we have chosen carefully matched samples of steep-radio-spectrum quasars with and without luminous EELRs. Our main goal was to measure or to place strong upper limits to star formation rates (SFRs) in both subsamples of quasars.\n\nA number of measures of SFR are available for galaxies in general. These include optical emission lines, such as H$\\alpha$ and [O\\,{\\sc ii}]\\,$\\lambda$3727 \\citep[e.g.,][and references therein]{Kew04}, mid-infrared (MIR) fine structure lines, notably [Ne\\,{\\sc ii}]\\,12.8$\\mu$m\\ \\citep{Ho07}, polycyclic aromatic hydrocarbon (PAH) emission in the MIR \\citep{Sch06,Shi07}, the far-infrared (FIR) continuum \\citep[e.g.,][]{Row97}, and the radio continuum \\citep[e.g.,][]{Con92}. Some of these measures are difficult or impossible to apply to host galaxies of luminous, radio-loud QSOs (i.e.,\\ quasars). For example, photoionization by the quasar UV continuum produces emission lines that will mask those from star formation, and radio emission from the non-thermal radio source can swamp that due to star formation. The PAH emission and the FIR continuum appear to offer the most promise for use with quasar host galaxies, and they have the additional advantage over any optical approaches that they can reveal even deeply obscured starbursts.\n\nIn this {\\it Paper}, we present and discuss {\\it Spitzer}\\ MIR spectra and FIR photometry of 13 steep-radio-spectrum and\/or FR\\,II \\citep{Fan74} quasars at $z \\sim 0.3$. Throughout we assume a cosmological model with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_m=0.3$, and $\\Omega_{\\Lambda}= 0.7$.\n\n\\begin{deluxetable*}{lcccccccccccc}\n\n\\setlength{\\tabcolsep}{.11cm}\n\\tablewidth{0pt}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{Quasar Sample \\label{tab1}}\n\\tablehead{ \n\\colhead{} & \\colhead{} &\\colhead{} &\n\\colhead{log $L_{\\rm [O\\,III]}^{\\rm eelr}$} & \\colhead{log $\\lambda L_{4861}$} & \\colhead{} & \n\\colhead{log $P_{\\rm 178}^{\\rm tot}$} & \\colhead{Size} & \n\\colhead{$F_{\\rm 24 \\mu m}$} & \\colhead{$F_{\\rm 70 \\mu m}$} & \\colhead{$F_{\\rm 160 \\mu m}$} & \n\\colhead{log $\\lambda L_{\\rm 60 \\mu m}$} & \\colhead{log $L_{\\rm PAH}$}\n\\\\\n\\colhead{Name} & \\colhead{Designation} & \\colhead{$z$} & \n\\colhead{(erg s$^{-1}$)} & \\colhead{(erg s$^{-1}$)} & \\colhead{$\\alpha_{\\nu}$} & \n\\colhead{(W Hz$^{-1}$)} & \\colhead{(kpc)} & \n\\colhead{(mJy)} & \\colhead{(mJy)} & \\colhead{(mJy)} &\n\\colhead{(erg s$^{-1}$)} & \\colhead{(erg s$^{-1}$)}\n\\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \n\\colhead{(4)} & \\colhead{(5)} & \\colhead{(6)} & \n\\colhead{(7)} & \\colhead{(8)} & \\colhead{(9)} & \n\\colhead{(10)} & \\colhead{(11)} & \\colhead{(12)} &\n\\colhead{(13)}  \n}\n\\startdata\n\\multicolumn{13}{c}{CSS Quasar with EELR}   \\\\\n\\hline\n        3C\\,48&0137$+$3309&0.367&~~\\,42.75&45.11&0.82&28.41&  \\hph3&   128.6&   754.6&  ~475.9&46.11&~~\\,43.97\\\\\n\\hline\n\\multicolumn{13}{c}{FR\\,II Quasars with EELRs}   \\\\\n\\hline\n     3C\\,249.1&1104$+$7658&0.312&~~\\,42.82&45.52&0.84&27.63&\\phn137&\\phn43.9&\\phn60.2& $<$14.0&44.84& $<$42.70\\\\\n      Ton\\,616&1225$+$2458&0.268&~~\\,42.59&44.63&0.80&26.67&\\phn279&\\phn11.9&\\phn12.6& $<$13.5&44.04& $<$42.40\\\\\n      Ton\\,202&1427$+$2632&0.366&~~\\,42.32&45.19&0.62&26.94&   1169&\\phn40.3&\\phn61.0& $<$11.5&44.99& $<$42.81\\\\\n     4C\\,37.43&1514$+$3650&0.371&~~\\,42.97&45.27&0.80&27.36&\\phn276&\\phn30.0&\\phn35.0& $<$13.0&44.76& $<$42.84\\\\\n     3C\\,323.1&1547$+$2052&0.264&~~\\,42.20&45.18&0.71&27.33&\\phn285&\\phn32.4&\\phn18.8& $<$14.1&44.19& $<$42.62\\\\\n  PKS\\,2251+11&2254$+$1136&0.326&~~\\,42.47&45.17&0.73&27.37& \\tph47&\\phn42.2&\\phn30.1& $<$26.6&44.59& $<$42.72\\\\\n\\hline\n\\multicolumn{13}{c}{FR\\,II Quasars without EELRs}   \\\\\n\\hline\n     4C\\,25.01&0019$+$2602&0.284& $<$40.95&45.11&0.66&26.82&\\phn204&\\phn67.8&   116.4&~~\\,53.6&45.06& $<$42.82\\\\\n     4C\\,13.41&1007$+$1248&0.241& $<$40.38&45.38&0.67&26.95&\\phn387&\\phn72.9&\\phn86.5& $<$22.1&44.78& $<$42.78\\\\\n       3C\\,246&1051$-$0918&0.344& $<$40.92&45.21&0.78&27.68&\\phn415&\\phn20.5&\\phn20.0& $<$15.8&44.46& $<$42.56\\\\\n     4C\\,49.22&1153$+$4931&0.334& $<$40.40&44.71&0.49&27.32& \\tph80&\\phn25.3&\\phn42.2& $<$13.6&44.75& $<$42.71\\\\\n       3C\\,351&1704$+$6044&0.372& $<$41.29&45.40&0.74&27.85&\\phn282&\\phn99.2&   173.7&~~\\,68.2&45.47& $<$43.08\\\\\n     4C\\,31.63&2203$+$3145&0.295& $<$40.94&45.41&0.16&26.99&\\phn352&\\phn67.2&\\phn83.3&~~\\,54.2&44.95& $<$42.81     \n\\enddata\n\\tablecomments{\nCol.\\,(1): Common name. \nCol.\\,(2): J2000.0 designation.\nCol.\\,(3): Redshift.\nCol.\\,(4): Extended {[O\\,{\\sc iii}]}\\ $\\lambda5007$ luminosity in logarithmic and\nupper limits for non-EELR quasars.\nCol.\\,(5): Optical continuum luminosity under H$\\beta$.\nCol.\\,(6): Radio spectral index ($f_{\\nu} \\propto \\nu^{-\\alpha_{\\nu}}$)\nbetween 0.4 and 2.7 GHz.\nCol.\\,(7): 178 MHz radio luminosity in logarithmic (The value for\nTon\\,202 was interpolated between 151 and 365 MHz fluxes).\nCol.\\,(8): Projected linear size of the radio source (References: \\citealt{Aku91,Bri94,Fen05,Gop00,Gow84,Kel94,Sto85}).\nCols.\\,(9)--(11): Observed flux densities in the MIPS 24, 70, and 160 $\\mu$m\\ bands.\nCol.\\,(12): Interpolated luminosity at rest wavelength 60 $\\mu$m\\ from MIPS photometry.\nCol.\\,(13): Observed luminosity and upper limits of the PAH 7.7$\\mu$m feature from IRS spectra.\nThe reference for Cols.\\,(4)--(6) is \\citet{Sto87}. Data in Col.\\,(7) are compiled from the NASA\/IPAC Extragalactic Database (NED).  \n}\n\\end{deluxetable*}\n\n\\section{Sample Selection and {\\it Spitzer}\\ Observations}\n\nOur sample of quasars has been selected from the \\citet{Sto87} survey for extended emission around low-redshift QSOs. We have a principal sample of 12 FR\\,II quasars (which we will refer to as ``the FR\\,II sample''), half of which have luminous EELRs and half of which do not, to fairly strong upper limits. The EELR and non-EELR quasars have been paired as closely as possible in both redshift and rest-frame optical continuum luminosity. Characteristics of all of the objects are tabulated in Table~\\ref{tab1}.  For the EELR and non-EELR quasars, the mean redshifts are 0.318 and 0.312, respectively. The continuum luminosities are measured at H$\\beta$, and the mean values for log $\\lambda L_{4861}$ (erg s$^{-1}$) are 45.23 and 45.26. The radio powers of the subsamples are also similar, with mean values for log $P_{178 {\\rm MHz}}$ (W Hz$^{-1}$) of 27.32 and 27.44. We also include 1 EELR quasar with a compact steep-spectrum (CSS) one-sided radio jet, 3C\\,48, which is a special object in a number of other ways (see, e.g., \\citealt{Can00a,Sto07}) and will be discussed apart from the FR\\,II sample.\n\nDeep MIR spectra of all 13 quasars were obtained with the {\\it Spitzer}\\ Infrared Spectrograph \\citep[IRS;][]{Hou04}  as part of our Cycle 4 GO program (PID 40001). All of the objects were observed in the standard staring mode with the Short Low 1st order module (SL1, 7.5$-$14.2 $\\mu$m). The slit width was 3.7\\arcsec\\ or 16.5 kpc at $z = 0.3$. Each observation consisted of a 60 s ramp for 8$-$16 cycles with two nod positions. The total integration time was about 16$-$32 minutes. We used the IDL procedure IRSCLEAN\\footnote{All of the data reduction software mentioned in this section is available at the {\\it Spitzer}\\ Science Center website at http:\/\/ssc.spitzer.caltech.edu\/.} (ver. 1.9) to remove bad pixels in the pipeline post-BCD coadded nod-subtracted images and extracted the spectra with SPICE (ver. 2.1.2) using the optimal extraction method for point sources. \n\nFIR photometry at 24, 70, and 160 $\\mu$m\\ was obtained with the Multiband Imaging Photometer for {\\it Spitzer}\\ \\citep[MIPS;][]{Rie04} either as part of our GO program or from the archive (PIDs 00049, 00082, 20084). All of the observations were obtained using the small-field default resolution photometry mode. Typical integration time per pixel was 42 s at 24 $\\mu$m\\ (3 s exposure for 1 cycle or 3 s$\\times$1), 700 s at 70 $\\mu$m\\ (10 s$\\times$7), and 80 s at 160 $\\mu$m\\ (10 s$\\times$4). Our data reduction started from the pipeline BCD files. For 24 $\\mu$m\\ observations, final mosaic images were constructed with the MOPEX software after self-flat-fielding and background correction. At 70 and 160 $\\mu$m, we used the pipeline filtered BCDs to construct mosaics for most of the sources. 3C\\,48 and 3C\\,351 are so bright at 70 $\\mu$m\\ that they have to be masked out before applying the spatial and temporal filters; we thus used the offline IDL filtering procedures by D.~Fadda and D.~Frayer to generate filtered BCDs. The non-filtered 160 $\\mu$m\\ BCDs of 3C\\,48 were used because they result in a better mosaic image than the filtered BCDs. Point sources were extracted with APEX, and point response function (PRF)-fitting photometry was carried out. All of the sources were detected at 24 and 70 $\\mu$m, but only 4 were detected at 160 $\\mu$m. Upper limits at 160 $\\mu$m\\ were estimated from the standard deviation images associated with the mosaics using an aperture of 48\\arcsec\\ and corrected for the finite aperture size by multiplying the values by 1.6 (MIPS Data Handbook).  The photometry results are listed in Table~\\ref{tab1}. The systematic uncertainties were estimated to be 4\\% at 24 $\\mu$m, 5\\% at 70 $\\mu$m, and 12\\% at 160 $\\mu$m\\ \\citep{Eng07,Gor07,Sta07}. Statistical errors from the PRF fitting are much smaller and therefore could be neglected.\n\n\\begin{figure*}[!tb]\n\\epsscale{1.1}\n\\plotone{f1.ps}\n\\caption{\nAverage IRS spectrum (thick black line) of ({\\it a}) six EELR quasars and ({\\it b}) six non-EELR quasars. Individual spectra have been normalized at 8.8 $\\mu$m\\ and are shown as light grey curves. Overplotted for comparison are the IRS spectrum of 3C\\,48 (blue), the average spectrum of 28 PG QSOs (red), and the average ``intrinsic\" QSO spectrum derived from 8 FIR-weak PG QSOs after subtraction of a starburst component (green; \\citealt{Net07}); all have been arbitrarily offset for clarity. Important emission features are labelled in panel {\\it b}.\n\\label{fig:irs}} \n\\end{figure*}\n\n\\begin{figure*}[!tb]\n\\epsscale{0.5}\n\\plotone{f2a.ps}\n\\hskip 0.2in\n\\plotone{f2b.ps}\n\\caption{\nMIR to FIR SEDs of FR\\,II quasars: ({\\it a}) EELR quasars and ({\\it b}) non-EELR quasars. The red curves are the IRS spectra and the data points (detections) and downward arrows (upper limits) are MIPS photometry. Overplotted for comparison is the average ``intrinsic\" QSO SED, scaled to the IRS luminosity of each source \\citep[grey dashed curve;][]{Net07}. In the panel for 3C\\,351, the most FIR luminous FR\\,II quasar in our sample, we also show the SED of 3C\\,48 as the blue dash-dot line (no scaling has been performed).  \n\\label{fig:sed}} \n\\end{figure*}\n\n\\begin{figure*}[!tb]\n\\epsscale{1.1}\n\\plottwo{f3a.ps}{f3b.ps}\n\\caption{\n({\\it a}) PAH 7.7 $\\mu$m\\ luminosity vs. redshift for various\nsamples, as indicated.  The solid curve indicates a constant flux limit of\n$6\\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$, approximating that of the PG\nsample. Note that the upper limits for our FR\\,II\nquasar sample (larger red and blue arrows) are below or comparable to\nthe detections for the PG QSO sample, in spite of the higher redshift\nof the FR\\,II sample. The symbols given in the legend for this panel\napply to panel $b$ as well.\n({\\it b}) PAH 7.7 $\\mu$m\\ luminosity vs. optical continuum luminosity.\nThe PG QSOs from \\citet{Sch06} are shown as filled black squares\n(detections) or downward black arrows (upper limits). Similarly, the\nhigh-redshift mm-bright QSOs from \\citet{Lut08} are shown as gray\nstars or large gray downward arrows, and FR\\,II 3CR quasars and radio\ngalaxies from \\citet{Shi07} are shown as orange diamonds or downward\narrows. Our FR\\,II quasars with EELRs are shown as downward red\narrows, and the ones without EELRs are downward blue arrows. 3C\\,48 is\nindicated by a green filled circle. The only FR\\,II quasar that is in the PG sample\nbut not in our sample, PG\\,2349$-$014, is marked by a filled black\nsquare with a white plus sign. The dotted and dashed lines show\nconstant ratios of star formation rates and black hole accretion\nrates.  The three common objects between our sample and the PG QSOs\nare connected with grey solid lines. The $\\sim4\\times$ reduced upper\nlimits of $L_{\\rm PAH}$ for the three common objects result from our\nmuch longer IRS integrations.\nThe FR\\,II quasars are clearly under-luminious in the PAH emission for\ntheir optical luminosity (which can be considered to be a proxy for\nblack-hole accretion rate) with respect to both the PG QSOs that are\nnot FR\\,II quasars and the mm-bright QSOs.\n\\label{fig:opt}}\n\\end{figure*}\n\n\\section{Results}\\label{results}\n\nWe did not detect PAH emission in any of the FR\\,II quasars, although strong PAH features were clearly seen at 6.2, 7.7, and 8.6 $\\mu$m\\ in the spectrum of the CSS quasar 3C\\,48. It was already known that the host galaxy of 3C\\,48 is undergoing intense star formation, with evidence from both its warm {\\it IRAS} FIR color and the young stellar populations revealed by deep optical spectra \\citep{Can00a}. The luminosity of the PAH 7.7 $\\mu$m\\ feature ($L_{\\rm PAH}$) is about $2.4\\times10^{10}$ $L_{\\odot}$, implying a SFR of 680 $M_{\\odot}$\\ yr$^{-1}$ (SFR[$M_{\\odot}$\\ yr$^{-1}$] = 283 $L_{\\rm PAH}$[10$^{10}$ $L_{\\odot}$], assuming $L_{\\rm PAH}\/L_{\\rm IR}$ = 0.0061 and SFR[$M_{\\odot}$\\ yr$^{-1}$] = 1.73 $L_{\\rm IR}$[10$^{10}$ $L_{\\odot}$]; \\citealt{Vei09,Ken98}. Here, and throughout this paper, $L_{\\rm IR}$ refers to the {\\it star-forming} luminosity in the rest-frame 8--1000 $\\mu$m range). The aromatic features remain undetected even in the stacked spectrum of the FR\\,II sample. In Fig.~\\ref{fig:irs} we show average spectra of two subsamples---FR\\,II quasars with and without EELRs, each containing six objects. The average spectra of the two subsamples do not differ: basically, one sees only narrow high-ionization emission lines superposed on a featureless continuum. \\citet{Net07} provided a composite spectrum of 28 PG QSOs with IRS spectra, as well as an ``intrinsic\" QSO spectral energy distribution (SED) from 1.2 to 70 $\\mu$m, which was constructed by combining 8 FIR-weak PG QSOs (presumably with little star formation) and subtracting a scaled starburst component. The MIR portion of these two SEDs is also shown in Fig.~\\ref{fig:irs} for comparison. Broad PAH emission is clearly seen in the composite PG QSO spectrum, but by design it is absent from the intrinsic QSO spectrum. The average spectra of FR\\,II quasars are consistent with being entirely intrinsic QSO emission. \n\nWe compute the upper limits to the PAH 7.7$\\mu$m\\ flux based on the noise in the region between 6.5 and 8.5 $\\mu$m\\ excluding the narrow [Ne\\,{\\sc vi}] line. A low-order polynomial fit to the continuum is subtracted from each individual spectrum and the standard deviation (i.e.,\\ noise) of the residual is determined. To obtain the upper limits, we assume that the undetected PAH has a Lorentzian profile with the same width (0.6 $\\mu$m\\ FWHM) as our best fit to the 3C\\,48 PAH 7.7$\\mu$m\\ feature and that its peak amplitude cannot be greater than 5 times the noise level. This peak-to-noise ratio of 5 corresponds to a total S\/N of $\\sim$14 if we use the optimal aperture of 1.45$\\times$ FWHM. The PAH upper limits reported in Table~\\ref{tab1} are consistent with the upper limits obtained by \\citet{Sch06} for the 3 quasars that are in both samples, once one takes into account the $\\sim$18$\\times$ longer integrations of our observations. Furthermore, we found that the PAH feature would have been clearly seen in both stacked spectra and individual spectra if synthetic PAH emission with these upper limits were to be added to every object. \n\nFigure~\\ref{fig:sed} shows the MIR--FIR SEDs of our FR\\,II quasars. A scaled version of the \\citet{Net07} intrinsic QSO SED is included in each panel for comparison. Most of the SEDs probed by IRS show steady rises towards both ends of the spectrum and a dip near 8 $\\mu$m; these features are readily seen in the intrinsic QSO SED and can be attributed to a combination of hot dust blackbody emission and silicate emission that peaks near $\\sim$10 $\\mu$m. As an example of starburst quasars, the SED of 3C\\,48 is overplotted in the panel of 3C\\,351, the most FIR luminous FR\\,II quasar in our sample. The SEDs of the FR\\,II quasars can be mostly explained as intrinsic QSO emission without a starburst component. The SEDs of quasars with EELRs and the ones without EELRs are essentially the same. These results are consistent with our PAH measurements, as described in the two preceding paragraphs.\n\nA luminosity correlation between the optical continuum and the PAH emission has been observed in low-redshift PG QSOs and high-redshift millimeter-bright QSOs \\citep{Net07,Lut08}. This correlation implies an intimate connection between QSO nuclear activity, as indicated by the strength of the continuum at rest-frame 5100 \\AA, and star formation in their host galaxies, as traced by the PAH emission. While the median redshift of our sample is significantly above that of the PG sample, our IRS exposures, which are typically $\\sim18$ times those for the PG sample, are sufficient to give luminosity upper limits that are lower than the detections or upper limits for the bulk of the PG sample (see Fig.~\\ref{fig:opt}$a$). Figure~\\ref{fig:opt}$b$ examines whether our FR\\,II quasars are consistent with this optical-continuum\/PAH correlation. We converted the continuum luminosities under H$\\beta$ from \\citet{Sto87} to $\\lambda L_{5100}$ by multiplying the values by a factor of 1.74 (after the correction for cosmology), which was determined by comparing the two continuum luminosities of the three quasars in both our sample and the \\citet{Net07} sample. \nAlso plotted in Figure~\\ref{fig:opt} are the 32 FR\\,II radio galaxies and 15 FR\\,II quasars from the 3CR sample of \\citet{Shi07}, which have both 7.7 $\\mu$m\\ PAH measurements from IRS spectra and {[O\\,{\\sc iii}]}\\,$\\lambda5007$ and\/or {[O\\,{\\sc ii}]}\\,$\\lambda3727$ fluxes from the literature (see \\citealt{Jac97} and references therein\\footnote{An electronic version of the catalog is available at http:\/\/www.science.uottawa.ca\/$\\sim$cwillott\/3crr\/3crr.html}). We converted their {[O\\,{\\sc iii}]}\\ or {[O\\,{\\sc ii}]}\\ luminosities into $ \\lambda L_{5100}$ based on the observed {[O\\,{\\sc iii}]}-continuum correlation for type-1 QSOs \\citep[$ \\lambda L_{5100} \\simeq 320\\times L_{\\rm [O III]}$;][]{Zak03,Hec04} and assuming {[O\\,{\\sc iii}]}\/{[O\\,{\\sc ii}]}\\ = 3.3, as measured from sources where fluxes of both lines are available. \nIt is evident that the FR\\,II quasars\/radio galaxies fall systematically below the correlation established by the PG QSOs and the \\citet{Lut08} mm-bright QSOs.\n\nFive of the QSOs in the \\citet{Net07} sample are radio-loud, among which three are also in our sample (4C\\,13.41, 4C\\,31.63, and PKS\\,2251+11). The remaining two are PG\\,1302$-$102 and PG\\,2349$-$014. PG\\,1302$-$102 has a flat radio spectrum and PAH was not detected ($L_{\\rm PAH} <  5.3\\times10^9$ $L_{\\odot}$; \\citealt{Sch06}). PG\\,2349$-$014 has a steep radio spectrum and an FR\\,II radio morphology. Although the aromatic 7.7$\\mu$m\\ feature has been claimed to be detected in PG\\,2349$-$014 ($L_{\\rm PAH} = 3.6\\times10^9$ $L_{\\odot}$; \\citealt{Sch06}), the low PAH luminosity places it at the transitional region between the FR\\,II sources and the radio-quiet QSOs in Fig.~\\ref{fig:opt}{\\it b} (the black square with a white plus sign). \n\nThe optical continuum luminosity at 5100 \\AA\\ is widely used to estimate the bolometric luminosity for type-1 AGN ($L_{bol} \\sim 10 \\lambda L_{5100}$), which in turn provides the black hole accretion rate (BHAR) given a typical radiative efficiency of $\\eta = 0.054$ \\citep{Mart08}. The PAH luminosity can be converted into a SFR. As shown in Fig.~\\ref{fig:opt}$b$, most of the PG QSOs in the \\citet{Net07} sample and the mm-bright QSOs in the \\citet{Lut08} sample are within a narrow range of SFR\/BHAR ratios, 15 $<$ SFR\/BHAR $<$ 150; while our FR\\,II quasars show significantly lower SFR at any given BHAR (we have conservatively assumed that $L_{\\rm PAH} = 0.0061 L_{\\rm IR}$; see discussion of this ratio in \\S \\ref{sf_eelr}). \nAs the referee has pointed out, the observed separation in Figure~\\ref{fig:opt}{\\it b} could also be explained by an on-average 10$\\times$ higher radiative efficiency, $\\eta$, for FR\\,II sources than for radio-quiet QSOs. However, the theoretically allowed range of $\\eta$ is only from 0.054 to 0.42 (non-rotating and maximally rotating black holes, respectively; \\citealt{Sha83}), and, to make it worse, the presence of a jet can {\\it reduce} the efficiency \\citep{Jol08}. Therefore, a difference in $\\eta$ could, at best, partially explain the large offset that we observe in Figure~\\ref{fig:opt}{\\it b}. \nWe note that to explain the Magorrian bulge mass$-$black hole mass relation, under the assumption that star formation and black hole accretion are always coeval, a SFR\/BHAR $\\simeq$ 700 is required \\citep{Har04}. The measured low values have been used to argue for a much longer duration for star formation than that for the AGN activity \\citep{Net07}, making the existence of a SFR-BHAR correlation questionable.\n\n\\section{Implications}\n\n\\subsection{Can Quasar EELRs be Produced by Starburst Superwinds?}\\label{sf_eelr}\n\nMost of our EELR quasars have upper limits to  $L_{\\rm PAH}$ of less than $2\\times10^9$ $L_{\\odot}$. The non-detection of PAH in the stacked spectrum of the entire sample of 12 FR\\,II quasars shows that such upper limits are quite conservative. These upper limits imply a SFR $<12$ $M_{\\odot}$\\ yr$^{-1}$. In the above calculation we used a higher $L_{\\rm PAH}\/L_{\\rm IR}$ ratio of 0.028 than in calculating the SFR of 3C\\,48 (0.0061, \\S~3), because it is both observed \\citep[e.g.,][]{Sch06} and theoretically predicted \\citep[e.g.,][]{Dal02} that $L_{\\rm PAH}\/L_{\\rm IR}$ increases with decreasing IR luminosity. Following \\citet{Shi07}, we determined $L_{\\rm PAH}\/L_{\\rm IR} \\simeq 0.028$ for $L_{\\rm PAH} = 2\\times10^9$ $L_{\\odot}$\\ from the star-forming templates of \\citet{Dal02}. However, as discussed in \\citet{Fu06}, to inject enough momentum into a typical EELR within a dynamical timescale of 10 Myr to explain the observed chaotic velocity field, the SFR of the central star burst must exceed $\\sim60$ $M_{\\odot}$\\ yr$^{-1}$ (eqs. [2]$-$[3] in \\citealt{Vei05}). More importantly, the absence of significant PAH emission in both FR\\,II quasars with EELRs and the ones without strongly suggests that starburst is not a critical factor in the formation of an EELR. This conclusion is also supported by the MIPS photometry, which indicates that the FIR SED is dominated by warm dust directly heated by the quasars (Fig.~\\ref{fig:sed}).\n\n\\subsection{Star Formation in FR\\,II Quasars}\n\nRelations among mergers, starbursts and nuclear activities have long been sought after by extragalactic researchers \\citep[e.g.,][]{San88}. Such connections are attractive because starbursts and AGN require many common triggering conditions, and gas-rich mergers can in principle provide these necessary conditions. Optical spectroscopy of QSO host galaxies, although technically challenging, has revealed diverse starforming properties. While active\/recent star formation ($< 300$ Myr) have been detected in the host galaxies of certain classes of QSOs, such as FIR-loud QSOs and post-starburst QSOs \\citep[e.g.,][]{Can01}, the majority of optically luminous QSOs show signs of past merger events and massive star formation that happened 1$-$2 Gyr ago (see \\citealt{Can06} for a review; also \\citealt{Can07,Ben08}). A similar result has been found for the FR\\,II radio galaxy 3C\\,79, which shows both a luminous EELR and evidence for a luminous quasar hidden from our direct view \\citep{Fu08}.\n\nAbout 40\\% of the PG QSOs studied by \\citet{Sch06} show PAH emission in individual {\\it Spitzer}\\ IRS spectra, and PAH is also present in the average spectrum of the remaining QSOs. In sharp contrast, our considerably deeper IRS spectra of steep-spectrum radio-loud quasars show no evidence for PAH emission, either individually or averaged, except in the case of the CSS quasar 3C\\,48. In fact, previous {\\it Spitzer}\\ observations have found that powerful radio galaxies and quasars from the 3CR catalog \\citep{Spi85} generally show FIR colors consistent with hot dust directly heated by the AGN \\citep{Shi05} and weak or undetectable PAH emission \\citep{Haa05,Ogl06,Cle07,Shi07}, suggesting a lower level of star formation compared to the QSOs from the PG and 2MASS samples \\citep{Sch83,Cut01}. The fraction of 3CR quasars\/radio galaxies that show detectable PAH features is only 18\\%, compared to $\\sim$48\\% for both PG and 2MASS QSOs in the same redshift range ($z < 0.5$) and observed with the same instrument.\n\nThere are substantial reasons for believing that radio-loud quasars, at least at low redshifts, are drawn from a population of massive elliptical galaxies that may have suffered mergers in the fairly recent past, while many radio-quiet QSOs are found either in disk galaxies or in ongoing gas-rich mergers \\citep{Sik07,Sik08,Wol08}. We can test this explanation by looking at the PG QSOs in the \\citet{Sch06} sample that also have morphological determinations by \\citet{Guy06} or elsewhere in the literature. Of the 9 objects which had individual PAH detections and for which morphological information is available (8 of which are classified by Guyon et al.), all are classified either as having a disk or as ``strongly interacting,'' a term that almost always implies a gas-rich interaction or merger. It is not at all surprising that such host galaxies should show substantial star formation. \n\n\\subsection{How Does 3C\\,48 Fit In?}\n\nAt first sight, the CSS quasar 3C\\,48 seems to be a counter-example to the trend both \\citet{Shi07} and we have found for steep-radio-spectrum quasars. In particular, 3C\\,48 has one of the most luminous EELRs among low-redshift quasars, yet its host galaxy is also undergoing star formation at a prodigious rate (\\S \\ref{results}). Is 3C\\,48 simply an anomaly, perhaps related to its status as a CSS source, or can it be understood within the framework we have described? Because of the small size of our sample and the rather large uncertainties in quasar lifetimes and duty cycles, we can do little more than explore some of the possibilities related to these options. First, however, we summarize some of the special characteristics of 3C\\,48 and its host galaxy (see \\citealt{Sto07} for more detail).\n\nBecause 3C\\,48 shows a one-sided jet, one might suspect that the jet is oriented almost along our line-of-sight and is strongly Doppler boosted; however, \\citet{Wil91} have argued from the weakness of the nuclear radio component that the jet is oriented closer to the plane of the sky and that the radio structure is intrinsically one sided; so the projected jet length of $\\sim3$ kpc is probably not too far from its actual length. 3C\\,48 is one of the only two CSS sources among powerful quasars with $z<0.5$. Such sources are rare, not because they are intrinsically uncommon, but because the CSS stage of an extended radio source likely lasts only a short time, typically $\\sim10^4$ years \\citep{deS99}. The host galaxy of 3C\\,48 is clearly undergoing a major merger \\citep{Sto87,Can00a,Sch04,Sto07}. The evidence suggests that the current episode of quasar activity has been triggered only relatively recently. \n\nWe consider first the possibility that 3C\\,48 is simply an earlier stage of the formation of an EELR like those in our FR\\,II EELR subsample. This picture has considerable appeal, because we do see a massive ($\\gtrsim10^9$ M$_{\\odot}$) outflow of ionized gas extending over a large solid angle from a region near the base of the CSS radio jet, with velocities ranging up to $\\sim1000$ {km s$^{-1}$} \\citep{Cha99,Can00a,Sto07}. This outflow is consistent with the sort of quasar-jet-driven wind we have supposed to be responsible for producing the large-scale EELRs seen in 3C\\,48 and other quasars. Nevertheless, there are difficulties with this simple evolutionary view. Since 3C\\,48 also has an EELR on scales much larger than that of the current radio jet, one would have to appeal to a previous episode of quasar activity in order to produce it according to the scenario we have outlined.  It is also not clear that the respective lifetimes of the EELR and the starburst would allow evidence for the starburst to die out quickly enough. EELR lifetimes are likely on the order of $10^7$ years, both from their close association with radio sources and from dynamical estimates. Vigorous star formation in 3C\\,48 is taking place not only in the nuclear region, but also over most of the host galaxy \\citep{Can00a}. While we do not have similarly detailed spectroscopy for any of the host galaxies in our FR\\,II sample, our deep spectroscopy of the FR\\,II EELR radio galaxy 3C\\,79 has shown that its last significant episode of star formation was at least 1 Gyr ago \\citep{Fu08}. This EELR, at least, could not have been produced in conjunction with a starburst like that occurring in 3C\\,48. While it is clear that some QSOs are triggered more-or-less simultaneously with massive starbursts \\citep[e.g.,][]{Can01}, many showing similar levels of QSO activity seem to have been triggered $\\sim1$ Gyr {\\it after} major starbursts \\citep{Can06,Can07,Ben08}, although we do not yet have a clear understanding why this should be so.\n\nIt seems, then, that 3C\\,48 is not likely to be an example of a direct precursor to objects like those in our FR\\,II EELR subsample. Nevertheless, it does seem to be a close cousin. Its EELR has a luminosity similar to those of our other EELR quasars, and the massive outflow of ionized gas from the base of the radio jet seems a striking confirmation of the sort of process we have inferred, from less direct evidence, to have produced the massive EELRs in our FR\\,II subsample \\citep[][and references therein]{Fu09}.  The crucial difference between 3C\\,48 and the rest is in whether a massive starburst is triggered approximately contemporaneously with the formation of the EELR. This difference, in turn, is likely to depend on the details of the mechanism by which gas is supplied to the system. In particular, the uniformly and abnormally low metallicities of the broad-line gas in the EELR quasars for which this measurement can be made \\citep{Fu07a} indicate an {\\it external} source for this gas and suggest the mergers of a gas-rich late-type galaxies with massive early-type galaxies with little cold gas of their own. Although we do not have direct metallicity information for the broad-line region in 3C\\,48, the morphology of the merger, the large amounts of molecular gas present \\citep{Win97}, and the fact that it has $L_{\\rm IR} > 10^{12}$ $L_{\\odot}$, qualifying it as an ``ultra-luminous infrared galaxy,'' all suggest that it is a merger between two roughly equal-mass gas-rich galaxies.\n\n\\subsection{Overview}\n\nOur deep {\\it Spitzer} IRS spectra and MIPS photometry of matched subsamples of FR\\,II quasars with and without luminous EELRs give tight upper limits to current SFRs for all of the quasars in both subsamples, supporting and extending earlier results on FR\\,II quasars and radio galaxies by \\citet{Shi07}. These upper limits indicate that SFRs, relative to BHARs, are generally much lower in FR\\,II quasars than they are in optically selected QSO samples. In the FR\\,II quasars, then, very little bulge mass is being added via star formation during the current episode of black-hole growth. Our star-formation upper limits are also sufficiently low that we can discount the possibility of galactic-scale superwinds resulting from star formation in the host galaxies of these quasars. As we mentioned in \\S \\ref{intro}, the most luminous EELRs are always associated with quasars with strong radio jets, yet there is little significant morphological correspondence between the EELRs and the radio structure. This fact indicates a mechanism connected with the production of the radio jet that is much less strongly collimated than the jet itself. We have suggested in this context that the initiation of an FR\\,II jet is accompanied by a nearly hemispherical blast wave \\citep{Fu07b}. In the usual case of two jets, these blast waves can be capable, at least in some cases, of clearing most of the interstellar medium from the host galaxy. Although, as we have emphasized, 3C\\,48 is atypical in several ways, it is not unreasonable to suppose that it gives us a fairly typical picture of a very young radio jet; and in this case we see a massive, high-velocity, wide-solid-angle outflow coming from a region near the base of the radio jet, tending to corroborate the picture we have suggested for the FR\\,II quasars with luminous EELRs.\n\n\\acknowledgments \nH. F. is grateful for the hospitality of the Purple Mountain Observatory, where part of this work was completed. We thank Luis Ho, David Rupke, Jong-Hak Woo, Yanling Wu, and Lin Yan for helpful discussions. We also thank the anonymous referee for helping us to improve the paper and clarify some obscure points. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL\/Caltech and by NSF under grant AST-0807900. 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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMassive stars, $M \\ga 8~M_{\\odot}$, can play a dominant role in the\nevolution of their host galaxies via feedback phenomena such as\nionizing radiation, strong stellar winds and supernova\nexplosions. These processes exert a large influence on the outcome of\nsubsequent star formation (SF). Massive SF can therefore be considered\nas an integral part of the overall SF process and a key ingredient for\na complete description of it. However, our comprehension of many\naspects of massive SF is still unsatisfactory. For example, it is\nstill not known whether massive SF is intrinsically different to the\ngenerally much better understood process of SF at low masses or not\n\\citep[see e.g.][]{ZinneckerandYorke2007}.\n\n\\smallskip\n\nThe uncertainties regarding massive SF are in part the result of the\nshort Kelvin-Helmholtz (KH) timescale associated with massive stars:\nthey contract rapidly to sustain a high luminosity while still\nacquiring their mass. If the accretion proceeds spherically, the\nensuing radiation pressure could limit the final stellar mass to\napproximately 40~$M_{\\odot}$\n\\citep[][]{Larson1971,Kahn1974,Wolfire1987}. This led to the\nsuggestion that massive SF is intrinsically different to the low mass\ncase. However, recent radiation-hydrodynamical simulations clearly\ndemonstrate that the radiation is actually channelled through\noptically thin polar cavities while accretion continues through an\nequatorial disc\n\\citep[][]{YorkeandSonnhalter2002,Krumholz2009,Kuiper2010}. Therefore,\ndeviations from spherical symmetry, which are suggested by the\nspectral energy distributions of young massive stars\n\\citep[][]{Harvey1977,Guertler1991}, remove the limit imposed by\nspherical accretion. Nonetheless, confirming that massive stars build\nup their final mass via disc accretion by direct observations is\nchallenging. Massive young stellar objects (MYSOs) are generally\nlocated at kpc distances and their AU-scale circumstellar environment\nremains unresolved with single dish telescopes at most wavelength\nregions \\citep[see e.g.][]{Cesaroni2007}.\n\n\\smallskip\n\nLong-baseline infrared interferometry is one of the few techniques\nwhich offers the milli-arcsecond (mas) resolution required to probe\nfor MYSO discs \\citep[see\ne.g.][]{Follert2010,Vehoff2010,deWit2011,Grellmann2011}. In\nparticular, an exemplary result is delivered by \\citet{Kraus2010}\nusing the Very Large Telescope Interferometer (VLTI) and the\nnear-infrared beam-combiner AMBER. These authors reconstruct a\n$K$-band synthesis image of a 20$M_{\\odot}$ MYSO with an angular\nresolution of 2.4~mas. The resultant image shows hot material in a\ndisc-like geometry of approximately 20~AU in size. As a result, a\nconsensus is beginning to emerge that the disc accretion scenario can\naccount for the formation of stars up to at least $\\sim\n30\\,M_{\\odot}$. This implies that the MYSO circumstellar environment\ndeviates significantly from spherical symmetry. In this scenario,\nrotation, accretion discs and outflow activity play a crucial role in\nproviding the asymmetric environment required to facilitate accretion\n\\citep[see e.g.][]{Beuther2002}.\n\n\n\\smallskip \n\nThe combination of a high stellar luminosity with accretion and\noutflow activity results in the circumstellar environment of MYSOs\nbeing shaped by several forces. Determining the relative importance of\nthese phenomena in sculpting the circumstellar environments of MYSOs\nrequires spatially resolved observations. In particular, the\nmid-infrared (MIR) wavelength region provides a wealth of information\non the geometry of the accretion environment. Furthermore, this\ninformation can be accessed by both interferometric techniques and\ndiffraction limited single-dish imaging with 8m class telescopes. MIR\ninterferometric observations reveal that the $N$-band emission of\nMYSOs on scales of 50 to 100~AU is dominated by warm dust located in\nthe envelope \\citep[][]{Dewit2007,Linz2009,Vehoff2010}. It is expected\nthat such envelopes will contain outflow cavities as there are many\ndetections of outflow activity associated with MYSOs\n(e.g. \\object{G35.20-0.74} in De Buizer 2006\\nocite{DeBuizer2006};\n\\object{IRAS 20126+4104} in De Buizer 2007\\nocite{DeBuizer2007};\n\\object{Cep\\,A HW2} in de Wit et al. 2009\\nocite{deWit2009}). In\n\\citet[][]{deWit2010}, we argue that, specifically, the warm dust\nresponsible for the $N-$band emission of W33A is located in the\nenvelope close to the walls of the cavities evacuated by the outflow.\n\n\\smallskip\n\nThe aim of this paper is twofold. We aim to assess the source of the\nMIR emission of a sample of MYSOs and thus test the premise that their\nMIR emission is dominated by warm dust located in outflow cavity\nwalls. This will then allow us to determine whether outflows play an\nimportant role in shaping the environments of MYSOs ({{as opposed to,\n    for example, rotation}}). We address these goals by comparing\nspatially resolved images of a sample (20 objects) to appropriate\nmodels. Our approach is similar to the work presented previously in\n\\citet[][hereafter DW09]{deWit2009} with the addition of 2D\nmodelling. We present MIR VLT images observed at $20\\,\\mu$m for a\nsample of MYSOs. We then compare the spatially resolved images to\ntwo-dimensional, axis-symmetric dust radiative transfer models that\nfeature rotating, collapsing envelopes and outflow cavities \\citep[the\nmodels of][]{W22003,W12003}. The paper is structured as follows. We\ndescribe the sample selection, the observations and data reduction\nprocedures in Sect. \\ref{obs}. The MIR images are described in\nSect. \\ref{data} and the radiative transfer model and its use are\ndetailed in Sect. \\ref{model}. The results of the modelling are\npresented in Sect. \\ref{mod_res} and discussed in\nSect. \\ref{disc}. Finally, our conclusions are given in\nSect. \\ref{conc}.\n\n\\section{Selection, observations and data reduction}\n\\label{obs}\n\nWe aim to perform our analysis on a representative sample of MYSOs, in\norder to draw general conclusions. Selecting such a sample is not\ntrivial. Initial attempts employed $IRAS$ data \\citep[see\ne.g.][]{Palla1991,Molinari1996,Sridharan2002}. As a result, such\ncatalogues suffered from source confusion due to the large beam of\n$IRAS$. To rectify this problem and generate a Galaxy-wide, unbiased,\nMYSO sample, we have conducted a survey to detect and characterise\nMYSOs: the RMS survey \\citep[][]{Lumsden2002,James-RMS}. The RMS is\nbased on data of the $MSX$ survey \\citep[][]{Egan2003}. $MSX$ data\noffers a significant improvement over $IRAS$ data in terms of\nresolution (e.g. arcsecond rather than arcminute resolution), and thus\nenables the selection of a representative sample of MIR bright MYSOs.\n\n\\smallskip\n\nFrom the RMS database\\footnote{http:\/\/www.ast.leeds.ac.uk\/RMS\/} we\nselected objects with distance estimates within 3~kpc\\footnote{Some of\n  distances estimates have since been revised to larger values.} and\nwith 21\\,$\\mu$m $MSX$ fluxes larger than 30\\,Jy. The flux limit was\nimposed in order to obtain a decent signal-to-noise ratio in the wings\nof the resolved profiles. In total, we observed 19 objects at\n20~$\\mu$m using the VLT Imager and Spectrometer for the Mid Infrared\n\\citep[VISIR,][]{VISIR} mounted at the Cassegrain focus of UT3 of the\nVLT (see Table\\ref{t1}). Observations were conducted using the imaging\nmode of the instrument and the Q3 filter which has a central\nwavelength of 19.5~$\\mu$m and a half-band-width of 0.4~$\\mu$m. During\nthe observations, VISIR was equipped with a DRS 256$\\times$256 Si:As\ndetector with an angular pixel size of 0.127\\arcsec. This\nconfiguration enabled us to obtain oversampled, diffraction limited\nimages, which with 8~m class telescopes results in an angular\nresolution of approximately 0.6\\arcsec.\n\n\\smallskip\n\nIn this paper, we also present the imaging observations and analysis\nof a key MYSO object \\object{W33A} (\\object{G012.9090-00.2607}). It\nwas observed at the slightly longer wavelength of 24.5~$\\mu$m with the\nCOMICS instrument \\citep[Cooled Mid Infrared Camera and\nSpectrometer][]{Kataza2000} mounted on the Cassegrain focus of the\nSubaru telescope. We employed the imaging facility of COMICS and used\na filter centred at 24.5~$\\mu$m (see DW09 for filter response\nfunctions). During the observations, COMICS was equipped with a\n320$\\times$240 Si:As IBC detector which provides oversampled\ndiffraction limited images with a pixel size of 0.13\\arcsec. A log of\nall the observations is presented in Table \\ref{t1}.\n\n\\smallskip\n\nFor both the VISIR and COMICS observations, standard stars were\nobserved to provide a reference point-spread-function (PSF). The\nstandard stars selected are MIR bright,$\\sim$100~Jy at 25~$\\mu$m ,\nisolated sources which are expected to be unresolved. Data reduction,\nconsisting of shifting and adding nodded and chopped images, was\nconducted using routines written in {\\sc{idl}}. The resultant images\nwere not astrometrically corrected. Therefore, the images are\npresented in terms of distance from the peak of the intensity\ndistribution of the target source. The azimuthally averaged intensity\nprofiles of the standards are displayed in Fig. \\ref{psfs}. The radial\nprofiles of the VISIR standard stars are generally consistent until a\nlevel of approximately 0.001 times the peak flux and appear to behave\nas point sources. The standard star of the COMICS data (HD 124897) is\nof lower signal-to-noise ratio but is similar out to 1 percent of the\npeak flux to the standards presented in DW09.\n\n\\begin{center}\n  \\begin{figure}[t]\n    \\begin{center}\n      \\includegraphics[width=0.4\\textwidth]{fig_1_a.ps} \n      \\caption{The azimuthally averaged intensity profiles of the PSF standard stars. HD\\,124897 is \n        the PSF standard taken with Subaru\/COMICS. The solid line with square points is the average VISIR PSF.\\label{psfs}}\n    \\end{center}\n  \\end{figure}\n\\end{center}\n\n\n\\begin{table*}\n  \\caption{The log of the observations.\\label{t1}}\n  \\begin{small}\n    \\begin{tabular}{llllccp{0.5cm} p{0.25cm} p{0.25cm} p{0.25cm}}\n      \\hline\n      \\vspace*{-2.5mm}\n      \\rule{0pt}{0.5mm}\\\\\n      Object & RA &DEC(J2000)  & date & Integration &  $\\overline{\\mathrm{AM}}$ & \\multicolumn{1}{c}{S\/N}& \\multicolumn{1}{c}{$d$} & \\multicolumn{1}{c}{$L$}&$F_{\\rm 21\\mu m}$ \\\\\n      & (h:m:s) & (\\degr:\\arcmin:\\arcsec)         &          &  (s) & & & \\multicolumn{1}{c}{(kpc)} & \\multicolumn{1}{c}{($\\mathrm{10^3}L_{\\odot}$)}&(Jy)\\\\\n      \\hline\n      \\hline\n      {\\bf{MYSOs:}}\\\\\n      {\\bf{VISIR}}\\\\\n      G263.7434+00.1161   &  08 48 48.64 &$-$43 32 29.1 & 09\/05\/11              & 760  \t          &1.1       & 900      &  2.0     & 4.1 & 93\\\\\n      G263.7759$-$00.4281 &  08 46 34.84 &$-$43 54 29.9 & 09\/05\/11+05\/12  & 1500           &1.1       & 150       &  1.5     & 4.0 & 65\\\\\n      G265.1438+01.4548   &  08 59 27.40 &$-$43 45 03.7 & 09\/06\/04+05\/05   & 1833           &1.4       & 150      &  2.5     & 8.6 & 42\\\\\n      G268.3957$-$00.4842 &  09 03 25.08 &$-$47 28 27.6 & 09\/06\/05+06\/22   & $2\\times$ 1305  & 1.6\/1.7& 250 &  1.6     & 3.7 & 50\\\\ \n      G269.1586$-$01.1383 &  09 03 31.76 &$-$48 28 45.5 & 09\/05\/12              & 700             &1.2       & 25        &  2.7     & 7.3 & 98\\\\\n      G310.0135+00.3892   &  13 51 37.85 &$-$61 39 07.5 & 09\/05\/26               & 120             &1.4       & 250      &  3.3     & 57 & 259\\\\\n      G314.3197+00.1125   &  14 26 26.28& $-$60 38 31.5 & 09\/05\/26               & 2000           &1.3       & 600      &  3.7     & 13 & 58\\\\\n      G318.0489+00.0854   &  14 53 42.99 &$-$59 08 56.5 & 09\/06\/01+06\/07   & $2\\times$ 1996  &1.2\/1.2 & 50    &  ?       & ?  & 41\\\\\n      G318.9480$-$00.1969 &  15 00 55.31 &$-$58 58 52.6 & 09\/05\/10              & 2000           &1.2       & 650      &  2.6     & 11 & 55\\\\\n      G326.4755+00.6947   &  15 43 18.97 &$-$54 07 35.6 & 09\/06\/07              & 1851            &1.2       & 650      &  2.8     & 8.8 & 42\\\\\n      G332.2941+02.2799   &  16 05 41.84 &$-$49 11 29.5 & 09\/06\/07              & 3199   \t  &1.2       & 75        &  1.9     & 0.8 & 32\\\\\n      G332.9868$-$00.4871 &  16 20 37.81 &$-$50 43 49.6 & 09\/06\/07             & 1500            &1.1       & 500      &  3.5     & 26 & 64\\\\\n      G333.1075$-$00.5020 &  16 21 14.22 &$-$50 39 12.6 & 09\/06\/07             & 1447            &1.4       & 50        &  3.5     & 2.9 & 48\\\\\n      G339.6221$-$00.1209 &  16 46 05.99 &$-$45 36 43.9 & 09\/06\/07             & 2225            &1.4       & 150      &  13.1   & 520 & 39\\\\\n      G341.1281$-$00.3466 &  16 52 33.19 &$-$44 36 10.8 & 09\/06\/08             & 2423  \t  &1.1       & 400      &  3.5     & 6.2 &37\\\\\n      G343.5024$-$00.0145 &  16 59 20.90 &$-$42 32 38.4 & 09\/06\/07             & 400              &1.4       & 25        &  3.0     & 18 & 131\\\\\n      G343.5213$-$00.5171 &  17 01 34.04 &$-$42 50 19.7 & 09\/05\/10             & 1497            &1.1       & 75        &  3.0     & 13 & 47\\\\\n      G345.0061+01.7944   &  16 56 46.37 &$-$40 14 26.7 & 09\/05\/10              & 280             &1.0       & 100      &  1.8      & 8.7 & 154\\\\\n      G349.7215+00.1203   &  17 18 11.11& $-$37 28 23.4 & 09\/06\/07              & 3520           &1.6       & 200      &  24.4    & 310 & 31\\\\\n      {\\bf{COMICS}}\\\\\n      G012.9090$-$00.2607 & 18:14:39.56 &$-$17:52:02.3 & 09\/08\/30 & 1405 & 1.3 & 200& 3.8& 54& 144\\\\\n      \\hline\n      {\\bf{Standards:}}\\\\\n      {\\bf{VISIR}}\\\\\n      HD 108903 & 12 31 09.96 &$-$57 06 47.6  & 09\/05\/28+06\/07+06\/08 & 60\/60     & 2.0\/1.4\/1.4 & 200\/400\/700& \\\\\n      HD 123139 & 14 06 40.95 &$-$36 22 11.8  & 09\/06\/23             & 2500      & 1.0         & 350&\\\\\n      HD 150798 & 16 48 39.90 &$-$69 01 39.8  & 09\/05\/10+05\/26       & 3600\/3600 & 1.4\/1.4     & 1800\/600&\\\\\n      HD 163376 & 17 57 47.80 &$-$41 42 58.7  & 09\/05\/26             & 3600      & 1.1         & 150&\\\\\n      HD 81797  & 09 27 35.24 &$-$08 39 31.0  & 09\/05\/12             & 2500      & 1.2         & 850&\\\\\n      {\\bf{COMICS}}\\\\\n      HD 124897  & 14 15 39.67 &+19 10 56.7 & 09\/08\/30& 856& 1.9 & 50 \\\\\n      \\hline\n    \\end{tabular}\n    \\tablefoot{The signal to noise ratio is defined as the ratio of the central peak to the root-mean-squared (rms) background noise. \n      The distances and luminosities are taken from the RMS database.}\n  \\end{small}\n\\end{table*}\n\n\\section{Observational results}\n\n\\label{data}\n\nWe assessed the MIR images of our target sources by comparing their\nazimuthally averaged intensity profiles to the instrumental PSF\ndetermined from the standard stars. We find that the majority of the\nMYSOs observed, 14 of 20, are spatially resolved. The images of the\ntarget MYSOs are presented in Figs. \\ref{unres_im} and \\ref{res_im},\nwhich contain the unresolved and resolved MYSOs respectively.\n\n\\smallskip\n\nThe resolved MYSOs generally appear as a single source within the\nVISIR field of view, $30\\arcsec\\times30\\arcsec$. Most display an\napproximately circular symmetric morphology at the resolution of the\nobservations ($\\sim$0.6\\arcsec), although some show clear signs of\nstructure (e.g G263.7759$-$00.4281, Fig. \\ref{res_im}: panel\nA). Several objects are associated with additional, distinct sources\nof localised emission, which often have a cometary type morphology\n(e.g. G269.1586$-$01.1385 \\& G349.7215+00.1203, Fig. \\ref{res_im}:\npanels D \\& M). This is suggestive of a H{\\sc{ii}} region\n\\citep[][]{Hoare2007PPV}. Indeed, both the objects mentioned above are\ndetected at radio wavelengths at coordinates consistent with the\nlocations of the cometary 20~$\\mu$m sources\n\\citep[see][]{Urquhart2007a}. It is of interest to note that\n24.5\\,$\\mu$m images of regions containing both MYSOs and ultracompact\nH{\\sc{ii}} regions show a clear morphological difference between the\ntwo types of source, i.e. compact vs extended (DW09).\n\n\\smallskip\n\nApproximately one third of the sample are unresolved. Using the\nKolmogorov-Smirnov test, we investigated whether the resolved and\nunresolved samples were drawn from intrinsically different\ndistributions of MYSOs in terms of distance and luminosity. There is\nno significant difference between the distance and luminosity\ndistributions of the two samples, as might be expected given that the\nobjects are drawn from a single catalogue. However, the angular size\nof a centrally heated source of flux is dependent upon both the\nluminosity of central object and the distance to it \\citep[angular\nsize $\\propto \\frac{\\sqrt{L}}{d}$, see e.g.][]{Vinkovic2007}. We find\nthat the resolved objects generally have a higher value of\n$\\frac{\\sqrt{L}}{d}$ than the unresolved objects. More specifically,\nthe hypothesis that the unresolved sources have the same\n$\\frac{\\sqrt{L}}{d}$ distribution as the resolved sources can be\ndiscarded at a level of 98 percent significance. Thus, we conclude\nthat resolved sources appear larger than the unresolved sources as the\nresolved sources are generally more luminous {\\emph{and}} less distant\nthan their unresolved counterparts.\n\n\\smallskip\n\n\\begin{center}\n  \\begin{figure*}[!h]\n    \\begin{center}\n      \\begin{tabular}{l l l}\n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_a.ps} & \n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_b.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_c.ps}\\\\\n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_d.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_e.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_2_f.ps}\\\\\n      \\end{tabular}\n      \\caption{Unresolved sources\\label{unres_im}. The images are\n        scaled logarithmically. North is to the top of the page and\n        East is to the Left. {{The contours typically represent 1, 5,\n            25 and 75 percent of the peak flux.}}}\n    \\end{center}\n  \\end{figure*}\n\\end{center}\n\n\\begin{center}\n  \\begin{figure*}[!h]\n    \\begin{center}\n      \\begin{tabular}{l l l}\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_a.ps} &\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_b.ps} &\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_c.ps} \\\\\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_d.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_e.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_f.ps}\\\\\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_g.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_h.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_i.ps}\\\\\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_j.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_k.ps}&\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_l.ps}\\\\\n      \\end{tabular}\n      \\caption{Resolved sources\\label{res_im}. The images are scaled\n        logarithmically. North is to the top of the page and East is to the\n        Left. {{The contours typically represent 2, 5, 10, 25 and 75 percent of\n            the peak flux.}}}\n    \\end{center}\n  \\end{figure*}\n\\end{center}\n\n\\addtocounter{figure}{-1}\n\n\\begin{center}\n  \\begin{figure*}[!h]\n    \\begin{center}\n      \\begin{tabular}{ l}\n        \\includegraphics[width=0.3\\textwidth]{new_fig_3_m.ps} \\\\\n      \\end{tabular}\n      \\caption{continued.}\n    \\end{center}\n  \\end{figure*}\n\\end{center}\n\n\n\\section{Radiative transfer modelling}\n\\label{model}\n\nTo investigate the nature of the circumstellar structure of the\nresolved sources, we compare the observed images to 2D axis-symmetric\nradiative transfer models appropriate for massive stellar objects\nsurrounded by in-falling, rotating envelopes. By simultaneously\ncomparing the resolved MIR images and SEDs to the radiative transfer\ncalculations, we aim to understand the source of the 20\\,$\\mu$m MIR\nemission of MYSOs. The models and the modelling procedure are\ndescribed in detail in the following sections.\n\n\n\n\\subsection{The radiation transfer code}\nWe employ the 2D, axis-symmetric dust radiation transfer code of\nWhitney et al. \\citep[for details see][]{W22003,W12003}. The code\ncalculates radiation transfer through a dusty structure which consists\nof a proto-stellar envelope surrounding a central star. A low density\npolar cavity can be inserted into the overall envelope structure. The\ncode also allows for a dusty circumstellar disc. The density\ndistribution of the envelope is described by the treatment of a\ncollapsing, rotating envelope of \\citet{Ulrich1976} and\n\\citet[][]{Terebey1984}. The key parameters determining the\nenvelope density are the mass infall rate and the centrifugal\nradius. The dust sublimation radius is calculated self-consistently\nfrom the stellar luminosity and the dust sublimation temperature. The\nshape of the polar cavities can be described in two ways: using a\npolynomial function or following streamlines. The streamline is\nconical on large scales, while the apparent opening angle of the\npolynomial function can be specified (the opening angle at the stellar\nsurface is 180$\\degr$). If included, the dust disc is flared and\nfollows the $\\alpha$ prescription \\citep[see][]{Shakura1973} to\naccount for flux due to accretion. The reader is referred to\n\\citet{W22003,W12003} for more details on the code and the possible\ngeometries. We have used this code extensively and the reader is\nreferred to \\citet{deWit2010,deWit2011} for more details on the\napplication of the code to model MYSOs and their environments.\n\n\\subsection{Modelling methodology}\n\n\nOur previous use of the Whitney et al. code consisted of constructing\na series of dedicated models in order to reproduce the SEDs,\ninterferometric and auxiliary spatial information of MYSOs. We will\nexploit these customised models appropriate for MYSO environments in\nour approach here, partially motivated by the lack of such models in\nthe well-known SED grid \\citep[][]{Robitaille2007}. In particular, we\nwill use a model geometry that provides a successful fit to the source\nW33A \\citep[][]{deWit2010}. Its defining feature is that the MIR\nemission in the $N$-band originates in warm dust located in the\nwalls of outflow cavities. The model proto-stellar envelope extends\ninwards down to the dust sublimation radius and it has low density\noutflow cavities as observed in several MIR observations of MYSOs\n\\citep[see e.g.][]{DeBuizer2007}. We note that the model\ndoes not explicitely include a dust disc (according to the\nprescription of the Whitney et al. code). This is motivated by the\naforementioned MIR interferometric observations. Adding a disc\ngenerally results in high visibilities in the MIR whereas MYSOs\ntypically exhibit low visibilities, indicative of a more extended dust\ndistribution \\citep[see e.g.][]{deWit2010}.\n\n\\smallskip\n\nWe illustrate the model 24.5\\,$\\mu$m image for the W33A model in panel\na) of Fig.\\,\\ref{test_image}. Also shown are the PSF convolved image\n(panel b), the COMICS image of W33A and the corresponding radial\nintensity profile, panels c) and d) respectively. The shape of the\noutflow cavities can clearly be delineated in the model image in panel\na). This demonstrates that the dust located in the outflow\ncavity walls is warmed-up and emits a significant fraction of the MIR\nflux. W33A is know to drive an outflow in the SE\/NW direction\n\\citep[see][and references therein]{Davies2010}. The orientation of\nthe outflow is traced by nebulosity in the NIR. While the 24.5~$\\mu$m\nimage of this object is resolved (panel c), it appears relatively\nsymmetric. However, the image is slightly extended towards the SE, in\nthe direction of the blue lobe of its outflow. In panel d) we compare\nthe model's radial intensity profile with the observed one and find\nthey are essentially identical. We underline that we did not perform a\nmodel fit to the 24.5\\,$\\mu$m image, we simply used the final W33A\nmodel presented in de Wit et al. (2010) to create the corresponding\nimage.\n\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.5\\textwidth]{temp.ps} \n  \\caption{{\\it Panel a:} Logarithmically scaled 24.5~$\\mu$m image of\n    the standard W33A model. {\\it Panel b:} The same image as panel a)\n    but convolved with the COMICS instrumental PSF. {\\it Panel c:} The\n    COMICS image of W33A. {\\it Panel d:} The azimuthally averaged\n    radial intensity profile at 24.5~$\\mu$m of W33A (points with error\n    bars) alongside the model radial profile and that of a PSF\n    standard (the lower line). The outflow axis of the model is\n    aligned with the SE\/NW orientation of W33A's\n    outflow.\\label{test_image}}\n\\end{figure}\n\n\nMotivated by the good match of the W33A model and image and the\nknowledge that the SEDs of MYSOs are known to be similar, we proceeded\nto fit the observations of the remaining MYSOs in the following\nway. The SEDs of the MYSOs observed were constructed from the\ncontinuum data assembled on the RMS database (which makes use of the\nwork of various authors) and the methodology of \\citet{M2011}. The SED\ncoverage differs from one source to the other. Generally one can\ndistinguish between sources with continuum flux measurements up to\n100\\,$\\mu$m and those with sub-mm observations. We note that we do not\nfit explicitly the 9.7\\,$\\mu$m silicate feature, contrary to the\nmodelling efforts in DW09. This is simply because this information is\nnot available for the majority of our sources.\n\n\\smallskip\n\nTo reproduce both the SEDs and 20\\,$\\mu$m intensity distributions of\nthe resolved sample, we used the W33A model geometry summarised\nabove. We only altered parameters that can be expected to vary between\nobjects: the inclination, the envelope infall rate (which scales the\ntotal dust mass in the envelope) and the opening angle of the outflow\ncavity (which may vary with time). We emphasise that we use one basic\nmodel to reproduce the observations. There may be differences between\nsources, for example the centrifugal radius and the properties of\ncircumstellar discs -- if any are present -- may vary from source to\nsource. The model will not account for such differences. However, it\nprovides a simple test of the hypothesis that the bulk of the $Q-$band\nemission of MYSOs can be attributed to warm dust in outflow cavity\nwalls. Here we outline the methodology followed in reproducing the\nobservations.\n\n\\smallskip\n\n\n\nFor a given MYSO, the stellar luminosity was initially set to the\nvalue in Table \\ref{t1} and the system inclination {{(the angle\n    between the polar axis and the line of sight)}} was estimated from\nits MIR image. The stellar luminosity and infall rate were then varied\nuntil the model SED was consistent with that observed. This set the\nluminosity and provided first estimates of the other parameters. Once\nthe SED was satisfactorily reproduced, the inclination and outflow\nopening angle were varied in an attempt to recreate the observed\nspatial intensity profile. This had an affect on the reproduction of\nthe SED. Therefore, the infall rate was allowed to vary to enable a\nfit to both the SED and the radial profile. The model fitting was done\nby hand rather than using a $\\chi^2$ minimisation technique as the\ncomputational time required to make a grid of model images for each\nobject was prohibitively large.\n\n \\smallskip\n\n We demonstrate the effect of varying the free parameters in\n Fig. \\ref{vary}. In general, the more inclined models appear more\n extended. This might be expected if much of the MIR emission\n originates in outflow cavity walls. At large inclinations, the projected\n area of the outflow cavity walls is larger and thus the resultant flux\n distribution is more extended. The cavity opening angle has a similar\n effect on the intensity distribution. This is because increasing the\n opening angle increases the distance at which the majority of the\n cavity wall is exposed to direct irradiation from the central\n object. This also increases the projected area of the outflow\n cavities and the extension seen in the azimuthally averaged intensity\n distribution. Since both parameters affect the observed extension,\n they are slightly degenerate. However, the two parameters have\n different effects on the gradient of the radial profile (see\n Fig. \\ref{vary}).\n\n\\smallskip\n\n\n\nThe final SEDs are presented in\nFig. \\ref{seds} and the associated azimuthally averaged intensity\nprofiles are displayed in Fig. \\ref{rad_profs}. The parameters of the\nselected models are presented in Table \\ref{pars}. Since the sample\nexhibits a varied morphology (see Fig. \\ref{res_im}), we discuss the\nmodelling results for each object individually in Appendix\n\\ref{notes}. We discuss the general results in the following section.\n\n\n\n\\begin{center}\n  \\begin{figure}\n    \\begin{center}\n      \\begin{tabular}{l l}\n        \\includegraphics[width=0.235\\textwidth]{fig_5_a.ps} & \n        \\includegraphics[width=0.235\\textwidth]{fig_5_b.ps}\\\\\n        \\includegraphics[width=0.235\\textwidth]{fig_5_c.ps}&\n        \\includegraphics[width=0.235\\textwidth]{fig_5_d.ps}\\\\\n        \\includegraphics[width=0.235\\textwidth]{fig_5_e.ps}&\n        \\includegraphics[width=0.235\\textwidth]{fig_5_f.ps} \\\\\n      \\end{tabular}\n      \\caption{An exploration of the parameter space immediately\n        surrounding the model of G310.0135+00.3892. The plots show the\n        effect of varying the free parameters (the inclination,\n        opening angle and infall rate). For comparison, a PSF standard\n        is also shown in the intensity distribution\n        figures.\\label{vary}}\n    \\end{center}\n  \\end{figure}\n\\end{center}\n\n\n\n\\begin{table}\n  \\begin{center}\n    \\caption{The key parameters of the individual models.\\label{pars}}\n    \\begin{tabular}{l  c c c p{0.5cm} p{0.5cm} }\n      \\hline\n      Object & $\\frac{L_{\\mathrm{Model}}}{L_{\\mathrm{RMS}}}$ & $i$ & $\\dot{m}$ &  \\multicolumn{1}{c}{$\\theta_{\\mathrm{op. ang.}}$}  &  \\multicolumn{1}{c}{$M_{\\rm{Env.}}$}\\\\\n      &                                          &$^{\\circ}$  & $M_{\\odot}{\\mathrm{yr^{-1}}}$ & \\multicolumn{1}{c}{$^{\\circ}$} & $10^3M_{\\odot}$\\\\\n      \\hline\n      \\hline\n\n      \\multicolumn{4}{l}{{\\bf{SED and intensity profile}}}\\\\\n\n      G263.7759$-$00.4281 & 1.0 & 87 & $\\mathrm{1.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{25} & 15.5\\\\\n      G265.1438+01.4548   & 1.2 & 32 & $\\mathrm{2.0\\times 10^{-4}}$  &  \\multicolumn{1}{c}{25} & 15.5\\\\\n      G268.3957$-$00.4842 & 0.5 & 30 & $\\mathrm{3.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.4\\\\\n      G310.0135+00.3892   & 1.0 & 32 & $\\mathrm{7.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.8\\\\\n      G332.2941+02.2799   & 1.4 & 87 & $\\mathrm{12.5\\times 10^{-5}}$ &  \\multicolumn{1}{c}{20} & 12.8\\\\\n      G332.9868$-$00.4871 & 1.0 & 15 & $\\mathrm{1.0\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.3\\\\\n      G339.6221$-$00.1209 & 1.2 & 60 & $\\mathrm{12.0\\times 10^{-4}}$ &  \\multicolumn{1}{c}{12} & 11.4\\\\\n      G345.0061+01.7944   & 1.0 & 63 & $\\mathrm{4.0\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.4\\\\\n      G349.7215+00.1203   & 1.4 & 65 & $\\mathrm{7.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{15} & 11.3\\\\\n\n      \\multicolumn{4}{l}{{\\bf{SED only}}}\\\\\n      G269.1586$-$01.1383 & 0.8 & 60 & $\\mathrm{7.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{45} & 32.2\\\\\n      G343.5024$-$00.0145 & 1.3 & 85 & $\\mathrm{3.0\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.3 \\\\\n      G343.5213$-$00.5171 & 1.0 & 57 & $\\mathrm{2.5\\times 10^{-4}}$  &  \\multicolumn{1}{c}{10} & 9.3\\\\\n\n      \\hline\n    \\end{tabular}\n    \\tablefoot{We note that the output luminosity is affected by\n      inclination angle, and is thus not necessarily identical to the\n      source luminosity. Parameters not listed were not varied. In the\n      case of several objects, particularly those associated with\n      H{\\sc{ii}} regions, e.g. G343.5024$-$00.0145, the model only\n      reproduces the objects' SED. {{$\\theta_{\\mathrm{op. ang.}}$\n          denotes half the full opening angle}}.}\n  \\end{center}\n\\end{table}\n\n\n\n\\begin{center}\n  \\begin{figure*}[!h]\n    \\begin{center}\n      \\begin{tabular}{l l l}\n        \\includegraphics[width=0.3\\textwidth]{fig_6_a} &\n        \\includegraphics[width=0.3\\textwidth]{fig_6_b} &\n        \\includegraphics[width=0.3\\textwidth]{fig_6_c} \\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_6_d}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_e}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_f}\\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_6_g}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_h}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_i}\\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_6_j}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_k}&\n        \\includegraphics[width=0.3\\textwidth]{fig_6_l} \\\\ \n      \\end{tabular}\n      \\caption{The SEDs of the resolved MYSOs compared to the final model SEDs\\label{seds}.}\n\n    \\end{center}\n  \\end{figure*}\n\\end{center}\n\n\\begin{center}\n  \\begin{figure*}[!h]\n    \\begin{center}\n      \\begin{tabular}{l l l}\n        \\includegraphics[width=0.3\\textwidth]{fig_7_a.ps} &\n        \\includegraphics[width=0.3\\textwidth]{fig_7_b.ps} &\n        \\includegraphics[width=0.3\\textwidth]{fig_7_c.ps} \\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_7_d.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_e.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_f.ps}\\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_7_g.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_h.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_i.ps}\\\\\n        \\includegraphics[width=0.3\\textwidth]{fig_7_j.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_k.ps}&\n        \\includegraphics[width=0.3\\textwidth]{fig_7_l.ps} \\\\ \n      \\end{tabular}\n      \\caption{The azimuthally averaged intensity profiles of the\n        resolved MYSOs (upper circles) alongside the model radial\n        profiles (squares) and the associated PSF standard (lower\n        circles). The lower limit to the $y$ axis is set to the\n        root-mean-square noise in the background of the MYSO\n        images. The upper limit of the $x$ is axis is set to the\n        distance at which the MYSO profiles fall to the level of the\n        background noise. The radial profile of G263.7759-00.4281 is\n        also shown averaged over the upper and lower half of its\n        bipolar morphology separately (short and long dashed lines\n        respectively). The error bars represent the rms within a given\n        annuli and thus represent an upper limit on the uncertainty in\n        the flux distribution.\\label{rad_profs}}\n    \\end{center}\n  \\end{figure*}\n\\end{center}\n\n\\section{Modelling results}\n\n\\label{mod_res}\n\n\nIn general, the observed SEDs could be reproduced relatively\nwell. There are several cases where the observed NIR fluxes could not\nbe reproduced exactly. However, we do not weight the NIR fluxes\nheavily as these can be strongly dependent on dust opacities and the\ncircumstellar geometry on small scales. In the majority of cases,\nbarring objects associated with H{\\sc{ii}} regions, the spatial\ndistribution of the 20\\,$\\mu$m emission of the MYSOs observed can also\nbe reproduced by our 2D axis-symmetric radiative transfer (RT)\nmodel. The individual model fitting results for the SEDs and the\n20\\,$\\mu$m intensity profiles are shown in Figs.\\,\\ref{seds} and\n\\ref{rad_profs}. Clearly, because of the unequal SED coverage from\nsource to source, the model fidelity differs. We do not attempt to\nprovide a unique model for each single source but evaluate whether the\nW33A model is capable of providing satisfactory fits, by changing a\nfew of the model's most basic parameters. From this starting point, we\nfind that in general, if the image exhibits a single source with an\napproximately symmetric morphology, both the object's SED and the\n20\\,$\\mu$m intensity profile are well reproduced. By extension of the\ncase for W33A, the majority of the 20\\,$\\mu$m emission in these\ninstances is very likely emission from warm dust in cavity walls.\n\n\\smallskip\n\nWhen an object's morphology appears either very complex or cometary, the\nmodel generally fails to reproduce the intensity profile. For example,\nthis is the case for G343.5024$-$00.0145, which is a known\nultracompact H{\\sc{ii}} region. For this object, a decent fit to the\nSED (panels I in Figs. \\,\\ref{seds} and \\ref{rad_profs}) predicts a\n``MYSO'' structure that is much more compact than the one observed,\neven with extreme values for the inclination and opening angle. A\ncometary appearance alone will often indicate an ionized nature for\nthe region, and this is generally confirmed via radio continuum\nobservations. That the model cannot reproduce the observations of\nMYSOs associated with H{\\sc{ii}} regions is perhaps to be expected as\nH{\\sc{ii}} regions represent a source of flux not considered by the\nmodel. Furthermore, the appearance of a H{\\sc{ii}} region likely\nsignifies the end of the embedded accretion phase of a MYSO, and it is\nthis phase the model is designed to represent.\n\n\\smallskip\n\nWe give an overview of the values for the varied parameters in\nTable\\,\\ref{pars}. The opening angles of the best-fitting models are\ngenerally narrow ($\\sim10 \\degr$). This implies that the outflows of\nthe MYSOs observed are relatively well collimated. The envelope infall\nrates of the models, a parametrisation of the density distribution,\nare generally of the order\n$\\mathrm{10^{-4}}M_{\\odot}\\mathrm{yr^{-1}}$. The corresponding values\nof visual extinction, $A_V$, are typically between 10 and 30, in\nagreement with the measured $A_V$s of MYSOs \\citep{Porter1998},\nalthough several of the more inclined models have $A_V$s in excess of\n100. The total masses of the envelopes are generally of the order\n10,000~$M_{\\odot}$. {\\color{black}We note that these are dependent on\n  the envelope outer radii, which are set at $\\mathrm{5 \\times\n    10^5}$~AU. In the case of the W33A model, reducing the outer\n  radius by a factor of 2 decreases the total mass by a factor of\n  approximately 5. This is a result of adopting a low but constant\n  density, $\\rho = 8 \\times 10 ^{-20} \\mathrm{g\\,cm^{-3}}$, within the\n  outflow cavity.} The envelope masses calculated are comparable to\nthe clouds that the objects are located in \\citep[see][and references\ntherein]{Urquhart2011}. Even allowing for an uncertainty of a factor\nof a few, this indicates that these objects are still heavily embedded\nin their natal material and their environment still contains a large\nreservoir of material available to form further stars.\n\n\\smallskip\n\n\n\n\\section{Discussion}\n\\label{disc}\n\\subsection{On the mid-IR emission of MYSOs}\n\nOver the past decade, observational evidence indicating that the\nformation of massive stars, up to a mass of at least $\\sim\n30\\,M_{\\odot}$, is accompanied by phenomena characteristic\nof low mass SF has accumulated. Such phenomena include:\ncircumstellar discs \\citep[see\ne.g.][]{Kraus2010,Davies2010,Masque2011,Goddi2011}, molecular outflows\n\\citep[see e.g.][]{Beuther2002,Cyganowski2009} and jet activity\n\\citep[see e.g.][]{Rodriguez1994,Guzman2010}. \n\n\\smallskip\n\nDespite the many common characteristics of low and high mass SF, some\nobservational phenomena seem to be exclusively associated with young,\nembedded high-mass stars. One example is the Class II 6.7-GHz methanol\nmaser, whose activity is uniquely associated with massive star forming\nregions \\citep[e.g.][]{Walsh1997}. These sites evidently produce\nsufficient IR radiation in order to pump this transition, unlike sites\nof low mass SF. It was initially thought that this maser activity\noriginated in circumstellar discs. As a result, a correlation between\nthe extension of several MYSOs in the MIR and their maser emission led\nto the suggestion that the MIR emission of MYSOs traces discs\n\\citep[see e.g.][]{DeBuizer2000}. However, observations with 8m class\ntelescopes were able to spatially resolve the MIR emission of a number\nof masing sources to reveal that their MIR emission is aligned with,\nrather than perpendicular to, their large CO molecular outflows\n\\citep[see e.g.][]{DeBuizer2006,DeBuizer2007}. These findings\nassociate the MIR emission of massive YSOs with outflow cavities,\nrather than circumstellar discs. Nevertheless, the kinematic diversity\nof methanol masers in a large sample of massive outflow sources does\nnot support a uniquely identifiable source within the close MYSO\nenvironment \\citep[][]{Cyganowski2009,Moscadelli2011}. Therefore, the\nsource of the MIR emission of MYSOs has yet to be conclusively\nestablished.\n\n\\smallskip\n\nRecently, several massive young stellar objects with compact MIR\nemission were investigated using long-baseline interferometry. In\nseveral cases, the spatially resolved $N$-band emission on scales of\n100~AU was found to be consistent with thermal envelope emission (de\nWit et al. 2007; Linz et al. 2009\\nocite{Linz2009}) and in particular\nwith emission arising from the cavity walls (de Wit et al. 2010).\nHowever, in some other cases, the mas-scale MIR emission of MYSOs is\nsuggested to be (at least partially) located in a circumstellar disc\n\\citep[see e.g.][]{Follert2010,deWit2011,Grellmann2011}. In other\nwords, there are hints that the $N$-band emission, resolved with\ninterferometry, is not completely dominated by envelope emission. The\nanalysis of the $Q$-band emission presented in this paper favours the\ncavity wall emission interpretation for a sample of MYSOs. Upon first\ninspection, this appears at odds with the scenario of a disc\ndominating the $N-$band emission. Here, we consider the possible\ncontribution to the $Q-$band emission by a circumstellar disc in order\nto assess whether our finding agrees with theoretical expectations.\n\n\\smallskip\n\nThe recent study by \\citet{Zhang2011} discusses in detail the\nappearance of embedded young massive stars in the IR wavelength\nregion. The authors use a dust radiative transfer code, as we\ndo. {{However, their geometry differs from the one employed in this\n    paper in that their code includes an accretion disc that extends\n    from the stellar surface to the centrifugal radius}}. Therefore,\ntheir results allow us to assess the possible disc contribution to the\n$Q-$band emission of MYSOs. The RT calculations of \\citet{Zhang2011}\nreveal that outflow cavities have a large influence on the shape of\nthe SED. Furthermore, in their calculation for a $60^{o}$ inclination\n(see e.g. their Fig. 9), the outflow cavities are clearly significant\nfeatures in the images, up to a wavelength of 70\\,$\\mu$m. In the NIR,\nscattered light from the cavities makes a significant contribution to\nthe near-IR emission. In the $N$-band, thermal emission from the disc\nand cental object are also evident. In the $Q$-band however, the\nemission from the base of the cavity clearly dominates the flux.\n\n\\smallskip\n\nThe relative contributions of the disc and envelope to the modelled\nSED depend critically on exactly those parameters with which we fit\nour observations, i.e.  opening angle, mass infall rate and\ninclination. We note that the opening angle applied by\n\\citet{Zhang2011} is rather wide ($2\\alpha \\approx 90^{o}$), which\nwould imply an object in a rather advanced phase of formation,\nfollowing the expected widening of the opening angle as function of\ntime \\citep[see e.g.][]{Kuiper2010}. Smaller opening angles increase\nthe extinction towards the central regions and thus favour the\ndominance of cavity wall emission in the total MIR emission. A similar\nreasoning applies for the mass infall rate. It is therefore not\nsurprising that the observed MYSOs, which are believed to be in a less\nevolved phase than the phase represented by the fiducial model of\n\\citet{Zhang2011}, are dominated by the cavity wall emission. This\nargues against a large effect, if any, by a circumstellar disc on our\nVISIR images. This conclusion can also be inferred from the spectral\nenergy distribution of an accretion disc, since it peaks at\nwavelengths much shorter than 20\\,$\\mu$m. Furthermore, the envelope\ncontains much more cool material than does any reasonable accretion\ndisc, whose sizes for MYSOs are estimated to be less than 500\\,AU\n\\citep[see\ne.g.][]{Patel2005,Hoare2006,Reid2007,Izas2007,deWit2011,Goddi_2011}. We\nconclude that it is reasonable to expect that dust emission from the\nenvelope (the cavity walls) dominates the continuum SED at\nwavelengths longward of the $N$-band. This is in agreement with our\nfinding that the observed extension of the $Q-$band flux can be\nmodelled as emission from outflow cavity walls.\n\n\n\\subsection{The structure of MYSO envelopes: rotation versus outflows}\n\nWe have shown that the morphology of the 20~$\\mu$m emission of MYSOs\nis consistent with that predicted by models incorporating outflow\ncavities. We followed a similar methodology to DW09 and have improved\non the modelling part of their analysis. DW09 compared their images to\n1-D radiative transfer models. This was motivated by the rather\nspherical appearance of most sources, which suggests that their MIR\nemission originates from a relatively symmetrical dusty envelope. They\nfound that, in general, their data were best fit by an envelope with a\ndensity gradient of $\\rho \\propto r^{-1}$. This density law is\nshallower than that obtained via a similar analysis performed in the\nsub-mm \\citep[see e.g.][]{Mueller2002}. It was therefore suggested\nthat this flattening of the density law towards smaller spatial scales\ncould be evidence for the onset of rotational support of the envelope\nat approximately 1000~AU. This suggestion is probably invalidated by\nthe results presented in this paper based on more realistic RT models.\n\n\\smallskip\n\nThermal emission from an envelope with evacuated outflow cavities fits\nthe SED and the $N$-band (de Wit et al. 2010) and $Q$-band (this\npaper) morphology of the luminous MYSO W33A. Additionally, in this\nparticular case, the envelope model also fits the observed\nmorphologies of the near-IR scattering nebula and the 350\\,$\\mu$m\nemission. This provides strong evidence in favour of this model and\nour explanation of the 20~$\\mu$m emission (warm dust in the cavity\nwalls). Moreover, the W33A image (Fig.\\,\\ref{test_image}) is slightly\nextended towards the SE which is the direction of the blue lobe of the\nlarge scale outflow, reinforcing this interpretation. Importantly, we\nnote that W33A is not the only case in our sample for which the\nextension at 20~$\\mu$m is in the direction of the outflow\nposition angle (see for example the sources G263.7759$-$00.4281 and\nG332.2941+02.2799). Therefore, our association of the observed MIR\nflux with outflow cavity walls can be substantiated.\n\n\\smallskip\n\nThe brightness of the cavity walls, as exemplified in panel a) of\nFig.\\,\\ref{test_image}, is a consequence of the density contrast\nbetween the in-falling envelope and the outflow cavities. The\nradiation of the central source cannot penetrate far in the dense\nenvelope, but can heat dust in the walls of the outflow cavity to\nlarge radii. Once the model images are convolved with an instrumental\nPSF, it is no longer obvious that the flux traces outflows, despite\nthe source being resolved. The flux from the outflow cavity walls can\nappear more extended than symmetric envelope emission. A centrally\nconcentrated density distribution (steep density law) in a spherical\nenvelope produces a small thermal emission region in the MIR. The\nshallower the density law the larger the (normalised) emission region\nbecomes. Therefore, the influence of the outflow cavities can mimic\nthe effect of a shallow density gradient. We verified that the sizes\nof the emission regions differ strongly between a $-2$ and $-1$\ndensity law, but the difference between a flat density law and a $-1$\ndensity law are relatively minor. Still the SEDs differ\nsignificantly. The steeper density law produces higher MIR fluxes. As\nan exercise, we fitted the standard 2D W33A model with a spherical\nmodel and find that the SED and intensity profiles are matched best by\na density law between $-0$ and $-1$. This explains the results\npresented in DW09. Given the more realistic nature of the 2D models we\nhave employed, we suggest that the preference for shallow density laws\nfound by DW09 is the result of much of the MIR emission of MYSO emanating from\nwarm dust in the cavity walls. An explanation in terms of rotational\nsupport on scales of $\\sim$1000~AU cannot be substantiated.\n\n\\subsection{SED fitting}\n\nFitting spectral energy distributions has long been used as a tool to\nprovide important constraints on the morphology of circumstellar\nenvironments \\citep[see e.g.][]{Guertler1991}. Still, the inclusion of\nspatially resolved observations at different wavelength regimes is\nindispensable for a correct interpretation of the observed emission\n\\citep[see e.g.][]{vdt2000}. This is because SED fitting can\nresult in degenerate solutions. To assess the added value provided by\nthe VISIR images presented in this paper, we use the results\npresented in \\citet{M2011}. These authors used the grid of models\nprovided by \\citet{Robitaille2007} to fit the SEDs of a large\npercentage of MYSOs from the RMS survey in order to determine their\nbolometric luminosity. The grid of \\citet{Robitaille2007} was created\nusing the same RT code of Whitney et al. which we use in this\npaper. Therefore, we used the code to create images for the best\nfitting models found in \\citet{M2011} (in the filter used) and assessed\nwhether the model images are consistent with our observations.\n\n\\smallskip\n\nTo illustrate the results of this exercise, we focus on the object\nG310.0135+00.3892. This is an object of particular interest as it is\nthe object studied by \\citet[][]{Kraus2010} via a high angular\nresolution VLTI aperture synthesis image in the $K$-band. We generate\n20\\,$\\mu$m images of the set of models that best fit the object's SED\nas found by \\citet{M2011}. The result is shown in Fig. \\ref{test}. The\nSED fitting tool preferentially returns models with large cavity\nopening angles ($\\sim30-50 \\degr$), but these all fail to reproduce\nthe observed intensity profile. The best fitting model of\n\\citet{Kraus2010}, which is also based on SED fitting with the model\ngrid of \\citet{Robitaille2007} and the code of Whitney et al.,\nprovides a better match. However, it still over-predicts the flux in\nthe central 1\\arcsec~from the source (see Fig. \\ref{test}). This is\nlikely due to the large opening angle of the Kraus et al. model\n($\\mathrm{40^{\\circ}}$) generating a MIR emission region which is too\nextended. The VISIR observations clearly demonstrate that, for the\nassumed intermediate inclination, a narrow opening angle is\nrequired. We conclude that the spatially resolved VLT-VISIR images at\n20~$\\mu$m provide valuable constraints on the nature of the envelopes\nsurrounding young massive stars.\n\n\n\n\\begin{center}\n  \\begin{figure}\n    \\begin{center}\n      \\includegraphics[width=0.4\\textwidth]{fig_8_a.ps}\n      \\caption{The azimuthally averaged intensity profile of G310.0135\n        (upper circles) compared to a PSF standard (lower circles) and\n        the models that fit the SED of this object from\n        \\citet[][dashed lines]{M2011}. The solid line is the radial\n        profile of the model of \\citet{Kraus2010}. The error bars\n        represent the rms within a given annuli and thus represent an\n        upper limit on the uncertainty in the flux\n        distribution.\\label{test}}\n    \\end{center}\n  \\end{figure}\n\\end{center}\n\n\n\n\\section{Conclusions}\n\n\\label{conc}\n\nIn this paper we present diffraction limited, MIR imaging of a sample\nof massive young stellar objects drawn from the RMS survey. By\ncomparing the spatially resolved images of the MYSOs to 2D radiative\ntransfer models, we constrain the structure of the envelopes that\nsurround them. Here we list the salient results.\n\n\\begin{itemize}\n\n\\item[--]{We spatially resolve the MIR (19.5 or 24.5~$\\mu$m) emission\n  of 14 MYSOs. The remainder of the MYSOs observed (6) are unresolved\n  at the current resolution ($\\sim$0.6\\arcsec).}\n\n\\item[--]{The model of the MYSO W33A that we have developed previously\n    can, in most cases, reproduce the images and SEDs of the MYSOs. This\n    is not the case for objects associated with a H{\\sc{ii}}\n    region. That the model can reproduce the infrared emission of a\n    sample of MYSOs suggests that the circumstellar environments\n    of MYSOs are relatively uniform. The large\n    envelope masses of the models are consistent with the\n    identification of MYSOs as objects in their accretion phase.}\n\n\n\\item[--]{It is found that the extent of the MIR emission of the\n    spatially resolved MYSOs can generally be attributed to warm dust\n    located in the walls of outflow cavities. We suggest that the\n    relatively shallow intensity profile gradients found by previous\n    MIR diffraction limited imaging of MYSOs is indicative of outflow\n    cavities (rather than alternative explanations such as rotational\n    flattening).}\n\n\n\\item[--]{We show that the varied morphology observed can be\n    attributed to outflow cavities seen at a variety of\n    inclinations. This emphasises that outflows are an ubiquitous\n    feature of massive star formation.}\n\n\\end{itemize}\n\n\n\\begin{acknowledgements}\n  HEW acknowledges the financial support of the MPIfR. WJdW is grateful\n  for the hospitality of G. Weigelt and the IR interferometry group at\n  the MPIfR where this paper was finalised. J. Mottram is thanked for\n  providing details of previous fits to the SEDs of many of the\n  sources in this paper. This paper made use of information from the\n  Red MSX Source survey database at www.ast.leeds.ac.uk\/RMS which was\n  constructed with support from the Science and Technology Facilities\n  Council of the UK.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{introduction_section}\n\nThe syntactic dependency structure of a sentence can be defined as a network where vertices are words and connections indicate syntactic dependencies, e.g., the relationship between the subject of a sentence and its verb (Fig. \\ref{syntactic_dependency_trees_figure}). These networks are typically trees and directed \\cite{tesniere59,hays64,melcuk88}. However, link direction is irrelevant for the present article and therefore omitted. Syntactic dependency networks can be seen as \nspanning trees on a lattice \\cite{Manna1992a,Barthelemy2006a} and are indeed a particular case of geographically embedded or spatial networks \\cite{Reuven2010a_Chapter8, Gastner2006a, Guillier2017a}\nin one dimension, i.e. the dimension defined by the linear order of the words of the corresponding sentence \\cite{Ferrer2004b}. \n\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{syntactic_dependency_trees}\n\\caption{\\label{syntactic_dependency_trees_figure} Syntactic dependency trees of sentences. Top: a tree without dependency crossings. \"John\" is the subject of the verbal form \"gave\". Center: a tree with one edge crossing (the crossing is formed by the edge linking \"ate\" and \"yesterday\" and the edge linking \"apple\" and \"which\"). Bottom: the same sentence after fronting \"yesterday\". Notice that the length of the dependency between \"ate\" and \"yesterday\" and the dependency between \"apple\" and \"which\" have reduced (while the length of other dependencies has remained constant). Top and center are adapted from \\cite{Ambati2008a}. }\n\\end{figure*}\n\nIn the context of syntactic dependency networks, the length of an edge is defined as an Euclidean distance, namely,\nthe linear distance between the words that are connected: adjacent words are at distance 1, words separated by one word are at distance 2, and so on \\cite{Ferrer2004b,Liu2017}. In the sentence at the top of Fig. \\ref{syntactic_dependency_trees_figure}, {\\em John} and {\\em gave} are at distance 1 while {\\em gave} and {\\em apple} are at distance 3. \n\nSyntactic dependency trees exhibit certain statistical patterns concerning the length of their dependencies and the variance of their degrees. First, edge lengths are biased towards low values \\cite{Ferrer2004b,Ferrer2003f} as it happens in other geographical networks \\cite{Gastner2006a}. The distribution of edge lengths decays exponentially \\cite{Ferrer2004b} as is the case of the distribution of projection lengths in real neural networks \\cite{Ercsey2013a}. Additionally, the mean edge length is smaller than expected by chance \\cite{Ferrer2004b, Ferrer2013c, Liu2008a, Futrell2015a, Liu2017}.  The simplest null hypothesis assumes a uniformly random permutation of the words of a sentence and predicts that the expected edge length is $(n + 1)\/3$, where $n$ is the number of vertices of the tree (the length of the sentence in words) \\cite{Ferrer2004b,Zornig1984a}.  \nSecond, their hubiness coefficient does not exceed $25\\%$ \\cite{Ferrer2017a}. $h$, the hubiness coefficient is a normalized variance of vertex degrees. $h$ is a number between 0 and 1 that is minimum for linear trees and maximum for star trees (Fig. \\ref{star_and_linear_trees_figure}). \nIndeed, the hubiness of real syntactic dependencies is close to trees from the ensemble of uniformly random trees, for which $h$ tends to zero as $n$ increases \\cite{Ferrer2017a}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale = 0.8]{star_and_linear_tree}\n\\caption{\\label{star_and_linear_trees_figure} Linear arrangements of trees with $n = 6$ vertices. Top: A star tree, a tree with a vertex of maximum degree \\cite{Ferrer2013d}. Bottom: A linear tree, a tree where vertex degrees do not exceed 2 \\cite{Ferrer2013d}. }\n\\end{center}\n\\end{figure}\n\nThe target of the present article are the edge crossings that can arise when drawing connections above the sentence. Fig. \\ref{syntactic_dependency_trees_figure} shows two planar sentences (a sentence is planar if it does not have crossings) and a sentence with one crossing. It is widely accepted that crossing dependencies are relatively uncommon in languages \\cite{lecerf60,hays64,melcuk88,Ferrer2006d,Park2009a,Liu2010a,Gildea2010a}. \nIndeed, the actual number of crossings per sentence does not reach $3.5$ across languages and is only above $1$ in a few of them \\cite{Ferrer2017a}. However, how small a number is depends on the scale of measurement and a null model is required. A rigorous demonstration that crossings are really scarce has been missing for decades. Recently, statistical evidence that crossings are significantly small has been provided \\cite{Ferrer2017a}. Furthermore, sentences where dependencies are shorter tend to have fewer crossings \\cite{Ferrer2015c}. Fig. \\ref{syntactic_dependency_trees_figure} (center and bottom) illustrates the tendency of crossings to reduce as dependency lengths reduce.\n\n\n\n\nResearch on syntactic dependency networks parallels research on non-spatial networks: as the statistical properties of many real networks have been compared against the predictions of null models, the Erd\\H{o}s-R\\'enyi graph being one of the most simple and popular examples \\cite{Newman2010a}, the statistical properties of syntactic dependency networks have been compared against the predictions of null models with increasing levels of complexity for the length of syntactic dependencies \\cite{Ferrer2004b, Liu2008a, Futrell2015a} or for the number of crossings \\cite{Ferrer2017a, Ferrer2014f, Ferrer2014c}.\n\nBeyond network theory, the issue of the presence and frequency of crossing dependencies in the syntax of natural languages has received considerable attention in the computational linguistics community, as supporting them makes parsing computationally harder \\cite{McDonald07,Hav07,GomCL2016}. Crossings are also relevant in biology, where they appear in networks of nucleotides whose vertices are occurrences of nucleotides $A$, $G$, $U$, and $C$ while edges are Watson-Crick ($A$-$U$, $G$-$C$) and $U$-$G$ base pairs \\cite{Chen2009a}. \n\nIn this context, a question naturally arises: what is the reason for the low frequency of crossing dependencies, consistently observed across languages? A traditional answer consists of postulating that there is some kind of grammatical ban on\ncrossing dependencies \\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07,Ninio2017}.\nHowever, this position fails to explain many linguistic phenomena involving crossings\n\\cite{Versley2014,Levy2012a}. Another option is to assume that crossing dependencies can be grammatical, but only if they follow certain patterns or hard constraints. However, while some classes of non-crossing dependency structures have a very good empirical coverage of real sentences \\cite{kuhlmann06,gomez2011cl,GomNiv2013,GomCL2016}, these proposals still face counterexamples that fall outside the restrictions \\cite{chen2010unavoidable,Bhat2012,ChenMain2014}.\n\nFrom the perspective of theoretical linguistics, the grammatical ban on crossings can be interpreted:\n\\begin{itemize}\n\\item\nAs a ban set independently from performance considerations, e.g., requiring some hidden parameter to be turned. In this case the ban can be seen as avoidable (e.g., it depends on whether the parameter is on or off for each given language). \n\\item\nAs a consequence of performance constraints associated directly to crossing dependencies. The ban would be inevitable if the cognitive pressures were strong enough but then it would not be properly a ban (a norm added on top of human cognition) but rather a side-effect of cognitive constraints. This view is challenged by psychological and graph theoretic research indicating that crossing dependencies can be easier to process (\\cite{deVries2012a} and \\cite{Ferrer2014f} and references therein).\n\\end{itemize}  \nSome researchers have adopted an apparently neutral position concerning the nature of the ban but assume that the low frequency of crossings derives from an independent and specific constraint on crossings: explicitly when postulating a principle of minimization of crossings \\cite{Liu2008a} or implicitly in a large body of research on dependency length minimization that takes for granted that syntactic dependencies should not cross \\cite{Liu2008a,Park2009a,GildeaTemperley10,Futrell2015a,Gulordava2015}. \n\nIf it turned out that non-crossing dependencies can be explained as a side-effect of some cognitive pressure that is not directly associated to crossings (e.g., dependency length minimization), could all these views be regarded as really neutral regarding the nature of the ban?  \n\nIn this article, we explore a simpler hypothesis: that in order to explain the scarcity of crossing dependencies in language, it is not necessary to assume any underlying rule or principle of human languages that is responsible directly for this fact (including the possibility of some cognitive cost associated directly to crossings). \nInstead, the low frequency of crossings may naturally arise, indirectly, from the actual length of dependencies \\cite{Ferrer2015c}, which are constrained by a well-known  psychological principle: dependency length minimization (see \\cite{Liu2017}, \\cite{Ferrer2013e} or \\cite{Tily2010a} for a review). That explanatory principle, which holds even in languages allowing for words to scramble freely \\cite{Futrell2015a}, could follow from more general constraints on language processing \\cite{Christiansen2015a}. \n\nAs dependency length minimization can be seen as particular case of minimization of the Euclidean distance between connected vertices in an $m$-dimensional space, our originally linguistic problem on crossings is related to the general problem of minimizing the cost of load transportation over a network in complex systems science \\cite{Guillier2017a} and the minimum linear arrangement problem of computer science \\cite{Ferrer2004b,Ferrer2016a}.\n\nTo investigate the origins of the scarcity of crossing dependencies, we use treebanks (collections of sentences with their corresponding syntactic dependency network) to provide statistical evidence that the amount of dependency crossings in a wide range of languages can be predicted with small error by a simple estimator based exclusively on dependency length information and information on which edges can potentially cross (edges that share a vertex cannot cross). \n\nWe will show that the estimator consistently delivers good predictions of the number of crossings, \nin two different collections of dependency treebanks with diverse annotations. An annotation is a set of criteria used to define the syntactic dependency structure of a sentence. We will argue that this is the best explanation for the low frequency of crossings when both psychological plausibility and parsimony at all levels (from a model of crossings to a general theory of language) are required. Our predictor is a null model in the sense that for every pair of edges that may potentially cross it assumes that the corresponding vertices take random positions in the linear sequence of the sentence. \n\nThe remainder of the article is organized as follows. Section \\ref{predictors_section} discusses various ways in which the crossings of a sentence could be predicted. \nSection \\ref{crossing_theory_section} presents the predictor of crossings chosen for this article and its theoretical background. The dependency trees used to test the predictor are presented in Section \\ref{methods_section}. Section \\ref{results_section} shows the results of the predictions, and Section \\ref{discussion_section} discusses some implications for computational linguistics and linguistic theory.    \n\n\\section{Possible predictors}\n\n\\label{predictors_section}\n\nHere we will examine various possibilities to predict the number of dependency crossings in a sentence. \nThe problem of the origins of non-crossing dependencies can be recast as problem of modeling: we want to find the best model\nfor predicting the number of crossings in a sentence. According to modern model selection, the best model is the one that has the best trade-off between quality of fit (predictive power) and parsimony \\cite{Burnham2002a}.\nWe complement this view\ninvolving further requirements: \n\\begin{itemize}\n\\item\nThe model must be psychologically realistic. A model that assumes orderings of words that are hard to produce by the human brain should be penalized with respect to one that is based on orderings that real speakers produce (or can rather easily produce). We are not only simply concerned about predicting the low number of crossings of a sentence but also understanding why that number is that low. Hiding the problem under the carpet of grammar or an ad-hoc principle of planarity\ndoes not help.\n\\item  \nIts assumptions must be valid. The predictions of a model may be compatible with real data and even be of high quality but its assumptions may not be supported by real data or inconsistent with the source that produced it.  \n\\item \nWe are not only concerned about the best model in a local sense but one that leads to a general theory of word order or even a comprehensive theory of language that is compact. A real scientific theory is more than a collection of disconnected ideas or models \\cite{Bunge2001a_French}.\nModels that lead to an unnecessarily fat general theory when integrated into it should also be penalized. \nModels that exploit assumptions from successful models in other domains should be favored. \n\nFor instance, a model that allows one to understand not only the scarcity of crossings but also why adjectives tend to be placed \nbefore the noun in SOV languages is preferable to one that requires an independent solution to explain the placement of adjectives \\cite{Ferrer2014f}. SOV languages are languages that tend to put the subject (S) before the object (O) and in turn, O before the verb (V) under some general conditions \\cite{wals-81}.\n\\end{itemize}\n\nIn what follows, we will use $C$ to refer to the number of dependency crossings in the parse of a sentence (i.e., the number of pairs of syntactic dependencies that cross). Our goal is, therefore, to find a suitable predictor for $C$. Note that $C = 0$ for a star tree \\cite{Ferrer2013d}. The sum of the lengths of all dependencies in a sentence will be denoted by $D$. \n\n\\subsection{Minimization of crossings}\n\nA principle of minimization of crossings \\cite{Liu2008a} leads to a simple deterministic predictor: $C = 0$, reflecting a grammatical ban on crossings\n\\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07}.\n\nThis predictor is problematic for various reasons:\n\\begin{itemize}\n\\item\nConcerning the validity of its assumptions, the model assumes that $C = 0$ independently from $D$, while $C$ and $D$ are positively correlated in many languages \\cite{Ferrer2015c}. \n\\item\nConcerning the accuracy of its predictions, this model fails because sentences with $C > 0$ are found in many languages \\cite{Ferrer2017a} and the likelihood of the model is minimum, which indicates that the model is among the worst possible models for crossing dependencies according to modern model selection \\cite{Burnham2002a} because its likelihood is zero. Furthermore, \nthe model fails to explain many linguistic phenomena involving crossings\n\\cite{Versley2014,Levy2012a}.  \n\\item\nIts psychological validity is unclear. If the model is interpreted as arising from processing difficulties inherent to crossing dependencies \\cite{Levy2012a} or computational tractability (as reviewed in Section \\ref{introduction_section}) then it is challenged by psychological and graph theoretic research indicating that sentences with $C >0$ can be easier to process than sentences with $C= 0$ (see \\cite{deVries2012a}, \\cite{Ferrer2014f}, \\cite{Chung1978a} and references therein). Another problem is how a language generation process could warrant that $C=0$. If $C=0$ is determined before the sentence is produced, how is it possible that sentence production does not introduce (many) crossings? Crossing theory indicates that a star tree is needed to keep a low number of crossings \\cite{Ferrer2014f}. \nIf $C=0$ is determined while the sentence is produced (linearized), how are crossings avoided on the fly as real language production is not a batch process \\cite{Christiansen2015a}? It looks simpler to consider that non-crossing dependencies are a side-effect of a principle of dependency length minimization \\cite{Ferrer2006d,Ferrer2014c,Ferrer2014f}.\n\\item\nConcerning the compactness of the whole theory, the model $C = 0$ leads to a fatter theory of language because the scarcity of crossings and also the positive correlation between $D$ and $C$ could be explained to a large extent by recycling the highly predictive principle of dependency length minimization \\cite{Ferrer2013e}, as we will see below. \n\\end{itemize}\n\nAnother option is to assume that crossing dependencies can be grammatical, but only if they follow certain patterns or hard constraints. However, while some\nclasses of dependency structures tolerating certain crossings\nhave a very good empirical coverage \\cite{kuhlmann06,gomez2011cl,GomNiv2013,GomCL2016}, these proposals still face counterexamples that fall outside the restrictions \\cite{chen2010unavoidable,Bhat2012,ChenMain2014}.\n\nOne possibility is to relax the simple deterministic predictor above so that on average $C = \\gamma$, where $\\gamma$ is a constant, e.g., $\\gamma = 3.3$ as in  ancient Greek \\cite{Ferrer2017a}. However, it has been shown that this is problematic because $C = \\gamma$ might be impossible to reach if $n$ is sufficiently small (see Appendix of \\cite{Ferrer2015c}). Therefore, a proper relaxation of this deterministic predictor is  $C = \\gamma(n)$, where $\\gamma$ is a function that only depends on $n$ \\cite{Ferrer2015c}.\nThis allows one to capture the variation in the number of crossings across languages, but adding extra parameters, and it is still problematic for the reasons of the case $\\gamma(n) = 0$ that we have examined above. Further arguments can be found in Section 4.3 of \\cite{Ferrer2014f}. \n\n\\subsection{Minimum linear arrangement}\n\nA minimum linear arrangement of a sentence is an ordering of the words of the sentence that minimizes the sum of dependency lengths.\nOne may predict the assumed number of crossings by calculating the minimum linear arrangements of a sentence \\cite{Ferrer2006d}. A possible predictor could be the mean number of crossings over all those arrangements. \n\nThe predictive power of the model is supported by the fact that solving the minimum linear arrangement problem reduces crossings to practically zero  \\cite{Ferrer2006d}, as in many languages. A potential problem of this model is that it has never been checked whether it predicts the actual number of crossings of real sentences, as far as we know.  \n\nPerhaps the major challenge for this predictor is the validity of the assumption of a minimum linear arrangement because:\n\\begin{itemize}\n\\item\nThe actual value of $D$ in real sentences is located between the minimum and that of a random ordering of vertices \\cite{Ferrer2004b,Ferrer2013c}. The ratio $\\Gamma = D\/D_{min}$ (where $D_{min}$ is the minimum value of $D$) is greater than 1.2 in Romanian for sufficiently long sentences \\cite{Ferrer2004b} and a similar lower bound on language efficiency has been found in English across centuries \\cite{Tily2010a}.\n\\item\nIt may not be valid also for theoretical reasons: word order is a multiconstraint satisfaction problem where the principle of dependency length minimization is in conflict with  other word order constraints \\cite{Ferrer2014a,Futrell2015a}.\nThus, a model based on minimum linear arrangements is not that simple: it has to explain why dependency length minimization dominates fully over other principles or provide evidence that the distortion caused by other principles can be neglected. Below we will present a model that does not have this problem because it works on true dependency lengths, which are expected to be determined by the interplay between dependency length minimization and other principles.  \n\\item\nThe full minimization of $D$ is cognitively unrealistic, as it is incompatible with the predictions of the now-or-never bottleneck \\cite{Christiansen2015a}. As for the latter, notice that the minimization of $D$ implies that the whole sentence must be available as input for some minimum linear arrangement algorithm, whereas actual language generation and processing is intrinsically online and heavily constrained by our fleeting memory \\cite{Christiansen2015a}.\n\\end{itemize}\n\n\\subsection{Random linear arrangement}\n\nIf the minimum linear arrangement is too restrictive, one could consider the opposite:  \npredicting the number of crossings assuming a random ordering of the words of the sentence \\cite{Ferrer2014f}. However, a random linear arrangement cannot explain the low numbers of crossings observed in real sentences. Empirically, the number of crossings of sentences is much smaller than the number of crossings expected by random linear arrangement \\cite{Ferrer2017a}. Theoretically, a constant low number of crossings requires a star tree \\cite{Ferrer2014f}. \n\nThe failure of a random linear arrangement is not surprising. First, it \nis cognitively unrealistic: even in languages with high word order flexibility, word order is constrained \\cite{wals-81,Lester2015a}. Second, the assumption that the ordering of sentences is arbitrary (unconstrained) is easily rejected by the fact that dependency lengths are below chance in real languages \\cite{Ferrer2004b,Ferrer2013c,Futrell2015a}. \nThus, this predictor is only useful as a random baseline for other predictors. Here we will compare it against a better predictor that is introduced next. \n  \n\\subsection{Random linear arrangement with some knowledge about dependency lengths}\n\nA stronger predictor can be built by focusing on the set of pairs of edges that may potentially cross and basing predictions on the actual length of the edges under the assumption of a random linear arrangement of the four vertices that are potentially involved in an edge crossing \\cite{Ferrer2014c}. So far, its predictive power is supported by its capacity to predict the actual number of crossings in random trees with an error of about $5\\%$ \\cite{Ferrer2014c}.\nA crucial goal of the present article is to test the accuracy of its predictions on real sentences. \nThis predictor is promising because actual dependency lengths are below chance, i.e. below $(n+1)\/3$ \\cite{Ferrer2004b,Ferrer2013c}, a domain where the probability theory behind this model indicates that shortening a dependency yields a reduction in the probability that it crosses a dependency of unknown length in a random linear arrangement of the two edges (Section 5 of \\cite{Ferrer2014f}).\n\nFor the reason above,\nthis predictor is fully compatible with the positive correlation between $D$ and $C$ \\cite{Ferrer2015c,Ferrer2014f}, in contrast with the deterministic predictor ($C=0$) and its generalization. \nConcerning assumptions, this model is simpler than the model based on minimum linear arrangements: this model does not assume an unrealistic ordering of the elements of the sentence but the true ordering.  \nIts psychological validity is greater than that of the minimum linear arrangement predictor because it can base its prediction on information from sentences that have actually been produced by a speaker or a writer.\nContrary to the minimum linear arrangement predictor, this model bases its prediction on true dependency lengths instead of ideal values of $D$.\n\nHowever, it can be argued that a fundamental assumption of the model, namely that vertices are arranged linearly at random, is not supported empirically, following the arguments against the random linear arrangement predictor. This is a fair criticism, but for this reason this model should be regarded as a null hypothesis rather than as a fully realistic model. \n\nHaving said that,\nmodeling requires a compromise between quality of fit, adequacy and parsimony. If this null hypothesis model provides predictions of sufficient quality on real sentences, do we really need to worry about providing a more realistic but also more complicated model? \nPut differently, suppose that the information provided by the lengths of edges suffices to predict reasonably well the low number of crossings of real sentences, even without assuming any additional constraint on the linear arrangement of the involved vertices that could help to minimize crossings, but instead modeling it under the weakest possible assumption (namely placing vertices at random). Then considering more fine-grained information or more realistic orderings is secondary to our particular goal. In the worst case, this predictor would be an inevitable baseline for an alternative model. \n\nBefore we proceed, it is worth noting that our article is not a mere application of an established model or theory to a concrete dataset, but the first test of a novel theory on a massive collection of networks from different languages and different annotation criteria, which has implications to our understanding of the faculty of language as such.  \nThe result of such a test is far from trivial, and thus its success is a relevant contribution, for two reasons.\nFirst, our model, which is a null model rather than a realistic model, assumes that vertices are arranged purely at random in a sequence (preserving the original edge lengths). However, real sentences are not random sequences of words, as research on long correlations in physics has been showing for more than a decade, e.g. \\cite{Montemurro2001b,Altmann2012a}. \nSecond, although such null model predictor has been tested previously on uniformly random trees \\cite{Ferrer2014c}, one cannot assume the predictor will work on real sentences given the substantial statistical differences between uniformly random trees and real syntactic dependency trees \\cite{Ferrer2017a}.\n\nThe next section introduces the mathematical definition of the promising predictor above and its theoretical background. \n\n\\section{Crossing theory}\n\n\\label{crossing_theory_section}\n\nHere we provide a quick overview of a crossing theory developed in a series of articles \\cite{Ferrer2013b,Ferrer2013d,Ferrer2014c,Ferrer2014f, Ferrer2017a}. \nIt is correct to state that $C$ cannot exceed the number of pairs of different edges, namely\n\\begin{equation}\nC \\leq {n - 1 \\choose 2}\n\\label{naive_maximum_number_of_crossings_equation}\n\\end{equation}  \nHowever, the truth is that \n\\begin{equation}\nC \\leq {n - 2 \\choose 2}\n\\label{maximum_number_of_crossings_equation}\n\\end{equation}\nwith equality in case of a linear tree (see \\cite{Ferrer2017a} for linear arrangements of linear tree that maximize $C$). The upper bound above is defined based only on knowledge of the size of the tree. Adding further properties of the tree, the upper bound can be refined.\n\nA central concept of crossing theory is $Q$, the set of pairs of edges of a tree that can potentially cross when their vertices are arranged linearly in some arbitrary order (edges sharing a vertex cannot cross).  \n$|Q|$, the cardinality of $Q$, is the potential number of crossings, i.e. \n\\begin{equation}\nC \\leq |Q| \\leq {n - 2 \\choose 2}.\n\\end{equation}\nWe have  \n\\begin{equation}\n|Q| = \\frac{n}{2} \\left(\\left< k^2 \\right>_{star} - \\left< k^2 \\right>\\right),\n\\label{potential_number_of_crossings_equation}\n\\end{equation}\nwhere \n$\\left< k^2 \\right>$ is the mean of the squared degrees of its vertices and $\\left< k^2 \\right>_{star}= n - 1$ is the value of $\\left< k^2 \\right>$ in a star tree of size $n$ \\cite{Ferrer2014c,Ferrer2013d}. $|Q| = 0$ if and only if the tree is a star tree \\cite{Ferrer2013d}.\n\nWith the theoretical background above, it is easy to see why $C$ cannot exceed the number of different pairs that can be formed out of $n-2$ elements (Eq. \\ref{maximum_number_of_crossings_equation}) instead of $n-1$, that coincides with the number of edges (Eq. \\ref{naive_maximum_number_of_crossings_equation}): that is the conclusion of computing the value of $\\left< k^2 \\right>$ for a linear tree, i.e. $\\left< k^2 \\right>_{linear} = 4 - 6\/n$, and then applying \n$\\left< k^2 \\right> \\geq 4 - 6\/n$ to Eq. \\ref{potential_number_of_crossings_equation} \n\\cite{Ferrer2017a}.\n\n$C$ denotes the number of crossings of the linear arrangement of a graph in general while  \n$C_{true}$ denotes the number of crossings of the syntactic dependencies of a real sentence. The relative number of crossings is $\\bar{C} = C\/|Q|$ or \n$\\bar{C}_{true} = C_{true}\/|Q|$ \\cite{Ferrer2014c}. $C$ can be expressed as a sum over $Q$, i.e. \n\\begin{equation}\nC = \\sum_{(e_1, e_2) \\in Q} C(e_1, e_2),\n\\label{number_of_crossings_equation}\n\\end{equation} \nwhere $C(e_1, e_2)$ is an indicator variable, $C(e_1, e_2)$ = 1 if the edges $e_1$ and $e_2$ cross and $C(e_1, e_2) = 0$ otherwise.  \nThe simplest prediction about $C$ than can be made departs from the null hypothesis that the vertices are arranged linearly at random (all possible orderings are equally likely). Following Eq. \\ref{number_of_crossings_equation}, the expected number of crossings under that null hypothesis turns out to be \n\\begin{eqnarray}\nE_0[C] & = & \\sum_{(e_1, e_2) \\in Q} E[C(e_1, e_2)]\\\\\n       & = & \\sum_{(e_1, e_2) \\in Q} p(C(e_1, e_2) = 1), \n\\label{expected_number_of_crossings_equation}\n\\end{eqnarray}\nwhere $p(C(e_1, e_2) = 1)$ is the probability that the edges $e_1$ and $e_2$ cross knowing that they belong to $Q$. \nUnder that null hypothesis, the probability that two edges of $Q$ cross is constant, i.e. $p(C(e_1, e_2) = 1) = 1\/3$, yielding \\cite{Ferrer2013d} $E_0[C] = |Q|\/3$.\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{probability_given_two_lengths_4_panels}\n\\caption{\\label{probability_given_two_lengths_figure} $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$\nas a function of $d(e_1)$ and $d(e_2)$ for different values of $n$ (tree size in vertices). \n$d(e)$ is the length of the edge $e$ and $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ is the probability that $e_1$ and $e_2$ (a pair of edges of $Q$) cross in a random linear arrangement of their vertices, knowing their lengths. Brightness is proportional to $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ (black for $p(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = 0$ and white for $p(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = 1$). The two panels on top show the value of the probability.} \n\\end{figure*}\n\nThe prediction offered by $E_0[C]$ can be improved by introducing knowledge about the length of the dependencies (e.g., edges of length 1 or $n - 1$ are not crossable). Suppose that $d(e)$ is the length of the edge $e$ and that $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ is the probability that $e_1$ and $e_2$ (two arbitrary edges of $Q$) cross in a random linear arrangement of their vertices knowing their lengths.\nThe predictor $E_2[C]$ is obtained when $p(C(e_1, e_2) = 1)$ is replaced by $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$\nin Eq. \\ref{expected_number_of_crossings_equation}, yielding \n\\begin{equation}\nE_2[C] = \\sum_{(e_1, e_2) \\in Q} p(C(e_1, e_2) = 1 | d(e_1), d(e_2)),\n\\label{predicted_number_of_crossings_equation}\n\\end{equation}\n$p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ depends only on $n$, $d(e_1)$ and $d(e_2)$ and\nis defined as  \n\\begin{equation}\np(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = \\frac{|\\alpha(d(e_1),d(e_2))|}{|\\beta(d(e_1),d(e_2))|},\n\\label{probability_of_crossing_given_two_lengths_equation}\n\\end{equation}\nwhere here $|..|$ is the cardinality operator, $\\alpha(d_1,d_2)$ is the set of valid pairs of initial positions of two edges of lengths $d_1$ and $d_2$ that involve a crossing and $\\beta(d_1,d_2)$ is the set of valid pairs of initial positions of edges of lengths $d_1$ and $d_2$, thus $\\alpha(d_1,d_2) \\subseteq \\beta(d_1,d_2)$. Fig. \\ref{probability_given_two_lengths_figure} shows a two-dimensional map of $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$. The perimeter of the map contains zeroes because an edge of minimum length ($1$) or maximum length ($n-1$) cannot cross any other edge. The map is symmetric with respect to the diagonal that crosses the top-left corner and the bottom-right corner by symmetry, namely\n\\begin{eqnarray}\np(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = \\notag \\\\  \n                                    p(C(e_2, e_1) = 1 | d(e_2), d(e_1)). \n\\end{eqnarray}\nThe map for $n = 4$, the minimum value of $n$ needed to have $|Q| > 0$, shows that only edges of length 2 can cross. The maps for $n = 8$, $n = 12$ and $n=16$ show that a reduction of the length of one of the edges causes the probability of crossing to reduce if edge lengths are sufficiently small. This reduction of the probability of crossings is likely to occur in real languages, where the mean length of dependencies is on average smaller than the random baseline $(n+1)\/3$ \\cite{Ferrer2004b, Ferrer2013c} and long edges would imply a cognitive cost that may not be afforded \\cite{Liu2017,Christiansen2015a,Ferrer2013e}.\n\nAlthough $E_0[C]$ and $E_2[C]$ are predictors of $C$ that have the same mathematical structure (they are sums of probabilities over pairs of edges of $Q$), $E_0[C]$ is a true expectation while $E_2[C]$ is not. \n\nThe relative error of a predictor is defined as \\cite{Ferrer2014c}\n\\begin{equation}\n\\Delta_x = E_x\\left[\\bar{C}\\right] - \\bar{C}_{true} = \\frac{1}{|Q|}\\left(E_x[C]-C_{true}\\right).\n\\label{error_equation}\n\\end{equation}\n$\\Delta_0$ will be used as a baseline for $\\Delta_2$. Interestingly,\n$\\Delta_0$ converges to $1\/3$ for sufficiently long sentences when $C_{true}$ is bounded by a constant and $|Q|$ is large enough. The reason is that $E_0[C]\/|Q| = 1\/3$ and then \n\\begin{equation}\n\\Delta_0 = \\frac{1}{3} - C_{true}\/|Q|.\n\\end{equation}\nThat explains why $\\Delta_0$ converges to $1\/3$ for sufficiently large $n$ in uniformly random trees where $C$ is bounded by a small constant \\cite{Ferrer2014c} because uniformly random trees have a high $|Q|$, or equivalently, a low hubiness \\cite{Ferrer2017a}. We also expect $\\Delta_0$ to converge to $1\/3$ in real syntactic dependency trees because $C_{true}$ is small and their hubiness is also low \\cite{Ferrer2017a}. \n\n\\section{Materials and methods}\n\n\\label{methods_section}\n\nWe considered the corpora in version 2.0 of the HamleDT collection of treebanks \\cite{HamleDTJournal,HamledTStanford}. This collection is a harmonization of existing treebanks for 30 different languages into two well-known annotation styles: Prague dependencies \\cite{PDT20} and Universal Stanford dependencies \\cite{UniversalStanford}. Therefore, this collection allows us to evaluate our predictions of crossings both across a wide range of languages and two popular annotation schemes. The latter is useful because observations like the number of dependency crossings present in treebank sentences do not only depend on the properties of languages themselves, but also on annotation criteria (\\cite{Ferrer2015c} lists some examples of how annotation criteria may affect $C$).\n\nEach of the syntactic dependency structures in the treebanks was preprocessed by removing nodes corresponding to punctuation tokens, as it is standard in research related to dependency length (e.g., \\cite{Ferrer2004b,Ferrer2015c,Futrell2015a}), which is only concerned with dependencies between actual words. To preserve the syntactic structure of the rest of the nodes, non-punctuation nodes that had a punctuation node as their head were attached as dependents of their nearest non-punctuation ancestor. Null elements, which appear in the Bengali, Hindi and Telugu corpora, were also subject to the same treatment as punctuation.\n\nAfter this preprocessing, a syntactic dependency structure was included in our analyses if (1) it defined a tree and (2) the tree was not a star tree. \nThe reason for (1) is that our theory (e.g., $Q$) assumes a tree structure \\cite{Ferrer2013d,Ferrer2014c} and that we wanted to avoid the statistical\nproblem of mixing trees with other kinds of graphs, e.g., the potential number\nof crossings depends on the number of edges \\cite{Ferrer2013b,Ferrer2014f,Ferrer2013d}. \nThe reason for (2) is that crossings are impossible in a star tree \\cite{Ferrer2013b}. Condition (2) implies that the syntactic dependency structure has at least four vertices (otherwise all the possible trees are star trees). By excluding star trees we are discarding trees where the prediction cannot fail.  \nAn additional reason for excluding star trees is that their relative number of crossings, $C\/|Q|$, is not defined because\n$C = |Q| = 0$.\n\nTable \\ref{relative_table} shows the number of sentences in the original treebanks and the number of sentences actually included in our analyses, after filtering by the criteria (1) and (2) above. The average number of crossings per sentence does not exceed 1 for most of the treebanks. See \\cite{Ferrer2017a} for further details on the statistical properties of crossings in our collections of dependency treebanks. \n\nHere we adopt the convention of sorting languages in tables not alphabetically but decreasingly by number of crossings, measured according to the average number of crossings (the average $C_{true}$) with Stanford dependencies. It can be observed that languages that are known for their word order freedom, e.g., Latin or Ancient Greek, stand out on top of Table \\ref{relative_table}. On the other hand, agglutinating languages like Basque, Japanese, Turkish, the Uralic languages Estonian and Finnish, and the Dravidian languages Tamil and Telugu, are placed rather to the bottom of the table. \nAgglutinating languages are languages where certain information is often integrated into words as morphemes (not leading to new vertices in the tree, except in the Turkish treebank) while non-agglutinating languages would instead place it in separate words (leading to separate vertices). Therefore, one expects fewer chances for dependency crossings in agglutinating languages, as equivalent information is expressed with fewer vertices, and the number of crossings tends to increase with the length of the sentence \\cite{Ferrer2017a}. \n\nOur ordering by crossings should be taken as an approximation. For the sake of space, we only employ an ordering by crossings based on Stanford dependencies. Furthermore, the potential number of crossings may depend on factors such as genre, topic, sentence length or treebank size (number of sentences) and other biases \\cite{Ferrer2015c,Ferrer2017a}. The collection of treebanks is heterogeneous in this respect. Therefore, the fact that one treebank has more crossings than another does not imply that the language of the former exhibits higher word order freedom than that of the latter. Other variables should be controlled for a more accurate ordering. Therefore, the focus of our article is on the power of the predictors in spite of the heterogeneity of the treebank collection. Linguistic distinctions such as agglutinating versus non-agglutinating languages are made to illustrate the potential of future linguistic research.\n \n\\section{Results}\n\n\\label{results_section}\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{error2}\n\\caption{\\label{relative2_figure} $\\Delta_2$, the error of the predictor based on the probability that two edges cross in a random linear arrangement preserving their original length, as a function of $n$, the size of the tree. Points and error bars indicate, respectively, mean values and $\\pm 1$ standard deviation over proportions in a collection of treebanks. Tree sizes represented by less than two treebanks are excluded. Top: Stanford annotations. Bottom: Prague annotations.}\n\\end{figure*}\n\nFigure \\ref{relative2_figure} shows that, on average across treebanks, $\\Delta_2$ increases as $n$ increases till $n = 10$ and decreases from that point onwards in both annotations. The maximum average $\\Delta_2$ that is reached at $n = 10$ is $0.052$ for Stanford annotations and $0.038$ for Prague annotations. \nThe predictor never fails for $n = 4$ and $n = 5$ ($\\Delta_2 = 0$ in both cases) and from $n = 6$ onwards it always overestimates (on average) the actual number of crossings (recall Eq. \\ref{error_equation}). Figure \\ref{relative0_figure} shows that, on average across treebanks, $\\Delta_0$ converges to $1\/3$ as expected. \n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{error0}\n\\caption{\\label{relative0_figure} $\\Delta_0$, the error of the random linear arrangement predictor, as a function of $n$, the size of the tree. The format is the same as in Fig. \\ref{relative2_figure}. A control line has been added to indicate 1\/3, namely, the $\\Delta_0$ that is expected when $C_{true}$ is bounded above by a small constant and $n$ goes to infinity. }\n\\end{figure*}\n\nTable \\ref{relative_table} shows that the average $\\Delta_2$, the relative error of the predictor $E_2[C]$,\nis small: it does not exceed $5\\%$. Thus, the average $\\Delta_2$ is at least 6 times smaller than the baseline $\\Delta_0 \\approx 33\\%$. \nThe averages presented in Table \\ref{relative_table} have been produced mixing measurements from sentences of different lengths. This is potentially problematic because the results might be heavily determined by the distribution of sentence lengths \\cite{Ferrer2013c}. \n\nTo control for sentence length, sentences were grouped by length and the average $\\Delta_2$ was computed for the sentences within each group. Table \\ref{relative_grouping_by_sentence_length_table} summarizes the statistical properties over the average $\\Delta_2$ of each group. Interestingly, the average over group averages of $\\Delta_2$ decreases with respect to the previous analysis: it does not exceed $4.3\\%$. Thus, the average $\\Delta_2$ is at least 7 times smaller than the baseline error, again $\\Delta_0 \\approx 33\\%$. The minimum size of a group is one sentence; the qualitative results are very similar if the minimum size is set to 2.\n      \n\\newcommand{\\specialcell}[2][c]{%\n  \\begin{tabular}[#1]{@{}c@{}}#2\\end{tabular}}\n\n\\begin{turnpage}\n\\begin{table*}\n\\caption{\\label{relative_table} Summary of results for each treebank: number of sentences before and after filtering, average values of $C_{true}$ and $\\Delta_0$, and average, median and standard deviation of the relative error $\\Delta_2$, over the trees of each treebank. \nRomanian (Prague) is the only treebank with no crossing dependencies. Languages are sorted decreasingly by average $C_{true}$ according to Stanford dependencies.}\n\n\\begin{small}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{lr|rccccc|rccccc} \n\\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Stanford annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Prague annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\phantom{x} \\\\ \\hline\nTreebank & \\#Sent & \\specialcell{\\#Sent\\\\ \\footnotesize{(filtered)}} &\n\\specialcell{$C_{true}$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_0$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(st. dev.)}} \n& \\specialcell{\\#Sent\\\\ \\footnotesize{(filtered)}} &\n\\specialcell{$C_{true}$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_0$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(st. dev.)}}\\\\ \\hline\nAnc. Greek & 21173 & 18713 & 3.2621 & 0.244 & 0.030 & 0.027 & 0.058  & 16237 & 3.3528 & 0.243 & 0.025 & 0.020 & 0.058 \\\\\nLatin & 3473 & 3036 & 2.1785 & 0.282 & 0.034 & 0.031 & 0.046         & 2833 & 1.8503 & 0.286 & 0.036 & 0.032 & 0.047 \\\\\nDutch & 13735 & 10974 & 1.3980 & 0.311 & 0.046 & 0.041 & 0.051       & 11131 & 0.9898 & 0.315 & 0.034 & 0.027 & 0.041 \\\\\nHungarian & 6424 & 6103 & 0.9720 & 0.326 & 0.036 & 0.031 & 0.033     & 5047 & 0.8675 & 0.326 & 0.034 & 0.030 & 0.032 \\\\\nArabic & 7547 & 2280 & 0.9807 & 0.328 & 0.019 & 0.016 & 0.021        & 2248 & 0.0881 & 0.333 & 0.013 & 0.010 & 0.016 \\\\\nGerman & 38020 & 33492 & 0.7826 & 0.325 & 0.050 & 0.046 & 0.036      & 32443 & 0.7230 & 0.326 & 0.043 & 0.039 & 0.033 \\\\   \nSlovenian & 1936 & 1719 & 0.7749 & 0.322 & 0.047 & 0.039 & 0.046     & 1581 & 0.3125 & 0.327 & 0.035 & 0.027 & 0.038 \\\\     \nDanish & 5512 & 4894 & 0.6800 & 0.324 & 0.047 & 0.040 & 0.038        & 4840 & 0.1643 & 0.331 & 0.027 & 0.022 & 0.027 \\\\        \nGreek & 2902 & 2584 & 0.6540 & 0.330 & 0.039 & 0.033 & 0.028         & 2543 & 0.2057 & 0.332 & 0.030 & 0.024 & 0.023 \\\\    \nCatalan & 14924 & 14520 & 0.6419 & 0.331 & 0.034 & 0.029 & 0.023     & 14556 & 0.0873 & 0.333 & 0.020 & 0.017 & 0.016 \\\\\nPortuguese & 9359 & 8621 & 0.6336 & 0.328 & 0.039 & 0.033 & 0.032    & 8596 & 0.2465 & 0.331 & 0.021 & 0.016 & 0.021 \\\\\nSpanish & 15984 & 15354 & 0.6218 & 0.331 & 0.034 & 0.029 & 0.024     & 15424 & 0.1105 & 0.333 & 0.020 & 0.016 & 0.017 \\\\\nPersian & 12455 & 11579 & 0.5914 & 0.326 & 0.027 & 0.023 & 0.031     & 11632 & 0.4024 & 0.329 & 0.030 & 0.024 & 0.033 \\\\      \nCzech & 87913 & 74843 & 0.5277 & 0.326 & 0.040 & 0.035 & 0.035       & 70023 & 0.3729 & 0.327 & 0.031 & 0.025 & 0.031 \\\\  \nEnglish & 18791 & 18275 & 0.5241 & 0.330 & 0.049 & 0.043 & 0.031     & 18369 & 0.1072 & 0.333 & 0.034 & 0.029 & 0.024 \\\\\nSwedish & 11431 & 10714 & 0.4871 & 0.328 & 0.043 & 0.039 & 0.034     & 10207 & 0.1946 & 0.332 & 0.034 & 0.029 & 0.029 \\\\   \nSlovak & 57408 & 47727 & 0.4559 & 0.324 & 0.044 & 0.036 & 0.044      & 44297 & 0.2688 & 0.326 & 0.034 & 0.026 & 0.039 \\\\     \nRussian & 34895 & 31581 & 0.4171 & 0.326 & 0.038 & 0.032 & 0.035     & 31900 & 0.1570 & 0.330 & 0.027 & 0.021 & 0.028 \\\\   \nItalian & 3359 & 2502 & 0.4153 & 0.329 & 0.035 & 0.029 & 0.032       & 2398 & 0.0621 & 0.333 & 0.020 & 0.014 & 0.024 \\\\     \nBulgarian & 13221 & 12119 & 0.3598 & 0.326 & 0.045 & 0.039 & 0.042   & 11947 & 0.1248 & 0.329 & 0.023 & 0.017 & 0.029 \\\\   \nFinnish & 4307 & 4078 & 0.3183 & 0.326 & 0.034 & 0.028 & 0.038       & 4011 & 0.1279 & 0.330 & 0.028 & 0.024 & 0.031 \\\\        \nHindi & 13274 & 12417 & 0.3043 & 0.332 & 0.027 & 0.025 & 0.017       & 12334 & 0.3875 & 0.330 & 0.015 & 0.012 & 0.015 \\\\   \nJapanese & 17753 & 4614 & 0.1641 & 0.326 & 0.024 & 0.019 & 0.032     & 4792 & 0.0002 & 0.333 & 0.006 & 0.000 & 0.013 \\\\    \nBasque & 11225 & 9072 & 0.1391 & 0.330 & 0.028 & 0.022 & 0.033       & 8717 & 0.1252 & 0.330 & 0.026 & 0.021 & 0.029 \\\\    \nRomanian & 4042 & 3145 & 0.1021 & 0.331 & 0.028 & 0.021 & 0.036      & 3193 & 0.0000 & 0.333 & 0.015 & 0.005 & 0.026 \\\\     \nBengali & 1129 & 678 & 0.1062 & 0.321 & 0.027 & 0.000 & 0.051        & 651 & 0.1244 & 0.320 & 0.025 & 0.000 & 0.052 \\\\      \nTurkish & 5935 & 3862 & 0.0984 & 0.330 & 0.031 & 0.025 & 0.038       & 3518 & 0.1373 & 0.327 & 0.015 & 0.000 & 0.026 \\\\\nEstonian & 1315 & 851 & 0.0376 & 0.331 & 0.016 & 0.000 & 0.037       & 843 & 0.0130 & 0.332 & 0.013 & 0.000 & 0.031 \\\\      \nTamil & 600 & 584 & 0.0240 & 0.333 & 0.026 & 0.022 & 0.025           & 585 & 0.0137 & 0.333 & 0.023 & 0.019 & 0.023 \\\\           \nTelugu & 1450 & 429 & 0.0140 & 0.322 & 0.016 & 0.000 & 0.045         & 373 & 0.0080 & 0.325 & 0.014 & 0.000 & 0.043 \\\\      \n\n\\end{tabular}\n\n\\end{ruledtabular}\n\\end{center}\n\\end{small}\n\\end{table*}\n\\end{turnpage}\n\n\n\\begin{table*}\n\\caption{\\label{relative_grouping_by_sentence_length_table} \nSummary of results for each treebank: number of distinct sentence lengths, \naverage $\\Delta_0$, and average, median and standard deviation of the average values of $\\Delta_2$ over the groups of sentences with the same length. Languages are sorted decreasingly by average \n$C_{true}$ according to Stanford dependencies as in Table \\ref{relative_table}.}\n\\begin{scriptsize}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{l|rcccc|rcccc} \n\\phantom{x} & \\multicolumn{2}{l}{\\emph{Stanford annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Prague annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} \\\\ \\hline\nTreebank & \\#Lengths & \\specialcell{$\\Delta_0$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(st. dev.)}} \n& \\#Lengths & \\specialcell{$\\Delta_0$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(st. dev.)}} \\\\ \\hline\n\nAnc. Greek & 66 & 0.293 & 0.025 & 0.024 & 0.019  & 65 & 0.292 & 0.021 & 0.021 & 0.020 \\\\   \nLatin & 59 & 0.309 & 0.031 & 0.029 & 0.018       & 59 & 0.313 & 0.031 & 0.030 & 0.017 \\\\        \nDutch & 54 & 0.319 & 0.037 & 0.035 & 0.019       & 54 & 0.323 & 0.027 & 0.024 & 0.016 \\\\        \nHungarian & 65 & 0.328 & 0.027 & 0.025 & 0.014   & 65 & 0.329 & 0.026 & 0.023 & 0.013 \\\\    \nArabic & 109 & 0.331 & 0.014 & 0.013 & 0.006     & 109 & 0.333 & 0.010 & 0.008 & 0.005 \\\\      \nGerman & 85 & 0.328 & 0.033 & 0.032 & 0.012      & 85 & 0.329 & 0.029 & 0.027 & 0.011 \\\\       \nSlovenian & 57 & 0.326 & 0.034 & 0.032 & 0.016   & 50 & 0.329 & 0.027 & 0.024 & 0.014 \\\\    \nDanish & 66 & 0.328 & 0.031 & 0.029 & 0.013      & 66 & 0.332 & 0.019 & 0.017 & 0.010 \\\\       \nGreek & 75 & 0.331 & 0.027 & 0.024 & 0.010       & 74 & 0.333 & 0.021 & 0.019 & 0.008 \\\\        \nCatalan & 98 & 0.332 & 0.023 & 0.021 & 0.008     & 98 & 0.333 & 0.014 & 0.012 & 0.006 \\\\      \nPortuguese & 88 & 0.331 & 0.024 & 0.023 & 0.009  & 88 & 0.332 & 0.013 & 0.012 & 0.006 \\\\   \nSpanish & 95 & 0.332 & 0.023 & 0.022 & 0.009     & 95 & 0.333 & 0.014 & 0.013 & 0.006 \\\\      \nPersian & 93 & 0.329 & 0.023 & 0.021 & 0.009     & 93 & 0.331 & 0.022 & 0.021 & 0.009 \\\\     \nCzech & 88 & 0.330 & 0.024 & 0.022 & 0.010       & 87 & 0.331 & 0.019 & 0.017 & 0.009 \\\\        \nEnglish & 74 & 0.331 & 0.033 & 0.031 & 0.013     & 75 & 0.333 & 0.023 & 0.021 & 0.010 \\\\      \nSwedish & 74 & 0.329 & 0.028 & 0.026 & 0.011     & 73 & 0.331 & 0.022 & 0.021 & 0.010 \\\\      \nSlovak & 92 & 0.330 & 0.024 & 0.022 & 0.010      & 87 & 0.331 & 0.020 & 0.018 & 0.010 \\\\       \nRussian & 80 & 0.330 & 0.024 & 0.022 & 0.010     & 80 & 0.332 & 0.017 & 0.016 & 0.008 \\\\      \nItalian & 69 & 0.331 & 0.024 & 0.022 & 0.010     & 68 & 0.333 & 0.014 & 0.012 & 0.008 \\\\      \nBulgarian & 64 & 0.330 & 0.029 & 0.027 & 0.012   & 63 & 0.332 & 0.016 & 0.014 & 0.009 \\\\    \nHindi & 69 & 0.332 & 0.020 & 0.018 & 0.008       & 69 & 0.331 & 0.012 & 0.010 & 0.007 \\\\        \nJapanese & 44 & 0.330 & 0.021 & 0.020 & 0.010    & 44 & 0.333 & 0.008 & 0.006 & 0.007 \\\\     \nFinnish & 41 & 0.329 & 0.028 & 0.025 & 0.016     & 41 & 0.331 & 0.024 & 0.021 & 0.013 \\\\      \nBasque & 35 & 0.331 & 0.026 & 0.022 & 0.017      & 35 & 0.331 & 0.024 & 0.021 & 0.015 \\\\       \nRomanian & 46 & 0.332 & 0.023 & 0.021 & 0.011    & 46 & 0.333 & 0.012 & 0.010 & 0.008 \\\\     \nBengali & 18 & 0.322 & 0.034 & 0.028 & 0.026     & 17 & 0.321 & 0.034 & 0.027 & 0.030 \\\\      \nTurkish & 51 & 0.332 & 0.030 & 0.027 & 0.013     & 49 & 0.331 & 0.015 & 0.013 & 0.010 \\\\     \nEstonian & 25 & 0.331 & 0.036 & 0.032 & 0.020    & 25 & 0.332 & 0.032 & 0.028 & 0.019 \\\\      \nTamil & 40 & 0.333 & 0.023 & 0.020 & 0.011       & 40 & 0.333 & 0.018 & 0.016 & 0.010 \\\\        \nTelugu & 10 & 0.330 & 0.043 & 0.030 & 0.030      & 10 & 0.331 & 0.037 & 0.029 & 0.033 \\\\       \n\\end{tabular}\n\\end{ruledtabular}\n\\end{center}\n\\end{scriptsize}\n\\end{table*}\n\n\\section{Discussion}\n\n\\label{discussion_section}\n\nWe have shown that $E_2[C]$ predicts $C_{true}$ with small error, much better than the baseline. \nThe positive results are not surprising given the previous success of $E_2[C]$ predicting crossings on uniformly random trees, where $\\Delta_2$ is about $5\\%$, i.e. about 6 times smaller than the baseline $\\Delta_0$, for sufficiently long sentences \\cite{Ferrer2014c}.  \nIt is also worth noting that $E_2[C]$ behaves well even in the treebanks with the lowest proportion of crossings, where one could argue that grammar would impose the heaviest constraints against crossings. For example, it achieves a particularly low relative error in the Romanian and Japanese Prague treebanks although they contain no or almost no crossings (Table \\ref{relative_table}).\n\n\nFrom a linguistic standpoint (recall Section \\ref{methods_section}), notice that $E_2[C]$ does not achieve its worst performance in languages known for their high word order freedom such as Ancient Greek and Latin (which are also the ones with the highest number of crossings according to Table \\ref{relative_table}) based on Stanford dependencies; however, its relative performance worsens for these languages when Prague dependencies are employed. \nIn Table \\ref{relative_table}, the average $\\Delta_2$ with Stanford dependencies indicates that $E_2[C]$ is able to make its best predictions in Estonian and Telugu, two agglutinating languages, with other agglutinating languages like Japanese or Tamil also showing better predictions than average. \nHowever, this may be an effect of the shorter sentences observed in these languages (Tables 5 and 6 of \\cite{Ferrer2017a}) and the tendency of the errors of the predictor to be smaller in sufficiently short sentences (Figure \\ref{relative2_figure}).\n\nIf we instead look at the table obtained by grouping by sentence lengths (Table \\ref{relative_grouping_by_sentence_length_table}), we observe that the predictor is remarkably robust across very dissimilar language types and families. As a representative example, if we focus on Stanford dependencies, the best prediction (average $\\Delta_2 = 0.014$) is obtained for Arabic: an Afro-Asiatic, non-agglutinating language whose treebank contains long sentences with a relatively high number of crossings; while the third best (average $\\Delta_2 = 0.021$) corresponds to Japanese: a Japonic, agglutinating language with short sentences and little observed crossings. The situation is very similar with Prague annotations, with Japanese exhibiting the best prediction, and Arabic the second best.\nThese simple and partial linguistic analyses are just reported to illustrate the potential of future linguistic research that explores in more depth the relationship between language traits and annotation criteria on the one hand, and crossings and predictions on the other.\n\nIt could be argued that the good predictions of $E_2[C]$ are not surprising at all \nbecause the syntactic dependency structures that we have analyzed could be the result of some sophisticated apparatus: a complex language faculty or external grammatical knowledge which could have produced, indirectly, a distribution of dependency lengths and vertex degrees that is favorable for $E_2[C]$. \nThen the input with which the predictor yields good predictions, e.g., dependency lengths, would be an indirect result of that complex device.\nHowever, $E_2[C]$ does not require such a device: $E_2[C]$ also makes accurate predictions on uniformly random trees with a small number of crossings \\cite{Ferrer2014c}. Therefore, the need of external grammatical knowledge to explain the origins of non-crossing dependencies is seriously challenged. \n\nThe high precision of $E_2[C]$ suggests that the actual number of crossings in sentences might be a side effect of the dependency lengths, which are in turn constrained by a general principle of dependency length minimization (see \\cite{Ferrer2013e,Liu2017} for a review of the empirical and theoretical backup of that principle).\nA ban on crossings by grammar (e.g., \\cite{hudson07,Tanaka97}),\na principle of minimization of crossings \\cite{Liu2008a} or a competence-plus \\cite{Hurford2012_Chapter3} limiting the number of crossings, may not be necessary to explain the low frequency of crossings in world languages. \n\n\nIn spite of the arguments in favor of a model predicting crossings based on dependency lengths reviewed and expanded in this article, other factors must be considered. First, chunks, i.e. subsequences of words that work as a unit, could also contribute to explain the scarcity of crossings: the number of crossings has been shown to reduce when chunks are sufficiently small in computer experiments \\cite{Lu2016a}. \nSecond, it looks difficult to rule out \nsome principle of minimization of crossings or planarity constraint. The reason is the positive correlation between crossings and dependency lengths that has been unveiled by this article and previous research combining both theory and experiment (see \\cite{Ferrer2014c}, \\cite{Ferrer2014f}, \\cite{Ferrer2015c} and references therein). The question is: what is the causal force for the scarcity of crossings: (a) a principle of minimization of crossings that explains why dependency lengths are short or (b) a principle of dependency length minimization that explains the scarcity of crossings? \\cite{Ferrer2014f}. A temporary solution to this dilemma is straightforward if we are seriously concerned about the construction of a general theory of language that is not only highly predictive but also parsimonious: a theory of language based on (b) is more parsimonious than one based on (a) \\cite{Ferrer2014f}.\n\nFrom a higher perspective, dependency length minimization follows from \nthe now-or-never bottleneck, a fundamental constraint on language processing \\cite{Christiansen2015a}, and then the scarcity of crossings could be a further prediction of such a fundamental constraint. The latter would imply that the now-or-never bottleneck and the theory of spatial\/geographical networks \\cite{Reuven2010a_Chapter8,Gastner2006b,Ferrer2004b,Chung1984} are the key for the development of a parsimonious theory of language.\n\nDespite the focus of this article on language, the article is relevant for research on other spatial networks. As researchers on dependency networks have been assuming that syntactic dependency trees tend to be planar or should be planar \\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07,Ninio2017}, research on infrastructure  networks, e.g. road networks, has been assuming that road networks are planar (see \\cite{Viana2013a} and references therein) while indeed crossings in road networks cannot be neglected \\cite{Eppstein2008a}. In the domain of infraestructure networks, we could borrow questions that have been formulated for syntactic dependency networks: is the number of crossings actually small? \\cite{Ferrer2017a}. Can the number of crossings of real infraestructure networks be explained as a result of pressure to reduce crossings directly or indirectly as a result of some principle of dependency length minimization? (this article). These are questions that may not have the same answer as in syntactic dependency trees and that could be illuminated with extensions or generalizations of the theoretical framework reviewed in this article for the two dimensional continuous case. We hope that our work stimulates further research in the field of spatial networks.\n\n\n\\begin{acknowledgments}\n\nWe thank Morten Christiansen for helpful discussions, Wolfgang Maier for comments on an earlier version of this manuscript, and Dan Zeman for help with data conversion.\n\nRFC is funded by the grants 2014SGR 890 (MACDA) from AGAUR (Generalitat de Catalunya) and also\nthe APCOM project (TIN2014-57226-P) from MINECO (Ministerio de Economia y Competitividad).\nCGR has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714150 - FASTPARSE) and from the TELEPARES-UDC project (FFI2014-51978-C2-2-R) from MINECO.\n\n\\end{acknowledgments}\n\n\n\n\n\\newcommand{\\beeksort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\\subsection{X-ray Binaries}\nX-ray binaries are among the brightest X-ray sources in the sky. An \nX-ray binary consists of a compact object (neutron star or black hole) \nand a normal companion star, orbiting around each other. At certain \nstage of the binary evolution, the separation of the two stars becomes \nso close that material from the upper atmosphere of the companion star \nbegins to flow toward the compact object under the influence of the \nlatter's gravity. This process is known\nas mass accretion due to ``Roche-lobe overflow''. The accreted \nmatter, carrying large amount of angular momentum from the orbital\nmotion, circulates around the compact object and forms an accretion \ndisk. As the matter spirals in toward the compact object, due to \nangular momentum losses from viscous processes, its gravitational \nenergy is converted into heat. Depending upon the mass accretion rate,\nthe temperature of the inner accretion disk can reach more than one \nmillion degrees so that X-rays are produced. Therefore, the X-ray \nobservation of such \nsources provides a valuable tool to probe regions very close \nto the compact object, where relativistic effects are strong thus \nimportant. It should be noted, however, that Roche-lobe overflow \ndoes not always occur. For some sources (especially \nthose with a massive companion star), the accretion process may simply \ninvolve the capture of stellar wind from the companion star by the \ncompact object. A small accretion disk can still form in such ``wind-fed'' \nsystems, due to any residual angular momentum of the captured matter. \n\n\\subsection{Black Hole Binaries}\nIn some cases, the companion star is visible optically. By carefully\nmonitoring its orbital motion, we can derive the so-called mass function, \nwhich is defined as\n\\begin{equation}\nf_c(M) \\equiv \\frac{M_x^3 \\sin^3 i}{(M_x + M_c)^2} = \\frac{P_{orb} K_c^3}{2\\pi\nG},\n\\label{eq:mf}\n\\end{equation}\nwhere $M_x$ and $M_c$ are the mass of the compact object and its\ncompanion star respectively, $i$ is the inclination angle of the\nbinary orbit with respect to the line-of-sight (with $i=0$ for face-on\nsystems), $P_{orb}$ is the binary orbital period, and $K_c$ is the\nsemi-amplitude of the optical radial velocity curve. Both $M_x\/M_c$ \nand $i$ can be further constrained by modeling ellipsoidal variation \nin the optical light curves~\\cite{av78}, and $M_c$ by the spectral \ntype of the companion star. For roughly a dozen X-ray binaries, \n$M_x$ is estimated to be greater than $\\sim 3 M_{\\odot}$, which is \nconsidered to be a reliable upper limit to the mass of a neutron\nstar~\\cite{rr74}, for any values of $M_c$ and $i$ within their\nrespective allowed parameter spaces. These sources are, therefore,\ngood candidates for black hole binaries (BHBs). Table~\\ref{tab:bh} \nlists all such systems known to date. It is clear from Eq.~\\ref{eq:mf} \nthat without making any assumptions the mass function sets a firm \nlower limit to the mass of the compact object. Therefore, those with \nthe measured mass function greater than $3 M_{\\odot}$ are the best \nBHB candidates (BHBCs).\n\\begin{table}[btp]\n\\caption{Stellar-Mass Black Hole Candidates$^{\\dag}$}\n\\vspace{0.4cm}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{|l|cccc|}\n\\hline\nSources & $M_x$ & $M_c$ & $i$ & $f_c(M)$ \\\\ \n & ($M_{\\odot}$) & ($M_{\\odot}$) & (degrees) & ($M_{\\odot}$) \\\\ \\hline \nCyg X-1~\\cite{her95} & 4.7--14.7 & 11.7--19.2 & 27--67& 0.25 \\\\ \nLMC X-3~\\cite{cow83} & 7--14 & 4--8 & 50--70 & 2.3 \\\\ \nLMC X-1~\\cite{hut87} & 4--8 & 17-24 & 40--63 & 0.14 \\\\ \nGS 2023+338~\\cite{sha94} & 10--15 & 0.5--1.0 & 52--60 & 6.26 \\\\ \nA 0620-00~\\cite{snc94} & 5.1--17.1 & 0.2--0.7 & 31--54 & 3.18 \\\\ \nGS 1124-68~\\cite{snc97} & 4--11 & 0.5--0.8 & 39--74 & 3.1 \\\\ \nGS 2000+25~\\cite{bee96} & 4.8--14.4 & $<$ 0.7 & 43--85 & 4.97 \\\\ \nGRO J1655-40~\\cite{ob97} & 6.8--7.2 & 2.2--2.5 & 69.5 & 3.24 \\\\ \nH 1705-250~\\cite{mar95,har97} & 4.9--7.9 & 0.07--0.42 & 48--51 &4.65 \\\\ \nGRO J0422+32~\\cite{bee97} & $>$ 9 & 0.2--0.4 & 13--31 & 1.21 \\\\ \n4U 1543-47~\\cite{oro98} & 2.7--7.5 & 2.3--2.6 & 24--36 & 0.22 \\\\ \\hline\n\\multicolumn{5}{l}{$^{\\dag}$In case of multiple measurements, the\nnumbers are taken from the} \\\\\n\\multicolumn{5}{l}{most recent papers. See references in these papers \nfor proper credits} \\\\\n\\multicolumn{5}{l}{on individual results.}\n\\end{tabular}\n\\end{center}\n\\label{tab:bh}\n\\end{table}\n\n\\subsection{Spin of Black Hole}\nBHBs are thought to be formed by the core collapse of very massive\nstars which are usually fast rotators. The conservation of stellar \nangular momentum would lead to extremely rotating black holes, if \nthe progenitor stars behave like rigid bodies. In reality, however, \nthe stellar core and envelop are likely decoupled, with the ejected \nenvelop carrying away significant amount of angular momentum. There \nmay also be other processes that can cause additional loss of angular \nmomentum during the formation. Therefore, while we expect that all\nblack holes rotate to some extent, it is not clear whether any can \nbe formed rapidly spinning.\n\nIn a binary configuration, the process of mass accretion can, in \nprinciple, result in the \naccretion of angular momentum by the black hole and, thus, the hole \ncan be subsequently spun up~\\cite{ba70,th74}. However, most \nBHBs are transient sources (see Table~\\ref{tab:bh}, except for the \ntop three sources) with a very low duty cycle of X-ray outbursts \n(when the mass accretion rate surges by orders of magnitude), \nso the lifetime-averaged accretion rate is extremely low. Consequently, \nthe accretion-induced spin-up of black holes is probably not effective \nin these sources (unlike in active galactic nuclei~\\cite{ba70}). Other \ntheoretical considerations, such as the presence of sub-Keplerian \naccretion flows in the vicinity of black holes~\\cite{ny95,cht95}, would \nonly make the mechanism less effective. \n\nRecently, evidence has been found for the presence of rapidly spinning \nblack holes in GRO J1655-40 and GRS 1915+105~\\cite{zcc97}, the only\ntwo Galactic sources that occasionally display spectacular radio jets \nwith superluminal motion, analogous to radio-loud quasars, which is\nwhy they are sometimes referred to as ``microquasars''. Also\ndiscovered are slowly or moderately spinning black holes in several\n``normal'' BHBCs. Not only has the study raised questions \nabout the formation of rapidly rotating black holes, it has also\nbrought into new focus possible\nobservational consequences of black hole rotation in BHBs. Subsequently, \na lot of excitement has been generated about the prospect of using\nBHBs to probe relativistic effects in strong gravity \nregime~\\cite{no97,zcc97,czc98}.\n\n\\section{GRO J1655-40: A Test Case}\n\\subsection{Existence of A Spinning Black Hole?}\n\\label{ssec:spin}\nThe binary parameters are known to such a high accuracy for GRO\nJ1655-40 (with uncertainties typically only a few percent; see \nTable~\\ref{tab:bh}) that makes the interpretation of the X-ray data\nfor this source much more reliable than for any other sources. Despite\nbeing a microquasar, GRO J1655-40 appears as a normal BHBC in terms of\nthe observed spectral properties. The X-ray spectrum can be well\ndescribed by an ultra-soft component at low energies and a power law \nat high energies~\\cite{zh97}, which is canonical for\nBHBCs~\\cite{tl95}. The ultra-soft component is generally attributed to \nthe emission from the hot inner region of an accretion disk, while \nthe power-law tail is thought to be the product of soft photons being \nCompton upscattered by energetic electrons in the region.\n\nIn practice, the ultra-soft component is often modeled by a\nmulti-color blackbody~\\cite{mi84}. In this model, the accretion disk \nis assumed to be geometrically thin, optically thick (i.e., the \nstandard $\\alpha$ disk~\\cite{ss73,nt73}). The emission from such \na disk is a blackbody, but the temperature of the blackbody is a\nfunction of the distance to the black hole. There are only two free \nparameters in the model: the radial distance of the inner disk edge \nto the black hole and the effective temperature at the inner edge, \nonce the distance of the source and the inclination angle of the disk \n(which is usually assumed to be equal to the inclination of the binary \norbit) are known. The model has worked remarkably well for a number of \nBHBCs (see a review by Tanaka and Lewin~\\cite{tl95}). One of the most \ninteresting results is that the inferred radius of the inner disk edge \nremains fairly stable and constant as the observed fluxes vary \ngreatly~\\cite{tl95}, strongly implying that the last (marginally) \nstable orbit is reached.\n\nSince the model assumes Newtonian gravity, appropriate corrections\nneed to be made to include relativistic\neffects~\\cite{pt74,ha89}. Also, gravitational shift and \ngravitational focusing cause both the observed color temperature \nand integrated flux to deviate from the local values, depending on \nthe inclination angle of the disk and the spin of the black \nhole~\\cite{cu75}. Furthermore, in the hot inner region of an \naccretion disk, electron scattering is likely to dominate over \nfree-free absorption, so the inner disk actually radiates \napproximately as a ``diluted'' blackbody with the peak effective \ntemperature lower than that derived from fitting the\nspectrum~\\cite{ehs84}. Unfortunately, this ``color correction factor'' \nis still poorly determined, although it appears to depend only weakly \non the properties of the black hole, mass accretion rate, or the \nlocation on the disk~\\cite{st95}. \n\nTaking into account these factors, we found that for several BHBCs the\ninferred radius of the inner disk edge is consistent with that of the \nlast stable orbit around non-rotating black holes~\\cite{zcc97}. When \nwe applied the same model to the spectrum of GRO J1655-40 during an \noutburst, however, we found that the inner edge of the disk is only\nabout 1.2 Schwarzschild radii away from the black hole~\\cite{zcc97}, \nwhich is, of course, impossible if the black hole does not rotate. \nTherefore, we were forced to conclude that GRO J1655-40 contains a \n{\\it rapidly} spinning black hole. The spin of the black hole was \ndetermined to be $\\sim$93\\% maximal rotation for this source.\n\nThe ``ordinary'' nature of GRO J1655-40 as a BHBC (in terms of its \nspectral properties) is strongly supported by \nrecent observations with the {\\it Rossi X-ray Timing Explorer}\n(RXTE). The source was monitored by RXTE extensively throughout its \nrecent (1996--1997) X-ray outburst. From these observations, Sobczak \net al.~\\cite{sob98} found that the observed X-ray spectrum can be \nwell characterized by the canonical spectral shape of BHBCs (soft \nmulti-color blackbody plus hard power law) throughout the entire\nperiod, which confirms the previous findings~\\cite{zh97} (although \nthe actual values of the parameters differ slightly). In addition, \nthey found that the inferred radius of the inner disk edge remained \nroughly constant (with occasional low points) as the X-ray flux varied \nby a factor of 3--4, similar to normal BHBCs~\\cite{tl95}. \n\nHowever, these authors mistakenly concluded that the corrections for\nrelativistic effects are negligible for this source (actually they\nmis-quoted Zhang et al.~\\cite{zcc97} on this issue). As a matter of\nfact, the general relativistic effects are very significant in this \ncase~\\cite{zcc97}. In particular, the inner boundary condition (i.e.,\ntorque free at the last stable orbit) can drastically change the \ntemperature profile of the inner portion of the accretion disk under \nNewtonian gravity~\\cite{pt74,ha89}. For GRO J1655-40, the\ncorrection amounts to a factor of $\\sim$2, after taking into account \nall the effects. Consequently, the radius of the inner disk edge in\nSobczak et al. was over-estimated by the same amount~\\cite{sob98}. \n\nOur interpretation of the same data would, therefore, put the last \nstable orbit roughly at 1.5 Schwarzschild radii, which confirms the \nneed for the presence of a rapidly rotating black hole in this system \n(although the spin of the black hole is $\\sim$78\\% maximal rotation \nin this case). We take note of occasional low values of the inferred \nradius when the measured X-ray luminosity is relatively\nhigh~\\cite{sob98}. We interpret them as the times when the adopted \nspectral model breaks down. It is likely that when accretion rate is \nhigh the inner portion of a standard $\\alpha$ disk may become unstable \nto radiation pressure and fragmentation~\\cite{le74,kro98}. During the \noutburst, GRO J1655-40 might be situated close to the critical point \nfor such instability, as in the case of Cyg X-1 (A. Zdziarski, private\ncommunication). \n\nIn contrast, Sobczak et al. proposed that the last stable orbit was\nreached by the accretion disk {\\it only} at the {\\it smallest} radius \n(i.e., the lowest point) obtained from their spectral analysis. To \nexplain why the inner edge of the disk stays far away from the last \nstable orbit during nearly the entire outburst, they invoked strong\nradiation drag. Intuitively, however, the radiation drag should be\nstronger in the higher luminosity state (when the ``low points'' \noccur), which would be opposite to their line of arguments. Moreover, \nit seems unlikely that the radiation drag is capable of significantly \naffecting gas dynamics in an {\\it optically thick} disk (which is \nthe most fundamental assumption in the model), since any external\nradiation fields (if present) can hardly penetrate the surface of the \ndisk and the internal emission percolates through the surface\npreferentially in the direction perpendicular to the thin\ndisk. Finally, from a\npractical point of view, it seems risky to base the entire argument on \na single data point which is barely allowed physically~\\cite{sob98}, \nespecially considering the fact that not all the ``low points'' reach \nthe same level.\n\nIn summary, while further studies are required to reveal important \ndetails about accretion processes in BHBCs, it seems highly plausible, \nas suggested by \nobservations~\\cite{tl95}, that during an X-ray outburst the accretion \ndisk is of the standard form~\\cite{ss73,nt73} in transient BHBCs \nand the inner disk edge reaches and remains at the last stable orbit. \nGRO J1655-40 is no exception. Quantitative modeling of its X-ray \nspectrum has led to the conclusion that a rapidly rotating black hole \nis likely to be present in this source. \n\n\\subsection{Gravitomagnetic Precession}\nA natural consequence of curved spacetime in the vicinity of a spinning \nblack hole is the gravitomagnetic precession of orbits that are tilted \nwith respect to the equatorial plane perpendicular to the spin\naxis~\\cite{lt18}. For circular orbits, which are probably most\nrelevant to accretion processes in X-ray binaries, the precession\nfrequency can be expressed as \n$\\nu_{FD} = \\nu_{\\theta} \\Delta \\Omega\/2\\pi$, \nwhere $\\nu_{\\theta}$ is the orbital frequency of the polar (or $\\theta$)\nmotion, and $\\Delta \\Omega$ is the angle by which the line of nodes of a\ncircular orbit are dragged during each orbital period. Note that this\ndefinition is slightly different form the one adopted by Cui et \nal.~\\cite{czc98}. As correctly pointed out by Merloni et al.~\\cite{mer98}, \nwith the new definition, $\\nu_{FD}$ approaches the rotation frequency \nof the black hole at the event horizon, which is physically expected. \nFig.~\\ref{fig:fd} shows $\\nu_{FD}$ as a function of black hole spin. \nComparing it with Fig. 1 of Cui et al.~\\cite{czc98}, we find that, as \nexpected, the difference becomes significant only in the case of \nextremely spinning black holes when the last stable orbit approaches \nthe event horizon. Therefore, the results of Cui et al.~\\cite{czc98}\nhardly change, even for microquasars, neither do any of the\nconclusions that they reached (see also discussion in the following \nsection). \n\n\\subsection{``Stable'' QPOs}\nQuasi-periodic oscillations (QPOs) have been observed in the X-ray light\ncurves of BHBCs over a wide frequency range, from mHz to roughly a few \nhundred Hz (reviews by van der Klis~\\cite{van95} and Cui~\\cite{cui98}). \nWhile it is clear that no single process can account for all QPOs, the \norigins of QPOs are not known. \n\\begin{figure}[t]\n\\centerline{\\epsfig{figure=freq.ps,width=3.0in}}\n\\caption{Gravitomagnetic precession frequency (multiplied by black\nhole mass), as a function of the distance from the last stable orbit, \nfor different black hole spin. The weak-field limits are also shown \nin dash-dotted line for comparison. }\n\\label{fig:fd}\n\\end{figure}\n\nOne type of QPOs in BHBCs is of particular interest here. Such QPO was \nfirst reliably established in GRS 1915+105 with the detection of a \n67 Hz QPO~\\cite{mrg97}. One unique characteristic of this QPO \nis the stability of its frequency against any variation in the X-ray flux. \nThe QPO appears to represent a transient event, and seems to be\npresent only in certain spectral states. Shortly after this discovery, \na similar \nsignal was detected at $\\sim$300 Hz for GRO J1655-40~\\cite{rem98},\n{\\it only} when the X-ray luminosity was relatively high~\\cite{sob98}\n(i.e., at the ``low points'', as discussed in \\S~\\ref{ssec:spin}). \nWe subsequently proposed that these ``stable'' QPOs are perhaps the\nobservational manifestation of gravitomagnetic precession of accreted,\norbiting matter around spinning black holes~\\cite{czc98}.\n\nTo facilitate comparison with observations, we have computed the\nexpected gravitomagnetic precession frequency of a disk annulus at\nwhich the effective disk temperature peaks. The results are plotted in\nFig.~\\ref{fig:fdspin}, as a function of black hole spin for three\nvalues of black hole mass.\nComparing this figure to Fig. 2 of Cui et al.~\\cite{czc98}, we find,\nagain as expected, significant difference exists only for large (and\npositive) black hole spin. Note that the results depend little on the\nvalue of Q~\\cite{czc98,mer98}, which is a constant of motion \nthat specifies the tilt of precessing orbits~\\cite{wil72}.\n\nFor GRO J1655-40 (where $M_{bh} = 7 M_{\\odot}$), the spin of the black \nhole would be about 97\\% maximal rotation were the observed 300\nHz QPO to be attributed to the gravitomagnetic precession (as shown \nin Fig.~\\ref{fig:fdspin}). This result assumes that the QPO originates \nin the modulation of the X-ray emission from the disk, which does not\nnecessarily have to be the case, as will be discussed in\n\\S~\\ref{ssec:xmm}. To\nderive a lower limit to the black hole spin, we computed the frequency\nof gravitomagnetic precession right at the last stable orbit, as a\nfunction of the spin, and set the result equal to the QPO\nfrequency. We found that in this case the spin would be $\\sim$87\\% \nmaximal rotation. The results are consistent with those obtained \nspectroscopically (see also Fig.~\\ref{fig:ms}). \n\n\\section{Application to Other Sources}\n\\label{sec:aos}\nAttempts have been made to apply the model to QPOs observed in ``normal''\nBHBCs~\\cite{czc98}. The closest examples are perhaps \nthe so-called ``very-high-state'' QPOs that were observed only twice,  \nin GX 339-4~\\cite{miy91} ($\\sim$6 Hz) and GS 1124-68~\\cite{bel97} \n(5--8 Hz), in a high luminosity state (i.e., the very-high state).\nThe frequency of such QPOs is fairly stable against flux variation, \nand any change in the QPO frequency can be explain by the variation \nin radiation pressure~\\cite{czc98}. The low values of the QPO\nfrequency would imply the presence of only slowly rotating black holes \nin these sources were these QPOs to be produced by the gravitomagnetic \nprecession of accreted matter. This would in turn support the\nspeculation that the relativistic jets observed in microquasars might\nsomehow related to the spin of black holes~\\cite{zcc97}.\n\\begin{figure}[t]\n\\centerline{\\epsfig{figure=fvsa.ps,width=3.0in}}\n\\caption{Gravitomagnetic precession frequency of a disk annulus, at\nwhich the effective disk temperature peaks, as a function of black \nhole spin (assuming $Q=1$). }\n\\label{fig:fdspin}\n\\end{figure}\n\nSimilar QPOs have also been observed in GS 1124-68~\\cite{bel97} and Cyg\nX-1~\\cite{cui97} during the transition between the low and high state.\nHowever, in these cases, the situation is much more complicated and,\nthus, the interpretation is much more uncertain, because of the\nvariable nature of the sources and the transient nature of the QPO\nphenomenon in general~\\cite{cui98}. For a particular source, a \ndifferent set of QPOs are often present at different times (or\nfluxes). It is, therefore, difficult to associate one QPO (if any)\nwith the gravitomagnetic precession of accreted matter. For instance, \na lower-frequency QPO (6--8 Hz) was often present in GRO J1655-40 \nduring the same outburst when the 300 Hz QPO was\ndiscovered~\\cite{men98}. To make further progress, an systematic \neffort is imperative to reliably characterize and classify QPOs \nin BHBCs. \n\n\\section{Unsettled Issues}\n\\label{sec:ui}\n\\subsection{Origin of Tilted Orbits} \nAccreted matter must be in a non-equatorial orbit, with respect to \nthe spin axis of the black hole, in order for the orbit to undergo \ngravitomagnetic precession. In principle, such orbits could be \nproduced during the formation of a black hole binary, due to the \nmisalignment of the spin axis of the black hole with the axis of the \nbinary motion~\\cite{hil83}. \n\nIf mass accretion proceeds through a flat viscous disk, the disk can\nstill be warped, due to irradiation by a central X-ray \nsource~\\cite{pri96}. However, in this case, differential \ngravitomagnetic precession generates twists in the disk and the inner \nregion of the disk tends to gradually realign with the equatorial \nplane due to the viscous torque~\\cite{bp75}. Once the realignment\ntakes place, gravitomagnetic precession modes are damped out and,\nthus, the QPO should simply disappear. On the other hand, this process \ncan help explain why the QPOs seem to be present {\\it only} during \ntransitional or unstable periods, when the inner region of the disk \nexperiences significant changes.\n\nAs mentioned in \\S~\\ref{ssec:spin}, the inner portion of the disk may\nbecome unstable~\\cite{le74,kro98}, when the accretion rate is\nrelatively high. Once this occurs, the disk might become fragmented. \nThe gravitomagnetic precession of the fragments is probably only \nweakly damped, so it might be capable of producing the observed QPOs.\n\n\\subsection{Excitation of Gravitomagnetic Modes}\nA recent theoretical development involves the discovery of a class of\nhigh-frequency gravitomagnetic modes that are only weakly damped, in \naddition to the strongly-damped low-frequency ones~\\cite{ml98}. These\nhigh-frequency modes are very localized spiral corrugations of the\ninner portion of the disk. They are perhaps most relevant to the\nobserved QPOs, because X-ray emission from BHBCs is\nlikely to originate very close to the central black hole. However, it\nremains to be seen whether the modes can be excited easily and whether\nthey are capable of modulating X-ray emission~\\cite{ml98}.\n\nIt should be pointed out that two critical assumptions are made in the\nstudy: small tilt angle and (pseudo) Newtonian potential. The former \nsimplifies the equation by ignoring the non-linear aspect of the\nproblem, which might affect the damping time scales of gravitomagnetic \nmodes in a significant way; non-linear calculations are required to \nshed light on this issue. The latter might still represent a good \napproximation for systems that contain a relatively slowly rotating \nneutron star, but is certainly invalid for rapidly rotating black\nholes in which the stable QPOs are observed. These assumptions,\ntherefore, tend to limit the scope of the applicability of the\nresults. At present, it would seem premature to draw any definitive \nconclusions about confirming or rejecting the gravitomagnetic origin \nof the stable QPOs in microquasars.\n\n\\subsection{X-ray Modulation Mechanisms}\n\\label{ssec:xmm} \nThe gravitomagnetic precession of accreted matter only provides a \nnatural frequency for the QPOs. To actually see them, certain physical \nprocesses are required to produce X-ray modulations. The processes \nare entirely unknown at present. Possibilities include (by no means\nexclusively): (1) variation in the X-ray emitting area of the \ndisk, due to precession; (2) Doppler or gravitational shift of photon \nenergy, due to the Keplerian motion of self-illuminating clumps or \n``hot spots'' in the accretion disk; (3) oscillation that modulates \nthe emission from the disk; and (4) occultation of a highly compact \nX-ray emitting region by disk fragments or the inner part of the \ndisk itself.\n\nThe transient nature of the QPOs seems to imply that the\nfirst possibility, if viable at all, must be only a small effect. Any\nprocesses that involve discrete clumps would unavoidably also produce \nX-ray modulation at the orbital (both polar and azimuthal)\nfrequencies~\\cite{ms98,mer98}. These QPOs have not been seen \nin BHBCs, despite the high quality of RXTE data (the QPOs of comparable \nstrength have been detected at around kHz, using the RXTE data, in \nmore than a dozen neutron star systems~\\cite{van97}). Therefore, such \nprocesses also seem unlikely to be\nresponsible for the observed QPOs. Hot spots or oscillations that\nmodulate the emission from the accretion disk can also modulate\nComptonized hard X-ray emission, if the seed photons originate in \nthe disk emission. However, it seems difficult for these processes \nto amplify the fractional amplitude of the QPOs toward high photon \nenergies, which is observed~\\cite{mrg97,rem98}. Such energy\ndependence of the QPO amplitude might, in our view, hold the key to\nour understanding of the origin of the QPOs.\n\nThe only remaining possibility in the above list is the occultation of\na highly compact hard X-ray emitting region by detached disk annuli \nor the inner disk itself. Interestingly, both\nobservations~\\cite{gro98} and theories~\\cite{na97,lt98} support the \npresence of such X-ray emitting \nregions in transient BHBCs during an X-ray outburst. As an\nexample, Laurent \\& Titarchuk~\\cite{lt98} recently showed that the \nobserved power-law type hard X-ray spectrum of BHBCs may be the result \nof soft photons from the accretion disk being inverse-Compton\nscattered by the bulk motion of relativistic electrons in the vicinity\nof central black holes. The Comptonizing electrons are spatially confined \nalmost entirely within the inner edge of the disk. In the context of \nthis model, harder X-rays should, on average, originate closer to the \nblack hole. Therefore, a precessing ring of matter occults a larger \nfraction of the harder X-ray emitting region, which might explain the \nobserved energy dependence of the QPO amplitude, although a definitive \nanswer still awaits detailed calculations based on this model.\n\n\\subsection{Alternative Models} \nThe stable QPOs in GRS 1915+105 and GRO J1655-40 can conceivably be \nassociated with the Keplarian motion of self-illuminating clumps \nright at the last stable orbit around the black \nhole~\\cite{mrg97,rem98}. In the case of GRO J1655-40 whose\nblack hole mass is accurately known, the model would be able to\nexplain the observed QPO frequency (300 Hz) only if the black hole \ndoes not rotate~\\cite{rem98}. This is clearly incompatible with the\nrequirement for the presence of a rapidly rotating black hole in this\nsource from modeling the observed X-ray spectrum. A more serious (and\nmodel independent) problem with this interpretation is due to the \nfact that general relativistic effects cause the temperature of the \naccretion disk to vanish at the last stable\norbit~\\cite{pt74,ha89}. Consequently, no X-ray emission is expected \nfrom the inner edge of the disk.\n\nAnother proposal has been made to associate the stable QPOs to \nepicyclic oscillation modes in the accretion disk~\\cite{now97}. Not \nonly can such modes be supported by the disk, they also become trapped \nin the innermost portion of the disk, purely due to relativistic\neffects~\\cite{kf80,nw92,nw93,ct95,per97}. It is also \nrealized that g-mode oscillations affect the largest area of a region \nin the disk where most X-rays are emitted and are, therefore, perhaps \nmost relevant to observations~\\cite{now97}. This model can explain \nthe measured frequencies of the stable QPOs in microquasars. However,\nlike other disk-based processes, it may have difficulty in explaining\nthe observed energy dependence of the QPO amplitude. Furthermore, it  \ncannot account for the very-high-state QPOs or other relatively \nlow-frequency QPOs in BHBCs (see \\S~\\ref{sec:aos}), although arguments \ncan be made that these are fundamentally different from the stable \nQPOs in microquasars.\n\nTo quantify the comparison among the three possible models\n(gravitomagnetic precession, Keplerian motion, and g-mode\noscillation), we have applied the models to both GRO J1655-40 and GRS\n1915+105. For each model, a relationship between the mass and spin of\nthe black hole is derived from the measured frequency of the QPO. The\nresults are shown in Fig.~\\ref{fig:ms}, for both sources. In the\n\\begin{figure}[th]\n\\centerline{\\epsfig{figure=summary.1655.ps,width=2.8in}}\n\\centerline{\\epsfig{figure=summary.1915.ps,width=2.8in}}\n\\caption{(top) Allowed parameter space for the mass and spin of the \nblack hole in GRO J1655-40. The solid line shows the spectroscopic\nresults of Zhang et al. (see \\S~2.1), with the \nlightly-shaded region indicating the estimated uncertainty. The \ndot-dashed, short-dashed, and long-dashed lines show, respectively, the \npredictions of three proposed models: Keplerian motion, g-mode\noscillation, and gravitomagnetic precession. For the last model,\ntwo limiting cases are shown for frequencies of gravitomagnetic \nprecession at the inner edge of the disk (upper curve) and at where \nthe effective disk temperature peaks (lower curve). The heavily-shaded \narea represents the confidence region of the measured black hole \nmass. (bottom) Similar to the top panel, but for GRS 1915+105. In \nthis case, the mass of the black hole is not known. }\n\\label{fig:ms}\n\\end{figure}\ngravitomagnetic precession model, two limiting cases are shown for \nfrequencies at the inner edge of the disk and at where the effective\ndisk temperature peaks, since we do not know the actual X-ray\nmodulation mechanism (see \\S~\\ref{ssec:xmm}). Also shown in\nthe figure is the same relationship derived from modeling the observed\nX-ray spectrum for each source~\\cite{zcc97}. The intersection between \nthe latter\nand each of the three model curves, therefore, yields a solution to\nthe mass and spin of the black hole, for a particular source, as \npredicted by that model. For GRO J1655-40, the mass of the black hole\nis known quite accurately, so it can be used to determine which models\nare compatible with the observed properties.\n\nThe uncertainty of the spectroscopic results\nis likely dominated by that of the color correction factor (up to\n$\\sim$12\\%~\\cite{st95}), which can contribute as much a uncertainty\nas $\\sim$24\\% in the derived radius of the last stable\norbit~\\cite{zcc97}. The next major source of uncertainty comes from \ndetermining the source distance, to which the radius of the last stable\norbit is directly proportional. It amounts to $\\sim$6\\% for \nGRO J1655-40~\\cite{hr95} and $\\sim$12\\% for GRS 1915+105~\\cite{mr94}. \nLikely minor contributions include uncertainties in the binary \ninclination angle, in the derived X-ray flux and the peak effective \ntemperature of the disk, and in various relativistic correction \nfactors~\\cite{zcc97}. To be relatively conservative, we have adopted \nan overall uncertainty of 40\\% on the radius of the last stable orbit\nfor both sources. The lightly-shaded area in Fig.~\\ref{fig:ms}\nreflects this measurement uncertainty.\n\nClearly, the Keplerian motion model is ruled out, since it is far \nfrom being consistent with the spectroscopic results. While the \ng-mode oscillation model seems to work for GRO J1655-40, it fails to \nyield any physical solutions for GRS 1915+105. The difference is so\nlarge that any reconciliation between the model and the data seems\nunlikely, unless the spectroscopic method has severely underestimated \nthe radius of the last stable orbit (or the method does not apply in \nthis case). On the other hand, the gravitomagnetic precession model\nseems to explain the observations well for both sources. For GRS\n1915+105, the model constrains the mass ($\\sim$3--14 $M_{\\odot}$) and \nspin ($\\sim$37--90\\% maximal rotation) of the black hole. Future\ninfrared observations of the source might be able to test the mass\nconstraint. \n\n\\section*{Acknowledgments} We would like to thank Andrea Merloni for\nsharing with us the results from his work prior to publication. W. Cui  \nis also grateful to Ron Remillard for keeping him updated of the\nresults on GRO J1655-40 and for many helpful discussions, despite \nsome difference in their interpretation of the data. This work is \nsupported in part by NASA through Contract NAS5-30612.\n\n\\section*{References}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCommunity structure \\cite{Mucha2010} is widely observed in various types of real networks such as social networks \\cite{Zachary1977}-\\cite{Chouchani2020}, biological networks \\cite{Jonsson2006}-\\cite{Cui2020}, technological networks \\cite{Newman2004a}, and so on. A community is a group of nodes with more edges connecting nodes within the group and less edges connecting nodes to different groups \\cite{Girvan2002}. Community detection is of significant importance in understanding organization principles and predicting dynamical behaviors of a network \\cite{Fortunato2010}, which has already found wide applications in interdisciplinary domains, such as to find the pathways between diseases and drugs \\cite{Pham2019}, to reveal the role of each part in a layered neural network \\cite{Watanabe2019}, to mine user opinions from social networks \\cite{Li2019}, and so on. \n\nSince the past decades, many methods of community detection have been proposed, such as graph partitioning \\cite{Shi2011}, spectral clustering \\cite{Donetti2004}-\\cite{Jin2015}, similarity expansion \\cite{Xiang2009}-\\cite{Pan2010}, statistical inference \\cite{Newman2007}, dynamic methods \\cite{Zhou2003}-\\cite{Hu2008}, and integer programming model \\cite{Srinivas2019}. Among them, greedy algorithms based on quality function optimization have attracted a lot of attention \\cite{Newman2004b}-\\cite{Shao2020}. \nIn a typical greedy algorithm, some nodes are selected as central nodes of communities (others are non-central nodes), from which the nearest neighbor search is usually adopted to expand provisional communities \\cite{Saha2016}. Obviously, the selection of central nodes are vital for the accuracy of a greedy algorithm. In an early work \\cite{Chen2013}, nodes either with global maximal degrees (GMD) or with local maximal degrees (LMD) are regarded as central nodes. The GMD takes nodes of the top-$k$ highest degrees as central nodes. However, the selection of the value of $k$ is often arbitrary. On the other hand, nodes with maximal degrees are not always central nodes \\cite{Blondel2008b,Lu2016}. In contrast, the LMD designates nodes with higher degree than all neighbors as central nodes. Empirical evidences \\cite{Chen2013} suggest that the LMD performs overall better than the GMD in identifying central nodes of provisional communities. \n\nIn this paper, we propose a novel local centrality index called the local superiority index (LSI) and combine it to the greedy algorithm to detect communities. In our proposed algorithm, the community expansion, aiming to maximize the value of the quality function, starts from central nodes selected by the LSI. After expansion, we allot residual nodes to suitable communities. Allocation of residual nodes may decrease the quality function, while it is found that the optimal community structure is not always with the largest quality function \\cite{Zhang2014}. Our algorithm is evaluated by normalized mutual information (NMI) \\cite{Kuncheva2004} on real networks with known communities and benchmark networks generated artificially \\cite{Lancichinetti2008}. \n\n\n\\section{Methods}\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.9\\textwidth]{fig1_new2_rev.eps\n    \\caption{Comparison among the three considered indices. The illustrated network is constructed by connecting two complete networks through a single edge. (a) According to the GMD, the red node (node $1$) is the top-$1$ highest-degree node, the pink nodes (nodes $2,3,4,5$) and node $1$ are the top-$5$ highest-degree nodes. (b) According to the LMD, only node $1$ is a central node, other nodes are all non-central nodes. (c) According to the LSI, nodes $1$ and $2$ are central nodes.}\n\\label{fig_Illustration}\n\\end{figure}\n\nAs shown in figure 1, we construct an illustration network by connecting two complete subnetworks through a single edge. Intuitively, this network has two communities (\\{1,3,4,5\\} and \\{2,6,7\\}) and nodes 1 and 2 should be treated as central nodes. However, neither GMD nor LMD can exactly identify those two central nodes. The top-$k$ highest-degree nodes may indeed belong to $\\textless k$ communities and thus are not all central nodes, and the LMD can't identify central nodes with relatively smaller degrees.\n\nInspired by these shortcomings, we propose the so-called LSI index. The LSI of an arbitrary node $i$ reads\n\n\\begin{equation}\nLSI_i = ({{k_i} - \\frac{1}{{{k_i}}}\\sum\\limits_{j \\in {\\Gamma _i}} {{k_j}} })\/ ({{k_i} + \\frac{1}{{{k_i}}}\\sum\\limits_{j \\in {\\Gamma _i}} {{k_j}} }),\n\\end{equation}\nwhere $k_i$ and $k_j$ represent the degree of node $i$ and node $j$, and $\\Gamma_i$ denotes the set of $i$'s neighbors. Obviously, the value of LSI lies between -1 and 1, and the larger LSI corresponds to the higher local centrality. Without loss of generality, in this paper we treat nodes with $LSI_i \\geq 0$ as central nodes for community detection, while other nodes are non-central nodes. As shown in Fig. 1(c), LSI identifies nodes $1$ and $2$ as central nodes.\n\nWe adopt the fitness function $F$ \\cite{Lancichinetti2009,Zhou2007} as the quality function. For a community $C_i$, its $F$ value is\n\n\\begin{equation}\nF(C_i) = \\frac{d_{in}(C_i)}{d_{in}(C_i) + d_{out}(C_i)}, \n\\end{equation}\nwhere $d_{in}(C_i)$ is the internal degree (i.e., the twice number of edges in $C_i$), and $d_{out}(C_i)$ is the external degree (i.e., the number of edges connecting nodes in $C_i$ with nodes outside $C_i$). The fitness function $F$ for the whole network is defined as the sum of $F(C_i)$ of all communities, as\n\\begin{equation}\nF = \\sum_{i=1}^m{F(C_i)},\n\\end{equation}\nwhere $m$ is the total number of communities.\n\n\nDetailed procedures of the greedy algorithm are described below. Firstly, we identify all central nodes with LSI values no less than zero. Then we start the community expansion procedure. Initially, we choose one central node randomly as the seed of a provisional community $C_1$. At each step, we search all unassigned nodes that are neighboring to at least one node in $C_1$. Our goal is to find one node whose addition can maximize $F(C_1)$ and add this node to $C_1$. In case more than one node can maximize $F(C_1)$, we randomly choose one of them. We repeat this step until any addition of one neighbor can't increase $F(C_1)$. If $F(C_1)\\leq 0.5$, the provisional community $C_1$ is disbanded and the current expansion becomes invalid, because the condition of weak community (i.e., $F(C_1)>0.5$) is not satisfied. At the same time, the central node is converted to an unassigned non-central node. We repeat this process until all provisional communities stop expanding. After the expansion, some nodes may still be unassigned and form one or more connected subgraphs. For each subgraph, we define its potential neighbors as all assigned nodes neighboring to at least one node of this subgraph. Then each subgraph is allotted to a provisional community who contains the potential neighbor with the highest LSI value. In case more than one provisional community satisfies such condition, we randomly choose one of them. Although the allocation of unassigned nodes may decrease the quality function, it is reasonable that the optimal community structure is not always with the largest quality function \\cite{Zhang2014}. In the above procedure, the different order in choosing the central nodes may lead to different results, so we carry out the algorithm many times and adopt the result with the highest $F$. \n\n\n\n\\begin{table} \n\\small\n\\centering  \n\\caption{Structural features of the six real networks under consideration. $N$, $L$ and $m$ represent the number of nodes, edges and communities respectively, and $\\langle k\\rangle$ represents the average degree. }\n\\begin{indented}\n\\item[]\\begin{tabular}  \n{>{\\columncolor{white}}rccccc}  \n\\hline\n\\rowcolor[gray]{0.9}   Networks &$N$ &$L$   &$m$ &$\\langle k \\rangle$ &references \\\\ \n\\hline\nkarate   &34   &78  &2  &4.5882 &\\cite{Zachary1977} \\\\  \ndolphins   &62 &159 &4   &5.129 &\\cite{Lusseau2004} \\\\   \nfootball   &115 &613    &12&10.6609 &\\cite{Girvan2002} \\\\  \npolblogs   &1222 &16714    &2&27.3552 &\\cite{Lada2005} \\\\  \npolbooks   &105 &441    &3&8.4 &\\cite{Krebs0000},\\cite{Newman0000} \\\\ \nhighschool   &248 &1004    &8    &8.0968 &\\cite{Moody2001} \\\\ \n\\hline  \n\\end{tabular} \n\\end{indented} \n\\end{table} \n\nIn this paper, we use two kinds of networks. One is the real networks  with known ground truth (i.e., recognized community structure), the other is the LFR benchmark networks generated artificially \\cite{Lancichinetti2008}. We consider six real networks, with detailed description as follows. (i) Karate network \\cite{Zachary1977}. A network of friendship in a karate club in an American university. The club split into two parts after a dispute between the coach and the treasurer. (ii) Dolphins network \\cite{Lusseau2003}. This network contains 62 dolphins living in Doubtful Sound, New Zealand. An edge exists between two dolphins if they are observed together more often than expected by chance from 1994 to 2001. This network contains two communities \\cite{Lusseau2004}. (iii) Football network \\cite{Girvan2002}. A network of American football games between Division IA colleges during regular season Fall 2000. The nodes denote the 115 teams and the edges represent 613 games played in the course of the year. Each conference form a community and games are more frequent between members of the same conferences. (iv) Polblogs network \\cite{Lada2005}. A directed network of hyperlinks between weblogs on US politics around the time of the 2004 presidential election. We convert this network to an undirected network by treating all directed links as undirected edges and reserve the largest component. Each weblog has a political leaning: blue (left or liberal) or red (right or conservative), and thus this network contains two communities. (v) Polbooks network \\cite{Krebs0000,Newman0000}. In this network, nodes represent books about US politics sold in Amazon.com, and edges connecting frequently co-purchasing book pairs \\cite{Lu2012}. Each book is assigned a political tag (liberal, neutral or conservative), and thus the network is consisted of three communities. (vi) Highschool network \\cite{Moody2001}. A directed network represents the friendship choices made by students from different grades. We also convert this network to an undirected network. Fundamental structural statistics of these six networks are shown in Table 1.\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig2_NMI_groundtruth_rev7.eps}   \n    \\caption{ The NMI values of the three indices on the six real networks with known ground truth.  }\n\\label{fig_Illustration}\n\\end{figure}\n\n\n\nLFR networks \\cite{Lancichinetti2008} are classical benchmark networks generated by computer. In an LFR network, degrees are distributed in a power law with exponent $2<\\gamma<3$, and the sizes of communities also follow a power-law distribution with exponent $1<\\beta< 2$. Besides, the community size $s$ and node degree $k$ satisfy the constraints $s_{min}>k_{min}$ and $s_{max}>k_{max}$. An important mixing parameter $\\mu$ represents the ratio between the external degree of an arbitrary node with respect to its community and the total degree of this node. With the increase of $\\mu $, the community structure of the network becomes ambiguous. \n\n\n\n\\section{Results}\n\n\\begin{comment}\nIn this section, we evaluate the performance of LSI and the greedy algorithm with the LSI on the ground truth networks and benchmark networks by the NMI \\cite{Kuncheva2004}. Given a confusion matrix $\\textbf{N}$, whose element $N_{ij}$ on the $i$th row and the $j$th column represents the number of nodes contained in both the  $i$th standard community and the $j$th community detected by the algorithm, the NMI is defined as\n\n\\begin{equation}\nNMI = \\frac{{ - 2\\sum\\limits_{i = 1}^{{C_r}} {\\sum\\limits_{j = 1}^{{C_f}} {{N_{ij}}N\/{N_{i.}}{N_{.j}}} } }}{{\\sum\\limits_{i = 1}^{{C_r}} {{N_{i.}}\\log ({N_{i.}}\/N) + \\sum\\limits_{j = 1}^{{C_f}} {{N_{.j}}\\log ({N_{.j}}\/N)} } }},\n\\end{equation}\nwhere $N$ is the number of nodes in a network. $C_r$ is the number of the standard communities. $C_f$ denotes the number of communities detected by the algorithm to be evaluated. $N_{i.}$ and $N_{.j}$ stand for the sum of all elements in the $i$th row and the $j$th column of $\\textbf{N}$ respectively. The value of the NMI is between 0 and 1. The higher its value, the more accurate identified communities. \n\\end{comment}\n\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig3_NMI_LFR_rev3.eps}  \n    \\caption{How NMI changes with varying parameters in LFR networks. The default values of the four parameters are $\\mu=0.4$, $\\gamma=2.5$, $\\beta=1.5$ and $\\langle k\\rangle=20$. In each plot, only one parameter is varying. The results are averaged over 10 networks with a fixed number of nodes $N=1000$. \n } \n\\label{fig_Illustration}\n\\end{figure}\n\nWe evaluate algorithms' performance by NMI \\cite{Kuncheva2004}: the higher the NMI is, the better the algorithm is. Figure 2 shows that LSI can obtain better community structure than LMD and GMD for the real networks except the polbooks network.  Figure 3 compares the performance of the three indices on LFR networks. It is shown that LSI remarkably outperforms GMD and LMD. \n\nFigure 4 shows the relationship between nodes' LSI values and degrees for a real network and an artificially generated network. The difference between LSI and degree is remarkable, namely LSI can identify some low-degree nodes as central nodes while reject some high-degree nodes. If one would like to correctly detect a real community, it is natural to expect that one or more nodes in this community should be first identified as central nodes. If none of nodes in this community are central nodes, the probability that this community could be perfectly detected is very tiny. We set the number of central nodes identified by GMD being equal to the number of central nodes identified by LSI, and then compare how many real communities having central nodes by the three indices. As shown in Tables 2 and 3, more real communities have at least one central node by LSI than by LMD or GMD. In particular, as shown in Table 3, for LFR benchmark networks, most communities have at least one central node according to LSI, however, more than a half communities do not have any central nodes by GMD and LMD. Indeed, LMD only identify very few central nodes and most communities do not have any central nodes accordingly. This may be the reason why LSI can outperform LMD and GMD in community detection.\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig4_LCI_degree_20220104.eps}   \n    \\caption{LSI versus degree. Each data point represents one node. Nodes with  $LSI\\geq0$ are colored by yellow, while nodes with  $LSI<0$ are colored by red. (a) The result for the polblogs network. (b) The result for the LFR network with parameters  $N=1000$, $\\langle k\\rangle=20$, $k_{max}=100$, $\\gamma=2.5$, $\\beta=1.5$ and $\\mu=0.2$, where $k_{max}$ is the maximal degree. } \n\\label{fig_Illustration}\n\\end{figure}\n\n\n\\begin{table}\n\\centering  \n\\caption{The number of real communities $m$ and the number of central nodes identified by the three indices on the six real networks. Each number in bracket is the number of real communities containing at least one central node. The number of central nodes for GMD is set to be equal to the number of central nodes identified by LSI.  }\n\\begin{indented}\n\\item[]\\begin{tabular}\n{>{\\columncolor{white}}ccccccc}  \n\\hline\n\\rowcolor[gray]{0.9}   indicators &karate &dolphins &football &polblogs &polbooks   &highshool\\\\ \n\\hline\n$m$  &2 &4  &12   &2  &3  &8   \\\\   \nLSI          &5(2)   &19(4)  &65(12)  &123(2) &24(3)  &80(7)  \\\\  \nLMD          &2(2)   &5(3)   &32(11)  &6(2)   &3(3)   &12(6)   \\\\  \nGMD          &5(2)   &19(3)  &65(12)  &123(2) &24(3)  &80(7) \\\\ \n\\hline\n\\end{tabular}\n\\end{indented}  \n\\end{table} \n\n\\begin{table}  \n\\centering  \n\\caption{The number of real communities $m$ and the number of central nodes identified by the three indices on the LFR networks. Each number in bracket is the number of real communities containing at least one central node. The number of central nodes for GMD is set to be equal to the number of central nodes identified by LSI.}\n\\begin{indented}\n\\item[]\\begin{tabular}  \n{>{\\columncolor{white}}ccccccccc}  \n\\hline\n\\rowcolor[gray]{0.9}   indicators &$\\mu=0.1$ &$\\mu=0.2$ &$\\mu=0.3$ &$\\mu=0.4$ &$\\mu=0.5$   \\\\  \n\\hline\n$m$  &31 &31  &30   &35  &32   \\\\    \nLSI          &242(29)   &200(29)  &187(25)  &176(27) &166(23)   \\\\ \nLMD          &9(8)   &6(4)   &5(3)  &5(4)   &4(4)    \\\\   \nGMD      &242(13)    &200(15)  &187(15)  &176(15) &166(15)   \\\\  \n\n\\hline\n\\end{tabular}  \n\\end{indented}\n\\end{table} \n\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig5_0211.eps}  \n    \\caption{The NMI values of the proposed greedy algorithm based on LSI and the Newman fast algorithm for the six real networks. }\n\\label{fig_IIlustration}\n\\end{figure}\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig6_0211.eps}   \n    \\caption{The NMI values of the proposed greedy algorithm based on LSI and the Newman fast algorithm for the LFR networks. The default values of the four parameters are $\\mu=0.4$, $\\gamma=2.5$, $\\beta=1.5$ and $\\langle k\\rangle=20$. In each plot, only one parameter is varying. The results are averaged over 10 networks with a fixed number of nodes $N=1000$.   }\n\\label{fig_IIlustration}\n\\end{figure}\n\n\nThe Newman fast algorithm is the most well-known greedy algorithm. It starts with every node being a single community, and then join a pair of provisional communities at each step to maximize the modularity $Q$, until all nodes agglomerated together to form one big community. The progress can be represented as a dendrogram, namely a tree that shows the order of all joins. The best community partition is obtained by cutting the dendrogram at the layer corresponding to the maximal $Q$. We compare the Newman fast algorithm with the proposed greedy algorithm based on LSI. As shown in figure 5, the proposed algorithm performs better than the Newman fast algorithm on real networks except the polbooks network. Figure 6 reports the comparison on LFR networks, indicating that the proposed algorithm outperforms the Newman fast algorithm and the superiority is robust.\n \n\n\\section{Conclusion and Discussion}\n\nIn this paper, we proposed a novel index LSI and a variant of greedy expanding algorithm that ensures all provisional communities are weak communities (i.e., internal degree is larger than external degree). The LSI performs remarkably better than the GMD and LMD in detecting communities if we apply those indices to identify central nodes for community expanding. In addition, the proposed greedy algorithm based on LSI outperforms the well-known Newman fast algorithm. \n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig7_b.eps}   \n    \\caption{Illustration of the community structure of the polbooks network. Red, blue and black nodes belong to conservative, liberal, and neutral communities, respectively.} \n\\label{fig_Illustration}\n\\end{figure}\n\nPolbooks is the only network where the proposed algorithm is slightly worse than benchmark algorithms. Figure 7 illustrates the community structure of the polbooks network, from which we can observe that the neutral community is not a weak community, with external edges far more than internal edges. As our algorithm avoids weak communities, it may not perform well if some ground-truth communities themselves are not weak communities. It is hard to say whether this is a disadvantage or an advantage, since if a ground-truth community is not a weak community, it usually contains some domain-specific reasons and thus may not be a typical community from the purely topological viewpoint.   \n\nThe proposed method can be extended to handle directed networks and weighted networks by slightly modifying the definition of LSI. If we allow non-central nodes to be merged to more than one community, the proposed algorithm can deal with the overlapping community structure. We can also add a free parameter $\\alpha$ to the quality function as $F(C_i) = {d_{in}(C_i)}\/{[d_{in}(C_i) + d_{out}(C_i)]^\\alpha}$, and then uncover the hierarchical community structure with different resolutions via tuning the parameter $\\alpha$.\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\\addcontentsline{toc}{section}{Acknowledgements}\nThis work was supported by the National Natural Science Foundation of China under Grant Nos. 71874172, 11975071, and the Thousand Talents Program of Sichuan Province under Grant No. 17QR003.\n\n\n\n\n\n\n\\section*{References}\n\\addcontentsline{toc}{section}{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nModeling 3D human pose and motion is a fundamental problem towards understanding human behaviour, with a wide range of applications in medical prognosis, 3D content production, autonomous driving and  human-robot interaction. One of the most studied  computer vision tasks is pose estimation either from single images~\\cite{toshev2014deeppose,moreno20173d,pavlakos2017coarse,guo2021pi}, monocular videos~\\cite{pfister2015flowing,pfister2015flowing,pavllo20193d}, or from multi-camera settings~\\cite{amin2013multi,rhodin2018learning,tu2020voxelpose}. This development is certainly due to the availability of large-scale datasets such as Human3.6M~\\cite{h36m_pami}. Beyond extracting 3D human pose information from various types of visual data, a number of works have been proposed around the development of machine learning methods allowing to forecast future motion~\\cite{fragkiadaki2015recurrent,jain2016structural,martinez2017human,liu2019towards,chiu2019action,butepage2017deep,butepage2018anticipating,holden2015learning,hernandez2019human,li2018convolutional,gui2018adversarial,mao2019learning,kipf2016semi,elickovic2018graph,mao2020history,li2021rain,Dang_2021_ICCV,Sofianos_2021_ICCV,adeli2020socially,Adeli_2021_ICCV,guo2021multi} and more recently on generating plausible future sequences of realistic human 3D pose data~\\cite{lin2018human,kundu2019bihmp,barsoum2018hp,mao2021gsps,mao2019learning,walker2017pose,yan2018mt,aliakbarian2020stochastic,yuan2020dlow,aliakbarian2021contextually,petrovich2021action,cai2021unified,guo2020action2motion}. \n\n\\input{fig_teaser}\n\n\nHuman motion generation has several challenges:\ni) diversity: different from deterministic prediction, the generation methods should not just learn average patterns, but need to faithfully reflect the intrinsic intra-class variability. \nii) dynamics: the generation process must inherently model the dynamics of the 3D human pose data, so that they can be transferred to the generated data and avoid collapsing to a stable motion. \niii) smoothness: the generated data should be smooth.\n\n\n\n\nSome of these challenges have been partially addressed in past studies, although up to our knowledge, there is no existing methodology designed to face all  three challenges.\nFor instance, MT-VAE~\\cite{yan2018mt} combined the RNN-based motion prediction model with conditional variational autoencoders (VAE), where the difference  between the observed and future poses was encoded into the latent variable, which was then  concatenated with the RNN's hidden state to account for the dynamics.\nDLow~\\cite{yuan2020dlow} proposed to generate a diverse set of motion sequences, by training fifty different encoders (and a single decoder), and obtaining fifty different instances of the latent variable, and thus fifty different generated motions. \nGSPS~\\cite{mao2021gsps} inherited the diversity loss from DLow, but utilized a more powerful motion prediction framework, based on graph convolutional network (GCN) rather than the RNN. Diversity was obtained by concatenating random noise to the observed sequences. Finally, ACTOR, a Transformer-VAE method, was proposed in~\\cite{petrovich2021action} to perform  sequential generation, as opposed to previous methods that  were designed for a fixed output length. \nACTOR~\\cite{petrovich2021action} learned an action class informed token as input to the encoder and decoder, thus conditioning the generation with manually annotated labels. \nAll the above methods encode the observed human poses into sequence-level embedding, meaning that the entire observed sequence is encoded into a single time-independent embedding.\n\n    \n\nThis motivates us to propose HiT-DVAE{} inspired by the recent literature on dynamical variational autoencoders (DVAE)~\\cite{girin2020dynamical}. \nOur method belongs to the very general family of variational autoencoders (VAE) and is therefore a probabilistic method inherently able to generate stochastic, and hence diverse output. \nMore precisely, using DVAE, the sequence of observations is encoded into a sequence of latent variables instead of a single latent variable, thus offering larger representation power to learn and exploit the motion dynamics, see Figure~\\ref{fig:data_space}. \nBesides, we model the generative process with auto-regressive dependencies, meaning that the generation of each frame depends on the previous ones and can be done sequentially. We can then train this model to generate sequences of arbitrary length. \nFinally, we implement these auto-regressive dependencies with a transformer-like (attention-based) encoder-decoder architecture, thus learning to automatically select which are the best frames to inform the generation of the next human 3D pose. \nOverall, HiT-DVAE{} implements auto-regressive probabilistic dependencies with transformer-like attention mechanisms enabling the learning of pose sequence dynamics as well as stochastic motion generation. Hence, the proposed solution deals with all the three challenges mentioned above.\n\n\nConsidering the evaluation of generated data, recent works either evaluate the generations directly on the joint location of poses~\\cite{yuan2020dlow,mao2021gsps}, or using the feature extracted from a pretrained feature extractor~\\cite{petrovich2021action,guo2020action2motion}. Both protocols have clear shortcomings: the former just evaluates the best generated sample and the diversity of all generations, ignoring the performance of the generations except the best one; while the latter depends on the quality of the feature extractor. To thoroughly evaluate the generation quality, we use both evaluation methods, and broaden the first one to take performance stability into consideration.\n\nWe thoroughly evaluate HiT-DVAE{} on HumanEva-I and Human3.6M datasets, using both explicit and implicit metrics to measure the quality of generated data.\nExperimental results show that our method achieves state-of-the-art on most of the metrics for both datasets, proving that the generation of HiT-DVAE{} has high quality (smaller errors, better features, correct action) with better performance stability. \n\n\n    \n\n\n\n\n\n\n\n\\section{Related Work}\n\n\n\\subsection{Modeling Future Human motion}\n\\label{sec:realated_work}\nThe forecasting of human motion has been addressed under two paradigms so far: deterministic motion prediction and stochastic human motion generation. \nThe former aims at using deterministic approaches to regress a single future motion from the past observation which is the most likely to the ground truth; while the latter focuses at generating various possibilities of the future to model the multi-modal nature of human motions.\\\\\n\n\\subsubsection{Deterministic human motion prediction}\nDue to the inherent sequential structure of human motion, 3D human motion prediction has been mostly addressed with recurrent models~\\cite{fragkiadaki2015recurrent,jain2016structural,martinez2017human}. \nHowever, although RNNs can achieve great success in motion prediction, they represent the entire past motion history with a fixed-size hidden state and tend to converge to a static pose. Some works alleviate this problem by using RNN variants~\\cite{liu2019towards,chiu2019action}, sliding windows~\\cite{butepage2017deep,butepage2018anticipating}, convolutional models~\\cite{holden2015learning,hernandez2019human,li2018convolutional} or adversarial training~\\cite{gui2018adversarial}.\nSince human body pose data are structured, directly encoding the whole body into a compact latent embedding neglects the spatial connectivity of human joints. To this end, recent work tend to leverage the forward graph convolutional network (GCN)~\\cite{kipf2016semi,elickovic2018graph} with a predefined or learnable adjacency matrix~\\cite{mao2019learning,mao2020history,Dang_2021_ICCV,Sofianos_2021_ICCV,li2021rain,adeli2020socially,Adeli_2021_ICCV}. While deterministic methods have achieved promising results on accurate predictions, they exhibit strong limitations when it comes to model the diversity of plausible human motion forecasts. Stochastic methods are promising tools to overcome these limitations.\n\n\n\n\\subsubsection{Stochastic human motion generation}  To generate multiple future outcomes given a sequence of past observations, two types of approaches have been studied in the recent past: (i) the enhancement of deterministic methods with stochastic variations, e.g., incorporating noise, and (ii) leveraging conditional variational architectures that learn a probability distribution. In the first category, early works include combining random noise with hidden states either by concatenation~\\cite{lin2018human,kundu2019bihmp} or addition~\\cite{barsoum2018hp}. More recently, Mao~\\textit{et al.}~\\cite{mao2021gsps} further investigated this paradigm with a GCN based motion prediction model~\\cite{mao2019learning}, and showed promising results with dedicated designed losses. In the second category, past observations are encoded to learn a posterior latent space, then a random variable will be sampled and then combined with observations to predict the future~\\cite{walker2017pose,yan2018mt,aliakbarian2020stochastic,cai2021unified,aliakbarian2021contextually}. Rencently, DLow~\\cite{yuan2020dlow} proposed to explicitly generate a large number of samples during training, then to use a energy function to promote the diverse generation. ACTOR~\\cite{petrovich2021action} first introduce a Transformer-VAE to obtain long term attention. Rather than modeling the whole observation into a single embedding, Action2Motion~\\cite{guo2020action2motion} and HuMor~\\cite{rempe2021humor} exploit a auto-regressive generative model that the current generation will depend on the past prediction. However, they do not model the entire sequence, but only on the last frame, thus resulting in non-smooth motion generation.\n\n\\subsection{Deep generative modeling}\nThe family of stochastic human motion generation methods are mostly based in the general paradigm of variational inference, and of variational autoencoders. VAEs model the joint distribution of an observation $\\mbf{x}$ and a latent variable $\\mbf{z}$. In stochastic human motion generation, the observations $\\mbf{x}$ often corresponds to a sequence of poses, rather than a single pose. However, up to our knowledge, most of the previous methods use a single latent variable $\\mbf{z}$ to encode the entire observed sequence. Alternatively, one could consider a sequence of latent variables and of observations, and use a VAE to model the relationship between between $\\mbf{x}_t$ and $\\mbf{z}_t$ without any time dependencies. However, the dynamics and any temporal relationships cannot be modeled in this case, which is obviously not desirable. Dynamical variational autoencoders (DVAEs)~\\cite{girin2020dynamical} offer the possibility to model data sequences within the general paradigm of variational inference. DVAE is a general class of models and different models are obtained when considering various dependencies between the variables, e.g., variational recurrent neural networks~\\cite{chung2015recurrent} or stochastic recurrent neural networks~\\cite{fraccaro2016sequential}. However, current DVAE models have a major limitation: the probabilistic dependencies between variables are always implemented with recurrent neural networks (or variants), thus avoiding the possibility to select which past frames are used to inform the generation of the current frame. In addition and up to our knowledge, the use of DVAEs for human motion forecasting has not been investigated so far. This motivates us to explore the use of attention mechanisms within the DVAE paradigm with applications to human motion forecasting, as explained in the following.\n\n\n\\section{Method}\n\n \nWe address the problem of 3D human motion generation that we formalise as follows. \nGiven a sequence of $O$ observed 3D poses of a person $\\mbf{x}_{1:O} = [\n\\mbf{x}_1, \\ldots, \\mbf{x}_O]$, our aim is to generate a sequence of $G$ 3D poses $\\mbf{x}_{O+1:O+G} = [\\mbf{x}_{O+1}, \\ldots \\mbf{x}_{O+G}]$, that follow the observations $\\mbf{x}_{1:O}$. Each pose vector  $\\mbf{x}_t \\in \\mathbb{R}^{J\\times3}$ encodes the location of the $J$ joints of a person at time $t$ in Cartesian coordinates. Different from deterministic human motion prediction, we intend to generate multiple plausible future motion sequences with arbitrary length. To this end, we propose a new method named Hierarchical Transformer Dynamical Variational AutoEncoder or HiT-DVAE{}. Our method is based on the recently reviewed family of dynamical variational autoencoders~\\cite{girin2020dynamical}, which formulates the generative process of time series in an autoregressive and time-dependent perspective.\nUp to our knowledge, this general methodology has never been combined with attention-based mechanisms. On the one hand, existing variants of DVAEs are always implemented with recurrent networks~\\cite{girin2020dynamical} (or standard variants such as LSTM and GRU). On the other hand, even if self-attention has been proven useful when combined with a Conditional VAE~\\cite{petrovich2021action}, the architectures proposed so far encode the entire sequence into a single latent variable $\\mbf{z}$, therefore potentially limiting the representation capabilities of temporal dynamics. We propose HiT-DVAE{} to get the best of both worlds, enabling stochastic motion generation together with dynamic sequence modeling via transformer-like attention mechanisms.\n\n\n\n\\subsection{HiT-DVAE{}}\n\n\n\nThe proposed method is based on the very general DVAE methodology (see~\\cite{girin2020dynamical} for an exhaustive presentation on the topic). The basic principle of DVAEs is that for every observation $\\mbf{x}_t$ there is a corresponding latent variable $\\mbf{z}_t$, as opposed to VAEs which would encode the entire observed sequence $\\mbf{x}_{1:O}$ into a single latent variable $\\mbf{z}$.\nThe sequence of observations and corresponding latent variables will be denoted by $\\mbf{x}_{1:T} = [\\mbf{x}_t]_{t=1}^{T}$ and $\\mbf{z}_{1:T} = [\\mbf{z}_t]_{t=1}^{T}$, respectively. For the time being, we will assume that $T=O+G$, as if all 3D poses were observed even if this is not our setting. We will discuss the impact of having hybrid half-observed half-generated sequences later on.\n\nIn addition to the time-dependent latent variable $\\mbf{z}_{1:T}$, and inspired by~\\cite{petrovich2021action,yingzhen2018disentangled}, we add a time-independent latent variable $\\mbf{w}$. Very differently from~\\cite{petrovich2021action}, $\\mbf{w}$ will be learned in an unsupervised manner within the DVAE methodology, see~\\cite{yingzhen2018disentangled}, thus without requiring action class labels. Formally, the proposed generative model writes:\n\\begin{align}\n    p_{\\bs{\\theta}}(\\mbf{x}_{1:T}, \\mbf{z}_{1:T}, \\mbf{w}) &= \\prod_{t=1}^{T} p_{\\bs{\\theta}}(\\mbf{x}_{t}, \\mbf{z}_{t}, \\mbf{w} | \\mbf{x}_{1:t-1},\\mbf{z}_{1:t-1}) \\\\\n    &= p_{\\bs{\\theta}_{\\mbf{w}}}(\\mbf{w}) \\prod_{t=1}^{T} p_{\\bs{\\theta}_{\\mbf{x}}}(\\mbf{x}_{t} | \\mbf{x}_{1:t-1},\\mbf{z}_t, \\mbf{w}) p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\mbf{x}_{t-1},\\mbf{z}_{1:t-1}, \\mbf{w}),\n\\label{eq:dvae_generation}\n\\end{align}\nmeaning that the generative processes of both the observed and latent variables are auto-regressive, with cross-dependencies. We set $\\bs{\\theta} = \\bs{\\theta}_\\mbf{w} \\cup \\bs{\\theta}_\\mbf{z} \\cup \\bs{\\theta}_\\mbf{x}$. In order to learn this generative model, we introduce an inference model ($\\bs{\\phi}=\\bs{\\phi}_\\mbf{w}\\cup\\bs{\\phi}_\\mbf{z}$):\n\\begin{equation}\n    q_{\\bs{\\phi}}(\\mbf{z}_{1:T}, \\mbf{w} | \\mbf{x}_{1:T}) = q_{\\bs{\\phi}_{\\mbf{w}}}(\\mbf{w} | \\mbf{x}_{1:T}) \\prod_{t=1}^T q_{\\bs{\\phi}_{\\mbf{z}}}(\\mbf{z}_t | \\mbf{x}_{1:T}, \\mbf{w}).\n\\label{eq:dvae_inference}\n\\end{equation}\nThe training objective is to maximize the evidence lower bound (ELBO):\n\\begin{equation}\n    \\mcal{L}(\\bs{\\theta},\\bs{\\phi}; \\mbf{x}_{1:T}) = \\mbb{E}_{q_{\\bs{\\phi}}(\\mbf{z}_{1:T},\\mbf{w} | \\mbf{x}_{1:T})} \\left[ \\ln p_{\\bs{\\theta}}(\\mbf{x}_{1:T},\\mbf{z}_{1:T},\\mbf{w} ) - \\ln q_{\\bs{\\phi}}(\\mbf{z}_{1:T},\\mbf{w} | \\mbf{x}_{1:T}) \\right].\n\\label{eq:dvae_elbo}\n\\end{equation}\n\nAlthough the above equations define the probabilistic dependencies between the different random variables, there are plenty of ways of implementing these dependencies. In this paper, we propose to use a hierarchical transformer-based architecture. In our ablation study, we discuss other --perhaps more conventional-- ways of implementing such dependencies, that exhibit lower performance and demonstrate the interest of having both attention and a hierarchical structure, and thus justify the proposed HiT-DVAE{}. Both the encoder and the decoder of the proposed method exploit a spatial graph convolutional network (SGCN) to extract pose features from the raw poses $\\mbf{x}_t$. We will denote this pose feature extraction operation as $f$, and we will let the encoder and decoder fine-tune their pose extractor leading to $f_\\textsc{E}$ and $f_\\textsc{d}$, see below for more details. Figure~\\ref{fig:HIT-DVAE} shows an overview of our proposed model. Specifically, we employ the transformer architecture~\\cite{vaswani2017attention} jointly with GCN-based feature extractors to formulate the inference and generation on the sequential human motion data.  \n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{pipeline_v2.4.png}\n    \\caption{Overview of HiT-DVAE{}. The Encoder (left) inputs the observed sequence $\\mbf{x}_{1:T}$ to estimate the posterior distribution of the time-dependent $\\mbf{z}_{1:T}$ and time-independent $\\mbf{w}$ latent variables. Then the Decoder (right) reconstructs the data and the prior of $\\mbf{z}$.\n    \\label{fig:HIT-DVAE}\n\\end{figure}\n\n\\subsubsection{Generative Model (HiT-DVAE{} Decoder)} The generation of both $\\mbf{x}_{1:T}$ and $\\mbf{z}_{1:T}$ is performed via the attention mechanisms proposed in the original transformer architecture~\\cite{vaswani2017attention}. Specifically, the generative model will use multi-head cross-attention, after the GCN-based feature extractor $f_\\textsc{d}$. The generative processes of $\\mbf{x}$ and $\\mbf{z}$ differ on what variables are used as queries, keys and values in the attention mechanism. The output of the two decoders will be the parameters of the respective probability distributions, defined in~(\\ref{eq:dvae_generation}). Both distributions are considered to be Gaussian. While we learn both the mean and covariance matrix of the generative distribution of the latent variable $\\mbf{z}_t$, the covariance matrix of the observations $\\mbf{x}_t$ is considered to be the identity as in previous works~\\cite{aliakbarian2021contextually,rempe2021humor,guo2020action2motion,petrovich2021action,yuan2020dlow}. In particular we have $p_{\\bs{\\theta}_{\\mbf{x}}}(\\mbf{x}_{t} | \\mbf{x}_{1:t-1},\\mbf{z}_t, \\mbf{w}) = \\mathcal{N} (\\mbf{x}_t;\\; \\bs{\\mu}_{\\bs{\\bs{\\theta}}_{\\mbf{x}}, t}, \\mathbf{I})$ and $p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\mbf{x}_{t-1},\\mbf{z}_{1:t-1}, \\mbf{w}) = \\mathcal{N} (\\mbf{z}_t;\\; \\bs{\\mu}_{\\bs{\\bs{\\theta}}_{\\mbf{z}}, t}, \\bs{\\Sigma}_{\\bs{\\theta}_{\\mbf{z}}, t})$ with \n\\begin{align}\n\\bs{\\mu}_{\\bs{\\bs{\\theta}}_{\\mbf{x}}, t} &= \\textrm{MaskedMultiHead} \\left(Q_{\\bs{\\bs{\\theta}_{\\mbf{x}}},t}, K_{\\bs{\\bs{\\theta}_{\\mbf{x}}}}, V_{\\bs{\\bs{\\theta}_{\\mbf{x}}}}\\right), \\\\\nQ_{\\bs{\\bs{\\theta}_{\\mbf{x}}},t} &= \\left[ \\begin{tabular}{c} $\\mbf{z}_t$ \\\\ $\\mbf{w}$ \\end{tabular} \\right], K_{\\bs{\\bs{\\theta}_{\\mbf{x}}}} = V_{\\bs{\\bs{\\theta}_{\\mbf{x}}}} = [f_\\textsc{d}(\\mbf{x}_1), \\ldots, f_\\textsc{d}(\\mbf{x}_{T})],\\\\\n\\left[ \\begin{tabular}{c} \n$\\bs{\\mu}_{\\bs{\\theta}_{\\mbf{z}}, t}$ \\\\\n$\\bs{\\Sigma}_{\\bs{\\theta}_{\\mbf{z}}, t}$ \\end{tabular} \\right]  &= \\textrm{MaskedMultiHead} \\left(Q_{\\bs{\\theta}_{\\mbf{z}}, t}, K_{\\bs{\\bs{\\theta}_{\\mbf{z}}}}, V_{\\bs{\\theta}_{\\mbf{z}}}\\right), \\\\\nQ_{\\bs{\\theta}_{\\mbf{z}}, t} &= \\left[ \\begin{tabular}{c} $f_\\textsc{d}(\\mbf{x}_{t-1})$ \\\\ $\\mbf{w}$ \\end{tabular} \\right], K_{\\bs{\\bs{\\theta}_{\\mbf{z}}}} = V_{\\bs{\\theta}_{\\mbf{z}}} = [\\mbf{z}_1, \\ldots, \\mbf{z}_{T}],\n\\end{align}\nwhere a mask is used to prevent $\\mbf{z}_{t}$ and $\\mbf{x}_{t}$ from being generated from future latent and observed variables \\cite{vaswani2017attention}. Finally, we have $p_{\\bs{\\theta}_{\\mbf{w}}}(\\mbf{w}) = \\mathcal{N}(\\mbf{w};\\mathbf{0},\\mathbf{I})$, where $\\mathbf{0}$ and $\\mathbf{I}$ are the zero vector and the identity matrix of appropriate dimensions. \n\n\n\n\\subsubsection{Inference Model (HiT-DVAE{} Encoder)} The inference of the latent variables $\\mbf{w}$ and $\\mbf{z}_{1:T}$ from $\\mbf{x}_{1:T}$ is performed via a temporal GCN and the multi-head self-attention mechanism of the transformer encoder (TE). The extracted pose features are fed into the temporal GCN with $T$ nodes, where each node indicates a time frame, and then into a fully connected (FC) layer to output the mean and variance of $\\mbf{w}$, namely $\\bs{\\mu}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}}$ and $\\bs{\\Sigma}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}}$. Samples are drawn from the corresponding posterior $q_{\\bs{\\phi}_{\\mbf{w}}}(\\mbf{w} | \\mbf{x}_{1:T}) = \\mathcal{N}(\\mbf{w};\\; \\bs{\\mu}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}}, \\bs{\\Sigma}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}})$, concatenated to all pose features and then fed into the transformer encoder such that $q_{\\bs{\\phi}_\\mbf{z}}(\\mbf{z}_t | \\mbf{x}_{1:T}, \\mbf{w}) = \\mathcal{N} (\\mbf{z}_t;\\; \\bs{\\mu}_{\\bs{\\phi}_{\\mbf{z}}, t}, \\bs{\\Sigma}_{\\bs{\\phi}_{\\mbf{z}}, t})$ with\n\\begin{align}\n\\left[ \\begin{tabular}{c} \n$\\bs{\\mu}_{\\bs{\\phi}_{\\mbf{z}}, t}$ \\\\\n$\\bs{\\Sigma}_{\\bs{\\phi}_{\\mbf{z}}, t}$ \\end{tabular} \\right]  &= \\textrm{MultiHead}\\left(Q_{\\bs{\\phi}_{\\mbf{z}}, t}, K_{\\bs{\\bs{\\phi}_{\\mbf{z}}}}, V_{\\bs{\\phi}_{\\mbf{z}}}\\right), \\\\\nQ_{\\bs{\\phi}_{\\mbf{z}}, t} &= \\left[ \\begin{tabular}{ccc} \n    $f_\\textsc{E}(\\mbf{x}_t)$ \\\\\n    $\\mbf{w},$ \\end{tabular} \\right] , K_{\\bs{\\bs{\\phi}_{\\mbf{z}}}} = V_{\\bs{\\phi}_{\\mbf{z}}} = \\left[ \\begin{tabular}{ccc} \n    $f_\\textsc{E}(\\mbf{x}_1),$ & $\\ldots,$ & $f_\\textsc{E}(\\mbf{x}_T)$  \\\\\n    $\\mbf{w}$ & $\\ldots,$ & $\\mbf{w}$ \\end{tabular} \\right].\n\\end{align}\n\n\n\n\n\\subsubsection{Feature Extractor on Human Poses} As mentioned above, we extract pose features via a spatial GCN $f$. This strategy was suggested in~\\cite{chung2015recurrent}. In HiT-DVAE{}, the spatial GCN is composed of $J$ nodes in the graph, each of them representing a joint in the pose skeleton. While the architecture of the spatial GCN is the same as that of the generative and inference models, these two spatial GCNs are trained separately (i.e., the weights are not shared).\n\n\n\n\n\n\n\n\n\\subsection{Training losses}\n\n\\subsubsection{ELBO.} Optimising the evidence lower bound~(\\ref{eq:dvae_elbo}) in the case of the proposed HiT-DVAE{}, boils down to (i) minimising the $L_2$ loss on the reconstructed poses while (ii) minimising the KL divergence between the posterior and prior distributions over the latent variables.\nBecause directly optimising the ELBO would not encourage diversity in our generative model, we inspire from~\\cite{mao2021gsps}, and we explicitly generate $K$ motion sequences $\\{\\hat{\\mbf{x}}_{1:T}^k\\}_{k=1}^K$ and compute the ground-truth reconstruction loss and multi-modal reconstruction loss:\n\\begin{equation}\n    \\mathcal{L}_{\\textsc{r}} = \\min_{k} || \\hat{\\mbf{x}}_{1:T}^k - \\mbf{x}_{1:T}||^2, \\qquad \\mathcal{L}_{\\textsc{mm}} = \\frac{1}{M}\\sum_{m=1}^M \\min_{k} || \\hat{\\mbf{x}}_{1:T}^k - \\mbf{x}_{1:T}^m||^2,\n    \\label{eq:recon}\n\\end{equation}\nwhere $\\mbf{x}_{1:T}$ is the ground-truth, and $\\mbf{x}_{1:T}^m$ are the pseudo-ground truth sequences, which are selected from the training set following the same protocol as in~\\cite{mao2021gsps}. In addition to the reconstruction losses, we need to minimise the KL divergence:\n\\begin{align}\n    \\mathcal{L}_{\\textsc{kl-z}} &= \\frac{1}{T}\\sum_{t=1}^T D_{KL} ( q_{\\bs{\\phi}_{\\mbf{z}}}(\\mbf{z}_t | \\mbf{x}_{1:T}, \\mbf{w}) || p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\mbf{x}_{t-1}, \\mbf{z}_{1:t-1}, \\mbf{w}))\\\\\n    \\mathcal{L}_{\\textsc{kl-w}} &= D_{KL} ( q_{\\bs{\\phi}_{\\mbf{w}}}(\\mbf{w} | \\mbf{x}_{1:T}) || p_{\\bs{\\theta}_{\\mbf{w}}}(\\mbf{w})).\n\\label{eq:loss_kl}\n\\end{align}\nThe final evidence lower bound (ELBO) writes:\n\\begin{equation}\n    \\mathcal{L}_{\\textsc{elbo}} = \\lambda_{\\textsc{r}}\\mathcal{L}_{\\textsc{r}} + \\lambda_{\\textsc{mm}}\\mathcal{L}_{\\textsc{mm}} + \\lambda_{\\textsc{kl-z}}\\mathcal{L}_{\\textsc{kl-z}} + \\lambda_{\\textsc{kl-w}}\\mathcal{L}_{\\textsc{kl-w}}.\n\\end{equation}\n\n\n\n\\subsubsection{Diversity loss.} As suggested by~\\cite{yuan2020dlow,mao2021gsps}, we add diversity promoting losses on upper body and lower body:\n\\begin{equation}\n    \\mathcal{L}_{\\textsc{div}} = \\sum_{p\\in\\{l,u\\}}\\lambda_{\\textsc{div-}p}\\frac{2}{K(K-1)} \\sum_{k=1}^K \\sum_{k'=k+1}^K \\exp\\left(-\\frac{|| \\hat{\\mbf{x}}_{1:T}^{k,p} - \\hat{\\mbf{x}}_{1:T}^{k',p} ||_1}{\\alpha_p}\\right),\n\\label{eq:loss_div}\n\\end{equation}\nwhere $l$ ($u$) indicates the lower (upper) body part and $\\alpha_p$ is a normalizing factor.\n\n\\subsubsection{Realistic pose loss.} Follow~\\cite{mao2021gsps}, we employ three extra losses to penalize for unrealistic poses, $\\mathcal{L}_{\\textsc{l}}$ for shifting limb length, $\\mathcal{L}_{\\textsc{a}}$ for aberrant joint angles and $\\mathcal{L}_{\\textsc{nf}}$ for negative prior pose probability from a pre-trained pose prior model based on normalizing flow~\\cite{rezende2015variational,dinh2016density}. \nAltogether, our final training loss writes:\n\\begin{equation}\n    \\mathcal{L} = \\mathcal{L}_{\\textsc{elbo}} + \\mathcal{L}_{\\textsc{div}} + \\lambda_{\\textsc{l}}\\mathcal{L}_{\\textsc{l}} + \\lambda_{\\textsc{a}}\\mathcal{L}_{\\textsc{a}} + \\lambda_{\\textsc{nf}}\\mathcal{L}_{\\textsc{nf}}.\n\\label{eq:loss_tot}\n\\end{equation}\n\n\\subsection{HiT-DVAE{} for diverse human motion generation}\nThe losses above allow to train the proposed HiT-DVAE{} to reconstruct full sequences. In practice, as stated in the problem formulation, HiT-DVAE{} must input $O$ observed frames $\\mbf{x}_{1:O}$ and generate the following $G$ frames $\\mbf{x}_{O+1:O+G}$. However, if we use the model trained with ground-truth input over the entire sequence $\\mbf{x}_{1:T}$ ($T=O+G$) we encounter severe difficulties since after $O$ frames the input distribution changes from the ground-truth to the generated one, and the generation fails. One alternative could be training with ground-truth input until $O$ $\\mbf{x}_{1:O}$ and then complete with generated data $\\hat{\\mbf{x}}_{O+1:O+G}$. Unfortunately, at the beginning of the training the generated data is pattern-less, and the training diverges. In order to overcome this issue, we use scheduled sampling~\\cite{bengio2015scheduled}: we start training only with ground-truth data, and we progressively add more generated frames (randomly) with a proportion starting at $0\\%$ and up to $100\\%$.\n\nOnce our model is trained, we can use it to generate various future motion sequences in arbitrary length. Given $O$ observations in our setting, we can get the posterior of $\\mbf{z}_{1:O}$ and $\\mbf{w}$ from the inference model. Then, we can generate the next $G$ frames $\\hat{\\mbf{x}}_{O+1:O+G}$ by recursively applying the generative function on $p_{\\bs{\\theta}_{\\mbf{x}}}(\\mbf{x}_{t} | \\mbf{x}_{1:t-1},\\mbf{z}_t, \\mbf{w})$ and $ p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\mbf{x}_{t-1},\\mbf{z}_{1:t-1}, \\mbf{w})$. The diversity comes simply from the different samples of $\\mbf{z}_{O+1:O+G}$ and $\\mbf{w}$.\n\n\n\n\n\n\n\\section{Experiments}\n\n\n\\subsection{Evaluation Protocols}\n\\label{sec:eval_metric}\n\\input{fig_vis}\n\\subsubsection{Datasets}\nFollowing~\\cite{mao2021gsps,yuan2020dlow}, we train and evaluate our methods on Human3.6M~\\cite{ionescu2013human3} and HumanEva-I~\\cite{sigal2006humaneva} dataset: \\textbf{Human3.6M} is the most commonly used dataset for motion tasks. It contains 7 actors (S1,5,6,7,8,9,11) performing 15 annotated actions recorded at 50 Hz. Human pose is represented in 32 skeletons, while we follow~\\cite{mao2021gsps} to use 17 of the skeletons in our training and all testing implementations, and we use S1,5,6,7,8 as training set and the other two subjects as test set.\n\\textbf{HumanEva} contains 5 actions (Box, Gesture, Jog, ThrowCatch, Walking) performed by 3 actors, recorded at 60 Hz. Each pose is represented by 15 skeletons. \nFor both datasets, we remove the global translation and set the root joint as zero.\n\n\\input{tab_gen_eva}\n\n\\subsubsection{Explicit evaluation metrics}\nFollowing \\cite{mao2021gsps,yuan2020dlow}, we evaluate error and diversity of our results with the following metrics, calculating directly on the joint locations of poses:\ni) Average Pairwise Distance (APD): average $L2$ distance of all pairs among the generated sequences: $\\frac{1}{K(K-1)}\\sum_{i=1}^K \\sum_{j\\neq i}^K \\|\\hat{\\mathbf{x}}^i_{O+1:O+G} - \\hat{\\mathbf{x}}^j_{O+1:O+G}\\|_2$. APD measures the capacity of the model to generate diverse samples without considering their quality.\nii) Average Displacement Error (ADE): $L2$ distance between the ground truth and the 'best' generated sample among all, taking the average over all frames of the sequence:$\\frac{1}{G}\\min_k \\|\\hat{\\mathbf{x}}^k_{O+1:O+G} - \\mathbf{x}_{O+1:O+G}\\|$. Here 'best' means the one which is closest to the ground truth. ADE evaluates the upper bound of the generation quality of the model among all the generation results.\niii) Final Displacement Error (FDE): Similar to ADE, FDE evaluates the distance between the ground truth and the best sample, but just on the final frame instead of the whole sequence: $\\min_k \\|\\hat{\\mathbf{x}}^k_{O+G} - \\mathbf{x}_{O+G}\\|$. \niv) Multi-Modal ADE (MMADE) and Multi-Modal FDE (MMFDE): multi-modal version of ADE and FDE. \n\nThe MPJPE based metrics are widely used for evaluating the quality of generated motion, considering the diversity of the generated data and accurancy of the best generated one. While the shortcoming of them is obvious. On one hand, when we use the generator to generate actions, we could not always have the groundtruth to judge which generation is the best one; on the other hand, as described in Sec~\\ref{sec:realated_work}, generating one best example is what deterministic motion prediction aims at, while stochastic methods should generate various good samples. For example, a batch of generated motions where only one sample is exact while the others are all super crazy will result in a large APD and tiny errors which seems perfect numerically, but this is certainly not a good generation we want. So just considering the above metrics is not comprehensively and proper.\n\nThus, we take two solutions: 1) instead of just evaluating ii-iv) on the best generated sample, we also evaluate these criteria on the 'medium' example, which holds the medium distance to the groundtruth among all the generated samples. 2) beside of this explicit measurement based on poses, we also consider implicit measurements based on a pretrained action classifier, as described below.\n\n\\subsubsection{Implicit evaluation metrics} \nFollowing \\cite{petrovich2021action,guo2020action2motion}, we use a GRU-based action classifier pre-trained on real data to evaluate the quality of generated data by: \\\\\ni) calculating Recognition Accuracy (Acc) of the classifier on generated data to evaluate if the generated data could be recognized as the correct action class.\\\\\nii) extracting features of the generated data and real data respectively by the action classifier, and calculate the Frechet Inception Distance (FID) of these two distributions to evaluate the overall quality of generated data.\nFor the two datasets, we trained a classifier for each of them on their training splits.\n\n\n\\subsection{Implementation details} \nWe set the dimension of $\\mbf{z}_t$ to 16 and $\\mbf{w}$ to 32, and employ the same GCN architecture described in~\\cite{mao2019learning}. We use 1 GCN block with hidden size of 8 for spatial GCN and 4 GCN blocks with hidden size of 64 for temporal GCN. For the Transformer encoder and the decoder for generating $\\mbf{z}_t$, we set the input feature dimension to 64, with 4 multi-head, followed by a FC layer with dimension of 256, whereas for the Transformer decoder to generate $\\mbf{x}_t$, we set those parameters to 256, 4, 1024 respectively. \n\nWe generate $K=50$ samples for each observation. We train the model for 500 epochs with 1000 training samples per epoch, using Adam optimizer, and set learning rate to 0.001, batch size to 64 for HumanEva and 32 for H3.6M. We applied a linear KL annealing~\\cite{sonderby2016ladder} for the first 20 epochs to warm-up the latent space, then we take 80 epochs to increase the probability of schedule sampling from 0 to 1. For HumanEva, we train with a sequence length of 75, where the inference of $\\mbf{w}$ only takes 15 frames with a random start point. The weights of different loss terms $(\\lambda_r, \\lambda_{mm}, \\lambda_{d, l}, \\lambda_{d, u}, \\lambda_{l}, \\lambda_{a}, \\lambda_{nf}, \\lambda_z , \\lambda_w)$ and the normalizing factors $(\\alpha_l, \\alpha_u)$ are set to (10, 5, 0.1, 0.2, 100, 1, 0.001, 0.5, 0.1) and (15, 50). For H3.6M,  we train with a sequence length of 125, where $\\mbf{w}$ is inferred from 25 frames. The weights of different loss terms and the normalizing factors are set to (20, 10, 0.1, 0.2, 100, 1, 0.01, 0.5, 0.1) and (100, 300) respectively.\n\n\n\\input{tab_gen_h36m}\n\\subsection{Quantitative results}\nWe evaluate the generated motions on HumanEva-I and Human3.6m dataset by the explicit and implicit metrics described in Sec~\\ref{sec:eval_metric}, and found that our methods outperforms the state of the art methods on most of the evaluation metrics.\n\n\\subsubsection{HumanEva-I}\nAs shown in Tab~\\ref{table:tab_res_eva}, our method achieves comparable results with state of the art on explicit evaluation of diversity (APD) and errors of 'best' sample (ADE$_b$, FDE$_b$, MMADE$_b$, MMFDE$_b$). As discussed in Sec~\\ref{sec:eval_metric}, we know that just considering these errors and diversity is not reliable, because these erros just evaluate the best sample among all generations and APD just evaluates diversity without considering the generation quality. And we should note that APD is not always better for larger values, because although we want the generated data to have diversity, crazy large diversity represents that some of the generated samples might totally fail and the quality of generation is not guarantied.\nSo in order to comprehensively measure the performance of generated data, we test the error of 'medium' samples, action-recognition based accuracy ACC and feature-based FID scores on our data, and also on other code-released state of the art methods~\\cite{mao2021gsps,yuan2020dlow}.\n\nWe find that our method achieves significantly better results than other state of the art methods on the implicit metrics ACC and FID, which means that our generated data is better in feature distribution, and could generate more reasonable results that could be recognized as the right action. And our method is also clearly better on errors of medium sample (ADE$_m$, FDE$_m$, MMADE$_m$, MMFDE$_m$), which indicates higher stability of our overall generation quality.\\\\\n\n\\subsubsection{Human3.6M}\nSimilar conclusion could be drawn for Human3.6M dataset, as shown in Tab~\\ref{table:tab_res_h36m}. \nWhen training the action classifier for Human3.6M dataset, \ninstead of training on all the 15 action labels, we group the 15 actions into 5 groups. This is because that some actions in Human3.6M dataset are not with much difference, so it is is not suitable for training the action classier. For example, we could not see the difference between 'eating' and 'smoking' with just the skeletons of the person. With this grouping, the average classification accuracy on real data of our classifier increases from 48.1\\% to 85.5\\%. Note that even on the 15-action classifier with low real-data-accuracy, our method still has higher Acc and lower FID comparing with other state of the art methods. While we report the 5-group classifier here because we believe that a better classifier is more reliable for calculating accuracy and extracting features for FID. More details about the action classifier could be found in supplementary material\n\n\n\n\n\\input{tab_abla}\n\\subsection{Ablation study}\nTab~\\ref{table:tab_abla} shows ablation studies on our method with different architecture design. We bold the best results and underlined the second best ones. We could find that, without schedule sampling, our method tends to generate more diverse results, but with worse quality either on explicit metrics and implicit metrics. \nLooking in the results of Human3.6M, the use of attention mechanism brings higher generation quality on explicit measurement. The global time-independent variable $\\mbf{w}$ bring more diversity on both two datasets (see the results w\/o $\\mbf{w}$). When we consider a vanilla DVAE model (w\/o Att. \\& $\\mbf{w}$), without the hierarchical transformer (HIT) architecture, it is very likely to collapse to a static state on the sequential latent space, which leads to moderate generation quality, and much worse diversity. The final setting of HiT-DVAE{} we used performs good on almost all the metrics and is a balance of different evaluations.\n\n\n\n\n\\subsection{Qualitative results}\nTo qualitatively evaluate our generated results, we visualize various generating samples of our methods in Fig~\\ref{fig:vis} comparing with other state of the art methods. we could see that other methods either generates very similar samples for all the generations, either result in some weird actions, while our method performs well on all the generations, with diverse but always reasonable results. More visualisations could be found in supplementary material.\n\n\n\n\n\\section{Probabilistic Dependencies via Masked MHA}\nThe temporal dependencies are implemented via the mask of the attention modules of the transformer decoder and encoder. The attention matrix in a Transformer layer, and is computed as follows:\n\\begin{equation}\n    \\textrm{Att}(\\bs{Q}, \\bs{K}, \\bs{V}) = \\textrm{Softmax}\\left( \\bs{\\mathcal{M}} \\circ \\frac{\\bs{Q}\\bs{K}^T}{\\sqrt{d_k}} \\right) \\bs{V},\n\\end{equation}\nwhere $\\bs{Q}, \\bs{K}, \\bs{V}$ represent the query, key and value, and $d_k$ represents the input feature dimension of query and key. $\\bs{\\mathcal{M}}$ is the attention mask and $\\circ$ denotes the element-wise multiplication. Obviously, an upper triangular mask without diagonal can prevent the model to see the future input. In this case, we can generate the entire sequence of ${\\bf x}_{1:T}$ or ${\\bf z}_{1:T}$ simultaneously. In practice, given an observed sequence with length $T$, we only generate  ${\\bf x}_{2:T}$ and ${\\bf z}_{2:T}$ to bypass the estimation of initial state ${\\bf x}_0$ and ${\\bf z}_0$. \n\n\\input{fig_mask}\n\nFig.~\\ref{fig:mask} shows three cases of probabilistic dependencies when using different mask in the Transformer layer. Note that Fig.~\\ref{fig:mask} (a) is a non-causal situation, thus we can not generate future motion via this dependencies. The mask in Fig.~\\ref{fig:mask} (c) will make the attention computed only on one element, thus the attention mask is meaningless in this case. We choose the mask shown in Fig.~\\ref{fig:mask} (b) in our proposed HiT-DVAE.\n\n\\section{Pseudo-code for HiT-DVAE}\nHere, we provide the pseudo-code for HiT-DVAE in training and generation:\n\\begin{algorithm}[h]\n\\caption{HiT-DVAE in training}\n\\begin{algorithmic}\n\\Inputs{}\\vspace{-4mm}\n\\State{$\\triangleright$ Observation on human sequence $\\mbf{x}_{1:T}$}\n\\For{epo in epochs}\n\\State{\\textbf{Inference:}}\n\\State{$\\triangleright$ Compute posterior of $\\mbf{w}$ and sample $ \\mbf{w} \\sim q_{\\bs{\\phi}_{\\mbf{w}}}(\\mbf{w} | \\mbf{x}_{1:T}) =  \\mathcal{N}(\\mbf{w};\\; \\bs{\\mu}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}}, \\bs{\\Sigma}_{\\bs{\\bs{\\phi}_{\\mbf{w}}}})$ }\n\\State{$\\triangleright$ Compute posterior $\\mbf{z}_{1:T}$ and sample $\\mbf{z}_t \\sim q_{\\bs{\\phi}_\\mbf{z}}(\\mbf{z}_t | \\mbf{x}_{1:T}, \\mbf{w}) = \\mathcal{N} (\\mbf{z}_t;\\; \\bs{\\mu}_{\\bs{\\phi}_{\\mbf{z}}, t}, \\bs{\\Sigma}_{\\bs{\\phi}_{\\mbf{z}}, t})$ for $t=1,...,T$ }\n\\State{\\textbf{Generation:}}\n\\State{$\\triangleright$ Compute the distribution of $\\mbf{x}_{2:T}$ via $p_{\\bs{\\theta}_{\\mbf{x}}}(\\mbf{x}_{t} | \\mbf{x}_{1:t-1},\\mbf{z}_t, \\mbf{w}) = \\mathcal{N} (\\mbf{x}_t;\\; \\bs{\\mu}_{\\bs{\\bs{\\theta}}_{\\mbf{x}}, t}, \\mathbf{I})$ for $t=2, ..., T$}\n\\State{$\\triangleright$ Compute the prior of $\\mbf{z}_{2:T}$ via  $p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\mbf{x}_{t-1},\\mbf{z}_{1:t-1}, \\mbf{w}) = \\mathcal{N} (\\mbf{z}_t;\\; \\bs{\\mu}_{\\bs{\\bs{\\theta}}_{\\mbf{z}}, t}, \\bs{\\Sigma}_{\\bs{\\theta}_{\\mbf{z}}, t})$ for $t=2, ..., T$}\n\\State{\\textbf{Compute loss and optimize via Adam}}\n\\EndFor\n\\end{algorithmic}\n\\label{algo:vem-enhancement}\n\\end{algorithm}\n\n\\vspace{9mm}\n\n\\begin{algorithm}[h]\n\\caption{HiT-DVAE in generation}\n\\begin{algorithmic}\n\\Inputs{}\\vspace{-4mm}\n\\State{$\\triangleright$ Observation on human sequence $\\mbf{x}_{1:O}$}\n\\Init{}\\vspace{-4mm}\n\\State{$\\triangleright$ Compute posterior of $\\mbf{w}$ and $\\mbf{z}_{1:O}$}\n\\For{t in range(O+1, O+G)}\n\\State{$\\triangleright$ Generate $\\hat{\\mbf{z}}_t$ via $\\mbf{z}_t \\sim p_{\\bs{\\theta}_{\\mbf{z}}}(\\mbf{z}_{t} | \\hat{\\mbf{x}}_{t-1},\\mbf{z}_{1:t-1}, \\mbf{w})$}\n\\State{$\\triangleright$ Generate $\\hat{\\mbf{x}}_t$ via $\\mbf{x}_t \\sim p_{\\bs{\\theta}_{\\mbf{x}}}(\\mbf{x}_{t} | \\mbf{x}_{1:O}, \\hat{\\mbf{x}}_{O+1:t-1},\\hat{\\mbf{z}}_t, \\mbf{w}) $}\n\\EndFor\n\\Outputs{}\\vspace{-4mm}\n\\State{$\\triangleright$ Generated human motion sequence $\\hat{\\mbf{x}}_{O+1:O+G}$}\n\\end{algorithmic}\n\\label{algo:vem-enhancement}\n\\end{algorithm}\n\\vspace{-9mm}\n\n\\section{Action-classifier}\nAs explained in Sec.~4.1 and Sec.~4.3 of the main paper, we trained a RNN-based classifier to calculate the implicit evaluation metrics ACC and FID following~\\cite{guo2020action2motion,petrovich2021action}.\nThe classifier we used is build upon 2 simple GRU layers with hidden size of 128. \nWhen training on Human3.6M dataset, we found that some action classes do not differ much from each other, which makes it difficult to train a good classifier. As our goal is to have a classifier which offers good feature, we believe the classifier with low accuracy on real data is not reliable enough, so we group the 5 similar actions, \nand trained the classifier on these 5 groups instead of the 15 original classes. The groups of actions are detailed in Tab~\\ref{table:group_15_class}.\\\\\nNote that even on the classifier trained on the 15 original classes, our method still performs better than others, as shown in Tab.~\\ref{table:res_15_class}.\n\n\\input{tab_supp_act_class_h36m}\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nIn this paper we have investigated the use of attention combined with temporal probabilistic models for human motion generation. In particular, we proposed HiT-DVAE{}, a variational method modeling temporal dependencies between the observations and the latent variables, and exploiting attention to select which observations will be used to inform the generation of the current frame. Up to our knowledge, this is the first time that models with temporal latent variables and the use of attention are proposed to handle the human motion generation task. \nWe exhaustively evaluated our method on two widely used datasets, HumanEva and Human3.6M, and reported state-of-the-art results\n\\bibliographystyle{splncs04}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{SecIntro}\n\nA defining prediction of hierarchically clustering models is that the Universe must be teeming with low-mass systems left over from the collapse of the early stages of the hierarchy \\citep{White1978}. The $\\Lambda$ cold dark matter ($\\Lambda$CDM) paradigm is no exception; indeed, the abundance of $\\Lambda$CDM halos massive enough, in principle, to host a galaxy is so high that they outnumber faint galaxies by a large factor \\citep[see, e.g.,][]{Klypin1999,Moore1999}.  For example, more than $1,000$ halos with virial\\footnote{We define virial quantities as those calculated within a radius where the mean inner density equals 200 times the critical density of the Universe, $\\rho_{\\rm crit}=3H^2\/8\\pi G$. Virial quantities are identified by a ``200'' subscript.} mass exceeding $10^8 \\rm \\ M_\\odot$ are expected within $\\sim 2 \\rm \\ Mpc$ from the barycentre of the Local Group (LG), a region that contains fewer than $100$ galaxies with baryonic masses exceeding $10^5 \\rm \\ M_\\odot$ \\citep[][and references therein]{Sawala2016}.\n\nThis discrepancy is usually explained by assuming that galaxies fail to form in halos below a certain halo mass, leaving a large number of systems essentially ``dark'', or free of stars.  The main culprit is cosmic reionization, which heats most baryons to $\\sim 10^4 \\rm \\ K$ at relatively high redshift and prevents them from settling and condensing into galaxies in the shallow potential wells of low-mass halos\\cite[e.g.,][]{Bullock2000}\n\nThe existence of these ``dark'' minihalos\\footnote{Throughout this paper we shall refer to halos in the mass range $10^8<M_{200}\/{\\rm M_\\odot}<5\\times 10^{9}$ as minihalos.} is a cornerstone prediction of $\\Lambda$CDM and their search has attracted great interest. Their presence could be inferred from their gravitational effects on dynamically cold structures, such as galaxy disks \\citep[see][and references therein]{Feldmann2015} or thin stellar streams \\citep{Ibata2002,Johnston2002,Carlberg2009}, or else from the distortions they may induce in gravitationally lensed images of distant galaxies \\citep{Mao1998,Dalal2002,Vegetti2010,Hezaveh2016}. High-energy physicists, on the other hand, seek them as potential sources of energetic gamma rays powered by dark matter particle annihilation \\citep{Diemand2007,Springel2008,Charles2016}.\n\nA more prosaic alternative is to look for direct signatures of their baryonic content (which should be a nearly pristine H+He gaseous mix, given the lack of internal enrichment sources) in redshifted absorption against the light of luminous distant objects. Indeed, minihalos were once hypothesized as responsible for the forest of Lyman-$\\alpha$ lines in the spectra of high redshift quasars \\citep{Rees1986,Ikeuchi1986}, until it was realized that the large coherence length of absorption features was more naturally explained by the density ripples induced by CDM-driven fluctuations on larger scales \\citep[see][for a review]{Rauch1998}.\n\n``Dark'' minihalos might also be detectable in 21 cm emission and are actively sought in \\ion{H}{I} surveys of the local Universe \\citep[see, e.g.,][for a recent review]{Giovanelli2016}. Indeed, minihalos were proposed early on as hosts of the ``high velocity'' clouds (HVCs) of neutral hydrogen seen in 21 cm surveys of large areas of the sky \\citep{Blitz1999,Braun1999}. \n\nThe large sizes of HVCs, however, were shown to be incompatible with that interpretation: current models predict that gas in minihalos should be highly ionized by the cosmic UV background, except for a small central ``core'' of neutral hydrogen \\citep{Sternberg2002}.  The mass and size of the neutral core depend sensitively on the mass of the halo and on the pressure of the surrounding medium. \\ion{H}{I} cores of $\\sim$ kpc size and mass $10^5$-$10^6 \\rm \\ M_\\odot$ are expected in halos with virial mass in the $10^9$-$10^{10} \\rm \\ M_\\odot$ range. At a putative distance of $\\sim 1$ Mpc, these clouds would be much smaller and fainter than the typical HVC but still within range of current surveys \\citep{Giovanelli2010}. \n\nThe most promising minihalo candidates are the Ultra Compact High Velocity Clouds (UCHVCs) detected in surveys such as ALFALFA \\citep{Adams2013} and GALFA \\citep{Saul2012}. Their sizes and fluxes are consistent with minihalos in the Local Group volume, a result that has prompted deep follow-up imaging of some of the most prominent UCHVCs without obvious luminous counterparts in existing galaxy catalogs \\citep[see, e.g.,][]{Sand2015,Bellazzini2015b}. These searches have revealed new dwarf galaxies, as illustrated by the discovery of Leo P, a gas-rich star forming dwarf at the edge of the Local Group \\citep{Giovanelli2013}. \n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{FigMasses}\n    \\caption{{\\it Top:} Baryonic mass content of simulated halos at $z=0$ as a function of halo virial mass for the L1 simulations. The blue curve indicates the median stellar mass of simulated galaxies (measured within the galactic radius, $r_{\\rm gal}$), whereas the green dashed curve shows the median total {\\it bound} baryonic mass of the same galaxies, measured within the virial radius, $r_{200}$. The baryon mass of ``dark'' minihalos (i.e., star free) is shown with open circles; red for  {\\small RELHICs} and grey for {\\small COSWEBs}. The magenta solid line indicates the predicted gas mass of {\\small RELHICs} according to the model of Appendix~\\ref{SecApp}. For comparison, we also show the stellar mass vs halo mass relation derived using the abundance-matching technique by~\\citet{Moster2013} (dot-dashed line) and~\\citet{Behroozi2013} (dotted line). Gas-free minihalos are indicated in the middle panel. {\\it Bottom:} Fraction of ``dark'' minihalos (thick dashed line). The other curves show the fraction of {\\small RELHICs} computed for the resolution levels L1 (red solid) and L2 (purple dot-dashed), as well as for EAGLE run Recal-L025N0752 (brown dotted). Note the excellent agreement of all these simulations at the high-mass end.}\n    \\label{FigMasses}\n\\end{figure}\n\nSome UCHVCs are thus clearly associated with faint galaxies, and, therefore, with minihalos. The converse, however, seems less clear. The sheer number of UCHVCs preclude many, if not most, of them from being associated with minihalos \\citep{Garrison-Kimmel2014}, but it is unclear what criteria might be used to discriminate true minihalo candidates from \\ion{H}{I} ``debris'' in the Galactic halo. \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{FigSpatialDistribution}\n    \\caption{Panels, from left to right, show the distribution of gas, dark matter, and stars in one of the simulated volumes (namely V01-L1). Top and bottom rows show different orthogonal projections of a $7$ Mpc cubic box centred at the barycentre of the two main galaxies. Projections are chosen respect to the \"sheet\" that cross the volume. Colours indicate projected density, on a logarithmic scale. The location of {\\small RELHICs} and {\\small COSWEBs} are indicated in the left and middle panels, respectively. Note that {\\small RELHICs} shun the high-density regions of the volume near the main galaxies, where cosmic web stripping is important and the population of {\\small COSWEBs} dominates. Arrows show the positions of the individual {\\small COSWEB} and {\\small RELHIC} shown in Fig.~\\ref{FigC} and Fig.~\\ref{FigR}, respectively.}\n    \\label{FigSpatialDistribution}\n\\end{figure*}\n\nWe examine these issues here using the cosmological hydrodynamical simulations of the {\\small APOSTLE}\/EAGLE projects. We focus, in particular, on the gas content of ``dark'' minihalos. Given their lack of stars, and therefore of any energetic ``feedback'', two main mechanisms play a role in setting the gaseous content of minihalos: (i) cosmic reionization, which should evaporate much, but not all, of the baryons from such shallow potential wells, and (ii) ram pressure stripping by the cosmic web, which may unbind the gaseous content from minihalos that travel through dense filaments or ``pancakes'' of gas. \\citet{Benitez-Llambay2013} show that the latter effect may reduce substantially the baryonic content of low-mass halos, especially in high-density regions such as groups of galaxies.\n\nThis paper is organized as follows. We begin in Section~\\ref{SecSims} with a brief summary of the {\\small APOSTLE}\/EAGLE suite of simulations, followed in Section~\\ref{SecRes} by a discussion of our main results. We analyse the baryon content of dark minihalos in Sec.~\\ref{SecMgasM200}. We identify two populations of dark minihalos in Sec.~\\ref{SecCR}; one where the properties of the gas component are mainly set by the ionizing background, and another where the gas has been nearly completely removed by cosmic web stripping. We discuss the properties of the former in Sec.~\\ref{SecR}, where we also present a simple model that reproduces their main structural properties. We use this model to make predictions for their \\ion{H}{I} content, column density profiles, and 21 cm line widths, and compare them with the properties of UCHVCs in Sec.~\\ref{SecObs}. We conclude with a brief summary of our main conclusions in Sec.~\\ref{SecConc}, and provide an appendix with an analytic model for the gas density and temperature profiles in minihalos.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{FigC}\n    \\caption{Evolution of a cosmic web stripped system from the V01-L1 simulations, with $ M_{200} \\sim 10^{9.1} \\rm M_{\\odot}$ at redshift $z=0$. The left panels show the evolution of (from top to bottom) the dark mass and gas mass within $r_{200}$; the logarithm of the virial temperature, $T_{200} \\sim 10^4 {\\rm K} \\left ( V_{200} \/ 17 {\\rm km \\ s^{-1} }\\right )^2$, and (bottom panel) the gas mass fraction ($M_{\\rm gas}\/M_{200}$) in units of the universal value, $f_{\\rm bar}=\\Omega_b\/(\\Omega_0+\\Omega_{\\rm b})$. The four panels on the right show snapshots of the evolution, at the times indicated by the solid circles in the left panels. Note how reionization heats up the gas at early times, reducing the halo baryon content. In this particular example, the system loses essentially all of its bound mass after passing through a dense region of the cosmic web at $z\\sim 0.6$. }\n    \\label{FigC} \n\\end{figure*}\n\n\\section{Numerical Simulations}\n\\label{SecSims}\n\n\\subsection{The {\\small APOSTLE} project}\n\nWe use a suite of cosmological hydrodynamical simulations from the {\\small APOSTLE}\\footnote{{\\small APOSTLE} stands for ``A Project Of Simulating The Local Environment''} project~\\citep{Fattahi2016a,Sawala2016}. These are zoom-in simulations that follow the formation of various Local Group-like cosmological environments. Twelve different realizations are followed at three different resolutions (L1, L2, and L3, in order of decreasing resolution). All volumes are selected from the DOVE N-body simulation~\\citep{Jenkins2013} using the criteria described in~\\cite{Fattahi2016a}. DOVE adopted a cosmological model with parameters consistent with WMAP7 measurements~\\citep{Komatsu2011}, listed in Table~\\ref{Tab:table1}.\n\n\n\n\\begin{table}\n\t\\centering\n\t\\caption{This table summarizes the main parameters of the simulations used in our analysis}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{ccccr}\n\t\t\\hline\n\t\t$\\rm H_{0}$ & $\\rm \\Omega_{0}$ & $\\Omega_{b}$ & $\\Omega_{\\Lambda}$ & $z_{\\rm reion}$\\\\\n\t\t\\hline\n\t\t$70.4 \\ \\rm (km\/s\/Mpc) $ & $0.272$ & $0.0455$ & $0.728$ & $11.5$ \\\\\n\t\\end{tabular}\n\t\n\t\\begin{tabular}{ccccc}\n\t\t\\hline\n\t\tID & $\\rm \\left (M_{gas}\/\\rm M_{\\odot}\\right )$ & $\\rm \\left ( M_{drk}\/\\rm M_{\\odot} \\right )$ & $\\rm n_{part}$ & $(\\epsilon_0\/\\rm pc)$ \\\\\n\t\t\\hline\n\t\t$\\rm V01-L1$ & $9.89 \\times 10^3$ & $4.92 \\times 10^4$ & $\\sim 2.6 \\times 10^8$ & $134$\\\\\n\t\t$\\rm V04-L1$ & $4.93 \\times 10^3$ & $2.45 \\times 10^4$ & $\\sim 5.4 \\times 10^8$ & $134$\\\\\n\t\t$\\rm V11-L1$ & $1.01 \\times 10^4$ & $5.02 \\times 10^4$ & $\\sim 2.4 \\times 10^8$ & $134$\\\\\n\t\t\n\t\t\\hline\t\t\n\t\\end{tabular}\n\t\\label{Tab:table1}\n\\end{table}\n\n\n\nEach simulated volume contains a relatively isolated pair of halos of combined virial mass in the range $10^{12.2}$-$10^{12.5} \\,\\rm M_{\\odot}$; the two halos (meant to represent the Milky Way and M31) have separations $\\sim 600$-$1000 \\rm \\ kpc$, approach with a relative radial velocity $-250$-$0$ km s$^{-1}$, and have tangential velocities that do not exceed $100$ km s$^{-1}$~\\cite[see ][]{Fattahi2016a}.\n\nDifferent resolution levels are chosen so that each level improves on the previous by a factor of $\\sim 12$ in particle mass, and a factor of $\\sim 12^{1\/3}$ in gravitational force resolution. The highest-resolution level (L1) has a baryon particle mass of $(5 - 10) \\times 10^3 \\rm \\ M_\\odot$ (dark matter particles are $\\sim 5$ times heavier), and a Plummer-equivalent gravitational softening of $\\epsilon_0 = 134$ pc. At the time of writing, all twelve volumes have been completed at L3 and L2 resolution but only three volumes (V01, V04, and V11) have also been completed at L1 resolution. We focus on those three volumes in the rest of this paper, although we use results from lower resolution runs to assess the sensitivity of our results to numerical resolution.\n\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{FigR}\n    \\caption{As Fig.~\\ref{FigC}, but for the case of {\\small RELHIC} with $M_{200} \\sim 10^{9.1} \\rm \\ M_{\\odot}$ . This halo has its baryon content substantially reduced by cosmic reionization, but it is not affected by cosmic web stripping, since it inhabits the low density outskirts of the Local Group volume. Its baryon content is essentially constant after $z\\sim 5$-$6$ and is set by the hydrostatic balance of UV-heated gas in the potential of the minihalo. (see Appendix~\\ref{SecApp}).}\n    \\label{FigR}\n\\end{figure*}\n\n{\\small APOSTLE} simulations were run with the same modified version of the {\\tt P-Gadget3} code, last described by ~\\cite{Springel2005}, which was used to run the simulations of the {\\tt EAGLE} project~\\citep{Schaye2015, Crain2015\nThe {\\tt EAGLE} code includes a set of subgrid prescriptions to account for the effects of radiative cooling, photoheating, star formation, energy feedback from star formation, and AGN feedback, among others. The adjustable numerical parameters of the subgrid modules were chosen to provide an approximate match to the galaxy stellar mass function and galaxy sizes over cosmological volumes, and correspond to that of the EAGLE 'Reference' runs.  As discussed by \\citet{Sawala2016} the three {\\small APOSTLE} resolutions yield very similar galaxy stellar mass functions (over the range resolved with more than 50 star particles), indicating good numerical convergence in the dwarf galaxy regime probed by {\\small APOSTLE}. In addition, the {\\small APOSTLE} galaxy mass-halo mass relation matches rather well the abundance-matching constraints for galaxies in the range $10^7 \\le M_{\\rm str}\/M_{\\odot} \\le 10^{10}$ (see Fig.~\\ref{FigMasses}). We conclude that, at least in the low-mass regime, our results are weakly sensitive to numerical resolution, and we therefore attempt no further parameter recalibration.\n\n\n\\subsection{Reionization and the UV background}\n\nCooling and photoheating processes in the {\\tt EAGLE} code are implemented following the procedures outlined in~\\cite{Wiersma2009a}. In brief, the thermodynamic state of the gas is modelled using {\\tt CLOUDY}~\\citep{Ferland1998}, assuming ionization equilibrium with the cosmic microwave background (CMB) and a  spatially uniform evolving UV\/X-ray background radiation field as calculated by ~\\cite{Haardt2001} (HM01 thereafter) in the optically thin regime. The simulation neglects self-shielding which could have an impact on regions with $n_{\\rm H} > 10^{-2} \\rm \\ cm^{-2}$, and temperatures below $\\sim 10^4 \\rm \\ K$~\\citep[see, e.g,][]{Schaye2001,Rahmati2013}\\footnote{Our study focuses on systems with gas densities $n_{\\rm H} \\le 10^{-1} \\rm \\ cm^{-2}$ and temperatures $T \\ge 10^4 \\rm \\ K$. Self-shielding could in principle change the properties of some of these systems since the temperature would be slightly reduced, thus increasing the \\ion{H}{I} mass predicted in Sec.~\\ref{SecHI}.}.\nReionization of the Universe is modelled by switching on the HM01 background radiation field at redshift $z_{\\rm reion} = 11.5$. The photoheating and photoionizing rates are kept fixed in the redshift range $z=9-11.5$. For redshift $z \\le 9$, the UV background is allowed to evolve, and reaches a maximum at redshift $z \\sim 2$. In addition, an extra heating of 2eV per proton mass is injected to the gas particles at $z_{\\rm reion}$, which accounts for a boost in the photoheating rates during reionization relative to the optically thin rates assumed here, ensuring that the photoionized gas is rapidly heated to a temperature of $\\sim 10^4 \\rm \\ K$. This is done instantaneously for \\ion{H}, but for \\ion{He}{II} the extra heat is distributed in redshift with a Gaussian of width $0.5$, centred at $z=3.5$. \n\nFor redshift $z>11.5$, the net cooling\\footnote{We refer to the net cooling of the gas as the difference between the radiative heating and cooling processes.} of the gas is computed by exposing it to the CMB and the photodissociating background obtained by cutting the HM01 spectrum at 1 Ryd. Note that the presence of photodissociating radiation and the finite resolution of our simulations imply that we cannot model the formation of Pop III stars via $\\rm H_{2}$ cooling in minihalos, which could play a role for isolated halos of mass $\\sim 10^5 \\rm \\ M_{\\odot}$.\n\n\n\\subsection{Halo finding}\n\nHalos are identified in the simulations using the group finder {\\tt SUBFIND}~\\citep{Springel2001, Dolag2009}, which identifies self-bound substructures within a catalogue of friends-of-friends (FoF) halos built with a linking length of 0.2 times the mean interparticle separation.  {\\tt SUBFIND} provides a list of  self-bound subhalos within each FoF halo, organized as a ``central'' halo and its respective ``satellites''. \n\n\nMost of our analysis is based on central halos identified at redshift $z=0$ within a spherical volume of radius $3.5 \\rm \\ Mpc$ centred at the barycentre of each simulated ``Local Group''. \nWe keep for analysis all central halos with $M_{200} \\ge 10^8 \\ \\rm M_{\\odot}$ (i.e., with typically more than $3000$ dark matter particles). These limits in volume and mass ensure that all selected halos are far enough from the boundaries of the high-resolution zoom-in region and that we are able to resolve them confidently. \n\n\\section{Results}\n\\label{SecRes}\n\n\\subsection{Baryonic content of {\\small APOSTLE} halos}\n\\label{SecMgasM200}\n\n\n\\begin{figure}\t\n\\includegraphics[width=\\columnwidth]{FigRhoT}\n\\caption{Temperature-density diagram for all gas particles bound to {\\small RELHICs}. All particles of all {\\small RELHICs} are shown, and compared with two curves indicating (i) the loci of particles where the photoheating timescale equals the age of the Universe, $t_{H} \\approx 13.76 \\rm \\ Gyr$ (green dashed curve); and (ii) the loci where photoheating and radiative cooling are in equilibrium (thick magenta curve). Gas in {\\small RELHICs} has been photoheated to a density-dependent temperature that matches one of those two regimes. The tight relation between density and temperature that results defines a temperature-density relation for gas in {\\small RELHICs} (red dashed line) that can be used to derive density and temperature profiles assuming that the gas is in hydrostatic equilibrium to the potential well of the dark matter.}\n    \\label{FigRhoT}\n\\end{figure}\n\nWe begin by analysing the baryonic content within the virial boundaries of the simulated halos. This is presented in the top panel of Figure~\\ref{FigMasses}, where we show the relation between virial mass and the mass of various baryonic components for the three ``high-resolution'' volumes. The oblique dashed line indicates, for reference, the theoretical maximum baryonic mass within the virial radius, $M_{\\rm bar}=f_{\\rm bar}M_{200}$, where $f_{\\rm bar}=\\Omega_{\\rm b}\/(\\Omega_{\\rm 0}+\\Omega_{\\rm b}=0.167$ is the universal baryon fraction.\n\nThe blue solid line indicates the median stellar mass \"bound\" to the central galaxies. Note that we only consider ``central'' galaxies in this figure; i.e., the most massive subhalo of each FoF halo. \n\nThe median stellar mass plummets below a virial mass of $\\sim 10^{10} \\rm \\ M_\\odot$, mainly because not all those low-mass halos harbour luminous galaxies. This may be seen from the thick black dashed line in the bottom panel of Fig.~\\ref{FigMasses}, which shows the fraction of central galaxies that do not have stars. All halos above $10^{10} \\rm \\ M_\\odot$ have luminous galaxies, but the fraction dips to $50\\%$ at $M_{200} \\sim 5 \\times 10^9 \\rm \\ M_\\odot$. Below $10^9 \\rm \\ M_\\odot$ essentially all halos are ``dark''\\footnote{Strictly speaking, these halos have galaxies less massive than a few $10^3 \\rm \\ M_\\odot$ in stars, the mass of one baryon particle at this resolution level.}. \n\nThe median total baryon mass bound to {\\it luminous} halos, measured within $r_{200}$, is shown by the green curve in Fig.~\\ref{FigMasses}, with a shading that indicates $\\pm1\\sigma$ dispersion. The total gas mass within $r_{200}$ bound\\footnote{We note that {\\tt SUBFIND} can at times underestimate these masses, especially in regions where the mean ambient gas density is comparable to the density in the outer regions of the minihalo or when a minihalo is embedded in hotter gas; the masses shown here have been carefully recomputed to take those issues into account.} to ``dark'' (i.e., star free) halos is shown with open circles. Two populations are clearly apparent: one where the bound gas mass is so small ($<10^5 \\rm \\  M_\\odot$, shown in grey) that it can barely be measured (the gas particle mass is $\\sim 10^4 \\rm \\ M_\\odot$ at resolution level L1); and another where the gas mass correlates tightly with virial mass (shown in red). We shall hereafter refer to the former as ``{\\small COSWEBs}'' (short for cosmic web-stripped halos) and to the latter as ``{\\small RELHICs}'' (for Reionization-Limited \\ion{H}{I} Clouds). \n\n\nBefore discussing the origin of these two populations, we show in the bottom panel of Fig.~\\ref{FigMasses} their relative fractions as a function of mass, as well as the dependence on numerical resolution. The thick solid red curve indicates the fraction of {\\small RELHICs} in the highest-resolution (L1 level) runs: {\\small RELHICs} inhabit halos spanning a small range in virial mass, $3\\times 10^{8}<M_{200}\/\\rm M_\\odot<5\\times 10^{9}$, making up about half of all systems at the mid point of that range. Lower-resolution simulations (L2 runs, shown with a dot-dashed line), as well as simulations of a larger volume\\footnote{EAGLE run Recal-L025N0752.} at L2 resolution yield {\\small RELHIC} fractions similar to that of L1 runs at the high-mass end of the range, but underestimate their abundance at the low-mass end, below a virial mass of $\\sim 2\\times 10^9 \\rm \\ M_\\odot$.  The drop in {\\small RELHIC} fractions below that mass is mainly a result of limited numerical resolution. As we discuss below, the main difference between {\\small RELHICs} and {\\small COSWEBs} is environmental, so we would expect the ratio of the two to approach a constant at low virial masses.  Interestingly, although not shown here, {\\small RELHICs} in all of these runs track the same gas mass-virial mass trend shown in the upper panel of Fig.~\\ref{FigMasses}, regardless of resolution.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{FigRadProf}\n        \\caption{Acceleration (top left), temperature (bottom left) and gas density (top right) profiles for one {\\small RELHIC} of virial mass $\\rm M_{200} \\sim 5\\times 10^{9} \\rm \\ M_{\\odot}$ at redshift $z=0$. Points indicate individual particles bound to the {\\small RELHIC}. The dashed red line in the top left panel is an NFW fit to the spherically averaged acceleration profile of the halo ($a(r)=GM(r)\/r^2$). Dashed curves in the other two panels are results of the model presented in Appendix~\\ref{SecApp}, where the gas is simply assumed to be in hydrostatic equilibrium. The inset in the bottom right panel shows, with red-dashed line, the temperature-density relation (Fig.~\\ref{FigRhoT}) used in the model (note that we have inverted the x-axis respect to Fig.~\\ref{FigRhoT} for clarity). For comparison, we also show with a black dot-dashed line the temperature-density relation expected from a stronger UV background, which in this case corresponds to the $z=1$ HM01 spectrum. Blue curves in the right hand panels show the neutral hydrogen profiles, number density in the top, and column density in the bottom.}\n\t\\label{FigRadProf}\n\\end{figure*}\n\n\n\\subsection{COSWEBs and RELHICs}\n\\label{SecCR}\n\nThe spatial distribution of the two populations ({\\small RELHICs} and {\\small COSWEBs}) are clearly different, as shown in Fig.~\\ref{FigSpatialDistribution}, hinting at an environmental origin of their distinction. This figure shows two orthogonal projections of one {\\small APOSTLE} volume run at the highest-resolution (V01-L1): the two main LG galaxies are clearly visible near the centre, surrounded by a two-dimensional ``sheet'' of gas that extends out to several Mpc from the LG barycentre. {\\small COSWEBs}, shown as open squares in the middle panels, cluster around the main galaxies and inhabit the mid plane of the sheet, whereas {\\small RELHICs} (open circles in the left panels) populate underdense regions in the periphery of the LG volume.\n\nThe evolution of a typical {\\small COSWEB} is shown in Fig.~\\ref{FigC}. The top left panel shows the evolution of the various mass components, measured within the virial radius of the most massive progenitor. As expected, the dark mass grows monotonically, with occasional jumps corresponding to merger events. The gas mass largely follows suit, except at late times, when it drops dramatically, in this example at $z\\sim 0.6$. The drop is more clearly seen in the bottom left panel, which tracks $\\tilde f_{\\rm bar}$, the baryon fraction of the halo expressed in units of the universal baryon fraction. \n\n\n\n\\begin{figure}\n\t\\includegraphics[scale=0.5]{FigModProf}\n    \\caption{Model gas density profiles for {\\small RELHICs}. The model, which is described in detail in Appendix~\\ref{SecApp}, solves the equations of hydrostatic equilibrium in NFW halos with the assumption that gas follows the temperature-density relation shown in Fig.~\\ref{FigRhoT} and the boundary condition that densities approach the mean density of the Universe (${\\bar n}_{\\rm H}$) at $r\\gg r_{200}$. The top panel shows gas densities, and the bottom panel column densities of \\ion{H}{I}. Neutral fractions are computed as in the appendix A1 of \\citet{Rahmati2013}. Two characteristic radii are indicated in the top panel; the virial radius of each halo, $r_{200}$, and the radius where the total enclosed gas mass equals $f_{\\rm bar}\\, M_{200}$, with $f_{\\rm bar}$ the universal baryon fraction. Curves in each panel correspond to the same halos, from top to bottom, and are coloured by either gas mass (top panel) or virial mass (bottom panel).}\n    \\label{FigModProf}\n\\end{figure}\n\n\nThe four panels on the right of Fig.~\\ref{FigC} show snapshots of the gas component at different stages of the evolution, at times marked by the solid circles in the left panels. The top two right panels show the {\\small COSWEB} just before and after reionization, when the gas is suddenly heated to $\\sim 10^4$K, a temperature that exceeds the virial temperature of the halo at that time (see the bottom green curve, which tracks $\\log_{10} T_{\\rm 200}\/$K). The gas is therefore too hot to remain bound, and evaporates from the potential well of the halo. By $z\\sim 2$, the halo has only been able to retain about $1\/20$th of its baryons.  Just after $z \\sim 1$, however, the {\\small COSWEB} travels through an overdense region of the cosmic web, which ram-pressure strips the remainder of the gas, reducing its bound gas content to a negligible amount. The {\\small COSWEB} is unable to re-accrete gas by redshift $z=0$, mainly because it is now moving relatively fast within the much hotter LG environment. {\\small COSWEBs} are thus star\/gas-free minihalos whose gaseous content has been removed by the cosmic web~\\citep[see][for a more extensive discussion of cosmic web stripping]{Benitez-Llambay2013}.\n\n\nBy contrast, {\\small RELHICs} are minihalos that have avoided dense regions of the cosmic web, and have therefore been able to retain more baryons. The evolution of a typical {\\small RELHIC} is shown in Fig.~\\ref{FigR}. Comparing this to that of the {\\small COSWEB} in Fig.~\\ref{FigC}, we see that the main difference is that the {\\small RELHIC} baryon fraction remains more or less constant after the initial drop caused by reionization. At $z=0$ this particular {\\small RELHIC} has been able to retain about $1\/30$th of the universal baryon fraction.  What sets the final baryon content of {\\small RELHICs} and, in particular, what is the origin of the strong correlation between bound gas mass and virial mass shown in Fig.~\\ref{FigMasses}? We address these questions next.\n\n\n\n\\section{RELHICs}\n\\label{SecR}\n\n\\subsection{Gas densities and temperatures}\n\\label{SecRhoT}\n\nWe start by considering the thermodynamic state of the gas in {\\small RELHICs}. This is shown in Fig.~\\ref{FigRhoT}, where we plot the density and temperature of all gas particles bound to the 249 {\\small RELHICs} identified in all three simulated volumes. We have excluded 9 {\\small RELHICs} (i.e., $3$ per volume) with masses $M_{200} \\gtrsim 10^{9.7} \\rm M_{\\odot}$, whose central gas densities reach values $n_{\\rm H} \\approx 10^{-1} \\rm \\ cm^{-3}$, and thus their central thermodynamic properties are expected to be governed by the effective equation of state imposed in the EAGLE code on the unresolved multiphase interstellar\/star forming medium. \n\nThe gas in the remaining {\\small RELHICs} spans nearly seven decades in density, from the mean baryon density of the Universe (${\\bar n_{\\rm H}}\\approx 10^{-6.7} \\rm \\ cm^{-3}$) to densities 100 times below the threshold chosen for star formation in EAGLE ($n_{\\rm H, th}\\approx 10 \\,$cm$^{-3}$ for a gas with primordial composition)\\footnote{We discuss the impact of the particular choice of the star formation density threshold on our results in Appendix~\\ref{SecApp2}.}. Gas temperatures are a non-monotonic function of density, first climbing to a maximum of $\\sim 4\\times 10^4 \\rm \\ K$ at $n_{\\rm H} \\sim 10^{-4.8} \\rm \\ cm^{-3}$, and then dropping gradually at higher densities, approaching $10^4 \\rm \\ K$.\n\nThe gas is, on average, hotter than the virial temperature of a typical {\\small RELHIC} ($T_{200}=10^{4.1} \\rm \\ K$ for a virial mass of $2\\times 10^9 \\rm \\ M_\\odot$), implying that the gas temperature is largely set by the ionizing background, and not by the gravitational collapse of the halo. This is further demonstrated by the green dashed curve, which indicates where the photoionizing heating time-scale equals the age of the Universe, $t_{H} \\approx 13.76 \\rm \\ Gyr$. Temperatures of low density gas in {\\small RELHICs} are clearly set by the ionizing background. \n\nAt densities $> 10^{-4.8}$cm$^{-3}$ radiative cooling induced by collisional effects becomes more important and the gas settles at the ``equilibrium'' temperatures where radiative cooling effects balance photoheating from the ionizing background, shown by the thick purple line in Fig.~\\ref{FigRhoT}~\\citep[see e.g.,][]{Haehnelt1996, Theuns1998}. As is clear from this figure, these two regimes describe very well the temperature-density relation of gas in {\\small RELHICs} (shown by the red dashed curve).\n\n\n\\subsection{Gas masses}\n\\label{SecGasM}\n\nThe $n_{\\rm H}$-$T$ relation followed by gas particles in {\\small RELHICs} effectively defines a pressure-density relation, $P=P(\\rho)$, that enables us to estimate the gas mass bound to a halo of given virial mass. This may be done by assuming that the gas is in hydrostatic equilibrium within the potential of the dark halo and solving: \n\\begin{equation}\n\\displaystyle\\frac{1}{\\rho}\\frac{dP}{dr} = -\\frac{GM(r)}{r^2},\n\\label{EqHydroEquil}\n\\end{equation}\nto give a density profile that may be integrated to compute the total gas mass within $r_{200}$, once a boundary condition (e.g., an external pressure) is chosen. \n\nThe simplest choice is to assume that far from the virial boundary the gas reaches the mean baryon density of the Universe, at the appropriate temperature set by the ionizing background: this specifies the external pressure that closes the set of equations, enabling a simple estimate of {\\small RELHIC} gas masses. \n\nWe present details of the calculation in Appendix~\\ref{SecApp} and show the main result by the thick purple line in Fig.~\\ref{FigMasses}. Despite its simplicity, the model predicts accurately the gas mass of {\\small RELHICs} for halos not exceeding virial masses of order $5\\times 10^9 \\rm \\ M_\\odot$. At higher masses gravitational heating becomes important and, in addition, the central densities become high enough for self-shielding and cooling processes to become important; in those halos the gas would not be able to stay in hydrostatic equilibrium, but will collapse into a rotationally supported disk where it may form stars. Indeed, very few, if any, halos above $5\\times 10^9 \\rm \\ M_\\odot$ remain ``dark'', as shown in the bottom panel of Fig.~\\ref{FigMasses}.\n\n\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{FigHI}\n    \\caption{Properties of {\\small RELHICs} (open circles), compared with those of the model presented in Appendix~\\ref{SecApp}. Left panels show, as a function of \\ion{H}{I} mass within $r_{200}$, the total gas mass (top) and the central \\ion{H}{I} column density $\\rm N_{\\ion{H}{I},0}$. Right panels show the total gas mass vs the velocity dispersion in bulk motions of the gas (top); and the central \\ion{H}{I} column density $\\rm N_{\\ion{H}{I},0}$ vs \\ion{H}{I} size, $R_{\\rm HI}$. Several characteristic radii for the latter are shown; from top to bottom the dashed lines indicate the radius of the iso column density contour of $10^{20}$, $10^{19}$, etc, in units of cm$^{-2}$. Each {\\small RELHIC} is shown at the column density immediately  below its central value; for example, the radius of the $10^{18}$ cm$^{-2}$ contour is shown for  those {\\small RELHICs} with central column densities in the range $10^{18}<N_{\\ion{H}{I},0}\/$cm$^{-2}<10^{19}$, and so on.}\n    \\label{FigHI}\n\\end{figure*}\n\n\\subsection{Gas and temperature profiles}\n\\label{SecRhoTvsR}\n\nWe may test further the simple hydrostatic model described in Appendix~\\ref{SecApp} by using it to predict the density and temperature profiles of the gas component of {\\small RELHICs} and comparing them with the simulation results. Figure~\\ref{FigRadProf} shows an example for a relatively massive {\\small RELHIC}; $M_{200} \\sim 5\\times 10^{9} \\rm \\ M_{\\odot}$ and $\\rm M_{gas} \\sim 3\\times 10^{7} \\rm \\ M_{\\odot}$. \n\nWe first measure the acceleration profile of the halo, assuming spherical symmetry: $a(r)=GM(r)\/r^2$, where $M(r)$ is the total enclosed mass within radius $r$. Baryons contribute so little mass that this is effectively equivalent to the dark matter acceleration profile. We show $a(r)$ in the top left panel of Fig.~\\ref{FigRadProf} by the solid black line. The red dashed curve is a fit to this acceleration profile, namely, a Navarro-Frenk-White \\citep[][hereafter NFW]{Navarro1996,Navarro1997} profile with concentration parameter $c=11.4$. \n\nWe then integrate Eq.~\\ref{EqHydroEquil} numerically, using a fit to the temperature-density relation (see the red dashed curve in the inset of the bottom right panel of Fig.~\\ref{FigRadProf}), and normalizing the profile at the radius that contains half of all gas particles\\footnote{This normalization procedure improves fits to individual halos, but its results are not very different from those obtained assuming the simple boundary condition that the gas profile should converge to the mean density of the Universe at large radii.}. The result for the density profile may be seen in the top right panel of Fig.~\\ref{FigRadProf}. Clearly the predicted profile is in excellent agreement with the simulation (dots correspond to individual gas particles in the simulation). \n\nThe model temperature profile is shown in the bottom left panel of the same figure and is also in excellent agreement with the results from the simulation. We have verified that the model works equally well for other {\\small RELHICs} of different masses. We have also verified that the overall properties of {\\small RELHICs} are relatively insensitive to our choice of UV background. This may be seen in the inset of Figure~\\ref{FigRadProf}, where the  dot-dashed curve shows the equilibrium temperature-density relation obtained if the intensity of the UV background is increased to match the $z=1$ HM01 spectrum. Using that relation leads to changes in the gas mass predicted for {\\small RELHICs} smaller than $15\\%$.\n\nThis analysis demonstrates that the gas in {\\small RELHICs} is in hydrostatic equilibrium with the halo potential and that a simple model allows us to predict the mass, structural parameters, and radial profiles of the gas component of these ``dark'' minihalos accurately. In particular, we may use the same model to predict the neutral hydrogen content of {\\small RELHICs}, an issue to which we turn next.\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{FigPhoto.pdf}\n    \\caption{\\ion{H}{I} column density maps of four {\\small RELHICs}, chosen to have central column densities exceeding $10^{18}$ cm$^{-2}$. Contours correspond to column densities of $(5\\times 10^{14}, 1\\times 10^{15},4\\times 10^{15}, 1\\times 10^{18})$ atoms per $\\rm cm^{2}$. Note that at relatively high column densities, {\\small RELHICs} appear essentially round, with axis ratios $b\/a>0.8$. This is consistent with the idea that {\\small RELHICs} are in hydrostatic equilibrium in the potential of mildly triaxial dark matter halos.}\n    \\label{FigPhoto}\n\\end{figure}\n\n\n\\subsection{HI masses and radial profiles}\n\\label{SecHI}\n\nThe blue solid lines in Fig.~\\ref{FigRadProf} show the density profiles of neutral hydrogen, derived using the fitting formula given in appendix A1 of~\\cite{Rahmati2013}. This model uses a simple but accurate fit to the photoionization rates, obtained from radiative transfer simulations, where the scaling of the characteristic self-shielding density is taken from the analytic model of~\\cite{Schaye2001}, and  computes neutral fractions as a function of density and temperature assuming ionization equilibrium. In the inner regions of the example {\\small RELHIC} shown in Fig.~\\ref{FigRadProf} the gas is dense and cold enough to be $\\sim 100\\%$ neutral; the neutral fraction drops rapidly from the center outwards. The \\ion{H}{I} column density profile is shown in the bottom right panel of Fig.~\\ref{FigRadProf}: the profile is quite steep, in part due to the onset of self-shielding in the model, and it drops from a well-defined central value of $10^{20}$ cm$^{-2}$ to $10^{18}$ cm$^{-2}$ at $\\sim 1.25$ kpc from the centre\\footnote{We compute the column density profiles by integrating the density profile along the line-of-sight within a sphere of radius $2\\times r_{200}$.}. \n\nWe show the gas density and \\ion{H}{I} column density profiles in Fig.~\\ref{FigModProf}, as a function of halo virial mass or, equivalently, as a function of the total gas mass. {\\small RELHICs} have density profiles that vary in shape as the halo mass decreases, and central densities that correlate strongly with mass. Although the most massive {\\small RELHICs} may reach central \\ion{H}{I} column densities $\\rm N_{\\ion{H}{I},0}$ of $10^{21}$ cm$^{-2}$ these drop steeply with decreasing mass, dipping below $10^{15}$ cm$^{-2}$ for halos below $2.5 \\times  10^9 \\rm \\ M_\\odot$. This suggests that only the most massive {\\small RELHICs} might be detectable in 21 cm surveys such as ALFALFA~\\citep[e.g.,][]{Haynes2011}, which only reaches column densities exceeding $\\sim 10^{18}$ cm$^{-2}$.\n\nWe provide further structural properties of the \\ion{H}{I} component of {\\small RELHICs} in Fig.~\\ref{FigHI}, where we show the total \\ion{H}{I} mass within $r_{200}$ as a function of central \\ion{H}{I} column density and as a function of the total gas mass (left panels). The dashed lines show the results of the model described in Appendix~\\ref{SecApp} ( Fig.~\\ref{FigModProf}), which agree very well with the simulation results.  Clearly, neutral hydrogen makes up a very small fraction of the gaseous content of minihalos, confirming the expectations of the analytic models of \\citet{Sternberg2002}: minihalos are essentially spheres of ionized gas in hydrostatic equilibrium and they have a small core of neutral hydrogen.\n\nThe bottom right-hand panel of Fig.~\\ref{FigHI} shows several characteristic radii, where the {\\small RELHIC} \\ion{H}{I} column density drops from its central value to $10^{20}$, $10^{19}$,...,$10^{12}$ cm$^{-2}$, respectively (top to bottom).  The dashed lines indicate the results of the model profiles shown in Fig.~\\ref{FigModProf}, whereas the open circles correspond to simulated {\\small RELHICs}, grouped so that each is plotted at the radius where the column density drops by about one decade (or less) from the center. For example, the radii shown for {\\small RELHICs} with central column densities between $10^{18}$ and $10^{19}$ cm$^{-2}$ is that where the column density drops to $10^{18}$ cm$^{-2}$, and so on. Note that, defined this way, most {\\small RELHICs} within reach of current \\ion{H}{I} surveys (i.e., $M_{\\rm HI}> 10^4 \\rm \\  M_\\odot$; $N_{\\rm HI}>10^{18}$ cm$^{-2}$) are expected to be compact, sub-kpc systems \\citep{Sternberg2002}.\n\n{\\small RELHICs} are near hydrostatic equilibrium, so the random bulk motions of the gas are quite small compared with the characteristic velocity dispersion of the halos they inhabit. This may be seen in the top right panel of Fig.~\\ref{FigHI}, where we show, as a function of the total gas mass within $r_{200}$, the velocity dispersion of {\\small RELHIC} gas particles, compared with the characteristic rms velocities of dark matter particles, $\\sigma_{200}\\approx V_{200}\/\\sqrt{2}$.\nTypical bulk motions in {\\small RELHICs} are below $5$ km\/s in essentially all cases, well below $\\sigma_{200}$, implying that the broadening of the 21 cm line should be mostly thermal.\n\nFinally, we examine the morphologies of {\\small RELHICs} in Fig.~\\ref{FigPhoto}, which shows \\ion{H}{I} column density maps for $4$ relatively massive {\\small RELHICs} (see masses in figure legends). Drawing attention to the $10^{18}$ cm$^{-2}$ contour (black thick inner line), which corresponds to the sensitivity limit of surveys such as ALFALFA, we see that {\\small RELHICs} would appear essentially round in such surveys. This is a direct consequence of the fact that the gas is in hydrostatic equilibrium in the dark halo potential. Indeed, although $\\Lambda$CDM halos are intrinsically triaxial, the axis ratios of the potential are much less aspherical than those of the mass distribution \\citep[see, e.g.,][]{Hayashi2007}.\n\n\n\n\\subsection{UCHVCs as Local Group RELHICs}\n\\label{SecObs}\n\nWe explore now the possibility that {\\small RELHICs} might have been detected already in existing \\ion{H}{I} surveys. Given their sizes and low \\ion{H}{I} masses, we compare {\\small RELHICs} with the population of Ultra Compact High Velocity Clouds (UCHVCs) first discussed by \\citet{Giovanelli2010} in the context of the ALFALFA survey. These are identified as high signal-to-noise \\ion{H}{I} sources with sizes less than $30'$, and velocities well outside the range expected for Galactic rotation. Note that $30'$ corresponds to $\\sim 2$ kpc at a distance of $250$ kpc, so these sources might include sub-kpc {\\small RELHICs} in the Local Group.\n\nWe begin by noting that, as shown in Fig.~\\ref{FigSpatialDistribution}, {\\small RELHICs} shun the region close to the Local Group barycentre and mainly populate the underdense regions of its outskirts. Indeed, we find no {\\small RELHIC} within $500$ kpc of any of the two main LG galaxies in any of the three \"high-resolution\" volumes we have analysed. This has two important consequences; one is that, coupled with the low \\ion{H}{I} masses expected of {\\small RELHICs}, their \\ion{H}{I} fluxes will be quite low, and another is that few {\\small RELHICs} will have negative Galactocentric radial velocities, as most will still be expanding away outside the LG turnaround radius. \n\nWe show this in Fig.~\\ref{FigHVCs}, where we show, as a function of the \\ion{H}{I} flux\\footnote{We compute \\ion{H}{I} fluxes using the total \\ion{H}{I} mass within the virial radius of a {\\small RELHIC}, $M_{\\ion{H}{I}}$, and its distance to the LG primary galaxies expressed in Mpc, $d_{\\rm Mpc}$: $M_{\\rm HI}\/{\\rm M_\\odot}=2.36 \\times 10^5 \\, S_{21}\\, \\left (d\/\\rm Mpc \\right )^2$, with $S_{21}$ given in units of Jy km\/s. Note that as we consider the two main galaxies of each simulated LG, every {\\small RELHIC} is shown twice  in Fig.~\\ref{FigHVCs}.}, $S_{21}$, in units of $\\rm Jy \\ km \\ s^{-1}$, the \\ion{H}{I} size of the {\\small RELHIC}, defined as the mean radius $(\\sqrt{ab})$, where $a$ and $b$ are the semiaxes of the best fitting ellipse to its $10^{18}$ cm$^{-2}$ isodensity contour (top panel),  Galactocentric radial velocity, $V_{\\rm gsr}$ (second from top), the FWHM line broadening parameter\\footnote{$W_{50}$ is computed by adding in quadrature the broadening due to the gas temperature and its bulk velocity dispersion. The former dominates, and is given by $T\/$K$=21.8\\, W_{50}^2$, with $W_{50}$ given in km\/s.}, $W_{50}$ (third from top), and the axis ratio of the limiting \\ion{H}{I} column density isocontour, $b\/a$ (bottom panel).\n\n\n\\begin{figure}\n\t\\includegraphics[scale=0.45]{FigHVCs}\n    \\caption{Simulated Local Group {\\small RELHICs} (red open circles), and simulated LG dwarfs (magenta points) compared with the ALFALFA Ultra Compact High Velocity Clouds (UCHVCs, black crosses) from the compilation of \\citet{Adams2013}. {\\small RELHICs} are ``observed'' from the centre of the two primary galaxies, so that each {\\small RELHIC} is shown twice. Only those with fluxes $S_{21}>0.1$ Jy km\/s are shown. From top to bottom we show, as a function of $S_{21}$, the size of the $10^{18}$ cm$^{-2}$ \\ion{H}{I} column density contour in arcmin; the Galactocentric radial velocity $V_{\\rm gsr}$; the linewidth $W_{50}$; and the axis ratio $b\/a$. Note that, in general, Local Group {\\small RELHICs} are smaller in size, fainter, rounder, and more homogeneous in their linewidths than currently observed UCHVCs. {\\small RELHICs} also have predominantly positive Galactocentric radial velocities, consistent with their large distances to the primaries ($d>500$ kpc). Magenta points indicate simulated galaxies with a non-zero stellar component ($M_{\\rm str}< 10^6 \\rm \\ M_{\\odot}$). These systems, in contrast, have properties resembling those of UCHVCs. Some are bigger in size, have higher fluxes, and exhibit a wider range of morphologies. Vertical dashed lines show the flux limit of the UCHVC compilation.\n}\n    \\label{FigHVCs}\n\\end{figure}\n\n\nFig.~\\ref{FigHVCs} compares simulated {\\small RELHICs} (open red circles) and luminous simulated dwarfs ($M_{\\rm str} < 10^6 \\rm \\  M_{\\odot}$; magenta circles), with the $59$ UCHVCs catalogued by \\citet{Adams2013} from ALFALFA data (see black crosses). The stellar mass limit roughly corresponds to that of Leo~P, which was discovered after follow-up imaging of an UCHVC ~\\citep{Giovanelli2013}. Note that {\\it all} {\\small APOSTLE} {\\small RELHICs} are just below the flux limit of the ALFALFA search, which is of $3$ Jy km\/s (shown by the vertical dashed line). UCHVCs are also much more numerous and heterogeneous as a population than expected for {\\small RELHICs}, which are rather small; fairly round ($b\/a>0.8$); have a narrow dispersion of linewidths about $W_{50}\\sim 20$ km\/s; and are almost exclusively moving away from the Galaxy. UCHVCs are, however, comparable to simulated dwarfs, which reach higher fluxes, exhibit a wider range of morphologies, and are bigger than {\\small RELHICs}. We do not analyse the distribution of $\\rm W_{50}$ for our simulated dwarfs as their temperature is set by a an effective equation of state imposed to model the ISM, which is set to $10^4 \\rm \\ K$. Because of this, the HI fluxes estimated for dwarfs must be regarded as lower limits. \n\nWe conclude that the \\citet{Adams2013} UCHVC catalogue does not contain the star-free ``dark'' minihalos we associate with {\\small RELHICs}, and that the properties of some UCHVCs might be consistent with very faint dwarf galaxies that have so far escaped detection in optical surveys. Indeed, as discussed in Sec.~\\ref{SecIntro}, some UCHVCs have already been identified as low surface brightness galaxies, some in the Local Volume \\citep{Sand2015} and some as far away as the Virgo Cluster \\citep{Bellazzini2015a}.\n\n\\section{Summary and Conclusions}\n\\label{SecConc}\n\nWe have used the {\\small APOSTLE} suite of cosmological hydrodynamical simulations of the Local Group to examine the gas content of $\\Lambda$CDM minihalos. We focussed our analysis on systems that are free of stars in our highest-resolution runs, since in such systems the bound gas content at $z=0$ should only depend on the effects of the UV ionizing background and on the ram pressure stripping that affects minihalos as they travel through the cosmic web. \n\n``Dark'' minihalos (or, more precisely, systems with stellar mass $M_{\\rm str}<10^4 \\rm \\ M_\\odot$, the mass resolution limit of our simulations) split into two well-defined groupings: one where the mass of bound gas is set by the ionizing background and correlates tightly with the minihalo virial mass ({\\small RELHICs}, for REionization Limited \\ion{H}{I} Clouds), and another where there is little or no bound gas left within the halo after stripping by the cosmic web ({\\small COSWEBs}, for COSmic WEb Stripped systems). The differentiation is thus mainly environmental; gas-free {\\small COSWEBs} populate the  high-density regions near the luminous galaxies of the Local Group, where gas densities are high and cosmic web stripping is important, whereas the relatively gas-rich {\\small RELHICs} inhabit the underdense outskirts. Few {\\small RELHICs} are found within $500$ kpc of either the Milky Way or the M31 analogues in the simulations.\n\nIn terms of halo virial mass, the transition between luminous galaxies and dark systems like {\\small RELHICs} and {\\small COSWEBs} happens relatively quickly. Dark minihalos have masses that do not exceed $M_{200} \\sim 10^{10} \\rm \\  M_\\odot$; their fraction increase rapidly with decreasing mass, and they make up essentially all halos below $10^9 \\rm \\ M_\\odot$. {\\small RELHICs} make up most of the more massive dark minihalos; their abundance peaks at roughly $50\\%$ for $M_{200}\\sim 2\\times 10^9 \\rm \\ M_\\odot$. The {\\small RELHIC} bound gas mass fraction decreases with decreasing mass; from $20\\%$ of the universal baryon fraction  at $M_{200} \\sim 5 \\times 10^9 \\rm \\ M_\\odot$ to  $0.3\\%$ ($10^5 \\rm \\ M_\\odot$, or ten particles in our highest-resolution runs) in $\\sim 3\\times 10^8 \\rm \\ M_\\odot$ minihalos.\n\nThe gas component in {\\small RELHICs} is in approximate hydrostatic equilibrium with the dark matter potential and in thermal equilibrium with the ionizing UV background. Their thermodynamic properties are therefore well understood, and their gas density and temperature profiles are in excellent agreement with a simple model where UV-heated gas is in thermal and hydrostatic equilibrium within NFW halos. Gas in {\\small RELHICs} is nearly pristine in composition and nearly fully ionized, with small (sub-kpc) neutral hydrogen cores that span a large range of \\ion{H}{I} masses and column densities. These cores have negligible Doppler broadening and nearly round morphologies. \n\nThe most massive {\\small RELHICs} have properties comparable to those of some Ultra Compact High Velocity Clouds (UCHVCs) but the bulk of the Local Group {\\small RELHIC} population should have \\ion{H}{I} fluxes just below $\\sim 3$ Jy km\/s, the limit of the ALFALFA UCHVC detection. \n\nOther differences between {\\small RELHICs} and UCHVCs are the following: (i)  the sheer number of UCHVCs implies that most UCHVCs are not {\\small RELHICs} (we expect fewer than 10 Local Group {\\small RELHICs} with $S_{21}>0.1$ Jy km\/s over the whole sky); (ii) {\\small RELHICs} should mostly reside beyond $\\sim 500$ kpc from the Milky Way, leading to low \\ion{H}{I} fluxes ($<3$ Jy km\/s), very small angular sizes ($<3'$), and predominantly positive Galactocentric radial velocities; (iii) {\\small RELHICs} should be nearly round on the sky ($b\/a>0.8$ at $10^{18}$ cm$^{-2}$) and (iv) have a very narrow distribution of thermally broadened line widths ($W_{50}\\sim 20$ km\/s). \n\nThe small overlap in properties between UCHVCs and {\\small RELHICs} suggest that the former are not part of the abundant dark minihalo population expected in the $\\Lambda$CDM models. UCHVCs are either \\ion{H}{I} ``debris'' in the Galactic halo, or else the \\ion{H}{I} component of more massive halos, most of whom are expected to host a luminous stellar component as well. Further work is underway that aims to clarify the overall abundance of {\\small RELHICs} in cosmological volumes; their contribution to the low-mass end of the \\ion{H}{I} mass function; their relation to ultra faint galaxies; and the best strategies to detect them. Although {\\small RELHICs} seem too faint to be a dominant source of \\ion{H}{I} detections in extant or planned surveys, they may be easier to detect and study in absorption against the light of luminous background objects at moderate redshifts. {\\small RELHICs} are a robust prediction of the $\\Lambda$CDM paradigm so their detection and characterization would offer a unique opportunity to shed light onto the ``dark'' side of a cold dark matter-dominated universe.\n\n\n\\section{Acknowledgements}\n\nWe acknowledge useful discussions with John Cannon, Luke Leisman, Manolis Papastergis, Antonino Marasco and Tom Osterloo. We also thank the anonymous referee for valuable comments that helped to improve the paper. We have benefited from the following public Python packages: {\\tt numpy} \\citep{van2011numpy}, {\\tt scipy} \\citep{Jones2001}, {\\tt matplotlib} \\citep{Hunter2007}, {\\tt Ipython} \\citep{Perez2007} and {\\tt py-sphviewer} \\citep{Benitez-Llambay2015b}. RCA is a Royal Society University Research Fellow. This work was supported by the Science and Technology Facilities Council (gran number ST\/L00075X\/1) and the European Research Council (grant numbers GA 267291 \"Cosmiway\"). It was also partially supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office ([AP P7\/08 CHARM]), and by the European Research Council under the European Union's Seventh Framework Programme (FP7\/2007-2013)\/ERC grant agreement 278594-GasAround Galaxies and by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.043.409.\nThis work used the DiRAC Data Centric system at Durham University,operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by\nBIS National E-infrastructure capital grant ST\/K00042X\/1, STFC capital grants ST\/H008519\/1 and ST\/K00087X\/1, STFC DiRAC Operations grant ST\/K003267\/1 and Durham University. DiRAC is part of the National E-Infrastructure.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStaggered fermions are an inexpensive and popular way of\ndiscretizing quarks on a space-time lattice.  They preserve a chiral\nsymmetry with a local action at the expense of containing extra modes\nthat quadruple the number of quark species being simulated.  The\nquadrupled species, called {\\it tastes}, are mixed at nonzero lattice\nspacing, but are expected to produce four independent quark flavors in\nthe continuum limit.  Evidence of this behavior can be seen in the low\neigenvalues of the staggered Dirac matrix.  At small lattice spacings\nthe eigenvalues tend to cluster into distinct quartets which represent\nthe four tastes.  As the lattice spacing is decreased, the eigenvalues\nwithin a quartet will become more degenerate \\cite{latev}.\n\nIn \\cite{srmt} a Random Matrix Theory (RMT) for staggered\nlattice fermions was introduced to describe the low eigenvalues of the\nstaggered Dirac operator.  The staggered RMT (SRMT) adds additional\nterms to the standard chiral Random Matrix Theory that have the\nappropriate symmetries of the lattice operator.  These additional\nterms in the SRMT reproduce the known $O(a^2)$ (ignoring any extra\nfactors of $\\alpha_s$ which we drop for convenience) taste breaking\nterms that appear in the staggered chiral Lagrangian.\n\nThe SRMT is equivalent to the staggered chiral Lagrangian at zero\nmomentum.  This equivalence has been demonstrated for the fermionic\npartition function, but is only conjectured for the partially quenched\ncase, which is related to the eigenvalues of the Dirac operator.  Here\nwe will directly compare estimates of the size of the taste breaking\nin lattice simulations determined from the staggered chiral Lagrangian\n(in the $p$-regime) with estimates obtained from comparing low\neigenvalues to the SRMT predictions (in the $\\epsilon$-regime).\nOur initial tests show good agreement with the predictions of SRMT\nin support of the conjectured equivalence of the partially quenched\ntheories.\n\n\n\\section{Staggered Chiral Lagrangian}\n\nThe effective chiral Lagrangian for staggered fermions at order $a^2$\n is given by \\cite{Lee:1999zxa,Aubin:2003mg}\n\\begin{eqnarray}\n\\label{cV}\n{\\cal L} = \\frac{F^2}{8} \\tr{\\partial_\\mu U \\partial_\\mu U^\\dagger} \n-\\frac{1}{2} \\Sigma_0 m \\tr{U + U^\\dagger}\n+ a^2 {\\cal V}\n\\end{eqnarray}\nwhere $\\tr{X}$ stands for the trace of $X$, and\n$F$ and $\\Sigma_0$ are the low energy constants (LECs) related to the\npion decay constant (with the convention that the physical value for\n $F \\approx 131$ MeV)\nand the magnitude of the chiral condensate, respectively.\nThe taste breaking terms can be divided into two parts ${\\cal V} =\n{\\cal V}_{1t} + {\\cal V}_{2t}$.  The first part contains the single-trace terms\n(with $\\xi_\\mu = \\gamma_\\mu^*$)\n\\begin{eqnarray}\n\\label{sL}\n-{\\cal V}_{1t} &=&\n  C_1 \\tr{ \\xi_5 U \\xi_5 U^\\dagger }\n+ C_3 \\frac{1}{2} \\sum_\\mu \\left[ \\tr{ \\xi_\\mu U \\xi_\\mu U } + h.c. \\right] \\nonumber\\\\\n&+& C_4 \\frac{1}{2} \\sum_\\mu \\left[ \\tr{ \\xi_{\\mu 5} U \\xi_{5 \\mu} U } + h.c. \\right]\n+ C_6 \\sum_{\\mu<\\nu} \\tr{ \\xi_{\\mu\\nu} U \\xi_{\\nu\\mu} U^\\dagger }\n\\end{eqnarray}\nand the second part has the two-trace terms\n\\begin{eqnarray}\n\\label{c25tr2}\n-{\\cal V}_{2t} &=& \n    C_{2V} \\frac{1}{4} \\sum_\\mu \\left[ \\tr{ \\xi_\\mu U } \\tr{ \\xi_\\mu U } + h.c. \\right]\n+ C_{2A} \\frac{1}{4} \\sum_\\mu \\left[ \\tr{ \\xi_{\\mu5} U } \\tr{ \\xi_{5\\mu} U } + h.c. \\right] \\nonumber\\\\\n&+& C_{5V} \\frac{1}{2} \\sum_\\mu \\left[ \\tr{ \\xi_\\mu U } \\tr{ \\xi_\\mu U^\\dagger } \\right]\n+ C_{5A} \\frac{1}{2} \\sum_\\mu \\left[ \\tr{ \\xi_{\\mu5} U } \\tr{ \\xi_{5\\mu} U^\\dagger } \\right]\n.\n\\end{eqnarray}\n\n\n\\section{Staggered Chiral Random Matrix Theory}\n\nThe (fermionic) staggered chiral random matrix theory partition\nfunction can be written as \\cite{srmt}\n\\begin{eqnarray}\n\\label{SRMT}\nZ_{SRMT} =\n\\int dW p_0(W) p_T(T) \\prod_{f=1}^{N_f} \\det(D+m_f)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nD = \\left( \\begin{array}{cc}\n0 & i W \\\\\ni W^\\dagger & 0\n\\end{array} \\right) \n\\otimes \\mathbb{I}_4 + a T  ~.\n\\end{eqnarray}\nwhere $W$ is a $(N+\\nu) \\times N$ complex matrix with $\\nu$ the absolute\n value of the topological charge and $T$ incorporates the taste breaking terms.\nThe matrix potential for $W$ is conveniently a Gaussian,\n\\begin{eqnarray}\np_0(W) = \\exp(-\\alpha N \\tr{W^\\dagger W})\n\\end{eqnarray}\nwith $\\sqrt{\\alpha} = \\Sigma_0 V\/ 2N$ ($V$ is the four volume).\n\nThe taste breaking contribution to the SRMT ($T$) has eight terms that correspond directly\nwith the eight taste breaking terms of the chiral Lagrangian.\nIts complete form was given in \\cite{srmt}.\nAs an example, the $C_4$ term corresponds to\n\\begin{eqnarray}\n\\label{t4}\nT_4 = \\sum_{\\mu} \\left( \\begin{array}{cc} A_\\mu & 0 \\\\ 0 & B_\\mu \\\\ \\end{array} \\right)\n\\otimes \\xi_{\\mu 5}\n\\end{eqnarray}\nwhere $A_\\mu$ and $B_\\mu$ are Hermitian matrices of size $(N+\\nu) \\times (N+\\nu)$ and \n$N\\times N$, respectively.\nFor convenience we can choose a Gaussian weight function for these matrices,\n\\begin{eqnarray}\np_{T_4} = \\exp\\left(-\\frac{\\alpha N^2}{2 V C_4} \\sum_\\mu \\tr{A_\\mu^2} + \\tr{B_\\mu^2} \\right) ~.\n\\end{eqnarray}\nOne can then show that the (fermionic) RMT with this extra matrix term is equivalent\nto the zero-momentum staggered chiral Lagrangian with just a $C_4$ taste breaking term.\n\nNote that while the correction to the RMT enters at order $a$, this still reproduces\na term of order $a^2$ in the chiral Lagrangian.  This is due to the taste breaking terms\nin the SRMT being traceless.  As demonstrated in \\cite{srmt}, when expanding the\ndeterminant of the SRMT Dirac matrix, the $O(a)$ term vanishes for this reason, and results\nin a partition function that has a leading corrections at $O(a^2)$ even\n though the SRMT Dirac operator has terms of $O(a)$.\n\nThe two-trace terms can be incorporated in the SRMT in a couple of ways.\nOne is to add terms such as\n\\begin{eqnarray}\n \\left( \\begin{array}{cc} b_\\mu \\mathbb{I}_{N+\\nu} & 0 \\\\\n 0 & \\pm b_\\mu \\mathbb{I}_{N} \\\\\n \\end{array} \\right) \\otimes \\xi_{\\mu 5}\n\\end{eqnarray}\nwith a Gaussian weight for the scalar $b_\\mu$.\nThis will give a contribution to the $C_{2A}$ and $C_{5A}$ terms.\nWhile this term will reproduce the correct term in the chiral Lagrangian,\nit has no analogue in the lattice Dirac matrix.  This term is of the\nform of a fluctuating taste-dependent mass, which is not present on\nthe lattice.\n\nAn alternative form for the two-trace terms is to simply modify the potentials\nfor the matrix terms corresponding to the one-trace terms.  By replacing the simple\nGaussian weight with one that also includes $\\tr{A_\\mu}^2$, $\\tr{B_\\mu}^2$ and\n$\\tr{A_\\mu}\\tr{B_\\mu}$ in the exponential, the coefficients can be tuned to give\nthe correct two-trace terms in the chiral Lagrangian \\cite{srmt}.\n\n\n\\section{Generalized Staggered Random Matrix Theory}\n\nIf we look at the final form of the SRMT Dirac matrix, we see that it\nis the most general matrix that is consistent with the symmetries of\nthe staggered Dirac operator; it is anti-Hermitian and anticommutes\nwith the staggered chiral symmetry matrix $\\gamma_5 \\otimes \\xi_5$.\nAs mentioned in the previous section, the single-trace terms from the\nchiral Lagrangian are reproduced by considering independent Gaussian\nweights for the remaining matrix elements.  Meanwhile the two-trace\nterms can be reproduced by adding two-trace terms to the RMT weights.\n\nIn this manner one could consider generalizing the SRMT to include\nmore terms in the weight function in an attempt to reproduce higher\norder terms in the chiral Lagrangian (at zero momentum).  One can then\ndraw an analogy between the formulation of a RMT and of an effective\nLagrangian.  For the Lagrangian one includes all terms, up to some\norder, that are consistent with the symmetries.  Likewise for the RMT\none can consider a matrix containing all elements consistent with the\nsymmetries of the Dirac operator, with a generalized weight function\nfor these matrix elements up to some level of complexity.  Of course the\nmapping from the RMT potential to the chiral Lagrangian at higher\norder may not be as simple as in the SRMT presented here, but one\ncould speculate that a generalized RMT could reproduce all higher\norder terms of the chiral Lagrangian.  Again this equivalence\nonly holds for the zero momentum Lagrangian, but it is also possible to\nreduce the full Lagrangian to an effective zero momentum Lagrangian\n(for a recent example see \\cite{Lehner:2010mv}),\nwhich then might be representable as a RMT.\n\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{nv3232.eps}\n    \\caption{\\label{nv}\n      Number variance of the lowest eigenvalues of the \n      $32^4$ lattice ensemble and the SRMT.\n    }\n  \\end{center}\n\\end{figure}\n\n\n\\section{Dominant form of taste breaking}\n\nMost of the taste breaking coefficients can be measured in lattice\nsimulations by comparing to staggered chiral perturbation theory\n(S$\\chi$PT) results.  The four single-trace parameters are uniquely\ndetermined from the splittings of the pion masses at leading order\n\\cite{Aubin:2003mg}.\nTypically the $C_4$ term is found to be the dominant contribution to\nthe pion spectrum \\cite{Lee:1999zxa}.  The two-trace terms enter in\nS$\\chi$PT{} formulas in the combinations\n$C_{V,A}^{\\pm} = (C_{2\\{V,A\\}} \\pm C_{5\\{V,A\\}})\/2$.\nThey don't contribute to the pion splittings at leading order,\nbut the $C_A^-$ and $C_V^-$ terms do appear in one-loop expressions,\nwhile the $C_A^+$ and $C_V^+$ terms do not \\cite{Aubin:2003mg}.\nIn lattice simulations $C_A^-$ has been found to be larger than $C_V^-$\n\\cite{Aubin:2004fs}.\n\nBoth of the dominant terms, $C_4$ and $C_A^-$, come from same term in\nthe SRMT with the more general form for the weight function.  This\nsupports the idea of constructing the SRMT from a single matrix with\nthe correct symmetries and with a generalized weight function.  Based\non the lattice measurements, the leading contribution to taste\nbreaking in the staggered Dirac matrix has an axial-vector taste\nstructure.  If one imagined rotating the staggered Dirac matrix into a\ntaste basis and then expanding in powers of $a$, the dominant\ncorrection at order $a$ would then have the same form as in\n(\\ref{t4}).  Of course the exact potential for the lattice Dirac\nmatrix would be much more complicated than in the SRMT, but the\nleading effects at low energy can be captured by the SRMT weight\nfunction considered here.  Among the terms in the SRMT potential\ncorresponding to the axial-vector matrix, we have no reason to favor\none over the other, so we would naively expect them to be of\nsimilar magnitude.  We would then expect the corresponding terms\nin the chiral Lagrangian to be of similar order, and also\ndominant over the corresponding terms with other taste symmetries,\nwhich is consistent with lattice measurements.\n\n\n\\section{Extracting LECs from RMT}\n\nThe RMT predictions for the eigenvalue correlations can be used to\nextract LECs from lattice simulations.\nFor example $\\Sigma_0$ can be obtained by fitting to the eigenvalue\ndensity.\nAdditionally $F$ can be obtained from the correlations of eigenvalues\nwith an imaginary chemical potential \\cite{rmtf}.\n\nIf the taste breaking is small enough then these methods can apply\ndirectly to staggered eigenvalues by replacing each quartet of\neigenvalues with its average \\cite{latev}.  In this case one could also try\nto extract the taste breaking parameters from the splittings of the\neigenvalues within the quartet.  In practice, it would likely be too\ndifficult to extract all the parameters, but if one assumes that the\n$C_4$ term is dominant, then it should be possible to estimate it from\ncomparison to the SRMT.\n\nHowever, if the taste breaking is large, then the higher order taste\nbreaking terms can become important.  In this case the effective\nchiral Lagrangian reduces to a single flavor for the remaining\nstaggered chiral symmetry with a new set of LECs that are in principle\nunrelated to the original ones \\cite{srmt}.  Thus extracting the LECs\nfrom low eigenvalues when there aren't clear quartets present may not\nyield the continuum LECs in chiral Lagrangian.\n\n\\begin{figure}\n  \\vspace{-7mm}\n  \\begin{center}\n    \\begin{minipage}[t]{.49\\textwidth}\n      \\includegraphics[clip,width=\\textwidth]{idq3232.eps}\n    \\end{minipage}\n    \\begin{minipage}[t]{.49\\textwidth}\n      \\includegraphics[clip,width=\\textwidth]{idq23232.eps}\n    \\end{minipage}\n    \\caption{\\label{id}\n      Individual integrated densities of the lowest four eigenvalues\n      from the $32^4$ lattice ensemble and the SRMT.  The dominant taste breaking\n      parameter in the SRMT is set to $a^2VC_4 = 0.3$ (left) and $0.2$ (right).\n    }\n  \\end{center}\n  \\vspace{-7mm}\n\\end{figure}\n\n\n\\section{Comparison to lattice simulations}\n\\vspace{-2mm}\n\nAs an initial test of the SRMT, we will compare the leading order taste breaking\nterm obtained from fitting the Dirac operator eigenvalues to the SRMT with that\nobtained from fitting the pion spectrum to the staggered chiral Lagrangian.\nSince it is difficult to get a single lattice ensemble that we could use for\nboth measurements we will use two ensembles with all parameters identical except\nfor the volume.\n\nFor the pion masses we use an ensemble from the MILC collaboration\n2+1+1 flavor HISQ runs \\cite{Bazavov:2010pi}.  The ensemble has a\nvolume of $32^3\\times 96$ with a lattice spacing of $a\\approx 0.09$ fm and\nwith a light quark mass $m_l = m_s\/5$.  From the pion mass splittings\nwe can get the single-trace taste breaking terms in the chiral\nLagrangian.  Using a value of $F=131$ MeV, this gives\n\\begin{eqnarray}\na^2 V C_1 =  0.03(8),~~\na^2 V C_3 = -0.03(4),~~\na^2 V C_4 =  0.84(4),~~\na^2 V C_6 =  0.03(3)  ~.\n\\end{eqnarray}\nWe can see here that $C_4$ is clearly dominant, as expected, and that\nthe other coefficients are all consistent with zero.\n\nWe generated a new ensemble of 430 lattices of size $32^4$ with all\nother parameters the same as the previous ensemble.  From the volume\nscaling we expect to find\n\\begin{eqnarray}\n\\label{c4}\na^2 V C_4 = 0.28(1)\n\\end{eqnarray}\non this new ensemble.\nWe then compare lattice eigenvalues with numerical simulations of the SRMT\nwith only the $C_4$ taste breaking term.  For this comparison we choose to\nuse the number variance statistic.  This shows the fluctuations\n(variance) in the number of eigenvalues in an interval starting at\nzero versus the average number in the interval.  We evaluated it\nnumerically from simulations of the SRMT with $N=400$.\n\nIn Figure \\ref{nv} we plot the number variance of the lattice\neigenvalues against the SRMT at different values of $a^2 V C_4$.  We\ncan see that the best agreement for small intervals (up to around\n2 to 4 eigenvalues) is at $a^2 V C_4 \\approx 0.3$.\nThis is in good agreement with our prediction (\\ref{c4})\nobtained from the pion mass splittings.\nAs the length of the interval grows,\nthe lattice results start to move away from the SRMT result.  This is\nlikely due to higher momentum modes entering on the lattice, that aren't\ncaptured in the RMT.  This happens at the QCD equivalent of the Thouless energy \\cite{te},\nwhich for this ensemble, in units of the average eigenvalue spacing,\nis $F^2 \\sqrt{V} \\approx 3.5$.\nThis is consistent with our observations from the number variance.\n\nAs a further check that the SRMT describes the low eigenvalues of the\nstaggered Dirac operator, we look at the individual integrated\ndensities of the lowest four eigenvalues.\nIn Figure \\ref{id} we see the integrated density from the $32^4$\nlattice along with that from the numerical simulations of the SRMT at\ntwo different values of the taste breaking parameter\n$a^2VC_4 = 0.2,0.3$.\nWe can see that the value of $0.3$ fits the lattice data much better\nthan at $0.2$, again confirming that the predictions of the SRMT\nfit the lattice data well and give estimates of the taste breaking\nparameters that are consistent with the staggered chiral Lagrangian.\n\n\n\\section{Summary}\n\nWe have shown a chiral RMT that incorporates all leading order taste\n breaking terms from staggered chiral Lagrangian.\nThe SRMT can be constructed by considering a RMT\n with same symmetries as the staggered Dirac operator and a generalized weight function.\nInitial tests show that the predictions of the SRMT are in good agreement with\n lattice simulations within the range of validity of the SRMT.\nAdditionally, using the predictions of the SRMT,\n the dominant taste breaking parameter can be extracted from\n the low eigenvalues of the staggered Dirac operator and gives consistent\n results with that obtained from the pion mass splittings.\n\n\n\\begin{acknowledgments}\n  We thank Doug Toussaint for providing the pion splitting data for\n  the MILC ensemble.  This research used resources of the Argonne\n  Leadership Computing Facility at Argonne National Laboratory, and\n  was supported by the U.S. DOE under contract DE-AC02-06CH11357.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConsider the set of natural numbers ${\\bf \\mathbb {N}^+} = \\{1,2,3,4,...\\} $, the Collatz mapping function $f$ maps each integer $n \\in {\\bf \\mathbb N^+}$ to another positive integer also in ${\\bf \\mathbb  N^+}$ by the configuration\n\\begin{equation}\nf(n)= \\frac{3^{b(n)}n + b(n)}{2}\n\\end{equation}\nwhere the binary flag function $b(n)=1$ is fixed if n is odd, otherwise $b(n)=0$ \\cite{Ter76}. \n\nNote that this representation is equivalent to the Collatz mapping function \n\\begin{equation}\nf(n) = \n\\begin{cases}\n\\frac{3n+1}{2} &\\text{if  $n \\equiv 1 $ (mod 2)}, i.e., ~[1]_2 \\\\\n\\frac{n}{2}  &\\text{if $n \\equiv 0$  (mod 2)}, i.e., ~ [0]_2 \\\\\n\\end{cases}\n\\end{equation}\noften used in literature. The iterative execution of $f(n)$ always produces a sequence that terminates at 1 for any integer input $n$, according to the mathematician Lothar Collatz. This conjecture is known as the Collatz conjecture (CC) \\cite{Lag85}, \\cite{And98}, \\cite{Van05} and was first proposed in 1937. It is commonly known as the \\textit{3n+1} or \\textit{3x+1} conjecture \\cite{Lag85}, \\cite{Marc96}, \\cite{Gar81}, \\cite{Sim99}, Ulam conjecture, Thwaites conjecture, the Kakutani's problem, the Hasse's algorithm, and the Syracuse problem \\cite{Mad97}, \\cite{Syr01}. This problem is easy to state but extremely difficult to prove. Many mathematicians have investigated and written articles about the conjetcure \\cite{Ste77}, \\cite{Bel06}, \\cite{Bel98}, \\cite{Sim05}, \\cite{Sin10}. We recommend the work of Lagarias \\cite{Lag85}, \\cite{Lag11}, \\cite{Lag12} for comprehensive and annotated bibliographies on the subject. \n\nThe principal aim of this paper is to document, collate and present the results of a newly proposed number system formalised for investigating the truth of the CC. \n\n\n\\subsection{A new covering system and congruence classes modulo 18}\nHere we seek to formulate a simple covering system for the odd integers such that every odd integer may be represented by exactly one of the residue classes $d_i \\equiv r_i \\mod18$ and each residue class is a finite (or infinite) collection (i.e., reordered set) of integers congruent to any member of the set $\\{u_{i1}(\\mathrm{mod}\\ {2^13^2}),\\ \\ldots,\\ u_{im}(\\mathrm{mod}\\ {2^m3^2})\\}$. \n\nThe covering system sought for the proposed Collatz number systems must be 1-cover, i.e., covers every integer exactly once. One of the objectives of this proposition is to develop a complete recategorisation of odd numbers based on both their residue classes and Collatz profiles (i.e., according to their $2^m $ divisibility properties after multiplying them by 3 and adding 1).\n\n\n\\subsection{Definitions}\nLet $\\sigma_{\\infty}(n)$ represent the total stopping time of an integer $n$ under the Collatz system, where the symbol ``$\\sigma_{\\infty}$'' refers to ``number of iterations it takes to get to 1'' starting from the input $n$ - this is often referred to as the total stopping time of $n$.\n\nDefine the term \\textit{Collatz profile} $f(d_i,m)$ as a representation of odd numbers $d_i$ congruent to $r_i \\mod 18$ whose Collatz result $3d_i+1$ is divisible by $2^m$ such that the optimised result $\\frac{3d_i+1}{2^m}$ is odd.  Hence, under the Collatz system the total stopping time relation $\\sigma_{\\infty}(d_i)=\\sigma_{\\infty}(\\frac{3d_i+1}{2^m})+(m+1)$ holds, i.e., the total stopping time $\\sigma_{\\infty}(d_i)$ is $(m+1)$ iterates more than that of the next odd number in the Collatz sequence: $\\sigma_{\\infty}(\\frac{3d+1}{2^m})=\\sigma_{\\infty}(d_i)-(m+1)$. \n\nFor example, $f(13,3) = \\frac{3(13)+1}{2^{\\bf 3}} = 5$ implies that if the total stopping time of number {\\bf 13} was $\\sigma_{\\infty}(13)$ under the Collatz system, then the total stopping time of number {\\bf 5} \\textit{would be} equal to $\\sigma_{\\infty}(5)=\\sigma_{\\infty}(13)-({\\bf 3}+1)=\\sigma_{\\infty}(13)-4$, i.e., $\\sigma_{\\infty}(13) = 9$ and $\\sigma_{\\infty}(5) = 5$. Such simple inference (concept) may be used to prove the CC by \\textit{reverse engineering}, i.e., the relative inference of total stopping times.\n\nDefine $r$ to be an odd number such that the following conditions are satisfied:\n\\begin{enumerate}\n\\item $1 \\le r <18$;\n\\item $d_i=18V_{r_i, m}(n)+r_i$, $n \\ge 0$;  and\n\\item $\\frac{3d_i+1}{2^m}$ is odd.\n \\end{enumerate}\nIf $V_{r, m}(n)$ is defined as the multiplicative set (function) that uniquely identifies numbers of same residue class attributable to similar Collatz-like pattern (profile), i.e., $V_{r, m}(n)$ is regarded as a parameter set dependent on actual values of $d_i$ and $r$ where $d_i$ is odd and congruent to $ r$ (mod 18), i.e., $r \\in S$; where $S={1, 5, 3, 13, 17,15, 7, 11, 9}$. S is a reordered set of $r$. This set S establishes the connection between certain sets of integer numbers of one residue class in number system to other sets of other residue classes. The rationale behind this \\textit{reordering} of the residue classes is of utmost importance (as demonstrated and explained in the next section). \n\nIf we further stratify or reclassify every congruence class $d_i$ according to the result $ \\frac{3d_i+1}{2^m}$, i.e., so that m is optimised and $ \\frac{3d_i+1}{2^m}$ yields an odd number result, how does $d_i$ relate to the variable $m$, the exponent of $2^m$? Are there scientific or theoretical evidences to support the claim that the metatheory behind such relations may be completely deterministic? Ultimately, what is the schema for the proposed generalised Collatz based number system? These are the theoretical questions considered in this paper. \n\nFrom a theoretical point of view, the task of constructing a generalised schema of the Collatz based number system seems difficult, time consuming, and daunting. In practical terms, the difficulty associated with such tasks often seem more discouraging, because there is almost no guarantee of little or no recompense for the time invested. This is no longer the case. Here we present fundamental results that facilitate new theoretical perspectives on the Collatz conjecture and the generalised Collatz based number system. \n\nThe next section (Section \\ref{4np1}) introduces a modified (optimised) Collatz function and briefly gives a gentle introduction about a new discovery that inspires better understanding and new perspectives on residue classes modulo 18. \n\nThe proposed metatheories of the Collatz based number system and its general proof are presented in section \\ref{Reorder}, including the proposed schemata comprising of a map of generalised Collatz based number system and corresponding map of (distinct and fundamental) total stopping time functions.\n\n\n\\section{The fundamental relations between certain odd numbers} \\label{4np1}\nIt is important to understand the fundamental relations between integers $d_i$ congruent to $r \\mod 18$ and these relations define the inferred collatz properties, i.e., the divisibility quantities: $2^m |  (3d_i+1)$ analogous to the result $[p]_{54}$ where $1 \\le p \\le 54$ and p must be odd. In other words, how can new covering subsystems be formulated for the residue classes of odd integers, classifying each subsystem according to the prescribed Collatz properties? This question requires a thorough understanding of: a) the mechanisms of Collatz sequence transformations from odd to odd integers; b) the formulation of a new covering subsystems for the entire set of odd integers; and c) the determination of (total) stopping time for every odd integer for the production of an irrefutable proof of the Collatz conjecture. The last point, which is not addressed completely in this foundational paper, requires a thorough analysis of the propose theoretical schema of the Collatz based number system, i.e., a complete understanding of the covering system of all odd integers. \n\n\n\\subsection{The modified Collatz function}Introduce the modified Collatz function\n\\begin{equation}\\label{fdm}\nf(d,m)= \\frac{3^{b(d)}d + b(d)}{2^m} = \\frac{3d+1}{2^m},\n\\end{equation}\nwhere $m$ is the maximum exponent such that the odd number $d \\in {\\bf N^+}$ and the result $f(n,m)$ is an odd integer, i.e.,  $2^m \\mid (3d+1)$ but $2^{m+1} \\ndiv (3d+1)$ \\cite{Ter76}.\n\n\\subsection{Residue classes modulo 18}\nGiven that $d$ is an odd number which belongs to only one of the following residue classes: \n\n\\begin{equation}\nd = \n\\begin{cases}\nd_1 &\\text{if  $ d\\equiv 1 $ (mod 18), e.g. 1, 349525, \\dots}\\\\\nd_2  &\\text{if $ d \\equiv 5$  (mod 18), e.g. 5}\\\\\n\\textcolor{red}{d_3  }&\\text{if $ d \\equiv 3$  (mod 18), e.g. \\textcolor{red}{21}}\\\\\nd_4  &\\text{if $ d \\equiv 13$  (mod 18), e.g. 85}\\\\\nd_5  &\\text{if $ d \\equiv 17$  (mod 18), e.g. 341}\\\\\n\\textcolor{red}{d_6  }&\\text{if $ d \\equiv 15$  (mod 18), e.g. \\textcolor{red}{1365}}\\\\\nd_7  &\\text{if $ d \\equiv 7$  (mod 18), e.g. 5461}\\\\\nd_8  &\\text{if $ d \\equiv 11$  (mod 18), e.g. 21845}\\\\\n\\textcolor{red}{d_9  }&\\text{if $ d \\equiv 9$  (mod 18), e.g. \\textcolor{red}{87381}}\\\\\n\\hline\n\\textcolor{gray}{d_{10} }&\\textcolor{gray}{\\text{if $ d \\equiv 1$  (mod 18), e.g. \\textcolor{gray}{349525}}: \\text{$ d_{10} = [1]_{18} = d_1$}} \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\nwhere $\\{d_1 ~\\cup ~d_2 ~\\cup ~d_4 ~\\cup ~d_5 \\cup ~d_7 ~\\cup ~d_8\\}$ is the set of all odd numbers indivisible by 3, \n$[1]_{18} \\in d_1$, \n$[5]_{18} \\in d_2$, \n$[3]_{18} \\in d_3$, \n$[13]_{18} \\in d_4$, \n$[17]_{18} \\in d_5$, \n$[15]_{18} \\in d_6$, \n$[7]_{18} \\in d_7$, \n$[11]_{18} \\in d_8$, \n$[9]_{18} \\in d_9$, and of course, \n$[0]_3 \\in \\{\\{d_3~ \\cup~d_6~ \\cup~d_9\\}$. This rearrangement enables one to capture cyclic recurrence relations between the odd integers using the relation $4d_i+1$, where $d_i \\in d$ as demonstrated in the next subsection.\n\n\\subsection{Cyclic recurrence relations between odd numbers}\nLet $r_i$ be a member of the finite set $S=\\{1,5,3,13,17,15,7,11, 9\\}$; $1 \\le i \\le 9$. This rearrangement of $[r_i]_{18}$ gives new insights into cyclic recurrence relations between the odd numbers:\n\n\\begin{equation}\nd =\n\\begin{cases}\n[r_i]_{18} &\\text{if  $ 4([r_{i+8}]_{18})+1 \\equiv r_{i} ~(mod ~ 18) $ \\dots}\\\\\n[r_{i+1}]_{18} &\\text{if  $ 4([r_i]_{18})+1 \\equiv r_{i+1} ~ (mod ~ 18) $ \\dots}\\\\\n\\vdots \\\\\n[r_{i+8}]_{18} &\\text{if  $ 4([r_{i+7}]_{18})+1 \\equiv r_{i+8} ~ (mod ~ 18)  $ }\n\\end{cases};\n\\end{equation}\n e.g. Let $ 19 \\in d_i = [r_i]_{18} = [1]_{18} \\equiv 1$ (mod 18), then  $ 77 = 4d_i+1 = [r_{i+1}]_{18} = [5]_{18} \\equiv 5$ (mod 18),  $ 309 = 4^2d_i+5 = [r_{i+2}]_{18} = 3$ (mod 18),  $ 4^3d_i+21 = [r_{i+3}]_{18} $, $\\dots$ , $ 1267029 = 4^8d_i+21845 =  [r_{i+8}]_{18}$, $ 5068117 = 4^9d_i+87381 =  [r_{i}]_{18}$, and so on. \n  \n\\section{The metatheory and properties of odd Integers }\\label{Reorder}\nWe formulate and show how the arrays of $d_i$ that correspond to certain integers congruent to $r_i$ modulo 18 are defined, i.e., the $d_i=18V_{r_i, m}(n)+r_i$, $n \\ge 0$ and $1 \\le i \\le 9$ that satisfy the required optimisation condition $\\frac{3d_i+1}{2^m}=odd$. \n\n\\subsection{New theories about certain odd numbers}\n\\begin{theorem}\\label{mainthm}\nLet the odd number transformation under the Collatz sequence system be defined as the function $f(d_i,m)= \\frac{3d_i+1}{2^m} = d_{{i,m}_{next}}$ where $d_{{i,m}_{next}}$ represents the next odd number after the odd number $d_i$. The fundamental rules governing the relations between sets of $d_i$ and corresponding (i.e. mapped) sets of $V_{r_i,m}(n)$ are found to be completely deterministic (i.e. non-chaotic) on the following conditions enumerated:\n\\begin{enumerate}\n\\item\n$d_i \\equiv S(i) \\mod 18$, i.e., $r_i = S(i)$;\n\\item\n$d_i = 18V_{r_i,m}(n)+S(i)$; and\n\\item\nthe Collatz result $d_{{i,m}_{next}} = \\frac{3(18~ V_{r_i,m}(n)+1)+S(i)}{2^m}$ must be odd.\n\\end{enumerate}\n\\end{theorem}\n\nThe proof of this major theorem requires enlisting all the possible $d_i$ and $V_{i,m}(n)$ values and demonstrating that the result $\\frac{3(18V_{i,m}(n)+1)+S(i)}{2^m}$ is always odd. This theorem alone requires 162 explicit proofs - one for every statement. For the purpose of brevity, the trivialities involved in the explicit proofs are all avoided. \n\n\\begin{proof}\nWhen $ i =1, r_1 = 1$:\n\n{\\small\n\\begin{equation}\nV_{1, m}(n) = \n\\begin{cases}\n\\text{$V_{1,1} \\in $ \\{1,3,5,7,...\\}} = & 2n + 1 \\\\\n\\text{$V_{1,2} \\in $ \\{0,4,8,12,...\\}} = & 4n \\\\\n\\text{$V_{1,3} \\in $ \\{6,14,22,30,...\\}} = & 8n + 6 \\\\\n\\text{$V_{1,4} \\in $ \\{2,18,34,50,...\\}} = & 16n + 2 \\\\\n\\text{$V_{1,5} \\in $ \\{10,42,74,106,...\\}} = & 32n + 10 \\\\\n\\text{$V_{1,6} \\in $ \\{58,122,186,250,...\\}} = & 64n + 58 \\\\\n\\text{$V_{1,7} \\in $ \\{26,154,282,410,...\\}} = & 128n + 26 \\\\\n\\text{$V_{1,8} \\in $ \\{90,346,602,858,...\\}} = & 256n + 90 \\\\\n\\text{$V_{1,9} \\in $ \\{218,730,1242,1754,...\\}} = & 512n + 218 \\\\\n\\text{$V_{1,10} \\in $ \\{474,1498,2522,3546,...\\}} = & 1024n + 474 \\\\\n\\text{$V_{1,11} \\in $ \\{2010,4058,6106,8154,...\\}} = & 2048n + 2010 \\\\\n\\text{$V_{1,12} \\in $ \\{986,5082,9178,13274,...\\}} = & 4096n + 986 \\\\\n\\text{$V_{1,13} \\in $ \\{7130,15322,23514,31706,...\\}} = & 8192n + 7130 \\\\\n\\text{$V_{1,14} \\in $ \\{11226,27610,43994,60378,...\\}} = & 16384n + 11226 \\\\\n\\text{$V_{1,15} \\in $ \\{3034,35802,68570,101338,...\\}} = & 32768n + 3034 \\\\\n\\text{$V_{1,16} \\in $ \\{52186,117722,183258,248794,...\\}} = & 65536n + 52186 \\\\\n\\text{$V_{1,17} \\in $ \\{84954,216026,347098,478170,...\\}} = & 131072n + 84954 \\\\\n\\text{$V_{1,18} \\in $ \\{150490,412634,674778,936922,...\\}} = & 262144n + 150490 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_1 = 18V_{1, m}(n)+1 =\n\\begin{cases}\n\\text{$d_{1,1} \\in $ \\{19,55,91,127,...\\}} = & 36n + 19 \\\\\n\\text{$d_{1,2} \\in $ \\{1,73,145,217,...\\}} = & 72n + 1 \\\\\n\\text{$d_{1,3} \\in $ \\{109,253,397,541,...\\}} = & 144n + 109 \\\\\n\\text{$d_{1,4} \\in $ \\{37,325,613,901,...\\}} = & 288n + 37 \\\\\n\\text{$d_{1,5} \\in $ \\{181,757,1333,1909,...\\}} = & 576n + 181 \\\\\n\\text{$d_{1,6} \\in $ \\{1045,2197,3349,4501,...\\}} = & 1152n + 1045 \\\\\n\\text{$d_{1,7} \\in $ \\{469,2773,5077,7381,...\\}} = & 2304n + 469 \\\\\n\\text{$d_{1,8} \\in $ \\{1621,6229,10837,15445,...\\}} = & 4608n + 1621 \\\\\n\\text{$d_{1,9} \\in $ \\{3925,13141,22357,31573,...\\}} = & 9216n + 3925 \\\\\n\\text{$d_{1,10} \\in $ \\{8533,26965,45397,63829,...\\}} = & 18432n + 8533 \\\\\n\\text{$d_{1,11} \\in $ \\{36181,73045,109909,146773,...\\}} = & 36864n + 36181 \\\\\n\\text{$d_{1,12} \\in $ \\{17749,91477,165205,238933,...\\}} = & 73728n + 17749 \\\\\n\\text{$d_{1,13} \\in $ \\{128341,275797,423253,570709,...\\}} = & 147456n + 128341 \\\\\n\\text{$d_{1,14} \\in $ \\{202069,496981,791893,1086805,...\\}} = & 294912n + 202069 \\\\\n\\text{$d_{1,15} \\in $ \\{54613,644437,1234261,1824085,...\\}} = & 589824n + 54613 \\\\\n\\text{$d_{1,16} \\in $ \\{939349,2118997,3298645,4478293,...\\}} = & 1179648n + 939349 \\\\\n\\text{$d_{1,17} \\in $ \\{1529173,3888469,6247765,8607061,...\\}} = & 2359296n + 1529173 \\\\\n\\text{$d_{1,18} \\in $ \\{2708821,7427413,12146005,16864597,...\\}} = & 4718592n + 2708821 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_i+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =2, r_2 = S(2) = 5$ (note that the index of V below is 5):\n{\\small\n\\begin{equation}\nV_{5, m}(n) = \n\\begin{cases}\n\\text{$V_{5,1} \\in $ \\{1,3,5,7,...\\}} = & 2n + 1 \\\\\n\\text{$V_{5,2} \\in $ \\{2,6,10,14,...\\}} = & 4n + 2 \\\\\n\\text{$V_{5,3} \\in $ \\{4,12,20,28,...\\}} = & 8n + 4 \\\\\n\\text{$V_{5,4} \\in $ \\{0,16,32,48,...\\}} = & 16n \\\\\n\\text{$V_{5,5} \\in $ \\{24,56,88,120,...\\}} = & 32n + 24 \\\\\n\\text{$V_{5,6} \\in $ \\{8,72,136,200,...\\}} = & 64n + 8 \\\\\n\\text{$V_{5,7} \\in $ \\{40,168,296,424,...\\}} = & 128n + 40 \\\\\n\\text{$V_{5,8} \\in $ \\{232,488,744,1000,...\\}} = & 256n + 232 \\\\\n\\text{$V_{5,9} \\in $ \\{104,616,1128,1640,...\\}} = & 512n + 104 \\\\\n\\text{$V_{5,10} \\in $ \\{360,1384,2408,3432,...\\}} = & 1024n + 360 \\\\\n\\text{$V_{5,11} \\in $ \\{872,2920,4968,7016,...\\}} = & 2048n + 872 \\\\\n\\text{$V_{5,12} \\in $ \\{1896,5992,10088,14184,...\\}} = & 4096n + 1896 \\\\\n\\text{$V_{5,13} \\in $ \\{8040,16232,24424,32616,...\\}} = & 8192n + 8040 \\\\\n\\text{$V_{5,14} \\in $ \\{3944,20328,36712,53096,...\\}} = & 16384n + 3944 \\\\\n\\text{$V_{5,15} \\in $ \\{28520,61288,94056,126824,...\\}} = & 32768n + 28520 \\\\\n\\text{$V_{5,16} \\in $ \\{44904,110440,175976,241512,...\\}} = & 65536n + 44904 \\\\\n\\text{$V_{5,17} \\in $ \\{12136,143208,274280,405352,...\\}} = & 131072n + 12136 \\\\\n\\text{$V_{5,18} \\in $ \\{208744,470888,733032,995176,...\\}} = & 262144n + 208744 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_2 = 18V_{5, m}(n)+5 = \n\\begin{cases}\n\\text{$d_{2,1} \\in $ \\{23,59,95,131,...\\}} = & 36n + 23 \\\\\n\\text{$d_{2,2} \\in $ \\{41,113,185,257,...\\}} = & 72n + 41 \\\\\n\\text{$d_{2,3} \\in $ \\{77,221,365,509,...\\}} = & 144n + 77 \\\\\n\\text{$d_{2,4} \\in $ \\{5,293,581,869,...\\}} = & 288n + 5 \\\\\n\\text{$d_{2,5} \\in $ \\{437,1013,1589,2165,...\\}} = & 576n + 437 \\\\\n\\text{$d_{2,6} \\in $ \\{149,1301,2453,3605,...\\}} = & 1152n + 149 \\\\\n\\text{$d_{2,7} \\in $ \\{725,3029,5333,7637,...\\}} = & 2304n + 725 \\\\\n\\text{$d_{2,8} \\in $ \\{4181,8789,13397,18005,...\\}} = & 4608n + 4181 \\\\\n\\text{$d_{2,9} \\in $ \\{1877,11093,20309,29525,...\\}} = & 9216n + 1877 \\\\\n\\text{$d_{2,10} \\in $ \\{6485,24917,43349,61781,...\\}} = & 18432n + 6485 \\\\\n\\text{$d_{2,11} \\in $ \\{15701,52565,89429,126293,...\\}} = & 36864n + 15701 \\\\\n\\text{$d_{2,12} \\in $ \\{34133,107861,181589,255317,...\\}} = & 73728n + 34133 \\\\\n\\text{$d_{2,13} \\in $ \\{144725,292181,439637,587093,...\\}} = & 147456n + 144725 \\\\\n\\text{$d_{2,14} \\in $ \\{70997,365909,660821,955733,...\\}} = & 294912n + 70997 \\\\\n\\text{$d_{2,15} \\in $ \\{513365,1103189,1693013,2282837,...\\}} = & 589824n + 513365 \\\\\n\\text{$d_{2,16} \\in $ \\{808277,1987925,3167573,4347221,...\\}} = & 1179648n + 808277 \\\\\n\\text{$d_{2,17} \\in $ \\{218453,2577749,4937045,7296341,...\\}} = & 2359296n + 218453 \\\\\n\\text{$d_{2,18} \\in $ \\{3757397,8475989,13194581,17913173,...\\}} = & 4718592n + 3757397 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_2+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =3, r_3 = S(3) = 3$:\n{\\small\n\\begin{equation}\nV_{3, m}(n) = \n\\begin{cases}\n\\text{$V_{3,1} \\in $ \\{0,2,4,6,...\\}} = & 2n \\\\\n\\text{$V_{3,2} \\in $ \\{3,7,11,15,...\\}} = & 4n + 3 \\\\\n\\text{$V_{3,3} \\in $ \\{5,13,21,29,...\\}} = & 8n + 5 \\\\\n\\text{$V_{3,4} \\in $ \\{9,25,41,57,...\\}} = & 16n + 9 \\\\\n\\text{$V_{3,5} \\in $ \\{17,49,81,113,...\\}} = & 32n + 17 \\\\\n\\text{$V_{3,6} \\in $ \\{1,65,129,193,...\\}} = & 64n + 1 \\\\\n\\text{$V_{3,7} \\in $ \\{97,225,353,481,...\\}} = & 128n + 97 \\\\\n\\text{$V_{3,8} \\in $ \\{33,289,545,801,...\\}} = & 256n + 33 \\\\\n\\text{$V_{3,9} \\in $ \\{161,673,1185,1697,...\\}} = & 512n + 161 \\\\\n\\text{$V_{3,10} \\in $ \\{929,1953,2977,4001,...\\}} = & 1024n + 929 \\\\\n\\text{$V_{3,11} \\in $ \\{417,2465,4513,6561,...\\}} = & 2048n + 417 \\\\\n\\text{$V_{3,12} \\in $ \\{1441,5537,9633,13729,...\\}} = & 4096n + 1441 \\\\\n\\text{$V_{3,13} \\in $ \\{3489,11681,19873,28065,...\\}} = & 8192n + 3489 \\\\\n\\text{$V_{3,14} \\in $ \\{7585,23969,40353,56737,...\\}} = & 16384n + 7585 \\\\\n\\text{$V_{3,15} \\in $ \\{32161,64929,97697,130465,...\\}} = & 32768n + 32161 \\\\\n\\text{$V_{3,16} \\in $ \\{15777,81313,146849,212385,...\\}} = & 65536n + 15777 \\\\\n\\text{$V_{3,17} \\in $ \\{114081,245153,376225,507297,...\\}} = & 131072n + 114081 \\\\\n\\text{$V_{3,18} \\in $ \\{179617,441761,703905,966049,...\\}} = & 262144n + 179617 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_3 = 18V_{3, m}(n)+3 =\n\\begin{cases}\n\\text{$d_{3,1} \\in $ \\{3,39,75,111,...\\}} = & 36n + 3 \\\\\n\\text{$d_{3,2} \\in $ \\{57,129,201,273,...\\}} = & 72n + 57 \\\\\n\\text{$d_{3,3} \\in $ \\{93,237,381,525,...\\}} = & 144n + 93 \\\\\n\\text{$d_{3,4} \\in $ \\{165,453,741,1029,...\\}} = & 288n + 165 \\\\\n\\text{$d_{3,5} \\in $ \\{309,885,1461,2037,...\\}} = & 576n + 309 \\\\\n\\text{$d_{3,6} \\in $ \\{21,1173,2325,3477,...\\}} = & 1152n + 21 \\\\\n\\text{$d_{3,7} \\in $ \\{1749,4053,6357,8661,...\\}} = & 2304n + 1749 \\\\\n\\text{$d_{3,8} \\in $ \\{597,5205,9813,14421,...\\}} = & 4608n + 597 \\\\\n\\text{$d_{3,9} \\in $ \\{2901,12117,21333,30549,...\\}} = & 9216n + 2901 \\\\\n\\text{$d_{3,10} \\in $ \\{16725,35157,53589,72021,...\\}} = & 18432n + 16725 \\\\\n\\text{$d_{3,11} \\in $ \\{7509,44373,81237,118101,...\\}} = & 36864n + 7509 \\\\\n\\text{$d_{3,12} \\in $ \\{25941,99669,173397,247125,...\\}} = & 73728n + 25941 \\\\\n\\text{$d_{3,13} \\in $ \\{62805,210261,357717,505173,...\\}} = & 147456n + 62805 \\\\\n\\text{$d_{3,14} \\in $ \\{136533,431445,726357,1021269,...\\}} = & 294912n + 136533 \\\\\n\\text{$d_{3,15} \\in $ \\{578901,1168725,1758549,2348373,...\\}} = & 589824n + 578901 \\\\\n\\text{$d_{3,16} \\in $ \\{283989,1463637,2643285,3822933,...\\}} = & 1179648n + 283989 \\\\\n\\text{$d_{3,17} \\in $ \\{2053461,4412757,6772053,9131349,...\\}} = & 2359296n + 2053461 \\\\\n\\text{$d_{3,18} \\in $ \\{3233109,7951701,12670293,17388885,...\\}} = & 4718592n + 3233109 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_3+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $ i =4, r_4 = S(4) = 13$ (note that the index of V below is 13):\n{\\small\n\\begin{equation}\nV_{13, m}(n) = \n\\begin{cases}\n\\text{$V_{13,1} \\in $ \\{1,3,5,7,...\\}} = & 2n + 1 \\\\\n\\text{$V_{13,2} \\in $ \\{2,6,10,14,...\\}} = & 4n + 2 \\\\\n\\text{$V_{13,3} \\in $ \\{0,8,16,24,...\\}} = & 8n \\\\\n\\text{$V_{13,4} \\in $ \\{12,28,44,60,...\\}} = & 16n + 12 \\\\\n\\text{$V_{13,5} \\in $ \\{20,52,84,116,...\\}} = & 32n + 20 \\\\\n\\text{$V_{13,6} \\in $ \\{36,100,164,228,...\\}} = & 64n + 36 \\\\\n\\text{$V_{13,7} \\in $ \\{68,196,324,452,...\\}} = & 128n + 68 \\\\\n\\text{$V_{13,8} \\in $ \\{4,260,516,772,...\\}} = & 256n + 4 \\\\\n\\text{$V_{13,9} \\in $ \\{388,900,1412,1924,...\\}} = & 512n + 388 \\\\\n\\text{$V_{13,10} \\in $ \\{132,1156,2180,3204,...\\}} = & 1024n + 132 \\\\\n\\text{$V_{13,11} \\in $ \\{644,2692,4740,6788,...\\}} = & 2048n + 644 \\\\\n\\text{$V_{13,12} \\in $ \\{3716,7812,11908,16004,...\\}} = & 4096n + 3716 \\\\\n\\text{$V_{13,13} \\in $ \\{1668,9860,18052,26244,...\\}} = & 8192n + 1668 \\\\\n\\text{$V_{13,14} \\in $ \\{5764,22148,38532,54916,...\\}} = & 16384n + 5764 \\\\\n\\text{$V_{13,15} \\in $ \\{13956,46724,79492,112260,...\\}} = & 32768n + 13956 \\\\\n\\text{$V_{13,16} \\in $ \\{30340,95876,161412,226948,...\\}} = & 65536n + 30340 \\\\\n\\text{$V_{13,17} \\in $ \\{128644,259716,390788,521860,...\\}} = & 131072n + 128644 \\\\\n\\text{$V_{13,18} \\in $ \\{63108,325252,587396,849540,...\\}} = & 262144n + 63108 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_4 = 18V_{13, m}(n)+13 =\n\\begin{cases}\n\\text{$d_{4,1} \\in $ \\{31,67,103,139,...\\}} = & 36n + 31 \\\\\n\\text{$d_{4,2} \\in $ \\{49,121,193,265,...\\}} = & 72n + 49 \\\\\n\\text{$d_{4,3} \\in $ \\{13,157,301,445,...\\}} = & 144n + 13 \\\\\n\\text{$d_{4,4} \\in $ \\{229,517,805,1093,...\\}} = & 288n + 229 \\\\\n\\text{$d_{4,5} \\in $ \\{373,949,1525,2101,...\\}} = & 576n + 373 \\\\\n\\text{$d_{4,6} \\in $ \\{661,1813,2965,4117,...\\}} = & 1152n + 661 \\\\\n\\text{$d_{4,7} \\in $ \\{1237,3541,5845,8149,...\\}} = & 2304n + 1237 \\\\\n\\text{$d_{4,8} \\in $ \\{85,4693,9301,13909,...\\}} = & 4608n + 85 \\\\\n\\text{$d_{4,9} \\in $ \\{6997,16213,25429,34645,...\\}} = & 9216n + 6997 \\\\\n\\text{$d_{4,10} \\in $ \\{2389,20821,39253,57685,...\\}} = & 18432n + 2389 \\\\\n\\text{$d_{4,11} \\in $ \\{11605,48469,85333,122197,...\\}} = & 36864n + 11605 \\\\\n\\text{$d_{4,12} \\in $ \\{66901,140629,214357,288085,...\\}} = & 73728n + 66901 \\\\\n\\text{$d_{4,13} \\in $ \\{30037,177493,324949,472405,...\\}} = & 147456n + 30037 \\\\\n\\text{$d_{4,14} \\in $ \\{103765,398677,693589,988501,...\\}} = & 294912n + 103765 \\\\\n\\text{$d_{4,15} \\in $ \\{251221,841045,1430869,2020693,...\\}} = & 589824n + 251221 \\\\\n\\text{$d_{4,16} \\in $ \\{546133,1725781,2905429,4085077,...\\}} = & 1179648n + 546133 \\\\\n\\text{$d_{4,17} \\in $ \\{2315605,4674901,7034197,9393493,...\\}} = & 2359296n + 2315605 \\\\\n\\text{$d_{4,18} \\in $ \\{1135957,5854549,10573141,15291733,...\\}} = & 4718592n + 1135957 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_4+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =5, r_5 = S(5) = 17$ (note that the index of V below is 17):\n{\\small\n\\begin{equation}\nV_{17, m}(n) = \n\\begin{cases}\n\\text{$V_{17,1} \\in $ \\{1,3,5,7,...\\}} = & 2n + 1 \\\\\n\\text{$V_{17,2} \\in $ \\{0,4,8,12,...\\}} = & 4n \\\\\n\\text{$V_{17,3} \\in $ \\{6,14,22,30,...\\}} = & 8n + 6 \\\\\n\\text{$V_{17,4} \\in $ \\{10,26,42,58,...\\}} = & 16n + 10 \\\\\n\\text{$V_{17,5} \\in $ \\{2,34,66,98,...\\}} = & 32n + 2 \\\\\n\\text{$V_{17,6} \\in $ \\{50,114,178,242,...\\}} = & 64n + 50 \\\\\n\\text{$V_{17,7} \\in $ \\{82,210,338,466,...\\}} = & 128n + 82 \\\\\n\\text{$V_{17,8} \\in $ \\{146,402,658,914,...\\}} = & 256n + 146 \\\\\n\\text{$V_{17,9} \\in $ \\{274,786,1298,1810,...\\}} = & 512n + 274 \\\\\n\\text{$V_{17,10} \\in $ \\{18,1042,2066,3090,...\\}} = & 1024n + 18 \\\\\n\\text{$V_{17,11} \\in $ \\{1554,3602,5650,7698,...\\}} = & 2048n + 1554 \\\\\n\\text{$V_{17,12} \\in $ \\{530,4626,8722,12818,...\\}} = & 4096n + 530 \\\\\n\\text{$V_{17,13} \\in $ \\{2578,10770,18962,27154,...\\}} = & 8192n + 2578 \\\\\n\\text{$V_{17,14} \\in $ \\{14866,31250,47634,64018,...\\}} = & 16384n + 14866 \\\\\n\\text{$V_{17,15} \\in $ \\{6674,39442,72210,104978,...\\}} = & 32768n + 6674 \\\\\n\\text{$V_{17,16} \\in $ \\{23058,88594,154130,219666,...\\}} = & 65536n + 23058 \\\\\n\\text{$V_{17,17} \\in $ \\{55826,186898,317970,449042,...\\}} = & 131072n + 55826 \\\\\n\\text{$V_{17,18} \\in $ \\{121362,383506,645650,907794,...\\}} = & 262144n + 121362 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_5 = 18V_{17, m}(n) +17 =\n\\begin{cases}\n\\text{$d_{5,1} \\in $ \\{35,71,107,143,...\\}} = & 36n + 35 \\\\\n\\text{$d_{5,2} \\in $ \\{17,89,161,233,...\\}} = & 72n + 17 \\\\\n\\text{$d_{5,3} \\in $ \\{125,269,413,557,...\\}} = & 144n + 125 \\\\\n\\text{$d_{5,4} \\in $ \\{197,485,773,1061,...\\}} = & 288n + 197 \\\\\n\\text{$d_{5,5} \\in $ \\{53,629,1205,1781,...\\}} = & 576n + 53 \\\\\n\\text{$d_{5,6} \\in $ \\{917,2069,3221,4373,...\\}} = & 1152n + 917 \\\\\n\\text{$d_{5,7} \\in $ \\{1493,3797,6101,8405,...\\}} = & 2304n + 1493 \\\\\n\\text{$d_{5,8} \\in $ \\{2645,7253,11861,16469,...\\}} = & 4608n + 2645 \\\\\n\\text{$d_{5,9} \\in $ \\{4949,14165,23381,32597,...\\}} = & 9216n + 4949 \\\\\n\\text{$d_{5,10} \\in $ \\{341,18773,37205,55637,...\\}} = & 18432n + 341 \\\\\n\\text{$d_{5,11} \\in $ \\{27989,64853,101717,138581,...\\}} = & 36864n + 27989 \\\\\n\\text{$d_{5,12} \\in $ \\{9557,83285,157013,230741,...\\}} = & 73728n + 9557 \\\\\n\\text{$d_{5,13} \\in $ \\{46421,193877,341333,488789,...\\}} = & 147456n + 46421 \\\\\n\\text{$d_{5,14} \\in $ \\{267605,562517,857429,1152341,...\\}} = & 294912n + 267605 \\\\\n\\text{$d_{5,15} \\in $ \\{120149,709973,1299797,1889621,...\\}} = & 589824n + 120149 \\\\\n\\text{$d_{5,16} \\in $ \\{415061,1594709,2774357,3954005,...\\}} = & 1179648n + 415061 \\\\\n\\text{$d_{5,17} \\in $ \\{1004885,3364181,5723477,8082773,...\\}} = & 2359296n + 1004885 \\\\\n\\text{$d_{5,18} \\in $ \\{2184533,6903125,11621717,16340309,...\\}} = & 4718592n + 2184533 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_5+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =6, r_6 = S(6) = 15$ (note that the index of V below is 15):\n{\\small\n\\begin{equation}\nV_{15, m}(n) = \n\\begin{cases}\n\\text{$V_{15,1} \\in $ \\{0,2,4,6,...\\}} = & 2n \\\\\n\\text{$V_{15,2} \\in $ \\{1,5,9,13,...\\}} = & 4n + 1 \\\\\n\\text{$V_{15,3} \\in $ \\{7,15,23,31,...\\}} = & 8n + 7 \\\\\n\\text{$V_{15,4} \\in $ \\{3,19,35,51,...\\}} = & 16n + 3 \\\\\n\\text{$V_{15,5} \\in $ \\{27,59,91,123,...\\}} = & 32n + 27 \\\\\n\\text{$V_{15,6} \\in $ \\{43,107,171,235,...\\}} = & 64n + 43 \\\\\n\\text{$V_{15,7} \\in $ \\{11,139,267,395,...\\}} = & 128n + 11 \\\\\n\\text{$V_{15,8} \\in $ \\{203,459,715,971,...\\}} = & 256n + 203 \\\\\n\\text{$V_{15,9} \\in $ \\{331,843,1355,1867,...\\}} = & 512n + 331 \\\\\n\\text{$V_{15,10} \\in $ \\{587,1611,2635,3659,...\\}} = & 1024n + 587 \\\\\n\\text{$V_{15,11} \\in $ \\{1099,3147,5195,7243,...\\}} = & 2048n + 1099 \\\\\n\\text{$V_{15,12} \\in $ \\{75,4171,8267,12363,...\\}} = & 4096n + 75 \\\\\n\\text{$V_{15,13} \\in $ \\{6219,14411,22603,30795,...\\}} = & 8192n + 6219 \\\\\n\\text{$V_{15,14} \\in $ \\{2123,18507,34891,51275,...\\}} = & 16384n + 2123 \\\\\n\\text{$V_{15,15} \\in $ \\{10315,43083,75851,108619,...\\}} = & 32768n + 10315 \\\\\n\\text{$V_{15,16} \\in $ \\{59467,125003,190539,256075,...\\}} = & 65536n + 59467 \\\\\n\\text{$V_{15,17} \\in $ \\{26699,157771,288843,419915,...\\}} = & 131072n + 26699 \\\\\n\\text{$V_{15,18} \\in $ \\{92235,354379,616523,878667,...\\}} = & 262144n + 92235 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_6 = 18V_{15, m}(n) +15 = \n\\begin{cases}\n\\text{$d_{6,1} \\in $ \\{15,51,87,123,...\\}} = & 36n + 15 \\\\\n\\text{$d_{6,2} \\in $ \\{33,105,177,249,...\\}} = & 72n + 33 \\\\\n\\text{$d_{6,3} \\in $ \\{141,285,429,573,...\\}} = & 144n + 141 \\\\\n\\text{$d_{6,4} \\in $ \\{69,357,645,933,...\\}} = & 288n + 69 \\\\\n\\text{$d_{6,5} \\in $ \\{501,1077,1653,2229,...\\}} = & 576n + 501 \\\\\n\\text{$d_{6,6} \\in $ \\{789,1941,3093,4245,...\\}} = & 1152n + 789 \\\\\n\\text{$d_{6,7} \\in $ \\{213,2517,4821,7125,...\\}} = & 2304n + 213 \\\\\n\\text{$d_{6,8} \\in $ \\{3669,8277,12885,17493,...\\}} = & 4608n + 3669 \\\\\n\\text{$d_{6,9} \\in $ \\{5973,15189,24405,33621,...\\}} = & 9216n + 5973 \\\\\n\\text{$d_{6,10} \\in $ \\{10581,29013,47445,65877,...\\}} = & 18432n + 10581 \\\\\n\\text{$d_{6,11} \\in $ \\{19797,56661,93525,130389,...\\}} = & 36864n + 19797 \\\\\n\\text{$d_{6,12} \\in $ \\{1365,75093,148821,222549,...\\}} = & 73728n + 1365 \\\\\n\\text{$d_{6,13} \\in $ \\{111957,259413,406869,554325,...\\}} = & 147456n + 111957 \\\\\n\\text{$d_{6,14} \\in $ \\{38229,333141,628053,922965,...\\}} = & 294912n + 38229 \\\\\n\\text{$d_{6,15} \\in $ \\{185685,775509,1365333,1955157,...\\}} = & 589824n + 185685 \\\\\n\\text{$d_{6,16} \\in $ \\{1070421,2250069,3429717,4609365,...\\}} = & 1179648n + 1070421 \\\\\n\\text{$d_{6,17} \\in $ \\{480597,2839893,5199189,7558485,...\\}} = & 2359296n + 480597 \\\\\n\\text{$d_{6,18} \\in $ \\{1660245,6378837,11097429,15816021,...\\}} = & 4718592n + 1660245 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_6+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =7, r_7 = S(7) = 7$:\n{\\small\n\\begin{equation}\nV_{7, m}(n) = \n\\begin{cases}\n\\text{$V_{7,1} \\in $ \\{0,2,4,6,...\\}} = & 2n \\\\\n\\text{$V_{7,2} \\in $ \\{1,5,9,13,...\\}} = & 4n + 1 \\\\\n\\text{$V_{7,3} \\in $ \\{3,11,19,27,...\\}} = & 8n + 3 \\\\\n\\text{$V_{7,4} \\in $ \\{7,23,39,55,...\\}} = & 16n + 7 \\\\\n\\text{$V_{7,5} \\in $ \\{31,63,95,127,...\\}} = & 32n + 31 \\\\\n\\text{$V_{7,6} \\in $ \\{15,79,143,207,...\\}} = & 64n + 15 \\\\\n\\text{$V_{7,7} \\in $ \\{111,239,367,495,...\\}} = & 128n + 111 \\\\\n\\text{$V_{7,8} \\in $ \\{175,431,687,943,...\\}} = & 256n + 175 \\\\\n\\text{$V_{7,9} \\in $ \\{47,559,1071,1583,...\\}} = & 512n + 47 \\\\\n\\text{$V_{7,10} \\in $ \\{815,1839,2863,3887,...\\}} = & 1024n + 815 \\\\\n\\text{$V_{7,11} \\in $ \\{1327,3375,5423,7471,...\\}} = & 2048n + 1327 \\\\\n\\text{$V_{7,12} \\in $ \\{2351,6447,10543,14639,...\\}} = & 4096n + 2351 \\\\\n\\text{$V_{7,13} \\in $ \\{4399,12591,20783,28975,...\\}} = & 8192n + 4399 \\\\\n\\text{$V_{7,14} \\in $ \\{303,16687,33071,49455,...\\}} = & 16384n + 303 \\\\\n\\text{$V_{7,15} \\in $ \\{24879,57647,90415,123183,...\\}} = & 32768n + 24879 \\\\\n\\text{$V_{7,16} \\in $ \\{8495,74031,139567,205103,...\\}} = & 65536n + 8495 \\\\\n\\text{$V_{7,17} \\in $ \\{41263,172335,303407,434479,...\\}} = & 131072n + 41263 \\\\\n\\text{$V_{7,18} \\in $ \\{237871,500015,762159,1024303,...\\}} = & 262144n + 237871 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_7 = V_{7, m}(n) +7 =\n\\begin{cases}\n\\text{$d_{7,1} \\in $ \\{7,43,79,115,...\\}} = & 36n + 7 \\\\\n\\text{$d_{7,2} \\in $ \\{25,97,169,241,...\\}} = & 72n + 25 \\\\\n\\text{$d_{7,3} \\in $ \\{61,205,349,493,...\\}} = & 144n + 61 \\\\\n\\text{$d_{7,4} \\in $ \\{133,421,709,997,...\\}} = & 288n + 133 \\\\\n\\text{$d_{7,5} \\in $ \\{565,1141,1717,2293,...\\}} = & 576n + 565 \\\\\n\\text{$d_{7,6} \\in $ \\{277,1429,2581,3733,...\\}} = & 1152n + 277 \\\\\n\\text{$d_{7,7} \\in $ \\{2005,4309,6613,8917,...\\}} = & 2304n + 2005 \\\\\n\\text{$d_{7,8} \\in $ \\{3157,7765,12373,16981,...\\}} = & 4608n + 3157 \\\\\n\\text{$d_{7,9} \\in $ \\{853,10069,19285,28501,...\\}} = & 9216n + 853 \\\\\n\\text{$d_{7,10} \\in $ \\{14677,33109,51541,69973,...\\}} = & 18432n + 14677 \\\\\n\\text{$d_{7,11} \\in $ \\{23893,60757,97621,134485,...\\}} = & 36864n + 23893 \\\\\n\\text{$d_{7,12} \\in $ \\{42325,116053,189781,263509,...\\}} = & 73728n + 42325 \\\\\n\\text{$d_{7,13} \\in $ \\{79189,226645,374101,521557,...\\}} = & 147456n + 79189 \\\\\n\\text{$d_{7,14} \\in $ \\{5461,300373,595285,890197,...\\}} = & 294912n + 5461 \\\\\n\\text{$d_{7,15} \\in $ \\{447829,1037653,1627477,2217301,...\\}} = & 589824n + 447829 \\\\\n\\text{$d_{7,16} \\in $ \\{152917,1332565,2512213,3691861,...\\}} = & 1179648n + 152917 \\\\\n\\text{$d_{7,17} \\in $ \\{742741,3102037,5461333,7820629,...\\}} = & 2359296n + 742741 \\\\\n\\text{$d_{7,18} \\in $ \\{4281685,9000277,13718869,18437461,...\\}} = & 4718592n + 4281685 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_7+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =8, r_8 = S(8) = 11$ (note that the index of V below is 11):\n{\\small\n\\begin{equation}\nV_{11, m}(n) = \n\\begin{cases}\n\\text{$V_{11,1} \\in $ \\{0,2,4,6,...\\}} = & 2n \\\\\n\\text{$V_{11,2} \\in $ \\{3,7,11,15,...\\}} = & 4n + 3 \\\\\n\\text{$V_{11,3} \\in $ \\{1,9,17,25,...\\}} = & 8n + 1 \\\\\n\\text{$V_{11,4} \\in $ \\{5,21,37,53,...\\}} = & 16n + 5 \\\\\n\\text{$V_{11,5} \\in $ \\{13,45,77,109,...\\}} = & 32n + 13 \\\\\n\\text{$V_{11,6} \\in $ \\{29,93,157,221,...\\}} = & 64n + 29 \\\\\n\\text{$V_{11,7} \\in $ \\{125,253,381,509,...\\}} = & 128n + 125 \\\\\n\\text{$V_{11,8} \\in $ \\{61,317,573,829,...\\}} = & 256n + 61 \\\\\n\\text{$V_{11,9} \\in $ \\{445,957,1469,1981,...\\}} = & 512n + 445 \\\\\n\\text{$V_{11,10} \\in $ \\{701,1725,2749,3773,...\\}} = & 1024n + 701 \\\\\n\\text{$V_{11,11} \\in $ \\{189,2237,4285,6333,...\\}} = & 2048n + 189 \\\\\n\\text{$V_{11,12} \\in $ \\{3261,7357,11453,15549,...\\}} = & 4096n + 3261 \\\\\n\\text{$V_{11,13} \\in $ \\{5309,13501,21693,29885,...\\}} = & 8192n + 5309 \\\\\n\\text{$V_{11,14} \\in $ \\{9405,25789,42173,58557,...\\}} = & 16384n + 9405 \\\\\n\\text{$V_{11,15} \\in $ \\{17597,50365,83133,115901,...\\}} = & 32768n + 17597 \\\\\n\\text{$V_{11,16} \\in $ \\{1213,66749,132285,197821,...\\}} = & 65536n + 1213 \\\\\n\\text{$V_{11,17} \\in $ \\{99517,230589,361661,492733,...\\}} = & 131072n + 99517 \\\\\n\\text{$V_{11,18} \\in $ \\{33981,296125,558269,820413,...\\}} = & 262144n + 33981 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_8 = 18 V_{11, m}(n) +11\n\\begin{cases}\n\\text{$d_{8,1} \\in $ \\{11,47,83,119,...\\}} = & 36n + 11 \\\\\n\\text{$d_{8,2} \\in $ \\{65,137,209,281,...\\}} = & 72n + 65 \\\\\n\\text{$d_{8,3} \\in $ \\{29,173,317,461,...\\}} = & 144n + 29 \\\\\n\\text{$d_{8,4} \\in $ \\{101,389,677,965,...\\}} = & 288n + 101 \\\\\n\\text{$d_{8,5} \\in $ \\{245,821,1397,1973,...\\}} = & 576n + 245 \\\\\n\\text{$d_{8,6} \\in $ \\{533,1685,2837,3989,...\\}} = & 1152n + 533 \\\\\n\\text{$d_{8,7} \\in $ \\{2261,4565,6869,9173,...\\}} = & 2304n + 2261 \\\\\n\\text{$d_{8,8} \\in $ \\{1109,5717,10325,14933,...\\}} = & 4608n + 1109 \\\\\n\\text{$d_{8,9} \\in $ \\{8021,17237,26453,35669,...\\}} = & 9216n + 8021 \\\\\n\\text{$d_{8,10} \\in $ \\{12629,31061,49493,67925,...\\}} = & 18432n + 12629 \\\\\n\\text{$d_{8,11} \\in $ \\{3413,40277,77141,114005,...\\}} = & 36864n + 3413 \\\\\n\\text{$d_{8,12} \\in $ \\{58709,132437,206165,279893,...\\}} = & 73728n + 58709 \\\\\n\\text{$d_{8,13} \\in $ \\{95573,243029,390485,537941,...\\}} = & 147456n + 95573 \\\\\n\\text{$d_{8,14} \\in $ \\{169301,464213,759125,1054037,...\\}} = & 294912n + 169301 \\\\\n\\text{$d_{8,15} \\in $ \\{316757,906581,1496405,2086229,...\\}} = & 589824n + 316757 \\\\\n\\text{$d_{8,16} \\in $ \\{21845,1201493,2381141,3560789,...\\}} = & 1179648n + 21845 \\\\\n\\text{$d_{8,17} \\in $ \\{1791317,4150613,6509909,8869205,...\\}} = & 2359296n + 1791317 \\\\\n\\text{$d_{8,18} \\in $ \\{611669,5330261,10048853,14767445,...\\}} = & 4718592n + 611669 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_8+1}{2^m} = odd\n\\end{equation}\n}\n\nWhen $i =9, r_9 = S(9) = 9$:\n{\\small\n\\begin{equation}\nV_{9, m}(n) = \n\\begin{cases}\n\\text{$V_{9,1} \\in $ \\{1,3,5,7,...\\}} = & 2n + 1 \\\\\n\\text{$V_{9,2} \\in $ \\{0,4,8,12,...\\}} = & 4n \\\\\n\\text{$V_{9,3} \\in $ \\{2,10,18,26,...\\}} = & 8n + 2 \\\\\n\\text{$V_{9,4} \\in $ \\{14,30,46,62,...\\}} = & 16n + 14 \\\\\n\\text{$V_{9,5} \\in $ \\{6,38,70,102,...\\}} = & 32n + 6 \\\\\n\\text{$V_{9,6} \\in $ \\{22,86,150,214,...\\}} = & 64n + 22 \\\\\n\\text{$V_{9,7} \\in $ \\{54,182,310,438,...\\}} = & 128n + 54 \\\\\n\\text{$V_{9,8} \\in $ \\{118,374,630,886,...\\}} = & 256n + 118 \\\\\n\\text{$V_{9,9} \\in $ \\{502,1014,1526,2038,...\\}} = & 512n + 502 \\\\\n\\text{$V_{9,10} \\in $ \\{246,1270,2294,3318,...\\}} = & 1024n + 246 \\\\\n\\text{$V_{9,11} \\in $ \\{1782,3830,5878,7926,...\\}} = & 2048n + 1782 \\\\\n\\text{$V_{9,12} \\in $ \\{2806,6902,10998,15094,...\\}} = & 4096n + 2806 \\\\\n\\text{$V_{9,13} \\in $ \\{758,8950,17142,25334,...\\}} = & 8192n + 758 \\\\\n\\text{$V_{9,14} \\in $ \\{13046,29430,45814,62198,...\\}} = & 16384n + 13046 \\\\\n\\text{$V_{9,15} \\in $ \\{21238,54006,86774,119542,...\\}} = & 32768n + 21238 \\\\\n\\text{$V_{9,16} \\in $ \\{37622,103158,168694,234230,...\\}} = & 65536n + 37622 \\\\\n\\text{$V_{9,17} \\in $ \\{70390,201462,332534,463606,...\\}} = & 131072n + 70390 \\\\\n\\text{$V_{9,18} \\in $ \\{4854,266998,529142,791286,...\\}} = & 262144n + 4854 \\\\\n\\vdots\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nd_9 = 18V_{9, m}(n)+9 =\n\\begin{cases}\n\\text{$d_{9,1} \\in $ \\{27,63,99,135,...\\}} = & 36n + 27 \\\\\n\\text{$d_{9,2} \\in $ \\{9,81,153,225,...\\}} = & 72n + 9 \\\\\n\\text{$d_{9,3} \\in $ \\{45,189,333,477,...\\}} = & 144n + 45 \\\\\n\\text{$d_{9,4} \\in $ \\{261,549,837,1125,...\\}} = & 288n + 261 \\\\\n\\text{$d_{9,5} \\in $ \\{117,693,1269,1845,...\\}} = & 576n + 117 \\\\\n\\text{$d_{9,6} \\in $ \\{405,1557,2709,3861,...\\}} = & 1152n + 405 \\\\\n\\text{$d_{9,7} \\in $ \\{981,3285,5589,7893,...\\}} = & 2304n + 981 \\\\\n\\text{$d_{9,8} \\in $ \\{2133,6741,11349,15957,...\\}} = & 4608n + 2133 \\\\\n\\text{$d_{9,9} \\in $ \\{9045,18261,27477,36693,...\\}} = & 9216n + 9045 \\\\\n\\text{$d_{9,10} \\in $ \\{4437,22869,41301,59733,...\\}} = & 18432n + 4437 \\\\\n\\text{$d_{9,11} \\in $ \\{32085,68949,105813,142677,...\\}} = & 36864n + 32085 \\\\\n\\text{$d_{9,12} \\in $ \\{50517,124245,197973,271701,...\\}} = & 73728n + 50517 \\\\\n\\text{$d_{9,13} \\in $ \\{13653,161109,308565,456021,...\\}} = & 147456n + 13653 \\\\\n\\text{$d_{9,14} \\in $ \\{234837,529749,824661,1119573,...\\}} = & 294912n + 234837 \\\\\n\\text{$d_{9,15} \\in $ \\{382293,972117,1561941,2151765,...\\}} = & 589824n + 382293 \\\\\n\\text{$d_{9,16} \\in $ \\{677205,1856853,3036501,4216149,...\\}} = & 1179648n + 677205 \\\\\n\\text{$d_{9,17} \\in $ \\{1267029,3626325,5985621,8344917,...\\}} = & 2359296n + 1267029 \\\\\n\\text{$d_{9,18} \\in $ \\{87381,4805973,9524565,14243157,...\\}} = & 4718592n + 87381 \\\\\n\\vdots\n\\end{cases}\n\\rightarrow \\frac{3d_9+1}{2^m} = odd\n\\end{equation}\n}\n\nThe proofs of these $d_i$ statement is not difficult, for example \n\\[\n\\begin{split}\n\\frac{3(36n+27)+1}{2^1}&=54n+41 \\\\\n \\frac{3(72n+9)+1}{2^2}&=54n+7 \\\\ \n&\\vdots \\\\\n\\frac{3(4718592n+87381)+1}{2^{18}}&=54n+1.\n \\end{split}\n \\]\n\\end{proof}\n\n\\begin{conjecture} \\label{conj1} On the boundedness of $\\frac{3d_i+1}{2^m}$.\nOne of the implications of theorem \\ref{mainthm} is that the next odd number $d_{i_{next}}=\\frac{3d_{i,m}+1}{2^m}$ is bounded between $54n$ and $54n+54$, i.e., $54n < d_{i_{next}} < 54(n+1)$  where $n$ is a variable dependent on $d_i$ (as also prescribed in the proposed Collatz map in \\ref{CollMachinery1}). \n\\end{conjecture} \n\nThe implication of conj \\ref{conj1} is $d_{i_{next}}$ must be congruent to residue $a$ modulo $54n$ where $ 1 \\le a \\le 53$ and  $a$ is odd. \n\n\\begin{conjecture} \\label{conj2}\nThese $d_i$ are values are the fundamental (principal) sets for all odd integers because they solely represent the sets from which all every odd number could be derived. \n\\end{conjecture}\n\nThe proof of this last conjecture is beyond the scope of the objectives of this foundational paper.\n\nAs a result of the proposed conjecture in \\ref{conj2} the map presented in \\ref{CollMachinery1} is constructed from the $d_i$ formulations and proposed as the generalised Collatz based number system. This map \\ref{CollMachinery1} consists of 3 major compartments: the top; middle; and the bottom sections representing the $d_i$, $3d_i+1$ and optimised $\\frac{3d_i+1}{2^m}$ results, respectively.\n\nLiewise, the map \\ref{CollHeightMap1} consists of 3 major compartments: the top; middle; and the bottom sections representing the corresponding total stopping time functions of $d_i$, $3d_i+1$ and optimsed $\\frac{3d_i+1}{2^m}$, respectively.\n\n\n\n\n\\begin{landscape}\n{\\fontsize{2.5}{4.0}\\selectfont\n\\begin{equation}\\label{CollMachinery1}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_1\\downarrow}} \\\\\n36n + 19*\\\\\n\\textcolor{blue}{ 72n + 1} \\\\\n\\textcolor{red}{ 144n + 109} \\\\\n288n + 37\\\\\n576n + 181\\\\\n1152n + 1045\\\\\n2304n + 469\\\\\n\\textcolor{blue}{ 4608n + 1621} \\\\\n9216n + 3925\\\\\n\\textcolor{blue}{ 18432n + 8533} \\\\\n36864n + 36181\\\\\n73728n + 17749\\\\\n\\textcolor{blue}{ 147456n + 128341} \\\\\n294912n + 202069\\\\\n\\textcolor{blue}{ 589824n + 54613} \\\\\n1179648n + 939349\\\\\n2359296n + 1529173\\\\\n4718592n + 2708821\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_1 \\downarrow}} \\\\\n108n + 58*\\\\\n\\textcolor{blue}{ 216n + 4} \\\\\n\\textcolor{red}{ 432n + 328} \\\\\n864n + 112\\\\\n1728n + 544\\\\\n3456n + 3136\\\\\n6912n + 1408\\\\\n\\textcolor{blue}{ 13824n + 4864} \\\\\n27648n + 11776\\\\\n\\textcolor{blue}{ 55296n + 25600} \\\\\n110592n + 108544\\\\\n221184n + 53248\\\\\n\\textcolor{blue}{ 442368n + 385024} \\\\\n884736n + 606208\\\\\n\\textcolor{blue}{ 1769472n + 163840} \\\\\n3538944n + 2818048\\\\\n7077888n + 4587520\\\\\n14155776n + 8126464\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{1_{next}} \\downarrow}} \\\\\n54n + 29*\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_2\\downarrow}} \\\\\n36n + 23*\\\\\n72n + 41\\\\\n144n + 77\\\\\n\\textcolor{blue}{ 288n + 5} \\\\\n\\textcolor{red}{ 576n + 437} \\\\\n1152n + 149\\\\\n2304n + 725\\\\\n4608n + 4181\\\\\n9216n + 1877\\\\\n\\textcolor{blue}{ 18432n + 6485} \\\\\n36864n + 15701\\\\\n\\textcolor{blue}{ 73728n + 34133} \\\\\n147456n + 144725\\\\\n294912n + 70997\\\\\n\\textcolor{blue}{ 589824n + 513365} \\\\\n1179648n + 808277\\\\\n\\textcolor{blue}{ 2359296n + 218453} \\\\\n4718592n + 3757397\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_2 \\downarrow}} \\\\\n108n + 70*\\\\\n216n + 124\\\\\n432n + 232\\\\\n\\textcolor{blue}{ 864n + 16} \\\\\n\\textcolor{red}{ 1728n + 1312} \\\\\n3456n + 448\\\\\n6912n + 2176\\\\\n13824n + 12544\\\\\n27648n + 5632\\\\\n\\textcolor{blue}{ 55296n + 19456} \\\\\n110592n + 47104\\\\\n\\textcolor{blue}{ 221184n + 102400} \\\\\n442368n + 434176\\\\\n884736n + 212992\\\\\n\\textcolor{blue}{ 1769472n + 1540096} \\\\\n3538944n + 2424832\\\\\n\\textcolor{blue}{ 7077888n + 655360} \\\\\n14155776n + 11272192\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{2_{next}} \\downarrow}} \\\\\n54n + 35*\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_3\\downarrow}} \\\\\n\\textcolor{blue}{ 36n + 3}* \\\\\n72n + 57\\\\\n144n + 93\\\\\n288n + 165\\\\\n576n + 309\\\\\n\\textcolor{blue}{ 1152n + 21} \\\\\n\\textcolor{red}{ 2304n + 1749} \\\\\n4608n + 597\\\\\n9216n + 2901\\\\\n18432n + 16725\\\\\n36864n + 7509\\\\\n\\textcolor{blue}{ 73728n + 25941} \\\\\n147456n + 62805\\\\\n\\textcolor{blue}{ 294912n + 136533} \\\\\n589824n + 578901\\\\\n1179648n + 283989\\\\\n\\textcolor{blue}{ 2359296n + 2053461} \\\\\n4718592n + 3233109\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_3 \\downarrow}} \\\\\n\\textcolor{blue}{ 108n + 10}* \\\\\n216n + 172\\\\\n432n + 280\\\\\n864n + 496\\\\\n1728n + 928\\\\\n\\textcolor{blue}{ 3456n + 64} \\\\\n\\textcolor{red}{ 6912n + 5248} \\\\\n13824n + 1792\\\\\n27648n + 8704\\\\\n55296n + 50176\\\\\n110592n + 22528\\\\\n\\textcolor{blue}{ 221184n + 77824} \\\\\n442368n + 188416\\\\\n\\textcolor{blue}{ 884736n + 409600} \\\\\n1769472n + 1736704\\\\\n3538944n + 851968\\\\\n\\textcolor{blue}{ 7077888n + 6160384} \\\\\n14155776n + 9699328\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{3_{next}} \\downarrow}} \\\\\n\\textcolor{blue}{ 54n + 5}* \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_4\\downarrow}} \\\\\n\\textcolor{blue}{ 36n + 31}* \\\\\n72n + 49\\\\\n\\textcolor{blue}{ 144n + 13} \\\\\n288n + 229\\\\\n576n + 373\\\\\n1152n + 661\\\\\n2304n + 1237\\\\\n\\textcolor{blue}{ 4608n + 85} \\\\\n\\textcolor{red}{ 9216n + 6997} \\\\\n18432n + 2389\\\\\n36864n + 11605\\\\\n73728n + 66901\\\\\n147456n + 30037\\\\\n\\textcolor{blue}{ 294912n + 103765} \\\\\n589824n + 251221\\\\\n\\textcolor{blue}{ 1179648n + 546133} \\\\\n2359296n + 2315605\\\\\n4718592n + 1135957\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_4 \\downarrow}} \\\\\n\\textcolor{blue}{ 108n + 94}* \\\\\n216n + 148\\\\\n\\textcolor{blue}{ 432n + 40} \\\\\n864n + 688\\\\\n1728n + 1120\\\\\n3456n + 1984\\\\\n6912n + 3712\\\\\n\\textcolor{blue}{ 13824n + 256} \\\\\n\\textcolor{red}{ 27648n + 20992} \\\\\n55296n + 7168\\\\\n110592n + 34816\\\\\n221184n + 200704\\\\\n442368n + 90112\\\\\n\\textcolor{blue}{ 884736n + 311296} \\\\\n1769472n + 753664\\\\\n\\textcolor{blue}{ 3538944n + 1638400} \\\\\n7077888n + 6946816\\\\\n14155776n + 3407872\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{4_{next}} \\downarrow}} \\\\\n\\textcolor{blue}{ 54n + 47}* \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_5\\downarrow}} \\\\\n36n + 35*\\\\\n72n + 17\\\\\n\\textcolor{blue}{ 144n + 125} \\\\\n288n + 197\\\\\n\\textcolor{blue}{ 576n + 53} \\\\\n1152n + 917\\\\\n2304n + 1493\\\\\n4608n + 2645\\\\\n9216n + 4949\\\\\n\\textcolor{blue}{ 18432n + 341} \\\\\n\\textcolor{red}{ 36864n + 27989} \\\\\n73728n + 9557\\\\\n147456n + 46421\\\\\n294912n + 267605\\\\\n589824n + 120149\\\\\n\\textcolor{blue}{ 1179648n + 415061} \\\\\n2359296n + 1004885\\\\\n4718592n + 2184533\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_5 \\downarrow}} \\\\\n108n + 106*\\\\\n216n + 52\\\\\n\\textcolor{blue}{ 432n + 376} \\\\\n864n + 592\\\\\n\\textcolor{blue}{ 1728n + 160} \\\\\n3456n + 2752\\\\\n6912n + 4480\\\\\n13824n + 7936\\\\\n27648n + 14848\\\\\n\\textcolor{blue}{ 55296n + 1024} \\\\\n\\textcolor{red}{ 110592n + 83968} \\\\\n221184n + 28672\\\\\n442368n + 139264\\\\\n884736n + 802816\\\\\n1769472n + 360448\\\\\n\\textcolor{blue}{ 3538944n + 1245184} \\\\\n7077888n + 3014656\\\\\n14155776n + 6553600\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{5_{next}} \\downarrow}} \\\\\n54n + 53*\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n54n + 25\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_6\\downarrow}} \\\\\n36n + 15*\\\\\n\\textcolor{blue}{ 72n + 33} \\\\\n144n + 141\\\\\n288n + 69\\\\\n\\textcolor{blue}{ 576n + 501} \\\\\n1152n + 789\\\\\n\\textcolor{blue}{ 2304n + 213} \\\\\n4608n + 3669\\\\\n9216n + 5973\\\\\n18432n + 10581\\\\\n36864n + 19797\\\\\n\\textcolor{blue}{ 73728n + 1365} \\\\\n\\textcolor{red}{ 147456n + 111957} \\\\\n294912n + 38229\\\\\n589824n + 185685\\\\\n1179648n + 1070421\\\\\n2359296n + 480597\\\\\n4718592n + 1660245\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_6 \\downarrow}} \\\\\n108n + 46*\\\\\n\\textcolor{blue}{ 216n + 100} \\\\\n432n + 424\\\\\n864n + 208\\\\\n\\textcolor{blue}{ 1728n + 1504} \\\\\n3456n + 2368\\\\\n\\textcolor{blue}{ 6912n + 640} \\\\\n13824n + 11008\\\\\n27648n + 17920\\\\\n55296n + 31744\\\\\n110592n + 59392\\\\\n\\textcolor{blue}{ 221184n + 4096} \\\\\n\\textcolor{red}{ 442368n + 335872} \\\\\n884736n + 114688\\\\\n1769472n + 557056\\\\\n3538944n + 3211264\\\\\n7077888n + 1441792\\\\\n14155776n + 4980736\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{6_{next}} \\downarrow}} \\\\\n54n + 23*\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n54n + 19\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_7\\downarrow}} \\\\\n36n + 7*\\\\\n\\textcolor{blue}{ 72n + 25} \\\\\n144n + 61\\\\\n\\textcolor{blue}{ 288n + 133} \\\\\n576n + 565\\\\\n1152n + 277\\\\\n\\textcolor{blue}{ 2304n + 2005} \\\\\n4608n + 3157\\\\\n\\textcolor{blue}{ 9216n + 853} \\\\\n18432n + 14677\\\\\n36864n + 23893\\\\\n73728n + 42325\\\\\n147456n + 79189\\\\\n\\textcolor{blue}{ 294912n + 5461} \\\\\n\\textcolor{red}{ 589824n + 447829} \\\\\n1179648n + 152917\\\\\n2359296n + 742741\\\\\n4718592n + 4281685\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_7 \\downarrow}} \\\\\n108n + 22*\\\\\n\\textcolor{blue}{ 216n + 76} \\\\\n432n + 184\\\\\n\\textcolor{blue}{ 864n + 400} \\\\\n1728n + 1696\\\\\n3456n + 832\\\\\n\\textcolor{blue}{ 6912n + 6016} \\\\\n13824n + 9472\\\\\n\\textcolor{blue}{ 27648n + 2560} \\\\\n55296n + 44032\\\\\n110592n + 71680\\\\\n221184n + 126976\\\\\n442368n + 237568\\\\\n\\textcolor{blue}{ 884736n + 16384} \\\\\n\\textcolor{red}{ 1769472n + 1343488} \\\\\n3538944n + 458752\\\\\n7077888n + 2228224\\\\\n14155776n + 12845056\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{7_{next}} \\downarrow}} \\\\\n54n + 11*\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_8\\downarrow}} \\\\\n36n + 11*\\\\\n72n + 65\\\\\n144n + 29\\\\\n\\textcolor{blue}{ 288n + 101} \\\\\n576n + 245\\\\\n\\textcolor{blue}{ 1152n + 533} \\\\\n2304n + 2261\\\\\n4608n + 1109\\\\\n\\textcolor{blue}{ 9216n + 8021} \\\\\n18432n + 12629\\\\\n\\textcolor{blue}{ 36864n + 3413} \\\\\n73728n + 58709\\\\\n147456n + 95573\\\\\n294912n + 169301\\\\\n589824n + 316757\\\\\n\\textcolor{blue}{ 1179648n + 21845} \\\\\n\\textcolor{red}{ 2359296n + 1791317} \\\\\n4718592n + 611669\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_8 \\downarrow}} \\\\\n108n + 34*\\\\\n216n + 196\\\\\n432n + 88\\\\\n\\textcolor{blue}{ 864n + 304} \\\\\n1728n + 736\\\\\n\\textcolor{blue}{ 3456n + 1600} \\\\\n6912n + 6784\\\\\n13824n + 3328\\\\\n\\textcolor{blue}{ 27648n + 24064} \\\\\n55296n + 37888\\\\\n\\textcolor{blue}{ 110592n + 10240} \\\\\n221184n + 176128\\\\\n442368n + 286720\\\\\n884736n + 507904\\\\\n1769472n + 950272\\\\\n\\textcolor{blue}{ 3538944n + 65536} \\\\\n\\textcolor{red}{ 7077888n + 5373952} \\\\\n14155776n + 1835008\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{8_{next}} \\downarrow}} \\\\\n54n + 17*\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\textcolor{red}{ 54n + 41} \\\\\n54n + 7\\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_9\\downarrow}} \\\\\n\\textcolor{red}{ 36n + 27}* \\\\\n72n + 9\\\\\n144n + 45\\\\\n288n + 261\\\\\n576n + 117\\\\\n\\textcolor{blue}{ 1152n + 405} \\\\\n2304n + 981\\\\\n\\textcolor{blue}{ 4608n + 2133} \\\\\n9216n + 9045\\\\\n18432n + 4437\\\\\n\\textcolor{blue}{ 36864n + 32085} \\\\\n73728n + 50517\\\\\n\\textcolor{blue}{ 147456n + 13653} \\\\\n294912n + 234837\\\\\n589824n + 382293\\\\\n1179648n + 677205\\\\\n2359296n + 1267029\\\\\n\\textcolor{blue}{ 4718592n + 87381} \\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even~_9 \\downarrow}} \\\\\n\\textcolor{red}{ 108n + 82}* \\\\\n216n + 28\\\\\n432n + 136\\\\\n864n + 784\\\\\n1728n + 352\\\\\n\\textcolor{blue}{ 3456n + 1216} \\\\\n6912n + 2944\\\\\n\\textcolor{blue}{ 13824n + 6400} \\\\\n27648n + 27136\\\\\n55296n + 13312\\\\\n\\textcolor{blue}{ 110592n + 96256} \\\\\n221184n + 151552\\\\\n\\textcolor{blue}{ 442368n + 40960} \\\\\n884736n + 704512\\\\\n1769472n + 1146880\\\\\n3538944n + 2031616\\\\\n7077888n + 3801088\\\\\n\\textcolor{blue}{ 14155776n + 262144} \\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{9_{next}} \\downarrow}} \\\\\n\\textcolor{red}{ 54n + 41}* \\\\\n54n + 7\\\\\n54n + 17\\\\\n54n + 49\\\\\n54n + 11\\\\\n\\textcolor{blue}{ 54n + 19} \\\\\n54n + 23\\\\\n\\textcolor{blue}{ 54n + 25} \\\\\n54n + 53\\\\\n54n + 13\\\\\n\\textcolor{blue}{ 54n + 47} \\\\\n54n + 37\\\\\n\\textcolor{blue}{ 54n + 5} \\\\\n54n + 43\\\\\n54n + 35\\\\\n54n + 31\\\\\n54n + 29\\\\\n\\textcolor{blue}{ 54n + 1} \\\\\n\\vdots \\\\\n\\end{cases}\n\\footnote{All rights reserved. $\\copyright$ Michael A. Idowu, 2014.}\n\\end{equation}\n}\n\\end{landscape}\n\n\\begin{landscape}\n{\\fontsize{2.5}{4.0}\\selectfont\n\\begin{equation}\\label{CollHeightMap1}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_1\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+29)+2\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+3} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+4} \\\\\n\\sigma_{\\infty}(54n+7)+5\\\\\n\\sigma_{\\infty}(54n+17)+6\\\\\n\\sigma_{\\infty}(54n+49)+7\\\\\n\\sigma_{\\infty}(54n+11)+8\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+9} \\\\\n\\sigma_{\\infty}(54n+23)+10\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+11} \\\\\n\\sigma_{\\infty}(54n+53)+12\\\\\n\\sigma_{\\infty}(54n+13)+13\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+14} \\\\\n\\sigma_{\\infty}(54n+37)+15\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+16} \\\\\n\\sigma_{\\infty}(54n+43)+17\\\\\n\\sigma_{\\infty}(54n+35)+18\\\\\n\\sigma_{\\infty}(54n+31)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_1 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+29)+1\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+2} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+3} \\\\\n\\sigma_{\\infty}(54n+7)+4\\\\\n\\sigma_{\\infty}(54n+17)+5\\\\\n\\sigma_{\\infty}(54n+49)+6\\\\\n\\sigma_{\\infty}(54n+11)+7\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+8} \\\\\n\\sigma_{\\infty}(54n+23)+9\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+10} \\\\\n\\sigma_{\\infty}(54n+53)+11\\\\\n\\sigma_{\\infty}(54n+13)+12\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+13} \\\\\n\\sigma_{\\infty}(54n+37)+14\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+15} \\\\\n\\sigma_{\\infty}(54n+43)+16\\\\\n\\sigma_{\\infty}(54n+35)+17\\\\\n\\sigma_{\\infty}(54n+31)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{1_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_2\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+35)+2\\\\\n\\sigma_{\\infty}(54n+31)+3\\\\\n\\sigma_{\\infty}(54n+29)+4\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+5} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+6} \\\\\n\\sigma_{\\infty}(54n+7)+7\\\\\n\\sigma_{\\infty}(54n+17)+8\\\\\n\\sigma_{\\infty}(54n+49)+9\\\\\n\\sigma_{\\infty}(54n+11)+10\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+11} \\\\\n\\sigma_{\\infty}(54n+23)+12\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+13} \\\\\n\\sigma_{\\infty}(54n+53)+14\\\\\n\\sigma_{\\infty}(54n+13)+15\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+16} \\\\\n\\sigma_{\\infty}(54n+37)+17\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+18} \\\\\n\\sigma_{\\infty}(54n+43)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_2 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+35)+1\\\\\n\\sigma_{\\infty}(54n+31)+2\\\\\n\\sigma_{\\infty}(54n+29)+3\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+4} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+5} \\\\\n\\sigma_{\\infty}(54n+7)+6\\\\\n\\sigma_{\\infty}(54n+17)+7\\\\\n\\sigma_{\\infty}(54n+49)+8\\\\\n\\sigma_{\\infty}(54n+11)+9\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+10} \\\\\n\\sigma_{\\infty}(54n+23)+11\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+12} \\\\\n\\sigma_{\\infty}(54n+53)+13\\\\\n\\sigma_{\\infty}(54n+13)+14\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+15} \\\\\n\\sigma_{\\infty}(54n+37)+16\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+17} \\\\\n\\sigma_{\\infty}(54n+43)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{2_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_3\\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+2} \\\\\n\\sigma_{\\infty}(54n+43)+3\\\\\n\\sigma_{\\infty}(54n+35)+4\\\\\n\\sigma_{\\infty}(54n+31)+5\\\\\n\\sigma_{\\infty}(54n+29)+6\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+7} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+8} \\\\\n\\sigma_{\\infty}(54n+7)+9\\\\\n\\sigma_{\\infty}(54n+17)+10\\\\\n\\sigma_{\\infty}(54n+49)+11\\\\\n\\sigma_{\\infty}(54n+11)+12\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+13} \\\\\n\\sigma_{\\infty}(54n+23)+14\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+15} \\\\\n\\sigma_{\\infty}(54n+53)+16\\\\\n\\sigma_{\\infty}(54n+13)+17\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+18} \\\\\n\\sigma_{\\infty}(54n+37)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_3 \\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+1} \\\\\n\\sigma_{\\infty}(54n+43)+2\\\\\n\\sigma_{\\infty}(54n+35)+3\\\\\n\\sigma_{\\infty}(54n+31)+4\\\\\n\\sigma_{\\infty}(54n+29)+5\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+6} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+7} \\\\\n\\sigma_{\\infty}(54n+7)+8\\\\\n\\sigma_{\\infty}(54n+17)+9\\\\\n\\sigma_{\\infty}(54n+49)+10\\\\\n\\sigma_{\\infty}(54n+11)+11\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+12} \\\\\n\\sigma_{\\infty}(54n+23)+13\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+14} \\\\\n\\sigma_{\\infty}(54n+53)+15\\\\\n\\sigma_{\\infty}(54n+13)+16\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+17} \\\\\n\\sigma_{\\infty}(54n+37)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{3_{next}} \\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_4\\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+2} \\\\\n\\sigma_{\\infty}(54n+37)+3\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+4} \\\\\n\\sigma_{\\infty}(54n+43)+5\\\\\n\\sigma_{\\infty}(54n+35)+6\\\\\n\\sigma_{\\infty}(54n+31)+7\\\\\n\\sigma_{\\infty}(54n+29)+8\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+9} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+10} \\\\\n\\sigma_{\\infty}(54n+7)+11\\\\\n\\sigma_{\\infty}(54n+17)+12\\\\\n\\sigma_{\\infty}(54n+49)+13\\\\\n\\sigma_{\\infty}(54n+11)+14\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+15} \\\\\n\\sigma_{\\infty}(54n+23)+16\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+17} \\\\\n\\sigma_{\\infty}(54n+53)+18\\\\\n\\sigma_{\\infty}(54n+13)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_4 \\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+1} \\\\\n\\sigma_{\\infty}(54n+37)+2\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+3} \\\\\n\\sigma_{\\infty}(54n+43)+4\\\\\n\\sigma_{\\infty}(54n+35)+5\\\\\n\\sigma_{\\infty}(54n+31)+6\\\\\n\\sigma_{\\infty}(54n+29)+7\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+8} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+9} \\\\\n\\sigma_{\\infty}(54n+7)+10\\\\\n\\sigma_{\\infty}(54n+17)+11\\\\\n\\sigma_{\\infty}(54n+49)+12\\\\\n\\sigma_{\\infty}(54n+11)+13\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+14} \\\\\n\\sigma_{\\infty}(54n+23)+15\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+16} \\\\\n\\sigma_{\\infty}(54n+53)+17\\\\\n\\sigma_{\\infty}(54n+13)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{4_{next}} \\downarrow}} \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_5\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+53)+2\\\\\n\\sigma_{\\infty}(54n+13)+3\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+4} \\\\\n\\sigma_{\\infty}(54n+37)+5\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+6} \\\\\n\\sigma_{\\infty}(54n+43)+7\\\\\n\\sigma_{\\infty}(54n+35)+8\\\\\n\\sigma_{\\infty}(54n+31)+9\\\\\n\\sigma_{\\infty}(54n+29)+10\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+11} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+12} \\\\\n\\sigma_{\\infty}(54n+7)+13\\\\\n\\sigma_{\\infty}(54n+17)+14\\\\\n\\sigma_{\\infty}(54n+49)+15\\\\\n\\sigma_{\\infty}(54n+11)+16\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+17} \\\\\n\\sigma_{\\infty}(54n+23)+18\\\\\n\\sigma_{\\infty}(54n+25)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_5 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+53)+1\\\\\n\\sigma_{\\infty}(54n+13)+2\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+3} \\\\\n\\sigma_{\\infty}(54n+37)+4\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+5} \\\\\n\\sigma_{\\infty}(54n+43)+6\\\\\n\\sigma_{\\infty}(54n+35)+7\\\\\n\\sigma_{\\infty}(54n+31)+8\\\\\n\\sigma_{\\infty}(54n+29)+9\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+10} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+11} \\\\\n\\sigma_{\\infty}(54n+7)+12\\\\\n\\sigma_{\\infty}(54n+17)+13\\\\\n\\sigma_{\\infty}(54n+49)+14\\\\\n\\sigma_{\\infty}(54n+11)+15\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+16} \\\\\n\\sigma_{\\infty}(54n+23)+17\\\\\n\\sigma_{\\infty}(54n+25)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{5_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\sigma_{\\infty}(54n+25) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_6\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+23)+2\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+3} \\\\\n\\sigma_{\\infty}(54n+53)+4\\\\\n\\sigma_{\\infty}(54n+13)+5\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+6} \\\\\n\\sigma_{\\infty}(54n+37)+7\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+8} \\\\\n\\sigma_{\\infty}(54n+43)+9\\\\\n\\sigma_{\\infty}(54n+35)+10\\\\\n\\sigma_{\\infty}(54n+31)+11\\\\\n\\sigma_{\\infty}(54n+29)+12\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+13} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+14} \\\\\n\\sigma_{\\infty}(54n+7)+15\\\\\n\\sigma_{\\infty}(54n+17)+16\\\\\n\\sigma_{\\infty}(54n+49)+17\\\\\n\\sigma_{\\infty}(54n+11)+18\\\\\n\\sigma_{\\infty}(54n+19)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_6 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+23)+1\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+2} \\\\\n\\sigma_{\\infty}(54n+53)+3\\\\\n\\sigma_{\\infty}(54n+13)+4\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+5} \\\\\n\\sigma_{\\infty}(54n+37)+6\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+7} \\\\\n\\sigma_{\\infty}(54n+43)+8\\\\\n\\sigma_{\\infty}(54n+35)+9\\\\\n\\sigma_{\\infty}(54n+31)+10\\\\\n\\sigma_{\\infty}(54n+29)+11\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+12} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+13} \\\\\n\\sigma_{\\infty}(54n+7)+14\\\\\n\\sigma_{\\infty}(54n+17)+15\\\\\n\\sigma_{\\infty}(54n+49)+16\\\\\n\\sigma_{\\infty}(54n+11)+17\\\\\n\\sigma_{\\infty}(54n+19)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{6_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\sigma_{\\infty}(54n+19) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_7\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+11)+2\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+3} \\\\\n\\sigma_{\\infty}(54n+23)+4\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+5} \\\\\n\\sigma_{\\infty}(54n+53)+6\\\\\n\\sigma_{\\infty}(54n+13)+7\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+8} \\\\\n\\sigma_{\\infty}(54n+37)+9\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+10} \\\\\n\\sigma_{\\infty}(54n+43)+11\\\\\n\\sigma_{\\infty}(54n+35)+12\\\\\n\\sigma_{\\infty}(54n+31)+13\\\\\n\\sigma_{\\infty}(54n+29)+14\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+15} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+16} \\\\\n\\sigma_{\\infty}(54n+7)+17\\\\\n\\sigma_{\\infty}(54n+17)+18\\\\\n\\sigma_{\\infty}(54n+49)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_7 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+11)+1\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+2} \\\\\n\\sigma_{\\infty}(54n+23)+3\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+4} \\\\\n\\sigma_{\\infty}(54n+53)+5\\\\\n\\sigma_{\\infty}(54n+13)+6\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+7} \\\\\n\\sigma_{\\infty}(54n+37)+8\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+9} \\\\\n\\sigma_{\\infty}(54n+43)+10\\\\\n\\sigma_{\\infty}(54n+35)+11\\\\\n\\sigma_{\\infty}(54n+31)+12\\\\\n\\sigma_{\\infty}(54n+29)+13\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+14} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+15} \\\\\n\\sigma_{\\infty}(54n+7)+16\\\\\n\\sigma_{\\infty}(54n+17)+17\\\\\n\\sigma_{\\infty}(54n+49)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{7_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_8\\downarrow}} \\\\\n\\sigma_{\\infty}(54n+17)+2\\\\\n\\sigma_{\\infty}(54n+49)+3\\\\\n\\sigma_{\\infty}(54n+11)+4\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+5} \\\\\n\\sigma_{\\infty}(54n+23)+6\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+7} \\\\\n\\sigma_{\\infty}(54n+53)+8\\\\\n\\sigma_{\\infty}(54n+13)+9\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+10} \\\\\n\\sigma_{\\infty}(54n+37)+11\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+12} \\\\\n\\sigma_{\\infty}(54n+43)+13\\\\\n\\sigma_{\\infty}(54n+35)+14\\\\\n\\sigma_{\\infty}(54n+31)+15\\\\\n\\sigma_{\\infty}(54n+29)+16\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+17} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+18} \\\\\n\\sigma_{\\infty}(54n+7)+19\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_8 \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+17)+1\\\\\n\\sigma_{\\infty}(54n+49)+2\\\\\n\\sigma_{\\infty}(54n+11)+3\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+4} \\\\\n\\sigma_{\\infty}(54n+23)+5\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+6} \\\\\n\\sigma_{\\infty}(54n+53)+7\\\\\n\\sigma_{\\infty}(54n+13)+8\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+9} \\\\\n\\sigma_{\\infty}(54n+37)+10\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+11} \\\\\n\\sigma_{\\infty}(54n+43)+12\\\\\n\\sigma_{\\infty}(54n+35)+13\\\\\n\\sigma_{\\infty}(54n+31)+14\\\\\n\\sigma_{\\infty}(54n+29)+15\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+16} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+17} \\\\\n\\sigma_{\\infty}(54n+7)+18\\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{8_{next}} \\downarrow}} \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\vdots \\\\\n\\end{cases}\n\\begin{cases}\n\\textcolor{black}{{\\bf Odd~d_9\\downarrow}} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+2} \\\\\n\\sigma_{\\infty}(54n+7)+3\\\\\n\\sigma_{\\infty}(54n+17)+4\\\\\n\\sigma_{\\infty}(54n+49)+5\\\\\n\\sigma_{\\infty}(54n+11)+6\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+7} \\\\\n\\sigma_{\\infty}(54n+23)+8\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+9} \\\\\n\\sigma_{\\infty}(54n+53)+10\\\\\n\\sigma_{\\infty}(54n+13)+11\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+12} \\\\\n\\sigma_{\\infty}(54n+37)+13\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+14} \\\\\n\\sigma_{\\infty}(54n+43)+15\\\\\n\\sigma_{\\infty}(54n+35)+16\\\\\n\\sigma_{\\infty}(54n+31)+17\\\\\n\\sigma_{\\infty}(54n+29)+18\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+19} \\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Even_9 \\downarrow}} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41)+1} \\\\\n\\sigma_{\\infty}(54n+7)+2\\\\\n\\sigma_{\\infty}(54n+17)+3\\\\\n\\sigma_{\\infty}(54n+49)+4\\\\\n\\sigma_{\\infty}(54n+11)+5\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19)+6} \\\\\n\\sigma_{\\infty}(54n+23)+7\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25)+8} \\\\\n\\sigma_{\\infty}(54n+53)+9\\\\\n\\sigma_{\\infty}(54n+13)+10\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47)+11} \\\\\n\\sigma_{\\infty}(54n+37)+12\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5)+13} \\\\\n\\sigma_{\\infty}(54n+43)+14\\\\\n\\sigma_{\\infty}(54n+35)+15\\\\\n\\sigma_{\\infty}(54n+31)+16\\\\\n\\sigma_{\\infty}(54n+29)+17\\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1)+18} \\\\\n\\vdots \\\\\n\\textcolor{black}{{\\bf Odd~d_{9_{next}} \\downarrow}} \\\\\n\\textcolor{red}{ \\sigma_{\\infty}(54n+41) } \\\\\n\\sigma_{\\infty}(54n+7) \\\\\n\\sigma_{\\infty}(54n+17) \\\\\n\\sigma_{\\infty}(54n+49) \\\\\n\\sigma_{\\infty}(54n+11) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+19) } \\\\\n\\sigma_{\\infty}(54n+23) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+25) } \\\\\n\\sigma_{\\infty}(54n+53) \\\\\n\\sigma_{\\infty}(54n+13) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+47) } \\\\\n\\sigma_{\\infty}(54n+37) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+5) } \\\\\n\\sigma_{\\infty}(54n+43) \\\\\n\\sigma_{\\infty}(54n+35) \\\\\n\\sigma_{\\infty}(54n+31) \\\\\n\\sigma_{\\infty}(54n+29) \\\\\n\\textcolor{blue}{ \\sigma_{\\infty}(54n+1) } \\\\\n\\vdots \\\\\n\\end{cases}\n\\end{equation}\n}\n\\footnote{All rights reserved. $\\copyright$ Michael A. Idowu, 2014.}\n\\end{landscape}\n\n\\begin{conjecture} \\label{conj3}\nAn irrefutable proof of the Collatz conjecture essentially requires 18 fundamental formulae that represent the total stopping time functions of the fundamental covering system of odd integers.\n\\end{conjecture}\n\n\\section{Conclusions}\\label{Concl}\nA proposed proof of the CC essentially requires deriving the formulae for fundamental total stopping time functions for all odd integers or proving that all the $[3]_{36n+4x}$ numbers eventually converge below these start points using the proposed Collatz based number system, which is both visually demonstrable and theoretically evident.\n\nThe proposed covering system of the generalised Collatz based number system requires about 162 distinct sets of odd numbers, from which any other integers could be derived. \nEach fundamental set of odd numbers corresponds to a single fundamental total stopping time function in the proposed schemata.\n\nAn irrefutable proof of the Collatz conjecture essentially requires 18 fundamental formulae that represent the total stopping time functions of the fundamental covering system of odd integers.\n\nThe Collatz map \\ref{CollMachinery1} has many applications. For example, a visual and ingenous method to classify odd numbers to appropriate residue classes modulo 18 is easy. For example, $\\{ 349525,1 \\} \\in [1]_{18}$: $3+4+9+5+2+5 = 28 \\equiv 2+8 \\equiv 1+0 \\in [1]_{18}$; $\\{341, 17\\} \\in [17]_{18}$: $341 \\equiv 3+4+1 \\equiv  1+7 \\in [17]_{18}$. The reader is encouraged to try out this simple technique.\n\nThis foundational paper may be regarded as a proposed ``new mathematics'' of the Collatz based number system.  \n\nThe whole idea may be used as a new theoretical framework for teaching and understanding elementary number system. \n\n\nThis novel theoretical framework is anticipated to open up new research and further development opportunities in number theory, dynamical systems, and discrete mathematics, including deterministic modelling, metamathematical and optimised integer factorisation, ergodic theory, dynamical systems, covering systems, cryptosystems and cryptography.\n\nOne of our aspirations is to further exploit and innovate the main results in visualisable algorithm development. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results\\label{Section: Introduction and Main Results}}\n\n\n\\input{ch1sec1.tex}\n\n\\section{A Handlebody-Theoretic Reverse to the Plus Construction\\label{Section: A Handlebody-Theoretic Reverse to the Plus Construction}}\n\\input{ch3sec1.tex}\n\n\\section{Some Preliminaries to Creating Pseudo-Collarable High-Dimensional Manifolds\\label{Section: Some Preliminaries to Creating Pseudo-Collarable High-Dimensional Manifolds}}\n\\input{ch4sec1.tex}\n            \n\\section{Some Algebraic Lemmas, Part 1\\label{Section: Some Algebraic Lemmas, Part 1}}\n\\input{ch4sec2.tex}\n        \n\\section{Some Algebraic Lemmas, Part 2\\label{Section: Some Algebraic Lemmas, Part 2}}\n\\input{ch4sec3.tex}\n        \n\\section{Some Algebraic Lemmas, Part 3\\label{Section: Some Algebraic Lemmas, Part 3}}\n\\input{ch4sec4.tex}\n        \n\\section{Manifold Topology\\label{Section: Manifold Topology}}\n\\input{ch4sec5.tex}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nRecently various methods for calculating correlation functions of topological\nSigma Models (A-Model) have been obtained from the analysis of their\nstructure as the twisted version of $N=2$ super conformal field theory.\nOne of the most\nstriking feature is Mirror Symmetry,which emerged from the analysis of\nspectrum of 2-types of rings (chiral,chiral) and (chiral,antichiral) of\n Topological Sigma Model on Calabi-Yau mfd and from the speculation that\nthere are pairs of Calabi-Yau manifold $M$ and $\\tilde{M}$ such that\n(chiral,chiral) ring of $M$ (resp.$\\tilde{M}$) and (chiral,antichiral)\nring of  $\\tilde{M}$ (resp.$M$) coincide. Another feature is recently found\nby Kontsevich, Manin \\cite{km} and Dubrovin \\cite{dub} ,which uses the\nassociativity of\nchiral algebra for Topological Sigma Models (coupled with gravity but\non small phase space).\nIn \\cite{km}, the authors defined Gromov-Witten classes as cohomology\nclasses of moduli space of punctured $CP^{1}$ and showed that associativity\ncondition\nis reduced to the characteristic of boundaries of the moduli space under the\nassumption of splitting axiom. Using the method of algebraic geometry numbers\nof rational curves in M are counted which pass through  Poincare duals\nof cohomology classes corresponding to inserted chiral primary fields. They\nshowed that for Fano\nvarieties $M$ whose cohomology are generated by $H^{2}(M)$, the associativity\ncondition is enough for\nthe construction of all correlation functions.\n\nWith these results available we explicitcy calculate correlation functions of\n$CP^{3}$,$CP^{4}$ and $Gr(2,4)$. In the case of $CP^{3}$ and $CP^{4}$ we obtain\na series of overdetermined differential equation for the free energy.\nWe assume the expansion of the free energy as a generating function of\ntopological amplitudes and impose ghost number selection rule. Then we pick\na subset of these equations with linear terms and derive recurtion relations\nwhich determine the correlation functions coming from moduli\nspace of maps of degree $d$ from the ones of lower degree. Then we observe\nthat all of them are determined and calculate them to several degree d. For\n$CP^{3}$,$d=6$,$CP^{4}$,$d=4$,and for $Gr(2,4)$,$d=4$. We checked\ncompatibility of all the equations in our calculation up to the degree\nwe calculated.\n\nIn section 1, we introduce the  interpretation of correlation functions\nas intersection numbers of holomorphic maps from $CP^{1}$ to the target\nspace $M$ originally derived by Witten \\cite{w2} and explain when\n coupled to gravity, they count the number of rational curves passing\nthrough Poincar\\'e duals of inserted operators. In section 2, assuming that\nthese models are twisted version of $N=2$ super conformal field theory\nand have stable property of chiral algebra under pertubation of chiral\nprimaary fields, we set up the calculation of the free energy $F_{M}$ using\nassociativity\nof this algebra. In section 3, we carry out the calculation for each case\n$CP^{3}$,$CP^{4}$ and $Gr(2,4)$. Recurtion relations which are\nused in determination of amplitudes are written out explicitly. All results\nare\ncollected in tables at the end of this paper.\n\n\\section{Meaning of the Correlation Function}\n\\setcounter{equation}{0}\n\n In the topological sigma model (A-model) which describes maps from $CP^1$\nto the\ntarget space $M$, BRST-closed observables are constructed from elements of\n$H^{*}(M)$ {\\cite{w1}. We denote the BRST-closed observable\nconstructed from\n$W \\in H^{*}(M)$ as ${\\cal O}_{W}$. Witten  showed in the pure matter case\n\\cite{w2}\n(without coupling to gravity) topological correlation functions\nare given in terms of intersection numbers of holomorphic maps from $CP^{1}$\nto $M$ as follows.\n\\begin{eqnarray}\n\\langle {\\cal O}_{W_{i_1}}(z_{1}){\\cal O}_{W_{i_2}}(z_{2}) \\cdots\n{\\cal O}_{W_{i_k}}(z_{k})\n\\rangle& =& \\int_{{\\cal M}_{d}(M)} \\chi(\\nu) \\prod_{j=1}^{k} \\phi_{j}^{*}\n(W_{i_j})\\\\\n \\phi_{j} : {\\cal M}_{d}(M)& \\mapsto &M\\nonumber\\\\\n               f \\in {\\cal M}_{d}(M) , z_j & \\mapsto &f(z_{j}) \\in M\\nonumber\n\\;\\;\\;\\;j=1,\\cdots,k\n\\label{a1}\n\\end{eqnarray}\n(${\\cal M}_{d}(M)$ is the moduli space of holomorphic maps from $CP^{1}$ to\n$M$ of degree d,and $(z_{1},\\ldots,z_{k})$ are ``fixed'' distinct points\non $CP^{1}$. Degree $d$ is related to the sum of $dim_{C}(W_{i_j})$ by the\ntopological selection rule which we will introduce later ).\n\n$\\nu$ is the additional degree of freedom which arises when f can be\ndecomposed as $f = \\tilde{f}\\circ \\alpha $ where $\\alpha$ is a map from\n$CP^{1}$ to $CP^{1}$ of degree $\\frac {d}{j}$ and $\\tilde{f}$ a map from\n$CP^{1}$ to $M$ of degree $j (d|j)$. But as we will discuss later, we have\nto consider $\\nu$ only when $M$ is C.Y manifold, i.e. $c_{1}(TM)=0$.\n\n Since $\\phi_{j}^{*}(W_{i_j})$ defines $dim_{C}(W_{i_j})$ form on\n${\\cal M}_{d}(M)$, in generic case when $\\nu$ is trivial, $\\langle\n{\\cal O}_{W_{i_1}} \\cdots {\\cal O}_{W_{i_k}}\\rangle$ doesn't vanish only\nwhen the following conditions are satisfied.\n\\begin{eqnarray}\n \\sum_{j=1}^{k}dim_C(W_{i_j}) & = & dim_{C}({\\cal  M}_{d}(M))\\nonumber\\\\\n                              & = & dim H^{0}(f^{*}(TM))\\nonumber\\\\\n                              & = & dc_1(TM) + dim_{C}(M)\n\\label{a2}\n\\end{eqnarray}\nIn deriving the third line from the second line, we used Rieman-Roch theorem\nand assumed $H^{1}(f^{*}(TM))=0$.\n\nIf we take $W_{i_j}$ as the form which has a  delta-function support on the\nPoincare dual of $W_{i_j}$, $PD(W_{i_j})$, we can interpret\n$\\phi_{i}^{*}(W_{i_j})$ as the following\nconstraint on ${\\cal M}_{d}(M)$.\n\\begin{equation}\n      f(z_j) \\in PD(W_{i_j})\n\\label{a3}\n\\end{equation}\nWe can easily see the above condition imposes $dim_{C}(W_i)$ independent\nconstraints on ${\\cal M}_{d}(M)$ (use count degrees of freedom in the complex\nsense). Since we have to use $(dim_{C}(M)-dim_{C}(f(CP^1))-\ndim_{C}(PD(W_{i_j}))$ degrees of freedom to let $f(CP^1) \\cap PD(W_{i_j})\n\\ne \\emptyset$ and in case $dim_{C}(f(CP^1))=1$, we have to use one further\ndegree of freedom to let $z_j$ to lie on\n$f(CP^{1}) \\cap PD(W_{i_j})$.\nCondition (\\ref{a3}) tells us that by imposing all the constraints\n$i=1,\\cdots,k$, we have zero degrees of freedom and topological correlation\nfunctions reduce to\n\\begin{eqnarray}\n\\lefteqn{\\langle {\\cal O}_{W_{i_1}}(z_1)\\cdots {\\cal O}_{W_{i_k}}(z_k)\n\\rangle_{generic}}\\nonumber\\\\\n&& = {}^{\\sharp}\\{ f: CP^{1}\n\\stackrel{hol.}{\\longrightarrow}  M|f(z_j) \\in PD(W_{i_j}),\n j=1,\\cdots,k \\}\n\\label{a4}\n\\end{eqnarray}\n At this point, we consider the case of multiple cover map.\n{}From the above argument, multiple cover map $f = \\tilde{f}\\circ \\alpha $\nalso has to satisfy the condition (\\ref{a3}) which restricts the motion\nof $f(CP^1)=\\tilde{f}(CP^1)$ in the target space M. But since $\\tilde{f}$ is\na map of degree j, it has as many as $jc_1(TM)+dim_{C}(M) (<dc_1(TM)+\ndim_{C}(M))$ freedom in M and this is imcompatible with (\\ref{a3}). Only\nwhen $c_1(TM)=0$ i.e. $M$ is C.Y manifold, compatibility of (\\ref{a2}) and\n(\\ref{a3}) holds in the case of multiple cover map and we have to integrate the\nadditional $\\nu$. Then we conclude that when $c_1(TM)>0$,we can neglect\n$\\chi(\\nu)$ and only consider the generic case.\n\nNext, let us consider what happens if we couple topological gravity to the\nabove\ntopological sigma model. Roughly speaking, we have to integrate over moduli\nspace of $CP^{1}$ with punctures. Since the moduli space of $CP^{1}$ with\n$k$-punctures are given by the position of $k$-distinct points on $CP^1$\ndivided by $SL(2,C)$, which is the internal symmetry group of $CP^{1}$, the\ndifinition (\\ref{a1}) is modified as follows.\n\\begin{eqnarray}\n\\langle {\\cal O}_{W_{i_1}}\\cdots{\\cal O}_{W_{i_k}} \\rangle& =&\n\\int _{{\\cal M}_{d,0,k}(M)} \\prod_{j=1}^{k} \\tilde{\\phi_{j}}^{*}(W_{i_{j}})\\\\\n \\tilde{\\phi_{j}} : {\\cal M}_{d,0,k}(M) &\\mapsto& M \\nonumber\\\\\n  (z_1,z_2,\\cdots,z_k,f)\/SL(2,C) &\\mapsto& (f(z_1),\\cdot\\cdot\\cdot,f(z_k))\n\\nonumber\n\\label{a5}\n\\end{eqnarray}\nwhere the action of $u \\in SL(2,C)$ is defined as follows.\n\\begin{equation}\nu\\circ(z_1,z_2,\\cdots,z_k,f)= (u(z_1),u(z_2),\\cdots,u(z_k),(u^{-1})^{*}\nf)\n\\label{a6}\n\\end{equation}\nThis action is natural in the sense that the image of $\\tilde{\\phi_{j}}$\nremains invariant under $SL(2,C)$. Main difference between (\\ref{a4}) and\n(\\ref{a6}) is that in the former case, we keep $z_{i}$ ``fixed'' on\n$CP^{1}$ but in the latter they move. Then we have\n$dim_{C}({\\cal M}_{d,0,k}(M))= k-3 + dc_{1}(TM)+dim_{C}(M)$ and modify\n(\\ref{a2}) as follows.\n\\begin{eqnarray}\n  \\sum_{j=1}^{k}dim_{C}(W_{i_j}) = dc_{1}(TM) + dim_{C}(M) +k-3\\nonumber\\\\\n \\Longleftrightarrow \\sum_{j=1}^{k}dim_{C}(W_{i_j}-1)=\n  dc_1(TM) + dim_{C}(M)-3\n\\label{a7}\n\\end{eqnarray}\n Integrating over the positions of $k$-punctures the codition (\\ref{a3})\nchanges into\n\\begin{equation}\n   f(CP^{1})\\cap PD(W_{i_j})\\neq \\emptyset .\n\\label{a8}\n\\end{equation}\nUnder the condition (\\ref{a7}), $f(CP^{1})\\cap PD(W_{i_{j}})$ must be a\nfinite point set for each $j$ and $z_i$ integration contributes\n$(f(CP^{1})\\cap PD(W_{i_{j}}))^{\\sharp}$ to the correlation function.\nThen we have\n\\begin{eqnarray}\n\\langle{\\cal O}_{W_{i_1}}\\cdots {\\cal O}_{W_{i_k}}\\rangle\n= \\sum _{f}\\prod_{j=1}^{k}(f(CP^{1}) \\cap PD(W_{i_{j}}))^{\\sharp}\\nonumber\\\\\n\\{ f: CP^{1} \\stackrel{hol.}{\\longrightarrow}  M |\n f(CP^{1})\\cap PD(W_{i_{j}})\\neq \\emptyset\\quad  j =1,2,\\cdots,k \\}\n\\label{a9}\n\\end{eqnarray}\n\n\\section{Set up of the calculation}\n\n\\setcounter{equation}{0}\n\nTopological Sigma Model (A-Model) can be constructed as the twisted version\nof $N=2$ super conformal field theory \\cite{ey}. We can perturb topological\nfield theory\nby adding the terms $\\sum_{i} t_{i}{\\cal O}_{W_i}$ to the lagrangian and\ncorrelation functions depend on variables $\\{t_i\\}$ \\cite{cv}.\n\\begin{eqnarray}\n\\langle {\\cal O}_{W_{i_{1}}} {\\cal O}_{W_{i_{2}}} \\cdots {\\cal O}_{W_{i_{k}}}\n(t_1,t_2,\\ldots,t_{dim(H^{*}(M))})\\rangle \\nonumber \\\\\n= \\int {\\cal D}X e ^{-L(X)+\\sum_{i} t_{i}{\\cal O}_{W_i}} {\\cal O}_{W_{i_{1}}}\n{\\cal O}_{W_{i_{2}}} \\cdots {\\cal O}_{W_{i_{k}}}\n\\label{b1}\n\\end{eqnarray}\nwhere X denotes the field variables of the A-Model. We set $D=dim(H^{*}(M))$.\nAs the twisted version of N=2 SCFT (coupled with gravity and on small\n phase space ),$\\{ {\\cal O}_{W_i} \\}$ has ring structure which can be\ndetermined from three point correlation functions.\n\\begin{eqnarray}\n    {\\cal O}_{W_i}{\\cal O}_{W_j}=C^{k}_{ij}(t_1,t_2,\\ldots,t_D){\\cal O}_{W_k}\\\\\n    \\mbox{where}\\quad C^{k}_{ij}= C_{ijl}\\eta^{lk}\\\\\n     C_{ijl} = \\langle {\\cal O}_{W_i} {\\cal O}_{W_j} {\\cal O}_{W_l} (t_1,t_2,\n\\ldots,t_D) \\rangle\\\\\n         \\eta^{kl}\\eta_{lm} = \\delta^{k}_{m}\\\\\n          \\eta_{lm} = C_{1lm}(t_1,t_2,\\ldots,t_D)\n\\label{b2}\n\\end{eqnarray}\nIn our notation $W_{1}$ corresponds to $1 \\in H^{*}(M)$ and we set $W_{2}$ to\nthe\nK\\\"ahler form of $M$ (in our case where $M$ is $CP^{3},CP^{4}$ or\n$Gr(2,4)$, $dim(H^{2}(M))=dim(H^{1,1}(M))=1$,this notation is O.K).\n We assume that $t_{i}$'s are flat coordinates and\n$\\eta_{lm}$ do not depend on them and determined by classical intersection\nnumber $\\int_{M} W_{l} \\wedge W_{m}$. Next, we impose associativity\ncondition on this algebra.(This relation is DWVV eq.)\n\\begin{eqnarray}\n     && ({\\cal O}_{W_i}{\\cal O}_{W_j}){\\cal O}_{W_k}={\\cal O}_{W_i}(\n{\\cal O}_{W_j} {\\cal O}_{W_k}) \\nonumber\\\\\n&\\Longleftrightarrow&  C^{l}_{ij}{\\cal O}_{W_l}{\\cal O}_{W_k}={\\cal O}_{W_i}\nC^{m}_{jk}{\\cal O}_{W_m}\\nonumber\\\\\n&\\Longleftrightarrow& C^{l}_{ij}C^{n}_{lk}{\\cal O}_{W_n} = C^{m}_{jk}C^{n}_{im}\n{\\cal O}_{W_n}\\nonumber\\\\\n&\\Longleftrightarrow& C^{l}_{ij}C^{n}_{lk}=C^{m}_{jk}C^{n}_{im}\\nonumber\\\\\n&\\Longleftrightarrow& C_{ijm}\\eta^{lm}C_{lkn}=C_{jkm}\\eta^{lm}C_{imn}\n\\label{b3}\n\\end{eqnarray}\nAnd there exists a free energy (prepotential) $F_{M}(t_1,\\ldots,t_D)$ which\nsatisfies following\nconditions.\n\\begin{eqnarray}\n   C_{ijk}(t_1,t_2,\\ldots,t_D)= \\partial_{i}\\partial_{j}\\partial_{k}F_{M}\\\\\n       (\\partial_{i} := \\frac{\\partial}{\\partial t_{i}})\n\\label{b4}\n\\end{eqnarray}\n Combining (\\ref{b3}) and (\\ref{b4}), we obtain a series of partial\ndifferential\nequations for $F_{M}$.\n\\begin{equation}\n \\eta^{lm}\\partial_{i}\\partial_{j}\\partial_{m} F_{M}\\partial_{l}\\partial_{k}\n\\partial_{n} F_{M} = \\eta^{lm}\\partial_{j}\\partial_{k}\\partial_{m} F_{M}\n\\partial_{i}\\partial_{l}\\partial_{n} F_{M}\n\\label{b5}\n\\end{equation}\n We can also consider prepotential as the generating function of all\ntopological correlation functions.\n\\begin{eqnarray}\n F_{M}(t_1,\\ldots,t_{D}) =\n\\sum_{n_1,\\cdots,n_{D} \\geq 0}\\langle {\\cal O}^{n_1}_{W_1}\n\\cdots {\\cal O}^{n_{D}}_{W_{D}}\\rangle\n\\frac{t_{1}^{n_{1}}}{n_{1}!}\\cdots\\frac{t_{D}^{n_{D}}}{n_{D}!}\n\\label{b6}\n\\end{eqnarray}\n(${\\cal O}^{n_{i}}_{W_{i}}$ represents $\\overbrace{{\\cal O}_{W_i}\\cdots\n{\\cal O}_{W_i}}^{\\mbox{$n_{i}$ times}}$ and should not be confused with\n${\\cal O}_{(W_{i})^{n_i}}$ ).\n At the topological point (i.e., all the $t_{i}$'s are set to zero)\ncorrelation functions become intersection numbers\non moduli spaces of holomorphic maps from $CP^{1}$ (with $k$-- marked points)\nto target space $M$.\n\n Holomorphic maps $f$ are characterized by their degree which equal\nthe intersection number of $f^{*}(CP^{1})$ with the K\\\"ahler form\nof target space $M$.\n Then $\\langle {\\cal O}^{n_1}_{W_1}\\cdots{\\cal O}^{n_D}_{W_{D}}\\rangle$\n remains non--zero only when the following topological selection rule\nis satisfied.\n\\begin{eqnarray}\n\\sum_{i=1}^{D} n_{i}dim_{C}(W_i) = \\sum_{i=1}^{D}n_{i} -3 +\ndim H^{0}(CP^{1},\\phi^{*}(TM)) \\nonumber \\\\\n\\Longleftrightarrow \\sum_{i=1}^{D}n_{i}(dim_{C}(W_{i})-1)= -3 + dc_1(TM)\n+ dim_{C}(M)\n\\label{b7}\n\\end{eqnarray}\nHere d is the degree of holomorphic map and we used Riemann--Roch theorem\nin deriving the second line from the first one. When d equals zero, $f$\nis  a constant map to the target space and moduli space becomes just the direct\nproduct of target space and moduli space of $CP^{1}$ with\n$\\displaystyle{\\sum_{i=1}^{D} n_i}$ punctures. Then selection rules (\\ref{b7})\ndecomposes to\n\\begin{equation}\n\\sum_{i=1}^{D}n_i dim_{C}(W_i) = dim_{C}(M)\n\\label{b8}\n\\end{equation}\n and\n\\begin{equation}\n\\sum_{i=1}^{D} n_i =3\n\\label{b9}\n\\end{equation}\n\n From (\\ref{b9}), we conclude that in $d = 0$ case only $3$-point functions\nsurvive and correlation functions are just classical intersection numbers\n$\\int _{M}W_{i}\\wedge W_{j}\\wedge W_{k}$.\n\n{}From the flat metric condition, insertion of $W_{1}$ remains non-zero only\nfor three point functions from constant maps because one and two point\nfunctions including $W_{1}$ cannot remein nonzero when $c_{1}(M) \\geq 1$\nand $d \\geq 1$ and if we suppose $n (\\geq3)$ point functions remain nonzero\nin $d \\geq 1$ sector, flat metric condition is broken(in three point functions\n in $d \\geq 1$ sector, we take into account of the\ninsertion of operator ${\\cal O}_{W_{2}}(dim_{C}(W_2)=1)$ which we will discuss\nlater).\n\n With these considerations, expansion of the free energy becomes\n\\begin{eqnarray}\nF_{M}(t_1,\\ldots ,t_D)\n=\\frac 16 \\int_{M}(\\sum_{i=1}^{D} t_i W_i )^{3} + \\sum_ {d=1}^{\\infty}\n \\sum _{n_2,\\cdots,n_D\\geq 0} \\frac{{t_{2}}^{n_{2}}}{n_{2}!}\\cdots\n\\frac{{t_{D}}^{n_{D}}}{n_{D}!}\\langle {\\cal O}^{n_2}_{W_{2}}\\cdots\n{\\cal O}^{n_D}_{W_{D}}\\rangle\\nonumber\\\\\n (\\sum_{i=2}^{D} n_{i}(dim_{C}(W_i)-1)= -3 + dc_{1}(TM)+dim_C(M))\n\\label{b10}\n\\end{eqnarray}\n (where the product of the first term of r.h.s. means taking the wedge\nproduct of $H^{*}(M)$).\n\n Next, we consider the insertion of the operator ${\\cal O}_{W_2}$ which\ncorresponds to the K\\\"ahler form of the target space . Since\n$ codim_{C}(PD(W_{2}))=1$, holomorphic map $f$ of degree $d$\nalways intersects with it in $d$-points,and the condition\n$f(CP^{1})\\cap PD(W_{2}) \\neq \\emptyset$ imposes no constraint. Then from\n(\\ref{a9}) the\n insertion of ${\\cal O}_{W_2}$ results in the multiplication by a factor d,\n\\begin{equation}\n\\langle {\\cal O}^{n_{2}}_{W_{2}}{\\cal O}^{n_{3}}_{W_{3}}\n\\cdots {\\cal O}^{n_{D}}_{W_{D}}\\rangle =\nd^{n_{2}}\\langle {\\cal O}^{n_{3}}_{W_{3}}\\cdots {\\cal O}^{n_{D}}_{W_{D}}\n\\rangle\n\\label{b11}\n\\end{equation}\n Combining (\\ref{b10}) and (\\ref{b11}) we obtain the following expansion\n\\begin{eqnarray}\nF_{M}(t_1,\\ldots ,t_D)\n= \\frac 16 \\int_{M}(\\sum_{i=1}^{D} t_i W_i )^{3} + \\sum_{d=1}^{\\infty}\n \\sum _{n_3,\\cdots,n_D\\geq 0} \\frac{{t_{3}}^{n_{3}}}{n_{3}!}\\cdots\n\\frac{{t_{D}}^{n_{D}}}{n_{D}!}\\langle {\\cal O}^{n_3}_{W_{3}}\\cdots\n{\\cal O}^{n_D}_{W_{D}}\\rangle e^{dt_{2}}\\nonumber\\\\\n(\\sum_{i=3}^{D}n_{i}(dim_{C}(W_{i})-1)=-3+dc_{1}(TM)+dim_{C}(M))\n\\label{b12}\n\\end{eqnarray}\n Then by combining (\\ref{b5}) and (\\ref{b12}), we determine the correlation\nfunctions in the case where target space is $CP^{3},CP^{4}$ and $Gr(2,4)$.\n\n\\section{The Calculations}\n\n\\setcounter{equation}{0}\n\n\nThe non zero Betti numbers of $CP^3$, $CP^4$ and $Gr(2,4)$ are\n\\begin{eqnarray}\n         b_{00} = b_{11} = b_{22} = b_{33} = 1\n\\end{eqnarray}\n\\begin{eqnarray}\n         b_{00} = b_{11} = b_{22} = b_{33} = b_{44}= 1\n\\end{eqnarray}\n\\begin{eqnarray}\n         b_{00} = b_{11} = b_{33} = b_{44} = 1, \\qquad   b_{22}=2\n\\end{eqnarray}\nrespectively. By means of wedge product we obtain an associative, commutative\nring $H^*(M,Q)$ for each manifold $M$.For $CP^3$, $CP^4$, and $Gr(2,4)$ we have\n the multiplication table as follows.\n\\clearpage\n\\begin{table}[h]\n\\caption{\\bf The ring of $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n     \\hline\n$ $  & $W_1$ & $W_2$ & $W_3$ & $W_4$ \\\\ \\hline\n$W_1$ & $W_1$ & $W_2$ & $W_3$ & $W_4$ \\\\ \\hline\n$W_2$ & $W_2$ & $W_3$ & $W_4$ & $0$ \\\\ \\hline\n$W_3$ & $W_3$ & $W_4$ & $0$ & $0$ \\\\ \\hline\n$W_4$ & $W_4$ & $0$ & $0$ & $0$ \\\\ \\hline\n\\multicolumn{5}{c}{$\\scriptstyle{dim_{\\cal C}(W_1)=0\\; dim_{\\cal C}(W_2)=1\\;}$}\n\\\\\n\\multicolumn{5}{c}{$\\scriptstyle{dim_{\\cal C}(W_3)=2\\; dim_{\\cal\nC}(W_4)=3\\;}$}\\\\\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\begin{table}[h]\n\\caption{\\bf The ring of $CP^4$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$ $  & $W_1$ & $W_2$ & $W_3$ & $W_4$ & $W_5$ \\\\ \\hline\n$W_1$ & $W_1$ & $W_2$ & $W_3$ & $W_4$ & $W_5$ \\\\ \\hline\n$W_2$ & $W_2$ & $W_3$ & $W_4$ & $W_5$ & $0$ \\\\ \\hline\n$W_3$ & $W_3$ & $W_4$ & $W_5$ & $0$ & $0$ \\\\ \\hline\n$W_4$ & $W_4$ & $W_5$ & $0$ & $0$ & $0$ \\\\ \\hline\n$W_5$ & $W_5$ & $0$ & $0$ & $0$ & $0$\\\\ \\hline\n\\multicolumn{6}{c}{$\\scriptstyle{dim_{\\cal C}(W_1)=0\\; dim_{\\cal\nC}(W_2)=1\\;}$}\\\\\n\\multicolumn{6}{c}{$\\scriptstyle{dim_{\\cal C}(W_3)=2\\; dim_{\\cal C}(W_4)=3\\;\ndim_{\\cal C}(W_5)=4\\;}$}\\\\\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\\begin{table}[h]\n\\caption{\\bf The ring of $Gr(2,4)$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|}\n     \\hline\n$ $  & $W_1$ & $W_2$ & $W_3$ & $W_4$ & $W_5$ & $W_6$ \\\\ \\hline\n$W_1$ & $W_1$ & $W_2$ & $W_3$ & $W_4$ & $W_5$ & $W_6$ \\\\ \\hline\n$W_2$ & $W_2$ & $W_3+W_4$ & $W_5$ & $W_5$ & $W_6$ & $0$ \\\\ \\hline\n$W_3$ & $W_3$ & $W_5$ & $W_6$ & $0$ & $0$ & $0$ \\\\ \\hline\n$W_4$ & $W_4$ & $W_5$ & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n$W_5$ & $W_5$ & $W_6$ & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n$W_6$ & $W_6$ & $0$ & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n\\multicolumn{7}{c}{$\\scriptstyle{dim_{\\cal C}(W_1)=0\\; dim_{\\cal C}(W_2)=1\\;\ndim_{\\cal C}(W_3)=2\\;}$}\\\\\n\\multicolumn{7}{c}{$\\scriptstyle{dim_{\\cal C}(W_4)=2\\; dim_{\\cal\nC}(W_5)=3\\;dim_{\\cal C}{W_6}=4\\;}$}\\\\\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\n\nDual to each cohomology class is a class of cycles (e.g. for the case of $CP^3$\n$W_3$ is dual to a point, $W_2$ is dual to a line, $W_1$ is dual to the\n$CP^3$).\n\nAs a point intersects the $CP^3$ in a point and a line intersects the plane by\na point. Thus we have for $CP^3$,\n\\begin{eqnarray}\n<W_1,W_4>=1, <W_2,W_3>=1\n\\end{eqnarray}\nFor $CP^4$\n\\begin{eqnarray}\n<W_1,W_5>=1, \\qquad <W_2,W_4>=1, \\qquad <W_3,W_3>=1\n\\end{eqnarray}\nand for Gr(2,4) it becomes\n\\begin{eqnarray}\n<W_1,W_6>=1, \\qquad <W_2,W_5>=1 \\\\\n<W_3,W_3>=1, \\qquad <W_4,W_4>=1\n\\end{eqnarray}\nAll other intersections on generators being zero. The $CP^3$, $CP^4$ ring can\nbe identified the ring of polynomials in one indeterminate C[x] modulo the\ngradient of\n\\begin{eqnarray}\nW(x)=x^4\/4, \\qquad W(x)=x^5\/5\n\\end{eqnarray}\n\nIne the case of Grassmanians their cohomology $H^*(Gr,Q)$ can't be generated by\n$H^2(Gr,Q)$.\nThe cohomology ring of Grassmanian $Gr(2,4)$ for instance, can be written as\nthe singularity ring generated by a single potential\\cite{va}\n\\begin{eqnarray}\n          W(x_i)={\\frac 1{5}}x^5_1-x^3_1x_2+x^2_2x_1\n\\end{eqnarray}\nWhere $x_1$ correspond to $W_2$ and $x_2$ correspond to ${\\frac\n1{2}}(W_3+W_4)$.\n\n\n\n{}From (3.6) one can split $F_M$ into a classical part and instanton correction\npart as\n\\begin{eqnarray}\n      F_M=f_{cl}+f_M\n\\end{eqnarray}\nSo for $ CP^3$, $CP^4$ and $Gr(2,4)$ we have\n\\begin{eqnarray}\n        F_{CP^3}= {\\frac 1{2}t_1^2t_4} + t_1t_2t_3 + {\\frac 1{6}t_2^3} +\nf_{CP^3}(t_2,t_3,t_4)\n\\end{eqnarray}\\begin{eqnarray}\n        F_{CP^4}= {\\frac 1{2}t_1^2t_5} + {\\frac 1{2}t_1t_4^2} + {\\frac\n1{2}t_3^2t_4} + t_1t_2t_5 + f_{CP^4}(t_2,t_3,t_4,t_5)\n\\end{eqnarray}\\begin{eqnarray}\n        F_{Gr(2,4)}= {\\frac 1{2}t_1^2t_6} + {\\frac 1{2}t_1t_3^2} + {\\frac\n1{2}t_1t_4^2}\n+ t_1t_1t_5 + {\\frac 1{2}t_2^2t_3} + {\\frac 1{2}t_2^2t_4}\\nonumber\\\\\n + f_{Gr(2,4)}(t_2,t_3,t_4,t_5,t_6)\n\\end{eqnarray}\n{}From (2.7) the Riemann-Roch theorem tell us\n\\begin{eqnarray}\ndimH^0(CP^1,f^*(TM))-3=(dimM-3)+dc_1(TM)\n\\end{eqnarray}\nOnce we specify the target space , we know its frist Chern class, then the\nabove formula give the dimension of its moduli space.\nIn case of $CP^3$\n$c_1(TCP^3)=4$\nso,\n\\begin{eqnarray}\ndimH^0(CP^1,f^*(TCP^3))-3=4d\n\\end{eqnarray}\nFor $CP^4$ and $Gr(2,4)$, the frist Chern class are\n\\begin{eqnarray}\nc_1(CP^4)=5, \\qquad  c_1(TGr(2,4))=4\n\\end{eqnarray}\nThus\n\\begin{eqnarray}\ndimH^0(CP^1,f^*(TCP^4))-3=5d+1\\\\  \\noalign{\\vskip10pt}\ndimH^0(CP^1,f^*(TGr(2,4)))-3=4d+1\n\\end{eqnarray}\n{}From (\\ref{b12}) we can expond $f_M$ further as follows\n\\begin{eqnarray}\nf_{CP^3}=\\sum_{d=1}^{\\infty}\\sum_{n_3+2n_4=4d}{\\frac{<{\\cal O}_{W_3}^{n_3}\n {\\cal O}_{W_4}^{n_4}>}{n_{3}! n_{4}!}}t_3^{n_3}t_4^{n_4}e^{dt_2}\\nonumber\\\\\\noalign{\\vskip10pt}\n=\\sum_{d=1}^{\\infty}\\sum_{n_4}{\\frac{<{\\cal O}_{W_3}^{4d-2n_4}\n {\\cal O}_{W_4}^{n_4}>}{(4d-2n_{4})! n_{4}!}}t_3^{4d-2n_4}t_4^{n_4}e^{dt_2}\n\\label{d}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nf_{CP^4}=\\sum_{d=1}^{\\infty}\\sum_{n_3+2n_4+3n_5=5d+1}{\\frac{<{\\cal\nO}_{W_3}^{n_3}\n {\\cal O}_{W_4}^{n_4}{\\cal O}_{W_5}^{n_5}>}{n_{3}!\nn_{4}!n_{5}!}}t_3^{n_3}t_4^{n_4}t_5^{n_5}e^{dt_2}\\nonumber\\\\\\noalign{\\vskip10pt}\n=\\sum_{d=1}^{\\infty}\\sum_{n_4,n_5}{\\frac{<{\\cal O}_{W_3}^{5d-2n_4-3n_5+1}\n {\\cal O}_{W_4}^{n_4}{\\cal O}_{W_5}^{n_5}>}{(5d-2n_{4}-3n_{5}+1)!\nn_{4}!n_{5}!}}t_3^{5d-2n_4-3n_{5}+1}t_4^{n_4}t_5^{n_5}e^{dt_2}\\label{e}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nf_{Gr(2,4)}=\\sum_{d=1}^{\\infty}\\sum_{n_3+n_4+2n_5+3n_6=4d+1}{\\frac{<{\\cal\nO}_{W_3}^{n_3}\n {\\cal O}_{W_4}^{n_4}{\\cal O}_{W_5}^{n_5}{\\cal\nO}_{W_6}^{n_6}>}{n_{3}!n_{4}!n_{5}!n_6!}}t_3^{n_3}t_4^{n_4}t_5^{n_5}t_6^{n_6}e^{dt_2}\\nonumber\\\\\\noalign{\\vskip10pt}\n=\\sum_{d=1}^{\\infty}\\sum_{n_4,n_5,n_6}{\\frac{<{\\cal O}_{W_3}^{4d-n_4-2n_5-3n_6+\n1}\n {\\cal O}_{W_4}^{n_4}{\\cal O}_{W_5}^{n_5}{\\cal\nO}_{W_6}^{n_6}>}{(4d-n_{4}-2n_{5}-3n_6+1)! n_{4}!n_{5}!}}\nt_3^{4d-n_4-2n_{5}-3n_6+1}t_4^{n_4}t_5^{n_5}t_6^{n_6}e^{dt_2}\\label{f}\n\\end{eqnarray}\n\nThen, we abbreviate the notion in the following calculation as\n\\begin{eqnarray}\n{<{\\cal O}_{W_3}^{4d-2n_4}{\\cal O}_{W_4}^{n_4}>}_{CP^3}=N_{n_4}^d\\\\\\noalign{\\vskip10pt}\n{<{\\cal O}_{W_3}^{5d-2n_4-3n_5+1}{\\cal O}_{W_4}^{n_4}{\\cal\nO}_{W_5}^{n_5}>}_{CP^4}=N_{n_4,n_5}^d\\\\\\noalign{\\vskip10pt}\n{<{\\cal O}_{W_3}^{4d-n_4-2n_5-3n_6+1}{\\cal O}_{W_4}^{n_4}{\\cal\nO}_{W_5}^{n_5}{\\cal O}_{W_6}^{n_6}>}_{Gr(2,4)}\n=N_{n_4,n_5,n_6}^d\n\\end{eqnarray}\nWe let $t_2=x,t_3=y,t_4=z$ for $CP^3$,$t_2=w,t_3=x,t_4=y,t_5=z$ for $CP^4$ and\n$t_2=v,t_3=w,t_4=x,t_5=y,\nt_6=z$ for $Gr(2,4)$.\nA deformation of the multiplication table (table 1, table 2, table 3) become\nthe fusion rules\nfor the quantum cohomology ring with ${\\cal O}_{W_i}$'s substituted for\n$t$'s as\n\\begin{eqnarray}\n{\\cal O}_{W_i} \\circ {\\cal O}_{W_j} =\n{\\partial}_i{\\partial}_j{\\partial}_lF_M\\eta^{lk}{\\cal O}_{W_k}\n\\end{eqnarray}\nThe structure constants of the quantum cohomology obey the so called WDVV\nequation which satisfying the requirements[3]\n\n(i)commutativity\n\n(ii)associativity\n\n(iii)existence of a unit ${\\cal O}_{W_1}$\n\nCommutativity follows from the definition, while condition(3.6)\n(equivalently(3.17)\nexpresses that ${\\cal O}_{W_1}$\nplays the role of unit. The crucial assumption is the associativity which\nimposes strong conditions on $f_M$. Now let us introduce\nsome more notations, by $f_{M,xyz}$ we mean\n${\\partial}_x{\\partial}_y{\\partial}_zf_M $. In the following we will simply\n omit the index ``M'', and just denote it\nas $f_{xyz}$.\n\n\nThe quantum ring of $CP^3$ is\n\\ai\n{\\cal O}_{W_2}{\\cal O}_{W_2}&=&f_{xxz}{\\cal O}_{W_1}+f_{xxy}{\\cal\nO}_{W_2}+(f_{xxx}+1){\\cal O}_{W_3},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_3}&=&f_{xyz}{\\cal O}_{W_1}+f_{xyy}{\\cal\nO}_{W_2}+(f_{xxy}){\\cal O}_{W_3}+{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_4}&=&f_{xzz}{\\cal O}_{W_1}+f_{xyz}{\\cal\nO}_{W_2}+(f_{xxz}){\\cal O}_{W_3},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_3}&=&f_{yyz}{\\cal O}_{W_1}+f_{yyy}{\\cal\nO}_{W_2}+(f_{xyy}){\\cal O}_{W_3},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_4}&=&f_{yzz}{\\cal O}_{W_1}+f_{yyz}{\\cal\nO}_{W_2}+(f_{xyz}){\\cal O}_{W_3},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_4}&=&f_{zzz}{\\cal O}_{W_1}+f_{yzz}{\\cal\nO}_{W_2}+(f_{xzz}){\\cal O}_{W_3}.\n\\bj\nThe quantum ring of $CP^4$ is\n\\ai\n{\\cal O}_{W_2}{\\cal O}_{W_2}&=&f_{wwz}{\\cal O}_{W_1}+f_{wwy}{\\cal\nO}_{W_2}+f_{wwx}{\\cal O}_{W_3}+(f_{www}+1){\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_3}&=&f_{wxz}{\\cal O}_{W_1}+f_{wxy}{\\cal\nO}_{W_2}+f_{wxx}{\\cal O}_{W_3}+f_{wwx}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_4}&=&f_{wyz}{\\cal O}_{W_1}+f_{wyy}{\\cal\nO}_{W_2}+f_{wxy}{\\cal O}_{W_3}+f_{wwy}{\\cal O}_{W_4}+{\\cal O}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_5}&=&f_{wzz}{\\cal O}_{W_1}+f_{wyz}{\\cal\nO}_{W_2}+f_{wxz}{\\cal O}_{W_3}+f_{wwz}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_3}&=&f_{xxz}{\\cal O}_{W_1}+f_{xxy}{\\cal\nO}_{W_2}+f_{xxx}{\\cal O}_{W_3}+f_{wxx}{\\cal O}_{W_4}+{\\cal O}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_4}&=&f_{xyz}{\\cal O}_{W_1}+f_{xyy}{\\cal\nO}_{W_2}+f_{xxy}{\\cal O}_{W_3}+f_{wxy}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_5}&=&f_{xzz}{\\cal O}_{W_1}+f_{xyz}{\\cal\nO}_{W_2}+f_{xxz}{\\cal O}_{W_3}+f_{wxz}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_4}&=&f_{yyz}{\\cal O}_{W_1}+f_{yyy}{\\cal\nO}_{W_2}+f_{xyy}{\\cal O}_{W_3}+f_{wyy}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_5}&=&f_{yzz}{\\cal O}_{W_1}+f_{yyz}{\\cal\nO}_{W_2}+f_{xyz}{\\cal O}_{W_3}+f_{wyz}{\\cal O}_{W_4},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_5}{\\cal O}_{W_5}&=&f_{zzz}{\\cal O}_{W_1}+f_{yzz}{\\cal\nO}_{W_2}+f_{xzz}{\\cal O}_{W_3}+f_{wzz}{\\cal O}_{W_4}.\n\\bj\nThe quantum ring of $Gr(2,4)$ is\n\\ai\n{\\cal O}_{W_2}{\\cal O}_{W_2}&=&f_{vvz}{\\cal O}_{W_1}+f_{vvy}{\\cal\nO}_{W_2}+(f_{vvw}+1){\\cal O}_{W_3}\\nonumber\\\\\\noalign{\\vskip10pt}\n&+&(f_{vvx}+1){\\cal O}_{W_4}+f_{vvv}{\\cal O}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_3}&=&f_{vwz}{\\cal O}_{W_1}+f_{vwy}{\\cal\nO}_{W_2}+f_{vww}{\\cal O}_{W_3}\\nonumber\\\\\\noalign{\\vskip10pt}\n&+&f_{vwx}{\\cal O}_{W_4}+(f_{vvw}+1){\\cal O}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_4}&=&f_{vxz}{\\cal O}_{W_1}+f_{vxy}{\\cal\nO}_{W_2}+f_{vwx}{\\cal O}_{W_3}+f_{vxx}{\\cal O}_{W_4}+(f_{vvx}+1){\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_5}&=&f_{vyz}{\\cal O}_{W_1}+f_{vyy}{\\cal\nO}_{W_2}+f_{vwy}{\\cal O}_{W_3}+f_{vxy}{\\cal O}_{W_4}+f_{vvy}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_2}{\\cal O}_{W_6}&=&f_{vzz}{\\cal O}_{W_1}+f_{vyz}{\\cal\nO}_{W_2}+f_{vwz}{\\cal O}_{W_3}+f_{vxz}{\\cal O}_{W_4}+f_{vvz}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_3}&=&f_{wwz}{\\cal O}_{W_1}+f_{wwy}{\\cal\nO}_{W_2}+f_{www}{\\cal O}_{W_3}\\nonumber\\\\\\noalign{\\vskip10pt}\n&+&f_{wwx}{\\cal O}_{W_4}+f_{vww}{\\cal O}_{W_5} +{\\cal O}_{W_6},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_4}&=&f_{wxz}{\\cal O}_{W_1}+f_{wxy}{\\cal\nO}_{W_2}+f_{wwx}{\\cal O}_{W_3}+f_{wxx}{\\cal O}_{W_4}+f_{vwx}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_5}&=&f_{wyz}{\\cal O}_{W_1}+f_{wyy}{\\cal\nO}_{W_2}+f_{wwy}{\\cal O}_{W_3}+f_{wxy}{\\cal O}_{W_4}+f_{vwy}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_3}{\\cal O}_{W_6}&=&f_{wzz}{\\cal O}_{W_1}+f_{wyz}{\\cal\nO}_{W_2}+f_{wwz}{\\cal O}_{W_3}+f_{wxz}{\\cal O}_{W_4}+f_{vwz}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_4}&=&f_{xxz}{\\cal O}_{W_1}+f_{xxy}{\\cal\nO}_{W_2}+f_{wxx}{\\cal O}_{W_3}\\nonumber\\\\\\noalign{\\vskip10pt}\n&+&f_{xxx}{\\cal O}_{W_4}+f_{vxx}{\\cal O}_{W_5} +{\\cal O}_{W_6},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_5}&=&f_{xyz}{\\cal O}_{W_1}+f_{xyy}{\\cal\nO}_{W_2}+f_{wxy}{\\cal O}_{W_3}+f_{xxy}{\\cal O}_{W_4}+f_{vxy}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_4}{\\cal O}_{W_6}&=&f_{xzz}{\\cal O}_{W_1}+f_{xyz}{\\cal\nO}_{W_2}+f_{wxz}{\\cal O}_{W_3}+f_{xxz}{\\cal O}_{W_4}+f_{vxz}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_5}{\\cal O}_{W_5}&=&f_{yyz}{\\cal O}_{W_1}+f_{yyy}{\\cal\nO}_{W_2}+f_{wyy}{\\cal O}_{W_3}+f_{xyy}{\\cal O}_{W_4}+f_{vyy}{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_5}{\\cal O}_{W_6}&=&f_{yzz}{\\cal O}_{W_1}+f_{yyz}{\\cal\nO}_{W_2}+f_{wyz}{\\cal O}_{W_3}+f_{xyz}{\\cal O}_{W_4}+f_{}vyz{\\cal\nO}_{W_5},\\\\\\noalign{\\vskip10pt}\n{\\cal O}_{W_6}{\\cal O}_{W_6}&=&f_{zzz}{\\cal O}_{W_1}+f_{yzz}{\\cal\nO}_{W_2}+f_{wzz}{\\cal O}_{W_3}+f_{xzz}{\\cal O}_{W_4}+f_{vzz}{\\cal O}_{W_5}.\n\\bj\nAssociativity condition (3.7) implies  the free energy of $CP^3$ must satifying\nthe following constraint equation\n\\ai\n&&-2f_{xyz}-f_{xyy}f_{xxy}+f_{yyy}f_{xxx}=0,\\\\ \\noalign{\\vskip10pt}\n&&-f_{xzz}-f_{xyy}f_{xxz}+f_{yyz}f_{xxx}=0,\\\\\\noalign{\\vskip10pt}\n&&f_{yzz}-f_{xxz}f_{yyy}+f_{xxy}f_{yyz}=0,\\\\\\noalign{\\vskip10pt}\n&&-2f_{xyz}f_{xxz}+f_{xzz}f_{xxy}+f_{yzz}f_{xxx}=0,\\\\\\noalign{\\vskip10pt}\n&&f_{zzz}-f_{xyz}^2+f_{xzz}f_{xyy}-f_{yyz}f_{xxz}+f_{yzz}f_{xxy}=0,\\\\\\noalign{\\vskip10pt}\n&&f_{yyy}f_{xzz}-2f_{yyz}f_{xyz}+f_{yzz}f_{xyy}=0.\n\\bj\n\nFor $CP^4$ there are 17 independent constraint equations. We just write down\nfive of them which are enough to determine the correlation functions of $CP^4$\n\\ai\n&&-f_{wwz}-f_{wwy}f_{wxx}+f_{wxx}^2+2f_{wwx}f_{xxx}-f_{www}f_{xxy}=0, \\\\\\noalign{\\vskip10pt}\n&&f_{wxy}^2 + f_{wwy}f_{wyy}+2f_{wyz}-f_{wwx}f_{xyy}-f_{www}f_{yyy}=0, \\\\\\noalign{\\vskip10pt}\n&&f_{wxy}f_{wxz}+f_{wwz}f_{wyy}+f_{wzz}-f_{wwx}f_{xyz}-f_{www}f_{yyz}=0,\\\\\\noalign{\\vskip10pt}\n&&-f_{wxz}f_{xyy}+f_{wxy}f_{xyz}-f_{wwz}f_{yyy}+f_{wwy}f_{yyz}+f_{yzz}=0,\\\\\\noalign{\\vskip10pt}\n&&f_{wxy}f_{xxy}+f_{wwy}f_{xyy}-f_{wxx}f_{xyy}+f_{xyz}-f_{wwx}f_{yyy}=0.\n\\bj\n\n\n\nFor the case of $Gr(2,4)$ there are fifty-six independent equations. We also\nwrite down nine of them that determine the corelation functions of $Gr(2,4)$\n\\ai\n&& -f_{vvz}-f_{vvy}f_{vww}+f_{vww}^2+f_{vwx}^2+2f_{vvw}f_{vwy}\\nonumber\\\\\n&& -f_{vvw}f_{www}-f_{vvx}f_{wwx}-f_{vvv}f_{wwy}=0,\\\\\\noalign{\\vskip10pt}\n&& -f_{vvz}-f_{vvy}f_{vxx}+f_{vwx}^2+f_{vxx}^2+2f_{vvx}f_{vxy} \\nonumber\\\\\n&& -f_{vvw}f_{wxx}-f_{vvx}f_{xxx}-f_{vvv}f_{xxy}=0,\\\\\\noalign{\\vskip10pt}\n&& -f_{vxz}-f_{vww}f_{vxy}+f_{vwx}f_{vwy}-f_{vwx}f_{www}+f_{vww}f_{wwx} \\nonumber\\\\\n&& -f_{vxx}f_{wwx}-f_{vvx}f_{wwy}+f_{vwx}f_{wxx}+f_{vvw}f_{wxy}=0,\\\\\\noalign{\\vskip10pt}\n&& -f_{wwz}-f_{xxz}-f_{vxx}f_{wwy}+f_{wwx}^2+f_{wxx}^2+2f_{vwx}f_{wxy} \\nonumber\\\\\n&& -f_{www}f_{wxx}-f_{wwx}f_{xxx}-f_{vww}f_{xxy}=0,\\\\\\noalign{\\vskip10pt}\n&&-f_{xyz}-f_{vxy}f_{wwy}+f_{wwx}f_{wwy}+f_{vwy}f_{wxy}+f_{wxx}f_{wxy} \\nonumber\\\\\n&&-f_{www}f_{wxy}+f_{vwx}f_{wyy}-f_{wwx}f_{xxy}-f_{vww}f_{xyy}=0,\\\\\\noalign{\\vskip10pt}\n&&-f_{xzz}-f_{vxz}f_{wwy}+f_{wwx}f_{wwz}+f_{vwz}f_{wxy}+f_{wxx}f_{wxz} \\nonumber\\\\\n&& -f_{www}f_{wxz}+f_{vwx}f_{wyz}-f_{wwx}f_{xxz}-f_{vww}f_{xyz}=0,\\\\\\noalign{\\vskip10pt}\n&& f_{wxy}+f_{vwy}f_{wwy}+f_{vxy}f_{wxy}+f_{vvy}f_{wyy} \\nonumber\\\\\n&& -f_{vww}f_{wyy}-f_{vwx}f_{xyy}-f_{vvw}f_{yyy}=0,\\\\\\noalign{\\vskip10pt}\n&&f_{wxz}-f_{vwy}f_{vxy}+f_{vwx}f_{vyy}+f_{vwy}f_{wwx}+f_{vxy}f_{wxx} \\nonumber\\\\\n&& -f_{vwx}f_{wwy}+f_{vvy}f_{wxy}-f_{vxx}f_{wxy}-f_{vvx}f_{wyy}=0,\\\\\\noalign{\\vskip10pt}\n&& -f_{yyz}+f_{vwz}f_{wyy}-f_{vwy}f_{wyz}+f_{vxz}f_{xyy} \\nonumber\\\\\n&& +f_{vvz}f_{yyy}-f_{vxy}f_{xyz}-f_{vvy}f_{yyz}=0.\n\\bj\n\nSubstituting the free energy (\\ref{d}--\\ref{f}) into (4.29),(4.30)\nand (4.31) one obtains the recursion relations of correlation functions. For\n$CP^3$ one has\n\n\\ai\n&&2dN^d_{m+1}-N^d_{m}=\\sum_{\\stackrel{f+g=d} {n+n'=m}}\\left(\\begin{array}{c}\n          m \\\\ n\\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n          &&\\bigg[ -\\left(\\begin{array}{c}\n          4d-2m-3 \\\\ 4f-2n-2 \\end{array}\\right)\nfN^f_{n} N^g_{n'}\n    +\\left(\\begin{array}{c}\n          4d-2m-3 \\\\ 4f-2n-3 \\end{array}\\right)\ng^3N^f_{n} N^g_{n'}     \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\n&&dN^d_{m+2}-N^d_{m+1}=\\sum_{\\stackrel{f+g=d} {n+n'=m}}\\left(\\begin{array}{c}\n          m \\\\ n\\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n          &&\\bigg[ \\left(\\begin{array}{c}\n          4d-2m-4 \\\\ 4f-2n-4 \\end{array}\\right)\ng^3N^f_{n+1} N^g_{n'}\n    -\\left(\\begin{array}{c}\n          4d-2m-4 \\\\ 4f-2n-2 \\end{array}\\right)\nfg^2N^f_{n} N^g_{n'+1}     \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\n&&N^d_{m+2}=\\sum_{\\stackrel{f+g=d} {n+n'=m}}\\left(\\begin{array}{c}\n          m \\\\ n\\end{array}\\right)\\nonumber\\\\\n          &&\\bigg[ \\left(\\begin{array}{c}\n          4d-2m-5 \\\\ 4f-2n-2 \\end{array}\\right)\nf^2N^f_{n+1} N^g_{n'}\n    -\\left(\\begin{array}{c}\n          4d-2m-5 \\\\ 4f-2n-1 \\end{array}\\right)\nf^2N^f_{n} N^g_{n'+1}     \\bigg].\n\\bj\nFor the case of $CP^4$ the recursion relations read as follows\n\\ai\n&&d^2 N^d_{m_1,m_2+1}-2dN^d_{m_1,m_2}+N^d_{m_1,m_2}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d} {n_1+n_1'=m_1, n_2+n_2'=m_2}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\\nonumber\\\\\n          &&\\bigg[ -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-2 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nf^2gN^f_{n_1+1,n_2} N^g_{n_1',n_2'}     \\nonumber\\\\\n&&  +\\left(\\begin{array}{c}\n          5d-2m_2-3m_3-2 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nfgN^f_{n_1,n_2} N^g_{n_1',n_2'}     \\nonumber\\\\\n&& +2\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-2 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2gN^f_{n_1,n_2} N^g_{n_1'+1,n_2'}     \\nonumber\\\\\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-2 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2N^f_{n_1,n_2} N^g_{n_1',n_2'}     \\nonumber\\\\\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-2 \\\\ 5f-2n_1-3n_2+1 \\end{array}\\right)\nf^3N^f_{n_1,n_2} N^g_{n_1',n_2'+1}     \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\n&&N^d_{m_1+1,m_2+1}-dN^d_{m_1,m_2+2}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d} {n_1+n_1'=m_1, n_2+n_2'=m_2}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\\nonumber\\\\\n          &&\\bigg[ \\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2-2 \\end{array}\\right)\nfgN^f_{n_1+1,n_2} N^g_{n_1',n_2'+1}     \\nonumber\\\\\n&&  +\\left(\\begin{array}{c}\n          5d-2m_2-3m_3-5 \\\\ 5f-2n_1-3n_2-2 \\end{array}\\right)\nf^2gN^f_{n_1,n_2+1} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2N^f_{n_1,n_2} N^g_{n_1'+1,n_2'+1}     \\nonumber\\\\\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2+1 \\end{array}\\right)\nf^3N^f_{n_1,n_2} N^g_{n_1'+2,n_2'+1}        \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\n&&N^d_{m_1+2,m_2}-2dN^d_{m_1+1,m_2+1}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d} {n_1+n_1'=m_1, n_2+n_2'=m_2}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n          &&\\bigg[ \\left(\\begin{array}{c}\n          5d-2m_1-3m_2-4 \\\\ 5f-2n_1-3n_2-2 \\end{array}\\right)\nfgN^f_{n_1+1,n_2} N^g_{n_1'+1,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          5d-2m_2-3m_3-4 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nf^2gN^f_{n_1+1,n_2} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-4 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2N^f_{n_1,n_2} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-4 \\\\ 5f-2n_1-3n_2+1 \\end{array}\\right)\nf^3N^f_{n_1,n_2} N^g_{n_1'+3,n_2'}        \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\n&&N^d_{m_1+1,m_2+2}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d} {n_1+n_1'=m_1, n_2+n_2'=m_2}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n          &&\\bigg[ \\left(\\begin{array}{c}\n          5d-2m_1-3m_2-7 \\\\ 5f-2n_1-3n_2-3 \\end{array}\\right)\nfN^f_{n_1,n_2+1} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          5d-2m_2-3m_3-7 \\\\ 5f-2n_1-3n_2-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2} N^g_{n_1'+1,n_2'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& +\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-7 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2N^f_{n_1,n_2+1} N^g_{n_1'+3,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-7 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nf^2N^f_{n_1+1,n_2} N^g_{n_1'+2,n_2'+1}        \\bigg].\n\\bj\nFor the case of $Gr(2,4)$ the recursion relation of becomes\n\\ai\n&&N^d_{m_1+3,m_2}-N^d_{m_1+1,m_2+1}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d} {n_1+n_1'=m_1, n_2+n_2'=m_2}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n          &&\\bigg[ \\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2} N^g_{n_1'+1,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          5d-2m_2-3m_3-5 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nf^2N^f_{n_1+1,n_2} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2-1 \\end{array}\\right)\nfN^f_{n_1,n_2} N^g_{n_1'+2,n_2'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&& -\\left(\\begin{array}{c}\n          5d-2m_1-3m_2-5 \\\\ 5f-2n_1-3n_2 \\end{array}\\right)\nf^2N^f_{n_1,n_2} N^g_{n_1'+3,n_2'}        \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& d^2 N^d_{m_1,m_2,m_3+1}-2d\nN^d_{m_1,m_2+1,m_3}+N^d_{m_1,m_2,m_3}+N^d_{m_1+1,m_2,m_3}  \\nonumber\\\\\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\n\\nonumber\\\\\n&&\\bigg[ - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf^2g N^f_{n_1,n_2+1,n_3} N^g_{n_1',n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1,n_2,n_3} N^g_{n_1',n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + 2 \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2g N^f_{n_1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2 N^f_{n_1,n_2,n_3} N^g_{n_1',n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3\\end{array}\\right)\nf^2 N^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3+1 \\end{array}\\right)\nf^3 N^f_{n_1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}  \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& d^2 N^d_{m_1,m_2,m_3+1}-2d\nN^d_{m_1+1,m_2+1,m_3}+N^d_{m_1+2,m_2,m_3}+N^d_{m_1+3,m_2,m_3}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf^2g N^f_{n_1,n_2+1,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1+2,n_2,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + 2 \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2g N^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2 N^f_{n_1,n_2,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3\\end{array}\\right)\nf^2 N^f_{n_1+1,n_2,n_3} N^g_{n_1'+3,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-2 \\\\ 4f-n_1-2n_2-3n_3+1 \\end{array}\\right)\nf^3 N^f_{n_1,n_2,n_3} N^g_{n_1'+2,n_2'+1,n_3'}    \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& dN^d_{m_1+1,m_2,m_3+1}+N^d_{m_1,m_2+1,m_3}-N^d_{m_1+1,m_2+1,m_3}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf N^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1+2,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3\\end{array}\\right)\nf^2 N^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf N^f_{n_1+1,n_2,n_3} N^g_{n_1'+2,n_2',n_3'}\\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-3 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2 N^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}    \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& N^d_{m_1,m_2,m_3+1}+N^d_{m_1+2,m_2,m_3+1}\\nonumber\\\\ \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1+2,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4\\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1,n_2,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+2,n_2,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   +2\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2\\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1'+3,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf N^f_{n_1,n_2,n_3} N^g_{n_1'+2,n_2'+1,n_3'}    \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& N^d_{m_1+1,m_2+1,m_3+1}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2+1,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nf N^f_{n_1,n_2+1,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+2,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-1\\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1'+2,n_2'+1,n_3'}\\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2'+2,n_3'}    \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& N^d_{m_1+1,m_2,m_3+2}\\nonumber\\\\\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-3 \\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3+1} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2',n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  + \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-3 \\end{array}\\right)\nf N^f_{n_1,n_2,n_3+1} N^g_{n_1'+1,n_2'+1,n_3'}    \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2',n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-1\\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nN^f_{n_1+1,n_2,n_3} N^g_{n_1'+2,n_2',n_3'+1}\\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-6 \\\\ 4f-n_1-2n_2-3n_3-1\\end{array}\\right)\nfN^f_{n_1,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'+1}    \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& N^d_{m_1,m_2+3,m_3}-N^d_{m_1,m_2+1,m_3+1}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1,n_2+1,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2+1,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf^2 N^f_{n_1,n_2+1,n_3} N^g_{n_1',n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1,n_2,n_3} N^g_{n_1',n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3} N^g_{n_1'+1,n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-5 \\\\ 4f-n_1-2n_2-3n_3\\end{array}\\right)\nf^2 N^f_{n_1,n_2,n_3} N^g_{n_1',n_2'+2,n_3'}         \\bigg],\\\\\\noalign{\\vskip10pt}\\noal\\noalign{\\vskip10pt}\n&& N^d_{m_1,m_2+2,m_3}-N^d_{m_1+1,m_2,m_3+1}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfg N^f_{n_1,n_2+1,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfg N^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nf N^f_{n_1,n_2+1,n_3} N^g_{n_1'+1,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nfN^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2+1,n_3} N^g_{n_1'+2,n_2',n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1\\end{array}\\right)\nf^2 N^f_{n_1,n_2+1,n_3} N^g_{n_1'+1,n_2'+1,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3-1 \\end{array}\\right)\nf N^f_{n_1+2,n_2,n_3} N^g_{n_1'+1,n_2'+1,n_3'}\\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-4 \\\\ 4f-n_1-2n_2-3n_3 \\end{array}\\right)\nf^2 N^f_{n_1+1,n_2,n_3} N^g_{n_1',n_2'+2,n_3'}    \\bigg],\\\\\\noalign{\\vskip10pt} \\noalign{\\vskip10pt}\\noal\n&& N^d_{m_1,m_2+1,m_3+2}\\nonumber\\\\  \\noalign{\\vskip10pt}\n&&=\\sum_{\\stackrel{f+g=d, n_1+n_1'=m_1} {n_2+n_2'=m_2,\nn_3+n_3'=m_3}}\\left(\\begin{array}{c}\n          m_1 \\\\ n_1 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_2 \\\\ n_2 \\end{array}\\right)\n\\left(\\begin{array}{c}\n          m_3 \\\\ n_3 \\end{array}\\right)\\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\bigg[ \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-3 \\end{array}\\right)\nfN^f_{n_1,n_2,n_3+1} N^g_{n_1',n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1,n_2+1,n_3} N^g_{n_1',n_2'+1,n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-3 \\end{array}\\right)\nf N^f_{n_1+1,n_2,n_3+1} N^g_{n_1'+1,n_2'+2,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&  -\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nfN^f_{n_1+1,n_2+1,n_3} N^g_{n_1'+1,n_2'+1,n_3'+1}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&   +\\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-2 \\end{array}\\right)\nf^2N^f_{n_1,n_2,n_3+1} N^g_{n_1',n_2'+3,n_3'}     \\nonumber\\\\\\noalign{\\vskip10pt}\n&&\\  - \\left(\\begin{array}{c}\n          4d-m_1-2m_2-3m_3-7 \\\\ 4f-n_1-2n_2-3n_3-1\\end{array}\\right)\nf^2 N^f_{n_1,n_2+1,n_3} N^g_{n_1',n_2'+2,n_3'+1}         \\bigg].\n\\bj\n\\normalsize\nIn there recursion relations, d, f, and g are all greater or equal to one. So\nwhen d equals one, r,h,s of these equations vanish since g+f $\\geq$ 2. Then\nwe have a set of linear relations for $N^1_*$'s. We can use these linear\nrelations to determine all the the\n$<{\\cal O}_{W_4}{\\cal O}_{W_4}>$=1 for $CP^3$, $<{\\cal O}_{W_5}{\\cal\nO}_{W_5}>$=1 for $CP^4$ and $<{\\cal O}_{W_5}{\\cal O}_{W_6}>$=1 for $Gr(2,4)$.\nThen we put\nthese degree 1 correlation functions to the r,h,s of (4.35),(4.36),(4.37)\nand obtain linear\nrelations for $N^2_*$'s. This time, these linear relations thoroughly determine\nthem. For higher degree, the process is the same as d=2 case. We observe that\nrecursion relations we have witten down are suffcient for determination.\nWe checked the compatiable condition in the case of d $\\leq$ 4. It seems that\nthe over determined system of WDVV equation work well in all degrees of maps in\nthe case of $CP^3, CP^4$ and $Gr(2,4)$. The intersection numbers of moduli\nspace of d $\\leq$ 4 are given in the following tables.\n\\footnotesize\n\\begin{table}[h]\n\\caption{\\bf D=1 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|}\n     \\hline\n$N_{0}=2\\;\\;\\;$ & $N_{1}=1\\;\\;\\;$ & $N_{2}=1\\;\\;\\;$ \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1.5cm\n\\begin{table}\n\\caption{D=2 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$N_{0}=92\\;\\;\\;$ & $N_{1}=18\\;\\;\\;$ & $N_{2}=4\\;\\;\\;$ & $N_{3}=1\\;\\;\\;$ &\n$N_{4}=0\\;\\;\\;$ \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1.5cm\n\\begin{table}\n\\caption{D=3 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|l|}\n     \\hline\n$N_{0}=80160$ & $N_{1}=9864$ & $N_{2}=1312$ & $N_{3}=190$ & $N_{4}=30$ &\n$N_{5}=5$ & $N_{6}=1$ \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1.5cm\n\\begin{table}\n\\caption{D=4 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n     \\hline\n$N_{0}=383306880\\;$ & $N_{1}=34382544\\;$ & $N_{2}=3259680\\;$ & $N_{3}=327888\\;$\n& $N_{4}=35104\\;$ \\\\ \\hline\n$N_{5}=4000\\;$ & $N_{6}=480\\;$ & $N_{7}=58\\;$ & $N_{8}=4\\;$ & \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1.5cm\n\\begin{table}\n\\caption{D=5 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|}\n     \\hline\n$N_{0}=6089786376960$ & $N_{1}=429750191232$ & $N_{2}=31658432256$ &\n$N_{3}=2440235712$  \\\\ \\hline\n$N_{4}=197240400$ & $N_{5}=16744080$ & $N_{6}=1492616$ & $N_{7}=139098$ \\\\\n\\hline\n $N_{8}=13354$ & $N_{9}=1265$ & $N_{10}=105$ & \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1.5cm\n\\begin{table}\n\\caption{D=6 $CP^3$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|}\n     \\hline\n$N_{0}=244274488980962304\\;$ & $N_{1}=14207926965714432\\;$ &\n$N_{2}=855909223176192\\;$ \\\\ \\hline\n$N_{3}=53486265350784\\;$ & $N_{4}=3472451647488\\;$ & $N_{5}=234526910784\\;$ \\\\\n\\hline\n$N_{6}=16492503552\\;$ & $N_{7}=1207260512\\;$ & $N_{8}=91797312\\;$ \\\\ \\hline\n$N_{9}=7200416\\;$ & $N_{10}=573312\\;$  & $N_{11}=44416\\;$ \\\\ \\hline\n$N_{12}=2576$ & &\\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n   \\clearpage\n\\begin{table}\n\\caption{D=1 $CP^4$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|}\n     \\hline\n$N_{00}=5\\;\\;$ & $N_{10}=3\\;\\;$ & $N_{20}=2\\;\\;$ & $N_{30}=1\\;\\;$ &\n$N_{01}=1\\;\\;$ & $N_{11}=1\\;\\;$ & $N_{02}=1\\;\\;$ \\\\ \\hline\n\\end{tabular}~\n      \\end{center}\n     \\end{table}\n\n\\begin{table}\n\\caption{D=2 $CP^4$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$N_{00}=6620\\;$ & $N_{10}=1734\\;$ & $N_{20}=473\\;$ & $N_{30}=132\\;$ &\n$N_{40}=36\\;$ & $N_{50}=10\\;$ \\\\ \\hline\n$N_{01}=219$ & $N_{11}=67$ & $N_{21}=21$ & $N_{31}=6$ & $N_{41}=2$ & \\\\ \\hline\n$N_{02}=11$ & $N_{12}=4$ & $N_{22}=1$ & & &\\\\ \\hline\n$N_{03}=1$ & $N_{13}=0$ & & & &\\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip-1cm\n\\begin{table}\n\\caption{D=3 $CP^4$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n     \\hline\n$N_{00}=213709980$ & $N_{01}=2770596$ & $N_{02}=45954$ & $N_{03}=1011$ &\n$N_{04}=30$  \\\\ \\hline\n$N_{05}=0$ & $N_{10}=35806494$ & $N_{11}=511012$ & $N_{12}=9386$ & $N_{13}=225$\n  \\\\ \\hline\n$N_{14}=5$ & $N_{20}=6165822$ & $N_{21}=96548$ & $N_{22}=1931$ & $N_{23}=45$\n\\\\ \\hline\n$N_{24}=1$ & $N_{30}=1085892$ & $N_{31}=18469$ & $N_{32}=385$ & $N_{33}=9$   \\\\\n\\hline\n$N_{40}=194024$ & $N_{41}=3512$ & $N_{42}=76$ & & \\\\ \\hline\n$N_{50}=34780$ & $N_{51}=664$ & $N_{52}=16$ & & \\\\ \\hline\n$N_{60}=6216$ & $N_{61}=128$ & & & \\\\ \\hline\n$N_{70}=1108$ & & &  &\\\\ \\hline\n$N_{80}=188$ & &  & &\\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\\vskip -1cm\n\\begin{table}\n\\caption{D=4 $CP^4$}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|}\n     \\hline\n$N_{00}=47723447905060$ & $N_{01}=327439797532$ & $N_{02}=2679044142$ &\n$N_{03}=26578256$ \\\\ \\hline\n$N_{04}=324764$ & $N_{05}=4830$ & $N_{06}=61$ & $N_{07}=1$  \\\\ \\hline\n$N_{10}=5876564125104$ & $N_{11}=43242657488$ & $N_{12}=380720598$ &\n$N_{13}=4063860$ \\\\ \\hline\n$N_{14}=52507$ & $N_{15}=732$ & $N_{16}=9$ & \\\\ \\hline\n$N_{20}=738764469204$ & $N_{21}=5823161346$ & $N_{22}=54948346$ &\n$N_{23}=622980$  \\\\ \\hline\n$N_{24}=8133$ & $N_{25}=107$ & &\\\\ \\hline\n$N_{30}=94605276228$ & $N_{31}=796460052$ & $N_{32}=7990720$ & $N_{33}=94104$\n\\\\ \\hline\n$N_{34}=1218$ & $N_{35}=14$ & &\\\\ \\hline\n$N_{40}=12302188692$ & $N_{41}=110031632$ & $N_{42}=1159218$ & $N_{43}=13962$\n\\\\ \\hline\n$N_{44}=178$ & & &\\\\ \\hline\n$N_{50}=1617593360$ & $N_{51}=15251816$ & $N_{52}=166936$ & $N_{53}=2056$ \\\\\n\\hline\n$N_{60}=213984472$ & $N_{61}=2110864$ & $N_{62}=23968$ & $N_{63}=320$ \\\\ \\hline\n$N_{70}=28346212$ & $N_{71}=291632$ & $N_{72}=3516$ & \\\\ \\hline\n$N_{80}=3748804$ & $N_{81}=40492$ & & \\\\ \\hline\n$N_{90}=343260$ & $N_{91}=5552$ & & \\\\ \\hline\n$N_{100}=63740$ & & &\\\\ \\hline\n       \\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\n\\begin{table}\n\\caption{D=1 Gr(2,4)}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$N_{000}=0\\;\\;\\;\\;\\;\\;\\;\\;$ & $N_{100}=0\\;\\;\\;\\;\\;\\;\\;\\;$ &\n$N_{200}=1\\;\\;\\;\\;\\;\\;\\;\\;$ & $N_{300}=1\\;\\;\\;\\;\\;\\;\\;\\;$ &\n$N_{400}=0\\;\\;\\;\\;\\;\\;\\;\\;$ & $N_{500}=0\\;\\;\\;\\;\\;\\;\\;\\;$ \\\\ \\hline\n$N_{001}=0$ & $N_{101}=1$ & $N_{201}=0$ & & & \\\\ \\hline\n$N_{010}=0$ & $N_{110}=1$ & $N_{210}=1$ & $N_{310}=0$ & & \\\\ \\hline\n$N_{011}=1$ & & & & & \\\\ \\hline\n$N_{020}=1$ & $N_{120}=1$ & & & & \\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\\vskip-30cm\n\n\\begin{table}\n\\caption{D=2 Gr(2,4)}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|l|}\n     \\hline\n$N_{000}=2$ & $N_{100}=6$ & $N_{200}=18$ & $N_{300}=34$ & $N_{400}=42$ &\n$N_{500}=42$ & $N_{600}=34$ & $N_{700}=18$ \\\\ \\hline\n$N_{800}=6$ & $N_{900}=2$ & & & & & &\\\\ \\hline\n$N_{001}=1$ & $N_{101}=3$ & $N_{201}=5$ & $N_{301}=5$ & $N_{401}=5$ &\n$N_{501}=3$ & $N_{601}=1$ &\\\\ \\hline\n$N_{002}=1$ & $N_{102}=1$ & $N_{202}=1$ & $N_{302}=1$ & & & &\\\\ \\hline\n$N_{003}=1$ & & & & & & & \\\\ \\hline\n$N_{010}=3$ & $N_{110}=9$ & $N_{210}=17$ & $N_{310}=21$ & $N_{410}=21$ &\n$N_{510}=17$ & $N_{610}=9$ & $N_{710}=3$\\\\ \\hline\n$N_{011}=2$ & $N_{111}=3$ & $N_{211}=3$ & $N_{311}=3$ & $N_{411}=2$ & & & \\\\\n\\hline\n$N_{012}=1$ & $N_{112}=1$ & & & & & & \\\\ \\hline\n$N_{020}=5$ & $N_{120}=9$ & $N_{220}=11$ & $N_{320}=11$ & $N_{420}=9$ &\n$N_{520}=5$ & & \\\\ \\hline\n$N_{021}=2$ & $N_{121}=2$ & $N_{221}=2$ & & & & & \\\\ \\hline\n$N_{030}=5$ & $N_{130}=6$ & $N_{230}=6$ & $N_{330}=5$ & & & & \\\\ \\hline\n$N_{031}=1$ & & & & & & & \\\\ \\hline\n$N_{040}=3$ & $N_{140}=3$ & & & & & & \\\\ \\hline\n       \\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\n\\begin{table}\n\\caption{D=3 Gr(2,4)}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n     \\hline\n\\vspace{-0.5mm}\n$N_{000}=504\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ & $N_{001}=100\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ &\n$N_{002}=25\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ & $N_{003}=6\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ &\n$N_{004}=2\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{100}=1824$ & $N_{101}=307$ & $N_{102}=55$ & $N_{103}=9$ & $N_{104}=2$  \\\\\n\\hline\n\\vspace{-0.5mm}\n$N_{200}=5159$ & $N_{201}=676$ & $N_{202}=83$ & $N_{203}=10$ & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{300}=11319$ & $N_{301}=1109$ & $N_{302}=101$ & $N_{303}=9$ & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{400}=19512$ & $N_{401}=1460$ & $N_{402}=101$ & $N_{403}=6$ & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{500}=27472$ & $N_{501}=1605$ & $N_{502}=83$ & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{600}=32517$ & $N_{601}=1460$ & $N_{602}=55$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{700}=32517$ & $N_{701}=1109$ & $N_{702}=25$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{800}=27472$ & $N_{801}=676$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{900}=19512$ & $N_{901}=307$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1000}=11319$ & $N_{1001}=100$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1100}=5159$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1200}=1824$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1300}=504$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{010}=538$ & $N_{011}=109$ & $N_{012}=23$ & $N_{013}=5$ &  \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{110}=1603$ & $N_{111}=246$ & $N_{112}=35$ & $N_{113}=5$ &  \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{210}=3607$ & $N_{211}=403$ & $N_{212}=42$ & $N_{213}=5$ &  \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{310}=6278$ & $N_{311}=528$ & $N_{312}=42$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{410}=8864$ & $N_{411}=579$ & $N_{412}=35$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{510}=10499$ & $N_{511}=528$ & $N_{512}=23$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{610}=10499$ & $N_{611}=403$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{710}=8864$ & $N_{711}=246$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{810}=6278$ & $N_{811}=109$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{910}=3609$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1010}=1603$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{1110}=538$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{020}=523$ & $N_{021}=94$ & $N_{022}=16$ & $N_{023}=2$ &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{120}=1203$ & $N_{121}=153$ & $N_{122}=18$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{220}=2100$ & $N_{221}=198$ & $N_{222}=18$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{320}=2960$ & $N_{321}=216$ & $N_{322}=16$ & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{420}=3501$ & $N_{421}=198$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{520}=3501$ & $N_{521}=153$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{620}=2960$ & $N_{621}=94$ & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{720}=2100$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{820}=1203$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{920}=523$ & & & & \\\\ \\hline\n\\vspace{-0.5mm}\n$N_{030}=420$ & $N_{031}=61$ & $N_{032}=7$ & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{130}=729$ & $N_{131}=76$ & $N_{132}=7$ & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{230}=1019$ & $N_{231}=82$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{330}=1200$ & $N_{331}=76$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{430}=1200$ & $N_{431}=61$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{530}=1019$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{630}=729$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{730}=420$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{040}=262$ & $N_{041}=28$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{140}=358$ & $N_{141}=30$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{240}=418$ & $N_{241}=28$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{340}=418$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{440}=358$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{540}=262$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{050}=124$ & $N_{051}=10$ & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{150}=144$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{250}=144$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{350}=124$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{060}=48$ & & & &\\\\ \\hline\n\\vspace{-0.5mm}\n$N_{160}=48$ & & & &\\\\ \\hline\n\\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\\begin{table}\n\\caption{D=4 Gr(2,4)}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$N_{000}=1044120$ & $N_{001}=93726$ & $N_{002}=9970$ & $N_{003}=1170$ &\n$N_{004}=138$ & $N_{005}=20$ \\\\ \\hline\n$N_{100}=3094440$ & $N_{101}=251402$ & $N_{102}=22570$ & $N_{103}=2058$ &\n$N_{104}=190$ & $N_{105}=20$ \\\\ \\hline\n$N_{200}=8093840$ & $N_{201}=570998$ & $N_{202}=4179$ & $N_{203}=2998$ &\n$N_{204}=214$ & $N_{205}=20$ \\\\ \\hline\n$N_{300}=18245976$ & $N_{301}=1086890$ & $N_{302}=64434$ & $N_{303}=3690$ &\n$N_{304}=214$ &\\\\ \\hline\n$N_{400}=35219976$ & $N_{401}=1752446$ & $N_{402}=84818$ & $N_{403}=3942$ &\n$N_{404}=190$ &\\\\ \\hline\n$N_{500}=58571280$ & $N_{501}=2434530$ & $N_{502}=96894$ & $N_{503}=3690$ &\n$N_{504}=138$ &\\\\ \\hline\n$N_{600}=84843440$ & $N_{601}=2951174$ & $N_{602}=96894$ & $N_{603}=2998$ & &\n\\\\ \\hline\n$N_{700}=108066120$ & $N_{701}=3143726$ & $N_{702}=84818$ & $N_{703}=2058$ & &\n\\\\ \\hline\n$N_{800}=121770480$ & $N_{801}=2951174$ & $N_{802}=64434$ & $N_{803}=1170$ & &\n\\\\ \\hline\n$N_{900}=121770480$ & $N_{901}=2434530$ & $N_{902}=41794$ & & & \\\\ \\hline\n$N_{1000}=108066120$ & $N_{1001}=1752446$ & $N_{1002}=22570$ & & & \\\\ \\hline\n$N_{1100}=84843440$ & $N_{1101}=1086890$ & $N_{1102}=9970$ & & & \\\\ \\hline\n$N_{1200}=58571280$ & $N_{1201}=570998$ & & & &\\\\ \\hline\n$N_{1300}=35219976$ & $N_{1301}=251402$ & & & &\\\\ \\hline\n$N_{1400}=18245976$ & $N_{1401}=93726$  & & & & \\\\ \\hline\n$N_{1500}=8093840$ & & & & &\\\\ \\hline\n$N_{1600}=3094440$ & & & & &\\\\ \\hline\n$N_{1700}=1044120$ & & & & &\\\\ \\hline\n$N_{010}=692760$ & $N_{011}=63904$ & $N_{012}=6528$ & $N_{013}=675$ &\n$N_{014}=74$ & $N_{015}=6$ \\\\ \\hline\n$N_{110}=1852184$ & $N_{111}=147070$ & $N_{112}=12060$ & $N_{113}=976$ &\n$N_{114}=80$ & \\\\ \\hline\n$N_{210}=4249660$ & $N_{211}=281764$ & $N_{212}=18506$ & $N_{213}=1181$ &\n$N_{214}=80$ &  \\\\ \\hline\n$N_{310}=8297556$ & $N_{311}=454858$ & $N_{312}=24196$ & $N_{313}=1254$ &\n$N_{314}=74$ & \\\\ \\hline\n$N_{410}=13886500$ & $N_{411}=631136$ & $N_{412}=27526$ & $N_{413}=1181$ & & \\\\\n\\hline\n$N_{510}=20177804$ & $N_{511}=764000$ & $N_{512}=27526$ & $N_{513}=976$ & & \\\\\n\\hline\n$N_{610}=25736664$ & $N_{611}=813396$ & $N_{612}=24196$ & $N_{613}=675$ & &\\\\\n\\hline\n$N_{710}=29015656$ & $N_{711}=764000$ & $N_{712}=18506$ & & & \\\\ \\hline\n$N_{810}=29015656$ & $N_{811}=631136$ & $N_{812}=12060$ & & & \\\\ \\hline\n$N_{910}=25736664$ & $N_{911}=454858$ & $N_{912}=6528$ & & & \\\\ \\hline\n$N_{1010}=20177804$ & $N_{1011}=281764$ & & & & \\\\ \\hline\n$N_{1110}=13886500$ & $N_{1111}=147070$ & & & & \\\\ \\hline\n$N_{1210}=8297556$ & $N_{1211}=63904$ & & & & \\\\ \\hline\n$N_{1310}=4249660$ & & & & & \\\\ \\hline\n$N_{1410}=1852184$ & & & & & \\\\ \\hline\n$N_{1510}=692760$ & & & & & \\\\ \\hline\n$N_{020}=440638$ & $N_{021}=39460$ & $N_{022}=3624$ & $N_{023}=332$ &\n$N_{024}=26$ & \\\\ \\hline\n$N_{120}=1025894$ & $N_{121}=75712$ & $N_{122}=5512$ & $N_{123}=338$ &\n$N_{124}=26$ & \\\\ \\hline\n$N_{220}=2019894$ & $N_{221}=121884$ & $N_{222}=7112$ & $N_{223}=408$ & &\\\\\n\\hline\n$N_{320}=3391958$ & $N_{321}=168332$ & $N_{322}=8032$ & $N_{323}=338$ & & \\\\\n\\hline\n$N_{420}=4932358$ & $N_{421}=203048$ & $N_{422}=8032$ & $N_{423}=332$ &  &\\\\\n\\hline\n$N_{520}=6290046$ & $N_{521}=215904$ & $N_{522}=7112$ & & & \\\\ \\hline\n$N_{620}=7089646$ & $N_{621}=203048$ & $N_{622}=5512$ & &  &\\\\ \\hline\n$N_{720}=7089646$ & $N_{721}=168332$ & $N_{722}=3624$ & & & \\\\ \\hline\n$N_{820}=6290046$ & $N_{821}=121884$ & & & &\\\\ \\hline\n$N_{920}=4932358$ & $N_{921}=75712$ & & &  &\\\\ \\hline\n$N_{1020}=3391958$ & $N_{1021}=39460$ & & & & \\\\ \\hline\n$N_{1120}=2019894$ & & & & &\\\\ \\hline\n$N_{1220}=1025894$ & & & & & \\\\ \\hline\n$N_{1320}=440638$ & & & & & \\\\ \\hline\n       \\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\\begin{table}\n\\caption{D=4 Gr(2,4)}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|}\n     \\hline\n$N_{030}=256946$ & $N_{031}=21072$ & $N_{032}=1695$ & $N_{033}=121$ & \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  \\\\ \\hline\n$N_{130}=508026$ & $N_{131}=33665$ & $N_{132}=2131$ & $N_{133}=126$ & &\\\\\n\\hline\n$N_{230}=852818$ & $N_{231}=46042$ & $N_{232}=2379$ & $N_{233}=121$ & &\\\\\n\\hline\n$N_{330}=1237234$ & $N_{331}=55181$ & $N_{332}=2379$ & & &\\\\ \\hline\n$N_{430}=1574370$ & $N_{431}=58548$ & $N_{432}=2131$ & & &\\\\ \\hline\n$N_{530}=1772374$ & $N_{531}=55181$ & $N_{532}=1695$ & & &\\\\ \\hline\n$N_{630}=1772374$ & $N_{631}=46042$ & & & &\\\\ \\hline\n$N_{730}=1574370$ & $N_{731}=33665$ & & & &\\\\ \\hline\n$N_{830}=1237234$ & $N_{831}=21072$ & & & &\\\\ \\hline\n$N_{930}=852818$ & & & & &\\\\ \\hline\n$N_{1030}=508026$ & & & & &\\\\ \\hline\n$N_{1130}=256946$ & & & & &\\\\ \\hline\n$N_{040}=131874$ & $N_{041}=9540$ & $N_{042}=626$ & $N_{043}=36$ & &\\\\ \\hline\n$N_{140}=220250$ & $N_{141}=12808$ & $N_{142}=690$ & & &\\\\ \\hline\n$N_{240}=317466$ & $N_{241}=15196$ & $N_{242}=690$ & & &\\\\ \\hline\n$N_{340}=402090$ & $N_{341}=16072$ & $N_{342}=626$ & & &\\\\ \\hline\n$N_{440}=451610$ & $N_{441}=15196$ & & & &\\\\ \\hline\n$N_{540}=451610$ & $N_{541}=12808$ & & & &\\\\ \\hline\n$N_{640}=402090$ & $N_{641}=9540$ & & & &\\\\ \\hline\n$N_{740}=317466$ & & & & &\\\\ \\hline\n$N_{840}=220250$ & & & & &\\\\ \\hline\n$N_{940}=131874$ & & & & &\\\\ \\hline\n$N_{050}=58170$ & $N_{051}=3544$ & $N_{052}=190$ & & &\\\\ \\hline\n$N_{150}=82790$ & $N_{151}=4156$ & $N_{152}=190$ & & &\\\\ \\hline\n$N_{250}=104070$ & $N_{251}=4380$ & & & &\\\\ \\hline\n$N_{350}=116486$ & $N_{351}=4156$ & & & &\\\\ \\hline\n$N_{450}=116486$ & $N_{451}=3544$ & & & &\\\\ \\hline\n$N_{550}=104070$ & & & & &\\\\ \\hline\n$N_{650}=82790$ & & & & &\\\\ \\hline\n$N_{750}=58170$ & & & & &\\\\ \\hline\n$N_{060}=21638$ & $N_{061}=1104$ & & & &\\\\ \\hline\n$N_{160}=26958$ & $N_{161}=1160$ & & & &\\\\ \\hline\n$N_{260}=30062$ & $N_{261}=1104$ & & & &\\\\ \\hline\n$N_{360}=30062$ & & & & &\\\\ \\hline\n$N_{460}=26958$ & & & & &\\\\ \\hline\n$N_{560}=21638$ & & & & &\\\\ \\hline\n$N_{070}=6888$ & $N_{071}=290$ & & & &\\\\ \\hline\n$N_{170}=7664$ & & & & &\\\\ \\hline\n$N_{270}=7664$ & & & & &\\\\ \\hline\n$N_{370}=6888$ & & & & &\\\\ \\hline\n$N_{080}=1916$ & & & & &\\\\ \\hline\n$N_{180}=1916$ & & & & &\\\\ \\hline\n       \\end{tabular}\n      \\end{center}\n     \\end{table}\n\n\n\n\n\n\n\\clearpage\n\\normalsize\n\\section{Conclusion}\n\n We have shown in this paper that the free energy of topological sigma models\non $CP^{3}$,$CP^{4}$ and $Gr(2,4)$ (coupled to gravity but on small phase\n space) can actually be evaluated using DWVV equation, some properties of\ntopological field theory and one initial condition($\\langle {\\cal O}_{W_4}\n{\\cal O}_{W_4} \\rangle =1$ for $CP^{3}$,$\\langle {\\cal O}_{W_5}\n{\\cal O}_{W_5} \\rangle =1$ for $CP^{4}$ and $\\langle {\\cal O}_{W_5}\n{\\cal O}_{W_6} \\rangle=1$ for $Gr(2,4)$). Although in $CP^{3}$ and $CP^{4}$\n case where classical cohomology ring of target space $M$ is generated by\n$H^{2}(M)$, this fact was already shown by Kontsevich and Manin, we find\nthis strategy also work for $Gr(2,4)$. We also observe that for $CP^{3}$ and\n$CP^{4}$, expansion\ncoefficients of the free energy acctually counts the number of rational curves\npassing through $PD(W)$ of insertion operator ${\\cal O}_{W}$ in $d=1$ case.\n(see Appendix A) From (\\ref{a9}) we see the contribution of one rational curve\nsatisfying ``passing through condition'' to correlation function is the\nproduct of intersection numbers between  the rational curve and corresponding\nPoincar\\'e duals, however, it seems this factor equals  $1$, unless $W$ is the\nK\\\"{a}hler form. If not so, most of correlation functions  in degree $d$\nsector have to be devisible by $d$, but our results do not support this\nspeculation.\n In $Gr(2,4)$ case, we find a rather interesting symmetry in the correlation\nfunctions. Interchange between the insertion of ${\\cal O}_{W_3}$ and\n${\\cal O}_{W_4}$ (in words of Schubert cycle of $Gr(2,4)$,$W_{3}$ and $W_{4}$\n correspond to $\\sigma_{1,1}$ and $\\sigma_{2}$) does not change\ncorrelation functions. This is natural because in classical geometry we can not\ndistinguish $\\sigma_{1,1}$ from $\\sigma_{2}$ in its algebraic structure.\nIn algebraic geometry, $Gr(2,4)$ describes lines in $CP^{3}$.\n$\\sigma_{2}$ correspond to the set of lines passing through a point $p_{0}$\nand $\\sigma_{1,1}$ to the set of lines contained fixed plane $h_{0}$. And\nthe symmeetry of the above two sets comes from duality, i.e we can see\n$(a_1:a_2:a_3:a_4)$ as a point of $CP^{3}$ or as a linear form on it.\nAfter the completion of this manuscript, a paper by Di Francesco and Itzykson\n``Quantum intersection rings'' has appeared which overlaps with our work.\n\n\\vskip1cm\n\\noindent{\\large\\bf Acknowledgement}\n\\vskip0.5cm\n\n\n We would like to thank Prof.T.Eguchi for suggesting this problem and kind\nencouragement.We also thank Dr.K.Hori for many useful discussions.\nWe are indebted to Dr.T.Hotta and Dr.T.Izubuchi for manipulation of computers.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Preparation}\nWe consider a superprocess which arises as the short life time and\nhigh density diffusion limit of a branching particle system, which\ncan be described as follows: in the $n^{\\mathrm{th}}$ approximation\nstep each particle has mass $1\/n$ and lives a random time which is\nexponential with mean $1\/n$. While a particle is alive, its motion\nis described by a diffusion process  corresponding to the operator\n$L$. At the end of its life, the particle dies and is replaced by a\nrandom number of particles situated at the parent particle's final\nposition. The distribution law of the number of descendants is\nspatially varying such that the mean number of descendants is\n$1+\\frac{\\beta(x)}{n}$, while the variance is assumed to be\n$2\\alpha(x)$. All these mechanisms are independent of each other.\nThe process is determined by the quadruple $(L,\\beta,\\alpha;D)$,\nwhere $L$ is a second order elliptic operator corresponding to the\nunderlying diffusion process on $D$. See Appendix A in Engl\\\"ander\nand Pinsky (1999) \\cite{EngPin99} for a precise statement on the\nparticle approximation.\n\n\nWe start by presenting a formal description of the model\nconsidered in this article. For convenience we first recall the\nbasic notation: let $D\\subseteq \\mathbb{R}^d$ be a domain and let\n${\\cal B}(D )$ denote the  Borel sets of $D$. We write ${\\cal\nM}_f(D)$ and ${\\cal M}_c(D)$ for the class of finite measures\nresp.\\ the class of finite measures with compact support on ${\\cal\nB}(D )$. For $\\mu\\in{\\cal M}_f(D )$, denote $ \\|\\mu\\|:=\\mu (D )$\nand let $C^+_b(D )$ and\n $C^+_c(D )$ be the class of non-negative bounded continuous\nresp.\\ non-negative continuous  functions $D \\rightarrow\\mathbb{R}$ having\ncompact support. Write $C^{k,\\eta}(D )$ for the usual H\\\"older\nspaces of index $\\eta\\in (0,1]$ including derivatives of order $k$,\nand set $C^{\\eta}(D):=C^{0,\\eta}(D)$.\n\n\nWe continue with the definition of the\n$(L,\\beta,\\alpha;D)$-superdiffusion, $X$. Let $L$ be an elliptic\noperator on the domain $D\\subseteq\\mathbb{R}^d$ of the form \\be{A3}\n   L:=\\frac{1}{2}\\nabla \\cdot a \\nabla +b\\cdot\\nabla,\n\\end{equation}\nwhere $a_{i,j},b_i\\in C^{1,\\eta}(D)$, $i,j=1,...,d$, for some\n$\\eta\\in (0,1]$, and the matrix $a(x):=(a_{i,j}(x))$ is symmetric,\nand positive definite for all $x\\in D$. In addition, let\n$\\alpha,\\beta\\in C^{\\eta}(D)$, and assume that $\\alpha$ is\npositive, and $\\beta$ is bounded from above.\n\n\\smallskip\nWe now present our model.\n\\begin{definition}[Time-homogeneous  superdiffusion]\\qquad\\qquad\n\nLet $\\left( X,\\mathbf{P}^{\\mu\\,},\\,\\mu\\in\\mathcal{M}_f(D)\\right) $\ndenote the $(L,\\beta,\\alpha;D )$-superdiffusion. That is, $X$ is\nthe unique $\\mathcal{M}_f(D)$-valued continuous (time-homogeneous)\nMarkov process which satisfies, for any $g\\in C^+_b(D)$ ,\n\\begin{equation}\n   {\\bf E}^{\\mu}\\exp\\left\\langle  X_{t\\,},-g\\right\\rangle\n =\\exp\\, \\langle\\mu,-u(\\cdot ,t)\\rangle,\n\\label{Laplace.functional}%\n\\end{equation}\nwhere $u$ is the minimal nonnegative solution to\n\\begin{equation}\n\\left.\n\\begin{array}\n[c]{c}%\nu_{t}=Lu+\\beta u-\\alpha u^{2}\\quad\\text{\\emph{on}}\\;D%\n\\times(0,\\infty),\\vspace{8pt}\\\\\n\\lim\\limits_{t\\downarrow 0}u(\\cdot,t)=g(\\cdot).\n\\end{array}\n\\;\\right\\}  \\label{cum.equ}%\n\\end{equation}\nAs usual, $\\left\\langle \\nu,f\\right\\rangle $ denotes the integral\n$\\int_{D}\\nu(\\mathrm{d} x)\\,f(x).$\n\\end{definition}\n\n(See Dynkin (1991, 2002) \\cite{Dyn91}, \\cite{Dy02} or Dawson\n(1993) \\cite{Daw93} for the definition of superprocesses in\ngeneral; see Engl\\\"ander and Pinsky (1999) \\cite{EngPin99} for\nmore on the definition in the particular setting above.)\n\n\\begin{remark}[Time-inhomogeneous\nsuperdiffusion] \\rm\nThe model under consideration is a time-homogeneous process.\nHowever, it  is important to point out that for the formulation of\nthe main theorem and the proof, the introduction of certain {\\it\ntime-inhomogeneous} superdiffusions is required. The previous\ndefinition will therefore be generalized for time-inhomogeneous\nsuperdiffusions in  Appendix B. (See Definition \\ref{D2}.)\n\\end{remark}\n\nLet \\be{A5q}\n\\begin{aligned}\n  \\lambda _{c}\n &:=\n  \\lambda _{c}(L+\\beta )\n  \\\\\n &:= \\inf \\{\\lambda \\in \\mathbb{R}\\ : \\\n  \\exists u>0\\ \\text{satisfying}\\ (L+\\beta -\\lambda )u=0\\ \\text{in}\\\n  D \\}\n\\end{aligned}\n\\end{equation}\ndenote the {\\it generalized principal eigenvalue }for $L+\\beta $\non $D$. Let $\\xi^{L}$ be the diffusion process on $D$\ncorresponding to $L$, and denote by $\\mathbb P^x$ the law\nof $\\xi^L$ starting at $x\\in D$.\nThen from a probabilistic point of view, the\ngeneralized principal eigenvalue can be equivalently expressed as\n\\be{A5qq}\n  \\lambda_{c}\n =\n  \\sup_{\\{A:\\ A\\subset \\subset D,\\ \\partial A\\ \\mathrm{is}\\ C^{2,\\eta}\\}}\n  \\lim_{t\\to \\infty }\\frac{1}{t}\\log{\\mathbb{E}^{x}\n  \\Big[\n  \\exp\\big[\\int_{0}^{t}\\beta (\\xi^{L}_s)\\,\\mathrm{d}s\\big];\\,\\tau^{A}>t\n  \\Big]},\n\\end{equation}\nfor any $x\\in D,$ where $\\tau ^{A}=\\inf \\{ t\\geq 0:\\xi^{L}(t)\\not\\in\nA\\}$, and the $C^{2,\\eta}$-boundary is defined with the help of\n$C^{2,\\eta}$-maps in the usual way. (See Section 4.4 in Pinsky\n(1995) \\cite{Pin95} on the subject). Hence, since $\\beta$ is bounded\nfrom above, $\\lambda_{c}<\\infty$; and it is known from standard\ntheory that for any $\\lambda \\geq \\lambda _{c}$, there exists a\nfunction $0<f \\in C^{2,\\eta }(D)$ such that $(L+\\beta )f =\\lambda f\n$ on $D$. (See Section 4.3 in Pinsky (1995) \\cite{Pin95})\n\nPinsky proved that $X$ {\\em exhibits local extinction}  (i.e., the\nsupport of $X$ leaves any given bounded set, ${\\bf P}^{\\mu}$-a.s.\nfor each $\\mu\\in{\\cal M}_c$) if and only if $\\lambda_c \\le 0$.\n(See Theorem 6 and Remark 1 in Pinsky (1996) \\cite{Pin96}.)\n\\smallskip\n\n\nFrom now on we are interested in the situation where $X$ does not\nexhibit local extinction. We therefore assume that $\\lambda_c>0$.\nWe get a first rough impression about the local growth rate  by\nthe following statement  taken from Theorem 7(b) in Pinsky (1996)\n\\cite{Pin96}: \\addtocounter{lemma}{-1} \\begin{lemma}[Local\nbehavior in expectation]\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nFor $\\mu\\in{\\cal M}_c(D)$, and $g\\in C_c^+(D)$, satisfying\n$||\\mu||\\neq 0$ and $g\\not\\equiv 0$, \\be{A6}\n   \\limsup_{t\\to\\infty}\\,\\exp[-\\rho t]{\\bf E}^{\\mu}[\\langle X_t,g\\rangle]\n =\n   \\left\\{\\begin{array}{cc}0&\\mbox{ if }\\rho >\\lambda_c\\\\\\infty&\n   \\mbox{ if }\\rho <\\lambda_c\\end{array}\\right..\n\\end{equation}\n(See also Appendix \\ref{Pinproblem}.)\n\\end{itemize}\n\n\\end{lemma}\n\n\n\\smallskip\n\nWe are therefore going to concentrate on scaling with the exponent\n$\\rho=\\lambda_c$. In addition to  the concept of the generalized\nprincipal eigenvalue we will then need some further ones  from the\nso-called {\\it criticality theory} of second order elliptic\noperators. In particular, we will use the concepts of {\\it critical}\nand {\\it product-critical} (or  product-$L^1$ critical) operators.\nRecall that  the operator $L+\\beta-\\lambda_c$ is called critical if\nthere exists a positive function $f$ satisfying that\n$(L+\\beta-\\lambda_c)f=0$ but there is no (minimal positive) Green's\nfunction for the operator $L+\\beta-\\lambda_c$. In this case $f$ is\nunique up to constant multiples and is called the {\\em ground\nstate}. The operator $L+\\beta-\\lambda_c$ is called product-critical\nif it is critical with  ground state $0<\\phi_c$, and $\\phi_c$ and\n$\\tilde{\\phi_c}$ (i.e.\\ the ground state for the formal adjoint of\n$L+\\beta-\\lambda_c$) satisfy $\\langle\n\\mathrm{d}x,\\phi_c\\tilde{\\phi_c}\\rangle<\\infty$. In this case we\nnormalize them by $\\langle \\mathrm{d}x,\\phi_c\\tilde{\\phi_c}\\rangle\n=1$.\n\nIf $L+\\beta-\\lambda_c$ possesses a Green's function, then it is\ncalled {\\it subcritical}.\n\nFor the reader, it will be handy to have Appendix 2 of\n\\cite{EngTur00} at hand, where a review on criticality theory is\ngiven. For a complete presentation of the theory, the reader is\nreferred to Chapter 4 in \\cite{Pin95}.\\smallskip\n\n\\subsection{Motivation}\n\nWhen $D=\\mathbb{R}^d$ and  $L+\\beta-\\lambda_c$ is\nproduct-critical, it is known from Theorem 1 in \\cite{EngTur00}\nthat if $\\parallel\\alpha\\phi_c\\parallel_{\\infty}<\\infty$ and the\ninitial state $\\mu$ is such that $\\langle \\mu,\\phi_c\\rangle\n<\\infty$, then  the following  holds in the vague topology:\n\n\\be{A7} \\lim_{t\\to\\infty}  \\exp[-\\lambda_c t] X_t\\, (\\mathrm{d}x)\n =\n    N^{\\mu}\\,\\tilde{\\phi_c}\\, \\mathrm{d}x\\ \\ \\mathrm{in\\ law},\n\\end{equation}\nwhere the limiting non-negative non-degenerate random variable\n$N^{\\mu}$ was identified with the help of a certain {\\em invariant\ncurve}.\n\nIt is important to point out that even though product-criticality\nis equivalent to the ergodicity of an auxiliary diffusion process\n(see next section), {\\it the original motion process\ncorresponding to $L$\ndoes not have to be ergodic}. In fact it can even be transient --\nsee Example 23 in \\cite{EngTur00}.\n\n\n(On the other hand, it follows from the discussion in Appendix A\nthat when $L+\\beta-\\lambda_c$ on $D\\subseteq \\mathbb R^d$ is not\nproduct-critical, then for $\\mu\\in{\\cal M}_c$, and $g\\in\nC_c^+(D)$, \\be{A6o} \\lim_{t\\to\\infty}\\,\\exp[-\\lambda_c t]\\langle\nX_t,g\\rangle\n = 0 \\ \\ \\mathrm{in\\ }L^1.)\n\\end{equation}\n\n\nThere are two disadvantages of the method used in \\cite{EngTur00}.\nFirst, the assumption that $D=\\mathbb{R}^d$ is essential. Second,\nthe proof does not yield any probabilistic insight.\n\nIn this paper our goal is to improve the statement in (\\ref{A7})\nand to provide a proof that is probabilistic in nature. We will\nshow that the `{\\it Law of Large Numbers}' holds, that is, that\none can replace the convergence in law by convergence in\nprobability\\footnote{Since the limit is not constant, therefore,\nunlike in classical probability theory, one has to distinguish\nbetween convergence in law (WLLN) and convergence in probability\n(LLN).}. Furthermore we will drop the assumption that\n$D=\\mathbb{R}^d$. In the proof we will replace the analytic\nreasoning given in \\cite{EngTur00} (which relies on dynamical\nsystems) by a more probabilistic one using space-time weighted\nsuperprocesses ($H$-transforms).\n\nWe suspect that in fact the Strong Law of Large Numbers holds,\nthat is that the convergence in probability can  be replaced by\nalmost sure convergence. However we could not upgrade the proof of\nthis paper to give the Strong Law.\n\nIn the recent paper \\cite{FleSwa 03} the authors study  a\nsupercritical superprocess taking values in the space of finite\nmeasures on $[0,1]$, whose underlying motion is the Wright--Fisher\ndiffusion corresponding to the operator $$L=\\frac{1}{2}\nx(1-x)\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2},$$ and whose branching\nmechanism is $\\gamma u(1-u)$ with $\\gamma\n>0$ (that is, $\\alpha=\\beta=\\gamma$). They establish a dichotomy in the\nlong-time behavior of this superprocess. For $\\gamma\\le 1$, the mass\nin the interior $(0,1)$ dies out after a finite random time, while\nfor $\\gamma >1$, the mass in $(0,1)$ grows exponentially with rate\n$\\gamma - 1$ (as $t\\to\\infty$ and with positive probability) and is\napproximately uniformly distributed over $(0,1)$.\n\nThis result is in line with that of \\cite{EngTur00} if one considers\nthe restriction of the superprocess on the (open) domain $(0,1)$. In\nfact it is easy to show that $\\lambda_c:=\\gamma-1$ is the principal\neigenvalue of the linearized elliptic operator $L+\\gamma$.\nHere is a possible argument:  the operator $L+\\gamma-\\lambda_c=L+1$\ncan be $h$-transformed ($h=v$, where $v$ is an explicitly given\nfunction in the paper) into a diffusion operator, which -- according\nto their Lemma 20 -- corresponds to a (positive) recurrent\ndiffusion. Consequently, this $h$-transformed operator  is {\\it\ncritical}, and thus  its principal eigenvalue is zero. By\n$h$-transform invariance, the same is then true for the original\noperator $L+\\gamma-\\lambda_c.$ (See again Chapter 4 in\n\\cite{Pin95}). Furthermore, the product-criticality and boundedness\nassumptions are automatically satisfied by the boundedness of\n$D=(0,1)$.\n\nFinally the fact that the limiting measure is the Lebesgue measure,\nis also in line with \\cite{EngTur00}. Indeed, according to\n\\cite{EngTur00}, the limiting density is a harmonic function with\nrespect to the adjoint of $L+\\gamma-\\lambda_c=L+1$, that is with\nrespect to $\\tilde L+1$, where $\\tilde L$ is the adjoint of $L$. An\neasy computation reveals that $$\\tilde L+1=\\frac{1}{2}\nx(1-x)\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2}+(1-2x)\\frac{\\mathrm{d}}{\\mathrm{d}x}.$$\nSince the adjoint of a critical operator is also critical, and since\n positive harmonic functions for a critical\noperator are unique up to constant multiples,  the limiting density\nmust be a properly normalized constant on the unit interval, that\nis, the limiting density is one.\n\nHowever, as the authors point out referring to \\cite{EngTur00},\n`\\it their methods use in an essential way the fact that their\nunderlying space is $\\mathbb{R}^d$ (and not an open subset of\n$\\mathbb{R}^d$, like $(0, 1)$), and therefore their results are\nnot applicable to our situation.'\\rm\n\nIn the present article, as already mentioned,  we manage to\novercome this difficulty, so the result of \\cite{FleSwa 03} will\nfit our main result (in \\cite{FleSwa 03} the result is somewhat\nstronger as they prove convergence in $L^2$).\n\n\n\n\n\\section{Main Result}\nRecall from (\\ref{A5q}) the definition of the principal eigenvalue\n$\\lambda_c$  of $L+\\beta$ on $D$ and the corresponding ground state\n$\\phi_c$, and that throughout the paper we assume that\n$\\lambda_c>0$. Also, $\\{{\\cal S}_s\\}_{s\\ge 0}$ will denote the\nsemigroup (`expectation semigroup') corresponding to the operator\n$L+\\beta$ on $D$. So far we have recalled (\\ref{A7}). In order to\nreplace in (\\ref{A7}) the convergence in law by convergence in\nprobability, we will assume the same conditions as in\n\\cite{EngTur00}, except that we work with a generic domain $D$.\n\n\n\\begin{assumption}\\label{assum}\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nIn addition to the assumption that $\\lambda_c>0$, also assume that\n$L+\\beta-\\lambda_c$ is product-critical, that $\\alpha\\phi_c$ is\nbounded and that $X$ starts in a state $\\mu$ with\n$\\langle\\mu,\\phi_c\\rangle<\\infty$.\n\\end{itemize}\n\\end{assumption}\nBefore   reading the remainder of this section, it is recommended\nthat the reader consults Appendix B regarding the definition of\ntime-inhomogeneous superdiffusions as well as the space-time\n$H$-transform (weighted superdiffusion) introduced there.\n\nLet $X$ be a $(L,\\beta,\\alpha;D)$-superdiffusion with $X_0=\\mu$.\n Let $H(x,t):=\\exp (-\\lambda_c t )\\phi_c(x),\\ x\\in D,\\\nt\\ge 0$. It turns out (see Lemma~\\ref{LH1} in Appendix B) that the\n(time-inhomogeneous) process $X^H$ defined by \\be{H1aa}\n  X_t^H:=H(\\cdot,t)X_t\\quad \\left(\\mbox{that is,}\\\n  \\frac{\\mathrm{d}X_t^H}{\\mathrm{d}X_t}=H(\\cdot,t)\\right), \\quad\n  t\\geq 0\n\\end{equation}\nis an $(L+a\\frac{\\nabla \\phi_c}{\\phi_c}\\cdot \\nabla,0,\\alpha \\phi_c\ne^{-\\lambda_c t};D)$-superdiffusion. In the sequel $\\widetilde{\\bf\nE}$ and $\\widetilde{\\mathrm{Var}}$ will denote expectation and\nvariance with respect to the law of $X^H$.\n\n\\beL{JA}{Bounded variance}\n\\begin{itemize}\n\n \\item[{}] \\hspace{.2cm} \\be{JL(A2)i}\n\\lim_{t\\to\\infty}\\widetilde{\\mathrm{Var}}^{\\phi_c\\mu}\n(\\|X_t^H\\|)=\\int_0^{\\infty}\\mathrm{d}s\\,e^{-2\\lambda_c s}\\,\n \\langle \\mu,{\\cal S}_s[\\alpha \\phi_c^2]\\rangle<\\infty,\\end{equation}\nand  $\\| X^H\\|$ is a uniformly integrable $\\widetilde{\\bf\nP}^{\\phi_c\\mu}$-martingale.\n\\end{itemize}\n\\end{lemma}\n\\bf{Proof:}\\rm\\ Let $\\overline{X}^H$ denote the {\\em total mass\nprocess}, i.e., \\be{Y11}\n   \\overline{X}^H:=\\| X^H\\|.\n\\end{equation}   Abbreviate \\be{LH}\n   L^{\\phi_c}_0:=L+a\\frac{\\nabla \\phi_c}{\\phi_c}\\cdot \\nabla\n\\end{equation}\nand note that in fact\n$$L^{\\phi_c}_0(u)=\\phi_c^{-1}(L+\\beta-\\lambda_c)(\\phi_c u)=\n   H^{-1}(L+\\beta+\\partial_t)(Hu).$$\n(Here $\\partial_t$ denotes differentiation with respect to time.)\nLet ${\\cal S}^{\\phi_c}$ denote the $h$-transformed semigroup with\n$h=\\phi_c$, that is ${\\cal S}^{\\phi_c}_s(\\cdot)=(\\phi_c)^{-1}{\\cal\nS}_s(\\phi_c \\cdot).$\n\nDefine ${\\cal S}^H_s:=\n   e^{-\\lambda_c s}{\\cal S}^{\\phi_c}_s$;\nthen the semigroup $\\{{\\cal S}^H_s\\}_{s\\ge 0}$ corresponds to the\noperator $L_0^{\\phi_c}$ that has no zeroth order part. In\nparticular then\n\\begin{equation}\\label{contraction} {\\cal S}^H_s\n1\\le 1.\n\\end{equation}\nFinally, the product-criticality assumption on $L+\\beta-\\lambda_c$\nguarantees that the diffusion process corresponding to\n$L^{\\phi_c}_0$ on $D$ is positive recurrent (ergodic) (see Section\n4.4. in Pinsky \\cite{Pin95}). (Since ergodicity implies\nconservativeness, thus in fact ${\\cal S}^H_s 1= 1;$ nonetheless, for\nus it will be enough to know (\\ref{contraction}).)\n\nBy Lemma~\\ref{LH1}(b) of Appendix B along with  Theorem A2 in\n\\cite{EngPin99}, we have that for all $f\\in\nC^2_{\\mathrm{const}}(D):=\\{f\\in C^2(D):\\ \\exists\n\\Omega\\subset\\subset D\\mathrm{\\ such\\ that}\\ f=\\mathrm{const}\\\n\\mathrm{on}\\ D\\setminus \\Omega \\}$, \\be{H3a} \\mathrm{d}\\langle\nX^H_t,f\\rangle =\\langle\nX^H_t,L^{\\phi_c}_0f\\rangle\\,\\mathrm{d}t+\\mathrm{d}M_t(f),\n\\end{equation}\nwhere $\\{M_t(f)\\}_{t\\ge 0}$ is a square-integrable $\\widetilde{{\\bf\nP}}^{\\phi_c\\mu}$-martingale,\nand its quadratic variation (i.e. the increasing process in the\nDoob-Meyer decomposition) $\\langle M(f)\\rangle$ is given by \\be{H4}\n   \\langle M(f)\\rangle_t\n =\n   \\int^t_0\\mathrm{d}s\\,e^{-\\lambda_c s}\\langle X^H_s,\\alpha\n   {\\phi_c}f^2\\rangle,\\ \\ t\\ge 0.\n\\end{equation}\n(One can take the function class $C^2_{\\mathrm{const}}(D)$ instead\nof just $C^2_c(D)$, because the diffusion process corresponding to\n$L^{\\phi_c}_0$ on $D$ is conservative, that is, it never leaves\nthe domain $D$ with probability one.)\n\nApplying (\\ref{H3a}) to the function $f\\equiv 1$, it follows that\n$\\overline{X}^H$ is a $\\widetilde{{\\bf P}}^{\\phi_c\\mu}$-martingale.\nFurthermore,  by (\\ref{H4}), \\be{H4f<}\n  \\widetilde{{\\bf E}}^{\\phi_c\\mu}\\left[\\langle X^H_t,1\\rangle^2\\right]\n =\n  \\langle\\mu, \\phi_c\\rangle^2+\\int_0^{t}\\mathrm{d}s\\,e^{-\\lambda_c s}\\,\n   \\langle \\phi_c\\mu,{\\cal S}^H_s[\\alpha \\phi_c]\\rangle.\n\\end{equation}\nThat is \\be{H4f<2} \\widetilde{\\mathrm{Var}}^{\\phi_c\\mu} (\\|X_t^H\\|)\n =\\int_0^{t}\\mathrm{d}s\\,e^{-\\lambda_c s}\\,\n   \\langle \\phi_c\\mu,{\\cal S}^H_s[\\alpha \\phi_c]\\rangle=\n   \\int_0^{t}\\mathrm{d}s\\,e^{-2\\lambda_c s}\\,\n \\langle \\mu,{\\cal S}_s[\\alpha \\phi_c^2]\\rangle.\n\\end{equation}\nLetting $t\\to\\infty$ we obtain the first statement of the lemma.\n\nReplacing $t$ by $\\infty$ in the first of the integrals in\n(\\ref{H4f<2}), we have from (\\ref{contraction}) and from our\n assumptions that\n$$\\widetilde{\\mathrm{Var}}^{\\phi_c\\mu} (\\|X_t^H\\|)\\le\\int_0^{\\infty}\\mathrm{d}s\\,e^{-\\lambda_c s}\\,\n   \\langle \\phi_c\\mu,{\\cal S}^H_s[\\alpha \\phi_c]\\rangle\n\\le{\\lambda_c }^{-1}\\parallel\\alpha \\phi_c\\parallel_{\\infty}\\,\n   \\langle \\mu, \\phi_c\\rangle<\\infty.$$\nHence, by (\\ref{H4f<}),\n$$\\sup\\nolimits_{t\\ge 0}\\widetilde{{\\bf E}}^{\\mu\\phi_c}\\left[\\langle\nX^H_t,1\\rangle^2\\right]<\\infty,$$ and consequently $\\overline{X}^H$\nis uniformly integrable. This completes the proof of the second\nstatement of the lemma. $\\hfill\\square$\n\n\n\n\\begin{remark}\\rm\\\n Our proof of LLN will indeed use the condition that\n $\\alpha \\phi_c$ is  bounded, however it is quite possible\n that this condition is not necessary and that assuming the finiteness of the integral in\n (\\ref{JL(A2)i}) (along with $\\langle\\mu,\n\\phi_c\\rangle<\\infty$) would suffice.\n\\end{remark}\n\nAn immediate consequence of uniform integrability is that\n$\\widetilde{{\\bf\nE}}^{\\phi_c\\mu}[\\overline{X}^H_\\infty]=\\langle\\mu,\\phi_c\\rangle$,\nwhich is finite by assumption, and positive for $\\mu\\not =0$. This\nyields that $\\widetilde{{\\bf\nP}}^{\\phi_c\\mu}[\\overline{X}^H_\\infty=0]<1$ for $\\mu\\not =0$. We\nrecord this in a lemma.\n \\beL{LH1q}{Limit of the total mass}\n\\begin{itemize}\n\\item[{}]\\hspace{2cm}\n\nThe martingale $\\overline{X}^H$ has a $\\widetilde{{\\bf\nP}}^{\\phi_c\\mu}$-a.s. limit\n$\\overline{X}^H_{\\infty}:=\\lim_{t\\to\\infty}\\overline{X}^H_t$ which\nis positive with positive probability.\n\\end{itemize}\n\\end{lemma}\n\\subsection{Heuristics for the Law of Large Numbers}\nBefore stating the {\\it Law of Large Numbers} for the class of\nsuperdiffusions under consideration,  in this subsection we give\nsome heuristic computations. These will justify why we call our main\nresult `the Law of Large Numbers'.\n\nWorking with the $H$-transformed superprocess and at the same time,\nhaving the particle approximation in mind, consider particles with\nunderlying motion $Y$ corresponding to the elliptic operator\n$L_0^{\\phi_c}$ (the  probabilities for\n$Y$ will be denoted by\n$\\{\\mathbb P^{x}, \\ x\\in D\\}$) and with critical binary branching at\nrate $\\exp [-\\lambda_c t]\\alpha(x)$ at position $x\\in D$ and time\n$t\\ge 0$. Furthermore let the system be started with initial\ndiscrete measure being ``close'' to $\\nu\\neq 0$.\n\nLet ${\\cal I}_t^n$ denote the collection of particles alive at\ntime $t$ in the $n^{\\mathrm{th}}$ approximation step.\nFinally, the event of `survival' is $\\{|{\\cal I}^n_t|>0\\ \\forall\nt>0\\}$.\n\nThe Law of Large Numbers would mean that if $0\\not\\equiv f\\in\nC_c^+(D)$, then as $t\\to\\infty$, (and without further specifying\nwhat ``$\\approx$'' means), \\be{AA1}\n \\|\\nu\\| \\frac{\\frac{1}{|{\\cal I}^n_t|}\\sum_{x\\in{\\cal I}_t^n}f(x)}\n  {\\langle \\nu,{\\mathbb E}^{x}[f(Y_t)]\\rangle}\\approx 1\\hspace{0.5cm}\n \\hspace{0.5cm}  \\mathrm{on}\\ \\ \\{|{\\cal I}^n_t|>0\\ \\forall t>0\\}.\n\\end{equation}\n Now recalling that in the $n^{th}$\napproximating step the individual particle mass is scaled down by\n$n$ and recalling also Lemma~\\ref{LH1q}, one has that (for $n$\nlarge), $|{\\cal I}^n_t|\\approx n\\overline{X}^H_{\\infty}$ as\n$t\\to\\infty$. Putting this together with (\\ref{AA1}), one gets\nformally, that for large $n$, \\be{AA1b}\n \\|\\nu\\| \\frac{\\frac{1}{n}\\sum_{x\\in{\\cal I}_t^n}f(x)}\n  {\\langle \\nu,{\\mathbb E}^{x}[f(Y_t)]\\rangle}\\approx \\overline{X}^H_{\\infty},\\ \\\n \\ \\mathrm{as}\\ t\\to\\infty.\n\\end{equation}\nNote that in fact\n$$\\langle \\nu,{\\mathbb\nE}^{x}[f(Y_t)]\\rangle=\\widetilde{\\mathbf E}^{\\nu}\\langle\nX^H_t,f\\rangle=e^{-\\lambda_c t}\\,\\mathbf E^{\\mu}\\langle\nX_t,f\\phi_c\\rangle$$ ($\\nu=\\phi_c\\mu$). (The first equality can be\nshown for instance by taking first the particular case\n$\\nu=\\delta_x,\\ x\\in D$, and using that the two expectations satisfy\nthe same parabolic problem; then integrating with respect to\n$\\nu(\\mathrm{d}x)$.) Furthermore, passing to the limit (as\n$n\\to\\infty$) formally, the numerator of the fraction on the left\nhand side of (\\ref{AA1b}) becomes\n$$\\langle X^H_t,f\\rangle=e^{-\\lambda_c t}\\langle X_t,f\\phi_c\\rangle.$$\nHence, for the new test function $0\\not\\equiv\\hat f:=f\\phi_c\\in\nC_c^+(D)$,\n$$\\frac{ \\langle X_t,\\hat f\\rangle}{\\mathbf{E}^{\\mu}\\langle X_t,\\hat\nf\\rangle}\\approx\\frac{\\overline{X}^H_{\\infty}}{\\|\\nu\\|}.$$\n\n\\subsection{Main theorem}\nMaking  the  intuition of the previous subsection precise, we now\nstate our main result:\n\\beT{JC1}{Law of Large Numbers}\n\\label{LLNproduct-critical}\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nLet $f\\in C_c^+(D)$. If $f\\not\\equiv 0$ and $||\\mu||\\neq 0$, then\n\\be{A71} \\lim_{t\\to\\infty}\\, \\frac{\\langle X_t,f\\rangle}{{\\bf\nE}^{\\mu}\\langle X_t,f\\rangle}\n=\\frac{\\overline{X}^H_{\\infty}}{\\langle\\mu,\\phi_c\\rangle}, \\qquad\n\\mathrm{in }\\;\\,{\\bf P^{\\mu}}\\mathrm{-probability}.\n\\end{equation}\n\\end{itemize}\n\\end{theorem}\\smallskip\n\nComparing our theorem with (\\ref{A7}), we can now identify the\nlimiting distribution: $N^{^\\mu}=\\overline{X}^H_{\\infty}$ in law.\n\\begin{remark}\\rm\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nOne has to be a bit careful though when making heuristic\ninferences  using the particle picture as in the previous\nsubsection.\n\nObviously, the {\\it discrete} system in the $n^{th}$ approximation\nstep is so that $\\liminf_{t\\to\\infty}|{\\cal I}^n_t|\\ge 1$ under\nsurvival. That is, $\\liminf_{t\\to\\infty}\\frac{1}{n}|{\\cal I}^n_t|>0$\nunder survival. Recall that, heuristically,  (for $n$ large),\n$\\frac{1}{n}|{\\cal I}^n_t|\\approx \\overline{X}^H_{\\infty}$ as\n$t\\to\\infty$.\n\nOn the other hand, in the recent paper \\cite{Eng04} an example of a\n{\\it superprocess} is given that satisfies the conditions of our\nprevious theorem and for which\n$${\\bf P}^{\\mu}(\\overline{X}^H_{\\infty}=0\\mid S)>0,$$\nwhere $S$ is the event of survival, $S:=\\{\\|X_t\\|>0,\\ \\forall t\\ge\n0 \\}.\\hfill\\Diamond$\n\\end{itemize}\n\\end{remark}\n\\begin{conjecture}\\rm\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nWe conjecture that convergence in probability in\n (\\ref{A71}) can be replaced by almost sure\nconvergence.\n\\end{itemize}\n\\end{conjecture}\nWe close this section with a remark concerning an old result.\n\\begin{remark}\\rm\n\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nA simple case of a superdiffusion is when $D=\\mathbb R^d,\\ d\\ge\n1,\\ L=\\frac{1}{2}\\Delta$, with $\\alpha,\\beta$ positive constants\n(supercritical super-Brownian motion). Here $\\lambda_c=\\beta$ and\n$$\\frac{1}{2}\\Delta+\\beta-\\lambda_c= \\frac{1}{2}\\Delta.$$ Since\n$\\phi_c=\\tilde \\phi_c\\equiv 1,\\,d\\ge 1,$ the operator\n$\\frac{1}{2}\\Delta$ is either critical but not product-critical\n($d\\le 2$), or subcritical ($d\\ge 3$). Therefore this case is not\nincluded in our setup. On the other hand, the corresponding\n(Strong) Law of large Numbers is well known -- at least for the\ndiscrete particle systems. Using techniques from Fourier transform\ntheory, Watanabe (\\cite{Wat67}) proved SLLN for branching-Brownian\nmotion in $\\mathbb R^d$ and in certain subdomains of it. It is not\nclear however if his method can be generalized for more general\nbranching diffusions. $\\hfill\\diamond$\n\\end{itemize}\n\\end{remark}\n\\section{Proof of the result}\nThe  proof is  based on two observations. The first one is that\nthe problem can be formulated in terms of $X^H$, that is, one can\nreduce the problem to the investigation of a {\\it critical}\nsuperdiffusion with {\\it ergodic motion component} and {\\it\nexponentially decaying branching rate} (again, recall that $X^H$\nis an $(L+a\\frac{\\nabla \\phi_c}{\\phi_c}\\cdot \\nabla,0,\\alpha\n\\phi_c e^{-\\lambda_c t};D)$-superdiffusion).\n\nThe second one is that by considering some large time $t+T$ (where\nboth $t$ and $T$ are large), the changes of $\\overline{X}^H$ are\nnegligible after time $t$, while  the remaining time  $T$ is still\nlong enough to distribute the produced mass according to the ergodic\nflow given by the $H$-transformed migration.\n\nTo simplify  notation, we will write $W:=X^H$ (and, accordingly,\n$\\overline {W}_{\\infty}:=\\overline {X}^H_{\\infty}$). Denote\n$\\nu:=W_0$. By assumption, $\\|\\nu\\|=\\langle \\mu\n,\\phi_c\\rangle<\\infty.$ We need to show that for all $\\epsilon>0$,\n\\begin{equation}\\label{W}\n\\lim_{t\\to\\infty}{\\bf P}^{\\mu}\\Big(\\Big|\\frac{\\langle\nX_t,f\\rangle} {{\\bf E}^{\\mu}\\langle\nX_t,f\\rangle}-\\frac{\\overline{W}_{\\infty}}{\\|\\nu\\|}\\Big|>\\epsilon\\Big)=0.\n\\end{equation}\nRecall that $\\widetilde{{\\bf P}}$ denotes the probabilities with respect\nto the law of $W$.\nDenoting $\\phi_c^{-1}f=:g\\in\nC_c^+(D)$, we rewrite (\\ref{W}) as\n$$\\lim_{t\\to\\infty}\\widetilde{{\\bf P}}^{\\nu}\\Big(\\Big|\\frac{\\langle W_t,g\\rangle}\n{\\widetilde{{\\bf E}}^{\\nu}\\langle\nW_t,g\\rangle}-\\frac{\\overline{W}_{\\infty}}{\\|\\nu\\|}\\Big|>\\epsilon\\Big)=0.$$\nIt is easy to check that the limiting invariant density for the\ndiffusion corresponding to $L_0^{\\phi_c}$ is $\\phi_c\\tilde\\phi_c$\n(recall the normalization\n$\\int_D\\mathrm{d}x\\,\\phi_c(x)\\tilde\\phi_c(x)=1$). Since $Z:=\\widetilde{{\\bf E}}^{\\nu}W$ is just the deterministic\n$L_0^{\\phi_c}$-flow starting from $\\nu$, therefore\n$$\\lim_{t\\to\\infty}\\widetilde{{\\bf E}}^{\\nu}\\langle\nW_t,g\\rangle=||\\nu||\\cdot\\langle \\phi_c\\tilde\\phi_c ,g \\rangle,$$\nand consequently our statement is tantamount to saying that for\nall $\\epsilon>0$,\n$$\\lim_{t\\to\\infty}\\widetilde{{\\bf P}}^{\\nu}\\left(\\left|\\langle W_t,g\\rangle-\n\\langle \\phi_c\\tilde\\phi_c ,g\n\\rangle\\overline{W}_{\\infty}\\right|>\\epsilon\\right)=0.$$  Let\n$T>0$ and let $Z_{W_t}$ denote the deterministic\n$L_0^{\\phi_c}$-flow starting from the (random) measure $W_t$. Then\n\\begin{equation}\\label{three}\n\\widetilde{{\\bf P}}^{\\nu}\\big(\\big|\\langle W_{t+T},g\\rangle-\\langle\n\\phi_c\\tilde\\phi_c ,g\n\\rangle\\overline{W}_{\\infty}\\big|>\\epsilon\\big)\\le\nS_1(t)+S_2(t,T)+S_3(t,T),\n\\end{equation} where\n\\begin{align}\nS_1(t)&:=\\widetilde{{\\bf P}}^{\\nu}\\Big(\\big|\\langle \\phi_c\\tilde\\phi_c ,g\n\\rangle\\overline{W}_{\\infty}-\\langle \\phi_c\\tilde\\phi_c ,g\n\\rangle\\|W_{t}\\|\\big|>\\epsilon\/3\\Big)\\nonumber\\\\\nS_2(t,T)&:=\\widetilde{{\\bf P}}^{\\nu}\\Big(\\big|\\langle \\phi_c\\tilde\\phi_c ,g\n\\rangle\\|W_{t}\\|-\\langle\nZ_{W_t}(T),g\\rangle\\big|>\\epsilon\/3\\Big)\\nonumber\\\\\nS_3(t,T)&:=\\widetilde{{\\bf P}}^{\\nu}\\big(\\big|\\langle\nZ_{W_t}(T),g\\rangle-\\langle\nW_{t+T},g\\rangle\\big|>\\epsilon\/3\\big).\\nonumber\n\\end{align}\nTake $\\limsup_{t\\to\\infty}\\limsup_{T\\to\\infty}$ on both sides of\n(\\ref{three}).  We have\n$$\\limsup_{t\\to\\infty}\\widetilde{{\\bf\nP}}^{\\nu}\\left(\\left|\\langle W_{t},g\\rangle -\\langle\n\\phi_c\\tilde\\phi_c ,g\n\\rangle\\overline{W}_{\\infty}\\right|>\\epsilon\\right)\\le I+II+III,$$\nwhere\n\\begin{align}\nI&:= \\limsup_{t\\to\\infty}S_1(t)\\\\\nII&:=\\limsup_{t\\to\\infty}\\limsup_{T\\to\\infty}S_2(t,T)\\\\\nIII&:=\\limsup_{t\\to\\infty}\\limsup_{T\\to\\infty}S_3(t,T).\n\\end{align}\nNow, since $\\|W_t\\|\\to\\overline{W}_{\\infty}$ as $t\\to\\infty$\na.s.,\n$$I=\\lim_{t\\to\\infty}S_1(t)=0.$$\nAlso $ II=0,$ because for all fixed $t\\ge 0,\\\n\\lim_{T\\to\\infty}S_2(t,T)=0.$ (Indeed, for {\\it all}\n$\\omega\\in\\Omega$, $\\lim_{T\\to\\infty} \\langle\nZ_{W_t(\\omega)}(T),g\\rangle=\\langle \\phi_c\\tilde\\phi_c ,g\n\\rangle\\|W_{t}(\\omega)\\|$.) Therefore, if we show that\n\\begin{equation}\\label{S3}\nIII=0,\n\\end{equation} then we are done.\n\nIn order to do this, use  at time $t$ that $W$ is a\ntime-inhomogeneous Markov-process, and then apply Chebysev's\ninequality:\n\\begin{equation}\n\\label{MarkovChebysev}\n\\begin{aligned}\n   S_3(t,T)\n &=\n   \\widetilde{{\\bf E}}^{\\nu}{\\bf \\hat P}^{W_t}\\left(\\left|\\langle Z_{W_t}(T),\n   g\\rangle-\\langle W_{t+T},g\\rangle\\right|>\\epsilon\/3\\right)\n  \\\\\n &\\le\n   9 \\epsilon^{-2}\\,\\widetilde{{\\bf E}}^{\\nu} \\mathrm{\\hat{\\sigma}^2}_{W_t}\n   \\langle W_{T},g\\rangle,\n\\end{aligned}\n\\end{equation}\nwhere ${\\bf \\hat P}$ is the law of the $(L+a\\frac{\\nabla\n\\phi_c}{\\phi_c}\\cdot \\nabla,0,\\alpha \\phi_c e^{-\\lambda_c\n(t+s)};D)$-superdiffusion (here $t$ is fixed and $s>0$ is time)\nand $\\hat{\\sigma}^2$ denotes variance.\n\nLet us now recall  how the formulae for the first two moments of\n$\\langle W_{T},g\\rangle$ are obtained: by writing\n$u_{\\theta}(t,x)$ for the solution of the semilinear parabolic\nevolution equation (corresponding to the superprocess) with\ninitial value $\\theta g$, one differentiates (repeatedly) with\nrespect to $\\theta$ and sets $\\theta=0$.\n\nFor  time-homogeneous processes with constant branching rate  this\nis written down in detail in \\cite{Etheridge 02}, p.37. Since the\nderivation of these `moment formulae' only uses differentiation with\nrespect to $\\theta$ (and not $t$ or $x$), therefore the proof goes\nthrough for the more general setting where coefficients are\nspace-time-dependent.\n\nIn our case, from these moment formulae and from\n(\\ref{MarkovChebysev}), one obtains (recall also\n(\\ref{contraction})) that for all $T>0$,\n\\begin{equation}\n\\label{moments}\n\\begin{aligned}\n   S_3(t,T)\n &\\le\n   \\frac{9}{\\epsilon^{2}}\\cdot\\widetilde{\\mathbf{E}}^{\\nu}\\int_0^T\\mathrm{d}s\\,\n   2 e^{-\\lambda_c(t+s)}\n   \\big\\langle W_t,\\mathcal{S}^H_s\\big[\\alpha\\phi_c\n   \\big(\\mathcal{S}^H_{T-s}g\\big)^2\\big]\\big\\rangle\n  \\\\\n &\\le\n   \\frac{C\\cdot\\widetilde{{\\bf E}}^{\\nu}\\|W_t\\|}\n   {\\epsilon^{2}\\cdot \\lambda_c e^{\\lambda_c t}},\n\\end{aligned}\n\\end{equation}\n with\n$$C=C(\\|g\\|,\\|\\alpha\\phi_c\\|):=18\\,\\|\n\\alpha\\phi_c\\|\\cdot\\|g\\|^2.$$ (Note that we have an extra factor $2$\nrelative to \\cite{Etheridge 02} --- indeed in \\cite{Etheridge 02}\nthe nonlinear term in the semilinear parabolic evolution equation is\n$\\frac{1}{2}\\gamma u^2$.) Recall that $\\|W\\|$ is a $\\widetilde{{\\bf\nP}}^{\\nu}$-martingale with mean $\\|\\nu\\|$ and continue\n(\\ref{moments}) with\n$$=\\frac{C\\cdot\\|\\nu\\|}{\\epsilon^{2}\\cdot \\lambda_c e^{\\lambda_c t}}.$$\n Since this holds for\nall $T> 0$, thus $$\\limsup_{T\\to\\infty}S_3(t,T)\\le\n\\frac{C\\cdot\\|\\nu\\|}{\\epsilon^{2}\\cdot \\lambda_c e^{\\lambda_c\nt}}.$$ Letting $t\\to\\infty$, one obtains (\\ref{S3}), completing\nour proof. $\\qquad\\qquad\\square$\n\n\\begin{appendix}\n\\section{The behavior of the process in expectation}\n\\label{Pinproblem}\\rm For the cases when  $L+\\beta-\\lambda_c$ is\nsubcritical, or critical but $\\langle\n\\mathrm{d}x,\\phi_c\\tilde{\\phi_c}\\rangle=\\infty$, Theorem 7(b)(ii)\nin \\cite{Pin96} states that for $g\\in C_c^{+}$, \\be{A8}\n \\lim_{t\\to\\infty}   e^{-\\lambda_c t}{\\bf E}^{\\mu}[\\langle X_t,g\\rangle]=0.\n\\end{equation}\nNote, however, that in the proof there is a glitch: the proof\nsimply refers to Theorem 4.9.9 in \\cite{Pin95} which deals with\nthe product-critical case only, and is therefore not applicable\nfor the cases mentioned.\n\nNevertheless,  for the subcritical case, and for\n$\\mu\\in\\mathcal{M}_c(D)$, the statement can be verified by a very\nsimple argument as follows (cf. Theorem 4.9.1. in \\cite{Pin95}).\nFirst, note that by the first moment formula,\n$$e^{-\\lambda_c t}{\\bf E}^{\\mu}[\\langle\nX_t,g\\rangle]=\\langle\\mu ,[\\mathcal{T}_t g](x)\\rangle,$$  where\n$\\{\\mathcal{ T}_t\\}_{t\\ge 0}=\\{e^{-\\lambda_c t}\\mathcal{\nS}_t\\}_{t\\ge 0}$ denotes the semigroup corresponding to the\noperator $L+\\beta-\\lambda_c$ on $D$.\n\nMake an $h$-transform now: $$[\\mathcal{T}_t\ng](x)=h(x)[\\mathcal{T}^h_t (\\hat g)](x),$$ where $\\hat {g}:=g\nh^{-1}$, and pick an $h>0$ satisfying $(L+\\beta-\\lambda_c)h=0$, to\nreduce the problem to the proof of \\be{A8a} \\lim_{t\\to\\infty}\n\\langle  h\\mu, \\mathbb E^{x} [\\hat g(\\xi^h_t)]\\rangle= 0,\n\\end{equation}\n where $\\xi^h$ denotes the diffusion corresponding to\n$\\mathcal{T}^h$, that is, to the operator $L_{0}^{h}$ (defined\nanalogously to $L_{0}^{\\phi_c}$ in (\\ref{LH}) with $\\phi_c$\nreplaced by $h$) and $\\mathbb  E$ denotes the corresponding\nexpectation. (Of course, $h\\mu\\in\\mathcal{M}_c(D)$).\n\nFurthermore, it is enough to show the statement with\n$\\mu:=\\delta_x,\\ x\\in D$, because once we know that,  the general\nstatement follows by bounded convergence: $\\mathbb E^{x} [\\hat\ng(\\xi^h_t)]\\le \\|\\hat g\\|$ for all $x\\in D$ and $t\\ge 0$.\n\nIn \\cite{Pin95}, Chapter 4 it is shown that subcriticality is\ninvariant under $h$-transforms, and that the transience of a\ndiffusion is equivalent to the subcriticality of the corresponding\nelliptic operator. Therefore, in our case, it follows that $\\xi^h$\nis transient. Since $g$ is compactly supported, by an obvious\ncomparison argument, it is enough to show (\\ref{A8a}) with $g$\nreplaced by the indicator $1_{B}$, where $B$ is a ball. By\ntransience $1_{B}(\\xi^h_t)\\to 0\\ \\mathrm{as}\\ t\\to\\infty$ a.s.,\nand the statement follows from this and bounded convergence.\n\nSimilarly, the critical but non-product critical case can be reduced\nto the analogous (but much subtler) problem of showing (\\ref{A8a})\nfor a null-recurrent $\\xi^h$. (Cf. the well known analogous limit\ntheorem for countable state space Markov chains -- see e.g.\nProposition 5.3 and Corollary 6.39 in \\cite{KemSneKna76}.) In fact,\nas mentioned in the notes at the end of Chapter 4 in \\cite{Pin95},\nthis result is known in the case when $L$ is  {\\it symmetric}   with\nrespect to some reference measure $\\rho\\mathrm{d}x$ (see\n\\cite{ChK91} or \\cite{Sim93}). (Recall that $L$ is symmetric if and\nonly if $b=a\\nabla Q$ for some $Q\\in C^{2,\\eta}(D)$, $\\eta\\in\n(0,1]$, and in this case $L$ possesses a self-adjoint extension due\nto the Friedrichs extension theorem -- see Chapter 4 in\n\\cite{Pin95}.) Recently Pinchover completed the result by  proving\nit  for the general (non-selfadjoint) setting (see \\cite{Pinch04}).\n\nConsequently,   the $\\rho >\\lambda_c$ (over-scaling) part of\n(\\ref{A6}) is  immediate. One does not need however the above deep\nresult for the  $\\rho >\\lambda_c$ part. Here is a simple\nalternative proof:  using an h-transform with an $h>0$ satisfying\n$(L+\\beta-\\lambda_c)h=0$, the statement is equivalent to \\be{A8b}\n\\lim_{t\\to\\infty} e^{(\\lambda_c -\\rho)t}\\mathbb\nE^{x}[g(\\xi^h_t)]=0,\n\\end{equation}\nfor each $x\\in D$, which is true in virtue of the boundedness of\n$g$.\n\nThe $\\rho <\\lambda_c$ (under-scaling) part of (\\ref{A6}) is\nharder, and we are only able to provide the rigorous proof of the\nsomewhat weaker assertion:\n\\be{A6b}\n   \\limsup_{t\\to\\infty}\\,\\exp[-\\rho t]{\\bf E}^{\\mu}[\\langle X_t,g\\rangle]\n = \\infty.\n\\end{equation}\n\n\nTo this end, denote by $p^{L+\\beta}(t;x,\\cdot),\\ x\\in D,$ the\nkernel corresponding to the operator $L+\\beta$ and note that since\n$g$ is compactly supported, by an obvious comparison argument, it\nis enough to prove that \\be{A8d}\n   \\limsup_{t\\to\\infty} \\frac{1}{t}\\log \\int_B \\mu(\\mathrm{d}x)\\, p^{L+\\beta}(t;x,B)\n =\n   \\lambda_{c},\n\\end{equation}\nfor each $x\\in D$, and Borel set $B\\subset\\subset D$. Clearly, we\nmay assume that $\\|\\mu\\|=1$. To verify (\\ref{A8d}),  make again an\n$h$-transform with an $h>0$ satisfying $(L+\\beta-\\lambda_c) h=0$ on\n$D$. Then $L+\\beta$ transforms into $L_0^h+\\lambda_c$. Moreover,\nsince the generalized principal eigenvalue is invariant under\n$h$-transforms and since\n$\\lambda_c(L_0^h+\\lambda_c)=\\lambda_c(L_0^h)+\\lambda_c$, it follows\nthat $\\lambda_c(L_0^h)=0$. Let  $p_0^h(t,x,\\cdot)$ denote the\ntransition measures corresponding to $L_0^h$. Fix $x\\in D$ and\n$B\\subset\\subset D$. Since $h( \\cdot\n)h^{-1}(x)p^{L+\\beta}(t,x,\\cdot)=e^{\\lambda_c t}p_0^h(t,x,\\cdot)$\n(see Theorem 4.1.1. in \\cite{Pin95}), and since $h$ is bounded\nbetween two positive constants on $B$, we have \\be{A8e}\n   \\limsup_{t\\to\\infty} \\frac{1}{t}\\log\\int_B \\mu(\\mathrm{d}x)\\, p_0^h(t;x,B)\n =\n   \\limsup_{t\\to\\infty} \\frac{1}{t}\\log\\int_B \\mu(\\mathrm{d}x)\\, p^{L+\\beta}(t;x,B)-\\lambda_c.\n\\end{equation}\nSince $\\|\\mu\\|=1$ and $p_0^h(t;x,B)\\le 1$, the left hand side of\n(\\ref{A8e}) is non-positive, giving immediately \\be{A8f}\n   \\lambda_{+}(x,B)\n :=\n   \\limsup_{t\\to\\infty} \\frac{1}{t}\\log\\int_B \\mu(\\mathrm{d}x)\\, p^{L+\\beta}(t;x,B)\\le \\lambda_c.\n\\end{equation}\n Suppose now that $\\lambda_{+}(x,B)< \\lambda_c$ and pick\n\\be{A8g}\n   c\n \\in (-\\lambda_c,-\\lambda_{+}(x,B)).\n\\end{equation}\nThen by (\\ref{A8e}) and (\\ref{A8g}), along with Fubini's theorem,\none obtains \\be{A8h1} \\int_B \\mu(\\mathrm{d}x)\\,\\int_0^{\\infty} e^{c\nt}p^{L+\\beta}(t,x,B)\\, \\mathrm{d}t<\\infty.\n\\end{equation}\nHence, for almost every $x\\in B$, \\be{A8h}  G^{(L+\\beta+c)}(x,B)\n :=\\int_0^{\\infty} e^{c t}p^{L+\\beta}(t,x,B)\\,\n\\mathrm{d}t<\\infty.\n\\end{equation}\nIt follows from general theory then  that\n$e^{ct}p^{L+\\beta}(t,x',B')$ is in fact integrable for {\\it  all }\n$x'\\in D$ and $B'\\subset\\subset D$, that is, that  the operator\n$L+\\beta+c$ possesses a (minimal positive) {\\it Green's function} on\n$D$; however\n this contradicts to the  well known fact that an  operator  with positive generalized\nprincipal eigenvalue does not  possess a Green's function. (In our\ncase $\\lambda_c(L+\\beta+c)=\\lambda_c+c>0.$)\n\n\n\n\\section{The $H$-transform of superdiffusions}\n\\label{Htransformsection}\\rm This section treats a generalization of\nthe $h$-transform for superdiffusions introduced in \\cite{EngPin99}.\nThe $h$-transform was used in the proofs in \\cite{EngTur00}. The\nmethod used in the present paper however requires  the spatial\nfunction $0<h=h(x)$ to  be replaced by a space-time function\n$0<H=H(x,t)$. (The  reader should not confuse with the space-time\nharmonic transformation yielding a Girsanov-type change of measure\n-- see  e.g. \\cite{Ove94}.)\n\nWe start with the more general definition of a  time-inhomogeneous\nsuperdiffusion. Let $\\tilde{L}$ be an elliptic operator on\n$D\\times\\mathbb{R}^+$ of the form \\be{A3n}\n   \\tilde{L}:=\\frac{1}{2}\\nabla \\cdot \\tilde{a} \\nabla +\\tilde{b}\\cdot\\nabla\n\\end{equation}\nwhere the functions\n$\\tilde{a}_{i,j},\\tilde{b}_i:D\\times\\mathbb{R}^+\\to\\mathbb{R}$, $i,j=1,...,d$ are\n$C^{1,\\eta}(D)$  (for some $\\eta\\in (0,1]$) in the space, and\ncontinuously differentiable in the time coordinate. Moreover\nassume that the symmetric matrix $\\tilde{a}(x,t):=(a_{i,j}(x,t))$\nis  positive definite for all $x\\in D$ and $t\\in\\mathbb{R}^+$.\n\nIn addition, let $\\tilde{\\alpha},\\tilde{\\beta}:D\\times\\mathbb{R}^+\\to\\mathbb{R}$,\nbe $C^{\\eta}(D)$ in the space, and continuously differentiable in\nthe time coordinate. Finally assume that $\\tilde{\\alpha}$ is\npositive, and $\\tilde{\\beta}$ is bounded from above.\n\n\n\\beD{D2}{Time-inhomogeneous\n$(\\tilde{L},\\tilde{\\beta},\\tilde{\\alpha};D)$-superdiffusion} \\rm\\\n\\begin{itemize}\n\n\\item[{}] {\\bf (i)} The\n{\\rm$(\\tilde{L},\\tilde{\\beta},\\tilde{\\alpha};D)$-superdiffusion}\nis a  measure-valued (inhomogeneous) Markov process, $(X,{\\bf\nP}^{\\mu,r};\\,\\mu\\in{\\cal M}_f(D),r\\ge 0)$,  that is, a family\n$\\{{\\bf P}^{\\mu,r}\\}$ of probability measures where ${\\bf\nP}^{\\mu,r}$ is a probability on $C([r,\\infty) )$ and the family is\nindexed by ${\\cal M}_f(D))\\times [0,\\infty)$, such that the\nfollowing holds: for each $g\\in C_b^+(D)$ and $\\mu\\in{\\cal\nM}_f(D)$, \\be{A4c}\n   {\\bf E}^{\\mu,r}[\\exp-\\langle X_t,g\\rangle]=\\exp-\\langle\\mu,u(\\cdot,r;t,g)\\rangle,\n\\end{equation}\nwhere $u=u(\\cdot,\\cdot;t,g)$ is a particular  non-negative solution\nto the backward equation \\be{A5s}\n\\begin{aligned}\n   -\\partial_r u&=\\tilde{L}u+\\tilde{\\beta} u-\\tilde{\\alpha} u^2\\hspace{1cm}\n  \\mbox{in }D\\times (0,t),\n  \\\\\n   \\lim_{r\\uparrow t}u(\\cdot,r;t,g)&=g(\\cdot).\n\\end{aligned}\n\\end{equation}\n{\\bf (ii)} To determine the solution $u$ uniquely, use the\nequivalent {\\it forward} equation along with the minimality of the\nsolution: fix $t>0$ and introduce the `time-reversed' operator\n$\\hat{L}$ on $D\\times (0,t)$ by \\be{A3n2}\n   \\hat{L}:=\\frac{1}{2}\\nabla \\cdot \\hat{a} \\nabla\n   +\\hat{b}\\cdot\\nabla,\n\\end{equation}\nwhere, for $r\\in[0,t]$,\n$$\\hat a(\\cdot,r):=\\tilde a(\\cdot,t-r)\\ \\mathrm{and}\\ \\hat b(\\cdot,r):=\\tilde b(\\cdot,t-r);$$\nfurthermore let\n$$\\hat \\beta(\\cdot,r):=\\tilde \\beta(\\cdot,t-r)\\ \\mathrm{and}\\ \\hat \\alpha(\\cdot,r):=\n\\tilde \\alpha(\\cdot,t-r).$$ Consider now $v$, the {\\it minimal}\nnon-negative solution to the {\\it forward} equation\\be{A5s2}\n\\begin{aligned}\n   \\partial_r v&=\\hat{L}v+\\hat{\\beta} v-\\hat{\\alpha} v^2\\hspace{1cm}\n  \\mbox{in }D\\times (0,t),\n  \\\\\n   \\lim_{r\\downarrow 0}v(\\cdot,r;t,g)&=g(\\cdot).\n\\end{aligned}\n\\end{equation}\nThen $$u(\\cdot,r;t,g)=v(\\cdot,t-r;t,g).$$ (See also the remark\nfollowing this definition.)\n\\end{itemize}\n\\end{definition}\n\\begin{remark}[Minimal non-negative solutions for forward equations]\\begin{itemize}\n\n\\item[{}]\\rm\nIn the time-homogeneous case, minimal non-negative solutions for\nforward equations have been constructed in Appendix\nA in \\cite{EngPin99}, and in Section 2 in \\cite{EngPin03}\n--- the construction uses the approximation of $D$ by compactly embedded subdomains\nwith Dirichlet condition on their boundaries. The construction goes\nthrough for the time-inhomogeneous setting.\n\nIn \\cite{EngPin99}\n and \\cite{EngPin03} the time interval is $[0,\\infty)$ rather than\n $[0,t]$. However that does not make any difference --- in fact the\n solution on the infinite time interval can be defined by first\n working on finite time intervals and then showing that they can be\n taken arbitrarily large without having the solution blown up.\n \\end{itemize}\n \\end{remark}\n\nAs we will see in Lemma \\ref{LH1} (b), one way of defining a\ntime-inhomogeneous superdiffusion is to start with a\ntime-homogeneous one, and then to apply an `{\\it $H$-transform}'. In\ngeneral, the $H$-transform of a time-inhomogeneous superdiffusion is\ndefined as follows. Let $0<H\\in C^{2,\\eta}(D)\\times\nC^{1,\\eta}(\\mathbb R^+)$ and let $X$ be a\n$(\\tilde{L},\\tilde{\\beta},\\tilde{\\alpha};D)$-superdiffusion. We\ndefine a new process $X^H$ by\n\\begin{equation}\\label{newprocess}\n  X_t^H:=H(\\cdot,t)X_t\\quad \\left(\\mbox{that is,}\\\n  \\frac{\\mathrm{d}X_t^H}{\\mathrm{d}X_t}=H(\\cdot,t)\\right), \\quad\n  t\\geq 0.\n\\end{equation}\nThis way one obtains a new superdiffusion, which, in general, {\\it\nis not finite measure-valued} but only $\\sigma$-finite\nmeasure-valued. That is, if $\\mathcal{M}(D)$ denotes the family of\nall (finite or infinite) measures on $D$, then\n$$X^H_t\\in\\mathcal{M}_H^{(t)}(D):=\\{\\nu\\in \\mathcal{M}(D)\\mid H(\\cdot,t)^{-1}\\nu\\in\n\\mathcal{M}_f(D)\\}$$ (c.f. \\cite{EngPin99}, p. 688.)\n\n\nIn \\cite{EngPin99}, Section 2, it was shown, that, from an\nanalytical point of view, the (spatial) $h$-transform of the\nsuperdiffusion is given by a certain transformation of the\ncorresponding semilinear operator. This remains the case for the\nspace-time $H$-transform.\n\\beL{LH1}{H-transform}\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nLet $X^H$ be defined by (\\ref{newprocess}). Then \\item[(a)] $X^H$\nis a $(\\tilde{L}+\\tilde{a}\\frac{\\nabla H}{H}\\cdot\\nabla,\n  \\tilde{\\beta}+\\frac{\\tilde{L}H}{H}+\\frac{\\partial_rH}{H},\n  \\tilde{\\alpha}H;D)$-superdiffusion.\n\\item[(b)] In particular, if $X$ is a time-homogeneous\n  $(L,\\beta,\\alpha;D)$-superdiffusion, and $H$ is of the form\n\\be{H}\n   H(x,t):=e^{-\\lambda_c t}h(x),\n\\end{equation}\nwhere $\\lambda_c$ is the principal eigenvalue of $L+\\beta$, and\n$h$ is a positive solution of $(L+\\beta)h=\\lambda_c h$, then $X^H$\nis a $(L+a\\frac{\\nabla h}{h}\\cdot \\nabla,0,\\alpha h e^{-\\lambda_c\nt};D)$-superdiffusion.\n\\end{itemize}\n\\end{lemma}\n\\begin{remark}[Unbounded $\\tilde \\beta$'s]\\begin{itemize}\n\\item[{}] \\hspace{2cm}\\rm\n\nAs it is already the case with the spatial $h$-transform for\nsuperdiffusions, it is possible that the coefficient $\\tilde\n\\beta$ transforms into a new coefficient that is no longer\nbounded. In fact this can be the very definition of\nsuperdiffusions for certain unbounded $\\tilde \\beta$'s (see\n\\cite{EngPin99}, Section 2 for explanation).\n\n\\end{itemize}\\end{remark}\n\n {\\bf Proof of\nLemma~\\ref{LH1}. } In order to avoid minor technical\ninconveniences, we implement the method in \\cite{EngPin99} (see\nthe second paragraph on p. 689.). Namely, we use that the Laplace\ntransition functional restricted to the family of measures ${\\cal\nM}_c(D)$ and the family of functions $C^+_c(D)$ uniquely\ndetermines a measure-valued Markov process, and we choose working\nwith these smaller spaces rather than replacing ${\\cal M}_f(D)$\nand $C^+_b(D)$ by $H$-dependent spaces.\n\nPick $\\nu\\in{\\cal M}_c(D)$, and $f\\in C_c(D)$. Define\n$\\mu^{(s)}:=\\nu\/H(\\cdot,s)\\in{\\cal M}_c(D)$, and\n$g^{(t)}(\\cdot):=H(\\cdot,t)f(\\cdot)\\in C^+_c(D)$. Obviously,\n \\be{H3b}\n  \\widetilde{{\\bf E}}^{\\nu,s}\\left[\\exp-\\langle X^H_t,f\\rangle\\right]\n  ={\\bf E}^{\\mu^{(s)},s}\\left[\\exp-\\langle\n  X_t,g^{(t)}\\rangle\\right].\n\\end{equation}\nBy the log-Laplace equation (\\ref {A4c}), we can continue with\n$$=\\exp-\\left\\langle\\frac{\\nu}{H(\\cdot,s)},u(\\cdot,s;t,g^{(t)})\n\\right\\rangle.$$ Consider the operator  $${\\cal A}:\nC^{2,\\eta}(D)\\times C^{1,\\eta}(\\mathbb R^+)\\mapsto C^{\\eta}(D)\\times\nC^{\\eta}(\\mathbb R^+)$$ defined by \\be{H4a}\n\\begin{aligned}\n   {\\cal A}(u)&:=\\partial_s\n   u+(\\tilde{L}+\\tilde{\\beta})u-\\tilde{\\alpha}u^2.\n\\end{aligned}\n\\end{equation}\nDefine the $H$-transformed operator  ${\\cal A}^H$ in the usual\n way:\n\\begin{equation}\\label{conjugate}\n {\\cal A}^H(u):=\n   \\frac{1}{H}{\\cal A}(Hu).\n\\end{equation}\n    Then a direct computation gives\n\\be{H5}\n\\begin{aligned}\n   {\\cal A}^H(u):=\n   \\frac{\\partial_sH}{H}u+\\partial_su+\\tilde{L}u+\\tilde{a}\\frac{\\nabla H}{H}\\cdot\n   \\nabla u+\\tilde{\\beta}u+\\frac{\\tilde{L}H}{H}u-\\tilde{\\alpha}Hu^2.\n\\end{aligned}\n\\end{equation}\nAnother, trivial computation yields that if $$\nv(\\cdot,\\cdot;t,f):=u(\\cdot,\\cdot;t,H(\\cdot,t)f)\/H(\\cdot,t),$$ then\n$v(\\cdot,\\cdot;t,f)$ is the solution in (\\ref{A5s}) with ${\\cal A}$\nreplaced by ${\\cal A}^H$, and with the property in Definition\n\\ref{D2}(ii).\n Thus the quadruple\n$(\\tilde{L},\\tilde{\\beta},\\tilde{\\alpha};D)$ transforms into the\nquadruple given in part (a).\\smallskip\n\n\\medskip\n\\noindent Part (b) is straightforward computation. $\\hfill\\Box}\\newcommand{\\bi}{\\bigskip$\n\\begin{remark}\\rm\n\\begin{itemize}\n\\item[{}] \\hspace{2cm}\n\nIt is precisely equation (\\ref{conjugate}) that justifies the name\n`$H$-transform'; the transformation on the semilinear operator works\nthe same way as Doob's $h$-transform would work on a linear\noperator.\n\\end{itemize}\n\\end{remark}\n\\end{appendix}\n\n\\noindent {\\bf Acknowledgment. }\\quad\nThe authors are grateful to the anonymous referee for\nseveral comments and suggestions\nthat helped to improve the presentation and correctness\nof the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Proof of Theorem \\ref{theorem_equilibrium}}\\label{proof_theorem}\n\\begin{enumerate}\n\\item[(a)] In regime $A$, since $p^\\t_l >\\Delta c^{\\theta}(0)$, from \\eqref{UlL}, we have $\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\H)] \\leq -c_{\\H}^{\\theta}(\\sigma^{t*}_{l})-p^\\t_l < -c_{\\L}^{\\theta}(\\sigma^{t*}_{l}) =\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\L)]$. Therefore, from \\eqref{travel_low_deviate}, we must have $\\sigma^{t*}_{l}=0$. From \\eqref{bayes}, we can check that $\\beta^{*}(h|\\H)=1$ and $\\beta^{*}(l|\\H)=0$. Following from Lemma \\ref{BR}, $\\sig^{d*}_\\H=0$. Thus, the PBE in \\eqref{eq_A} is the unique equilibrium. \n\\item[(b)] In regime $B$, we first prove by contradiction that $\\sigma^{t*}_{l} \\in (0, 1)$. Assume that $\\sigma^{t*}_{l}=0$, then from \\eqref{eq_operator_rational} and \\eqref{bayes}, $\\beta^{*}$ and $\\sig^{d*}_\\L$ must be in \\eqref{eq_A}. Then, from \\eqref{UlL}, $\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\L)]=-c_{\\L}^{\\theta}(0)$. However, if type $l$ agents deviate to choose $\\H$, the utility is $-c_{\\H}^{\\theta}(0)-p^\\t_l$. Since in regime $B$, $p^\\t_l <\\Delta c^{\\theta}(0)$, type $l$ agents has incentive to deviate to $\\H$, which contradicts $\\sigma^{t*}_{l}=0$. Additionally, from Proposition \\ref{no_pool}, $\\sigma^{t*}_{l}<1$. Therefore, in this regimes $\\sigma^{t*}_{l} \\in (0,1)$, i.e. type $l$ agents take both servers in equilibrium, which implies the follows:\n\\begin{align}\\label{equal_l}\n\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\L)]=\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\H)].\n\\end{align}\nFurthermore, there are three cases for $\\sig^{d*}_\\H$: $\\sig^{d*}_\\H=0$, $\\sig^{d*}_\\H \\in (0, 1)$ and $\\sig^{d*}_\\H=1$. It turns out that these three cases correspond to subregime $\\B_1$, $\\B_2$ and $\\B_3$, respectively.\n\n\\begin{enumerate}\n\\item[($\\B_2$)] $\\sig^{d*}_\\H \\in (0, 1)$:\nIn this case, from Lemma \\ref{BR}, we know that $\\beta^{*}(l|\\H)$ must be $p^\\d\/F_l$, and  $\\sigma^{t*}_{l}=\\widehat{\\sigma}_{\\l}^\\t$ in \\eqref{siglbar} is the unique equilibrium strategy.\nAdditionally, from \\eqref{equal_l}, the operator's strategy $\\sig^{d*}_\\L$ should satisfy $-c_{\\L}^{\\theta}(\\sigma^{t*}_{l})=-c_{\\H}^{\\theta}(\\sigma^{t*}_{l})-p^\\t_l-F_l \\sig^{d*}_\\L$. Thus, $\\sig^{d*}_\\L$ is in \\eqref{eq_B_2}. Furthermore, we can check that when $p^\\d$ and $p^\\t_l$ satisfy \\eqref{B_2_condition}, the strategies and the beliefs in \\eqref{eq_B_2_general} are all feasible. Therefore, PBE in \\eqref{eq_B_2_general} is the unique equilibrium.\n\n\\item[($\\B_1$)] $\\sig^{d*}_\\H=0$: In this case, From \\eqref{equal_l}, we must have $\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\H)]=-c_{\\H}^{\\theta}(\\sigma^{t*}_{l})-p^\\t_l =-c_{\\L}^{\\theta}(\\sigma^{t*}_{l})=\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\L)]$, which leads to $\\Delta c^{\\theta}(\\sigma^{t*}_{l})=p^\\t_l$. From \\eqref{bayes}, $\\beta^{*}$ is in \\eqref{b_B_1}. \n\nWe now argue that the strategy in \\eqref{eq_B_1_general} is indeed a PBE if the cost parameters satisfy \\eqref{B1}. We have argued that the condition $p^\\t_l <\\Delta c^{\\theta}(0)$ is needed to ensure that $\\sigma^{t*}_{l} \\in (0, 1)$. \n\nIf $p^\\d>(1-\\theta)F_l$, then from Lemma \\ref{BR}, we know that $\\sig^{d*}_\\H=0$. If $p^\\d<(1-\\theta)F_l$, then again from Lemma \\ref{BR}, as long as $\\sigma^{t*}_{l} < \\widehat{\\sigma}_{\\l}^\\t$, $\\sig^{d*}_\\H=0$. Since $\\Delta c^{\\theta}(\\sigma^t_l)$ decreases in $\\sigma^t_l$ and $\\Delta c^{\\theta}(\\sigma^{t*}_{l})=p^\\t_l$, we must have $p^\\t_l =\\Delta c^{\\theta}(\\sigma^{t*}_{l})> \\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$, which leads to constraints in \\eqref{B1}. \n\n\n\n\n\n\n\\item[($\\B_3$)] $\\sig^{d*}_\\H =1$:\nIn this case, from \\eqref{equal_l}, we obtain $\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\L)]=-c_{\\L}^{\\theta}(\\sigma^{t*}_{l})=-c_{\\H}^{\\theta}(\\sigma^{t*}_{l})-p^\\t_l-F_l=\\mathbb{E}_{\\sigma^{*}}[U^\\t_l(\\H)]$. Therefore, $\\sigma^{t*}_{l}$ satisfies $\\Delta c^{\\theta}(\\sigma^{t*}_{l})=p^\\t_l+F_l$. From \\eqref{bayes}, $\\beta^{*}$ is obtained from \\eqref{b_B_3}. To ensure that the action $\\mathrm{I}$ is a dominant strategy for the operator, from Lemma \\ref{BR}, we need $p^\\d<(1-\\theta)F_l$, and $\\sigma^{t*}_{l}>\\widehat{\\sigma}_{\\l}^\\t$. Besides, $\\Delta c^{\\theta}(\\sigma^{t*}_{l})$ decreases in $\\sigma^{t*}_{l}$ and $\\Delta c^{\\theta}(\\sigma^{t*}_{l})=p^\\t_l+F$, we can conclude that $p^\\t_l+F=\\Delta c^{\\theta}(\\sigma^{t*}_{l})<\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$, i.e. $p^\\t_l$ satisfies \\eqref{condition_B_3}. \\QEDA\n\\end{enumerate} \n\\end{enumerate}\n\\end{appendices}\n\n\n\n\\section*{Acknowledgments}\nThis work was supported in part by the Singapore National Research Foundation through the Singapore MIT Alliance for Research and Technology (SMART) Center for Future Mobility (FM), and US National Science Foundation (NSF) grants.\n\n\n\\bibliographystyle{ieeetr}\n\n\n\\section{Concluding remarks}\nIn this article, we introduced a signaling game to study the effects of operator\ninspection strategy on the strategic misbehavior in a V2I-based highway system. Our model captures three key features: (a) Travelers' ability to exploit the vulnerabilities of V2I communications; (b) The incomplete and asymmetric information on the part of operator resulting from lack of direct observability of travelers' true type; (c) Congestion externality on the compliant travelers resulting from misbehavior. We provided a complete characterization of PBE, and derived comparative statics with respect to practically relevant parameters such as fine, costs of multi-priority lanes, and technological costs of inspection and misbehavior. These results suggest guidelines for the design and deployment of inspection technologies for smart transportation systems, and are relevant to a class of strategic integrity attacks, where deterrence via inspection is socially desirable.\n\n\n\\section{Equilibrium Characterization}\\label{sec_equilibrium}\nIn this section, we characterize the PBE of the signaling game. In Sec. \\ref{properties}, we show three properties of PBE that are crucial for equilibrium analysis. In Sec. \\ref{regimes}, we focus on analyzing the equilibrium regimes, where the properties of PBE are qualitatively distinct. \n\n\\subsection{General properties of PBE}\\label{properties}\nFrom (A1), we know that the costs of both servers are finite when no agent misbehaves. The following lemma guarantees stability in PBE, i.e. the costs of both servers are also finite in equilibrium. \n\\begin{lemma}\\label{finite_equilibrium}\nIn any PBE, $c_{\\H}^{\\theta}(\\sigma^{t*})<\\infty$ and $c_{\\L}^{\\theta}(\\sigma^{t*})<\\infty$. \n\\end{lemma}\n\\begin{proof}\nWe prove by contradiction. If $c_{\\H}^{\\theta}(\\sigma^{t*})=\\infty$, the aggregate amount of agents taking server $\\H$ in equilibrium must be higher than that without misbehavior. Hence, the amount of agents on server $\\L$ is lower than that without misbehavior, which ensures $c_{\\L}^{\\theta}(\\sigma^{t*})<\\infty$. Given any operator's strategy $\\sigma^d \\in [0,1]$, from \\eqref{expected_cost_agent}, we must have $\\mathbb{E}_{\\sigma^{*}}[U^t_h(\\H)] < \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\L)]$ and $\\mathbb{E}_{\\sigma^{*}}[U^t_l(\\H)]< \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\L)]$. From \\eqref{travel_high} and \\eqref{travel_low_deviate}, we have $\\sigma^{t*}_{h}=1$ and $\\sigma^{t*}_{l}=0$, i.e. no agents use server $\\H$ in equilibrium. This contradicts the claim that $c_{\\H}^{\\theta}(\\sigma^{t*})=\\infty$. Analogously, we argue that $c_{\\L}^{\\theta}(\\sigma^{t*})<\\infty$.  \n\\end{proof}\nThe next proposition shows that type $h$ agents do not misbehave in equilibrium. Consequently, the operator does not inspect agents that choose to access the server $\\L$. \n\\begin{proposition}\\label{h_non_deviate}\nIn any PBE, $(\\sigma^{*}, \\beta^{*})$ satisfies:\\vspace{-0.1cm}\n\\[\\sigma^{t*}_{h}=0, \\quad \\sig^{d*}_\\L=0, \\quad \\beta^{*}(\\l|\\L)=1, \\quad \\beta^{*}(\\h|\\L)=0.\\]\n\\end{proposition}\n\\begin{proof}\nWe first prove $\\sigma^{t*}_{h}=0$ by contradiction. Assume that there exists a PBE such that $\\sigma^{t*}_{h}>0$, i.e. there exists a fraction of type $h$ agents using server $\\L$.  From \\eqref{UhL} and \\eqref{travel_high_deviate}, we must have $-c_\\L^\\theta(\\sigtwe)-p^\\t_h \\stackrel{\\eqref{UhL}}{\\geq} \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\L)] \\stackrel{\\eqref{travel_high_deviate}}{\\geq} \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\H)]\\stackrel{\\eqref{UhL}}{=} -c_\\H^\\theta(\\sigtwe)$. Thus, $c_\\L^\\theta(\\sigtwe) \\leq c_\\H^\\theta(\\sigtwe) -p^\\t_h$. Since $p^\\t_h \\geq 0$, $c_\\L^\\theta(\\sigtwe) \\leq c_\\H^\\theta(\\sigtwe)$. From \\eqref{UlL}, we have $\\mathbb{E}_{\\sigma^{*}}[U^t_l(\\L)] \\stackrel{\\eqref{UlL}}{=} -c_\\L^\\theta(\\sigtwe) \\geq -c_\\H^\\theta(\\sigtwe) \\stackrel{\\eqref{UlL}}{>} \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\H)]$. Hence, from \\eqref{travel_low_deviate}, we must have $\\sigma^{t*}_{l}=0$, i.e. no agents of type $l$ take server $\\H$. Additionally, since $c_\\L(\\sigt)$ is increasing in $\\sigma^t_h$ and decreasing in $\\sigma^t_l$, when $\\sigma^{t*}_{h}>0$ and $\\sigma^{t*}_{l}=0$, we have $c_\\L^\\theta(\\sigtwe) > c_{\\L}^{\\theta}(0,0)$. Analogously, $c_\\H(\\sigt)$ is increasing in $\\sigma^t_l$ and decreasing in $\\sigma^t_h$, and thus $c_\\H^\\theta(\\sigtwe) < c_{\\H}^{\\theta}(0,0)$.  Consequently, we derive $c_{\\H}^{\\theta}(0,0) > c_\\H^\\theta(\\sigtwe) \\geq c_\\L^\\theta(\\sigtwe) > c_{\\L}^{\\theta}(0,0)$, which contradicts (A1). Therefore, $\\sigma^{t*}_{h}=0$. \n\nNext, from \\eqref{bayes}, we can check that the belief updated by Bayes' rule satisfies $\\beta^{*}(\\l|\\L)=1$ and $\\beta^{*}(\\h|\\L)=0$. \n\nFinally, since $\\beta^{*}(l|\\L)=1$ implies that only type $l$ agents take server $\\L$. From \\eqref{cost_L}, the action $\\mathrm{I}$ is strictly dominated by the action $\\mathrm{N}$. Hence, $\\sig^{d*}_\\L=0$.  \\end{proof}\n\nIn addition, the next proposition ensures that not all type $l$ agents misbehave in equilibrium. \n\\begin{proposition}\\label{no_pool}\nIn any PBE, $\\sigma^{t*}_{l}<1$. \n\\end{proposition}\n\\begin{proof}\nAgain we prove this claim by contradiction. Assume that $\\sigma^{t*}_{l}=1$, i.e. all the agents of type $l$ uses server $\\H$. From Proposition \\ref{h_non_deviate}, agents of type $h$ do not use server $\\L$ in equilibrium. Therefore, in PBE, no agents use server $\\L$. From (A1), we know that $\\mathbb{E}_{\\sigma^{*}}[U^t_l(\\L)]=-c_{\\L}^{\\theta}(0,1)>-c_{\\H}^{\\theta}(0,1) \\geq \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\H)]$, which contradicts the equilibrium condition in \\eqref{travel_low_deviate}. Hence, $\\sigma^{t*}_{l}<1$.   \\end{proof}\n\nFrom Propositions \\ref{h_non_deviate} -- \\ref{no_pool}, we can conclude that both servers are used in equilibrium. This implies that ``pooling'' equilibrium, in which agents of different types send identical signals, does not exist in our game. Consequently, we can drop $\\sigma^{t*}_{h}$, $\\beta^{*}(\\cdot|\\L)$ and $\\sig^{d*}_\\L$ in our further analysis. For simplicity, we abuse the notation and use $c_{\\H}^{\\theta}(\\sigma^t_l)$ (resp. $c_{\\L}^{\\theta}(\\sigma^t_l)$) to denote the cost of server $\\H$ (resp. $\\L$) when type $l$ agents' strategy is $\\sigma^t_l$, and $\\sigma^t_h=0$. Additionally, we define $\\Delta c^{\\theta}(\\sigma^t_l)$ as the cost difference between $\\L$ and $\\H$ when the strategy of type $l$ agents is $\\sigma^t_l$: \n\\begin{align*}\n\\Delta c^{\\theta}(\\sigma^t_l) \\deleq c_{\\L}^{\\theta}(\\sigma^t_l)-c_{\\H}^{\\theta}(\\sigma^t_l).\n\\end{align*} \nThe function $\\Delta c^{\\theta}(\\sigma^t_l)$ evaluates the incentive of a type $l$ agent to misbehave given that the fraction of misbehaving type $l$ agents is $\\sigma^t_l$. From Assumptions (A2)-(A3), we know that this incentive strictly decreases in type $l$'s misbehavior rate $\\sigma^t_l$ and the fraction of type $h$ agents $\\theta$. We can thus properly define the function $\\(\\dc\\)^{-1}$, which is the inverse of $\\Delta c^{\\theta}$.\n\n\\subsection{Equilibrium regimes}\\label{regimes}\nWe now provide a full characterization of PBE. From Propositions \\ref{h_non_deviate}-\\ref{no_pool}, we know that in general, there are two possible cases in equilibrium: In the first case, no agent of type $l$ takes server $\\H$, which means misbehavior is completely deterred; and in the other case, a fraction of type $l$ agent population takes server $\\H$. Indeed, we find that there exist two equilibrium regimes, each corresponding to one of the two cases. Furthermore, the second regime (i.e. a positive fraction of type $l$ agents take server $\\H$) can be divided into three subregimes depending on whether the inspection rate of the operator on server $\\H$ is zero, positive or one. \n\nBefore presenting the equilibrium structure, we introduce the following misbehavior threshold rate:\n\\begin{align}\\label{siglbar}\n\\widehat{\\sigma}_{\\l}^\\t \\deleq \\frac{p^\\d \\theta}{(1-\\theta) (F_l-p^\\d)}.\n\\end{align}\nNote that the threshold $\\widehat{\\sigma}_{\\l}^\\t$ is dependent on the inspection cost $p^\\d$, the fine $F_l$, and the relative class size $\\theta$.\n\nThe next lemma provides the best response correspondence $\\sig^{d*}_\\H$ in equilibrium.\n\\begin{lemma}\\label{BR}\nGiven any PBE, if $0<p^\\d \\leq (1-\\theta) F_l$, and \n\\begin{itemize}\n\\item[-] If $\\sigma^{t*}_{l}<\\widehat{\\sigma}_{\\l}^\\t$, then $\\beta^{*}(l|\\H)<p^\\d\/F_l$ and $\\sig^{d*}_\\H=0$\n\\item[-] If $\\sigma^{t*}_{l}=\\widehat{\\sigma}_{\\l}^\\t$, then $\\beta^{*}(l|\\H)=p^\\d\/F_l$ and $\\sig^{d*}_\\H\\in [0,1]$\n\\item[-] If $\\sigma^{t*}_{l}>\\widehat{\\sigma}_{\\l}^\\t$, then $\\beta^{*}(l|\\H)>p^\\d\/F_l$ and $\\sig^{d*}_\\H=1$.\n\\end{itemize}\nAdditionally, if $p^\\d>(1-\\theta) F_l$, then $\\sig^{d*}_\\H=0$.\n\\end{lemma}\n\\begin{proof}\nFirst, we can check that if $0 < p^\\d \\leq (1-\\theta) F_l$, then $\\widehat{\\sigma}_{\\l}^\\t \\in [0, 1]$. From \\eqref{bayes}, we know that if $\\sigma^{t*}_{l}=\\widehat{\\sigma}_{\\l}^\\t$, then $\\beta^{*}(l|\\H)= p^\\d\/F_l$. In this case, $-p^\\d+F_l \\beta^{*}(l|\\H)=0$, and thus any $\\sig^{d*}_\\H \\in [0,1]$ maximizes $\\mathbb{E}_{\\sig}[U_\\H^d|\\beta^{*}]$ in \\eqref{cost_H}. Additionally, since $\\beta^{*}(l|\\H)$ increases in $\\sigma^{t*}_{l}$, if $\\sigma^{t*}_{l}<\\widehat{\\sigma}_{\\l}^\\t$, we must have $\\beta^{*}(l|\\H)< p^\\d\/F_l$. Consequently, $-p^\\d+F_l \\beta^{*}(l|\\H)<0$, and from \\eqref{cost_H} and \\eqref{eq_operator_rational}, $\\sig^{d*}_\\H=0$. The case for $\\sigma^{t*}_{l}>\\widehat{\\sigma}_{\\l}^\\t$ can be argued analogously. \n\nAdditionally, we argue for the case in which $p^\\d>(1-\\theta)F_l$. Following from \\eqref{bayes} and the fact that $\\sigma^{t*}_{l} \\in [0, 1]$ as well as $\\sigma^{t*}_{h}=0$, we have $\\beta^{*}(l|\\H) =1-\\frac{\\theta}{\\theta+(1-\\theta) \\sigma^{t*}_{l}} \\leq 1-\\theta.$ Hence, if $p^\\d>(1-\\theta)F_l$, then $-p^\\d + F_l \\beta^{*}(l|\\H) <0$. From \\eqref{cost_H} and \\eqref{eq_operator_rational}, we know that $\\sig^{d*}_\\H=0$. \n\\end{proof}\n\n\nLemma \\ref{BR} shows that the probability of detecting a misbehavior on server $\\H$ is no higher than $(1-\\theta)$, which is achieved only when all type $l$ agents misbehave. Therefore, the maximum expected fine is $(1-\\theta)F_l$. If the inspection cost is higher than the maximum expected fine, then the operator does not inspect any agent. On the other hand, if the inspection cost is lower than the maximum expected fine, then following the belief update rule \\eqref{bayes}, $\\widehat{\\sigma}_{\\l}^\\t$ leads to the belief $\\beta^{*}(l|\\H)=p^\\d\/F_l$, which is the threshold belief such that the operator is indifferent between the action $\\mathrm{I}$ and $\\mathrm{N}$ in equilibrium. If $\\sigma^{t*}_{l}$ is higher (resp. lower) than $\\widehat{\\sigma}_{\\l}^\\t$, then the operator inspects the agents taking the server $\\H$ with probability one (resp. zero). \n\nAdditionally, note that as the fraction of $h$ type goes to zero (i.e. $\\theta \\to 0$), the threshold $\\widehat{\\sigma}_{\\l}^\\t \\to 0$, which implies that the defender will tend to inspect with probability 1. This is intuitive because when the fraction of type $h$ is small, even if the misbehavior rate of type $l$ agent is low, the operator still has a high chance of detecting a misbehavior by inspecting agents on the server $\\H$. \n\nWe next introduce the equilibrium regimes in terms of the misbehavior cost $p^\\t_l$ and the inspection cost $p^\\d$:\\footnote{Due to space limitations, we only discuss generic cases, where the game parameters are in the interior of each regime. }\n\\begin{enumerate}\n\\item In regime $A$,  $p^\\t_l$ satisfies $p^\\t_l >\\Delta c^{\\theta}(0)$.\n\\item In regime $B$, $p^\\t_l$ satisfies $p^\\t_l <\\Delta c^{\\theta}(0)$.\nThere are three subregimes of regime $B$. \\\\\nSubregime $\\B_1$: \n\n\\begin{footnotesize}\n\\begin{equation}\\label{B1}\n\\begin{split}\n&\\left\\{\\(p^\\t_l, p^\\d\\) \\left\\vert \n\\begin{array}{l}\n \\max \\{\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t), 0\\} < p^\\t_l <  \\Delta c^{\\theta}(0), \\text{and}\\\\\n 0<p^\\d<(1-\\theta)F_l\n \\end{array}\n \\right.\\right\\} \\bigcup \\left\\{\\(p^\\t_l, p^\\d\\) \\left\\vert \n\\begin{array}{l}\n0 < p^\\t_l <  \\Delta c^{\\theta}(0), \\text{and}\\\\\np^\\d> (1-\\theta)F_l\n \\end{array}\n \\right.\\right\\}\n\\end{split}\n\\end{equation}\n\\end{footnotesize}Subregime $\\B_2$:\n\\begin{footnotesize}\n\\begin{align}\\label{B_2_condition}\n\\noindent\\left\\{\\(p^\\t_l, p^\\d\\) \\left\\vert\\begin{array}{l}\n \\max \\{\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-F_l, 0\\} < p^\\t_l\\\\\n  <  \\max\\{\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t), 0\\}, \\text{and}\\\\\n  0<p^\\d<(1-\\theta)F_l.\n  \\end{array}\n  \\right.\\right\\}\n\\end{align}\n\\end{footnotesize}Subregime $\\B_3$:\n\\begin{footnotesize}\n\\begin{align}\\label{condition_B_3}\n\\left\\{\\(p^\\t_l, p^\\d\\) \\left\\vert \n\\begin{array}{l}\n 0< p^\\t_l < \\max \\{\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-F_l, 0\\},\\text{and}\\\\\n 0<p^\\d<(1-\\theta)F_l.\n \\end{array} \\right.\\right\\}\n\\end{align}\n\\end{footnotesize}\n\\end{enumerate}\nNote that the boundaries of subregimes are non-linear since the threshold $\\widehat{\\sigma}_{\\l}^\\t$ is non-linear in the inspection cost $p^\\d$. The interpretations of regime boundaries will be straightforward after we present the PBE in each regime. Note that the regime A, and subregimes B1 and B2 are non-empty, but subregime $\\B_3$ can be empty if the fine $F_l$ is sufficiently high.\n\n\n\nWe are now ready to fully characterize the PBE.\n\\begin{theorem}\\label{theorem_equilibrium}\nPBE is unique in each regime, and can be written as follows:\n\\begin{enumerate}\n\\item[(a)] Regime $A$: \n\\begin{footnotesize}\n\\begin{align}\\label{eq_A}\n\\sigma^{t*}_{l}=0, \\quad \\sig^{d*}_\\H=0, \\quad \\beta^{*}(h|\\H)=1, \\quad \\beta^{*}(l|\\H)=0.\n\\end{align}\n\\end{footnotesize}\n\\item[(b)] Regime $B$:\\\\\n\\underline{Subregime $\\B_1$}:\n\\begin{footnotesize}\n\\begin{subequations}\\label{eq_B_1_general}\n\\begin{alignat}{2}\n&\\sigma^{t*}_{l}=\\(\\dc\\)^{-1}(p^\\t_l), && \\quad \\sig^{d*}_\\H=0, \\label{eq_B_1}\\\\%\\Delta c^{\\theta}(\\sigma^{t*}_{l})&=p^\\t_l\\\\\n&\\beta^{*}(h|\\H)=\\frac{\\theta}{\\theta+ (1-\\theta) \\sigma^{t*}_{l}}, &&\\quad \\beta^{*}(l|\\H)=\\frac{(1-\\theta) \\sigma^{t*}_{l}}{\\theta+ (1-\\theta) \\sigma^{t*}_{l}} \\label{b_B_1}\n\\end{alignat}\n\\end{subequations}\n\\end{footnotesize}\n\\underline{Subregime $\\B_2$}:\n\\begin{footnotesize}\n\\begin{subequations}\\label{eq_B_2_general}\n\\begin{alignat}{2}\n&\\sigma^{t*}_{l}= \\widehat{\\sigma}_{\\l}^\\t, &&\\quad \\sig^{d*}_\\H=\\frac{\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-p^\\t_l}{F_l}, \\label{eq_B_2}\\\\\n&\\beta^{*}(h|\\H)=\\frac{F_l-p^\\d}{F_l},&& \\quad \\beta^{*}(l|\\H)=\\frac{p^\\d}{F_l} \n\\end{alignat}\n\\end{subequations}\n\\end{footnotesize}\n\\underline{Subregime $\\B_3$}:\n\\begin{footnotesize}\n\\begin{subequations}\n\\begin{alignat}{2}\n&\\sigma^{t*}_{l}=\\(\\dc\\)^{-1}(p^\\t_l+F_l), \\quad && \\sig^{d*}_\\H=1,\\label{eq_B_3}\\\\%\\Delta c^{\\theta}(\\sigma^{t*}_{l})&=p^\\t_l+F_l\n&\\beta^{*}(h|\\H)=\\frac{\\theta}{\\theta+ (1-\\theta) \\sigma^{t*}_{l}}, \\quad && \\beta^{*}(l|\\H)=\\frac{(1-\\theta) \\sigma^{t*}_{l}}{\\theta+ (1-\\theta) \\sigma^{t*}_{l}}\\label{b_B_3}\n\\end{alignat}\n\\end{subequations}\n\\end{footnotesize}\n\\end{enumerate}\n\\end{theorem}\nThe proof of this theorem is in Appendix A.\n\nNow we discuss the equilibrium structure in detail. First, we interpret the regime boundaries:\n\\begin{enumerate}\n\\item[(i)] Regimes $A$ and $B$ are distinguished by the threshold $\\Delta c^{\\theta}(0)$, which is the travel cost reduction that a type $l$ agent can enjoy by misbehaving given that all the other agents are complaint.\n\\item[(ii)] In regime $B$, the threshold $(1-\\theta) F_l$ is the maximal expected fine obtained by the operator. We say that the inspection cost is high if $p^\\d$ is higher than $(1-\\theta) F_l$, and low otherwise. \n\nAdditionally, the threshold $\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$ is the gain from misbehavior when the misbehavior rate is $\\widehat{\\sigma}_{\\l}^\\t$ and the operator does not inspect any agent, and the threshold $\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-F_l$ is the gain when the operator inspects all agents on the server $\\H$. We say that the misbehavior cost $p^\\t_l$ is \\emph{relatively high} compared to $p^\\d$, if $p^\\t_l> \\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$; \\emph{relatively medium} if $\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-F_l<p^\\t_l < \\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$, and \\emph{relatively low} if $p^\\t_l < \\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)-F_l$.\n\\end{enumerate} \n\n\t\nWe can relate the equilibrium strategy profiles and the conditions determining the regime boundaries:\n\\begin{basedescript}{\\desclabelstyle{\\pushlabel}}\n\\item{ [Regime $A$]: Misbehavior cost $p^\\t_l > \\Delta c^{\\theta}(0)$. Misbehavior is fully deterred, and no inspection is needed.}\n\\item{[Regime $B$]: Misbehavior cost $p^\\t_l<\\Delta c^{\\theta}(0)$. Misbehavior occurs with positive probability. }\n\\end{basedescript}\n\\begin{itemize}\n\\item[-] $\\B_1$: \\emph{Misbehavior cost is relatively high or the inspection cost is high.} The operator does not inspect. The misbehavior rate is such that server $\\H$ and $\\L$ incur identical utility for type $l$ agents given that no agent is inspected.\n\\item[-] $\\B_2$: \\emph{Misbehavior cost is relatively medium and the inspection cost is low.} The operator inspects a positive fraction of agents. Misbehavior rate is equal to the threshold $\\widehat{\\sigma}_{\\l}^\\t$ in \\eqref{siglbar}. \n\\item[-] $\\B_3$: \\emph{Misbehavior cost is relatively low and the inspection cost is low.} The operator inspects all the agents. The misbehavior rate is such that server $\\H$ and $\\L$ incur identical utility for type $l$ agents given that all agents on server $\\H$ is inspected. Moreover, this rate is higher than the threshold $\\widehat{\\sigma}_{\\l}^\\t$. \n\\end{itemize}\n\n\n\n\n\n\n\n\nWe summarize how PBE changes with the misbehavior and inspection costs in table \\ref{comparison}:\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{} & $A$ &  $\\B_1$ & $\\B_2$ & $\\B_3$\\\\\n\\hline\n\\multirow{2}{*}{$p^\\t_l$ increases} & $\\sigma^{t*}_{l}$ & $-$ & $\\downarrow$& $-$ &$\\downarrow$ \\\\\n\\cline{2-6}\n&$\\sig^{d*}_\\H$ & $-$ & $-$ & $-$ & $-$\\\\\n\\hline\n\\multirow{2}{*}{$p^\\d$ increases} & $\\sigma^{t*}_{l}$ & $-$ & $-$ & $\\uparrow$  & $-$ \\\\\n\\cline{2-6}\n&$\\sig^{d*}_\\H$ & $-$ & $-$ & $\\downarrow$ & $-$\\\\\n\\hline\n\\end{tabular}\n\\caption{Qualitative properties of PBE}\n\\label{comparison}\n\\end{table}\n\\vspace{-0.4cm}\nThe main implications of our equilibrium analysis are as follows: \n\\begin{itemize}\n\\item[-] The misbehavior is completely deterred only when the misbehavior cost is sufficiently high.\n\\item[-] In subregime $\\B_2$, the belief $\\beta^{*}(\\H)$ does not depend on $\\theta$. The intuition is that in this subregime, both the agents and the operator use mixed strategies in equilibrium, thus, the threshold strategy $\\widehat{\\sigma}_{\\l}^\\t$ in \\eqref{siglbar} increases in $\\theta$ to ensure that the belief $\\beta^{*}(l|\\H)$ (resp. $\\beta^{*}(h|\\H)$) is maintained at the threshold value $p^\\d\/F_l$ (resp. $1-p^\\d\/F_l$), which makes the operator indifferent between $\\mathrm{I}$ and $\\mathrm{N}$. \n\\item[-] One can verify the intuitive property that the utility of the type $l$ (resp. $h$) agents is non-increasing (resp. non-decreasing) in $p^\\t_l$ and non-decreasing (resp. non-increasing) in $p^\\d$. Similarly, the operator's utility is non-decreasing in $p^\\t_l$ and non-increasing in $p^\\d$. \n\\item[-] In general, the misbehavior rate $\\sigma^{t*}_{l}$ is non-increasing in $p^\\t_l$, and non-decreasing in $p^\\d$. The inspection rate $\\sig^{d*}_\\H$ is non-decreasing in $p^\\t_l$, and non-increasing in $p^\\d$\n\\end{itemize}\n\n\nFinally, from (A3), we know that the minimal technology cost needed to deter misbehavior, $\\Delta c^{\\theta}(0)$, decreases in $\\theta$. Also, $\\Delta c^{\\theta}(\\widehat{\\sigma}_{\\l}^\\t)$ decreases in $\\theta$. Therefore, as $\\theta$ increases, the sizes of the regime $A$ and the subregime $\\B_1$ increase, and the sizes of the two other subregimes decrease. This implies that the misbehavior rate is lower and the inspection is less needed when more agents are of type $h$.\n\nMoreover, as the fine $F_l$ increases, the size of $\\B_2$ increases, and the size of $\\B_3$ decreases or becomes empty. However, $F_l$ has no effect on $A$ and $\\B_1$, where inspection is not needed. This observation implies that (i) Fine is effective in reducing the misbehavior rate when the inspection cost is relatively high compared to the misbehavior cost, but cannot fully deter misbehavior, and (ii) If the fine is higher than $\\Delta c^{\\theta}(0)$, then the operator will not inspect all agents\n\n\n\n\\section{Toll Evasion Example}\\label{sec_implication}\nWe apply our equilibrium results to a specific example of Electronic Toll Collection (ETC) system. In the ETC setting, server $\\H$ (resp. $\\L$) represents the tolled (resp. toll-free) lanes. Type $h$ are the agents that are willing to pay the toll, and type $l$ are the agents that are not willing to pay. The total arrival rate of both types of agents is $\\lambda=2400$ veh\/hr. The fraction $\\theta=0.3$ is the fraction of type $h$ agents.\nTherefore, the arrival rate of type $h$ (resp. $l$) agents is $\\theta\\lambda$ (resp. $(1-\\theta)\\lambda$). \n\nThe toll is collected electronically, and the access to the tolled lanes is granted to the paying agents after the RSU obtains their reported identifier. \nSuch an operation is technologically feasible; see e.g. the European DSRC Toll Collection systems \\cite{papadimitratos09}. \n\nWe model the highway as two parallel $M\/M\/1$ queuing systems, one representing the tolled lanes ($\\H$), and the other representing the toll-free lanes ($\\L$). For background on modeling highway traffic with stochastic queuing models, see \\cite{newell13}.\nBoth the tolled lanes and the toll-free lanes have a capacity (service rate) of 1700 veh\/hr, i.e. $\\mu_\\H=\\mu_\\L=1700$ veh\/hr.\nThe travel cost on a server is the product of the expected system time and the value of time $\\mathrm{VoT}=50$ USD\/hr. The fine is $F_l=100$ USD. Following standard results in queuing theory, the expected cost functions are as follows:\n\n\\begin{footnotesize}\n\\begin{subequations}\\label{queue}\n\\begin{align*}\n&c_{\\H}^{\\theta}(\\sigma)=\\left\\{\\begin{array}{ll}\n\\frac{\\mathrm{VoT}}{\\mu_\\H-\\theta\\lambda(1-\\sigma^t_h)-(1-\\theta)\\lambda\\sigma^t_l}, &\\quad  \\text{if }\\theta\\lambda(1-\\sigma^t_h)+(1-\\theta)\\lambda\\sigma^t_l<\\mu_\\H,\\\\\n\\infty,&\\quad  o.w.\n\\end{array}\\right.\\\\%\\label{queue_h}\\\\\n&c_{\\L}^{\\theta}(\\sigma)=\\left\\{\\begin{array}{ll}\n\\frac{\\mathrm{VoT}}{\\mu_\\L-(1-\\theta)\\lambda(1-\\sigma^t_l)-\\theta\\lambda\\sigma^t_h},  & \\quad \\text{if } (1-\\theta)\\lambda(1-\\sigma^t_l)+\\theta\\lambda\\sigma^t_h<\\mu_\\L,\\\\\n\\infty, & \\quad o.w.\n\\end{array}\\right\n\\end{align*}\n\\end{subequations}\n\\end{footnotesize}\nWe can check that the cost functions satisfy Assumptions (A1) - (A3). Fig. \\ref{fig:regime} illustrates the regimes of PBE.\n\\begin{figure}[H]\n    \\centering\n        \\includegraphics[width=0.5\\textwidth]{regime_plot.jpg}\n    \\caption{PBE regimes.}\\label{fig:regime}\n\\end{figure}\n\\vspace{-0.3cm}\nIn this example, the minimum $p^\\t_l$ that deters misbehavior is $\\Delta c^{\\theta}(0)=2.35$ USD. Note that this is the technology cost per signal. A device that is used to manipulate the message sent to the RSU can be expensive, but if the device is repeatedly used, the average cost can be low.\n\nAdditionally, since the fine $F_l=100>\\Delta c^{\\theta}(0)=2.35$, the sub-regime $\\B_3$ is empty. This implies that given any $p^\\t_l$ and $p^\\d$, the operator will not inspect all agents. Given parameters in $\\B_2$, $p^\\t_l=0.5$ USD and $p^\\d=5$ USD, the equilibrium misbehavior rate is $\\sigma^{t*}_{l}=\\widehat{\\sigma}_{\\l}^\\t=2.15\\%$, and the inspection rate is $\\sig^{d*}_\\H=0.34 \\%$. \n\n\n\\section{Introduction}\nVehicle-to-Infrastructure (V2I) and Vehicle-to-Vehicle (V2V) communications are commonly regarded as integral features of smart highway systems \\cite{varaiya93,papadimitratos09}. With the projected growth of V2I and V2V capabilities, it is expected that they will support important operations such as safety-preserving maneuvers (overtaking), lane management, intersection control, and also enable traffic management with connected\/autonomous vehicles \\cite{kurzhanskiy15, bergenhem12}, \\cite{jin18hscc}. These applications typically require the presence of road-side units (RSUs) that are capable of receiving messages from individual vehicles (i.e., their on-board units (OBUs)), authenticating these messages, and providing relevant information to the neighboring vehicles and\/or actuators (e.g., traffic signals). This message exchange is typically supported by the Dedicated Short Range Communications (DSRC) technology, which enables the RSU to gather information such as vehicle identifier, vehicle class, and safety-related data. In recent years, numerous security concerns have been identified in this context \\cite{petit15}, \\cite{lawson08}, \\cite{hubaux04}. Prior research in security of vehicular communications has focused on the identification of vulnerabilities and design of defense solutions to prevent, detect, and respond to various security threats \\cite{alpcan2010network}. However, an aspect that has received relatively little attention is the modeling of strategic misbehavior by vehicles that can negatively impact highway operations. \n\nIn this paper, we focus on a generic setting of lane management operation enabled by V2I communications, and develop a model of strategic misbehavior using a signaling game formulation. To motivate the setting, let us consider a highway segment with a downstream bottleneck; see Fig.\\ref{toll_lanes}. The highway is equipped with a RSU and all incoming vehicles have OBUs. The highway section has two classes of lanes: the high-priority lanes are meant to serve the travelers with preferential access to the system, and the low-priority (or general purpose) lanes are meant to serve all other travelers. We consider the two sets of lanes as parallel servers. The RSU receives and authenticates the messages from the incoming vehicles. A vehicle is provided access to a server if the RSU is able to authenticate its message and adjudge it to be a vehicle belonging to that class. The travel cost incurred in accessing each server increases with the aggregate number of vehicles routed through that server due to the congestion effects. \nWe assume that the two classes are pre-established using well-known economic principles.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{toll_lanes}\n\\caption{A V2I-based highway operations.}\n\\label{toll_lanes}\n\\end{figure}\n\\vspace{-0.25cm}\nThe main feature that we consider in the abovementioned setting is the ability of travelers to manipulate the communication between their vehicles' OBU and the RSU, and obtain access to the server that does not correspond to their vehicle class. Such an attack can be realized if the vehicles (henceforth referred to as ``agents'') can misreport their identity information to the RSU. We consider this type of attack as an instance of \\emph{strategic misbehavior}, which needs to be deterred by the system operator because it imposes additional congestion externalities on the other travelers. Misbehavior can be deterred\nusing technological means such as inspecting the messages for their integrity, checking the vehicle's identity using video cameras, number plate recognition, or manual inspection. On successful detection, a suitable fine can be imposed by the operator. It is important to include the following features in modeling misbehavior: (i) The operator has incomplete information about the priority class of incoming vehicles, in that the true class of vehicle can be known only after inspection (which is costly); (ii) Each misbehaving agent incurs a technological cost for manipulating its message, and is subject to a fine if inspected. These features naturally lead us to pose our model as a signaling game \\cite{spence1974market}.  \n\n\nIn our game, the agents are non-atomic and each agent has private information about its type; i.e., each agent privately knows whether she is a high- or low-priority agent. The operator has the technological capability of message collection (via the RSU), inspection, and fine collection. We say that an agent misbehaves if she sends a signal that is different from her true type and obtains access to the lane (server) that does not correspond to her true type.\nAll agents are subject to inspection by the operator. A misbehaving agent incurs a non-negative cost, and if detected, is charged a non-negative fine.\n\t\n\tThe equilibrium concept that we use is the Perfect Bayesian Equilibrium (PBE) \\cite{fudenberg1991perfect}. In the PBE, (a) the players satisfy sequential rationality, and (b) the operator's belief is consistent with the prior distribution of agent types and the agents' strategy. The specific features that distinguish our game from the classical models of signaling games are: non-atomic agent populations, and congestion externality imposed by an agent on other agents sending the same signal (i.e. accessing the same server). Under mild assumptions on the cost functions of both servers, we provide a complete characterization of the PBE for our signaling game. \n\t\n\tIn particular, we show that in equilibrium (i) a high priority agent does not have the incentive to misbehave for gaining access to the low-priority server; (ii) not every low-priority agent misbehaves. Moreover, we distinguish two regimes based on how the technological cost of misbehavior compares with the maximum gain from misbehavior (evaluated as the difference in travel costs of two servers when no agent misbehaves). In the first regime, the misbehavior is completely deterred as the technological cost of misbehavior is high and the operator does not need to inspect any agent. In the second regime, the low-priority agents misbehave with a positive probability. The operator's inspection strategy in the second regime can be further distinguished by sub-regimes that correspond to zero, partial, and complete inspection.  \n\t\nOur equilibrium analysis can be used to study the comparative statics with respect to practically relevant parameters such as the fraction of each priority class, the inspection cost, and the fine. Firstly, for given cost of misbehavior and cost of inspection, the equilibrium misbehavior rate (and the hence the rate of inspection) decreases as the fraction of high-priority agents increases. Secondly, fine can be effective for decreasing misbehavior rate even when the inspection cost is high (relative to the cost of misbehavior), but cannot achieve complete deterrence. Thirdly, if the fine is sufficiently high, then there is no need to inspect all agents in equilibrium. These insights are relevant for the design and deployment of inspection technologies to achieve higher security levels of V2I-enabled highway operations. Finally, we illustrate these effects in the setting of Electronic Toll Collection (ETC), where the servers are modeled as M\/M\/1 queueing systems, and the fraction of high-priority travelers (and the toll that they need to pay for priority access) is exogenously known. \n\n\n  \n\n\n  \n\n\n\\section{Introduction}\n\nVehicle-to-infrastructure (V2I) communications are increasingly applied to modern highways in order to improve efficiency, reliability, and safety \\cite{varaiya93}.\nAn important application of the V2I communications is multi-priority lane management operations for heterogeneous traffic.\nFor example, electronic toll collection is enabled by onboard device and road side units (RSU) that exchange data \\cite{papadimitratos09}.\nOther examples of lane management operations include high-occupancy vehicle (HOV) and high occupancy tolled (HOT) lanes \\cite{kurzhanskiy15}, and, as a proposed operation, dedicated lanes for vehicle platoons \\cite{bergenhem12,jin18hscc}.\n\n\n\n\nWe consider V2I communications enabled by the cyber infrastructure (e.g. road-side units, RSU) that receives messages from the individual vehicles for authentication purposes and monitors the traffic as it gets segmented into different lanes. \nNaturally, this capability assumes the presence of an onboard unit (OBU) on individual vehicles that access the highway system.\nTechnologies commonly used V2I communications include Dedicated Short Range Communications (DSRC), Radio Frequency Identification (RFID), and Global Positioning System (GPS) \\cite{papadimitratos09,depalma11}.\nUpon arrival to the system, each vehicle (i.e. its OBU) sends a message to the RSU through one or more channels; this message can include information such as vehicle identifier, vehicle class, toll debit information, safety-related information (e.g. speed\/acceleration), etc.\n\nAll of these technologies to some extent rely on the information\/data reported by the vehicles.\nMotivated by previously reported vulnerabilities in the V2I communications technology \\cite{hubaux04,lawson08,petit15}, we consider that individual travelers may manipulate the message, albeit at a cost, so that the RSU receives false information and incorrectly authorizes such agents.\nIn this paper, we focus on the incentive of agents to misreport their vehicle identity information to gain illegal access to the lanes dedicated to other traffic classes. Such misbehavior can result in losses such as lost toll revenue, operational inefficiencies, or even safety risks; hence, it should be inspected by the system operator.\n\nDue to the large number of devices involved in V2I communications, full prevention of misbehavior is neither technologically nor economically feasible.\nHowever, there are several inspection schemes in response to potential misbehaviors.\nWithin V2I communications, internal learning-based verification algorithms may detect\nBeyond V2I communications, the system operator may have access to other sources of information for cross verification \\cite{}.\nFor example, Automated Number Plate Recognition (ANPR) is also a commonly used vehicle identification technology \\cite{depalma11}.\nIf a highway is equipped with both V2I communications and ANPR, then the system operator may have the capability of cross-checking data retrieved via both technologies.\nAnother example is police patrol with random inspection of misbehavior \\cite{borndorfer13}.\nNote that both inspection schemes may require additional investment in infrastructure or labor, and may induce additional congestion.\n\nIn this article, we develop a framework for quantitative estimate of the incentive for a class of misbehavior and design of effective and economically efficient inspection schemes.\nWe also account for the main tradeoff faced by the system operator in inspecting the agents for misbehavior.\nFor potential misbehaving vehicles, we evaluate the benefit (travel time reduction) resulting from misbehavior, and study the probability of misbehavior of a strategic driver that is aware of the cost of conducting misbehavior (technological investment and potential fine).\nFor the system operator, we study the optimal inspection strategy in terms of inspection probability\/rate and the fine for misbehavior.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{.\/figures\/toll_lanes}\n\\caption{An automated highway with V2I-based lane management operations.}\n\\label{toll_lanes}\n\\end{figure}\n\nOur analysis considers an automated highway section with a downstream bottleneck (Sec. \\ref{sec_physical}).\nThe highway section has a set of high-priority (e.g. tolled or HOV) lanes and a set of low-priority (e.g. general-purpose) lanes; see Fig.~\\ref{toll_lanes}.\nWe consider the two sets of lanes as two parallel servers.\nThe highway faces two types of vehicles: high-priority vehicles and low-priority vehicles.\nUpon arrival, each vehicles reports its types to the system operator via V2I communications.\nThen, the system operator authorizes the vehicle access to the high-priority server if the vehicle is reportedly a high-priority-type vehicle.\nEvery lane has a travel cost that increases with the number of vehicles taking this lane.\nOur formulation is compatible with a broad class of travel cost functions with such monotonicity.\nWhile we do not specialize our model to capture losses such as lost toll revenue or safety risk, we do capture the key contributing factor: the congestion externality induced by misbehaving agents on the other travelers.\n\n\nTo study the interaction between misbehaving vehicles and the system operator, we consider a signaling game formulation (Sec. \\ref{sec_misbehavior}). \nSignaling game is a natural choice for our purpose, since we particularly focus on vehicles strategically abusing the V2I communications.\nThe vehicles are modeled as a set of non-atomic agents, and the system operator is considered as the defender. Each agent privately knows its type, i.e. high- or low-priority.\nIn addition, each agent sends a signal of its type to the system operator (defender), which may not match its true type. \nWe say that the agent misbehaves if it sends a signal different from its true type, and thus uses a server that does not match its type.\nFor a low-priority agent, conducting a misbehavior induces a strictly positive technological cost.\nFor the sake of generality, we also consider a high-priority vehicle sending a low-priority signal as ``misbehaving'' (the technological cost for such a ``misbehavior'' can be made zero.)\nThis construction allows us to consider the possible incentive of a high-priority agent to switch to the low-priority server.\nThe defender observes the signal that an agent sends but does not know the agent's true type. \nThe defender then forms a belief about the fraction of agents who misbehave, and determines the inspection rate on each server. If an agent misbehaves, and is inspected, then the misbehavior is detection. A fine is charged to the agent by the defender. \n\nThe equilibrium concept that we use is the Perfect Bayesian Equilibrium (PBE), where the agents and the defender are sequentially rational, and the defender's belief is consistent with the prior distribution and the agents' strategy according to Bayes' rule. Our game is different from the classical signaling game (see \\cite{spence1974market,fudenberg1991perfect,cho1987signaling,banks1987equilibrium}) in two aspects: (1) The agents are considered as non-atomic players; (2) Both lanes are congested since the cost of each lane increases in the aggregate demand of agents on it. In fact, these two features make the equilibrium properties of our game different from those classical models. \n\nBased on the signaling game formulation, we are able to characterize the equilibrium behavior of agents and the system operator (Sec. \\ref{sec_equilibrium}). Our analysis is based on a set of mild assumptions regarding the baseline scenario where no agent misbehaves (Assumptions (A1)). We show that in equilibrium (i) a high-priority agent never has the incentive to switch to the low-priority server (Proposition \\ref{h_non_deviate}), and (ii) not every low-priority agent misbehaves (Proposition \\ref{no_pool}). In general, there are two regimes based on the magnitude of misbehavior cost compared to the maximum misbehavior incentive, which is the travel cost difference on two servers when no agents misbehave: in the first regime, denoted by $A$, the misbehavior cost is higher, and thus misbehavior is deterred, whereas in the second regime, denoted by $B$, the misbehavior cost is lower, and thus the misbehavior rate is always positive in equilibrium. Furthermore, three sub-regimes exist in regime $B$, where the defender chooses to inspect no agents, a fraction of agents and all agents, respectively (Theorem \\ref{theorem_equilibrium}).\n\n\\textcolor{violet}{We also demonstrate practical implication from our analysis for cyber-physical security in smart highways (Sec. \\ref{sec_implication}). For illustration purpose, we consider a specific setting of electronic toll collection (ETC). We show that given any misbehavior cost and inspection cost, as the fraction of agents who are willing to pay the toll increases, the misbehavior rate is lower and the inspection is less needed. Moreover, when the inspection cost is relatively high comparing to the misbehavior cost, fine is effective in decreasing the misbehavior rate, but does not deter misbehavior completely. \nFinally, if the fine is higher than the maximum misbehavior incentive, then the defender does not have to inspect every agent (i.e. the regime of complete inspection is empty).\n}\n\n\n\\section{Signaling game for misbehavior inspection}\\label{sec_misbehavior}\nWe now model the strategic interaction between the agent populations (travelers $\\t$) that are prone to misbehavior and the system operator (defender $\\d$) who decides to inspect them based on the received messages. We consider that the agents are capable of compromising the integrity of messages sent to the RSU in order to obtain access to the server that does not correspond to their true type. Recall that\nthe operator cannot know an agent's true type unless she inspects the agent. This information asymmetry between the agents and the operator naturally leads to a signaling game formulation. \n\nIn the game, each agent type is modeled as a population of non-atomic players. The cost of misbehavior is non-negative for each type $h$ agent, denoted $p^\\t_h \\in \\mathbb{R}_{\\geq 0}$, and strictly positive for each type $l$ agent, denoted $p^\\t_l \\in \\mathbb{R}_{> 0}$.\\footnote{We make this technical assumption to avoid triviality in equilibrium analysis. It is consistent with our setting of differentiated priority system.}\n\nAs mentioned before, the operator does not know the type of each agent, but can observe the agent's signal, i.e. the server taken by the agent. The signal space is the set of servers $\\{\\H,\\L\\}$. We say that a type $l$ (resp. type $h$) agent misbehaves if she chooses the server $\\H$ (resp. $\\L$). The operator forms a belief of the true type given the observed signal. We denote $\\beta(\\H) \\deleq \\(\\beta(\\h|\\H), \\beta(\\l|\\H)\\)$ (resp. $\\beta(\\L) \\deleq \\(\\beta(\\h|\\L), \\beta(\\l|\\L)\\)$) as the operator's belief given the signal $\\H$ (resp. $\\L$), where $\\beta(\\h|\\H)$ and $\\beta(\\l|\\H)$ (resp. $\\beta(\\h|\\L)$ and $\\beta(\\l|\\L)$) are the posterior probabilities that an agent on the server $\\H$ (resp. $\\L$) is in fact a type $h$ and $l$ agent, respectively. Based on the signal and the belief, the operator chooses to inspect an agent ($\\mathrm{I}$), or not to inspect ($\\mathrm{N}$). The inspection incurs a positive cost on the operator, denoted $p^\\d \\in \\mathbb{R}_{>0}$. We denote $\\sigd_\\H$ (resp. $\\sigd_\\L$) as the probability of inspecting an agent on the server $\\H$ (resp. $\\L$). Then, the operator's inspection strategy is $\\sigma^d \\deleq \\(\\sigd_\\H, \\sigd_\\L\\)$, and the strategy profile is $\\sigma \\deleq \\(\\sigma^t, \\sigma^d\\)$. Furthermore, for simplicity, we assume that if an agent misbehaves, and if he is inspected, then the misbehavior is detected with probability 1.\\footnote{If the probability of\ndetection is smaller than 1, then the effective inspection rate is simply the total inspection rate scaled by the detection probability. Our analysis approach can be straightforwardly extended to this case.} A fine $F_h \\in \\mathbb{R}_{\\geq 0}$ (resp. $F_l \\in \\mathbb{R}_{\\geq 0}$) is charged to the type $h$ (resp. $l$) agent if his misbehavior is detected. \n\nWe are now ready to discuss the utility functions of the agents and the operator. The utility of each agent is the summation of three parts: (i) $-c_{\\H}^{\\theta}(\\sigma^t)$ (resp. $-c_{\\L}^{\\theta}(\\sigma^t)$), which is the travel cost if the agent chooses the server $\\H$ (resp. $\\L$); (ii) $-p^\\t_h$ (resp. $-p^\\t_l$), which is the technology cost of misbehavior for a type $h$ (resp. $l$) agent; (iii) $-F_h$ ($-F_l$), which is the fine if the misbehavior is detected upon inspection of a type $h$ (resp. $l$) agent. The utility of the operator is the summation of two parts: (i) $-p^\\d$, which is the inspection cost; (ii) $F_h$ (resp. $F_l$), which is collected fine when the misbehavior of a type $h$ (resp. $l$) agent is detected.\n\nThe game is played in the following steps; see Fig. \\ref{game_diagram}. First, the type of each agent is chosen by the fictitious player ``Nature'' according to the probability distribution $\\mathrm{Pr}(h)=\\theta$ and $\\mathrm{Pr}(l)=1-\\theta$. Then the agents send the signal (choose the server) according to strategy $\\sigma^t$ based on their type. Next, the operator observes the signal, and the belief $\\b$ is updated based on the observed signal. The operator then chooses to inspect or not according to $\\sigma^d$. All the game parameters are common knowledge, except that each agent privately knows his type. \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{game_diagram.png}\n\\caption{Game Tree with the agent's utility (top) and the operator's utility (bottom) indicated at each leaf node.}\n\\label{game_diagram}\n\\end{figure}\n\n\nGiven strategy profile $\\sigma$, we denote the expected utility of type $h$ agents choosing the server $\\H$ (resp. $\\L$) as $\\mathbb{E}_{\\sig}[U^t_h(\\H)]$ (resp. $\\mathbb{E}_{\\sig}[U^t_h(\\L)]$). The expected utilities for type $l$ agents are similarly denoted as $\\mathbb{E}_{\\sig}[U^t_l(\\H)]$ and $\\mathbb{E}_{\\sig}[U^t_h(\\H)]$, respectively. The expected utilities of agents can be written as follows:\n\n\\begin{footnotesize}\n\\begin{subequations}\\label{expected_cost_agent}\n\\begin{align}\n\\mathbb{E}_{\\sig}[U^t_h(\\H)]&=-c_{\\H}^{\\theta}(\\sigma^t),\\text{   } \\mathbb{E}_{\\sig}[U^t_h(\\L)]=-c_{\\L}^{\\theta}(\\sigma^t)-p^\\t_h-F_h \\sigd_\\L, \\label{UhL}\\\\\n\\mathbb{E}_{\\sig}[U^t_l(\\H)]&=-c_{\\H}^{\\theta}(\\sigma^t)-p^\\t_l-F_l \\sigd_\\H, \\text{   } \\mathbb{E}_{\\sig}[U^t_l(\\L)]=-c_{\\L}^{\\theta}(\\sigma^t). \\label{UlL}\n\\end{align}\n\\end{subequations}\n\\end{footnotesize}Given any strategy profile $\\sigma$ and any belief $\\b$, the expected utility of the operator when observing signal $\\H$ (resp. $\\L$) is denoted as $\\mathbb{E}_{\\sig}[U_\\H^d|\\beta]$ (resp. $\\mathbb{E}_{\\sig}[U_\\L^d|\\beta]$), which can be written as follows:\n\\begin{footnotesize}\n\\begin{subequations}\\label{expected_cost_operator}\n\\begin{align}\n\\mathbb{E}_{\\sig}[U_\\H^d|\\beta] &= \\(-p^\\d+ F_l  \\beta(\\l|\\H) \\) \\sigd_\\H, \\label{cost_H}\\\\\n\\mathbb{E}_{\\sig}[U_\\L^d|\\beta]&= \\(-p^\\d + F_h \\beta(\\h|\\L)\\) \\sigd_\\L.\\label{cost_L}\n\\end{align}\n\\end{subequations}\n\\end{footnotesize}Note that in all the cost functions \\eqref{UhL}-\\eqref{UlL} and \\eqref{cost_H}-\\eqref{cost_L}, the travel cost is the expected travel time multiplied with the value of time, and hence can be treated as a monetary cost, similar to the fine and misbehavior cost.\n\n\nThe equilibrium concept in this game is the perfect Bayesian equilibrium (PBE), see \\cite{fudenberg1991perfect}:\n\\begin{definition}\nA pair $(\\sigma^{*}, \\beta^{*})$ of strategy profile $\\sigma^{*}$ and belief assessment $\\beta^{*}$ is a PBE if it satisfies both sequential rationality and consistency:\n\\begin{itemize}\n\\item \\emph{Sequential rationality}: (i) The servers that are used by each type of agents incur the highest expected utility: \n\\begin{footnotesize}\n\\begin{subequations}\\label{rational_traveler}\n\\begin{align}\n\\sigma^{t*}_{h} >0 &\\quad \\Rightarrow \\quad \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\L)] \\geq \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\H)], \\label{travel_high_deviate}\\\\\n\\sigma^{t*}_{h} <1 &\\quad \\Rightarrow \\quad \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\L)] \\leq \\mathbb{E}_{\\sigma^{*}}[U^t_h(\\H)], \\label{travel_high}\\\\\n\\sigma^{t*}_{l} >0 &\\quad  \\Rightarrow \\quad \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\H)] \\geq \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\L)], \\label{travel_low_deviate}\\\\\n\\sigma^{t*}_{l} <1 &\\quad  \\Rightarrow \\quad \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\H)] \\leq \\mathbb{E}_{\\sigma^{*}}[U^t_l(\\L)]. \\label{travel_low}\n\\end{align} \n\\end{subequations}\n\\end{footnotesize}\n(ii) The operator maximizes expected utility:\n\\begin{footnotesize}\n\\begin{equation}\\label{eq_operator_rational}\n\\begin{split}\n\\sig^{d*}_\\H &= \\argmax_{\\sigd_\\H \\in [0, 1]} \\mathbb{E}_{\\sig}[U_\\H^d|\\beta^{*}], \\text{  } \\sig^{d*}_\\L=\\argmax_{\\sigd_\\L \\in [0, 1]} \\mathbb{E}_{\\sig}[U_\\L^d|\\beta^{*}]\n\\end{split}\n\\end{equation}\n\\end{footnotesize}\n\\item \\emph{Consistency}: $\\beta^{*}$ is updated according to the agent's strategy $\\sigma^{*}$ using the Bayes' rule:\n\\begin{footnotesize}\n\\begin{subequations}\\label{bayes}\n\\begin{align}\n\\beta^{*}(\\h|\\H) &= \\frac{\\theta  \\(1-\\sigma^{t*}_{h}\\)}{\\theta  \\(1-\\sigma^{t*}_{h}\\)+ \\(1-\\theta\\) \\sigma^{t*}_{l}},  \\\\\n\\beta^{*}(\\h|\\L) &= \\frac{\\theta  \\sigma^{t*}_{h}}{\\theta  \\sigma^{t*}_{h}+\\(1-\\theta\\) \\(1-\\sigma^{t*}_{l}\\)},\n\\end{align} \n\\end{subequations}\n\\end{footnotesize}\nand $\\beta^{*}(\\l|\\H)=1-\\beta^{*}(\\h|\\H)$, $\\beta^{*}(\\l|\\L)=1-\\beta^{*}(\\h|\\L)$. \n\\end{itemize}\n\\end{definition}\n\nWe offer two remarks on PBE. First, with regard to the sequential rationality of agents, the rationality constraints \\eqref{travel_high_deviate} and \\eqref{travel_low_deviate} ensure that if the agents of a given type misbehave with positive probability in equilibrium, then the expected utility in choosing to access the other server is no less than the expected utility in choosing to access the server corresponding to their own type. On the other hand, constraints \\eqref{travel_high} and \\eqref{travel_low} ensure that if agents use the server for their true type with positive probability in equilibrium, then the utility of choosing the other server is not strictly higher.\n\n\nSecond, the consistency of beliefs requires that the operator's updated belief of each type given the received signal is consistent with the prior distribution and the agents' strategy in accordance with the Bayes' rule. The operator then chooses the optimal inspection rate based on her belief.\n\n\n\\section{Physical model for highway traffic} \\label{sec_physical}\nIn this section, we discuss three physical models of highway traffic. In these models, two classes of traffic are given different priorities: The class of traffic with high priority is permitted to use the lane with lower travel time, whereas the class with low priority can only use the other lane with higher travel time. Travelers in the low priority class have incentive to sneak in the high priority lane by \\textcolor{red}{to be filled in}. Therefore, the issue of misbehavior and how to inspect them naturally arise. \n\\subsection{Toll system with ordinary traffic}\\label{sec_toll}\n\\subsection{Mixed traffic with CAVs and ordinary traffic}\\label{sec_mixed_traffic}\n\\subsection{Two classes of CAV traffic}\\label{sec_two_classes}\n\n\\section{Modeling Misbehavior} \\label{sec_physical}\n\nIn this section, we consider a model of lane management operations on a highway section equipped with vehicle-to-infrastructure (V2I) communications capability, and propose a model of strategic misbehavior for this setting.\n\nSuppose that the highway system faces a fixed traffic demand comprised of two types of agent populations: a high priority type, denoted $h$, and a low priority type, denoted $l$. \nThe fraction of type $h$ agents is $\\theta \\in (0,1)$, and the fraction of type $l$ agents is $1-\\theta$. \nThroughout this article, we assume that $\\theta$ is exogenous and independent of potential misbehavior; see Remark \\ref{r1}.\nThere are two sets of lanes on the highway, $\\H$ and $\\L$, which we model as two parallel servers. \nIn the absence of any misbehavior, server $\\H$ is only assessed by $h$ agents, and server $\\L$ by the type $l$ agents.\n\\begin{remark}\\label{r1}\n Admittedly, the assumption of fixed fractions $\\theta$ precludes us from considering situations where the agent populations would choose their priority type (routing behavior) in anticipation of the potential misbehavior that they may face. However, it allows us to identify the effect of any given $\\theta$ on the misbehavior rate. \n \\end{remark} \n\n\nTo reduce his\/her travel cost, an agent may manipulate his reported signal, and choose to take the server that is not meant for his type. \nWe use $\\sigma^t_l$ (resp. $\\sigma^t_h$) to denote the fraction of $l$ (resp. $h$) agents that misbehave.\nWe denote the misbehavior strategy as $\\sigma^t=(\\sigma^t_h,\\sigma^t_l)$. \nSince the aggregate demand of agents using a server is determined by the relative size of agent populations ($\\theta$), and misbehavior strategy profile ($\\sigma^t$), we will use the notations $c_{\\H}^{\\theta}(\\sigma^t)$ (resp. $c_{\\L}^{\\theta}(\\sigma^t)$) to denote the cost of server $\\H$ (resp. $\\L$).\n\nWe make the following assumptions on the cost functions:\n\\begin{enumerate}\n\\item[(A1)] $c_{\\H}^{\\theta}(0,0)<\\infty$, $c_{\\L}^{\\theta}(0,0)<\\infty$, $c_{\\H}^{\\theta}(0,0)<c_{\\L}^{\\theta}(0,0)$, and $c_{\\H}^{\\theta}(0, 1) > c_{\\L}^{\\theta}(0, 1)$\n\\item[(A2)] $c_{\\H}^{\\theta}(\\sigma^t_h, \\sigma^t_l)$ (resp. $c_{\\L}^{\\theta}(\\sigma^t_h, \\sigma^t_l)$) decreases in $\\sigma^t_h$ (resp. $\\sigma^t_l$), and increases in $\\sigma^t_l$ (resp. $\\sigma^t_h$) \n\\item[(A3)] $c_{\\H}^{\\theta}(\\sigma^t_h, \\sigma^t_l)$ increases in $\\theta$. $c_{\\L}^{\\theta}(\\sigma^t_h, \\sigma^t_l)$ decreases in $\\theta$.\n\\end{enumerate}\n(A1) reflects that server $\\H$ has higher priority than server $\\L$, and ensures that both servers face stable queues. (A2) and (A3) captures the congestion nature of the highway system. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA \\emph{cusp} of a Fuchsian group $\\Gamma\\subset\\operatorname{PSL}_2(\\mathbb{R})$ is an\n$x\\in\\mathbb{P}^1(\\mathbb{R})$ that is the unique fixed point of an element of $\\Gamma$\n(see~\\cite{Sh} Ch. 1). The modular group $\\operatorname{PSL}_2(\\mathbb{Z})$ is a finite coarea Fuchsian\ngroup whose cusps are $\\mathbb{P}^1(\\mathbb{Q}) = \\mathbb{Q} \\cup \\{\\infty\\}$. In~\\cite{LR}, Long and\nReid show that there exist finite coarea Fuchsian subgroups of $\\operatorname{PSL}_2(\\mathbb{Q})$ that are\n\\emph{not} commensurable with $\\operatorname{PSL}_2(\\mathbb{Z})$ (i.e., not arithmetic) and whose cusp set\nequals $\\mathbb{P}^1(\\mathbb{Q})$. They call such groups \\emph{pseudomodular}.\n\nLong and Reid studied a particular family $\\Delta(u^2,2t)$ of Fricke groups as candidates\nfor pseudomodularity. Fricke groups uniformize one-cusped hyperbolic tori and all such\ntori are uniformized by Fricke groups; see~\\cite{Ab} and~\\cite{Ke}. Among Long and Reid's\nstated open problems in~\\cite{LR} is the determination of the values\n$(u^2,2t)\\in\\mathbb{Q}\\times\\mathbb{Q}$ for which $\\Delta(u^2,2t)$ is pseudomodular. Recall\nthat if $\\widehat{\\mathbb{Z}}$ is the profinite completion of $\\mathbb{Z}$, then $\\mathbb{A}_{\\fieldQ,f} =\n\\mathbb{Q}\\otimes_\\mathbb{Z}\\widehat{\\mathbb{Z}}$ is an additive topological group having a basis of open\nneighborhoods of 0 consisting of $m\\widehat{\\mathbb{Z}}$ for $m\\in\\mathbb{Q}$. Then our chief result\nis:\n\n\\begin{theorem}\\label{thm:ad}\n  The Fuchsian group $\\Delta(u^2,2t)$ is pseudomodular or arithmetic if and only if its\n  finite cusps are dense in the ring $\\mathbb{A}_{\\fieldQ,f}$ of finite adeles over $\\mathbb{Q}$.\n\\end{theorem}\n\n\\noindent Theorem~\\ref{thm:ad} is in fact true for arbitrary zonal Fuchsian subgroups of\n$\\operatorname{PSL}_2(\\mathbb{Q})$. We remark that for a given $t$ only finitely many $u^2$ yield\narithmetic groups. We shall use refinements of Theorem~\\ref{thm:ad} to give explicit,\ninfinite families of $\\Delta(u^2,2t)$ whose cusps are proper subsets of\n$\\mathbb{P}^1(\\mathbb{Q})$. For example:\n\n\\begin{theorem}\\label{thm:infnonpsm}\n  Let $p$ be a prime and $t$ an integer at least 2. Then $\\Delta(p^{-2},2t)$ is neither\n  pseudomodular nor arithmetic. In particular, infinitely many $\\Delta(u^2,2t)$ are\n  neither pseudomodular nor arithmetic.\n\\end{theorem}\n\nOur method is to find $\\Delta(u^2,2t)$-invariant subsets of $\\mathbb{Q}$ that are defined\nnumber-theoretically, either adelically or $p$-adically. In~\\cite{LR}, Long and Reid\nexhibit finitely many $\\Delta(u^2,2t)$ that are neither pseudomodular nor arithmetic. For\neach such group, they provide a rational number fixed by a hyperbolic element of\n$\\Delta(u^2,2t)$. Such fixed points cannot also be cusps; see~\\cite{Be}, p. 199. We do not\nknow whether rational hyperbolic fixed points exist for all non-pseudomodular $\\Delta$,\nand in any case, our proofs do not require or produce them. So far, our results disprove\npseudomodularity with a finite amount of data, in terms of either finitely many primes or\n(finite-index) congruence subgroups of the modular group $\\operatorname{PSL}_2(\\mathbb{Z})$. In the final\nsection of this paper we discuss some further questions and possible generalizations.\n\n\n\\section{Results}\n\nThe numerator and denominator of a rational number will be taken coprime with the\ndenominator positive. Let $v_p:\\mathbb{Q}^\\times\\to\\mathbb{Z}$ be the usual discrete valuation at\nprime $p$. If $p$ does not divide the denominator of a rational number $x$ (i.e.,\n$v_p(x)\\geq 0$) then say $x$ is \\emph{integral} at $p$. Integral $x$ have representatives\nin the ring of $p$-adic integers $\\mathbb{Z}_p$ and we write $x\\equiv m\\ (\\bmod\\ p)$ to mean\nthat the residue of $x$ mod $p\\mathbb{Z}_p$ is $m$ in $\\mathbb{Z}_p\/p\\mathbb{Z}_p\\cong\n\\mathbb{Z}\/p\\mathbb{Z}$. Finally, for nonzero rational $x$, we say $p\\mid x$ or $p$ \\emph{divides}\n$x$ when $v_p(x) > 0$.\n\nDenote by $\\mathcal{C}_\\infty(G)$ the cusp set of a Fuchsian group $G$ and let the finite cusps\nbe denoted by $\\mathcal{C}(G):=\\mathcal{C}_\\infty(G)\\setminus\\{\\infty\\}$. For rationals $u^2$ and\n$t$ with $0 < u^2 < t-1$, let $\\Delta(u^2,2t)$ be the subgroup of $\\operatorname{PSL}_2(\\mathbb{R})$\ngenerated by the hyperbolic elements\n\\[\ng_1 = \\frac{1}{\\sqrt{-1+t-u^2}}\\left(\n  \\begin{array}{cc}\n    t-1 & u^2 \\\\\n    1 & 1 \\\\\n  \\end{array}\n\\right),\\quad g_2 = \\frac{1}{u\\sqrt{-1+t-u^2}}\\left(\n  \\begin{array}{cc}\n    u^2 & u^2 \\\\\n    1 & t-u^2 \\\\\n  \\end{array}\n\\right).\n\\]\nAs in~\\cite{LR}, $\\Delta$ is a zonal Fricke group freely generated by $g_1$ and $g_2$\nwith $\\mathcal{C}_\\infty(\\Delta)$ a nonempty subset of $\\mathbb{P}^1(\\mathbb{Q})$. In fact, by\nconsidering the traces of $g_1$, $g_2$ and $g_1g_2$ and using results in \\S1 of~\\cite{CS}\non varieties of group representations, we can show that every Fricke group with cusps in\n$\\mathbb{P}^1(\\mathbb{Q})$ is conjugate in $\\operatorname{PGL}_2(\\mathbb{Q})$ to some $\\Delta(u^2,2t)$. Therefore,\nwe can study all such Fricke groups by looking at the groups $\\Delta(u^2,2t)$. We are\ninterested in when $\\mathcal{C}_\\infty(\\Delta)=\\mathbb{P}^1(\\mathbb{Q})$ or, equivalently, when\n$\\mathcal{C}(\\Delta)=\\mathbb{Q}$. Let $\\Lambda(u^2,2t)$ be the kernel of the homomorphism\n$\\Delta(u^2,2t) \\to \\mathbb{Z}\/2\\oplus\\mathbb{Z}\/2$ given by $g_1\\mapsto (1,0)$ and $g_2\\mapsto\n(0,1)$. Then $\\Lambda(u^2,2t)\\subset\\operatorname{PSL}_2(\\mathbb{Q})$ and $\\mathcal{C}(\\Lambda) =\n\\mathcal{C}(\\Delta)$. If $\\mathcal{C}(\\Lambda)$ is not dense in $\\mathbb{A}_{\\fieldQ,f}$, or in some finite\nproduct $\\prod_i\\mathbb{Q}_{p_i}$ with $\\mathbb{Q}$ embedded diagonally, then\n$\\mathcal{C}(\\Lambda)\\neq\\mathbb{Q}$ since $\\mathbb{Q}$ is dense in each of those sets.\n\nGiven this obstruction to pseudomodularity, we ask three questions: (i) when are the\nfinite cusps of $\\Delta(u^2,2t)$ dense in a specified finite product\n$\\prod_i\\mathbb{Q}_{p_i}$, (ii) is cusp density in all such products a sufficient condition\nfor pseudomodularity, and (iii) how do the answers to the former two questions vary as\n$u^2$ and $t$ are themselves varied $p$-adically? The propositions below\naddress these questions and, in particular, prove Theorem~\\ref{thm:infnonpsm}.\n\n\\begin{proposition}\\label{prop:square}\n  Let $p$ be prime. If $v_p(t)\\geq 0$ and $v_p(u^2)\\leq -2$, or if $v_p(t) < 0$ and\n  $v_p(u^2)\\leq 2(v_p(t)-1)$, then $\\mathcal{C}(\\Delta(u^2,2t))$ is not dense in $\\mathbb{Q}_p$.\n\\end{proposition}\n\n\\noindent Considering conditions at two primes, we can show:\n\n\\begin{proposition}\\label{prop:2primes}\n  If $p$ and $q$ are prime, $v_p(u^2) = -1 = v_q(u^2)$, and $t$ is integral at $p$ and at\n  $q$, then $\\mathcal{C}(\\Delta(u^2,2t))$ is not dense in $\\mathbb{Q}_p\\times\\mathbb{Q}_q$.\n\\end{proposition}\n\n\\noindent Results similar to Proposition~\\ref{prop:2primes} hold for $t$ that are\nnon-integral at $p$ or at $q$; we omit their statements for brevity. As a corollary to\nthe above two propositions, whenever $t$ is an integer and the denominator of $u^2$ is\ncomposite, $\\Delta(u^2,2t)$ is not pseudomodular. We have an even stronger result for\nintegral $t$:\n\n\\begin{proposition}\\label{prop:tisint}\n  Let $t$ be an integer and suppose $\\Delta(u^2,2t)$ is pseudomodular. Then\n  \\begin{enumerate}\n  \\item[\\emph{(a)}] $u^2$ has prime or unit denominator, say $p$,\n  \\item[\\emph{(b)}] if this $p$ is an odd prime, then $p$ does not divide $t$, and\n  \\item[\\emph{(c)}] for all odd primes $q$ dividing $t$, $u^2$ (necessarily in $\\mathbb{Z}_q$)\n    is equivalent to $0$ or $-1\\bmod q$.\n  \\end{enumerate}\n\\end{proposition}\n\nTo prove each of the above propositions, we give a proper nonempty $\\Delta$-invariant\nsubset $U$ of $\\mathbb{Q}$ that is open in the topology induced by that of $\\mathbb{Q}_p$ or\n$\\mathbb{Q}_p\\times\\mathbb{Q}_q$. For example, under the hypotheses of\nProposition~\\ref{prop:2primes}, the set of nonzero rationals $x$ for which exactly one of\n$v_p(x)$ and $v_q(x)$ is negative is $\\Delta$-invariant. To prove that our $U$ are\n$\\Delta$-invariant, we show that the generators $g_i^{\\pm 1}$ of $\\Delta$ map $U$ into\nitself.\n\nThe difficulty of our approach lies in finding such sets $U$. To do this, we search\ncomputed data for patterns in valuations and congruences. We build a data set by taking a\ntuple $(x_1,\\ldots,x_m)$ of rational numbers and applying to it several elements of\n$\\Delta$. Useful values for the $x_i$ are rational hyperbolic (or \\emph{special}) fixed\npoints, such as those tabulated in~\\cite{LR}. In some cases, a tuple of length one yields\nsome information, but typically we let the tuple be given by multiple pairs of special\nfixed points, with each pair fixed by a single hyperbolic element of $\\Delta$. When we\nhave no data on special fixed points, we select the $x_i$ with particular valuation or\ncongruence properties. In any case, a successful search suggests a candidate $U$, which we\nthen confirm as described above. Varying parameters such as the denominator of $u^2$, in\nsuch a way as to preserve the proof of the $\\Delta$-invariance, allows us to eliminate\ninfinite families of candidates for pseudomodularity and divorce our results from computed\ndata.\n\nIn the other direction, to show that $\\mathcal{C}(\\Delta)$ is dense in a finite product\n$H=\\prod_i\\mathbb{Q}_{p_i}$, we show that for each rational $x$, we can move $x$ arbitrarily\nclose to the cusp 0 with elements of (a group commensurable with)\n$\\Delta$. We use this argument to prove that for $t$ a prime integer, we have identified\nabove \\emph{all} cases for which $\\mathcal{C}(\\Delta(u^2,2t))$ is not dense in some $H$:\n\n\\begin{proposition}\n  If $t$ is prime, $u^2$ has prime denominator not equal to $t$ and $u^2\\equiv 0$ or\n  $-1\\bmod t$, then $\\mathcal{C}(\\Delta(u^2,2t))$ is dense in every finite product\n  $\\prod_i\\mathbb{Q}_{p_i}$ and hence is dense in the product $\\prod_p\\mathbb{Q}_p$ over all\n  primes.\n\\end{proposition}\n\n\\noindent There are groups with special fixed points to which this proposition applies,\nsuch as $\\Delta(6\/11,6)$ with a special fixed point of $1\/4$. Consequently:\n\n\\begin{corollary}\\label{cor:density}\n  Density of $\\mathcal{C}(\\Delta(u^2,2t))$ in the product $\\prod_p\\mathbb{Q}_p$ of all $p$-adic\n  fields is not a sufficient condition for pseudomodularity.\n\\end{corollary}\n\n\\noindent Both this corollary and Theorem~\\ref{thm:ad} can be interpreted in terms of congruence\ndata with respect to all primes. While density in the adelic topology implies\npseudomodularity, density in the product topology on all $p$-adic fields does not.\nTheorem~\\ref{thm:ad} holds because an orbit of a rational under a fixed translation\n$x\\mapsto x + t$ with $t\\in\\mathbb{Q}$ is open in $\\mathbb{Q}$ in the topology induced by\n$\\mathbb{A}_{\\fieldQ,f}$.\n\n\n\n\n\\section{Questions}\n\nSince $\\Lambda(u^2,2t)$ is a finitely generated subgroup of $\\operatorname{PSL}_2(\\mathbb{Q})$, it is\nalways a subgroup of $\\operatorname{PSL}_2(\\mathbb{Z}_S)$, where $\\mathbb{Z}_S =\n\\mathbb{Z}[p_1^{-1},\\ldots,p_r^{-1}]$ for some minimal set of primes $p_1,\\ldots,p_r$. By\nRiemann-Hurwitz, $\\Lambda(u^2,2t)$ always has exactly four orbits of cusps, so if we can\nshow that $\\Lambda(u^2,2t)$ has more than four orbits in its action on $\\mathbb{P}^1(\\mathbb{Q})$,\nthen some rationals cannot be cusps, whence $\\Lambda(u^2,2t)$ is not pseudomodular.\n\n\\begin{question}\n  For a non-pseudomodular $\\Lambda(u^2,2t)\\subset\\operatorname{PSL}_2(\\mathbb{Z}_S)$ as above, can we find a\n  subgroup $K$ of (finite index in) $\\operatorname{PSL}_2(\\mathbb{Z}_S)$ such that $\\Lambda(u^2,2t)\\subset\n  K$ and $K$ has more than four orbits in its action on $\\mathbb{P}^1(\\mathbb{Q})$?\n\\end{question}\n\nFor the groups already eliminated by our earlier propositions, the answer is yes. We also\nhave a positive answer for some groups whose cusps are dense in the product of all\n$\\mathbb{Q}_p$'s, such as $\\Lambda(6\/11,6)$. In fact, we can construct a\n$K\\subset\\operatorname{PSL}_2(\\mathbb{Z}[2^{-1}])$ containing $\\Lambda(6\/11,6)$ that has eight orbits in its\naction on $\\mathbb{P}^1(\\mathbb{Q})$. An explicit description of the orbits of $K$ gives us a\n$\\Delta(6\/11,6)$-invariant, nonempty, proper subset $X$ of $\\mathbb{P}^1(\\mathbb{Q})$ that is open\nin the 33-congruence topology, as defined here.\n\n\\begin{definition}\\label{def:congtop}\n  Denote by $\\Gamma(M)$ the principal congruence subgroup of $\\operatorname{PSL}_2(\\mathbb{Z})$ of level\n  $M$.  For a finite set of primes $p_1,\\ldots,p_r$, let $N=p_1\\ldots p_r$ and let\n  $\\mathcal{V}_N$ be the set of orbits $\\Gamma(N^j)\\cdot 0$ as $j$ ranges over positive\n  integers. The \\emph{$N$-congruence topology} on $\\mathbb{P}^1(\\mathbb{Q})$ is the topology\n  generated by the $\\operatorname{PSL}_2(\\mathbb{Z})$-translates of $\\mathcal{V}_N$.\n\\end{definition}\n\n\\noindent If we replace $\\Gamma(N^j)$ with $\\Gamma^0(N^j)$ (matrices lower-triangular mod\n$N^j$) in this definition then we recover the (coarser) topology on $\\mathbb{P}^1(\\mathbb{Q})$\ninduced by inclusion in $\\prod_{i=1}^r \\mathbb{P}^1(\\mathbb{Q}_{p_i})$. Thus, in\nPropositions~\\ref{prop:square} and \\ref{prop:2primes}, we also have cusp sets failing to\nbe dense in some congruence topology.\n\n\\begin{question}\n  If the cusps of $\\Delta$ are dense in $\\mathbb{P}^1(\\mathbb{Q})$ in all congruence topologies, is\n  $\\Delta$ necessarily pseudomodular or arithmetic? Equivalently, if $\\Delta$ is neither\n  pseudomodular nor arithmetic, is there a congruence subgroup $\\Gamma$ of\n  $\\operatorname{PSL}_2(\\mathbb{Z})$ so that $\\mathcal{C}(\\Delta)$ misses some orbit $\\Gamma\\cdot\n  x\\in\\mathbb{P}^1(\\mathbb{Q})$?\n\\end{question}\n\nMore generally, we can allow non-congruence subgroups $\\Gamma$ and we can ask analogous\nquestions for larger families of Fuchsian groups or for Kleinian groups whose cusps lie in\nan imaginary quadratic number field.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nFuture large-scale surveys such as \\textit{Euclid} \\citep{laureijs+11} and the Large Synoptic Survey Telescope \\citep{ivezic+08} will rely on accurate redshifts ($z$) measured to a significant portion of observed galaxies in order to measure the dark energy equation-of-state to target goals of a few percent accuracy using weak lensing \\citep{albrecht+06,bordoloi+12,newman+15}. Due to the challenge of obtaining spectroscopic redshifts (spec-z's) to the majority galaxies in these samples, many of these studies will heavily rely on accurate photometric redshifts (photo-z's) derived from galaxy broad- and\/or narrow-band spectral energy distributions (SEDs).\n\nTwo main techniques are used to derive photo-z's and the associated redshift probability distribution function (PDF) $P(z|\\rom{\\mathbf{F}}{obs})$ for a given set of observed fluxes $\\rom{\\mathbf{F}}{obs}$. Template fitting approaches \\citep[e.g.,][]{benitez00,bolzonella+00,feldmann+06,ilbert+06,cool+13,johnson+13,speagle+15} rely on deriving a set of forward mappings and their associated likelihoods from a collection of templates and associated model parameters to observed color space. Machine learning approaches \\citep[e.g.,][]{collisterlahav04,abazajian+09,liyee08,gerdes+10,carrascokindbrunner14,bonnett15,hoyle15,elliott+15,sadeh+15,almosallam+15}, on the other hand, use training data to derive the best inverse mapping from observed color space to redshift.\\footnote{See \\citet{hildebrandt+10}, \\citet{dahlen+13}, and \\citet{sanchez+14} for recent overviews and comparisons, and \\citet{menard+13} as well as related work for new applications of clustering-based redshifts (which will not be discussed here).}\n\nThe majority of photo-z developments in the last few years has been dominated by the introduction of numerous machine learning-based approaches. These are attractive relative to template fitting methods for several reasons:\n\\begin{enumerate}\n\t\\item \\textit{Single parameter estimation}: While template-fitting approaches are physically motivated, they involve fitting a number of extraneous parameters that must be marginalized over afterwards to derive $P(z|\\rom{\\mathbf{F}}{obs})$. Machine learning methods by design tend to focus on single-parameter estimation, making them somewhat more attractive.\n\t\\item \\textit{Computational speed}: Most template codes either resort to brute-force sampling \\citep{ilbert+06,cool+13} or Markov Chain Monte Carlo (MCMC) techniques \\citep{johnson+13,speagle+15} run over a large pre-generated grid of model photometry to derive $P(z|\\rom{\\mathbf{F}}{obs})$, both of which tend to take significantly more time than the majority of machine learning architectures.\n\t\\item \\textit{Empirically driven}: Most template fitting codes suffer from serious systematic uncertainties involved with modeling the impact of dust attenuation and emission line variation on an often limited set of templates \\citep[e.g.,][]{coleman+80,kinney+96,polletta+07} along with the need for good relative photometric calibrations. Machine learning methods are not subject to these uncertainties.\\footnote{If photo-z's are used to fit stellar population synthesis models at a later period, however, then good relative photometric calibrations still are required to avoid introducing biases.}\n\\end{enumerate}\n\nWhile machine learning approaches have been shown to be effective within the bounds of their training sets \\citep{sanchez+14}, attempts to extend their predictions beyond them have only been partially successful \\citep{hoyle+15}. Due to the limited regions of color space spanned by spec-z's training sets today \\citep{masters+15} and the difficulty of obtaining reliable spectra for higher redshift sources, it is likely that template-based photo-z's will continue to serve a crucial role in upcoming imaging surveys.\n\nThis paper is the latest in a series that attempt to make template fitting-based approaches more competitive in order to keep pace with the rapid development in machine learning applications.\nIn \\citet{speagle+15}, we investigated the general features of typical pre-generated model ``grids'' and were able to develop a simulated annealing and ensemble MCMC \\citep{goodmanweare10,foremanmackey+13} driven approach to exploring those grids more efficiently in an attempt to improve computational speed and efficiency. However, this did not affect the way that these grids of model photometry were created or improve on the modeling assumptions involved.\n\nIn an attempt to fix this glaring drawback, we introduced a new framework in Speagle \\& Eisenstein (2015a) (henceforth Paper I) that allows an algorithm to explore a large number of ``fuzzy archetypes'' using clustering methods such as Self Organizing Maps (SOMs). This approach significantly reduces systematics from dust and emission-line modeling and allows for an adaptive exploration of the space in a computationally efficient manner.\n\nIn this paper, we investigate the performance of several extensions of this new hybrid approach against a mock LSST and \\textit{Euclid} catalog. As the relevant likelihood surface is challenging to search, with many potential local minima that are often widely-separated \\citep{speagle+15}, we limit our focus in this paper to the case the same model parameters has been used to generate both the templates and observed galaxies. This enables us to better investigate the relative performance of each algorithm and avoid issues relating to template-mismatch systematics.\n\nThe paper is organized as follows. In \\S\\ref{sec:formalism}, we briefly summarize the basics of template-fitting approaches and the framework outlined in Paper I. In \\S\\ref{sec:methods}, we outline the eight basic methods we use to derive photo-z's using our fuzzy templates and SOMs. In \\S\\ref{sec:results}, we describe our mock catalog and examine the performance of each of our individual approaches. We conclude in \\S\\ref{sec:conc}.\n\nThroughout this work, boldface font ($\\mathbf{x}$, $\\mathbf{\\Theta}$) is used to represent vectors\/matrices while normal font ($x$, $\\Theta$) is used for singular variables. All observations are generated from 9 bands of data (LSST $ugrizY$ and \\textit{Euclid} $YJH$) with error properties based on $5\\sigma$ imaging depths of 26.5\\,mag (24.5 in $u$) for LSST and 24\\,mag for \\textit{Euclid} along with an assumed calibration uncertainty of 0.01\\,mag ($\\approx$\\,1\\%).\n\n\n\\section{Formalism}\n\\label{sec:formalism}\n\n\\subsection{Fuzzy Templates}\n\\label{subsec:fuzzy}\n\nModel photometric fluxes $\\rom{\\mathbf{F}}{model}$ for a given series of templates are generated via\n\\begin{equation}\\label{eq:phot}\n\\rom{\\mathbf{F}}{model}=\\frac{\\int_{\\nu_z}  S_{\\nu,\\textrm{model}}(\\nu) R_{\\textrm{model}}(\\nu) \\mathbf{T}(\\nu)\\nu^{-1}d\\nu}{\\int_{\\nu_z} \\mathbf{T}(\\nu) \\nu^{-1} d\\nu},\n\\end{equation}\nwhere the model template\n\\begin{equation}\nS_{\\nu,\\textrm{model}}(\\nu)=\\sum_{\\textrm{gal}} c_{\\textrm{gal}} \\times  S_{\\nu,\\textrm{gal}}(\\nu) + \\sum_{\\textrm{lines}} \\textrm{EW}_{\\textrm{lines}} \\times  S_{\\nu,\\textrm{lines}}(\\nu)\n\\end{equation}\nis a linear combination of ``basis'' galaxy \\citep[see, e.g.,][]{coleman+80,kinney+96,polletta+07} and emission line templates \\citep[e.g.,][]{kennicuttevans12} scaled by their equivalent widths (EWs), the dust (i.e. reddening) model\n\\begin{equation}\n-2.5\\log \\left(R_{\\textrm{model}}(\\nu)\\right) = \\sum_{\\textrm{dust}}\\rom{\\ensuremath{E(B\\,\\textrm{--}\\,V)}}{dust}\\rom{k}{dust}^\\prime(\\nu) + \\rom{A}{IGM}(\\nu,z)\n\\end{equation}\nis the sum of a linear combination of dust templates for a given $\\mathbf{E}(B-V)$ that are applied as a uniform screen, with the rest-frame dust screen $\\rom{k}{dust}^\\prime(\\nu)=\\rom{k}{dust}+c_{b,\\textrm{dust}}\\rom{k}{bump}$ composed of a combination of an underlying dust continuum and the 2175\\,{\\AA} bump \\citep{fitzpatrickmassa07} and the IGM reddening template given by the parameterization outlined in \\citet{madau95}. This combined template is convolved with the corresponding set of filter curves $\\mathbf{T}(\\nu)$ and over the redshifted set of frequencies $\\nu_z$ probed by the respective filters.\n\nAs many template-fitting approaches only explore a single basis galaxy and dust template at a given time, we henceforth ignore linear combinations of those respective components. In addition, for convenience we will re-parameterize our full set of parameters into two vectors: $\\boldsymbol{\\theta}=\\left\\lbrace z,\\textrm{gal},\\textrm{dust} \\right\\rbrace$, which contains the parameters surrounding the baseline archetype, and $\\boldsymbol{\\phi}=\\left\\lbrace \\ensuremath{E(B\\,\\textrm{--}\\,V)}, c_b^\\prime\\equiv c_b \\times \\ensuremath{E(B\\,\\textrm{--}\\,V)}, \\boldsymbol{\\Delta} \\textrm{\\textbf{EW}} \\right\\rbrace$, which contains the associated ``fuzzy'' parameters.\n\nFor computational reasons \\citep[see][]{speagle+15}, most codes opt to use large \\textit{pre-generated grids} of photometry from combinations of parameters sampled at a given granularity in each dimension during the fitting process. However, most of these grids are constructed using only a small collection of galaxy ($\\lesssim$\\,$30$) and dust ($\\lesssim$\\,$5$) templates taken from low-$z$ observations and then subsequently applied to the high-$z$ universe with large modifications.\n\nInstead of starting with a small number of templates in well-understood regions of color space and trying to simulate less-understood regions of color space, we instead attempt to sample the color-space manifold occupied by galaxies directly using a much larger set of galaxy spectral ``archetypes''. In order to allow for possible variation away from our discrete set of starting archetypes, we generate a small level of intrinsic ``fuzziness'' using sets of individualized, independent priors on dust attenuation and emission line strengths that allows for small deviations away from each baseline archetype. This transforms an individual galaxy's position in color space into a multidimensional PDF centered at its original location. By keeping the perturbations relatively small, we remain in the regime where the uniform dust screen approximation is likely to remain valid while also allowing emission line strengths to vary independently of one another.\n\n\nAs Paper I, we use the set of 129 high-quality UV\\,--\\,IR spectra from \\citet{brown+14} to serve as our baseline set of archetypes and measure the corresponding EWs of $\\lbrace$H$\\alpha$+N{\\scriptsize[II]}, H$\\beta$, H$\\gamma$, O{\\scriptsize[II]}, and the O{\\scriptsize[III]} doublet to derive appropriate emission line templates. These are combined with a series of 8 dust curves of the form $k(x)=a_x(\\beta)(x)^{-\\beta}$+$b_x(\\beta)+c_b\\rom{k}{bump}(x|x_0,\\gamma)$, where $\\beta=\\lbrace 0.3,0.45,0.6,0.75,0.9,1.05,1.2,1.35 \\rbrace$, $a_x$ and $b_x$ are chosen such that each dust curve is normalized to $A(V)=3.0$ and $A(2.5\\,\\textrm{$\\mu$m})=0$, and the mean amplitude ($c_b$), position ($x_0$), and width ($\\gamma$) of the 2175\\,{\\AA} bump are taken from the mean fits presented in \\citet{fitzpatrickmassa07}. We then choose $P(\\boldsymbol{\\phi})=P[\\Delta\\ensuremath{E(B\\,\\textrm{--}\\,V)}] \\times P[\\Delta c_b^\\prime] \\times P[\\boldsymbol{\\Delta} \\rom{\\textrm{\\textbf{EW}}}{line}]$ such that\n\\begin{eqnarray}\nP[\\Delta\\ensuremath{E(B\\,\\textrm{--}\\,V)}]=\\mathcal{N}[0,\\Delta A(1500\\textrm{\\,{\\AA}})=0.05\\,\\textrm{mag}], \\nonumber \\\\\nP[\\Delta c_b^\\prime]= \\mathcal{N}[0,1.5] \\times P[\\Delta\\ensuremath{E(B\\,\\textrm{--}\\,V)}],\\,\\,\\textrm{and} \\\\\nP[\\Delta \\rom{\\textrm{EW}}{line}]=\\mathcal{N}[0,0.2 \\times \\max(\\textrm{EW},0.005\\rom{\\lambda}{line})], \\nonumber\n\\end{eqnarray}\nwhere $\\mathcal{N}(\\mu,\\sigma)$ indicates a normal (i.e. Gaussian) distribution with mean $\\mu$ and standard deviation $\\sigma$. These form the underlying set of priors for our fuzzy archetypes. \n\nOur decision to fix $P[\\Delta\\ensuremath{E(B\\,\\textrm{--}\\,V)}]$ to $A(1500\\,{\\AA})$ limits the amount of dust extinction in the UV to the approximately linear regime ($\\sim$\\,$5\\%$ $1\\sigma$ variation) across the entire wavelength range for each dust curve. Likewise, our choice of prior for $\\Delta c_b^\\prime$ allows fluctuations to $0$ to take place at the $2\\sigma$ level, which enables us to fit dust curves without the impact of the 2175\\,{\\AA} bump while still disfavoring possible non-physical solutions. Finally, the 20\\% variation ($\\sim 0.1$\\,dex) in EW allowed by  $P(\\Delta \\rom{\\textrm{EW}}{line})$ mimics the observed scatter among emission line scaling relations \\citep{kennicutt98,kennicuttevans12}, while the EW ``floor'' leaves open the possibility for small variation at the $\\lesssim 1$\\,--\\,$2$\\% level.\n\n\n\\subsection{Self-Organizing Maps}\n\\label{subsec:som}\n\nThe Self-Organizing Map \\citep[SOM;][]{kohonen82,kohonen01} is an unsupervised machine learning algorithm that projects high-dimensional data onto a lower-dimensional space using competitive training of a large set of ``cells'' in a way that preserves general topological features and correlations present in the higher-dimensional data. Each cell in the grid is assigned a position $\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}$ and contains a cell model $\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}(t)$ that is the same size as a vector from the training data ($\\rom{N}{filter}$) and initialized (most often randomly) at iteration $t=0$. Training then proceeds as follows:\n\\begin{enumerate}\n\\item Draw (with replacement) a random object $\\rom{\\mathbf{F}}{obs}^{i}$ from the input dataset.\n\\item Compute $\\chi^2\\left(\\rom{\\mathbf{F}}{obs}^{i},\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}(t),\\rom{s}{cell}(t)\\right)$ for every cell on the map.\n\\item Select the best-matching (i.e. lowest $\\chi^2$) cell $\\rom{\\mathbf{x}}{som}^{\\textrm{best}}$ from the corresponding set of $\\chi^2$'s.\n\\item Update the cell models within the map according to an evolving \\textit{learning rate} $\\mathcal{A}(t)$ and \\textit{neighborhood function} (i.e. kernel density) $\\mathcal{H}(\\rom{\\mathbf{x}}{som}^{\\textrm{best}},\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}|t)$ such that\n\\begingroup\\makeatletter\\def\\f@size{8.5}\\check@mathfonts\n\\begin{equation}\n\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}(t+1)=\\rom{s}{cell}(t)\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}(t)+\\mathcal{A}(t)\\mathcal{H}(t)\\left[\\rom{\\mathbf{F}}{obs}^{i}-\\rom{s}{cell}(t)\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}\\right].\n\\end{equation}\n\\endgroup\n\\item Increment $t$ and repeat steps (i) through (iv) while $t<\\rom{N}{iter}$.\n\\end{enumerate}\nAfter training, objects are ``sorted'' onto the map by repeating steps (ii) and (iii) for every object in the input dataset and assigning them to the best-matching cell.\n\nFollowing Paper I, we set\n\\begin{eqnarray}\n\\mathcal{A}(t)=a_0\\left(\\frac{a_1}{a_0}\\right)^{t\/\\rom{N}{iter}}, \\nonumber \\\\\n\\mathcal{H}(t|\\rom{\\mathbf{x}}{som}^{\\textrm{best}},\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}) = \\exp\\left(-\\frac{||\\rom{\\mathbf{x}}{som}^{\\textrm{best}}-\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}||^2}{\\sigma^2(t)}\\right), \\textrm{ and} \\\\\n\\sigma(t)=\\sigma_0\\left(\\frac{1}{\\sigma_0}\\right)^{t\/\\rom{N}{iter}}, \\nonumber\n\\end{eqnarray}\nwhere $\\mathcal{A}(t)$ and $\\mathcal{H}(t)$ are now functions of the hyper-parameters $a_0$, $a_1$, and $\\sigma_0$.\n\nWe use the SOM to organize both rest-frame and observed-frame colors using ``model catalogs'' generated from numerous Monte Carlo realizations of our fuzzy templates and a given $P(\\boldsymbol{\\theta})$. These are generated as follows:\n\\begin{enumerate}\n\t\\item \\textit{Rest-frame SOM } (2-D): We construct a model catalog of 9,675 objects using 75 Monte Carlo realizations from $P(\\boldsymbol{\\phi})$ and uniform sampling of $\\rom{k}{dust}$ for each fuzzy archetype, which are assigned $1$\\% errors to in order to ensure the SOM training proceeds in logarithmic rather than linear space. We construct a two-dimensional $30 \\times 30$ SOM and initialize the cell models randomly on the uniform interval $[0,1]$. We set $[a_0,a_1,\\sigma_0]=[1,0.5,30]$ and allow the map to train for $\\rom{N}{iter}=10,000$ iterations. The corresponding set of mappings from the model catalog onto the SOM are then recorded.\n\t\\item \\textit{Observed-frame SOM} (3-D): We construct a model catalog of 3,228,225 objects using 25 Monte Carlo realizations from $P(\\boldsymbol{\\phi})$ with uniform sampling from $\\rom{k}{dust}$ for each fuzzy archetype sampled on every point of an input $z=0$\\,--\\,$10$ ($\\Delta z=0.01$) redshift grid. Each object is assigned $1$\\% errors as well as a baseline error floor of $0.01$, which are added in quadrature. We construct a three-dimensional $30 \\times 30 \\times 30$ SOM and initialize the cell models according to pre-defined color gradients in $u-r$, $r-z$, and $z-H$. We set $[a_0,a_1,\\sigma_0]=[0.5,0.2,10]$ and allow the map to train for $\\rom{N}{iter}=1,000,000$ iterations. The corresponding set of mappings from the model catalog onto the SOM are then recorded.\n\\end{enumerate}\n\n\n\n\n\\section{Methods}\n\\label{sec:methods}\n\n\nTo test whether our use of (linearized) fuzzy templates and Self-Organizing Maps (SOMs) can be used to derive photo-z's, we decide to compare the following eight different methods:\n\\begin{enumerate}\n\t\\item {\\tt{BruteForce}}: A standard brute-force approach that involves comparing against all 3,228,225 models used to create the 3-D observed-frame SOM.\n\t\\item {\\tt{BruteForce\\_LinearFuzzy}}: A modified brute-force approach that explores all $N_z \\times \\rom{N}{gal} \\times \\rom{N}{dust} = 1,033,032$ possible combinations of our non-linear parameters $\\boldsymbol{\\theta}$, where at each step we optimize over all linear parameters $\\boldsymbol{\\phi}$.\n\t\\item {\\tt{SOM\\_MCMC\\_RestFrame}}: An MCMC-based approach that explores a 2-D SOM constructed with rest-frame realizations of our fuzzy templates along with changes in photometry as a function of redshift.\n\t\\item {\\tt{SOM\\_MCMC\\_ObservedFrame}}: An MCMC-based approach that explores a 3-D SOM constructed from \\textit{observed-frame} realizations (i.e. redshift already included) of our fuzzy templates.\n\t\\item {\\tt{SOM\\_Hierarchical\\_MonteCarlo}}: A hierarchical sampling approach that uses a fixed number of randomly sampled objects drawn from each SOM cell.\n\t\\item {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}: A hierarchical sampling approach that draws objects randomly from SOM cells based on an adaptive SOM importance density.\n\t\\item {\\tt{SOM\\_CellModel\\_LimitedSum}}: A limited reconstruction using all models contained within a small subset of SOM cells selected based on their SOM cell models.\n\t\\item {\\tt{SOM\\_CellModel\\_Average}}: A weighted redshift average across the SOM based on the SOM cell models.\n\\end{enumerate}\n\nWe discuss each of these in turn.\n\n\\subsection{{\\tt{BruteForce}} (gold standard)}\n\\label{subsec:BruteForce}\n\nDeriving $P(z|\\rom{\\mathbf{F}}{obs})$ for any individual object $\\rom{\\mathbf{F}}{obs}$ in a brute-force approach involves computing the likelihood $P(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs})$ to the entire collection $\\rom{\\mathbf{F}}{model}(\\boldsymbol{\\theta},\\boldsymbol{\\phi})$'s. Each $\\rom{\\mathbf{F}}{model}$ has a corresponding redshift attached to it, and the final $P(z|\\rom{\\mathbf{F}}{obs})$ is constructed by marginalizing over all the other input parameters such that\n\\begin{equation}\\label{eq:pz}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs}) \\propto \\sum_{\\textrm{dust}}\\sum_{\\textrm{gal}}\\sum_{\\boldsymbol{\\phi}} w(\\boldsymbol{\\theta},\\boldsymbol{\\phi}) P(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs}),}\n\\end{equation}\nwhere $w(\\boldsymbol{\\theta},\\boldsymbol{\\phi})$ is an associated model weight (from, e.g., priors on $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$) and $P(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs})$ is the computed likelihood between the data and the model. For the purposes of this paper, we take our collection of model photometry to be the 3,228,225 objects included in our observed-frame model catalog (see \\S\\ref{subsec:som}) and assign all models equal weight such that $w(\\boldsymbol{\\theta},\\boldsymbol{\\phi})=1$.\n\n\\subsection{{\\tt{BruteForce\\_LinearFuzzy}}}\n\\label{subsec:BruteForce_linear}\n\nIn order to analytically marginalize over the ``fuzziness'' in our fuzzy templates, we first need to render them in a more statistically-friendly and computationally-convenient form. As outlined in Paper I, we can accomplish this by approximating the changes in the underlying archetype as being linear in $\\boldsymbol{\\phi}$ and recasting the fitting procedure as a (conditional) linear algebra problem that can be solved using iterative least squares. We detail the full derivation in Appendix~\\ref{app:linear_fuzzy} and give a brief outline below.\n\nTaylor expanding around the baseline archetype and ignoring all terms higher than first order, $\\rom{\\mathbf{F}}{model}(z)$ can be written as\n\\begin{eqnarray}\ns\\rom{\\mathbf{F}}{model}(z) = s\\rom{\\mathbf{F}}{gal}(z) + \\Delta\\ensuremath{E(B\\,\\textrm{--}\\,V)} \\,s\\rom{\\mathbf{F}}{dust}(z) \\nonumber \\\\\n+ \\Delta c_b^\\prime \\,s\\rom{\\mathbf{F}}{bump}(z) + \\sum_{\\textrm{lines}} \\Delta \\textrm{EW}_{\\textrm{lines}}^\\textrm{gal} \\,s\\rom{\\mathbf{F}}{lines}^{\\textrm{gal}}(z),\n\\end{eqnarray}\nwhere $\\rom{\\mathbf{F}}{gal}(z)$ is the flux of the baseline galaxy archetype, $\\rom{\\mathbf{F}}{dust}(z)$ is the ``dust flux'', $\\rom{\\mathbf{F}}{bump}(z)$ is the ``2175\\,{\\AA} bump flux'', and $\\rom{\\mathbf{F}}{lines}^{\\textrm{gal}}(z)$ is the corresponding emission line flux for a given archetype and emission line.\n\nAssuming that our priors are Gaussian, the corresponding log-likehood is\n\\begingroup\\makeatletter\\def\\f@size{8.2}\\check@mathfonts\n\\begin{equation}\\label{eq:chi2_mod}\n-2\\log P(\\boldsymbol{\\theta},\\boldsymbol{\\phi},s|\\rom{\\mathbf{F}}{obs}) \\equiv \\sum_i \\sigma_{i}^{-2} \\left[\\Delta F_i(\\boldsymbol{\\theta},\\boldsymbol{\\phi},s)\\right]^2 + \\sum_j \\left(\\frac{\\phi_j}{\\sigma_{\\phi_j}(\\boldsymbol{\\theta})}\\right)^2\n\\end{equation}\n\\endgroup\nwhere $\\boldsymbol{\\sigma}^2$ is the observed variances (we take all model errors to be 0), $\\boldsymbol{\\Delta}\\mathbf{F}(\\boldsymbol{\\theta},\\boldsymbol{\\phi},s) = \\mathbf{F}_{\\textrm{obs}} - s\\mathbf{F}_{\\textrm{model}}(\\boldsymbol{\\theta},\\boldsymbol{\\phi})$ are the associated flux residuals, $\\boldsymbol{\\sigma}_{\\phi}(\\boldsymbol{\\theta})$ are the standard deviations of the corresponding Gaussian priors for a given $\\boldsymbol{\\theta}$, the sum over $i$ is taken over all observed bands, and the sum over $j$ is taken over all relevant nuisance parameters. This allows us to re-write our expression for $\\rom{\\mathbf{F}}{model}$ as\n\\begin{equation}\\label{eq:linear_phot}\nsF_{\\textrm{model},i}(\\boldsymbol{\\theta},\\boldsymbol{\\phi})=sF_{\\textrm{gal},i}(\\boldsymbol{\\theta})+\\sum_j s\\mathbf{X}_{ij}(\\boldsymbol{\\theta}) \\boldsymbol{\\phi}_j,\n\\end{equation}\nwhere $\\mathbf{X}(\\boldsymbol{\\theta})$ is an $\\rom{N}{filt} \\times (2+\\rom{N}{lines})$ matrix of pre-computed coefficients for a given $\\boldsymbol{\\theta}$. \n\nFor fixed $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$, we can marginalize over $s$ to minimize $\\rom{\\chi^2}{mod}(s|\\boldsymbol{\\theta},\\boldsymbol{\\phi})$ via\n\\begin{equation}\\label{eq:chi_s}\ns = \\left. {\\sum_i {\\sigma_i^{-2}}} F_{\\textrm{obs},i}F_{\\textrm{model},i} \\middle\/ {\\sum_i \\sigma_i^{-2} F_{\\textrm{model},i}F_{\\textrm{model},i}} \\right. ,\n\\end{equation}\nwhile for fixed $s$ and $\\boldsymbol{\\theta}$, we can marginalize over $\\boldsymbol{\\phi}$ by solving \n\\begin{equation}\\label{eq:chi_lparams}\n\\left(\\mathbf{X}(\\boldsymbol{\\theta})^T\\rom{\\mathbf{W}}{obs}\\right)\\Delta\\rom{\\mathbf{F}}{gal}(\\boldsymbol{\\theta})=\\left(\\mathbf{X}(\\boldsymbol{\\theta})^{T}\\rom{\\mathbf{W}}{obs}\\mathbf{X}(\\boldsymbol{\\theta})+\\mathbf{W}_\\phi(\\boldsymbol{\\theta})\\right)\\boldsymbol{\\phi},\n\\end{equation}\nwhere $\\Delta\\rom{\\mathbf{F}}{gal}(\\boldsymbol{\\theta}) = \\mathbf{F}_{\\textrm{obs}} - s\\rom{\\mathbf{F}}{gal}(\\boldsymbol{\\theta})$ is the baseline galaxy flux residual, $\\rom{\\mathbf{W}}{obs}=\\textrm{diag}(\\dots,\\sigma_i^{-2},\\dots)$ is the associated observational weight matrix, $\\mathbf{W}_\\phi=\\textrm{diag}(\\dots,\\sigma_{\\phi_j}^{-2},\\dots)$ is the prior weight matrix, and $T$ is the transpose operator.\n\nThis gives us an simple iterative scheme for computing $\\rom{P}{max}(\\boldsymbol{\\theta}|\\rom{\\mathbf{F}}{obs})$ with respect to $s$ and $\\boldsymbol{\\phi}$ for a given choice of $\\boldsymbol{\\theta}$. Using this approach, we execute a brute-force search over all $N_z \\times \\rom{N}{gal} \\times \\rom{N}{dust} = 1,033,032$ possible combinations of our non-linear parameters $\\boldsymbol{\\theta}$. This gives us\n\\begin{equation}\\label{eq:pz_linear}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs}) \\propto \\sum_{\\textrm{dust}}\\sum_{\\textrm{gal}} \\rom{P}{max}(\\boldsymbol{\\theta}|\\rom{\\mathbf{F}}{obs}).}\n\\end{equation}\n\n\\subsection{{\\tt{SOM\\_MCMC\\_RestFrame}}}\n\\label{subsec:mcmc_restframe}\n\n\nIn Paper I, we introduced a method to derive estimates to $P(z|\\rom{\\mathbf{F}}{obs})$ in a more exact sense using Markov Chain Monte Carlo (MCMC) methods running ``over'' the SOM whose performance only depends on the ability of the SOM to effectively cluster the space in a topologically smooth fashion. We summarize our approach here and direct the reader to Paper I for additional details.\n\nIn brief, we implement a hybrid version of MCMC sampling that transitions to new regions based on comparisons between individual sets of model photometry but with chains that live ``on top of'' the SOM. Chains propose new models in a hierarchical fashion by first proposing a new cell position\\footnote{If a proposed cell has no objects, the process is repeated from the proposed cell until a valid cell is reached.} and then selecting a random model within that cell. Each chain then transitions to the proposed cell with an associated transition probability that satisfies detailed balance.\n\nAfter the chains have converged to the stationary target distribution (the ``burn-in'' phase), they are allowed to sample for a set number of iterations. The final $P(z|\\rom{\\mathbf{F}}{obs})$ can then be reconstructed via\n\\begingroup\\makeatletter\\def\\f@size{8.7}\\check@mathfonts\n\\begin{equation}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs})\\approx\\sum_{\\textrm{chain}} \\rom{w}{chain} P(z|\\textrm{chain},\\rom{\\mathbf{F}}{obs}).}\n\\end{equation}\n\\endgroup\nwhere $\\rom{w}{chain} = \\rom{P}{max}(\\rom{\\mathbf{F}}{obs}|\\rom{\\mathbf{F}}{model},\\textrm{chain})$ is a maximum-likelihood weight used to suppress chains that happen to converge to spurious minima and\n\\begin{equation}\nP(z|\\textrm{chain},\\rom{\\mathbf{F}}{obs})={\\sum_{\\textrm{dust}}}^{\\textrm{chain}} {\\sum_{\\textrm{gal}}}^{\\textrm{chain}} {\\sum_{\\boldsymbol{\\phi}}}^{\\textrm{chain}} w(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs})\n\\end{equation}\nis the sum over the trials from each individual chain with $w(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs})$ the associated weight of each trial. This is usually taken to be $w=1$ if the trial was accepted and $w=0$ if the trial was rejected, although other weighting schemes are possible \\citep{bernton+15}.\n\nWe implement this sampling scheme on the 2-D rest-frame SOM with where new cells are proposed according to a multivariate Gaussian with $\\boldsymbol{\\Sigma} = \\textrm{diag}\\left[(\\rom{\\sigma}{som}^\\textrm{dim\\_1})^2, (\\rom{\\sigma}{som}^\\textrm{dim\\_1})^2\\right]$ where $\\rom{\\sigma}{som}^\\textrm{dim\\_1}=\\rom{\\sigma}{som}^\\textrm{dim\\_2}=2$. Once a specific archetype, dust template, and $\\boldsymbol{\\phi}$ have been chosen, we generate the associated redshifted photometry based on a separate Gaussian proposal with $\\sigma_z=2.5\\%$ of the corresponding redshift grid.\n\nTo improve performance and reduce the dependence on our initial choice of parameters, we include a short learning step during every MCMC run after the burn-in phase is complete where the corresponding size of each component of the proposal ($\\lbrace\\rom{\\sigma}{som}^\\textrm{dim\\_i}\\rbrace_i,\\sigma_{z}$) is adjusted to better approximate the underlying shape of the PDF. We set a fixed number of iterations for burn-in, learning, and sampling of $5000$, $1000$, and $5000$ trials, respectively, and sample each object using 32 chains run in parallel.\n\n\\subsection{{\\tt{SOM\\_MCMC\\_ObservedFrame}}}\n\\label{subsec:mcmc_observedframe}\n\nIn \\S\\ref{subsec:mcmc_restframe}, we used a 2-D SOM with 900 cells to sort 9675 rest-frame relations of our fuzzy archetypes that were then allowed to vary as a function of redshift. However, we can instead attempt to reorganize the entire fuzzy archetype-redshift \\textit{observed-frame} color space using a single SOM. We use this approach to construct an 3-D observed-frame SOM composed of 27,000 cells based on $\\sim$\\,$\\sn{3}{6}$ individual realizations of our model photometry from $z=0$\\,--\\,$10$.\n\nTo explore this observed-frame SOM, we again implement the MCMC sampling scheme described above on the 3-D observed-frame SOM. New cells are proposed according to the multivariate Gaussian distribution with $\\boldsymbol{\\Sigma} = \\textrm{diag}\\left[(\\rom{\\sigma}{som}^\\textrm{dim\\_1})^2, (\\rom{\\sigma}{som}^\\textrm{dim\\_2})^2,(\\rom{\\sigma}{som}^\\textrm{dim\\_3})^2\\right]$ where $\\rom{\\sigma}{som}^\\textrm{dim\\_1}=\\rom{\\sigma}{som}^\\textrm{dim\\_2}=\\rom{\\sigma}{som}^\\textrm{dim\\_3}=2$. As before, we set a fixed number of iterations for burn-in, learning, and sampling of $5000$, $1000$, and $5000$ trials, respectively, and sample each object using 32 chains run in parallel.\n\n\\subsection{{\\tt{SOM\\_Hierarchical\\_MonteCarlo}}}\n\\label{subsec:stratifiedMC}\n\nRather than taking advantage of the topologically smooth nature of the SOM to ``explore'' its corresponding reduced-dimensional manifold, we can instead treat the SOM as simply a way to partition the model space into a series of  clusters (i.e. cells). Rather than using the relationship between these clusters (i.e. the ``graph'' they form) to define some sampling algorithm, we can then treat each cluster as an independent entity. Thus, instead of sampling from a local neighborhood function in SOM space to determine what quasi-contiguous regions of the SOM contain the majority of the likelihood, we instead wish to determine which isolated clusters contain significant amounts of likelihood to merit further exploration and\/or sampling.\n\nIn other words, we want to break down the problem into a \\textit{hierarchical sampling} approach, where we first consider which clusters (stored in some arbitrary, hierarchical structure) to explore before considering the individual models themselves. In the case of the SOM, where the clusters are organized along a single manifold, this is a simple two-tiered problem involving the individual cells and the objects contained within them.\n\nThe process of determining how likely a particular $\\rom{\\mathbf{F}}{obs}$ (along with associated errors $\\rom{\\boldsymbol{\\sigma}}{obs}$) is matched to a particular cluster is an open problem. As clusters can have very different distributions of galaxy colors and occupation rates depending of its location in color space (Paper I), the use of simple summary statistics will often introduce biases depending on the properties data in question (see \\S\\ref{sec:results}).\n\nAlternately, we can directly sample from the PDF spanned by each cluster using Monte Carlo techniques. Unlike the case of summary statistics, this approach is guaranteed to give a good approximation to the likelihood contained within each cluster when the number of samples $\\rom{N}{MC}\\sim N(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$. \n\nIn addition to giving us information on the likelihood contained within each cluster, such an approach naturally introduces a \\textit{stratified Monte Carlo} sampling scheme to uncover the underlying PDF, where random model photometry drawn from partitioned regions of color space can be used to stochastically reconstruct the color-redshift relation probed by $\\rom{\\mathbf{F}}{obs}$ and established by the individual SOM cells. In the case outlined above where $\\rom{N}{MC}\\sim N(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ for each cell, this approach simply reduces to a Monte Carlo version of a brute-force approach.\n\nHowever, the power of this stratified Monte Carlo approach is especially apparent in the sparse-sampling regime where only a small amount of objects is drawn from each cluster. While such an approach might not be effective in determining the precise likelihood contained within individual cluster, it does provide a fair statistical representation of a sample galaxy's distribution over the entire region of color space spanned by the collection of SOM cells. Sparse sampling of this sort using single-draw Monte Carlo approaches has been shown to be quite effective at reconstructing $N(z)$ distributions (and relevant summary statistics) to ensembles of objects, as shown in, e.g., \\citet{wittman09}, \\citet{carrascokindbrunner14c}, and \\citep{masters+15}.\n\nDue to the small number of occupied cells in our observed-frame SOM ($\\sim$\\,21,600 out of 27,000), we decide to simply draw an equal number of samples from occupied cells on the SOM. The corresponding $P(z|\\rom{\\mathbf{F}}{obs})$ can then be directly calculated via\n\\begin{equation}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs})\\approx\\sum_{\\textrm{cell}} \\sum_{\\mathbfcal{R}(\\textrm{cell})} N(\\mathbf{x}_{\\textrm{som}}^{\\textrm{cell}}) P(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}|\\rom{\\mathbf{F}}{obs}),}\n\\end{equation}\nwhere $\\sum_{\\textrm{cell}}$ is taken over all occupied cells, $\\mathbfcal{R}(\\textrm{cell})$ is a collection of $\\rom{N}{MC}$ random numbers $\\lbrace\\mathcal{R}_i\\rbrace_i$ that select the corresponding indices of model fluxes $\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}$ contained within a given cell, and $N(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ accounts for the different occupation rates of each cell. We set $\\rom{N}{MC}=5$ to allow for more robust sampling than single-draw approaches while still keeping the number of total draws ($\\sim$\\,110,000) a factor of $\\sim$\\,30 below the total number of available models.\n\n\\subsection{{\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}}\n\\label{subsec:importancesampling}\n\nWhile the stratified Monte Carlo approach is easy to implement and relatively resilient against random misses (since every cell on the SOM is sampled evenly), most objects with well-localized colors are strongly associated with a very small set of clusters. As a result, a significant amount of computation time will be wasted evaluating clusters containing model fluxes with $P(\\rom{\\mathbf{F}}{obs}|\\rom{\\mathbf{F}}{model})\\approx 0$. Rather than drawing an equal number of samples from every SOM cell, we could use preliminary approximations of the likelihood across the SOM to instead take advantage of adaptive \\textit{importance sampling} to favor target cells that are more likely to contain the majority of the likelihood. \n\nIn the scheme of the hierarchical interpretation outlined above in \\S\\ref{subsec:stratifiedMC}, we can approximate the likelihood contained in each individual cluster $P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})$ via the identity\n\\begin{eqnarray}\nP(\\rom{\\mathbf{F}}{obs})=\\int P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})P(\\rom{\\mathbf{F}}{obs}|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})d\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}\\nonumber \\\\\n=\\int\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})\\frac{P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})P(\\rom{\\mathbf{F}}{obs}|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})}{\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})}d\\rom{\\mathbf{x}}{som}^{\\textrm{cell}},\n\\end{eqnarray}\nwhere $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ is a generic approximation to $P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})$. For a series of $\\rom{N}{IS}$ samples drawn from $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$, the $n$th posterior moment of $P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})$ can then be estimated as\n\\begingroup\\makeatletter\\def\\f@size{8.7}\\check@mathfonts\n\\begin{equation}\n\\int (\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})^nP(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})d\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}=\\frac{\\int (\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})^n \\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}) w(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})d\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}}{\\int \\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})w(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})d\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}}\\nonumber\n\\end{equation}\n\\endgroup\n\\begin{equation}\n\\approx \\left. {\\sum_{\\rom{N}{IS}} (\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})^n  w(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})} \\middle\/ {\\sum_{\\rom{N}{IS}} w(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})} \\right. ,\n\\end{equation}\nwhere $w(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})=\\frac{P(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})P(\\rom{\\mathbf{F}}{obs}|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})}{\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})}$ is the associated \\textit{importance weight} -- the overall likelihood of a given trial formed by a combination of the initial prior and the computed likelihood normalized by the proposal distribution. Estimates of the relevant marginalized distributions can be constructed by assigning each sampled location an appropriate \\textit{kernel density} with an amplitude proportional to its importance weight.\n\nUsing the same logic as \\S\\ref{subsec:stratifiedMC}, implementing a Monte Carlo sampling scheme within each cluster allows us to separate out the proposal distribution $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ which operates on the set of clusters and the associated importance weights $w(\\rom{\\mathbf{F}}{model})$ assigned to each particular chosen model. Assuming $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ is properly normalized, the corresponding importance weight for a random model $\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}$ located at cell $\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}$ is then\n\\begin{equation}\nw(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i})=N(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})\\frac{P(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}|\\rom{\\mathbf{F}}{obs})}{\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})}.\n\\end{equation}\n\nFor a number of $\\rom{N}{IS}$ samples drawn randomly from cells drawn from $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$, we can then estimate $P(z|\\rom{\\mathbf{F}}{obs})$ as\n\\begin{equation}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs})\\approx \\sum_{\\mathbfcal{R}(\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}))} w(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}) P(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}|\\rom{\\mathbf{F}}{obs}).}\n\\end{equation}\nIn this paper, we have avoided the use of kernels to avoid additional computational overhead.\n\nFor the implementation used in this paper, we further build on this general approach using an adaptive importance sampling scheme equivalent to a crude form of Sequential\/Population Monte Carlo \\citep{doucet+01} and related methods:\n\\begin{enumerate}\n\t\\item Compute $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ using the stratified Monte Carlo scheme outlined in \\S\\ref{subsec:stratifiedMC} with $\\rom{N}{MC}$ random models sampled in each cell.\n\t\\item Draw $\\rom{N}{batch}$ cells from $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$.\n\t\\item Draw a random model $\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i}$ from each cell.\n\t\\item Use the corresponding set of $w(\\rom{\\mathbf{F}}{model}^{\\mathcal{R}_i})$'s to update $\\pi(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$.\n\t\\item Repeat steps (ii) through (iv) for $\\rom{N}{iter}$ iterations.\n\t\\item Use the total number of objects sampled from steps (i) through (v) to compute $P(z|\\rom{\\mathbf{F}}{obs})$.\n\\end{enumerate}\nWe set $\\rom{N}{MC}=1$, $\\rom{N}{batch}=2500$, and $\\rom{N}{iter}=20$ in this paper, giving us a total of $\\sim$\\,70,000 models comparisons (slightly less than the amount used in \\S\\ref{subsec:stratifiedMC}). In addition, we impose a ``likelihood floor'' for each computed weight of $10^{-3}$ when updating the proposal distribution both to prevent multi-valued inverse functions when drawing from the proposal and to ensure each cell has a small (but non-negligible) probability of being selected with each draw in order to ensure great robustness during sampling. Note that this likelihood floor is \\textit{not} used when constructing $P(z|\\rom{\\mathbf{F}}{model})$.\n\n\\subsection{{\\tt{SOM\\_CellModel\\_LimitedSum}}}\n\\label{subsec:som_limited}\n\nInstead of sampling over the entire SOM, we can instead attempt to limit the total number of terms included in the sum to only cells which carry significant weight, after which we can just compute the reduced set of $P(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$'s directly. In other words, after computing the full set of $P(\\rom{\\mathbf{F}}{obs},\\rom{\\mathbf{F}}{som}^{\\textrm{cell}})$'s across the 3-D observed-frame SOM for a particular object, we select a collection of $k$ cells such that\n\\begin{equation}\nP(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})>c_\\eta \\textrm{max}\\left[P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})\\right],\n\\end{equation}\nwhere $c_\\eta$ is a specified acceptance threshold which we take to be $10^{-3}$ and $\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}$ is the associated SOM cell model (see \\S\\ref{subsec:som}), which we assume to be a reasonable proxy of the data in a given cell. As noted in \\S\\ref{subsec:stratifiedMC}, while this assumption is not always true (especially in sparely populated and\/or highly variable regions of color space), it is a good first-order approximation to the problem.\n\nAfter using the cell models to select our best-fitting cells, we can then compute $P(z|\\rom{\\mathbf{F}}{obs})$ as\n\\begingroup\\makeatletter\\def\\f@size{8.4}\\check@mathfonts\n\\begin{equation}\n\\boxed{P(z|\\rom{\\mathbf{F}}{obs}) \\approx \\sum_k P(z|\\mathbf{x}_k) = \\sum_k {\\sum_{\\textrm{dust}}}^{\\mathbf{x}_k} {\\sum_{\\textrm{gal}}}^{\\mathbf{x}_k} {\\sum_{\\boldsymbol{\\phi}}}^{\\mathbf{x}_k} P(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rom{\\mathbf{F}}{obs}),}\n\\end{equation}\n\\endgroup\nwhere ${\\sum}^{\\mathbf{x}_k}$ indicates the sum is taken over all objects sorted into the $k$th cell. As the number of terms included in the sum over $k$ should only be a small fraction of the total number of cells, it should be reasonably quick to compute.\n\n\\subsection{{\\tt{SOM\\_CellModel\\_Average}}}\n\\label{subsec:som_average}\n\nInstead of comparing individual models and\/or fuzzy templates, we now wish to take advatange of the quantization the SOM provides by assigning the models to a discrete number of cells. In particular, every cell in our 3-D observed-frame SOM has an associated redshift number distribution $N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$, whose combined sum would be proportional to the input redshift prior which all the objects in our model catalog were drawn from. For any individual object, this $N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ distribution corresponds to a specific redshift probability $P(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}},\\rom{\\mathbf{F}}{obs})$ for the given cell through a redshift-dependent weight function $w(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}},\\rom{\\mathbf{F}}{obs})$. This allows us to write $P(z|\\rom{\\mathbf{F}}{obs})$ as\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\begin{equation}\nP(z|\\rom{\\mathbf{F}}{obs})=\\sum_{\\textrm{cell}}P(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}},\\rom{\\mathbf{F}}{obs})=\\sum_{\\textrm{cell}} w(z|\\rom{\\mathbf{F}}{obs},\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}),\n\\end{equation}\n\\endgroup\nwhere unoccupied cells again are excluded from the sum.\n\nAssuming that the SOM has clustered objects efficiently relative to the observed errors on $\\rom{\\mathbf{F}}{obs}$, we roughly can approximate the likelihood of a given cell as simply being proportional to its intrinsic redshift distribution such that $P(z|\\rom{\\mathbf{F}}{obs},\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}) \\sim N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$. If we further assume (as in \\S\\ref{subsec:som_limited}) that the cell model is a good proxy for the data in a given cell, we can further approximate the cell weight as being the likelihood of the corresponding cell model $P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})$. This gives, up to a normalization factor, \n\\begin{equation}\nP(z|\\rom{\\mathbf{F}}{obs}) \\approx \\sum_{\\textrm{cell}} P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}).\n\\end{equation}\n\nAs each $N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ is an array of $N_z$ elements, this sum is computationally taxing due to the large number of cells used in the construction of the observed-frame SOM. As a result, we instead opt to use a point estimate for $N(z|\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$ based on its zeroth moment (i.e. mean) $\\bar{z}(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$, which can be pre-computed and saved to memory. The mean weighted redshift across the SOM for a given object can then be written as\n\\begin{equation}\n\\boxed{\\rom{z}{avg}(\\rom{\\mathbf{F}}{obs}) = \\left . \\sum_{\\textrm{cell}} P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})\\bar{z}(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}}) \\middle \/ \\sum_{\\textrm{cell}} P(\\rom{\\mathbf{F}}{obs},\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}) \\right . .}\n\\end{equation}\nWhile the full $P(z|\\rom{\\mathbf{F}}{model})$ distribution could be recovered using kernel density estimation centered on the corresponding collection of $\\bar{z}(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})$'s and $P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})$'s (see \\S\\ref{subsec:importancesampling}), for computational convenience we instead opt to compute only the first moment\n\\begingroup\\makeatletter\\def\\f@size{8.5}\\check@mathfonts\n\\begin{equation}\n\\sigma_{z,\\textrm{avg}}^2(\\rom{\\mathbf{F}}{obs}|\\rom{z}{avg}) = \\frac{\\sum_{\\textrm{cell}}  P(\\rom{\\mathbf{F}}{obs},\\rom{\\mathbf{F}}{som}^{\\textrm{cell}})\\left[\\bar{z}(\\rom{\\mathbf{x}}{som}^{\\textrm{cell}})-\\rom{z}{avg}(\\rom{\\mathbf{F}}{obs})\\right]^2}{\\sum_{\\textrm{cell}} P(\\rom{\\mathbf{F}}{som}^{\\textrm{cell}}|\\rom{\\mathbf{F}}{obs})}.\n\\end{equation}\n\\endgroup\nGiven that this method is the ``crudest'' approach that makes the strongest assumptions concerning the data and the SOM, we take this as a suitable estimate of the shape of the corresponding PDF.\n\n\\subsection{Summary}\n\\label{subsec:pz_list}\n\nWe have outlined eight separate methods to derive photo-z's for individual objects. Each pair of methods (brute-force, MCMC, hierarchical, cell model-dependent) represent four different ways to use templates to derive $P(z|\\rom{\\mathbf{F}}{obs})$. In our brute-force approaches, all available models and\/or model parameters are compared to the data in order to derive the appropriate redshift PDF. In our MCMC-based approaches, we attempt to explore the set of models contained within contiguous regions on the SOM. In our hierarchical sampling approaches, we randomly sample galaxies according to the SOM's partitions in color space to stochastically construct estimates to the specific portion of the SOM-based color-redshift relation for each individual galaxy. Finally, in our cell model-dependent approaches, we attempt to ``triage'' the space spanned by the SOM using a series of summary statistics in order to derive limited estimates of $P(z|\\rom{\\mathbf{F}}{obs})$. Each take advantage of different aspects of our initial template set and the final SOM outputs and represent a variety of ways to take advantage of the SOM in order to maximize the utility of our fuzzy templates.\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{BruteForce}} (gold standard)]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_gridsearch_all_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{BruteForce\\_LinearFuzzy}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_gridsearch_linear_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_RestFrame}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_restframe_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_ObservedFrame}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_observedframe_y24.png}}\n\t\\caption{\n\t\tThe input ($\\rom{z}{spec}$) versus fitted ($\\rom{\\bar{z}}{phot}$) redshifts for our {\\tt{BruteForce}} (our ``gold standard''; black), {\\tt{BruteForce\\_LinearFuzzy}} (yellow-green), {\\tt{SOM\\_MCMC\\_RestFrame}} (purple), and {\\tt{SOM\\_MCMC\\_ObservedFrame}} (teal) methods (see \\S\\ref{sec:methods}) to the 10,000 objects in our 9-band LSST $ugrizY$ and \\textit{Euclid} $YJH$ mock catalog consisting of $\\rom{Y}{LSST}=24$\\,mag objects. $P(z|\\rom{\\mathbf{F}}{obs})$ distributions with redshift-normalized standard deviations $\\sigma_{z}^\\prime > 0.15$ are flagged as insecure and plotted as red points. The redshift-normalized mean bias $\\bar{\\Delta z}^\\prime$, median bias ${\\Delta z}_{50}^\\prime$, 1$\\sigma$ median absolute deviation (MAD) $\\sigma_{z,\\textrm{MAD}}^\\prime$, and the catastrophic outlier fraction $\\rom{f}{cat}$ for the full and secure samples are listed in each plot.\n\t\tSee Figure~\\ref{fig:photz_2} for more details.}\n\n\t\\label{fig:photz}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_MonteCarlo}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_montecarlo_5_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_importancesampling_2500-20_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_LimitedSum}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_partialcellsum_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_Average}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_zszp_som_average_y24.png}}\n\t\\caption{\n\t\tAs Figure~\\ref{fig:photz}, but for our {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} (dark red), {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}} (pink), {\\tt{SOM\\_CellModel\\_LimitedSum}} (gray), and {\\tt{SOM\\_CellModel\\_Average}} (dark yellow) methods (see \\S\\ref{sec:methods}). \n\t\tWhile all of our methods are able to reproduce the one-to-one relationship between spec-z and photo-z quite well with $\\sigma_{z,\\textrm{MAD}}^\\prime\\lesssim0.015$, our  {\\tt{SOM\\_CellModel\\_LimitedSum}} and {\\tt{SOM\\_CellModel\\_Average}} approaches consistently fail to identify unreliable photo-z's due to subtle biases introduced from their reliance on SOM cell models. Once insecure photo-z's have been removed, we find the majority of methods achieve accuracies comparable to our brute-force approaches with $|\\bar{\\Delta z}^\\prime|\\approx0.5\\%$, $\\sigma_{z,\\textrm{MAD}}^\\prime\\approx1.1\\%$, and catastrophic outlier rates of $\\lesssim 0.4$\\%, with our {\\tt{BruteForce\\_LinearFuzzy}}, {\\tt{SOM\\_MCMC\\_ObservedFrame}}, and {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} giving the best overall results.}\n\n\t\\label{fig:photz_2}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{BruteForce}} (gold standard)]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_gridsearch_all_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{BruteForce\\_LinearFuzzy}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_gridsearch_linear_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_ObservedFrame}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_som_observedframe_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_RestFrame}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_som_restframe_y24.png}}\n\t\\caption{\n\t\tThe two-dimensional stacked $P(z|\\rom{\\mathbf{F}}{obs})$ distributions for our {\\tt{BruteForce}} (gold standard), {\\tt{BruteForce\\_LinearFuzzy}}, {\\tt{SOM\\_MCMC\\_RestFrame}}, and {\\tt{SOM\\_MCMC\\_ObservedFrame}} methods (see \\S\\ref{sec:methods}) generated using 500 Monte Carlo samples drawn from the redshift PDF of each object and plotted using the same color scheme as Figure~\\ref{fig:photz}. \n\t\tSee Figure~\\ref{fig:photz_pzstack_2} for more details.}\n\t\\label{fig:photz_pzstack}\n\n\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_MonteCarlo}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_som_montecarlo_5_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_som_importancesampling_2500-20_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_LimitedSum}}]{\\includegraphics[scale=0.25]{new_figures\/v2_lsst_euclid_Pzstack_som_partialcellsum_y24.png}}\n\t\\caption{\n\t\tAs Figure~\\ref{fig:photz_pzstack}, but for our {\\tt{SOM\\_Hierarchical\\_MonteCarlo}}, {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}, and {\\tt{SOM\\_CellModel\\_LimitedSum}} methods (see \\S\\ref{sec:methods}). \n\t\tAll methods recover the general degeneracies present across the objects in our catalog -- including redshift-reddening degeneracies and confusion between the 1216\\,{\\AA} and 4000\\,{\\AA} breaks. We find that the increased flexibility of {\\tt{BruteForce\\_LinearFuzzy}} not only sharpens degeneracies as compared to {\\tt{BruteForce}} but also creates new ones, such as a thin band of probability at lower redshift for sources at $z \\sim 4$\\,--\\,$6$. In addition, we confirm that the large band of flagged outliers at low redshift from {\\tt{SOM\\_MCMC\\_RestFrame}} (see Figure~\\ref{fig:photz}) is due to an overabundance of probability at higher redshifts for low-$z$ sources, illustrating the impact of the redshift gradient noted in Speagle \\& Eisenstein (2015a). Finally, while {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} gives good overall performance, its PDFs are noticeably ``rougher'' than the other approaches, likely due to sparse sampling in the small fraction of cells where the models with the highest likelihoods are contained.}\n\t\\label{fig:photz_pzstack_2}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{BruteForce}} (gold standard)]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_gridsearch_all_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{BruteForce\\_LinearFuzzy}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_gridsearch_linear_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_RestFrame}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_restframe_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_ObservedFrame}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_observedframe_y24.png}}\n\t\\caption{\n\t\tThe mean redshift bias $\\rom{\\Delta \\bar{z}}{bin} = \\langle\\rom{z}{spec}\\rangle_{\\textrm{bin}} - \\langle\\rom{\\bar{z}}{phot}\\rangle_{\\textrm{bin}}$ in a given $\\rom{\\bar{z}}{phot}$-selected redshift tomographic bin, plotted as a function of redshift for our {\\tt{BruteForce}} (gold standard), {\\tt{BruteForce\\_LinearFuzzy}}, {\\tt{SOM\\_MCMC\\_RestFrame}}, and {\\tt{SOM\\_MCMC\\_ObservedFrame}} methods (see \\S\\ref{sec:methods}). The results for the full sample are colored using the same color scheme as Figure~\\ref{fig:photz}, while those from the secure subsample are shown in red. For comparison, the results from a ``cleaned'' subsample where all catastrophic outliers (i.e. objects with $\\Delta z^\\prime > 0.15$) have been removed are shown in blue. Errors for each sample have been derived through bootstrap resampling, with 1$\\sigma$ errors plotted. The \\textit{Euclid} photo-z accuracy requirements for weak-lensing of $\\Delta \\bar{z} < 0.002(1+z)$ from \\citet{laureijs+11} are over-plotted as dotted-dashed black lines, with the relevant redshift range highlighted in light blue.\n\t\tSee Figure~\\ref{fig:photz_zbias_2} for more details.}\n\t\\label{fig:photz_zbias}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_MonteCarlo}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_montecarlo_5_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_importancesampling_2500-20_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_LimitedSum}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_partialcellsum_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_Average}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_ztomographicbias_som_average_y24.png}}\n\t\\caption{\n\t\tAs Figure~\\ref{fig:photz_zbias}, but for our {\\tt{SOM\\_Hierarchical\\_MonteCarlo}}, {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}, {\\tt{SOM\\_CellModel\\_LimitedSum}}, and {\\tt{SOM\\_CellModel\\_Average}} methods (see \\S\\ref{sec:methods}). \n\t\tFor the full sample, numerous outliers lead to significant biases for most redshift bins below $z \\lesssim 4$. While variation of $\\rom{\\Delta \\bar{z}}{bin}(\\rom{z}{bin})$ appears to be symmetrical for the {\\tt{BruteForce}} approach and the sparse-sampling {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} approaches, in the majority of cases the overabundance of up-scattered redshifts tend to drive variation in $\\rom{\\Delta \\bar{z}}{bin}(\\rom{z}{bin})$ to be positive (with the exception of {\\tt{BruteForce\\_LinearFuzzy}}, which introduces new low-$z$ degeneracies; see Figure~\\ref{fig:photz_pzstack}). After restricting to secure photo-z's only, the bias decreases substantially (except for {\\tt{SOM\\_CellModel}}-based approaches; see Figure~\\ref{fig:photz_2}), with the majority of tomographic bins for the secure subsample meeting the \\textit{Euclid} photo-z accuracy requirements above $z \\sim 0.8$.}\n\t\\label{fig:photz_zbias_2}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{BruteForce}} (gold standard)]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_gridsearch_all_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{BruteForce\\_LinearFuzzy}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_gridsearch_linear_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_RestFrame}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_restframe_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_MCMC\\_ObservedFrame}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_observedframe_y24.png}}\n\t\\caption{\n\t\tAs Figure~\\ref{fig:photz_zbias}, but for the redshift-normalized mean scatter $\\sigma(\\Delta z^\\prime)$ for our {\\tt{BruteForce}}, {\\tt{BruteForce\\_LinearFuzzy}}, {\\tt{SOM\\_MCMC\\_RestFrame}}, and {\\tt{SOM\\_MCMC\\_ObservedFrame}} methods (see \\S\\ref{sec:methods}). The \\textit{Euclid} photo-z accuracy requirements for weak-lensing of $\\sigma_z^\\prime < 0.05$ from \\citet{laureijs+11} are over-plotted as dotted-dashed black lines, with the relevant redshift range highlighted in light blue.\n\t\tSee Figure~\\ref{fig:photz_etastd_2} for more details.}\n\t\\label{fig:photz_etastd}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_MonteCarlo}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_montecarlo_5_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_importancesampling_2500-20_y24.png}}\\\\\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_LimitedSum}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_partialcellsum_y24.png}}\n\t\\qquad\n\t\\subfloat[][{\\tt{SOM\\_CellModel\\_Average}}]{\\includegraphics[scale=0.19]{new_figures\/v2_lsst_euclid_etastd_som_average_y24.png}}\n\t\\caption{\n\t\tAs Figure~\\ref{fig:photz_etastd}, but for our {\\tt{SOM\\_Hierarchical\\_MonteCarlo}}, {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}, {\\tt{SOM\\_CellModel\\_LimitedSum}}, and {\\tt{SOM\\_CellModel\\_Average}} methods (see \\S\\ref{sec:methods}).\n\t\tSimilar to Figure~\\ref{fig:photz_zbias}, numerous outliers present in the full sample lead to significant biases for most redshift bins below $z \\lesssim 3.6$, while the majority of tomographic bins for our secure photo-z's above $z \\sim 0.8$ meet the accuracy requirements (with the exception of the {\\tt{SOM\\_CellModel}}-based approaches). Due to our small sample size and the sensitivity of $\\sigma(\\Delta z^\\prime)$ to outliers, isolated failures in tomographic bins at $z>0.8$ are often due to only a small number of outliers.}\n\t\\label{fig:photz_etastd_2}\n\\end{figure*}\n\n\\section{Performance Tests Against Mock LSST and \\textit{Euclid} Data}\n\\label{sec:results}\n\n\nAlthough we would ideally like to test each of these approaches using real data, preliminary tests with a number of individual objects taken from Cosmological Origins Survey \\citep[COSMOS;][]{scoville+07} data indicate that the limited range spanned by portions of the \\citet{brown+14} archetypes are unrepresentative of a number of actively star-forming galaxies at higher redshift. As noted in Paper I, similar issues exist among all currently used template sets \\citep{coleman+80,kinney+96,polletta+07}. Rather than supplementing the \\citet{brown+14} galaxies with synthetic spectra created from stellar population synthesis models, we instead use a mock catalog drawn from our fuzzy templates to serve as performance benchmarks for our approach. While this does not give us a fully representative idea of how each of our individual approaches work in practice, it does give us an idealized comparison as to their \\textit{intrinsic effectiveness} assuming no template mismatch and similar modeling assumptions, which can help to identify the most useful approaches out of the eight proposed here.\n\nWe create a mock catalog of 10,000 objects scaled to $24$\\,mag in the LSST $Y$-band to examine performance at a reasonable given magnitude. We generate each object by sampling uniformly from $\\boldsymbol{\\theta}$ over the redshift interval $z=0$\\,--\\,$6$ ($\\Delta z = 0.01$) and the given set of priors $P(\\boldsymbol{\\phi})$ on our fiducial set of templates (see \\S\\ref{subsec:fuzzy}). Each object is then assigned Gaussian errors (added in quadrature) according to the estimated calibration uncertainty (1\\%) and observational errors based on the expected imaging depths, and the photometry is jittered according the error bars.\n\nWe use each of the methods detailed in \\S\\ref{sec:methods} to derive photo-z's to the entire mock catalog. Although we save the full $P(z|\\rom{\\mathbf{F}}{obs})$ for each object when available (i.e. excluding {\\tt{SOM\\_CellModel\\_Average}}), we choose to use the zeroth moment $\\rom{\\bar{z}}{phot}$ and the square root of the first moment $\\sigma_z$ as our fiducial photo-z point estimates and their associated errors.\n\nWe report five summary statistics for each sample: the redshift-normalized \\textit{mean bias},\n\\begin{equation}\n\\bar{\\Delta z}^\\prime = \\left\\langle\\frac{|\\rom{z}{spec} - \\rom{\\bar{z}}{phot}|}{1+\\rom{z}{spec}}\\right\\rangle,\n\\end{equation}\nthe redshift-normalized \\textit{median bias},\n\\begin{equation}\n{\\Delta z}_{50}^\\prime = \\textrm{median}\\left(\\frac{|\\rom{z}{spec} - \\rom{\\bar{z}}{phot}|}{1+\\rom{z}{spec}}\\right),\n\\end{equation}\nthe redshift-normalized \\textit{mean scatter},\n\\begin{equation}\n\\sigma(\\Delta z^\\prime) = \\left[\\textrm{Var}\\left(\\frac{|\\rom{z}{spec} - \\rom{\\bar{z}}{phot}|}{1+\\rom{z}{spec}}\\right)\\right]^{-\\frac{1}{2}}\n\\end{equation}\nthe $1\\sigma$ \\textit{median absolute deviation} (MAD),\n\\begin{equation}\n\\sigma_{z,\\textrm{MAD}}^\\prime = \\textrm{68.3 percentile}\\left(\\left|\\frac{|\\rom{z}{spec} - \\rom{\\bar{z}}{phot}|}{1+\\rom{z}{spec}} - {\\Delta z}_{50}^\\prime\\right|\\right),\n\\end{equation}\nand the \\textit{catastrophic outlier fraction},\n\\begin{equation}\n\\rom{f}{cat}=N(\\Delta z^\\prime > 0.15)\/N.\n\\end{equation}\n$\\rom{z}{spec}$ in all cases is taken to be the input redshift, while $\\rom{\\bar{z}}{phot}$ is the estimated redshift from each of the different methods outlined in \\S\\ref{sec:methods}.\n\nIn order to remove multimodal and\/or unconstrained fits, we classify our objects into ``secure'' and ``insecure'' redshifts based on their mean redshift-normalized \\textit{standard deviation},\n\\begin{equation}\n\\sigma_{z}^\\prime = \\sigma_z\/(1+\\bar{z}).\n\\end{equation}\nObjects are flagged as insecure if $\\sigma_z^\\prime > 0.15$ (i.e. if the standard deviation of the redshift PDF is larger than the catastrophic error boundaries), and otherwise are classified as secure. These are then used to construct ``full'' and ``secure'' samples. This is done to ensure that we have an unbiased view of the effectiveness of our redshift methods to not only derive accurate photo-z's but also to flag outliers. As weak lensing approaches are extremely sensitive to such outliers \\citep{cunha+14}, being able to accurately detect and remove badly constrained photo-z's are key to their successful application.\\footnote{We note that median-based point estimates $z_{50,\\textrm{phot}}$ give comparable results to $\\rom{\\bar{z}}{phot}$ in terms of overall accuracy. However, median-based estimates of the relative \\textit{spread} $\\sigma_{z,\\textrm{MAD}}$ are less effective at flagging possible catastrophic errors because the majority of the multi-modal PDFs tends to be concentrated near one mode.}\n\nThe generally quality of the photo-z point estimates derived from each method is illustrated in Figures~\\ref{fig:photz} and~\\ref{fig:photz_2}, along with several summary statistics. We find that all of the methods are able to recover the general one-to-one relation between the input and output redshift relatively well with $\\sigma_{z,\\textrm{MAD}}^\\prime\\lesssim1.15$\\%. However, we find that both our {\\tt{SOM\\_CellModel\\_LimitedSum}} and {\\tt{SOM\\_CellModel\\_Average}} methods are unable to consistently flag insecure photo-z's, impeding their ability to remove catastrophic outliers that can severely bias weak lensing analyses. We find this is mainly due to the assumption that the SOM cell models are a reliable proxy for the data clustered around them: while this assumption is true in general, it introduces a color-dependent bias during the sampling process that distorts the derived redshift PDF. Even in the worst case, however, we find that our catastrophic outlier fraction is $\\lesssim 3.5$\\%.\n\nAfter limiting each of our other six methods to their secure subsamples, we find their accuracy noticeably improves, with redshift-normalized mean biases $\\lesssim 0.1$\\% and catastrophic outlier fractions of $\\sim 0.35$\\%, a reduction of a factor of $\\sim$\\,5\\,--\\,10. While both {\\tt{SOM\\_CellModel\\_LimitedSum}} and {\\tt{SOM\\_CellModel\\_Average}} also display improved performance, they are only able to reduce their catastrophic outlier fractions by a factor of 2\\,--\\,3.\n\nAmong our ensemble of methods, we find that {\\tt{BruteForce\\_LinearFuzzy}}, {\\tt{SOM\\_MCMC\\_ObservedFrame}}, and {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} give the best results relative to {\\tt{BruteForce}} before and after the removal of insecure redshifts. In addition, although our {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}} approach gives a slightly higher catastrophic outlier fraction after removing photo-z's flagged as insecure, we find its general overall performance is also similar to {\\tt{BruteForce}}. Finally, although our {\\tt{SOM\\_MCMC\\_RestFrame}} run displays a noticeably high concentration of misfit objects scattered from low to high redshift, when comparing only the secure subsamples we find its performance is nearly identical to the previous two methods. We thus find both our MCMC-driven and hierarchical sampling methods can give comparable results relative to a brute-force search and are effective at locating and characterizing $P(z|\\rom{\\mathbf{F}}{obs})$.\n\n\nTo examine the redshift PDFs from our individual methods more closely, in Figure~\\ref{fig:photz_pzstack} we plot their two-dimensional stacked $P(z|\\rom{\\mathbf{F}}{obs})$ distributions using 500 Monte Carlo realizations drawn from the redshift PDF of each object. As can be clearly seen, all of our approaches recover the broad features and degeneracies observed in the {\\tt{BruteForce}} run, including the low-$z$ redshift-reddening degeneracy (as well as several others at higher redshifts) as well as confusion between the 1216\\,{\\AA} and 4000\\,{\\AA} breaks. We confirm that the {\\tt{SOM\\_MCMC\\_RestFrame}} case displays an overabundance of objects and corresponding probability at higher redshifts, which is likely due to the impact of the redshift gradient observed in first noted in Paper I, which tends to increase the likelihood that chains converge to higher redshift solutions. In addition, we find that while our {\\tt{BruteForce\\_LinearFuzzy}} run performs comparably to {\\tt{BruteForce}} run, the increased flexibility actually causes existing degeneracies to sharpen while simultaneously creating new degeneracies not present in any of the other runs. This indicates that while marginalizing over the linear parameters of our fuzzy templates is generally effective, it occasionally can enhance the likelihoods of degenerate solutions.\n\nIn order to measure the dark energy equation-of-state, LSST and \\textit{Euclid} will rely on weak lensing to determine the cosmic shear in given redshift tomographic bins to sub-percent accuracy \\citep{ivezic+08,laureijs+11}. This approach depends on having extremely accurate measurements of the mean redshift of a given bin. To illustrate the accuracy of each approach more clearly in the context of measuring cosmic shear, we plot the evolution in mean bias $\\Delta\\bar{z}=\\langle\\rom{z}{spec}\\rangle_{\\textrm{bin}}-\\langle\\rom{\\bar{z}}{phot}\\rangle_{\\textrm{bin}}$ and the normalized standard deviation $\\sigma_z^\\prime$ in fixed $\\Delta z = 0.2$ redshift tomographic bins (selected using $\\rom{z}{phot}$) from $z=0$\\,--\\,$6$ in Figures~\\ref{fig:photz_zbias},~\\ref{fig:photz_zbias_2},~\\ref{fig:photz_etastd}, and~\\ref{fig:photz_etastd_2}. To establish an appropriate baseline comparison for each method, we compare both the full and secure samples to an explicitly ``cleaned'' sample where all catastrophic outliers (i.e. objects with $\\Delta z^\\prime > 0.15$) have been removed.\n\nWe find that for all of our methods (excluding our {\\tt{SOM\\_CellModel}}-based approaches), the secure subsamples are able to meet the \\textit{Euclid} mean photo-z bias accuracy requirements for weak-lensing of $\\Delta\\bar{z} < 0.002(1+z)$ for most redshift bins above $z \\sim 0.8$. Without removing insecure objects, however, we find the full sample fails to meet the accuracy requirements for almost all redshift bins where $z \\lesssim 4$. After this point, the redshift identification for most objects becomes more secure due to the prominence of the 1216\\,{\\AA} break combined with \\textit{Euclid}'s NIR coverage in the $YJH$ bands.\n\nLikewise, the observed redshift-normalized scatter $\\sigma_z^\\prime$ for the secure subsample successfully meets the \\textit{Euclid} photo-z accuracy requirements of $\\sigma_z^\\prime < 0.05$ for most redshift bins above $z \\sim 1$ with a comfortable margin for error (although betwen $z \\sim 3$\\,--\\,$4$ there is some tension seen in our {\\tt{SOM\\_MCMC\\_RestFrame}} and {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}} results), while the full sample fails for $z \\lesssim 3.6$, regardless of the method used.\n\nTogether, these results illustrate the power of using the clustering methods such as the SOM to enhance photo-z searches, as well as the danger in relying heavily on summary statistics rather than the underlying collection of model photometry. Ultimately, our findings suggest that the intrinsic ability of the SOM to capture variable information content and widely-separated\/broad degeneracies in parameter space and project it into a reduced-dimensional manifold in a topologically smooth fashion lends a significant amount of flexibility to the model-fitting process. This feature is clearly apparent in our 3-D {\\tt{SOM\\_MCMC\\_ObservedFrame}} implementation, which displays performance noticeably better performance than the 2+1-D {\\tt{SOM\\_RestFrame}} counterpart that is on par with our {\\tt{BruteForce}} ``gold standard''. In addition, our hierarchical sampling approaches demonstrate the more generally ability granted by clustering methods to partition the large amount of model photometry into useful partitions in color space that can be quickly explored.\n\nHowever, while our results emphasize the SOM is a useful tool for clustering and re-organizing the space, using simple summary statistics that describe individual SOM cells to derive photo-z estimates is inherently risky. As individual cell models (and cell means) can display significant variation from the model data assigned to them, either due to high intrinsic variability, sparse sampling, or both (Paper I), relying on them directly will almost guarantee that certain regions of color space will become distorted during the sampling process. This can lead to occasional biases in the final photo-z estimates that make it more difficult to estimate the quality of the final results. While these approaches are reasonable for deriving quick photo-z estimates, they should not be used in place of more rigorous approaches if identifying and removing catastrophic errors from the overall sample are important. It thus remains important to use these approaches judiciously, especially when abstracting away from the underlying set of models.\n\nIn addition, it is important to note that all the results derived here have utilized the same set of templates and modeling assumptions, and thus have avoided possible systematic errors due to template mismatches with observed photometry. This will most likely increase the mean bias, scatter, and catastrophic outlier fraction within a given tomographic redshift bin in a redshift-dependent manner. It is thus important to recognize these results as emulating the best-case performance of each of these sampling techniques as applied to upcoming large-scale surveys.\n\n\\section{Conclusion}\n\\label{sec:conc}\n\nDue to the extremely difficult task of obtaining spectroscopic redshifts (spec-z's) to the billions of objects that will be observed in future large-scale photometric surveys, fast and accurate photometric redshifts (photo-z's) remain a necessary tool to enable ``big data'' science. Although significant progress has been made using machine learning to derive photo-z's \\citep{carrascokindbrunner13,carrascokindbrunner14,sadeh+15,bonnett15,almosallam+15,hoyle15}, improvements in template-fitting methods have in general lagged behind, with codes today still suffering from many of the same model deficiencies and computational issues that plagued them years ago. While template-fitting methods remain effective (see \\citealt{hildebrandt+10,dahlen+13,sanchez+14} for recent comparisons and \\citealt{cool+13,ilbert+13,tomczak+14} for recent applications), they are in general under-equipped to handle the enormous number of objects LSST, \\textit{Euclid}, and other similar surveys are expected to observed. Because template-fitting methods will likely continue to serve a necessary function in extragalactic science for the upcoming decade, this issue is worrisome: while machine learning approaches are effective within the bounds of their training sets \\citep{sanchez+14}, the limited training set provided by spec-z's today \\citep{masters+15} and the difficulty of getting reliable spectra to higher redshift sources essentially guarantees that template-based photo-z's will continue to serve a crucial role in high-$z$ science.\n\nWe have attempted to test the performance of the new hybrid template-fitting and machine-learning framework outlined in Paper I that alleviates both of these issues by allowing a large set of  ``fuzzy archetypes'' to be fit \\textit{en masse} in approximately constant time. While the observational requirements are more intensive than those used by current techniques, our method significantly reduces existing model dependencies and takes advantage of machine learning techniques such as Self Organizing Maps (SOMs) to improve the scope of parameter space that can be quickly yet robustly explored by different sampling techniques. \n\nUsing a mock set of galaxies with LSST $ugrizY$ and \\textit{Euclid} $YJH$ photometry with associated calibration uncertainties and observational errors, we quantify the performance of eight different approaches to deriving photo-z's -- a standard brute-force approach ({\\tt{BruteForce}}), a fuzzy template-based brute-force approach ({\\tt{BruteForce\\_LinearFuzzy}}), two approaches based on Markov Chain Monte Carlo (MCMC) sampling over the SOM ({\\tt{SOM\\_MCMC\\_ObservedFrame}} and {\\tt{SOM\\_MCMC\\_RestFrame}}), two approaches based on hierarchical sampling over the SOM ({\\tt{SOM\\_Hierarchical\\_MonteCarlo}} and {\\tt{SOM\\_Hierarchical\\_ImportanceSampling}}), and two ``quick estimate'' approaches based on summary statistics derived from the SOM ({\\tt{SOM\\_CellModel\\_LimitedSum}} and {\\tt{SOM\\_CellModel\\_Average}}). We find that all of our approaches give good photo-z estimates with a redshift-normalized 1$\\sigma$ median absolute deviation (MAD) of $\\sigma_{z,\\textrm{MAD}}\\approx 1.1$\\% and small catastrophic outlier fractions of $\\sim$\\,$2\\%$. Although we find our {\\tt{SOM\\_CellModel}}-based approaches are not able to consistently flag insecure photo-z's, our remaining MCMC-driven methods are able to robustly flag outliers and recover the underlying PDF relative to our brute-force approaches, with the results of {\\tt{SOM\\_MCMC\\_ObservedFrame}} and {\\tt{SOM\\_Hierarchical\\_MonteCarlo}} virtually indistinguishable from our {\\tt{BruteForce}} ``gold standard''. \n\nIn addition, we find our methods are able to meet the stringent \\textit{Euclid} photo-z accuracy requirements for the weak lensing analysis of cosmic shear in the majority of redshift bins in our sample over $z = 0$\\,--\\,$6$, both in terms of the mean redshift bias as well as the general accuracy over the corresponding ensemble of objects. We thus confirm that the fundamental framework is applicable to future photo-z applications to wide-field surveys and can generate accurate predictions assuming a representative sample of underlying fuzzy archetypes can be obtained\/generated.\n\nThere are a number of future applications of this work, some of which have been suggested in Paper I and others which have been partially explored by previous authors. In particular, while our {\\tt{SOM\\_CellModel}}-based  approaches can be computed extremely rapidly, their dependence on individual SOM cell models tends to bias the results for a number of obejcts. Some of this dependence might be substantially reducd or even removed with a recourse to a ``Random Atlas'' of SOMs \\citep{carrascokindbrunner14}, which we could use to derive a combined ensemble prediction as outlined in \\citet{carrascokindbrunner14}. Alternately, we could move away from discrete mappings of objects onto their best-matching cells onto the SOM (\\S\\ref{subsec:som}) in favor of general PDFs over the entire collection of SOM cells. These could (along with some set of summary statistics) then be used as a type of non-linear dimensionality reduction whose components would be fed into other machine-learning architectures in an attempt to predict the color-redshift relationship at a given location.\n\nFinally, one could investigate taking the general framework outlined and tested here and applying it to the large regions of parameter space spanned by stellar population synthesis models in an effort to achieve similar results. As the large numbers of parameters used in these models render direct Monte Carlo sampling methods intractable, such an approach would have to involve iterative training steps where a small amount of objects (fit using more computationally expensive methods) would be used to approximate the manifold (and degeneracies) where galaxies live, which would followed by successive SOM training until the entire relevant regions of model space have been subsequently mapped using the minimum number of objects. This approach would then serve as a way of interpolating between sparse sampling of the relevant manifold, which could be used to improve our understanding of galaxy properties and accelerate SED fitting techniques. We are currently looking into a number of these possibilities both as they apply to improving photometric redshifts as well as SED fitting more generally.\n\n\\section*{Acknowledgements}\n\nJSS would like to thank Michael Brown, Peter Capak, and Daniel Masters for insightful discussions as well as Charles Alcock for supervising the senior thesis course where a portion of this work was completed. JSS is grateful for financial support from the CREST program, which is funded by the Japan Science and Technology (JST) Agency. This work has benefited extensively from access to Harvard University's Odyssey computing cluster, which is supported by the FAS Division of Science's Research Computing Group.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe systematic study of the higher finiteness properties of groups\nwas initiated forty years ago by Wall \\cite{wall} and Serre \\cite{serre}.\nIn 1963, Stallings \\cite{stall} constructed the first example\nof a finitely presented group $\\Gamma$ with $H_3(\\Gamma;\\mathbb Q)$ infinite\ndimensional; his example was a subgroup of a direct product of three\nfree groups. \nThis was  the first indication of the\ngreat diversity to be found amongst the finitely presented subgroups\nof direct products of free groups, a theme developed in \\cite{bieri}.\n\nIn contrast, Baumslag and Roseblade \\cite{BR}\n proved that in a direct product of two\nfree groups the only finitely presented subgroups are the \n{\\em{obvious ones}}: \nsuch a subgroup is either free or has a subgroup of finite index\nthat is a direct product of free groups. \nIn \\cite{bhms} the present\nauthors explained this contrast  by proving that\n the exotic behaviour among the finitely presented\nsubgroups of direct products of free groups is accounted for entirely \nby the failure of higher\nhomological-finiteness conditions. In particular, we\nproved that the only subgroups $S$ of type ${\\rm FP}_n$ in a \ndirect product of $n$ free\ngroups are the obvious ones: if $S$ intersects each of the direct factors\nnon-trivially, it virtually splits as the direct product of these intersections.\nWe also proved that this splitting phenomenon persists when one replaces\nfree groups by the fundamental groups of compact surfaces \\cite{bhms};\nin the light of the work of Delzant and Gromov \\cite{DG}, \nthis has significant implications for  the structure of \nK\\\"ahler groups.\n\nExamples show that the splitting phenomenon for ${\\rm FP}_\\infty$\nsubgroups does not extend to products of more general\n2-dimensional hyperbolic groups or\nhigher-dimensional Kleinian groups \\cite{mb:haef}. But recent work\nat the confluence of logic, group theory and topology has brought to\nthe fore a class of groups that is more profoundly tied to surface\nand free groups than either of the above classes, namely {\\em{limit groups}}.\n\nLimit groups arise naturally from several points of view. They are\nperhaps most easily described as the finitely generated\ngroups $L$ that are {\\em{fully residually free}} (or \n{\\em{$\\omega$--residually free}}):\n for any finite subset $T\\subset L$ there exists a homomorphism\nfrom $L$ to a free group that is injective on $T$. \nIt is in this guise that limit groups were studied extensively by \nKharlampovich and Myasnikov \\cite{KM1,KM2,KM3}.\nThey are also known as {\\em{$\\exists$-free groups}} \\cite{res}, reflecting the\nfact that these are precisely the finitely generated groups that have\nthe same existential theory as a free group.\n\nMore geometrically,  limit groups are the\nfinitely generated groups that have a Cayley graph in which each\nball of finite radius is isometric to a ball of the same radius in\nsome Cayley graph of a free group of fixed rank.\n\nThe name {\\em{limit\ngroup}} was introduced by Sela. His original definition\ninvolved a certain limiting action on an $\\mathbb R$-tree, but \nhe also emphasized that these are\nprecisely the groups that arise when one takes limits of \nstable sequences of homomorphisms $\\phi_n:G\\to F$, where $G$ is an arbitrary\nfinitely generated group and $F$ is a free group;  {\\em{stable}} means that\nfor each $g\\in G$ either $I_g=\\{n\\in\\mathbb{N} : \\phi_n(g)=1\\}$ or \n $J_g=\\{n\\in\\mathbb{N} : \\phi_n(g)\\neq 1\\}$ is finite, and the {\\em{limit}} of $(\\phi_n)$\nis the quotient of $G$ by $\\{g\\mid |I_g|=\\infty\\}$.\n\n\nIn his account \\cite{S9}\nof the outstanding problems concerning limit groups, Sela\nasked whether the main theorem of \\cite{bhms} could be extended to cover\nlimit groups. The present article represents the culmination of a project\nto prove that it can. Building on ideas and results from \\cite{bhms,\nbh1, bh2, bh3, BM1} we prove:\n\n\\begin{thmA}\\label{main} If $\\Gamma_1,\\dots,\\Gamma_n$ are limit\ngroups and $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ is a subgroup\nof type ${\\rm FP}_n(\\mathbb{Q})$, then $S$ is virtually a direct product of  $n$\nor fewer limit groups.\n\\end{thmA}\n\nCombining this result with the fact that every finitely generated residually\nfree group can be embedded into a direct product of finitely many limit groups\n(\\cite[Corollary 2]{KM2}, \\cite[Claim 7.5]{S1}), we obtain:\n\n\\begin{corollary}\\label{FPinfty} Every residually free group of type\n${\\rm FP}_\\infty(\\mathbb{Q})$ is virtually a direct product of a finite number of limit groups.\n\\end{corollary}\n\nB.~Baumslag \\cite{benjy} proved that a \nfinitely generated,\nresidually free group is fully residually free (i.e.~a limit group)\nunless it contains a subgroup isomorphic to $F\\times\\mathbb{Z}$, where $F$\nis a free group of rank 2. Corollary\n\\ref{FPinfty} together with the methods used to prove Theorem \\ref{main}\nyield the following generalization of Baumslag's result:\n\n\\begin{corollary} Let $\\Gamma$ be a residually free group of type ${\\rm FP}_n(\\mathbb{Q})$\nwhere $n\\ge 1$, let $F$ be a free group of rank 2 and let\n$F^n$ denote the direct product of $n$ copies of $F$. Either $\\Gamma$\ncontains a subgroup isomorphic to $F^n\\times\\mathbb{Z}$ or else $\\Gamma$ is\nvirtually a direct product of $n$ or fewer limit groups.\n\\end{corollary}\n\nWe also prove that if a subgroup of a direct product \nof $n$ limit groups \nfails to be of type ${\\rm FP}_n(\\mathbb{Q})$, then\none can detect this failure in the homology of a subgroup of finite\nindex. \n\n\\begin{thmA}\\label{split}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be limit groups \nand  let $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ be a \nfinitely generated subgroup\nwith \n$L_i=\\Gamma_i\\cap S$ non-abelian for $i=1,\\dots,n$.\n\nIf $L_i$ is finitely generated for $1\\le i\\le r$ and not finitely\ngenerated for $i>r$, then there is a subgroup of finite index $S_0\\subset S$\nsuch that $S_0=S_1\\times S_2$, where $S_1$ is the direct\nproduct of the limit groups $S_0\\cap\\Gamma_i,\\ i\\le r$ and (if $r<n$) \n$S_2=S_0\\cap(\\Gamma_{r+1}\\times\\cdots\\times\\Gamma_n)$ has\n$H_k(S_2;\\mathbb Q)$  infinite dimensional for some $k\\le n-r$.\n\\end{thmA}\n\nIn Section \\ref{s:last} we shall prove a more technical version\nof Theorem \\ref{split} and account for abelian intersections.\n  \n Theorems \\ref{main} and \\ref{split} are \nthe exact analogues  of  Theorems A and B of \\cite{bhms}. \nIn Section  \\ref{reductions} \nwe introduce a sequence of reductions that will allow\nus to deduce both theorems from the following result \n(which, conversely, is \nan easy consequence of Theorem \\ref{split}). We remind the reader\nthat a subgroup of a direct\nproduct is called a {\\em{subdirect product}} if its projection to each \nfactor is surjective. \n\n\\begin{thmA}\\label{main3} Let $\\Gamma_1,\\dots,\\Gamma_n$ be non--abelian limit groups \nand  let $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ be a \nfinitely generated subdirect product which intersects each factor non-trivially. \nThen either :\n\\begin{enumerate}\n\\item\n$S$ is of finite index; {\\em{or}}\n\n\\item $S$ is of infinite index and has a finite-index subgroup \n$S_0 < S$ such that $H_j(S_0;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension\nfor some $j\\leq n$.\n\\end{enumerate}\n\\end{thmA} \n  \n \nFor simplicity of exposition, the homology of a group $G$ in this\npaper will almost always be with coefficients in a $\\mathbb{Q}\\,G$-module -- typically the\ntrivial module $\\mathbb{Q}$.  But with minor modifications, \nour arguments also apply\nwith other coefficient modules, giving corresponding results under\nthe finiteness conditions ${\\rm FP}_n(R)$ for other suitable rings $R$. \n\nA notable aspect of the proof of the above theorems is that following\na raft of reductions based on geometric methods, the proof takes\nan unexpected twist in the direction of nilpotent groups.\nThe turn of events that leads us in\nthis direction is explained in Section \\ref{s:elements}\n-- it begins with a\nsimple observation about higher commutators from \\cite{BM1}\nand proceeds via a spectral sequence argument.\n\n\n\nSeveral of our results shed light on the nature of arbitrary finitely\npresented subgroups of direct products of limit groups, most notably\nTheorem 4.2. These results suggest that there is\na real prospect of understanding all such subgroups, i.e.~all\nfinitely presented residually free groups. We  take up this\nchallenge in  \\cite{bhms3}, where we describe precisely which subdirect\nproducts of limit groups are finitely presented and present a solution\nto the conjugacy and membership problems for these subgroups\n(cf.~\\cite{BM1}, \\cite{BW}). But\nthe isomorphism problem for finitely presented residually free groups remains\nopen, and beyond that lie many further challenges.\nFor example, with applications of the surface-group case to K\\\"ahler \ngeometry in mind \\cite{DG},\none would like to know if all finitely presented subdirect\nproducts of limit groups satisfy a polynomial isoperimetric inequality. \n\n\\iffalse\nSuch an\nunderstanding certainly entails a calculation of the \nBieri-Neumann-Strebel invariants\nof direct products of limit groups -- see for example \\cite{meinert,MMvW,gehrke}.\nThe BNS invariants, however, apply only to subgroups that contain\nthe commutator subgroup, so further methods will need to be developed\nto cope with the general case, and many challenges remain. \n\\fi \n\n\n\\smallskip\n\nWe are grateful to the many colleagues with whom we have had useful\ndiscussions at various times about aspects of this work, particularly\nEmina Alibegovi\\'c, Mladen Bestvina, Karl Gruenberg, Peter Kropholler, Zlil Sela\nand Henry Wilton. We are also grateful to the anonymous referee\nfor his\/her helpful comments.\n\n\\section{Limit groups and their decomposition}\n\nSince this is the fourth in a series of papers on limit groups\n(following \\cite{bh1,bh2,bh3}), we shall only recall the minimal\nnecessary amount of information about them. The reader unfamiliar\nwith this fascinating class of groups should consult \nthe introductions in \\cite{AB,BF}, the original papers of\nSela \\cite{S1,S2,S6},  or those of\n Kharlampovich and Myasnikov \\cite{KM1,KM2,KM3} where\nthe subject is approached from a perspective more in keeping with\ntraditional combinatorial group theory; a further\n perspective is developed in \\cite{CG}.\n \n\\subsection{Limit groups}\n\nOur results rely  on  the fact that limit groups\nare the finitely generated subgroups of \n$\\omega$-residually free tower ($\\omega$-rft)\ngroups \n\\cite[Definition 6.1]{S1} (also known as NTQ-groups\n\\cite{KM2}).  A useful summary of Sela's proof\nof this result was given by Alibegovi\\'c and Bestvina in\nthe appendix to \\cite{AB} (cf.~\\cite[(1.11), (1.12)]{S2}).\nThe equivalent result of Kharlampovich and Myasnikov\n \\cite[Theorem 4]{KM2} is presented in a more algebraic manner. \n\nAn $\\omega$-rft group is the fundamental group of a\ntower space  assembled from graphs, tori\nand surfaces in a hierarchical manner. The number of\nstages in the process of assembly is the {\\em{height}}\nof the tower. Each stage in the construction involves\nthe attachment of an orientable surface along its boundary, or\nthe attachment of an $n$-torus $T$  along an\nembedded circle representing a primitive element\nof $\\pi_1T$. (There are additional constraints in each\ncase.)\n\nThe {\\em{height}} \nof a limit group $\\Gamma$ is the minimal height of an \n$\\omega$-rft group\nthat has a subgroup isomorphic to $\\Gamma$. Limit\ngroups of height $0$ are free products of \nfinitely many free abelian groups (each of finite rank)\nand surface\ngroups of Euler characteristic at most $-2$.\n\n\nThe splitting described in the following proposition is obtained as follows:\nembed $\\Gamma$ in an $\\omega$-rft group $G$ of minimal height, take  the\ngraph of groups decomposition of $G$  that   the Seifert-van Kampen\nTheorem associates to the addition of the final block in the tower, \nthen apply Bass-Serre theory to get an induced graph of \ngroups decomposition  of $\\Gamma$.\n\n\nRecall that a graph-of-groups decomposition is termed\n{\\em{$k$-acylindrical}} if in the action on the associated\nBass-Serre tree, the stabilizer of each geodesic edge-path of length\ngreater than $k$ is trivial; if the value of $k$ is unimportant,\none says simply that the decomposition is {\\em{acylindrical}}.\n\n\\begin{prop}\\label{graph} If $\\Gamma$ is a freely-indecomposable\nlimit group of height $h\\ge 1$, then it is the fundamental group of a\nfinite graph of groups that has infinite cyclic edge groups and has a\nvertex group that is a non-abelian limit group of height $\\le h-1$.\nThis decomposition \nmay be chosen to be 2-acylindrical.\n\\end{prop}\n \nNote also that any non-abelian limit group of height $0$  splits\nas $A\\ast_C B$ with $C$ infinite-cyclic or trivial, and this splitting\nis 1-acylindrical for surface groups, and 0-acylindrical for free products.\n \n\\subsection{The class of groups ${\\mathcal C}$}\n\nWe define a class of finitely presented groups\n${\\mathcal C}$ in a hierarchical manner; it is the union of the\nclasses ${\\mathcal C}_n$ defined as follows.\n\n\nAt {\\em{level $0$}} we have the class ${\\mathcal C}_0$ consisting\nof free products $A\\ast B$ of non-trivial,\nfinitely presented groups, where at least one of $A$ and $B$\nhas cardinality at least $3$ -- in other words, all finitely\npresented non-trivial free products with the exception of $\\mathbb{Z}_2*\\mathbb{Z}_2$. \n \nA group lies in ${\\mathcal C}_n$ if and only if it is the fundamental group\nof a finite, acylindrical graph of finitely presented groups, where all of the\nedge groups are cyclic, and at least one of the vertex groups\nlies in ${\\mathcal C}_{n-1}$.\n\n\\bigskip\n\nThe following is an immediate consequence of Proposition \\ref{graph}.\n\n\\begin{cor}\\label{LinC}\nAll non-abelian limit groups lie in ${\\mathcal C}$.\n\\end{cor}\n\n\\subsection{Other salient properties}\\label{ss:props}\n\nIn the proof of Theorems \\ref{main} and \\ref{split}, the\nonly properties of limit groups $\\Gamma$ that will be needed are the\nfollowing.\n\\begin{enumerate}\n\\item Limit groups are finitely presented, \nand their finitely generated subgroups\nare limit groups.\n\\item \nIf $\\Gamma$ is non-abelian, it lies in ${\\mathcal C}$ (Corollary \\ref{LinC}).\n\\item Cyclic subgroups are\nclosed in the profinite topology on $\\Gamma$.\n (This is true for\nall finitely generated subgroups \\cite{wilt}.)\n\\item If a subgroup $S$ of $\\Gamma$ has finite dimensional $H_1(S;\\mathbb Q)$,\nthen  $S$ is finitely generated (and hence is a \nlimit group) \\cite[Theorem 2]{bh1}.\n\\item Limit groups are of type ${\\rm FP}_\\infty$ (in fact $\\mathcal F_\\infty$).\nThis follows directly from the fact that the class of limit groups\ncoincides with the class of constructible limit groups, \\cite[Definition 1.14]{BF}.\n\\end{enumerate} \n\n\n\n\\subsection{Notation}\n Throughout this paper, we consistently use the notational convention that\n$S$ is a subgroup of the\ndirect product of the limit groups $\\Gamma_i$ ($1\\le i\\le n$),  that\n$L_i$ denotes the intersection $S\\cap\\Gamma_i$, and that\n$p_i : \\Gamma_1\\times\\dots\\times \\Gamma_n\\to\\Gamma_i$ is the coordinate\nprojection.\n\n\n\\subsection{Subgroups of finite index} \\label{ss:fi}\nThroughout the proof Theorems \\ref{main}, \\ref{split}\nand \\ref{main3}\nwe shall repeatedly pass to subgroups of finite\nindex $H_i\\subset\\Gamma_i$. When we do so, we shall assume\nthat our original subgroup\n$S$\nis replaced by $p_i^{-1}(H)\\cap S$ and \nthat each $\\Gamma_j$ ($j\\neq i$) is replaced\nby $p_jp_i^{-1}(H_i)$. This does not affect the intersections\n$L_j= S \\cap \\Gamma_j$ ($j\\neq i$). \n\n\nRecall \\cite[VIII.5.1]{ksbrown} that the property ${\\rm FP}_n$ \nis inherited by finite-index subgroups and persists in finite\nextensions. In the proof of Theorem \\ref{main3}\nwe detect the failure of property\n${\\rm FP}_n$ by considering the homology of subgroups of finite\nindex: if $H_k(S_1;\\mathbb{Q})$ is infinite dimensional for some $S_1<S$\nof finite index, then neither $S$ nor $S_1$ is of type ${\\rm FP}_k$.\n\nSome care is required here because one cannot conclude\nin the previous sentence that $S$ has an infinite-dimensional\nhomology group: the finite-dimensionality\nof homology groups is a property that persists in finite extensions \nbut is not, in general, inherited by finite-index subgroups. \nIn  the context of the proof\nof Theorem \\ref{main3}, care has been taken to ensure\nthat each passage to a finite-index subgroup\nrespects this logic.\n\n\n\n\\section{Reductions of the main theorem}\\label{reductions}\n\nThe following proposition reduces Theorem \\ref{main} \nto Theorem \\ref{main3}.\n\n\\begin{prop}\\label{assume}\nTheorem \\ref{main} is true if and only if it holds under the following\nadditional assumptions.\n\\begin{enumerate}\n\\item $n\\ge 2$.\n\\item Each projection $p_i:S\\to\\Gamma_i$ is surjective.\n\\item Each intersection $L_i=S\\cap\\Gamma_i$ is non-trivial.\n\\item Each $\\Gamma_i$ is a non-abelian limit group.\n\\item Each $\\Gamma_i$ splits as an HNN-extension over a cyclic subgroup $C_i$\nwith stable letter $t_i\\in L_i$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{pf}\n\\noindent (1) The case $n=0$ of Theorem \\ref{main} is trivial.  \n\nIn the case $n=1$, $S<\\Gamma_1$ has type ${\\rm FP}_1(\\mathbb{Q})$, so is finitely generated.\nBut a finitely generated subgroup of a limit group is again a limit group,\n and there is nothing more to prove.\n(The case $n=2$ was proved in \\cite{bh3} but an independent proof is\ngiven below.)\n \n\\smallskip\\noindent (2)  Since $S$ has type ${\\rm FP}_n(\\mathbb{Q})$ it is finitely generated,\nhence so is $p_i(S)$  and we can  replace  each $\\Gamma_i$ by $p_i(S)$.\n \n\\smallskip\\noindent (3) If, say, $L_n$ is trivial, then the projection map\n$q_n:S\\to\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}$ is injective, and $S$\nis isomorphic to a subgroup $q_n(S)$ of $\\Gamma_1\\times\\cdots\n\\times\\Gamma_{n-1}$.\nAfter iterating  this argument,  we may assume that each\n$L_i$ is non-trivial.\n\n\\smallskip\\noindent (4) Suppose that one or more of the $\\Gamma_i$ is abelian. \nA group is an abelian limit group if and only if it is free abelian of finite\nrank.  Hence a direct product of finitely many abelian limit groups is again\nan abelian limit group.  This reduces us to the case where precisely\none of the $\\Gamma_i$ -- say $\\Gamma_n$ -- is abelian.\n\nNow, replacing $\\Gamma_n$ by a finite index subgroup if necessary, we may\nassume that $L_n\\subset\\Gamma_n$ is a direct factor of $\\Gamma_n$:\nsay $\\Gamma_n=L_n\\oplus M$.  Since $M\\cap S$ is trivial, the projection\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\Gamma_1\\times\\cdots\n\\times\\Gamma_{n-1}\\times L_n$ \nwith kernel $M$ maps $S$ isomorphically onto a subgroup $T$\nof $\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}\\times L_n$.  Since \n$L_n\\subset T$, it follows that $S\\cong T=U\\times L_n$ for some\nsubgroup $U$ of $\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}$.  \nBut then $U$\nhas type ${\\rm FP}_n(\\mathbb{Q})$, since $S$ does, and if Theorem \\ref{main} holds\nin the case where all the $\\Gamma_i$ are non-abelian, then $U$ is virtually\na direct product of $n-1$ or fewer limit groups.  But then $S\\cong U\\times L_n$\nis virtually a direct product of $n$ or fewer limit groups, so Theorem \\ref{main}\nholds in full generality.\n\n\\smallskip\\noindent (5)  \nThe subgroup $L_i$ of $\\Gamma_i$ is normal by (2) and non-trivial by (3).\nHence it contains an element\n$t_i$ that acts hyperbolically on the tree of the splitting described in \nProposition \\ref{graph}\n(see \\cite[Section 2]{bh2}).  \nThen by\n\\cite[Theorem 3.1]{bh2}, $t_i$ is the stable letter in some HNN decomposition\n(with cyclic edge-stabilizer)  of a finite-index subgroup $\\Delta_i\\subset\\Gamma_i$.\n\nReplacing each $\\Gamma_i$ by the corresponding subgroup $\\Delta_i$, and\n$S$ by $S\\cap (\\Delta_1\\times\\cdots\\times\\Delta_n)$, gives us the desired\nconclusion. \n\n(The above argument extends to all groups in ${\\mathcal C}$ under the additional hypothesis that\nthe edge groups in the splittings defining ${\\mathcal C}$  are all closed in the profinite topology.)\n \\end{pf}\n\n\\section{The elements of the proof of Theorem \\ref{main3}}\\label{s:elements}\n\nWe have seen that Theorem \\ref{main} follows  from\nTheorem \\ref{main3}.\nThe proof of Theorem \\ref{main3} extends from Section   \\ref{fgnormal} to \nSection \\ref{finalstep}. \nIn the present section we give an overview of the contents of these sections\nand indicate how they will be assembled to complete the proof.\n\nIn Section \\ref{fgnormal} we prove the following extension of the basic result that\nnon-trivial, finitely-generated normal subgroups of non-abelian limit groups have\nfinite index \\cite{bh1}.\n\n\\begin{theorem}\\label{index}\nLet $\\Gamma$ be a group in ${\\mathcal C}$, and $1\\ne N<G<\\Gamma$ with $N$ normal \nin $\\Gamma$ and $G$ finitely generated. \n Then $|\\Gamma:G|<\\infty$. \n\\end{theorem}\n\nUsing this result, together with the HNN decompositions of the\n$\\Gamma_i$ described in Proposition \\ref{assume}, \nwe deduce (Section \\ref{vna}):\n\n\\begin{theorem}\\label{propvna}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups.\nIf $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup with $H_2(S_1;\\mathbb{Q})$ finite dimensional\nfor all finite-index subgroups $S_1<S$,\nand if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen:\n\n\\begin{itemize}\n\\item the image of each projection $S\\to\\Gamma_i\\times\\Gamma_j$ \nis of finite index in $\\Gamma_i\\times\\Gamma_j$;\n\\item the quotient groups\n$\\Gamma_i\/L_i$ are virtually nilpotent\nof  class at most $n-2$.\n\\end{itemize}\n\\end{theorem}\n\nWe highlight the case $n=2$. \n\n\n\\begin{cor}\\label{cor2factor} \nIf $\\Gamma_1$ and $\\Gamma_2$ are non-abelian limit groups, \nand $S<\\Gamma_1\\times\\Gamma_2$\nis a subdirect product  intersecting each factor non-trivially, \nwith $H_2(S_1;\\mathbb{Q})$ finite dimensional\nfor all finite-index subgroups $S_1<S$,\n then $S$ has finite index in $\\Gamma_1\\times\\Gamma_2$.\n\\end{cor}\n\nAn important special case of Theorem \\ref{main3},\nconsidered in Section \\ref{kernelZ},\narises where $S$ is the kernel of an epimorphism\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\mathbb{Z}$.\n\n\\begin{theorem}\\label{theoremkernelZ}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups, and\n$N$ the kernel of an epimorphism\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\mathbb{Z}$.  \nThen there is a subgroup of finite index $N_0\\subset N$\nsuch that \nat least one of the homology groups $H_k(N_0;\\mathbb{Q})$ ($0\\le k\\le n$)\nhas infinite $\\mathbb{Q}$-dimension. \n\\end{theorem}\n\n\nWe complete the proof of Theorem \\ref{main3} in Section \\ref{finalstep}.\nWe have seen that each of the $\\Gamma_i\/L_i$ is virtually nilpotent.\nSetting $\\Gamma=\\Gamma_1\\times\\cdots\\times\\Gamma_n$ and \nnoting that $S$ contains the product $L=L_1\\times\\cdots\\times L_n$,\nwe argue by induction on the difference in Hirsch lengths $d=h(\\Gamma\/L)-h(S\/L)$\nto prove that $H_k(S;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension for some $k\\le n$\nif $d>0$. \nThe initial step of the induction is provided by Theorem \\ref{theoremkernelZ},\nand the inductive step is established using the LHS spectral sequence.\nSection \\ref{s:last} contains a proof of Theorem \\ref{split}.\n\n\n\\section{Subgroups containing normal subgroups}\\label{fgnormal}\nIn this section we prove Theorem \\ref{index}.\nWe assume that the reader is familiar with Bass-Serre theory\n\\cite{Serre2},\nwhich we shall use freely. \nAll our actions on trees are without inversions.\n\n\\begin{lemma}\\label{lemmacocompact}\nLet $\\Delta$ be a group acting $k$-acylindrically, \ncocompactly and minimally on a tree $X$.\nLet $H$ be a finitely generated subgroup of $\\Delta$.\nSuppose that $M<H$ is a non-trivial subgroup which is normal in $\\Delta$.\nThen the action of $H$ on $X$ is cocompact.\n\\end{lemma}\n\n\\begin{proof}\nIf $X$ is a point there is nothing to prove, so we may assume that\n$X$ has at least one edge.\nBy hypothesis, $\\Delta$ has no global fixed point in its action on $X$.\nBy \\cite[Corollary 2.2]{bh2}, the non-trivial normal subgroup $M<\\Delta$ contains\nelements which act hyperbolically on $X$, and the union of the axes of all such elements\nis the unique minimal $M$-invariant subtree $X_0$ of $X$.\nSince $M$ is normal in $\\Delta$, the $M$-invariant subtree $X_0$ is also\ninvariant under the action of $\\Delta$.  But $X$ is minimal as a $\\Delta$-tree,\nso $X_0=X$.\n\nWe have shown that $M$ acts minimally on $X$. Since $M<H$, it\nfollows that $H$ acts minimally on $X$, so the quotient graph of\ngroups $\\mathcal G$ has no proper sub-(graph of groups) such that the inclusion induces an\nisomorphism on $\\pi_1$. A standard argument in Bass-Serre theory shows that\nsince $H$ is finitely\ngenerated, the topological graph underlying $\\mathcal G$ is compact, as claimed.\n\\end{proof}\n\n\\begin{prop}\\label{propindex}\nLet $\\Gamma\\in{\\mathcal C}$, and let $C,G$ be \nsubgroups of $\\Gamma$ with $C$ cyclic and $G$ finitely\ngenerated.  If $|G\\backslash\\Gamma\/C|<\\infty$,\nthen $|\\Gamma:G|<\\infty$.\n\\end{prop}\n\n\\begin{pf}\nLet $\\Gamma$ be a group in ${\\mathcal C}$.\nWe argue by induction on the level $\\ell=\\ell(\\Gamma)$ in the hierarchy \n$\\mathcal C=\\cup_n{\\mathcal C}_n$ where $\\Gamma$ first appears. \nBy definition,  $\\Gamma$ has a non-trivial,\n$k$-acylindrical, cocompact\naction on a tree $T$, with cyclic edge stabilizers.\nWithout loss of generality we can suppose that this action is minimal.\n \nIf $\\ell=0$ there is a single orbit of edges, the edge stabilizers are trivial and the vertex stabilizers are non-trivial.\nIf $\\ell>0$ the edge-stabilizers are non-trivial and the stabilizer of some vertex $w$ is in ${\\mathcal C}_{\\ell-1}$.\n\nLet $c$ be a generator for $C$.\nWe treat the initial and inductive stages of the argument simultaneously, \nbut distinguish two cases according to the action\nof $c$.\n\n\\smallskip\\noindent{\\bf Case 1.} Suppose that $c$ fixes a vertex $v$ of $T$.\n\nThen, by our double-coset hypothesis, the $\\Gamma$-orbit of $v$ consists of only\nfinitely many $G$-orbits $Gv_i$.  Since the action of $\\Gamma$ on $T$\nis cocompact, there is a constant $m>0$ such that $T$ is the $m$-neighbourhood\nof $\\Gamma v$, and hence the quotient graph $X=G\\backslash T$ is the \n$m$-neighbourhood of the finitely many vertices $Gv_i$. In other\nwords,  $X$ has finite diameter.  \n\nNote also that $\\pi_1X$ has finite rank, because it is a retract\nof $G$ which is finitely\ngenerated.\n\nFinally, note that $X=G\\backslash T$ has only finitely many valency 1 vertices.  \nFor otherwise, we can deduce a contradiction\nas follows.  Since $G$ is finitely generated, if there are infinitely\nmany vertices of valency 1, then the induced graph-of-groups\ndecomposition of $G$ is degenerate, in the sense that there\nis a valency 1 vertex $\\bar u$ with $G_{\\bar u}=G_{\\bar e}$,\nwhere $\\bar e$ is the unique edge of $G\\backslash T$\nincident at $\\bar u$.\n\nNow $\\bar u=Gu$ for some vertex $u$ in $T$,\n and $\\bar e=Ge$ for an edge $e$ incident at $u$.\nThe group $G_{\\bar u}$\nis the stabilizer of $u$ in $G$, and $G_{\\bar e}$ is the stabilizer\nof $e$ in $G$.  The fact that $\\bar u=Gu$ has valency $1$ in \n$G\\backslash T$ means that $G_{\\bar u}$ acts transitively on the link\n{\\rm{Lk}} of $u$ in $T$.  Hence $|{\\rm{Lk}}|=|G_{\\-u}:G_{\\-e}|=1$, so\n$u$ is a valency\n$1$ vertex of $T$.  But this contradicts the fact that $T$ is minimal\nas a $\\Gamma$-tree.\n\n\nWe have shown that $X=G\\backslash T$ has finite diameter, finite rank,\nand only finitely many vertices of valency 1.  It follows that $X$\nis a finite graph. \n\nIn the case where $\\Gamma$ has level $\\ell=0$, the stabilizer\n$\\Gamma_e$ of any edge $e$ of $T$ is trivial. The number of edges\nin $X=G\\backslash T$ that are images of edges $\\gamma e\\in\\Gamma e$\ncan therefore be counted as $|G\\backslash\\Gamma\/\\Gamma_e|=|G\\backslash\\Gamma|=|\\Gamma:G|$.\nHence, in this case, $|\\Gamma:G|<\\infty$, as required.\n\nIn the case where $\\ell>0$, there is a vertex $w$ of $T$ whose\nstabilizer $\\Gamma_w$ in $\\Gamma$ is a group in ${\\mathcal C}_{\\ell-1}$.\nLet $\\Gamma_e$ denote the stabilizer of some edge\n$e$ incident at $w$.  Then\n$|(G\\cap\\Gamma_w)\\backslash\\Gamma_w\/\\Gamma_e|$ is bounded above\nby the finite number of edges of\n$X=G\\backslash T$ incident at $Gw\\in G\\backslash T$\nthat are images of edges $\\gamma e\\in \\Gamma e$. \nBy inductive hypothesis, $G\\cap\\Gamma_w$ has finite index in\n$\\Gamma_w$.  Similarly, for each $\\gamma\\in\\Gamma$,\n$G\\cap\\gamma\\Gamma_w\\gamma^{-1}$ has finite index in\n$\\gamma\\Gamma_w\\gamma^{-1}$.  \nConsider the action of $\\Gamma_w$ by right multiplication on $G\\backslash\\Gamma$: the orbits\nare\nthe double cosets $G\\backslash\\Gamma\/ \\Gamma_w$ and hence are finite in number\nbecause they index a subset of the vertices of $X=G\\backslash T$; moreover the\nstabilizer of $G\\gamma$ is $\\gamma^{-1} G \\gamma \\cap \\Gamma_w$, which we have just\nseen has finite index in $\\Gamma_w$.  Thus $G\\backslash\\Gamma$ is finite.\n\n\n\\smallskip\\noindent{\\bf Case 2.}\nSuppose that $c$ acts hyperbolically on $T$, with\naxis $A$ say. \n\nThen the double coset hypothesis implies that the axes\n$\\gamma(A)$, for $\\gamma\\in\\Gamma$, belong to only finitely many \n$G$-orbits.   On the other hand, the convex hull of \n$\\bigcup_{\\gamma\\in\\Gamma}\\gamma(A)$ is a $\\Gamma$-invariant subtree of $T$, \nand hence by minimality is the whole of $T$.  \n\n\nLet $T_0$ be the minimal $G$-invariant subtree of $T$.  If $T_0=T$\nthen $X=G\\backslash T$ is finite since $G$ is finitely generated, and so\n$|G\\backslash\\Gamma\/\\Gamma_e|<\\infty$ for any edge-stabilizer \n$\\Gamma_e$ in $\\Gamma$.  \nIf $\\ell=0$, then $\\Gamma_e$ is trivial, so $|\\Gamma:G|<\\infty$.  \nOtherwise, choose  \n $e$ incident at a vertex $w$ whose stabilizer $\\Gamma$\n is in ${\\mathcal C}_{\\ell-1}$ and apply the inductive hypothesis as above\n to deduce that $|\\Gamma:G|<\\infty$.\n\nIt remains to consider the case $T_0\\ne T$.\n\nNow, for any subgraph $Y$ of $T$, and any $g\\in G$, \nwe have $$d(g(Y),T_0)=d(g(Y),g(T_0))=d(Y,T_0).$$\nSince the $\\Gamma$-orbit of $A$ contains only finitely many\n$G$-orbits, there is a global upper bound $K$, say, on\n$d(\\gamma(A),T_0)$ as $\\gamma$ varies over $\\Gamma$.\n\nSince $T\\neq T_0$ and $T$ is spanned by the $\\Gamma$-orbit of $A$, \nthere is a \ntranslate $\\gamma(A)$ of $A$ that is not contained in $T_0$.  \nRecall that the action is $k$--acylindrical.\nChoose a vertex $u$ on $\\gamma(A)$ with $d(u,T_0)>K+k+2$\nand let $\\Gamma_u$ denote its stabiliser in $\\Gamma$. Let $p$ be the\nvertex a distance $K$ from $T_0$ on the unique shortest \npath from $T_0$ to $u$. Since $d(\\gamma(A),T_0)\\le K$,\nthe geodesic $[p,u]$ is contained in $\\gamma(A)$. Similarly,\n$[p,u]$ is contained\nin any translate of $A$ that passes through $u$.\nIn particular, if $\\delta\\in\\Gamma_u$ \nthen $[p,u]\\subset\\delta\\gamma(A)$, and since $\\delta$\nfixes $u$ we have $\\delta(p)=p$ or $\\delta(p')=p$,\nwhere $p'$ is the unique point of $\\gamma(A)$ other than\n$p$ with $d(u,p)=d(u,p')$. \n\nIf $\\delta$ fixes the edge of $[p,u]$\nincident at $u$, then $\\delta(p)=p$ hence $\\delta$ fixes\n$[p,u]$ pointwise, which  contradicts the $k$-acylindricality of\nthe action unless $\\delta=1$. Thus the stabiliser of this edge\nis trivial, which is a contradiction  unless $\\ell=0$.\n\nIf $\\ell=0$ then, replacing $u$ by an adjacent vertex if\nnecessary, we may assume that $|\\Gamma_u|>2$.\nChoose distinct non-trivial elements $\\delta_1,\\delta_2\\in\\Gamma_u$.\nIt cannot be that all three of $\\delta_1,\\delta_2,\\delta_1\\delta_2^{-1}$\nsend $p'$ to $p$. Thus one of them fixes $p$, hence\n$[p,u]$, which again contradicts the $k$-acylindricality of\nthe action.\n\n\n\\end{pf}\n\nWe are now able to complete the proof of Theorem \\ref{index}.\n\n\\begin{pfof}{Theorem \\ref{index}}\nSuppose that $\\Gamma\\in{\\mathcal C}$, $G<\\Gamma$ is finitely generated, and $N$ is a non-trivial\nnormal subgroup of $\\Gamma$ that is contained in $G$. \n Then by definition of ${\\mathcal C}$, $\\Gamma$ acts non-trivially,\n cocompactly and $k$-acylindrically on\na tree $T$ with cyclic edge stabilizers.  There is no loss of generality\nin assuming the action is minimal, so we may apply \nLemma \\ref{lemmacocompact} to see that the \naction of $G$ is cocompact.  \nThe stabilizer $\\Gamma_e$ in $\\Gamma$\nof an edge $e$ is cyclic, and the finite number of edges in\n$G\\backslash T$ is an upper bound on $|G\\backslash\\Gamma\/\\Gamma_e|$.\nIt follows from Proposition \\ref{propindex} that $|\\Gamma:G|<\\infty$, as claimed.\n\\end{pfof}\n\n\\section{Nilpotent quotients}\\label{vna}\n\nIn this section we prove Theorem \\ref{propvna}, which steers us\naway from the study of groups acting on trees and into the realm\nof nilpotent groups.\n\nWe first prove a general lemma (from \\cite{BM1})\nabout a subdirect product $S$ of $n$ arbitrary (not necessarily limit) groups $\\Gamma_1,\\dots,\\Gamma_n$.  As before, we write $L_i$ for\nthe normal subgroup $S\\cap\\Gamma_i$ of $\\Gamma_i$.\nWe also introduce the following notation.  We write $K_i$ for the\nkernel of the $i$-th projection map $p_i:S\\to\\Gamma_i$, \nand $N_{i,j}$ for the image of $K_i$ under the $j$-th projection \n$p_j:S\\to\\Gamma_j$.  \nThus $N_{i,j}$ is a normal subgroup of $\\Gamma_j$.\n\n\nWe shall denote by  $[x_1,x_2,\\dots,x_n]$ the  left-normed $n$-fold commutator   \n$[[..[x_1,x_2],x_3],\\dots],x_n]$.\n \n\n\\begin{lemma}\\label{multicomm} \n$[N_{1,j},N_{2,j},\\dots,N_{j-1,j},N_{j+1,j},\\dots,N_{n,j}]\\subset L_j$.\n\\end{lemma}\n\n\\begin{pf} Suppose that $\\nu_{i,j}\\in N_{i,j}$ for a fixed $j$ and for all $i\\ne j$.\nThen\n there exist $\\sigma_i\\in S$ with $p_i(\\sigma_i)=1$ and $p_j(\\sigma_i)=\\nu_{i,j}$.  \nLet $\\sigma$ denote the $(n-1)$-fold commutator\n$[\\sigma_1,\\dots,\\sigma_{j-1},\\sigma_{j+1},\\dots,\\sigma_n]\\in S$.  Then\n$p_j(\\sigma)$ is the $(n-1)$-fold commutator\n$$[\\nu_{1,j},\\dots,\\nu_{j-1,j},\\nu_{j+1,j},\\dots,\\nu_{n,j}]\\in\\Gamma_j.$$\n\nOn the other hand, for $i\\ne j$, we have $p_i(\\sigma)=1$ since $p_i(\\sigma_i)=1$.\nHence $\\sigma\\in L_j$, and $p_j(\\sigma)=\\sigma\\in L_j$.\n\nSince the choice of $\\nu_{i,j}\\in N_{i,j}$ was arbitrary, we have \n$$[N_{1,j},N_{2,j},\\dots,N_{j-1,j},N_{j+1,j},\\dots,N_{n,j}]\\subset L_j$$\nas claimed.\n\\end{pf}\n\n\nWe now consider a finitely generated\nsubdirect product $S$ of non-abelian\nlimit groups $\\Gamma_1,\\dots,\\Gamma_n$\nsuch that  $H_2(S_1;\\mathbb{Q})$ is finite dimensional for every\nfinite-index subgroup $S_1<S$.\n  \nLet $L_i,C_i$ and $t_i$ be as in Proposition \\ref{assume}.\nWe consider the image $A_{i,j}:=p_j(p_i^{-1}(C_i))$ \nunder the projection\n$p_j$ of the preimage under $p_i$ of the cyclic group $C_i$.  \nClearly $N_{i,j}<A_{i,j}<\\Gamma_j$.\n\nIn the remainder of this section we shall prove that $N_{i,j}\\subset \\Gamma_j$\nis of finite index for all $i$ and $j$. Lemma \\ref{multicomm}\n then implies that\n$\\Gamma_i\/L_i$ is virtually nilpotent of class at most $n-2$,\nas is claimed in Theorem \\ref{propvna}. \n\nAs a first step towards showing that $N_{i,j}\\subset \\Gamma_j$\nis of finite index, we prove the following lemma.\n\n\\begin{lemma}\\label{comms}\nLet $\\Gamma_1,\\dots,\\Gamma_n$  be non-abelian limit groups.\nIf $S < \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup\nwith $H_2(S;\\mathbb{Q})$ finite dimensional, and if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen for all $i,j$:\n\\begin{enumerate}\n\\item $|\\Gamma_j:A_{i,j}|<\\infty$;\n\\item $A_{i,j}\/N_{i,j}$ is cyclic.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{pf}\n(1) It suffices to consider the case $i=1$.\nThe HNN decomposition $\\Gamma_1= B_1*_{C_1}$ described in\nProposition \\ref{assume} (5) pulls back to an HNN\ndecomposition of $S$ with stable letter $\\widehat t_1=(t_1,1,\\dots,1)$, \nbase group $\\widehat {B_1}={p_1}^{-1}(B_1)$, and\n amalgamating subgroup $\\widehat C_1={p_1}^{-1}(C_1)$.\nAs $C_1$ is cyclic, $\\widehat C_1=K_1\\rtimes\\<\\widehat c_1\\>$\nwhere $\\widehat c_1$ is a choice of a lift of a generator of $C_1$.\nConsider the Mayer-Vietoris sequence for the HNN decomposition \nof $S$.\n$$\\cdots\\to H_2(S;\\mathbb{Q})\\to H_1(\\widehat C_1;\\mathbb{Q}) \\overset{\\phi}\\to \nH_1(\\widehat{B_1};\\mathbb{Q}) \\to H_1(S;\\mathbb{Q})\\to\\cdots$$\nThe map $\\phi$ is the difference between the map \ninduced by inclusion and the map  induced by the \ninclusion twisted by the action of $\\widehat t_1$ by conjugation.\nNotice that $\\widehat t_1$ commutes with $K_1$ \nand so acts trivially on $H_*(K_1;\\mathbb{Q})$. Thus $\\phi$\nfactors through the map $H_1(\\widehat C_1;\\mathbb{Q})\\to H_1(\\langle\\widehat c_1\\rangle;\\mathbb{Q})$,\nin particular the image of $\\phi$ has dimension at most 1. \nSince $H_2(S;\\mathbb{Q})$ is finite dimensional by hypothesis, \nit follows that $H_1(\\widehat C_1;\\mathbb{Q})$ is finite dimensional.\nFor each $j$, $A_{1,j}=p_j(\\widehat C_1)$ is a homomorphic image\nof $\\widehat{C_1}$, so $H_1(A_{1,j};\\mathbb{Q})$ is finite-dimensional.\nSince $A_{1,j}$ is a subgroup of the non-abelian limit group $\\Gamma_j$,\nit follows that it is finitely generated.  Since it\ncontains the non-trivial normal subgroup $L_j$,\nTheorem \\ref{index} now implies that $A_{1,j}$\nhas finite index in $\\Gamma_j$, as claimed.\n\n(2) As $p_j$ is surjective,\n  $A_{i,j}\/N_{i,j}=p_j(\\widehat C_i)\/p_j(K_i)$ is a homomorphic\nimage of $\\widehat C_i\/K_i$, so it is also cyclic, as claimed.\n\\end{pf}\n\nThe other crucial ingredient in the proof of Theorem \\ref{propvna}\nis the following proposition.\n \n\\begin{prop}\\label{surprising}\n Let $G$ be an HNN extension of the form\n$B\\ast_C$ with stable letter $t$, finitely generated base-group\n$B$ and infinite-cyclic edge group $C$. Suppose that $G$\nhas normal subgroups $L$ and $N$ such that $t\\in L$,\n$C\\cap N=\\{1\\}$ and $G\/N$ is infinite-cyclic. Suppose further\nthat $H_1(N;\\mathbb{Q})$ is infinite dimensional. Let $\\Delta\\subset G$ be the\nunique subgroup of index 2 that contains $B$. Then, there\nexists an element $x\\in L\\cap B\\cap N$ such that \n$R\\overline x\\subset H_1(N\\cap \\Delta;\\mathbb{Q})$ is a free $R$-module\nof rank 1, where $R=\\mathbb{Q}[\\Delta\/(N\\cap \\Delta)]$ and $\\overline x$\nis the homology class determined by $x$.\n\\end{prop}\n\n\\begin{proof}  Let $T$ be the Bass-Serre tree of the splitting\n$G=B\\ast_C$ and consider the graph of groups decomposition\nof $N_2:=N\\cap \\Delta$ with underlying graph $X=N_2\\backslash T$; since $N_2C$\nhas finite index in $G$, this is a finite graph. Each vertex\ngroup in this decomposition is a conjugate of $B\\cap N_2$,\nand the edge groups are trivial since $C\\cap N_2=\\{1\\}$.\n\nThus, as an abelian group, $H_1(N_2;\\mathbb{Q})$ is the direct sum of\n$H_1(X;\\mathbb{Q})$ and $p$ copies of $H_1(B\\cap N_2;\\mathbb{Q})$,\nwhere $p$ is the \nindex of $BN_2$ in $G$. The first of these summands is finite-dimensional, and hence $H_1(B\\cap N_2;\\mathbb{Q})$ is\ninfinite-dimensional (since $H_1(N;\\mathbb{Q})$ is infinite-dimensional,\nimplying that $H_1(N_2;\\mathbb{Q})$ is too). \n\nLet $\\tau$ be a generator of $G\/N$. Then $M:=H_1(B\\cap N_2;\\mathbb{Q})$ is a\n$\\mathbb{Q}[\\tau^{\\pm p!}]$-module, which is finitely generated\nbecause $B$ is finitely generated and $B\/(B\\cap N_2)$\nis finitely presented. \nSince $\\mathbb{Q}[\\tau^{\\pm p!}]$ is a principal ideal domain, the\nmodule $M$ has a free direct summand. We fix $z\\in B\\cap N_2$\nso that $\\overline z\\in M$ generates this free summand. \nIt follows that $R\\overline z$ has infinite $\\mathbb{Q}$-dimension, and so\nis a free submodule of \nthe $R$-module $H_1(N_2;\\mathbb{Q})$.\n \n Since $t\\notin \\Delta$,\n $z_1:=z$ and $z_2:=tzt^{-1}$ belong to distinct vertex groups\n in $X$.  Hence $x:=[z,t]=z_1z_2^{-1}\\in L\\cap N\\cap \\Delta$\n is such that $\\overline x=\\overline z_1-\\overline z_2$\n generates a free $\\mathbb{Q}[\\tau^{\\pm p!}]$-submodule of $H_1(N_2;\\mathbb{Q})$, and hence also a free $R$-submodule.\n \\end{proof}\n\nThe following proposition completes the proof of Theorem \\ref{propvna}.\n\n\\begin{prop} \nLet $\\Gamma_1,\\dots,\\Gamma_n$  be non-abelian limit groups.\nIf $S < \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup\nwith $H_2(S_1;\\mathbb{Q})$ finite dimensional for each subgroup\n$S_1$ of finite index in $S$, and if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen (in the notation of Lemma\n\\ref{comms})\n$N_{i,j}\\subset \\Gamma_j$\nis of finite index for all $i$ and $j$.\n\\end{prop}\n\n\\begin{pf} It suffices to consider the case $(i,j)=(2,1)$.\nLet $T$ be the projection of $S$ to $\\Gamma_1\\times\\Gamma_2$, and define\n$M_i=T\\cap\\Gamma_i$ for $i=1,2$.  Notice that $M_1 = N_{2,1}$, the projection to $\\Gamma_1$\nof the kernel of the projection $p_2:S\\to\\Gamma_2$,\nand similarly $M_2=N_{1,2}$.  \n\nSince $S$ projects onto each of $\\Gamma_1$ and $\\Gamma_2$, the same is\ntrue of $T$.  Hence we have isomorphisms\n$$\\frac{\\Gamma_1}{M_1}\\cong\\frac{T}{M_1\\times M_2}\\cong\\frac{\\Gamma_2}{M_2}.$$\nWe will assume that these groups are infinite, and obtain a contradiction.\n\nBy Lemma \\ref{comms}, $T\/(M_1\\times M_2)$ is virtually cyclic,\nso we may choose a finite index subgroup $T_0<T$ containing\n$M_1\\times M_2$ such that $T_0\/(M_1\\times M_2)$ is infinite\ncyclic.  Hence $G_i:=p_i(T_0)$ is a finite-index subgroup\ncontaining $M_i$ for $i=1,2$, such that $G_i\/M_i$ is infinite\ncyclic.  Choose $\\tau\\in T_0$ such that $\\tau.(M_1\\times\nM_2)$ generates $T_0\/(M_1\\times M_2)$, and let \n$\\tau_i=p_i(\\tau)\\in G_i$\nfor $i=1,2$.\n\nThe HNN-decomposition of $\\Gamma_i$ from Proposition \\ref{assume} (5)\ninduces\n an HNN decomposition $G_i=B'_i\\ast_{C'_i}$\nwith stable letter $t'_i\\in L_i$, where $C'_i=C_i\\cap G_i$\nand $t'_i$ an appropriate power of the stable letter $t_i$ of\n$\\Gamma_i$.\nNotice that, by Lemma \\ref{comms}, $C'_i\\cap M_i=\\{1\\}$.\nFor each $i=1,2$, Proposition \\ref{surprising} \n(with $G=G_i$, $N=M_i$, $L=L_i$, $t=t'_i$, $B=B'_i$,\n$C=C'_i$) provides an index $2$ subgroup $\\Delta_i$\nin $G_i$ and an element $ x_i\\in M_i\\cap\\Delta_i\\cap L_i$ such\nthat $\\overline x_i$ generates a free $\\mathbb{Q}[\\tau_i^{\\pm 1}]$-submodule\nof $H_1(M_i\\cap\\Delta_i;\\mathbb{Q})$.\n\nNow define $M'_i:=M_i\\cap\\Delta_i$.\nIt follows that  $\\overline x_1\\otimes \\overline x_2$ generates a free $\\mathbb{Q}[\\tau_1^{\\pm 1},\\tau_2^{\\pm 1}]$-submodule of \n$$H_1(M'_1;\\mathbb{Q})\\otimes_\\mathbb{Q} H_1(M'_2;\\mathbb{Q})\\subset H_2(M'_1\\times M'_2;\\mathbb{Q}).$$\n\nLet $T_1$ be the finite-index subgroup of $T_0$ defined by\n$T_1:=(M'_1\\times M'_2)\\rtimes\\<\\tau\\>$, and let $S_1<S$ be the \npreimage of $T_1$ under the projection $S\\to T$.\nUsing the LHS spectral sequence for the short exact sequence\n$M'_1\\times M'_2\\to T_1\\to\\<\\tau\\>$, we see that \n$$H_0(\\langle \\tau\\rangle;H_2(M'_1\\times M'_2;\\mathbb{Q}))\\subset H_2(T_1;\\mathbb{Q})$$\nhas an infinite dimensional $\\mathbb{Q}$-subspace generated by the images of\n$$\\{(\\tau_1^mx_1\\tau_1^{-m})\\otimes (\\tau_2^nx_2\\tau_2^{-n});~m,n\\in\\mathbb{Z}\\}.$$\nIn particular, the image of the map\n$H_2(L_1\\times L_2;\\mathbb{Q})\\to H_2(T_1;\\mathbb{Q})$ induced by inclusion is infinite-dimensional.\nBut this contradicts the hypothesis that $H_2(S_1;\\mathbb{Q})$ is finite dimensional,\n since the inclusion $(L_1\\times L_2)\\to T_1$ factors\nthrough $S_1$. This is the desired contradiction which completes the proof.\n\\end{pf}\n\n \n\n\\section{Normal subgroups with cyclic quotient}\\label{kernelZ}\n\n\n\n\\begin{prop} If $\\Gamma_1,\\dots, \\Gamma_n$ are groups of\ntype ${\\rm FP}_n(\\mathbb{Z})$ and $\\phi:\\Gamma_1\\times\\cdots\\times\n\\Gamma_n\\to\\mathbb{Z}$ has non-trivial restriction to each\nfactor, then $H_{j}(\\ker\\phi;\\mathbb{Z})$\nis finitely generated for $j\\le n-1$. \n\\end{prop}\n\n\\begin{proof} We first prove the result in the special\ncase where the restriction of\n$\\phi$ to each factor is epic.\nThus we may write $\\Gamma_i = L_i\\rtimes\\langle t_i\\rangle$ where\n$S=\\ker\\phi$, $L_i=S\\cap\\Gamma_i$ is the kernel of $\\phi|_{\\Gamma_i}$ and $\\phi(t_i)$ is a fixed\ngenerator of $\\mathbb{Z}$.\n\nIf $n\\ge 2$\nand we fix a finite set $A_i\\subset L_i$ such that $\\Gamma_i=\\langle A_i,t_i\\rangle$,\nthen $S$ is generated by $A_1\\cup\\dots\\cup A_n\\cup\n\\{ t_1t_2^{-1},\\dots,t_1t_n^{-1}\\}$. \n\n We proceed by induction on $n$\n(the initial case $n=1$ being trivial), considering\nthe LHS spectral sequence in homology for\nthe projection of $S$ to $\\Gamma_n$,\n$$\n1\\to S_{n-1}\\to S \\overset{p_n}\\to \\Gamma_n\\to 1,\n$$\nwhere $S_{n-1}$ is the kernel of the restriction of\n$\\phi$ to $\\Gamma_1\\times\\cdots\\times \\Gamma_{n-1}$.  In particular,\nthe inductive hypothesis applies to $S_{n-1}$.\n\nSince $\\Gamma_n$ is of type ${\\rm FP}_n(\\mathbb{Z})$ and $H_q(S_{n-1};\\mathbb{Z})$ is\nfinitely generated for $q\\le n-2$, by induction, \non the $E^2$ page of the spectral sequence\nthere are only finitely generated groups in the rectangle \n $0\\le p\\le n$ and $0\\le q\\le n-2$. It follows that all of the\ngroups on the $E^\\infty$ page that contribute to $H_j(S;\\mathbb{Z})$\nwith $j\\le n-1$ are finitely generated, with the possible exception\nof that in position $(0,n-1)$.\n\nOn the $E^2$ page, the group in position $(0,n-1)$ is\n$H_0(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Z}))$, which is the quotient of \n$H_{n-1}(S_{n-1};\\mathbb{Z})$ by the action of $\\Gamma_n$. \nThis action\nis determined by taking a section of $p_n:S \\to \\Gamma_n$\nand using the conjugation action of $S$. The section\nwe choose is that with image $L_n\\rtimes\\langle t_1t_n^{-1}\\rangle$.\nSince $L_n$ and $t_n$ commute with $S_{n-1}$, we have\n$$H_0(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Z})) = H_0(\\langle t_1\\rangle;H_{n-1}(S_{n-1};\\mathbb{Z})) \\ .$$\nThe latter group is the $(0,n-1)$ term on the $E^2$ page\nof the spectral sequence for the extension \n$$1\\to S_{n-1}\\to \\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}\\overset{\\phi}\\to \\mathbb{Z}\\to 1.$$\nThis is a 2-column spectral sequence, so the $E^2$ page\ncoincides with the $E^\\infty$ page. \nSince $\\Gamma_1\\times\\cdots\\times \\Gamma_{n-1}$\nis of type ${\\rm FP}_{n-1}$ (indeed of type ${\\rm FP}_{n}$), it follows that   \n$H_0(\\langle t_1\\rangle;H_{n-1}(S_{n-1};\\mathbb{Z}))$ is finitely generated, and\nthe induction is complete.\n\nFor the general case, replace $\\mathbb{Z}$ by the finite index subgroup\n$\\phi(\\Gamma_1)\\cap\\cdots\\cap\\phi(\\Gamma_n)$ ($=m\\mathbb{Z}$, say); replace each $\\Gamma_i$\nby the finite-index subgroup $\\Delta_i=\\Gamma_i\\cap\\phi^{-1}(m\\mathbb{Z})$,\nand replace $S$ by the finite-index subgroup $T=S\\cap(\\Delta_1\\times\\cdots\\times\\Delta_n)$.  Since $\\phi(\\Delta_i)=m\\mathbb{Z}$ for each $i$,\nthe above special-case argument applies to $T$, to show that\n$H_j(T;\\mathbb{Z})$ is finitely generated for each $0\\le j\\le n-1$.  Moreover,\n$T$ is normal in $S$, and we may consider the LHS spectral sequence of the short exact sequence\n$$1\\to T\\to S\\to S\/T\\to 1.$$\nOn the $E^2$ page of this spectral sequence, the terms $E^2_{pq}$\nin the region $0\\le q\\le n-1$ are homology groups of the finite\ngroup $T\/S$ with coefficients in the finitely generated modules\n$H_q(T;\\mathbb{Z})$, and so they are finitely generated abelian groups.\nBut all the terms that contribute to $H_j(S;\\mathbb{Z})$ for $0\\le j\\le n-1$\nlie in this region, so $H_j(S;\\mathbb{Z})$ is finitely generated for $j\\le n-1$,\nas required.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{thm14}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups and\nlet $S$ be the kernel of an epimorphism $\\phi:\\Gamma_1\\times\\cdots\\times\n\\Gamma_n\\to\\mathbb{Z}$. \nIf the restriction of $\\phi$ to each of the $\\Gamma_i$\nis epic, then  $H_n(S;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is by induction on $n$. \nThe case $n=1$ was established in \\cite{bh1}: the group $S=\\ker\\phi$ is a normal subgroup of the \nnon-abelian limit group $\\Gamma_1$, and if $H_1(S,\\mathbb{Q})$ were finite dimensional\nthen $S$ would be finitely generated, and hence would have finite\nindex in $\\Gamma_1$. \n\nThe preceding proposition shows that $H_j(S;\\mathbb{Z})$ is finitely generated, and hence $H_j(S;\\mathbb{Q})$  is\nfinite dimensional for $j<n$. As in the proof of that proposition, we\nconsider  the LHS spectral\nsequence for\n$$\n1\\to S_{n-1}\\to S\\overset{p_n}\\to \\Gamma_n\\to 1,\n$$\nnow with $\\mathbb{Q}$-coefficients. There are now \nonly finitely generated $\\mathbb{Q}$-modules in the region $0\\le q\\le n-2$. \n(Recall that $\\Gamma_i$ is of type ${\\rm FP}_\\infty$.)\nIn particular, the terms on the $E^2$ page\ninvolved in the calculation of $H_n(S;\\mathbb{Q})$ are all finitely generated\nexcept for \n$$H_0(\\Gamma_n;H_{n}(S_{n-1};\\mathbb{Q})) = H_0(\\langle t_1\\rangle;H_{n}(S_{n-1};\\mathbb{Q}))$$\nand\n$$H_1(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Q})).$$\n\nIt suffices to prove that the latter is infinite dimensional\nover $\\mathbb{Q}$. (The former is actually finite dimensional, but this\nis irrelevant.) \n   \nThe module $M=H_{n-1}(S_{n-1};\\mathbb{Q})$ is a homology group of \nthe kernel of a map from an ${\\rm FP}_\\infty$ group to $\\mathbb{Z}$.\nIt is thus a homology group of a chain complex of\nfree $R=\\mathbb{Q}[t,t^{-1}]$ modules of finite rank. \nThe ring $R$ is Noetherian, so such a homology\ngroup is finitely generated as an $R$-module.\nBy the inductive hypothesis, $M$ has infinite $\\mathbb{Q}$-dimension. \nSo by the classification of finitely generated modules over a principal\nideal domain, \n$M$ has a free direct summand, that is $M=M_0\\oplus R$.\n\nThe $\\Gamma_n$-action on $M$ factors through \n the quotient $\\Gamma_n\\to\\Gamma_n\/L_n = \\<t_n\\>$, since\n$L_n$ acts trivially, so the direct sum decomposition \n passes to $M$ considered as a $\\mathbb{Q}\\Gamma_n$ module.\nHence $H_1(\\Gamma_n;M)=H_1(\\Gamma_n;M_0) \\oplus H_1(\\Gamma;R)$.\n\nFinally, as a $\\mathbb{Q}\\Gamma_n$ module, \n$R=\\mathbb{Q}\\Gamma_n \\otimes_{\\mathbb{Q} L_n} \\mathbb{Q}$, so by Shapiro's Lemma \n$H_1(\\Gamma_n;R) \\cong H_1(L_n;\\mathbb{Q})$ \n(see for instance \\cite[III.6.2. and III.5]{ksbrown}) . \n  \n As $L_n$ is an infinite index normal subgroup of a non-abelian\n limit group, it is not finitely generated, and therefore neither is\nthe $\\mathbb{Q}$-module  $H_1(L_n;\\mathbb{Q})$ \\cite{bh1}.\n \\end{proof} \n\nTheorem \\ref{theoremkernelZ} follows immediately from Theorem \\ref{thm14}\nin the\nlight of the K\\\"unneth formula, after one has passed to a subgroup \nof finite index to ensure that whenever $\\Gamma_i\\to \\mathbb{Z}$ is non-trivial it\nis onto.\n\n\\section{Completion of the proof of the Main Theorem}\\label{finalstep}\n\nThe following lemma and its corollary provide an extension to the \nvirtual context of known results about finitely generated nilpotent\ngroups.\nWe shall apply them to direct products of the  \nvirtually nilpotent quotients of $\\Gamma_i\/L_i$ resulting from \n Theorem \\ref{propvna}.\n\n\\begin{lemma}\\label{nilp}\nLet $G$ be a finitely generated virtually nilpotent group and \nlet $\\overline S$ be\na subgroup of infinite index.  Then there exists a subgroup $K$\nof finite index in $G$ and an epimorphism $f:K\\to\\mathbb{Z}$ such that \n$(\\overline S\\cap K)\\subset {\\mathrm{ker}}(f)$.\n\\end{lemma}\n\n\\begin{pf}\nWe argue by induction on the Hirsch length\n$h(G)$, which is strictly positive, since $G$ is infinite. \n\nIn the initial case, $h(G)=1$ means that $G$ has an infinite cyclic\nsubgroup $K$ of finite index.  \nSince $\\overline S$ has infinite index in $G$,\n$\\overline S$ is finite, so $(\\overline S\\cap K)$ is trivial, \nand we can take $f:K\\to\\mathbb{Z}$ to be an isomorphism.\n\n\\medskip\nFor the inductive step, let $H$ be a finite index torsion-free\nsubgroup of $G$, and $C$ an infinite cyclic central subgroup of\n$H$.  \nIf $C\\-S$ has infinite index in $G$, then the inductive hypothesis\napplies to $H\/C$ and we are done.  \nOtherwise, $\\overline S$ has infinite index in $C\\-S$, \nso $C\\cap \\overline S$ has infinite index in $C\\cong\\mathbb{Z}$.  \nBut then $C\\cap \\overline S=\\{1\\}$, and since $C<H$, it follows that \n$C\\overline S\\cap H=C\\times (\\overline S\\cap H)$.  \nPut $K=C\\overline S\\cap H$ and let $f$ be the projection\n$K\\to C$ with kernel $\\overline S\\cap H$. \n\\end{pf}\n\nWe note that Lemma \\ref{nilp}\nwould not remain true if one assumed only that $G$ were polycyclic. For example, it fails for\nlattices $G=\\mathbb{Z}^2\\rtimes\\langle t \\rangle$ \nin the 3-dimensional Lie group $\\rm{Sol}$\nif one takes $S=\\langle t\\rangle$.\n\n\\smallskip\nRepeated applications of Lemma \\ref{nilp} yield the following.\n \n\n\n\\begin{cor}\\label{chain}\nLet $G$ be a finitely generated, virtually nilpotent group and\nlet $\\overline S$ be a subgroup of $G$.  \nThen there is a subnormal chain\n$\\-S_0<\\-S_1<\\cdots <\\-S_r=G$, \nwhere $\\-S_0$ is a subgroup of finite index\nin $\\-S$ and for each $i$ the quotient group \n$\\-S_{i+1}\/\\-S_i$ is either finite or cyclic.\n\\end{cor}\n\n \n For the benefit of topologists, we should note that the following algebraic argument is\nmodelled on the geometric proof of the Double Coset Lemma in \\cite{bh2}.\n\n\\begin{pfof}{Theorem \\ref{main3}}\n\nLet $\\Gamma=\\Gamma_1\\times\\cdots\\times\\Gamma_n$.\nRecall that \nthe $\\Gamma_i$ are non-abelian, the projections $p_i:S\\to\\Gamma_i$\nare surjective, and the intersections $L_i=S\\cap\\Gamma_i$\nare non-trivial. \nBy passing to a subgroup of finite index we may assume that each $\\Gamma_i$\nsplits as in Proposition \\ref{assume}(5).\nLet $L=L_1\\times\\dots\\times L_n$.\n\nWe only need consider the \ncase when $S$ has infinite index in $\\Gamma$.\nWe shall \nderive a contradiction from the assumption that for\nall subgroups\n$S_0<S$ of finite index  and for all $0\\le j\\le n$,\n$H_j(S_0,\\mathbb{Q})$ is finite-dimensional.\n \nFrom Theorem \\ref{propvna} we know\nthat each of the quotient groups $\\Gamma_i\/L_i$ is virtually\nnilpotent, and hence so is $\\Gamma\/L$.\n\nSince $L\\subset S$ and $S$ has infinite index in $\\Gamma$,\n the image $\\-S$ of $S$ in $\\Gamma\/L$ is of infinite index\nand we may apply\nLemma \\ref{nilp} with $\\Gamma\/L$ in the role of $G$.\nLet $\\Lambda<\\Gamma$ be the preimage of the subgroup $K$\nprovided by the lemma. \nNote that $\\Lambda$ has finite index in $\\Gamma$,\ncontains $L$, and admits an epimorphism $f:\\Lambda\\to\\mathbb{Z}$\nsuch that $S\\cap \\Lambda\\subset\\mathrm{ker}(f)$.\nAs in (\\ref{ss:fi}), we may replace the groups\n$\\Gamma_i$ and $S$ by  finite-index \nsubgroups so as to ensure that $L\\subset S\\subset N$, \nwhere $N$  is the\nkernel of an epimorphism $\\Gamma\\to\\mathbb{Z}$.\nBy Theorem  \\ref{theoremkernelZ}, there is a finite index subgroup\n$N_0<N$ and an integer $j\\le n$ such that  $H_j(N_0;\\mathbb{Q})$ is infinite\ndimensional.  \n\nBy Corollary \\ref{chain} (applied to the image of $S\\cap N_0$ in\n$\\Gamma\/L$) there is a \nsubgroup $S_0$ contained in $S\\cap N_0$, \nwhich has finite index in $S$, \nand a subnormal chain of subgroups\n$S_0\\triangleleft S_1\\triangleleft\\cdots\\triangleleft S_k=N_0$\nwith $S_{i+1}\/S_i$ either finite or cyclic for each $i$.  \nWe now use the following lemma to contradict  the\nassumption that $H_j(S_0;\\mathbb{Q})$ is finite-dimensional. \n\n\\begin{lemma}\\label{chain2}\nLet $S_0\\triangleleft S_1$ be groups with\n $S_1\/S_0$ \nfinite or cyclic.\nIf $H_j(S_0;\\mathbb{Q})$ is finite dimensional for $0\\le j\\le n$,\nthen $H_j(S_1;\\mathbb{Q})$ is finite dimensional for  $0\\le j\\le n$.\n\\end{lemma}\n\n\\begin{proof}\nIn the LHS spectral sequence for the\ngroup extension $S_0\\to S_1\\to (S_1\/S_0)$ we have\n$E^2_{p,q}=H_p(S_1\/S_0;H_q(S_0;\\mathbb{Q}))$.\nBy hypothesis,\n$E^2_{p,q}$ has finite $\\mathbb{Q}$-dimension for $q\\le n$.  \nMoreover, $E^2_{p,q}=0$ for $p>1$, since $S_1\/S_0$ \nhas homological dimension at most $1$ over $\\mathbb{Q}$.\nThus the derivatives on the $E^2$ page all vanish and the spectral sequence stabilizes at the $E^2$ page.  \nHence, for $0\\le j\\le n$, we have\n$$\\mathrm{dim}_\\mathbb{Q}(H_j(S_1;\\mathbb{Q}))=\\mathrm{dim}_\\mathbb{Q}(E^2_{0,j})+\\mathrm{dim}_\\mathbb{Q}(E^2_{1,j-1})<\\infty,$$\nas required.\n\\end{proof}\n\nRepeatedly applying this lemma to the subnormal sequence \n$S_0\\triangleleft S_1\\triangleleft\\cdots\\triangleleft S_k=N_0$\nimplies that $H_j(N_0;\\mathbb{Q})$ is finite dimensional for all $j\\le n$, contradicting Theorem \\ref{theoremkernelZ}.\n\n\\end{pfof}\n\n\nThis completes the proof of Theorem \\ref{main3}, \nfrom which Theorem \\ref{main} follows immediately.\n\n \\section{From Theorem \\ref{main3} to Theorem \\ref{split}} \\label{s:last}\n\nLet $\\Gamma_i,L_i$ and $S$ be as in the statement of Theorem \\ref{split},\nbut without necessarily assuming that the $L_i$ are non-abelian for all $i$.  We first discuss how this situation differs from the special case\nstated in Theorem \\ref{split}.\n\nIf some $L_i$ is trivial, then $S$ is isomorphic to a subgroup\nof the direct product of the $\\Gamma_j$ with $j\\ne i$, as in Proposition\n\\ref{assume} (3).  We now assume that $L_i\\ne \\{1\\}$ for each $i$.\n\nAs in Proposition \\ref{assume} (2), we may replace each $\\Gamma_i$ by\n$p_i(S)$, where $p_i:S\\to\\Gamma_i$ is the projection, and\nhence assume that $p_i$ is surjective, and so each $L_i$ is normal in\n$\\Gamma_i$.\n\nIf some $L_i$ is non-trivial and abelian, then it is free abelian of finite\nrank, by \\cite[Corollary 1.23]{BF}.  Since $L_i$ is normal, it has finite\nindex in $\\Gamma_i$, and it follows immediately from the $\\omega$-residually\nfree property that $\\Gamma_i$ is itself abelian.\n\nArguing as in Proposition \\ref{assume} (4), we may assume that only one of\nthe $\\Gamma_i$ is abelian, say $\\Gamma_1$,  and that $L_1$ is the only\nnon-trivial abelian $L_i$.   We may also assume that $L_1$ is a direct \nfactor of $\\Gamma_1$; say $\\Gamma_1=L_1\\times M_1$.  But then $S$ virtually\nsplits as a direct product $L_1\\times S'$, where $S'=S\\cap (\\Gamma_2\\times\\cdots \\Gamma_n)$.\n\nNote that the above reduction involved only one passage to a finite index subgroup,\nand that was within the abelian factor $\\Gamma_1$.  The other $\\Gamma_i$ and $L_i$\nare left unchanged.  In particular, the $L_i$ remain non-abelian.\n\nWe have now reduced to the situation of the statement of Theorem \n\\ref{split}, with the additional hypothesis that each $p_i:S\\to\\Gamma_i$\nis surjective.\n\nIn particular, each  $L_i$ is normal in   $\\Gamma_i$, and hence\nis of finite index   for $i=1,\\dots,r$. \n\nLet\n $\\Pi_r:\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to \\Gamma_1\\times\\cdots\\times\\Gamma_r$\n be the natural projection,\n let $\\Lambda =  L_1\\times\\cdots\\times L_r$\n and let $\\hat S_0=S\\cap\\Pi_r^{-1}(\\Lambda)$.\n Then $\\hat S_0$ has finite index \n in $S$ and $\\hat S_0=\\Lambda\\times \\hat S_2$, where\n  $\\hat S_2 = \\hat S_0\\cap(\\Gamma_{r+1}\\times\\cdots\\times\\Gamma_n)$.\nTheorem \\ref{main3} now says that  that $\\hat S_2$ has\n a subgroup of finite index $S_2$ with\n$H_k(S_2;\\mathbb Q)$  infinite dimensional for some $k\\le n-r$.   \n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nA recent class of (mostly medical) imaging modalities, called hybrid, coupled-physics, or multi-wave modalities offers the possibility to reconstruct high-contrast parameters of interest with high resolution. High contrast is important to discriminate between, say, healthy and non-healthy tissues. Resolution is important to detect anomalies at an early stage. \n\nSuch hybrid modalities typically involve two steps. In the first step, not considered in this paper, a high resolution modality takes as an input measurements performed at the boundary of a domain of interest and provides as an output internal functionals of the parameters of interest and of specific solutions of underlying partial differential equations describing the probing (medical imaging) mechanism. This paper is concerned with the second step, involving the quantitative reconstruction of the parameters from knowledge of said internal functionals. For recent books and reviews on hybrid inverse problems, we refer the reader to, e.g., \\cite{A-Sp-08,AS-IP-12,B-IO-12,KK-EJAM-08,S-SP-2011,WW-W-07}.\n\nMost practically used hybrid inverse problems involve internal functionals that are polynomials in the parameters of interest and the specific solutions mentioned above. Combined with the equations describing the latter solutions, we observe that all available information represents a coupled, often redundant, system of nonlinear partial differential equations.\n\nIn some instances, local algebraic manipulations allow us to solve such a system explicitly. In the framework of functionals of solutions to second-order equations (and not of their derivatives), we refer the reader to, e.g., \\cite{BR-IP-11,BRUZ-IP-11,BU-CPAM-13,BU-IP-10,BU-AML-12}. Such theories find applications in the quantitative step of the imaging modalities Photo-acoustic tomography, Thermo-acoustic tomography, Transient Elastography, and Magnetic Resonance Elastography; see also \\cite{CAB-IP-07,CLB-SPIE-09,MY-IP-04,MZM-IP-10,PS-IP-07,SU-IO-12}. In the framework of functionals of the gradients of solutions, which find applications in Ultrasound Modulated tomography and in Current Density Imaging, we refer the reader to, e.g., \\cite{ABCTF-SIAP-08,B-APDE-13,BBMT-13,BGM-IP-13,BGM-IPI-13,BM-LINUMOT-13,BS-PRL-10,CFGK-SJIS-09,GS-SIAP-09,KK-AET-11,MB-aniso-13,MB-IP-12,MB-IPI-12}.\n\nIn many cases, explicit algebraic inversions may not be known or may not be applicable because not enough information is available. This paper proposes a framework to address several such problems when the linearization of the coupled system is {\\em elliptic}. Hybrid inverse problems need not be elliptic; see the example of the $0$-Laplacian in \\cite{B-APDE-13,CFGK-SJIS-09} (also recalled below in section \\ref{sec:zerolap}) or the Photo-acoustic problem as treated in, e.g., \\cite{BR-IP-11,BU-IP-10}. However, when the number of internal functionals increases, the resulting hybrid system becomes more redundant and hence more likely to be elliptic. We consider such a setting in section \\ref{sec:ellipticity}. We recall that elliptic systems augmented with boundary conditions that satisfy the Lopatinskii conditions admit left-parametrices. This follows from the theory of Agmon-Douglis-Nirenberg \\cite{ADN-CPAM-59,ADN-CPAM-64} and the extensions to redundant systems by Solonnikov \\cite{S-JSM-73}. The existence of parametrices allows us to solve the linear problem up to possibly a finite dimensional space. Along with the construction of a parametrix, elliptic regularity theory provides optimal stability results for the linearization of the hybrid inverse problem. \n\nThe analysis of elliptic hybrid inverse problems was first addressed in \\cite{KS-IP-12} by means of systems of pseudo-differential operators that were shown to be elliptic in the sense of Douglis and Nirenberg. The differential systems considered in this paper simplify the analysis of boundary conditions and hence of injectivity for the linearized and nonlinear hybrid inverse problems as we now describe.\n\n\n\nThe possible existence of a finite dimensional kernel for the linearized hybrid inverse problem prevents us from determining whether the available internal functionals uniquely determine the coefficients of interest. Moreover, the dimension of the finite dimensional kernel is not stable with respect to small perturbations, which prevents us from analyzing the uniqueness and stability properties of the nonlinear hybrid problem. A powerful methodology to obtain uniqueness results in the framework of elliptic systems of equations is the notion of {\\em unique continuation}. In section \\ref{sec:ucp}, we revisit two classical notions of unique continuation. One is based on the Holmgren theorem, which we generalize to the setting of redundant systems considered in this paper. The second one is based on the use of Carleman estimates as they are formulated in Calder\\'on's uniqueness theorem. See \\cite{C-AJM-58,C-PSFD-62,H-SP-63,H-I-SP-83,H-III-SP-94,L-AM-57,N-AMS-73,Z-Birk-83} for references on these unique continuation results. Several extensions of these results, following the presentation in \\cite{N-AMS-73}, are given in the setting of redundant systems in section \\ref{sec:ucp} with proofs postponed to the appendix.\n\nOnce a reasonable uniqueness result has been obtained for the linearization of the nonlinear hybrid inverse problem, several statements about uniqueness and iterative reconstruction procedures can be formulated for the nonlinear hybrid inverse problem. A constructive fixed point iteration method and a non-constructive local uniqueness result for the nonlinear problem are presented in section \\ref{sec:nonlinear}.\n\nAs an application of the conditions of ellipticity including boundary conditions and the conditions for unique continuation, we consider the case of power density internal functionals $H_j(x)=\\gamma(x)|\\nabla u_j|^2(x)$, where $u_j$ is the solution to $\\nabla\\cdot\\gamma\\nabla u_j=0$ on an open domain $X\\subset\\Rm^n$ with boundary conditions $u_j=f_j$ on $\\partial X$ for $1\\leq j\\leq J$. For such a problem, we characterize the conditions under which the redundant problem is elliptic (for $J=2$ in dimension $n=2$ and $J=3$ in higher dimension) and analyze cases in which a unique continuation principle (UCP) applies.\n\n\n\n\\section{Inverse Problems with local internal functionals.}\n\\label{sec:ellipticity}\n\n\\subsection{Systems of nonlinear partial differential equations}\n\nLet $\\gamma$ be a set of constitutive parameters in (linear or nonlinear, scalar or systems of) partial differential equations of the form\n\\begin{equation}\n\\label{eq:pde} \n \\mL(\\gamma,u_j) =0  \\quad \\mbox{ in } X,\\qquad \\tilde \\mB u_j = f_j\\quad\\mbox{ on } \\partial X,\n\\end{equation}\nwhere $\\mL$ is a polynomial in the derivatives of the solution $u_j$ and those of $\\gamma$ on the open domain $X\\subset\\Rm^n$, with $u_j$ augmented with boundary conditions on $\\partial X$ for $1\\leq j\\leq J$. \n\nLet us now assume knowledge of the functionals\n\\begin{equation}\n\\label{eq:fct}\n\\mM(\\gamma,u_j) = H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{equation}\nwhere $\\mM$ is a polynomial in the derivatives of the solution $u_j$ and those of $\\gamma$.\n\nSeveral hybrid inverse problems may be recast in this general framework. More generally, we could have knowledge of functionals of the form $\\mM(\\gamma,u_i,u_j) = H_{ij}$, or functionals $\\mM$ depending on more than two solutions $u_j$. We restrict ourselves to \\eqref{eq:fct} to simplify notation.\n\nThe above problem may thus be recast as a system of nonlinear partial differential equations for $(\\gamma,\\{u_j\\})$:\n\\begin{equation}\n\\label{eq:syst}\n\\begin{array}{rcl}\n\\mL(\\gamma,u_j) &=&0  \\quad \\mbox{ in } X,\\qquad \\tilde \\mB u_j = f_j\\quad\\mbox{ on } \\partial X, \\qquad 1\\leq j\\leq J \\\\\n\\mM(\\gamma,u_j) &=& H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{array}\n\\end{equation}\n\nThe first relevant question for the inverse problem is whether the above system admits a {\\em unique solution}. Since the solutions $u_j$ are uniquely determined by knowledge of $\\gamma$, we are primarily interested in finding a unique solution to the parameters $\\gamma$. The strategy followed in \\cite{KS-IP-12} consists of writing a system of pseudo-differential equations for $\\gamma$. Considering the higher-dimensional coupled system of equations for $(\\gamma,\\{u_j\\})$ allows us to simplify the analysis of the uniqueness question for \\eqref{eq:syst}.\n\nThe second question pertains to the stability properties of the reconstruction. Provided that the solution to the inverse problem is unique, we wish to understand how perturbations in the information $\\{H_j\\}$ propagates to the reconstruction of $\\gamma$.\n\nThe uniqueness and stability properties of the system depend on the number of acquired internal functionals $J$ and on the way the medium was probed via the boundary conditions $\\{f_j\\}$. Understanding how the uniqueness and stability properties are affected by changes in $J$ and the boundary conditions $\\{f_j\\}$ is the third question we wish to (very partially) answer.\n\n\\subsection{Linearization}\n\nSome problems of the form \\eqref{eq:syst} can directly be solved as non-linear systems. For instance, when $\\mL$ is a linear second-order equation in $u_j$ and $\\mM(\\gamma,u_j)=u_j$ the solution itself, the full non-linear problem is analyzed in \\cite{BU-CPAM-13,BU-AML-12}.\n\nFor many problems in which direct reconstruction procedures may not readily be available, it is fruitful to analyze the linearization of \\eqref{eq:syst}. Neglecting boundary conditions at first, this yields\n\\begin{equation}\n\\label{eq:linsyst}\n\\begin{array}{rcl}\n\\partial_\\gamma \\mL(\\gamma,u_j) \\delta\\gamma + \\partial_u \\mL(\\gamma,u_j) \\delta u_j&=&0  \\quad \\mbox{ in } X, \\qquad 1\\leq j\\leq J\\\\\n\\partial_\\gamma \\mM(\\gamma,u_j) \\delta\\gamma + \\partial_u \\mM(\\gamma,u_j) \\delta u_j &=& \\delta H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{array}\n\\end{equation}\nNote that the above differential operators may all be of different orders. With $v = (\\delta\\gamma,\\{\\delta u_j\\})$, we may recast the above system as\n\\begin{equation}\n\\label{eq:linear}\n\\mA v = \\mS,\n\\end{equation}\nfor an implicitly defined source $\\mS$. Let us assume that each $u_j$ is a scalar solution and that each $H_j$ is also a scalar information. Then $\\mA$ is a system of differential operators of size $2J\\times(J+M)$, where $M$ is the number of scalar functions describing $\\gamma$. Note that for $J<M$, the above system is under-determined. We consider here the case $J\\geq m$. In several practical problems, $J=M$ gives rise to a determined system $\\mA$ that is either not invertible, or invertible with non-optimal stability properties. It is therefore also fruitful to consider the setting with $J>M$.\n\n\n\\subsection{Ellipticity}\n\nIn applications, $\\mL$ is often a linear, elliptic, operator in the variables $\\{u_j\\}$. Adding the constraints \\eqref{eq:fct}, however, may render the coupled system \\eqref{eq:syst} or its linear version \\eqref{eq:linsyst} non-elliptic. One fruitful strategy to solve \\eqref{eq:linear} on the whole domain $X$ (with appropriate boundary conditions) is therefore to ``ellipticize\" $\\mA$, i.e., to find a number of constraints $J$ sufficiently large so that $\\mA$ is elliptic, provided that such a $J$ exists.\n\nWhat we mean by elliptic is defined as follows. For each $x\\in X$, $\\mA(x,D)_{ij}$ is a polynomial in $D=(\\partial_{x_1},\\ldots,\\partial_{x_n})$ for $1\\leq i\\leq 2J$ and $1\\leq j\\leq J+M$. We define the {\\em principal part} $\\mA_0$ of $\\mA$ in the sense of Douglis and Nirenberg \\cite{DN-CPAM-55}. For each row $1\\leq i\\leq 2J$ of the system, we associate an integer $s_i$ and for each column $1\\leq j\\leq J+M$ of the system an integer $t_j$. We normalize these integers by assuming that ${\\rm max}(s_i)=0$.\n\nWe assume that $\\mA_{ij}(x,D)$ is a polynomial in $D$ of degree not greater than $s_i+t_j$.  Then $\\mA_{0,ij}(x, D)$ is the part of the polynomial in $\\mA_{ij}(x,D)$  of degree exactly equal to $s_i+t_j$. \n\nWhen all differential operators in \\eqref{eq:linsyst} are of the same order $t$, then we may choose $s_i=0$ and $t_j=t$, in which case $\\mA_0(x,D)$ is composed of entries that are homogeneous polynomials of degree $t$ in $D$. Many practical problems arise in forms in which the differential operators in \\eqref{eq:linsyst} have different orders.\n\nWe say that $\\mA$ is {\\em elliptic} when the matrix $\\mA_0(x,\\xi)=\\{\\mA_{0,ij}(x,\\xi)\\}$, the symbol of the operator $\\mA_0$, is {\\em full-rank} (i.e., of rank $J+M$ here) for all $\\xi\\in\\Sm^{n-1}$ the unit sphere and all $x\\in\\bar X$.\n\nBeing full-rank is ``more likely\" when $J$ is large, i.e., when $\\mA$ is over-determined. It is then useful to acquire redundant information $H_j$ until \\eqref{eq:linear} above is elliptic because elliptic systems enjoy more favorable (and in fact optimal) stability estimates than non-elliptic systems.\n\n\\subsection{Lopatinskii boundary conditions}\n\\label{sec:Lop}\n\nLet us assume that we have been able to prove that $\\mA$ was a redundant elliptic system of equations. Then the system can be solved, up to possibly a finite dimensional subspace, when it is augmented by boundary conditions that satisfy the Lopatinskii criterion. This is defined as follows; see \\cite{S-JSM-73}.\n\nWe consider the system\n\\begin{equation}\n\\label{eq:linbc}\n\\mA v = \\mS\\quad\\mbox{ in } X,\\qquad \\mB v = \\phi\\quad\\mbox{ on } \\partial X,\n\\end{equation}\nwhere $\\mB(x,D)$ is a $Q\\times (J+M)$ matrix with entries $\\mB_{ij}(x,D)$ that are polynomial in $D$ for each $x\\in\\partial X$.  We denote by $b_{ij}$ the order of $\\mB_{ij}$ and by $\\sigma_i=\\max_j (b_{ij}-t_j)$. Then $\\mB_0(x,D)$ is the principal part of $\\mB$ and consists of entries $\\mB_{0,ij}(x,D)$ defined as the polynomials of $\\mB_{ij}(x,D)$ of degree exactly equal to $\\sigma_i+t_j$. \n\nThe Lopatinskii conditions are defined as follows. For each $x\\in\\partial X$, we denote by $\\nu(x)$ the outward unit normal to $X$ at $x\\in\\partial X$. We then think of $z$ as the parameterization of the half line $x-z\\nu(x)$ for $z\\geq0$. Let $\\zeta\\in\\Sm^{n-1}$ with $\\zeta\\cdot\\nu(x)=0$ and consider the system of ordinary differential equations\n\\begin{equation}\n\\label{eq:lopat}\n\\begin{array}{rcl}\n \\mA_0 (x,i\\zeta + \\nu(x) \\dr{}z) u(z) &=& 0 \\quad \\mbox{ in } z>0 \\\\[2mm]\n \\mB_0 (x,i\\zeta + \\nu(x) \\dr{}z) u(z) &=& 0 \\quad \\mbox{ at } z=0.\n\\end{array}\n\\end{equation}\nWe assume that for each $x\\in\\partial X$, the only solution to the above system such that $u(z)\\to 0$ as $z\\to\\infty$ is $u\\equiv0$. This is the Lopatinskii condition for $(\\mA,\\mB)$. We then also say that $\\mB$ covers $\\mA$.\n\nThe above conditions need to be verified for the specific problems being considered. In some situations, the boundary conditions provided by \\eqref{eq:pde} (or their linearization) generate a cover of $\\mA$. In other situations, they need to be augmented with additional boundary conditions for $\\delta u_j$ as well as for $\\delta\\gamma$ as we shall see.\n\nWhen $\\mA$ is elliptic and $\\mB$ covers $\\mA$, we say that $(\\mA,\\mB)$ is an elliptic system.\n\n\\subsection{Parametrices and stability estimates.}\n\\label{sec:param}\n\nFollowing work in \\cite{ADN-CPAM-59,ADN-CPAM-64,DN-CPAM-55} on determined systems, the case of overdetermined elliptic systems was treated in \\cite{S-JSM-73}. The salient feature of these works is that the operator $A=(\\mA,\\mB)$ admits a left-parametrix (a left-regularizer) in the following sense. Let $(\\mS,\\phi)$ in \\eqref{eq:linbc} be in the space\n\\begin{displaymath}\n \\mR(p,l) = W_p^{l-s_1}(X)\\times\\ldots\\times  W_p^{l-s_{2J}}(X) \\times W_p^{l-\\sigma_1-\\frac1p}(\\partial X) \\times \\ldots \\times W_p^{l-\\sigma_Q-\\frac1p}(\\partial X),\n\\end{displaymath}\nfor some $l\\geq0$ and $p>1$ and let us assume that $(\\mA,\\mB)$ is a bounded operator from \n\\begin{displaymath}\n v\\in \\mU(p,l) = W_p^{l+t_1}(X)\\times\\ldots\\times W_p^{l+t_{J+M}}(X)\n\\end{displaymath}\nto $(\\mA,\\mB)v=(\\mS,\\phi)\\in\\mR(p,l)$. Such is the case when the coefficients of $\\mA$ and $\\mB$ are sufficiently regular. More precisely, with $l$ sufficiently large so that $p(l-s_i)>n$ for all $1\\leq i\\leq 2J$ (to simplify; see \\cite{S-JSM-73} for slight generalizations), we assume that $\\mA_{ij}$ is a sum of homogeneous operators of degree $s_i+t_j-\\kappa$ for $0\\leq \\kappa\\leq s_i+t_j$ and that the coefficients of these operators are of class $W^{l-s_i}_p(X)$. Moreover, assuming $l$ large enough so that $p(l-\\sigma_q)>n$ as well for $1\\leq q\\leq Q$, we assume that $\\mB_{qj}$ is a sum of homogeneous operators of degree $\\sigma_q+t_j-\\kappa$ for $0\\leq \\kappa\\leq \\sigma_q+t_j$ and that the coefficients of these operators are of class $W^{l-\\sigma_q-\\frac 1p}(\\partial X)$.\n Here $W_p^s(X)$ is the standard Sobolev space of functions with $s$ derivatives that are $p$-integrable in $X$ with standard extensions for $s$ not an integer \\cite{adams}.\n\nThe main result in \\cite{S-JSM-73} is the existence of a bounded operator $R$ from $\\mR(p,l)$ to $\\mU(p,l)$ such that \n\\begin{equation}\n\\label{eq:leftparam}\nRA=I-T,\n\\end{equation}\nwhere $I$ is the indentity operator and $T$ is compact in $\\mU(p,l)$. \nWhen $1$ is not in the spectrum of $T$ so that $I-T$ is invertible, then $A$ is invertible with bounded inverse $(I-T)^{-1}R$.\nHowever, $1$ could very well be in the spectrum of $T$, in which case ${\\rm dim Ker}A$ is finite but positive.\n\nMoreover, we have the following stability estimate\n\\begin{equation}\n\\label{eq:stabell}\n\\dsum_{j=1}^{J+M} \\|v_j\\|_{W_p^{l+t_j}(X)} \\leq C \\Big( \\dsum_{i=1}^{2J} \\|\\mS_i\\|_{W_p^{l-s_i}(X)} + \\dsum_{i=1}^Q \\| \\phi_i\\|_{W_p^{l-\\sigma_i-\\frac 1p}(\\partial X)} \\Big) + C_2 \\dsum_{t_j>0} \\|v_j\\|_{L^p(X)},\n\\end{equation}\nfor some constants $C>0$ and $C_2>0$. \n\nThe presence of $C_2>0$ indicates the possibility that $A$ may not be invertible. The presence of finite dimensional kernels is a serious difficulty in the analysis of the nonlinear problem \\eqref{eq:syst} because such a dimension is not stable with respect to perturbations. What we can ensure is that for $A_1$ sufficiently small, then ${\\rm dim Ker}(A+A_1)\\leq{\\rm dim Ker}A$; see, e.g., \\cite{H-III-SP-94}.\n\nWhether we can choose $C_2=0$ above, i.e., whether $A$ is invertible, depends on lower-order terms that are not captured by the principal part $(\\mA_0,\\mB_0)$. Their analysis can prove quite complicated in practical settings and we do not follow that route here. Instead, our aim is to modify $A$ so that a unique continuation principle may be applied. In section \\ref{sec:ucp}, we augment the properly modified system $(\\mA,\\mB)v=(\\mS,\\phi)$ with additional boundary conditions, which in some cases allow us to obtain injectivity results. \n\nNote that the parametrix $R$ is clearly not unique. It is theoretically constructive, as can be seen by following the proof in \\cite{ADN-CPAM-64,S-JSM-73}. However, its practical, for instance numerical, implementation is not straightforward. The modified, higher-order, systems proposed later in the section offer a more direct numerical inversion procedure.\n\n\\subsection{Example of power-density measurements}\n\\label{sec:pdm}\nTo illustrate the theoretical result of this paper, we consider the example of the reconstruction of a scalar coefficient from knowledge of the so-called power density measurements. Consider the scalar elliptic equation\n\\begin{equation}\n\\label{eq:ellj}\n\\mL(\\gamma, u_j) := \\nabla\\cdot \\gamma\\nabla u_j =0 \\quad \\mbox{ in } \\quad X,\\qquad u_j = f_j \\quad \\mbox{ on } \\partial X,\\qquad 1\\leq j\\leq J.\n\\end{equation}\nHere, $X$ is an open domain in $\\Rm^n$ for $n\\geq2$ with smooth boundary $\\partial X$.\nThe objective is to reconstruct the scalar coefficient $\\gamma$, uniformly bounded above and below by positive constants, from knowledge of the power densities\n\\begin{equation}\n\\label{eq:pdj}\nH_j(x) = \\mM(\\gamma,u_j) := \\gamma(x) |\\nabla u_j|^2 ,\\qquad x\\in X,\\qquad 1\\leq j\\leq J,\n\\end{equation}\nwhere $u_j$ is the solution to \\eqref{eq:ellj}.\n\nThis problem and some variations have received significant theoretical and numerical analyses in recent years; see, e.g., \\cite{ABCTF-SIAP-08,B-APDE-13,BBMT-13,BS-PRL-10,CFGK-SJIS-09,GS-SIAP-09,KK-AET-11,KS-IP-12,MB-IPI-12}; generalizations for anisotropic coefficients $\\gamma$ can be found in \\cite{BGM-IP-13,BGM-IPI-13,MB-IP-12,MB-aniso-13}. Explicit reconstruction procedures exist when the number of internal functionals $J$ is sufficiently large; see \\cite{BBMT-13,BGM-IP-13,CFGK-SJIS-09,MB-IPI-12,MB-aniso-13}. The case $J=1$, which does not correspond to an elliptic system, was analyzed in \\cite{B-APDE-13}. The main features of this analysis are recalled in section \\ref{sec:zerolap}. For intermediate values of $J$, the above hybrid inverse problem may not have an explicit reconstruction but may still be modeled by a redundant elliptic system. Such a problem was also analyzed in \\cite{KK-AET-11,KS-IP-12}. The conditions of ellipticity of the system $(\\mA,\\mB)$ are described in detail in section \\ref{sec:linell}. A modified system is presented in \\ref{sec:elimell}, whereas optimal stability estimates of the form \\eqref{eq:stabell} are presented in section \\ref{sec:stabestim} for the power density measurement problem.\n\n\\subsubsection{The $0-$Laplacian when $J=1$}\n\\label{sec:zerolap}\n\nWhen $J=1$, the $2\\times2$ system of nonlinear partial differential equations is formally determined with two unknown coefficients $(\\gamma,u_1)$. The elimination of $\\gamma$ from such a system is in fact straightforward and we obtain the equation for $u:=u_1$ (with $H:=H_1$) given by\n\\begin{equation}\n\\label{eq:0Lap}\n \\nabla\\cdot \\dfrac{H(x)}{|\\nabla u|^2}\\nabla u =0 \\quad \\mbox{ in } \\quad X,\\qquad u = f \\quad \\mbox{ on } \\partial X.\n\\end{equation} \nThe above equation may be transformed as \n\\begin{equation}\n  \\label{eq:Cauchy2}\n  (I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2 u + \\nabla \\ln H\\cdot\\nabla u =0\\,\\mbox{ in } X,\\qquad u=f \\,\\,\\mbox{ and } \\,\\, \\pdr{u}{\\nu} = j \\,\\, \\mbox{ on } \\partial X.\n\\end{equation}\nHere $\\widehat{\\nabla u}=\\frac{\\nabla u}{|\\nabla u|}$ and we introduced Cauchy data on $\\partial X$ anticipating the fact that \\eqref{eq:Cauchy2} is a quasilinear strictly hyperbolic equation, at least provided that $\\widehat{\\nabla u}$ is defined. Indeed, we observe that the operator $(I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2$ is hyperbolic with respect to the (unknown) direction $\\widehat{\\nabla u}$.\n\nThe above problem is analyzed in \\cite{B-APDE-13}. The two salient features of that analysis are: (i) unique reconstructions of $u$, and hence $\\gamma$, are guaranteed only on part of the domain $X$; see \\cite{B-APDE-13}; and (ii) the stability estimates are {\\em sub-elliptic}: first-order derivatives of $u$ are controlled by the gradient of $H$ rather than second-order derivatives as would be the case if $(I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2$ was replaced by an elliptic operator. For $\\gamma$, this translates into an inequality of the following form. Let $H$ and $\\tilde H$ be two measurements corresponding to the pairs $(u,\\gamma)$ and $(\\tilde u,\\tilde\\gamma)$, respectively. Assume that the Cauchy data of $u$ and $\\tilde u$ agree on $\\partial X$. Then we find that on an appropriate (see \\cite{B-APDE-13}) subdomain $\\mO\\subset X$, we have the following stability estimate:\n\\begin{equation}\n\\label{eq:subelliptic}\n \\|\\gamma-\\tilde\\gamma\\|_{L^2(\\mO)} \\leq C \\| \\nabla H-\\nabla \\tilde H\\|_{L^2(\\mO)}.\n\\end{equation}\nAs we shall see below, this estimate is sub-optimal (with a loss of one derivative) when compared to elliptic estimates of the form \\eqref{eq:stabell}. It is however optimal for (principally normal) operators of principal type \\cite{H-SP-63}.\n\n\\subsubsection{Linearization and ellipticity}\n\\label{sec:linell}\n\nThe linearization of the above problem with $J=1$ is a hyperbolic equation. We now wish to show that redundancy in the data ($J\\geq2$) allows us to render the system elliptic under some conditions. We consider two ways to obtain elliptic systems of equations.\n\nWe first linearize the coupled system \\eqref{eq:ellj}-\\eqref{eq:pdj} about solutions $(\\gamma,u_j)$ and obtain\n\\begin{equation}\n\\label{eq:pdlinsyst}\n\\begin{array}{rcll}\n\\nabla\\cdot \\delta\\gamma\\nabla u_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j &=& 0 &\\quad \\mbox{ in } \\, X\\\\\n\\delta\\gamma|\\nabla u_j|^2 + 2 \\gamma \\nabla u_j\\cdot\\nabla\\delta u_j &=& \\delta H_j &\\quad \\mbox{ in } \\, X\\\\\n\\delta u_j &=& 0 & \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nWe define \n\\begin{equation}\n\\label{eq:Fj}\nF_j=\\nabla u_j\n\\end{equation}\nand {\\em assume} that $|F_j|\\geq c_0>0$ is bounded from below by a positive constant uniformly. Such an assumption is valid for and appropriate open set of boundary conditions $f_j$; see, e.g., \\cite{B-IO-12,BBMT-13,BU-CPAM-13,BU-IP-10} for details of constructions based on complex geometric optics solutions or unique continuation principles, which we do not reproduce here.\n\nLet $\\mA_J$ be the operator applied to $\\delta v=(\\delta \\gamma,\\{\\delta u_j\\})$ in the above system. Its principal part $\\mP_J$ has for (principal) symbol $\\mp_j$ a $2J\\times(J+1)$ matrix given by\n\\begin{equation}\n\\label{eq:pdpJ}\n\\mp_J(x,\\xi) =  \\left(\\begin{matrix}\n |F_1|^2  & 2 \\gamma F_1\\cdot i\\xi & \\ldots & 0 \\\\\nF_1\\cdot i\\xi  & -\\gamma|\\xi|^2 & \\ldots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n |F_J|^2 & 0 & \\ldots & 2 \\gamma F_J\\cdot i\\xi\\\\\nF_J\\cdot i\\xi  & 0 & \\ldots & -\\gamma|\\xi|^2\n\\end{matrix}\\right).\n\\end{equation}\nWhen $J=1$, the determinant of $\\mp_J$ is given by $\\gamma|F_1|^2(2(\\hat{F_1}\\cdot\\xi)^2-|\\xi|^2)$, which is hyperbolic with respect to $F_1$ as seen in section \\ref{sec:zerolap} above. The above system is in Douglis-Nirenberg form for $s_{2k}=1$, $s_{2k+1}=0$, $t_1=0$, $t_j=1$ for $j\\geq2$.\n\nWhen $J\\geq1$, we observe that the sub-determinants with the largest number of powers of $|\\xi|^2$ are of the form\n\\begin{displaymath}\n   \\gamma^{J}|\\xi|^{2(J-1)} |F_j|^2 q_j(x,\\xi),\\qquad q_j(x,\\xi) := 2(\\hat{F_j}(x)\\cdot\\xi)^2-|\\xi|^2.\n\\end{displaymath}\nWe thus obtain that $\\mp_J(x,\\xi)$ is injective if\nthe quadratic forms $q_j(x,\\xi)=0$ for all $1\\leq j\\leq J$ imply that $\\xi=0$. \n\\begin{definition}\n\\label{def:ellcondpj}\nDefine the quadratic forms and operators\n\\begin{equation}\n\\label{eq:qjPj}\nq_j(x,\\xi) = 2 \\big(\\hat F_j\\cdot \\xi)^2 - |\\xi|^2 ,\\qquad P_j(x,D) = \\Delta - 2 \\hat F_j\\otimes\\hat F_j : \\nabla\\otimes \\nabla.\n\\end{equation}\nHere $\\hat F_j(x)$ are vector fields of unit vectors defined on $\\bar X$. We say that the family $\\{q_j\\}$ or $\\{P_j\\}$ is elliptic at $x$ if\n\\begin{equation}\n\\label{eq:condellip}\n q_j(x,\\xi) =0 \\mbox{ for all }1\\leq j\\leq J \\quad\\mbox{ implies } \\quad \\xi=0.\n\\end{equation}\nWe say that such families are elliptic in $X$ if they are elliptic in at all points $x\\in X$. \n\\end{definition}\nWe can then prove the \n\\begin{lemma}\n\\label{lem:ellipAJ}\nWe assume that $F_j:=\\nabla u_j$ is such that $|F_j|$ is bounded from below by a positive constant on $\\bar X$ for all $1\\leq j\\leq J$.\n\nThe operator $\\mA_J$ defined in \\eqref{eq:pdlinsyst} with principal symbol given in \\eqref{eq:pdpJ} is elliptic in $\\bar X$ if and only if the above family of quadratic forms $\\{q_j\\}$ is elliptic in $\\bar X$.\n\\end{lemma}\n\\begin{proof}\n  We have already seen the sufficiency of the condition. Let us prove its necessity and assume that $\\mp_J(x,\\xi)$ is maximal rank for $x\\in X$ and $\\xi\\not=0$. This means that one determinant of $(J+1)\\times(J+1)$ sub-matrices of $\\mp_J$ is non-vanishing. Each column of $\\mp_J$ beyond the first one has two non-vanishing entries. For all $1\\leq j\\leq J$ except for one entry $j_0$, then either $-\\gamma(x)|\\xi|^2$ or $2\\gamma F_j\\cdot i\\xi$ appears as a multiplicative factor in the determinant of the sub-matrix. Since $\\gamma(x)|\\xi|^2$ never vanishes, we may discard the determinant involving $2\\gamma F_j\\cdot i\\xi$. We thus obtain that if one determinant of a sub-matrix does not vanish, then that determinant may be chosen as $\\gamma^{J}|\\xi|^{2(J-1)} |F_{j_0}|^2 q_{j_0}(x,\\xi)$. Since by assumption $\\gamma^{J}|\\xi|^{2(J-1)} |F_{j_0}|^2$ is bounded away from $0$, we observe that the injectivity of $\\mp_J$ implies that (at least) one of the quadratic forms $q_{j}(x,\\xi)$ does not vanish. Thus, $\\mA_J$ being elliptic implies that $\\{q_j\\}$ is elliptic in the sense of definition \\ref{def:ellcondpj}.\n\\end{proof}\n\nThe ellipticity of $\\mA_J$ is thus a consequence of the fact that the null cones of quadratic forms intersect only at $0$. \nWe have the following properties:\n\\begin{proposition}\n\\label{prop:ell}\n (i) Let $x\\in\\Rm^2$ and assume that $\\hat F_1(x)$ and $\\hat F_2(x)$ are neither parallel nor orthogonal. Then the corresponding $(q_1,q_2)$ in \\eqref{eq:qjPj} form an elliptic family at $x$. \n \n (ii) In dimension $n\\geq2$, let $\\hat F_1(x)$ and $\\hat F_2(x)$ be two different directions and define $\\hat F_3(x)=\\alpha\\hat F_1(x)+\\beta\\hat F_2(x)$ with $\\alpha\\beta\\not=0$ such that $|\\hat F_3(x)|=1$. Then the corresponding $(q_1, q_2, q_3)$ in \\eqref{eq:qjPj} form an elliptic family at $x$. \n \n (iii) In dimension $n\\geq3$, a family $(q_1,q_2)$ is never elliptic at a given point $x$ independent of the choice of $\\hat F_1,\\hat F_2$.\n\\end{proposition}\n\\begin{proof}\n  (i) In dimension $n=2$, it is clear that the null cones (where $q_j$ vanishes, two lines of vectors $\\xi$ in $\\Rm^2$) coincide if and only if $\\hat F_1$ and $\\hat F_2$ are either parallel or orthogonal.\n  \n(iii) The result is obvious if $\\hat F_1=\\pm \\hat F_2$. Assume otherwise and let $F_3$ be a unit vector orthogonal to $\\hat F_1$ and $\\hat F_2$. Assume $\\hat F_1\\cdot\\hat F_2\\geq0$ for otherwise change the sign of $\\hat F_2$ (which does not modify the quadratic forms $q_j$). $\\xi$ belongs to the intersection of the null cones $\\{q_j(x,\\xi)=0\\}$ for $j=1,2$ if $2(\\hat F_1\\cdot\\xi)^2=2(\\hat F_2\\cdot\\xi)^2=|\\xi|^2$. Define $\\xi=\\hat F_1+\\hat F_2+\\lambda F_3$. The first constraint is satisfied when $2(1+\\hat F_1\\cdot\\hat F_2)^2 = |\\hat F^1+\\hat F^2|^2+\\lambda^2$, i.e., when  $\\lambda=\\pm \\big(2\\hat F_1\\cdot\\hat F_2+ 2(\\hat F_1\\cdot\\hat F_2)^2)^{\\frac12}$. But then it is clear that this $\\xi\\not=0$ also belongs to the second cone so that $(q_1,q_2)$ is not elliptic.\n\n(ii) Let us assume that $\\hat F_3=\\alpha \\hat F_1+\\beta \\hat F_2$ with $\\alpha\\beta\\not=0$. Then $\\xi$ belongs to the three null cones if \n\\begin{displaymath}\n \\Big(\\dfrac{\\alpha \\hat F_1+\\beta \\hat F_2}{|\\alpha \\hat F_1+\\beta \\hat F_2|}\\cdot\\xi\\Big)^2 = \\dfrac12 |\\xi|^2 = (\\hat F_1\\cdot \\xi) ^2 = (\\hat F_2\\cdot\\xi)^2.\n\\end{displaymath}\nExpanding the first constraint, we get\n\\begin{displaymath}\n \\alpha^2(\\hat F_1\\cdot\\xi)^2 + \\beta^2(\\hat F_2\\cdot\\xi)^2 + 2\\alpha\\beta \\hat F_1\\cdot\\xi\\hat F_2\\cdot\\xi = \\frac12|\\xi|^2(\\alpha^2+\\beta^2+2\\alpha\\beta \\hat F_1\\cdot\\hat F_2).\n\\end{displaymath}\nThe last two constraints imply that $2\\hat F_1\\cdot\\xi\\hat F_2\\cdot\\xi=\\eps|\\xi|^2$ with $\\eps=\\pm1$. Combined with the latter, they also imply that\n\\begin{displaymath}\n\\alpha\\beta \\eps = \\alpha\\beta \\hat F_1\\cdot\\hat F_2.\n\\end{displaymath}\nSince $\\alpha\\beta\\not=0$, this can only occur if $\\hat F_1=\\pm\\hat F_2$, which is a contradiction.\n\\end{proof}\n\nFrom the practical point of view, this result says that if $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are not parallel, then the internal functionals $H_j$ for $u_1$, $u_2$, and $u_1+u_2$ (with $F_3=\\nabla (u_1+u_2)=\\nabla u_1+\\nabla u_2$) generate three quadratic forms $q_j$, $1\\leq j\\leq 3$ that form an elliptic family. Such internal functionals are obtained by choosing three boundary conditions of the form $f_1$, $f_2$, and $f_3=f_1+f_2$. This result holds for all $n\\geq2$.\n\n\\subsubsection{Sufficient conditions for ellipticity} We have thus obtained the following result. In dimension $n=2$, $J\\geq2$ is necessary for $\\mA_J$ to be elliptic. Moreover, $J=2$ is sufficient when $n=2$ if $\\nabla u_1$ and $\\nabla u_2$ are nowhere parallel or orthogonal. In dimension $n\\geq3$, $J\\geq3$ is necessary for $\\mA_J$ to be elliptic. Moreover, $J=3$ is sufficient for $\\mA_J$ to be elliptic in all dimensions $n\\geq2$ by choosing as boundary conditions, e.g.,  $(f_1,f_2,f_1+f_2)$ provided that $\\nabla u_1$ and $\\nabla u_2$ are nowhere parallel.\n\n\\subsubsection{Boundary conditions and Lopatinskii condition.} In order to obtain an optimal theory of stability estimates, the system needs to be augmented with boundary conditions that satisfy the Lopatinskii condition. Dirichlet conditions on $\\delta u_j$ and no condition on $\\delta\\gamma$ satisfy such conditions. Indeed, we need to show that $v(z)=(\\delta\\gamma(z),\\ldots,\\delta u_J(z))\\equiv 0$ is the only solution to\n \\begin{equation}\\label{eq:pdLP}\n \\begin{array}{l}\n\\delta u_j(0)=0,\\qquad    (iF_j\\cdot\\zeta+F_j\\cdot N \\partial_z) \\delta\\gamma + \\gamma (i\\zeta+\\partial_z)^2 \\delta u_j =0 ,\\,\\\\|F_j|^2 \\delta\\gamma +2\\gamma(iF_j\\cdot\\zeta+ F_j\\cdot N \\partial_z) \\delta u_j=0,\\,\\, z>0\n\\end{array}\n\\end{equation}\nwith $v(z)$ vanishing as $z\\to\\infty$ for $N=\\nu(x)$ at $x\\in\\partial X$ and $z$ coordinate along $-N$. Eliminating $\\delta\\gamma$ as earlier, we deduce that \n\\begin{displaymath}\n \\Big( |F_j|^2 (i\\zeta+\\partial_z)^2 - 2 (iF_j\\cdot\\zeta+F_j\\cdot N\\partial_z)^2 \\Big) \\delta u_j=0\n\\end{displaymath}\nThe leading term of the above second order equation with constant coefficients is $|F_j|^2q_j(x,N)\\partial_z^2$.  If $q_j(x,N)\\not=0$ for some $j=j_0$, which is the condition for joint ellipticity described in definition \\ref{def:ellcondpj}, then the same proof showing that Dirichlet conditions cover the Laplace operator show that $\\delta u_{j_0}=0$. We then deduce that $\\delta\\gamma=0$ from the second line in \\eqref{eq:pdLP} and by ellipticity on the first line in \\eqref{eq:pdLP} that all $\\delta u_j=0$ and hence $v\\equiv0$.\n\n\nLet us define the spaces $\\mX=W^l_{p}(X)\\times W^{l+1}_p(X;\\Rm^J)$ and $\\mY^i=W^l_p(X;\\Rm^{2J})$ with $l$ large enough so that the latter spaces are all algebras; i.e., $pl>n$. We also define $\\mY^\\partial = W^{l+1-\\frac 1p}_p(X;\\Rm^{J})$ for the traces of $\\delta u_j$ on $\\partial X$, which vanish by construction. Then we observe that $(\\mA,\\mB)$ in \\eqref{eq:pdlinsyst} maps $\\mU(p,l)=\\mX$ to $\\mR(p,l)=\\mY^i\\times\\mY^\\partial$. Moreover, the coefficients $\\gamma$ and $u_j$ appearing in the definition of $\\mA$ belong to $W^l_{p}(X)$ and $W^{l+1}_p(X;\\Rm^J)$, respectively, by assumption for $\\gamma$ and by elliptic regularity for $u_j$ solution of the elliptic problem \\eqref{eq:ellj}.\n\n\\subsubsection{Elimination and ellipticity}\n\\label{sec:elimell}\n\nThe above system involves $J+1$ unknowns. Strategies to lower the dimension of the system include: (i) eliminating $\\gamma$ as we did in obtaining \\eqref{eq:0Lap}; or (ii) eliminating all $u_j$. The second strategy follows from the observation\n\\begin{displaymath}\n   \\delta u_j = L_\\gamma^{-1} (\\nabla\\cdot \\delta\\gamma \\nabla u_j),\\qquad L_\\gamma = -\\nabla\\cdot\\gamma \\nabla,\n\\end{displaymath}\nwith $L_\\gamma^{-1}$ defined by solving $L_\\gamma$ with vanishing Dirichlet conditions.\n\nThe result is a redundant system of the form $P_j\\delta\\gamma=\\delta H_j$ where $P_j$ is a pseudo-differential operator with principal symbol given by $|F_j|^2q_j(x,\\xi)$. The redundant system for $\\delta\\gamma$ is therefore elliptic under the same conditions as those for \\eqref{eq:pdpJ} above. The main difficulty is that $P_j$ is no longer local (no longer a (system of) partial differential equation). It is not clear how one may approach the question of uniqueness for such a system. The unique continuation principles presented in section \\ref{sec:ucp} do not apply directly. See \\cite{KS-IP-12} for an analysis of such a method.\n\nThe first strategy based on the elimination of $\\delta\\gamma$ preserves the differential structure of the original system since $\\delta\\gamma$ appears undifferentiated in the second equation in \\eqref{eq:pdlinsyst}. We find that\n\\begin{equation}\n\\label{eq:deltauj}\n\\nabla\\cdot\\gamma\\big(-\\nabla \\delta u_j + 2 \\hat F_j \\hat F_j\\cdot\\nabla \\delta u_j\\big) = \\nabla\\cdot \\dfrac{\\delta H_j}{|F_j|^2} F_j.\n\\end{equation}\nThe symbol for such an equation is then given by $\\gamma q_j(x,\\xi)$ so that the above equation is not elliptic, as we already know. However, the elimination of $\\delta\\gamma$ also provides the constraint\n\\begin{equation}\n\\label{eq:constraintjk}\n2\\gamma\\dfrac{1}{|F_j|^2} F_j\\cdot\\nabla \\delta u_j - \\dfrac{\\delta H_j}{|F_j|^2}  =  2\\gamma\\dfrac{1}{|F_k|^2} F_k\\cdot\\nabla \\delta u_k - \\dfrac{\\delta H_k}{|F_k|^2} \\quad 1\\leq j<k\\leq J.\n\\end{equation}\nIt turns out that the combination of \\eqref{eq:deltauj} with \\eqref{eq:constraintjk} makes the redundant system for the $\\{u_j\\}$ elliptic provided $q_j(x,\\xi)=0$ for all $j$ implies that $\\xi=0$ as above. To see this, assume that \n\\begin{displaymath}\n  q_j(x,\\xi) \\delta u_j =0,\\qquad |F_k|^2F_j\\cdot\\xi \\delta u_j = |F_j|^2 F_k\\cdot\\xi \\delta u_k.\n\\end{displaymath}\nLet $\\xi\\not=0$. Then, not all $q_j(x,\\xi)$ vanish. Assume that $q_1(x,\\xi)\\not=0$. Then $\\delta u_1=0$ so that $F_j\\cdot\\xi \\delta u_j=0$ for all $j\\geq2$. However, $F_j\\cdot\\xi$ and $q_j(x,\\xi)$ cannot vanish at the same time so that $\\delta u_j=0$. This shows the injectivity of the symbol of the redundant system, which is easily found to be in Douglis-Nirenberg form. \n\nMoreover, unlike the redundant system of strategy (ii), the above system \\eqref{eq:deltauj}-\\eqref{eq:constraintjk} is differential. It may therefore be augmented with boundary conditions that satisfy the Lopatisnkii conditions. We leave the details to the reader to verify, as we did in \\eqref{eq:pdLP}, that such conditions are satisfied for Dirichlet conditions $\\delta u_j=0$ on $\\partial X$ under the same conditions guaranteeing ellipticity in $X$. The above system for the $(\\{\\delta u_j\\})$ is therefore elliptic when the family of quadratic forms $\\{q_j\\}$ is elliptic.\n\n\\subsubsection{Stability estimates}\n\\label{sec:stabestim}\n\nBoth systems of differential equations for $(\\delta\\gamma,\\{\\delta u_j\\})$ and for $(\\{\\delta u_j\\})$ after elimination of $\\delta\\gamma$ are elliptic with Dirichlet conditions for $\\delta u_j$ under the conditions stated, e.g., in Proposition \\ref{prop:ell}. We may therefore apply the general theory of elliptic redundant systems described above and obtain a parametrix for both systems. Moreover, in both cases, we obtain the following stability estimates\n\\begin{equation}\n\\label{eq:pdstab}\n  \\|\\delta\\gamma-\\delta\\tilde\\gamma\\|_{W_p^l(X)} + \\dsum_{j=1}^J \\|\\delta u_j-\\delta\\tilde u_j\\|_{W_p^{l+1}(X)} \\leq C \\dsum_{j=1}^J \\|\\delta H_j-\\delta \\tilde H_j\\|_{W_p^l(X)} +C_2 \\dsum_{j=1}^J \\|\\delta u_j-\\delta\\tilde u_j\\|_{L_p(X)}.\n\\end{equation}\nIn other words, $\\delta\\gamma$ and $\\nabla \\delta u_j$ are reconstructed from $\\delta H_j$ with {\\em no loss} of derivative, unlike the construction in \\eqref{eq:subelliptic}. However, we are not guaranteed that any of the linear systems is indeed invertible, and hence the presence of the last term on the right-hand-size in \\eqref{eq:pdstab}. Proving the injectivity of the above systems is much more delicate than proving their ellipticity. The following section proposes some generic strategies to do so. \n\n\\subsubsection{Generalization to similar models}\n\nThe results presented above generalize to the setting where the internal functionals are of the form\n\\begin{equation}\n\\label{eq:alphaH}\nH_j(x) = \\gamma^{\\alpha} |\\nabla u_j|^2,\n\\end{equation}\nwhere $\\alpha\\geq0$. The case $\\alpha=1$ was treated above. The case $\\alpha=2$ corresponds to the setting of CDII and MREIT \\cite{NTT-Rev-11,SW-SR-11}. As shown, e.g., in \\cite{KS-IP-12,MB-IPI-12}, the coupled problem is elliptic with $J=1$ for $\\alpha>2$, is degenerate elliptic for $\\alpha=2$, and is hyperbolic for $\\alpha<2$. The cases $\\alpha\\geq2$ become elliptic for $J$ chosen sufficiently large. We do not consider these extensions further here.\n\n\\subsection{Sufficient conditions for ellipticity}\n\\label{sec:ellipticitycond}\n\nTo summarize the above derivation, we have modeled the nonlinear hybrid inverse problem as a coupled system of nonlinear partial differential equations \\eqref{eq:syst}. Its linearization is then given by \\eqref{eq:linsyst}. These systems have $J+M$ unknowns for $2J\\geq J+M$ equations. When $J$ is large compared to $M$, we expect the principal part $\\mA_0$ of $\\mA$ in the Douglis-Nirenberg sense to be elliptic since a redundant matrix is ``more likely\" to be full-rank than a less elongated matrix. \n\n\nThe matrix $\\mA_0$ is full rank if we can prove that, for well chosen boundary conditions $f_j$, the internal functionals $H_j$ are sufficiently independent. Such a result is problem-dependent. In the setting of power-density measurements, we have obtained that the boundary conditions $f_j$ were well-chosen if the quadratic form $q_j(x,\\xi)$ were jointly elliptic at each point $x\\in \\bar X$. A sufficient condition to do so is to choose two boundary conditions $f_1$ and $f_2$ such that $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are nowhere co-linear (for the quadratic forms corresponding to the three boundary conditions $f_1$, $f_2$, and $f_1+f_2$ are then jointly elliptic).\n\nStrategies to ensure that, e.g., $\\nabla u_1$ and $\\nabla u_2$ are nowhere co-linear have been presented in, e.g., \\cite{B-APDE-13,BBMT-13,BU-IP-10,BU-CPAM-13}, see also the review \\cite{B-IO-12}. Such strategies are based on the use of Complex Geometrical Optics when the latter are available, or on the use on local construction and unique continuation properties of the operators in \\eqref{eq:pde} as described in \\cite{BU-CPAM-13}. In both settings, it is proved that qualitative properties of solutions to elliptic equations, such as for instance the independence of gradients of solutions, hold for an open set of boundary conditions $f_j$. More explicit constructions of such boundary conditions are also proposed in \\cite{BC-JDE-13}.\n\n\\section{Injectivity results for the linearized problem}\n\\label{sec:ucp}\n\nIn this section, we consider two methods to obtain injectivity of an elliptic operator $\\mA$ augmented with appropriate boundary conditions $\\mB$.\nBoth are based on replacing the redundant system $\\mA$ by its normal, determined, form $\\mA^t\\mA$, and augmenting it with appropriate Dirichlet boundary conditions that ensure its injectivity under certain assumptions. The first method invokes a Holmgren unique continuation principle while the second method is based on unique continuation principles that are consequences of Carleman estimates.\n\n\nHere, the operator $\\mA^t$ is defined such that $(\\mA^t)_{ki}=(\\mA_{ik})^t=:\\mA^t_{ik}$, where $(\\mA_{ik})^t$ is the formal adjoint to $\\mA_{ik}$ for the usual inner product $(\\cdot,\\cdot)$ on $L^2(X)$.\n\nOne difficulty with operators $\\mA$ that are elliptic in the Douglis-Nirenberg (DN) sense is that the normal operator $\\mA^t\\mA$ need not be elliptic, even in the DN sense. Consider for instance the $2\\times2$ system in one independent variable defined by $\\mA_{11}=a$, $\\mA_{12}=\\mA_{21}=\\partial_x$ and $\\mA_{22}=\\partial^2_x$, which is DN elliptic with $s_1=t_1=0$ and $s_2=t_2=1$ when $a\\not=1$. Defining $\\mC=\\mA^T\\mA$, the principal term in $\\mC$ is $\\mC_{11}=\\partial_x^2$, $\\mC_{12}=\\mC_{21}=\\partial_x^3$, $\\mC_{22}=\\partial_x^4$, which is independent of $a$ and not elliptic.\n\nOne way to ensure that $\\mA^t\\mA$ is elliptic when $\\mA$ is elliptic is to assume that $\\tau=t_j$ independent of $j$ and $s_i=0$ independent of $i$.  We then verify that the leading term in $(\\mA^t\\mA)_{kl}$ is a differential operator of degree equal to $2\\tau$ and that  $\\mA^t\\mA$ is a strongly elliptic system of size $(J+M)\\times(J+M)$; see, e.g., \\cite[Proposition 4.1.16]{T-AV-95}. The principal part of ${\\rm Det}(\\mA^t\\mA)(x,\\xi)$ is equal to ${\\rm Det}(\\mA_0^t\\mA_0)(x,\\xi)$ and is a polynomial of degree $2(J+M)\\tau$ that is uniformly bounded from below for all $x\\in \\bar X$ and $\\xi\\in\\Sm^{n-1}$ by assumption of ellipticity.\n\nThe results in \\cite{C-JMAA-91} show that an elliptic system in DN form can always be transformed into an elliptic overdetermined system of first-order equations. The procedure increases the order of the system and differentiates any row involving a term with $s_i+t_j=0$. This forces us to impose boundary conditions on the parameters $\\delta\\gamma$ and the solutions $\\delta u_j$ that may not be necessary in the definition of $(\\mA,\\mB)$ above. \n\nFor the rest of the section, we {\\em assume} that $\\mA$ has been recast into a form where $\\tau=t_j$ is independent of $j$ and $s_i=0$ is independent of $i$; for instance with $\\tau=1$ or $\\tau=2$ as described in \\cite{C-JMAA-91}. We still keep the notation $2J$ and $J+M$ for the size of the system $\\mA$ so that with this notation, $\\mA^t\\mA$ is a system of size $(J+M)\\times(J+M)$.\n\n\n\n\n\n\n\n\nLet us augment $\\mA^t\\mA$ with the Dirichlet boundary conditions\n\\begin{equation}\n\\label{eq:Dirichlet} \n\\big(\\pdr{}\\nu\\big)^q v_j =\\phi_{qj}\\quad\\mbox{ on } \\partial X, \\qquad 0\\leq q\\leq \\tau-1,\\quad 1\\leq j\\leq J+M.\n\\end{equation}\nWe recast the above constraints as $\\mD v=\\phi$ on $\\partial X$. It is proved in \\cite[p.43-44]{ADN-CPAM-64} that such boundary conditions cover $\\mA^t\\mA$, i.e., that the Lopatinskii conditions are satisfied. We thus consider the problem\n\\begin{equation}\n\\label{eq:normal}\n\\mA^t\\mA v = \\mA^t\\mS\\quad\\mbox{ in } X,\\qquad \\mD v = \\phi\\quad\\mbox{ on } \\partial X.\n\\end{equation}\nSince $N:=(\\mA^t\\mA,\\mD)$ is elliptic, the above system admits a left parametrix $G$ such that $GN=I-T$ with $T$ compact  in $\\mU(p,l)$. Moreover, we have the stability estimate\n\\begin{equation}\n\\label{eq:stabnormal}\n\\dsum_{j=1}^{J+M} \\|v_j\\|_{W_p^{l+\\tau}(X)} \\leq C \\Big( \\dsum_{j=1}^{J+M} \\|(\\mA^t\\mS)_j\\|_{W_p^{l-\\tau}(X)} + \\dsum_{j,q} \\| \\phi_{qj}\\|_{W_p^{l-\\tau+q-\\frac 1p}(\\partial X)} \\Big) + C_2 \\dsum_{j=1}^{J+M} \\|v_j\\|_{L^p(X)},\n\\end{equation}\nWe verify that \n\\begin{displaymath}\n  \\|\\sum_{i=1}^{2J} \\mA^t_{ji} S_i \\|_{W_p^{l-\\tau}(X)} \\leq C \\sum_{i=1}^{2J} \\|S_i\\|_{W_p^{l}(X)},\n\\end{displaymath}\nso that \\eqref{eq:stabnormal} may also be seen as an analog of \\eqref{eq:stabell}.\n\nOur objective is to find sufficient conditions under which the above system is injective, and hence invertible, so that $C_2=0$ in the above estimates. We start with the following simple lemma:\n\\begin{lemma}\n\\label{lem:Normal}\nLet us assume that $v$ is a solution of \n\\begin{equation}\n\\label{eq:normal0}\n\\mA^t\\mA v = 0 \\quad\\mbox{ in } X,\\qquad \\mD v = 0\\quad\\mbox{ on } \\partial X.\n\\end{equation}\nThen $v$ is a solution of \n\\begin{equation}\n\\label{eq:syst0}\n\\mA v = 0 \\quad\\mbox{ in } X,\\qquad \\mD v = 0\\quad\\mbox{ on } \\partial X.\n\\end{equation}\n\\end{lemma}\nNote that the Dirichlet conditions associated to $\\mA^t\\mA$ are now redundant for \\eqref{eq:syst0}. For example, consider $\\mA=\\Delta+c(x)$ a scalar operator. For some choices of $c(x)$ (for instance $c(x)=\\lambda$ an eigenvalue of $-\\Delta$ on $X$ with Dirichlet conditions), $\\mA$ is not invertible. However, \\eqref{eq:syst0} corresponds to $(\\Delta+c(x))v=0$ with both $v=0$ and $\\partial_\\nu v=0$ on $\\partial X$. It is then known that the unique solution to the above constraints is $v=0$ and hence $(\\mA^t\\mA,\\mD)$ is injective.\n\\begin{proof}\n We observe that the differential operator $(\\mA^t)_{ki}=(\\mA_{ik})^t=:\\mA^t_{ik}$ is of the same order $\\tau$ as $\\mA_{ik}$. Since $\\partial_\\nu^q v_k=0$ for $0\\leq q\\leq \\tau-1$, we obtain by integrations by parts that\n \\begin{displaymath}\n 0=  \\sum_{ijk}(\\mA^t_{ik}\\mA_{ij} v_j,v_k) = \\sum_i (\\sum_j \\mA_{ij} v_j, \\sum_k \\mA_{ik} v_k) =\\sum_i \\|(\\mA v)_i\\|^2.\n\\end{displaymath}\nThis implies the result.\n\\end{proof}\n\\subsection{Holmgren unique continuation and generic results}\n\\label{sec:ucph}\n\nThe above result shows that the injectivity of \\eqref{eq:normal} is a consequence of the injectivity of \\eqref{eq:syst0} with redundant boundary conditions. The operator $\\mA$ is therefore augmented with more boundary conditions than is necessary to render it elliptic. This is a possible price to pay to guarantee that the solution to the normal system \\eqref{eq:normal} is uniquely defined.\n\nIn general, however, even with redundant boundary conditions, it is not entirely straightforward to ensure that $v=0$ is the only solution to \\eqref{eq:syst0}. In this section, we consider the case where $\\mA$ is well-approximated by an operator with analytic coefficients.  We have the following result adapted from \\cite{H-I-SP-83}.\n\\begin{proposition}\\label{prop:WFanalytic}\n Let $\\mA_A(x,D)$ be a $2J\\times (J+M)$ system of differential equations with analytic coefficients in $X$ such  that $\\mA_A^t \\mA_A(x,D)$ is an elliptic operator for all $x\\in X$. Assume that $\\mA_A v=0$ in $X$. Then $v$ is analytic in $X$.\n\\end{proposition}\n\\begin{proof}\n  Since $\\mA_Av=0$, we also have that\n  \\begin{displaymath}\n   ^{co}(\\mA_A^t\\mA_A) (\\mA_A^t\\mA_A) v = {\\rm Diag}\\big({\\rm det} (\\mA_A^t \\mA_A),\\ldots,{\\rm det} (\\mA_A^t \\mA_A) \\big) v =0,\n\\end{displaymath}\nwhere $^{co}(\\mA_A^t\\mA_A)$ is the matrix of co-factors of $\\mA_A^t\\mA_A$. Since $\\mA_A^t \\mA_A$ is elliptic (a consequence of the fact that the leading term in $\\mA_A(x,\\xi)$ is of full rank $J+M$ for all $(x,\\xi)\\in\\bar X\\times\\Sm^{n-1}$), then $P={\\rm det} (\\mA_A^t \\mA_A)$ is an elliptic operator such that ${\\rm Char}P=\\emptyset$, where the characteristic set of $P$ is $\\{(x,\\xi)\\in \\bar X\\times \\Sm^{n-1},\\, P_m(x,\\xi)=0\\}$ and $P_m$ is the principal part of $P$. We deduce  from \\cite[Theorem 6.1]{H-I-SP-83} that $WF_A(v_j)\\subset {\\rm Char}P \\cup WF_A(Pv_j)=\\emptyset$ so that each $v_j$ is analytic in $X$.\n\\end{proof}\n\nFrom this, we deduce the following unique continuation result:\n\\begin{theorem}[Holmgren]\n\\label{thm:holmgren}\nLet $\\mA_A$ be as in Proposition \\ref{prop:WFanalytic} and let us assume that $\\mA_A v=0$ in $X$. Then we have:\n\n(i) Assume that $v=0$ in an open set $\\Omega\\subset\\subset X$. Then $v=0$ in $X$.\n\n(ii) Assume that $\\mD v=0$ on an open set $\\Sigma$ of $\\partial X$. Then $v=0$ in $X$.\n\\end{theorem}\n\\begin{proof}\n  Let us first prove (i). We know from Proposition \\ref{prop:WFanalytic} that the functions $v_j$ are analytic. Since they vanish on an open set, they have to vanish everywhere.\n  \n  Now (ii) is a standard consequence of (i), which we write in detail for systems. Let $x_0\\in \\Sigma$ and $V$ a sufficiently small open ball around $x_0$ where the coefficients of $\\mA_A$ are analytic and where $\\mD v=0$ on $\\Sigma\\cap V\\subset\\Sigma$. Since $\\mA_A$ is injective, we deduce that $(\\partial_n)^{\\tau_j} v_j=0$ on $\\Sigma\\cap V$ as well, as for any higher-order derivative in fact. Let us extend $v$ by $0$ on $V$. Then we verify that $\\mA_A v=0$ in $V$ (all that needed verification was that the equation was satisfied point-wise on $\\Sigma\\cap V$). But now (i) implies that $v=0$ on $V\\cup X$.\n\\end{proof}\n\nThe above result shows that $A_A=\\mA_A^t\\mA$ is injective as well. Since $R_AA_A=I-T_A$ for some left-parametrix $R_A$ and compact operator $T_A$, we deduce that $1$ is not in the spectrum of $T_A$ so that $A_A^{-1}=(I-T_A)^{-1}R_A$. As a consequence, any operator $A$ sufficiently close to $A_A$ is also invertible with a bounded inverse. We  summarize this result as:\n\\begin{corollary}\n\\label{cor:generic} Let $\\mA=\\mA_A+\\mA_1$ with $\\mA_1$ sufficiently small in operator norm from $\\prod_i W_p^{l}(X)$ to $\\prod_j W_p^{l+\\tau}(X)$. Then $\\mA$ augmented with $\\mD v=0$ is injective and $\\mA^t\\mA$ augmented with the same boundary conditions is invertible with bounded inverse. In other words, \\eqref{eq:stabnormal} holds with $C_2=0$.\n\\end{corollary}\nThe invertibility of $\\mA^t\\mA$ is therefore {\\em generic}, i.e., holds for an open and dense set of coefficients in the definition of the elliptic operator $\\mA^t\\mA$; see \\cite{SU-JFA-09}. Note, however, that the size of norm of $\\mA_1$ for which $\\mA$ is invertible depends on $\\mA_A$.\n\n\\medskip\n\nAnother similar result states that the invertibility of $\\mA^t\\mA$ is guaranteed when $X$ is a sufficiently small domain.\n\\begin{theorem}\n\\label{thm:small} Let us assume that $\\mA(0,D)$ is an elliptic operator. Then for $X$ sufficiently small, we have that $\\mA$ is injective and that $\\mA^t\\mA$ is invertible when augmented with Dirichlet conditions $\\mD v=0$. As a consequence, \\eqref{eq:stabnormal} holds with $C_2=0$.\n\\end{theorem}\n As in the previous corollary, the size of the domain $X$ depends on the bound for $(\\mA(0,D)^t\\mA(0,D))^{-1}$ with Dirichlet conditions on $\\partial X$, which is independent of $X$ as we shall see. It is therefore possible to estimate the size of the domain $X$ for which the above theory applies; see also \\cite{BM-LINUMOT-13} for a similar result with a different method of proof.\n\\begin{proof}\n We assume that all coefficients of $\\mA$ are sufficiently smooth and that $0\\in X$. Let $P=\\mA(0,D)$ be the operator of order $m=\\tau$ with coefficients frozen at $x_0=0$. Then $P^tP$ with Dirichlet conditions on $\\partial X$  is invertible as an application of Theorem \\ref{thm:holmgren}. For constant coefficients, a more precise theory applies. Let $u\\in H^m(X)$ (we restrict ourselves to Hilbert spaces to simplify notation) be such that $u=\\partial^j_\\nu u=0$ for $1\\leq j\\leq m-1$, i.e., $\\mD u=0$. We assume $\\partial X$ sufficiently smooth that we can extend $u$ by $0$ outside of $X$ and obtain a function in $H^m(\\Rm^n)$ with the same norm. Let $Y$ be an open set such that $\\bar X\\subset Y\\subset\\Rm^n$. \n \n The elliptic theory for constant coefficient operators \\cite{H-I-SP-83} provides the existence of a fundamental solution $E$ such that\n\\begin{displaymath}\n  E * P^*Pu = u .\n\\end{displaymath}\nThis implies that \n\\begin{displaymath}\n   \\|u\\|_{H^{m}(X)} = \\|u\\|_{H^{m}(Y)} = \\|E * P^*Pu\\|_{H^{m}(Y)} \\leq\\|E*P^*\\|_{{\\mathcal L}(L^2(X),H^m(Y))} \\|Pu\\|_{L^2(X)}.\n\\end{displaymath}\nThis proves the existence of a constant $C$ independent of $X$ such that \n\\begin{displaymath}\n   \\|Pu\\|_{L^2(X)} \\geq C  \\|u\\|_{H^{m}(X)}.\n\\end{displaymath}\nThis holds for any operator with constant coefficients. \n\nNow let $Q$ be an operator of order at most $m$ and of small norm $\\eps$ from $L^2(X)$ to $H^m(X)$. Then\n\\begin{displaymath}\n \\|(P+Q)u\\|_{L^2(X)} \\geq (C -\\eps) \\|u\\|_{H^m(X)}\\geq C_1 \\|u\\|_{H^m(X)},\n\\end{displaymath}\nwhich implies that $P+Q$ is injective for $\\eps$ sufficiently small.\n\nThis is applied as follows. Let $P=\\mA(x,D)$ be an elliptic operator with not-necessarily constant coefficients and define $P_0=\\mA(0,D)$ the operator with coefficients frozen at $x_0=0$. Then $P-P_0$ is an operator bounded from $H^m(X_\\eps)$ to $L^2(X_\\eps)$ with bound $C_1\\eps$ for $C_1$ independent of $\\eps$ and $X_\\eps\\subset B(x_0,\\eps)$. From the above, we deduce that\n\\begin{displaymath}\n  \\|P_0 u\\|_{L^2(X_\\eps)}\\geq C \\|u\\|_{H^m(X)}\n\\end{displaymath}\nfor all function $u\\in H^m(X)$ such that $\\mD u=0$. For $C_1\\eps<C$, we find that\n\\begin{displaymath}\n  \\|P u\\|_{L^2(X_\\eps)}\\geq C_2 \\|u\\|_{H^m(X)}\n\\end{displaymath}\nfor some $C_2>0$. This proves that $P=\\mA$ is injective on sufficiently small domains.\n\\end{proof}  \n  \n  \n\\subsection{Unique continuation principle}\n\\label{sec:ucpc}\nWhen the domain $X$ is large or when the operator $\\mA$ is not sufficiently close to an operator $\\mA_A$ with analytic coefficients, then the unique continuation results must rely on an other principle than analyticity. \n\nUnique continuation in the absence of analyticity in the coefficients is not always guaranteed, even for scalar elliptic operators, although the construction of counter-examples is not straightforward. For general references on unique continuation principles (UCP) for mostly scalar, not necessarily elliptic, operators, as well as counter-examples, see \\cite{H-III-SP-94,N-CPAM-57,N-AMS-73,Z-Birk-83} and their references.\n\nIn this section, we revisit Calder\\'on's \\cite{C-AJM-58,C-PSFD-62} result of uniqueness from Cauchy boundary data closely following the presentation in \\cite{N-AMS-73} and extend it to redundant systems. Our objective is to adapt \\cite[Theorem 5]{N-AMS-73} to specific settings of redundant systems. We first present local uniqueness results, whose proofs are postponed to the appendix, and then use classical arguments to extend them to global uniqueness results. \n\n\\subsubsection{Local uniqueness result}\n\\label{sec:lur}\n\nLet $x$ be a point in $\\Rm^{n+1}$ and $N$ be a unit vector equal to $(0,\\ldots,0,1)$ in an appropriate system of coordinates. We want to address the local uniqueness of the Cauchy problem. Assume that $L_q$ for $1\\leq q\\leq Q$ are differential operators of order $m$ and that\n\\begin{equation}\n\\label{eq:redundsyst}\n L_q u =0 \\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq q\\leq Q$ and $1\\leq j\\leq m-1$, \nwhere $V$ is a neighborhood of $0$. We assume that $N$ is non characteristic (at $x$) for all $L_q$, i.e., $p_q(x,N)\\not=0$ with $p_q$ the principal symbol of $L_q$. This and \\eqref{eq:redundsyst} imply that all derivatives of $u$ vanish on $\\{x_{n+1}=0\\}$ and that $u$ can be extended by $0$ on $V\\cap \\{x_{n+1}<0\\}$. The uniqueness problem may therefore be recast as: if $u$ satisfies\n\\begin{equation}\n\\label{eq:UCPsyst}\n L_q u =0 \\mbox{ in } V \\quad \\mbox{ for } 1\\leq q\\leq Q\\qquad \\mbox{ and } \\qquad u=0 \\mbox{ in } \\{x_{n+1}<0\\},\n\\end{equation}\nthen $u\\equiv0$ in a full neighborhood of $0$. \n\nWhen $Q=1$, it is known that $L=L_1$ needs to satisfy several restrictive assumptions in order for the result to hold; see \\cite{N-AMS-73}. The main advantage of the redundancy in the above system is that such assumptions need to be valid only locally (in the Fourier variable $\\xi$) for each operator, and globally collectively.\n\nChanging notation, we define $t=x_{n+1}$ and still call $x=(x_1,\\ldots,x_n)$. Then $p_q=p_q(x,t,\\xi,\\tau)$ is the principal symbol of $L_q$ for this choice of coordinates. Sufficient conditions for the uniqueness to the above Cauchy problem involve the properties of the roots $\\tau$ of the above polynomials $\\tau\\mapsto p_q(\\cdot,\\tau)$ as a function of $\\xi$. In the setting of redundant measurements, these conditions may hold for different values of $q$ for different values of $\\xi$. This justifies the following definition of our assumptions. \n\nWe assume the existence of a finite covering $\\{\\Omega_\\nu\\}$  of the unit sphere $\\Sm^{n-1}$ (corresponding to $|\\xi|=1$) such that the following holds. For each $\\nu$, there exists $q=q(\\nu)$ and $\\eps>0$ such that for each $(x,t)$ close to $0$ and each $\\xi\\in\\Omega_\\nu$, we have:\n\\begin{equation*}\n\\label{eq:condCalderon}\n\\begin{array}{rl}\n(i) & p_q(x,t,\\xi,\\tau) \\mbox{  has at most simple real roots $\\tau$ and at most double complex roots} \\\\\n(ii) & \\mbox{distinct roots $\\tau_1$ and $\\tau_2$ satisfy } |\\tau_1-\\tau_2|\\geq\\eps>0\\\\\n(iii) & \\mbox{non-real roots $\\tau$ satisfy } |\\Im \\tau|\\geq\\eps.\n\\end{array}\n\\end{equation*}\nThen we have the following result:\n\\begin{theorem}\n\\label{thm:calderonsystem}[Calder\\'on's result for redundant systems.]\nAssume that $N$ is non characteristic for the operators $L_q$ at the origin and that for a finite covering $\\{\\Omega_\\nu\\}$ of the unit sphere, (i)-(ii)-(iii) above are satisfied.   Then \\eqref{eq:UCPsyst} implies that $u=0$ in a full neighborhood of $0$. \n\\end{theorem}\nThe proof, which  closely follows that of Theorem 5 in \\cite{N-AMS-73}, is presented in the appendix.\n\nIn the above theorem, $u$ is scalar. The above proof extends to vector-valued functions $u$ when the system for $u$ is diagonally dominant. For a determined system given by a matrix $L_{ij}$ of operators of order $m\\geq1$, this means that the operators $L_{ii}$ are of order $m$ and satisfy the hypotheses of the above theorem and the operators $L_{ij}$ for $i\\not=j$ are of order at most $m-1$.  This generalizes to the setting of redundant systems $L_{ij}u_j=S_i$ where for each $i$, only one operator $L_{ij}$ is of order $m$ and for each $j$, the operators $L_{kj}$ of order $m$ collectively satisfy the hypotheses of the above theorem; see Theorem \\ref{thm:calderonsystemred:app} in the appendix, which we do not reproduce here.\n\n\nThe above theorem may not apply directly in applications. However, it serves as a component to obtain more general results. We consider one such result that finds applications in the framework of power density measurements. \n\nWe consider a setting where the leading term in the system may not be diagonal but rather upper-triangular. Unique continuation properties may still be valid provided that (a sufficiently large number of) the diagonal operators are elliptic. However, the corresponding complex roots may no longer be double. Instead, we need the stronger condition for some $\\eps>0$:\n\\begin{equation*}\n\\label{eq:condCalderon2}\n\\begin{array}{rl}\n(iv) & p_q(x,t,\\xi,\\tau) \\mbox{ has at most simple roots $\\tau$} \\mbox{ that satisfy } |\\Im \\tau|\\geq\\eps.\n\\end{array}\n\\end{equation*}\n\nThen we have the following result.\n\\begin{theorem}\n\\label{thm:2by2system}\nConsider the redundant system of equations\n\\begin{equation}\n\\label{eq:2by2syst}\n\\left(\\begin{matrix} L_1 & L_0 \\\\ L_3 & L_2 \\end{matrix} \\right) \n\\left(\\begin{matrix} u_1 \\\\ u_2\\end{matrix} \\right) = 0\\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u_k=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq j\\leq m-1$ and $1\\leq k\\leq 2$, with the following assumptions. The operators $L_1$ and $L_0$ are (vector-valued) $Q_1\\times1$ operators of order $m$,  where $L_1$ satisfies the hypotheses of Theorem \\ref{thm:calderonsystem} with $Q=Q_1$. The operators $L_2$ and $L_3$ are $Q_2\\times1$ operators of order $m$ and at most $m-1$, respectively. Moreover, $L_2$ satisfies the ellipticity hypothesis (i)-(ii)-(iv) with $Q=Q_2$.\nThen $(u_1,u_2)=0$ in a full neighborhood of $0$.\n\nThe same result extends to systems of the form\n\\begin{equation}\n\\label{eq:RbyRsyst}\n\\left(L_{ij} \\right)_{1\\leq i,j\\leq R} u = 0\\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u_k=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq k\\leq R$, where $L_{ij}$ is (a vector-valued operator) of order $m-1$ for $i>j$, $L_{ii}$  is (a vector-valued operator) of order $m$ that satisfies the hypotheses of Theorem \\ref{thm:calderonsystem} with an appropriate value of $Q$ when all $L_{ik}$ are of order $m-1$ for $k\\not=i$ and the ellipticity hypothesis (i)-(ii)-(iv)  with an appropriate value of $Q$ when at least one operator $L_{ik}$ for $k<i$ is of order $m$.\n\\end{theorem}\nThe proof of this theorem is also given in the appendix. Note that {\\em (i)-(ii)-(iv)}, as opposed to {\\em (i)-(ii)-(iii)}, may be seen as an ellipticity condition since the symbol $p_{q(\\nu)}$ does not vanish on $\\Omega_\\nu$. It is known that Carleman estimates are sharper in such a setting; see, e.g., \\cite{LR-JPCS-11,N-AMS-73}.\n\n\\begin{remark}\n\\label{rem:LC}\\rm Let $\\mQ$ be an invertible $Q_1\\times Q_1$ matrix and define the linear operators $\\tilde L_1=\\mQ L_1$ and $\\tilde L_0=\\mQ L_0$. Then the conclusions of Theorem \\ref{thm:2by2system} are the same as those obtained when $L_1$ and $L_0$ are replaced by $\\tilde L_1$ and $\\tilde L_0$. In other words, the results of Theorem \\ref{thm:calderonsystem} hold if $L_q$ is replaced by an equivalent linear combination. The condition that $N$ be non characteristic for $L_q$ may then be replaced by the condition that it be non characteristic for the  linear combinations $\\tilde L_q=\\mQ_{q,q'} L_{q'}$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:extensions} \\rm\n If $L_0$ is an operator of order $m-1$ in the latter theorem so that the above system is diagonally dominant, then it is sufficient to assume that  $L_1$ and $L_2$ satisfy the less constraining hypotheses of Theorem \\ref{thm:calderonsystem}; see Theorem \\ref{thm:calderonsystemred:app} and the discussion below Theorem \\ref{thm:calderonsystem}.\n\\end{remark}\n\n\\subsubsection{Global uniqueness result}\n\nThe above local results extend to global unique continuation results such as \\cite[Sections 8.5-8.6]{H-I-SP-83}:\n\\begin{theorem}\n\\label{thm:globalUCP}\n Let $\\mA$ be a (redundant) system of operators of order $m$, let $R_{\\mA}\\subset\\{(x,N)\\in \\bar X\\times \\Sm^{n-1}\\}$ be the set of points where the hypotheses of either Theorem \\ref{thm:calderonsystem} or Theorem \\ref{thm:2by2system}, (and possibly their extensions in Remark \\ref{rem:extensions}) are satisfied and let us define $\\Sigma_{\\mA}= \\bar X\\times\\Sm^{n-1} \\backslash R_{\\mA}$. Let\n \\begin{displaymath}\n \\mA u =0 \\quad \\mbox{ in } X,\\qquad \\partial^j_\\nu u =0 \\quad \\mbox{ on } \\partial X, \\,\\, 0\\leq j\\leq m-1.\n\\end{displaymath}\nLet $\\bar N({\\rm supp}(u))$ be the subset of  $(\\bar x,\\xi)\\in X\\times\\Sm^{n-1}$ composed of the closure of the normal set of the support of $\\{u_j\\}$; see \\cite{H-I-SP-83}. Heuristically, that set may be seen as the points $(x,\\xi)$ where $x$ belongs to the boundary of ${\\rm supp (u)}$ and $\\xi$ is a normal (inward or outward) unit vector to that boundary.\n\nThen $\\bar N({\\rm supp}(u))\\subset \\Sigma_{\\mA}$. When the latter set is empty, then $u\\equiv0$ in $X$.\n\\end{theorem}\nThe proof follows from the local results noting that the boundary to the support of $u$ cannot occur at a point and for a direction where a local UCP principle holds;  see \\cite[Sections 8.5-8.6]{H-I-SP-83} for the definitions and derivations. \n\n\n\\begin{remark}\\rm\n Let $\\mA$ be a $p\\times q$ system with $p\\geq q$ and let us assume that $\\mA u=0$ in the above theorem is replaced by \n\\begin{equation}\n\\label{eq:globalucpconst}\n|(\\mA u)_k|(x) \\leq C \\dsum_{|\\alpha|\\leq m-1} \\dsum_{l=1}^q |D^\\alpha u_l|(x),\n\\end{equation}\nfor a constant $C$ independent of $x\\in \\bar X$ and $1\\leq k\\leq p$. Let, as above,  $\\Sigma_\\mA$ be the set of points where the hypotheses of Theorem \\ref{thm:calderonsystem} do not hold (with obvious modifications for  Theorems \\ref{thm:2by2system}).  Then the conclusions of the above theorem, namely that $\\bar N({\\rm supp}(u))\\subset \\Sigma_{\\mA}$, still hold; see, e.g., \\cite{H-III-SP-94} and the proofs in the appendix.\n\\end{remark}\n\n\n\\subsection{Application to power-density measurements}\n\\label{sec:pduniq}\n\nWe now apply the unique continuation results of sections \\ref{sec:ucph} and \\ref{sec:ucpc} to the setting of power density measurements described in section \\ref{sec:pdm}.\n\n\\subsubsection{Second-order linear systems of equations}\n\nBoth unique continuation results require that the system $\\mA$ be elliptic in the regular sense, i.e., with $\\tau=t_j$ independent of $j$ and $s_i=0$ for all $i$. The systems \\eqref{eq:pdlinsyst} and \\eqref{eq:deltauj}-\\eqref{eq:constraintjk} are not in this form. We propose two modifications of the latter systems for which uniqueness results can be proven.\n\n\\subsubsection{System after elimination.}\nLet us first consider the system \\eqref{eq:deltauj}-\\eqref{eq:constraintjk}. The latter constraints are first-order. However, they easily become second-order by differentiation. We thus consider the system\n\\begin{equation}\n\\label{eq:elim2nd}\n\\begin{array}{rcl}\n  \\nabla\\cdot\\gamma\\big(-\\nabla \\delta u_j + 2 \\hat F_j \\hat F_j\\cdot\\nabla \\delta u_j\\big) &=& \\nabla\\cdot \\dfrac{\\delta H_j}{|F_j|^2} F_j.\\\\[3mm]\n\\nabla \\big( 2\\gamma\\dfrac{1}{|F_j|^2} F_j\\cdot\\nabla \\delta u_j - 2\\gamma\\dfrac{1}{|F_k|^2} F_k\\cdot\\nabla \\delta u_k\\big)  &=&   \\nabla \\big( \\dfrac{\\delta H_j}{|F_j|^2} - \\dfrac{\\delta H_k}{|F_k|^2}\\big) \\quad 1\\leq j<k\\leq J,\n\\end{array}\n\\end{equation}\nin $X$ with boundary conditions $\\delta u_j=\\partial_\\nu \\delta u_j=0$ on $\\dX$. The above system is clearly elliptic under the same hypotheses as  \\eqref{eq:deltauj}-\\eqref{eq:constraintjk}. Moreover, it satisfies the above ellipticity condition with $\\tau=2$.\n\n\\subsubsection{System in triangular form.}\nLet us come back to \\eqref{eq:pdlinsyst}. We apply the operator $K_j=2\\gamma |F_j|^{-2}F_j\\cdot\\nabla$ to the first line and $L_\\gamma |F_j|^{-2}$ to the second line with $L_\\gamma:=\\nabla\\cdot \\gamma\\nabla$ to obtain after subtraction\n\\begin{equation}\n\\label{eq:pdlinsyst2}\n\\begin{array}{rcl}\n\\nabla\\cdot \\delta\\gamma F_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j &=& 0 \\\\\nL_\\gamma \\delta\\gamma - K_j \\nabla\\cdot \\delta\\gamma F_j +[L_\\gamma K_j,K_j L_\\gamma] \\delta u_j &=& L_\\gamma \\dfrac{\\delta H_j}{|F_j|^2}.\n\\end{array}\n\\end{equation}\nSince $L_\\gamma$ is second order and $K_j$ is first-order, the commutator $[L_\\gamma K_j,K_j L_\\gamma]$ is a second-order differential operator. We thus again obtain a system that is elliptic when \\eqref{eq:pdlinsyst} is (we leave this proof to the reader; see also \\eqref{eq:triangsyst} below) and that is in the appropriate form with $\\tau=2$.\n\n\n\\subsubsection{Holmgren unique continuation result}\n\nThe results of section \\ref{sec:ucph} apply to the power density measurement systems \\eqref{eq:elim2nd} and \\eqref{eq:pdlinsyst2}. Let $\\mA$ denote the linear operator in one of the aforementioned systems. Then $\\mA^t\\mA$ augmented with Dirichlet conditions is injective provided that $\\gamma$ and $u_j$ are sufficiently close to analytic coefficients or provided that the problem is posed on a sufficiently small domain, and provided of course that the quadratic forms $\\{q_j\\}$ are collectively elliptic on $\\bar X$ in the sense of Definition \\ref{def:ellcondpj}.\n\nThe main difference between \\eqref{eq:elim2nd}  and \\eqref{eq:pdlinsyst2} pertains to the boundary conditions. In the case of \\eqref{eq:pdlinsyst2}, vanishing Dirichlet and Neumann boundary conditions are imposed on $\\delta\\gamma$ and $\\delta u_j$. This means that the value of $(\\gamma,\\{u_j\\})$ and of its (normal) derivatives is known on $\\partial X$. In the case of \\eqref{eq:elim2nd}, only the values of $(\\{u_j\\})$ and of its (normal) derivatives need to be known. \n\nIn fact, the boundary conditions for $\\delta\\gamma$ at $x\\in\\partial X$ may be deduced from those for $\\delta u_1$, say, provided that $\\nu(x)$ is non characteristic for $P_1$. Indeed, we observe that \n\\begin{displaymath}\n  \\delta H_1 = \\delta\\gamma |\\nabla u_1|^2 + 2 \\gamma \\nabla u_j\\cdot\\nabla \\delta u_1.\n\\end{displaymath}\nThis provides boundary conditions for $\\delta\\gamma$ when $\\delta u_1$ and $\\partial_\\nu  \\delta  u_1$ are known on $\\partial X$. The derivation of $\\partial_\\nu \\delta\\gamma$ is obtained as follows. Using the equation\n\\begin{displaymath}\n   \\nabla\\cdot \\delta\\gamma \\nabla u_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j =0,\n\\end{displaymath}\nand replacing $\\delta\\gamma$ above using the expression of $\\delta H_1$, we obtain after straightforward eliminations that\n\\begin{displaymath}\n   (1-2 (\\widehat{\\nabla u_1} \\cdot \\nu(x))^2) \\partial^2_\\nu u_1 =f_1,\n\\end{displaymath}\nwhere $f_1$ is a known expression involving $\\partial_\\nu u_1$ and tangential derivatives of $u_1$ of degree up to $2$ at $x$, which are known since $u_1$ is known on $\\partial X$. Since $\\nu$ is non characteristic for $P_1$, this provides an expression for $\\partial^2_\\nu u_1$, and hence for $\\partial_\\nu\\delta\\gamma$ using the above expression for $\\delta H_1$.\n\nSo knowledge of Cauchy data for $\\delta\\gamma$ is in fact a consequence of knowledge of Cauchy data for the solutions $\\delta u_j$. Thus, the major requirement in solving $\\mA^t\\mA v=\\mA^t\\mS$ is that we impose that $\\partial_\\nu\\delta u_j=g_j$ for a known function $g_j$, which implies that $\\partial_\\nu u_j$ is known on $\\dX$.\n\n\n\n\\subsubsection{Calder\\'on unique continuation result}\nThe symbol of the operator \\eqref{eq:pdlinsyst} is elliptic in the Douglis-Nirenberg sense. Recall that  the results of unique continuation principle (UCP) of the preceding sections apply to operators that are elliptic in a classical sense. We thus consider the  modification of the system \\eqref{eq:pdlinsyst2}. We were not able to obtain a satisfactory global UCP using Carleman-type estimates for the system \\eqref{eq:elim2nd}.\n   \nUCP for such  the modified system \\eqref{eq:pdlinsyst2} is based on proving a collective UCP for quadratic forms, which we now address in detail.\n\\begin{lemma}\n\\label{lem:ucphyper}\nLet $P_j(x,D)$ be second-order operators with principal symbols given by $q_j(x,\\xi)=2(\\hat F_j\\cdot\\xi)^2-|\\xi|^2$, where $\\hat F_j(x)\\in\\Sm^{n-1}$ for $1\\leq j\\leq J$. \nLet $(x,N)\\in\\bar X\\times\\Sm^{n-1}$. For $\\xi'\\in\\Rm^n$ such that $\\xi'\\cdot N=0$, we define\n\\begin{displaymath}\n  q_{j;N}(x,\\xi') = 2(\\hat F_j\\cdot\\xi')^2 - \\big(1-2(\\hat F_j\\cdot N)^2\\big) |\\xi'|^2.\n\\end{displaymath}\nLet us assume that the quadratic forms are independent in the following UCP sense:\n\\begin{equation}\\label{eq:UCPN}\n  q_{j;N}(x,\\xi') =0 \\quad \\mbox{ for all } 1\\leq j\\leq J\\mbox{ s.t. } q_j(x,N)\\not=0\\qquad \\mbox{ implies that } \\qquad \\xi'=0.\n\\end{equation}\n\nThen $\\{P_j\\}$ collectively satisfies a UCP for $(x,N)$ in the sense that we can find a partition $\\{\\Omega_\\nu\\}$ such that  (i)-(ii)-(iii) holds.\n\\end{lemma}\n\n\\begin{proof}\nThe proof in dimension $n=2$ is trivial: $P_j$ satisfies UCP except for four directions on the null cone, i.e., for $\\xi$ such that $q_j(x,\\xi)=0$. Since \\eqref{eq:UCPN} implies that $N$ is not in the intersection of the null cones, then UCP holds for one of the operators $P_j$ and we can choose $\\Omega_1=\\Sm^{1}$ in the partition.\n\n Let $(x,N)\\in\\bar X\\times\\Sm^{n-1}$ and $n\\geq3$. Let $\\xi'\\in\\Sm^{n-1}$ such that $\\xi'\\cdot N=0$. Then\n \\begin{displaymath}\n  q_j(x,\\xi'+\\tau N) = 2(\\hat F_j\\cdot (\\xi'+\\tau N))^2- |\\xi'+\\tau N|^2 = \\big(2(\\hat F_j\\cdot N)^2-1)\\tau^2 + 4 \\hat F_j\\cdot N\\hat F_j\\cdot \\xi' \\tau + 2 (\\hat F_j\\cdot \\xi')^2-1.\n\\end{displaymath}\nThis quadratic polynomial (because $q_j(x,N)=2(\\hat F_j\\cdot N)^2-1\\not=0$) with real-valued coefficients has a double root when the determinant\n\\begin{equation}\\label{eq:Deltaprime}\n \\Delta'_j = -1+2 \\big((\\hat F_j\\cdot N)^2 + (\\hat F_j\\cdot \\xi')^2\\big) = q_{j;N}(x,\\xi')=0.\n\\end{equation}\nFor $\\xi'$ away from such points, then the UCP condition holds for $P_j$ since the roots cannot be simple and complex roots (which come in conjugate pairs) cannot be close to each-other. By assumption, $\\Delta'_j=0$ for all $j$ is not possible and by continuity, $\\max_{j}|\\Delta'_j|$ is bounded away from $0$ so that different roots $\\tau$ are separated by some $\\eps>0$ (as well as the imaginary part of such roots since they come in conjugate pairs). This generates a finite partition $\\{\\Omega_\\nu\\}$. On each patch, one $P_j$ satisfies the UCP properties {\\em (i)-(ii)-(iii)}.\n\\end{proof}\n\n\nIn dimension $n=2$, ellipticity of $(P_1,P_2)$ is equivalent to UCP of $(P_1,P_2)$ at each $(x,N)$. In dimension $n\\geq3$, collective ellipticity and collective UCP are different notions. As soon as there is one $j$ such that $2(\\hat F_j\\cdot N)^2\\geq1$, the UCP is satisfied at $(x,\\xi)$ while ellipticity does not necessarily holds.\nHowever, we have the result:\n\\begin{lemma}\n\\label{lem:ElltoUCP}\n In dimension $n=2$, collective UCP and collective ellipticity of operators of the form $P_j$ above is equivalent. In dimension $n=3$, collective ellipticity of $\\{P_j\\}$ such that $q_j(N)\\not=0$ implies collective UCP at $(x,N)$.\n\\end{lemma}\n\\begin{proof}\n The case $n=2$ is handled as in the proof of Lemma \\ref{lem:ucphyper}. Consider now the case $n=3$. Non-UCP at $(x,N)$ implies from \\eqref{eq:Deltaprime} and the fact that $N$ is non characteristic for $P_j$ that\n \\begin{displaymath}\n   (\\hat F_j\\cdot N)^2 + (\\hat F_j\\cdot \\xi')^2 = \\frac12.\n\\end{displaymath}\nIf we decompose $\\hat F_j=\\hat F_j\\cdot N N + \\hat F_j\\cdot \\xi' \\xi' + \\hat F_j\"$ with $\\hat F_j\"$ orthogonal to ${\\rm span}(N,\\xi')$, then we find that \n\\begin{displaymath}\n  |\\hat F_j\"|^2 = \\frac 12.\n\\end{displaymath} \nIn dimension $n=3$, we find that $\\hat F_j\"=\\pm|\\hat F_j| N\\times \\xi'$. This implies that $N\\times\\xi'$ belongs to the null cone of $q_j$, i.e., $q_j(x,N\\times\\xi')=0$. This holds for every $j$, which implies that $N\\times \\xi'=0$, a contradiction.\n\\end{proof}\n\n\\begin{definition}\\label{def:UCP}\nWe say that the family of operators $P_j(x,D)$ or of quadratic forms $q_j(x,\\xi)$ collectively satisfy a global UCP on $\\bar X$ if  for every $(x,N)\\in \\bar X\\times \\Sm^{n-1}$, we can find a family $\\tilde q_j(x,\\xi)=\\mQ_{jk}q_k(x,\\xi)$ with $\\mQ$ an invertible matrix such that $\\{\\tilde q_j(x,\\xi)\\}$ collectively satisfies a UCP at $(x,N)$ in the sense of Lemma \\ref{lem:ucphyper}.\n\\end{definition}\n\nThe above definition involves a linear change of quadratic forms for each $(x,N)$ that allows us to obtain a family of modified forms $\\tilde q_j$ with similar properties to those of $q_j$ and such that $\\tilde q_j(N)\\not=0$; see Remark \\ref{rem:LC}. This will prove useful in the analysis of reconstruction from  power density functionals. We will also need the following Lemma in dimension $n=3$.\n\\begin{lemma}\n\\label{lem:3dUCP}\nLet $F_1$ and $F_2$ be three-dimensional vector fields on $\\bar X$ such that for each $x\\in\\bar X$, ${\\rm rank}(F_1,F_2)=2$. Let $F_3=F_1+F_2$. We denote by $\\{P_j\\}$ and $\\{q_j\\}$ the corresponding operators and quadratic forms defined in \\eqref{eq:qjPj} for $1\\leq j\\leq 3$. Then $\\{q_j\\}$ satisfies a global UCP on $\\bar X$.\n\\end{lemma}\n\\begin{proof}\n  Let us fix $(x,N)\\in \\bar X\\times \\Sm^2$ and choose a basis of $\\Rm^3$ such that  $\\hat F_1(x)=e_1$ and ${\\rm span}(F_1,F_2)=(e_1,e_2)$. The three quadratic forms $q_j$ allow us to obtain by linear combination any $q_c$ corresponding to $F_c=c e_1+se_2$ with $s=\\sqrt{1-c^2}$. It is then not difficult to find three values of $c$ such that the corresponding $\\tilde q_j$ for $1\\leq j\\leq 3$ form an elliptic family and such that $\\tilde q_j(N)\\not=0$. From Lemma \\ref{lem:ElltoUCP}, we obtain that $\\{\\tilde q_j\\}$ satisfies a UCP at $(x,N)$. The hypotheses of global UCP in Definition \\ref{def:UCP} are met.\n\\end{proof}\n\n\n\\subsubsection{Elliptic system in triangular form.} We now present a modification of the linear system for $(\\delta\\gamma,\\{\\delta u_j\\})$ for which a global UCP result as described in the above definition can be obtained.\n\nWe recast \\eqref{eq:pdlinsyst2} as\n\\begin{equation}\n\\label{eq:triangsyst}\n\\left(\\begin{matrix} P_j & \\tilde P_j \\\\ 0 & \\Delta \\end{matrix} \\right) \\left(\\begin{matrix} \\delta\\gamma \\\\ \\delta u_j \\end{matrix} \\right) = l.o.t.\\left(\\begin{matrix} \\delta\\gamma \\\\ \\delta u_j \\end{matrix} \\right)  + S_j,\n\\end{equation}\nwhere $l.o.t.$ means a system of differential operators of order at most $1$. Let $P$ and $\\tilde P$ denote the $J\\times 1$ columns of the second-order, homogeneous, operators $P_j$ and $\\tilde P_j$, respectively. The symbol of $P_j$ is proportional to $q_j(x,\\xi)$.  The operator $\\Delta$ satisfies hypothesis {\\em (iv)} of Theorem \\ref{thm:2by2system} while the operator $\\tilde P$ is of order $2$. If the operators $P$ collectively satisfy the hypotheses of Theorem \\ref{thm:calderonsystem}, then the above system \\eqref{eq:pdlinsyst2} satisfies a UCP, as does the fourth-order operator $\\mA^t\\mA$ as constructed in the preceding section:\n\n\\begin{theorem}\n\\label{thm:uniqpowdens}\nLet $\\mA v= \\mS$ in $X$ be the system for $v=(\\delta\\gamma,\\{\\delta u_j\\})$ described in \\eqref{eq:triangsyst} and augmented with boundary conditions $v=\\partial_\\nu v=0$ on $\\partial X$. Assume that the coefficients in $\\mA$ are sufficiently smooth (see proof of Theorem \\ref{thm:calderonsystem:app}). Let the operators $P_j$ above satisfy a UCP on $\\bar X$ as described in Definition \\ref{def:UCP} and Lemma \\ref{lem:ucphyper}. Then any solution to that system, or to the system $\\mA^t\\mA v=\\mA^t \\mS$ with the same boundary conditions, is unique.\n\\end{theorem}\n\\begin{proof}\n  The proof is a direct application of Theorem \\ref{thm:2by2system} since $\\Delta$ satisfies hypothesis {\\em (iv)}, $\\tilde P$ is of order $2$, and $P$ collectively satisfies a UCP at each $(x,N)$ in $\\bar X\\times \\Sm^{n-1}$. \n\\end{proof}\n\n\nIn dimensions $n=2$ and $n=3$, we have the following sufficient conditions on the internal functionals to guaranty injectivity of $\\mA^t\\mA$:\n\\begin{corollary}\\label{cor:UCP}\n\\label{cor:3d} Assume that $(u_1,u_2)$ are solutions such that $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are such that ${\\rm rank}(F_1,F_2)=2$ for each $x\\in\\bar X$.  \n\nIn dimension $n=2$, assume moreover that  $F_1\\cdot F_2\\not=0$ for each $x\\in\\bar X$. Then $\\{P_1,P_2\\}$ satisfies a global UCP property on $\\bar X$ and the results of Theorem \\ref{thm:uniqpowdens} hold.\n\nIn dimension $n=3$, define $F_3=F_1+F_2$. Then $\\{P_1,P_2,P_3\\}$ satisfies a global UCP property on $\\bar X$ and the results of Theorem \\ref{thm:uniqpowdens} hold.\n\\end{corollary}\n\nThe proof is immediate using Lemma \\ref{lem:3dUCP}, remark \\ref{rem:LC} and the preceding Theorem.\n\nWe have thus exhibited a system of equations  $\\mA^t\\mA v=\\mA^t \\mS$ with boundary conditions $v$ and $\\partial_\\nu v$ prescribed on $\\partial X$, which admits a unique solution that verifies the stability estimate \\eqref{eq:pdstab} (generalized as in \\eqref{eq:stabnormal} for non-homogeneous boundary conditions) with $C_2=0$.\n\nAs in the preceding section devoted to the Holmgren uniqueness result, this result comes at the cost of having to impose boundary conditions of the form $v$ and $\\partial_\\nu v$ to both $v=\\delta u_j$ and $v=\\delta\\gamma$. As we saw in the preceding section, the boundary conditions for $\\delta\\gamma$ at $x\\in\\partial X$ may be deduced from those for $\\delta u_j$.\n\nComparing ellipticity and uniqueness criteria for $\\mA$ in \\eqref{eq:triangsyst}, we observe that ellipticity is obtained by imposing conditions $\\delta u_j=g_j$ on $\\dX$ and ensuring that $\\{q_j\\}$ is an elliptic family of quadratic forms. UCP is obtained by imposing the additional boundary conditions $\\partial_\\nu\\delta u_j=\\psi_j$ and by ensuring that $\\{q_j\\}$ satisfy a global UCP on $\\bar X$. In dimension $n=2$, global ellipticity and global UCP for $\\{q_j\\}$ are equivalent. In dimension $n=3$, both are satisfied for the family of three internal functionals described in Corollary \\ref{cor:UCP}.\n\n\n\\section{Local uniqueness for nonlinear inverse problem}\n\\label{sec:nonlinear}\n\nOnce the injectivity of a linear system can be established, standard theories may be applied to obtain local uniqueness results for the nonlinear inverse problems. Let us recast the nonlinear system of PDE \\eqref{eq:syst} as\n\\begin{equation}\n\\label{eq:nonlinearnotmod}\n\\begin{array}{rcl}\n \\tilde\\mF (v) &=& \\tilde \\mH \\quad \\mbox{ in } X \\\\\n \\tilde \\mB v &=& \\tilde g \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nHere $v=(\\gamma,\\{u_j\\}_{1\\leq j\\leq J})$ and $\\tilde \\mB$ is an operator that maps $v$ to the trace of $\\{u_j\\}$ on $\\partial X$. We assume here, as is the case for hybrid inverse problems, that the boundary condition operator $\\tilde \\mB$ is linear.\n\nThe linearization of the above system involves the operator $\\tilde\\mF'(v_0)$ for a fixed $v_0$. The analysis of the preceding sections did not allow us to show that the differential operator $\\tilde \\mF'(v_0)$ augmented with the boundary conditions $\\tilde \\mB$ was invertible. Rather, we obtained that for an appropriate linear differential operator $\\mT(v_0)$, then $\\mA:=\\mT(v_0)\\tilde \\mF'(v_0)$ augmented with the augmented boundary conditions $\\mB$ was invertible. More precisely, we obtained that\n\\begin{equation}\n\\label{eq:linearmod}\n\\begin{array}{rcl}\n  \\mA w := \\mT(v_0) \\tilde \\mF'(v_0) w &=& \\mS  \\quad \\mbox{ in } X \\\\\n \\mB w &=&  h\\quad \\mbox{ on } \\partial X,\n\\end{array}\n\\end{equation}\nadmitted a unique solution with a stability estimate given by\n\\begin{equation}\n\\label{eq:stablinear}\n  \\|w\\|_\\mX \\leq C \\|(\\mS,h) \\|_\\mY.\n\\end{equation}\n\nThe above linearized operator is the linearization of the nonlinear operator\n\\begin{equation}\n\\label{eq:nonlinearmF}\n\\mF(v_0+w):=\\mT(v_0)\\tilde \\mF(v_0+w).\n\\end{equation}\n We thus modify the original inverse problem and replace it with an inverse problem that is necessarily satisfied by the exact solution $v=v_0+w$ we wish to reconstruct and is given by\n\\begin{equation}\n\\label{eq:nonlinearmod2}\n\\begin{array}{rcl}\n \\mF (v_0+w) &=& \\mH:=\\mT(v_0) \\tilde\\mH \\quad \\mbox{ in } X \\\\\n \\mB (v_0+w) &=& g \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nBorrowing the notation of section \\ref{sec:param}, we observe that the nonlinear operator $(\\mF(\\cdot),\\mB \\cdot)$ maps $\\mX=\\mU(p,l)$ to $\\mY=\\mR(p,l)$ for appropriately defined spaces $\\mU$ and $\\mR$ for $(\\mA,\\mB)=(\\mF'(v_0),\\mB)$.\n\nLet us pause on the definition of the boundary condition $g$ for $v_0+w$. We cannot expect $\\mB v_0 =g$ with $g$ known so that $\\mB w =0$ on $\\partial X$. The reason is that $v_0$ is typically constructed by guessing $\\gamma_0$ and solving the linear elliptic problems for $\\{u_{j,0}\\}$ with imposed Dirichlet conditions. It is for such a construction of $v_0$ that we were able to show that $A=(\\mA,\\mB)$ above was injective for $\\mA=\\mF'(v_0)$. The boundary condition $g_0:=\\mB v_0$ thus partially depends on solving the above problem and is not in general given by the measurements $g$ (unless $v_0$ is the solution $v$). We thus recast the (modified) nonlinear hybrid inverse problem as\n\\begin{equation}\n\\label{eq:nonlinearmod}\n\\begin{array}{rcl}\n \\mF (v_0+w) &=& \\mH:=\\mT(v_0) \\tilde\\mH \\quad \\mbox{ in } X \\\\\n \\mB w &=& g-g_0 \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nThe objective of the following section is to provide an iterative algorithm to reconstruct $w$ provided that $v_0$ is sufficiently close to the solution we wish to obtain in the sense that $g-g_0$ and $\\mH-\\mH_0$ are sufficiently small with $\\mH_0:=\\mF(v_0)$. In section \\ref{sec:injectivity}, we obtain an injectivity result stating that if $\\mF(v_0+w)=\\mF(v_0+\\tilde w)$ and $\\mB w=\\mB \\tilde w$, then $w=\\tilde w$.\n\n\\begin{remark}\n\\label{rem:powerdensity}\n\nIn the power density setting, we recast \\eqref{eq:ellj} and \\eqref{eq:pdj} as\n\\begin{displaymath}\n   \\mF_{2j-1}(v_0+w) = 0,\\qquad \\tilde \\mF_{2j}(v_0+w)=H_j,\\qquad 1\\leq j\\leq J,\n\\end{displaymath}\nrespectively. Let then $L_\\gamma$, $F_j$, and $M_j$ constructed from $v_0=(\\gamma,\\{u_j\\})$ as in the derivation of \\eqref{eq:pdlinsyst2}. Finally, let us define\n\\begin{displaymath}\n  \\mF_{2j}(v_0+w) = L_\\gamma \\tilde \\mF_{2j-1}(v_0+w) - M_j \\mF_{2j-1}(v_0+w) = |F_j|^{-2} H_j =: K_j,\\quad 1\\leq j\\leq J.\n\\end{displaymath}\nThe system $\\mF_{2j-1}(v_0+w)=0$ and $\\mF_{2j}(v_0+w)=K_j$ for $1\\leq j\\leq J$ is recast as \n\\begin{displaymath}\n  \\mF(v_0+w) = \\mH,\n\\end{displaymath}\nin the notation of \\eqref{eq:nonlinearmod} and implicitly defines the linear operator $\\mT(v_0)$. We denote by $\\mA=\\mF'(v_0)$ the linearization of $\\mF$ at $v_0$, which agrees with the differential operator defined in \\eqref{eq:pdlinsyst2}.\n\nWhereas the operator $\\tilde \\mB$ maps $v_0$ to the traces of $\\{u_j\\}$ on $\\partial X$, the extended operator $\\mB$ maps $v_0$ to the traces of $v_0$ and $\\nu\\cdot \\nabla v_0$ on $\\partial X$.\n\n\\end{remark}\n\n\\subsection{Iterative fixed point and reconstruction procedure}\n\\label{sec:fixedpoint}\n\nLet us define\n\\begin{equation}\n\\label{eq:taylormF}\n\\mF(v_0+w) = \\mF(v_0) + \\mF'(v_0) w + \\mG(w;v_0),\n\\end{equation}\nwhere $\\mG(w;v_0)$ is quadratic in the first variable in the sense that \n\\begin{equation}\n\\label{eq:quadG}\n  \\|(\\mG(w;v_0),0)\\|_\\mY \\leq C \\|w\\|_\\mX^2.\n\\end{equation}\nThe latter estimate comes from the fact that $\\mF(v)$ is polynomial in $v$ and the partial derivatives $D$ and that $\\mX$ is an algebra. We may thus recast the nonlinear system of equations for $w$ as\n\\begin{equation}\n\\label{eq:nonlinearw}\n\\begin{array}{rcl}\n \\mA w &=& \\mH - \\mH_0 - \\mG(w;v_0) \\quad \\mbox{ in } X \\\\\n \\mB w &=& g-g_0 \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nSince the linear operator $A=(\\mA,\\mB)$ is invertible, we may recast the above equation as\n\\begin{equation}\n\\label{eq:integralw}\n w = \\mI(w) :=A^{-1}(\\mH - \\mH_0, g-g_0) - A^{-1}(\\mG(w;v_0),0) .\n\\end{equation}\nFrom the polynomial structure of $\\mF$and the boundedness of $A^{-1}$ from $\\mY$ to $\\mX$, we deduce in addition to \\eqref{eq:quadG} that\n\\begin{equation}\n\\label{eq:constants}\n\\begin{array}{rcl}\n  \\|A^{-1}(\\mG(w;v_0),0)\\|_{\\mX} &\\leq & C_1 \\|w\\|_\\mX^2\\\\\n  \\|\\mI(w)-\\mI(\\tilde w)\\|_\\mX &\\leq& C_2 (\\|w\\|_{\\mX} + \\|\\tilde w\\|_{\\mX}) \\|w-\\tilde w\\|_{\\mX}.\n\\end{array}\n\\end{equation}\nAs a consequence, for $\\|w\\|_\\mX\\leq R$ and $\\|A^{-1}(\\mH-\\mH_0,g-g_0)\\|_\\mX\\leq \\eta$ sufficiently small so that $\\eta+C_1R^2\\leq R$ and $2C_2 R<1$, we find that $\\mI$ is a contraction from the ball $B=\\{w,\\, \\|w\\|_{\\mX}\\leq R\\}$ onto itself.\n\nThis shows that for $(\\mH-\\mH_0,g-g_0)$ sufficiently small, then the solution to \\eqref{eq:nonlinearw} is unique and is obtained by the converging algorithm $w^{k+1}=\\mI(w^k)$ initialized with $w^0=0$.\n\n\\subsection{Injectivity results}\n\\label{sec:injectivity}\n\nThe fixed point algorithm of the preceding section provides us with an explicit local reconstruction procedure for the nonlinear hybrid inverse problem. A similar methodology allows us to obtain local uniqueness results that are more general but not constructive. Using the same notation as in the preceding section, let us assume that \n\\begin{equation}\n\\label{eq:uniqF}\n\\mF(v_0+w) = \\mF(v_0+\\tilde w) = \\mH\\quad\\mbox{ in } X,\\qquad \\mB w=\\mB \\tilde w\\mbox{ on }\\partial X,\n\\end{equation}\nfor a fixed $v_0\\in\\mX$. In other words, the measurements associated with $v_0+w$ and $v_0+\\tilde w$ are identical. Injectivity of the nonlinear problem $\\mF$ locally in the vicinity of $v_0$ then means proving that $w=\\tilde w$.\n\nSince $\\mF$ is a polynomial in $w$, then there is a polynomial function in $w$ and $\\tilde w$ such that \n\\begin{equation}\n\\label{eq:diffF}\n\\mF(v_0+\\tilde w) - \\mF(v_0+w) = \\mJ(w,\\tilde w) \\cdot (\\tilde w -w).\n\\end{equation}\n\nWhen $\\tilde w-w$ is sufficiently small, then $\\mJ(w,\\tilde w)$ satisfies the same properties as $\\mF'(v_0)$, the Fr\\'echet derivative of $\\mF$. For instance, the ellipticity and unique continuation properties of $\\mF'(v_0)$ still hold for $\\mJ(w,\\tilde w)$ for $w$ and $\\tilde w$ small in $\\mX$. As a consequence, we obtain that \\eqref{eq:uniqF} implies that $w=\\tilde w$ and hence that $\\mF$ is injective. More generally, $\\mJ(w,\\tilde w)$ may still admit a left-inverse for $w$ and $\\tilde w$ not necessarily close to each-other, in which case we also deduce that $w=\\tilde w$ for $w$ and $\\tilde w$ not necessarily small.\n\n\nLet us assume more generally that we have two measurements\n\\begin{displaymath}\n   \\mF(v_0+w) = \\mH,\\quad \\mB w=g-g_0 ;\\qquad \\mF(v_0+\\tilde w) = \\tilde\\mH,\\quad \\mB\\tilde w=\\tilde g-g_0.\n\\end{displaymath}\nAssume that $\\mF'(v_0)$ is elliptic and injective. Then, $\\mJ(w,\\tilde w)$ is elliptic and hence admits a left-inverse (since it is injective), at least for $w$ and $\\tilde w$ sufficiently close to $0$. As a consequence,  we obtain as above the stability estimate\n\\begin{equation}\n\\label{eq:stabnonlinear}\n  \\|w-\\tilde w\\|_{\\mX} \\leq C \\|(\\mH-\\tilde\\mH,g-\\tilde g)\\|_{\\mY}.\n\\end{equation}\n\n\nNote that the nonlinear problem \\eqref{eq:uniqF} is invertible generically. Indeed, for $v_0$ analytic, $\\mA^t\\mA$ augmented with Dirichlet boundary conditions $\\mD$ is invertible. As a consequence, the above result shows that \\eqref{eq:uniqF} may be inverted for $(v_0+w)$ in an open set including $v_0$. Since analytic coefficients $v_0$  in $X_0$ restricted to $X$ are dense in the set of sufficiently smooth coefficients on $X$, we obtain that the inverse problem may be inverted {\\em generically}; see \\cite{SU-JFA-09}.\nWhen $v_0$ is not analytic, then we need to find a unique continuation principle based on Carleman estimates to obtain an estimate of the form \\eqref{eq:stabnonlinear} for the fully nonlinear hybrid inverse problem.\n\n\\cout{\nOLD OLD \n\\subsection{Iterative fixed point and reconstruction procedure}\n\\label{sec:fixedpoint}\n\nLet us consider a setting where $(\\mA,\\mB)$ is injective or $(\\mA^t\\mA,\\mD)$ is invertible so that the linearized problem \\eqref{eq:linear} is solved by either $v=(I-T)^{-1}RA(\\mS,0)$ or by $v=(\\mA^t\\mA)^{-1}(\\mA^t\\mS,0)$. We now present standard results concerning the nonlinear hybrid inverse problem \\eqref{eq:syst}.\n\nLet $v_0=(\\gamma,\\{u_j\\})\\in\\mX=\\mU(p,l)$ the point about which \\eqref{eq:syst} was linearized to give \\eqref{eq:linsyst}. We recast both equations in \\eqref{eq:syst} as \n\\begin{equation}\n\\label{eq:inX}\n\\mF(v_0+v) = \\mH,\n\\end{equation}\nwhere $\\mH\\in\\mY$ collects the available data $\\{H_j\\}$ and $\\mY$ is defined as the appropriate subset of $\\mR(p,l)$ defined in section \\ref{sec:param}. We also define $\\mH_0:=\\mF(v_0)$. \n\nIn the procedure of construction of a left-parametrix to $A$, we impose in addition to \\eqref{eq:inX} the boundary condition $\\mB(v_0+v)=\\mB(v_0)$ so that $\\mB(v)=0$. Note that in the setting of power-density measurements, we do not have injectivity of such an operator $A$. Let us assume nonetheless that $1$ is not an eigenvalue of the operator $T$ so that $(I-T)^{-1}R$ is a bounded left-inverse for $A$. \n\nIn the inversion of $(\\mA^t\\mA,\\mD)$, we impose in addition to \\eqref{eq:inX} the boundary condition $\\mD(v_0+v)=\\mD(v_0)$ so that $\\mD(v)=0$. We have obtained in the preceding section several unique continuation results that apply to the differential operator $\\mA$ given in \\eqref{eq:elim2nd} or \\eqref{eq:pdlinsyst2}.\n\nWe recast \\eqref{eq:inX} as \n\\begin{displaymath}\n  \\mF'(v_0) v = \\mH -\\mF(v_0) - \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big).\n\\end{displaymath}\nWe are in a setting where $\\mF'(v_0)$ augmented with appropriate vanishing boundary conditions admits a left-parametrix, which we denote by $(\\mF')^{-1}(v_0)$, and which takes the expression $(I-T)^{-1}R$ or $(\\mA^t\\mA)^{-1}\\mA^t$ with boundary conditions $\\mB v=0$ and $\\mD v=0$, respectively.\n\nWe then have that\n\\begin{equation}\n\\label{eq:contraction}\nv = \\mG(v) := (\\mF')^{-1}(v_0) ( \\mH-\\mH_0) - (\\mF')^{-1}(v_0)   \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big).\n\\end{equation}\n\n\nSince $\\mF(w)$ is polynomial in $w$ with smooth coefficients and we have the stability estimate \\eqref{eq:stabell} with $C_2=0$,  we obtain that \n\\begin{displaymath}\\begin{array}{rcl}\n   \\|\\mG(v)-\\mG(w)\\|_{\\mX} &\\leq& C_1 (\\|v\\|_{\\mX}+\\|w\\|_{\\mX}) \\|v-w\\|_{\\mX} \\\\\n     \\| (\\mF')^{-1}(v_0) \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big) \\|_{\\mX} &\\leq & C_3 \\|v\\|_{\\mX}^2,\n     \\end{array}\n\\end{displaymath}\nfor $\\|v\\|_{\\mX}$ sufficiently small since $\\mY$ forms an algebra. \n\nFor $\\|v\\|_{\\mX}<R$ sufficiently small and $\\| (\\mF')^{-1}(v_0) ( \\mH-\\mH_0)\\|_{\\mX}<\\eta$ sufficiently small so that $\\eta+C_3R^2\\leq R$ and $2C_1R<1$, we find that $\\mG$ is a contraction from $B=\\{v,\\, \\|v\\|_{\\mX}\\leq R\\}$ to itself. This proves that for $\\mH$ sufficiently close to $\\mH_0$, then the solution to \\eqref{eq:inX} is unique and may be obtained by the converging algorithm $v^{k+1}=\\mG(v^k)$.\n\nProvided that $(\\mathcal F'(v))^{-1}$ is shown to exist and to be uniformly bounded as an operator from $\\mY$ to $\\mX$ for $v$ in the vicinity of $v_0$, then a Newton scheme with faster convergence properties may be introduced:\n\\begin{equation}\n\\label{eq:Newton}\n v^{k+1}-v^k = (\\mathcal F'(v_0+v^k))^{-1} \\big(\\mH-\\mF(v_0+v^k)\\big).\n\\end{equation}\nThe analysis of the scheme and its very fast (quadratic) convergence rate are then classical.\n\n\\subsection{Application to power density measurements.}\n\nUnfortunately, the framework we just described does not directly apply to the setting of power density measurements. The reason, as described in section \\ref{sec:pduniq}, is that the operator $\\mA$ introduced in \\eqref{eq:pdlinsyst} is elliptic in the DN sense but $\\mA^t\\mA$ is not elliptic. The uniqueness results presented in the preceding section thus do not apply directly to \\eqref{eq:pdlinsyst}. We concentrate here on \\eqref{eq:pdlinsyst2} for which we were able to prove a unique continuation principle and for which the Holmgren theory of injectivity applies.\n\nWe first realize that \\eqref{eq:pdlinsyst2} is no longer the linearization of the \\eqref{eq:ellj}-\\eqref{eq:pdj}. We need to construct a nonlinear problem for $v$ whose linearization at $v_0=(\\gamma,\\{u_j\\})$ would be given by \\eqref{eq:pdlinsyst2}. This is done as follows. We recast \\eqref{eq:ellj} and \\eqref{eq:pdj} as\n\\begin{displaymath}\n   \\mF_{2j-1}(v_0+v) = 0,\\qquad \\tilde \\mF_{2j}(v_0+v)=H_j,\\qquad 1\\leq j\\leq J,\n\\end{displaymath}\nrespectively. Let then $L_\\gamma$, $F_j$, and $M_j$ constructed from $v_0=(\\gamma,\\{u_j\\})$ as in the definition of \\eqref{eq:pdlinsyst2}. Finally, let us define\n\\begin{displaymath}\n  \\mF_{2j}(v_0+v) = L_\\gamma \\tilde \\mF_{2j-1}(v_0+v) - M_j \\mF_{2j-1}(v_0+v) = |F_j|^{-2} H_j =: K_j,\\quad 1\\leq j\\leq J.\n\\end{displaymath}\nThe system $\\mF_{2j-1}(v_0+v)=0$ and $\\mF_{2j}(v_0+v)=K_j$ for $1\\leq j\\leq J$ is recast as \n\\begin{displaymath}\n  \\mF(v_0+v) = \\mH,\n\\end{displaymath}\nto fit the notation in \\eqref{eq:inX}. We then denote by $\\mA=\\mF'(v_0)$ the linearization of $\\mF$ at $v_0$, which then agrees with the differential operator defined in \\eqref{eq:pdlinsyst2}.\n\n\nIt is for this problem that the preceding theory (almost) applies. The main practical difficulty resides in the imposition of redundant boundary conditions in the equation \n\\begin{displaymath}\n  \\mF(v_0) = \\mH_0, \\quad \\mbox{ with } v_0 \\mbox{ and } \\partial_\\nu v_0 \\mbox{ prescribed on } \\partial X\n\\end{displaymath}\nIt is indeed unlikely that an initial guess $v_0$, obtained for instance from an initial guess for $\\gamma$ and then solving elliptic equations for $u_j$, may satisfy predetermined Neumann conditions. To remedy this problem, we have to leave open the possibility that $\\partial_\\nu v_0$ may not be the measured $\\partial_\\nu (v+v_0)$; in other words $\\partial_\\nu v=g$ may not vanish. \n\n\\medskip\n\n{\\bf OLD OLD}\n\n\\medskip\n\nNote that $v_0$ and $v_0+v$ and their first derivatives need to agree on $\\partial X$ to uniquely solve \\eqref{eq:pdlinsyst2}. Since $\\mF$ is a second-order differential operator, it may be difficult to find an initial guess $v_0$ satisfying the above hypotheses. We therefore replace the above equation for $v_0$ by\n\\begin{displaymath}\n  \\mA^t\\mF(v_0) = \\mA^t \\mH_0\\quad \\mbox{ with } v_0 \\mbox{ and } \\partial_\\nu v_0 \\mbox{ prescribed on } \\partial X.\n\\end{displaymath}\nThis is a fourth-order problem with appropriate Dirichlet conditions. \n\n\nConsider the scheme introduced in \\eqref{eq:pdlinsyst2} or \\eqref{eq:elim2nd}, which we recast as $\\mA v=\\mS$. Then $\\mF'(v_0)$ is an operator from $\\mX=W^l_p(X)\\times W^{l+1}_p(X;\\Rm^J)$ to $\\mY=W^l_p(X;\\Rm^{2J})$, which to $\\mS$ associates $v$ solution of\n\\begin{equation}\n\\label{eq:vquad}\n \\mA^t\\mA v = \\mA^t \\mS\\quad\\mbox{ in }\\, X,\\qquad v=\\partial_\\nu v =0 \\quad\\mbox{ on } \\, \\partial X.\n\\end{equation}\nWe have presented sufficient conditions in the preceding section for the above problem to be uniquely solvable and hence for $(\\mF'(v_0))^{-1}$ to be bounded from $\\mY$ to $\\mX$. \n\n\nWe then find that $\\mF'(v)$ is invertible from $\\mY$ to $\\mX$ for $\\|v-v_0\\|_{\\mX}$ sufficiently small so that the nonlinear iteration schemes in \\eqref{eq:contraction} or \\eqref{eq:Newton} converge. \nThis is because in the setting of power density measurements, elliptic regularity for $v_0\\in \\mX$ shows that $\\delta v\\in\\mX$ as well provided that $\\delta H\\in \\mY$. More precisely, for $\\gamma\\in W^l_{p}(X)$, then $u_j\\in W^{l+1}_p(X;\\Rm^J)$ by elliptic regularity of \\eqref{eq:ellj} \\cite{S-JSM-73} and then $(\\delta\\gamma,\\{\\delta u_j\\})\\in\\mX$ by elliptic regularity of either \\eqref{eq:pdlinsyst2} or \\eqref{eq:elim2nd} since power density measurements for $\\gamma\\in W^l_p(X)$ and $u\\in W^{l+1}_p(X)$ of the form $\\gamma|\\nabla u|^2$ are indeed in $W^l_p(X)$. \n\n\nWhen one of the unique continuation results applies (which may require that $l$ above be quite large so that coefficients are sufficiently smooth; see the proof of Theorem \\ref{thm:calderonsystem:app} in the appendix), then the nonlinear hybrid inverse problem may be solved locally using \\eqref{eq:contraction} or \\eqref{eq:Newton}. Indeed, for \\eqref{eq:Newton}, $v^{k+1}-v^k$ belongs to $\\mX$ as soon as $(\\mF'(v_0))^{-1}$ is bounded from $\\mY$ to $\\mX$ and $\\mH-\\mH_0$ is sufficiently small in $\\mY$. This provides a local uniqueness result and an explicit reconstruction procedure for the power density measurement hybrid inverse problem.\n\nWe refer the reader to \\cite{BNSS-JIIP-13} for the implementation of a similar scheme that displays very fast convergence rates but for which we are unable to prove a unique continuation principle.\n\\subsection{Injectivity results}\n\\label{sec:injectivity}\n\nThe fixed point algorithm of the preceding section provides us with an explicit local reconstruction procedure for the nonlinear hybrid inverse problem. A similar methodology allows us to obtain local uniqueness results that are more general but not constructive. Using the same notation as in the preceding section, let us assume that \n\\begin{equation}\n\\label{eq:uniqF}\n\\mF(v_0+v) = \\mF(v_0) = \\mH,\n\\end{equation}\nin other words the measurements associated with $v_0$ and $v_0+v$ are identical. Injectivity of the nonlinear problem $\\mF$ locally in the vicinity of $v_0$ then means proving that $v=0$.\n\nSince $\\mF$ is a polynomial in $v$ (or more generally a differentiable function in $v$), then there is a polynomial function in $v$ and $\\tilde v$ such that \n\\begin{equation}\n\\label{eq:diffF}\n\\mF(v_0+\\tilde v) - \\mF(v_0+v) = \\mG(v,\\tilde v) \\cdot (\\tilde v -v).\n\\end{equation}\n\nWhen $\\tilde v-v$ is sufficiently small, then $\\mG(v,\\tilde v)$ satisfies the same properties as $\\mF'(v)$, the Fr\\'echet derivative of $\\mF$. For instance, the ellipticity and unique continuation properties of $\\mF'(v)$ still hold for $\\mG(v,\\tilde v)$ for $v-\\tilde v$ small in the appropriate topology. As a consequence, we obtain that \\eqref{eq:uniqF} implies that $v=0$ and hence that $\\mF$ is injective. However, $\\mG(v,\\tilde v)$ may still admit a left-inverse for $v$ and $\\tilde v$ not necessarily close to each-other, in which case we also deduce that $v=\\tilde v$ for $v-\\tilde v$ not necessarily small.\n\n\nLet us assume more generally that we have two measurements\n\\begin{displaymath}\n   \\mF(v_0+v) = \\mH,\\qquad \\mF(v_0+\\tilde v) = \\tilde\\mH.\n\\end{displaymath}\nAssume that we have been able to prove that $\\mF'(v_0)$ is elliptic and injective. Then, $\\mG(v,\\tilde v)$ is elliptic and hence admits a left-inverse (since it is injective), at least for $v$ and $\\tilde v$ sufficiently close to $v_0$. As a consequence, if we assume that the boundary conditions that need to be imposed for $v$ and $v_0$ agree on $\\partial X$, then we obtain the stability estimate\n\\begin{equation}\n\\label{eq:stabnonlinear}\n  \\|v-\\tilde v\\|_{\\mX} \\leq C \\|\\mH-\\tilde\\mH\\|_{\\mY}.\n\\end{equation}\nErrors in the boundary conditions $\\mB(v)$ or $\\mD(v)$ may be accounted for in a similar manner. Using the notation in section \\ref{sec:param}, this is\n\\begin{equation}\n\\label{eq:stabnonlin2}\n\\dsum_{j=1}^{J+M} \\|v_j-\\tilde v_j\\|_{W_p^{l+t_j}(X)} \\leq C  \\dsum_{i=1}^{2J} \\|\\mH_i-\\mH_{0,i}\\|_{W_p^{l-s_i}(X)}.\n\\end{equation}\n\n\nNote that the nonlinear problem \\eqref{eq:uniqF} is invertible generically. Indeed, for $v_0$ analytic, the inversion of $\\mA^t\\mA$ augmented with Dirichlet boundary conditions $\\mD$ is guaranteed. As a consequence, the above result shows that \\eqref{eq:uniqF} may be inverted for $(v+v_0)$ in an open set including $v_0$. Since analytic coefficients $v_0$  in $X_0$ restricted to $X$ are dense in the set of sufficiently smooth coefficients on $X$, we obtain that the inverse problem may be inverted {\\em generically}; see \\cite{SU-JFA-09}.\n\nWhen $v_0$ is not analytic, then we need to find a unique continuation principle based on Carleman estimates; see the paragraph on applications to power density measurements in section \\ref{sec:fixedpoint}, which extends to the setting of injectivity results and not-necessarily local stability estimates of the form \\eqref{eq:stabnonlinear}.\n\n}\n\n\\section*{Acknowledgment} \nI would like to thank the organizers of the Irvine conference for their invitation to a meeting that was a befitting tribute to the exceptional contributions to analysis and inverse problems of Gunther Uhlmann. I am glad to acknowledge many insightful discussions with Shari Moskow on the content of section 4, which borrows some ideas of our joint work in \\cite{BM-LINUMOT-13}. I am also indebted to  Thomas Widlak for noticing and proposing solutions to inconsistencies in the presentation of the Lopatinskii conditions in section 2.6.2.\nThis paper was partially funded by NSF grant DMS-1108608 and AFOSR Grant NSSEFF- FA9550-10-1-0194.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nWe know many things about the structure of our Galaxy, and we have an approximate idea\nof the functional shape of its components: thin and thick disc, bulge, long bar, halo\nor spheroid, spiral arms, ring. In the past decades, different large-area surveys in visible\nor near-infrared wavelengths have allowed us to know our Galaxy much better.\nHowever there are some parts of the Galaxy that are not well known; one of them is the\nouter part of the disc, with Galactocentric distances beyond $R=15$ kpc.\nWe know this component is warped and flared, but the\ndetails of its shape are still a topic to develop further.\n\nWe are interested in analyse this outer part of stellar disc in our Galaxy in this paper,\nusing available visible data of the Sloan Digital Sky Survey (SDSS) in low Galactic latitude regions.\nSeveral authors (Bilir et al. 2006, 2008; Juri\\'c et al. 2008; Jia et al. 2014) \nhave previously used the SDSS to measure the parameters of the disc, but they have explored the high Galactic latitudes, so they could not access the outer disc. Other authors have analyzed the disc \nusing the Two Micron All Sky Survey (2MASS) either at low latitudes \n(L\\'opez-Corredoira et al. 2002, 2004) or at high latitudes using red-clump giants (Cabrera-Lavers et al. 2005, 2007; Chang et al. 2011) or within the whole sky using a given luminosity function (Polido et al. 2013), but, even using near-plane regions, the depth of 2MASS is not enough to analyze the\nouter disc; it is dominated by stars with smaller distances than the outer disc we\nwould like to analyze.\n\nOur purpose here is to use a survey like the SDSS (\\S \\ref{.data}), deeper than 2MASS, \nin near-plane regions, although avoiding the plane itself because of the high extinction. From this we separate a type of dwarf stars as\nstandard candles (\\S \\ref{.method}).\nWe are particularly interested in modelling the flare of the outer disc. There\nis some evidence of its existence in the different components of the Galaxy, but\nthe stellar flare at $R\\gtrsim 15$ kpc is still poorly known.\nThe flare of HI was investigated for instance by Nakanishi \\& Sofue (2003), Levine et al. (2006), or Kalberla et al. (2007). It was modelled by Kalberla et al. (2007) in terms of a dark matter ring, by Saha et al. (2009) in terms of a lopsided halo,\nor by L\\'opez-Corredoira \\& Betancort-Rijo (2009) in terms of accretion of intergalactic matter onto the disc.\nAnother example of modelling is given by Narayan \\& Jog (2002), who\nused a three-component (HI, H$_2$ and stars) disc gravitationally coupled to calculate the scale height of each component and their flares, and they derived a good prediction for the gas flare;\nthe result of mild stellar flaring up to $R=12$ kpc was obtained in the model by Narayan \\& Jog (2002), and it agreed well with the  observational data on flaring provided by Kent et al. (1991). \nThe stellar flare was also observed by\nAlard (2000), L\\'opez-Corredoira et al. (2002), Yusifov (2004), Momany et al. (2006), or Reyl\\'e et al. (2009) for the outer disc, but limited to $R\\lesssim 20$ kpc and with large uncertainties over 15 kpc or for the differences between the flare of the thin and thick disc.\n\nA flare in the thick disc is \nposited (Hammersley \\& L\\'opez-Corredoira 2011; Mateu et al. 2011; L\\'opez-Corredoira et al. 2012), \nand this may constitute an explanation for the Monoceros ring instead of the tidal stream\nhypothesis. This is an additional reason to pursue this research: to know whether we are able\nto fit our star counts without needing some new extragalactic component. The fact that\na flared thin+thick disc alone fits our data (\\S \\ref{.fit}, \\S \\ref{.explor}) gives a positive answer,\nwhich is discussed and compared with other works of the literature in \\S \\ref{.compar}, \\S \\ref{.discu}, together with the considerations derived from the morphological information obtained in this paper.\n\n\\section{Data} \n\\label{.data}\n\nThe SDSS (Sloan Digital Sky Survey)-DR8 release (Aihara et al. 2011) contains imaging of \n14\\,555 deg$^2$ of the sky in five filters (u,g,r,i,z), with a completeness higher than 95\\% \nfor $m_g\\le 22.2$. Within this survey, the subsample SEGUE (Sloan Extension for Galactic Understanding \nand Exploration) includes many Galactic plane regions.\nFor this paper, in which we are interested to explore the outer disc, we used the regions with \n$|\\ell |> 50^\\circ $ (to avoid the regions of the inner Galaxy), \n$|b|\\le 23^\\circ $ (dominated by disc stars), taking all the point-like SDSS sources with\nthe flags and constraints for the $g$ and $r$ filters: ((flags\\_r\\,\\& \\,0x10000000)\\, != 0),\n((flags\\_r\\, \\& \\,0x8100000c00a4)\\, = 0), (((flags\\_r\\, \\& \\,0x400000000000)\\, = 0)\\, or\\, \n(psfmagerr\\_r <= 0.2)),\n(((flags\\_r\\, \\& \\,0x100000000000)\\, = 0)\\, or\\, (flags\\_r\\, \\& \\,0x1000)\\, = 0), plus the same thing for \nflags\\_g;\nthis avoids the stars that are too close to the borders, are too small to determine\nthe radial profile, have saturated pixels or too many interpolated pixels\nto derive a correct flux, stars for which the deblend algorithm finds two or more\ncandidates in cases with a magnitude error in $r$ larger than 0.2, stars with poor detection, or cases with cosmic rays.\nAll the magnitudes were corrected for extinction by the SDSS team using the \nGalactic extinction model of Schlegel et al. (1998). \nIn total, we cover an area of 1\\,745 deg$^2$.\nAs we explain in the next section, we reduced this area by avoiding the regions very close to the plane with high extinction to reduce the errors due to the correction of extinction.\n\n\\section{Method}\n\\label{.method}\n\nThe simplest method of determining the stellar density along a \nline of sight in the disc is by isolating a group of stars with the same colour and absolute magnitude \n$M$ within a colour magnitude diagram. This allows the luminosity function to\nbe replaced by a constant in the stellar statistics equation and the differential star counts for each line of sight, $A(m)$, can be immediately converted into density $\\rho (r)$:\n\n\\begin{equation}\n\\label{diffsc}\nA(m)\\equiv \\frac{dN(m)}{dm}=\\frac{ln\\ 10}{5}\\omega \\rho\n[r(m)]r(m)^3  \n,\\end{equation}\\[\nr(m)=10^{[m-M+5]\/5}\n,\\]\nwhere $\\omega $ is the area of the solid angle in radians and $r$ is the\ndistance in parsecs.\n\nIn the near-infrared, red-clump giants have been successfully used as  standard candles, particularly for the innermost 15 kpc from the Galactic centre\n(L\\'opez-Corredoira et al. 2002). However, red-clump stars in the outer disc at the distance of interest would appear at $m_k\\ge 14$, where the local dwarfs with the same colour would completely dominate the counts \n(L\\'opez-Corredoira et al. 2002). Therefore, we have to use something different here.\n\nFor our extinction-corrected areas, it is possible to use star counts in visible. An examination of the HR diagram shows that when the extinction is low, the late F and early G dwarfs can be isolated using colour with only minimal contamination from other sources with the same colour, but different\nabsolute magnitudes. For this work we selected the \nsources between F8V and G5V. \nThere will be sufficient stars detected in the outer Galaxy to give meaningful statistics, \nwhich would not be the case if a smaller range in absolute magnitudes were used. Earlier sources were\nnot included as these sources would belong to a younger population with a \nfar lower scale height; the absolute magnitude also changes\nfar more rapidly with colour. Later sources would have significant giant contamination, \nand again the absolute magnitude changes more rapidly with colour. \n\n   \\begin{figure}\n\\vspace{1cm}\n   \\centering\n   \\includegraphics[width=8cm]{CMdiagram.eps}\n   \\caption{Extinction-corrected colour-magnitude diagram for the region $\\ell=183^\\circ $, $b=21^\\circ$. The region between the dashed lines contains F8V-G5V dwarfs.}\n   \\label{Fig:CMdiagram}\n   \\end{figure}\n\nThe F8V-G5V dwarfs have a range of $g-r$ of 0.36 to 0.49 \nand a range of absolute magnitudes $M_g$=4.2 to 5.3 (Bilir et al. 2009),\nwhich makes the sources approximately $m_g\\lesssim 21.5$ \nat the distance of interest up to 22 kpc. See the selection example in Fig. \\ref{Fig:CMdiagram}. We adopted a constant absolute magnitude for all of them\n$M_g$=4.8. For a range of absolute magnitudes, when the counts are converted into \ndensity vs. distance, there is some smoothing which is not included in a model that assumes a single absolute magnitude. This smoothing has little effect,  \nand the above approximation remains valid: see L\\'opez-Corredoira et al. 2002, \\S 3.3.1\nfor a calculation of the difference between a narrow Gaussian distribution and a Dirac delta: it leads to an error in the scale length of the order of 2\\% assuming an r.m.s. in the Gaussian distribution of 0.3 mag.; although the application of L\\'opez-Corredoira et al. is for red-clump giants, \nit is valid for any kind of population.\nWe did not take into account the possibility that some of these stars might indeed be binary systems.\nSee Siegel et al. (2002) or Bilir et al. (2009, \\S 3.4) for a calculation of \nthe effects of a high ratio of\nbinary stars to derive the parameters of the disc: they may produce \n$\\Delta M_g\\sim 0.2$ mag.\nThere are some radial and vertical gradients of metallicity for\nthin and thick disc (Rong et al. 2001; Ak et al. 2007; Andreuzzi et al. 2011); \nhere, like in Juri\\'c et al. (2008),\nwe did not take into account the variation of the absolute magnitude due to the variation of metallicity: \nsee Siegel et al. (2002) or Juri\\'c et al. (2008, \\S 2.2.1) for a discussion on it. \nA rough estimate of the maximum difference for the most extreme cases of \nlower metallicity for the highest values of $R$ and $z$ ($[Fe\/H]\\sim 1$ dex\nlower than in the solar neighbourhood) is $\\Delta M_R\\approx 0.4$ [Siegel et al. 2002, using $R-I=0.38$, which is the corresponding transformation from SDSS to Johnson filters (Jordi et al. 2006) of the\ncolor of our population with average $(r-i)=0.13$ (Bilir et al. 2009)]. The variation of $M_g$ is\nsimilar (Juri\\'c et al. 2008, Fig. 3). Therefore, \nthe fact that we did not take this metallicity gradient into account \nmay produce an overestimate of\nthe distance of the stars at the highest $R$ or $z$ of $\\sim 20$\\% . Its effects are explored in \\S \n\\ref{.explor}.\n \nIn the 1\\,745 deg$^2$ of our SDSS data are 6\\,506\\,360 stars with $g-r$ (corrected for extinction) between 0.36 and 0.49. The source densities in these regions are expected to be very low, therefore we require regions of more than 0.5 square degrees of sky to be covered to provide sufficient counts to give reasonable statistics. \nWe divided them into multiple space bins with $\\Delta \\ell \\times \\Delta b=2^\\circ \\times 2^\\circ $, $\\Delta m_g=0.2$ for $15.25<m_g<21.85$, and,\nsince we know their distances and coordinates (hence, we know their position in 3D space), given the differential star counts $A(m)$, we can derive through Eq. (\\ref{diffsc}) the density $\\rho (R,\\phi ,z)$ \n(in cylindrical Galactocentric coordinates). This average stellar density is plotted in Fig.\n\\ref{Fig:dens}, excluding the bins with $|b|<8^\\circ $.\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8.5cm]{bins_phi.eps}\n\\caption{Average stellar density of F8V-G5V stars from available SDSS data as a function of the \ncylindrical coordinates $R$ (kpc), $\\phi $ (deg.), $z$ (kpc)\nfor the range $0<R<30$ kpc, $|z|<15$ kpc, constrained within Galactic latitudes\n$8^\\circ \\le |b|\\le 22^\\circ $. The circle stands for $R=8$ kpc and $z=0$ (like the\nSun position) for any value of $\\phi $ (for the Sun, it would be $\\phi =0 $).\nThe axes in the bottom left panel apply to all panels in the figure.}\n\\label{Fig:dens}\n\\end{figure}\n\nWe excluded these in-plane regions (bins with $|b|<8^\\circ $), which reduced \nour area to 1\\,396 deg$^2$, to reduce the errors\nin the extinction correction: with extinctions of $\\langle A(g)\\rangle \\lesssim 1.5$ mag \nin filter g at most for $|b|\\ge 8^\\circ $ (Schlegel et al. 1998), $E(g-r)=0.27A(g)$ \nand a relative uncertainty of the reddening of \n$\\sim 10$\\% (Schlegel et al. 1998; M\\\"ortsell 2013), we derive relative\nerrors of $\\Delta (g-r) \\lesssim 0.04$ for each region. With a $\\frac{dM_g}{d(g-r)}\\approx\n\\frac{5.4-4.2}{0.49-0.36}=9.2$, we compute uncertainties in the distance determination\nof $\\lesssim 18$\\%. Although this error is still large, when combining all the bins,\nthe errors compensate for each other and are mostly cancelled on except for some systematic\nones that may lead to some variation of the scales. We discuss this point in \\S \\ref{.discu}. At present, we do not carry out a deconvolve the line of sight distribution with the spread of distances because of the r.m.s. of the reddening, which would provide a better determination of the morphology, because one would need a very accurate expression of the r.m.s. of the extinction, which is not available (see also discussion in \\S \\ref{.discu}). Because the regions are all well off the plane we assumed that \nthis extinction is local.\n\n\n\\section{Fitting free parameters of a disc model}\n\\label{.fit}\n\n\\subsection{Disc model}\n\nAfter deriving the stellar density $\\rho (R,\\phi ,z)$, we can\nfit a disc model to obtain its best free parameters. We\nused the following model, which contains flared axysymmetric thin+thick discs:\n             \n\\begin{equation}\n\\label{rho}\n\\rho _{\\rm disc}(R,z)=\\rho _{\\rm thin}(R,z)+\\rho _{\\rm thick}(R,z)\n,\\end{equation}\n\\[\n\\rho_{\\rm thin} (R,z)= \\rho_\\odot \\frac{h_{\\rm z, thin}(R_\\odot)}{h_{\\rm z, thin}(R)}\\exp\\left( \\frac{R_\\odot}{h_{\\rm r, thin}} + \\frac{h_{\\rm r, hole}}{R_\\odot} \\right)\n\\]\\[\\ \\ \\ \\ \\times \n\\exp\\left( -\\frac{R}{h_{\\rm r, thin}}-\\frac{h_{\\rm r, hole}}{R} \\right) \\exp\\left( -\\frac{|z|}{h_{\\rm z, thin}(R)} \\right)\n,\\]\\[\n\\rho_{\\rm thick} (R,z)= f_{\\rm thick}\\,\\rho_\\odot \\frac{h_{\\rm z, thick}}{h_{\\rm z, thick}(R)}\\exp \\left( \\frac{R_\\odot}{h_{\\rm r, thick}} + \\frac{h_{\\rm r, hole}}{R_\\odot} \\right)\n\\]\\[\\ \\ \\ \\ \\times \\exp \\left( -\\frac{R}{h_{\\rm r, thick}}-\\frac{h_{\\rm r, hole}}{R} \\right) \\exp\n\\left( -\\frac{|z|}{h_{\\rm z, thick}(R)} \\right)\n,\\]\nwhere the respective scale heights are\n\\begin{equation}\nh_{\\rm z, thin}(R) = h_{\\rm z, thin}(R_\\odot)\\left( 1 + \\sum_{i=1}^2 k_{\\rm i, thin}(R-R_\\odot )^i \\right)\n,\\end{equation}\\[\nh_{\\rm z, thick}(R) = \\left\\{ \\begin{array}{lcl}\n      h_{\\rm z, thick}(R_{\\rm ft})&  R<R_{\\rm ft}\\\\\n      h_{\\rm z, thick}(R_{\\rm ft})\\left( 1 + \\sum_{i=1}^2 k_{\\rm i, thick}(R-R_{\\rm ft})^i\\right) & R \\geq R_{\\rm ft}\\\\\n   \\end{array}\n   \\right.\n.\\]\nThis expresses an exponentially flared disc (L\\'opez-Corredoira et al. 2002) with a hole or deficit of\nstars in the inner in-plane region (L\\'opez-Corredoira et al. 2004). Since we did not examine the inner\ndisc, we kept $h_{\\rm r, hole}$ constant: $h_{\\rm r, hole}=3.74$ kpc (L\\'opez-Corredoira et al. 2004).\nWe also kept constant the ratio of thick to thin disc stars in the solar neighbourhood,\n$f_{\\rm thick}=0.09$; and the rest of the parameters were left free in our fit: \nthe scale length of the thin disc $h_{\\rm r, thin}$, the scale height of the thin disc at solar Galactocentric\ndistance $h_{\\rm z, thin}(R_\\odot)$, the scale length of the thick disc $h_{\\rm r, thick}$,\nthe scale height of the thin disc at solar Galactocentric\ndistance $h_{\\rm z, thick}(R_\\odot)$, the two parameters defining the flare of the thin disc\n$k_{\\rm 1, thin}$ and $k_{\\rm 2, thin}$, the two parameters defining the flare of the thick disc $k_{\\rm 1, thick}$, $k_{\\rm 2, thick}$, and the scale at which the thick disc flare starts, $R_{\\rm ft}$.\n\nBovy et al. (2012) suggested that there is no thick disc that sensibly can be \ncharacterized as a distinct component\nbecause mono-abundance sub-populations, defined in the [alpha\/Fe]-[Fe\/H] space, are well described by single-exponential spatial-density profiles in both the radial and the vertical direction; therefore, any star of a given abundance would be associated with a sub-population of a given scale height and there would be a continuous and monotonic distribution of disc thicknesses. This result is contradicted by other authors however (e.g., Haywood et al. 2013), who showed a bimodal distribution in the [alpha\/Fe]-[Fe\/H] space, or that thin and thick disc can be distinguished because of the rotation and vertical dispersion, age, or other indicators of metallicity. Here we assumed that there are morphologically two discs and do not engage in the discussion about their populations.\n\nWe also included a halo, but did not try to fit any of its parameters; we just took\nthe fixed density distribution provided by Bilir et al. (2008) using SDSS data. \nIts contribution to our counts is lower than the disc contribution (20\\% for the highest $R$ and $z$ of\nour range).\n\\begin{equation}\n\\rho_{\\rm halo} (R,z)= 1.4\\times 10^{-3}\\,\\rho_\\odot \\frac{\\exp\\left[10.093\\left( 1 - \\left(\\frac{R_{\\rm sp}}{R_\\odot} \\right)^{1\/4} \\right) \\right]}{(R_{\\rm sp}\/R_\\odot )^{7\/8}} \n,\\end{equation}\\[\nR_{\\rm sp} = \\sqrt{R^2+2.52z^2}\n.\\]\n\nOur model does not take the stellar warp into account.\nThe warp moves the position of the Galactic plane: the plane can be \nshifted away from the expected position of $b=0$. It can be clearly\nseen in the counts within a few kpc of the Sun (L\\'opez-Corredoira et al. 2002, \nMomany et al. 2006, Reyl\\'e et al. 2009) and its effect is stronger towards \nthe outer edge of the disc.\nThe models of the warp are normally simple, based on tilted rings, and in general \nrepresent the star count well. Nonetheless, its average effect is not important for\n$|b|\\ge 8^\\circ $ (see Fig. 15\/top of L\\'opez-Corredoira et al. 2002 with ratio of positive\/negative counts\nclose to unity): some regions are overdense in the positive latitudes with respect to the negative latitude for the northern warp region (the southern warp is not visible with SDSS), but on average for our fit both positive and negative latitude data cancel each other out and only a \nminor second-order effect may remain in the fit, which we neglected. \nIn Fig. \\ref{Fig:dens}, no significant signs of the northern warp are detected\n(the southern warp is not visible in our SDSS data). A more detailed \ndiscussion on the influence of the warp is given in the\nlast paragraph of the next subsection and in \\S \\ref{.explor}.\n\n\\subsection{Best fit}\n\n\\begin{figure*}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=18cm]{fitres.eps}\n\\caption{Best fit (solid line) for the data of $\\rho (R,z)$ extracted from the star counts of\nF8-G5V stars in the SDSS. The dashed line stands for the best fit of the disc within $R<15$ kpc\nwithout any flare, and extrapolated for $R\\ge 15$ kpc.\nError bars stand only for Poissonian errors in the star counts and\ndo not include other possible factors such as errors in the extinction.\nNote that in the lowest $|z|$ bins there are no \npoints at high $R$ because of our constraint of \n$|b|>8^\\circ $ (see Fig. \\ref{Fig:dens}).}\n\\label{Fig:fitres}\n\\end{figure*}\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=9cm]{err_theta_zpos.eps}\\\\\n\\vspace{1cm}\n\\includegraphics[width=9cm]{err_theta_zneg.eps}\n\\caption{Residuals of $\\rho $ with respect to the best fit for the data extracted from the star counts of\nF8-G5V stars in the SDSS. $R>7.5$ kpc, $|z|<3.5$ kpc. Top: $z>0$. Botton: $z<0$.}\n\\label{Fig:err_theta}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Disc parameters of the best fit (see text). Note that there are nine independent parameters\nin the fit; the last six parameters depend on the previous ones. The constraint \n$|\\phi |\\le 30^\\circ $ explores the region where the warp amplitude is very low. The metallicity gradient\nis modelled following Eq. (\\ref{gradmet}).}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\textrm{parameter} & \\textrm{All data, no grad. metal.} & $|\\phi |\\le 30^\\circ $, no grad. metal. & \\textrm{All data, with grad. metal.}  \\\\\n\\hline\n$h_{\\rm r, thin}$ (kpc) & $2.0^{+0.3}_{-0.4}$  & 2.1  & 2.0  \\\\\n$h_{\\rm z, thin}(R_\\odot)$ (kpc) & $0.24^{+0.12}_{-0.01}$  & 0.28  & 0.21  \\\\\n$h_{\\rm r, thick}$ (kpc) & $2.5^{+1.2}_{-0.3}$  & 2.5  & 2.7  \\\\\n$h_{\\rm z, thick}(R_\\odot)$ (kpc) & $0.71^{+0.22}_{-0.02}$  & 0.60  & 0.55 \\\\ \\hline\n$k_{\\rm 1, thin}$ (kpc$^{-1}$) & -0.037  & 0.090 & -0.190 \\\\\n$k_{\\rm 2, thin}$ (kpc$^{-2}$) & 0.052  & 0.043 & 0.070 \\\\\n$k_{\\rm 1, thick}$ (kpc$^{-1}$) & 0.021  & 1.448 & 0.000 \\\\\n$k_{\\rm 2, thick}$ (kpc$^{-2}$) & 0.006  & -0.055 & 0.010 \\\\\n$R_{\\rm ft}$ (kpc) & $6.9^{+10.1}_{-1.9}$ & 15.8 & 6.9 \\\\ \\hline\n$h_{\\rm z, thin}(15\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 3.3$^{+1.8}_{-1.7}$    & 3.7   & 3.1 \\\\\n$h_{\\rm z, thin}(20\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 8.1$^{+5.4}_{-5.4}$    & 8.3   & 8.8 \\\\\n$h_{\\rm z, thin}(25\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 15.5$^{+10.8}_{-11.3}$  & 14.9  & 17.8 \\\\\n$h_{\\rm z, thick}(15\\,{\\rm kpc})\/h_{\\rm z, thick}(R_\\odot)$ & 1.5$^{+4.8}_{-0.4}$   & 1.0  & 1.6 \\\\\n$h_{\\rm z, thick}(20\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$  & 2.3$^{+12.9}_{-0.8}$  & 6.1   & 2.7 \\\\\n$h_{\\rm z, thick}(25\\,{\\rm kpc})\/h_{\\rm z, thick}(R_\\odot)$ & 3.4$^{+25.4}_{-1.7}$   & 9.6  & 4.2 \\\\ \\hline\n\\label{Tab:bestfit}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nWe used the selected regions for the fit, excluding bins with a density lower than\n$3\\times 10^{-9}$ star pc$^{-3}$. For the fit of the scale lengths and scale \nheights ($h_{\\rm r, thin}$,\n$h_{\\rm z, thin}(R_\\odot)$, $h_{\\rm r, thick}$, $h_{\\rm z, thick}(R_{\\rm ft})$), we used \nthe regions with $|z|\\le 3$ kpc, $R<15$ kpc, i.e., we neglect the flare; after we obtained these four parameters we fitted the rest of them for the flare in the regions with $1.5<|z({\\rm kpc})|\\le 3.5$ kpc, $7.5<R({\\rm kpc})<30$. Within these ranges, the Galaxy is dominated by the disc rather than the halo star counts.\n\nThe parameters that produce the best weighted fit are given in Table \\ref{Tab:bestfit}.\nFigs. \\ref{Fig:fitres} and \\ref{Fig:err_theta} show how this model fits the data.\nThe error bars (1$\\sigma $) of these parameters were derived using the method of Avni (1976) for four free\nparameters (the first fit of the scales) and\nfive free parameters (the fit of the flare): $\\Delta \\chi^2=4.72$ and 5.89 respectively; \nwhere we normalized $\\chi ^2$ such that \n$\\chi _{\\rm min}^2=N$ (number of data points) to take into account that the Poissonian\nerror is only a small part of the total error.\n\nIt can be observed how the flare becomes strong for high $R$ and $z$. In Fig. \\ref{Fig:flare}, we plot the functional shape of $h_{\\rm z, thin}(R)$ and $h_{\\rm z, thick}(R)$ in this best-fit model.\nThe error bars for the flare amplitude are high, but they exclude the non-flared solution.\nNote that the errors for the thin disc include the free variation of the thick disc to compensate for the\nrespective variations within those errors and vice versa. If we fixed one of the discs, the errors bars of\nthe other disc would be much lower than the case with its variation. When we combine both discs, we also expect\nto reduce the error of the average disc (the errors of the average are also derived through the\n$\\chi ^2$ analysis), from which we conclude that we need a flare to fit our data, although distinguishing between thin and\nthick disc component is only possible with low significance. \nIt is clear from Fig. \\ref{Fig:fitres} that an extrapolation of the\ndisc fitted at $R<15$ kpc without flare does not work at larger $R$: the shape of the density presents\na more complex form than the dashed line in the log-linear plot of Fig. \\ref{Fig:fitres}.\n\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8cm]{flare.eps} \n\\caption{Scale height of the thin and thick discs according to our best fit.\nThe average is defined as the $-\\left(\\frac{d\\,ln\\,\\rho _{\\rm disc}(R,z=0)}{d|z|}\\right)^{-1}$\nand takes into account the increasing ratio of thick disc stars outwards.}\n\\label{Fig:flare}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8cm]{flare2.eps} \n\\caption{Scale height of the average defined as in Fig. \\protect{\\ref{Fig:flare}} for the best fits \nof Table \\protect{\\ref{Tab:bestfit}}.}\n\\label{Fig:flare2}\n\\end{figure}\n\nWe can see how the variation of the density with $z$ for\nhigh $R$ becomes quite weak: green extends to almost all regions with high $R$ in Fig. \\ref{Fig:dens}, and the few blue areas (very low densities) represent the few fluctuations due possibly\nto some errors in the extinction or warp presence\n(because the lowest latitudes are more affected by these\nfluctuations). This is precisely what we obtain in our fit, \nwhich indicates the existence of a conspicuous\nflare: a very significant increase of the scale height of the disc both for the thin and thick discs.\nIn Fig. \\ref{Fig:fitres}, we also observe the effect of the flare: at $R\\approx 15$ kpc and high $z$ the\ndensity becomes almost constant with little decrease with $R$ \nbecause of a combination of the exponential decrease with radius and the increase of the density at high $z$ due to the increase of the scaleheight.\n\nFig. \\ref{Fig:err_theta} shows some residuals in the observed stellar \ndensity with respect to the best fit. If the northern warp effect were significant, it would produce an excess density at $60^\\circ \\lesssim \\phi \\lesssim 100 ^\\circ $ at $z>0$ and a deficit of stars in the same azimuths at $z<0$, and we do not observe that.\nIn general, some deviations from null residuals may be due to different effects such as metallicity gradients, a different\ncontamination of the halo than expected (at high $|z|$), some irregularities in the distribution of the extinction,\nsome degree of lopsidedness, or a variation of $h_z$ with $\\phi $ (L\\'opez-Corredoira \\& Betancort-Rijo 2009).\n\n\\section{Exploring the effects of the warp and metallicity gradients}\n\\label{.explor}\n\nWe carried out two numerical experiments to determine the effects of \nthe warp and the metallicity gradients. Since \nthere are large uncertainties for both, we do not expect to derive direct\nconclusions from the results of the following best fits, but they should serve as an estimate of the typical variation in our parameters under these changes. \n\n\\begin{description}\n\n\\item[Warp:] instead of the whole data set, we used only those with $|\\phi |\\le 30^\\circ $, where the amplitude\nof the warp is lowest. We fitted $\\rho (R,z)$ in the same way as before and the parameters obtained are those given in Table \\ref{Tab:bestfit} and Fig. \\ref{Fig:flare2}. \nThey are compatible with the previous values, confirming our guess that the warp \ndoes not strongly change our results.\n\n\\item[Metallicity gradient:] we attributed to each star a Galactocentric distance $R'$, $\\phi '$ and vertical position $z'$ corresponding to their coordinates and a distance $r'(m)$ given by\n\\begin{equation}\n\\label{gradmet}\nr'(m)=10^{[m-M'+5]\/5}\n,\\end{equation}\\[\nM'=4.8-0.4\\Delta[Fe\/H]\n,\\]\\[\n\\Delta[Fe\/H](R'[{\\rm kpc}],z'[{\\rm kpc}])=\n\\]\\[\n\\left\\{ \\begin{array}{lcl}\n      -0.078(R'-R_\\odot)-0.220|z'| &  , R'\\le f(z') \\\\\n      -1 & , R'>f(z') \\\\\n   \\end{array}\n   \\right.\n,\\]\\[\nf(z')=R_\\odot +12.82-2.82|z'|\n.\\]\n\nAs said in \\S \\ref{.method}, this stems from the variation of the \nabsolute magnitude of F8V-G5V stars \nestimated by Siegel et al. (2002) for a variation of matallicity with respect to the Sun \n[$\\Delta M_R\\approx 0.4$ for $\\Delta[Fe\/H]=-1$ using $R-I=0.38$, which is the corresponding transformation from SDSS to Johnson filters (Jordi et al. 2006) of the\ncolor of our population with average $(r-i)=0.13$ (Bilir et al. 2009); $\\Delta M_g\\approx \\Delta M_R$ \n(Juri\\'c et al. 2008, Fig. 3)], and considering a variation of metallicity from the combination of radial and vertical gradients of metallicity for the disc given by Rong et al. (2001) and Ak et al. (2007); assuming the\nsame gradient for thin and thick discs and that \nit remains constant for the farthest disc for a metallicity lower than the solar one.\nThis is probably an overestimation of $|\\Delta[Fe\/H]|$, which should be lower than unity at least for the\nthin disc (Andreuzzi et al. 2011), but it serves as a limit of the strongest \neffect of this gradient.\nGiven that $\\Delta[Fe\/H](R',z')$ depends on the position and the position depends on this variation of metallicity, we carried out the calculation with an iterative process.\n\nThen, we fitted $\\rho (R',z')$ in the same way as before and the parameters obtained are\nthose given in Table \\ref{Tab:bestfit} and Fig. \\ref{Fig:flare2}. The results for the flare parameters\nare totally compatible within the error bars with those obtained without taking into account any gradient\nof metallicity.\nAs said in \\S \\ref{.method}, a lower metallicity in the farthest parts of the disc may overestimate\nthe distance of the stars (unfortunately, we do not know by how much since there are no \naccurate measurements of the metallicity of stars at those Galactocentric distances; we estimate an error of no more\nthan 20\\% [\\S \\ref{.method}]), but our numerical experiment shows that \nthe variation of the scale lengths and scale heights is slight and\nthe need of the flare is beyond these uncertainties with the metallicity.\n\n\\end{description}\n\n\n\\section{Comparison with other works}\n\\label{.compar}\n\nThe scale lengths and scale heights can be compared with those in other publications, although mainly for low $R$. Table \\ref{Tab:compar} gives some of the values of the literature.\nJia et al. (2014, Table 1) reported many other values.\nIn general, the relative trends between thin and thick disc\nare similar in all these papers and our results, but there\nis some variation in the absolute scales that might arise because of different observed populations, different techniques of distance estimation, or different regions of application, apart, of course, from\npossible systematic errors. Jia et al. (2014) have shown how the parameters, especially the \nscale height of the thin disc, depend on the absolute magnitude of the main-sequence stars used, indicating\nthat different populations have different velocity dispersions. The numbers of\nJuri\\'c et al. (2008) are somewhat higher than ours, possibly because they represent a range of smaller $R$, or possibly because\nof their method of distance determination. Bilir et al. (2006) showed that  \nthe scale length depends on Galactic longitude. Here, we derived its average value but did\nnot examine any dependence on Galactic coordinates. \n\n\\begin{table*}\n\\caption{Some values of the scale lengths and scale heights from the literature (units in kpc), \nderived either with SDSS (visible) or 2MASS (near infrared red).}\n\\begin{tabular}{ccccccc}\nReference & source & spatial range & $h_{\\rm r, thin}$ & $h_{\\rm z, thin}(R_\\odot)$ & $h_{\\rm r, thick}$ & $h_{\\rm z, thick}(R_\\odot)$ \\\\\n\\hline\nL\\'opez-Corredoira et al. (2002) & 2MASS & low Gal. latitudes & 3.3 & 0.28 & -- & -- \\\\ \nCabrera-Lavers et al. (2005) & 2MASS & high Gal. latitudes & -- & 0.27 & -- & 1.1  \\\\\nBilir et al. (2006) & SDSS & interm. Gal. latitudes & 1.9 & 0.22 & -- & --  \\\\\nCabrera-Lavers et al. (2007) & 2MASS & interm. Gal. latitudes & -- & 0.19 & -- & 0.96  \\\\\nBilir et al. (2008) & SDSS & high Gal. latitutes & -- & 0.19 & -- & 0.63  \\\\\nJuri\\'c et al. (2008) & SDSS & $r<1.5$ kpc & 2.6 & 0.30 & 3.6 & 0.90  \\\\\nChang et al. (2011) & 2MASS & interm.-high Gal. latitudes & 3.7 & 0.36 & 5.0 & 1.0  \\\\\nPolido et al. (2013) & 2MASS & whole sky & 2.1 & 0.21 & 3.0 & 0.64 \\\\\nJia et al. (2014) & SDSS+SCUSS & interm. Gal. latitude & -- & 0.20 & -- & 0.60 \\\\ \n\\label{Tab:compar}\n\\end{tabular}\n\\end{table*}\n\nThe flare was previously observed by Alard (2000), \nL\\'opez-Corredoira et al. (2002), Yusifov (2004), Momany et al. (2006), or Reyl\\'e et al. (2009).\nPolido et al. (2013) also introduced a flare in their model, but did no fit their parameters.\nThe values of the scale height at high $R$ from our fit (see Fig. \\ref{Fig:flare}) are high: with\na thin disc scaleheight around 0.8 kpc at R=15 kpc, lower than the\nextrapolation of L\\'opez-Corredoira et al.\n(2002) or that of Yusifov (2004), and\nhigher than the values of Alard (2000), Momany et al. (2006) and Reyl\\'e et al. (2009). Between 20-25 kpc we derive a\nthin disc scale height of 2-4 kpc, which is also higher than the values by\nMomany et al. (2006) and Reyl\\'e et al. (2009)  \n(for the rest of the authors, there are no values at such \nhigh values of $R$).\nFor the thick disc, there are few or no studies to compare our studies with: \nthere is a hint with unconclusive results  \nby Cabrera-Lavers et al. (2007), which was constrained within $R<10$ kpc and\nobtained the opposite sign in the increase of scale height for the solar neighbourhood;\nit is interesting that the values of the flare in the thick disc \nthat we obtained are those that are needed to explain the Monoceros ring in terms of \nGalactic structure (Hammersley \\& L\\'opez-Corredoira 2011).\n\nNote that in our results at $R\\gtrsim 17$ kpc \nthe thin disc becomes ``thicker'' than the so-called thick disc.\nThis should motivate us to change the nomenclature: maybe instead of thin disc + thick disc\nwe should speak of disc 1 and disc 2. In any case, our model is just an exercise of fitting\nstellar densities and, within the error bars we are unable to see which disc has a larger scale height\nat large Galactocentric radius. We do not aim here to distinguish among different populations.\nWe see from our results that we need a flare to interpret the global density, but, with\nthe present analysis, we are not able to distinguish  the populations\nthat are flared at high $R$. We expect that at high $R$ there should be no significant difference between both discs\nthicknesses and we have only one mixed component. An average disc as plotted in \nFig. \\ref{Fig:flare} represents this average old population of type F8V-G5V stars.\nIn this average disc, at high $R$ \nthe thick disc (or better: disc 2) has a higher ratio of stars than at $R_\\odot $:\nwhile at $R_\\odot $ it is 9\\% of the stars of the thin disc (or better: disc 1), at $R=25$ kpc it is 50\\% of the stars of the thin disc, because the scale length of the thick disc is larger than that of the thin disc, and consequently the fall-off of the density is slower.\n \nThe theoretical explanation of the observed features is beyond the scope of this paper.\nThe origin of the thick disc and its flare need to be\npredicted by a model that aims to understand these observations. One of the hypotheses for the formation of thick discs is through minor mergers, which predicts a scale length of the thick disc larger than the scale length of the thin disc, a flare of increasing scale height in the thick disc, and a constant scale height for the stellar excess added by the merger (Qu et al.\n2011). If we assumed that the mixture of the thin disc of the primary galaxy \nplus the stellar excess due to the accreted minor galaxy produces what we observe as the thick disc, our results would be fitted by those predictions. Nonetheless, our analysis is very rough, and without a necessary study of the populations we are unable to confirm this scenario.\n\n\\section{Discussion and conclusions}\n\\label{.discu}\n\nOur method of deriving the 3D stellar distribution is quite straightforward, although it may contain some errors due mainly to inappropriate extinction estimate (some systematic error in the scales may be produced, but we do not expect it to exceed 18\\%; see\n\\S \\ref{.method}), metallicity gradients, or the effect of the warp. \nEven taking into account these factors, \nwe have not observed features that suggested that the derived morphology\nmight be very different: the scales might change slightly, \nbut the presence of the flare is unavoidable.\n\nOur results show that the stellar density distribution of the outer disc (up to $R=30$ kpc) is well fitted by a component of thin+thick disc with flares (increasing scale height outwards). From our diagrams, it is clear that there is no a cut-off of the stellar component at $R=14-15$ kpc\nas stated by Ruphy et al. (1996) or Minniti et al. (2011); we only examined off-plane regions \n($|b|\\ge 8^\\circ $) so we cannot judge what occurs in the in-plane regions, but from our results and by interpolating the results in the $z$-direction one can clearly conclude the reason why Ruphy et al. or Minniti et al. appreciated a significant drop-off of stars at $R=14-15$ kpc: the flare becomes important at those galactocentric distances, and consequently, the stars are distributed in a much wider range of heights, producing this apparent depletion of in-plane stars. Indeed, our Galactic disc does not present a cut-off there but the stars are spread in off-plane regions, even at $z$ of several kpc up to \na Galactocentric distance of 15 scale lengths. Assuming that our fit is correct, for a constant luminosity function along the disc, the flux of the Milky Way seen observed face-on would follow a dependence\n(from Eq. (\\ref{rho}), neglecting the hole of the inner disc, which is totally insignificant for $R>R_\\odot $)\n\\begin{equation}\nF(R)\\propto \\int _{-\\infty}^\\infty dz\\,\\rho_{\\rm disc}(R,z)\n\\end{equation}\\[\\,\\,\\,\\,\\,\\,\n\\approx \\rho _\\odot \\left[\\exp\\left(-\\frac{R}{h_{\\rm r, thin}}\\right)+f_{\\rm thick}\\exp\\left(-\\frac{R}{h_{\\rm r, thick}}\\right)\\right]\n.\\]\nIt is clear that in $F(R)$ there is no radial truncation in the explored range, and if this $F(R)$ did not represent our Galaxy, we would not derive as good a fit as we did. This does not mean that radial truncations are not possible in spiral galaxies: there are other galaxies in which it is observed (van der Kruit \\& Searle 1981; Pohlen et al. 2000), but the Milky Way is not one of them.\n\nThe smoothness of the observed stellar distribution also suggests that there is a continuous structure\nand not a combination of a Galactic disc plus some other substructure or extragalactic component. The\nhypothesis of interpreting the Monoceros ring in terms of a tidal stream of a putative accreted dwarf galaxy\n(Sollima et al. 2011; Conn et al. 2012; Meisner et al. 2012; Li et al. 2012)\nis not only unnecessary (as stated by Momany et al. 2006; Hammersley \\& L\\'opez-Corredoira 2011; L\\'opez-Corredoira et al. 2012), but appears to be quite inappropriate:\nwe see in Fig. \\ref{Fig:dens} no structure overimposed on the Galactic disc.\nInstead, the observed flare explains the overdensity in the Monoceros ring\nobserved in the SDSS fields (Hammersley \\& L\\'opez-Corredoira 2011).\n\nGiven these results, it would be interesting for future works if the dynamicists explained the existence of the observed flares in the Galactic disc, and further observational research on the spectroscopical\nfeatures of the disc stars at very high $R$ and high $z$ is necessary \nto know more about the origin and evolution of this component.\n\n\\begin{acknowledgements}\nThanks are given to A. Cabrera-Lavers, S. Zaggia and the anonymous referee for useful comments that helped us to improve this paper. MLC was supported by the grant AYA2012-33211 of the Spanish Ministry of Economy and Competitiveness (MINECO). Thanks are given to Astrid Peter (language editor of A\\&A) for proof-reading of the text.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/.\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. \n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of the details of the fate of a false vacuum plays a key role in\nunderstanding the properties of a variety of systems. It extends from the\nunderstanding the characteristics of various phase transitions in condensed\nmatter and particle physics all the way to quantifying the human angst that\nour universe itself may actually be in a state of a false vacuum. The gross\nfeatures of the decay mechanism, the formation of bubbles and the quantum\nunder the barrier tunneling involve both the classical and quantum aspects\nof the problem. The description of tunneling in the non-relativistic quantum mechanics\nfor a particle moving in many dimensions was considered in \\cite{Banks}.  In quantum field\ntheory the problem of a false vacuum instability was first addressed in \\cite%\n{Kobzarev}, assuming that the thickness of the wall is small compared to the\nsize of the bubble. Coleman \\cite{Coleman} has generalized the results of \n\\cite{Kobzarev} and suggested how to encapsulate and calculate the\nnon-perturbative quantum part of the process by identifying Euclidean\nclassical configurations which give the leading contribution to the\nprobability of the false vacuum decay. In this paper we study cases where\nthe method suggested by Coleman, as is, needs to be reexamined and modified.\nIn particular, we investigate how the theory must be modified by taking into\naccount the inevitable quantum fluctuations of the scalar field.\n\nWe start by a short review of the Coleman theory for the scalar field\npotential $V\\left( \\varphi \\right) ,$ which has a shape shown in Fig. \\ref%\n{Figure1} (for details see, for example, \\cite{Weinberg}, \\cite{Mukhanov1}%\n)). The false vacuum configuration $\\ \\varphi \\left( \\mathbf{x}\\right)\n=\\varphi _{0}$ is unstable and decays via bubble nucleation. The field\ninside the emerged expanding bubble either tends to its value in the true\nminimum or $\\varphi $ $\\rightarrow $ $-\\infty $ for the unbounded potential.\n\n\\vspace{1.0cm}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_1.eps}\n\\end{center}\n\\par\n\\vspace*{-1.11cm} \\hspace*{7.8cm} - $V_{min}$%\n\\par\n\\vspace*{-3.2cm} \\hspace*{5.3cm} $\\varphi_{min}$%\n\\par\n\\vspace*{-3.8cm} \\hspace*{7.85cm}$V$%\n\\par\n\\vspace*{2.8cm} \\hspace*{9.35cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-2.0cm} \\hspace*{8.05cm}$V_{bar}$%\n\\par\n\\vspace*{0.8cm} \\hspace*{13.0cm}$\\varphi$%\n\\par\n\\vspace*{3cm}\n\\caption{}\n\\label{Figure1}\n\\end{figure}\n\n\\vspace{0.5cm}\n\nThe scalar field $\\varphi \\left( \\mathbf{x},t\\right) $ is a system with\ninfinitely many degrees of freedom with spatial coordinates $\\mathbf{x=}%\n\\left( x^{1},x^{2},x^{3}\\right) $ enumerating them, that is, $\\varphi \\left( \n\\mathbf{x},t\\right) =\\varphi _{\\mathbf{x}}\\left( t\\right) $. The scalar\nfield action can be written as \n\\begin{equation}\nS=\\int (\\mathcal{K-V)}\\,dt\\,,  \\label{1}\n\\end{equation}%\nwhere \n\\begin{equation}\n\\mathcal{K=}\\frac{1}{2}\\int \\left( \\frac{\\partial \\varphi }{\\partial t}%\n\\right) ^{2}\\,d^{3}x\\,,  \\label{2}\n\\end{equation}%\nand \n\\begin{equation}\n\\mathcal{V=}\\int \\left( \\frac{1}{2}\\left( \\frac{\\partial \\varphi }{\\partial\nx^{i}}\\right) ^{2}+V\\left( \\varphi \\right) \\right) \\,d^{3}x  \\label{3}\n\\end{equation}%\nare, correspondingly, the kinetic and interaction energies of the field\nconfiguration $\\varphi \\left( \\mathbf{x}\\right) $ and its first time\nderivatives$.$ Note that the functional $\\mathcal{V}\\left( \\varphi \\left( \n\\mathbf{x}\\right) \\right) $ (\\textit{but not} $V\\left( \\varphi \\right) )$\nplays the role of the potential energy when we are considering the\nsub-barrier tunneling in quantum field theory. For the scalar field\npotential $V\\left( \\varphi \\right) ,$ shown in Fig. \\ref{Figure1}, the field\nconfiguration $\\varphi \\left( \\mathbf{x}\\right) =\\varphi _{0}$ has zero\npotential energy, $\\mathcal{V}\\left( \\varphi _{0}\\right) =0,$ while the\nother homogeneous static solution $\\varphi \\left( \\mathbf{x}\\right) =\\varphi\n_{\\min }$ has a lower energy $\\mathcal{V}\\left( \\varphi _{\\min }\\right)\n=V_{\\min }\\times $\\textit{volume }$<$ $0.$ Therefore the local minimum at $%\n\\varphi _{0}$ is unstable (false vacuum) and decays via sub-barrier\ntunneling as a result of which the bubbles of the true vacuum are formed. To\ncalculate the decay rate Coleman has assumed that in the semiclassical\napproximation the dominant contribution to the tunneling rate comes from the\ninstanton - Euclidean solution of the scalar field equation, which matches\nthe metastable vacuum $\\varphi \\left( \\mathbf{x}\\right) =\\varphi _{0}$ to\nsome classically allowed configuration $\\varphi \\left( \\mathbf{x}\\right) $\nwith $\\mathcal{V}\\left( \\varphi \\left( \\mathbf{x}\\right) \\right) =0,$\ndescribing the emerging bubble in Minkowski space. On symmetry grounds it\nwas shown \\cite{Coleman1} that the minimal action has a $O\\left( 4\\right) $%\n-invariant solution of the scalar field equation \n\\begin{equation}\n\\frac{\\partial ^{2}\\varphi }{\\partial \\tau ^{2}}+\\Delta \\varphi -\\frac{dV}{%\nd\\varphi }=0\\,,  \\label{4}\n\\end{equation}%\nfor which $\\varphi $ depends only on $\\varrho =\\sqrt{\\tau ^{2}+\\mathbf{x}^{2}%\n}$ and where $\\tau =it$ is the Euclidean time. This reduces to the ordinary\ndifferential equation \n\\begin{equation}\n\\ddot{\\varphi}(\\varrho )+\\frac{3}{\\varrho }\\,\\dot{\\varphi}(\\varrho )-\\frac{dV%\n}{d\\varphi }\\,=0\\,,  \\label{5}\n\\end{equation}%\nwhere dot denotes the derivative with respect to $\\varrho .$ If we assume\nthat the field was initially in the false vacuum state one of the boundary\nconditions for $\\left( \\ref{5}\\right) $ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\rightarrow \\infty \\right) =\\varphi _{0}\\,.  \\label{6}\n\\end{equation}%\nAs a second condition Coleman has suggested to use \n\\begin{equation}\n\\dot{\\varphi}(\\varrho =0)=0  \\label{7}\n\\end{equation}%\nto avoid the singularity in the center of the bubble. Equation $\\left( \\ref%\n{5}\\right) $ with \\textit{these boundary conditions} has an unambiguous\nsolution $\\varphi \\left( \\varrho \\right) $ called the Coleman instanton. The\naction for this instanton is given by%\n\\begin{equation}\nS_{I}\\,=\\,2\\pi ^{2}\\,\\int_{0}^{+\\infty }d\\varrho \\,\\varrho ^{3}\\,\\left( \n\\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right)  \\label{8}\n\\end{equation}%\nand the false vacuum decay rate \\textit{per unit time per unit volume} can\nbe estimated as%\n\\begin{equation}\n\\Gamma \\simeq \\varrho _{0}^{-4}\\exp \\left( -S_{I}\\right) \\,,  \\label{9a}\n\\end{equation}%\nwhere $\\varrho _{0}$ is the size of the bubble. The pre-exponential factor\nis based on dimensional grounds. Calculating the potential energy $%\n\\left( \\ref{3}\\right) $ at the moment of Euclidean time $\\tau ,$ we infer\nthat $\\mathcal{V}\\left( \\tau \\right) >0$ for $0<\\tau <\\infty .$ Since the\ntotal energy is normalized to zero this means that in this range of $\\tau $\nthe instanton describes the sub-barrier tunneling in the Euclidean time. As $%\n\\tau \\rightarrow \\infty ,$ $\\mathcal{V}\\rightarrow 0,$ corresponding to the\nfalse vacuum state. The potential energy also vanishes at $\\tau =0$ and\nhence at this instant\\ the bubble emerges from under the barrier in\nMinkowski space. To prove that $\\mathcal{V}\\left( \\tau =0\\right) =0$ we\nfirst note that for $\\varphi \\left( \\mathbf{x},\\tau \\right) =\\varphi \\left(\n\\varrho \\right)$ the expression $\\left( \\ref{3}\\right)$, calculated at $\\tau\n=0,$ reduces to \n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =4\\,\\pi \\int_{0}^{+\\infty }d\\varrho \\,\\varrho\n^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) \\,.\n\\label{10b}\n\\end{equation}%\nNext we find that the first integral of $\\left( \\ref{5}\\right) $ can be\nwritten as%\n\\begin{equation}\n\\frac{1}{2}\\dot{\\varphi}^{2}-V=\\int_{\\rho }^{\\infty }\\frac{3}{\\tilde{\\varrho}%\n}\\left( \\frac{d\\varphi }{d\\tilde{\\varrho}}\\right) ^{2}d\\tilde{\\varrho}\\,,\n\\label{11a}\n\\end{equation}%\nwhere the boundary condition $\\left( \\ref{6}\\right) $ has been taken into\naccount. Finally integrating this equation one gets\\footnote{%\nThe second equality in (\\ref{12b}) is the result of integration by parts\ntaking into account (\\ref{6}) and (\\ref{7}) as well as requiring that $%\n\\varphi(\\varrho)$ decays faster than $\\varrho^{-\\frac{1}{2}}$ at $%\n\\varrho\\rightarrow \\infty$.} \n\\begin{equation}\n\\int_{0}^{\\infty }\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}-V\\right) \\varrho\n^{2}d\\varrho =\\int_{0}^{\\infty }d\\left( \\varrho ^{3}\\right) \\left(\n\\int_{\\rho }^{\\infty }\\frac{1}{\\tilde{\\varrho}}\\left( \\frac{d\\varphi }{d%\n\\tilde{\\varrho}}\\right) ^{2}d\\tilde{\\varrho}\\right) =\\int_{0}^{\\infty }\\dot{%\n\\varphi}^{2}\\varrho ^{2}d\\varrho  \\label{12b}\n\\end{equation}%\nand hence%\n\\begin{equation}\n\\int_{0}^{\\infty }\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}+V\\right) \\varrho\n^{2}d\\varrho =0  \\label{13a}\n\\end{equation}%\nimplying that $\\mathcal{V}\\left( \\tau =0\\right) $ $\\left( \\ref{10b}\\right) $\nreally vanishes.\n\nThere is a class of potentials for which the results obtained using the\nColeman boundary conditions can either lead to a questionable outcome or\ncannot be applied at all, namely,\n\n\\textit{a}) for the very steep unbounded potentials (see Fig. \\ref{Figure3})\nColeman's instantons lead to nearly instantaneous instability of the false\nvacuum, the so called zero size instanton problem,\n\n\\textit{b}) for some unbounded potentials, when false vacuum must be\nunstable, the Coleman instanton does not exist.\n\nWe will show that both problems have common origin and can be resolved by\ntaking into account inevitable quantum fluctuations which induce the\nultraviolet cutoff scale. In that situation the remedy suggested by Coleman\nto avoid a singularity at the origin needs to be and is replaced by an\nultraviolet cutoff scale induced by those very quantum fluctuations. In\nturn, this allows us to abandon the very restrictive boundary condition $%\n\\left( \\ref{7}\\right) $ and obtain a whole class of new instantons which\nwould be singular in the absence of this quantum ultraviolet cutoff.\n\nIn this paper we will consider an unbounded linear potential and clarify\nthe role of quantum fluctuations in resolving small instanton problem. The\ncase of a quartic unbounded potential for which the Coleman instanton does not\nexist is analyzed in the companion paper $\\cite{MRS2}.$\n\n\\section{The model}\n\nLet us consider the classically exactly solvable potential \n\\begin{equation}\nV\\left( \\varphi \\right) =\\left\\{ \n\\begin{array}{cc}\n\\lambda _{-}\\,\\varphi _{0}^{3}\\,\\varphi +\\frac{\\lambda _{+}}{4}\\,\\varphi\n_{0}^{4} & \\text{for }\\varphi <0\\,, \\\\ \n\\frac{\\lambda _{+}}{4}\\,\\left( \\varphi -\\varphi _{0}\\right) ^{4} & \\text{for \n}\\varphi >0\\,.%\n\\end{array}%\n\\right.  \\label{14a}\n\\end{equation}%\nIt has a local minimum, corresponding to the false vacuum at $\\varphi _{0} $%\n. To avoid the problem with one loop quantum corrections, we will assume\nthat the dimensionless coupling constant $\\lambda _{+}$ is always much\nsmaller than unity. At $\\varphi =0$ the $\\varphi ^{4}-$potential is matched\nto a linear unbounded potential for which the dimensionless constant $%\n\\lambda _{-}$ can be taken to be large.\n\nThe results obtained in this paper can also be applied to estimate the\nprobability of tunneling for some potentials with a second true minimum at\nsome negative $\\varphi _{\\min },$ which for $\\varphi <0$ can well be\napproximated by the linear potential $\\left( \\ref{14a}\\right) $ nearly up to \n$\\varphi _{\\min }$ (see dotted line in Fig. \\ref{Figure2}).\n\n\\vspace*{0.9cm}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_2.eps}\n\\end{center}\n\\par\n\\vspace*{-4.4cm} \\hspace*{11.33cm}$V_{min}$%\n\\par\n\\vspace*{1.18cm} \\hspace*{11.2cm} $-\\frac{4V_{bar}}{\\beta }\\left( 1-\\frac{%\n\\beta }{4}-{\\nu}^{2\/3}\\right)$%\n\\par\n\\vspace*{-3.4cm} \\hspace*{10.58cm}$V_{bar}$%\n\\par\n\\vspace*{-0.243cm} \\hspace*{11.56cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-1.8cm} \\hspace*{3.3cm}$(a)$%\n\\par\n\\vspace*{-0.48cm} \\hspace*{7.13cm}$(b)$%\n\\par\n\\vspace*{-1.0cm} \\hspace*{11.1cm}$V$%\n\\par\n\\vspace*{1.6cm} \\hspace*{12.9cm}$\\varphi$%\n\\par\n\\vspace*{4cm}\n\\caption{}\n\\label{Figure2}\n\\end{figure}\n\n\\section{The Coleman Instanton}\n\nFirst we find the explicit exact solution for the Coleman instanton and show\nhow the problem mentioned in the introduction arises in this simple\nparticular case. For positive $\\varphi $ equation $\\left( \\ref{5}\\right) $\ntakes the following form%\n\\begin{equation}\n\\ddot{\\varphi}+\\frac{3}{\\varrho }\\dot{\\varphi}-\\lambda _{+}\\left( \\varphi\n-\\varphi _{0}\\right) ^{3}=0  \\label{15}\n\\end{equation}%\nand its solution with the boundary condition $\\left( \\ref{6}\\right) $ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\varphi _{0}\\frac{\\varrho ^{2}-\\varrho\n_{0}^{2}}{\\varrho ^{2}-\\varrho _{0}^{2}\/(1+\\delta )}\\,,  \\label{16}\n\\end{equation}%\nwhere%\n\\begin{equation}\n\\delta =\\frac{4\\left( 1+\\sqrt{1+\\lambda _{+}\\,\\varphi _{0}^{2}\\,\\varrho\n_{0}^{2}\/2}\\right) }{\\lambda _{+}\\,\\varphi _{0}^{2}\\,\\varrho _{0}^{2}}\n\\label{17}\n\\end{equation}%\nand the constant of integration $\\varrho _{0}$ (the size of the bubble) yet\nremains to be fixed with the help of the second boundary condition $\\left( %\n\\ref{7}\\right) .$ The solution $\\left( \\ref{16}\\right) $ is valid only for $%\n\\varrho _{0}<\\varrho <\\infty .$ At $\\varrho =\\varrho _{0}$ the field $%\n\\varphi $ vanishes and then becomes negative. For $\\varphi <0$ the potential\nis linear and equation $\\left( \\ref{5}\\right) $ simplifies to%\n\\begin{equation}\n\\ddot{\\varphi}+\\frac{3}{\\varrho }\\,\\dot{\\varphi}-\\lambda _{-}\\,\\varphi\n_{0}^{3}=0\\,.  \\label{18}\n\\end{equation}%\nThe solution of this equation, which vanishes at $\\varrho _{0}$ and\nsatisfies $\\left( \\ref{7}\\right) ,$ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\frac{\\lambda _{-}\\,\\varphi _{0}^{3}}{8}%\n\\,\\left( \\varrho ^{2}-\\varrho _{0}^{2}\\right) \\,.  \\label{19}\n\\end{equation}%\nThe derivative of the field $\\varphi $ at $\\varrho =\\varrho _{0}$ must be\ncontinuous. This allows us to express the size of the bubble in terms of the\nparameters $\\lambda _{+},\\lambda _{-}$ and $\\varphi _{0}.$ Equating the\nderivatives of solutions $\\left( \\ref{16}\\right) $ and $\\left( \\ref{19}%\n\\right) $ at $\\varrho _{0}$ we obtain the following equation \n\\begin{equation}\n1+\\frac{1}{\\delta }=\\frac{\\lambda _{-}\\,\\varphi _{0}^{2}}{8}\\,\\varrho\n_{0}^{2}\\,,  \\label{20}\n\\end{equation}%\nwith $\\delta $ given in $\\left( \\ref{17}\\right) .$ Solving this equation for \n$\\varrho _{0}^{2}$ one gets \n\\begin{equation}\n\\varrho _{0}^{2}=\\frac{8}{\\varphi _{0}^{2}\\,\\lambda _{-}\\,\\left( 1-\\beta\n\\right) }\\,,  \\label{21}\n\\end{equation}%\nwhere we have introduced the parameter \n\\begin{equation}\n\\beta =\\frac{\\lambda _{+}}{\\lambda _{+}+\\lambda _{-}}  \\label{21a}\n\\end{equation}%\ninstead of $\\lambda _{+}.$\n\nAs one can see from $\\left( \\ref{16}\\right) $ for $\\delta \\ll 1$ the bubble\nhas a thin wall. Substituting $\\varrho _{0}^{2}$ from $\\left( \\ref{21}%\n\\right) $ into equation $\\left( \\ref{20}\\right) $ we find \n\\begin{equation}\n\\delta =\\frac{1-\\beta }{\\beta }  \\label{22a}\n\\end{equation}%\nand therefore the thin-wall approximation is valid only if $1-\\beta \\ll 1,$\nthat is, for rather flat potential $\\left( \\lambda _{-}\\ll \\lambda\n_{+}\\right) $ at negative $\\varphi $. As it follows from $\\left( \\ref{19}%\n\\right) $ the value of the scalar field in the center of the bubble is equal\nto%\n\\begin{equation}\n\\varphi \\left( \\varrho =0\\right) =-\\frac{\\varphi _{0}}{1-\\beta }\\,,\n\\label{23}\n\\end{equation}%\nwhich corresponds to \n\\begin{equation}\nV\\left( \\varphi \\left( \\varrho =0\\right) \\right) =-\\frac{\\lambda\n_{-}\\,\\varphi_{0}^{4}}{1-\\beta } \\,\\left( 1-\\frac{\\beta }{4}\\right) =-\\frac{%\n4\\,V_{bar}}{\\beta}\\, \\left( 1-\\frac{\\beta }{4}\\right) ,  \\label{24}\n\\end{equation}%\nwhere%\n\\begin{equation}\nV_{bar}=\\lambda _{+}\\,\\varphi _{0}^{4}\/4  \\label{24a}\n\\end{equation}%\nis the height of the barrier. The instanton action can be easily calculated\nand is equal to%\n\\begin{equation}\nS_{I}=\\frac{8\\,\\pi ^{2}}{3\\,\\lambda _{-}\\left( 1-\\beta \\right) ^{3}}\\,\\left(\n2-\\beta \\right)\\, (2-2\\beta +\\beta ^{2}) \\,.  \\label{25}\n\\end{equation}\nWe note that if one would decide to consider $\\lambda_-\\rightarrow \\infty$, $%\n\\beta\\rightarrow 0$ the value of the scalar field in the center of the\nbubble is equal to $\\varphi(\\varrho=0)\\rightarrow -\\varphi_0$ and $V\\left(\n\\varphi \\left( \\varrho =0\\right) \\right)\\rightarrow -\\infty$.\n\n\\textit{Instantaneous vacuum decay via small bubbles.} For a very steep\nunbounded potential shown in Fig. \\ref{Figure3} and obtained by taking the\nlimit $\\lambda _{-}\\rightarrow \\infty ,$ the bubble size given in $\\left( %\n\\ref{21}\\right) $ shrinks to zero as $\\varrho _{0}\\propto \\lambda\n_{-}^{-1\/2} $ and the action $\\left( \\ref{25}\\right) $ vanishes$.$ The false\nvacuum decay rate $\\left( \\ref{9a}\\right) $ becomes infinite irrespective\nhow high is the potential barrier $V_{bar}$. This instantaneous false vacuum\ndecay happens via infinitely small bubbles, which according to $\\left( \\ref%\n{24}\\right) $ emerge with infinitely large negative potential in the center.\nSuch a conclusion does not sound physically acceptable. Moreover, even for\nfinite but large enough $\\lambda _{-}$ the problem of small instantons still\nremains because they lead to unexpectedly efficient decay with the rate\npractically independent on the shape of the potential at positive $\\varphi $%\n. The situation starts to look even more strange if one assumes that the\npotential in Fig. \\ref{Figure3} gets a second rather sharp minimum (see\ndotted line). Then, irrespective of the depth of this second minimum, it\nlooks like the Coleman instanton ceases to exist and, thus, nearly\nirrelevant modification of the potential abruptly changes the decay rate\nfrom infinity to zero.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_3.eps}\n\\end{center}\n\\par\n\\vspace*{-4.08cm} \\hspace*{5.55cm}$V_{min}$%\n\\par\n\\vspace*{1.6cm} \\hspace*{5.32cm} -$-\\frac{128\\,V_{bar}}{\\sigma^2\\,\\lambda_+}$%\n\\par\n\\vspace*{-4.1cm} \\hspace*{5.58cm}$V_{bar}$%\n\\par\n\\vspace*{0.5cm} \\hspace*{7.6cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-2.5cm} \\hspace*{2.88cm}$(a)$%\n\\par\n\\vspace*{-0.46cm} \\hspace*{3.78cm}$(b)$%\n\\par\n\\vspace*{-0.9cm} \\hspace*{5.4cm}$V$%\n\\par\n\\vspace*{1.7cm} \\hspace*{13.0cm}$\\varphi$%\n\\par\n\\vspace*{4.0cm}\n\\caption{{}}\n\\label{Figure3}\n\\end{figure}\n\\textbf{\\ }\n\nBelow we will show how these problems are resolved if we take into account\nquantum fluctuations.\n\n\\section{Quantum Fluctuations}\n\nThe instanton is a \\textit{classical} solution and it is reliable only if\nboth the field and its derivative exceed the level of the minimal quantum\nfluctuations. The bubble emerges in Minkowski space at $\\tau =0$. At this\ninstant the distance to the center of the bubble is equal to $\\varrho $ and\nthe \\textquotedblleft typical\\textquotedblright\\ amplitude of the quantum\nfluctuations in corresponding scales is about (see for example, \\cite%\n{Mukhanov2}): \n\\begin{equation}\n\\left\\vert \\delta \\varphi _{q}\\right\\vert \\simeq \\frac{\\sigma}{\\varrho }\\,,\n\\label{27}\n\\end{equation}%\nand, respectively, its time derivative is of order%\n\\begin{equation}\n\\left\\vert \\delta \\dot{\\varphi}_{q}\\right\\vert \\simeq \\frac{\\sigma }{\\varrho\n^{2}}\\,,  \\label{28}\n\\end{equation}%\nwhere $\\sigma $ is the numerical coefficient of order unity (we set $\\hbar=1$%\n). The quantum fluctuations grow very fast as $\\varrho \\rightarrow 0$ and,\nhence, Coleman's boundary condition $\\dot{\\varphi}_{\\varrho =0}=0,$\nformulated in the deep ultraviolet limit, must be re-analyzed. The potential \n$V\\left( \\varphi \\right) $ can be arbitrarily shifted along the $\\varphi$%\n--axis. Therefore, to estimate the magnitude of quantum fluctuations it is\nmore appropriate to consider either the typical change of the classical\nfield $\\Delta \\varphi \\simeq \\dot{\\varphi}\\times \\varrho $ in scales $%\n\\varrho $ or it's time derivative. In both cases we obtain (up to a\nnumerical coefficient) the same result and to be concrete we will use $%\n\\left( \\ref{28}\\right) $ to determine when quantum fluctuations become\nrelevant. Equating the derivative of the classical solution $\\left( \\ref{19}%\n\\right) $ to the amplitude of quantum fluctuations $\\left( \\ref{28}\\right) $\nwe find the following ultraviolet cutoff scale%\n\\begin{equation}\n\\varrho _{uv}=\\left( \\frac{4\\,\\sigma }{\\lambda _{-}\\,\\varphi _{0}^{3}}%\n\\right) ^{1\/3}=\\left( \\frac{\\sigma ^{2}}{32}\\right) ^{1\/6}\\lambda\n_{-}^{1\/6}\\,\\left( 1-\\beta \\right) ^{1\/2}\\varrho _{0}\\,,  \\label{29}\n\\end{equation}%\nwhere $\\beta $ is defined in $\\left( \\ref{21a}\\right) $ and $\\varrho _{0}$\nis given in $\\left( \\ref{21}\\right) .$ The classical solution with the\nColeman boundary condition $\\left( \\ref{7}\\right) $ is obviously valid only\nfor $\\varrho >\\varrho _{uv}$ and it is completely spoiled by quantum\nfluctuations on scales smaller than $\\varrho _{uv}$. Let us notice that for\nthe very steep potentials with $\\lambda _{-}\\gg 1$ the ultraviolet cutoff\nscale exceeds the size of the bubble and therefore the instantons which lead\nto the large decay rate cannot be trusted anymore.\n\nThe ultraviolet cutoff $\\varrho _{uv}$ resulted from the fact that the\nquantum fluctuations are characterized by the field derivative of order $%\n1\/\\varrho ^{2}$ for all values of $\\varrho $ while the contribution of the\nclassical solution at small $\\varrho $ decreases as $\\varrho $. Thus, for\nsmall enough $\\varrho $ the quantum fluctuations dominate. For the large\nvalues of $\\varrho $ the quantum fluctuations continue to exhibit a $%\n1\/\\varrho ^{2}$ decay while the classical solution decreases as $1\/\\varrho\n^{3}$. Thus, the quantum fluctuations take over both in the deep\nultra-violet and the large infrared scales carving a range of values of $%\n\\varrho $ for which the classical solutions can be valid.\n\nNext we estimate the infrared cutoff scale above which (in length) quantum\nfluctuations dominate. To find this scale we have to equate the derivative\nof solution $\\left( \\ref{16}\\right) ,$ valid for $\\varrho >\\varrho _{0},$ to \n$\\left( \\ref{28}\\right) .$ As a result one gets%\n\\begin{equation}\n\\varrho _{ir}\\simeq \\frac{2}{\\sigma }\\frac{\\delta }{1+\\delta }\\,\\varphi\n_{0}\\,\\varrho _{0}^{2}\\simeq \\frac{4\\sqrt{2}}{\\sigma }\\lambda\n_{-}^{-1\/2}\\left( 1-\\beta \\right) ^{1\/2}\\varrho _{0}\\,.  \\label{30}\n\\end{equation}%\nTo simplify the result we have assumed that $\\varrho _{ir}\\gg \\varrho _{0}.$\nAs one can check a posteriori this assumption is really valid when both $%\n\\lambda _{+}$ and $\\lambda _{-}$ are much smaller than unity.\n\nFor $\\lambda _{-}>1$ we have $\\varrho _{ir}<\\varrho _{0}$ and moreover $%\n\\varrho _{uv}>\\varrho _{ir}$. Thus, the range \n\\begin{equation}\n\\varrho _{uv}<\\varrho <\\varrho _{ir}  \\label{31}\n\\end{equation}%\nwhere the classical instanton solution is supposed to be valid completely\ndisappears. This tells us that for $\\lambda _{-}>1$ the classical instanton\nwith the Coleman boundary condition can not be applied.\n\nConcluding this section we would like to stress that both the \\textit{%\nultraviolet and infrared cutoff scales} \\textit{are entirely determined by\nthe parameters characterizing the corresponding classical solution.}\n\n\\section{New Instantons}\n\n\\label{section5}\n\nThe existence of the cutoff scales allows us to obtain a new class of $%\nO\\left( 4\\right) $ instantons, which otherwise would be singular and having\nan infinite action would not contribute to the decay rate. Let us first\nconsider the new solutions emerging thanks to the existence of the\nultraviolet cutoff scale. Later on we will estimate how the infrared cutoff\ncorrections influence these new solutions. Thus, we will assume that for $%\n\\varrho >\\varrho _{0}$ solution $\\left( \\ref{16}\\right) -\\left( \\ref{17}%\n\\right) $ is still valid, but abandon the boundary condition $\\dot{\\varphi}%\n_{\\varrho =0}=0.$ Then the most general solution of equation $\\left( \\ref{18}%\n\\right) $, which vanishes at $\\varrho _{0}=0$ and valid for $\\varphi <0$ is%\n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\frac{\\lambda_{-}\\,\\varphi _{0}^{3}}{8}\\left(\n\\varrho ^{2}-\\varrho _{0}^{2}\\right) \\left( 1-\\frac{\\varrho _{1}^{2}}{%\n\\varrho ^{2}}\\right) ,  \\label{32}\n\\end{equation}%\nwhere the constant of integration $\\varrho _{1}<\\varrho _{0}$ parametrizes\nour new instantons. These instantons are singular at $\\varrho \\rightarrow 0.$\nHowever, as we have shown above the solution $\\left( \\ref{32}\\right) $ can\nonly be trusted for $\\varrho >\\varrho _{uv}$. This allows us to avoid a\nsingularity in the classical solution. When $\\varrho _{1}\\neq 0$ there is a\nbounce at \n\\begin{equation}\n\\varrho _{b}=\\sqrt{\\varrho_{0}\\,\\varrho _{1}}\\,,  \\label{33}\n\\end{equation}%\nwhere $\\dot{\\varphi}(\\varrho _{b})=0,$ and after that the field $\\varphi $\nmust go back and vanish at $\\varrho _{1},$ which is smaller than $\\varrho\n_{b}.$ However, before this happens the quantum fluctuations become relevant\nat $\\varrho _{uv}>\\varrho _{b}$ and as we will show the classical bubble\nwith a quantum core emerges in Minkowski space$.$ To calculate $\\varrho\n_{uv} $ we equate the derivative of $\\left( \\ref{32}\\right) $ to $\\left( \\ref%\n{28}\\right) $ and obtain the following equation $\\qquad $\n\\begin{equation}\n1-\\frac{\\varrho _{b}^{4}}{\\varrho _{uv}^{4}} =\\frac{4\\,\n\\sigma}{\\lambda_{-}\\varphi _{0}^{3}}\\,\\frac{1}{\\varrho_{uv}^{3}}\\,,  \\label{34}\n\\end{equation}%\nwhich can be solved exactly for $\\varrho _{uv}$. Because we will not need\nthe explicit solution for $\\varrho _{uv}$ we skip it here, but instead let\nus \\ rewrite equation $\\left( \\ref{34}\\right) $ in more convenient form as%\n\\begin{equation}\n\\frac{\\varrho _{uv}^{2}-\\varrho _{b}^{2}}{\\varrho _{uv}^{2}}=\\nu \\left( \n\\frac{\\bar{\\varrho}}{\\varrho _{uv}}\\right) ^{3},  \\label{35}\n\\end{equation}%\nwhere%\n\\begin{equation}\n\\bar{\\varrho}^{2}\\equiv \\frac{8}{\\varphi_{0}^{2}\\,\\lambda _{-}\\,\\left( 1-\\beta\n\\right) }  \\label{35a}\n\\end{equation}%\nand \n\\begin{equation}\n\\nu \\equiv \\frac{\\sigma }{4\\,\\sqrt{2}\\,\\kappa\\, \\left( \\varrho _{uv}\\right) }%\n\\,\\lambda_{-}^{1\/2}\\,\\left( 1-\\beta \\right) ^{3\/2}.  \\label{36}\n\\end{equation}%\nHere, $\\kappa \\left( \\varrho _{uv}\\right) =1+\\varrho _{b}^{2}\/\\varrho_{uv}^{2} $ and since $\\varrho _{b}^{2}\/\\varrho _{uv}^{2}<1$ it\nimplies that $1<\\kappa \\left( \\varrho _{uv}\\right) <2.$ The scale $\\bar{%\n\\varrho}$ entering in $\\left( \\ref{35}\\right) $ is equal to the bubble size $%\n\\varrho _{0}$ (see (\\ref{21})) only for $\\varrho _{1}=0,$ corresponding to\nthe Coleman instanton. To determine $\\varrho _{0}$ in general case we require\nthat $\\dot{\\varphi}(\\varrho )$ is continuous at $\\varrho _{0}$ and equating\nthe derivatives of $\\left( \\ref{16}\\right) $ and $\\left( \\ref{32}\\right) $\nat this point we obtain the following equation \n\\begin{equation}\n1+\\frac{1}{\\delta }=\\frac{\\lambda _{-}\\,\\varphi_{0}^{2}}{8}\\,(\\varrho_{0}^{2}-\\varrho _{1}^{2})\\,,  \\label{38}\n\\end{equation}%\nwhere $\\delta $ is given in $\\left( \\ref{17}\\right) .$ Solving this equation\nfor $\\varrho _{0}^{2}$ one gets \n\\begin{equation}\n\\varrho _{0}^{2}=\\frac{1}{2}\\left( 1+\\sqrt{1+4\\,\\beta \\,\\frac{\\varrho _{1}^{2}}{%\n\\bar{\\varrho}^{2}}}~\\right) \\bar{\\varrho}^{2}+\\varrho_{1}^{2}\\,.  \\label{39}\n\\end{equation}\n\nThus we have found the new instantons with finite action, which are\nparametrized by $\\varrho _{1}$. The main contribution to the decay rate\ncomes from those instantons which have the minimal action while describing\nthe required transition. Let us stress that $\\varrho _{1}^{2}$ must be\npositive because otherwise the growing mode $\\varphi \\propto 1\/\\varrho ^{2}$\nwould be increasing faster than the quantum fluctuations as $\\varrho\n\\rightarrow 0$ and we would end up at a singularity, where $\\varphi\n\\rightarrow -\\infty .$ For positive $\\varrho _{1}^{2}$ the classical field $%\n\\varphi $ bounces at $\\varrho _{b}$ and evolves towards the positive values.\nHowever, as we have noticed above, before the bounce is reached the\nclassical solution fails at $\\varrho _{uv}>\\varrho _{b}$ and the bubble\nemerges from under the barrier materializing in Minkowski space.\n\nTo simplify the formulae it is convenient to parametrize the instantons by\nthe dimensionless parameter $\\chi \\left( \\varrho _{1}\\right) ,$ related to $%\n\\varrho _{1}^{2}$ as%\n\\begin{equation}\n\\varrho _{1}^{2}=\\chi \\left( 1+\\beta \\,\\chi \\right) \\bar{\\varrho}^{2}\\,,\n\\label{40}\n\\end{equation}%\nwhere $\\bar{\\varrho}^{2}$ is defined in $\\left( \\ref{35a}\\right) $. Then, as\nit follows from $\\left( \\ref{39}\\right) $ \n\\begin{equation}\n\\varrho _{0}^{2}=\\left( 1+\\chi \\right) \\left( 1+\\beta\\, \\chi \\right) \\bar{%\n\\varrho}^{2}  \\label{42}\n\\end{equation}%\nand%\n\\begin{equation}\n\\varrho _{b}^{2}=\\varrho _{0}\\,\\varrho _{1}=\\left( 1+\\beta\\, \\chi\\, \\right) \\sqrt{%\n\\chi \\,(1+\\chi )}\\,\\bar{\\varrho}^{2}\\,.  \\label{43}\n\\end{equation}%\nThe expression $\\left( \\ref{32}\\right) $ for $\\varphi \\left( \\varrho \\right) \n$ can be rewritten in the following more convenient form: \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =-\\frac{\\varphi _{0}}{1-\\beta }\\left( \\frac{%\n\\left( \\varrho _{0}-\\varrho _{1}\\right) ^{2}}{\\bar{\\varrho}^{2}}-\\frac{%\n\\left( \\varrho ^{2}-\\varrho _{b}^{2}\\right) ^{2}}{\\bar{\\varrho}^{2}\\varrho\n^{2}}\\right)  \\label{44a}\n\\end{equation}%\nand using the formulae above as well as equation $\\left( \\ref{35}\\right) $\nwe obtain%\n\\begin{equation}\n\\varphi _{uv}\\equiv \\varphi \\left( \\varrho _{uv}\\right) =-\\frac{\\varphi_{0}%\n}{1-\\beta }\\left( \\frac{1+\\beta \\,\\chi }{\\left( \\sqrt{1+\\chi }+\\sqrt{\\chi }%\n\\right) ^{2}}-\\nu ^{2}\\left( \\frac{\\bar{\\varrho}}{\\varrho_{uv}}\\right)^{4}\\right) \\,.  \\label{45}\n\\end{equation}%\nThe value of the potential at $\\varphi _{uv}$ is \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =V\\left( \\varphi _{uv}\\right) =-\\frac{4V_{bar}}{%\n\\beta }\\left( \\frac{1+\\beta\\, \\chi }{\\left( \\sqrt{1+\\chi }+\\sqrt{\\chi }\\right)^{2}}\n-\\frac{\\beta }{4}-\\nu^{2}\\,\\left( \\frac{\\bar{\\varrho}}{\\varrho _{uv}}%\n\\right)^{4}\\right) \\,.  \\label{46}\n\\end{equation}%\nAssuming that $\\lambda _{-}\\ll 1$ we find that when $\\chi $ varies from zero\nto infinity $V_{uv}$ changes within the range \n\\begin{equation}\n-\\frac{4V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4}-\\nu ^{2\/3}\\right)\n<V_{uv}\\left( \\chi \\right) <0\\,,  \\label{46a}\n\\end{equation}%\nvanishing at $\\chi \\rightarrow \\infty $. The last term in the brackets is\ndue to the quantum corrections and it is much smaller than unity for $%\n\\lambda _{-}\\ll 1$ (the case of $\\lambda _{-}>1$ will be considered\nseparately).\n\nThus, we conclude that as a result of quantum regularization there appears a\nwhole class of new nonsingular instantons which all contribute to the decay\nof the false vacuum. Depending on $\\chi $, which parametrizes these\ninstantons, the value of the potential in the central part of the bubble\ndominated by quantum fluctuations, after subtracting the energy of zero\npoint fluctuations, is equal to $V_{uv}$ and takes its value within the\ninterval $\\left( \\ref{46a}\\right) .$ If there exists a second true vacuum\nwith $V_{\\min }$ in the range $\\left( \\ref{46a}\\right) $ (see curve (b) in\nFig. \\ref{Figure2}), we expect that the dominant contribution to the false\nvacuum decay is given by the instanton with $\\chi $ determined by equation $%\nV_{uv}\\left( \\chi \\right) \\simeq V_{\\min }.$\n\nTo verify that the new instanton really emerges from under the potential\nbarrier in Minkowski space at $\\tau =0$ we have to check that%\n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =4\\,\\pi \\int_{\\varrho _{uv}}^{+\\infty\n}d\\varrho \\,\\varrho ^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}%\n^{2}\\,+\\,V(\\varphi )\\right) +\\frac{4\\,\\pi }{3}\\,\\varrho_{uv}^{3}V_{uv},\n\\label{47}\n\\end{equation}%\nvanishes up to a possible quantum correction not exceeding the contribution\nof a single quantum. The second term in $\\left( \\ref{47}\\right) $ accounts\nfor the shifted energy inside the bubble quantum core, which remains after\nnormalizing the energy of quantum fluctuations to zero in the false vacuum\nstate. Substituting solutions $\\left( \\ref{16}\\right) $ and $\\left( \\ref{32}%\n\\right) $ in $\\left( \\ref{47}\\right) $ we obtain%\n\\begin{eqnarray}\n&& 4\\,\\pi \\int_{\\varrho _{uv}}^{+\\infty }d\\varrho \\,\\varrho ^{2}\\,\\left( \\frac{1%\n}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right)  \\notag \\\\\n&=&\\frac{\\pi\\, \\lambda_{-}^{2}\\,\\varphi _{0}^{6}}{24}\\left[ \\frac{\\left(\n\\varrho _{uv}^{4}-\\varrho _{b}^{4}\\right) ^{2}}{\\varrho _{uv}^{3}}-4\\varrho\n_{uv}\\left( \\left( \\varrho _{0}^{2}-\\varrho _{uv}^{2}\\right) \\left( \\varrho\n_{1}^{2}-\\varrho _{uv}^{2}\\right) +\\frac{2\\lambda _{+}}{\\lambda\n_{-}^{2}\\varphi _{0}^{2}}\\varrho _{uv}^{2}\\right) \\right]  \\notag \\\\\n&=&\\frac{4\\,\\pi }{3}\\varrho_{uv}^{3}\\left( \\frac{1}{2}\\left( \\dot{\\varphi}%\n\\left( \\varrho _{uv}\\right) \\right) ^{2}-V\\left( \\varphi \\left( \\varrho\n_{uv}\\right) \\right) \\right) ,  \\label{48}\n\\end{eqnarray}%\nand hence%\n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =\\frac{2\\,\\pi }{3}\\varrho _{uv}^{3}\\left( \n\\dot{\\varphi}\\left( \\varrho _{uv}\\right) \\right) ^{2}=\\frac{2\\,\\pi\\, \\sigma ^{2}%\n}{3\\,\\varrho _{uv}}\\,,  \\label{49}\n\\end{equation}%\nwhich corresponds to one quantum of energy in scales $\\varrho _{uv}.$ Thus,\nthe energy balance is satisfied within an accuracy dictated by the\ntime-energy uncertainty relation and the bubble emerges from under the\nbarrier.\n\nTo determine the decay rate we calculate the action%\n\\begin{equation}\nS_{I}\\,=\\,2\\pi ^{2}\\,\\int_{\\varrho _{uv}}^{+\\infty }d\\varrho \\,\n\\varrho^{3}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) +%\n\\frac{\\pi ^{2}}{2}\\varrho _{uv}^{4}V_{uv}\\,,  \\label{50}\n\\end{equation}%\nwhere the last term accounts for the contribution of the bubble quantum\ncore. The result is%\n\\begin{eqnarray}\nS_{I}\\, &=&\\frac{\\pi \\,\\varphi_{0}^{2}}{96}\\left[\\lambda_{-}^{2}\\,\\varphi_{0}^{4}\\,\n\\left( \\varrho_{uv}^{6}+\\frac{3\\varrho_{b}^{8}}{\\varrho _{uv}^{2}}%\n-\\varrho_{0}^{6}-\\frac{3\\,\\varrho_{b}^{8}}{\\varrho_{0}^{2}}\\right) +\\frac{%\n32\\,\\left( \\lambda _{-}\\varphi _{0}^{2}\\left( \\varrho _{0}^{2}-\\varrho\n_{1}^{2}\\right) -8\\right) }{\\lambda _{+}\\varphi _{0}^{2}}\\right.  \\notag \\\\\n&&\\left. +\\left( 16\\,\\lambda _{-}\\,\\varphi _{0}^{2}\\left( \\left( 1+\\frac{%\n3\\,\\lambda _{+}}{4\\lambda_{-}}\\right) \\varrho _{0}^{2}\n-\\varrho_{1}^{2}\\right) \\right) \\varrho_{0}^{2}+32\\,\\varrho_{0}^{2}\\right]\\,.\n\\label{51}\n\\end{eqnarray}%\nSubstituting here the expression for $\\varrho _{0},\\varrho _{1}$ and $%\n\\varrho _{b}$ from $\\left( \\ref{40}\\right) -\\left( \\ref{43}\\right) $ and\nusing equation $\\left( \\ref{35}\\right) $ for $\\varrho _{uv}$ this action can\nbe represented in the following form:%\n\\begin{eqnarray}\nS_I &=&\\frac{8\\pi ^{2}}{3\\lambda _{-}\\left( 1-\\beta \\right) ^{3}}\\Bigg[ %\n\\left( 2-\\beta \\right) (2-2\\beta +\\beta ^{2})+\\frac{4\\,\\chi^{3\/2}\\left(\n4+3\\,\\chi \\right) \\,\\left( 1+\\beta \\,\\chi \\right) ^{3}}{ 2\\,\\left( 1+\\chi\n\\right) ^{3\/2}+\\chi^{1\/2}\\left( 3+2\\,\\chi \\right) }  \\notag \\\\\n&-&\\beta\\, \\chi^{2}\\left( 1+2\\,\\left( 1+\\beta \\,\\chi \\right) +3\\left( 1+\\beta\\,\n\\chi \\right)^{2}\\right) \\Bigg] +\\frac{\\pi^{2}\\,\\sigma ^{2}}{6}\\left(\n1+2\\,\\left( \\frac{\\varrho_{b}^{2}}{\\varrho_{uv}^{2}+\\varrho_{b}^{2}}\\right)\n^{2}\\right) \\,.  \\label{52}\n\\end{eqnarray}%\nThe last term here is always of order unity and can be neglected.\n\n\\textbf{Remarks on infrared cutoff}\\textit{. } Considering instantons we\nhave ignored the infrared cutoff. Let us estimate how the results above are\nchanged if we take it into account. For $\\chi \\neq 0$ the expression $\\left( %\n\\ref{30}\\right) $ for $\\varrho _{ir}$ is modified as \n\\begin{equation}\n\\varrho _{ir}\\simeq \\frac{4\\sqrt{2}}{\\sigma }\\lambda _{-}^{-1\/2}\\left(\n1-\\beta \\right) ^{1\/2}\\left( \\frac{1+\\chi }{1+\\beta \\chi }\\right)\n^{1\/2}\\varrho _{0}\\,.  \\label{53}\n\\end{equation}%\nIf both $\\lambda _{+},\\lambda _{-}\\ll 1$ the bubble size $\\varrho _{0}$ is\nalways much smaller than $\\varrho _{ir}$ and the infrared effects do not\ninfluence much the instantons for any $\\chi $. However, if $\\lambda _{-}\\gg\n1,$ it follows from $\\left( \\ref{53}\\right) $ that only if \n\\begin{equation}\n\\chi \\gg \\chi _{\\min }=4\\,\\nu ^{2}\\,,  \\label{54}\n\\end{equation}%\nwhere $\\nu $ is defined in $\\left( \\ref{36}\\right) ,$ we have $\\varrho\n_{ir}\\gg \\varrho _{0},$ otherwise $\\varrho _{ir}$ can becomes even smaller\nthan $\\varrho _{uv}$ and, hence, the classical solutions with $\\chi <\\chi\n_{\\min }$ do not make sense. Thus, for the steep potentials with $\\lambda\n_{-}\\gg 1$ the value of $\\chi $ must always exceed $\\chi _{\\min }\\approx\n\\lambda _{-}.$\n\nCalculating the corrections to the potential energy $\\left( \\ref{49}\\right) $\nand action (\\ref{50}) due to the infrared cutoff, we find that%\n\\begin{equation}\n\\Delta \\mathcal{V}_{ir}=-4\\pi \\int_{\\varrho _{ir}}^{+\\infty }d\\varrho \\,\n\\varrho^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) =%\n\\frac{2\\,\\pi \\,\\sigma ^{2}}{3}\\left( 1+\\frac{\\sigma^{2}\\,\\lambda_{+}}{60}\\right) \n\\frac{1}{\\varrho _{ir}}  \\label{55}\n\\end{equation}%\nand \n\\begin{equation}\n\\Delta S_{I}=-\\frac{\\pi ^{2}\\,\\sigma^{2}}{32}\\left( 1+\\frac{\\sigma^{2}\\,\\lambda _{+}}{4}\\right) \\,,  \\label{56}\n\\end{equation}%\nrespectively. If we would decide to normalize $V\\left( \\varphi \\left(\n\\varrho _{ir}\\right) \\right) $ to zero we would have to subtract from the\naction \n\\begin{equation}\n\\Delta S=-\\frac{\\pi ^{2}\\,\\sigma^{4}\\,\\lambda_{+}}{128} \\,.  \\label{57}\n\\end{equation}%\nThus, as it follows from $\\left( \\ref{55}\\right) -\\left( \\ref{57}\\right) $\nall infrared corrections are at the level of one quantum with the frequency\ncorresponding to the infrared cutoff scale. Hence, except the important\nbound $\\left( \\ref{54}\\right)$ which has to be respected for $\\lambda\n_{-}\\gg 1$, the quantum infrared corrections can be ignored.\n\n\\section{Implications}\n\nWe will now apply the results obtained above in several limiting cases. We\nconsider separately the instantons with $\\chi \\ll 1$ and $\\chi \\gg 1.$ The\naction $\\left( \\ref{52}\\right) $ simplifies to \n\\begin{equation}\nS_{I}\\left( \\chi \\right) =\\frac{8\\,\\pi ^{2}}{3\\,\\lambda _{-}\\left( 1-\\beta\n\\right) ^{3}}\\left[ \\left( 2-\\beta \\right) (2-2\\,\\beta +\\beta^{2})\n+8\\,\\chi^{3\/2}+O\\left(\\chi ^{2}\\right) \\right]  \\label{57a}\n\\end{equation}%\nfor $\\chi \\ll 1$ and up to corrections of order $\\chi ^{3\/2}$ coincides with\nthe Coleman action, while for $\\chi \\gg 1$ it becomes%\n\\begin{equation}\nS_{I}=\\frac{8\\,\\pi^{2}\\,\\chi }{\\lambda_{-}\\left(1-\\beta \\right) ^{3}}\\,\n\\left[\\frac{1}{3}\\left( 1-\\frac{\\beta}{2}\\right) \\left(\\beta \\,\\chi \\right)\n^{2}+\\left( 1-\\frac{\\beta}{4}\\right)^{2}\\,\\beta\\, \\chi +\\left( 1-\\frac{\\beta }{%\n4}\\right) \\left( 1-\\frac{\\beta }{4}+\\frac{\\beta ^{2}}{8}\\right) +O\\left( \n\\frac{1}{\\chi }\\right) \\right] .~  \\label{57b}\n\\end{equation}%\nLet us recall that in order to ignore the quantum loop corrections $\\lambda\n_{+}$ must always be smaller than unity, while $\\lambda _{-}$ can be\n large and therefore we investigate the potentials with $\\lambda\n_{-}\\ll 1$ and $\\lambda _{-}\\gg 1$ separately.\n\n\\textbf{A)} For $\\lambda _{-}\\ll 1$ the parameter $\\nu $ defined in $\\left( %\n\\ref{36}\\right) $ is much smaller than unity and thus from $\\left( \\ref{46}%\n\\right) $ one obtains \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =-\\frac{4\\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4}%\n-2\\,\\sqrt{\\chi }-\\nu^{2\/3}+O\\left( \\chi \\right) \\right)  \\label{58}\n\\end{equation}%\nfor $\\chi \\ll 1.$ These instantons with the action $\\left( \\ref{57a}\\right) $\ngive the main contribution to the decay rate for unbounded potential and for\nthe potentials with the second true minimum of depth $V_{\\min }<V_{uv}\\left(\n\\chi =0\\right) .$ They are different from the Coleman instanton ($\\chi =0)$ only\nby higher order corrections. As it follows from $\\left( \\ref{35}\\right) $\nthe size of quantum core%\n\\begin{equation}\n\\varrho _{uv}\\simeq \\left( \\frac{\\sigma ^{2}}{32}\\right) ^{1\/6}\\,\n\\lambda_{-}^{1\/6}\\,\\left( 1-\\beta \\right)^{1\/2}\\varrho _{0}  \\label{60}\n\\end{equation}%\nis much smaller than the bubble size \n\\begin{equation}\n\\varrho _{0}=\\left( 1+\\frac{1+\\beta}{2}\\,\\chi +O\\left( \\chi ^{2}\\right)\n\\right) \\bar{\\varrho}\\,.  \\label{60a}\n\\end{equation}%\nTo calculate the contribution of instantons with the value of the potential\nin the quantum core%\n\\begin{equation}\nV_{uv}\\left( \\chi _{\\varepsilon }\\right) =\\varepsilon \\,V_{uv}\\,\n\\left(\\chi=0\\right) \\simeq -\\frac{4\\,\\varepsilon \\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4%\n}\\right) \\,,  \\label{60b}\n\\end{equation}%\nwhere $\\varepsilon \\ll 1,$ to the decay rate we have to consider solutions\nwith $\\chi \\gg 1.$ In this case the expression $\\left( \\ref{46}\\right) $\nsimplifies to%\n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =-\\frac{V_{bar}}{\\beta }\\left( \\left( 1-\\frac{%\n\\beta }{2}\\right) \\frac{1}{\\chi }+O\\left( \\frac{1}{\\chi ^{2}}\\right) \\right)\n,  \\label{63}\n\\end{equation}%\nand as it follows from $\\left( \\ref{60b}\\right) $ \n\\begin{equation}\n\\chi _{\\varepsilon }\\simeq \\frac{1}{2}\\left( \\frac{2-\\beta }{4-\\beta }%\n\\right) \\frac{1}{\\varepsilon }\\gg 1\\,.  \\label{64}\n\\end{equation}%\nThe expressions for $\\varrho _{0}^{2}$ becomes%\n\\begin{equation}\n\\varrho _{0}^{2}\\left(\\chi_{\\varepsilon }\\right) \\simeq \\frac{4}{\\left(\n1-\\beta \\right) }\\left( \\frac{2-\\beta }{4-\\beta }\\right) ^{2}\\frac{1}{%\n\\varepsilon \\lambda _{-}\\varphi _{0}^{2}}\\left( \\left( \\frac{4-\\beta }{%\n2-\\beta }\\right) +\\frac{1}{2}\\,\\frac{\\beta }{\\varepsilon }\\right) ,  \\label{65}\n\\end{equation}%\nand the action $\\left( \\ref{57b}\\right) $ is equal to%\n\\begin{equation}\nS_{\\varepsilon }=\\frac{\\pi ^{2}\\left( 2-\\beta \\right) }{\\varepsilon \\,\\lambda\n_{-}\\left( 1-\\beta \\right) ^{3}}\\left[ \\frac{1}{6}\\left( \\frac{2-\\beta }{%\n4-\\beta }\\right) ^{3}\\frac{\\beta^{2}}{\\varepsilon^{2}}+\\frac{\\left(\n2-\\beta \\right) }{8}\\frac{\\beta }{\\varepsilon }+\\left( 1-\\frac{\\beta }{4}+%\n\\frac{\\beta ^{2}}{8}\\right) +O\\left( \\varepsilon \\right) \\right] \\,.\n\\label{67}\n\\end{equation}%\nThe contribution of these $\\chi _{\\varepsilon }$-instantons to the decay\nrate is given by%\n\\begin{equation}\n\\Gamma \\sim \\varrho _{0}^{-4}\\left( \\chi _{\\varepsilon }\\right) \\exp \\left(\n-S_{\\varepsilon }\\right) \\,.  \\label{67a}\n\\end{equation}%\nFor the unbounded potential the instantons with $\\chi \\ll 1$ give the main\ncontribution to the overall decay rate. However, even in this case the whole\nspectrum of instantons is present when we consider the false vacuum decay.\nThe formula $\\left( \\ref{67a}\\right) $ can also be applied to estimate the\nleading contribution to the decay rate for a broad class of potentials with\ntwo minima in the case when the potential is well approximated by the linear\npotential nearly up to the second true minimum of depth\n\\begin{equation}\nV_{\\min }\\simeq -\\frac{4\\,\\varepsilon \\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta \n}{4}-\\nu ^{2\/3}\\right)\n\\end{equation}%\n(see, for example, curve (b) in Fig. \\ref{Figure2}).\n\nThe expression $\\left( \\ref{17}\\right) $ for the parameter $\\delta ,$\ncharacterizing the thickness of the bubble wall for $\\chi \\neq 0,$ is\nmodified as \n\\begin{equation}\n\\delta =\\frac{1-\\beta }{\\beta \\left( 1+\\chi \\right) },  \\label{70a}\n\\end{equation}%\nand hence the bubble has a thin wall $\\delta \\ll 1$ for all $\\chi $ if $%\n\\lambda _{-}\\ll \\lambda _{+}$ $\\left( 1-\\beta \\ll 1\\right) .$ In the case $%\n\\lambda _{+}\\ll \\lambda _{-}<1$ only bubbles with $\\chi _{\\varepsilon }\\sim\n\\varepsilon ^{-1}\\gg \\lambda _{-}\/\\lambda _{+}$ have a thin wall.\n\nWhen we have two nearly degenerate minima, that is $\\varepsilon \\rightarrow\n0 $ ($\\varepsilon \\ll \\beta $), the probability of the vacuum decay, for\nexample, for $\\beta \\ll 1$ vanishes as \n\\begin{equation}\n\\Gamma \\sim4\\, \\lambda _{+}^{2}\\left( \\varphi _{0}\\frac{\\varepsilon }{\\beta }%\n\\right) ^{4}\\exp \\left[ -\\frac{\\pi ^{2}}{24\\lambda _{+}}\\left( \\frac{\\beta }{%\n\\varepsilon }\\right) ^{3}\\right]  \\label{70b}\n\\end{equation}%\nin complete agreement with our expectations.\n\n\\textbf{B) }$\\lambda _{-}\\gg 1$ (\\textit{zero size instanton problem}).\nFinally let us consider the most interesting case of a very steep unbounded\npotential (see Fig. \\ref{Figure3} where $\\lambda _{-}\\rightarrow \\infty $).\nSince $\\lambda _{+}$ is always smaller than unity $\\beta \\ll 1$ and the\nparameter $\\nu ^{2}$ in $\\left( \\ref{36}\\right) $ is \n\\begin{equation}\n\\nu ^{2}\\simeq \\frac{\\sigma ^{2}}{128}\\,\\lambda _{-}\\gg 1\\,.  \\label{77}\n\\end{equation}%\nAs one can find by inspecting $\\left( \\ref{45}\\right) $ and $\\left( \\ref{46}%\n\\right) ,$ in this case the value of $\\chi $ must be larger than unity\nbecause otherwise the quantum fluctuations dominate over the classical\ninstanton solution everywhere. Therefore, the Coleman instanton ($\\chi =0$)\nis never trustable$.$ The solution of equation $\\left( \\ref{35}\\right) $ is \n\\begin{equation}\n\\varrho _{uv}=\\varrho _{b}\\left( 1+\\frac{1}{2}\\frac{\\nu }{\\left( \\left(\n1+\\beta \\chi \\right) \\chi \\right) ^{3\/2}}+O\\left( \\frac{\\nu ^{2}}{\\chi ^{3}}%\n\\right) \\right)  \\label{77a}\n\\end{equation}%\nand to the leading order the equations $\\left( \\ref{45}\\right) $ and $\\left( %\n\\ref{46}\\right) $ simplify to%\n\\begin{equation}\n\\varphi_{uv}\\left(\\chi \\right) \\simeq -\\varphi_{0}\\,\\left( \\frac{1+\\beta\\,\n\\chi }{4\\,\\chi }-\\frac{\\nu ^{2}}{\\left( 1+\\beta\\, \\chi \\right)^{2}\\chi ^{2}}%\n\\right)  \\label{78}\n\\end{equation}%\nand \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) \\simeq -\\frac{4\\,V_{bar}}{\\beta}\\left( \\frac{1}{%\n4\\,\\chi }-\\frac{\\nu^{2}}{\\left( 1+\\beta\\, \\chi \\right)^{2}\\,\\chi ^{2}}\\right) \\,.\n\\label{79}\n\\end{equation}%\nIt follows from here that only if \n\\begin{equation}\n\\chi >4\\,\\nu ^{2}  \\label{80}\n\\end{equation}%\nboth $\\varphi _{uv}$ and $V_{uv}$ are negative and hence the tunneling\nbecomes possible. This lower bound on $\\chi $ is in complete agreement with\nthe bound $\\left( \\ref{54}\\right) $ obtained by inspecting the relevance of\nthe infrared cutoff scale. Because of this bound the minimal value of the\npotential in the center of the bubble must be always larger than \n\\begin{equation}\nV_{c}\\simeq -\\frac{32\\,\\varphi_{0}^{4}}{\\sigma ^{2}}\\,.  \\label{81}\n\\end{equation}\nLet us consider the $\\chi_{\\varepsilon }$ instanton for which \n\\begin{equation}\nV_{uv}\\left( \\chi _{\\varepsilon }\\right) =\\varepsilon\\, V_{c}\\,,  \\label{82}\n\\end{equation}%\nwhere $\\varepsilon <1.$ We obtain the corresponding $\\chi _{\\varepsilon }$\nsubstituting expansion $\\left( \\ref{79}\\right) $ into (\\ref{82}), where we\nkeep only the first term,%\n\\begin{equation}\n\\chi _{\\varepsilon }\\simeq \\frac{\\sigma^{2}\\,\\lambda _{-}}{128\\,\\varepsilon }\\,.\n\\label{82aa}\n\\end{equation}%\nThe size of the bubble is given by \n\\begin{equation}\n\\varrho_{0}^{2}\\,\\varphi _{0}^{2}\\simeq \\frac{\\sigma ^{2}}{16\\,\\varepsilon }%\n\\left( 1+\\frac{\\sigma ^{2}\\,\\lambda _{+}}{128\\,\\varepsilon }+O\\left( \\frac{%\n\\varepsilon}{\\lambda_{-}}\\right) \\right) .  \\label{82bb}\n\\end{equation}%\nand this size is always larger than some minimal size, that is, \n\\begin{equation}\n\\varrho _{0}\\geq \\frac{\\sigma }{4\\,\\varphi _{0}},  \\label{82cc}\n\\end{equation}%\nirrespectively of the value of $\\lambda _{-}>1.$ This resolves the problem\nof small size instantons for the steep potentials. To calculate the action\nwe have to substitute $\\left( \\ref{82aa}\\right) $ in $\\left( \\ref{57b}%\n\\right) .$ As a result one gets%\n\\begin{equation}\nS\\left( \\chi_{\\varepsilon }\\right) \\simeq \\frac{\\pi^{2}\\sigma ^{2}}{%\n16\\,\\varepsilon }\\left[ 1+\\frac{\\sigma^{2}\\,\\lambda_{+}}{128\\,\\varepsilon }+%\n\\frac{1}{3}\\left(\\frac{\\sigma^{2}\\,\\lambda_{+}}{128\\,\\varepsilon }\\right)^{2}\n+O\\left( \\frac{\\varepsilon}{\\lambda_{-}}\\right) \\right] \\,.\n\\label{82dd}\n\\end{equation}%\nNotice that both the bubble size or the action do not depend on $\\lambda\n_{-} $ in the leading order and therefore we can take a limit $\\lambda\n_{-}\\rightarrow \\infty $ (see Fig. \\ref{Figure3}), when all $\\lambda _{-}$%\n-dependent corrections proportional to $\\varepsilon \/\\lambda _{-}$ vanish.\nAlthough all instantons contribute to the vacuum decay rate, that major\ncontribution to this rate for the unbounded potential and the potentials\nwith the second very deep minimum $V_{\\min }<V_{c}$ (see curve (a) in Fig. %\n\\ref{Figure3}) give instantons with the minimal possible size and action,\ncorresponding to $\\varepsilon \\simeq 1,$ so that%\n\\begin{equation}\n\\Gamma \\sim \\varphi_{0}^{4}\\,\\exp \\left[ -\\frac{\\pi ^{2}\\sigma ^{2}}{16}%\n\\right] \\,.  \\label{83}\n\\end{equation}%\nOne has to stress that in this case we are working on the border of\nsemiclassical approximation and the emerging bubbles contain only one\nquantum. Therefore equation $\\left( \\ref{83}\\right) $ gives only a very rough\nestimate for the probability of decay. To determine the leading contribution\nto the false vacuum decay rate in case when the second minimum of depth $%\nV_{\\min }$ is above $V_{c}$ (see curve (b) in Fig. \\ref{Figure3}) one has to\nuse the formulae $\\left( \\ref{82bb}\\right) $ and $\\left( \\ref{82dd}\\right) $\nwith $\\varepsilon $ determined by equation $V_{\\min }\\simeq \\varepsilon\nV_{c}.$\n\n\\section{Conclusions}\n\nIn this paper we have analyzed the role of quantum fluctuations for the\nclassical $O\\left( 4\\right) $ solutions which describe the false vacuum\ndecay. It was shown that the quantum fluctuations induce the ultraviolet and\ninfrared cutoff scales, beyond which the classical instanton solutions\ncannot be trusted anymore because the level of quantum fluctuations exceed\nthe contribution of the classical field. The cutoff scales are entirely\ndetermined by the parameters characterizing the corresponding classical\nsolutions. This allows one to abandon the boundary condition imposed by\nColeman on the derivative of the field at $\\rho=0$, which is outside\nthe range of validity of the approximation. The regularized solutions are\nindeed not singular. As a result there emerges the whole spectrum of new\ninstantons, which all contribute to the false vacuum decay. The largest\ncontribution comes from the instantons with the minimal possible action. In\nmany cases these instantons (up to small corrections) lead to the same\nresult as instanton with the Coleman boundary conditions. However, in several\nimportant cases the Coleman approach fails. In particular, we have shown\nthat for the very steep unbounded from below potential, shown in Fig. \\ref%\n{Figure3}, the instanton with the Coleman boundary conditions would lead to\nnearly instantaneous vacuum instability via formation of infinitely small\nbubbles. To the contrary, as the size of the new instantons always exceeds\nsome minimal scale determined by the parameters of the original potential,\nthe decay probability obtained by these new instantons remains finite even\nfor case where the unbounded potential is very steep. This removes the\ndifficulties that were associated with the application of \nsmall instantons\ufffd. We have also checked that the results in the\nmethod we suggest agrees with the standard method when they overlap.\n\n\\vspace*{1.0cm}\n\n\\textbf{Acknowledgments}\n\n\\bigskip\n\nThe work of V. M. was supported under Germany's Excellence\nStrategy---EXC-2111---Grant No. 39081486.\n\nThe work of E. R. is partially supported by the Israeli Science Foundation\nCenter of Excellence. E. R. would also like to thank NHETC at Rutgers Physics\nDepartment, the IHES and the CCPP at NYU for hospitality.\n\nA. S. would like express a gratitude to the Racah Institute of Physics of\nthe Hebrew University of Jerusalem and Ludwig Maxmillian University of\nMunich for the hospitality during his visits. The work of A. S. was\npartially supported by RFBR grant No. 20-02-00411.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAdversarial patches \\citep{brown_2017} are one of the most relevant threat models for attacks on autonomous systems such as highly automated cars or robots. In this threat model, an attacker can freely control a small subregion of the input (the ``patch'') but needs to leave the rest of the input unchanged. This threat model is relevant because it corresponds to a physically realizable attack \\citep{lee_2019}: an attacker can print the adversarial patch pattern, place it in the physical world, and it will become part of the input of any system whose field of view overlaps with the physical patch. Moreover, once an attacker has generated a successful patch pattern, this pattern can be easily shared, will be effective against all systems using the same perception component, and an attack can be conducted without requiring access to the individual system. This makes for instance attacking an entire fleet of cars of the same vendor feasible.\n\nWhile several empirical defenses were proposed \\citep{Hayes18,NaseerKP19,selvaraju_grad-cam:_2019,wu2020defending}), these only offer robustness against known attacks but not necessarily against more effective attacks that may be developed in the future \\citep{chiang2020certified}. In contrast, certified defenses for the patch threat model \\citep{chiang2020certified, levine,zhang_clipped_2020, xiang_patchguard_2020} allow guaranteed robustness against all possible attacks for the given threat model. Ideally, a certified defense should combine high certified robustness with efficient inference while maintaining strong performance on clean inputs. Moreover, the training objective should be based on the certification problem to avoid post-hoc calibration of the model for certification.\n\nExisting defenses do not satisfy all of these conditions:  \\citet{chiang2020certified} proposed an approach that extends interval-bound propagation \\citep{Gowal_2019_ICCV} to the patch threat model. In this approach, there is a clear connection between training objective and certification problem. However, certified accuracy is relatively low and clean performance severely affected (below $50\\%$ on CIFAR10). Moreover, inference requires separate forward passes for all possible patch positions and is thus computationally very expensive. Derandomized smoothing \\citep{levine} achieves much higher certified and clean performance on CIFAR10 and even scales to ImageNet. However, inference is computationally expensive since it is based on separately propagating many differently ablated versions of a single input. Moreover, training and certification are disconnected and a separate tuning of parameters of the post-hoc certification procedure on some hold-out data is required, a drawback shared also by Clipped BagNet \\citet{zhang_clipped_2020} and PatchGuard \\citep{xiang_patchguard_2020}.\n\nIn this work, we propose \\textsc{BagCert}, which combines high certified accuracy ($60\\%$ on CIFAR10 for $5\\times 5$ patches) and  clean performance ($86\\%$ on CIFAR10), efficient inference (43 seconds on a single GPU for the $10.000$ CIFAR10 test samples), and end-to-end training for robustness against patches of varying size, aspect ratio, and location. \\textsc{BagCert} is based on the following contributions:\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]\n\t\\item We propose three different conditions that can be checked for certifying robustness. One of these corresponds to the  condition proposed by \\citet{levine}. However, we show that an alternative condition improves certified accuracy of the same model typically by roughly $3$ percent points while remaining broadly applicable.\n\t\\item We derive a loss function that directly optimizes for certified accuracy against a uniform distribution of patch sizes at arbitrary positions. This loss corresponds to a specific type of the well known class of margin losses.\n\t\\item Similarly to \\citet{levine}, we classify images via a majority voting over a large number of predictions that are based on small local regions of a single input. However, the proposed model achieves this via a single forward-pass on the unmodified input, by utilizing a neural network architecture with very small receptive fields, similar to BagNets \\citep{brendel2018approximating}. This enables efficient inference with surprisingly high clean accuracy and was concurrently proposed by \\citet{zhang_clipped_2020} and \\citet{xiang_patchguard_2020}.\n\\end{itemize}\n\n\\section{Related Work}\n\n\\paragraph{Adversarial Patch Attacks}\n\nVulnerability of image classifiers to adversarial patch attacks was first demonstrated by \\citet{brown_2017}. They show that a specifically crafted physical adversarial patch is able to fool multiple ImageNet models into predicting the wrong class with high confidence. Numerous patch attacks were proposed for object detection \\citep{LiuYLSCL19, lee_2019, ThysRG19, huang2019adversarial} and optical flow estimation \\citep{ranjan2019attacking}. The versatility of these attacks allows them to perform efficiently in the black box setup \\citep{abs-2006-12834} as well as suppressing detected objects in a scene without overlapping any of them \\citep{lee_2019}. Following \\citet{athalye_synthesizing_2017}, adversarial patches can be printed out and placed in the physical world to fool different models independently from the scaling, rotation, brightness and other visual transformations. These factors make adversarial patch attacks a non-negligible threat for the safety-critical perception systems \\citep{ThysRG19}.   \n\n\\paragraph{Heuristic Defenses Against Patch Attacks}\n\nSeveral heuristic defenses against adversarial patches such as digital watermarking \\citep{Hayes18} or local gradient smoothing \\citep{NaseerKP19} have been proposed. However, similarly to the results obtained for the norm-bounded adversarial attacks \\citep{AthalyeC018}, it was demonstrated that these defenses can be easily broken by white-box attacks which account for the pre-processing steps in the optimization procedure \\citep{chiang2020certified}. The role of spatial context in the object detection algorithms which makes them vulnerable to the patch attacks was investigated by \\citet{Saha_2019} and an empirical defense based on Grad-CAM \\citep{selvaraju_grad-cam:_2019} was proposed. Existing augmentation techniques based on adding Gaussian noise patch \\citep{lopes2020improving} or a patch from a different image \\citep{Yun_2019_ICCV} increase robustness against occlusions caused by adversarial patches. \\citet{wu2020defending} propose a defense that uses adversarial training to increase robustness against occlusion attacks.\n\n\\paragraph{Certified Defenses}\n\nEvaluating defense methods using their performance against empirical attacks can lead to the false sense of security since stronger adversaries might be developed in the future that break the defenses \\citep{AthalyeC018,UesatoOKO18}. Therefore, it is important to have guarantees of robustness. Numerous works were proposed in the field of certified robustness ranging from complete verifiers finding the worst-case adversarial examples exactly \\citep{HuangKWW17, tjeng2017verifying} to faster but less accurate incomplete methods that provide an upper bound on the robust error \\citep{GehrMDTCV18, WongK18,WongSMK18,Gowal_2019_ICCV}. \nAnother line of work is based on Randomized Smoothing \\citep{8835364,NIPS2019_9143, pmlr-v97-cohen19c}, which exhibits strong empirical results and scales to ImageNet, however at the cost of increasing inference time by orders of magnitude. Certified defenses crafted for the patch attacks were first proposed by \\citet{chiang2020certified}. They adapt the IBP method \\citep{Gowal_2019_ICCV} to the patch threat model. Although their approach allows to obtain robustness guarantees, it only scales to small patches and causes a significant drop in clean accuracy. \\citet{levine} proposed (de)randomized smoothing: they train a base classifier for classifying images where all but a small local region is ablated. At inference time, many (or even all) possible ablations are classified and a majority vote determines the final classification. If this majority vote is with sufficient margin, the decision is provable robust against patch attacks because a patch will be completely ablated in most of the inputs and can thus only influence a minority of the votes. This method provides significant accuracy improvement when compared to \\citep{chiang2020certified} and allows training and certifying ImageNet models. However, its inference in block-smoothing mode is computationally expensive. A last line of work is based on using models with small receptive fields such as BagNets \\citep{brendel2018approximating}: \\citet{zhang_clipped_2020} apply a clipping function while \\citet{xiang_patchguard_2020} apply a ``detect-and-mask'' filter to the logits of pretrained BagNets before global averaging. Using small receptive fields limits the number of region scores affected by a local patches while clipping and masking ensure that few very large region scores cannot dominate the global average. We note that these approaches do not train models directly for certified robustness but rather achieve it by applying post-hoc procedures that come with additional hyperparameters that require careful tuning.\n\t\n\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{graphics\/concept}\n\t\\caption{Illustration of \\textsc{BagCert} training for a 1D input and two classes. An input $X$ is processed by region scorer $f_\\theta$, consisting of a 3-layer CNN with kernel sizes 3, 1, and 3. The resulting continuous region scores are passed through a Heaviside step function (replaced by a sigmoid in the backward pass) to obtain binary region scores $s$ for every class. The differences $\\Delta$ between true and non-true class scores are then processed by spatial aggregation $g$, in this case simply summing them via $g=g_{\\sum}$. The resulting value is maximized by passing it into margin loss $L$.}\n\t\\label{Figure:Framework}\n\\end{figure}\n\n\\section{Method}\nWe introduce \\textsc{BagCert}, a framework which consists of novel conditions for certifying robustness, a specific model architecture, and a new end-to-end training procedure. \\textsc{BagCert} allows end-to-end training of classifiers whose robustness against adversarial patch attacks can be certified efficiently. We outline our approach for the task of image classification but note that it can be extended to other tasks with grid-structured inputs. We refer to Figure \\ref{Figure:Framework} for an illustration of the training phase and to Figure \\ref{Figure:Certification} in the supplementary material for an illustration of certification of \\textsc{BagCert}.\n\n\\paragraph{Threat Model} \\label{sec:threat_model}\nWe consider a threat model in which an attacker can conduct an image-dependent patch attack. Let $x \\in [0, 1]^{w_{in}\\times h_{in} \\times c_{in}}$ be an input image of resolution $w_{in}\\times h_{in}$ with $c_{in}$ channels. Let $p$ be a patch and $l$ be a region of an image $x$ having the same size as patch $p$. We denote a set of feasible regions $l$ as $\\mathcal{L}$. For example, for a patch $p \\in [0, 1]^{n \\times c_{in}}$  consisting of $n$ pixels, $\\mathcal{L}$ could be the set of all $w_p \\times h_p$ rectangular regions $l$ of an image $x$ with $w_p\\cdot h_p=n$.  We define an operator $A$ such that $A(x, p, l)$ is the result of placing a patch $p$ onto an image $x$ over a region $l$. We assume that the attacker has white-box knowledge of the model and conducts an input-dependent attack, that is attack region $l$ and inserted patch $p$ can be chosen for every input independently.\n\n\n\\subsection{Certification} \\label{sec:certification}\nWe base our method for certification on assuming a certain structure of the classifier. More specifically, we decompose the classifier into two components:\n\\begin{itemize}\n\t\\item A region scorer $f_\\theta$ that maps from inputs $x$  to region scores $s \\in \\{0, 1\\}^{w_{out}\\times h_{out} \\times c_{out}}$, where $w_{out}\\times h_{out}$ is the output resolution, $c_{out}$ is the number of classes, and $\\theta$ are trainable parameters. Please note that we allow $\\sum_{c} s_{i,j, c} \\neq 1$. \n\t\\item A spatial aggregator $g$ that maps from region scores $s$ to (global) class scores $S \\in [0, 1]^{c_{out}}$. In this work, we restrict $g$ to be monotonically increasing, that is: for class $c$ and two patch score maps $s^{(1)}$ and $s^{(2)}$  with $s^{(1)}_{i, j, c} \\geq s^{(2)}_{i, j, c} \\,\\forall i, j$, we require $g(s^{(1)})_c \\geq g(s^{(2)})_c \\,\\forall c$.\n\\end{itemize}\n\nGenerally, we base certification on upper bounding the effect of an actual attack in the threat model. For this, we only exploit architectural properties of $f$ and $g$ that are valid for any choice of model parameters $\\theta$. More specifically, we only exploit the output dependency map $R$ of $f$, which we define as $R(l) = \\{(i, j) \\mid \\exists x, \\theta, p: f_\\theta(A(x, p, l))_{i, j} \\neq f_\\theta(x)_{i, j}\\}$. Informally, $R(l)$ is the set of all indices of the score map that can be affected by a patch applied at region $l$, for any choice of input $x$, patch $p$, and parameters $\\theta$. That is: the set of all outputs of $f_\\theta$ whose receptive fields overlap with $l$. We discuss options for $f$ and the resulting $R$ in Section \\ref{sec:model}. \n\nFor input $x$ with class label $c_t$ and $s= f_\\theta(x)$, we define the \"worst-case\" score map $s^{wc}(s, l)$ as\n$$s^{wc}_{i, j, c}(s, l) = \n\\begin{cases} s_{i, j, c} &\\text{ if } (i, j) \\notin R(l) \\\\ \n1           &\\text{ if }  (i, j) \\in R(l) \\wedge c \\neq c_t \\\\\n0           &\\text{ if }  (i, j) \\in R(l) \\wedge c = c_t \\\\\n\\end{cases}$$\nMoreover, we define $\\Delta_{i, j, c} = s_{i, j, c_t} - s_{i, j, c}$ and similarly  $\\Delta^{wc}_{i, j, c} = s^{wc}_{i, j, c_t} - s^{wc}_{i, j, c}$. It follows directly that $\\Delta^{wc}_{i, j, c} = \\Delta_{i, j, c} \\,\\forall (i,j) \\notin R(l)$ and $\\Delta^{wc}_{i, j, c} = -1 \\,\\forall (i,j) \\in R(l), c \\neq c_t$.\n\nFor certifying robustness in the threat model for input $x$ with class label $c_t$, we need to show $g(f_\\theta(A(x, p, l)))_{c_t} > g(f_\\theta(A(x, p, l)))_c \\, \\forall c \\neq c_t \\, \\forall l \\in \\mathcal{L} \\, \\forall p$. For this, it suffices to check \n\\begin{condition} \\label{expensive_condition}\n\t$g(s^{wc}(s, l))_{c_t} > g(s^{wc}(s, l))_{c} \\; \\forall c \\neq c_t, \\forall l \\in \\mathcal{L}$\n\\end{condition}\n\\begin{proof}\nConsider arbitrary $l \\in \\mathcal{L}$ and $p$ and let $s^{adv} = f_\\theta(A(x, p, l))$. With $s^{adv}_{i,j,c} \\in \\{0, 1\\}$  we obtain\\footnote{We would like to note that these are ``trivial'' lower and upper bounds for $s^{adv}$ and we see the potential to improve upon these bounds in future work, for instance by relaxing $s \\in [0, 1]^{w_{out}\\times h_{out} \\times c_{out}}$ and applying interval bound propagation \\citep{Gowal_2019_ICCV}. However, the proposed simple bounds have the advantage of not requiring additional forward passes through the model and thus being computationally efficient.}\n $$\n s^{adv}_{i,j, c}\n \\begin{cases} = s^{wc}_{i, j, c}(s, l) = s_{i, j, c}(s, l)         & \\text{ if } (i, j) \\notin R(l) \\\\ \n \t\t       \\leq s^{wc}_{i, j, c}(s, l) = 1  & \\text{ if } (i, j) \\in R(l) \\wedge c \\neq c_t \\\\\n\t\t       \\geq s^{wc}_{i, j, c}(s, l) = 0  & \\text{ if } (i, j) \\in R(l) \\wedge c = c_t \\\\\n \\end{cases}\n $$\nWith $g$ being monotonically increasing, we obtain $g(s^{adv})_{c_t} \\geq g(s^{wc}(l))_{c_t}$ and for all $c \\neq c_t$ $g(s^{wc}(l))_{c} \\geq g(s^{adv})_{c}$. Condition \\ref{expensive_condition} implies $g(s^{adv})_{c_t} > g(s^{adv})_{c} \\, \\forall c \\neq c_t$. \n\\end{proof}\n\nChecking the Condition \\ref{expensive_condition} requires one forward-pass through $f_\\theta$ to obtain $s=f_\\theta(x)$ and $\\vert \\mathcal{L}\\vert$ times the construction $s^{wc}(s, l)$ and the evaluation of $g$. We now consider a special case where this can be implemented very efficiently.\n\n\\subsubsection{Spatial Sum Aggregation}\nFor the case $g = g_{\\sum}(s) = \\sum_{i=1,j=1}^{w_{out}, h_{out}} s_{i, j}$, Condition \\ref{expensive_condition} simplifies to\n\\begin{condition} \\label{expensive_condition_sum}\n\t$\\min\\limits_{c \\neq c_t} \\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} > \\vert R(l) \\vert \\quad \\forall l \\in \\mathcal{L}$\n\\end{condition}\n\\begin{proof} For all $c \\neq c_t$, we exploit $ \\forall (i, j) \\in R(l): \\Delta^{wc}_{i, j, c} = -1$. With Condition \\ref{expensive_condition_sum}, we obtain\n\t\\begin{align*}\n\tg_{\\sum}(s^{wc}(l))_{c_t} - g_{\\sum}(s^{wc}(l))_{c} \n\t  &= \\sum_{i=1,j=1}^{w_{out}, h_{out}} s^{wc}_{i, j, c_t} -  \\sum_{i=1,j=1}^{w_{out}, h_{out}} s^{wc}_{i, j, c}\n\t  = \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta^{wc}_{i, j, c} \\\\\n \t  &= \\sum_{i,j \\notin R(l)} \\Delta^{wc}_{i, j, c} + \\sum_{i,j \\in R(l)} \\Delta^{wc}_{i, j, c} \\\\\n \t  &= \\sum_{i,j \\notin R(l)} \\Delta_{i, j, c} - \\vert R(l) \\vert > 0. \\qedhere\n   \\end{align*}\n   \\vspace*{-.5cm}\n\\end{proof}\n\nWe note that $\\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} = \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} - \\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c_t} $.  For the special case that all $R(l)$ are rectangular, \n $\\sum_{i,j \\in R(l)} \\Delta_{i, j, c}$ can be computed efficiently for all $l \\in \\mathcal{L}$ simultaneously via integral images\/summed-area tables \\citep{Crow_1984}. For instance, $R(l)$ is rectangular for $l$ being rectangular input patches and the $R$ resulting from an CNN with grid-aligned kernels.\n\nFor the case that the $R(l)$ are not all rectangular and $\\vert \\mathcal{L}\\vert$ becomes large, checking Condition \\ref{expensive_condition_sum} can become prohibitively expensive.\nFor this case, we derive a condition that corresponds to an upper bound on Condition \\ref{expensive_condition_sum} and can be evaluated in constant time with respect to $\\vert \\mathcal{L}\\vert$:\n\\begin{condition} \\label{cheap_condition}\n\t$\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} > 2 R^{max}(\\mathcal{L}) \\text{  with  } R^{max}(\\mathcal{L}) = \\max\\limits_{l \\in \\mathcal{L}}\\vert R(l) \\vert$\n\\end{condition}\n\\begin{proof}\n$\\Delta_{i, j, c} \\leq 1$ implies $\\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c} \\leq \\vert R(l) \\vert \\leq  R^{max}(\\mathcal{L})$. For all $c \\neq c_t$, using Condition \\ref{cheap_condition}:  $\\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} = \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} - \\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c_t} >  2 R^{max}(\\mathcal{L}) -  R^{max}(\\mathcal{L}) \\geq  \\vert R(l) \\vert$  \\end{proof}\n\nWe note that Condition \\ref{cheap_condition} corresponds to the condition proposed by \\citet{levine}. It is, however, a strictly weaker condition than Condition \\ref{expensive_condition_sum}. Thus, Condition \\ref{expensive_condition_sum} is preferable if all $R(l)$ are rectangular or $\\vert \\mathcal{L}\\vert$ is of moderate size. We refer to Figure \\ref{Figure:Certification} in the supplementary material for an illustration of Condition \\ref{expensive_condition_sum} and \\ref{cheap_condition}.\n\n\n\\subsection{Model} \\label{sec:model}\nCrucially, the quality of the certification depends on $R^{max}(\\mathcal{L}) = \\max_{l \\in \\mathcal{L}}\\vert R(l) \\vert$: the larger this quantity becomes, the larger the left-hand side of Condition \\ref{expensive_condition_sum} or Condition \\ref{cheap_condition} needs to be to fulfill the condition. We focus on the specific case where $f_\\theta$ is realized by a convolutional neural network (CNN). In that case, $\\vert R(l) \\vert$ is determined fully by $l$ and the receptive field of the CNN. More specifically, we obtain $R(l) = \\{(i, j) \\mid \\exists (\\tilde{i},\\tilde{j}) \\in l : \\vert i - \\tilde{i}\\vert \\leq \\left \\lfloor w_{rf}\/2 \\right \\rfloor \\wedge \\vert j - \\tilde{j}\\vert \\leq \\left \\lfloor h_{rf}\/2 \\right \\rfloor \\}$ for a receptive field size of $w_{rf} \\times h_{rf}$ and ignoring operation strides.\n\nReceptive field sizes of CNNs are determined by the shapes of the convolutional kernels as well as operation strides. We propose using standard CNN architectures such as ResNets but replacing most $3\\times 3$ convolutions by $1 \\times 1$ convolutions, using stride 1 in (nearly) all operations, and removing all dense layers. This results in a network with very small receptive field sizes and thus small $R(l)$. We note that the proposed architecture is similar to BagNets \\citep{brendel2018approximating} and using this type of model was concurrently proposed for certifying robustness against patch attacks by \\citet{zhang_clipped_2020} and \\citet{xiang_patchguard_2020}. BagNets obtain surprisingly high classification accuracy despite small receptive field sizes \\citep{brendel2018approximating}. Importantly, in contrast to BagNets, we do not apply a global average pooling on the final feature layer. This results in a dense output of shape ${w_{out}\\times h_{out} \\times c_{out}}$. The ratios $w_{in} \/ w_{out}$ and $h_{in} \/ h_{out}$ depend on the strides applied in the network and control mostly the computational overhead. We note that the cost for forward\/backward passes in BagNets are in the same order of magnitude as those of a corresponding residual network. Because of the small receptive fields of BagNets, $\\vert R(l) \\vert$ is small if $l$ is a small contiguous region of the input, such as a rectangular patch.\n\t\nWe apply a Heaviside step function $H(x) =\\begin{cases} 0, \\text{ for } x < 0 \\\\ 1, \\text{ for } x \\geq 0 \\end{cases}$ as final layer of $f_\\theta$, which ensures $f_\\theta(X) \\in \\{0, 1\\}^{w_{out}\\times h_{out} \\times c_{out}}$. Similar to clipping \\citet{zhang_clipped_2020} and masking \\citep{xiang_patchguard_2020} this also ensures that a patch cannot flip the global classification by perturbing a local score so strongly that it dominates the globally aggregated score. However, since $H$ is constant nearly everywhere, it does not provide useful gradient information and thereby precludes end-to-end training. We address this by applying a \"straight-through\" type trick \\citep{Bengio_2013} where we replace $H$ in the backward pass by its smooth approximation, the logistic sigmoid function $s(x) = \\frac{1}{1 + e^{-x}}$. That is, we use $H(x)$ in the forward pass but replace the true gradient of $H$ with $H'(x):=s'(x) = s(x)(1 - s(x))$. We explore alternatives to the Heaviside step function in Section \\ref{section:score_functions} in the appendix.\n\nWhile the proposed model computes $f_\\theta(X)$ in a single forward-pass and controls $\\vert R(l) \\vert$ indirectly via the architecture of $f$, we note that alternative models are compatible with \\textsc{BagCert}. For instance, one could compute every element of the output $s_{i, j}$ via a separate forward pass of an arbitrary model on an ablated \\citep{levine} or cropped version of the input similar to Mask-DS-ResNet \\citep{xiang_patchguard_2020}. This also ensures that a specific element of the output depends only on the cropper\/non-ablated part of the input. While these works are more flexible in terms of model architecture, they require a number forward passes proportional to the resolution of the output $s$, which would make inference (and end-to-end) training computationally much more expensive.\n\n\\subsection{End-to-End Training} \\label{sec:training}\nHaving derived conditions that can be used for certifying robustness against patch attacks in Section \\ref{sec:certification} as well as differentiable model for the region scorer $f$ in Section \\ref{sec:model}, we now define a loss function for end-to-end training. We restrict ourselves to the case of a spatial sum aggregation $g_{\\sum}$.\n\nWe recall Condition \\ref{cheap_condition}: $\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} >2 R^{max}(\\mathcal{L})$. The corresponding loss for this can be defined as $L_H(\\Delta, c_t, R^{max}) = H(\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} \\leq 2 R^{max})$, that is: the loss is 1 if there is a target class $c$ such that $ \\sum_{i=1,j=1}^{w_{out}, h_{out}}  \\Delta_{i, j, c}$ becomes smaller\/equal two times the size of the maximum affected patch score region. However, this requires choosing $\\mathcal{L}$ and the resulting $R^{max}(\\mathcal{L})$ before training, which is undesirable. Instead, we stay agnostic with respect to the specific $\\mathcal{L}$ and simply assume a uniform distribution\\footnote{We note that other choices than the uniform distribution would be an interesting direction for future work, in particular if the defender has prior knowledge about more likely patch sizes and shapes.} for $R^{max}(\\mathcal{L})$,  that is  $R^{max}(\\mathcal{L}) \\sim \\mathcal{U}(0, R)$. Here, $R$ corresponds to the maximum patch size (in region score space) we consider. This results in the loss \n\\begin{align*} L_R(\\Delta, c_t) \n&= \\int\\limits_0^R p(\\tilde{R}) L_H(\\Delta, c_t, \\tilde{R}) d\\tilde{R}\n= \\int\\limits_0^R \\frac{1}{R}H(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} \\leq 2 \\tilde{R}) d\\tilde{R} \\\\\n&= 1 - \\frac{1}{R} \\int\\limits_0^R H(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} > 2 \\tilde{R}) d\\tilde{R} \\\\\n&= 1 -\\frac{1}{R} \\min(\\frac{1}{2}\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c}, R) = 1 -\\frac{1}{2R} \\min(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c}, 2R).\n\\end{align*}\nIn practice, we minimize $\\tilde{L}_R(\\Delta, c_t) = -\\min(\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\frac{\\Delta_{i, j, c}}{w_{out} \\cdot h_{out}}, M)$ with $M=\\frac{2R}{w_{out} \\cdot h_{out}}$. This loss can be interpreted as a margin loss with margin $M$, where the margin corresponds to twice the maximum patch size in region score space against which we want to become certifiably robust. \n\n\\paragraph{One-hot penalty} While we do not strictly enforce $\\sum_{c} s_{i,j, c} = 1$, we sometimes found it beneficial to add a term in the loss that encourages $S=g_{\\sum}(s)$ being approximately ``one-hot'', that is $L_{oh}(S) = \\max_{c \\neq c_{max}} S_c - S_{c_{max}}$ with $c_{max} = \\arg\\max_c S_c$. Since $S_c \\in [0, 1]$, it holds that $L_{oh}(S) \\in [-1, 0]$ and $L_{oh}(S) = -1$ iff $S_{c_{max}} = 1$ and $S_c = 0 \\;\\forall c \\neq c_{max}$. The term $L_{oh}(S)$ prevents training from prematurely converging to a solution where $s_{i,j, c}$ is approximately constant for all $i, j, c$, which we observed otherwise for tasks with many classes (e.g. ImageNet). The total loss becomes $L_{total} = \\tilde{L}_R(\\Delta, c_t) + \\sigma L_{oh}(S)$, where $\\sigma$ controls the strength of this one-hot penalty.\n\n\n\\section{Experiments}\nWe perform an empirical evaluation of \\textsc{BagCert} on CIFAR10 \\citep{krizhevsky_cifar10_2009} and ImageNet \\citep{ILSVRC15}. We report clean and certified accuracy and compare to Interval Bound Propagation (IBP) \\citep{chiang2020certified}, Derandomized Smoothing (DS) \\citep{levine}, Clipped BagNet (CBN) \\citep{zhang_clipped_2020}, and PatchGuard \\citep{xiang_patchguard_2020}. For DS, we focus on block-smoothing and for PatchGuard, we focus on the masked BagNet (Mask-BN) because column smoothing for DS (and the derived Mask-DS for PatchGuard) perform poorly for non-square patches that are ``short-but-wide'' (see Figure \\ref{figure:patch_aspect_ratios}). We notice that column smoothing and Mask-DS perform better than column smoothing and Mask-BN against square-patches; however, there is no reason an attacker should prefer square over non-square rectangular patches. Details on the \\textsc{BagCert} model architecture and training can be found in Appendix \\ref{section:experimental_details}. Moreover, we focus on certified accuracy, a lower bound on the actual robustness of a model. Results for accuracy against a strong adversarial patch attack, corresponding to an upper bound on actual robustness, are discussed in Section \\ref{section:patch_attacks} in the appendix.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/scatter_plot}\n\t\\caption{Clean versus certified accuracy on CIFAR10 and ImageNet for \\textsc{BagCert} with different receptive fields and train margins ($M \\in \\{0.25, 0.5, 0.75, 1.0\\}$ for CIFAR10, $M=0.25$ for ImageNet) when certifying via Condition \\ref{cheap_condition} (circles) and Condition \\ref{expensive_condition_sum} (stars), same setting connected by thin line. Smaller $M$ generally corresponds to larger clean accuracy for CIFAR10. Baselines are Derandomized Smoothing (DS) \\citep{levine}, Masked BagNet (Mask-BN) and Masked DS-ResNet (Mask-DS) \\citep{xiang_patchguard_2020}, and Clipped BagNet (CBN) \\citep{zhang_clipped_2020}. Results for these baselines are taken from the respective papers.\n\t}\n\t\\label{figure:clean_vs_certified}\n\\end{figure}\n\nFigure \\ref{figure:clean_vs_certified} shows results for different methods against $5\\times 5$ patches for CIFAR10 corresponding to $2.4\\%$ of the image size and patches of $2\\%$ of the image size for ImageNet. For CIFAR10, when certifying accuracy via Condition \\ref{cheap_condition}, the Pareto frontier of \\textsc{BagCert} follows closely the one reported for DS with block smoothing and $\\theta=0.3$. This is somewhat surprising given that both model and training procedure are very different and only the condition for certifying robustness is identical. We hypothesize that both approaches have reached close to optimal Pareto frontiers when certifying robustness via  Condition \\ref{cheap_condition}. However, as Table \\ref{table:resource_requirements} shows, \\textsc{BagCert} requires (depending on its receptive field size) only between $39.0$ and $48.5$ seconds for certifying all $10.000$  test examples on a single Tesla V100 SXM2 GPU while DS with block smoothing requires $788$ seconds. \\textsc{BagCert} also clearly dominates Mask-BN and CBN, which utilize a similar model architecture, as well as IBP (not shown) which reaches $47.8\\%$ clean and $30.3\\%$ certified accuracy.  Moreover, when applying Condition \\ref{expensive_condition_sum} for certification, certified accuracy is increased by approx. 3 percent points without changes in clean accuracy or any noticeable increase in certification time. In summary, the strongest \\textsc{BagCert} model with receptive field $7\\times 7$ and margin $M=0.5$ can certify all $10.000$ test examples in $43.2$ seconds, reaching clean accuracy of $86\\%$ and certified accuracy of $60\\%$.\n\n\\begin{table}[tb]\n\t\\begin{tabular}{l|rrrrr|rr}\n\t\t\\toprule\n\t\tRF of \\textsc{BagCert}   \t\t\t           &  5   &  7   &  9   &  11  & 13 & DS (BS) & DS (CS) \\\\\n\t\t\\midrule\n\t\tCertification time (seconds)            & 39.0 & 40.6 & 43.2 & 45.9 & 48.5 & 788.0   & 28.0 \\\\\n\t\tNumber of parameters                           & 28M  & 38M  & 47M  & 57M  & 66M  & 11M    &  11M\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Certification time for 10.000 CIFAR10 test examples and number of model parameters.}\n\t\\label{table:resource_requirements}\n\\end{table}\n\nOn ImageNet, \\textsc{BagCert} also dominates all baselines in terms of certified accuracy, reaching $18.9\\%$ via Condition \\ref{cheap_condition} and $22.9\\%$ via Condition \\ref{expensive_condition_sum} for receptive field size $17$ and margin $M=0.25$. Running certification for the entire validation set of $50.000$ images takes roughly $7$ minutes.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/patch_sizes}\n\t\\caption{Certified accuracy against square patches of different sizes on CIFAR10. Shown is the performance for different receptive fields of \\textsc{BagCert} (left) and train margins (right). Lines correspond to the same model (without retraining), evaluated against patches of different size.}\n\t\\label{figure:patch_sizes}\n\\end{figure}\n\nFigure \\ref{figure:patch_sizes} shows accuracy of \\textsc{BagCert} certified via Condition \\ref{expensive_condition_sum} for square patches of different sizes on CIFAR10. Again, baselines are dominated for both $2\\times 2$ and $5 \\times 5$ patches. Moreover, a single configuration of $\\textsc{BagCert}$ with receptive field size 7 and margin $M=0.5$ performs close to optimal for all patch sizes and can certify non-trivial performance for up to $10 \\times 10$ patch size. This implies that a single model can be used for a broad range of threat models. Figure \\ref{figure:patch_aspect_ratios} shows a similar analysis for non-square patches of a total size of 24 pixels. While \\textsc{BagCert} with the same configuration as above achieves a certified accuracy of $40\\%$ or more for any patch aspect ratio, performance of DS with column smoothing varies greatly with aspect ratio. In particular, ``short-but-wide'' patches of shape $24 \\times 1$ or $12 \\times 2$ reduce certified accuracy of column smoothing close to $0\\%$. Since there is no reason to assume attackers will restrict themselves to square patches, we do not consider DS with column smoothing or Mask-DS \\citep{xiang_patchguard_2020} general patch defenses, despite good performance for square patches and efficient certification according to Table \\ref{table:resource_requirements}.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/patch_aspect_ratios}\n\t\\caption{Certified accuracy against non-square patches of total size 24 pixels on CIFAR10. Shown is the performance for different receptive fields of \\textsc{BagCert} (left) and train margins (right) compared to Derandomized Smoothing with Column-Smoothing \\citep{levine}. Lines correspond to the same model (without retraining), evaluated against patches of different aspect rations.}\n    \\label{figure:patch_aspect_ratios}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\nWe have introduced a novel framework \\textsc{BagCert} that combines efficient certification with end-to-end training for certified robustness. The main contributions are a model architecture based on a CNN with small receptive field, certification conditions that are applicable to a broad range of models, and a margin-loss based objective that is derived from the certification condition. The resulting model achieves high certified robustness against patches with a broad range of sizes, aspect ratios, and locations on CIFAR10 and ImageNet. Promising directions for future work are the exploration of other choices for the spatial aggregation function $g$ (such as ones using the ``detect-and-mask'' mechanism from PatchGuard \\citep{xiang_patchguard_2020}) and corresponding certification conditions and losses that can be used for end-to-end training. Moreover, the development of alternative choices for models with small receptive fields could be promising, such as ones based on learnable receptive fields or based on self-attention. Moreover, applying \\textsc{BagCert} to other modalities than images would be an exciting avenue.\n\n\\nocite{he_resnet_2016}\n\\nocite{ioffe_batch_2015}\n\\nocite{gastaldi_shake_2017}\n\\nocite{loshchilov_sgdr_2017}\n\\nocite{madry2018towards}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n{ Correlating various channels in dark matter indirect\ndetections is crucial to reduce the astrophysical uncertainties and\nextract information from the dark matter annihilation or decay\nprocesses.}\n\n\n{  TeV cosmic ray electrons (CRE) and neutrinos provide ideal\nprobes of potential signatures of dark matter particle annihilation\nor decay in the vicinity of the solar system. TeV electrons cool\nvery fast while propagating in the Milky Way, thus the sources of\nsuch high energy electrons must be within 1 kpc.  The direction of\nthe source is largely erased in the electron flux. On the other\nhand, the propagation of neutrinos is barely affected and provides\nexcellent directional information. Furthermore, high energy\nneutrinos and charged leptons are generically correlated with each\nother due to the electroweak symmetry. If a hint of an excess is\nsuggested in one channel, searching for the corresponding signal in the other channel can solidly test such a hypothesis.}\n\n\n\nThe CRE spectrum has recently been directly measured up to 5 TeV\nusing the spaceborne DArk Matter Particle Explorer (DAMPE;\n\\cite{2017APh....95....6C}).  The DAMPE measurement of the CRE\nspectrum has unprecedented high energy resolution, low background,\nand well controlled instrumental systematics. Although the majority\nof the spectrum can be fitted by a smoothly broken power-law model\nwith a spectral break at E$\\sim$0.9 TeV, a tentative peak at\n$\\sim$1.4 TeV in $e^+e^-$ total spectrum has been claimed\n\\cite{Ambrosi:2017wek}. The excess of $e^+e^-$ pairs at 1.4 TeV is\napproximately $2.5\\times10^{-8}~\\rm{GeV^{-1}~s^{-1}~sr^{-1}~m^{-2}}$\ncompared with the best continuum fit of the electron-positron energy\nspectrum of DAMPE \\cite{Yuan:2017ysv}. Structures around TeV are\nalso evident in the electron-positron spectrum measured by the\nCalorimetric Electron Telescope (CALET), although more statistics\nand refined data analysis are needed to reach a conclusion\n\\citep{2017PhRvL.119r1101A}.\n\nA sharp peak in the $e^+e^-$ spectrum is hard to explain with a distant source.  Electrons and positrons with TeV energy quickly lose energy via synchrotron radiation and inverse Compton scattering processes. Even if the initial spectrum is monoenergetic, propagation introduces a dispersion in the energy. If the peak is confirmed by future data, the  $1.5\\%$ energy resolution of DAMPE  in the TeV regime would require that  the source of the TeV electrons and positrons  has to be within 0.3 kpc \\cite{Yuan:2017ysv,2017arXiv171200370G,2017arXiv171200037C,2017arXiv171111579L,2017arXiv171111563D,  2017arXiv171111452C,2017arXiv171111376A,2017arXiv171111333G,2017arXiv171111182C,2017arXiv171110995F,2017arXiv171200362J,   2017arXiv171200011C, 2017arXiv171200005H, 2017arXiv171111052Z, 2017arXiv171110989Y,  2017arXiv171110996F, Yang:2017cjm}.\n\nTwo types of scenarios have been proposed to explain the DAMPE electron-positron flux. In an astrophysical scenario, an isolated young pulsar could produce such a sharp peak if it rotates relatively slowly and has a mild magnetic field (e.g., \\citep{Yuan:2017ysv, 2017arXiv171200011C}). In a non-astrophysical scenario, small dark matter (DM) substructure, such as a nearby clump, can produce a large $e^+e^-$ flux due to its enhanced DM density (e.g., \\citep{2017arXiv171111563D, 2017arXiv171111579L}). It is thus crucial to explore correlated multi-messenger signals that may help to discriminate between possible explanations \\citep{2017arXiv171200370G}. If an association is established, the directional information carried by weakly-interacting particles such as neutrinos will be crucial to finding the source.\n\nFrom a particle physics viewpoint, dark matter models fall into two\ngeneric groups according to the final state electron chirality. The\nleft-handed electrons are fundamentally different from right-handed\nelectrons. If the $e^+e^-$ produced by DM are left-handed, one\ngenerically expects a comparable neutrino flux simultaneously. This\nis because the electroweak symmetry breaking (EWSB) scale is at\nO(100) GeV, thus induces negligible difference between left-handed\nelectrons and neutrinos at an energy as high as 1.4 TeV. The\nassociated neutrinos should be almost monochromatic and may carry\nimportant directional information if they come from a nearby DM\nclump.\n\nIn this work, using the $e^\\pm$ flux at 1.4 TeV measured by DAMPE as\na reference of the TeV electron excess, we illustrate that searching\nfor associated neutrino signals can provide strong tests on DM\ninterpretations. We first demonstrate that the accompanying\nneutrinos, with a monochromatic energy of 1.4~TeV, are naturally\nexpected in a generic class of models. Then we briefly discuss the\nflavors of neutrinos when they reach the earth. Based on the\nneutrino background levels in the IceCube one-year public data, we\ndemonstrate that the existing $\\sim$8-year IceCube data is\nsufficient to decisively test the possibility that the DAMPE $e^\\pm$\nexcess is left-handed and produced from the annihilation or decay of\nDM.\n\n{ We emphasize that, in this paper, we use the tentative peak\nappearing in DAMPE's measurement as a demonstration to show the\nbenefit on studying the correlation between electron and neutrino\nchannels. We are not limited by this particular choice of benchmark\nscenario and similar studies can be applied in more general cases.}\n\n\n\\section{Classification of Models}\nIf a left-handed electron is involved\nin the DM annihilation process, generically a comparable neutrino\nflux is also generated due to $SU(2)_L$ symmetry. We present a\nclassification of models and demonstrate that a neutrino flux is\nnaturally produced. To quantify the relative ratio, we define\n$\\eta_i$ as\n \\begin{align}\\label{eqn:eta}\n\\eta_i=L_{\\nu_i}\/L_{e} \\end{align} where $L$ is the injection\nluminosity and $i$ labels the lepton flavor. We do not distinguish\n$e^+$ and $e^-$ because they are indistinguishable using DAMPE at\nhigh energy. Similarly, IceCube data cannot distinguish neutrinos\nfrom anti-neutrinos. We have not yet included the effects of\nneutrino oscillation. If only electrons and positrons are produced,\n$\\eta_{\\mu,\\tau}=0$. For the flavor universal scenario,\n$\\eta_{e,\\mu,\\tau}$ are all equal.\n\nThere are two ways to link dark matter with a standard model (SM)\nlepton: s or t-channel exchange of a mediator. $Z'$ is a typical\ns-channel mediator, which is the gauge boson of a new $U(1)$ gauge\ngroup. Label the charges of left and right-handed leptons as\n$q'_{L,i}$ and $q'_{R,i}$, $\\eta_i$ is\n$\\eta_i={{q'}^2_{L,i}}\/{\\left({q'}^2_{L,e}+{q'}^2_{R,e}\\right)}$. If\n$Z'$ does not couple to right handed leptons, then $\\eta_i$ is 1.\nWith comparable charge assignments, $\\eta_i$ is O(1). In\n\\cite{Athron:2017drj}, the s-channel models were studied and we\ndiscuss the classification of the t-channel models in this section.\n\n{\\it (i) $SU(2)_L$ singlet.} If dark matter is a singlet of\n$SU(2)_L$ and the leptons in the final states are a $SU(2)_L$\ndoublet, the mediator should be an $SU(2)_L$ doublet,\n\\begin{align}\nL \\supset \\lambda_{i} \\phi \\Psi_{L,1} L_i + M_{\\Psi}\n\\Psi_{L,1}\\Psi_{L,2} + h.c.\n\\end{align}\n$\\Psi_{L,1}$ and $\\Psi_{L,2}$ are heavy vector-like fermions in the\nfundamental representation of $SU(2)_L$ with hypercharge +1 and -1.\nDM annihilation is mediated by $\\Psi$. EWSB may introduce a mass\nsplitting between $SU(2)_L$ components of $\\Psi$. However EWSB\nhappens around 100 GeV, much smaller than the hypothesized 1.4~TeV\nDM mass. The induced mass splitting is small, $<10\\%$, and\ncomparable neutrino flux should be produced, i.e. $|1-\\eta_e|< 0.1$.\n\n\nIf DM is a real scalar, the leading annihilation is p-wave due to\nchirality suppression. Comparing with complex scalar DM, to achieve\nthe same $e^+e^-$ production rate, the DM energy density in the\nclump needs to be much larger, $O(\\sim 10^3)$, assuming similar\ncoupling constants and $M_{\\Psi}$. DM particle may be fermionic. For\na gauge singlet, it can pair with itself and form a Majorana\nfermion.  DM can also be a Dirac fermion composed of two Weyl\nspinors. For Majorana DM, the dominant annihilation channel is again\np-wave suppressed.\n\n{\\it (ii) Non-trivial representation under $SU(2)_L$.} Let us first\nconsider the fundamental representation; we comment on higher\nrepresentations later. If a DM particle is an $SU(2)_L$ doublet, the\nheavy mediator transforms under trivial or adjoint representation.\nWe introduce two sets of Weyl spinors, $\\chi_1$ and $\\chi_2$, which\nare $SU(2)_L$ doublets with hypercharge +1 and -1. With a\nvector-like mass, their EM neutral components form a Dirac fermion\nas DM.\n\nIf the mediators,  $\\phi_n$ and $\\phi_c$, are $SU(2)_L$ singlets.\nThey carry 0 and +2 hypercharge respectively. The Lagrangian is\n\\begin{equation}\nL \\supset \\lambda_{1,i}\\chi_1 L_i \\phi_n + \\lambda_{2,i}\\chi_2\nL_i\\phi_c\\nonumber+M_\\chi \\chi_1\\chi_2\n+m_n^2\\phi_n^2+m_c^2\\phi_c^2 +h.c.\n\\end{equation}\n$M_\\chi$ determines the DM mass scale.  The relative ratio is\ndetermined by $\\lambda_{1,i}$, $\\lambda_{2,i}$ and $|m_{n,c}|$. With\ngeneric choices, $\\eta_e\\sim O(1)$.\n\n$\\eta_e$ is a free parameter here because no symmetry relates\n$\\lambda_{1,i}$, $\\lambda_{2,i}$ and $|m_{n,c}|$. Such freedom is\ngone if the heavy mediator transforms non-trivially under $SU(2)_L$,\ne.g., an adjoint representation when DM is a doublet or when DM is\nin higher representation of $SU(2)_L$. Then $\\eta_e$ is expected to\ndiffer from unity by at most 10\\%.\n\nIn the $SU(2)_L$ doublet case, DM cannot be a Dirac fermion, because\nof strong dark matter direct detection constraints. Similar to the\nhiggsinos in Minimal Supersymmetric Standard Model, a small mixing\nafter EWSB with other fermions, such as the Wino or Bino, breaks a\nDirac fermion into two Majorana fermions. The DM direct detection\nconstraints can then be evaded. Since DM is effectively Majorana,\nits annihilation is p-wave suppressed.\n\n\n\nWe now comment briefly on the DM decay which can be realized when\nmediators are lighter than DM. { The lifetime of the dark matter\nparticle can be cosmologically long if the coupling between the DM\nfield and the mediator is very weak. This can be naturally realized\nif such vertex violates an approximate global symmetry.} The energy\nof the electron and neutrino is\n \\begin{align}\nE_{e,\\nu} = \\frac{m_{DM}^2-m_{med}^2}{2m_{DM}}\n \\end{align}\n\nSome components of the mediator are charged, but the collider\nconstraints are weak. If the heavy charged particles are long-lived,\ntheir mass can still be O(100) GeV\n\\cite{ATLAS:2014fka,Khachatryan:2016sfv}. The charged heavy\nparticles may lose its charge by decaying to charged SM leptons and\nneutral particles. This evades potential problems in cosmology. The\ncharged leptons from the secondary decay is softer and can hide in\nthe continuous cosmic-ray background.\n\n\n{\\bf Neutrino Oscillation} The flavor of a neutrino changes during\npropagation. We present the detailed calculation for neutrino\noscillations in the Appendix. For a sizable DM clump considered\nhere, i.e., O(10) pc,  the observed neutrino fraction, $\\kappa_i$,\nis independent on the DM clump size and the distance to us  as shown in Figure~\\ref{fig:eleconly}. In the\nAppendix, we show the flavor fraction in two scenarios,\nelectron-only and flavor universal.  The undetermined CP violating\nangle in PMNS matrix $\\delta_{CP}$ only affects the results by O(1),\nthus it does not change our conclusion qualitatively.\n\n\n\n\n\n\\section{Neutrino Flux}\nAfter leaving a source, TeV electrons diffuse in the Galactic magnetic field while neutrinos travel in a straight path to reach the earth. We now describe how the electron and the neutrino fluxes are connected.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{atmFlux_ps}\n\\includegraphics[width=.7\\textwidth]{atmFlux_ex}\n\\end{center}\n\\caption{The flux of neutrinos from a fiducial dark matter source\nthat may produce { the 1.4 TeV excess} in the electron spectrum\nmeasured using DAMPE (mean and a hypothetical 5$\\sigma$ downward\nfluctuation in blue lines), compared with the flux of the\natmospheric neutrino background around 1.4 TeV (kaon and pion\ncomponents are indicated by grey dash-dotted and dashed lines, and\nthe total is shown by the black solid line). The top and bottom\npanels assume that the source is point-like and extended\nrespectively. In the top (bottom) panel, the blue cross data points\npresent the average number of TeV events in a\n$0.5^\\circ\\times0.5^\\circ$ ($2^\\circ\\times2^\\circ$) sky patch in the\nfive-year (ten-year) IceCube data, which is obtained by scaling that\nin the IceCube-86 2011-2012 data by the corresponding number of\nyears \\cite{2016PhRvL.117g1801A}. The colored contours show the flux\nneeded for the source to stand out of the atmospheric background\nwith 1 to 6$\\sigma$ significance levels. We find that the existing\n$\\sim$8-year IceCube data can decisively test a generic class of\ndark matter models that produce neutrinos and the DAMPE TeV\nelectrons simultaneously.   } \\label{fig:excess}\n\\end{figure}\n\nThe density of particles $n$ at a location $\\vec{x}$ from a source at $\\vec{x}_s$ follows the transport equation \\citep{1990acr..book.....B}\n\\begin{equation}\n    \\frac{\\partial n(\\vec{x}, E)}{\\partial t} = D(E)\\,\\nabla^2\\,n+Q_s(E)\\delta(\\vec{x} - \\vec{x}_s)\\,,\n\\end{equation}\nassuming that there is no energy loss and that the source is stationary.\nHere $D(E)=D_0\\,\\left(E\/E_0\\right)^\\alpha$ is the spatial diffusion coefficient, with $D_0=10^{28}\\,\\rm cm^2\\,s^{-1}$, $\\alpha=1\/3$ and $E_0=3$~GeV corresponding to the diffusion coefficient in the interstellar medium \\citep{1990acr..book.....B}, and $Q_s(E)$ is the particle injection rate.\nThe solution can be written as \\footnote{When there is more than one DM clump nearby, our calculation would still apply if a single clump makes the dominant contribution to the electron and neutrino fluxes. }\n \\begin{align}\n    n_e(E) = \\frac{Q_e(E)}{4\\pi\\,R_s\\,D(E)}\n \\end{align}\n\nAssuming that the DAMPE excess is due to a  monochromatic electron population  with an observed energy density $w_e = n_e\\,E_e= 9.8\\times10^{-19}\\,\\rm erg\\,cm^{-3}$ \\citep{Yuan:2017ysv}, the electron injection power is $L_e = 7.6\\times10^{32}\\,\\left({R_s}\/{0.2\\,\\rm kpc}\\right)\\left({D(E)}\/{10^{29}\\,\\rm cm^2\\, s^{-1}}\\right)\\,\\rm erg\\,s^{-1}$.\n\n\nA population of neutrinos is produced at the same time as the\nelectrons. The injection powers of the two species are connected by\nEq.~\\ref{eqn:eta}. The neutrino flux is then\n\\begin{eqnarray}\\label{eqn:F_nu}\nF_{\\nu_j} &=&   \\frac{\\sum_i\\,\\eta_i\\,\\kappa_j\\,L_e}{\\delta \\left(\\log E_\\nu\\right)\\,4\\pi\\,R_s^2} \\\\\n&=& 8.3\\times10^{-8}\\,\\eta\\,\\left(\\frac{R_s}{0.2\\,\\rm kpc}\\right)^{-1}\\left(\\frac{D(E)}{10^{29}\\,\\rm cm^2\\, s^{-1}}\\right) \\nonumber \\\\\n&&\\left(\\frac{\\sigma\\left(\\log\nE_\\nu\\right)}{1.2}\\right)\\left(\\frac{\\sum_i\\,\\eta_i\\,\\kappa_j}{1}\\right)\\,\\rm\nGeV\\,cm^{-2}\\,s^{-1} \\nonumber \\\\ \\nonumber\n\\end{eqnarray}\nwhere\n$\\delta\\left(\\log E_\\nu\\right)$ is the energy resolution of neutrino\nevents, which is $\\delta\\left(\\log E_{\\nu_\\mu}\\right)\\sim 1.2$ for\nthe IceCube $\\nu_\\mu$ events based on the reconstruction of the muon\nenergy in the IceCube detector   \\citep{2016PhRvL.117g1801A}. $\\kappa_i$ is the\nfraction of neutrino in each flavor at the time of detection. More\ndetails about neutrino oscillation can be found in the Appendix.\n\nThe neutrino flux depends on the source distance. A DM clump with a total mass of $10^7-10^8\\,M_\\odot$ with a distance of $0.1-0.3\\,\\rm kpc$ has been suggested to account for the DAMPE TeV data \\citep{Yuan:2017ysv}. The source is unlikely to be more distant than O(1) kpc, as electrons would have suffered from significant energy loss and not present a feature as narrow as in the DAMPE spectrum. We thus choose $0.2\\,\\rm kpc$ as a default distance for the following calculations.\n\nFor comparison, at 1.4~TeV, the averaged conventional muon neutrino flux per solid angle, including both pion and kaon contributions, is $3\\times 10^{-5}\\,\\rm GeV \\, cm^{-2}\\,s^{-1}\\,sr^{-1}$ \\citep{2002ARNPS..52..153G}.  The averaged conventional electron neutrino flux per solid angle is $2\\times 10^{-6}\\,\\rm GeV\\,cm^{-2}\\,s^{-1}\\,sr^{-1}$ \\citep{2002ARNPS..52..153G}.\nIf the observed electrons arrive from a preferred direction, which for example may be indicated by an anisotropy in the electron data, one can search for coincident neutrinos in the associated sky location. The average angular resolution of the IceCube detector is $0.5^\\circ$ for $\\nu_\\mu$ events and a few degrees for $\\nu_e$ events at $\\sim 1$~TeV. The background flux in the sky patch surrounding the source direction would thus be\n$F^{\\rm bg}_{\\nu_\\mu} = 2.3\\times10^{-9}\\,\\left({\\delta\\theta}\/{0.5^\\circ}\\right)^2\\,\\rm GeV\\,cm^{-2}\\,s^{-1}$\nand\n$F^{\\rm bg}_{\\nu_e} = 1.5\\times10^{-8}\\,\\left({\\delta\\theta}\/{5^\\circ}\\right)^2\\,\\rm GeV\\,cm^{-2}\\,s^{-1}$\n\nUsing the IceCube effective area of $\\sim 5000\\,\\rm cm^2$ at 1.4~TeV, Eq. ~\\ref{eqn:F_nu} corresponds to an average of $7.7$ events per year of IceCube data. Therefore it will be easy to use IceCube to identify a bright source if we know its location in the sky.\nHowever, no specific directional information has been provided by the current data of DAMPE \\citep{2017APh....95....6C}. This could be due to a limit of the statistics, or because electrons started from a relatively distant location and have lost most of their angular information during a diffusive propagation. Below we investigate the feasibility of a blind search for the associated neutrino signal using the IceCube muon neutrinos events. We do not use cascade events due to their poor angular resolutions, but note that they may be useful for searches for very extended sources as shown in Ref.~\\cite{2017arXiv170502383I}.\n\n\nWe analyze the public IceCube data from the full 86-string detector configuration taken during 2011-2012, using only up-going neutrinos from the northern sky to eliminate background muon events \\citep{2016PhRvL.117g1801A}. The sample contains a total of 20,145 neutrino events with reconstructed energy in the approximate 320~GeV to 20~TeV range. We select 18,722 events that may have a deposition energy at  1.4~TeV according to a $\\delta \\left(\\log E_{\\nu_\\mu}\\right) \\sim 1.2$ energy resolution \\citep{2016PhRvL.117g1801A} (between 320~GeV and 3080~GeV). The public data only have the zenith angle of the reconstructed events, rather than the full two-dimensional angular location, so we cannot perform an analysis with these data alone.  In the following, however, we project an analysis based on an extrapolation of the neutrino numbers to a 5-year and a 10-year data set, which we assume will include both the zenith angle and the azimuthal angle for each reconstructed event.\n\nThe events are pixelized based on a HEALPix71 \\citep{2005ApJ...622..759G} pixelization scheme for spatial binning.\nTwo source scenarios are considered. In the first scenario, we assume that the dark matter source is point-like.  We choose   a bin size of approximately $0.5^\\circ \\times 0.5^\\circ$ (with the HEALPix parameter $\\rm Nside=128$, corresponding to the angular solution of the IceCube $\\nu_\\mu$ events \\citep{2016PhRvL.117g1801A}. In the second scenario, we assume that the source is extended. We choose   a spatial bin size of $\\sim 2^\\circ\\times2^\\circ$ ($\\rm Nside = 32$), corresponding to  a dark matter clump with a size of $\\sim 10\\,\\rm pc$ and a distance of $\\sim 0.2-0.3$~kpc as suggested  for e.g. by the benchmark case in Ref.~\\citep{Yuan:2017ysv}. We divide the\ndata into 6 bins according to the zenith angle of the reconstructed\nevents, and let all bins have equal solid angles.  We verify that the event number in pixels in all zenith angle bands  follows the Poisson distribution.\n\n\n\nFigure~\\ref{fig:excess} presents the flux of TeV neutrinos from the\natmospheric background in an element sky patch (a pixel in our analysis) in the point-source and the extended-source scenarios,\ncomparing to the flux of neutrinos from a fiducial  source that produces the DAMPE TeV electrons.\nThe\nsolid black line corresponds to the atmospheric model of\n\\cite{2002ARNPS..52..153G}, including both  pion and kaon\ncomponents. The blue cross data points and their values  show  the average number of events in a pixel at the corresponding  zenith angle expected in  five-year data (top) and ten-year data (bottom). The values are obtained by scaling the average event number in one pixel in the 2011-2012 data by the number of observational years. The dependence of the  event number on the zenith angle is consistent with that of the total   atmospheric neutrino flux. The  scaling between the two is obtained by fitting the data to the model, and its physical meaning is the effective area multiplied by the observational time and the angular size of the pixel.\nThe colored contours show the flux levels that are needed to reach a 1 to 6 $\\sigma$ deviation from a background-only hypothesis. Specifically, the local probability of  deviation is calculated using the Poisson distribution with a mean determined by the average number of atmospheric neutrinos in a pixel. The local probability is then corrected by a trial factor that equals to the number of independent pixels used in the analysis. Finally, the global probability is quoted using the corresponding number of standard deviations for a Gaussian distribution.\n\nThe  solid blue line indicates the expected number of neutrino events from a dark matter source that may explain the TeV excess of electrons measured by DAMPE (as described by equation~\\ref{eqn:F_nu}). We find that in both point-source and extended source scenarios, the mean flux is high enough to reject a background-only hypothesis with high significance. We additionally  consider a pessimistic scenario in which the number of neutrino events from the dark matter source is 5-$\\sigma$ below its mean value, integrated over the uncertainty of the DAMPE electron flux at 1.4~TeV, as indicated by the dashed blue line in both plots. Even in this case, the five-year data is able to reveal a dark matter source in most part of the northern sky with high confidence levels.\n\n\\section{Conclusion}\nIn this letter, we demonstrate the feasibility of testing TeV dark\nmatter models by neutrino observations. We focus on a benchmark\nscenario in which the tentative TeV electron excess measured by\nDAMPE is explained by the DM annihilation or decay in a nearby\nsubhalo. In a generic class of dark matter models where $e^\\pm$ from\nDM annihilation or decay are in a $SU(2)_L$ doublet, neutrinos of\ncomparable flux are simultaneously generated.  We have shown that\nthe existing $\\sim$8-year IceCube data is sufficient to identify the\nassociated neutrinos with high significance, and decisively test any\ndark matter models using left-handed leptons to explain the DAMPE\nTeV { peak}.\n\n\nFor a point-like subhalo in the northern sky, our results are robust even in a pessimistic scenario where due to fluctuations the neutrino flux is $5\\sigma$  lower than expected. For an extended source, ten years of IceCube data can reveal its existence in most parts of the northern sky. But if the source is much more extended than an angular size of  $\\sim 2^\\circ$, the prospects for using IceCube to detect or constrain the dark matter source would not be as good.  We do note that there are two refinements to our analysis that could improve those prospects significantly.  First, we assumed for simplicity that the neutrino flux from an extended source would be uniform over the whole solid angle.  In reality, the flux would be centrally concentrated in a way specific to the halo properties, so a search for that concentration would yield a stronger signal.  Second, we likewise assumed that the reconstructed background atmospheric neutrino flux is equally spread over the entire $\\sim 300-3000$~GeV range, but in fact that flux drops sharply with increasing energy.  As a result, the excess signal at higher reconstructed energies is likely to be substantially larger than in our current conservative estimates.\n\n\n\n\nThe propagation of electrons depends on the diffusion coefficient of\nthe region between the source and the earth. If the  diffusion\ncoefficient is well below the average value of the ISM ( as it is in\nthe region surrounding the Geminga pulsar \\cite{2017PhRvD..96j3013H,\n2017arXiv171106223A}), electrons would be confined to be near the\nsource for longer. To maintain a non-broadened { peak} feature,\nthe source would then need to be closer than that in our benchmark\ncase. Therefore the neutrino flux (in equation~\\ref{eqn:F_nu}) does\nnot strongly depend on the diffusion coefficient.\n\n\nWe have used the IceCube detector to test DM models in this work. Other high-energy neutrino experiments, including the Super-Kamiokande \\citep{2016PhRvD..94e2001R} and the ANTARES Telescope \\citep{2017arXiv171101496A}  may provide additional sky coverages to find or constrain associated neutrino signals. Future  experiments  such as IceCube-Gen2 \\citep{2014arXiv1412.5106I} and  KM3NeT \\citep{2016JPhG...43h4001A}   will provide improved sensitivity to examine very extended or extremely faint dark matter halos.\n\n\n\\section*{Neutrino oscillations and injection channels}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{elec-only.eps}\n\\caption{ The fraction of each flavor in the measured neutrino flux\naround the earth. The blue lines correspond to the scenario in which\nonly electron neutrinos are produced by DM annihilation. Solid,\ndashed and dot-dashed lines are for electron, muon and tau neutrinos\nrespectively.  The black dotted line is for the flavor universal\nscenario, in which the fractions for each flavor is equal to 1\/3.}\n\\label{fig:eleconly}\n\\end{center}\n\\end{figure}\n\n\nMixing among neutrinos can be characterized by the PMNS matrix, $U$.\nNeglecting the possible CP phases from the Majorana mass terms,\nthere are four parameters in PMNS matrix: three mixing angles and\none CP phase, i.e.\n$\\{\\theta_{12},\\theta_{13},\\theta_{23},\\delta_{CP}\\}$. $\\theta_{12}$\nand $\\theta_{13}$ have been measured with good accuracy,\n$\\textrm{sin}^2(2\\theta_{13})=0.093\\pm0.008$ and\n$\\textrm{sin}^2(2\\theta_{12})=0.846\\pm0.021$. $\\theta_{23}$ has\nlarger uncertainty, $\\textrm{sin}^2(2\\theta_{23})>0.92$. In the\nfollowing calculation, we take the central values of\n$\\textrm{sin}^2(2\\theta_{12})$ and $\\textrm{sin}^2(2\\theta_{13})$,\nwhile $\\textrm{sin}^2(2\\theta_{23})$ is taken to be 0.97. The\nuncertainties in these values do not change our results\nqualitatively.  $\\delta_{CP}$ remains to be determined, and we treat\nit as a free parameter.\n\nThe oscillation of neutrinos is then written as\n\\begin{eqnarray}\nP_{\\alpha\\to\\beta}=&\\delta_{\\alpha\\beta}&-4\\sum_{i>j}Re (U^*_{\\alpha\ni} U_{\\beta i}U_{\\alpha j} U^*_{\\beta j})\n\\textrm{sin}^2\\bigg(\\frac{\\Delta m_{ij}^2\nL}{4E}\\bigg)\\nonumber\\\\\n&+&2\\sum_{i>j}Im (U^*_{\\alpha i} U_{\\beta i}U_{\\alpha j} U^*_{\\beta\nj}) \\textrm{sin}\\bigg(\\frac{\\Delta m_{ij}^2 L}{4E}\\bigg)\n\\end{eqnarray}\nHere $\\Delta m_{ij}^2$ is the mass square splitting among neutrino\nmass eigenstates. In a vacuum, $\\Delta m_{21}^2 =\n(7.53\\pm0.18)\\times 10^{-5}\\textrm{eV}^2$ and $|\\Delta\nm_{31}^2|\\simeq |\\Delta m_{32}^2| = (2.44\\pm0.06)\\times\n10^{-3}\\textrm{eV}^2$. The size of the DM clump which generates the monochromatic electron flux can be large, for example O(10) pc as considered in our\nbenchmark case. For neutrinos with energy O(TeV), this is much larger\nthan the distance to have one oscillation, i.e.  $\\bigg(\\frac{\\Delta m_{ij}^2 10 \\textrm{pc}}{4 \\textrm{TeV}}\\bigg) \\gg 1$.  After averaging the whole clump,\n\n\\begin{eqnarray}\n\\langle\nP_{\\alpha\\to\\beta}\\rangle=\\delta_{\\alpha\\beta}&-&2\\sum_{i>j}Re\n(U^*_{\\alpha i} U_{\\beta i}U_{\\alpha j} U^*_{\\beta j})\n\\end{eqnarray}\n\n\n\n{(a) Electron-only channel:}\nthis is the scenario where DM annihilation only produces\nelectrons and electron neutrinos. The flavor fraction of neutrino flux, $\\kappa_i$, are labeled as blue lines in Fig. 1.\n\n\n\n{(b) Flavor-universal channel:}\nif the DM annihilation is universal in flavor, i.e. the initial flux\nfor each flavor is the same, the flux observed on earth remains\nflavor universal, guaranteed by the unitarity of the mixing matrix. This is represented by the black dotted line in Fig. 1.\n\n\n\n\n\n\\section*{Acknowledgements}\nYZ is also supported by US Department of Energy under\ngrant de-sc0007859. KF acknowledges the support of a Joint\nSpace-Science Institute prize postdoctoral fellowship at the\nUniversity of Maryland. YZ thank the support of grant from the\nOffice of Science and Technology, Shanghai Municipal Government (No.\n16DZ2260200).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDriven by potential Internet-of-Things (IoT) applications, the fifth-generation (5G) mobile communication systems need to support short-packet communication applications with strict latency and throughput constraints; see, e.g., \\cite{shortpackets1}. A slow fading channel model serves as a simple and relevant setup that captures the non-ergodic nature of delay-limited information transmission. A special example is the so-called quasi-static fading model, where the channel fading remains constant during a codeword, but varies from codeword to codeword with respect to a certain probability distribution.\n\nOutage probability is a fundamental performance indicator for such quasi-static fading channels. It characterizes the non-negligible probability that the instantaneous achievable rate is lower than the chosen code rate. The outage behavior of quasi-static fading channels has been thoroughly studied under the ideal assumption of perfect channel state information (CSI) at the receiver. With perfect CSI, the optimal decoder follows a nearest neighbour decoding rule, which selects the (faded) codeword that has the smallest Euclidean distance to the received signal.\n\nIn practical systems, however, the CSI is imperfect. It is typically supplied to the receiver via transmitting a prescribed pilot, and the receiver estimates the CSI from its received (faded and noisy) pilot. Nevertheless, owing to its simplicity, the nearest neighbor decoding rule is usually adopted by the decoder even under imperfect CSI, and therefore the decoder is mismatched to the actual channel; see, e.g., \\cite{Lapidoth-Fading} \\cite{Weingarten-weighted}.\n\nBased on the generalized mutual information (GMI), the generalized outage probability has been defined and studied \\cite{Taufiq-CSIR} \\cite{Taufiq-CSIT&CSIR}, which characterizes the asymptotic outage behavior of the nearest neighbor decoding rule in the high signal-to-noise ratio (SNR) limit, for multiple-input multiple-output (MIMO) slow fading channels with imperfect CSI.\\footnote{Quasi-static fading channels with imperfect CSI is a very realistic issue frequently met in practice, because a coherence block is generally not long enough and many existing standards insert pilots only sparingly within a coherence block. Nevertheless, some oddness exists for such a model: treating the coding block length as an asymptotically large quantity is necessary for information-theoretic analysis, whereas this mathematical assumption would in term imply adequate resource for channel training so as to render the CSI perfect. So this model should be viewed as a compromise between mathematical modeling and engineering practice.} Performance analysis in the finite length regime under imperfect CSI has also been conducted in \\cite{Potterr-finite} \\cite{Schiessl}.\n\nA recent study reveals that, by generalizing the nearest neighbor decoding rule to incorporate output processing and codeword scaling, the achievable performance in terms of GMI can be substantially improved, for ergodic fading channels with imperfect CSI \\cite{Wang-GNNDR} \\cite{Pang-GNNDRMIMO}. This observation motivates us to investigate the potential impact of generalized nearest neighbor decoding for slow fading channels, where the performance is measured in terms of outage probability.\n\nIn this paper, we conduct a preliminary study along this direction. Instead of solving for the optimal nonlinear output processing and codeword scaling functions as in \\cite{Wang-GNNDR}, we turn to a simple linear shrinkage technique: rather than using the linear minimum mean-squared error (LMMSE) estimate of the CSI, we further linearly shrink the LMMSE estimate, when conducting the nearest neighbor decoding. The shrinkage coefficient is selected to minimize the outage probability.\n\nThe concept of shrinkage estimator has been used to estimate the covariance matrix in high dimensions. Well-conditioned estimators that combine the sample covariance matrix and certain more stable statistics are referred to as linear shrinkage or weighted estimators \\cite{Yuki-Linearshrinkage}. A shrinkage estimator linearly combining the sample covariance matrix and a diagonal (or spherical) matrix may achieve a desirable tradeoff between bias and variance \\cite{ledoit-well} \\cite{Chen-Shrinkage}. The situation is somewhat similar in our context. For a non-ergodic slow fading channel with imperfect CSI, the achievable rate, i.e., GMI in our study, is in fact a random variable induced by both the true channel fading and the CSI, and its exact value is unknown to the receiver. Although the LMMSE estimator minimizes the mean-squared error of estimated CSI, due to the highly nonlinear relationship between the CSI estimation error and the GMI, the LMMSE estimator does not necessarily achieve the most desirable tradeoff between mean and variance of the GMI, and hence does not necessarily lead to the minimum outage probability. Heuristically, it is risky for the receiver to over-estimate the CSI, because that would lead to an overly optimistic belief about the achievable rate. So a linear shrinkage receiver is potentially helpful in reducing such risk.\n\nOur numerical results suggest that, compared with the LMMSE estimator, the proposed linear shrinkage estimator usually achieves an SNR gain of nearly $1.5$ dB in Rayleigh fading channels with moderate number of receive antennas. On the other hand, in the asymptotic regime where the number of receive antennas grows without bound, we establish that the linear shrinkage estimator degenerates into the LMMSE estimator.\n\nThroughout the paper, the following notation is used. Uppercase bold letters $\\mathbf{X}$ represent random vectors and lowercase bold letters $\\mathbf{x}$ represent their realizations; uppercase letters $\\mathrm{X}$ and lowercase letters $\\mathrm{x}$ represent random scalars and their realizations, respectively. Expectation is denoted by $\\boldsymbol{E}[\\cdot]$. The $n \\times n$ identity matrix is $\\boldsymbol{I}_{n}$. Symbols $^{*}$ and $\\|\\|$ represent conjugate transpose and a vector's Euclidean norm, respectively. Logarithm $\\log ()$ uses base $e$.\n\n\\section{System Model and GMI}\n\nConsider information transmission over a slow fading channel with a single transmit antenna and $N_{r}$ receive antennas, disturbed by independent and identically distributed (i.i.d.) complex additive white Gaussian noise\n$\\mathbf{Z} \\sim \\mathcal{CN}\\left(\\bm{0},\\sigma^{2}\\boldsymbol{I}_{N_{r}}\\right)$. The input-output relationship of the channel is\n\\begin{equation}\n    \\mathbf{Y}=\\mathbf{S}\\mathrm{X}+\\mathbf{Z},\n    \\label{E1}\n\\end{equation}\nwhere $\\mathrm{X} \\in \\mathbb{C}$ and $\\mathbf{S} \\in \\mathbb{C}^{N_{r} \\times 1}$ are the scalar input and the vector channel state (i.e., fading coefficients), respectively. Denote the CSI supplied to the receiver by $\\mathbf{V} \\in \\mathbb{C}^{N_{r} \\times 1}$, which is correlated with $\\mathbf{S}$. Under the slow fading assumption, $\\mathbf{S}$ and $\\mathbf{V}$ remain constant over a length-$N$ coding block. Therefore, the conditional probability distribution satisfies\n\\begin{equation}\n    p_{\\mathbf{Y} | \\mathrm{X}, \\mathbf{S}}\\left(\\mathbf{y}^{N} | \\mathrm{x}^{N}, \\mathbf{s}\\right) = \\prod_{n=1}^{N} p_{\\mathbf{Y} | \\mathrm{X}, \\mathbf{S}} \\left(\\mathbf{y}_{n} | \\mathrm{x}_{n}, \\mathbf{s}\\right).\n    \\label{E2}\n\\end{equation}\n\nAt the transmitter, when performing coding of a prescribed rate $R$ and block length $N$, the source selects a message $\\mathrm{M}$ from $\\mathcal{M} = \\left\\{1,2, \\ldots,\\left[e^{N R}\\right]\\right\\}$ uniformly randomly and the encoder maps $\\mathrm{M}$ to a transmitted codeword $x^N(\\mathrm{M})$. We adopt an i.i.d. Gaussian codebook ensemble, for which each codeword obeys $\\mathcal{CN}(\\bm{0}, P\\boldsymbol{I}_N)$, where $P$ is the average transmit power.\n\nAt the receiver, let there be (possibly vector-valued) mappings $g$ and $f$ as the output processing function and the codeword scaling function, respectively \\cite{Wang-GNNDR}. The decoding metric of the generalized nearest neighbor decoding rule is hence given by\\footnote{In \\cite{Wang-GNNDR} $g$ and $f$ are scalar-valued, but its derivation of the GMI remains essentially the same for vector-valued $g$ and $f$, which will be adopted in our linear shrinkage estimator.}\n\\begin{equation}\n    D(m)= \\frac{1}{N} \\sum_{n=1}^{N} \\|g\\left(\\mathbf{y}_{n}, \\mathbf{v}\\right) - f\\left(\\mathbf{y}_{n}, \\mathbf{v}\\right) \\mathrm{x}_{n}(m)\\|^{2}.\n    \\label{E3}\n\\end{equation}\n\nFor an i.i.d. codebook ensemble and a prescribed decoding metric, an achievable rate is given by the GMI, which is a lower bound of the mismatch capacity, and is indeed the maximum information rate to guarantee that the decoding error probability, averaged over the specified i.i.d. codebook ensemble, decays to zero as the coding block length grows without bound \\cite{Lapidoth-Fading} \\cite{Weingarten-weighted}. Below we provide an outlined derivation of the GMI in our problem setup, and for further details, see, e.g., \\cite{Wang-GNNDR}.\n\nThe decoding error probability averaged over the i.i.d. Gaussian codebook ensemble is identical for all messages due to the symmetry. Therefore, it suffices to study the decoding error probability conditioned upon message $m = 1$ being transmitted. For $m = 1$, we have\n\\begin{equation}\n    \\begin{aligned}\n    \\lim _{N \\rightarrow \\infty} D(1) &=\\lim _{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^{N}\\|g\\left(\\mathbf{y}_{n}, \\mathbf{v}\\right) - f\\left(\\mathbf{y}_{n}, \\mathbf{v}\\right) \\mathrm{x}_{n}(1)\\|^{2}\\\\\n    &=\\boldsymbol{E}\\left[\\|g(\\mathbf{Y}, \\mathbf{V}) - f(\\mathbf{Y}, \\mathbf{V}) \\mathrm{X}\\|^{2} \\big| \\mathbf{S}, \\mathbf{V}\\right],\n    \\label{E4}\n    \\end{aligned}\n\\end{equation}\nalmost surely, according to the law of large numbers. Note that here a key difference between the counterpart analysis in \\cite{Wang-GNNDR} is that since the channel is non-ergodic, the expectation needs to be conditioned upon $(\\mathbf{S}, \\mathbf{V})$; in other words, the limit of $D(1)$ is a random variable induced by the channel fading and the CSI encountered.\n\nThe GMI is the asymptotic exponent of the probability that an incorrect codeword accumulates a decoding metric smaller than \\eqref{E4}, and is obtained from a large deviations analysis, as\n\\begin{equation}\n    I_{\\mathrm{GMI}} = \\sup_{\\theta < 0} \\left\\{\\theta \\boldsymbol{E}\\left[\\|g(\\mathbf{Y}, \\mathbf{V}) - f(\\mathbf{Y}, \\mathbf{V}) \\mathrm{X}\\|^{2} \\big| \\mathbf{S}, \\mathbf{V}\\right] - \\Lambda(\\theta)\\right\\},\n    \\label{E5}\n\\end{equation}\n\\begin{equation}\n    \\begin{gathered}\n    \\Lambda(\\theta) =\\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\Lambda_{N}(N \\theta), \\\\\n    \\Lambda_{N}(N \\theta) =\\log \\boldsymbol{E}\\left[e^{N \\theta D(m)} \\big| \\mathbf{Y}, \\mathbf{V}\\right], \\quad \\forall m \\neq 1.\n    \\label{E6}\n    \\end{gathered}\n\\end{equation}\n\nConditioned upon the channel output $\\mathbf{Y}$ and the CSI $\\mathbf{V}$, $\\boldsymbol{E}\\left[e^{N \\theta D(m)} \\mid \\mathbf{Y}, \\mathbf{V}\\right]$ becomes finite product series of the conditional expectations of conditionally independent random variables, wherein $\\|g(\\mathbf{Y}, \\mathbf{V}) - f(\\mathbf{Y}, \\mathbf{V}) \\mathrm{X}\\|^{2}$ obeys a non-central chi-squared distribution. We can hence derive that\\footnote{Here the expression is slightly more general than that in \\cite{Wang-GNNDR} due to the fact that $g$ and $f$ are vector-valued.}\n\\begin{equation}\n    \\begin{aligned}\n    I_{\\mathrm{GMI}}&=\\sup_{\\theta<0} \\{ \\theta \\boldsymbol{E}\\left[\\|g(\\mathbf{Y}, \\mathbf{V}) - f(\\mathbf{Y}, \\mathbf{V}) \\mathrm{X}\\|^{2} \\big| \\mathbf{S}, \\mathbf{V}\\right] \\\\\n    &-\\theta \\boldsymbol{E}\\left[\\|g(\\mathbf{Y}, \\mathbf{V})\\|^{2} + \\frac{\\theta |g^*(\\mathbf{Y}, \\mathbf{V}) f(\\mathbf{Y}, \\mathbf{V})|^{2} P}{1-\\theta\\|f(\\mathbf{Y}, \\mathbf{V})\\|^{2} P} \\Big| \\mathbf{S}, \\mathbf{V}\\right] \\\\\n    &\\left.+\\boldsymbol{E}\\left[\\log \\left(1-\\theta\\|f(\\mathbf{Y}, \\mathbf{V})\\|^{2} P\\right) \\big| \\mathbf{S}, \\mathbf{V}\\right]\\right\\}.\n    \\label{E7}\n    \\end{aligned}\n\\end{equation}\nAgain, note that due to non-ergodic fading, the expectations in \\eqref{E7} need to be conditioned upon $(\\mathbf{S}, \\mathbf{V})$, and consequently, the GMI $I_{\\mathrm{GMI}}$ is a random variable induced by $\\mathbf{S}$ and $\\mathbf{V}$. For a code rate $R$, the outage probability can then be written as\n\\begin{equation}\n    P_\\mathrm{out} = p\\left(I_{\\mathrm{GMI}} < R\\right).\n\\end{equation}\n\nIt is interesting to emphasize that, since $I_{\\mathrm{GMI}}$ depends upon both $\\mathbf{S}$ and $\\mathbf{V}$, its exact value is unknown even to the receiver, unless $\\mathbf{S}$ can be determined by $\\mathbf{V}$, i.e., the receiver having perfect CSI.\n\nA natural optimization problem is to solve for the optimal $g$ and $f$ that minimize $P_\\mathrm{out}$, i.e., for a given $R$,\n\\begin{equation}\n    \\min_{g, f} p\\left(I_{\\mathrm{GMI}} < R\\right).\n    \\label{E8}\n\\end{equation}\nBut this problem appears elusive, partly because the probability distribution of $I_{\\mathrm{GMI}}$ is hardly tractable. In the subsequent analysis, we turn to a linear shrinkage estimator of the CSI, which is a special form of $(g, f)$, i.e., $g(\\mathbf{Y}, \\mathbf{V})$ degenerating into $\\mathbf{Y}$ and $f(\\mathbf{Y}, \\mathbf{V})$ degenerating into ${b}\\mathbf{V}$, where ${b}$ is the linear shrinkage coefficient.\\footnote{The resulting decoder is different from the weighted nearest neighbor decoder in \\cite{Weingarten-weighted}: here only $f$ is scaled, whereas for the weighted nearest neighbor decoder both $g$ and $f$ are simultaneously scaled.}\n\n\\section{Linear Shrinkage Receiver}\\label{sec:LSR}\n\nIn this section, we consider the scenario where the CSI is supplied to the receiver via transmitting a prescribed pilot, as\n\\begin{equation}\n    \\mathbf{V} = \\mathbf{Y}_{p} = \\mathbf{S} X_{p} + \\mathbf{Z}_{p},\n    \\label{E9}\n\\end{equation} \nwhere $\\mathbf{Y}_{p}$ is the received pilot signal and $X_{p}$ is the transmitted pilot known to both transmitter and receiver. For simplicity, we assume that the covariance matrix of $\\mathbf{S}$ is $\\eta^{2}\n\\boldsymbol{I}_{N_{r}}$ and the noise is white Gaussian, $\\mathbf{Z}_{p} \\sim \\mathcal{CN}\\left(\\bm{0}, \\sigma_{p}^{2}\n\\boldsymbol{I}_{N_{r}}\\right)$. According to standard linear estimation theory \\cite{Poor1994An}, the LMMSE estimate of $\\mathbf{S}$ is given by\n\\begin{equation}\n    \\hat{\\mathbf{S}} = \\frac{\\eta^{2} X_{p}^{*} \\mathbf{Y}_{p}}{\\eta^{2}\\left|X_{p}\\right|^{2}+\\sigma_{p}^{2}}.\n    \\label{E10}\n\\end{equation}\nIf we further assume $\\mathbf{S}$ to be complex Gaussian, i.e., Rayleigh fading, then the above LMMSE estimate is also the MMSE estimate $\\boldsymbol{E}\\left[\\mathbf{S} | \\mathbf{Y}_{p}\\right]$.\n\nLet us denote ${\\frac{\\eta^{2} X_{p}^{*} }{\\eta^{2}\\left|X_{p}\\right|^{2}+\\sigma_{p}^{2}}}$ in the LMMSE estimate by $a$, so that $\\hat{\\mathbf{S}}_\\mathrm{LMMSE} = a \\mathbf{Y}_p$. On the other hand, the linear shrinkage estimator with shrinkage coefficient ${b}$ is simply $\\hat{\\mathbf{S}}_\\mathrm{LS} = {b} \\mathbf{Y}_p$. Clearly, when ${b} = a$, the linear shrinkage estimator degenerates into the LMMSE estimator. The decoding rule becomes\n\\begin{equation}\n   \\hat{m}=\\operatorname{argmin}_{m \\in \\mathcal{M}}\\frac{1}{N} \\sum_{n=1}^{N}\\left\\|\\mathbf{y}_{n}-{b} \\mathbf{v}_{n}\\mathrm{x}_{n}(m)\\right\\|^{2}.\n   \\label{E11}\n\\end{equation}\n\nApplying the GMI analysis in the previous section, we obtain the GMI achieved by a receiver employing linear shrinkage estimator.\n\n\\begin{theorem}\\label{thm:GMI}\n    For i.i.d. Gaussian codebook ensemble and generalized nearest neighbor decoding using linear shrinkage estimator, the GMI is given by\n    \\begin{equation}\n        I_{\\mathrm{GMI}}=\\sup _{\\theta<0} K_{\\text {LS}},\n        \\label{E12}\n    \\end{equation}\n    \\begin{equation}\n        \\begin{aligned}\n        K_{\\text {LS }}&=\\theta\\left(P\\|\\mathbf{S}-{b} \\mathbf{V}\\|^{2}-P\\|\\mathbf{S}\\|^{2}\\right)+\\log \\left(1-P \\theta\\|{b} \\mathbf{V}\\|^{2}\\right)\\\\\n        &-\\frac{P \\theta^{2}\\left(\\|{b} \\mathbf{V}\\|^{2} \\sigma^2 + P\\left|\\mathbf{S}^{*} {b} \\mathbf{V}\\right|^{2}\\right)}{1-P \\theta\\|{b} \\mathbf{V}\\|^{2}}.\n        \\label{E13}\n        \\end{aligned}\n    \\end{equation}\n\\end{theorem}\n\n{\\it Proof:} Specializing $g(\\mathbf{y}, \\mathbf{v}) = \\mathbf{y}$ and $f(\\mathbf{y}, \\mathbf{v}) = {b} \\mathbf{v}$ in \\eqref{E7} leads to \\eqref{E12} and \\eqref{E13}. $\\Box$\n\nWe remark that $K_\\mathrm{LS}$ is a random variable induced by $\\mathbf{S}$ and $\\mathbf{V}$. Nevertheless, the choice of the parameter $\\theta < 0$ can depend upon $\\mathbf{S}$ and $\\mathbf{V}$. This is because this parameter is from a large deviations analysis and can be any negative real number. The numerator of $\\frac{d K_{\\text {LS}}}{d \\theta}$ is a quadratic function of $\\theta$ whose parabola opens upward. Letting $\\frac{d K_{\\text {LS}}}{d \\theta}$ be zero, we can find that either there is a unique negative solution given by the corresponding quadratic formula and is the optimal $\\theta$ to maximizing $K_\\mathrm{LS}$, or there is no solution and then we should set $\\theta \\rightarrow 0$ (leading to $I_\\mathrm{GMI} = 0$). A special case is when ${b} \\mathbf{v} = \\mathbf{s}$, i.e., perfect CSI, and then $\\theta = -1\/\\sigma^2$ is the optimal solution, leading to the familiar result of $I_{\\mathrm{GMI}} = \\log \\left(1+P\\|\\mathbf{S}\\|^{2}\/\\sigma^2\\right)$.\n\nHaving obtained the optimal $\\theta$ for a fixed ${b}$, the linear shrinkage coefficient ${b}$ should be further optimized to minimize the outage probability for each given code rate $R$. That is, the criterion of designing ${b}$ is\n\\begin{equation}\n    \\min_{b} p\\left(\\sup _{\\theta<0} K_{\\text {LS }} < R\\right).\n    \\label{E14}\n\\end{equation}\nThis optimization can be solved numerically via a standard line search procedure; see the discussion in Section \\ref{sec:experiment}.\n\n\\begin{figure*}[ht]\n\t\\subfigure[$N_r = 4$,  $R = 1$ (bit\/channel use)]{\n\t\t\\begin{minipage}[t]{0.3\\linewidth}\\label{fig1a}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=2.4in]{NR=4,R=1.eps}\n\t\\end{minipage}}\n\t\\subfigure[$N_r = 8$,  $R = 2$ (bits\/channel use)]{\n\t\t\\begin{minipage}[t]{0.3\\linewidth}\\label{fig1b}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=2.4in]{NR=8,R=2.eps}\n\t\\end{minipage}}\n\t\\subfigure[$N_r = 16$,  $R = 3$ (bits\/channel use)]{\n\t\t\\begin{minipage}[t]{0.3\\linewidth}\\label{fig1c}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=2.45in]{NR=16,R=3.eps}\n\t\\end{minipage}}\n\t\\caption{Outage probability versus SNR under different receive antennas and code rates.}\n\t\\label{fig1}\n\\end{figure*}\n\n\\section{Massive-antenna Asymptotic Analysis}\n\\label{sec:Massive}\n\nIn this section, we investigate how the linear shrinkage estimator behaves asymptotically in the massive-antenna regime. Our main finding is the following result.\n\n\\begin{theorem}\n    \\label{thm:Massive}\n    When $b = a$, for any rate $R$, as ${N_{r} \\rightarrow \\infty}$, the outage probability asymptotically decreases to zero; for any $b \\neq a$, there exists a finite threshold rate $\\underline{R}$, such that for any rate $R > \\underline{R}$, as ${N_{r} \\rightarrow \\infty}$, the outage probability is bounded away from zero.\n\\end{theorem}\n\n{\\it Proof:} In the massive-antenna regime, we can apply the central limit theorem to obtain the following convergence property: for any $\\epsilon > 0$, there exists a corresponding finite $\\gamma_\\epsilon$, such that for all sufficiently large $N_r$, in\n\\begin{equation}\n    \\begin{aligned}\n    \\|\\mathbf{S}\\|^{2} = \\eta^2 N_r + \\eta^2 \\sqrt{N_r} \\mathrm{W}_\\mathbf{s},\n    \\end{aligned}\n\\end{equation}\n\\begin{equation}\n    \\begin{aligned}\n    \\|\\mathbf{Z}_p\\|^{2} = \\sigma_p^2 N_r + \\sigma_p^2 \\sqrt{N_r} \\mathrm{W}_\\mathbf{z},\n    \\end{aligned}\n\\end{equation}\n\\begin{equation}\n    \\begin{aligned}\n    \\mathbf{S}^{*} \\mathbf{Z}_{p} = \\sqrt{\\sigma_p^2 \\eta^2 N_r} \\mathrm{W}_\\mathbf{sz},\n    \\end{aligned}\n\\end{equation}\nthe bounds $- \\gamma_\\epsilon < \\mathrm{W}_\\mathbf{s}, \\mathrm{W}_\\mathbf{z}, \\mathrm{W}_\\mathbf{sz} < \\gamma_\\epsilon$ simultaneously hold with probability at least $1 - \\epsilon$. Denote the event that the above convergence property holds by $\\mathcal{T}$.\n\nFor the expression of the GMI in Theorem \\ref{thm:GMI}, as mentioned in Section \\ref{sec:LSR}, the optimal $\\theta$ exists and can be given by a quadratic formula. Its exact form is as follows:\\footnote{Throughout the proof we set $\\sigma^2 = 1$.}\n\\begin{equation}\n    \\begin{aligned}\n    \\theta^{*} = \\frac{-\\mathrm{B}-\\sqrt{\\mathrm{B}^{2}-4 \\mathrm{AC}}}{2\\mathrm{A}},\n    \\label{E15}\n    \\end{aligned}\n\\end{equation}\nwhere $\\mathrm{A}=P^{2} \\mathrm{U}_1\\|{b}\\mathbf{V}\\|^{4}+\\mathrm{U}_2 P\\|{b}\\mathbf{V}\\|^{2}$, $\\mathrm{B}=P\\|{b}\\mathbf{V}\\|^{4}-2 \\mathrm{U}_2-2 P \\mathrm{U}_1\\|{b}\\mathbf{V}\\|^{2}$, $\\mathrm{C}=\\mathrm{U}_1-\\|{b}\\mathbf{V}\\|^{2}$,\n$\\mathrm{U}_1=\\|\\mathbf{S}-{b}\\mathbf{V}\\|^{2}-\\|\\mathbf{S}\\|^{2}$, and $\\mathrm{U}_2=\\|{b}\\mathbf{V}\\|^{2}+P\\left|\\mathbf{S}^{*} {b}\\mathbf{V}\\right|^{2}$. Utilizing the expression of $\\mathbf{V}$ in terms of $\\mathbf{S}$ and $\\mathbf{Z}_{p}$, we have\n\\begin{equation}\n    \\begin{aligned}\n    \\|\\mathbf{V}\\|^{2}=\\|\\mathbf{S}\\|^{2}\\left|X_{p}\\right|^{2}+\\left\\|\\mathbf{Z}_{p}\\right\\|^{2}+\\mathbf{S}^{*}X_{p}^{*} \\mathbf{Z}_{p}+\\mathbf{Z}_{p}^{*} \\mathbf{S}X_{p},\n    \\end{aligned}\n\\end{equation}\n\\begin{equation}\n    \\begin{aligned}\n    \\mathbf{S}^{*} {b} \\mathbf{V}+{b}^{*} \\mathbf{V}^{*} \\mathbf{S}=\\|\\mathbf{S}\\|^{2}\\left({b}X_{p}+{b}^{*} X_{p}^{*}\\right)+{b}\\mathbf{S}^{*} \\mathbf{Z}_{p}+{b}^{*} \\mathbf{Z}_{p}^{*} \\mathbf{S}.\n    \\end{aligned}\n\\end{equation}\nThat is, $\\theta^*$ can be represented in terms of $\\|\\mathbf{S}\\|^2$, $\\|\\mathbf{Z}_p\\|^2$, and $\\mathbf{S}^* \\mathbf{Z}_p$.\n\nNow, we let ${b}=a=\\frac{\\eta^{2}X_{p}^{*}}{\\eta^{2}\\left|X_{p}\\right|^{2}+\\sigma_{p}^{2}}$, and proceed to analyze the asymptotic behavior of the resulting $\\theta^*$, conditioned on event $\\mathcal{T}$.\n\nA long and tedious calculation shows that, when ${b} = a$, the $N_r^3$ and $N_r^{2.5}$ terms in the denominator $2\\mathrm{A}$ of $\\theta^*$ vanish, and the remaining positive highest order term is $O(N_r^2)$. On the other hand, the numerator of $\\theta^*$ is dominated by $-2\\mathrm{B}$, and its leading term is also $O(N_r^2)$. Consequently, when ${b} = a$, $\\theta^*$ is strictly negative, and the dominant term in the corresponding $K_\\mathrm{LS}$, i.e., \\eqref{E13}, is given by\n\\begin{equation}\n    \\begin{aligned}\n    K_{\\text {LS}} = \\log \\left(1-P \\theta^{*} \\frac{\\eta^{4}\\left|X_{p}\\right|^{2}}{\\eta^{2}\\left|X_{p}\\right|^{2}+\\sigma_{p}^{2}} N_{r}\\right),\n    \\end{aligned}\n\\end{equation}\nwhich is $O\\left(\\log{N_{r}}\\right)$.\n\nThen, consider any ${b} \\neq a$, and investigate the asymptotic behavior of $\\theta^*$, also conditioned on $\\mathcal{T}$. Now the denominator $2\\mathrm{A}$ of $\\theta^*$ has a nonzero leading term of order $N_r^3$, and consequently, $\\theta^* = O(1\/N_r)$. Evaluating $K_\\mathrm{LS}$ in \\eqref{E13}, we can show that there exists a constant $\\overline{K}$ independent of $N_r$ upper bounding $K_\\mathrm{LS}$, i.e., $K_\\mathrm{LS} \\leq \\overline{K} = O(1)$.\n\nSo, in order to conclude the proof, we have for any given $R$ and $\\epsilon > 0$:\n\n(a) If ${b} = a$,\n\\begin{equation}\n    \\begin{aligned}\n        p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R) &= p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R | \\mathcal{T}) p(\\mathcal{T})\\\\\n        &+ p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R | \\mathcal{T}^c) p(\\mathcal{T}^c)\\\\\n        &\\leq p(O(\\log N_r) < R) + p(\\mathcal{T}^c) \\rightarrow 0\n    \\end{aligned}\n\\end{equation}\nas $N_r \\rightarrow \\infty$.\n\n(b) If ${b} \\neq a$,\n\\begin{equation}\n    \\begin{aligned}\n        p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R) &= p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R | \\mathcal{T}) p(\\mathcal{T})\\\\\n        &+ p(\\sup_{\\theta < 0} K_\\mathrm{LS} < R | \\mathcal{T}^c) p(\\mathcal{T}^c)\\\\\n        &\\geq (1 - \\epsilon) p(\\overline{K} < R),\n    \\end{aligned}\n\\end{equation}\nas $N_r \\rightarrow \\infty$, which is bounded away from zero for all rates $R > \\overline{K}$. This completes the proof. $\\Box$\n\nTheorem \\ref{thm:Massive} holds in the massive-antenna limit.\\footnote{Theorem \\ref{thm:Massive} may appear natural from a heuristic view, but we feel that its underlying mechanism is different from the well known channel hardening effect. Because the CSI is noisy over each individual antenna, even with an infinite number of antennas it is still impossible to decrease the channel estimation error to zero. So according to our GMI analysis, it is not obvious to assert that the LMMSE coefficient is the outage minimizer.} But for a finite number of receive antennas, the optimal shrinkage coefficient ${b}$ may noticeably deviate from $a$, and its reduction in outage probability may be evident, as our numerical experiments suggest, in the next section.\n\n\\begin{figure}[ht]\n    \\centerline{\\includegraphics[scale=0.5]{amplitudevalue.eps}}\n    \\caption{Shrinkage coefficients versus SNR, under different values of $N_r$. Note that the LMMSE estimate coefficient $a$ does not depend upon $N_r$.}\n    \\label{fig2a}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centerline{\\includegraphics[scale=0.5]{NR=8,R=2,SNR=5,HIST.eps}}\n    \\caption{Comparison of histograms of GMI, under $N_r = 8$, $R = 2$ (bits\/channel use), $\\mathrm{SNR} = 5$dB.}\n    \\label{fig2b}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centerline{\\includegraphics[scale=0.5]{BER-8-1-3.eps}}\n    \\caption{BER comparison on 5G NR LDPC, under $N_{r} = 16$, $N = 544$, and $R = 1\/3$ (bits\/channel use).}\n    \\label{fig3}\n\\end{figure}\n\n\\section{Numerical Simulations}\n\\label{sec:experiment}\n\nIn this section, numerical simulations are conducted to validate the effectiveness of our proposed linear shrinkage receiver. We begin by comparing the outage probability of the LMMSE estimator and the linear shrinkage estimator, and then demonstrate the bit-error-rate (BER) decoding performance on the 5G NR LDPC. Spatially independent Rayleigh fading is adopted throughout, for simplicity.\n\n\\subsection{Outage Probability}\n\nFor each experiment, we fix the code rate and vary the SNR. We assume that the pilot noise variance $\\sigma^2_p$ is identical to the noise variance during information transmission $\\sigma^2$, and that the pilot is $X_p = \\sqrt{P}$. The LMMSE estimate coefficient $a$ is hence real positive, and we further restrict our search of optimal ${b}$ to be along the real positive line as well.\\footnote{This is a simplification based upon our numerical experiment, which shows that allowing ${b}$ to have imaginary part does not appear rewarding.}\n\nFigure \\ref{fig1} shows the outage probabilities with SNR, under three settings. Therein, ``LSR'' denotes the performance achieved by a receiver using linear shrinkage estimator. Compared with the LMMSE estimator, the linear shrinkage estimator usually has an SNR gain of nearly $1.5$ dB.\n\nThe optimal shrinkage coefficients ${{b}}$ are depicted in Figure \\ref{fig2a} for the three settings in Figure \\ref{fig1}, and we also plot the LMMSE coefficient $a$. The trend of ${{b}}$ is roughly consistent with that of ${a}$ with SNR. For a fixed SNR, the value of ${{b}}$ is closer to ${a}$ as the number of receive antennas increases. This is in line with our theoretic analysis in Section \\ref{sec:Massive}. Figure \\ref{fig2b} compares the histograms of the GMI between the linear shrinkage estimator and the LMMSE estimator, under $N_r = 8$, $R = 2$ bits\/channel use, and $\\mathrm{SNR} = 5$ dB. We observe that the linear shrinkage estimator achieves a more desirable mean-variance tradeoff compared to the LMMSE estimator: the linear shrinkage estimator induces a GMI distribution with a slightly decreased mean but a ``thinned'' tail in the low rate region, so the overall effect is a smaller outage probability.\n\n\\subsection{Decoding Performance on 5G NR LDPC}\n\nWe evaluate the performance of the linear shrinkage estimator for 5G New Radio low-density-parity-check (LDPC) short codes \\cite{Richardson-LDPCNR}. We use QPSK modulation and transmit coded symbols over a slow fading channel. The received signal is decoded via the belief propagation algorithm based on soft decision. The linear shrinkage coefficient affects the log-likelihood ratio (LLR). Figure \\ref{fig3} shows the BER performance for a code rate $1\/3$ bits\/channel use, codeword length $544$, with $16$ receive antennas. The results demonstrate that the linear shrinkage technique is capable of improving the reliability for short codes.\n\n\\section{Conclusion}\n\nIn this paper, we have proposed to apply a linear shrinkage technique to improve the outage behavior for slow fading channels with imperfect CSI. By linearly shrinking the LMMSE estimate of the CSI, the probability distribution of the GMI can be tuned to yield an improved mean-variance tradeoff, and consequently the outage probability is reduced. This effect is evident when the number of receive antennas is not too large. On the other hand, in the asymptotic regime of massive antennas, the LMMSE estimator is the only linear shrinkage estimator that achieves asymptotically vanishing outage probability at all rates. For future research, an interesting direction is to investigate whether some form of nonlinear processing, beyond the linear shrinkage receiver here, can lead to further performance improvement.\n\n\\section*{Acknowledgment}\nThis work was supported by the National Key Research and Development Program of China under Grant 2018YFA0701603.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\nEmbedding microscopic sensors, computers and actuators into\nmaterials allows physical systems to actively monitor and respond to\ntheir environments in precisely controlled ways. This is particularly so\nfor microelectromechanical systems (MEMS)~\\cite{berlin95,bryzek94,web.mems96} \nwhere the devices are\nfabricated together in single silicon wafers. Applications include\nenvironmental monitors, drag reduction in fluid flow, compact data\nstorage and improved material properties. \n\nIn many such applications the relevant mechanical processes are\nslow compared to sensor, computation and communication speeds. This\ngives a {\\em smart matter} regime, where control programs\nexecute many steps within the time available for responding to\nmechanical changes. A key difficulty in realizing smart matter{'}s\npotential is developing the control programs. This is due to the need to\nrobustly coordinate a physically distributed real-time response with\nmany elements in the face of failures, delays, an unpredictable\nenvironment and a limited ability to accurately model the system{'}s\nbehavior. This is especially true in the mass production of smart\nmaterials where manufacturing tolerances and occasional defects will\ncause the physical system to differ somewhat from its nominal\nspecification. These characteristics limit the effectiveness of\nconventional control algorithms, which rely on a single global processor\nwith rapid access to the full state of the system and detailed knowledge\nof its behavior. \n\nA more robust approach for such systems uses a collection of\nautonomous agents, that each deal with a limited part of the overall\ncontrol problem. Individual agents can be associated with each sensor or\nactuator in the material, or with various aggregations of these devices,\nto provide a mapping between agents and physical location. This leads to\na community of computational agents which, in their interactions,\nstrategies, and competition for resources, resemble natural\necosystems~\\cite{Huberman88Eco}. Distributed\ncontrols allow the system as a whole to adapt to changes in the\nenvironment or disturbances to individual components~\\cite{hogg91a}.\n\nMultiagent systems have been extensively studied in the context of\ndistributed problem solving~\\cite{durfee91,gasser89,lesser95}. They have also been\napplied to problems involved in acting in the physical world, such as\ndistributed traffic control~\\cite{nagel94},\nflexible manufacturing~\\cite{upton92}, the design\nof robotic systems~\\cite{sanderson83,williams96a},\nand self-assembly of structures~\\cite{semela95}.\nHowever, the use of multiagent systems for controlling smart matter is a\nchallenging new application due to the very tight coupling between the\ncomputational agents and their embedding in physical space.\nSpecifically, in addition to computational interactions between agents\nfrom the exchange of information, there are mechanical interactions\nwhose strength decreases with the physical distance between them.\n\nIn this paper we present a novel control strategy for unstable\ndynamical systems based on market mechanisms. This is a particularly\nchallenging problem, for in the absence of controls, the physics of an\nunstable system will drive it rapidly away from the desired\nconfiguration. This is the case, for example, for a structural beam\nwhose load is large enough to cause it to buckle and break. In such\ncases, weak control forces, if applied properly, can counter departures\nfrom the unstable configuration while they are still small. Successful\ncontrol leads to a virtual strengthening and stiffening of the material.\nIntentionally removing this control also allows for very rapid changes\nof the system into other desired configurations. Thus an effective way\nof controlling unstable systems opens up novel possibilities for making\nstructures extremely adaptive.\n\n\n\n\\section{ Dynamics of Unstable Smart Matter}\nThe devices embedded in smart matter are associated with\ncomputational agents that use the sensor information to determine\nappropriate actuator forces. The overall system dynamics is a\ncombination of the behavior at the location of these agents and the\nbehavior of the material between the agent locations. In mechanical\nsystems, displacements associated with short length scales involve\nrelatively large restoring forces, high frequency oscillations and rapid\ndamping. Hence, they are not important for the overall\nstability~\\cite{hogg96b}. Instead, stability is\nprimarily determined by the lowest frequency modes. We assume that there\nare enough agents so that their typical spacing is much smaller than the\nwavelengths associated with these lowest modes. Hence, the lower\nfrequency dynamics is sufficiently characterized by the displacements at\nthe locations of the agents only. The high-frequency dynamics of the\nphysical substrate between agents serves only to couple the agents{'}\ndisplacements.\n\nThe system we studied, illustrated in Fig.~\\ref{x:pendulum}a, consists \nof {\\it n} mass\npoints connected to their neighbors by springs. In addition a\ndestabilizing force proportional to the displacement acts on each mass\npoint. This force models the behavior of unstable fixed points: the\nforce is zero exactly at the fixed point, but acts to amplify any small\ndeviations away from the fixed point. This system can be construed as a\nlinear approximation to the behavior of a variety of dynamical systems\nnear an unstable fixed point, such as the inverted pendulae shown in the\nFig.~\\ref{x:pendulum}b. In the absence of\ncontrol, any small initial displacement away from the vertical position\nrapidly leads to all the masses falling over. In this case, the lowest\nmode consists of all the pendulae falling over in the same direction and\nis the most rapidly unstable mode of behavior for this system. By\ncontrast, higher modes, operating at shorter length scales, consist of\nthe masses falling in different directions so that springs between them\nact to reduce the rate of falling.\n\n\\begin{figure}\n\\hspace*{\\fill}\n\\epsfig{file=pend1a.eps, width=2.5in}\n\\hspace*{\\fill}\n\\epsfig{file=pend1b.eps, width=2.5in}\n\\hspace*{\\fill}\n\\caption{\\label{x:pendulum}\\small An unstable dynamical system. \na) The unstable chain with the mass points displaced from the unstable \nfixed point which is indicated by the horizontal dashed line. \nThe masses are coupled to their neighbors with springs, and those \nat the end of the chain are connected to a rigid wall. \nb) A chain of upward-pointing pendulae connected by springs as an \nexample of an unstable spatially extended system.}\n\\end{figure}\n\nThe system{'}s physical behavior is described by\n\\begin{enumerate}\n\\item the number of mass points {\\it n}\n\\item the spring constant {\\it k} of the springs\n\\item a destabilizing force coefficient {\\it f}\n\\item a damping force coefficient {\\it g}\n\\end{enumerate} \nWe also suppose the mass of each point is equal to one. The resulting\ndynamics of the unstable chain is given by\\footnote{We used a\nstandard ordinary-differential-equation solver~\\cite{shampine75} to \ndetermine the controlled system{'}s\nbehaviors.}~\\cite{goldstein80}:\n\\begin{equation}\\label{x:motion}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{{{dx_{i}}\\over{dt}}}$\\hfilneg&$\\displaystyle{{}=\nv_{i}\\hskip 0.265em }$\\cr \n$\\displaystyle{{{dv_{i}}\\over{dt}}}$\\hfilneg&$\\displaystyle{{}=\nk{\\left( x_{i-1}\\hskip -0.167em -\\hskip -0.167em \nx_{i}\\right) }+k{\\left( x_{i+1}\n\\hskip -0.167em -\\hskip -0.167em x_{i}\\right) }\n+fx_{i}-gv_{i}+H_{i}}$\\cr \n}}\\end{equation}\nwhere \\(\nx_{i}\\) is the displacement of mass point\n{\\it i}, \\(\nv_{i}\\) is the corresponding velocity, and \\(\nx_{0}=x_{n+1}=0\\) is the boundary condition. The \\(\nH_{i}\\) term in Eq.~(\\ref{x:motion})\nis the additional control force produced by the actuator attached to\nmass point {\\it i}. We suppose the\nmagnitude of this control force is proportional to the power\n\\(\nP_{i}\\) used by the actuator. For reasons of simplicity we\nuse a proportionality factor of 1.\n\nFor these systems, the long time response to any initial condition\nis determined by the eigenvalues of the matrix corresponding to the\nright hand side of Eq.~(\\ref{x:motion}).\nSpecifically, if the control force makes all eigenvalues have negative\nreal parts, the system is stable~\\cite{hogg96b}.\nThe corresponding eigenvectors are the system{'}s modes. Thus to evaluate\nstability for {\\em all} initial conditions, we can use\nany single initial condition that includes contributions from all modes.\nIf there are any unstable modes, the displacements will then grow. We\nused this technique to evaluate stability in the experiments described\nbelow.\n\n\n\n\\section{ A Power Market for Control}\nThe control problem is  how hard to push on the various mass points\nto maintain them at the unstable fixed point. This problem can involve\nvarious goals, such as maintaining stability in spite of perturbations\ntypically delivered by the system{'}s environment, using only weak control\nforces so the actuators are easy and cheap to fabricate, continuing to\noperate even with sensor noise and actuator failures, and being simple\nto program, e.g., by not requiring a detailed physical model of the\nsystem.\n\nComputational markets are one approach to this control\nproblem~\\cite{clearwater96,Ferguson88,huberman95b,Kurose89,Malone88,sutherland68,\nwaldspurger92,wellman93}.\nAs in economics, the use of prices provides a flexible mechanism for\nallocating resources, with relatively low information\nrequirements~\\cite{hayek78}: a single price\nsummarizes the current demand for each resource.\n\nIn designing a market of computational agents, a key issue is to\nidentify the consumers and producers of the goods to be traded. Various\npreferences and constraints are introduced through the definition of the\nagents{'} utilities. This ability to explicitly program utility functions\nis an important difference from the situation with human markets.\nFinally, the market mechanism for matching buyers and sellers must be\nspecified.\n\nIn the market control of smart matter treated here, actuators, or\nthe corresponding mass points to which they are attached, are treated as\nconsumers. The external power sources are the producers and as such are\nseparate from consumers. All consumers start with a specified amount of\nmoney. All the profit that the producers get from selling power to\nconsumers is equally redistributed to the consumers. This funding policy\nimplies that the total amount of money in the system will stay\nconstant.\n\nIn the spirit of the smart matter regime, where control\ncomputations are fast compared to the relevant mechanical time scales,\nwe assume a market mechanism that rapidly finds the equilibrium point\nwhere overall supply and demand are equal. Possible mechanisms include a\ncentralized auction or decentralized bilateral trades or arbitrage. This\nequilibrium determines the price and the amount of power traded. Each\nactuator gets the amount of power that it offers to buy for the\nequilibrium price and uses this power to push the unstable chain.\n\nThe utility function for using power\n{\\it P} reflects a trade-off between using\npower to act against a displacement and the loss of wealth involved.\nWhile a variety of utility functions are possible, a particularly simple\none for agent {\\it i}, expressed in terms\nof the price of the power, {\\it p}, and the\nagent{'}s wealth, \\(\nw_{i}\\), is:\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{U_{i}=-{{1}\\over{2w_{i}}}\npP^{2}+bP{\\left| X_{i}\\right| }\n}$\\cr \n}}\\end{equation}\nwhere\n\\begin{equation}\\label{x:structure}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{X_{i}=\\sum _{j=1}^{n}\na_{ij}x_{j}}$\\cr \n}}\\end{equation}\nis a linear combination of the displacements of all mass points that\nprovides information about the chain{'}s state. The parameter\n{\\it b} determines the relative importance\nto an agent of responding to displacements compared to conserving its\nwealth for future use.\n\nActuator {\\it i} always pushes in the\nopposite direction of \\(\nX_{i}\\), i.e., it acts to reduce the value of\n\\(\nX_{i}\\). In this paper we focus on the simple case of purely\nlocal control where \\(\na_{ij}=1\\) when \\(\ni=j\\) and is 0 otherwise. Thus, consumer\n{\\it i} considers only its own displacement\n\\(\nx_{i}\\). For simplicity, we use an ideal competitive market\nin which each consumer and producer acts as though its individual choice\nhas no affect on the overall price, and agents do not account for the\nredistribution of profits via the funding policy. Thus a consumer{'}s\ndemand function is obtained by maximizing its utility function as a\nfunction of power:\n\\begin{equation}\\label{x:demand}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{{{dU_{i}}\\over{dP}}}$\\hfilneg&$\\displaystyle{{}=\n-p{{P}\\over{w_{i}}}+b{\\left| X_{%\ni}\\right| }=0\\hskip 0.212em {\\ifmmode\\Rightarrow\\else$\\Rightarrow$\\fi}\\hskip 0.212em \nP_{i}{\\left( p\\right) }=b{\\left| \nX_{i}\\right| }{{w_{i}}\\over{%\np}}}$\\cr \n}}\\end{equation}\nThis demand function causes the agent to demand more power when the\ndisplacement it tries to control is large. It also reflects the\ntrade-off in maintaining wealth: demand decreases with increasing price\nand when agents have little wealth. The overall demand function for the\nsystem is just the sum of these individual demands, giving\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{P^{{\\rm demand}}{\\left( p\\right) }\n={{b}\\over{p}}\\sum {\\left| \nX_{i}\\right| }w_{i}}$\\cr \n}}\\end{equation}\n\n\nSimilarly, each producer tries to maximize its profit\n\\(\n\\rho \\) given by the difference between its revenue from\nselling power and its production cost \\(\nC{\\left( P\\right) }\\): \\(\n\\rho =pP-C{\\left( P\\right) }\\). To provide a constraint on the system to minimize\nthe power use, we select a cost function for which the cost per unit of\npower, \\(\nC{\\left( P\\right) }\/P\\) increases with the amount of power. A simple example\nof such a cost function is\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{C{\\left( P\\right) }={{1}\\over{2\na}}P^{2}}$\\cr \n}}\\end{equation}\nThe parameter {\\it a} reflects the relative\nimportance of conserving power and maintaining stability. We obtain the\nproducer{'}s supply function by maximizing its profit:\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{{{d\\rho }\\over{dP}}=p-{{dC}\\over{%\ndP}}=0\\hskip 0.212em {\\ifmmode\\Rightarrow\\else$\\Rightarrow$\\fi}\\hskip 0.212em P{\\left( \np\\right) }=ap}$\\cr \n}}\\end{equation}\nThis is the same for all producers, so the overall supply function is\nthen just\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{P^{{\\rm supply}}{\\left( p\\right) }\n=nap}$\\cr \n}}\\end{equation}\n\n\nFrom this the price and amount of traded power is determined by the\npoint where the overall supply and demand curves intersect, i.e.,\n\\(\nP^{{\\rm demand}}{\\left( p\\right) }\n=P^{{\\rm supply}}{\\left( p\\right) }\n\\). For our choices of the utility and cost functions,\nthis condition can be solved analytically to give\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{p_{trade}=\\Rad{{{b}\\over{na}}\n\\sum _{i=1}^{n}{\\left| X_{%\ni}\\right| }w_{i}}\\DoRad }$\\cr \n}}\\end{equation}\nGiven this equilibrium price, agent {\\it i}\nthen gets an amount of power equal to \\(\nP_{i}{\\left( p_{trade}\\right) }\n\\) according to Eq.~(\\ref{x:demand}) and the resulting control force is directly proportional\nto received power.\n\nWe can also consider the case where the amount of power available\nto the system is limited to \\(\nP^{{\\rm global}}_{{\\rm max}}\\). This hard constraint has the effect of limiting the\noverall supply function when the price is high so it becomes\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{P^{{\\rm supply}}{\\left( p\\right) }\n=\n\\left\\{\\matrix{nap&{\\rm if}\\hskip 0.212em p{\\ifmmode<\\else$<$\\fi}P^{%\n{\\rm global}}_{{\\rm max}}\/na\\cr \nP^{{\\rm global}}_{{\\rm max}}&\n{\\rm otherwise}\\cr }\\right.}$\\cr \n}}\\end{equation}\n\n\nThe final aspect of the market dynamics is how the wealth changes\nwith time. This is given by\n\\begin{equation}\\vcenter{\\halign{\\strut\\hfil#\\hfil&#\\hfil\\cr \n$\\displaystyle{{{dw_{i}}\\over{dt}}}$\\hfilneg&$\\displaystyle{{}=\n-pP_{i}{\\left( p\\right) }+{{1\n}\\over{n}}pP^{{\\rm demand}}{\\left( \np\\right) }}$\\cr \n$\\displaystyle{}$\\hfilneg&$\\displaystyle{{}=-b{\\left| X_{i}\\right| }\nw_{i}+{{b}\\over{n}}\\sum _{%\nj=1}^{n}{\\left| X_{j}\\right| }\nw_{j}}$\\cr \n}}\\end{equation}\nbecause we use the funding policy that all expenditures are returned\nequally to the agents in the system.\n\n\n\n\\section{ Comparing with Local Controls}\nAs a simple comparison for the market behavior, we also study a\nlocal control method. In this case, each actuator\n{\\it i} pushes with a strength that is\nproportional to the displacement of its respective mass point, and\nignores the displacements of all other mass points. Specifically, the\nlocal controls are simply given by \\(\nH_{i}=-c\\hskip 0.212em x_{i}\\).  Other control strategies~\\cite{hogg96b} \ncan try to estimate the amplitude of the\nlowest modes and push only against these modes, since these are the ones\nmost important for stability.\n\nFor comparison with the market, we restrict ourselves to the case\nwhere the amount of available power is limited. This is useful for\nevaluating the ability of different control methods to maintain\nstability using only weak forces. We distinguish two ways the power\ncould be limited for the local control. In the first, each actuator is\nseparately limited to use no more than \\(\nP_{{\\rm max}}\\) power (local control 1), which corresponds to a\nsituation where each actuator has a separate power source such as its\nown battery. Any actuator that requests more power than this maximum has\nits control force reduced to require only \\(\nP_{{\\rm max}}\\), i.e., \\(\n{\\left| H_{i}\\right| }=P_{{\\rm max}\n}\\). The second local control allows available power to\nbe moved among the different actuators and is limited only in that all\nactuators together cannot use more than \\(\nP^{{\\rm global}}_{{\\rm max}}=nP_{{\\rm \nmax}}\\) power (local control 2). This overall limit is\nimplemented by comparing the total power requested according to the\nlocal control, i.e., \\(\nP_{{\\rm request}}=c\\sum _{i}\n{\\left| x_{i}\\right| }\\) to the maximum available. If the requested amount\nexceeds the maximum, each agent has its power reduced by the factor\n\\(\nP^{{\\rm global}}_{{\\rm max}}\/P_{%\n{\\rm request}}\\) so that the overall used power equals the global\nlimit. The corresponding market has a total available power of\n\\(\nP^{{\\rm global}}_{{\\rm max}}=nP_{{\\rm \nmax}}\\).\n\n\n\n\\section{ Results}\nWe studied a chain composed of 27 mass points, all of them having\nunit mass and connected by springs with a spring constant of value 1 and\ndamping coefficient 0.1. The destabilizing force coefficient is 0.2,\nwhich is sufficient to make the system unstable when there is no control\nforce. All agents start with an initial wealth of 50 money units and we\nare using the values \\(\na=0.05\\) and \\(\nb=0.001\\) in the cost and utility functions. For definiteness,\nwe chose an initial condition where the single element in the middle of\nthe chain had a unit displacement and all other values were at zero.\nThis configuration includes a contribution from all the modes of the\nsystem, which are just sinusoidal waves in this chain with uniform\nmasses and spring constants~\\cite{goldstein80}.\nFor the local control, we used \\(\nc=0.2\\), which is more than sufficient to ensure stable\ncontrol when power is unlimited~\\cite{hogg96b}.\n\nWith \\(\nP_{{\\rm max}}=0.012\\), Fig.~\\ref{x:limit} compares\nthe performances of both local and market controls. We show both the\ntotal power use \\(\n\\sum _{i}P_{i}\\) and the average displacement of the chain\n\\(\n\\sum _{i}{\\left| x_{i}\\right| }\n\/n\\). As can be seen, for the chosen parameter values the\nmarket is able to control the unstable chain in spite of the fact that\nthe power is limited to a global maximum. This limit is reached several\ntimes. The local controls (1 and 2), on the other hand, fail in both\ncases, as seen in the figure. These results were obtained in a\nsimulation run that lasted 20 time units. A longer simulation shows that\nthe overall power usage and average displacements decrease with time for\nthe market control while displacement continues increasing for the local\ncontrols.\n\n\\begin{figure}\n\\hspace*{\\fill}\n\\epsfig{file=powerUse.ps, width=3in}\n\\hspace*{\\fill}\n\\caption{\\label{x:limit}\\small a) Time development of the overall power usage \nfor a market control (solid) and local controls 1 and 2 (dashed) in the \ncase of limited available power. With the same power limit in all three \ncases, the market is the only one that can control the unstable chain. \nPoints I, II and III mark the times at which the supply and demand curves \nintersections are shown in Fig.~\\ref{x:curves}. b) Corresponding time \ndevelopment of the average displacement for a market control and local \ncontrols 1 and 2. The market reduces the average displacement with time \nwhereas the local control is not able to prevent it from growing.}\n\\end{figure}\n\nSince the power cost function \\(\nC{\\left( P\\right) }\\) does not change, the overall supply curve never\nchanges, as shown in Fig.~\\ref{x:curves} which\ndisplays the supply curve and some demand curves for different times.\nThe demand curves depend on the displacements and wealth of the agents.\nSince these are dynamical variables, overall demand curve changes in\ntime. In addition to the times I, II and III marked in\nFig.~\\ref{x:limit}a, we also plot the overall\ndemand curves for later times IV, V and VI. This shows that the amount\nof traded power decreases with time while the unstable chain is\ncontrolled by the market.\n\n\\begin{figure}\n\\hspace*{\\fill}\n\\epsfig{file=supplyDemand.ps, width=4in}\n\\hspace*{\\fill}\n\\caption{\\label{x:curves}\\small Overall supply curve (dashed) and the overall \ndemand curves (solid) at times I=1.6, II=8.0, III=18.4, IV=40.0, \nV=60.0 and VI=90.0 for the market example of Fig.~\\ref{x:limit}.}\n\\end{figure}\n\nTo demonstrate how robust the market mechanism is, we show in\nFig.~\\ref{x:failure} the system{'}s response when\nan actuator breaks down. In this case we slightly increase the amount of\npower available compared to the simulation used for\nFig.~\\ref{x:limit} to \\(\nP_{{\\rm max}}=0.015\\), so that the local controller 2 can also control the\nunbroken system. With the system initially functioning properly, we\nturned off the actuator in the middle of the chain after 10 time units\nand observed the consequent evolution. As can be seen, the market is\nstill able to control the system whereas the local control fails to do\nso.\n\n\\begin{figure}\n\\hspace*{\\fill}\n\\epsfig{file=actfailure.ps, width=3in}\n\\hspace*{\\fill}\n\\caption{\\label{x:failure}\\small Comparison of a market control and a local \ncontrol in the case where one actuator breaks after 10 time units. \nBoth control strategies would be able to control the system when all \nactuators would work perfectly. The market is still able to control \nthe system although one actuator is broken but the local controller fails. \na) Overall used power vs.~time. b)  Average displacement vs.~time.}\n\\end{figure}\n\n\n\n\\section{ Discussion}\nIn this paper we presented a novel mechanism of controlling\nunstable dynamical systems by means of a multiagent system using a\nmarket mechanism. We described how we defined consumers{'} and producers{'}\nutility functions that lead to the overall supply and demand curves and\nevaluated the price and amount of traded power within the system.\n\nWe showed that the market approach is able to control an unstable\ndynamical system in the case of limited power whereas a traditional\nlocal control strategy fails under the same assumptions. We also\ndemonstrated that a market control adapts better to cases when an\nactuator breaks during the controlling process. These results show that\na market control can be more robust than a local control when operating\nwith given power constraints by focussing the power in those parts of\nthe system where it is most needed. This not only reduces total power\nuse but, more importantly, also allows control with weaker, and thus\neasier to fabricate, actuators.\n\nThe power of market approaches to control lies in the fact that\nrelatively little knowledge of the system to be controlled is needed.\nThis is in stark contrast to traditional AI approaches, which use\nsymbolic reasoning with extremely detailed models of the physical\nsystem. However, while providing a very robust and simple design\nmethodology, a market approach suffers from the lack of a high level\nexplanation for its global behavior. An interesting open issue is to\ncombine this approach with the more traditional AI one.\n\nAlthough we have chosen particular forms of utility, supply and\ndemand functions, there are many other functional forms that can also\ncontrol the system. These could include additional goals, such as faster\nrecovery from sudden changes and minimizing the number of active\nactuators. Furthermore, different funding strategies are possible, where\nprofits are shared unequally among agents or the funds are allocated by\nan external agent. A very promising approach is the possibility of\nimproving the performance of the system by having different market\norganizations that change in time. In our system, this corresponds to\nthe agents learning to use information on the displacements or\nvelocities of their neighbors when making their control decisions. In\nthis way the multiagent system would take advantage of the fact that\nmarkets are a simple and powerful discovery process: new methods for\nselecting trades can be tried by a few consumers or producers and, if\nsuccessful relative to existing approaches, gradually spread to other\nagents. Such a learning mechanism could help the system discover those\norganizational structures that lead to improved performance and\nadaptability.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince Ohtomo and Hwang reported the existence of a high mobility\nelectron gas at the $n$-type (LaO)$^{+}$\/(TiO$_{2}$)$^{0}$\ninterface between two band-gap insulators LAO and\nSTO\\cite{ohtomo04}, many novel properties related to the electron\ngas at the interface, such as insulator-metal transition,\nsuperconductivity, and ferromagnetism were\nfound\\cite{thiel06,cen08,rijnders08,reyren07,Caviglia08,brinkman07}.\nThe mechanism of conductivity and the dimensionality of the\ninduced carrier at the interface were intensively studied. In the\n$n$-type interface structure a polar electrical field along LAO\n[001] direction is arisen from the alternating stack of\n(LaO)$^{+}$ and (AlO$_{2}$)$^{-}$ charged layers. Charge\nreconstruction at the interface was proposed as a way to avoid\ndiverging electrostatic potential with thick LAO, which\ncompensates the polar electric field in LAO and induces electron\ngas at the interface\\cite{ohtomo04,nakaga06}. The densities and\ndistributions of the induced carriers charge  were measured in\nvarious experiments\n\\cite{ohtomo04,thiel06,brinkman07,basletic08,nakaga06,eckstein07,\nherranz07,siemons07,kalabukhov07,willmott07,yoshimatsu08,sing09}.\nTwo different origins of the induced carriers charge, i.e.\nintrinsic and extrinsic doping, have been proposed. Intrinsic\ncharge doping at the interface mainly happens in stoichiometrical\nstructure without oxygen\nvacancy\\cite{thiel06,brinkman07,basletic08, willmott07,sing09}.\nWith thick LAO overlayer the charge transfers from valence band of\nLAO surface to conduction band of STO at the interface, leading to\na metallic interface\\cite{ pentcheva09,son09, li09}. While\nextrinsic charge doping was found to be ascribed to oxygen\nvacancies in many experiments\\cite{ohtomo04,brinkman07,\nherranz07,kalabukhov07,Maurice08}.\n\n\n\nNoticeably,  the conducting properties at intrinsic and extrinsic\ndoped interfaces measured in the experiments are very different.\nIn the ideal interface of thin LAO overlayer without oxygen\nvacancy the carrier densities measured by Hall effect and x-ray\nphotoelectron spectroscopy (XPS) are even less than 0.1 electron\nper two-dimensional unit cell\n(2-d.u.c.)\\cite{thiel06,basletic08,sing09}. Recent\ndensity-functional-theory (DFT) calculations showed that ionic\npolar distortions  partially compensate polar electric field in\nLAO, reducing carrier density at the\ninterface\\cite{pentcheva06,pentcheva08,pentcheva09,ishibashi,lee08,li09}.\nFor thin LAO overlayer the carrier charge was found in XPS\nexperiment to distribute in a few layers of STO at the\ninterface\\cite{sing09}. Latest DFT study found that for less than\n7 layers of LAO the carrier charge distributes in 3 layers of STO\nbut for thicker LAO the carrier could accumulate in deep STO\nlayers about 3 nm away from the interface. However, with oxygen\nvacancies in the interface structure  the carrier densities\nmeasured in experiments vary from 0.5 to three orders of magnitude\ne\/2-d.u.c. dependent on the oxygen pressure in preparing\nprocess\\cite{ohtomo04,brinkman07,herranz07,kalabukhov07}. The\nexperiment in Ref.[\\onlinecite{herranz07}] explicitly verified\nthat high carrier density is ascribed to high concentration of\noxygen vacancies in STO substrate which were generated under lower\noxygen pressure during preparing the sample.\n\n\nBesides the carrier density, another issue which people have been\ndebating extensively is the distribution of carrier charge induced\nby oxygen vacancy. In some sample produced in high oxygen pressure\ntwo-dimensional  distributions were\nobserved\\cite{thiel06,brinkman07,basletic08,sing09}, but in others\nproduced in low oxygen pressure obvious 3-dimensional distribution\nin STO were observed \\cite{basletic08,herranz07}. These\nexperimental hints indicated the carrier distribution is related\nto the concentration of oxygen vacancies in some way. Up to now,\npeople have realized that oxygen vacancy play an important role in\nn-type LAO\/STO interface, however the locations of oxygen\nvacancies and the dependence of two-dimensional and\nthree-dimensional carrier distributions on the locations of oxygen\nvacancies are still not clear.\n\n\n\nIn this paper we studied the n-type interface with oxygen\nvacancies by using density-functional-theory (DFT) calculations.\nIt should be noted that the interface composed of sandwich\nstructure was also investigated in experiment but the structure of\nLAO overlayer on STO substrate has attracted the most of\nattentions because of its abundant conducting properties. In this\npaper we focused on the structure of LAO overlayer on STO. We\ncalculated and compared the total energies of the configurations\nwith varying locations and concentrations of oxygen vacancies.  We\nfound that the locations of oxygen vacancies are related to the\nconcentration. From the calculations of electronic structures we\nalso unveiled that the distribution of carriers is strongly\ndependent on the locations of oxygen vacancies.  Based on the\nresults from our calculations, energy band structures at the\ninterface and localized surface conductivity were discussed.\n\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{AtomicStructures.eps}\n    \\caption{(Color online) Atomic structures of (2$\\times$2) supercells\n     (a) without oxygen vacancy, and (b) with oxygen vacancy in LAO surface.\n      Oxygen vacancy is denoted by black ball.}\\label{Struc}\n\\end{figure}\n\n\\section{Computational method}\n\nWe carried out DFT calculations by using the Vienna \\textit{ab\ninitio} Simulation Package (VASP) \\cite{kresse96} within a\ngeneralized gradient approximation \\cite{wang91} together with the\nprojector augmented wave pseudopotentials \\cite{blochl94,kresse96}\nand the cut-off energy of 400 eV for the plane wave basis.  We\nmodeled the LAO\/STO interface by a slab consisting of 2 to 7  LAO\nlayers on top of  STO(001) substrate and a vacuum region of 14\n{\\AA} along the $c$-axis in a supercell geometry. Dipole\ncorrections were used to correct the errors of electrostatic\npotential, forces, and total energy caused by periodic boundary\ncondition \\cite{makov95}. To simulate various concentrations of\noxygen vacancies the in-plane unit-cells were taken as\n(2$\\times$2), (3$\\times$2), and (3$\\times$3). $\\Gamma$-centered\n(5$\\times$5), (3$\\times$5), and (3$\\times$3) \\textbf{k}-point\nmeshes were used to sample the Bullion zone, respectively.\nFig.~\\ref{Struc} (a) and (b) present relaxed (2$\\times$2)\nsupercells  without oxygen vacancy and with oxygen vacancy in LAO\nsurface, respectively. The in-plane lattice constant of the slab\nwas constrained at the calculated equilibrium lattice constant\n$a=3.942$ {\\AA} of the STO substrate. All coordinates of atomic\npositions were fully relaxed with forces less than 0.02eV\/{\\AA}\nexcept for the atoms in the bottom two layers of STO, which were\nfixed in their bulk positions.\n\n\n\n\\section{Location of oxygen vacancies of low concentration}\nIn this paper we divide the concentrations of oxygen vacancies in\nthe interface structure into two regions, i.e. low concentration\nand high concentration. In the former the concentrations of oxygen\nvacancies is less than or equal to 1\/4 per 2-d.u.c. and the\ndensity of induced carriers is less than or equal to 0.5 electron\nper 2-d.u.c.. While in the latter the concentrations of oxygen\nvacancies is larger than 1\/4 vacancy per 2-d.u.c. and the density\nof induced carriers is larger than 0.5 electron per 2-d.u.c.. To\nsimulate low concentration of oxygen vacancy in the interface\nstructure only one vacancy is involved in the supercells. While\nfor higher concentration more vacancies are involved in the\nsupercells. For convenient we use n$_{V}$ and n$_{c}$ to denote\nthe concentration of oxygen vacancies and carrier density in\n2-d.u.c., respectively.\n\n\\subsection{Binding energies of oxygen atom} \n\nOxygen vacancies were explicitly confirmed to exist in STO\nsubstrate of the interface structure produced under low oxygen\npressure\\cite{basletic08,herranz07}. However, up to now  there has\nbeen no report about whether oxygen vacancy exits in LAO. To\ninvestigate the location of oxygen vacancy we calculated the total\nenergies of oxygen vacancy in varying locations under various\nconcentrations. We used (3$\\times$2) and (2$\\times$2) supercells\nwith one vacancy as the representatives of n$_{V}$ less than and\nequal to 1\/4 per 2-d.u.c., respectively. To compare them in same\nplot we calculated the binding energy of one oxygen atom\ncorresponding to the vacancy defined as:\n$$E_{b}=(E_{V}+1\/2E_{O_{2}})-E_{0} , $$ in which $E_{V}$, $E_{O_{2}}$,\nand $E_{0}$ are total energies of the system with oxygen vacancy,\noxygen molecule and ideal system without vacancy, respectively.\nFigure~\\ref{BindingE1} presents the binding energies of  one\noxygen atom corresponding to the vacancy at varying locations in\n(3$\\times$2) and (2$\\times$2) supercell. Oxygen atom in LAO\nsurface has the lowest binding energy, which is far less than that\nin STO substrate. This indicates that oxygen vacancy is most\neasily formed in LAO surface rather than in STO substrate.\nMoreover the binding energy of oxygen atom in LAO surface\ndecreases with the thickness of LAO. Such results are strongly\nrelated to polar electric field in LAO. In the ideal interface\nstructure the polar  field shifts the energy bands of LAO toward\nhigher energy layer by layer, substantially raising the total\nenergy. While oxygen vacancy in LAO  dopes  electron carriers at\nthe interface, which partially or completely screen the polar\nfield in LAO, remarkably lowering the total energy. In contrast to\nthe vacancy in the surface, the vacancy inside LAO just screens\npart of polar field from the layer with the vacancy to the\ninterface. Therefore corresponding configuration has higher total\nenergy, as illustrated in Fig.~\\ref{BindingE1}.\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{BindingEnergy1.eps}\n    \\caption{(Color online) Binding energies of oxygen\n        atom corresponding to one vacancy at varying locations in (3$\\times$2) supercell\n        consisting of STO substrate and  4 LAO layers and\n        (2$\\times$2) supercells consisting of STO substrate and 2 or 3 LAO layers.}\n        \\label{BindingE1}\n\\end{figure}\n\n\\subsection{Formation energy of oxygen vacancy}\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{FormationEnergy.eps}\n    \\caption{(Color online) Formation energies of one oxygen vacancy\n     in LAO surface as a function of oxygen chemical potential.\n     The thicknesses of LAO vary from 2 to 5 layers for\n     (2$\\times$2) supercells, and only 4 layers of LAO for\n     (3$\\times$2) supercell. Formation energy of ideal structure\n      without vacancy is taken as zero.}\\label{FormationE}\n\\end{figure}\nTo investigate dependence of the formation of oxygen vacancy in\nthe surface on the thickness of LAO, we calculated  the formation\nenergies of one vacancy in the surface of (2$\\times$2) supercells\nconsisting of 2 to 5 layers of LAO according to the formula\n$$E_{f}=E_{V}-(E_{0}-\\mu_{O}) ,$$ in which  $\\mu_{O}$ is chemical potential of oxygen\natom, $E_{V}$, and $E_{0}$ are total energies of the system with\none oxygen vacancy in the surface and ideal system without\nvacancy, respectively. As shown in  Fig.~\\ref{FormationE}\n the formation energies of (2$\\times$2) supercells with one vacancy in LAO\nsurface decrease with the thickness of LAO overlayer and trend to\nconverge for those of thickness of LAO more than 4 layers. As\naforementioned, in the ideal structure the electrostatic energy\ninduced by polar field increases with the thickness of LAO.\nFollowing electronic calculations show that for (2$\\times$2)\nsupercells with one oxygen vacancy in the surface the density of\ndoping charge at the interface is 0.5 e\/2-d.u.c., which exactly\ncompensates polar electric field.  This implies that oxygen\nvacancy in the surface of thicker LAO overlayer eliminates more\nelectrostatic energy than that of thinner LAO overlayer.  While\nonce the LAO overlayer exceeds 4 layers, as demonstrated in\nprevious DFT studies, in ideal structure without vacancy the\nintrinsic doping charge at the interface partially screens the\npolar field, preventing the increasing of electrostatic energy\nwith the thickness of LAO\\cite{pentcheva09,son09, li09}. Therefor,\nthe energy difference between vacant and ideal structures does not\nincrease with thickness of LAO. So we can predict that under same\npreparing condition oxygen vacancy can be formed more easily in\nthe sample with thicker LAO film, while this tendency does not\nstrengthen any more once LAO exceeds 4 layers. Moreover, as shown\nin Fig.~\\ref{FormationE}, for two unitcells (2$\\times$2) and\n(3$\\times$2) which have the same 4 layers of LAO the lower\nconcentration of vacancy is more stable and therefor more easily\nbe formed. As well known, oxygen pressure is a dominant factor of\nthe formation of oxygen vacancy in LAO\/STO interface structure. In\nFig.~\\ref{FormationE} at oxygen rich limit the ideal structure has\nthe lowest formation energy. This is consistent with the\nexperimental results that vacancy-free structures were fabricated\nunder high oxygen\npressure\\cite{thiel06,brinkman07,basletic08,sing09}. While in the\nmiddle or near the poor limit vacant structures have lower\nformation energies.\n\n\n\\section{Electronic structures of oxygen vacancies of low concentration}\n\\subsection{Oxygen-vacancy state in LAO surface and induced carrier}\nAbove binding-energy calculations have shown that in the\nconfigurations of n$_{V}<=$1\/4 per 2-d.u.c. oxygen vacancy lies in\nLAO surface. To investigate the electronic properties of oxygen\nvacancy in the surface we calculated (2$\\times$2), (3$\\times$2),\n(3$\\times$3), and (4$\\times$4) supercells with one vacancy,\nrespectively. As a representative of n$_{V}<$1\/4 per 2-d.u.c., the\nelectronic structure and local atomic structure of (3$\\times$3)\nsupercell are presented in Fig.~\\ref{figDOS:1}. One can see in\nFig.~\\ref{figDOS:1} (a) and (b) that the empty state of oxygen\nvacancy lies a little bit below the conduction band minimum (CBM)\nof LAO and is localized in the surface layer. Obviously, the level\nof oxygen vacancy is higher than CBM of STO. Charge transfer from\nthe state of oxygen vacancy in the surface to conduction band of\nSTO at the interface lowers the total energy of the system. Our\ncalculation showed that 2 electrons accumulate at the interface,\ni.e. n$_{c}$= 0.22 e\/2-d.u.c., which is the twice of n$_{V}$.  The\nsame relation between n$_{c}$ and n$_{V}$ were also found in other\nconfigurations of n$_{V}<=$1\/4 per 2-d.u.c..\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{figDOS1.eps}\n    \\caption{(Color online) (a) Layer-projected DOS for the (3$\\times$3)\n        supercell with one oxygen vacancy in LAO surface. (b) Spacial\n        distribution of oxygen-vacancy state in the (3$\\times$3) surface.\n        (c) Side view of polar distortions of cations and anions in upper\n        3 layers of LAO.} \\label{figDOS:1}\n\\end{figure}\n\n\\subsection{Lattice polarization dependence on the concentration of vacancy in LAO surface}\nFor the configurations of n$_{V}<$1\/4 per 2-d.u.c. the carrier\ndensity at the interface is less than 0.5 e\/2-d.u.c.. The polar\nelectrical field in LAO overlayer is screened partly and then the\npotential gradient in LAO overlayer still remains. In\nFig.~\\ref{figDOS:1} (a) one can see that the valence and\nconduction bands shift towards higher energy layer-by-layer from\nthe interface to the surface. The polar distortions in LAO induced\nby residual polar electrical field are shown in\nFig.~\\ref{figDOS:1} (c). Summarizing for configurations of\nn$_{V}<=$1\/4  we found that with increasing the concentration of\noxygen vacancy from (4$\\times$4) to (2$\\times$2) the density of\ncarriers at the interface increases till 0.5e\/2-d.u.c and the\npotential gradient and the polar distortions reduce till vanish.\n\nFor the configurations of n$_{V}<$1\/4, the thickness of LAO is\nanother factor which affects the density of carriers at the\ninterface. For a given concentration of oxygen vacancies, with\nincreasing the thickness of LAO the VBM of LAO shifts upwards\nuntil it touches CBM of STO. This critical thickness of LAO,\ndenoted as t$_{c}$, is dependent on the concentration of vacancies\nin the surface n$_{V}$. Once the thickness of LAO overlayer\noverruns t$_{c}$ the VBM of LAO does not shift up anymore,\ninstead, more charge transfers from valence band of LAO surface to\nthe interface. According to the electrostatics we give an\nexpression of carrier density n$_{c}$ as a function of n$_{V}$ and\nt (thickness of LAO overlayer)\n\\begin{eqnarray}\n  \\label{eq:1}\n  n_{c}= \\left\\{ \\begin{array} {r@{\\quad \\quad}l}\n    \\frac{1}{2} - \\frac{A}{t-t_{0}} & t>t_{c} \\\\ \\\\   2n_{v} & t\\leq\\ t_{c}\n    \\end{array} \\right. \\\\\n  t_{c}= \\frac{A}{1\/2-2n_{V}}+t_{0} ,\n\\end{eqnarray}\nwhere A=1.97, t$_{0}$=0.053 obtained  from our previous DFT study.\nThe scale of t and t$_{0}$ is unit-cell layer. For the\nconfiguration of (3$\\times$3) with one vacancy in the surface the\ncritical thickness equals 7 u.c in our DFT calculation and 7.1\nu.c. from above formula.\n\nFigure~\\ref{figDOS:2} (a) presents the layer-projected DOSs of\n(2$\\times$2) supercell with one vacancy in LAO surface. The state\nof vacancy in LAO surface lies a little bit below the CBM of LAO\nand is localized in the surface layer. Calculated density of\ncarriers at the interface is equal to 0.5 e\/2-d.u.c., which is\nexactly enough to completely screen the polar electric field in\nLAO. One can see this point from straight-aligned VBMs and CBMs of\nLAO layers in Fig.~\\ref{figDOS:2} (a). Similar to what is shown in\nFig.~\\ref{Struc} (b),  oxygen vacancy in the surface induces\nstrong tilt distortion which even extend into a few layers of STO,\nbut average polar displacement in each LAO layer is zero. It means\nno net electrostatic field in LAO. For comparison\nFig.~\\ref{figDOS:2} (b) presents the layer-projected DOSs of\n(2$\\times$2) supercell with one vacancy in STO substrate. Oxygen\nvacancy in STO produces a shallow energy level in the CBM of STO\nand generates two electron carriers. However, strong potential\ngradient appears in LAO overlayer because no charge transfers from\nthe LAO surface to the interface.  We can conclude that carriers\ngenerated by oxygen vacancies in STO hardly have any effect on\nscreening the polar electric field in LAO overlayer.\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{figDOS2.eps}\n    \\caption{Layer-projected DOS for the (2$\\times$2)\nsupercell (a) with one oxygen vacancy in LAO surface, and (b) with\none oxygen vacancy in STO-1 layer.} \\label{figDOS:2}\n\\end{figure}\n\n\\section{Distribution of oxygen vacancies of high concentration}\n\n\\subsection{Binding energies}\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{BindingEnergy2.eps}\n    \\caption{(Color online) Binding energies of  one oxygen\n        atom at varying locations in (2$\\times$2) supercell\n        consisting of 4 u.c. layers of LAO under the condition\n         of one oxygen vacancy fixed in LAO surface.}\n        \\label{BindingE2}\n\\end{figure}\nWe modeled the configurations of n$_{V}>$1\/4 per 2-d.u.c. by\n(2$\\times$2) supercell with two oxygen vacancies. As we pointed\nout above that when n$_{V}<=$1\/4 per 2-d.u.c. oxygen vacancy\nfavors to lie in LAO surface. Considering this point in the\ncalculations of (2$\\times$2) supercell with two oxygen vacancies\none is always fixed in LAO surface and the others varies from LAO\nsurface to STO subxtrate. Fig.~\\ref{BindingE2} illustrates binding\nenergies of the oxygen atom corresponds to the second vacancy.\nAlthough oxygen atom in surface still has the lowest binding\nenergy, in contrast to the cases of n$_{V}<=$1\/4 per 2-d.u.c.  the\nenergy differences with other configurations decrease largely. The\nenergy difference between the configurations of the second vacancy\nin LAO surface and in STO substrate is about 0.2 eV, which is far\nless than that for the first vacancy. This implies that second\nvacancy has large probability to appear in STO. Following\nelectronic structure calculations show that for the configurations\nof n$_{V}>$1\/4 per 2-d.u.c. the second vacancy does not contribute\nany energy gain by screening the electrostatic field in LAO\nbecause the first vacancy already completely screens it.\n\n\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{figDOS3.eps}\n    \\caption{Layer-projected DOSs for (2$\\times$2)\n        supercells (a) with  two oxygen vacancies in LAO surface, and (b)\n        with one vacancy in LAO surface and one in LAO-3 layer.}\n\\label{figDOS:3}\n\\end{figure}\n\n\\subsection{Electronic structure of  oxygen vacancies of high concentration}\n\nFigure~\\ref{figDOS:3} (a) presents the layer-projected DOS of\n(2$\\times$2) supercell with two oxygen vacancies in surface\nAlO$_{2}$ layer.  The state of vacancies is widened due to the\ninteraction of two vacancies, and only half is occupied. Two\nelectrons in this state transfer to the interface. Another typical\nconfiguration is shown in Fig.~\\ref{figDOS:3}(b), in which one\nvacancy lies in surface AlO$_{2}$ layer  and the other in the\n (LAO-3)-AlO$_{2}$ layer. The state of oxygen\nvacancy in LAO-3 layer extends into the surface layer and is fully\noccupied, while the state of oxygen vacancy in the surface layer\nis empty. Our calculated amount of carriers at the interface is\nstill 2. In these two configurations the former has a metallic\nsurface, while the latter has an insulating surface. Both dope two\nelectron carrier at the interface, although two oxygen vacancies\nexist in LAO. Several other configurations with two oxygen\nvacancies in LAO also show the same carrier density 0.5e\/2-d.u.c\nat the interface. This indicates that oxygen vacancies in LAO can\ncontribute at most 0.5 electron carrier per 2-d.u.c. at the\ninterface, and extra vacancies apart from 1\/4 per 2-d.u.c. have no\ncontribution to the conductivity at the interface. It can be\nunderstood from the sight of polar electric field in LAO. Charge\ntransfer of 0.5 e\/2-d.u.c from the LAO surface to the interface\ncompletely compensate the polar field in LAO, lowering the total\nenergy. While Charge transfer of more than 0.5 e\/2-d.u.c would set\nup an inverse electric field in LAO, which could raise the total\nenergy.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{figDOS4.eps}\n    \\caption{Layer-projected DOSs for the (2$\\times$2)\nsupercells (a) with one oxygen vacancy in LAO surface and one in\nSTO, and (b) with one oxygen vacancy in LAO surface and two in\nSTO.} \\label{figDOS:4}\n\\end{figure}\n\nSince a great amount of oxygen vacancies were found in STO\nsubstrate in the samples produced under lower oxygen pressure, the\ncontribution of vacancies in STO to density and distribution of\nthe carriers should be investigated.  Fig.~\\ref{figDOS:4}(a) shows\nthe Layer-projected DOSs for the (2$\\times$2) supercell with one\noxygen vacancy in LAO surface and one in STO. The carrier density\nis 1.0 e\/2-d.u.c. in our calculation. With one more vacancy in\nSTO, as shown in Fig.~\\ref{figDOS:4}(b), the carrier density\nincreases  0.5 e\/2-d.u.c. more. In contrast with the upper limit\ncontribution of the vacancies in LAO every vacancy in STO\ngenerates two electron carriers in STO. Considering the vacancies\ndistributing uniformly in whole STO substrate under low oxygen\npressure in experiments, a 3-dimensional  distribution of carriers\ncan be formed. In respect of band structure no potential gradient\nappears in LAO, while valence and conduction bands of STO have\nevident bending at the interface.\n\n\n\n\n\n\\section{Discussion}\n\\subsection{Band diagrams}\n\nAccording to above calculated electronic structures we plot\nschematic band diagrams of the interface. Since oxygen vacancies\nin STO do not cause the band bending at the interface, only the\nband structure with various concentrations of oxygen vacancies in\nLAO are plotted.  Fig.~\\ref{figband} presents the band structures\nof oxygen vacancies in or near LAO surface. At the interface in\nSTO side strong band bending happens in all configurations. The\nreason is that net positive-charged LAO overlayer, produced by\nelectrons transfer from LAO surface to STO, generates an\nattractive potential to electrons in STO, lowing the potential at\nthe interface in STO side. In Fig.~\\ref{figband} (a) n$_{V}<$1\/4\nper 2-d.u.c. and n$_{c}<$ 0.5 e\/2-d.u.c.. The polar electrical  in\nLAO is partially screened and the residual field results in bands\nsloping in LAO. In Fig.~\\ref{figband} (b) n$_{V}=$1\/4 per 2-d.u.c.\nand n$_{c}=$ 0.5 e\/2-d.u.c.. The polar field is completely\nscreened, leading to flat bands in LAO. In contrast with\nn$_{V}<$1\/4 per 2-d.u.c. in the case of n$_{V}=$1\/4 per 2-d.u.c.\nmore charge transfers from the surface to the interface, leading\nto larger band bending at the interface in STO side. In\nFig.~\\ref{figband} (c) n$_{V}>$1\/4 per 2-d.u.c. but still n$_{c}=$\n0.5 e\/2-d.u.c.. The band structure is the same as the case of\nn$_{V}=$1\/4 but the surface is metallic. In Fig.~\\ref{figband} (d)\nvacancies are in surface and the layer below surface and the\nsurface is insulating. In previous XPS experiment a flat valence\nband was observed\\cite{yoshimatsu08}. According to above analysis\none can easily realize that such a band structure is arisen from\nthe oxygen vacancies of n$_{V}>=$1\/4 per 2-d.u.c. in LAO surface.\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{BandDiagram.eps}\n    \\caption{Band diagram of the interface of LAO overlayer on STO  with oxygen vacancies\n    in LAO surface for  (a) n$_{V}<$ 1\/4, (b) n$_{V}=$ 1\/4, (c) n$_{V}>$ 1\/4,\n    (d) n$_{V}>$ 1\/4 but vacancies in both surface layer\n    and the layer below surface. } \\label{figband}\n\\end{figure}\n\n\\subsection{Charge carriers distribution: 2-dimensional or 3-dimensional}\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{CarrierDistribution.eps}\n    \\caption{(a) Layer-resolved carrier density in STO\n        for (LAO)$_{3}$\/(STO)$_{17}$-(2$\\times$1) with one oxygen vacancy in LAO surface.\n        (b) Charge density plot of the induced carrier.} \\label{Carrier}\n\\end{figure}\n\n\nTo clarify the distribution of the carriers induced by oxygen\nvacancy we carried out the calculations of electronic structure\nfor (LAO)$_3$\/(STO)$_{17}$-(2$\\times$1) supercell by employing 17\nSTO layer. With one oxygen vacancy in LAO surface the integrated\ncarrier density in STO is 0.5 e\/2-d.u.c.. While the carriers in\nSTO consist of two separate components, as illustrated in\nFig.~\\ref{Carrier}(a). The interface component contains the main\npart of carriers, and the extended part with a broad peak locates\nnear the 14th layer, i.e. about 5 nm, from the interface. The\npotential profile in the STO side illustrated in\nFig.~\\ref{figband} resembles the case of the inversion layer in\nmetal-oxide-semiconductor field-effect transistor and\nsemiconductor heterostructures\\cite{ando82}. Within this\ntriangle-like potential, the carriers in the lowest bound state\nwhich can be approximately described by the Airy function\naccumulate at a few nm away from the interface\\cite{ando82}.\nHowever, the interface state is originated from different\nmechanism. To understand the nature of two different components of\nthe carriers, we plotted the charge density of the induced carrier\nin the STO side in Fig.~\\ref{Carrier}(b). The interface component\nconsists mostly of the Ti $d_{xy}$ orbitals, but the extended\ncomponent has contributions from all the Ti $t_{2g}$ orbitals,\ni.e., $d_{xy}$, $d_{yz}$, and $d_{zx}$ states. The character at\nthe interface stems from a strong compressive distortion of the\nTiO$_{6}$ octahedron at the interface, which produces a strong\ntetragonal field, lowering of the $d_{xy}$ state. Similar\ncharacter of carriers was found in the ideal interface of LAO\/STO\nwithout vacancy, and the mechanism was detailedly analyzed in our\nprevious work\\cite{li09}.\n\nFrom Fig.~\\ref{Carrier} one can see that most of carriers\naccumulate at the interface in four layers of STO although there\nis extended state. Such a two-dimensional distribution of carriers\ninduced by oxygen vacancy in LAO surface is consistent with the\ntwo-dimensional electron gas measured in LAO\/STO samples produced\nunder higher oxygen pressure\\cite{ohtomo04,basletic08}. While\nunder low oxygen pressure oxygen vacancies can be formed uniformly\nin STO besides LAO surface. As well known, oxygen vacancies in STO\ngenerate a shallow level in conduction band and Fermi level lies\nin the CBM of STO. Considering band-bending at the interface the\ncarriers first fill in the potential valley at the interface and\nthen in STO inside. This implies that large amount of oxygen\nvacancies in LAO\/STO system would generate a three dimensional\ndistribution of carriers in STO but with higher carrier density\nnear the interface. Above picture about carrier distribution was\nmeasured in previous experiment by using conducting atomic forcer\nmicroscopy (AFM), in which the carrier density was found to decays\nexponential near the interface and extends a few $\\mu$m in\nSTO\\cite{basletic08}.\n\n\n\\subsection{Localized surface conductivity}\nAnother finding in our calculations which we like to emphasize is\nthe surface conductivity in the system of LAO\/STO. As shown in\nFig.~\\ref{figDOS:3}(a), oxygen vacancies of more than 1\/4 per\n2-d.u.c. generated a metallic surface. In contrast to nearly\nuniform distribution of the carriers at the interface, the\ncarriers in the surface is relatively localized. In the case of\nFig.~\\ref{figDOS:3}(a), the states of two close vacancies overlap.\nA part of electrons in the states transfer to the interface and\nthe left form localized surface carriers. This localized surface\ncarriers may be responsible for the few-nanometer-size conducting\nislands in the surface of LAO\/STO found in recent\nexperiment\\cite{cen08}.\n\n\\section{Summary}\nWe investigated the n-type LAO overlayer on STO(001) substrate\nwith oxygen vacancies by using DFT calculation. We found that\noxygen vacancies favor to appear first in LAO surface, which\ngenerate a two-dimensional carriers at the interface. The density\nof carriers induced by vacancies in LAO surface has an upper limit\n0.5 e\/2-d.u.c.. For the concentration of vacancies in LAO surface\nless than 1\/4 per 2-d.u.c., the density of induced carriers at the\ninterface is less than 0.5 e\/2-d.u.c. and the energy bands in LAO\nslope up from interface to surface. While for that of equal to or\nmore than 1\/4 per 2-d.u.c. the density of induced carriers is\nequal to 0.5 e\/2-d.u.c. and no slope of the energy bands occurs in\nLAO. We found that for the case of the concentration of oxygen\nvacancies more than 1\/4 per 2-d.u.c. the surface presents\nmetallicity. We also investigated the role of oxygen vacancy in\nSTO. We found that every oxygen vacancy in STO generates two\nelectron carrier, but this carrier charge has no effect on\nscreening the polar electric field in LAO. We predict that when a\nlarge amount of oxygen vacancies present in LAO\/STO system the\ncarriers in STO show a three-dimensional distribution with higher\ndensity at the interface.\n\n$  $\n\n\\acknowledgements\nThis work was supported by BK21 and KOSEF\nthrough the ARP (R17-2008-033-01000-0).  We also acknowledge the\nsupport from KISTI under the Supercomputing Application Support\nProgram.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent Let $d \\ge 2$ and $2d\/(d+2)<p<\\infty$ and $N \\ge 1$. Consider the following parabolic system of $p$-Laplace type:\n\\begin{equation}\n\\label{pparatype.eq}\n\\frac{\\partial u_i}{\\partial t} = \\mathop{\\operatorname{div}}\\nolimits A_i(t,x, \\nabla u) + B_i(t,x,\\nabla u), \\quad i = 1, \\dots, N, \\quad \\text{ in $I \\times Q$},\n\\end{equation}\nwhere $I \\subset \\mathbb{R}$ is an interval, $Q \\subset \\mathbb{R}^d$ a cube, and $A_i$ and $B_i$ satisfy certain structural conditions. These are the same as in \\cite{KL} and do not require any smoothness of $A_i$ and $B_i$, see Section~\\ref{subsec.structure}. In a celebrated paper, Kinnunen and Lewis have obtained the higher integrability of the gradient of weak solutions. \n\n\\begin{theorem}[Theorem~2.8 in {\\cite{KL}}]\n\\label{T2.8KL} \nThere exists $\\delta >0$ depending on $p$, $d$ and the structural constants $c_1$, $c_2$ and $c_3$ such that if $u \\in L^2(I \\times Q) \\cap L^p(I;W^{1,p}(Q))$ is a weak solution to \\eqref{pparatype.eq}, $I' \\Subset I$ and $Q' \\Subset Q$, then \n\\[\nu \\in L^{p + \\delta}(I';W^{1,p + \\delta}(Q')).\n\\]\nThe norm of $u$ in $L^{p + \\delta}(I';W^{1,p + \\delta}(Q'))$ depends on the same constants, on $N$, $I$, $I'$, $Q$, $Q'$, the structural constant $c_4$ and the norms $\\|u\\|_{L^2(I \\times Q)}$ and $\\|u\\|_{L^p(I; W^{1,p}(Q))}$.\n\\end{theorem}\n\nThe case $p=2$ is due to earlier work of Giaquinta and Struwe \\cite{GS1982}. The significance of these results is highlighted by the otherwise lacking regularity for solutions to parabolic systems, which can be essentially discontinuous.\n\nIn this short note we prove the following in-time H\\\"older continuity as an addendum to the Kinnunen--Lewis result.\n\n\n\\begin{theorem}\n\\label{mainthrm.thrm} \nLet $\\alpha := \\frac{1}{2} (\\tfrac{1}{p} - \\tfrac{1}{p+\\delta})$ and $q := \\frac{2}{1-2\\alpha}$, where $\\delta>0$ is from Theorem~\\ref{T2.8KL}. If $u \\in L^2(I \\times Q) \\cap L^p(I;W^{1,p}(Q))$ is a weak solution to \\eqref{pparatype.eq}, $I' \\Subset I$ and $Q' \\Subset Q$, then there is a representative\n\\[\nu \\in C^\\alpha(I'; L^q(Q')).\n\\]\nThe norm of $u$ in $C^\\alpha(I'; L^q(Q'))$ has the same dependencies as in Theorem \\ref{T2.8KL}. Moreover, for a.e.\\ $x \\in Q'$ the restriction $u|_{I' \\times \\{x\\}}$ is $\\alpha$-H\\\"older continuous.\n\\end{theorem}\n\nExperts in interpolation theory of vector-valued Triebel--Lizorkin type spaces (see \\cite{Denk-Kaip}) will realize that Theorem~\\ref{mainthrm.thrm} can be obtained from midway complex interpolation of the smoothness properties\n\\begin{align*}\nu \\in L^{p + \\delta}(I';W^{1,p + \\delta}(Q')) \\quad \\& \\quad \nu \\in W^{1,p'}(I';W^{-1,p'}(Q')),\n\\end{align*}\nwhere the second one follows from the equation \\eqref{pparatype.eq}. Since $2\/q = 1\/(p+\\delta) + 1\/p'$, we find $u \\in H^{1\/2,q}(I'; L^q(Q'))$. As time is a one-dimensional variable, this breaks the threshold $1-q\/2 < 0$ in embeddings of such vector-valued Bessel potential spaces and leads to H\\\"older continuity. Still, we believe that an elementary and self-contained proof to deduce Theorem~\\ref{mainthrm.thrm} from Theorem~\\ref{T2.8KL} will be of interest for a broader audience and the purpose of our note is to provide such an argument. \n\nThe abstract strategy sketched above is our guide in doing so. First, we smooth and localize the weak solution $u$ and use the equation to write the $t$-derivative of the approximant as a global negative order Bessel potential (Lemma \\ref{tdervbound.lem}). Second, we use the scalar-valued Mihlin Fourier multiplier theorem and Hadamard's three lines theorem from complex analysis to bound a fractional order potential (Proposition~\\ref{besselinterp.prop}). Third, a Fourier analytic characterization of H\\\"older continuity can be used to obtain the desired regularity of the approximant and further that of the local solution itself (Section~\\ref{proof.sec}).\n\nWe close this introduction with a brief comparison to our previous work with P.~Auscher in \\cite{ABES1}, where we obtained regularity as in Theorem~\\ref{mainthrm.thrm} for linear operators and $p=2$ by a more involved approach. See also \\cite{zaton} for a generalization to higher order systems.\nThe flexibility in the definition of the structure functions $A$ and $B$, allows us to use Theorem~\\ref{mainthrm.thrm} for inhomogenous linear systems of the form\n\\[\\frac{\\partial u_i}{\\partial t} - \\mathop{\\operatorname{div}}\\nolimits A_i(t,x, \\nabla u) - B_i(t,x,\\nabla u) = \\mathop{\\operatorname{div}}\\nolimits F_i + f_i, \\quad i = 1, \\dots, N, \\quad \\text{ in $I \\times Q$},\\]\nwhere $F, f \\in L^{2 + \\eta}$ for some $\\eta > 0$. The condition on $f$ is more restrictive than in \\cite{ABES1}. This is needed here -- as in the classical Lions theory~\\cite{Li57} -- because we use $u \\in W^{1,p'}(I'; W^{-1,p'}(Q'))$ as \\emph{a priori} information.\n\\bigskip\n\n\\noindent \n\\textbf{Acknowledgement.} This research was supported by the CNRS through a PEPS\nJCJC project and by DFG through DFG SFB 1060 and DFG EXC 2047.\n\n\n\\section{Preliminaries}\n\\label{sec:prelim}\n\n\n\\subsection{Structural assumptions}\n\\label{subsec.structure}\nWe summarize the assumptions of \\cite{KL}. The matrix-valued function $A: I \\times Q \\times ({\\mathbb{R}^d})^N  \\to \\mathbb{R}^{d \\times N}$ has columns given by\n\t\\[A_i = A_i(t,x,V), \\quad i = 1, \\ldots, N \\]\nand the vector-valued function $B : I \\times Q \\times ({\\mathbb{R}^d})^N   \\to \\mathbb{R}^N $ has scalar entries\n\t\\[B_i = B_i(t,x, V), \\quad i = 1, \\ldots, N.\\]\nBoth are (Lebesgue) $(d+1)$-measurable functions on $I \\times Q$, whenever $V = V(t,x)$ is $(d+1)$-measurable on $I \\times Q$. For example, $A$ and $B$ could be of Carath\\'{e}odory type. We also assume there are positive constants $c_j$, $j=1,2,3$, such that for almost every $(t,x) \\in I \\times Q$ and every $V \\in ({\\mathbb{R}^d})^N $,\n\\begin{align*}\n\t|A_i(t,x, V)| &\\le c_1|V|^{p-1} + h_1(t,x),\\\\\n\t|B_i(t,x, V)| &\\le c_2|V|^{p-1} + h_2(t,x),\n\\intertext{for $i = 1,\\dots N$ and}\n\t\\sum_{i = 1}^N\\langle A_i(t,x, V), V_i &\\rangle  \\ge c_3 |V|^p - h_3(t,x).\n\\end{align*}\nHere $\\langle \\cdot, \\cdot \\rangle$ is the standard inner product on ${\\mathbb{R}^d}$ and $h_j$, $j = 1,2,3,$ are measurable functions on $I \\times Q$ satisfying\n\\[\\|(|h_1| + |h_2|)^{p\/(p-1)} + |h_3|\\|_{L^{\\hat{q}}(I \\times Q)} = c_4 < \\infty,\\]\nfor some $\\hat{q} > 1$.\n\n\n\\subsection{Weak solutions}\n\\label{subsec.weak}\nThe space $L^{p}(I; W^{1,p}(Q))$ consists of all functions $f \\in L^{p}(I \\times Q)$ so that for almost every $t \\in I$ the function $f(t, \\cdot)$ is in the usual Sobolev space $W^{1,p}(Q)$ and \n\\[\\|f\\|_{L^{p}(I; W^{1,p}(Q))} := \\|f\\|_{L^{p}(I \\times Q)} + \\||\\nabla f|\\|_{L^{p}(I \\times Q)} < \\infty. \\]\nWe use the same notation for $\\mathbb{R}^{N}$ valued functions with the obvious interpretation. We then say that $u$ is a weak solution to \\eqref{pparatype.eq} if $u \\in L^2(I \\times Q) \\cap L^p(I;W^{1,p}(Q))$ and if \n\\[\\int_{I} \\int_{Q} \\sum_{i=1}^N \\left(-u_i\\frac{\\partial \\phi_i}{\\partial t}  +\\langle A_i(t,x,\\nabla u), \\nabla \\phi_i \\rangle - B_i(t,x,\\nabla u)\\phi_i \\right) \\; dx \\, dt = 0\\]\nholds for all $\\phi = (\\phi_1, \\dots, \\phi_N) \\in C^\\infty_c(I \\times Q)$.\n\n\n\\subsection{Potential spaces}\n\\label{subsec.potential}\nWe define the Fourier transform on the Schwartz space $\\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C})$ as usual by\n\\begin{align*}\n\\mathcal{F} f(\\tau, \\xi) = \\iint e^{-\\i \\tau t - \\i\\langle \\xi, x \\rangle} f(t,x) \\; d x \\, d t\n\\end{align*}\nand extend it to the tempered distributions by duality. The partial Fourier transforms with respect to only space or time variables are denoted by the subscripts $x$ and $t$. We define the Bessel potentials of order $s \\in \\mathbb{C}$ through\n\\begin{align*}\nJ^{s} f & = \\mathcal{F}^{-1} ( (1+ |\\cdot|^{2} )^{-s\/2} \\mathcal{F} f).\n\\end{align*}\nAgain, a subscript $x$ or $t$ tells with respect to which variable the potential is taken. If $s > 0$ and $f \\in L^p({\\mathbb{R}^d})$ for some $p \\in (1,\\infty)$, then $J_x^s f$ is given as a convolution with an integrable function\n\\begin{align}\n\\label{potential-formula.eq}\nJ_x^s f = G_x^s \\ast_x f, \\qquad G_x^s (x) = \\frac{1}{(4 \\pi)^{\\frac{s}{2}} \\Gamma(\\frac{s}{2})} \\int_0^{\\infty} \\delta^{\\frac{s-d}{2}} \\mathrm{e}^{\\frac{-\\pi |x|^2}{\\delta} - \\frac{\\delta}{4\\pi}} \\; \\frac{d \\delta}{\\delta}.\n\\end{align}\nSee Section V.3.1 in \\cite{Stein} for this classical formula. The Bessel potential space $H^{1,p}({\\mathbb{R}^d}) = \\{J_x^1 f: f \\in L^{p}({\\mathbb{R}^d})\\}$ with norm $g \\mapsto \\| J_x^{-1}g \\|_{L^{p}({\\mathbb{R}^d})}$ coincides with $W^{1,p}({\\mathbb{R}^d})$ up to equivalence of norms. See Section V.3.3 in \\cite{Stein}. \n\n\n\\subsection{Mollification}\n\\label{subsec.molli}\nThe definition of weak solutions does not imply any \\emph{a priori} regularity in time direction. This causes technical problems, which in the context of this paper can be overcome through a mollification argument. Let $\\varphi \\in C_c^\\infty(\\mathbb{R}^{d+1})$ be an even function with integral one that we fix from this point on. \nFor $ \\epsilon >0$ we denote the mollification of a function $g$ with $\\varphi$ by\n\\[g_\\epsilon(t,x) := \\iint \\frac{1}{\\epsilon^{n+1}} \\varphi \\left( \\frac{t - s}{\\epsilon}, \\frac{x-y}{\\epsilon} \\right) g(s,y) \\; dy \\, ds .  \\] \n\n\n\n\\section{A priori potential estimate}\n\\label{tdersec.sec}\n\n\n\\noindent We rephrase integrability and differentiability of localized weak solutions to \\eqref{pparatype.eq} using Bessel potentials. The first inequality below is a reference to Theorem \\ref{T2.8KL} whereas the second one expresses the regularity of the time derivative of the localized solution that follows from the equation. We call a constant \\emph{admissible} if it depends on $p$, $d$, $N$, $I$, $Q$, $c_1,\\ldots, c_4$, $\\|u\\|_{L^2(I \\times Q)}$, $\\|u\\|_{L^p(I; W^{1,p}(Q))}$ and the chosen cut-off function $\\chi \\in C_c^{\\infty}( I\\times Q; \\mathbb{R})$. \n\n\\begin{lemma}\n\\label{tdervbound.lem} \nLet $\\chi \\in  C_c^{\\infty}( I\\times Q )$. Let $u$ be a weak solution to \\eqref{pparatype.eq} in $I \\times Q$ and define $v := \\chi u$. Then there is an admissible constant $C$ such that for any $\\epsilon > 0$,\n\\begin{align*}\n\\| J_{x}^{-1} (v_\\epsilon) \\|_{L^{p+\\delta}(\\mathbb{R}^{d+1})} &\\leq C, \\\\\n\\| J_{t}^{-1} J_{x}^{1} (v_\\epsilon) \\|_{L^{p'}(\\mathbb{R}^{d+1})} &\\leq C.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe use the symbol $\\lesssim$ for inequalities that hold up to a multiplicative admissible constant. We obtain from Young's inequality, the choice of $\\chi$ and Theorem~\\ref{T2.8KL} that\n\\begin{align*}\n\\|v_\\epsilon\\|_{L^{p+\\delta}(\\mathbb{R}; W^{1,p+\\delta}({\\mathbb{R}^d}))} \n\\lesssim \\|u\\|_{L^{p+\\delta}(I; W^{1,p+\\delta}(Q))} \n\\leq C.\n\\end{align*}\nBy coincidence of Sobolev and potential spaces, the left-hand side is comparable to $\\| J_{x}^{-1} (v_\\epsilon) \\|_{L^{p+\\delta}(\\mathbb{R}^{d+1})}$. Hence, we have the first estimate.\n\nTo prepare the second estimate, we fix $\\phi \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{R}^{N})$ normalized such that $\\|\\phi\\|_{L^p(\\mathbb{R}; W^{1,p}({\\mathbb{R}^d}))} = 1$. By H\\\"older's inequality we have that\n\\begin{align*}\n\\bigg|\\iint \\langle v_\\epsilon, \\phi \\rangle  \\; dx \\, dt \\bigg|\n= \\bigg|\\iint \\langle v, \\phi_\\epsilon \\rangle  \\; dx \\, dt \\bigg|\n&\\lesssim \\|u \\|_{L^\\infty(I; L^2(Q))} \\| \\phi_{\\epsilon }\\|_{L^p(I; L^{2}(Q))}.\n\\end{align*}\nThe Caccioppoli inequality (Lemma~3.2 in \\cite{KL} with $a=0$) yields\n\\begin{align*}\n\\|u \\|_{L^\\infty(I; L^2(Q))} \\lesssim \\|u \\|_{L^2(I \\times Q)} + \\|u \\|_{L^p(I; W^{1,p}(Q))}.\n\\end{align*}\nSince $p \\geq \\frac{2d}{d + 2}$, we have the Sobolev embedding $W^{1,p}(Q) \\subseteq L^{2}(Q)$. Thus,\n\\begin{align*}\n\\|\\phi_{\\epsilon }\\|_{L^p(I; L^{2}(Q))} \\lesssim \\|\\phi_{\\epsilon }\\|_{L^p(I; W^{1,p}(Q))} \\leq \\|\\phi\\|_{L^p(\\mathbb{R}; W^{1,p}({\\mathbb{R}^d}))} = 1.\n\\end{align*}\nAltogether, we have found that\n\\begin{align}\n\\label{tdervbound.lem.eq1}\n\\bigg|\\iint \\langle v_\\epsilon, \\phi \\rangle  \\; dx \\, dt \\bigg| \\lesssim \\|u \\|_{L^2(I \\times Q)} + \\|u \\|_{L^p(I; W^{1,p}(Q))}.\n\\end{align}\nNext, we get from the equation for $u$, using the summation convention for $i=1,\\ldots, N$ and omitting the variable of integration $dx\\, dt$ for the sake of readability, \n\\begin{align*}\n-\\iint \\langle \\partial_t (v_\\epsilon), \\phi \\rangle \n\t&= \\iint \\bigg(\\langle u, \\partial_t (\\chi   \\phi_\\epsilon)\\rangle -  \\langle u, (\\partial_t\\chi)   \\phi_{\\epsilon} \\rangle \\bigg)  \\\\\n\t&= \\iint \\bigg(\\langle A_i(t,x,\\nabla u), \\nabla (\\chi  \\phi_{\\epsilon})_i \\rangle - B_i(t,x,\\nabla u)(\\chi   \\phi_{\\epsilon} )_i \n\t\t\t- \\langle u, (\\partial_t\\chi) \\phi_{\\epsilon}\\rangle \\bigg)  \\\\\n    &= \\iint \\langle  (\\chi A_i(t,x,\\nabla u))_{\\epsilon}, \\nabla \\phi_i \\rangle\n\t\t\t+ \\iint \\langle A_i(t,x,\\nabla u), (\\nabla \\chi)  (\\phi_{\\epsilon})_i \\rangle  \\\\ \n    &\\quad - \\iint B_i(t,x,\\nabla u)(\\chi \\phi_{\\epsilon})_i \n \t\t\t- \\iint \\langle u, (\\partial_t\\chi)   \\phi_{\\epsilon} \\rangle  \\\\ \n \t&=: \\mathrm{I} + \\mathrm{II} - \\mathrm{III} - \\mathrm{IV}.\n\\end{align*}\nUsing H\\\"older's inequality and the upper bound for $A$, we have\n\\[\n|\\mathrm{I}| \n\t\\lesssim (c_4 + c_1\\|\\nabla u\\|_{L^p(I \\times Q)}^{p-1}) \\| \\nabla \\phi\\|_{L^p(I \\times Q)} \\leq c_4 + c_1\\|\\nabla u\\|_{L^p(I \\times Q)}^{p-1}.\n\\]\nSimilarly, replacing $\\nabla \\phi$ by $\\phi$, we get\n\\[\n|\\mathrm{II}| + |\\mathrm{III}| \\lesssim c_4 + (c_1+c_2)\\|\\nabla u\\|_{L^p(I \\times Q)}^{p-1}.\n\\]\nFor $\\mathrm{IV}$, we can argue as in \\eqref{tdervbound.lem.eq1} with $\\phi_\\epsilon$ replaced by $(\\partial_t \\chi)\\phi_\\epsilon$, in order to give \n\\[\n|\\mathrm{IV}| \\lesssim \\|u \\|_{L^2(I \\times Q)} + \\|u \\|_{L^p(I; W^{1,p}(Q))}.\n\\]\nSummarizing these estimates, we get\n\\[\n\\bigg|\\iint \\langle v_\\epsilon, \\phi \\rangle  \\; dx \\, dt \\bigg| + \\bigg|\\iint \\langle \\partial_t (v_\\epsilon), \\phi \\rangle  \\; dx \\, dt \\bigg| \\leq C\n\\]\nThis is true for any $\\phi \\in \\mathcal{S}(\\mathbb{R}^{d+1})$ normalized in $L^p(\\mathbb{R}; W^{1,p}({\\mathbb{R}^d}))$. In view of the equivalence of Sobolev and potential spaces on ${\\mathbb{R}^d}$, this is the same as taking $\\phi = J_x^1 \\psi$, where $\\psi \\in \\mathcal{S}(\\mathbb{R}^{d+1})$ is normalized in $L^p(\\mathbb{R}^{d+1})$. Hence, we get\n\\[\n\\bigg|\\iint \\langle J_x^{1}v_\\epsilon, \\psi \\rangle  \\; dx \\, dt \\bigg| + \\bigg|\\iint \\langle \\partial_t (J_x^1 v_\\epsilon), \\psi \\rangle  \\; dx \\, dt \\bigg| \\leq C\n\\]\nSince  $\\mathcal{S}(\\mathbb{R}^{d+1})$ is dense in $L^{p'}(\\mathbb{R}^{d+1})$, we obtain\n\\[\n\\|J_x^1 v_\\epsilon\\|_{L^p(\\mathbb{R}^{d+1})} + \\|\\partial_t (J_x^1 v_\\epsilon)\\|_{L^p(\\mathbb{R}^{d+1})} \\leq C\n\\]\nNow, we invoke the equivalence of Sobolev and potential spaces in $t$ and apply Fubini's theorem to conclude the bound for $\\|J_t^{-1} J_x^1 (v_\\epsilon)\\|_{L^p(\\mathbb{R}^{d+1})}$.\n\\end{proof}\n\n\n\\section{Interpolation estimate of a mixed potentials}\n\n\n\\noindent We begin by recalling (a simple version of) the Mihlin multiplier theorem.\n\n\n\\begin{proposition}[Theorem 8.2 in {\\cite{M-S}}]\n\\label{multthrm.thrm}\nLet $n\\geq 1$, let $m: \\mathbb{R}^{n} \\to \\mathbb{C}$ satisfy, for all multi-index of order $|\\alpha| \\leq n+2$ and all $\\xi \\neq 0$,\n\\[\n|\\partial^\\alpha_{\\xi}m(\\xi)| \\le M  |\\xi|^{-|\\alpha|}.\n\\]\nThen, for any $q \\in (1,\\infty)$ there is a constant $C=C(n,q)$, such that for all $\\phi \\in \\mathcal{S}(\\mathbb{R}^n)$ and for $\\mathcal{F}$ the Fourier transform on $\\mathcal{S}(\\mathbb{R}^n)$,\n\\[\n\\|\\mathcal{F}^{-1}(m \\mathcal{F} \\phi)\\|_{L^q(\\mathbb{R}^n)} \\leq CM \\|\\phi\\|_{L^q(\\mathbb{R}^n)}.\n\\]\n\\end{proposition}\n\nThe multiplier theorem entails quantitative bounds for Bessel potentials.\n\n\\begin{lemma}\n\\label{BesselBound.lem}\nLet $a \\in [0,\\infty)$ and $b \\in \\mathbb{R}$. For all $q \\in (1,\\infty)$ there is a constant $C = C(d,q)$ such that for all $\\phi \\in \\mathcal{S}({\\mathbb{R}^d})$,\n\\[\n\\|J_x^{2a+2\\i b} \\phi \\|_{L^q({\\mathbb{R}^d})} \\leq C (1+a+|b|)^{d+2} \\|\\phi\\|_{L^q({\\mathbb{R}^d})}.\n\\]\nThe same holds for $J_t^{2a + 2\\i b}$ upon replacing $d$ by $1$.\n\\end{lemma}\n\n\\begin{proof}\nWe put $h(\\sigma) = (1+\\sigma)^{-a-\\i b}$. According to the Mihlin multiplier theorem, we have\n\\[\n\\|J_x^{2a+2\\i b} \\phi \\|_{L^q({\\mathbb{R}^d})} \\leq C M \\|\\phi\\|_{L^q({\\mathbb{R}^d})},\n\\]\nwhere\n\\[\nM = \\max_{|\\alpha| \\leq d+2} \\sup_{\\xi \\neq 0} |\\xi|^{|\\alpha|} |\\partial_\\xi^\\alpha(h(|\\xi|^2))|.\n\\]\nLet $\\alpha$ be any multi-index. By induction on the length of $\\alpha$, we find numerical constants $c_\\beta(\\alpha)$, one for each multi-index $\\beta$ with $2 \\beta_i \\leq \\alpha_i$, $i=1,\\ldots,d$, such that\n\\[\n\\partial_\\xi^\\alpha(h(|\\xi|^2)) = \\sum_\\beta c_\\beta(\\alpha) h^{(|\\alpha|-|\\beta|)}(|\\xi|^2) \\xi^{\\alpha-2\\beta}.\n\\]\nSince the higher order derivatives of $h$ satisfy \n\\begin{align*}\n|h^{(k)}(\\sigma)|\n\\leq c(k)(1+a+|b|)^k(1+\\sigma)^{-a-k},\n\\end{align*}\nwe can take $M= C(d)(1+a+|b|)^{d+2}$. The one dimensional case is clearly included in the computation.\n\\end{proof}\n\nWe deduce the following interpolation inequality. In the proof we shall use the notion of holomorphic functions $f: \\Omega \\subset \\mathbb{C} \\to L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$. Holomorphy is defined via convergence of difference quotients. If $f$ is locally bounded, then it is equivalent to holomorphy of $z \\mapsto \\iint f(z) \\overline{\\phi} \\; dx \\, dt$ for all $\\phi \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$. The reader can refer to Appendix~A of \\cite{ABHN} for further background.\n\n\\begin{proposition}\n\\label{besselinterp.prop}\nLet $f \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$, let $\\theta \\in (0,1)$ and let $q_0,q_{\\theta}, q_1 \\in (1,\\infty)$ be such that $1\/{q_{\\theta}} = (1-\\theta)\/{q_0} + \\theta\/{q_1}$. Then there is a constant $C = C(d,N,q_0,q_1)$ such that\n\\[ \n\\| J^{2\\theta - 1}_xJ_t^{-\\theta} f \\|_{L^{q_{\\theta}}(\\mathbb{R}^{d+1})} \\leq  C\\| J_x^{-1}f \\|_{L^{q_0}(\\mathbb{R}^{d+1})}^{1-\\theta} \\|J_x^{1} J_t^{-1} f  \\|_{L^{q_1}(\\mathbb{R}^{d+1})}^{\\theta}.\n\\]\n\\end{proposition}\n\\begin{proof}\nBy duality, we have\n\\begin{align}\n\\| J^{2\\theta - 1}_xJ_t^{-\\theta} f \\|_{L^{q_{\\theta}}(\\mathbb{R}^{d+1})} = \\sup_\\phi \\bigg| \\iint \\langle ( J^{2\\theta - 1}_xJ_t^{-\\theta}f)(x,t), \\overline{\\phi(x,t)} \\rangle \\; dx \\, dt \\bigg|,\n\\end{align}\nwhere the supremum is taken over all $\\phi \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$ normalized in $L^{q_{\\theta}'}(\\mathbb{R}^{d+1})$.\nThe idea of proof, coming from the Riesz--Thorin theorem, is to use a holomorphic parametrization of the duality pairing for fixed $\\phi$ via functions defined on the strip $S:= \\{a+\\i b: a \\in (0,1),\\, b \\in \\mathbb{R}\\}$. More precisely, we define whenever $z \\in \\overline{S}$,\n\\begin{align*}\nF(z) := \\mathrm{e}^{(z-\\theta)^2} J_x^{2z-1} J_t^{-z} f, \\qquad\nG(z) := |\\phi|^{\\frac{(1-z)q_{\\theta}'}{q_0'}+ \\frac{zq_{\\theta}'}{q_1'}} \\frac{\\overline{\\phi}}{|\\phi|},\n\\end{align*}\nwhere the expression for $G(z)$ is interpreted as $0$ on the set where $\\phi$ vanishes.\n\n\nWe derive properties of $F$. For $z = a + \\i b$ we have\n\\begin{align}\n\\label{besselinterp.prop.eq1}\nF(z) = \\Big(\\mathrm{e}^{(a-\\theta)^2-b^2} \\mathrm{e}^{\\i 2b(a-\\theta)}\\Big) J_x^{2a + 2 \\i b} J_t^{(1-a)-\\i b}(J_x^{-1} J_t^{-1} f).\n\\end{align}\nSince $J_x^{-1} J_t^{-1} f$ is a Schwartz function, Lemma~\\ref{BesselBound.lem} applied componentwise in combination with Fubini's theorem yields that $F$ is {\\it qualitatively} bounded on $S$ with values in $L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$. (The polynomial growth in $b$ is compensated by the exponential function.) Again using Lemma~\\ref{BesselBound.lem}, we have the following quantitative bounds on $\\partial S$:\n\\begin{align}\n\\label{besselinterp.prop.eq2}\n\\begin{split}\n\\|F(\\i b)\\|_{L^{q_0}(\\mathbb{R}^{d+1})} \n&\\leq C(d,N,q_0) \\|J_x^{-1} f\\|_{L^{q_0}(\\mathbb{R}^{d+1})},\\\\\n\\|F(1+ \\i b)\\|_{L^{q_1}(\\mathbb{R}^{d+1})} \n&\\leq C(d,N,q_1) \\|J_t^{-1}J_x^{1} f\\|_{L^{q_1}(\\mathbb{R}^{d+1})}.\n\\end{split}\n\\end{align}\nNext, it follows from $\\mathcal{F}f \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$ and dominated convergence, that we have a continuous function\n\\begin{align*}\n\\mathcal{F}F: \\overline{S} \\to L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N), \\qquad z \\mapsto \\mathrm{e}^{(z-\\theta)^2} (1+|\\xi|^2)^{\\frac{1-2z}{2}} (1+|\\tau|^2)^{\\frac{z}{2}} \\mathcal{F}f(\\tau,\\xi).\n\\end{align*}\nSince the Fourier transform is isometric on $L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$, the same follows for $F$.\nFinally, $F$ is holomorphic in $S$. Indeed, for any $\\psi \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$ we can use Parseval's formula to give\n\\begin{align*}\n\\iint \\langle F(z), \\overline{\\psi} \\rangle \\; dx \\, dt = \\iint \\big \\langle e^{(z - \\theta)^2} (1 +|\\xi|^2)^{\\frac{1-2z}{2}}(1 +|\\tau|^2)^{\\frac{z}{2}} \\mathcal{F}f, \\overline{\\mathcal{F}\\psi} \\big \\rangle \\; d\\xi \\, d\\tau\n\\end{align*}\nand the integral in $z$ along any triangle $\\triangle \\Subset S$ vanishes by Fubini's theorem and holomorphy of the integrand for fixed $(\\tau,\\xi)$.\n\nThe function $G: \\overline{S} \\to L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$ enjoys the same kind of properties. Here, boundedness follows directly from $\\phi \\in \\mathcal{S}(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$, continuity and holomorphy are obtained as before, and on $\\partial S$ we get from H\\\"older's inequality and the normalization of $\\phi$ that\n\\begin{align}\n\\label{besselinterp.prop.eq3}\n\\begin{split}\n\\|G(\\i b)\\|_{L^{q'_0}(\\mathbb{R}^{d+1})} \n\\leq 1, \\qquad\n\\|G(1+ \\i b)\\|_{L^{q'_1}(\\mathbb{R}^{d+1})} \n\\leq 1.\n\\end{split}\n\\end{align}\n\nNow, define a scalar-valued function on $\\overline{S}$ by\n\\begin{align*}\nH(z) := \\iint \\langle F(z), G(z) \\rangle \\; d x \\, dt.\n\\end{align*}\nThe $L^2(\\mathbb{R}^{d+1}; \\mathbb{C}^N)$-valued properties for $F$ and $G$ above imply that $H$ is bounded and continuous on $\\overline{S}$ and holomorphic in $S$. (The inner product preserves continuity and holomorphy by nearly the same proof as for products of scalar functions.)\nOn the boundary, we conclude from \\eqref{besselinterp.prop.eq2}, \\eqref{besselinterp.prop.eq3} and H\\\"older's inequality that\n\\begin{align*}\n |H(\\i b)| &\\leq C(d,N,q_0) \\|J_x^{-1} f\\|_{L^{q_0}(\\mathbb{R}^{d+1})} =: M_0,\\\\\n |H(1+\\i b)| &\\leq C(d,N,q_1) \\|J_t^{-1}J_x^{1} f\\|_{L^{q_1}(\\mathbb{R}^{d+1})}=:M_1.\n\\end{align*}\nHadamard's three lines theorem yields $|H(\\theta)| \\leq M_0^{1-\\theta} M_1^{\\theta}$. This means that\n\\begin{align*}\n\\bigg|\\iint \\langle J^{2\\theta - 1}_xJ_t^{-\\theta}f, \\overline{\\phi} \\, \\rangle \\; dx \\, dt \\bigg|  \\leq C \\|J_x^{-1} f\\|_{L^{q_0}(\\mathbb{R}^{d+1})}^{1-\\theta} \\|J_t^{-1}J_x^{1} f\\|_{L^{q_1}(\\mathbb{R}^{d+1})}^{\\theta}\n\\end{align*}\nand the  claim follows since $\\phi$ was arbitrary and normalized in $L^{q_{\\theta}'}(\\mathbb{R}^{d+1})$.\n\\end{proof}\n\n\n\n\\section{Proof of Theorem~\\ref{mainthrm.thrm}}\n\\label{proof.sec}\n\n\t\n\\noindent Let $u$ be a weak solution to \\eqref{pparatype.eq} in $I \\times Q$. We define nested intervals and cubes $I'' \\Subset I' \\Subset I$ and $Q'' \\Subset Q' \\Subset Q$ and pick a smooth function $1_{I'' \\times Q''} \\le \\chi \\le 1_{I'\\times Q'}$. Define a localized version $v = \\chi u$ as in Lemma~\\ref{tdervbound.lem}. Since the nested sets are arbitrary, it suffices to obtain the continuity statements on $I'' \\times Q''$.\n\n\\noindent \\emph{Step 1: H\\\"older continuity with values in spatial $L^q$}. Lemma~\\ref{tdervbound.lem} justifies applying Proposition~\\ref{besselinterp.prop} to $f=v_\\epsilon$ with $q_0 = p+\\delta$ and $q_1 = p'$. Thus, setting $\\theta =\\frac{1}{2}$ in Proposition~\\ref{besselinterp.prop}, we have\n\\begin{align}\n\\label{half-order-exponent.eq1}\n\\| J_t^{-1\/2} v_\\epsilon \\|_{L^{q}(\\mathbb{R}^{d+1})} \\leq C,\n\\end{align}\nfor some admissible $C$. The exponent $q>2$ is given by\n\\begin{align}\n\\label{half-order-exponent.eq2}\n \\frac{1}{q} = \\frac{1}{2} + \\frac{1}{2} \\left(\\frac{1}{p+\\delta} - \\frac{1}{p} \\right).\n\\end{align}\nWe have $(J_t^{-1\/2} v_\\epsilon)(\\cdot \\,,x) = J_t^{-1\/2} (v_\\epsilon(\\cdot \\,,x)) \\in L^q(\\mathbb{R})$ for every $x \\in {\\mathbb{R}^d}$ since $v_\\epsilon$ is a Schwartz function, but \\eqref{half-order-exponent.eq1} gives a quantitative bound. \n\nFix $x \\in {\\mathbb{R}^d}$. As $q' < 2$, we obtain from Minkowski's inequality that $G_t^{1\/2} \\in L^{q'}(\\mathbb{R})$. This Bessel kernel was defined in \\eqref{potential-formula.eq}. Hence, we have by Young's convolution inequality that\n\\begin{align}\n\\label{half-order-exponent.eq3}\n\\|v_\\epsilon(\\cdot \\,,x)\\|_{L^\\infty(\\mathbb{R})}\n\\leq \\|G_t^{1\/2}\\|_{L^{q'}(\\mathbb{R})} \\|J_t^{-1\/2} v_\\epsilon(\\cdot \\,,x)\\|_{L^q(\\mathbb{R})}.\n\\end{align}\nNow, we appeal to a Fourier analytic characterization of H\\\"older continuity. This uses a smooth function $\\psi$ with support in the set $\\{\\frac{1}{2} \\leq t \\leq 4\\}$ with the property that $\\sum_{j \\in \\mathbb{Z}} \\psi_j(t) = 1$ for all $t \\neq 0$, where $\\psi_j(t) = \\psi(2^{-j}t)$. For one construction see Lemma 8.1 in \\cite{M-S}.\n\n\\begin{lemma}[Lemma~8.6 in {\\cite{M-S}}]\n\\label{holder.lem}\nLet $f \\in \\mathcal{S}(\\mathbb{R})$ and let $0< \\alpha <1$. There is a constant $C = C(\\alpha)$ such that for all $t \\neq s$,\n\\begin{align*}\n\\frac{|f(t) - f(s)|}{|t-s|^\\alpha} \\leq C \\sup_{j \\in \\mathbb{Z}} 2^{j\\alpha} \\|\\mathcal{F}_t^{-1}(\\psi_j \\mathcal{F}_t f)\\|_{L^\\infty(\\mathbb{R})}.\n\\end{align*}\n\\end{lemma}\n\nWe put $f := v_\\epsilon(\\cdot \\,,x)$. Since $\\mathcal{F}_t^{-1} (\\psi_j \\mathcal{F}_t f)$ has a Fourier transform with support in $(-2^{j+2}, 2^{j+2})$, Bernstein's inequality (Lemma 4.13 in \\cite{M-S}) yields\n\\begin{align*}\n\\|\\mathcal{F}_t^{-1}(\\psi_j \\mathcal{F}_t f)\\|_{L^\\infty(\\mathbb{R})} \n&\\leq c 2^{j\/q} \\|\\mathcal{F}_t^{-1}(\\psi_j \\mathcal{F}_t f)\\|_{L^q(\\mathbb{R})} \\\\\n&\\leq c 2^{j\/q} C(p,\\psi) 2^{\\mathrm{min}\\{-j\/2,1 \\}} \\|J_t^{-1\/2} v_\\epsilon(\\cdot\\, ,x)\\|_{L^q(\\mathbb{R})},\n\\end{align*}\nwhere $c$ is a numerical constant and the second step is due to the Mihlin multiplier theorem applied to $m(\\tau) = \\psi_j(\\tau) (1+|\\tau|^2)^{-1\/4}$. The computation of the Mihlin norm is done \\emph{verbatim} as in the proof of Lemma~\\ref{BesselBound.lem}, taking into account that on the support of $\\psi_j$ we have $2^{j-1} \\leq |\\tau| \\leq 2^{j+2}$ in order to obtain the decay in $j$.\n\n\nThe assumptions of Theorem \\ref{multthrm.thrm} are hence satisfied. Consequently, we can take $\\alpha = \\frac{1}{2} - \\frac{1}{q} = \\frac{1}{2} (\\tfrac{1}{p} - \\tfrac{1}{p+\\delta})$ in Lemma~\\ref{holder.lem}. In view of \\eqref{half-order-exponent.eq3} we find for all $t \\neq s$ and all $x \\in {\\mathbb{R}^d}$ that\n\\begin{align}\n\\label{half-order-exponent.eq4}\n|v_\\epsilon(t,x)| + \\frac{|v_\\epsilon(t,x) - v_\\epsilon(s,x)|}{|t-s|^\\alpha} \\leq C \\|J_t^{-1\/2} v_\\epsilon(\\cdot \\,,x)\\|_{L^q(\\mathbb{R})}.\n\\end{align}\nWe fix a representative for $v$, a subsequence of $\\epsilon$ and a set $E$ of $(d+1)$-measure zero such that $v_\\epsilon(t,x) \\to v(t,x)$\nas $\\epsilon \\to 0$, whenever $(t,x) \\in \\mathbb{R}^{d+1} \\setminus E$. Then we integrate the $q$-th power in $x$, use \\eqref{half-order-exponent.eq1} and pass to the limit via Fatou's lemma, to get\n\\begin{align*}\n\\sup_{t \\in \\mathbb{R}} \\left(\\int_{{\\mathbb{R}^d}} |v(t,x)|^{q}\\right)^{1\/q} + \\sup_{\\substack{t,s \\in \\mathbb{R} \\\\ t \\neq s}}\\left(\\int_{{\\mathbb{R}^d}}\\frac{|v(t,x) -v(s,x)|^{q}}{|t - s|^{\\alpha q}} \\right)^{1\/q} \\leq C.\n\\end{align*}\nSince $v=u$ on $I'' \\times Q''$, we obtain $u \\in C^\\alpha(I''; L^q(Q''))$ as required.\n\n\\noindent \\emph{Step 2: H\\\"older continuity on almost all segments in time}. Since the Fourier transform turns convolutions into products, we obtain for all $\\varphi \\in \\mathcal{S}(\\mathbb{R}^{d+1})$ that\n\\begin{align}\n\\label{half-order-exponent.eq5}\n\\langle J_t^{-1\/2} v_\\epsilon, \\varphi \\rangle\n= \\langle J_t^{-1\/2} v, \\varphi_\\epsilon \\rangle,\n\\end{align}\nwhere we use the duality pairing on $\\mathcal{S}'(\\mathbb{R}^{d+1})$. In the limit as $\\epsilon \\to 0$, we have $\\varphi_\\epsilon \\to \\varphi$ in $\\mathcal{S}(\\mathbb{R}^{d+1})$ and therefore $J_t^{-1\/2} v_\\epsilon \\to J_t^{-1\/2} v$ in $\\mathcal{S}'(\\mathbb{R}^{d+1})$. On the other hand, this sequence is bounded in $L^q(\\mathbb{R}^{d+1})$ by \\eqref{half-order-exponent.eq1} and hence admits a weakly convergent subsequence. Identifying the limits, we get $J_t^{-1\/2} v \\in L^q(\\mathbb{R}^{d+1})$. Now, we can use \\eqref{half-order-exponent.eq5} to write $J_t^{-1\/2} v_\\epsilon = (J_t^{-1\/2} v)_\\epsilon$ and obtain strong convergence in $L^q(\\mathbb{R}^{d+1})$. This implies that $\\|J_t^{-1\/2} v_\\epsilon\\|_{L^q(\\mathbb{R})} \\to \\|J_t^{-1\/2} v\\|_{L^q(\\mathbb{R})}$ in $L^q({\\mathbb{R}^d})$. Hence, we can pass to a subsequence such that for almost every $x \\in {\\mathbb{R}^d}$,\n\\begin{align*}\n\\|J_t^{-1\/2} v_\\epsilon(\\cdot \\,,x)\\|_{L^q(\\mathbb{R})} \\to \\|J_t^{-1\/2} v(\\cdot \\,,x)\\|_{L^q(\\mathbb{R})}.\n\\end{align*}\nFix $x$ with this property and such that $E_x := \\{t : (t,x) \\in E\\}$ has $1$-measure zero, where $E$ is as in Step~1. Passing to the limit in \\eqref{half-order-exponent.eq4}, we obtain that $v(\\cdot\\,,x)$ satisfies the $\\alpha$-H\\\"older condition on $E_x$. Hence, we can re-define $v(\\cdot\\,,x)$ so that it is $\\alpha$-H\\\"older continuous. Since all modifications take place in $E$, we obtain a representative for $v$ that is $\\alpha$-H\\\"older continuous on $\\mathbb{R} \\times \\{x\\}$ for a.e.\\ $x \\in {\\mathbb{R}^d}$. We conclude again since $v=u$ on $I'' \\times Q''$. \\hfill {\\qedsymbol}\n\\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Special values of the Lommel functions $s_{\\protect\\mu ,\\protect\\nu\n}(z)$ and $S_{\\protect\\mu ,\\protect\\nu }(z)$}\n\n\\ \\ \\ \\ The Lommel functions $s_{\\mu ,\\nu }(z)$ and $S_{\\mu ,\\nu }(z)$ with\nunrestricted $\\mu ,\\nu $ satisfy the relations \\cite{Magnus1}\n\n\\begin{quote}\n\\begin{equation}\n\\frac{2\\nu }{z}S_{\\mu ,\\nu }(z)=(\\mu +\\nu -1)S_{\\mu -1,\\nu -1}(z)-(\\mu -\\nu\n-1)S_{\\mu -1,\\nu +1}(z),  \\label{eq1}\n\\end{equation}\n\n\\begin{equation}\n\\lbrack (\\mu +1)^{2}-\\nu ^{2}]S_{\\mu ,\\nu }(z)+S_{\\mu +2,\\nu }(z)=z^{\\mu +1},\n\\label{eq2}\n\\end{equation\n\\begin{equation}\n\\frac{dS_{\\mu ,\\nu }(z)}{dz}+\\frac{\\nu }{z}S_{\\mu ,\\nu }(z)=(\\mu +\\nu\n-1)S_{\\mu -1,\\nu -1}(z),  \\label{eq3}\n\\end{equation\nand the symmetry propert\n\\begin{equation*}\nS_{\\mu ,-\\nu }(z)=S_{\\mu ,\\nu }(z).\n\\end{equation*}\n\nIn the case where $\\mu $ is an integer $k$ and $\\nu =1\/2$, the recurrence\nrelation (2) viewed as a second-order difference equation is \n\\begin{equation}\n(2k+1)(2k+3)S_{k,1\/2}(z)+4S_{k+2,1\/2}(z)=4z^{\\,k+1},  \\label{eq4}\n\\end{equation\nand can be reduced to first-order equations in the cases of even or odd\nvalues of $k.$ \\ \n\n\\bigskip\n\\end{quote}\n\n\\subsection{Case $k=2m$}\n\n\\begin{quote}\nWith $k=2m$ equation (4) can be written \n\\begin{equation}\n(4m+1)(4m+3)\\,f_{m}(z)+4\\,f_{m+1}(z)=4z^{\\,2m+1},  \\label{eq5}\n\\end{equation}\n\nwhere $f_{m}(z)=S_{2m,1\/2}(z)$ or $s_{2m,1\/2}(z).$ It has the solutio\n\\begin{equation*}\nf_{m}(z)=(-1)^{m}{\\Gamma (2}m+1\/2{)}\\left[ \\frac{f_{0}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\frac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}\\right] ,\n\\end{equation*\n\\ the initial values of $\\ f_{0}(z)$ being either $S_{0,1\/2}(z)$ or \ns_{0,1\/2}(z).$ \\ Using those relations the solutions to (5) become\n\n\\begin{subequations}\n\\label{0}\n\\begin{align}\nS_{2m,1\/2}(z)& =(-1)^{m}\\Gamma (2m+1\/2)[\\tfrac{S_{0,1\/2}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}],  \\label{eq6a} \\\\\ns_{2m,1\/2}(z)& =(-1)^{m}\\Gamma (2m+1\/2)[\\tfrac{s_{0,1\/2}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}],  \\label{eq6b}\n\\end{align\nwhere $S_{0,1\/2}(z),$ and $s_{0,1\/2}(z)$ have been given by Magnus, \\textit\net al} \\cite{Magnus2} as \n\\end{subequations}\n\\begin{eqnarray*}\n\\frac{1}{\\sqrt{\\pi }}S_{0,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\cos (z)\n\\frac{1}{2}-S(\\chi )]-\\sin (z)[\\frac{1}{2}-C(\\chi )]\\right\\} , \\\\\n\\frac{1}{\\sqrt{\\pi }}s_{0,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\sin\n(z)C(\\chi )-\\cos (z)S(\\chi )\\right\\} ,\n\\end{eqnarray*\nwit\n\\begin{equation*}\n\\chi =\\sqrt{\\frac{2z}{\\pi }},\n\\end{equation*}\n\nand where $S$ and $C$ are the Fresnel sine and cosine integrals \\cite{NBS\n\\begin{eqnarray*}\nS(z) &=&\\int_{0}^{z}\\sin (\\frac{1}{2}\\pi t^{2})\\,dt, \\\\\nC(z) &=&\\int_{0}^{z}\\cos (\\frac{1}{2}\\pi t^{2})\\,dt.\n\\end{eqnarray*}\n\\end{quote}\n\n\\subsection{Case $k=2m+1$}\n\n\\begin{quote}\n\\ \\ \\ \\ In the case where $k=2m+1,$ the difference equation (2) become\n\\begin{equation}\n(4m+3)(4m+5)\\,f_{m}(z)+4\\,f_{m+1}(z)=4z^{\\,2m+2},  \\label{eq7}\n\\end{equation}\n\nwhere $f_{m}(z)=S_{2m+1,1\/2}(z)$ or $s_{2m+1,1\/2}(z).$ Here the solution is \n\\begin{equation*}\nf_{m}(z)=(-1)^{m}\\Gamma (2m+3\/2)\\left[ \\frac{2\\,f_{0}(z)}{\\sqrt{\\pi }\n-z^{2}\\sum_{j=0}^{m-1}\\frac{(-z^{2})^{\\,j}}{\\Gamma (2j+7\/2)}\\right] .\n\\end{equation*\nIn the special case $\\mu =-1$ and $\\nu =1\/2$ in (2) the functions\\thinspace\\ \n$f_{0}(z)\n\\begin{equation*}\nf_{0}(z)=S_{1,1\/2}(z)=1+\\frac{1}{4}S_{-1,1\/2}(z),\n\\end{equation*\no\n\\begin{equation*}\nf_{0}(z)=s_{1,1\/2}(z)=1+\\frac{1}{4}s_{-1,1\/2}(z).\n\\end{equation*\nWe hav\n\\begin{eqnarray*}\nS_{1,1\/2}(z) &=&1+\\sqrt{\\frac{\\pi }{2z}}\\{\\cos (z)[\\frac{1}{2}-C(\\chi\n)]+\\sin (z)[\\frac{1}{2}-S(\\chi )]\\}, \\\\\ns_{1,1\/2}(z) &=&1-\\sqrt{\\frac{\\pi }{2z}}\\{\\sin (z)S(\\chi )+\\cos (z)C(\\chi\n)\\},\n\\end{eqnarray*\nwhere values of $S_{-1,1\/2}(z),$ $s_{-1,1\/2}(z)$ have been given by Magnus, \n\\textit{et al }as\n\n\\begin{eqnarray*}\n\\frac{\\,1}{2\\sqrt{\\pi }}S_{-1,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\cos (z)\n\\frac{1}{2}-C(\\chi )]+\\sin (z)[\\frac{1}{2}-S(\\chi )]\\right\\} , \\\\\n\\frac{\\,1}{2\\sqrt{\\pi }}s_{-1,1\/2}(z) &=&-\\sqrt{\\frac{2}{z}}\\left\\{ \\sin\n(z)S(\\chi )+\\cos (z)C(\\chi )\\right\\} .\n\\end{eqnarray*}\n\\end{quote}\n\nUsing those relations, the solutions to (7) become after resumming \n\\begin{subequations}\n\\begin{eqnarray}\nS_{2m+1,1\/2}(z) &=&(-1)^{m}\\Gamma (2m+3\/2)[\\frac{S_{-1,1\/2}(z)}{2\\sqrt{\\pi }\n+\\sum_{j=0}^{m}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+3\/2)}],  \\label{eq8a} \\\\\ns_{2m+1,1\/2}(z) &=&(-1)^{m}\\Gamma (2m+3\/2)[\\frac{s_{-1,1\/2}(z)}{2\\sqrt{\\pi }\n+\\sum_{j=0}^{m}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+3\/2)}].  \\label{eq8b}\n\\end{eqnarray}\n\n\\subsection{Integrals with values containing the Fresnel functions}\n\nThe integrals \n\\end{subequations}\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz,\n\\end{eqnarray*\nwhich contain even powers of the variable, can be expressed in terms of the\nLommel functions. \\ \n\nIn the first instance, Maple gives \n\\begin{eqnarray}\n\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz &=&[1-\\frac{s_{k+1,1\/2(\\lambda )}}\n\\lambda ^{k}}]\\frac{\\cos (\\lambda )}{(2k+1)}  \\label{eq9} \\\\\n&&+[(2k-1)s_{k,3\/2}+(2\/\\lambda )s_{k+1,1\/2}]\\frac{\\sin (\\lambda )}{2\\lambda\n^{k}(2k+1)}.  \\notag\n\\end{eqnarray\nUsing (1) and (2) we get the simplified for\n\\begin{equation}\n\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{k}}[(2k-1)\\cos\n(\\lambda )\\,s_{k-1,1\/2}(\\lambda )+2\\sin (\\lambda )\\,s_{k,1\/2}(\\lambda )].\n\\label{eq10}\n\\end{equation\nThe values of the integral in the case where $k=2m$ can be obtained from the\nLommel expressions above in (6b) and (8b) with $m$ replaced with $m-1$. \\ We\nge\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+1\/2)}\n2\\lambda ^{2m}}\\{\\sqrt{\\frac{2}{\\lambda }}C(\\chi ) \\\\\n&&-\\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}{\\Gamma\n(2j+3\/2)} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}\n\\Gamma (2j+5\/2)}\\}.\n\\end{eqnarray*\nIn the case $k=2m+1$ we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m+1}\\Gamma (2m+3\/2\n}{2\\lambda ^{2m+1}}\\{\\sqrt{\\frac{2}{\\lambda }}S(\\chi ) \\\\\n&&+\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}\n\\Gamma (2j+5\/2)} \\\\\n&&-\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-\\lambda ^{2})^{\\,j}}{\\Gamma (2j+3\/2)\n\\}.\n\\end{eqnarray*\nWith these results we see that all cases of cosine integrals with even\npowers of the variable have been obtained in terms containing the Fresnel $S$\nfunction. \\ \n\nFor the corresponding sine integrals, integration by parts give\n\\begin{equation*}\n\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz=\\frac{\\sin (\\lambda )}{2k+1}-\\frac\n2\\lambda }{2k+1}\\int_{0}^{1}z^{2(k+1)}\\cos (\\lambda z^{2})dz.\n\\end{equation*\nUsing (10) we hav\n\\begin{equation*}\n\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{k}}[(2k-1)\\sin\n(\\lambda )s_{k-1,1\/2}(\\lambda )-2\\cos (\\lambda )s_{k,1\/2}(\\lambda )].\n\\end{equation*\nWith $k=2m$ we ge\n\\begin{equation*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m}}[(4m-1)\\sin\n(\\lambda )s_{2m-1,1\/2}(\\lambda )-2\\cos (\\lambda )s_{2m,1\/2}(\\lambda )],\n\\end{equation*\nwhich then become\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+1\/2)}\n2\\lambda ^{2m}}\\{\\sqrt{\\frac{2}{\\lambda }}S(\\chi ) \\\\\n&&-\\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{\\Gamma\n(2j+3\/2)} \\\\\n&&+\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n\\Gamma (2j+5\/2)}\\}.\n\\end{eqnarray*\nFor $k=2m+1$ the sine integrals are given by Maple a\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{2m+1}}[\\sin\n(\\lambda )(4m+1)s_{2m,1\/2}(\\lambda )-2\\cos (\\lambda )s_{2m+1,1\/2}(\\lambda )].\n\\end{equation*\nUsing the values of Lommel functions given above we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+3\/2)}\n2\\lambda ^{2m+1}}\\{\\sqrt{\\frac{2}{\\lambda }}C(\\chi ) \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n\\Gamma (2j+5\/2)} \\\\\n&&-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{\\Gamma\n(2j+3\/2)}\\}.\n\\end{eqnarray*\nWith these results we see that all values of the sine integrals which\ncontain \\textit{even} powers of $z$ requires the presence of the Fresnel\nintegrals $C(\\chi )$.\n\n\\section{Integrals with values containing elementary functions}\n\nWe next consider integrals which contain \\textit{odd} powers of $z$ i.e\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{2k+1}\\sin (\\lambda z^{2})dz.\n\\end{eqnarray*\nIn the first instance Maple give\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz &=&\\frac{1}{2(k+1)}\\{(1-\\frac\ns_{k+3\/2,1\/2}(\\lambda )}{\\lambda ^{k+1\/2}})\\cos (\\lambda ) \\\\\n&&+(\\,k\\,s_{k+1\/2,3\/2}(\\lambda ))+\\frac{s_{k+3\/2,1\/2}(\\lambda )}{\\lambda }\n\\frac{\\sin (\\lambda )}{\\lambda ^{k+1\/2}}\\},\n\\end{eqnarray*\nwhich reduces t\n\\begin{equation}\n\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{k+1\/2}}[k\\cos\n(\\lambda )s_{k-1\/2,1\/2}(\\lambda )+\\sin (\\lambda )s_{k+1\/2,1\/2}(\\lambda )]\n\\label{eq11}\n\\end{equation\nusing (1) and (2). \\ Where $k=2m$ we hav\n\\begin{equation}\n\\int_{0}^{1}z^{4m+1}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+1\/2}\n[2m\\cos (\\lambda )s_{2m-1\/2,1\/2}(\\lambda )+\\sin (\\lambda\n)s_{2m+1\/2,1\/2}(\\lambda )].  \\label{eq12}\n\\end{equation\nWe see that this expression requires the Lommel functions \ns_{2m-1\/2,1\/2}(\\lambda )$ and $s_{2m+1\/2,1\/2}(\\lambda )$. \\ In the first\ncase we have from (2) and the initial condition $s_{3\/2,1\/2}(\\lambda )=\\sqrt\n\\lambda }\\left[ 1-\\sin (\\lambda )\/\\lambda \\right] $, the Lommel function \ns_{2m-1\/2,1\/2}(\\lambda )$ as \n\\begin{equation*}\ns_{2m-1\/2,1\/2}(\\lambda )=\\frac{(-1)^{m}(2m-1)!}{\\sqrt{\\lambda }}\\left[ \\sin\n(\\lambda )-\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j+1}}{(2j+1)!}\\right] .\n\\end{equation*\n\\ The function $s_{2m+1,1\/2}(\\lambda )$ then can be obtained from the\ndifferential-difference equation in (3). \\ That is to sa\n\\begin{equation}\n\\frac{d\\,s_{2m+1\/2,1\/2}(\\lambda )}{d\\lambda }+\\frac{1}{2\\lambda \ns_{2m+1\/2,1\/2}(\\lambda )=2m\\,s_{2m-1\/2,1\/2}(\\lambda ).  \\label{eq13}\n\\end{equation\nWe get with the initial condition $s_{2m+1\/2,1\/2}(0)=0,$ the solution to\n(13) a\n\\begin{equation}\ns_{2m+1\/2,1\/2}(\\lambda )=\\frac{2m}{\\sqrt{\\lambda }}\\int_{0}^{\\lambda }\\sqrt{\n}s_{2m-1\/2,1\/2}(z)\\,dz,  \\label{eq14}\n\\end{equation\nwhich immediately gives \n\\begin{equation*}\ns_{2m+1\/2,1\/2}(\\lambda )=\\frac{(-1)^{m+1}(2m)!}{\\sqrt{\\lambda }}\\left[ \\cos\n(\\lambda )-\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\\right] .\n\\end{equation*\nWe have obtained in these cases closed forms for the Lommel functions which\ncontain only elementary functions. As a result, we ge\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+1}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m)!}{2\\lambda\n^{2m+1}}\\{\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\n\\\\\n&&-\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nThe cosine integral containing powers $4m+3$ is given b\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+3}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+3\/2}\n[(2m+1)\\cos (\\lambda )\\,s_{2m+1\/2,1\/2}(\\lambda )+\\sin (\\lambda\n)s_{2m+3\/2,1\/2}(\\lambda )].\n\\end{equation*\nIn the latter expression the quantity $s_{2m+3\/2,1\/2}(\\lambda )$ can be\nobtained from (3) and we have \n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+3}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m+1}(2m+1)!}\n2\\lambda ^{2m+2}}\\{1-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda\n^{2j}}{(2j)!} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nAn expression for the sine integrals with odd powers i.e\n\\begin{equation*}\n\\int_{0}^{1}z^{2k+1}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{k+1\/2}}[k\\sin\n(\\lambda )\\,\\,s_{k-1\/2,1\/2}(\\lambda )-\\cos (\\lambda )s_{k+1\/2,1\/2}(\\lambda\n)],\n\\end{equation*\nhas been obtained using integration by parts of the corresponding $\\cos\n(\\lambda z^{2})$ integral together with the latter's integrated form. \\ Then\nin the case where $\\ k=2m$ we have \n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+1}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m)!}{2\\lambda\n^{2m+1}}\\{1-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)\n} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nWhere $k=2m+1$ we hav\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+3}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+3\/2}\n[(2m+1)\\sin (\\lambda )\\,\\,s_{2m+1\/2,1\/2}(\\lambda )-\\cos (\\lambda\n)s_{2m+3\/2,1\/2}(\\lambda ).\n\\end{equation*\nIn the latter expression the quantity $s_{2m+3\/2,1\/2}(\\lambda )$ can also be\nobtained from (3) and the integral being sought has the value\n\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+3}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m+1)!}\n2\\lambda ^{2m+2}}\\{-\\lambda \\cos (\\lambda )\\sum_{j=0}^{m}\\frac\n(-1)^{\\,j}\\lambda ^{2j}}{(2j+1)!} \\\\\n&&+\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\\}.\n\\end{eqnarray*\n\\ \n\n\\section{Asymptotic forms for the integrals containing $S(\\protect\\chi )$\nand $C(\\protect\\chi )$}\n\nIt is useful to provide values of the integrals evaluated above in section 2\nwhere the parameter $\\lambda $ is large. \\ Initially we consider the case of\nthe integrals with even powers of the integration variable $z$ i.e\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz,\n\\end{eqnarray*\nwhich we have seen contain the Fresnel integrals. The asymptotic forms for\nthe functions $S\\left( \\chi \\right) $ and $C(\\chi )$ can be obtained from\nthe expressions \\cite{NIST} \n\\begin{eqnarray*}\nS\\left( \\sqrt{\\frac{2\\lambda }{\\pi }}\\right) &=&\\frac{1}{2}-f\\,(\\sqrt{\\frac\n2\\lambda }{\\pi }})\\cos (\\lambda )-g(\\sqrt{\\frac{2\\lambda }{\\pi }})\\sin\n(\\lambda ), \\\\\nC\\left( \\sqrt{\\frac{2\\lambda }{\\pi }}\\right) &=&\\frac{1}{2}+f\\,(\\sqrt{\\frac\n2\\lambda }{\\pi }})\\sin (\\lambda )-g(\\sqrt{\\frac{2\\lambda }{\\pi }})\\cos\n(\\lambda ),\n\\end{eqnarray*\nwhere $f(z)$ and $g(z)$ are the Fresnel auxiliary functions. \\ Their\nasymptotic forms with (cut of\\thinspace f $N\\geq 1$) are given b\n\\begin{eqnarray*}\nf\\,(\\sqrt{\\frac{2\\lambda }{\\pi }}) &\\thicksim &\\frac{1}{\\sqrt{2}\\,\\pi\n\\lambda ^{1\/2}}\\sum_{j=0}^{N-1}\\frac{(-1)^{j}}{\\lambda ^{2j}}\\Gamma (2j+1\/2),\n\\\\\ng\\,(\\sqrt{\\frac{2\\lambda }{\\pi }}) &\\thicksim &\\frac{1}{\\sqrt{2}\\,\\pi\n\\lambda ^{3\/2}}\\sum_{j=0}^{N-1}\\frac{(-1)^{j}}{\\lambda ^{2j}}\\Gamma (2j+3\/2).\n\\end{eqnarray*\nWe get for the integrals in question, having used the relatio\n\\begin{equation*}\n\\frac{1}{\\Gamma (-z)}=-\\frac{\\Gamma (z+1)}{\\pi }\\sin (\\pi z),\n\\end{equation*\nin the sine and cosine sums, the expression\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+1\/2)}{\n}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+1\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j}}\\Gamma (2j-2m-1\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-3\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+1\/2)}{\n}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+1\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-3\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j}}\\Gamma (2j-2m-1\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+3\/2\n}{2}\\{\\frac{(-1)^{m+1}}{\\sqrt{2}\\lambda ^{2m+3\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j}}\\Gamma (2j-2m-3\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N+1}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-5\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+3\/2\n}{2}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+3\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j+1}}\\Gamma (2j-2m-1\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j}}\\Gamma (2j-2m-3\/2)\\}.\n\\end{eqnarray*\nWe see that the terms in these asymptotic expressions contain the sine and\ncosine sums along with the additional and significant contributions from the\nFresnel integrals.\n\n\\subsection{Asymptotic values for Fresnel related integrals}\n\n\\bigskip\n\nIn extensions of the Schwinger-Englert semi-classical theory \\cite{englert}\\\nof atomic structure, asymptotic forms for the integral\n\\begin{eqnarray*}\n&&\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz, \\\\\n&&\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz,\n\\end{eqnarray*\narise with $\\lambda \\rightarrow \\infty $ and $0<a<1.$ \\ Here we give\nestimates for those forms. \\ Expanding the denominators of the integrals\nabove we have \n\\begin{equation*}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz=\\sum_{k=0}^{\\infty\n}a^{2k}\\int_{0}^{1}z^{4k}\\sin (\\lambda z^{2})-a\\sum_{k=0}^{\\infty\n}a^{2k}\\int_{0}^{1}z^{4k+2}\\sin (\\lambda z^{2}),\n\\end{equation*\nusing the results obtained above we get (where $z!!$ is the double factorial\nfunction and $N$ is the order of truncation\n\\begin{eqnarray*}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz &\\thicksim &\\frac{1}\n2}\\sqrt{\\frac{\\pi }{2\\lambda }}\\sum_{k=0}^{\\infty }(-1)^{k}\\left( \\frac{a}\n2\\lambda }\\right) ^{2k}[(4k-1)!!-\\frac{a}{2\\lambda }(4k+1)!!] \\\\\n&&+\\frac{\\cos (\\lambda )}{2}\\{\\sum_{k=0}^{\\infty\n}a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac{(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda\n\\right) ^{2j-1}} \\\\\n&&+\\frac{a}{\\lambda }\\sum_{k=0}^{\\infty }a^{2k}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda \\right) ^{2j}}\\} \\\\\n&&+\\frac{\\sin (\\lambda )}{2}\\{\\sum_{k=0}^{\\infty\n}a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac{(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda\n\\right) ^{2j}} \\\\\n&&+\\frac{a}{\\lambda }\\sum_{k=0}^{\\infty }a^{2k}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j-1}}\\}.\n\\end{eqnarray*\nIn lowest order in $a$ and $1\/\\lambda $ we hav\n\\begin{equation}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac{1}{4\n\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1-\\frac{a}{2\\lambda }\\right) -\\frac{\\cos\n(\\lambda )}{2\\lambda }(1-a)-\\frac{\\sin (\\lambda )}{\\left( 2\\lambda \\right)\n^{2}}(1+a)+\\cdots ,  \\label{eq15}\n\\end{equation\nand in the case of the cosine integral we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz &\\thicksim &\\frac{1}\n2}\\sqrt{\\frac{\\pi }{2\\,\\lambda }}\\sum_{k=0}^{\\infty }(-1)^{k}\\left( \\frac{a}\n2\\lambda }\\right) ^{2k}[(4k-1)!!+\\frac{a}{2\\lambda }(4k+1)!!] \\\\\n&&+\\cos (\\lambda )\\{\\sum_{k=0}^{\\infty }a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda \\right) ^{2j}} \\\\\n&&+\\sum_{k=0}^{\\infty }a^{2k+1}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j}}\\} \\\\\n&&-\\sin (\\lambda )\\{\\sum_{k=0}^{\\infty }a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j-1}} \\\\\n&&+\\sum_{k=0}^{\\infty }a^{2k+1}(4k+1)!!\\sum_{j=1}^{k+N+1}\\frac\n(-1)^{j}(4j-4k-7)!!}{\\left( 2\\lambda \\right) ^{2j-1}}\\}.\n\\end{eqnarray*\nIn lowest order in $a$ and $1\/\\lambda $ we hav\n\\begin{equation}\n\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac{1}{4\n\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1+\\frac{a}{2\\lambda }\\right) -\\frac{\\cos\n(\\lambda )}{\\left( 2\\lambda \\right) ^{2}}(1+a)+\\frac{\\sin (\\lambda )}\n2\\lambda }(1-a)+\\cdots  \\label{eq16}\n\\end{equation\nIt is interesting to note that the integrals with infinite range i.e\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}=-\\frac{\\pi }{\n\\sqrt{a}}\\sin (\\lambda \/a)+\\frac{\\pi }{2\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\sin (\\frac{\\lambda }{a})+\\cos \n\\frac{\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\sin (\\frac{\\lambda }{a})-\\cos \n\\frac{\\lambda }{a})\\right\n\\end{array\n\\right\\} ,  \\label{eq17}\n\\end{equation\nan\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}=\\frac{\\pi }{\n\\sqrt{a}}\\cos (\\lambda \/a)-\\frac{\\pi }{2\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\cos (\\frac{\\lambda }{a})-\\sin \n\\frac{\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\cos (\\frac{\\lambda }{a}))+\\sin \n\\frac{\\lambda }{a})\\right\n\\end{array\n\\right\\} ,  \\label{eq18}\n\\end{equation\nwhich have the exact values shown above, posses\\ the same forms for large \n\\lambda $ as the integrals (albeit without oscillations) with finite range.\n\\ That is to say \n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}\\thicksim \\frac{\n}{4}\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1-\\frac{a}{2\\lambda }\\right) +\\cdots\n,  \\label{eq19}\n\\end{equation\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac\n1}{4}\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1+\\frac{a}{2\\lambda }\\right) +\\cdots\n\\label{eq20}\n\\end{equation\n\\ \\ \\ \n\nIt is also worth noting that asymptotic values of the integrals which\ncontain higher powers of the trigonometric function described above and with\nhigher powers of the denominators i.e. \n\\begin{equation*}\n\\int_{0}^{1}\\frac{\\sin ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\\hspace\n0.25in}\\int_{0}^{1}\\frac{\\cos ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\nan\n\\begin{equation*}\n\\int_{0}^{\\infty }\\frac{\\sin ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }}\n\\hspace{0.25in}\\int_{0}^{\\infty }\\frac{\\cos ^{n}(\\lambda z^{2})}\n(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\ncan be expressed in terms of the forms given in (15,16) and (19,20). \\ By\nway of example, writing \n\\begin{equation*}\nI_{\\nu }^{\\,(\\eta )}(a,\\lambda )=\\int_{0}^{\\infty }\\frac{\\cos ^{\\,\\eta\n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\n(noting the scaling property\n\\begin{equation*}\nI_{\\nu }^{\\,(\\eta )}(a,\\lambda )=\\frac{1}{\\sqrt{a}}I_{\\nu }^{\\,(\\eta\n)}(1,\\lambda \/a),\n\\end{equation*\nwe get the differential-difference equation\n\n\\begin{equation}\nI_{\\,\\nu +1\\,}^{\\,(\\eta )}(a,\\lambda )=I_{\\nu }^{\\,(\\eta )}(a,\\lambda\n)+\\left( \\frac{a}{\\nu }\\right) \\frac{d\\,I_{\\nu }^{\\,(\\eta )}(a,\\lambda )}{da\n.  \\label{eq21}\n\\end{equation\nIn the case where $\\eta =1$ and $\\nu =1\/2$ we hav\n\\begin{eqnarray*}\nI_{1\/2}^{\\,(1)}(a,\\lambda ) &=&\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}\n(1+a\\,z^{2})^{\\,1\/2}} \\\\\n&=&\\frac{\\pi }{4\\sqrt{a}}\\left\\{ \\sin (\\frac{\\lambda }{2a})\\,J_{0}(\\frac\n\\lambda }{2a})-\\cos (\\frac{\\lambda }{2a})Y_{0}(\\frac{\\lambda }{2a})\\right\\} ,\n\\end{eqnarray*\na closed form expression where $J_{0}(z)$ and $Y_{0}(z)$ are Bessel\nfunctions of the first and second kind. \\ The value for this integral for\nlarge $\\frac{\\lambda }{2a}$ using Hankel's \\cite{Hankel}\\ \\ asymptotic\nexpressions for the Bessel functions i\n\\begin{equation*}\nI_{1\/2}^{\\,(1)}(a,\\lambda )\\thicksim \\frac{1}{4}\\sqrt{\\frac{2\\pi }{\\lambda }\n\\left\\{ 1+\\frac{1}{4}\\left( \\frac{a}{\\lambda }\\right) -\\frac{9}{32}\\left( \n\\frac{a}{\\lambda }\\right) ^{2}-\\frac{75}{128}\\left( \\frac{a}{\\lambda \n\\right) ^{3}+\\cdots \\right\\} .\n\\end{equation*\nIn the case of the integral $I_{1}^{\\,(2)}(a,\\lambda )$, its exact value is \n\\begin{eqnarray*}\nI_{1}^{\\,(2)}(a,\\lambda ) &=&\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2}\n}{(1+a\\,z^{2})} \\\\\n&=&\\frac{\\pi }{2\\sqrt{a}}\\cos ^{2}(\\lambda \/a)-\\frac{\\pi }{4\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{4\\lambda }{\\pi a}})\\left[ \\cos (\\frac{2\\lambda }{a})-\\sin \n\\frac{2\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{4\\lambda }{\\pi a}})\\left[ \\cos (\\frac{2\\lambda }{a}))+\\sin \n\\frac{2\\lambda }{a})\\right\n\\end{array\n\\right\\} ,\n\\end{eqnarray*\nor rewriting it in terms of the Fresnel auxiliary functions \\cite{Aux}$\\\nf\\,(\\chi ^{\\prime })$ and $g(\\chi ^{\\prime })$ referred to above (here with \n\\chi ^{\\prime }=2\\sqrt{\\lambda \/\\pi a}$ ) we get \n\\begin{equation*}\n\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}{(1+a\\,z^{2})}=\\frac{\\pi }{\n\\sqrt{a}}\\left\\{ 1+f\\,(\\chi ^{\\prime })+g(\\chi ^{\\prime })\\right\\} .\n\\end{equation*\nThe asymptotic values of $\\ f\\,(\\chi ^{\\prime })$ and $\\ g(\\chi ^{\\prime })$\nfor large $\\lambda $ \\cite{asymptotic} give the resul\n\\begin{eqnarray*}\n\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}{(1+a\\,z^{2})} &\\thicksim \n\\frac{\\pi }{4\\sqrt{a}}+\\frac{1}{8}\\sqrt{\\frac{\\pi }{\\lambda }}\\left\\{ 1\n\\frac{3}{16}\\left( \\frac{a}{\\lambda }\\right) ^{2}+\\cdots \\right\\} \\\\\n&&+\\frac{1}{32}\\sqrt{\\frac{\\pi }{\\lambda }}\\left\\{ \\frac{a}{\\lambda }-\\frac\n15}{16}\\left( \\frac{a}{\\lambda }\\right) ^{3}+\\cdots \\right\\} .\n\\end{eqnarray*\nThe first of the higher order integrals i.e. $\\ I_{2}^{\\,(2)}$ has the\nclosed form expression \n\\begin{eqnarray*}\nI_{2}^{\\,(2)} &=&\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}\n(1+a\\,z^{2})^{2}} \\\\\n&=&\\frac{\\pi }{8\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\n2\\cos ^{2}(\\frac{\\lambda }{a})+4(\\frac{\\lambda }{a})\\sin (\\frac{2\\lambda }{a\n)+\\sqrt{\\frac{4\\lambda }{\\pi a}} \\\\ \n+C(\\sqrt{\\frac{4\\lambda }{\\pi a}})[\\sin (\\frac{2\\lambda }{a})(1-4\\frac\n\\lambda }{a})-\\cos (\\frac{2\\lambda }{a})(1+4\\frac{\\lambda }{a})] \\\\ \n-S(\\sqrt{\\frac{4\\lambda }{\\pi a}})[\\sin (\\frac{2\\lambda }{a})(1+4\\frac\n\\lambda }{a})+\\cos (\\frac{\\lambda }{a})(1-4\\frac{2\\lambda }{a}\n\\end{array\n\\right\\} .\n\\end{eqnarray*}\n\nThe higher order integrals i.e. those containing powers of the cosine\nfunction i.e. $\\cos ^{\\,\\eta }(\\lambda x^{2})$ are expressible in terms of\nthe integrals in (18). \\ Using the trigonometric relations \n\\begin{equation*}\n\\cos ^{\\eta }(\\lambda x^{2})=\\frac{1}{2^{\\eta }}\\sum_{j=0}^{\\eta }\\binom\n^{\\eta }}{j}\\,\\cos ([\\eta -2j]\\lambda x^{2}),\n\\end{equation*\nwe have \n\\begin{eqnarray*}\nI_{\\nu }^{\\,(2\\eta )}(a,\\lambda ) &=&\\sqrt{\\frac{\\pi }{a}}\\frac{\\Gamma\n(2\\eta -1\/2)}{2^{\\,2\\eta }\\,\\eta !\\,(\\eta -1)!}+\\frac{1}{2^{2\\eta -1}\n\\sum_{j=1}^{\\eta }\\binom{2\\,\\eta }{\\eta -j}\\,I_{\\nu }^{\\,(1)}(a,2j\\,\\lambda\n), \\\\\nI_{\\nu }^{\\,(2\\eta +1)}(a,\\lambda ) &=&\\frac{1}{2^{2\\eta }}\\sum_{j=0}^{\\eta \n\\binom{2\\eta +1}{\\eta -j}\\,I_{\\nu }^{\\,(1)}(a,(2j+1)\\,\\lambda ).\n\\end{eqnarray*\nWith results from Maple, it is possible using an incomplete form of\ninduction to give general expressions for the integrals $I_{\\nu\n}^{\\,(1)}(a,\\lambda )$ with even and odd values of $\\nu .$ The integrals are\nfound to contain the Anger functions \\cite{Anger} $\\mathbf{J}_{1\/2}(z)$ and \n\\mathbf{J}_{3\/2}(z)$ i.e\n\\begin{equation*}\nI_{2n}^{\\,(1)}(a,\\lambda )=\\frac{1}{\\sqrt{a}\\,2^{\\,\\epsilon (n)+n}}\\left\\{ \n\\begin{array}{c}\n2\\sqrt{\\frac{2\\pi a}{\\lambda }}\\sum_{k=0}^{2n-1}a_{k,2n}\\left( \\frac{\\lambda \n}{a}\\right) ^{k} \\\\ \n+2\\pi \\cos (\\lambda \/a)\\sum_{k=0}^{n-1}c_{k,2n}\\left( \\frac{\\lambda }{a\n\\right) ^{2k} \\\\ \n+4\\pi \\sin (\\lambda \/a)\\sum_{k=0}^{n-1}d_{k,2n}\\left( \\frac{\\lambda }{a\n\\right) ^{2k+1} \\\\ \n-\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{1\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}e_{k,2n}\\left( \\frac{\\lambda }{a}\\right) ^{2k} \\\\ \n+\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{3\/2}(\\lambda\n\/a)\\sum_{k=0}^{n-1}f_{k,2n}\\left( \\frac{\\lambda }{a}\\right) ^{2k+1}\n\\end{array\n\\right\\}\n\\end{equation*\nwhere the Greubel eta function $\\epsilon (n)$ is defined here as \n\\begin{equation*}\n\\epsilon (n)=\\lfloor \\sqrt{2}n\\rfloor +\\lfloor \\sqrt{3\/2}n\\rfloor \\,\\ ,\n\\end{equation*\nand \\ $\\lfloor z\\rfloor $ is the floor function. \\ The sequence of integers\nproduced by the eta function has been studied by G. C. Greubel and others\n\\cite{Greubel}. The expression for $I_{2n+1}^{\\,(1)}(a,\\lambda )$ i\n\\begin{equation*}\nI_{2n+1}^{\\,(1)}(a,\\lambda )=\\frac{1}{\\sqrt{a}\\,2^{\\,\\epsilon (n+1)+n}\n\\left\\{ \n\\begin{array}{c}\n2\\sqrt{\\frac{2\\pi a}{\\lambda }}\\sum_{k=0}^{2n-1}a_{k,2n+1}\\left( \\frac\n\\lambda }{a}\\right) ^{k} \\\\ \n+2\\pi \\cos (\\lambda \/a)\\sum_{k=0}^{n}c_{k,2n+1}\\left( \\frac{\\lambda }{a\n\\right) ^{2k} \\\\ \n+2\\pi \\sin (\\lambda \/a)\\sum_{k=0}^{n-1}d_{k,2n+1}\\left( \\frac{\\lambda }{a\n\\right) ^{2k+1} \\\\ \n-\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{1\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}e_{k,2n+1}\\left( \\frac{\\lambda }{a}\\right) ^{2k} \\\\ \n+\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{3\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}f_{k,2n+1}\\left( \\frac{\\lambda }{a}\\right) ^{2k+1\n\\end{array\n\\right\\} .\n\\end{equation*\nAlternatively, in keeping with results given above the Anger functions can\nbe written in terms of the Fresnel functions appearing in $I_{1}^{\\,(2)},$\nand $\\ I_{2}^{\\,(2)}$, using the relation\n\\begin{eqnarray*}\n\\mathbf{J}_{1\/2}(z) &=&\\sqrt{\\frac{2}{\\pi z}}[C(\\sqrt{\\frac{2z}{\\pi }\n)\\{\\sin (z)+\\cos (z)\\}+S(\\sqrt{\\frac{2z}{\\pi }})\\{\\sin (z)-\\cos (z)\\}], \\\\\n\\mathbf{J}_{3\/2}(z) &=&-\\frac{2}{\\pi z}+\\sqrt{\\frac{2}{\\pi z}}C(\\sqrt{\\frac\n2z}{\\pi }})\\{\\sin (z)\\,(1+1\/z)-\\cos (z)\\,(1-1\/z)\\} \\\\\n&&-\\sqrt{\\frac{2}{\\pi z}}S(\\sqrt{\\frac{2z}{\\pi }})\\{\\sin (z)\\,(1-1\/z)+\\cos\n(z)\\,(1+1\/z)\\}.\n\\end{eqnarray*}\n\nThe coefficients $a_{k,\\nu },c_{k,\\nu },d_{k,\\nu },e_{k,\\nu },f_{k,\\nu }$\noccurring in the $I_{\\nu }^{\\,(1)}$ integrals are interrelated by (21) from\nwhich we obtain the connection formula\n\\begin{eqnarray*}\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2n,2n+1} &=&-e_{n,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2k,2n+1}+(4k-4n)\\,a_{2k,2n}\n&=&-e_{k,2n},\\hspace{0.25in}(k<n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2k+1,2n+1}+(4k+2-4n)\\,a_{2k+1,2n}\n&=&f_{k,2n},\\hspace{0.25in}(k<n), \\\\\n&& \\\\\n\\frac{n}{2^{\\Delta \\,\\epsilon (n)}}c_{n,2n+1} &=&-d_{n-1,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}c_{k,2n+1}+(4k+1-4n)\\,c_{k,2n}\n&=&-4\\,d_{k-1,2n},\\hspace{0.25in}(k<n), \\\\\n&& \\\\\n2n\\,d_{k,2n+1}+(4k+3-4n)\\,d_{k,2n} &=&c_{k,2n,}\\hspace{0.25in}(k\\leq n-1), \\\\\n&& \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}e_{0,2n+1} &=&(4n-1)e_{0,2n},\\hspace\n0.25in}(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}e_{k,2n+1}+(4k+1-4n)\\,e_{k,2n}\n&=&2f_{k-1,2n},\\hspace{0.25in}(k>0), \\\\\n&& \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}f_{n,2n+1} &=&-2e_{n,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}f_{k,2n+1}+(4k-1-4n)\\,f_{k,2n}\n&=&-2e_{k,2n},\\hspace{0.25in}(k<n),\n\\end{eqnarray*\nwith $\\Delta \\epsilon (n)=\\epsilon (n+1)-\\epsilon (n).$ \\ These relations do\nnot appear to be useful except as an internal check on the values of the\ncoefficients.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIndividual neutral atoms trapped in optical tweezers\\,\\cite{Saffman10,Saffman2016,barredo_atom-by-atom_2016,endres_atom-by-atom_2016,barredo_synthetic_2018,Browaeys20,Henriet2020quantum,Morgado20,Beterov20,Wu21} have emerged as a powerful platform for quantum information processing. In those systems, one uses the strong interaction between atoms excited to a \\emph{Rydberg state} to generate entanglement. This interaction can be exploited for various tasks, including the creation of fast and robust quantum gates\\,\\cite{Jaksch00,Isenhower10,levine_high-fidelity_2018,Levine19} for digital quantum information processing, the generation of tunable spin models for the study of many-body quantum physics\\,\\cite{labuhn2016tunable,Bernien17,Lienhard18,leseleuc_observation_2019,omran_generation_2019}, or the deterministic creation of large highly entangled GHZ states with up to 20 atoms\\,\\cite{omran_generation_2019}.\n\n\nQuantum computing and quantum simulation schemes for neutral-atom Quantum Processing Units (QPUs) are extremely flexible, in part due to the fact that, unlike for most other platforms, the QPU (and therefore its qubit connectivity) is not hard-wired during the manufacturing process. In fact, in neutral-atom optical traps, the connectivity of the QPU can be reprogrammed at \\textit{every single run}. This opportunity opens up new avenues for exploration on these devices but also requires providing the user with a lower-level description of the physical setup.\n\nWe here introduce \\emph{Pulser}~\\cite{Pulser}, an open-source software library for designing pulse sequences for neutral-atom QPUs. The main goal of this library is to serve as an interface between experienced users and neutral-atom quantum hardware. Using Pulser, users can control all the relevant physical parameters of a pulse-level quantum program. This program can subsequently be sent to QPUs via external tools and executed on the hardware after a device-specific compilation step. Providing users with a pulse-level access to QPUs will allow them to explore advanced pulse-shaping techniques facilitating the design of new quantum gate protocols, the implementation of optimal control and error mitigation techniques, or the exploration of mixed digital-analog algorithms\\,\\cite{Parra-rodriguez_20}. Exposing the low-level details of the hardware will also enable algorithm developers to design software procedures while keeping the particularities of the hardware in mind. Besides providing an interface to a QPU, Pulser includes a built-in emulator that faithfully reproduces the hardware behavior. This is intended to make the design of experiments easier for both the theoretician and the experimentalist.\n\n\nPulser is an open-source Python library licensed under the Apache License 2.0 and intended for the community of quantum scientists and engineers working on neutral-atom devices. The development and use of open-source software has enabled the flexible and efficient simulation of quantum systems \\cite{Johansson_2012,Johansson_2013}, the connection of quantum processors to the cloud \\cite{Zeng_2017_Nature, Karalekas_2020} and their use to launch jobs running quantum hardware. These tasks appear as quantum circuits expressed in intermediate representation, or in the more recent framework of hybrid classical-quantum algorithms, which require a fast and tailored interoperability between quantum and classical computing units with classical optimization routines \\cite{McClean_2016_NJP,Bharti_2021,Endo_2021}. More recently, specific tools and frameworks have also put more focus on the pulse-level description of quantum controls \\cite{Alexander_2020_QST,Ball_2020,Wittler_2021, QUA-libs, li2021qutip-qip}, and their optimization to mitigate the impact of noise on the coherence of the intended protocols \\cite{Goerz_2019_SciPost} in the so-called era of noisy-intermediate scale quantum computing (NISQ) \\cite{Preskill_2018_Quantum}. However, no existing open-source tool has yet specialized its features to the physics of neutral-atom quantum processors, which, due to the peculiarities of their physical properties, provide largely untapped opportunities both for quantum simulation and quantum computing tasks.\n\nIn this paper, we start by briefly reviewing in Section \\ref{section:description} the main physical ingredients that determine the dynamics of Rydberg atom arrays. We then present in Section \\ref{section:sequences} a guided walk-through for designing pulse sequences with Pulser, followed by examples on how Pulser can be leveraged for some notable use-cases in Section \\ref{section:usecases}. Finally, we conclude and comment on the future directions and extensions for this tool.\n\n\\section{Description of the physical system}\n\\label{section:description}\nIn neutral-atom devices, atoms are trapped by arrays of optical tweezers\\,\\cite{Nogrette14} in arbitrary, customizable patterns. In Pulser, we call this ensemble of atoms and their specific positions the \\texttt{Register}. For each atom in a \\texttt{Register}, the quantum information is encoded in specific electronic energy levels, which are reached through optical addressing techniques. In Pulser, the components responsible for driving these transitions are called \\texttt{Channel}s and they do so through the emission of \\texttt{Pulse}s.\n\n\n\\subsection{Driving two-level transitions}\n\nWe define a pulse as the modulation of a channel's output amplitude, detuning, and phase over a finite duration $\\tau$. For a channel targeting the transition between energy levels $a$ and $b$, with resonance frequency $\\omega_{ab} = |E_{a}-E_{b}|\/\\hbar$, the output amplitude determines the \\textit{Rabi frequency}\\footnote{The Rabi frequency determines the amplitude of a signal (in frequency units), not the frequency itself.} $\\Omega(t)$, and the detuning $\\delta(t)$ is defined relatively to $\\omega_{ab}$ and the frequency of the channel's output signal $\\omega(t)$, as $\\delta(t) = \\omega(t) - \\omega_{ab}$. Additionally, the phase $\\varphi$ of a pulse can be set to an arbitrary, constant value.\n\nA pulse-driven transition between two energy levels can be mapped to a spin-$1\/2$ system through the drive Hamiltonian:\n\\begin{equation}\\label{drive_ham}\nH^D(t)= \\frac{\\hbar}{2} \\mathbf{\\Omega}(t)\\cdot\\bm{\\sigma},\n\\end{equation}\nwhere $\\bm{\\sigma} = (\\sigma^x,\\sigma^y,\\sigma^z)^T$ is the Pauli vector and $\\bm{\\Omega}(t) =\n(\\Omega(t) \\cos(\\varphi),-\\Omega(t) \\sin(\\varphi),-\\delta(t))^T$ the rotation vector. Since there can be multiple transitions being addressed overall, we explicitly define the Pauli matrices for a transition between states $\\ket{a}$ and $\\ket{b}$ (where $E_b > E_a$) to be:\n\\begin{align}\n    \\sigma^x &= \\op{a}{b} + \\op{b}{a}\\\\\n    \\sigma^y &= i\\op{a}{b} - i\\op{b}{a} \\\\\n    \\sigma^z & = \\op{b}{b} - \\op{a}{a}\n\\end{align}\nIn the Bloch sphere representation, for each instant $t$, this Hamiltonian describes a rotation around the axis $\\mathbf{\\Omega}$ with angular velocity $\\Omega_{eff} = |\\bm{\\Omega}| = \\sqrt{\\Omega^2 + \\delta^2}$, as illustrated in Figure \\ref{fig:bloch_rotation}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\columnwidth, trim=0 0.5cm 0 0.8cm, clip]{figures\/bloch_rotation.png}\n\\caption{Representation of the drive Hamiltonian's dynamics as a rotation in the Bloch sphere.}\n\\label{fig:bloch_rotation}\n\\end{figure}\n\n\n\n\\subsection{Rydberg states}\n\\label{sec:rydberg}\n\nIn neutral-atom devices, atoms are driven to Rydberg states as a way to make them interact over large distances. Depending on the electronic levels that are involved in the process, the atoms experience different types of interactions, translating into different Hamiltonians~\\cite{Browaeys20}.\\\\\n\nIn the so-called ``Ising'' configuration, which is obtained when the spin states are one of the ground states $\\ket{g}$ and a Rydberg state $\\ket{r}$~\\cite{schauss2015crystallization,labuhn2016tunable,Bernien17,leseleuc2018accurate}  (the \\texttt{ground-rydberg} basis in Pulser, see Fig. \\ref{Ising_config}.), the interaction adds a term to the drive Hamiltonian describing the transition between those states: \n\n\\begin{equation}\n\\mathcal H^{gr}(t) = \\sum_i \\left(H^D_i(t) + \\sum_{j<i}\\frac{C_6}{(R_{ij})^6}\\hat n_i \\hat n_j \\right),\n\\label{eq:ising_ham}\n\\end{equation}\n\n\\noindent where $\\hat n_i$ denotes the projector $\\ketbra{r}{r}_i$ on the $i$-th atom, $R_{ij}$ is the distance between atoms $i$ and $j$, and $C_6$ is a constant depending on the specific Rydberg level $\\ket{r}$. \n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.4]{figures\/two-level.png}\n\\caption{The excitation between a ground and Rydberg state is taken as two-level transition with  Rabi frequency $\\Omega$ and detuning $\\delta$. In Pulser, it is called the \\texttt{ground-rydberg} basis and is addressed by \\texttt{Rydberg} channels.}\n\\label{Ising_config}\n\\end{figure}\n\nThe effect of this interaction is expressed in the so-called \\textit{Rydberg blockade}, illustrated in Figure \\ref{blockade}. When driving two atoms from $\\ket{g}$ to $\\ket{r}$ with a resonant pulse ($\\delta=0$) of Rabi frequency $\\Omega$, the interaction prevents the simultaneous excitation of the two atoms in the state $ \\ket{r}$ if $\\hbar\\Omega \\ll C_6\/R^6$. Instead, the system evolves between the $\\ket{gg}$ and the entangled state $\\ket{\\psi_+}=(\\ket{gr}+\\ket{rg})\/\\sqrt{2}$\\footnote{Note that the state $\\ket{\\psi_-}=(\\ket{gr}-\\ket{rg})\/\\sqrt{2}$ is not coupled to $\\ket{gg}$ by the laser}, with an effective Rabi frequency of $\\sqrt{2}\\Omega$. Similarly, an atom's excitation from $\\ket{g}$ to $\\ket{r}$ is suppressed if there is another atom in the Rydberg state at a distance $R \\ll R_b$, where \n\\begin{equation}\\label{eq:blockade_radius}\n    R_b=\\left(\\frac{C_6}{\\hbar\\Omega}\\right)^{1\/6}\n\\end{equation}\nis the \\textit{Rydberg blockade radius}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/rydberg_computing_blockade.pdf}\n\\caption{Two nearby atoms interact via a van der Waals potential which shifts the energy level for the doubly excited state $|rr\\rangle$. If the laser excites simultaneously both atoms, the lower-energy entangled state $(|gr\\rangle + |rg\\rangle)\/\\sqrt{2}$ is produced instead.}\n\\label{blockade}\n\\end{figure}\n\n\n\n\nBy acting on all atoms with the same global pulse, addressing the $\\ket{g} \\leftrightarrow \\ket{r}$ transition with $\\varphi=0$, one obtains the Hamiltonian:\n\\begin{equation}\n    \\mathcal H(t) = \\sum_i \\left(\\frac{\\hbar\\Omega(t)}{2} \\sigma_i^x \n    -\\hbar \\delta(t) \\hat n_i +\\sum_{j<i} \\frac{C_6}{(R_{ij})^6}\\hat n_i \\hat n_j \\right)\n\\label{eq:global_ising}\n\\end{equation}\nSince $\\hat n_i=(1+\\sigma_i^z)\/2$, Eq. (\\ref{eq:global_ising}) represents a quantum Ising Hamiltonian with transverse field $\\Omega(t)$ and Ising couplings $J_{ij} \\propto 1\/R_{ij}^6$.\\\\\n\nOther configurations exist, but the corresponding channels are currently not implemented in Pulser. In the so-called ``XY'' configuration, which is obtained when the spin states $\\ket{\\downarrow}$ and $\\ket{\\uparrow}$ are two Rydberg states that are dipole-coupled, such as $\\ket{nS}$ and $\\ket{nP}$~\\cite{barredo2015coherent,orioli2018relaxation,leseleuc_observation_2019,Scholl2021}, the dipolar interaction generates an exchange-like term of the form\n\\begin{equation}\n\\mathcal{H}_{int}=  2\\sum_{i \\neq j}\\frac{C_3}{R_{ij}^3} \\left(\\sigma_i^x \\sigma_j^x + \\sigma_i^y \\sigma_j^y \\right),\n\\label{eq:XY_hamiltonian}\n\\end{equation}\nwhere $C_3$ is a coefficient which depends on the selected Rydberg states. In this configuration, the driving is induced by a microwave field instead of an optical laser field.\\\\\n\nThe continuous manipulation of a system's Hamiltonian is referred to as the \\textit{analog approach} to quantum computing. The very high level of control in this analog mode has several applications, the most direct one being the quantum simulation of interacting many-body systems. This includes observing the out-of-equilibrium dynamics of the system after a quench or a drive\\,\\cite{Bernien17,Bluvstein21}, or determining the ground-state of a specific Hamiltonian through adiabatic variation of its parameters\\,\\cite{Lienhard18,scholl2020programmable,ebadi2020quantum}. Additionally, one can use the blockade effect to reproduce a connectivity structure among the atoms, which can be exploited to solve combinatorial problems that can be mapped onto the Ising Hamiltonian\\,\\cite{pichler2018quantum,henriet2020robustness, dalyac2020qualifying}. Those applications will be studied in more detail in Section \\ref{section:usecases}.\\\\\n\n\n\n\\subsection{Digital addressing and quantum gates}\n\nIn opposition to the analog approach stands the \\textit{digital approach}, in which a system's state evolves through a series of discrete manipulations of its qubits' states, known as quantum gates. This is the underlying approach in quantum circuits and can be replicated on neutral-atom devices at the pulse-level. To achieve this, the qubit states are encoded in two hyperfine ground states of the system, named \\textit{ground}, $\\ket{g}\\equiv\\ket{0}$, and \\textit{hyperfine}, $\\ket{h}\\equiv\\ket{1}$. In Pulser, these states form the \\texttt{digital} basis, which is addressed by \\texttt{Raman} channels.\n\nFor these energy levels, the Rydberg blockade effect is not present, so the resulting dynamics from a pulse addressing the $\\ket{h} \\leftrightarrow \\ket{g}$ transition of an atom will be dictated by the driving Hamiltonian of eq. (\\ref{drive_ham}). For a pulse of duration $\\tau$, the time evolution of this system is dictated by the operator\n\\begin{equation}\nU(\\bm \\Omega, \\tau) = T \\exp \\left[ -  \\frac{i}{2} \\int_0^\\tau \\bm \\Omega(t) \\cdot \\bm \\sigma \\mathrm dt\\right],\n\\end{equation}\nwhere $T$ denotes the time-ordering operator. The unitary $U$ describes a rotation around the time-dependent axis $\\bm\\Omega(t)$. For a resonant pulse (i.e. one with $\\delta = 0$) of phase $\\varphi$, we get a rotation angle of\n\\begin{equation}\\label{rot_angle}\n    \\theta = \\int_0^\\tau \\Omega(t) \\mathrm dt\n\\end{equation}\naround the fixed axis \n\\begin{equation}\\label{rot_axis}\n   \\bm e(\\varphi) = (\\cos\\varphi, -\\sin\\varphi, 0),\n\\end{equation}\nsituated on the equator of the Bloch sphere. The corresponding unitary operator is\n\\begin{equation}\n\\begin{split}\\label{res_rotation}\n    R_{\\bm e(\\varphi)}(\\theta) &= e^{-i\\frac{\\theta}{2}(\\cos(\\varphi) \\sigma_x - \\sin(\\varphi) \\sigma_y)}\\\\\n    &=  e^{i\\frac{\\varphi}{2}\\sigma_z}e^{-i\\frac{\\theta}{2}\\sigma_x}e^{-i\\frac{\\varphi}{2}\\sigma_z}\\\\\n    &= R_{ z}(-\\varphi)R_{ x}(\\theta)R_{ z}(\\varphi)\n\\end{split}\n\\end{equation}\nwhich, as its decomposition shows, can be thought of as a rotation around the Bloch sphere's $x$-axis, conjugated by $z$-rotations (i.e. phase gates). By following this gate with another $z$-rotation (which can be achieved virtually through a shift in the phase reference frame \\cite{IBM_VZ-gates}), we can then construct any arbitrary single-qubit gate,\n\\begin{equation}\n\\begin{split}\n    U(\\gamma, \\theta, \\varphi) &= R_z(\\gamma + \\varphi) R_{\\bm e(\\varphi)}(\\theta) \\\\\n    &= R_z(\\gamma)R_x(\\theta)R_z(\\varphi),\n\\end{split}\n\\end{equation}\nwhich relies solely on resonant pulses and phase reference frame changes. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/CZ_full}\n    \\caption{CZ implementation protocol, where three consecutive pulses address the $\\ket{g} \\leftrightarrow \\ket{r}$ transition. Full (dashed) red lines illustrate if a given pulse is effective (not effective).}\n    \\label{fig:cz_protocol}\n\\end{figure}\n\nWith the ability to perform arbitrary single-qubit gates, all that is missing in order to have a universal quantum gate set is an appropriate two-qubit gate. In neutral-atom devices, the native gate is the controlled-Z (CZ); its implementation leverages the Rydberg blockade effect to flip the phase of a two-qubit subsystem, conditionally on its initial state. To achieve this, the atoms, depending on their initial state, are sequentially brought to an auxiliary Rydberg state and back by a sequence of three consecutive laser pulses. The effect of this pulse sequence on the four computational states $\\ket{00}$, $\\ket{01}$, $\\ket{10}$ and $\\ket{11}$ is shown in Figure \\ref{fig:cz_protocol}. The first $\\pi$-pulse attempts to bring the \\textit{control} atom into the Rydberg state, which will condition whether the \\textit{target} will be able to go to $\\ket{r}$ and back with the second $2\\pi$-pulse, due to the Rydberg blockade. In particular, the $\\ket{00}$ state does not experience two consecutive phase flips as it would if the atoms did not interact (see top right panel of Fig. \\ref{fig:cz_protocol}, where the Rydberg energy level of the target atom experiences a shift in energy $\\Delta E$ due to the control atom being in $\\ket{r}$). Finally, the third pulse brings the \\textit{control} back to its original state, in case it was excited. In the end, all states but $\\ket{11}$ experience a phase flip (see bottom left panel of Fig. \\ref{fig:cz_protocol}), and the process corresponds to an application of a CZ within a global phase of $\\pi$ (see table \\ref{tab:cz}).\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c||c | c |c | c}\n       \n     \n        Initial state & $\\ket{00}$ & $\\ket{01}$ & $\\ket{10}$ & $\\ket{11}$ \\\\\n        \\hline\n        Final state &  $-\\ket{00}$ & $-\\ket{01}$ & $-\\ket{10}$ & $\\ket{11}$ \\\\\n    \\end{tabular}\n    \\caption{Effect of the CZ pulse sequence in the computational basis.}\n    \\label{tab:cz}\n\\end{table}\n\n\nAltogether, the space in which the atom arrays can evolve is here composed of three states: two non-interacting hyperfine states, $\\{ |g\\rangle , |h\\rangle \\}$ and an auxiliary Rydberg state $|r\\rangle $ which is only populated in the transient dynamics of the system during the application of multi-qubit gates. The Rydberg state thus plays both the role of an excited state in the \\texttt{ground-rydberg} basis in the analog approach, and of an auxiliary state for realizing multi-qubit gates in the \\texttt{digital} basis.\n\nSince the most general sequence, with channels addressing both transitions, involves three energy levels, the size of the Hilbert space is $3^N$, where $N$ is the number of atoms in the array. Typically, a personal computer can efficiently simulate the evolution of systems of size below $N \\sim 20$.\n\n\\subsection{Measurement}\n\nIn neutral-atom devices, the measurement process encompasses all the atoms in the register and necessarily terminates the computation. The process is based on the fluorescence imaging of the atom array, where ``bright'' and ``dark'' sites are linked to the measured states\\,\\cite{Fuhrmanek11}. Due to there being multiple addressable transitions, the final state of each atom can be in more than two levels and, therefore, the measurement's outcome will be basis-dependent. In Pulser, the convention is to associate the state that is exclusive to a given basis to $1$ and all other outcomes to $0$, as summarized in table \\ref{tab:measures}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c||c | c}\n        \\cline{2-3}\n     \n        &  \\lstinline[] $ground-rydberg$ & \\lstinline[] $digital$\\\\ \\hline\n        $\\ket{r}$ & 1 & 0 \\\\\n        $\\ket{g}$ & 0 & 0 \\\\\n        $\\ket{h}$ & 0 &  1 \\\\\n    \\end{tabular}\n    \\caption{Corresponding measurement outcome of each state, depending on the measurement basis.}\n    \\label{tab:measures}\n\\end{table}\n\n\n\n\\section{Designing sequences with Pulser}\n\\label{section:sequences}\n\nPulser is written in the Python language to facilitate adoption by a widespread audience. It is also Open Source Software and can be accessed and developed by any interested contributor. To install the latest stable version of the Pulser package, type into a terminal (where Python version 3.7 or greater is installed):\n\\begin{lstlisting}[numbers=none]\npip install pulser\n\\end{lstlisting}\n\n\n\\subsection{Code Architecture}\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/Pulser_Architecture_v2.png}\n\\caption{Relationship between the main Pulser classes. The central object is the \\texttt{Sequence}, which is linked to a \\texttt{Device}. The \\texttt{Device} holds the available \\texttt{Channels} --- which are selected and declared in the \\texttt{Sequence} --- and information of the hardware constraints. These constraints are enforced upon the \\texttt{Register}, where the neutral-atom array is defined, and upon the \\texttt{Pulses}. Each \\texttt{Pulse}, defined by its amplitude and detuning \\texttt{Waveforms} and a fixed phase, populates the declared channels alongside other commands like \\texttt{target} --- which points local addressing channels to specific qubits --- and \\texttt{delay} --- which idles the channel. The resulting \\texttt{Sequence} can then be sent for execution on the neutral-atom QPU or emulated through Pulser's \\texttt{Simulation} class.}\n\\label{fig:architecture}\n\\end{figure*}\n\nThe main components of Pulser are:\n\n\\begin{itemize}\n    \\item[(\\emph{i})] The \\texttt{Device}, which consists of a series of specifications that characterize the hardware, including the chosen Rydberg level, the ranges of the amplitudes and frequencies of the lasers, the minimal and maximal distance between the atoms, and the different channels that can be declared (see (\\emph{iii}) below).\n    \\item[(\\emph{ii})] The \\texttt{Register} stores the information about the coordinates of the atoms and their respective ID's, which serve to identify them when targeting specific operations.\n    \\item[(\\emph{iii})] The \\texttt{Channel}s represent the action of the lasers and are organized by addressing (local or global) and the type of transition (Rydberg or Raman).\n    \\item[(\\emph{iv})] \\texttt{Waveform}s are the basic building blocks of a pulse. They can have custom or predetermined shapes, like a ramp waveform or a Blackman waveform, all of them indicating the specific profile of the waveform and its duration.\n    \\item[(\\emph{v})] \\texttt{Pulse}s consist of waveforms for the amplitude and the detuning. They can be further shifted by a phase. Once a \\texttt{Pulse} is constructed it has to be added to the sequence indicating which atom(s) are targeted and what channel will implement it.\n    \\item[(\\emph{vi})] The \\texttt{Sequence} contains the schedule of the pulses in each channel. It is also linked with a \\texttt{Register} and the \\texttt{Device} in which it is to be executed. This is the information that can be sent to a real neutral-atom QPU or simulated on a classical computer.\n    \\item[(\\emph{vii})] \\texttt{Simulation} is included to emulate results for the application of sequences on small instances. In Pulser, we have made use of the QuTiP libraries\\,\\cite{Johansson_2012,Johansson_2013} for the simulation of quantum systems. Each simulation run returns a specialized object that holds the results and features methods for post-processing them.\n\\end{itemize}\n\n\\subsection{Pulse Sequence Creation: A guided walkthrough}\n\nWe will show, step-by-step, how a basic pulse sequence is created with Pulser by going through the paradigmatic protocol for creating the Bell state $\\ket{\\Phi^+} = \\frac{1}{\\sqrt{2}}(\\ket{00} + \\ket{11})$. \n\nStarting from the archetypal circuit in Figure \\ref{fig:bell_circuit}(a), we need to adapt it for execution on neutral-atom devices. The major and necessary change is the decomposition of the CNOT gate, which is not native in neutral-atom devices, to an equivalent construction based on the CZ gate. Additionally, we replace the Hadamard with an equivalent gate (when starting from $\\ket{0}$) that can be executed with a resonant pulse, which reduces the number of free parameters in the pulse, making it less error-prone. These changes result in the equivalent circuit of Figure \\ref{fig:bell_circuit}(b).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1.\\columnwidth]{figures\/bell_eq.pdf}\n\\caption{ (a) Typical quantum circuit for generating the Bell state $\\ket{\\Phi^+}.$ (b) Equivalent circuit to that of Figure \\ref{fig:bell_circuit}(a), adapted for execution on a neutral-atom device. }\n\\label{fig:bell_circuit}\n\\end{figure}\n\nFrom this quantum circuit, we can create the corresponding pulse sequence in Pulser. For this effect, the following modules will be needed:\n\n \\begin{lstlisting}[firstnumber=1]\n import numpy as np\n import pulser\n from pulser import Pulse, Sequence, Register\n \\end{lstlisting}\n \n\n\\subsubsection{Creating the Register}\n\nThe \\texttt{Register} defines the positions of the atoms and their respective names. There are multiple ways of defining a \\texttt{Register}, the most customizable one being to create a dictionary that associates a name (the key) to a coordinate in micrometers (the value). In this case, we only need two qubits, so we will create a \\texttt{Register} with two atoms, 4 $\\mu$m apart, which we will name \\lstinline{'c'} and \\lstinline{'t'}.\n\n\\begin{lstlisting}[firstnumber=4]\n# Coordinates in micrometers\nqubits = {'c': (-2, 0), 't': (2, 0)}\nreg = Register(qubits)\n\\end{lstlisting}\n\nNote that, although we can arbitrarily specify the position of each qubit, the \\texttt{Device} with which we will associate it will impose some restrictions, limiting in particular the minimal distance between any two qubits and the maximal distance of each qubit from the center of the array.\n   \n\n\n\n\n\n\\subsubsection{Initializing the Sequence}\n\nTo create a \\texttt{Sequence}, one has to provide it with the \\texttt{Register} instance and the \\texttt{Device} in which the sequence will be executed. Each \\texttt{Device} object is associated with a physical device and holds its particular constraints. When linked to a \\texttt{Sequence}, it dictates whether the chosen register is valid and, as the sequence is created, validates all instructions.\nWe import the device (in this case, \\texttt{Chadoq2}, which is Pasqal's research and development prototype) from \\texttt{pulser.devices} and initialize our sequence with the freshly created register:\n\n\\begin{lstlisting}[firstnumber=7]\nfrom pulser.devices import Chadoq2\nseq = Sequence(reg, Chadoq2)\n\\end{lstlisting}\n\n\\subsubsection{Channel declaration}\n\nEach device has a set of available channels, which can be consulted through \\texttt{seq.available\\_channels}. Since we need local addressing on both the \\texttt{ground-rydberg} basis (for the CZ gate) and the \\texttt{digital} basis (for the single-qubit gates), we will declare the \\lstinline{'rydberg_local'} and \\lstinline{'raman_local'} channels, which we will name \\lstinline{'rydberg'} and \\lstinline{'digital'}, respectively.\n\n\\begin{lstlisting}[firstnumber=9]\nseq.declare_channel('digital', 'raman_local')\nseq.declare_channel('rydberg', 'rydberg_local',\n    initial_target='c')\n\\end{lstlisting}\n\nOn real devices, each channel can only be declared once and will no longer appear in the list of available channels after declaration, appearing instead in \\texttt{seq.declared\\_channels}.\n\n\\subsubsection{Making the pulses}\nThe different channels will be populated by \\textit{pulses}, which are specific objects that need to be created. A pulse is made up of an amplitude (i.e. Rabi frequency) waveform, a detuning waveform and a fixed phase, although special convenience methods exist when one or both waveforms are constant. In our example, all pulses will be resonant, so we will use \\texttt{Pulse.ConstantDetuning()} to conveniently set $\\delta=0$ without defining its waveform. \n\nFor the amplitude waveform, eq. (\\ref{rot_angle}) tells us that what defines the angle of rotation is simply the integral of the waveform. This ideal scenario does not distinguish between a rectangular waveform or something more complex, as long as its integral is the same. In practice, a properly-shaped waveform can mitigate unwanted modifications to the modulated signal, stemming from spectral leakage or noise sensitivity. Here, we will use the \\textit{Blackman} waveform, which is a tapering function designed to have close to minimum spectral leakage.\n\nStarting with the single-qubit gates, all of those involved have $\\theta=\\pi\/2$, so we declare a Blackman waveform with an area of $\\pi\/2$ and duration of $200$ ns, making up a relatively short pulse without exceeding the amplitude range allowed by the channel where it will be executed. \n\n\\begin{lstlisting}[firstnumber=12]\nfrom pulser.waveforms import BlackmanWaveform\nhalf_pi_wf = BlackmanWaveform(200, np.pi\/2)\n\\end{lstlisting}\n\nThe rotation axis is defined from eq. (\\ref{rot_axis}), from which we gather that a rotation around the $y$-axis corresponds to a phase $\\varphi = -\\pi\/2$. Inversely, since $R_y^\\dag(\\pi\/2) = R_{-y}(\\pi\/2)$, we will have $\\varphi=\\pi\/2$ defining a rotation around $-y$. The corresponding pulse declarations are:\n\n\n\\begin{lstlisting}[firstnumber=14]\nry_pulse = Pulse.ConstantDetuning(\n    amplitude=half_pi_wf,\n    detuning=0,\n    phase=-np.pi\/2,\n)\nry_dag_pulse = Pulse.ConstantDetuning(\n    amplitude=half_pi_wf,\n    detuning=0,\n    phase=np.pi\/2,\n)\n\\end{lstlisting}\n\nMoving on to the CZ gate, which follows the protocol of Figure \\ref{fig:cz_protocol}, we will need a $\\pi$-pulse and a $2\\pi$-pulse, which we can shape with a Blackman waveform as well. However, because we are now dealing with Rydberg interactions, we have to make sure that the Rydberg blockade radius resulting from the chosen waveform is well above the spacing between the atoms. This concern does not extend to all the pulses assigned to the \\texttt{Rydberg} channel, because a Rydberg blockade is only meaningful if multiple atoms are being excited to the Rydberg state simultaneously or if an atom is being excited after its neighbors. Thus, in this case, only the $2\\pi$-pulse will determine a relevant Rydberg blockade radius, which means the $\\pi$-pulses surrounding it can be kept short:\n\n\\begin{lstlisting}[firstnumber=24]\npi_wf = BlackmanWaveform(200, np.pi)\npi_pulse = Pulse.ConstantDetuning(pi_wf, 0, 0)\n\\end{lstlisting}\n\nTo figure out what is the Rabi frequency upper bound that corresponds to a certain Rydberg blockade radius, as dictated by eq. (\\ref{eq:blockade_radius}), we use the \\texttt{rabi\\_from\\_blockade()} method of our chosen device. We set a conservative Rydberg blockade radius of $8 \\mu$m, double the distance between the atoms. The corresponding maximum Rabi frequency value is then\n\n\\begin{lstlisting}[firstnumber=26]\nmax_val = Chadoq2.rabi_from_blockade(8)\n# Result: max_val = 19.10672378540039 rad\/us\n\\end{lstlisting}\nConversely, it is also possible to figure out what the Rydberg blockade radius is for a given Rabi frequency value through the \\texttt{rydberg\\_blockade\\_radius()} method.\n\nWith an upper bound on the amplitude, we can create the appropriate Blackman waveform and corresponding pulse using the following formulation (which leverages the fact that there is a simple relationship between the duration, area and maximum value of a Blackman waveform):\n\\begin{lstlisting}[firstnumber=28]\ntwo_pi_wf = BlackmanWaveform.from_max_val(\n    max_val=max_val,\n    area=2*np.pi,\n)\ntwo_pi_pulse = Pulse.ConstantDetuning(\n    amplitude=two_pi_wf,\n    detuning=0,\n    phase=0,\n)\n\\end{lstlisting}\n\nAll the pulse objects necessary to program the circuit of Figure \\ref{fig:bell_circuit}(b) have been created. As a remark, note that every \\texttt{Pulse} and \\texttt{Waveform} can be plotted by calling its \\texttt{draw()} method, which can be useful to visualize the shape of the waveforms, as well as inspect the range of values they span. As an example, Figure \\ref{fig:two_pi_draw} displays the plot of \\lstinline{two_pi_pulse}, showing both the amplitude and detuning waveforms in the same picture.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/two_pi_draw}\n    \\caption{The outcome of calling \\lstinline{two_pi_pulse.draw()}. Both the amplitude and detuning waveforms are displayed.}\n    \\label{fig:two_pi_draw}\n\\end{figure}\n\n\n\\subsubsection{Composing the Sequence}\n\nAs previously mentioned, the workings of neutral-atom devices ultimately influence the sequences designed in Pulser. In many aspects --- like the way single-qubit gates are realized as pulses --- neutral-atom devices are very similar to other popular implementations. However, when compared to superconducting qubit architectures for instance, there are some marked differences, the most striking being in the connection between channels and qubits. \n\nIn superconducting qubit devices, each qubit has multiple dedicated channels, each serving a specific function, like that of manipulating or measuring the state of the qubit. This means that if one is to apply a pulse on a given qubit, that pulse will be allocated to the channel exclusively serving that qubit. \n\nOn the other hand, a channel in a neutral-atom QPU can serve all the qubits in the register, be it simultaneously or sequentially. In practice, this means that a channel needs to know on which qubit (or qubits) to apply the pulses to, and while there are some channels --- specifically, those of the \\texttt{Global} type --- whose targets are fixed to be the entire register, the channels of type \\texttt{Local} have the ability to change their target throughout the sequence. When the state of a single-qubit is changed, a \\texttt{Local} channel targets it before applying the pulse, after which it can move on to target a different qubit onto which the next pulse will be applied. In this way, a \\texttt{Local} channel can individually manipulate the state of all the qubits in a register, though it will do it sequentially rather than in parallel.\n\nThus, every channel needs to start with a target. For \\texttt{Global} channels, all the atoms are the target by definition, but for \\texttt{Local} channels, their target has to be explicitly defined. This initial target can be set at channel declaration (see how \\lstinline{'rydberg'} was set to target the \\lstinline{'c'} atom), or it can be done by using the standard \\lstinline{target} method of \\lstinline{seq} --- the \\texttt{Sequence} instance previously created.\n\n\\begin{lstlisting}[firstnumber=37]\nseq.target('c', 'digital')\n\\end{lstlisting}\n\nNow both channels have an initial target, so we can compose the sequence by allocating the \\texttt{Pulse}s to the declared \\texttt{Channel}s.\n\nWe start with the single-qubit gates, $R_y(\\pi\/2)$ and $R_y^\\dag(\\pi\/2)$. Recall that these transitions are addressed by the \\texttt{Raman}-type channel, which we called \\lstinline{'digital'}. Thus, the first two gates of circuit \\ref{fig:bell_circuit} are added to the same channel as follows:\n\n\\begin{lstlisting}[firstnumber=38]\nseq.add(ry_pulse, 'digital')\nseq.target('t', 'digital')\nseq.add(ry_dag_pulse, 'digital')\n\\end{lstlisting}\nNotice how the target is changed to \\lstinline{'t'} in between additions, so that the $R_y^\\dag(\\pi\/2)$ gate acts on the correct qubit. \n\nNext, we implement the CZ gate by replicating the protocol of Figure \\ref{fig:cz_protocol}. This will involve the other declared channel, so we must clarify how pulses in different channels are aligned in time. By default, when adding a pulse to a channel (which is targeting one or more qubits), it will be added as soon as all targeted qubits are free. This means that, if existing pulses on other channels are acting on one of the targeted qubits, the channel will be idle until they are done. On the other hand, if there is no overlap between the different channels' targets, the pulses will be played in parallel; this is called the \\lstinline{'min-delay'} protocol. The other addition protocols are the \\lstinline{'wait-for-all'}, in which the pulse will only play after all previously added pulses are finished, and the \\lstinline{'no-delay'}, where a delay is never added. \n\nThere is an additional command, called \\texttt{align()}, which introduces delays that align a subset of channels with the one that finished the latest, such that the next pulse added to any of them will start right after the latest channel has finished. When all the declared channels are aligned, the outcome is identical to that of using the \\lstinline{'wait-for-all'} protocol; otherwise, when only some of the channels are aligned, it corresponds to a ``wait-for-some'' protocol.\n\nAs illustrated in Fig.\\,\\ref{fig:cz_protocol}, the CZ implementation starts with a $\\pi$-pulse on channel \\lstinline{'rydberg'}, which is targeted to the \\lstinline{'c'} qubit. The channel starts empty, but the pulse will not be added at $t=0$ by default because it would overlap with the first pulse of channel \\lstinline{'digital'}, which is also targeting \\lstinline{'c'}. Instead, it would wait for this first pulse to end and then start playing in parallel with the second pulse in the \\lstinline{'digital'} channel. Although this would be a valid implementation, we will instead use the \\lstinline{align} command to make the pulse start only after the \\lstinline{'digital'} channel is done with both pulses (the \\lstinline{'wait-for-all'} would work too because we only declared two channels, but this way can be generalized to the case were other channels are declared). This will avoid dead-times in the CZ implementation, thus keeping the atoms in the Rydberg state no longer than necessary (which is desirable, since the atoms are more sensitive to noise when in the Rydberg state). \n\nThus, the CZ is implemented through\n\\begin{lstlisting}[firstnumber=41]\nseq.align('digital', 'rydberg')\nseq.add(pi_pulse, 'rydberg')\nseq.target('t', 'rydberg')\nseq.add(two_pi_pulse, 'rydberg')\nseq.target('c', 'rydberg')\nseq.add(pi_pulse, 'rydberg')\n\\end{lstlisting}\n\nFinally, there is still a single-qubit gate on the \\lstinline{'t'} qubit that must be applied after the CZ is done (recall that channel \\lstinline{'digital'} was left targeting \\lstinline{'t'}),\n\n\\begin{lstlisting}[firstnumber=47]\nseq.align('digital', 'rydberg')\nseq.add(ry_pulse, 'digital')\n\\end{lstlisting}\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/bell_seq.pdf}\n\\caption{The final Bell state creation sequence, displayed by calling \\lstinline{seq.draw()}. The current target of each channel is displayed in the orange boxes and is part of the original output, but the gates corresponding to each part of the circuit were annotated on top of it. The blue boxes on the right specify the basis in which the system is measured at the end of the sequence.}\n\\label{fig:cz_seq}\n\\end{figure*}\n\n\\subsubsection{Measurement}\n\nAt last, the measurement signals the end of a sequence, so after it no more changes are possible. We can measure a sequence by calling:\n\\begin{lstlisting}[firstnumber=49]\nseq.measure(basis='digital')\n\\end{lstlisting}\nWhen measuring, one has to select the desired measurement basis\\footnote{Here, we choose the \\lstinline{'digital'} basis, not to be confused with the channel we named \\lstinline{'digital'} for addressing this transition.}. The available options depend on the device and can be consulted by calling the chosen device's \\texttt{supported\\_bases} attribute.\n\n\\subsubsection{Visualization}\n\nThroughout its creation process, the sequence can be visualized through the \\texttt{Sequence.draw()} method.\nAs an example, the final sequence is shown in Figure \\ref{fig:cz_seq}. Note that the phase of each pulse is not represented, only the amplitude and detuning modulations of each channel's output --- the latter being always zero in this example. To inspect the phase of each pulse, one can print out the sequence instead, through\n\\begin{lstlisting}[firstnumber=50]\nprint(seq)\n\\end{lstlisting}\nThis will list all the contents of each channel, alongside its start and finish times.\n\n\n\n\\subsubsection{Simulation}\nThe primary goal of Pulser is to serve as a front-end framework for designing pulse sequences for neutral-atom quantum processors. However, it is also possible to emulate the output of a sequence locally on a classical device through the \\texttt{Simulation} class. Pulser's emulator relies on the open-source package QuTiP. Incorporating an emulator in Pulser is valuable for several reasons. First, it enables users to carefully design their pulse sequences and understand their effects before sending the instructions to a real device. It also enables them to gain intuition with simple setups, and explore new possible applications on limited-size systems. Coming back to our example, we use the emulator on our sequence by calling:\n\\begin{lstlisting}[firstnumber=51]\nfrom pulser.simulation import Simulation\nsim = Simulation(seq)\nres = sim.run()\n\\end{lstlisting}\nEvery time the \\texttt{run()} method is called, a \\texttt{SimulationResults} object instance is returned, which holds the results of the simulation.\n\nThe simulation performed is a time evolution of the system at specific times, such that \\lstinline{res} holds the state of the system at each instant. All the states can be accessed through the \\texttt{states} attribute, the final entry being the state of the system at the end of the sequence (before measurement). \n\nIn this case, since the three energy levels of the two atoms were involved, the final state will be a nine-entry vector. However, the population of the Rydberg levels should be negligible, so it is possible to reduce the final state to the \\lstinline{'digital'} basis of interest. We do so by calling\n\\begin{lstlisting}[firstnumber=54]\nres.get_final_state(reduce_to_basis='digital')\n\\end{lstlisting}\nwhich returns a \\texttt{qutip.Qobj} instance holding the final state in the \\lstinline{'digital'} basis. For this example, the outcome is\n$$\n\\begin{pmatrix} \n    0.707+i0.006 \\\\\n    -3.88 \\times 10^{-05}+i0.006 \\\\\n    -2.99\\times 10^{-05}      \\\\\n     0.707       \n\\end{pmatrix},\n$$\nwhich is very close to what we expect for $\\ket{\\Phi^+}$. \n\nAnother way to probe the results is to emulate a real experiment, in which case the only information about the final state of the system is obtained in the measurement. For a given number \\texttt{N\\_samples} of repetitions of the experiment, we call\n\\begin{lstlisting}[firstnumber=55]\nres.sample_final_state(N_samples=1e4)\n\\end{lstlisting}\nDue to finite sampling errors, the outcome will vary slightly for each call. An example of an outcome is \\lstinline[] ${'00': 5005, '11': 4995}$.\n\n\\section{Pulser in action : exploration of usecases with the emulator}\n\\label{section:usecases}\n\nIn this section, we reproduce some results obtained experimentally using Pulser's built-in emulator. These examples, among several others, can be  accessed online at Pulser's documentation site\\footnote{https:\/\/pulser.readthedocs.io}, which includes the code for the optimization of parameters and the generation of the figures.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/afm_seq_ph_diag.pdf}\n\\caption{(a) Schematic phase diagram of a two-dimensional quantum Ising system, with respect to the ratio of applied Rabi frequency $\\hbar \\Omega\/U$ and detuning $\\hbar \\delta\/U$. The blue region indicates the antiferromagnetic (AFM) phase, and outside of this region a paramagnetic (PM) phase is found. (b) The pulse sequence for the adiabatic preparation of the antiferromagnetic state in the Ising model, displayed by calling \\lstinline{seq.draw()}.}\n\\label{AFM:phase_diagram_and_seq}\n\\end{figure*}\n\n\n\\subsection{Preparing an antiferromagnetic state and observing the produced correlations}\n\\label{adiabatic_preparation}\n\n\nFirst we illustrate how to build a sequence for adiabatically preparing an antiferromagnetic state from the Ising-like model of eq.\\eqref{eq:ising_ham}, as well as studying its correlations. These types of states have been studied in neutral-atom setups up to hundreds of qubits, showing remarkable precision and allowing measurements that are beyond the best available classical simulation algorithms \\cite{Bernien17,Lienhard18, scholl2020programmable, ebadi2020quantum}.\n\nThe atoms will be placed in a square array of interatomic distance $R$. We will shine a laser globally with a pulse sequence representing a path in the phase diagram of the quantum Ising model, that starts in the paramagnetic phase and ends in the antiferromagnetic dome, as shown in Fig. \\ref{AFM:phase_diagram_and_seq}(a). We shall evaluate the quality of the resulting state by calculating the spin-spin correlation function, defined as:\n\\begin{equation}\\label{eq:corr_fct}\ng^{(2)}(k,l) = \\frac{1}{N_{k,l}}\\sum_{\\{i,j\\}} \\Big(\\langle \\hat n_i \\hat n_j \\rangle - \\langle \\hat n_i \\rangle \\langle \\hat n_j \\rangle \\Big).\n\\end{equation}\nThe function $g^{(2)}$ measures the correlation between local operators $\\hat n_i, \\hat n_j$ at different positions, averaged over all pairs of atoms $i,j$ separated by a vector $(kR,lR)$ ($N_{k,l}$ is the total number of such pairs). The pulses will be constructed by indicating the values for the maximum Rabi frequency, and the initial and final detuning (as reported in \\cite{lienhard2018observing}). \n\\begin{lstlisting}[firstnumber=1]\n# Load basic packages and classes\nimport numpy as np\nimport pulser\nfrom pulser import Register, Pulse, Sequence\nfrom pulser.devices import Chadoq2\nfrom pulser.waveforms import RampWaveform\nfrom pulser.simulation import Simulation\n# Define setup parameters (in rad\/us and ns)\nOmega_max = 2.3 * 2*np.pi\nU = Omega_max \/ 2.3\ndelta_0 = -6 * U\ndelta_f = 2 * U\nt_rise = 250\nt_fall = 500\nt_sweep = (delta_f-delta_0)\/(2*np.pi*10)*1000\n\\end{lstlisting}\nSince we want to use the blockade effect for nearest-neighbors only, we tune $\\Omega_\\text{max}$ to be of the same order as the van der Waals interaction magnitude for nearby atoms, $\\Omega_{\\text{max}} \\sim U$. This will fix an interatomic distance $R_{\\text{interatomic}}$, which we use to define our register:\n\n\\begin{lstlisting}[firstnumber=16]\nR_interatomic = Chadoq2.rydberg_blockade_radius(U)\nreg = Register.square(3, R_interatomic, prefix='q')\n\\end{lstlisting}\n\n\n\\subsubsection{Creating the sequence}\n\nWe compose our pulse using the \\texttt{Pulse} and \\texttt{Sequence} classes:\n\n\\begin{lstlisting}[firstnumber=18]\n# Define each pulse\nrise = Pulse.ConstantDetuning(\n        RampWaveform(t_rise, 0., Omega_max),\n        delta_0,\n        0.)\nsweep = Pulse.ConstantAmplitude(\n        Omega_max,\n        RampWaveform(t_sweep, delta_0, delta_f),\n        0.)\nfall = Pulse.ConstantDetuning(\n        RampWaveform(t_fall, Omega_max, 0.),\n        delta_f,\n        0.)\n# Add to sequence\nseq = Sequence(reg, Chadoq2)\nseq.declare_channel('ising', 'rydberg_global')\nseq.add(rise, 'ising')\nseq.add(sweep, 'ising')\nseq.add(fall, 'ising')\n\\end{lstlisting}\nThe sequence is readily plotted with the \\texttt{draw()} method, as illustrated in Figure \\ref{AFM:phase_diagram_and_seq}(b).\n\n\n\\subsubsection{Simulation: Spin-Spin Correlation Function}\n\nWe run a simulation of the sequence with the \\texttt{run()} method of the \\texttt{Simulation} class. For simple pulse sequences like the one we are using here, we can use a reduced sampling rate for faster emulation. A progress estimate can also be included if desired, with the parameter \\lstinline{progress_bar=True}. Once the final state is obtained, we may sample from it using the \\texttt{sample\\_final\\_state()} method. Since there was no measurement basis (\\lstinline{'ground-rydberg'} or \\lstinline{'digital'}) specified during the composition of the sequence, we include it as an argument with \\texttt{meas\\_basis}:\n\n\\begin{lstlisting}[language=python, firstnumber=37]\nsimul = Simulation(seq, sampling_rate=0.02)\nresults = simul.run(progress_bar=True)\ncount = results.sample_final_state(\n                   meas_basis='ground-rydberg')\nmost_freq = {k:v for k,v in count.items() if v>10}\n\\end{lstlisting}\n\nFrom the output of the simulation, we can plot the histogram of the sampled results, shown in Figure \\ref{histogram_results}. Notice the peak corresponding to the antiferromagnetic state.\\\\\n\n\\begin{figure}[h]\\label{fig:AFM_hist}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/AFM_hist}\n\\caption{Most frequent results from the sampling of the final state.}\n\\label{histogram_results}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/figure_AFM_pulser_paper_2.pdf}\n\\caption{(a) Simulation results for the $g^{(2)}$ correlation function, eq.\\eqref{eq:corr_fct}. (b) Sweep results for the $S_{\\text{N\u00e9el}}$ score function, eq.\\eqref{eq:neel}.}\n\\label{fig:AFM}\n\\end{figure}\n\nFrom \\lstinline{results}, one can access the state of the system throughout the simulation (as a list of \\texttt{qutip.Qobj}s) by calling \\lstinline{results.states}. This can be useful for post-processing the simulation results and computing observables. Alternatively, one can also call \\lstinline{results.expect()}  to get the exact expectation value of a list of observables. For example, \\lstinline{results.expect( [qutip.basis(9, 0).proj()])} will return the probability of measuring the all-excited state $|r\\rangle^{\\otimes 9}$ for every state in \\lstinline{results}.\n\nWe show in Figure \\ref{fig:AFM}(a) the output of the simulation for the correlation function $g^{(2)}$, eq. \\eqref{eq:corr_fct}. Note that the correlation function is expected to decay exponentially (modulo finite-size effects), which is best observed at larger system sizes~\\cite{scholl2020programmable}.\n\nChanging the endpoint $\\delta_f$ of the pulse's path results in different qualities of the final state. In order to evaluate how well it represents an antiferromagnetic state, we will explore the $\\Omega = 0$ line on the phase diagram, Fig. \\ref{AFM:phase_diagram_and_seq}(a). Since the value of correlation function $g^{(2)}(k,l)$ (eq. \\eqref{eq:corr_fct}) can be positive or negative, we compensate with alternating signs to construct a single score for the state \\cite{lienhard2018observing}:\n\\begin{equation}\nS_{\\text{N\u00e9el}} = \\sum_{(k,l)\\neq (0,0)} (-1)^{|k| + |l|} g^{(2)}(k,l),\n\\label{eq:neel}\n\\end{equation}\nwhich should be largest when the state is closest to an antiferromagnetic product state. By sweeping over different values of detuning $\\delta_f$, we examine the region $ 0 < \\hbar \\delta_f\/U < 4$. In a few seconds, Pulser gives the results shown in Figure \\ref{fig:AFM} (b), showing where the antiferromagnetic phase is more strongly present.\n\n\n\\subsection{Variational algorithms with Pulser: solving a graph problem with neutral atoms}\n\n\nThe implementation of variational algorithms, such as the Quantum Approximation Optimization Algorithm\\,\\cite{Farhi14} (QAOA), is facilitated by the use of a \\textit{parametrized sequence} in Pulser. We illustrate this aspect here, by solving the \\emph{Maximum Independent Set} (MIS) problem on \\emph{Unit Disk} graphs, which can be naturally studied on neutral-atom devices \\cite{pichler2018quantum, henriet2020robustness, dalyac2020qualifying}. \n\n\n\\subsubsection{From a graph to an atomic register} \n\nAn \\emph{independent set} of a graph is a subset of vertices where any two elements of this subset are not connected by an edge. The MIS corresponds to the largest of such subsets. The so-called \\emph{Unit Disk} (UD) version of this problem corresponds to the instances where the graph under consideration lives in 2D and displays an edge between two nodes if they are within a unit length of each other.\\\\\n\nInterestingly, ensembles of interacting Rydberg atoms in 2D can be naturally represented by Unit Disk graph structures: the atoms are the nodes of the graphs and an edge in the graph corresponds to two atoms being within the blockade radius of each other. In this example, we will take $\\Omega$ fixed to a frequency of 1 $\\text{rad}\/\\mu$s, hence the blockade radius can be obtained by using \\lstinline{Chadoq2.rydberg_blockade_radius(1.)}. The following code block instantiates the atomic register, and displays the graph induced by their interactions with the \\lstinline{draw_graph=True} and \\lstinline{draw_half_radius=True} arguments passed to the \\lstinline{reg.draw()} method, as illustrated in Figure \\ref{fig:MIS_reg}.\n\n\n\\begin{lstlisting}[firstnumber=1]\n# Load basic packages and classes\nimport numpy as np\nimport pulser \nfrom pulser import Register, Pulse, Sequence\nfrom pulser.devices import Chadoq2\nfrom pulser.simulation import Simulation\n# Set Register\npos = np.array([[0., 0.], [-4, -7], [4,-7], [8,6], [-8,6]])\nreg = Register.from_coordinates(pos)\nRb = Chadoq2.rydberg_blockade_radius(1.)\n\nreg.draw(\n    blockade_radius=Rb,\n    draw_graph=True,\n    draw_half_radius=True\n    )\n\\end{lstlisting}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.6]{figures\/MIS_reg}\n\\caption{The register for the MIS graph problem. The shaded circles show the blockade half-radius, such that the intersecting regions indicate existing links.}\n\\label{fig:MIS_reg}\n\\end{figure}\n\nThe graph $G$ has two maximal independent sets: $(1,3,4)$ and $(2,3,4)$, respectively \\texttt{01011} and \\texttt{00111} in binary. One could try to prepare those states using an adiabatic approach such as the one illustrated in section \\ref{adiabatic_preparation}. Another approach is to use QAOA, as we illustrate below.\n\n\\subsubsection{Building the parametrized sequence}\n\nQAOA exploits both a quantum and a classical processor that exchange information within a feedback loop to solve an optimization problem. The role of the neutral-atom processor is to prepare and measure an $N$-atom parametrized wavefunction. The outcome of the measurement is then used by the classical processor in a standard classical optimization procedure, that updates the parameters for the next iteration.\\\\\n\nMore specifically, the preparation of the parametrized wavefunction is achieved through the successive application of two non-commutative Hamiltonians, with all atoms initially starting in the ground state $|00\\dots0\\rangle$ of the \\lstinline{ground-rydberg} basis. The first one is realized by taking $\\Omega = 1$ $\\text{rad}\/\\mu s$, and $\\delta = 0 $ $\\text{rad}\/\\mu s$ in Eq.(\\ref{eq:global_ising}). The second Hamiltonian is realized with $\\Omega = \\delta = 1$ $\\text{rad}\/\\mu s$. These two Hamiltonians are applied successively, for durations $t_j$ and $s_j$, respectively. The dimension of $\\bm t$ and $\\bm s$, i.e. the number of layers applied to the system, is referred to as the depth of the algorithm. The outcome of the measurement is then used as the objective function in a standard classical optimization procedure, that updates the parameters for the next iteration.\\\\\n\nWhen implementing a variational procedure, multiple sequences are created that differ only by the durations $t_j$ and $s_j$ of the pulses. This is conveniently handled in Pulser by turning a regular sequence into a \\textit{parametrized} sequence. A \\texttt{Sequence} is said to be parametrized once a \\textit{variable} is declared and used in a sequence-building call, from which point the sequence building process can continue as usual but the sequence is no longer built on the fly. Instead, all calls to a parametrized sequence are stored as a \\textit{blueprint} for generating a new \\texttt{Sequence}, which may depend on the value of the declared variables. Consequently, it is not possible to progressively monitor the creation process of a sequence that is parametrized (e.g. by drawing the state of a sequence as new pulses are added). The building of the sequence itself only happens when \\texttt{Sequence.build()} is called, at which point specific values for all the declared variables have to be specified as arguments.\\\\\n\nThe \\texttt{Variable} objects are obtained by calling \\texttt{Sequence.declare\\_variable()} and can be used throughout the \\texttt{Sequence} creation process (i) as parameters when creating new \\texttt{Waveform}s and \\texttt{Pulses}, or (ii) as arguments for standard sequence creation methods like \\texttt{add}, \\texttt{target}, \\texttt{align} and \\texttt{delay}. Moreover, they support basic arithmetic operations and, when they are of \\texttt{size}$>1$, iteration and item accessing. \\\\\n\nThe notion of parametrized sequence can become very handy with real world constraints such as those we find in cloud-based platforms. First, the bandwidth allocation of a program, variational or not, is greatly reduced thanks to this factorization. A user only needs to send the parametrized sequence that describes his program once, and then each new iteration only requires the associated set of parameters. Thanks to the high flexibility of Pulser, users can have a very fine-grained control over the waveforms that define their pulses. This results in needing larger objects to define these pulses, which makes such a factorization even more useful.\n\nIn addition, one area of improvement for cold atom platforms is the large calibration times for atom register configuration changes. This means that running randomly independent sequences on the same QPU can be very inefficient. The sequences of a given user's program on the other hand share some common parameters, including the register configuration. On the QPU side, we can treat the parametrized sequences as the basic block of the QPU scheduling to greatly reduce the latencies due to the calibration processes, make sure the different user programs are run efficiently and that all the sequences in a given program are run on the same QPU.\\\\\n\nThe example below shows how \\texttt{Sequence} can be used to create a parametrized QAOA sequence with two layers, with the variable duration of each layer stored in the \\lstinline{t_list} and \\lstinline{s_list} arrays. Notice that no value is given to the contents of \\lstinline{t_list} and \\lstinline{s_list}.\n \n\n\\begin{lstlisting}[firstnumber=17]\n# Parametrized sequence\nseq = Sequence(reg, Chadoq2)\nseq.declare_channel('ch0', 'rydberg_global')\n\nt_list = seq.declare_variable('t_list', size=2)\ns_list = seq.declare_variable('s_list', size=2)\n\nfor t, s in zip(t_list, s_list): \n    pulse_1 = Pulse.ConstantPulse(1000*t, 1., 0., 0) \n    pulse_2 = Pulse.ConstantPulse(1000*s, 1., 1., 0)\n    \n    seq.add(pulse_1, 'ch0')\n    seq.add(pulse_2, 'ch0')\n\\end{lstlisting}\n\nThe sequence \\lstinline{seq} above will only be built once the user provides specific values for \\lstinline{t_list} and \\lstinline{s_list} while calling the \\texttt{Sequence.build()} method. It thus enables to build a variety of sequences that share the same structure:\n\n\\begin{lstlisting}[numbers=none]\n# Build sequences with specific values\nmy_seq_1 = seq.build(t_list=[2,4], s_list=[3,6])\nmy_seq_2 = seq.build(t_list=[1,3], s_list=[2,5])\n\\end{lstlisting}\n\n\\subsubsection{Classical optimization and QAOA results}\n\nThe parametrized sequence above can then be used in conjunction with a  classical optimizer in order to determine a set of good values for \\lstinline{t_list} and \\lstinline{s_list}. When running the full QAOA procedure in closed loop, the optimizer is responsible for iteratively selecting the next set of parameters to be tested. The parametrized sequence can then be updated in Pulser and the new program sent externally to the quantum hardware. \n\nThe procedure can also be emulated locally: to try out the algorithm, we applied the non-gradient method Nelder-Mead for a few dozen function evaluations, initializing the parameters in a convenient point (the implementation can be found at Pulser's online documentation). \nThis is already sufficient in order to find some acceptable parameters $\\bm t$ and $\\bm s$. The performance of QAOA can be tested by sampling from the final state $|\\psi(t_f)\\rangle$ which returns both MISs of the graph with high probability. We show in Figure \\ref{fig:MIS_optim} the histogram of the recorded bitstrings.  \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/MIS_optim}\n\\caption{Histogram of most frequent bitstrings after the optimization loop. The MIS graphs (in red) are the ones that are observed most frequently.}\n\\label{fig:MIS_optim}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nWith recent advances in the field, neutral-atom processors now provide unique opportunities for the exploration of uncharted territories in many-body quantum physics with hundreds of interacting quantum units\\,\\cite{ebadi2020quantum,scholl2020programmable,Bluvstein21}. By using distinct electronic transitions for encoding the quantum information, one can implement various Hamiltonians depending on the task at hand. Beyond analog computing, recent developments in qubit addressing have opened the door to high-fidelity quantum gates\\,\\cite{levine_high-fidelity_2018,Levine19}. In addition to the flexibility in terms of information encoding and processing, the geometry of the quantum register itself is also highly programmable\\,\\cite{barredo_synthetic_2018,Schymik20}. This holds great promises for the implementation of graph-related algorithms\\,\\cite{Pichler18,zhou20,henriet2020robustness,dalyac2020qualifying,Henry21}, the exploration of lattice models in condensed-matter physics\\,\\cite{ebadi2020quantum,scholl2020programmable,leseleuc_observation_2019}, or the reduction of the gate-count for quantum circuits\\,\\cite{Henriet2020quantum}. \n\nIn this paper, we introduced Pulser, an open-source Python library allowing users to pilot such powerful devices at the pulse level. This tool will enable the exploration of new avenues in physics and quantum information science requiring an advanced control of the system. After their creation in Pulser, pulse sequences can be translated and read by an arbitrary waveform generator for implementation on a quantum processor. The architecture of the software makes it possible to use as an interface for any neutral-atom QPU as the specifics of the back-end can be encapsulated by creating new \\texttt{Device} objects. Pulser currently enables the control of laser pulses during the quantum information processing phase, but it could be extended in the future to include other controls either in the processing phase, during the assembly of the register or the measurement phase. Being a low-level tool, Pulser could be interfaced with higher-level application-oriented libraries and frameworks for digital quantum information processing, such as Cirq\\,\\cite{cirq_developers_2021_4586899}, Qiskit\\,\\cite{Qiskit}, Atos myQLM, Pennylane\\,\\cite{Pennylane} or tket\\,\\cite{tket}, which abstract away many hardware features. \n\nThe primary purpose of Pulser's built-in emulator is to reproduce faithfully what the output of a real experiment would look like, in order to facilitate the design of hardware runs. In that spirit, some developments are currently underway for the inclusion of realistic error models\\,\\cite{sylvain18}. Some work could also be done to extend the simulations capabilities in terms of the number of qubits that can be efficiently simulated, including the development of tensor-network based methods and GPU acceleration. \n\n\\section*{Acknowledgments}\nWe thank Lucas Beguin, Julien Br\u00e9mont, Antoine Browaeys, Mauro d'Arcangelo, Eric Giguere, Christophe Jurczak, Thierry Lahaye, Boxi Li, Georges-Olivier Reymond, Pascal Scholl, Adrien Signoles, and William J. Zeng for useful discussions.\n\n\n\n\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe condition (on the autocovariances $\\gamma_k$ of a stationary time series)\n\\begin{equation}\n\\lim_{k \\rightarrow \\infty} |\\gamma_k| \\; \\ln k = 0\n\\label{eq:berman}\n\\end{equation}\nwas introduced by \\citet[Theorem 3.1, page 510]{ber}.  It appears to\nhave been adopted as a fundamental sufficient condition in proving\nresults about extreme value distributions for correlated data.  It is\ncited for instance in \\citet[2.5.1, p.~444]{lead1}, \\citet[5.1,\np.~248]{lin}, \\citet[4.1.1, p.~80]{lead2}, \\citet[Theorem 3.8.2,\np.~169; see also p.~198]{gal}, and in \\citet[Theorem 4.4.8,\np.~217]{emb}, where it is described as being ``very weak''.  It\nappears to be effectively the weakest condition that one can assume\nand still obtain positive results in this context.\n\nIn \\citet*{cha} the authors found it necessary to assume, in addition\nto the Berman condition, another condition\n\\begin{equation}\n\\sum_{k=1}^{\\infty} \\frac{|\\gamma_k|}{k^{\\epsilon}} < \\infty\n\\mbox{~~for some~~} \\epsilon < 1.\n\\label{eq:sum.cond}\n\\end{equation}\nThis is given as condition (7) on page 598 of \\citeauthor{cha}.  In that\npaper the authors find it expedient to deal with the residuals from\nfitting an ARMA model to the time series $X_t$ under consideration.\nThey are thereby concerned with the innovation terms of such a model.\nSuppose that $X_t$ and $a_t$ are related via an ARMA model in which\nthe $a_t$ play the role of the innovations.  \\citeauthor{cha} remark\nthat if the time series $X_t$ is a fractionally integrated ARMA\n(``FARIMA'') time series (whence it satisfies the two conditions\nof interest (\\ref{eq:berman}) and (\\ref{eq:sum.cond})) then the\ninnovations $a_t$ also form a FARIMA series provided that the\nmodel is invertible.  Hence the $a_t$ satisfy the two conditions\nof interest as well.\n\n\\citeauthor{cha} assert that more is true:  if $X_t$ is \\emph{any}\nstationary time series satisfying (\\ref{eq:berman}) and\n(\\ref{eq:sum.cond}) and if $X_t$ and a series of innovations $a_t$\nare related by an invertible ARMA model, then the $a_t$ will also\nsatisfy these conditions.  In this note we present the proof of\nthat claim.\n\nWe now remark that interest is focussed on the $a_t$ and these\nquantities are thought of as being the output of a filter, with\nthe $X_t$ being the input.  However the phrasing of the claim,\nwith the $a_t$ being the innovations of an ARMA model, makes it\nappear as if the $a_t$ are the \\emph{input} to a filter, which is\nrather confusing.  The required condition of invertibility\nof the ARMA model is also somewhat disconcerting.  Finally,\nit turns out that a slightly stronger claim may be established.\nWe therefore re-phrase the assertion to be proven, in a stronger\nand less confusing form, and state the original claim as a corollary\nof the re-phrased assertion.\n\n\\section{The Main Result}\n\\label{sec:result}\n\nWe state the result to be proven as follows:\\\\\n\n\\noindent\n{\\bf Theorem:} Suppose that $X_t$ is a stationary time series with\nautocovariances $\\gamma_k$ satisfying conditions (\\ref{eq:berman})\nand (\\ref{eq:sum.cond}) and that the series $Y_t$ is the output of\na linear filter with input $X_t$ given as follows:\n\\[\nY_t = \\sum_{n=0}^{\\infty} \\psi_n X_{t-n}\n\\]\nSuppose that the $\\psi_n$ are summable (whence the $Y_t$ form a\nstationary time series).  Furthermore suppose that the $\\psi_n$\nsatisfy the condition\n\\begin{equation}\n|\\gamma^W_k| \\leq C r^{|k|} \\mbox{~~for all~~} k\n\\label{eq:bound}\n\\end{equation}\nfor some constants $C$ and $r$, $0 < r < 1$, where\n\\[\n\\gamma^W_k = \\sum_{n = -\\infty}^{\\infty} \\psi_n \\psi_{n+k}\n\\]\nand where we set $\\psi_n = 0$ for $n < 0$ (to simplify the notation).\nThen the autocovariances $\\gamma^Y_k$ of the series $Y_t$ satisfy\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) as well.\\\\\n\n\\noindent {\\bf Proof:}\\\\\n\\begin{quote}\n\nWe remark that the $\\gamma^W_k$ are in fact the autocovariances of\na time series $W_t$ defined by\n\\[\nW_t = \\sum_{n=0}^{\\infty} \\psi_n b_{t-n}\n\\]\nwhere $b_t$ is white noise with variance 1.\n\nObserve that\n\\begin{eqnarray*}\n\\gamma^Y_k & = & \\sum_{n=-\\infty}^{\\infty} \\sum_{m=-\\infty}^{\\infty}\n                 \\psi_n \\psi_m \\gamma_{m-n+k} \\\\\n           & = & \\sum_{h=-\\infty}^{\\infty} \\sum_{n=-\\infty}^{\\infty}\n                 \\psi_n \\psi_{n+h} \\gamma_{k+h} \\\\\n           & = & \\sum_{h=-\\infty}^{\\infty} \\gamma^W_h \\gamma_{k+h}\n\\end{eqnarray*}\n\nTo show that the $\\gamma^Y_k$ satisfy condition\n(\\ref{eq:berman}) we write\n\\begin{eqnarray*}\n\\gamma^Y_k\n & = & \\sum_{h=-\\infty}^{-k-1} \\gamma^W_h \\gamma_{k+h} +\n                 \\sum_{h=-k}^{-1} \\gamma^W_h \\gamma_{k+h} +\n                 \\sum_{h=0}^\\infty \\gamma^W_h \\gamma_{k+h} \\\\\n & = & \\sum_{j=1}^{\\infty} \\gamma^W_{k+j} \\gamma_j +\n       \\sum_{j=0}^{k-1} \\gamma^W_{k-j} \\gamma_j +\n       \\sum_{j=0}^{\\infty} \\gamma^W_j \\gamma_{k+j} \\\\\n & = & \\xi_1(k) + \\xi_2(k) + \\xi_3(k) \\mbox{~~(say).}\n\\end{eqnarray*}\n\nTo deal with $\\xi_1(k)$ we observe that\n\\[\n|\\xi_1(k)| \\ln k\n\\leq \\sum_{j=1}^{\\infty} |\\gamma^W_{k+j}| |\\gamma_j| \\; \\ln k\n\\leq C \\; \\gamma_0 \\; r^k \\ln k \\; \\frac{r}{1-r}\n\\]\nusing (\\ref{eq:bound}).  This quantity $\\rightarrow 0$\nas $k \\rightarrow \\infty$ since $r < 1$.\n\nSimilarly\n\\[\n|\\xi_3(k)| \\ln k \\leq \n\\sum_{j=0}^{\\infty} |\\gamma^W_j| |\\gamma_{k+j}| \\ln (k+j) \\; \\leq\nC \\sum_{j=0}^{\\infty} r^j |\\gamma_{k+j}| \\ln (k+j)\n\\]\nTake $\\delta > 0$; for sufficiently large $k$, $|\\gamma_{k+j}| \\ln (k+j)\n\\leq \\delta$ for all $j \\geq 0$.  Hence\n\\[\n|\\xi_3(k)| \\ln k \\leq \\frac{\\delta \\times C}{1-r}\n\\]\nfor sufficiently large $k$, and since $\\delta$ is arbitrary,\n$ |\\xi_3(k)| \\ln k \\rightarrow 0 $ as $k \\rightarrow \\infty$.\n\nTo deal with the middle term $\\xi_2(k)$ we note that\n\\begin{eqnarray*}\n|\\xi_2(k)| & \\leq & C \\sum_{j=0}^{k-1} |\\gamma_j| r^{k-j} \\\\\n & = & C \\left [ \\sum_{j=0}^{[k\/2]} |\\gamma_j| r^{k-j}\n               + \\sum_{j=[k\/2]+1}^{k-1} |\\gamma_j| r^{k-j}\n         \\right ] \\\\\n & \\leq & C \\sum_{j=0}^{[k\/2]} \\gamma_0 r^{k-j}\n               + C \\sum_{j=[k\/2]+1}^{k-1} |\\gamma_j| r^{k-j} \\\\\n & \\leq & C \\gamma_0 \\frac{r^{k\/2}}{1-r}\n               + C \\gamma_{j^*(k)} \\frac{r}{1-r}\n\\end{eqnarray*}\nwhere $j^*(k) = \\mbox{argmax} \\{|\\gamma_j| : [k\/2]+1 \\leq j \\leq k-1 \\}$.\n\nHence\n\\begin{eqnarray*}\n\\ln k \\times |\\xi_2(k)| & \\leq & C \\left [ \\gamma_0 \\ln k \\frac{r^{k\/2}}{1-r}\n                            + (\\ln k\/2 + \\ln 2) \\left ( \\gamma_{j^*(k)}\n                               \\frac{r}{1-r} \\right ) \\right ] \\\\\n & \\leq & C \\left [ \\gamma_0 \\ln k \\frac{r^{k\/2}}{1-r}\n                            + (\\ln j^*(k) + \\ln 2) \\left ( \\gamma_{j^*(k)}\n                               \\frac{r}{1-r} \\right ) \\right ] \\\\\n\\end{eqnarray*}\nwhich $\\rightarrow 0$ as $k \\rightarrow \\infty$.\n\nWe have thus established that the autocovariances $\\gamma^Y_k$\nsatisfy (\\ref{eq:berman}).  We now proceed to show that condition\n(\\ref{eq:sum.cond}) is satisfied:\n\n\\begin{eqnarray*}\n\\sum_{k=1}^{\\infty} \\frac{|\\gamma^Y_k|}{k^{\\epsilon}}\n & = & \\sum_{k=1}^{\\infty} \\left | \\sum_{h=-\\infty}^{\\infty} \\gamma^W_h\n       \\frac{\\gamma_{k+h}}{k^{\\epsilon}} \\right |\n       \\leq \\sum_{h=-\\infty}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & = & \\sum_{h=-\\infty}^{-1} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & = & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & \\leq & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{(k+h)^{\\epsilon}}\n       \\left (\\frac{k+h}{k}\\right)^{\\epsilon} \\\\\n & \\leq & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\zeta \\sum_{h=0}^{\\infty} |\\gamma^W_h| (1+h)\n       \\mbox{, where~~} \\zeta = \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k}|}{k^{\\epsilon}}\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\phantom{\\sum_{k=1}^{\\infty} \\frac{|\\gamma^Y_k|}{k^{\\epsilon}}}\n & = & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\left [ \\sum_{k=1}^{j}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} + \\sum_{k=j+1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} \\right ] +\n       \\zeta \\sum_{h=0}^{\\infty} |\\gamma^W_h| (1+h) \\\\\n & \\leq & \\sum_{j=1}^{\\infty} C r^j \\left [ \\gamma_0 \\sum_{k=1}^{j}\n       \\frac{1}{k^{\\epsilon}} +\n       \\sum_{\\ell=1}^{\\infty} \\frac{|\\gamma_{\\ell}|}{\\ell^{\\epsilon}}\n       \\left ( \\frac{\\ell}{\\ell+j} \\right)^\\epsilon\n       \\right ] + \\zeta \\sum_{h=0}^{\\infty} C r^h \\; (1+h) \\\\\n & \\leq & C \\sum_{j=1}^{\\infty} r^j \\left [ j \\gamma_0 +\n       \\sum_{\\ell=1}^{\\infty} \\frac{|\\gamma_{\\ell}|}{\\ell^{\\epsilon}}\n       \\right ] + \\zeta C \\sum_{h=0}^{\\infty} r^h \\;(1+h) \\\\\n & \\leq & C \\sum_{j=1}^{\\infty} r^j [ j \\gamma_0 + \\zeta] +\n            \\zeta C \\sum_{h=0}^{\\infty} r^h \\;(1+h) \\\\\n & = & C \\left [ \\gamma_0 \\sum_{j=1}^{\\infty} jr^j +\n         \\zeta \\sum_{j=1}^{\\infty}  r^j + \\zeta \\sum_{j=0}^{\\infty} r^j\n         + \\zeta \\sum_{j=1}^{\\infty} j r^j \\right ] \\\\\n & = & C \\left [ (\\gamma_0 + \\zeta) \\sum_{j=1}^{\\infty} j r^j +\n       \\zeta \\; \\frac{1+r}{1-r} \\right ] < \\infty\n\\end{eqnarray*}\nsince $\\sum_{j=1}^{\\infty} jr^j$ converges.  (The radius of convergence\nof this power series is 1, and by assumption $0 < r < 1$.)\n\\hfill \\rule{2mm}{3.5mm}\\\\\n\\end{quote}\n\n\\section{The Original Claim}\n\\label{sec:orig.claim}\n\nWe state the original claim as:\\\\\n\n\\noindent\n{\\bf Corollary:}  Suppose that $X_t$ satisfies conditions\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) and that $X_t$ and $a_t$\nare related by the invertible ARMA model\n\\[\n\\phi(B)X_t = \\theta(B) a_t\n\\]\nwhere $\\phi(z)$ and $\\theta(z)$ are polynomials and $B$ is the\n``backshift'' operator.  Then the $a_t$ satisfy conditions\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) as well.\\\\\n\n\\noindent {\\bf Proof:}\\\\\n\\begin{quote}\nThe invertibility of the model tells us that $a_t$ can be\nexpressed as\n\\[\na_t = \\sum_{n=0}^{\\infty} \\psi_n X_{t-n}\n\\]\nwhere\n\\[\n\\frac{\\phi(z)}{\\theta(z)} = \\psi(x) = \\sum_{n=0}^{\\infty} \\psi_n z^n\n\\]\nwith the coefficients $\\psi_n$ being summable.  (It is more usual in\nsuch a context to denote the coefficients of the series expansion\nas ``$\\pi_n$'' rather than ``$\\psi_n$'', but to make clear the\nrelationship to the main result we eschew the $\\pi_n$ notation.)\n\nNow if we set\n\\[\nW_t = \\sum_{n=0}^{\\infty} \\psi_n b_{t-n}\n\\]\nwhere $b_t$ is white noise (with variance 1) then basic results\nabout ARMA time series (see e.g. \\citet[Chapter 3, problem 3.11]{bro}) tell\nus that the autocovariances $\\gamma^W_k$ of $W_t$ satisfy\n(\\ref{eq:bound}).  Hence the claim follows by the theorem proven\nin section~\\ref{sec:result}.\\\\\n\\mbox{ } \\hfill \\rule{2mm}{3.5mm}\\\\\n\\end{quote}\n\n\\section*{Acknowledgments}\nThe authors express their warm thanks to an anonymous referee whose\ncomments substantially improved the exposition in this paper.  The\nfirst author's research is supported by a grant from the Natural\nSciences and Engineering Research Council of Canada.\n\n\\bibliographystyle{melvin}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main goals of the LEP experiments is to search for new\nparticles predicted by theories beyond the Standard\nModel. In this letter we report on searches for \nunstable  charginos and neutralinos.\nThese particles are predicted by supersymmetric theories (SUSY)\n\\cite{susy}. In SUSY theories with minimal particle\ncontent (MSSM) \\cite{mssm}, in addition to the ordinary particles, there is\na supersymmetric spectrum of particles with spins which differ by one\nhalf with respect to their Standard Model partners.\n\nCharginos (\\charginos{\\pm}), the supersymmetric partners of \\Wpm\\, \nand \\Hpm, are pair produced via $s$-channel \n$\\gamma\/\\mathrm{Z}$ exchange. The production cross section can be\nreduced by an order of magnitude when the $t$-channel scalar neutrino\n(\\susy{\\nu}) exchange is important.\nNeutralinos, the supersymmetric partners of Z, \\gam, and neutral\nHiggs bosons, are pair produced  \n$\\ee \\rightarrow \\neutralino{i}\\neutralino{j}~ (i,j=1, \\ldots ,4$;\nordered by their masses) via $s$-channel $\\mathrm{Z}$ exchange and their\n production cross section can be enhanced by $t$-channel exchange of a \nscalar electron (\\selectron{\\pm}). \n\nShort-lived supersymmetric particles are expected\nin R-parity conserving SUSY models. \nThe R-parity is a quantum number which \ndistinguishes ordinary particles from supersymmetric particles.\nIf R-parity is conserved, supersymmetric particles are \npair-produced and the lightest supersymmetric particle, \nthe lightest neutralino, \\neutralino{1}, is stable. The neutralino is weakly-interacting \nand escapes detection.\nIn this letter we assume R-parity conservation, which implies \nthat the decay chain of supersymmetric particles always contain,\nbesides standard particles, two invisible neutralinos\ncausing the missing energy signature.\n\nWhen the masses of the scalar leptons and the charged Higgs bosons (\\Hpm) are very large,\nthe \\chargino{\\pm} decays via \\ensuremath{\\mathrm{W^\\ast}}:  \n$\\chargino{\\pm} \\rightarrow \\neutralino{1}\\ensuremath{\\mathrm{W^\\ast}}\n                \\rightarrow \\neutralino{1}\\, f\\bar{f}^\\prime$.\nIf the \\slepton{\\pm} and \\susy{\\nu}\\, masses are comparable to \n$M_{\\W}$ the chargino also \ndecays via virtual scalar lepton or scalar neutrino \nand the leptonic branching fraction is enhanced.\nFinally for \\slepton{\\pm} and \\susy{\\nu}\\, lighter than the chargino, \nthe decay modes \n$\\chargino{\\pm} \\rightarrow \\slepton{\\pm} \\nu$ or\n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^\\pm$ become dominant. \nWhen the masses of the neutral SUSY Higgs bosons (\\ho, \\Ao) and of\nthe scalar leptons are very large, the heavier neutralinos \n($\\neutralino{j},\\ j \\ge 2$) decay via \\ensuremath{\\mathrm{Z^\\ast}}:\n$\\neutralino{j} \\rightarrow \\neutralino{k}\\ensuremath{\\mathrm{Z^\\ast}}\n                \\rightarrow \\neutralino{k}\\, f\\bar{f}$  with $k < j$.\nFor a chargino lighter than neutralinos, the latter decay via\n$\\ensuremath{\\mathrm{W^\\ast}}$ such as $\\neutralino{j} \\rightarrow \\chargino{\\pm} f\\bar{f}^\\prime$.\nIf the scalar lepton masses are comparable to the Z mass,\nthe neutralino decays also via a virtual scalar lepton, enhancing the leptonic \nbranching fraction.\nFinally, for \\susy{\\nu}\\, and \\slepton{\\pm} lighter than neutralinos the\ntwo-body decays \n$\\neutralino{j} \\rightarrow \\slepton{\\pm} \\ell^\\mp$ or\n$\\neutralino{j} \\rightarrow \\susy{\\nu} \\nu$ (j $\\ge 2$)\nbecome dominant.    \nThe radiative decays $\\neutralino{j} \\rightarrow \\neutralino{k} \\gamma $ \nare also possible via higher-order diagrams.\n\nPrevious results on chargino and neutralino searches have been \nreported by L3~\\cite{reflep1.5,susy_96,susy_97} \nand other LEP experiments~\\cite{charlep98}.\nIn this letter, new limits are presented on chargino \nand neutralino production cross sections.\nThese experimental results are interpreted in \nthe framework of the constrained MSSM. \nWithin these models lower limits on the masses of \nsupersymmetric particles are derived.\nFor these limits present experimental results are combined \nwith those obtained previously by L3 \nat the Z peak \\cite{oldsusyl3} and at energies up to\n$\\sqrt{s} = 183 \\gev{}$ \\cite{reflep1.5,susy_96,susy_97,nota99slep}.\n\n\\section{Data Sample and Simulation} \\label{dtsmcs}\n\nWe present the analysis of data collected by\nthe L3 detector \\cite{l3-detector} in 1998, \ncorresponding to an integrated luminosity of 176.3 pb$^{-1}$ at an\naverage centre-of-mass energy of 188.6 \\gev{}, denoted hereafter as \n$\\sqrt{s} = 189 \\gev{}$.\n\nStandard Model reactions are simulated with the following \nMonte Carlo generators:\n{\\tt PYTHIA}~\\cite{PYTHIA} for \n  $\\ee \\rightarrow \\mathrm{q\\bar{q}}$,\n  $\\ee \\rightarrow \\mathrm{Z}\\,\\ee$ and \n  $\\ee \\rightarrow \\gamma\\!\/\\mathrm{Z}\\,\\gamma\\!\/\\mathrm{Z} $;\n{\\tt EXCALIBUR}~\\cite{EXCALIBUR} for\n   $\\ee \\rightarrow \\mathrm{W^\\pm\\, e^\\mp \\nu}$;\n{\\tt KORALZ}~\\cite{KORALZ} for\n   $\\ee \\rightarrow \\mu^+\\mu^-$ and\n   $\\ee \\rightarrow \\tau^+\\tau^-$;\n{\\tt BHWIDE}~\\cite{BHWIDE} for \n   $\\ee \\rightarrow \\ee$;\n{\\tt KORALW}~\\cite{KORALW} for\n   $\\ee \\rightarrow \\mathrm{W^+ W^-}$;\ntwo-photon interaction processes have been simulated using \n{\\tt DIAG36}~\\cite{DIAG} ($\\ee \\rightarrow \\ee \\ell^+\\ell^-$) and\n{\\tt PHOJET}~\\cite{PHOJET} ($\\ee \\rightarrow \\ee\\, \\mathrm{hadrons}$), \nrequiring at least 3 \\gev{} for the invariant mass of the two-photon system.\nThe number of simulated events for each background process is \nequivalent to more than 100 times the statistics of the collected \ndata sample except for two-photon interactions for which it is more \nthan two times the data statistics.\n\nSignal events are generated with the Monte Carlo program \n{\\tt SUSYGEN}~\\cite{susygen2.2}, for masses of SUSY particles \n($M_{\\rm SUSY}$) ranging from 45 \\gev{} up to the kinematic limit and for\n$\\DELTAM$ values \n($\\ensuremath{\\Delta M} = M_{\\rm SUSY} - M_{\\neutralino{1}}$) between 3 \\gev{} and\n$M_{\\rm SUSY}-1 \\GeV{}$. \nThe explicit two-body decay branching ratios for charginos \n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^\\pm, \\slepton{\\pm} \\nu $ \nor  $\\neutralino{2,3,4} \\rightarrow \\susy{\\nu} \\nu , \\slepton{} \\ell$\nhave been estimated with {\\tt SUSYGEN}.\n\nThe detector response is simulated using the {\\tt GEANT} \npackage~\\cite{geant}. It takes into account effects of energy loss,\nmultiple scattering and showering in the detector materials and\nin the beam pipe. Hadronic interactions are simulated with the\n{\\tt GHEISHA} program~\\cite{gheisha}. Time dependent inefficiencies\nof the different subdetectors are also taken into account\nin the simulation procedure.\n\n\n\\section{Analysis Procedure}\n\n\\subsection{Signal topologies and optimisation procedure}\n\\label{sec:optimization}\n\n\nBesides the main characteristic of missing transverse momentum,\nsupersymmetric particle signals can be further specified according\nto the number of leptons or the multiplicity of hadronic jets in the\nfinal state. As mentioned in the introduction, chargino pair\nproduction gives final states similar to \\W\\W\\, production.\nFor neutralinos, we distinguish two classes of detectable processes: \n$\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$ and \n$\\ee \\rightarrow \\neutralino{2}\\neutralino{2}$.\nFor these signals, final states are given by the Z branching ratios.\nBoth for charginos and neutralinos, the event energy\nis directly related to \\ensuremath{\\Delta M}\\, ($\\ensuremath{\\Delta M} = M_{\\rm SUSY} - M_{\\neutralino{1}}$).\n\nWe devise five types of selection criteria oriented to all decays \nof charginos, as follows:\nat least two acoplanar leptons (e,$\\mu$);  hadrons and at least one isolated lepton;\n at least two acoplanar taus;  hadrons and at least one isolated tau;\n purely hadronic final states with high multiplicity.\n$\\neutralino{2}\\neutralino{2}$ production gives rise to final states very similar\nto those of chargino pair production, even if with very different \nbranching ratios. Hence, chargino selections based on these five topologies are also effective\nto select $\\neutralino{2}\\neutralino{2}$ events.\n\nThe two-acoplanar-jets final state on the other hand deserves a\ndedicated selection since it accounts for 70\\% of the decays in\n$\\neutralino{1}\\neutralino{2}$ events, and 28\\% in\n$\\neutralino{2}\\neutralino{2}$ events.\n\nThe signal topologies and the associated background sources depend strongly \non \\ensuremath{\\Delta M}. Therefore all five selections are optimised separately for four\ndifferent \\ensuremath{\\Delta M}\\, ranges:  \nthe very low \\ensuremath{\\Delta M}\\, range at $3-5 \\gev{}$, \nthe low \\ensuremath{\\Delta M}\\, range at $10-30 \\gev{}$, \nthe medium \\ensuremath{\\Delta M}\\, range at $40-70 \\gev{}$ and the high range\n\\ensuremath{\\Delta M}\\, at $80-94 \\gev{}$.\nIn the low and very low \\ensuremath{\\Delta M}\\, ranges, the expected topologies for the signal are \ncharacterised by a low multiplicity and a low visible energy, and\n the background is dominated by two-photon interactions.\nFor medium and high \\ensuremath{\\Delta M}\\, ranges, the signal signatures are very similar to those of\nW-pair production; in particular for $\\ensuremath{\\Delta M} > 80 \\gev{}$ on-shell Ws are produced.\n\n\nThe cut values of each selection are {\\it a priori} optimised using\nMonte Carlo signal and background events. \nThe optimisation procedure varies all cuts simultaneously to maximise the\nsignal efficiency and the background rejection.\nIn fact, the average limit ($\\kappa^{-1}$) is minimised for an infinite  \nnumber of tries, \nassuming only background contributions. This is expressed mathematically\nby the following formula:\n\\begin{equation}  \n\\kappa=\\epsilon\/ \\Sigma_{n=0}^{\\infty} k(b)_{n} P(b,n)\n\\end{equation}\nwhere $k(b)_{n}$ is the 95\\% confidence level Bayesian upper limit, \n$P(b,n)$ is the Poisson distribution \nfor $n$ events with an expected background of $b$ events, and \n$\\epsilon$ is the signal efficiency.\n\n\n\n\\subsection{Event selection}\n\nLepton and photon identification, and isolation criteria in hadronic events\nare unchanged compared to our previous analysis \\cite{susy_96}.\nThe Durham algorithm~\\cite{durham} is used for the clustering of\nhadronic jets.\n\nEvents are first selected by requiring at least 3 \\gev{} of visible energy\nand 3 \\gev{} of transverse momentum. Beam-gas events are rejected by\nrequiring the visible energy in a cone of $30^\\circ$ \naround the beam\npipe ($E_{30^0}$) to be less than 90\\% of the total and the missing momentum\nvector to be at least $10^\\circ$ away from the beam pipe.\nTagged two-photon interactions are rejected by requiring the sum of\nthe energies measured in the lead-scintillator\nring calorimeter and in the luminosity monitors~\\cite{l3-detector}\nto be less than 10 \\GeV{}.\nThese two detectors cover the polar angle range \n$1.5^\\circ < \\theta <9^\\circ$ on both sides of the interaction \npoint.\n\n\n\\subsubsection{Leptonic final states} \\label{sec:selection_lep}\n\nFor the pure leptonic final states, dedicated selections have been\noptimised for the charginos, where the two leptons \nmay have a different flavour. Those selections are very  \nsimilar to the scalar lepton selections which are described in \nReference \\cite{slep99ref}. At the end, a combination of all\nthe leptonic selections, providing the optimal sensitivity, is done for\nthe chargino and the neutralino leptonic decays.\n\n\\subsubsection{Lepton plus hadrons final states} \\label{sec:selection_lephad}\n\nWe select events with at least one isolated electron, muon or tau for which \nthe energy, not associated to the lepton, in a cone of $30^\\circ$ \nhalf-opening angle around its direction is less than 2 \\GeV{}. \nThe following quantities are defined:\nthe energy depositions\n($E^{\\perp}_{25}$ and $E_{25}$)\nwithin $\\pm25^\\circ$ around the missing \nenergy direction in the R--$\\phi$ plane \n or in space, respectively. \nWe apply cuts on the number of tracks in the hadronic system\n($N_{tk} - N_{lep}$) and the number of calorimetric clusters ($N_{cl}$).\nFurthermore, cuts are applied on the missing energy direction isolation\n($\\theta_{miss}$ and $E^{\\perp}_{25}$),\nthe total transverse momentum ($p_{\\perp}$),\nthe energy of the isolated lepton ($E_{lep}$), \nthe recoil mass ($M_{rec}$),\nas well as on the acoplanarity angle between\nthe jet and the lepton. A cut is applied on the visible energy ($E_{vis}$)\nand $E_{TTJL}$\nwhich is defined as the absolute\nvalue of the projection of the total momentum of the jet and the \nlepton onto the\ndirection perpendicular to the lepton-jet thrust computed in the R-$\\phi$ \nplane. \nA cut on the invariant mass of the hadronic system ($M_{had}$) \nremoves most of the WW background.\n\nThe cut values at $\\sqrt{s} = 189 \\gev{}$, are shown in Table~\\ref{tab2} for the \ndifferent \\ensuremath{\\Delta M}\\ ranges.\n\n\n\\subsubsection{Purely hadronic final states} \\label{sec:selection_hadrons}\n\nThe list of cuts used at $\\sqrt{s} = 189 \\gev{}$ is reported in Table~\\ref{tab3} for the different \n\\ensuremath{\\Delta M}\\ ranges. \nAgain, we apply cuts on $N_{cl}$, $N_{tk}$, $p_{\\perp}$, $E_{vis}$,\nacollinearity and acoplanarity as well as on the missing energy \npolar angle ($\\theta_{miss}$) and isolation ($E^{\\perp}_{25}$, $E_{25}$).\nThe absolute value of the total momentum of the event \nalong the beam line normalised to the visible energy ($p_\\parallel$),\nthe recoil mass ($M_{rec}$) and the visible mass ($M_{vis}$) are also used\nin the selections.\n\nIn all the selections, but the very low \\ensuremath{\\Delta M},\na cut on the width of the two jets \nis applied. We define $y_{\\perp}$ as the ratio between the scalar sum of the\nparticle momenta transverse to the jet direction and the jet energy.  \nWe require $y_{\\perp}$ to be large in order to select four-jet-like events.\nIn the low \\ensuremath{\\Delta M}\\ range a cut on the ratio $E_{TTJ}$\/$p_\\perp$ \nis applied. \n$E_{TTJ}$ is equivalent to $E_{TTJL}$ using the momenta and the directions of\nthe two jets.    \n\n\n\\section{Results}\n\nThe results at $\\sqrt{s} = 189 \\gev{}$, for \nthe eighteen chargino selections and the four neutralino\nselections are shown in Table \\ref{tab8}. The results for the very low and low \\ensuremath{\\Delta M}\\, selections are shown together.\nA good agreement between the expected background from Standard Model\nprocesses and the selected data is observed.\n\nThe eighteen chargino selections\nfind 147 candidates in the data when expecting 148 events\nfrom the Standard Model processes. \nIn the low and\nvery low \\ensuremath{\\Delta M}\\ regions\n72 events are selected, 11 events in the medium \\ensuremath{\\Delta M}\\, region and\n67 events in the high \\ensuremath{\\Delta M}\\, region. \nIn the four neutralino selections\n50 candidates are found \nwhereas 48.1 events are expected\nfrom the Standard Model processes, \nmost of those events are selected by the \nlow $\\ensuremath{\\Delta M}$ selections.\n\nEach selection is parametrised as a function of a single parameter, $\\xi$,\nin the following manner: \ngiven a lower edge, $X_{loose}^i$, and an upper \nedge, $X_{tight}^i$,\nfor the cut on the variable $i$, the parameter $\\xi$ \nis equal to 0 when this cut is at the lower edge\n(many background events satisfy the selection)\nand 100 when it is at the upper edge\n(no or few background events pass the selection). All cuts  \n($i = 1, . . ., N$) are related to the parameter $\\xi$ as follows:\n$$X_{cut}^i=\nX_{loose}^i +\n(X_{tight}^i- X_{loose}^i)\n\\times \\frac{\\xi}{100}.\n$$\nThe parameter $\\xi$ is scanned around the optimal value ($\\xi$=50)\nto check the agreement between data and Monte Carlo at different background \nrejection stages.\nAs illustrated in Figure~\\ref{fig:xi_chaneu} \nfor the lepton and hadrons final state in chargino decays and \nthe pure hadronic final state in the neutralino decays, the data and\nMonte Carlo simulations are \nin good agreement for all the \\ensuremath{\\Delta M}\\ selections.\nThe vertical arrows show the $\\xi$ value corresponding to\nthe optimised cuts.\n\nFor intermediate \\ensuremath{\\Delta M}\\, values different from those chosen for\noptimisation we choose the combination of selections \nproviding the highest sensitivity \\cite{susy_96}.\nIn this combination procedure, we take into account the overlap\namong the selections within the \ndata and Monte Carlo samples.\n\n\nTypical selection efficiencies, as well as the number of background events\nexpected for a chargino mass of 94 \\gev{} for the purely leptonic final state\n(LL) or\nfor the $\\ensuremath{\\mathrm{W^\\ast}}\\neutralino{1}$ decay mode, are displayed in\nTable \\ref{tab4}. In \nthe latter case, a maximum efficiency of 47\\% is reached for a background\ncontamination of 7.5 events for $\\ensuremath{\\Delta M} = 30 \\gev{}$.\nIn the low \\ensuremath{\\Delta M}\\ region the efficiency decreases due to\nthe large contamination of two-photon interactions and due to the\nlower trigger acceptance.\nFor large \\ensuremath{\\Delta M}\\ it decreases because of the WW background.\n\nThe selection efficiencies, as well as the number of background events\nexpected for a sum of neutralino masses \n$M_{\\neutralino{1}} + M_{\\neutralino{2}} = 188 \\gev{}$ \nfor the pure leptonic decays and for the $\\ensuremath{\\mathrm{Z^\\ast}}\\neutralino{1}$ \ndecay mode are displayed in Table~\\ref{tab5b}.\nCompared to the chargino selection, the efficiencies are\nlower due to the invisible decays of the \\ensuremath{\\mathrm{Z^\\ast}}.\n\nSystematic errors on the signal efficiencies are evaluated as in\nReference \\citen{reflep1.5}, and they are typically 5\\% relative, \ndominated by Monte Carlo statistics. These errors\nare taken into account following the procedure explained \nin Reference \\citen{cal_limit}.\n\n\n\\section{Model independent upper limits on production cross sections}\nNo excess of events is observed\nand we set upper limits on the \nchargino and neutralino production cross sections \nin the framework of the MSSM.\nExclusion limits at 95\\% C.L. are derived taking into account background\ncontributions.\n\nTo derive the upper limits on production cross sections \nand for interpretations in the MSSM we combine the \n$\\sqrt{s} = 189 \\gev{}$ data sample\nwith those collected by L3 at lower centre-of-mass energies \n\\cite{reflep1.5,susy_96,susy_97}.\n\nThe contours of upper limits on the production cross sections for the process\n$\\ee \\rightarrow \\chargino{\\pm}\\chargino{\\mp}$ are shown in \nFigure~\\ref{fig:xsection_chargino} assuming \n$\\chargino{\\pm} \\rightarrow \\ensuremath{\\mathrm{W^\\ast}}\\neutralino{1}$ for the chargino decay \nwith standard W branching fractions, \nand for purely leptonic W decays.\nIn most of the kinematically accessible region, cross sections larger than \n0.2~pb are excluded for both scenarios.\n\nSimilarly, cross section limits for associated neutralino production \n$\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$ are derived as shown in\nFigure~\\ref{fig:xsection_neutralino}\nassuming  $\\neutralino{2} \\rightarrow \\ensuremath{\\mathrm{Z^\\ast}}\\neutralino{1}$, \nwith standard Z branching fractions \nand for purely leptonic Z decays.\nIn most of the kinematically accessible region, cross sections larger than \n0.3~pb are excluded for both scenarios.\n\n\n\n\\section{Interpretation in the MSSM}\n\nIn the MSSM, with Grand Unification assumptions~\\cite{MSSM_GUT},\nthe masses and couplings of the SUSY\nparticles as well as their production cross sections,\nare entirely described~\\cite{mssm} once five parameters are fixed:\n$\\tan\\beta$, the ratio of the vacuum expectation values of the two Higgs \ndoublets, $M \\equiv M_2$, the gaugino mass parameter,\n$\\mu$, the higgsino mixing parameter,\n$m_0$, the common mass for scalar fermions at the GUT scale, and $A$, \nthe trilinear coupling in the Higgs sector.\nThe following MSSM parameter space is investigated: \n$$\n   \\begin{array}{rclcrcl}\n         0.7 \\leq& \\tan\\beta & \\leq 60 ,     &&\n0 \\leq &M_2 &\\leq 2000 \\gev{} ,\\\\\n -2000 \\gev{} \\leq& \\mu       & \\leq 2000 \\gev{} ,&& \n0 \\leq& m_0         &\\leq  500 \\gev{} .\n  \\end{array}\n$$\n\nTo derive the absolute limits on the masses \nof the lightest neutralino and of\nthe lightest chargino,\na scan in the MSSM parameter space is performed \nin steps of $0.2 \\gev{}$ for $M_2$,\n$1.0 \\gev{}$ for $\\mu$ and $0.5 \\gev{}$ for $m_0$. \n\nMass eigenstates of scalar quarks and leptons are in general a \nmixture of the weak eigenstates $\\susy{f}_R$ and $\\susy{f}_L$.\nThe mixing between these two states\nis proportional to the mass of the partner fermion.\nHence the mixing can be sizable only for particles of the\nthird generation. The mixing is governed by the parameters $A$, \n$\\mu$ and $\\tan\\beta$. Besides $\\mu$ and $\\tan\\beta$, also\na scan on $A$ is performed to check the validity of the following results.\n\nAll the limits on the cross sections previously shown, combined with\nthe results obtained at lower centre-of-mass energies and with the\nresults of scalar lepton searches obtained at $\\sqrt{s} = 189 \\gev{}$ \n\\cite{slep99ref}, can be \ntranslated into exclusion regions in the MSSM parameter space.\nTo derive limits in the MSSM, \nwe optimise the global selection for any different\npoint in the parameter space. This is obtained, choosing every time the\ncombination of selections providing the highest sensitivity, given \nthe production cross sections and the decay branching fractions which are \ncalculated with the generator {\\tt SUSYGEN}. When the mixing in the\nscalar tau sector is considered, masses and decay branching fractions\nare calculated with the generator {\\tt ISAJET} \\cite{isajet}.\n\n\n\\subsection{Limits on chargino and neutralino masses} \n\nIn the MSSM, while the cross sections and decay branching fractions\nof the\ncharginos and neutralinos depend on the masses of the scalar leptons, \ntheir masses depend only on \n$M_2$, $\\mu$ and $\\tan\\beta$. \nThe exclusions in the\nhigh $m_0$ range are derived from chargino and\nneutralino searches, while for low $m_0$ the searches for scalar\nleptons \\cite{slep99ref}, and for photons and\nmissing energy final states \\cite{papho97}, also contribute.\nWe also take into account all chargino and neutralino cascade decays:\n\\begin{itemize}\n\\item\n $\\chargino{\\pm} \\rightarrow \\neutralino{2}\\, \\ensuremath{\\mathrm{W^\\ast}}$: \n we observe a slight decrease of the efficiency relative to\n $\\chargino{\\pm} \\rightarrow \\neutralino{1}\\, \\ensuremath{\\mathrm{W^\\ast}}$\n depending on the masses of \\neutralino{2}, \\chargino{\\pm} and \n \\neutralino{1}.\n The lowest efficiency is then used for cascade decays.\n\\item\n $\\neutralino{3,4} \\rightarrow \\neutralino{2}\\, \\ensuremath{\\mathrm{Z^\\ast}}$:\n the efficiency is found to be larger than the efficiency obtained for the\n $\\neutralino{3,4} \\rightarrow \\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$ channel, \n especially in the high \\ensuremath{\\Delta M}\\ region. \n The efficiencies obtained in the latter\n channel are used. \n\\item \n\n$\\neutralino{3,4} \\rightarrow \\susy{\\nu} \\nu$: when the $ \\susy{\\nu}$\nbecomes detectable through its cascade decays into $\\neutralino{2}$ or \n  $\\chargino{\\pm}$. This is especially relevant in the mixed region\n($\\mu\\sim -M_2$) for\nthe low $\\tan\\beta$ values.\n\\end{itemize}\n\n\nDepending on the neutralino-chargino field content, \none distinguishes the following \ncases for the determination of lower limits on the neutralino and chargino \nmasses:\n\\begin{itemize}\n\\item   \n Higgsino-like  \\neutralino{2} and \\chargino{\\pm}  ($M_2 \\gg |\\mu|$):\n in this case, the production cross sections do not depend on the scalar lepton\n masses, \\ensuremath{\\Delta M}\\ is low and decreases with increasing $M_2$.\nConsequently, the limits on the masses of the next-to-lightest neutralino\n and the lightest chargino decrease with $M_2$ as depicted in \n Figure~\\ref{fig:mass_m2}.\n For $\\tan\\beta = \\sqrt{2}$ and $M_2$ less than $500 \\gev{}$, \n$M_{\\neutralino{2}} \\leq 101 \\GeV{}$ and\n $M_{\\chargino{\\pm}} \\leq 93 \\GeV{}$\n are excluded.\n\\item\n Gaugino-like $\\chargino{\\pm}$  ($|\\mu| \\gg M_2$):\n the chargino cross section depends strongly on the scalar neutrino mass. \n For $50 \\gev{} \\leq M_{\\susy{\\nu}} \\leq 80 \\gev{}$ \nthe cross section is reduced by one order of magnitude compared to what is\n expected for $M_{\\susy{\\nu}} \\geq 500 \\gev{}$.\n  When the two body decay \n $\\chargino{\\pm} \\rightarrow \\ell^\\pm \\, \\susy{\\nu}$ \n is dominant, the relevant \\ensuremath{\\Delta M}\\ becomes \n $\\ensuremath{\\Delta M} = M_{\\chargino{\\pm}} - M_{\\susy{\\nu}}$.\nIf the $\\susy{\\nu}$ is mass degenerate with the \\chargino{\\pm} the \nacceptance is substantially reduced. However, when this occurs scalar \nleptons are light and the experimental sensitivity is recovered with these\nchannels.\n\n\\end{itemize}\n\nThe mass limit of the lightest\nchargino is shown in Figure \\ref{fig:mass_cha} as a function of \n$\\tan\\beta$ for all the different chargino field contents. \nAt large $\\tan\\beta$ values,\nthe lower mass limit of the lightest chargino is obtained\nwhen the lightest chargino and the $\\susy{\\nu}$ are mass degenerate\n(gaugino region). At \nlow $\\tan\\beta$ values the lower mass limit on the lightest \nchargino is obtained when the lightest chargino and \nthe $\\neutralino{1}$ (LSP) are mass\ndegenerate (higgsino region). Finally, for $M_2<2$ \\TeV, \nfor $\\tan\\beta \\leq 60$ and for any $m_0$ values, \nthe lower mass limit of the lightest chargino is:\n$$   M_{\\chargino{\\pm}} \\geq 67.7 \\gev{} .$$\n\nThe scalar tau can be much lighter than\nthe scalar electron and muon. This mass splitting \noccurs in particular for large $\\tan\\beta$ and $A$ values.\nWhen this happens, chargino and next-to-lightest neutralino decays \nare affected. Therefore, detection efficiencies are estimated for chargino and\nnext-to-lightest neutralino decays with 100\\% branching ratio\ninto $\\stau{}_{1}\\nu$ and $\\stau{\\pm}_{1}\\tau^{\\mp}$, respectively.\nIn particular, when the \n$\\stau{}_{1}$ and the LSP are mass degenerate the efficiencies \ndecrease substantially.\nHowever, the experimental sensitivity can be partially recovered\ntaking into account also the process\n$\\epem\\ra \\susy{\\nu}\\susy{\\nu}$, where \nthe $\\susy{\\nu}$ is visible through its cascade decays.\nIn particular, the limit on the chargino mass \nholds for any value of the mixing if $\\tan\\beta < 20$.\nFor higher $\\tan\\beta$ values this limit can be decreased\nat most by 10 \\gev. \n\nIndirect limits on the mass of the lightest neutralino\nare also derived as a function of $m_0$ and as a function of tan$\\beta$.\nIn the\nlow  $m_0$ region ($\\leq 65 \\gev{}$) the mass limit on the LSP\ncomes mainly from the scalar lepton searches.\nFor large $m_0$ values ($\\geq 200 \\gev{}$), only  \nthe chargino and neutralino searches contribute. At \nlow $\\tan\\beta$ the processes\n$\\epem\\ \\rightarrow \\neutralino{2} \\neutralino{3,4}$\ncontribute significantly and they are taken into account.\nThe lower mass limit is found at $\\tan\\beta = 1$, $\\mu = -70 \\gev{}$\nand $m_0 = 500 \\gev{}$, as shown in Figure~\\ref{fig:neutralino_mass_a}.\nFor these values of the parameters, \nthe chargino mass is at the kinematic limit and \nthe mass difference between the chargino \nand the LSP is maximal.\n\nFor intermediate $m_0$ values ($65 \\gev{} \\leq m_0 \\leq 95 \\gev{}$)\nthe production cross section for charginos is minimal\nand the $\\susy{\\nu}$ is light enough to allow\nthe following decay modes:\n$\\neutralino{2,3,4} \\rightarrow \\susy{\\nu}\\nu$ and\n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^{\\pm}$. \nThis is the region where the exclusion is due to the interplay of many\ndifferent searches. The limit on the lightest neutralino as a function\nof $m_0$, and for two extreme values of $\\tan\\beta$, \nis shown in Figure~\\ref{fig:neutralino_mass_b}. \nFor low $\\tan\\beta$ values\n($\\le \\sqrt{2}$), the minimum is found for\n$\\mu \\sim -70 \\gev{}$ and large $m_0$ values. \nBetter limits are obtained  \nfor intermediate $m_0$ values, where \nthe neutralino production cross sections are large and the \ntwo body decays of the $\\neutralino{3,4}$ into $\\susy{\\nu} \\nu $   \nare visible through the cascade decays of the\n$\\susy{\\nu}$. \nFor larger $\\tan\\beta$ values, the minimum is found in the gaugino \nregion ($-2000 \\gev{}  < \\mu < -200 \\gev{}$) and for \n$70 \\gev{} \\leq m_0 \\leq 80 \\gev{}$. In this region of the parameter space,   \nthe $\\susy{\\nu}$ and the chargino are \nmass degenerate, the heavier neutralinos  \ndecay invisibly and the experimental sensitivity \nis entirely due to the scalar lepton searches.\n\nFinally in Figure~\\ref{fig:neutralino_any_a}, the mass limit on the\nlightest neutralino as a function of $\\tan\\beta$ for any $m_0$ value\nis shown. For $\\tan\\beta \\geq 0.7$, the \nlower mass limit of the lightest neutralino is \n$$\n   M_{\\neutralino{1}} \\geq 32.5 \\gev{} .\n$$\n\nThe mass limit on the\nlightest neutralino is very little affected\nby the mixing in the scalar tau sector.\nThe limit holds for any value of the mixing if $\\tan\\beta < 20$\nand it can be reduced at most by 1.5 \\gev{} for higher $\\tan\\beta$ values.\nNevertheless, the absolute mass limit for the lightest neutralino\ndoes not change since the lowest value is still found at $\\tan\\beta =1$.\n\n\nThe mass limit on the lightest neutralino is also translated in\nan absolute limit on $M_2$. This is shown as a function \nof $\\tan\\beta$ for any $m_0$ and $\\mu$ as depicted \nin Figure ~\\ref{fig:neutralino_any_b}. \nValues of $M_2$ lower than $54.8 \\gev{}$ are now excluded at 95$\\%$ C.L.\n\n\n\\section*{Acknowledgments}\n\nWe  express our gratitude to the CERN accelerator divisions for the\nexcellent performance of the LEP machine. We also acknowledge\nand appreciate the effort of the engineers, technicians and support staff \nwho have participated in the construction and maintenance of this experiment.\n\n\\newpage\n\n\n\\bibliographystyle{l3stylem}\n\\begin{mcbibliography}{10}\n\n\\bibitem{susy}\nY.A. Golfand and E.P. Likhtman, \\JETP {\\bf 13} (1971) 323; \\\\ D.V. Volkhov and\n  V.P. Akulov, \\PL {\\bf B 46} (1973) 109; \\\\ J. Wess and B. Zumino, \\NP {\\bf B\n  70} (1974) 39;\\\\ P. Fayet and S. Ferrara, \\PRep {\\bf C 32} (1977) 249;\\\\ A.\n  Salam and J. Strathdee, \\FortP {\\bf 26} (1978) 57\\relax\n\\relax\n\\bibitem{mssm}\nH. P. Nilles, \\PRep {\\bf 110} (1984) 1;\\\\ H. E. Haber and G. L. Kane, \\PRep\n  {\\bf 117} (1985) 75;\\\\ R. Barbieri, \\NCim {\\bf 11} No. 4 (1988) 1\\relax\n\\relax\n\\bibitem{reflep1.5}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 377} (1996) 289\\relax\n\\relax\n\\bibitem{susy_96}\nL3 Collab., M. Acciarri \\etal, Eur. Phys. Journal {\\bf C 4} (1998) 207\\relax\n\\relax\n\\bibitem{susy_97}\nL3 Collab., M. Acciarri \\etal, contributed paper n. 493 to {\\it ICHEP98},\n  Vancouver, July 1998\\relax\n\\relax\n\\bibitem{charlep98}\nALEPH Collab., R. Barate \\etal, CERN-EP-99-014 (1999);\\\\ DELPHI Collab., P.\n  Abreu \\etal, \\PL {\\bf B 446} (1999) 75;\\\\ OPAL Collab., G. Abbiendi \\etal,\n  Eur. Phys. Journal {\\bf C 8} (1999) 255\\relax\n\\relax\n\\bibitem{oldsusyl3}\nL3 Collab., O. Adriani \\etal, \\PRep {\\bf 236} (1993) 1; \\\\ L3 Collab., M.\n  Acciarri \\etal, \\PL {\\bf B 350} (1995) 109\\relax\n\\relax\n\\bibitem{nota99slep}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 456} (1999) 283\\relax\n\\relax\n\\bibitem{l3-detector}\nL3 Collab., B. Adeva \\etal, Nucl. Instr. and Meth. {\\bf A 289} (1990) 35; \\\\ M.\n  Chemarin \\etal, Nucl. Instr. and Meth. {\\bf A 349} (1994) 345; \\\\ M. Acciarri\n  \\etal, Nucl. Instr. and Meth. {\\bf A 351} (1994) 300; \\\\ G. Basti \\etal,\n  Nucl. Instr. and Meth. {\\bf A 374} (1996) 293; \\\\ I.C. Brock \\etal, Nucl.\n  Instr. and Meth. {\\bf A 381} (1996) 236; \\\\ A. Adam \\etal, Nucl. Instr. and\n  Meth. {\\bf A 383} (1996) 342\\relax\n\\relax\n\\bibitem{PYTHIA}\nT. Sj{\\\"o}strand, ``PYTHIA~5.7 and JETSET~7.4 Physics and Manual'', \\\\\n  CERN--TH\/7112\/93 (1993), revised August 1995;\\\\ T. Sj{\\\"o}strand, \\CPC {\\bf\n  82} (1994) 74\\relax\n\\relax\n\\bibitem{EXCALIBUR}\n{\\tt EXCALIBUR} version 1.11 is used.\\\\ F.A.~Berends, R.~Kleiss and R.~Pittau,\n  Nucl. Phys. {\\bf B 424} (1994) 308; Nucl. Phys. {\\bf B 426} (1994) 344; Nucl.\n  Phys. (Proc. Suppl.) {\\bf B 37} (1994) 163; Phys. Lett. {\\bf B 335} (1994)\n  490; Comp. Phys. Comm. {\\bf 83} (1994) 141\\relax\n\\relax\n\\bibitem{KORALZ}\n{\\tt KORALZ} version 4.02 is used.\\\\ S. Jadach, B.F.L. Ward and Z. W\\c{a}s,\n  \\CPC {\\bf 79} (1994) 503\\relax\n\\relax\n\\bibitem{BHWIDE}\n{\\tt BHWIDE} version 1.01 is used.\\\\ S. Jadach \\etal, \\PL {\\bf B 390} (1997)\n  298\\relax\n\\relax\n\\bibitem{KORALW}\n{\\tt KORALW} version 1.33 is used.\\\\ M.~Skrzypek \\etal, \\CPC {\\bf 94} (1996)\n  216;\\\\ M.~Skrzypek \\etal, \\PL {\\bf B 372} (1996) 289\\relax\n\\relax\n\\bibitem{DIAG}\nF.A.~Berends, P.H.~Daverfeldt and R. Kleiss,\n\\newblock  Nucl. Phys. {\\bf B 253}  (1985) 441\\relax\n\\relax\n\\bibitem{PHOJET}\n{\\tt PHOJET} version 1.10 is used. \\\\ R.~Engel, \\ZfP {\\bf C 66} (1995) 203; \\\\\n  R.~Engel and J.~Ranft, \\PR {\\bf{D 54}} (1996) 4244\\relax\n\\relax\n\\bibitem{susygen2.2}\n{\\tt SUSYGEN} version 2.2 is used.\\\\ S. Katsanevas and P. Morawitz, \\CPC {\\bf\n  112} (1998) 227\\relax\n\\relax\n\\bibitem{geant}\nThe L3 detector simulation is based on GEANT Version 3.15.\\\\ See R. Brun \\etal,\n  ``GEANT 3'', CERN DD\/EE\/84-1 (Revised), September 1987\\relax\n\\relax\n\\bibitem{gheisha}\nH. Fesefeldt, RWTH Aachen Preprint PITHA 85\/02 (1985)\\relax\n\\relax\n\\bibitem{durham}\nS. Catani \\etal, \\PL {\\bf B 269} (1991) 432;\\\\ S. Bethke \\etal, \\NP {\\bf B 370}\n  (1992) 310\\relax\n\\relax\n\\bibitem{slep99ref}\nL3 Collab., M. Acciarri \\etal, {\\it Search for Scalar leptons in $\\ee$\n  collisions at $\\sqrt{s}$=189 \\gev}, contributed paper n. 7-46 to {\\it\n  EPS-HEP99}, Tampere, July 1999, and also submitted to Phys. Lett\\relax\n\\relax\n\\bibitem{cal_limit}\nR.D. Cousins and V.L. Highland, \\NIM {\\bf A 320} (1992) 331\\relax\n\\relax\n\\bibitem{MSSM_GUT}\nSee for instance:\\\\ L. Ibanez, Phys. Lett. {\\bf B 118} (1982) 73;\\\\ R.\n  Barbieri, S. Farrara and C. Savoy, Phys. Lett. {\\bf B 119} (1982) 343\\relax\n\\relax\n\\bibitem{isajet}\n{\\tt ISAJET} version 7.44 is used. \\\\ H.~Baer \\etal, BNL-HET-98-39, (1998),\n  hep-ph\/9810440\\relax\n\\relax\n\\bibitem{papho97}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 444} (1998) 503\\relax\n\\relax\n\\end{mcbibliography}\n\n \n\\newpage\n\\typeout{   }     \n\\typeout{Using author list for paper 186 -?}\n\\typeout{$Modified: Fri Sep 10 08:43:14 1999 by clare $}\n\\typeout{!!!!  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\nP.Molnar\\rlap,\\tute\\berlin\\\nB.Monteleoni\\rlap,\\tute{\\florence,\\dag}\\ \nT.Moulik\\rlap,\\tute\\tata\\\nG.S.Muanza\\rlap,\\tute\\lyon\\\nF.Muheim\\rlap,\\tute\\geneva\\\nA.J.M.Muijs\\rlap,\\tute\\nikhef\\\nM.Musy\\rlap,\\tute\\rome\\ \nM.Napolitano\\rlap,\\tute\\naples\\\nF.Nessi-Tedaldi\\rlap,\\tute\\eth\\\nH.Newman\\rlap,\\tute\\caltech\\ \nT.Niessen\\rlap,\\tute\\aachen\\\nA.Nisati\\rlap,\\tute\\rome\\\nH.Nowak\\rlap,\\tute\\zeuthen\\                    \nY.D.Oh\\rlap,\\tute\\korea\\\nG.Organtini\\rlap,\\tute\\rome\\\nR.Ostonen\\rlap,\\tute\\seft\\\nC.Palomares\\rlap,\\tute\\madrid\\\nD.Pandoulas\\rlap,\\tute\\aachen\\ \nS.Paoletti\\rlap,\\tute{\\rome,\\cern}\\\nP.Paolucci\\rlap,\\tute\\naples\\\nR.Paramatti\\rlap,\\tute\\rome\\ \nH.K.Park\\rlap,\\tute\\cmu\\\nI.H.Park\\rlap,\\tute\\korea\\\nG.Pascale\\rlap,\\tute\\rome\\\nG.Passaleva\\rlap,\\tute{\\cern}\\\nS.Patricelli\\rlap,\\tute\\naples\\ \nT.Paul\\rlap,\\tute\\ne\\\nM.Pauluzzi\\rlap,\\tute\\perugia\\\nC.Paus\\rlap,\\tute\\cern\\\nF.Pauss\\rlap,\\tute\\eth\\\nD.Peach\\rlap,\\tute\\cern\\\nM.Pedace\\rlap,\\tute\\rome\\\nS.Pensotti\\rlap,\\tute\\milan\\\nD.Perret-Gallix\\rlap,\\tute\\lapp\\ \nB.Petersen\\rlap,\\tute\\nymegen\\\nD.Piccolo\\rlap,\\tute\\naples\\ \nF.Pierella\\rlap,\\tute\\bologna\\ \nM.Pieri\\rlap,\\tute{\\florence}\\\nP.A.Pirou\\'e\\rlap,\\tute\\prince\\ \nE.Pistolesi\\rlap,\\tute\\milan\\\nV.Plyaskin\\rlap,\\tute\\moscow\\ \nM.Pohl\\rlap,\\tute\\eth\\ \nV.Pojidaev\\rlap,\\tute{\\moscow,\\florence}\\\nH.Postema\\rlap,\\tute\\mit\\\nJ.Pothier\\rlap,\\tute\\cern\\\nN.Produit\\rlap,\\tute\\geneva\\\nD.O.Prokofiev\\rlap,\\tute\\purdue\\ \nD.Prokofiev\\rlap,\\tute\\peters\\ \nJ.Quartieri\\rlap,\\tute\\salerno\\\nG.Rahal-Callot\\rlap,\\tute{\\eth,\\cern}\\\nM.A.Rahaman\\rlap,\\tute\\tata\\ \nP.Raics\\rlap,\\tute\\debrecen\\ \nN.Raja\\rlap,\\tute\\tata\\\nR.Ramelli\\rlap,\\tute\\eth\\ \nP.G.Rancoita\\rlap,\\tute\\milan\\\nG.Raven\\rlap,\\tute\\ucsd\\\nP.Razis\\rlap,\\tute\\cyprus\nD.Ren\\rlap,\\tute\\eth\\ \nM.Rescigno\\rlap,\\tute\\rome\\\nS.Reucroft\\rlap,\\tute\\ne\\\nT.van~Rhee\\rlap,\\tute\\utrecht\\\nS.Riemann\\rlap,\\tute\\zeuthen\\\nK.Riles\\rlap,\\tute\\mich\\\nA.Robohm\\rlap,\\tute\\eth\\\nJ.Rodin\\rlap,\\tute\\alabama\\\nB.P.Roe\\rlap,\\tute\\mich\\\nL.Romero\\rlap,\\tute\\madrid\\ \nA.Rosca\\rlap,\\tute\\berlin\\ \nS.Rosier-Lees\\rlap,\\tute\\lapp\\ \nJ.A.Rubio\\rlap,\\tute{\\cern}\\ \nD.Ruschmeier\\rlap,\\tute\\berlin\\\nH.Rykaczewski\\rlap,\\tute\\eth\\ \nS.Saremi\\rlap,\\tute\\lsu\\ \nS.Sarkar\\rlap,\\tute\\rome\\\nJ.Salicio\\rlap,\\tute{\\cern}\\ \nE.Sanchez\\rlap,\\tute\\cern\\\nM.P.Sanders\\rlap,\\tute\\nymegen\\\nM.E.Sarakinos\\rlap,\\tute\\seft\\\nC.Sch{\\\"a}fer\\rlap,\\tute\\aachen\\\nV.Schegelsky\\rlap,\\tute\\peters\\\nS.Schmidt-Kaerst\\rlap,\\tute\\aachen\\\nD.Schmitz\\rlap,\\tute\\aachen\\ \nH.Schopper\\rlap,\\tute\\hamburg\\\nD.J.Schotanus\\rlap,\\tute\\nymegen\\\nG.Schwering\\rlap,\\tute\\aachen\\ \nC.Sciacca\\rlap,\\tute\\naples\\\nD.Sciarrino\\rlap,\\tute\\geneva\\ \nA.Seganti\\rlap,\\tute\\bologna\\ \nL.Servoli\\rlap,\\tute\\perugia\\\nS.Shevchenko\\rlap,\\tute{\\caltech}\\\nN.Shivarov\\rlap,\\tute\\sofia\\\nV.Shoutko\\rlap,\\tute\\moscow\\ \nE.Shumilov\\rlap,\\tute\\moscow\\ \nA.Shvorob\\rlap,\\tute\\caltech\\\nT.Siedenburg\\rlap,\\tute\\aachen\\\nD.Son\\rlap,\\tute\\korea\\\nB.Smith\\rlap,\\tute\\cmu\\\nP.Spillantini\\rlap,\\tute\\florence\\ \nM.Steuer\\rlap,\\tute{\\mit}\\\nD.P.Stickland\\rlap,\\tute\\prince\\ \nA.Stone\\rlap,\\tute\\lsu\\ \nH.Stone\\rlap,\\tute{\\prince,\\dag}\\ \nB.Stoyanov\\rlap,\\tute\\sofia\\\nA.Straessner\\rlap,\\tute\\aachen\\\nK.Sudhakar\\rlap,\\tute{\\tata}\\\nG.Sultanov\\rlap,\\tute\\wl\\\nL.Z.Sun\\rlap,\\tute{\\hefei}\\\nH.Suter\\rlap,\\tute\\eth\\ \nJ.D.Swain\\rlap,\\tute\\wl\\\nZ.Szillasi\\rlap,\\tute{\\alabama,\\P}\\\nT.Sztaricskai\\rlap,\\tute{\\alabama,\\P}\\ \nX.W.Tang\\rlap,\\tute\\beijing\\\nL.Tauscher\\rlap,\\tute\\basel\\\nL.Taylor\\rlap,\\tute\\ne\\\nC.Timmermans\\rlap,\\tute\\nymegen\\\nSamuel~C.C.Ting\\rlap,\\tute\\mit\\ \nS.M.Ting\\rlap,\\tute\\mit\\ \nS.C.Tonwar\\rlap,\\tute\\tata\\ \nJ.T\\'oth\\rlap,\\tute{\\budapest}\\ \nC.Tully\\rlap,\\tute\\prince\\\nK.L.Tung\\rlap,\\tute\\beijing\nY.Uchida\\rlap,\\tute\\mit\\\nJ.Ulbricht\\rlap,\\tute\\eth\\ \nE.Valente\\rlap,\\tute\\rome\\ \nG.Vesztergombi\\rlap,\\tute\\budapest\\\nI.Vetlitsky\\rlap,\\tute\\moscow\\ \nD.Vicinanza\\rlap,\\tute\\salerno\\ \nG.Viertel\\rlap,\\tute\\eth\\ \nS.Villa\\rlap,\\tute\\ne\\\nM.Vivargent\\rlap,\\tute{\\lapp}\\ \nS.Vlachos\\rlap,\\tute\\basel\\\nI.Vodopianov\\rlap,\\tute\\peters\\ \nH.Vogel\\rlap,\\tute\\cmu\\\nH.Vogt\\rlap,\\tute\\zeuthen\\ \nI.Vorobiev\\rlap,\\tute{\\moscow}\\ \nA.A.Vorobyov\\rlap,\\tute\\peters\\ \nA.Vorvolakos\\rlap,\\tute\\cyprus\\\nM.Wadhwa\\rlap,\\tute\\basel\\\nW.Wallraff\\rlap,\\tute\\aachen\\ \nM.Wang\\rlap,\\tute\\mit\\\nX.L.Wang\\rlap,\\tute\\hefei\\ \nZ.M.Wang\\rlap,\\tute{\\hefei}\\\nA.Weber\\rlap,\\tute\\aachen\\\nM.Weber\\rlap,\\tute\\aachen\\\nP.Wienemann\\rlap,\\tute\\aachen\\\nH.Wilkens\\rlap,\\tute\\nymegen\\\nS.X.Wu\\rlap,\\tute\\mit\\\nS.Wynhoff\\rlap,\\tute\\aachen\\ \nL.Xia\\rlap,\\tute\\caltech\\ \nZ.Z.Xu\\rlap,\\tute\\hefei\\ \nB.Z.Yang\\rlap,\\tute\\hefei\\ \nC.G.Yang\\rlap,\\tute\\beijing\\ \nH.J.Yang\\rlap,\\tute\\beijing\\\nM.Yang\\rlap,\\tute\\beijing\\\nJ.B.Ye\\rlap,\\tute{\\hefei}\\\nS.C.Yeh\\rlap,\\tute\\tsinghua\\ \nAn.Zalite\\rlap,\\tute\\peters\\\nYu.Zalite\\rlap,\\tute\\peters\\\nZ.P.Zhang\\rlap,\\tute{\\hefei}\\ \nG.Y.Zhu\\rlap,\\tute\\beijing\\\nR.Y.Zhu\\rlap,\\tute\\caltech\\\nA.Zichichi\\rlap,\\tute{\\bologna,\\cern,\\wl}\\\nF.Ziegler\\rlap,\\tute\\zeuthen\\\nG.Zilizi\\rlap,\\tute{\\alabama,\\P}\\\nM.Z{\\\"o}ller\\rlap.\\tute\\aachen\n\\newpage\n\\begin{list}{A}{\\itemsep=0pt plus 0pt minus 0pt\\parsep=0pt plus 0pt minus 0pt\n                \\topsep=0pt plus 0pt minus 0pt}\n\\item[\\aachen]\n I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG$^{\\S}$\\\\\n III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG$^{\\S}$\n\\item[\\nikhef] National Institute for High Energy Physics, NIKHEF, \n     and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands\n\\item[\\mich] University of Michigan, Ann Arbor, MI 48109, USA\n\\item[\\lapp] Laboratoire d'Annecy-le-Vieux de Physique des Particules, \n     LAPP,IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, France\n\\item[\\basel] Institute of Physics, University of Basel, CH-4056 Basel,\n     Switzerland\n\\item[\\lsu] Louisiana State University, Baton Rouge, LA 70803, USA\n\\item[\\beijing] Institute of High Energy Physics, IHEP, \n  100039 Beijing, China$^{\\triangle}$ \n\\item[\\berlin] Humboldt University, D-10099 Berlin, FRG$^{\\S}$\n\\item[\\bologna] University of Bologna and INFN-Sezione di Bologna, \n     I-40126 Bologna, Italy\n\\item[\\tata] Tata Institute of Fundamental Research, Bombay 400 005, India\n\\item[\\ne] Northeastern University, Boston, MA 02115, USA\n\\item[\\bucharest] Institute of Atomic Physics and University of Bucharest,\n     R-76900 Bucharest, Romania\n\\item[\\budapest] Central Research Institute for Physics of the \n     Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary$^{\\ddag}$\n\\item[\\mit] Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n\\item[\\debrecen] Lajos Kossuth University-ATOMKI, H-4010 Debrecen, Hungary$^\\P$\n\\item[\\florence] INFN Sezione di Firenze and University of Florence, \n     I-50125 Florence, Italy\n\\item[\\cern] European Laboratory for Particle Physics, CERN, \n     CH-1211 Geneva 23, Switzerland\n\\item[\\wl] World Laboratory, FBLJA  Project, CH-1211 Geneva 23, Switzerland\n\\item[\\geneva] University of Geneva, CH-1211 Geneva 4, Switzerland\n\\item[\\hefei] Chinese University of Science and Technology, USTC,\n      Hefei, Anhui 230 029, China$^{\\triangle}$\n\\item[\\seft] SEFT, Research Institute for High Energy Physics, P.O. Box 9,\n      SF-00014 Helsinki, Finland\n\\item[\\lausanne] University of Lausanne, CH-1015 Lausanne, Switzerland\n\\item[\\lecce] INFN-Sezione di Lecce and Universit\\'a Degli Studi di Lecce,\n     I-73100 Lecce, Italy\n\\item[\\lyon] Institut de Physique Nucl\\'eaire de Lyon, \n     IN2P3-CNRS,Universit\\'e Claude Bernard, \n     F-69622 Villeurbanne, France\n\\item[\\madrid] Centro de Investigaciones Energ{\\'e}ticas, \n     Medioambientales y Tecnolog{\\'\\i}cas, CIEMAT, E-28040 Madrid,\n     Spain${\\flat}$ \n\\item[\\milan] INFN-Sezione di Milano, I-20133 Milan, Italy\n\\item[\\moscow] Institute of Theoretical and Experimental Physics, ITEP, \n     Moscow, Russia\n\\item[\\naples] INFN-Sezione di Napoli and University of Naples, \n     I-80125 Naples, Italy\n\\item[\\cyprus] Department of Natural Sciences, University of Cyprus,\n     Nicosia, Cyprus\n\\item[\\nymegen] University of Nijmegen and NIKHEF, \n     NL-6525 ED Nijmegen, The Netherlands\n\\item[\\caltech] California Institute of Technology, Pasadena, CA 91125, USA\n\\item[\\perugia] INFN-Sezione di Perugia and Universit\\'a Degli \n     Studi di Perugia, I-06100 Perugia, Italy   \n\\item[\\cmu] Carnegie Mellon University, Pittsburgh, PA 15213, USA\n\\item[\\prince] Princeton University, Princeton, NJ 08544, USA\n\\item[\\rome] INFN-Sezione di Roma and University of Rome, ``La Sapienza\",\n     I-00185 Rome, Italy\n\\item[\\peters] Nuclear Physics Institute, St. Petersburg, Russia\n\\item[\\salerno] University and INFN, Salerno, I-84100 Salerno, Italy\n\\item[\\ucsd] University of California, San Diego, CA 92093, USA\n\\item[\\santiago] Dept. de Fisica de Particulas Elementales, Univ. de Santiago,\n     E-15706 Santiago de Compostela, Spain\n\\item[\\sofia] Bulgarian Academy of Sciences, Central Lab.~of \n     Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria\n\\item[\\korea] Center for High Energy Physics, Adv.~Inst.~of Sciences\n     and Technology, 305-701 Taejon,~Republic~of~{Korea}\n\\item[\\alabama] University of Alabama, Tuscaloosa, AL 35486, USA\n\\item[\\utrecht] Utrecht University and NIKHEF, NL-3584 CB Utrecht, \n     The Netherlands\n\\item[\\purdue] Purdue University, West Lafayette, IN 47907, USA\n\\item[\\psinst] Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland\n\\item[\\zeuthen] DESY, D-15738 Zeuthen, \n     FRG\n\\item[\\eth] Eidgen\\\"ossische Technische Hochschule, ETH Z\\\"urich,\n     CH-8093 Z\\\"urich, Switzerland\n\\item[\\hamburg] University of Hamburg, D-22761 Hamburg, FRG\n\\item[\\taiwan] National Central University, Chung-Li, Taiwan, China\n\\item[\\tsinghua] Department of Physics, National Tsing Hua University,\n      Taiwan, China\n\\item[\\S]  Supported by the German Bundesministerium \n        f\\\"ur Bildung, Wissenschaft, Forschung und Technologie\n\\item[\\ddag] Supported by the Hungarian OTKA fund under contract\nnumbers T019181, F023259 and T024011.\n\\item[\\P] Also supported by the Hungarian OTKA fund under contract\n  numbers T22238 and T026178.\n\\item[$\\flat$] Supported also by the Comisi\\'on Interministerial de Ciencia y \n        Tecnolog{\\'\\i}a.\n\\item[$\\sharp$] Also supported by CONICET and Universidad Nacional de La Plata,\n        CC 67, 1900 La Plata, Argentina.\n\\item[$\\diamondsuit$] Also supported by Panjab University, Chandigarh-160014, \n        India.\n\\item[$\\triangle$] Supported by the National Natural Science\n  Foundation of China.\n\\item[\\dag] Deceased.\n\\end{list}\n}\n\\vfill\n\n\n\n\n\n\n\n\\newpage\n\n\n\n\n\\begin{table} [htbp]\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|} \\hline  \n\\multicolumn{5}{|c|}{Electron\/Muon plus hadrons selections  } \\\\  \\hline\n           & Very Low \\ensuremath{\\Delta M} & Low \\ensuremath{\\Delta M} & Medium \\ensuremath{\\Delta M} & Large \\ensuremath{\\Delta M} \\\\  \\hline\nNo. of isolated leptons   $\\ge$ &   1   &   1     & 1      &  1    \\\\  \\hline\n$N_{tk} - N_{lep}$       $\\ge$ &  2    &  4      & 5      &  4    \\\\  \\hline\n$N_{cl}$                 $\\ge$ &  6    &  10     & 10     &  10   \\\\  \\hline\n$\\sin(\\theta_{miss})$    $\\ge$ &  0.74 &  0.38   & 0.23   &  0.28 \\\\  \\hline\n $E^{\\perp}_{25}$ (\\gev{})  $\\le$ &  0.52 &  --     & --     & 11.6  \\\\  \\hline\n $p_{\\perp}$ (\\gev{})       $\\ge$ &  3.24 &  5.62   & 8.65   & 9.84  \\\\  \\hline\n $E_{lep} $  (\\gev{})       $\\ge$ &  1.51 & 2.59    & 6.17   & 25.9  \\\\  \\hline\n $E_{lep} $  (\\gev{})       $\\le$ &  9.12 &  27.5   & 31.2   & 43.8  \\\\  \\hline\n $E_{TTJL} $  (\\gev{})       $\\ge$ &  1.27 &  0.95  & 1.44   &  --   \\\\  \\hline\n $M_{had} $  (\\gev{})       $\\le$ &  5.0   &  28.2  & 39.1   & 89.0  \\\\  \\hline\n $M_{rec} $  (\\gev{})      $\\ge$ &  144   &  130    & 107    & 57.0  \\\\  \\hline\n $E_{vis} $  (\\gev{})      $\\ge$ &  4.02   & 8.90   & 31.5   & 65.3  \\\\  \\hline\n $E_{vis} $  (\\gev{})     $\\le$ &  11.0  &  59.1   & 93.6   & 118   \\\\  \\hline\n\\end{tabular} \n\\caption{Values of the cuts for the lepton plus hadrons selections; they\n         are determined with    the optimisation procedure described in \n         Section~\\protect\\ref{sec:optimization}.\\label{tab2}}\n\\end{center}\n\\end{table}\n\\begin{table}[!]\n\\vspace{-0.1 cm}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|} \\hline  \n\\multicolumn{5}{|c|}{Chargino Hadronic selections  } \\\\  \\hline\n                  & Very Low \\ensuremath{\\Delta M}    & Low \\ensuremath{\\Delta M} & Medium \\ensuremath{\\Delta M} & Large \\ensuremath{\\Delta M} \\\\  \\hline\n$N_{cl}$             $\\ge$ &  14    & 14      &  14    &  14      \\\\  \\hline\n$N_{tk}$             $\\ge$ &  5     & 5       & 5      &  5       \\\\  \\hline\n $p_{\\perp} $ (\\gev{})   $\\ge$ &  3.72  & 10.0    & 11.5   & 11.4     \\\\  \\hline\n $p_{\\perp}\/E_{vis}$  $\\ge$ &  --    & 0.20    & 0.15   & 0.10     \\\\  \\hline\n$E_{vis}$ (\\gev{})      $\\le$ & 12.0   &  68.0   & 76.0   & 149    \\\\  \\hline\n Acollinearity (rad)  $\\le$ & 2.00   & --      & --     &  3.02  \\\\  \\hline\n Acoplanarity (rad)   $\\le$ & 2.18   & 2.89    &  2.92  &  3.11 \\\\  \\hline\n sin($\\theta_{miss}$) $\\ge$ &  0.56  & 0.46    &  0.20  &  0.61  \\\\  \\hline\n$E^{\\perp}_{25} $ (\\gev{}) $\\le$ &  0.21& 5.80    &  5.80  & 3.25     \\\\  \\hline\n $E_{25}$ (\\gev{})        $\\le$ &  --  & --      &  --    & 2.53     \\\\  \\hline\n $p_{\\parallel}\/E_{vis}$  $\\le$ &  --  & 0.53    &  0.95  &  0.55  \\\\  \\hline\n $E_{lep}^{max}$ (\\gev{}) $\\le$ &  9.12 &  27.5   & 31.2   & 43.8  \\\\  \\hline\n $M_{vis}$ (\\gev{})       $\\ge$ &  2.85& 9.3    &  35.4  &  --      \\\\  \\hline\n $M_{rec}$ (\\gev{})       $\\ge$ &   -- & 124     &   67.2 & --     \\\\  \\hline\n $E_{vis}\/\\sqrt{s}$     $\\ge$ &   -- & --      &   --   & 0.60     \\\\  \\hline\n $E_{30^0}\/E_{vis}$     $\\le$ &  --  & 0.22    & 0.40   & 0.65      \\\\  \\hline\n $E_{TTJ}\/p_\\perp$      $\\ge$ &  0.24& --      &  0.24  & --       \\\\  \\hline\n $y_{\\perp}$            $\\ge$ &  --  & 0.28    &  0.28  &  0.40   \\\\  \\hline\n\\end{tabular} \n\\caption{Values of the cuts for the purely hadronic selections\n         which  are determined with the optimisation procedure\n         described in Section~\\protect\\ref{sec:optimization}.\\label{tab3}}\n\n\\end{center}\n\\end{table}\n\n\\begin{table} [htbp]\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \\hline  \n & \\multicolumn{2}{|c|}{Low \\ensuremath{\\Delta M}}& \\multicolumn{2}{|c|}{Medium \\ensuremath{\\Delta M}}& \n\\multicolumn{2}{|c|}{High \\ensuremath{\\Delta M}}& \\multicolumn{2}{|c|}{Combined} \\\\  \\hline\n  & $N_{data}$ & $N_{exp}$ & $N_{data}$ & $N_{exp}$ & $N_{data}$ & $N_{exp}$ \n& $N_{data}$ & $N_{exp}$ \\\\  \\hline\n\\chinop & 72 & 66.9 & 11 & 10.9 & 67 & 76.7 & 147 & 148. \\\\ \\hline\n\\chinonn & 43 & 39.3 & 6 & 7.78 & 3 & 2.45 & 50 & 48.1 \\\\ \\hline\n\\end{tabular} \n\\caption[cascade]{Results for charginos and neutralinos:\n           $N_{data}$ is the number of observed events and\n           $N_{exp}$ is the number of expected events from Standard \n           Model processes for the total integrated\n           luminosity collected at $\\sqrt{s} = 189 \\gev{}$.\n\\label{tab8}}\n\\end{center} \n\\end{table} \n\\begin{table} [htbp] \n\\begin{center} \n\\begin{tabular}{|c|r|r|r|r|} \n\\hline  \n{ }   &\n            \\multicolumn{2}{|c|}{~~LL~~} &\n            \\multicolumn{2}{|c|}{~~$\\neutralino{1}\\, \\ensuremath{\\mathrm{W^\\ast}}$~~} \\\\ \\hline\n{ \\ensuremath{\\Delta M}{} (\\gev{}) }  & $\\epsilon $ (\\%) & $~N_{exp}$ &\n                    $\\epsilon $ (\\%) & $N_{exp}$   \\\\\n\\hline \n3   &  1.6  & 20.3           &  1.9   & 39.4 \\\\ \n5   &  6.5   & 20.3          &  16.8  & 76.8  \\\\\n10  &  19.2   & 27.7            &  8.5   & 2.9  \\\\\n20  &  25.0 &   7.3           &  38.2  & 10.4  \\\\\n30  &  30.0   & 7.3            &  46.6  & 7.5  \\\\ \n40  & 28.9   & 7.3            &  44.2  & 4.9  \\\\\n50  & 26.9   & 7.3            &  25.3  & 4.9   \\\\ \n60  & 21.6   & 7.3            &  16.6  & 4.9  \\\\\n75  & 34.5   & 34.6            &  20.8  & 55.5  \\\\\n90  & 31.8   & 34.6            &  7.8   & 55.5  \\\\\n\\hline\n\\end{tabular} \n\\caption[cascade]{Optimised chargino efficiencies ($\\epsilon$)  \n        for the purely leptonic (LL) \n        and for the\n        $\\neutralino{1}\\,\\ensuremath{\\mathrm{W^\\ast}}$ decay mode.\n        $N_{exp}$ is the number of events expected from  \n        Standard Model processes.\n        Results are given as a function of \n        $\\ensuremath{\\Delta M}$\n        for $M_{\\chargino{\\pm}} = 94 \\gev{}$ at $\\sqrt{s} = 189 \\gev{}$.\n\\label{tab4}}\n\\end{center}\n\\end{table}\n\n\\begin{table} [htbp]\n\\begin{center}\n\\begin{tabular}{|c|r|r|r|r|}\n\\hline  \n{ }   &\n            \\multicolumn{2}{|c|}{~~LL~~} &\n            \\multicolumn{2}{|c|}{~~$\\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$~~} \\\\ \\hline\n{ \\ensuremath{\\Delta M}{} (\\gev{}) }  & $\\epsilon $ (\\%)  &~$N_{exp}$ &\n                    $\\epsilon $ (\\%) & $N_{exp}$ \n                    \\\\\n\n\\hline\n6  & 9.1    & 9.5    & 3.5   & 35.9   \\\\\n10  & 10.9  & 10.2   & 10.9  & 35.9    \\\\\n20  & 27.3  & 4.2    & 9.2   & 3.4    \\\\\n40  & 30.8  & 2.8    & 25.9  & 3.4    \\\\\n60  & 34.1  & 19.2   & 35.2  & 7.7     \\\\\n80  & 35.5  & 19.2   & 35.2  & 7.7    \\\\\n100 & 29.3  & 21.4   & 20.7  & 7.7    \\\\\n140 & 17.7  & 25.5   & 9.4  & 2.4    \\\\\n180 & 10.5  & 25.5   & 8.7   & 2.4  \\\\\n\\hline \n\\end{tabular} \n\\caption[cascade]{Optimised neutralino efficiencies ($\\epsilon$)\n        for the purely leptonic (LL)\nfinal states and for the $\\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$ decay mode. \n         Results are given as a function of         $\\ensuremath{\\Delta M}$\n        for $M_{\\neutralino{2}} + M_{\\neutralino{1}} = 188 \\gev{}$\nat $\\sqrt{s} = 189 \\gev{}$.\n \n\\label{tab5b}}\n\\end{center} \n\\end{table} \n\n\\pagebreak \n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\psfig{file=xi_ljjl.eps,width=8.0cm} &\n\\psfig{file=xi_ljjm.eps,width=8.0cm} \\\\\n\\psfig{file=xi_nhhv.eps,width=8.0cm} &\n\\psfig{file=xi_nhhh.eps,width=8.0cm} \n\\end{tabular}\n \\caption{\nNumber of events selected in data (points),\n in Monte Carlo simulation of standard processes (solid line)\nand signal sensitivity (dashed line)\nas a function of selection cuts with increasing background rejection\npower. The vertical arrows show the $\\xi$ value corresponding to\nthe optimised cuts.\nThe distributions are shown for the \nchargino lepton-jets low \\ensuremath{\\Delta M}\\ a), the \nchargino lepton-jets medium \\ensuremath{\\Delta M}\\ b), the\nneutralino jet-jet very low \\ensuremath{\\Delta M}\\ c) and the \nneutralino jet-jet high  \\ensuremath{\\Delta M}\\ d)\nselections, respectively\n         }\n \\label{fig:xi_chaneu}\n\\end{figure}\n\n\\begin{figure}[hbtp]\n\\begin{center}\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{upper_char_w_pap.eps}} \\\\ \n  \\mbox{\\epsfxsize=10.0cm \\epsffile{upper_char_ll_pap.eps}}\n  \\caption{Upper limits on the $\\ee \\rightarrow \\chargino{+}\\chargino{-}$\n production cross section\nup to $\\sqrt{s} = 189 \\gev{}$ in \nthe $M_{\\neutralino{1}} - M_{\\chargino{\\pm}}$ plane.\n           Exclusion limits are obtained assuming\n  standard W branching\nratios in the chargino decay a) or purely leptonic W \n           decays b), \n           $\\chargino{\\pm} \\rightarrow  \\neutralino{1} \\ell^\\pm \\nu $\n           ($\\ell=$e, $\\mu$, $\\tau$).\n          }\n  \\label{fig:xsection_chargino}\n \\end{center}\n\\end{figure} \n\\begin{figure}[hbtp]\n \\begin{center}\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{pap_neu.eps}} \\\\\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{pap_neu_lep.eps}}\n  \\caption{Upper limits on the $\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$\n production cross section\nup to $\\sqrt{s} = 189 \\gev{}$ \n           in the $M_{\\neutralino{1}} - M_{\\neutralino{2}}$ plane.\n           Exclusion limits are obtained \nassuming standard Z branching\n ratios in the next-to-lightest neutralino decay \n$\\neutralino{2} \\rightarrow \\ensuremath{\\mathrm{Z^\\ast}} \\neutralino{1}$ a)\nor assuming purely leptonic Z \n           decays b),   \n            $ \\neutralino{2}\\rightarrow  \\neutralino{1} \\ell^+ \\ell^- $\n           ($\\ell=$e, $\\mu$, $\\tau$).\n          }\n  \\label{fig:xsection_neutralino}\n \\end{center}\n\\end{figure} \n\n\n\\begin{figure}[hbtp]\n \\begin{center} \\vspace*{-1.5cm}\n \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.50cm \\epsffile{higgsino_189_neu2.eps}} \\\\\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.50cm \\epsffile{higgsino_189_char.eps}}\n\\vspace{-.5cm}\n \\caption {Lower mass limits as a function of $M_2$ for\n           the next-to-lightest neutralino a)\nand the lightest chargino b).\n The limits are shown for $\\tan\\beta = \\sqrt{2}$ and for $\\mu>0$ and $\\mu<0$. \n          } \n  \\label{fig:mass_m2}\n \\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}[hbtp]\n \\begin{center}\n\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=18cm \\epsffile{limchar_189_last.eps}}\n  \\caption{\nLower limit on $M_{\\chargino{\\pm}}$ as a function of\n           $\\tan\\beta$ and for any value of $m_0$.\nThe solid line (gaugino region) shows the lower limit obtained for\nlight scalar neutrinos (also small $M_2$ values), which corresponds to the \nabsolute lower limit for large $\\tan\\beta$ values. The \ndashed line (higgsino region) shows the lower limit obtained for very small $\\ensuremath{\\Delta M}$ values. This line corresponds to the absolute lower limit for \nsmall $\\tan\\beta$ values.\n  \\label{fig:mass_cha}}\n \\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}\n \\begin{center}\n\\vspace{-1.5cm}\n  \\hspace*{-2.0cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limneu_m0500_189_last.eps}}\n\\vspace{-0.5cm}\n \\caption{ Lower limit on the lightest neutralino mass, $M_{\\neutralino{1}}$,\n           as a function of $\\tan\\beta$ for  \n           $m_0=500 \\gev{}$, when combining all chargino and neutralino searches.}\n \\label{fig:neutralino_mass_a}\n\\vspace{-.5cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{tan_beta_pap.eps}}\n\\vspace{0.0cm}\n  \\caption{Lower limit on the lightest neutralino mass, $M_{\\neutralino{1}}$,\n            as a function of\n            $m_0$ for two values of $\\tan\\beta$. Scalar lepton searches\ncontribute in the low $m_0$ region. Chargino searches contribute mainly\nin the high $m_0$ region. For the low $\\tan\\beta$ values, the neutralino\nsearches give additional contribution in the intermediate $m_0$ region.\n          }\n  \\label{fig:neutralino_mass_b}\n \\end{center}\n\\end{figure} \n\n\\begin{figure}\n \\begin{center}\n\\vspace{-1.cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limneu_anym0_189_last.eps}}\n\\vspace{-.5cm}\n  \\caption{Lower limit on $M_{\\neutralino{1}}$\n            as a function of\n  $\\tan\\beta$ and for any value of $m_0$,  when combining\nthe chargino, neutralino and scalar lepton searches.\n          }\n  \\label{fig:neutralino_any_a}\n\\vspace{-.5cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limm2_anym0_189_last.eps}}\n  \\caption{Lower limit on $M_2$\n            as a function of\n           $\\tan\\beta$ and for any value of $m_0$,  when combining\nthe chargino neutralino and scalar lepton searches.          }\n  \\label{fig:neutralino_any_b}\n \\end{center}\n\\end{figure} \n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nRobots that can operate in unconstrained environments and collaborate with humans must be capable of learning about new objects they may encounter. Pointing at an object with our hand is a natural way to communicate with the robot about a new object. In this paper, we consider the problem of teaching robots novel objects by pointing at the new object. Once we show the robot a new object, it can generate and store a feature vector corresponding to that object and then re-use it for one-shot localization of the object in new scenes (Fig.~\\ref{fig:hero}).\n\nNeural networks have been used in recent years to learn fully differentiable visuomotor policies that directly map pixels to actuator commands~\\cite{levine2016end,zhang2018deep,yu2018one,rahmatizadeh2018virtual,giusti2015machine}. The neural network architecture typically used for such policies can be decomposed into vision layers and control layers. The vision layers comprise of several convolutional and pooling layers followed by a spatial attention mechanism that attends to the objects of interest in the image. We propose modulating the spatial attention so as the network is able to attend to the object that the hand is pointing at (see Fig.~\\ref{fig:hero}) while ignoring the other distracting objects in the scene.\n\nIn this work, we assume that only the location of the object of interest is available for training and that the position and orientation of the pointing hand are unavailable. So, this is a weakly supervised learning problem where the neural network must figure out as part of the learning process that the pointing hand in the image is salient and then learn to attend to the object being pointed at. On the other hand, this assumption makes the process of data acquisition with a real robot easier by reducing the labeling effort\n\nUnlike other papers on object detection~\\cite{redmon2016you,ren2015faster}, we are primarily interested in teaching robots new objects. This means that we are interested in objects not seen by the neural network during training. We accomplish this using Siamese networks~\\cite{koch2015siamese,venkatesh2019one,bertinetto2016fully}, which are twin neural networks with shared weights. The idea is to use the neural network to obtain from the image a feature vector representing the object of interest rather than classifying the contents of the image as is usually done (Fig.~\\ref{fig:hero}). This vector can be subsequently used in new environments to find the novel object of interest.\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/hero.PNG}\n    \\caption{One-shot localization of novel object selected by the pointing hand. The feature vector of the object (blue bottle cap) that the hand is pointing at is extracted using the proposed attention modulation mechanism. This is then used to localize the object in new scenes. Note that the pointing finger is at considerable distance from the object of interest.}\n    \\label{fig:hero}\n\\end{figure}\n\nOur contributions are as follows:\n\\begin{itemize}\n    \\item We propose a spatial attention modulation mechanism that endows the neural network with the ability to selectively attend to the object that is being pointed at while ignoring other distracting objects in the scene.\n    \\item We show that the proposed method can be combined with Siamese networks to teach robots novel objects.\n\\end{itemize}\n\nThe proposed network architecture is trained on synthetic data constructed from a dataset of emojis. We demonstrate the proposed method on the Dobot Magician robot arm. We show that the robot learns new objects that we point at and can find them in new scenes.\n\nThe rest of this paper is organized as follows. In the next section, papers related to this work are discussed. In Section~3, the proposed model architecture is described. Section~4 details experimental results, and Section~5 concludes the paper.\n\n\n\\section{RELATED WORK}\n\\subsection{Hand Recognition}\nOne of the ways to design a system that can infer the object of interest in a similar scenario is to use a pipelined approach. For example, one could employ deep learning models trained to localize human hands in an image~\\cite{openpose, doosti2019hand}, extract the most relevant keypoints of the hand (say the joints of the index finger) and then fit a line that passes through these keypoints. An object recognition module could then be used to localize each object in the scene, project these points on the line and pick the object corresponding to the closest point to the the hand as the object of interest. However, training such a system requires a strong level of supervision such as the positions of all objects in the scene, and possibly even the keypoints of the hand if one would like to fine-tune the hand localization models. However, this approach will not be feasible in a weak supervision setting as outlined in this paper where only the location of the object of interest is given. Moreover, it has been shown across a wide range of problems that using an end-to-end approach leads to better performance as compared to using a pipelined approach for a given task~\\cite{gupta2016synthetic, end_to_end_speech, zhang2018deep}.\n\n\\subsection{Spatial Attention}\nThe architecture of typical end-to-end networks for visuomotor tasks can be broadly grouped into two sets of layers. The initial group of layers form the vision layers that help in localizing the relevant objects in the image. The remaining layers form what is known as the control layers which are responsible for coming up with the appropriate control actions required to perform the task at hand. A key component in such end-to-end networks is some form of a spatial attention mechanism that learns to attend to the relevant object of interest in the scene. The work presented in~\\cite{zhang2018deep} demonstrates the use of imitation learning for teaching a PR2 robot to perform simple tasks such as pick-and-place. The authors developed a virtual reality based system to teleoperate the robot and collect training data. The data was then used to train an end-to-end network that maps image pixels directly to robot joint velocities. The network consists of an initial set of convolution layers that generates a feature map. The feature map is passed through a spatial softargmax layer to output a feature vector. The resulting feature vector is then passed through a few fully connected layers to predict the joint velocities of the robot. The spatial softargmax layer serves as a simple spatial attention mechanism where the attention weight corresponding to each pixel of the feature map depends on the degree of activation.\n\n\\subsection{One Shot Learning}\nApart from inferring the object of interest in an image in the presence of other objects, another goal of this paper is to enable robots to recognize objects that they have never encountered before by training it on only a few examples involving the novel object. Broadly speaking, meta learning and Siamese networks are two approaches one can take to achieve this. We review both approaches below.\n\n\\subsubsection{Meta Learning}\nIn Meta learning, also known as learning to learn, a distribution of tasks are provided as training data. Typically only a few examples for each task are provided in the training data. The weights of the network after the training process completes serves as a good initialization for the network to learn to perform any new previously unseen task. Only a few training examples involving the novel task and a few gradient descent steps are required for the network to converge to an optimal set of weights~\\cite{maml}. Recently, meta learning and imitation learning has been combined to enable robots to perform novel tasks such as pick-and-place by training on just a single example \\cite{osiml, daml}.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/archext.PNG}\n    \\caption{The proposed neural network architecture for one-shot localization of the object selected by the pointing hand. The convolutional layers in the ``Conv\" block are Conv3x3(16)-ELU-Conv3x3(32)-ELU-Conv3x3(64)-ELU-MaxPool2x2-Conv3x3(64)-ELU-Conv3x3(128)-ELU-MaxPool2x2-Conv3x3(256)-ELU-Conv3x3(512)-ELU-Conv3x3(1024)-ELU. All convolutions are ``valid\" convolutions that do not use padding so that the feature vector for the object is the same regardless of whether it is near the edge of the image or at the center. The receptive field of the ``Conv\" block is 34~px.}\n    \\label{fig:arch}\n\\end{figure*}\n\n\\subsubsection{Siamese Networks}\nSiamese networks are used to address the similarity learning problem where it is desirable to infer if a pair of images (referred to exemplar and search images) are similar to each other or not. This is done by using twin convolution neural networks with shared weights that transform the images $x_1$ and $x_2$ into feature embeddings $\\phi(x_1)$ and $\\phi(x_2)$, respectively. The embedding pair is then combined using a transformation $g$ that can be used to make suitable predictions depending on the task at hand. For example, in the context of image classification~\\cite{koch2015siamese} the transformation $g$  is a distance metric that can be used to measure the similarity score between the object in the exemplar image and the search image. The system is trained on several examples of similar and dissimilar pairs of images. Once training is complete a database of images is built with one image corresponding to each object of interest. At test time the similarity of the search image is tested against each image in the database to determine the object class of the search image. Siamese networks have been used in face recognition systems as well~\\cite{deepface}. However, in both these papers the comparison is possible only if the exemplar and search images are of the same dimension. \n\nThe authors of~\\cite{bertinetto2016fully} used fully convolutional neural networks to enable comparison of images of dissimilar dimensions with Siamese networks. Their architecture was adapted successfully for object tracking in videos. Here the user provides the exemplar image by cropping out the object of interest from the first frame of the video which is then compared against each subsequent frame using the Siamese network. More recently, the authors in~\\cite{venkatesh2019one} combined fully convolutional Siamese networks with spatial attention to enable object localization for robot pick-and-place tasks. The paper explores specifying the object of interest by using visual cues instead of requiring the user to provide a cropped image of the object. Given a group of objects in a scene the user indicates to the robot the object of interest by shining a laser beam directly on it. Although the authors talk of localizing novel objects using a laser beam as a visual cue, the network designed by them should work for any other kind of visual cue (such as a stick or even a hand) so long as it is in very close proximity with the object of interest. However, a human merely has to point at an object from a distance to convey that it is of interest and an observer infers and localizes the object being referred to by looking in the direction of the pointing hand. We would like to design systems that communicate intent to robots much like how humans communicate with each other via visual cues or using natural language~\\cite{blocks, touchdown}. Learning to localize an object of interest from natural language instructions requires a different architecture design compared to the one presented in this paper. We will restrict our focus to learning to localize the object of interest by using a visual cue provided such as a hand pointing at the object from a distance.\n\n\n\n\n\\section{NETWORK ARCHITECTURE}\n\n\n\\subsection{Localizing the Object of Interest}\n\n\n\n\n\n\n\n\nThe proposed neural network for one-shot localization is shown in Fig.~\\ref{fig:arch}. Let the exemplar image (denoted as $x$) correspond to the image that contains the novel object that is being pointed at by the hand. Let the search image (denoted as $\\hat{x}$) be the image of the new scene in which the same object must be localized. The network outputs the locations of the object in the exemplar image and the search image which are denoted as $(p_x, p_y)$ and $(\\hat{p}_x, \\hat{p}_y)$, respectively. The mean squared error loss is used to train the network.\n\nThe localization of the object is performed in a similar fashion as described in \\cite{venkatesh2019one} except for the attention modulation block. The exemplar image is passed through the CNN to obtain a feature map $x_f$. This is then passed through a bottleneck convolutional layer (conv$1\\times1$) to obtain $x_o$. Let us ignore for now how the attention modulation map $x_m$ is generated. We will describe the generation of the attention map $x_m$ in Section~\\ref{sec:attn-mod}. A spatial attention map $x_{o^*}$ is generated  by adding the attention modulation map $x_m$ to $x_o$ and the resulting sum is passed through a spatial soft-argmax layer whose output is the predicted location of the object of interest $(p_x, p_y)$ in the exemplar image $x$ (see Eqns. \\eqref{eqn:x_ostart}, \\eqref{eqn:softargmax_x} and \\eqref{eqn:softargmax_y}). The spatial attention map $x_{o^*}$ is used to obtain the feature vector $f$ corresponding to the object of interest from the feature map $x_f$ (see Eqn. \\eqref{eqn:feat_vec}). Note that $\\odot$ in Fig.~\\ref{fig:arch} corresponds to the element-wise multiplication operation (with the appropriate broadcasting done to account for the different number of channels present in $x_f$ and $x_{o^*}$).\n\n\n\\begin{equation}\n    x_{o^*_{i, j}} = softmax_{i, j} \\big(x_{o_{i, j}} + x_{m_{i, j}}\\big)\n    \\label{eqn:x_ostart}\n\\end{equation}\n\n\\begin{equation}\n    p_x = \\sum_{i, j} x_{o^*_{i, j}} i\n    \\label{eqn:softargmax_x}\n\\end{equation}\n\\begin{equation}\n    p_y = \\sum_{i, j} x_{o^*_{i, j}} j\n    \\label{eqn:softargmax_y}\n\\end{equation}\n\n\\begin{equation}\n    f = \\sum_{i, j} x_{o^*_{i, j}} x_{f_{i, j}}\n    \\label{eqn:feat_vec}\n\\end{equation}\n\nThe localization of the object in the search image $\\hat{x}$ is done by first passing $\\hat{x}$ through the CNN to obtain $\\hat{x}_f$. Then the feature vector $f$ is used like a matched filter (or equivalently as a conv1$\\times$1 layer with $f$ as the weights) to generate $\\hat{x}_{o^*}$. The location of the object $(\\hat{p}_x,\\hat{p}_y)$ in the search image $\\hat{x}$ is then determined by passing $\\hat{x}_{o^*}$ through a spatial soft-argmax layer (similar to the operations in Eqns.~\\eqref{eqn:softargmax_x} and \\eqref{eqn:softargmax_y}).\n\n\n\n\\subsection{Generating the Attention Modulation Map}\n\\label{sec:attn-mod}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/attn_mod.PNG}\n    \\caption{Beam \/ Cone like attention modulation maps for different positions and orientations of the pointing hand. The dark regions correspond to a value of -2.0 which suppresses peaks corresponding to objects in $x_o$ (Fig.~\\ref{fig:arch}), whereas the bright regions correspond to value 0.0 which allows values in that area in $x_o$ to pass through unchanged. The beam width is \\ang{30}, and the step size is \\ang{15}.}\n    \\label{fig:attn_mod}\n\\end{figure}\n\nWhen there are multiple objects in the scene, we expect multiple bright spots each corresponding to an object in $x_o$ (Fig.~\\ref{fig:sample_non_siamese}). We would like to suppress the peaks in $x_o$ corresponding to objects that are not being pointed at. Had we known the location and orientation of the hand, we could have directly suppressed the irrelevant peaks. However, since we do not have labels corresponding to the pose of the pointing hand in the scene, the neural network must learn to attend to the hand and then use this to  suppress the irrelevant peaks in $x_o$. To enable this, we use a ``soft\" or differentiable way to compute the position and orientation of the hand which is then employed to suppress irrelevant objects in $x_o$.\n\nThe feature map $x_f$ is passed through two independent bottleneck layers to produce maps $x_{hp}$ and $x_{ho}$ corresponding to the position and orientation of the pointing hand respectively in $x$. Similar to Eq.~\\eqref{eqn:feat_vec}, spatial attention is used to attend to the pointing hand and to obtain the orientation of the hand $x_{ho^*}$. The final position and orientation of the hand ($x_h$) is used to ``soft select\" a pre-defined attention modulation map $x_m$ (see Fig.\\ref{fig:attn_mod}). The set of pre-defined attention modulation maps include beams from all possible locations and orientations of the pointing hand as shown in Fig.~\\ref{fig:attn_mod}. Each modulation map is constructed by drawing a beam emanating from the position of the hand and in the direction the hand is pointing at. We use an orientation step size of $\\ang{15}$ with a beam width of $\\ang{30}$. Thus, there are 30$\\times$30$\\times$24 $=$ 21600 such maps. Note that no explicit loss function is used to learn $x_{hp}$ and $x_{ho}$. Rather, network learns to predict appropriate values for $x_{hp}$ and $x_{ho}$ that result in the ``selection\" of an appropriate attention map, which is possible only by correctly recognizing the position and orientation of the hand. The modulation map thus obtained, $x_m$, is added to $x_o$ to highlight the object being pointed at while suppressing the irrelevant ones. Thus, the pixels in $x_o$ that lie inside the beam are passed as is whereas the pixels that lie outside the beam are suppressed. Note that the entire attention modulation scheme is differentiable and hence can be learned through back-propagation.\n\nThe proposed way of creating attention modulation maps is most suitable for top view images (Fig.~\\ref{fig:sample_non_siamese}). For perspective views where the depth of the object is more relevant, it may be necessary to generate maps in 3D by casting a cone of rays and using the perspective projection. We leave this for future work.\n\n\n\n\n\\section{EXPERIMENTAL RESULTS}\n\n\n\nTo evaluate the proposed neural network, we first train it on a synthetic dataset and compare it with alternative architectures. The trained network is deployed on a robot arm to demonstrate its real world performance.\n\n\\subsection{Localization Performance}\nA dataset of 5000 training images and 1000 test images is created by placing emojis (Fig.~\\ref{fig:emoji}) at non-overlapping positions against a backdrop as shown in Fig.~\\ref{fig:sample_non_siamese}. A hand emoji (Fig.~\\ref{fig:hand_emoji}) is placed at a random location pointing to an object. One or more distracting objects are placed at random locations not on the line segment between the pointing hand and the object. The label for each sample is the position of the object that the hand is pointing at.\n\nTo evaluate the proposed spatial attention modulation mechanism, we first consider only localization of the object in the image containing the pointing hand ($x$) while ignoring the other input ($\\hat{x}$) and the output of Siamese network $(\\hat{p}_x, \\hat{p}_y)$. Table~\\ref{table:compare} compares the proposed approach with two baselines. The FC layers baseline refers to using fully connected layers\\footnote{The fully connected layers used are FC1024-ELU-FC256-ELU-FC2.} to predict $(p_x, p_y)$ from $x_f$. The Conv layers baseline uses convolutional layers\\footnote{Conv3x3(2048)-ELU-MaxPool2x2-Conv3x3(2048)-ELU-Conv3x3(2048)-ELU-MaxPool2x2-Conv3x3(2048)-ELU-Conv3x3(2048)-ELU-FC2.} to predict the position of the object. The networks are trained with mean squared error loss with weight decay 1e-8 using the Adam optimizer\\cite{kingma2014adam} with learning rate 1e-4. The evaluation metric is accuracy where we consider the prediction to be accurate if there is sufficient overlap between the ground truth and predicted bounding box. Specifically, the IOU (intersection-over-union) between the ground truth bounding box and the predicted bounding box has to be at least 0.5. All the three networking achieve low training error (accuracy over 99\\%), but the test error varies, and the proposed approach generalizes the best. A sample output is shown in Fig.~\\ref{fig:sample_non_siamese} where we observe that the spatial attention modulation mechanism is working as one might expect.\n\nA second dataset containing images corresponding to a new environment ($\\hat{x}$) where the object highlighted by the pointing is present along with distracting objects is constructed as before (Fig.~\\ref{fig:sample_siamese_synth}). The proposed architecture in its entirety with the Siamese network to process $\\hat{x}$ and predict $(\\hat{p}_x, \\hat{p}_y)$ is trained on this dataset. The accuracy on this dataset drops only marginally to 95.31\\%. Table~\\ref{table:compare2} compares performance on this dataset and shows that attention modulation is essential to localize the desired object. The sample output in Fig.~\\ref{fig:sample_siamese_synth} shows that the desired object in $\\hat{x}$ is being attended to.\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/emoji.png}\n      \\caption{A few sample objects used for training and evaluation. The set of emojis is divided into 2075 for training and 703 for testing.}\n      \\label{fig:emoji}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.9\\linewidth]{figures\/hand_emoji.png}\n      \\caption{A few sample hand images used for training and evaluation. The set of hand emojis is divided into 47 for training and 8 for testing.}\n      \\label{fig:hand_emoji}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_nonsiamese2.png}\n      \\caption{Sample prediction of the proposed architecture. The network has properly localized the pointing hand and chosen a suitable attention modulation map. The activation corresponding to the object that is not pointed at has been appropriately suppressed in $x_{o^*}$.}\n      \\label{fig:sample_non_siamese}\n\\end{figure}\n\n\\begin{table}[!t]\n\\caption{Comparison of the proposed approach with different baselines\n\\label{table:compare}\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nNeural Network Architecture & Accuracy\\\\\n\\hline\nFC layers & 11.72\\%\\\\\nConv layers & 41.41\\%\\\\\nProposed approach & 96.88\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!t]\n\\caption{Comparison of localization performance of the Siamese network on novel objects with and without attention modulation\n\\label{table:compare2}\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nNeural Network Architecture & Accuracy\\\\\n\\hline\nWithout Attention Modulation\\cite{venkatesh2019one} & 12.5\\%\\\\\nProposed approach (with modulation) & 95.31\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_siamese_synth.png}\n      \\caption{A sample prediction from the proposed architecture on synthetic data.}\n      \\label{fig:sample_siamese_synth}\n\\end{figure}\n\n\\subsection{Evaluation on Robot Arm}\nWe demonstrate the proposed neural network using the Dobot Magician, a 3-DoF robot arm (Fig.~\\ref{fig:dobot}). The objects used for evaluation with the robot are shown in Fig.~\\ref{fig:real_objs}. To convert the localized object in pixel space to the robot co-ordinate space, a chessboard calibration pattern is used (Fig.~\\ref{fig:calib}), and OpenCV is used for calibration. Figure~\\ref{fig:sample_siamese} shows a sample predicted from the proposed neural network. We see that the pointing hand has been localized, and the network has learnt to predict an appropriate attention modulation map that selects the object being pointed at (blue bottle cap in Fig.~\\ref{fig:sample_siamese}). We also see that the activation corresponding to the distracting object in $x_{o^*}$ has been successfully suppressed. With the feature vector $f$  corresponding to the bottle cap extracted, the Siamese net successfully attends to the same bottle cap in a new scene ($\\hat{x}$). In this manner, 20 trials were performed. The proposed network localizes the desired object to within 1~cm in all the trials. A video of the robot in operation is available at \\url{https:\/\/youtu.be\/bJ5HKllhqLg}.\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/dobot_1.png}\n      \\caption{A sample demonstration with the Dobot Magician robot arm.}\n      \\label{fig:dobot}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/real_objs.jpg}\n      \\caption{Objects used for evaluating the proposed approach on the Dobot arm.}\n      \\label{fig:real_objs}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/calib2.png}\n      \\caption{The chessboard calibration pattern used to convert pixels to robot co-ordinates.}\n      \\label{fig:calib}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_siamese.png}\n      \\caption{A sample prediction from the proposed architecture.}\n      \\label{fig:sample_siamese}\n\\end{figure}\n\\section{CONCLUSIONS}\n\nWe have proposed a spatial attention modulation method that endows a neural network with the ability to attend to a hand pointing at an object in an image and to focus on the object that is being pointed at. The proposed approach generalizes significantly better compared to architectures that use only fully connected or convolutional layers for localization. Furthermore, this approach can be combined with a Siamese network to localize objects that were not present in the training dataset. This network architecture can be used in building robots that can interact naturally with humans and learn about new objects over time.\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\\section*{ACKNOWLEDGMENT}\n\nThis project was supported by the Robert Bosch Center for Cyber-Physical Systems.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Preliminaries}\n\\subsection{Forcing}\nFirst we review some essential facts about forcing.  We refer the reader to \\cite{jechbook} and \\cite{kunen} for background and details.\n\nA partial order $\\mathbb{P}$ is said to be \\emph{separative} when $p \\nleq q \\Rightarrow (\\exists r \\leq p) r \\perp q$.  Every partial order $\\mathbb{P}$ has a canonically associated equivalence relation $\\sim_s$ and a separative quotient $\\mathbb{P}_s$, which is isomorphic to $\\mathbb{P}$ if $\\mathbb{P}$ is already separative.  In most cases we will assume our partial orders are separative.  For every separative partial order $\\mathbb{P}$, there is a canonical complete boolean algebra $\\mathcal{B}(\\mathbb{P})$ with a dense set isomorphic to $\\mathbb{P}$.\n\nA map $e: \\mathbb{P} \\to \\mathbb{Q}$ is an \\emph{embedding} when it preserves order and incompatibility.  An embedding is said to be \\emph{regular} when it preserves the maximality of antichains.  A order-preserving map $\\pi : \\mathbb{Q} \\to \\mathbb{P}$ is called a \\emph{projection} when $\\pi(1_\\mathbb{Q}) = 1_\\mathbb{P}$, and $p \\leq \\pi(q) \\Rightarrow (\\exists q' \\leq q) \\pi(q') \\leq p$.\n\n\n\\begin{lemma}Suppose $\\mathbb{P}$ and $\\mathbb{Q}$ are partial orders. \n\n\\begin{enumerate}[(1)]\n\\item $G$ is a generic filter for $\\mathbb{P}$ iff $\\{ [p]_s : p \\in G \\}$ is a generic filter for $\\mathbb{P}_s$.\n\n\\item $e : \\mathbb{P} \\to \\mathbb{Q}$ is a regular embedding iff for all $q \\in \\mathbb{Q}$, there is $p \\in \\mathbb{P}$ such that for all $r \\leq p$, $e(r)$ is compatible with $q$.\n\n\\item The following are equivalent:\n\\begin{enumerate}[(a)]\n\\item There is a regular embedding $e : \\mathbb{P}_s \\to \\mathcal{B}(\\mathbb{Q}_s)$.\n\\item There is a projection $\\pi : \\mathbb{Q}_s \\to \\mathcal{B}(\\mathbb{P}_s)$.\n\\item There is a $\\mathbb{Q}$-name $\\dot{g}$ for a $\\mathbb{P}$-generic filter such that for all $p \\in \\mathbb{P}$, there is $q \\in \\mathbb{Q}$ such that $q \\Vdash p \\in \\dot{g}$.\n\\end{enumerate}\n\n\\item Suppose $e : \\mathbb{P} \\to \\mathbb{Q}$ is a regular embedding.  If $G$ is a filter on $\\mathbb{P}$ let $\\mathbb{Q}\/G = \\{ q : \\neg \\exists p \\in G( e(p) \\perp q) \\}$.  The following are equivalent:\n\\begin{enumerate}[(a)]\n\\item $H$ is $\\mathbb{Q}$-generic over $V$.\n\\item $G = e^{-1}[H]$ is $\\mathbb{P}$-generic over $V$, and $H$ is $\\mathbb{Q} \/ G$-generic over $V[G]$.\n\\end{enumerate}\n\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{lemma}Suppose $\\mathbb{P}$ and $\\mathbb{Q}$ are partial orders.  $\\mathcal{B}(\\mathbb{P}_s) \\cong \\mathcal{B}(\\mathbb{Q}_s)$ iff the following holds.  Letting $\\dot{G}, \\dot{H}$ be the canonical names for the generic filters for $\\mathbb{P},\\mathbb{Q}$ respectively, there is a $\\mathbb{P}$-name for a function $\\dot{f}_0$ and a $\\mathbb{Q}$-name for a function $\\dot{f}_1$ such that:\n\\begin{enumerate}[(1)]\n\\item $\\Vdash_\\mathbb{P} \\dot{f}_0(\\dot{G})$ is a $\\mathbb{Q}$-generic filter,\n\\item $\\Vdash_\\mathbb{Q} \\dot{f}_1(\\dot{H})$ is a $\\mathbb{P}$-generic filter,\n\\item $\\Vdash_\\mathbb{P}  \\dot{G} = \\dot{f}_1^{\\dot{f}_0(\\dot{G})} (\\dot{f}_0(\\dot{G}))$, and $\\Vdash_\\mathbb{Q}  \\dot{H} = \\dot{f}_0^{\\dot{f}_1(\\dot{H})} (\\dot{f}_1(\\dot{H}))$.\n\\end{enumerate}\nAn isomorphism is given by $p \\mapsto || \\check{p} \\in \\dot{f}_1(\\dot{H}) ||_{\\mathcal{B}(\\mathbb{Q}_s)}$.\n\\end{lemma}\n\nFor a broader notion of ``forcing equivalence,'' the best that can be said in general is the following:\n\n\\begin{lemma}Suppose $\\mathbb{P}$ and $\\mathbb{Q}$ are partial orders.\n\\begin{enumerate}[(1)]\n\\item If $e : \\mathbb{P} \\to \\mathbb{Q}$ is a regular embedding, and any $\\mathbb{Q}$-generic $H$ yields $V[H] = V[e^{-1}[H]]$, then there is a predense set $A \\subseteq \\mathcal{B}(\\mathbb{Q}_s)$ such that $\\mathcal{B}(\\mathbb{P}_s) \\cong  \\mathcal{B}(\\mathbb{Q}_s) \\restriction a$ for all $a \\in A$.\n\n\\item $\\mathbb{P}$ and $\\mathbb{Q}$ yield the same generic extensions iff for a dense set of $p \\in \\mathbb{P}$, there is $q \\in  \\mathcal{B}(\\mathbb{Q}_s )$ such that  $\\mathcal{B}(\\mathbb{P}_s) \\restriction p  \\cong \\mathcal{B}(\\mathbb{Q}_s )\\restriction  q$.\n\\end{enumerate}\n\\end{lemma}\n\nA partial order $\\mathbb{P}$ is said to be \\emph{$\\kappa$-distributive} if for any collection of maximal antichains in $\\mathbb{P}$, $\\{ A_\\alpha : \\alpha < \\beta < \\kappa \\}$, there is a maximal antichain $A$ such that $A$ refines $A_\\alpha$ for all $\\alpha < \\beta$.  $\\mathbb{P}$ is called \\emph{$(\\kappa,\\lambda)$-distributive} if the same holds restricted to antichains of size $\\leq \\lambda$.  Forcing with $\\mathbb{P}$ adds adds no new functions from any $\\alpha<\\kappa$ to $\\lambda$ iff $\\mathcal{B}(\\mathbb{P})$ is $(\\kappa,\\lambda)$-distributive.\n\nA strictly stronger property than distributivity is strategic closure.  For a partial order $\\mathbb{P}$ and an ordinal $\\alpha$, we define a game $G_\\alpha(\\mathbb{P})$ with two players \\emph{Even} and \\emph{Odd}.  \\emph{Even} starts by playing some element $p_0 \\in \\mathbb{P}$.  At successor stages $\\beta+1$, the next player must play some element $p_{\\beta+1} \\leq p_\\beta$.  \\emph{Even} plays at limit stages $\\beta$ if possible, by playing a $p_\\beta$ that is $\\leq p_\\gamma$ for all $\\gamma < \\beta$.  If \\emph{Even} cannot play at some stage below $\\alpha$, the game is over and \\emph{Odd} wins; otherwise \\emph{Even} wins.  We say that $\\mathbb{P}$ is \\emph{$\\alpha$-strategically closed} if for every $p \\in \\mathbb{P}$, \\emph{Even} has a winning strategy with first move $p$.  Note that under this definition, every partial order is trivially $\\omega$-strategically closed.\n\nA stronger property that $\\kappa$-strategic closure is $\\kappa$-closure.  $\\mathbb{P}$ is \\emph{$\\kappa$-closed} when any descending chain of length less than $\\kappa$ has a lower bound.  $\\mathbb{P}$ is \\emph{$\\kappa$-directed closed} when any directed set of size ${<} \\kappa$ has a lower bound.\n\nFor any partial order $\\mathbb{P}$, the \\emph{saturation} of $\\mathbb{P}$, $\\sat(\\mathbb{P})$, is the least cardinal $\\kappa$ such that every antichain in $\\mathbb{P}$ has size less than $\\kappa$.  Erd\\H{o}s and Tarski~\\cite{ET} proved that $\\sat(\\mathbb{P})$ is always regular.  The \\emph{density} of $\\mathbb{P}$, $\\den(\\mathbb{P})$, is the least cardinality of a dense subset of $\\mathbb{P}$.  Clearly $\\sat(\\mathbb{P}) \\leq \\den(\\mathbb{P})^+$ for any $\\mathbb{P}$.  We say $\\mathbb{P}$ is \\emph{$\\kappa$-saturated} if $\\sat(\\mathbb{P}) \\leq \\kappa$, and $\\mathbb{P}$ is \\emph{$\\kappa$-dense} if $\\den(\\mathbb{P}) \\leq \\kappa$.  A synonym for $\\kappa$-saturation is the \\emph{$\\kappa$-chain condition ($\\kappa$-c.c.).}\n\nThe properties of distributivity, strategic closure, saturation, and density are robust in the sense that they are absolute between $\\mathbb{P}$ and $\\mathcal{B}(\\mathbb{P})$ for any separative partial order $\\mathbb{P}$, and often inherited by intermediate forcings:\n\\begin{lemma}Suppose $e : \\mathbb{P} \\to \\mathbb{Q}$ is a regular embedding and $\\kappa$ is a cardinal.\n\\begin{enumerate}[(1)]\n\\item If $\\mathbb{Q}$ is $\\kappa$-strategically closed, then so is $\\mathbb{P}$.\n\\item $\\mathbb{Q}$ is $\\kappa$-distributive iff $\\mathbb{P}$ is $\\kappa$-distributive and $\\Vdash_{\\mathbb{P}} \\mathbb{Q} \/ \\dot{G}$ is $\\kappa$-distributive.\n\\item $\\mathbb{Q}$ is $\\kappa$-saturated iff $\\mathbb{P}$ is $\\kappa$-saturated and $\\Vdash_{\\mathbb{P}} \\mathbb{Q} \/ \\dot{G}$ is $\\kappa$-saturated.\n\\item $\\mathbb{Q}$ is $\\kappa$-dense iff $\\mathbb{P}$ is $\\kappa$-dense and $\\Vdash_{\\mathbb{P}} \\mathbb{Q} \/ \\dot{G}$ is $\\kappa$-dense.\n\\end{enumerate}\n\\end{lemma}\n\n\nFor any forcing $\\mathbb{P}$ and any $\\mathbb{P}$-name $\\dot{X}$ for a set of ordinals, there is a canonically associated complete subalgebra $\\mathcal{A}_{\\dot{X}} \\subseteq \\mathcal{B}(\\mathbb{P})$ that captures $\\dot{X}$.  It is the smallest complete subalgebra containing all elements of the form $|| \\check{\\alpha} \\in \\dot{X} ||$ for $\\alpha$ an ordinal.  $\\mathcal{A}_{\\dot{X}}$ has the property that whenever $G \\subseteq \\mathbb{P}$ is generic, $\\dot{X}^G$ and $G \\cap \\mathcal{A}_{\\dot{X}}$ are definable from each other using the parameters $\\mathcal{B}(\\mathbb{P})$ and its powerset, as computed in the ground model.  In this case, we have $V[\\dot{X}^G] = V[G \\cap \\mathcal{A}_{\\dot{X}}]$.  See \\cite[p. 247]{jechbook} for details. \n\n\n\n\\subsection{Ideals}\n\nLet $Z$ be any set.  An \\emph{ideal} $I$ on $Z$ is a collection of subsets of $Z$ closed under taking subsets and pairwise unions.  If $\\kappa$ is a cardinal, $I$ is called \\emph{$\\kappa$-complete} if it is also closed under unions of size less than $\\kappa$.  \\emph{``Countably complete''} is taken as synonymous with ``$\\omega_1$-complete.''  $I$ is called \\emph{nonprincipal} if $\\{z \\} \\in I$ for all $z \\in Z$, and \\emph{proper} if $Z \\notin I$.  Hereafter we will assume all our ideals are nonprincipal and proper.\n\nLet $X = \\bigcup Z$.  $I$ is called \\emph{fine} if for all $x \\in X$, $\\{ z : x \\notin z \\} \\in I$.  $I$ is called \\emph{normal} if for any sequence $\\langle A_x : x \\in X \\rangle \\subseteq I$, the \\emph{``diagonal union''} $\\{ z : \\exists x(x \\in z \\in A_x) \\}$ is in $I$.  It is well-known that $I$ is normal iff for any $A \\in \\p(Z) \\setminus I$ and any function $f$ on $A$ such that $f(z) \\in z$ for all $z \\in A$, there is an $x$ such that $f^{-1}(x) \\notin I$.\n\nTo fix notation, let $I^* = \\{ Z \\setminus A : A \\in I \\}$ (the \\emph{$I$-measure one sets}), $I^+ = \\p(Z) \\setminus I$ (the \\emph{$I$-positive sets}), $\\hat{x} = \\{ z : x \\in z \\}$, and denote diagonal unions by $\\nabla_{x \\in X} A_x$.  Note that $\\nabla_{x \\in X} A_x = \\bigcup_{x \\in X} \\hat{x} \\cap A_x$.\n\nThe following basic fact seems to have been previously overlooked--see, for example, the hypotheses of several theorems in \\cite{foremanhandbook} and \\cite{foremanduality}.\n\n\\begin{proposition}All normal and fine ideals are countably complete.\n\\end{proposition}\n\\begin{proof}Let $I$ be a normal and fine ideal on $Z \\subseteq \\p(X)$.  If $\\{ x_\\alpha : \\alpha < \\kappa \\}$ is an enumeration of distinct elements of $X$, and $\\bigcap_{\\alpha<\\kappa} \\hat{x}_\\alpha \\in I^*$, then $I$ is $\\kappa^+$-complete.  For suppose that $\\{ A_\\alpha : \\alpha<\\kappa \\} \\subseteq I$, but $A =  \\bigcup_{\\alpha<\\kappa} A_\\alpha \\in I^+$.  Then by hypothesis, $A \\cap (\\bigcap_{\\alpha < \\kappa} \\hat{x}_\\alpha) \\in I^+$.  Let $f : A \\to X$ be defined by $f(z) = x_\\alpha$, where $\\alpha$ is the least ordinal such that $z \\in A_\\alpha$.  By normality, there is some $A_\\alpha \\in I^+$, a contradiction.  So it suffices to find an infinite set $\\{x_n : n < \\omega \\} \\subseteq X$ such that $\\bigcap_{n<\\omega} \\hat{x}_n \\in I^*$.  Since we assume $I$ is proper and nonprincipal, $X$ is infinite.  We show that any infinite set of distinct elements of $X$ suffices.\n\nLet $\\{x_n : n < \\omega \\}$ be distinct elements of $X$, and suppose the contrary, that $B = \\{ z : \\{x_n : n < \\omega \\} \\nsubseteq z \\} \\in I^+$.  By fineness, $B \\cap \\hat{x}_0 \\in I^+$.  For each $z \\in B \\cap \\hat{x}_0$, let $n_z$ be the largest integer such that $\\{x_0,...,x_{n_z} \\} \\subseteq z$.  Let $f : B \\cap \\hat{x}_0 \\to X$ be defined by $f(z) = x_{n_z}$.  By normality, there is an $n$ such that $f^{-1}(x_n) \\in I^+$.  Then for all $z \\in f^{-1}(x_n)$, $x_{n+1} \\notin z$.  This contradicts fineness. \n\\end{proof}\n\n\\begin{proposition}If $I$ is a normal, fine, $\\kappa$-complete ideal on $Z \\subseteq \\p(\\kappa)$, then $\\kappa \\in I^*$.\n\\end{proposition}\n\\begin{proof}Suppose $A = \\{ z \\in Z : z$ is not an ordinal$\\} \\in I^+$.  Let $f : A \\to \\kappa$ be such that $f(z)$ is the least $\\alpha \\in z$ such that $\\alpha \\nsubseteq z$.  Then for some $\\alpha$, $f^{-1}(\\alpha) \\in I^+$.  However, $\\{ z : \\alpha \\subseteq z \\} \\in I^*$ by fineness and $\\kappa$-completeness.   \n\\end{proof}\n\nProofs of the following facts can be found in \\cite{foremanhandbook}.  If $I$ is an ideal on $Z$, say $A \\sim_I B$ if the symmetric difference $A \\Delta B$ is in $I$.  Let $[A]_I$ denote the equivalence class of $A$ mod $\\sim_I$.  The equivalence classes form a boolean algebra under the obvious operations, which we denote by $\\p(Z) \/ I$.  Normality ensures a certain amount of completeness of the algebra:\n\n\\begin{proposition}\n\\label{sums}\nSuppose $I$ is a normal and fine ideal on $Z \\subseteq \\p(X)$.  If $\\{ A_x : x \\in X \\} \\subseteq \\p(Z)$, then $\\nabla A_x$ is the least upper bound of $\\{ [A_x]_I : x \\in X \\}$ in $\\p(Z)\/I$.\n\\end{proposition}\n\nIf we force with this algebra, we get a generic ultrafilter $G$ on $Z$ extending $I^*$.  We can form the ultrapower $V^Z \/ G$.  If this ultrapower is well-founded for every generic $G$, then $I$ is called precipitous.  A combinatorial characterization of precipitousness is given by the following:\n\n\\begin{theorem}[Jech-Prikry]\n\\label{prec}\n$I$ is a precipitous ideal on $Z$ iff the following holds: For any sequence $\\langle A_n : n < \\omega \\rangle\\subseteq \\p(I^+)$, such that for each $n$, \n\\begin{enumerate}[(1)]\n\\item $B_n = \\{ [a]_I : a \\in A_n \\}$ is a maximal antichain in $\\p(Z) \/ I$,\n\\item $B_{n+1}$ refines $B_n$,\n\\end{enumerate}\nthere is a function $f$ with domain $\\omega$ such that for all $n$, $f(n) \\in A_n$, and $\\bigcap_{n<\\omega} f(n) \\not= \\emptyset$.\n\\end{theorem}\n\nFor an ideal $I$, the saturation, density, distributivity, and strategic closure of $I$ refers to that of the corresponding boolean algebra.  The next proposition is immediate from Theorem~\\ref{prec}:\n\n\\begin{proposition}\nIf $I$ is an $\\omega_1$-complete, $\\omega_1$-distributive ideal, then $I$ is precipitous.\n\\end{proposition}\n\n\\begin{proposition}Suppose $I$ is a $\\kappa$-complete precipitous ideal on $Z$, and there is no $A \\in I^+$ such that $I {\\restriction} A$ is $\\kappa^+$-complete.  Let $G$ be $\\p(Z)\/I$-generic, and let $j : V \\to M$ be the associated elementary embedding, where $M$ is the transitive collapse of $V^Z\/G$.  Then the critical point of $j$ is $\\kappa$.\n\\end{proposition}\n\n\\begin{proposition}\nLet $I$ be an ideal  $Z \\subseteq \\p(X)$.  Then $I$ is normal and fine iff $1 \\Vdash_{\\p(Z)\/I} [\\id]_{\\dot G} = j[X]$.\n\\end{proposition}\n\n\\begin{proposition}\nSuppose $I$ is an ideal on $Z \\subseteq \\p(X)$.  If $I$ is $\\kappa$-complete and $\\kappa^+$-saturated, or if $I$ is normal, fine, and $|X|^+$-saturated, then every antichain in $\\p(Z)\/I$ has a system of pairwise disjoint representatives. \n\\end{proposition}\n\n\\begin{proof}\nIf $I$ is $\\kappa$-complete, and $\\{ A_\\alpha : \\alpha < \\kappa \\}$ is an antichain, replace each $A_\\alpha$ with $A_\\alpha \\setminus \\bigcup_{\\beta<\\alpha} A_\\beta$.  If $I$ is normal and fine, and $\\{ A_x : x \\in X \\}$ is an antichain, replace $A_x$ by $A_x \\cap \\hat{x} \\setminus \\bigcup_{y \\not= x} A_y \\cap \\hat{y}$.   \n\\end{proof}\n\n\\begin{theorem}\n\\label{disj}\nSuppose $I$ is a countably complete ideal on $Z$, and every antichain in $\\p(Z)\/I$ has a system of pairwise disjoint representatives.  Then:\n\\begin{enumerate}[(1)]\n\\item $I$ is precipitous.\n\\item $\\p(Z)\/I$ is a complete boolean algebra.\n\\item If $G$ is generic over $\\p(Z)\/I$, $j : V \\to M$ is the associated embedding, and $ j[\\lambda] \\in M$, then $M$ is closed under $\\lambda$-sequences from $V[G]$.\n\\end{enumerate}\n\\end{theorem}\n\nIf $\\kappa=\\mu^+$ and $I$ is a normal, fine, $\\kappa$-complete ideal on $\\p_\\kappa(\\lambda)$, then $I$ is not $\\lambda$-saturated.  For otherwise, let $j : V \\to M \\subseteq V[G]$ be a generic embedding arising from $I$.  Then $M \\models |[\\id]| = \\mu$, and $[\\id] =  j[\\lambda]$, so $\\lambda$ has cardinality $\\mu$ in $V[G]$.  So the smallest possible density of such an ideal is $\\lambda$.\n\n\\subsection{Elementary embeddings}\nProofs of the following can be found in \\cite{kanamori}.\n\n\\begin{lemma}Suppose $M$ and $N$ are models of $ZF^-$, $j : M \\to N$ is an elementary embedding, $\\mathbb{P} \\in M$ is a partial order, $G$ is $\\mathbb{P}$-generic over $M$, and $H$ is $j(\\mathbb{P})$-generic over $N$.  Then $j$ has a unique extension $\\hat{j} : M[G] \\to N[H]$ with $\\hat{j}(G) = H$ iff $j[G] \\subseteq H$.\n\\end{lemma}\n\n\\begin{lemma}Suppose $M$, $N$ are transitive models of ZFC with the same ordinals, and $j : M \\to N$ is an elementary embedding.  Then either $j$ has a critical point, or $j$ is the identity and $M = N$.\n\\end{lemma}\n\n\\section{Dense ideals from large cardinals}\n\nHere we show that it is consistent relative to an almost-huge cardinal that there is a normal, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$, where $\\kappa$ is the successor of a regular cardinal $\\mu$, and $\\lambda \\geq \\kappa$ is regular, for many particular choices for $\\mu,\\lambda$.  We also show that relative to a super-almost-huge cardinal, there can exist a successor cardinal $\\kappa$ such that for every regular $\\lambda \\geq \\kappa$, there is a normal, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$.  This generalizes a theorem of Woodin about the relative consistency of an $\\aleph_1$-dense ideal on $\\aleph_1$, and has the following additional advantages: (1) An explicit forcing extension is taken, rather than an inner model of an extension.  (2) Careful constructions within a model where the axiom of choice fails, as presented in~\\cite{foremanhandbook}, are avoided.\n\nLet us first recall the essential facts about almost-huge cardinals (see~\\cite{kanamori}, 24.11).  A cardinal $\\kappa$ is almost-huge if there is an elementary embedding $j : V \\to M$ with critical point $\\kappa$, such that $M^{<j(\\kappa)} \\subseteq M$.\n\n\\begin{theorem}The following are equivalent:\n\\begin{enumerate}[(1)]\n\\item $\\kappa$ carries an almost-huge embedding $j$ such that $j(\\kappa) = \\delta$.\n\\item $\\delta$ is inaccessible, and there is a sequence $\\langle U_\\alpha : \\kappa \\leq \\alpha < \\delta \\rangle$ such that:\n\\begin{enumerate}\n\\item each $U_\\alpha$ is a normal, fine, $\\kappa$-complete ultrafilter on $\\p_\\kappa(\\alpha)$,\n\\item for $\\alpha < \\beta$, $U_\\alpha = \\{ A \\subseteq \\p_\\kappa(\\alpha) : \\{ z \\in \\p_\\kappa(\\beta) : z \\cap \\alpha \\in A \\} \\in U_\\beta \\}$, and\n\\item for all $\\alpha < \\delta$ and all $f : \\p_\\kappa(\\alpha) \\to \\kappa$ such that $\\{ z : f(z) \\geq \\ot(z) \\} \\in U_\\alpha$, there is $\\beta$ such that  $\\alpha \\leq \\beta < \\delta$ and $\\{ z : f (z \\cap \\alpha) = \\ot(z) \\} \\in U_\\beta$.\n\\end{enumerate}\n\\end{enumerate}\nFurthermore, if a system as in (2) is given, the direct limit model and embedding witness the almost-hugeness of $\\kappa$ with target $\\delta$.\n\\end{theorem}\n\nA system as in (2) will be called an \\emph{almost-huge tower}.  Almost-huge towers capture almost-hugeness in a minimal way:\n\n\\begin{corollary}\nIf $\\kappa$ has an almost-huge tower of height $\\delta$, and $j : V \\to M$ is the embedding derived from the tower, then we have $\\delta < j(\\delta) < \\delta^+$, and $j[\\delta]$ is cofinal in $j(\\delta)$.\n\\end{corollary}\n\\begin{proof}\nFor each $\\alpha < \\delta$, let $M_\\alpha$ be the transitive collapse of $V^{\\p_\\kappa(\\alpha)} \/ U_\\alpha$, and let $j_\\alpha : V \\to M_\\alpha$ and $k_\\alpha : M_\\alpha \\to M$ be the canonical embeddings, with $j = k_\\alpha \\circ j_\\alpha$.  Since $\\delta$ is inaccessible, $j_\\alpha(\\kappa) < \\delta$ and $j_\\alpha(\\delta) = \\delta$ for each $\\alpha < \\delta$.\n\nIf $\\gamma < j(\\delta)$, then there are some $\\alpha,\\beta < \\delta$ such that $k_\\alpha(\\beta) = \\gamma$.  Thus there are only $\\delta$ ordinals below $j(\\delta)$.  Also, there is $\\eta < \\delta$ such that $j_\\alpha(\\eta) > \\beta$, so $j(\\eta) > \\gamma$, and thus $j[\\delta]$ is cofinal in $j(\\delta)$.  \n\\end{proof}\n\nA \\emph{super-almost-huge} cardinal is a cardinal $\\kappa$ such that for all $\\lambda \\geq \\kappa$, there is an almost huge tower of height $\\geq \\lambda$.  The next result follows from considering the set of closure points under witnesses to property (c) in the tower characterization.\n\n\\begin{corollary}\nIf $\\kappa$ has an almost-huge tower of Mahlo height $\\delta$,  then for stationary many $\\alpha < \\delta$, $V_\\alpha \\models ZFC +  \\kappa$ is super-almost-huge.\n\\end{corollary}\n\nThere is a vast gap in strength between almost-huge and huge:\n\n\\begin{theorem}\n\\label{tourney}\nIf $\\kappa$ is a huge cardinal, then there is a stationary set $S \\subseteq \\kappa$ such that for all $\\alpha < \\beta$ in $S$, $\\alpha$ has an almost-huge tower of height $\\beta$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $j : V \\to M$ is an elementary embedding with critical point $\\kappa$, $j(\\kappa) = \\delta$, and $M^\\delta \\subseteq M$.  Then $\\kappa$ carries an almost-huge tower $\\vec{U}$ of length $\\delta$, and $\\vec{U} \\in M$.  Let $F$ be the ultrafilter on $\\kappa$ defined by $F = \\{ X \\subseteq \\kappa: \\kappa \\in j(X) \\}$.  Let $A = \\{ \\alpha < \\kappa : \\alpha$ carries an almost-huge tower of height $\\kappa \\}$.  Since $\\kappa \\in j(A)$, $A \\in F$.  Now let $c: \\kappa^2 \\to 2$ be defined by $c(\\alpha,\\beta) = 1$ if $\\alpha$ carries an almost-huge tower of height $\\beta$, and $c(\\alpha,\\beta) = 0$ otherwise.  By Rowbottom's theorem, let $H \\in F$ be homogeneous for $c$.  We claim $c$ takes constant value 1 on $H$.  For if $\\alpha \\in A \\cap H$, then $\\{ \\alpha,\\kappa \\} \\in [j(A \\cap H)]^2$, and $j(c)(\\alpha,\\kappa) = 1$.  \n\\end{proof}\n\n\\subsection{Layering and absorption}\n\n\\begin{definition}\nWe will call a partial order $\\mathbb{P}$ \\emph{$(\\mu,\\kappa)$-nicely layered} when there is a collection $\\mathcal{L}$ of atomless regular suborders of $\\mathbb{P}$ such that:\n\\begin{enumerate}[(1)]\n\\item for all $\\mathbb{Q} \\in \\mathcal{L}$, $\\mathbb{Q}$ is $\\mu$-closed and has size $< \\kappa$,\n\\item for all $\\mathbb{Q}_0, \\mathbb{Q}_1 \\in \\mathcal{L}$, if $\\mathbb{Q}_0 \\subseteq \\mathbb{Q}_1$, then $\\Vdash_{\\mathbb{Q}_0} \\mathbb{Q}_1 \/ \\dot{G}$ is $\\mu$-closed, and\n\\item for all $\\mathbb{P}$-names $\\dot{f}$ for a function from $\\mu$ to the ordinals, and all $\\mathbb{Q}_0 \\in \\mathcal{L}$, there is an $\\mathbb{Q}_1 \\in \\mathcal{L}$ and an $\\mathbb{Q}_1$-name $\\dot{g}$ such that $\\mathbb{Q}_0 \\subseteq \\mathbb{Q}_1$, and $\\Vdash_\\mathbb{P} \\dot{f} = \\dot{g}$.\n\\end{enumerate}\n\nWe will say $\\mathbb{P}$ is \\emph{$(\\mu,\\kappa)$-nicely layered with collapses,} $(\\mu,\\kappa)$-NLC, when additionally for all $\\alpha < \\kappa$ and all $\\mathbb{Q}_0 \\in \\mathcal{L}$, there is $\\mathbb{Q}_1 \\in \\mathcal{L}$ such that $\\mathbb{Q}_0 \\subseteq \\mathbb{Q}_1$ and $\\Vdash_{\\mathbb{Q}_1} |\\mathbb{Q}_1| = \\mu$.\n\\end{definition}\n\n\\begin{proposition}\nIf $\\mathcal{L}$ witnesses that $\\mathbb{P}$ is $(\\mu,\\kappa)$-nicely layered, then $\\mathbb{P}$ is $\\kappa$-c.c.\\ and $\\bigcup \\mathcal{L}$ is dense in $\\mathbb{P}$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that $\\{ p_\\alpha : \\alpha < \\eta \\} \\subseteq \\mathbb{P}$, $\\eta \\geq \\kappa$, is a maximal antichain.  Let $\\dot{f}$ be a name of a function with domain $\\{ 0 \\}$ such that $f(0) = \\alpha$ iff $p_\\alpha \\in G$.  There cannot be a regular suborder $\\mathbb{Q}$ of size $< \\kappa$ and a $\\mathbb{Q}$-name $\\dot{g}$ that is forced to be equal to $\\dot{f}$, since such a $\\dot{g}$ would have $<\\kappa$ possible values for its range.\n\nSimilarly, let $p \\in \\mathbb{P}$ be arbitrary, and let $\\{ p_\\alpha : \\alpha < \\delta \\}$  be a maximal antichain with $p = p_0$.  Let $\\dot{f}$ be a name of a function with domain $\\{ 0 \\}$ such that $f(0) = \\alpha$ iff $p_\\alpha \\in G$. If $\\mathbb{Q}$ is a regular suborder and $\\dot{g}$ is a $\\mathbb{Q}$-name such that $\\Vdash_\\mathbb{P} \\dot{f} = \\dot{g}$, then there is some $q \\in \\mathbb{Q}$ forcing $\\dot{g}(0) = 0$, so $q \\leq p$.\n\\end{proof}\n\n\\begin{proposition}If there exists a $(\\mu,\\kappa)$-NLC poset, then $\\alpha^{<\\mu} < \\kappa$ for all $\\mu < \\kappa$.\n\\end{proposition}\n\\begin{proof}\nLet $\\mathbb P$ be $(\\mu,\\kappa)$-NLC with layering $\\mathcal L$, and let $\\alpha < \\kappa$.  If $\\mathbb Q \\in \\mathcal L$ collapses $\\alpha$ to $\\mu$, then we can build a $\\mu$-closed tree $T \\subseteq \\mathbb Q$ of height $\\mu$ such that each level is an antichain of size $\\geq \\alpha$.  $\\alpha^{<\\mu} \\leq |T| < \\kappa$. \n\\end{proof}\n\n\\begin{lemma}[Folklore]\n\\label{folk}\nIf $\\mathbb{P}$ is a $\\mu$-closed partial order such that $\\Vdash_\\mathbb{P} |\\mathbb{P}| = \\mu$, then $\\mathcal{B}(\\mathbb{P}) \\cong \\mathcal{B}(\\col(\\mu,|\\mathbb{P}|))$.\n\\end{lemma}\n\n\\begin{proof}\nPick a $\\mathbb{P}$-name $\\dot{f}$ for a bijection from $\\mu$ to $\\dot{G}$.  We build a tree $T \\subseteq \\mathbb{P}$ that is isomorphic to a dense subset of $\\col(\\mu,|\\mathbb{P}|)$, and show that it is dense in $\\mathbb{P}$.  Each level will be a maximal antichain in $\\mathbb{P}$.  Let the first level $T_0 = \\{ 1_\\mathbb{P} \\}$.  If levels $\\{ T_\\beta : \\beta < \\alpha + 1 \\}$ are defined, below each $p \\in T_\\alpha$, pick a $|\\mathbb{P}|$-sized maximal antichain of conditions deciding $\\dot{f}(\\alpha)$, and let $T_{\\alpha+1}$ be the union of these antichains.  If $\\{ T_\\beta : \\beta < \\lambda \\}$ is defined up to a limit $\\lambda$, pick for each descending chain $b$ through the previous levels, a $|\\mathbb{P}|$-sized maximal antichain of lower bounds to $b$, and set $T_\\lambda$ equal to the union of these anithchains.  It is easy to check that $T_\\lambda$ is a maximal antichain.  Let $T = \\bigcup_{\\alpha<\\mu} T_\\alpha$.  To show $T$ is dense, let $p \\in \\mathbb{P}$.  Let $q \\leq p$ be such that for some $\\alpha < \\mu$, $q \\Vdash \\dot{f}(\\alpha) = p$.   $q$ is compatible with some $r \\in T_{\\alpha+1}$.  Since $r$ decides $\\dot{f}(\\alpha)$ and forces it in $\\dot{G}$, $r \\leq p$.   \n\\end{proof}\n\n\n\\begin{lemma}\n\\label{rearrange}\nSuppose $\\mu < \\kappa$ are regular, and $\\mathbb{P}$ is $(\\mu,\\kappa)$-NLC.  If $G$ is $\\mathbb{P}$-generic over $V$, then there is a forcing $\\mathbb{R} \\in V[G]$ such that $\\mathbb{R}$ adds a filter $H \\subseteq \\break \\col(\\mu,{<}\\kappa)$ which is generic over $V$ and such that $(\\ord^\\mu)^{V[G]} = (\\ord^\\mu)^{V[H]}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{L}$ witness the $(\\mu,\\kappa)$-NLC property.  In $V[G]$, let $\\mathbb{R}$ be the collection of filters $h \\subseteq \\col(\\mu,{<}\\alpha)$ for $\\alpha< \\kappa$ which are generic over $V$, such that for some $\\mathbb{Q} \\in \\mathcal{L}$, $V[h] = V[G \\cap \\mathbb{Q}]$.  The ordering is end-extension.  \n\n\nLet $h \\in \\mathbb{R}$ with $\\mathbb{Q}_0 \\in \\mathcal{L}$ a witness, and let and $\\alpha < \\kappa$ be arbitrary.  Let $\\alpha < \\beta < \\kappa$ and $\\mathbb{Q}_1 \\supseteq \\mathbb{Q}_0$ in $\\mathcal{L}$ be such that in $V[h]$, $|\\mathbb{Q}_1 \/ (G \\cap \\mathbb{Q}_0 ) | = |\\beta|$, and $\\mathbb{Q}_1$ collapses $\\beta$ to $\\mu$.  By the definition and Lemma~\\ref{folk}, $\\mathbb{Q}_1 \/ (G \\cap \\mathbb{Q}_0 )$ is equivalent in $V[h]$ to $\\col(\\mu,\\beta)$, which is equivalent to the ${<}\\mu$-support product of $\\col(\\mu,\\gamma)$ for $\\alpha \\leq \\gamma \\leq \\beta$.  The filter $G \\cap \\mathbb{Q}_1$ therefore gives a filter $h' \\supseteq h$ on $\\col(\\mu,<\\beta+1)$ that is generic over $V$, with $V[h'] = V[\\mathbb{Q}_1 \\cap G]$.  \n\nLet $h \\in \\mathbb{R}$ with $\\mathbb{Q}_0 \\in \\mathcal{L}$ a witness, and let $f: \\mu \\to \\ord$ in $V[G]$ be arbitrary.  By the definition of $(\\mu,\\kappa)$-NLC, we can find some $\\mathbb{Q}_1 \\supseteq \\mathbb{Q}_0$ in $\\mathcal{L}$ such that $f \\in V[G \\cap \\mathbb{Q}_1]$.  By the previous paragraph, we may find $\\mathbb{Q}_2 \\supseteq \\mathbb{Q}_1$ in $\\mathcal{L}$ equivalent to some $\\col(\\mu,<\\alpha)$, and some filter $h' \\subseteq \\col(\\mu,{<}\\alpha)$ generic over $V$, extending $h$, and such that $V[G \\cap \\mathbb{Q}_2] = V[h']$.\n\nIf $F$ is generic over $\\mathbb{R}$, let $H = \\bigcup_{h \\in F} h$. Since $\\col(\\mu,{<}\\kappa)$ is $\\kappa$-c.c., $H$ is generic, since any maximal antichain from $V$ intersects some $h \\in F$.  By the above arguments, any $f : \\mu \\to \\ord$ in $V[G]$ is in $V[H]$.  Conversely, any $f : \\mu \\to \\ord$ in $V[H]$ lives in some $V[h]$ with $h \\in \\mathbb{R}$, so is in $V[G]$.   \n\\end{proof}\n\n\\subsubsection{The anonymous collapse}\n\nLet $\\kappa$ be a regular cardinal whose regularity is preserved by a forcing $\\mathbb{P}$.  Let $A(\\mathbb{P})$ be the complete subalgebra of $\\mathcal{B} ( \\mathbb{P} * \\add(\\kappa))$ generated by the canonical name for the $\\add(\\kappa)$-generic set.  More precisely, if $e : \\mathbb{P}*\\add(\\kappa) \\to  \\mathcal{B} ( \\mathbb{P} * \\add(\\kappa))$ is the canonical dense embedding, $A(\\mathbb{P})$ is completely generated by the elements of the form $e(\\langle 1, \\dot{\\{ \\langle \\alpha,1 \\rangle \\} } \\rangle)$.  By \\cite[p. 247]{jechbook}, we have a canonical correspondence between such $\\add(\\kappa)$-generic sets $X$ which come after forcing with $\\kappa$-preserving posets $\\mathbb P$, and $A(\\mathbb P)$-generic filters $H$.  We will move between the two by writing, for example, $X_H$ and $H_X$.\n\nIn the case that $\\alpha^{<\\mu} < \\kappa$ for all $\\alpha < \\kappa$ and $\\mathbb{P} = \\col(\\mu,{<}\\kappa)$, denote $A(\\mathbb{P})$ by $A(\\mu,\\kappa)$, and write $B(\\mu,\\kappa)$ for $\\mathcal{B}(\\col(\\mu,{<}\\kappa)*\\add(\\kappa))$.\n\n\\begin{lemma}\n\\label{quotdist}\nIf $\\mathbb{P}$ is $(\\mu,\\kappa)$-NLC, and $H \\subseteq A(\\mathbb{P})$ is generic over $V$, then \\break  $\\mathcal{B}(\\mathbb{P} * \\add(\\kappa))\/H$ is $\\kappa$-distributive in $V[H]$.\n\\end{lemma}\n\n\\begin{proof}\n$V[H] = V[X_H]$ for the canonically associated $X_H \\subseteq \\kappa$, and by forcing with $\\mathcal{B}(\\mathbb{P} * \\add(\\kappa))\/H$ over $V[H]$, we recover a filter $G * X_H$ for $\\mathbb{P} * \\add(\\kappa)$, generic over $V$.\n \nIf $G * X$ is $\\mathbb{P} * \\add(\\kappa)$-generic over $V$, then $X$ codes all subsets of $\\mu$ that live in $V[G]$.  By the definition of $(\\mu,\\kappa)$-NLC, every $z \\in (\\ord^\\mu)^{V[G]}$ occurs in some submodel of the form $V[G \\cap \\mathbb{Q}]$, where $\\mathbb{Q}$ is isomorphic to $\\col(\\mu,\\alpha)$ for some $\\alpha < \\kappa$.  Thus $z \\in V[y]$ for some $y \\subseteq \\mu$ in $V[G]$, so $(\\ord^\\mu)^{V[X]} \\supseteq (\\ord^\\mu)^{V[G]}$.  Since $\\add(\\kappa)$ adds no $\\mu$-sized sets of ordinals, $(\\ord^\\mu)^{V[G]} = (\\ord^\\mu)^{V[G*X]}  \\supseteq (\\ord^\\mu)^{V[X]}$.  Thus $\\mathcal{B}(\\mathbb{P} * \\add(\\kappa))\/H$ is $\\kappa$-distributive.  \n\\end{proof}\n\n\\begin{lemma}\n\\label{cheat}\nLet $V$ be a countable transitive model of ZFC (or just assume generic extensions are always available), and assume $\\Vdash^V_\\mathbb{P} \\kappa$ is regular.  If $X \\subseteq \\kappa$, the following are equivalent:\n\\begin{enumerate}[(1)]\n\\item $X$ is $A(\\mathbb{P})$-generic over $V$.\n\\item There is $G \\subseteq \\mathbb{P}$ such that $G$ is generic over $V$, and $X$ is $\\add(\\kappa)$-generic over $V(P_0)$, where $P_0 = \\p_\\kappa(\\kappa)^{V[G]}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nIf $X$ is $A(\\mathbb{P})$-generic then force with $\\mathcal{B}(\\mathbb{P} * \\add(\\kappa))\/H_X$ over $V[X]$, obtaining $G$ such that $G*X$ is $\\mathbb{P}*\\add(\\kappa)$-generic over $V$.  Then $X$ is $\\add(\\kappa)$-generic over $V[G]$, and since $\\add(\\kappa)^{V[G]} = \\add(\\kappa)^{V(P_0)}$, $X$ is $\\add(\\kappa)$-generic over $V(P_0)$.\n\nSuppose $G \\subseteq \\mathbb{P}$ is generic over $V$, and $X$ is $\\add(\\kappa)$-generic over $V(P_0)$, but not $A(\\mathbb{P})$-generic over $V$.  Then some $p \\in \\add(\\kappa)^{V(P_0)}$ forces this with $\\dom(p) = \\alpha < \\kappa$, and $X \\restriction \\alpha = p$.  Take $Y \\subseteq \\kappa$ such that $Y \\restriction \\alpha = p$ that is $\\add(\\kappa)$-generic over the larger model $V[G]$.  Then $Y$ is $A(\\mathbb{P})$-generic over $V$, and $V(P_0)[Y]$ can see this, but this contradicts the property of $p$.  So $X$ was $A(\\mathbb{P})$-generic over $V$.   \n\\end{proof}\n\n\n\\begin{theorem}\n\\label{forget}\nFor any $\\mathbb{P}$ that is is $(\\mu,\\kappa)$-NLC, there is an isomorphism \\break $\\iota : A(\\mathbb{P}) \\to A(\\mu,\\kappa)$ such that $\\iota(|| \\alpha \\in \\dot{X} ||_{A(\\mathbb{P})}) = || \\alpha \\in \\dot{X} ||_{A(\\mu,\\kappa)}$ for all $\\alpha < \\kappa$.\n\\end{theorem}\n\n\\begin{proof}\nLet $X$ be $A(\\mathbb{P})$-generic over $V$.  There is a $\\kappa$-distributive forcing over $V[X]$ to get $G$ such that $G * X$ is $\\mathbb{P} * \\add(\\kappa)$-generic over $V$.  By Lemma~\\ref{rearrange}, we can do further forcing to obtain $H \\subseteq \\col(\\mu,{<}\\kappa)$ generic over $V$ such that $(\\ord^\\mu)^{V[H]} = (\\ord^\\mu)^{V[G]}$.  By Lemma~\\ref{cheat}, $X$ is also $A(\\mu,\\kappa)$-generic over $V$.\n\nConversely, every $A(\\mu,\\kappa)$-generic $X$ is $A(\\mathbb{P})$-generic.  For suppose $X$ is a counterexample.  Then there is some $(p,\\dot{q}) \\in \\col(\\mu,{<}\\kappa) * \\add(\\kappa)$ such that $(p,\\dot{q}) \\Vdash \\dot{X}$ is not $A(\\mathbb{P})$-generic over $V$.  Let $Y$ be any $A(\\mathbb{P})$-generic set, and let $P_0 = \\p(\\mu)^{V[Y]}$.   By the above, $Y$ is $A(\\mu,\\kappa)$-generic over $V$.  Thus we can force over $V[Y]$ to get $H \\subseteq \\col(\\mu,{<}\\kappa)$ such that $H *Y$ is $B(\\mu,\\kappa)$-generic over $V$.  By the homogeneity of the Levy collapse, there is some automorphism $\\pi \\in V$ such that $p \\in \\pi[H] = H^\\prime$.   By the homogeneity of Cohen forcing, there is some automorphism $\\sigma$ in $V(P_0)$ such that $\\sigma[Y]$ is a generic $Y^\\prime$ such that $Y^\\prime \\restriction \\dom(\\dot{q}^{H^\\prime}) = \\dot{q}^{H^\\prime}$.  $Y^\\prime$ is also $A(\\mathbb{P})$-generic over $V$.  However, $(p,\\dot{q}) \\in H^\\prime * Y^\\prime$, so we have a contradiction.\n\n\nThis implies that we have a canonical correspondence between $A(\\mathbb{P})$- and $A(\\mu,\\kappa)$-generic filters, i.e. definable functions $f,g$ such that for any generic $H$ for $A(\\mathbb{P})$, $f(H)$ is the generic for $A(\\mu,\\kappa)$ computed from $X_H$, and vice versa, and $g(f(H)) = H$.  For $p \\in A(\\mathbb{P})$, put $\\iota(p) = || p \\in g(\\dot{H}) ||_{A(\\mu,\\kappa)}$.  It is easy to see that $\\iota$ is a complete embedding.  For any $q \\in A(\\mu,\\kappa)$, there is $p \\in A(\\mathbb{P})$ forces that $q \\in f(\\dot{H})$.  Thus if $H$ is generic for $A(\\mu,\\kappa)$ and $\\iota(p) \\in H$, then $p \\in g(H)$, so $q \\in f(g(H)) = H$, hence $\\iota(p) \\leq q$.  The range of $\\iota$ is dense, so it is an isomorphism.  By the way we construct $f$ and $g$, $\\iota(|| \\alpha \\in \\dot{X} ||_{A(\\mathbb{P})}) = \\break || \\alpha \\in \\dot{X} ||_{A(\\mu,\\kappa)}$.   \n\\end{proof}\n\nThis machinery has some interesting applications to the absoluteness of some properties of a given powerset.  First, it is easy to see for regular $\\mu < \\kappa$ such that $\\alpha^{<\\mu} < \\kappa$ for all $\\alpha < \\kappa$, $\\col(\\mu,{<}\\kappa) \\times \\add(\\mu,\\lambda)$ is $(\\mu,\\kappa)$-NLC for every $\\lambda$.  Thus if $X$ is $A(\\mu,\\kappa)$-generic, then for any $\\lambda$, we may further force to obtain a model which is a $(\\col(\\mu,{<}\\kappa) \\times \\add(\\mu,\\lambda)) * \\add(\\kappa)$-generic extension with the same $\\ord^\\mu$.  Taking inner models given by such $\\col(\\mu,{<}\\kappa) \\times \\add(\\mu,\\lambda)$-generic sets, we produce many models with the same cardinals and same $\\p(\\mu)$, each assigning a different cardinal value for $2^\\mu$.  For example, if we add $\\omega_1$ Cohen reals to any model of $M$ of ZFC, this is the same as forcing with $\\col(\\omega,{<}\\omega_1)$.  There is for each uncountable ordinal $\\alpha \\in M$, a generic extension with the same reals and same cardinals, in which it appears we have added $\\alpha$ many Cohen reals.\n\nBy using weakly compact cardinals, we can get even more dramatic examples.  If $\\kappa$ is weakly compact, every $\\kappa$-c.c.\\ partial order captures small sets in small factors.  To show this, first consider a partial order $\\mathbb{P}$ of size $\\kappa$.  We can code $\\mathbb{P}$ as $A \\subseteq \\kappa$, and by weak compactness, there is some transitive elementary extension $(V_\\kappa,\\in,A) \\prec (M,\\in,B)$.  If $\\mu < \\kappa$, then any $\\mathbb{P}$-name for function $f : \\mu \\to \\ord$ has an equivalent name $\\tau \\in V_\\kappa$ by the $\\kappa$-c.c.  Since $A \\in M$ and $M$ sees $A$ as a regular suborder of $B$, $M$ thinks that $\\tau$ is a $\\mathbb{Q}$-name for some regular suborder $\\mathbb{Q}$ of $B$.  By elementarity, $V_\\kappa$ thinks that $\\tau$ is a $\\mathbb{Q}$-name for some regular $\\mathbb{Q}$ of $A$.  For $\\mathbb{P}$ of arbitrary size, let $\\tau$ be a $\\mathbb{P}$-name of size $<\\kappa$, take some regular $\\theta$ such that $\\mathbb{P},\\tau \\in H_\\theta$, and take an elementary $M \\prec H_\\theta$ with $\\mathbb{P},\\tau \\in M$ such that $|M| = \\kappa$ and $M^{<\\kappa} \\subseteq M$.  It is easy to see that $M \\cap \\mathbb{P}$ is a regular suborder of $\\mathbb{P}$, and so the above considerations apply to show that there is some regular $\\mathbb{Q} \\subseteq \\mathbb{P} \\cap M \\subseteq \\mathbb{P}$ of size $<\\kappa$ such that $\\tau$ is a $\\mathbb{Q}$-name.\n\nTherefore, if $\\kappa$ is weakly compact and $\\mathbb{P}$ is $\\kappa$-c.c.,\\ the collection $\\mathcal{L}$ of all regular suborders of $\\mathbb{P}$ of size $<\\kappa$ witnesses that $\\mathbb{P}$ is $(\\omega,\\kappa)$-nicely layered.  If $\\mathbb{P}$ also forces $\\kappa = \\aleph_1$, then this collection also witnesses that $\\mathbb{P}$ is $(\\omega,\\kappa)$-NLC.  To check this, take any $\\mathbb{Q}_0 \\in \\mathcal{L}$, any $\\mathbb{P}$-name $\\tau$ of size $<\\kappa$, and $\\alpha < \\kappa$.  Let $H \\subseteq \\mathbb{Q}_0$ be generic.  Since $\\kappa$ is still weakly compact in $V[H]$, there is some regular $\\mathbb{Q}_1 \\subseteq \\mathbb{P}\/H$ of size $<\\kappa$ in $V[H]$ such that the $(\\mathbb{P}\/H)$-name associated to $\\tau$ is a $\\mathbb{Q}_1$-name.  Let $\\beta \\geq \\max \\{ \\alpha, | \\mathbb{Q}_0 * \\dot{\\mathbb{Q}}_1| \\}$.  Since $\\mathbb{P} \/ (\\mathbb{Q}_0  * \\dot{\\mathbb{Q}_1})$ adds a generic for $\\col(\\omega,\\beta)$, we have $\\mathbb{Q}_2 \\in \\mathcal{L}$ extending $\\mathbb{Q}_0 * \\dot{\\mathbb{Q}}_1$ such that $\\mathbb{Q}_2 \\sim \\col(\\omega,\\beta)$.\n\nIn particular, if $\\kappa$ is weakly compact, then $\\col(\\omega,<\\kappa) * \\dot{\\mathbb{Q}}$, where $\\dot{\\mathbb{Q}}$ is forced to be c.c.c.,\\ is $(\\omega,\\kappa)$-NLC.  Thus an extremely wide variety of forcing extensions with very different theories can be obtained, each sharing the same reals and same cardinals.\n\n\n\n\\subsubsection{An unfortunate reality}\n\nDespite the universality of $A(\\mu,\\kappa)$, it is difficult to characterize its combinatorial structure.  While it absorbs all of the small sets added by a $(\\mu,\\kappa)$-NLC forcing, no such forcing completely embeds into it.  The reader may opt to skip this section, as later results will not depend it.\n\nTo show this, we first isolate two properties of a forcing extension that depend on two regular cardinals $\\mu <\\kappa$.  The author is grateful to Mohammad Golshani for bringing these properties to his attention.\n\\begin{enumerate}[(1)]\n\\item \\emph{Levy($\\mu,\\kappa$):} $(\\exists A \\in [\\kappa]^\\kappa ) ( \\forall y \\in [ \\kappa ]^\\mu \\cap V ) y \\nsubseteq A$.\n\\item \\emph{Silver($\\mu,\\kappa$):} $(\\exists A \\in [\\kappa]^\\kappa) (\\forall X \\in [\\kappa]^\\kappa \\cap V)(\\exists y \\in [X]^\\mu \\cap V) y \\cap A = \\emptyset$.\n\\end{enumerate}\n\nNote that these are both $\\Sigma_1$ properties of the parameters $([ \\kappa ]^\\mu)^V$ and $([ \\kappa ]^\\kappa)^V$.  For any partial order $\\mathbb{P}$, and collection of dense subsets $\\mathcal{D} \\subseteq \\p(\\mathbb{P})$ the statement, ``There is a filter $G \\subseteq \\mathbb{P}$ that is $\\mathcal{D}$-generic,'' is also a $\\Sigma_1$ property of $\\mathbb{P}$ and $\\mathcal{D}$.  Now  the following proposition either holds or fails for a given partial order $\\mathbb{P}$ and cardinals $\\mu < \\kappa$:\n\\[\n(*)_{\\mu,\\kappa} : (\\forall X \\in [\\mathbb{P} ]^\\kappa ) ( \\exists y \\in [X]^\\mu ) y \\mbox{ has a lower bound in }\\mathbb{P}.\n\\]\n\n\n\\begin{lemma}\nIf $\\mathbb{P}$ is a separative partial order that satisfies $(*)_{\\mu,\\kappa}$, preserves the regularity of $\\kappa$, and such that $\\den(\\mathbb{P} \\restriction p) = \\kappa$ for all $p \\in \\mathbb{P}$, then $\\mathbb{P}$ forces $Silver(\\mu,\\kappa)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\{ p_\\alpha : \\alpha < \\kappa \\}$ be a dense subset of $\\mathbb{P}$.  Inductively build a dense $D \\subseteq \\{ p_\\alpha : \\alpha < \\kappa \\}$, putting $p_\\alpha \\in D$ just in case there is no $\\beta < \\alpha$ such that $p_\\beta \\in D$ and $p_\\beta \\leq p_\\alpha$.  $D$ has the property that for all $p \\in D$, $| \\{ q \\in D : p \\leq q \\} | < \\kappa$.  Fixing a bijection $f : D \\to \\kappa$, we claim that if $G  \\subseteq \\mathbb{P}$ is generic, $A = f[G]$ witnesses $Silver(\\mu,\\kappa)$.  Note that since $\\mathbb{P}$ is nowhere $<\\kappa$-dense, $A$ is an unbounded subset of $\\kappa$.  Now let $p \\in D$ and $X = \\{ q_\\alpha : \\alpha < \\kappa \\} \\in [D]^\\kappa$ be arbitrary.  There is some $B \\in [\\kappa]^\\kappa$ such that for all $\\alpha \\in B$, $p \\nleq q_\\alpha$.  For each $\\alpha \\in B$, choose $r_\\alpha \\leq p$ such that $r_\\alpha \\perp q_\\alpha$.  By $(*)$, there is some $y \\in [B]^\\mu$ such that $\\{ r_\\alpha : \\alpha \\in y \\}$ has a lower bound $r$.  We have $r \\Vdash \\{ q_\\alpha : \\alpha \\in \\check{y} \\} \\cap \\dot{G} = \\emptyset$.  As $p$ and $X$ were arbitrary, $Silver(\\mu,\\kappa)$ is forced.\n\\end{proof}\n\n\n\n\\begin{lemma}\nIf $\\mathbb{P}$ is a $\\kappa$-c.c.\\ separative partial order of size $\\kappa$ satisfying $\\neg (*)_{\\mu,\\kappa}$, then some $p \\in \\mathbb{P}$ forces $Levy(\\mu,\\kappa)$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $X \\in [\\mathbb{P}]^\\kappa$ witnesses $\\neg (*)_{\\mu,\\kappa}$.  By the $\\kappa$-c.c., there is some $p$ such that $p \\Vdash | \\check{X} \\cap \\dot{G} | = \\kappa$.  If $y \\in [X]^\\mu$, then $1 \\Vdash \\check{y} \\nsubseteq \\dot{G}$, since otherwise some $q$ is a lower bound to $y$.  Hence $p$ forces that $X \\cap G$ witnesses $Levy(\\mu,\\kappa)$.   \n\\end{proof}\n\n\\begin{lemma}Suppose $\\mu < \\kappa$, $\\mu$ is regular for all $\\alpha < \\kappa$, $\\alpha^\\mu < \\kappa$.  There are two $(\\mu,\\kappa)$-NLC partial orders $\\mathbb{P}_0$ and $\\mathbb{P}_1$ such that $\\mathbb{P}_0$ forces $Levy \\wedge \\neg Silver$, and $\\mathbb{P}_1$ forces $\\neg Levy \\wedge Silver$.\n\\end{lemma}\n\n\\begin{proof}Let $\\mathbb{P}_0$ be the Levy collapse $\\col(\\mu, {<}\\kappa)$, and let $\\mathbb{P}_1$ be the Silver collapse,\n\\[ \\{ p : (\\exists \\alpha < \\mu)(\\exists x \\in [\\kappa]^\\mu) p : x \\times \\alpha \\to \\kappa, \\text{ and } p(\\beta,\\gamma) < \\beta \\text{ for all } (\\beta,\\gamma) \\in \\dom p \\} \\]\nIt is easy to see that $\\mathbb{P}_1$ satisfies $(*)_{\\mu,\\kappa}$, while $\\mathbb{P}_0$ fails this property, as witnessed by $X = \\mathbb{P}_0$.  Hence by the previous lemmas, $\\mathbb{P}_0$ forces $Levy(\\mu,\\kappa)$, and $\\mathbb{P}_1$ forces $Silver(\\mu,\\kappa)$.  We must show that the respective negations are also forced.\n\nLet $\\dot{A}$ be a $\\mathbb{P}_0$-name such that $1 \\Vdash \\dot{A} \\in [\\kappa]^\\kappa$.  Let $p \\in \\mathbb{P}$ be arbitrary, and let $\\gamma< \\kappa$ be such that $\\supp(p) \\subseteq \\gamma$.  Let $X_0 = \\{ \\alpha < \\kappa : p \\nVdash \\alpha \\notin \\dot{A} \\}$.  For each $\\alpha \\in X_0$, pick some $q_\\alpha \\leq p$ such that $q_\\alpha \\Vdash \\alpha \\in \\dot{A}$.  By a delta-system argument, let $X_1 \\in [X_0]^\\kappa$ be such that there is $r \\leq p$ such that for all $\\alpha \\in X_1$,  $q_\\alpha \\restriction \\gamma = r$, and for $\\alpha \\not= \\beta$ in $X_1$, $(\\supp(q_\\alpha) \\setminus \\gamma) \\cap (\\supp(q_\\beta) \\setminus \\gamma) = \\emptyset$.  For any $q \\leq r$ and $y \\in [X_1]^\\mu$, $q \\nVdash \\check{y} \\cap \\dot{A} = \\emptyset$.  This is because for such $q$, there is some $\\alpha \\in y$ such that $(\\supp(q_\\alpha) \\setminus \\gamma) \\cap \\supp(q) = \\emptyset$, so $q$ is compatible with $q_\\alpha$.  Hence $r \\Vdash (\\exists X \\in [\\kappa]^\\kappa \\cap V)(\\forall y \\in [X]^\\mu \\cap V) y \\cap \\dot{A} \\not= \\emptyset$.  As $\\dot{A}$ and $p$ were arbitrary, $\\neg Silver(\\mu,\\kappa)$ is forced.\n\nNow let $\\dot{A}$ be a $\\mathbb{P}_1$-name such that $1 \\Vdash \\dot{A} \\in [\\kappa]^\\kappa$, and let $p \\in \\mathbb{P}_1$ be arbitrary.  Form $X_0$, $\\{q_\\alpha : \\alpha \\in X_0 \\}$, and $X_1$ like above.  We can take a $y \\in [X_1]^\\mu$ such that $\\bigcup_{\\alpha \\in y} q_\\alpha = q \\in \\mathbb{P}_1$.  Then $q \\Vdash \\check{y} \\subseteq \\dot{A}$, so $q$ forces $\\neg Levy(\\mu,\\kappa)$.   \n\\end{proof}\n\n\n\\begin{corollary}Suppose $\\mu$, $\\kappa$, $\\mathbb{P}_0$, and $\\mathbb{P}_1$ are as above.  Let $G$ be $\\mathbb{P}_0$-generic and $H$ be $\\mathbb{P}_1$-generic over $V$.  Let $\\mathbb{Q} \\in V$ be a partial order.  If $\\mathbb{Q}$ forces $Levy(\\mu,\\kappa)$, then $V[H]$ has no $\\mathbb{Q}$-generic, and if $\\mathbb{Q}$ forces $Silver(\\mu,\\kappa)$, then $V[G]$ has no $\\mathbb{Q}$-generic.  If $\\mathbb{Q}$ is $\\kappa$-c.c.\\ and of size $\\kappa$, then no $\\kappa$-closed forcing extension of $V[G]$ or $V[H]$ can introduce a generic for $\\mathbb{Q}$.\n\\end{corollary}\n\n\\begin{proof}\nSince $V[H]$ satisfies $\\neg Levy$, and $Levy$ is a $\\Sigma_1$ property with parameters in $V$, no inner model of $V[H]$ containing $V$ can satisfy $Levy$.  Likewise, no inner model of $V[G]$ containing $V$ can satisfy $Silver$.  To see that the non-existence of $\\mathbb{Q}$-generics is preserved by $\\kappa$-closed forcing, suppose that for some such forcing $\\mathbb{R} \\in V[G]$, $r \\Vdash^{V[G]}_\\mathbb{R} \\dot{K}$ is $\\mathbb{Q}$-generic over $V$.  Since $\\mathbb{Q}$ has size $\\kappa$, we can build a descending sequence $\\{ r_\\alpha : \\alpha < \\kappa \\}$ below $r$ such that for all $q \\in \\mathbb{Q}$, there is $r_\\alpha$ deciding whether $q \\in K$.  Let $K' = \\{ q : (\\exists \\alpha < \\kappa) r_\\alpha \\Vdash q \\in \\dot{K} \\}$.  Any maximal antichain $A \\in V$ contained in $\\mathbb{Q}$ has size $< \\kappa$, thus some $r_\\alpha$ completely decides $A \\cap K$.  Since $r_\\alpha \\Vdash \\check{A} \\cap \\dot{K} \\not= \\emptyset$, we must have $K' \\cap A \\not= \\emptyset$, so $K'$ is $\\mathbb{Q}$-generic over $V$.  The argument for $\\kappa$-closed forcing over $V[H]$ is the same.   \n\\end{proof}\n\n\\begin{theorem}Suppose $\\mu < \\kappa$ are regular and $\\alpha^\\mu<\\kappa$ for all $\\alpha<\\kappa$.  No $(\\mu,\\kappa)$-NLC forcing regularly embeds into $A(\\mu,\\kappa)$.  Further, a generic extension by $A(\\mu,\\kappa)$ has no generic filters for any $\\kappa$-c.c.\\ forcing $\\mathbb{Q}$ such that $\\den(\\mathbb{Q} \\restriction q) \\geq \\kappa$ for all $q \\in \\mathbb{Q}$.\n\\end{theorem}\n\\begin{proof}\nFirst note that we only need to consider $\\mathbb{Q}$ such that $\\den(\\mathbb{Q} \\restriction q) = \\kappa$ for all $q \\in \\mathbb{Q}$.  For if $p \\in A(\\mu,\\kappa)$ is such that $p \\Vdash \\dot{K}$ is $\\mathbb{Q}$-generic, then there would be some $q \\in \\mathcal{B}(\\mathbb{Q})$ and some $p' \\leq p$ such that $\\mathcal{B}(\\mathbb{Q}) \\restriction q$ completely embeds into $A(\\mu,\\kappa) \\restriction p'$.  Since $d(A(\\mu,\\kappa) = \\kappa$, this implies $\\mathcal{B}(\\mathbb{Q}) \\restriction q \\leq \\kappa$.\n\nLet $\\mathbb{Q}$ be any $\\kappa$-c.c.\\ forcing such that $\\den(\\mathbb{Q} \\restriction q) = \\kappa$ for all $q \\in \\mathbb{Q}$.  For any $p \\in \\mathbb{Q}$, if $(*)$ holds for $\\mathbb{Q} \\restriction p$, then $p \\Vdash Silver$, and otherwise for some $q \\leq p$, $q \\Vdash Levy$.  Thus $\\Vdash_\\mathbb{Q} Levy \\vee Silver$.  Suppose $K$ is $\\mathbb{Q}$-generic over $V$, and $X$ is $A(\\mu,\\kappa)$-generic over $V$.  There are two further forcings $\\mathbb{R}_0,\\mathbb{R}_1$ over $V[X]$ that respectively get filters $G,H$ such that $V[G][X]$ is $\\mathbb{P}_0 * \\add(\\kappa)$-generic, and $V[H][X]$ is $\\mathbb{P}_1 * \\add(\\kappa)$-generic.  If $V[K] \\models Levy$, then $K \\not \\in V[H][X]$, and if $V[K] \\models Silver$, then $K \\notin V[G][X]$.  Thus $V[X]$ has no $\\mathbb{Q}$-generics.\n\\end{proof}\n\n\n\\subsection{Construction of a dense ideal}\n\nFirst we will define a useful strengthening of ``nicely layered.''\n\n\\begin{definition}\n$\\mathbb{P}$ is \\emph{$(\\mu,\\kappa)$-very nicely layered (with collapses)} when there is a sequence $\\langle \\mathbb{Q}_\\alpha : \\alpha < \\kappa \\rangle = \\mathcal{L}$ such that:\n\\begin{enumerate}[(1)]\n\\item $\\mathcal{L}$ witnesses that $\\mathbb{P}$ is $(\\mu,\\kappa)$-nicely layered (with collapses),\n\\item $\\mathcal{L}$ is $\\subseteq$-increasing,\n\\item every subset of $\\mathbb{P}$ of size $<\\mu$ with a lower bound has an infimum, and\n\\item there is a system of continuous projection maps $\\pi_\\alpha : \\mathbb{P} \\to \\mathbb{Q}_\\alpha$ such that for each $\\alpha$, $\\pi_\\alpha \\restriction \\mathbb{Q}_\\alpha = \\id$, and for $\\beta < \\alpha < \\kappa$, $\\pi_\\beta = \\pi_\\beta \\circ \\pi_\\alpha$.\n\n(By continuous, we mean that for any $X \\subseteq \\mathbb P$, if $\\inf(X)$ exists, then for all $\\alpha<\\kappa$, $\\inf(\\pi_\\alpha[X]) = \\pi_\\alpha(\\inf(X))$.)\n\\end{enumerate}\n\\end{definition}\n\nA typical example is the Levy collapse $\\col(\\mu,{<}\\kappa)$.  In the general case, we will usually abbreviate the action of the projection maps $\\pi_\\alpha(q)$ by $q {\\restriction} \\alpha$.  In applying clause (3), we will use the next proposition, proof of which is left to the reader.\n\n\\begin{proposition}\nIf $\\mathbb{P}$ is a partial order such that every descending chain of length $< \\mu$ has an infimum, then every directed subset of size $<\\mu$ has an infimum.\n\\end{proposition}\n\n\n\\begin{theorem}\n\\label{maindense}\nAssume $\\kappa$ carries an almost-huge tower of height $\\delta$, and let $j : V \\to M$ be given by the tower.  Let $\\mu,\\lambda$ be regular such that $\\mu < \\kappa \\leq \\lambda < \\delta$.  Suppose $\\Vdash_{A(\\mu,\\kappa)}$ `` $\\dot{ \\mathbb{P} }$ is $(\\kappa,\\delta)$-very nicely layered and forces $\\delta = \\lambda^+$.''  If $X * H$ is $A(\\mu,\\kappa) * \\dot{\\mathbb{P}}$-generic, then in $V[X][H]$, there is a normal, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $H_X$ be the $A(\\mu,\\kappa)$-generic filter computed from $X$. Let $K \\times C$ be $B(\\mu,\\kappa)\/H_X \\times \\col(\\mu,\\lambda)$-generic over $V[X][H]$, and for brevity let $W = V[X][H][K][C]$.  Note that $V[X][K] = V[G][X]$, where $G * X$ is some \\break $\\col(\\mu,{<}\\kappa) * \\add(\\kappa)$-generic filter over $V$.  Let $\\langle \\mathbb{Q}_\\alpha : \\alpha < \\delta \\rangle$ witness that $\\mathbb{P}$ is $(\\kappa,\\delta)$-nicely layered.  By the distributivity of $B(\\mu,\\kappa)\/H_X$ in $V[X]$, $\\mathbb{P}$ and its layers $\\mathbb{Q}_\\alpha$ are still $\\kappa$-closed in $V[G][X]$.  For $\\alpha < \\beta$, the relation $\\Vdash_{\\mathbb{Q}_\\alpha}$ ``$\\mathbb{Q}_\\beta \/ \\mathbb{Q}_\\alpha$ is $\\kappa$-closed'' holds in $V[G][X]$ because in $V[X]$, $B(\\mu,\\kappa)\/H_X \\times \\mathbb{Q}_\\alpha$ is $\\kappa$-distributive.  Furthermore, since no sequences of length $<\\mu$ are added, the forcing given by the definition of $\\col(\\mu,\\lambda)$ is the same between $V$, $W$, and intermediate models.\n\n\nThe forcing to get from $V[G]$ to $W$ is equivalent to $(\\add(\\kappa) \\times \\col(\\mu,\\lambda)) *  \\mathbb{P}$.  Let $\\mathcal{L}$ be the collection of subforcings of the form $(\\add(\\kappa) \\times \\col(\\mu,\\lambda)) * \\mathbb{Q}_\\alpha$ for $\\alpha < \\delta$.  This sequence then witnesses the $(\\mu,\\delta)$-NLC property in $V[G]$.  The closure properties are evident, and since the whole forcing has the $\\delta$-c.c., functions from $\\mu$ to ordinals are indeed captured by these factors.\n\n\n\nLet $P_0 = \\p(\\mu)^W$, and consider the submodel $M(P_0)$.  In $W$, $Q_0 = \\break \\p(\\add(\\delta))^{M(P_0)}$ has cardinality $\\delta$.  To show this, let $Y \\subseteq \\delta$ be $\\add(\\delta)$-generic over $W$.  By Theorem~\\ref{forget}, $Y$ is $A(\\mu,\\delta)$-generic over $V$, and hence over $M$ since $(\\col(\\mu,{<}\\delta)*\\add(\\delta))^M = (\\col(\\mu,{<}\\delta)*\\add(\\delta))^V$ by the closure of $M$.  Since $M[Y]$ thinks $j(\\delta)$ is inaccessible, $M[Y] \\models |Q_0| < j(\\delta)$, so $W[Y] \\models |Q_0| = \\delta$ since $j(\\delta) < (\\delta^+)^V$.  Since $W \\models 2^\\mu = \\delta$, $W$ and $W[Y]$ have the same cardinals, so $W \\models |Q_0| = \\delta$.   Therefore, working in $W$, we can inductively build a set $\\hat{X} \\subseteq \\delta$ that is $\\add(\\delta)$-generic over $M(P_0)$ with $\\hat{X} \\cap \\kappa = X$.  By Lemma~\\ref{cheat}, $\\hat{X}$ is $A(\\mu,\\delta)$-generic over $M[G]$.  A further forcing produces $G^\\prime \\supseteq G$, such that $G^\\prime * \\hat{X}$ is $\\col(\\mu,{<}\\delta)*\\add(\\delta)$-generic over $M$, so we have an elementary $\\hat{j} : V[G][X] \\to M[G^\\prime][\\hat{X}]$ extending $j$.  By elementarity, for the corresponding filters $H_X$ and $H_{\\hat{X}}$ on the respective algebras $A(\\mu,\\kappa)^V$ and $A(\\mu,\\delta)^M$, we have $j[H_X] \\subseteq H_{\\hat{X}}$.  Hence we can define in $W$ the restricted elementary embedding $\\hat{j} : V[X] \\to M[\\hat{X}]$.\n\n\n\n\nNow we wish to extend $\\hat{j}$ to have domain $V[X][H]$.  As in the argument for Lemma~\\ref{quotdist}, every element of $(\\ord^\\mu)^W$ is coded by some element of $M$ and some $y \\subseteq \\mu$ coded in $\\hat{X}$, so $M[\\hat{X}]$ is closed under ${<}\\delta$-sequences from $W$.  Consequently, $H \\cap \\mathbb{Q}_\\alpha$ and $\\hat{j}[ H \\cap \\mathbb{Q}_\\alpha]$ are in $M[\\hat{X}]$ for all $\\alpha < \\delta$.  Also, $M[\\hat{X}] \\vDash$ ``$\\hat{j}(\\mathbb{P})$ is $(\\delta,j(\\delta))$-very nicely layered.''  Each $\\hat{j}[ H \\cap \\mathbb{Q}_\\alpha ]$ is a directed set of size $\\mu$ in $M[\\hat{X}]$, so it has an infimum $m_\\alpha \\in \\hat{j}(\\mathbb{Q}_\\alpha)$.\n\nLet $\\langle A_\\alpha : \\alpha < \\delta \\rangle \\in W$ enumerate the maximal antichains of  $\\hat{j}(\\mathbb{P})$ from $M[\\hat{X}]$.  (There are only $\\delta$ many because $M[\\hat{X}]$ thinks this partial order has inaccessible size $j(\\delta)$ and is $j(\\delta)$-c.c.)  Inductively define an increasing sequence of ordinals $\\langle \\alpha_i \\rangle_{i < \\delta} \\subseteq \\delta$, and a corresponding decreasing sequence of conditions $\\langle p_i \\rangle_{i < \\delta} \\subseteq \\hat{j}(\\mathbb{P})$ as follows.\n\nAssume as the induction hypothesis that we have defined the sequences up to $i$, and for all $\\xi < i$ and all $\\alpha < \\delta$, $p_\\xi$ is compatible with $m_\\alpha$, and for all $\\xi < i$, there is some $a \\in A_\\xi$ such that $p_\\xi \\leq a$.  Let $q_i = \\inf_{\\xi<i} p_\\xi$.  This is compatible with all $m_\\alpha$ because for all $\\alpha$, $\\langle p_\\xi \\wedge m_\\alpha : \\xi < i \\rangle$ is a descending chain in $\\hat{j}(\\mathbb{P})$.  Let $\\alpha_i \\geq \\sup_{\\xi<i} \\alpha_\\xi$ be such that $A_i \\subseteq \\hat{j}(\\mathbb{Q}_{\\alpha_i})$ and $q_i \\in \\hat{j}(\\mathbb{Q}_{\\alpha_i})$.  This is possible by the chain condition and because $j[\\delta]$ is cofinal in $j(\\delta)$.  Choose $p_i \\in \\hat{j}(\\mathbb{Q}_{\\alpha_i})$ below $q_i \\wedge m_{\\alpha_i}$ and some $a \\in A_i$.  $p_i$ is compatible with all $m_\\alpha$, because for any $\\alpha > \\alpha_i$, $m_\\alpha \\restriction j(\\alpha_i) = m_{\\alpha_i}$.  This is because for any $\\alpha < \\beta < \\delta$, \n\\begin{align*}\nm_\\beta \\restriction j(\\alpha) = & (\\inf \\{ j(p) : p \\in H {\\restriction} \\beta \\}) \\restriction j(\\alpha) = \\inf \\{ j(p) {\\restriction} j(\\alpha) : p \\in H {\\restriction} \\beta \\} \\\\\n= & \\inf \\{ j(p {\\restriction} \\alpha) : p \\in H {\\restriction} \\beta \\} = \\inf \\{ j(p) : p \\in H {\\restriction} \\alpha \\} = m_\\alpha. \n\\end{align*}\n\nThe upward closure of the sequence $\\langle p_i \\rangle_{i < \\delta}$ is a filter $\\hat{H}$ which is $\\hat{j}(\\mathbb{P})$-generic over $M[\\hat{X}]$.  For all $p \\in H$, $\\hat{j}(p) \\in \\hat{H}$ since there is some $m_\\alpha \\leq \\hat{j}(p)$.  Thus we get an extended elementary embedding $\\hat{j} : V[X][H] \\to M[\\hat{X}][\\hat{H}]$.  In $W$, we define an ultrafilter $U$ over $(\\p(\\p_\\kappa \\lambda ))^{V[X][H]}$: let $A \\in U$ iff $j[\\lambda] \\in \\hat{j}(A)$.  Note that $j[\\lambda] \\in \\p_{j(\\kappa)}(j(\\lambda))^{M[\\hat{X}][\\hat{H}]}$.  $U$ is $\\kappa$-complete and normal with respect to functions in $V[X][H]$.  If $f : \\p_\\kappa(\\lambda) \\to \\lambda$ is a regressive function in $V[X][H]$ on a set $A \\in U$, then $\\hat{j}(f)(j[\\lambda]) = j(\\alpha)$ for some $\\alpha < \\lambda$, so $\\{ z \\in A : f(z) = \\alpha \\} \\in U$.\n\nNow the forcing to obtain $U$ was $\\mathbb{Q} = B(\\mu,\\kappa)\/H_X \\times \\col(\\mu,\\lambda)$, the product of a $\\kappa$-dense and a $\\lambda$-dense partial order.  In $V[X][H]$, let $e : \\p(\\p_\\kappa \\lambda) \\to \\mathcal{B}(\\mathbb{Q})$ be defined by $e(A) = || \\check{A} \\in \\dot{U} ||$.  Let $I$ be the kernel of $e$.  $I$ is clearly a normal, $\\kappa$-complete ideal.  $e$ lifts to a boolean embedding of $\\p(\\p_\\kappa \\lambda) \/ I$ into $\\mathcal{B}(\\mathbb{Q})$.  Since $\\mathbb{Q}$ is $\\lambda^+$-c.c., $I$ is $\\lambda^+$-saturated.  If $\\langle [A_\\alpha] : \\alpha < \\lambda \\rangle$ is a maximal antichain in $\\p_\\kappa(\\lambda) \/ I$, then $\\nabla A_\\alpha$ is the least upper bound and is in the dual filter to $I$. $e(\\nabla A_\\alpha) = || \\nabla A_\\alpha \\in \\dot{U} || = 1$, and this is the least upper bound in $\\mathcal{B}(\\mathbb{Q})$ to $\\{ e(A_\\alpha) : \\alpha <\\lambda \\}$.  This is because if there were a generic extension in which all $A_\\alpha \\notin U$, then $\\nabla A_\\alpha \\notin U$ as well since $U$ is normal with respect to sequences from $V[X][H]$.  Therefore $e$ is a complete embedding, and thus $I$ is $\\lambda$-dense.  \n\\end{proof}\n\nWe can also characterize the exact structure of $\\p(\\p_\\kappa \\lambda) \/ I$.  First note the following about the ground model embedding $j : V \\to M$.  $M$ is the direct limit of the coherent system of $\\alpha$-supercompactness embeddings $j_\\alpha : V \\to M_\\alpha$ for $\\alpha < \\delta$.  Every member of $M_\\alpha$ is represented as $j_\\alpha(f)(j_\\alpha[\\alpha])$ for some function $f \\in V$ with domain $\\p_\\kappa(\\alpha)$.  If $k_\\alpha : M_\\alpha \\to M$ is the factor map such that $j = k_\\alpha \\circ j_\\alpha$, then the critical point of $k_\\alpha$ is above $\\alpha$, so $k_\\alpha(x) = k_\\alpha[x]$ when $M_\\alpha \\vDash |x| \\leq |\\alpha|$.  Since $M$ is the direct limit, for any $x \\in M$, there is some $\\alpha < \\delta$ and some $f \\in V$ such that\n\\[ x = k_\\alpha([f]) = k_\\alpha(j_\\alpha(f)(j_\\alpha[\\alpha])) = j(f)(k_\\alpha(j_\\alpha[\\alpha])) = j(f)(j[\\alpha])).\n\\]\n\nLet $U \\subseteq \\p(\\p_\\kappa \\lambda) \/ I$ be generic over $V[X][H]$, and let $j_U : V \\to N$ be the generic ultrapower embedding.  Since $e :  \\p(\\p_\\kappa \\lambda) \/ I \\to \\mathcal{B}(\\mathbb{Q})$ is a complete embedding, forcing with $\\mathcal{B}(\\mathbb{Q}) \/ e[U]$ over $V[X][H][U]$ produces a model $W$ as above.  Notice that the definition of $e$ and $U$ makes $A \\in U$ iff $j[\\lambda] \\in \\hat{j}(A)$.  Hence we can define an elementary embedding $k : N \\to M[\\hat{X}][\\hat{H}]$ by $k([f]) = \\hat{j}(f)(j[\\lambda])$, and we have $\\hat{j} = k \\circ j_U$.\n\nWhat is the critical point of $k$?  Since $N \\vDash \\mu^+ = \\delta$, certainly it must be at least $\\delta$.  Let $\\beta$ be any ordinal.  There is some $\\alpha$ such that $\\lambda \\leq \\alpha < \\delta$ and some $f \\in V$ such that $\\beta = j(f)(j[\\alpha])$.  Let $b : \\lambda \\to \\alpha$ be a bijection in $V[X][H]$.  Then $\\beta = j(f)(\\hat{j}(b)[j[\\lambda]])$.  Furthermore, $j[\\lambda] = k(j_U[\\lambda])$.  Therefore, $\\beta = k( j_U(f)(j_U(b)[j_U[\\lambda]]))$.  Thus $\\beta \\in \\ran(k)$, and so $k$ does not have a critical point.\n\nTherefore, $N = M[\\hat{X}][\\hat{H}]$.  By the closure of $M[\\hat{X}][\\hat{H}]$, the generic $K \\times C$ for $\\mathbb{Q}$ is in $M[\\hat{X}][\\hat{H}] = N \\subseteq V[X][H][U]$.  So the quotient $\\mathcal{B}(\\mathbb{Q}) \/ e[U]$ is trivial and $\\p(\\p_\\kappa \\lambda) \/ I \\cong \\mathcal{B}(\\mathbb{Q}) \\restriction q$ for some $q$.   \n\nThe generic embeddings coming from $I$ extend the original almost-hugeness embedding.  In particular, $j[\\delta]$ is cofinal in $j(\\delta)$.  This can also be deduced from the assumption that there is some $A \\in I^*$ of size $\\lambda$, which of course follows from $\\lambda^{<\\kappa} = \\lambda$.  In contrast, Burke and Matsubara~\\cite{BM} proved that if there is a normal, fine, $\\kappa$-complete, $\\lambda^+$-saturated ideal on $\\p_\\kappa(\\lambda)$ and $\\cf(\\lambda)<\\kappa$, then it is forced that $\\sup (j[\\lambda^+]) < j(\\lambda^+)$.  It seems to be unknown whether it is consistent to have saturated ideals on $\\p_\\kappa(\\lambda)$ for successor $\\kappa$ and singular $\\lambda$, and this result suggests that quite different methods will be needed for an answer.\n\n\\subsubsection{Minimal generic supercompactness}\nGeneralizing supercompactness, we will say cardinal $\\kappa$ is \\emph{generically supercompact} when for every $\\lambda \\geq \\kappa$, there is a forcing $\\mathbb{P}$ such that whenever $G \\subseteq \\mathbb{P}$ is generic, there is an elementary embedding $j : V \\to M$, where $M$ is a transitive class in $V[G]$, $\\crit(j) = \\kappa$, $j(\\kappa) > \\lambda$, and $M^\\lambda \\cap V[G] \\subseteq M$.  We note that unlike in the case of non-generic supercompactness, the condition that $j[\\lambda] \\in M$ does not imply that $M$ is closed under $\\lambda$-sequences from $V[G]$.  Whenever a supercompact $\\kappa$ is turned into a successor cardinal by a $\\kappa$-c.c.\\ forcing, we'll have that for all $\\lambda \\geq \\kappa$, there is a normal, fine, precipitous ideal on $\\p_\\kappa(\\lambda)$ whose generic embeddings always extend the original supercompactness embedding.  But if $j : V \\to M$ is an embedding coming from a normal ultrafilter on $\\p_\\kappa(\\lambda)$, then $2^{\\lambda^{<\\kappa}} < j(\\kappa) < (2^{\\lambda^{<\\kappa}})^+$.  If $\\kappa = \\mu^+$ in a generic extension $V[H]$, and a further extension gives $\\hat{j} : V[H] \\to M[\\hat{H}] \\subseteq V[H][G]$ extending $j$, then $M[\\hat{H}]$ is not closed under $\\lambda$-sequences from $V[H][G]$.  This is because $| \\lambda | = | j(\\kappa)| = \\mu$ in $V[H][G]$, while $M[\\hat{H}]$ thinks $j(\\kappa)$ is a cardinal.\n\nStronger properties of ideals on $\\p_\\kappa(\\lambda)$ are needed to give genuine generic supercompactness.  One such property is $\\lambda^+$-saturation, which is implied by $\\lambda$-density.  We now sketch how to get a model in which there is a successor cardinal $\\kappa$ such that for all regular $\\lambda \\geq \\kappa$, there is a normal, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$.  Start with a super-almost-huge cardinal $\\kappa$ and a regular $\\mu < \\kappa$.  The first part of the forcing is $A(\\mu,\\kappa)$.  Then we do a proper class iteration, which we prefer to describe instead as an iteration up to an inaccessible $\\delta > \\kappa$ such that $V_\\delta \\vDash \\kappa$ is super-almost-huge.\n\nLet $T = \\{ \\alpha < \\delta : \\kappa$ carries an almost-huge tower of height $\\alpha \\}$.  Let $C$ be the closure of $T$, and let $\\langle \\alpha_\\beta \\rangle_{\\beta < \\delta}$ be its continuous increasing enumeration.  Over $V^{A(\\mu,\\kappa)}$, let $\\mathbb{P}_\\delta$ be the Easton-support limit of the following:\n\\begin{itemize}\n\\item Let $\\mathbb{P}_0 = \\col(\\kappa,<\\alpha_0)$.\n\\item If $\\beta$ is zero or a successor ordinal, let $\\mathbb{P}_{\\beta+1} = \\mathbb{P}_{\\beta} * \\col(\\alpha_\\beta,<\\alpha_{\\beta+1})$.\n\\item If $\\beta$ is a limit ordinal such that $\\alpha_\\beta$ is singular, let $\\mathbb{P}_{\\beta+1} = \\col(\\alpha_\\beta^+,<\\alpha_{\\beta+1})$.\n\\item If $\\beta$ is a limit ordinal such that $\\alpha_\\beta$ is regular, let $\\mathbb{P}_{\\beta+1} = \\col(\\alpha_\\beta,<\\alpha_{\\beta+1})$.\n\\end{itemize}\n\nIt is routine to verify that this iteration preserves the regularity of the members of $T$, the successors of the singular limit points of $T$, and the regular limit points of $T$.  Further, the set of non-limit-points of $T$ becomes the set of successors of regular cardinals between $\\kappa$ and $\\delta$.\n\nLet $X \\subseteq \\kappa$ be $A(\\mu,\\kappa)$-generic over $V$, and let $H \\subseteq \\mathbb{P}_\\delta$ be generic over $V[X]$. Suppose $\\kappa \\leq \\lambda < \\delta$, and $\\lambda$ is regular in $V[X][H]$.  Then there is some successor ordinal $\\beta<\\delta$ such that $\\alpha_\\beta \\in T$ and $\\alpha_\\beta = \\lambda^+$.  Consider the subforcing $A(\\mu,\\kappa) * \\mathbb{P}_\\beta = (A(\\mu,\\kappa) * \\mathbb{P}_{\\beta-1}) * \\col(\\lambda,<\\alpha_\\beta)$.  The forcing $\\mathbb{P}_\\beta$ is $(\\kappa,\\alpha_\\beta)$-very nicely layered in $V[X]$.\n\nIf $j : V \\to M_\\beta$ is an almost-huge embedding with critical point $\\kappa$ and $j(\\kappa) = \\alpha_\\beta$, then by Theorem~\\ref{maindense}, there is a normal, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$ in $V[X][H_\\beta]$.  Now note that the tail-end forcing $\\mathbb{P}_{\\beta,\\delta}$ is $\\alpha_\\beta$-closed.  Since $\\lambda^{<\\kappa} = \\lambda$ in $V[X][H_\\beta]$, no new subsets of $\\p_\\kappa(\\lambda)$ are added by the tail.  The collection $\\{ A_\\alpha : \\alpha < \\lambda \\}$ witnessing the $\\lambda$-density of $I$ retains this property, as this is a local property of the boolean algebra $\\p_\\kappa(\\lambda) \/ I$ and $\\{ A_\\alpha : \\alpha < \\lambda \\}$.  Normality and completeness of $I$ are likewise preserved.  \n\nThis method is quite flexible, and can done by iterating collapsing posets other than the Levy collapse, or by using products rather than iterations.\n\n\\subsubsection{Dense ideals on successive cardinals?}\n\nAt the time of this writing, it is unknown whether there can exist simultaneously a normal $\\kappa$-dense ideal on $\\kappa$ and a normal $\\kappa^+$-dense ideal on $\\kappa^+$.  The following is the current best approximation.\n\nSuppose $\\langle \\kappa_n : n < \\omega \\rangle$ is a sequence of cardinals such that for all $n$, $\\kappa_n$ carries an almost-huge tower of height $\\kappa_{n+1}$.  Such a sequence will be called an \\emph{almost-huge chain}.  Obviously, extending this to sequences of length longer than $\\omega$ requires an extra idea; perhaps we just stack one $\\omega$-chain above another, or maybe postulate some relationship between the $\\omega$-chains.  By Theorem~\\ref{tourney}, such chains occur quite often below a huge cardinal.\n\n\nSuppose $\\langle \\kappa_n : 0 < n < \\omega \\rangle$ is an almost-huge chain, and $\\mu < \\kappa_1$ is regular.  Consider the full-support iteration $\\mathbb{P}$ of $\\langle \\mathbb{P}_n : n < \\omega \\rangle$, where $\\mathbb{P}_0 = A(\\mu,\\kappa_1)$, and for all $n < \\omega$, $\\mathbb{P}_{n+1} = \\mathbb{P}_n * A(\\kappa_n,\\kappa_{n+1})$.  The stage $\\mathbb{P}_1 = A(\\mu,\\kappa_1) * A(\\kappa_1,\\kappa_2)$ regularly embeds into $A(\\mu,\\kappa_1) * (\\col(\\kappa_1,< \\! \\kappa_2) * \\add(\\kappa_2))$.  The first two stages here add a normal $\\kappa_1$-dense ideal on $\\kappa_1$ and make $\\kappa_1=\\mu^+$, $\\kappa_2 = \\mu^{++}$.  The third stage preserves this since it adds no subsets of $\\kappa_1$.  By Lemma~\\ref{quotdist}, the quotient forcing $\\mathbb{Q}$ to get from $V^{\\mathbb{P}_1}$ to this three-stage extension is $\\kappa_2$-distributive.  Now the tail-end forcing $\\mathbb{P} \/ \\mathbb{P}_1$ is $\\kappa_2$-strategically closed.  Since $\\mathbb{Q}$ does not add any plays of the relevant game of length $< \\! \\kappa_2$, $\\mathbb{P} \/ \\mathbb{P}_1$ remains $\\kappa_2$-strategically closed in $V^{\\mathbb{P}_1 * \\mathbb{Q}}$, so forcing with it preserves the $\\kappa_1$-dense ideal on $\\kappa_1$.   Also, $\\mathbb{Q}$ remains $\\kappa_2$-distributive in $V^{\\mathbb{P}}$, since $\\mathbb{Q} \\times (\\mathbb{P} \/ \\mathbb{P}_1)$ is $\\kappa_2$-distributive in $V^{\\mathbb{P}_1}$.  It thus remains the case in $V^\\mathbb{P}$ that there is a $\\kappa_2$-distributive forcing adding a normal $\\kappa_1$-dense ideal on $\\kappa_1$.\n\n\nSimilarly, consider $V^{\\mathbb{P}_n}$ for $n >1$.  $\\mathbb{P}_n = \\mathbb{P}_{n-2} * (A(\\kappa_{n-1}, \\kappa_n) * A(\\kappa_n,\\kappa_{n+1}))$.  Since $|\\mathbb{P}_{n-2}| = \\kappa_{n-1}$ (or $\\mu$ for $n=2$), $\\kappa_{n}$ retains an almost-huge tower of height $\\kappa_{n+1}$ in $V^{\\mathbb{P}_{n-2}}$.  Thus the same argument applies: In $V^{\\mathbb{P}_n}$, there is a $\\kappa_{n+1}$-distributive forcing adding a normal $\\kappa_n$-dense ideal on $\\kappa_n$, and this remains true in $V^\\mathbb{P}$.  Therefore, we obtain a model in which for all $n >0$, there is a $\\mu^{+n+1}$-distributive forcing adding a normal $\\mu^{+n}$-dense ideal on $\\mu^{+n}$.  By repeating this with a tall enough stack of almost-huge chains, we obtain the consistency of ZFC with the statement, ``For all regular cardinals $\\kappa$, there is a $\\kappa^{++}$-distributive forcing adding a normal $\\kappa^+$-dense ideal on $\\kappa^+$.''\n\n\\section{Structural constraints}\n\nSaturated ideals have a strong influence over the combinatorial structure of the universe in their vicinity.  Phenomena of this type may also be viewed as the universe imposing constraints on the structural properties of ideals.  Below are some of the most interesting known results to this effect.  Proofs can be found in~\\cite{foremanhandbook}.\n\n\\begin{enumerate}[(1)]\n\\item (Tarski) If $I$ is a nowhere-prime ideal which is $\\kappa$-complete and $\\mu$-saturated for some $\\mu < \\kappa$, then $2^{<\\mu} \\geq \\kappa$.\n\\item (Jech-Prikry) If $\\kappa = \\mu^+$, $2^\\mu = \\kappa$, and there is a $\\kappa$-complete, $\\kappa^+$-saturated ideal on $\\kappa$, then $2^\\kappa = \\kappa^+$.\n\\item (Jech-Prikry) If $\\kappa = \\mu^+$, and there is a $\\kappa$-complete, $\\kappa^+$-saturated ideal on $\\kappa$, then there are no $\\kappa$-Kurepa trees.\n\\item (Woodin) If there is a countably complete, $\\omega_1$-dense ideal on $\\omega_1$, then there is a Suslin tree.\n\\item (Woodin) If there is a countably complete, uniform, $\\omega_1$-dense ideal on $\\omega_2$, then $2^\\omega = \\omega_1$.  (Uniform means that all sets of size $<\\omega_2$ are in the ideal--equivalent to fineness.)\n\\item (Shelah) If $2^\\omega < 2^{\\omega_1}$, then $NS_{\\omega_1}$ is not $\\omega_1$-dense.\n\\item (Gitik-Shelah) If $I$ is a $\\kappa$-complete, nowhere-prime ideal, then $\\den(I) \\geq \\kappa$.\n\\end{enumerate}\n\nWe note that result (2) easily generalizes to the following: If $\\kappa = \\mu^+$, $2^\\mu = \\kappa$, and there is a normal, fine, $\\kappa$-complete, $\\lambda^+$-saturated ideal on $\\p_\\kappa(\\lambda)$, then $2^\\lambda = \\lambda^+$.\n\nIf no requirements are made for the ideal $I$ and the set $Z$ on which it lives, almost no structural constraints on quotient algebras remain.  The following strengthens a folklore result, probably known to Sikorski.  The argument was supplied by Don Monk in personal correspondence.\n\n\\begin{proposition}\nLet $\\mathbb{B}$ be a complete boolean algebra, and let $\\kappa$ be a cardinal such that $2^\\kappa \\geq |\\mathbb{B}|$.  There is a uniform ideal $I$ on $\\kappa$ such that $\\mathcal{B} \\cong \\p(\\kappa)\/I$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\kappa$, $\\mathbb{B}$ be as hypothesized.  By the theorem of Fichtenholz-Kantorovich and Hausdorff (see \\cite[Lemma 7.7]{jechbook}), there exists a family $F$ of $2^\\kappa$ many subsets of $\\kappa$ such that for any $x_1,...,x_n,y_1,...,y_m \\in F$, $x_1 \\cap ... \\cap x_n \\cap (\\kappa \\setminus y_1) \\cap ... \\cap (\\kappa \\setminus y_m)$ has size $\\kappa$.  $F$ generates a free algebra: closing $F$ under finitary set operations gives a family of sets $G$ such that any equation holding between elements of $G$ expressed as boolean combinations of elements of $F$ holds in all boolean algebras.  If we pick any surjection $h_0 : F \\to \\mathbb{B}$ and extend it to $h_1 : G \\to \\mathbb{B}$ in the obvious way, then $h_1$ will be a well-defined homomorphism.\n\nLet $I_{bd}$ be the ideal of bounded subsets of $\\kappa$.  Since all elements of $G$ are either empty or have cardinality $\\kappa$, $G \\cong G\/I_{bd}$, so $h_1$ has an extension $h_2$ from the algebra generated by $G \\cup I_{bd}$ to $\\mathbb{B}$, where $h_2(x) = 0$ for all $x \\in I_{bd}$.  Finally, by Sikorski's extension theorem, there is a further extension to a homomorphism $h_3 : \\p(\\kappa) \\to \\mathbb{B}$.  The kernel of $h_3$ is an ideal $I$ such that $\\p(\\kappa)\/I \\cong \\mathbb{B}$.   \n\\end{proof}\n\n\n\\subsection{Cardinal arithmetic and ideal structure}\n\nA careful examination of the proof of Woodin's theorem (5) shows that $\\omega_2$ can be replaced by any $\\omega_n$, $2 \\leq n < \\omega$.  Aside from that, Woodin's argument is rather specific to the cardinals involved.  In \\cite{foremanhandbook}, Foreman asked (Open Question 27) whether the analogous statement holds one level up:\n\n\\begin{question}[Foreman]Does the existence of an $\\omega_2$-complete, $\\omega_2$-dense, uniform ideal on $\\omega_3$ imply that $2^{\\omega_1} = \\omega_2$?\n\\end{question}\n\nTo answer this, we invoke an easy preservation lemma about ideals under small forcing.  If $I$ is an ideal, $\\mathbb{P}$ is a partial order, and $G \\subseteq \\mathbb{P}$ is generic, then $\\bar{I}$ denotes the ideal generated by $I$ in $V[G]$, i.e. $\\{ X : (\\exists Y \\in I) X \\subseteq Y \\}$.\n\n\n\\begin{lemma}\n\\label{pres}\nSuppose $I$ is a $\\kappa$-complete ideal on $Z \\subseteq \\p(X)$, $\\mathbb{P}$ is a partial order, and $G$ is $\\mathbb{P}$-generic.\n\\begin{enumerate}[(1)]\n\\item If $\\sat(\\mathbb{P}) \\leq \\kappa$, then $\\bar{I}$ is $\\kappa$-complete in $V[G]$.\n\\item If $\\den(\\mathbb{P}) < \\kappa$, then $\\den(\\bar{I})^{V[G]} \\leq \\den(I)^V$.  \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor (1), let $\\dot{s}$ be a $\\mathbb{P}$-name for a sequence of elements of $\\bar{I}$ of length less than $\\kappa$.  By $\\kappa$-saturation, let $\\beta < \\kappa$ be such that $1 \\Vdash dom(\\dot{s}) \\leq \\beta$.  For each $\\alpha < \\beta$, let $A_\\alpha$ be a maximal antichain such that for $p \\in A_\\alpha$, $p \\Vdash \\dot{s}(\\alpha) \\subseteq \\check{b}^p_\\alpha$, where ${b}^p_\\alpha \\in I$.  Then $B = \\bigcup_{p, \\alpha} b^p_{\\alpha} \\in I$, and $1 \\Vdash \\bigcup \\dot{s} \\subseteq \\check{B}$.\n\n\nFor (2), let $D \\subseteq \\mathbb{P}$ be a dense set of size less than $\\kappa$, and let $A \\in \\bar{I}^+$.  Then $A = \\bigcup_{d \\in D \\cap G} \\{ z : d \\Vdash z \\in \\dot{A} \\}$.  By (1), there is some $d \\in D$ such that $\\{ z : d \\Vdash z \\in \\dot{A} \\} \\notin I$.  This shows that $(\\p(Z)\/I)^V$ is dense in $(\\p(Z)\/\\bar{I})^{V[G]}$, and the conclusion follows.  \n\\end{proof} \n\n\\begin{corollary}\n\\label{wg}\nIf there is a $\\kappa^+$-complete, $\\kappa^+$-dense, uniform ideal on $\\kappa^{++}$, then $2^\\kappa = \\kappa^+$.\n\\end{corollary}\n\n\\begin{proof}\nSuppose for a contradiction that $f: \\p(\\kappa) \\to \\kappa^{++}$ is a surjection.  Let $\\mathbb{P} = \\col(\\omega,\\kappa)$, and let $G$ be $\\mathbb{P}$-generic.  Since $\\den(\\mathbb{P}) = \\kappa$, Lemma~\\ref{pres} implies that $\\bar{I}$ is $\\kappa^+$-complete and $\\kappa^+$-dense in $V[G]$.  Furthermore, in $V[G]$, $\\kappa^+ = \\omega_1$ and $\\kappa^{++}= \\omega_2$.  Thus Woodin's theorem implies that $V[G] \\vDash$ CH.  However, $f$ witnesses the failure of CH, a contradiction.  \n\\end{proof}\n\n\nAnother interesting constraint can be derived from the following:\n\n\\begin{theorem}[Shelah~\\cite{shelahproper}]\n\\label{shelahcof}\nSuppose $V \\subseteq W$ are models of ZFC.  If $\\kappa$ is a regular cardinal in $V$, and $\\cf(\\kappa) \\not= \\cf(|\\kappa|)$ in $W$, then $(\\kappa^+)^V$ is not a cardinal in $W$.\n\\end{theorem}\n\n\\begin{corollary}[Burke-Matsubara \\cite{bmclub}]\n\\label{cofcon}\nIf $\\kappa = \\mu^+$, $\\lambda \\geq \\kappa$ is regular, and $I$ is a normal, fine, $\\kappa$-complete, $\\lambda^+$-saturated ideal on $\\p_\\kappa(\\lambda)$, then $\\{ z : \\cf(z) = \\cf(\\mu) \\} \\in I^*$.\n\\end{corollary}\n\\begin{proof}\nLet $G$ be a generic ultrafilter extending $I^*$.  Since $\\crit(j) = \\kappa$ and $\\lambda^+$ is preserved, $j(\\kappa) = \\lambda^+$, and $|\\lambda| = \\mu$ in $V[G]$.  By Shelah's theorem, $\\cf(\\lambda) = \\cf(\\mu)$ in $V[G]$ and in the ultrapower $M$ since $M^\\mu \\cap V[G] \\subseteq M$.  Since $1 \\Vdash [\\id] = j[\\lambda]$, \\L o\\'{s}'s theorem gives $\\{ z : \\cf(z) = \\cf(\\mu) \\} \\in I^*$.   \n\\end{proof}\n\n\\begin{theorem}\n\\label{dist}\nSuppose $\\kappa = \\mu^+$, and $I$ is a normal, fine, $\\kappa^+$-saturated ideal on $\\kappa$.  Then $\\p(\\kappa) \/ I$ is $\\cf(\\mu)$-distributive iff $\\mu^{<\\cf(\\mu)} = \\mu$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\p(\\kappa) \/ I$ is $\\cf(\\mu)$-distributive, and let $\\{ f_\\alpha : \\alpha < \\delta \\}$ be an enumeration of $[\\mu]^{<\\cf(\\mu)}$, where $\\delta$ is a cardinal.  If $\\mu < \\delta$, then for any $\\p(\\kappa) \/ I$-generic $G$, $([\\mu]^{<\\cf(\\mu)})^V$ is a proper subset of $([\\mu]^{<\\cf(\\mu)})^{V[G]}$, since $j[\\delta] \\not= j(\\delta)$.  This contradicts the distributivity of  $\\p(\\kappa) \/ I$.\n\nSince $\\p(\\kappa)\/I$ is $\\kappa^+$-saturated, it is $\\cf(\\mu)$-distributive iff it is $(\\cf(\\mu), \\kappa)$- \\break distributive.  Let $G$ be $\\p(\\kappa) \/ I$-generic and let $M$ be the generic ultrapower.  Let $\\beta < \\cf(\\mu)$, and suppose $f \\in V[G]$ is a function from $\\beta$ to $\\kappa$.   By Theorem~\\ref{disj}, $f \\in M$.  By Corollary~\\ref{cofcon}, $M \\vDash \\cf(\\kappa) = \\cf([\\id]) = \\cf(\\mu)$.  Thus there is a $\\gamma < \\kappa$ such that $\\ran(f) \\subseteq \\gamma$.  Observe that $j(^\\beta \\gamma) = (^\\beta \\gamma)^M = (^\\beta \\gamma)^V$, since $\\mu^\\beta < \\kappa$.  Hence $f \\in V$.  \n\\end{proof}\n\n\n\\subsection{Stationary reflection}\nA stationary subset $S$ of a regular cardinal $\\kappa$ is said to reflect if there is some $\\alpha < \\kappa$ of uncountable cofinality such that $S \\cap \\alpha$ is stationary in $\\alpha$.  A collection of stationary subsets $\\{ S_i : i < \\delta \\}$ of $\\kappa$ is said to reflect simultaneously if there is some $\\alpha < \\kappa$ if $S_i \\cap \\alpha$ is stationary for all $i < \\delta$.  It is well known that if $\\kappa = \\mu^+$ and $X$ is a set of regular cardinals below $\\mu$, then the statement that every stationary subset of $\\{ \\alpha < \\kappa : \\cf(\\alpha) \\in X \\}$ reflects contradicts $\\square_\\mu$, and the statement that every pair of stationary subsets of $\\{ \\alpha < \\kappa : \\cf(\\alpha) \\in X \\}$ reflect simultaneously contradicts the weaker principle $\\square(\\kappa)$.\n\n\\begin{theorem}\nSuppose there is a $\\kappa^{+}$-complete, $\\kappa^{++}$-saturated, uniform ideal on $\\kappa^{+n}$ for some $n \\geq 2$.  Then for $2 \\leq m \\leq n$, every collection of $\\kappa$ many stationary subsets of $\\kappa^{+m}$ contained in $\\cof(\\leq \\kappa)$ reflects simultaneously. \n\\end{theorem}\n\n\\begin{proof}\nSuppose $I$ is such an ideal and $j : V \\to M \\subseteq V[G]$ is a generic embedding arising from the ideal.  The critical point of $j$ is $\\kappa^+$, and all cardinals above $\\kappa^{+}$ are preserved.  Since $I$ is uniform, and there is a family of $\\kappa^{+n+1}$ many almost-disjoint functions from $\\kappa^{+n}$ to $\\kappa^{+n}$, $j(\\kappa^{+n}) \\geq (\\kappa^{+n+1})^V$.  The first $n-1$ cardinals in $V$ above $\\kappa$ must map onto the first $n-1$ cardinals in $M$ above $\\kappa$.  But in $M$, there are at least $n-1$ cardinals in the interval $(\\kappa,(\\kappa^{+n+1})^V)$ since all cardinals above $\\kappa^+$ are preserved.  Thus if $j(\\kappa^{+n}) > (\\kappa^{+n+1})^V$, then $\\kappa^{+n+1}$ would be collapsed.  So for $1 \\leq m \\leq n$, $j(\\kappa^{+m}) = (\\kappa^{+m+1})^V$.\n\nLet $\\{ S_\\alpha : \\alpha < \\kappa \\}$ be stationary subsets of $\\kappa^{+m}$ concentrating on $\\cof(\\leq \\kappa)$, where $2 \\leq m \\leq n$.  By the $\\kappa^{++}$-chain condition, these sets remain stationary in $V[G]$.  By the above remarks, $\\gamma = \\sup (j[\\kappa^{+m}]) < j(\\kappa^{+m})$.  For each $\\alpha$, $j \\restriction S_\\alpha$ is continuous since $\\kappa < \\crit(j)$.  For each $\\alpha$, let $C_\\alpha$ be the closure of $S_\\alpha$.  In $V[G]$, we can define a continuous increasing function $f : C_\\alpha \\to \\gamma$ extending $j \\restriction S_\\alpha$ by sending $\\sup (S_\\alpha \\cap \\beta)$ to $\\sup (j[S_\\alpha \\cap \\beta])$ when $\\beta$ is a limit point of $S_\\alpha$. This shows that $j[S_\\alpha]$ is stationary in $\\gamma$.  Now $M$ may not have $j[S_\\alpha]$ as an element, but it satisfies that $j(S_\\alpha) \\cap \\gamma$ is stationary in $\\gamma$.  Furthermore, $j( \\{ S_\\alpha : \\alpha < \\kappa \\}) = \\{ j(S_\\alpha) : \\alpha < \\kappa \\}$, and $M$ sees that these all reflect at $\\gamma$.  By elementarity, the $S_\\alpha$ have a common reflection point.  \n\\end{proof}\n\n\\begin{proposition}\n\\label{Z-refl}\nSuppose $\\mu,\\kappa,\\lambda$ are regular cardinals such that $\\omega < \\mu < \\kappa = \\mu^+ < \\lambda$, and $I$ is an ideal on $\\p_\\kappa(\\lambda)$ as in Theorem~\\ref{maindense}.  Then every collection $\\{ S_i : i < \\mu \\}$ of stationary subsets of $\\lambda \\cap \\cof( \\omega)$ reflects simultaneously.\n\\end{proposition}\n\n\\begin{proof}The algebra $\\p(\\p_\\kappa(\\lambda)) \/ I$ is isomorphic to $\\mathcal{B}(\\mathbb{P} \\times \\mathbb{Q})$, where $\\mathbb{P}$ is $\\kappa$-dense and $\\Vdash_\\mathbb{P} ``\\mathbb{Q}$ is $\\mu$-closed.''  Forcing with $\\mathbb{P} \\times \\mathbb{Q}$ thus preserves the stationarity of any subset of $\\lambda \\cap \\cof(\\omega)$.  If $j : V \\to M \\subseteq V[G]$ is a generic embedding arising from the ideal, then since $j[\\lambda] \\in M$ and $M$ thinks $j(\\lambda)$ is regular, $\\gamma = \\sup(j[\\lambda])< j(\\lambda)$.  The restriction of $j$ to each $S_i$ is continuous, and as above we may define in $V[G]$ a continuous increasing function from the closure of $S_i$ into $\\gamma$, showing $j[S_i]$ is stationary in $\\gamma$ for each $i$.  Thus $M \\models (\\forall i < \\mu) j(S_i) \\cap \\gamma$ is stationary, so by elementarity, the collection reflects simultaneously.   \n\\end{proof}\n\n\n\\subsection{Nonregular ultrafilters}\nThe computation of the cardinality of ultrapowers is an old problem of model theory.  Originally, it was conjectured that if $\\mu,\\kappa$ are infinite cardinals, and $U$ is a countably incomplete uniform ultrafilter on $\\kappa$, then $| \\mu^\\kappa \/ U | = \\mu^\\kappa$ \\cite{ck}.  It was shown by Donder~\\cite{donder} that this conjecture holds in the core model below a measurable cardinal.  A key tool in computing the size of ultrapowers is the notion of regularity:\n\n\\begin{definition}\nAn ultrafilter $U$ on $Z$ is called \\emph{$(\\mu, \\kappa)$-regular} if there is a sequence $\\langle A_\\alpha : \\alpha < \\kappa \\rangle \\subseteq U$ such that for any $Y \\subseteq \\kappa$ of order type $\\mu$, $\\bigcap_{\\alpha \\in Y} A_\\alpha = \\emptyset$.\n\\end{definition}\n\n\\begin{theorem}[Keisler \\cite{keisler}]\n\\label{keisler}\nSuppose $U$ is a $(\\mu, \\kappa)$-regular ultrafilter on $Z$, witnessed by $\\langle A_\\alpha : \\alpha < \\kappa \\rangle$.  For each $z \\in Z$, let $\\beta_z = \\ot( \\{ \\alpha : z \\in A_\\alpha \\}) < \\mu$.  Then for any sequence of ordinals $\\langle \\gamma_z : z \\in Z \\rangle$, we have $| \\prod \\gamma_z^{\\beta_z} \/ U | \\geq | \\prod \\gamma_z \/ U |^\\kappa$.\n\\end{theorem}\n\nObviously any uniform ultrafilter on a cardinal $\\kappa$ is $(\\kappa,\\kappa)$-regular.  Also, any fine ultrafilter on $\\p_\\kappa(\\lambda)$ is $(\\kappa,\\lambda)$-regular, as witnessed by $\\langle \\hat{\\alpha} : \\alpha < \\lambda \\rangle$.  Much can be seen by exploiting a connection between dense ideals and nonregular ultrafilters.\n\n\\begin{lemma}[Huberich \\cite{huberich}]\n\\label{huberich}\nSuppose $\\mathbb{B}$ is a complete boolean algebra of density $\\kappa$, where $\\kappa$ is regular.  Then there is an ultrafilter $U$ on $\\mathbb{B}$ such that whenever $X \\subseteq \\mathbb{B}$ and $\\sum X \\in U$, then there is $Y \\subseteq X$ such that $|Y| < \\kappa$ and $\\sum Y \\in U$.\n\\end{lemma}\n\n\\begin{proof}\nLet $D = \\{ d_\\alpha : \\alpha < \\kappa \\}$ be dense in $\\mathbb{B}$.  For any maximal antichain $A \\subseteq \\mathbb{B}$, let $\\gamma_A > 0$ be least such that for all $\\alpha < \\gamma_A$, there are $\\beta < \\gamma_A$ and $a \\in A$ such that $d_\\beta \\leq d_\\alpha \\wedge a$.  Let $C_A = \\{ d \\in D \\restriction \\gamma_A : (\\exists a \\in A) d \\leq a \\}$.  Let $F = \\{ \\sum C_A : A$ is a maximal antichain$\\}$.\n\nWe claim $F$ has the finite intersection property.  Let $A_1,...,A_n$ be maximal antichains.  We may assume $\\gamma_{A_1} \\leq ... \\leq \\gamma_{A_n}$.  Let $d_{\\alpha_1} \\leq d_0 \\wedge a_1$ for some $a_1 \\in A_1$, where $\\alpha_1 < \\gamma_{A_1}$.  Let $d_{ \\alpha_2} \\leq d_{\\alpha_1} \\wedge a_2$ for some $a_2 \\in A_2$, where $\\alpha_2 < \\gamma_{A_2}$.  Proceeding inductively, we get a descending chain $d_{\\alpha_1} \\geq ... \\geq d_{\\alpha_n}$, where each $d_{\\alpha_i} \\in C_{A_i}$.  Thus $d_{\\alpha_n} \\leq \\sum C_{A_1} \\wedge ... \\wedge \\sum C_{A_n}$.\n\nLet $U \\supseteq F$ be any ultrafilter.  If $\\sum X \\in U$, then we can find an antichain $A$ that is maximal below $\\sum X$ such that $(\\forall a \\in A)(\\exists x \\in X) a \\leq x$.  Extending $A$ it to a maximal antichain $A'$, we have $\\sum C_{A'} \\in F$.  Since $|C_{A'}| < \\kappa$, the conclusion follows.   \n\\end{proof}\n\nIf $I$ is an ideal on $Z$ and $U'$ is an ultrafilter on $\\p(Z)\/I$, then $U'$ generates an ultrafilter $U \\supseteq I^*$ on $Z$ by taking $U = \\{ X : [X]_I \\in U' \\}$.\n\n\n\\begin{lemma}\n\\label{singreg}\nSuppose $\\kappa=\\mu^+$, $\\lambda$ is regular, and $I$ is a normal and fine, $\\kappa$-complete, $\\lambda$-dense ideal on $Z \\subseteq \\p_\\kappa(\\lambda)$.  Then any ultrafilter $U \\supseteq I^*$ given by Lemma \\ref{huberich} is $(\\cf(\\mu)+1,\\lambda)$-regular.\n\\end{lemma}\n\n\\begin{proof}\nLet $U'$ be an ultrafilter on $\\p(Z)\/I$ given by Lemma \\ref{huberich} and let $U$ be the corresponding ultrafilter on $Z$.  By Corollary~\\ref{cofcon}, $\\{ z : \\cf(z) = \\cf(\\mu) \\} \\in I^*$.  For such $z$, choose $A_z \\subseteq z$ of order type $\\cf(\\mu)$ that is cofinal in $z$.    We will inductively build a sequence of intervals $\\{ (x_\\alpha,y_\\alpha) : \\alpha < \\lambda \\}$, each contained in $\\lambda$, such that $y_\\alpha < x_\\beta$ when $\\alpha < \\beta$, and such that for all $\\alpha$, $\\{ z : A_z \\cap (x_\\alpha,y_\\alpha) \\not= \\emptyset \\} \\in U$.\n\nSuppose we have constructed the intervals up to $\\beta$.  Let $\\lambda > x_\\beta > \\sup \\{ y_\\alpha : \\alpha < \\beta \\}$.  For $z \\in \\hat{x}_\\beta$, let $y_\\beta(z) \\in z$ be such that $A_z \\cap (x_\\beta,y_\\beta(z)) \\not= \\emptyset$.  Since $I$ is normal, there is a maximal antichain $A$ of $I$-positive sets such that for all $a \\in A$, $y_z(\\beta)$ is the same for all $z \\in a$.  There is some $A' \\subseteq A$ of size $< \\lambda$ such that $\\sum A' \\in U'$.  Let $y_\\beta > x_\\beta$ be such that for $z \\in a \\in A'$, $y_\\beta(z) < y_\\beta$.\n\nFor $\\alpha < \\lambda$, let $X_\\alpha = \\{ z : A_z \\cap (x_\\alpha,y_\\alpha) \\not= \\emptyset \\}$.  Since each $A_z$ has order type $\\cf(\\mu)$ and the intervals $(x_\\alpha,y_\\alpha)$ are disjoint and increasing, each $A_z$ cannot have nonempty intersection with all intervals in some sequence of length greater than $\\cf(\\mu)$.  Thus if $s \\subseteq \\lambda$ and $z \\in \\bigcap_{\\alpha \\in s} X_\\alpha$, then $\\ot(s) \\leq \\cf(\\mu)$. \n\\end{proof}\n\n\\begin{lemma}\n\\label{ultsize}\nSuppose $I$ is a $\\lambda$-dense, $\\kappa$-complete ideal on $Z$, and $\\p(Z)\/I$ is a complete boolean algebra.  Then any $U \\supseteq I^*$ given by Lemma \\ref{huberich} has the property that for all $\\alpha < \\kappa$, $| \\alpha^Z \/ U | \\leq 2^{<\\lambda}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $U'$ be an ultrafilter on $\\p(Z)\/I$ given by Lemma \\ref{huberich} and let $U$ be the corresponding ultrafilter on $Z$.\nTo compute a bound on $|\\alpha^Z\/U|$ for $\\alpha < \\kappa$, we identify a small subset of $\\alpha^Z$ and show that it contains representative of every equivalence class modulo $U$. Let $D$ witness $\\lambda$-density. Choose an antichain $A \\subseteq D$ of size $< \\! \\lambda$, and choose $f : A \\to \\alpha$.  There are $\\sum_{\\gamma < \\lambda} \\lambda^\\gamma \\cdot \\alpha^\\gamma = 2^{<\\lambda}$ many choices.  Using $\\kappa$-completeness, let $\\{ B_\\beta : \\beta < \\alpha \\}$ be pairwise disjoint and such that each $[B_\\beta]_I = \\sum f^{-1}(\\beta)$.  Let $g_f : Z \\to \\alpha$ be defined by $g_f(z) = \\beta$ if $z \\in B_\\beta$ and $g_f(z) = 0$ if $z \\notin \\bigcup_{\\beta<\\alpha} B_\\beta$.\n\nNow let $g : Z \\to \\alpha$ be arbitrary.  By $\\kappa$-completeness, $A = \\{ g^{-1}(\\beta) :\\beta < \\alpha$ and $g^{-1}(\\beta) \\in I^+ \\}$ forms a maximal antichain.  Let $A' \\subseteq D$ be a maximal antichain refining $A$.  There is some $A'' \\subseteq A'$ of size $< \\! \\lambda$ such that $\\sum A'' \\in U'$.  Let $f : A'' \\to \\alpha$ be defined by $f(a) = \\beta$ iff $a \\leq_I g^{-1}(\\beta)$.  If $[B]_I = \\sum A''$, then $\\{ z \\in B : g(z) \\not= g_f(z) \\} \\in I$, so $g =_U g_f$.\n\\end{proof}\n\nThe following contrasts with the consistency results of Section 2:\n\n\\begin{corollary}Suppose $\\mu$ is a singular cardinal such that $2^{\\cf(\\mu)} < \\mu$, $\\lambda$ is regular, and $2^{<\\lambda} < 2^\\lambda$.  Then there is no normal and fine, $\\lambda$-dense ideal on $\\p_{\\mu^+}(\\lambda)$.  Furthermore, there is a proper class of such $\\lambda$.\n\\end{corollary}\n\n\\begin{proof}\nSuppose such an ideal exists, and let $U \\supseteq I^*$ be given by Lemma \\ref{huberich}.  Then $| \\prod \\mu \/U| \\leq 2^{<\\lambda}$, and $U$ is $(\\cf(\\mu)+1,\\lambda)$-regular.  Theorem~\\ref{keisler} implies that $| \\prod 2^{\\cf(\\mu)} \/ U | \\geq 2^\\lambda$, a contradiction.   \n\n\nAssume for a contradiction that $\\alpha$ is such that $2^{<\\lambda} = 2^\\lambda$ for all regular $\\lambda \\geq \\alpha$.  Let $\\kappa = 2^\\alpha$.  We will show by induction the impossible conclusion that $2^\\beta = \\kappa$ for all $\\beta \\geq \\alpha$.  Suppose that this holds for all $\\gamma < \\beta$. If $\\beta$ is regular, $2^{<\\beta} = 2^\\beta$ by assumption, so $2^\\beta = \\kappa$.  If $\\beta$ is singular, then by \\cite[Theorem 5.16]{jechbook} $2^\\beta = (2^{<\\beta})^{\\cf(\\beta)} = \\kappa^{\\cf(\\beta)}$.  If $\\cf(\\beta) < \\gamma < \\beta$, then $\\kappa = 2^\\gamma = (2^\\gamma)^{\\cf(\\beta)} = \\kappa^{\\cf(\\beta)}$. \n\\end{proof}\n\n\\begin{corollary}\n\\label{incon1}\nIf $\\kappa$ is singular such that $2^{\\cf(\\kappa)} < \\kappa$, then there is no uniform, $\\kappa^+$-complete, $\\kappa^+$-dense ideal on $\\kappa^{+n}$ for $n \\geq 2$.\n\\end{corollary}\n\n\\begin{proof}Assume $I$ is a uniform, $\\kappa^+$-complete, $\\kappa^+$-dense ideal on $\\kappa^{+n}$ for some $n \\geq 2$.  Define $\\phi : \\p(\\kappa^+) \\to {\\p(\\kappa^{+n})\/I}$ by $X \\mapsto || \\kappa^+ \\in j(X) ||_{\\p(\\kappa^{+n})\/I}$.  Let $J = \\ker \\phi$.  $\\phi$ lifts to an embedding of $\\p(\\kappa^+)\/J$ into $\\p(\\kappa^{+n})\/I$.  Since $J$ is clearly normal and $\\kappa^{++}$-saturated, the embedding is regular, since for a maximal antichain $\\{ A_\\alpha : \\alpha < \\kappa^+ \\}$, $\\Vdash \\kappa^+ \\in j(\\nabla_{\\alpha < \\kappa^+} A_\\alpha)$, so it is forced that for some $\\alpha<\\kappa^+$, $\\phi(A_\\alpha)$ is in the generic filter.  Thus $J$ is a normal $\\kappa^+$-dense ideal on $\\kappa^+$.  We have $2^{<\\kappa^+} < 2^{\\kappa^+}$ by Corollary~\\ref{wg}, so $\\kappa$ cannot be singular such that $2^{\\cf(\\kappa)} < \\kappa$.   \n\\end{proof}\n\nThese methods can also be used to deduce more cardinal arithmetic consequences of dense ideals.  First we need a few more lemmas:\n\n\\begin{theorem}[Kunen-Prikry \\cite{kp}]\n\\label{kp}\nIf $\\kappa$ is regular and $U$ is a $(\\kappa^+,\\kappa^+)$-regular ultrafilter, then $U$ is $(\\kappa,\\kappa)$-regular.\n\\end{theorem}\n\n\\begin{lemma}\n\\label{linear}\nSuppose $( L, < )$ is a linear order such that for all $x \\in L$, \\break $| \\{ y \\in L : y < x \\} | \\leq \\kappa$.  Then $|L| \\leq \\kappa^+$.\n\\end{lemma}\n\n\\begin{corollary}\n\\label{wg2}\nSuppose there is a $\\kappa^+$-complete, $\\kappa^+$-dense ideal on $\\kappa^{+n}$, where $n \\geq 2$.  Then for $0 \\leq m \\leq n$, $2^{\\kappa^{+m}} = \\kappa^{+m+1}$.\n\\end{corollary}\n\n\\begin{proof}\nLet $I$ be such an ideal, and let $U \\supseteq I^*$ be given by Lemma~\\ref{huberich}.  By Lemma~\\ref{ultsize}, $|\\kappa^{\\kappa^{+n}} \/ U | \\leq 2^\\kappa$, which is $\\kappa^+$ by Corollary~\\ref{wg}.  Note that for any cardinal $\\mu$, any ultrafilter $V$ on a set $Z$, and any $g : Z \\to \\mu^+$,  $\\{ [f]_V : f <_V g \\}$ has cardinality at most $| \\mu^Z \/ V |$.   Thus, applying Lemma~\\ref{linear} inductively, we get that $| (\\kappa^{+m})^{\\kappa^{+n}} \/ U | \\leq \\kappa^{+m+1}$ for all $m < \\omega$.\n\n$U$ is $(\\kappa^{+n},\\kappa^{+n})$-regular, so by Theorem~\\ref{kp}, it is $(\\kappa^{+m},\\kappa^{+m})$-regular for $m \\leq n$.  Assume for induction that $2^{\\kappa^{+r}} = \\kappa^{+r+1}$ for $r < m \\leq n$; note the base case $m = 1$ holds.  Let $\\{ X_\\alpha : \\alpha < \\kappa^{+m} \\}$ witness $(\\kappa^{+m},\\kappa^{+m})$-regularity, and let $\\beta_z = \\ot(\\{ \\alpha : z \\in X_\\alpha \\})$.  By Theorem~\\ref{keisler} and the above observations, we have:\n\\[ 2^{\\kappa^{+m}} \\leq |\\prod 2^{\\beta_z} \/ U| \\leq |\\prod 2^{\\kappa^{+m-1}} \/ U| = |\\prod \\kappa^{+m} \/ U| \\leq \\kappa^{+m+1} \\mbox{. } \\] \\end{proof}\n\n\n\nWe note that if the hypothesis of Corollary~\\ref{wg2} is consistent, then no cardinal arithmetic above $\\kappa^{+n}$ can be deduced from it, since any forcing which adds no subsets of $\\kappa^{+n}$ will preserve the relevant properties of the ideal.\n\n\n\nBy combining this technique with the results of Section 2, we can answer the following, which was Open Question 16 from \\cite{foremanhandbook}:\n\n\n\\begin{question}[Foreman]\nIs it consistent that there is a uniform ultrafilter $U$ on $\\omega_3$ such that $\\omega^{\\omega_3} \/U$ has cardinality $\\omega_3$? Is it consistent that there is a uniform ultrafilter $U$ on $\\aleph_{\\omega+1}$ such that $\\omega^{\\aleph_{\\omega+1}} \/U$ has cardinality $\\aleph_{\\omega+1}$? Give a characterization of the possible cardinalities of ultrapowers.\n\\end{question}\n\n\\begin{theorem}Assume ZFC is consistent with a super-almost-huge cardinal.  Then it is consistent that every regular uncountable cardinal $\\kappa$ carries a uniform ultrafilter $U$ such that $|\\omega^\\kappa \/ U| = \\kappa$. \n\\end{theorem}\n\nThis follows from Section 2 and the next result.\n\n\\begin{lemma}Suppose $\\kappa = \\mu^+$, GCH holds at cardinals $\\geq \\mu$, and for all regular $\\lambda \\geq \\kappa$, there is a normal and fine, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$.  Then for every regular $\\lambda$, there is a uniform ultrafilter $U$ on $\\lambda$ such that $|\\mu^\\lambda \/ U| = \\lambda$.\n\\end{lemma}\n\n\\begin{proof}\nLet $I$ be a normal and fine, $\\kappa$-complete, $\\lambda$-dense ideal on $Z = \\p_\\kappa(\\lambda)$, where $\\kappa= \\mu^+$ and $\\lambda$ is regular.  Note that $|Z| = \\lambda$, and every $Y \\subseteq Z$ of size $<\\lambda$ is in $I$.   Let $U \\supseteq I^*$ be given by Lemma~\\ref{ultsize}, so that $| \\mu^Z \/ U | \\leq 2^{<\\lambda}$.  If $2^{<\\lambda} = \\lambda$, then $| \\mu^Z \/ U | \\leq \\lambda$.  Since $2^\\mu = \\kappa$ and any ultrafilter extending $I^*$ is $(\\kappa,\\lambda)$-regular, Theorem~\\ref{keisler} implies that $| \\kappa^Z \/ U | > \\lambda$, and Lemma~\\ref{linear} implies that $| \\kappa^Z \/ U | \\leq | \\mu^Z \/ U |^+$.  Thus $| \\mu^Z \/ U | = \\lambda$.   \n\\end{proof}\n\nThe following extra conclusion can be immediately deduced in the case of $\\mu < \\aleph_\\omega$ and $\\lambda = \\rho^+$, where $\\cf(\\rho) = \\omega$.  Suppose $\\mu = \\omega_n$.  Since $| \\omega_{n+1}^Z \/ U | > \\lambda$, we cannot have $| \\omega_m^Z \/ U | < \\rho$ for any $m$, since by Lemma~\\ref{linear}, we would have $| \\omega_r^Z \/ U | < \\rho$ for all $r < \\omega$.  Also, $U$ is $(\\omega,\\omega)$-regular, so Theorem~\\ref{keisler} implies that $|\\omega^Z \/U| \\geq |\\omega^Z \/U|^\\omega$.  Thus $| \\omega_m^Z \/ U | = \\lambda$ for all $m \\leq n$.\n\n\\section{Compatibility with square}\nSolovay~\\cite{solovay} showed that $\\square_\\delta$ fails when $\\delta \\geq \\kappa$ and $\\kappa$ is strongly compact.  In contrast, we will show $(\\forall \\delta \\geq \\kappa)\\square_\\delta$ is consistent with the kind of generically supercompact $\\kappa$ constructed in Section 2.  The key difference is that nontrivial forcings may be absorbed into the quotient algebras of the ideals in the generic case.\n\nWe start with a model given by Section 2, force $\\square$, and show that dense ideals still exist.  We will use the following variation on Foreman's duality theorem~\\cite{foremanduality}.\n\n\\begin{lemma}\n\\label{dualabsorb}\nSuppose $I$ is a precipitous ideal on $Z \\subseteq \\p(X)$ and $e : \\mathbb{P} \\to \\mathcal{B}(\\p(Z)\/I)$ is a regular embedding.  Suppose that for all generic $G \\subseteq \\mathcal{B}(\\p(Z)\/I)$, if $j : V \\to M \\subseteq V[G]$ is the associated embedding and $H = e^{-1}[G]$, there is a filter $\\hat{H} \\in V[G]$ that is $j(\\mathbb{P})$-generic over $M$ and such that $j[H] \\subseteq \\hat{H}$.  Then there is a $\\mathbb{P}$-name for a precipitous ideal $J$ on $Z$ such that $\\mathcal{B}( \\mathbb{P} * \\dot{\\p(Z)\/J}) \\cong \\mathcal{B}(\\p(Z)\/I)$.  Furthermore, $J$ has the same completeness and normality properties as $I$.\n\\end{lemma}\n\n\\begin{proof}\nLet $H$ be $\\mathbb{P}$-generic over $V$, and let $G$ be $\\mathcal{B}(\\mathcal{P}(Z)\/I)$-generic over $V$ with $e[H] \\subseteq G$.  Let $j : V \\to M$ be the generic ultrapower embedding, and let $\\hat{H}$ be as hypothesized.  Then $j$ is uniquely extended to $\\hat{j} : V[H] \\to M[\\hat{H}]$.  In $V[H]$, let $\\mathbb{Q} = \\mathcal{B}(\\mathcal{P}(Z)\/I) \/ e[H]$, and let $J = \\{ X \\subseteq Z : 1 \\Vdash_{\\mathbb{Q}} [\\id]_M \\notin \\hat{j}(X) \\}$.  Let $\\iota : \\mathbb{P} * \\dot{\\p(Z)\/J}$ be defined by $\\iota(p,\\dot{X}) = e(p) \\wedge || [\\id] \\in \\hat{j}(\\dot{X}) ||$.\n\nIt is easy to see that $\\iota$ is order and antichain preserving.  To see that the range of $\\iota$ is dense, let $A \\in I^+$ be arbitrary.  Take $G$ with $A \\in G$, so that $[\\id] \\in j_G(A)$.  If $H = e^{-1}[G]$, then some $p \\in H$ forces $A \\in J^+$.  Let $\\dot{X}$ be a $\\mathbb{P}$-name such that $p \\Vdash \\dot{X} = \\check{A}$ and $q \\Vdash \\dot{X} = Z$ whenever $q \\perp p$; this makes sure $\\dot{X}$ is forced to be in $J^+$.  Then $\\iota(p,\\dot{X}) \\leq A$, since any $G$ with $e(p) \\wedge || [\\id] \\in \\hat{j}(\\dot{X}) || \\in G$ must have $[\\id] \\in j(A)$ and thus $A \\in G$.\n\nSuppose $H * \\bar{G} \\subseteq  \\mathbb{P} * \\p(Z) \/ \\bar{I}$ is generic, and let $G * \\hat{H} = \\iota[H * \\bar{G}]$.  For $A \\in J^+$, $A \\in \\bar{G}$ iff $[\\id]_M \\in \\hat{j}(A)$.  If $i : V[H] \\to N = V[H]^Z \/ \\bar{G}$ is the canonical ultrapower embedding, then there is an elementary embedding $k : N \\to M[\\hat{H}]$ given by $k([f]_N) = \\hat{j}(f)([\\id]_M)$, and $\\hat{j} = k \\circ i$.  Thus $N$ is well-founded, so $J$ is precipitous.  If $f : Z \\to \\ord$ is a function in $V$, then $k([f]_N) = j(f)([\\id]_M) = [f]_M$.  Thus $k$ is surjective on ordinals, so it must be the identity, and $N = M[\\hat{H}]$.  Since $i = \\hat{j}$ and $\\hat{j}$ extends $j$, $i$ and $j$ have the same critical point, so the completeness of $J$ is the same as that of $I$.  Finally, since $[\\id]_N = [\\id]_M$, $I$ is normal in $V$ iff $J$ is normal in $V[H]$, because $j \\restriction X = \\hat{j} \\restriction X$, and normality is equivalent to $[\\id] = j[X]$.\n\\end{proof}\n\n\n\nFor a cardinal $\\delta$, let $\\mathbb{S}_\\delta$ be the collection of bounded approximations to a $\\square_\\delta$ sequence.  That is, a condition is a sequence $\\langle C_\\alpha : \\alpha \\in \\eta \\cap \\mathrm{Lim} \\rangle$ such that $\\eta < \\delta^+$ is a successor ordinal, each $C_\\alpha$ is a club subset of $\\alpha$ of order type $\\leq \\delta$, and whenever $\\beta$ is a limit point of $C_\\alpha$, $C_\\alpha \\cap \\beta = C_\\beta$.  For proof of the following lemma, we refer the reader to \\cite{sssr}.\n\n\\begin{lemma}For every cardinal $\\delta$, $\\mathbb{S}_\\delta$ is countably closed and $(\\delta+1)$-strategically closed and adds a $\\square_\\delta$ sequence $\\langle C_\\alpha : \\alpha \\in \\delta^+ \\cap \\mathrm{Lim} \\rangle = \\bigcup G$, where $G \\subseteq \\mathbb{S}_\\delta$ is the generic filter.  For every regular $\\lambda \\leq \\delta$, there is a $\\mathbb{S}_\\delta$-name for a ``threading'' partial order $\\mathbb{T}_\\delta^\\lambda$ that adds a club $C \\subseteq (\\delta^+)^V$ of order type $\\lambda$ and such that whenever $\\alpha$ is a limit point of $C$, $C \\cap \\alpha = C_\\alpha$.  Furthermore, $\\mathbb{S}_\\delta * \\mathbb{T}_\\delta^\\lambda$ has a $\\lambda$-closed dense subset of size $2^\\delta$.\n\\end{lemma}\n\n\\begin{theorem}\nSuppose $\\kappa$ is super-almost-huge and $\\mu < \\kappa$ is regular.  Then there is a $\\mu$-distributive forcing extension in which $\\kappa = \\mu^+$, $\\square_\\lambda$ holds for all cardinals $\\lambda \\geq \\kappa$, and for all regular $\\lambda \\geq \\kappa$ there is a normal, fine, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$.\n\\end{theorem}\n\n\\begin{proof}\nBy Section 2, we may pass to a $\\mu$-distributive forcing extension in which $\\kappa = \\mu^+$ and for all regular $\\lambda \\geq \\kappa$ there is a normal, fine, $\\kappa$-complete, $\\lambda$-dense ideal on $\\p_\\kappa(\\lambda)$, and GCH holds above $\\mu$.  Over this model, force with $\\mathbb{P}$, the Easton support product of $\\mathbb{S}_\\lambda$ where $\\lambda$ ranges over all cardinals $\\geq \\kappa$.  For every cardinal $\\lambda$, $\\mathbb{P}$ naturally factors into $\\mathbb{P}_{<\\lambda} \\times \\mathbb{P}_{\\geq \\lambda}$.   Note that if $\\lambda \\geq \\kappa$, $\\mathbb{P}_{\\geq \\lambda}$ is $(\\lambda + 1)$-strategically closed.\n\nFirst we show that for each regular $\\lambda \\geq \\kappa$, $\\mathbb{P}_{\\geq \\lambda}$ is $\\lambda^+$-distributive in $V^{\\mathbb{P}_{<\\lambda}}$.  Suppose that $H_0 \\times H_1$ is $(\\mathbb{P}_{<\\lambda} \\times \\mathbb{P}_{\\geq \\lambda})$-generic, and $f : \\lambda \\to \\ord$ is in $V[H_0][H_1]$.  Then in $V[H_1]$, there is a $\\mathbb{P}_{<\\lambda}$-name $\\tau$ for $f$.  By GCH and the fact that we take Easton support, $|\\mathbb{P}_{<\\lambda}| = \\lambda$, so it is $\\lambda^+$-c.c.\\ in $V[H_1]$.  Thus $\\tau$ may be assumed to be a subset of $V$ of size $\\lambda$.  By the strategic closure of $\\mathbb{P}_{\\geq \\lambda}$, $\\tau \\in V$.  Thus $f = \\tau^{H_0} \\in V[H_0]$, establishing the claim.\n\nNext we show that $\\mathbb{P}$ preserves all regular cardinals.  First note that since $\\mathbb{P}$ is $(\\kappa+1)$-strategically closed, $\\mathbb{P}$ cannot change the cofinality of any regular $\\delta$ to some $\\lambda \\leq \\kappa$.  If $\\mathbb{P}$ does not preserve regular cardinals, then in some generic extension $V[G]$, there are $\\lambda < \\delta$ which are regular in $V$ with $\\kappa < \\lambda$, such that $V[G] \\models \\cf(\\delta) = \\lambda$.  Let $H = H_0 \\times H_1$, where $H_0 \\subseteq \\mathbb{P}_{<\\lambda}$ and $H_1 \\subseteq \\mathbb{P}_{\\geq \\lambda}$.  By the $\\lambda^+$-c.c.\\ of $\\mathbb{P}_{<\\lambda}$, $V[H_0] \\models \\cf(\\delta) > \\lambda$, and by the $\\lambda^+$-distributivity of $\\mathbb{P}_{\\geq \\lambda}$ in $V[H_0]$, $V[H] \\models \\cf(\\delta) > \\lambda$, a contradiction.  Since a square sequence is upwards absolute to models with the same cardinals and $\\mathbb{S}_\\lambda$ regularly embeds into $\\mathbb{P}$ for all $\\lambda \\geq \\kappa$, $\\mathbb{P}$ forces $(\\forall \\lambda \\geq \\kappa) \\square_\\lambda$.\n\n\nFor each regular $\\lambda \\geq \\kappa$, let $Z_\\lambda = \\p_\\kappa(\\lambda)$.  We want to show that in $V^{\\mathbb{P}}$, for each regular $\\lambda \\geq \\kappa$, there is a normal, fine, $\\lambda$-dense ideal on $Z_\\lambda$.  It suffices to show that such an ideal exists in $V^{\\mathbb{P}_{<\\lambda}}$, since $\\mathbb{P}_{\\geq \\lambda}$ adds no subsets of $\\lambda$, and $|Z_\\lambda | = \\lambda$.  First note that by the strategic closure of $\\mathbb{P}$, the dense ideal on $\\kappa$ is unaffected.\n\nLet $\\mathbb{Q}$ be the Easton support product of $\\mathbb{S}_\\lambda * \\mathbb{T}_\\lambda^\\mu$, where $\\lambda$ ranges over all cardinals. There is a coordinate-wise regular embedding of $\\mathbb{P}$ into $\\mathbb{Q}$.  When $\\lambda$ is regular, $\\mathbb{Q}_{<\\lambda}$ has a dense $\\mu$-closed subset of size $\\lambda$.  Hence it regularly embeds into $\\mathcal{B}(\\col(\\mu,\\lambda))$.  The dense ideal $I_\\lambda$ on $Z_\\lambda$ in $V$ has quotient algebra isomorphic to $\\mathcal{B}(\\mathbb{R} \\times \\col(\\mu,\\lambda))$ for some small $\\mathbb{R}$, and so $\\mathbb{Q}_{<\\lambda}$ regularly embeds into this forcing.\n\nIf $G \\subseteq \\p(Z_\\lambda)\/I_\\lambda$ is generic, let $H$ be the induced generic for $\\mathbb{P}_{<\\lambda}$, and let $j : V \\to M \\subseteq V[G]$ be the ultrapower embedding.  Recall that $\\crit(j) = \\kappa$, $j(\\kappa) = \\lambda^+$, $\\lambda^{++}$ is a fixed point of $j$, and $j[\\lambda] \\in M$.  First note that $j[\\lambda] \\setminus j(\\kappa)$ is an Easton set in $M$.  If $j(\\kappa) \\leq \\delta \\leq j(\\lambda)$ and $\\delta$ is regular in $M$, then since $\\ot(j[\\lambda] \\cap \\delta) \\leq \\lambda < \\delta$, $\\sup(j[\\lambda] \\cap \\delta) < \\delta$.\n\nFor each cardinal $\\delta$ such that $\\kappa \\leq \\delta < \\lambda$, let $\\langle C_\\alpha^\\delta : \\alpha < \\delta^+ \\rangle$ be the $\\square_\\delta$ sequence and let $t_\\delta$ be the ``thread'' of order type $\\mu$, both given by $H \\restriction ( \\mathbb{S}_\\delta * \\mathbb{T}^\\mu_\\delta)$.  By the $\\mu$-distributivity of $\\mathbb{S}_\\delta * \\mathbb{T}_\\delta^\\mu$, all initial segments of $t_\\delta$ are in $V$, and since they are small, $j(t_\\delta \\cap \\alpha) = j[t_\\delta \\cap \\alpha]$ for $\\alpha < \\delta^+$, and $j$ is continuous at all limit points of $t_\\delta$.  Let $\\gamma_\\delta = \\sup(j[\\delta^+]) < j(\\delta^+)$, and in $M$ consider $m_\\delta = \\bigcup_{\\alpha < \\delta^+} j(\\langle C_\\alpha^\\delta : \\beta < \\alpha \\rangle) \\cup \\{ (\\gamma_\\delta, j[t_\\delta] ) \\} $.  Each $m_\\delta$ is a condition in $(\\mathbb{S}_{j(\\delta)})^M$, and the sequence $m = \\langle m_\\delta : \\delta \\in j[\\lambda] \\setminus \\kappa \\cap \\card^M \\rangle$ is a condition in $(\\mathbb{P}_{<j(\\lambda)})^M$.\n\n$M$ thinks $\\mathbb{P}_{<j(\\lambda)}$ is $j(\\kappa)$-strategically closed, and this is true in $V[G]$ as well since these models share the same $\\lambda$-sequences, and $j(\\kappa) = \\lambda^+$ in $V[G]$.  Since $j(\\lambda^+) < (\\lambda^{++})^V$, $\\p(\\mathbb{P}_{<j(\\lambda)})^M$ has cardinality $\\lambda^+$ in $V[G]$.  Thus we may use the winning strategy to build a filter $\\hat{H} \\subseteq (\\mathbb{P}_{<j(\\lambda)})^M$ that is generic over $M$, with $m \\in \\hat{H}$.  Since $m$ is a lower bound to $j(p)$ for all $p \\in H$, we have $j[H] \\subseteq \\hat{H}$.\n\nTherefore, the hypotheses of Lemma~\\ref{dualabsorb} are satisfied with respect to $I_\\lambda$, $Z_\\lambda$, and $\\mathbb{P}_{<\\lambda}$.  Thus we have a $\\mathbb{P}_{<\\lambda}$-name for a normal and fine ideal $J_\\lambda$ on $Z_\\lambda$ such that $\\mathcal{B}(\\mathbb{P}_{<\\lambda} * \\dot{\\p(Z_\\lambda)\/J_\\lambda)} \\cong \\mathcal{B}(\\p(Z_\\lambda)\/I_\\lambda)$.  Hence $J_\\lambda$ is $\\lambda$-dense in $V[H]$.\n\\end{proof}\n\n\nWe note that in the case $\\kappa = \\omega_1$, $\\p(Z_\\lambda)\/J_\\lambda \\cong \\mathcal{B}(\\col(\\omega,\\lambda))$ for all regular $\\lambda$.  But for higher cardinals, Proposition~\\ref{Z-refl} shows the quotient algebras must differ from those given by Theorem~\\ref{maindense}.  The crux is that the ``threading'' forcings are left over as regular suborders.\n\nIt is not possible to improve this result to get the consistency of, for example, ``For all cardinals $\\lambda \\geq \\omega_1$, $\\square_\\lambda$ holds and there is a normal, fine, $\\lambda$-dense ideal on $\\p_{\\omega_1}(\\lambda)$.''  Burke and Matsubara~\\cite{BM} showed that if $\\cf(\\lambda) < \\kappa$ and there is normal, fine, $\\kappa$-complete, $\\lambda^+$-saturated ideal on $\\p_\\kappa(\\lambda)$, then every stationary subset of $\\lambda^+ \\cap \\cof({<}\\kappa)$ reflects.\n\n\n\\bibliographystyle{amsplain.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Plan of the talk}\n\\begin{itemize}\n\\item\nSummary of what happend since 1968\\\\\n(when I first met Ivan Todorov)\n\\item\nThe relation between representation theory and harmonic analysis \nfor groups $SO(D,2)$ and $SO(D+1,1)$\\\\ $\\   $ i.e. a group theoretical\n version of the familiar relation between field theory in Minkowski and Euklidean space\n\\item recent developments: Are we close to the {\\em analytic} solution of the 3-dimensional \n(conformal) Ising model?\n\\end{itemize}\nThe second point is important: positions and residues of poles in the \n partial wave amplitude $g(\\chi=[l,\\delta])$ which is obtained by partial wave analysis of the {\\bf Euklidean} 4-point Green functions are restricted by the requirement that they are associated with unitary positive energy ray representations of\n the {\\bf Minkowskian} conformal group $SO(D,2)$. \n\nAnd it can also be a source of interesting mathematics.\n\\section{What happened since 1968?\n}\nI met Ivan Todorov in 1968 at the ICTP Trieste.\nI had been invited by Abdus Salam, who had studied my PhD-thesis on dilatation- and conformal symmetry \\cite{Mack0} of February 1967.The collaboration with Abdus Salam resulted in a joined paper \\cite{MackSalam}.  Abdus Salam also pointed out Dirac's early work on conformal symmetry \\cite{Dirac} which introduced the conformal covariant description of ($D=4$-dimensional) space time using lightlike $D+2$ - vectors $\\xi$\n\nIvan had worked at the  research center in Dubna in the USSR, which also hosted scientist from Bulgaria and other states of the Warsaw pact. It was Ivans\n first visit to the west.\nWe began our collaboration on conformal symmetry with a paper \\cite{MackTodorov}\n on the ladder reprentation of the conformal group $SO(4,2)$. This is a unitary positive energy representation \n which also serves to label the states of the $H$-atom.\n\nWe agreed to meet again at the Vienna conference on High Energy physics that year. Shortly before the conference, the Soviet Union's troops invaded \nCzechoslovakia, and I got worried whether Ivan would come. But never before \nhad there been so many scientists from eastern Europe at a conference in the\n west.\n I met Ivan and our paper got finished and published.\n\nThis was the beginning of a collaboration lasting 25 years. Our next paper demonstrated freedom of Conformal invariant Green functions from\n ultraviolet divergences \\cite{MackTodorovUV}\n\nWilsons 1969 paper on operator product expansions (OPE) \\cite{Wilson} cited me honorably and created a burst of activities:\\\\\n Polyakov \\cite{Polyakov} (1970),  Migdal \\cite{Migdal}(1971), \nd'Eramo, Parisi and Peliti \\cite{ParisiPeliti, DEramoParisiPeliti} (1971), \nSymanzik \\cite{Symanzik, Symanzik1, MackSymanzik} (1971), Ferrara, Gatto and Grillo, \\cite{FerraraGattoGrillo,FerraraGrilloGatto} (1972), same and Parisi \\cite{FerraraGattoGrilloParisi} (1972), Tonin et al.\\cite{BonoraCiccarielloSartoriTonin}, Brezin, Itzykson, Zuber  and Zinn Justin.\n\nI began working on a group theoretical approach to conformal field theory in \n1972. It used group representation theory and harmonic analysis on groups in the spirit of Eugen Wigner \\cite{Mack1,Mack2,Mack3}\n\nThe work was completed by Dobrev, Petkova, Petrova and Todorov \\cite{DobrevPetkovaPetrovaTodorov}\nand published jointly as a book \\cite{DobrevMackPetkovaPetrovaTodorov}:\n{\\em Harmonic Analysis\non the $n$-dimensional Lorentz Group and its Application to Conformal Quantum Field Theory}, Lecture Notes im Physics {\\bf 63}, Springer 1977\n\nI had met Eugen Wigner at the Universiy of Princeton. My PhD adviser Hans \nKastrup had been invited by Wigner who had studied Kastrups papers on\n conformal symmetry \\cite{KastrupAP,KastrupNPB} This visit resulted in further papers on the subject \\cite{KastrupPR}. Hans Kastrup  paved the way to my visit. \nThe visit was financed by the\n Studienstiftung des Deutschen Volkes.\n\nTo prepare, I had studied Wigners book on group theory in quantum mechanics. \nHe became my hero and influences  my work until now\n\nEugen Wigner and his wife invited Arthur Jaffe and me to a long weekend in \ntheir winter home in Vermont. And I met him repeatedly again later on. At the conference of Nobel Laureates in Trieste which was organized by Abdus Salam. \nAnd during later visits in Princeton. The last encounter was at the conference\n in Baltimore which celebrated Wigners 90th birthday. Arthur Jaffe was also \nthere. A greeting message by Abdus Salam was presented  in which he explained how much he had learned from Wigner. \n\nIn this talk I will only speak about conformal field theory in more than 2 dimensions. A major breakthrough in 2-dimensional conformal field theory occurred in 1984 with the paper \\cite{BelavinPolyakovZamolodchikov} of Belavin, Polyakov and Zamolodchikov on \n{\\em Infinite conformal symmetry in two-dimensional quantum field theory}. L\\\"uscher \nand myself had discovered before \\cite{mackUnpub} that the 2-dimensional traceless stress \nenergy tensor is a Lie field; they also found the bound $c\\geq 1\/2$ on the coefficient of the central term. This paper remained unpublished, but I \npresented it at a seminar in Copenhagen which Polyakov attended.\n\n\\begin{figure}[t!]\n\\psset{unit=0.7cm}\n\n\\psset{dotsize=0.1 0}\n\\psset{linewidth=.05}\n\\pspicture(0,0)(10,15.2)\n\\psline(0,1)(0,14)\n\\psline(7.4,1)(7.4,14)\n\\rput(4.7,9.2){$p_\\infty$}\n\\psdot(2.5,4) \n\\psdot(2.5,11.7)\n\\psdot(4.7,8.7)\n\\pscurve(2.5,4)(0,6.5)(4.5,8.66)(4.7,8.7)(4.9,8.74)(7.4,10)(2.5,11.7)\n\\pscurve(2.5,4)(7.4,6)(4.7,8.7)(1,9.4)(0.5,9.55)(0,10)(2.5,11.7\n\\psellipse(3.7,1)(3.7,1)\n\\psellipse(3.7,14)(3.7,1)\n\\endpspicture\n\\caption{The universal covering $\\tilde{M}$ of conformally compactified Minkowski space time is a tube with  point(s) $p_{\\infty}$ at spatial infinity of Minkowski space located on the back of the tube. The boundary of Minkowski space passes through it. The two corner points of the boundary of Minkowski space are on the front of the tube.\n \\label{fig:spaceTime}\n}\n\\end{figure}\n\n\n\n\n\n\\section{The space time manifold $\\tilde{M}$}\nConformal field theory satisfies locality \\cite{LuescherMack}.\n\nIt lives on the infinite sheeted covering $\\tilde{M}$ - a tube (Figure 1) -  of Minkowski space which admits a conformal invariant causal structure. Minkowski space is imbedded into $\\tilde{M}$ and is characterized by its point(s) at spatial infinity\n$p_\\infty$.\n\n\\subsection{Lagrangean approach}\nThere exists a coupled set of integral equations for all the n-point functions \nof $\\phi^3$-theory \\cite{Symanzik}. They can also be used for $\\phi^4$-theory by considering it as a theory with the fundamental fields $\\phi(x)$ and $:\\phi^2:(x)$.\nThe equations for propagator and 3-vertex require renormalization; \nrenormalized versions have been derived in the 60's \\cite{Symanzik}. The 4-point function furnishes also the Bethe Salpeter kernel.\n\\subsection{Bethe-Salpeter equations $\\mapsto$ define kernels}\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\n\\def\\OvalShortFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\n\n\\def\\DoubleCircleUp{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.49)(-.3,1.0)\n\\psline(.5,.49)(.3,1.0)\n}\n\n\n\\def\\DoubleCircleFeet {\n\\psline(-1.,-1.)(-0.75,-0.5)\n\\psline(1.,-1.)(0.75,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\\pspicture(-1,0)(15,7)\n\\rput(0,3.7){\n\\rput(0,1.5){\\BSKern} \n\\rput(0,1.5){\\OvalShortFeet}\n\\rput(0,1.5){\\OvalUpFeet}\n\\rput(1.3,1.5){{\\large 2i}}\n\\rput(2,1.5){=}\n\\rput(3.5,1.5){\\BSKern} \n\\rput(3.5,1.5){\\OvalShortFeet}\n\\rput(3.5,1.5){\\OvalUpFeet}\n\\rput(4.8,1.5){{\\large 1i}}\n\\rput(5.7,1.5){\\ -  $\\frac 1 2$}\n\\rput(7.5,1.){\\BSKern} \n\\rput(7.5,1.){\\OvalShortTwoFeet}\n\\rput(7.5,2.5){\\BSKern}\n\\rput(7.5,2.5){\\large B} \n\\rput(7.5,2.5){\\OvalUpFeet}\n\\rput(8.8,1.){{\\large 1i}}\n\\psline(7.0,1.5)(7.0,2.0)\n\\psline(8.0,1.5)(8.0,2.0)\n\\rput(9.7,1.5){-\\ $\\sum$}\n\\rput(11.5,2.5){\\BSKern}\n\\rput(11.5,2.5){\\OvalUpFeet}\n\\rput(11.5,2.5){\\large B}\n\\rput(11.5,1.){\\DoubleCircleUp}\n\\rput(11.5,1.){\\DoubleCircleFeet}\n}\n\n\\rput(0,0){\n\\rput(0,1.5){\\BSKern} \n\\rput(0,1.5){\\OvalShortTwoFeet}\n\\rput(0,1.5){\\OvalUpFeet}\n\\rput(0,1.5){{\\large B}}\n\\rput(2,1.5){=}\n\\rput(3.5,1.5){\\BSKern} \n\\rput(3.5,1.5){\\OvalShortTwoFeet}\n\\rput(3.5,1.5){\\OvalUpFeet}\n\\rput(4.8,1.5){{\\large 1i}}\n\\rput(5.7,1.5) {\\ -  $\\frac 1 2$}\n\\rput(7.5,1.){\\BSKern} \n\\rput(7.5,1.){\\OvalShortTwoFeet}\n\\rput(7.5,2.5){\\BSKern}\n\\rput(7.5,2.5){\\large B} \n\\rput(7.5,2.5){\\OvalUpFeet}\n\\rput(8.8,1.){{\\large 1i}}\n\\psline(7.0,1.5)(7.0,2.0)\n\\psline(8.0,1.5)(8.0,2.0)\n}\n\n\n\n\\endpspicture\n\n\n\n\\noindent The 1-particle irreducible graphs are obtained by subtracting Born terms.\n\\subsection{Dynamical equations}\nfor $\\phi^3$-theory; can be generalized to $\\phi^4$-theory by considering fields $\\phi$ and\n$:\\phi^2$:\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\n\\def\\Ellipse{\n\\psellipse(0,0)(1.4,0.7)\n}\n\n\\def\\LineUp {\n\\psline(0,0)(0,1.5)\n}\n\\def\\Circle {\n\\pscircle(0,0){.7}\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\CircleUpAmp {\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,.7)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\\def\\OvalShortFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\\def\\DoubleCircleFeet {\n\\psline(-1.,-1.)(-0.75,-0.5)\n\\psline(1.,-1.)(0.75,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\n\\def\\OvalShortInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\\def\\OvalShortIPIInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n\\psline[linestyle=dotted](-1.,0)(1.,0) \n}\n\n\\def\\OvalShortIPICircle {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\pscircle(0,1.5){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n\\psline(0,2.0)(0,2.2) \n}\n\n\n\\def\\DoubleCircleInt{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\n\\def\\DoubleCircleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\n\\pscircle(0,1.5){.5}\n\\psline(0,2.0)(0,2.2)\n}\n\n\n\\def\\DoubleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n}\n\n\n\n\\pspicture(-1,0)(13,3.8)\n\\rput(0,1.5){\\CircleUpAmp}\n\\rput(0,1.5){\\CircleFeet}\n\\rput(-1,.1){$x_1$}\n\\rput(1,.1){$x_n$}\n\\rput(1.5,1.5){=}\n\\rput(2.2,1.5){$\\quad \\frac 1 2 $}\n\\rput(4.5,1.5){\\OvalShortIPICircle}\n\\rput(5.8,1.5){$ 2i $}\n\\rput(4.5,1.5){\\OvalShortFeet}\n\\rput(7,1.5){$\\quad + \\sum$}\n\\rput(9.5,1.5){\\DoubleCircleCircle}\n\\rput(9.5,1.5){\\DoubleCircleFeet}\n\n\n\\endpspicture\n\n\n\n The  equations for $n\\geq3$ can be solved iteratively by skeleton graph \nexpansions, producing also skeleton graph expansions for the Bethe Salpeter\n kernel. After that it remains to solve the renormalized Schwinger Dyson \nequations for 3-vertex and propagator:\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\n\\def\\Ellipse{\n\\psellipse(0,0)(1.4,0.7)\n}\n\n\\def\\LineUp {\n\\psline(0,0)(0,1.5)\n}\n\\def\\Circle {\n\\pscircle(0,0){.7}\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n\\rput(1,0.5){${}^\\prime$}\n}\n\\def\\CircleUpAmp {\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,.7)\n\\rput(1,0.5){${}^\\prime$}\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\\def\\CircleTwoFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShortFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n\\rput(0,0){{\\large B}}\n}\n\\def\\DoubleCircleFeet {\n\\psline(-1.,-1.)(-0.75,-0.5)\n\\psline(1.,-1.)(0.75,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\n\\def\\OvalShortInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\\def\\OvalShortIPIInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n\\psline[linestyle=dotted](-1.,0)(1.,0) \n}\n\n\\def\\OvalShortIPICircle {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\pscircle(0,1.5){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n\\psline(0,2.0)(0,2.2) \n}\n\n\n\\def\\DoubleCircleInt{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\n\\def\\DoubleCircleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\n\\pscircle(0,1.5){.5}\n\\psline(0,2.0)(0,2.2)\n}\n\n\n\\def\\DoubleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n}\n\n\n\\pspicture(-1.5,1)(13,3.7)\n\\rput(0,1.5){\\CircleUpAmp}\n\\rput(0,1.5){\\CircleTwoFeet}\n\\rput(1.5,1.5){=}\n\\rput(2.2,1.5){$\\quad \\frac 1 2 $}\n\\rput(4.5,1.5){\\OvalShortIPICircle}\n\\rput(5.5,3.5){${}^\\prime$}\n\\rput(4.5,1.5){\\BSKern}\n\\rput(4.5,1.5){\\OvalShortFeet}\n\n\\endpspicture\n\n\n\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\n\\def\\Ellipse{\n\\psellipse(0,0)(1.4,0.7)\n}\n\n\\def\\LineUp {\n\\psline(0,0)(0,1.5)\n}\n\\def\\Circle {\n\\pscircle(0,0){.7}\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\BSKernArrow {\n\\psline[linewidth=3pt]{->}(.45,-.5)(.45,.5) \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\CircleUpAmp {\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,.7)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\\def\\OvalShortFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\\def\\DoubleCircleFeet {\n\\psline(-1.,-1.)(-0.75,-0.5)\n\\psline(1.,-1.)(0.75,-0.5)\n\\psline[linewidth=.1,linestyle=dotted,dotsep=.3](-.5,-1)(.5,-1)\n}\n\n\n\\def\\OvalShortInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\\def\\OvalShortIPIInt{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n\\psline[linestyle=dotted](-1.,0)(1.,0) \n}\n\n\\def\\OvalShortIPICircle {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\pscircle(0,1.5){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n\\psline(0,2.0)(0,2.2) \n}\n\n\n\\def\\DoubleCircleInt{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(0,1.5)\n\\psline(.5,.5)(0,1.5)\n}\n\n\\def\\DoubleCircleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n\\psline(-.5,.5)(-.2,1.1)\n\\psline(.5,.5)(.2,1.1)\n\n\\pscircle(0,1.5){.5}\n\\psline(0,2.0)(0,2.2)\n}\n\n\n\\def\\DoubleCircle{\n\\pscircle(-.75,0){.5}\n\\pscircle(.75,0){.5}\n}\n\n\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\n\\def\\OvalShortTwoShortFeet {\n\\psline(-0.6,-0.6)(-0.5,-0.5)\n\\psline(.6,-0.6)(0.5,-0.5)\n}\n\n\n\\def\\OvalShortOneShortFoot {\n\\psline(0,-0.6)(0,-0.5)\n}\n\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\n\\def\\OvalUpShortFeet {\n\\psline(-.6,.6)(-.5,.5)\n\\psline(.6,.6)(.5,.5)\n}\n \\def\\OvalUpShortFoot {\n\\psline(0,.6)(0,.5)\n}\n\n\n\n\\pspicture(-1,-0.5)(12,5)\n\\rput(1.2,0){\n \\psline[linewidth=3pt,linecolor=black]{->}(-1.4,3.3)(-.7,3.3)\n \\rput(1.5,3.3){= \\ $x^\\mu - y^\\mu$}\n \\rput(-.7,2.75){$y$}\n \\rput(-1.4,2.8){$x$}\n}\n \\rput(0.,1.5){0}\n \\rput(1.5,1.5){ = \\ \\ $\\frac 1 2 $}\n \\rput(-3.5,-0.25){\n\\rput(7.5,1.){\\OvalShort} \n\\rput(7.5,1.){\\OvalShortOneShortFoot}\n\\rput(7.5,2.5){\\OvalShort}\n\\rput(7.5,2.5){\\OvalUpShortFoot}\n\\psline(7.0,1.5)(7.0,2.0)\n\\psline[linewidth=3pt]{->}(8.0,1.5)(8.0,2.0)\n}\n\\rput(5.7,1.5){+ \\ $\\frac 1 4$\n\\rput(0,-1.){\n\\rput(7.5,1.){\\OvalShort} \n\\rput(7.5,1.){\\OvalShortOneShortFoot}\n\\rput(7.5,2.5){\\BSKernArrow}\n\\rput(7.5,2.5){\\large B}\n\n\\psline(7.0,1.5)(7.0,2.0)\n\\psline(8.0,1.5)(8.0,2.0)\n\\rput(7.5,4){\\OvalShort} \n\\rput(7.5,4){\\OvalUpShortFoot} \n\\psline(7.0,3.0)(7.0,3.5)\n\\psline(8.0,3.0)(8.0,3.5)\n\n}\n\\endpspicture\n\nThe equation for the inverse propagator is actually equivalent to an equation for the 3-point function of the stress enegy tensor \\cite{MackSymanzik}.\n\\subsection{Skeleton graph expansions}\nThe equations for $n\\geq 4$ point functions can be solved iteratively, producing skeleton graph expansions for Green functions and also for the Bethe Salpeter kernel \\cite{MackSymanzik}:\n\\psset{unit=.8cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\BSKern{\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n\\rput(0,0.05){$B$}\n}\n\n\\def\\BSKernWithFeet{\n  \\rput(0,0){\n  \\BSKern\n  \\OvalUpFeet \n  \\OvalShortTwoFeet \n }\n}\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\PartialWaveFeetDown{\n\\rput(0,0){\\CGKern}\n\\rput(2.2,0){\\CGKern}\n\\psline(.5,0)(1.7,0)\n}\n\\def\\CGKern{\n\\pscircle(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\Vertex{\n\\pscircle(0,0){.5}\n}\n\\def\\BornTerm{\n\\rput(0,0){\\Vertex}\n\\rput(2.2,0){\\Vertex}\n\\psline(.5,0)(1.7,0)\n\\psline(-.3,-.3)(-1.,-1.\n\\psline(2.5,-.3)(3.2,-1.\n\\psline(-.3,.3)(-1.,1)  \n\\psline(2.5,.3)(3.2,1.).\n}\n\\def\\BornTermDownFeet\n{\n\\rput(0,0){\\Vertex}\n\\rput(2.2,0){\\Vertex}\n\\psline(.5,0)(1.7,0)\n\\psline(-.3,-.3)(-1.,-1.\n\\psline(2.5,-.3)(3.2,-1.\n}\n\\def\\BornTermCrossed{\n\\BornTermDownFeet\n\\psline(0.3,0.3)(3.2,1.)\n\\psline(1.9,0.3)(-1.,1)\n}\n\\def\\SecondGraph{\n\\rput(0,0){\\Vertex}\n\\rput(2.2,0){\\Vertex}\n\\rput(0,1.5){\\Vertex}\n\\rput(2.2,1.5){\\Vertex}\n\\psline(0,0.4)(0,1.1)\n\n\\psline(2.2,.4)(2.2,1.1\n\\psline(0.3,0.3)(1.9,1.2\n\\psline(1.9,0.3)(0.3,1.2\n\\psline(-.3,-.3)(-1.,-1.)\n\\psline(2.5,-.3)(3.2,-1.)\n\\psline(-.3,1.8)(-1.,2.5)\n\\psline(2.5,1.8)(3.2,2.5)\n}\n\n\n\\pspicture(-1.2,-1)(11,3.5)\n\\rput(0,1){\n\\rput(0,.0){\\BSKernWithFeet}\n\\rput(2,0){$=$}\n\\rput(3.3,0){\\BornTerm}\n\\rput(6.6,0){$+$}\n\\rput(8.0,0){\\BornTermCrossed}\n\\rput(11.3,0){$+$}\n\\rput(12.5,0){\\SecondGraph}\n\\rput(16.4,0){$+ ...$}\n}\n\\endpspicture\n\nIn $\\phi^4$-theory, Born terms involve $:\\phi^2:$ propagators\n\\subsection{$\\epsilon$-expansions}\nIn $D=6+\\epsilon$ dimensions for $\\varphi^3$-theory (and in $D=4-\\epsilon$ dimensions for $\\phi^4$-theory), the dressed vertices are of order $\\epsilon$. Therefore only a finite number of skeleton graphs contribute to any order in $\\epsilon$. Inserting into the equations for dressed vertex and propagator, these can be solved by power series expansion in $\\epsilon$. For $\\phi^3$-theory in \n$6+\\epsilon$ dimensions one finds to leading orders \\cite{Mack3}: The \nfundamental field $\\phi$ has dimension\n$$ d= \\frac{D}{2} -1 + \\Delta,\\qquad \\qquad \\Delta= \\frac {1}{18}\\epsilon + ...$$ \nand a trajectory of traceless symmetric tensor fields of even rank $s\\geq 2$ with  dimension $d_s= D-2 + s +\\sigma_s$,\n$$ \\half \\sigma_s= \\left[ \\frac {1}{18} - \\frac{2}{3(s+2)(s+1)}\\right]\n \\epsilon + ..\n$$\n\n\\section{\\large The relation between reprentations and harmonic analysis for covering groups of $SO(D,2)$ and $SO(D+1,1)$}\n Basic: Elementary representations $\\chi=[l,\\delta]$\n{\\bf are representations of the complex Lie algebra}.\n Partial wave amplitudes $g(\\chi)$ depend on $\\chi$. Their use is a crucial feature of the group theoretical approach \\cite{Mack1,Mack2,Mack3,DobrevPetkovaPetrovaTodorov,DobrevMackPetkovaPetrovaTodorov}.\n The (partial) equivalence of representations $\\chi=[l,\\delta]$ and $\\tilde{\\chi}=[\\tilde{l}, D-\\delta]$ ($\\tilde{l}=l$ for completely symmetric tensor \nrepresentations) is the basis of the shadow operator formalism introduced by Ferrara et al. \\cite{FerraraGattoGrillo}\n\\\\\n\\subsection{Euklidean partial wave expansion}\nfor the 4-point function $<\\phi(x_4)...\\phi(x_1)>$.\n\\begin{figure}[h]\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\PartialWaveUp{\n\\rput{90}(0,0){\\PartialWave}\n}\n\\def\\FourPointFct{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(0.60,0.60)(1.25,1.25)\n\\psline(1.25,-1.25)(0.60,-0.60)\n\\psline(-0.60,0.60)(-1.25,1.25)\n}\n\n\\def\\FourPointMellin{\n\\rput(-2.9,0){$=\\int d\\delta \\ M(\\{ \\delta_{ij}\\})$}\n\\psline(-1.25,-1.25)(-1.25,1.25)\n\\psline(-1.25,-1.25)(1.25,-1.25)\n\\psline(-1.25,-1.25)(1.25,1.25)\n\\psline(1.25,1.25)(-1.25,1.25)\n\\psline(1.25,1.25)(1.25,-1.25)\n\\psline(1.25,-1.25)(-1.25,1.25)\n}\n\n\n\n\\pspicture(0,0)(11,3.0)\n\n\\rput(7,0){\n\\rput(-2.5,1.25){ $ = \\int d\\chi  g(\\chi) $}\n\\rput(-.0,0){\n\\rput(-1.2,0){$3$}\n\\rput(-1.2,2.3){$1$}\n\\rput(3.4,0){$4$}\n\\rput(3.4,2.3){$2$}\n\\rput(0.0,1.25){\\PartialWave}\n\\rput(1.3,1.7){$\\chi$}\n}\n}\n\\rput(1.2,1.25){\\FourPointFct}\n\\rput(11,1.25){$=\\  ...$\n\\endpspicture\n\\caption{Euklidean partial wave expansion for the 4-point function \\newline\n $<\\phi(x_4)...\\phi(x_1)>$.\nThe dots stand for expansions in the two other channels $(12)\\mapsto (34) $ and  $(13)\\mapsto (24)$ of the same amplitude, implying the equality shown in section 4.2\n \\label{fig:EuklideanPartialWave.tex}}\n\\end{figure}  \n\n\n\n\nThis formula {\\em is quite trivial, simply expressing the orthogonality} (and completeness!) {\\em of the partial waves} say Simmons-Duffin, Stanford, and Witten \\cite{SimmonsDuffinStanfordWitten}.\n\n indeed: Physical positivity implies constraints on the partial waves $g(\\chi)$!\nBut also\n completeness is not trivial, because presence of a Born term requires contributions from the {\\em complementary} series in addition to the \n{\\em principal} series.\n\n They can\n be included by choosing a proper path of the $c$-integration\n$$\\int d\\chi .... = \\sum_l\\int_C dc \\rho_l(c)...,\\qquad \\chi=[l,\\half D + c]$$\n\\setlength{\\unitlength}{0.65mm}\n\\begin{picture}(100,60)(-2,0)\n\\thinlines\n\\put(0,10){\n\\put(38,25){\\line(1,0){4}}\n\\thicklines\n\\put(40,0){\\vector(0,1){50}}\n\\put(32,25){\\circle*{3}}\n\\put(30,15){$c_f$}\n\\put(48,25){\\circle{3}}\n\\put(32,25){\\circle{9}}\n\\put(36.4,24.5){\\vector(0,-1){1}}\n\\put(48,25){\\circle{9}}\n\\put(43.6,24.5){\\vector(0,-1){1}}\n\n\\put(62,25){\\circle*{3}}\n\\put(74,25){\\circle*{3}}\n\\put(84,25){\\circle*{3}}\n\n\\put(18,25){\\circle{3}}\n\\put(6,25){\\circle{3}}\n\\put(-4,25){\\circle{3}}\n}\n\\put(0,3){a) Path $C$ of the  c-integration for $l=0$}\n\\end{picture}\n\n\n{\\bf Positivity constraints} were examined already in the 7o's. \nThey restrict the position of the\n poles and fix the sign of of the residues of the partial wave amplitudes $g(\\chi)$ defined by\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\PartialWaveUp{\n\\rput{90}(0,0){\\PartialWave}\n}\n\\def\\FourPointFct{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(0.60,0.60)(1.25,1.25)\n\\psline(1.25,-1.25)(0.60,-0.60)\n\\psline(-0.60,0.60)(-1.25,1.25)\n}\n\\def\\FourPointFctLeftOnly{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(-1.25,1.25)(-0.60,0.60)\n}\n\n\n\n\\def\\FourPointMellin{\n\\rput(-2.9,0){$=\\int d\\delta \\ M(\\{ \\delta_{ij}\\})$}\n\\psline(-1.25,-1.25)(-1.25,1.25)\n\\psline(-1.25,-1.25)(1.25,-1.25)\n\\psline(-1.25,-1.25)(1.25,1.25)\n\\psline(1.25,1.25)(-1.25,1.25)\n\\psline(1.25,1.25)(1.25,-1.25)\n\\psline(1.25,-1.25)(-1.25,1.25)\n}\n\\def\\rhspw{\n  \\FourPointFctLeftOnly\n  \\rput(2,0){\\rotatebox{-90}{\\CGKern}}\n  \\psline(.6,.6)(2.0,0)\n  \\psline(.6,-.6)(2.0,0)\n  \\rput(3.5,0){$\\chi$}\n}\n\n\n\\pspicture(0,0)(11,2.5)\n\\rput(0.5,1){\n \\rput(0.1,0){$g(\\chi)$}\n \\rput(1.3,0){\\rotatebox{-90}{\n   \\CGKern\n   \\CircleFeet\n  } \n }\n \\rput(2.8,0){$\\chi =$}\n \\rput(4.5,0){\\rhspw}\n}\n\\endpspicture\n\n\n\\subsection{Crossing relations}\n\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\n\\pspicture(-1,0)(11,3.5)\n\\rput(1,0.5){\n\n\\rput(0.5 ,1.25){ $ \\int d\\chi \\quad g(\\chi) $}\n\\rput(3,0){\n\\rput(-1.2,0){$3$}\n\\rput(-1.2,2.3){$1$}\n\\rput(3.4,0){$4$}\n\\rput(3.4,2.3){$2$}\n\\rput(0,1.25){\\PartialWave}\n\\rput(1.3,1.7){$\\chi$}\n}\n\\rput(2,0){\n\\rput(6,1.25){$\\quad =\\int d\\chi^\\prime g(\\chi^\\prime)$}\n\\rput{90}(9,0.5){\\PartialWave}\n\\rput(9.75,1.75){$\\chi^\\prime$}\n\\rput(7.8,3.8){$1$}\n\\rput(10.2,3.8){$2$}\n\\rput(10.2,-0.4){$4$}\n\\rput(7.8,-0.4){$3$}\n }\n}\n\\endpspicture\n\n\n\n\\noindent The crossing relation is a linear relation for the partial wave, \nof the form $$g(\\chi)= \\int d\\chi^\\prime C(\\chi^\\prime, \\chi)g(\\chi^\\prime).$$ Using orhogonality of the CG-kernels, the crossing kernel \n$C(\\chi,\\chi^\\prime)$ is defined by\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleLeftFoot{\n \\psline(-1.,-1.)(-0.3,-0.3)\n}\n\\def\\CircleLongLeftFoot{\n \\psline(-1.,-1.)(1.,1.)\n}\n\\def\\CircleRightFoot{\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\CircleLongRightFoot{\n\\psline(1.,-1.)(-1.,1.)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\PartialWaveFeetDown{\n\\rput{-90}(0,0){\n\\CGKern\n  }\n \\CircleLongLeftFoot\n\\rput{90}(2.2,0){\n \\CGKern\n  }\n \\rput(2.0,0){\\CircleLongRightFoot}\n}\n\\def\\PartialWaveUp{\n\\rput{90}(0,0){\\PartialWave}\n}\n\\def\\FourPointFct{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(0.60,0.60)(1.25,1.25)\n\\psline(1.25,-1.25)(0.60,-0.60)\n\\psline(-0.60,0.60)(-1.25,1.25)\n}\n\n\\def\\FourPointMellin{\n\\rput(-2.9,0){$=\\int d\\delta \\ M(\\{ \\delta_{ij}\\})$}\n\\psline(-1.25,-1.25)(-1.25,1.25)\n\\psline(-1.25,-1.25)(1.25,-1.25)\n\\psline(-1.25,-1.25)(1.25,1.25)\n\\psline(1.25,1.25)(-1.25,1.25)\n\\psline(1.25,1.25)(1.25,-1.25)\n\\psline(1.25,-1.25)(-1.25,1.25)\n}\n\n\n\\pspicture(-1,0)(5,3.4)\n\n\\rput(7,0){\n \\rput(-.5,0){\n \\rput(0,1.25){\\PartialWaveFeetDown} \n \\rput(1.2,0.7){$\\chi$}\n \\rput(1.6,3.4){$\\chi^\\prime$\n }\n\\rput(0.3,2.5){\\rotatebox{90}(\\CGKern}\n}\n\\rput(0.5,2.0){\\CGKern\n\\CircleFeet\n\\rput(0.5,1.2){$\\chi^\\prime$}\n}\n\\rput(3.5,2.){$C(\\chi,\\chi^\\prime)\\ = \\ \\frac {1}{2}$}\n\\endpspicture\n\n\n\n\\subsection{\\large Derivation of Operator product expansions \n from Euklidean partial wave expansions}\n\\subsubsection{Split of the partial waves}\nOperator product expansions (OPE) have the structure \\\\\n$$ \\phi^1(-\\half x_1)\\phi^2(\\half x_1)=\\sum_n C_n(x_1)O^n(0)$$\nwhere $O^n$ are fields or derivatives of fields.\nThe derivatives of each nonderivative field can be summed up. The resulting \nterms are known as {\\bf confornal blocks}.\n To obtain this expansion, \\\\ \n{\\bf the Clebsch Gordan coefficients need to be split.}\n\n\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\n\\def\\zigzagTwoHor{\n\\psline(0,0)(0.1,0.1)(0.3,-0.1)(0.5,0.1)(0.6,0)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\CGhalf{\n\\psarc*(0,0){.5}{180}{0}\n\\psarc(0,0){.5}{0}{180}\n}\n\\def\\CGhalfHor{\n\\psarc*(0,0){.5}{90}{-90}\n\\psarc(0,0){.5}{-90}{90}\n}\n\\def\\QKern{\n\\psarc*(0,0){.5}{90}{-90}\n\\psarc(0,0){.5}{-90}{90}\n\\psline(-1.,-1)(-.3,-.3)\n\\psline(-1,1)(-.3,.3)\n\\rput(0.5,0){\\zigzagTwoHor}\n\\rput(0.7,0.5){$\\chi$}\n}\n\n\n\\def\\QKerntilde{\n\\psarc*(0,0){.5}{90}{-90}\n\\psarc(0,0){.5}{-90}{90}\n\\psline(-1.,-1)(-.3,-.3)\n\\psline(-1,1)(-.3,.3)\n\\rput(0.5,0){\\zigzagTwoHor}\n\\rput(0.7,0.5){$\\tilde{\\chi}$}\n\\rput(1.2,0.0){\\zigzaglongHor}\n}\n\\def\\zigzaglongHor{\n\\psline(0,0)(0.1,0.1)(0.3,-0.1)(0.5,0.1)(0.7,-0.1)(0.8,0)\n}\n\\def\\SplitCGrhs{\n \\QKern \n \\rput(1.5,0){$+$}\n \\rput(2.7,0){\\QKerntilde}\n}\n\\def\\SplitCGlhs{\n  \\rotatebox{-90}{\\CGKern}\n  \\psline(-1.,-1)(-.3,-.3)\n  \\psline(-1,1)(-.3,.3)\n}\n\\pspicture(0,0)(11,2.)\n\\rput(1,1){\n \\rput(0,0){\\SplitCGlhs}\n \\rput(1.7,0.0){$=$\n \\rput(3.0,0){\\SplitCGrhs}\n}\n\\endpspicture\n\n\nThis split exploits the equivalence of representations $\\chi=[l,h+c]$ and \n$\\tilde{\\chi}=[l,h-c]$ $(h=\\half D)$ of the Euklidean conformal group to split\n the three point function into Clebsch-Gordan kernels of the second kind in a \nway  modeled after the split of Legendre functions $P_l$ into Legendre\n functions of the second kind,  $Q_l$,\n\\ba\n&& \\mathfrak{V}(\\x_0,\\tilde{\\chi}, \\x_2 , \\chi_2, \\x_1, \\chi_1)\n= \\  \\label{kernel2ndKind}\\nn  \\\\ && \\frac {\\pi}{\\sin(l+c)}\n[\\mathfrak{Q}(\\x,\\tilde{\\chi} ,\\x_2,\\chi_2,\\x_1, \\chi_1)-  \\nn \\\\\n&& \\qquad \\qquad - \\int d^D\\y \\Delta^{\\tilde{\\chi}}(\\x,\\y )\\mathfrak{Q}(\\y,\\chi,\\x_2,\\chi_1,\\x_1, \\chi_2)] \n\\nn\n\\ea\nsuch that the partial Fourier transform \\\\\n$\\mathfrak{Q}(p,\\tilde{\\chi} ,\\x_2,\\chi_2,\\x_1, \\chi_1)$\\\\\n{\\bf is an entire holomorphic function of $p$.}\n\\subsection{Continuation to Minkowski apace}\nAfter the split, one can\ndeform the path of the $c$-integration in the partial wave expansion involving\n$g(\\chi)$, $\\chi=[l,h+c])$, $h=D\/2$\\\\picking up a sum over residues as shown in the following picture\\\\[4mm]\n\\setlength{\\unitlength}{0.65mm}\n\\begin{picture}(100,95)(-2.6,0)\n\\thinlines\n\\put(0,50){\n\\put(38,25){\\line(1,0){4}}\n\\thicklines\n\\put(40,0){\\vector(0,1){50}}\n\\put(32,25){\\circle*{3}}\n\\put(30,15){$c_f$}\n\\put(48,25){\\circle{3}}\n\\put(32,25){\\circle{9}}\n\\put(36.4,24.5){\\vector(0,-1){1}}\n\\put(48,25){\\circle{9}}\n\\put(43.6,24.5){\\vector(0,-1){1}}\n\n\\put(62,25){\\circle*{3}}\n\\put(74,25){\\circle*{3}}\n\\put(84,25){\\circle*{3}}\n\n\\put(18,25){\\circle{3}}\n\\put(6,25){\\circle{3}}\n\\put(-4,25){\\circle{3}}\n\n\\put(0,2.5){a) Original path of c-integration} \n}\n\n\\put(-100,0){\n\\thinlines\n\\put(138,25){\\line(1,0){4}}\n\\put(140,23){\\line(0,1){4}}\n\\thicklines\n\\put(132,25){\\circle*{3}}\n\\put(130,15){$c_f$}\n\\put(139,15){$0$}\n\n\\put(132,25){\\circle{9}}\n\\put(136.4,24.5){\\vector(0,-1){1}}\n\\put(148,25){\\circle{3}}\n\n\\put(162,25){\\circle*{3}}\n\\put(162,25){\\circle{9}}\n\\put(166,24.5){\\vector(0,-1){1}}\n\n\\put(174,25){\\circle*{3}}\n\\put(174,25){\\circle{9}}\n\\put(178,24.5){\\vector(0,-1){1}}\n\n\\put(184,25){\\circle*{3}}\n\\put(184,25){\\circle{9}}\n\\put(188,24.5){\\vector(0,-1){1}}\n\n\\put(118,25){\\circle{3}}\n\\put(106,25){\\circle{3}}\n\\put(96,25){\\circle{3}}\n\n\\put(100,3){b) Path after closure} \n}\n\\end{picture}\n\n\n\nThis results in a discrete sum of conformal blocks\nas shown before, i.e. the expected OPE.\n\\begin{figure}[h!\n\\psset{unit=1.2cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleLeftFoot{\n \\psline(-1.,-1.)(-0.3,-0.3)\n}\n\\def\\CircleLongLeftFoot{\n \\psline(-1.,-1.)(1.,1.)\n}\n\\def\\CircleRightFoot{\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\CircleLongRightFoot{\n\\psline(1.,-1.)(-1.,1.)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n \\rput{-90}(0,0){\n \\CGKern\n \\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\PartialWaveQ{\n \\rput{-90}(0,0){\n \\QKern\n \\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n}\n\n}\n\\def\\PartialWaveFeetDown{\n\\rput{-90}(0,0){\n\\CGKern\n  }\n \\CircleLongLeftFoot\n\\rput{90}(2.2,0){\n \\CGKern\n  }\n \\rput(2.0,0){\\CircleLongRightFoot}\n}\n\\def\\PartialWaveUp{\n\\rput{90}(0,0){\\PartialWave}\n}\n\\def\\FourPointFct{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(0.60,0.60)(1.25,1.25)\n\\psline(1.25,-1.25)(0.60,-0.60)\n\\psline(-0.60,0.60)(-1.25,1.25)\n}\n\n\\def\\FourPointMellin{\n\\rput(-2.9,0){$=\\int d\\delta \\ M(\\{ \\delta_{ij}\\})$}\n\\psline(-1.25,-1.25)(-1.25,1.25)\n\\psline(-1.25,-1.25)(1.25,-1.25)\n\\psline(-1.25,-1.25)(1.25,1.25)\n\\psline(1.25,1.25)(-1.25,1.25)\n\\psline(1.25,1.25)(1.25,-1.25)\n\\psline(1.25,-1.25)(-1.25,1.25)\n}\n\n\\def\\QKern{\n\\psarc*(0,0){.5}{90}{-90}\n\\psarc(0,0){.5}{-90}{90}\n\\psline(-1.,-1)(-.3,-.3)\n\\psline(-1,1)(-.3,.3)\n\\rput(0.5,0){\\zigzagTwoHor}\n\\rput(0.7,0.5){$\\chi$}\n}\n\n\\def\\QKerntilde{\n\\psarc*(0,0){.5}{90}{-90}\n\\psarc(0,0){.5}{-90}{90}\n\\psline(-1.,-1)(-.3,-.3)\n\\psline(-1,1)(-.3,.3)\n\\rput(0.5,0){\\zigzagTwoHor}\n\\rput(0.7,0.5){$\\tilde{\\chi}$}\n\\rput(1.2,0.0){\\zigzaglongHor}\n}\n\\def\\zigzaglongHor{\n\\psline(0,0)(0.1,0.1)(0.3,-0.1)(0.5,0.1)(0.7,-0.1)(0.8,0)\n}\n\n\\def\\zigzagTwoHor{\n\\psline(0,0)(0.1,0.1)(0.3,-0.1)(0.5,0.1)(0.6,0)\n}\n\n\\def\\SplitCGrhs{\n \\QKern \n \\rput(1.5,0){$+$}\n \\rput(2.7,0){\\QKerntilde}\n}\n\\pspicture(0,0)(3,7.0)\n\\rput(0,5){\n \\rput(1,0){\\FourPointFct}\n \\rput(4.1,0){$=\\ \\sum_{poles\\ in \\chi} res_\\chi g(\\chi)$} \n \\rput(8,0){\\QKern}\n \\rput(10,0){\\rotatebox{90}{\\CGKern \\CircleFeet} }\n}\n\\rput(6.5,1.0){\n \\rotatebox{90}{\\QKern}\n  \\rput(0,2){\n   {\\rotatebox{180}{\\CGKern \\CircleFeet}}\n }\n}\n\\rput(5.7,1.5){$=\\ \\sum_{poles\\ in \\chi} res_\\chi g(\\chi) \\qquad \\qquad \\qquad \\qquad  = \\ \\  . . .$}\n\\endpspicture\n\\caption{\ndiscrete version of the crossing relation for partial wave $g(\\chi)$\n}\n\\end{figure}\n\n\n\n\n\\section{\\large Recent progress}\n\\subsection{Conformal Regge theory}\nFamilies of poles of the partial wave $g(\\chi=[l, \\delta])$ labelled by \n(even or odd) $l$ are analogues of Regge trajectories \\cite{Mack6,CostaGoncalvesPenedones,SimmonsDuffin}. But in contrast to conventional Regge theory the pole ``approximation'' is exact!\n\\subsection{Analysis of the conformal blocks}\nDolan and Osborn \\cite{DolanOsborn} determined the conformal blocks for even dimension $D$. Recursion relations were studied by Penedones, Trevisani and Yamazaki \\cite{PenedonesTrevisaniYamazaki}\n\nEl-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi \n\\cite{ElShowkPaulosPolandRychkovSimmonsDuffinVichi}\\\\\ngeneralized the result to arbitrary $D$ and use numerical evaluation to \nbound and find critical indices for the {\\bf Ising model}\n\n\\subsection{Mellin representation}\nI proposed to \n consider bootstrap in Mellin space in 2009 \\cite{Mack6}. The Mellin representation was advocated as a natural language by Fitzpatrick, Kaplan, Penedones, Raju and van Rees\n\\cite{FitzpatrickKaplanPenedones} \n  and further studied by  Dey, Ghosh and Sinha \\cite{DeyGhoshSinha}(2018)\\\\\n\\begin{figure}[t]\n\\psset{unit=0.6cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\OvalShort { \n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n}\n\\def\\OvalLong { \n\\psline(-.75,-.5)(.75, -.5)\n\\psline(-.75,.5)(.75,.5)\n\\psarc(-.75,0){.5}{90}{270}\n\\psarc(.75,0){.5}{270}{90}\n}\n\\def\\BSKern {\n\\psline(-.5,-.5)(.5, -.5)\n\\psline(-.5,.5)(.5,.5)\n\\psarc(-.5,0){.5}{90}{270}\n\\psarc(.5,0){.5}{270}{90}\n\\psline[linestyle=dotted](-1.,0)(1.,0)\n}\n\n\\def\\CircleUp{\n\\pscircle(0,0){.5}\n\\psline(0,.5)(0,1.5)\n}\n\\def\\OvalShortTwoFeet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\n\\def\\OvalUpFeet {\n\\psline(-1.,1.)(-.5,.5)\n\\psline(1,1)(.5,.5)\n}\n\n\\def\\OvalShorFteet {\n\\psline(-1.,-1.)(-0.5,-0.5)\n\\psline(1.,-1.)(0.5,-0.5)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\n\\def\\PartialWave{\n\\rput{-90}(0,0){\n\\CGKern\n\\CircleFeet\n }\n\\rput{90}(2.2,0){\n\\CGKern\n\\CircleFeet\n }\n}\n\\def\\FourPointFct{\n\\pscircle(0,0){0.9}\n\\psline(-1.25,-1.25)(-0.60,-0.60)\n\\psline(0.60,0.60)(1.25,1.25)\n\\psline(1.25,-1.25)(0.60,-0.60)\n\\psline(-0.60,0.60)(-1.25,1.25)\n}\n\n\\def\\FourPointMellin{\n\\rput(-6,0){$=\\int d\\delta \\ M(\\{ \\delta_{ij}\\})$}\n\\psline(-1.25,-1.25)(-1.25,1.25)\n\\psline(-1.25,-1.25)(1.25,-1.25)\n\\psline(-1.25,-1.25)(1.25,1.25)\n\\psline(1.25,1.25)(-1.25,1.25)\n\\psline(1.25,1.25)(1.25,-1.25)\n\\psline(1.25,-1.25)(-1.25,1.25)\n}\n\n\n\\pspicture(-1,0)(15,4.5\n\n\n\\rput(0,4){\n \\rput(0,1.25){\\FourPointFct}\n \\rput(4,1.25){ $\\ = \\int d\\chi \\ g(\\chi) $}\n \\rput(8.5,1.25){\\PartialWave}\n \\rput(9.5,1.7){$\\chi$}\n}\n\\rput(12,1.25){\\FourPointMellin}\n\\endpspicture\n\\caption{ Partial wave expansion and Mellin representation for the 4-point function.$<\\phi(x_4)...\\phi(x_1)>$.  The lines in the Mellin representation represent propagators \n$\\Gamma(\\delta_{ij}(\\xi_i\\xi_j)^{-\\delta_ij}$ in covariant language\nIntegration is over $\\delta_{ij}=\\delta_{ji}$ such that $\\sum_{i}\\delta_{ij}=d_i$ when the scalar field $\\phi(x_i)$ has dimension $d_i$\n \\label{fig:partialWaveAndMellin.tex}}\n\\end{figure} \n\n The lines in the Mellin representation represent propagators \n$\\Gamma(\\delta_{ij})(\\xi_i\\xi_j)^{-\\delta_{ij}}$ \n in Dirac's covariant language.\nIntegration is over $\\delta_{ij}=\\delta_{ji}$  such that\n $\\sum_{i}\\delta_{ij}=d_i$  when the scalar field $\\phi(x_i)$\n has dimension $d_i$\n\\subsection{Pismaks simplified version of the crossing relation}\nPismak \\cite{Pismak} proposed to study a simplified version of the crossing relation for the partial wave $g(\\chi)$ as shown in Figure 5.\n\\begin{figure}[h]\n\\psset{unit=1cm}\n\\psset{dotsize=0.2 0}\n\\psset{linewidth=.05}\n\\def\\CircleRightFoot{\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\CircleFeet{\n\\psline(-1.,-1.)(-0.3,-0.3)\n\\psline(1.,-1.)(0.3,-0.3)\n}\n\\def\\CircleWLeftFootLong\n{\n\\pscircle(0,0){.5}\n\\psline(1.7,-1.7)(0.3,-0.3)\n}\n\\def\\CircleWRightFootLong\n{\n\\pscircle(0,0){.5}\n\\psline(-1.7,-1.7)(-0.3,-0.3)\n}\n\n\\def\\CircleWFeetLong\n{\n\\pscircle(0,0){.5}\n\\psline(-1.7,-1.7)(-0.3,-0.3)\n\\psline(1.7,-1.7)(0.3,-0.3)\n}\n\\def\\zigzagOne{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0,0.4)\n}\n\\def\\zigzagTwo{\n\\psline(0,0)(0.1,0.1)(-0.1,0.3)(0.1,0.5)(0,0.6)\n}\n\\def\\CGKern{\n\\pscircle*(0,0){.5}\n\\rput(0,0.5){\\zigzagTwo}\n}\n\\def\\PartialWave{\n \\rput{-90}(0,0){\n \\CGKern\n \\CircleFeet\n  }\n \\rput{90}(2.2,0){\n \\CGKern\n\\CircleFeet\n }\n}\n\n\\def\\PartialWaveHorizontalBottomLong{\n\\rput(1.15,.5){$\\chi$}\n \\rput{-90}(0,0){\n \\CGKern\n\n\\CircleWLeftFootLong \n }\n \\rput{90}(2.2,0){\n \\CGKern\n\\CircleWRightFootLong\n }\n\\rput(-1.7,-2.1){$3$}\n\\rput(3.9,-2.1){$4$}\n}\n\\def\\PartialWaveUp{\n\\rput{90}(0,0){\n \\rput{-90}(0,0){\n \\CGKern\n \\CircleWFeetLong\n  }\n \\rput{90}(2.2,0){\n \\CGKern\n\\CircleRightFoot\n  }\n }\n\\rput(-0.9,3.55){$p_1=0$}\n\\psline[linearc=.25](1.7,-1.65)(1.0,3.0)(.3,2.55)\n\\rput(1.5,2.0){$m$}\n\\rput(1.5,1.5){$\\alpha$}\n\\rput(0.65,1.0){$\\chi^\\prime$}\n\\psline(1.4,1.75)(1.1,1.75)\n\\rput(-2,0.5){$= \\int g(\\chi^\\prime)$}\n}\n\n\n\\def\\PartialWaveHorizontalBottomLongP10{\n\\PartialWaveHorizontalBottomLong\n\\psline(-.3,.3)(-1.,1)\n\\rput(-1.3,1.3){$p_1=0$}\n\\rput(2.2,0){\n\\psline[linearc=.25](.3,.3)(1.,1.)(1.7,-1.7)\n\\psline(1.1,0.0)(1.4,0.0)\n\\rput(1.6,.3){$m$}\n\\rput(1.6,-.4){$\\alpha$}\n }\n}\n\\pspicture(-1,0)(15,5.0)\n\\rput(10,1.5){\n\\PartialWaveUp\n}\n\\rput(2,2,0.5){\n\\rput(-1.5,0){$\\int  g(\\chi)$}\n\\PartialWaveHorizontalBottomLongP10\n}\n\\endpspicture\n\\caption{ Pismaks version of the crossing symmetry of the partial wave $g(\\chi)$. In the Ising model with fields $\\phi(x)$ and $\\phi^2(x)$, $g(\\chi)$\nis a $2\\times 2$ matrix.\n \\label{fig:crossingKernelPismak}\n}\n\\end{figure}\n\n\\subsection{Spacetime derivation of the Lorentzian OPE inversion formula}\nCaron-Huot \\cite{CaronHuot}, and Simmons-Duffin, Stanford, Witten \\cite{SimmonsDuffinStanfordWitten}\n derive a formula for the partial wave \n$g(\\chi=[\\Delta, J] ) = n_{\\Delta,J}I_{\\Delta,J}$  expressed in terms of vacuum \nvacuum expectation value of commutators of the  four scalar fields\n $\\phi_i(x_i)$ of dimension $d_i$ in Minkowski space,  $n_{\\Delta,J}$\nis  an explicitly given normalization factor\n\\ba\nI_{\\Delta,J} &=& -\\hat{C}_J(1)\n\\left[ \\int_{3>1,2>4} \\frac {d^D x_{3}d^Dx_{4}}{vol(SO(D-1)}\\right.\\nn \\\\\n&&\n \\frac{ <\\Omega, [\\phi_4(x_4),\\phi_2(x_2)][\\phi_1(x_1),\\phi_3(x_3)]]|\\Omega>}{ |x_{34}|^{J+2D-d_3-d_4-\\Delta}}\n\\nn \\\\\n&&\n\\left.\n(m\\cdot x_{34})^J \\theta(m\\cdot x_{34}) \\right.\\nn \\\\\n &+& (-1)^J  \\int _{4>1,2>3}\n  \\frac{d^D x_3 d^D x_4}{vol(SO(D-1))}\\nn \\\\\n&&\n \\frac{<\\Omega|[\\phi_3,(x_3),\\phi_2(x_2)][\\phi_1(x_1),\\phi_4(x_4)]|\\Omega>}{|x_{34}|^{J+2D-d_3-d_4-\\Delta}}\n\\nn \\\\\n&& \\left.\n (-m\\cdot x_{34})^J  \\theta(-m\\cdot x_{34}) \n\\right]\n\\nn \n\\ea\nwhere $m$ is the nullvector $m^\\mu=(1,1,0,...,0)$ where the second component is the time direction, and coordinates $x_1,x_2$\nhave been fixed to \n$x_1=(1,0,0,...), $, $x_2=0$. The notation $i>j$ means that $x_i$ is in the future light cone of $x_j$.\n\nAn explicit expression for\n $\\hat{C}_J(1)$  is given.\n\nThe formula has the following advantages: It can be {\\bf analytically continued in the spin $J$}, and for real dimension and spin, the integrand satisfies {\\bf positivity conditions}.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nRecent technological advances in machine learning have transformed healthcare in many ways from expediting research and discovery, to making diagnostic decisions or treatment recommendations for individual patients. Particularly, a growing body of research has explored clinical decision-making in critical settings \n(e.g. \\cite{ghassemi2015multivariate, desautels2016prediction, awad2017early}). Critical care practitioners need to make time sensitive and accurate decisions when treating patients, often in settings where there is a high degree of uncertainty and patients have little physiologic reserve \\cite{lighthall}. Recent algorithms have demonstrated the ability to provide alerts for impending events\n, and in some cases \neven make better treatment recommendations than clinicians \\cite{ghassemi2017predicting, outperforms}. \n\nDespite these advances however, machine learning models are still prone to critical errors, often failing to generalise across different care settings or different institutions (cite?). Translating these models for safe use in clinical practice thus presents a unique set of challenges. An emerging line of research in machine interpretability, defined by Doshi-Velez and Kim \\cite{fdv-rigorous} as ``the ability to explain or to present in understandable terms to a human,'' provides an alternative way to ensure systems preserve properties such as safety or nondiscrimination.  If a model is interpretable to human stakeholders, then clinical experts can inspect and verify that a model's reasoning is sound prior to integration in a care setting.\n\nIn this paper, we propose a method for learning summaries of patient patient physiological time-series data that are \\emph{human-simulatable}, or can be easily calculated by a human when given access to the time-series data \\cite{lipton-mythos}. Importantly, the parameters of our model are \\emph{decomposable}, and can be described using a meaningful and short text description \\cite{lipton-mythos}. We show that by using our model to first derive human-simulable summaries from a patient's physiological timeseries, and combining these with other patient data in a logistic regression model, we can make accurate predictions that are easy to interpret.  By learning summaries that are human-interpretable functions of the patient's time-series, our work explicitly optimizes for human-interpretability. Optimizing for interpretability is particularly meaningful in healthcare settings, where models are used to inform high-risk and high-uncertainty decisions.  \n\nOur technical contributions are twofold.  First, we introduce several functions used to summarize the dynamics of physiological time-series data.  \nIn our summary definitions, we also formalize several design choices of clinical significance, such as how much of each patient's time-series data should be used to compute summary features.  Second, we propose an generalizable optimization procedure to search over these design choices automatically.  Our approach simultaneously enables optimization over both the human-interpretable time-series summaries and a linear model to make predictions for any intensive care classification task.  We achieve performance comparable to complex models for two intensive care prediction tasks while preserving the human-interpretability of the learned model and representations.\n\n\\section{Related Work}\n\n\\paragraph{Black-box prediction methods}\nA number of works develop ways to use timeseries data to make predictions.  Black-box deep models such as recurrent neural networks have been used for a variety of intensive care tasks including predicting in-hospital mortality \\cite{benchmarking-deep, ng-deep-learning-ehr} or vasopressor onset \\cite{intervention-cnns}. Models which learn hidden representations from the patient's time-series such as switching dynamical systems models \\cite{hughes-vaso, fdv-vaso-sam} or variations of Hidden Markov models \\cite{gp-timeseries-acuity} have also been used for \nintensive care prediction tasks.  While these approaches typically get better prediction accuracy, their representations are not easily interpreted by clinicians.  Unlike these approaches, our work explicitly learns human-interpretable summaries of the time series for performing prediction. \n\n\\paragraph{Prediction methods that use expert-derived features}\nSeveral common risk stratification models have also been deployed in intensive care units, such as SAPS-II \\cite{saps-ii} and APACHE \\cite{apache}. These methods use simple score-card algorithms to evaluate patient acuity.  These algorithms take as input features such as the patient's ``average'' or ``worst'' values contained in the time-series to compute score.  While easily computed and understood, their predictive power is limited.  Such scores have also typically been targeted toward a single end point, such as in-hospital mortality or mortality 28 days post-discharge rather than on-going risk stratification such as deterioration in the next several hours.\n\nCloser to our work, there exists a growing literature on methods that, like the score-card approaches, first compute statistics of a timeseries, and then use those to make predictions \\cite{NIPS2012_3cec07e9,Harutyunyan19,DBLP:journals\/midm\/GuoLC20}.\nThese studies have included statistics such as the maximum, minimum, mean, and variance of a particular timeseries dimension.  While somewhat more flexible than the traditional score-card approaches, this set of statistics is \nstill limited and does not capture clinically important patterns such as whether a value is increasing or decreasing, nor do they consider the appropriate duration over which the statistics should be computed. Moreover, the majority of these studies \nfail to justify which summary statistics can achieve the best performances for physiological time series. In contrast to these methods, we focus on learning representations that are simple simulable functions of a patient's timeseries.\n\n\\paragraph{Methods for feature ranking and selection.}\nCharacterising time-series using summary statistics can often contain redundant information if features are coupled, or non-representative features are selected for making predictions. As a result, several approaches have been developed to rank features according to their importance, and select only the most relevant among these \\cite{WEI201911}. Recently, Tonekaboni et al. \\cite{NEURIPS2020_08fa4358} propose a method for quantifying the importance of features \\emph{over time} based on their contribution to time-dependent shifts of a model's output distribution. While the method allows the authors to evaluate the aggregate importance of subsets of features over time to promote post-hoc interpretability \\cite{lipton-mythos}, they do not explicitly optimise for interpretability as we propose in our work. \n\n\n\n\\end{document}\n\n\\section{Introduction}\nAccurate predictions of patient risk in critical care units can aid clinicians in making more effective decisions.  Specifically, early identification of patients at high risk for in-hospital mortality is critical to assess patient disease acuity and inform life-saving interventions \\cite{saps-ii, escobar2020}.  To predict in-hospital mortality risk, researchers have developed algorithms ranging from simple score-cards \\cite{saps-ii, apache} to statistical machine learning (ML) models.  Recent advances in ML have led to the development of models with vast improvements in predictive accuracy for patient in-hospital mortality risk \\cite{ awad2017early, gp-timeseries-acuity, ng-deep-learning-ehr, benchmarking-deep}.  \n\nDespite these improvements, however, ML models are still prone to critical errors, often failing to generalize across different care settings or institutions \\cite{futoma2020myth}.  An emerging line of research in interpretability, defined by  \\cite{fdv-rigorous} as ``the ability to explain or to present in understandable terms to a human,'' provides an alternative way to ensure systems preserve properties such as safety or nondiscrimination.  If a model is interpretable to stakeholders, then clinical experts can inspect the model and verify that its reasoning is sound.  This ability to audit and validate is especially important when the models are used to inform critical decisions affecting patient health.  \n\nIn this work, we present a novel ML method to learn clinical timeseries summaries that are interpretable and predictive.  We introduce functions to compute summaries that align with simple and intuitive concepts, such as whether the timeseries is decreasing or spikes above a critical threshold. In contrast to prior work, our method learns how much of the timeseries should be used to calculate these summaries, discarding earlier timesteps that may be irrelevant for a specific prediction task.  Importantly, we introduce relaxations of our summary definitions to enable differentiable optimization, allowing summary parameters to be learned jointly with those of a downstream model.  \nWe show that with our method, we can achieve accuracies comparable with state-of-the-art baselines without sacrificing interpretability.\n\n\\section{Related Work}\n\nPrior work on explaining clinical timeseries models fall into two categories: learning simple models that are inherently interpretable, and generating explanations of complex black box models.  We summarize a few key examples below.\n\n\\textbf{Interpreting Deep Models.} \nOne popular strategy for explaining ML models is learning a second post-hoc explanation model to explain the first black box model \\cite{ribeiro2016should}.  Many post-hoc explanation techniques for clinical timeseries models train explanation models that quantify the relative importance of each clinical variable \\cite{NEURIPS2020_08fa4358, lundberg2018explainable}.  However, several works argue against the use of post-hoc explanation techniques, as explanation models are not always faithful or representative of the true underlying black boxes \\cite{rudin2019stop, lakkaraju2020fooling}. \nOur method avoids these problems by design, instead explicitly optimizing for interpretability so that a second explanation model is not needed.\n\nAnother line of research proposes attention mechanism models specifically designed for timeseries.  \\cite{choi2017retain, sha2017} present neural attention architectures for clinical timeseries and argue that attention scores measure feature importance. However, attention methods are often highly complex and nonlinear. Furthermore, \\cite{serrano2019} shows attention scores do not always reflect true importance.  Instead of approximating importance, our study uses linear models over richer features, where importance does not need to be approximated but can be directly read off model coefficients.\n\n\\textbf{Expert Systems and Expert Features.}  Our work extends a long tradition of clinical experts hand-crafting features to create interpretable clinical decision-support algorithms.  Two expert systems widely used in ICUs are SAPS-II \\cite{saps-ii} and APACHE \\cite{apache}, which use simple score-card algorithms to evaluate patient acuity.  These systems use input features such as the patient's average or worst lab or vital values over time to compute mortality risk.  While SAPS-II and APACHE are simple and simulable to clinicians, their predictiveness is limited, as they cannot capture how labs or vitals change across a patient's stay.\n\nA similar line of research proposes manual construction of expert features from clinical data, which are then used as input to ML models \\cite{sun2012, roe2020}.  One limitation of this approach is that expert feature derivation is expensive and requires clinical expertise. Rather than relying on expert knowledge to identify which summary features will be the most predictive for a given task, our work instead uses optimization with a sparsity constraint to automatically learn which summary features are the most predictive.\n\n\\textbf{Summarizing Clinical Timeseries.}  A number of works have proposed a wide range of summary statistics for patient timeseries data.  Many works such as \\cite{awad2017early, harutyunyan2019} train clinical models using the minimum value, maximum value, first measured value, or skew of clinical timeseries data.  \\cite{DBLP:journals\/midm\/GuoLC20} proposes a more comprehensive set of 14 summary statistics to characterize the central tendency, dispersion tendency, and distribution shape of clinical timeseries data.  Our work extends this research, and is the first to our knowledge to use the slope of the timeseries or proportion of time above or below critical thresholds.  Our work is also novel in that we explicitly model and optimize for the duration over which we compute each summary feature.\n\n\\section{Cohort and Problem Set-Up}\nOur goal is to learn summaries of patient timeseries data that are both human-interpretable and predictive for a downstream classification task.  Our approach will define how to calculate these human-interpretable summaries, and describe how both summary and classification model parameters are learned through optimization. In what follows, we detail each of these processes.\n\n\\textbf{Prediction Task.} Our work examines early prediction of in-hospital mortality.  We use the patient's first 24 hours of data to predict if they would later expire over the course of the remainder of their admission.  Patients who expired in the first 24 hours of their stay were excluded from our cohort.\n\n\\textbf{Cohort Selection.}\nWe use data from the MIMIC-III critical care database \\cite{mimic-iii}, which contains deidentified health data from patients in critical care units of the Beth Israel Deaconess Medical Center between 2001 and 2012.  All data was extracted from MIMIC-III PhysioNet version 1.4, which contains 30,232 patients. We exclude patients under 18 years of age and patients whose weight is not measured.  We include data from each patient's first hospitalization only, and only patients with stays between 24-72 hours \\cite{awad2017early}. After applying these criteria, our final cohort contained 11,035 patients, 15.23\\% of whom died in-hospital. Cohort characteristics and demographics are summarized in Table \\ref{table:mimic-summary-stats}.   \n\\begin{table}[h!]\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c | c  c  | c  c  c  c | c  c  c  c|} \n \\hline\n Cohort & Age & \\% Female & \\% Urgent & \\% Emergency & \\% Elective & \\% MICU & \\% SICU & \\% CCU & \\% CSRU &   \\\\ \n \\hline\n All & 64.7 & 43.8 &  1.12 & 84.46 & 14.43 & 42 & 18 & 12 & 16 &\\\\\\hline\n + & 70.9 & 46.5  & 0.77 & 96.25 & 2.97 & 53 & 18 & 13 & 4 &\\\\ \\hline\n - & 64.1 & 43.6  & 1.14 & 83.22 & 15.64 & 41 & 18 & 12 & 17 & \\\\ \\hline\n\\end{tabular}}\n\\caption{Mean statistics for the population cohort, and for cohorts of positive versus negative patients for in-hospital mortality.  Abbreviations: MICU, medical care unit; SICU, surgical care unit; CCU, cardiac care unit; CSRU, cardiac-surgery recovery\nunit.}\n\\label{table:mimic-summary-stats}\n\\end{table}\n\n\\subsection{Extracting Inputs and Outputs}\nFor each patient $n$ in our $N$ patient cohort, we extracted static observations and physiological data including labs and vital signs sampled hourly. All clinical variables were separately normalized to have zero mean and unit variance.  Figure \\ref{figure:feature-extraction} shows how features are extracted for patients that are positive versus negative for in-hospital mortality.  \n\\begin{figure}[]\n\\centering\n    \\includegraphics[width=0.6\\textwidth]{images\/feature_extraction_mortality.png}\n\\caption{Example positive and negative time-series to illustrate feature extraction.  The two trajectories have input data extracted from time 0 to time of prediction $T = 24$.  Figure inspired by Sherman et al \\cite{sherman}.}\n\\label{figure:feature-extraction}\n\\end{figure}\n\n\\textbf{Static observations $\\bm{S}$.} Matrix $\\bm{S}: (N \\times 8)$ contains $8$ demographic variables for each patient $n$: their age at admission, gender, and other information about their ICU stay (their first ICU service type, and whether their admission was urgent, emergency, or elective).\n\n\\textbf{Per-timestep clinical observations $\\bm{X}$.}  The clinical variable tensor $\\bm{X}: (N \\times D \\times T)$ contains $D = 28$ measurements of clinical variables for each of the $N$ patients at time $t$, discretized by hour.  These 28 measurements consist of vital signs and labs: diastolic blood pressure, fio2, GCS score, heartrate, mean arterial blood pressure, systolic blood pressure, SRR, oxygen saturation, body temperature,  urine output, blood urea nitrogen, magnesium, platelets, sodium, ALT and AST, hematocrit, po2, white blood cell count, bicarbonate, creatinine,  lactate, pco2, glucose, INR, hemaglobin, and bilirubin.  Missing values at timestep $t$ were imputed using either the most recent measurement of the variable, or the population median if the variable had not yet been measured during the patient's stay.  We use the patient's first $T = 24$ hours of data to predict in-hospital mortality.  We use subscripts to index into the tensor: for example, $\\bm{X}_t$ indicates the $(N \\times D)$ matrix of measurements taken at time $t$.\n\n\n\\textbf{Per-timestep measurement indicators $\\bm{M}$.}\nThe measurement indicator tensor $\\bm{M}: (N \\times D \\times T)$ contains indicator elements  $\\bm{M}_{n, d, t}$ which are 1 if their corresponding clinical variable in $\\bm{X}_{n, d, t}$ was measured at time $t$, 0 otherwise.\n\n\\textbf{Outcome labels $\\bm{y}$.} Label vector $\\bm{y}$ contains indicators $\\bm{y}_n$ which are 1 if patient $n$ expired in-hospital, 0 otherwise.\n \n\n\\section{Methods}\nGiven a cohort of $N$ training examples $\\{ \\bm{X}, \\bm{M}, \\bm{S}, \\bm{y} \\}$, we propose a novel procedure for learning predictive human-interpretable timeseries summaries. Concretely, we first compute summaries $\\bm{H}$ from clinical timeseries data $(\\bm{X}, \\bm{M})$. We then use the summaries $\\bm{H}$ in addition to static data $\\bm{S}$ and clinical variables $\\bm{X}$ as input to a Logistic Regression model to predict labels $\\bm{y}$.  This process of using human-interpretable summaries for prediction is shown in Figure \\ref{figure:model-diagram}.\n\nIn Section \\ref{section:model-def}, we motivate and introduce our novel summary features.  In Section \\ref{section:relaxations}, we give continuous relaxations of summary feature definitions to enable efficient inference of summary parameters.  In Section \\ref{section:learning-process}, we discuss how predictive summary features can be jointly learned with downstream classification model parameters.\n\n\\begin{figure}[]\n\\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/model_diagram.png}\n\\caption{Illustrates summary extraction for prediction from timeseries data.  First, non-static clinical variables $\\{\\bm{X}, \\bm{M}\\} $ are used to compute interpretable summary features $\\bm{H}$.  Then, summary features $\\bm{H}$, static features $\\bm{S}$, and the non-static variables $\\{\\bm{X}, \\bm{M}\\}$ at the time of prediction are given as input to a Logistic Regression model $g$, which predicts output labels $\\bm{\\hat{y}}$.  Figure inspired by Ghassemi, Szolovits et al \\cite{hughes-vaso}.}\n\\label{figure:model-diagram}\n\\end{figure}\n\n\\subsection{Model: Defining human-interpretable summaries}\\label{section:model-def}\nMeasures of central tendency and dispersion have commonly been used to summarize timeseries (see survey in \\cite{DBLP:journals\/midm\/GuoLC20}).  Our key modelling innovations include adding additional features that correspond to how clinicians themselves describe timeseries.  Our novel contributions include explicitly modelling the overall trend of a lab\/vital and the number of hours that a lab\/vital dips above or below a threshold, as well as allowing different features to be computed over different periods of time (e.g. the most recent 6 hours vs. the most recent 24 hours).\n\nThe $I = 13$ summary statistics used in this study are listed in Table \\ref{table:summaries-weighted}.  Each of the summary statistics takes into account measurement indicators $\\bm{M}$ so that clinical variable summaries are computed only using timesteps where the variable is measured. Each summary statistic is applied to each of the $D$ clinical variables to create the summary feature tensor $\\bm{H}: (N \\times D \\times I)$.\n\nBelow, we expand on the parameterization of our summary features.  Next, we describe how we enable efficient, automated search over summary feature parameters.  Importantly, our approach automates many processes associated with summary design, enabling optimization over summary parameters. \n\n\\textbf{Incorporating Duration.}\nMany prior works in clinical timeseries modelling do not use all timesteps for the patient, but instead only the most recent available data, such as the six or twelve hours before the time of prediction.  In contrast to prior works, we explicitly model how much of each timeseries  should be used to compute each of the $I$ summaries in $\\bm{H}$.  For example, we may wish to exclude earlier measurements of a particular clinical variable if only the variable's recent measurements before time of prediction $T$ are significant for a prediction task.  Specifically, for each variable $d$ and for each summary function $i$, we define a duration time $\\bm{C}_{d, i}$.  Only the variable's timeseries data that occurred in the immediately previous $\\bm{C}_{d, i}$ hours before the time of prediction is used to calculate summary $i$.  We organize all the duration time parameters $\\bm{C}_{d,i}$ into a $(D \\times I)$ matrix $\\bm{C}$.\n\nTo exclude data that occurs before time $(T -\\bm{C}_{d, i})$ when computing summary features, we multiply each of the original timeseries variables by indicator variables for whether the measurements occurred within $\\bm{C}_{d, i}$ hours before time of prediction $T$.  For example, a mean summary statistic would be computed using indicator variables $\\mathbbm{1}(\\cdot)$ as:\n\\begin{equation}\\label{non_diff_cutoff_indicators} \n\\bm{H}_{i = mean}: (N \\times D) = \\left( \\textstyle\\sum_{t=1}^{T}  \\mathbbm{1}(t > T - \\bm{C}_{i = mean}) \\odot \\bm{X}_{ t} \\odot \\bm{M}_{ t} \\right) \/ \\left( \\textstyle\\sum_{t = 1}^{T} \\bm{M}_{ t} \\odot \\mathbbm{1}(t > T - \\bm{C}_{i = mean})  \\right)\n\\end{equation}\n\nwhere $\\odot$ is the element-wise multiplication operator and division is performed element-wise.  Our objective is to learn duration parameters $\\bm{C}$ that maximize the predictiveness of their corresponding summary features $\\bm{H}$.\n\n\\textbf{Threshold Parameters.}\nSome of the summary functions $f_i$ in Table \\ref{table:summaries-weighted} have additional parameters such as thresholds.  For example, one of the summaries is the proportion of the patient's measured timeseries where their measured clinical variables are above some $D$-dimensional critical threshold parameter vector $\\bm{\\phi}^{+}$ for each variable:\n\\begin{equation}\\label{thresholds-non-diff}\n    \\left( \\textstyle\\sum_{t = 1}^{T}  \\bm{M}_t \\odot \\mathbbm{1} (\\bm{X}_t > \\bm{\\phi}^{+}) \\right) \/ \\left( \\textstyle\\sum_{t = 1}^{T}  \\bm{M}_t \\right)\n\\end{equation}\nThese summaries correspond to clinically-intuitive ideas, such as whether the patient been mostly well or sick.  As with durations, we learn threshold parameters automatically to avoid burdening experts and to assist in prediction.\n\n\\subsection{Continuous Relaxations for Efficient Inference}\\label{section:relaxations}\n\\textbf{Learning Duration Parameters.}\nIn Equation \\ref{non_diff_cutoff_indicators}, we showed how summary functions $f_i$ that only use the most recent $\\bm{C}$ hours of data can be calculated using indicator variables $\\mathbbm{1}(t > T - \\bm{C}_{d, i}) $.  These indicator variables, however, do not have informative gradients and are not differentiable.  To enable differentiable optimization for duration time parameters $\\bm{C}$, we introduce weight parameters $\\bm{W}$ by relaxing the indicator random variables using the sigmoid function $\\sigma(x) = \\frac{1}{1+e^{-x}}$.  Using the duration parameter matrix $\\bm{C}$, we define $D$-dimensional vectors $\\bm{w}_{t, i}$ that compose weight tensor $\\bm{W}: (T \\times I \\times D)$: \n\n\\begin{equation}\\label{w_def}\n    \\bm{w}_{t, i}  = \\sigma \\left((t - T + \\bm{C_i})\/ \\tau \\right) \n\\end{equation}\n\nFor each feature $d$ and summary $i$, clinical observations $\\bm{X}_{ d, t}$ where $t > T - \\bm{C}_{d, i}$ will have corresponding weights $\\bm{w}_{t, i, d}$ near 1.  Timesteps $t$ where $t < T - \\bm{C}_{d, i}$ will have corresponding weights $\\bm{w}_{t, i, d}$ near 0.  Temperature parameter $\\tau$ controls the harshness of the weight matrix: small temperatures push the sigmoid function towards its edges, learning weights that are closer to exactly 0 for timesteps before $T -\\bm{C}_{d, i}$ and 1 for timesteps after $T - \\bm{C}_{d, i}$, effectively functioning as the indicator variables in Equation \\ref{non_diff_cutoff_indicators}.\n\nWeighted summary functions $f_i$ used to derive human-interpretable summaries $\\bm{H}$ can be found in Table \\ref{table:summaries-weighted}. Duration parameters $\\bm{C}$ that determine weight tensor $\\bm{W}$ are included in $\\beta_{\\bm{H}}$, the set of all parameters necessary to compute the summaries.\n\\begin{table}[h!]\n\\centering\n\n\\begin{tabular}{|c| c|} \n \\hline\n Description & Function \\\\ [0.5ex] \n \\hline\n \n Mean of the time-series &   $\\left(\\sum_{t=1}^{T} \\left( \\bm{w}_{t, i} \\odot \\bm{X}_t \\odot \\bm{M}_t \\right)\\right) \/  \\left(\\sum_{t = 1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i} \\right) $\\\\ \\hline\n \n Variance of the time-series &   $ \\frac{(\\sum_{t = 1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i})^2}{(\\sum_{t=1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i})^2 - \\sum_{t=1}^{T} \\bm{M}_t \\odot \\bm{W}^2_t}   \\odot \\sum_{t = 1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i} \\odot  \\left( \\bm{X}_t {-} \\bar{\\bm{X}} \\right)^2$ \\\\ \\hline\n \n\n \n Indicator if feature was ever measured &  $\\sigma \\left( \\left( \\sum_{t = 1}^T \\bm{w}_{t, i} \\odot \\bm{M}_t \\right) \/ \\left(\\tau \\odot \\sum_{t = 1}^T \\bm{w}_{t, i}  \\right)\\right)$\\\\ \\hline\n \n\n \n Mean of the indicator sequence &    $ \\left(\\sum_{t = 1}^{T} \\bm{w}_{t, i} \\odot \\bm{M}_t \\right) \/ \\left(\\sum_{t = 1}^{T} \\bm{w}_{t, i} \\right)$\\\\ \\hline\n \nVariance of the indicator sequence &  $ \\left( \\frac{(\\sum_{t=1}^{T} \\bm{w}_{t, i})^2}{(\\sum_{t=1}^{T} w_t)^2 - \\sum_{t=1}^{T} \\bm{W}^2_t} \\right)  \\sum_{t = 1}^{T} \\bm{w}_{t, i} \\odot ( \\bm{M}_t - \\bar{\\bm{M}})^2$\\\\ \\hline\n \n \\# switches from missing to measured &  $\\left(\\sum_{t = 1}^{T - 1} \\bm{w}_{t, i} \\odot  | \\bm{M}_{t + 1} - \\bm{M}_t|\\right) \/ \\left(\\sum_{t=1}^{T} \\bm{w}_{t, i} \\right) $\\\\ \\hline\n \n First time the feature is measured &  $ \\min t$ s.t. $\\bm{M}_t = 1$\\\\ \\hline\n \n Last time the feature is measured &  $ \\max t$ s.t. $\\bm{M}_t = 1$\\\\ \\hline\n \n Proportion of time above threshold $\\bm{\\phi}^{+}$ &  $  \\left(  \\sum_{t = 1}^{T} \\bm{w}_{t, i} \\odot \\bm{M}_t \\odot \\sigma( \\frac{\\bm{X}_t- \\bm{\\phi}^{+}}{\\tau}) \\right) \/ \\left( \\sum_{t = 1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i} \\right)$ \\\\ \\hline\n \n Proportion of time below threshold $\\phi^{-}$ &  $   \\left( \\sum_{t = 1}^{T} \\bm{w}_{t, i} \\odot \\bm{M}_t \\odot \\sigma( \\frac{\\bm{\\phi}^{-} - \\bm{X}_t }{\\tau}) \\right) \/ \\left( \\sum_{t = 1}^{T} \\bm{M}_t \\odot \\bm{w}_{t, i} \\right)$\\\\ \\hline\n \n Slope of a L2 line & $\\frac{\\sum_{t= 1}^T \\bm{w}_{t, i} (t - \\bar{t}_{\\bm{w}}) (\\bm{X}_t - \\bar{\\bm{X}}_{\\bm{w}})}{\\sum_{t = 1}^T \\bm{w}_{t, i} (t - \\bar{t}_{\\bm{w}})^2}$, where $\\bar{t}_{\\bm{w}} = \\frac{\\sum_t \\bm{w}_{t, i} \\cdot t}{\\sum_t \\bm{w}_{t, i}}$ and $\\bar{\\bm{X}}_{\\bm{w}} = \\frac{\\sum_t \\bm{w}_{t, i} \\odot \\bm{X}_t}{\\sum_t \\bm{w}_{t, i}}$ \\\\ \\hline\n \n\n\n \n Standard error of the L2 line slope &   $1 \/ \\left( \\sum_t \\bm{w}_{t, i} \\odot (t - \\bar{t}_{\\bm{w}})^2 \\right)$  \\\\ \\hline\n\n \n\n \n\n \n\n\n \\hline\n\\end{tabular}\n\\caption{Table of functions $f_i$ used to calculate human-interpretable summaries $\\bm{H}$.  For each of the $D$ clinical variables, all $I$ of the above functions are applied to each of the $N$ patients.  Each of the $I$ summary features $i$ is defined with respect to $D$-dimensional weight vectors $\\bm{w}_{t, i}$ defined in Section \\ref{section:relaxations}.  Paramater $\\tau$ is a temperature parameter for the sigmoid function.   $\\mathbbm{1}(\\cdot)$ denotes indicator variables for events inside the parentheses, $\\odot$ indicates element-wise matrix multiplication and division is done element-wise.  Additionally, $\\bar{\\bm{X}} = \\sum_t^T \\bm{M}_t \\odot \\bm{X}_t \/\\sum_t^T \\bm{M}_t$, and  $\\bar{\\bm{M}}_t = \\frac{1}{T} \\sum_t^T \\bm{M}_t$.}\n\\label{table:summaries-weighted}\n\\end{table}\n\n \\textbf{Learning Threshold Parameters.}\nOur work relaxes summary definitions to enable differentiable optimization to learn summary parameters.\nThe indicator variables used to define the proportion of hours that a patient's timeseries is above thresholds $\\bm{\\phi}^{+}$ in Equation \\ref{thresholds-non-diff} are non-differentiable with respect to $\\bm{\\phi^{+}}$. To enable differentiable optimization, our work defines our threshold summary features using the sigmoid function $\\sigma$ with temperature parameter $\\tau$:\n\\begin{equation}\n    f_{threshold}(\\bm{X}, \\bm{M}, \\bm{W}) =   \\left(\\textstyle\\sum_{t = 1}^{T} \\bm{w}_{t, i = threshold} \\odot \\bm{M}_t \\odot \\sigma\\left( \\frac{\\bm{X}_t - \\bm{\\phi}^{+}}{\\tau}\\right)\\right) \/ \\left( \\textstyle\\sum_{t = 1}^{T} \\bm{M}_t \\cdot \\bm{w}_{t, i = threshold} \\right) \n\\end{equation}\n\nThreshold parameters $\\{\\bm{\\phi}^{+}, \\bm{\\phi}^{-}\\}$ are included in $\\beta_{\\bm{H}}$, the set of all parameters necessary to compute the summaries.\n\n\\subsection{Learning Process}\\label{section:learning-process}\nOur study uses summary features $\\bm{H}$, along with static variables $\\bm{S}$ and timeseries variables $\\{\\bm{X}, \\bm{M}\\}$ at prediction time $T = 24$ as input to a Logistic Regression model $g$ with coefficients $\\bm{\\beta}_g$. Logistic Regression model $g$ outputs predicted probabilities $\\hat{\\bm{y}} = p(\\bm{y} = \\bm{1} | \\bm{X},\\bm{S}, \\bm{M})$.  Our objective is to learn optimal summary and model parameters $\\bm{\\beta} = \\{\\bm{\\beta}_{\\bm{H}}, \\bm{\\beta}_g\\}$.  We jointly learn parameters $\\bm{\\beta}$ by minimizing the loss function\n\\begin{equation}\n    \\mathcal{L}(\\bm{\\beta}; \\bm{X}, \\bm{M}, \\bm{S}, \\bm{y}) = - \\frac{1}{N} \\sum_{n = 1}^N \\omega_n \\left( \\bm{y}_n \\cdot \\log[g(\\bm{X}_n, \\bm{M}_n, \\bm{S}_n, \\bm{\\beta})] + (1 - \\bm{y}_n) \\log[1 - g(\\bm{X}_n, \\bm{M}_n, \\bm{S}_n, \\bm{\\beta})] \\right)  + \\Omega(\\bm{\\beta}_g)\n\\end{equation} \nOur loss function is the sum of the weighted binary cross-entropy loss using predictive model $g$ and the regularization penalty $\\Omega(\\bm{\\beta}_g)$.  To account for class imbalance, we reweight each training example $n$'s loss contribution by $\\omega_n$, the inverse of its class frequency in the training dataset.\n\n\\textbf{Horseshoe Regularization on Coefficients.}\nWe explicitly optimize for sparsity in model parameters $\\bm{\\beta}_g$ via our regularization penalty in our training objective.  We use a Horseshoe regularization penalty $\\Omega(\\bm{\\beta}_g)$ with shrinkage parameter $1$ to encourage sparsity in the learned regression coefficients\\cite{horseshoe-bhadra}.    \n\n\\section{Experiments}\nWe compare models learned using our summaries to other interpretable models as well as deep learning baselines. We show that our models outperform other traditional human-interpretable model classes and achieve performance comparable to deep models on the in-hospital mortality task.\n\n\\textbf{Model configurations.} Our baseline models are: Ridge Logistic Regression models that take as input only the patient timeseries measured at the time of prediction $T$, and Ridge Logistic Regression and LSTM models that take as input all of the patient timeseries.  We used an LSTM as our deep baseline architecture as prior works document their superior performance at mortality prediction from clinical timeseries data \\cite{Harutyunyan19}.\n\nOur LSTM models were trained on the sequential timeseries data $\\{\\bm{X}, \\bm{M}\\}$ with a step size of 1 hour.  The LSTM hidden states at each timestep were then used to predict both the next timestep of the patient timeseries $\\bm{X}$, and to predict outcome labels $\\bm{y}$.  All $T$ of the output hidden states $\\bm{X}_t$ were input to a fully-connected layer, which output predictions of the next timestep $\\bm{X}_{t + 1}$.  The last output hidden state at time $T$ was also input with static features $\\bm{S}$ to a fully-connected layer to predict outcome labels $\\bm{y}$.  The LSTM models were trained to minimize both the mean-squared error of the next state prediction, and the binary cross entropy loss of the classification prediction.  ReLU activation functions were applied to both fully-connected layers.\n\n\\textbf{Training Details.}\nFor training and testing, we split the cohort of $N$ patients into train and test sets, where all data associated with each patient is either in train or test.  All performance metrics are averaged across five train-test splits.  Ridge baseline models were implemented using RidgeCV from \\texttt{scikit-learn} \\cite{scikit-learn}. LSTMs as well as our summary-based Logistic Regression models were implemented with PyTorch, and trained with the Adam optimizer\\cite{adam} at a batch size of 256.  We trained all of our Logistic Regression models for 30,000 epochs and LSTM models for 10,000 epochs using early stopping.  All hyperparameters (including the LSTM hidden state and layer dimensions, optimizer learning rate, and regularization parameters) were selected via random hyperparameter search \\cite{bergstra2012random}.  All temperature parameters $\\tau$ were set to $0.1$. \n\nOur final learned LSTM models have hidden state dimension 32, 1024 nodes in the layer to predict the next timestep, and 64 nodes in the layer to predict labels $\\bm{y}$.  They are trained using a learning rate of $1e-05$.  Our final learned models with summaries use $\\alpha = 1e-05$, Horseshoe shrinkage parameter $1.0$, and learning rate $1e-05$.  \n\n\\section{Results}\n\\textbf{Our learned models achieve performance comparable to state-of-the-art baselines.} Table \\ref{table:mortality-auc} compares the performance of our learned models with linear and deep baselines for the in-hospital mortality prediction task. Applying a linear model to the learned summary features $\\bm{H}$ consistently improves AUCs in comparison to using a linear model on clinical timeseries and static data alone. Our models achieve an AUC performance comparable to state-of-the-art LSTM models with test AUCs of $0.9000 \\pm 0.0223$. Notably, our models that allow differentiable optimization to learn duration times outperform models that compute all summary features using the entire duration of the patient timeseries.  This implies that there is predictive value in explicitly modelling how much of each variable's clinical timeseries should be considered for a specific prediction task.\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c | c| c |} \n \\hline\n \\textbf{Model} & Train set AUC & Test set AUC\\\\ \\hline\n LR trained on $\\{\\bm{S}, \\bm{M}\\}$ and $\\bm{X}$ at time $T$ only & $0.8653 \\pm 0.0013$ & $0.8626 \\pm 0.0079$ \\\\ \\hline\n LR trained on $\\{\\bm{S}, \\bm{M}, \\bm{X}\\}$ & $0.8931 \\pm 0.0015$\t& $ 0.8668 \\pm 0.0122$ \\\\ \\hline\n LSTM trained on $\\{\\bm{S}, \\bm{M}, \\bm{X}\\}$ & $\\bf{0.9101 \\pm 0.0018}$ & $\\bf{0.9000 \\pm 0.0223}$ \\\\ \\hline\n Our model, trained on $\\{\\bm{S}, \\bm{M}, \\bm{X}\\}$, non-differentiable durations $\\bm{C}$ & $\\bf{0.9065 \\pm 0.0019}$ & $\\bf{0.8818 \\pm 0.0063}$ \\\\ \\hline \n  Our model, trained on $\\{\\bm{S}, \\bm{M}, \\bm{X}\\}$, differentiable durations $\\bm{C}$ & $\\bf{0.9074 \\pm 0.0016}$ & $\\bf{ 0.8867 \\pm 0.0061}$ \\\\ \\hline\n\\end{tabular}\n\n\\caption{Performance of learned models on the in-hospital mortality prediction task.  AUCs are averaged over five train-test splits with their standard error.  Abbreviations: LR, Logistic Regression; LSTM, long short-term memory. $\\bm{C}$ refers to the duration parameters defined in Section \\ref{section:relaxations}.}\n \n\\label{table:mortality-auc}\n\\end{table}\n\n\\textbf{Our learned models use fewer features to achieve higher accuracy in comparison to other interpretable baselines.} To evaluate the sparsity of each model, we performed a set of ablation experiments where we zeroed all but the $N$ coefficients with the largest magnitudes for each of the learned Logistic Regression models.  In Figure \\ref{figure:ablation-sparsity}, we show the average test set AUC for Logistic Regression baseline versus our models using only $N$ coefficients.  Our models consistently outperform baseline models when all but $N$ coefficients are zeroed, suggesting that our models learn a smaller and more predictive set of important features. \n\\begin{figure}[]\n\\centering\n    \\includegraphics[width=0.5\\textwidth]{images\/ablation_mortality.png}\n\\caption{Prediction quality (mean test AUC) vs. model complexity (number of non-zero features) for baseline Ridge Regression trained on all timesteps of the patient timeseries, versus our models.  The dark blue horizontal line shows our model's average test set AUC using all $401$ derived features, and the orange horizontal line shows the average test set AUC of the Logistic Regression model trained only on the $65$ features extracted at the time of prediction.}\n\\label{figure:ablation-sparsity}\n\\end{figure}\n\n\\textbf{Our learned models are interpretable.} \nTable \\ref{table:key-summary-features} shows 15 key summary features that consistently have the largest learned Logistic Regression coefficients across train-test splits. The corresponding coefficients for each feature can be interpreted as a measure of the feature's contribution to the final classification label.  For example, because the mean of the patient's GCS has a large negative coefficient, this means that patients with higher mean GCS scores will be assigned lower predicted probabilities for in-hospital mortality.  Therefore our models are \\emph{decomposable} \\cite{lipton-mythos}, as each of the model's features and coefficients has an intuitive clinical explanation.\n\n\n\\textbf{Our learned summary features are clinically sensible.} \nThe vast majority of the key summary features learned by our models shown in Table \\ref{table:key-summary-features} are supported by studies in medical literature. For instance, it is widely accepted that patients who are older tend to have lower chances of survival in ICU settings  \\cite{nielsen2019survival,fuchs2012icu}. Similarly, patients with lower GCS scores of below 6 tend to have severe injuries and higher chances of mortality \\cite{bastos1993glasgow}. Notably a lower GCS score in the later hours of a patient's hospitalisation significantly reduces a patient's chances of survival \\cite{10.1001\/archneur.1990.00530110035013,settervall2011hospital}. Finally, the normal range of features such as the blood oxygen saturation (SPO$_2$) is between 95\\% and 100\\%. An SPO$_2$ consistently below 90\\% indicates hypoxaemia or potential respiratory distress. These patients have to be mechanically ventilated in ICU and frequently have lower chances of survival \\cite{lazzerini2015hypoxaemia,vold2015low}.\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c | l| l | r | } \n \\hline\n \\textbf{Feature} & \\textbf{Aggregation} & \\textbf{Time} & \\textbf{Coefficient} \\\\ \\hline\n Age & static value & - &  14.9 \\\\ \\hline\n BUN & value at & hour 24 & 6.2 \\\\ \\hline\n GCS & mean over & hours 5 - 24 & - 5.59 \\\\ \\hline\n HR & value at & hour 24 & 4.69 \\\\ \\hline\n FiO$_2$ & value at & hour 24 & 4.31 \\\\\\hline\n Hct & times measured over & hours 2 - 24 & - 3.81 \\\\  \\hline\n HR, & mean over & hours 2 - 24 & 3.78 \\\\ \\hline\n GCS & value at & hour 24 & - 3.62 \\\\ \\hline\n GCS & hours below 6.08 & hours 7 - 24 & 3.37 \\\\\\hline\n Creatinine & hours below 0.35 mg\/dL  & hours 5 - 24 & 2.76 \\\\ \\hline\n FiO$_2$ & hours above 62.96\\% & hours 16 - 24 & 2.59 \\\\ \\hline\n SpontaneousRR & mean over & hours 2 - 24 & 2.29 \\\\ \\hline\n GCS & times measured over & hours 5 - 24 & 2.23 \\\\ \\hline\n Sodium & hours below 131.57 mEq\/L & hours 1 - 24 & 2.16 \\\\\\hline\n WBC & hours below 0.78 cells\/mL &  hours 6 - 24 & 2.10 \\\\\\hline\n SPO$_2$ & hours below 92.36\\% & hours 10 - 24 & 2.04 \\\\  \\hline\n\\end{tabular}\n\n\\caption{ Key summary features, sorted from largest to smallest coefficient magnitudes, from learned models.  }\n \n\\label{table:key-summary-features}\n\\end{table}\n\n\\textbf{Initialization sensitivity.}\nIn general, we observed that our optimization procedure is stable, learning the same 15 key summary features across different stochastic parameter initializations and train-test splits.  However, there are cases where we observed that the learned duration parameters $\\bm{C}$ varied depending on their initialization. As such, we recommend that practitioners incorporate prior knowledge about the clinical prediction task when initializing the duration time parameters.  For example, if examining the entire duration of the patient's timeseries is necessary for a prediction task, then the duration parameters should be initialized to include the entire timeseries by default.  \n\n\\section{Discussion \n\\& Conclusion}\nIn this work, we defined functions to compute interpretable, parameterizable summaries of clinical timeseries, and developed relaxations so that our summary parameters could be jointly learned with a downstream predictive model.  In our experiments, we used Logistic Regression to make predictions because its coefficients are easily decomposable \\cite{lipton-mythos}.   However, because our learned summaries are inherently interpretable, any other interpretable architecture could be used instead.  Our methodology is generalizable and enables the efficient learning of intuitive and predictive timeseries summaries without placing any assumptions on the downstream model architecture.\n\n\\textbf{Future work.} Our study poses many interesting directions for future work.  One avenue would be to conduct a user study to validate the human-interpretability and decomposability of our proposed summary features.  Another would be to evaluate whether the summary features learned for particular critical care prediction tasks remain predictive for a wider set of critical care prediction tasks. \nFinally, we could also develop additional summary statistic functions, or expand our framework to consider sharing duration parameters across features or across summaries to better model dependencies between clinical labs and vitals---as many physiological events are characterized by several simultaneous changes to multiple labs and vitals \\cite{hotchkiss2017septic}.\n\n\\textbf{Conclusion.}  In this paper, we propose a new method to learn interpretable and predictive summary features from clinical timeseries data.  In addition to introducing novel summary statistics including slope and threshold features, our work differs from prior work by learning the duration of timeseries data that should be used to compute each summary.  We demonstrate that our learned timeseries summaries achieve performance quality comparable to state-of-the-art deep models when trained to predict early patient mortality risk on real patient data.  We also qualitatively validate our models to confirm their interpretability and sensibility. Our work is an important step towards optimizing for  representations of clinical timeseries data that are both highly predictive and interpretable.\n\n\\textbf{Acknowledgements:} NJ and FDV acknowledge support from NIH R01 MH123804-01A1. SP acknowledges support from the Miami Foundation and SNSF P2BSP2-184359.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNeutrino oscillation measurements have become more and more\naccurate. The latest combined results \\cite{Lisi} read $\\D\nm^2_{\\odot} = (7.9 \\pm 0.7) \\times 10^{-5} \\ {\\rm eV}^2$and $\\D\nm^2_{\\rm atm} = (2.4^{+0.5}_{-0.6})\\times 10^{-3} \\ {\\rm eV}^2$,\nfor the mass squared differences of the large mixing angle\nsolution (LMA) to the solar neutrino problem \\cite{solar1,solar2},\nand for atmospheric neutrinos \\cite{atm1,atm2,atm3}, respectively.\nBoth mass differences are sub-eV, but the neutrino mass scale is\nnot yet certain.\n\nNeutrino data also hint at the possibility of more than three\nmassive, mostly active neutrinos. The Liquid Scintillator Neutrino\nDetector (LSND) Collaboration \\cite{LSND1} has reported evidence\nfor a $\\bar{\\nu}_e$ flux $30$ meters away from a source of\n$\\bar{\\nu}_\\mu$, produced in $\\pi^+ \\rightarrow \\mu^+ \\nu_\\mu$\nwith subsequent $\\mu^+ \\rightarrow \\bar{\\nu}_\\mu + e^+ + \\nu_e$\ndecay. The unexpected flux can be explained if there is a small\nprobability $P(\\bar{\\nu}_\\mu \\rightarrow \\bar{\\nu}_e)=(0.26 \\pm\n0.08)\\%$ for a neutrino produced as a $\\bar{\\nu}_\\mu$ to be\ndetected as a $\\bar{\\nu}_e$ \\cite{LSND2}.\nThis LSND anomaly has yet to be confirmed, and the MiniBooNE\nexperiment at Fermilab \\cite{MiniBooNE} should very soon give a\ndefinitive confirmation or refutation. Numerous attempts to solve\nthe LSND puzzle, however, have already been proposed\n\\cite{Strumia}. It has been shown that oscillations with extra\nsterile neutrinos can fit the LSND anomaly \\cite{sterile}. But it\nhas also been pointed out that extra sterile neutrinos could be in\nconflict with Big Bang Nucleosynthesis (BBN) \\cite{Cirelli}, as\nwell as SN1987A supernova neutrino events \\cite{CPT}.\n\nIn a recent work \\cite{Gouvea}, it has been argued that, if the\nright-handed neutrino Majorana scale $m_R$ is of ${\\cal O}({\\rm\neV})$, adequate fits to the LSND data can be obtained. This ``eV\nseesaw\" scenario runs against theoretical arguments in favor of a\nvery large $m_R$. To name a few such arguments: the canonical\nseesaw mechanism \\cite{seesaw1,seesaw2,seesaw3} with $m_R \\sim\n10^{14}\\ {\\rm GeV}$ can elegantly explain why neutrino masses are\nso small, even with lepton Yukawa couplings that are of order one;\nthermal leptogenesis \\cite{leptogenesis1} points to $m_R \\gtrsim\n10^{10}\\ {\\rm GeV}$ \\cite{leptogenesis2}. However, as stressed in\nRef. \\cite{Gouvea}, nothing {\\it experimental} is really known\nabout the magnitude of $m_R$, except perhaps the LSND result,\nwhich is at eV scale.\n\nThe purpose of this letter is to show that the eV seesaw proposed\nin Ref.~\\cite{Gouvea} can be straightforwardly incorporated in a\nfour generation scenario.\n\nThe Standard Model (SM) with a sequential fourth generation (SM4)\nis not ruled out by electroweak precision measurements, if one\nallows the extra active neutrino to have mass close to $50$ GeV\n\\cite{Maltoni,HPS}. To avoid bounds from direct search at LEP II\n\\cite{LEPII}, mixing of the fourth heavy neutrino with the three\nlight neutrinos should be small $(\\lesssim 10^{-6})$. It is clear\nthat in standard seesaw with $m_R \\sim 10^{15}$ GeV, an extra\ngeneration is hard to accommodate (A different approach to predict\na light sterile neutrino in the presence of a fourth generation is\nthe so called ``flipped seesaw\" \\cite{GZ}). All four (mostly)\nactive neutrinos will then be light, contradicting the invisible\n$Z$ width which measures only three light neutrinos. But taking\n$m_R$ at scale ${\\cal O}(\\rm{eV})$, one can now have a\nsufficiently heavy fourth neutrino.\n\nIn the following we will show that, by taking $m_R \\sim {\\cal\nO}(\\rm{eV})$, the fourth neutrino is pseudo-Dirac and heavy. It\nwill not affect the invisible $Z$ width, and largely decouples\nfrom lower generations. Aside from three mostly active light\nneutrinos, three sterile neutrinos with mass $\\gtrsim {\\rm eV}$ is\npredicted. A numerical analysis gives results consistent with the\nLSND as well as solar and atmospheric data. It seems that\n$\\sin^2\\theta_{13}$ cannot be large.\n\n\n\\section{Pseudo-Dirac Fourth Neutrino}\n\nFollowing Ref. \\cite{Gouvea} but allowing for a possible 4th\ngeneration, the $8 \\times 8$ neutrino mass matrix $M$ is given by\n\\begin{equation}\nM = M_D + \\Delta M_R + \\delta M_D,\n \\label{massmatrix}\n\\end{equation}\nin a form suggestive of mass hierarchies.\nIn the basis where the $4 \\times 4$ Dirac mass matrix is diagonal,\nthe dominant Dirac mass for the 4th generation arises from\n\\begin{equation}\nM_D = m_D\\left(\n\\begin{array}{cc}\n0 & I_4 \\\\\nI_4 & 0\n\\end{array} \\right),\n\\end{equation}\nwhere $m_D \\sim 50$ GeV, $0$ and $I_4$ are ${4\\times4}$ matrices\nwith zero elements, except 1 in 44 element of $I_4$.\nThe right-handed Majorana mass matrix is given by\n\\begin{equation}\n\\Delta M_R=m_R \\left(\n\\begin{array}{cc}\n0 & 0 \\\\\n0 & r  \\\\\n\\end{array} \\right),\n\\label{MR}\n\\end{equation}\nwhere $m_R \\sim$ eV \\cite{Gouvea}, and $r$ is a ${4\\times4}$\nsymmetric matrix with elements $r_{ij} \\sim 1$.\nThe third matrix is\n\\begin{equation}\n\\delta M_D = m_R \\left(\n\\begin{array}{cc}\n0 & \\varepsilon \\\\\n\\varepsilon & 0\n\\end{array} \\right),\n\\end{equation}\nwhich is pinned more to the $m_R$ scale, with\n\\begin{equation}\n{ \\varepsilon =\\left(\n\\begin{array}{cccc}\n\\ep_1 & 0 & 0 & 0  \\\\\n0 & \\ep_2 & 0 & 0  \\\\\n0 & 0 & \\ep_3 & 0  \\\\\n0 & 0 &    0  & 0  \\\\\n\\end{array} \\right)\n},\n\\end{equation}\nwhere $\\ep_4$ has been absorbed into $m_D$. Clearly, with $m_R\/m_D\n\\equiv x \\sim 10^{-10}$ and $\\ep_i$ considerably less than 1,\n$\\Delta M_R$ and $\\delta M_D$ can be treated as perturbations to\n$M_D$.\n\nWe note that the neutrino mass matrix of Eq.~(\\ref{massmatrix})\ncould arise from very small deviations from a democratic structure\nfor the Dirac contribution \\cite{Silva}, and lepton number is\nassumed to be only slightly violated. The latter requires $m_R\n\\sim 0$ \\cite{Gouvea}, i.e. symmetry is enhanced in the limit of\n$\\Delta M_R \\rightarrow 0$. Having smaller elements in $\\delta\nM_D$ compared to $m_R$ is purely phenomenological \\cite{Gouvea}.\n\nThe dominant $M_D$ has six null eigenvalues, plus two eigenvalues\n$\\pm m_D$. For the eigenvectors associated with zero eigenvalues,\none can choose $e_i^{(0)} = (0,\\ldots,0,1,0\\ldots,0)$ with $1$ in\nthe $i^{th}$ position for $i=1,2,3$ and $i=5,6,7$. For the\neigenvalues $-m_D$ and $m_D$, the corresponding eigenvectors are\n$e_{4,8}^{(0)}=(0,0,0,\\mp1\/\\sqrt{2},0,0,0,1\/\\sqrt{2})$.\nAt zeroth order in $x$, the states $e_4^{(0)}$ and $e_8^{(0)}$\ncombine into a pure Dirac state of mass $m_D$. When linear\ncorrections in $x$ are considered, the perturbed states $e_4$ and\n$e_8$ with $e_i= e_i^{(0)} + x e_i^{(1)}$ have masses which differ\nby ${\\cal O}(x)$, and they now correspond to a pseudo-Dirac\nneutrino with mass $\\sim m_D$, which we denote as $N$ (the charged\npartner is denoted $E$, with $m_E \\gtrsim 100$ GeV\n\\cite{PDG2004}).\n\nFor the six null eigenvalues of $M_D$, one can apply perturbation\ntheory with degeneracies. A linear combination of the degenerate\nunperturbed states $e_i^{(0)}$ diagonalizes the $\\Delta M_R +\n\\delta M_D$ perturbation. That is,\nby diagonalizing the $6\\times6$ perturbation matrix\n\\begin{equation}\n{\\scriptsize M^{(3)}=m_R\\left(\n\\begin{array}{cccccc}\n0 & 0 & 0 & \\ep_1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & \\ep_2 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & \\ep_3 \\\\\n\\ep_1 & 0 & 0 & r_{11} & r_{12} & r_{13} \\\\\n0 & \\ep_2 & 0 & r_{21} & r_{22} & r_{23} \\\\\n0 & 0 & \\ep_3 & r_{31} & r_{32} & r_{33} \\\\\n\\end{array} \\right)},\n\\label{M3x3}\n\\end{equation}\none obtains the corrections of ${\\cal O}(x)$ to the mass\neigenvalues, and the correct eigenstates at order zero. For the\neffect of the 4th generation neutrino through the right-handed\nsector, $r_{ij}$, one has linear corrections in $x$ proportional\nto $r_{44}$ to the eigenvalues $\\pm m_D$ for $e_4$ and $e_8$ as\nalready stated, and ${\\cal O}(x^2)$ corrections to all the other\neigenvalues. The big hierarchy between the matrix elements\n$M_{48}$, $M_{84} \\cong m_D$ and all the others allow the fourth\ngeneration to largely decouple from the other three.\nAs stated in the Introduction, this is also required by direct\nsearch limits that demand very small mixings between $N$ and the\nlight neutrino flavors.\n\nHaving reduced the problem to a $6\\times 6$ case, the analysis\nperformed in \\cite{Gouvea} suggests that one could find a solution\nto the LSND puzzle. Our main goal here is to confirm the\npossibility of an existing solution, and to gain some insight on\nwhat could be a plausible scenario.\n\n\\\n\n\\section{3+3 Neutrino Model }\n\nWe set to zero all phases for simplicity, since there are already\ntoo many parameters. We define $U^{\\prime}$ to be the rotation\nmatrix which diagonalizes $M^{(3)}$. Having started in the basis\nwhere the Dirac neutrino mass matrix $M_D + \\delta M_D$ is\ndiagonal, one still has the freedom to perform a rotation\n$U^{\\prime \\prime}$ in the left sector\n\\begin{equation}\n{\\scriptsize U^{\\prime \\prime}=\\left(\n\\begin{array}{cccccc}\nc_1 c_3 & s_1 c_3 & s_3 & 0 & 0 & 0 \\\\\n-s_1 c_2 - c_1 s_2 s_3 & c_1 c_2 - s_1 s_2 s_3 & s_2 c_3 & 0 & 0 & 0  \\\\\ns_1 s_2 - c_1 c_2 s_3 & -c_1 s_2 - s_1 c_2 s_3 & c_2 c_3 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{array} \\right)}.\n\\label{Uprpr}\n\\end{equation}\nWe assume no mixing between the fourth and first three generation\ncharged leptons, as already discussed.\nFor the right sector, a rotation will just change $r_{ij}$ to\n$r_{ij}^{\\prime}$, resulting in no change to our numerical\nanalysis.\n\nThe probability for a neutrino, produced with flavor $\\alpha$ and\nenergy $E$, to be detected as a neutrino of flavor $\\beta$ after\ntravelling a distance $L$ is \\cite{PDG2004}\n\\begin{equation}\nP(\\nu_\\a \\rightarrow \\nu_\\b) = \\delta_{\\a \\b}-4\\sum_{j>i}^n U_{\\a\nj} U_{\\b j}U_{\\a i}U_{\\b i} \\, \\sin^2{x_{ji}},\n \\label{Palphabeta}\n\\end{equation}\nwhere $\\a=e, \\mu, \\tau, s_i$ with $s_i$ the sterile neutrino\nflavors, $U=U^{\\prime \\prime} U^{\\prime}$, and $x_{ji}=1.27\\D\nm_{ji}^2 \\, L\/E$ with $\\D m_{ji}^2 \\equiv m_j^2-m_i^2$.\nApplying Eq.~(\\ref{Palphabeta})\n\\cite{SCS} in the ``3 active plus 3 sterile neutrino\"\n $(3+3)$ case, using the approximations $x_{21}=x_{31}=x_{32}=0$ and\n$x_{i1}=x_{i2}=x_{i3}$ for $i=4,5,6$, one obtains\n\\begin{widetext}\n\\begin{eqnarray}\nP(\\nu_\\a \\rightarrow \\nu_\\b) &= & \\d_{\\a\\b} +\n 4[ U_{\\a 4}^2(U_{\\b 4}^2-\\d_{\\a\\b})\\sin^2{x_{41}} + U_{\\a 5}^2(U_{\\b 5}^2-\\d_{\\a\\b}) \\sin^2{x_{51}}\n  + U_{\\a 6}^2(U_{\\b 6}^2 -\\d_{\\a\\b}) \\sin^2{x_{61}} \\nonumber \\\\\n&& + U_{\\a 4}U_{\\b 4}U_{\\a 5}U_{\\b\n5}(\\sin^2{x_{41}}+\\sin^2{x_{51}} -\\sin^2{x_{54}})\n + U_{\\a 4}U_{\\b 4}U_{\\a 6}U_{\\b 6}(\\sin^2{x_{41}}+\\sin^2{x_{61}}\n -\\sin^2{x_{64}})\\nonumber \\\\\n &&+ U_{\\a 5}U_{\\b 6}U_{\\a 5}U_{\\b 6}(\\sin^2{x_{51}}+\\sin^2{x_{61}} -\\sin^2{x_{65}})]\\, ,\n \\label{Pmue}\n\\end{eqnarray}\n\\end{widetext}\nwhere orthogonality of $U$ has been used.\nExpressions for the mixing angles are given by \\cite{Gouvea2}\n\\begin{eqnarray}\n&&\\tan^2{\\theta_{12}}\\equiv\\frac{|U_{e2}|^2}{|U_{e1}|^2}\\, , \\,\\,\n\\tan^2{\\theta_{23}}\\equiv\\frac{|U_{\\mu 3}|^2}{|U_{\\tau 3}|^2} \\, ,\\nonumber \\\\\n&& \\ \\sin^2{\\theta_{13}}\\equiv{|U_{e3}|^2}\\, . \\label{mixingangle}\n\\end{eqnarray}\n\n\\begin{table}\n\\caption{Best fit values, $2\\s$ and  $3\\s$  intervals for\nthree-flavor neutrino oscillation parameters from global data,\nincluding solar, atmospheric, reactor (KamLAND and CHOOZ) and\naccelerator (K2K) experiments, taken from Ref.~\\cite{Maltoni2}.  }\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n& BEST FIT & $2\\s$ & $3\\s$\n\\\\ \\hline \\\\\n$\\D m_{21}^2\\,(10^{-5} \\; {\\rm eV}^2)$ & $8.1$  & $7.5-8.7$  & $7.2-9.1$    \\\\\n$\\D m_{31}^2\\,(10^{-3} \\; {\\rm eV}^2)$ & $2.2$  & $1.7-2.9$  & $1.4-3.3$      \\\\\n$\\sin^2{\\theta_{12}}$ & $0.30$  & $0.25-0.34$  & $0.23-0.38$    \\\\\n$\\sin^2{\\theta_{23}}$ & $0.50$  & $0.38-0.64$  & $0.34-0.68$     \\\\\n$\\sin^2{\\theta_{13}}$ & $0.0$  & $\\leq 0.028$  & $\\leq 0.047$      \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{center}\n\\end{table}\n\nTo perform our numerical analysis, we build the $\\chi^2$ by using\nthe three-flavor neutrino oscillation parameters $\\D m_{ji}^2$ and\n$\\sin^2{\\theta_{ij}}$ taken from Ref.~\\cite{Maltoni2}, which is\ncompiled from global data including solar, atmospheric, reactor\n(KamLAND and CHOOZ) and accelerator (K2K) experiments. These are\ngiven in Table 1. We also include the LSND result of\n$P(\\bar{\\nu}_\\mu \\rightarrow \\bar{\\nu}_e)=(0.26\\pm0.08)\\%$, and\nrequire $\\D m^2_{41} \\sim 1 \\, {\\rm eV}^2$, for a total of seven\ninputs.\nAs one clearly has too many parameters to make a proper fit, we\njust minimize the $\\c^2$ built with these quantities by letting\nthe twelve parameters of the model vary.\n\nAs an illustration, we find $s_1=-0.57,\\,s_2=0.98,\\, s_3=0.80$,\nwhich give\n\\begin{equation}\n{\\scriptsize\nU^{\\prime \\prime}=\\left(\n\\begin{array}{cccccc}\n0.49 & -0.34 & 0.80 & 0 & 0 & 0 \\\\\n-0.54 & 0.60 & 0.58 & 0 & 0 & 0  \\\\\n-0.69 & -0.72 & 0.11 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{array} \\right) \\, ,\n} \\label{Udoubleprime}\n\\end{equation}\nand\n\\begin{equation}\n{\\scriptsize  M^{(3)}=5\\,{\\rm eV} \\left(\n\\begin{array}{cccccc}\n0 & 0 & 0 & 0.051 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0.004 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0.031 \\\\\n0.051 & 0 & 0 & 1.195 & 0.861 & 1.038 \\\\\n0 & 0.004 & 0 & 0.861 & 0.968 & 0.878 \\\\\n0 & 0 & 0.031 & 1.038 & 0.878 & 1.264 \\\\\n\\end{array} \\right),\n} \\label{M3x3num}\n\\end{equation}\nfrom which, together with Eq.~(\\ref{M3x3}), one can read the\nvalues for the rest of the parameters coming from the minimization\nprocess. The eigenvalues of $M^{(3)}$ are\n\\begin{eqnarray}\n(m_1,\\,m_2,\\,m_3) &=& -(7 \\times 10^{-5}, \\, 9\\times 10^{-3}, \\,\n0.048) \\;{\\rm eV},\n\\nonumber \\\\\n(m_4, \\, m_5, \\, m_6) &=& (1, \\, 1.2, \\, 15) \\; {\\rm eV},\n\\label{masses}\n\\end{eqnarray}\nand the associated rotation matrix is\n\\begin{equation}\n{\\scriptsize U^{\\prime}=\\left(\n\\begin{array}{cccccc}\n-0.06 & -0.99 & -0.11 & 0 & 0 & 0 \\\\\n-0.38 & 0.12 & -0.91 & 0.01 & -0.06 & 0.05  \\\\\n0.90 & -0.02 & -0.37 & -0.17 & 0.05 & 0.12 \\\\\n-0.19 & 0 & 0.09 & -0.75 & 0.16 & 0.60 \\\\\n0.05 & -0.01 & 0.07 & 0.22 & -0.83 & 0.49 \\\\\n0.01 & 0 & 0 & 0.60 & 0.52 & 0.61 \\\\\n\\end{array} \\right).\n} \\label{Uprime}\n\\end{equation}\n\nFrom Eq.~(\\ref{masses}) one gets $\\D m_{21}^2=8.1\\times\n10^{-5}\\,{\\rm eV}^2$ and $\\D m_{31}^2=2.3\\times 10^{-3}\\,{\\rm\neV}^2$, which of course is in good agreement with data. The\nrequirements for the neutrino mass splitting applied in\nRef.~\\cite{SCS} for the $3+2$ case, $0.1\\, {\\rm eV}^2 \\leq \\D\nm_{41}^2 \\leq \\D m_{51}^2\\leq 100 \\, {\\rm eV}^2$, are also\nsatisfied. This guarantees that the approximations used to derive\nEq.~(\\ref{Pmue}) are valid.\n\n\nFrom Eqs.~(\\ref{Udoubleprime}) and (\\ref{Uprime}), we obtain the\nfull rotation matrix $U = U''U'$, i.e.\n\\begin{equation}\n{\\scriptsize\nU=\\left(\n\\begin{array}{cccccc}\n0.82 & -0.54 & -0.04 & -0.14 & 0.06 & 0.08 \\\\\n0.33 & 0.60 & -0.71 & -0.09 & -0.01 & 0.10  \\\\\n0.42 & 0.59 & 0.69 & -0.03 & 0.05 & -0.03 \\\\\n-0.19 & 0 & 0.09 & -0.75 & 0.16 & 0.60 \\\\\n0.05 & -0.01 & 0.07 & 0.22 & -0.84 & 0.49 \\\\\n0.01 & 0 & 0.01 & 0.60 & 0.52 & 0.61 \\\\\n\\end{array} \\right).\n} \\label{U}\n\\end{equation}\nUsing $x_{ji}=1.27\\D m_{ji}^2({\\rm eV}^2) L({\\rm m})\/E({\\rm MeV})$\nwith $L\/E \\sim 1$, together with Eq.~(\\ref{Pmue}) for the $\\mu\n\\rightarrow e$ case, Eqs.~(\\ref{masses}) and~(\\ref{U}), one\nobtains $P(\\bar{\\nu}_\\mu \\rightarrow \\bar{\\nu}_e)=0.15\\%$, which\nis within 2$\\sigma$ from the LSND central value.\n\n\n$M^{(3)}$ in Eq.~(\\ref{M3x3num}) can be viewed as deviating from\n\\begin{equation}\n{\\scriptsize M^{(3)} = m_R \\left(\n\\begin{array}{cccccc}\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & 1 \\\\\n0 & 0 & 0 & 1 & 1 & 1 \\\\\n0 & 0 & 0 & 1 & 1 & 1 \\\\\n\\end{array} \\right) },\n\\label{M3x3app}\n\\end{equation}\nwhich has five zero eigenvalues and one nonzero eigenvalue equal\nto $3m_R \\sim 15$ eV, and diagonalized by\n\\begin{equation}\n{\\scriptsize U^{\\prime }=\\left(\n\\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{\\sqrt{2}} & 0 & \\frac{1}{\\sqrt{2}} \\\\\n0 & 0 & 0 & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} \\\\\n\\end{array} \\right) }.\n\\end{equation}\nTogether with $U''$ of Eq.~(\\ref{Uprpr}), one has\n\\begin{equation}\n{\\scriptsize U=\\left(\n\\begin{array}{cccccc}\nc_1 c_3 & s_1 c_3 & s_3 & 0 & 0 & 0 \\\\\n-s_1 c_2 - c_1 s_2 s_3 & c_1 c_2 - s_1 s_2 s_3 & s_2 c_3 & 0 & 0 & 0  \\\\\ns_1 s_2 - c_1 c_2 s_3 & -c_1 s_2 - s_1 c_2 s_3 & c_2 c_3 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{\\sqrt{2}} & 0 & \\frac{1}{\\sqrt{2}} \\\\\n0 & 0 & 0 & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} \\\\\n\\end{array} \\right) }.\n\\label{Uapp}\n\\end{equation}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.3\\textwidth]{chiSQ_s2fixed.eps}\n  \\vskip 0.5cm\n  \\caption{\n  Contour-plot of $\\chi^2$ vs the mixing angles $s_1$ and\n  $s_3$, with $\\ep_i$ and $r_{ij}$ as in\n  Eq.~(\\ref{M3x3num}), and $s_2 = 0.98$ held fixed.\n  The regions in different shades are only indicative, and should\n  not be interpreted as the $1\\s$, $2\\s$ and $3\\s$ regions,\n  as the rest of the parameters are fixed at the best fit values.\n  }\n  \\label{fig:chiSQ23}\n\\end{figure}\n\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth,height=0.45\\textwidth]{plotall_s2fixed.eps}\n  \\caption{\n  $P(\\bar{\\nu}_\\mu \\rightarrow \\bar{\\nu}_e)$, $\\sin^2{\\theta_{12}}$,\n  $\\sin^2{\\theta_{23}}$ and $\\sin^2{\\theta_{13}}$ vs $s_1$ and $s_3$,\n  corresponding to the lower right solution in Fig. 1, with\n  $\\ep_i$ and $r_{ij}$ fixed as in Eq.~(\\ref{M3x3num}), and $s_2=0.98$.}\n  \\label{fig:allplot}\n\\end{figure}\n\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth,height=0.45\\textwidth]{plotall_s2fixed2nd.eps}\n  \\caption{\n  Same as Fig. 2, but corresponding to upper left solution in Fig. 1.}\n  \\label{fig:allplot}\n\\end{figure}\n\nApplying this to Eqs.~(\\ref{Pmue}) for the $\\mu \\rightarrow e$\ncase, one sees that the transition probability $P(\\nu_\\mu\n\\rightarrow \\nu_e)$ would vanish. Deviations from\nEq.~(\\ref{M3x3app}) as realized in Eq.~(\\ref{M3x3num}) not only\nproduces the needed neutrino mass spectrum, finite transition\nprobability $P(\\nu_\\mu \\rightarrow \\nu_e)$ is also achieved.\n\nIn our analysis we did not take into account the null\nshort-baseline experiments (NSBL). Let us compare our results with\nthe ones obtained by doing a full analysis in Ref~\\cite{SCS} for\nthe (3+2) case. Using the rotation matrix elements $U_{e4}=-0.14$,\n$U_{\\mu4}=-0.09$, $U_{e5}=0.06$, $U_{\\mu5}=-0.01$, $U_{e6}=0.08$\nand $U_{\\mu6}=0.10$ as one can read from Eq.~(\\ref{U}), together\nwith Eq.~(\\ref{Pmue}) respectively for $\\mu \\rightarrow e$, $e\n\\rightarrow e$ and $\\mu \\rightarrow \\mu$, we obtain $P(\\nu_\\mu\n\\rightarrow \\nu_e)=0.0015$, $P(\\nu_e \\rightarrow \\nu_e)=0.89$ and\n$P(\\nu_\\mu \\rightarrow \\nu_\\mu)=0.93$ in the $L\/E \\sim 1$\napproximation. Our calculated values for oscillation appearance\nand disappearance probabilities can now be compared with the ones\nobtained by using the best fit values for the rotation matrix\nelements in the (3+2) case as in Ref.~\\cite{SCS}, $U_{e4}=0.121$,\n$U_{\\mu4}=0.204$, $U_{e5}=0.036$ and $U_{\\mu5}=0.224$. In the $L\/E\n\\sim 1$ approximation these give $P(\\nu_\\mu \\rightarrow\n\\nu_e)=0.0021$, $P(\\nu_e \\rightarrow \\nu_e)=0.95$ and $P(\\nu_\\mu\n\\rightarrow \\nu_\\mu)=0.84$. We remark that, although a full\nanalysis is needed to tell if our model is able to accommodate\nboth NSBL and LSND data, our predictions seem to be not too far\naway from the results of Ref.~\\cite{SCS}.\n\n\n\n\nFrom Eqs.~(\\ref{mixingangle}) and~(\\ref{U}), we find\n$\\sin^2{\\theta_{12}}=0.30$, $\\sin^2{\\theta_{23}}=0.52$, and\n\\begin{equation}\n\\sin^2{\\theta_{13}}=0.0018. \\label{theta13}\n\\end{equation}\nAs expected, the values for $\\sin^2{\\theta_{12}}$ and\n$\\sin^2{\\theta_{23}}$ are in good agreement with data, but the as\nyet unmeasured $\\sin^2{\\theta_{13}}$ turns out to be rather small.\n\n\n\n\n\\section{In Search of Sizable \\boldmath $\\sin^2{\\theta_{13}}$}\n\nTo investigate the possibility for a bigger value of\n$\\sin^2{\\theta_{13}}$, we restrict the $\\chi^2$ to the four inputs\nof $\\sin^2{\\theta_{ij}}$ and $P(\\bar{\\nu}_\\mu \\rightarrow\n\\bar{\\nu}_e)$. The mass spectrum is not affected by the rotation\nof Eq.~(\\ref{Uprpr}). With $\\ep_i$ and $r_{ij}$ given as in\nEq.~(\\ref{M3x3num}), we first fix $s_1=-0.57$ and perform a\n$\\chi^2$ fit vs $s_2$ and $s_3$. We iterate with fixing $s_2 =\n0.98$ ($s_3 = 0.8$) and minimize $\\chi^2$ vs $s_1$ and $s_3$\n($s_1$ and $s_2$). We find for both cases of fixing $s_1$ and\n$s_3$ to the values found in previous section,\n$\\sin^2{\\theta_{23}}$ is quite strongly dependent on $s_2$, and\nthe value around 0.98 is preferred. We thus illustrate with fixing\n$s_2 = 0.98$.\n\nIn Fig. 1 we show the contour plot of $\\chi^2$ vs $s_1, s_3$. The\nthree different shaded regions should not be interpreted as the\n$1\\s$, $2\\s$ and $3\\s$ regions, since we have fixed the rest of\nthe parameters to the best fit values. But they still give an\nindication of variations around the best fit region under the\nabove assumptions.\n\n\n\n\nIn Fig. 2 we plot the four quantities $P(\\bar{\\nu}_\\mu \\rightarrow\n\\bar{\\nu}_e)$, $\\sin^2{\\theta_{12}}$, $\\sin^2{\\theta_{23}}$ and\n$\\sin^2{\\theta_{13}}$ vs $s_1$ and $s_3$, for the solution on the\nlower right of Fig. 1. The same is plotted in Fig.~3 for the upper\nleft solution. Again, $\\ep_i$ and $r_{ij}$ are fixed as in\nEq.~(\\ref{M3x3num}), and $s_2$ is held fixed at 0.98. We see that\n$P(\\bar{\\nu}_\\mu \\rightarrow \\bar{\\nu}_e)$ can reach the one\n$\\sigma$ region and $\\sin^2{\\theta_{12}}$ is well within range.\nHowever, to push $\\sin^2{\\theta_{13}}$ beyond 0.01,\n$\\sin^2{\\theta_{23}}$ seems to wander away from maximal mixing of\n0.5, and values at $\\sim 0.4$ or 0.6 has to be tolerated. We note\nfurther that the sensitivity of $\\sin^2{\\theta_{23}}$ is to $s_1$,\nrather than $s_3$.\n\nWe conclude that $\\sin^2{\\theta_{13}}$ greater than 0.01 is\npossible, but seemingly not preferred. It is not clear whether\nthis is an artefact of not being able to do a real fit.\nNote that we have not checked explicitly whether constraints from\nshort baseline disappearance experiments are fully satisfied.\n\n\n\n\n\\section{Discussion and Conclusion}\n\nIt is tempting to consider whether mixing between the fourth and\nthe first three light charged lepton generations could modify the\nsituation with $\\sin^2{\\theta_{13}}$. But as already mentioned in\nthe Introduction, one needs to satisfy both bounds from direct\nsearch at LEP II and electroweak precision measurements. We have\npursued a numerical study, but find that, if we wish to keep the\nmixings sufficiently small so that the fourth active heavy\nneutrino will be semi-stable, no important change with respect to\nthe no-mixing case is observed. We note that the heavy neutrino\ncould be heavier than 50 GeV and still with suppressed mixing to\nlower generations, but then one would have to face electroweak\nprecision constraints. We note in passing that semi-stable heavy\nneutrinos are still of interest~\\cite{Rybka} to dark matter search\nexperiments, as a fourth heavy lepton was once a leading dark\nmatter candidate.\n\nIn summary, we have extended the eV seesaw scenario to four lepton\ngenerations. Taking the LSND scale as the right-handed seesaw\nscale $m_R \\sim$ eV, one has a heavy pseudo-Dirac neutrino with\nmass $m_N \\sim 50$ GeV, which largely decouples from other\ngenerations, and is relatively stable. One effectively has a $3+3$\nsolution to the LSND anomaly, where we illustrate with numerical\nsolutions. As a possible outcome, our numerical study indicates\nthat the third mixing angle, $\\sin^2{\\theta_{13}}$, seems to be\nless than 0.01.\n\n\n\\vskip 0.3cm \\noindent{\\bf Acknowledgement}.\\ \\ This work is\nsupported in part by NSC-94-2112-M-002-035 and HPRN-CT-2002-00292.\nWe thank G. Raz, T. Volansky and R. Volkas for useful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{A}{} traditional approach for transmitting digital\ninformation over a communication link with high spectral efficiency\n($\\eta\\geq2$ bit\/s\/Hz) is to combine good error-correcting code with\nhigh-order modulation. Over last two decades, several capacity-approaching\ncoded modulation techniques have been proposed, including low-density\nparity check (LDPC) coded modulation {[}1, 2{]}, turbo-coded modulation\nand turbo trellis-coded modulation {[}3{]}. In this paper, we propose\nan alternative bandwidth-efficient modulation technique, which does\nnot rely on any error correction coding in the traditional sense,\nwhile still achieving error rates close to the best known coded modulation\nschemes. Our method is based on serial concatenation of an orthogonal\nlinear transform, such as DFT or WHT, with memoryless nonlinearity.\nWe demonstrate that such a simple signal construction may exhibit\nproperties of random code ensemble, as a result approaching channel\ncapacity if decoded using a maximum-likelihood (ML) decoder. \n\nWhen the proposed modulation scheme relies on DFT transform, it can\nbe viewed as an OFDM signal distorted by very strong memoryless nonlinearity.\nIn the context of OFDM, any nonlinearity is usually considered a highly\nundesirable phenomenon. In fact, the sensitivity of OFDM technology\nto nonlinear distortions is usually cited as a major drawback of OFDM.\nNonetheless, in {[}4{]}, the authors theoretically proved that the\nbit error rate (BER) performance of OFDM transmission subjected to\nstrong memoryless nonlinearity can outperform linear transmission,\nprovided that an optimal ML-receiver is used. Moreover, in {[}5{]},\nwe proposed a practical message-passing receiver for nonlinearly distorted\nOFDM signals, showing that the BER performance of hard-clipped OFDM\nsignals can be up to 2 dB better than linear OFDM transmission. In\nthis paper, we go one step further to demonstrate that if the uncoded\nOFDM signal is distorted by memoryless nonlinearity with certain properties,\nthe resultant signal waveform may achieve BER performance close to\nthe best known capacity\\textendash approaching coded modulation techniques.\nThe same performance can also be achieved by replacing DFT in the\ntransmitter with WHT, resulting in simpler hardware\/software implementation\nof the encoder and the decoder.\n\n\\section{Proposed modulation technique}\n\n\\subsection{Achieving capacity by serial concatenation of orthogonal transform\nand memoryless nonlinearity}\n\nConsider the modulation scheme illustrated in Figure \\ref{fig:direct_modulator}.\nIn the transmitter, $N$ uncoded real or complex baseband modulation\nsymbols $\\ensuremath{\\boldsymbol{\\mathbf{x}}=\\left\\{ x_{k}\\right\\} }$\n(e.g. $M$-QAM, or $M$-PAM) are first transformed by means of orthogonal\nblock transform and then passed through memoryless nonlinearity block.\nThe signal at the modulator output can be expressed as:\n\n\\begin{equation}\n\\ensuremath{s_{n}=f\\left(z_{n}\\right),\\quad n=0,1,...,N-1}\\label{eq: main_modulator}\n\\end{equation}\nwhere $f(z)$ is a memoryless nonlinear function, and\n\n\\begin{equation}\n\\ensuremath{{\\bf z}={\\bf Fx},}\n\\end{equation}\nwhere $\\mathbf{z}=\\left\\{ z_{n}\\right\\} $, and $\\mathbf{F}$ is a\n$N\\times N$ (unitary) orthogonal transform matrix. In this paper,\nwe consider two types of transforms: Walsh-Hadamard transform, and\ndiscrete Fourier transform (real-valued or complex).\n\nLet us now assume that the memoryless nonlinear function $f(z)$ can\nbe represented as $\\ensuremath{f\\left(z\\right)=\\underbrace{g\\left(...g\\left(g\\left(z\\right)\\right)\\right)}_{l-times}}$,\nwhere $g(z)$ is a deterministic chaotic map. Thus, $f(z)$ has certain\nproperties of chaotic iterated functions, namely, \\emph{sensitive\ndependence on initial conditions} {[}6{]}. Under such assumptions,\nand if $N\\rightarrow\\infty$ and $l\\rightarrow\\infty$, the ensemble\nof waveforms generated by (\\ref{eq: main_modulator}) possesses all\nmajor properties of a random code ensemble, because even small modifications\nin the message sequence $\\ensuremath{\\left\\{ x_{k}\\right\\} }$ (e.g.,\nin a single bit) lead to: \n\\begin{description}\n\\item [{a)}] small modifications, at least, for all samples of the intermediate\nsignal $z_{n}$ due to the spreading properties of the orthogonal\ntransform operation, and \n\\item [{b)}] large (and pseudo-random) modifications for all samples of\nwaveform $s_{n}$ due to the properties of the nonlinear function\n$f(z)$. \n\\end{description}\nTherefore, we conjecture that where $f(z)$ has the above-mentioned\nproperties if the waveforms generated by (\\ref{eq: main_modulator}) are\ndemodulated using a ML decoder, system performance may approach channel\ncapacity. Unfortunately, brute-force, maximum-likelihood decoding\nof (\\ref{eq: main_modulator}) is prohibitively complex, generally\nof the order $O(M^{N})$, where $M$ is the modulation order. However,\nas in many practical applications, we can rely on belief-propagation\ntechniques to approximate ML-decoding. A perfect candidate for decoding\nthe signals generated by (\\ref{eq: main_modulator}) is the generalized\napproximate message passing (GAMP) algorithm {[}7{]}. Having already\nbeen applied to the decoding of clipped OFDM signals in {[}5{]}, it\nhas demonstrated good performance and reasonably good convergence\nbehavior. \n\n\\begin{figure}[tbh]\n\\includegraphics[scale=1.1]{modulator}\\caption{\\label{fig:direct_modulator}Proposed modulation scheme}\n\\end{figure}\n\n\n\\subsection{GAMP algorithm and its modifications}\n\nIn the additive white Gaussian noise (AWGN) channel, the model of\nthe received signal can be expressed as\n\n\\begin{equation}\n\\ensuremath{y_{n}=f\\left(z_{n}\\right)+w_{n},}\\quad n=0,1,...,N-1\\label{eq:gamp_model}\n\\end{equation}\nwhere $w_{n}$ is the AWGN term with zero mean and variance $\\mu^{w}$.\nModel (\\ref{eq:gamp_model}) is equivalent to a general problem formulation\nfor the GAMP algorithm {[}7{]}, which belongs to a class of Gaussian\napproximations of loopy belief propagation for dense graphs. In our\ndecoder implementation, we use the sum-product variant of the GAMP\nalgorithm, which approximates minimum mean-squared error estimates\nof $\\mathbf{x}$ and $\\mathbf{z}$. \n\nDuring simulation study, we discovered that the convergence behavior\nand bit and frame error rate (FER) performance of the conventional\nGAMP algorithm applied to the decoding of (\\ref{eq:gamp_model}) may\nbe improved by several simple algorithm modifications. In particular,\nwe used damped version of the GAMP algorithm, which is known to improve\nconvergence for certain mixing matrices {[}8{]}, and introduced scaling\nfactor for AWGN term variance $\\mu^{w}\\rightarrow\\alpha\\mu^{w}$.\nIn addition, we implemented the final decision selection and post-processing\nscheme as will be explained in the next section that helped us overcome\ncertain deficiencies of a conventional GAMP algorithm. To reduce computational\ncomplexity we opted to use a version of GAMP algorithm with scalar\nstep-sizes {[}7{]}. Note that the scalar step-size version of the\nGAMP algorithm is optimal for WHT and complex DFT since in both cases\n$\\left|F_{n,k}\\right|^{2}=1$. Moreover,  our experimental study revealed\nthat the performance loss for real DFT case ($\\left|F_{n,k}\\right|^{2}\\neq1$)\nwas also marginal. \n\nThe GAMP algorithm adapted to our problem is summarized below. The\nalgorithm generates a sequence of estimates ${\\bf \\hat{x}}\\left(t\\right)$,\n${\\bf \\hat{z}}\\left(t\\right)$, for $t=1,2,...$ through the following\nrecursions:\n\n\\emph{Parameters:}\n\n$t_{max}$ the maximum number of iterations, \n\n$\\alpha$ the noise scaling factor, \n\n$\\beta$ the damping factor.\n\n\\emph{Step 1) Initialization:}\n\n$t=1$, ${\\bf \\hat{x}}\\left(1\\right)=\\boldsymbol{0}$, $\\tilde{\\mathbf{x}}\\left(1\\right)=\\boldsymbol{0}$,\n${\\bf \\boldsymbol{\\mu}}^{x}\\left(1\\right)=\\boldsymbol{1}$, ${\\bf \\hat{s}}\\left(0\\right)=\\boldsymbol{0}$\n\n\\emph{Step 2) Estimation of output nodes:}\n\n\\begin{equation}\n\\mu_{n}^{p}\\left(t\\right)=\\frac{1}{N}\\sum\\limits _{k=0}^{N-1}\\mu_{k}^{x}\\left(t\\right),\\forall n\\label{eq:gamp_algo_start}\n\\end{equation}\n\n\\begin{equation}\n\\hat{p}_{n}\\left(t\\right)=\\sum\\limits _{k=0}^{N-1}F_{n,k}\\hat{x}_{k}\\left(t\\right)-\\mu_{n}^{p}\\left(t\\right)\\hat{s}_{n}\\left(t-1\\right),\\forall n\n\\end{equation}\n\n\\begin{equation}\n\\hat{z}_{n}\\left(t\\right)=\\frac{1}{C}\\int\\limits _{-\\infty}^{\\infty}ze^{-\\frac{\\left(y_{n}-f\\left(z\\right)\\right)^{2}}{2\\alpha\\mu^{w}}-\\frac{\\left(\\hat{p}_{n}\\left(t\\right)-z\\right)^{2}}{2\\mu_{n}^{p}\\left(t\\right)}}dz,\\forall n\\label{eq:Integral_start}\n\\end{equation}\n\n\\begin{equation}\n\\mu_{n}^{z}\\left(t\\right)=\\frac{1}{C}\\int\\limits _{-\\infty}^{\\infty}z^{2}e^{-\\frac{\\left(y_{n}-f\\left(z\\right)\\right)^{2}}{2\\alpha\\mu^{w}}-\\frac{\\left(\\hat{p}_{n}\\left(t\\right)-z\\right)^{2}}{2\\mu_{n}^{p}\\left(t\\right)}}dz-\\left(\\hat{z}_{n}\\left(t\\right)\\right)^{2},\\forall n\n\\end{equation}\nwhere \n\n\\begin{equation}\nC=\\int\\limits _{-\\infty}^{\\infty}e^{-\\frac{\\left(y_{n}-f\\left(z\\right)\\right)^{2}}{2\\alpha\\mu^{w}}-\\frac{\\left(\\hat{p}_{n}\\left(t\\right)-z\\right)^{2}}{2\\mu_{n}^{p}\\left(t\\right)}}dz,\\forall n\\label{eq:Integral_end}\n\\end{equation}\n\n\\begin{equation}\n\\hat{s}_{n}\\left(t\\right)=\\left(1-\\beta\\right)\\hat{s}_{n}\\left(t-1\\right)+\\beta\\dfrac{\\hat{z}_{n}\\left(t\\right)-\\hat{p}_{n}\\left(t\\right)}{\\mu_{n}^{p}\\left(t\\right)},\\forall n\n\\end{equation}\n\n\\begin{equation}\n\\mu_{n}^{s}\\left(t\\right)=\\left(1-\\beta\\right)\\mu_{n}^{s}\\left(t-1\\right)+\\beta\\frac{1-\\frac{\\mu_{n}^{z}\\left(t\\right)}{\\mu_{n}^{p}\\left(t\\right)}}{\\mu_{n}^{p}\\left(t\\right)},\\forall n\n\\end{equation}\n\n\\emph{Step 3) Estimation of input nodes}: \n\n\\begin{equation}\n\\tilde{x}_{k}\\left(t\\right)=\\left(1-\\beta\\right)\\tilde{x}_{k}\\left(t-1\\right)-\\beta\\hat{x}_{k}\\left(t\\right)\n\\end{equation}\n\n\\begin{equation}\n\\mu_{k}^{r}\\left(t\\right)=\\left(\\frac{1}{N}\\sum\\limits _{n=0}^{N-1}\\mu_{n}^{s}\\left(t\\right)\\right)^{-1},\\forall k\n\\end{equation}\n\n\\begin{equation}\n\\hat{r}_{k}\\left(t\\right)=\\tilde{x}_{k}\\left(t\\right)+\\mu_{k}^{r}\\left(t\\right)\\sum\\limits _{n=0}^{N-1}F_{n,k}^{*}\\hat{s}_{k}\\left(t\\right),\\forall k\n\\end{equation}\n\n\\begin{equation}\n\\hat{x}_{k}\\left(t+1\\right)=\\sum\\limits _{m=1}^{M}d_{m}P_{m,k},\\forall k\n\\end{equation}\n\n\\begin{equation}\n\\mu_{k}^{x}\\left(t+1\\right)=\\sum\\limits _{m=1}^{M}\\left(d_{m}-\\hat{x}_{k}\\left(t+1\\right)\\right)^{2}P_{m,k},\\forall k\\label{eq:gamp_algo_end}\n\\end{equation}\nwhere\n\n\\begin{equation}\nP_{m,k}=\\frac{e^{-\\frac{\\left(d_{m}-\\hat{r}_{k}\\left(t\\right)\\right)^{2}}{2\\mu_{k}^{r}\\left(t\\right)}}}{\\sum\\limits _{l=1}^{M}e^{-\\frac{\\left(d_{l}-\\hat{r}_{k}\\left(t\\right)\\right)^{2}}{2\\mu_{k}^{r}\\left(t\\right)}}},\n\\end{equation}\n$M$ is the number of points in the signal constellation, and $\\left\\{ d_{m}\\right\\} $\nis the vector of constellation points. For example, for 2-PAM modulation,\n$\\left\\{ d_{m}\\right\\} =\\left[-1{\\rm \\quad+}1\\right]$, and for 4-PAM\nmodulation $\\left\\{ d_{m}\\right\\} =\\left[-3\\mathord{\\left\/\\vphantom{-3{\\sqrt{5}}}\\right.\\kern -\\nulldelimiterspace}\\sqrt{5}\\quad-1\\mathord{\\left\/\\vphantom{-1{\\sqrt{5}}}\\right.\\kern -\\nulldelimiterspace}\\sqrt{5}{\\rm \\quad+}1\\mathord{\\left\/\\vphantom{1{\\sqrt{5}}}\\right.\\kern -\\nulldelimiterspace}\\sqrt{5}{\\rm \\quad+}{\\rm 3}\\mathord{\\left\/\\vphantom{{\\rm 3}{\\sqrt{5}}}\\right.\\kern -\\nulldelimiterspace}\\sqrt{5}\\right]$. \n\nSteps (\\ref{eq:gamp_algo_start})\\textendash (\\ref{eq:gamp_algo_end})\nare repeated with $t\\rightarrow t+1$ until $t_{max}$ iterations\nhave been performed. \n\nWe presented here the GAMP algorithm version for the real-valued modulation\nand real orthogonal transform (e.g. $M$-PAM and WHT or real-DFT).\nNonetheless, the extension to the complex case is straightforward.\nMoreover, in case of Cartesian-type nonlinearity $f(z)$ the complexity\nof the decoding algorithm with $N$ complex input symbols is essentially\nthe same as the complexity of the real-valued algorithm with $2N$\ninput symbols.\n\nMore details on the GAMP algorithm and its thorough analysis can be\nfound in {[}7{]}. \n\n\\subsection{Choosing optimal nonlinearity $f(z)$}\n\nChoosing the optimal shape of nonlinearity $f(z)$ is not easy, requiring\na balance between two conflicting requirements. Firstly, the nonlinear\nfunction should be reasonably \\textquotedbl{}chaotic\\textquotedbl{}\nto guarantee sensitivity to initial conditions and, therefore, random-like\nproperties of the coded waveforms. On the other hand, our experimental\nstudy implies that a \\textquotedbl{}truly chaotic\\textquotedbl{} shape\nof nonlinearity $f(z)$ precludes GAMP algorithm from converging to\nthe ML-solution. Therefore, we adopted an \\emph{ad hoc} procedure\nto select and optimize the shape of the memoryless nonlinearity $f(z)$.\nOur choice of nonlinearity $f(z)$ relies on two key ideas: \n\\begin{itemize}\n\\item $f(z)$ should contain a linear or almost linear region around $f(0)$\nto allow the message passing decoder to converge. \n\\item $f(z)$ should \\emph{resemble} a chaotic iterated function in other\nregions to guarantee sensitivity to initial conditions. \n\\end{itemize}\nFor a general representation of $f(z)$, we suggest using the flexible,\npiece-wise linear model:\n\n\\begin{equation}\n\\ensuremath{f\\left(z\\right)=\\left\\{ \\begin{array}{l}\n{\\mathop{\\rm sgn}}\\left(z\\right)\\left(a_{0}\\left|z\\right|+b_{0}\\right),{\\rm \\quad if}\\;0\\le\\left|z\\right|<T_{1}\\\\\n{\\mathop{\\rm sgn}}\\left(z\\right)\\left(a_{1}\\left|z\\right|+b_{1}\\right),{\\rm \\quad if\\;}T_{1}\\le\\left|z\\right|<T_{2}\\\\\n...\\\\\n{\\mathop{\\rm sgn}}\\left(z\\right)\\left(a_{i-1}\\left|z\\right|+b_{i-1}\\right),{\\rm \\quad if\\;}T_{i-1}\\le\\left|z\\right|<T_{i}\\\\\n{\\mathop{\\rm sgn}}\\left(z\\right)\\left(a_{i}\\left|z\\right|+b_{i}\\right),{\\rm \\quad if\\;}\\left|z\\right|\\ge T_{i}\n\\end{array}\\right.}\\label{eq:Nonlinearity equation}\n\\end{equation}\nwith some parameters $\\boldsymbol{a}=G_{0}\\left\\{ a_{0},a_{1},...,a_{i}\\right\\} $,\n$\\boldsymbol{b}=\\left\\{ b_{0},b_{1},...,b_{i}\\right\\} $, and $\\boldsymbol{T}=G_{0}^{-1}\\left\\{ 0,T_{1},T_{2},...,T_{i}\\right\\} $,\nwhere $G_{0}$ is a scaling factor that does not affect the ``shape''\nof nonlinearity $f(z)$. For complex modulation (i.e. complex DFT),\nwe apply function (\\ref{eq:Nonlinearity equation}) separately to\nthe real and imaginary components of $z$ (Cartesian-type nonlinearity).\nTo select the parameters of (\\ref{eq:Nonlinearity equation}), we\ndevised dozens of different functions $f(z)$ that satisfy the above\nmentioned heuristic requirements, and then optimized parameter $G_{0}$\nat our target BER and FER (namely, BER$\\leq10^{-5}$, FER$\\leq$1\\%).\nThis non-exhaustive search procedure yielded several ``good'' sets\nof parameters $\\boldsymbol{a}$, $\\boldsymbol{b}$, and $\\boldsymbol{T}$,\nsome of which are summarized in Table 1 and illustrated in Figure\n\\ref{fig:Nonlinearities}.\n\n\\begin{table}[tbh]\n\\caption{\\label{tab:Nonlinearities}Example parameters of nonlinearity $f(z)$}\n\n\\centering{\n\\begin{tabular}{|c|>{\\raggedright}p{0.85\\columnwidth}|}\n\\hline \n\\emph{No.} & \\emph{Parameters}\\tabularnewline\n\\hline \n1 & $\\boldsymbol{a}G_{0}^{-1}$= \\{1 2 2 -2 -2 2 2 -2 -2 -0.5\\}\\newline$\\boldsymbol{b}$\n= \\{0 -2 -2.5 4 4.5 -4 -4.5 6 6.5 2.5\\}\\newline$\\boldsymbol{T}G_{0}$=\n\\{0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3\\}; $G_{0}=0.53$\\tabularnewline\n\\hline \n2 & $\\boldsymbol{a}G_{0}^{-1}$= \\{1 2 2 -2 -2 2 2 -2 -2 -0.5\\}\\newline$\\boldsymbol{b}$\n= \\{0 -2 -3.5 4 3.5 -4 -4.5 6 6.5 2.5\\}\\newline$\\boldsymbol{T}G_{0}$=\n\\{0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3\\}; $G_{0}=0.5125$\\tabularnewline\n\\hline \n3 & $\\boldsymbol{a}G_{0}^{-1}$= \\{1.25 2 2 -2 -2 2 2 -2 -2 -0.5\\}\\newline$\\boldsymbol{b}$\n= \\{0 -1.6 -3.1 3.6 3.1 -3.6 -4.1 5.6 6.1 2.4\\}\\newline$\\boldsymbol{T}G_{0}$=\n\\{0 0.8 1.05 1.3 1.55 1.8 2.05 2.3 2.55 2.8\\}; $G_{0}=0.415$\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.61]{nonlinearity2}\n\n\\caption{\\label{fig:Nonlinearities}Three examples of nonlinear functions $f(z)$:\nnonlinearity 1 (top), nonlinearity 2 (middle), and nonlinearity 3\n(bottom).}\n\\end{figure}\n\n\n\\section{Simulation results}\n\nThe performance of the proposed modulation scheme with the GAMP-based\ndecoder was studied by means of Monte-Carlo simulation. Our initial\nsimulation-based study revealed that the conventional GAMP algorithm\n(with $\\alpha=1$, and $\\beta=1$) exhibits several deficiencies,\nin particular, relatively high error-floor, slow convergence and occasional\ninstability problems (divergence with the increase of number of iterations).\nIt turns out that the error-floor issue was, at least, in part due\nto the decoder algorithm. To alleviate these deficiencies we implemented\nseveral enhancement techniques. Firstly, we appended the cyclic-redundancy\ncheck (CRC) block to each data frame and used it for the early stopping\ncriterion. Secondly, after each decoding iteration we calculated the\nEuclidean distance $E(t)$ between the received vector $\\{y_{n}\\}$\nand the reconstructed waveform $f\\left(\\sum\\limits _{k=0}^{N-1}F_{n,k}\\hat{x}_{k}\\left(t\\right)\\right)$,\nand if a CRC error was detected after $t_{max}$ iterations the final\ndecision was based on the vector $\\left\\{ \\hat{x}_{k}\\left(t\\right)\\right\\} $\nthat corresponded to the minimum Euclidean distance $E(t)$. Although\nthis procedure did not improve FER, it minimized the BER for incorrectly\ndecoded frames. Thirdly, we experimentally discovered that the decoder\nconvergence speed can be significantly improved by using noise scaling\nfactor $\\alpha<1$ simultaneously with damping ($\\beta<1$). However,\nsuch a technique could result in occasional algorithm divergence,\ntherefore to  achieve better performance we implemented the following\nprocedure: $t_{max}\/2$ iterations were performed using the modified\nGAMP algorithm ($\\alpha<1$, $\\beta<1$), and if the decoder could\nnot converge within the first $t_{max}\/2$ iterations, all internal\nvariables were reset and the subsequent $t_{max}\/2$ iterations were\nperformed with the conventional GAMP settings ($\\alpha=1$, $\\beta=1$).\nThe overall decoding algorithm flowchart is depicted in Figure \\ref{fig:Decoding-algorithm-flowchart}.\n\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.53]{algorithm}\\caption{\\label{fig:Decoding-algorithm-flowchart}Decoding algorithm flowchart}\n\\end{figure}\n\nOur simulation results indicate that the performance of the proposed\nscheme with WHT, real DFT or complex DFT with Cartesian-type nonlinearity\nis almost identical. Therefore, from complexity point of view it seems\npreferable to use WHT, and, hence, most of our simulation results\nreported here were obtained for WHT-based scheme.\n\nFigure  \\ref{fig:BER-vs-EbNo-capacity} illustrates the BER vs $E_{b}\/N_{0}$\ncurves for  2-PAM input modulation with three nonlinearities (Table\n\\ref{tab:Nonlinearities}\/Figure \\ref{fig:Nonlinearities}) and different\nvalues of $N$. In all simulations, the variance of $\\{z_{n}\\}$ was\nnormalized to 1, the maximum number of iterations ($t_{max}$) was\nset to 100, and integrals (\\ref{eq:Integral_start})\\textendash (\\ref{eq:Integral_end})\nwere approximated using numerical summation. At the initialization\nstep the parameters $\\alpha$, $\\beta$ were set to $\\alpha=0.71$,\n$\\beta=0.875$. \n\nRemarkably, for the selected nonlinearity parameters $G_{0}$, $\\boldsymbol{a}$,\n$\\boldsymbol{b}$, $\\mathbf{T}$ the proposed modulation scheme exhibits\nbehavior similar to random-like codes with the presence of a waterfall\nregion and error-floor, and, as expected, performance improves for\nlarger frame sizes. Moreover, the proposed modulation scheme with\nnonlinearity 3 and $N$=16384 achieves target BER=$10^{-5}$ at $E_{b}\/N_{0}=3.3$\ndB, which represents 6.3 dB gain over uncoded 2-PAM or 4-QAM modulation\nand is only about 1.5 dB away from the \\emph{unconstrained} AWGN channel\ncapacity. These results are better or within $0.1\\div0.3$ dB of the\nperformance of the capacity-approaching, bandwidth-efficient modulation\nschemes with $\\eta=2$ bit\/s\/Hz reported in {[}1-3, 9, 10{]}. BER\ncurves for some of these advanced coded modulation schemes with comparable\nframe sizes are reproduced in Figure \\ref{fig:BER-vs-EbNo} for comparison. \n\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.57]{n1K_16K_capacity}\\caption{\\label{fig:BER-vs-EbNo-capacity}BER vs $E_{b}\/N_{0}$ for the proposed\nmodulation scheme with different nonlinearities and frame sizes in\nAWGN channel ($\\eta=2$ bit\/s\/Hz)}\n\\end{figure}\n\nAlthough we set $t_{max}=100$, the average number of decoder iterations\nwas significantly lower. For example, at $BER=10^{-5}$ the average\nnumber of decoder iterations was about 26 for the frame size $N$=16384,\nand about 17 for the frame size $N$=4096, and only 7 iterations for\nthe frame size $N$=1024.\n\nA considerable performance improvement over uncoded modulation was\nalso observed for 4-PAM or 16-QAM input modulation ($\\eta=4$ bit\/s\/Hz).\nHowever, in case of 4-PAM\/16-QAM, the  GAMP algorithm is sub-optimal\nin terms of BER performance, and the example nonlinearities that we\nuse for 4-QAM\/2-PAM (Table \\ref{tab:Nonlinearities}) are not optimal\neither. Therefore, application of the proposed technique to high-order\nmodulation formats ($\\eta\\geq4$ bit\/s\/Hz) is an open research topic. \n\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.46]{7a_n16384_compare_w_papers_light}\n\n\\caption{\\label{fig:BER-vs-EbNo}Performance comparison of the proposed technique\n(WHT, Nonlinearity 3, $N$=16384) with state-of-the-art coded modulation\nschemes ($2$ bit\/s\/Hz):\\protect \\\\\n1) ARJ4A LDPC, $N$=16384, 16-APSK {[}9{]}, \\protect \\\\\n2) ARJ4A LDPC, $N$=16384, 8-PSK {[}9{]}, \\protect \\\\\n3) Regular LDPC, BICM, $d_{v}$=3, $n$=20000, 4-PAM {[}1{]}\\protect \\\\\n4) Regular LDPC, BICM, $d_{v}$=3, $n$=10000, 4-PAM {[}1{]}\\protect \\\\\n5) Irregular LDPC, BICM, $d_{v}$=15, $n$=20000, 4-PAM {[}1{]}\\protect \\\\\n6) Irregular LDPC, BICM, $d_{v}$=15, $n$=10000, 4-PAM {[}1{]}\\protect \\\\\n7) eIRA LDPC, BICM, $n$=10000, 4-PAM {[}10{]}\\protect \\\\\n8) eIRA LDPC, BICM, $n$=9000, 8-PSK {[}10{]}\\protect \\\\\n9) Turbo TCM, $N$=10000, 8-PSK {[}3{]}}\n\\end{figure}\n\n\n\\section{Discussion}\n\nWe believe that the proposed joint coding-modulation technique might\nbe useful in some wired and wireless communication applications, since\nit has several interesting properties:\n\\begin{itemize}\n\\item Good BER performance: As illustrated in the previous section the BER\nperformance is on par with that of the best known coded modulation\nschemes.\n\\item Relatively low decoder complexity: The GAMP-based decoder complexity\nper iteration is dominated by two orthogonal transform operations,\nwhich, in case of fast WHT, require only $N\\log_{2}(N)$ real additions\nor subtractions. The input\/output nonlinear step is generally a scalar\noperation with complexity of order $O(N)$. Although, in our simulation\nmodel, we used numerical integration to approximate integrals (\\ref{eq:Integral_start})\\textendash (\\ref{eq:Integral_end}),\nthese can be expressed in closed-form using tabulated Gauss error\nfunction, and, consequently, can be simplified or implemented via\nlookup tables. It is also possible to use simpler max-sum version\nof the GAMP algorithm to trade-off performance and complexity {[}7{]}. \n\\item The choice of nonlinearity $f(z)$ may be tailored to the requirements\nof the transmission system in order to improve overall system efficiency.\nFor example, $f(z)$ may be selected to improve power conversion efficiency,\nor illumination-to-communication conversion efficiency in visible\nlight communication applications {[}11{]}.\n\\item An interesting application of the proposed modulation scheme is to\nuse it in combination with a conventional, linear OFDM transmitter,\nas illustrated in Figure \\ref{fig:precoder_for_ofdm}. Such a system\narrangement may result in the OFDM signal with reduced PAPR. Due to\nthe presence of nonlinearity $f(z)$, the distribution of signal samples\nat the output of the modulator illustrated in Figure \\ref{fig:precoder_for_ofdm}\nresembles the distribution of $M$-QAM\/$M$-PAM signal affected by\nadditive Gaussian noise. It can be shown that if the nonlinearity\n$f(z)$ has a relatively large linear region the PAPR of such a signal\nwill be much lower than that of a conventional OFDM signal. Figure\n\\ref{fig:CCDF} compares the complementary cumulative distribution\nfunctions (CCDF) of PAPR for signals generated by the transmission\nsystem illustrated in Figure \\ref{fig:precoder_for_ofdm} with nonlinearity\n1 (complex-valued OFDM with Cartesian-type nonlinearity), and a conventional\nOFDM system. We analyzed CCDF for Nyquist sampled waveforms and four-times\noversampled waveforms, since four-times oversampling provides a good\napproximation of the continuous-time PAPR {[}12{]}. As may be seen,\nat probability $10^{-4}$ the PAPR of the continuous signal generated\nby the transmission system illustrated in Figure \\ref{fig:precoder_for_ofdm}\nis approximately 2.3 dB lower than the PAPR of a conventional OFDM\nsignal. The difference is increased to 3.9dB for Nyquist-sampled waveforms.\n\\end{itemize}\nIt should be noted that the proposed method has several limitations\nand open issues. Firstly, in this paper, we focused primarily on the\ncase $\\eta=2$ bit\/s\/Hz. One possible way to apply this technique\nto systems with higher spectral efficiency is to use the output waveform\npuncturing. This approach (i.e. random puncturing) seems to work well\nup to $\\eta=2.5\\div3$ bit\/s\/Hz. It is also possible to use higher-order\ninput modulations (16-QAM\/4-PAM), however, as we mentioned earlier,\nextension of this technique to spectral efficiency $\\eta\\geq4$ bit\/s\/Hz\nis still an open research topic. Secondly, the GAMP algorithm is based\non Gaussian and quadratic approximations, and, therefore, it is apparently\nsub-optimal, especially, for small frame sizes. We were able to improve\nperformance of the GAMP-based decoder using several \\emph{ad hoc}\ntechniques. However, it is still unclear, how close the GAMP decoder\nperformance is to the theoretical ML performance. Finally, our choice\nof nonlinearity $f(z)$ is generally based on a heuristic and experimental\napproach. The problem here is that the performance of the proposed\nscheme is limited not only by distance distribution of encoded waveforms\nbut also by non-idealities of the decoding algorithm, and this fact\ngreatly complicates optimization strategy. \n\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.8]{ofdm_precoder}\n\n\\caption{\\label{fig:precoder_for_ofdm}Proposed modulation scheme as a pre-coder\nfor a conventional OFDM transmitter}\n\\end{figure}\n\\begin{figure}[tbh]\n\\includegraphics[scale=0.5]{ccdf_papr}\n\n\\caption{\\label{fig:CCDF}CCDF of PAPR for the proposed pre-coded OFDM (Figure\n\\ref{fig:precoder_for_ofdm}) with $N$=1024 and nonlinearity \\#1:\\protect \\\\\n1) Conventional OFDM (Nyquist sampling)\\protect \\\\\n2) Conventional OFDM (four-times oversampling)\\protect \\\\\n3) OFDM with the proposed modulation (Nyquist sampling)\\protect \\\\\n4) OFDM with the proposed modulation (four-times oversampling)}\n\\end{figure}\n\n\\balance\n\n\\section{Conclusions}\n\nIn this paper, we proposed a novel joint coding-modulation technique\nbased on serial concatenation of orthogonal linear transformation\n(e.g., WHT or DFT) with memoryless nonlinearity. We demonstrated that\nsuch a simple signal construction may exhibit properties of a random\ncode ensemble, as a result approaching channel capacity. Our computer\nsimulations confirmed that if the decoder relies on the approximate\nmessage passing algorithm, the proposed modulation technique exhibits\nperformance on par with state-of-the-art coded modulation schemes\nthat use capacity-approaching component codes. The proposed technique\ncould be extended to modulation formats with higher spectral efficiency\n($\\eta\\geq4$ bit\/s\/Hz) and other types of orthogonal transformations,\noffering one possible direction for our future research.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Musical Organization: Strings \\& Schemes}\nIn the context of expressive communication systems like natural language or tonal music, it might help to clarify what the term `organization' really means, at least as it is intended here. In the simplest sense, musical organization refers to the relationships between events on the musical surface, be they notes, chords, motives, phrases, or any other coherent `units' of that organization. Following the classes of mental structure described by \\textcite{Mandler1979}, we might represent the connections between these events as unordered (or \\textit{coordinate}) relations, such as the members of a major triad, temporally ordered (or \\textit{proordinate}) relations, such as the progression from dominant to tonic at the end of a phrase (i.e., V--I), or hierarchical (or \\textit{superordinate}\/\\textit{subordinate}) relations, such as the prolongation of a given harmony through other (subordinate) harmonies (e.g., I--V$^4_3$--I$^6$). Bearing these relational types in mind, tonal harmony is thus an \\textit{emergent} organizational system in that superordinate structures emerge out of the coordinate and proordinate relations among events on the surface. The real challenge here, then, is to model these relational types using strings.\n\nAmong computer scientists, a \\textit{string} is a finite set of discrete symbols---a database of nucleic acid sequences, a dictionary of English words, or for the purposes of this study, a corpus of Haydn string quartets. In the first two cases, the mapping between the individual character or word in a printed text and its symbolic representation in a computer database is essentially one-to-one. Music encoding is considerably more complex. Notes, chords, phrases, and the like are characterized by a number of different features, so digital encodings of individual events must concurrently represent multiple properties of the musical surface. To that end, many symbolic formats encode standard music notation as a series of discrete event sequences (i.e., strings) in an $m \\times n$ matrix, where $m$ denotes the number of events in the symbolic representation (e.g., notes as notated in a score), and $n$ refers to the number of encoded features or attributes (e.g., pitch, onset time, rhythmic duration, etc.).\n\nTo model the coordinate relations (i.e., vertical sonorities) associated with tonal harmony using unidimensional strings, corpus studies typically limit the investigation to a particular chord typology from music theory (e.g., Roman numerals, figured bass nomenclature, or pop chord symbols), and then identify chord events using either human annotators \\autocite{Burgoyne2012,Declercq:2011,Tymoczko:2011}, or rule-based computational classifiers \\autocite{Temperley:1999,Rowe:2001}. Yet unfortunately, existing typologies depend on a host of assumptions about the sorts of simultaneous relations the researcher should privilege (e.g., triads and seventh chords), and may also require additional information about the underlying tonal context, which again must be inferred either during transcription \\autocite{Margulis2008}, or using some automatic (key-finding) method. \\textcite{White2015} distinguishes this `top-down' approach from the `bottom-up', data-driven methods that build composite representations of chord events from simpler representations of note events \\autocite{Cambouropoulos2015,Conklin2002,Quinn2010,Quinn2011, Sapp2007}.\n\nWith a representation scheme in place, researchers then divide the corpus into contiguous sequences of $n$ events (called \\textit{n}-grams) to model the proordinate relations between harmonies. The resulting \\textit{n}-gram distributions serve as input for tasks associated with pattern discovery \\autocite{Conklin2002}, classification \\autocite{Conklin2013}, automatic harmonic analysis \\autocite{Taube1999}, and prediction \\autocite{Sears2018}. And yet since much of the world's music is hierarchically organized such that certain events are more central (or prominent) than others \\autocite{Bharucha1983}, non-contiguous events often serve as focal points in the sequence \\autocite{Gjerdingen2014}. For this reason,  corpus studies employing string-based methods often suffer from the \\textit{contiguity fallacy}---the assumption that note or chord events on the musical surface depend only on their immediate neighbors \\autocite{Sears2017}.\n\nBy way of example, consider the closing measures of the main theme from the final movement of Haydn's string quartet Op. 50, No. 2, shown in Example \\ref{ex:haydn_ex}a. The passage culminates in a perfect authentic cadence that features a conventional harmonic progression and a falling upper-voice melody. In the music theory classroom, students are taught to reduce this musical surface to a succession of chord symbols, such as the Roman numeral annotations shown below. Yet despite the ubiquity of these harmonies throughout the history of Western tonal music, existing string-based methods generally fail to retrieve this sequence of chords due to the presence of intervening embellishing tones (shown in gray), a limitation one study has called the \\textit{interpolation problem} \\autocite{Collins2014}.\n\n\\begin{example}[t!]\n\t\\centering\n\t\\input{Fig1.pdf_tex}\n\t\\caption{(a) Haydn, String Quartet in C major, Op. 50\/2, iv, mm. 48--50. Embellishing tones are shown with gray noteheads, and Roman numeral annotations appear below. (b) Expansion. Downbeat chord onsets are annotated with the chromatic scale-degree combination (csdc) scheme for illustrative purposes.}\n\t\\label{ex:haydn_ex}\n\\end{example}\n\nThus, the following sections consider whether string-based computational methods can discover (1) the most recurrent harmonies on the musical surface (i.e., coordinate relations); (2) the syntactic progressions that characterize a given idiom (i.e., proordinate relations); and (3) the recursive hierarchy by which certain harmonies are more central (or prominent) than others (i.e., superordinate\/subordinate relations). To that end, I have developed a representation scheme that loosely approximates Roman numeral symbols using a corpus of 50 expositions from Haydn string quartets \\autocite{Sears2016}. In addition to the symbolic encodings, the corpus includes accompanying text files with manual annotations for the key, mode, modulations, and pivot boundaries in each movement. Thus, I will sidestep the key-finding problem, which has already received considerable attention elsewhere (e.g., \\citeauthor{Temperley2008} \\citeyear{Temperley2008}). What interests me here, and consequently provides the impetus for the following pages, are the methods we use to discover the syntactic or recursive structures described in many theories of harmony. Hence, the corpus will serve as a toy dataset, with the hope that we might apply these methods to larger datasets in future work.\n\n\\section*{Coordinate Relations: Representation \\& Recurrence}\n\nCorpus studies in music research often treat the \\textit{note} event as the unit of analysis, examining features like chromatic pitch \\autocite{Pearce2004}, melodic interval \\autocite{Vos1989}, and chromatic scale degree \\autocite{Margulis2008}. Using computational methods to identify \\textit{composite} events like triads and seventh chords in complex polyphonic textures is considerably more complex, since the number of distinct $n$-note combinations associated with any of the above-mentioned features is enormous. Thus, many music analysis software frameworks derive chord progressions from symbolic corpora by first performing a \\textit{full expansion} of the symbolic encoding \\autocite{Conklin2002}, which duplicates overlapping note events at every unique onset time.\\footnote{In the Humdrum toolkit, this technique is called \\textit{ditto} \\autocite{Huron1993}, while \\textit{Music21} calls it \\textit{chordifying} \\autocite{Cuthbert2010}.} Shown in Example \\ref{ex:haydn_ex}b, expansion produces 14 distinct onset times. This partitioning method is admittedly too fine-grained to resemble the Roman numeral analysis in Example \\ref{ex:haydn_ex}a, but provides a useful starting point for the reduction methods that follow.\n\nTo relate the chord event at each onset to an underlying tonic, some studies use the opening key signature, with the researcher determining the mode from the score, resulting in chord distributions that often fail to control for modulations or changes in modality \\autocite{Margulis2008}. Key-finding algorithms have also become more common in recent decades, allowing researchers to automatically identify the key of a passage with high degrees of accuracy ($>90\\%$) \\autocite{Albrecht2013}. Nevertheless, the lack of available annotated corpora indicating modulations and changes of mode makes testing these algorithms quite difficult. Since in this case the corpus includes annotations for the key, mode, modulations, and pivot boundaries in each movement, we can simply map each note event to a chromatic scale degree (or \\texttt{csd}) modulo 12. This scheme consists of twelve distinct symbols numbered from 0 to 11, where 0 denotes the tonic, 7 the dominant, and so on. Absent instrumental parts for each distinct onset receive an undefined symbol $\\perp$.\n\nWe may now represent each chord onset as a chromatic scale-degree combination (or \\texttt{csdc}) to examine the recurrence of sonorities on the musical surface. Each onset contains between one and four note events, so the initial vocabulary consists of $13^4$ (or $28,561$) possibilities. To reduce the vocabulary of possible chord types, \\textcite{Quinn2010} excluded voice doublings and allowed permutations between the upper parts, so we can adopt that approach here. Thus, the major triads $\\langle0, 4, 4, 7\\rangle$ and $\\langle0, 7, 4, 0\\rangle$ would reduce to $\\langle0, 4, 7, \\perp\\rangle$. These exclusion criteria decrease the size of the potential vocabulary to 2784 distinct types, though in this corpus the vocabulary of \\texttt{csdc} consisted of just 688 types.\\footnote{Ideally, we would reduce the vocabulary to less than, say, 100 symbols, but given the number of combinatorial possibilities for three- and four-note chords, I will instead introduce a novel reduction method in the final section of this chapter.}\n\nIn total, 38\\% of the $19,570$ onsets in the corpus consisted of fewer than three distinct chromatic scale degrees (e.g., $<$$0,4,\\perp,\\perp$$>$), so I have omitted those onsets in order to examine the most common chord types. The multi-level pie plot in Figure \\ref{fig:csdc_pie} presents the onsets consisting of at least three chromatic scale degrees from major-mode passages in the corpus, with the proportions weighted by the rhythmic duration of each onset (see \\textcite{Sears2016} for further details). The inner pie plot represents the diatonic harmonies with Roman numeral notation, with upper and lower case Roman numerals denoting major and minor triads, respectively. The outer concentric circle represents each inversion (root position, first, second, and third), with the inversions appearing in clockwise order for each harmony beginning in root position.\n\nIn major-mode passages, tonic harmony appeared most frequently, followed by dominant harmony, the predominant harmonies IV and ii, and finally vii, vi, and iii. In the outer concentric circle, root position chords predominated for harmonies like I, IV, V, and vi, but unsurprisingly, first inversion chords appeared more frequently for ii, iii, and vii. What is more, the 49 \\texttt{csdc} types representing diatonic harmony---triads and seventh chords for every diatonic harmony in every inversion---accounted for approximately 62\\% of the three- and four-note combinations in major-mode passages of the corpus.\n\nTogether, these findings suggest that (1) the most central sonorities in most theories of harmony are also the most frequent in this corpus, and (2) like their natural language counterparts, the chromatic scale-degree combinations follow a power-law distribution between frequency and rank, with the most frequent (top-ranked) types---the diatonic harmonies of the tonal system---accounting for the vast majority of the three- and four-note combinations in the corpus. Of course, these claims are by no means new. Frequency distributions of both words and chords often display power-law (or \\textit{Zipfian}) distributions \\autocite{Zipf1935, Rohrmeier2008, Sears2017}. What would be new is to provide evidence that the statistical regularities characterizing a tonal corpus also reflect the \\textit{order} in which its constituent harmonies occur. To that end, the next section introduces string-based methods for the identification and ranking of recurrent temporal patterns. \n\n\\begin{figure}[t!]\n\t\\centering\n\t\\input{Fig2.pdf_tex}\n\t\\caption{Multi-level pie plot of the diatonic chromatic scale degree combinations consisting of at least three chromatic scale degrees from the major mode (chord onsets located within the boundaries of a pivot were excluded). The inner pie and outer ring represent diatonic harmony (triads and seventh chords) and inversion (root position, first, second, third), respectively. Inversions appear in clockwise order for each harmony beginning in root position (labels only provided for dominant harmony). Roman numerals iii and IV did not appear in third inversion. $N = 11,253$.}\n\t\\label{fig:csdc_pie}\n\\end{figure}\n\n\\section*{Proordinate Relations: Syntax \\& Skip-grams}\n\nIn corpus linguistics, researchers often discover recurrent multi-word expressions (sometimes called \\textit{collocations}) by dividing the corpus into sub-sequences of $n$ events (or $n$-grams), and then determining the number of instances (or \\textit{tokens}) associated with each unique $n$-gram \\textit{type}. \\textit{N}-grams consisting of one, two, or three events are often called \\textit{unigrams}, \\textit{bigrams}, and \\textit{trigrams}, respectively, while longer \\textit{n}-grams are typically represented by the value of \\textit{n}. The previous discussion represented the corpus using unigrams, for example, but to discover the most conventional (i.e., syntactic) harmonic progressions using the current representation scheme, we need only increase the value of \\textit{n}.\n\nEach piece $m$ consists of a contiguous sequence of combinations, so let $k$ represent the length of the sequence in each movement, and let $C$ denote the total number of movements in the corpus. The number of contiguous \\textit{n}-gram tokens in the corpus is\n\\begin{equation}\\label{eq:1}\n\\displaystyle\\sum_{m=1}^{C} k_m-n+1\n\\end{equation}\n\nTable \\ref{tab:top10} presents the top ten contiguous bigram types ranked by count. The combinations in each type are represented with the \\texttt{csdc} scheme, but for most types I also include Roman numeral annotations. In short, nine of the top ten types repeat tonic or dominant harmony, or scale degrees $\\hat{1}$ or $\\hat{5}$, with the top-ranked type, I--I, featuring 440 tokens. The seventh-ranked type, V$^7$--I, is in fact the only non-repeating bigram to crack the top ten. Thus, the musical surface contains considerable repetition, thereby obscuring the kinds of patterns we might hope to study (e.g., harmonic progressions containing more than one distinct harmony). Perhaps worse, recall that a significant portion of the chromatic scale degree combinations in the corpus feature fewer than three distinct chromatic scale degrees.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\caption{Top ten contiguous bigram types ranked by count.}\n\t\\begin{threeparttable}\n\t\t\\renewcommand{\\TPTminimum}{\\columnwidth}\n\t\t\\makebox[\\textwidth]{\n\t\t\t\\begin{tabular}{S[table-format=2.0]ccllccrcllcc}\n\t\t\t\t\\toprule\n\t\t\t\t& & \\multicolumn{5}{c}{Without Exclusion Criteria} &       & \\multicolumn{5}{c}{With Exclusion Criteria} \\\\\n\t\t\t\t\\cmidrule{3-7}\\cmidrule{9-13}\n\t\t\t\t\\multicolumn{1}{c}{Rank} & & \\textit{N} & \\multicolumn{2}{c}{\\texttt{csdc} (mod 12)} & \\multicolumn{2}{c}{RN} &       & \\textit{N} & \\multicolumn{2}{c}{\\texttt{csdc} (mod 12)} & \\multicolumn{2}{c}{RN} \\\\\n\t\t\t\t\\midrule\n\t\t\t\t1   &  & 440   & $0,4,7,\\perp$ & $0,4,7,\\perp$ & I     & I     &       & 147   & $7,2,5,11$ & $0,4,\\perp,\\perp$   & V$^7$    & I \\\\\n\t\t\t\t2    & & 212   & $7,0,4,\\perp$ & $7,0,4,\\perp$ & I$^6_4$   & I$^6_4$   &       & 59    & $7,0,4,\\perp$ & $7,2,11,\\perp$ & I$^6_4$     & V \\\\\n\t\t\t\t3    & & 212   & $0,4,\\perp,\\perp$   & $0,4,\\perp,\\perp$   & I     & I     &       & 56    & $11,2,5,7$ & $0,4,7,\\perp$ & V$^6_5$   & I \\\\\n\t\t\t\t4    & & 182   & $4,0,7,\\perp$ & $4,0,7,\\perp$ & I$^6$    & I$^6$    &       & 52    & $0,2,5,11$ & $0,4,\\perp,\\perp$   & (vii) & I \\\\\n\t\t\t\t5    & & 154   & $0,4,\\perp,\\perp$   & $0,4,7,\\perp$ & I     & I     &       & 42    & $7,0,4,\\perp$ & $7,2,5,11$ & I$^6_4$   & V$^7$ \\\\\n\t\t\t\t6    & & 153   & $7,2,11,\\perp$ & $7,2,11,\\perp$ & V     & V     &       & 39    & $7,2,5,\\perp$ & $7,0,4,\\perp$ & V$^7$   & I$^6_4$ \\\\\n\t\t\t\t7    & & 147   & $7,2,5,11$ & $0,4,\\perp,\\perp$   & V$^7$    & I     &       & 31    & $7,0,4,\\perp$ & $7,2,5,\\perp$ & I$^6_4$    & V$^7$ \\\\\n\t\t\t\t8    & & 139   & $7,2,5,11$ & $7,2,5,11$ & V$^7$    & V$^7$    &       & 30    & $5,2,7,11$ & $4,0,7,\\perp$ & V$^4_2$   & I$^6$ \\\\\n\t\t\t\t9    & & 137   & $7,\\perp,\\perp,\\perp$     & $7,\\perp,\\perp,\\perp$     &       &       &       & 28    & $5,2,9,\\perp$ & $7,0,4,\\perp$ &  ii$^6$     & I$^6_4$ \\\\\n\t\t\t\t10   & & 105   & $0,\\perp,\\perp,\\perp$     & $0,\\perp,\\perp,\\perp$     &       &       &       & 27    & $5,0,9,\\perp$ & $4,0,7,\\perp$ & IV   & I$^6$ \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular\n\t\t}\n\t\t\\begin{tablenotes}[flushleft]\n\t\t\t\\small\n\t\t\t\\item \\textit{Note.} Exclusion criteria: (1) either chord contains only one distinct chromatic scale degree (\\textit{monophony}); (2) neither chord contains at least three distinct chromatic scale degrees (\\textit{polyphony}); (3) chords share the same chromatic scale degrees regardless of inversion (\\textit{identity}); (4) chords share the same chromatic scale degree in the bass and subsets or supersets of chromatic scale degrees in the upper parts (\\textit{similarity}). Parentheses suggest a change of harmony from one chord to the other, but with a pedal in the bass.\n\t\t\\end{tablenotes}\n\t\\end{threeparttable\n\t\\label{tab:top10}\n\\end{table}\n\nCorpus linguists typically solve problems like this by removing (or \\textit{filtering}) bigram types reflecting parts of speech (or syntactic categories) ``that are rarely associated with interesting linguistic expressions'' \\autocite[31]{Manning1999}. For instance, researchers often exclude types containing articles like `the' or `a' in natural language corpora to ensure that adjective-noun and noun-noun expressions will receive higher ranks in the distribution. For our purposes, one could easily extend this sort of thinking to harmonic corpora by excluding \\textit{n}-grams according to the temporal periodicity or proximity of their constituent members. \\textcite{Symons2012}, for example, discovered recurrent contrapuntal patterns in a corpus of two-voice solfeggi by sampling events at regular temporal intervals. Similarly, I increased the ranking of conventional cadential progressions like ii$^6$-I$^6_4$-V$^7$-I by privileging patterns with temporally proximal members \\autocite{Sears2016}.\\footnote{In this instance, I$^6_4$ refers to a double suspension above the cadential dominant, which is more commonly notated as V$^6_4$, as is the case in Example \\ref{ex:haydn_ex}. Unfortunately, this dominant embellishment may only be determined from the immediate harmonic context (e.g., V$^{6-5}_{4-3}$ vs. I$^6_4$-I$^6$), so the present encoding scheme cannot distinguish six-four inversions of the tonic from six-four embellishments of the dominant. Thus, I have retained the I$^6_4$ annotation for instances of $<$7,0,4,$\\perp$$>$ in the analyses that follow.}\n\nThere are, of course, a number of reasons to exclude patterns, but for the present study, I will exclude bigram types if they fail to represent what \\textcite[231]{Meyer2000b}  referred to as ``forceful harmonic progressions\": progressions featuring a genuine harmonic (i.e., pitch) change between primarily tertian sonorities.\\footnote{\\textcite[231]{Meyer2000b} explains, ``... the perception and cognition of patterns (and hence the formation of schemata) are dependent on clear differentiation between successive stimuli. More specifically, forceful harmonic progression depends in part on the amount of pitch change between successive triads.''} To that end, I have excluded bigram types if (1) either chord contains only one distinct chromatic scale degree (\\textit{monophony}, e.g., $<$0,$\\perp$,$\\perp$,$\\perp$$>$); (2) neither chord contains at least three distinct chromatic scale degrees (\\textit{polyphony}, e.g., $<$0,$\\perp$,$\\perp$,$\\perp$$>$$\\rightarrow$$<$0,2,$\\perp$,$\\perp$$>$); (3) chords share the same chromatic scale degrees regardless of inversion (\\textit{identity}, e.g., $<$0,4,7$\\perp$$>$$\\rightarrow$$<$4,0,7,$\\perp$$>$); and (4) chords share the same chromatic scale degree in the bass and subsets or supersets of chromatic scale degrees in the upper parts (\\textit{similarity}, e.g., $<$7,5,11,$\\perp$$>$$\\rightarrow$$<$7,2,5,11$>$). The first two criteria ensure that the filtered bigram types will feature tertian sonorities in some way, while the latter two criteria emphasize the importance of pitch change from one sonority to the next.\n\nShown in the right-most columns of Table \\ref{tab:top10}, 34\\% of the 5378 bigram types in the corpus met these exclusion criteria. With exclusion, progressions deemed `cadential' in most theories of harmony rose to the top of the table, with five of the top ten progressions featuring a six-four embellishment of the dominant. Along with these progressions, the table also includes typical tonic-prolongational progressions like V$^6_5$--I, V$^4_2$--I$^6$, and IV--I$^6$. The appeal of filtering in this way is thus that potentially meaningful progressions emerge out of distributional statistics. Nevertheless, by only including the counts for contiguous bigram types, we necessarily omit syntactic progressions with intervening embellishing tones from the final count. In a previous study, for example, I found that the progression I$^6$--ii$^6$--V$^7$--I \\textit{never} appears contiguously in this corpus despite the apparent ubiquity of the pattern in the classical style \\autocite{Sears2017}. Of the thousands of chord onsets examined here, it therefore seems unreasonable to assume that the conventional progressions in the right-most columns of Table \\ref{tab:top10} should feature so few tokens. In this case, the commitment to contiguous \\textit{n}-grams---the standard method in musical corpus research---has effectively tied our hands.\n\nTo discover associations lying beneath (or beyond) the musical surface, we might simply relax the contiguity assumption to ensure potentially relevant bigram types appear in the distribution. Shown in Figure \\ref{fig:non_contiguous}, the top plot depicts the contiguous and non-contiguous bigram tokens for a 5-event sequence with solid and dashed arcs, respectively. According to Equation \\eqref{eq:1}, the number of contiguous tokens in a 5-event sequence is $k-n+1$, or four tokens. If we also include all possible non-contiguous relations, the number of tokens is given by the combination equation:\n\\begin{equation}\n{k \\choose n} = \\frac{k!}{n!(k-n)!} = \\frac{k(k-1)(k-2) \\ldots (k-n+1)}{n!}\n\\end{equation}\n\nThe notation ${k \\choose n}$ denotes the number of possible combinations of $n$ events from a sequence of $k$ events. By including the non-contiguous associations, the number of tokens for a 5-event sequence increases to 10. As \\textit{n} and \\textit{k} increase, the number of patterns can very quickly become unwieldy: a 20-event sequence, for example, contains 190 possible tokens. To overcome the combinatoric complexity of counting tokens in this way, researchers in natural language processing have limited the investigation to what I have called \\textit{fixed-skip} \\textit{n}-grams \\autocite{Sears2017,Guthrie2006}, which only include \\textit{n}-gram tokens if their constituent members occur within a fixed number of skips $t$. Shown in the bottom plot in Figure \\ref{fig:non_contiguous}, $ac$ and $bd$ constitute one-skip tokens (i.e., $t=1$), while $ad$ and $be$ constitute two-skip tokens. Thus, up to 10 tokens appear in a 5-event sequence when $t=3$.\n\nBy relaxing the restrictions on the possible associations between events in a sequence, the skip-gram method has increased our chances of including potentially meaningful \\textit{n}-grams in the final distribution. It does not ensure that the most frequent types will feature genuine harmonic progressions, however. To be sure, the skip-gram method aggregates the counts for bigram types whose members occur \\textit{within} a certain number of skips, and so is just as susceptible to the most repetitive patterns on the surface. As a result, repeating bigram types like I--I tend to retain their approximate ranking regardless of the permitted number of skips.\n\nTo resolve this issue, we could again exclude patterns that do not represent a genuine harmonic change between adjacent members, but corpus linguists also frequently employ alternative ranking functions whose goal is to discover recurrent multi-word expressions using other relevance criteria. In this case, the skip-gram method provides counts for each \\textit{n}-gram type at a number of possible skips. If the harmonies in conventional harmonic progressions tend to appear in strong metric positions and feature intervening embellishing tones, we might instead rank patterns using a statistic that characterizes the \\textit{depth} at which conventional harmonic progressions tend to emerge.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\def\\svgwidth{.5\\textwidth}\n\t\\input{Fig3.pdf_tex}\n\t\\caption{Top: A five-event sequence, with arcs denoting all contiguous (solid) and non-contiguous (dashed) bigram tokens. Bottom: All tokens, with $t$ indicating the number of skips between events.}\n\t\\label{fig:non_contiguous}\n\\end{figure}\n\nFigure \\ref{fig:poly_plot} presents scatter plots of the counts measured from zero to ten skips for the cadential six-four progression, I$^6_4$--V$^7$, and the top-ranked type in the corpus, I--I. Recall that I--I features 440 tokens on the surface (see Table \\ref{tab:top10}). As the number of skips $t$ between the members of this type increase, the associated counts decrease. In other words, repetitive patterns like I--I appear far more prevalently at (or near) the surface. This result seems unsurprising---presumably over enough skips, \\textit{all} bigram types become less frequent. Yet since conventional harmonic progressions often appear in strong metric positions and feature intervening embellishing tones, we might assume that the counts for these patterns should actually \\textit{increase} with $t$ up to a certain point, and then decrease for more distal relations. This is in fact exactly what we find for the I$^6_4$--V$^7$ progression in Figure \\ref{fig:poly_plot}, which features its highest count when $t=3$, but decreasing counts when $t>3$.\n\nSo how might we privilege patterns like I$^6_4$--V$^7$ in the final ranking? If conventional harmonic progressions tend to appear beneath the surface, the counts across skips should increase as $t$ increases. We could then model this assumption by fitting a first-order polynomial (or \\textit{linear}) trend to the distribution of counts for each $n$-gram type. The best-fit line modeled by the equation $y_i = \\beta_1x_i +\\beta_0$ minimizes the error between the predicted count at each skip and its actual count in Cartesian space (called \\textit{linear regression}), where the leading coefficient $\\beta_1$ characterizes the shape of the trend (i.e., the slope of the line). Thus, positive estimates of $\\beta_1$ would indicate an increasing trend, whereas negative estimates would indicate a decreasing trend.\n\nIn this model, ranking bigram types using $\\beta_1$ privileges patterns whose counts increase as $t$ increases. As a result, conventional progressions like I--V$^7$ and V--I appear in the top ten, but so do syntactic retrogressions like ii$^6$--I (table not shown). In this case, permitting such large skips ensures that the final distribution will features types whose members skip over syntactically meaningful harmonies \\textit{within} the progression. Thus, we could revise the linear model by assuming that the counts should increase as $t$ initially increases (e.g., up to $t=3$), and then \\textit{decrease} for larger skips. A second-order polynomial trend---which would produce a U-shape if the leading coefficient $\\beta_2$ is positive, and an inverted U-shape if $\\beta_2$ is negative---would peak near the center of the distribution (i.e., at around $t=5$), so I have ranked each bigram type using the leading coefficient $\\beta_3$ of a third-order polynomial trend. Patterns that increase from zero to approximately three skips, and then decrease for larger numbers of skips, will produce the positive trend found for the I$^6_4$--V$^7$ progression in Figure \\ref{fig:poly_plot}. Conversely, patterns that decrease exponentially from zero to ten skips will produce the negative trend found for the I--I progression. In point of fact, these two patterns are the highest- and lowest-ranked types in the distribution.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\input{Fig4.pdf_tex}\n\t\\caption{Scatter plots of counts at each skip for the highest- and lowest-ranked bigram types: $<$7,0,4,$\\perp$$>$ $\\rightarrow$ $<$7,2,5,11$>$ (left) and $<$0,4,7,$\\perp$$>$ $\\rightarrow$ $<$0,4,7,$\\perp$$>$ (right). Counts in both plots are fit with a third-order polynomial trend (solid line).}\n\t\\label{fig:poly_plot}\n\\end{figure}\n\nTable \\ref{tab:top10poly} presents the top ten bigram types ranked by $\\beta_3$. Even without exclusion criteria, the top ten types include several conventional harmonic progressions, such as V$^7$--I, a pre-dominant-to-I$^6_4$ progression, and a tonic-prolongational progression from I to IV$^6_4$. With exclusion criteria, a few other interesting patterns emerge, such as the progression from I$^6$ to IV, or the tonic pedal supporting a progression from vii to I. Perhaps more importantly, few of the patterns discovered using this method appear frequently on the surface. The sixth-ranked V$^7$--I progression, for example, includes just three tokens on the surface. Even across the top one hundred patterns in the distribution, the median count is just four tokens. Thus, it seems that patterns appearing just beneath the surface---i.e., whose counts peak at around $t=3$---feature many of the conventional (or syntactic) progressions described in most theories of harmony.\n\nYet despite the success of this ranking function to privilege patterns representing a genuine harmonic change of some sort, a persistent problem remains: namely, four of the top ten progressions in Table \\ref{tab:top10poly} are variants of the same V$^7$--I progression. This finding suggests that the vocabulary, at 688 symbols, is simply too large to serve as a suitable proxy for the chord vocabularies in theories of harmony. Discovering the most recurrent, syntactic progressions---or reducing the musical surface to a sequence of its most central (or salient) harmonies---requires a novel vocabulary reduction method, a problem I turn to in the next section.\n\n\\section*{Superordinate\/Subordinate Relations: Recursion \\& Reduction}\n\nThe appeal of the scheme selected for this study is that the most common chromatic scale degree combinations will have analogues in most theories of harmony. Frankly, this is not surprising given that the scheme relies on human annotations about the keys and modes for each movement. Nevertheless, the \\texttt{csdc} representation is more promiscuous than traditional definitions of `chord' would embrace. Whereas theorists tend to assign chordal status only to those vertical sonorities featuring stacked intervals of a third, the \\texttt{csdc} scheme makes no distinctions between chord tones and non-chord tones, consonant and dissonant intervals, or diatonic and chromatic scale degrees \\autocite{Quinn2010}. Hence, the vocabulary of \\texttt{csdc} types is enormous.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\caption{Top ten bigram types ranked by leading coefficient of third-order polynomial trend.}\n\t\\begin{threeparttable}\n\t\t\\renewcommand{\\TPTminimum}{\\columnwidth}\n\t\t\\makebox[\\textwidth]{\n\t\t\t\\begin{tabular}{S[table-format=2.0]ccllccrcllcc}\n\t\t\t\t\\toprule\n\t\t\t\t& & \\multicolumn{5}{c}{Without Exclusion Criteria} &       & \\multicolumn{5}{c}{With Exclusion Criteria} \\\\\n\t\t\t\t\\cmidrule{3-7}\\cmidrule{9-13}\n\t\t\t\t\\multicolumn{1}{c}{Rank} & & \\textit{$\\beta_3$} & \\multicolumn{2}{c}{\\texttt{csdc} (mod 12)} & \\multicolumn{2}{c}{RN} &       & \\textit{$\\beta_3$} & \\multicolumn{2}{c}{\\texttt{csdc} (mod 12)} & \\multicolumn{2}{c}{RN} \\\\\n\t\t\t\t\\midrule\n\t\t\t\t1   &  & .429  & $7,0,4,\\perp$ & $7,2,5,11$ & I$^6_4$   & V$^7$    &       & .429  & $7,0,4,\\perp$ & $7,2,5,11$ & I$^6_4$   & V$^7$ \\\\\n\t\t\t\t2    & & .240  & $5,\\perp,\\perp,\\perp$     & $0,\\perp,\\perp,\\perp$ & &    &       & .181  & $0,4,7,\\perp$   & $0,5,9,\\perp$ & I     & IV$^6_4$ \\\\\n\t\t\t\t3    & & .220  & $7,\\perp,\\perp,\\perp$     & $2,\\perp,\\perp,\\perp$ & &    &       & .171  & $7,5,11,\\perp$ & $0,4,\\perp,\\perp$     & V$^7$    &  I  \\\\\n\t\t\t\t4    & & .181  & $0,4,7,\\perp$   & $0,5,9,\\perp$ & I     & IV$^6_4$    &       & .166  & $5,9,\\perp,\\perp$ & $7,0,4,\\perp$ & PrD    & I$^6_4$ \\\\\n\t\t\t\t5    & & .173  & $2,\\perp,\\perp,\\perp$     & $7,\\perp,\\perp,\\perp$     &       &       &       & .162  & $0,4\\perp,\\perp$   & $0,5,9,\\perp$ & I     & IV$^6_4$ \\\\\n\t\t\t\t6    & & .171  & $7,5,11,\\perp$ & $0,4,\\perp,\\perp$     & V$^7$    &  I        &       & .155  &  $7,2,5,11$ & $0,4,\\perp,\\perp$     & V$^7$    &  I \\\\\n\t\t\t\t7    & & .166  & $5,9,\\perp,\\perp$ & $7,0,4,\\perp$ & PrD    & I$^6_4$    &       & .144  &  $7,2,5,11$ & $0,4,7,\\perp$     & V$^7$    &  I \\\\\n\t\t\t\t8    & & .162  & $0,4\\perp,\\perp$   & $0,5,9,\\perp$ & I     & IV$^6_4$   &       & .141  & $4,0,7,\\perp$   & $5,0,9,\\perp$ & I$^6$     & IV \\\\\n\t\t\t\t9    & & .161  & $0,4,7,\\perp$& $0,\\perp,\\perp,\\perp$     & I    &       &       & .135  & $7,2,5,\\perp$ & $0,4,\\perp,\\perp$ & V$^7$    &  I \\\\\n\t\t\t\t10   & & .155  & $7,2,5,11$ & $0,4,\\perp,\\perp$     & V$^7$    &  I     &       & .122  & $0,2,5,11$ & $0,4,7,\\perp$ & (vii)    &  I \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular\n\t\t}\n\t\t\\begin{tablenotes}[flushleft]\n\t\t\t\\small\n\t\t\t\\item \\textit{Note.} A third-order polynomial trend fit to the counts from zero to ten skips for each bigram type (i.e., $y_i = \\beta_3x^3_i + \\beta_2x^2_i + \\beta_1x_i +\\beta_0$). PrD denotes predominant function. See Table \\ref{tab:top10} for exclusion criteria.\n\t\t\\end{tablenotes}\n\t\\end{threeparttable\n\t\\label{tab:top10poly}\n\\end{table}\n\nHow, then, do we solve the harmonic reduction problem when the relations between note events explode in combinatorial complexity for complex polyphonic textures? To demonstrate that these sorts of organizational systems emerge out of distributional statistics, let us consider again the two ranking functions from the previous section: (1) \\textit{count}, which assumes that the most relevant or meaningful types are the most frequent; and (2) a \\textit{polynomial trend}, which assumes that the most relevant types tend to appear just beneath the surface (e.g., at $t=3$). The former is a simple statistic that represents the sum of the instances for each type, whereas the latter exploits domain-specific knowledge about the corpus under investigation. There are limitations to both ranking functions, however. One could argue, for example, that since the top-ranked types in Table \\ref{tab:top10poly} are by no means the most frequent, the assumption that they are somehow relevant (or conventional) is unwarranted. Similarly, corpus linguists have argued that count is not a sufficient indicator for a strong attraction between words \\autocite[5]{Evert2008}, since two highly frequent words are also likely to co-occur quite often just by chance. V$^7$ and I appear frequently in the corpus (see Figure \\ref{fig:csdc_pie}), for example, so it is possible that their appearance in Table \\ref{tab:top10} simply reflects the joint probability of their co-occurrence.\n\nTo resolve this issue, corpus linguists have developed a large family of \\textit{association} (or \\textit{attraction}) \\textit{measures} that quantify the statistical association between co-occurring words \\autocite{Evert2008}. The majority of these measures use contingency tables. Table \\ref{tab:contingency} presents the contingency table for the most common bigram type associated with the V$^7$--I progression: $<$7,2,5,11$>$ $\\rightarrow$ $<$0,4,$\\perp$,$\\perp>$. The counts reflect tokens with up to five skips between bigram members. The table has four cells for tokens containing both chord$_1$ and chord$_2$ (a), tokens containing chord$_1$ but not chord$_2$ (i.e., any other chord) (b), tokens containing chord$_2$ but not chord$_1$ (c), and tokens containing neither chord (d). The \\textit{marginal frequencies}, so called because they appear at the margins of the table, represent the sum of each row and column. Thus, the co-occurence frequency for V$^7$--I is 581.\n\nAgain, there are dozens of available association measures \\autocite{Pecina2009}, but \\textit{Fisher's exact test} is perhaps the most appropriate (or mathematically rigorous) significance test for the analysis of contingency tables \\autocite{Agresti2002}. The mathematical formalism need not concern us here, but in short, Fisher's exact test computes the total probability of all possible outcomes that are similar to or more extreme than the observed contingency table. The resulting probability (or \\textit{p-value}) will be large if the two chords of a given bigram are statistically independent, but very small if the two chords are unlikely to co-occur at the estimated frequency just by chance. In this case, the \\textit{p}-value is vanishingly small ($p < .0001$), suggesting that V$^7$ and I are statistically dependent.\n\nEssentially, association measures produce empirical statements about the \\textit{statistical attraction} between chords. They do not measure the potential asymmetry of this association, however. This is to say that in many cases chord$_1$ could be more (or less) predictive of chord$_2$ than the other way around, so \\textcite{Gries2013} and \\textcite{Nelson2014} have suggested alternative association measures based on the predictive asymmetry between the members of each bigram. According to \\textcite{Firth1957}, association measures should quantify the statistical influence an event exerts on its neighborhood, where some events exert more influence than others. Given such a measure, we could reduce the chord vocabulary by privileging chords that exert the greatest `attractional force' on their neighbors.\n\nAsymmetric (or \\textit{directional}) association measures typically compute the conditional probabilities between the members of each bigram \\autocite{Michelbacher2007}. In Table \\ref{tab:contingency}, for example, the probability that I follows V$^7$ can be computed from the frequencies in the first row. In this case, $P(\\text{I}|\\text{V}^7)$ is $\\frac{a}{a+b}$, with $a$ representing all of the instances in which I follows V$^7$, and $b$ representing all of the instances in which some other harmony follows V$^7$. Thus, $\\frac{581}{3873}=.15$, which tells us that I follows V$^7$ roughly 15\\% of the time. This estimate does not represent the probability that V$^7$ \\textit{precedes} I, however. To compute this statistic, we can use the frequencies in the first column of Table \\ref{tab:contingency}. Here, $P(\\text{V}^7|\\text{I})$ is $\\frac{a}{a+c}$, or $\\frac{581}{6216}=.09$. Thus, for this particular variant of the V$^7$--I progression, V$^7$ is a better predictor of I than the other way around. Or put another way, I exerts the greater attractional force.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\caption{Contingency table for the bigram $<$7,2,5,11$>$ $\\rightarrow$ $<$0,4,$\\perp$,$\\perp>$ (V$^7$--I). Counts reflect tokens with up to five skips between chord events.}\n\t\\setlength{\\tabcolsep}{16pt}\n\t\\def\\arraystretch{1.5}\n\t\\begin{tabular}{r|S[table-format=4.0] S[table-format=6.0]r|S[table-format=6.0]}\n\t\t& {I}     & {Not I}    &       & Totals \\\\\n\t\t\\midrule\n\t\tV$^7$    & 581 \\enspace {(a)}   & 3292 \\enspace {(b)}  &       & 3873 \\\\\n\t\tNot V$^7$   & 5635 \\enspace {(c)}  & 106862 \\enspace {(d)} &       & 112497 \\\\\n\t\t\\midrule\n\t\tTotals & 6216  & 110154 &       & 116370 \\\\\n\t\\end{tabular\n\t\\label{tab:contingency\n\\end{table\n\nI have formalized this statistical inference in the following way:\n\\begin{equation}\n\\textsc{asym} = P(\\text{chord}_2|\\text{chord}_1) - P(\\text{chord}_1|\\text{chord}_2) = \\frac{a}{a+b} - \\frac{a}{a+c}\n\\end{equation}\nIn this equation, \\textsc{asym} is simply the arithmetic difference between the two conditional probabilities. The estimates of \\textsc{asym} fall in a range between $-1$ and 1, where positive values indicate that chord$_2$ is the attractor, negative values indicate that chord$_1$ is the attractor, and 0 indicates bidirectionality, since both harmonies exert equivalent attractional force. For the bigram in Table \\ref{tab:contingency}, the positive directional asymmetry of .06 tells us that I exerts more influence on V$^7$, and so serves as the statistical attractor within the bigram.\n\nIt bears mentioning here that directional asymmetries differ from the \\textit{temporal} asymmetries described in theories of harmony. Whereas temporal asymmetries refer to the conventionality or syntactic plausibility of harmonies according to their temporal order (e.g., ii$^6$--V$^7$ compared to V$^7$--ii$^6$), directional asymmetries attempt to capture the attractional force between two harmonies in a \\textit{specified} temporal relationship. Thus, V$^7$--ii$^6$ may be much less likely---and thus, less syntactic---than ii$^6$--V$^7$, but our notion of directional asymmetry simply indicates which of the two harmonies is the stronger attractor in both progressions.\n\nTo reduce the vocabulary of chord types, we could simply calculate the number of bigram types in which each unigram type is the attractor. Shown in Table \\ref{tab:asym_ranks}, \\textit{N}$_{\\text{attractor}}$ indicates that $<$$0,4,7,\\perp$$>$ serves as the attractor in 604 distinct bigram types in the corpus. Unsurprisingly, \\%$_{\\text{attractor}}$ also tells us that this variant of I is the attractor for every bigram in which it appears. Finally, the table also presents an alternative asymmetric measure based on the sum of the asymmetries for each bigram type in which the indicated unigram type is a member. In this case, I exerts the greatest attractional force of any of unigram type in the corpus $\\sum_{\\textsc{asym}}=56.41$, with other harmonies like V, V$^7$, I$^6$, and ii$^6$ also making the top ten. Vocabulary reduction methods could then use a table like this one to create an \\textit{$n$-best list}, which uses a specified threshold $n$ to determine the members (and non-members) of the vocabulary \\autocite{Evert2008}. We would then assimilate harmonies appearing below this threshold into those appearing above using a kind of incremental clustering method.\n\nA discussion of clustering methods for directional asymmetry data deserves its own study, but for the sake of illustration, I have presented one possible method in Example \\ref{ex:reduction}. In this case, reducing the musical surface to a sequence of its most central harmonies is a specific case of the more general vocabulary reduction problem considered thus far. Starting with the attractional force rankings represented in Table \\ref{tab:asym_ranks}, a simple harmonic reduction algorithm could reduce a sequence of harmonies---and thus, the size of the overall vocabulary---by linking the chord exerting the least attractional force in the sequence, denoted by $c_i$, to the left or right chord neighbor exerting the greater attractional force ($c_{i-1}$ or $c_{i+1}$). The algorithm would then remove $c_i$ from the sequence and repeat the process until all of the chords have been linked.\n\nShown in Example \\ref{ex:reduction}, the harmonic reduction algorithm just described could be used to produce a tree diagram not unlike those found in Lerdahl and Jackendoff's (1983) approach. In this case, the chord in onset three, $<$$9,0,7,\\perp$$>$, features the least attractional force of the chords in the sequence, so the algorithm linked onset three to the stronger attractor, which in this case was the right neighbor, $<$$9,0,5,\\perp$$>$ (i.e., IV$^6$). The algorithm then removed onset three from the sequence and started the process again, at each step linking the combination of scale degrees exerting the least attractional force to the stronger adjacent attractor, and then removing that combination from the sequence. Branches reaching above the horizontal dashed line produce the reduction shown in the bottom system. Thus, for this passage the algorithm pruned all of the chords containing embellishing tones, resulting in the sequence of chords corresponding to the Roman numeral annotations provided below.\n\n\\begin{table}[t!]\n\t\\centering\n\t\\caption{Top ten unigram types, ranked according to the number of bigram types in which each unigram type is the attractor.}\n\t\\begin{threeparttable}\n\t\t\\renewcommand{\\TPTminimum}{\\columnwidth}\n\t\t\\makebox[\\textwidth]{\n\t\t\t\\begin{tabular}{S[table-format=2.0]S[table-format=3.0]S[table-format=3.1]clc}\n\t\t\t\t\\toprule\n\t\t\t\t\\multicolumn{1}{c}{Rank} & \\textit{N}$_{\\text{attractor}}$ & \\%$_{\\text{attractor}}$ & $\\sum_{\\textsc{asym}}$ &  \\multicolumn{1}{l}{\\texttt{csdc} (mod 12)} & RN \\\\\n\t\t\t\t\\midrule\n\t\t\t\t1     & 604   & 100   & 56.41 &  $0,4,7,\\perp$ & I \\\\\n\t\t\t\t2     & 560   & 99.6  & 38.33 &  $0,4,\\perp,\\perp$   & I \\\\\n\t\t\t\t3     & 555   & 99.3  & 31.35 &  $7,\\perp,\\perp,\\perp$     &  \\\\\n\t\t\t\t4     & 509   & 98.1  & 36.38 &  $7,2,11,\\perp$ & V \\\\\n\t\t\t\t5     & 508   & 98.5  & 29.10  &  $7,2,5,11$ & V$^7$ \\\\\n\t\t\t\t6     & 508   & 97.7  & 23.89 &  $0,\\perp,\\perp,\\perp$     &  \\\\\n\t\t\t\t7     & 497   & 98.8  & 35.34 &  $4,0,7,\\perp$ & I$^6$ \\\\\n\t\t\t\t8     & 440   & 96.9  & 32.27 &  $7,0,4,\\perp$ & I$^6_4$ \\\\\n\t\t\t\t9     & 425   & 95.6  & 27.36 &  $5,2,9,\\perp$ & ii$^6$ \\\\\n\t\t\t\t10   & 412 & 96.3   & 13.98  & $2,\\perp,\\perp,\\perp$ & \\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular\n\t\t}\n\t\t\\begin{tablenotes}[flushleft]\n\t\t\t\\small\n\t\t\t\\item \\textit{Note.} \\textit{N}$_{\\text{attractor}}$ denotes the number of bigram types in which each unigram type is the attractor, and \\%$_{\\text{attractor}}$ indicates the percentage of bigram types in which each unigram type appears as the attractor.\n\t\t\\end{tablenotes}\n\t\\end{threeparttable\n\t\\label{tab:asym_ranks\n\\end{table\n\nPresumably the pronounced directional asymmetries in the distribution of counts for each bigram type allow this algorithm to distinguish genuine harmonies from chords containing embellishing tones. Nevertheless, a toy example like this one tends to paper over the cracks of what is in fact a very difficult problem. Note, for example, that the highest branches of the tree only partly reflect how an analyst might parse this particular passage. The three harmonies at the very top of the tree seem reasonable enough (I--V$^7$--I), but this algorithm identified the next most important harmony as I$^6$ rather than IV, producing the progression, I--I$^6$--V$^7$--I. If Meyer's (\\citeyear{Meyer2000b}) preference for genuine harmonic change is reasonable, then IV would be the more fitting member in the final progression even though I$^6$ obtained the higher rank in Table \\ref{tab:asym_ranks}.\n\nWe could perhaps solve this problem by applying the algorithm not just to the passage in question, but to the entire corpus of movements simultaneously. At each step, the algorithm would assimilate variants of harmonies like $<$$9,0,7,\\perp$$>$  into more stable attractors like $<$$9,0,5,\\perp$$>$, and then adjust the ranks in Table \\ref{tab:asym_ranks} before starting the process again. In this way, the attractional force for each harmony in the distribution would change from one hierarchical level to the next. One could imagine that by incrementally assimilating variants into their more stable attractors, the resulting vocabulary might better reflect the functional categories described in theories of harmony (e.g., tonic, predominant, and dominant), and so produce greater attractional force estimates for harmonies like IV relative to those like I$^6$ at higher levels of the hierarchy.\n\n\\begin{example}[t!]\n\t\\centering\n\t\\input{Fig5.pdf_tex}\n\t\\caption{Tree diagram produced by the harmonic reduction algorithm for Op. 50\/2, iv, mm. 48--50. Branches reaching above the horizontal dashed line produce the reduction shown in the bottom system.}\n\t\\label{ex:reduction}\n\\end{example}\n\nDespite these limitations, the important point here is that when the vocabulary is larger than, say, 5 to 10 symbols, the admittedly simple algorithm just described produces surface-to-mid-level parsings not unlike those found in a Roman numeral analysis. Thus, it seems reasonable to suggest that the statistical associations between events near the surface reflect many of the organizational principles captured by most theories of harmony.\n\n\\section*{Conclusions}\n\nThis chapter adapted string-based methods from corpus linguistics to examine the coordinate, proordinate, and superordinate\/subordinate relations characterizing tonal harmony. In doing so, I have assumed that three of the organizational principles associated with natural languages---recurrence, syntax, and recursion---might also appear in tonal music. To that end, I began by examining the distribution of chromatic scale degree combinations across a corpus of Haydn string quartets. Unsurprisingly, the diatonic harmonies of the tonal system accounted for the vast majority of the combinations in the corpus. To model progressions of these harmonies over time, I then employed skip-grams, which include sub-sequences in an \\textit{n}-gram distribution if their constituent members occur within a certain number of skips. After applying filtering measures and ranking functions of various types, the most relevant (or meaningful) harmonic progressions emerged at the top of the \\textit{n}-gram distribution. Finally, to reduce the musical surface in Example \\ref{ex:haydn_ex}a to a sequence of its most central harmonies, I presented a simple harmonic reduction algorithm (and tree diagram) based on an asymmetric probabilistic measure of attractional force. In this case, the algorithm pruned all of the chords containing embellishing tones.\n\nTo examine the potential of string-based methods in corpus studies of music, I made certain simplifying assumptions about the principles mentioned above. Perhaps the most obvious of these was to restrict the purview of coordinate relations to temporally coincident scale degrees. This restriction seems reasonable for homorhythmic textures, but much less so for string quartets, piano sonatas, and the like, which often feature accompanimental textures that prolong harmonies over time (e.g., an Alberti bass pattern). This problem was at least partly resolved by the harmonic reduction algorithm, which assimilates variants into nearby attractors (e.g., I$^6$ into I), but note that it cannot replace variants of a given harmonic category---say, for example, $<$$0,4,\\perp,\\perp$$>$---if their more central (or prototypical) attractors---$<$$0,4,7,\\perp$$>$---do not appear nearby. Thus, the current algorithm would sometimes fail to produce convincing harmonic reductions for passages featuring complex polyphonic textures.\\footnote{For an innovative string-based solution to this problem, see \\textcite{White2013b}.}\n\nNevertheless, because these methods are ambivalent about the organizational systems they model, one need only revise the simplifying assumptions above to suit the needs of the research program. Indeed, since these methods proceed from the bottom up, we could just as easily use skip-grams or attraction measures to study melody, rhythm, or meter. Similarly, corpus studies in cognitive linguistics or systematic musicology might argue that the syntactic properties of natural language or tonal music should reflect limitations of human auditory processing, so it seems reasonable to impose similar restrictions on the sorts of contiguous and non-contiguous relations the skip-gram method should model \\autocite{Sears2017}. This claim seems especially relevant if we assume that the recursive hierarchy described throughout this chapter is nonuniform and discontinuous \\autocite{Meyer1973}, in that the statistics operating at relatively surface levels of musical organization---the syntax of harmonic progressions (e.g., I--ii$^6$--V--I)---might differ from those operating at deeper levels---long-range key relationships (e.g., I--III--V--I).\n\nThere are, indeed, many possible solutions for the computational problems described here. My goal was not to offer definitive results, but to demonstrate that the most recent methods developed in corpus linguistics and natural language processing have much to offer for corpus studies of music. Indeed, if \\textcite{Patel2008} is right that language and music share certain fundamental design features, then skip-grams, contingency tables, and association measures represent invaluable tools for the study of tonal harmony.\n\n\\pagebreak\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Section: Introduction}\nFor object recognition to be done correctly, a model must preserve the hierarchical relationships between object parts and their corresponding wholes. It is not sufficient that all the pieces of an object are present in an image, they must also be oriented correctly with respect to one another. Convolutional Neural Networks (CNNs) are limited in their ability to model the spatial hierarchy between objects. Even if sub-sampling operations (e.g., max-pooling) are removed, the representation of data in CNNs do not take into account the relationships between object parts and wholes.\n\nCapsule Networks \\cite{Sabour_2017}, \\cite{Hinton_2018} learn such hierarchies by grouping feature map channels together to form a vector of features (i.e., a capsule) that captures the instantiation parameters of objects and by learning transformation matrices that encode viewpoint \\textit{invariant} information. These networks generalize to novel viewpoints by incorporating the viewpoint changes directly into the activities of the capsules. The capsules can represent various properties of an object ranging from its size and shape to more subtle features such as texture and orientation. Since their introduction, CapsNets have produced state-of-the-art accuracies on datasets such as MNIST \\cite{Zhao_2019} and smallNORB using a network with far fewer parameters compared with their CNN counterparts \\cite{Sabour_2017}.\n\nAt their core, CapsNets \\cite{Sabour_2017} make use of a dynamic routing algorithm to ensure that the output of a lower-level capsule is routed to the most appropriate higher-level capsule. This is done by multiplying the lower-level capsules with learned viewpoint invariant transformation matrices to produce prediction vectors. These matrices make sense of the spatial relationships between object features. The scalar product between the prediction vectors and each of the higher-level capsules governs the agreement between a part and a whole. Large values imply that a part has a high likelihood of belonging to a whole and vice versa. The combination of the scalar products and the transformation matrices ultimately decides which whole is most suited for each part. This ``routing-by-agreement'' is a more effective way of sending spatial information between layers in a network than the routing implemented by max-pooling, since the former maintains the exact spatial relationships between neurons in each layer of the network, regardless of the network depth.\n\nAlthough CapsNets have shown promising results, there are some key limitations that make them unsuitable for real world use. One such issue is inference using CapsNets is significantly slower compared with CNNs. This is primarily due to the dynamic routing procedure which requires several iterations to produce the output vectors of a capsule layer. This limitation prevents deeper CapsNet architectures from being used in practice.\n\nWe present a method for speeding up inference for CapsNets, with potential applications for training. This is accomplished by making use of the accumulated information in the dynamically calculated routing coefficients computed using the training dataset. Analyses of intra-class routing coefficients show that they form a unique signature for each class. Intuitively, this is because parts of the same object should generally be similar and, more importantly, distinct from the parts of a different object. In practice, the network is trained to produce prediction vectors (using the lower-level capsules and learned transformation matrices) that closely correlate with the higher-level capsule associated with its own class. This observation allows creation of a set of \\textit{master} routing coefficients using the individual routing coefficients associated with each training example. At inference time, the master routing coefficients are used instead of dynamically computing the routing coefficients for each input to the network. Our method for fast routing at inference effectively replaces the \\boldmath{$r$} iterations in the routing procedure with a single matrix-multiply operation, allowing the network to be parallelized at run-time. On the MNIST dataset and its variants, fast inference decreases the test time accuracies by less than $0.5 \\%$ and by approximately $5 \\%$ for CIFAR10.\n\nSection \\ref{Section: Capsule Network Architecture} describes the three-layer network architecture that was used. Section \\ref{Section: Comparison Between Dynamic and Fast Routing Procedures} compares the differences in the routing procedure at inference between the dynamic routing algorithm and the fast routing procedure. Section \\ref{Section: Analyses of Dynamically Calculated Routing Coefficients} analyses the dynamic routing coefficients from the MNIST and CIFAR10 training datasets. In Section \\ref{Section: Creation of Master Routing Coefficients}, we detail the procedure for creating a set of master routing coefficients and compare the master set of routing coefficients with the dynamically calculated ones in Section \\ref{Section: Analyses of Master Routing Coefficients}. Section \\ref{Section: Results} compares the test accuracies between the dynamic and fast routing procedures for the five datasets. Discussions on the use and applicability of master routing coefficients are given in Section \\ref{Section: Discussion}. The general procedure for creating master routing coefficients is detailed in Appendix \\ref{App. General Approach for Creating Master Routing Coefficients}.\n\n\\section{Capsule Network Architecture}\n\\label{Section: Capsule Network Architecture}\nThe CapsNet architecture shown in Fig. \\ref{Fig: CapsNet Architecture} follows the three layer network from \\cite{Sabour_2017}.  For the MNIST \\cite{MNIST}, Background MNIST (bMNIST) and Rotated MNIST (rMNIST) \\cite{R_and_BG_MNIST}, and Fashion MNIST (fMNIST) \\cite{F_MNIST} datasets, the input to the network is a $28 \\times 28$ grayscale image that is operated on by a convolutional layer to produce a $20 \\times 20 \\times 256$ feature map tensor (for CIFAR10, the $32 \\times 32 \\times 3$ image is randomly cropped so that its spatial dimensions are $24 \\times 24$). The second convolutional layer outputs a $6 \\times 6 \\times 256$ feature map tensor. Each group of $8$ neurons ($4$ for CIFAR10) in this feature map tensor is then grouped channel-wise and forms a single lower-level capsule, \\boldmath{$i$}, for a total of $6 \\times 6 \\times (256 \\div 8) = 1152$ lower-level capsules ($1024$ for CIFAR10).\n\nThe lower-level capsules are fed to the routing layer, where the dynamic routing procedure converts these capsules to the $10 \\times 16$ DigitCaps matrix, where $10$ is the number of higher-level capsules (also the number of object classes) and $16$ is the dimensionality of the capsules. Here, we use Max-Min normalization \\cite{Zhao_2019} as opposed to Softmax to convert the raw logits into routing coefficients. Details on the dynamic routing procedure are given in Section \\ref{Section: Comparison Between Dynamic and Fast Routing Procedures}.\n\nEach row in the DigitCaps matrix represents the instantiation parameters of a single object class and the length of the row vector represents the probability of the existence of that class. During training, non-ground-truth rows in the DigitCaps matrix are set to zero and the matrix is passed to a reconstruction network composed of two fully-connected layers of dimensions $512$ and $1024$ with ReLU activations, and a final fully-connected layer of dimension $28 \\times 28 \\times 1 = 784$ ($24 \\times 24 \\times 3 = 1728$ for CIFAR10) with a sigmoid activation. During inference, the reconstruction network is not used. Instead, the row in the DigitCaps matrix with the largest L2-norm is taken as the predicted object class for the input.\n\nOur implementation uses TensorFlow \\cite{TensorFlow} with training conducted using the Adam optimizer \\cite{Adam_Optimizer} with TensorFlow's default parameters and an exponentially decaying learning rate. Unless otherwise noted, the same network hyperparameters in \\cite{Zhao_2019} were used here for training as well. Original code is adapted from \\cite{Sabour_Code}.\n\n\\begin{figure}[htp]\n\\centering\n{\\includegraphics[width = 3.5 in]{CapsNet_Architecture}}\n\\caption{(Left) Three-layer CapsNet architecture adapted from Sabour et al. \\cite{Sabour_2017}. The PrimaryCaps layer consists of $6 \\times 6 \\times 32 = 1152$ $8$-D vector capsules for the MNIST dataset and its variants ($1024$ $4$-D vector capsules for CIFAR10). The routing procedure produces the $10 \\times 16$ DigitCaps layer, which is used to calculate the margin loss. The DigitCaps output is also passed to the reconstruction network where the non-ground-truth rows are masked with zeros. The network takes the masked $10 \\times 16$ DigitCaps matrix as input and learns to reproduce the original image. Margin and reconstruction loss functions follow those from \\cite{Sabour_2017}.}\n\\label{Fig: CapsNet Architecture}\n\\end{figure}\n\n\\section{Comparison Between Dynamic and Fast Routing Procedures}\n\\label{Section: Comparison Between Dynamic and Fast Routing Procedures}\nThe dynamic routing procedure for CapsNets using Max-Min normalization is given by Max-Min Routing Procedure below. This algorithm is used during normal training and inference. The prediction vectors to the routing layer, \\boldmath{$\\hat{u}_{j|i}$}, are created by multiplying each of the lower-level capsules, \\boldmath{$u_i$}, in PrimaryCaps by their respective transformation weight matrix, \\boldmath$W_{ij}$. The higher-level capsules \\boldmath{$s_j$} are computed as a sum over all the prediction vectors, weighted by the routing coefficients, \\boldmath{$c_{ij}$}. The routing coefficients are initialized with a value of $1.0$ and can be viewed as independent probabilities representing the likelihood of a lower-level capsule being assigned to a higher-level capsule \\cite{Zhao_2019}. For a given input to the routing layer, its routing coefficient matrix has shape $N_i \\times N_j$, where $N_i$ and $N_j$ are the number of lower and higher-level capsules, respectively. The higher-level capsules, \\boldmath{$s_j$}, are then squashed using a non-linear function so that the vector length of the resulting capsule, \\boldmath{$v_j$}, is between $0$ and $1$. These operations are shown in Eq. \\ref{Eqs. u_hat, s_J, and v_J}.\n\n\\begin{equation}\n\t\\label{Eqs. u_hat, s_J, and v_J}\n\t\\hat{u}_{j|i} = W_{ij}u_i, \\quad s_j = \\sum_{i} {c_{ij}\\hat{u}_{j|i}}, \\quad v_j = \\frac{||s_j||^2}{1 + ||s_j||^2}\\frac{s_j}{||s_j||}\n\\end{equation}\n\nDuring the procedure, the update to the routing coefficients, \\boldmath{$b_{ij}$}, is computed as the dot product between the prediction vectors and the current state of the higher-level capsules, \\boldmath{$v_j$}. The update to the routing coefficients is then normalized via Max-Min normalization over the object classes as given by Eq. \\ref{Eq: Max-Min Normalization}, where $p$\/$q$ are the lower\/upper bounds of the normalization. For the first iteration, the routing coefficients are initialized to $1.0$ outside of the main routing for-loop.\n\n\\begin{equation}\n\t\\label{Eq: Max-Min Normalization}\n\tc_{ij} = p + \\frac{b_{ij} - min(b_{ij})}{max(b_{ij}) - min(b_{ij})} * (q-p)\n\\end{equation}\n\n\\begin{table}[h!]\n    \\begin{tabular}{l}\n    \\hline\n    \\textbf{Max-Min Routing Procedure} \\\\\n    \\hline\n    1: Input to Routing Procedure: ({$\\bm{\\hat{u}_{j|i}}$}, $r$, $l$) \\\\\n    2: \\quad for all capsules $i$ in layer $l$ and capsule $j$ in layer ($l$ + 1): $c_{ij}$ $\\leftarrow$ 1.0 \\\\\n    3: \\quad \\textbf{for} $r$ iterations: \\\\\n    4: \\quad \\quad for all capsule $j$ in layer ($l$ + 1): $\\bm{s_j}$ $\\leftarrow$ $\\sum_{i} c_{ij} \\bm{\\hat{u}_{j|i}}$ \\\\\n    5: \\quad \\quad for all capsule $j$ in layer ($l$ + 1): $\\bm{v_j}$ $\\leftarrow$ Squash($\\bm{s_j}$) \\\\\n    6: \\quad \\quad for all capsule $i$ in layer $l$ and capsule $j$ in layer ($l$ + 1): $b_{ij} \\leftarrow b_{ij} + \\bm{\\hat{u}_{j|i}} \\cdot$ $\\bm{v_j}$ \\\\\n    7: \\quad \\quad for all capsule $i$ in layer $l$: $\\bm{c_i}$ $\\leftarrow$ Max-Min ($b_{ij}$) $\\Longrightarrow$ $\\mathrm{Given~in~Eq.}$ ~\\ref{Eq: Max-Min Normalization} \\\\\n       \\quad \\quad \\textbf{return} $\\bm{v_j}$ \\\\\n    \\hline\n    \\end{tabular}\n    \\label{Procedure: Max-Min Routing}\n\\end{table}\n\nFor fast inference, the routing coefficients no longer need to be dynamically calculated for each new input. Instead, the prediction vectors are simply multiplied with the precomputed master routing coefficient tensor ($C_{ij}$), summed, and squashed to form the higher-level capsules. Classification using the parent-level capsules, \\boldmath{$v_j$}, is made in the usual way afterwards. Details on the master routing coefficients are given in the following sections.\n\n\\begin{table}[h!]\n    \\begin{tabular}{l}\n    \\hline\n    \\textbf{Fast Routing Procedure for Inference} \\\\\n    \\hline\n    1: Input to Fast Routing Procedure: ($C_{ij}$, \\boldmath$\\hat{u}_{j|i}$, $l$) \\\\\n    2: \\quad for all capsule $j$ in layer ($l$ + 1): $s_j$ $\\leftarrow$ $\\sum_{i} C_{ij}\\boldmath\\hat{u}_{j|i}$ \\\\\n    3: \\quad for all capsule $j$ in layer ($l$ + 1): $v_j$ $\\leftarrow$ Squash($s_j$) \\\\\n       \\quad \\quad \\textbf{return} $v_j$ \\\\\n    \\hline\n    \\end{tabular}\n    \\label{Procedure: Fast Routing for Inference}\n\\end{table}\n\n\\section{Analyses of Dynamically Calculated Routing Coefficients}\n\\label{Section: Analyses of Dynamically Calculated Routing Coefficients}\nFor each input to the routing layer, an $N_i \\times N_j$ routing coefficient matrix is initialized with a value of $1.0$ and then iteratively updated in the routing procedure. By design, the $i^{th}$ column of routing coefficients is responsible for linking the lower and higher-level capsules for the $i^{th}$ object class. Intuitively, one would expect intra-class objects to have similarly activated routing coefficients compared with those from inter-class objects since the routing coefficients are simply the agreement between prediction vectors and higher-level capsules. This can be shown quantitatively by computing the correlations between the dynamically calculated routing coefficients for the different object classes.\n\nWe choose to compute the correlations between only the ground-truth (GT) columns in the routing coefficient matrices rather than between entire routing coefficient matrices, since the GT columns are used for the correct classification of each image. In other words, for a network that has been properly trained, the routing coefficients in the GT columns are the ones that provide the unique signatures for each object class. The routing coefficients in the other columns generally do not evolve to form unique signatures in the same manner as the GT routing coefficients. This can be observed in the tuning curves associated with the higher-level capsules (c.f. Section \\ref{Section: Analyses of Master Routing Coefficients}).\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 4.5 in]{GT_Routing_Coefficient_CH}}\n\\caption{(a) Heatmap showing the correlations between the GT columns for the first $100$ images in the MNIST training set. The heatmap is created by calculating the correlation coefficient between GT columns in the $1152 \\times 10$ routing coefficient matrix associated with each image. (b) Heatmap showing the correlations between the GT columns for the first $100$ images in the CIFAR10 training set. The heatmap is created by calculating the correlation coefficient between GT columns in the $1024 \\times 10$ routing coefficient matrix associated with each image. Only half of the computed values are shown for each heatmap since the correlations are symmetric.}\n\\label{Fig: GT Routing Coefficient CH}\n\\end{figure}\n\nFigure \\ref{Fig: GT Routing Coefficient CH} (a) shows the correlation heatmap between the GT columns of the routing coefficient matrices for the first $100$ images in the MNIST training dataset. For each image, the GT column in its routing coefficient matrix refers to the column that corresponds to its object class. For example, an image of the digit $5$, the GT column in that image's routing coefficient matrix is the sixth column (due to zero-indexing). Likewise, Fig. \\ref{Fig: GT Routing Coefficient CH} (b) shows the correlation heatmap for the first $100$ images in the CIFAR10 training dataset calculated in the same way. For MNIST, high correlation is observed between routing coefficients for intra-class objects. For a more complicated dataset such as CIFAR10, the distinction between inter-class routing coefficients is not as clear since the same part can often be associated with more than one object class in the dataset. For example, trucks and automobiles will have often multiple similar parts (e.g., wheels, headlights, body frame, etc.) and airplanes and ships will often appear against the same color background.\n\nFigures \\ref{Fig: Mean Class Routing Coefficient CH} (a) and (b) show the correlation heatmaps between each of the object classes for the MNIST and CIFAR10 datasets, respectively, averaged over \\textit{all} of their training images. For example, element $(0, 0)$ in the heatmaps is the mean correlation between the GT columns of all objects in class $0$ for the dataset and element $(0, 1)$ is the mean correlation between the GT columns of all objects in classes $0$ and $1$, and so on. The same trends exist when the correlation is computed across all training images as those in Fig. \\ref{Fig: GT Routing Coefficient CH} for the case of $100$ images for each dataset. In particular, it is interesting to note the second highest correlations for each object class. For the MNIST dataset, the second highest correlation with the digit $0$ is the digit $6$ and the digit $9$ has high correlations with the digits $4$ and $7$. These digit classes are often most similar to one another for the MNIST dataset (c.f. \\cite{Zhao_2019} for examples). For CIFAR10, high inter-class correlations exist between airplanes (class $0$) and ships (class $8$) and automobiles (class $1$) and trucks (class $9$). These classes often present the most challenging examples for classification. For MNIST, intra-class correlations are significantly higher compared with inter-class correlations and thus, the network is able to properly distinguish between the digit classes. For CIFAR10, intra-class correlations are lower and this leads to difficulties in classifying new images.\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 4.5 in]{Mean_Class_Routing_Coefficient_CH}}\n\\caption{(a) Heatmap showing the average class correlations for \\textit{all} images in the MNIST training set. For example, the value in element $(0, 0)$ is the mean correlation between the GT columns in the routing coefficient matrix for all objects in class $0$ and the value in element $(0, 1)$ is the mean correlation between the GT columns in the routing coefficient matrix for all objects in class $0$ and $1$, etc. (b) Heatmap showing the average class correlations for \\textit{all} images in the CIFAR10 training set. Only half of the computed values are shown for each heatmap since the correlations are symmetric.}\n\\label{Fig: Mean Class Routing Coefficient CH}\n\\end{figure}\n\n\\section{Creation of Master Routing Coefficients}\n\\label{Section: Creation of Master Routing Coefficients}\nSince the intra-class routing coefficients from the training dataset form a unique signature for each object class, they can be used to create a set of master routing coefficients that can generalize well to new examples. There are several ways in which a set of master routing coefficients can be created. We detail the procedure shown in Fig. \\ref{Fig: Creation of Master Coefficients}, which was used to generate the master routing coefficients used for fast inference. A general approach for creating master routing coefficients is given in Appendix \\ref{App. General Approach for Creating Master Routing Coefficients}.\n\nDuring training, the routing coefficient matrix is initialized to $1.0$ for each input to the routing layer and then iteratively updated to reflect the agreement between the input prediction vectors to the routing layer and the final output vectors of the routing layer. At the start of training, the routing coefficients can change for the same input since the prediction vectors are being updated by the (trainable) network weights, \\boldmath$W_{ij}$. However, once training has converged, the routing coefficients naturally converge since they are computed in a bootstrap manner; i.e., the update to the routing coefficients are calculated using just the prediction vectors and an initial set of routing coefficients. After training is completed, the routing coefficient matrix associated with each training image can be extracted.\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 4.5 in]{Creation_of_Master_Coefficients}}\n\\caption{Procedure for creating a single master routing coefficient  matrix for fast inference. After a network has been properly trained, the routing coefficient matrix associated with each training image is extracted and used to form a single master routing coefficient matrix via three steps: 1) summation, 2) normalization, and 3) dimension reduction. Details are given in Section \\ref{Section: Creation of Master Routing Coefficients}.}\n\\label{Fig: Creation of Master Coefficients}\n\\end{figure}\n\nFor training images, fast inference can be conducted on the \\textit{training} dataset by simply using the individual routing coefficient matrix associated with each image. For new images, however, there is no apriori method of determining which routing coefficient matrix to use for each image since the class label is unknown (if such a method existed, then the classification problem is essentially solved without the need for the network to even make the prediction). As a result, fast inference on new images must rely on the use of a \\textit{single} routing coefficient matrix. This matrix has the same shape as the individual routing coefficient matrices and each column is assigned to a known object class.\n\nTo create the master routing coefficient matrix, we accumulate the information contained in the individual routing coefficient matrices from the training dataset. This process involves three main steps: 1) summation, 2) normalization, and 3) dimension reduction. First, we train the CapsNet to convergence in the usual manner, run inference on the training images, and save the routing coefficients associated with the training images at the last step of the routing procedure (for MNIST, this results in $60,000$ $1152 \\times 10$ matrices). Then, we initialize $N$ matrices as containers to hold the accumulated class-specific routing coefficients for each of the $N$ object classes. For each training example, its individual routing coefficient matrix is summed in the appropriate container matrix for its class. After all training images have been processed, the set of $N$ container matrices holds the sum of all routing coefficients at the last routing iteration, one for each class.\n\nEach container matrix is then normalized by their respective class frequency, followed by a row-wise Max-Min normalization. At this point, each container matrix can be viewed as a master routing coefficient matrix for that class. However, the network expects a single routing coefficient matrix and, without additional information, there is no straightforward method to point to the correct container matrix when the network is presented with a new example.\n\nThus, in order to present a single routing coefficient matrix to the network at inference, the set of $N$ container matrices must be reduced to a single matrix. This is done by transferring only the GT column from each of the $N$ container matrices to its corresponding column in the master routing coefficient matrix. In other words, only the first column from the first container matrix (which holds the accumulated routing coefficients for the first object class) is transferred to the first column of the master routing coefficient matrix, and so on for the columns in the other container matrices. The end result of the dimension reduction is a single routing coefficient matrix that can be used for new examples during inference.\n\n\\section{Analyses of Master Routing Coefficients}\n\\label{Section: Analyses of Master Routing Coefficients}\nIn order for the master routing coefficient matrix to generalize well to new images, each column of routing coefficients in the matrix should only correlate highly with its own object class. In other words, the first column of routing coefficients in the master routing coefficient matrix should correlate highly with routing coefficients in the first columns of the individual routing coefficient matrices for images associated with the first object class. The second column of routing coefficients in the master routing coefficient matrix should correlate highly with routing coefficients in the second columns of the individual routing coefficient matrices for images associated with the second object class, and so on. If the first column in the master routing coefficient matrix correlated highly with the fifth column in an individual routing coefficient matrix associated with an image from the fifth object class, this would imply that the first column of master routing coefficients does \\textit{not} form an unique signature for the first object class---it can likely classify an image that belongs to the first object class as that of the fifth object class (and vice versa).\n\nTo quantify the degree to which each column of the master routing coefficient matrix is representative of its object class, we compute the correlation between each column in the master routing coefficient matrix and the GT columns in each individual routing coefficient matrix. This is shown in Figs. \\ref{Fig: Mean CH Between GT Master and GT Individual Routing Coefficients} (a) and (b) for the MNIST and CIFAR10 datasets, respectively. These correlations are between a single column of routing coefficients from the \\textit{master} routing coefficient matrix and the GT column of routing coefficients from the \\textit{individual} routing coefficient matrix associated with each training image for the dataset. Thus, the correlations are not symmetric (i.e., the correlation for element $(0, 1)$ in the heatmap is not the same as the correlation for element $(1, 0)$).\n\nFrom Figs. \\ref{Fig: Mean CH Between GT Master and GT Individual Routing Coefficients} (a) and (b), we see that each column of routing coefficients in the master routing coefficient matrix has higher intra-class correlations than inter-class correlations with the individual GT routing coefficients from each image in the training dataset. This is effectively why the master routing coefficients are able to generalize well to new images during inference---they are able to uniquely route the lower-level capsules to the correct higher-level capsules for new images by using the accumulated information from the training data. This relationship between master and individual routing coefficients is stronger for simpler datasets such as MNIST and its variants than for a more complicated dataset such as CIFAR10. For CIFAR10, intra-class correlations are still higher compared with inter-class correlations; however, as mentioned above, inter-class correlations can also be high for objects belonging to similar classes.\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 4.5 in]{Mean_CH_Between_GT_Master_and_GT_Individual_Routing_Coefficients}}\n\\caption{(a) Correlation heatmap between the columns in the master routing coefficients matrix and the GT columns from the individual routing coefficient matrices associated with the MNIST training images. This shows that creating the master routing coefficients as outlined in Section \\ref{Section: Creation of Master Routing Coefficients} results in a set of routing coefficients that correlates highly within its own class. (b) Correlation heatmap between the columns in the master routing coefficients matrix and the GT columns from the individual routing coefficient matrices associated with the CIFAR10 training images. For CIFAR10, high correlations can also exist between \\textit{inter}-class routing coefficients. For example, the correlation between the routing coefficients in column $1$ (associated with the object class ``car'') of the master routing coefficient matrix and GT columns $9$ (associated with the object class ``truck'') from individual routing coefficient matrices is almost as high as the \\textit{intra}-class correlations for classes $1$ and $9$, individually.}\n\\label{Fig: Mean CH Between GT Master and GT Individual Routing Coefficients}\n\\end{figure}\n\nThe effectiveness of the master routing coefficients can also be examined by looking at the output capsules, \\boldmath{$v_j$}, from the DigitCaps layer. Figures \\ref{Fig: DigitCaps Routing Examples} (a) and (b) compare the digit class probabilities for the same set of test images from the MNIST dataset between the dynamic and fast inference routing procedures. Likewise, Figs. \\ref{Fig: DigitCaps Routing Examples} (c) and (d) compare the probabilities for the same set of test images from the CIFAR10 dataset. For MNIST, the master routing coefficients produce similarly peaked values for the digit class probabilities compared with the use of dynamically calculated routing coefficients---each column in the master routing coefficient matrix is a unique signature for that digit class. On the other hand, the digit class probability comparison for CIFAR10 is noticeably different. Although the master routing coefficients are able to correctly classify three out of the five test examples shown, the classification is not particularly robust compared with the use of dynamically calculated routing coefficients (the dynamically calculated routing coefficients correctly classifies four out of the five test image examples shown, and have lower probabilities for the non-ground-truth classes).\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 5.5 in]{DigitCaps_Routing_Examples}}\n\\caption{Output class probabilities for the same set of test images from the MNIST and CIFAR10 datasets. The class probabilities are obtained from networks that use dynamically calculated routing coefficients ((a) and (c)) and the master routing coefficients ((b) and (d)). For MNIST, the master routing coefficients produce nearly identical class probabilities compared with the use of dynamically calculated routing coefficients. For CIFAR10, the use of master routing coefficients correctly classify three out of the five test image examples shown and there is strong inter-class competition.}\n\\label{Fig: DigitCaps Routing Examples}\n\\end{figure}\n\nThe digit class probabilities can also be class-averaged over the test dataset as shown in Fig. \\ref{Fig: Mean Vector Length Per Class}. These ``tuning curves'' show how well the network is able to distinguish between the different object classes in the dataset. For MNIST, the tuning curves resulting from the fast inference procedure is similar to those from the dynamic routing procedure, suggesting that the master routing coefficients provides an accurate reflection of the dynamically calculated routing coefficients. The tuning curves for CIFAR10 from the dynamic routing procedure show that, for certain object classes, the discriminability is not as robust compared with MNIST---multiple peaks exist for each object class. For fast inference, each tuning curve is still peaked at the correct object class; however, they are also highly peaked around the other classes as well.\n\n\\begin{figure}[h]\n\\centering\n{\\includegraphics[width = 5.5 in]{Mean_Vector_Length_Per_Class}}\n\\caption{Class-averaged output probabilities for the MNIST and CIFAR10 test datasets obtained with the use of dynamically calculated routing coefficients ((a) and (c)) and the master routing coefficients ((b) and (d)). The master routing coefficients for the MNIST dataset produce tuning curves similar to those obtained using dynamically calculated routing coefficients. For CIFAR10, the tuning curve for each object class obtained using dynamically calculated routing coefficients exhibit multiple peaks around non-ground-truth classes for each object class. In addition, multiple high-valued peaks are observed for the tuning curves resulting from the use of master routing coefficients.}\n\\label{Fig: Mean Vector Length Per Class}\n\\end{figure}\n\n\\section{Results}\n\\label{Section: Results}\nTable \\ref{Table: Routing Method Accuracies} provides the test accuracies for five datasets for the two different routing methods at inference. For fast routing, the master routing coefficients are created using the steps detailed in Section \\ref{Section: Creation of Master Routing Coefficients} and the procedure is given by Fast Routing Procedure for Inference in Section \\ref{Section: Comparison Between Dynamic and Fast Routing Procedures}. With Max-Min normalization, fast routing decreases the test accuracy by approximately $0.5 \\%$ for the MNIST dataset and its variants and by $5 \\%$ for CIFAR10.\n\nThe master routing coefficients can also be created from a network trained using Softmax as the normalization in the routing layer as opposed to Max-Min. In this approach, the routing procedure is exactly the same as in \\cite{Sabour_2017} and the master routing coefficients are created in the same manner as that detailed in Section \\ref{Section: Creation of Master Routing Coefficients} except the Softmax function is used to normalize the rows in the container matrices instead of Max-Min. For networks trained with Softmax in the routing layer, the majority of the routing coefficients are grouped around their initial value of $0.1$ after three routing iterations \\cite{Zhao_2019}. Since the container matrices for each class are created via summation, class frequency averaging, and normalizing via Softmax, the resulting master routing coefficient matrix contains values close to $0.1$ (for a dataset with $10$ classes). This amounts to a uniform distribution for the master routing coefficients and results in a significantly reduced test time performance across all five datasets.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Mean of the maximum test accuracies and their standard deviations on five datasets for the different routing methods at inference. Five training sessions were conducted for each dataset. The conditions under which the datasets were trained are the same as \\cite{Zhao_2019}. The master routing coefficients used for fast inference is created using the procedure outlined in Section \\ref{Section: Creation of Master Routing Coefficients}.}\n    \\begin{tabular}{cccccc}\n    \\hline\n    \\textbf{Routing Method} & \\textbf{MNIST {[}\\%{]}} & \\textbf{bMNIST {[}\\%{]}} & \\textbf{fMNIST {[}\\%{]}} & \\textbf{rMNIST {[}\\%{]}} & \\textbf{CIFAR10 {[}\\%{]}} \\\\\n    \\hline\n    Dynamic, Max-Min & 99.55 $\\pm$ 0.02 & 93.09 $\\pm$ 0.04 & 92.07 $\\pm$ 0.12 & 95.42 $\\pm$ 0.03 & 75.92 $\\pm$ 0.27 \\\\\n    Fast, Max-Min & 99.43 $\\pm$ 0.08 & 92.93 $\\pm$ 0.10 & 91.52 $\\pm$ 0.20 & 95.04 $\\pm$ 0.04 & 70.33 $\\pm$ 0.36 \\\\\n    Dynamic, Softmax & 99.28 $\\pm$ 0.07 & 89.08 $\\pm$ 0.21 & 90.52 $\\pm$ 0.16 & 93.72 $\\pm$ 0.09 & 73.65 $\\pm$ 0.10 \\\\\n    Fast, Softmax & 98.92 $\\pm$ 0.30 & 84.34 $\\pm$ 4.37 & 80.16 $\\pm$ 7.35 & 84.13 $\\pm$ 4.28 & 47.11 $\\pm$ 8.70 \\\\\n    \\hline\n    \\end{tabular}\n    \\label{Table: Routing Method Accuracies}\n\\end{table}\n\n\\section{Discussion}\n\\label{Section: Discussion}\nIn CapsNets, routing coefficients form the link between capsules in adjacent network layers. Throughout the dynamic routing procedure, the updates to the routing coefficients come from the agreement between the prediction vectors and the higher-level capsules calculated using the prediction vectors. Since the prediction vectors are computed by a convolutional layer and are learned by the network, they capture the lower-level features for that object class. Using this information, we create a set of master routing coefficients from the training data that generalize well to test images.\n\nIf the network is properly trained (i.e., it is able to adequately distinguish between each object class) then the prediction vectors are unique and, as a result, the routing coefficients have high intra-class correlations. This is the case for MNIST and its variants. For a more complex dataset such as CIFAR10, the network does not learn sufficiently different prediction vectors for each object class during training. This is evident by comparing the tuning curves in Fig. \\ref{Fig: Mean Vector Length Per Class} between MNIST and CIFAR10 for the case of \\textit{dynamic} routing. As a result, the master routing coefficients created for CIFAR10 do not perform as well.\n\nA better process for creating a master routing coefficient matrix is also possible. In this paper, the approach taken to create the master routing coefficients uses \\textit{all} of the training data. For networks that can sufficiently distinguish between each object class, using all of the data makes sense. For CIFAR10, not all individual routing coefficient matrices are equally useful since some have high inter-class correlations. Therefore, a method of selecting the routing coefficients that have high intra-class and low inter-class correlations can be helpful. This can be done in several ways. For example, clustering analyses can be used to group routing coefficients that have high intra-class properties and remove training examples that result in ``outlier'' routing coefficients for each class. A similarity measure (e.g., correlation, dot product, etc.) can also be used to exclude outliers. Filtering methods will be taken up in future work.\n\nGiven that a single set of routing coefficients can be used for inference, a question to ask is whether or not it can also be useful for training. At least two approaches can be implemented for training using the master routing coefficients. First, a CapsNet can be retrained using the single master routing coefficient matrix to see if the network can learn to recognize the object classes better when it is able to use the cumulative knowledge (contained in the master routing coefficients) from all training data at once. Second, training can be expedited by initially training a CapsNet using dynamic routing on a carefully selected subset (or all) of the training data for a few epochs. Afterwards, the master routing coefficient matrix can be created from this training session and used to retrain the network (using all of the training data) to convergence. Both of these approaches will be taken up in future work.\n\n\\section{Summary}\n\\label{Section: Summary}\nCapsule Networks have the ability to learn part-whole relationships for objects and can potentially generalize to novel viewpoints better than conventional CNNs. In contrast to CNNs, they can maintain the exact spatial relationships between the different components of an object by discarding sub-sampling layers in the network. State-of-the-art performance has already been demonstrated for the MNIST dataset by \\cite{Zhao_2019}. However, CapsNets are slow to train and run in real-time due to the dynamic routing algorithm. In addition, state-of-the-art performance on more complicated datasets still present some challenges, possibly due to the prohibitively high cost of constructing deeper CapsNet models. In this work, we focused on methods to improve the speed of CapsNets for inference while still maintaining the accuracy obtained using the dynamic routing algorithm. To this end, we have implemented a method that allows for fast inference while keeping the test accuracy comparable to the dynamic implementation.\n\n\\medskip\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe concerns about the potential health risks caused by the X-ray radiation of computed tomography (CT) have led to the concept of ALARA (As Low As Reasonably Achievable)~\\cite{international19921990},~\\cite{slovis2003children},~\\cite{brenner2007hall},~\\cite{SARMA2012750}. \nThe concept aims to regulate the delivery of excessive X-ray radiation to the patients. \nAlthough low-dose CT (LDCT) imaging decreases the risks, it yields images of deteriorated quality because of the increased noise background and the pertinent appearance of artifacts that could affect the diagnostic decisions~\\cite{yu2009radiation}.\n\nFor mitigating the excessive noise in LDCT, different methods were proposed. One of the approaches is to apply iterative reconstruction techniques. These methods aim to improve the quality of CT slices through optimization of an objective function that incorporates a system model, a statistical noise model, and prior information in the image domain. Although these iterative techniques gave an improvement in the quality of the reconstructed CT slices, they are not efficient because they require multiple forward and back-projection operations.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=2.5in]{figures\/schemes\/n2ntd_scheme.pdf}\n\\caption{The scheme of the proposed self-supervised Noise2NoiseTD-ANM approach. $y$ denotes noisy pixels, $\\Omega_y$ is neighborhood of noisy pixels, $\\lambda$ and $\\sigma_e^2$ are estimated noise parameters. Green arrows correspond to the training process and purple arrows correspond to inference.}\n\\label{fig:approach}\n\\end{figure}\n\nThe other approach is to employ image processing: either traditional or deep-learning-based. Because these methods were adapted from denoising images in the natural domain, they are not overly dependent on the CT scanner and its acquisition geometry. Compared to iterative reconstruction techniques, they can separately denoise raw CT projections or already reconstructed CT slices. These methods can serve as an auxiliary tool for noise reduction before or after reconstruction. Thus, they, especially deep-learning-based methods, are of high research interest.\nThe traditional methods are not very adaptive to image content and often contain many parameters which require careful case-by-case tuning. Besides, they provide inferior quality than deep-learning-based methods. \nSpecifically, supervised deep learning methods (Noise2Clean) that assume model training on the big data set of paired noisy-clean images show the best performance~\\cite{3DCT},~\\cite{RED},~\\cite{ResCT}. However, their requirement for the paired noisy-clean images may not always be accomplished.  \nIn a clinical setting, for example, in a radiology suite, the acquisition of such paired data can take at least twice longer than the regular CT exam and would increase exposure to the X-ray dose. Furthermore, obtained images may not ideally coincide due to the patient breathing and movements.\nGenerating the noisy data artificially is an option to alleviate the lack of the image pairs, but the addition of the noise to the images, especially the medical ones, can bias the predictions of a convolutional neural network (CNN)~\\cite{green2018learning} unless a very accurate noise simulation is used, which requires deep insights into the actual CT system,~\\cite{zabic2013low} and is not always possible.\n\nBecause of the problems with the data availability, different unsupervised and self-supervised methods for image denoising were proposed. However, unsupervised methods, which are mostly based on generative adversarial networks~\\cite{park2019unpaired},~\\cite{wolterink2017generative},~\\cite{kang2019cycle}, sill require a set of clean images. Besides, this set, as well as the set of noisy images, should be well representative of the diversity of cases found in reality. Thus, these methods imply the need to irradiate some patients with rather high doses of CT X-rays. \nSelf-supervised methods, which do not require clean images at all, appear to be more suitable for denoising medical images than supervised and unsupervised methods. \n\nSome of the previously proposed self-supervised CT denoising methods~\\cite{n2n_ct2020},~\\cite{wu2019consensus},~\\cite{hendriksen2020noise2inverse},\\cite{yuan2020half2half} require additional data generation that complicates the denoising process. The other methods~\\cite{xu2021deformed2self},~\\cite{choi2021self},~\\cite{unal2021self} do not have this requirement. However, the previous self-supervised CT denoising methods do not get advantage from all the available information, such as CT noise properties, and some of them do not consider similarity of adjacent projections. A more detailed description of related work on self-supervised denoising will be given in the section~\\ref{sec:rel_work}.\nIn this paper, we propose a new self-supervised approach for denoising that takes into account the mentioned properties.\n\nUnlike most previously proposed methods that denoise already reconstructed CT slices, we propose to work in the projection domain. Although radiologists work with CT slices, image processing methods can be directly applied to projections. Working with CT projections may be more advantageous than processing CT slices. Firstly, CT reconstruction is an ill-posed inverse problem. There are a lot of different reconstruction methods, which have many parameters influencing quality of reconstructed CT slices. A method developed for a one set of parameters can be unsuitable for other sets leading to the necessity of stack of methods designed for each set. For the projection data, there is no such a problem. Secondly, after the reconstruction CT slices may contain artefacts, e.g., streaks, caused by the noise in projections, which are more difficult to be removed than the initial noise. \nThirdly, in the case of deep learning methods, they require many data for training. Here is one more benefit to working in the projection domain, as the projection data of one patient contains more images than the reconstructed image data of this patient. Because the medical images are difficult to be obtained, it is a crucial property. \nFinally, CT projections share almost the same content, and the noise is spatially uncorrelated\non projections that can be useful for denoising, especially in the absence of clean images. One more point about noise is that its pixel-wise independent distribution on projections is one of the main assumptions for the possibility to apply self-supervised denoising methods. In the image domain, this assumption is violated because of the reconstruction process. \n\nIn this paper, we propose the self-supervised Noise2NoiseTD-ANM (ANM stands for ``adaptable noise model'') approach for denoising low-dose CT projections that is an extension of our earlier proposed method in~\\cite{zainulina2021n2ntd}. Like our previous method, it uses information contained in sequences of images, whereas most other self-supervised denoising methods denoise each image separately, which can give sub-optimal quality of denoised images. In addition, the developed approach takes into account the theoretical distribution of the CT noise. This makes denoising models better generalize to different noise levels, even relatively high ones.\nThe scheme of the proposed approach is illustrated in Fig.~\\ref{fig:approach}.\n\n\n\\subsection{Contributions}\nThe contribution of this paper is in the following:\n\\begin{enumerate}\n   \n   \n    \\item Novelty. The approach incorporates actual CT noise model. The noise model improves the capability of our denoising framework to generalize to previously unseen noise levels.\n    \\item Flexibility. Our approach simplifies model adaptation to different CT scan settings because it is designed to pre-estimate parameters of the noise model and to optimize the denoising and the noise models separately. \n    \\item Validation. We performed the comparison of the proposed approach, the Noise2Clean, the Half2Half technique~\\cite{yuan2020half2half}, and one of the best self-supervised methods introduced in the paper ``High-Quality Self-Supervised Deep Image Denoising'' \\cite{laine2019hqss}, adapted to CT projection denoising. We used both simulated data (three different noise levels) and a dataset with the real noise. Our method outperformed state-of-the-art algorithms.\n\\end{enumerate}\n\n\n\\section{Related work}\n\\label{sec:rel_work}\nThe first attempt to train a neural network using only noisy images was made by J. Lehtinen et al. Their work~\\cite{Noise2Noise} gave rise to self-supervised denoising methods. Their method consists in training a neural network to predict one noisy image from another. The approach is based on the properties of the loss functions, which allow the model to converge in the limit to the same state as in the case of supervised training. For the applicability of the approach, the noise have to be zero-mean and noise components of different pixels have to be independent to each other.\nFor medical data, this approach does not seem appropriate because of the requirement of paired noisy images. However, different methods were introduced that extend the Noise2Noise approach and adopt it to denoise CT images. Some of them give solutions on how to generate an acceptable dataset for model training. \n\nIn~\\cite{n2n_ct2020} the Noise2Noise approach was applied for denoising X-ray projections and CT images and compared to the Noise2Clean. The Noise2Noise approach has shown rather good results compared to the Noise2Clean. However, they used an artificially generated dataset for the experiments. \n\nIn the works~\\cite{wu2019consensus} and~\\cite{hendriksen2020noise2inverse}  the ``data splitting'' methods were proposed for generation of a dataset for the Noise2Noise approach. These methods consist in reconstruction of paired CT slices from non-intersecting sets of CT projections. Although these methods showed rather good results, they may introduce reconstruction artifacts that can violate the noise assumptions. The other possible negative outcome of the ``data splitting'' is the drop in resolution of reconstructed images. These methods allow denoising only already reconstructed slices. \n\n\\cite{yuan2020half2half} introduced a Half2Half method consisting in the obtainment of two half-dose projections from the given projection using the theoretical distribution of count-domain data for Noise2Noise training. The method gave acceptable results in the experiments carried out by the authors and provided a reasonable approach for the dataset generation. \nHowever, the Half2Half approach needs additional knowledge about acquisition parameters for noise simulation that is not always possible to obtain. \n\nThus, the Noise2Noise-based methods have the main bottleneck consisting in accurate data generation satisfying all assumptions of the Noise2Noise. This bottleneck makes the models less flexible. \n\nThe drawback with the dataset generation is overcome by self-supervised methods based on the usage of blind spots. These methods assume model training using only original noisy images as input and target. Nevertheless, the application of the blind-spot-methods for denoising of low-dose CT projections is little researched. Some of the methods use pixel masking during training~\\cite{krull2019noise2void},~\\cite{xie2020noise2same}, which makes them computationally inefficient, and other use special neural network architectures~\\cite{laine2019hqss},~\\cite{batson2019noise2self},~\\cite{lee2020noise2kernel} that restricts their flexibility. The main weakness of most blind-spot-based methods is that they exclude information about the noisy pixel to be denoised that makes them prone to loss of small details and blurring and cause worse performance compared to the Noise2Clean and Noise2Noise approaches. One of the most efficient solutions for this problem \nwas introduced in~\\cite{laine2019hqss}, where the authors proposed to include information about the excluded noisy pixel during inference by Bayes rule.\nThis approach was adopted by us for denoising of low-dose CT projections in~\\cite{zainulina2021n2ntd}. Also, in~\\cite{zainulina2021n2ntd} we proposed a new method Noise2NoiseTD, which uses information from adjacent projections that can help prevent the model from over-smoothing and losing edges.\n\nRecent works~\\cite{xu2021deformed2self},~\\cite{choi2021self},~\\cite{unal2021self} also attempt to perform self-supervised denoising of CT data, however they do not take the true noise model into account.\n\nIn the paper, we propose the extension of our previous approach. \nWe will compare it to the approach from~\\cite{laine2019hqss} adopted for denoising of CT projections, which will be referred to as Noise2Void-4R ($4$R stands for $4$ rotations), and to the Half2Half approach~\\cite{yuan2020half2half}.\n\n\n\\section{Methods}\nIn this section, we present the description of our approach. Firstly, we give an overview of the CT noise properties. Then, we describe how we use the adjacency in the approach and how we incorporate it in a neural network. After that, we describe the train-inference schemes of the model depending on the CT noise distribution. We introduce a noise model for self-supervised training that incorporates the CT noise properties. The description of the usage of the adjacency and the noise model completes the description of the Noise2NoiseTD-ANM approach. Additionally, in the section, we present the neural network architecture used as the backbone for the Noise2NoiseTD-ANM, Noise2Void-4R, Half2Half, and Noise2Clean approaches for comparison. There will be given details about modifications of the neural network corresponding to the applied approaches.\n\n\\subsection{CT noise properties}\n\\label{sev:CT_noise}\nCT projections are the line integrals of the linear attenuation coefficients of the body. They are obtained after normalization applied to the data measured by detectors.  \n\nAccording to~\\cite{buzug2011computed}, the measured data can be described by Lambert-Beer's law:\n\\begin{equation}\n    I = I_0\\exp(-p),\n\\end{equation}\nwhere $I$ is the number of detected photons, $I_0$ is the incident number of photons, $p$ is the line integral of linear attenuation coefficients, i.e., the projection.\nLet $\\exp(-p)$ be the transmission data $T$. Then, $I=I_0T$.\n\nIn practice, the actual value of $I$ is not available: it is corrupted by noise. The detected number of photons $I$ is a random variable that can be described by Poisson distribution ($\\mathcal{P}$)~\\cite{buzug2011computed}. Besides, the electronic noise inherent in the CT scanner contributes to the overall noise level. The electronic noise can be modeled by Gaussian distribution with parameters $\\mu_e$ and $\\sigma_e^2$, the mean and variance of the electronic noise, $\\mathcal{N}\\left(\\mu_e, \\sigma_e^2\\right)$~\\cite{la2006penalized}. In CT systems, $\\mu_e$ is usually calibrated to be $0$. Thus, the detected number of photons obey the following mixed Poisson-Gaussian distribution:\n\\begin{equation}\n    I = \\mathcal{P}\\left(I_0T_{hd}\\right) + \\mathcal{N}\\left(0, \\sigma_e^2\\right).\n\\end{equation}\n$T_{hd}=\\exp(-p_{hd})$ and $p_{hd}$ are the transmission and projection data, respectively, which do not contain noise ($hd$ corresponds to high-dose data). \n\nThen, for the noisy projection $\\hat{p}$, the transmission data $\\hat{T}$ can be modeled by the distribution:\n\\begin{equation}\n    \\hat{T} = \\exp(-\\hat{p}) = \\frac{I}{I_0} = \\frac{1}{I_0}\\left(\\mathcal{P}\\left(I_0T_{hd}\\right) + \\mathcal{N}\\left(0, \\sigma_e^2\\right)\\right).\n\\end{equation}\n\nApproximating Poisson noise as signal-dependent Gaussian noise and defining high-dose transmission data as $x=T_{hd}=\\exp(-p_{hd})$, we can express the transmission data for the noisy projection using Gaussian distribution:\n\\begin{equation} \\label{eq:distr}\n    \\hat{T} = \\exp(-\\hat{p}) = \\mathcal{N}\\left(\\mu_x, \\sigma_x^2+\\frac{\\mu_x}{I_0}+\\frac{\\sigma_e^2}{I_0^2}\\right).\n\\end{equation}\n\nWhereas $\\mu_x$ and $\\sigma_x^2$ characterize clean data, $I_0$ and $\\sigma_e^2$ are the noise parameters depending on the properties of a CT scanner. The commonly employed in CT scanners bowtie filtering modulates an X-ray beam as a function of the angle to balance the photon flux on a detector array~\\cite{liu2014dynamic}. It reduces the radiation at the periphery of the field of view resulting in a bell shape distribution of incident flux levels. Besides, recent CT scanners use a dose modulation technique,\nwhich regulates the tube current for each projection angle depending on the properties of organs to not expose to excessive radiation. Because the flux level is proportional to the tube current, dose modulation influences its distribution. Thus, the incident flux level and, hence, the noise variance depends on the position of the detector column and the tube current.\nAs for the variance of electronic noise $\\sigma_e^2$, it is the characteristic of the detector and does not depend on the X-ray beam. \n\n\\subsection{Using adjacent projections in a time-distributed denoising model}\nAs in our previous work~\\cite{zainulina2021n2ntd}, we propose to restore the projection depending on its noisy adjacent projections. However, the previously proposed model was not completely blind causing its over-fitting to the noise at some moment during training and requiring early stopping. In this work we exclude the projection to be denoised from the consideration of the denoising model.\nThe problem can be considered as the prediction of the middle frame in the sequence depending on its past and future frames. Because the neighboring projections share almost the same content, while the noise is independently distributed on each projection, the prediction will be mostly dependent on the structural features of the projections, excluding the noise features. It will allow recovering noise-free middle frame. Thus, we restore a denoised version of a projection $p_i$ from projections $p_{i\\pm 1},\\dots,p_{i\\pm k}$. The choice of the number of the adjacent frames $k$ depends on how much the content on two consequent frames differs and the computational and the memory capacity of the device. In this study $k=3$ is chosen.\n\nIn order to predict the frame from the sequence of past and future frames, we propose using the convolutional long short-term memory (ConvLSTM) units because LSTM was shown to be effective in the time-series tasks. ConvLSTM units included in the model are based on the ConvLSTM layers introduced in~\\cite{shi2015convlstm}. Compared to~\\cite{shi2015convlstm}, we do not use the past cell status for gates calculation, reducing the number of model parameters without really affecting prediction quality. Also, according to~\\cite{bias-free}, all the convolutions are made bias-free. In this study, the ConvLSTM unit consisted of one ConvLSTM cell.\n\nThis unit processes features extracted by some CNN for each frame independently.  \nIt makes a prediction in one direction, i.e. the middle projection $p_i$ is predicted from the combination of the features extracted from the sequence $p_{i-k},\\dots,p_{i-1}$ and from the sequence $p_{i+1},\\dots,p_{i+k}$. The features extracted from both sequences are combined using attention~\\cite{SE} along the time axis and concatenation along the channel axis. Then, these features are combined by a fusing CNN to obtain the denoised middle projection.\n\n\\subsection{Train-inference scheme}\n\nAs in~\\cite{zainulina2021n2ntd}, we use the train-inference scheme proposed in~\\cite{laine2019hqss} that assumes model training and inference depending on the Gaussian approximation of the data distribution.\nAccording to the description of the CT data distribution, the noisy and clean transmission data can be modeled by Gaussian distribution with rather explainable parameters. Therefore, the training of the models for the self-supervised approaches is carried out on the transmission data.\n\nLet  $y=\\left(y_1,\\dots,y_N\\right)$ denote the pixels of noisy transmission data. Each noisy pixel has a neighborhood $\\Omega_{y_i}$ that consists of noisy pixels surrounding the pixel $y_i$ on its adjacent projections. Let $\\Omega_y=\\left(\\Omega_{y_1},\\dots,\\Omega_{y_N}\\right)$ be the neighborhoods of the noisy pixels.\nWe want to restore the pixels of clean data $x=\\left(x_1,\\dots,x_n\\right)$.\nThe parameters of the Gaussian distribution of clean data $p(x|\\Omega_y)$ are predicted by neural networks. Because only the noisy data $\\mathcal{D}=\\left\\{y_i, \\Omega_{y_i}\\right\\}_{i=1}^N$ is available, then the networks are trained using a loss function that maximizes the log-likelihood of the distribution of the noisy data $p(y|\\Omega_y)$:\n\\begin{equation}\n    \\mathcal{L}=-\\sum_{i=1}^{n}\\log{p(y_i|\\Omega_{y_i})}.\n\\end{equation}\nAfter substituting the distribution from~\\ref{eq:distr}, we get the following loss function:\n\\begin{equation} \n\\label{eq:loss}\n\\begin{split}\n    \\mathcal{L}=\\sum_{i=1}^{n}\\left(\\frac{(y_i-\\mu_{x_i})^2}{2\\sigma_{y_i}^2} + \\frac{1}{2}\\log\\sigma_{y_i}^2\\right),\\\\\n    \\sigma_{y_i}^2 = \\sigma_{x_i}^2+\\sigma_{n_i}^2,\\; \n    \\sigma_{n_i}^2 = \\frac{\\mu_{x_i}}{\\lambda}+\\frac{\\sigma_e^2}{\\lambda^2}.\n\\end{split}\n\\end{equation}\nThe $\\lambda$ denotes the parameter approximating the actual incident flux level $I_0$. The $\\sigma_e^2$ denotes approximation of the electronic noise variance.\n\nWith this loss function, the model is trained to predict estimations of the parameters of the clean data distribution $\\mu_x$ and $\\sigma_x^2$. Then we get the prediction of the clean data incorporating the information about the noisy pixels $y$ itself by Bayes rule:\n\\begin{equation}\n    p(x|y, \\Omega_y) \\sim p(y|x)p(x|\\Omega_y).\n\\end{equation}\nGiven $p(y|x)=\\mathcal{N}_y(x, \\sigma_n^2)=\\mathcal{N}_x(y, \\sigma_n^2)$ (the equation used the symmetry of Gaussian distribution to swap $x$ and $y$) and $p(x|\\Omega_y)=\\mathcal{N}(\\mu_x,\\sigma_x^2)$ are normal distributions, the distribution $p(x|y, \\Omega_y)$ is a scaled normal with the mean~\\cite{bromiley2003products}:\n\\begin{equation} \\label{eq:prediction}\n    \\mathbb{E}_x[p(x|y, \\Omega_y)] = \\frac{y\\sigma_x^2+\\mu_x\\sigma_n^2}{\\sigma_x^2+\\sigma_n^2}.\n\\end{equation}\nThis mean coincides with the posterior mean estimation of the clean data that is used for prediction.\n\n\n\\subsection{Noise model}\n We propose to make the neural network representing the noise model with the parameters $\\lambda$ and $\\sigma_e^2$ that approximate the incident flux level and the electronic noise variance, respectively. The parameters of the noise model can be pre-estimated independently and optimized together with the main denoising model. The noise model parameters can be pre-estimated depending on the CT properties described earlier.\n \nAs for the parameter $\\lambda$, if the bowtie filtering was applied, the $\\lambda$ parameter should differ across detector columns, i.e., it should depend on the column position of the pixel to be denoised. Otherwise, some parts of the projection would be over-smoothed, and others would stay noisy because of the under- or overestimation of the Poisson parameter of the noise. \nBecause the modern CT scanners use dose modulation, the $\\lambda$ parameter should also depend on the used tube current. Thus, $\\lambda=\\lambda(i, mA)$, where $i$ is a detector column number and $mA$ is a tube current.\nAs for the variance of electronic noise $\\sigma_e^2$, we assume that $\\sigma_e^2$ is the same for all pixels. It probably differs between the detector cells, but we assume that this difference is relatively small. The possible error is negligible compared to the cost caused by the increase in the number of parameters by the number of detector cells.\n\nSummarizing all the above properties, we present the noise model that allows accounting for the parameters of the acquisition process. \nFirstly, we made the model able to predict different values for different detector columns by introducing an embedding layer that maps the column number of the pixel into the value that characterizes the number of incident photons influenced by the bowtie filter. \nThen, this value is normalized by the tube current. The normalization is needed to\ntransform the obtained values into the distribution of incident photons for the specific current and potential.\nThe normalization consists of two steps. The first step is to map the tube current to the slope and bias coefficients. The mapping is made by the linear layers with $2$ output channels and ReLU activation between them. The second step is to multiply the value by the slope coefficient and add the bias coefficient. This type of normalization was chosen according to the theoretical dependency between incident flux levels of different tube currents~\\cite{zeng2015simulation}. \nIn this way, the proposed noise model allows predicting the approximation of the incident flux level $I_0$  depending only on the column of the pixel and acquisition parameters. There is no need to know photon distribution for the test data.\nAs for the electronic noise variance $\\sigma_e^2$, it is represented as the parameter with the shape $1\\times 1$ in the model.\nThe scheme of the proposed noise model is presented in Fig.~\\ref{fig:arch_noise}.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=2.5in]{figures\/schemes\/Noise_net.pdf}\n\\caption{The architecture of the noise model for self-supervised model training.}\n\\label{fig:arch_noise}\n\\end{figure}\n\n\nIf train and test projections were obtained using different CT scanner settings, for example, if the bowtie filter is removed or changed, the noise model can be tuned for the new settings and retrained on the new dataset in a self-supervised mode, and the main denoising model can be frozen during training to speed up the process. If some noise parameters ($I_0$ or $\\sigma_e^2$) are known, they can be used instead of the trained ones. \n\n\n\\subsection{Comparison approaches}\nWe decided to benchmark our approach Noise2NoisTD-ANM against the Noise2Clean, Half2Half and Noise2Void-4R approaches. The Noise2Clean model is trained using pairs of noisy and clean projections. The Half2Half approach assumes creation of pairs of noisy projections from the original projections and training the model in a supervised manner using one noisy projection as input and the other as target. The Noise2Void-4R approach is an adaptation of the approach from~\\cite{laine2019hqss} for training CT projections using the proposed noise model.\nAll approaches used the same neural network architecture as a backbone, with some modifications caused by the requirements of the approaches.\nWe decided to use a relatively simple neural network that shows good results for denoising that is DnCNN (Denoising Convolutional Neural Network)~\\cite{zhang2017dncnn}. In our previous work~\\cite{zainulina2021n2ntd}, our model was based on the U-Net architecture~\\cite{ronneberger2015u} but in this study the U-Net architecture did not show significant improvements over the DnCNN.\n\nFor the Half2Half approach, we initially created the half-dose pairs as it was described in the original paper~\\cite{yuan2020half2half}. Then, we trained the model on the transmission data using MSE loss function.\nUnlike the self-supervised approaches, the Noise2Clean model is trained using MSE loss function on the projection, not transmission data. Also, we used residual learning, i.e., we trained the model to predict the noise because it was shown to be effective for supervised denoising~\\cite{zhang2017dncnn}. Because the noise model applied for the self-supervised approaches allows to account for the noise level depending on the used tube current automatically, we decided to include the analog of the noise level map into the input of the Noise2Clean model~\\cite{zhang2018ffdnet}. \nAs was shown in~\\cite{zainulina2021n2ntd}, the usage of adjacent projections enhances results for the supervised training and does not give any improvements for the Noise2Void-4R approach. Therefore, we used adjacent projections as additional channels for the Noise2Clean approach and did not use them for the Noise2Void-4R approach.\n\nThe model architectures for each approach are depicted in Fig.~\\ref{fig:arch_dncnn}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{figures\/schemes\/DnCNN_net.pdf}\n\\caption{Model architectures of the compared approaches.}\n\\label{fig:arch_dncnn}\n\\end{figure*}\n\n\\section{Experiments and results}\n\\subsection{Data preparation}\nThe study used the publicly available dataset of CT projections that is Low Dose CT Image and Projection data (LDCT-and-Projection-data)~\\cite{LDCT}. The experiments were carried out on abdomen projections obtained with the fixed tube voltage $100$ kVp. During acquistion of these projections, dose modulation and a bowtie filter were used. In order to evaluate the performance of the projection-domain denoising approaches in the image domain, the reconstruction was performed by the open TIGRE toolbox~\\cite{TIGRE}. Also, the dataset has information about photon distribution ($I_0$) for each CT projection that will be used for the noise model estimation.\n\nFor each patient, CT projection data are provided for both full and simulated lower dose levels. The provided low dose level is $25\\%$ of the routine dose. For testing the approaches on the more severe cases of $10\\%$ and $5\\%$ of the routine dose, new low-dose data were simulated according to the algorithm from~\\cite{zeng2015simulation}. \nBecause the simulated data may differ from real data and the testing results may be different for simulated and real data, we also performed the comparison of the approaches on the data with real noise. It was possible because the dose differs for each projection due to the properties of organs and patient-specific requirements and therefore the full-dose projections having high noise levels can be found. These projections can be considered as real noisy projections. \n\n\\subsubsection{Data selection}\nFor the experiments, we selected projections of several patients. The selection was based on noise levels of projections. Because noise levels of projections correlate with used tube currents (higher noise levels correspond to lower tube currents), the projections were chosen depending on the distributions of the tube currents. The noise levels of the selected projections were verified using the algorithm proposed in~\\cite{liu2014noise_level} because patient anatomies can also have an impact on noise levels. \nThe distribution of noise levels correlated with the distribution of the tube currents.\n\n\nFor the train dataset, we selected projections of one patient, whose low-dose projections have the wide range of the estimated noise levels.\nWe randomly picked up $21800$ projections from them so that at least $100$ projections are adjacent to be able to use the connections between neighboring frames. We denote the train dataset projections as TM (the train set, middle noise levels). Paired full-low-dose projections were used to train the Noise2Clean model, and low-dose projections were used to train the self-supervised models. For the test datasets, we chose projections of two patients with low and high noise levels of low-dose projections and picked up $6000$ consecutive projections with the highest noise levels. We denoted these sets as SL and SH, respectively, where S means that low-dose projections were simulated and the second letter shows the noise level: low and high. Their low-dose projections serve as noisy inputs to the denoising approaches, and their full-dose projections serve as reference images. Then we found full-dose projections with the highest noise level comparable to noise levels of low-dose projections of other patients. We chose $10000$ consecutive projections with the highest noise level from them to be the test dataset with real data. It was denoted as RH (real noise, high level).\nFig.~\\ref{fig:hist_mAs} presents the distributions of the tube currents (mA) for the patients, whose projections were selected for the experiments. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=2.5in]{figures\/plots\/mA_low.pdf}\n\\caption{Distribution of the tube currents (mA) used to acquire projections for different patients. The figure illustrates noise levels of the train and test sets used in the experiments because noise levels correlate with used tube currents (higher tube currents correspond to lower noise levels).}\n\\label{fig:hist_mAs}\n\\end{figure}\n\n\\subsubsection{Low-dose data simulation}\nUsing the algorithm from~\\cite{zeng2015simulation}, we simulated low-dose projections with doses equal to $5\\%$ and $10\\%$ of the routine dose to test the denoising approaches on the projections with higher noise levels. We created $5\\%$- and $10\\%$-low-dose projections from the full-dose projections of the SL and SH sets. Also, we simulated $5\\%$ TM low-dose projections to test the approaches being trained on data with the high noise level and their adaptability. \n\n\n\\subsection{Learning settings}\n\\subsubsection{Pre-estimation of the $\\lambda$ parameter of the noise model}\nThe noise model parameters $\\lambda$ (the embedding and mapping layers) were pre-estimated using photon distributions and the information about used tube currents provided for each projection in the LDCT dataset.\nWe took the subset of the TM dataset and extracted information about photon distributions and corresponding values of tube currents. We trained the embedding layer and the mapping layers to predict a photon distribution from its tube current. The training was performed using the MSE loss function, Adam optimizer with learning rate $10^{-2}$ for $1000$ epochs. The optimized parameters were tested on full-dose and low-dose photon distributions and tube currents of the validation subset of the TM dataset. RMSRE (root mean square relative error)~\\cite{zeng2015simulation} is equal to $0.16\\pm0.11\\%$ for the full-dose dataset, and it is equal to $0.53\\pm0.40\\%$ for the low-dose dataset. Thus, the error does not exceed $1\\%$ and can be considered insignificant.\nFig.~\\ref{fig:pred_ph_dist} shows predicted and original photon distributions for different tube currents.\nThe output values of the pre-estimated model embedding layer form a bell-shaped curve, this coincides with the theoretical shape of the photon distribution after the bowtie filter.\nThen, this shape is well stretched out to fit the photon distribution corresponding to the certain tube current value.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=2.5in]{figures\/plots\/pred_ph_dist.pdf}\n\\caption{Comparison of the predicted and actual photon distributions (Noise Equivalent Quanta) for different tube currents (mA). The solid line is predicted values, and the dashed line is true values. The figure shows that the noise estimation module successfully estimates Poisson parameter of the noise.}\n\\label{fig:pred_ph_dist}\n\\end{figure}\n\nExperiments with a joint training of the noise model and main denoising model showed that this pre-estimation makes models converge faster and give better results. Besides, freezing the pre-estimated embedding and mapping layers during joint training also leads to slightly better results. Because of that,\nthe pre-estimated and frozen embedding and mapping layers were used for all denoising models dependent on this noise model.\n\n\\subsubsection{Model training}\nAll models were trained in Pytorch using Adam optimizer with default parameters and learning rate $10^{-4}$. The minibatch used for training supervised and self-supervised models consisted of $64$ and $16$ randomly cropped $64\\times 64$ patches of frames, correspondingly. The Noise2Clean model was trained using MSE loss on the projection data. The Half2Half model was trained using MSE loss on the transmission model. The Noise2Void-4R and Noise2NoiseTD-ANM models were trained using the loss function~(\\ref{eq:loss}) on the transmission data. For all models, the training continued until the training curve reached a plateau. \n\n\\subsection{Results on the simulated data}\n\\subsubsection{Models trained on the $25\\%$-dose data}\n\nWe tested the models trained on the $25\\%$-dose TM dataset on the SL and SH datasets. Low-dose projections provided by the LDCT dataset with the $25\\%$ routine dose and $10\\%$ and $5\\%$ low-dose projections simulated by us were denoised and compared to the corresponding full-dose projections.\nThe quantitative comparison in the projection domain is presented in Table~\\ref{tab:sim}. Additionally to the usually applied methods of evaluation of denoised images SSIM and PSNR, we used GMSD (gradient magnitude similarity deviation)~\\cite{xue2013gmsd}. It considers image over-blurring unintentionally introduced by denoising, which can be overlooked by PSNR or SSIM.\n\nThe Noise2Clean approach shows the best results for the $25\\%$ low-dose projections. However, the difference between the Noise2Clean and the proposed Noise2NoiseTD-ANM approach is minor, especially in the image domain, although the Noise2NoiseTD-ANM method uses less information than the supervised approach.\nFor the lower percents of the routine dose levels, the Noise2Clean approach gives the worse results. It shows the lowest quality for the $5\\%$ low-dose projections. Although the Noise2Clean approach used the noise level map as an additional input channel, it failed to generalize to lower noise levels as the self-supervised approach did. \n\nAs for the self-supervised approaches, the difference between them is not big. It becomes more prominent for lower doses. Although the proposed approach does not show a great advantage over other self-supervised methods in the projection domain, it gives the best results in the image domain for all doses and all image quality assessment methods.\n\n\\begin{table*}[!t]\n\\caption{Quantitative comparison of the denoising approaches, simulated data}\n\\label{tab:sim}\n\\centering\n\\begin{tabular}{c| c| c c c | c c c}\\hline\\hline\n &  & \\multicolumn{3}{|c|}{projection domain} & \\multicolumn{3}{c}{image domain}\\\\ \\cline{3-8}\nDataset & Approach & SSIM$\\uparrow$ & PSNR$\\uparrow$ & GMSD$\\downarrow$ & SSIM$\\uparrow$ & PSNR$\\uparrow$ & GMSD$\\downarrow$\\\\\n\\hline\\hline\n\\multicolumn{8}{c}{$25\\%$ of the routine dose} \\\\\\hline\n& Low dose &\t$0.905 \\pm 0.036$ & \t$39.2 \\pm 2.0$ & \t$0.013 \\pm 0.006$ \n& $0.957\\pm0.022$ & $43.9\\pm3.0$ & $0.003\\pm0.002$ \\\\\n& Noise2Clean & $\\mathbf{0.975 \\pm 0.010}$ & \t$\\mathbf{45.6 \\pm 1.6}$ & \t$\\mathbf{0.004 \\pm 0.002}$ \n& $\\mathbf{0.987\\pm0.005}$ & $\\mathbf{48.7\\pm2.0}$ & $\\mathbf{0.002\\pm0.001}$ \\\\ \nSL & Half2Half & $0.973 \\pm 0.010$\t& $44.9 \\pm 1.5$\t& $0.006 \\pm 0.002$ \n& $0.985\\pm0.005$ & $47.3\\pm1.6$ & $0.006\\pm0.002$ \\\\\n& Noise2Void-4R &\t$0.973 \\pm 0.010$ & \t$44.9 \\pm 1.7$ & \t$0.006 \\pm 0.002$\t\n& $0.986\\pm0.005$ & $47.8\\pm1.9$ & $0.004\\pm0.002$ \\\\\n& Ours &\t$0.972 \\pm 0.010$ & \t$44.4 \\pm 1.3$ & \t$0.006 \\pm 0.002$\n& $\\mathbf{0.987\\pm0.005}$ & $48.6\\pm2.1$ & $\\mathbf{0.002\\pm0.001}$ \\\\\n \\hline\n& Low dose &\t$0.873 \\pm 0.023$ & \t$37.1 \\pm 1.3$ & \t$0.016 \\pm 0.005$\n& $0.832\\pm0.030$ & $35.6\\pm1.3$ & $0.015\\pm0.004$ \\\\\n& Noise2Clean &\t$\\mathbf{0.965 \\pm 0.007}$ & \t$\\mathbf{43.7 \\pm 1.1}$ & \t$\\mathbf{0.006 \\pm 0.002}$\n& $\\mathbf{0.944\\pm0.012}$ & $\\mathbf{41.0\\pm1.4}$ & $\\mathbf{0.009\\pm0.003}$ \\\\\nSH & Half2Half & $0.960 \\pm 0.009$\t& $42.8 \\pm 1.2$\t& $0.010 \\pm 0.003$ \n& $0.935\\pm0.014$ & $39.4\\pm1.8$ & $0.021\\pm0.007$ \\\\\n& Noise2Void-4R &\t$0.960 \\pm 0.009$ & \t$42.6 \\pm 1.3$ & \t$0.010 \\pm 0.004$ \n& $0.938\\pm0.013$ & $39.8\\pm1.7$ & $0.016\\pm0.005$ \\\\\n& Ours &\t$0.961 \\pm 0.008$ & \t$42.3 \\pm 1.0$ & \t$0.009 \\pm 0.004$ \n& $0.941\\pm0.011$ & $40.5\\pm1.4$ & $\\mathbf{0.009\\pm0.003}$ \\\\\n \\hline\\hline\n\\multicolumn{8}{c}{$10\\%$ of the routine dose} \\\\\\hline\n& Low dose &\t$0.787 \\pm 0.075$ & \t$33.9 \\pm 2.8$ & \t$0.044 \\pm 0.025$ \n& $0.845\\pm0.103$ & $37.3\\pm5.0$ & $0.024\\pm0.024$ \\\\\n& Noise2Clean &\t$0.958 \\pm 0.029$ & \t$41.6 \\pm 4.3$ & \t$0.023 \\pm 0.023$ \n& $0.939\\pm0.057$ & $42.5\\pm5.7$ & $0.016\\pm0.017$ \\\\\nSL & Half2Half & $0.957 \\pm 0.018$\t& $42.7 \\pm 1.9$\t& $0.011 \\pm 0.004$ \n& $0.979\\pm0.008$ & $45.8\\pm1.8$ & $0.008\\pm0.002$ \\\\\n& Noise2Void-4R &\t$0.961 \\pm 0.015$ & \t$\\mathbf{43.1 \\pm 1.6}$ & \t$0.010 \\pm 0.003$ \n& $0.980\\pm0.007$ & $46.1\\pm1.9$ & $0.007\\pm0.002$ \\\\\n& Ours &\t$\\mathbf{0.963 \\pm 0.010}$ & \t$42.8 \\pm 1.0$ & \t$\\mathbf{0.007 \\pm 0.002}$ \n& $\\mathbf{0.981\\pm0.006}$ & $\\mathbf{46.7\\pm1.8}$ & $\\mathbf{0.004\\pm0.002}$ \\\\\n\\hline\n& Low dose &\t$0.860 \\pm 0.026$ & \t$36.0 \\pm 1.9$ & \t$0.023 \\pm 0.012$\n& $0.813\\pm0.037$ & $34.6\\pm1.9$ & $0.021\\pm0.013$ \\\\\n& Noise2Clean &\t$\\mathbf{0.963 \\pm 0.012}$ & \t$42.5 \\pm 3.2$ & \t$0.012 \\pm 0.015$ \n& $0.938\\pm0.020$ & $40.2\\pm2.3$ & $0.013\\pm0.009$ \\\\\nSH & Half2Half & $0.962 \\pm 0.008$\t& $43.0\\pm 1.2$\t& $0.010 \\pm 0.003$ \n& $0.938\\pm0.013$ & $39.5\\pm1.8$ & $0.022\\pm0.007$ \\\\\n& Noise2Void-4R &\t$0.962 \\pm 0.008$ & \t$\\mathbf{43.0 \\pm 1.1}$ & \t$0.011 \\pm 0.003$ \n& $0.935\\pm0.015$ & $39.2\\pm1.9$ & $0.024\\pm0.008$ \\\\\n& Ours &\t$0.961 \\pm 0.009$ & \t$41.4 \\pm 0.8$ & \t$\\mathbf{0.009 \\pm 0.003}$ \n& $\\mathbf{0.942\\pm0.013}$ & $\\mathbf{40.4\\pm1.6}$ & $\\mathbf{0.012\\pm0.004}$ \\\\\n \\hline\\hline\n\\multicolumn{8}{c}{$5\\%$ of the routine dose} \\\\\\hline\n& Low dose &\t$0.632 \\pm 0.106$ & \t$28.2 \\pm 3.3$ & \t$0.108 \\pm 0.050$\n& $0.565\\pm0.262$ & $28.1\\pm7.2$ & $0.113\\pm0.081$ \\\\\n& Noise2Clean &\t$0.863 \\pm 0.103$ & \t$31.6 \\pm 5.9$ & \t$0.099 \\pm 0.059$ \n& $0.707\\pm0.228$ & $32.5\\pm7.9$ & $0.086\\pm0.062$ \\\\\nSL & Half2Half & $0.918 \\pm 0.039$\t& $39.2 \\pm 2.4$\t& $0.025 \\pm 0.012$\n& $0.961\\pm0.019$ & $43.5\\pm2.4$ & $0.010\\pm0.003$ \\\\\n& Noise2Void-4R &\t$0.934 \\pm 0.027$ & \t$40.5 \\pm 1.9$ & \t$0.017 \\pm 0.004$ \n& $0.968\\pm0.014$ & $44.0\\pm2.2$ & $0.011\\pm0.004$ \\\\\n& Ours &\t$\\mathbf{0.963 \\pm 0.012}$ & \t$\\mathbf{41.9 \\pm 1.1}$ & \t$\\mathbf{0.011 \\pm 0.003}$ \n& $\\mathbf{0.980\\pm0.007}$ & $\\mathbf{45.9\\pm1.8}$ & $\\mathbf{0.007\\pm0.003}$ \\\\\n \\hline\n& Low dose &\t$0.742 \\pm 0.055$ & \t$31.1 \\pm 2.9$ & \t$0.058 \\pm 0.028$ \n& $0.569\\pm0.118$ & $27.5\\pm3.5$ & $0.087\\pm0.046$ \\\\\n& Noise2Clean &\t$0.935 \\pm 0.034$ & \t$36.3 \\pm 5.9$ & \t$0.048 \\pm 0.038$\n& $0.779\\pm0.123$ & $32.3\\pm4.5$ & $0.060\\pm0.037$ \\\\\nSH & Half2Half & $0.944 \\pm 0.016$\t& $40.8 \\pm 1.7$\t& $0.017 \\pm 0.006$\n& $0.916\\pm0.020$ & $38.1\\pm1.8$ & $0.027\\pm0.008$ \\\\\n& Noise2Void-4R &\t$\\mathbf{0.956 \\pm 0.009}$ & \t$\\mathbf{41.9 \\pm 1.4}$ & \t$0.016 \\pm 0.006$ \n& $0.924\\pm0.017$ & $38.1\\pm2.1$ & $0.033\\pm0.011$ \\\\\n& Ours &\t$\\mathbf{0.956 \\pm 0.009}$ & \t$40.7 \\pm 0.8$ & \t$\\mathbf{0.012 \\pm 0.004}$ \n& $\\mathbf{0.930\\pm0.015}$ & $\\mathbf{39.2\\pm1.7}$ & $\\mathbf{0.019\\pm0.006}$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure*}[!t]\n\\centering\n\\subfloat[from the SL dataset]{\\includegraphics[width=0.5\\linewidth]{figures\/results\/L033_prj.jpg}%\n}\n\\hfil\n\\subfloat[from the SH dataset]{\\includegraphics[width=0.5\\linewidth]{figures\/results\/L134_prj.jpg}%\n}\n\\caption{The example parts of CT projections of different doses ($25\\%$, $10\\%$, $5\\%$) denoised by the approaches. N2NTD-ANM stands for Noise2NoiseTD-ANM. The arrow points to the parts that the Noise2Clean model failed to restore due to high noise levels.}\n\\label{fig:sim_prj}\n\\end{figure*}\n\nThe approaches were also evaluated qualitatively. Fig.~\\ref{fig:sim_prj} shows the example parts of the denoised low-dose CT projections. The CT slices reconstructed from these projections are presented in Fig.~\\ref{fig:sim_img}. It can be found from the figures that although the Noise2Clean approach preserves structural details rather good, being trained on the data with lower noise levels, it fails to restore pixels corrupted by more severe noise. This led to streaks and noise on the reconstructed CT slices.\nAt the same time, the $25\\%$ low-dose projections denoised by the Noise2NoiseTD-ANM and the reconstructed from them CT slices have the quality comparable to the CT projections and the reconstructed slices of the Noise2Clean approach.\nThe figures also confirm that the Noise2NoiseTD-ANM approach preserves edges and small details better than the Noise2Void-4R and Half2Half models. \n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/results\/L033_img.jpg}\n\\caption{The comparison of CT slices reconstructed from the denoised $25\\%$, $10\\%$, $5\\%$ low-dose projections (from the SL dataset). N2V-4R and N2NTD-ANM stand for Noise2Void-4R and Nois2NoiseTD-ANM, respectively.}\n\\label{fig:sim_img}\n\\end{figure*}\n\n\n\n\\subsubsection{Cross-test}\nFor testing the capability of the approaches to make a neural network able to generalize to different noise levels, we performed cross-testing. Additionally to the models trained on the $25\\%$-dose TM dataset, we trained the models of each approach on the $5\\%$-dose TM datasets using the same training setting. The trained models were tested on the five sets of projections ($6000$ projections in each set) including the SL and SH datasets. Three other datasets were selected such as their noise levels were between the noise levels of the SL and SH datasets. Each testing dataset was made at $25\\%$, $10\\%$, and $5\\%$ of the routine dose. The results of the cross-testing are presented in Fig.~\\ref{fig:cross_test}.\n\n\\begin{figure*}[!t]\n\\centering\n\\subfloat{\\includegraphics[width=0.45\\linewidth]{figures\/plots\/Cross_test_prj_dom_SSIM_5_sets_h2h.pdf}%\n}\n\\hfil\n\\subfloat{\\includegraphics[width=0.45\\linewidth]{figures\/plots\/Cross_test_prj_dom_GMSD_5_sets_h2h.pdf}%\n}\n\\caption{The quantitative comparison of the denoising models trained on $25\\%$ and $5\\%$ low-dose projections using box plots in the projection domain.}\n\\label{fig:cross_test}\n\\end{figure*}\n\nThe box plots show that the difference between the Noise2Clean models trained on the datasets made at $25\\%$ and $5\\%$ doses is significant. The difference is less between the Half2Half models but it is more pronounced than for the Noise2Void-4R and Noise2NoiseTD-ANM models. Thus, the Noise2NoiseTD-ANM and Noise2Void-4R models generalize better to various noise levels. They show approximately the same tendency in the differences between models trained at dose levels of $25\\%$ and $5\\%$. The both methods used the proposed noise model that improved their generalization capability.\nThe advantage of the noise model in making the denoising adaptable to various noise levels will also be further proven by comparing between using the noise model and the simple MSE-loss function on the projection data for training.\n\nIn the comparison of the self-supervised models, the Noise2NoiseTD-ANM model shows slightly better results, this is more pronounced on the GMSD box plot and may become more visible in the image domain. Although the difference between the approaches is not prominent for $25\\%$ of the routine dose, it increases with the decrease of the dose and becomes clear for $5\\%$ of the routine dose.\nThe advantage of the Noise2NoiseTD-ANM model over the Noise2Void-4R model is that it leverages information from adjacent projections that helps to preserves edges and some small details. The Noise2Void-4R restores a pixel depending on the neighborhood of this pixel from one image to be denoised. If this neighborhood contains a big fraction of the corrupted pixels, it is less likely to restore the pixel correctly. At the same time, the usage of the adjacent projections by the Noise2NoiseTD-ANM approach decreases the error.\n\n\\subsection{Results on the real data}\nWe tested the Noise2Clean and Noise2NoiseTD-ANM approaches on the RH dataset. \nBecause the proposed method assumes training without high-quality reference images, we also trained the Noise2NoiseTD-ANM model directly on the RH dataset with the same learning settings. We denoted this model as Noise2NoiseTD-ANM$^*$.\nThe comparison was performed only qualitatively because there are no high-quality images for assessment of the methods using full-reference quality measures and there are no reference-free measures proven to be representative. \n\nThe comparison is presented in Fig.~\\ref{fig:real_prj}. The figure demonstrates that the Noise2NoiseTD-ANM model removed the noise slightly worse than the Noise2Clean, but it kept some structural details better. The Noise2NoiseTD-ANM$^*$ model trained on the data with real noise suppressed the noise better than the Noise2NoiseTD-AMNM model while maintaining the same level of detail preservation.\nThe parts of the CT slices reconstructed from the denoised projections are shown in Fig.~\\ref{fig:real_img}. The orange row indicates the small area in the lung where for the Noise2Clean model the light band occurred as a result of over-smoothing while it does not exist in the original noisy image and did not appear for the Noise2NoiseTD-ANM models.\n\nThe Noise2NoiseTD-ANM$^*$ model demonstrated higher visual quality. Thus, the model training directly on the data with real noise can be advantageous and the application of the Noise2NoiseTD-ANM approach on the real data can result in more accurate denoising than the application of the Noise2Clean model trained on the simulated data.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/results\/L006_prj.jpg}\n\\caption{The comparison of the parts of the denoised CT projections with real noise. The arrow shows that the Noise2NoiseTD-ANM models better preserve structures than the Noise2Clean model.}\n\\label{fig:real_prj}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/results\/L006_img.jpg}\n\\caption{The comparison of the example parts of CT slices reconstructed from the denoised projections that had real noise. The arrow shows that the Noise2Clean model almost connected structures that were originally disconnected, unlike the Noise2NoiseTD-ANM models, due to stronger smoothing.}\n\\label{fig:real_img}\n\\end{figure}\n\n\n\\subsection{Experiments with the noise model}\nThe experiment with cross-testing showed that the noise model could help the self-supervised Noise2Void-4R and Noise2NoiseTD-ANM models better perform on the data with unseen noise levels.\nTo ensure that the better adaptation of the self-supervised models to different noise levels is the advantage of the used noise model we tested the used noise model against the simple MSE loss function. We trained the Noise2NoiseTD-ANM model (without ReLU activation at the end) using the MSE loss function on the projection data using the earlier defined learning settings. The model trained using the MSE-loss function was compared to the Noise2NoiseTD-ANM model trained using the noise model on the five test datasets used for the cross-testing in the projection domain.\nThe results (Fig.~\\ref{fig:mse_vs_noise}) prove that the proposed noise model enhances generalization capabilities of the denoising model. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/plots\/mse_vs_noise.pdf}\n\\caption{The quantitative comparison of the noise model and the MSE-loss function in the projection domain.}\n\\label{fig:mse_vs_noise}\n\\end{figure}\n\nAdditionally, we tested whether considering CT scanner properties gives improvements. We compared the proposed noise model ($\\lambda=\\lambda(i, mA)$) with the model that assumes Gaussian approximation of the mixed Poisson-Gaussian distribution of the noise with the constant parameters $\\lambda$ ($\\lambda=const$) and $\\sigma_e^2$ of the noise variance. These parameters can hardly be pre-estimated in this case. \nFor comparison, we trained the Noise2NoiseTD-ANM model in the transmission domain with the noise variance's $\\lambda$ parameter depending on neither a pixel position nor the tube current ($\\lambda=const$). \nThe results of the comparison in the projection domain are presented in Table~\\ref{tab:noise}. Fig.~\\ref{fig:noise_prj} gives the visual comparison of the example denoised projection parts and the obtained CT slices.\nThe results show that varying parameter $\\lambda$ of the noise variance allows models to remove noise better while preserving edges and fine details. Besides, the independence of the parameter $\\lambda$ from the tube current would not allow the models easily adapt to different noise levels, the parameter would need to be retrained or it would worsen the results.\n\n\\begin{table}[!t]\n\\caption{The quantitative evaluation of the noise model with the Poisson parameter $\\lambda$ dependent on the detector column position and tube current ($\\lambda=\\lambda(i, mA)$) and the constant Poisson parameter $\\lambda$ ($\\lambda=const$) in the projection domain.}\n\\label{tab:noise}\n\\centering\n\\begin{tabular}{c| c c c}\\hline\\hline\nNoise model & SSIM$\\uparrow$ & PSNR$\\uparrow$ & GMSD$\\downarrow$ \\\\\n\\hline\\hline\n\\multicolumn{4}{c}{SL dataset}\\\\\\hline\n$\\lambda=const$ &\t$0.967 \\pm 0.011$ & \t$44.0 \\pm 1.3$ & \t$0.008 \\pm 0.002$ \\\\ \n$\\lambda=\\lambda(i, mA)$ &\t$0.972 \\pm 0.010$ & \t$44.4 \\pm 1.3$ & \t$0.006 \\pm 0.002$ \\\\\n\\hline\\hline \n\\multicolumn{4}{c}{SH dataset}\\\\\\hline\n$\\lambda=const$ &\t$0.956 \\pm 0.009$ & \t$42.5 \\pm 1.2$ & \t$0.010 \\pm 0.004$  \\\\ \n$\\lambda=\\lambda(i, mA)$ &\t$0.961 \\pm 0.008$ & \t$42.3 \\pm 1.0$ & \t$0.009 \\pm 0.004$  \\\\ \\hline\n\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[!t]\n\\centering\n\\subfloat[parts of CT projections]{\\includegraphics[width=\\linewidth]{figures\/results\/L033_nm_prj_wo_pl.jpg}}\n\\vfil\n\\subfloat[parts of CT slices]{\\includegraphics[width=0.95\\linewidth]{figures\/results\/L033_nm_img.jpg}\n}\n\\caption{The example parts of CT projections from the SL dataset denoised by the Noise2NoiseTD-ANM with different noise models and the CT slices reconstructed from them.}\n\\label{fig:noise_prj}\n\\end{figure}\n\n\n\\section{Discussion and conclusions}\nIn the study, we considered self-supervised deep-learning methods for denoising low-dose CT projections. Although supervised methods were shown to perform better than the self-supervised approaches for the denoising task, they require the representative dataset of paired clean-noisy images, which is difficult to obtain in the clinical setting. \nAs for synthetic approaches (e.g., using GANs \\cite{prokopenko2019unpaired}), it is not always possible to obtain the complete information about the CT machine necessary for the accurate simulation. Besides, a closer proximity of the distributions of training and testing data is known to lead to better results, making the training on real data preferable.\nTherefore, self-supervised approaches for denoising low-dose CT images are of particular value. Early self-supervised approaches mostly built upon the Noise2Noise model, with the generation of the dataset complicating the denoising challenge and making the solution less adaptable.\n\nHerein, we proposed a new self-supervised approach Noise2NoiseTD-ANM that uses only the original noisy projections. This method is based on restoring a certain CT projection from its adjacent projections, as in the problem of frame prediction from the frame sequence, and modeling data distributions for model training and inference. \nFor this, we included ConvLSTM units in the network and developed the noise model that takes into account the theoretical physics-based noise distribution of the CT projections. This noise model incorporates the effect of bowtie filtering, adapting to the dose modulation.\nIt should be noted that the train-inference scheme was taken from~\\cite{laine2019hqss} and adopted for denoising CT projections using the developed noise model.  \n\nUsing the same backbone neural network architecture, we compared our approach with the Noise2Clean, the Half2Half, and the state-of-the-art blind-spot network-based approach~\\cite{laine2019hqss}, using the adapted train-inference scheme.  We tested the models on the simulated test data with both high and low noise levels. The results showed that in the case of training and testing on the data with approximately the same noise levels, the Noise2Clean approach gives slightly better results. However, the Noise2Clean model, even with given noise level maps, fails to properly denoise projections of different noise levels, whereas, our physics-based self-supervised approach with realistic noise model is more successful in generalizing to any noise levels. These experiments also emphasize the advantage of the self-supervised technique over the supervised training regime. \nThe noise model, capable of estimating the noise properties separately, also allows to easily adapt to various denoising scenarios. For example, if the test noise distribution model differs from the noise model of the trained model, it can be easily tuned, and then the main denoising model and the noise model can be retrained together on the new data, or only the parameters of the noise model can be optimized.\n\nIn the comparison of the self-supervised approaches, our approach marginally outperformed the others. While the approach from~\\cite{laine2019hqss} relies only on the neighborhood of a pixel to be denoised, our approach looks at the adjacent projections that contain almost the same content but different noise realization. Therefore, our approach is less prone to losing some object details, even in the cases of severe noise. \n\nBesides, we compared the approaches on test projections with real noise. Even trained on the simulated data, the proposed approach turned out to be better than the supervised approach. Nevertheless, our approach can be directly trained on the data to be denoised. As was expected, training of the Noise2NoiseTD-ANM model on the test data with real noise gave improvements. However, the comparison was performed only qualitatively. Similarly to other medical imaging modalities, the quantitative comparison could not be performed because of the lack of representative no-reference image quality assessment (IQA) methods \\cite{Kastryulin}. The no-reference IQA methods as well as full-reference IQA methods, appropriate for the assessment of CT projections and CT images, are yet to be developed.\n\nThus, the proposed self-supervised approach Noise2NoiseTD-ANM outperformed the other considered approaches in terms of adaptability and the quality of the denoised images. We believe this approach is promising given it can readily denoise LDCT images in a clinical setting. Further development of the self-supervised approach could decrease the dose of X-ray radiation even more, especially if combined with other low-photon image recovery methods \\cite{pronina2020microscopy}. However, the proposed model still requires additional testing using more projections with real noise in a controlled manner. Lastly, the opinion of the radiologists about the projections denoised by our approach should also be analyzed.\n\n\\section*{Compliance with Ethical Standards}\nThis research study was conducted retrospectively using human subject data made available in open access by The Cancer Imaging Archive (TCIA). The usage of this dataset has been approved by the Philips internal committee for biomedical experiments.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe thank Dr. Frank Bergner, Dr. Thomas Koehler (Philips GmbH Innovative Technologies) and Dr. Kevin M.Brown (Philips Healthcare) for supporting this research and providing feedback for this article.\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn \\cite{AG2} we explored the question what symmetric pairs are\nGelfand pairs. We introduced the notion of regular symmetric pair\nand conjectured that all symmetric pairs are regular. This\nconjecture would imply that many symmetric pairs are Gelfand\npairs, including all connected symmetric pair over ${\\mathbb C}$.\n\nIn this paper we show that the pairs $$(GL(V),O(V)), \\,\n(GL(V),U(V)), \\, (U(V),O(V)), \\, (O(V \\oplus W),O(V) \\times O(W)),\n\\, (U(V \\oplus W),U(V) \\times U(W))$$ are regular where $V$ and\n$W$ are quadratic or hermitian spaces over arbitrary local field\nof characteristic zero. We deduce from this that the pairs\n$(GL_n({\\mathbb C}),O_n({\\mathbb C}))$ and $(O_{n+m}({\\mathbb C}),O_n({\\mathbb C}) \\times O_m({\\mathbb C}))$\nare Gelfand pairs.\n\nIn general, if we would know that all symmetric pairs are regular,\nthen in order to show that a given symmetric pair $(G,H)$ is a\nGelfand pair it would be enough to check the following condition\nthat we called \"goodness\":\\\\\n(*) Every closed $H$-double coset in $G$ is invariant with respect\nto $\\sigma$. Here, $\\sigma$ is the anti-involution defined by\n$\\sigma(g):= \\theta(g^{-1})$ and $\\theta$ is an involution (i.e.\nautomorphism of order 2) of $G$ such that $H = G^{\\theta}$.\n\nThis condition always holds for connected symmetric pairs over\n${\\mathbb C}$.\n\nMeanwhile, before the conjecture is proven, in order to show that\na given symmetric pair is a Gelfand pair one has to verify that\nthe pair is good, to prove that it is regular and also to compute\nits \"descendants\" and show that they are also regular. The\n\"descendants\" are certain symmetric pairs related to centralizers\nof semisimple elements.\n\n\nIn this paper we develop further the tools from \\cite{AG2} \nfor proving regularity of symmetric pairs. We also\nintroduce a systematic way to compute descendants of classical\nsymmetric pairs.\n\nBased on that we show that all the descendants of the above\nsymmetric pairs are regular.\n\n\\subsection{Structure of the paper} $ $\\\\\nIn section \\ref{PrelNot} we introduce the notions that we discuss\nin this paper. In subsection \\ref{GelPairs} we discuss the notion\nof Gelfand pair and review a classical technique for proving\nGelfand property due to Gelfand and Kazhdan. In subsection\n\\ref{SymPar} we review the results of \\cite{AG2}, introduce the\nnotions of symmetric pair, descendants of a symmetric pair, good\nsymmetric pair and regular symmetric pair mentioned above and\ndiscuss their relations to Gelfand property.\n\nIn section \\ref{MainRes} we formulate the main results of the\npaper. We also explain how they follow from the rest of the paper.\n\nIn section \\ref{GradRepDef} we introduce terminology that enables\nus to prove regularity for symmetric pairs in question.\n\nIn section \\ref{SecReg} we prove regularity for symmetric pairs in\nquestion.\n\nIn section \\ref{CompDes} we compute the descendants of those\nsymmetric pairs.\n\n\n\\subsection{Acknowledgements}\nWe are grateful to \\textbf{Herve Jacquet} for a suggestion to\nconsider the pair $(U_{2n}, U_n \\times U_n)$ which inspired this\npaper. We also thank \\textbf{Joseph Bernstein}, \\textbf{Erez\nLapid}, \\textbf{Eitan Sayag} and \\textbf{Lei Zhang} for fruitful\ndiscussions and \\textbf{Gerard Schiffmann} for  useful remarks.\n\nBoth authors were partially supported by a BSF grant, a GIF grant, and an ISF Center\nof excellency grant. A.A was also supported by ISF grant No. 583\/09 and \nD.G. by NSF grant DMS-0635607. 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 National Science Foundation.\n\n\n\\section{Preliminaries and notations} \\label{PrelNot}\n\\setcounter{lemma}{0}\n\\begin{itemize}\n\\item Throughout the paper we fix an arbitrary local field $F$ of characteristic zero.\n\\item All the algebraic varieties and algebraic\ngroups that we will consider will be defined over $F$.\n\\item For a group $G$ acting on a set $X$ and an element $x \\in X$\nwe denote by $G_x$ the stabilizer of $x$.\n\\item By a reductive group we mean an algebraic reductive group.\n\\end{itemize}\n\nIn this paper we will refer to distributions on algebraic\nvarieties over archimedean and non-archimedean fields. In the\nnon-archimedean case we mean the notion of distributions on\n$l$-spaces from \\cite{BZ}, that is linear functionals on the space\nof locally constant compactly supported functions. In the\narchimedean case one can consider the usual notion of\ndistributions, that is continuous functionals on the space of\nsmooth compactly supported functions, or the notion of Schwartz\ndistributions (see e.g. \\cite{AG1}). It does not matter here which\nnotion to choose since in the cases of consideration of this\npaper, if there are no nonzero equivariant Schwartz distributions\nthen there are no nonzero equivariant distributions at all (see\nTheorem 4.0.2 in \\cite{AG2}).\n\n\n\\begin{notation}\nLet $E$ be an extension of $F$. Let $G$ be an algebraic group\ndefined over $F$. We denote by $G_{E\/F}$ the canonical algebraic\ngroup defined over $F$ such that $G_{E\/F}(F)=G(E)$.\n\\end{notation}\n\n\n\\subsection{Gelfand pairs} \\label{GelPairs}\n$ $\\\\\nIn this section we recall a technique due to Gelfand and Kazhdan\n(\\cite{GK}) which allows to deduce statements in representation\ntheory from statements on invariant distributions. For more\ndetailed description see \\cite{AGS}, section 2.\n\n\\begin{definition}\nLet $G$ be a reductive group. By an \\textbf{admissible\nrepresentation of} $G$ we mean an admissible representation of\n$G(F)$ if $F$ is non-archimedean (see \\cite{BZ}) and admissible\nsmooth {Fr\\'{e}chet \\,} representation of $G(F)$ if $F$ is archimedean.\n\\end{definition}\n\nWe now introduce three notions of Gelfand pair.\n\n\\begin{definition}\\label{GPs}\nLet $H \\subset G$ be a pair of reductive groups.\n\\begin{itemize}\n\\item We say that $(G,H)$ satisfy {\\bf GP1} if for any irreducible\nadmissible representation $(\\pi,E)$ of $G$ we have\n$$\\dim Hom_{H(F)}(E,\\mathbb{C}) \\leq 1$$\n\n\n\\item We say that $(G,H)$ satisfy {\\bf GP2} if for any irreducible\nadmissible representation $(\\pi,E)$ of $G$ we have\n$$\\dim Hom_{H(F)}(E,\\mathbb{C}) \\cdot \\dim Hom_{H}(\\widetilde{E},\\mathbb{C})\\leq\n1$$\n\n\\item We say that $(G,H)$ satisfy {\\bf GP3} if for any irreducible\n{\\bf unitary} representation $(\\pi,\\mathcal{H})$ of $G(F)$ on a\nHilbert space $\\mathcal{H}$ we have\n$$\\dim Hom_{H(F)}(\\mathcal{H}^{\\infty},\\mathbb{C}) \\leq 1.$$\n\\end{itemize}\n\n\\end{definition}\nProperty GP1 was established by Gelfand and Kazhdan in certain\n$p$-adic cases (see \\cite{GK}). Property GP2 was introduced in\n\\cite{Gross} in the $p$-adic setting. Property GP3 was studied\nextensively by various authors under the name {\\bf generalized\nGelfand pair} both in the real and $p$-adic settings (see e.g.\n\\cite{vD,Bos-vD}).\n\nWe have the following straightforward proposition.\n\n\\begin{proposition}\n$GP1 \\Rightarrow GP2 \\Rightarrow GP3.$\n\\end{proposition}\n\nWe will use the following theorem from \\cite{AGS} which is a\nversion of a classical theorem of Gelfand and Kazhdan.\n\n\\begin{theorem}\\label{DistCrit}\nLet $H \\subset G$ be reductive groups and let $\\tau$ be an\ninvolutive anti-automorphism of $G$ and assume that $\\tau(H)=H$.\nSuppose $\\tau(\\xi)=\\xi$ for all bi $H(F)$-invariant distributions\n$\\xi$ on $G(F)$. Then $(G,H)$ satisfies GP2.\n\\end{theorem}\n\nIn the cases we consider in this paper GP2 is equivalent to GP1 by\nthe following proposition.\n\n\\begin{proposition} \\label{GP2GP1}\n$ $\\\\\n(i) Let $V$ be a quadratic space (i.e. a linear space with a\nnon-degenerate quadratic form) and let $H \\subset GL(V)$ be any\ntranspose invariant subgroup.\n Then $GP1$ is equivalent to $GP2$ for the pair\n$(\\mathrm{GL}(V),H)$.\\\\\n(ii) Let $V$ be a quadratic space and let $H \\subset O(V)$ be any\nsubgroup. Then $GP1$ is equivalent to $GP2$ for the pair\n$(O(V),H)$.\n\\end{proposition}\nIt follows from the following 2 propositions.\n\n\\begin{proposition} \\label{GKCor}\nLet $H \\subset G$ be reductive groups and let $\\tau$ be an\nanti-automorphism of $G$ such that\\\\\n(i) $\\tau^2\\in Ad(G(F))$\\\\\n(ii) $\\tau$ preserves any closed conjugacy class in $G(F)$\\\\\n(iii) $\\tau(H)=H$.\\\\\nThen $GP1$ is equivalent to $GP2$ for the pair $(G,H)$.\n\\end{proposition}\n\nFor proof see \\cite{AG2}, Corollary 8.2.3.\n\n\\begin{proposition} $ $\\\\\n(i) Let $V$ be a quadratic space and let $g \\in GL(V)$. Then $g$\nis conjugate to $g^{t}$.\\\\\n(ii) Let $V$ be a quadratic space and let $g \\in O(V)$. Then $g$\nis conjugate to $g^{-1}$ inside $O(V)$\n\\end{proposition}\nPart (i) is well known. For the proof of (ii) see \\cite{MVW},\nProposition I.2 in chapter 4.\n\n\\subsection{Symmetric pairs} \\label{SymPar}\n$ $\\\\\nIn this subsection we review some tools developed in \\cite{AG2}\nthat enable to prove that a symmetric pair is a Gelfand pair. The\nmain results discussed in this subsection are Theorem\n\\ref{LinDes}, Theorem \\ref{GoodHerRegGK} and Proposition\n\\ref{SpecCrit}.\n\n\\begin{definition}\nA \\textbf{symmetric pair} is a triple $(G,H,\\theta)$ where $H\n\\subset G$ are reductive groups, and $\\theta$ is an involution of\n$G$ such that $H = G^{\\theta}$. We call a symmetric pair\n\\textbf{connected} if $G\/H$ is connected.\n\nFor a symmetric pair $(G,H,\\theta)$ we define an antiinvolution\n$\\sigma :G \\to G$ by $\\sigma(g):=\\theta(g^{-1})$, denote ${\\mathfrak{g}}:=Lie\nG$, ${\\mathfrak{h}} := LieH$, $\\g^{\\sigma}:=\\{a \\in {\\mathfrak{g}} | \\theta(a)=-a\\}$. Note that\n$H$ acts on $\\g^{\\sigma}$ by the adjoint action. Denote also\n$G^{\\sigma}:=\\{g \\in G| \\sigma(g)=g\\}$ and define a\n\\textbf{symmetrization map} $s:G \\to G^{\\sigma}$ by $s(g):=g\n\\sigma(g)$.\n\nIn case when the involution is obvious we will omit it.\n\\end{definition}\n\n\\begin{remark}\nLet $(G,H,\\theta)$ be a symmetric pair. Then ${\\mathfrak{g}}$ has a ${\\mathbb Z}\/2{\\mathbb Z}$\ngrading given by $\\theta$.\n\\end{remark}\n\n\n\\begin{definition}\nLet $(G_1,H_1,\\theta_1)$ and $(G_2,H_2,\\theta_2)$ be symmetric\npairs. We define their \\textbf{product} to be the symmetric pair\n$(G_1 \\times G_2,H_1 \\times H_2,\\theta_1 \\times \\theta_2)$.\n\\end{definition}\n\n\\begin{definition}\nWe call a symmetric pair $(G,H,\\theta)$ \\textbf{good} if for any\nclosed $H(F) \\times H(F)$ orbit $O \\subset G(F)$, we have\n$\\sigma(O)=O$.\n\\end{definition}\n\n\\begin{proposition} \\label{GoodCrit}\nEvery connected symmetric pair over ${\\mathbb C}$ is good.\n\\end{proposition}\nFor proof see e.g. \\cite{AG2}, Corollary 7.1.7.\n\n\\begin{definition}\nWe say that a symmetric pair $(G,H,\\theta)$ is a \\textbf{GK pair}\nif any $H(F) \\times H(F)$ - invariant distribution on $G(F)$ is\n$\\sigma$ - invariant.\n\\end{definition}\n\n\\begin{remark}\nTheorem \\ref{DistCrit} implies that any GK pair satisfies GP2.\n\\end{remark}\n\\subsubsection{Descendants of symmetric pairs}\n\\begin{proposition} \\label{PropDescend}\nLet  $(G,H,\\theta)$ be a symmetric pair. Let $g \\in G(F)$ such\nthat $HgH$ is closed. Let $x=s(g)$. Then $x$ is a semisimple\nelement of $G$.\n\\end{proposition}\nFor proof see e.g. \\cite{AG2}, Proposition 7.2.1.\n\\begin{definition}\nIn the notations of the previous proposition we will say that the\npair $(G_x,H_x,\\theta|_{G_x})$ is a \\textbf{descendant} of\n$(G,H,\\theta)$.\n\\end{definition}\n\n\n\\subsubsection{Tame symmetric pairs}\n\\begin{definition}\nLet $\\pi$ be an action of a reductive group $G$ on a smooth affine\nvariety $X$.\nWe say that an algebraic automorphism $\\tau$ of $X$ is \\textbf{$G$-admissible} if \\\\\n(i) $\\pi(G(F))$ is of index at most 2 in the group of\nautomorphisms of $X$\ngenerated by $\\pi(G(F))$ and $\\tau$.\\\\\n(ii) For any closed $G(F)$ orbit $O \\subset X(F)$, we have\n$\\tau(O)=O$.\n\\end{definition}\n\n\\begin{definition}\nWe call an action of a reductive group $G$ on a smooth affine\nvariety $X$ \\textbf{tame} if for any $G$-admissible $\\tau : X \\to\nX$, every $G(F)$-invariant distribution on $X(F)$ is\n$\\tau$-invariant.\n\nWe call a symmetric pair $(G,H,\\theta)$ \\textbf{tame} if the\naction of $H\\times H$ on $G$ is tame.\n\\end{definition}\n\n\\begin{remark}\nEvidently, any good tame symmetric pair is a GK pair.\n\\end{remark}\n\n\\begin{notation}\nLet $V$ be an algebraic finite dimensional representation over $F$\nof a reductive group $G$. Denote $Q(V):=V\/V^G$. Since $G$ is\nreductive, there is a canonical embedding $Q(V) \\hookrightarrow\nV$.\n\\end{notation}\n\n\\begin{notation}\nLet $(G,H,\\theta)$ be a symmetric pair. We denote by\n$\\mathcal{N}_{G,H}$ the subset of all the nilpotent elements in\n$Q(\\g^{\\sigma})$. Denote $R_{G,H}:=Q(\\g^{\\sigma}) - \\mathcal{N}_{G,H}$.\n\\end{notation}\nNote that our notion of $R_{G,H}$ coincides with the notion\n$R(\\g^{\\sigma})$ used in \\cite{AG2}, \nNotation 2.3.10. This follows from\nLemma 7.1.11 in \\cite{AG2}.\n\n\\begin{definition}\nWe call a symmetric pair $(G,H,\\theta)$ \\textbf{weakly linearly\ntame} if for any $H$-admissible transformation $\\tau$ of $\\g^{\\sigma}$\nsuch that every $H(F)$-invariant distribution on $R_{G,H}$ is also\n$\\tau$-invariant,\nwe have\\\\\n(*) every $H(F)$-invariant distribution on $Q(\\g^{\\sigma})$ is also\n$\\tau$-invariant.\n\\end{definition}\n\n\\begin{theorem} \\label{LinDes}\nLet $(G,H,\\theta)$ be a symmetric pair. Suppose that all its\ndescendants (including itself)  are weakly linearly tame. Then\n$(G,H,\\theta)$ is tame.\n\\end{theorem}\nFor proof see Theorem 7.3.3 in \\cite{AG2}.\n\nNow we would like to formulate a criterion for being weakly linearly\ntame. For it we\nwill need the following lemma and notation.\n\n\\begin{lemma}\nLet $(G,H,\\theta)$ be a symmetric pair. Then any nilpotent element\n$x \\in \\g^{\\sigma}$ can be extended to an $sl_2$ triple $(x,d(x),x_-)$\nsuch that $d(x) \\in {\\mathfrak{h}}$ and $x_- \\in \\g^{\\sigma}$.\n\\end{lemma}\nFor proof see e.g. \\cite{AG2}, Lemma 7.1.11.\n\n\\begin{notation}\nWe will use the notation $d(x)$ from the last lemma in the future.\nIt is not uniquely defined but whenever we will use this notation\nnothing will depend on its choice.\n\\end{notation}\n\n\\begin{proposition} \\label{SpecCrit}\nLet $(G,H,\\theta)$ be a symmetric pair. Suppose that for any\nnilpotent $x \\in \\g^{\\sigma}$ we have\n$$\\operatorname{Tr}(ad(d(x))|_{{\\mathfrak{h}}_x}) < \\dim Q(\\g^{\\sigma}) .$$\nThen the pair $(G,H,\\theta)$ is weakly linearly tame.\n\\end{proposition}\nThis proposition follows from \\cite{AG2} (Propositions 7.3.7 and\n7.3.5).\n\n\\subsubsection{Regular symmetric pairs}\n\\begin{definition}\nLet $(G,H,\\theta)$ be a symmetric pair. We call an element $g \\in\nG(F)$ \\textbf{admissible} if\\\\\n(i) $Ad(g)$ commutes with $\\theta$ (or, equivalently, $s(g)\\in Z(G)$) and \\\\\n(ii) $Ad(g)|_{{\\mathfrak{g}}^{\\sigma}}$ is $H$-admissible.\n\\end{definition}\n\n\\begin{definition}\nWe call a symmetric pair $(G,H,\\theta)$ \\textbf{regular} if for\nany admissible $g \\in G(F)$ such that every $H(F)$-invariant\ndistribution on $R_{G,H}$ is also $Ad(g)$-invariant,\nwe have\\\\\n(*) every $H(F)$-invariant distribution on $Q(\\g^{\\sigma})$ is also\n$Ad(g)$-invariant.\n\\end{definition}\n\nThe following two propositions are evident.\n\\begin{proposition} \\label{TrivReg}\nLet $(G,H,\\theta)$ be symmetric pair. Suppose that any $g \\in\nG(F)$ satisfying $\\sigma(g)g \\in Z(G(F))$ lies in $Z(G(F))H(F)$.\nThen $(G,H,\\theta)$ is regular. In particular if the normalizer of\n$H(F)$ lies inside $Z(G(F))H(F)$ then $(G,H,\\theta)$ is regular.\n\\end{proposition}\n\n\\begin{proposition} $ $\\\\\n(i) Any weakly linearly tame pair is regular. \\\\\n(ii) A product of regular pairs is regular (see \\cite{AG2},\nProposition 7.4.4).\n\\end{proposition}\n\nIn section \\ref{GradRepDef} we will introduce terminology that will\nhelp to verify the condition of Proposition \\ref{SpecCrit}.\n\nThe importance of the notion of regular pair is demonstrated by\nthe following theorem.\n\n\\begin{theorem} \\label{GoodHerRegGK}\nLet $(G,H,\\theta)$ be a good symmetric pair such that all its\ndescendants (including itself) are regular. Then it is a GK pair.\n\\end{theorem}\nFor proof see \\cite{AG2}, Theorem 7.4.5.\n\\section{Main Results} \\label{MainRes}\nHere we formulate the main results of the paper and explain how\nthey follow from the rest of the paper.\n\n\\setcounter{lemma}{0}\n\n\\begin{definition}\nA quadratic space is a linear space with a fixed non-degenerate\nquadratic form.\n\nLet $F'$ be an extension of $F$ and $V$ be a quadratic space over\nit. We denote by $O(V)$ the canonical algebraic group such that\nits $F$-points form the group of orthogonal transformations of\n$V$.\n\\end{definition}\n\n\n\\begin{definition}\nLet $D$ be a field with an involution $\\tau$. A hermitian space\nover $(D,\\tau)$  is a linear space over $D$ with a fixed\nnon-degenerate hermitian form.\n\nSuppose that $D$ is an extension of $F$ and $F \\subset D^{\\tau}$.\nLet $V$ be a hermitian space over $(D,\\tau)$. We denote by $U(V)$\nthe canonical algebraic group such that its $F$-points form the\ngroup of unitary transformations of $V$.\n\\end{definition}\n\n\\begin{definition}\nLet $G$ be a reductive group and $\\varepsilon \\in G$ be an element of\norder 2. We denote by $(G,G_{\\varepsilon})$ the symmetric pair defined by\nthe involution $x \\mapsto \\varepsilon x \\varepsilon$.\n\\end{definition}\n\nThe following lemma is straightforward.\n\\begin{lemma}\nLet $V$ be a quadratic space.\\\\\n(i) Let $\\varepsilon \\in GL(V)$ be an element of order 2. Then\n$GL(V)_{\\varepsilon} \\cong GL(V_1) \\times GL(V_2)$ for some decomposition\n$V=V_1 \\oplus V_2$.\\\\\n(ii) Let $\\varepsilon \\in O(V)$ be an element of order 2. Then\n$O(V)_{\\varepsilon} \\cong O(V_1) \\times O(V_2)$ for some orthogonal\ndecomposition $V=V_1 \\oplus V_2$.\\\\\n(iii) Let $V$ be a hermitian space.\\\\\n Let $\\varepsilon \\in U(V)$ be an element of order 2. Then\n$U(V)_{\\varepsilon} \\cong U(V_1) \\times U(V_2)$ for some orthogonal\ndecomposition $V=V_1 \\oplus V_2$.\n\\end{lemma}\n\n\\begin{theorem}\nLet $V$ be a quadratic space over $F$. Then all the descendants of\nthe pair $(O(V),O(V)_{\\varepsilon})$ are regular.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{CompDesO_OtO} below, the descendants of the pair\n$(O(V),O(V)_{\\varepsilon})$ are products of pairs of the types\\\\\n(i) $(GL(W),O(W))$ for some quadratic space $W$ over some field\n$F'$ that extends $F$\\\\\n(ii) $(U(W_E) , O(W))$ for some quadratic space $W$ over some\nfield $F'$ that extends $F$, and some quadratic extension $E$ of\n$F'$. Here, $W_E:=W \\otimes _{F'}E$ is the extension of scalars\nwith the corresponding hermitian structure.\\\\\n(iii) $(O(W),O(W)_{\\varepsilon})$ for some quadratic space $W$ over $F$.\n\nThe pair (i) is regular by Theorem \\ref{GL_O} below. The pair (ii)\nis regular by subsection \\ref{U_reg} below. The pair (iii) is\nregular by subsection \\ref{O_OtO} below.\n\\end{proof}\n\\begin{corollary}\nSuppose that $F={\\mathbb C}$ and Let $V$ be a quadratic space over it. Then\nthe pair $(O(V),O(V)_{\\varepsilon})$ satisfies GP1.\n\\end{corollary}\n\\begin{proof}\nThis pair is good by Proposition \\ref{GoodCrit} and all its\ndescendants are regular. Hence by Theorem \\ref{GoodHerRegGK} it is\na GK pair. Therefore by Theorem \\ref{DistCrit} it satisfies GP2.\nNow, by Proposition \\ref{GP2GP1}, it satisfies GP1.\n\\end{proof}\n\\begin{theorem}\nLet $D\/F$ be a quadratic extension and $\\tau \\in Gal(D\/F)$ be the\nnon-trivial element. Let $V$ be a hermitian space over $(D,\\tau)$.\nThen all the descendants of the pair $(U(V),U(V)_{\\varepsilon})$ are\nregular.\n\\end{theorem}\n\\begin{proof}\nBy theorem \\ref{CompDesU_UtU} below, the descendants of the pair\n$(U(V),U(V)_{\\varepsilon})$ are products of pairs of the types\\\\\n(a) $(G \\times G, \\Delta G)$ for some reductive group $G$.\\\\\n(b) $(GL(W),U(W))$ for some hermitian space $W$ over some\nextension $(D',\\tau')$ of $(D,\\tau)$\\\\\n(c) $(G_{E\/F},G)$ for some reductive group $G$ and some quadratic\nextension $E\/F$.\\\\\n(d) $(GL(W),GL(W)_{\\varepsilon})$ where $W$ is a linear space over $D$\nand $\\varepsilon \\in GL(W)$ is an element of order $\\leq 2$.\\\\\n(e) $(U(W),U(W)_{\\varepsilon})$ where $W$ is a hermitian space over\n$(D,\\tau)$.\n\nThe pairs (a) and (c) are regular by Theorem \\ref{2RegPairs}\nbelow. The pairs (b) and (e) are regular by subsection \\ref{U_reg}\nbelow. The pair (d) is regular by Theorem \\ref{RJR} below.\n\\end{proof}\n\\begin{theorem} \\label{Main_GL_O}\nLet $V$ be a quadratic space over $F$. Then all the descendants of\nthe pair $(GL(V),O(V))$ are weakly linearly tame. In particular,\nthis pair is tame.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{CompDesGL_O} below, the descendants of the pair\n$(GL(V),O(V))$ are products of pairs of the type $(GL(W),O(W))$\nfor some quadratic space $W$ over some field $F'$ that extends\n$F$. By Theorem \\ref{GL_O} below, these pairs are weakly linearly\ntame. Now, the pair $(GL(V),O(V))$ is tame by Theorem\n\\ref{LinDes}.\n\\end{proof}\n\\begin{corollary}\nSuppose that $F={\\mathbb C}$ and Let $V$ be a quadratic space over it. Then\nthe pair $(GL(V),O(V))$ is GP1.\n\\end{corollary}\n\n\\begin{theorem}\nLet $D\/F$ be a quadratic extension and $\\tau \\in Gal(D\/F)$ be the\nnon-trivial element. Let $V$ be a hermitian space over $(D,\\tau)$.\nThen all the descendants of the pair $(GL(V),U(V))$ are weakly\nlinearly tame. In particular, this pair is tame.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{CompDesGL_U} below, all the descendants of the\npair\n$(GL(V),U(V))$ are products of pairs of the types\\\\\n(i) $(GL(W) \\times GL(W), \\Delta GL(W))$ for some linear space $W$\nover some field $D'$ that extends $D$\\\\\n(ii)  $(GL(W),U(W))$ for some hermitian space $W$ over some\n$(D',\\tau')$ that extends $(D,\\tau)$.\n\nThe pair (i) is weakly linearly tame by Theorem \\ref{2RegPairs}\nbelow and the pair (ii) is weakly linearly tame by subsection\n\\ref{U_reg} below. Now, the pair $(GL(V),U(V))$ is tame by Theorem\n\\ref{LinDes}.\n\\end{proof}\n\\begin{theorem}\nLet $V$ be a quadratic space over $F$. Let $D\/F$ be a quadratic\nextension and $\\tau \\in Gal(D\/F)$ be the non-trivial element. Let\n$V_D:=V \\otimes_F D$ be its extension of scalars with the\ncorresponding hermitian structure. Then all the descendants of the\npair $(U(V_D),O(V))$ are weakly linearly tame. In particular, this\npair is tame.\n\\end{theorem}\n\\begin{proof}\n\nBy Theorem \\ref{CompDesU_O} below, all the descendants of the pair\n$(U(V_D),O(V))$ are products\nof pairs of the types\\\\\n(i) $(GL(W),O(W))$ for some quadratic space $W$ over some field\n$F'$ that extends $F$.\\\\\n(ii) $(U(W_{D'}),O(W))$ for some extension $(D',\\tau')$ of\n$(D,\\tau)$ and some quadratic space $W$ over $D'^{\\tau'}$.\n\nThe pair (i) is weakly linearly tame by Theorem \\ref{GL_O} below\nand the pair (ii) is weakly linearly tame by subsection\n\\ref{U_reg} below. Now, the pair $(GL(V),U(V))$ is tame by Theorem\n\\ref{LinDes}.\n\\end{proof}\n\n\\section{${\\mathbb Z}\/2{\\mathbb Z}$ graded representations of $sl_2$ and their\ndefects} \\label{GradRepDef}\n\nIn this section we will introduce terminology that will help to\nverify the condition of Proposition \\ref{SpecCrit}.\n\n\\subsection{Graded representations of $sl_2$}\n\n\\begin{definition}\nWe fix standard basis $e,h,f$ of $sl_2(F)$. We fix a grading on\n$sl_2(F)$ given by $h \\in sl_2(F)_0$ and $e,f \\in sl_2(F)_1$. A\n\\textbf{graded representation of $sl_2$} is a representation of\n$sl_2$ on a graded vector space $V=V_0 \\oplus V_1$ such that\n$sl_2(F)_i(V_j) \\subset V_{i+j}$ where $i,j \\in {\\mathbb Z}\/2{\\mathbb Z}$.\n\\end{definition}\n\nThe following lemma is standard.\n\\begin{lemma}$ $\\\\\n(i) Every graded representation of $sl_2$ which is\nirreducible as a graded representation is irreducible just as a representation.\\\\\n(ii) Every irreducible representation $V$ of $sl_2$ admits exactly\ntwo gradings. In one highest weight vector lies in $V_0$ and in\nthe other in $V_1$.\n\\end{lemma}\n\n\\begin{definition}\nWe denote by $V_{\\lambda}^{w}$ the irreducible graded\nrepresentation of $sl_2$ with highest weight $\\lambda$ and highest\nweight vector of parity $p$ where $w = (-1)^p$.\n\\end{definition}\n\nThe following lemma is straightforward.\n\n\\begin{lemma} \\label{GradedTensors}\n\\setcounter{equation}{0}\n\\begin{align}\n& (V_{\\lambda}^w)^* = V_{\\lambda}^{w(-1)^{\\lambda}}\\\\\n& V_{\\lambda_1}^{w_1} \\otimes V_{\\lambda_2}^{w_2} =\n\\bigoplus_{i=0}^{\\min(\\lambda_1, \\lambda_2)}\nV_{\\lambda_1+\\lambda_2 - 2i}^{w_1w_2(-1)^i}\\\\\n& \\Lambda^2(V_{\\lambda}^w) = \\bigoplus_{i=0}^{\\lfloor\n\\frac{\\lambda-1}{2} \\rfloor} V_{2\\lambda-4i-2}^{-1}.\n\\end{align}\n\\end{lemma}\n\n\\subsection{Defects\n\\begin{definition}\nLet $\\pi$ be a graded representation of $sl_2$. We define the\n\\textbf{defect} of $\\pi$ to be\n$$def(\\pi)=\\operatorname{Tr}(h|_{(\\pi^e)_0})-\\dim(\\pi_1)$$\n\\end{definition}\nThe following lemma is straightforward\n\n\\begin{lemma} \\label{Defects}\n\n\\begin{align}\n&def(\\pi \\oplus \\tau)=def(\\pi) + def(\\tau)\\\\\n&def(V_{\\lambda}^w) =\\frac{1}{2}(\\lambda w + w( \\frac{1\n+(-1)^{\\lambda}}{2}) -1)= \\frac{1}{2} \\left\\{%\n\\begin{array}{ll}\n   \\lambda w + w -1 & \\lambda \\text{ is even} \\\\\n    \\lambda w -1 & \\lambda \\text{ is odd}\n\\end{array}%\n\\right.\n\\end{align}\n\n\\end{lemma}\n\\begin{definition}\nLet ${\\mathfrak{g}}$ be a $({\\mathbb Z}\/2{\\mathbb Z})$ graded Lie algebra. We say that ${\\mathfrak{g}}$\n\\textbf{is of negative defect} if for any graded homomorphism\n$\\pi: sl_2 \\to {\\mathfrak{g}}$, the defect of ${\\mathfrak{g}}$ with respect to the adjoint\naction of $sl_2$ is negative. \n\nWe say that  ${\\mathfrak{g}}$\n\\textbf{is of negative normalized defect} if the semi-simple part of ${\\mathfrak{g}}$ (i.e. the quotient of ${\\mathfrak{g}}$ by its center) \nis of negative defect.\n\\end{definition}\n\\begin{remark}\nClearly,  ${\\mathfrak{g}}$\nis of negative normalized defect if and only if for any graded homomorphism\n$\\pi: sl_2 \\to {\\mathfrak{g}}$, the defect of ${\\mathfrak{g}}$ with respect to the adjoint\naction of $sl_2$ is less than the dimension of the odd part of the center of  ${\\mathfrak{g}}$.\n\\end{remark}\n\\begin{definition}\nWe say that a symmetric pair $(G,H,\\theta)$ \\textbf{is of negative \nnormalized defect} if the Lie algebra ${\\mathfrak{g}}$ with the grading defined by\n$\\theta$ is of negative \nnormalized defect.\n\\end{definition}\n\n\\begin{lemma}\nLet $(G,H,\\theta)$  be a symmetric pair. Assume that ${\\mathfrak{g}}$ is semi-simple. Then\n$Q(\\g^{\\sigma})=\\g^{\\sigma}.$\n\\end{lemma}\n\\begin{proof}$ $\\\\\nAssume the contrary: there exist $0 \\neq x \\in \\g^{\\sigma}$ such that\n$Hx=x$. Then $\\dim (CN_{Hx,x}^{\\g^{\\sigma}}) = \\dim \\g^{\\sigma}$, hence\n$CN_{Hx,x}^{\\g^{\\sigma}} = \\g^{\\sigma}$. On the other hand, $CN_{Hx,x}^{\\g^{\\sigma}} \\cong [{\\mathfrak{h}}, x]^\\bot=(\\g^{\\sigma})^x$ (here $(\\cdot)^\\bot$ means the ortogonal compliment w.r.t. the kiling form). Therefore $\\g^{\\sigma}=(\\g^{\\sigma})^x$ and hence $x$\nlies in the center of ${\\mathfrak{g}}$, which is impossible.\n\\end{proof}\n\nProposition \\ref{SpecCrit} can be rewritten now in the following\nform\n\\begin{theorem}\nA symmetric pair of negative \nnormalized defect is weakly linearly tame.\n\\end{theorem}\n\nEvidently, a product of pairs of negative normalized defect is again of\nnegative normalized defect.\n\nThe following lemma is straightforward.\n\\begin{lemma} \\label{NegDefAlg}\nLet $(G,H,\\theta)$ be a symmetric pair. Let $F'$ be any field\nextending $F$. Let $(G_{F'},H_{F'},\\theta)$ be the extension of\n$(G,H,\\theta)$ to $F'$. Suppose that it is of negative normalized defect (as\na pair over $F'$) . Then $(G,H,\\theta)$ and\n$(G_{F'\/F},H_{F'\/F},\\theta)$ are of negative normalized defect (as pairs over\n$F$).\n\\end{lemma}\n\nIn \\cite{AG2} we proved the following (easy) proposition (see\n\\cite{AG2}, Lemma 7.6.6).\n\\begin{proposition} \\label{pi_pibarAG2}\nLet $\\pi$ be a representation of $sl_2$. Then\n$\\operatorname{Tr}(h|_{(\\pi^e)})<\\dim(\\pi)$.\n\\end{proposition}\nWe would like to reformulate it in terms of defect. For this we\nwill need the following notation.\n\\begin{notation} $ $\\\\\n(i) Let $\\pi$ be a representation of $sl_2$. We denote by ${\\overline{\\pi}}$\nthe representation of $sl_2$ on the same space defined by\n${\\overline{\\pi}}(e):=-\\pi(e)$, ${\\overline{\\pi}}(f):=-\\pi(f)$ and ${\\overline{\\pi}}(h):=\\pi(h)$. \\\\\n(ii) We define grading on $\\pi \\oplus {\\overline{\\pi}}$ by the involution $s(v\n\\oplus w):=  w \\oplus v$.\n\\end{notation}\nProposition \\ref{pi_pibarAG2} can be reformulated in the following\nway.\n\\begin{proposition} \\label{pi_pibar}\nLet $\\pi$ be a representation of $sl_2$. Then $def(\\pi \\oplus\n{\\overline{\\pi}})<0$.\n\\end{proposition}\nIn \\cite{AG2} we also deduced from this proposition the following\ntheorem (see \\cite{AG2}, 7.6.2).\n\\begin{theorem} \\label{2RegPairs}\nFor any reductive group $G$, the pairs $(G \\times G,\\Delta G)$ and\n$(G_{E\/F},G)$ are of negative normalized defect and hence weakly linearly\ntame. Here $\\Delta G$ is the diagonal in $G \\times G$.\n\\end{theorem}\nIn \n\\cite[\\S\\S 2.7]{AG2} we\nproved the following theorem.\n\\begin{theorem} \\label{RJR}\nThe pair $(GL(V \\oplus V),GL(V) \\times GL(V))$ is of negative\nnormalized defect and hence regular.\n\\end{theorem}\nNote that in the case $dimV \\neq dimW$ the pair $(GL(V \\oplus\nW),GL(V) \\times GL(W))$ is obviously regular by Proposition\n\\ref{TrivReg}.\n\n\n\\section{Proof of regularity and tameness} \\label{SecReg}\n\n\\subsection{The pair $(GL(V),O(V))$}  $ $\\\\\\\\\nIn this subsection we prove that the pair $(GL(V),O(V))$ is weakly\nlinearly tame.  \nFor $\\dim V \\leq 1$  it is obvious.\nHence it is enough to prove the following\ntheorem.\n\n\\begin{theorem} \\label{GL_O}\nLet $V$ be a quadratic space \nof dimension at least $2$. Then the pair $(GL(V),O(V))$ has\nnegative normalized defect.\n\\end{theorem}\nWe will need the following notation.\n\n\\begin{notation}\nLet $\\pi$ be a representation of $sl_2$. We define grading on $\\pi\n\\otimes {\\overline{\\pi}}$ by the involution $s(v \\otimes w):= - w \\otimes v$.\n\\end{notation}\n\nTheorem \\ref{GL_O} immediately follows from the following one.\n\n\\begin{theorem}\nLet $\\pi$ be a representation of $sl_2$ of dimension  at least 2. Then $def(\\pi \\otimes {\\overline{\\pi}})<-1.$\n\\end{theorem}\n\nThis theorem in turn follows from the following lemma.\n\n\\begin{lemma}\nLet $V_{\\lambda}$ and $V_{\\mu}$ be irreducible\nrepresentations of $sl_2$. Then\\\\\n(i) $def(V_{\\lambda} \\otimes \\overline{V_{\\lambda}}) = -(\\lambda +\n1)(\\frac{\\lambda}{2}+1)$.\\\\\n(ii) $def(V_{\\lambda} \\otimes \\overline{V_{\\mu}} \\oplus V_{\\mu}\n\\otimes \\overline{V_{\\lambda}})<0$.\n\\end{lemma}\n\\begin{proof} $ $\\\\\n(i) Follows from the fact that $V_{\\lambda} \\otimes\n\\overline{V_{\\lambda}} = \\bigoplus_{i=0}^{\\lambda} V_{2\\lambda -\n2i}^{-1}$ and from Lemma \\ref{Defects}.\\\\\n(ii) Follows from Proposition \\ref{pi_pibar}.\n\\end{proof}\n\n\\subsection{The pair $(O(V_1 \\oplus V_2), O(V_1) \\times O(V_2))$} \\label{O_OtO}\n $ $\\\\\\\\\nIn this subsection prove that the pair $(O(V_1 \\oplus V_2), O(V_1)\n\\times O(V_2))$ is regular. For that it is enough to prove the\nfollowing theorem.\n\n\\begin{theorem}\nLet $V_1$ and $V_2$ be quadratic spaces. Assume $\\dim V_1 = \\dim\nV_2$. Then the pair $(O(V_1 \\oplus V_2), O(V_1) \\times O(V_2))$\nhas negative normalized defect.\n\\end{theorem}\n\nThis theorem immediately follows from the following one.\n\n\\begin{theorem} \\label{Lambda2neg}\nLet $\\pi$ be a (non-zero) graded representation of $sl_2$ such\nthat $\\dim \\pi_0 = \\dim \\pi_1$ and $\\pi \\simeq \\pi^*$. Then\n$\\Lambda^2(\\pi)$ has negative defect.\n\\end{theorem}\n\nFor this theorem we will need the following lemma.\n\n\\begin{lemma}\nLet $V_{\\lambda_1}^{w_1}$ and $V_{\\lambda_2}^{w_2}$ be irreducible\ngraded\nrepresentations of $sl_2$. Then\\\\\n(i) $def(V_{\\lambda_1}^{w_1} \\otimes V_{\\lambda_2}^{w_2}) =$\n$$= -\\frac{1}{2} \\left\\{%\n\\begin{array}{lll}\n    \\min(\\lambda_1,\\lambda_2)+1 - \\frac{w_1w_2}{2}(\\lambda_1+\\lambda_2+1+(-1)^{\\min(\\lambda_1,\\lambda_2)}(|\\lambda_1-\\lambda_2|-1)), & \\lambda_1 \\neq \\lambda_2 & \\mod 2; \\\\\n    \\min(\\lambda_1,\\lambda_2)+1 - w_1w_2(\\max(\\lambda_1,\\lambda_2)+1), & \\lambda_1 \\equiv \\lambda_2 \\equiv 0 & \\mod 2; \\\\\n    \\min(\\lambda_1,\\lambda_2)+1 - w_1w_2(\\min(\\lambda_1,\\lambda_2)+1), & \\lambda_1 \\equiv \\lambda_2 \\equiv 1 & \\mod 2;\n\\end{array}%\n\\right.$$\n(ii) $def(\\Lambda^2(V_{\\lambda_1}^{w_1}))= -\\frac{\\lambda^2}{4} -\n\\frac{\\lambda}{2} - \\frac{1 + (-1)^{\\lambda+1}}{8}$\n\\end{lemma}\n\\begin{proof}\nThis lemma follows by straightforward computations from Lemmas\n\\ref{GradedTensors} and \\ref{Defects}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Lambda2neg}]\nSince $\\pi \\simeq \\pi^*$, $\\pi$ can be decomposed to a direct sum\nof irreducible graded representations in the following way\n$$ \\pi = (\\bigoplus_{i=1}^l V_{\\lambda_i}^{1})  \\oplus\n(\\bigoplus_{j=1}^m V_{\\mu_j}^{-1}) \\oplus (\\bigoplus_{k=1}^n\nV_{\\nu_k}^{1} \\oplus V_{\\nu_k}^{-1}).$$\n\nHere, all $\\lambda_i$ and $\\mu_j$ are even and $\\nu_k$ are odd.\nSince $\\dim \\pi_0 = \\dim \\pi_1$, $l=m$.\n\nBy the last lemma, $def(V_{\\lambda_i}^1 \\otimes (V_{\\nu_k}^{1}\n\\oplus V_{\\nu_k}^{-1})) = -(\\min(\\lambda_1,\\lambda_2)+1)<0$.\nSimilarly, $def(V_{\\mu_j}^{-1} \\otimes (V_{\\nu_k}^{1} \\oplus\nV_{\\nu_k}^{-1}))<0$. Also, $def((V_{\\nu_{k_1}}^{1} \\oplus\nV_{\\nu_{k_1}}^{-1}) \\otimes (V_{\\nu_{k_2}}^{1} \\oplus\nV_{\\nu_{k_2}}^{-1}))<0$ and $def(\\Lambda^2(V_{\\lambda}^w)) \\leq 0$\nfor all $\\lambda$ and $w$.\n\nHence if $l=0$ we are done. Otherwise we can assume $n=0$. Now,\n\n\\begin{multline*}\ndef(\\Lambda^2(\\pi)) \\leq \\sum_{1 \\leq i<j \\leq\nl}|\\lambda_i-\\lambda_j| + \\sum _{1 \\leq i<j \\leq l}|\\mu_i-\\mu_j| -\n\\sum_{1 \\leq i,j \\leq l}(\\lambda_i+\\mu_j+2)<\\\\\n< \\sum_{1 \\leq i<j \\leq l}(\\lambda_i+\\lambda_j) + \\sum _{1 \\leq\ni<j \\leq l}(\\mu_i+\\mu_j) - \\sum_{1 \\leq i,j \\leq\nl}(\\lambda_i+\\mu_j)=-\\sum_{i=1}^l (\\lambda_i+\\mu_i) \\leq 0.\n\\end{multline*}\n\n\\end{proof}\n\n\\subsection{The pairs $(GL(V),U(V)), \\quad (U(V_1 \\oplus V_2), U(V_1) \\times U(V_2)) \\text{ and } (U(V_D),O(V))$\n} \\label{U_reg}\n$ $\\\\\n\nIn this subsection prove that the pairs $(GL(V),U(V))$ and\n$(U(V_D),O(V))$ are weakly linearly tame and the pair $(U(V_1\n\\oplus V_2), U(V_1) \\times U(V_2))$ is regular.\\\\\n\nLet $V$ be a hermitian space. Note that $(GL(V),U(V))$ is a form\nof $(GL(W) \\times GL(W), \\Delta GL(W))$ for some $W$ and $(U(V\n\\oplus V), U(V) \\times U(V))$ is a form of $(GL(W \\oplus W),\nGL(W)) \\times GL(W))$ for some $W$.\n\nAlso, for any quadratic space $V$ of dimension at least 2 and any quadratic extension\n$D\/F$, the pair $(U(V_D),O(V))$ is a form of $(GL(W),O(W))$ for\nsome quadratic space $W$.\n\nHence by Lemma \\ref{NegDefAlg} and Theorems \\ref{2RegPairs},\n\\ref{RJR} and \\ref{GL_O} those 3 pairs are of negative normalized defect and\nhence are weakly linearly tame.\nIf $\\dim V \\leq 1$ then the pair $(U(V_D),O(V))$ is obviously linearly tame.\n\nIf $V_1$ and $V_2$ are non-isomorphic hermitian spaces then\n$(U(V_1 \\oplus V_2), U(V_1) \\times U(V_2))$ is regular by\nProposition \\ref{TrivReg}.\n\n\\section{Computation of descendants} \\label{CompDes}\nIn this section we compute the descendants of the pairs we\ndiscussed before. For this we use a technique of computing\ncentralizers of semisimple elements of orthogonal and unitary\ngroups, which is described in \\cite{SpSt}. The proofs in this\nsection are rather straightforward but technically involved. The\nmost important things in this section are the formulations of the\nmain theorems: Theorems \\ref{CompDesGL_O}, \\ref{CompDesGL_U},\n\\ref{CompDesU_O}, \\ref{CompDesO_OtO}, \\ref{CompDesU_UtU}. Those\ntheorems are summarized graphically in subsection \\ref{SecDiag}.\n\n\\subsection{Preliminaries and notation for orthogonal and unitary groups}\n\n\\subsubsection{Orthogonal group}\n\\begin{notation}\nLet $V$ be a linear space over $F$. Let $x\\in GL(V)$ be a\nsemi-simple element and let $Q = \\sum_{i=0}^{n} a_i \\xi^i \\in\nF[\\xi]$ (where $a_n \\neq 0$) be an irreducible polynomial.\n\\itemize{\n\\item Denote $F_Q:= F[\\xi]\/Q$\n\\item Denote $inv(Q):= \\sum_{i=0}^n a_{n-i}\\xi^i$\n\\item Denote $V_{Q,x}^{0}:=Ker(Q(x)) \\text{ and }\nV_{Q,x}^{1}:=Ker(inv(Q)(x))$\\\\\nWe define an $F_Q$-linear space structure on $V_{Q,x}^{i}$ by\nletting $\\xi$ act on $V_{Q,x}^{0}$ by $x$ and on $V_{Q,x}^{1}$ by\n$x^{-1}$. We will consider $V_{Q,x}^{i}$ as linear spaces over\n$F_Q$.\n\n\\item In case $Q$ is proportional to $inv(Q)$ we define an involution $\\mu$ on $F_Q$\nby $\\mu(P(\\xi)):=P(\\xi^{-1})$ .\n\\item For a linear space $W$ over $F_Q$ we can consider its dual\nspace $W^*$ over $F_Q$ and the dual space of $W$ over $F$ which we\ndenote by $W_F^*$. The space $W_F^*$ has a canonical structure of\na linear space over $F_Q$. The spaces $W_F^*$ and $W^*$ can be\nidentified as linear spaces over $F_Q$. For this identification\none has to choose an $F$-linear functional $\\lambda:F_Q \\to F$. We\nwill fix such functional $\\lambda$ such that\n$\\lambda(\\mu(d))=\\lambda(d)$ if $\\mu$ is defined.\\\\ From now on we\nwill identify $W_F^*$ and $W^*$. }\n\\end{notation}\n\nThe following two lemmas are straightforward.\n\\begin{lemma}\nLet $V$ be a quadratic space over $F$. Let $x\\in GL(V)$\n and let $P,Q \\in F[\\xi]$ be irreducible\npolynomials. Suppose that either \\\\\n(i) $x=x^t$ and $P$ is not proportional to $Q$ or\\\\\n(ii) $x \\in O(V)$ and $P$ is not proportional to $inv(Q)$\\\\\nThen $Ker(Q(x))$ is orthogonal to $Ker(P(x))$.\n\\end{lemma}\n\n\\begin{lemma}\nLet $(V,B)$ be a quadratic space over $F$. Let $x\\in GL(V)$ be a\nsemi-simple element and let $Q \\in F[\\xi]$ be an\nirreducible polynomial. Then\\\\\n(i) If $x=x^t$ then $B$ defines an $F_Q$-linear isomorphism\n$V_{Q,x}^i \\cong\n(V_{Q,x}^i)^*$.\\\\\n(ii) If $x\\in O(V)$ then $B$ defines an $F_Q$-linear isomorphism\n$V_{Q,x}^i \\cong (V_{Q,x}^{1-i})^*$.\n\\end{lemma}\n\n\n\\subsubsection{Unitary group}\n$ $\\\\\nFrom now and till the end of the paper we fix a quadratic\nextension $D$ of $F$ and denote by $\\tau$ the involution that\nfixes $F$.\n\n\\begin{notation}\nLet $V$ be a hermitian space over $(D,\\tau)$. Let $x\\in GL(V)$ be\na semi-simple element and let $Q = \\sum_{i=0}^{n} a_i \\xi^i \\in\nD[\\xi]$ (where $a_n \\neq 0$) be an irreducible polynomial.\n\\itemize{\n\\item Denote $D_Q:= D[\\xi]\/Q$\\\\\n\\item Denote $$inv(Q):= \\sum_{i=0}^n a_{n-i}\\xi^i, \\quad\nQ^*:=\\tau(inv(Q))$$\n\\item Denote $$V_{Q,x}^{00}:=Ker(Q(x)), \\quad\nV_{Q,x}^{01}:=Ker(Q^*(x)) , \\quad V_{Q,x}^{10}:=Ker(inv(Q)(x)) ,\n\\quad V_{Q,x}^{11}:=Ker(\\tau(Q)(x)).$$\nWe twist the action of $D$ on $V_{Q,x}^{i1}$ by $\\tau$. We define\n$D_Q$-linear space structure on $V_{Q,x}^{ij}$ by letting $\\xi$\nact on $V_{Q,x}^{0j}$ by $x$ and on $V_{Q,x}^{1j}$ by $x^{-1}$. We\nwill consider $V_{Q,x}^{ij}$ as linear spaces over $D_Q$.\n\n\\item If $Q$ is proportional to $Q^*$ we define an involution $\\mu^{01}$ on $D_Q$ by\n$\\mu^{01}(P(\\xi)):=\\tau(P)(\\xi^{-1})$.\\\\\nIf $Q$ is proportional to $inv(Q)$ we define an involution\n$\\mu^{10}$ on $D_Q$ by\n$\\mu^{10}(P(\\xi)):=P(\\xi^{-1})$.\\\\\nIf $Q$ is proportional to $\\tau(Q)$ we define an involution\n$\\mu^{11}$ on $D_Q$ by $\\mu^{11}(P(\\xi)):=\\tau(P)(\\xi)$.\n\n\\item For a linear space $W$ over $D_Q$ we can consider its dual\nspace $W^*$ over $D_Q$ and the dual space of $W$ over $D$ which we\ndenote by $W_D^*$. The space $W_D^*$ has a canonical structure of\na linear space over $D_Q$. The spaces $W_D^*$ and $W^*$ can be\nidentified as linear spaces over $D_Q$. For this identification\none has to choose a $D$-linear functional $\\lambda:D_Q \\to D$. We\nwill fix such functional $\\lambda$ such that\n$$ \\lambda(\\mu^{ij}(d))=\\tau^{j}(\\lambda(d)) \\quad \\text{ if } \\mu^{ij}\n\\text{ is defined}.$\nFrom now on we will identify $W_D^*$ and $W^*$. }\n\\end{notation}\n\nThe following two lemmas are straightforward.\n\\begin{lemma}\nLet $V$ be a hermitian space over $(D,\\tau)$. Let $x\\in GL(V)$ and\nlet $P,Q \\in D[\\xi]$ be irreducible\npolynomials. Suppose that either \\\\\n(i) $x=x^*$ and $P$ is not proportional to $\\tau(Q)$ or\\\\\n(ii) $x \\in U(V)$ and $P$ is not proportional to $Q^*$\\\\\nThen $Ker(Q(x))$ is orthogonal to $Ker(P(x))$.\n\\end{lemma}\n\n\\begin{lemma}\nLet $(V,B)$ be a hermitian space over $(D,\\tau)$. Let $x\\in GL(V)$\nbe a semi-simple element and let $Q \\in D[\\xi]$ be an\nirreducible polynomial. Then\\\\\n(i) If $x=x^*$ then $B$ defines a $D_Q$-linear isomorphism\n$V_{Q,x}^{ij} \\cong\n(V_{Q,x}^{1-i,1-j})^*$.\\\\\n(ii) If $x\\in U(V)$ then $B$ defines a $D_Q$-linear isomorphism\n$V_{Q,x}^{ij} \\cong (V_{Q,x}^{i,1-j})^*$.\n\\end{lemma}\n\\subsection{The pair $(GL(V),O(V))$}\n\\begin{theorem} \\label{CompDesGL_O}\nLet $V$ be a quadratic space over $F$. Then all the descendants of\nthe pair $(GL(V),O(V))$ are products of pairs of the type\n$(GL(W),O(W))$ for some quadratic space $W$ over some field $F'$\nthat extends $F$.\n\\end{theorem}\n\\begin{proof}\nNote that in this case the anti-involution $\\sigma$ is given by\n$\\sigma(x)=x^t$. Let $x\\in GL(V)^{\\sigma} $ be a semi-simple\nelement. Let $P$ be the minimal polynomial of $x$. We will now\ndiscuss a special case and then deduce the general case from\nit.\\\\\\\\\nCase 1. $P$ is irreducible over $F$.\\\\\nClearly $GL(V)_x \\cong GL(V_{P,x}^0)$. The isomorphism $V_{P,x}^0\n\\cong (V_{P,x}^0)^*$ gives a quadratic structure on $V_{P,x}^0$.\nNow\n $O(V)_x \\cong O(V_{P,x}^0)$.\\\\\\\\\nCase 2. General case\\\\\nLet $P=\\prod_{i \\in I}P_i$ be the decomposition of $P$ to\nirreducible polynomials. Clearly $V = \\bigoplus V_{P_i,x}^0$ and\n$V_{P_i,x}^0$ are orthogonal to each other. Hence the pair\n$(GL(V)_x,O(V)_x)$\n is a product of pairs from Case 1.\n\\end{proof}\n\n\\subsection{The pair $(GL(V),U(V))$}\n\n\\begin{theorem} \\label{CompDesGL_U}\nLet $(V,B)$ be a hermitian space over $(D,\\tau)$. Then all the\ndescendants\nof the pair $(GL(V),U(V))$ are products of pairs of the types\\\\\n(i) $(GL(W) \\times GL(W), \\Delta GL(W))$ for some linear space $W$\nover some field $D'$ that extends $D$\\\\\n(ii)  $(GL(W),U(W))$ for some hermitian space $W$ over some\n$(D',\\tau')$ that extends $(D,\\tau)$.\n\\end{theorem}\n\\begin{proof}\nNote that in this case the anti-involution $\\sigma$ is given by\n$\\sigma(x)=x^*$. Let $x\\in GL(V)^{\\sigma} $ be a semi-simple\nelement. Let $P$ be the minimal polynomial of $x$. Note that\n$\\tau(P)$ is proportional to $P$. We will now discuss 2 special\ncases and then deduce the general case from\nthem.\\\\\\\\\nCase 1. $P=Q\\tau(Q)$ where $Q$ is irreducible over $D$.\\\\\nClearly $GL(V)_x \\cong GL(V_{Q,x}^{00}) \\times GL(V_{Q,x}^{11})$.\nRecall that $B$ gives a non-degenerate pairing between\n$V_{Q,x}^{00}$ and $V_{Q,x}^{11}$, and the spaces $V_{Q,x}^{ii}$\nare isotropic. Therefore\n$$GL(V_{Q,x}^{00})\\cong GL(V_{Q,x}^{11}), \\quad GL(V)_x \\cong\nGL(V_{Q,x}^{00})^2 \\text{ and } U(V)_x \\cong \\Delta GL(V_{Q,x}^{00}) < GL(V_{Q,x}^{00})^2. $$\\\\\nCase 2. $P$ is irreducible over $D$.\\\\\nClearly $GL(V)_x \\cong GL(V_{P,x}^{00})$ and $V_{P,x}^{00}$ is\nidentical to $V_{P,x}^{11}$ as $F$-linear spaces but the actions\nof $D_P$ differ by a twist by $\\mu^{11}$. Hence the isomorphism\n$V_{P,x}^{00} \\cong (V_{P,x}^{11})^*$ gives a hermitian structure\non $V_{P,x}^{00}$ over $(D_P,\\mu^{11})$. Now\n$U(V)_x \\cong U(V_{P,x}^{00}) < GL(V_{P,x}^{00})$.\\\\\\\\\nCase 3. General case\\\\\nLet $P=\\prod_{i \\in I}P_i$ be the decomposition of $P$ to\nirreducible polynomials. Then $\\tau(P_i)$ is proportional to\n$P_{s(i)}$ where $s$ is some permutation of $I$ of order $\\leq$ 2.\nLet $I = \\bigsqcup I_{\\alpha}$ be the decomposition of $I$ to\norbits of $s$. Denote $V_{\\alpha}:=Ker (\\prod _{i \\in \\alpha}\nP_i(x))$. Clearly $V = \\bigoplus V_{\\alpha}$ and $V_{\\alpha}$ are\northogonal to each other. Hence the pair $(GL(V)_x,U(V)_x)$ is a\nproduct of pairs from the first 2 cases.\n\\end{proof}\n\n\\subsection{The pair $(U(V_D),O(V))$}\n\\begin{theorem} \\label{CompDesU_O}\nLet $(V,B)$ be a quadratic space over $F$. Let $V_D:=V \\otimes_F\nD$ be its extension of scalars with the corresponding hermitian\nstructure.\n\nThen all the descendants of the pair $(U(V_D),O(V))$ are products\nof pairs of the types\\\\\n(i) $(GL(W),O(W))$ for some quadratic space $W$ over some field\n$F'$ that extends $F$.\\\\\n(ii) $(U(W_{D'}),O(W))$ for some extension $(D',\\tau')$ of\n$(D,\\tau)$ and some quadratic space $W$ over $D'^{\\tau'}$.\n\\end{theorem}\n\nFor the proof of this theorem we will need the following notation\nand lemma.\n\n\\begin{notation}\nLet $(V,B)$ be a quadratic space over $F$. The involution $\\tau$\ndefines an involution $\\widetilde{\\tau}$ on $V_D$. The form $B$\ndefines a quadratic form $B_D$ on $V_D$ and a hermitian form\n$B_D^{\\tau}$ on $V_D$.\n\\end{notation}\n\n\n\\begin{lemma}\nLet $(V,B)$ be a quadratic space over $F$. Let $P$ be an\nirreducible polynomial. Let $x \\in U(V_D)$ be a semi-simple\nelement such that $x = x^t$ (where $x^t$ is defined by $B_D$).\nThen the involution $\\widetilde{\\tau}$ gives a $D_{P}$-linear\nisomorphism $V_{P,x}^{ij} \\cong V_{P,x}^{i,1-j}$.\n\\end{lemma}\n\\begin{proof}\nWe will show that $\\widetilde{\\tau}$ maps $V_{P,x}^{00}$ to\n$V_{P,x}^{11}$, and the other cases are done similarly. Let $v \\in\nV_{P,x}^{ij}$. We have\n$$ P^*(x)(\\widetilde{\\tau}(v))=\n\\widetilde{\\tau}(inv(P)(x^*)(v))=\n\\widetilde{\\tau}(inv(P)(x^{-1})(v))=\\widetilde{\\tau}(x^{-degP}P(x)(v))=0.$$\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{CompDesU_O}]\nNote that in this case the anti-involution $\\sigma$ is given by\n$\\sigma(x)=x^t$. Let $x\\in U(V_D)^{\\sigma} $ be a semi-simple\nelement. Let $P$ be the minimal polynomial of $x$. Then $P$ is\nproportional to $P^*$. We will now discuss 2 special cases and\nthen deduce the general case from\nthem.\\\\\\\\\nCase 1. $P=QQ^*$ where $Q$ is irreducible over $D$.\\\\\nClearly $GL(V)_x \\cong GL(V_{Q,x}^{00}) \\times GL(V_{Q,x}^{01})$.\nRecall that $B_D^{\\tau}$ gives a non-degenerate pairing between\n$V_{Q,x}^{00}$ and $V_{Q,x}^{01}$, and the spaces $V_{Q,x}^{0i}$\nare isotropic. Therefore $$GL(V_{Q,x}^{00})\\cong GL(V_{Q,x}^{01}),\n\\quad GL(V)_x \\cong GL(V_{Q,x}^{00})^2, \\quad  U(V)_x \\cong \\Delta\nGL(V_{Q,x}^{00}) < GL(V_{Q,x}^{00})^2$$\nCompose the isomorphism $V_{Q,x}^{00} \\cong V_{Q,x}^{01}$ given by\n$\\widetilde{\\tau}$ with the isomorphism $V_{Q,x}^{01} \\cong\n(V_{Q,x}^{00})^*$ given by $B_D^{\\tau}$. This gives a quadratic\nstructure on $V_{Q,x}^{00}$. Now\n$$O(V)_x \\cong\n\\Delta O(V_{Q,x}^{00}) < \\Delta GL(V_{Q,x}^{00}). $$\\\\\nCase 2. $P$ is irreducible over $D$.\\\\\nClearly $GL(V)_x \\cong GL(V_{P,x}^{00})$ and $V_{P,x}^{00}$ is\nidentical to $V_{P,x}^{01}$ as $F$-linear spaces but the actions\nof $D_P$ on them differ by a twist by $\\mu^{01}$. Hence the\nisomorphism $V_{P,x}^{00} \\cong (V_{P,x}^{01})^*$ given by\n$B_D^{\\tau}$ gives a hermitian structure on $V_{P,x}^{00}$ over\n$(D_P,\\mu^{01})$ and the isomorphism $V_{P,x}^{00} \\cong\nV_{P,x}^{01}$ given by $\\widetilde{\\tau}$ gives an antilinear\ninvolution of $V_{P,x}^{00}$. Now\n$$U(V)_x \\cong U(V_{P,x}^{00}) < GL(V_{P,x}^{00}) \\text{ and }O(V)_x \\cong O(V_{P,x}^{00}) < U(V_{P,x}^{00})\n.$$\\\\\nCase 3. General case\\\\\nLet $P=\\prod_{i \\in I}P_i$ be the decomposition of $P$ to\nirreducible polynomials. Then $P_i^*$ is proportional to\n$P_{s(i)}$ where $s$ is some permutation of $I$ of order $\\leq$ 2.\nLet $I = \\bigsqcup I_{\\alpha}$ be the decomposition of $I$ to\norbits of $s$. Denote $V_{\\alpha}:=Ker (\\prod _{i \\in \\alpha}\nP_i(x))$. Clearly $V_D = \\bigoplus V_{\\alpha}$, $V_{\\alpha}$ are\northogonal to each other and each $V_{\\alpha}$ is invariant with\nrespect to $\\widetilde{\\tau}$. Hence the pair $(GL(V)_x,U(V)_x)$\nis a product of pairs from the first 2 cases.\n\\end{proof}\n\n\\subsection{The pair $(O(V_1 \\oplus V_2), O(V_1) \\times O(V_2))$}\n\n\\begin{theorem} \\label{CompDesO_OtO}\nLet $(V,B)$ be a quadratic space over $F$.\\\\\nLet $\\varepsilon \\in O(V)$ be an element of order 2. Then all the\ndescendants of the pair $(O(V),O(V)_{\\varepsilon})$ are products of pairs\nof the types\\\\\n(i) $(GL(W),O(W))$ for some quadratic space $W$ over some field\n$F'$ that extends $F$\\\\\n(ii) $(U(W_E) , O(W))$ for some quadratic space $W$ over some\nfield $F'$ that extends $F$, and some quadratic extension $E$ of\n$F'$.\\\\\n(iii) $(O(W),O(W)_{\\varepsilon})$ for some quadratic space $W$ over $F$.\n\\end{theorem}\n\nFor the proof of this theorem we will need the following\nstraightforward lemma.\n\n\\begin{lemma}\nLet $(V,B)$ be a quadratic space over $F$.\\\\\nLet $\\varepsilon \\in O(V)$ be an element of order 2. Let $x \\in O(V)$\nsuch that $\\varepsilon x \\varepsilon = x^{-1}$. Let $Q$ be an irreducible\npolynomial. Then $\\varepsilon$ gives an $F_Q$- linear isomorphism\n$V_{Q,x}^i \\cong V_{Q,x}^{1-i}$.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Theorem \\ref{CompDesO_OtO}]\nNote that the involution $\\sigma$ on $O(V)$ is given by $x \\mapsto\n\\varepsilon x^{-1}\\varepsilon$. Let $x\\in O(V)^{\\sigma}$ be a semi-simple\nelement and let $P$ be its minimal polynomial.\n\nNote that the minimal polynomial of $x^{-1}$ is $inv(P)$ and hence\n$P$ is proportional to $inv(P)$. We will now discuss 3 special\ncases and then deduce the general case from\nthem.\\\\\\\\\nCase 1. $P=Qinv(Q)$, where $Q$ is an irreducible polynomial.\\\\\nNote that $GL(V)_x \\cong \\prod _{i} GL(V_{Q,x}^{i})$.\n\nSince $B$ defines a non-degenerate pairing $V_{Q,x}^{0} \\cong\n(V_{Q,x}^{1})^*$, and $V_{Q,x}^{i}$ are isotropic, we have\n$$O(V)_x \\cong \\Delta GL(V_{Q,x}^{0}) < GL(V_{Q,x}^{0})^2.$$\nNow, compose the isomorphism $V_{Q,x}^{i} \\cong V_{Q,x}^{1-i}$\ngiven by $\\varepsilon$ with the isomorphism $V_{Q,x}^{1-i} \\cong\n(V_{Q,x}^{i})^*$. This gives a quadratic structure on\n$V_{Q,x}^{0}$. Clearly, $\\varepsilon$ gives an isomorphism $V_{Q,x}^{0}\n\\cong V_{Q,x}^{1}$ as quadratic spaces and hence\n$$(O(V)_{\\varepsilon})_x \\cong  \\Delta O(V_{Q,x}^{0}) < \\Delta\nGL(V_{Q,x}^{0}).$$\\\\\nCase 2. $P$ is irreducible and $x \\neq x^{-1}$\\\\\nIn this case $GL(V)_x \\cong GL(V_{P,x}^{0})$. Also, $V_{P,x}^{0}$\nand $V_{P,x}^{1}$ are identical as $F$-vector spaces but the\naction of $F_P$ on them differs by a twist by $\\mu$. Therefore the\nisomorphism $V_{P,x}^{0} \\cong (V_{P,x}^{1})^*$ gives a hermitian\nstructure on $V_{P,x}^{0}$ over $(F_P,\\mu)$ and $\\varepsilon$ gives an\n$(F_P,\\mu)$-antilinear automorphism of $V_{P,x}^{0}$. Now\n$$O(V)_x \\cong U(V_{P,x}^{0}).$$\nDenote $W:= (V_{P,x}^{0})^{\\varepsilon}$. It is a linear space over\n$(F_P)^{\\mu}$. It has a quadratic structure. Now\n$$(O(V)_{\\varepsilon})_x \\cong O(W) < U(V_{P,x}^{0}).$$\\\\\nCase 3. $P$ is irreducible and $x = x^{-1}$.\\\\\nAgain, $GL(V)_x \\cong GL(V_{P,x}^{0})$. However, in this case\n$F_P=F$ and $V_{P,x}^{0}=V$. Also $O(V)_x \\cong O(V_{P,x}^{0}).$\nNow, $\\varepsilon$ commutes with $x$ and hence $\\varepsilon \\in O(V)_x \\cong\nO(V_{P,x}^{0})$. Hence\n$$(O(V)_{\\varepsilon})_x \\cong (O(V_{P,x}^{0}))_{\\varepsilon} < O(V_{P,x}^{0}).$$\\\\\nCase 4. General case\\\\\nLet $P = \\prod_{i \\in I} P_i$ be the decomposition of $P$ to\nirreducible multiples. Since $P$ is proportional to $inv(P)$,\nevery $P_i$ is proportional to $P_{s(i)}$ where $s$ is some\npermutation of $I$ of order $\\leq 2$.\n\nLet $I = \\bigsqcup I_{\\alpha}$ be the decomposition of $I$ to\norbits of $s$. Denote $V_{\\alpha}:=Ker(\\prod_{i \\in \\alpha}\nP_i(x))$. Clearly $V = \\bigoplus V_{\\alpha}$ and $V_{\\alpha}$ are\northogonal to each other and $\\varepsilon$-invariant. Hence the pair\n$(O(V)_x,(O(V)_{\\varepsilon})_x)$ is a product of pairs from the first 3\ncases.\n\n\\end{proof}\n\n\\subsection{The pair $(U(V_1 \\oplus V_2), U(V_1) \\times U(V_2))$}\n\n\n\\begin{theorem} \\label{CompDesU_UtU}\nLet $(V,B)$ be a hermitian space over $(D,\\tau)$.\\\\\nLet $\\varepsilon \\in U(V)$ be an element of order 2. Then all the\ndescendants of the pair $(U(V),U(V)_{\\varepsilon})$ are products of pairs\nof the types\\\\\n(i) $(GL(W) \\times GL(W), \\Delta GL(W))$ for some linear space $W$\nover some field $D'$ that extends $D$\\\\\n(ii) $(U(W) \\times U(W), \\Delta U(W))$ for some hermitian space\n$W$ over some extension $(D',\\tau')$ of $(D,\\tau)$\\\\\n(iii) $(GL(W),U(W))$ for some hermitian space $W$ over some\nextension $(D',\\tau')$ of $(D,\\tau)$\\\\\n(iv) $(GL(W_{D'}), GL(W))$ where $F'$ is a field extension of $D$,\n$D'\/F'$ is a quadratic extension, $W$ is a linear space over $F'$\nand $W_{D'}:= W \\otimes_{F'} D'$ is its extension of scalars to\n$D'$ \\\\\n(v) $(GL(W),GL(W)_{\\varepsilon})$ where $W$ is a linear space over $D$\nand $\\varepsilon \\in GL(W)$ is an element of order $\\leq 2$.\\\\\n(vi) $(U(W_E) , U(W))$ where $W$ is a hermitian space over some\nextension $(D',\\tau')$ of $(D,\\tau)$, $(E,\\tau'')$ is some\nquadratic extension of $(D',\\tau')$ and $W_E=W \\otimes_{D'}E$ is\nan extension of scalars with the corresponding hermitian\nstructure.\\\\\n(vii) $(U(W),U(W)_{\\varepsilon})$ where $W$ is a hermitian space over\n$(D,\\tau)$.\n\\end{theorem}\n\nFor the proof of this theorem we will need the following\nstraightforward lemma.\n\n\\begin{lemma}\nLet $(V,B)$ be a hermitian space over $(D,\\tau)$.\\\\\nLet $\\varepsilon \\in U(V)$ be an element of order 2. Let $x \\in U(V)$\nsuch that $\\varepsilon x \\varepsilon = x^{-1}$. Let $Q$ be an irreducible\npolynomial. Then $\\varepsilon$ gives an $D_Q$- linear isomorphism\n$V_{Q,x}^{ij} \\cong V_{Q,x}^{1-i,j}$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{CompDesU_UtU}]\nLet $x\\in U(V)^{\\sigma}$ be a semi-simple element and let $P$ be\nits minimal polynomial.\n\nNote that the minimal polynomial of $x^*$ is $P^*$ and hence $P^*$\nis proportional to $P$. Since $x\\in U(V)^{\\sigma}$, we have\n$x^{-1} = \\varepsilon x \\varepsilon$ and hence its minimal polynomial is $P$.\nHence $P$ is proportional to $inv(P)$. We will now discuss 7\nspecial cases and then deduce the general case from\nthem.\\\\\\\\\nCase 1. $P=QQ^*inv(Q)\\tau(Q)$, where $Q$ is an irreducible\npolynomial.\\\\\nNote that $GL(V)_x \\cong \\prod _{ij} GL(V_{Q,x}^{ij}) \\cong\nGL(V_{Q,x}^{00})^4 $. This identifies $U(V)_x$ with a diagonal\n$\\Delta GL(V_{Q,x}^{00})^2 < GL(V_{Q,x}^{00})^4$ and\n$(U(V)_{\\varepsilon})_x$ with a diagonal $\\Delta\nGL(V_{Q,x}^{00}) < GL(V_{Q,x}^{00})^4$. \\\\\\\\\nCase 2. $P=Qinv(Q)$, where $Q$ is an irreducible polynomial and\n$Q^*=Q$.\\\\\nNote that $GL(V)_x \\cong \\prod _{i} GL(V_{Q,x}^{i0}) \\cong\nGL(V_{Q,x}^{00})^2$. Note also that in this case $V_{Q,x}^{i0}$\nand $V_{Q,x}^{i1}$ are identical as sets and $F$-vector spaces but\nthe actions of $D_Q$ on them differ by a twist by $\\mu^{01}$. Now\nthe isomorphism $V_{Q,x}^{i0} \\cong (V_{Q,x}^{i1})^*$ gives a\n$(D_Q, \\mu^{01})$-hermitian structure on $V_{Q,x}^{i0}$.\nTherefore, $U(V)_x \\cong U(V_{Q,x}^{00})\\times U(V_{Q,x}^{10})$.\nNote that $\\varepsilon$ gives an isomorphism of $(D_Q,\n\\mu^{01})$-hermitian spaces between $V_{Q,x}^{00}$ and\n$V_{Q,x}^{01}$. Hence $$U(V)_x \\cong U(V_{Q,x}^{00})^2 \\text{ and\n}\n(U(V)_{\\varepsilon})_x \\cong \\Delta U(V_{Q,x}^{00}) < U(V_{Q,x}^{00})^2.$$\\\\\nCase 3. $P=Qinv(Q)$, where $Q$ is an irreducible polynomial and\n$Q^* = inv(Q)$.\\\\\nNote that $GL(V)_x \\cong \\prod _{i} GL(V_{Q,x}^{i0}) \\cong \\prod\n_{j} GL(V_{Q,x}^{0j})$.\n\nSince $B$ defines a non-degenerate pairing $V_{Q,x}^{00} \\cong\n(V_{Q,x}^{01})^*$ and $V_{Q,x}^{0i}$ are isotropic, we have\n$$U(V)_x \\cong \\Delta GL(V_{Q,x}^{00}) < (GL(V_{Q,x}^{00}))^2.$$\nNote that in this case $V_{Q,x}^{ij}$ and $V_{Q,x}^{1-i,1-j}$ are\nidentical as sets and as $F$-vector spaces but the action of $D_Q$\non them differs by a twist by $\\mu^{11}$.\n\nNow, compose the isomorphism $V_{Q,x}^{00} \\cong V_{Q,x}^{10}$\ngiven by $\\varepsilon$ with the isomorphism $V_{Q,x}^{10} \\cong\n(V_{Q,x}^{11})^*$. This gives a $(D_Q,\\mu^{11})$ unitary structure\non $V_{Q,x}^{00}$. Similarly we get a unitary structure on\n$V_{Q,x}^{10}$. Finally, $\\varepsilon$ gives an isomorphism $V_{Q,x}^{00}\n\\cong V_{Q,x}^{10}$ as unitary spaces and hence\n$$(U(V)_{\\varepsilon})_x \\cong  \\Delta U(V_{Q,x}^{00}) < \\Delta\nGL(V_{Q,x}^{00}).$$\\\\\nCase 4. $P=QQ^{*}$, where $Q$ is an irreducible polynomial, $Q=\ninv(Q)$ and $x \\neq x^{-1}$.\\\\\nNote that $GL(V)_x \\cong \\prod _{j} GL(V_{Q,x}^{0j})$ and as\nbefore\n$$U(V)_x \\cong  \\Delta GL(V_{Q,x}^{00}) < (GL(V_{Q,x}^{00}))^2.$$\nIn this case $V_{Q,x}^{0j}$ and $V_{Q,x}^{1j}$ are identical as\nsets and as $F$-vector spaces but the action of $D_Q$ on them\ndiffers by a twist by $\\mu^{10}$. Hence $\\varepsilon$ gives a\n$(D_Q,\\mu^{10})$ anti-linear automorphism of $V_{Q,x}^{0j}$. Let\n$W_j:=(V_{Q,x}^{0j})^{\\varepsilon}$. This is a linear space over\n$(D_Q)^{\\mu^{10}}$. Therefore,\n$$(U(V)_{\\varepsilon})_x \\cong \\Delta GL(W_0) <  \\Delta GL(V_{Q,x}^{00}).$$\\\\\nCase 5. $P=QQ^{*}$, where $Q$ is an irreducible polynomial, $Q=\ninv(Q)$ and $x = x^{-1}$.\\\\\nAs in the previous case, $$GL(V)_x \\cong \\prod _{j}\nGL(V_{Q,x}^{0j}) \\text{ and } U(V)_x \\cong  \\Delta\nGL(V_{Q,x}^{00}) < (GL(V_{Q,x}^{00}))^2.$$\nIn this case $D_Q = D$ and $\\mu^{10}$ is trivial. Hence\n$V_{Q,x}^{0j}$ and $V_{Q,x}^{1j}$ are identical as $D_Q$-linear\nspaces.\n\nAlso, $\\varepsilon$ gives a $D_Q$-linear automorphism of $V_{Q,x}^{0j}$.\nSo we can interpret $\\varepsilon$ as an element in $GL(V_{Q,x}^{00})$.\nTherefore,\n$$(U(V)_{\\varepsilon})_x \\cong \\Delta (GL(V_{Q,x}^{00}))_{\\varepsilon} <  \\Delta\nGL(V_{Q,x}^{00}).$$\\\\\nCase 6. $P$ is irreducible and $x \\neq x^{-1}$\\\\\nIn this case $GL(V)_x \\cong GL(V_{P,x}^{00})$. Also,\n$V_{P,x}^{00}$ and $V_{P,x}^{01}$ are identical as $F$-vector\nspaces but the action of $D_P$ on them differs by a twist by\n$\\mu^{01}$. Again, the isomorphism $V_{P,x}^{00} \\cong\n(V_{P,x}^{01})^*$ gives a $(D_P,\\mu^{01})$ hermitian structure on\n$V_{P,x}^{00}$ and\n$$U(V)_x \\cong U(V_{P,x}^{00}).$$\nNote that $V_{P,x}^{00}$ and $V_{P,x}^{10}$ are identical as\n$F$-vector spaces but the action of $D_P$ on them differs by a\ntwist by $\\mu^{10}$. Hence, $\\varepsilon$ gives a $(D_P,\\mu^{10})$\nanti-linear automorphism of $V_{P,x}^{00}$. Denote $W:=\n(V_{P,x}^{00})^{\\varepsilon}$. It is a linear space over\n$(D_P)^{\\mu^{10}}$. It has a\n$((D_P)^{\\mu^{10}},\\mu^{01}|_{(D_P)^{\\mu^{10}}})$ hermitian\nstructure. Now\n$$(U(V)_{\\varepsilon})_x \\cong U(W) < U(V_{P,x}^{00}).$$\\\\\nCase 7. $P$ is irreducible and $x = x^{-1}$.\\\\\nAgain, $$GL(V)_x \\cong GL(V_{P,x}^{00}) \\text{ and } U(V)_x \\cong\nU(V_{P,x}^{00}).$$ In this case $D_P=D$ and $\\mu^{01}=\\tau$. Also,\n$\\varepsilon$ commutes with $x$ and hence $\\varepsilon \\in U(V)_x \\cong\nU(V_{P,x}^{00})$. Hence\n$$(U(V)_{\\varepsilon})_x \\cong U(V_{P,x}^{00})_{\\varepsilon} < U(V_{P,x}^{00}).$$\\\\\nCase 8. General case\\\\\nLet $P = \\prod_{i \\in I} P_i$ be the decomposition of $P$ to\nirreducible multiples. Since $P$ is proportional to $inv(P)$,\nevery $P_i$ is proportional to $P_{s_1(i)}$ for some permutation\n$s_1$ of $I$ of order $\\leq 2$. Since $P$ is proportional to\n$P^*$, every $P_i$  is proportional to some $P_{s_2(i)}$. This\ngives rise to an action of ${\\mathbb Z} \/ 2{\\mathbb Z} \\times {\\mathbb Z} \/ 2{\\mathbb Z}$ on $I$.\n\nLet $I = \\bigsqcup I_{\\alpha}$ be the decomposition of $I$ to\norbits of this action. Denote $V_{\\alpha}:=Ker(\\prod_{i \\in\n\\alpha} P_i(x))$. Clearly $V = \\bigoplus V_{\\alpha}$ and\n$V_{\\alpha}$ are orthogonal to each other and $\\varepsilon$-invariant.\nHence the pair $(U(V)_x,(U(V)_{\\varepsilon})_x)$ is a product of pairs\nfrom the first 7 cases.\n\\end{proof}\n\n\\subsection{Genealogical trees of the symmetric pairs considered in this paper} \\label{SecDiag}\n$ $\\\\\nThe following diagrams sum up the results of this section.\n\nAn arrow $\"(G_1,H_1) \\to (G_2,H_2)\"$ means that pairs of type\n$(G_1,H_1)$ may have descendants with factor of the type\n$(G_2,H_2)$. We will not draw the obvious arrows $\"(G,H) \\to\n(G,H)\"$ and when we draw $\"(G_1,H_1) \\to (G_2,H_2) \\to (G_3,H_3)\"$\nwe mean also $\"(G_1,H_1) \\to (G_3,H_3)\"$.\n\n{\\small\n$$\\framebox{\\parbox{470pt}{\n\n\\xymatrix{\n & \\framebox{\\parbox{80pt}{$ \\quad \\,\\, (U(V),U(V)_{\\varepsilon})$}}\\ar@{->}[d]\\ar@{->}[dl]\\ar@{->}[ddr]\\ar@{->}[ddrr] & &\n\\\\\n \\framebox{\\parbox{64pt}{$(GL(V),U(V))$}}\\ar@{->}[d] & \\framebox{\\parbox{80pt}{$\\,\\,\\,\\, (GL(V),GL(V)_{\\varepsilon})$}}\\ar@{->}[d]\\ar@{->}[dl] & &\\\\\n\\framebox{\\parbox{115pt}{$(GL(V)\\times GL(V), \\Delta GL(V))$}} &\n\\framebox{\\parbox{80pt}{$(GL(V)_{E\/F},GL(V))$}} &\n\\framebox{\\parbox{67pt}{$(U(V)_{E\/F}, U(V))$}} &\n\\framebox{\\parbox{92pt} {$(U(V) \\times U(V), \\Delta U(V))$}}}\n$ $\\\\\n\\text{Here }V\\text{ is a linear or hermitian space over some\nfinite field extension of }F \\text{ and }E \\text{ is} \\text{some\nquadratic extension of } F. }}$$\n\\framebox{\\parbox{475pt}{\n$$ \\xymatrix{ \\framebox{\\parbox{63pt}{$(O(V),O(V)_{\\varepsilon})$}}\\ar@{->}[d]\\\\\n \\framebox{\\parbox{63pt}{$(U(V_E),O(V))$}}\\ar@{->}[d]\\\\\n\\framebox{\\parbox{63pt}{$(GL(V), O(V))$}}}$$\n$ $\\\\\n\\text{Here }V\\text{ is a quadratic space over some finite field\nextension of }F \\text{ and }E \\text{ is} \\text{some quadratic\nextension of } F. }}}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBimanual gestures are central to everyday life, and constitute a fundamental ground for the study of basic principles of human behaviour.\nTraditionally, the study of bimanual gestures in humans focus on very simple motions involving fingers and hands, and including coordination, symmetry, in-phase and anti-phase behaviours.\nThese studies are aimed at understanding the dynamics associated with bimanual movements and target aspects of motor control, as exemplified by the Haken-Kelso-Bunz (HKB) model for self-organisation in human movement pattern formation \\cite{Kelso1984}\\cite{Hakenetal1985}.\n\nIn this paper, we are interested in determining how the study of bimanual gestures can lead to automated systems for their detection and classification in unconstrained, everyday environments.\nIn the context of assistive systems for fragile populations, including elderly, people with disabilities and other people with special needs, the need arises to provide caregivers, medical staff or simply relatives with a tool able to assess the ability of assisted people to perform bimanual gestures in their \\textit{natural} environment. \nSuch approach is in line with the \\textit{ageing in place} paradigm, a recent healthcare position which acknowledges and focuses on the role that a person's surroundings (the home, the neighbourhood) play for his well-being in older ages \\cite{Wiles11}. \nA familiar environment brings about a sense of security, independence and autonomy which has a positive impact on routines and activities, and ultimately on quality of life.\n\nThere is a big gap between clinical studies involving the coordination of finger movements and the recognition of such general-purpose bimanual gestures as \\textit{opening and closing curtains}, \\textit{sweeping the floor}, or \\textit{filling a cup with water}.  \nHowever, the first step to take is to determine how current understanding of bimanual movements and their representation in the brain can lead to better detection and classification systems in real-world environments.\nThree factors must be considered when designing such a system.\n\n\\textit{Factor 1. It is debated whether bimanual gestures are controlled in intrinsic or extrinsic coordinates, or rather multiple coordination strategies co-occur}. \n\nBimanual movements tend to motion symmetry and stabilisation \\cite{Kelso1984}.\nThis has been typically explained using co-activation of homologous muscles in neuronal motor structures, due to bilateral cross-talk, suggesting that bimanual coordination is mostly done using intrinsic (i.e., proprioceptive) coordinates.\nMechsner \\textit{et al} suggest that, instead, such a coordination is due to spatial, perceptual symmetry only, i.e., using extrinsic (i.e., exteroceptive) visual coordinates \\cite{Mechsneretal2001}. \nIf this were true, there would be no need to map visual representations to motor representations (and \\textit{viceversa}), and voluntary movements could be organised on the basis of perceptual goals.\nThe role of different coordinates and their interplay for bimanual coordination mechanisms has been studied by Sakurada and colleagues \\cite{Sakuradaetal2015}.\nStarting from studies relating temporal and spatial couplings in bilateral motions, including the adaptation exhibited by two hands having to perform motions of different speed (i.e., the fastest becoming slightly slower and viceversa) \\cite{Heueretal1998}, and the fact that the movement of a non-dominant hand is likely to be assimilated by the dominant one \\cite{Byblowetal2000}, they demonstrate a relatively stronger contribution of intrinsic components in bimanual coordination, although both components are flexibly regulated according to the specific bimanual task.\nFurthermore, they argue that the central nervous system regulates bilateral coordination at different levels, as hypothesised by Swinnen and Wenderoth \\cite{SwinnenandWenderoth2004}.\nThe importance of both intrinsic and extrinsic coordinates seems to be confirmed by recent studies in interpersonal coordination \\cite{Kodamaetal2015}.\nIt is suggested that coordinated motion is informed by a full perception-action coupling, including visual and haptic sensorimotor loops, which propagates to the neuromuscular system.\n\nWe derive two requirements for our analysis:\n\\begin{itemize}\n\\item[$R_1$)] we must consider models \\textit{agnostic} with respect to an explicit coordination at the motor level between the two hands\/arms;\n\\item[$R_2$)] classification techniques must be robust to variation in speed, both for the bimanual gesture as a whole and for the single hand\/arm. \n\\end{itemize} \n\n\\textit{Factor 2. Ageing affects the way we move, and therefore coordination in bimanual gestures varies over time}.\n\nAccording to the \\textit{dedifferentiation} paradigm, ageing is now considered a parallel and distributed process occurring at various levels in the human's body.\nThe dedifferentiation can be defined as ``the process by which structures, mechanisms of behaviour that were specialised for a given function lose their specialisation and become simplified, less distinct or common to different functions'' \\cite{BaltesandLindenberger1997}.\nAs a consequence, ageing affects not only individual body \\textit{subsystems} (i.e., the muscular system or the brain), but also their interactions.\nIt is argued by Sleimen-Malkoun and colleagues that such a process can lead to common and intertwined causes for \\textit{cognitive ageing}, i.e., a general slowing down of information processing, including the information related to procedural memory and -- therefore -- movement and coordination \\cite{SleimenMalkounetal2014}.\nIt is posited that the ageing brain undergoes anatomical and physiological changes, for the reorganising activation patterns between neural circuits.\nAs far as motor task complexity is concerned, a generalised increased activation of brain areas is even more evident, which reflects a greater involvement of processes related to executive control.\n\nAlso in this case, we derive an important requirement:\n\\begin{itemize}\n\\item[$R_3$)] we must consider models which can be adapted over time and which follow the evolution of the musculoskeletal system, at least implicitly, thus requiring the use of forms of machine learning techniques.\n\\end{itemize}  \n\n\\textit{Factor 3. Different mental representations of sensorimotor loops and action, involving discrete and continuous organisation principles, are under debate}.\n\nBeside the models aimed at representing bimanual gestures assuming a motor control framework, much work has been carried out in the past few years to devise building blocks for mental action representation \\cite{Kelso1984}\\cite{Hakenetal1985}\\cite{Mechsneretal2001}\\cite{SwinnenandWenderoth2004}\\cite{Sakuradaetal2015}\\cite{Kodamaetal2015}.\nAssuming a goal-directed cognitive perspective, it has been shown how movements can be represented as a serial and functional combination of goal-related body postures, or goal postures (i.e., key frames), as well as their transitional states.\nFurthermore, it has been posited that movements can be expressed as incremental changes between goal postures, which reduces the amount of effortful attention needed for their execution \\cite{Rosenbaumetal2007}.\nOn these premises, Basic Action Concepts (BACs) have been proposed as building blocks of mental action representations.\nBACs represent chucked body postures related to common functions to realise goal-directed actions.\nSchack and colleagues posit that complex (including bimanual) actions are mentally represented as a combination of executed action and intended or observed effects \\cite{Schacketal2014}.\nFurthermore, they argue that the map linking motion and perceptual effects is bi-directional and stored hierarchically in long-term memory, in structures resembling \\textit{dendograms} \\cite{Schack2012}.\nThis is a specific case of what has been defined by Bernstein as the \\textit{degrees of freedom problem} \\cite{Bernstein1967}.\nThe problem is related to how the various parts of the motor system can become harnessed so as to generate coordinated behaviour when needed.\nAs Bernstein theorised, a key role is played by muscular-articular links (i.e., synergies) in constraining how many degrees of freedom lead to dexterous behaviour.\nHarrison and Stergiou argue that dexterity and motion robustness are enabled by multi-functional (degenerate) body parts able to assume context-dependent roles.\nAs a consequence, task-specific human-environment interactions can flexibly generate adaptable motor solutions.\n\nWe derive two requirements:\n\\begin{itemize}\n\\item[$R_4$)] although motion models are intrinsically continuous, we need to derive a discrete representation able to provide action labels which, in principle, can lead to more complex organisations;\n\\item[$R_5$)] models must capture dexterity in everyday environments and robustness to different executions of the same gesture, which leads to models obtained by human demonstration.\n\\end{itemize}\n\nOn the basis of these requirements, we propose a bimanual wearable system able to detect and classify bimanual gestures using the inertial information provided by two wrist-mounted smartwatches.\nThe system builds up and significantly extends previous work \\cite{Bruno12}, and adheres to the \\textit{wearable sensing} paradigm, which envisions the use of sensors located on a person's body, either with wearable devices such as smartwatches, or with purposely engineered articles of clothing \\cite{Lara13}, to determine a number of important parameters, in our case motion.\nSince sensors are physically carried around, the monitoring activity can virtually occur in any place and it is usually focused on the detection of movements and gestures.\n\nThe contribution is four-fold: (i) an analysis of two procedures for modelling bimanual gestures, respectively explicitly and implicitly taking the correlation between the two hands\/arms into account; (ii) an analysis of two procedures for the classification of run-time data, respectively relying on the probability measure and the Mahalanobis distance to compute the similarity between run-time data and previously stored models of bimanual gestures; (iii) a performance assessment of the developed techniques with the standard statistical metrics of accuracy, precision and recall over the collected dataset, as well as under real-life conditions; (iv) a dataset of $60$ recordings of five bimanual gestures, performed by ten volunteers, to support reproducible research.\n\nThe paper is structured as follows.\nSection \\ref{sec:related_work} describes the theoretical background of the proposed modelling and recognition procedures, as well as related work, in view of the requirements outlined above.\nSection \\ref{sec:system_architecture} provides a thorough description of the system's architecture and insights on the five bimanual gestures considered for the analysis; the performance of such system are presented and discussed in Section \\ref{sec:experimental_evaluation}.\nConclusions follow.\n\n\\section{Related Work}\n\\label{sec:related_work}\n\nWearable systems for the automatic recognition of human gestures and full-body movements typically rely on inertial information.\nAccelerometers prove to be the most informative sensor for the task \\cite{Lester05}.\nTo comply with end users' constraints related to the impact of the monitoring system on their appearance and freedom of motion, most solutions adopt a single sensing device, either located at the waist \\cite{Mathie04} or, as it is becoming more and more common, at the wrist \\cite{Dietrich14}.\n\nDue to the similarities in the input data and in the operating conditions, most systems adopt a similar architecture \\cite{Lara13}, sketched in Figure \\ref{fig:system_architecture}.\nThe architecture identifies two stages, namely a training phase (on the left hand side in the Figure) and a testing phase (on the right hand side).\nThe \\textit{training} phase, typically executed off-line without strict computational constraints, is devoted to the creation of a compact representation of a gesture\/movement of interest, on the basis of an informative set of examples.\nThis also complies with requirements $R_3$ and $R_5$ above.\nThe \\textit{testing} phase, which may be subject to real-time and computational constraints, is responsible for the analysis of an input data stream to detect the gesture, among the modelled ones, which more closely matches it, if any, and label it accordingly.\nPlease note that the word ``testing'' is used here with respect to the data stream to analyse, with no reference to the stage of development of the monitoring system.\nSpecifically, we denote with the term ``validation'' the development stage in which we assess the performance of the system, and with the term ``deployment'' the stage in which the system is actually adopted by end users.\nDuring validation, the testing phase executes an off-line analysis of labelled gesture recordings, while during deployment the testing phase executes an on-line analysis of a continuous data stream, in unsupervised conditions.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=11cm]{generic_architecture}\n\\caption{The typical architecture of wearable sensing systems for the recognition of human gestures. The left hand side lists the tasks of the training phase, while the right hand side lists the tasks executed during the testing phase.}\n\\label{fig:system_architecture}\n\\end{figure}\n\nDuring the training phase (see Figure \\ref{fig:system_architecture} on the left hand side), it is first necessary to acquire and build the training set of measured attributes for the gestures of interest.\nTwo approaches are possible.\nThe \\textit{specialised} approach envisions the creation of a training set exclusively composed of gesture recordings performed by the person to be monitored during the deployment stage.\nThis approach maximises the recognition accuracy at the expenses of a long setup for each new installation.\nHowever, it enforces requirements $R_3$ and $R_5$, in that it allows someone to periodically retrain the system if necessary.\nConversely, the \\textit{generalised} approach envisions the creation of a training set composed of a large number of gesture recordings, performed by a number of volunteers (not necessarily including the person to monitor).\nUsing gestures provided by different individuals maximises the likelihood that the model is able to capture a much varied dexterity, as posited by $R_5$. \nThis approach, albeit more prone to errors, greatly reduces the setup costs and is to be preferred in the case of Ambient Assisted Living applications, in which the perceived system's ease of use is crucial for its success \\cite{Bruno12, Bulling14}.\n\nOnce the training set is available, it is typically filtered for noise reduction and\/or formatted for later processing (\\textit{data pre-processing} stage).\nThen, the purpose of the \\textit{feature extraction} procedure is to determine relevant information (in the form of features) from raw signals.\nFeatures are expected to (i) enhance the differences between gestures while being invariant when extracted from data patterns corresponding to the same gesture, (ii) lead to a compact representation of the gesture and (iii) require limited computational time and resources for the extraction and processing, since these operations are subject to real-time constraints during deployment.\nLiterature discriminates between \\textit{statistical} features, extracted using methods such as the Fourier and the Wavelet transform on the basis of quantitative characteristics of the data, and \\textit{structural} features, aiming at enhancing the interrelationship among data \\cite{Lara13}.\nMost human activity recognition systems based on inertial measurements make use of statistical features, usually defined in the time- or frequency-domain \\cite{Krassnig10}.\nAlternative feature extraction methods include the use of Principal Component Analysis \\cite{Mashita12} and autoregressive models \\cite{Lee11}.\nWith respect to time-domain features exclusively, which minimise the computational load introduced by the feature extraction procedure, gravity and body acceleration are among the most commonly adopted \\cite{Karantonis06, Krassnig10, Bruno12}.\nDiscriminating between gravity and body acceleration is a non-trivial operation, for the two features overlap both in the time and frequency domains.\nTwo approaches are typically adopted to separate them.\nThe former exploits additional sensors, such as gyroscopes \\cite{Chul09} or magnetometers \\cite{Bonnet07}, to compute the orientation or attitude of another body part, usually the torso.\nThe latter exploits the known features of gravity and uses either low-pass filters to isolate the gravitational component \\cite{Karantonis06, Bruno12}, or high-pass filters to isolate the body acceleration components \\cite{Sharma08}.\n\nOnce gravity and body acceleration components are isolated, the need arises to model them as features.\nSix different 1-dimensional sampled signals are available, i.e., the three $g_x, g_y, g_z$ gravity and the three $b_x, b_y, b_z$ body acceleration components along the $x$, $y$ and $z$ axes.\nAgain, two possibilities are discussed in the Literature.\nThe first is to assume the signals to be pairwise uncorrelated, which yields six separate 2-dimensional features, each feature being composed of timing information and the corresponding signal value on a given axis, i.e., $(t, g_i)$ and $(t, b_i)$, where $i \\in \\{x, y, z\\}$.\nThe second is to assume the $x$, $y$ and $z$ components of gravity and body acceleration to be correlated, which yields two separate 4-dimensional features, i.e., $(t, g_x, g_y, g_z)$, $(t, b_x, b_y, b_z)$, each feature being composed of timing information and the corresponding signal values on all axes.\nThe explicit use of correlation among tri-axial acceleration data has been proved to lead to better results in terms of classification rate and computational time \\cite{Cho08, Krassnig10, Bruno12}.\nIt is noteworthy that, in case of bimanual gestures, it is possible to explicitly model the correlation among inertial data originating from the two hands\/arms, or to consider them as separate signals.\nIn this way, we can comply with requirements $R_1$ and (in part) $R_2$ above. \n\nFinally, the \\textit{modelling} procedure is devoted to the creation of a compact and informative representation of the considered gestures in terms of available sensory data.\nTwo classes of approaches have been traditionally pursued.\nIn \\textit{logic-based} solutions each gesture to monitor and recognise is encoded through sound and well-defined rules, which are based on ranges of admissible values for a set of relevant parameters.\nRecognition is carried out by analysing run-time sensory values to progressively converge towards the encoded gesture more closely matching run-time data.\nDecision trees, which allow for a fast and simple classification procedure, are the most adopted solution in logic-based approaches \\cite{Lee02, Mathie04, Karantonis06, Krassnig10}.\n\\textit{Probability-based} solutions assume instead each gesture to be represented by a model encoding relevant moments of the training set, and to be identified using non ambiguous labels.\nIn this case, recognition is typically performed by comparing run-time sensory data with the stored models through probabilistic distance measures.\nCommonly adopted techniques include Neural Networks \\cite{Krassnig10}, Hidden Markov Models \\cite{Minnen05, OlguinOlguin06, Choudhury08} and Gaussian Mixture Models \\cite{Bruno12}.\nIn our work, we exploit probability-based models to comply with requirement $R_4$.\n\nDuring the testing phase (see Figure \\ref{fig:system_architecture} on the right hand side), analogously to what happens in the training phase, a number of steps are sequentially executed.\nThe feature extraction step executes the same algorithms of the training phase. Once the testing stream has been processed (typically focusing on a time window), it is possible to evaluate its features against the previously trained models (\\textit{recognition}), generating a predicted label.\nIn the testing phase, we exploit specific distance metrics relating the stored models with the run-time data stream. \nIn this way, we can account for requirement $R_2$.\n\nMost wearable sensing systems based on a single sensing point (e.g., the right wrist) focus on the recognition of gestures which are either one-handed, e.g., \\textit{bringing a cigarette to the lips to smoke} \\cite{Dietrich14}, or, albeit involving both hands, such as \\textit{cutting meat with fork and knife} \\cite{Bruno13} or even the full-body, e.g., \\textit{climbing stairs} \\cite{Bruno12}, correspond to a unique and generalised pattern at the considered sensing point.\nThe presented work relaxes this assumption, by evaluating a wearable sensing system based on two sensing points (the left and right wrists) which allows for the modelling and recognition of \\textit{generic} bimanual gestures.\n\n\\section{System Architecture}\n\\label{sec:system_architecture}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10.8cm]{modelling_architecture}\n\\caption{System architecture (training phase).}\n\\label{fig:modelling_architecture}\n\\end{figure}\nFigure \\ref{fig:modelling_architecture} shows a schematic view of the training phase of the proposed system, while Figure \\ref{fig:testing_architecture} details the operations performed during the testing phase.\nThe blocks devoted to \\textit{data synchronisation} and \\textit{data pre-processing}, as well as the \\textit{feature extraction} block, are the same in the two phases.\nWe consider two approaches for the modelling stage: (i) \\textit{explicit} modelling of the correlation of the two hands ($2\\times7D$ approach, see Figure \\ref{fig:modelling_architecture} on the left hand side), presupposing the stress on intrinsic coordinates in motor control studies, and (ii) \\textit{implicit} modelling of the correlation ($4\\times4D$ approach, see Figure \\ref{fig:modelling_architecture} on the right hand side), which assumes the importance of extrinsic coordinates.\nWe also consider two approaches for the comparison of testing data with the available models, respectively based on the probability measure and the Mahalanobis distance.\nThe former takes into account the likelihood of a model \\textit{as a whole}, whereas the second weights more the contribution of body degeneracy in robustness and dexterity.\n\n\\subsection{Pre-processing \\& Feature extraction}\n\nThe proposed system relies on the inertial information collected by two tri-axial accelerometers, respectively located at a person's left and right wrists.\nTo properly manage the two data streams, they should be synchronised, and the \\textit{data synchronisation} procedure heavily depends on the choices made related to hardware solutions.\nAs it will be described in Section \\ref{sec:experimental_evaluation}, in the case of the devices we adopt, the synchronisation is guaranteed by the manufacturer.\n\nAll the trials of a gesture in the training set (i.e., all the couples of left- and right-wrist data streams associated with a single execution of the gesture) are initially synchronised with each other manually, so that the starting moment of the gesture is the same across all recordings, and trimmed to have equal length.\nThen, the \\textit{data pre-processing} stage filters each acceleration stream with a median filter of size $3$ to reduce noise.\n\nLet us assume that we have $M$ different bimanual gestures.\nFor each gesture $m$ to learn, where $m=1, \\ldots, M$, let us assume that the training set includes $S_m$ trials and $s$ is one of them. After the pre-processing stage, all the $S_m$ trials are synchronised and truncated as to be composed of the same number $K_m$ of observations.\nA trial is defined as:\n\\begin{equation}\ns = \\{g_{l,k}, b_{l,k}, g_{r,k}, b_{r,k}\\} \\quad k = 1, \\ldots, K_m\n\\label{eq:4Dtrial}\n\\end{equation}\nwith:\n\\begin{equation}\n\\begin{split}\ng_{l,k}&=(t_k, g_{l,x,k}, g_{l,y,k}, g_{l,z,k}), \\\\\nb_{l,k}&=(t_k, b_{l,x,k}, b_{l,y,k}, b_{l,z,k}), \\\\\ng_{r,k}&=(t_k, g_{r,x,k}, g_{r,y,k}, g_{r,z,k}), \\\\\nb_{r,k}&=(t_k, b_{r,x,k}, b_{r,y,k}, b_{r,z,k}), \\\\\n\\end{split}\n\\label{eq:4Dfeatures}\n\\end{equation}\nwhere $l$ and $r$ denote, respectively, the acceleration streams provided by the sensing device on the left and on the right wrist, $g_{l,k}$ includes the time and $x, y$ and $z$ components of the gravity on the left acceleration stream, $b_{l,k}$ includes the time and $x, y$ and $z$ components of the body acceleration on the left acceleration stream, $g_{r,k}$ includes the time and $x, y$ and $z$ components of the gravity on the right acceleration stream and $b_{r,k}$ includes the time and $x, y$ and $z$ components of the body acceleration on the right acceleration stream.\nThe \\textit{feature extraction} stage separates the $l$ and $r$ tri-axial acceleration streams provided by the sensing devices into their gravity and body acceleration components \\cite{Bruno12}, by applying a low pass Chebyshev I $5^{\\circ}$ order filter ($F_{cut}=0.25 {Hz}, A_{pass}= 0.001 {dB}, A_{stop}= -100 {dB},F_{stop}= 2 {Hz}$).\n\n\\subsection{Modelling}\n\nThe goal of the \\textit{modelling} stage is to combine the $S_m$ trials in the training set to obtain a generalised version, i.e., a \\textit{model}, of gesture $m$, defined in terms of the two features of gravity and body acceleration.\nTwo approaches are possible.\n\nIn the \\textit{explicit} correlation modelling ($2\\times7D$ approach, see Figure \\ref{fig:modelling_architecture} on the left hand side), we merge the left and right components of each trial $s$ to create $7$-dimensional features, i.e.:\n\\begin{equation}\ns = \\{G_{k}, B_{k}\\} \\quad k = 1, \\ldots, K_m\n\\label{eq:7Dtrial}\n\\end{equation}\nwith:\n\\begin{equation}\n\\begin{split}\nG_{k} &= (t_k, g_{l,x,k}, g_{l,y,k}, g_{l,z,k}, g_{r,x,k}, g_{r,y,k}, g_{r,z,k}), \\\\\nB_{k} &= (t_k, b_{l,x,k}, b_{l,y,k}, b_{l,z,k}, b_{r,x,k}, b_{r,y,k}, b_{r,z,k}), \\\\\n\\end{split}\n\\label{eq:7Dfeatures}\n\\end{equation}\nand the model of gesture $m$ is then defined in terms of $G$ and $B$.\n\nConversely, in the \\textit{implicit} correlation modelling ($4\\times4D$ approach, see Figure \\ref{fig:modelling_architecture} on the right hand side), we keep the left and right hand streams independent, thus considering the four features defined in \\eqref{eq:4Dfeatures}.\nThe model of gesture $m$ is then defined in terms of $g_l, b_l, g_r$ and $b_r$.\n\nThe first approach corresponds to assuming that the motion of the two hands is \\textit{fully constrained} by the performed gesture, while the latter approach leaves to later stages the responsibility of correlating the two data streams.\nAt the same time, in the first case we assume the contribution of the left and right hands\/arms as correlated, whereas in the second case we do not pose such an assumption.\nAlbeit introducing an additional step, the possibility of tuning the correlation of the two data streams opens interesting scenarios for the recognition stage.\nConsider the gesture of rotating a tap's handle with the left hand, which can occur in a number of situations (for example, when filling a glass with tap water, or when washing a dish): by varying the importance given to this hand we can either have a more flexible system, which is able to recognise many situations in light of the common traits in the left hand stream, or a more specialised one, which is focused on one situation only and is able to filter out all the others in light of the differences in the right hand stream.\n\nThe modelling procedure in itself is the same for both approaches and, in particular, it relies on Gaussian Mixture Modelling (GMM) and Gaussian Mixture Regression (GMR) for the retrieval of the \\textit{expected} curve and covariance matrix of each considered feature, on the basis of a given training set. The procedure has been first introduced in the field of Human-Robot Interaction \\citep{Calinon10} and later used for the purposes of Human Activity Recognition with a wrist-placed inertial device for a single arm \\cite{Bruno12}. We point to the references for its thorough description.\n\nWe denote with $f$ the generic feature of interest, i.e., $f$ can either correspond to gravity $G$ or body acceleration $B$ in the $2\\times7D$ approach, or to one among $g_l, b_l, g_r$ and $b_r$ in the $4\\times4D$ approach.\nWe assume the following definitions.\n\\begin{itemize}\n\\item $f_{s,k}\\in\\mathbb{R}^n$ is the data point $k$ of feature $f$ of trial $s$, defined as:\n\\begin{equation}\nf_{s,k} = \\{f_{t,k}, f_{a,s,k}\\},\n\\label{eq:xiks}\n\\end{equation}\nwhere $f_{t,k}$ stores the time information and $f_{a,s,k}$ includes the acceleration components.\nThe dimension $n$ depends on the modelling approach.\n\\item $F^f\\in\\mathbb{R}^{n\\times O_m}$, with $O_m = S_mK_m$, is the ordered set of data points generating the feature curve $f$ for all the $S_m$ trials, defined as:\n\\begin{equation}\nF^f = \\{f_1, \\ldots, f_k, \\ldots, f_{O_m}\\},\n\\label{eq:Xixi}\n\\end{equation}\nwhere\n\\begin{equation}\nf_k = \\{f_{t,k}, f_{a,k}\\}\n\\label{eq:xik}\n\\end{equation}\nis a generic data point taken from the whole training set, i.e., by hiding the information about the trial $s$ to which it belongs.\n\\end{itemize}\n\nThe purpose of GMM+GMR is to build the expected version of all features of gesture $m$, i.e.:\n\\begin{equation}\n\\hat{F}^{f} = \\{\\hat{f}_1, \\ldots, \\hat{f}_k, \\ldots, \\hat{f}_{K_m}\\}, \n\\label{eq:hatXifeat}\n\\end{equation}\nwith:\n\\begin{equation}\n\\hat{f}_k = \\{f_{t,k},\\hat{f}_{a,k},\\hat{\\Sigma}_{a,k}\\}, \n\\label{eq:hatxir}\n\\end{equation}\nwhere $\\hat{f}_{a,k}$ is the conditional expectation of $f_{a,k}$ given $f_{t,k}$ and $\\hat{\\Sigma}_{a,k}$ is the conditional covariance of $f_{a,k}$ given $f_{t,k}$.\nThe model $\\hat{F}^m$ of gesture $m$ is then defined as the set of the feature models $\\hat{F}^{f}$.\nPlease note that the number of data points in the expected curve may not be necessarily the same as the number $K_m$ of data points in the trials of the training set.\nThe equality is imposed here for the clarity of the description.\n\n\\begin{figure}\n\\centering\n\\subfigure[Left hand]\n{\\includegraphics[width=12.1cm]{WO_left4x4HD}}\n\\subfigure[Right hand]\n{\\includegraphics[width=12.1cm]{WO_right4x4HD}}\n\\caption{Model of the gesture \\textit{open a wardrobe} in the implicit correlation approach ($4\\times4D$), which is defined in terms of: (i) left hand gravity component (a)-left, (ii) left hand body acceleration component (a)-right, (iii) right hand gravity component (b)-left, (iv) right hand body acceleration component (b)-right.}\n\\label{fig:WO_4times4D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[$l_x$, $l_y$, $l_z$ axes]\n{\\includegraphics[width=10.4cm]{GRAPH_OpenCloseCurtains_2x7_1}}\n\\subfigure[$r_x$, $r_y$, $r_z$ axes]\n{\\includegraphics[width=10.4cm]{GRAPH_OpenCloseCurtains_2x7_2}}\n\\caption{Model of the gesture \\textit{open and close curtains} in the explicit correlation approach ($2\\times7D$), which is defined in terms of: (i) gravity component (a,b)-left, (ii) body acceleration component (a,b)-right.}\n\\label{fig:OCC_2times7D}\n\\end{figure}\n\nFigure \\ref{fig:WO_4times4D} shows the $2D$ projections of the model of the gesture \\textit{open a wardrobe}, computed from the full dataset of $60$ recordings with the $4\\times4D$ approach.\nThe four modelled features are $g_l$ (a)-left, $b_l$ (a)-right, $g_r$ (b)-left, $b_r$ (b)-right.\nFigure \\ref{fig:OCC_2times7D} shows the $2D$ projections of the model of the gesture \\textit{open and close curtains}, computed from the full dataset of $60$ recordings with the $2\\times7D$ approach.\nThe two modelled features are $G$ (a,b)-left and $B$ (a,b)-right.\nIn both cases, the solid red line represents the projection of the expected curve $\\hat{f}_a$ on one time-acceleration space, while the pink area surrounding it represents the conditional covariance $\\hat{\\Sigma}_a$.\n\nThe modelling procedure requires in input the number of Gaussian functions to use, which varies both with the gestures and with the features.\nThe \\textit{modelling parameters estimation} stage implements a procedure based on the k-means clustering algorithm and the silhouette clustering quality metric \\cite{Rousseeuw87} for the estimation of the number of Gaussian functions to use and their initialisation \\cite{Bruno12}.\nOther choices are equally legitimate.\nWe again point to the references for the details of the procedure.\n\n\\subsection{Comparison}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=11.5cm]{testing_architecture}\n\\caption{System architecture (testing phase).}\n\\label{fig:testing_architecture}\n\\end{figure}\nAs it is shown in Figure \\ref{fig:testing_architecture}, the testing phase executes the same procedures for data synchronisation, data pre-processing (noise reduction) and feature extraction of the training phase.\nThen, in accordance with the chosen modelling approach, the features are either expressed in the form of \\eqref{eq:4Dfeatures} or \\eqref{eq:7Dfeatures}.\n\nThe recognition procedure is composed of a comparison stage, devoted to ranking the similarity between the testing data and each previously learned model, and a classification stage, responsible for the final labelling of the testing data on the basis of comparison results. \n\nWe propose two \\textit{comparison} procedures.\nThe \\textit{distance}-based comparison computes the Mahalanobis distance between the testing features and each model features, while the \\textit{probability}-based comparison computes the likelihood of the occurrence of the testing features given each model features.\nIn both cases, the \\textit{classification} stage identifies the model with most prominent distance\/probability, if any, and labels the testing data accordingly.\nBoth comparison techniques are applied for the two considered modelling approaches, thus yielding four combinations of training and testing procedures.\nThe comparison of their performance is the topic of the experimental evaluation presented in Section \\ref{sec:experimental_evaluation}.\n\nLet us consider a moving horizon window of length $K_w$ on the testing streams and let us denote with $F^w$ the set of features $F^{w,f}$ extracted from the window in accordance with the chosen modelling approach.\n\nIn the distance-based approach, on the basis of the $M$ available models, we compute $M$ distances $d(\\hat{F}^m,F^w)$.\nThe Mahalanobis distance is a probabilistic distance measure used to compute the similarity between sets of random variables whose means and variances are known \\cite{Mahalanobis36}.\nThe Mahalanobis distance $d$ between a generic element $\\hat{f}_k \\in \\hat{F}^{f}$ defined by \\eqref{eq:hatxir} and a generic element $f_{k} \\in F^{w,f}$ defined in accordance with \\eqref{eq:xik}, is computed as:\n\\begin{equation}\nd(\\hat{f}_k,f_{k}) = \\sqrt{(\\hat{f}_{a,k}-f_{a,k})^T (\\hat{\\Sigma}_{a,k})^{-1}(\\hat{f}_{a,k}-f_{a,k})}.\n\\label{eq:dxiMcj}\n\\end{equation}\nThe accumulated distance between $\\hat{F}^f$ and $F^{w,f}$ is computed as:\n\\begin{equation}\nd(\\hat{F}^{f},F^{w,f}) = {\\sum_{k=1}^{K_m}}d(\\hat{f}_k,f_{k}),\n\\label{eq:dphixy}\n\\end{equation}\nthat is, by integrating the distance between each element in the run-time stream and its corresponding element in the model.\nThe overall distance $d(\\hat{F}^m,F^w)$ is computed as a weighted sum of all feature distances.\nIn our experiments we choose the weights to be equal for all features.\nThis approach weights more the precision associated with gesture production, and emphasises dexterity and robustness of bimanual motions.\n\nIn the probability-based approach, the probability of the window feature $F^{w,f}$ to be an occurrence of the model $\\hat{F}^{f}$ is computed as:\n\\begin{equation}\n\\begin{split}\np(\\hat{F}^{f},F^{w,f})&=\\mathcal{N}(f_k,\\hat{f}_{a,k},\\hat{\\Sigma}_{a,k}) \\quad\\quad\\quad \\forall k \\in 1...K_m\\\\\n&=\\frac{1}{\\sqrt{2\\pi^n|\\hat{\\Sigma}_{a,k}|}}e^{-\\frac{1}{2}(f_{a,k}-\\hat{f}_{a,k})^T(\\hat{\\Sigma}_{a,k})^{-1}(f_{a,k}-\\hat{f}_{a,k})}.\n\\end{split}\n\\label{eq:probability}\n\\end{equation}\n\nThe overall probability $p(\\hat{F}^m,F^w)$ is computed as a weighted sum of all feature probabilities, i.e., a \\textit{mixture}.\nWe again choose the weights to be equal for all features.\nWhen we use probabilities, we consider the gesture as a whole, and therefore we account for small variations in the gesture execution speed.\n\n\\section{Experimental Evaluation}\n\\label{sec:experimental_evaluation}\n\n\\subsection{Experimental Setup}\n\nIn all the experiments, we adopt two smartwatches LG G watch R W110 as sensing devices (Android Wear 1.0, CPU Quad-Core 1.2GHz Qualcomm Snapdragon 400, 4GB\/512MB RAM) equipped with a tri-axial accelerometer.\nThe sampling frequency is 40Hz.\nThe smartwatches automatically sync on startup with the smartphone they are paired with.\nBy pairing the two smartwatches with the same smartphone, we ensure that they are synchronised with each other with a precision satisfying the requirements of our application.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=12cm]{curtains}}\n\\caption{Open and close curtains.}\n\\label{fig:OCC}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=12cm]{sweeping}}\n\\caption{Sweep the floor.}\n\\label{fig:Swp}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=12cm]{filling_cup}}\n\\caption{Fill a cup with tap water.}\n\\label{fig:FCoT}\n\\end{figure}\n\nWe consider five bimanual gestures:\n\\begin{itemize}\n\\item \\textit{Open and close curtains} (OCC).\nExtend and retract lateral-sliding curtains by pulling the connecting cords with an alternated up-and-down movement of the hands.\nKeep pulling until the curtain is fully closed or opened (see Figure \\ref{fig:OCC}).\n\\item \\textit{Sweep the floor} (SWP).\nPull a conventional broom from left to right, to sweep the floor.\nThree strokes are required (see Figure \\ref{fig:Swp}).\n\\item \\textit{Fill a cup with tap water} (FCOT).\nWith the right hand, take a cup from the sink and hold it below the tap, while rotating the tap's handle with the left hand to fill the cup with water (see Figure \\ref{fig:FCoT}).\n\\item \\textit{Take a bottle from the fridge} (RFF).\nWith the right hand, open the door of a small fridge, then, with the left one, take a bottle from it.\nLastly, close the fridge door with the right hand (see Figure \\ref{fig:RfF}).\n\\item \\textit{Open a wardrobe} (WO).\nOpen a two-doors small wardrobe moving the two hands concurrently (see Figure \\ref{fig:WO}).\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=12cm]{removing_from_fridge}}\n\\caption{Take a bottle from the fridge.}\n\\label{fig:RfF}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=12cm]{wardrobe_opening}}\n\\caption{Open a wardrobe.}\n\\label{fig:WO}\n\\end{figure}\n\nIntuitively, the gestures \\textit{open and close curtains}, \\textit{sweep the floor} and \\textit{open a wardrobe} fully constrain the movement of the two hands, while the gestures \\textit{fill a cup with tap water} and \\textit{take a bottle from the fridge} allow for more freedom in their execution, as far as synchronisation between the arms is concerned.\nMoreover, the gestures \\textit{sweep the floor} and \\textit{open a wardrobe} imply that the two hands are moved concurrently, while the gestures \\textit{open and close curtains}, \\textit{fill a cup with tap water} and \\textit{take a bottle from the fridge} mostly require the two hands to be moved in sequence.\nLastly, the gestures \\textit{open and close curtains} and \\textit{sweep the floor} are recursive, i.e., composed of a number of repetitions of simpler movements, while the gestures \\textit{open a wardrobe}, \\textit{fill a cup with tap water} and \\textit{take a bottle from the fridge} are non-recursive.\n\nFigures \\ref{fig:OCC}-\\ref{fig:WO} show pictures of an execution of the gesture on the left hand side and the acceleration measured at the two wrists during the execution of the same gesture on the right hand side.\nThe impact of the aforementioned gesture characteristics (constrained\/not constrained, concurrent\/sequential, recursive\/not recursive) on the accelerations measured at the wrist, and therefore on the considered modelling and comparison procedures, is not known: finding it is one of the goals of the experiments we have conducted.\n\nFor each gesture, we collected a dataset of $60$ recordings from $10$ volunteers with age ranging from $22$ to $30$ years old.\nThe volunteers, wearing the smartwatches, have been asked to autonomously start and stop the recordings and, once an experimenter described the gesture, to perform it in a natural way.\nAll repetitions were supervised.\nIn addition to this dataset, we asked a number of volunteers to take some recordings in real-life conditions.\nThey have been asked to clean a room and, amidst the other activities, to perform the five bimanual gestures of interest.\nVolunteers could freely choose the timing and sequence of the gestures, and the choices have been annotated by an experimenter.\n\n\\subsection{Performance Analysis}\n\nWe tested the four combinations of modelling and comparison procedures in terms of the standard statistical measures of accuracy, precision and recall by using k-fold cross validation on the collected dataset.\nFor all gestures, we split the dataset in $6$ groups of $10$ recordings each and iteratively used $5$ groups as training dataset and the remaining group for validation.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=11cm]{confusion_4D_prob_crop.eps}}\n\\caption{Results of k-fold cross validation for the $4\\times4D$ modelling approach and probability-based comparison. The bottom row reports the \\textit{recall} measures, while the rightmost column reports the \\textit{precision} measures. The purple cell reports the overall \\textit{accuracy} of the system.}\n\\label{fig:validation_4Dprob}\n\\end{figure}\n\nThe results of the k-fold cross validation on the four combinations of modelling and comparison procedures are shown in Figures \\ref{fig:validation_4Dprob}-\\ref{fig:validation_7Ddist}.\nIn all figures, the first five rows\/columns refer to the gestures \\textit{open and close curtains} (OCC), \\textit{sweep the floor} (SWP), \\textit{fill a cup with tap water} (FCOT), \\textit{take a bottle from the fridge} (RFF) and \\textit{open a wardrobe} (WO).\nThe yellow row\/column collectively represents all the gestures which are not among the modelled ones (N.A.).\nThe columns denote the true labels of all validation recordings (i.e., the gestures actually performed during each of them), while the rows denote the labels assigned by the recognition system.\nSince, collectively, the validation dataset is composed of $60$ recordings per modelled gesture, a perfect recognition system would show $60$ recordings in the first five green cells (i.e., with the predicted label matching the true label) and none in the red cells (meaning that the recording of a gesture has been classified as an occurrence of another gesture) or in the yellow cells (meaning that the recording of a gesture has been classified as an occurrence of an unknown gesture).\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=11cm]{confusion_4D_dist_crop.eps}}\n\\caption{Results of k-fold cross validation for the $4\\times4D$ modelling approach and distance-based comparison. The bottom row reports the \\textit{recall} measures, while the rightmost column reports the \\textit{precision} measures. The purple cell reports the overall \\textit{accuracy} of the system.}\n\\label{fig:validation_4Ddist}\n\\end{figure}\n\nColumn $1$ of Figure \\ref{fig:validation_4Dprob} reports that, out of $60$ validation recordings actually referring to the gesture \\textit{open and close curtains}, $59$ were correctly labelled as occurrences of that gesture and one was labelled as an occurrence of the gesture \\textit{take a bottle from the fridge}.\nThis analysis allows for computing the \\textit{recall} performance of the system, which corresponds to the ratio between the number of recordings of gesture $m$ correctly labelled as occurrences of gesture $m$ and the overall number of recordings of gesture $m$.\nThe recall values for all gestures are listed in the bottom row of the confusion matrix. \n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=11cm]{confusion_7D_prob_crop.eps}}\n\\caption{Results of k-fold cross validation for the $2\\times7D$ modelling approach and probability-based comparison. The bottom row reports the \\textit{recall} measures, while the rightmost column reports the \\textit{precision} measures. The purple cell reports the overall \\textit{accuracy} of the system.}\n\\label{fig:validation_7Dprob}\n\\end{figure}\n\nA similar analysis of the rows of the confusion matrix allows for assessing the \\textit{precision} performance of the system.\nAs an example, row $2$ of Figure \\ref{fig:validation_4Dprob} reports that, out of $59$ recordings labelled as occurrences of the gesture \\textit{sweep the floor}, $58$ were true recordings of that gesture, while one was a recording of the gesture \\textit{fill a cup with tap water}.\nThe precision metric measures the ratio between number of recordings of gesture $m$ correctly labelled as occurrences of gesture $m$ and the overall number of recordings labelled as occurrences of gesture $m$. The precision values for all gestures are listed in the rightmost column of the confusion matrix.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=11cm]{confusion_7D_dist_crop.eps}}\n\\caption{Results of k-fold cross validation for the $2\\times7D$ modelling approach and distance-based comparison. The bottom row reports the \\textit{recall} measures, while the rightmost column reports the \\textit{precision} measures. The purple cell reports the overall \\textit{accuracy} of the system.}\n\\label{fig:validation_7Ddist}\n\\end{figure}\n\nLastly, the aggregated analysis of the number of correct labels over the total number of recordings, i.e., the \\textit{accuracy} performance of the system, is reported in the purple cell at the bottom-right corner.\nAs an example, the recognition system adopting the implicit correlation modelling approach ($4\\times4D)$ and probability-based comparison, whose confusion matrix in shown in Figure \\ref{fig:validation_4Dprob}, has an overall accuracy of $82\\%$.\n\n\\subsection{Real-life Conditions}\n\n\\begin{table}\n\\centering\n\\caption{Experiment scripts for the real-life tests.}\n\\label{tab:real-life}\n\\begin{tabular}{@{ }c@{ } @{ }c@{ }}\n\\hline\nScenario \\#1 & Scenario \\#2 \\\\\n\\hline \n\\multirow{2}{*}{SWP} & WO \\\\ \n & RFF \\\\ \n\\multirow{2}{*}{SWP} & FCOT \\\\ \n & OCC \\\\ \n\\hline\n\\end{tabular} \n\\end{table}\n\nAs an additional test for assessing the system's performance, we asked two volunteers to wear the smartwatches at the wrists while carrying out cleaning chores of their choice in a room.\nThe first volunteer was instructed to perform twice, amidst the other activities, the gesture \\textit{sweep the floor}, while the second volunteer was instructed to perform each of the other gestures once, as summarised in Table \\ref{tab:real-life}.\nBoth tests were supervised by an experimenter, annotating the time when each gesture of interest was performed.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{t1_measurements_}\n\\caption{Acceleration streams registered during the execution of the real-life scenario 1. Red circles denote the two executions of the gesture \\textit{sweep the floor}.}\n\\label{fig:scenario1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{t2_measurements_}\n\\caption{Acceleration streams registered during the execution of the real-life scenario 2. Red circles denote, from left to right, the execution of the gestures \\textit{open a wardrobe}, \\textit{take a bottle from the fridge}, \\textit{fill a cup with tap water} and \\textit{open and close curtains}.}\n\\label{fig:scenario2}\n\\end{figure}\n\nFigure \\ref{fig:scenario1} shows the accelerations registered at the wrists of the first volunteer, while performing the real-life scenario 1, while Figure \\ref{fig:scenario2} shows the accelerations registered at the wrists of the second volunteer, while performing the real-life scenario 2.\nThe red circles denote all occurrences of the modelled bimanual gestures.\nFor these tests we used the whole dataset of $60$ recordings per gesture previously described as training set for the creation of the models.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{4x4_POSS_PROB_1}\n\\caption{Output of the system for the recording of scenario 1 shown in Figure \\ref{fig:scenario1}. The left hand graph refers to the $4\\times4D$ modelling approach and probability-based comparison, while the right hand side refers to the $4\\times4D$ modelling approach and distance-based comparison.}\n\\label{fig:validation_scenario1_4D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{2x7_POSS_PROB_1}\n\\caption{Output of the system for the recording of scenario 1 shown in Figure \\ref{fig:scenario1}. The left hand graph refers to the $2\\times7D$ modelling approach and probability-based comparison, while the right hand side refers to the $2\\times7D$ modelling approach and distance-based comparison.}\n\\label{fig:validation_scenario1_7D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{4x4_POSS_PROB_2}\n\\caption{Output of the system for the recording of scenario 2 shown in Figure \\ref{fig:scenario2}. The left hand graph refers to the $4\\times4D$ modelling approach and probability-based comparison, while the right hand side refers to the $4\\times4D$ modelling approach and distance-based comparison.}\n\\label{fig:validation_scenario2_4D}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{2x7_POSS_PROB_2}\n\\caption{Output of the system for the recording of scenario 2 shown in Figure \\ref{fig:scenario2}. The left hand graph refers to the $2\\times7D$ modelling approach and probability-based comparison, while the right hand side refers to the $2\\times7D$ modelling approach and distance-based comparison.}\n\\label{fig:validation_scenario2_7D}\n\\end{figure}\n\nFigures \\ref{fig:validation_scenario1_4D}-\\ref{fig:validation_scenario2_7D} show the output of the recognition system, for the first (Figures \\ref{fig:validation_scenario1_4D} and \\ref{fig:validation_scenario1_7D}) and the second scenario (Figures \\ref{fig:validation_scenario2_4D} and \\ref{fig:validation_scenario2_7D}), for the four combinations of modelling and comparison procedures.\nIn all figures, the x-axis denotes time and the y-axis denotes the probability or inverse distance value of the models.\nThe \\textit{TP} box marks the true positive recognitions, i.e., all the occurrences of a modelled gesture which are correctly recognised.\nThe \\textit{FP} box marks the false positive recognitions, i.e., all the occurrences of gestures which have been assigned the wrong label.\n\n\\subsection{Discussion}\n\nAs Figures \\ref{fig:validation_4Dprob}-\\ref{fig:validation_7Ddist} show, the combination allowing for the best recognition performance relies on the \\textit{implicit} correlation modelling approach and the \\textit{probability-based} comparison: this system achieves an overall accuracy of $82\\%$ (see Figure \\ref{fig:validation_4Dprob}) and it retains good recognition performance, especially in terms of precision, for all modelled gestures (the gesture with worst recall is \\textit{open a wardrobe}, with $48.3\\%$, while the gesture with worst precision is \\textit{fill a cup with tap water}, with $70.9\\%$).\nInterestingly enough, the combination resulting in the worst recognition performance ($38.3\\%$ overall accuracy) relies on the explicit correlation modelling approach and the same probability-based comparison (see Figure \\ref{fig:validation_7Dprob}), thus confirming that there is a tight relation between the modelling and comparison procedures.\nThe same modelling approach, with the distance-based comparison procedure, significantly improves its performance ($69.7\\%$ overall accuracy, see Figure \\ref{fig:validation_7Ddist}).\n\nFor all four considered combinations, the gestures \\textit{open and close curtains} and \\textit{sweep the floor} consistently achieve the highest precision and recall values, which seems to suggest that recursive motions (i.e., composed of the repeated execution of simple movements), regardless whether they involve the concurrent or sequential motion of the two arms, are easier to model and recognise.\nIn other words, repeated gestures produce more stable patterns as far as the detection and classification system is concerned.\nConversely, the gestures \\textit{open a wardrobe} and \\textit{take a bottle from the fridge} consistently achieve the lowest precision and recall values, with the latter performing especially poorly with the implicit correlation modelling approach.\nThese results, albeit preliminary, seem to confirm our intuition that the explicit modelling approach is much more sensitive to small differences between run-time streams and the corresponding model than the implicit correlation modelling approach.\n\nIn accordance with the k-fold validation analysis, also in the real-life tests the combinations achieving the best performance are the implicit correlation modelling approach with probability-based comparison and the explicit correlation modelling approach with distance-based comparison, which, on the whole, successfully recognise respectively three and four of the six gestures of interest.\n\nReal-life tests highlight a difference in the pattern of the recognition labels between the probability-based and distance-based comparison procedures.\nAs Figures \\ref{fig:validation_scenario1_4D}-\\ref{fig:validation_scenario2_7D} show, in the second case the labels follow a smoother pattern, which reduces the number of false positive recognition.\nThe adoption of reasoning techniques analysing label patterns to increase the recognition accuracy, has been proved effective in the case of a single stream \\cite{Bruno14b} and may lead to significant improvements also in the case of bimanual gestures.\n\nWhat these results have to say about the way we currently understand bimanual gestures in humans?\nNot surprisingly, the best results we achieve involve implicit correlation with probability-based comparison.\nWhilst with implicit correlation we do not assume the motion of the two hands\/arms to be correlated explicitly, probability-based comparison is more robust with respect to small variations in the phases of the two hands\/arms.\nIf we had assumed the two limbs to be explicitly correlated in motor space, i.e., through intrinsic coordinates, we would have constrained the phases of the two hands\/arms to be perfectly tuned.\nThis is not what happens in practice, as it is exemplified in the experiments by Heuer and colleagues \\cite{Heueretal1998} and Byblow \\textit{et al} \\cite{Byblowetal2000}.\nSince, in real-world gesture execution, perfect synchronisation seldom happens, a better result is obtained -- on average -- when we allow for the maximum flexibility in gesture execution.\nProbability-based measures enforce such flexibility even more, because they do not constraint the distance metric to be tied to a specific time instant of the gesture execution. \n\nFinally, it is noteworthy that the use of Gaussian mixtures allows for determining what parts of the motion are more relevant, for those are characterised by lower amplitudes of the covariance.\nAs a consequence, the covariance matrix associated with each element in the model assures to give a proper weight to the sample itself. \nThis allows to consider the differing importance of the various gesture's phases in the recognition phase, therefore taking into account motion dexterity and robustness. \n\nThe combination of the two methods for modelling and recognition allows us to support the Bernstein's intuition about motion constraints \\cite{Bernstein1967}, i.e., (i) variations in degrees of freedom affecting motion performance are constrained (in our case, by low covariance values in intra-arm correlation of inertial data); (ii) variation in degrees of freedom that \\textit{do not affect} task performance can be unconstrained (again, by larger values in the covariance in intra-arm correlation); (iii) co-variations between gesture-relevant degrees of freedom not impacting on performance are permitted (by considering implicit correlations among the two hands\/arms).  \n\n\n\n\n\n\n\n\t\t\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThis paper discusses design choices involved in the detection and classification of bimanual gestures in unconstrained environments.\nThe assumption we make is the availability of inertial data originating from two distinct sensing points, conveniently located at the wrist, given the availability of such COTS sensors as smartwatches.\nOur models are grounded with respect to Gaussian Mixture Modelling (GMM) and Gaussian Mixture Regression (GMR), which we use as basic modelling procedure.\nStarting from these two positions, we compare different modelling (i.e., explicit and implicit correlations between the two hands\/arms) and classification techniques (i.e., based on distance and probability-related considerations), which are inspired by literature about the representation of bimanual gestures in the brain. \nOur architecture allows for different combinations of modelling and classification techniques. \nFurthermore, it can be extended as a framework to support reproducible research.\n\nExperiments show results related to $5$ generic bimanual activities, which have been selected on the basis of three main parameters: (not) constraining the two hands by a physical tool, (not) requiring a specific sequence of single-hand gestures, being recursive (or not). \nThe best results are obtained when considering an implicit coordination among the two hands\/arms (i.e., the two motions are modelled separately) and using a probability-based distance for classification (i.e., the specific timing characteristics of the trajectories are considered only to a limited extent). \nThis seems to confirm a few insights from the literature related to motor control of bimanual gestures, and opens up a number of interesting research questions for the upcoming future.\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{ Nomenclature}\n\\input{Introduction}\n\\input{SecondOrderDSMCTheory}\n\\input{CaseStudies2}\n\\input{Conclusion}\n\\small\n\\section*{ACKNOWLEDGMENT}\n This material is based upon the work supported by the National Science Foundation under Grant No. 1434273. Dr. Ken Butts from Toyota Motor Engineering $\\&$ Manufacturing North America is gratefully acknowledged for his technical comments during the course of this study. \\vspace{-0.3cm}\n\\bibliographystyle{unsrt}\n\n\\section{Case Study: Automotive Engine Control} \n\\label{sec:CS}\\vspace{-0.15cm}\nHere, application of the proposed method in Section~\\ref{sec:UncertaintyPrediction} is demonstrated for a physics-based spark ignition (SI) combustion engine model~\\cite{Shaw} during cold start. Our proposed algorithm fits the requirements of this automotive control problem well, as it contains complicated plant model dynamics prone to uncertainty in slowly-fluctuating environments, yet require uncertainty mitigation to achieve tracking of desired trajectory behavior\n\nThe engine model~\\cite{Shaw} is parameterized for a 2.4-liter, 4-cylinder, DOHC 16-valve Toyota 2AZ-FE engine. The engine rated power is 117kW $@$ 5600~RPM, and it has a rated torque of 220 Nm $@$ 4000~RPM. The experimental validation of different components of the engine model is available in~\\cite{Sanketi}. The nonlinear model has four states including the exhaust gas temperature~($T_{exh}$), fuel mass flow rate into the cylinders (${\\dot{m}_f}$), the engine speed~(${\\omega_e}$), and the mass of air inside the intake manifold ($m_{a}$). The control problem is defined to steer $T_{exh}$, ${\\omega_e}$, and air-fuel ratio ($AFR$) to their pre-defined desired values. A set of four SISO DSMCs is designed to achieve this objective. Four states of the model and corresponding dynamics and controllers will be discussed in the following sections. Details of the functions and constants in the engine model are found in the Appendix and~\\cite{Sanketi}. \n\n$\\bullet {\\textbf{~~Exhaust~Gas~Temperature~Controller:}}$~Discretized model for exhaust gas temperature ($T_{exh}$) is:\n\\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_1}\nT_{exh}(k+1)=(1-\\frac{T}{\\tau_e})T_{exh}(k)\\\\\n+\\frac{T}{\\tau_e}(7.5\\Delta(k)+600)AFI(k)  \\nonumber\n\\end{gather}\nwhere $\\Delta(k)$ is the control input. The sliding surface for $T_{exh}$ controller is defined to be the error in tracking the desired exhaust gas temperature ($s_{1}=T_{exh}-{T_{exh,d}}$). The dynamics of the exhaust gas temperature ($f_{{T_{exh}}}$) with multiplicative unknown term ($\\alpha_{T_{exh}}$) is: \\vspace{-0.20cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_Texh}\nf_{{T_{exh}}}=\\alpha_{T_{exh}}\\Big(\\frac{1}{\\tau_e}[600AFI-T_{exh}]\\Big)\n\\end{gather}\nThe exhaust gas time constant ($\\tau_{e}$) has a significant role in the exhaust gas temperature dynamics (Eq.~(\\ref{eq:Engine_discretized_Texh})). This means that any error in estimating the time constant ($\\tau_{e}$) directly affects the dynamics and causes deviation from the nominal model. Multiplicative uncertainty term ($\\alpha_{T_{exh}}$) is assumed to represent any error in estimating $\\tau_{e}$. The error in the modeled $T_{exh}$ dynamics is removed by using the following adaptation law with respect to Eq.~(\\ref{eq:D2SMC_14}): \\vspace{-0.15cm}\n\\begin{gather}\n\\label{eq:adaptive_Texh}\n\\hat{\\alpha}_{T_{exh}}(k+1)=\\hat{\\alpha}_{T_{exh}}(k)+\\frac{T(s_1(k))}{\\tau_e \\rho_{\\alpha_1}}(600AFI-T_{exh}(k))\n\\end{gather}\nBy incorporating Eq.~(\\ref{eq:Engine_discretized_1}) and $\\hat{\\alpha}_{T_{exh}}$ from Eq.~(\\ref{eq:adaptive_Texh}) into Eq.~(\\ref{eq:D2SMC_6}), the second order adaptive DSMC for exhaust gas temperature becomes: \\vspace{-0.25cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_1}\n\\Delta(k)=\\frac{\\tau_e}{7.5\\,.\\,AFI\\,.\\,T}[-\\hat{\\alpha}_{T_{exh}}(k)\\frac{T}{\\tau_e}(600\\,.\\,AFI\\\\\n-T_{exh}(k))-(\\beta_1+1)s_1(k)+T_{exh,d}(k+1)-T_{exh,d}(k)] \\nonumber\n\\end{gather} \\vspace{-0.75cm}\n\n$\\bullet { \\textbf{~~Fuel~Flow~Rate~Controller:}}$~The discretized difference equation for the fuel flow rate is: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_2}\n\\dot{m}_f(k+1)=\\dot{m}_f(k)+\\frac{T}{\\tau_f}[\\dot{m}_{fc}(k)-\\dot{m}_f(k)]\n\\end{gather}\nThe fuel flow dynamic ($f_{\\dot{m}_f}$) with multiplicative uncertainty term ($\\alpha_{\\dot{m}_f}$) is as follows: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_mdotf}\nf_{\\dot{m}_f}=-\\alpha_{\\dot{m}_f}\\Big(\\frac{1}{\\tau_f}\\dot{m}_f(k)\\Big) \n\\end{gather}\nThe sliding variable for the fuel flow controller is defined to be the error in tracking the desired fuel mass flow ($s_{2}={\\dot{m}_{f}}-{\\dot{m}_{f,d}}$). In a similar manner to $T_{exh}$ dynamics, the fuel evaporation time constant $\\tau_{f}$ dictates the dynamics of the fuel flow into the cylinder. Consequently, any error in estimating $\\tau_{f}$ leads to a considerable deviation from the nominal model. $\\alpha_{\\dot{m}_f}$ is introduced to the fuel flow dynamics to represent the uncertainty in estimating $\\tau_{f}$. The adaptation law for $\\alpha_{\\dot{m}_f}$ is:  \n\\vspace{-0.4cm}\n\\begin{gather}\n\\label{eq:adaptive_mdotf}\n\\hat{\\alpha}_{\\dot{m}_f}(k+1)=\\hat{\\alpha}_{\\dot{m}_f}(k)-\\frac{T(s_2(k))}{\\tau_f \\rho_{\\alpha_2}}\\dot{m}_f(k)\n\\end{gather}\nwhere, $\\dot{m}_{f,d}$ in $s_2$ is calculated according to desired AFR. The adaptive control law for $\\dot{m}_{fc}$ is: \\vspace{-0.15cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_2}\n\\dot{m}_{fc}(k)=\\frac{\\tau_f}{T}[\\hat{\\alpha}_{\\dot{m}_f}(k)\\frac{T}{\\tau_f}\\dot{m}_f(k) \\\\-(\\beta_2+1)s_2(k)+\\dot{m}_{f,d}(k+1)-\\dot{m}_{f,d}(k)] \\nonumber\n\\end{gather} \\vspace{-0.5cm}\n\n$\\bullet { \\textbf{~~Engine~Speed~Controller:}}$~The rotational dynamics of the engine is described by using the following equation:\\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_3}\n\\omega_e(k+1)=\\omega_e(k)+\\frac{T}{J}T_E(k) \n\\end{gather}\nwhere $T_E(k)$ is the engine torque and is found by $30000~m_a(k)-(0.4~\\omega_e(k)+100)$. There is no direct control input for modulating the engine speed, therefore $m_a$ is considered as the synthetic control input. The calculated $m_a$ from engine speed controller will be used as the desired trajectory in intake air mass flow rate controller. $f_{\\omega_e}$ of the engine with multiplicative uncertainty ($\\alpha_{\\omega_e}$) is as follows: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_we}\nf_{\\omega_e}=-\\alpha_{\\omega_e}\\Big(\\frac{1}{J}T_{loss}\\Big)\n\\end{gather} \nwhere $T_{loss}=0.4\\omega_e+100$. $T_{loss}$ represents the torque losses on the crankshaft. Thus, the multiplicative uncertainty $\\alpha_{\\omega_e}$ compensates for any error in estimated torque loss. The sliding variable for the engine speed controller is defined to be $s_3=\\omega_e-{\\omega_{e,d}}$. $\\alpha_{\\omega_e}$ is driven to ``1'' using the following adaptation law: \\vspace{-0.3cm}\n\\begin{gather}\n\\label{eq:adaptive_we}\n\\hat{\\alpha}_{\\omega_e}(k+1)=\\hat{\\alpha}_{\\omega_e}(k)-\\frac{T(s_3(k))}{J.\\rho_{\\alpha_3}}(0.4\\omega_e(k)+100)\n\\end{gather}\nFinally, the desired synthetic control input ($m_{a,d}$) is:\\vspace{-0.3cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_3}\nm_{a,d}(k)=\\frac{J}{30,000\\,T}[\\hat{\\alpha}_{\\omega_e}(k)\\frac{T}{J}(100+0.4\\omega_e(k))\\\\\\nonumber-(\\beta_3+1)s_3(k)  \n+\\omega_{e,d}(k+1)-\\omega_{e,d}(k)]\n\\end{gather}\n\n$\\bullet { \\textbf{~~Air~Mass~Flow~Controller:}}$\nThe following state difference equation describes the air mass flow behaviour: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_3}\nm_a(k+1)=m_a(k)+T[\\dot{m}_{ai}(k)-\\dot{m}_{ao}(k)]\n\\end{gather}\nThe calculated $m_{a,d}$ from Eq.~(\\ref{eq:Engine_DSMC_Final_3}) is used as the desired trajectory to obtain $\\dot{m}_{ai}$ as the control input of $m_a$ controller. The last sliding surface for the air mass flow controller is defined to be $s_4=m_a-{m_{a,d}}$. The intake air manifold mass dynamic with the unknown term ($\\alpha_{m_a}$) is: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_ma}\nf_{m_a}=-\\alpha_{m_a}\\dot{m}_{ao}(k)\n\\end{gather}  \\vspace{-0.35cm}\nwhere, air mass flow into the cylinder is determined by~\\cite{Shaw}: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_ma2}\n\\dot{m}_{ao}=k_1\\eta_{vol}m_a\\omega_e \n\\end{gather}\n$\\eta_{vol}$ is the volumetric efficiency. As can be seen from Eq.~(\\ref{eq:Engine_discretized_ma}) and~(\\ref{eq:Engine_discretized_ma2}), the multiplicative uncertainty term in the intake air manifold dynamics ($\\alpha_{m_a}$) represents the uncertainty in $\\dot{m}_{ao}$ that is extracted from $\\eta_{vol}$ map. $\\alpha_{m_a}$ is updated using the following adaptation law:  \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:adaptive_ma}\n\\hat{\\alpha}_{m_a}(k+1)=\\hat{\\alpha}_{m_a}(k)-\\frac{T(s_4(k))}{\\rho_{\\beta_4}}\\dot{m}_{ao}\n\\end{gather}\nFinally, the controller input is:  \\vspace{-0.2cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_4}\n\\dot{m}_{ai}(k)=\\frac{1}{T}[\\hat{\\alpha}_{m_a}(k)\\dot{m}_{ao}(k)T-(\\beta_4+1)s_4(k)\\\\\n+m_{a,d}(k+1)-m_{a,d}(k)] \\nonumber\n\\end{gather}\nIn the absence of model uncertainties ($\\alpha_{T_{exh}}=\\alpha_{\\dot{m}_f}=\\alpha_{\\omega_e}=\\alpha_{{m}_a}=1$), Figures~\\ref{fig:Engine_2DSMC_SamplingComparison_10ms} and~\\ref{fig:Engine_2DSMC_SamplingComparison_40ms} show the results of tracking the desired $AFR$, $T_{exh}$, and engine speed trajectories, using the first and second order DSMCs for sampling times of $10~ms$ and $40~ms$, respectively. The mean tracking errors for both controllers are listed in Table~\\ref{table:tracking_Results}. It can be observed from Fig.~~\\ref{fig:Engine_2DSMC_SamplingComparison_10ms} and Table~\\ref{table:tracking_Results} when the signals at the controller I\/O are sampled every $10~ms$, both first and second order DSMCs illustrate acceptable tracking performances, while the second order controller is 50\\% more robust on average in terms of the tracking errors. As long as the Shannon's sampling theorem criteria, which states that the sampling frequency must be at least twice the maximum frequency of the measured analog signal, is satisfied, increasing the sampling time helps to reduce the computation cost. Upon increasing the sampling rate from $10~ms$ to $40~ms$, the first order DSMC performance degrades significantly. On the other side, despite the increase in the sampling time, the second order DSMC still presents smooth and accurate tracking results. By comparing the first and second order DSMC results at $T=40~ms$, it can be concluded that the proposed second order DSMC outperforms the first order controller by up to 85\\% in terms of the mean tracking errors. \\vspace{-0.5cm}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_SamplingComparison_10ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_SamplingComparison_10ms}Engine tracking results by the first and second order DSMCs with $T=10~ms$.} \\vspace{-0.8cm}\n\\end{center}\n\\end{figure}\n\\begin{table} [htbp!]\n\\small\n\\begin{center}\n\\caption{Mean ($\\bar{e}$) of Tracking Errors. Values Inside the Parentheses Show the Resulting Improvement from the Second Order DSMC Compared to the First Order DSMC.} \\linespread{1.15}\n\\label{table:tracking_Results}\\vspace{-0.15cm}\n\\begin{tabular}{lccccc}\n        \\hline\\hline\n\\multicolumn{1}{c}{} & \\multicolumn{2}{c}{$\\bar{e}~(T=10~ms)$}                                                        &  & \\multicolumn{2}{c}{$\\bar{e}~(T=40~ms)$} \\\\\n\\cline{2-3} \\cline{5-6}\n\\textbf{}& {$1^{st}$-Order} & {$2^{nd}$-Order} &  & {$1^{st}$-Order}  & {$2^{nd}$-Order} \\\\\n\\textbf{}& {DSMC} & {DSMC} &  & {DSMC}  & {DSMC} \\\\\n\\textbf{}   & {\\textcolor{blue}{Reference}}& \\textbf{}  &  & {\\textcolor{blue}{Reference}} & \\textbf{}  \\\\ \\hline\nAFR& 0.028  &  0.010   &    &  0.126   &  0.019       \\\\ \\vspace{0.05cm}\n[-]        &        & \\textcolor{blue}{(-64\\%)} &  &  & \\textcolor{blue}{(-84.9\\%)} \\\\ \\hline \\vspace{0.05cm} \n$T_{exh}$ & 0.2  & 0.1  &  & 1.8 & 0.2 \\\\ \\vspace{0.05cm}\n[$^o$C]                 &      & \\textcolor{blue}{(-50\\%)} &  &  & \\textcolor{blue}{(-88.9\\%)}\\\\ \\hline \\vspace{0.05cm} \n$N$  & 0.1  &   0.06   &  &   1.9    & 0.3\\\\ \\vspace{0.05cm}\n [RPM]                &      & \\textcolor{blue}{(-40\\%)} &  &  & \\textcolor{blue}{(-84.2\\%)} \\\\ \n\\hline\\hline \n\\end{tabular} \\vspace{-0.5cm}\n\\end{center}\n\\end{table}\n\\linespread{1} \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_SamplingComparison_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_SamplingComparison_40ms}Engine tracking results by the first and second order DSMCs with $T=40~ms$.} \\vspace{-0.8cm}\n\\end{center}\n\\end{figure} \n\nThe effect of the unknown multiplicative terms (up to 25\\%) on the engine plant's dynamics ($f$) is shown in Fig.~\\ref{fig:Engine_2DSMC_DynamicImpact_40ms}. The uncertainty terms in the model introduce a permanent error in the estimated dynamics compared to the nominal model. If these errors are not removed in the early seconds of the controller operation, the tracking performance will be affected adversely. Upon activation of the adaptation mechanism, as it can be observed from Fig.~\\ref{fig:Engine_2DSMC_DynamicImpact_40ms}, the model with error is steered towards the nominal model in less than 2 $sec$. Consequently, the errors in the model are removed. Fig.~\\ref{fig:Engine_2DSMC_ParamConv_40ms} shows the results of unknown multiplicative uncertainty term ($\\hat{\\alpha}$) estimation against the actual (nominal) values ($\\alpha$). \\vspace{-0.35cm}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_DynamicImpact_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_DynamicImpact_40ms}The effect of the model uncertainty terms on the engine dynamics when using the second order DSMC and how the adaptation mechanism drives the model with error to its nominal value: (a) $T_{exh}$, (b) $\\dot{m}_f$, (c) $\\omega_e$, and (d) $m_a$ ($T=40~ms$).} \\vspace{-0.75cm}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_ParamConv_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_ParamConv_40ms}Estimation of unknown multiplicative parameters in adaptive DSMC ($T=40~ms$).} \\vspace{-0.7cm}\n\\end{center}\n\\end{figure}\n\nFig.~\\ref{fig:Engine_2DSMC_AdaptiveTracking_40ms} shows the comparison between the tracking performances of the non-adaptive and adaptive second order DSMCs. As expected, the non-adaptive DMSC fails to track the desired trajectories, which explains the importance of handling the model uncertainties in the body of the DSMC. On the other hand, once the adaptation algorithm is enabled and the convergence period of the unknown parameters is over, the adaptive DSMC tracks all the desired trajectories smoothly with the minimum error under $T=40~ms$\n\\vspace{-0.4cm}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_AdaptiveTracking_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_AdaptiveTracking_40ms}Comparison between adaptive and non-adaptive second order DSMCs for the engine model with uncertainties: (a) $AFR$, (b) $T_{exh}$, and (c) $N$~($T=40~ms$).} \\vspace{-0.9cm}\n\\end{center}\n\\end{figure}\n\n\n\\section{Case Studies} \\label{sec:CS}\\vspace{-0.15cm}\nHere, application of the proposed method in Section~\\ref{sec:UncertaintyPrediction} is demonstrated for one linear (DC motor) and one nonlinear (SI engine) case studies. Our proposed algorithm fits the requirements of these automotive control problems well, as they contain complicated plant model dynamics prone to uncertainty in slowly-fluctuating environments, yet require uncertainty mitigation to achieve tracking of desired trajectory behavior.\\vspace{-0.15cm}\n\\subsection{Linear Case Study: DC Motor Speed Control} \\label{CS_DCMotor} \\vspace{-0.10cm}\nDC motors are common automotive actuators for control applications that require rotary and transitional motions. For speed regulation of a DC motor, the control input is voltage ($V$) to the motor's armature and the output is rotation speed (${\\theta}$) of the shaft. Assuming a constant magnetic field and linear relationship between motor torque and armature current ($\\mathcal{I}$), by choosing the rotor speed and current as the state variables, the following discretized linear time-invariant state-space representation can be used to describe the dynamics of the DC motor~\\cite{DCMotor_ref}, in which $\\alpha_{pq}$ represents the unknown multiplicative term on the plant's parameters: \\vspace{-0.25cm}\n\\begin{subequations}\n\\begin{gather}\\label{eq:DC_motor_linear_DSMC_Uncertain}\n\\theta(k+1)=T\\left([-{\\alpha}_{11}\\frac{k_f}{J}]\\theta(k)\\right) \\\\\n+T\\left(\\frac{k_m}{J}{\\mathcal{I}}(k)+\\frac{1}{J} \\Gamma\\right) +\\theta(k) \\nonumber \\\\\n{\\mathcal{I}}(k+1)=T\\left([-\\alpha_{21}\\frac{k_b}{L}]\\theta(k)+[-\\alpha_{22}\\frac{R}{L}]{\\mathcal{I}}(k)\\right)\\\\\n+\\frac{T}{L}V(k)+{\\mathcal{I}}(k) \\nonumber\n\\end{gather}\n\\end{subequations}\nwhere $J$ is the rotor's moment of inertia, $R$ is the electrical resistance, $L$ is the electric inductance, $\\Gamma$ is the generated torque on the rotor, $k_f$ is the mechanical damping, $k_m$ is the motor torque constant, and $k_b$ is the electromotive force constant. The DC motor model constants are listed in\nthe Appendix. Three multiplicative ($\\alpha_{pq}$) unknown parameters are derived to their nominal values by solving the adaptation law in Eq.~(\\ref{eq:D2SMC_14}). A second order DSMC is designed to regulate the DC motor rotational speed with respect to the desired speed profile ($\\theta_d$). The first sliding surface is defined as the error in tracking the desired speed profile ($s_{1}=\\theta-\\theta_d$). Since there is no direct control input on DC motor rotational speed, $\\mathcal{I}_d$ is defined as the synthetic control input for controlling the shaft speed. $\\mathcal{I}_d$ is used to define the second sliding surface ($s_{2}=\\mathcal{I}-\\mathcal{I}_d$) in which the control input is voltage. The final synthetic ($\\mathcal{I}_d$) and physical ($V$) control inputs are calculated as follows: \\vspace{-0.25cm}\n\\begin{subequations} \\label{eq:DC_motor_adaptiveDSMC_modified}\n\\begin{gather}\n{\\mathcal{I}}_d(k)=\\frac{J}{k_m}\\left(\\frac{1}{T}\\left[-\\beta_1(\\theta(k)-\\theta_d(k))+\\theta_d(k+1)\\right.\\right.\\\\\n\\left.\\left.-\\theta(k)\\right]+[\\hat{\\alpha}_{11}\\frac{k_f}{J}\\theta(k)]-\\frac{1}{J}\\Gamma\\right) \\nonumber \\\\\nV(k)=L\\left(\\frac{1}{T_s}\\left[-\\beta_2({\\mathcal{I}}(k)-{\\mathcal{I}}_d(k))+{\\mathcal{I}}_d(k+1)\n\\right.\\right.\\\\\n\\left.\\left.-{\\mathcal{I}}(k)\\right] \n+[\\hat{\\alpha}_{21}\\frac{k_b}{L}]\\theta(k)+\\hat{\\alpha}_{22}\\frac{R}{L}{\\mathcal{I}}(k)\\right) \\nonumber\n\\end{gather}\n\\end{subequations}\nSimulations are done in MATLAB Simulink$^{\\textregistered}$~which allows for testing the controller in a model-in-the-loop (MIL) platform for different sampling rates. In the absence of model uncertainties ($\\alpha_{pq}=1$), Fig.~\\ref{fig:DC_2DSMC_SamplingComparison} shows the comparison between the first~\\cite{Pan_Discrete} and second order DSMCs for different sampling rates. Although at lower sampling rates (e.g., $200~ms$) both controllers show similar performances, by increasing the sampling rate to $800~ms$, the higher robustness characteristics of the second order DSMC in comparison with the first order controller are revealed. The comparison results show that the second order DSMC is improving the tracking errors by 69\\% on average for different sampling rates, compared to the first order DSMC.\\vspace{-0.35cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{DC_2DSMC_SamplingComparison.eps} \\vspace{-0.85cm}\n\\caption{\\label{fig:DC_2DSMC_SamplingComparison}Comparison between the first and second order DSMCs in tracking the desired speed trajectory of the DC motor for different sampling rates. (a) First order DSMC, (b) Second order DSMC. Values inside the parentheses show the resulting improvement ($\\downarrow$) from the second order DSMC compared to the first order DSMC.} \\vspace{-0.55cm}\n\\end{center}\n\\end{figure}\n\nThe convergence results of the unknown parameters are shown in Fig.~\\ref{fig:DC_2DSMC_ParamConv}. This is done by simultaneously solving Eq.~(\\ref{eq:D2SMC_14}) and Eq.~(\\ref{eq:DC_motor_adaptiveDSMC_modified}). The performance of the first~\\cite{Amini_DSCC2016}, and second order adaptive DSMCs are shown in Fig.~\\ref{fig:DC_2DSMC_SpeedTracking} for tracking the desired speed profile under 200 $ms$ of sampling time, and up 50\\% uncertainty on the plant's parameters. As can be observed, the second order DSMC is able to significantly improve the tracking performance by 60\\% compared to the first order adaptive DSMC.\\vspace{-0.45cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{DC_2DSMC_ParamConv.eps} \\vspace{-0.85cm}\n\\caption{\\label{fig:DC_2DSMC_ParamConv}Convergence results of the unknown multiplicative ($\\alpha$) terms in the DC motor model (sampling time=200 ms).} \\vspace{-0.75cm}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{DC_2DSMC_SpeedTracking.eps} \\vspace{-0.85cm}\n\\caption{\\label{fig:DC_2DSMC_SpeedTracking}DC motor speed control using the first and second order \\textit{adaptive} DSMCs (sampling time=200 ms).} \\vspace{-1cm}\n\\end{center}\n\\end{figure}\n\\subsection{Nonlinear Case Study: Automotive Engine Control}\\label{subsec:Engine_Control}\\vspace{-0.10cm}\nIn this section, the performance of the proposed adaptive second order DSMC is studied for a physics-based spark ignition (SI) combustion engine model~\\cite{Shaw} during cold start. The engine model~\\cite{Shaw} is parameterized for a 2.4-liter, 4-cylinder, DOHC 16-valve Toyota 2AZ-FE engine. The engine rated power is 117kW $@$ 5600~RPM, and it has a rated torque of 220 Nm $@$ 4000~RPM. The experimental validation of different components of the engine model is available in~\\cite{Sanketi}. The nonlinear model has four states including the exhaust gas temperature~($T_{exh}$), fuel mass flow rate into the cylinders (${\\dot{m}_f}$), the engine speed~(${\\omega_e}$), and the mass of air inside the intake manifold ($m_{a}$). The control problem is defined to steer $T_{exh}$, ${\\omega_e}$, and air-fuel ratio ($AFR$) to their pre-defined desired values. A set of four SISO DSMCs is designed to achieve this objective. Four states of the model and corresponding dynamics and controllers will be discussed in the following sections. Details of the functions and constants in the engine model are found in the Appendix and~\\cite{Sanketi}. \n\n$\\bullet {\\textbf{~~Exhaust~Gas~Temperature~Controller:}}$~Discretized model for exhaust gas temperature ($T_{exh}$) is:\n\\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_1}\nT_{exh}(k+1)=(1-\\frac{T}{\\tau_e})T_{exh}(k)\\\\\n+\\frac{T}{\\tau_e}(7.5\\Delta(k)+600)AFI(k)  \\nonumber\n\\end{gather}\nwhere $\\Delta(k)$ is the control input. The sliding surface for $T_{exh}$ controller is defined to be the error in tracking the desired exhaust gas temperature ($s_{1}=T_{exh}-{T_{exh,d}}$). The dynamics of the exhaust gas temperature ($f_{{T_{exh}}}$) with multiplicative unknown term ($\\alpha_{T_{exh}}$) is: \\vspace{-0.20cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_Texh}\nf_{{T_{exh}}}=\\alpha_{T_{exh}}\\Big(\\frac{1}{\\tau_e}[600AFI-T_{exh}]\\Big)\n\\end{gather}\nThe exhaust gas time constant ($\\tau_{e}$) has a significant role in the exhaust gas temperature dynamics (Eq.~(\\ref{eq:Engine_discretized_Texh})). This means that any error in estimating the time constant ($\\tau_{e}$) directly affects the dynamics and causes deviation from the nominal model. Multiplicative uncertainty term ($\\alpha_{T_{exh}}$) is assumed to represent any error in estimating $\\tau_{e}$. The error in the modeled $T_{exh}$ dynamics is removed by using the following adaptation law with respect to Eq.~(\\ref{eq:D2SMC_14}): \\vspace{-0.15cm}\n\\begin{gather}\n\\label{eq:adaptive_Texh}\n\\hat{\\alpha}_{T_{exh}}(k+1)=\\hat{\\alpha}_{T_{exh}}(k)+\\frac{T(s_1(k))}{\\tau_e \\rho_{\\alpha_1}}(600AFI-T_{exh}(k))\n\\end{gather}\nBy incorporating Eq.~(\\ref{eq:Engine_discretized_1}) and $\\hat{\\alpha}_{T_{exh}}$ from Eq.~(\\ref{eq:adaptive_Texh}) into Eq.~(\\ref{eq:D2SMC_6}), the second order adaptive DSMC for exhaust gas temperature becomes: \\vspace{-0.25cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_1}\n\\Delta(k)=\\frac{\\tau_e}{7.5\\,.\\,AFI\\,.\\,T}[-\\hat{\\alpha}_{T_{exh}}(k)\\frac{T}{\\tau_e}(600\\,.\\,AFI\\\\\n-T_{exh}(k))-(\\beta_1+1)s_1(k)+T_{exh,d}(k+1)-T_{exh,d}(k)] \\nonumber\n\\end{gather} \\vspace{-0.75cm}\n\n$\\bullet { \\textbf{~~Fuel~Flow~Rate~Controller:}}$~The discretized difference equation for the fuel flow rate is: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_2}\n\\dot{m}_f(k+1)=\\dot{m}_f(k)+\\frac{T}{\\tau_f}[\\dot{m}_{fc}(k)-\\dot{m}_f(k)]\n\\end{gather}\nThe fuel flow dynamic ($f_{\\dot{m}_f}$) with multiplicative uncertainty term ($\\alpha_{\\dot{m}_f}$) is as follows: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_mdotf}\nf_{\\dot{m}_f}=-\\alpha_{\\dot{m}_f}\\Big(\\frac{1}{\\tau_f}\\dot{m}_f(k)\\Big) \n\\end{gather}\nThe sliding variable for the fuel flow controller is defined to be the error in tracking the desired fuel mass flow ($s_{2}={\\dot{m}_{f}}-{\\dot{m}_{f,d}}$). In a similar manner to $T_{exh}$ dynamics, the fuel evaporation time constant $\\tau_{f}$ dictates the dynamics of the fuel flow into the cylinder. Consequently, any error in estimating $\\tau_{f}$ leads to a considerable deviation from the nominal model. $\\alpha_{\\dot{m}_f}$ is introduced to the fuel flow dynamics to represent the uncertainty in estimating $\\tau_{f}$. The adaptation law for $\\alpha_{\\dot{m}_f}$ becomes:  \n\\vspace{-0.4cm}\n\\begin{gather}\n\\label{eq:adaptive_mdotf}\n\\hat{\\alpha}_{\\dot{m}_f}(k+1)=\\hat{\\alpha}_{\\dot{m}_f}(k)-\\frac{T(s_2(k))}{\\tau_f \\rho_{\\alpha_2}}\\dot{m}_f(k)\n\\end{gather}\nwhere, $\\dot{m}_{f,d}$ is calculated according to desired AFR. The adaptive control law for $\\dot{m}_{fc}$ is: \\vspace{-0.15cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_2}\n\\dot{m}_{fc}(k)=\\frac{\\tau_f}{T}[\\hat{\\alpha}_{\\dot{m}_f}(k)\\frac{T}{\\tau_f}\\dot{m}_f(k) \\\\-(\\beta_2+1)s_2(k)+\\dot{m}_{f,d}(k+1)-\\dot{m}_{f,d}(k)] \\nonumber\n\\end{gather} \\vspace{-0.75cm}\n\n$\\bullet { \\textbf{~~Engine~Speed~Controller:}}$~The rotational dynamics of the engine is described by using the following difference equation:\\vspace{-0.4cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_3}\n\\omega_e(k+1)=\\omega_e(k)+\\frac{T}{J}T_E(k) \n\\end{gather}\nwhere $T_E(k)$ is the engine torque and is found by $30000~m_a(k)-(0.4~\\omega_e(k)+100)$. There is no direct control input for modulating the engine speed, therefore $m_a$ is considered as the synthetic control input. The calculated $m_a$ from engine speed controller will be used as the desired trajectory in intake air mass flow rate controller. $f_{\\omega_e}$ of the engine with multiplicative uncertainty ($\\alpha_{\\omega_e}$) is as follows: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_we}\nf_{\\omega_e}=-\\alpha_{\\omega_e}\\Big(\\frac{1}{J}T_{loss}\\Big)\n\\end{gather} \nwhere $T_{loss}=0.4\\omega_e+100$. $T_{loss}$ represents the torque losses on the crankshaft. Thus, the multiplicative uncertainty $\\alpha_{\\omega_e}$ compensates for any error in estimated torque loss. The sliding variable for the engine speed controller is defined to be $s_3=\\omega_e-{\\omega_{e,d}}$. $\\alpha_{\\omega_e}$ is driven to ``1'' using the following adaptation law: \\vspace{-0.3cm}\n\\begin{gather}\n\\label{eq:adaptive_we}\n\\hat{\\alpha}_{\\omega_e}(k+1)=\\hat{\\alpha}_{\\omega_e}(k)-\\frac{T(s_3(k))}{J.\\rho_{\\alpha_3}}(0.4\\omega_e(k)+100)\n\\end{gather}\nFinally, the desired synthetic control input ($m_{a,d}$) is:\\vspace{-0.3cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_3}\nm_{a,d}(k)=\\frac{J}{30,000\\,T}[\\hat{\\alpha}_{\\omega_e}(k)\\frac{T}{J}(100+0.4\\omega_e(k))\\\\\\nonumber-(\\beta_3+1)s_3(k)  \n+\\omega_{e,d}(k+1)-\\omega_{e,d}(k)]\n\\end{gather}\n\n$\\bullet { \\textbf{~~Air~Mass~Flow~Controller:}}$\nThe following state difference equation describes the air mass flow behaviour in discrete time: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_3}\nm_a(k+1)=m_a(k)+T[\\dot{m}_{ai}(k)-\\dot{m}_{ao}(k)]\n\\end{gather}\nThe calculated $m_{a,d}$ from Eq.~(\\ref{eq:Engine_DSMC_Final_3}) is used as the desired trajectory to obtain $\\dot{m}_{ai}$ as the control input of $m_a$ controller. The last sliding surface for the air mass flow controller is defined to be $s_4=m_a-{m_{a,d}}$. The intake air manifold mass dynamic with the unknown term ($\\alpha_{m_a}$) is: \\vspace{-0.25cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_ma}\nf_{m_a}=-\\alpha_{m_a}\\dot{m}_{ao}(k)\n\\end{gather}  \\vspace{-0.35cm}\nwhere, air mass flow into the cylinder is determined by~\\cite{Shaw}: \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:Engine_discretized_ma2}\n\\dot{m}_{ao}=k_1\\eta_{vol}m_a\\omega_e \n\\end{gather}\n$\\eta_{vol}$ is the volumetric efficiency. As can be seen from Eq.~(\\ref{eq:Engine_discretized_ma}) and~(\\ref{eq:Engine_discretized_ma2}), the multiplicative uncertainty term in the intake air manifold dynamics ($\\alpha_{m_a}$) represents the uncertainty in $\\dot{m}_{ao}$ that is extracted from $\\eta_{vol}$ map. $\\alpha_{m_a}$ is updated using the following adaptation law:  \\vspace{-0.2cm}\n\\begin{gather}\n\\label{eq:adaptive_ma}\n\\hat{\\alpha}_{m_a}(k+1)=\\hat{\\alpha}_{m_a}(k)-\\frac{T(s_4(k))}{\\rho_{\\beta_4}}\\dot{m}_{ao}\n\\end{gather}\nFinally, the controller input is:  \\vspace{-0.2cm}\n\\begin{gather}\\label{eq:Engine_DSMC_Final_4}\n\\dot{m}_{ai}(k)=\\frac{1}{T}[\\hat{\\alpha}_{m_a}(k)\\dot{m}_{ao}(k)T-(\\beta_4+1)s_4(k)\\\\\n+m_{a,d}(k+1)-m_{a,d}(k)] \\nonumber\n\\end{gather}\nIn the absence of model uncertainties ($\\alpha_{T_{exh}}=\\alpha_{\\dot{m}_f}=\\alpha_{\\omega_e}=\\alpha_{{m}_a}=1$), Fig.~\\ref{fig:Engine_2DSMC_SamplingComparison_10ms} and~\\ref{fig:Engine_2DSMC_SamplingComparison_40ms} show the results of tracking the desired $AFR$, $T_{exh}$, and engine speed trajectories, using the first and second order DSMCs for sampling times of $10~ms$ and $40~ms$, respectively. The mean tracking errors for both controllers are listed in Table~\\ref{table:tracking_Results}. It can be observed from Fig.~~\\ref{fig:Engine_2DSMC_SamplingComparison_10ms} and Table~\\ref{table:tracking_Results} when the signals at the controller I\/O are sampled every $10~ms$, both first and second order DSMCs illustrate acceptable tracking performances, while the second order controller is 50\\% more robust on average in terms of the tracking errors. Upon increasing the sampling rate from $10~ms$ to $40~ms$, the first order DSMC performance degrades significantly. On the other side, despite the increase in the sampling time, the second order DSMC still presents smooth and accurate tracking results. By comparing the first and second order DSMC results at $T=40~ms$, it can be concluded that the proposed second order DSMC outperforms the first order controller by up to 85\\% in terms of the mean tracking errors. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_SamplingComparison_10ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_SamplingComparison_10ms}Engine tracking results by the first and second order DSMCs with $T=10~ms$.} \\vspace{-0.55cm}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_SamplingComparison_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_SamplingComparison_40ms}Engine tracking results by the first and second order DSMCs with $T=40~ms$.} \\vspace{-0.40cm}\n\\end{center}\n\\end{figure} \n\\begin{table} [htbp!]\n\\small\n\\begin{center}\n\\caption{Mean ($\\bar{e}$) of Tracking Errors. Values Inside the Parentheses Show the Resulting Improvement from the Second Order DSMC Compared to the First Order DSMC.} \\linespread{1.15}\n\\label{table:tracking_Results}\\vspace{-0.15cm}\n\\begin{tabular}{lccccc}\n        \\hline\\hline\n\\multicolumn{1}{c}{} & \\multicolumn{2}{c}{$\\bar{e}~(T=10~ms)$}                                                        &  & \\multicolumn{2}{c}{$\\bar{e}~(T=40~ms)$} \\\\\n\\cline{2-3} \\cline{5-6}\n\\textbf{}& {$1^{st}$-Order} & {$2^{nd}$-Order} &  & {$1^{st}$-Order}  & {$2^{nd}$-Order} \\\\\n\\textbf{}& {DSMC} & {DSMC} &  & {DSMC}  & {DSMC} \\\\\n\\textbf{}   & {\\textcolor{blue}{Reference}}& \\textbf{}  &  & {\\textcolor{blue}{Reference}} & \\textbf{}  \\\\ \\hline\nAFR& 0.028  &  0.010   &    &  0.126   &  0.019       \\\\ \\vspace{0.05cm}\n[-]        &        & \\textcolor{blue}{(-64\\%)} &  &  & \\textcolor{blue}{(-84.9\\%)} \\\\ \\hline \\vspace{0.05cm} \n$T_{exh}$ & 0.2  & 0.1  &  & 1.8 & 0.2 \\\\ \\vspace{0.05cm}\n[$^o$C]                 &      & \\textcolor{blue}{(-50\\%)} &  &  & \\textcolor{blue}{(-88.9\\%)}\\\\ \\hline \\vspace{0.05cm} \n$N$  & 0.1  &   0.06   &  &   1.9    & 0.3\\\\ \\vspace{0.05cm}\n [RPM]                &      & \\textcolor{blue}{(-40\\%)} &  &  & \\textcolor{blue}{(-84.2\\%)} \\\\ \n\\hline\\hline \n\\end{tabular} \\vspace{-0.5cm}\n\\end{center}\n\\end{table}\n\\linespread{1} \n\nThe effect of the unknown multiplicative terms (up to 25\\%) on the engine plant's dynamics ($f$) is shown in Fig.~\\ref{fig:Engine_2DSMC_DynamicImpact_40ms}. The uncertainty terms in the model introduce a permanent error in the estimated dynamics compared to the nominal model. If these errors are not removed in the early seconds of the controller operation, the tracking performance will be affected adversely. Upon activation of the adaptation mechanism, as it can be observed from Fig.~\\ref{fig:Engine_2DSMC_DynamicImpact_40ms}, the model with error is steered towards the nominal model in less than 2 $sec$. Consequently, the errors in the model are removed. Fig.~\\ref{fig:Engine_2DSMC_ParamConv_40ms} shows the results of unknown multiplicative uncertainty term ($\\hat{\\alpha}$) estimation against the actual (nominal) values ($\\alpha$).\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_DynamicImpact_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_DynamicImpact_40ms}The effect of the model uncertainty terms on the engine dynamics when using the second order DSMC and how the adaptation mechanism drives the model with error to its nominal value: (a) $T_{exh}$, (b) $\\dot{m}_f$, (c) $\\omega_e$, and (d) $m_a$ ($T=40~ms$).} \\vspace{-0.55cm}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_ParamConv_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_ParamConv_40ms}Estimation of unknown multiplicative parameters in adaptive DSMC ($T=40~ms$).} \\vspace{-0.55cm}\n\\end{center}\n\\end{figure}\n\nFig.~\\ref{fig:Engine_2DSMC_AdaptiveTracking_40ms} shows the comparison between the tracking performances of the non-adaptive and adaptive second order DSMCs. As expected, the non-adaptive DMSC fails to track the desired trajectories, which explains the importance of handling the model uncertainties in the body of the DSMC. On the other hand, once the adaptation algorithm is enabled and the convergence period of the unknown parameters is over, the adaptive DSMC tracks all the desired trajectories smoothly with the minimum error under $40~ms$ sampling time\n\\vspace{-0.4cm}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle=0,width= \\columnwidth]{Engine_2DSMC_AdaptiveTracking_40ms.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:Engine_2DSMC_AdaptiveTracking_40ms}Comparison between adaptive and non-adaptive second order DSMCs for the engine model with uncertainties: (a) $AFR$, (b) $T_{exh}$, and (c) $N$~($T=40~ms$).} \\vspace{-0.75cm}\n\\end{center}\n\\end{figure}\n\n\n\\section{Summary and Conclusions}   \\label{sec:Conclusion} \\vspace{-0.1cm}\nA new adaptive second order discrete sliding mode controller (DSMC) formulation for nonlinear uncertain systems was introduced in this paper. Based on the discrete Lyapunov stability theorem, an adaptation law was determined for removing generic unknown multiplicative uncertainty terms within the nonlinear difference equation of the plant's model. The proposed controller was examined for a spark ignition combustion engine control problem to track desired air-fuel ratio, engine speed, and exhaust gas temperature trajectories. Comparing to the first order DSMC, the second order DSMC shows significantly better robustness against data sampling imprecisions, and can provide up to 80\\% improvement in terms of the tracking errors. The better performance of the second order DSMC can be traced in driving the higher order derivatives (difference functions) of the sliding variable to zero. In the presence of the model uncertainties, it was shown that the adaptation mechanism is able to remove the errors in the modeled dynamics quickly, and steer the dynamics towards their nominal values. Increasing the sampling time raises the required time for the adaptation law to compensate for the uncertainties in the models. This required time was increased by two times, when the sampling time was increased from $10~ms$ to $40~ms$ in the engine tracking control problem, though the adaptation mechanism still could remove the model uncertainties in less than two seconds.\\vspace{-0.25cm}\n\n\n\n\n\n\n\n\n\n\\section{INTRODUCTION} \\label{Sec:Intro} \\vspace{-0.2cm}\nThe key feature of sliding mode control (SMC) is converting a high dimensional tracking control problem into a lower dimensional stabilization control problem~\\cite{Slotine}. SMC is well known for its robust characteristics against model uncertainty\/mismatch and external disturbances, while it requires low computational efforts. However, there are challenging issues that arise during digital implementation of SMC, among which chattering phenomenon has been widely reported in the literature~\\cite{utkin2013sliding}. One effective approach for reducing the oscillation due to chattering is the use of higher order SMC for continuous-time systems. This approach was first introduced in the 1980s~\\cite{nollet2008observer}. The basic idea of the higher order SMC is to drive all the higher order derivatives of the sliding variable to the sliding manifold, in addition to the zero convergence condition of the sliding variable. In this approach, the chattering caused by the discontinuity is transferred to the higher derivatives of the sliding variable. The final control input is calculated by integrating the $r-1$ derivatives of the input for $r-1$ times, and the result would be a continuous chattering-free signal of a $r^{th}$-order SMC~\\cite{nollet2008observer}. Higher order SMC leads to less oscillations; however, it adds complexity to the calculations.\n\nIn addition to the high frequency oscillations issue, it is shown in the literature~\\cite{Amini_DSC,AminiSAE2016,Hansen_DSCC} that upon digital implementation of the baseline SMC software, controller performance degrades from its expected behaviour significantly. The gap between the designed and implemented SMCs is mostly created due to data sampling errors that are introduced by the analog-to-digital (ADC) converter unit at the controller input\/output (I\/O).\nDiscrete sliding mode control (DSMC) was shown to be an effective approach to mitigate the ADC implementation imprecisions and enhancing the controller robustness against ADC uncertainties~\\cite{Pan_Discrete,Amini_ACC2016}. However, the chattering phenomenon due to the discontinuous nature of the discrete controller is more problematic for the DSMC and can even lead to instability since the\nsampling rate is far from infinite~\\cite{sira1991non}. \n\nSimilar to continuous-time SMC, it was shown in~\\cite{mihoub2009real} that a second order DSMC shows less oscillations compared to a first order DSMC. The second order DSMC in~\\cite{mihoub2009real} is formulated for linear systems without consideration of the uncertainties in the model. Moreover, the study in~\\cite{mihoub2009real} lacks the stability analysis of the closed-loop system. In this paper, a new second order DSMC formulation is developed for a general class of single-input single-output (SISO) \\textit{uncertain nonlinear systems}. Moreover, the asymptotic stability of the new controller is guaranteed via a Lyapunov stability argument. \n\nSimilar to implementation imprecisions, any uncertainty in the plant model, which is used for designing the model-based controller, results in a significant gap between the designed and implemented controllers.\nThe previous works in the literature that aimed to handle uncertainties in the model via an adaptive SMC structure are limited to continuous-time domain~\\cite{Slotine}, and linear systems~\\cite{Chan_Automatica}. The adaptive DSMC formulation from our previous works in~\\cite{Pan_DSC,Amini_DSCC2016,Amini_CEP} presents a generic solution for removing the model uncertainties for nonlinear systems based on a first order DSMC formulation. The proposed second order DSMC formulation from this paper allows us to derive the adaptation laws via a Lyapunov stability argument to remove the uncertainty in the plant's model quickly. \n\nThe contribution of this paper is threefold. First, a new second order DSMC is formulated for a general class of nonlinear affine systems. Second, the proposed controller is extended to handle the multiplicative type of model uncertainty using a discrete Lyapunov stability argument that also guarantees the asymptotic stability of the closed-loop system. Third, this paper presents the first application of the second order DSMC for an automotive combustion engine control problem. The proposed second order DSMC not only demonstrates robust behavior against data sampling imprecisions compared to a first order DSMC, but it also removes the uncertainties in the model quickly and steers the dynamics to their nominal values.\n\\vspace{-0.15cm}\n\n\\section{Second Order Sliding Mode Control} \\vspace{-0.15cm} \\label{sec:UncertaintyPrediction}\n\n\\subsection{Continuous Second Order Sliding Mode Control}\\vspace{-0.10cm} \\label{sec:ContinousTimeSecondSMC}\nA general class of continuous-time SISO nonlinear systems can be expressed as follows:\n\\vspace{-0.25cm}\n\\begin{gather}\\label{eq:C2SMC_1}\n\\dot{x}=f(t,x,u)\n\\end{gather}\nwhere $x{\\in{\\mathbb{R}^{n}}}$ and $u{\\in{\\mathbb{R}}}$ are the state and the input variables, respectively. The sliding mode order is the number of continuous successive derivatives of the differentiable sliding variable $s$, and it is a measure of the degree of smoothness of the sliding variable in the vicinity of the sliding manifold. For the continuous-time systems, the $r^{th}$ order sliding mode is determined by the following equalities~\\cite{salgado2004robust}:\n\\vspace{-0.22cm}\n\\begin{gather}\\label{eq:C2SMC_2}\ns(t,x)=\\dot{s}(t,x)=\\ddot{s}(t,x)=...=s^{r-1}(t,x)=0\n\\end{gather}\nThe sliding variable ($s$) is defined as the difference between desired ($x_d$) and measured signal ($x$):\n\\vspace{-0.22cm}\n\\begin{gather}\\label{eq:C2SMC_3}\n    s(t,x)=x(t)-x_d(t) \n\\end{gather}\nFor the second order SMC design,\na new sliding variable ($\\xi$) is defined according to $s$ and $\\dot{s}$\n\\vspace{-0.22cm}\n\\begin{gather}\\label{eq:C2SMC_4}\n\\xi(t,x)=\\dot{s}(t,x)+\\lambda s(t,x),~\\lambda>0\n\\end{gather}\nEq.~(\\ref{eq:C2SMC_4}) describes the sliding surface of a system with a relative order equal to one, in which the input is $\\dot{u}$ and output is $\\xi(t,x)$~\\cite{sira1990structure}. The control input is obtained according to the following law:\n\\vspace{-0.22cm}\n\\begin{gather}\\label{eq:C2SMC_5}\n\\dot{\\xi}(t,x)=0 \\Rightarrow \\ddot{s}(t,x)+\\lambda \\dot{s}(t,x)=0\n\\end{gather}\nwhich according to the sliding variable definition needs the second derivative of the state variable ($\\ddot{x}(t)$).~\nBy substituting Eq.~(\\ref{eq:C2SMC_3}) and $\\ddot{x}(t)$ into Eq.~(\\ref{eq:C2SMC_5}), $\\dot{u}$ is calculated as follows:\n\\vspace{-0.25cm}\n\\begin{gather}\\label{eq:C2SMC_9}\n\\dot{u}(t)=\\frac{1}{\\frac{\\partial}{\\partial u}f(t,x,u)}\\Big(-\\frac{\\partial}{\\partial t}f(t,x,u) \\\\-\\big(\\frac{\\partial}{\\partial x}f(t,x,u)\\big)f(t,x,u)+\\ddot{x}_d(t)-\\lambda\\dot{s}(t,x)\\Big) \\nonumber\n\\end{gather}\nand finally the control input is:\n\\vspace{-0.25cm}\n\\begin{gather}\\label{eq:C2SMC_10}\nu(t)=\\int\\dot{u}(t)dt\n\\end{gather}\n\nThis approach guarantees asymptotic convergence of the sliding variable and its derivative to zero~\\cite{mihoub2009real}. \\vspace{-0.15cm}\n\\subsection{Discrete Adaptive Second Order Sliding Mode Control} \\label{sec:DiscreteTimeSecondDSMC}\nThe affine form of the nonlinear system in Eq.~(\\ref{eq:C2SMC_1}) with an unknown multiplicative term ($\\alpha$) can be presented using the following state space equation:\n\\vspace{-0.15cm}\n\\begin{gather}\\label{eq:D2SMC_1}\n\\dot{x}(t)=\\alpha f(x(t))+g(x(t))u(t)\n\\end{gather}\nwhere $g(x(t))$ is a non-zero input coefficient and $f(x(t))$ represents the dynamics of the plant and does not depend on the inputs. $\\alpha$ is an unknown constant, and represents the errors in the modeled plant's dynamic. By applying the first order Euler approximation the continuous model in Eq.~(\\ref{eq:D2SMC_1}) is descritized as follows:\n\\vspace{-0.15cm}\n\\begin{gather}\\label{eq:D2SMC_2}\nx(k+1)=T\\alpha f(x(k))+Tg(x(k))u(k)+x(k)\n\\end{gather}\nwhere $T$ is the sampling time. Similar to Eq.~(\\ref{eq:C2SMC_4}), a new discrete sliding variable is defined:\n\\vspace{-0.25cm}\n\\begin{gather}\\label{eq:D2SMC_3}\n\\xi(k)={s}(k+1)+\\beta s(k),~\\beta>0\n\\end{gather}\nwhere $s(k)=x(k)-x_d(k)$, and $\\beta$ is the new sliding variable gain. Substituting Eq.~(\\ref{eq:D2SMC_2}) into Eq.~(\\ref{eq:D2SMC_3}) yields:\n\\vspace{-0.2cm}\n\\begin{gather}\\label{eq:D2SMC_4}\n\\xi(k)=T\\alpha f(x(k))+Tgu(k)+x(k)-x_d(k+1)+\\beta s(k)\n\\end{gather}\nThe second order discrete sliding law is defined as~\\cite{mihoub2009real}:\n\\vspace{-0.25cm}\n\\begin{gather}\\label{eq:D2SMC_5}\n\\xi(k+2)=\\xi(k+1)=\\xi(k)=0\n\\end{gather}\nApplying Eq.~(\\ref{eq:D2SMC_5}) to the nonlinear system in Eq.~(\\ref{eq:D2SMC_2}) results in the following control input:\n\\vspace{-0.15cm}\n\\begin{gather}\\label{eq:D2SMC_6}\nu(k)=\\frac{1}{gT}\\Big(-T\\hat{\\alpha}(k)f(x(k))-x(k)+x_d(k+1)-\\beta s(k)\\Big) \n\\end{gather}\nwhere $\\hat{\\alpha}$ is the estimation of the unknown multiplicative uncertainty term in the plant's model. By incorporating the control law ($u$) into the second order sliding variable ($\\xi$), we have:\n\\vspace{-0.3cm}\n\\begin{gather}\\label{eq:D2SMC_7}\n\\xi(k)=Tf(\\alpha-\\hat{\\alpha}(k))=Tf\\tilde{\\alpha}(k)\n\\end{gather}\nwhere ($\\tilde{\\alpha}$) is the difference between the unknown and estimated multiplicative uncertainty terms ($\\tilde{\\alpha}(k)=\\alpha-\\hat{\\alpha}(k)$). In order to determine the stability of the closed-loop system, and derive the adaptation law to remove the uncertainty in the model, a Lyapunov stability analysis is employed here. The following Lyapunov candidate function is proposed: \n\\begin{gather}\\label{eq:D2SMC_8}\nV(k)=\\frac{1}{2}\\Big({s}^2(k+1)+\\beta{s}^2(k)\\Big)\\\\\n+\\frac{1}{2}\\rho_{\\alpha}\\Big(\\tilde{\\alpha}^2(k+1)+\\beta \\tilde{\\alpha}^2(k)\\Big) \\nonumber\n\\end{gather}\nwhere $\\rho_{\\alpha}>0$ is a tunable parameter (adaptation gain) chosen for the numerical sensitivity of the unknown parameter estimation. The proposed Lyapunov function in Eq.~(\\ref{eq:D2SMC_8}) is positive definite and quadratic with respect to the sliding variable ($s(k)$) and the unknown parameter estimation error ($\\tilde{\\alpha}(k)$). In the discrete time domain, the negative semi-definite condition is required for the difference function of $V$ to guarantee the asymptotic stability of the closed-loop system~\\cite{Pan_DSC,Amini_CEP}. The Lyapunov difference function is calculated using a Taylor series expansion:  \\vspace{-0.20cm}\n\\begin{gather}\\label{eq:D2SMC_9}\nV(k+1)=V(k)+\\frac{\\partial V(k)}{\\partial s(k)}\\Delta s(k)\\\\\n+\\frac{\\partial V(k)}{\\partial s(k+1)}\\Delta s(k+1)+\\frac{\\partial V(k)}{\\partial \\tilde{\\alpha}(k)}\\Delta \\tilde{\\alpha}(k) \\nonumber \\\\\n+\\frac{\\partial V(k)}{\\partial \\tilde{\\alpha}(k+1)}\\Delta \\tilde{\\alpha}(k+1)  \n+\\frac{1}{2} \\frac{\\partial^2 V(k)}{\\partial {s}^2(k)}\\Delta {s}^2(k) \\nonumber \\\\\n+\\frac{1}{2} \\frac{\\partial^2 V(k)}{\\partial {s}^2(k+1)}\\Delta {s^2(k+1)}+\n\\frac{1}{2} \\frac{\\partial^2 V(k)}{\\partial {\\tilde{\\alpha}}^2(k)}\\Delta {\\tilde{\\alpha}}^2(k) \\nonumber \\\\\n+\\frac{1}{2} \\frac{\\partial^2 V(k)}{\\partial {\\tilde{\\alpha}}^2(k+1)}\\Delta {\\tilde{\\alpha}}^2(k+1)+... \\nonumber\n\\end{gather\nwhere $\\Delta s(k)\\equiv s(k+1)-s(k)$ and $\\Delta \\tilde{\\alpha}(k) \\equiv \\tilde{\\alpha}(k+1)-\\tilde{\\alpha}(k)$. \nNext, the Lyapunov difference function ($\\Delta V(k)=V(k+1)-V(k)$) is calculated by substituting the values of the partial derivatives into Eq.~(\\ref{eq:D2SMC_9}): \\vspace{-0.15cm}\n\\begin{gather}\\label{eq:D2SMC_11}\n\\Delta V(k)=\\beta s(k)\\Delta s(k)+s(k+1)\\Delta s(k+1) \\\\\n+\\rho_{\\alpha}\\beta\\tilde{\\alpha}(k)\\Delta\\tilde{\\alpha}(k)+\\rho_{\\alpha}\\tilde{\\alpha}(k+1)\\Delta\\tilde{\\alpha}(k+1) \\nonumber \\\\\n+\\frac{1}{2} \\beta\\Delta {s}^2(k)+\\frac{1}{2}\\Delta{s}^2(k+1)\\nonumber \\\\\n+\\frac{1}{2}\\rho_{\\alpha}\\beta\\Delta{\\tilde{\\alpha}}^2(k)+\\frac{1}{2}\\rho_{\\alpha}\\Delta{\\tilde{\\alpha}}^2(k+1)+... \\nonumber\n\\end{gather}\nwhere cross term second order derivatives are zero. Eq.~(\\ref{eq:D2SMC_11}) can be simplified after substituting Eq.~(\\ref{eq:D2SMC_7}) at $k$ and $k+1$ time steps: \\vspace{-0.2cm}\n\\begin{gather}\\label{eq:D2SMC_12}\n\\Delta V(k)=-\\beta(\\beta+1)s^2(k)-(\\beta+1)s^2(k+1)\\\\\n+\\beta s(k)Tf\\tilde{\\alpha}(k)+\\rho_{\\alpha}\\beta \\tilde{\\alpha}(k)\\Delta \\tilde{\\alpha}(k)\\nonumber \\\\\n+s(k+1)Tf\\tilde{\\alpha}(k+1)+\\rho_{\\alpha}\\tilde{\\alpha}(k+1)\\Delta\\tilde{\\alpha}(k+1) \\nonumber \\\\\n+O\\left(\\Delta {s}^2(k),\\Delta{s}^2(k+1),\\Delta\\tilde{\\alpha}^2(k),\\Delta\\tilde{\\alpha}^2(k+1)\\right)+... \\nonumber \n\\end{gather}\nwhich yields: \\vspace{-0.25cm}\n\\begin{gather}\\label{eq:D2SMC_13}\n\\Delta V(k)=-(\\beta+1)\\big(s^2(k+1)+\\beta s^2(k)\\big)\\\\\n+\\rho_{\\alpha}\\beta\\tilde{\\alpha}(k)\\Big(\\frac{s(k)Tf}{\\rho_{\\alpha}}+\\Delta\\tilde{\\alpha}(k)\\Big)  \\nonumber \\\\\n+\\rho_{\\alpha}\\tilde{\\alpha}(k+1)\\Big(\\frac{s(k+1)Tf}{\\rho_{\\alpha}}+\\Delta\\tilde{\\alpha}(k+1)\\Big) \\nonumber \\\\\n+O\\left(\\Delta {s}^2(k),\\Delta{s}^2(k+1),\\Delta\\tilde{\\alpha}^2(k),\\Delta\\tilde{\\alpha}^2(k+1)\\right)+... \\nonumber \n\\end{gather}\nin which the higher order ($>2$) terms are zero. As can be seen from Eq.~(\\ref{eq:D2SMC_13}), the first term is negative definite when $\\beta>0$. To guarantee the asymptotic stability of the closed-loop system, and minimize the tracking errors, the Lyapunov difference function should be at least negative semi-definite~\\cite{Amini_DSC}. To this end, the second and third terms in Eq.~(\\ref{eq:D2SMC_13}) should become zero, which leads to the following adaptation law:\\vspace{-0.35cm}\n\\begin{gather}\\label{eq:D2SMC_14}\n\\tilde{\\alpha}(k+1)=\\tilde{\\alpha}(k)-\\frac{s(k)Tf}{\\rho_{\\alpha}}  \n\\end{gather} \nBy using Eq.~(\\ref{eq:D2SMC_14}) to estimate the unknown uncertainty term, the Lyapunov difference function becomes: \\vspace{-0.20cm}\n\\begin{gather}\\label{eq:D2SMC_15}\n\\Delta V(k)=-(\\beta+1)\\big(s^2(k+1)+\\beta s^2(k)\\big)\\\\\n+O\\left(\\Delta {s}^2(k),\\Delta{s}^2(k+1),\\Delta\\tilde{\\alpha}^2(k),\\Delta\\tilde{\\alpha}^2(k+1)\\right) \\nonumber\n\\end{gather}\n\nLet us assume that by using Eq.~(\\ref{eq:D2SMC_14}), the uncertainty in the model will be removed. This means that the error in estimating the unknown parameter converges to zero ($\\tilde{\\alpha}(k+1)=\\tilde{\\alpha}(k)=0$). Thus, by expanding the second order terms ($O(.)$), and assuming a small enough sampling time ($T$), which means all terms that contain $T^2$ can be neglected, Eq.~(\\ref{eq:D2SMC_11}) can be re-arranged as follows: \\vspace{-0.20cm}\n\\begin{gather}\\label{eq:D2SMC_16}\n\\Delta V(k)=\\beta s(k)(s(k+1)-s(k))\\\\\n+s(k+1)(s(k+2)-s(k+1))+\\frac{1}{2} \\beta(s(k+1)-s(k))^2 \\nonumber \\\\\n+\\frac{1}{2}(s(k+2)-s(k+1))^2+... \\nonumber\n\\end{gather}\nSince it was assumed that the uncertainty in the model is compensated by Eq.~(\\ref{eq:D2SMC_14}), $s(k+1)$ and $s(k+2)$ can be replaced by $-\\beta s(k)$ and $\\beta^2 s(k)$, respectively, according to Eq.~(\\ref{eq:D2SMC_5}). Thus, Eq.~(\\ref{eq:D2SMC_16}) can be simplified as: \\vspace{-0.25cm}\n\\begin{gather}\\label{eq:D2SMC_18}\n\\Delta V(k)=-\\frac{1}{2}\\beta\\big(-{\\beta}^3-{\\beta}^2+\\beta+1\\big)s^2(k)\n\\end{gather}\n$-{\\beta}^3-{\\beta}^2+\\beta+1$ is positive if $1>\\beta>0$. In other words, if $1>\\beta>0$, then $\\Delta V(k)\\leq 0$, which guarantees the asymptotic stability of the system: \\vspace{-0.15cm}\n\\begin{gather}\\label{eq:D2SMC_19}\nV\\rightarrow 0 \\Rightarrow s(k+1)~\\&~s(k) \\rightarrow 0 \\xRightarrow[]{1>\\beta>0} \\xi(k) \\rightarrow 0\n\\end{gather}\n\nIt was shown that the second order sliding mode (Eq.~(\\ref{eq:D2SMC_5})) and the adaptation law from Eq.~(\\ref{eq:D2SMC_14}) guarantee the negative semi-definite condition of the Lyapunov difference function. This means that the sliding variable ($s$) and the error in estimating the unknown parameter ($\\tilde{\\alpha}$) converge to zero in finite time. Moreover, since the second order DSMC steers both first and second derivatives (difference functions) of the sliding variable to the origin, it provides better tracking performance, lower chattering, and higher robustness against data sampling imprecisions, compared to the first order DSMC. Fig.~\\ref{fig:AdaptiveDSMC_Schematic} shows the overall schematic of the proposed second order adaptive DSMC along with the adaptation mechanism. \\vspace{-0.4cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{AdaptiveDSMC_Schematic.eps} \\vspace{-0.75cm}\n\\caption{\\label{fig:AdaptiveDSMC_Schematic} Schematic of the proposed second order adaptive DSMC.} \\vspace{-0.7cm}\n\\end{center}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpintronics aims at using the information carried by the electrons' spin in solids.\nFor this purpose, establishing reliable methods to create, transfer and detect the spin current is an urgent task.\nCompared to the charge transports, spin transports have one serious fundamental difficulty.\nThat is the non-conservation of the spin in solids.\nThis limits the range of the spin transmission to be less than the spin diffusion length, which is typically $\\mu$m scale in metals.\n\n\nThe non-conservation of the spins is expressed by a source term in the continuity equation for the spin \n\\begin{align}\n\\dot{s}^\\alpha+\\nabla \\cdot\\bm{j}_{\\rm s}^\\alpha={\\cal T}^\\alpha.\n  \\label{sconteq}\n\\end{align}\nHere $s$ and $\\bm{j}_{\\rm s}$ are the spin density and spin current, respectively, $\\alpha=x,y,z$ is the spin direction and ${\\cal T}$ is the spin relaxation torque resulting in the non-conservation of the spin.\nIn the most cases in metals, the dominant origin of ${\\cal T}$ is the spin-orbit interaction.\n\nAlthough the relaxation torque term is essential in spin transports, it has so far been treated only on the phenomenological ground.\nThe continuity equation is equivalent to the Boltzmann equation, which is useful in discussing the spin transports.\nThe Boltzmann equation for the distribution function of each spin channel was discussed by Son et al. \\cite{Son87} and later by Valet and Fert \\cite{Valet93} in the context of the giant magnetoresistance in multilayer systems.\nIn their analysis, they approximated the spin relaxation torque as proportional to the inverse of a spin relaxation time $\\tau_{\\rm sf}$ \nand to some unknown function representing a driving force for the spin accumulation.\nThe driving force was written in terms of what they called the spin chemical potential $\\mu_{\\rm s}$.\nThe relaxation torque was approximated as ${\\cal T} ^z =\\mu_{\\rm s}\/\\tau_{\\rm sf}$.\nThey argued that $\\mu_{\\rm s}$ satisfies the diffusion equation,\n$\\nabla^2 \\mu_{\\rm s} =-\\ell_{\\rm sf}^{-2} \\mu_{\\rm s}$,  with the diffusion length $\\ell_{\\rm sf}\\propto \\sqrt{\\tau_{\\rm sf}}$.\nMicroscopic calculation for $\\mu_{\\rm s}$ has not been done so far.\n\n\n\nThe diffusion equation for the spin has been widely used to discuss recent spin transports in metallic junctions \\cite{Takahashi08}.\nThe decay of spin transport has been confirmed in non-local spin injection experiments \\cite{Jedema01,Kimura07,Seki08}, which\n indicate that the spin diffusion decays with a decay length of 350-500nm in Cu and  100nm in Au at room temperature.\nAlthough the spin diffusion equation appears to be so far successful, the phenomenological treatment of the spin relaxation term and the spin chemical potential must be improved \nto consider the spin transport seriously.\n\n\nBesides  diffusive spin current, there is another spin current that is driven by an effective field.\nIn contrast to the diffusive one, this field-driven contribution  \nshould not decay in uniform (single domain) ferromagnets, since the ratio of the spin current and the charge current is determined by the spin polarization ratio of the material, which is a statistical mechanical quantity. \nThe field-driven (local) spin current and the diffusive spin current behave differently, as was recently  demonstrated theoretically  in the case of the inverse spin Hall effect \\cite{Takeuchi10}. \n\n\nIn the field of the current-driven magnetization dynamics,  the spin relaxation torque has been studied from the microscopic viewpoint \\cite{KTS06,TKS_PR08,TE08}.\nIn this context, Eq. (\\ref{sconteq}) gives the expression for the torque acting on the spin density ${{\\bm s}}$ as \n\\begin{align}\n\\tau^\\alpha = -\\nabla \\cdot\\bm{j}_{\\rm s}^\\alpha+{\\cal T}^\\alpha.\n\\end{align}\nIn the adiabatic limit, i.e., slowly varying magnetization, and under uniform current, the first term reduces to \n$\\nabla \\cdot\\bm{j}_{\\rm s}^\\alpha=(P\/2e)(\\bm{j}\\cdot\\nabla){{\\bm s}}^\\alpha$, where $P$ is the spin polarization of the current \\cite{TKS_PR08,TE08}, namely to the adiabatic spin-transfer torque.\nWhen the spin-relaxation sets in, the conduction electron no longer  follows the magnetization profile, and new contribution to the torque arises from the ${\\cal T}$ term.\nThis torque was shown to be\n\\begin{align}\n{\\cal T} =\n-\\beta \\frac{P}{es^2} \n({{\\bm s}}\\times (\\bm{j}\\cdot\\nabla){{\\bm s}}),  \\label{betatorque}\n\\end{align}\nwhere $\\beta$ is a coefficient inversely proportional to the spin relaxation time $\\tau_{\\rm s}$ \\cite{KTS06,TE08}.\nThis torque, called $\\beta$ term, turned out to be essential in determining the efficiency of the current-driven domain wall motion \\cite{TK04,Zhang04,Thiaville05}.\nThe magnitude of the parameter $\\beta$ has recently been intensively studied experimentally by measuring the domain wall speed under current \\cite{Thomas06,Moore09}.\nTheoretical formulation for estimating $\\beta$ in the first-principles calculations was carried out recently \\cite{Garate09}.\n\n\n\nThe spin relaxation torque has been studied also from the viewpoint of how to define the spin current.\nIt was discussed that the spin relaxation torque contains a term written as a divergence of the torque dipole density, ${\\bm P}$ \\cite{Culcer04}.\nGeneralized argument was given by Shi et al. \\cite{Shi06}, where they discussed that the $z$ component of the relaxation torque is written as a divergence, \n\\begin{align}\n{\\cal T}^z= -\\nabla\\cdot{\\bm P}, \n\\end{align}\nif the system has the inversion symmetry. \nThis means that the total torque integrated over the system should vanish.\nShi et al. also argued that if the relaxation torque is a divergence of ${\\bm P}$, one can define a spin current that is conserved. \nIn fact, defining \n$\\tilde{\\bm{j}_{\\rm s}}\\equiv \\bm{j}_{\\rm s}+{\\bm P}$, the continuity equation (\\ref{sconteq}) reduces to $\\dot{s}^z+\\nabla\\cdot \\tilde{\\bm{j}_{\\rm s}}=0$.\nThe explicit form of the torque dipole density was not calculated in Ref. \\cite{Shi06}.\nObviously, in the presence of the inhomogeneity of the magnetization, the $\\beta$ torque (\\Eqref{betatorque}) cannot be written as a divergence, and thus it indeed represents the spin angular momentum lost by the spin relaxation.\n\n\nThe result that the spin relaxation torque is given by a derivative of the applied electric field ${\\bm E}$ is understood as follows.\nThe spin relaxation torque should of course vanish when ${\\bm E}=0$.\nIt cannot be directly proportional to ${\\bm E}$, since field-driven spin current in uniform ferromagnets should not decay. \nTherefore, the simplest expression for the relaxation torque is a derivative of the field.\nIt is not, however, obvious whether it should be always written in a rotationally invariant way, or if it can be\nanisotropic, since the rotational invariance is broken in uniform ferromagnets because of the magnetization.\n\n\n\nThe first aim of the present paper is to calculate the spin relaxation torque microscopically in the presence of the applied electric field. \nThe spin relaxation mechanism we take into account is the spin-orbit interaction  due to random impurities. \nOur explicit calculation reveals that the spin relaxation torque is not always rotationally symmetric in uniform ferromagnets, but is generally given by\n\\begin{align}\n{\\cal T}^z = \\gamma (\\nabla\\cdot {\\bm E})\n+\\delta\\gamma(\\partial_z E_z),\n\\label{TauE}\n\\end{align}\nwhere $z$ axis is along the magnetization, \n $\\gamma$ and  $\\delta\\gamma$ are coefficient proportional to the inverse spin relaxation time.\nThe spin relaxation torque is, therefore, anisotropic.\nRelaxation torque of \\Eqref{TauE} indicates that the torque dipole density is given by\n\\begin{align}\n{\\bm P} = -(\\gamma  {\\bm E} +\\delta\\gamma ({\\bm n}\\cdot {\\bm E}){\\bm n}),\n\\end{align}\nwhere ${\\bm n}$ represents the direction of the magnetization.\nThese anisotropic behaviors of the transport quantities is common when spin-orbit interaction exists, as is well-known in charge transport as the anisotropic magnetoresistance (AMR) \\cite{Mcguire75}. \n\nThe second aim of the paper is to study the spin current on the same microscopic footing as the relaxation torque.\nWe show that the spin current is made up of a field-driven contribution which is local and the diffusive one with nonlocality.\nThe field-driven contribution is anisotropic like the AMR effect for the charge current \n(we call the effect as spin AMR effect).\nThe diffusive contribution is given as a gradient of a spin chemical potential, $\\mu_{\\rm s}$.\nWe will derive the linear-response expression for the spin chemical potential. \nOur microscopic study on the spin current demonstrates the validity of the half-phenomenological treatments \\cite{Valet93}.\n\n\n\n\n\n\n\\section{Model}\n\nWe consider the conduction electron system taking account of the spin-orbit interaction, the impurity scattering without spin flip, and the applied electric field. \nThe Hamiltonian of the system is given as\n$H=H_0+ {H_{\\rm so}}+H_{\\rm em}+H_{\\rm imp}$, where \n$H_0$\n is the free electron Hamiltonian including the uniform magnetization, ${H_{\\rm so}}$ is the spin-orbit interaction, $H_{\\rm em}$ is the interaction with the gauge field representing the applied electric field, and $H_{\\rm imp}$ is the spin-independent impurity scattering.\nThe free part reads \n\\begin{align}\nH_0\\equiv \\sum_{{\\bm k}\\sigma}\\epsilon_{{\\bm k}\\sigma} c^\\dagger_{{\\bm k}\\sigma} c_{{\\bm k}\\sigma} ,\n\\end{align}\n where \nthe electron creation and annihilation operators are denoted by ${c^{\\dagger}}$ and $c$, respectively, $\\epsilon_{{\\bm k}\\sigma}\\equiv \\frac{k^2}{2m}-{\\epsilon_F}-\\sigma {M}$, ${\\epsilon_F}$ is the Fermi energy, ${M}$ is the spin splitting due to the magnetization and $\\sigma\\equiv \\pm$ represents the spin.\nThe spin-orbit interaction is represented by\n${H_{\\rm so}}={H_{\\rm so}}^0+{H_{\\rm so}}^{A}$,\n where \n($\\sigma_k$ ($k=x,y,z$) is the Pauli matrix)\n\\begin{align}\n{H_{\\rm so}}^0 &=  -\\frac{i}{2}\n\\sum_{ijk} \\epsilon_{ijk}\\int\\! {d^3x}  (\\nabla_i v_{\\rm so}^{(k)}) \n({c^{\\dagger}} \\vvec{\\nabla}_j \\sigma_k c)\n\\\\\n{H_{\\rm so}}^{A} &= - e\\sum_{ijk} \\epsilon_{ijk}\\int\\! {d^3x}  \n (\\nabla_i v_{\\rm so}^{(k)}) A_j({\\bm x},t) ({c^{\\dagger}} \\sigma_k c).\n\\end{align}\n(We suppress the spin index when obvious,  namely, $c=(c_+,c_-)$. )\nThe spin-orbit potential $v_{\\rm so}^{(k)}$ is assumed to arise from random impurities and to depend on the spin direction ($k$).\nThe averaging over the spin-orbit potential is carried out as \n\\begin{align}\n\\average{ v_{\\rm so}^{(k)}({\\bm p})  v_{\\rm so}^{(\\gamma)}(-{\\bm p}')}_{\\rm i} =n_{{\\rm so}}{\\lambda_{\\rm so}}^2 \\delta_{{\\bm p},{\\bm p}'}\\delta_{k\\gamma}, \n\\label{vsav}\n\\end{align} \nwhere $n_{{\\rm so}}$ and ${\\lambda_{\\rm so}}$ are the concentration of the spin-orbit impurities and the strength of the interaction, respectively.\nThe average of the spin-orbit potential at the linear order is zero in our model, and thus we do not take account of the the anomalous Hall and spin Hall effects. \nWe consider a case where electric field \n${\\bm E}({\\bm x},t)(\\equiv -\\dot{{\\bm A}}({\\bm x},t))$ is position and time dependent. \nThe electromagnetic interaction is written as\n\\begin{align}\nH_{\\rm em} = -\\frac{e}{m} \\sum_{{\\bm k},{\\bm q}}\n\\sum_i k_i A_i({\\bm q},\\Omega) ({c^{\\dagger}}_{{\\kv}-\\frac{{\\bm q}}{2}}c_{{\\kv}+\\frac{{\\bm q}}{2}}),\n\\end{align}\nwhere $\\Omega$ is the frequency of the electric field.\nWe will consider the limit of small $\\Omega$ and small $q$.\nThe scattering by the normal impurities is represented by\n\\begin{align}\nH_{\\rm imp} &= \n\\sum_{i=1}^{N_{\\rm imp}}\\sum_{{\\bm k}\\kv'} \n\\frac{v_{\\rm imp}}{N} e^{i({\\bm k}-{\\bm k}')\\cdot{\\bm R}_i} {c^{\\dagger}}_{{\\bm k}'}c_{{\\bm k}},\n\\end{align}\nwhere $v_{\\rm imp}$ represents the strength of the impurity potential, ${\\bm R}_i$ represents the position of random impurities, $N_{\\rm imp}$ is the number of impurities, and \n$N\\equiv V\/a^3$ is number of sites.\nTo estimate physical quantities, we take the random average over impurity positions in a standard manner \\cite{TKS_PR08}.\n\n\nTo derive the spin continuity equation, \\Eqref{sconteq}, we derive the equation of motion for the spin density, \n${\\bm \\se}({\\bm x},t)\\equiv \\average{{c^{\\dagger}}({\\bm x},t){\\bm \\sigma} c({\\bm x},t)}$\n($\\average{\\ }$ represents the quantum average).\nThe time development of the spin density reads \n\\begin{align}\n\\dot{s}^\\alpha &= \ni\\average { [H,c^\\dagger]\\sigma^\\alpha c+ c^\\dagger\\sigma^\\alpha [H,c] },\\label{eqcommutators}\n\\end{align}\nwhere $H$ is the total Hamiltonian of the system.\nThe commutators are calculated as in Appendix \\ref{APP:eqofmo}, and \\Eqref{eqcommutators}\nturns out to be \\Eqref{sconteq}, namely \n\\begin{align}\n\\dot{s}^\\alpha = -\\nabla \\cdot\\bm{j}_{\\rm s}^\\alpha+{\\cal T}^\\alpha, \\nonumber\n\\end{align}\n with the spin current given as\n\\begin{align}\nj_{\\rm s}^{\\alpha} \\equiv j_{\\rm s}^{{\\rm (n)},\\alpha}+j_{\\rm s}^{{\\rm so},\\alpha}\n, \n\\end{align}\nwhere \n\\begin{align}\n \\jspin{,i}{{\\rm (n)},\\alpha} \n  &\\equiv -\\frac{i}{2m} \n\\average{ {c^{\\dagger}} \\sigma^\\alpha \\vvec{\\nabla}_i c }\n -\\frac{e}{m}A_i \n\\average{ {c^{\\dagger}} \\sigma_\\alpha c }\n\\nonumber\\\\\n&\\equiv \n \\jspin{,i}{(0),\\alpha} + \\jspin{,i}{A,\\alpha}  ,\n\\label{spincurrentsdefs}\n\\end{align}\nand \n\\begin{align}\n \\jspin{,i}{{\\rm so}, \\alpha} \n  \\equiv \n-\\sum_{j} \\epsilon_{ij\\alpha} \n(\\nabla_jv_{\\rm so}^{(\\alpha)}) \\average{ {c^{\\dagger}} c}. \\label{jssodef}\n\\end{align}\nThe relaxation torque reads \n\\begin{align}\n{\\cal T}^{\\alpha}\n\\equiv {\\cal T}_{{\\rm so}}^{\\alpha} +{\\cal T}_{{\\rm so}}^{A,\\alpha}, \\label{totaltaudef}\n\\end{align}\n where \n\\begin{align}\n{\\cal T}_{{\\rm so}}^{\\alpha} \n&\\equiv \ni \\sum_{ijkl} \\epsilon_{ijk} \\epsilon_{\\alpha lk} (\\nabla_iv_{\\rm so}^{(k)})\n\\average{ {c^{\\dagger}} \\sigma_l \\vvec{\\nabla}_j c},\n\\label{Tsodef} \n\\\\\n{\\cal T}_{{\\rm so}}^{A,\\alpha} \n& \\equiv \n2e \\sum_{ijkl} \\epsilon_{ijk} \\epsilon_{\\alpha lk} (\\nabla_iv_{\\rm so} ^{(k)} ) A_j\n\\average{ {c^{\\dagger}} \\sigma_l c}.\\label{TsoAdef}\n\\end{align}\n\n\nThe spin relaxation torque depends on the definition of the spin current. \nFor instance, if we redefine the spin current as\n ${\\bm{j}_{\\rm s}' } ^\\alpha \\equiv {\\bm{j}_{\\rm s}^\\alpha }-{\\bm C}^\\alpha$, where ${\\bm C}$ is a vector, the continuity equation (\\ref  {sconteq}) becomes \n$\\dot{s}^\\alpha+\\nabla \\cdot{ \\bm{j}_{\\rm s}' } ^\\alpha={{\\cal T}'} ^\\alpha$,  where \nthe relaxation torque reads ${{\\cal T}'} ^\\alpha\\equiv {{\\cal T}} ^\\alpha+\\nabla\\cdot {\\bm C}^\\alpha$.\nThis ambiguity of spin current definition of course does not affect physical quantities such as the total torque acting on the spin density, which is given by $\\dot{{{\\bm s}}}$.\n\n\n\n\n\\section{Spin relaxation torque}\n\n\nWe calculate the spin relaxation torque as a linear response to the applied electric field. \nThe uniform magnetization is chosen as along $z$ axis.\nThe spin-orbit interaction is included to the second order.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=0.45\\hsize]{spinrelaxationdiag6.eps}\n\\caption{ \nFeynman diagrams representing the relaxation torque.\nSolid lines represent the electron Green's function with the lifetime  ($\\tau_\\sigma$) included, \n$v_{\\rm so}$ represents the spin-orbit interaction (double dashed line)  and dotted line represents the interaction with the gauge field ($A$).\nThe first two diagrams are the contributions to ${\\cal T}_{{\\rm so}}^{\\alpha}$\nand the last diagram is the contribution to ${\\cal T}_{{\\rm so}}^{A,\\alpha}$.\nThe vertex marked by cross represents the relaxation torque.\n\\label{FIGspinrelxationdiag1}\n}\n\\end{center}\n\\end{figure}\nThe contributions to the relaxation torque, \\Eqref{totaltaudef}, are shown in Fig. \\ref{FIGspinrelxationdiag1}.\nThe leading contribution for small $1\/({\\epsilon_F}\\tau)$ and $q\\ell$ \n($\\ell$ is the electron mean free path)\nturns out to be the first diagram in Fig. \\ref{FIGspinrelxationdiag1}, which reads \n(see Appendix \\ref{SEC:appA} for details)\n\\begin{align}\n{\\cal T}^{\\alpha} \n&= \\delta_{\\alpha,z}\n\\frac{2}{45\\pi}\nn_{{\\rm so}} {\\lambda_{\\rm so}}^2 \\frac{e}{m^2}\n(3\\nabla\\cdot\\dot{{\\bm A}}+\\nabla_z\\dot{A}_z)\n\\nonumber\\\\\n&\\times \n\\sum_{{\\bm k}\\kv'}\nk^2 (k')^4 \n\\sum_{\\sigma=\\pm} \\sigma  \\green{{\\bm k},\\sigma}{{\\rm r}} \n\\green{{\\bm k}',-\\sigma}{{\\rm r}} (\\green{{\\bm k}',-\\sigma}{{\\rm a}})^2 \n+{\\rm c.c.},\n\\label{Tauzdominant}\n\\end{align}\nwhere $\\green{{\\bm k}\\sigma}{{\\rm r}}$ and $\\green{{\\bm k}\\sigma}{{\\rm a}}$ are the retarded and advanced electron Green's functions, respectively, carrying the wave vector ${\\bm k}$ and spin $\\sigma$ with zero frequency. \nAs we see, only $z$ component of the torque is finite.\nThe Green's functions include the lifetime arising from the self-energy process due to normal impurities and the spin-orbit interaction.\nThe inverse lifetime for the electron with spin $\\sigma(=\\pm)$ is given as\n\\begin{align}\n{\\tau_\\sigma}^{-1}\n=\n2\\pi n_{\\rm imp}v_{\\rm imp}^2 \\nu_\\sigma\n(1+\\kappa_{z,\\sigma}+\\kappa_{\\perp} \\gamma_\\sigma),\n\\end{align}\nwhere $\\nu_\\sigma$ is the spin-resolved electron density of states, $n_{\\rm imp}$ and $v_{\\rm imp}$ are the concentration and the potential strength of the impurities,\n$\\kappa_{z,\\sigma}\\equiv \\frac{1}{3}\\frac{n_{{\\rm so}}{\\lambda_{\\rm so}}^2}{n_{\\rm imp}v_{\\rm imp}^2}k_{F\\sigma}^4 $\nand  \n$\\kappa_{\\perp} \\equiv \\frac{2}{3}\\frac{n_{{\\rm so}}{\\lambda_{\\rm so}}^2}{n_{\\rm imp}v_{\\rm imp}^2}k_{F+}^2 k_{F-}^2 $\nare dimensionless ratios of the spin-orbit interaction to the normal impurity scattering\n($k_{F\\sigma}$ is the spin-dependent Fermi wavelength), and \n$\\gamma_\\sigma\\equiv\\frac{\\nu_{-\\sigma}}{\\nu_\\sigma}$.\nThe total relaxation torque  is therefore given by (\\Eqref{TauE}),\n\\begin{align}\n{\\cal T}^z = \\gamma (\\nabla\\cdot {\\bm E})\n+\\delta\\gamma(\\partial_z E_z),\n\\end{align}\nwhere  \n\\begin{align}\n\\gamma\n&\\equiv\n \\frac{8\\pi e}{15m^2}\nn_{{\\rm so}} {\\lambda_{\\rm so}}^2 \n\\dos_{+} \\dos_{-} k_{F+}^2 k_{F-}^2 \n\\sum_{\\sigma=\\pm} \\sigma k_{F\\sigma}^2\\tau_{\\sigma}^2\n\\end{align}\nand $\\delta\\gamma\\equiv \\gamma\/3$.\nThe parameter $\\gamma$ is proportional to the spin flip rate due to the spin-orbit interaction.\nOur result indicates that the \nrelaxation torque is zero in uniform ferromagnet when uniform electric field is applied.\nThus the spin current does not decay in this case.\nIn fact,  the static solution of \\Eqref{sconteq} with ${\\cal T}^z=0$ is \n$j_{\\rm s}^z={\\rm constant}$. \n\nThe degree of the asymmetry, $\\delta\\gamma\/\\gamma$, is not universal but is model dependent.\nFor instance, in the case of junction with weak electron hopping at point-like leads, $\\delta \\gamma$ vanishes \\cite{Takezoe10}. \n\n\n\n\n\\section{Spin current}\n\nWe here calculate the spin current within the same formalism.\nWithin the linear response theory, the spin current is calculated by estimating the Feynman diagrams shown in Figs. \\ref{FIGspincurrent} and \\ref{FIGdiffusion}, which correspond to the field-induced contribution and the effect of the diffusive electron motion, respectively.\n\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=0.6\\hsize]{spincurrentdiag6.eps}\n\\caption{ \nFeynman diagrams representing the local contribution to the spin current\n(the first two terms in \\Eqref{jsexpression2}).\nThe first three diagrams correspond to $ \\jspin{}{{\\rm (n)}}$, and the last two diagrams represent the contribution from the anomalous velocity,\n $\\jspin{}{{\\rm so}}$.\nDotted and double dashed lines denote the interaction with the applied electric field and the spin-orbit interaction, respectively.\nThe vertex marked by cross represents the spin current.\n\\label{FIGspincurrent}\n}\n\\end{center}\n\\end{figure}\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=0.65\\hsize]{spincurrentdiffusive3.eps}\n\\caption{ \nLeft: The vertex correction contribution to $j_{\\rm s}^{{\\rm (n)},z}$,\nresulting in the diffusive spin current\n(the last term of \\Eqref{jsexpression2}).\nRight:\n$\\Gamma_{\\sigma\\sigma'}$, which is a ladder process of the successive electron scattering by the normal impurity and the spin-orbit interaction (represented by thick dotted lines)\nconnecting the  spin indices $\\sigma$ and $\\sigma'$.\n\\label{FIGdiffusion}\n}\n\\end{center}\n\\end{figure}\n\n\nWe first estimate the normal part of the spin current, \n$\\jspin{,i}{{\\rm (n)},z}$ (\\Eqref{spincurrentsdefs}), shown in the first two diagrams in Fig. \\ref{FIGspincurrent}.\nThe contribution $\\jspin{,i}{{\\rm (n)},z}$ is defined including the anomalous contribution from the electromagnetic gauge field, $\\jspin{,i}{A,z}$.  \nIts dominant contribution in the limit of small $\\Omega$ and small $q$  is calculated as \n(see Appendix \\ref{SEC:appB} for detail)\n\\begin{align}\n\\jspin{,i}{{\\rm (n)},z} \n&=\n -i\\frac{1}{2\\pi} \\frac{e^2}{m^2}\n\\sum_{{\\bm k}{\\bm q}}e^{-i{\\bm q}\\cdot{\\bm x}}\\int\\!\\frac{d\\Omega}{2\\pi} e^{i\\Omega t}\n\\sum_{j}\n \\Omega A_j({\\bm q},\\Omega) \n\\sum_{\\sigma} \\sigma \\left[\nk_i k_j\\green{{\\kv}-\\frac{{\\bm q}}{2},\\sigma}{{\\rm r}} \\green{{\\kv}+\\frac{{\\bm q}}{2},\\sigma}{{\\rm a}} \n\\right.\n\\nonumber\\\\\n&\n\\left. +\n\\sum_{{\\bm k}'\\sigma'}\nk_i k'_j\n\\green{{\\kv}-\\frac{{\\bm q}}{2},\\sigma}{{\\rm r}} \\green{{\\kv}+\\frac{{\\bm q}}{2},\\sigma}{{\\rm a}} \n\\green{{\\kvp}-\\frac{{\\bm q}}{2},\\sigma'}{{\\rm r}} \\green{{\\kvp}+\\frac{{\\bm q}}{2},\\sigma'}{{\\rm a}} \nn_{\\rm imp}v_{\\rm imp}^2 \\Gamma_{\\sigma\\sigma'}({\\bm q},\\Omega)\n\\right]\n\\label{spincurrent1}\n\\end{align}\nIn \\Eqref{spincurrent1}, the first term is the contribution shown in the left of Fig. \\ref{FIGspincurrent},\nand the second term is the contribution from the vertex correction (Fig. \\ref{FIGdiffusion}).\n\nThe factor $\\Gamma_{\\sigma\\sigma'}({\\bm q},\\Omega)$ contains all the \nvertex corrections due to the normal impurities and the spin-orbit interaction shown in Fig. \\ref{FIGdiffusion}.\nThe equation of motion for $\\Gamma_{\\sigma\\sigma'}$ is derived in the same manner as in Ref. \\cite{Hikami80} carried out in the context of quantum correction  (the diffusion without the spin-orbit interaction was considered in Ref. \\cite{Ban09}). \nThe equation is obtained as\n\\begin{align}\n\\Gamma_{\\sigma\\sigma} \n&= (1+\\kappa_z)(1+\\Pi_\\sigma\\Gamma_{\\sigma\\sigma})\n+\\kappa_\\perp \\Pi_{-\\sigma}\\Gamma_{-\\sigma,\\sigma}\n\\nonumber\\\\\n\\Gamma_{\\sigma,-\\sigma} \n&= \\kappa_\\perp(1+\\Pi_{-\\sigma}\\Gamma_{-\\sigma,-\\sigma})\n+(1+\\kappa_z)  \\Pi_{\\sigma}\\Gamma_{\\sigma,-\\sigma},\n\\label{Gammaeqs}\n\\end{align}\nwhere \n\\begin{align}\n\\Pi_\\sigma({\\bm q},\\Omega)\n&\\equiv n_{\\rm imp}v_{\\rm imp}^2 {\\sum_{\\kv}} \n \\green{{\\kv}-\\frac{{\\bm q}}{2},\\sigma}{{\\rm r}} \\green{{\\kv}+\\frac{{\\bm q}}{2},\\sigma}{{\\rm a}} \\nonumber\\\\\n&\\simeq \n [1-(D_\\sigma q^2 \\tau_\\sigma+\\kappa_z+\\kappa_\\perp\\gamma_\\sigma)],\n\\label{Pires}\n\\end{align}\nwhere $D_\\sigma \\equiv \\frac{(k_{F\\sigma})^2}{3m^2}\\tau_\\sigma$ is the diffusion constant.\nNeglecting quantities of order of $(\\kappa_z,\\kappa_\\perp)^2$, \\Eqref{Gammaeqs} is solved as\n\\begin{align}\n\\Gamma_{\\sigma\\sigma} \n&= \\frac{1+\\kappa_z-(1+2\\kappa_z)\\Pi_{-\\sigma}}\n{[1-(1+\\kappa_z)\\Pi_+][1-(1+\\kappa_z)\\Pi_-]}\n\\nonumber\\\\\n\\Gamma_{\\sigma,-\\sigma} \n&= \\frac{\\kappa_\\perp}\n{[1-(1+\\kappa_z)\\Pi_+][1-(1+\\kappa_z)\\Pi_-]}.\n\\label{Gammaeqs2}\n\\end{align}\nUsing \\Eqref{Pires}, we obtain $\\Gamma_{\\sigma\\sigma'}$  as  (assuming the rotational symmetry for the wave vectors when averaging over the spin-orbit potential)\n\\begin{align}\n\\Gamma_{\\sigma\\sigma} \n&= \\frac{1}\n{D_\\sigma q^2 \\tau_\\sigma+\\kappa_\\perp\\gamma_\\sigma}\n\\nonumber\\\\\n\\Gamma_{\\sigma,-\\sigma} \n&= \\frac{\\kappa_\\perp}\n{[D_+ q^2 \\tau_++\\kappa_\\perp\\gamma_+]\n [D_- q^2 \\tau_-+\\kappa_\\perp\\gamma_-]}.\n\\label{Gammaeqs3}\n\\end{align}\nBy use of \\Eqref{Gammaeqs3} and summing over the wave vectors,\nthe normal spin current, \\Eqref{spincurrent1}, reads\n\\begin{align}\n\\jspin{,i}{{\\rm (n)},z} \n&=\n\\sigma_{{\\rm s}}^0 E_i\n- \\nabla_i \\mu_{\\rm s},\n\\label{jsexpression0}\n\\end{align}\nwhere \n$\\sigma_{{\\rm s}}^0 \\equiv e \\sum_{\\pm} (\\pm) D_\\pm \\nu_\\pm$ is the bare spin conductivity divided by $e$.\nThe first term of \\Eqref{jsexpression0} is the field-driven contribution.\nThe second gradient term is a diffusive contribution (vertex corrections), arising from the spin accumulation.\nThe effective potential describing the spin accumulation, $\\mu_{\\rm s}$, \nreads\n\\begin{align}\n\\mu_{\\rm s} \\equiv\n\\int\\! {d^3x}' \\chi({\\bm x}-{\\bm x}') (\\nabla\\cdot {\\bm E})({\\bm x}'), \\label{muspin}\n\\end{align}\nwhere $\\chi$ is a correlation function arising from the electron diffusion, given as ($V$ is the system volume)\n\\begin{align}\n\\chi({\\bm x})\n&\\equiv \n- \\sum_{\\pm}  (\\pm)  \\sigma_{\\pm}\n\\frac{1}{V}{\\sum_{\\qv}} \\frac{e^{-i{\\bm q}\\cdot{\\bm x}}}\n{q^2+(\\ls{,\\pm})^{-2}}.   \\label{chidef}\n\\end{align}\nHere\n$\\sigma_\\pm \\equiv e D_\\pm \\nu_\\pm$ is the spin-resolved Boltzmann conductivity divided by $e$, and the correlation length is given as\n\\begin{align}\n\\ls{,\\sigma} \n= \\sqrt{D_\\sigma\\tau_{{\\rm s},\\sigma}}.\n\\end{align}\nThe lifetime of the spin $\\sigma$ electron reads \n$\\tau_{{\\rm s},\\sigma}\n\\equiv {\\tau_\\sigma}\/{(\\kappa_{\\perp} \\gamma_\\sigma)}$.\nDefining $\\mu_{\\rm s}=\\mu_+-\\mu_-$, \nwe see that spin-resolved effective potential satisfies\n\\begin{align}\n(-\\nabla^2 + (\\ls{,\\sigma})^{-2})\n\\mu_\\sigma \n= -\\sigma_\\sigma (\\nabla\\cdot{\\bm E})\n.\\label{mudiffusioneq}\n\\end{align}\n\n\nIn three-dimensions, the correlation function reads \n\\begin{align}\n\\chi({\\bm x})=\\frac{1}{4\\pi |{\\bm x}|}\\sum_{\\pm} (\\pm){\\sigma_{\\pm}}e^{-|{\\bm x}|\/\\ls{,\\pm}}.\n\\end{align}\n\n\n\n\n\n\n \n\n\nThe local part of the spin current arises also from the anomalous spin current due to the spin-orbit interaction, defined in \\Eqref{jssodef}.\nThis contribution is calculated by evaluating the last two diagrams in Fig. \\ref{FIGspincurrent}\nas (see Appendix \\ref{SEC:appB})\n\\begin{align}\n\\jspin{,i}{{\\rm so},z} \n&=\n\\delta \\sigma_{{\\rm s}} (1-\\delta_{i,z}) E_i,\n\\label{jssores2}\n\\end{align}\nwhere\n\\begin{align}\n\\delta \\sigma_{{\\rm s}} \n&\\equiv \n\\frac{\\pi}{9}\nn_{{\\rm so}}{\\lambda_{\\rm so}}^2 \\frac{e}{m}  \n\\sum_{\\pm}(\\pm) (k_{F\\pm})^4 (\\nu_{\\pm})^2 \\tau_\\pm .\n\\label{jssores3}\n\\end{align}\nThis spin-orbit correction to the spin conductivity is anisotropic, resulting in a spin version of the anisotropic magnetoresistance (AMR) effect, namely, spin AMR effect. \n\n\nFrom Eqs. (\\ref{spincurrent1})(\\ref{Gammaeqs3})(\\ref{jssores2}), the leading contribution to the spin current for small ${\\bm q}$ and $\\Omega$ is obtained as the sum of the local part driven by the electric field and the diffusive part as\n\\begin{align}\n\\bm{j}_{{\\rm s}}^{z}=\\sigma_{{\\rm s}} {\\bm E} -\\delta\\sigma_{\\rm s} ({\\bm n}\\cdot{\\bm E}) {\\bm n} \n- \\nabla \\mu_{\\rm s} , \n\\label{jsexpression2}\n\\end{align}\nwhere \n$\\sigma_{{\\rm s}} \\equiv \\sigma_{{\\rm s}}^{0}   +\\delta \\sigma_{{\\rm s}} $ and ${\\bm n}$ is the unit vector along the magnetization.\nIn terms of the angle $\\theta$ defined by\n$\\cos\\theta\\equiv ({\\bm n}\\cdot{\\bm E})\/E$, the magnitude of the field-driven (local) current reads\n\\begin{align}\nj_{{\\rm s}}^{{\\rm loc},z}=\\sqrt{ (\\sigma_{{\\rm s}\\parallel})^2 \n + ((\\sigma_{{\\rm s}\\perp })^2-(\\sigma_{{\\rm s}\\parallel})^2) \\sin^2\\theta},\n\\label{jsmag}\n\\end{align}\nwhere $\\sigma_{{\\rm s}\\parallel}\\equiv \\sigma_{{\\rm s}}^{0}$ and $\\sigma_{{\\rm s}\\perp}\\equiv \\sigma_{{\\rm s}}$.\nWhen the degree of the anisotropy is small, the spin current becomes \n\\begin{align}\n\\frac{j_{{\\rm s}}^{{\\rm loc},z}}{E}\n=\\sigma_{{\\rm s}\\parallel}\\left(1+ \n \\frac{1}{2}\\left( \\frac{\\sigma_{{\\rm s}\\perp }}{\\sigma_{{\\rm s}\\parallel}} \\right)^2 \\sin^2\\theta \\right).\n\\label{jsmag2}\n\\end{align}\nWe define the magnitude of the spin AMR as \n\\begin{align}\n\\frac{\\Delta\\rho_{{\\rm s}}}{\\rho_{{\\rm s}\\perp}}\n& \\equiv \\frac{\\rho_{{\\rm s}\\parallel}-\\rho_{{\\rm s}\\perp}} {\\rho_{{\\rm s}\\perp}} \n= \\frac{ \\delta \\sigma_{{\\rm s}} \/ \\sigma_{{\\rm s}} }{1-\\delta \\sigma_{{\\rm s}} \/ \\sigma_{{\\rm s}}},\n\\end{align}\nwhere $\\rho_{{\\rm s}\\alpha} \\equiv (\\sigma_{{\\rm s}\\alpha})^{-1}$ ($\\alpha=\\parallel,\\perp$).\n\n\n\n\n\n\n\n\\section{Spin injection}\n\nWe have thus derived the explicit expression for the spin chemical potential within the linear response theory.\nLet us apply \\Eqref{muspin} to a ferromagnetic-normal metal junction with an insulating barrier, used in the nonlocal spin injection experiments  \\cite{Kimura07}, depicted in Fig. \\ref{FIGspininjection}(a).\nWhen the voltage is applied perpendicular to the interface (we choose the $x$ axis in this direction), the electric field is uniform inside the ferromagnet and the normal metal except at the interface. \nWriting the voltage drop at the interface (chosen as at $x=0$) by $V_{\\rm FN}$, we obtain \n\\begin{align}\n\\nabla\\cdot{\\bm E} \\simeq \\delta(x) V_{\\rm FN}\/d,\n\\end{align}\n where $d$ is the width of the interaface, which is treated as small enough compared with the electron mean free path, resulting in the delta function in $\\nabla\\cdot{\\bm E}$. \nIn totally unpolarized non-magnetic metals, namely, if $\\sigma_+=\\sigma_-$ and $D_+=D_-$, the correlation function in \\Eqref{chidef} always vanishes.\nAs is naively guessed, therefore, spin injection thus requires an effective spin polarization close to the interface, induced by the exchange interaction with the ferromagnet.\nThis spin polarization is expected to be localized within a short distance of a few lattice constants from the interface. \nLet us approximate the interface polarization by introducing spin-dependent diffusion constant and the density of states, $ \\overline{D}_\\sigma$ and $\\overline\\nu_\\sigma$, respectively, at the interface.\nThe long-range behavior of the spin correlation function in the non-magnetic side is then obtained as\n\\begin{align}\n\\chi^{\\rm (N)}({\\bm x}) &=\n-\\frac{e}{4\\pi} (\\sum_\\sigma \\sigma \\overline{D}_\\sigma\\overline\\nu_\\sigma)\\frac{e^{-|{\\bm x}|\/\\ls{}}}{|{\\bm x}|}, \n\\end{align}\nwhere $\\ls{}$ in the spin diffusion length in the normal metal \n( $\\ls{}$ is a long ($\\sim \\mu$m) length scale and thus does not depend on the spin).\nWe therefore obtain from \\Eqref{muspin} the chemical potential as\n\\begin{align}\n\\mu_{\\rm s}^{\\rm (N)} ({\\bm x}) &=\n\\frac{q_{\\rm s} }{4\\pi |{\\bm x}|}e^{-|{\\bm x}|\/\\ls{}},\n\\end{align}\n where \n$q_{\\rm s}\\equiv \n(\\sum_\\sigma \\sigma \\overline{D}_\\sigma \\overline\\nu_\\sigma) eV_{\\rm FN})A_{\\rm FN}\/d\n$,\n is the spin accumulation rate at the interface (per unit time), and $A_{\\rm FN}$ is the area of the junction.\nThis result of $\\mu_{\\rm s}$ is consistent with intuitive and phenomenological results of the spin injection in the perpendicular structure shown in Fig. \\ref{FIGspininjection}(a). \nIn contrast, when the voltage is applied parallel to the ideal interface as shown in Fig. \\ref{FIGspininjection}(b), spin injection does not occur since $\\nabla\\cdot{\\bm E}=0$ at the interaface and thus $ \\mu_{\\rm s}^{\\rm (N)}=0$.\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=0.5\\hsize]{spininjection.eps}\n\\caption{ \n(a) Creation of diffusive spin current (spin injection) by applying \nthe electric voltage perpendicular to the F-N interface. \n(b) When the voltage is applied parallel to an ideal interface, no spin current is induced since $\\nabla\\cdot{\\bm E}=0$.\n\\label{FIGspininjection}\n}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Total torque and asymmetric $\\beta$ term}\n\nThe continuity equation (\\ref{sconteq}), indicates that the spin polarization (magnetization) changes due to the spin relaxation.\n(Change of the magnetization magnitude is a feature of the itinerant magnetism.)\nBy use of Eqs. (\\ref{TauE})(\\ref{mudiffusioneq})(\\ref{jsexpression2}), we see that \\Eqref{sconteq}\nresults in\n\\begin{align}\n\\dot{s}^z \n&= \\gamma_{\\tau}  \\nabla\\cdot{\\bm E}+\\delta \\gamma_{\\tau}  \\nabla_z E_z\n+\\sum_{\\pm}(\\pm)\\frac{\\mu_{\\pm}}{(\\ls{,\\pm})^2}, \\label{dotsz}\n\\end{align}\nwhere \n$\\gamma_{\\tau} \\equiv \\gamma-\\delta \\sigma_{{\\rm s}}$\nand \n$\\delta \\gamma_{\\tau} \\equiv  \\delta\\gamma+\\delta \\sigma_{{\\rm s}}$.\nGeneral case with uniform magnetization along any unit vector ${\\bm n}$ is given by (${{\\bm s}}\\equiv s{\\bm n}$) \n\\begin{align}\n\\dot{{{\\bm s}}} \n\n=\n{\\bm n} \\left( \\gamma_{\\tau} \\nabla\\cdot{\\bm E} + \\delta \\gamma_{\\tau}  \\nabla_\\parallel E_\\parallel\n+\\sum_{\\pm}(\\pm)\\frac{\\mu_{\\pm}}{(\\ls{,\\pm})^2}  \\right), \\label{dotsnv}\n\\end{align}\nwhere $E_\\parallel\\equiv {\\bm n}\\cdot{\\bm E}$ and $\\nabla_\\parallel\\equiv {\\bm n}\\cdot\\nabla$.\n\n\nIn addition to the change of the magnitude, \\Eqref{dotsnv}, there is a torque, which is perpendicular to ${\\bm n}$.\nSuch torque arises when the magnetization is not homogeneous, and plays important roles in current-induced magnetization dynamics. \nWe have carried out the calculation of the current-induced torque done in Ref. \\cite{TE08} on the same footing as the derivation of \\Eqref{TauE}.\nAs a result, we found that the $\\beta$ term becomes asymmetric as (see Appendix \\ref{APP:torque} for details of the calculation)\n\\begin{align}\n{\\cal T}^{(\\beta),\\alpha}_{{\\rm so}} \n& =\n-\\frac{P}{es^2} \n[\\beta {{\\bm s}}\\times (\\bm{j}\\cdot\\nabla){{\\bm s}} \n+\\delta \\beta {{\\bm s}}\\times (j_\\parallel \\nabla_\\parallel){{\\bm s}}]^\\alpha,\n\\end{align} \nwhere $j_\\parallel\\equiv {\\bm n}\\cdot\\bm{j}$ is the current along the local magnetization  \nand $\\delta \\beta\/\\beta=-1\/5$ in the present model.\nThe spin transfer torque due to the spin-orbit interaction is thus different from that due to the spin-flip scattering.\nThe expression of the total torque allowing for the spatially varying current density and the magnetization is therefore obtained as \n\\begin{align}\n\\dot{{{\\bm s}}} \n&=  \n-\\frac{P}{2e}\n(\\nabla\\cdot\\bm{j}){\\bm n} \n-\\frac{P}{e}\\left[ \\beta {\\bm n}\\times (\\bm{j}\\cdot\\nabla){\\bm n}\n+\\delta \\beta {\\bm n}\\times (j_\\parallel \\nabla_\\parallel){\\bm n})\n\\right]\n\\nonumber\\\\\n& \n+ {\\bm n}\n\\left(\n\\gamma_{\\tau} (\\nabla\\cdot{\\bm E})\n+\\delta\\gamma_{\\tau}   (\\partial_\\parallel E_\\parallel)\n+\\sum_{\\pm} (\\pm) (\\ls{,\\pm})^{-2}\\mu_\\pm\n\\right).\n\\label{torquesum}\n\\end{align}\nThis expression clearly demonstrates that the spin relaxation torque requires some inhomogeneity either of the applied current or the spin structure,  in addition to the spin-orbit (or spin flip) interaction.\nTotally homogeneous system does not relax.\nThe last term in Eq. (\\ref{torquesum}) gives useful information for measuring the spin accumulation induced by the spin current.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe have carried out a microscopic calculation of the spin relaxation torque and the spin current induced in disordered ferromagnetic metals by the applied electric field. \nThe spin-orbit interaction arising from the random impurities is included as a source of spin relaxation, and inhomogeneity of the applied electric field is taken into account.\nWe found that the spin relaxation torque in the uniform magnetization case is written as a divergence of the electric field plus an anisotropic term. \nThe spin current was shown to be made up of field-driven (local) and diffusive (nonlocal) contributions, the latter written as a gradient of a spin chemical potential.\nWe have derived a general linear response expression for the spin chemical potential.\nThe spin injection effect was briefly discussed based on our results.\nWhen the analysis is applied to the inhomogeneous magnetization case, we argued that the $\\beta$ torque in the current-induced magnetization dynamics can be anisotropic.\n\n\nBefore finishing, we emphasize that the expression for the spin current and $\\mu_{\\rm s}$ are meaningless without specifying the physical observable to be measured.\nIn the inverse spin Hall effect, which was originally proposed as \\cite{Hirsch99,Takahashi08,Saitoh06}\n$j_\\mu \\propto \\epsilon_{\\mu\\nu\\rho}\\jspin{,\\nu}{\\rho}$, \nit has recently been demonstrated that the charge current is not directly proportional to the spin current \\cite{Hosono10,Takeuchi10}.\nSolving for the spin current only does not therefore provide physical information. \n\n\n\n\n\n\n\n\n\\section*{Acknowledgment}\nThe authors thank E. Saitoh, J. Shibata, H. Kohno, S. Murakami for valuable discussions. \nThis work was supported by a Grant-in-Aid for Scientific Research in Priority Areas, \"Creation and control of spin current\" (1948027), the Kurata Memorial Hitachi Science and Technology Foundation and the Sumitomo Foundation. \n\n\n\n\n\n\n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\\label{sec:intro}\nThe optimal allocation of communication resources is part of the constant race for better performance in mobile communications.\nAs new technologies such as high resolution video streaming and vehicular communications emerge, the demands to be fulfilled by a scheduler are becoming increasingly heterogenous, resulting in a complex optimization task that must be solved in real time.\n\nIn recent years, deep learning~(\\deeplearning) methods have distinguished themselves with successes in complex tasks, including image classification~\\cite{krizhevsky2012imagenet} and control~\\cite{mnih2015human}, among others.\n\\deeplearning, as a subset of machine learning~(\\machinelearning), follows a different paradigm compared to classic model-based approaches. Instead of designing an optimal algorithm for a given goal, an algorithm is iteratively approximated based on training data. Without the need to explicitly model the underlying processes, the issue of modeling complexity is sidestepped.\nThis property has sparked an interest for research applying \\deeplearning to the highly performance driven field of communication systems.\n\nIn particular, increasing the performance and capabilities of mobile communication systems promotes new prospects in emergency patient care.\nFor example, video, vitals, or specialist input can be exchanged wirelessly between on-site medical professionals and hospital staff, yielding a significant headstart in time-sensitive treatments~\\cite{zanatta2016pre}.\nNevertheless, a deliberate approach is required due to the low tolerance for delays and outliers in medical tasks.\n\nVarious forays have been made into integrating \\deeplearning methods with communications resource allocation tasks, mostly based on deep Q-Networks~(\\dqn)~\\cite{joseph2019towards, ye2019deep, al2020learn} and actor-critic methods~\\cite{huang2020deep}.\nHowever, model-based approaches have long thrived thanks to the valuable expert knowledge that shapes their design. Ideally, incorporating this expert knowledge into data-based methods could govern and stabilize the learning process and produce more tractable algorithms~\\cite{shlezinger2020model}.\n\nTherefore, we investigate a combined approach that offers some of the advantages of both \\deeplearning and model-based approaches.\nFor the task of scheduling discrete resources in vehicle-to-base-station communication, we implement a group of simple, model-based scheduling algorithms.\nWe then train a \\dqn to select the best model-based algorithm in a given situation, optimizing the long term effect on typical performance indicators: time-outs, sum capacity, and packet rate.\nParticular attention is paid to the performance of Emergency Vehicles~(\\textsc{\\small \\emergencyvehicle}\\xspace) as a priority class of users.\n\n\n\n\\section{Setup \\& Notation}\nIn the following, we introduce our system model. We describe the communication channel between user and base station, the resources available for allocation, and the generation of jobs. We then formulate the resource allocation problem.\n\n\\subsection{Resource Allocation System Model}\nIn our system model, a number of \\( \\numusers \\)~user vehicles are connected to a base station, moving on a 2D plane in a grid akin to the Manhattan-model of movement~\\cite{aschenbruck2008survey}.\nAt any discrete time instance~\\( \\timeindex \\), vehicles~\\( \\userindex \\) will take a fixed size step in a direction of movement that is randomly drawn, with a \\SI{98}{\\percent} chance to re-select their prior direction of movement, \\SI{0}{\\percent} chance to select the direction opposite to their prior movement, and uniform chance to draw a \\ang{90} turn left, right, or stopping.\n\nThe communication channel between base station and user vehicle~\\( \\userindex \\) is characterized by its power gain\n\\begin{align}\n\t\\channelquality_\\userindex[\\timeindex]\n\t=\n\t| \\tilde{\\channelquality}_{\\userindex}[\\timeindex] |^{\\num{2}} \\cdot \\pathloss_{\\userindex}[\\timeindex],\n\\end{align}\nknown to the base station. For each simulation time step~\\( \\timeindex \\), fading channel amplitudes~\\( | \\tilde{\\channelquality}_{\\userindex}[\\timeindex] | \\) are randomly drawn from a {Rayleigh distribution} and multiplied with a distance proportional path loss factor\n\\begin{align}\n\t\\pathloss_\\userindex[\\timeindex]\n\t=\n\t\\min\\left(\n\t1,\\,\\left(\\distanceuser_\\userindex[\\timeindex]\\right)^{\\num{-1}}\n\t\\right),\n\\end{align}\nwhere \\( \\distanceuser_{\\userindex}[\\timeindex] \\)~is the vehicles distance from the base station.\nThe path loss factor ensures a degree of correlation of the power gain~\\( \\channelquality_{\\userindex}[\\timeindex] \\) over time.\nWhile exponents of~\\( \\num{-2} \\) or lower are more typically assumed in modeling real-life path loss, we selected an exponent of~\\( \\num{-1} \\) to reduce the spread of path loss values encountered. For the purpose of this paper, this lessens the computation required in the subsequent learning task.\n\nThe base station has access to a limited number~\\( \\numresourcesavailable \\) of discrete resource blocks available for allocation, as shown in \\reffig{fig:resourceblocks}.\nThese resources can be filled with jobs~\\( \\jobindex \\) from the job queue.\nAt each time step~\\( \\timeindex \\), new jobs are generated at a probability~\\( \\probability_{\\jobindex} \\) per user.\nA job~\\( \\jobindex \\) is assigned to a specific user~\\( \\userindex \\) and is defined by two attributes, a request size~\\( \\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex] \\) in discrete resource blocks, and a time-to-timeout~\\( \\timetotimeout_{\\jobindex}[\\timeindex] \\) in discrete simulation steps.\nWe define the set~\\( \\jobsset[\\timeindex] \\) of all jobs in the job queue in time step~\\( \\timeindex \\), and the set~\\( \\jobsset_{\\userindex}[\\timeindex] \\) of jobs in queue assigned to user~\\( \\userindex \\) in~\\(\\timeindex \\).\n\nThe job attributes' initial values are governed by a user profile that declares a maximum job size~\\( \\resourceblock_{\\userindex,\\,\\text{max}} \\) and initial time-to-timeout~\\( \\timetotimeout_{\\userindex,\\,\\text{init}} \\) for users~\\( \\userindex \\) of that profile.\nAt generation, the job is assigned a size~\\( \\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex] \\) drawn from a discrete uniform distribution, \\( \n\t{\\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex] \\sim \\mathbb{U}\\{1,\\, \\resourceblock_{\\userindex,\\,\\text{max}}\\}} \n\\). When a number of the base stations resources~\\( \\numresourcesavailable \\) is allocated to a job~\\( \\jobindex \\), that jobs remaining size is decreased accordingly, while a lifetime count~\\( \\resourceblock_{\\userindex,\\,\\text{sx}}[\\timeindex] \\) of resources scheduled to user~\\( \\userindex \\) is increased.\nThe jobs initial time-to-timeout~\\( {\n\t\\timetotimeout_{\\jobindex}[\\timeindex] \\gets \\timetotimeout_{\\userindex,\\,\\text{init}}\n} \\) is decremented for each time step~\\( \\timeindex \\) that passes without the job being fully scheduled. Once the time-to-timeout~\\( \\timetotimeout_{\\jobindex}[\\timeindex] \\) reaches zero, the job is discarded from the job queue and added to a set~\\( \\jobsset_{\\text{fail}}[\\timeindex] \\) for that time step~\\( \\timeindex \\), for use in performance metric calculation.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\input{figures\/adaptiveresourceblocks.pgf}\n\t\\caption\n\t{%\n\t\tJobs consisting of a number of discrete resource blocks arrive in the job queue.\n\t\tA scheduler is tasked with assigning them to a limited number~\\( \\numresourcesavailable \\) of resources.\n\t}%\n\t\\label{fig:resourceblocks}\n\\end{figure}\n\n\\subsection{Problem Statement}\nA scheduler is tasked with assigning the limited number~\\( \\numresourcesavailable \\) of resource blocks to the jobs in queue.\nThree metrics are selected to gauge the performance of a scheduling algorithm:\n\\begin{enumerate}\n\t\\item \\( \\reward_L[\\timeindex] \\): Resource blocks discarded from timeout\n\t\\item \\( \\reward_P[\\timeindex] \\): Global packet rate achieved\n\t\\item \\( \\reward_C[\\timeindex] \\): Global channel capacity achieved\n\\end{enumerate}\n\nFirstly, the \\emph{sum of resource blocks discarded} due to timing out\n\\begin{align}\n\t\\reward_L[\\timeindex]\n\t=\n\t\\sum_{\\jobindex \\in \\jobsset_{\\text{fail}}[\\timeindex]} \\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex] \n\\end{align}\nfor all users should be minimized.\n\nSecondly, a packet rate is introduced as the lifetime ratio of resources requested and scheduled for a user~\\( \\userindex \\).\nWe define the set~\\( \\jobsset_{\\userindex,\\,\\text{new}}[\\timeindex] \\) as the set of new jobs assigned to user~\\( \\userindex \\), generated in time step~\\( \\timeindex \\). Using the lifetime sum of discrete resources~\\( \\resourceblock_{\\userindex,\\,\\text{sx}}[\\timeindex] \\) scheduled to jobs of user~\\( \\userindex \\), we calculate the \\emph{sum packet rate} over all users:\n\\begin{align}\n\t\\reward_P[\\timeindex]\n\t= \t\\sum_{\\userindex=\\num{1}}^{\\numusers}\n\t\t\\frac{\n\t\t\t\\resourceblock_{\\userindex,\\,\\text{sx}}[\\timeindex]\n\t\t}{\n\t\t\t\\sum_{\\tilde{\\timeindex}=\\num{1}}^{\\timeindex}\n\t\t\t\\sum_{\\jobindex \\in \\jobsset_{\\userindex,\\,\\text{new}}\\left[\\tilde{\\timeindex}\\right]}\n\t\t\t\\resourceblock_{\\jobindex,\\,\\text{req}}\\left[\\tilde{\\timeindex}\\right]\n\t\t}\n\t= \t\\sum_{\\userindex=\\num{1}}^{\\numusers}\n\t\t\\packetrate_{\\userindex}[\\timeindex]\n\t.\n\\end{align}\nA strong sum packet rate performance is achieved by a scheduler that does not neglect any single user.\n\nLast, the \\emph{sum rate capacity} achieved by each transmission is calculated using the Signal-to-Noise ratio~(\\SNR) resulting from the selected channels instantaneous fading characteristics as\n\\begin{align}\n\t\\reward_C[\\timeindex]\n\t=\n\t\\sum_{\\userindex=1}^{\\numusers}\n\t\\log\n\t\\left(\n\t\t1 +\n\t\t\\channelquality_{\\userindex}[\\timeindex]\n\t\t\\frac{\\signalpower}{\\noisepower}\n\t\\right) \n\t=\n\t\\sum_{\\userindex=1}^{\\numusers}\n\t\\log\n\t\\left(\n\t\t1 + \\SNR_{\\userindex}[\\timeindex]\n\t\\right) \n\\end{align}\nfor a Gaussian input alphabet, where signal power~\\( \\signalpower \\) and expected noise power~\\( \\noisepower \\) are fixed for all vehicles~\\( \\userindex \\).\n\nThe scheduler is be tasked with balancing all performance metrics, thus, we collect all target metrics in a weighted sum utility\n\\begin{align}\n\t\\tilde{\\reward}[\\timeindex] = \n\t  \\weight_C\\reward_C[\\timeindex]\n\t+ \\weight_P \\reward_P[\\timeindex]\n\t- \\weight_L \\reward_L[\\timeindex],\n\\end{align}\nwith respective tunable weights~\\( \\weight_C, \\weight_P, \\weight_L \\). Additionally, we designate a priority class of \\textsc{\\small \\emergencyvehicle}\\xspace-type users. Their significance is communicated to the optimization process by adding \\textsc{\\small \\emergencyvehicle}\\xspace timeouts~\\( \\reward_{L,\\, \\emergencyvehicleformula}[\\timeindex] \\) to the weighted sum utility with their own tunable weight~\\( \\weight_{L,\\, \\emergencyvehicleformula} \\):\n\\begin{align}\\label{eq:reward}\n\t\\reward[\\timeindex]\n\t&=\t  \\tilde{\\reward}[\\timeindex]\n\t\t- \\weight_{L,\\, \\emergencyvehicleformula} \\reward_{L,\\, \\emergencyvehicleformula}[\\timeindex]\\nonumber\\\\\n\t&=\t  \\weight_C \\reward_C[\\timeindex]\n\t\t+ \\weight_P \\reward_P[\\timeindex]\n\t\t- \\weight_L \\reward_L[\\timeindex]\n\t\t- \\weight_{L,\\, \\emergencyvehicleformula} \\reward_{L,\\, \\emergencyvehicleformula}[\\timeindex].\n\\end{align}\n\n\n\\section{Algorithm Selection Approach}\\label{sec:deep}\nWhere model-based approaches struggle to find an optimal solution, deep Reinforcement Learning~(\\reinforcementlearning) may be applied to learn a function that approximates the optimal algorithm along the domain of reasonable input data.\n\\reffig{fig:RL_Training} schematically depicts the desired learning process.\nAt the same time, some applications require boundary conditions to be met, which tend to be easy to formulate in model-based algorithms but cannot be enforced directly in standard \\reinforcementlearning approaches.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\input{figures\/RL_training.pgf}\n\t\\caption\n\t{%\n\t\tIn the desired RL loop, the parametrized scheduler interacts with the system by selecting an action \\( \\actionsca \\) and receiving a resulting reward~\\( \\reward[\\timeindex] \\) and new system state~\\( \\statevec[\\timeindex+1] \\).\n\t\tBased on these experiences, the scheduler updates its parametrization~\\( \\parameters \\) to promote high-reward actions and demote low reward actions.\n\t}%\n\t\\label{fig:RL_Training}\n\\end{figure}\n\nOur scheduler learns to adaptively switch between a selection of model-based algorithms to tackle this problem.\nIn order to select the best algorithm, the scheduler makes use of a deep Q-Network~(\\dqn)~\\cite{sutton2018reinforcement} to learn the long-term benefit of selecting each operation mode in a given situation.\nAs a result, model-based design benefits, such as hard performance guarantees and human interpretability, are provided by the pool of model-based algorithms.\nMeanwhile, the superposition of algorithms allows for greater performance than each individual algorithm on flexible goal metrics.\nAs an additional benefit, the individual model-based algorithms do not have to be sophisticated enough to perform well in every circumstance, so long as the overall selection is rich enough to serve any problem.\n\n\\subsection{Model-Based Algorithms}\nFour model-based scheduling algorithms are implemented for the \\dqn to select from:~\\cite{schmidt2018analyse}\n\n(1) A \\textbf{Maximum Throughput\\,(\\textsc{\\small MT}\\xspace)} scheduler that allocates as many resources as requested by order of descending channel power gains.\n\n(2) A \\textbf{Max-Min-Fair\\,(\\textsc{\\small MMF}\\xspace)} scheduler looks to distribute the available resources~\\( \\numresourcesavailable \\) performantly but fairly among the number~\\( \\numusers_{\\text{req}}[\\timeindex] \\) of users that have jobs assigned to them in the job queue in time step~\\( \\timeindex \\). It allocates by order of priority\n\\( \n\t{\\channelquality_{\\userindex}[\\timeindex]\\,\n\t\/\\,\n\t\\sum_{\\jobindex \\in \\jobsset_{\\userindex}[\\timeindex]}\n\t\\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex]}\n \\), favoring good channels and small requests, but allocates at most an equal share\n\\( \n\t\\left\\lfloor\n\t\t\\numresourcesavailable\n\t\t\/\n\t\t\\numusers_{\\text{req}}[\\timeindex]\n\t\\right\\rfloor\n\\).\n\n(3) A \\textbf{Delay Sensitive\\,(\\textsc{\\small DS}\\xspace)} scheduler assigns a channel priority \n\\begin{align*}\n\t{p}_{c, \\userindex}[\\timeindex]\n\t=\n\t\\frac{\n\t\t\\packetrate_{\\userindex}[\\timeindex]\n\t}\n\t{\n\t\t\\sum_{q=1}^{\\numusers} \\packetrate_{q}[\\timeindex]\n\t}\n\t\\frac{\n\t\t\\channelquality_{\\userindex}[\\timeindex]\n\t}\n\t{\n\t\t\\sum_{q=1}^{\\numusers} \\channelquality_{q}[\\timeindex]\n\t}\n\\end{align*}\ngiven each users relative channel power gain~\\( \\channelquality_{\\userindex}[\\timeindex] \\) and packet rate~\\( \\packetrate_{\\userindex}[\\timeindex] \\).\nThe \\textsc{\\small DS}\\xspace scheduler further draws on the users sum timeouts~\\(\n\t{\\timeouts_{\\userindex}[\\timeindex] = \\sum_{\\tilde{\\timeindex}=\\num{1}}^{\\timeindex} \\reward_{L, \\userindex}\\left[\\tilde{\\timeindex}\\right]} \n\\) and lowest remaining time~\\(\n\t{\\lowesttimetotimeout_{\\userindex}[\\timeindex] = \\min_{\\jobindex \\in \\jobsset_{\\userindex}[\\timeindex]} \\timetotimeout_{\\jobindex}[\\timeindex]}\n\\) for a timeout urgency\n\\begin{align*}\n\t{p}_{l, \\userindex}[\\timeindex]\n\t=\n\t\\frac{\n\t\t\\timeouts_{\\userindex}[\\timeindex]\n\t\t\/\n\t\t\\lowesttimetotimeout_{\\userindex}[\\timeindex]\n\t}\n\t{\n\t\t\\sum_{q=1}^{\\numusers}\n\t\t\\timeouts_{q}[\\timeindex]\n\t\t\/\n\t\t\\lowesttimetotimeout_{q}[\\timeindex]\n\t}\n\t.\n\\end{align*} \nEach user is allotted a share of the available resources according to the normalized, weighted priority vector\n\\begin{align*}\n\t\\mathbf{p}[\\timeindex]\n\t=\n\t(\n\tw_1\\mathbf{p}_{c}[\\timeindex]\n\t+\n\tw_2\\mathbf{p}_{l}[\\timeindex]\n\t)\n\t\/\n\t(\n\tw_1\n\t+\n\tw_2\n\t)\n\t.\n\\end{align*}\nUniquely, this scheduler skips jobs that are about to time out if the allotted discrete resources are not sufficient to complete the job. In this case, the resources are freed for the next highest priority.\n\n(4) An \\textbf{\\textsc{\\small \\emergencyvehicle}\\xspace Priority} scheduler assigns as many resources as requested to any \\textsc{\\small \\emergencyvehicle}\\xspace{}s and distributes remaining resources one-by-one, randomly assigning them to requesting users.\n\n\n\\subsection{Deep Algorithm Selection}\nOur deep learning scheduler consists of multiple elements: A pre- and post-processor, a \\dqn, an~\\( \\argmax \\) module, a memory module, and a learning algorithm that tunes the \\dqn. In this section, we will introduce these modules and illustrate their interconnection.\n\nFirst, to be able to make informed decisions, a small preprocessor prepares information about the current state of job queue and communication link as a system state vector~\\( \\statevec[\\timeindex] \\). Per user~\\( \\userindex \\), the preprocessor summarizes\n\\begin{itemize}\n\t\\item current queue length \\(\n\t\t\\statesca_{\\userindex, 1}[\\timeindex]\n\t\t\t= \\sum_{\\jobindex \\in \\jobsset_{\\userindex}[\\timeindex]} \\resourceblock_{\\jobindex,\\,\\text{req}}[\\timeindex]\n\t\t\\),\n\t\\item channel power gains \\(\n\t\t\\statesca_{\\userindex, 2}[\\timeindex]\n\t\t\t= \\channelquality_\\userindex[\\timeindex]\n\t\t\\),\n\t\\item average remaining time \\(\n\t\t\\statesca_{\\userindex, 3}[\\timeindex]\n\t\t\t= \\frac{1}{\\left| \\jobsset_{\\userindex}[\\timeindex] \\right|} \\sum_{\\jobindex \\in \\jobsset_{\\userindex}[\\timeindex]} \\timetotimeout_{\\jobindex}[\\timeindex]\n\t\t\\),\n\t\\item minimum remaining time \\(\n\t\t\\statesca_{\\userindex, 4}[\\timeindex]\n\t\t\t= \\min_{\\jobindex \\in \\jobsset_{\\userindex}[\\timeindex]} \\timetotimeout_{\\jobindex}[\\timeindex]\n\t\t\\),\n\t\\item and past packet rate \\(\n\t\t\\statesca_{\\userindex, 5}[\\timeindex]\n\t\t\t= \\packetrate_\\userindex[\\timeindex]\n\t\t\\),\n\\end{itemize}\nfor a length of \\( 5\\numusers \\) features for \\( \\numusers \\) users.\n\nFor a given system state~\\( \\statevec[\\timeindex] \\) and choice of model-based algorithm~\\( \\actionsca_{\\actionindex} \\), we define the long term expected rewards\n\\begin{align}\n\t\\label{eq:def_q}\n\t\\longrewardsca(\\statevec[\\timeindex], \\actionsca_{\\actionindex})\n\t=\n\t\\expectation\n\t\\left[\n\t\\sum_{z=\\timeindex}^{\\infty}\n\t\\rewarddiscount^{z-\\timeindex}\n\t\\reward[\\timeindex]\n\t\\;\\middle|\\;\n\t\\statevec = \\statevec[\\timeindex],\\,\n\t\\actionsca = \\actionsca_{\\actionindex}\n\t\\right],\n\\end{align}\nwith reward~\\( \\reward \\) as defined in~\\refeq{eq:reward}. The long term rewards are discounted by an exponential factor~\\(\n\t{\\num{0} \\leq \\rewarddiscount \\leq \\num{1}}\n\\).\nAs depicted in \\reffig{fig:dqn}, a \\dqn is set up to output an estimate~\\( \\longrewardestimsca \\) of the long term expected rewards~\\( \\longrewardsca \\) for each choice of model-based algorithm~\\( \\actionsca_{\\actionindex} \\).\nGiven a perfect approximation~\\(\n\t{\\longrewardestimsca = \\longrewardsca}\n\\), we maximize the long term expected rewards by simply selecting whichever action~\\(\n\t\\argmax_{\\actionindex} \\longrewardestimsca(\\statevec[\\timeindex],\\,\\actionsca_{\\actionindex})\n\\) has the highest \\emph{estimated} long term rewards.\nTherefore, the learning goal is to update the \\dqn parameters~\\( \\parameters \\) such that the estimate is approximately close to the true long term reward~\\( \\longrewardsca \\) for all possible~\\(\n\t{(\\statevec, \\actionsca_{\\actionindex})}\n\\), relying on the universal approximator property of neural networks~\\cite{sutton2018reinforcement}.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\input{figures\/DQN.pgf}\n\t\\caption\n\t{%\n\t\tThe DQN module estimates expected long term rewards~\\(\n\t\t{\\longrewardestimsca(\\statevec[\\timeindex], \\actionsca_\\actionindex)}\n\t\t\\) for a given state~\\( \\statevec[\\timeindex] \\) and all available actions~\\( \\actionsca_{\\actionindex} \\), according to its current parametrization~\\( \\parameters \\).\n\t\tIn this case, actions~\\( \\actionsca_{\\actionindex} \\) are the available model-based algorithms. The scheduler then selects the algorithm~\\( \\actionsca_{\\actionindex} \\) with the highest expected long term reward.\n\t}%\n\t\\label{fig:dqn}\n\\end{figure}\n\nFirst, the scheduler must make experiences to learn from.\nUsing an \\( \\explorationchance \\)-greedy exploration scheme, the scheduler initially explores the simulation by taking random actions, \\ie selecting a random model-based algorithm for a given state.\nStates, decisions, and their direct outcome are recorded and stored in a replay buffer in the form of tuples\n\\begin{align}\n\t\\experience\n\t=\n\t\\left(\n\t\t\\statevec[\\timeindex],\\,\n\t\t\\actionsca[\\timeindex],\\,\n\t\t\\reward[\\timeindex],\\,\n\t\t\\statevec[\\timeindex+1]\n\t\\right)\n\t.\n\\end{align}\n\nWe highlight that an experience only contains the immediate reward~\\( \\reward[\\timeindex] \\), not the desired long-term rewards~\\(\n\t{\\longrewardsca(\\statevec[\\timeindex], \\actionsca[\\timeindex])}\n\\), \\ie~\\refeq{eq:def_q}.\nHowever, we can extract additional information from~\\( \\statevec[\\timeindex+1] \\), the state that following the action. By again using our \\dqn, we estimate the rewards following~\\( \\statevec[\\timeindex+1] \\) to construct a learning target\n\\begin{align}\n\t\\label{eq:bootstrap}\n\t\\longrewardestimsca_{\\text{target}}(\\experience)\n\t=\n\t\\reward[\\timeindex]\n\t+\n\t\\rewarddiscount\n\t\\max_\\actionindex\n\t\t\\longrewardestimsca(\\statevec[\\timeindex+1], \\actionsca_\\actionindex)\n\t,\n\\end{align}\nthat incorporates the information of~\\( \\reward[\\timeindex] \\) and ~\\( \\statevec[\\timeindex + \\num{1}] \\) from the experience.\nUsing this target~\\( {\\longrewardestimsca_{\\text{target}}(\\experience)} \\), an estimation error~\\cite{sutton2018reinforcement}\n\\begin{align}\n\t\\tderror\n\t=\n\t\\longrewardestimsca_{\\text{target}}\\left( \\experience \\right)\n\t-\n\t\\longrewardestimsca \\left( \\experience \\right)\n\\end{align}\ngiven the networks current parametrization can be calculated for any experience tuple from the buffer.\n\nFollowing the principle of stochastic gradient descent~(\\stochasticgradientdescent), the networks parameters~\\( \\parameters \\) are then adjusted by sampling a minibatch of~\\( \\batchsize \\) experiences from the buffer to minimize a mean square error cost\n\\begin{align}\n\t\\cost\n\t=\n\t\\frac{\n\t\t1\n\t}\n\t{\n\t\t\\batchsize\n\t}\n\t\\sum_{\\batchindex=1}^{\\batchsize}\n\t(\n\t\t\\tderror_{\\batchindex}\n\t)^2\n\\end{align}\nfor the batch.\nMinibatch parameter updates are carried out every time a new experience is made.\n\nAs training progresses and the schedulers understanding of the simulation environment improves, the probability~\\( \\explorationchance \\) of selecting random exploration actions is gradually decreased, relying more and more on the network to make decisions. When prompted, at the beginning of a time step~\\( \\timeindex \\), the network will estimate the expected long term rewards~\\( \\longrewardestimsca \\) for selecting any of the available model-based algorithms~\\( \\actionsca_{\\actionindex} \\), given the current state~\\( \\statevec[\\timeindex] \\) and the networks current parameters~\\( \\parameters \\). The scheduler then selects the model-based algorithm~\\( \\actionsca_{\\actionindex} \\) with the highest expected long term rewards.\n\nTo increase training efficiency, we implement optimizations to the base \\dqn learning method: \n\\begin{itemize}\n\t\\item An infrequently updated network copy is used for the bootstrapping in \\refeq{eq:bootstrap} to increase training stability (\\emph{Target Network},~\\cite{mnih2015human})\n\t\\item Sampling of experiences from the buffer is weighted proportional to the experiences' estimation error magnitude (\\emph{Prioritized Replay},~\\cite{hessel2018rainbow})\n\t\\item The neural network structure is altered to seperately learn the contribution of state and action to the reward estimate (\\emph{DuelingDQN},~\\cite{hessel2018rainbow})\n\\end{itemize}\n\n\n\\section{Performance Evaluation}\\label{sec:performance}\n\\subsection{Implementation Details}\nWe configure the simulation with \\(\n\t{\\numresourcesavailable = \\num{16}}\n\\)~available resources and \\(\n\t{\\numusers = \\num{10}}\n\\) users.\nUser profiles are set up according to \\reftab{tab:user_profiles}, with five~'Normal', two~'High Datarate', two~'Low Latency' users, and one~'Emergency Vehicle'.\nFor the given configuration, a job creation probability~\\(\n\t{\\probability_{\\jobindex} = \\SI{20}{\\percent}}\n\\) for each simulation step and user puts a high expected load of \\( \\num{1.6} \\)~requests per available resource on average on the schedulers.\nWe run a total of \\( \\num{10000} \\)~episodes at \\( \\num{50} \\)~time steps~\\( \\timeindex \\) per episode for training and evaluation each, with the mini batch size set to~\\( {\\batchsize = 64} \\) sampled experiences per training step.\nParameter optimization via \\stochasticgradientdescent is carried out by the Adam optimizer~\\cite{kingma_adam_2015} with default settings and a learning rate of~\\( \\num{1e-4} \\).\nReward weightings are set to \\( {\\weight_C = \\weight_P = \\num{0.25}} \\), \\( {\\weight_L = \\weight_{L,\\, \\emergencyvehicleformula} = \\num{1.0}} \\), giving roughly equal significance to each goal metric with the average expected magnitude of the respective rewards.\n\n\\begin{table}[tb]\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\caption{User Profiles}\n\t\\label{tab:user_profiles}\n\t\\begin{center}\n\t\t\\begin{tabular}{c|c|c}\n\t\t\t\\hline\n\t\t\t& \\textbf{Delay} & \\textbf{Max Job Size} \\( \\resourceblock_{\\text{\\text{max}}} \\) \\\\ \n\t\t\t& in sim. steps\t\t& in res. blocks\t\t\\\\\\hline\\hline\n\t\t\tNormal\t\t\t\t& \\( \\num{20} \\)\t\t\t\t& \\( \\num{30} \\) \\\\\n\t\t\tHigh Packet Rate\t& \\( \\num{20} \\)\t\t\t\t& \\( \\num{40} \\) \\\\\n\t\t\tLow Latency\t\t\t& \\( \\num{2} \\)\t\t\t\t\t& \\( \\num{8} \\) \\\\\n\t\t\tEmergency Vehicle\t& \\( \\num{1} \\)\t\t\t\t\t& \\( \\num{16} \\) \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\nFor the algorithm selection \\dqn, a five layer feed-forward network is selected.\nLayers have \\( \\num{300} \\)~nodes each, except for the last layer that was split in two branches with \\( \\num{200} \\)~nodes each according to the DuelingDQN structure~\\cite{hessel2018rainbow}.\nExploration is done by selecting random actions at an initial chance~\\(\n\t{\\explorationchance = \\SI{99}{\\percent}}\n\\), decaying linearly to~\\( \\SI{0}{\\percent} \\) after~\\( \\SI{80}{\\percent} \\) of training episodes.\nThe exponential future reward decay factor is set to~\\( {\\rewarddiscount = \\num{0.9}} \\) in order to put significance on only a low number of future steps. User TX-SNR~\\( {\\signalpower \/ \\noisepower} \\) is fixed to \\( \\SI{13}{\\dB} \\).\n\nWe implement the simulation in python using the tensorflow library primarily. The full implementation is made available in~\\cite{gracla2020code}.\n\n\\subsection{Results}\n\nAs the simulation model contains stochastic components, the results achievable on each metric have an inherent variance depending on the specific realizations.\nFor example, a spike in generated jobs will result in timeouts irrespective of scheduling method.\nFor this reason, results achieved during testing are displayed in cumulative histograms.\n\n\\begin{figure*}[tb]\n\t\\centering\n\t\\subfloat[]{\n\t\t\\input{figures\/scheduling_testing_capacities.pgf}\n\t\t\\label{fig:results_capacity}\n\t}\n\t\\hfil\n\t\\subfloat[]{\n\t\t\\input{figures\/scheduling_testing_datarate_satisfaction.pgf}\n\t\t\\label{fig:results_packet_rate}\n\t}\n\t\\\\\n\t\\subfloat[]{\n\t\t\\input{figures\/scheduling_testing_latency_violations.pgf}\n\t\t\\label{fig:results_latency_violations}\n\t}\n\t\\hfil\n\t\\subfloat[]{\n\t\t\\input{figures\/scheduling_testing_timeouts_ev_only.pgf}\n\t\t\\label{fig:results_ev_latency_violations}\n\t}\n\t\\caption{Cumulative histograms of performance of Maximum Throughput~(MT), Max-Min-Fair~(MMF) and Delay Sensitive~(DS) schedulers as well as DQN adaptive scheduler on submetrics of capacity~(a), packet rate~(b), timeouts~(c) and EV timeouts~(d).}\n\t\\label{fig:results_submetrics}\n\\end{figure*}\n\n\\reffig{fig:results_reward} shows each schedulers performance on the combined reward metric~\\( \\reward \\).\nOn this metric, the \\dqn adaptive scheduler is able to find a strategy that outperforms any single model-based algorithm.\nBreaking the sum metric~\\( \\reward \\) down into its constituent parts sheds light on how this is achieved. As shown in \\reffig{fig:results_submetrics}, the \\dqn scheduler balances the submetrics against each other, yielding some performance on each of them compared to the benchmark.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\input{figures\/scheduling_testing_rewards.pgf}\n\t\\caption{Cumulative histogram of performance of Maximum Throughput~(MT), Max-Min-Fair~(MMF) and Delay Sensitive~(DS) schedulers as well as DQN adaptive scheduler on the weighted sum reward metric. For each episode, all achieved rewards \\( \\reward \\) are summed. Achieved reward sums are normalized by the highest achieved reward sum.}\n\t\\label{fig:results_reward}\n\\end{figure}\n\nOf particular interest are the \\textsc{\\small \\emergencyvehicle}\\xspace-specific timeouts in \\reffig{fig:results_ev_latency_violations}.\nThis metric is not specifically targeted by any of the model-based algorithms depicted.\nA modest double-weighting of \\textsc{\\small \\emergencyvehicle}\\xspace specific timeouts within the goal metric~\\( \\reward \\), combined with the introduction of the otherwise suboptimal \\textsc{\\small \\emergencyvehicle}\\xspace Priority scheduling algorithm to the selection pool, has enabled the \\dqn based scheduler to significantly suppress timeouts in Emergency Vehicles even compared to the otherwise timeout focused Delay Sensitive algorithm without hurting overall performance overmuch.\nAs \\reffig{fig:results_algorithm_selections} shows, the \\textsc{\\small \\emergencyvehicle}\\xspace Priority algorithm was only selected a comparably low amount of times to achieve this goal.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\input{figures\/scheduling_testing_algorithm_selection.pgf}\n\t\\caption{Relative selection rate of EV Priority~(EV P), Delay Sensitive~(DS), Max-Min-Fair~(MMF) and Maximum Throughput~(MT) schedulers during testing.}\n\t\\label{fig:results_algorithm_selections}\n\\end{figure}\n\nThe \\dqn{}s learned behaviour can also be monitored to reveal underlying features of the simulation.\nAs \\reffig{fig:results_algorithm_selections} highlights, for the given reward weighting, the \\textsc{\\small MMF}\\xspace algorithm was only selected a low number of times.\nFurther investigation could reveal whether these are, for example, high impact outlier cases that could be better served with an additional model-based algorithm that is specifically targeted to them, or whether the \\textsc{\\small MMF}\\xspace algorithm is not fit for the given scenario.\n\nThe \\dqn decision making does however add another layer of computation to the scheduling process.\nFurther, while the ensemble method can relax the sophistication required from each model-based part of the ensemble, the composite scheduling function can only assume the function space spanned by the group of model-based algorithms.\nIn other words, the \\dqn adaptive method is unable to discover strategies that go beyond combining the available models.\nDetermining whether the selection of model-based algorithms provided is rich enough to serve the problem therefore remains a burden on the designer.\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we presented a \\reinforcementlearning-based communications resource scheduler that constructs a scheduling paradigm based on an ensemble of model-based algorithms.\nWe achieve this by adaptively switching to whichever algorithm promises the highest expected long term benefit based on the current queue state.\nThis approach combines the flexible goal optimization of \\reinforcementlearning methods with the rigid predictability of model-based algorithms, which makes it noteworthy for applications such as scheduling \\textsc{\\small \\emergencyvehicle}\\xspace data that demand high performance, but have low tolerance for negative outliers as they may occur in classic \\reinforcementlearning approaches.\n\nFor the simulation presented, the adaptive model-switching scheduler is able to outperform single, model-based algorithms on a weighted sum utility metric.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\subsection{Background and Significance of Hyperspectral Remote Sensing}\n\\IEEEPARstart{I}{maging} spectroscopy, which was first-ever to be conceptualized by Goetz1 \\textit{et al.} in 1980's \\cite{goetz1985imaging}, is a seminal hyperspectral (HS) imaging technique of truly achieving the integration of the 1-D spectrum and the 2-D image for earth remote sensing (RS). Imaging spectroscopy is a typical ``passive'' RS technique, which assembles spectroscopy and digital photography into a unified system. Fig. \\ref{fig:GRSM_Imaging} shows the data acquisition process of two different imaging patterns: ``active'' RS and ``passive'' RS \\cite{tsang1985theory}. The resulting HS image collects hundreds of 2-D images finely sampled from the approximately contiguous wavelength across the whole electromagnetic spectrum \\cite{turner2003remote} (see Fig. \\ref{fig:electromagnetic}). This enables the recognition and identification of the materials, particularly for those that have extremely similar spectral signatures in visual cues (e.g., RGB) \\cite{hong2015novel}, at a more accurate and finer level. As a result, HS RS has been significantly advanced and widely applied in many challenging tasks of earth observation \\cite{bioucas2013hyperspectral}, such as fine-grained land cover classification, mineral mapping, water quality assessment, precious farming, urban planning and monitoring, disaster management and prediction, and concealed target detection.\n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\\includegraphics[width=0.48\\textwidth]{GRSM_Imaging}\n        \\caption{An illustration to clarify the data acquisition process of two different imaging patterns, i.e., ``active'' RS and ``passive'' RS. }\n\\label{fig:GRSM_Imaging}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\t  \\centering \n\t\t\\includegraphics[width=0.8\\textwidth]{electromagnetic}\n        \\caption{A showcase to clarify the electromagnetic spectrum: the order from low to high frequency is Long-waves, Radio-waves, Micro-waves, Infrared, Visible, Ultraviolet, X-rays, and Gamma-rays, where four widely-concerned intervals, e.g., Radio-waves, Infrared, Visible, and X-rays, are finely partitioned.}\n\\label{fig:electromagnetic}\n\\end{figure*}\n\nMore specifically, characterized by the distinctive 3-D signaling structure, the advantages of the HS image over conventional 1-D or 2-D signal products can be summarized as\n\\begin{itemize}\n    \\item compared to the common 1-D signal system, the HS 2-D spatial pattern provides us more structured information, enabling the discrimination of underlying objects of interest at a more semantically meaningful level;\n    \\item Beyond the 2-D natural images, the rich spectral information in HS images is capable of detecting materials through tiny discrepancies in the spectral domain, since the HS imaging instruments exploit the sensors to collect hundreds or thousands of wavelength channels with an approximately continuous spectral sampling at a subtle interval (e.g., 10nm). \n\\end{itemize}\n\nFurthermore, the significance of HS images compared to other RS imaging techniques with a lower spectral resolution, e.g., multispectral (MS) or RGB imaging, mainly embodies in the following three aspects:\n\\begin{itemize}\n    \\item[1)] HS images are capable of finely discriminating the different classes that belong to the same category, such as \\textit{Bitumen} and \\textit{Asphalt}, \\textit{Stressed Grass} and \\textit{Synthetic Grass}, \\textit{Alunite} and \\textit{Kaolin}. For those optical broadband imaging products (e.g., MS imagery), they can only identify certain materials with the observable differences in the spectral signatures, e.g., \\textit{Water}, \\textit{Trees}, \\textit{Soil}.\n    \\item[2)] The higher spectral resolution creates the possibilities to some challenging applications that can be hardly achieved by only depending on formerly imaging techniques, e.g., parameter extraction of biophysics and biochemistry, biodiversity conservation, monitoring and management of the ecosystem, automatic detection of food safety, which provides new insight into RS and geoscience fields.\n    \\item[3)] Due to the limitations of image resolution either in spectral or spatial domains, physically and chemically atmospheric effects, and environmental conditions (e.g., the interference of soil background, illumination, the uncontrolled shadow caused by clouds or building occlusion, topography change, complex noises), those traditional RS imaging techniques were to a great extent dominated by qualitative analysis. As HS RS arises, quantitative or semi-quantitative analysis becomes increasingly plausible in many practical cases.\n\\end{itemize}\n\n\\subsection{An Ever-Growing Relation between Non-convex Modeling and Interpretable AI in Hyperspectral Remote Sensing}\n\nIn recent years, a vast number of HS RS missions (e.g., MODIS, HypSEO, DESIS, Gaofen-5, EnMap, HyspIRI, etc.) have been launched to enhance our understanding and capabilities to the Earth and environment, contributing to the rapid and better development in a wide range of relevant applications, such as land cover land use classification, spectral unmixing, data fusion, image restoration, and multimodal data analysis. With the ever-growing availability of RS data sources from both satellite and airborne sensors on a large scale and even global scale, expert system-centric data processing and analysis mode has run into bottlenecks and can not meet the demand of the big data era. For this reason, data-driven signal and image processing, machine learning (ML), AI models have been garnering growing interest and attention from the researchers in the RS community.\n\nSupported by well-established theory and numerical optimization, convex models have been proven to be effective to model a variety of HS tasks under highly idealized assumptions. However, there exist unknown, uncertain, and unpredictable factors in the complex real scenes. Due to these factors that lead to the lack of sound understanding and modeling capability to the scene, convex models fail to work properly. The specific reasons could be two-fold. On the one hand, integrating the benefits of 1-D and 2-D signals, the 3-D structurized HS images offer greater potential and better solutions (compared to natural images) to deal with the varying situation, but simultaneously increase the model's complexity and uncertainty to some extent. On the other hand, due to unprecedented spatial, spectral, and temporal resolutions of HS images in remotely sensed HS imaging, the difficulties and challenges in the sophisticated HS vision approaches are mainly associated with the volume of the HS data, complex material (spectral) mixing behavior, and uncontrolled degradation mechanisms in data acquisition caused by illumination, noise, and atmospheric effects. \n\nThe aforementioned factors, to a great extent, limit convex models to be intelligent approaches for fully understanding and interpreting the real-life scenario. Therefore, this naturally motivates us to investigate the possibility of processing and analyzing the HS data in a non-convex modeling fashion. In the following, we briefly make a qualitative comparison between convex and non-convex models to clarify that non-convex modeling might be an optimally feasible solution towards interpretable AI models in HS RS.\n\n\\begin{itemize}\n    \\item Convex models are theoretically guaranteed to converge to the global optimal solution, yet most tasks related to HS RS are complex in reality and hardly simplified to an equivalent and perfect convex formulation. This to some extent makes convex models inapplicable to practical tasks, due to the lack of interpretability and completeness for problem modeling.\n    \\item Rather, non-convex models are capable of characterizing the complex studied scene in HS RS more finely and completely, thereby more tending to achieve automatization and intelligentization in the real world. Moreover, by excavating the intrinsic properties from the HS data effectively to yield physically meaningful priors, the solution space of non-convex models can be shrunk to a ``good'' region bit by bit.\n    \\item Although non-convex models are complex by considering more complicated prior knowledge, possibly leading to the lack of stable generalization ability, they hold the higher potential that convex models do not have, particularly in explaining models, understanding scenes, and achieving intelligent HS image processing and analysis. Furthermore, this might be able to provide researchers with a broader range of HS vision related topics, making it applicable for more real cases in a variety of HS RS-related tasks.\n\\end{itemize}\n\n\\subsection{Contributions}\nWith the advent of the big data era, an ever-increasing data bulk and diversity brings rare opportunities and challenges for the development of HS RS in earth observation. Data-driven AI approaches, e.g., ML-based, deep learning (DL)-based, have occupied a prominent place in manifold HS RS applications. Nevertheless, how to open the ``model'' and give them interpretability remains unknown yet. In this article, we raise a bold and understandable standpoint, that is, non-convex modeling might be an effective means to bridge the gap between interpretable AI models and HS RS. To support the opinion, this article provides a detailed and systematic overview by reviewing advanced and latest literature with an emphasis on non-convex modeling in terms of five classic and burgeoning topics related to HS RS. More specifically, \n\\begin{itemize}\n    \\item We present a comprehensive discussion and analysis related to non-convex modeling in five well-noticed and promising HS RS-related applications, such as HS image restoration, dimensionality reduction and classification, spectral unmixing, data fusion and enhancement, and cross-modality learning. \n    \\item For each topic, a few representative works are emphatically and detailedly introduced by the attempts to make a connection between non-convex modeling and intelligent\/interpretable models. Moreover, the example experiments (qualitatively or quantitatively) are subsequently performed after the detailed method description. Those selected methods that engage in the comparative experiments are accompanied by available code and data links for the sake of reproducibility. Finally, the remaining challenges are highlighted to further clarify the gap between interpretable ML\/AI models and practical HS RS applications. \n    \\item Regarding the three aspects of non-convex modeling, interpretable AI, and HS RS, the end of this article concludes with some remarks, makes summary analysis, and hints at plausible future research work.\n\\end{itemize}\n\nWe need to point out and emphasize, however, that this paper features food for thoughts for advanced readers, and it is not an introduction for beginners entering the field. The goal of this paper is to provide a cutting-edge survey rather than a real tutorial. As a result, readers are expected to have some prior knowledge across multidisciplinary, such as HS RS, convex optimization, non-convex modeling, ML, and AI, where some basic principle, definitions, and deductions need to be mastered. For beginners who are willing to start with new researches on non-convex modeling for HS RS applications, we recommend reading and learning following materials and references to get to know, for example, \n\\begin{itemize}\n    \\item what is the HS imaging or HS RS (e.g., principles, superiority, practicability) and its relevant applications (topics) \\cite{bioucas2013hyperspectral};\n    \\item convex optimization and its solutions (including a detailed description of physical meaningful priors, e.g., low-rank, sparsity, graph regularization, non-negativity, sum-to-one, etc.) as well as its relationship with non-convex modeling \\cite{boyd2004convex};\n    \\item a general guideline on why and how to build non-convex models and how to solve non-convex minimization problems \\cite{ekeland1979nonconvex};\n    \\item classic and advanced ML algorithms, including essential ideas, designing thought, implementation process \\cite{mohri2018foundations};\n    \\item a big picture about what is the AI and how to build the basic AI models \\cite{nilsson2014principles}.\n\\end{itemize}\n\nMoreover, we hope that this paper can be also regarded as a good starting point and evolve many novels, interesting, and noteworthy research issues around the fields of non-convex modeling, interpretable ML\/AI, and HS RS, serving to more application cases in reality. \n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=1\\textwidth]{SPM_Motivation}\n        \\caption{Illustration of five promising topics in HS RS, including image restoration, dimensionality reduction and classification, data fusion and enhancement, spectral unmixing, and cross-modality learning.}\n\\label{fig:motivation}\n\\end{figure*}\n\n\\section{Outline and Basic Preparation}\n\nThis paper starts with a brief introduction to a general non-convex minimization problem in signal or image processing, and then specifies the non-convex modeling with each topic in HS RS from an ML perspective. According to the characteristics of HS imaging, these different issues may bring fresh challenges to researches on non-convex modeling and optimization, which contributes to boosting the development of both HS RS and ML for intelligent data processing and analysis. Very recently, some successful showcases have garnered growing attention from researchers who engage in HS data processing and analysis, ML, or statistical optimization, leading to many newly-developed non-convex models and their solutions. They can be roughly categorized into the following several groups in a wide range of real applications related to HS RS. Fig. \\ref{fig:motivation} gives the illustration for each topic.\n\n\\subsection{A Brief Introduction from Convex to Non-convex Models}\nAs the name suggests, in the convex model, the shape of the area represented by the function (or model) is convex. Accordingly, let the function $f: \\mathbb{R}_{n}\\rightarrow\\mathbb{R}$ be convex, if the domain of $f$ is a convex set, and for the variable $\\theta\\in [0,1]$, any two points (e.g., $x$ and $y$) meet the following condition:\n\\begin{equation}\n\\label{g_eq1}\n\\begin{aligned}\n      f(y)\\geq \\theta f(x) + (1-\\theta) f(y),\n\\end{aligned}\n\\end{equation}\nthen the function $f$ is convex, whose necessary and sufficient condition is $f(y)\\geq f(x)+\\bigtriangledown f(x)(x-y)$.\n\nWithin the domain, the globally optimal solution of the convex model can be obtained by common convex optimization techniques, such as linear programming \\cite{chvatal1983linear}, quadratic programming \\cite{frank1956algorithm}, second order cone programming \\cite{lobo1998applications}.\n\nInformally, although convex methods have been widely used to model various tasks, owing to the existence and uniqueness of solutions and the relatively low model complexity, the real scenes are complex and changeable, inevitably leading to many uncertainties and difficulties in the modeling process. For this reason, non-convex modeling is capable of providing stronger modeling power to the algorithm designer and can fully meet the demand of characterizing the real complex scenes well. This naturally motivates us to shift our emphases on some key issues related to non-convex modeling.\n\nA general non-convex model usually consists of a smooth objective function (\\textit{e.g.,} Euclidean loss, negative log-likelihood) with Lipschitz gradient $f(\\mathbf{X})$ and the non-convex constraints $\\mathcal{C}$, which can be generalized by optimizing the following minimization problem:\n\n\\begin{equation}\n\\label{g_eq2}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{X}}\\frac{1}{2}f(\\mathbf{X}) \\;\\; {\\rm s.t.}\\; \\mathbf{X}\\in \\mathcal{C},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{X}$ is the to-be-estimated variable, which can be defined as a vector (1-D signal), a matrix (2-D image), or an unfolded matrix (3-D HS image). The constraints $C$ could be sparsity-promoting variants, low-rank, TV, and others, which need to be determined by the specific tasks.\n\nUnlike convex models, there exist many local minimums in non-convex models. This poses a big challenge on finding globally optimal solutions. For possible solutions of non-convex methods, one strategy is to relax the non-convex problem to an approximately convex model \\cite{jain2017non}. Another can break the non-convex problems down into several convex subproblems and solve them in parallel by the means of convex ways \\cite{boyd2011distributed}.\n\nIn the light of the different research goals, the general model in Eq. (\\ref{g_eq2}) can be extended to the task-driven variants covering a broad scope within the HS RS, including image restoration, dimensionality reduction, data fusion and enhancement, spectral unmixing, and cross-modality learning. It should be noted that in the following sections, some representative methods will be introduced with a focus on non-convex modeling, while the alternating direction method of multipliers (ADMM) optimization framework is recommended for use as a general solver to solve these non-convex models. For specific solutions of these models in each topic, please refer to the cited references.\n\n\\subsection{Main Abbreviation}\n\\vspace{5pt}\n\\begin{supertabular}{ll}\nAI: & artificial intelligence.\\\\\nANN: & artificial neural network.\\\\\nCML: & cross-modality learning.\\\\\nDL: & deep learning.\\\\\nDR: & dimensionality reduction.\\\\\nGSD: & ground sampling distance.\\\\\nHS: & hyperspectral.\\\\\nKNN: & $k$ nearest neighbors.\\\\\nLDA: & linear discriminant analysis.\\\\\nLMM: & linear mixing model.\\\\\nMA: & manifold alignment.\\\\\nML: & machine learning.\\\\\nMML: & multimodality learning.\\\\\nMS: & multispectral.\\\\\nNMF: & non-negative matrix factorization.\\\\\nRS: & remote sensing.\\\\\nSAR: & synthetic aperture radar.\\\\\nSOTA: & state-of-the-art.\\\\\nSSL: & shared subspace learning.\\\\\nSU: & spectral unmixing.\\\\\n1-D: & one-dimensional.\\\\\n2-D: & two-dimensional.\\\\\n3-D: & three-dimensional.\\\\\n\\end{supertabular}\n\n\\subsection{Nomenclature}\n\\vspace{5pt}\n\\begin{supertabular}{ll}\n$\\tensor{X}$: & to-be-estimated 3-D HS image.\\\\\n$\\tensor{Y}$: & observed 3-D HS image.\\\\\n$\\tensor{N}_{G}$: & 3-D Gaussian noise. \\\\\n$\\tensor{N}_{S}$: & 3-D sparse noise.\\\\\n$\\tensor{O}$: & core tensor.\\\\\n$r$: & rank of matrix.\\\\\n$\\mathbf{X}$: & unfolded 2-D matrix of $\\tensor{X}$.\\\\\n$\\mathbf{x}_{i}$: & the $i$-th pixel (1-D vector) of $\\mathbf{X}$.\\\\\n$\\mathbf{Y}$: & unfolded 2-D matrix of $\\tensor{Y}$.\\\\\n$\\mathbf{N}_{S}$: & unfolded 2-D matrix of $\\tensor{N}_{S}$.\\\\\n$\\mathbf{H}$: & first-order difference matrix.\\\\\n$\\mathcal{C}$: & model constraint set.\\\\\n$\\Phi$: & to-be-estimated variable set.\\\\\n$f$: & transformation functions.\\\\\n$\\mathbf{Q}$: & combination coefficients of NMF.\\\\\n$\\mathbf{W}$: & graph or manifold structure.\\\\\n$\\mathbf{L}$: & Laplacian matrix.\\\\\n$\\mathbf{D}$: & degree matrix.\\\\\n$\\mathbf{U}$: & subspace projection matrix.\\\\\n$\\mathbf{I}$: & identity matrix.\\\\\n$\\mathbf{d}$: & distance or similarity matrix.\\\\\n$\\sigma$: & standard derivation.\\\\\n$C_{k}$: & sample set of the $k$-th class.\\\\\n$\\mathbf{M}$: & one-hot encoded matrix.\\\\\n$\\mathbf{P}$: & regression matrix.\\\\\n$\\mathbf{E}$: & endmember matrix.\\\\\n$\\mathbf{E}_{0}$: & reference endmember matrix.\\\\\n$\\mathbf{A}$: & abundance matrix.\\\\\n$\\mathbf{S}$: & scaling factors (matrix).\\\\\n$\\mathbf{V}$: & spectral variability dictionary (matrix).\\\\\n$\\mathbf{J}$: & coefficients corresponding to $\\mathbf{V}$.\\\\ \n$\\mathbf{R}$: & spatial degradation function.\\\\\n$\\mathbf{G}$: & spectral response function.\\\\\n$\\mathbf{N}_{H}$: & HS noise.\\\\\n$\\mathbf{N}_{H}$: & MS noise.\\\\\n$\\mathbf{Z}$: & high spatial resolution MS image.\\\\\n$m$: & the number of the considered modality.\\\\\n$c$: & scaling constant.\\\\\n\\end{supertabular}\n\n\\subsection{Notation}\n\\begin{supertabular}{ll}\n$\\norm{\\mathbf{X}}_{\\F}$ & Forbenius norm of $\\mathbf{X}$, obtained by $\\sqrt{\\sum_{i,j}\\mathbf{X}_{i,j}^{2}}$\\\\\n$\\norm{\\mathbf{X}}_{1,1}$ & $\\ell_{1}$ norm of $\\mathbf{X}$, obtained by $\\sum_{i,j}|\\mathbf{X}_{i,j}|$\\\\\n$\\norm{\\mathbf{X}}_{2,1}$ & $\\ell_{2,1}$ norm of $\\mathbf{X}$, obtained by $\\sum_{i}|\\sqrt{\\sum_{j}\\mathbf{X}_{i,j}^{2}}|$\\\\\n$\\norm{\\mathbf{X}}_{1\/2}$ & $\\ell_{1\/2}$ norm of $\\mathbf{X}$, obtained by $\\sum_{i,j}\\mathbf{x}_{j}(i)^{1\/2}$\\\\\n$\\norm{\\mathbf{X}}_{q}$ & $\\ell_{q}$ norm of $\\mathbf{X}$, obtained by $\\sum_{i,j}\\mathbf{x}_{j}(i)^{q}$\\\\\n$\\norm{\\mathbf{X}}_{{\\rm TV}}$ & ${\\rm TV}$ norm of  $\\mathbf{X}$, obtained by $\\norm{\\mathbf{H}_{h}\\mathbf{X}+\\mathbf{H}_{v}\\mathbf{X}}_{2,1}$\\\\\n$\\norm{\\mathbf{X}}_{0}$ & $\\ell_{0}$ norm of $\\mathbf{X}$, obtained by $\\lim_{p\\rightarrow0}\\sum_{i,j}|\\mathbf{X}_{i,j}|^{p}$\\\\\n$\\tr(\\mathbf{X})$ & trace of $\\mathbf{X}$, obtained by $\\sum_{i}\\mathbf{X}_{i,i}$\\\\\n$\\norm{\\mathbf{X}}_{*}$ & nuclear norm of $\\mathbf{X}$, obtained by $\\tr(\\sqrt{\\mathbf{X}^{\\top}\\mathbf{X}})$\\\\\n$\\norm{\\mathbf{x}}_{2}$ & $\\ell_{2}$ norm of $\\mathbf{x}$, obtained by $\\sqrt{\\sum_{j}\\mathbf{x}_{j}^{2}}$\\\\\n$\\odot$ & the element-wise multiplication operator\\\\\n$\\phi_{i}$ & the neighbouring pixels of the target pixel $i$\\\\\n\\end{supertabular}\n\n\\section{Hyperspectral Image Restoration}\nOwing to the wealthy spectral information, the HS image has been widely used in different kinds of applications, including urban planning, agriculture, forestry, target detection, and so on. However, due to the limitation of hyperspectral imaging systems and the weather conditions, HS images are always suffering from the pollution of various noise. Fig.~\\eqref{fig:noiseType} illustrates different noise types of HS images observed from airborne and spaceborne sensors. Therefore, HS image denoising and restoration is a necessary pre-processing for the subsequent applications to assist the noise.\n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\\includegraphics[width=0.48\\textwidth]{Noise_type}\n        \\caption{Examples for different noise types of HS Images observed from airborne and spaceborne sensors.}\n\\label{fig:noiseType}\n\\end{figure}\n\nThe statistical distribution of hyperspectral noise is complicated. For instance, the readout noise, which is assumed to obey the Gaussian distribution, is produced by the imaging device (Charge-coupled Device) during the conversation from electrons to the final image~\\cite{martin2007anisotropic}. The stripes are generated in the hyperspectral data collected by the pushroom hyperspectral sensors~\\cite{datt2003preprocessing}. Due to the weather environment, the obtained HS data are always suffering from the cloud and cloud shadow~\\cite{gomez2005cloud}. Besides, the HS images are also suffering from the signal-dependent noise~\\cite{acito2011signal}, multiplicative noise, impulsive noise, Laplace noise and so on. Furthermore,  In this paper, we follow the mainstream and focus on the Gaussian noise removal~\\cite{Chang1999,yuan2012hyperspectral} and mixed noise removal problem~\\cite{zhang2013hyperspectral,he2015total}.\n\nSince 1980s, researchers have paid attention to the HS noise analysis. For example, maximum noise fraction (MNF) transformation ~\\cite{green1988transformation} noise-adjusted principal component analysis~\\cite{Chang1999} are utilized to extract high-quality components for the subsequent classification and reject the low-quality components. Following the mainstream of gray\/color image denoising in computer vision society, various state-of-the-art technologies, such as wavelets~\\cite{rasti2014wavelet}, sparse representation~\\cite{qian2012hyperspectral,li2016noise}, TV~\\cite{yuan2012hyperspectral,wu2017structure}, non-local means processing~\\cite{CVPR2014Meng,he2018non}, low-rank matrix representation~\\cite{zhang2013hyperspectral,he2015hyperspectral,xie2016hyperspectral,he2015total}, tensor representation~\\cite{xie2017kronecker,chen2020TCYB,chen2020TGRS}, DL~\\cite{chang2018hsi,yuanHSID-CNN2018} and so on.\n\nThe non-convex regularized methods have also been developed for HS image restoration. From~\\cite{zhang2013hyperspectral} the HS images are assumed to be corrupted by the additive noise, including Gaussian noise, stripes, deadlines, pixel missing, impulse noise and so on. The observation model is formulated as:\n\\begin{equation}\n\\label{eq:ob}\n\\tensor{Y} = \\tensor{X} + \\tensor{N}_{G} + \\tensor{N}_{S},\n\\end{equation}\nwhere $\\tensor{Y}$ represents the observed noisy image, $\\tensor{X}$ stands for the latent clean image, $\\tensor{N}_{G}$ is the Gaussian noise, and $\\tensor{N}_{S}$ is the sparse noise, including stripes, deadlines, pixel missing and impulse noise. Typically, when sparse noise $\\tensor{N}_{S}$ is omitted, the model \\eqref{eq:ob} is degraded to the Gaussian noise removal problem. From~\\cite{zhang2013hyperspectral}, the low-rank property exploration of clean image $\\tensor{X}$ has attracted much attention and achieved state-of-the-art HS image denoising performance~\\cite{he2018non,chen2020TGRS}. Generally speaking, the mainstreams are from two points. Firstly, how to explore the low-rank property of clean image $\\tensor{X}$. Until now, spectral low-rank property~\\cite{zhang2013hyperspectral,he2015hyperspectral}, spatial low-rank property~\\cite{liu2012denoising,chen2020TCYB} and non-local low-rank property~\\cite{xie2017kronecker,he2018non} have been well studied. Furthermore, how to balance the contribution of different low-rank properties is also a key problem~\\cite{chang2017hyper,he2018non}. Secondly, the low-rank constraint of $\\tensor{X}$ formulates a non-convex optimization problem. Therein, how to efficiently solve the rank constraint non-convex problem is another key problem. In the next subsection, we review several outstanding works and illustrate how these works formulate the low-rank modeling, and how to solve the non-convex problem.\n\n\\begin{table}[!t]\n  \\centering\n  \\caption{Prior properties of the selected methods. One, two, and three $\\bullet$ denote the low, medium, and high intensity of prior information, respectively.}\n  \\footnotesize\n    \\begin{tabular}{c||c|c|c||c|c}\n     \\toprule[1.5pt]\n     \\multirow{2}{*}{Methods} & \\multicolumn{3}{c||}{Low-rankness} & \\multicolumn{2}{c}{Local smoothness}\\\\\n     \\cline{2-6}& spatial & spectral & non-local & spatial & spectral   \\\\\n    \\hline\n    LRTA      & $\\bullet \\bullet$   & $\\bullet \\bullet$     &           &             &        \\\\\n    NAILRMA   &           & $\\bullet \\bullet$     &           &             &          \\\\\n    TDL       &           & $\\bullet$            & $\\bullet \\bullet \\bullet$   &             &       \\\\\n    FastHyDe  &           & $\\bullet \\bullet$     & $\\bullet$   &             &        \\\\\n    NGmeet    &           & $\\bullet \\bullet$     & $\\bullet \\bullet$   &             &       \\\\\n    \\hline\n    LRMR      &           & $\\bullet \\bullet$      &           &             &        \\\\\n    LRTV      &           & $\\bullet \\bullet$      &           & $\\bullet \\bullet$      &        \\\\\n    LRTDTV    & $\\bullet$    & $\\bullet \\bullet$      &           & $\\bullet \\bullet$      & $\\bullet \\bullet$      \\\\\n    LRTDGS    & $\\bullet$   & $\\bullet \\bullet \\bullet$     &           & $\\bullet \\bullet \\bullet$     &      \\\\\n    LRTF-FS   &           & $\\bullet \\bullet \\bullet$     &           & $\\bullet \\bullet \\bullet$     & $\\bullet \\bullet$     \\\\\n    \\bottomrule[1.5pt]\n    \\end{tabular}%\n  \\label{tab:Denoising_property}%\n\\end{table}%\n\n\n\\begin{table*}[!t]\n  \\centering\n  \\caption{The restoration results of different selected methods on Gaussian noise and mixed noised, respectively.}\n  \\footnotesize\n    \\begin{tabular}{c||ccccc|ccccc}\n     \\toprule[1.5pt]\n    \\multirow{2}{*}{Index} & \\multicolumn{5}{c|}{Gaussian noise removal} & \\multicolumn{5}{c}{Mixed noise removal} \\\\\n\\cline{2-11} & LRTA  & NAILRMA & TDL   & FastHyDe & NGmeet & LRMR  & LRTV  & LRTDTV & LRTDGS & LRTF-FS \\\\\n    \\hline\n    PSNR  & 25.99 & 32.81 & 32.11 & 33.51 & 33.82 & 32.22 & 33.05 & 32.34      &33.33       & 33.26 \\\\\n    SSIM  & 0.7095 & 0.9519 & 0.9443 & 0.9601 & 0.9607 & 0.9401 & 0.9459 &0.9335       &0.9596       & 0.9549 \\\\\n    MSA   & 11.35 & 4.75  & 4.72  & 4.42  & 4.38  & 4.95  & 5.06  & 4.09 & 4.24     & 4.44 \\\\\n    \\bottomrule[1.5pt]\n    \\end{tabular}%\n  \\label{tab:DenoisResults}%\n\\end{table*}%\n\n\\subsection{Gaussian Noise Removal}\nIn this section, five methods are selected to represent the state-of-the-art HS image Gaussian noise removal approaches. These methods utilize different low-rank matrix\/tensor decomposition models to exploit the spatial, spectral, or non-local low-rank properties of the original clean HS image. The properties of these five methods are summarized in Table \\ref{tab:Denoising_property}. The five methods are briefly described in the following.\n\n\\subsubsection{LRTA}\\footnote{\\url{https:\/\/www.sandia.gov\/tgkolda\/TensorToolbox\/}}\nOn the basis of the observation model~\\eqref{eq:ob} with ignoring sparse noise $S$, low-rank tensor approximation (LRMA)~\\cite{renard2008denoising} tries to restore the HS image from the following objective function\n\\begin{equation}\n\\label{LRTA}\n\\min_{\\tensor{X}} \\|\\tensor{Y}-  \\tensor{X}\\|_{\\F}^2\\;\\;\n{\\rm s.t.}\\; \\tensor{X} = \\tensor{O} \\times_1 \\mat{A} \\times_2 \\mat{B} \\times_3 \\mat{C},\n\\end{equation}\nwhere $\\tensor{X} = \\tensor{O} \\times_1 \\mat{A} \\times_2 \\mat{B} \\times_3 \\mat{C},$ is the Tucker decomposition,\n$\\tensor{O} \\in  \\mathbb{R}^{r_1 \\times r_2 \\times r_3}$ stands for the core tensor, and $\\mat{A} \\in \\mathbb{R}^{M \\times r_1}, \\mat{B}  \\in  \\mathbb{R}^{N \\times r_2}, \\mat{C}  \\in  \\mathbb{R}^{B \\times r_3}$ are the factors related to different dimensions. With the rank $(r_1, r_2, r_3)$ of Tucker decomposition set in advance, the LRTA model \\eqref{LRTA} can simultaneously capture the global spatial and spectral low-rank properties. \\eqref{LRTA} provides a simple general model for different kinds of low-rank matrix\/tensor decomposition based HS image denoising methods, that's to say, we can change the Tucker decomposition constraint of $\\tensor{X}$ to different kinds of matrix\/tensor decomposition, such as canonical polyadic (CP) decomposition, tensor train decomposition, tensor ring decomposition and so on.\n\n\\subsubsection{NAILRMA}\\footnote{\\url{https:\/\/sites.google.com\/site\/rshewei\/home}}\nNoise-adjusted iterative LRMA (NAILRMA)~\\cite{he2015hyperspectral} method assumes that the spectral low-rank property is more important than that of spatial ones, and therefore simply spectral low-rank regularizer is utilized to restrict the original spectral image $\\tensor{X}$. From other works~\\cite{zhang2020}, it also indicates that the spatial TV regularizer is more important than that of the spatial low-rank regularizer. In HS images, the similar signatures representing the same class also appear in the nearby spatial location. To enhance the spectral low-rank property, NAILRMA segmented the HS image into spatial overlapping patches and process each patch individually. The noise intensity in different bands of the HS image is different, which is a big challenge appearing in the HS image Gaussian noise removal, is mitigated by the noise-adjusted iterative strategy~\\cite{he2015hyperspectral}. At last, the randomized singular value decomposition (RSVD) is utilized to solve the non-convex low-rank approximation problem.\n\n\\subsubsection{TDL}\\footnote{\\url{http:\/\/gr.xjtu.edu.cn\/web\/dymeng\/}}\nTensor dictionary learning (TDL) combines the non-local regularization and low-rank tensor approximation. The noisy HS image is firstly segmented into spatial overlapping patches, and the similar patches are clustered together to formulate a higher-order tensor. In this way, the non-local spatial information is collected. Then the higher-order tensors are denoised as the same of~\\eqref{LRTA}, and finally the denoised higher-order tensors are utilized to formulate the final denoised HS image. TDL represents the first method to exploit non-local low-rank property, and the subsequent methods LLRT~\\cite{chang2017hyper}, KBR~\\cite{xie2017kronecker}, and NLTR~\\cite{chen2020TGRS} also achieve remarkable HS image Gaussian noise removal results.\n\n\\subsubsection{FastHyDe}\\footnote{\\url{https:\/\/github.com\/LinaZhuang\/FastHyDe_FastHyIn}}\nThe main difference between HS images and color\/multispectral images is the number of spectral bands. To eliminate this difference and utilize well developed color\/multispectral denoising methods for HS image denoising, Zhuang \\textit{et.al} proposed the fast HS denoising (FastHyDe)~\\cite{zhuang2018fast} method by translating the HS image to low-dimensional reduced image via SVD. By this translation, various state-of-the-art color\/multispectral image denoising methods, such as wavelets~\\cite{Chen2011TGRS} and BM3D~\\cite{zhuang2018fast} are used to denoise the reduced image. Finally, the denoised reduced image is translated back to the denoised HS image via inverse SVD. Generally speaking, under the framework of FastHyDe, the HS image noise removal task is linked to the development of color\/multispectral image noise removal tasks.\n\n\\subsubsection{NGmeet}\\footnote{\\url{https:\/\/github.com\/quanmingyao\/NGMeet}}\nSpatial non-local low-rank regularizer can produce state-of-the-art HS image noise removal performance. However, as the increase of spectral number, the time cost of non-local related methods also increase incredibly~\\cite{chang2017hyper,xie2017kronecker}. Non-local meets global (NGmeet) method also tries to translate the HS image to the reduced image, and utilizes a non-local low-rank method to denoise the reduced image. Different from FastHyDe, NGmeet tries to perfect the framework by iteratively eliminating the error caused by SVD on the noisy HS image, and automatically estimating the spectral rank of the reduced image.\n\n\n\\subsection{Mixed Noise Removal}\nIn this section, we select five representative methods for the HS image mixed noise removal. These methods are on the basis of the observation model~\\eqref{eq:ob}. We focus on the non-convex low-rank regularizer of original image $\\tensor{X}$. The properties of these five methods are summarized in Table \\ref{tab:Denoising_property}.\n\n\\subsubsection{LRMR}\nZhang \\textit{et al.} firstly introduced the observation model \\eqref{eq:ob} to analysis of complex HS noise~\\cite{zhang2013hyperspectral}. LRMR tries to restore the original clean image $\\tensor{X}$ from the noisy image via low-rank and sparse decomposition model as follows:\n\\begin{equation}\n\\label{LRMR}\n\\min_{\\mat{X}} \\norm{\\mat{Y}-  \\mat{X} - \\mat{N}_{S}}_{\\F}^2+ \\lambda_{1} rank(\\mat{X}) + \\lambda_{2} card(\\mat{N}_{S}),\n\\end{equation}\nwhere $\\mat{Y}, \\mat{X}, \\mat{N}_{S}$ are the reshaped matrices of $\\tensor{Y}, \\tensor{X}, \\tensor{N}_{S}$ along the spectral dimension, respectively, $\\lambda_1$ and $\\lambda_2$ are the parameters to trade-off the contributions of $rank(\\mat{X})$ and non-zero elements $card(\\mat{N}_{S})$. LRMR utilizes ``GoDec'' algorithm~\\cite{zhou2011godec} to alternatively update non-convex constraint $\\mat{X}$ and $\\mat{N}_{S}$, and finally obtains the restored image. To improve the efficiency of the optimization to~\\eqref{LRMR}, several non-convex substitutions, such as reweighted nuclear norm~\\cite{xie2016hyperspectral}, $\\gamma$-norm~\\cite{chenyongyong2017TGRS}, smooth rank approximation~\\cite{Ye2019TGRS} and normalized $\\epsilon$-Penalty~\\cite{Xie2020TIP}, are further developed to exploit the spectral low-rank property of $\\mat{X}$.\n\n\\subsubsection{LRTV}\nLow-rank total variation (LRTV) claimed that spectral low-rank property is not enough to describe the property and HS images, and therein introduced TV to explore the spatial smoothness via TV. Generally, low-rank regularization and TV are the two most studied regularizers, and the combination of them to produce the state-of-the-art HS image mixed noise removal is becoming popular. Most of the following works try to either improve the low-rank modeling~\\cite{Fan2018TGRS,Wang2018} or the smoothness modeling~\\cite{he2017Jstars,chen2020TCYB,ZengTGRS2020} of the HS image to further improve the restoration accuracy. To further combine low-rank and TV, the low-rank exploration of the HS difference image is also developed~\\cite{Sun2017letter,Peng2020TIP}.\n\n\\subsubsection{LRTDTV}\nLow-rank tensor decomposition with TV (LRTDTV)~\\cite{Wang2018} tries to improve LRTV by utilizing low-rank tensor decomposition to exploit the low-rank property of HS images, meanwhile spatial-spectral TV (SSTV) to explore the spatial and spectral smoothness simultaneously. Although LRTDTV achieved better mixed noise removal results as reported in~\\cite{Wang2018}, the spatial rank utilized in LRTDTV is much larger than that of spectral rank. This is mainly because the spatial low-rank property of HS images is not so important compared to the spectral low-rank property. From another side, the spatial non-local low-rank regularization is proved to be more efficient~\\cite{zhang2020} than spatial low-rank property for HS restoration problem.\n\n\\subsubsection{LRTDGS}\\footnote{\\url{ https:\/\/chenyong1993.github.io\/yongchen.github.io\/}}\nLow-rank tensor decomposition with group sparse regularization (LRTDGS)~\\cite{chen2020TCYB} also utilizes low-rank rank tensor decomposition to exploit the low-rank property of HS images. Differently, LRTDGS explores the group sparsity of the difference image instead of SSTV in LRTDTV. From the mathematical modeling, LRTDGS utilizes weighted $ell_{2,1}$ norm regularization to fulfill the row-group sparsity of the difference image.\n\n\\subsubsection{LRTF-FR}\\footnote{\\url{ https:\/\/yubangzheng.github.io\/homepage\/}}\nFollowing the idea of NGmeet~\\cite{he2018non}, factor-regularized low-rank tensor factorization (LRTF-FR)~\\cite{ZhengTGRS2020} also utilizes matrix decomposition to decouple the spatial and spectral priors. From one side, the spectral signatures of the HS image are assumed to be of smooth structure. From another side, the reduced image is assumed to have a group sparse structure in the difference domain. The optimization model of LRTF-FR is\n\\begin{equation}\n\\label{LTRF-FR}\n\\begin{aligned}\n    \\min_{\\tensor{X},\\tensor{N}_{S}}&\\norm{\\tensor{Y}-  \\tensor{X} - \\tensor{N}_{S}}_{\\F}^2+ \\lambda_1\\norm{\\tensor{X} \\times_3 \\mat{H}_3}_{2,1}\\\\\n     &+ \\lambda_2 \\sum_{k=1}^{2} \\norm{\\tensor{X} \\times_k \\mat{H}_k}_{\\F}^{2} + \\lambda_3 \\norm{\\tensor{N}_{S}}_{1,1},\n\\end{aligned}\n\\end{equation}\nwhere $\\mat{H}_k, k=1,2,3$ are the first-order difference matrices. Furthermore, in the optimization to \\eqref{LTRF-FR}, the reweighted strategy is utilized to update $\\ell_{2,1}$ norm and $L\\ell_1$ norm to further improve the restored results.\n\n\\subsection{Experimental Study}\nWe choose the HS image from DLR Earth Sensing Imaging Spectrometer (DESIS) installed on the International Space Station (ISS)\\cite{alonso2019data} for the experimental study to compare different methods on the Gaussian and mixed noise removal tasks. We remove the noisy bands and select a sub-image of size $400 \\times 400 \\times 200$ as the clean reference image, which is normalized to $[0,1]$. Firstly, we add Gaussian noise of noise variance $0.1569$ to simulate the Gaussian noisy image, and apply different Gaussian noise removal methods to remove the Gaussian noise. Furthermore, we add salt \\& pepper noise and stripes to simulate the mixed noisy image, and apply mixed noise removal methods to remove the mixed noise. As similar in~\\cite{chen2020TCYB}, we choose the mean of peak signal-to-noise rate (PSNR) over all bands, the mean of structural similarity (SSIM) over all bands, and the mean of spectral angle mapping (MSA) overall spectral vectors to evaluate the restored results.\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=1\\textwidth]{HyDe}\n        \\caption{The illustration of different methods on the noise removal results. The first row shows the results of different methods on the Guassian noise remove experiments (R:70, G:100, B:36). The second row displays the results of different methods on the mixed noise remove experiments (R:31, G:80, B:7).}\n\\label{fig:restoration}\n\\end{figure*}\n\nTable \\ref{tab:DenoisResults} presents the evaluation values of different methods on Gaussian noise and mixed noise removal results, respectively. For the Gaussian noise removal task, NGmeet achieves the best values of three evaluation indices. However, the gap between NGmeet and FastHyDe is limited. For the mixed noise removal task, LRTDGS achieves the best accuracy in PSNR and SSIM values, meanwhile LRTDTV achieved the best MSA value. Combining Tables \\ref{tab:Denoising_property} and \\ref{tab:DenoisResults}, we can conclude that, firstly, the spectral low-rank prior information is important for HS restoration. Secondly, the contribution of spatial low-rank prior information for HS restoration is limited. Thirdly, on the basis of spectral low-rank regularization, spatial and spectral smoothness prior can further improve the final HS restoration results.\n\n\\subsection{Remaining Challenges}\nUp to date, many non-convex regularized methods have been proposed to develop the low-rank priors and local smoothness priors, and achieve remarkable HS restoration results for Gaussian and mixed noise removal. However, these methods still face several challenges for further work. We summary these challenges as the following.\n\n\\begin{itemize}\n    \\item \\textbf{Efficiency.} Although low-rank related methods have achieved state-of-the-art restoration results, they are time-consuming. For instance, NGmeet and LRTVGS speed more than 10 minutes to process the HS image of size $400 \\times 400 \\times 200$. Furthermore, the related state-of-the-art restoration methods always exploit multiple priors of the HS image, resulting in the confusion of the parameter chosen. Therein, how to reduce the model complexity and improve the optimization efficiency of the HS image restoration is the key challenge.\n    \\item \\textbf{Scalability.} Previous non-convex related methods always focus on the small HS image processing. However, HS images are used to observe the earth and the spatial size of one scene is usually very large. How to improve the scalability of the restoration approaches is the key challenge. DL provides the possibility for fast and large scale processing of HS images. Whereas DL approaches always rely on the quality of training samples, and the applicable scope of the test data is always limited. To improve the scalability, how to embed well studied non-convex regularizers to the DL architecture should also be further analyzed.\n\n    \\item \\textbf{Real Application.} Until now, most HS image restoration methods are evaluated on the simulated experiments. However, in most cases, the evaluation indices fail to predict the accuracy of the real HI image restoration results. From another side, the noise distribution in the real noisy HS images is complex. How to testify the related methods on the real HS images should be also further analyzed. From another side, the training samples in the real application are always limited. The blind and unsupervised approaches will become the mainstream of future real HS image restoration.\n\\end{itemize}\n\n\n\\section{Dimensionality Reduction}\nThe HS dimensionality reduction (DR) and feature extraction have long been a fundamental but challenging research topic in HS RS \\cite{li2018discriminant,rasti2020feature}. The main reasons mainly lie in the following aspects. Due to the highly-correlated characteristic between spectral bands, the HS images are subjected to information redundancy, which could hurt the ability to discriminate the materials under the certain extremely-conditioned cases (\\textit{curse of dimensionality}). Plus, as the HS dimension gradually increases along with the spectral domain, large storage capability and high-performance computing are needed. Furthermore, these dimension-reduced features are usually applied for the high-level classification or detection task \\cite{wu2019orsim,wu2020fourier}. Recently, many works based on non-convex modeling have shown to be effective for automatically extracting dimension-reduced features of HS images. Linking with Eq. (\\ref{g_eq2}), the DR task can be generalized to the following optimization problem:\n\\begin{equation}\n\\label{DR_eq1}\n\\begin{aligned}\n      \\mathop{\\min}_{f_{\\Phi},\\mathbf{X}}\\frac{1}{2}\\norm{f_{\\Phi}(\\mathbf{Y})-\\mathbf{X}}_{\\F}^{2}\\;\\; {\\rm s.t.}\\; \\mathbf{X},f_{\\Phi}\\in \\mathcal{C},\n\\end{aligned}\n\\end{equation}\nwhere $f_{\\Phi}(\\bullet)$ denotes the transformation from the original HS space to dimension-reduced subspaces with the respect to the variable set $\\Phi$, and $\\mathbf{X}$ is the low-dimensional representations of $\\mathbf{Y}$. Revolving around the general form in Eq. (\\ref{DR_eq1}), we review currently advanced DR methods from three different aspects: unsupervised, supervised, and semi-supervised models.\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=0.9\\textwidth]{GRSM_SupervisedDR}\n        \\caption{An illustration for supervised DR models in HS images with two different groups: discriminant analysis based DR and regression-induced DR.}\n\\label{fig:GRSM_SupervisedDR}\n\\end{figure*}\n\n\\subsection{Unsupervised Model}\nNon-negative matrix factorization (NMF) \\cite{lee2001algorithms} is a common unsupervised learning tool, which has been widely applied in HS DR. These works can be well explained by Eq. (\\ref{DR_eq1}), the NMF-based DR problem can be then formulated as\n\\begin{equation}\n\\label{DR_eq2}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{Q}\\geq \\mathbf{0},\\mathbf{X}\\geq \\mathbf{0}}\\frac{1}{2}\\norm{\\mathbf{Y}-\\mathbf{X}\\mathbf{Q}}_{\\F}^{2} + \\Psi(\\mathbf{X}) + \\Omega(\\mathbf{Q}),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{Q}$ denotes the combination coefficients, $\\Phi(\\mathbf{X})$ and $\\Omega(\\mathbf{Q})$ are the potential regularization terms for the variables $\\mathbf{X}$ and $\\mathbf{Q}$, respectively. Until the current, there have been some advanced NMF-based works in HS DR. Gillis \\textit{et al.} \\cite{gillis2013sparse} used sparse NMF under approximations for HS data analysis. Yan \\textit{et al.} \\cite{yan2018novel} proposed a graph-regularized orthogonal NMF (GONMF) model with the application to spatial-spectral DR of HS images. Wen \\textit{et al.} \\cite{wen2016orthogonal} further extended the GONMF with combining multiple features for HS DR. Rasti \\textit{et al.} \\cite{rasti2016hyperspectral} designed an orthogonal total variation component analysis (OTVCA) approach for HS feature extraction. Moreover, the HS data are directly regarded as a high-dimensional tensor structure in \\cite{an2018tensor}, where the low-rank attribute is fully considered in the process of low-dimensional embedding. In detail, we summarize the regularization and constraints of the above methods as follows:\n\\begin{itemize}\n    \\item Sparsity \\cite{gillis2013sparse}: $\\Omega(\\mathbf{Q})=\\norm{\\mathbf{Q}}_{0}$;\n    \\item Graph Regularization \\cite{yan2018novel}:\\\\ $\\Psi(\\mathbf{X})=\\tr(\\mathbf{X}\\mathbf{L}\\mathbf{X}^{\\top}),\\; \\rm{s.t.}\\; \\mathbf{X}\\mathbf{X}^{\\top}=\\mathbf{I}$;\n    \\item Multi-graph Regularization \\cite{wen2016orthogonal}:\\\\ $\\Psi(\\mathbf{X})=\\sum_{i=1}^{s}\\tr(\\mathbf{X}\\mathbf{L}^{i}\\mathbf{X}^{\\top}),\\; \\rm{s.t.}\\; \\mathbf{X}\\mathbf{X}^{\\top}=\\mathbf{I}$;\n    \\item Total Variation \\cite{rasti2016hyperspectral}:\\\\ $\\Psi(\\mathbf{X})=\\sum_{i=1}^{r}\\norm{\\sqrt{(\\mathbf{H}_{h}\\mathbf{X}_{i})^{2}+(\\mathbf{H}_{v}\\mathbf{X}_{i})^{2}}}_{1},\\\\\n    \\rm{s.t.}\\; \\mathbf{Q}\\mathbf{Q}^{\\top}=\\mathbf{I}$;\n    \\item Low-rank Graph \\cite{an2018tensor}: $\\Psi(\\mathbf{X})=\\norm{\\mathbf{X}}_{*}+\\tr(\\mathbf{X}\\mathbf{L}\\mathbf{X}^{\\top})$.\n\\end{itemize}\n$\\mathbf{L}=\\mathbf{D}-\\mathbf{W}$ is the Laplacian matrix, where $\\mathbf{D}_{i,i}=\\sum_{j}\\mathbf{W}_{i,j}$ is the degree matrix and $\\mathbf{W}$ is the graph (or manifold) structure of $\\mathbf{X}$ \\cite{belkin2003laplacian}. $\\norm{\\bullet}_{0}$, $\\norm{\\bullet}_{2,1}$, and $\\norm{\\bullet}_{*}$ denote the $\\ell_{0}$-norm \\cite{aharon2006k}, $\\ell_{2,1}$-norm \\cite{nie2010efficient}, and nuclear norm \\cite{recht2010guaranteed}, respectively.\n\nAnother type of unsupervised DR approaches is \\textit{graph embedding}, also known as \\textit{manifold learning}, which can be also grouped into Eq. (\\ref{DR_eq1}) well (according to \\cite{wu2017joint}):\n\\begin{equation}\n\\label{DR_eq3}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{U},\\mathbf{X}}\\frac{1}{2}\\norm{\\mathbf{X}-\\mathbf{U}\\mathbf{Y}}_{\\F}^{2} + \\Psi(\\mathbf{X}) + \\Omega(\\mathbf{U})\\; {\\rm s.t.}\\; \\mathbf{X}\\mathbf{X}^{\\top}=\\mathbf{I},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{U}$ denotes the to-be-estimated projection matrix that bridges the high-dimensional data $\\mathbf{Y}$ with the low-dimensional embedding $\\mathbf{X}$. The regularization term for the variable $\\mathbf{U}$ can be usually expressed as\n\\begin{equation}\n\\label{DR_eq4}\n\\begin{aligned}\n      \\Omega(\\mathbf{U})=\\tr(\\mathbf{U}\\mathbf{Y}\\mathbf{L}\\mathbf{Y}^{\\top}\\mathbf{U}^{\\top})+\\norm{\\mathbf{U}}_{\\F}^{2},\n\\end{aligned}\n\\end{equation}\nwhile the regularizer with respect to $\\mathbf{X}$ can be given by\n\\begin{equation}\n\\label{DR_eq5}\n\\begin{aligned}\n      \\Psi(\\mathbf{X})=\\tr(\\mathbf{X}\\mathbf{L}\\mathbf{X}^{\\top}).\n\\end{aligned}\n\\end{equation}\nThe main difference between these \\textit{manifold learning}-based DR approaches lies in the graph construction, i.e., $\\mathbf{W}$. Ma \\textit{et al.} \\cite{ma2010local} integrated the KNN classifier with several representative \\textit{manifold learning} algorithms, i.e., locally linear embedding \\cite{roweis2000nonlinear}, Laplacian eigenmaps \\cite{belkin2003laplacian}, and local tangent space alignment \\cite{zhang2004principal}, for HS image classification. Huang \\textit{et al.} \\cite{huang2015dimensionality} embedded the sparse graph structure, which is performed by solving a $\\ell_{1}$-norm optimization problem, for the DR of HS images. He \\textit{et al.} \\cite{he2016weighted} extended the work of \\cite{huang2015dimensionality} by generating a weighted sparse graph. Hong \\textit{et al.} \\cite{hong2017learning} developed a new spatial-spectral graph for the DR of HS images, called RLMR, by jointly taking neighbouring pixels of a target pixel in spatial and spectral domains into account. An \\textit{et al.} \\cite{an2018patch} attempted to learn the low-dimensional tensorized HS representations on a sparse and low-rank graph. To sum up, core graphs of the aforementioned methods can be obtained by\n\\begin{itemize}\n    \\item Sparse Graph \\cite{huang2015dimensionality}: $\\min_{\\mathbf{W}}\\norm{\\mathbf{W}}_{1,1}, \\;\\rm{s.t.}\\; \\mathbf{Y}=\\mathbf{Y}\\mathbf{W}$;\n    \\item Weighted Sparse Graph \\cite{he2016weighted}:\\\\ $\\min_{\\mathbf{W}}\\norm{\\mathbf{d}\\odot \\mathbf{W}}_{1,1}, \\;\\rm{s.t.}\\; \\mathbf{Y}=\\mathbf{Y}\\mathbf{W},$\\\\\n    where $\\mathbf{d}$ denotes a weighted matrix on $\\mathbf{W}$ and $\\odot$ is the element-wise multiplication operator;\n    \\item Spatial-spectral Graph \\cite{hong2017learning}: \\\\\n    $\\mathop{\\min}_{\\mathbf{w}_{i,0}}\\sum_{j\\in \\phi_{i}^{spa}}\\norm{\\mathbf{y}_{i,j}-\\sum_{k\\in \\phi_{i}^{spe}}\\mathbf{y}_{i,k}w_{i,k,j}}_{2}^{2} \\\\\n    {\\rm s.t.} \\; \\norm{\\sum_{k\\in \\phi_{i}^{spe}}\\mathbf{y}_{i,k}(4w_{i,k,0}-\\sum_{k=1}^{4}w_{i,k,j})}_{2}^{2}\\leq \\eta, \\\\\n    \\qquad \\mathbf{w}_{i,j}^{\\T}\\mathbf{w}_{i,j}=1,$\\\\\n    where $\\phi_{i}^{spa}$ and $\\phi_{i}^{spe}$ denote the neighbouring pixels in spatial and spectral spaces, respectively;\n    \\item Sparse and Low-rank Graph \\cite{an2018patch}:\\\\ $\\min_{\\mathbf{W}}\\norm{\\mathbf{W}}_{1,1}+\\norm{\\mathbf{W}}_{*}, \\;\\rm{s.t.}\\; \\mathbf{Y}=\\mathbf{Y}\\mathbf{W}$.\n\\end{itemize}\n\n\\subsection{Supervised Model}\n\nUnlike unsupervised DR that relies on embedding various priors to reduce the dimension of HS data, supervised models are capable of better learning class-separable low-dimensional representations via the use of label information. The supervised DR models can be described from two different categories in this subsection, as shown in Fig. \\ref{fig:GRSM_SupervisedDR}. A typical group is the \\textit{discriminant analysis} \\cite{li2018discriminant} closely related to \\textit{graph embedding} and \\textit{manifold learning}. Intuitively speaking, these methods belong to a special case of unsupervised \\textit{graph embedding}, which means they can be well explained by Eq. (\\ref{DR_eq3}). The main difference lies in that the labels are used for constructing the graph structure, i.e., $\\mathbf{W}$, thereby yielding a more discriminative subspace.\n\nIn the supervised DR, a direct graph structure is written as\n\\begin{equation}\n\\label{DR_eq6}\n    {\\bf W}_{ij}=\n    \\begin{cases}\n      \\begin{aligned}\n      1, \\; \\; & \\text{if ${\\bf y}_{i}$ and ${\\bf y}_{j} \\in C_{k}$;}\\\\\n      0, \\; \\; & \\text{otherwise,}\n      \\end{aligned}\n    \\end{cases}\n\\end{equation}\nwhere $C_{k}$ means the sample set of the $k$-th class. Furthermore, more advanced supervised graphs have been developed to better represent the HS data in a low-dimensional subspace, such as sparse graph discriminant analysis \\cite{ly2013sparse}, collaborative graph discriminant analysis \\cite{ly2014collaborative}, feature space discriminant analysis (FSDA) \\cite{imani2015feature}, spatial-spectral local discriminant embedding \\cite{huang2019spatial}. These approaches sought to construct a \\textit{soft} graph instead of a \\textit{hard} graph in Eq. (\\ref{DR_eq6}). That is, the graph is built by using radial basis function (RBF) to measure the similarity between samples belonging to the same class \\cite{hong2020graph1}:\n\\begin{equation}\n\\label{DR_eq7}\n    {\\bf W}_{ij}=\n    \\begin{cases}\n      \\begin{aligned}\n      \\exp\\frac{-\\norm{\\mathbf{y}_{i}-\\mathbf{y}_{j}}_{2}^{2}}{2\\sigma^{2}}, \\; \\; & \\text{if ${\\bf y}_{i}$ and ${\\bf y}_{j} \\in C_{k}$;}\\\\\n      0, \\; \\; & \\text{otherwise,}\n      \\end{aligned}\n    \\end{cases}\n\\end{equation}\nor by solving $\\ell_{1}$-norm or $\\ell_{2}$-norm optimization functions in the same class set, e.g., \\cite{ly2013sparse}, \\cite{ly2014collaborative}.\n\nThe DR behavior can be also modeled from a regression perspective by directly connecting samples and labels \\cite{hong2019regression}, which provides a new insight into the research of the supervised HS DR. A general form for the regression-based supervised DR model can be formulated as\n\\begin{equation}\n\\label{DR_eq8}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{P}, \\mathbf{U}}&\\frac{1}{2}\\norm{\\mathbf{M}-\\mathbf{P}\\mathbf{X}}_{\\F}^{2}+\\Psi(\\mathbf{P})+\\Omega(\\mathbf{U})\\\\\n      &{\\rm s.t.}\\; \\mathbf{X}=\\mathbf{U}\\mathbf{Y},\\; \\mathbf{U}\\mathbf{U}^{\\top}=\\mathbf{I},\n\\end{aligned}\n\\end{equation}\nwhere the variable $\\mathbf{P}$ denotes the regression coefficients or basis signals, and $\\mathbf{M}$ is the one-hot encoded matrix obtained by labels. Eq. (\\ref{DR_eq8}) can be, to some extent, regarded as an interpretable linearized artificial neural network (ANN) mode (shallow network). Ji \\textit{et al.} \\cite{ji2009linear} jointly performed DR and classification, which is a good fit for Eq. (\\ref{DR_eq8}) with $\\Psi(\\mathbf{P})=\\norm{\\mathbf{P}}_{\\F}^{2}$. To enhance the spectrally discriminative ability, Hong \\textit{et al.} \\cite{hong2018joint} employed a LDA-like graph on the basis of \\cite{ji2009linear} to regularize the low-dimensional representations in a Laplacian matrix form, i.e., $\\Omega(\\mathbf{U})=\\tr(\\mathbf{U}\\mathbf{Y}\\mathbf{L}\\mathbf{Y}^{\\top}\\mathbf{U}^{\\top})$. In the same work \\cite{hong2018joint}, Hong \\textit{et al.} further extended their model to a deep version, called JPlay, with a $k$-layered linear regression: \n\\begin{equation}\n\\label{DR_eq9}\n\\begin{aligned}\n      &\\mathop{\\min}_{\\mathbf{P}, \\{\\mathbf{U}_{i}\\}_{i=1}^{k}}\\frac{1}{2}\\norm{\\mathbf{M}-\\mathbf{P}\\mathbf{X}_{i}}_{\\F}^{2}+\\Psi(\\mathbf{P})+\\Omega(\\{\\mathbf{U}_{i}\\}_{i=1}^{k})\\\\\n      &{\\rm s.t.}\\; \\mathbf{X}_{i}=\\mathbf{U}_{i}\\mathbf{X}_{i-1},\\; \\mathbf{X}_{1}=\\mathbf{U}_{1}\\mathbf{Y},\\; \\mathbf{X}_{i}\\geq \\mathbf{0},\\; \\norm{\\mathbf{x}_{i}}_{2}\\leq 1,\n\\end{aligned}\n\\end{equation}\nwith $\\Psi(\\mathbf{P})=\\norm{\\mathbf{P}}_{\\F}^{2}$ and\n\\begin{equation*}\n\\begin{aligned}\n\\Omega(\\{\\mathbf{U}_{i}\\}_{i=1}^{k})&=\\sum_{i=1}^{k}\\tr(\\mathbf{U}_{i}\\mathbf{X}_{i-1}\\mathbf{L}\\mathbf{X}_{i-1}^{\\top}\\mathbf{U}_{i}^{\\top}) \\\\\n &+\\sum_{i=1}^{k}\\norm{\\mathbf{X}_{i-1}-\\mathbf{U}_{i}^{\\top}\\mathbf{U}_{i}\\mathbf{X}_{i-1}}_{\\F}^{2}.\n\\end{aligned}\n\\end{equation*}\nThe J-Play attempts to open the ``black box'' of deep networks in an explainable way by multi-layered linearized modeling. With explicit mappings and physically meaningful priors, the non-convex J-Play takes a big step towards the interpretable AI model.\n\n\\subsection{Semi-supervised Model}\nDue to the fact that labeling samples is extremely expensive, particularly for RS images covering a large geographic region, the joint use of labeled and unlabeled information then becomes crucial in DR and classification. \n\nA simple and feasible strategy for semi-supervised learning is to integrate supervised and unsupervised techniques, e.g., LDA and locality preserving projections \\cite{he2004locality}. By simultaneously constructing graphs of labeled and unlabeled samples (e.g., using Eqs. (\\ref{DR_eq6}) and (\\ref{DR_eq7}), respectively), Eq. (\\ref{DR_eq3}) can be easily extended to a semi-supervised version, leading to semi-supervised discriminant analysis (SSDA) \\cite{liao2012semisupervised}. Zhao \\textit{et al.} \\cite{zhao2014general} further improved the SSDA performance by using ``soft'' (or ``pseudo'') labels predicted by label propagation instead of directly using unsupervised similarities between unlabeled samples. Similarly, Wu \\textit{et al.} \\cite{wu2018semi} generated pseudo-labels using the Dirichlet process mixing model and achieved a novel SSDA approach to learn the low-dimensional HS embedding. These above-mentioned methods are performed surrounding various hand-crafted graph structures ($\\mathbf{W}$). \n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=0.35\\textwidth]{GRSM_graph}\n        \\caption{An example to clarify the graph structure of JPSA method, where $\\mathbf{W}^{p}$ and $\\mathbf{W}^{sp}$ denote the pixel-wise and superpixel-wise subgraphs as well as $\\mathbf{W}^{a}$ is the aligned graph between pixels and superpixels.}\n\\label{fig:graph}\n\\end{figure}\n\n\nA different idea is to simulate the brain-like or human-like behaviors in the semi-supervised DR task. It is well known that the feedback reward is a key component that forms the intelligent information processing system. Inspired by it, \\cite{hong2019learning} developed an iterative multitask learning (IMR) framework by adaptively learning the label propagation (LP) on graphs to simulate the feedback mechanism, thereby achieving the HS DR process more effectively and efficiently. The IMR is a semi-supervised extension of Eq. (\\ref{DR_eq8}) with graph learning, which can be generally modeled as \n\\begin{equation}\n\\label{DR_eq10}\n\\begin{aligned}\n       \\mathop{\\min}_{\\mathbf{P},\\mathbf{U},\\mathbf{L}}&\\sum_{j=1}^{2}\\norm{\\mathbf{M}_{j}-\\mathbf{P}\\mathbf{U}\\mathbf{Y}_{j}}_{\\F}^{2}+\\Psi(\\mathbf{P})+\\Omega(\\mathbf{U})\\\\\n       &\\mathrm{s.t.} \\; \\mathbf{U}\\mathbf{U}^{\\top}=\\mathbf{I}, \\; \\mathbf{L}\\in \\mathcal{C},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{Y}_{1}$ and $\\mathbf{Y}_{2}$ denote the labeled and unlabeled samples from $\\mathbf{Y}$, respectively. $\\Psi(\\mathbf{P})=\\norm{\\mathbf{P}}_{\\F}^{2}$ and $\\Omega(\\mathbf{U})=\\tr(\\mathbf{U}\\mathbf{Y}\\mathbf{L}\\mathbf{Y}^{\\top}\\mathbf{U}^{\\top})$. The non-convex constraint $\\mathcal{C}$ with the respect to the variable $\\mathbf{L}$ can be summarized as\n\\begin{equation*}\n\\begin{aligned}\n       \\mathcal{C}:=\\{\\mathbf{L}=\\mathbf{L}^{\\top}, \\; \\mathbf{L}_{p,q,p\\neq q}\\preceq 0, \\;\\mathbf{L}_{p,q,p=q}\\succeq 0, \\;\\tr(\\mathbf{L})=c\\},\n\\end{aligned}\n\\end{equation*}\nwhere $c>0$ is a scaling constant. Eq. (\\ref{DR_eq10}) is a typical data-driven graph learning model, which is capable of automatically learning the graph structure from the data without any hand-crafted priors. By using the iterative strategy to simulate the feedback system, $\\mathbf{M}_{2}^{(t+1)}$ in the $t$$+$$1$-step can be updated by the graph-based LP on the learned graph of the $t$-step $\\mathbf{W}^{(t)}$:\n\\begin{equation}\n\\label{DR_eq11}\n\\begin{aligned}\n\\cdots\\cdots\\mathbf{M}_{2}^{(t+1)}\\leftarrow\\mathbf{W}^{(t)}\\leftarrow\\mathbf{M}_{2}^{(t)}\\cdots\\cdots.\n\\end{aligned}\n\\end{equation}\n\nBesides, another intelligent feature extraction algorithm, named JPSA, which is extended from \\cite{hong2018joint}, was presented in \\cite{hong2020joint} by the attempts to align pixels and superpixels for spatial-spectral semi-supervised HS DR. JPSA basically follows the JPlay framework and the major difference is the graph structure $\\mathbf{W}$. The graph in JPSA consists of not only pixel-wise and superpixel-wise similarities but also aligned components between pixels and superpixels. Fig. \\ref{fig:graph} gives an example to clarify the graph structure of JPSA. Note that the JPSA's graph can be seen as a full data-driven structure, which can, to a great extent, self-express the intrinsic properties of HS data and further achieves intelligent information extraction and DR.\n\n\\begin{table}[!t]\n\\centering\n\\caption{Quantitative comparison of different DR algorithms in terms of OA, AA, and $\\kappa$ using the NN classifier on the Indian Pines dataset. The best one is shown in bold.}\n\\resizebox{0.45\\textwidth}{!}{\n\\begin{tabular}{c||c|ccc}\n\\toprule[1.5pt] Methods & dimension & OA (\\%) & AA (\\%) & $\\kappa$ \\\\\n\\hline \\hline\nOSF & 220 & 65.89 & 75.71 & 0.6148\\\\\nOTVCA \\cite{rasti2016hyperspectral} & 16 & 74.18 & 77.61 & 0.7228\\\\\nRLMR \\cite{hong2017learning} & 20 & 83.75 & 86.90 & 0.8147\\\\\nFSDA \\cite{imani2015feature} & 15 & 64.14 & 74.52 & 0.5964\\\\\nJPlay \\cite{hong2018joint} & 20 & 83.92 & 89.35 & 0.8169\\\\\nIMR \\cite{hong2019learning} & 20 & 82.80 & 86.27 & 0.8033\\\\\nJPSA \\cite{hong2020joint} & 20 & \\bf 92.98 & \\bf 95.40 & \\bf 0.9197\\\\\n\\bottomrule[1.5pt]\n\\end{tabular}}\n\\label{tab:DR}\n\\end{table}\n\n\\subsection{Experimental Study}\nClassification is explored as a potential application to evaluate the performance of state-of-the-art (SOTA) DR algorithms, including original spectral features (OSF), OTVCA\\footnote{\\url{https:\/\/github.com\/danfenghong\/HyFTech}} \\cite{rasti2016hyperspectral}, RLMR\\footnote{\\url{https:\/\/github.com\/danfenghong\/IEEE_JSTARS_RLML}} \\cite{hong2017learning}, FSDA \\cite{imani2015feature}, JPlay\\footnote{\\url{https:\/\/github.com\/danfenghong\/ECCV2018_J-Play}} \\cite{hong2018joint}, IMR \\cite{hong2019learning}, and JPSA \\cite{hong2020joint}. Experiments are performed on the Indian Pine data using the nearest neighbor (NN) classifier in terms of three indices: \\textit{Overall Accuracy (OA)}, \\textit{Average Accuracy (AA)}, and \\textit{Kappa Coefficient} ($\\kappa$). The scene consists of $145 \\times 145$ pixels and $220$ spectral bands ranging from $0.4 \\mu m$ to $2.5 \\mu m$. More details regarding training and test samples can be found in \\cite{hang2019cascaded}.\n\nTable \\ref{tab:DR} lists the quantitative results of different DR methods. Overall, OSF without feature extraction or DR yields the worst classification performance, compared to those SOTA DR methods. This, to a great extend, demonstrates the effectiveness and necessity of DR in the HS image classification task. It is worth noting that the approaches with spatial-spectral modeling, e.g., OTVCA, RLMR, JPSA, tend to obtain better classification results. The performance of RLMR is superior to that of OTVCA, owing to the full consideration of the neighboring information in a graph form rather than the smoothing operation only modeled by the TV regularization. As a linearized deep model, supervised JPlay obviously performs better than others, especially FSDA that is also a supervised DR model. More importantly, the JPSA with a semi-supervised learning strategy dramatically outperforms other competitors, since it can jointly learn richer representations from both pixels and superpixels by means of spatial-spectral manifold alignment and deep regressive regression.\n\n\\subsection{Remaining Challenges}\nAlthough extensive SOTA methods have recently shown the effectiveness and superiority in the HS DR and classification, there is still a long way to go towards the AI-guided intelligent information processing. We herein summarize the potential remaining challenges briefly.\n\\begin{itemize}\n    \\item \\textbf{Optimal Subspace Dimension.} Subspace dimension is a crucial parameter in DR, which is determined experimentally and empirically in most of existing methods. Despite some parameter estimation algorithms, e.g., intrinsic dimension \\cite{levina2005maximum}, subspace identification \\cite{bioucas2008hyperspectral}, they fail to avoid the pre-survey of various prior knowledge and human intervention in the dimension estimation process.\n    \\item \\textbf{Effects of Noises.} HS images usually suffer from noise degradation in remotely sensed imaging. These noises are complex and closely associated with spectral signatures. Therefore, separating noises from HS data effectively and reducing the noise sensitivity (or preserving spectral discrimination) in the DR process remains challenging. \n    \\item \\textbf{Robustness and Generalization.} Robust estimation and advanced non-convex regularizers have been widely applied to model the DR behavior, yet the complex noise type, the limited training samples, and noisy labels hinder the robustness and generalization ability to be further improved. For this reason, more robust and intelligent models should be developed in either theory or practice emphatically in the next generation DR technique.\n\\end{itemize}\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=0.9\\textwidth]{MaterialMixture}\n        \\caption{A showcase in a real HS scene (Pavia City Centre) to have a quick look at the 3-D HS cube, spectral signals, and material mixture as well as pure pixels (i.e., \\textit{endmember}) and mixed pixels. In the studied scene, the pure pixels correspond to two spectral reflectance curves of \\textit{vegetation} and \\textit{water}, respectively, while the examples of mixed pixels explain the case of spectral mixing, e.g., the two mixed pixels comprise of three pure components (\\textit{endmembers}) in varying proportions. In addition, the figure located in the right upper gives two toy examples to illustrate the material miscibility.}\n\\label{fig:MaterialMixture}\n\\end{figure*}\n\n\\section{Spectral Unmixing}\n\nSpectral unmixing (SU) can be usually seen as a special case of blind source separation (BSS) problem in ML, referring to a procedure that decomposes the observed pixel spectrum of the HS image into a series of constituent spectral signals (or \\textit{endmembers}) of pure materials and a set of corresponding abundance fractions (or \\textit{abundance maps}) \\cite{bioucas2012hyperspectral}. Due to the meter-level ground sampling distance (GSD) of HS imaging, the spectral signatures for most pixels in HS images are acquired in the form of a complex mixture that consists of at least two types of materials. Fig. \\ref{fig:MaterialMixture} gives a showcase to visualize the HS cube, spectral signatures, and material mixing process as well as pure and mixed pixels. Different from general signals, e.g., digital signals, speech signals, there are specific absorption properties in the spectrum signals of different materials. Plus, HS images suffer from miscellaneous unknown degradation, either physically or chemically, in the remotely sensed imaging, inevitably bringing many uncertainties in SU. Therefore, SU plays a unique role in HS RS, yielding many challenging researchable tasks compared to BSS in ML. \n\nIdeally, a linear mixing model (LMM) can be used to accurately describe the SU process \\cite{keshava2002spectral}, which is modeled as the following constrained optimization problem:\n\\begin{equation}\n\\label{SU_eq1}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{E},\\mathbf{A}}\\frac{1}{2}\\norm{\\mathbf{Y}-\\mathbf{E}\\mathbf{A}}_{\\F}^{2}\\;\\;\\rm{s.t.}\\; \\mathbf{E},\\mathbf{A}\\in \\mathcal{C}.\n\\end{aligned}\n\\end{equation}\nThe variables $\\mathbf{E}$ and $\\mathbf{A}$ in Eq. (\\ref{SU_eq1}) stand for the endmembers and abundance maps in the SU issue, respectively. According to the endmembers ($\\mathbf{E}$) that are available (given) or not in the process of SU, existing SU models can be loosely divided into \\textit{blind SU} and \\textit{endmember-guided SU}.\n\n\\subsection{Blind Spectral Unmixing}\nNMF is a baseline model in a wide range of applications, and the same is true in SU. Up to the present, NMF-based interpretable models have been developed extensively for pursing the intelligent SU with the consideration of physically meaningful priors with respect to $\\mathbf{E}$ and $\\mathbf{A}$, e.g., the abundance non-negative constraint (ANC), the abundance sum-to-one constraint (ASC). The resulting basic blind SU model can be written as \n\\begin{equation}\n\\label{SU_eq2}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{E},\\mathbf{A}}\\frac{1}{2}\\norm{\\mathbf{Y}-\\mathbf{E}\\mathbf{A}}_{\\F}^{2} + \\Phi(\\mathbf{E}) + \\Omega(\\mathbf{A})\\;\\;\\rm{s.t.}\\; \\mathbf{E},\\mathbf{A}\\in \\mathcal{C},\n\\end{aligned}\n\\end{equation}\nwhere the constraint $\\mathcal{C}$ is \n\\begin{equation*}\n\\begin{aligned}\n      \\mathcal{C}:=\\{\\mathbf{E}\\geq \\mathbf{0},\\; \\mathbf{A}\\geq \\mathbf{0},\\; \\mathbf{1}^{\\top}\\mathbf{A}=\\mathbf{1}\\}.\n\\end{aligned}\n\\end{equation*}\n\nOn the basis of the model (\\ref{SU_eq2}), Yang \\textit{et al.} \\cite{yang2010blind} proposed sparse NMF for SU with a well-designed S-measure sparseness. Qian \\textit{et al.} \\cite{qian2011hyperspectral} imposed the sparsity constraint on abundances and used $\\ell_{1\/2}$-regularized NMF for blind SU, which has shown to be more effective than $\\ell_{0}$- and $\\ell_{1}$-norm terms. In \\cite{sigurdsson2014hyperspectral}, Sigurdsson \\textit{et al.} relaxed $\\ell_{1\/2}$-norm to $\\ell_{q}$-norm ($0\\leq q\\leq1$) for a better estimation of abundances. Thouvenin \\textit{et al.} \\cite{thouvenin2015hyperspectral} developed an improved LMM, called perturbed LMM (PLMM), by the attempts to model spectral variabilities as perturbed information that simply meets the Gaussian distribution. A similar work is presented in \\cite{drumetz2016blind}, where the scaling factor, as a major spectral variability (SV), is modeled into LMM to yield an extended LMM (ELMM) for the blind SU task. He \\textit{et al.} \\cite{he2017total} employed total variation (TV) and  weighted $\\ell_{1}$-norm terms to further enhance the smoothness and sparseness of abundances. Yao \\textit{et al.} \\cite{yao2019nonconvex} sought to explain the NMF-based SU model by simulating human observations on HS images, e.g., sparsity, non-local, smooth properties, in a non-convex modeling fashion. Another type of interesting SU strategy is to embed the graph or topological structure of the HS data. The local neighboring relation is introduced into the NMF model, showing robust SU results \\cite{liu2012enhancing}. Similarly, Lu \\textit{et al.} \\cite{lu2012manifold} enforced the abundances to follow the manifold structure of spectral signatures in the form of Laplacian regularization form for HS unmixing. Wang \\textit{et al.} \\cite{wang2016hypergraph} used a structuralized hypergraph regularization in sparse NMF to better depict the underlying manifolds of the HS data. Very recently, Qin \\textit{et al.} \\cite{qin2020blind} proposed a novel graph TV regularization to estimate endmembers and abundances more effectively. There are still other variants that directly unmix the 3-D HS tensor by preserving spatial structure information as much as possible. For that, Qian \\textit{et al.} \\cite{qian2016matrix} proposed a matrix-vector non-negative tensor factorization framework for blind SU. Imbiriba \\textit{et al.} \\cite{imbiriba2019low} modeled the low-rank properties in the HS tensor to address the SV for robust SU. A further modified work based on \\cite{imbiriba2019low} is proposed via weighted non-local low-rank tensor decomposition for sparse HS unmixing.\n\nBroadly, these key non-convex priors of the above models can be briefly summarized as follows:\n\\begin{itemize}\n    \\item $\\ell_{1\/2}$-NMF \\cite{qian2011hyperspectral}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{1\/2}=\\sum_{k,n=1}^{K,N}\\mathbf{a}_{n}(k)^{1\/2}$;\n    \\item $\\ell_{q}$-NMF \\cite{sigurdsson2014hyperspectral}:\n    $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{q}=\\sum_{k,n=1}^{K,N}\\mathbf{a}_{n}(k)^{q}$;\n    \\item PLMM \\cite{thouvenin2015hyperspectral}:\n    $\\Phi(\\mathbf{E})=\\frac{1}{2}\\norm{\\mathbf{E}-\\mathbf{E}_{0}}_{\\F}^{2}$,\\;\n    $\\Omega(\\mathbf{A})=\\frac{1}{2}\\norm{\\mathbf{A}\\mathbf{H}}_{\\F}^{2}$,\\; $\\Psi(\\Delta)=\\frac{1}{2}\\sum_{n=1}^{N}\\norm{\\Delta_{n}}_{\\F}^{2}$,\n    where $\\mathbf{E}_{0}$, $\\mathbf{H}$, and $\\Delta$ denote the reference endmembers, the matrix differences in spatial four nearest neighbors, and pixel-wise perturbed information, respectively. \n    \\item ELMM \\cite{drumetz2016blind}: $\\Phi(\\mathbf{E})=\\sum_{n=1}^{N}\\norm{\\mathbf{E}_{n}-\\mathbf{E}_{0}\\mathbf{S}_{n}}_{\\F}^{2}$,\\; $\\Omega(\\mathbf{A})=\\norm{\\mathbf{H}_{h}(\\mathbf{A})}_{2,1}+\\norm{\\mathbf{H}_{v}(\\mathbf{A})}_{2,1}$,\\; $\\Psi(\\mathbf{S})=\\norm{\\mathbf{H}_{h}(\\mathbf{S})}_{\\F}^{2}+\\norm{\\mathbf{H}_{v}(\\mathbf{S})}_{\\F}^{2}$, where $\\mathbf{H}_{h}$ and $\\mathbf{H}_{v}$ are the horizontal and vertical gradients;\n    \\item TV-RSNMF \\cite{he2017total}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{d}\\odot\\mathbf{A}}_{1,1}+\\norm{\\mathbf{A}}_{\\rm TV}$;\n    \\item NLHTV \\cite{yao2019nonconvex}:\\\\ $\\Omega(\\mathbf{A})=\\sum_{n=1}^{N}\\norm{J_{w}\\mathbf{a}_{n}}_{\\mathcal{S}_{1}}+\\sum_{i,j}\\log(|x_{i,j}|+\\epsilon)$, where $J_{w}$ and $\\norm{\\bullet}_{\\mathcal{S}_{1}}$ are defined as non-local Jacobian operator and the Schatten-1 norm, respectively.\n    \\item Graph $\\ell_{1\/2}$-NMF \\cite{lu2012manifold}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{1\/2}+\\tr(\\mathbf{A}\\mathbf{L}\\mathbf{A}^{\\top})$;\n    \\item Graph TV \\cite{qin2020blind}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{\\rm TV}+\\tr(\\mathbf{A}\\mathbf{L}\\mathbf{A}^{\\top})$.\n\\end{itemize}\n\nOwing to the powerful data fitting ability, DL-based SU approaches have recently been paid increasing attention and achieved better unmixing results \\cite{palsson2018hyperspectral,su2018stacked,palsson2019spectral,han2020deep}. Although these methods still suffer from the effects of ``black box'', i.e., the lack of model interpretability, yet their performances have preliminary shown the effectiveness and feasibility in unmixing the HS data more accurately.\n\n\\subsection{Endmember-Guided Spectral Unmixing}\nA mass of blind SU methods has been developed and shown to be effective to simultaneously obtain endmembers and abundance maps. However, these blind methods tend to extract physical meaningless endmembers, e.g., noisy signals, spectral signatures corresponding to non-existent materials, due to the lack of certain interpretable model guidance or prior knowledge. A straightforward solution is to provide nearly real endmembers extracted from HS images. This naturally leads to the researches on endmember-guided SU. As the name suggests, the SU process is performed with given reference endmembers or the guidance of extracted endmembers from the HS image. That is, the endmembers $\\mathbf{E}$ in Eq. (\\ref{SU_eq2}) are known. Accordingly, the endmember-guided SU can be implemented in a three-stage way.\n\\begin{itemize}\n    \\item Firstly, the number of endmembers can be estimated by subspace estimation algorithms, e.g., HySime \\cite{bioucas2008hyperspectral};\n    \\item Secondly, the endmembers can be extracted based on    geometric observations of HS data structure. Several well-known methods are vertex component analysis (VCA) \\cite{nascimento2005vertex}, pixel purity index (PPI) \\cite{chang2006fast}, and fast autonomous endmember extraction (N-FINDER) \\cite{winter1999n}.\n    \\item Lastly, the abundances of materials are estimated using regression-based methods, which can generally written as \n    \\begin{equation}\n        \\label{SU_eq3}\n        \\begin{aligned}\n              \\mathop{\\min}_{\\mathbf{A}}\\frac{1}{2}\\norm{\\mathbf{Y}-\\mathbf{E}\\mathbf{A}}_{\\F}^{2} + \\Omega(\\mathbf{A})\\;\\; {\\rm s.t.}\\; \\mathbf{A}\\in \\mathcal{C}.\n        \\end{aligned}\n    \\end{equation}\n\\end{itemize}\n\nFollowing the three steps, many well-working non-convex models have been successfully developed to estimate the abundance maps of different materials at a more accurate level. Heinz \\textit{et al.} \\cite{heinz2001fully} thoroughly analyzed the spectral mixture in the SU issue, yielding a fully constrained least-squares unmixing (FCLSU) algorithm. Due to the hard ASC, the abundances can not be fully represented in a simplex. For this reason, a partial constraint least-squares unmixing (PCLSU) \\cite{heylen2011fully} model emerges as required without ASC. Bioucas-Dias \\textit{et al.} \\cite{bioucas2010alternating} relaxed the strong $\\ell_{0}$-norm to the solvable $\\ell_{1}$-norm in the sparse HS unmixing model and designed a fast and generic optimization algorithm based on the ADMM framework \\cite{boyd2011distributed}, called sparse unmixing by variable splitting and augmented Lagrangian (SUnSAL). In \\cite{iordache2012total}, a TV spatial regularization is considered to further enhance the unmixing performance. Iordache \\textit{et al.} \\cite{iordache2013collaborative} extended the sparse regression model to the collaborative version regularized by $\\ell_{2,1}$-norm for SU. Fu \\textit{et al.} \\cite{fu2016semiblind} proposed a semi-blind HS unmixing model by correcting the mismatches between estimated endmembers and pure spectral signatures from the library. Huang \\textit{et al.} \\cite{huang2018joint} jointly imposed sparsity and low-rank properties on the abundances for better estimating abundance maps. Hong \\textit{et al.} \\cite{hong2018sulora} devised an interesting and effective subspace-based abundance estimation model. The model neatly sidesteps to directly decompose the HS data in the complex high-dimensional space instead of projecting the HS data into a more robust subspace, where the SV tends to be removed in a more generalized way with low-rank attribute embedding. Beyond the current framework, Hong \\textit{et al.} \\cite{hong2019augmented} further augmented the basic LMM by fully modeling SVs, e.g., principal scaling factors and other SVs that should be incoherent or low-coherent with endmembers, in order to yield an interpretable and more intelligent SU model, called augmented LMM (ALMM). \n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=0.47\\textwidth]{GRSM_SV}\n        \\caption{A visual example to clarify SVs in a real HS scene (Pavia City Centre). An image patch cropped from the scene is select to show the spectral bundles involving spectral variations of trees in (a). (b) shows a pure spectral signature (i.e., \\textit{endmember}) of trees acquired from the laboratory. (c) represents the differences between (a) and (b), which is seen as SVs.}\n\\label{fig:SV}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=1\\textwidth]{SU_AB}\n        \\caption{Visualization of abundance maps estimated by different SOTA SU algorithms on the Urban data, where SAM is computed to generate the classification-like maps regarded as the GT to measure the shape similarity of abundance maps obtained by different SU methods.}\n\\label{fig:SU_AB}\n\\end{figure*}\n\nThe non-convexity of these methods on priors, constraints, or modeling can be summarized as follows:\n\\begin{itemize}\n    \\item FCLSU \\cite{heinz2001fully}: $\\mathbf{A}\\geq \\mathbf{0}$, $\\mathbf{1}^{\\top}\\mathbf{A}=\\mathbf{1}$;\n    \\item PCLSU \\cite{heylen2011fully}: $\\mathbf{A}\\geq \\mathbf{0}$;\n    \\item SUnSAL \\cite{bioucas2010alternating}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{1,1}$, $\\mathbf{A}\\geq \\mathbf{0}$, $\\mathbf{1}^{\\top}\\mathbf{A}=\\mathbf{1}$;\n    \\item SUnSAL-TV \\cite{iordache2012total}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{1,1}+\\norm{\\mathbf{A}}_{\\rm TV}$, $\\mathbf{A}\\geq \\mathbf{0}$;\n    \\item CSR \\cite{iordache2013collaborative}: $\\Omega(\\mathbf{A})=\\norm{\\mathbf{a}}_{2,1}=\\sum_{n=1}^{N}\\norm{\\mathbf{a}_{n}}_{2}$, $\\mathbf{A}\\geq \\mathbf{0}$;\n    \\item DANSER \\cite{fu2016semiblind}: $\\Omega(\\mathbf{A})=\\sum_{n=1}^{N}(\\norm{\\mathbf{a}_{n}}_{2}^{2}+\\tau)^{p\/2}$, $\\mathbf{A}\\geq \\mathbf{0}$, $\\Phi(\\mathbf{E})=\\norm{\\mathbf{E}-\\mathbf{E}_{0}}_{\\F}^{2}$;\n    \\item SULoRA \\cite{hong2018sulora}: $\\Psi(\\mathbf{U})=\\norm{\\mathbf{Y}-\\mathbf{U}\\mathbf{Y}}_{\\F}^{2}+\\norm{\\mathbf{U}}_{*}$, $\\Omega(\\mathbf{A})=\\norm{\\mathbf{A}}_{1,1}$, $\\mathbf{A}\\geq \\mathbf{0}$, where $\\mathbf{U}$ denotes the subspace projection and $\\norm{\\bullet}_{*}$ is the nuclear norm that approximates the rank property of the matrix $\\bullet$;\n    \\item ALMM \\cite{hong2019augmented}: $\\Phi(\\mathbf{A})=\\norm{\\mathbf{A}}_{1,1}$, $\\mathbf{A}\\geq \\mathbf{0}$, $\\Gamma(\\mathbf{J})=\\norm{\\mathbf{J}}_{\\F}^{2}$, $\\Psi(\\mathbf{V})=\\norm{\\mathbf{A}^{\\top}\\mathbf{V}}_{\\F}^{2}+\\norm{\\mathbf{V}^{\\top}\\mathbf{V}-\\mathbf{I}}_{\\F}^{2}$, where $\\mathbf{V}$ and $\\mathbf{J}$ denote the SV dictionary and corresponding coefficients, respectively.\n\\end{itemize}\n\n\\subsection{Experimental Study}\nA real urban HS data acquired by the HYDICE over the urban area, Texas, USA, in 2015 (the latest version\\footnote{\\url{http:\/\/www.tec.army.mil\/Hypercube}}) is used to evaluate the performance of several selected SOTA unmixing methods qualitatively, including $\\ell_{1\/2}$-NMF \\cite{qian2011hyperspectral}, PLMM\\footnote{\\url{https:\/\/pthouvenin.github.io\/unmixing-plmm\/}} \\cite{thouvenin2015hyperspectral}, ELMM\\footnote{\\url{https:\/\/openremotesensing.net\/knowledgebase\/spectral-variability-and-extended-linear-mixing-model\/}} \\cite{drumetz2016blind}, NLHTV \\cite{yao2019nonconvex}, FCLSU \\cite{heinz2001fully}, SUnSAL\\footnote{\\url{http:\/\/www.lx.it.pt\/~bioucas\/}} \\cite{bioucas2010alternating}, SULoRA\\footnote{\\url{https:\/\/github.com\/danfenghong\/IEEE_JSTSP_SULoRA}} \\cite{hong2018sulora}, and ALMM\\footnote{\\url{https:\/\/github.com\/danfenghong\/ALMM_TIP}} \\cite{hong2019augmented}. The HS image consists of $307\\times 307$ pixels and 162 spectral bands after removing noisy bands in the wavelength range of $0.4 \\mu m$ to $2.5 \\mu m$ at a $2m$ GSD. Moreover, four main materials (or endmembers) are investigated in the studied scene, i.e., \\textit{asphalt}, \\textit{grass}, \\textit{trees}, and \\textit{roof}. Furthermore, HySime \\cite{bioucas2008hyperspectral} and VCA \\cite{nascimento2005vertex} algorithms are adopted to determine the number of endmembers and extract endmembers from the HS image (as the initialization for blind SU methods) for all compared algorithms, respectively.\n\nFig.~\\ref{fig:SU_AB} shows the visual comparison between different SOTA unmixing algorithms in terms of abundance maps. Owing to the consideration of real endmembers extracted from the HS scene, the last four endmember-guided SU methods perform evidently better than the blind SU ones. ELMM models the scaling factors, tending to better capture the distributions of different materials. The embedding of non-local spatial information makes the NLHTV method obtain a more similar shape of abundance maps to the GT, yielding comparable unmixing performance with ELMM. Remarkably, the unmixing results with regard to abundance maps of SULoRA and ALMM algorithms are superior to those of other methods, since the SVs can be fully considered by robustly embedding low-rank attributes in a latent subspace using SULoRA and characterizing complex real scenes more finely using ALMM.\n\n\\subsection{Remaining Challenges}\nSU has long been a challenging and widely concerned topic in HS RS. Over the past decades, tons of SU works have been proposed by the attempts to unmix these mixed spectral pixels more effectively. Yet, some key and essential issues and challenges still remain to be solved.\n\n\\begin{itemize}\n    \\item \\textbf{Benchmark Data.} Unlike classification, recognition, and detection tasks, the ground truth of material abundances is able to be hardly collected, due to the immeasurability of abundance values in reality. On the other hand, spectral signatures (i.e., endmembers) of pure materials are often acquired in the lab. This usually leads to uncertain mismatches between real endmembers and lab ones. It turns to be urgent to establish benchmark datasets for SU by drawing support from more advanced imaging techniques or developing interpretable ground truth generation models or processing chain.\n    \\item \\textbf{Evaluation Criteria.} Reconstruction errors (RE) or spectral angle mapper (SAM) are the two commonly used evaluation indices in SU. It should be noted, however, that the results of RE or SAM are not equivalent to those of unmixing. Linking to the issue of benchmark data, the measurement between real results and estimated ones is the optimal choice, if we have the ground truth for abundances and endmembers. If not, developing meaningful and reasonable evaluation indices (e.g., classification accuracy) should give the top priority in future work.\n    \\item \\textbf{Spectral Variability.} Spectral signatures inevitably suffer from various SVs caused by illumination and topography change, noise effects from external conditions and internal equipment, atmospheric interference, and complex mixing of materials in the process of imaging. Fig. \\ref{fig:SV} shows a visual example to specify the SVs (e.g., trees) in a real HS scene. Considerable uncertainties brought by these factors have a big negative impact on accurate estimation of abundances and endmembers in SU.\n    \\item \\textbf{Nonlinearity.} The complex interactions (e.g., intimate mixing, multilayered mixing \\cite{bioucas2012hyperspectral}) between multiple materials, also known as nonlinearity, inevitably occur in the process of HS imaging. The nonlinearity in SU is a longstanding and pending challenge. Most of existing nonlinear unmixing models only attempt to consider certain special cases \\cite{heylen2014review}, e.g., bilinear mixing, intimate mixtures, etc. Consequently, there is still lack of a general and powerful model that can robustly address various nonlinearities in SU.\n    \\item \\textbf{Model Explainability.} The non-negativity and the sum-to-one constraint considered in LMM are the basic priors for spectral signals in HS images. However, only the two constraints fail to model the complex unmixing process in an explainable fashion. To further enhance the explainability, new spectral mixture models should be developed beyond the classic LMM by fully excavating the intrinsic attribute knowledge that lies in the HS image. \n\\end{itemize}\n\n\\section{Data Fusion and Enhancement}\nThe high spectral resolution of HS images enables the identification and discrimination of materials, meanwhile the high spatial resolution can provide the possibility of the derivation of surface parameters~\\cite{yokoya2017hyperspectral}. However, due to the equipment limitation, there is usually a trade-off between the spatial and spectral resolutions, and the HS images obtained by the spaceborne imaging spectrometers are usually with a moderate ground sampling distance~\\cite{yokoya2017hyperspectral}. To enhance the spatial resolution, one popular way is to fuse the HS images with high spatial MS images to generate new high spatial-spectral HS (HrHS) images. In particular, enormous effects have been recently made to enhance the spatial or spectral resolutions of HS images by means of ML techniques. Fig.~\\ref{fig:datafusion} illustrates the fusion process of HS-MS images to generate the HrHS image. \nSuppose we have the low-spatial resolution HS image $\\tensor{Y} \\in  \\mathbb{R}^{m \\times n \\times B}$ and high-spatial resolution MS image $\\tensor{Z}\\in  \\mathbb{R}^{M \\times N \\times b}$ with $M \\gg m$, $N \\gg n$ and $B \\gg b$, the fusion purpose is to generate the high-spatial resolution HS image $\\tensor{X}\\in \\mathbb{R}^{M \\times N \\times B}$. The degradation models from $\\tensor{X}$ to $\\tensor{Y}$ and $\\tensor{Z}$ are formulated as\n\\begin{align}\n\\mat{Y} = \\mat{X}\\mat{R}+\\mat{N}_{H}\n\\\\\n\\mat{Z} = \\mat{G}\\mat{X}+\\mat{N}_{M}\n\\label{eq:Fusi}\n\\end{align}\nwhere $\\mat{X}, \\mat{Y}, \\mat{Z}$ are the reshaped matrices along the spectral dimension of $\\tensor{X}, \\tensor{Y}, \\tensor{Z}$, respectively, $\\mat{R}$ is the mixed cyclic convolution and downsampling operator, $\\mat{G}$ is the spectral response function (SRF) of the MS image sensor, $\\mat{N}_{H}$ and $\\mat{N}_{M}$ are the corresponding MS-HS noise. To unify different observation models~\\cite{Eismann2004,yokoya2012coupled,QiWei2015TGRS,Simoes2015,yokoya2017hyperspectral,he2020hyperspectral}, $\\mat{N}_{H}$ and $\\mat{N}_{M}$ are assumed to be the independent identically distributed Gaussian noise. Via the maximum a posteriori (MAP) estimation method and Bayes rule~\\cite{Eismann2004,QiWei2015TGRS,Simoes2015}, the following non-convex optimization model is obtained\n\\begin{align}\n&\\min_{\\mat{X}} \\|\\mat{Y} - \\mat{X}\\mat{R}\\|_{\\F}^2+\\|\\mat{Z} - \\mat{G}\\mat{X}\\|_{\\F}^2,\n\\label{eq:FusiOpt}\n\\end{align}\nwhere $\\mat{R}$ and $\\mat{G}$ are assumed to be known (in~\\cite{QiWei2015TGRS,yokoya2017hyperspectral}, $\\mat{R}$ and $\\mat{G}$ are estimated in advance of the optimization). and As mentioned in~\\cite{QiWei2015TGRS,Simoes2015}, the optimization of \\eqref{eq:FusiOpt} is a NP-hard problem, and over-estimation of $\\mat{Z}$ will result in the unstable fusion results. Therein, additional property of $\\mat{X}$ and prior regularizers should be exploited in the optimization model \\eqref{eq:FusiOpt}. It should be noted, however, that the two functions $\\mathbf{R}$ and $\\mathbf{G}$ can be given according to known sensors and also can be learned or automatically estimated from the data itself.\n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\\includegraphics[width=0.4\\textwidth]{DataFusion}\n        \\caption{Illustration of MS-HS image fusion to generate the HrHS image.}\n\\label{fig:datafusion}\n\\end{figure}\n\nHS pansharpening is a heuristic way to perform the HS-MS fusion \\cite{loncan2015hyperspectral}, which has been widely applied in the HS image enhancement task. Component substitution (CS) and multiresolution analysis (MRA) are the two main types of pansharpening techniques. The former one aims to inject detailed information of MS images into the low-resolution HS image, thereby generating the high-resolution HS product. The latter one is to pansharpen the HS image by linearly combining MS bands to synthesize a high-resolution HS band using regression techniques. Another group for the HS-MS fusion task is the subspace-based model, which roughly consists of Bayesian and unmixing based methods (see \\cite{yokoya2017hyperspectral}). Different from pansharpening, the subspace-based approaches project the to-be-fused MS and HS images to a new space where the dimension is generally smaller than that of the unknown high-resolution HS image, by the means of the probability-driven Bayesian estimation (Bayesian-based methods) or SU-guided matrix joint factorization (unmixing-based methods).\n\nIn the following, we focus on the subspace methods, and review the related HS-MS image fusion methods from the non-convex modeling perspective. A more detailed review can be referred to~\\cite{yokoya2017hyperspectral,loncan2015hyperspectral}.\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\t\\includegraphics[width=1\\textwidth]{DFE}\n        \\caption{The fusion results of different methods with Chikusei image. The color is composed of bands 70, 100, 36. An enlarged region is framed in {\\bf green} and the corresponding residual image between the fused image and MS-GT is framed in {\\bf red}.}\n\\label{fig:Chikusei_reconstruct}\n\\end{figure*}\n\n\\subsection{Unmixing based methods}\nHyperspectral unmixing (HU)~\\cite{bioucas2013hyperspectral,he2017total} assumes that the mixed class of an HS image can be decomposed to the collection of constitute spectra (endmembers), and their corresponding proportions (abundances). With LMM assumption, the different endmembers do not interfere with each other~\\cite{bioucas2013hyperspectral}. By embedding the LMM model into \\eqref{eq:FusiOpt}, we can obtain the following general unmixing based approaches\n\\begin{align}\n\\label{eq:FusiOptMix}\n&\\min_{\\mat{X},\\mat{E},\\mat{A}} \\|\\mat{Y} - \\mat{X}\\mat{R}\\|_{\\F}^2+\\|\\mat{Z} - \\mat{G}\\mat{X}\\|_{\\F}^2,\\\\\n& {\\rm s.t.} \\;\\; \\mat{X} = \\mat{E}\\mat{A}, \\; \\mat{E},\\mat{A} \\geq 0,\\; \\mat{1}_{R}^{\\top}\\mat{A}=\\mat{1}_{MN},\n\\notag\n\\end{align}\nwhere $\\mat{E}$, $\\mat{A}$ are the endmember matrix and abundance matrix, which are assumed to obey the non-negative and abundance sum-to-one constraints. Generally, nonlinear unmixing models~\\cite{bioucas2013hyperspectral} can be also utilized for the fusion task of HS-MS images. However, due to the generality of the LMM model, we focus on the review of LMM based fusion approaches.\n\nEismann \\textit{et al.} proposed a maximum a posteriori estimation method to deduce the cost objective function, and introduced a stochastic mixing model (MAP-SMM) to embed LMM into the cost function~\\cite{Eismann2004}. MAP-SMM method tries to estimate the prior probabilities for all the mixture classes, including the mean vectors and covariance matrices of the endmember classes. The learned prior probabilities are passed to the cost function to help the reconstruction of the final HrHS image $\\mat{X}$.\n\nYokoya \\textit{et al.,} regarded \\eqref{eq:FusiOptMix} as the coupled NMF (CNMF) problem~\\cite{yokoya2012coupled} and introduced the multiplicative update rules to optimize \\eqref{eq:FusiOptMix}. Firstly, CNMF utilizes $\\|\\mat{Y} - \\mat{E}\\mat{A}\\mat{R}\\|_{\\F}^2$ as the cost function to update $\\mat{E}$ and $\\mat{A}_h$ with the endmember matrix $\\mat{E}$ initialized by vertex component analysis (VCA). Here, $\\mat{A}_h = \\mat{A} \\mat{R}$ is the abundance matrix from HS images. Secondly, by initializing $\\mat{E}_m = \\mat{G}\\mat{E}$ which is the endmember matrix from the MS image, CNMF again utilizes the multiplicative update rules to update $\\mat{E}$ from the cost function $\\|\\mat{Z} - \\mat{G}\\mat{E}\\mat{A}\\|_{\\F}^2$. Finally, the HrHS image $\\mat{X}$ is reconstructed from $\\mat{E}\\mat{A}$. The following works~\\cite{akhtar2014sparse,akhtar2015bayesian,dong2016hyperspectral} also utilize the CNMF framework to fuse HS-MS images. Differently,~\\cite{akhtar2014sparse,dong2016hyperspectral} introduced a non-negative dictionary learning strategy, while \\cite{akhtar2015bayesian} proposed the proximal alternating linearized minimisation algorithm to update $\\mat{E}$ and $\\mat{A}$.\n\nOn the basis of \\eqref{eq:FusiOptMix}, Wang \\textit{et al.,} further regularized $\\mat{X}$ with non-local low-rank Tucker decomposition~\\cite{Wang2020TGRS}. The improved non-local Tucker decomposition regularized CNMF model~\\cite{Wang2020TGRS} was solved by the multi-block ADMM, and achieved remarkable fusion results. It indicates that additional regularizers on $\\mat{X}$ can further improve the fusion accuracy. From another side, it is necessary to make a trade-off between the complex models with higher accuracy and the computation efficiency for real large scale HS-MS image fusion task.\n\n\\subsection{Orthogonal subspace based methods}\nAnother common assumption in HS-MS fusion is that the spectral information of $\\mat{X}$ underlies a orthogonal subspace, whose dimension is much smaller than the number of bands $B$~\\cite{QiWei2015TGRS,he2017total}, $\\textit{i.e.,}$ $\\mat{X} = \\mat{E}\\mat{A}$ with $\\mat{E}\\in  \\mathbb{R}^{B \\times k}$, $\\mat{A}\\in  \\mathbb{R}^{k \\times MN}$, and $k\\ll B$.  $\\mat{E}^{\\top}$ is an orthogonal matrix with $\\mat{E}^{\\top}\\mat{E} = \\mat{I}_{k}$. Therefore, the subspace based model is formulated as\n\\begin{align}\n\\label{eq:FusiOpt2}\n&\\min_{\\mat{X},\\mat{E},\\mat{A}} \\|\\mat{Y} - \\mat{X}\\mat{R}\\|_{\\F}^2+\\|\\mat{Z} - \\mat{G}\\mat{X}\\|_{\\F}^2, \\\\\n& {\\rm s.t.}\\;\\; \\mat{X} = \\mat{E}\\mat{A}, \\; \\mat{E}^{\\top}\\mat{E} = \\mat{I}_{k}.\n\\notag\n\\end{align}\nAlthough additional spectral subspace prior is exploited, the optimization of \\eqref{eq:FusiOpt2} still faces several challenges. Firstly, if $k\\gg b$, that's to say, the dimension number of the subspace is larger than the bands' number of the MS image, the optimization of \\eqref{eq:FusiOpt2} is the under-estimate problem. Therefore, to ensure a reasonable solution, prior information of coefficient $\\mat{A}$ need to be exploited.~\\cite{QiWei2015TGRS} pre-trains a dictionary to represent $\\mat{A}$, and updates $\\mat{A}$ via sparse representation. Hyperspectral super-resolution (HySure) in~\\cite{Simoes2015} assumes that $\\mat{A}$ appears the spatial smoothness structure and regularize $\\mat{A}$ with band-by-band TV. \\cite{wei2015fast} translates the optimization of $\\mat{A}$ to a Sylvester equation and proposes a fast fusion method for  \\eqref{eq:FusiOpt2} (FUSE).\n\nSecondly, the optimization of orthogonal matrix $\\mat{E}^{\\top}$ is another challenge due to the non-convex of \\eqref{eq:FusiOpt2}. One appearing approach~\\cite{QiWei2015TGRS,Simoes2015,wei2015fast} is to pre-estimate $\\mat{E}$ from $\\mat{Y}$ in advance, and fix the variable $\\mat{E}$ during the optimization of \\eqref{eq:FusiOpt2}. Specially, FUSE~\\cite{wei2015fast} adopted principal component analysis (PCA), meanwhile, HySure utilized VCA to extract $\\mat{E}$ from $\\mat{Y}$. Another strategy is to regularize the update of $\\mat{E}$ and $\\mat{A}$ as a coupled matrix factorization problem, and blind dictionary learning strategy is utilized to update $\\mat{E}$~\\cite{kawakami2011high}. A hybrid inexact block coordinate descent~\\cite{wu2020hybrid} is introduced to exactly estimate $\\mat{E}$.\n\n\\subsection{Tensor based methods}\nThe above subspace based methods utilize low-rank matrix decomposition to exploit the low-rank property of the reshaped high-spatial resolution HS image $\\mat{X}$. However, the original HS image is a 3-D tensor, and therein, the researchers introduce the tensor decomposition to simultaneously capture the spatial and spectral low-rank property. The coupled sparse tensor factorization (CSTF) approach~\\cite{li2018fusing} utilized Tucker decomposition, presented as follows:\n\\begin{align}\n\\label{eq:TucD}\n&\\tensor{X} = \\tensor{O}\\times_1 \\mat{E}_1\\times_2 \\mat{E}_2\\times_3 \\mat{E}_3, \\\\\n&{\\rm subject \\;to} \\;\\; \\mat{E}_i^{\\top}\\mat{E}_i = \\mat{I},\\; \\|\\tensor{O}\\|_0 \\leq \\mathcal{C},\n\\notag\n\\end{align}\nto regularize the high-spatial resolution HS image $\\tensor{X}$. In \\eqref{eq:TucD}, the core tensor $\\tensor{O}$ is assumed to obey the sparse property, and $\\mat{E}_i$ is the orthogonal matrix of the $i$-th dimension. Subsequently, CP decomposition~\\cite{kanatsoulis2018hyperspectral}, tensor train decomposition~\\cite{dian2019learning}, tensor ring decomposition~\\cite{xu2020hyperspectral,he2020hyperspectral}, and so on, are utilized to regularize $\\tensor{X}$. Furthermore, non-local LRTD is also investigated for the fusion task~\\cite{dian2017hyperspectral,wang2017hyperspectral,xu2019nonlocal}.\n\nIt is worth noting that unmixing, orthogonal subspace, and tensor based methods share the common idea that spectral space of $\\mat{X}$ should lie in the low-dimensional space. Unmixing based approaches interpret the low-rank property as the endmembers and abundances, which are assumed to be non-negative, meanwhile Orthogonal subspace and tensor based methods ignore the non-negative restrict. Unmixing based approaches are interpretable from the physical meaning, but suffering from the unstable convergence in the optimization. Orthogonal subspace and tensor based methods lose physical meaning, but can be optimized more elegantly. \n\nVery recently, there are some preliminary works to perform the fusion task by means of DL-based methods \\cite{mei2017hyperspectral,haut2018new,liu2019stfnet,liu2019efficient,zheng2020coupled,uezato2020guided,yao2020cross} and show the effective and competitive fusion performance. A similar problem existed in these methods is the model interpretability and rationality. Clearly explaining the intrinsic meaning in each layer of deep networks would contribute to better modeling the fusion task and further obtaining higher-quality products.\n\n\\begin{table}[!t]\n\\centering\n\\caption{Quantitative comparison of different algorithms on the HS-MS image fusion experiments. The best one is shown in bold.}\n\\resizebox{0.4\\textwidth}{!}{\n\\begin{tabular}{c||cccc}\n\\toprule[1.5pt] Methods & RMSE & ERGAS & SA & SSIM \\\\\n\\hline \\hline\nCNMF \\cite{yokoya2012coupled} & 6.404 & 0.715 & 4.89 & 0.8857\\\\\nICCV'15 \\cite{lanaras2015hyperspectral} & \\bf 5.203 & \\bf 0.589 & \\bf 4.64 & \\bf 0.9139\\\\\nHySure \\cite{Simoes2015} & 8.537 & 0.812 & 9.45 & 0.8527\\\\\nFUSE \\cite{wei2015fast} & 8.652 & 0.869 & 9.51 & 0.8401\\\\\nCSTF \\cite{li2018fusing} & 8.32 & 0.841 & 8.34 & 0.8419\\\\\nSTEREO \\cite{kanatsoulis2018hyperspectral} & 9.4425 & 0.891 & 9.78  & 0.8231\\\\\nNLSTF \\cite{dian2017hyperspectral} & 8.254 & 0.819 & 8.36 & 0.8424\\\\\n\\bottomrule[1.5pt]\n\\end{tabular}}\n\\label{tab:fusion}\n\\end{table}\n\n\n\\begin{figure*}[!t]\n\t  \\centering\n\t\t\\subfigure[Training for MML and CML]{\n\t\t\t\\includegraphics[width=0.315\\textwidth]{MML_TR}\n\t\t}\n\t\t\\subfigure[Testing for MML]{\n\t\t\t\\includegraphics[width=0.315\\textwidth]{MML_TE}\n\t\t}\n\t\t\\subfigure[Testing for CML]{\n\t\t\t\\includegraphics[width=0.315\\textwidth]{CML_TE}\n\t\t}\n\t\t\\caption{An illustration for the model's training and testing in MML- and CML-based classification tasks (take the bi-modality as an example). (a) They share the same training process, i.e., two modalities are used for model training. The main difference lies in the testing phase, (b) MML still needs the input of two modalities, (c) while one modality is absent in CML.}\n\\label{fig:CML}\n\\end{figure*}\n\n\\subsection{Experimental Study}\nIn this section, we select  unmixing based methods: CNMF\\footnote{\\url{ http:\/\/naotoyokoya.com\/Download.html}}~\\cite{yokoya2012coupled}, ICCV'15\\footnote{\\url{https:\/\/github.com\/lanha\/SupResPALM}} \\cite{lanaras2015hyperspectral}; subspace based methods: HySure\\footnote{\\url{ https:\/\/github.com\/alfaiate}}~\\cite{Simoes2015}, FUSE\\footnote{\\url{ https:\/\/github.com\/qw245\/BlindFuse}}~\\cite{wei2015fast}; tensor decomposition regularized methods: STEREO\\footnote{\\url{ https:\/\/sites.google.com\/site\/harikanats\/}}~\\cite{kanatsoulis2018hyperspectral}, CSTF~\\cite{li2018fusing}; finally non-local tensor decomposition regularized method: NLSTF\\footnote{\\url{ https:\/\/sites.google.com\/view\/renweidian\/}}~\\cite{dian2017hyperspectral} for the comparison and analysis. We use evaluation indices, including the root mean square error (RMSE), relative dimensional global error in synthesis (ERGAS) \\cite{wald2000quality}, MSA, and SSIM \\cite{wang2004image} as evaluation criteria for the fusion results of different methods.\n\nThe selected dataset for the experiment is the Chikusei dataset obtained at Chikusei, Ibaraki, Japan, on 29 July 2014~\\cite{yokoya2017hyperspectral}. The selected high-spatial resolution HS image is of size $448 \\times 448 \\times 128$, and the simulated HS-MS images are of size $448 \\times 448 \\times 3$ and $14 \\times 14 \\times 128$, respectively. Tab.~\\eqref{tab:fusion} presents the quantitative comparison results of different algorithms on the HS-MS image fusion, meanwhile Fig.~\\eqref{fig:Chikusei_reconstruct} presents the visual illustration. From the results, it can be observed that even the HS image is spatially degraded by 32 times, the fusion methods can efficiently reconstruct the spatial details with the help of a 3 band MS image. On this tested toy dataset, ICCV'15 performed the best. However, different datasets need different kinds of regularizers. The fusion of HS-MS images for efficient and large scale applications is still a challenge for further research. \n\n\\subsection{Remaining Challenges}\nSubspace based non-convex methods for the fusion of HS-MS images have been well developed. However, most remarkable results are achieved on the simulated experiments. For the real applications with HS-MS images from two different satellite sensors, there still remain several challenges.\n\n\\begin{itemize}\n    \\item \\textbf{Blind.} Most fusion methods assume the linear spatial and spectral downsampling from HR-HS image to HS-MS images. However, in real applications, the degradation is complex and unknown in advance. how to blindly reconstruct the HR-HS image is a challenge in future research. \n    \\item \\textbf{Regularizer.} We reviewed the subspace based fusion methods from unmixing, orthogonal subspace, and tensor decomposition perspectives. Different assumptions are suitable for the exploited for the different structure of the HS image. How to mine the essence of HS images and develop efficient regularizers for large scale processing still remains a challenge.\n    \\item \\textbf{Evaluation.} In the real cases, the enhanced HrHS images from HS and MS images are not existed as the reference images in the real scenario. How to evaluate the final enhanced HrHS images is also a key problem for the future fusion approach development of HS-MS. \n\\end{itemize}\n\n\\section{Cross-modality Learning for Large-scale Land Cover Mapping}\nWith the ever-growing availability of diverse RS data sources from both satellite and airborne sensors, multimodal data processing and analysis in RS \\cite{dalla2015challenges,hong2020more} can provide potential possibilities to break the performance bottleneck in many high-level applications, e.g., land cover classification.\n\nHS data are featured by rich spectral information, which enables the high discrimination ability for material recognition at a more accurate and fine level. It should be noted, however, that the HS image coverage from space is much narrow compared to MS imaging due to the limitations of imaging principles and devices. That means the HS-dominated multimodal learning (MML) fails to identify the materials on a large geographic coverage and even global scale \\cite{hong2020x}. But fortunately, large-scale MS or synthetic aperture radar (SAR) images are openly available from e.g., Sentinel-1, Sentinel-2, Landsat-8. This, therefore, drives us to ponder over a problem: \\textit{can HS images acquired only in a limited area improve the land cover mapping performance using a larger area covered by the MS or SAR images?} This is a typical issue of cross-modality learning (CML) from a ML's point of view.\n\nTake the bi-modality as an example, CML for simplicity refers to that training a model using two modalities and one modality is absent in the testing phase, or \\emph{vice versa} (only one modality is available for training and bi-modality for testing) \\cite{ngiam2011multimodal}. Such a CML problem that exists widely in a variety of RS tasks is more applicable to real-world cases. Fig. \\ref{fig:CML} illustrates the differences between MML and CML in terms of training and testing process. The core idea of CML is to find a new data space, where the information can be exchanged effectively across different modalities. Thereupon, we formulate this process in a general way as follows: \n\\begin{equation}\n\\label{CML_eq1}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{X},\\{\\mathbf{U}_{s}\\}_{s=1}^{m}}\\sum_{s=1}^{m}\\frac{1}{2}\\norm{\\mathbf{X}-\\mathbf{U}_{s}\\mathbf{Y}_{s}}_{\\F}^{2} \\;\\; {\\rm s.t.}\\; \\mathbf{X},\\{\\mathbf{U}_{s}\\}_{s=1}^{m}\\in \\mathcal{C},\n\\end{aligned}\n\\end{equation}\nwhere $m$ is the number of input modality. For simplicity, we only consider the bi-modality case in this topic, i.e., $m=2$. According to different learning strategies on modalities, CML can be roughly categorized into two groups: manifold alignment (MA) and shared subspace learning (SSL). The differences between the two types of approaches mainly lie in \n\\begin{itemize}\n    \\item MA learns the low-dimensional embedding by preserving the aligned manifold (or graph) structure between different modalities. In the process of graph construction, the similarities between samples (unsupervised MA) and indirect label information (supervised or semi-supervised MA) are used. Despite the competitive performance obtained by MA-based approaches for the CML task, the discrimination ability of learned features remains limited due to the lack of directly bridging low-dimensional features with label information.\n    \\item SSL, as the name suggests, aims to find a latent shared subspace, where the features of different modalities are linked via a manifold alignment regularizer. Also, the learned features are further connected with label information. The two steps are jointly optimized in a SSL model, tending to yield more discriminative feature representations.\n\\end{itemize}\nMore specifically, we will briefly review and detail some representative approaches belonging to the aforementioned two groups as follows.\n\n\\subsection{Manifold Alignment based Approach}\nAs the name suggests, MA is capable of aligning multiple modalities on manifolds into a latent subspace, achieving a highly effective knowledge transfer \\cite{wang2009general}. Due to the interactive learning ability, MA has a good fit for large-scale RS image classification. In \\cite{matasci2011transfer}, the domain adaptation was investigated to reduce the gap between the source and target domains of HS data for land cover classification. By simultaneously considering labeled and unlabeled samples, Tuia \\textit{et al.} \\cite{tuia2014semisupervised} used semi-supervised MA (SSMA) techniques \\cite{wang2011heterogeneous} to align the multi-view RS images onto the manifold space by the attempts to eliminate the effects of image variants caused by different views. Matasci \\textit{et al.} \\cite{matasci2015semisupervised} modified the classic transfer component analysis \\cite{pan2011domain}, making it applicable to land cover classification of RS images. Moreover, a kernelized MA approach presented in \\cite{tuia2016kernel} projected the multimodal RS data to a higher dimensional space and aligned them in a nonlinear way. Hu \\textit{et al.} \\cite{hu2019comparative} deeply reviewed the semi-supervised MA methods with respect to the fusion classification of HS and polarimetric SAR images. Based on the work in \\cite{hu2019comparative}, the same investigators made full use of topological data analysis and designed a new graph structure for optical (e.g., HS) and SAR data fusion \\cite{hu2019mima}. \n\nMathematically, the MA idea can be implemented by solving the following non-convex model:\n\\begin{equation}\n\\label{CML_eq2}\n\\begin{aligned}\n      \\mathop{\\min}_{\\{\\mathbf{U}\\}_{s=1}^{m}}\\frac{A+C}{B},\n\\end{aligned}\n\\end{equation}\nwhere $A$, $B$, and $C$ are \n\\begin{equation*}\n\\begin{aligned}\n      &A = \\frac{1}{2}\\sum_{p=1}^{m}\\sum_{q=1}^{m}\\sum_{i=1}^{n}\\sum_{j=1}^{n}\\norm{\\mathbf{U}_{p}\\mathbf{y}_{p}^{i}-\\mathbf{U}_{q}\\mathbf{y}_{q}^{j}}_{2}^{2}\\mathbf{W}_{sim}^{i,j},\\\\\n      &B = \\frac{1}{2}\\sum\\limits_{p=1}^{m}\\sum\\limits_{q=1}^{m}\\sum\\limits_{i=1}^{n}\\sum\\limits_{j=1}^{n}\\norm{\\mathbf{U}_{p}\\mathbf{y}_{p}^{i}-\\mathbf{U}_{q}\\mathbf{y}_{q}^{j}}_{2}^{2}\\mathbf{W}_{dis}^{i,j},\\\\\n      &C = \\frac{1}{2}\\sum_{t=1}^{m}\\sum_{i=1}^{n}\\sum_{j=1}^{n}\\norm{\\mathbf{U}_{t}\\mathbf{y}_{t}^{i}-\\mathbf{U}_{t}\\mathbf{y}_{t}^{j}}_{2}^{2}\\mathbf{W}_{t}^{i,j}.\n\\end{aligned}\n\\end{equation*}\nBy minimizing the problem (\\ref{CML_eq2}), the $\\{\\mathbf{U}\\}_{s=1}^{m}$ can be estimated via generalized eigenvalues decomposition. We then have $\\mathbf{X}=\\mathbf{U}_{s}\\mathbf{Y}_{s}$. Three different graphs need to be pre-computed in Eq. (\\ref{CML_eq2}), including the similarity graph, i.e., $\\mathbf{W}_{sim}$:\n\\begin{equation}\n\\label{CML_eq3}\n\\begin{aligned}\n\t  \\mathbf{W}_{sim}=\n\t      \\left[\n        \t \\begin{matrix}\n        \t  \\mathbf{W}_{sim}^{1,1}&\\mathbf{W}_{sim}^{1,2}&\\cdots&  \\mathbf{W}_{sim}^{1,m}\\\\\n        \t  \\mathbf{W}_{sim}^{2,1}&\\mathbf{W}_{sim}^{2,2}&\\cdots&  \\mathbf{W}_{sim}^{2,m}\\\\\n        \t  \\vdots&\\vdots&\\ddots&\\vdots\\\\\n        \t  \\mathbf{W}_{sim}^{m,1}&\\mathbf{W}_{sim}^{m,2}&\\cdots&  \\mathbf{W}_{sim}^{m,m}\\\\\n             \\end{matrix}\n         \\right],\n\\end{aligned}\n\\end{equation}\nthe dissimilarity matrix, i.e., $\\mathbf{W}_{dis}$:\n\\begin{equation}\n\\label{CML_eq4}\n\\begin{aligned}\n\t  \\mathbf{W}_{dis}=\n\t      \\left[\n        \t \\begin{matrix}\n        \t  \\mathbf{W}_{dis}^{1,1}&\\mathbf{W}_{dis}^{1,2}&\\cdots&  \\mathbf{W}_{dis}^{1,m}\\\\\n        \t  \\mathbf{W}_{dis}^{2,1}&\\mathbf{W}_{dis}^{2,2}&\\cdots&  \\mathbf{W}_{dis}^{2,m}\\\\\n        \t  \\vdots&\\vdots&\\ddots&\\vdots\\\\\n        \t  \\mathbf{W}_{dis}^{m,1}&\\mathbf{W}_{dis}^{m,2}&\\cdots&  \\mathbf{W}_{dis}^{m,m}\\\\\n             \\end{matrix}\n         \\right],\n\\end{aligned}\n\\end{equation}\nand the topology structure for each single modality obtained by $knn$ graph, i.e., $\\mathbf{W}_{t}$:\n\\begin{equation}\n\\label{CML_eq5}\n\\begin{aligned}\n\t  \\mathbf{W}_{t}=\n\t      \\left[\n        \t \\begin{matrix}\n        \t  \\mathbf{W}_{t}^{1,1}&\\mathbf{0}&\\cdots&  \\mathbf{0}\\\\\n        \t  \\mathbf{0}&\\mathbf{W}_{d}^{2,2}&\\cdots&  \\mathbf{0}\\\\\n        \t  \\vdots&\\vdots&\\ddots&\\vdots\\\\\n        \t  \\mathbf{0}&\\mathbf{0}&\\cdots&  \\mathbf{W}_{t}^{m,m}\\\\\n             \\end{matrix}\n         \\right].\n\\end{aligned}\n\\end{equation}\nIn Eqs. (\\ref{CML_eq3})$-$(\\ref{CML_eq5}), $\\mathbf{W}_{sim}^{i,j}$, $\\mathbf{W}_{dis}^{i,j}$, and $\\mathbf{W}_{t}^{i,j}$ are given, respectively, by\n\\begin{equation*}\n    \\mathbf{W}_{sim}^{i,j}=\n    \\begin{cases}\n      \\begin{aligned}\n          1, \\; \\; & \\text{if $\\mathbf{y}_{p}^{i}$ and $\\mathbf{y}_{q}^{j} \\in C_{k}$}\\\\\n          0, \\; \\; & \\text{otherwise,}\n      \\end{aligned}\n    \\end{cases}\n\\end{equation*} \n\\begin{equation*}\n    \\mathbf{W}_{dis}^{i,j}=\n    \\begin{cases}\n      \\begin{aligned}\n          1, \\; \\; & \\text{if $\\mathbf{y}_{p}^{i}$ and $\\mathbf{y}_{q}^{j}\\notin C_{k}$ }\\\\\n          0, \\; \\; & \\text{otherwise,}\n      \\end{aligned}\n    \\end{cases}\n\\end{equation*} \n\\begin{equation*}\n    \\mathbf{W}_{t}^{i,j}=\n    \\begin{cases}\n      \\begin{aligned}\n      \\exp\\frac{\\norm{\\mathbf{y}^{i}-\\mathbf{y}^{j}}_{2}^{2}}{2\\sigma^{2}}, \\; \\; & \\text{if $\\mathbf{y}_{p}^{i}\\in\\phi_{k}(\\mathbf{y}_{q}^{j})$;}\\\\\n      0, \\; \\; & \\text{otherwise,}\n      \\end{aligned}\n    \\end{cases}\n\\end{equation*} \nwhere $\\phi_{k}(\\bullet)$ denotes the $k$ nearest neighbors of $\\bullet$.\n\n\\subsection{Shared Subspace Learning based Approach}\nDue to the lack of the direct relational modeling between the learned features and label information, MA-based approaches fail to activate the connections across modalities effectively \\cite{ngiam2011multimodal}, thereby yielding the relatively weak transferability between different modalities, particularly heterogeneous data. There have been some tentative works in recent years, providing potential solutions to overcome the aforementioned challenges. For example, Hong \\textit{et al.} \\cite{hong2019cospace} for the first time proposed a supervised CoSpace model to learn a latent discriminative subspace from HS-MS correspondences for the CML-related classification problem. Beyond it, the same authors \\cite{hong2019learnable} fully tapped the potential of the CoSpace by learning the data-driven graph structure from both labeled and unlabeled samples, yielding a learnable manifold alignment (LeMA) approach. Moreover, \\cite{hong2020learning} deeply investigated and analyzed different regression techniques, i.e., $\\ell_2$-norm ridge regression, $\\ell_1$-norm sparse regression, in CoSpace. In \\cite{hong2020graph}, a semi-supervised graph-induced aligned learning (GiAL) was developed by jointly regressing labels and pseudo-labels.\n\nAccordingly, these methods can be generalized to be a unified model \\cite{hong2019cospace} to address the CML's problem in a regression-based fashion:\n\\begin{equation}\n\\label{CML_eq6}\n\\begin{aligned}\n      \\mathop{\\min}_{\\mathbf{P}, \\{\\mathbf{U}_{s}\\}_{s=1}^{m}}&\\frac{1}{2}\\norm{\\mathbf{M}-\\mathbf{P}\\mathbf{U}_{s}\\mathbf{Y}_{s}}_{\\F}^{2}+\\Psi(\\mathbf{P})+\\Omega(\\{\\mathbf{U}_{s}\\}_{s=1}^{m})\\\\\n      & {\\rm s.t.}\\;\\; \\mathbf{U}_{s}\\mathbf{U}_{s}^{\\top}=\\mathbf{I}, \\;s=1,\\cdots,m,\n\\end{aligned}\n\\end{equation}\nwhere $\\{\\mathbf{U}_{s}\\}_{s=1}^{m}$ denote the projections linking to the shared features for different modalities. To avoid the over-fitting of the model and stabilize the learning process, $\\mathbf{P}$ can be regularized by the Frobenius-norm \\cite{hong2019cospace} or $\\ell_{1,1}$-norm \\cite{hong2020learning}:\n\\begin{equation}\n\\label{CML_eq7}\n\\begin{aligned}\n      \\Psi(\\mathbf{P})=\\norm{\\mathbf{P}}_{\\F}^{2}, \\; \\text{or}\\; \\norm{\\mathbf{P}}_{1,1},\n\\end{aligned}\n\\end{equation}\nand $\\Omega(\\{\\mathbf{U}_{s}\\}_{s=1}^{m})$ is specified as a manifold alignment term on the multimodal data, which is written as\n\\begin{equation}\n\\label{CML_eq8}\n\\begin{aligned}\n      \\Omega(\\{\\mathbf{U}_{s}\\}_{s=1}^{m})=\\tr(\\mathbf{U}\\mathbf{Y}\\mathbf{L}\\mathbf{Y}^{\\top}\\mathbf{U}^{\\top}),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{U}=[\\mathbf{U}_{1},\\mathbf{U}_{2},\\cdots,\\mathbf{U}_{m}]$ and\n\\begin{equation*}\n\\begin{aligned}\n   \\mathbf{Y}=\n\t      \\left[\n        \t \\begin{matrix}\n        \t  \\mathbf{Y}_{1} & \\mathbf{0} & \\cdots &  \\mathbf{0}\\\\\n        \t  \\mathbf{0} & \\mathbf{Y}_{2} & \\cdots &  \\mathbf{0}\\\\\n        \t  \\vdots&\\vdots&\\ddots&\\vdots\\\\\n        \t  \\mathbf{0} & \\mathbf{0} & \\cdots &  \\mathbf{Y}_{m}\\\\\n             \\end{matrix}\n         \\right].    \n\\end{aligned}\n\\end{equation*}\nSimilar to Fig. \\ref{fig:graph}, $\\mathbf{L}$ is a joint Laplacian matrix. \n\nUsing the general model in Eq. (\\ref{CML_eq6}), \n\\begin{itemize}\n    \\item \\cite{hong2019cospace} considers the HS-MS correspondences that exist in an overlapped region as the model input. The learned shared representations (e.g., $\\mathbf{X}=\\mathbf{U}_{s}\\mathbf{Y}_{s}$) can be then used for classification on a larger area, even though only MS data are available in the inference phase;\n    \\item Differently, \\cite{hong2019learnable} inputs not only the labeled HS-MS pairs but also unlabeled MS data in large quantity. With the graph learning, i.e., the variable $\\mathbf{L}$ is to be learned from the data rather than fixed by a given RBF, the unlabeled information can be made use of to find a better decision boundary. According to the equivalent form of Eq. (\\ref{CML_eq8}), we then have\n    \\begin{equation}\n        \\label{CML_eq9}\n        \\begin{aligned}\n             \\tr(\\mathbf{U}\\mathbf{Y}\\mathbf{L}\\mathbf{Y}^{\\top}\\mathbf{U}^{\\top})=\\frac{1}{2}\\tr(\\mathbf{W}\\mathbf{d})=\\frac{1}{2}\\norm{\\mathbf{W}\\odot\\mathbf{d}}_{1,1},\n        \\end{aligned}\n    \\end{equation}\n    where $\\mathbf{d}_{i,j}=\\norm{\\mathbf{x}_{i}-\\mathbf{x}_{j}}_{2}^{2}$ denotes the pair-wise distance in Euclidean space. Using Eq. (\\ref{CML_eq9}), the resulting optimization problem with respect to the variable $\\mathbf{W}$ is \n    \\begin{equation}\n        \\label{CML_eq10}\n        \\begin{aligned}\n              \\frac{1}{2}&\\norm{\\mathbf{W}\\odot\\mathbf{d}}_{1,1}\\\\\n              &{\\rm s.t.} \\; \\mathbf{W}=\\mathbf{W}^{\\top},\\; \\mathbf{W}_{i,j}\\geq 0,\\; \\norm{\\mathbf{W}}_{1,1}=c.\n        \\end{aligned}\n    \\end{equation}\n    \\item Inspired by the brain-like feedback mechanism presented in \\cite{hong2019learning}, a more intelligent CML model was proposed  \\cite{hong2020graph}. With the joint use of labels and pseudo-labels updated by the graph feedback in each iteration, more representative features can be also learned (even if a certain modality is absent, i.e., the CML case). \n\\end{itemize}\n\n\\begin{table}[!t]\n\\centering\n\\caption{Quantitative comparison of SOTA algorithms related to the CML's issue in terms of OA, AA, and $\\kappa$ using the NN classifier on the Houston2013 datasets. The best one is shown in bold.}\n\\resizebox{0.35\\textwidth}{!}{\n\\begin{tabular}{c||ccc}\n\\toprule[1.5pt] Methods & OA (\\%) & AA (\\%) & $\\kappa$ \\\\\n\\hline \\hline\nO-Baseline & 62.12 & 65.97 & 0.5889\\\\\nUSMA \\cite{he2004locality} & 65.54 & 68.81 & 0.6251\\\\\nSMA \\cite{wang2009general} & 68.01 & 70.50 & 0.6520\\\\\nSSMA \\cite{wang2011heterogeneous} & 69.29 & 72.00 & 0.6659\\\\\nCoSpace \\cite{hong2019cospace} & 69.38 & 71.69 & 0.6672\\\\\nLeMA \\cite{hong2019learnable} & 73.42 & 74.76 & 0.7110\\\\\nGiAL \\cite{hong2020graph} & \\bf 80.66 & \\bf 81.31 & \\bf 0.7896\\\\\n\\bottomrule[1.5pt]\n\\end{tabular}}\n\\label{tab:CML}\n\\end{table}\n\n\\subsection{Experimental Study}\nWe evaluate the performance of several SOTA algorithms related to the CML's issue both quantitatively and qualitatively. They are O-Baseline (i.e., using original image features), unsupervised MA (USMA) \\cite{he2004locality}, supervised MA (SMA)\\footnote{\\url{https:\/\/sites.google.com\/site\/changwangnk\/home\/ma-html}} \\cite{wang2009general}, SSMA \\cite{wang2011heterogeneous}, CoSpace\\footnote{\\url{https:\/\/github.com\/danfenghong\/IEEE_TGRS_CoSpace}} \\cite{hong2019cospace}, LeMA\\footnote{\\url{https:\/\/github.com\/danfenghong\/ISPRS_LeMA}} \\cite{hong2019learnable}, and GiAL \\cite{hong2020graph}. Three common indices, e.g., \\textit{OA}, \\textit{AA}, and $\\kappa$, are adopted to quantify the classification performance using the SVM classifier on the Houston2013 HS-MS datasets that have been widely used in many researches \\cite{hong2019cospace,hong2019learnable,hong2020learning,hong2020graph}. \n\nTable \\ref{tab:CML} gives the quantitative comparison between the above-mentioned methods for the CML-related classification, while Fig. \\ref{fig:CML_CM_H} visualizes a region of interest (ROI) of classification maps. By and large, the classification accuracy of O-Baseline, i.e., only using MS data, is much lower than other methods. By aligning multimodal data on manifolds, MA-based approaches perform better than O-Baseline with the approximated increase of $3\\%$ OA in USMA, $6\\%$ OA in SMA, and $7\\%$ OA in SSMA. As expected, the classification performance of SSL-based models, e.g., CoSpace, LeMA, and GiAL, is obviously superior to that of MA-based ones. In particular, GiAL dramatically outperforms other competitors, owing to the use of the brain-like feedback mechanism and graph-driven pseudo-label learning. Visually, shared learning methods tend to capture more robust spectral properties and achieve more realistic classification results. As can be seen from Fig. \\ref{fig:CML_CM_H}, the shadow region covered by clouds can be finely classified by CoSpace, LeMA, and GiAL, while MA-based models fail to identify the materials well in the region.\n\n\\begin{figure}[!t]\n\t  \\centering\n\t\t\\subfigure{\n\t\t\t\\includegraphics[width=0.42\\textwidth]{CML_CM_H}\n\t\t}\n        \\caption{ROI visualization of classification maps using different SOTA methods related to the CML's issue.}\n\\label{fig:CML_CM_H}\n\\end{figure}\n\n\\subsection{Remaining Challenges}\nCML has drawn growing interest from researchers in computer vision and ML, yet it is rarely investigated in the RS community. In other words, CML is a relatively emerging topic in RS, which means there are lots of difficulties (or challenges) to be overcome. In detail,\n\n\\begin{itemize}\n    \\item \\textbf{Data Preparation.} Since multimodal data are acquired by different contexts, sensors, resolutions, etc., this inevitably poses a great challenge to data collection and processing. For example, the errors caused by interpolation of different resolutions, registration methods of geographical coordinates, pixel-wise biases of different sensors, and uncertainties of image degradation in the imaging process easily generate unregistered multimodal data to a great extent.\n    \\item \\textbf{Model Transferability.} Due to different imaging mechanisms and principles, the viscosity between pixels from the same modality is much stronger than from the different modalities. This might lead to difficulties in fusing multimodal information at a deep level, particularly heterogeneous data (e.g., HS and SAR data), further limiting the model's transferability.\n    \\item \\textbf{Labeling.} Unlike natural images or street view images that are relatively easy and accurate to be labeled manually, labeling RS scenes (field trips needed) is extremely expensive and time-consuming. Consequently, a limited number of labeled samples are available for training and even worse, there are many noisy labels in these samples. These problems will be noteworthy and to-be-solved key points of the next generation interpretable AI models in the RS-related CML task.\n\\end{itemize}\n\n\\section{Conclusion and Future Prospect}\nCharacterized by the nearly continuous spectral profile that is capable of sampling and representing the whole electromagnetic spectrum, HS images play an important role in both promoting developments of new techniques and accelerating the practical applications, not only limiting to the fields of RS, geoscience, signal and image processing, numerical optimization and modeling, ML, and AI. However, there still exist severe difficulties and challenges that need to be carefully considered in the development and application of HS RS techniques. One sign reveals that HS data analysis methods dominated by expert systems have been unable to meet the demand of an ever-growing volume of HS data whether in performance gain or in processing efficiency. Another sign is that despite the currently unprecedented progress made on computer vision, ML, and AI techniques, the model compatibility and interpretability for HS RS applications remain limited.\n\nDue to the SVs of HS data caused by various degradation mechanisms (e.g., environmental condition, atmospheric effects, spectral nonlinear mixing, etc.), the redundancy of high-dimensional HS signals, and the complex practical cases underlying in the HS products (e.g., low spatial resolution, narrow imaging range, instrumental noises), convex models under ideal circumstances usually fail to extract useful and diagnostic information from HS images (especially those products that are corrupted seriously) and thereby understand our environment. Considering that non-convex modeling is capable of characterizing more complex real scenes and better providing the model interpretability, in this article we present a comprehensive and technical survey over five promising and representative research topics related to HS RS with a focus on non-convex modeling, such as HS image restoration, dimensionality reduction and classification, data fusion and enhancement, spectral unmixing, and cross-modality learning. Among these topics, we review the current state-of-the-art methods with illustrations, show the significance and superiority of non-convex modeling to bridge the gap between HS RS and interpretable AI, and point out remaining challenges and future research directions.\n\nIt is well-known that the HS image processing and analysis chain is wide-ranging. Apart from the five topics covered in this paper, we are not able to detailedly report all the important and promising applications related to HS RS missions. There are several very noteworthy and active fields that should be paid more attention in future work, including target\/change detection, time-series analysis, multitemporal fusion\/classification, physical parameter inversion, image quality assessment, and various practical applications (e.g., precious farming, disaster management and response). Moreover, some crucial steps for the HS image pre-processing algorithms are also missing, such as atmospheric and geometric corrections, geographic coordinate registration, etc. Furthermore, the methodologies summarized and reported in this article mainly focus on the survey of shallow non-convex models. Undeniably, deep models, e.g., DL-based methods, are capable of excavating deeper and intrinsic properties of HS data. There is, therefore, room for improvement in the development of more intelligent DL-related non-convex modeling with application to HS RS. For example, embedding more physically meaningful priors and devising advanced and novel deep unfolding \\cite{hershey2014deep} or unrolling \\cite{diamond2017unrolled} strategies to closely integrate data-driven DL and theoretically-guaranteed optimization technique is to open and interpret the so-called ``black box'' in DL models.\n\nFinally, we have to admit that non-convex modeling and optimization is a powerful tool across multidisciplinary and the relevant studies along the direction have been made tremendous progress theoretically and technically. This provides the possibility of creating new methodologies and implementing interpretable AI for various HS RS applications. In this paper, we attempt to ``intellectualize'' these models by introducing more interpretable and physically meaningful knowledge to meet the actual needs in a non-convex modeling fashion. In other words, we hope that non-convex modeling can play the role as a bridge to connect interpretable AI models and various research topics in HS RS. Our efforts in this paper are made to foster curiosity and create a good starting point for post-graduate, Ph.D. students, and senior researchers working in the HS-related fields, thereby further looking for new and advanced research directions in the interdisciplinary involving signal and image processing, ML, AI, and RS.\n\n\\section*{Acknowledgement}\nThe authors would like to thank Prof. D. Landgrebe from\nPurdue University for providing the AVIRIS Indian Pines data, Prof. P. Gamba from the University of Pavia for providing the ROSIS-3 Pavia University and Centre data, the Hyperspectral Image Analysis group at the University of Houston for providing the CASI University of Houston dataset used in the IEEE GRSS DFC2013 and DFC2018, and the Hyperspectral Digital Imagery Collection Experiment (HYDICE) for sharing the urban dataset free of charge. \n\nThis work from the D. Hong and X. Zhu sides is jointly supported by the German Research Foundation (DFG) under grant ZH 498\/7-2, by the Helmholtz Association through the framework of Helmholtz Artificial Intelligence (HAICU) - Local Unit ``Munich Unit @Aeronautics, Space and Transport (MASTr)'' and Helmholtz Excellent Professorship ``Data Science in Earth Observation - Big Data Fusion for Urban Research'', by the German Federal Ministry of Education and Research (BMBF) in the framework of the international AI future lab ``AI4EO -- Artificial Intelligence for Earth Observation: Reasoning, Uncertainties, Ethics and Beyond''. This work from the L. Gao side is supported by the National Natural Science Foundation of China under Grant 42030111 and Grant 41722108. This work from the W. He and N. Yokoya sides is supported by the Japan Society for the Promotion of Science under KAKENHI 19K20308 and KAKENHI 18K18067. This work from the J. Chanussot side has been partially supported by MIAI@Grenoble Alpes, (ANR-19-P3IA-0003) and by the AXA Research Fund. \n\nThe corresponding authors of this paper are Dr. Wei He and Prof. Lianru Gao.\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Intro.sec} In our $\\Lambda$CDM Universe,\nmost of the mass is made up of dark matter. On large scales baryons\ntrace the dark matter and its gravitational potential. Baryonic gas\nfalls into galaxies at the centres of dark matter haloes, where it\ncools radiatively and collapses to form stars. By naive gravitational\narguments, star formation should be a fast affair, consuming the gas\nover local free-fall times. However, from observations we know that it\nis a slow and inefficient process, taking $\\sim20-100$ free-fall\ntimes, depending on the scale under consideration\n\\citep[e.g.][]{Zuckerman1974, Krumholz2007,\n  EvansNealJ2009}. Also, while observers have a notoriously hard time\nconfirming the existence of gas flowing into galaxies\n\\citep{Crighton2013}, they instead routinely detect oppositely\ndirected \\emph{outflows} at velocities of hundreds of km\/s \\citep[see\nreview by][]{Veilleux2005}.\n\nTo understand the non-linear problem of galaxy formation and\nevolution, theorists use cosmological simulations of dark matter,\ndescribing the flow and collapse of baryonic star-forming gas either\nwith directly coupled hydrodynamics or semi-analytic models. Strong\nfeedback in galaxies is a vital ingredient in any model of galaxy\nevolution, be it hydrodynamical or semi-analytic, that comes even\nclose to reproducing basic observables, such as the star formation\nhistory of the Universe, the stellar mass function of galaxies, the\nKennicutt-Schmidt relation, rotational velocities, and outflows\n\\citep[e.g.][]{Vogelsberger2013, Dubois2014, Hopkins2014,\n  Schaye2015, Wang2015, Somerville2015}.\n\nIn order to capture the inefficient formation of stars, the first\ngeneration of galaxy evolution models included core collapse (or type\nII) supernova (SN) feedback, where massive stars ($\\ga 8 \\ \\rm{M}_{\\odot}$) end\ntheir short lives with explosions which inject mass, metals, and\nenergy into the inter-stellar medium (ISM). In early hydrodynamical\nsimulations, the time-integrated type II SN energy of a stellar\npopulation, $10^{51}$ ergs per SN event, was dumped thermally into the\ngas neighbouring the stellar population \\citep{Katz1992}. However,\nsuch thermal dump feedback had little impact on star formation,\nresulting in an over-abundance of massive and compact galaxies. This\nso-called over-cooling problem is partly numerical in nature, and a\nresult of low resolution both in time and space. As discussed by\n\\cite{DallaVecchia2012}, the energy is injected into too large a\ngas mass, typically resulting in much lower temperatures than those at\nwork in sub-pc scale SN remnants. The relatively high cooling rates at\nthe typical initial temperatures attained in the remnant, of\n$10^5-10^6$ K, allow a large fraction of the injected energy to be\nradiated away before the gas reacts hydrodynamically, resulting in\nsuppressed SN blasts and hence weak feedback.  Gas cooling is,\nhowever, also a real and physical phenomenon, and while it is\nover-estimated in under-resolved simulations, a large fraction of the\nenergy in SN remnants may in fact be radiated away instead of being\nconverted into large-scale bulk motion \\citep{Thornton1998}.\n\nA number of sub-resolution SN feedback models have been developed\nover the last two decades for cosmological simulations, with the\nprimary motivation of reproducing large-scale observables, such as the\ngalaxy mass function, by means of efficient feedback. The four main\nclasses of these empirically motivated SN feedback models are i)\nkinetic feedback \\citep{Navarro1993}, where a fraction of the SN\nenergy is injected directly as momentum, often in combination with\ntemporarily disabling hydrodynamical forces \\citep{Springel2003},\nii) delayed cooling \\citep[e.g.][]{Gerritsen1997, Stinson2006},\nwhere radiative cooling is turned off for some time in the SN remnant,\niii) stochastic feedback \\citep{DallaVecchia2012}, where the SN\nenergy is re-distributed in time and space into fewer but more\nenergetic explosions, and iv) multiphase resolution elements that\nside-step unnatural `average' gas states at the resolution limit\n\\citep{Springel2003, Keller2014}.\n\nIn principle, a physically oriented approach to implementing SN\nfeedback with sub-grid models is desirable. The goal is then to\ninject the SN blast as it would emerge on the smallest resolved scale,\nby making use of analytic models and\/or high-resolution simulations\nthat capture the adiabatic phase, radiative cooling, the momentum\ndriven phase, and the interactions between different SN\nremnants. However, these base descriptions usually include simplified\nassumptions about the medium surrounding the SN remnant, and fail to\ncapture the complex inhomogeneities that exist on unresolved scales\nand can have a large impact on cooling rates. In addition, even if the\nSN energy is injected more or less correctly at resolved scales, it\nwill generally fail to evolve realistically thereafter because the\nmulti-phase ISM of simulated galaxies is still at best marginally\nresolved. Hence there remains a large uncertainty in how efficiently\nthe SN blast couples to the ISM. This translates into considerable\nfreedom, which requires SN feedback models to be calibrated to\nreproduce a set of observations \\citep[see discussion\nin][]{Schaye2015}.\n\nThe most recent generation of cosmological simulations has been\nrelatively successful in reproducing a variety of observations, in\nlarge part thanks to the development of subgrid models for efficient\nfeedback and the ability to calibrate their parameters, as well as the\ninclusion of efficient active-galactic nucleus (AGN) feedback in\nhigh-mass galaxies.  However, higher-resolution simulation works\n\\citep[e.g.][]{Hopkins2012b, Agertz2013} suggest that SNe alone\nmay not provide the strong feedback needed to produce the inefficient\nstar formation we observe in the Universe.\n\nAttention has thus been turning towards complementary forms of stellar\nfeedback, which provide additional support to the action of\nSNe. Possible additional feedback mechanisms include stellar winds\n\\citep[e.g.][]{Dwarkadas2007, Rogers2013, Fierlinger2016},\nradiation pressure (e.g. \\citealt{Haehnelt1995, Thompson2005,\n  Murray2010}, but see \\citealt{Rosdahl2015}), and cosmic rays\n\\citep[e.g.][]{Booth2013, Hanasz2013, Salem2014,\n  Girichidis2016}.\n\nNone the less, SN explosions remain a powerful source of energy and\nmomentum in the ISM and a vital ingredient in galaxy evolution. For\nthe foreseeable future a sub-resolution description of them will\nremain necessary in cosmological simulations and even in most feasible\nstudies of isolated galaxies. The true efficiency of SN feedback is\nstill not well known, and hence we do not know to what degree we need\nto improve our SN feedback sub-resolution models versus appealing to\nthe aforementioned complementary physics.\n\nRather than introducing a new or improved sub-resolution SN feedback\nmodel, the goal of this paper is to study existing models, using\ncontrolled and relatively inexpensive numerical experiments of\nisolated galaxy discs modelled with gravity and hydrodynamics in the\nEulerian (i.e. grid-based) code {\\sc Ramses}{} \\citep{Teyssier2002}. We\nuse those simulations to assess each model's effectiveness in\nsuppressing star formation and generating galactic winds, the main\nobservational constraints we have on feedback in galaxies.\n\nWe study five subgrid prescriptions for core-collapse SN feedback in\nisolated galaxy discs. We explore the `maximum' and `minimum' effects\nwe can get from SN feedback using these models, and consider how they\nvary with galaxy mass, resolution, and feedback parameters where\napplicable. The simplest of those models is the `classic'\n\\emph{thermal dump}, where the SN energy is simply injected into the\nlocal volume containing the stellar population. Three additional\nmodels we consider have been implemented and used previously in\n{\\sc Ramses}{}. These are, in chronological order, \\emph{kinetic feedback},\ndescribed in \\cite{Dubois2008} and used in the Horizon-AGN\ncosmological simulations \\citep{Dubois2014}, \\emph{delayed\n  cooling}, described in \\cite{Teyssier2013}, and \\emph{mechanical\n  feedback}, described in \\cite{Kimm2014} and\n\\cite{Kimm2015}. In addition, for this work we have implemented\n\\emph{stochastic feedback} in {\\sc Ramses}{}, adapted from a previous\nimplementation in the smoothed particle hydrodynamics (SPH) code\n{\\sc Gadget}{}, described in \\citet[][henceforth\n{DS12}{}]{DallaVecchia2012}.\n\n\\begin{table*}\n  \\centering\n  \\caption\n  {Simulation initial conditions and parameters for the two disc\n    galaxies modelled in this paper. The listed parameters are, from \n    left to right:\n    Galaxy acronym used throughout the paper, $\\vcirc$: circular\n    velocity at the virial radius, $R_{\\rm vir}$: halo virial radius\n    (defined as the radius within which the DM density is $200$ times the\n    critical density at redshift zero), $L_{\\rm box}$:\n    simulation box length, $M_{\\rm halo}$: DM halo mass, $M_{\\rm disc}$: disc\n    galaxy mass in baryons (stars+gas), $f_{\\rm gas}$: disc gas\n    fraction, $M_{\\rm bulge}$: stellar bulge mass, $N_{\\rm part}$:\n    Number of DM\/stellar particles, $m_{*}$: mass of\n    stellar particles formed during the simulations, $\\Delta x_{\\rm max}$:\n    coarsest cell resolution, $\\Delta x_{\\rm min}$: finest cell resolution,\n    $Z_{\\rm disc}$: disc metallicity.}\n  \\label{sims.tbl}\n  \\begin{tabular}{l|rrrrrrrrrrrr}\n    \\toprule\n    Galaxy  & $\\vcirc$ & $R_{\\rm vir}$ & $L_{\\rm box}$   & $M_{\\rm halo}$ \n            & $M_{\\rm disc}$ & $f_{\\rm gas}$ & $M_{\\rm bulge}$ & $N_{\\rm part}$   \n            & $m_{*}$ & $\\Delta x_{\\rm max}$& $\\Delta x_{\\rm min}$  &  $Z_{\\rm disc}$ \\\\ \n    acronym & [$\\kms$]   & [kpc]   & [kpc]     & [$\\rm{M}_{\\odot}$]\n            & [$\\rm{M}_{\\odot}$]&         & [$\\rm{M}_{\\odot}$] &   \n            & [$\\rm{M}_{\\odot}$]& [kpc]   & [pc]      & [$\\Zsun$]  \\\\\n    \\midrule\n    {\\sc g9}   & $65$       & $89$    & $300$     &$10^{11}$ \n           &$3.5 \\times 10^9$& $0.5$   &$3.5 \\times 10^8$& $10^6$     \n           & $2.0 \\times 10^3$ & $2.3$   & $18$      & 0.1\\\\\n     {\\sc g10}   & $140$      & $192$   & $600$     & $10^{12}$\n           &$3.5 \\times 10^{10}$ & $0.3$ & $3.5 \\times 10^9$ & $10^6$ \n           & $1.6 \\times 10^4$ & $4.7$   & $36$ & $1.0$ \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nThe layout of this paper is as follows. First, we describe the setup\nof our isolated galaxy disc simulations in \\Sec{simulations.sec}. We\nthen describe the SN feedback models in \\Sec{models.sec}. In\n\\Sec{comparison.sec} we compare results for each of these models\nusing their fiducial parameters in galaxy discs of two different\nmasses, focusing on the suppression of star formation and the\ngeneration of outflows. In \\Sec{res_conv.sec} we compare how these\nresults converge with numerical resolution, both in terms of physical\nscale, i.e. minimum gas cell size, and also in terms of stellar\nparticle mass. In Sections \\ref{fb_stoch.sec} - \\ref{fb_kin.sec} we\ntake a closer look at the stochastic, delayed cooling, and kinetic\nfeedback models respectively, and study how varying the free\nparameters in each model affects star formation, outflows and gas\nmorphology. The reader can skip those sections or pick out those of\ninterest, without straying from the thread of the paper. We discuss\nour results and implications in \\Sec{Discussion.sec}, and, finally, we\nconclude in \\Sec{Conclusions.sec}.\n\n\\section{Simulations} \\label{simulations.sec}\nBefore we introduce the SN feedback models compared in this paper, we\nbegin by describing the default setup of the simulations common to all\nruns.\n\nWe run controlled experiments of two rotating isolated disc galaxies,\nconsisting of gas and stars, embedded in dark matter (DM) haloes. The\nmain difference between the two galaxies is an order of magnitude\ndifference in mass, both baryonic and DM. We use the AMR code\n{\\sc Ramses}{} \\citep{Teyssier2002}, which simulates the interaction of\ndark matter, stellar populations and baryonic gas, via gravity,\nhydrodynamics and radiative cooling. The equations of hydrodynamics\nare computed using the HLLC Riemann solver \\citep{Toro1994} and the\nMinMod slope limiter to construct variables at cell interfaces from\nthe cell-centred values. We assume an adiabatic index of\n$\\gamma = 5\/3$ to relate the pressure and internal energy, appropriate\nfor an ideal monatomic gas. The trajectories of collisionless DM and\nstellar particles are computed using a particle-mesh solver with\ncloud-in-cell interpolation (\\citealt{Guillet2011}; the resolution\nof the gravitational force is the same as that of the hydrodynamical\nsolver).\n\n\\subsection{Initial conditions}\nThe main parameters for the simulated galaxies and their host DM haloes\nare presented in \\Tab{sims.tbl}. We focus most of our analysis on the\nlower-mass galaxy, which we name {\\sc g9}{}. It has a baryonic mass of\n$M_{\\rm bar} = M_{\\rm disc} + M_{\\rm bulge} = 3.8 \\times 10^9 \\ \\rm{M}_{\\odot}$, with an initial\ngas fraction of $f_{\\rm gas}=0.5$, and it is hosted by a DM halo of mass\n$M_{\\rm halo}=10^{11} \\ \\rm{M}_{\\odot}$.  We also compare a less detailed set of\nresults for feedback models in a more massive galaxy, {\\sc g10}{}, similar\n(though somewhat lower) in mass to our Milky-Way (MW), with\n$M_{\\rm bar}= 3.8 \\times 10^{10} \\ \\rm{M}_{\\odot}$, $f_{\\rm gas}=0.3$, and\n$M_{\\rm halo}=10^{12} \\ \\rm{M}_{\\odot}$. Each simulation is run for $250$ Myr, which\nis $2.5$ orbital times (at the scale radii) for both galaxy masses,\nand enough for star formation and outflows to settle to quasi-static\nstates.\n\nThe initial conditions are generated with the {\\sc MakeDisk} code by\nVolker Springel \\citep[see][]{Springel2005a,Kim2014}, which has been\nadapted to generate {\\sc Ramses}{}-readable format by Romain Teyssier and\nDamien Chapon. The DM halo follows an NFW density profile\n\\citep{Navarro1997} with concentration parameter $c=10$ and spin\nparameter $\\lambda=0.04$ \\citep{Maccio2008}. The dark matter in each\nhalo is modelled by one million collisionless particles, hence the\n{\\sc g9}{} and {\\sc g10}{} galaxies have DM mass resolution of $10^5$ and\n$10^6 \\ \\rm{M}_{\\odot}$, respectively. The initial disc consists of stars and\ngas, both set up with density profiles which are exponential in radius\nand Gaussian in height from the mid-plane (scale radii of $1.5$ kpc\nfor {\\sc g9}{} and $3.2$ kpc for {\\sc g10}{}, and scale heights one tenth of the\nscale radius in both cases). The galaxies contain stellar bulges with\nmasses and scale radii both one tenth that of the disc.  The initial\nstellar particle number is $1.1\\times 10^6$ , a million of which are\nin the disc and the remainder in the bulge. The mass of the initial\nstellar particles is $1.7 \\times 10^3$ and $10^4 \\ \\rm{M}_{\\odot}$ for the\n{\\sc g9}{} and {\\sc g10}{} galaxies, respectively, close to the masses of\nstellar particles formed during the simulation runs, which are shown\nin \\Tab{sims.tbl}. While contributing to the dynamical evolution and\ngravitational potential of the rotating galaxy disc, the initial\nstellar particles do not explode as SNe. This initial lack of feedback\nresults in over-efficient early star formation and a subsequent strong\nfeedback episode which typically then suppresses the star formation to\na semi-equilibrium state within a few tens of Myr (see e.g. star\nformation rate plots in \\Sec{sf.sec}). To overcome this shortcoming,\nfuture improvements should include sensible age assignments to the\ninitial stellar particles, which could be used to perform SN feedback\nright from the start of the simulations.\n\nThe temperature of the gas discs is initialised to a uniform $T=10^4$\nK and the ISM metallicity $Z_{\\rm disc}$ is set to $0.1$ and $1$\n$\\Zsun$ for the {\\sc g9}{} and {\\sc g10}{} galaxies, respectively. The\ncircum-galactic medium (CGM) initially consists of a homogeneous hot\nand diffuse gas, with $n_{\\rm{H}}=10^{-6} \\ \\cci$, $T=10^6 $ K, and zero\nmetallicity. The cutoffs for the gas discs are chosen to minimize the\ndensity contrast between the disc edges and the CGM. The square box\nwidths for the {\\sc g9}{} and {\\sc g10}{} galaxies are $300$ and $600$ kpc,\nrespectively, and we use outflow (i.e. zero gradient) boundary\nconditions on all sides.\n\nThe same initial conditions and similar simulation settings were used\nin \\cite{Rosdahl2015}, where we studied stellar radiation feedback\nin combination with thermal dump SNe. The main differences from the\nsetup of the previous work, apart from not including stellar\nradiation, is that here we include a homogeneous UV background, we\nform stellar particles that are about a factor three more massive, and\nthe previous work included a bug, now fixed\\footnote{Thanks to Sylvia\n  Ploeckinger for finding and fixing the issue.}, in metal cooling,\nwhere the contribution of hydrogen and helium was double-counted at\nSolar metallicity\\footnote{We have checked and verified that the metal\n  cooling bug has a negligible effect on the results of both this and\n  our previous work.}. The most significant of these changes is the\nlarger stellar particle mass, which boosts the efficiency of thermal\ndump SN feedback in suppressing star formation and, to a lesser\nextent, in generating outflows.\n \n\\subsection{Adaptive refinement}\nEach refinement level uses half the cell width of the next coarser\nlevel, starting at the box width at the first level. Our simulations\nstart at level $7$, corresponding to a coarse resolution of\n$2^7=128^3$ cells, and adaptively refine up to a maximum level 14,\ncorresponding to an effective resolution of $16384^3$ cells. This\ncorresponds to an optimal physical resolution of $18$ pc and $36$ pc\nin the less and more massive galaxies, respectively. Refinement is\ndone on mass: a cell is refined if it is below the maximum refinement\nlevel, if its total mass (DM+stars+gas) exceeds $8 \\, m_{*}$\n(see mass values in \\Tab{sims.tbl}), or if its width exceeds a quarter\nof the local Jeans length\n \n\\subsection{Gas thermochemistry}\nThe gas temperature and the non-equilibrium ionization states of\nhydrogen and helium are evolved with the method presented in\n\\cite{{Rosdahl2013}}, which includes collisional\nionization\/excitation, recombination, bremsstrahlung, di-electronic\nrecombination, and Compton electron scattering off cosmic microwave\nbackground photons. We include hydrogen and helium photo-ionization\nand heating of diffuse gas from a redshift zero\n\\cite{Faucher-Giguere2009} UV background, and enforce an exponential\ndamping of the UV radiation above the self-shielding density of\n$n_{\\rm{H}}=10^{-2} \\ \\cci$.  \n\nAbove $10^4$ K, the contribution to cooling from metals is added using\n{\\sc Cloudy}{} \\citep[][version 6.02]{Ferland1998} generated tables,\nassuming photoionization equilibrium with a redshift zero\n\\cite{Haardt1996} UV background. Below $10^4$ K, we use fine\nstructure cooling rates from \\cite{Rosen1995}, allowing the gas to\ncool radiatively to $10$ K. \n\n\\subsection{Star formation}\\label{sf_model.sec}\nWe use a standard star formation (SF) model which follows a Schmidt\nlaw. In each cell where the hydrogen number density is above the star\nformation threshold, $n_{*} = 10 \\ \\cci$, gas is converted into stars at\na rate $\\dot \\rho_{*} = \\epsilon_{*} \\rho \/ t_{\\rm ff}$, where $\\rho$ is the\ngas (mass) density and $\\epsilon_{*}$ is the star formation efficiency per\nfree-fall time, $t_{\\rm ff} = \\left[ 3 \\pi\/(32 G \\rho) \\right]^{1\/2}$,\nwhere $G$ is the gravitational constant.  Stellar populations are\nrepresented by collisionless stellar particles that are created\nstochastically using a Poissonian distribution \\citep[for details\nsee][]{Rasera2006}, which returns the stellar particle mass as an\ninteger multiple of $m_*$ (see \\Tab{sims.tbl}). We use $\\epsilon_{*}=2\\%$ in\nthis work \\citep[e.g.][]{Krumholz2007}. In future work we will\nconsider how varying the details of star formation affects the\nefficiency of SN feedback, but that is beyond the scope of the present\npaper. The stellar particle masses are given in \\Tab{sims.tbl}, and\nare equal to the SF density threshold, $n_{*}$, times the volume of a\nmaximally refined gas cell\\footnote{We do not allow more than $90 \\%$\n  of the cell gas to be removed when forming stars. Thus, stellar\n  particles actually do not form below a density of $1.11 \\ n_{*}$.}.\n\n\\subsection{Artificial Jeans pressure} \nTo prevent numerical fragmentation of gas below the Jeans scale\n\\citep{Truelove1997}, an artificial `Jeans pressure' is maintained\nin each gas cell in addition to the thermal pressure. In terms of an\neffective temperature, the floor can be written as\n$\\TJeans = T_0 \\ n_{\\rm{H}} \/ n_{*}$, where we have set $T_0=500$ K (and $n_{*}$\nis the aforementioned star formation threshold), to ensure that the\nJeans length is resolved by a constant minimum number of cell widths\nat any density -- 7 and 3.5 cell widths in the smaller and larger\ngalaxy simulations, respectively \\citep[see Eq. 3\nin][]{Rosdahl2015}.  The pressure floor is non-thermal, in the\nsense that the gas temperature which is evolved in the thermochemistry\nis the difference between the total temperature and the floor --\ntherefore we can have $T << T_{\\rm J}$.\n\n\\section{SN feedback} \\label{models.sec} Supernova feedback is\nperformed with single and instantaneous injections of the cumulative\nSN energy per stellar population particle. Each stellar particle has\nan energy and mass injection budget of\n\\begin{align}\nE_{\\rm SN} &= 10^{51} \\ {\\rm{erg}} \\ \\ \\eta_{\\rm SN} \\ \\frac{m_{*}}{m_{\\rm SN}}, \n       \\label{Esn.eq} \\\\\nm_{\\rm ej} &=\\eta_{\\rm SN} \\ m_*, \\label{msn.eq}\n\\end{align}\nrespectively, where $\\eta_{\\rm SN}$ is the fraction of stellar mass that is\nrecycled into SN ejecta\\footnote{Note that we will neglect the mass\n  that ends up in stellar remnants of SNe.}, $m_{\\rm SN}$ is the average\nstellar mass of a type II SN progenitor, and, as a reminder, $m_{*}$\nis the mass of the stellar particle.  We assume a\n\\cite{Chabrier2003} initial mass function (IMF) and set\n$\\eta_{\\rm SN}=0.2$ and $m_{\\rm SN}=10 \\ \\rm{M}_{\\odot}$, giving at least\n$40 \\times 10^{51}$ ergs per particle in the {\\sc g9}{} galaxy and\n$320 \\times 10^{51}$ ergs in the {\\sc g10}{} galaxy.  We neglect the metal\nyield associated with stellar populations, i.e. the stellar particles\ninject no metals into the gas, and the metallicity of the gas disc\nstays at roughly the initial value of $0.1$ solar (which is negligibly\ndiluted due to mixing with the pristine CGM). The time delay for the\nSN event is 5 Myr after the birth of the stellar particle.\n\nThe model for SN energy and mass injection, and how it affects the\ngalaxy properties and its environment, is the topic of this paper. We\nexplore five different SN models, which we now describe.\n\n\\subsection{Thermal dump feedback}\nThis is the most simple feedback model, and one which is well known\nto suffer from catastrophic radiative losses at low resolution\n\\citep[e.g.][]{Katz1992}. The (thermal) energy and mass of the\nexploding stellar particle are dumped into the cell hosting it, and\nthe corresponding mass is removed from the particle. Unless the\nSedov-Taylor phase is well resolved in both space and time, the\nthermal energy radiates away before it can adiabatically develop into\na shock wave. The primary aim of each of the SN models that follow is\nto overcome this `overcooling' problem. \n\nNote that in SPH simulations, the energy in thermal dump feedback is\ntypically distributed over $\\sim 10^2$ neighbouring gas particles,\nwhereas in our implementation all the energy is injected into a single\ncell. Consequently, in SPH simulations with similar resolution, the\namount of gas that is heated is typically larger. This can lead to\nlower temperatures and larger radiative losses in SPH, but in the case\nof strong density gradients around the feedback event, it can also\nenhance feedback efficiency if SPH particles with a low density\nreceive part of the SN energy.\n\n\\subsection{Stochastic thermal feedback}\nWhile the other SN models described in this paper existed previously\nin {\\sc Ramses}{} and have been described and studied individually in\nprevious publications, we have for this work adapted to {\\sc Ramses}{} the\nstochastic SN feedback model presented in\n\\citet[][a.k.a. {DS12}{}]{DallaVecchia2012}, which has so far only\nbeen used in SPH. The idea is to heat the gas in the cell hosting the\nstellar particle to a temperature high enough that the cooling time is\nlong compared with the sound crossing time across the cell. The energy\nthen has the chance to do significant work on the gas before being\nradiated away, and overcooling is reduced.\n\nAs argued in {DS12}{}, a single SN energy injection should heat the gas\nenough that the ratio between the cooling time and sound crossing time\nacross a resolution element is $t_{\\rm{c}}\/t_{\\rm{s}} \\ga 10$. Given a local\ngas density $n_{\\rm{H}}$, a physical resolution $\\Delta x$, and assuming cooling\nis dominated by Bremsstrahlung (true for $T\\ga 10^7$ K), an expression\nfrom {DS12}{} (their eq. 15) can be used to derive an approximate\nrequired temperature increase, $\\Delta T_{\\rm stoch}$, to enforce this minimum\nratio and thus avoid catastrophic cooling, resulting in the condition\nthat\n\\begin{align}\\label{DT_min.eq}\n  \\Delta T_{\\rm stoch} \\ga 1.1 \\times 10^{7} \\ {\\rm K} \n  \\ \\left( \\frac{n_{\\rm{H}}}{10 \\ \\cci} \\right)\n  \\ \\left( \\frac{\\Delta x}{100 \\ {\\rm pc}} \\right).\n\\end{align}\n\nSome specified time delay after the birth of the stellar population\nparticle ($5$ Myr in this work), it injects its total available\nenergy, $E_{\\rm SN}$, into the hosting gas cell. Since $E_{\\rm SN}$ may be smaller\nthan what is needed for the required temperature increase $\\Delta T_{\\rm stoch}$,\nthe feedback event is done stochastically, with a probability\n\\begin{align}\\label{pSN.eq}\n\\begin{split}\n  p_{\\rm SN} & = \\frac{E_{\\rm SN}}{\\Delta \\epsilon \\ m_{\\rm cell}} \\\\\n   &= 1.6 \\\n  \\left( \\frac{\\eta_{\\rm SN}}{0.2}  \\right)\n  \\left( \\frac{m_{*}}{2 \\times 10^3 \\ {\\rm \\rm{M}_{\\odot}}}  \\right) \\\\\n  & \\qquad {} \\left( \\frac{\\Delta x}{18 \\ {\\rm pc}}  \\right)^{-3}\n  \\left( \\frac{n_{\\rm{H}}}{10 \\ \\cci}  \\right)^{-1}\n  \\left( \\frac{\\Delta T_{\\rm stoch}}{10^{7.5} \\ {\\rm K}}  \\right)^{-1}\n  ,\n\\end{split}\n\\end{align}\nwhere $m_{\\rm cell}$ is the gas mass of the host cell (including the SN\nejecta) and \n\\begin{align}\n\\Delta \\epsilon=\\frac{k_{\\rm B} \\Delta T_{\\rm stoch}}{(\\gamma-1)m_{\\rm p} \\mu}\n\\end{align}\nis the required specific energy, with $k_{\\rm B}$ the Boltzmann constant,\n$m_{\\rm p}$ the proton mass, and $\\mu$ the mean particle mass in units of\n$m_{\\rm p}$\\footnote{We use $\\mu=0.6$, assuming the gas to be\n  ionized.}. When a stellar particle is due to inject SN energy,\n$p_{\\rm SN}$ is calculated via \\Eq{pSN.eq}. If $p_{\\rm SN}\\ge 1$, the available\nenergy is sufficient to meet the cooling time constraint and it is\nsimply injected into the host cell. On the other hand, if $p_{\\rm SN} <1$, a\nrandom number $r$ between 0 and 1 is drawn: only if $r<p_{\\rm SN}$ is\nthe energy $\\Delta \\epsilon \\ m_{\\rm cell}$ injected into the cell,\notherwise no energy injection takes place. With this approach, the\nenergy averaged over the whole simulation box and sufficiently long\ntime-scales is close to the available SN energy budget, as we have\nconfirmed in our runs -- it is just distributed unevenly in space and\ntime in order to overcome the cooling catastrophe. Note that the mass\n(and metal) yield from the stellar particles is \\emph{not} subject to\nthe stochastic process, but is always injected into the host cell,\nregardless of whether or not the energy injection takes\nplace\\footnote{This results in slight cooling of gas in those cells\n  where stellar particles inject mass but not energy.}.\n\nWe note that our stochastic feedback implementation in AMR differs\nsignificantly from the original SPH implementation from {DS12}{} in two\nregards. First, whereas the probability for a feedback event varies\nwith the local density in AMR, the event probability is a constant\nover a simulation run in SPH, since each candidate for energy\ninjection is a gas particle and all gas particles typically have\nidentical mass. Second, thermal dump explosions in SPH are normally\ninjected into a number of (typically $\\sim 50$) neighbouring gas\nparticles, and the objective of the stochastic feedback model is then\nto reduce the number of neighbours receiving the feedback energy in\neach event (in effect making it more similar to our implementation of\nthermal dump feedback). However, in AMR, the energy is only released\ninto the gas cell hosting the exploding stellar particle, so our\nstochastic feedback model redistributes feedback events such that\nsome stellar particles explode with boosted energy, and some not at\nall.\n\nNaively, our stochastic feedback implementation may be presumed to\nre-distribute SN energy towards lower gas densities, as the\nprobability for each SN event scales inversely with the density via\n$m_{\\rm cell}$. This is not the case: there is no (average) re-distribution\nover densities, since the injected energy scales inversely with the\nprobability, and hence the average energy injected per `candidate'\nfeedback event, at any density, is\n\\begin{align}\\label{pSN2.eq}\n  p_{\\rm SN} \\ \\Delta \\epsilon \\ m_{\\rm cell} = E_{\\rm SN}.\n\\end{align}\n\nWe study the effects of varying the $\\Delta T_{\\rm stoch}$ parameter in\nSec. \\Sec{fb_stoch.sec}. For the comparison of feedback models, we\nuse the fiducial value of $\\Delta T_{\\rm stoch}=10^{7.5}$ K, because: i) it is\nroughly the minimum value given by \\Eq{DT_min.eq} using our star\nformation density threshold, ii) in our comparison of $\\Delta T_{\\rm stoch}$\nvalues in Sec. \\Sec{fb_stoch.sec} we find this gives a similar star\nformation and Kennicutt-Schmidt relation as higher $\\Delta T_{\\rm stoch}$ values\n(though note that higher values of $\\Delta T_{\\rm stoch}$ do produce stronger\noutflows), and iii) it is the fiducial value used in {DS12}{} and in the\nEAGLE simulations \\citep{Schaye2015}, allowing us to qualitatively\ncompare the efficiency of our AMR version of the stochastic feedback\nmodel to that of previous SPH works. Assuming a resolution of\n$\\Delta x=18$ pc and that stellar particles typically explode close to the\nstar-formation density threshold of $n_{*}=10 \\ \\cci$, \\Eq{DT_min.eq}\ngives $\\Delta T_{\\rm stoch} \\ga 10^6$ K, so our chosen fiducial value is well\nabove the estimated requirement from {DS12}{}\\footnote{Indeed, with\n  $\\Delta T_{\\rm stoch} =10^{7.5}$, $n_{\\rm{H}}=10 \\ \\cci$, and $\\Delta x=18$ pc, the sound\n  crossing time is $t_{\\rm{s}}\\approx 2 \\times 10^4$ yr (eq. 9 in {DS12}{})\n  and the cooling time is $t_{\\rm{c}} \\approx 3.3 \\times 10^6$ yr\n  $\\approx 165 \\ t_{\\rm{s}}$ (eq. 13 in {DS12}).}. Using the same values,\n\\Eq{pSN.eq} shows that the probability for feedback events is below\nunity at densities $n_{\\rm{H}} \\ga 16 \\ \\cci$, i.e. slightly above our\nadopted star formation density threshold, for both the {\\sc g9}{} and\n{\\sc g10}{} galaxies we simulate (the lower resolution in {\\sc g10}{} is exactly\ncounter-balanced by the larger stellar particle mass).\n\n\\subsection{Delayed cooling thermal feedback}\nAnother widely used method for overcoming the numerical overcooling\nproblem in galaxy formation simulations is to turn off radiative\ncooling in SN heated gas for a certain amount of time. This method,\nusually referred to as \\emph{delayed cooling}, has been used in SPH\nsimulations \\citep[e.g.][]{Gerritsen1999, Stinson2006,\n  Governato2010}, giving both a strong suppression in star\nformation and enhancement in outflows.\n\n\\cite{Teyssier2013} introduced an AMR version of this method, which\nwe use in this paper. Here, a specific energy tracer, $\\epsilon_{\\rm turb}$, is\nstored on the grid in the form of a passive scalar, and typically\nassociated with an unresolved turbulent energy. It is advected with\nthe gas and decays on a time-scale $t_{\\rm{delay}}$ as\n\\begin{align}\n  \\frac{D\\epsilon_{\\rm turb}}{D t} = - \\frac{\\epsilon_{\\rm turb}}{t_{\\rm{delay}}}.\n\\end{align}\nFor every feedback event, the SN energy of the stellar particle,\n$E_{\\rm SN}$, is injected as thermal energy into the host cell, as in\nthermal dump feedback, but at the same time it is added to the\nnon-thermal energy density $\\rho \\, \\epsilon_{\\rm turb}$ in the same cell. As long\nas the local ``turbulent velocity'' is above a certain limit in a\ngiven cell, $\\sigma_{\\rm turb} = \\sqrt{2 \\epsilon_{\\rm turb}} > \\sigma_{\\rm min}$, radiative\ncooling is disabled in that location, mimicking the non-thermal nature\nof turbulent energy. When the local turbulent velocity has fallen\nbelow $\\sigma_{\\rm min}$, via decay, diffusion, and mixing, radiative\ncooling is enabled again.\n\nThe main free parameter in the model is $t_{\\rm{delay}}$, which determines\nhow quickly the turbulent energy disappears. $\\sigma_{\\rm min}$ is\nalso an adjustable parameter, but it has more or less the same effect\nas $t_{\\rm{delay}}$, so we keep it fixed at $\\sigma_{\\rm min}=100 \\, \\kms$\n(corresponding to about $0.1\\%$ of the injected specific energy of a\nSN, or about $1\/30$th of the velocity in its unloaded remnant). The\nvalue of $t_{\\rm{delay}}$ can be motivated by an underlying physical\nmechanism, e.g. the crossing time over a few cell widths, after which\nthe resolved hydrodynamics should take over the unresolved advection\nof energy. The appendix of \\cite{Dubois2015} derives an expression\nfor the choice of an appropriate $t_{\\rm{delay}}$, given the local SN rate,\ndensity, and resolution (their eq. A8), for which our {\\sc g9}{}\nsimulation settings ($\\epsilon_{*}=0.02$, $\\eta_{\\rm SN}=0.2$, $n_{\\rm{H}}=10 \\ \\cci$,\n$\\Delta x=18$ pc) give $t_{\\rm{delay}} \\approx 1.3$ Myr.  However, in this paper we\nfollow the literature \\citep{Teyssier2013, Roskar2014,\n  Mollitor2015, Rieder2016}, and use a much larger fiducial\nvalue of $t_{\\rm{delay}}=10$ Myr for the {\\sc g9}{} galaxy, and $t_{\\rm{delay}}=20$ Myr in\nlow-resolution versions of {\\sc g9}{} and in the {\\sc g10}{} galaxy. Assuming\ndecay dominates over diffusion and mixing, and assuming the SN\nremnants travel at $\\sim 100$ ($1,000$) km\/s, our fiducial $t_{\\rm{delay}}=10$\nMyr corresponds to a delay length scale of $\\sim 1$ ($10$) kpc.  We\nexplore variations of $t_{\\rm{delay}}$ in \\Sec{fb_dc.sec} (including values\nclose to that derived by \\citealt{Dubois2015}).\n\nThe disadvantage of delayed cooling is that while overcooling is in\npart a numerical problem, radiative cooling is a real and physical\nprocess, without which stars would not form at all. By neglecting\nradiative cooling altogether, even if for a relatively short time,\ndelayed cooling is likely to result in over-efficient type II SN\nfeedback (but, at the same time, it perhaps compensates for the\nneglect of other feedback processes which may be important in galaxy\nevolution). In addition, delayed cooling can result in the gas\npopulating parts of the temperature-density diagram where the cooling\ntime is short, which may yield unrealistic predictions for absorption\nand emission diagnostics.\n\n\\subsection{Kinetic feedback}\nWe use the kinetic feedback model presented in\n\\citet[][]{Dubois2008}.  Here, the trick to overcoming numerical\novercooling is to skip the unresolved Sedov-Taylor phase and directly\ninject the expected collective result of that phase for a stellar\npopulation, which is an expanding momentum-conserving shock wave (or\nsnowplow). Note, however, that the injected kinetic energy may\nsubsequently be converted into thermal energy if shocks develop.\n\nSN mass and momentum is injected into gas within a bubble radius of\nthe exploding stellar particle. The free parameters for the method are\n$f_{\\rm k}$, the fraction of $E_{\\rm SN}$ which is released in kinetic form,\n$r_{\\rm bubble}$, the radius of the bubble, and $\\eta_{\\rm W}$, the sub-resolution mass\nloading factor of the Sedov-Taylor phase, describing how much mass,\nrelative to the stellar mass, is redistributed from the cell at the\nbubble centre to the bubble outskirts.\n\nThe redistributed mass consists of two components: one is the SN\nejecta, $m_{\\rm ej}=\\eta_{\\rm SN} m_{*}$, removed from the stellar particle, the\nother is the swept up mass, $m_{\\rm sw}=\\eta_{\\rm W}m_{*}$, removed from the\ncentral host cell (no more than $25\\%$ of the central cell mass is\nremoved, hence for individual feedback events at relatively low\ndensities it may happen that $m_{\\rm sw}$ is smaller than\n$\\eta_{\\rm W}m_{*}$). The total wind mass is thus $m_{\\rm W}=m_{\\rm ej}+m_{\\rm sw}$, which is\nredistributed uniformly (i.e. uniform density) to all cells inside the\nbubble.\n\nThe kinetic energy, $f_{\\rm k} E_{\\rm SN}$, is likewise distributed to the bubble\ncells, but with an injected velocity (directed radially away from the\nstellar particle) that increases linearly with distance from the\ncentre, such as to approximate the ideal Sedov-Taylor solution:\n\\begin{align}\n  {\\bf v}(\\Delta m_{\\rm cell}) = f_{\\rm N} v_{\\rm W} \n  \\frac{\\bf{r}_{\\rm cell}}{r_{\\rm bubble}},\n\\end{align}\nwhere $\\Delta m_{\\rm cell}$ is the mass added to the cell,\n$r_{\\rm cell}$ is the position of the centre of the cell relative to\nthe stellar particle, $f_{\\rm N}\\sim 1$ is a bubble normalisation\nconstant\\footnote{The normalisation constant is the volume-weighted\n  average distance from the centre, for each volume element in the\n  bubble. In the ideal case of infinitely small cells, the factor is\n  $1.29$.} required to ensure that the total redistributed energy is\nequal to $f_{\\rm k} E_{\\rm SN}$, and\n\\begin{align} \\label{kin_v.eq}\n  v_{\\rm W} = \\sqrt{\\frac{2 f_{\\rm k} E_{\\rm SN}}{m_{\\rm W}}} \n  = 3,162 \\ {\\rm km \/ s} \\ \\sqrt{\\frac{f_{\\rm k}}{1+\\eta_{\\rm W}\/\\eta_{\\rm SN}}}\n\\end{align}\nis the unnormalised wind velocity, where we used \\Eq{Esn.eq} for the\nlatter equality. Note that this is the velocity of the added mass,\ni.e. each cell gains momentum\n\\begin{align}\n  \\Delta {\\bf p} =  {\\bf v}(\\Delta m_{\\rm cell}) \\Delta m_{\\rm cell}\n  \\propto \\sqrt{f_{\\rm k} \\eta_{\\rm SN} (\\eta_{\\rm SN} + \\eta_{\\rm W})},\n\\end{align}\nso if the mass already in the cell is substantial compared to the\nadded mass, the resulting velocity change can be small. The injection\nis performed in the mass-weighted frame of the SN particle (with\n$m_{\\rm ej}$) and host cell (with $m_{\\rm sw}$). The remaining thermal energy,\n$(1-f_{\\rm k}) E_{\\rm SN}$, is then distributed uniformly between the bubble\ncells.\n\nIn this work, we use fiducial parameters $f_{\\rm k}=1$, $\\eta_{\\rm W}=1$, and\n$r_{\\rm bubble}=150$ pc, a size comparable to galactic super-bubbles (note that\nit is also comparable to the initial scale height of the stellar and\ngas disc in our simulations, which is $150$ pc and $320$ pc for the\n{\\sc g9}{} and {\\sc g10}{} galaxies, respectively). These values give a velocity\nfor the gas ejected from the central cell (from Eq. \\ref{kin_v.eq}) of\n$v_{\\rm W} \\approx 1,300$ km\/s. Our choice of $f_{\\rm k}=1$ implies that\nthere have been neither radiative losses nor momentum cancellation\nfrom the set of unresolved individual SNe inside the bubble. We\nexplore the effects of a smaller bubble and higher mass loading in\n\\Sec{fb_kin.sec}.\n\n\\subsection{Mechanical feedback}\nThis model was introduced to the {\\sc Ramses}{} code by \\citet[][see also\n\\citealt{Kimm2015}]{Kimm2014}, and an analogue SPH scheme was earlier\ndescribed independently in \\cite{Hopkins2014}. Here, momentum is\ndeposited into the neighbour cells of a SN hosting cell, with the\nmagnitude adaptively depending on whether the adiabatic phase of the\nSN remnant is captured by this small bubble of cells and the mass\nwithin it, or whether the momentum-conserving (snowplow) phase is\nexpected to have already developed on this scale. In the first case,\nthe momentum is given by energy conservation, while in the latter\ncase, the final momentum, which depends via the cooling efficiency on\nthe density and metallicity, is given by theoretical works\n\\citep{Blondin1998, Thornton1998}.\n\nIn a single SN injection event, SN momentum (and any excess energy) is\ndistributed over the nearest neighbours (sharing at least two\nvertices) of the SN host cell. The number of such cells can vary,\ndepending on the cell refinement, but given the extreme limit where\nall the neighbours are at a finer level (i.e. half the cell width) of\nthe SN host cell, the maximum number of neighbours is $N_{\\rm nbor}=48$. When\na neighbouring cell is at the same level as the host, or one level\ncoarser, it is given an integer weight $\\wc$ corresponding to how many\nof the $N_{\\rm nbor}$ finer level cell units it contains ($4$ if sharing a\nplane with the host, $2$ if sharing a line). The SN host cell has a\nweight of $\\wc=4$, so the total number of cell units receiving direct\nSN energy injection is $N_{\\rm inj}=N_{\\rm nbor}+4$.\n\nThe goal is to inject into each neighbour cell a momentum $\\Delta p$,\ncorresponding to that generated during the energy conserving phase if\nthat is resolved, but to let $\\Delta p$ converge towards that of the\nmomentum-conserving snowplow phase in the limit that the energy\nconserving phase is unresolved. In each SN neighbour cell, this limit\n(energy vs momentum conserving) depends on the local mass-loading,\ni.e. the ratio of the local wind mass, to the SN ejecta given to that\ncell,\n\\begin{align}\n  \\chi \\equiv \\frac{\\Deltam_{\\rm W}}{\\Deltam_{\\rm ej}},\n\\end{align}\nwith $\\Deltam_{\\rm ej}=\\frac{\\wc m_{\\rm ej}}{N_{\\rm inj}}$,\n$\\Deltam_{\\rm W}=m_{\\rm nbor}+\\frac{\\wc m_{\\rm cen}}{N_{\\rm inj}}+\\Deltam_{\\rm ej}$, $m_{\\rm cen}$ the\nmass initially contained in the SN host cell, and\n$m_{\\rm nbor}=\\wc\\rho_{\\rm nbor}\\left( \\Delta x_{\\rm cen}\/2 \\right)^3$ the initial neighbouring\ngas mass, with $\\rho_{\\rm nbor}$ and $\\Delta x_{\\rm cen}$ the gas neighbour cell gas\ndensity and host cell width, respectively.\n\n\nThe momentum injected into each neighbour cell, radially from the\nsource, is\n\\begin{align}\n  \\Delta p = \\frac{\\wc}{N_{\\rm inj}} \\! \\left\\{ \\! \\! \\!\n  \\begin{array}{lr}\n    \\sqrt{2 \\, \\chi \\, m_{\\rm ej} \\, f_{\\rm e} \\, E_{\\rm SN}} & {\\rm if}\\,  \n                         \\chi < \\chi_{\\rm tr} \\label{dp_mech.eq},\\\\\n    3 \\! \\times \\! 10^{5} \\, \\rm{M}_{\\odot} \\, \\frac{\\rm km}{\\rm s} \\, \n    E_{51}^{\\frac{16}{17}} \\, n_0^{-\\frac{2}{17}} \\ Z'^{-0.14} \\! \\! \\! \n                                            & {\\rm otherwise.}\n  \\end{array}\n          \\right.\n\\end{align} \nHere, the upper expression represents the resolved energy conserving\nphase, and comes from assuming the (final) cell gas mass of\n$\\Delta m_{\\rm W}$ receives a kinetic energy $\\frac{\\wc E_{\\rm SN}}{N_{\\rm inj}}$ (we\nignore $f_{\\rm e}$ for the moment).  The lower expression represents the\nasymptotic momentum reached in the snowplow phase, with $E_{51}$ is the\ntotal SN energy (i.e. $E_{\\rm SN}$) in units of $10^{51}$ erg, $n_0$ the\nlocal hydrogen number density in units of $1 \\ \\cci$, and\n$Z'=\\max \\left(Z\/\\Zsun,0.01\\right)$. The Solar metallicity form of the\nexpression was derived from analytic arguments, and confirmed with\nnumerical experiments, in \\cite{Blondin1998}, and the $Z$\ndependency was added in the numerical work of \\cite{Thornton1998}.\n\nThe phase transition ratio, $\\chi_{\\rm tr}$, is found by equating the\nsnowplow expression in \\Eq{dp_mech.eq} with\n$\\sqrt{2 \\, \\chi_{\\rm tr} \\, m_{\\rm ej} \\, f_{\\rm tr} \\, E_{\\rm SN}}$, where $f_{\\rm tr}=2\/3$ is the\nfraction of the SN energy assumed to be in kinetic form at the\ntransition. This gives\n\\begin{align}\n\\begin{split}\n  \\chi_{\\rm tr} =& \\frac{900}{m_{\\rm SN}\/\\rm{M}_{\\odot} \\ f_{\\rm tr}} \\ E_{51}^{-2\/17} \\ n_0^{-4\/17} \n            \\ Z'^{-0.28} \\\\\n  =& \\, 97 \\, \\left( \\frac{m_{\\rm SN}}{10 \\, \\rm{M}_{\\odot}}\\right)^{-\\frac{15}{17}}\n     \\left( \\frac{\\eta_{\\rm SN}}{0.2}\\right)^{-\\frac{2}{17}}\n     \\left( \\frac{m_{*}}{2 \\! \\times \\! 10^3 \\, \\rm{M}_{\\odot}}\\right)\n     ^{-\\frac{2}{17}} \\label{chi_tr.eq}\n  \\\\\n   & \\qquad {} \\left( \\frac{n_{\\rm{H}}}{10\\, \\cci}\\right)^{-\\frac{4}{17}}\n    \\left( \\frac{Z}{0.1 \\, \\Zsun}\\right)^{-0.28},\n\\end{split}\n\\end{align} \nwhere we used \\Eq{Esn.eq}, $E_{51}=m_{\\rm ej}\/m_{\\rm SN}$, and normalised to\ntypical values for the {\\sc g9}{} galaxy in the latter equality. The\nfunction\n\\begin{align}\n  f_{\\rm e}=1-(1-f_{\\rm tr}) \\frac{\\chi-1}{\\chi_{\\rm tr}-1}\n\\end{align}\nensures a smooth transition between the two expressions in\n\\Eq{dp_mech.eq}.\n\n\nIf the momentum injection results in removal of total energy in a\ncell, due to cancellation of velocities, the surplus energy (initial\nminus final) is added to the cell in thermal form. As it has no\npreferred direction, the SN host cell receives only thermal energy. In\n\\cite{Kimm2014} and \\cite{Kimm2015}, due to wrong book-keeping\nof the surplus energy, the thermal energy injection during the\nadiabatic phase was overestimated (by a factor $\\sim 2 - 4$) in\nregions where the swept up mass is large compared to the SN ejecta (by\na similar factor), but the correct momentum and energy was used during\nthe snowplow phase and the adiabatic phase with little mass loading\n($\\chi\\sim 1$). This bug has since been corrected.\n\nIf we assume that all the cells receiving the SN energy have the same\nrefinement level (i.e. the same cell width) and density and that the\ndensity is at least as high as the threshold for star formation, then\nthe initial mass of the neighbour dominates $\\Delta m_{\\rm W}$ and we can\nget a rough estimate for the local mass loading,\n\\begin{align}\n\\begin{split} \\label{chi_mech_approx.eq}\n  \\chi \\approx & \\, \\frac{m_{\\rm nbor}}{\\Delta m_{\\rm ej}}  \\\\\n  =& \\, \\frac{N_{\\rm inj}}{\\wc} \\frac{\\rho \\Delta x^3}{\\eta_{\\rm SN} m_{*}} \\\\\n  =& \\, 60.8 \\ \\ \\left( \\frac{\\wc}{4} \\right)^{-1}\n     \\left( \\frac{n_{\\rm{H}}}{10 \\, \\cci} \\right)\n     \\left( \\frac{\\Delta x}{18 {\\, \\rm pc}} \\right)^3 \\\\\n   & \\qquad {} \\qquad {} \\left( \\frac{\\eta_{\\rm SN}}{0.2} \\right)^{-1} \n    \\left( \\frac{m_{*}}{2 \\! \\times \\! 10^3 \\, \\rm{M}_{\\odot}} \\right)^{-1} \\\\\n  =& \\, 0.63 \\ \\chi_{\\rm tr} \\ \\ \\left( \\frac{\\wc}{4} \\right)^{-1}\n     \\left( \\frac{n_{\\rm{H}}}{10 \\, \\cci} \\right)^{\\frac{21}{17}}\n     \\left( \\frac{\\Delta x}{18 {\\, \\rm pc}} \\right)^3 \\\\\n   &  \\qquad {} \\qquad {}\\left( \\frac{\\eta_{\\rm SN}}{0.2} \\right)^{-\\frac{15}{17}} \n    \\left( \\frac{m_{*}}{2 \\! \\times \\! 10^3 \\, \\rm{M}_{\\odot}} \n         \\right)^{-\\frac{15}{17}} \\\\\n    & \\qquad {} \\qquad {} \n      \\left( \\frac{m_{\\rm SN}}{10 \\, \\rm{M}_{\\odot}}\\right)^{\\frac{15}{17}}\n      \\left( \\frac{Z}{0.1 \\, \\Zsun}\\right)^{0.28}.\n\\end{split}\n\\end{align}\nHere, the last equality, which comes from comparing with\n\\Eq{chi_tr.eq} and normalising to the {\\sc g9}{} simulation parameters,\nshows that mechanical feedback events are marginally resolved, with\nthe momentum injection being done using the upper expression in\n\\Eq{dp_mech.eq} for $n_{\\rm{H}} \\la 1.6 \\, n_{*}$, but switching to the final\nsnowplow momentum, i.e. the lower expression in \\Eq{dp_mech.eq}, for\nhigher gas densities. For the {\\sc g10}{} galaxy, where the resolution is\nlower ($\\Delta x=36$ pc), the stellar mass higher\n($1.6 \\times 10^4 \\, \\rm{M}_{\\odot}$), and the metallicity higher ($\\Zsun$),\nthe SN blasts are slightly worse resolved, with\n$\\chi \\approx 1.53 \\, \\chi_{\\rm tr}$ (at $n_{*}$) for the same\nassumptions. Here the effects of lower resolution and higher\nmetallicity, towards worse-resolved SN blasts, are counter-weighted by\nthe higher stellar particle mass.\n\n\\section{SN feedback model comparison}\\label{comparison.sec}\n\nWe begin by comparing all SN feedback models using the fiducial\nsettings. Later in this paper we will study each feedback model in\nmore detail and show how the results vary with the values of the free\nparameters. We focus on star formation, outflows, and galaxy\nmorphologies. Unless stated otherwise, our analysis will be restricted\nto the lower-mass {\\sc g9}{} galaxy.\n\n\\subsection{Galaxy morphologies}\n\\begin{figure}\n  \\centering\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_n01.pdf}}\n  \\hspace{-0.6mm}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_t01.pdf}} \\\\\n  \\vspace{-3.7mm}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_s08.pdf}}\n  \\hspace{-0.6mm}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_d02.pdf}}  \\\\\n  \\vspace{-3.7mm}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_y04.pdf}}\n  \\hspace{-0.6mm}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/maps_1_250_k01.pdf}}\n  \\caption\n  {\\label{maps_Fid.fig}Maps of gas column densities in the {\\sc g9}{}\n    galaxy at $250$ Myr, for the different SN feedback models. Each\n    panel shows face-on and edge-on views, with the model indicated\n    in the bottom left corner. The top left panel includes the\n    physical length scale and the colour scale for the gas column\n    density.}\n\\end{figure}\n\nIn \\Fig{maps_Fid.fig}, we show the total hydrogen column density\nface-on and edge-on at the end of the $250$ Myr run for each feedback\nmodel. The maps illustrate how the gas morphology is affected by the\nSN feedback models. Without feedback (top left panel), the galaxy\nbecomes clumpy, containing dense star-forming clouds which accrete\ngas, thus creating large `holes'. The gas outside the thin edge-on\ndisc is diffuse and featureless.\n\nCompared to the no feedback case, thermal dump feedback (top right\npanel) significantly changes the gas morphology, reducing the gas\nclumpiness and thickening the disc. In fact, comparing to other panels\nin \\Fig{maps_Fid.fig}, it has here a very similar morphological effect\nas the stochastic and mechanical feedback models (middle left and\nbottom right panels, respectively). We will come back to this\nsimilarity in later sections.\n\nDelayed cooling (middle right panel) and kinetic feedback (bottom\nleft), on the other hand, produce quite different morphologies from\nother models in \\Fig{maps_Fid.fig}. Delayed cooling diffuses the gas\nmore, with less obvious spiral structure, and the disc becomes\nthicker, indicating increasing feedback efficiency. In stark contrast,\nkinetic feedback results in a very thin disc plane, and thin, well\ndefined spiral filaments. Judging qualitatively from these images of\ncolumn density, delayed cooling appears most efficient in terms of\nsmoothing out the gas, thickening the disc, and creating outflows,\nwhile kinetic feedback visually appears weakest, with relatively dense\nand filamentary gas in the disc and low column densities out of the\ndisc plane. However, as we will see in what follows, kinetic feedback\nactually has the strongest and fastest (but relatively diffuse)\noutflows. We note that these distinct features of kinetic feedback are\nsensitive to the radius of momentum and mass injection, i.e. the $r_{\\rm bubble}$\nparameter. As we will argue in \\Sec{SD.sec}, with our fiducial bubble\nsize of $150$ pc, the momentum injection is essentially\nhydrodynamically decoupled from the galactic disc, and as we show in\n\\Sec{fb_kin.sec}, a considerably smaller bubble leads to kinetic\nfeedback behaving similarly to thermal dump, stochastic, and\nmechanical feedback.\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/sfr.pdf}\n  \\caption\n  {\\label{SFR.fig}Star formation rates in the {\\sc g9}{} galaxy for the SN\n    feedback models indicated in the legend using their fiducial\n    parameters. Thermal dump, stochastic, and mechanical feedback\n    produce nearly identical SFRs, while kinetic feedback produces a\n    steadily declining SFR, and delayed cooling is by far the most\n    efficient at suppressing star formation.}\n\\end{figure}\n\\subsection{Star formation} \\label{sf.sec} The feedback efficiencies\ncan be quantified and compared via the star formation rates (SFRs),\nwhich we show for the {\\sc g9}{} galaxy in \\Fig{SFR.fig}. The SFRs are\ncalculated by binning the stellar mass formed over time intervals of\n$1.2$ Myr. They vary by almost two orders of magnitude, depending on\nthe feedback model utilised, and one order of magnitude at the end of\nthe simulation runtime, by which time the rate of evolution has\nsettled down after the initial collapse of the disc (due to radiative\ncooling and lack of initial feedback) and burst of star formation\naround the $20$ Myr mark.\n\nFocusing on the star formation around $250$ Myr, we find that the\nfeedback models separate roughly into the same three groups as in our\nassessment of the morphologies. Thermal dump, stochastic, and\nmechanical feedback all perform almost identically in terms of star\nformation, indicating that thermal dump is not strongly affected by\novercooling (see \\Sec{resolved.sec}). The SFR is suppressed by about a\nfactor of $3-4$ compared to the no feedback case (labelled NoFB in the\nplot). This may seem an inefficient suppression, compared to the\ninferred $1-2\\%$ average efficiency of star formation observed in the\nUniverse, but it should be kept in mind that the star formation model\nalready has a built in sub-resolution efficiency of only $\\epsilon_{*}=2\\%$\n(see \\Sec{sf_model.sec}). We comment further on the choice and effect\nof $\\epsilon_{*}$ in the discussion (\\Sec{sfe.sec}).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/sfr_mw.pdf}\n  \\caption\n  {\\label{SFR_MW.fig}Star formation rates, as in \\Fig{SFR.fig}, but\n    for the more massive and lower-resolution {\\sc g10}{} galaxy (note that\n    the y-axis is scaled up by a factor of ten). Here, we find\n    larger differences between thermal dump, stochastic, and\n    mechanical feedback, while kinetic feedback and delayed cooling\n    remain qualitatively the same as in the less massive galaxy.}\n\\end{figure}\n\nAt $250$ Myr, kinetic feedback has a SFR fairly close to those three\naforementioned models. The difference is that the star formation rate\nhas not stabilised, but is declining steadily. As we will see, this is\ndue to the strong outflow removing gas from the star-forming ISM.\n\nDelayed cooling is by far the most effective at suppressing star\nformation. The feedback from the initial peak in the SFR almost blows\napart the gas disc, but once it has settled again the SFR stabilises\naround $0.1-0.2 \\, \\msunyr$, though it remains somewhat bursty. The\nfinal SFR at $250$ Myr is almost an order of magnitude lower than for\nthe other feedback models.\n\n\\subsubsection{Star formation in the more massive galaxy}\n\nIn \\Fig{SFR_MW.fig} we show the SFRs in the ten times more massive\n(and lower resolution) {\\sc g10}{} galaxy simulations.\n\nDue to the combination of the deeper gravitational potential, stronger\n(metal) cooling, lower resolution, and the SN events happening at\nhigher gas densities (typically by $0.5-1$ dex) we find more\ndifferences between the feedback models in their ability to suppress\nstar formation than for the {\\sc g9}{} galaxy. Thermal dump feedback is\nweak, with the star formation stabilising at the same rate as for the\ncase of no feedback. With stochastic and mechanical feedback the star\nformation is suppressed by about a factor of two compared to thermal\ndump, with mechanical feedback being somewhat stronger.  Kinetic\nfeedback shows the same qualitative behaviour as in the lower-mass\ngalaxy, with an initially high SFR that declines steadily due to gas\noutflows. Again, delayed cooling gives SFRs that are much lower than\nfor the other models.\n\n\\subsubsection{The Kennicutt-Schmidt relation}\n\nIn the local Universe, SFR surface densities, $\\Sigma_{\\rm SFR}$, are observed\non large scales to follow the Universal Kennicutt-Schmidt (KS)\nrelation, $\\Sigma_{\\rm SFR} \\propto \\Sigma_{\\rm gas}^{1.4}$, where $\\Sigma_{\\rm gas}$ is the gas\nsurface density \\citep{Kennicutt1998}. We plot in \\Fig{KS_Fid.fig}\nthe relation between SFR and gas surface densities in our simulations\nat $250$ Myr for the different feedback models, and compare it with\nthe empirical relation shown as a solid line (normalised for a\nChabrier IMF, see \\citealt{DallaVecchia2012}). In this plot we\ninclude results from both the low-mass {\\sc g9}{} and high-mass {\\sc g10}{}\ngalaxies, in order to show a wide range of surface densities, and to\ndemonstrate how the feedback efficiency changes for each model with\ngalaxy mass, metallicity, and physical resolution.  Results for the\n{\\sc g9}{} galaxy are shown with smaller opaque symbols, while the {\\sc g10}{}\ngalaxy is represented by larger and more transparent symbols. The gas\nand SFR surface densities are averaged along annulli around the galaxy\ncentre, with equally spaced azimuthal bins of $\\Delta r=500$ pc, and\nwe only include gas within a height of $2$ kpc from the disc plane\n($4$ kpc in the case of the {\\sc g10}{} disc).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/ks_500pc_fid_py.pdf}\n  \\caption\n  {\\label{KS_Fid.fig}The Kennicutt-Schmidt relation for different\n    feedback models at $250$ Myr. Small opaque symbols indicate the\n    {\\sc g9}{} galaxy, while larger and more transparent symbols are for\n    the {\\sc g10}{} galaxy. The values are averages within equally spaced\n    azimuthal bins of $\\Delta r=500$ pc. The grey solid line shows the\n    empirical \\protect\\cite{Kennicutt1998} law (see text).}\n\\end{figure}\n\nAll feedback models, and even the case of no feedback, produce slopes\nin the KS relation in rough accordance with observations at gas\nsurface densities substantially above the `knee' at\n$\\Sigma_{\\rm gas} \\approx 10 \\, \\msunpc$, though the slopes tend to be slightly\nsteeper than observed. The similarity to the observed slope is in\nlarge part a result of the built-in star formation model,\n$\\dot \\rho_{*} \\propto \\rho^{1.5}$.  However, even though all\nsimulations have the same sub-resolution local star formation\nefficiency of $\\epsilon_{*} = 2\\%$, the $\\Sigma_{\\rm SFR}$ normalisation varies by\nabout an order of magnitude, with delayed cooling being most efficient\nat suppressing the star formation for any given gas surface density,\nowing to the large scale height of the disc. At high gas surface\ndensities ($\\Sigma_{\\rm gas} \\gg 10 \\, \\msunpc$), all methods predict too high\n$\\Sigma_{\\rm SFR}$, except delayed cooling which predicts too low values.\n\nFor the lower-mass {\\sc g9}{} galaxy (smaller solid symbols), thermal\ndump, stochastic, mechanical feedback, and delayed cooling are all\nsimilar in the KS plot, though delayed cooling does not produce as\nhigh gas surface densities as the other models. Kinetic feedback has\nsignificantly higher SFR surface densities for given gas surface\ndensities (but relatively low maximum gas surface densities), owing to\nthe very thin disc produced by the almost decoupled injection of\nmomentum.\n\nFor the more massive {\\sc g10}{} galaxy, which was simulated with lower\nresolution, the picture is quite different (large transparent\nsymbols). With thermal dump feedback, the SFR surface densities shift\nsignificantly upwards and the relation is quite similar to the no\nfeedback case. Stochastic feedback, and to a lesser extent, mechanical\nfeedback, also shift upwards, away from the observed relation. For\nkinetic feedback, the relation is however almost unchanged in the more\nmassive galaxy (except for low gas surface densities, where it is\nhigher), but consistently remains about a factor two above the\nobserved relation. With delayed cooling, the gas surface densities\nbecome much higher than in the lower mass galaxy, but the SFR surface\ndensities are significantly lower than observed.\n\nFor delayed cooling, we can calibrate the available free parameter to\nimprove the comparison to observations. Halving the delayed cooling\ntime-scale in the {\\sc g10}{} galaxy, to the same value as used for the\n{\\sc g9}{} galaxy, results in a KS relation which is very close to the\nobserved one. For the other models, we cannot calibrate the feedback\nparameters to close in on the observed relation, and other measures\nare required, such as increasing the feedback energy per unit stellar\nmass. Another option is to reduce the star formation efficiency\nparameter, $\\epsilon_{*}$, in which case a fair match to observations can be\nproduced, but at the cost of making the feedback insignificant\ncompared to the no feedback case in terms of morphology, total SFR,\nand outflows (the feedback essentially all becomes captured inside\n$\\epsilon_{*}$).\n\n\\subsection{Outflows} \\label{outflows.sec} Galactic outflows are a\nvital factor in delaying the conversion of gas into stars. Feedback\nprocesses in the ISM are thought to eject large quantities of gas from\nthe galaxy, some of the gas escaping the gravitational pull of the\ngalactic halo altogether. Most of the gas, however, is expected to be\nejected at velocities below the halo escape velocity and to be\nrecycled into the disc. Galactic outflows are routinely detected in\nobservations \\citep[e.g.][]{Steidel2010, Heckman2015}, and while\nthe outflow speed of cold material can be fairly accurately\ndetermined, other properties of the outflows are not well constrained,\nincluding the mass outflow rate, the fraction of gas escaping the\nhalo, the density, and thermal state of the gas.\n\nOutflows are often characterised in terms of the mass loading factor,\nwhich is the ratio of the outflow rate and the rate of star formation\nin the galaxy. Its definition is somewhat ambiguous, as it depends on\nthe geometry and distance from the galaxy at which the outflows are\nmeasured, which is hard to determine in observations.  Observational\nworks have inferred outflow mass loading factors well exceeding unity\n\\citep[see e.g.][]{Bland-Hawthorn2007a, Schroetter2015}, and many\ntheoretical models require mass loading factors of $1-10$ in sub-L$_*$\ngalaxies to reproduce observable quantities in the Universe\n\\citep[e.g.][]{Puchwein2013,Vogelsberger2013,Barai2015,Mitra2015}.\n\nIt is therefore important to consider outflow properties when\nevaluating SN feedback models. Models that produce weak or no\noutflows, with mass loading factors well below unity, could be at odds\nwith current mainstream theories of galaxy evolution (although it is\nnot known whether SN feedback is directly responsible for outflows --\ne.g. cosmic rays could play a major role; \\citealt{Booth2013,\n  Hanasz2013, Salem2014, Girichidis2016}).\n\nIn \\Fig{OFtime.fig} we compare the time-evolution of outflows from the\n{\\sc g9}{} galaxy with the different SN feedback schemes\\footnote{We show\n  outflow plots for the {\\sc g9}{} galaxy only, but we comment on outflows\n  in the more massive {\\sc g10}{} galaxy (which have similar properties) at\n  the end of this subsection.}. We measure the gross gas outflow\n(i.e. ignoring inflow) across planes parallel to the galaxy disc, at a\ndistance of $2$ kpc in the left panels and further out at $20$ kpc in\nthe right panels. The top row of panels shows the mass outflow rate\nacross those planes ($\\dot M_{\\rm out}$), the middle row shows the mass loading\nfactor ($\\betaOut$), and the bottom row shows the mass-weighted\naverage of the outflow velocity perpendicular to the outflow plane\n($\\vzout$).\n\nIn terms of outflows $2$ kpc above the disc plane (left panels of\n\\Fig{OFtime.fig}), kinetic feedback is strongest, with\n$\\dot M_{\\rm out}\\approx 1 \\, \\msunyr$ and $\\betaOut$ slightly above\nunity. Delayed cooling produces a substantially lower outflow rate,\nbut since the SFR is also much lower, the mass loading factor is\nhigher. The other feedback models give much lower outflow rates, and\nhave mass loading factors $\\sim 10^{-2}-10^{-1}$. At a larger distance\nfrom the disc of $20$ kpc (right panels of \\Fig{OFtime.fig}), the\nsituation is quite similar. All models except for kinetic feedback\nhave declining outflow rates and mass loading factors, owing to the\nstrong initial starburst, which can be seen to result in an outflow\nrate peaking around $50$ Myr.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/oftime.pdf}\n  \\caption\n  {\\label{OFtime.fig}Gross mass outflow rates ($\\dot M_{\\rm out}$), mass loading\n    factors ($\\betaOut$), and mass-weighted average outflow velocities\n    ($\\vzout$), across planes $2$ and $20$ kpc above the disc plane of\n    the {\\sc g9}{} galaxy (left and right columns, respectively), for the\n    SN feedback models and their fiducial parameters. The colour\n    coding and linestyles are the same as in \\Fig{SFR.fig}. The thin\n    horizontal lines in the bottom panels indicate the escape\n    velocity.}\n\\end{figure}\n\nIn the bottom panels of \\Fig{OFtime.fig} we compare the average\noutflow velocities to the DM halo escape velocity\\footnote{The escape\n  velocity estimate ignores the contribution of baryons. Hence, it is\n  an underestimate, that is likely non-negligible close to the disc,\n  but insignificant at $20$ kpc.},\n\\begin{align}\n  \\vesc(h) \\approx 1.16 \\ \\vcirc \\\n  \\sqrt{\\frac{\\ln \\left(1+c x \\right)}{x}},\n\\end{align}\nwhere $x=h\/R_{\\rm vir}$ \\citep[e.g.][]{Mo2004}, which has been marked\nwith horizontal grey solid lines.  Close to the disc, the average\nvelocity for kinetic feedback is marginally higher than escape, but\nslowly declining due to the declining SFR. For the other feedback\nmodels, the outflow velocity is well below escape. Ten times further\nout, the mean outflow velocities are considerably higher for all\nfeedback models. This is to some degree due to the initial starburst,\nwhich ejects high-velocity outflows early in the simulation runs, and\nto some degree due to gas at lower velocities not having reached $20$\nkpc and thus not contributing to the velocity average. In any case,\nthese (average) velocities are still below the escape velocity, again\nwith the exception of kinetic feedback. This implies that for all\nmodels except kinetic feedback, most of the outflowing gas will not\nescape to infinity, but instead fall back on the galaxy where it will\neventually produce stars. Kinetic feedback does give the gas high\nenough velocity so that in principle it can escape the halo entirely,\nwhile in practice this may be complicated by CGM and IGM gas which\nstands in the way and needs to be swept out.\n\nThe main message of \\Fig{OFtime.fig}, however, is not the escape\nvelocity, but the low mass loading factors for thermal dump,\nstochastic, and mechanical feedback, far below the order unity\ninferred from observations of local galaxies. As before, we see a\nstrong similarity between the results produced by thermal dump,\nstochastic, and mechanical feedback.\n\n\\begin{figure}\n  \\includegraphics[width=0.49\\textwidth]\n    {figures\/sqo_fid_one.pdf}\n  \\caption\n  {\\label{SQO_Fid.fig}Local outflow properties at $250$ Myr across\n    planes $2$ kpc from the {\\sc g9}{} disc as a function gas surface\n    density, sampled from a $10$ kpc wide square grid of $100$ by\n    $100$ squares in the xy-plane of the galaxy disc.  All curves are\n    binned by gas surface density, with shaded regions showing\n    standard deviations within each bin.  {\\bf Top panel:} gross\n    outflow rates per unit area. {\\bf Middle panel:} mass loading\n    factors, i.e. average mass outflow fluxes divided by star\n    formation surface densities.  {\\bf Bottom panel:} mass-weighted\n    average gross outflow velocities (with the escape velocity shown\n    by a horizontal solid line).}\n\\end{figure}\n\nIn \\Fig{SQO_Fid.fig}, we study how the outflow properties $2$ kpc from\nthe disc scale with the local gas surface density. The panels show,\nfrom top to bottom, gross outflow rate per unit area ($\\Sigma_{\\rm out}$), mass\nloading factor $\\betaOut \\equiv \\Sigma_{\\rm out}\/\\Sigma_{\\rm SFR}$, and mass-weighted\ngross outflow velocity.  We split the face of the disc into a $10$ kpc\nwide grid of $100$ pc squares (that is, $100^2$ squares), and extract\nthe outflow properties in each square. In the plots, the outflow\nproperties are binned by the gas surface density, and the shaded\nregions show the logarithmic standard deviation in each bin.\n\nFor each model, the general trend is that higher gas surface\ndensities correspond to higher outflow rates and velocities, but lower\nmass loading factors.  The outflow velocities are noticeably higher\nthan those (at $250$ Myr) in \\Fig{OFtime.fig}. The reason is that\n\\Fig{OFtime.fig} shows the mass-weighted average of all outflowing\ngas, while \\Fig{SQO_Fid.fig} is restricted to a $10$ kpc wide square\nplane directly above and below the disc, hence capturing the more\ncollimated part of the outflows.  Kinetic feedback clearly stands out\nas having the highest outflow velocities, peaking close to $10^3$ km\/s\n(at the peak surface densities), with little scatter. With kinetic\nfeedback, almost all of the outflowing gas directly above or below the\ndisc is moving faster than the escape velocity (indicated with a\nhorizontal solid line). The other feedback models produce outflow\nrates and velocities that are alike (within roughly a factor of two),\nand much lower than for kinetic feedback, with the exception of\ndelayed cooling, which has a massive, slow outflow, and the highest\nmass loading factor.\n\n\\begin{figure*}\n  \\centering\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_n01.pdf}}\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_t01.pdf}}\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_s08.pdf}} \\\\\n  \\vspace{-4mm}\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_d02.pdf}}\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_y04.pdf}}\n  \\subfloat\n  {\\includegraphics[width=0.3\\textwidth]\n    {figures\/mapsof_2_250_k01.pdf}}\n  \\caption\n  {\\label{mapsOF_fid.fig}Slices along the $xz$-axes (at $y=0$; the\n    disc is seen edge-on), for the {\\sc g9}{} galaxy at $250$ Myr. Each\n    panel of two images shows the hydrogen number density (left) and\n    the gas temperature (right) for a given model. The models are,\n    clockwise from the top left (as indicated in the lower left corner\n    of each density map): no feedback, thermal dump, stochastic,\n    mechanical, kinetic, delayed cooling. In each map, dotted\n    horizontal lines mark planes at which we measure the outflow\n    properties shown in Figures \\ref{OFtime.fig} and\n    \\ref{SQO_Fid.fig}, i.e. at $2$ and $20$ kpc from the plane of the\n    disc. Thermal dump, stochastic, and mechanical feedback produce\n    qualitatively similar multi-phase outflows. Delayed cooling\n    produces outflows that are dense, cold, and slow, whereas those\n    produced by kinetic feedback are diffuse, hot, and fast.}\n\\end{figure*}\n\nIn \\Fig{mapsOF_fid.fig} we show images of slices along the $xz$-plane\n(at $y=0$) at $250$ Myr, with each set of two panels showing the\nhydrogen density (left) and temperature (right) for a given feedback\nmodel. Delayed cooling and kinetic feedback clearly stand out here.\nThe former yields dense and cold ($T\\la10^4$ K) outflows. The outflows\nfor the latter are diffuse, hot\n($10^6 \\ {\\rm K} \\, \\la T \\la 10^8 \\ {\\rm K}$), and the most extended\n(which is expected since \\Fig{SQO_Fid.fig} shows they are by far the\nfastest). The remaining three feedback models produce qualitatively\nsimilar multiphase ($10^4 \\ {\\rm K} \\, \\la T \\la 10^6$ K)\noutflows. The clear distinction between the most effective feedback\nmodel, i.e. delayed cooling, and the other, less effective models,\nin the outflow properties, could be used in future work as an\nobservational probe into how accurately those models represent actual\nfeedback in galaxies.\n\n\\subsubsection{Outflows from the more massive galaxy}\nFor the more massive {\\sc g10}{} galaxy, the outflow differences, which we\ndo not plot, are qualitatively similar to those in the above\nanalysis. Kinetic feedback gives the highest mass loading factor,\nwhich is again of order unity both at $2$ and $20$ kpc. All the other\nmodels give similar mass loading factors $2$ kpc above the disc as\nfor {\\sc g9}{}, but in contrast to {\\sc g9}{} the mass loading drops by $1-2$\norders of magnitude at $20$ kpc (the biggest drop occurring for\ndelayed cooling), due to the stronger gravitational pull. The outflow\nvelocities are slightly higher for all models, but they are still\nmuch (marginally for kinetic feedback) lower than the ($\\approx 500$\nkm\/s) escape velocity.\n\n\\subsection{Gas properties}\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]\n    {figures\/ph_fid.pdf}\n  \\caption\n  {\\label{PH_Fid.fig}Phase diagrams at $250$ Myr for {\\sc g9}{} runs with\n    various feedback models and fiducial settings. The shaded grey\n    region marks where the temperature is below the Jeans temperature\n    which is added artificially for pressure support. Dashed\n    red horizontal and vertical red lines enclose gas that is\n    star-forming. The solid vertical red line in each plot marks the\n    (mass-weighted) mean density, and the red solid curve shows the\n    mean temperature as a function of density. The diagrams are almost\n    identical for thermal dump, stochastic, and mechanical\n    feedback. For delayed cooling, we find a lot of dense gas at\n    `forbidden' temperatures ($\\sim 10^5$ K), where the cooling rate\n    peaks.}\n\\end{figure*}\nIn \\Fig{PH_Fid.fig} we compare gas properties for runs with the\ndifferent SN feedback models, using phase diagrams of gas temperature\nversus density at $250$ Myr. The colour scheme represents the mass of\ngas in each temperature-density bin. The mass-weighted mean density in\neach diagram is represented by a solid vertical line, while the\nmass-weighted mean temperature in each density bin is shown by solid\nred curves. Star-forming gas is enclosed by a dotted box, while gas\nwith temperatures below the artificial Jeans temperature (which has\nbeen subtracted from the `thermal' temperature plotted here) is\nindicated by the shaded diagonal region in the bottom right corner of\neach diagram.\n\nWe continue to see the same qualitative picture as before: delayed\ncooling and kinetic feedback stand out, while the remaining three\nmodels look similar. Delayed cooling yields by far the lowest mean\ndensity, $\\left< n_{\\rm{H}} \\right> \\approx 3 \\, \\cci$, which is well below\nthe star formation density threshold of $n_{*}=10 \\ \\cci$, and almost\ntwo orders of magnitude below the mean density without feedback. The\nother models all yield mean densities near $n_{*}$.\n\nDelayed cooling produces the highest mean temperatures at intermediate\ndensities of $n_{\\rm{H}} = 10^{-2}-10 \\ \\cci$ with a lot of gas at\ntemperatures $10^{4.5}-10^6$ K, which is probably unphysical given the\nshort radiative cooling times in this regime. Curiously, in more\ndiffuse gas, $n_{\\rm{H}} = 10^{-5}-10^{-3} \\, \\cci$ delayed cooling produces\nby far the lowest temperatures, $T\\sim 10^3-10^4$ K, while in the same\ndensity regime the gas is typically at $\\sim 10^5$ K with other\nfeedback models (even in the absence of feedback). From identical\nphase diagrams excluding the galaxy disc, we have confirmed that this\ndiffuse gas is primarily `CGM' gas outside the disc. In the case of\ndelayed cooling, these outflows, i.e. gas outside the disc, span a\nwide range of densities,\n$n_{\\rm{H}} \\sim 10^{-6} - 3 \\times 10^{-1} \\, \\cci$, in stark contrast to\nthe other feedback models, where the CGM gas reaches maximum\ndensities of a few times $10^{-3} \\, \\cci$.  With kinetic feedback,\nthe CGM has a clear bi-modality not seen for other models, with some\nof the gas following an adiabat starting around\n$n_{\\rm{H}}\\sim 10^{-2.5} \\, \\cci$, $T\\sim 10^{7.5}$ K, and extending towards\nmuch lower densities, and the remainder at $T \\ga 10^4$ K and spanning\ndensities of $n_{\\rm{H}} \\sim 10^{-5} - 3 \\times 10^{-3} \\, \\cci$ (the latter\nis flowing 'diagonally' from the disc, i.e. at a steep angle from the\naxis of disc rotation).\n\nAll feedback models produce hot and relatively diffuse gas,\npopulating the region $T\\sim10^4-10^8$ K,\n$n_{\\rm{H}}\\sim 3 \\times 10^{-4} - 10 \\, \\cci$ in \\Fig{PH_Fid.fig}. One might\nexpect this to belong to the outflowing CGM. However, if we exclude\nthe disc (out to $2$ kpc in height and $10$ kpc in radius), all of\nthis hot diffuse gas disappears from the phase diagrams, indicating\nthat it in fact belongs to the ISM. In the case of thermal dump,\nstochastic, and mechanical feedback, the outflowing CGM is indeed warm\nto hot, but dilute, with $n_{\\rm{H}}\\sim 10^{-6}-10^{-4} \\, \\cci$, while\nkinetic feedback produces the aforementioned bi-modality in the\noutflowing CGM, and delayed cooling produces circum-galactic outflows\nthat are predominantly at temperatures between $10^4$ and $10^5$ K.\n\n\\subsection{Impact of feedback on the local\n  environment} \\label{SD.sec} A major factor in any feedback model is\nhow efficiently it clears away those dense clouds where stars can\nform. When dense regions are quickly cleared by early SN explosions in\na stellar cluster, this can also boost the efficiency of subsequent SN\nexplosions which then take place at lower densities where cooling is\nless efficient and the momentum obtained in the SN remnant increases\n\\citep[Eq. \\ref{dp_mech.eq}; see also][]{Kim2014, Martizzi2015}.\nSNe exploding in the diffuse ISM have been suggested to prevent the\nformation of star-forming clouds \\citep[e.g.][]{Iffrig2015}, to\nmaintain the hot volume-filling ISM, and to generate fast outflows\n\\citep[e.g.][]{Ceverino2009}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/sd_cum.pdf}\n  \\caption\n  {\\label{SD_cum.fig}Local densities at which stellar particles are\n    formed over the run-time of the {\\sc g9}{} galaxy (dashed curves) and\n    at which subsequent SN events take place 5 Myr later (solid\n    curves), colour coded by feedback model as indicated in the upper\n    legend. For all of the feedback models studied, star formation\n    takes place at densities close to the density threshold for star\n    formation, $n_{*}=10 \\ \\cci$, which is significantly lower than the\n    densities at which stars form without feedback. Except for\n    kinetic feedback, most SN events happen at much lower densities,\n    indicating that the local environment is significantly altered by\n    feedback.}\n\\end{figure}\n\nIn \\Fig{SD_cum.fig} we show the gas densities at which stellar\nparticles are born (dashed curves) and the densities at which the SN\nevents take place (solid curves) for each of the feedback models in\nthe {\\sc g9}{} galaxy. Focusing first on the star formation densities,\nthey are almost indistinguishable for all the models, and differ only\nfor the case of no feedback, for which the stars typically form at\nsignificantly higher densities. This shows, as we have already seen\nfrom the morphological comparison in \\Fig{maps_Fid.fig}, that all the\nfeedback models are efficient at preventing and\/or destroying\nstar-forming clouds in this {\\sc g9}{} galaxy, and almost all the stellar\nparticles are formed within one dex of the star formation density\nthreshold of $n_{*}=10 \\ \\cci$. For the no feedback case, the clouds can\ncollapse to higher densities, impeded only by the density-dependent\npressure floor.\n\nFor the densities at which the SN events take place, there are larger\ndifferences between the feedback models. Thermal dump, stochastic,\nand mechanical feedback are similar, with a non-negligible\n$\\approx 10$ per cent of the SN energy injected below $0.1 \\ \\cci$\n($20-40$ per cent below $1 \\ \\cci$), and SN events consistently taking\nplace at lower densities than star formation.\n\nDelayed cooling stands out, in having more SNe than those\naforementioned models at densities $\\ga 10^{-1} \\ \\cci$, but fewer\nSNe for lower densities. This comes from the efficiency of delayed\ncooling in diffusing and thickening the ISM disc, resulting in a\nmass-weighted density distribution of the ISM (not shown) which peaks\nat $n_{\\rm{H}} \\approx 1 \\ \\cci$, about a dex lower than for thermal dump,\nstochastic, and mechanical feedback, but a volume-weighted\ndistribution (also not shown) which peaks at\n$n_{\\rm{H}} \\approx 10^{-2} \\ \\cci$, about a dex \\emph{higher} than for those\nother models. Delayed cooling hence smooths out not just the density\npeaks in the ISM, but also the density throughs, such that stars form\nat lower average densities, but SNe explode at higher minimum\ndensities than for the other models.\n\nStanding out much more distinctly, kinetic feedback SNe explode in gas\nwith almost exactly the same densities (and even higher) as the stars\nare formed, with almost no SNe exploding at lower densities. This\nsignature of kinetic feedback suggests a decoupling between the\ninjected momentum and the immediate environment surrounding the\nexploding stellar particle. Instead of quickly dispersing the sites of\nstar formation, the gas is gradually transported away from those\nsites, out of the ISM, without coupling to the immediately surrounding\ngas. This explains the thin bubble-free gas disc (\\Fig{maps_Fid.fig})\nand the gradual decline of the SFR (\\Fig{SFR.fig}), which is due to\nslow gas depletion. These distinct features are however strongly\ndependent on the size of the bubble, $r_{\\rm bubble}$, into which the SN momentum\nand mass is injected. The fiducial size $r_{\\rm bubble}=150$ pc results in this\ndecoupling between the SNe and the surrounding gas. In\n\\Sec{fb_kin.sec} we experiment with a smaller bubble size ($r_{\\rm bubble}=40$\npc) and find a very different behaviour for kinetic feedback, with\nresults resembling those for thermal dump, stochastic, and mechanical\nSNe, i.e. much lower outflow rates, a flatter star formation rate with\ntime, and a thicker disc.\n\n\\subsection{Similarity of three models in the low-mass\n  galaxy} \\label{resolved.sec}\n\nFor the low-mass {\\sc g9}{} galaxy (but not for the more massive {\\sc g10}{}\ngalaxy), we have seen that the results for thermal dump, stochastic,\nand mechanical feedback are near identical in terms of morphologies,\nstar formation, and gas properties.\n\nIn \\Eq{pSN.eq} we showed that the probability for stochastic feedback\nevents is above unity at the density threshold for star formation\n($n_{*}=10 \\, \\cci$) in the {\\sc g9}{} galaxy, and in\n\\Eq{chi_mech_approx.eq} we saw that, also at $n_{*}$, mechanical\nfeedback remains in the resolved adiabatic phase. In addition, we\nfound in \\Sec{SD.sec} that SN events do typically take place at\ndensities close to $n_{*}$. Hence there appears to be no significant\nnumerical overcooling issue in the {\\sc g9}{} runs, and it is then no\nsurprise to find similar results for thermal dump, stochastic, and\nmechanical feedback. It can be argued that the adiabatic phase of\nthermal dump feedback is resolved. Note that this may imply that\ndelayed cooling and kinetic feedback, for the fiducial parameters we\nhave chosen, converge to wrong results for the effects of SN feedback.\n\nFor the more massive {\\sc g10}{} galaxy, this is not the case: these\naforementioned feedback models give quite different results\n(\\Fig{SFR_MW.fig}), and thermal dump does not do much to suppress star\nformation compared to the no feedback case. Purely from Equations\n\\ref{pSN.eq} (stochastic probability) and \\ref{chi_mech_approx.eq}\n(mechanical feedback phase), one might expect the adiabatic phase to\nbe resolved here as well, since the changes in stellar mass and\nresolution, compared to the {\\sc g9}{} galaxy, cancel out. However, in\npart due to stronger metal cooling, but more importantly due to the\nlarger gravitational potential of the {\\sc g10}{} galaxy, stars form and\nexplode at significantly higher densities than in the {\\sc g9}{}\ngalaxy. Hence the probability for stochastic feedback events becomes\nlower than unity (on average $0.35$ in the stochastic feedback run),\nmost mechanical feedback events become pure snowplow momentum\ninjections, and thermal dump becomes a victim of numerical\novercooling.\n\n\\cite{Kim2014} derived a density limit at which the momentum\ncreated by single thermal dump type II SN explosions is\nresolution-converged with grid-hydrodynamics. They found that\nconvergence is maintained with a cell width\n$\\Delta x \\la 10 \\, {\\rm pc} \\ n_0^{-0.46}$, where\n$n_0=\\frac{n_{\\rm{H}}}{1 \\, \\cci}$. Taking $n_{*}=10 \\, \\cci$ we would need a\nresolution of $\\approx 3.5$ pc for converged thermal dump feedback.\nThat is a considerably higher resolution than ours ($18$ pc), so\nnaively we would expect overcooling to be significant for the {\\sc g9}{}\nthermal dump simulation.  In light of the above finding, that thermal\ndump feedback appears more or less resolved in the {\\sc g9}{} galaxy, the\nlack of resolution according to \\cite{Kim2014} is mitigated by the\nfact that each stellar particle in our {\\sc g9}{} simulations releases the\nequivalent of 40 type II SN explosions instantaneously, instead of\none, as assumed in \\cite{Kim2014}.\n\n\n\\subsection{SN model comparison summary}\n\n\\begin{itemize}\n\\item For the low-mass {\\sc g9}{} galaxy, the results for thermal dump,\n  stochastic, and mechanical feedback are near identical in terms of\n  morphologies, star formation, and gas properties. This is an\n  indication that the adiabatic phase of SN explosions is\n  resolved. For the more massive {\\sc g10}{} galaxy, thermal dump is\n  significantly weaker than stochastic and mechanical feedback in\n  suppressing star formation (though not so much in generating\n  outflows).\n\\item Delayed cooling is by far most efficient at suppressing star\n  formation and yields results closest to the observed\n  Kennicutt-Schmidt relation (at least for our assumed star formation\n  efficiency.\n\\item Thanks to a large fiducial `bubble radius' of $150$ pc for\n  momentum and mass injection, kinetic feedback has the highest\n  outflow rates and a mass loading factor of order unity. Delayed\n  cooling follows, with weaker outflow rates but a slightly higher\n  (but declining) mass loading factor. The other feedback models\n  produce much lower outflow rates and mass loading factors than those\n  two more efficient models. In the more massive ($\\sim$ MW) {\\sc g10}{}\n  galaxy, the mass loading factor is similar to that of {\\sc g9}{} close\n  to the galaxy plane, but drops by $1-2$ orders of magnitude ten\n  times further out at $20$ kpc, for all models except kinetic\n  feedback, which maintains a mass loading factor of unity.\n\\item The feedback models producing the lowest outflow rates and mass\n  loading factors produce hot and dilute outflows, while delayed\n  cooling yields distinctively cold and dense outflows. For kinetic\n  feedback the outflows have a clear bi-modal phase structure, with\n  relatively cold and dense outflows close to the disc, and hot and\n  diffuse outflows further out following a temperature-density\n  relation suggesting adiabatic cooling.\n\\item Given the large fiducial bubble radius, which effectively\n  decouples feedback from the ISM, kinetic feedback produces by far\n  the fastest outflow, some of it above the escape velocity. All other\n  models produce outflow velocities about an order of magnitude\n  lower, well below the escape velocity.\n\\end{itemize}\n\n\\section{Resolution convergence} \\label{res_conv.sec} Resolution\nconvergence is an important factor in assessing SN feedback\nmodels. Ideally, the effects of feedback should remain constant if the\nresolution is increased, or at least if it is varied within reasonable\nlimits, i.e. within a factor of a few\\footnote{Convergence with a\n  dramatic change in resolution, on the other hand, is usually not a\n  desired goal. With much lower resolution, a sub-grid model becomes\n  meaningless as the structures with which the model is supposed to\n  interact become completely unresolved and a `lower level' of\n  sub-grid physics must take over the current ones. With much higher\n  resolution, some of the real physics become resolved, and the\n  sub-grid model becomes irrelevant (though ideally it should then\n  converge towards a `first-principles' methodology).}. In practice\nsuch constancy, while desirable, is hard to obtain without making\nsignificant sacrifices, such as disabling physical processes like\nhydrodynamical interactions or radiative cooling in the ISM. A second\nbest choice is a small and predictable change with varying resolution,\nso the feedback parameters can be easily calibrated for different\nsetups in order to achieve ``weak convergence'' \\citep{Schaye2015}.\n\nIn this section we aim to understand how and to what extent measurable\ngalaxy properties change with resolution for the different feedback\nmodels. For this purpose, we use a lower-resolution version of the\n{\\sc g9}{} galaxy, which we call {\\sc g9lr}{}. The setup is identical to\n{\\sc g9}{}, except that the minimum cell width is two times larger,\ni.e. $36$ pc, and the particle mass (in the initial conditions as well\nas for new stellar particles) is eight times higher (i.e.\n$1.6 \\times 10^4 \\ \\rm{M}_{\\odot}$ for stellar particles,\n$\\approx 8 \\times 10^5 \\ \\rm{M}_{\\odot}$ for DM particles). For simplicity, and\nbecause the Jeans length is already resolved by $7$ cell widths in the\nhigher-resolution runs (and hence by $3.5$ cell widths in the\nlower-resolution ones), we leave the pressure floor and star formation\nthreshold unchanged.\n\nIn the left column of panels in \\Fig{SFR_res.fig}, we plot, for each\nfeedback model, the ratios of SFRs (upper panels) and mass loading\nfactors $2$ kpc from the disc (lower panels) for {\\sc g9lr}{} over {\\sc g9}{}\nruns.\n\nFor delayed cooling, the resolution has a significant effect on the\nrelative SFR, although it should be kept in mind that the SFR for\ndelayed cooling is quite small in the first place (and hence the\nabsolute change is low compared to the SFRs of other models). For\nother models, the SFRs change insignificantly with resolution, though\nthere is a systematic tendency of slightly lower SFR with lower\nresolution.\n\nThe lower resolution has a larger effect on the outflow rates, shown\nin the bottom left panel of \\Fig{SFR_res.fig}. Decreasing the\nresolution systematically increases the outflow rates for mechanical\nfeedback (by up to a factor four), thermal dump, stochastic feedback,\nand delayed cooling, (by roughly a factor two), while for kinetic\nfeedback the outflow rate is almost unchanged. The outflow rate\nincrease is likely connected to the winds being launched on larger\nscales, due to the larger cell width and mass (note that the specific\nenergy, i.e. the ratio between the SN energy and receiving gas mass is\nsimilar to the higher resolution run, since the particle mass is $8$\ntimes larger).  The obvious exception is kinetic feedback, where the\nenergy is distributed within a bubble radius which we have kept\nconstant, and indeed the outflow rate remains unchanged.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/sfr_of_ratios_res_x2.pdf}\n  \\caption\n  {\\label{SFR_res.fig}Resolution convergence tests. The upper (lower)\n    panels show, for the different feedback models, ratios of SFRs\n    (mass loading factors at 2 kpc from the disc) of {\\sc g9}{} runs at\n    low resolution and at the fiducial resolution. The left (right)\n    panels shows ratios where the stellar particle mass is increased\n    by a factor of eight (kept fixed) in the low-resolution runs. For\n    all plots, we have averaged the SFRs over intervals of $20$\n    Myr. For low resolution with more stellar massive particles, the\n    SFRs are well converged except for delayed cooling, though with a\n    trend of marginally lower SFRs. The mass loading factors do change\n    (except for kinetic feedback), generally showing an increase with\n    lower resolution. With a fixed stellar particle mass (i.e. lowered\n    SN specific energy), thermal dump feedback becomes weaker with\n    lower resolution, while other feedback models maintain similar\n    SFRs but higher mass loading factors.}\n\\end{figure}\n\nWe also consider the effect of `SN' resolution, where we lower the\ngrid resolution (and that of the initial conditions particles), just\nas in the {\\sc g9lr}{} runs, but keep the mass of newly formed stellar\nparticles fixed compared to the {\\sc g9}{} runs. These runs, which we call\n{\\sc g9lr\\_m$_*$}{}, give another measure of resolution convergence for the\nfeedback models, as the specific energy per feedback event (i.e. SN\nenergy over the local gas mass) is reduced by a factor of eight\ncompared to {\\sc g9}{} runs, while it was kept constant in\n\\Fig{SFR_res.fig}.  We compare those to the {\\sc g9}{} runs in the right\ncolumn of panels in \\Fig{SFR_res.fig}, showing the ratios of SFRs and\nmass loading factors for the feedback models.  For the SFR, the\nlargest difference occurs for thermal dump feedback, which becomes\nmuch less efficient at suppressing star formation rates due to the\nlower particle masses. The adiabatic phase becomes severely\nunresolved, and there is no mechanism built into the model to\ncompensate.  Other feedback models maintain similar average star\nformation with the lower specific energies.  The mass loading factors\nincrease somewhat if we decrease the spatial resolution and the\nspecific SN energies (bottom right panel of \\Fig{SFR_res.fig}), but at\n$250$ Myr the increase is smaller than for fixed specific energies\n(bottom left panel). Delayed cooling is an exception, showing a\ndecrease in mass loading at some time intervals, but an increase in\nothers, which is likely just caused by the relatively large variations\nin the SFRs.\n\nWith thermal dump feedback, resolution non-convergence is a well known\nproblem. With lower resolution, a larger gas mass is heated in a\nsingle feedback event, resulting in lower initial temperatures given a\nfixed SN energy. This would normally lead to higher SFRs, but in the\nleft panels of \\Fig{SFR_res.fig} the effect is counter-balanced by the\nuse of more massive stellar particles, slightly increasing the\nfeedback efficiency due to the higher SN energy per feedback event.\n\nFor the case of stochastic feedback, the fairly good convergence of\nthe SFR with both spatial resolution and stellar particle mass is not\na big surprise, since the stochasticity is built in to ensure that the\nheated gas receives a fixed specific energy, regardless of cell size,\ndensity, and particle mass. The outflow rates are relatively poorly\nconverged for stochastic feedback and low stellar particle mass, which\nlikely comes from the aforementioned tendency for the outflow rate to\nincrease with lower resolution, and hence larger launching scales,\neven if the specific SN energy is constant.  Mechanical feedback was\nshown by \\cite{Kimm2014} to converge well with resolution in terms\nof the final momentum reached, and indeed the whole point of the model\nis to maintain the same momentum injection regardless of whether the\nmomentum buildup is captured numerically or not. While we confirm that\nthe convergence is good for the SFR, the outflow rates are not very\nwell converged, neither in terms of spatial resolution nor stellar\nparticle mass. For delayed cooling, resolution convergence is not an\nobvious property, but if cooling is turned off long enough, the energy\n(i.e. cooling) losses should become insignificant and hence\nindependent of the resolution. The fact that the SFR (and to a lesser\nextent the mass loading factor) is poorly converged for delayed\ncooling hints that cooling losses are still significant, but we remind\nthat the absolute change is actually lower than for the other feedback\nmodels.  As long as kinetic feedback is sufficiently decoupled from\nthe ISM surrounding the SN explosion due to the use of a large SN\nbubble radius, it is not surprising to see good resolution\nconvergence, since the main effect of the feedback is then simply to\nslowly deplete the disc of gas mass.\n\nIn summary, except for thermal dump feedback, the feedback models\nconverge fairly well in terms of SFRs. However, none except for\nkinetic feedback (with a bubble radius larger than the disc height)\nconverge well in terms of the outflow mass loading factor, with the\nmass loading factor decreasing with higher resolution.\n\n\n\\section{Varying the temperature jump for stochastic feedback} \\label{fb_stoch.sec}\n\nWe now examine the effect of varying the stochastic heating,\nparameter, $\\Delta T_{\\rm stoch}$. We do not go lower than the fiducial value of\n$\\Delta T_{\\rm stoch}=10^{7.5}$ K, because already here the feedback is not very\nstochastic: in the {\\sc g9}{} galaxy the average probability for SN\ncandidates (which, as a reminder, is the ratio of the available SN\nenergy of the stellar particle to the energy required to heat the host\ngas cell by $\\Delta T_{\\rm stoch}$) is $p_{\\rm SN}\\approx 50\\%$ ($\\approx 35\\%$ in\n{\\sc g10}{}). Lower $\\Delta T_{\\rm stoch}$ would lead to order unity probabilities for\nSN explosions, converging towards thermal dump feedback.\n\nWe thus consider three values for $\\Delta T_{\\rm stoch}$ in addition to the\nfiducial one, each time increasing the injected specific energy by\nhalf a dex, i.e. $\\Delta T_{\\rm stoch}=10^{7.5}, 10^{8}, 10^{8.5}$, and $10^{9}$\nK.\n\n\\begin{figure}\n  \\centering\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/fb_stoch\/maps_1_250_sdt80.pdf}}\n  \\subfloat\n  {\\includegraphics[width=0.23\\textwidth]\n    {figures\/fb_stoch\/maps_1_250_sdt90.pdf}}\n  \\caption\n  {\\label{maps_stoch.fig}Maps of total hydrogen column density for the\n    {\\sc g9}{} galaxy at $250$ Myr, for variations in the stochastic\n    heating, with $\\Delta T_{\\rm stoch}=10^8$ K on the left and $\\Delta T_{\\rm stoch}=10^9$ K\n    on the right. Each panel shows face-on and edge-on views.}\n\\end{figure}\n\\Fig{maps_stoch.fig} shows maps of the hydrogen column density for\n$\\Delta T_{\\rm stoch}=10^8$ and $10^9$ K. For the case of $\\Delta T_{\\rm stoch}=10^8$ K, there\nis not a significant difference from the fiducial run (the middle left\npanel in \\Fig{maps_Fid.fig}), though the gas disc becomes slightly\nthicker and clumpier. For $\\Delta T_{\\rm stoch}=10^9$ K, the difference is much\nclearer. Here the face-on disc is more diffuse overall, but at the\nsame time it contains more dense clumps and filaments, especially\nclose to the centre and at the disc outskirts. This is due to the\nincreased stochasticity of SN explosions: the low probability for SN\nevents, on average $p_{\\rm SN}\\approx 3 \\%$, allows dense star-forming\nclumps to live longer before they are hit by the first SN\nexplosion\\footnote{We note that \\cite{DallaVecchia2012} advise\n  against such high values of $\\Delta T_{\\rm stoch}$ that probabilities for\n  feedback events become $\\ll 1$, which is clearly the case\n  here.}. This effect is amplified at the outskirts, where there is\nrelatively little star formation and thus a low rate of SN explosions\nper unit volume. Looking at the disc edge-on, there is much more\nstructure in the CGM for our maximum value of $\\Delta T_{\\rm stoch}$.\n\nIn \\Fig{OF_stoch.fig} we zoom out and look at the large-scale outflows\nin the `maximum' case of $\\Delta T_{\\rm stoch}=10^9$ K. The outflows are\ndramatically different from the fiducial setting for stochastic\nfeedback shown in the top right panel of \\Fig{mapsOF_fid.fig}: they\nare much denser, clumpier, (mostly) hotter, and more extended. Indeed,\nlooking at the dashed lines in \\Fig{SFR_stoch.fig}, we see that the\noutflow rate at $20$ kpc increases almost linearly with $\\Delta T_{\\rm stoch}$,\nand is about $3$ times higher than the SFR (solid) at the end of the\nrun for $\\Delta T_{\\rm stoch}=10^9$ K. The average outflow velocity (not shown)\nincreases by a factor $2-3$ at $2$ kpc for the highest $\\Delta T_{\\rm stoch}$\nconsidered (compared to the lowest), but is unaffected at $20$ kpc.\n\n\\begin{figure}\n  \\centering\n  {\\includegraphics[width=0.45\\textwidth]\n    {figures\/fb_stoch\/mapsof_2_250_sdt90.pdf}}\n  \\caption\n  {\\label{OF_stoch.fig}Edge-on views of the {\\sc g9}{} galaxy at $250$\n    Myr, for the run with the strongest stochastic heating\n    ($\\Delta T_{\\rm stoch}=10^9$ K). The two maps show slices of the hydrogen\n    number density (left) and temperature (right). Dotted horizontal\n    lines mark planes $2$ and $20$ kpc from the disc.}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/fb_stoch\/sfr_stoch.pdf}\n  \\caption\n  {\\label{SFR_stoch.fig}SFRs (solid curves) and gross outflow rates\n    $20$ kpc from the disc (dashed curves) for the {\\sc g9}{} galaxy, with\n    variations in $\\Delta T_{\\rm stoch}$ for the stochastic SN feedback model.\n    Increasing $\\Delta T_{\\rm stoch}$ within reasonable limits has little effect\n    on the SFR, but does increase the outflow rate (and hence the mass\n    loading factor) significantly.}\n\\end{figure}\n\nVarying $\\Delta T_{\\rm stoch}$ has a much weaker effect on the SFR than on the\noutflow, as shown in \\Fig{SFR_stoch.fig}. The `first' two increases in\n$\\Delta T_{\\rm stoch}$ have almost no effect on the SFR, while the highest value\nproduces an initially higher SFR which then declines gradually, much\nlike in the case of kinetic feedback, and ends up significantly lower\nthan for lower $\\Delta T_{\\rm stoch}$ values. As with kinetic feedback, the\ndecline in SFR is likely due to the strong outflow depleting the\ngalaxy of fuel for star formation. Also, due to the lower heating\nprobability, star-forming clumps can continue to form stars longer\nwithout having strong SN events disrupting them.\n\nWe recall that the SN energy is not directly redistributed to lower\ngas densities with stochastic feedback (see Eq. \\ref{pSN2.eq}). On the\ncontrary, we find that increasing $\\Delta T_{\\rm stoch}$ indirectly results in the\nSN energy being deposited at higher gas densities due to the increased\nclumpiness of the gas (not shown; $\\Delta T_{\\rm stoch} = 10^9$ K results in\nenergy being deposited at $\\approx 0.5-1$ dex higher densities,\ncompared to the fiducial case). Hence the stochasticity increases\nfeedback efficiency purely by increasing the injected energy of a\ngiven SN event \\emph{at a given density} (while the total SN energy\nover the galaxy is unchanged), and hence also the local cooling time.\n\nIn summary, increasing the stochasticity of feedback by increasing\n$\\Delta T_{\\rm stoch}$ strongly increases the outflows. The SFRs are insensitive\nto these stochasticity variations except for the highest value of\n$\\Delta T_{\\rm stoch}$ considered. With very large stochasticity, the disc also\nbecomes increasingly clumpy.\n\n\\section{Varying the time-scale in delayed cooling\n  feedback} \\label{fb_dc.sec} Of the feedback models we have compared\nin \\Sec{comparison.sec}, with their fiducial setup parameters, delayed\ncooling feedback suppresses star formation most strongly and yields\nthe highest mass loading factors for the outflows (see Figures\n\\ref{SFR.fig}, \\ref{SFR_MW.fig}, \\ref{KS_Fid.fig}, and\n\\ref{OFtime.fig}).\n\nSince delayed cooling feedback in its fiducial setup is strong, we\nexamine here what happens if we reduce the value of its free\nparameter, which is the cooling delay time-scale, $t_{\\rm{delay}}$. The\nfiducial value is $t_{\\rm{delay}}=10$ Myr, so here we compare with two runs\nwith significantly lower delay time-scales of $2$ and $0.5$ Myr. We\nfind that the results are highly sensitive to variations in\n$t_{\\rm{delay}}$. In \\Fig{SFR_dcool.fig} we show the {\\sc g9}{} SFRs and gross\noutflow rates across planes $20$ kpc from the disc, for those\nvariations. As expected, a shorter delay time-scale strongly decreases\nthe feedback efficiency, producing higher SFRs and lower outflow\nrates. However, even for the shortest delay time-scale considered here,\nthe SN feedback is still more efficient in terms of suppressing star\nformation than any of the other feedback models (and second only to\nkinetic feedback in terms of outflow rates). Note that the mass\nloading factor is particularly sensitive to the value of the free\nparameter. At $250$ Myr it decreases from $\\betaOut \\approx 2$ for\n$t_{\\rm{delay}} =10$ Myr to $\\betaOut \\sim 10^{-1}$ for $t_{\\rm{delay}} =0.5$ Myr\n(both at $2$ and $20$ kpc from the disc). The outflow velocities,\nwhich we do not show here, are unaffected at $20$ kpc, and are fairly\ninsensitive to $t_{\\rm{delay}}$ at $2$ kpc (few tens of percent increase in\noutflow velocity from smallest to highest $t_{\\rm{delay}}$).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/fb_dcool\/sfr_dcool.pdf}\n  \\caption\n  {\\label{SFR_dcool.fig}Star formation and gross outflow rates for the\n    {\\sc g9}{} galaxy for variations in the delay time-scale, $t_{\\rm{delay}}$, for\n    delayed cooling feedback. The solid lines show SFRs, while dashed\n    lines show outflow rates $20$ kpc from the disc. Decreasing\n    $t_{\\rm{delay}}$ from the fiducial value of $10$ Myr makes the SN feedback\n    weaker. However, even at the lowest dissipation time-scale shown,\n    the SFR at $250$ Myr is still low and the outflow rates are still\n    high compared to the other feedback models (for their fiducial\n    parameters).}\n\\end{figure}\n\nWe analysed the KS relation for the same variations in $t_{\\rm{delay}}$ (not\nshown). Decreasing $t_{\\rm{delay}}$ produces a KS relation which looks\nincreasingly like that of mechanical feedback (for the {\\sc g9}{} galaxy)\nin \\Fig{KS_Fid.fig}, with a similar slope and maximum gas surface\ndensity of $7 \\times 10^{2} \\ \\msunpc$ for $t_{\\rm{delay}}=0.5$\nMyr. Morphologically, the galaxy with the shortest dissipation\ntime-scale also looks quite similar (though a bit more diffuse) as the\nruns with mechanical, stochastic, and thermal dump feedback in\n\\Fig{maps_Fid.fig}.\n\nWhile the outflow rates decrease for shorter delay times-cales, the\nmorphological and phase structure of the outflows remain qualitatively\nsimilar.\n\n\\section{Variations in kinetic SN feedback} \\label{fb_kin.sec}\n\nIn \\Fig{SFR_kin.fig}, we show variations in the SFRs and gross mass\noutflow rates at $20$ kpc for the {\\sc g9}{} galaxy, for variations in the\nfree kinetic feedback parameters, which are the bubble radius\n(fiducially $150$ pc) and the sub-resolution mass loading parameter\n$\\eta_{\\rm W}$ (fiducially set to $1$).\n\nDecreasing the bubble radius, $r_{\\rm bubble}$, to about two times the minimum\ncell width (of $18$ pc), has a significant effect on the SFR and\ndramatically suppresses the gas outflow rate, which both (and also the\noutflow speed) become similar to those produced by thermal dump,\nstochastic, and mechanical feedback. Morphologically (not shown), the\nruns with the small bubble radius also resemble the runs with these\nother feedback models.\n\nThe smaller bubbles are less efficient at driving large-scale\noutflows, because a larger fraction of the energy is now deposited\ninto dense ISM gas. For the same reason, the SFR at early times\ndecreases for smaller bubbles.  At late times the SFR is higher\nbecause there remains substantially more gas in the galaxy as a\nconsequence of the weaker large-scale winds.\n\nIncreasing the sub-resolution mass loading, $\\eta_{\\rm W}$, by an order of\nmagnitude has little effect on star formation (\\Fig{SFR_kin.fig}),\noutflows (rates and speeds), and morphology (not shown). This is\nbecause most SN events occur quite close to the density threshold for\nstar formation (see \\Fig{SD_cum.fig}), meaning that usually there is\nnot much more mass available in the host cell than roughly matches the\nstellar particle mass. Hence, an initial mass loading of more than\n$\\approx 1$ is not possible, since it would mean removing more mass\nfrom the host cell than is available for re-distribution to the SN\nbubble. To investigate the effect of lower stellar particle masses, or\nstar formation happening well above the density threshold, we\nperformed runs where we used a three times lower stellar particle\nmass, $m_{*}=600 \\ \\rm{M}_{\\odot}$. Here increasing $\\eta_{\\rm W}$ \\emph{reduces} the\noutflow rates significantly (as it does to a smaller extent in\n\\Fig{SFR_kin.fig} for the smaller bubble radius). Moreover, the\nfeedback becomes more coupled to the disc, which becomes much\nthicker. This can be understood from the fact that with a higher mass\nloading, i.e. a larger ejected mass, the velocity of the ejected gas\nmust decrease to conserve momentum, and hence it is less likely to\nescape from the galaxy.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics\n    {figures\/fb_kin\/sfr_kin.pdf}\n  \\caption\n  {\\label{SFR_kin.fig}Star formation and outflow rates for variations\n    in the bubble radius ($r_{\\rm bubble}$) and sub-resolution mass loading\n    ($\\eta_{\\rm W}$) for kinetic feedback in the {\\sc g9}{} galaxy. The solid\n    lines show SFRs, while dashed lines show outflow rates $20$ kpc\n    from the disc. Reducing the bubble radius (thicker curves) from\n    the fiducial value of $150$ pc has a significant effect on the SFR\n    and dramatically reduces the outflow rates. Changes in $\\eta_{\\rm W}$\n    have little effect, but this is because the order unity ratio of\n    the stellar particle mass to the local cell mass allows little\n    room for increased sub-resolution mass loading.}\n\\end{figure}\n\nWe also looked at the KS relation for these variations in kinetic\nfeedback (not shown). Here, again, varying $\\eta_{\\rm W}$ has a negligible\neffect, while the smaller bubble radius gives a relation very similar\nto e.g. mechanical feedback.\n\n\\section{Discussion} \\label{Discussion.sec}  \n\nThe effects of the SN feedback models we have studied are sensitive\nto the resolution and\/or mass of the simulated galaxies. For our\nlower-mass {\\sc g9}{} galaxy, thermal dump, stochastic, and mechanical\nfeedback give very similar results in terms of SFRs, the\nKennicutt-Schmidt relation, and outflow mass loading factors, and we\nhave argued that this is an indication that the adiabatic phase of\nthermal SN feedback is `resolved' in this galaxy, with the caveat that\nwe inject the equivalent of $40$ individual SN explosions\ninstantaneously. For our (ten times) more massive {\\sc g10}{} galaxy,\nhowever, this no longer applies, and thermal dump feedback is\nsignificantly weaker than any of the other models.\n\nIf we compare our results to observations of SFRs in the local\nuniverse, delayed cooling comes closest to reality. Observed SFRs for\nlocal star-forming galaxies of similar stellar masses to {\\sc g9}{} are\ntypically in the range $\\approx 0.05-0.4 \\ \\msunyr$ \\citep[Figure 11\nin][]{Chang2015}. In \\Fig{SFR.fig}, we see that delayed cooling is\nthe only feedback model giving SFRs matching those observations,\nwhile the other models give values well above the upper limits. For\nstellar masses similar to our {\\sc g10}{} galaxy ($10^{10} \\, \\rm{M}_{\\odot}$),\n\\cite{Chang2015} give SFRs in the one-sigma range of\n$\\approx 0.3-1.5 \\, \\msunyr$, which actually falls slightly below the\ndelayed cooling SFR in \\Fig{SFR_MW.fig}.  While such a comparison to\nobservations gives some indication of what is required to produce a\nrealistic suppression of star formation, we stress that comparing the\nresults from our isolated and somewhat idealised setup to observed\nstar formation rates has many caveats. Perhaps most significantly, the\ngas fractions in our galaxies are high compared to those of local\ngalaxies \\citep[see e.g. the compilations in][]{Bahe2016,\n  Sales2016}.\n\nSuch caveats matter less if we consider the Kennicutt-Schmidt\nrelation. Here, all models except delayed cooling produce a slope\nthat is too steep compared to observations and\/or SFR surface\ndensities that are significantly too high for a given gas surface\ndensity. Delayed cooling does relatively well in terms of slope but\nfor our fiducial parameter choice it undershoots the SFR surface\ndensity. However, unlike for the other models, this can be calibrated\ntowards a reasonable fit to the observed KS relation by tuning the\ndelayed cooling time-scale.\n\nIn addition, delayed cooling and `decoupled' kinetic feedback are the\nonly models able to produce mass loading factors exceeding unity,\nwhile for other models with their fiducial parameters the mass\nloading factors are at best an order of magnitude below unity.\n\nWhile one can argue that these factors make delayed cooling for these\nobservables empirically most successful, the model produces\nunrealistic thermal signatures in the ISM and CGM, where large amounts\nof gas occupy a region in temperature-density space where the cooling\ntime is very short. Moreover, the convergence between thermal dump,\nstochastic, and mechanical feedback suggest the adiabatic phase is\nresolved and hence that the results from delayed cooling and kinetic\nfeedback may be unphysical.  \n\nMechanical feedback could be argued to be most physically motivated,\nbeing based on analytic calculations and high (sub-pc) resolution\nsimulations of the final momentum attained in various environments in\nterms of gas density and metallicity. Yet it does not come near\nobservations in suppressing star formation and produces weak outflows\nin our simulations.  Perhaps this discrepancy can be traced to the\nidealistic assumptions made when deriving the mechanical momentum\n(Eq. \\ref{dp_mech.eq}). While the momentum is realistic and converged\nfor a homogeneous medium, such a medium is not a good description of a\nmulti-phase and porous feedback-regulated star-forming ISM, and may\nlead to an under-prediction of the generated momentum\n\\citep[e.g.][]{Kimm2015}.  We also neglect the preprocessing of the\nlocal environment by stellar radiation, which lowers the surrounding\ngas densities and has been shown to increase the momentum injection\nfrom stellar populations, typically by a factor $\\sim 2$\n\\citep[e.g.][]{Geen2015}.\n\nIt was recently reported by \\cite{Gentry2016} that idealised\nexperiments of SN feedback in the literature have under-predicted the\nfinal momentum by up to an order of magnitude, due to i) the neglect\nof multiple successive SN explosions, ii) a lack of resolution, and\niii) a preference for Eulerian hydrodynamical solvers, which are\nargued to suffer from over-mixing and hence over-cooling. If these\nresults are confirmed, we may have much more momentum to play with.\n\nAnother culprit may be the simplified setup of our simulations. For\ncontrol and to reduce the computational cost, we used isolated galaxy\nsimulations, assuming an initial state of the galaxy and its dark\nmatter halo rather than letting it evolve naturally in a cosmological\ncontext. Ignoring environmental factors such as mergers and gas\naccretion, may change the behaviour of SN feedback.\n\nFinally, there are other feedback processes at play in galaxies that\nwe have neglected, such as AGN (thought to be important at high,\nlarger than $\\approx$ MW masses; e.g. \\citealt{Crain2015}), radiation\npressure \\citep[e.g.][]{Rosdahl2015}, and cosmic rays\n\\citep[e.g.][]{Booth2013, Hanasz2013, Salem2014, Girichidis2016,\n  Simpson2016}. The efficiency and interplay of each of those\nprocesses is not well constrained, but they likely provide an\nadditional suppression of the SFR on top of SN feedback. If they turn\nout to play an important role in galaxy evolution, some empirically\ncalibrated sub-resolution models for SN feedback may be re-interpreted\nto also represent other feedback processes at play in galaxies.\n\n\\subsection{Dependence on factors other than SN feedback} \\label{sfe.sec} \n\nGalaxy evolution involves a complex interplay of many physical\nprocesses, and the SN feedback efficiencies that we have reported may\nbe sensitive to factors other than the SN feedback models.\n\nOne of the most important choices in our simulations aside from the\nimplementation of feedback is the local star formation efficiency,\nwhich we have set to $\\epsilon_{*}=2\\%$. While this is a normal choice in\nsimulations of galaxy evolution, \\cite{Agertz2015} argued that such\na low value artificially suppresses the effects of feedback, and that\nhigher local SF efficiencies of $\\epsilon_{*} \\approx 0.1$ are needed to\nallow feedback to self-regulate star formation.  We have experimented\nwith $\\epsilon_{*}=0.1$ in the {\\sc g9lr}{} galaxy (while keeping the other\nparameters fixed). While a full analysis is beyond the scope of the\npresent work, it is worth mentioning the qualitative effects. For all\nSN feedback models considered, it results in a high early SFR peak,\nfollowed by a dip and then convergence towards similar but somewhat\nlower SFRs compared to those obtained for the fiducial\n$\\epsilon_{*}=0.02$. However, the SFR surface density moves in the wrong\ndirection, i.e. away from the observed KS relation, both in terms of\nslope and normalisation.\n\nWe also investigated the effect of reducing $\\epsilon_{*}$, to find what\ncalibration is required to reproduce the observed KS relation. A value\nof $\\epsilon_{*}=0.002$, i.e. ten times lower than our fiducial value,\nproduces a good match to the observed relation, but at the cost of\nmaking the self-regulating effect of SN feedback negligible. In other\nwords, the $\\epsilon_{*}$ parameter takes over as the dominant feedback\nmechanism when set to a very low value.\n\nWhile our non-comprehensive probes of the effect of a varying\nnumerical star formation efficiency $\\epsilon_{*}$ are discouraging, we have\nthus far not studied variations in $\\epsilon_{*}$ in combination with other\nfactors. For example, in combination with a high local star formation\nefficiency, early feedback may play a significant role by suppressing\nrunaway star formation for the $5$ Myr from the birth of the first\nstars until the onset of SN explosions in a given star-forming region\n\\citep[e.g.][]{Hopkins2012b, Stinson2013, Agertz2015}. Also\nthe locality of star formation may be important for the efficiency of\nfeedback, i.e. it may matter whether the distribution of star\nformation, and hence SN explosions, is scattered smoothly in space and\ntime, or happens in short localised bursts\n\\citep[e.g.][]{Federrath2013, Hopkins2013b, Walch2015a}.\n\nWe will investigate the role of $\\epsilon_{*}$ in more detail in future\nwork, where we combine higher local SF efficiencies with more\nstringent SF criteria and early feedback in the form of stellar\nradiation.\n\nThere are many more variations which may or may not affect the\nfeedback efficiency, and again we have made limited explorations,\nwhich we summarise below.\n\\begin{itemize}\n\\item We varied the density threshold for star formation, $n_{*}$, by a\n  factor of ten in either direction, for stochastic feedback. Higher\n  $n_{*}$ does give lower initial SFRs, but after $250$ Myr the values\n  are the same to within a few percent for two orders of magnitude\n  variations in $n_{*}$. Outflow rates remain nearly unchanged.\n\\item In the more massive galaxy we ran identical simulations with\n  $0.1$ Solar metallicity, instead of the default Solar\n  metallicity. The reduced metallicity has a marginal effect on the\n  outflows and on the SFRs in the case of delayed cooling and kinetic\n  feedback. However, for thermal dump, stochastic, and mechanical\n  feedback, it reduces the SFRs by a few tens of percent and increases\n  the outflow mass loading factor by about a factor of two.\n\\item Switching from non-equilibrium hydrogen and helium\n  thermochemistry to an equilibrium assumption in the {\\sc g9}{} galaxy\n  significantly increases outflow rates for thermal dump (factor\n  $\\sim 5$), stochastic feedback (factor $\\sim 4$), and mechanical\n  feedback (factor $\\sim 5$), while the SFRs are only marginally\n  reduced (and there is almost no effect on either SFRs or outflows\n  with kinetic feedback or delayed cooling). In a forthcoming paper,\n  we will explore the physical effects of equilibrium versus\n  non-equilibrium thermochemistry in the context of SN feedback.\n\\item With thermal dump and stochastic feedback we scaled the Jeans\n  pressure floor by a factor of three in each direction (i.e. three\n  times higher and lower pressure at a given gas density). The disc\n  morphology becomes slightly but noticeably smoother with a higher\n  Jeans pressure, but the effect on star formation and outflows is\n  negligible. A very large increase of the pressure floor suppresses\n  the effect of SN feedback with any model, since it overtakes the\n  role of SN feedback in suppressing the collapse of gas to\n  star-forming densities.\n\\item With mechanical feedback, we explored the effect of runaway OB\n  stars in the {\\sc g9}{} galaxy, giving a kick to each newborn stellar\n  particle with a random direction and random speed of $0-50 \\, \\kms$.\n  This has negligible effect on the SFRs, but the outflow rates are\n  enhanced by almost a factor of ten, due to SN explosions taking\n  place at significantly lower densities on average.\n\\item Also with mechanical feedback, we simulated individual $10^{51}$\n  erg SN explosions, stochastically sampled for each stellar particle\n  over the $5-50$ Myr lifetimes assumed for massive stars in the\n  (Chabrier) IMF \\citep[see details in][]{Kimm2015}. The effect is\n  similar to the above test with runaway OB stars: the SFRs are\n  marginally higher, while the outflow rates are increased\n  significantly, though somewhat less than for the runaway OB\n  stars. The reason for the increased outflow rates is that for a\n  given particle, early SN explosions can clear away the dense\n  environment, leading to late SN explosions taking place at reduced\n  densities. We also combined runaway OB stars and individual SN\n  explosions in the same run (again for the {\\sc g9}{} galaxy). This\n  produces SFRs marginally higher than individual SNe only and\n  outflows marginally higher than runaway OB stars only, i.e. it does\n  not give an extra boost to the outflows.\n\\item Changing the time delay for stochastic SN feedback (zero to $20$\n  Myr) has similar effects as runaway OB stars: The SFRs are only\n  marginally affected, while an increased delay increases the outflow\n  mass loading factor (it is halved for zero delay and doubles for a\n  $20$ Myr delay).\n\\end{itemize}\n\n\\subsection{Comparison with stochastic heating in\n  GADGET}\\label{DS-comp.sec}\nThe stochastic feedback model included in this work is based on the\nscheme introduced in {DS12}{} and used in the {\\sc Gadget}{} code. Similarly\nto this work, {DS12}{} explore the effects of their stochastic feedback\nmodel using rotating isolated galaxy discs of two masses, the main\ndifference being that their lower mass disc is roughly ten times less\nmassive than our {\\sc g9}{} disc (while their higher mass disc is\ncomparable in mass to our {\\sc g10}{}). They compare different values for\nstochastic heating and find $\\Delta T_{\\rm stoch}=10^{7.5}$ K to be efficient in\nsuppressing star formation (in the massive disc) and generating strong\noutflows (in both discs). In {DS12}{} this value, which we also choose as\nour fiducial value for stochastic heating, gives outflow mass loading\nfactors (at $20 \\%$ of the virial radii) of $\\approx 40$ and\n$\\approx 2$ for the low- and high-mass galaxies, respectively. This is\n$\\approx 40-200$ times larger than our mass loading factors, pointing\nto a significant difference between our simulations and {DS12}{} in the\nefficiency of stochastic feedback in suppressing star formation and\ngenerating outflows.\n\nIn \\App{ds.app} we discuss the differences between our simulations and\nthose of {DS12}{} and attempt to more closely reproduce their setup. We\nconclude that there is a major difference between the two versions\n(AMR and SPH) of stochastic feedback, with the SPH version being much\nmore efficient at generating outflows, for a given SFR. Pinpointing\nthe reason(s) for this difference, however, and whether it is due to\nsubtle differences in setup or resolution, or to more fundamental\ndifferences between AMR and SPH, remains a challenge for follow-up\nwork.\n\n\\section{Conclusions} \\label{Conclusions.sec} \n\nWe used simulations of isolated galaxy discs with the {\\sc Ramses}{} code\nto assess sub-resolution models for SN feedback in AMR simulations,\nin particular their efficiency in suppressing star formation and\ngenerating outflows. We focused our analysis on a dwarf galaxy, ten\ntimes less massive than the MW, using a spatial resolution of $18$ pc\nand a stellar (DM) mass resolution of $2 \\times 10^3 \\ \\rm{M}_{\\odot}$\n($10^5 \\ \\rm{M}_{\\odot}$), but also included a more limited analysis of a MW\nmass galaxy (using a resolution of $36$ pc, $1.6 \\times 10^4 \\ \\rm{M}_{\\odot}$\nstellar particles and $10^6 \\ \\rm{M}_{\\odot}$ DM particles).\n\nWe studied five SN feedback models: i) thermal dump of SN energy into\nthe host cell of the star particle, ii) stochastic thermal feedback,\nwhere the SN energy is re-distributed into fewer but more energetic\nexplosions, iii) kinetic feedback, where momentum is deposited\ndirectly into a bubble around the star particle, iv) delayed cooling,\nwhere cooling is suppressed temporarily in the expanding SN remnant,\nand v) mechanical feedback, which injects energy or momentum depending\non the resolution. Three of those models can be calibrated with\nadjustable parameters, which are the minimum local heating temperature\nfor stochastic feedback, the bubble size and local mass loading for\nkinetic feedback, and the cooling suppression time for delayed\ncooling). The mechanical feedback model has no free parameters (once\nthe SN energy has been decided) and the injected momentum is based on\nanalytic derivations and high-resolution simulations of cooling losses\nin expanding SN blasts.  We compared the results produced using these\nmodels with their fiducial settings, and for those models with\nadjustable parameters we studied the effects of parameter\nvariations. Our main results are as follows.\n\nFor our low-mass, high-resolution galaxy, thermal dump, stochastic,\nand mechanical feedback produce nearly identical results (Figs.\n\\ref{SFR.fig} and \\ref{OFtime.fig}). We showed that at our current\nresolution and star formation densities, stochastic feedback is\nactually not that stochastic, and mechanical feedback is still mostly\nin the adiabatic phase. Hence those feedback models are converged in\nthat setup, and thermal dump feedback adequately resolves the energy\ninjection (by multiple SNe in a single event). For our more massive\ngalaxy, stochastic and mechanical feedback become significantly\nstronger than thermal dump feedback, but are still weak compared to\ndelayed cooling and kinetic feedback (\\Fig{SFR_MW.fig}).\n\nStrong outflows are not easily generated in our AMR simulations. Mass\nloading factors of unity or above require extreme measures, such as\nturning off cooling for a prolonged time, or kinetic feedback that is\nin effect hydrodynamically decoupled due to the bubble radius\nexceeding the disc height (\\Fig{OFtime.fig}). The outflows produced by\ndelayed cooling and kinetic feedback are very distinct, the former\nbeing cold, dense, and slow, while the latter are hot, diffuse, fast,\nand featureless (Figs. \\ref{SQO_Fid.fig}, \\ref{mapsOF_fid.fig}, and\n\\ref{PH_Fid.fig}). The other models produce slow and remarkably\nsimilar outflows at intermediate densities and temperatures.\n\nSave for thermal dump feedback, all models do well in terms of\nresolution convergence when considering SFRs, while, with the\nexception of kinetic feedback, they produce significantly higher\noutflow rates at lower resolution (\\Fig{SFR_res.fig}).\n\nAlthough a direct comparison is difficult, stochastic feedback appears\nto produce much weaker outflows than in the similar disc runs with the\noriginal SPH version of the model of {DS12}{}.  This discrepancy is\nperhaps a result of subtle setup differences between our discs and\nthose of {DS12}{}, but we cannot rule out a more fundamental AMR versus\nSPH difference. Stochastic feedback does become efficient at\ngenerating massive outflows in our AMR discs if we use very high\nvalues for the stochastic heating temperature (up to $10^9$ K), but\nthis comes at the cost of strong stochasticity due to low SN\nprobabilities (Figs. \\ref{SFR_stoch.fig} and \\ref{maps_stoch.fig}).\n\nThe major handle on the generation of outflows appears to be how well\nthe SN feedback model circumvents gas cooling, directly or\nindirectly. Delayed cooling is the only model which succeeds at\ngenerating outflows with mass loading factors exceeding unity\n(\\Fig{OFtime.fig}), at reproducing the observed main sequence SFRs\n(Figs. \\ref{SFR.fig} and \\ref{SFR_MW.fig}), and the Kennicutt-Schmidt\nrelation (with appropriate calibration; \\Fig{KS_Fid.fig}). The other\nmodels fail to produce SN feedback strong enough to reproduce these\nobservations. This is discouraging, as delayed cooling retains too\nmuch energy for too long, which in reality is partly lost to radiative\ncooling, while the other feedback models are arguably more physically\nmotivated. Moreover, for the low-mass galaxy we argued that thermal\ndump, stochastic, and mechanical feedback converge because we resolve\nthe adiabatic phase of the feedback events. This implies that in this\ncase delayed cooling and kinetic feedback yield incorrect answers.  In\nparticular, delayed cooling results in gas occupying regions of\ntemperature-density space where the cooling time is very short, which\ncompromises predictions for observational diagnostics.\n\nPossible reasons for the disconnect between observations and our\nresults are: i) a lack of additional feedback physics, such as\nradiation feedback or cosmic rays, ii) an incomplete setup, i.e. an\ninsufficiently realistic description of galaxies, iii) other aspects\nof the subgrid physics, such as star formation, are unrealistic\n\\citep[e.g.][]{Agertz2015, Semenov2016}, iv) overcooling on\ngalactic scales is still an issue at our resolution, even if different\nfeedback models converge to the same results, and a significantly\nhigher resolution is required.\n\nThe current analysis will serve as a foundation for future studies of\nfeedback in galaxies, where we will use a sub-set of these models to\nstudy the interplay of SN feedback with different sub-grid methods for\nstar formation and with feedback in the form of stellar radiation.\n\n\\section*{Acknowledgements}\nWe thank J\\'er\\'emy Blaizot, L\\'eo Michel Dansac, Julien Devriendt,\nSylvia Ploeckinger, and Maxime Trebitsch for useful discussions, and\nthe anonymous referee for constructive comments. This work was funded\nby the European Research Council under the European Union's Seventh\nFramework Programme (FP7\/2007-2013) \/ ERC Grant agreement\n278594-GasAroundGalaxies. TK was supported by the ERC Advanced Grant\n320596 ``The Emergence of Structure during the Epoch of\nReionization''. The simulations were in part performed using the DiRAC\nData Centric system at Durham University, operated by the Institute\nfor Computational Cosmology on behalf of the STFC DiRAC HPC Facility\n(www.dirac.ac.uk). This equipment was funded by BIS National\nE-infrastructure capital grant ST\/K00042X\/1, STFC capital grant\nST\/H008519\/1, and STFC DiRAC Operations grant ST\/K003267\/1 and Durham\nUniversity. DiRAC is part of the National E-Infrastructure. We also\nacknowledge PRACE for awarding us access to the ARCHER resource\n(http:\/\/www.archer.ac.uk) based in the UK at the University of\nEdinburgh (PRACE-3IP project FP7 RI-312763).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nGraphite is an allotrope of carbon with layered structure. \nThe structure results in strongly anisotropic thermal and electrical transports~\\cite{PhysRev.127.694} along with other unique properties such as self-lubrication~\\cite{PhysRevLett.92.126101} and high stability up to 4000~K~\\cite{JGRB:JGRB3482}. \nWith added characteristics of flexibility, graphite sheets are industrial products making use of those properties. \nGrafoil~\\cite{Grafoil} is one of the best known commercial products based on this material, and is used in a wide variety of applications, e.g., sealing gaskets, thermal insulators, electrodes, and as a chemical reagent~\\cite{Chung1987}. \nIt consists of small natural graphite crystals ($=10$--20~nm)~\\cite{Takayoshi2009} which are first powdered, then exfoliated at high temperature, and finally rolled under high pressure. \nIt is also widely used as an adsorption substrate for basic research of two dimensional physical and chemical properties of adsorbate thin films~\\cite{RevModPhys.79.1381} because of its atomically flat surface of microcrystallites and large specific surface area. \nFor this purpose, there is another type of exfoliated graphite called ZYX~\\cite{ZYX}, synthesized from HOPG (highly oriented pyrolytic graphite) under rather moderate exfoliation and re-compression conditions, with larger platelet size ($=$100--200~nm). \n\nRecently, a new flexible graphite sheet, pyrolytic graphite sheet (PGS)~\\cite{pgs,ADEM:ADEM201300418}, has been invented. \nPGS is a thin graphite sheet of 10--100~$\\mu$m in thickness with a single-crystal-like structure, synthesized by heat decomposition of polymeric films. \nBecause of its extremely high in-plane thermal conductivity (2--5 times higher than that of copper at room temperature) and low density ($\\approx80$\\% of aluminum), PGS and its composites are being used for thermal management in electronic devices like smartphones~\\cite{ADEM:ADEM201300418}. \nIt is potentially useful for cryogenic applications, especially in space engineering.\nOne recent example is its use in a vibration isolated thermal link for cryocoolers~\\cite{McKinley2016174}. \nHowever, so far only little is known about physical properties of PGS, including thermal transport at cryogenic temperatures.\n\nIn this article, we report results of crystal analysis and electrical and thermal transport measurements at 2--300\\,K for PGS. \nHere we focused on an uncompressed version of PGS since it has higher crystallinity and thus higher in-plane conductivity than compressed commercial PGS. \nBy measuring both electrical and thermal conductivities, we could deduce a general relationship between them for graphite family materials where the standard Wiedemann-Franz law~\\cite{ANDP:ANDP18531650802} is not applicable. \nOther characteristics important for application as an adsorption substrate, such as nitrogen adsorption isotherm and real space imaging of morphology with various microscopes, will be published elsewhere~\\cite{part2_tbp}.\n\n\\section{Pyrolytic Graphite Sheet (PGS)}\nCommercial PGS~\\cite{pgs} (hereafter, cPGS) is made first by carbonizing a stack of polymer films of a few $\\mu$m thick at $T \\lesssim 1000$~K, then by graphitizing the resultant foamed carbon precursor at $T \\approx3000$~K, and finally by compression (rubbing) which reduces the thickness by 30--50\\%. \nCompared to chemical vapor deposition, which is used to synthesize HOPG, this is a convenient mass production method for thin graphite sheets of good crystallinity. \nPrevious transmission electron microscopy (TEM) observations~\\cite{10012555627} show that the cross-sectional structure of a similar kind of graphite to that used in the present study is laminar of ultra-thin crystalline graphite layers of 6--7~nm thick which corresponds to 16--20 graphenes.\nThe average lateral size of each layer is determined to be 10--100~$\\mu$m from electron channeling contrast  imaging with scanning electron microscope (SEM)~\\cite{10031183877}. \nIn general, thinner cPGS has higher crystallinity because, in the graphitization process, the liberated gas can escape more easily and the temperature distribution is more uniform. \n\nThe final compression procedure makes cPGS flexible (like paper) so as to be more useful in practical applications. \nHowever, it may break the lateral crystalline structure on large scales. \nTherefore, in this work, we mainly studied physical properties of uncompressed PGS (hereafter, uPGS) which is produced by exactly the same method as cPGS except the absence of the final compression. \nAs a trade off, uPGS is mechanically brittle and inflexible. \nThus, to shape it precisely, it is recommended to use a punch designed for cutting thin metal films~\\cite{nogamigk}. \n\nThe nominal thicknesses of uPGS studied here are 10, 17, 25, and 100~$\\mu$m.\nWe denote them as uPGS-10$\\mu$m, for example, in the following.\nTheir actual thicknesses measured by micrometer are 19$\\pm$2, 29.7$\\pm$0.6, 56$\\pm$3, and 145$\\pm$4~$\\mu$m, respectively.\nFor comparison we also studied properties of cPGS-10$\\mu$m whose measured thickness is 13$\\pm$2~$\\mu$m. \n\n\\section{Crystalline structure and defects}\nOut-of-plane X-ray diffraction was measured for uPGS-17$\\mu$m with a powder X-ray diffractometer~\\cite{xrd} using Cu K$\\alpha_1$ emission. \nThe uPGS sample of $10\\times10$ mm$^2$ was glued onto a glass holder with GE~7031 varnish. \nSharp diffraction peaks from graphite are observed indicating that PGS is made purely of graphite crystals (see Fig.~\\ref{xrdresults}).\nInterplanar spacing $d_{002}$ is determined as $0.33583(7)$~nm from peaks indexed as (002), (004), and (006) using Nelson-Riley function~\\cite{doi:10.1080\/14786444508520959}.\nFrom the full width at half maximum (FWHM) of the rocking curve of the (002) peak at $2\\theta=26.346$~deg, the mosaic angle spread is determined as 8.2$\\pm$0.1~deg as shown in the inset of Fig.~\\ref{xrdresults}. \nThis value is consistent with a mosaic angle spread ($=10\\pm$3~deg) roughly estimated from real space SEM imaging of a cross section of uPGS-17$\\mu$m~\\cite{part2_tbp}.\nIn-plane X-ray diffraction was also carried out (the data not shown here).\nThe sample was a stack of 29 uPGS-100$\\mu$m sheets (13$\\times$5~mm$^2$ each) fixed with epoxy glue (Stycast 1266) each other.\nIn addition to the peaks from regular spacing between graphene layers such as (002), those from in-plane honeycomb lattice like (110) and peaks indicative of three-dimensional graphite lattice indexed by (101), (102), (103), and (112) are observed.\n$d_{002}$ is determined as $0.33592(6)$~nm from the (002), (004), and (006) peaks, and the in-plane lattice parameter $a$ is determined as $0.2463$~nm from the (100) and (110) peaks. \nAll these diffraction results agree very well with the previous study for pyrographite films~\\cite{doi:10.1063\/1.96827}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{xrd_1}%\n\\caption{Out-of-plane X-ray diffraction spectrum for uPGS-17$\\mu$m. \n(Inset) Rocking curve of (002) peak at $2\\theta=26.346$~deg where the FWHM is 8.2~deg.\n\\label{xrdresults}}%\n\\end{figure}\n\nRaman spectra of cleaved surfaces of uPGS of 17, 25, and 100~$\\mu$m thick were measured at a wavelength of 532~nm using a laser Raman microscope~\\cite{raman} with a fixed exposure time of  8~seconds. \nFor comparison, cleaved surfaces of HOPG, Grafoil, and ZYX were also measured. \nThe most intensive features in Raman spectroscopy for graphite are G ($\\approx$1580~cm$^{-1}$) and G' ($\\approx$2710~cm$^{-1}$) peaks. \nThe relative intensity of G peak to G' one, $I$(G)\/$I$(G') is a good representative of the number $n$ of graphene layers for $n \\lesssim 6$~\\cite{PhysRevLett.97.187401}. \nMeasured $I$(G)\/$I$(G') values for uPGS are similar to those of other graphites, which confirms that they are thick enough graphites (see Table~\\ref{ratio}). \nIt is consistent with that all of them have similar FWHM values of G' band ($\\approx60$~cm$^{-1}$: not shown in the table). \nD band is known to appear if the surface contains edges or defects where the three-fold symmetry of honeycomb lattice is broken~\\cite{rohtua}.\nThe D band signal was not detected in uPGS and HOPG within experimental errors indicating high crystallinity with immeasurably small amounts of domains and defects~\\cite{doi:10.1063\/1.369027}. \n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Intensity ratios of $I$(G)\/$I$(G') and $I$(D)\/$I$(G) in Raman spectra for various graphite materials. All the surfaces were cleaved before the measurements.}\\label{ratio}\n\\begin{tabular}{rccc}\n\\hline \\hline\nmaterial & nominal thickness & $I$(G)\/$I$(G')   & $I$(D)\/$I$(G)       \\\\ \\hline\nuPGS& $100~\\mu$m  & 3.2(1) & $<10^{-3}$ \\\\\n& $25~\\mu$m   & 3.1(1) & $<10^{-3}$ \\\\\n& $17~\\mu$m  & 3.1(1) & $<10^{-3}$  \\\\\n\\\\\nHOPG    &    ---     & 3.3(2) & $<10^{-3}$   \\\\\n\\\\\nZYX     &     ---    & 3.2(1) & 0.003(2)   \\\\\n\\\\\nGrafoil & 130~$\\mu$m & 3.3(1) & 0.015(4) \\\\\n& 250~$\\mu$m & 3.3(1) & 0.033(4)  \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Electrical resistivity measurement}\nWe made in-plane ($\\rho_{\\parallel}$) and out-of-plane ($\\rho_{\\perp}$) electrical resistivity measurements for uPGS samples of four different thicknesses, i.e., 10, 17, 25, and 100~$\\mu$m, in the temperature range between 2 and 300\\,K.\nThey were carried out by the 4-terminal method using the AC transport and resistance options of Physical Properties Measurement System (PPMS) of Quantum Design, Inc.\nThe typical sample size is $0.5\\times9$~mm$^{2}$ for the $\\rho_{\\parallel}$ measurement and $3\\times3$~mm$^{2}$ for the $\\rho_{\\perp}$ one. \nGold lead wires of 50~$\\mu$m in diameter were glued to the samples with rubber-based carbon paste~\\cite{ucc} which adheres strongly to graphite. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.92\\columnwidth]{logrhopara}%\n\\caption{Temperature dependencies of in-plane electrical resistivity $\\rho_{\\parallel}$ of uPGS with various thicknesses (solid circles) and cPGS-10$\\mu$m (open circles). The arrows indicate peak temperatures. Data of Grafoil, ZYX~\\cite{niimi:4448} and natural graphite~\\cite{PhysRev.95.22} are also plotted.\\label{logrhopara}}%\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.97\\columnwidth]{logrhoperp}%\n\\caption{Temperature dependencies of out-of-plane electrical resistivity $\\rho_{\\perp}$ of uPGS with various thicknesses (solid circles) and cPGS-10$\\mu$m (open circles). Data of Grafoil and ZYX are also plotted~\\cite{niimi:4448}.\\label{logrhoperp}}%\n\\end{figure}\n\nResults of the $\\rho_{\\parallel}$ measurement are shown in Fig.~\\ref{logrhopara}. \n$\\rho_{\\parallel}$ values of the uPGS samples are in between those of exfoliated graphites (Grafoil and ZYX) and natural graphite.\nImportantly, thinner uPGS has lower $\\rho_{\\parallel}$ in order of thickness. \nThis is consistent with the fact that thinner uPGS has better crystallinity. \nNote that the variation of $\\rho_{\\parallel}$ with thickness, 3--5 times, is larger than the variation of density, twice at most. \n\nIn exfoliated graphites it is known that there are two different conduction mechanisms.\nThe first is metallic which dominates the low temperature behavior of $\\rho_{\\parallel}$ and $\\rho_{\\perp}$, and the second is variable range hopping (VRH) which dominates high temperature behavior~\\cite{PhysRevB.25.4167,niimi:4448}.\nAs a result, there exists a peak temperature ($T_{\\mathrm{peak}}$) in $\\rho$ vs. $T$ around 20~K that separates the two behaviors as shown in Fig.~\\ref{logrhopara}.\nWe observed a similar $T$ dependence for uPGS but with higher $T_{\\mathrm{peak}}$ (indicated by the arrows in the figure) again in order of thickness. \nAs the thickness decreases, the $T$ dependence of $\\rho_{\\parallel}$ becomes weaker above $T_{\\mathrm{peak}}$ and stronger below $T_{\\mathrm{peak}}$.\nThe measured in-plane conductivity $\\sigma_{\\parallel}$ ($= 1\/\\rho_{\\parallel}(T)$) is well described by the following equation below 150\\,K: \n\\begin{equation}\n\\label{sigmaeq}\n\\sigma=\\frac{1}{\\rho_{0}+AT}+\\sigma_{0}^{\\mathrm{h}}\\exp\\left(-\\frac{T_{0}}{T}\\right)^{\\alpha},\n\\end{equation}\nwhere the first term corresponds to the metallic channel ($\\sigma_{\\mathrm{metal}}$) and the second term to the hopping one. \nWithin the VRH model,  \n\\begin{gather*}\nT_{0}=\\lambda^{3}\/D_{\\mathrm{F}}k_{\\mathrm{B}},\\\\\n\\alpha=1\/(d+1),\n\\end{gather*}\nfor hopping in $d$ spatial dimensions.\nHere $\\lambda$ and $D_{\\mathrm{F}}$ are the decay length of the localized electronic wave function and the density of states at the fermi energy, respectively.\nThe hopping is presumably between neighboring microcrystallites across domain boundaries.\nThinner uPGS has longer $\\lambda$ and smaller $\\rho_{0}$, the residual resistivity, presumably because of larger microcrystalline size and less crystalline defects. \nThis explains the thickness dependence of $T_{\\mathrm{peak}}$.\nIt is noted that in our analysis $\\alpha$ is a fitting parameter unlike the previous works~\\cite{PhysRevB.25.4167, niimi:4448}.\nThe fittings give $\\alpha \\approx 1.0$, which corresponds to a simple Arrhenius type conduction, for all the samples.\n\nFigure~\\ref{logrhoperp} shows results of the $\\rho_{\\perp}$ measurement.\nAgain $\\rho_{\\perp}$ at a fixed temperature varies in order of thickness but with an opposite sign to $\\rho_{\\parallel}$, i.e., thinner uPGS has larger $\\rho_{\\perp}$.\nThis is naturally understood as follows.\nGraphite is a layered material with very weak interlayer coupling based on the van der Waals interaction.\nThus the anisotropy of resistivity ($\\eta = \\rho_{\\perp}\/\\rho_{\\parallel} > $$10^2$~\\cite{ANDP:ANDP19153531806,PhysRev.95.22}) is so large that it can easily be reduced if the system has a mosaic angle spread or the wavy laminar structure. \nThe $\\rho_{\\perp}$ data  below 150\\,K can also be well fitted by Eq.~\\ref{sigmaeq} with $\\alpha\\approx 1\/2$, nearly one dimensional hopping, being consistent with the above argument on large $\\eta$ (see Fig.~\\ref{vrhperp2}).\nIf we make the same analysis for the previously reported $\\rho_{\\perp}$ data of Grafoil and ZYX in Refs.~\\cite{PhysRevB.25.4167, niimi:4448}, we obtain a similar result ($\\alpha=0.42$), although in those papers the data were analyzed assuming $\\alpha=1\/4$ (three dimensional hopping). \n\nWe note that uPGS of any thickness has a kink in the $T$ dependence of $\\rho_{\\perp}$ at $T \\approx$ 250~K above which the dependence changes randomly in every thermal cycle.\nThis results in a $\\pm$5\\% difference in resistivity at 300~K.\nThe mechanism behind this curious behavior is not known at present. \nThe morphology of uPGS with many microscopic inaccessible voids may be changed either by thermal expansion or by desorption\/adsorption of gas confined in the voids in every cooling and warming cycle. \nThe $T$ dependence of $\\rho_{\\perp}$ is excellently reproducible at $T < 250$~K for uPGS and in the whole $T$ range we studied for cPGS. \n\nWe comment on effects caused by the final compression to produce cPGS from uPGS from the viewpoint of electrical conductance. \nThe open symbols in Figs.~\\ref{logrhopara}~and~\\ref{logrhoperp} are results of $\\rho_{\\parallel}$ and $\\rho_{\\perp}$ measurements for cPGS-10$\\mu$m. \nCompared to uPGS-10$\\mu$m, the $\\rho_{\\parallel}$ value at room temperature is only slightly larger.\nHowever, it has a steeper variation down to $T_{\\mathrm{peak}}$, and $T_{\\mathrm{peak}}$ itself is lower.\nTherefore, the overall $T$ dependence of cPGS-10$\\mu$m is rather similar to those of the exfoliated graphites.\nThis would be a result of mixing between $\\rho_{\\parallel}$ and $\\rho_{\\perp}$ caused by the compression. \nThe same is true for $\\rho_{\\perp}$ where the cPGS-10$\\mu$m samples have much weaker $T$ variations.\nThis is again similar to the behavior of exfoliated graphite. \nIt should be noted that the magnitude of $\\rho_{\\perp}$ and sometimes even its $T$ dependence differ from sample to sample in the case of cPGS (two different samples are shown in Fig.~\\ref{logrhoperp}), presumably because the compression damages the laminar structure.  \nAlso, $\\alpha$ values scatter to a large extent from 0.26 to 0.42. \n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.97\\columnwidth]{vrhperp2}%\n\\caption{Temperature dependencies of the out-of plane conductivity $\\alpha_{\\perp}$ of various uPGS samples plotted as logarithm of $\\alpha_{\\perp}$ vs. $1\/T$. \nThe data can be fitted to Eq.~\\ref{sigmaeq}, a straight line with a slope of $\\alpha$ in this plot, much better with $\\alpha=1\/2$ (thin solid lines) than $\\alpha=1\/4$ expected from the 3D VRH model (see main text). \nThe inset is a schematic diagram of the one dimensional inter-plane hopping.\n\\label{vrhperp2}}%\n\\end{figure}\n\n\\section{Thermal conductivity measurement}\nUsually, for metallic samples, it is possible to estimate thermal conductivity ($\\kappa_\\mathrm{WF}$) from the electrical resistivity $\\rho$, which can more easily be measured, through the Wiedemann-Franz (WF) law~\\cite{ANDP:ANDP18531650802}: \n\\begin{equation}\\label{WFeq}\n\\kappa_{\\mathrm{WF}}=\\frac{L_0 T}{\\rho}, \\quad L_0=2.44\\times10^{-8}\\,\\mathrm{W}\\Omega\\mathrm{K}^{-2}.\n\\end{equation}\nHowever, in the case of semimetal such as graphite, Eq.~\\ref{WFeq} underestimates the true thermal conductivity ($\\kappa$) by several orders of magnitude at temperatures where the thermal conduction by phonons plays an important role~\\cite{PhysRevB.31.6721}.\nThus we have measured in-plane $\\kappa$ of uPGS-10$\\mu$m directly using Thermal Transport option of PPMS. \nThe sample of 7\\,mm wide and 8\\,mm long was glued on 4 gold-plated copper electrodes with silver paste.\nThe electrodes are fixed to a thin rectangular support rod made of Stycast 2850FT ($2\\times0.5\\times8$\\,mm$^{3}$).\nThe thermal conductance of the support rod is negligibly small at temperatures above 50\\,K and is less than half of the total conductance with sample at lower temperatures.\nIt was carefully measured beforehand and subtracted from the total.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.97\\columnwidth]{PGSTT2ai}%\n\\caption{Measured in-plane thermal conductivities $\\kappa$ of uPGS-10$\\mu$m (this work), HOPG~\\cite{PhysRevB.31.6721}, natural graphite~\\cite{PhysRev.95.1095}, Grafoil~\\cite{UHER1980445}, and ZYX~\\cite{UHER1980445}. \nShadow areas represent $\\kappa$ of copper with residual resistance ratios between 50--500 and aluminum alloys of different kinds (Aluminum 1100, 3003-F, 5083-O, 6061-T6, 6063-T5)~\\cite{marquardt2002cryogenic}.\nThermal conductivity estimated from electric conductivity using the Wiedemann-Franz law $\\kappa_\\mathrm{WF}$ is indicated by the broken or dash-dotted lines for each specimen~\\cite{PhysRev.95.22,PhysRevB.30.1080,niimi:4448}.  \n\\label{PGSTT2ai}}%\n\\end{figure}\n\nIn Fig.~\\ref{PGSTT2ai}, measured $\\kappa$ data of uPGS-10$\\mu$m (closed circles) are shown with those of natural graphite (open circles)~\\cite{PhysRev.95.1095}, HOPG (solid line)~\\cite{PhysRevB.31.6721}, Grafoil~\\cite{UHER1980445}, and ZYX ~\\cite{UHER1980445}. \n$\\kappa$ of uPGS-10$\\mu$m is more than one order of magnitude higher than those of ZYX and Grafoil in the whole $T$ range between 2 and 300\\,K.\nThis is a great advantage of this material indicating the longer mean-free path of phonons and thus the higher crystallinity. \nRemarkably, $T$ dependencies of the three kinds of graphite are quite similar to each other.\nA peak in $\\kappa (T)$ around 150\\,K for uPGS-10$\\mu$m corresponds to the onset of Umklapp scattering.\nIt is in between the peak temperature ($T_{\\mathrm{peak}} = 100$\\,K) of natural graphite and HOPG and those ($T_{\\mathrm{peak}} \\approx 200$\\,K) of Grafoil and ZYX.\nAt $30 \\leq T \\leq 100$\\,K, the $T$ dependence is $\\kappa \\propto T^{2.01(2)}$ as expected from two-dimensional phonon conductivity.\nIt turns to $\\kappa \\propto T^{2.55(2)}$ at lower temperatures ($2 \\leq T \\leq 30$\\,K) where the conductivity is lower than that of natural graphite by a factor of 4--10. \n\nCompared to typical metallic thermal conductors, uPGS-10$\\mu$m has larger $\\kappa$ than copper at $T>60$\\,K and than aluminum alloys at $T>40$\\,K. \nThe $T$ bounds, above which uPGS can transfer heat faster, are extended down to 40\\,K and 20\\,K, respectively, if we consider the thermal diffusivity $\\kappa\/C_\\mathrm{vol}$ owing to the small density of graphite. \nHere $C_\\mathrm{vol}$ is the volumetric specific heat. \nBecause the density of graphite is 25\\% of copper and 80\\% of aluminum, $C_\\mathrm{vol}$ is always lower than those metals at any $T$ below 300\\,K~\\cite{doi:10.1021\/ja01623a006,marquardt2002cryogenic}. \nThus uPGS can be advantageous even at lower $T$ for specific purposes which requires lighter weight and\/or smaller heat capacity. \n\nIn Fig.~\\ref{PGSTT2ai}, we also plotted $\\kappa_{\\mathrm{WF}}$ estimated from the measured $\\rho_{\\parallel}$ through the WF law (broken and dash-dotted lines). \nFor all types of the graphite materials, $\\kappa$ is much higher than $\\kappa_{\\mathrm{WF}}$. \n$\\kappa\/\\kappa_{\\mathrm{WF}}$ is $\\approx$500 at $T\\approx100$\\,K and slowly decreases with decreasing $T$ down to 3--4 at the lowest temperature. \nIn addition, $T$ dependencies of $\\kappa_{\\mathrm{WF}}$ are quite different from the measured ones.\n\nFinally, it is interesting to note that the ratios $\\kappa \/ \\kappa_{\\mathrm{WF}}$ are rather similar for different graphites, particularly among uPGS-10$\\mu$m, Grafoil, and ZYX through the whole $T$ range as shown in Fig.~\\ref{ratioplot}.\nIn this sense, electrical resistance measurement still provides a useful ``rough\" estimation of thermal conductance for a variety of graphite materials. \nFrom the fact, we believe that uPGS-10$\\mu$m should have the highest in-plane thermal conductivity among other PGSs though we did not directly measure their $\\kappa$. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.97\\columnwidth]{ratioplot}%\n\\caption{Ratio between the measured in-plane thermal conductivity and that estimated from the measured electrical resistivity through the Wiedemann-Franz law plotted as a function of temperature for uPGS-10$\\mu$m, HOPG~\\cite{PhysRevB.31.6721,PhysRev.95.22}, natural graphite~\\cite{PhysRev.95.1095,PhysRevB.30.1080}, Grafoil~\\cite{UHER1980445,niimi:4448}, and ZYX~\\cite{UHER1980445}.\n\\label{ratioplot}}%\n\\end{figure}\n\n\\section{Conclusions}\nPGS is a recently developed mass-producible pyrolytic graphite sheet with several potential applications including a light-weight and highly conducting thermal link at cryogenic temperatures. \nWe made electrical and thermal conductivity measurements at 2$\\leq T \\leq$300\\,K along with X-ray diffraction and Raman spectroscopy characterizations of uncompressed PGSs (uPGSs) of various thicknesses between 10 and 100\\,$\\mu$m. \nThe thinnest uPGS (uPGS-10$\\mu$m) has the highest in-plane thermal conductivity ($\\kappa$) because of the highest crystallinity. \nBy the same reason, uPGS-10$\\mu$m and its compressed version (cPGS-10$\\mu$m) have more than one order of magnitude higher $\\kappa$ than that of Grafoil, a commonly used flexible exfoliated graphite sheet, in the whole temperature range. \nIt is a better thermal conductor than copper at $T>$40--60\\,K and aluminum alloys at $T>$20--40\\,K as natural graphite (NG) crystal and highly oriented pyrolytic graphite (HOPG) are so due to large thermal conduction by phonons. \nSince it is difficult to machine NG and HOPG into arbitrary thickness, uPGS\/cPGS have a great advantage over any other materials for application as a thin and inflexible\/flexible cryogenic thermal link. \n\nWe also found that there is a general relationship between the thermal conductivity estimated from in-plane electrical conductivity through the Wiedemann-Franz law ($\\kappa_\\mathrm{WF}$) and $\\kappa$ in graphite family materials. \nThis is useful so that one can conveniently estimate thermal conductance of a cryogenic part made of those materials from the more easily obtained electrical conductance.\n\n\\section{Acknowledgements}\nWe are grateful to Yoshiya Sakaguchi, Hiroyuki Hase, and Makoto Nagashima of Automotive \\& Industrial Systems Company of Panasonic corporation for providing us the uPGS and cPGS samples. \n\nThis work was financially supported by Grant-in-Aid for Scientific Research (B) (Grant No.~15H03684), and Challenging Exploratory Research (Grant No.~15K13398) from JSPS.\nThe laser Raman microscope used in this work was supplied by MERIT program, The University of Tokyo.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n\\newcommand{\\mysubsection}[1]{\\subsection{#1}}\n\\newcommand{\\mysubsubsection}[1]{\\subsubsection{#1}}\n\n\\date{March 27, 2012}\n\n\\author{Martin Pelikan \n\\and\nMark W. Hauschild \n\\and\nPier Luca Lanzi}\n\n\\begin{document}\n\n\\begin{titlepage}\n\\setlength{\\parindent}{0pt}\n\n\\noindent\n\\includegraphics[width=5in]{medal2.eps}\n\\vspace*{0.075in}\n{\\color{myblue}\n\\hrule height 2pt\n}\n\\vspace*{0.5in}\n\n{\\bf\n\\textsf{{\\large\nTransfer Learning, Soft Distance-Based Bias, and the Hierarchical BOA}}\n}\n\n\\vspace*{0.25in}\n\n\\textsf{Martin Pelikan, Mark W. Hauschild, and Pier Luca Lanzi}\n\n\\vspace*{0.25in}\n\n\\textsf{MEDAL Report No. 2012004}\n\n\\vspace*{0.25in}\n\n\\textsf{March 2012}\n\n\\vspace*{0.25in}\n\n{\\bf \\textsf{Abstract}}  \n\n\\vspace*{0.075in}\n\n{\\small \\textsf{An automated technique has recently been proposed to transfer learning in the hierarchical Bayesian optimization algorithm (hBOA) based on distance-based statistics. The technique enables practitioners to improve hBOA efficiency by collecting statistics from probabilistic models obtained in previous hBOA runs and using the obtained statistics to bias future hBOA runs on similar problems. The purpose of this paper is threefold: (1)~test the technique on several classes of NP-complete problems, including MAXSAT, spin glasses and minimum vertex cover; (2)~demonstrate that the technique is effective even when previous runs were done on problems of different size; (3)~provide empirical evidence that combining transfer learning with other efficiency enhancement techniques can often yield nearly multiplicative speedups.}}\n\n\\vspace*{0.25in}\n\n{\\bf \\textsf{Keywords}}\n\n\\vspace*{0.075in}\n{\\small \\textsf{Transfer learning, inductive transfer, learning from experience, estimation of distribution algorithms, hierarchical Bayesian optimization algorithm, decomposable problems, efficiency enhancement.}}\n\n\\vfill\n\n\\noindent\n\\begin{minipage}{6in}\n{\\small \\textsf{Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)\\\\\nDepartment of Mathematics and Computer Science, 321 ESH\\\\\nUniversity of Missouri--St. Louis\\\\\nOne University Blvd.,\nSt. Louis, MO 63121\\\\\nE-mail: \\url{medal@medal-lab.org}\\\\\nWWW: \\url{http:\/\/medal-lab.org\/}\\\\}}\n\\end{minipage}\n\n\\end{titlepage}\n\n\\author{\n{\\bf Martin Pelikan}\\\\\nMissouri Estimation of Distribution Algorithms Laboratory (MEDAL)\\\\\nDept. of Mathematics and Computer Science, 320 ESH\\\\\nUniversity of Missouri in St. Louis\\\\\nOne University Blvd., St. Louis, MO 63121\\\\\n\\url{martin@martinpelikan.net}\\\\\n\\url{http:\/\/martinpelikan.net\/}\n\\and\n{\\bf Mark W. Hauschild}\\\\\nMissouri Estimation of Distribution Algorithms Laboratory (MEDAL)\\\\\nDept. of Mathematics and Computer Science, 321 ESH\\\\\nUniversity of Missouri in St. Louis\\\\\nOne University Blvd., St. Louis, MO 63121\\\\\n\\url{mwh308@umsl.edu}\n\\and\n{\\bf Pier Luca Lanzi}\\\\\nDipartimento di Elettronica e Informazione\\\\\nPolitecnico di Milano\\\\\nPiazza Leonardo da Vinci, 32\\\\\nI-20133 Milano, Italy\\\\\n\\url{pierluca.lanzi@polimi.it}\n}\n\n\\maketitle\n\n\\begin{abstract}\nAn automated technique has recently been proposed to transfer learning in the hierarchical Bayesian optimization algorithm (hBOA) based on distance-based statistics. The technique enables practitioners to improve hBOA efficiency by collecting statistics from probabilistic models obtained in previous hBOA runs and using the obtained statistics to bias future hBOA runs on similar problems. The purpose of this paper is threefold: (1)~test the technique on several classes of NP-complete problems, including MAXSAT, spin glasses and minimum vertex cover; (2)~demonstrate that the technique is effective even when previous runs were done on problems of different size; (3)~provide empirical evidence that combining transfer learning with other efficiency enhancement techniques can often yield nearly multiplicative speedups.\n\\end{abstract}\n\n{\\bf Keywords:} Transfer learning, inductive transfer, learning from experience, estimation of distribution algorithms, hierarchical Bayesian optimization algorithm, decomposable problems, efficiency enhancement.\n\n\n\\mysection{Introduction}\nEstimation of distribution algorithms (EDAs)~\\cite{Hauschild:11c,Larranaga:02,Pelikan:02,Pelikan:12b} guide the search for the optimum by building and sampling probabilistic models of candidate solutions. The use of probabilistic models in EDAs provides a basis for incorporating prior knowledge about the problem and learning from previous runs in order to solve new problem instances of similar type with increased speed, accuracy and reliability~\\cite{Pelikan:book,Hauschild:12}. However, much prior work in this area was based on hand-crafted constraints on probabilistic models~\\cite{Muhlenbein:99a,Muhlenbein:02,Baluja:06,Schwarz:00*} which may be difficult to design or even detrimental to EDA efficiency and scalability~\\cite{Hauschild:09c}. Recently, Pelikan and Hauschild~\\cite{Pelikan:12} proposed an automated technique capable of learning from previous runs of  the hierarchical Bayesian optimization algorithm (hBOA) in order to improve efficiency of future hBOA runs on problems of similar type. The basic idea of the approach was to (1)~design a distance metric on problem variables that correlates with the expected strength of dependencies between the variables, (2)~collect statistics on hBOA models with respect to the values of the distance metric, and (3)~use the collected statistics to bias model building in hBOA when solving future problem instances of similar type. While the distance metric is strongly related to the problem being solved, the aforementioned study~\\cite{Pelikan:12} described a rather general metric that can be applied to practically any problem with the objective function represented by an additively decomposable function. However, the prior study~\\cite{Pelikan:12}  evaluated the proposed technique on only two classes of problems and it did not demonstrate several key features of this technique.\n\n\\begin{sloppy}\nThe purpose of this paper is threefold: (1)~Demonstrate the technique from ref.~\\cite{Pelikan:12} on other classes of challenging optimization problems, (2)~demonstrate the ability of this technique to learn from problem instances of one size in order to introduce bias for instances of another size, and (3)~demonstrate the potential benefits of combining this technique with other efficiency enhancement techniques, such as sporadic model building~\\cite{DBLP:journals\/gpem\/PelikanSG08}. As test problems the paper considers several classes of NP-complete additively decomposable problems, including MAXSAT, three-dimensional Ising spin glass, and minimum vertex cover. The new results together with the results published in prior work~\\cite{Pelikan:12} provide strong evidence of the broad applicability and great potential of this technique for learning from experience (transfer learning) in EDAs.\n\\end{sloppy}\n\nThe paper is organized as follows. Section~\\ref{section-hboa} outlines hBOA. Section~\\ref{section-transfer-learning} discusses efficiency enhancement of estimation of distribution algorithms using inductive transfer with main focus on hBOA and the distance-based bias~\\cite{Pelikan:12}. Section~\\ref{section-experiments} presents and discusses experimental results. Section~\\ref{section-conclusions} summarizes and concludes the paper.\n\n\\mysection{Hierarchical BOA}\n\\label{section-hboa}\nThe hierarchical Bayesian optimization algorithm (hBOA)~\\cite{Pelikan:book,Pelikan:01*} works with a population of candidate solutions represented by fixed-length strings over a finite alphabet. In this paper, candidate solutions are represented by $n$-bit binary strings. The initial population of binary strings is generated at random according to the uniform distribution over candidate solutions. Each iteration starts by selecting promising solutions from the current population; here binary tournament selection without replacement is used. Next, hBOA (1)~learns a Bayesian network with local structures~\\cite{Chickering:97} for the selected solutions and (2)~generates new candidate solutions by sampling the distribution encoded by the built network. To maintain useful diversity in the population, the new candidate solutions are incorporated into the original population using restricted tournament selection (RTS)~\\cite{Harik:95a}. The run is terminated when termination criteria are met. In this paper, each run is terminated either when the global optimum is found or when a maximum number of iterations is reached. \n\nhBOA represents probabilistic models of candidate solutions by Bayesian networks with local structures~\\cite{Chickering:97,Friedman:99}. A Bayesian network is defined by two components: (1)~an acyclic directed graph over problem variables specifying direct dependencies between variables and (2)~conditional probabilities specifying the probability distribution of each variable given the values of the variable's parents. A Bayesian network encodes a joint probability distribution as $p(X_1,\\ldots,X_n)=\\prod_{i=1}^n p(X_i|\\Pi_i)$ where $X_i$ is the $i$th variable (string position) and $\\Pi_i$ are the parents of $X_i$ in the underlying graph. \n\nTo represent conditional probabilities of each variable given the variable's parents, hBOA uses decision trees~\\cite{Pelikan:01*,Chickering:97}. Each internal node of a decision tree specifies a variable, and the subtrees of the node correspond to the different values of the variable. Each leaf of the decision tree for a particular variable defines the probability distribution of the variable given a condition specified by the constraints given by the path from the root of the tree to this leaf (constraints are given by the assignments of the variables along this path).  \n\n\nTo build probabilistic models, hBOA typically uses a greedy algorithm that initializes the decision tree for each problem variable $X_i$ to a single-node tree that encodes the unconditional probability distribution of $X_i$. In each iteration, the model building algorithm tests how much a model would improve after splitting each leaf of each decision tree on each variable that is not already located on the path to the leaf. The algorithm executes the split that provides the most improvement, and the process is repeated until no more improvement is possible.\nModels are evaluated using the Bayesian-Dirichlet (BDe) metric with penalty for model complexity, which estimates the goodness of a Bayesian network structure given data $D$ and background knowledge $\\xi$ as $p(B|D,\\xi) = c p(B|\\xi) p(D|B,\\xi),$\nwhere $c$ is a normalization constant ~\\cite{Chickering:97,Cooper:92}. The Bayesian-Dirichlet metric estimates the term $p(D|B,\\xi)$ by combining the observed and prior statistics for relevant combinations of variables~\\cite{Chickering:97}.  \nTo favor simpler networks to the more complex ones, the prior probability $p(B|\\xi)$ is often set to decrease exponentially fast with respect to the description length of the network's parameters~\\cite{Pelikan:book,Friedman:99}.\n\n\\mysection{Learning from Experience using Distance-Based Bias}\n\\label{section-transfer-learning}\n\n\nIn hBOA and other EDAs based on complex probabilistic models, building an accurate probabilistic model is crucial to the success~\\cite{Larranaga:02,Pelikan:02,Hauschild:09c,Lima:11}. However, building complex probabilistic models can be  time consuming and it may require rather large populations of solutions~\\cite{Larranaga:02,Pelikan:02}. That is why much effort has been put into enhancing efficiency of model building in EDAs and improving quality of EDA models even with smaller populations~\\cite{Hauschild:12,Muhlenbein:02,Baluja:06,Hauschild:08,Hauschild:09b}.\nLearning from experience~\\cite{Pelikan:book,Hauschild:12,Pelikan:12,Hauschild:08,Hauschild:09b} represents one approach to addressing this issue.\n\nThe basic idea of learning from experience is to gather information about the problem by examining previous runs of the optimization algorithm and to use the obtained information to bias the search on new problem instances. The use of bias based on the results of other learning tasks is also commonplace in machine learning where it is referred to as {\\em inductive transfer} or {\\em transfer learning}~\\cite{Pratt:91,Caruana:97}. \n Since learning model structure is often the most computationally expensive task in model building, learning from experience often focuses on identifying regularities in model structure and using these regularities to bias structural learning in future runs. \n\nAnalyzing probabilistic models built by hBOA and other EDAs is straightforward. The more challenging facet of implementing learning from experience in practice is that one must make sure that the collected statistics are meaningful with respect to the problem being solved. \nThe key to make the learning from experience work is to ensure that the pairs of variables are classified into a set of {\\em categories} so that the pairs in each category have a lot in common and can be expected to be either correlated or independent simultaneously~\\cite{Pelikan:12}. This section describes one approach to doing that~\\cite{Pelikan:12}, in which pairs of variables are classified into categories based on a predefined distance metric on variables.\n\n\\vspace*{-1.5ex}\n\\mysubsection{Distance Metric for Additively Decomposable Functions}\n\\vspace*{-0.5ex}\nFor many optimization problems, the objective function (fitness function) can be expressed as an additively decomposable function (ADF):\n\n\\vspace*{-1.8ex} \n\\begin{equation}\nf(X_1,\\ldots,X_n) = \\sum_{i=1}^m f_i(S_i),\n\\end{equation}\n\\vspace*{-1.8ex}\n\n\\noindent\nwhere $(X_1,\\ldots,X_n)$ are problem's decision variables (string positions), $f_i$ is the $i$th subfunction, and $S_i\\subset \\{X_1,X_2,\\ldots,X_n\\}$ is the subset of variables contributing to $f_i$. While there may often exist multiple ways of decomposing the problem using additive decomposition, one would typically prefer decompositions that minimize the sizes of subsets $\\{S_i\\}$. Note that the difficulty of ADFs is not fully determined by the order of subproblems, but also by the definition of the subproblems and their interaction; even with subproblems of order only 2 or 3, the problem can be NP-complete. \n\nThe definition of a distance between two variables of an ADF used in this paper as well as ref.~\\cite{Pelikan:12} follows the work of Hauschild et al.~\\cite{Hauschild:12,Hauschild:09c,Hauschild:08}. Given an ADF, we define the distance between two variables using a graph $G$ of $n$ nodes, one node per variable. For any two variables $X_i$ and $X_j$ in the same subset $S_k$, we create an edge in $G$ between the nodes $X_i$ and $X_j$. Denoting by $l_{i,j}$ the number of edges along the shortest path between $X_i$ and $X_j$ in $G$ (in terms of the number of edges), we define the distance between two variables as\n\n\\vspace*{-1.6ex}\n\\[\nD(X_i,X_j) = \n\\left\\{\n\\begin{array}{ll}\nl_{i,j} & \\mbox{if a path between $X_i$ and $X_j$ exists,}\\\\\nn & \\mbox{otherwise.}\n\\end{array}\n\\right.\n\\]\n\\vspace*{-1.8ex}\n\n\\noindent\nThe above distance measure makes variables in the same subproblem close to each other, whereas for the remaining variables, the distances correspond to the length of the chain of subproblems that relate the two variables. The distance is maximal for variables that are completely independent (the value of a variable does not influence the contribution of the other variable in any way). \n\nSince interactions between problem variables are encoded mainly in the subproblems of the additive problem decomposition, the above distance metric should typically correspond closely to the likelihood of dependencies between problem variables in probabilistic models discovered by EDAs. Specifically, the variables located closer with respect to the metric should more likely interact with each other. This observation has been confirmed with numerous experimental studies across a number of important problem domains from spin glasses distributed on a finite-dimensional lattice~\\cite{Hauschild:09c,Pelikan:12} to NK landscapes~\\cite{Pelikan:12}. \n\n\\vspace*{-0.7ex}\n\\mysubsection{Distance-Based Bias Based on Previous Runs of hBOA}\n\\begin{sloppy}\nThis section describes the approach to learning from experience developed by Pelikan and Hauschild~\\cite{Pelikan:12} inspired mainly by the work of Hauschild et al.~\\cite{Hauschild:12,Hauschild:08,Hauschild:09b}. \nLet us assume a set $M$ of hBOA models from prior hBOA runs on similar problems. Before applying the bias based on prior runs in hBOA, the models in $M$ are first processed to generate data that will serve as the basis for introducing the bias. The processing starts by analyzing the models in $M$ to determine the number $s(m,d,j)$ of splits on any variable $X_i$ such that $D(X_i,X_j)=d$ in a decision tree $T_j$ for variable $X_j$ in a model $m\\in M$. Then, the values $s(m,d,j)$ are used to compute the probability $P_k(d,j)$ of a $k$th split on a variable at distance $d$ from $X_j$ in a dependency tree $T_j$ given that $k-1$ such splits were already performed in $T_j$:\n\n\\vspace*{-1.3ex}\n\\begin{equation}\nP_k(d,j) = \\frac{\\left|\\{m\\in M: s(m,d,j)\\geq k\\}\\right|}{\\left|\\{m\\in M: s(m,d,j)\\geq k-1\\}\\right|}\\cdot\n\\end{equation}\n\\vspace*{-1.3ex}\n\\end{sloppy}\n\n\n\\noindent\nRecall that the BDe metric for evaluating the quality of probabilistic models in hBOA contains two parts: (1) the prior probability $p(B|\\xi)$ of the network structure $B$, and (2) the posterior probability $p(D|B,\\xi)$ of the data (population of selected solutions) given $B$. Pelikan and Hauschild~\\cite{Pelikan:12} proposed to use the prior probability distribution $p(B|\\xi)$ to introduce a bias based on distance-based statistics from previous hBOA runs represented by $P_k(d,j)$ by setting\n\\vspace*{-1.3ex}\n\\begin{equation}\n\\label{eq-pb-bias}\np(B|\\xi) = c \\prod_{d=1}^n \\prod_{j=1}^n \\prod_{k=1}^{n_s(d,j)} P^{\\kappa}_k(d,j),\n\\end{equation}\n\\vspace*{-1.8ex}\n\n\\noindent \nwhere $n_s(d,j)$ denotes the number of splits on any variable $X_i$ in $T_j$ such that $D(X_i,X_j)=d$, $\\kappa>0$ is used to tune the strength of bias (the strength of bias increases with $\\kappa$), and $c$ is a normalization constant.\nSince log-likelihood is typically used to evaluate model quality, when evaluating the contribution of any particular split, the change of the prior probability of the network structure can still be done in constant time. \n\n\n\\mysection{Experiments}\n\\label{section-experiments}\n\n\\mysubsection{Test Problems and Experimental Setup}\nThe experiments were done for three problem classes known to be difficult for most genetic and evolutionary algorithms:\n\\begin{inparaenum}[(1)]\n\\item Three-dimensional Ising spin glasses were considered with $\\pm J$ couplings and periodic boundary conditions~\\cite{Pelikan:06,young1998}; two problem sizes were used, $n=6\\times6\\times6=216$ spins and $n=7\\times7\\times7=343$ spins with 1,000 unique problem instances for each $n$. \n\\item Minimum vertex cover was considered for random graphs of fixed ratio $c$ of the number of edges and number of nodes~\\cite{Pelikan:07b,Weigt:01}; two ratios ($c=2$ and $c=4$) and two problem sizes ($n=150$ and $n=200$) were used with 1,000 unique problem instances for each combination of $c$ and $n$. \n\\item MAXSAT was considered for mapped instances of graph coloring with graphs created by combining regular ring lattices (with probability $1-p$) and random graphs (with probability $p$)~\\cite{Pelikan:03*,Gent:99}; 100 unique problem instances of $n=500$ bits (propositions) were used for each considered value of $p$, from $p=2^{-8}$ (graphs nearly identical to a regular ring lattice) to $p=2^{-1}$ (graphs with half of the edges random).\n\\end{inparaenum}\nFor more information about the test problems, we refer the reader to refs.~\\cite{Pelikan:06,Pelikan:07b,Pelikan:03*}.\n\nThe maximum number of iterations for each problem instance was set to the number of bits in the problem; according to preliminary experiments, this upper bound was sufficient.\nEach run was terminated either when the global optimum was found, when the population consisted of copies of a single candidate solution, or when the maximum number of iterations was reached. For each problem instance, we used bisection~\\cite{Pelikan:book,Sastry:01c} to ensure that the population size was within $5\\%$ of the minimum population size to find the optimum in 10 out of 10 independent runs. \nBit-flip hill climbing (HC)~\\cite{Pelikan:book} was incorporated into hBOA to improve its performance on all test problems except for the minimum vertex cover; HC was used to improve every solution in the population. For minimum vertex cover, a repair operator based on ref.~\\cite{Pelikan:07b} was incorporated instead.\nThe strength of the distance-based bias was tweaked using $\\kappa\\in\\{1,3,5,7,9\\}$.\n\n\nTo ensure that the same problem instances were not used for defining the bias as well as for testing it, 10-fold crossvalidation was used when evaluating the effects of distance-based bias derived from problem instances of the same size. \nFor each set of problems (by a set of problems we mean a set of random problem instances generated with one specific set of parameters), problem instances were randomly split into 10 equally sized subsets. In each round of crossvalidation, 1 subset of instances was left out and hBOA was run on the remaining 9 subsets of instances. The runs on the 9 subsets produced models that were analyzed in order to obtain the probabilities $P_k(d,j)$ for all $d$, $j$, and $k$. The bias based on the obtained values of $P_k(d,j)$ was then used in hBOA runs on the remaining subset of instances. The same procedure was repeated for each subset; overall, 10 rounds of crossvalidation were performed for each set of instances. When evaluating the effects of distance-based bias derived from problem instances of {\\em smaller} size, we did not use crossvalidation because in this case all runs had to be done on different problem instances (of different size). Most importantly, in every experiment, models used to generate statistics for hBOA bias were obtained from hBOA runs on {\\em different} problem instances.\nWhile the experiments were performed across a variety of computer architectures and configurations, the base case with no bias and the case with bias were always both run on the same computational node; the results of the two runs could therefore be compared against each other with respect to the actual CPU (execution) time. \n\n\n\nTo evaluate hBOA performance, we focus on the multiplicative speedup with respect to the execution time per run; the speedup is defined as a multiplicative factor by which the execution time improves with the distance-based bias compared to the base case. For example, an execution-time speedup of $2$ indicates that the bias allowed hBOA to find the optimum using only half the execution time compared to the base case without the bias. We also report the percentage of runs for which the execution time was strictly improved (shown in parentheses after the corresponding average multiplicative speedup). \n\nIn addition to the speedups achieved for various values of $\\kappa$, we examine the ability of the distance-based bias based on prior runs to apply across a range of problem sizes; this is done by using previous runs on instances of one size to bias runs on instances of another size. Since for MAXSAT, we only used instances of one size, this facet was only examined for the other two problem classes. \n\nFinally, we examine the combination of the distance-based bias based on prior runs and the sporadic model building~\\cite{DBLP:journals\/gpem\/PelikanSG08}. Specifically, we apply sporadic model building on its own using the model-building delay of $\\sqrt{n}\/2$ as suggested by ref.~\\cite{DBLP:journals\/gpem\/PelikanSG08}, and then we carry out a similar experiment using both the distance-based bias as well as the sporadic model building, recording the speedups with respect to the base case. Ideally, we would expect the speedups from the two sources to multiply. Due to the time requirements of solving MAXSAT, the combined effects were studied only for the remaining two problem classes. \n\n\n\n\\vspace*{-0.7ex}\n\\mysubsection{Results}\n\\vspace*{-0.48ex}\n\nThe results presented in tables~\\ref{table-sg-results}, \\ref{table-mvc-results} and~\\ref{table-maxsat-results} confirm the observation from ref.~\\cite{Pelikan:12} that the stronger the bias the greater the benefits, at least for the examined range of $\\kappa\\in\\{1,3,5,7,9\\}$ and most problem settings; that is why in the remainder of this discussion we focus on $\\kappa=9$. In all cases, the distance-based bias yielded substantial speedups of about~$1.2$ to~$3.1$. Best speedups were obtained for the minimum vertex cover. In all cases, performance on at least about $70\\%$ problem instances was strictly improved in terms of execution time; in most cases, the improvements were observed in a much greater majority of instances. The speedups were substantial even when the bias was based on prior runs on problem instances of different, smaller size; in fact, the speedups obtained with such a bias were nearly identical to the speedups with the bias based on the instances of the same size. The results thus provide clear empirical evidence that the distance-based bias is applicable even when the problem instances vary in size, which was argued~\\cite{Pelikan:12} to be one of the main advantages of the distance-based bias over prior work in the area but was not demonstrated. Finally, the results show the nearly multiplicative effect of the distance-based bias and sporadic model building, providing further support for the importance of the distance-based bias; the combined speedups ranged from about 4 to more than 11. \n\n\\input{tables}\n\n\\mysection{Summary and Conclusions}\n\\label{section-conclusions}\nThis paper extended the prior work on efficiency enhancement of the hierarchical Bayesian optimization algorithm (hBOA) using a distance-based bias derived from prior hBOA runs~\\cite{Pelikan:12}.  \nThe paper demonstrated that (1) the distance-based bias yields substantial speedups on several previously untested classes of challenging, NP-complete problems, (2) the approach is applicable even when prior runs were executed on problem instances of different size, and (3) the approach can yield nearly multiplicative speedups when combined with other efficiency enhancement techniques. In summary, the results presented in this paper together with the prior work~\\cite{Pelikan:12} provide clear evidence that learning from experience using a distance-based bias has a great potential to improve efficiency of hBOA in particular and estimation of distribution algorithms (EDAs) in general.\n\n\nSeveral topics are of central importance for future work. \nThe approach should be adapted to other model-directed optimization techniques, including other EDAs\nand genetic algorithms with linkage learning. The approach should also be modified to introduce\nbias on problems that cannot be formulated using an additive decomposition in a straightforward\nmanner or such a decomposition is not practical. \nFinally, it is important to study the limitations of the proposed approach, and create\ntheoretical models to automatically tune the strength of the bias and predict expected speedups.\n\n\\section*{Acknowledgments} \nThis project was sponsored by the National Science Foundation under grants ECS-0547013 and IIS-1115352, and by the Univ. of Missouri--St. Louis through the High Performance Computing Collaboratory sponsored by Information Technology Services. Most experiments were performed on the Beowulf cluster maintained by ITS at the Univ. of Missouri in St. Louis and the HPC resources at the University of Missouri Bioinformatics Consortium. 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 National Science Foundation.\n\n\\vspace*{-1.5ex}\n\\bibliographystyle{splncs}\n\\begin{small}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHiggs inflation \\cite{Bezrukov:2007ep,Salopek:1988qh}\n\\cite{Bezrukov:2008ej,Barvinsky:2008ia,DeSimone:2008ei,Bezrukov:2009db,Barvinsky:2009fy,Barvinsky:2009ii,Bezrukov:2013fka} is perhaps the most\neconomical approach to cosmological inflation: it identifies the only\nknown scalar field in the Standard Model (SM) -- the Higgs boson -- with\nthe inflaton, the scalar field that drives the inflationary expansion\nin the very early universe. The action for Higgs inflation\nfeatures a non-minimal coupling of the Higgs field to\nthe Ricci scalar. If the non-minimal coupling $\\xi$ is large, it leads to\na successful period of chaotic inflation, producing primordial\npower spectra that fit the observational bounds~\\cite{Ade:2013uln}.\nHence, Higgs inflation can be considered as a \"natural\" scenario,\nsince it does not seem to require the introduction of new\nphysics to explain the inflationary expansion of the universe.\n\nHowever, the naturalness of the Higgs inflation scenario has been\nunder debate. Refs.~\\cite{Barbon:2009ya,Burgess:2009ea} used\npower-counting techniques to determine the energy scale $\\Lambda$\nat which perturbation theory breaks down, thus determining the range\nof validity of Higgs inflation. It was claimed that this cutoff scale\n$\\Lambda$ lies dangerously close to the energy scale of inflation,\nthereby questioning the naturalness of Higgs inflation. Since then\nmany works have appeared claiming that Higgs inflation is\n\"natural\"~\\cite{Lerner:2009na,Ferrara:2010in}\nor \"unnatural\"~\\cite{Burgess:2010zq,Hertzberg:2010dc}.\nPerhaps the most complete treatment has been done in Ref.~\\cite{Bezrukov:2010jz}.\nIt was found that $\\Lambda$ is generally field dependent and lies above the\ntypical energy scales in different regions, such that the perturbative\n(semiclassical) expansion is valid in Higgs inflation.\n\nAlthough there seems to be a consensus about the cutoff scale\nas computed in Ref.~\\cite{Bezrukov:2010jz} (however,\nsee the recent work~\\cite{George:2013iia}~\\footnote{\nThe results presented in Ref.~\\cite{George:2013iia}\nuse different techniques and arrive at results that are consistent with\nthose presented in this paper and earlier in Ref.~\\cite{Weenink:2013oqa}.}),\nwe revisit the computation of the cutoff in this work.\nThe reason is that there are some important aspects\nthat have not been fully taken into account. The most important\naspect is that General Relativity contains a large diffeomorphism symmetry,\nwhich when truncated resembles the symmetry of a gauge theory.\nThis means that, like in QED, some of the degrees of freedom in the action are\nactually not physical. As a consequence, some of\nthe interaction vertices obtained after a na\\\"\\i ve perturbative\nexpansion of the action are gauge dependent, and any conclusion that one arrives at\nby using these vertices can be a gauge artifact.\nMoreover, it is possible that there are\nadditional vertices that conspire to cancel dominant\nperturbative contributions. Accounting for these aspects can raise the cutoff scale.\nEven though it is possible to determine the physical cutoff scale\nwithin a gauge dependent formulation,\nby far the simplest and most reliable way\nto determine this scale is to use the physical vertices,\nwhich can be obtained from the perturbative\naction in a manifestly gauge invariant way (an alternative is to completely fix the gauge freedom\nand take account of contributions from {\\it all} vertices).\nThis formulation of the action in terms of \\textit{gauge invariant perturbations}\nhas been found for a non-minimally coupled scalar field\nup to third order in perturbations\nin Refs.~\\cite{Weenink:2010rr,Prokopec:2012ug,Prokopec:2013zya}.\nIn this work we use these previously found results\nin order to demonstrate that the cutoff scale for physical, that is, gauge\ninvariant perturbations is always $\\geq M_P$.\nTo be more precise, we find that\n\\begin{eqnarray}\n\\left(\\frac{\\Lambda}{a}\\right)_J \\gtrsim \\sqrt{M_P^2+\\xi\\phi^2}\n\\,,\\qquad\n\\left(\\frac{\\Lambda}{a}\\right)_E  \\gtrsim M_P\n\\,,\n\\label{cutoffscalesJordanEinstein}\n\\end{eqnarray}\nwhere $a_J$ and $\\phi$ are the background scale factor and scalar field\nin the Jordan frame, and $a_E$ is the Einstein frame scale factor.\nThe extra scale factors have been overlooked in previous computations\nof the cutoff scale. They appear since $\\Lambda^{-1}$ is a comoving scale,\nand therefore in an expanding universe the corresponding physical length scale,\n$a\/\\Lambda$, always includes a scale factor $a$.\nThe cutoffs in Jordan and Einstein frame are different, simply\nbecause $\\Lambda$ is a dimensionful quantity which differs between\nthe frames, just like the effective Planck mass.\nIf $M_P$ is the energy scale where quantum gravity kicks in\nin the Einstein frame, then $M_{P,J}\\equiv \\sqrt{M_P^2+\\xi\\phi^2}$ can\nbe identified with the scale of quantum gravity in the Jordan frame.\nThus in either frame the perturbative (semiclassical) treatment is\nvalid all the way up to the scale at which gravity becomes strong.\nThis means that Higgs inflation is perfectly natural and that\nno new physics is necessary to explain the inflationary expansion\nof the universe and the anisotropies in the CMB.\n\nIn order to arrive at the physical cutoff scales~\\eqref{cutoffscalesJordanEinstein},\nwe first briefly discuss Higgs inflation and physically equivalent\nframes in section~\\ref{sec:Higgs inflation}. In section~\\ref{sec: the naturalness debate}\nwe review the naturalness debate. In section~\\ref{sec: gauge invariance}\nwe discuss the concept of gauge and frame dependence in cosmology\nand quote from Ref.~\\cite{Prokopec:2013zya}\nthe action for gauge and frame invariant cosmological perturbations.\nFinally we compute the cutoff scale for physical\nperturbations in section~\\ref{sec: cutoffscale} and conclude in section~\\ref{sec:Discussion}.\n\n\n\\section{Higgs inflation}\n\\label{sec:Higgs inflation}\n\nUnfortunately, inflation in the pure SM\nhas been ruled out by observations of the CMB, which have put tight constraints on\nthe inflationary potential. However, these constraints\ncan be greatly relaxed when the Higgs field $\\mathcal{H}$ is quadratically coupled\nto the Ricci scalar $R$ by a large non-minimal coupling $\\xi \\sim 10^4$\n\\cite{Futamase:1987ua,Fakir:1990eg,Komatsu:1999mt,Tsujikawa:2004my}.\nThe pure scalar and gravitational part of the action reads\n\\begin{equation}\nS = \\int d^4x \\sqrt{-g}\n\\biggl\\{\n\\frac12 M_P^2 R +\\xi\\mathcal{H}^{\\dagger}\\mathcal{H} R\n - g^{\\mu\\nu} (\\partial_{\\mu}\\mathcal{H})^{\\dagger}(\\partial_{\\nu}\\mathcal{H})\n -\\lambda \\left(\\mathcal{H}^{\\dagger}\\mathcal{H}-\\frac{v^2}{2}\\right)^{\\!2}\n\\biggr\\}\n\\,.\n\\label{SMHiggsaction}\n\\end{equation}\nHere the metric signature is ${\\rm sign}[g_{\\mu\\nu}]=(-,+,+,+)$, the Higgs self-coupling is\n$\\lambda$ and the Higgs vacuum expectation value (vev) is $v=246~\\rm{GeV}$.\nAlthough the complex doublet $\\mathcal{H}$ contains four degrees of freedom,\nthree of those can be absorbed by the gauge bosons (not shown in Eq.~\\eqref{SMHiggsaction}).\nIn unitary gauge the remaining scalar degree of freedom with non-zero vev\ncan be parametrized by $\\mathcal{H}=(0,\\Phi\/\\sqrt{2})$,\nsuch that the scalar action takes the Jordan frame form\n\\begin{equation}\nS = \\frac12 \\int d^4x \\sqrt{-g}\n\\Bigl\\{\nR F(\\Phi) -g^{\\mu\\nu} (\\partial_{\\mu}\\Phi)(\\partial_{\\nu}\\Phi)\n-\\frac{\\lambda}{2}(\\Phi^2-v^2)^2\n\\Bigr\\}\n\\,,\n\\label{Jordanframeaction}\n\\end{equation}\nwith $F(\\Phi)=M_P^2+\\xi\\Phi^2$. The claim is now that\na successful period of (chaotic) inflation is possible\nwhen the non-minimal coupling parameter is large, $\\xi \\sim 10^4$.\nSuccessful means that the model's predictions for the primordial\npower spectra of scalar and tensor perturbations\nagree with the observational constraints on these spectra.\nThe current 1$\\sigma$ constraints on the scalar spectral index\nand the tensor-to-scalar ratio are $n_s = 0.9603\\pm 0.0073$\nand $r < 0.11$, respectively~\\cite{Ade:2013uln}. On the theory\nside, it is known that in a minimally coupled model ($\\xi=0$)\nboth $n_s-1$ and $r$ are proportional to slow-roll parameters.\nMoreover, since the Hubble parameter $H$ does not directly couple\nto the time dependent expectation value of the scalar field\n$\\langle\\Phi\\rangle =\\phi(t)$ in the Friedmann equations,\nit is possible to express the slow-roll parameters in terms\nof the slope of the potential. Hence, in a minimally coupled\nmodel the shape of the scalar potential provides an intuition\nfor the successful realization of inflation. For instance,\na quartic potential is excluded by CMB observations~\\cite{Ade:2013uln}\nbecause it is insufficiently flat in the inflationary regime,\nwhich supports the previously mentioned statement that inflation is\nnot possible in the pure (\\textit{i.e.} minimally coupled) SM.\n\nUnfortunately, we do not have the luxury of such an intuition\nin the case of (large) non-minimal coupling $\\xi \\gg 1$.\nThe mixing of the gravitational and matter part of the action\nprevents us from expressing the slow-roll parameters in terms\nof the potential. Worse, it is not clear what are the slow-roll\nparameters in a non-minimally coupled model,\nthat is, what are the parameters that remain small\nduring the inflationary period, and neither is it clear how\nthe primordial power spectra depend on these small parameters.\nThe most straightforward way to see whether or not Higgs inflation\nworks is therefore to compute the primordial power spectra\nfor non-minimal coupling from scratch, which involves a derivation of the quadratic\naction for scalar and tensor perturbations in the Jordan frame~\\cite{Weenink:2010rr}.\nFortunately we can avoid this exercise\nby making use of a clever trick. If we redefine the metric,\nscalar field and scalar potential in the action as follows~\\cite{Weenink:2010rr},\n\\begin{align}\ng_{\\mu\\nu,E}&=\\frac{F}{M_P^2} g_{\\mu\\nu}\n\\nonumber\\\\\n\\left(\\frac{d\\Phi_E}{d\\Phi}\\right)^2\n&=\\frac{M_P^2}{F}\\left(1+\\frac{3}{2}\\frac{F^{\\prime 2}}{F}\\right)\n\\nonumber\\\\\nV_E(\\Phi_E)&=\\frac{M_P^4}{F^2}V(\\Phi)\n\\,,\n\\label{Conformaltransformationmetricscalar}\n\\end{align}\nwhere $F=F(\\Phi)$, $F'=dF\/d\\Phi$ and $V(\\Phi)=\\frac14 \\lambda (\\Phi^2-v^2)^2$,\nwe find that the action \\eqref{Jordanframeaction} becomes\n\\begin{equation}\nS_E = \\frac12 \\int d^4x \\sqrt{-g_E}\n\\Bigl\\{\nM_P^2 R_E -g_E^{\\mu\\nu} \\partial_{\\mu}\\Phi_E\\partial_{\\nu}\\Phi_E-2V_E(\\Phi_E)\n\\Bigr\\}\n\\,.\n\\label{Einsteinframeaction}\n\\end{equation}\nSince the metric has been rescaled by a factor it is commonly said\nthat we are in another conformal frame; the specific frame for which\nthe scalar field is minimally coupled to the Ricci scalar\nis called the Einstein frame, and quantities in this frame are indicated by a subscript $E$.\nThus the action has been rewritten\nto the more familiar minimally coupled form. The crucial point\nis that the Jordan and Einstein frame are related by\nfield redefinitions \\eqref{Conformaltransformationmetricscalar}, and hence the\ntwo formulations are physically equivalent. Nature is indifferent\nto whether we use one set of variables to describe her phenomena,\nor a different, but related, set. Physical observables should\ntherefore be invariant with respect to the frame in which you compute\nthem~\\cite{Flanagan:2004bz}. Specifically, the primordial power spectra for scalar and tensor\nperturbations (or the predicted values of $n_s$ and $r$) are the\nsame whether you compute them in the Jordan frame using variables\n$g_{\\mu\\nu}$ and $\\Phi$, or in the Einstein frame using $g_{\\mu\\nu,E}$\nand $\\Phi_E$ related to the first set by Eqs.~\\eqref{Conformaltransformationmetricscalar}.\nThis was first demonstrated in Refs.~\\cite{Makino:1991sg,Fakir:1992cg},\nsee also Ref.~\\cite{Weenink:2010rr}. Hence, the more familiar and more intuitive\nEinstein frame formulation can be used to check the successfulness of\nHiggs inflation \\cite{Bezrukov:2007ep,Salopek:1988qh}.\nIn the Einstein frame it is intuitively clear that Higgs inflation\nworks: the potential becomes exponentially flat in the large field limit.\nPredictions for the spectral index and scalar-to-tensor ratio in Higgs inflation\nare $n_s\\simeq 0.97$ and $r \\simeq 0.0032$, and are\ntherefore well within the observational bounds.\n\n\\section{The naturalness debate}\n\\label{sec: the naturalness debate}\n\n\nAt first sight Higgs inflation is a very attractive scenario, because\nit does not seem to require the introduction of new physics\nto explain the exponential expansion of the early universe.\nHowever, this has been questioned by Refs.~\\cite{Burgess:2009ea,Barbon:2009ya},\nwho looked at the ultraviolet cutoff scale $\\Lambda$ in Higgs inflation.\n$\\Lambda$ indicates the energy scale above which scattering amplitudes\nviolate the unitarity bound, and therefore determines when perturbation\ntheory breaks down. Such a cutoff can be found by power-counting techniques\nfor which one can use the following recipe:\n\\begin{enumerate}\n\\item Perform a perturbative expansion of the fields in the action.\n\\item Rescale the fields such that their kinetic terms are canonically normalized.\n\\item Read off the cutoff scale from the interaction terms with $D>4$ which\nare suppressed by $\\Lambda^{4-D}$.\n\\end{enumerate}\nIn Refs.~\\cite{Burgess:2009ea,Barbon:2009ya} the cutoff scale was obtained\nby expanding the scalar field around its vev, $\\Phi= v+\\varphi$,\nand the metric around Minkowski space, $g_{\\mu\\nu}=\\eta_{\\mu\\nu} + h_{\\mu\\nu}\/M_P$,\nwhere the normalization $M_P^{-1}$ is chosen such that the gravitational\nterms are canonically normalized.\nIn the Jordan frame, the cutoff follows from the expansion of the\n$\\xi \\mathcal{H}^{\\dagger} \\mathcal{H} R$ term, which gives the\n5-dimensional interaction $(\\xi\/M_P) \\varphi^2 \\Box h$. Reading off\nthe cutoff scale gives $\\Lambda = M_P\/\\xi$. In the Einstein frame\nthe cutoff follows from a small-field expansion of the potential,\nwhich gives $V_E=\\frac14\\lambda \\varphi_E^4-\\lambda(\\xi\/M_P)^2\\varphi_E^6+\\ldots$.\nThe dimension 6 terms are due to the small-field relation between Jordan and Einstein\nframe fields $\\Phi \\simeq \\Phi_E[1 - (\\xi\/M_P)^2]$.\nAgain, the cutoff scale from the dimension 6 term is $\\Lambda=M_P\/\\xi$.\nThe breakdown of perturbation theory may signal the appearance of new\nphysics, for example higher dimensional operators suppressed by $\\Lambda$,\nwhich solve the unitarity problems at energy scales above $\\Lambda$.\nNow, the point is that the cutoff scale $\\Lambda$ lies\ndangerously close to the energy scale of inflation, characterized\nby the Einstein frame Hubble parameter $H_E \\simeq \\sqrt{\\lambda}M_P\/\\xi$.\nThese higher dimensional operators may therefore enter the inflationary\npotential and affect inflationary predictions, or worse, spoil\nHiggs inflation. Thus we need knowledge of the ultraviolet completion\nof Higgs inflation in order to find out if inflation is successful,\nwhich obviously clashes with the original attractiveness of the scenario.\n\nSince the appearance of the works~\\cite{Burgess:2009ea,Barbon:2009ya}\nthere has been a debate in literature about\nthe presence or absence of unitarity problems in Higgs inflation.\nRefs.~\\cite{Lerner:2009na,Ferrara:2010yw,Ferrara:2010in}\nconsidered the Einstein frame analysis\nand objected, correctly, that the dimension 6 terms\nin the \\textit{small field} expansion of the potential\nonly give rise to unitarity problems in the \\textit{large field} regime\n($\\langle\\Phi\\rangle\\gg M_P\/\\xi$ or $\\langle \\Phi_E \\rangle \\geq M_P$), where the small field\nexpansion is no longer valid. Instead, one should perform a\nperturbative expansion around the field expectation value,\nwhich results in a field dependent cutoff scale.\n\nThen, by making use of the equivalence\nbetween the Jordan and Einstein frame, it was argued that there are also\nno unitarity problems in the Jordan frame for single field inflation.\nRefs.~\\cite{Burgess:2010zq,Hertzberg:2010dc} showed\nsubsequently that the unitarity bound $M_P\/\\xi$ again appears when\nthe Goldstone bosons are taken into account, both in the Jordan and Einstein\nframe. Ref.~\\cite{Burgess:2010zq} added that even in unitary gauge,\nwhere the Goldstone bosons are eaten by the gauge bosons, the cutoff\nscale appears in Higgs-gauge interactions. Ref.~\\cite{Ferrara:2010yw,Ferrara:2010in}\nargued that, instead of expanding around a small Higgs vev, the perturbative\nexpansion should be performed around a large expectation value $\\phi\\gg M_P\/\\xi$,\nwhich is the background relevant for inflation. Here $\\phi=\\langle \\Phi \\rangle$,\nwith $\\Phi$ the Higgs field in unitary gauge, $\\mathcal{H}=(0,\\Phi)$.\nIt was shown that the cutoff scale following from the Einstein frame\npotential is $M_P$ in the inflationary regime,\nbut undergoes a transition to $M_P\/\\xi$ for small field values $\\phi \\ll M_P\/\\xi$. The authors\nthen argued that this post-inflation unitary bound does not affect our ability\nto describe physical processes during inflation. Arguably the most\ncomplete treatment of the cutoff so far has been performed\nin Ref.~\\cite{Bezrukov:2010jz}. There the metric was expanded around\nan expanding background $g_{\\mu\\nu} =\\bar{g}_{\\mu\\nu}+\\delta g_{\\mu\\nu}$\nand likewise the scalar field was expanded around a time dependent\nbackground $\\Phi=\\phi+\\varphi$. Next, the Jordan frame action was\nput into canonical form by a redefinition of the metric and\nscalar field perturbations.\nIn the Jordan frame the cutoff originates from the $\\xi \\Phi^2 R$ term\nand was found to be $M_P\/\\xi$ for $\\phi \\ll M_P\/\\xi$,\n$\\xi\\phi^2\/M_P$ for $M_P\/\\xi < \\phi < M_P\/\\sqrt{\\xi}$ and $\\sqrt{\\xi}\\phi$\nfor $\\phi > M_P\/\\sqrt{\\xi}$. In the Einstein frame the same results\nwere obtained from the potential \\footnote{The cutoffs in the Jordan and Einstein\nframes are related by the conformal factor $\\sqrt{1+\\xi\\phi^2\/M_P^2}$. This\ncan be understood by realizing that the cutoff is some energy scale, or length\nscale, which is changed by the conformal transformation. In Ref. \\cite{Bezrukov:2010jz}\nthe cutoffs are only found to be equivalent\n(note: up to factors of the self-coupling $\\lambda$)\nonce this factor is taken into\naccount, see also \\cite{Bezrukov:2008ej,Bezrukov:2009db}.}.\n\n\n\\section{Gauge invariance}\n\\label{sec: gauge invariance}\n\nIn this work we revisit the computation of the cutoff scale\nin Higgs inflation. The reason is that all of the papers\nmentioned in the previous section have overlooked\na crucial aspect of the computation of quantum corrections in general relativity.\nThis crucial aspect is the fact that general relativity contains\na large diffeomorphism symmetry. Since the diffeomorphism symmetry\nresembles many aspects of gauge symmetries, general relativity\nis often called a gauge theory and its degrees of freedom (dofs)\nare generally gauge dependent. Due to gauge dependence not all\ndofs are physical. Moreover, general relativity is a constrained\ntheory, such that some of the dofs do not participate\nin the dynamics, but instead impose constraints on the system.\nTherefore, computing a cutoff by means of\nexpanding $g_{\\mu\\nu}=\\bar{g}_{mu\\nu}+\\delta g_{\\mu\\nu}$\nand $\\Phi = \\phi +\\varphi$ and using the corresponding vertices,\nwill generally give non-physical and incorrect answers.\nInstead one should first determine what are the truly physical\ndegrees of freedom, and subsequently find their interaction\nvertices and the ultraviolet cutoff.\n\nAs a small sidestep, let us illustrate the above by looking\nat an analogy: the simpler and more familiar case of QED.\nQED is described by a vector field $A_{\\mu}$,\nwhich naively contains 4 dofs. However, one of the 4 dofs\n(the longitudinal component of the spatial vector field) is\nnot physical due to the gauge symmetry\n$A_{\\mu} \\rightarrow A_{\\mu}+\\partial_{\\mu}\\Lambda$.\nMoreover one of the components of the vector field is\nnot dynamical: there are no time derivatives acting on $A_0$\nin the Maxwell action. This becomes particularly clear\nwhen the theory is written in Hamiltonian form.\nThe non-dynamical component\n(the Coulomb potential) can in fact be decoupled from the\ndynamical part of the action. Thus, out of 4 dofs in QED,\nthere are only 2 remaining dynamical dofs, which correspond to\nthe transverse components of the vector field, \\textit{i.e.}\nthe two polarizations of the photon.\n\nLet us now turn to Higgs inflation.\nThe action for Higgs inflation, which basically constitutes of\nthe Einstein-Hilbert action for general relativity, a scalar\nfield in a potential and a coupling of the scalar field to\nthe EH-action, contains naively $10+1$ dynamical degrees of freedom (dofs)\nin the metric and scalar field.\nHowever, due to the diffeomorphism symmetry (invariance\nunder coordinate reparametrizations $x^{\\mu}\\rightarrow x^{\\mu}+\\xi^{\\mu}$),\n4 of these dofs are not physical. The analogy with QED becomes\nmore apparent when we look at linearized perturbations. The action\nfor these first order perturbations is invariant under\nthe transformations $\\delta g_{\\mu\\nu}\\rightarrow \\delta g_{\\mu\\nu} +2\\nabla_{\\left(\\mu\\right.}\\xi_{\\left.\\nu\\right)}$\nand $\\varphi \\rightarrow \\varphi +\\dot{\\phi}\\xi^{0}$. This closely\nresembles the gauge symmetry of electrodynamics, only in this case\nthere are 4 gauge parameters, and thus 4 non-physical components\nof the metric.\n\nAlso, analogous to QED, 4 degrees of freedom\nin the metric are not dynamical. These are the $00$ and $0i$\ncomponents of the ADM metric,\n\\begin{equation}\nds^2=-N^2dt^2+g_{ij}(dx^i+N^idt)(dx^j+N^jdt)\n\\,.\n\\label{3rdg:ADMlineelement}\n\\end{equation}\nHere $g_{ij}$ is the spatial metric and $N$ and $N^{i}$\nare the lapse and shift functions, respectively.\nIn terms of the ADM metric the action \\eqref{Jordanframeaction}\ncan be written as\n\\begin{align}\nS=\\frac12 & \\int d^3xdt \\sqrt{g}\\Biggl\\{\nN R F(\\Phi)+\\frac{1}{N}\\left(E^{ij}E_{ij}-E^2\\right)F(\\Phi)\n-\\frac{2}{N} E F'(\\Phi)\\left(\\partial_t{\\Phi}\n- N^{i}\\partial_i\\Phi\\right)\n\\nonumber\\\\\n&+2g^{ij}\\nabla_iN \\nabla_jF(\\Phi)\n+\\frac{1}{N}\\left(\\partial_t{\\Phi}-N^{i}\\partial_i\\Phi\\right)^2\n-Ng^{ij}\\partial_i\\Phi\\partial_j\\Phi-2 N V(\\Phi)\\Biggr\\}\n\\,,\n\\label{ADMactionnonminimalEij}\n\\end{align}\nwhere $F'(\\Phi)=dF(\\Phi)\/d\\Phi$ and\nthe measure $\\sqrt{g}$, the Ricci scalar $R$\nand covariant derivatives $\\nabla_i$ are composed\nof the spatial part of the metric $g_{ij}$ alone.\nThe quantities $E_{ij}$ and $E$ are related\nto the extrinsic curvature $K_{ij}$ as $E_{ij}=-NK_{ij}$,\nwith\n\\begin{align}\nE_{ij}&=\\frac{1}{2}\\left(\\partial_tg_{ij}-\\nabla_iN_j-\\nabla_jN_i\\right)\\\\\nE&=g^{ij}E_{ij}\n\\,.\n\\end{align}\nFrom Eq.~\\eqref{ADMactionnonminimalEij} it can clearly be seen\nthat the lapse and shift functions are not dynamical (there are no kinetic terms\nfor them). This becomes even clearer in a Hamiltonian formulation~\\cite{Weenink:2010rr}, \nin which the lapse and shift functions\nmultiply the energy and momentum constraints. Because the lapse and shift functions\nare related to the constraints in general relativity and are non-dynamical,\nthey are also called constraint, or auxiliary fields. Thus, out of the\n$10+1$ degrees of freedom (dofs) in the action \\eqref{Jordanframeaction}, 4 are non-dynamical\nand related to the constraints, and 4 others are gauge dofs. We can therefore\nalready argue that there are only 3 dynamical dofs in the perturbed action,\nand we should take this into account when computing the physical cutoff.\n\nAs a first example of how incorrect results are obtained\nby not taking into account the gauge freedom and\nconstraints in general relativity, let us consider\nthe case of quantum corrections in Higgs inflation\nby expanding around a Minkowsky background, as done\nin Refs.~\\cite{Burgess:2009ea,Barbon:2009ya}. The dominant\ncontribution from the term $\\xi \\Phi^2 R$ is found to be\n$(\\xi\/M_P) \\varphi^2 \\Box h$. However, it is well known\nthat the only dynamical dofs in Minkowsky space are\nthe transverse traceless part of the metric, the graviton $\\gamma_{ij}$, and\nthe scalar field fluctuation $\\varphi$~~\\footnote{Commonly, in Minkowski space the $00$ and $0i$\ncomponents of the metric fluctuation $h_{\\mu\\nu}$ are zero\n(the solution for the auxiliary fields is zero), and -- in the absence of matter fields\n-- the gauge freedom\nis used to set the scalar and vectorial components of the spatial metric\nfluctuation $h_{ij}$ to zero. The only remaining degree of\nfreedom is the traceless and traceless $\\gamma_{ij}$.}. In that case,\na term such as $h = \\rm{Tr}[h_{\\mu\\nu}]$ disappears from\nthe action. The next most dominant term is then\n$(\\xi\/M_P^2) \\varphi^2 \\partial \\gamma_{ij}\\partial \\gamma_{ij}$,\nwhich is a dimension 6 term with a cutoff scale $M_P\/\\sqrt{\\xi}$,\nwhich is higher than the naive cutoff scale $M_P\/\\xi$.\nAlthough this is still an incorrect derivation of the cutoff\n-- one should perform an expansion around an expanding universe\nand a time dependent background field -- it already shows that\ntaking into account the gauge symmetry can raise the cutoff scale.\n\nSo let us continue by considering fluctuations on top of an expanding\nuniverse. In order to deal with the non-dynamical parts of the\nmetric and the gauge freedom, it is convenient to separate the\nmetric into different components and expand each component separately.\nWe thus insert into the action \\eqref{Jordanframeaction} (or Eq. \\eqref{ADMactionnonminimalEij})\n\\begin{align}\ng_{ij}&=a^2(t){\\rm e}^{2\\zeta}({\\rm e}^\\alpha)_{ij}\n\\nonumber\\\\\n\\Phi&=\\phi+\\varphi\n\\nonumber\\\\\nN&=\\bar{N}\\left(1+n\\right)\n\\nonumber\\\\\nN^i&=a^{-1}\\bar{N}(t)(a^{-1}\\partial_i s+n_{i}^{T})\n\\label{perturbations}\n\\,,\n\\end{align}\nwhere $a(t)$ is the scale factor and\n\\begin{equation}\n\\alpha_{ij}=a^{-2}\\partial_i\\partial_j \\tilde h + a^{-1}\\partial_{(i}h_{j)}^T+\\gamma_{ij}\n\\,,\n\\end{equation}\nwith $\\gamma_{ii}=0$, $\\partial_j\\gamma_{ji}=0$, $\\partial_j h^T_{j}=0$ and $\\partial_i n_i^T=0$.\nNow, all fluctuations of the metric and scalar field\nare by themselves not invariant under the gauge transformations (coordinate\nreparametrizations) of general relativity. Thus interaction\nvertices for the fluctuations in Eq. \\eqref{perturbations} are\ngenerally not gauge invariant, and thus not physical. In\norder to find the physical interaction vertices, the best way\nto go is to derive the action for \\textit{gauge invariant perturbations}.\nThese are variables that are by themselves invariant\nunder coordinate reparametrizations, and hence their\ninteraction vertices are physical.\nIn Refs.~\\cite{Weenink:2010rr,Prokopec:2012ug,Prokopec:2013zya} we have precisely done this:\nwe have derived the quadratic and cubic actions for gauge\ninvariant cosmological perturbations. We have found that,\nwhen written in terms of physical (gauge invariant) fields,\nthe scalar-tensor theory~(\\ref{Jordanframeaction})\ncan be separated into dynamical and constraint parts, such\nthat the constraint and dynamical fields\ndecouple.\nThe dynamical part contains one real scalar degree of freedom\nand one (traceless, transverse) tensor field with two degrees of freedom\n(polarizations).\nThe non-dynamical part contains gauge invariant versions\nof the constraint fields in the action.\nThere is one gauge invariant scalar lapse function\nand one gauge invariant shift vector field (with three components),\nwhich is usually split into a transverse vector and a (longitudinal) scalar.\nOn-shell all four gauge invariant constraint fields are zero\n(zero gauge invariant constraint fields solve the corresponding equations of motion).\nThis can also be understood by viewing the lapse and shift fields\nas auxiliary fields that can be solved for, order by order in perturbation\ntheory, and their solution inserted in the action \\cite{Prokopec:2013zya}\n(see also Ref.~\\cite{Maldacena:2002vr} for a gauge fixed approach). The\ngauge invariant lapse and shift functions being zero on-shell\ncorresponds to solving the equation of motion for the lapse\nand shift function. Solving for the auxiliary fields is hence analogous\nto decoupling the fields in the action.\nBecause of this decoupling, the considerations of the gauge invariant\nscalar-tensor theories significantly simplify, in the sense that\nit suffices to consider the action for the dynamical\ndegrees of freedom only (truncated to a certain order).\nIn Refs.~\\cite{Weenink:2010rr,Prokopec:2012ug,Prokopec:2013zya}.\nsuch an action was constructed up to cubic order in the fields\nfor a class of scalar-tensor theories with general $F(\\Phi)$ term\nin Eq.~(\\ref{Jordanframeaction}).\nIn order to do that, we had to make a choice for gauge invariant variables.\nThe potential complication is the fact that, at second order in gauge\ntransformations there are infinitely many such variables.\nHowever, one can show that\nthe form of the cubic actions, when expressed in terms of different\ngauge invariant variables, differs only by surface\nterms~\\cite{Prokopec:2012ug,Prokopec:2013zya}, which are irrelevant for\nthe (bulk) evolution. Nevertheless, these surface terms can be of physical relevance\nfor (the non-Gaussian part of) the initial state. One set of gauge invariant\nvariables is special, in that the variables are also frame independent:\nthey are invariant under a transformation from Jordan to Einstein frame.\nWhen expressed in terms of these variables,\nthe cubic actions in the Einstein and Jordan frame are\nmanifestly {\\it equal}\n(when one takes account of the trivial frame transformation of the time dependent background\nquantities that appear as prefactors in front of the cubic vertices).\n\nTherefore, in order to simplify the unitarity cutoff discussion\nit is natural to work with these unique frame (and gauge) independent\nvariables. They are the gauge invariant scalar\npertubation $W_\\zeta$ and the gauge invariant tensor (transverse, traceless graviton) perturbation\n$\\tilde\\gamma_{ij}$. These are the second order generalizations of\nthe gauge invariant Sasaki-Mukhanov field, $w_\\zeta=\\zeta-(H\/\\dot\\phi)\\varphi$,\nand the graviton perturbation $\\gamma_{ij}$,\nwhere $H(t)=\\dot a\/a=\\bar N(t)^{-1}d\\ln(a)\/dt$ is the Hubble parameter,\nand $\\phi(t)$ the background (inflaton) field.\nFor the precise definitions of $W_\\zeta$ and $\\tilde\\gamma_{ij}$ -- which are given by\n$w_\\zeta$ and $\\gamma_{ij}$ plus corrections that are quadratic in perturbations --\nwe refer to Ref.~\\cite{Prokopec:2013zya}. We have also\ngiven an explicit proof that these variables are frame independent,\nthus $W_{\\zeta,E}=W_{\\zeta}$ and $\\tilde{\\gamma}_{ij,E}=\\tilde{\\gamma}_{ij}$.\n\nThe complete action for these gauge and frame independent\ndynamical fields up to the cubic order\nfor these variable can be found in\nRefs.~\\cite{Weenink:2010rr,Prokopec:2012ug,Prokopec:2013zya},\nwhich we shall repeat here.\nThe quadratic action is a sum of the scalar and tensor parts,\n\\begin{eqnarray}\nS^{(2)}= S^{(2)}_{W_{\\zeta}} + S^{(2)}_{\\tilde\\gamma}\n\\,,\n\\label{quadratic action:total}\n\\end{eqnarray}\nwhere in the Jordan frame (see~(3.23) of Ref.~\\cite{Prokopec:2013zya}),\n\\begin{eqnarray}\nS^{(2)}_{W_{\\zeta}} &=& \\int d^{3}xdt \\bar{N} a^{3}\nz^2\\left[\\frac12\\dot{W}_{\\zeta}^2-\\frac12\\left(\\frac{\\partial_i W_{\\zeta}}{a}\\right)^2\\right]\n\\,,\n\\label{nmaction:2:wzetawzeta:GI}\\\\\nS^{(2)}_{{\\tilde\\gamma}_\\zeta} &=& \\int d^3xdt \\bar{N} a^3\\frac{F}{4}\n\\left[\\frac12(\\dot{\\tilde\\gamma}_{\\zeta ij})^2\n-\\frac12\\left(\\frac{\\partial_l \\tilde\\gamma_{\\zeta\\,ij}}{a}\\right)^2\\right]\n\\,,\n\\label{nmaction:2:gammagamma}\n\\end{eqnarray}\nwhere $\\bar N(t)$ is the background shift function\n(for conformal time $t\\rightarrow \\eta$, $\\bar N = a(\\eta)$,\nwhile for physical time $t$, $\\bar N(t)= 1$).\nNote that the prefactor in the integrand of Eq.~(\\ref{nmaction:2:wzetawzeta:GI})\nis in fact\n\\begin{equation}\nz^2 = \\frac{\\dot{\\phi}^2+\\frac{3}{2}\\frac{\\dot{F}^2}{F}}\n{\\big(H+\\frac12\\frac{\\dot{F}}{F}\\big)^2}\n\\,.\n\\label{z:Jordan}\n\\end{equation}\nThe cubic action can be conveniently split into the scalar-scalar-scalar,\nscalar-scalar-tensor, scalar-tensor-tensor and tensor-tensor-tensor vertices,\n\\begin{eqnarray}\nS^{(3)}= S^{(3)}_{W_{\\zeta}W_{\\zeta}W_{\\zeta}} + S^{(3)}_{W_{\\zeta}W_{\\zeta}\\tilde\\gamma}\n     + S^{(3)}_{W_{\\zeta}\\tilde\\gamma\\tilde\\gamma} + S^{(3)}_{\\tilde\\gamma\\tilde\\gamma\\tilde\\gamma}\n\\,.\n\\label{cubic action:total}\n\\end{eqnarray}\nUp to boundary terms,\nthe scalar-scalar-scalar part of the action is (see Ref.~\\cite{Prokopec:2012ug}\nand Eq.~(6.70) of Ref.~\\cite{Weenink:2013oqa})\n\\begin{eqnarray}\nS^{(3)}_{W_{\\zeta}W_{\\zeta}W_{\\zeta}}\n &=& \\int d^3x dt\\bar{N} a^3 F \\Biggl\\{\n\\frac{1}{4}\\frac{z^4}{F^2}\\left[W_{\\zeta}\\left(\\dot{W}_{\\zeta}^2\n+\\left(\\frac{\\partial_i W_{\\zeta}}{a}\\right)^2\\right)\n-2\\dot{W}_{\\zeta}\\left(\\frac{\\partial_i}{\\nabla^2}\n\\dot{W}_{\\zeta}\\right)\\partial_i W_{\\zeta}\\right]\n\\nonumber\\\\\n&&-\\,\\frac{1}{16}\\frac{z^6}{F^3}W_{\\zeta}\\left[\\dot{W}_{\\zeta}^2\n-\\left(\\frac{\\partial_i\\partial_j}{\\nabla^2}\\dot{W}_{\\zeta}\\right)\n\\left(\\frac{\\partial_i\\partial_j}{\\nabla^2}\\dot{W}_{\\zeta}\\right)\\right]\n+\\frac12 \\frac{z^2}{F}\\left[\\frac{\\frac{\\dot{z}}{z}-\\frac12\\frac{\\dot{F}}{F}}\n{H+\\frac12\\frac{\\dot{F}}{F}}\\right]^{\\cdot}\n \\dot{W}_{\\zeta}W_{\\zeta}^2\n\\Biggr\\}\n\\,,\n\\label{3rd:Jordan: Cubic GI Action Wzeta}\n\\end{eqnarray}\nthe scalar-scalar-tensor part of the action is (see Ref.~\\cite{Prokopec:2013zya}\nand Eqs.~(7.12) and~(7.47)  of~\\cite{Weenink:2013oqa}),\n\\begin{eqnarray}\nS^{(3)}_{W_{\\zeta}W_{\\zeta}\\tilde{\\gamma}_{\\zeta}}\n&=&\\int d^{3}xdt \\bar{N} a^{3}\n\\Biggl\\{\n\\frac12z^2\\tilde\\gamma_{\\zeta,ij}\\bigg(\\frac{\\partial_i}{a}W_\\zeta\\bigg)\n                            \\bigg(\\frac{\\partial_j}{a}W_\\zeta\\bigg)\n-\\frac{z^4}{8F}W_\\zeta\\dot{\\tilde\\gamma}_{\\zeta,ij}\\frac{\\partial_i\\partial_j}{\\nabla^2}{\\dot W}_\\zeta\n\\nonumber\\\\\n&&\\hskip 2.3cm\n+\\,\\frac{z^4}{8F}\\bigg(\\frac{\\partial_i\\partial_j}{\\nabla^2}{\\dot W}_\\zeta\\bigg)\n\\bigg(\\frac{\\partial_k}{\\nabla^2}\\dot W_\\zeta\\bigg)\n\\partial_k\\tilde\\gamma_{\\zeta,ij}\n\\Biggr\\}\n\\,,\\qquad\n\\label{3rdg:nmaction:3:ssg:GI:Wzeta}\n\\end{eqnarray}\nthe scalar-tensor-tensor part of the action is (see Ref.~\\cite{Prokopec:2013zya}\nand Eq.~(7.49) of~\\cite{Weenink:2013oqa}),\n\\begin{eqnarray}\nS^{(3)}_{W_{\\zeta}\\tilde{\\gamma}_{\\zeta}\\tilde{\\gamma}_{\\zeta}}\n&=&\\int d^{3}xdt \\bar{N} a^{3}\n\\Biggl\\{\n\\frac{z^2}{16}W_\\zeta\\bigg[\\dot{\\tilde\\gamma}_{\\zeta,ij}\\dot{\\tilde\\gamma}_{\\zeta,ij}\n   +\\bigg(\\frac{\\partial_l}{a}\\tilde\\gamma_{\\zeta,ij}\\bigg)\n   \\bigg(\\frac{\\partial_l}{a}\\tilde\\gamma_{\\zeta,ij}\\bigg)\\bigg]\n-\\frac{z^2}{8}\\dot{\\tilde\\gamma}_{\\zeta,ij}\\bigg(\\partial_l\\tilde\\gamma_{\\zeta,ij}\\bigg)\n    \\frac{\\partial_l}{\\nabla^2}\\dot W_\\zeta\n\\Biggr\\}\n\\,,\\qquad\n\\label{3rdg:nmaction:3:sgg:GI:Wzeta}\n\\end{eqnarray}\nand finally the tensor-tensor-tensor part of the action is (see Ref.~\\cite{Prokopec:2013zya}\nand Eq.~(7.39) of~\\cite{Weenink:2013oqa}),\n\\begin{eqnarray}\nS^{(3)}_{\\tilde{\\gamma}_{\\zeta}\\tilde{\\gamma}_{\\zeta}\\tilde{\\gamma}_{\\zeta}}\n&=&\\int d^3xdt \\bar{N} a^3 \\frac{F}{8}\n\\Biggl\\{\n\\tilde{\\gamma}_{\\zeta,ij}\n\\frac{\\partial_i\\tilde{\\gamma}_{\\zeta,kl}}{a}\n\\frac{\\partial_j\\tilde{\\gamma}_{\\zeta,kl}}{a}\n+\\tilde{\\gamma}_{\\zeta,kl}\n\\frac{\\partial_i\\tilde{\\gamma}_{\\zeta,kj}}{a}\n\\frac{\\partial_j\\tilde{\\gamma}_{\\zeta,il}}{a}\n-\\tilde{\\gamma}_{\\zeta,ik}\n\\frac{\\partial_i\\tilde{\\gamma}_{\\zeta,jl}}{a}\n\\frac{\\partial_j\\tilde{\\gamma}_{\\zeta,kl}}{a}\n\\Biggr\\}\n\\,.\\qquad\n\\label{3rdg:nmaction:3:ggg:GI:Wzeta}\n\\end{eqnarray}\n\nWhen written in terms of the frame independent variables $W_\\zeta$ and $\\tilde\\gamma_{\\zeta ij}$,\nwe proved in~\\cite{Prokopec:2013zya}\nthat the action~(\\ref{quadratic action:total}--\\ref{3rdg:nmaction:3:ggg:GI:Wzeta})\nis frame independent.\nSince the frame independent perturbations themselves coincide in Jordan\nand Einstein frame, $W_\\zeta=W_{\\zeta,E}$ and $\\tilde\\gamma_{\\zeta ij}=\\tilde\\gamma_{\\zeta ij,E}$,\nthe Einstein frame and Jordan frame actions coincide as well.\nThe frame independence becomes manifest when\none expresses the background quantities in the Jordan frame $a,H,z$ in\nterms of their Einstein frame counterparts\nin Eqs.~(\\ref{quadratic action:total}--\\ref{3rdg:nmaction:3:ggg:GI:Wzeta})\nby using the relations \\eqref{Conformaltransformationmetricscalar}\nat background level. This gives the following identities\n\\begin{eqnarray}\nF^{1\/2}\\bar N &=& M_{\\rm P}\\bar{N}_E\n\\,,\\quad\nF^{1\/2}a= M_{\\rm P}a_E\n\\,,\\quad\nF^{-1\/2}\\dot W_\\zeta  = M_{\\rm P}^{-1}\\dot W_{\\zeta,E}\n\\,,\\quad\n\\frac{H+\\frac{\\dot F}{2F}}{\\sqrt{F}}= \\frac{H_E}{M_{\\rm P}}\n\\nonumber\\\\\n\\frac{\\dot\\phi^2+\\frac32\\frac{\\dot F^2}{F}}{F^2}&=& \\frac{\\dot{\\phi}_E^2}{M_{\\rm P}^4}\n\\,,\\qquad\n\\frac{z}{\\sqrt{F}}= \\frac{z_E}{M_{\\rm P}}\n\\,,\\qquad\n\\frac{\\frac{\\dot z}{z}-\\frac12\\frac{\\dot F}{F}}{H+\\frac{\\dot F}{2F}}\n                 = \\frac{1}{M_{\\rm P}}\\frac{\\dot{z}_E}{z_E}\n\\,,\n\\label{from Jordan to Einstein}\n\\end{eqnarray}\nwhere all subscripts $E$ denote quantities in the Einstein frame, $z_E\\equiv\\dot{\\phi}_E\/H_E$\nand a dotted derivative denotes $\\dot{X}=dX\/(\\bar{N}dt)$ and $\\dot{X}_E=dX_E\/(\\bar{N}_Edt)$\nfor Jordan and Einstein frame quantities respectively.\nHence there is no need to quote the action in the Einstein frame\n(in which $z^2\/F=z_E^2\/M_P^2=\\dot{\\phi}_E^2\/(M_P^2H_E^2)$).\n\n\n\nIn order to compute the cutoff in a frame independent formulation,\nit is convenient to transform to canonically normalized variables\n\\begin{equation}\nV_\\zeta  =  a z W_{\\zeta} = a \\sqrt{2\\epsilon_E F } W_{\\zeta}\n\\,,\\qquad\n\\Gamma_{\\zeta,ij}  =  \\frac12 a \\sqrt{F} \\tilde{\\gamma}_{\\zeta,ij}\n\\,,\n\\label{3rdg:canonicalvariables}\n\\end{equation}\nwhere we have used\n\\begin{equation}\n\\epsilon_E = -\\frac{\\dot{H}_E}{H_E^2} = \\frac{z_E^2}{2M_P^2} = \\frac{z^2}{2F}\n\\,,\n\\end{equation}\nwhich relates $z$ (defined in Eq.~\\eqref{z:Jordan})\nto the slow-roll parameter $\\epsilon_E$ in the Einstein frame.\nIn terms of the canonical fields the\nsecond order action~(\\ref{quadratic action:total}--\\ref{nmaction:2:gammagamma})\nin conformal time $\\tau$ ($\\bar{N}(t)\\rightarrow a(\\tau)$) become\n\\begin{eqnarray}\nS^{(2)}_{V_\\zeta}&=&\\frac12\\int d^3xd\\tau\n\\left[V_\\zeta^{\\prime 2}-(\\partial_i V_\\zeta)^2+\\frac{(az)^{\\prime\\prime}}{az}V_\\zeta\\right]\n-\\frac12\\int d^3 x \\bigg[\\frac{(az)^{\\prime}}{az}V_\\zeta^2\n                   \\bigg]_{\\tau_{\\rm in}}^{\\tau_{\\rm fin}}\n\\label{scalar quadratic action:canonical}\\\\\nS^{(2)}_{\\Gamma_{\\zeta,ij}}&=&\\frac12\\int d^3x d\\tau\n\\left[\\Gamma_{\\zeta,ij}^{\\prime 2}-(\\partial_l \\Gamma_{\\zeta,ij})^2\n        +\\frac{(a\\sqrt{F})^{\\prime\\prime}}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}^2\\right]\n-\\frac12\\int d^3 x \\bigg[\\frac{(a\\sqrt{F})^{\\prime}}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}^2\n                   \\bigg]_{\\tau_{\\rm in}}^{\\tau_{\\rm fin}}\n\\,,\n\\label{tensor quadratic action:canonical}\n\\end{eqnarray}\nwhere a prime denotes a derivative with respect to conformal\ntime $\\tau$. The boundary terms\nin~(\\ref{scalar quadratic action:canonical}--\\ref{tensor quadratic action:canonical})\ndo not contribute to the propagator equation of motion and thus they can be discarded.\nFurthermore, the transformation to the canonically normalized fields has led to generation of\ntime dependent (negative) mass terms\nin~(\\ref{scalar quadratic action:canonical}--\\ref{tensor quadratic action:canonical}).\nHowever, these terms can be neglected on energy and momentum scales far above the Hubble scale.\nNamely, $(az)^\\prime\/(az)=aH+(1\/2)\\phi^\\prime [d\\ln(F)\/d\\phi]+(1\/2)[\\epsilon_E^\\prime\/\\epsilon_E]$.\nNow, since the latter two terms are suppressed by slow roll parameters,\nthe first term, $aH=\\cal{H}$ (here and subsequently ${\\cal H}$ denotes a conformal\nHubble rate) is the dominant term. Likewise, one can show that the dominant term\nin $(az)^{\\prime\\prime}\/(az) = 2a^2H^2$ plus slow roll suppressed terms, such that when\nthe (conformal) energy scale, $E_c\\gg {\\cal H}$, this term can be neglected.\nBecause $(a\\sqrt{F})^{\\prime\\prime}\/(a\\sqrt{F})= 2a^2 H^2$ plus slow roll suppressed terms,\nthe same consideration applies to canonically normalized gravitons.\nThis means that the canonically normalized scalar and graviton propagator behave in the\nultraviolet (on scales $E_c = |k^0|\\gg {\\cal H}$ and $\\|\\vec k\\|\\gg {\\cal H}$) as,\n$\\sim 1\/(\\eta_{\\mu\\nu}k^\\mu k^\\nu)$ plus corrections of the order ${\\cal H}^2$, which\nwe shall neglect in the following considerations.\n\n When the cubic action~(\\ref{3rd:Jordan: Cubic GI Action Wzeta}--\\ref{3rdg:nmaction:3:ggg:GI:Wzeta})\nis expressed in terms of the canonical variables~(\\ref{3rdg:canonicalvariables}),\none gets for the  pure scalar cubic action\n\\begin{eqnarray}\nS^{(3)}_{V_{\\zeta}V_{\\zeta}V_{\\zeta}} &\\simeq& \\int d^3xd\\tau \\sqrt{\\frac{\\epsilon_E}{8a^2F}}\n\\bigg\\{\n  V_\\zeta\\bigg[\n            (V_\\zeta^\\prime)^2-2\\frac{(az)^\\prime}{az}V_\\zeta^\\prime V_\\zeta\n             +\\frac{[(az)^\\prime]^2}{(az)^2}V_\\zeta^2 +(\\partial_iV_\\zeta)^2\n         \\bigg]\n\\label{pure scalar cubic:canonical}\n\\\\\n   && \\hskip 2.9cm  -\\, 2\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n \\bigg[\\frac{\\partial_i}{\\nabla^2}\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\\bigg]\n          (\\partial_iV_\\zeta)\n\\nonumber\\\\\n   && \\hskip 2.9cm\n -\\,\\frac{\\epsilon_E}{2}V_\\zeta\\bigg[\n            \\bigg(\\!V_\\zeta^\\prime\\!-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)^2\n\\!-\\frac{\\partial_i\\partial_j}{\\nabla^2}\\bigg(V_\\zeta^\\prime\\!-\\!\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n-\\frac{\\partial_i\\partial_j}{\\nabla^2}\\bigg(V_\\zeta^\\prime\\!-\\!\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n         \\bigg]\n\\nonumber\\\\\n   && \\hskip 2.9cm\n+\\frac{1}{\\epsilon_E}\\bigg(\n                         \\frac{\\epsilon_E^\\prime\/\\epsilon_E}{2{\\cal H}+(\\partial_\\tau F)\/F}\n                     \\bigg)^\\prime\n                  V_\\zeta^2\\bigg(\\!V_\\zeta^\\prime\\!-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n\\bigg\\}\n\\,.\n\\nonumber\n\\end{eqnarray}\nNext, the scalar-scalar-tensor cubic action~(\\ref{3rdg:nmaction:3:ssg:GI:Wzeta})\ncan be written as,\n\\begin{eqnarray}\nS^{(3)}_{V_{\\zeta}V_{\\zeta}\\Gamma_\\zeta}\n&=&\\int d^{3}xd\\tau\n\\Biggl\\{\n\\frac{1}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}\\big(\\partial_iV_\\zeta\\big)\\big(\\partial_jV_\\zeta\\big)\n\\nonumber\\\\\n&&\\hskip 1.5cm\n-\\,\\frac{z^2}{4aF^{3\/2}}V_\\zeta\\Bigg(\\Gamma_{\\zeta,ij}^\\prime-\\frac{(a\\sqrt{F})^\\prime}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}\\bigg)\n    \\frac{\\partial_i\\partial_j}{\\nabla^2}\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n\\nonumber\\\\\n&&\\hskip 1.5cm\n+\\,\\frac{z^2}{4aF^{3\/2}}\\frac{\\partial_i\\partial_j}{\\nabla^2}\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n\\frac{\\partial_l}{\\nabla^2}\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n\\partial_l\\Gamma_{\\zeta,ij}\n\\Biggr\\}\n\\,,\\qquad\n\\label{dominant ssg}\n\\end{eqnarray}\nthe scalar-tensor-tensor part of the action~(\\ref{3rdg:nmaction:3:sgg:GI:Wzeta}) becomes,\n\\begin{eqnarray}\nS^{(3)}_{V_{\\zeta}\\Gamma_{\\zeta}\\Gamma_{\\zeta}}\n&=&\\int d^{3}xd\\tau\n\\Biggl\\{\n\\frac{z}{4aF}V_\\zeta\\bigg[\\bigg(\\Gamma_{\\zeta,ij}^\\prime\n                       -\\frac{(a\\sqrt{F})^\\prime}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}\\bigg)\n    \\bigg(\\Gamma_{\\zeta,ij}^\\prime-\\frac{(a\\sqrt{F})^\\prime}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}\\bigg)\n   +\\big(\\partial_l\\Gamma_{\\zeta,ij}\\big)\\big(\\partial_l\\Gamma_{\\zeta,ij}\\big)\\bigg]\n\\nonumber\\\\\n&&\\hskip 1.5cm\n-\\,\\frac{z}{2aF}\\bigg(\\Gamma_{\\zeta,ij}^\\prime\n                    -\\frac{(a\\sqrt{F})^\\prime}{a\\sqrt{F}}\\Gamma_{\\zeta,ij}\\bigg)\n   \\big(\\partial_l\\Gamma_{\\zeta,ij}\\big)\n    \\frac{\\partial_l}{\\nabla^2}\\bigg(V_\\zeta^\\prime-\\frac{(az)^\\prime}{az}V_\\zeta\\bigg)\n\\Biggr\\}\n\\,,\\qquad\n\\label{dominant sgg}\n\\end{eqnarray}\nand finally the pure tensor part of the action~(\\ref{3rdg:nmaction:3:ggg:GI:Wzeta}) becomes\n\\begin{eqnarray}\nS^{(3)}_{\\Gamma_{\\zeta}\\Gamma_{\\zeta}\\Gamma_{\\zeta}}\n\\!=\\!\\int\\! \\frac{d^3xd\\tau}{a\\sqrt{F}}\n\\biggl\\{\n\\Gamma_{\\zeta,ij}\\big(\\partial_i\\Gamma_{\\zeta,kl}\\big)\\big(\\partial_j\\Gamma_{\\zeta,kl}\\big)\n\\!+\\!\\Gamma_{\\zeta,kl}\\big(\\partial_i\\Gamma_{\\zeta,kj}\\big)\\big(\\partial_j\\Gamma_{\\zeta,il}\\big)\n\\!-\\!\\Gamma_{\\zeta,ik}\\big(\\partial_i\\Gamma_{\\zeta,jl}\\big)\\big(\\partial_j\\Gamma_{\\zeta,kl}\\big)\n\\!\\biggr\\}\n\\,.\\quad\n\\label{dominant ggg}\n\\end{eqnarray}\n\nWe shall use the cubic action~(\\ref{pure scalar cubic:canonical}--\\ref{dominant ggg})\nin section~\\ref{sec: cutoffscale} to analyse the unitarity cutoff\nimplied by the $2\\to 2$ tree-level scattering processes.\nOf course, to complete the analysis,\none would also need the gauge invariant quartic vertices, which are currently not\navailable. We do not expect however, that quartic vertices\nwill change in any way the qualitative discussion we present in the next section.\n\n\\section{The cutoff scale revisited}\n\\label{sec: cutoffscale}\n\nA usual requirement for renormalizable theories\nis that the unitarity bound is not violated in\nperturbation theory, \\textit{i.e.} scattering\namplitudes should not become bigger than unity.\nIn particular there is the requirement of tree unitarity~\\cite{Cornwall:1974km},\nwhich states that $N$ particle tree amplitudes\n$\\mathcal{A}_N$ should not grow more rapidly than\n$E^{4-N}$, where $E$ is the center of mass energy.\nIf the amplitude grows faster, perturbation theory\nfails at some cutoff scale $\\Lambda$. This usually\nmeans that some new physics should enter at this energy\nscale. When the relevant energy scales of the theory\nunder consideration are well below the cutoff scale,\nthe theory can be considered 'natural', in the sense\nthat perturbation theory is valid and there is no\nneed for new physics. Conversely, there is a naturalness problem\nif typical energy scales are higher than the cutoff scale.\n\nThe cutoff is most easily computed when the perturbative\naction is written in canonical form, such that\nthe propagator goes as $1\/k^2$, where $k^2 = \\eta_{\\mu\\nu}k^\\mu k^\\nu$\nand $k^\\mu$ is a conformal 4-momentum.\nNext, the cutoff can be read off from the vertices of dimension\nhigher then 4, which should be suppressed as $\\Lambda^{4-D}$.\nThis has been done for Higgs inflation in Ref.~\\cite{}\nin both the Einstein and Jordan frame, and we\noutline the derivation here. In the Jordan frame the starting point is\nthe action~\\eqref{Jordanframeaction} with $F(\\Phi)=M_P^2+\\xi\\Phi^2$,\nwhich is the action for Higgs inflation in unitary gauge\nand gauge interactions are neglected.\nNext, following Ref.~\\cite{Bezrukov:2011sz},\nand analogously as it was done in~(\\ref{3rdg:canonicalvariables}),\ngeneric (non-invariant) perturbations $\\delta g_{\\mu\\nu}=g_{\\mu\\nu}-\\bar g_{\\mu\\nu}$\nand $\\varphi=\\Phi(x)-\\phi(t)$ can be rescaled to canonical variables $\\delta\\hat g_{\\mu\\nu}$\nand $\\hat\\varphi$ for which the corresponding quadratic actions are canonically normalized.\nHere $\\bar g_{\\mu\\nu}$ denotes a background metric (which in the UV can be approximated\nby Minkowski metric) and $\\phi(t)$ is a background field.\nThe dominant term in the action with dimension higher than $4$ and\nin the Jordan frame is of the order\n\\begin{equation}\n\\xi \\varphi^2 \\Box \\delta g\n\\,,\n\\label{dominant vertex: dim > 4}\n\\end{equation}\nwhere $\\delta g = \\bar{g}^{\\mu\\nu} \\delta g_{\\mu\\nu}$.\nWhen reexpressed in terms of the canonically normalized fields  $\\hat g_{\\mu\\nu}$\nand $\\hat\\varphi$, this term becomes\n\\begin{equation}\n \\frac{\\xi\\sqrt{M_P^2+\\xi\\phi^2}}{M_P^2+\\xi\\phi^2+6\\xi^2\\phi^2}\n\\hat{\\varphi}^2 \\Box {\\delta\\hat g}\n\\,.\n\\label{dominant vertex: dim > 4:2}\n\\end{equation}\nAt high energies this vertex scales as $E^2$, where $E=|k^0|$ denotes the energy scale. If\nwe consider a $2\\rightarrow 2$ scattering process of $\\hat{\\varphi}$\nvia exchange of a gravitational scalar $\\delta\\hat g$, we see that\nthe total amplitude scales as $E^2\/\\Lambda^2$ at high energies. The cutoff scale\nis precisely the inverse of the operator above. The cutoff in\nthe Jordan frame is thus\n\\begin{equation}\n\\Lambda \\sim \\frac{M_P^2+\\xi\\phi^2+6\\xi^2\\phi^2}{\\xi\\sqrt{M_P^2+\\xi\\phi^2}}\n\\,.\n\\label{cutoff scale:from paper}\n\\end{equation}\nIn Ref.~\\cite{Bezrukov:2010jz} the cutoff\nwas also computed \\textit{via} the Einstein frame. In that frame\nthe non-minimal coupling term is absent and the gravitational\nand field kinetic terms are canonical. Still, the cutoff\nscale reappears in the non-polynomial potential and shows\na similar  behavior as above (though not exactly equal).\nFrom~(\\ref{cutoff scale:from paper}) one sees that\n$\\Lambda \\propto \\sqrt{\\xi} \\phi$ in the regime where $\\phi\\gg M_P\/\\sqrt{\\xi}$,\n$\\Lambda \\sim \\xi \\phi^2\/M_P$ for $M_P\/\\sqrt{\\xi} \\gg \\phi \\gg M_P\/\\xi$ and\n$\\Lambda \\sim M_P\/\\xi$ when $\\phi\\ll M_P\/\\xi$. The authors\nof Ref. \\cite{Bezrukov:2011sz} then argue that all relevant\nenergy scales in these regimes are lower than the cutoff scale,\nsuch that the perturbative expansion is valid\nand Higgs inflation is natural.\n\nThese results are interesting, but the question arises whether they can be trusted.\nIn the following we point at the principal potential problems with\nthe analysis outlined above, which put into question the reliability of\nthe cutoff scale in Eq.~(\\ref{cutoff scale:from paper})\n\n\\begin{itemize}\n\n\\item[{\\bf a)}] {\\bf The computation of the cutoff is gauge dependent.}\n\nBoth the metric fluctuations $\\delta g_{\\mu\\nu}$ and scalar field\nperturbation $\\varphi$ are gauge dependent, hence the\nvertex~(\\ref{dominant vertex: dim > 4}) is also gauge dependent, and has on\nits own no physical meaning. Indeed, it is well known that,\nwhen working with gauge dependent quantities, one can reach realiable conclusions\nonly when one takes account of all terms up to this order.\nNamely, different gauge dependent terms can cancel each other, which\nwas not accounted for in Ref.~\\cite{Bezrukov:2011sz}, see {\\it e.g.} Ref.~\\cite{George:2013iia}.\nFurthermore, when written in terms of gauge invariant variables,\ngauge dependent vertices (such as $\\varphi^2\\Box \\delta g$)\ncan be absorbed into the lower order action, such that they simply `disappear' from\na gauge invariant action. Next, some metric perturbations are non-dynamical\nin the sense that they act as auxiliary (constraint) fields and should be solved for,\nwhich generate additional vertices that may cancel\nthe problematic vertex. In fact, this already\nhappens at the level of the quadratic action. Na\\\"\\i vely,\nthe field perturbation has an effective mass term\n$m^2_{\\rm eff}\\varphi^2=(-\\xi \\bar{R} + V'')\\varphi^2+..$,\nwhere $\\bar{R}=6(2H^2+\\dot{H})$ is the background Ricci scalar. Such a mass term\nis huge, in the sense that $|\\xi \\bar{R}| \\gg H^2$.\nHowever, only a light inflaton field, with $m_{\\rm eff}^2 \\ll H^2$\ncan generate a nearly scale invariant power spectrum.\nThus, such a mass term is disastrous for the model\nand na\\\"\\i vely rules out Higgs inflation. But, when\ncontributions from the auxiliary fields are taken into\naccount, the problematic $\\xi \\bar R$ contribution gets canceled, leaving\nonly a light effective mass for the inflaton field.\nSimilar cancellations occur at higher orders in a gauge\ninvariant formulation.\n\n\\item[{\\bf b)}] {\\bf The canonical redefinition mixes up frames.}\n\nA crucial step in the computation of the cutoff\nwas the definition of new perturbations $\\hat{\\varphi}$\nand ${\\delta \\hat g}$ which canonically normalize\nthe kinetic terms. However, this redefinition\nis in fact nothing more than a transformation\nfrom the Jordan to the Einstein frame at the level\nof perturbations~\\footnote{This can be explicitly checked\nby comparing Eqs. (2.9) and (2.10) in Ref.~\\cite{Bezrukov:2010jz}\nto the transformations\n\\[\n\\varphi_E=\\frac{d\\phi_E}{d\\phi}\\varphi+{\\cal O}(\\varphi^2)\n\\,,\\qquad\n\\zeta_E=\\zeta+\\frac{1}{2F}\\frac{dF}{d\\phi}\\varphi+{\\cal O}(\\varphi^2)\n\\,,\\qquad\n\\gamma_{ij,E}=\\gamma_{ij}\n\\]\nand the expansion of $g_{\\mu\\nu,E}=\\Omega^2 g_{\\mu\\nu}$ to first order\nin perturbations.}!\nOn the other hand, the cutoff is computed\nfrom the term $\\xi\\Phi^2 R$ in the Jordan frame action. So somehow one\ncomputes a cutoff from a Jordan frame vertex using\nEinstein frame perturbations, which is very bizarre.\nMoreover, we would like to emphasize that, although\nthe Einstein frame is often referred to as the frame\nin which both the gravitational action and scalar field action\nare written in canonical form, \\textit{the Einstein frame is not canonical}.\nCanonical formulation means that the kinetic sectors for\ndifferent fields are decoupled (and canonically normalized), such\nthat one can straightforwardly extract the canonical\nmomentum and quantize the theory. However, in the \"canonical\" Einstein\nframe the gravitational field still couples to\nthe scalar sector as $\\sqrt{-g_E}g^{\\mu\\nu}_E \\partial_{\\mu}\\Phi_E\\partial_{\\nu}\\Phi_E$.\nConversely, the scalar field still couples to the kinetic term for\nthe metric perturbations in the $\\sqrt{-g_E}R_E$ term via\nthe auxiliary fields in the metric (which contain\n$\\varphi_E$ in its first order solution). Hence,\nthe Einstein frame is formulated in a non-canonical\nway, just like the Jordan frame. The true\ncanonical formulation is only reached once one inserts\nperturbations and decouples physical and non-physical\ndegrees of freedom, which was done in Ref.~\\cite{Prokopec:2013zya},\nand the results of which are summarized in section~\\ref{sec: gauge invariance}.\nWhen using the frame independent variables $W_\\zeta$ and $\\tilde\\gamma_{\\zeta,ij}$\nthe Jordan frame quadratic and cubic actions\nbecomes manifestly equal to the Einstein frame actions, see Eq.~(\\ref{from Jordan to Einstein}).\nThus in a frame independent formulation the notion of frames becomes meaningless.\n\n\\item[{\\bf c)}] {\\bf Inequivalence of Jordan and transformed Einstein cutoff.}\n\nAs we have mentioned, the cutoffs computed directly\nin the Jordan frame, or $\\textit{via}$ the Einstein frame,\nare similar, but not exactly the same. However, they should\ncoincide, because no physical content is lost in the frame\ntransformation. Of course the origin of this problem is related\nto the already mentioned problems, namely a non-invariant formulation\nand a mixing-up of different frames.\n\n\\end{itemize}\n\n\n\\subsection{Frame independent computation of cutoff}\n\\label{Frame independent computation of cutoff}\n\n This subsection we show that the above problems become all obsolete\nonce the theory is written in a manifestly gauge invariant and frame\nindependent way. Firstly, when one makes use\nof gauge invariant variables, all vertices are physical\nvertices. Moreover, the gauge invariant perturbations\ndecouple in the quadratic action, which makes it very\neasy to write the theory in canonically normalized form.\nAnd obviously, when frame independent perturbations are\nused, results in the Jordan and Einstein frames become manifestly equivalent.\n\n As mentioned above, a simple way of estimating the unitarity cutoff scale for\ntree tree-level scattering amplitudes is to work with the canonically normalized\nfields~(\\ref{3rdg:canonicalvariables}), for which the propagator in the\nultraviolet acquires a simple form: $\\sim (\\eta_{\\mu\\nu}k^\\mu k^\\nu)^{-1}$\nplus corrections that are suppressed by ${\\cal H}^2=(aH)^2$,\nwhere $k^\\mu=(E_c\/c,\\vec k\\,)$ denotes a conformal energy and momentum\n(the corresponding physical energy and momentum are given by $E=E_c\/a$ and $\\vec k\/a$).\nUp to boundary terms, the cubic actions for various combinations of scalar and tensor\nvertices are given in Eqs.~(\\ref{pure scalar cubic:canonical}--\\ref{dominant ggg}).\n\n Let us first consider the scalar cubic action~(\\ref{pure scalar cubic:canonical}).\nThe cubic vertex ${\\cal V}$ from the first two lines is of the order,\n\\begin{equation}\n{\\rm Scalar\\; cubic\\; vertex:}\\quad\n{\\cal V}_{V_\\zeta^3} \\sim \\frac{\\epsilon_E^{1\/2}{\\rm max}\\big[E_c^2,\\|\\vec k\\|^2\\,\\big]}{a\\sqrt{F}}\n\\,,\n\\label{scalar cubic vertex}\n\\end{equation}\nwhere we made use of $\\partial_\\tau\\rightarrow E_c$,\n$\\partial_i\\rightarrow k^i$ and we took ultraviolet limit in which\n$(az)^\\prime\/(az)\\sim {\\cal H}\\ll E_c, \\|\\vec k\\,\\|$.\nThe terms in the third line in~(\\ref{pure scalar cubic:canonical}) lead to a vertex that is\nin addition suppressed by $\\epsilon_E$, which is during inflation less than unity, and hence\ncan be neglected. Finally, the vertex contribution from the fourth line is of the order\n$\\epsilon_E^{-1\/2}[\\epsilon_E^\\prime\/(\\epsilon_E{\\cal H})]^\\prime E_c$, that means it is\nproportional to a third order slow roll parameter,\n$\\epsilon^{(3)}=[\\epsilon_E^\\prime\/(\\epsilon_E{\\cal H})]^\\prime\/(\\epsilon_E{\\cal H})$,\nand which can be assumed to be small during inflation. More precisely,\nthis vertex contribution is suppressed with respect to~(\\ref{scalar cubic vertex}) if\n${\\rm min}\\big[E_c,\\|\\vec k^2\\|\/E_c]\\gg [\\epsilon_E^\\prime\/(\\epsilon_E{\\cal H})]^\\prime\/\\epsilon_E\n=\\epsilon^{(3)}{\\cal H}$, which one can safely assume to be the case throughout inflation.\nOn the other hand, we know that in four space-time dimensions a 2-to-2 scattering aplitude\nscales as,\n\\begin{equation}\n\\frac{{\\rm max}[E_c^2,\\|\\vec k\\,\\|^2]}{\\Lambda^2}\n           \\sim \\frac{{\\cal V}^2}{{\\rm max}[E_c^2,\\|\\vec k\\|^2]}\n\\,.\n\\label{general cutoff for 2-2 scattering}\n\\end{equation}\nUpon combining this with~(\\ref{scalar cubic vertex}) we finally get for the physical cutoff\nin the Jordan frame,\n\\begin{equation}\n{\\rm Scalar\\; cubic\\; vertex:}\\quad\n\\left(\\frac{\\Lambda}{a}\\right)_J \\sim \\sqrt{\\frac{M_{\\rm P}^2+\\xi\\phi^2}{\\epsilon_E}}\n           \\gtrsim \\sqrt{M_{\\rm P}^2+\\xi\\phi^2}\n\\,,\n\\label{cutoff Jordan: pure scalar}\n\\end{equation}\nwhere we took account of $F=M_{\\rm P}^2+\\xi\\phi^2$\nand $\\epsilon_E\\lesssim 1$ during inflation.\nIn conclusion, we have found that during entire Higgs inflation\nfor scatterings mediated by the scalar cubic interactions the unitarity bound is\nabove the Planck scale. In fact, the cutoff in Higgs inflation is higher than the\nunitarity cuttoff in the minimally coupled inflation, $\\sim M_{\\rm P}$.\nwhich is obtained by simply setting $\\xi\\rightarrow 0$ in~(\\ref{cutoff Jordan: pure scalar}).\n\n Now making use of~(\\ref{from Jordan to Einstein}),\nfrom which we see that\n\\begin{equation}\n a_E=a_J \\frac{F^{1\/2}}{M_{\\rm P}}\n\\,,\n\\label{scale factor in two frames}\n\\end{equation}\nthe cutoff~(\\ref{cutoff Jordan: pure scalar}) can be written in the Einstein frame as\n\\begin{equation}\n\\bigg(\\frac{\\Lambda}{a}\\bigg)_E \\sim \\frac{M_{\\rm P}}{\\epsilon_E} \\gtrsim M_{\\rm P}\n\\,.\n\\label{dominant scalar cubic vertex:4}\n\\end{equation}\nThe difference between the frames can be attributed to the frame dependence of the\nphysical cutoff scale $\\Lambda\/a$, and has no physical meaning.\nTherefore, to make a meaningful comparison of the Einstein and Jordan frame cutoffs,\nthe rescaling~(\\ref{scale factor in two frames}) has to be taken account of.\nHence, we conclude that there is a perfect agreement in the cutoff scale in two different frames,\nand in both frames the physical cutoff is above the Planck scale $M_{\\rm P}= 1\/(8\\pi G_N)^{1\/2}$.\n\n\nLet us now turn our attention to other cubic vertices.\nFrom Eqs.~(\\ref{dominant ssg}--\\ref{dominant ggg}))\nwe obtain the dominant contributions for the other types of vertices,\n\\begin{eqnarray}\n&{\\rm Scalar-scalar-tensor\\; vertex:}\\quad &\n{\\cal V}_{V_\\zeta^2\\Gamma_\\zeta}\\sim\n            \\frac{1}{aF^{1\/2}}\\times{\\rm max}\\Big[\\epsilon_EE_c^2, \\|\\vec k\\,\\|^2\\Big]\n\\nonumber\\\\\n&{\\rm Scalar-tensor-tensor\\; vertex:}\\quad &\n{\\cal V}_{V_\\zeta\\Gamma_\\zeta^2}\\sim\n          \\frac{\\epsilon_E^{1\/2}}{aF^{1\/2}}\\times{\\rm max}\\Big[E_c^2, \\|\\vec k\\,\\|^2\\Big]\n\\nonumber\\\\\n&{\\rm Pure\\; tensor\\; vertex:} \\qquad\\qquad\\quad\\quad\\;&\n{\\cal V}_{\\Gamma_\\zeta^3}\\quad\\sim \\frac{\\|\\vec k\\,\\|^2}{aF^{1\/2}}\n\\label{dominant other cubic vertices}\n\\end{eqnarray}\nWhen these are inserted into~(\\ref{general cutoff for 2-2 scattering})\none obtains the following results for the cutoff scales from different types of vertices,\n\\begin{eqnarray}\n&{\\rm Scalar-scalar-tensor\\; vertex:}\\quad &\n\\left(\\frac{\\Lambda}{a}\\right)_J\n\\sim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\\times{\\rm min}\\Big[\\epsilon_E^{-2},1\\Big]\n    \\gtrsim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\n\\nonumber\\\\\n&{\\rm Scalar-tensor-tensor\\; vertex:}\\quad &\n\\left(\\frac{\\Lambda}{a}\\right)_J \\sim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\n        \\times{\\rm min}\\Big[\\epsilon_E^{-1},1\\Big]\\gtrsim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\n\\label{cutoffs individual vertices}\n\\\\\n&{\\rm Pure\\; tensor\\; vertex:} \\qquad\\qquad\\quad\\quad\\;&\n\\left(\\frac{\\Lambda}{a}\\right)_J \\sim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\n       \\times{\\rm min}\\Big[E_c^2\/\\|\\vec k\\|^2,1\\Big]\\gtrsim \\sqrt{M_{\\rm P}+\\xi\\phi^2}\n\\,,\\qquad\n\\nonumber\n\\end{eqnarray}\nwhere the first expression in square brackets gives a cutoff for the\ncase when $E_c\\gg \\|\\vec k\\|$, and the latter for the case when  $E_c\\ll \\|\\vec k\\|$.\nAnalogous conclusions as in~(\\ref{cutoffs individual vertices})\nare reached when one considers $2-to-2$ scatterings composed by two classes of \nvertices, {\\it e.g.} a combination of a scalar-scalar-graviton and a pure graviton vertex.\n\nIn summary, we have shown that in all cases, for all kinds of vertices and throughout inflation\nthe physical cutoff in the Jordan frame is\nabove the scale $\\sqrt{M_{\\rm P}+\\xi\\phi^2}$, while in the Einstein frame it is\nabove $M_{\\rm P}$ (the difference has no physical meaning and it is attributed to the\nnon-invariant definition of the physical cutoff in the two frames)\n\\footnote{It comes as no surprise that the canonically normalized\nscalar-scalar-graviton and pure graviton vertices are not suppressed\nby a factor of $\\sqrt{\\epsilon_E}$. The reason is that in the\nde Sitter limit $\\epsilon_E \\rightarrow 0$ the curvature perturbation $\\zeta$\nbecomes a pure gauge mode, and is completely absorbed by the gauge invariant\nlapse and shift perturbations. The only remaining dynamical perturbations\nare the scalar $\\varphi$ and the graviton $\\gamma_{ij}$.\nThe term $g^{\\mu\\nu}\\partial_{\\mu}\\Phi\\partial_{\\nu}\\Phi$ in the original action\nthen gives the interaction term\n$\\gamma_{ij}\\partial_i\\varphi\\partial_j\\varphi$,\nwhich is not $\\epsilon_E$ suppressed.\nLikewise, the pure graviton vertices are always present\nand are not suppressed by powers of $\\epsilon_E$.}.\n\n\n\\begin{figure}[ttt]\n\\begin{center}\n\n\\includegraphics[width=0.8\\textwidth]{cutoffJordan.pdf}\\\\\n  \\caption{Physical cutoff $(\\Lambda\/a)_J=\\sqrt{M_P^2+\\xi\\phi^2}$ in the Jordan frame.}\n\\label{fig:cutoffJordan}\n\\end{center}\n\\end{figure}\nThe Jordan frame cutoff is shown in figure\n\\ref{fig:cutoffJordan}.\nThe important point is that also\nhere the cutoff is never smaller than\nthe Planck scale. This means that\nthe perturbative expansion is valid\nat least up to an energy scale of the order the Planck\nscale for a theory with some non-minimal coupling to gravity, such as Higgs\ninflation. Thus, the above analysis strongly suggests that,\nat least for the class of tree level $2\\rightarrow 2$ scattering\nprocesses considered here, there is no naturalness problem in Higgs inflation.\n\nThere are caveats to this statement however. Firstly,\nwe have only considered the scalar-gravity sector\nof Higgs inflation, but neglected interactions\nwith \\textit{e.g.} gauge fields. In Ref.~\\cite{Burgess:2010zq} it was stated that\nthe (low) cutoff scale of $M_P\/\\xi$ also appears in\nthe Higgs-gauge interactions. However, these vertices\nare gauge dependent as well, so the problem may be absent in a gauge invariant\nformulation that includes the gauge fields.\nSecondly, for the computation of the cutoff\nwe have used the partially integrated\nactions~(\\ref{pure scalar cubic:canonical}--\\ref{dominant ggg}).\nHad we not made use of partial integrations, we would\nhave found disastrous terms in the action.\nFor example, before any partial integration, the leading term in\nthe pure cubic scalar action for $V_\\zeta=W_{\\zeta}\/(az)$ from Ref.~\\cite{Prokopec:2013zya}\ncontributes (in the limit $E_c\\gg \\|\\vec k\\|$) to the cubic vertex as,\n$(\\epsilon_E F\/a)\\times{\\rm max}[E_c^2,\\|\\vec k\\|^2]$, implying a physical\ncutoff (in the Jordan frame),\n$(\\Lambda\/a)_J\\sim \\sqrt{\\epsilon_E(M_{\\rm P}^2 +\\xi\\phi^2)}\\times {\\cal H}\/E_c$, which\nis much below the Planck scale, and hence disastrous.\nSimilar problems occur when the scalar-graviton-graviton\nvertices before partial integrations are considered.\nTherefore, for a more complete understanding of naturalness\nit is of crucial importance to understand the role of\nthe boundary terms (on equal time hypersurfaces).\n\n\n\\section{Discussion}\n\\label{sec:Discussion}\n\n We have used the frame and gauge independent formalism\nfor scalar and tensor cosmological perturbations of Ref.~\\cite{Prokopec:2013zya}\nto show that the physical cutoff for 2-to-2 tree level scatterings in Higgs inflation\nis above the Planck scale $M_{\\rm P}=1\/\\sqrt{8\\pi G_N}$ throughout inflation.\nMore precisely, we found that in the Jordan frame, the physical cutoff scale\nis $(\\Lambda\/a)_J\\gtrsim\\sqrt{M_{\\rm P}^2 +\\xi\\phi^2}$, while in\nthe Einstein frame it is $(\\Lambda\/a)_E\\gtrsim M_{\\rm P}$, where $\\xi$ is the nonminimal coupling\nand $\\phi(t)$ denotes the Higgs {\\it vev}. The physical cutoff in the Jordan frame is illustrated\nin figure~\\ref{fig:cutoffJordan}. The difference between the two frames\nis immaterial in that it can be fully attributed to the frame dependence of\nthe (physical) cutoff, see Eq.~(\\ref{scale factor in two frames}).\nOur results are incomplete, in that we have not discussed the relevance of:\n\\begin{itemize}\n\\item[$\\bullet$] quartic vertices and loops,\n\\item[$\\bullet$] vertices containing gauge and fermionic fields,\n\\item[$\\bullet$] boundary terms on equal time hypersurfaces (that result from partial integrations),\n\\end{itemize}\nfor the question of naturalness in Higgs inflation.\nWe do however believe that the principal conclusion reached in this paper will not change\nwhen these contributions are fully accounted for.\n\n\n\\section*{Acknowledgements}\n\n We thank Damien George, Sander Mooij and Marieke Postma\nfor useful discussions. This work was in part supported by Nikhef, by the D-ITP consortium, a\nprogram of the NWO that is funded by the Dutch Ministry of Education,\nCulture and Science (OCW) and by the Institute for Theoretical Physics of Utrecht University.\n\n\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBars, or non-axisymmetric central light distributions, are found in about\n2\/3 of the disc galaxies in the present-day universe (Sellwood \\&\nWilkinson \\shortcite{SW93}).\n Mechanisms for bar formation have been explored quite extensively\n(Noguchi \\shortcite{NO88},  \\shortcite {NO96A}, \\shortcite\n{NO96B}, Shlosman \\& Noguchi \\shortcite{SN93}). As a result of cooling, gas discs are subject\nto gravitational instability and bars can arise spontaneously.\nIn galaxies of earlier types, due the stabilizing effect of the\ngrowing bulge, the disc is much less prone to spontaneous\nbar formation. A stronger perturbation is then needed, like an interaction\nwith a companion, a merger or the tidal forces of  a galaxy cluster.\nThe bar will not last forever because it may dissolve \nwhen the core mass becomes important, and will contribute to\nthe growth of the bulge (\\cite{NS96}). This implies that some bars are young, and\nthat some bulges could contain relatively young stars.\n\nThe fact that the heavy element abundance distribution\nis flatter in barred galaxies as compared with normal spirals of\nsimilar type can be explained by the action of inward\nand outward radial flows of interstellar gas induced by the non-axisymmetric potential\nof bars (Roberts et al. \\shortcite{RO79}; Sellwood \\& Wilkinson \\shortcite{SW93};\nFriedli et al. \\shortcite{FR94}). As a consequence,\nthe slope of any pre-existing radial\ngaseous or stellar abundance gradient decreases\nwith time. The long term effect is such\nthat the stronger the bar is, the flatter the abundance\ngradient becomes with time (Friedli \\& Benz \\shortcite{FB95}). The finding\nby Martin \\& Roy \\shortcite{MR94} of a relationship between the slopes of the global\n abundance gradient and the strength of bars provides support for the scenario\nof recently formed bars (Roy \\shortcite{RO96B}). \nThus bars may not necessarily  be primordial\nfeatures (Combes \\& Elmegreen \\shortcite{CE93}), but could also form at any time\nduring the lifetime of a galaxy (Friedli \\& Benz \\shortcite{FB95}; \nMartinet \\shortcite{MAR95}; Martin \\& Roy\n\\shortcite{MR95}).\n\nIn late-type galaxies (SBc, SBb), a young bar ($\\leq$1 Gyr) will be characterized as being\na gas-rich structure. Such bars can be the site of vigorous  \nstar formation;\nif the star formation process is not inhibited by the high cloud velocities,\nthe radial abundance distribution will present\na steep inner gradient (homogenizing effects being compensated by\nchemical enrichment due to vigorous star formation),\ncombined with a flatter gradient beyond corotation  due\nto dilution by the outward radial flows, as shown by Friedli et al. \\shortcite{FR94}\nand Friedli \\& Benz \\shortcite{FB95}.\nOn the other hand, if the level of star formation in the bar is modest or absent\ndue to inhibiting forces, the radial abundance gradient\nwould be weak both in the bar and in the disc. In both cases, with\nyoung bars, one should see breaks in the radial abundance\ndistribution, which move outward in the disc as times evolves.\nSome bulges, because they result in part from the stars in the bar being\nscattered out of the galaxy plane, should be polluted by `young' stars\naged between 0.5 and 1.0 Gyr. Although one cannot exclude the \npresence of young stars in bulges, they are very difficult to detect.\nThe interstellar gas is a better laboratory to search for young bars.\nThe best candidate found so far for\na galaxy with a young bar is NGC 3359 where Martin \\& Roy \\shortcite{MR95}\nhave observed an abrupt O\/H  gradient in the central region (which \nhas an intense episode of star formation), and a flat gradient beyond\ncorotation. NGC 3319, observed by Zaritsky\net al. \\shortcite{ZA94}, appears to be a case similar to NGC 3359.\n\nIn this paper we wish to explore the O\/H radial distribution in\nthe prominent southern bar galaxy NGC 1365 in order to test \nthe scenario of recent bar formation. NGC 1365 is as close to an archetype \nbarred galaxy as one can find in the nearby universe. Its bar is very\nrich in gas and it has moderate star forming activity.\n\n\n\n\\section{Observations and data reduction}\n\n\\subsection{The galaxy NGC 1365}\n\nNGC 1365, a dominant galaxy of the Fornax cluster, is probably the most spectacular\nof the nearby barred galaxies. Classified as SB(s)b I-II by de Vaucouleurs et al. \n\\shortcite{dV91} (RC3), it has a Seyfert 1.5 type\nnucleus and  a `hot spot' nuclear region\n(Sersic \\& Pastoriza \\shortcite{SP65}; Anatharamaiah et al. \\shortcite{AN93};\n Sandqvist et al. \\shortcite{SA95}). A distance for NGC 1365 of\n18 Mpc, as recently determined by a HST-WFPC2 Cepheid study  \n\\cite{MA96}, is adopted. The galaxy has clearly defined offset dust lanes \non the leading edge of the bar and along its two main spiral arms. A  velocity jump has\nbeen observed across the eastern and western dust lanes (\\cite{LJ87}) indicative of strong shocks in the gas flow (Athanassoula \\shortcite{AT92}).\nThe general properties of NGC 1365 are listed in Table 1.\n\n\\begin{table}\n\\caption{Global properties of NGC 1365} \n\\begin{tabular}{lc}\n\\hline \\hline\nParameter & Value \\\\ \n\\hline \n$\\alpha$ (J2000) & 3$^{\\rm h}$33$^{\\rm m}$37.$^{\\rm s}$37 \\\\\n$\\delta$ (J2000) &  -36$^\\circ$08$'$25.$''$5 \\\\\nMorphological type & SB(s)b I-II\\\\\nInclination & 40$^\\circ$\\\\\nPosition angle & 220$^\\circ$ \\\\\nGalactic extinction ($A_{\\rm B})$ & 0.21\\\\\nSystemic velocity (km s$^{-1}$) & 1640 \\\\\n$\\rho_{0}$  & 5.$'$61 \\\\\n$b\/a(i)$ & 0.51 \\\\\nD (Mpc) & 18.2\\\\\nScale (pc arcsec$^{-1}$) & 88 \\\\\n\\hline\n\\end{tabular}\\\\\nNotes: (a) Positions, angles and velocity\nare from J\\\"ors\\\"ater \\& van Moorsel \\shortcite{JM95}; (b) type, extinction\nand isophotal radius are from\nde Vaucouleurs et al. \\shortcite{dV91}; stellar bar axis\nratio corrected for inclination is from Martin \\shortcite{MA95};\n(d) Distance is from Madore et al. \\shortcite{MA96}\n\\end{table}  \n\n\nSpectrophotometry of several {H~\\sc ii}\\  regions in the galaxy was obtained by\nPagel et al. \\shortcite{PA79}, Alloin et al. \\shortcite{AL81} and  \nRoy \\& Walsh \\shortcite{RW88}; the\nearlier observations, although limited, suggested relatively\nhigh extinction, and a shallow global \nO\/H abundance gradient, a now well-established feature of barred\ngalaxies (Vila-Costas \\& Edmunds \\shortcite{VE92}; Zaritsky et al. \\shortcite{ZA94};\nMartin \\& Roy \\shortcite{MR94}). Star formation activity is moderate or weak\nin the bar, except in the nuclear ring (radius $\\sim$ 7$''$).\n\nNGC 1365 has been studied in {H~\\sc i}\\  (e.g. Ondrechen \\& van der Hulst \\shortcite{OV89}), but\nit is only recently that a detailed {H~\\sc i}\\  investigation with the VLA  \nhas been completed and published by\nJ\\\"ors\\\"ater \\& van Moorsel \\shortcite{JM95}; these authors show that\nNGC 1365 has a unique dropping rotation curve. Sandqvist et al. \n\\shortcite{SA95} performed\nradio continuum and CO observations of the central parts of the galaxy,\nfinding a radio jet and enhanced CO in the nucleus\nand dust lanes; references to earlier radio work can also be found in\nthat paper. Sandqvist et al. \\shortcite{SA95} estimated that the amount\nof molecular gas in the nuclear and bar region, is equal to the\ntotal amount of neutral atomic hydrogen in the galaxy,\nthat is 15 $\\times$ 10$^9$ M$_\\odot$.\nThe {H~\\sc i}\\  distribution shows a hole in the central region where CO is the strongest and\nthe neutral hydrogen is predominantly located in the spiral arm.\nThe bar region is a very gas-rich feature. \n\n\n\\subsection{Selection of H~{\\sc II} regions}\n\nAn AAT prime focus plate of NGC~1365 taken in the R band (RG 610 filter + \nO94-04 emulsion) on 1990 December 16 was scanned on the ESO PDS machine.\nThe triplet corrector was employed giving a scale of 15.$''$3 mm$^{-1}$ and the\nscanning aperture was 50 $\\mu$m and sampling 25 $\\mu$m (0.$''$381). This\nimage is shown in Figure 1; 163 potential {H~\\sc ii}\\  regions were identified and\ntheir X,Y positions determined by either fitting a 2-D Gaussian or more\noften simply by centroiding the image by eye. Given that there was no\noff-emission band image with which to check whether the identified regions\nwere indeed {H~\\sc ii}\\  regions, some contamination by stars, globular clusters\nor distant galaxies was anticipated. In addition to the {H~\\sc ii}\\  regions candidates, \nthe X,Y positions of HST GSC stars were measured by fitting 2-D Gaussians and a six\ncoefficient astrometric solution was made for 23 GSC stars. The RA and Dec\nof the potential {H~\\sc ii}\\  regions was then calculated. 'Sky' positions were also\nselected away from the galaxy and free of stars. In selecting {H~\\sc ii}\\  regions\nto observe, a number of considerations were taken in account: \n\n\\begin{enumerate}[(iv)]\n\\item about 55 object and sky fibres can be observed per fibre bundle per \naperture plate; \n\\item the full range of {H~\\sc ii}\\  region apparent brightness should be covered in \norder to avoid any bias; \n\\item fibres cannot be closer than 19$''$. Thus only one {H~\\sc ii}\\  region could\nbe selected in a crowded region; a nearby {H~\\sc ii}\\  region could of course\nbe selected for a second aperture plate.\nSome bright {H~\\sc ii}\\  regions should be included in common between\ndifferent aperture plates to check the observation quality and \nreproducibility of calibration; \n\\item the faintest {H~\\sc ii}\\  regions could be included on more than one aperture plate\nto increase the exposure time.\n\\end{enumerate}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig1.eps,height=18.0cm,clip=}}\n\\caption{R band image of NGC 1365 obtained by David Malin\nat the f\/3.3 prime focus of the 3.9-m Anglo-Australian Telescope.\nThe numbers correspond to the {H~\\sc ii}\\  regions of Table 2; the\n{H~\\sc ii}\\  regions are the brightest object seen closest to the number. G-1, G-2 and\nG-3 are more distant galaxies discovered serendipitously. North is\nup and east is left.}\n\\label{fig1}\n\\end{figure*}\n\n  The final choice of {H~\\sc ii}\\  regions selected is indicated by the numbered\nregions on Figure 1. These were divided between two aperture plates which\nwere observed on different nights. Both plate A (observed on 1994 December 09)\nand plate B (observed on 1994 December 10 and 11) had 49 fibres on {H~\\sc ii}\\  regions \nand 7 on sky positions. There were 30 regions in common between the two plates:\n8 had no detectable emission; 7 were so faint that the better exposure on\n1993 December 11 only was used for analysis; the remaining 15 spectra\nwere combined: 5 had very faint emission and 5 were among the\nbrightest {H~\\sc ii}\\  regions observed. For targets not in common, \n9 objects were rejected which had insufficient signal-to-noise in the emission\nlines. Three objects (G-1, G-2 and G-3 on Figure 1) were found to be\ndistant galaxies (See Appendix). Judging the \nstrength of line emission from the appearance on a broad band image is clearly\nnot sufficient since some regions have stronger continuum, particularly\nat smaller galactocentric radius.  The final number of distinct {H~\\sc ii}\\  region\nspectra obtained with well measured lines was 55. \n\n\\subsection{Fibre observations}\nLow dispersion spectra of the selected targets in NGC 1365 were obtained\nusing the RGO spectrograph (25 cm camera) and the Tektronix \\#2 1024 $\\times$ 1024\nCCD (24 $\\mu$m pixels) on the 3.9 m Anglo-Australian Telescope during the nights of 1993\nDecember 9-12. The FOCAP system with its bundle of about 55 fibres\nof 300 $\\mu$m core diameter (2.0$''$ on sky) was employed to \nfeed the spectrograph; the FOCAP system was used because it allowed\ncloser positioning of fibres than AUTOFIB.\nA 250 lines per mm grating was used in the\nfirst-order blaze to collimator configuration for a dispersion of 156 \\AA\\ mm$^{-1}$.\nMost of the {H~\\sc ii}\\  regions observed in NGC 1365 have\ndiameters in the range of 5$''$ -- 10$''$, much bigger\nthan the fibre apertures. To increase the area sampled by each fibre,\nthe telescope was set in a continuous scan motion, by moving it on a circle\nof 1$''$ radius at a rate of one full circle per 30 s. Convolved\nwith seeing of 1$''$ -- 2$''$, the effective sampling area was\n$\\sim$ 3$''$ -- 4$''$ in diameter. The  Tektronix CCD is a thinned device and was used\nin the XTRASLOW mode (394 sec readout time) to achieve the lowest readout noise \nof 2.3 e$^-$ rms.\nThe CCD  blue response is good, being  greater than 40\\% at 3727 \\AA. \n\n  Six 2000s exposures were taken with a first plate on 1993 December 09-10, one of which was\naffected by cloud. A second aperture plate was used for observations on the\nthird night (1993 Dec 11-12) when conditions were good, seeing $\\sim$1$''$,\nand seven 2000s exposure were performed. The {H~\\sc ii}\\  regions indicated on Figure 1 are only those for which the emission\nlines were analysed, the spectra containing at least detectable H$\\alpha$ and \nH$\\beta$ emission. Three of the objects rejected as having weak emission line spectra\nwere found to have high redshifts. They are indicated by a letter G prefix in\nFigure 1 and are discussed in the Appendix.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig2.eps,height=18.0cm,clip=}}\n\\caption{Examples of spectra of {H~\\sc ii}\\  regions in NGC 1365 obtained with FOCAP. \nThe region numbers refer to those in Figure 1 and Table 2. No correction for interstellar \nextinction has been applied.}\n\\label{fig2}\n\\end{figure*}\n\n\n\\subsection{Spectrophotometry with optical fibers}\n\nSpectroscopy using optical fibers for multiplexing is now a widely-used \ntechnique in astronomy. A great number of spectra are being\nobtained but, unfortunately, little attention has been given\nto the promising astrophysical potential of having usable spectral energy\ndistributions for so many sources in addition to radial velocities.\nThere is a general consensus that spectrophotometry cannot be done with fibres.\nHowever in the multi-fibre spectroscopy of 49 {H~\\sc ii}\\  regions in NGC~2997 (Walsh \\& Roy \\shortcite{WR89}),\nit was\nshown that with suitable care, spectrophotometry to {\\bf better} than 20\\% can be \nachieved. Five problem areas for fibre spectrophotometry were outlined in \nWalsh \\& Roy \\shortcite{WR89}. The use of a CCD with high quantum efficiency,\nas opposed to the IPCS for the NGC~2997 observations, together with the multi-night\ncoverage of one galaxy, have changed some of the emphasis of the problem areas\nand led to improved understanding of the limitations. The problem areas can be\nrestated thus: dealing with differential atmospheric refraction; \neffect on spectra of cross talk between fibres; efficacy of sky subtraction; \nreliability of spectral extraction.\n\n\\begin{enumerate}[]\n\n\\item Atmospheric refraction\n\nThe fibres were 300$\\mu$m in diameter (2.0$''$) so that at zenith distances\nabove 30$^\\circ$ the differential atmospheric refraction between 3727 and 6730\\AA\\\n(the extent of the most useful optical wavelength range for {H~\\sc ii}\\  regions) is 1.0$''$\n(see Walsh \\& Roy \\shortcite{WR90}).\nThe maximum zenith distance at which exposures were made of NGC~1365 is \n40$^\\circ$ so that differential refraction will have a modulating effect on\nthe spectrum. However the telescope was moved in a circular pattern of 1$''$\nradius to improve the sampling of the extended {H~\\sc ii}\\  regions, and this also has the\neffect of averaging out the effects of differential atmospheric refraction\nfor modest airmasses.\n\nComparison of spectra of the same {H~\\sc ii}\\  region taken at zenith distances of 6\nand 35$^\\circ$ and corrected for airmass, show only small differences in\nline ratios between the blue and red ends. However for the observations\nof the spectrophotometric standard stars (L745-46A and L870-2 \\cite{OK74}) the\ntelescope was not circled during the exposure. Depending on the exact position of the\nstar image over the fibre and the airmass, then the input spectra will differ,\nmost obviously in the blue where the differential atmospheric refraction\nchanges rapidly with wavelength. As for the case for NGC~2997, four {H~\\sc ii}\\ \nregions were in common between the fibre spectroscopy and the scanned long\nslit spectroscopy (Roy \\& Walsh \\shortcite{RW87}). However for NGC~2997, the \nproblems of relying on\nthe spectrophotometric standard for the flux calibration proved insurmountable\nand a sensitivity curve was deduced by comparing the fibre spectra with the\ndata from a long slit over identical regions. Whilst the same method could have been\nrelied on for the NGC~1365 observations, it is clearly preferable to rely on an \nindependent spectrophotometric calibration. \n\n  A number of precautions were taken in the observations of the standards.\nAt least three, and sometimes five, separate observations, each of 300s \nduration, were made on the standards and at low to moderate airmass (zenith\ndistance $\\leq$ 27$^\\circ$). At each exposure the standard star was recentered in the\nguide fibre and then offset to the appropriate fibre. During data reduction \nthe individual spectra of the standards were compared in\nterms of absolute level and shape. The differences in absolute level\ncould be attributed to the star not being centered in the fibre combined with\nthe seeing, which was around 1$''$-1.5$''$. Differences\nin spectral distribution were also apparent, in particular a relative in(de)crease\nin the blue when comparing different exposures. This must be attributed to \ndifferent centering of the star within the fibre: a higher blue response occurring\nwhen the fibre was more centered to lower altitude zenith distance. \nSome conspiracy between shifts in the parallactic direction and \nperpendicular to it could produce such effects, although no attempt was made\nto model this behaviour. \n\nIn forming the mean spectrum of the standard for a\ngiven night, a weighted mean of the strongest spectra was made; however if\na spectrum differed markedly in shape from the others, it was not included in the mean.\nIt was suggested in Walsh \\& Roy \\shortcite{WR89} that improved spectrophotometry\nof standard stars could be achieved with very good seeing\nby scanning the telescope in an elongated\npattern in the direction of the parallactic angle. This was not attempted during\nthe observations described here. Such a strategy will of course \nimpair absolute spectrophotometry (which can only be achieved with good seeing\nfor fibres), but this is usually not a serious issue and\ncertainly was not of major concern here.\nThe agreement between spectra taken of the same {H~\\sc ii}\\  region on different nights\nboth with the same fibre and with a different fibre suggest that the strategy of \nscanning a small circle is sufficient to ensure repeatable spectrophotometry\nto within 5\\% and could also be applied to the observation of the standard\nstars. Incidentally the level of agreement implies that the colour\ntransmission of the fibres is similar, at least to tolerances lower than the \nspectrophotometric ones.\n\n  Inverse sensitivity curves were formed for L745-46A (first and third nights)\nand for L870-2 (second, third and fourth nights). The two curves for L745-46A were\nvery significantly different in the blue, the curve derived for the third night\nshowing a sharp minimum centred at 4000\\AA. For L870-2 the curves had the same general\nshape but with differences in absolute level; for the second and third nights\nthe relative differences did not exceed about 6\\%. The sensitivity curves\nwere used to calibrate the {H~\\sc ii}\\  region spectra and the extinction, derived from\ncomparison of the hydrogen line ratios with the Case B values, was computed.\nIt was found that the L870-2 sensitivity curves gave a more consistent \nvalue of the extinction (the Seaton \\shortcite{SE79} Galactic curve\nwas employed as parametrized by Howarth \\shortcite{HO83}) for the different \nhydrogen line ratios (e.g. H$\\alpha$\/H$\\beta$ vs. H$\\delta$\/H$\\beta$). It was \ntherefore decided to use a mean weighted spectrum of\nthe L870-2 data from three nights to produce the calibration curve. On the\nhighest signal-to-noise {H~\\sc ii}\\  spectrum  (\\#42, night 3) the \nextinction is the\nsame (within the errors of measurement) on all lines ratios from H$\\alpha$\/H$\\beta$ to\nH$\\epsilon$\/H$\\beta$. Thus the calibration produced by the adopted L870-2 \nspectrum is at least adequate over the range 3950 to 6600\\AA. \n\nComparison of the spectra with those of identical {H~\\sc ii}\\  regions\nobserved by Roy \\& Walsh \\shortcite{RW88} and Alloin et al. value \\shortcite{AL81}\nwas also made. There are 9 {H~\\sc ii}\\  regions in common with Alloin et al\n(L4, L6, L7, L10, L15, L21, L24, L28 and L33). There is poor agreement with\nthe IDS spectra of Alloin et al, where low extinctions ($c$=0) were derived; for the\nIPCS spectra, for L33 the derived extinction is much higher than Alloin et al.\nvalue, for L10 it is higher whilst for the others there is satisfactory \nagreement. The {\\rm \\mbox{[O \\sc ii]}\\ }\/H$\\beta$ ratios presented by Alloin et al. are \nsystematically lower than presented here.\nThe spectra were also compared with those of Roy \\& Walsh \\shortcite{RW88} for \n{H~\\sc ii}\\  region emission summed over much larger areas than the fibre data.\nThe fibre data indicated higher extinction and higher dereddened [O~II]\/H$\\beta$ \nratios than the summed {H~\\sc ii}\\  region emission. The higher extinction is most\nprobably a real effect whereby the bright cores have higher extinction, whilst\nthe higher {\\rm \\mbox{[O \\sc ii]}\\ }\/H$\\beta$ ratios may be affected by the\ndifficulties of fibre spectrophotometric calibration at lower wavelengths.\n\n\\item Cross-talk between fibres\n\n  On account of the much higher signal-to-noise\nof the CCD data in comparison with the previous IPCS data (Walsh \\& Roy \n\\shortcite{WR89}), a better\nmeasurement of the cross-talk from fibre to fibre could be made. For an observation\nof the standard star, the counts in the two adjacent fibres along the slit yielded\ncross talk of 1.9 and 1.0\\%, with an increase to 2.5\\% in the red ($>$7000\\AA) in\nthe fibre with the higher level of cross-talk. For the other fibre, with lower\ncross talk, the spectrum had the same shape as that in the star fibre. A related\neffect was noticed in that some spectra after flux calibration were very blue.\nThe effect appeared strongest on weaker spectra\nalthough was not confined to them, and was also not simply correlated to the strength\nof adjacent fibre spectra. It is suggested that the effect could arise\nfrom the flat fielding, since the flat-field lamp has a low response in the blue.\nThis would make the flat field in the blue very sensitive to small changes\nin packing between fibres along the slit and small irregularities would \ncause differential cross-talk. However it is not clear why this cross-talk should be\ndifferent between the flat field and the sky data. Sky flats might assist in\nbetter understanding of this problem. It is clear that the fibres should be well\nseparated at the spectrograph entrance slit and on the detector in order to allow \nwell-defined minima to be distinguished between the spectra.\n\n\\item Sky subtraction\n\nAs suggested by  Wyse \\& Gilmore \\shortcite{WG92}, at least six\nfibres were employed for determination of the sky - in fact 7 were used. After \nnormalising the\nfibre to fibre response by the flat field spectra integrated in wavelength, the\nmean sky spectrum was formed and subtracted from the object spectra. The efficacy of\nsky subtraction was tested by examining the residuals on the mean sky-subtracted \nsky spectra. Examining the goodness of subtraction of the strong sky emission \nlines alone is not very reliable since small wavelength shifts ($\\leq$0.3 pixels) \nbetween the fibre spectra can result \nin improperly subtracted sky lines, although the mean difference across the line\ncan be zero. Such small shifts can result from slight\ndifferences in the polynomial fit to the comparison lines along the\nslit. Comparison \nof the fibre-to-fibre response between the flat field and the [O~{\\sc i}]5577\\AA\\\nline shows good agreement, suggesting that the method of correcting for the\nrelative fibre transmissions is satisfactory.\n\n\\item Spectral extraction\n\n The packing of the fibres is such that the separation\nof the spectra peak-to-peak is 5 or 6 CCD pixels, and there is not\na true zero intensity between fibres. The spectra were extracted simply by summing \nas many fibre spectra per object as possible; for the better separated spectra\none signal-free pixel column was left between fibre spectra. A full extraction\nwould require solving for all the extracted spectra simultaneously. Shifting the\nextent of the extracted spectrum by one pixel causes an error of up to 5\\%. \nAssigning a column from the edge of one spectrum into the adjacent spectrum\ngives rise to an error of about 2\\%. The simple linear extraction could be\nimproved by fitting the cross-dispersion profile and by optimal extraction, but\nwould not substantially affect the results presented here.\n\nComparing the spectra of the same {H~\\sc ii}\\  \nregions taken on different nights and with different fibres shows that these\nsystematic effects dominate the photon statistical errors associated with\nthe line fitting. For the bright lines, e.g. {\\rm \\mbox{[O \\sc iii]}\\ } and H$\\alpha$, the typical\ndifferences in the line ratios (expressed as a fraction of H$\\beta$) are \n10\\%; for {\\rm \\mbox{[O \\sc ii]}\\ } this difference is around 20\\%, with values up to 30\\%.\nFor spectra taken on\ndifferent nights, but in the same fibre, the differences are only slightly larger\nthan the  photon statistical errors, although conditions were not photometric\non the first and second nights. It is clearly the problems of spectral\nextraction and spectrophotometric calibration that dominate the uncertainties in\nthe derived line fluxes.\n\n\\end{enumerate}\n\n\n\\subsection{Reduction of the spectra}\n\nAll the data frames were bias subtracted and the spectra and flat fields formed\nfor each fibre. Wavelength calibration was achieved by fitting third order\npolynomials to the integrated spectra of each fibre for a Th-Ar lamp. Corrections\nof each individual exposure for airmass were made and the data summed. Following\nflux calibration by the merged L870-2 spectrum, the spectrum of each target was\nextracted and \nan interactive procedure for fitting continuum and emission lines was used\nto derive the line fluxes $F(\\lambda)$ expressed  in units of \n$F(H\\beta)$ = 100. Photon noise errors on the fitted line fluxes were computed \nand propagated. The magnitude of the \ninterstellar reddening was determined by the H$\\alpha$\/H$\\beta$ \\ ratios;\ncomparison was done to  the\ntheoretical decrement as given by Brocklehurst \\shortcite{BR71} for a temperature\nof 8,000 K and a density of 100 cm$^{-3}$, but after adding 2 \\AA\\\nof equivalent width to the H$\\beta$ \\ emission line to compensate for\nthe underlying Balmer absorption (cf. McCall, Rybski \\& Shields \\shortcite{MC85};\nRoy \\& Walsh \\shortcite{RW87}). If the underlying stellar absorption\nat H$\\beta$\\ is greater than 2 \\AA\\, then this correction would lead\nto an overestimation of the reddening correction, especially in {H~\\sc ii}\\  regions\nof low H$\\beta$\\ emission equivalent width. The spectrum was corrected\nin detail as a function of wavelength using the \nstandard reddening law of Seaton \\shortcite{SE79}, as\nspecified by Howarth \\shortcite{HO83}, assuming R = 3.1.\n\n\\section{Results}\n\nFigure 2 shows some examples typical of the best spectra, uncorrected for \ninterstellar reddening. The region\nnumbers refer to the {H~\\sc ii}\\  region identification numbers in Figure 1\nand Table 2. The difference in the relative strengths of the collisionally excited\nlines of {\\rm \\mbox{[O \\sc iii]}\\ } , {\\rm \\mbox{[N \\sc ii]}\\ } and {\\rm \\mbox{[S \\sc ii]}\\ } illustrate the change in excitation across\nthe galaxy. Region \\#30 is close to the nucleus.\n Table 2 lists the line intensities (H$\\beta$ = 100) corrected for reddening for the\nmain emission lines of the 55 {H~\\sc ii}\\  regions observed. $X$ and $Y$ refer\nto RA and DEC offsets in arcsec relative to the centre of the galaxy given in Table 1. The logarithmic\nextinction at H$\\beta$, $c$, includes both Galactic reddening (contribution about 0.07, \nsee Table 1) and extragalactic extinction. $\\rho\/\\rho_0$ is the radial distance\nexpressed in terms of the fractional isophotal radius (Table 1).\nWe made only a marginal detection of the WC9 star\nin region \\#30  (L4 of \\cite{AL81}) found by Phillips \\& Conti \\shortcite{PC92};\nthis is in contrast to NGC~2997 where we observed 49 {H~\\sc ii}\\  regions\nand found signatures of Wolf-Rayet stars in three (and possibly four) regions.\n\n\\subsection{Diagnostic diagrams}\n\nSeveral line ratios have been calculated after correction for reddening.\nDefining {\\rm \\mbox{[O \\sc iii]}\\ }  as 1.34$\\times I${\\rm \\mbox{[O \\sc iii]}\\ } 5007\\AA, {\\rm \\mbox{[N \\sc ii]}\\ }\nas 1.34$\\times I${\\rm \\mbox{[N \\sc ii]}\\ } 6584\\AA\\ and {\\rm \\mbox{[S \\sc ii]}\\ } as $I${\\rm \\mbox{[S \\sc ii]}\\ } 6713$+$6730\\AA, and using\n{\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } as a sequencing parameter, Figure 3 shows {\\rm \\mbox{[O \\sc ii]}}\/{{\\rm \\mbox {[O \\sc iii]}}\\ },\n{\\rm \\mbox{[N \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } , {\\rm \\mbox{[S \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } and {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } versus this parameter. The\ntight relationships shown by these sequences imply that\nthe majority of the nebulae are ionization-bound; these trends\nare characteristic of gas volumes photoionized by massive stars\n(McCall et al. \\shortcite{MC85}), and very similar, for example, to those measured in \nM101 (Kennicutt \\& Garnett \\shortcite{KG96}). The tight sequence shown by \n{\\rm \\mbox{[N \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } indicates that the\nfibre spectrophotometry indeed produces reliable line fluxes.  The change\nof {\\rm \\mbox{[N \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } and {\\rm \\mbox{[S \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } versus {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } is driven by the thermal\nproperties of the {H~\\sc ii}\\  regions, and not by change in the nitrogen\nand sulfur abundance\nratios (Garnett \\& Shields \\shortcite{GS87}). The abundance of oxygen mainly controls\nthe thermal equilibrium in {H~\\sc ii}\\  regions (McCall et al. \\shortcite{MC85}).\n\n\n\\begin{table*}\n\\caption{H~{\\sc ii} regions in NGC 1365 -- Reddening-corrected line fluxes (H$\\beta$ = 100)}\n\\begin{tabular}{rrrrccccccc}\n\\hline \\hline\n\nRW &  X &    Y  &    [OII]  &    [OIII]    &    HeI  &      [NII] &   [SII]  &  c(H$\\beta$)    & $\\rho\/\\rho_0$ \\\\\n\\# & $''$ &  $''$ &   3727 &      5007    &     5876 &       6584   & 6717-30 &           &   \\\\\n(1) & (2)   &  (3)  &    (4)    &   (5)     &   (6)       &   (7)    &  (8)   &     (9)   &     (10)  \\\\ \n\\hline \n1 & 210  &  250  &  222$\\pm$8 & 158$\\pm$5 & 14.4$\\pm$2.6 & 54$\\pm$3 & 49$\\pm$3 & 0.51$\\pm$0.10 & 0.97 \\\\\n2 & 151  &  235  &  368$\\pm$17  &    70$\\pm$4   &  12.9$\\pm$3.0 &  81$\\pm$4 &   80$\\pm$4 & 0.66$\\pm$0.13 & 0.84 \\\\\n3 & 101  &   254  &   351$\\pm$9  &     66$\\pm$2 &  10.8$\\pm$2.0 &  78$\\pm$2  &  68$\\pm$3 &  0.75$\\pm$0.07 & 0.84 \\\\\n4 &  76  &   248  &   606$\\pm$51   &    63$\\pm$9  &       & 108$\\pm$10 & 137$\\pm$12 & 0.80$\\pm$0.25 & 0.81 \\\\\n5 &  23  &   216  &   249$\\pm$10  &    187$\\pm$6  &    11.7$\\pm$2.0 &  42$\\pm$2  &  56$\\pm$2 & 0.66$\\pm$0.10 & 0.72 \\\\\n6 &  26  &   207   &  364$\\pm$26  &    100$\\pm$8  &        &  99$\\pm$7  &  90$\\pm$9 & 0.74$\\pm$0.18 & 0.68 \\\\\n7 & -14   &  160   &  329$\\pm$8  &   61$\\pm$2  &    13.5$\\pm$1.4 &  89$\\pm$2  &  57$\\pm$1 & 0.90$\\pm$0.06 & 0.56 \\\\\n8 & -22  &   158  &   476$\\pm$28 &    45$\\pm$4   &    &  98$\\pm$6 &   71$\\pm$6 & 0.97$\\pm$0.16 & 0.56 \\\\\n9 & -21  &   139   &  174$\\pm$7  &     37$\\pm$2  &    10.6$\\pm$1.8 &  94$\\pm$3 &   77$\\pm$3 & 0.95$\\pm$0.09 & 0.50 \\\\\n10 & -31  &   131  &   262$\\pm$14  &     35$\\pm$4  &      & 103$\\pm$5  &  79$\\pm$5 & 0.89$\\pm$0.12 & 0.48 \\\\\n11 & -50  &   104  &   228$\\pm$2   &    24$\\pm$1   &    9.9$\\pm$0.6 & 102$\\pm$1  &  71$\\pm$1 &  0.86$\\pm$0.02 & 0.53 \\\\\n12 & -14  &   106  &   222$\\pm$5  &    123$\\pm$3   &   10.7$\\pm$1.1 &  73$\\pm$2  &  49$\\pm$1 &  0.84$\\pm$0.06 & 0.37 \\\\\n13 & -26  &    99  &   255$\\pm$4  &     64$\\pm$1  &    10.4$\\pm$0.7 &  91$\\pm$1  &  60$\\pm$1 & 0.84$\\pm$0.03 & 0.37 \\\\\n14 & -63  &    88  &   221$\\pm$6  &     82$\\pm$2  &       & 71$\\pm$2  &  51$\\pm$2 & 0.73$\\pm$0.06 & 0.42 \\\\\n15 & -79  &   160  &   306$\\pm$27  &  129$\\pm$13  &     &  52$\\pm$8  &  85$\\pm$16 & 0.27$\\pm$0.27 & 0.67 \\\\\n16 & -111  &    87  &   264$\\pm$25  &     53$\\pm$9   &    &  80$\\pm$8 &  114$\\pm$12 & 0.58$\\pm$0.25 & 0.55 \\\\\n17 & -57  &    77  &   225$\\pm$5  &     30$\\pm$1   &    9.9$\\pm$1.0 &  96$\\pm$2  &  77$\\pm$1 & 0.84$\\pm$0.05 & 0.37 \\\\\n18 & -112 &     50 &    614$\\pm$18  &    115$\\pm$4  &    12.6$\\pm$2.3  & 84$\\pm$3 &  105$\\pm$5 & 0.77$\\pm$0.08 & 0.47 \\\\  \n19 & -114  &     2   &  366$\\pm$5   &   134$\\pm$2  &     9.6$\\pm$0.7 &  65$\\pm$1  &  63$\\pm$2 & 0.76$\\pm$0.04 & 0.41 \\\\\n20 & -94  &     6   &  326$\\pm$3   &    59$\\pm$1  &    10.3$\\pm$0.4 &  90$\\pm$1  &  53$\\pm$1 & 0.78$\\pm$0.02 & 0.34 \\\\\n21 & -91  &   -11  &   268$\\pm$7  &     24$\\pm$2  &    10.0$\\pm$1.5 &  99$\\pm$3  &  62$\\pm$2 & 0.83$\\pm$0.06 & 0.32 \\\\\n22 & -41  &   -16  &   105$\\pm$9  &     20$\\pm$3  &     9.5$\\pm$1.9 & 111$\\pm$4  &  62$\\pm$4 & 1.55$\\pm$0.09 & 0.14 \\\\\n23 & -57  &   -29  &   222$\\pm$12  &     21$\\pm$3  &     8.0$\\pm$2.1 & 101$\\pm$4  &  73$\\pm$5 & 1.01$\\pm$0.10 & 0.20 \\\\\n24 & -89  &   -39  &   381$\\pm$10  &     56$\\pm$2  &     9.1$\\pm$1.1 &  96$\\pm$2  &  59$\\pm$2 & 1.05$\\pm$0.06 & 0.31 \\\\\n25 & -146  &   -50  &   274$\\pm$6  &    117$\\pm$3  &    13.0$\\pm$1.5 &  64$\\pm$2 &   62$\\pm$2 & 0.53$\\pm$0.06 & 0.50 \\\\\n26 & -223   &  -12   &  234$\\pm$27  &    136$\\pm$16   &      & 78$\\pm$12 & 115$\\pm$24 & 0.07$\\pm$0.31 & 0.78 \\\\\n27 & -133  &   -74   &  291$\\pm$9   &   158$\\pm$5   &   13.5$\\pm$1.9 &  62$\\pm$2  &  62$\\pm$3 &  0.69$\\pm$0.09 & 0.47 \\\\\n28 & -67  &   -86  &   291$\\pm$5   &    67$\\pm$2    &  13.6$\\pm$1.2 &  81$\\pm$2  &  56$\\pm$2 & 0.81$\\pm$0.04 & 0.33 \\\\\n29 & -81   &  -64  &   158$\\pm$8  &     40$\\pm$3   &      & 106$\\pm$4 &   93$\\pm$4 & 0.78$\\pm$0.10 & 0.32 \\\\\n30 &  16   &   15   &   48$\\pm$2   &     4$\\pm$.5   &      &  95$\\pm$1&   38$\\pm$2 & 1.19$\\pm$0.02 & 0.07 \\\\\n31 &  29  &    18  &   163$\\pm$11  &     29$\\pm$3   &       & 101$\\pm$4  &  61$\\pm$4 & 1.18$\\pm$0.11 & 0.11 \\\\\n32 &  40   &   64  &   354$\\pm$13  &  34$\\pm$3   &       & 107$\\pm$4 &   91$\\pm$3 & 0.87$\\pm$0.08 & 0.23 \\\\\n33 &  65   &   53 &    182$\\pm$2  &     55$\\pm$1  &    10.7$\\pm$0.4 &  87$\\pm$1 &   50$\\pm$1 & 0.88$\\pm$0.02 & 0.26 \\\\\n34 &  93  &    35  &   269$\\pm$6  &    119$\\pm$3  &    16.0$\\pm$1.5  & 71$\\pm$2 &   51$\\pm$2 & 0.47$\\pm$0.06 & 0.32 \\\\\n35 & 127   &   21 &    305$\\pm$12  &     92$\\pm$4  &     &  80$\\pm$4 &   86$\\pm$3 & 0.61$\\pm$0.11 & 0.44 \\\\\n36 &  78   &   17   &   80$\\pm$3   &    14$\\pm$2  &    8.7$\\pm$1.2  & 96$\\pm$2  &  57$\\pm$2 & 0.67$\\pm$0.06 & 0.27 \\\\\n37 &  87   &    7   &  222$\\pm$12  &     34$\\pm$3   &       & 108$\\pm$4  &  94$\\pm$6 & 0.78$\\pm$0.11 & 0.31 \\\\\n38 &  85   &  -23  &   164$\\pm$4   &    27$\\pm$1  &     9.4$\\pm$0.6 & 105$\\pm$2 &   76$\\pm$2 & 1.11$\\pm$0.04 & 0.33 \\\\\n39 &  81  &   -39 &    321$\\pm$20   &    47$\\pm$5   &   14.4$\\pm$2.5 & 102$\\pm$5 &   75$\\pm$4 &  1.37$\\pm$0.15 & 0.34 \\\\\n40 & 78   &  -75   &  275$\\pm$5 &      38$\\pm$1  &     9.3$\\pm$0.8 &  93$\\pm$2 &   66$\\pm$2 & 1.10$\\pm$0.05 & 0.42 \\\\\n41 &  72  &   -85  &   126$\\pm$6   &    44$\\pm$2   &   13.6$\\pm$1.8 &  88$\\pm$3  &  59$\\pm$4 & 0.92$\\pm$0.09 & 0.43 \\\\\n42 &  47  &  -105  &   309$\\pm$2   &    66$\\pm$1  &    10.6$\\pm$0.3 &  95$\\pm$1  &  44$\\pm$1 & 1.14$\\pm$0.02 & 0.43 \\\\\n43 & 104   & -113  &   516$\\pm$10  &    122$\\pm$3  &     9.1$\\pm$1.6 &  74$\\pm$3  & 104$\\pm$3 & 0.53$\\pm$0.06 & 0.60 \\\\\n44 & 100  &  -120  &   276$\\pm$8  &    240$\\pm$6  &    13.3$\\pm$2.3 &  52$\\pm$3 &   59$\\pm$4 & 0.40$\\pm$0.08 & 0.61 \\\\\n45 & 10  &  -122  &   227$\\pm$5   &   103$\\pm$2   &   11.1$\\pm$1.7  & 76$\\pm$2 &   59$\\pm$2 & 0.61$\\pm$0.05 & 0.42 \\\\\n46 & 26  &  -144  &   357$\\pm$9  &     82$\\pm$3   &   13.3$\\pm$1.7 &  82$\\pm$2 &   61$\\pm$4 & 0.63$\\pm$0.07 & 0.52 \\\\\n47 &  9  &  -165  &   374$\\pm$20   &    91$\\pm$6    &       & 89$\\pm$5 &  109$\\pm$7 & 0.58$\\pm$0.14 & 0.57 \\\\\n48 & 46  &  -185  &   324$\\pm$13   &   107$\\pm$5   &    9.9$\\pm$2.8  & 73$\\pm$4 &   80$\\pm$5 & 0.63$\\pm$0.11 & 0.69 \\\\\n49 & -13 &   -200  &   541$\\pm$53  &     63$\\pm$9  &        &  78$\\pm$9 &   81$\\pm$12 & 0.80$\\pm$0.29 & 0.67 \\\\\n50 & -13 &   -230   &  428$\\pm$4   &   228$\\pm$2  &    11.5$\\pm$0.4 &  44$\\pm$1 &   35$\\pm$1 & 0.69$\\pm$0.02 & 0.77 \\\\\n51 &  -61 &   -240  &   380$\\pm$5  &     87$\\pm$2   &    8.6$\\pm$0.7  & 78$\\pm$2  &  66$\\pm$2 & 0.55$\\pm$0.03 & 0.78 \\\\\n52 & -123 &   -259  &   360$\\pm$15  &    121$\\pm$6   &   10.7$\\pm$2.7 &  90$\\pm$4  & 100$\\pm$7 & 0.77$\\pm$0.12 & 0.87 \\\\\n53 &  -142  &  -258 &    180$\\pm$12  &     76$\\pm$6   &      & 86$\\pm$6 &   76$\\pm$7 & 0.34$\\pm$0.16 & 0.89 \\\\\n54 &  -181  &  -241  &   395$\\pm$15  &    168$\\pm$7   &      & 63$\\pm$4  &  78$\\pm$6 & 0.38$\\pm$0.11 & 0.90 \\\\\n55 &  -193  &   -242  &   532$\\pm$51  &     60$\\pm$9   &     &  77$\\pm$9 &  110$\\pm$13 & 0.74$\\pm$0.28 & 0.92 \\\\\n\\hline\n\\end{tabular} \n\\end{table*}\n\n\\subsection{Radial gradients in NGC 1365}\n\nGalactocentric distances and azimuthal positions of the\n{H~\\sc ii}\\  regions were calculated by assuming a 40$^\\circ$ inclination of the galaxy disc to the \nplane of sky, and a position angle of 220$^\\circ$ from J\\\"ors\\\"ater \\& \nvan Moorsel \\shortcite{JM95}. As a normalizing radius, we used the value of\nthe isophotal radius $\\rho_0$ of 5.$'$61 (RC3); the\npossible choices of a normalizing radial parameter are discussed\nin detail by Vila-Costas \\& Edmunds \\shortcite{VE92} and Zaritsky et al. \\shortcite{ZA94}.\nLine ratios which show systematic radial variations are\nthe excitation {\\rm \\mbox{[O \\sc iii]\/{H$\\beta$}}\\ }, the abundance indicators {\\rm \\mbox{[N \\sc ii]}}\/{{\\rm \\mbox{[O \\sc iii]}}\\ } and\n{\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } and reddening (Fig. 4). The radial positions\nof the {H~\\sc ii}\\  regions are given in Table 2 in terms of\nthe fractional isophotal radius. Despite a large scatter, there \nappears indeed to be a systematic\nvariation of $c$(H$\\beta$), the logarithmic extinction at H$\\beta$, as a function\nof radius; such a radial trend has been observed so far only in the galaxy\nM101 (Kennicutt \\& Garnett \\shortcite{KG96}). The linear coefficient\nof correlation between $c$(H$\\beta$) and $\\rho\/\\rho_0$ is\nR = --0.46. The value of $c$(H$\\beta$)\\ is high\non average across the disc, dropping from $\\sim$1.2 near\nthe centre to 0.6--0.8 in the outer disc; the dust lanes of NGC 1365 are\nremarkably strong and indicate the rich dust content. Despite the relatively high\nextinction (the mean of $c$ = 0.75$\\pm$0.27), one can see through the galaxy;\nin particular, a distant elliptical  (object G-1 on Figure 1) was found just \nsouthwest of the bar.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig3.eps,height=16.0cm,clip=}}\n\\caption{Diagnostic diagrams of log {\\rm \\mbox{[O \\sc ii]}}\/{{\\rm \\mbox {[O \\sc iii]}}\\ }, log {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ }, log {\\rm \\mbox{[N \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ }\nand  log {\\rm \\mbox{[S \\sc ii]}}\/{{\\rm \\mbox {[O \\sc ii]}}\\ } vs. the sequencing index log {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } are shown for the \nobserved {H~\\sc ii}\\  regions in NGC 1365.}\n\\label{fig3}\n\\end{figure*}\n\nAs seen from the spectra of Figure 2, the level of excitation in NGC 1365\nis generally low, and no direct measurement of the electron temperatures\nis possible. Therefore semi-empirical methods must be relied on to derive abundances.\nThese methods have been widely discussed and used (e.g.\nEdmunds \\& Pagel \\shortcite{EP84}; McCall et al. \\shortcite{MC85}; Evans \\shortcite{EV86};\nGarnett \\& Shields \\shortcite{GS87}; Vilchez et al. \\shortcite{VI88}; Walsh \\& Roy \n\\shortcite{WR89};\nBelley \\& Roy \\shortcite{BR92}; Martin \\& Roy \\shortcite{MR94}; Zaritsky et al. \n\\shortcite{ZA94}). They are based on the result that larger values of {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } or \n{\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } are correlated with higher electron temperatures and lower abundances of \noxygen. These calibrations have been refined by Edmunds and Pagel \\shortcite{EP84},\nMcCall et al. \\shortcite{MC85}, and Dopita \\& Evans \\shortcite{DE86}; Zaritsky et al.\n\\shortcite{ZA94} \nhave produced a synthetic calibration for {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by merging the\nthree previous ones. It must be recalled that factors other than\nheavy element abundances also modify nebular\nline strengths. The limitations and uncertainties  of the\nsemiempirical methods due to various effects have been also discussed by\nMcGaugh \\shortcite{MG91} for ionization parameter and effective temperature\nof the ionizing radiation, by Henry \\shortcite{HE93} and Shields \\& Kennicutt\n\\shortcite{SK95} for dust, by Oey \\& Kennicutt \\shortcite{OK93} for density, by \nKinkel and Rosa \\shortcite {KR94}\nfor high Z effects and by Roy et al. \\shortcite{RO96A} for moderately low Z behavior.\n\nRadial gradients in {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } and {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } reflect primarily abundances\ngradients wherever those line ratios are sampled over a significant\nfraction of the {H~\\sc ii}\\  region volume. The absolute O\/H abundance predicted by the different\nsemi-empirical calibrations of these ratio can vary depending on the \ncalibration employed, but the relative trends can be probed reliably\n(see for example Henry \\& Howard \\shortcite{HH95}; Zaritsky et al. \\shortcite{ZA94}; \nMartin \\& Roy \\shortcite{MR94}; Kennicutt \\& Garnett \\shortcite{KG96}).\nThe radial abundance variation in oxygen abundances \nin NGC 1365 was determined\nfrom three calibrations: i) {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } as calibrated by Edmunds \\& Pagel \\shortcite{EP84};\nii) {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } also calibrated by Edmunds \\& Pagel \\shortcite{EP84}; and\n{\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } as calibrated by Zaritsky et al \\shortcite{ZA94}. The reader can\nthen compare the results with similar abundance data on the few other\ngalaxies which have been thoroughly sampled with spectroscopic\ntechniques, e.g. NGC 2997 (Walsh \\& Roy \\shortcite{WR89}) and\nM101 (Kennicutt \\& Garnett \\shortcite{KG96}). The overall trend in NGC 1365 is\na rather shallow abundance gradient at about --0.50 dex $\\rho_0$$^{-1}$, \nor --0.02 dex\/kpc;\nthe exact parameters of the correlation depends on the calibration used as shown\nin Table 3, where $R$ is the coefficient of correlation.  The three calibrations, \n{\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } by Edmunds \\& Pagel \\shortcite{EP84}, {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by\nZaritsky et al. \\shortcite{ZA94} and {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by Edmunds \\& Pagel \n\\shortcite{EP84}, give rather\nsimilar results within the errors. The calibration of {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by Edmunds \\& Pagel\n\\shortcite{EP84} tend to give a larger amplitude for the variation of O\/H across \nthe disc. A shallow gradient is expected for  strongly\nbarred galaxies, and NGC 1365 with a deprojected bar\naxis ratio $b\/a(i)$ = 0.51, has a strong bar \n(\\cite{MA95}). The central intercept is at about 12 + log O\/H = 9.15\n(depending on the calibration used).\n\n In addition, one observes\na rather obvious break at R $\\sim$ 185 - 200$''$ ($\\rho = 0.55 - 0.60 \\rho_0$); inside this\nradius the O\/H gradient is moderately steep (--0.80 dex $\\rho_0$$^{-1}$, i.e. --0.05 dex kpc$^{-1}$), while\noutside it is flat.  We have chosen $\\rho = 0.55 \\rho_0$ as the galactic\nradial distance where the break occurs and have calculated the\nappropriate equations and coefficient of correlation (Table 3);\nthis position for the  break ensures a reasonable number of sample points\n(34 in the inner part and 21 in the outer part)\n while corresponding to small residuals for the fits. \nThe correlations for the inner zone are almost as strong as\nfor all the points. Comparisons\nbetween the three abundance calibrations can be made; it is obvious from Table 3, that\nthere is {\\it no} abundance gradient (within the uncertainties) beyond the break.\nWe note that the values of O\/H derived by using\nthe {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } calibration by Edmunds \\& Pagel \\shortcite{EP84}\nare very similar to those given by the synthetic calibration\nof Zaritsky et al. \\shortcite{ZA94}; this applies to individual\npoints and to the correlations (Table 3). The break is also well seen\nin the radial plot of {\\rm \\mbox{[O \\sc iii]\/{H$\\beta$}}\\ } (not shown); thus\neffects related to dust or a erroneous reddening determination  can therefore \nbe excluded.\n\nPossible azimuthal asymmetries in the O\/H distribution were searched for,\nbut no evidence was found for such asymmetry. The significant assymetry in O\/H\ndistribution found by\nKennicutt \\& Garnett \\shortcite{KG96} in M101 remains a rather unusual and\nsignificant feature.\nThe apparent dispersion in abundance ($\\sim$0.2 dex) at fixed radius is probably \nmainly due to variations in nebular ionization as discussed\nby Kennicutt \\& Garnett \\shortcite{KG96}.\n  \n\\begin{table*}\n\\caption{Equations of the O\/H radial distribution in NGC 1365} \n\\begin{tabular}{lcrc}\n\\hline \\hline\nRadius & Equation & $R$ & Calibration \\\\\n\\hline \nR $\\leq$ 1.0 $\\rho_0$ & 12 $+$ log O\/H = 9.10$\\pm$0.04 - 0.42$\\pm$0.07 $\\rho_0$ & -0.62 &{\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } (EP84)\\\\\nR $\\leq$ 1.0 $\\rho_0$ & 12 $+$ log O\/H = 9.17$\\pm$0.05 - 0.53$\\pm$0.09 $\\rho_0$ & -0.63 & {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (ZA94)\\\\\nR $\\leq$ 1.0 $\\rho_0$ & 12 $+$ log O\/H = 9.12$\\pm$0.06 - 0.68$\\pm$0.11 $\\rho_0$ & -0.65 &{\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (EP84)\\\\\n\\hline \\hline\nR $\\leq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 9.23$\\pm$0.06 - 0.78$\\pm$0.17 $\\rho_0$ &  -0.63 &{\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } (EP84)\\\\\nR $\\leq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 9.26$\\pm$0.08 - 0.77$\\pm$0.21 $\\rho_0$ & -0.54 & {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (ZA84)\\\\\nR $\\leq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 9.28$\\pm$0.10 - 1.10$\\pm$0.28 $\\rho_0$ & -0.57 & {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (EP84)\\\\\n\\hline\nR $\\geq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 8.85$\\pm$0.15 - 0.07$\\pm$0.20 $\\rho_0$ & -0.09 & {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } (EP84)\\\\\nR $\\geq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 8.66$\\pm$0.17 + 0.13$\\pm$0.23 $\\rho_0$ & 0.13 & {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (ZA84)\\\\\nR $\\geq$ 0.55 $\\rho_0$ & 12 $+$ log O\/H = 8.50$\\pm$0.18 + 0.16$\\pm$0.24 $\\rho_0$ & 0.15 & {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } (EP84)\\\\\n\\hline \n\\end{tabular}\\\\\n\\end{table*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig4.eps,height=14.0cm,clip=}}\n\\caption{(a) The extinction $c$(H$\\beta$) is plotted against the normalized isophotal\nradius in NGC 1365. (b) The sequencing index log {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } vs. the normalized\nisophotal radius is shown.}\n\\label{fig4}\n\\end{figure*}\n\n\n\\section{Discussion}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig5.eps,height=16.0cm,clip=}}\n\\caption{(a) The gradient in oxygen abundance across NGC 1365 using the\nthe calibration of {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by Edmunds \\& Pagel (1984). (b) The\nsame gradient using the calibration of {\\rm \\mbox{[O \\sc iii]}}\/{{\\rm \\mbox{[N \\sc ii]}}\\ } by Edmunds \\& Pagel\n(1984). (c) The gradient in oxygen abundance using the synthetic\ncalibration of {\\rm ([O \\sc ii] + [O \\sc iii])\/{H$\\beta$}\\ } by Zaritsky et al. (1994). CR and -4\/1 indicates the\nradial positions of the galaxy corotation  and -4\/1 resonance respectively\nas found by J\\\"ors\\\"ater \\& van Moorsel (1995).}\n\\label{fig5}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=n1365fig6.eps,height=10.0cm,clip=}}\n\\caption{ Comparison of the O\/H abundance gradients in the interstellar gas\nof two barred galaxies (NGC 1365 and NGC 3359) and two\nnormal spirals (M 101 and NGC 2997). The\nnumber in parenthesis indicates the sample size of  {H~\\sc ii}\\  regions\nin each galaxy.}\n\\label{fig6}\n\\end{figure*}\n\n\nThe presence of a break in the radial abundance\ndistribution is one outstanding signature that a bar\nmay have recently ($\\leq$ 1 Gyr) formed in a spiral galaxy. \nFollowing Friedli et al. \\shortcite{FR94} and\nFriedli \\& Benz \\shortcite{FB95} who have\ndeveloped this scenario, two breaks may actually appear\nin the radial abundance distribution: (i) a ``steep-shallow''\nbreak, near corotation, due to vigorous enrichment by\nstar formation in the bar, followed by a much flatter portion\ndominated by the dilution effect of the outward\nflow, and (ii) an outer ``shallow-steep'' break defining\nwhere the outflow has had time to penetrate \nthe outer disc. This ``shallow-steep'' outer\nbreak moves to larger radius as time evolves,\nand as the extent of the radial homogenization reaches further out\nin the disc (Friedli et al. \\shortcite{FR94}; Friedli \\& Benz \\shortcite{FB95}).\nAs the bar ages, it dissolves and runs out of gas. Star formation then\nweakens, and dilution effects in the interstellar gas come to dominate;\nafter a few Gyr,\nthe resultant global abundance gradient ends up shallow, the break\nbecomes indistinguishable and the gradient displays\na monotonic decrease. The slope of any\npre-existing radial gaseous or stellar abundance gradient  decreases\nfrom the start of the first phase. To have a break, one needs vigorous\nstar formation in the bar, and grand design spiral arms for\nstrong outward flows.\nNGC 1365 is a barred galaxy with moderate star formation along the bar\nbut an intense nuclear starburst (Seyfert nucleus and\nring of {H~\\sc ii}\\  regions); this is evidence for an age of $\\sim$1 Gyr\n(see discussion in Martin \\& Roy \\shortcite{MR95}).\n\n A ``steep-shallow'' inner break is found in the abundance\ndistribution of NGC 1365 and it is located quite a bit further out\nthan the corotation radius (Figure 5).\nIt is assumed that due to an interaction or a merger,\nthe disc of NGC 1365 became unstable and developed a bar in a\nrecent epoch.\nOn account of angular momentum transfer via the arms, the bar induced large-scale\nradial flows of interstellar gas. First,\na quick phase of intense star formation occured along the\nbar major axis as well as along spiral arms; galaxies\nshowing this property (e.g. NGC 3359) have rather young bars. \nAt a later time (closer to our epoch), a starburst was triggered when\ngas had collapsed to the centre in a nuclear\nring as observed by Sandqvist et al. \\shortcite{SA95}. Star formation now continues \nmainly along the spiral arms.\nWith the fuel  progressively consumed, star formation\nis essentially limited to the ring and the spiral arms.  \nThus in this framework of a recently formed bar,\n NGC 1365 would be a disc spiral with a strong,\ngas rich, but moderately young bar (age $\\sim$1 Gyr). As indicated by the absence of a rich\npopulation of {H~\\sc ii}\\  regions in the bar, the star formation\nrate is now weak, except in the center and at the bar ends. \n\n From the neutral hydrogen\nkinematics, corotation (at $r = 145''$ or at $\\rho \\sim 0.43 \\rho_0$)\nis at 1.21 times the bar semi-major axis\n(see Linblad, Linblad \\& Athanassoula \\shortcite{LLA96}.\nThere is a clear break in the O\/H abundance radial distribution \nat $\\rho \\sim 0.55\\rho_0$ or $r = 185''$ (Figure 5). Although\nthere is some uncertainties ($\\pm 10\\%$) in the locations of corotation\nand of resonances (dependence on the model)\nand of the break, the latter is beyond corotation; \nthe break in the O\/H gradient is close to the -4\/1\nresonance at $r = 180''$. The position of the abundance gradient break\n$\\sim 30''$ beyond corotation is puzzling, since this\nis a region of strong radial mixing. However mixing always competes\nwith enrichment due to star formation in changing the local abundance.\nThe intense star forming activity in NGC 1365 observed at the bar ends and \njust beyond (betrayed by clumps of several bright {H~\\sc ii}\\  regions) may\ncompensate dilution and maintain high abundances, pushing the\nlocation of the break further where star formation is reduced.\n\nThe apparent dispersion of the O\/H points, at\na fixed radius, does not vary with galactocentric distance.\nThis is to be contrasted with the behavior of another\nbarred galaxy, NGC 3359, which has also a break in \nits radial abundance distribution (see Fig. 6). NGC 3359 has vigorous star\nformation in its bar and  a steep inner abundance gradient; \nthere the break is observed at\ncorotation. The abundance fluctuations, at a fixed radius, are much\nlarger in the outer parts of the disc -- where the gradient is flat --\nthan in the inner regions where the gradient is steep.\nMartin \\& Roy \\shortcite{MR95} have interpreted this behavior \nof the O\/H abundances in NGC 3559\nas due to a recently formed  bar $\\sim$400 Myr old, an age less than\nthe mean azimuthal mixing time in galaxy discs \\cite{RK95}. \nWe suggest that the homogenizing action of the bar in NGC 1365 may have acted on a timescale longer\nthan the characteristic azimuthal mixing time, that is the\nbar is older than 500 Myr. The high mass of gas accumulated in the centre,\nas implied by the presence of a nuclear starburst and a Seyfert nucleus,\nalso indicates an age $\\geq$ 1 Gyr for the bar of NGC 1365. Overall,\nseveral features are consistent with a bar formed\nabout 1 Gyr ago.\n\nIn Figure 6, the abundance gradients of four galaxies which\nhave had extensive spectrophotometry across their discs are shown:\ntwo normal spiral galaxies, NGC 2997 (Walsh \\& Roy \\shortcite{WR89}) and M 101\n(Kennicutt \\& Garnett \\shortcite{KG96}); two barred \ngalaxies, NGC 3359 (Martin \\& Roy \\shortcite{MR95}) and NGC 1365 (this work).\nThe shallower gradients and the presence of breaks in the radial\nabundance distribution is\neasily seen in the two barred galaxies. The difference between\nbarred and normal spirals in radial\nabundance distributions is striking. If the recently formed bar hypothesis\nis correct, the numerical simulation of Friedli et al. \\shortcite{FR94}\nof a disc with a recently formed bar and star formation\nshow that the pre-bar O\/H radial distribution in NGC 1365 would have \nbeen very similar to that observed nowadays in M~101.\n\n\nFinally, the models of Friedli and collaborators\npredict two breaks in the radial abundance distribution of galaxies\nwith young bars. So far no observational evidence\nfor the ``shallow-steep'' break in the outer disc has been found.\nIn order to ascertain the scenario of secular evolution of\ndisc galaxies (Martinet \\shortcite{MAR95}), by the recurrent action of bars, it will be\nnecessary to find this outer break. This more definite test of the ``young bar'' hypothesis appears to be difficult however. As \ngas density falls with radial distance and the star formation\nrate weakens, there are fewer and smaller {H~\\sc ii}\\  regions,\nnot unlike the interarm {H~\\sc ii}\\  regions. Long integration times will be required\nin order to determine O\/H abundances and  there may be only a few galaxies having a sufficient number of outer\n{H~\\sc ii}\\  regions to provide statistically significant samples.\n\n\n\\noindent\n{\\bf Acknowledgments} \\\\\nWe acknowledge the technical support of J. Pogson, the AAT telescope \noperator. Discussions with Daniel Friedli, Pierre Martin, and\nLaurent Drissen were most helpful.\nWe thank the PATT Committee for assigning time on the AAT for this\nproject. This investigation was funded in part by the Natural\nSciences and Engineering Research Council of Canada, the Fonds FCAR\nof the Government of Quebec and by\nthe Visitor Program of the European Southern Observatory through financial\nsupport of JRR.\n\n\\vspace{1cm}\n\n\\noindent\n{\\Large \\bf Appendix A: Serendipitously discovered galaxies} \\\\\n\nAs mentioned in section 2.2 three of the targets chosen as {H~\\sc ii}\\  regions were \nidentified from their spectra as galaxies. They are indicated as G-1 to G-3 on\nFigure 1. Brief details of the targets and their spectra are included here.\n\n\\noindent\n{\\bf G-1:} This bright galaxy is quite round (ellipticity on the B image $\\sim$ 0) so \nis probably of type around E0-E1. Its major axis has a half width of about 4$''$,\ncorrected for seeing. The spectrum shows a strong red continuum with absorption\nlines of Ca~{\\sc II} H and K, H$\\gamma$ and G-band and Mg~{\\sc I}. From the wavelengths\nof the Ca~{\\sc II} H and K lines the red-shift is 0.103. There is a \ndetection of Ca~{\\sc II} H and K absorption at the redshift of NGC 1365, making the\ngalaxy a very useful probe for studying the line of sight velocity dispersion\nin an interesting region near the bar.\n\n\\noindent\n{\\bf G-2:} This galaxy has a resolved almost circular core of half width $\\sim$2$''$\nand shows two spiral arms of total extent $\\sim$15$''$. The spectrum shows\na redward rising continuum with Ca~{\\sc II} H and K and Mg~{\\sc I} absorption lines.\nFrom the observed wavelength of the Ca~{\\sc II} H and K lines, the redshift is 0.294.\n\n\\noindent\n{\\bf G-3:} This galaxy displays a strong disc with a possible knot at its SE edge,\npossibly indicating an interacting system. The spectrum shows a rather flat continuum\nwith strong emission lines of {\\rm \\mbox{[O \\sc ii]}\\ }, H$\\beta$, and {\\rm \\mbox{[O \\sc iii]}\\ }. The redshift is\n0.131. The {\\rm \\mbox{[O \\sc iii]}\\ } 5007\/H$\\beta$ ratio is 2.4 and {\\rm \\mbox{[O \\sc ii]}\\ }\/H$\\beta$\\ ratio is 4.5.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince their inception in 2001 \\cite{Aussillous2001}, liquid marbles (LMs) have been a source of growing interest across fields as diverse as medicine \\cite{Sarvi2015,Ledda2016}, engineering \\cite{Fujii2017,Tian2010a} and chemistry \\cite{Paven2016,Wei2016}. LMs are constructed from microlitre droplets of water, supported by a layer of hydrophobic particles on the surface. In this manner, the hydrophobic particles minimise the comparatively high surface energy of water by encapsulating the droplet, and keeping it near spherical. This permits the water droplet to remain non-wetting on many (traditionally wettable) surfaces. \n\nThere are two major variables affecting the properties of a LM: the core and the coating. Traditionally the encapsulated liquid is water, although there are a range of both common (glycerol \\cite{Aussillous2001}) and uncommon (petroleum \\cite{Bormashenko2015a}) alternatives. The coating provides the largest affect on the mechanical properties of the marble, as it is the coating that interacts with the surface the LM is resting on. Coating parameters that can be modified to impart the desired properties include the composition, grain size, and mix ratio. By varying these, it is possible to adapt a liquid marble to many different situations.\n\nLiquid marbles have previously been investigated as fluidic transport devices. They are ideally suited to the transport of microlitre quantities of liquid, due to their non-wetting nature. Recent progress have been made in this area, with LM movement initiated by magnets \\cite{Khaw2016}, electrostatic fields \\cite{Aussillous2006}, gravity \\cite{Aussillous2001}, lasers \\cite{Paven2016} and the Maran\\-goni effect \\cite{Ooi2015} all reported. The movement of LMs can be exploited for chemical reactions, by controlling the time and place of reagent mixing. This has been demonstrated both with the coalescing of LMs \\cite{Chen2017}, and in the controlled destruction of LMs once they arrive at a chosen location \\cite{dupin2009stimulus,Fujii2010}.\n\n\nIn interaction gates, Boolean values of inputs and outputs are represented by presence of physical objects at given site at a given time. If an object is present at input\/output we assume that logical value of the input\/output is {\\sc True}; if the object is absent the logical value is {\\sc False}. The signal-object realise a logical function when they pass through a collision site. The objects might fuse, annihilate or deflect on impact. \n\nThe fusion gate (figure~\\ref{fig:gate}(a)) was first implemented in fluidic devices in the 1960s. The  gate is the most well known (on a par with the bistable amplifier) device in fluidics~\\cite{peter1965and, belsterling1971fluidic}: two nozzles are placed at right angles to each other, when there are jet flows in both nozzle they collide and merge  into a single jet entering the central outlet.  If the jet flow is present only in one of the input nozzles it goes into the vent. The central outlet represents {\\sc and} and the vent represent {\\sc and-not}. The fusion-based gate was also employed in designs of computing circuits in Belousov-Zhabotinsky medium \\cite{adamatzky2004collision, adamatzky2007binary, toth2010simple, adamatzky2011towards}, where excitation wave-fragments merge when collide; in the actions of slime mould \\cite{Mayne2015Vesicles}, when distributing vesicles collide; and a crab-based gate \\cite{gunji2011robust}, were swarms of solider crabs merge into a single swarm.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.3]{fusion_collision}}\n\\subfigure[]{\\includegraphics[scale=0.3]{annihilation_collision}}\\\\\n\\subfigure[]{\\includegraphics[scale=0.3]{elastic_collision}}\n\\subfigure[]{\\includegraphics[scale=0.3]{nonelastic_collision}}\n\\caption{Interaction-based gates. \n(a)~Fusion of signals.\n(b)~Annihilation of signals.\n(c)~Elastic deflection of signals.\n(d)~Non-elastic deflection of signals.}\n\\label{fig:gate}\n\\end{figure}\n\nIn the annihilation gate (figure~\\ref{fig:gate}b) signals disappear on impact. This gates has two-inputs and two-outputs, each of the outputs represents {\\sc and-not}. Computational universality of the Conway's Game of Life cellular automata was demonstrated using annihilation based collisions between gliders~\\cite{berlekamp1982winning}. We can also implement the annihilation gate by colliding excitation wave-fragments at certain angles~\\cite{adamatzky2011polymorphic}. \n\nKey deficiency of the fusion and annihilation gates is that, when implemented in media other than excitable spatially-extended systems, they do not preserve physical quantity of signals, e.g. when two signals merge the output signal will have a double mass of a signal input signal. This deficiency is overcome in the conservative logic, proposed by Fredkin and Toffoli in 1978~\\cite{fredkin2002conservative}. The logical value are represented by solid elastic bodies, aka billiard balls, which deflect when made to collide with one another (figure~\\ref{fig:gate}c). Intact output trajectories of the balls represent {\\sc and-not} function, output trajectories of deflected balls represent {\\sc and} function. The gates based on elastic collision led to development of a reversible (both logically and physically) gate:  Fredkin~\\cite{fredkin2002conservative} and Toffoli~\\cite{toffoli1980reversible} gates, which are the key elements of low-power  computing circuits \\cite{de2011reversible, bennett1988notes}, and amongst the key components of quantum~\\cite{berman1994quantum, barenco1995elementary, smolin1996five, zheng2013implementation} and optical~\\cite{kostinski2009experimental} computing circuits. \n\nThe soft-sphere collision gate proposed by Margolus~\\cite{margolus2002universal} gives us a rather realistic representation of interaction gates with real-life physical and biological bodies (figure~\\ref{fig:gate}d). Logical value $x=1$ is given by a ball presented in input trajectory marked $x$, and $x=0$ by the absence of the ball in the input trajectory $x$; the same applies to $y=1$ and $y=0$, respectively. When the two balls, approaching the collision gate along paths $x$ and $y$ collide, {they} compress but then spring back and reflect. As a result, the balls come out along the paths marked $xy$. If only one ball approaches the gate, that is for inputs $x=1$ and $y=0$ or $x=0$ and $y=1$, the balls exit the gate via path $x\\overline{y}$ (for input $x=1$ and $y=0$) or $\\overline{x}y$ (for input $x=0$ and $y=1$). Soft-sphere-like gates have  been implemented using microlitre sized water droplets on a superhydrophobic copper surface \\cite{Mertaniemi2012}. Using channels cut into the surface, {\\sc not-fanout}, {\\sc and-or} and {\\sc flip-flop} gates were demonstrated. The water droplets only rebounded in a very narrow collision property window, and a thorough \\& complete superhydrophobic surface treatment was required.\n\nIn this paper, we report the first exploitation of liquid marbles for implementation of interaction gates. The gate realised in the experimental laboratory conditions is a combination of the fusion and the Margolus gate: output trajectories of collided liquid marbles are so close that they can be interpreted as a single output. That said, if required, two liquid marbles at the output can be diverted along different paths to conserve a number of signals. By taking advantage of the liquid marbles' inherently low hysteresis, high tunability and capacity for enhanced versatility, we demonstrate the first step toward liquid marble facilitated, collision-based computing.\n\n\\section{Materials \\& Method}\n\\subsection{Regular \\& Reliable Liquid Marble Formation}\n\nWe first developed a technique for the regular and automatic formation of invariable LMs. This was achieved by programming a syringe driver (CareFusion Alaris GH) to feed a 21 gauge needle (\\SI{0.8}{\\milli\\metre} diameter) at a typical rate of \\SI{7.0}{\\milli\\litre\\per\\hour}. The rate can be easily increased or decreased, and this rate gave sufficiently fast LM formation for our purpose. The produced droplets (\\SI[separate-uncertainty = true,multi-part-units = single]{11.6 +- 0.16}{\\micro\\litre}) were permitted to fall onto a sheet of acrylic, slanted to \\ang{20} from horizontal, and surface-treated with a commercial hydrophobic spray (Rust-Oleum\\textsuperscript{\\textregistered} NeverWet\\textsuperscript{\\textregistered}). This formed beads of water, which were allowed to roll over a bed of appropriate hydrophobic powder. The result was a continuous `stream' of LMs with the same volume, coating and coating thickness. It should be noted that whilst the forming of LMs by running droplets down a powder slope has been separately developed by another group \\cite{Fujii2010}, our system prevents premature destruction of the powder bed by initially preforming the droplet on a treated hydrophobic surface.\n\n\\subsection{Maintaining Timing for Collisions}\n\nIn collision based computing, accurate timing is essential. As signals propagate through the system they must remain in sync, or the operation of many logic gates fails. In order to address this, an innovative system of electromagnets (EMs) was implemented. This was possible due to the generation of novel LMs with a mix of ultra-high density polyethylene (UHDPE) (Sigma-Aldrich, \\numrange[range-phrase = --]{3}{6e6}~\\si{\\gram\\per\\mole}, grain size approximately \\SI{100}{\\micro\\metre}) and nickel (GoodFellow Metals, \\SI{99.8}{\\percent}, grain size \\SIrange{4}{7}{\\micro\\metre}). A typical Ni\/UHDPE coating was \\SI{2.5}{\\mg}. The use of UHDPE provides strength and durability, and the inclusion of ferromagnetic nickel allows for a versatile magnetic LM.\n\nBy positioning an electromagnet (\\SI{100}{\\newton}, \\SI{12.0}{\\volt} DC, \\SI{29 x 22}{\\mm}) behind the acrylic slope, the rolling LM can be captured and released at will, by the switching on and off of said electromagnet. By controlling multiple, spatially-isolated electromagnets in series, non-concurrent LMs can be easily synchronised. To our knowledge, this is the first time electromagnets have been used to provide timing control with liquid marbles.\n\n\\subsection{Gate Design for Liquid Marble Collisions}\n\nThe collision gate was designed to allow for the colliding LMs to have a free path post-collision. This enabled the monitoring of the LM paths, and the future design and implementation of exiting pathways, creating a logic gate.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{LM-collider}\n  \\caption{Schematic of our LM collider. Labelled numbers are: (1)~syringe needle, (2)~uncoated water droplet, (3)~acrylic ramp, (4)~hydrophobic powder bed, (5)~liquid marble, and (6)~electromagnet. Droplets form and fall out of the two syringe needles, landing on a superhydrophobic surface. They then roll over a bed of Ni\/UHDPE powder, before being stopped and held stationary by the electromagnets. These electromagnets are then deactivated simultaneously, allowing the LMs to roll off and collide.}\n  \\label{fig:LM-collider}\n\\end{figure}\n\nTwo \\SI{16.0}{\\centi\\metre} acrylic pathways were slanted towards each other at \\ang{20}, affixed to an acrylic base sheet (\\SI{3.0}{\\mm} thick). The acrylic base sheet was then aligned with a pitch of \\ang{38} from horizontal, giving a final LM pathway slope of \\ang{16} from the horizontal plane. This gave reliable LM rolling without extreme angles. The gap between the two slanted pathways was set at \\SI{1.6}{\\centi\\metre}, after empirical testing. A \\SI{2.0}{\\centi\\metre}, length at the top of each pathway was made hydrophobic, as discussed above. Parallel auto-formation of hybrid LMs was achieved using the syringe driver, delivering \\SI{11.6}{\\micro\\litre} of water per syringe per drop. Each droplet of water was permitted to land on the treated section of each slope, before rolling across the powder beds of UHDPE and Ni to form LMs. The two rolling LMs were then captured using the electromagnets, allowing for any slight timing deviations to be accounted for. On controlled synchronous (or asynchronous) release of the electromagnets, the LMs simultaneously roll off the acrylic ramps on collision trajectories. Collisions were recorded at 120 fps using a Nikon Coolpix P900, and played back frame-by-frame for analysis. A schematic of our LM collider can be seen in figure \\ref{fig:LM-collider}, and photographs of our LM collider can be seen in figure~\\ref{fig:LM-collider-photo}.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.48\\textwidth]{Collider-photo1}}\n\\subfigure[]{\\includegraphics[width=0.23\\textwidth]{Collider-photo2}}\n\\subfigure[]{\\includegraphics[width=0.23\\textwidth]{Collider-photo3}}\n\\caption{Photographs of our LM collider, showing \n(a)~the overall layout, \n(b)~an upclose of the magnetic breaking\/release area, and\n(c)~an upclose of the droplet formation area. Labels are: (1)~syringe needle, (2)~acrylic ramp, (3)~hydrophobic powder bed, and (4)~electromagnet.}\n\\label{fig:LM-collider-photo}\n\\end{figure}\n\n\\section{Discussion}\n\\subsection{Liquid Marble Lifetime}\n\nFor a LM to be useful in a computing device, it has to have an appreciable lifetime. This is problematic for water based LM, as the gas permeability of LMs has previously been both established and exploited \\cite{Eshtiaghi2009,Tian2010a,Tian2010,Tian2013}. As such, lifetime experiments were conducted on UHDPE LMs, Ni LMs, water droplets, and our new Ni\/UHDPE hybrid LMs. Evaporation studies were conducted under ambient conditions, using \\SI{10.0}{\\micro\\litre} of DI water. UHDPE LMs were generated by rolling a droplet of water on a powder bed of UHDPE. Nickel LMs were made using a superhydrophobic surface (see above) to pre-form the droplet sphere, before rolling in an appropriate powder. A magnified view of both a Ni\/UHDPE hybrid LM and a nickel LM can be seen in figure~\\ref{fig:LM-photo}.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.4\\linewidth]{UHDPE-Ni-LM-photo-2}}\n  \\subfigure[]{\\includegraphics[width=0.4\\linewidth]{Ni-LM-photo}}\n  \\caption{Magnified photographs of (a) a Ni\/UHDPE LM, nickel can clearly been seen between the UHDPE particles; and (b) a pure nickel LM.  Scale bars are \\SI{1.0}{\\mm}. }\n  \\label{fig:LM-photo}\n\\end{figure}\n\nFrom the data shown in table~\\ref{tab:evap}, it can be seen that both the nickel and UHDPE LMs have slower evaporation rates than a pure water droplet. This is expected, as the solid particles on the surface of the liquid form an (incomplete) barrier to evaporation. It also supports previous studies \\cite{Dandan2009}. Experiments also indicated that pure UHDPE LMs evaporate at a comparable rate to pure nickel LMs. This is due to a balancing act between the short narrow pores of the nickel LM, the long wide channels of the UHDPE LM, and the much larger contact angle of UHDPE compared to nickel \\cite{Trevoy1958,Ammosova2015}. The larger grain size of the UHDPE creates longer channels for water vapour to traverse. This  results in a smaller water vapour concentration gradient. \n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Comparison of the evaporation rates for \\SI{10}{\\micro\\litre} LMs and an uncoated water droplet. }\n    \\begin{tabular}{cc}\n    \\toprule\n    \\textbf{LM Coating} & \\textbf{Initial Evaporation Rate} \\\\\n     & \/ \\si{\\mg\\per\\minute} \\\\\n    \\midrule\n    (Water Droplet) & 0.1392 \\\\\n    Ni    & 0.1133 \\\\\n    UHDPE & 0.1107 \\\\\n    Ni\/UHDPE & 0.0998 \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\label{tab:evap}%\n\\end{table}%\n\nIt is noteworthy that the new hybrid LM offers the best protection, with the lowest rate of evaporation. It is suggested that this is due to differences in the nickel and UHDPE particle sizes. This difference is clearly visible in the magnified photograph shown in figure~\\ref{fig:LM-photo}(a). The larger UHDPE particles offer good resistance to evaporation, due to their thickness. However, this also leads to large gaps between the particles, due to poor packing. This problem is alleviated by the nickel particles filling the available space. The use of two differently sized spheres for 3D spherical packing is well documented \\cite{Yamada2011,Sohn1968}, and has been shown to increase the maximum perfect packing density beyond the \\num{0.74} limit of a single-sized 3D sphere packing, to \\num{0.93} for an ideal binary system \\cite{Kansal2002}. In this instance, due to the multilayer nature of the LM coating \\cite{dupin2009stimulus}, it is more appropriate to relate to 3D sphere packing than 2D circle packing.\n\n\n\n\\subsection{Liquid Marble Collisions}\n\nIf the gate timings are not accurate, then the LMs will continue on their separate paths and shall not collide. If the timings are accurate, and it is taken that the two LMs are identical, then there are three possible outcomes of the collision. Firstly, that the LMs collide with some elastic property, and then continue on two distinct and new paths. Secondly, that the LMs collide with no elastic property, then continue vertically as two adjacent, but distinct, LMs. Thirdly, that on collision the LMs coalesce into a larger single LM with zero lateral velocity, which continues vertically down.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{single}}\n\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{bounce}}\n\\caption{Overlaid still frames of (a) a single LM, with frames at \\SIlist{0;142;209;242;267}{\\milli\\second}; \nand (b) two colliding LMs, with frames at \\SIlist{0;125;200;217;225;250;275}{\\milli\\second}.}\n\\label{fig:single-bounce}\n\\end{figure}\n\nVideo snapshots showing both a single uninterrupted and a colliding pair of LMs, can be seen in figure~\\ref{fig:single-bounce}. Our experiments demonstrate that LMs collide in an elastic manner. This is unsurprising, due to their previously reported soft-shell and compressible nature \\cite{Aussillous2001,Bormashenko2015}. It also supports the previously published, linear, non-coalescing collision of LMs \\cite{Bormashenko2015}. By monitoring the collisions at \\SI{120}{fps}, it was observed that LMs behave like two soft balls, acting in a manner described in the Soft Sphere Model (SSM), known as a Margolus gate \\cite{margolus2002universal}. A video of a typical collision can be seen in the supporting information. The important distinction between the SSM and the better known Billiard Ball Model (BBM) \\cite{fredkin2002conservative}, is the exit points of the colliding particles compared to the non-colliding particles. In the BBM, the particles are taken to be hard spheres, which instantly rebound off each other --- leading to the \\textbf{AB} paths being outside the corresponding \\textbf{\\={A}B} and \\textbf{A\\={B}} paths. In contrast, the SSM accounts for the finite and appreciable amount of time required for real-world soft spheres to rebound. The result is that the AB paths move to lie inside the unchanged \\textbf{\\={A}B} and \\textbf{A\\={B}} paths. The BBM and SSM pathways can be seen in figures \\ref{fig:BBM-SSM}(a) and \\ref{fig:BBM-SSM}(b), respectively.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.4\\linewidth]{BBM}}\n\\subfigure[]{\\includegraphics[width=0.4\\linewidth]{SSM}}\n\\subfigure[]{\\includegraphics[width=0.7\\linewidth]{steelballs}}\n\\caption{Showing the colliding and non-colliding routes for (a) BBM and (b) SSM pathways. (c) A collision of steel balls, following BBM under gravity (cf.\\ SSM LMs above). Frame times are \\SIlist{0;175;242;284;325}{\\ms}.}\n\\label{fig:BBM-SSM}\n\\end{figure}\n\nIt was possible to break the SSM analogy by increasing the speed of the LMs. The speed of the LMs was calculated by measuring the distance travelled by the LM in a certain number of frames, and knowing the recording frames per second. When the collision happens at \\SI{0.21}{\\metre\\per\\second}, the LMs bounce elastically following SSM paths. However, when the speed of collision is increased to \\SI{0.29}{\\metre\\per\\second}, the two LMs coalesce. This can be seen in the video snapshots in figure~\\ref{fig:coalesce}.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{coalesce}\n  \\caption{Overlaid still frames showing the coalescence of two colliding LMs. Frames shown at \\SIlist{0;117;159;184;209;234;250}{\\milli\\second}.}\n  \\label{fig:coalesce}\n\\end{figure}\n\nGrowth by coalescence of LMs is a commonly observed effect \\cite{Bhosale2012}. For a computing device, the physical nature of the input and output signals should be exactly the same. When two LMs coalesce, the output mass is double a single input mass. Consequently, if colliding LMs at this higher speed, a splitting device would be required to reduce the mass of the output LM. The facile splitting of a LM using a superhydrophobic treated scalpel has previously been reported \\cite{Bormashenko2011scalpel}.\n\nBy analysing the output paths of the LM collider, it becomes apparent that the gate could be modified to act as a 1-bit half-adder, with the possible outcomes demonstrated in figure \\ref{fig:half-adder}. When a single LM traverses the system from the \\textbf{A} or \\textbf{B} channel, it finishes at the left or right extremes, the \\textbf{A\\={B}} or \\textbf{\\={A}B} path, respectively. Once the exit pathways are combined, this is analogous to the sum output on a half-adder. An initial trial confirming feasibility of this is shown in figure~\\ref{fig:half-adder}(a), where a single LM enters from the right channel, crosses the gap, and is reflected to exit on the right side.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.7\\linewidth]{LM-reflect}}\n\\subfigure[]{\\includegraphics[width=0.35\\linewidth]{half-adder-a}}\n\\subfigure[]{\\includegraphics[width=0.35\\linewidth]{half-adder-b}}\n\\subfigure[]{\\includegraphics[width=0.35\\linewidth]{half-adder-c}}\n\\subfigure[]{\\includegraphics[width=0.35\\linewidth]{half-adder-d}}\n\\subfigure[]{\\includegraphics[width=0.4\\linewidth]{half-adder-e}}\n\\caption{(a) Overlaid frames showing the successful reflection of a LM, frames are timed at \\SIlist{0;234;334;375;400;434;476;517}{\\ms}.\n(b) \\& (c) The outcomes of a single unreflected LM passing through the adder. \n(d) The outcome of the adder when two LMs collide and coalesce. \n(e) The outcome of the adder when two LMs collide according to the SSM. \n(f) The electronic representation of a 1-bit half-adder.}\n  \\label{fig:half-adder}\n\\end{figure}\n\nWhen two synchronised LMs pass through the collider, they either rebound or coalesce, according to their velocity at impact. If the LMs coalesce, then the new LM travels straight down the only \\textbf{AB} path, which can be considered to be the carry output. Alternatively if the LMs rebound, as in the SSM, then there are two \\textbf{AB} paths. One of these paths is then considered to be the carry output, and the other is discarded (the choice between the two \\textbf{AB} paths is arbitrary in this case).\n\n\\subsection{One-Bit Full-Adder Proposed}\n\nBy using this design (complete with magnetic timing control) and an intuitive {\\sc xor} gate, we can adapt the model of the one-bit full-adder proposed originally for Belousov-Zhabotinsky medium \\cite{adamatzky2015binary}. The {\\sc xor} gate can be replaced with an {\\sc or} gate without loss of logic, as there is no situation where two LMs will arrive simultaneously. The design schematic can be seen in figure~\\ref{fig:full-adder1}. There are two sets of electromagnets, which cycle on-off in pairs; first EM1 releases, then shortly after EM2 releases. This maintains synchronisation between LMs across the two collision gates.\n\nFor this design iteration, we have used channels for the passage of LMs. This is a deviation from pure collision-based computing, where free-space is used as momentary `wires' on an ad hoc basis. However, in this case, we believe the use of channels to be an important intermediate step towards this goal.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{full-adder1-3}\n  \\caption{The general design schematic for a 1-bit full-adder, operated using liquid marbles.}\n  \\label{fig:full-adder1}\n\\end{figure}\n\nFor the example operations visualised in figure \\ref{fig:full-adder2}, if a LM travels down the \\textbf{A} and \\textbf{B} channel, then they will collide and travel straight to the carry output. If a LM travels down the \\textbf{B} and \\textbf{C\\textsubscript{in}} paths, then the \\textbf{B} LM crosses the first gate, before colliding with the \\textbf{C\\textsubscript{in}} LM and travelling straight down to join the carry output. If a single LM travels down the \\textbf{B} path, it will cross the first and second gate, finishing on the sum output. If a LM travels down the \\textbf{A}, \\textbf{B} and \\textbf{C\\textsubscript{in}} paths, then \\textbf{A} and \\textbf{B} will collide at the first gate and go straight to the carry output, whilst the \\textbf{C\\textsubscript{in}} LM will cross its gate and finish on the sum output.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.494\\linewidth]{fuller-adder2-a}}\n\\subfigure[]{\\includegraphics[width=0.494\\linewidth]{fuller-adder2-b}}\n\\subfigure[]{\\includegraphics[width=0.494\\linewidth]{fuller-adder2-c}}\n\\subfigure[]{\\includegraphics[width=0.494\\linewidth]{fuller-adder2-d}}\n\\caption{Example operations of the 1-bit full-adder to sum \n(a)~${1+1+0=10}$, \n(b)~${0+1+1=10}$,\n(c)~${0+1+0=01}$, and \n(d)~${1+1+1=11}$.}\n\\label{fig:full-adder2}\n\\end{figure}\n\nBased on the observations of our collision gate, we note that such a full-adder could be implemented with dimensions of approximately \\SI{15 x 15}{\\cm}, using LMs with a volume of approximately \\SI{10}{\\micro\\litre}, and a LM pre-collision run length of \\SI{2}{\\cm}. This gate could then be cascaded as required to produce an n-bit full-adder. Empirical testing has shown that reliable and manipulable LMs can be formed down to \\SI{1.0}{\\micro\\litre}, meaning that the device could then be scaled down appropriately. Using a single set of syringes, the automatic marble maker can up to eight LMs per needle per second. At this speed, synchronisation of the electromagnets becomes both more crucial and more tricky --- potentially requiring computer control. Initial investigations into the collision lifetime of the LM are promising, with six \\SI{3}{\\mm} LMs confined to a \\SI{2.5 x 2.5}{\\cm} space on an orbital shaker at 100\\,rpm, showing no signs of wear after three hours.\n\n\n\n\\section{Conclusions}\n\nIn summary, this demonstration of a collision interaction gate represents the first computing device operated by LMs. A new automatic technique for the easy and reproducible synthesis of LMs enhances the reliability of gate operations. The novel electromagnetic synchronisation of the LM collisions was made possible by the development of a new magnetic hybrid LM, with a coating composed of nickel and UHDPE, used in conjunction with electromagnets for breaking, holding, and synchronised release of the LMs. This collision gate would operate as a 1-bit half-adder, once the sum outputs are combined. A design schematic for a 1-bit full-adder was proposed.\n\n\nThe use of LMs for collision-based computation has many advantages (additional degrees of freedom) over previous approaches. Due to their nature, it is possible to carry cargo in the LMs, which adds an additional dimension to the calculations. It is also possible to initiate chemical reactions within marbles by their coalescence \\cite{Chen2017}. By varying the diameters of the LMs, different sizes can represent different values, and will have different relative trajectories --- removing the limitations of a binary system. Use of a magnet can remove the coating from magnetic marbles, which (if done on a superhydrophobic surface) can roll freely down a slope as droplets, before being reformed using a different coating. Compared to droplet computing \\cite{Mertaniemi2012,Katsikis2015}, only a tiny portion of the circuit needs to be treated hydrophobic, making larger and more complicated circuits easier and cheaper to construct. These points, combined with LM's ability to be easily merged, levitated \\cite{Chen2017}, divided \\cite{Bormashenko2011scalpel}, opened\/closed \\cite{Zhao2010}, and easily propelled by a variety of methods make LMs a fascinating and potentially prosperous addition to the unconventional computing family.  \n\nWe envision the continued development of LM arithmetic circuits, and are currently working on producing working models of cascading standard gates.\n\n\n\\section*{Acknowledgements}\n\nThis research was supported by the EPSRC with grant EP\/P016677\/1 ``Computing with Liquid Marbles''.\nThe authors thank Dr Richard Mayne for his help with microscope imaging. \n\n\n\\section*{Supporting Information}\n\nVideo footage of the liquid marble collisions is available at \\url{youtu.be\/Lt1VWBtRk6E}, \\url{youtu.be\/isufSrhW_8M}, \\url{youtu.be\/sFEtVaFfxKI}, and \\url{youtu.be\/H8Si883lCw4}.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and results}\nOne of the fundamental topics in extremal graph theory is the study of minimum degree conditions that force a graph to contain a given spanning substructure. Perhaps the best known result in the area is Dirac's theorem~\\cite{dirac}, which\nstates that any graph $G$ on $n\\geq 3$ vertices with minimum degree $\\delta (G) \\geq n\/2$ contains a Hamilton cycle. \nThe P\\'osa--Seymour conjecture (see~\\cite{posa} and~\\cite{seymour}) states that any graph $G$ on $n$ vertices with $\\delta (G) \\geq\nrn\/(r+1)$ contains the $r$th power of a Hamilton cycle. (The $r$th power of a Hamilton cycle $C$ is obtained from $C$ by adding an edge\nbetween every pair of vertices of distance at most $r$ on $C$.)  Koml\\'os, S\\'ark\\\"ozy and Szemer\\'edi~\\cite{kss} proved this conjecture\n for sufficiently large graphs.\n\nA decade ago, B\\\"ottcher, Schacht and Taraz~\\cite{bot} proved a very general minimum degree result, the so-called \\emph{Bandwidth  theorem}. \nA graph $H$ on $n$ vertices is said to have \\emph{bandwidth at most $b$}, if there exists a \nlabelling of the vertices of $H$ by the numbers $1, \\dots ,n$ such that for every edge $ij\\in E(H)$ we have $|i-j|\\leq b$.\nClearly every graph $H$ has bandwidth at most $|H|-1$. \nFurther, a Hamilton cycle has bandwidth $2$, and in general the $r$th power of a Hamilton cycle has bandwidth at most $2r$.\nB\\\"ottcher, Preussmann,  Taraz and  W\\\"urfl~\\cite{BPTW10} proved that every planar graph  $H$ on $n$ vertices with bounded maximum degree has bandwidth at most $O(n\/\\log n)$.\nThe Bandwidth theorem gives a condition on the minimum  degree of a graph $G$ on \n$n$ vertices\nthat ensures $G$ contains \\emph{every} $r$-chromatic graph on $n$ vertices of bounded degree and of bandwidth $o(n)$.\n\\begin{theorem}[The Bandwidth  theorem, B\\\"ottcher, Schacht and Taraz~\\cite{bot}]\\label{bst}\nGiven any $r,\\Delta \\in \\mathbb N$ and any $\\gamma >0$, there exist constants $\\beta >0$ and $n_0 \\in \\mathbb N$\nsuch that the following holds. Suppose that $H$ is an $r$-chromatic graph on $n \\geq n_0$ vertices with $\\Delta (H) \\leq \\Delta$ \nand bandwidth at most $\\beta n$. If $G$ is a graph on $n$ vertices with\n$$\\delta (G) \\geq \\left( \\frac{r-1}{r}+\\gamma \\right)n,$$\nthen $G$ contains a copy of $H$. \n\\end{theorem}\nWe remark that Theorem~\\ref{bst} had been conjectured by  Bollob\\'as and Koml\\'os~\\cite{komlos}.\nSince the Bandwidth  theorem was proven, a number of variants of the result have been obtained including for arrangeable graphs~\\cite{arrange}, degenerate graphs~\\cite{lee} and in the setting of random and pseudorandom graphs~\\cite{abet, bkt, hls}, as well as\nfor robustly expanding graphs~\\cite{knox}.\nVery recently a Bandwidth  theorem for approximate decompositions was proven by Condon, Kim, K\\\"uhn and Osthus~\\cite{ckko}, whilst Glock and Joos~\\cite{gj} proved a $\\mu n$-bounded edge colouring extension of Theorem~\\ref{bst}.\nA general embedding result of B\\\"ottcher, Montgomery, Parczyk and Person~\\cite{bmpp} also implies a bandwith theorem in the setting of randomly perturbed graphs.\n\nFor many graphs $H$, the minimum degree condition in Theorem~\\ref{bst} is best-possible up to the term $\\gamma n$. \nFor example, suppose that $H$ is a \\emph{$K_r$-factor} (that is, we seek a collection of vertex-disjoint copies of $K_r$ in $G$ that together cover\nall the vertices in $G$). So $\\chi(H)=r$, $\\Delta(H) =r-1$ and $H$ has bandwidth $r-1$. \nSuppose that $G$ is obtained from two disjoint vertex classes $A$ and $B$ of sizes $n\/r+1$ and $(r-1)n\/r-1$ respectively so that $G$ contains all edges other than those with both endpoints in $A$.\n Then it is easy to see\nthat $G$ does not contain a $K_r$-factor, however, $\\delta (G) = \\left( \\frac{r-1}{r} \\right)n-1$. In fact, note that the famous Hajnal--Szemer\\'edi theorem~\\cite{hs} asserts that an $n$-vertex graph $G$ contains a $K_r$-factor provided\n$r|n$ and $\\delta (G)\\geq \\left( \\frac{r-1}{r} \\right)n$. Thus, this extremal example is sharp.\n(Note though for many $r$-partite graphs $F$, a significantly lower minimum degree condition than that in Theorem~\\ref{bst} ensures an $F$-factor, see~\\cite{kuhn2}.)\n\nAs for many other problems in the area, this extremal example has the characteristic that it contains a \\emph{large} independent set.\nThere has thus been significant interest in seeking variants of classical results in extremal graph theory, where one now forbids the host graph from containing a large independent set.\nIndeed, nearly 50 years ago, Erd\\H{o}s and S\\'os~\\cite{sos} initiated the study of the Tur\\'an problem under the additional assumption of small independence number. That is, they considered the\nnumber of edges in an $n$-vertex $K_r$-free graph with independence number $o(n)$. This topic is now known as \\emph{Ramsey-Tur\\'an theory} and has been extensively studied by numerous authors (see e.g. \\cite{lenz,ehss,mum, sim}).\nMore recently, there has been interest in similar questions but where now one seeks a $K_r$-factor in an $n$-vertex graph with independence number $o(n)$ and large minimum degree (see~\\cite{triangle, bms,  han}).\n\nA stronger notion of a graph not containing a large independent set, is that of being \\emph{locally dense}. More precisely, given $\\rho,d >0$, we say that an $n$-vertex graph $G$ is \\emph{$(\\rho,d)$-dense} \nif every $X \\subseteq V(G)$  satisfies $e(G[X]) \\geq d\\binom{|X|}{2}-\\rho n^2$. \nNote that the property of being locally dense is weaker than being dense and (pseudo)random and stronger than having sublinear independence number.\nLocally dense graphs have been considered in a number of previous papers. For example, there have been several papers on a question of Erd\\H{o}s, Faudree, Rousseau, and Schelp~\\cite{efrs}; there they considered a  variant of the notion of $(\\rho,d)$-dense, and asked for the values of $\\rho$ and $d$ that\nguarantee a $(\\rho,d)$-dense graph contains a triangle.\nOne can view the notion of locally dense as a parameter that ensures a graph is in some sense `random-like'. Therefore, there has been interest in determining the number of (homomorphic) copies of a fixed graph $H$ in a $(\\rho,d)$-dense graph $G$, and in particular whether this count is close to the value obtained if $G$ were a random graph; the study of this topic (for graphs {and} hypergraphs) was initiated by\nKohayakawa,  Nagle,  R\\\"odl and Schacht~\\cite{knrs}.\n\n\nThe aim of this paper is to prove the following locally dense version of the Bandwidth  theorem.\n\\begin{theorem}\\label{main}\nFor all $\\Delta \\in \\mathbb{N}$ and $d,\\eta>0$, there exist constants $\\rho,\\beta,n_0>0$ such that for every $n \\geq n_0$, the following holds.\nLet $H$ be an $n$-vertex graph with $\\Delta(H) \\leq \\Delta$ and bandwidth at most $\\beta n$.\nThen any $(\\rho,d)$-dense graph $G$ on $n$ vertices with $\\delta(G)\\geq (1\/2+\\eta)n$ contains a copy of $H$.\n\\end{theorem}\nIn the case when $H$ corresponds to a $K_r$-factor, Theorem~\\ref{main} had been proven by Reiher and Schacht (see~\\cite{bms}).\nNote that in the case when $H$ is connected, the minimum degree in Theorem~\\ref{main} is best-possible, up to the $\\eta n$ term. Indeed, if $G$ consists of two vertex-disjoint cliques each of size $n\/2$ then\n$G$ trivially does not contain $H$ though $G$ is locally dense and $\\delta (G)=n\/2-1$.\n\nA striking feature of Theorem~\\ref{main} is that, unlike Theorem~\\ref{bst}, the minimum degree condition does not depend on the chromatic number of $H$.\nIn particular, when $\\chi (H)=2$, the minimum degree condition in Theorem~\\ref{main} is the same as that in Theorem~\\ref{bst}. Thus, in the case of bipartite $H$, there is no benefit in adding the condition that $G$ is locally dense. However, when $\\chi (H)>2$, the minimum degree in Theorem~\\ref{main} is substantially reduced compared to the Bandwidth  theorem. \n\nIt would also be extremely interesting to prove a version of Theorem~\\ref{main} for graphs of sublinear independence number.\nNote though that examples in~\\cite{bms} show that the statement of Theorem~\\ref{main} is far from true if we require that $G$ has sublinear independence number instead of the locally dense condition.\n Indeed, the minimum degree necessary for the existence of a $K_r$-factor in such a graph is at least $(\\frac{r-2}{r}+o(1))n$ for every $r \\geq 4$.\nSo these two problems are genuinely different.\n\n\n\nThe proof of Theorem~\\ref{main} draws on ideas from~\\cite{3partite, bot}, and our approach makes use of the Regularity--Blow-up method.\nWe also employ several new ideas (particularly with regard to dealing with so-called exceptional vertices). In the next section we give an overview of the proof of Theorem~\\ref{main}.\nIn Section~\\ref{sec3} we introduce some notation, as well as several fundamental properties of locally dense graphs. The Regularity and Blow-up lemmas\nare presented in Section~\\ref{secrl}. A key step in the proof of Theorem~\\ref{main} is to show that the hypothesis of this theorem ensures $G$ contains the $r$th power of a Hamilton cycle; we prove this in Section~\\ref{cyclesec}.\nThe proof of Theorem~\\ref{main} then breaks into two main parts: the proof of two so-called Lemmas for $H$ (presented in Section~\\ref{sec6}) and the Lemma for $G$ (presented in Section~\\ref{sec7}). In Section~\\ref{sec8} we combine all these results\nto prove Theorem~\\ref{main}. We give some concluding remarks in Section~\\ref{sec9}.\n\n{\\bf Additional note.} Since this paper was first submitted,\n Ebsen, Maesaka,  Reiher, Schacht and Sch\\\"ulke~\\cite{newpaper} have built on our work to generalise Theorem~\\ref{main}. Indeed, they replace the minimum degree condition on $G$ with an inseparability condition.\n\n\\section{Overview of the proof of Theorem~\\ref{main}}\\label{sketch}\nThe overall strategy follows in the same spirit as the proof of the Bandwidth theorem in~\\cite{bot}, though the precise details of the proofs of the key steps in the argument turn out to be quite different.\nIn particular, the setting of locally dense graphs both smooths over some aspects of the proof, as well\nas introducing additional difficulties. \nOften in problems involving embedding a spanning structure, the most challenging aspect of the proof is dealing with so-called exceptional vertices (i.e. either trying to cover the remaining last few vertices in the host graph or those few vertices that do not fit in some general structure in the host graph). In this paper, we take a novel approach to dealing with such vertices.\nBelow we outline the key steps in our proof and highlight\nsome of the main novelties in our approach.\n\n\n\\smallskip\n\n\\noindent\n{\\bf Obtaining structure in $G$.} \nSuppose that $H$ and $G$ are as in the statement of the theorem where $\\chi (H)=r$.\nThe first step in the proof is to apply the Regularity lemma (Lemma~\\ref{reg}) to $G$ to obtain the reduced graph $R$ of $G$. The reduced graph $R$ is locally dense (with somewhat different parameters compared to $G$) and \n`inherits' the minimum degree of $G$ (i.e. $\\delta (R) > (1\/2+o(1))|R|$). These properties ensure that $R$ contains an almost spanning subgraph $Z^{2r}_{\\ell}$ which has the following  properties:\n\\begin{itemize}\n\\item $Z^{2r}_{\\ell}$ covers all but at most $2r$ of the vertices in $R$;\n\\item $Z^{2r}_{\\ell}$ consists of $\\ell$ vertex-disjoint copies $K^1,\\dots, K^{\\ell}$ of $K_{2r}$ (in particular $|Z^{2r}_{\\ell}|=2r\\ell$);\n\\item For each $1\\leq i \\leq \\ell$, there are all possible edges between $K^{i}$ and $K^{i+1}$ except that we miss a perfect matching between the two. (Note here $K^{\\ell+1}:=K^{1}$.)\n\\end{itemize}\nThe existence of $Z^{2r}_{\\ell}$ in $R$ can be guaranteed by finding a sufficiently large power of a Hamilton cycle in $R$. This is achieved in Section~\\ref{cyclesec} (see Theorem~\\ref{rcycle}).\nUsing this, one can easily deduce that $G$ contains an \\emph{almost} spanning structure $\\mathcal C$ that looks like the `blow-up' of $Z^{2r}_{\\ell}$. More precisely, if $V(Z^{2r}_{\\ell})=\\{1,\\dots, 2r\\ell\\}$ and\n$V_1,\\dots, V_{2r\\ell}$ are the corresponding clusters in $G$, then\n\\begin{itemize}\n\\item[(i)] $V(\\mathcal C)= V_1 \\cup \\dots \\cup V_{2r\\ell}$;\n\\item[(ii)] $\\mathcal C [V_i,V_j]$ is ${\\varepsilon}$-regular whenever $ij \\in E(Z^{2r}_{\\ell})$;\n\\item[(iii)] If $jk$ is an edge in one of the cliques $K^i$ then $\\mathcal C [V_j,V_k]$ is superregular.\n\\end{itemize}\nWe refer to $\\mathcal C$ as a \\emph{cycle structure}.\n\nSuppose that in fact $\\mathcal C$ is a spanning subgraph of $G$. In this case, ideally, one would now like to take the following approach. Let $x_1,\\dots, x_n$ denote the bandwidth ordering of $H$. Partition $V(H)$ into $\\ell$ classes $C_1,\\dots, C_{\\ell}$ so that \n\\begin{itemize}\n\\item $c_i:=|C_i|=|\\cup  _{j \\in V(K^i)} V_j|$ for all $1\\leq i \\leq \\ell$;\n\\item $C_1$ contains the vertices $x_1,\\dots, x_{c_1}$; $C_2$ contains the vertices $x_{c_1+1},\\dots, x_{c_1+c_2}$ and so forth.\n\\end{itemize}\nThen embed the vertices from $C_1$ into the clusters in $G$ corresponding to the clique $K^1$, embed the vertices from $C_2$ into the clusters in $G$ corresponding to the clique $K^2$, and so on.\n\nAt first sight this seems like a plausible strategy: since the partition of $V(H)$ respected the bandwidth ordering of $H$ (and as $H$ has small bandwidth), most edges in $H$ lie in the induced subgraphs $H[C_i]$; all remaining edges lie in the bipartite graphs $H[C_i,C_{i+1}]$. Suppose one could map each $C_i$ onto the clusters corresponding to $K^i$, so that each such cluster $V_j$ receives precisely $|V_j|$ vertices from $C_i$,\nand crucially, all edges $xy$ in $H[C_i]$ are such that $x$ and $y$ are mapped to different clusters in $K^i$.\nThat is, suppose\n we have a graph homomorphism $\\phi _i$ between $H[C_i]$ and $K^i$ that maps precisely $|V_j|$ vertices to each $V_j$.\nFurther, suppose the $\\phi _i$ together combine to give a graph homomorphism $f$ from $H$ to $Z^{2r}_{\\ell}$ (so the edges in each $H[C_i,C_{i+1}]$ are mapped to edges in $R[V(K^i),V(K^{i+1})]$).\nSet $G_i:=G[\\cup  _{j \\in V(K^i)} V_j]$. \nThen (iii) above ensures we could  apply the Blow-up lemma to each graph $G_i$ so as to embed $H[C_i]$ into $G_i$. Further, (ii) ensures that we can achieve this embedding so all edges in the graphs $H[C_i,C_{i+1}]$ are also present. That is, we would obtain an embedding of $H$ into $G$.\n\nThis naive approach fails though as there is no guarantee one can map each $C_i$ onto the clusters corresponding to $K^i$ so that each such cluster $V_j$ receives precisely $|V_j|$ vertices from $C_i$.\nFurthermore, in the above approach we assumed that $\\mathcal C$ is a spanning subgraph of $G$; in reality we have a small exceptional set $V_0$ of vertices in $G$ uncovered by $\\mathcal C$.\n\n\\smallskip\n\n\\noindent\n{\\bf The Basic Lemma for $H$ and the Lemma for $G$.} \nInstead of the above, we prove the so-called Basic~Lemma~for~$H$ (Lemma~\\ref{lemmaforH1}). Here we show that one can find a graph homomorphism $f$ from $H$ into $Z^{2r}_{\\ell}$ so that for every cluster $V_i$ of $R$, \\emph{approximately} $|V_i|$ vertices are mapped to it.\nThis therefore `almost' gives us the desired graph homomorphism $f$ from $H$ into $Z^{2r}_{\\ell}$. \nIn the proof of Lemma~\\ref{lemmaforH1} we rely on the fact that the $K^i$ in $Z^{2r}_{\\ell}$ are copies of $K_{2r}$; note that in the analogous structure in the proof of the  Bandwidth theorem~\\cite{bot}, the $K^i$ are copies of $K_r$.\nTo see why our  condition is helpful for us, note that whilst in general an $r$-partite graph $H'$ does not have an `almost balanced' graph homomorphism into $K_r$ (since $H'$ may have colour classes of wildly different sizes), for  $r$-partite graphs $H'$ of bounded degree and sublinear bandwidth\none can \nalways find an almost balanced graph homomorphism from $H'$ into $K_{2r}$.\n\nNext, in the Lemma~for~$G$ (Lemma~\\ref{lemmaforG}) we prove that, if one does not have an  exceptional set $V_0$, then we can move vertices around  the cycle structure $\\mathcal C$ in such a way to ensure  that now each cluster $V_i$ in $\\mathcal C$ has size precisely corresponding \nto the number of vertices mapped to $V_i$ by $f$. This is at the expense of weakening condition (iii): after applying Lemma~\\ref{lemmaforG} we only have that each clique $K^i$ splits into two cliques $K^i _1$ and $K^i _2$ of size $r$ such that \nif $jk$ is an edge in one of the cliques $K_1^i$ or $K_2 ^i$ then $\\mathcal C [V_j,V_k]$ is superregular.\nHowever, this is still good enough to apply the Blow-up lemma to find our desired embedding of $H$ into $G$.\n\n\n\\smallskip\n\n\\noindent\n{\\bf Special Lemma for $H$.} So far we have been assuming that there is no exceptional set $V_0$; further, in the the proof of the  Bandwidth theorem~\\cite{bot}, B\\\"ottcher, Schacht and Taraz were able to utilise the large minimum degree to incorporate exceptional vertices into (their analogue of the cycle structure) $\\mathcal C$. We have significantly smaller minimum degree, so are unable to do this in our setting.\n\nInstead, given the bandwidth ordering $x_1,\\dots, x_n$ of $H$, we reserve a short initial segment $x_1,\\dots, x_t$; and let $H'$ denote the subgraph of $H$ induced by $x_1,\\dots, x_t$. \nHere $t$ will be significantly bigger than $\\beta n$ (recall $H$ has bandwidth at most $\\beta n$), but still $H'$ will only be a small fraction of $H$.\nVia the Special Lemma for $H$ (Lemma~\\ref{lemmaforH}) we will embed $H'$ into $G$ in such a way that crucially all of $V_0$ is covered by $H'$, and equally importantly, we do not cover more than a small proportion of each cluster $V_i$ in $\\mathcal C$.\n\nIn the proof of Lemma~\\ref{lemmaforH},\nsince $V_0$ may only contain very few (or even no edges) we must assign an independent set $I$ in $H'$ of size $|V_0|$ to be embedded onto $V_0$.\nWe then must connect up $I$ through the rest of $G$ to obtain a copy of $H'$. In particular, since $H'$ is much smaller than $H$, the distance between two vertices $x, y \\in I$ in $H'$ may also be small. So it is crucial that $G$ is `highly connected'.\nThe Connecting lemma (Lemma~\\ref{connect}) ensures this is the case. (Lemma~\\ref{connect} is also applied in the proof of Theorem~\\ref{rcycle}.)\n\nCare is also needed to ensure that Lemma~\\ref{lemmaforH} is compatible with the Basic~Lemma~for~$H$ (Lemma~\\ref{lemmaforH1}). That is, we use Lemma~\\ref{lemmaforH} to embed $H'$ in $G$ and Lemma~\\ref{lemmaforH1} to embed $H\\setminus H'$ in $G$. Thus, we need to ensure the copies of $H'$ and $H\\setminus H'$ can be positioned in $G$ in such a way that they `glue' together  to form a copy of $H$.\n\n\n\\smallskip\n\nNote that the reader should view the above overview as an idealisation of the proof.\nIndeed, when we prove Theorem~\\ref{main} in Section~\\ref{sec8}, some of the details will be a little different. For example, for technical reasons it is in fact important that we find a spanning copy of $Z^{r^*}_{\\ell}$ in $R$ for some $r^* \\gg r$ rather than $Z^{2r}_{\\ell}$.\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec3}\n\n\\subsection{Notation}\nGiven a set $X$ and $k \\leq |X|$, write $\\binom{X}{k}$ for the set of $k$-element subsets of $X$. Given $r \\in \\mathbb{N}$, write $[[2r]]^2 := [r]^2 \\cup ([2r]\\setminus[r])^2$.\nGiven a function $f : X \\rightarrow Y$ and $A \\subseteq X$, we write $f|_A$ for the restriction of $f$ to $A$ and $f(A) := \\lbrace f(a) : a \\in A\\rbrace$.\n\n\nGiven a graph $G$, we write $V(G)$ and $E(G)$ for the  vertex and edge sets respectively, and $|G| := |V(G)|$ and $e(G) := |E(G)|$.\nThe \\emph{degree} of a vertex $x \\in V(G)$ is denoted by $d_G(x)$ and its neighbourhood  by $N_G(x)$. The \\emph{degree} of a subset $X \\subseteq V(G)$ is $d_G(X) := |\\bigcap_{x \\in X}N_G(x)|$.\nA subgraph $H \\subseteq G$ is \\emph{$s$-extendable} if $d_G(V(H)) \\geq s$.\nGiven  vertices $x_1,\\dots, x_k$ we write $N_G(x_1,\\dots,x_k):=\\bigcap_{1\\leq i\\leq k}N_G(x_i)$.\nIf $A \\subseteq V(G)$ we write $N_G(x,A):=N_G(x) \\cap A$ and $d_G(x,A) := |N_G(x) \\cap A|$.\nWe say that $A$ is \\emph{$k$-independent} if every vertex in $A$ is at distance at least $k+1$ in $G$, i.e.~the shortest path in $G$ between any pair of elements in $A$ has length at least $k+1$.\nGiven $X,Y \\subseteq V(G)$ (not necessarily disjoint), define $e_G(X,Y)$ to be the number of edges $xy \\in E(G)$ with $x \\in X$ and $y \\in Y$.\nIf $X$ and $Y$ are disjoint then $G[X,Y]$ is the bipartite graph with vertex classes $X$ and $Y$ whose edge set consists of all those edges in $G$ with one endpoint in $X$, the other in $Y$.\n\nGiven two graphs $G$ and $H$, we say that $f : V(H) \\rightarrow V(G)$ is a \\emph{graph homomorphism} if $f(x)f(y) \\in E(G)$ whenever $xy \\in E(H)$.\nIf $f$ is additionally injective, we say that $f$ is an \\emph{embedding (of $H$ into $G$)}. Then $H \\subseteq G$.\n\n\\smallskip\n\nThroughout the paper we will ignore floors and ceilings wherever they do not affect the argument.\nThe constants in the hierarchies used to state our results are chosen from right to left.\nFor example, if we claim that a result holds whenever $0<1\/n\\ll a\\ll b\\ll c\\le 1$ (where $n$ is the order of the graph), then \nthere are non-decreasing functions $f:(0,1]\\to (0,1]$, $g:(0,1]\\to (0,1]$ and $h:(0,1]\\to (0,1]$ such that the result holds\nfor all $0<a,b,c\\le 1$ and all $n\\in \\mathbb{N}$ with $b\\le f(c)$, $a\\le g(b)$ and $1\/n\\le h(a)$. \nNote that $a \\ll b$ implies that we may assume in the proof that e.g. $a < b$ or $a < b^2$.\n\nGiven numbers  $a,b,c$, we write $a=b\\pm c$ to mean $a \\in [b-c,b+c]$.\n\n\\subsubsection{Named graphs}\\label{notation}\nGiven a graph $H$, the graph $H^r$, called the \\emph{$r$th power of $H$}, is obtained from $H$ by adding an edge between every pair of vertices of distance at most $r$ in $H$.\nIn particular:\n\\begin{itemize}\n\\item $P^r_k = P = v_1\\ldots v_k$ is an \\emph{$r$-path} if $V(P) = \\lbrace v_1,\\ldots,v_k\\rbrace$ and $E(P) = \\bigcup_{j \\in [r]}\\lbrace v_iv_{i+j} : 1 \\leq i \\leq k-j\\rbrace$; and\n\\item $C^r_k = C = w_1\\ldots w_k$ is an \\emph{$r$-cycle} if $V(C) = \\lbrace w_1,\\ldots,w_k\\rbrace$ and \n$E(C) = \\bigcup_{j \\in [r]} \\lbrace w_iw_{i+j} : 1 \\leq i \\leq k\\rbrace$, where addition is modulo $k$.\n\\end{itemize}\nAdditionally,\n\\begin{itemize}\n \\item $F$ is an \\emph{$r$-trail (of length $s$)} if there exists an ordered sequence of not necessarily distinct vertices $v_1,\\ldots,v_s$ such that $V(F) = \\lbrace v_1,\\ldots,v_s\\rbrace$ and $E(F) = \\bigcup_{j \\in [r]}\\lbrace v_iv_{i+j} : 1 \\leq i \\leq s-j\\rbrace$.\nObserve that $P^r_k$ is an $r$-trail, and $F \\cong P^r_s$ if and only if $|F|=s$. \n\n\\item A \\emph{$K$-tiling} is a collection of vertex disjoint copies of $K$. If it contains $t$ copies, we denote it by $t\\cdot K$. If $H \\subseteq G$ is a  $K$-tiling which is also spanning, we say that $H$ is a \\emph{$K$-factor} of $G$.\n\\end{itemize}\nDefine\n\\begin{itemize}\n\\item  $Z^r_\\ell$ to be the graph with vertex set $[\\ell]\\times[r]$ in which $(i,j)(i',j')$ is an edge whenever (i) $|i-i'| \\leq 1$ and $j \\neq j'$ and when (ii) $i=\\ell$, $i'=1$ and $j \\neq j'$.\n\\end{itemize}\nThus, $Z^r _\\ell$ is obtained from a cycle on $\\ell$ vertices by replacing each vertex with a clique on $r$ vertices and replacing every edge with a complete bipartite graph minus a certain perfect matching.\nAs indicated in Section~\\ref{sketch}, $Z^{2r}_\\ell$  will be used in the proof of Theorem~\\ref{main} as a framework for embedding (most of) $H$ into $G$.\n Note that B\\\"ottcher, Schacht and Taraz~\\cite{bot} used a very similar structure in their proof of the Bandwidth theorem.\n\n\\smallskip\n\nObserve that\n\\begin{equation}\\label{CZ}\n2\\ell\\cdot K_r \\subseteq C^{r-1}_{2r\\ell} \\subseteq Z^r_{2\\ell} \\subseteq C^{2r-1}_{2r\\ell} \\subseteq Z^{2r}_{\\ell},\n\\end{equation}\nand the lexicographic ordering of $V(Z^{r}_\\ell)$ (i.e.~$(1,1)(1,2),\\ldots,(1,r),(2,1),\\ldots,(\\ell,r)$) is an $(r-1)$-cycle ordering of $C^{r-1}_{r\\ell}$.\\COMMENT{AT: is there a better way to express this last statement?}\n\n\\medskip\nGiven $A,B \\subseteq V(G)$ and $x_1,\\ldots,x_k \\in V(G)$, when we say that e.g. $ABx_1\\ldots x_k$ is an $r$-path (respectively $r$-trail, $r$-cycle), we mean that any ordering $a_1,\\ldots,a_{|A|}$ of $A$ and any ordering $b_1,\\ldots,b_{|B|}$ of $B$ are such that $a_1\\ldots a_{|A|}b_1\\ldots b_{|B|} x_1 \\ldots x_k$ is an $r$-path (respectively $r$-trail, $r$-cycle). An $r$-path (respectively $r$-trail, $r$-cycle), $Ax_1\\ldots x_kB$ or $x_1\\ldots x_kAB$ is defined analogously.\n\nSuppose $X$ and $Y$ are disjoint sets of vertices of size $r$.\nWe say that an $r$-path $P$ is \\emph{between} $X$ and $Y$ if $P = Xx_1\\ldots x_kY$ for some vertices $x_1,\\ldots,x_k$.\nObserve that $P[X],P[Y] \\cong K_r$.\nFurther, $P$ \\emph{avoids} a set $W \\subseteq V(G)$ if $V(P)\\cap W = \\emptyset$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Properties of locally dense graphs}\n\nIn this section we prove some simple properties of locally dense graphs $G$: that $G$ induced on a large vertex subset is still locally dense; after removing a small set of vertices, $G$ is still locally dense; and $G$ contains many copies of cliques of a fixed size which additionally have a large common neighbourhood.\n\nA fact that we shall use often throughout the paper is that if $0<\\rho<\\rho'$ and $0<d'<d$, then a $(\\rho,d)$-dense graph is also $(\\rho',d')$-dense.\n\n\n\\begin{lemma}\\label{nbrhd}\nLet $r,n \\in \\mathbb{N}$ and $0 < 1\/n \\ll \\rho \\ll d,1\/r$, and $0 < d, \\alpha < 1$.\n Let $G$ be a $(\\rho,d)$-dense graph on $n$ vertices and let $U \\subseteq V(G)$ where $|U|=\\alpha n$.\nThen\n\\begin{itemize}\n\\item[(i)] $G[U]$ is $(\\rho\/\\alpha^2,d)$-dense;\n\\item[(ii)] $G\\setminus U$ is $(\\rho\/(1-\\alpha)^2,d)$-dense;\n\\item[(iii)] $G$ contains at least $dn\/2$ vertices of degree at least $dn\/2$;\n\\item[(iv)] $G$ contains at least $(d\/2)^{\\binom{r+1}{2}}n^r\/r!$ copies of $K_r$, each of which is $d^rn\/2^r$-extendable.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nThe proof of~(i) is clear and (ii) follows immediately from (i). \nFor~(iii), let $Y := \\lbrace v \\in V(G) : d_G(v) \\geq dn\/2\\rbrace$.\nThen\n$$\n2d\\binom{n}{2} - 2\\rho n^2 \\leq 2e(G) \\leq (n-|Y|) \\frac{dn}{2} + |Y| n\n$$\nand so\n$$\n|Y| \\geq \\frac{dn-2d-4\\rho n}{2-d} \\geq \\frac{dn}{2},\n$$\nproving~(iii). \n\n\nIt remains to prove~(iv).\nWe claim that for each $i \\leq r$, there is a set $\\mathcal{T}_i$ of (ordered) tuples $\\V x = (x_1,\\ldots,x_i)$ such that $G[\\lbrace x_1,\\ldots,x_i \\rbrace] \\cong K_i$ and $d_G(\\lbrace x_1,\\ldots,x_i\\rbrace) \\geq d^in\/2^i$ for all $\\V x \\in \\mathcal{T}_i$, and $|\\mathcal{T}_i| \\geq (d\/2)^{\\binom{i+1}{2}}n^i$.\nThis will immediately imply (iv) as $\\mathcal{T}_r$ gives rise to at least $(d\/2)^{\\binom{r+1}{2}}n^r\/r!$ (unlabelled) copies of $K_r$ each of which is $d^rn\/2^r$-extendable.\n\nWe will prove this by induction on $i$. Part (iii) implies that $G$ contains a set $\\mathcal{T}_1$ of $dn\/2$ copies of $K_1$ which are all $dn\/2$-extendable.\nSuppose that we have obtained $\\mathcal{T}_{i-1}$ with the required properties for some $2 \\leq i \\leq r$.\n\nFix $\\V x = (x_1,\\ldots,x_{i-1}) \\in \\mathcal{T}_{i-1}$.\nThe graph $G_{\\V x} := G[N_G(x_1,\\ldots,x_{i-1})]$ induced by its neighbourhood contains at least $d^{i-1}n\/2^{i-1}$ vertices and so (i) implies that it is $(2^{2i-2}\\rho\/d^{2i-2},d)$-dense and hence $(\\sqrt{\\rho},d)$-dense.\nNow, using the fact that $1\/n \\ll \\sqrt{\\rho} \\ll d,1\/r$,~(iii) implies that $G_{\\V x}$ contains at least $(d\/2)\\cdot d^{i-1}n\/2^{i-1} = d^in\/2^i$ vertices each of degree at least $d^in\/2^i$.\nEach such vertex $y$ gives rise to an $r$-tuple $\\V x(y) := (x_1,\\ldots,x_{i-1},y)$.\nCertainly $G[\\lbrace x_1,\\ldots,x_{i-1},y\\rbrace] \\cong K_i$, and further $d_G(\\lbrace x_1,\\ldots,x_{i-1},y\\rbrace) \\geq d^in\/2^i$ since $y$ has at least this many neighbours in the common neighbourhood of $\\V x$.\nLet $\\mathcal{T}_i$ be the collection of all these tuples $\\V x(y)$ formed from each $\\V x \\in \\mathcal{T}_{i-1}$.\nObserve that they have the required properties, and are all distinct, so\n$$\n|\\mathcal{T}_i| \\geq d^in\/2^i \\cdot |\\mathcal{T}_{i-1}| \\geq (d\/2)^{\\binom{i}{2}+i}n^i = (d\/2)^{\\binom{i+1}{2}}n^i.\n$$\nThis completes the proof of the lemma.\n\\end{proof}\n\n\n\n\n\n\n\\begin{comment}\n\n\nThe next result\nstates that for every $r \\in \\mathbb{N}$, every large locally dense graph on $n$ vertices with minimum degree at least $n\/2+o(n)$ contains the $r$th power of a Hamilton cycle.\nThis is a very special case of our main result, Theorem~\\ref{main}.\n\n\\begin{theorem}\\label{rcycle}\nFor all $r\\in \\mathbb{N}$ and $d,\\eta>0$, there exist $\\rho,n_0>0$ such that every $(\\rho,d)$-dense graph $G$ on $n \\geq n_0$ vertices with $\\delta(G)\\geq (1\/2+\\eta)n$ contains a copy of $C^r_n$. Moreover, this copy contains many extendable copies of $K_r$.\n\\end{theorem}\nNote that Theorem~\\ref{rcycle} is an important tool the  proof of Theorem~\\ref{main}, in the same way that (an approximate version of) the result in~\\cite{kss} was used in the proof of Theorem~\\ref{bst}.\nIndeed, Theorem~\\ref{rcycle}  ensures that the reduced graph $R$ of a graph $G$ (as in Theorem~\\ref{main}) will contain a spanning $(4r-1)$-cycle.\nBy (\\ref{CZ}) this implies $R$ contains an almost spanning copy of $Z^2r _{\\ell}$. As outlined in Section~\\ref{sketch}, this copy of $Z^2r _{\\ell}$ will be used as a `guide' for embedding $H$ into $G$.\n\nWe prove Theorem~\\ref{rcycle} via the connecting-absorbing method.\nFor this, we need a \\emph{Connecting lemma} for locally dense graphs\n The lemma states that any sufficiently extendable $r$-cliques can be joined by a short $r$-path (in fact a short $2r$-path; this stronger statement will be used in the proof of Theorem~\\ref{main}).\n\\end{comment}\n\nWe will need a \\emph{Connecting lemma} to find a short $r$-path between two `extendable' copies of $K_{r}$ in a locally dense graph $G$ with $\\delta(G) > (1\/2+o(1))n$.\n\\COMMENT{Replaced $\\Omega(1)$ with $o(1)$ here to be consistent. I think now with $>$ notation this is ok}\nThe heart of the proof is the following lemma, which is the only part of the proof of Theorem~\\ref{main} which requires $\\delta(G) > (1\/2+o(1))n$ (elsewhere, linear minimum degree suffices). Somewhat similar lemmas have been used elsewhere in other settings e.g.~\\cite{ferb, st1}.\n\n\n\\begin{lemma}\\label{ingredient}\nLet $0 < 1\/n \\ll \\rho \\ll d,\\eta, 1\/r < 1$ where $n,r \\in \\mathbb{N}$.\nLet $G$ be an $n$-vertex graph and let $U \\subseteq V(G)$ be a subset of size $n' \\geq \\eta n\/2$ such that $G[U]$ is $(\\rho,d)$-dense and $d_G(x,U) \\geq (1\/2+\\eta)n'$ for all $x \\in V(G)$.\nLet $X,Y,W$ be pairwise disjoint subsets of $V(G)$ such that $|X|=|Y| = \\lceil 4r\/\\eta \\rceil$ and $|W| \\leq \\eta n'\/2$. Then there is $Z \\subseteq U$ such that\n\\begin{itemize}\n\\item[(i)] $G[Z] \\cong K_r$;\n\\item[(ii)] $Z \\cap (X\\cup Y \\cup W)=\\emptyset$;\n\\item[(iii)] there exist $X' \\subseteq X$ and $Y' \\subseteq Y$ with $|X'|=|Y'|=r$ such that $N_G(Z) \\supseteq X' \\cup Y'$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nLet $C := \\lceil 4r\/\\eta \\rceil$ and let $U' := U\\setminus (X\\cup Y \\cup W)$.\nThen\n$$\ne_G(U',X\\cup Y) \\geq (|X|+|Y|)((1\/2+\\eta)n' - |X|-|Y|-|W|) \\geq 2C(1\/2+\\eta\/4)n' \\geq (1+\\eta\/2)Cn'.\n$$\nLet $U''$ be the collection of those vertices in $U'$ which each have at least $C+r$ neighbours in $X\\cup Y$.\nThen $(1+\\eta\/2)Cn' \\leq (C+r-1)(n'-|U''|) + 2C|U''|$ and so\n$$\n|U''| \\geq \\frac{(1+\\eta\/2)Cn' - (C+r-1)n'}{C-r+1} \\geq \\frac{\\eta n'}{4},\n$$\nwhere the final inequality follows from the fact that $C \\geq 4r\/\\eta$.\nThere are not more than $2^{2C}$ ways a vertex can attach to $X \\cup Y$, so there is $U^* \\subseteq U''$ such that $N_G(v,X\\cup Y)$ is identical for all $v \\in U^*$ and $|U^*| \\geq \\eta n'\/2^{2C+2}$.\nNote further that, since each such $v$ has at least $C+r$ neighbours in $X \\cup Y$, there is $X' \\subseteq X$ and $Y' \\subseteq Y$ such that $|X'|=|Y'|=r$ and $X' \\cup Y' \\subseteq N_G(U^*)$.\nLemma~\\ref{nbrhd}(i) now implies that $G[U^*]$ is $(2^{4C+4}\\rho\/\\eta^2,d)$-dense and hence $(\\sqrt{\\rho},d)$-dense.\nBut then by~Lemma~~\\ref{nbrhd}(iv) there is $Z \\subseteq U^*$ which spans a $K_{r}$.\nThe desired properties~(i)--(iii) are immediate.\n\\end{proof}\nAs well as being applied in the proof of the Connecting lemma below, Lemma~\\ref{ingredient} is also a key tool in the proof of Theorem~\\ref{rcycle} in Section~\\ref{cyclesec}, which in turn is a crucial tool for the proof of Theorem~\\ref{main}.\n\n\n\n\n\\begin{lemma}[Connecting lemma]\\label{connect}\nLet $0 < 1\/n \\ll \\rho \\ll d,\\eta \\leq 1\/r$ where $r \\in \\mathbb{N}$ and let $G$ be a $(\\rho,d)$-dense graph on $n$ vertices with $\\delta(G)\\geq (1\/2+\\eta)n$.\nLet $W,X,Y$ be subsets of $V(G)$ such that $|W| \\leq \\eta n\/4$ and $X,Y$ induce $r$-cliques in $G$, and each one either\n\\begin{itemize}\n\\item lies in a copy of $K_{\\lceil 9 r\/\\eta\\rceil}$ which is disjoint from $W$; or\n\\item is $\\eta n$-extendable.\n\\end{itemize}\nThen $G$ contains a copy of $P^{r}_{3r} = x_1\\ldots x_{3r}$ avoiding $W$\nsuch that $Xx_1\\ldots x_{3r}$ induces a copy of $P^{r}_{4r}$, and $x_1\\ldots x_{3r}Y$ induces a copy of $P^{r}_{4r}$.\n\\end{lemma}\nThe Connecting lemma will ensure that the reduced graph $R$ of a graph $G$ (as in Theorem~\\ref{main}) is `highly connected'. This property will be exploited when embedding a part of $H$ into $G$ so as to cover all of the exceptional set $V_0$ (specifically, we make use of Lemma~\\ref{connect} in Section~\\ref{special}).\n\n\\begin{proof}\nSuppose that $X$ is $\\eta n$-extendable.\nLet $C := \\lceil 9r\/\\eta \\rceil$ and $c := \\lceil 4r\/\\eta\\rceil$, and also let $G_X := G[N_G(X)\\setminus W]$. Then Lemma~\\ref{nbrhd}(i)  implies that $G_X$ is a $(16\\rho\/(9\\eta^2),d)$-dense graph on at least $3\\eta n\/4$ vertices.\nBut $4\/(3\\eta n) \\ll 16\\rho\/(9\\eta^2) \\ll d, 1\/C$, so Lemma~\\ref{nbrhd}(iv) implies that $G_X$ contains a copy of $K_{C}$.\nTherefore $X$ lies in a copy of $K_{C+r}$ which does not intersect $W$.\n\nThis implies that we may assume  both $X,Y$ lie in a copy of $K_{C}$ which does not intersect $W$.\nLet $X^*$ be the vertex set of the $K_{C}$ containing $X$ and define $Y^*$ analogously for $Y$.\nChoose $X' \\subseteq X^*$ of size $c$ which is disjoint from $X$.\nSince $|Y^*|-|Y|-|X|-|X'| = C - 2r - c \\geq c$,\nwe can choose $Y' \\subseteq Y^*$ of size $c$ which is disjoint from $X,Y,X'$.\nApply Lemma~\\ref{ingredient} with $n,r,\\eta,V(G),X',Y',X \\cup Y \\cup W$ playing the roles of $n,r,\\eta,U,X,Y,W$ to obtain $Z \\subseteq V(G)$ which induces a copy of $K_{r}$; is disjoint from $X' \\cup Y'\\cup X \\cup Y \\cup W$, and there exist $X'' \\subseteq X'$ and $Y'' \\subseteq Y'$ such that $|X''|=|Y''|=r$ and $X'' \\cup Y'' \\subseteq N_G(Z)$.\nNotice that, by construction, each of $X \\cup X''$, $X'' \\cup Z$, $Z \\cup Y''$ and $Y'' \\cup Y$ induce cliques, and the overlap of each consecutive pair induces a clique of size at least $r$. Further, none of these sets intersect with $W$.\nThus $XX''ZY''Y$ induces an $r$-path. Thus there is an $r$-path with vertex set $X''\\cup Z \\cup Y''$ (of length $3r$) which has the required property.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Regularity and Blow-up lemmas and associated tools}\\label{secrl}\n\n\\subsection{Regularity}\nWe will apply Szemer\\'edi's Regularity lemma in the proof  of Theorem~\\ref{main}. For this we need the following definitions. Given a bipartite graph $G$ with vertex classes $A$ and $B$ and parameters ${\\varepsilon},\\delta \\in (0,1)$,\n\\begin{itemize}\n\\item let $d_G(A,B) := \\frac{e_G(A,B)}{|A||B|}$ be the \\emph{density} of $G$; and say that $G$ is\n\\item \\emph{${\\varepsilon}$-regular} if, for every $X \\subseteq A$ and $Y \\subseteq B$ with $|X| \\geq {\\varepsilon}|A|$ and $|Y| \\geq {\\varepsilon}|B|$ we have that $|d_G(A,B)-d_G(X,Y)| \\leq {\\varepsilon}$;\n\\item \\emph{$({\\varepsilon},\\delta)$-regular} if $G$ is ${\\varepsilon}$-regular and additionally $d_G(A,B) \\geq \\delta$; \n\\item \\emph{$({\\varepsilon},\\delta)$-superregular} if $G$ is $({\\varepsilon},\\delta)$-regular and additionally $d_G(a,B) \\geq \\delta|B|$ for every $a \\in A$ and $d_G(b,A) \\geq \\delta|A|$ for every $b \\in B$.%\n\\end{itemize} \n\nIt will be convenient to use the degree form of the Regularity lemma; this can be derived from the standard version~\\cite{reglem}.\n\n\\begin{lemma}[Degree form of the Regularity lemma]\\label{reg}\nFor all ${\\varepsilon} \\in (0,1)$ and $M' \\in \\mathbb{N}$, there exist $M,n_0\\in \\mathbb{N}$ such that the following holds for all graphs $G$ on $n \\geq n_0$ vertices and $\\delta \\in (0,1)$.\nThere is a partition $V(G) = V_0 \\cup V_1 \\cup \\ldots \\cup V_L$ and a spanning subgraph $G' \\subseteq G$ such that\n\\begin{itemize}\n\\item[(i)] $M' \\leq L \\leq M$;\n\\item[(ii)] $|V_0| \\leq {\\varepsilon} n$;\n\\item[(iii)] $|V_1|=\\ldots = |V_L| =: m$;\n\\item[(iv)] $d_{G'}(x) \\geq d_G(x)-(\\delta+{\\varepsilon})n$ for all $x \\in V(G)$;\n\\item[(v)] for all $i \\in [L]$ the graph $G'[V_i]$ is empty;\n\\item[(vi)] for all $i \\in [L]$ the graph $G'[V_i,V_j]$ is either empty or $({\\varepsilon},\\delta)$-regular.\n\\end{itemize}\n\\end{lemma}\n\nWe call $V_1,\\ldots,V_L$ the \\emph{clusters} of $G$ and the vertices in $V_0$ the \\emph{exceptional vertices}.\nThe graph $G'$ is the \\emph{pure graph}.\nNote that the $({\\varepsilon},\\delta)$-regular pairs may have very different densities.\nThe \\emph{reduced graph} $R$ of $G$ with parameters ${\\varepsilon},\\delta$ and $M'$ has vertex set $[L]$ and contains $ij$ as an edge precisely when $G'[V_i,V_j]$ is $({\\varepsilon},\\delta)$-regular.\n\nThe next lemma states that the reduced graph $R$ of a locally dense graph $G$ is still locally dense (with worse parameters), and further $R$ inherits the minimum degree of $G$.\n\n\\begin{lemma}\\label{inherit}\nLet $0<1\/n \\ll  1\/M' \\ll {\\varepsilon}\\ll\\delta  \\ll d \\leq 1$; $1\/M' \\ll \\rho \\ll d$; $\\delta \\ll \\eta$. Define $\\rho ^*:= \\max \\{ 3 \\rho, 3 \\delta \\}$.\nLet $G$ be a $(\\rho,d)$-dense graph of order $n$ with $\\delta(G) \\geq (1\/2+\\eta)n$.\nApply Lemma~\\ref{reg} with parameters ${\\varepsilon},\\delta$ and $M'$ to obtain a pure graph $G'$ and a reduced graph $R$ of $G$ with $V(R)=[L]$.\nThen $R$ is $(\\rho ^*,d)$-dense with $\\delta(R) \\geq (1\/2+\\eta\/2)L$.\n\\end{lemma}\n\n\\begin{proof}\nHere~(i)--(vi) will refer to the conclusions of Lemma~\\ref{reg}.\nParts~(ii) and~(iii) imply that\n\\begin{equation}\\label{boring}\n(1-{\\varepsilon})n \\leq mL \\leq n.\n\\end{equation}\nLet $X \\subseteq [L]$ and let $Y := \\bigcup_{i \\in X}V_i \\subseteq V(G)$. So $|Y|=m|X|$.\nThen\n$$\nd\\binom{|Y|}{2} - \\rho n^2 - |Y|(\\delta+{\\varepsilon})n \\leq e(G[Y]) - |Y|(\\delta+{\\varepsilon})n \\stackrel{(iv)}{\\leq} e(G'[Y]) \\stackrel{(v)}{\\leq} e(R[X])\\cdot m^2\n$$\nand so, dividing by $m^2$,\n$$\ne(R[X]) \\stackrel{(\\ref{boring})}{\\geq} d\\cdot\\frac{|X|^2 - \\frac{|X|}{m}}{2} - \\rho \\left(\\frac{L}{1-{\\varepsilon}}\\right)^2 - |X|(\\delta+{\\varepsilon})\\frac{L}{1-{\\varepsilon}} \\geq d\\binom{|X|}{2} - \\rho ^* L^2,\n$$\nas required.\n\nLet $i \\in [L]$ and $x_i \\in V_i$. Then $d_{G'}(x_i) \\geq d_G(x_i) - (\\delta+{\\varepsilon})n \\geq (1\/2+\\eta-\\delta-{\\varepsilon})n$ by~(iv).\nThe number of clusters $V_k$ of $G$ containing some $y \\in N_{G'}(x_i)$ is therefore at least\n$$\n\\frac{(1\/2 + \\eta-\\delta-{\\varepsilon})n - |V_0|}{m} \\geq \\frac{(1\/2+\\eta\/2)n}{m} \\geq (1\/2+\\eta\/2)L.\n$$\nBut then~(vi) implies that $i$ is adjacent to each of the vertices corresponding to these clusters in $R$. So $d_R(i) \\geq (1\/2+\\eta\/2) L$, as required. \n\\end{proof}\nNote that in the case when $\\rho \\ll \\delta$ in Lemma~\\ref{inherit}, $R$ only inherits the property being locally dense with a significantly worse parameter playing the role of $\\rho$.\nThat is, now $R$ is $(3\\delta, d)$-dense rather than $(\\rho,d)$-dense. \n\nThe next well-known proposition states that (super)regular pairs are robust in the sense of adding or removing a small number of vertices. This version appears as Proposition~8 in~\\cite{3partite}.\n\n\\begin{proposition}\\label{newsuperslice}\nLet $G$ be a graph with $A,B \\subseteq V(G)$ disjoint.\nSuppose that $G[A,B]$ is $({\\varepsilon},\\delta)$-regular and let $A',B' \\subseteq V(G)$ be disjoint such that $|A \\triangle A'| \\leq \\alpha|A'|$ and $|B \\triangle B'| \\leq \\alpha|B'|$ for some $0 \\leq \\alpha < 1$.\nThen $G[A',B']$ is $({\\varepsilon}',\\delta')$-regular, with\n$$\n{\\varepsilon}' := {\\varepsilon} + 6\\sqrt{\\alpha} \\ \\ \\text{ and }\\ \\ \\delta' := \\delta-4\\alpha.\n$$\nIf, moreover, $G[A,B]$ is $({\\varepsilon},\\delta)$-superregular and each vertex $x \\in A'$ has at least $\\delta'|B'|$ neighbours in $B'$ and each vertex $x \\in B'$ has at least $\\delta'|A'|$ neighbours in $A'$, then $G[A',B']$ is $({\\varepsilon}',\\delta')$-superregular with ${\\varepsilon}'$ and $\\delta'$ as above.\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe following lemma is well known in several variations.\nThe version here \nfollows immediately from \\cite[Lemma 4.6]{starxiv}.\n\\begin{lemma}\\label{superslice}\nLet $L \\in \\mathbb{N}$ and suppose that $0 < 1\/m  \\ll {\\varepsilon} \\ll \\delta, 1\/\\Delta, 1\/L \\leq 1$.\nLet $R$ be a graph with $V(R) = [L]$ and $\\Delta (R)\\leq \\Delta$.\nLet $G$ be a graph with vertex partition $V_1, \\ldots, V_{L}$ such that $|V_i|=m$ for all $1 \\leq i \\leq L$, and in which $G[V_i,V_j]$ is $({\\varepsilon},\\delta)$-regular whenever $ij \\in E(R)$.\nThen for each $i \\in V(R)$, $V_i$ contains a subset $V_i'$ of size $(1-\\sqrt{{\\varepsilon}})m$ such that for every edge $ij$ of $R$, the graph $G[V_i',V_j']$ is $(4\\sqrt{{\\varepsilon}},\\delta\/2)$-superregular.\n\\end{lemma}\n\\COMMENT{AT: modified the statement of this lemma, as it didn't quite work correctly for us before (didn't have $L$ huge in our application)}\n\n\n\n\n\n\n\n\\subsection{Embedding lemmas}\n\n\n\n\nThe next lemma is similar to a  \\emph{Partial embedding lemma} from~\\cite[Lemma 10]{3partite} which in turn is similar to an embedding lemma \ndue to Chv\\'atal, R\\\"odl, Szemer\\'edi and Trotter~\\cite{trot}.\nGiven a homomorphism from a graph $H$ into the reduced graph $R$ of $G$  such that every pre-image is small, the lemma yields an embedding of some vertices of $H$ into $G$, while finding large candidate sets for the remaining vertices.\nFurther (deviating from~\\cite{3partite}), we would like to ensure that certain vertices of $H$ are embedded into given target sets of large size.\n\n\\begin{lemma}[Embedding lemma with target sets]\\label{target}\nLet $0 < 1\/n  \\ll 1\/L \\ll {\\varepsilon} \\ll c \\ll \\delta  \\ll 1\/\\Delta$ where $n,L \\in \\mathbb{N}$.\nLet $G$ be an $n$-vertex graph, $R$ an $L$-vertex graph and $H$ a graph on at most ${\\varepsilon} n$ vertices such that\n\\begin{itemize}\n\\item $G$ has partition $\\lbrace V_a : a \\in V(R)\\rbrace$ where $|V_a| \\geq m \\geq (1-{\\varepsilon})n\/L$ for all $a \\in V(R)$ and $G[V_a,V_{a'}]$ is $({\\varepsilon},\\delta)$-regular whenever $aa' \\in E(R)$.\n\\item $\\Delta(H)\\leq\\Delta$ and there is a graph homomorphism $\\phi: V(H)\\rightarrow V(R)$ such that $|\\phi^{-1}(a)|\\leq 2{\\varepsilon} m$ for all $a \\in V(R)$.\n\\item Let $X \\cup Y$ be a partition of $V(H)$ and suppose that there is \n$W \\subseteq X$ such that for each $w \\in W$, there is a set $S_w \\subseteq V_{\\phi(w)}$ with $|S_w| \\geq cm$.\n\\end{itemize}\nThen there is an embedding $f$ of $H[X]$ into $G$ such that\n\\begin{itemize}\n\\item[(i)] $f(x) \\in V_{\\phi(x)}$ for all $x \\in X$;\n\\item[(ii)] $f(w) \\in S_w$ for all $w \\in W$;\n\\item[(iii)] for all $y \\in Y$ there exists $C_y \\subseteq V_{\\phi(y)}\\setminus f(X)$ such that $C_y \\subseteq N_G(f(x))$ for all $x \\in N_H(y) \\cap X$, and $|C_y| \\geq cm$.\n\\end{itemize} \n\\end{lemma\nSince the proof of Lemma~\\ref{target} is essentially identical to that of Lemma~10 from ~\\cite{3partite}, we omit the proof.\n\n\nWe will also use the Blow-up lemma of Koml\\'os, S\\'ark\\\"ozy and Szemer\\'edi~\\cite{kss2}, which states that, for the purposes of embedding a spanning $k$-partite graph $H$ of bounded degree, a graph $G$ with a vertex partition into $k$ classes, each pair of which is superregular, in fact behaves like a complete $k$-partite graph.\nFurther, as in Lemma~\\ref{target}, one can ensure that a small fraction of the vertices of $H$ are embedded into some given target sets.\n\n\\begin{lemma}[Blow-up lemma~\\cite{kss2}]\\label{blowlem}\nFor every $d,\\Delta,c > 0$ and $k \\in \\mathbb{N}$ there exist constants ${\\varepsilon}_0$ and $\\alpha$ such that the following holds.\nLet $n_1, \\ldots, n_k$ be positive integers,  $0 < {\\varepsilon} < {\\varepsilon}_0$, and $G$ be a $k$-partite graph with vertex classes $V_1, \\ldots, V_k$ where $|V_i|=n_i$ for $i \\in [k]$.\nLet $J$ be a graph on vertex set $[k]$ such that $G[V_i,V_j]$ is $({\\varepsilon},d)$-superregular whenever $ij \\in E(J)$.\nSuppose that $H$ is a $k$-partite graph with vertex classes $W_1, \\ldots, W_k$ of size  at most $n_1, \\ldots, n_k$ respectively with $\\Delta(H) \\leq \\Delta$.\nSuppose further that there exists a graph homomorphism $\\phi : V(H) \\rightarrow V(J)$ such that $|\\phi^{-1}(i)| \\leq n_i$ for every $i \\in [k]$.\nMoreover, suppose that in each class $W_i$ there is a set of at most $\\alpha n_i$ special vertices $y$, each of them equipped with a set $S_y \\subseteq V_i$ with $|S_y| \\geq cn_i$.\nThen there is an embedding of $H$ into $G$ such that every special vertex $y$ is mapped to a vertex in $S_y$.\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Finding the power of a Hamilton cycle}\\label{cyclesec}\nThe next result\nstates that for every $r \\in \\mathbb{N}$, every large locally dense $n$-vertex graph $G$  with minimum degree at least $(1\/2+o(1))n$ contains the $r$th power of a Hamilton cycle.\nThis is a very special case of our main result, Theorem~\\ref{main}.\n\n\n\\begin{theorem}\\label{rcycle}\nFor all $r,s\\in \\mathbb{N}$ and $d,\\eta>0$, there exist $\\rho,n_0>0$ such that every $(\\rho,d)$-dense graph $G$ on $n \\geq n_0$ vertices with $\\delta(G)\\geq (1\/2+\\eta)n$ contains the $r$th power of a  Hamilton cycle.\nIn fact, for every $n'\\in \\mathbb N$ such that $n-s\\leq n' \\leq n$, $G$ contains the $r$th power of a cycle covering precisely $n'$ vertices.\n\\end{theorem}\nNote that Theorem~\\ref{rcycle} is an important tool in the  proof of Theorem~\\ref{main}, in the same way that (an approximate version of) the result in~\\cite{kss} was used in the proof of Theorem~\\ref{bst}.\nIndeed, Theorem~\\ref{rcycle}  ensures that the reduced graph $R$ of a graph $G$ (as in Theorem~\\ref{main}) will contain a spanning $(4r-1)$-cycle.\nBy (\\ref{CZ}) this implies $R$ contains a spanning copy of $Z^{2r} _{\\ell}$.\n\\COMMENT{Actually in this application of Theorem~\\ref{rcycle}, we will instead obtain a spanning $r^*$-cycle in $R$ where $r^*$ is significantly bigger than $4r$. This ensures we obtain an extra useful property, in addition to the main desired property that \n$Z^{2r} _{\\ell}$ lies in $R$.}\nAs outlined in Section~\\ref{sketch}, this copy of $Z^{2r} _{\\ell}$ will be used as a `guide' for embedding $H$ into $G$.\n\n\nWe remark  that one can give a significantly shorter proof of Theorem~\\ref{rcycle} if one only seeks the $r$th power of a  cycle covering (say) at least  $(1-\\eta )n $ vertices in $G$. However, for our application to Theorem~\\ref{main} we (rather subtly) require that\nwe have a $(4r-1)$-cycle in $R$ covering all but a very small number of vertices (much fewer than $\\rho|R|$ vertices in $R$ can be left uncovered). So such a weaker version of Theorem~\\ref{rcycle} is not sufficient.\n\nThe proof of Theorem~\\ref{rcycle} is an application of the \\emph{Connecting--Absorbing method}, a technique first developed by R\\\"odl,  Ruci\\'nski and  Szemer\\'edi~\\cite{rrs2}.\nThe first step in the proof is to find a short \\emph{absorbing} $2r$-path $P_{abs}$ in $G$ which has the property that $V(P_{abs})\\cup Z$ spans an $r$-path in $G$ (with the same start- and endpoints as $P_{abs}$) for any very small set of vertices $Z$.\nWe then reserve a small pot of vertices $V'$ (known as  a \\emph{reservoir}), that will allow us to connect up pairs of paths into longer paths.\nNext we (via an application of the Regularity lemma) find a collection $\\mathcal P$ of a constant number of vertex-disjoint $2C$-paths that together cover almost all of the remaining vertices in $G$ (here $C$ is chosen to be significantly bigger than $r$).\nUsing vertices from the reservoir, we are then able to connect together all the paths in $\\mathcal P$ together with $P_{abs}$ to form a single $r$-cycle  covering almost all the vertices in $G$. The remaining uncovered vertices in $G$ are absorbed \nby $P_{abs}$ to obtain the $r$th power of a Hamilton cycle.\n\n\\smallskip\n\n\n{\\noindent \\it Proof of Theorem~\\ref{rcycle}.}\nNote that if $n$ is sufficiently large then any $n'$-vertex induced subgraph $G'$ of an $n$-vertex graph $G$ as in the theorem must be $(2\\rho,d)$-dense with  $\\delta (G')\\geq (1\/2+\\eta\/2)n'$.\nSo as the $r$th power of a Hamilton cycle in $G'$ corresponds to an $r$-cycle  of length $n'$ in $G$,\nit suffices to prove the first part of the statement of the theorem.\n \nFurther, it suffices to prove the theorem under the additional assumption that $d \\ll \\eta, 1\/r$.\nDefine constants $ \\rho, {\\varepsilon}, \\delta, d_1, \\eta _0,\\eta _1,\\eta_2, \\eta _3>0$ and $M' \\in \\mathbb N$, and apply the Regularity lemma (Lemma~\\ref{reg}) \nwith inputs ${\\varepsilon}$ and $M'$ to obtain some $M=M({\\varepsilon},M')$ so that we have\n\\begin{align}\\label{hier}\n0<  1\/M \\leq 1\/M' \\ll {\\varepsilon} \\ll \\delta \\ll \\rho   \\ll \\eta _3  \\ll \\eta _2 \\ll \\eta _1 \\ll \\eta _0 \\ll d_1 \\ll d \\ll \\eta, 1\/r.\n\\end{align}\nLet $n$ be sufficiently large, and consider any $n$-vertex graph $G$ that is $(\\rho,d)$-dense  with $\\delta(G)\\geq (1\/2+\\eta)n$.\n\n\nOur initial aim is to construct a small absorbing $2r$-path $P_{abs}$. The next claim provides the building blocks for this absorbing path.\n\\begin{claim}\\label{block}\nThere exists a collection $\\mathcal K$ of at most $\\eta  _0n\/8r$ vertex-disjoint copies of $K_{2r}$ in $G$ such that:\n\\begin{itemize}\n\\item[(i)] Each $K \\in \\mathcal K$ is $d_1n$-extendable in $G$.\n\\item[(ii)] Given any vertex $x\\in V(G)$, there are at least $2\\eta _2 n$ copies $K$ of $K_{2r}$ in $\\mathcal K$ so that $V(K) \\subseteq N_G(x)$.\n\\end{itemize}\n\\end{claim}\n\\begin{claimproof}\nLet $\\mathcal C$ denote the set of all copies of $K_{2r}$  that are $d_1n$-extendable in $G$. So certainly $|\\mathcal C|\\leq n^{2r}$.\nConsider any $x \\in V(G)$. Since $d_G(x)\\geq n\/2$, Lemma~\\ref{nbrhd}(i) implies that $G[N(x)]$ is $(4 \\rho, d)$-dense. Thus, Lemma~\\ref{nbrhd}(iv) implies that\nthere are at least $(d\/2)^{\\binom{2r+1}{2}} (n\/2)^{2r}\/(2r)!$ copies $K$ of $K_{2r}$ in $\\mathcal C$ so that $V(K) \\subseteq N_G(x)$. \n(Here we use the property that $d^{2r}\/2^{2r} \\geq d_1$ by (\\ref{hier}).) Let $L_x$ denote the set of these copies of $K_{2r}$.\n\nLet $\\mathcal C_p$ be obtained from $\\mathcal C$ by selecting each $K \\in \\mathcal C$ independently with probability\n$$p:= \\frac{\\eta _1}{n^{2r-1}}.$$\nHence, \n$$\\mathbb E (|\\mathcal C_p|)\\leq\\eta _1 n \\ \\ \\text{ and } \\ \\ \\mathbb E( |\\mathcal C_p \\cap L_x |) \\geq (d\/2)^{\\binom{2r+1}{2}} \\frac{ (n\/2)^{2r}}{(2r)! } \\times  \\frac{\\eta _1}{n^{2r-1}} \\stackrel{(\\ref{hier})}{\\geq} 2 d_1 \\eta _1 n $$\nfor each $x \\in V(G)$.\nThus, a Chernoff bound implies that, with high probability,\n\\begin{align}\\label{nicebounds}\n|\\mathcal C_p| \\leq 2\\eta _1 n \\ \\ \\text{ and } \\ \\ |\\mathcal C_p \\cap L_x |\\geq  d_1 \\eta _1 n \n\\end{align}\nfor all $x \\in V(G)$.\nLet $Y$ denote the number of pairs of copies of $K_{2r}$ from $\\mathcal C_p$ that share at least one vertex.\nThen\n$$\\mathbb E( Y) \\leq p^2 \\binom{n}{2r} 2r \\binom{n}{2r-1} \\leq \\eta ^2 _1 n.$$ \nBy Markov's inequality the probability that $|Y|\\leq 2\\eta ^2 _1 n$ is at least $1\/2$. Therefore, there is a choice of $\\mathcal C_p$ such that this condition holds together with\n(\\ref{nicebounds}). Fix such a choice of $\\mathcal C_p$; then for each intersecting pair of cliques in $\\mathcal C_p$, remove one of them to obtain a new collection $\\mathcal K$.\nNote that the definition of $\\mathcal C_p$ and (\\ref{nicebounds}) implies that $\\mathcal K$ is a collection of at most $\\eta _0 n\/8r$ vertex-disjoint copies of $K_{2r}$ in $G$.\nFurther, since $d_1 \\eta _1 n -  2\\eta ^2 _1 n\\geq d_1 \\eta _1 n\/2 \\geq 2 \\eta _2 n$, we see that (ii) is satisfied, as desired.\n\\end{claimproof}\n\n\\smallskip\n\nWith Claim~\\ref{block} at hand, it is straightforward to obtain our desired absorbing $2r$-path $P_{abs}$.\n\n\\begin{claim}\\label{pabs}\n$G$ contains a $2r$-path $P_{abs}$ on at most $\\eta _0 n$ vertices such that the following conditions hold. \n\\begin{itemize}\n\\item[(i)] Both the set of the first and last $2r$ vertices on $P_{abs}$ induce $K_{2r}$s in $G$ that are $d_1n$-extendable. (Denote these sets by $S$ and $E$ respectively.)\n\\item[(ii)] Given any set $Z\\subseteq V(G)\\setminus V(P_{abs})$ of size at most $\\eta _2 n$, there is an $r$-path $P$ in $G$ with vertex set $V(P_{abs}) \\cup Z$ whose first $2r$ vertices are the elements of $S$ (ordered as in $P_{abs}$) and\nthe last $2r$ vertices are the elements of $E$ (ordered as in $P_{abs}$).\n\\end{itemize}\n\\end{claim}\n\\begin{claimproof}\nLet $\\mathcal K$ be as in Claim~\\ref{block}, and enumerate its elements by $K^1, \\dots, K^t$ (so $t \\leq \\eta  _0n\/8r$).\nApply Lemma~\\ref{connect} to $G$ with $d_1,2r, V(K^1), V(K^2), V(\\mathcal K)$ playing the roles of $\\eta, r, X,Y,W$. (Note we can indeed apply this lemma by Claim~\\ref{block}(i) and as $|V(\\mathcal K)|\\leq d_1n\/4$.)\nWe thus obtain a copy $P_1=x^1_1\\dots x^1_{6r}$ of $P^{2r}_{6r}$ in $G$ avoiding $V(\\mathcal K)$ such that $V(K^1)x^1_1\\dots x^1_{6r}$ and $x^1_1\\dots x^1_{6r}V(K^2)$ both induce copies of $P^{2r}_{8r}$.\nRepeating this process iteratively we obtain a collection  $P_1, \\dots , P_{t-1}$ of vertex-disjoint copies of $P^{2r}_{6r}$ in $G$ so that\n$V(K^i)x^i_1\\dots x^i_{6r} V(K^{i+1})$ induces a copy of $P^{2r}_{8r}$ in $G$ for each $1 \\leq i \\leq t-1$. (Here we have written $P_i=x^i_1\\dots x^i_{6r}$.)\nNote that to ensure the $P_i$s are vertex-disjoint, at every step we update $W$; so at step $i$, $W$ contains $V(\\mathcal K)$ and the vertices from $P_1,\\dots, P_{i-1}$ (so $|W|\\leq d_1n\/4$). \n\nLet $P_{abs}$ denote the $2r$-path\nobtained by the following concatenation:\n$$P_{abs}:=V(K^1)P_1V(K^2)P_2V(K^3)\\dots V(K^{t-1})P_{t-1}V(K^t).$$\n\nNotice that $P_{abs}$ contains $(t-1)8r+2r \\leq 8rt \\leq \\eta _0 n$ vertices. Further (i) follows since both $K^1$ and $K^{t}$ are $d_1n$-extendable in $G$ by definition of $\\mathcal K$.\n Consider any set $Z=\\{z_1,\\dots, z_{\\ell}\\}\\subseteq V(G)\\setminus V(P_{abs})$ of size at most $\\eta _2 n$. For each $1\\leq i \\leq \\ell$, by Claim~\\ref{block}(ii), there are at least $\\eta _2 n$ choices for $j_i$ such that:\n\\begin{itemize}\n\\item $2\\leq j_i \\leq t-1$;\n\\item $V(K^{j_i})\\subseteq N_G (z_i)$.\n\\end{itemize}\nIn particular, writing $V(K^{j_i})=\\{y_1,\\dots, y_{2r}\\}$, notice that \n\\begin{align}\\label{con}\nP_{j_i-1}y_1\\dots y_r z_i y_{r+1}\\dots y_{2r} P_{j_i}\n\\end{align}\nis an $r$-path in $G$.\n\nSince we have at least $\\eta _2 n$ choices, we may define $j_1,j_2,\\dots, j_{\\ell}$ to be distinct.\n We can then insert each $z_i$ into $P_{abs}$ as indicated by~(\\ref{con}) to obtain the  desired $r$-path $P$ on $V(P_{abs}) \\cup Z$.\n\\end{claimproof}\n\n\n\n\n\\medskip\n\nLet $S$ be as in Claim~\\ref{pabs}. Then $|N_G (S) \\setminus V(P_{abs})| \\geq d_1 n\/2$. Lemma~\\ref{nbrhd}(i) implies that $G_S:=G[N_G(S)\\setminus V(P_{abs})]$ is $(4\\rho\/d ^2_1,d)$-dense and therefore \n$(\\rho ^{1\/2},d)$-dense.\nSet \n\\begin{align}\\label{Cdef1}\nC:= \\lceil 4r\/\\eta _3 \\rceil.\n\\end{align}\nNote that $ \\rho ^{1\/2} \\ll d , 1\/C$. Thus, Lemma~\\ref{nbrhd}(iv) implies that $G_S$ contains a copy $K^S_{2C+1}$ of $K_{2C+1}$. Similarly, we find a copy $K^E _{2C+1}$ of $K_{2C+1}$ in $G$ that is disjoint from\n$K^S_{2C+1}$ and $P_{abs}$ so that $V(K^E_{2C+1}) \\subseteq N_G (E)$. We will view both $K^S_{2C+1}$ and $K^E _{2C+1}$ as $2C$-paths of length $2C+1$.\n\n\\smallskip\n\nSet $G_0:=G\\setminus (V(P_{abs}) \\cup V(K^S_{2C+1}) \\cup V(K^E_{2C+1}) )$. Certainly, $|G_0|\\geq (1-2\\eta _0)n$ and \n\\begin{align*}\nd_G (x,V(G_0))\\geq (1\/2+3\\eta\/4)n \\text{ for all } x \\in V(G).\n\\end{align*}\n\n\nBy selecting vertices randomly (and\napplying a Chernoff bound), one can obtain a set $V'\\subseteq V(G_0)$ of $n':=\\eta _3 n$ vertices such that \n\\begin{align}\\label{degV'}\nd_G (x,V')\\geq (1\/2+\\eta\/2)n' \\text{ for all } x \\in V(G).\n\\end{align}\nSet $G_1:=G[V']$ and $G_2:=G_0\\setminus V'$. Lemma~\\ref{nbrhd}(i) implies that $G_1$ is $(\\rho\/\\eta _3^2,d)$-dense and thus, $(\\rho ^{1\/2},d)$-dense.\nSimilarly $G_2$ is $(2\\rho,d)$-dense.\n\nApply Lemma~\\ref{reg} to  $G_2$ with parameters ${\\varepsilon}, \\delta$ and $M'$ to obtain a partition $V_0,V_1,\\dots, V_\\ell$ of $V(G_2)$, pure graph $G'_2$ and the reduced graph $R$ of $G_2$.\nHere $V_0$ is the exceptional set on at most ${\\varepsilon} n$ vertices and $M'\\leq \\ell \\leq M$.\nSet $m:=|V_1|=\\dots=|V_\\ell|$.\nThen Lemma~\\ref{inherit} implies that $R$ is $(6\\rho,d)$-dense. In particular, Lemma~\\ref{nbrhd}(i) implies that $R'$ is $(6\\rho\/\\eta _3 ^2, d)$-dense for any $R'\\subseteq R$ on $\\eta _3\\ell$ vertices.\n\n\nNote that $1\/\\ell \\ll 6\\rho\/\\eta _3 ^2 \\ll d ,1\/C$. Thus, Lemma~\\ref{nbrhd}(iv) implies that every $R'\\subseteq R$ on $\\eta _3 \\ell$ vertices contains a copy of $K_{2C+1}$. \nIn particular, $R$ contains a $K_{2C+1}$-tiling $\\mathcal T$\ncovering all but at most $\\eta _3 \\ell $ vertices.\n\nConsider any copy $K$ of $K_{2C+1}$ in $\\mathcal T$. The vertices of $K$ correspond to clusters $V_{i_1},\\dots, V_{i_{2C+1}}$ in $G_2$; let $G_K$ denote the  subgraph of $G'_2$ induced by the vertices in these clusters combined.\nEvery tuple $(V_{i_j},V_{i_k})$ of such clusters  forms an ${\\varepsilon}$-regular pair of density at least $\\delta$ in $G_K$.\nMoreover, Lemma~\\ref{superslice} implies that for each such cluster $V_{i_j}$ there is a subset $V'_{i_j}\\subseteq V_{i_j}$ of size $(1-{\\varepsilon} ^{1\/2})m$ so that $(V'_{i_j},V'_{i_k})$ forms an $(4{\\varepsilon} ^{1\/2},\\delta\/2)$-superregular pair in $G_K$ (for each $1 \\leq j \\not = k \\leq 2C+1$).\nThe Blow-up lemma (Lemma~\\ref{blowlem}) now implies that $G_K$ contains a $2C$-path  covering all but at most $(2C+1){\\varepsilon} ^{1\/2}m $ vertices in $G_K$.\n\nOverall, this implies that $G_2$ contains a collection $\\mathcal P$ of at most $\\ell\/(2C+1) \\leq M$ vertex-disjoint $2C$-paths, that together cover all but at most \n\\begin{align}\n\\label{cover}\\left ( (2C+1){\\varepsilon} ^{1\/2}m \\times \\frac{\\ell}{2C+1} \\right )+ \\left ({\\eta _3 \\ell} \\times m \\right )+|V_0| \\leq {\\varepsilon} ^{1\/2} n +\\eta _3 n+{\\varepsilon} n\\stackrel{(\\ref{hier})}{\\leq} {2 \\eta _3 n}\n\\end{align}\nvertices in $G_2$.\n\n\n\nWe will now use vertices in $G_1$ to connect together all of the $2C$-paths in $\\mathcal P\\cup \\{K^S_{2C+1} , K^E_{2C+1}\\}$ to obtain an $r$-path in $G$ whose  first $2C+1$ vertices are the vertices of $K^E_{2C+1}$ and whose last \n$2C+1$ vertices are the vertices of $K^S_{2C+1}$.\nNote that we will have to reorder some of the vertices in the $2C$-paths in $\\mathcal P$, so that is one reason why we `drop' from $2C$-paths to an $r$-cycle.\nLabel the $2C$-paths in $\\mathcal P\\cup \\{K^S_{2C+1} , K^E_{2C+1}\\}$ by $P_1, \\dots, P_t$, where $P_1:=K^E_{2C+1}$ and $P_t := K^S_{2C+1}$.\nIn particular, note  $M'\/4C \\leq t \\leq M+2$\n\nFor each $P_i$, let $S_i$ denote the copy of $K_C$ induced by the first $C$ vertices on $P_i$; let $E_i$ denote the copy of $K_C$ induced by the last $C$ vertices on $P_i$; and let $P'_i$ denote the $2C$-path obtain from $P_i$ by deleting all vertices from $S_i$ and $E_i$. (Note that $P'_i$ is certainly non-empty.)\n \n\\begin{claim}\\label{helpful}\nLet $W\\subseteq V(G_1)$ be arbitrary so that $|W|\\leq {\\varepsilon} n'$. Given any $1 \\leq i \\leq t-1$, there is an $r$-path $P$ in $G$ so that:\n\\begin{itemize} \n\\item[(i)] $V(P)\\cap V(G_2) =E_i \\cup S_{i+1}$;\n\\item[(ii)] $|V(P)\\cap V(G_1)|=r $;\n\\item[(iii)] The first $C$ vertices on $P$ are precisely the vertices from $E_i$; \n\\item[(iv)] The last $C$ vertices on $P$ are precisely the vertices from $S_{i+1}$;\n\\item[(v)] $P$ is disjoint from $W$.\n\\end{itemize}\n\\end{claim}\n\n\\begin{claimproof}\nApply Lemma~\\ref{ingredient} with $G,V',n',\\eta _3,\\sqrt{\\rho},d,E_i,S_{i+1},W,r$ playing the roles of $G,U,n',\\eta,\\rho,d,X,$ $Y,W,r$ to obtain a copy $K$ of $K_r$ in $G_1=G[V']$ such that $V(K) \\cap W = \\emptyset$ (recall that $E_i \\cup S_{i+1}$ is disjoint from $V'$), and there exist $E'_i \\subseteq E_i$ and $S'_{i+1} \\subseteq S_{i+1}$ such that $|E'_i|=|S'_{i+1}|=r$ and $E'_i \\cup S'_{i+1} \\subseteq N_G(K)$.\n\n\\begin{comment}\nBy (\\ref{degV'}), every vertex in $E_i \\cup S_{i+1}$ has  at least $(1\/2+\\eta\/2)n' $ neighbours into $V(G_1)=V'$ in $G$.\nThus,\n$$\ne_G(V',E_i \\cup S_{i+1}) \\geq (|E_i |+| S_{i+1}|)(1\/2+\\eta\/2)n'  = Cn' + C\\eta n'.\n$$\nLet $V''$ be the collection of those vertices in $V'$ which each have at least $C+2r$ neighbours in $E_i \\cup S_{i+1}$.\nThen $Cn' + C\\eta n' \\leq e_G(V',E_i \\cup S_{i+1}) \\leq  (C+2r-1)(n'-|V''|) + 2C|V''|$ and so\n$$\n|V''| \\geq \\frac{Cn'(1+\\eta) - (C+2r-1)n'}{C-2r+1} \\stackrel{(\\ref{Cdef1})}{\\geq} \\frac{\\eta n'}{2}.\n$$\nIn particular, $|V''\\setminus W|\\geq \\eta n'\/4$.\nThere are not more than $4^C$ ways a vertex can `attach' to $E_i \\cup S_{i+1}$, so there is a set $V^* \\subseteq V''\\setminus W$ such that $N_G(v,E_i \\cup S_{i+1})$ is identical for all $v \\in V^*$ and $|V^*| \\geq \\eta n'\/4^{C+1}$.\nNote further that, since each such $v$ has at least $C+r$ neighbours in $E_i \\cup S_{i+1}$, there is $E'_i \\subseteq E_i$ and $S'_{i+1} \\subseteq S_{i+1}$ such that $|E'_i|=|S'_{i+1}|=r$ and $E'_{i} \\cup S'_{i+1} \\subseteq N_G(v)$ for all $v \\in V^*$.\n\nAs $G_1$ is $(\\rho ^{1\/2},d)$-dense, Lemma~\\ref{nbrhd}(i) implies that $G_1[V^*]$ is $(4^{C+1}\\rho ^{1\/2}\/\\eta,d)$-dense. As $\\rho \\ll \\eta, 1\/r$, the definition of $C$ implies that $\\rho ^{1\/4} \\gg 4^{C+1}\\rho ^{1\/2}$ and so $G_1[V^*]$ is $(\\rho ^{1\/4},d)$-dense. Therefore, as $\\rho ^{1\/4} \\ll d \\ll 1\/r$ and $n$, and thus $|V^*|$, is sufficiently large, Lemma~\\ref{nbrhd}(iv) implies that $G_1[V^*]$ contains a copy $K$ of $K_r$.\n\\end{comment}\nAltogether this implies that $G_1$ contains the desired $r$-path $P$. Indeed, we construct $P$ so that the first $C-r$ vertices on $P$ are those vertices in $E_i\\setminus E'_i$ (in an arbitrary order); the next $r$ vertices are the elements from $E'_i$; after that we take the vertices from $K$; then from $S'_{i+1}$; the final $C-r$ vertices on $P$ are from $S_{i+1}\\setminus S'_{i+1}$.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nWith Claim~\\ref{helpful} to hand it is now easy to complete the proof of the theorem.\nSuppose for some $ j < t-1$ we have defined vertex-disjoint $r$-paths $P^*_1,\\dots, P^* _j$ such that, for each $i\\leq j$, $P=P^*_i$ satisfies (i)--(iv) in Claim~\\ref{helpful}.\nThen define $W$ to be all those vertices in an $r$-path $P^*_1,\\dots, P^* _j$ that lie in $G_1$. So $|W|=jr\\leq (M+2)r \\leq {\\varepsilon} n'$. Claim~\\ref{helpful} then implies there is an $r$-path $P^*_{j+1}$ in $G$ that satisfies the conclusion of Claim~\\ref{helpful} (where\n$j+1$ plays the role of $i$ and $P^*_{j+1}$ the role of $P$).\n\nThus, we obtain vertex-disjoint $r$-paths $P^*_1,\\dots, P^* _t$ such that, for each $i\\leq t$, $P=P^*_i$ satisfies (i)--(iv) in Claim~\\ref{helpful}.\nConsider the concatenation \n$$P^*:= S_1P'_1P^*_1P'_2P^*_2\\dots P'_{t-1} P^*_{t-1}P'_tE_t.$$\nThis induces an $r$-path in $G$ (with many additional edges).\nFurther, note that by the definition of $P_1$ (and thus $S_1$), the first $C$ vertices on $P^*$ lie in $K^E_{2C+1}$, and so are adjacent in $G$ to every vertex in $E$.\nSimilarly, the last $C$ vertices on $P^*$ lie in $K^S_{2C+1}$, and so are adjacent in $G$ to every vertex in $S$.\nThus, if we concatenate $P^*$ together with $P_{abs}$ we obtain an $r$-cycle $C^*$ in $G$ (with many additional edges).\n\nNote that, by~(\\ref{cover}), $C^*$ covers every vertex in $G$ except for at most $2\\eta _3 n$ vertices in $G_2$ and at most $n'=\\eta _3 n$ vertices in $G_1$. \nSince $3 \\eta _3 n< \\eta _2 n$, we may use the absorbing property (Claim~\\ref{pabs}(ii)) of $P_{abs}$ to obtain the $r$th power of a Hamilton cycle in $G$, as required.\\qed\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Lemmas for $H$}\\label{sec6}\nOur rough aim is to find `compatible' partitions of the vertex sets of $G$ and of $H$ that allow us to apply the embedding lemmas (Lemmas~\\ref{target} and~\\ref{blowlem}) to complete the embedding of $H$ into $G$.\nIn this section we state and prove the so-called \\emph{Lemmas for $H$}, whose input is some information about the structure of $G$, and whose output is a suitable partition of $H$.\n\n\n\\subsection{Partitioning a graph of low bandwidth: the basic lemma for $H$}\nAt some stage of the proof, $G$ will return some `ideal' part sizes $\\lbrace m_{i,j} : (i,j) \\in [\\ell]\\times[2r]\\rbrace$, where $\\chi(H) \\leq r$. We would then like to find a suitable partition of $H$ the parts of which are close to these ideal sizes (equivalently, a mapping $f$ from $V(H)$ into $[\\ell]\\times[2r]$ whose pre-images have controlled size). This is the purpose of the next lemma. \nIt guarantees that $f$ is a graph homomorphism into $Z^{2r}_\\ell$ and produces a small set $B$ such that $f$ restricted to $V(H)\\setminus B$ is a graph homomorphism into a $K_{2r}$-factor (this is~$(\\mathscr{B}3)$).\nFurther,~$(\\mathscr{B}4)$ says that for  the first few vertices of $H$ (with respect to the bandwidth ordering of $H$), we have control of their images.\n\nBefore stating and proving Lemma~\\ref{lemmaforH1}, we would like to compare it to Lemma~8 in~\\cite{bot}, the Lemma for $H$ in the Bandwidth theorem. There, the assumptions on $H$ are the same (in fact slightly weaker), and the graph $Z^{2r}_\\ell$ mentioned above is replaced by a given graph $R$ of large minimum degree which contains a spanning subgraph $S$ (very similar to $Z^{r}_\\ell$), which in turn contains a $K_r$-factor. Most edges are (and must be) mapped to the $K_r$-factor, which is much sparser than the $K_{2r}$-factor we have at our disposal. This means that the proof of Lemma~8 in~\\cite{bot} is much harder to prove than our Lemma~\\ref{lemmaforH1}. Despite this, our lemma does not follow from the statement of Lemma~8 in~\\cite{bot}, so we prove it here. \n\n\n\n\n\\begin{lemma}[Basic Lemma for $H$]\\label{lemmaforH1}\nLet $n,r,\\ell,\\Delta \\geq 1$ be integers and let $\\beta>0$ be such that $0 < 1\/n \\ll 1\/r,1\/\\ell,1\/\\Delta,\\beta$.\nLet $H$ be a graph on $n$ vertices with $\\Delta(H) \\leq \\Delta$ and assume that $H$ has a labelling $x_1,\\ldots,x_n$ of bandwidth at most $\\beta n$ and $\\chi(H) \\leq r$.\nFurthermore, suppose $\\lbrace m_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ is such that $\\sum_{(i,j) \\in [\\ell]\\times[2r]}m_{i,j}=n$; $m_{i,j} \\geq 10\\beta n$ for all $(i,j)\\in[\\ell]\\times[2r]$; and $|m_{i,j}-m_{i,j'}| \\leq 1$ whenever $i \\in [\\ell]$ and $j,j'\\in[2r]$.\nLet $\\chi : V(H) \\rightarrow [r]$ be a proper colouring of $H$.\nThen there exists a mapping $f:V(H)\\rightarrow [\\ell]\\times[2r]$ and a set of special vertices $B \\subseteq V(H)$ with the following properties:\n\\begin{itemize}\n\\item[$(\\mathscr{B}1)$] $B \\cap \\lbrace x_1,\\ldots,x_{\\beta n}\\rbrace = \\emptyset$ and $|B| \\leq 2\\ell\\beta n$;\n\\item[$(\\mathscr{B}2)$] $\\left| |f^{-1}(i,j)| - m_{i,j}\\right| \\leq 10\\beta n$ for every $(i,j) \\in [\\ell]\\times[2r]$;\n\\item[$(\\mathscr{B}3)$] for every edge $uv\\in E(H)$, writing $f(u)=:(i,j)$ and $f(v)=:(i',j')$, we have $|i-i'|\\leq 1$ and $j \\neq j'$. If additionally $u,v \\notin B$, then $i=i'$;\n\\item[$(\\mathscr{B}4)$] for all $s \\leq \\beta n$ we have $f(x_s)=(1,\\chi(x_s))$.\n\\end{itemize}\nIn particular, $f$ yields a homomorphism from $H$ to $Z^{2r}_{\\ell}$.\n\\end{lemma}\nNote that the graph $Z^{2r}_\\ell$ which appears in Lemma~\\ref{lemmaforH1} will be found in the reduced graph $R$ of $G$: since $G$ is locally dense, $R$ is also locally dense (see Lemma~\\ref{inherit}) and thus, by Theorem~\\ref{rcycle}, we can find a spanning $(4r-1)$-cycle in $R$, which contains $Z^{2r}_\\ell$ (see~(\\ref{CZ})).\n\n\nRecall that each vertex in  $R$ corresponds to a unique cluster in $G$.\nIn the proof of Theorem~\\ref{main}, the homomorphism $f$ from $H$ to $Z^{2r}_{\\ell}\\subseteq R$ will be a guide  as to which cluster in $G$ we should embed a vertex $x$ into for \\emph{most} vertices $x\\in V(H)$.\nThat is, roughly speaking, if $f(x)=(i,j)\\in V(R)$, we embed $x$ into the cluster in $G$ corresponding to $(i,j)$. \nNote though that $f$ does not `guide' us as to which vertices from $H$ we should embed into the exceptional set $V_0$ of $G$.\nSo in the proof of Theorem~\\ref{main} we in fact apply Lemma~\\ref{lemmaforH1} to an almost spanning subgraph of $H$, rather than $H$ itself; the remaining part of $H$ is then embedded into $G$ \nvia an additional Lemma for $H$ (Lemma~\\ref{lemmaforH} in Section~\\ref{special}). In particular, Lemma~\\ref{lemmaforH} governs which vertices from $H$ are embedded into $V_0$.\nProperty $(\\mathscr{B}4)$ of the homomorphism $f$ is used to ensure we can `fit' the two Lemmas for $H$ together to complete the embedding of $H$ into $G$.\n\n\nThe idea of the proof of Lemma~\\ref{lemmaforH1} is to first obtain a proper $2r$-colouring $\\chi '$ of $H$ such that in any initial segment $x_1,\\dots, x_t$ of the bandwidth ordering of $H$, every colour is used roughly the same number of times in $\\chi '$.\nThis then allows us to define $f$ in a sequential way. That is, for some $t_1$ we map each $x_j$ in $\\{x_1,\\dots,x_{t_1} \\}$ to $(1,\\chi '(x_j))$; then for some $t_2$ we map each $x_j$ in $\\{x_{t_1+1},\\dots,x_{t_2}\\}$ to $(2,\\chi '(x_j))$, and so on.\n\n\\smallskip\n\n{\\noindent \\it Proof of Lemma~\\ref{lemmaforH1}.}\nLet $N := \\lceil 1\/(2\\beta)\\rceil$ and\npartition the ordered vertices $x_1,\\ldots,x_n$ into consecutive intervals $A_1,A_2,\\ldots,A_{2N}$ each of length $\\beta n$ (except possibly $A_{2N}$ which could be smaller).\n\\COMMENT{AT: deleted definition of $A_i^j$ as don't think it is used}\nWe view each interval as being ordered with the inherited bandwidth ordering.\n\nWe will first define a (proper) $2r$-colouring $\\chi' : V(H) \\rightarrow [2r]$ by iteratively defining colourings $\\chi'_i$ for $i \\in [N]$ with the following properties:\n\\begin{itemize}\n\\item[$\\mathscr{P}_1(i)$] $\\chi'_i : \\bigcup_{2 \\leq t \\leq 2i}A_t \\rightarrow [2r]$ is a proper colouring of $H[\\bigcup_{2 \\leq t \\leq 2i}A_t]$;\n\\item[$\\mathscr{P}_2(i)$] for all odd $2 \\leq t \\leq 2i$ we have $\\chi'_i(A_t) \\subseteq [r]$ and for all even $2 \\leq t \\leq 2i$ we have $\\chi'_i(A_t) \\subseteq [2r]\\setminus[r]$;\n\\item[$\\mathscr{P}_3(i)$] writing $b_i^j(s) := |\\lbrace x \\in \\bigcup_{2 \\leq t \\leq 2s}A_t : \\chi'_i(x)=j\\rbrace|$ for all $j \\in [2r]$ and $s \\in [i]$, we have $|b_i^j(s)-b_i^{j'}(s)| \\leq \\beta n$ for all $(j,j')\\in[[2r]]^2$ and $s \\in [i]$.\n\\end{itemize}\nFor $\\mathscr{P}_3(i)$ recall that $[[2r]]^2:=[r]^2\\cup ([2r]\\setminus [r])^2$.\nDefine $\\chi'_1 : A_2 \\rightarrow [2r]$ by setting $\\chi'_1(x) = \\chi(x)+r$. Clearly this satisfies $\\mathscr{P}_1(1)$--$\\mathscr{P}_3(1)$, in particular as $|A_2| \\leq \\beta n$.\nSuppose we have defined $\\chi'_i$ for some $i < N$ satisfying $\\mathscr{P}_1(i)$--$\\mathscr{P}_3(i)$.\nBy permuting the sets of colours $[r]$ and $[2r]\\setminus[r]$, we can obtain a new proper $2r$-colouring $c_1$ of $H[\\bigcup_{2 \\leq t \\leq 2i} A_t]$ satisfying $\\mathscr{P}_1(i)$--$\\mathscr{P}_3(i)$ and with the additional property that\n\\begin{equation}\\label{c1}\n|c_1^{-1}(1)| \\geq \\ldots \\geq |c_1^{-1}(r)|\\quad\\text{and}\\quad |c_1^{-1}(r+1)| \\geq \\ldots \\geq |c_1^{-1}(2r)|.\n\\end{equation}\n\nDefine $k : A_{2i+1} \\cup A_{2i+2} \\rightarrow [2r]$ by setting\n\\begin{equation*}\\label{kdef99} \nk(x) = \\begin{cases} \\chi(x) &\\mbox{if } x \\in A_{2i+1} \\\\\n\\chi(x)+r & \\mbox{if } x \\in A_{2i+2}. \\end{cases} \n\\end{equation*}\nClearly $k$ is a proper colouring of $H[A_{2i+1}\\cup A_{2i+2}]$ since $\\chi$ is.\nBy permuting the sets of colours $[r]$ and $[2r]\\setminus[r]$, we can obtain a new proper colouring $c_2$ of $H[A_{2i+1}\\cup A_{2i+2}]$ from $k$ such that\n\\begin{equation}\\label{c2}\n|c_2^{-1}(1)| \\leq \\ldots \\leq |c_2^{-1}(r)|\\quad\\text{and}\\quad |c_2^{-1}(r+1)| \\leq \\ldots \\leq |c_2^{-1}(2r)|\n\\end{equation}\n(note that the ordering is reversed compared to~(\\ref{c1})).\nFinally, define $\\chi'_{i+1}$ by setting\n\\begin{equation}\\label{chidef}\n\\chi'_{i+1}(x) = \\begin{cases} c_1(x) &\\mbox{if } x \\in \\bigcup_{2 \\leq t \\leq 2i}A_t \\\\\nc_2(x) & \\mbox{if } x \\in A_{2i+1}\\cup A_{2i}. \\end{cases} \n\\end{equation}\nThe fact that $\\mathscr{P}_2(i+1)$ holds is clear from $\\mathscr{P}_2(i)$ and the definitions of $c_1$, $k$, $c_2$ and $\\chi'_{i+1}$.\n\nTo see that $\\mathscr{P}_1(i+1)$ holds, let $x,y \\in \\bigcup_{2 \\leq t \\leq 2i+2}A_t$ where $xy \\in E(H)$.\nWe need to show that $\\chi'_{i+1}(x) \\neq \\chi'_{i+1}(y)$.\nLet $2 \\leq t,t' \\leq 2i+2$ be such that $x \\in A_t$ and $y \\in A_{t'}$. Then $|t-t'|\\leq 1$ since the intervals $A_j$ respect the bandwidth ordering and each one (except perhaps $A_{2N}$) has size $\\beta n$. If $|t-t'|=1$, then $\\mathscr{P}_2(i+1)$ implies that one of $\\chi'_{i+1}(x),\\chi'_{i+1}(y)$ lies in $[r]$ and the other in $[2r]\\setminus[r]$, as required.\nSo we may assume that $t=t'$. If $2 \\leq t \\leq 2i$, then $(\\chi'_{i+1}(x),\\chi'_{i+1}(y))=(c_1(x),c_1(y))$. But $c_1$ is a proper colouring since it was obtained from $\\chi'_i$ by permuting colours, and $\\chi'_i$ is a proper colouring by $\\mathscr{P}_1(i)$.\nSuppose that $t \\in \\lbrace 2i+1,2i+2\\rbrace$. Then similarly $(\\chi'_{i+1}(x),\\chi'_{i+1}(y))=(c_2(x),c_2(y))$, and $c_2$ is a proper colouring since it was obtained from the proper colouring $k$ by permuting colours.\nThus $\\mathscr{P}_1(i+1)$ holds.\n\nFor $\\mathscr{P}_3(i+1)$, define for $j \\in [2r]$ and $s \\in [i+1]$\n$$\nb_{i+1}^j(s) := \\left|\\left\\lbrace x \\in \\bigcup_{2 \\leq t \\leq 2s}A_t : \\chi'_{i+1}(x)=j\\right\\rbrace\\right|\n$$\nand let $b_{i+1}^j := b_{i+1}^j(i+1) = |(\\chi'_{i+1})^{-1}(j)|$. Then (\\ref{chidef}) implies that $b_{i+1}^j=|c_1^{-1}(j)|+|c_2^{-1}(j)|$ for all $j \\in [2r]$. \nNow let $(j,j')\\in[[2r]]^2$.\nClearly $|b_{i+1}^j(s)-b_{i+1}^{j'}(s)| \\leq \\beta n$ for all $s \\in [i]$ since this is true for $\\chi'_i$ and hence $c_1$.\nSo it remains to show that $|b_{i+1}^j-b_{i+1}^{j'}| \\leq \\beta n$.\nEquations~(\\ref{c1}) and~(\\ref{c2}) imply that\n the quantities $|c_1^{-1}(j)|-|c_1^{-1}(j')|$ and $|c_2^{-1}(j)|-|c_2^{-1}(j')|$ are never both positive, and never both negative, since $j$ and $j'$ are in different orders. This implies that\\COMMENT{please check my changes here}\n\\begin{align*}\n|b_{i+1}^j-b_{i+1}^{j'}| & = \\left| |c_1^{-1}(j)|-|c_1^{-1}(j')| + |c_2^{-1}(j)|-|c_2^{-1}(j')|\\right| \\\\ & \\leq \\max \\left \\{ \\left| |c_1^{-1}(j)|-|c_1^{-1}(j')|\\right|,\n\\left| |c_2^{-1}(j)|-|c_2^{-1}(j')|\\right|\n \\right \\}.\n\\end{align*}\nNote that $\\left| |c_2^{-1}(j)|-|c_2^{-1}(j')|\\right|\\leq \\beta n$.\nFurther, $c_1$ was obtained from $\\chi'_i$ by permuting colours in $[r]$ and in $[2r]\\setminus[r]$, so there is some $(q,q') \\in [[2r]]^2$ for which $c_1^{-1}(j)=(\\chi'_i)^{-1}(q)$ and $c_1^{-1}(j')=(\\chi'_i)^{-1}(q')$.\nThus $\\left| |c_1^{-1}(j)|-|c_1^{-1}(j')|\\right| = |b_i^q(i)-b_i^{q'}(i)|$ which is at most $\\beta n$ by $\\mathscr{P}_3(i)$.\nThus $\\mathscr{P}_3(i+1)$ holds.\n\nTherefore we can obtain a colouring $\\chi'_N : V(H) \\setminus A_1 \\rightarrow [2r]$ satisfying $\\mathscr{P}_1(N)$--$\\mathscr{P}_3(N)$.\nFinally, define $\\chi' : V(H) \\rightarrow [2r]$ by setting\n\\begin{equation}\\label{chiprimedef}\n\\chi'(x) = \\begin{cases} \\chi'_N(x) &\\mbox{if } x \\in V(H)\\setminus A_1 \\\\\n\\chi(x) & \\mbox{if } x \\in A_{1}. \\end{cases} \n\\end{equation}\nThe following properties hold:\n\\begin{itemize}\n\\item[(i)] $\\chi' : V(H) \\rightarrow [2r]$ is a proper colouring;\n\\item[(ii)] for all odd $t \\in [2N]$ we have $\\chi'(A_t) \\subseteq [r]$ and for all even $t\\in[2N]$ we have $\\chi'(A_t) \\subseteq [2r]\\setminus[r]$;\n\\item[(iii)] writing $d^j(s) := |\\lbrace x \\in \\bigcup_{t \\in [s]}A_t : \\chi'(x)=j\\rbrace|$\\COMMENT{note change in definition}\n for all $j \\in [2r]$ and $s \\in [2N]$, we have $|d^j(s)-d^{j'}(s)| \\leq 2\\beta n$ for all $(j,j')\\in[[2r]]^2$ and $s \\in [2N]$.\n\\end{itemize}\n\n\\smallskip\n\nLet $M_0=n_0 := 0$.\nFor all $i \\in [\\ell]$, let $M_i := \\sum_{j \\in [2r]}m_{i,j}$; and $n_i := \\sum_{t \\in [i]}M_t$.\n(Note that $n_\\ell =n$.)\nLet $B_i := \\lbrace x_{n_{i-1}+1},\\ldots, x_{n_i}\\rbrace$.\nSo $B_1,\\ldots,B_{\\ell}$ is a partition of $V(H)$ which respects the bandwidth ordering, and each interval inherits the bandwidth ordering.\nLet\n$$\nB := \\bigcup_{2 \\leq i \\leq \\ell}\\lbrace x_{n_{i-1}+1},\\ldots, x_{n_{i-1}+\\beta n}\\rbrace \\cup \\bigcup_{1 \\leq i \\leq \\ell-1} \\lbrace x_{n_{i}-\\beta n+1},\\ldots, x_{n_i}\\rbrace,\n$$\nand define $f : V(H) \\rightarrow [\\ell]\\times[2r]$ by setting\n\\begin{equation}\\label{fmapdef}\nf(x) := (i,\\chi'(x)) \\text{ if } x \\in B_i. \n\\end{equation}\nWe claim that $f$ is the required mapping.\nNote $|B| = 2(\\ell-1)\\beta n$, \nand if $t \\leq n_1-\\beta n$, then $x_t \\notin B$.\nBut $n_1-\\beta n \\geq 9\\beta n$, so certainly $\\lbrace x_1,\\ldots,x_{\\beta n}\\rbrace \\cap B = \\emptyset$.\nHence, $(\\mathscr{B}1)$ holds.\nTo show~$(\\mathscr{B}2)$, fix $i \\in [\\ell]$. Choose the smallest $p^- \\in [2N]$ such that the first element of $A_{p-}$ lies in $B_i$, and the largest $p^+ \\in [2N]$ such that the last element of $A_{p^+}$ lies in $B_i$.\nSo $B_i$ is the union of $\\bigcup_{p^- \\leq t \\leq p^+}A_t$ together with a proper subset of $A_{p^--1}$ and a proper subset of $A_{p^++1}$.\nThus,\n\\begin{equation}\\label{f1eq}\n|f^{-1}(i,j)| = |\\lbrace x \\in B_i : \\chi'(x)=j\\rbrace| = d^j(p^+) - d^j(p^--1) \\pm (|A_{p^--1}|+|A_{p^++1}|)\n\\end{equation}\nfor all $j \\in [2r]$.\nLet $(j,j')\\in[[2r]]^2$. Then the sizes of $f^{-1}(i,j)$ and $f^{-1}(i,j')$ do not differ much:\n\\begin{equation}\\label{fdiff}\n|f^{-1}(i,j)-f^{-1}(i,j')| \\stackrel{(\\ref{f1eq})}{\\leq} \\left| d^j(p^+)  - d^{j'}(p^+) \\right| + \\left|  d^j(p^--1) - d^{j'}(p^--1)\\right| + 4\\beta n \\stackrel{(iii)}{\\leq} 8\\beta n.\n\\end{equation}\nFor any fixed $i \\in [\\ell]$,\n$$\nS_1 := \\sum_{j \\in [r]}|f^{-1}(i,j)| \\stackrel{(ii),(\\ref{fmapdef})}{=} \\sum_{t \\text{ odd}}|B_i \\cap A_t|\\quad\\text{and}\\quad S_2 := \\sum_{j \\in [2r]\\setminus[r]}|f^{-1}(i,j)|=\\sum_{t \\text{ even}}|B_i \\cap A_t|.\n$$ \nTherefore $S_1+S_2=|B_i|=M_i$ and $\\left|S_1-S_2\\right| \\leq \\beta n$. So $S_1,S_2 = M_i\/2 \\pm \\beta n$. By definition of the $m_{i,j}$ and $M_i$ we have that $|m_{i,j}-M_i\/(2r)| \\leq 1$ for all $j \\in [2r]$.\nNow let $j \\in [r]$. We have\n\\begin{align*}\n\\left| |f^{-1}(i,j)|-m_{i,j}\\right| &\\leq \\left| |f^{-1}(i,j)|-M_i\/(2r)\\right| + 1 \\leq \\frac{1}{r}\\left| r|f^{-1}(i,j)| - S_1 \\right|  + 2\\beta n\\\\\n&\\leq \\frac{1}{r}\\sum_{j' \\in [r]}\\left| |f^{-1}(i,j)|-|f^{-1}(i,j')|\\right| + 2\\beta n \\stackrel{(\\ref{fdiff})}{\\leq} 10\\beta n,\n\\end{align*}\nas required. The case when $j \\in [2r]\\setminus[r]$ is almost identical.\nThus~$(\\mathscr{B}2)$ holds.\n\nNow let $uv \\in E(H)$ and write $f(u)=:(i,j)$ and $f(v)=:(i',j')$ for $i,i'\\in[\\ell]$ and $j,j'\\in[2r]$.\nSince $|B_t| > \\beta n$ for all $t \\in [\\ell]$ and $u \\in B_i$ and $v \\in B_{i'}$, we have that $|i-i'| \\leq 1$ by consideration of the bandwidth ordering.\nWe also have $j=\\chi'(u)$ and $j'=\\chi'(v)$, and $\\chi'$ is a proper colouring of $H$, so $j \\neq j'$.\nSuppose additionally that $u,v \\notin B$.\nIf $i \\neq i'$, then $u$ and $v$ are separated by at least $2\\beta n$ in the bandwidth ordering, so $uv \\not\\in E(H)$, a contradiction.\n\nFinally, if $s \\leq \\beta n$, then $x_s \\in A_1 \\cap B_1$. So $f(x_s)=(1,\\chi'(x))=(1,\\chi(x))$ by~(\\ref{chiprimedef}). So~$(\\mathscr{B}4)$ holds.\n\\qed\n\n\n\\subsection{Covering exceptional vertices: the second lemma for $H$}\\label{special}\n\nThe second lemma for $H$ will be used to find an embedding of a short initial segment of $H$ (in bandwidth ordering) into $G$ such that the exceptional set $V_0$, obtained after applying the Regularity lemma, lies in the image of this embedding.\nIn fact the pre-image of $V_0$ will be a $2$-independent set, which exists because $H$ has small maximum degree and bandwidth.\nAs well as embedding this initial segment, we would like to find target sets for its neighbours so that eventually we can extend this embedding to the whole of $H$.\n\n\n\n\\begin{lemma}[Special Lemma for $H$]\\label{lemmaforH}\nLet $n,r,L \\geq 1$ be integers and let \n$0 < 1\/n \\ll \\beta \\ll 1\/L \\ll {\\varepsilon}  \\ll \\rho \\ll \\eta \\ll d, 1\/r,1\/\\Delta$.\\COMMENT{AT: removed $\\delta$ from hierarchy -- don't think it is used. Added $1\/\\Delta$}\n\\COMMENT{AT: Important for Lemma 15 that it is $\\eta \\ll d,1\/r$ here}\nLet $G$ be an $n$-vertex graph, $R$ an $L$-vertex graph and $\\lbrace b_1,\\ldots,b_r\\rbrace \\subseteq V(R)$ be such that\n\\begin{itemize}\n\\item[$(\\mathscr{G}1)$] $G$ has vertex partition $\\lbrace V_0\\rbrace  \\cup \\lbrace V_a : a \\in V(R)\\rbrace$ where $|V_0| \\leq {\\varepsilon} n$ and $|V_a|=:m$ for all $a \\in V(R)$;\n\\item[$(\\mathscr{G}2)$] each $v \\in V_0$ is equipped with a subset $N_v \\subseteq V(R)$ with $|N_v| \\geq \\eta L\n\\COMMENT{AT: changed notation as $N(v)$ could get confused with $N_G (v)$}\n\\item[$(\\mathscr{G}3)$] $R$ is $(\\rho,d)$-dense and $\\delta(R) \\geq (1\/2+\\eta)L$;\n\\item[$(\\mathscr{G}4)$] $R[\\lbrace b_1,\\ldots,b_r\\rbrace] \\cong K_r$ and $\\lbrace b_1,\\ldots,b_r\\rbrace$ lies in a copy of $K_{18 r\/\\eta ^2}$ in $R$.\\COMMENT{AT: Important for Lemma 15 that it is $\\eta ^2$ here}\n\\end{itemize}\nThen there exists an integer $s \\leq {\\varepsilon}^{1\/4}n$ such that the following holds.\nLet $H$ be a graph on $s+\\beta n$ vertices with $\\Delta(H) \\leq \\Delta$ and assume that $H$ has a labelling $x_1,\\ldots,x_{s+\\beta n}$ of bandwidth at most $\\beta n$ and $\\chi(H) \\leq r$.\nLet $X := \\lbrace x_1,\\ldots,x_{s}\\rbrace$ and $Y := \\lbrace x_{s+1},\\ldots,x_{s+\\beta n}\\rbrace$.\nLet $\\chi : V(H) \\rightarrow [r]$ be a proper colouring of $H$.\nThen there exists a mapping $f:V(H)\\rightarrow V(R) \\cup V_0$ with the following properties:\n\\begin{itemize}\n\\item[$(\\mathscr{D}1)$] setting $I := f^{-1}(V_0)$, we have that $I$ is a subset of $X$ which is $2$-independent in $H$, and each  vertex in $V_0$ is mapped onto from a unique vertex in $H$ (so $|I|=|V_0|$);\n\\COMMENT{AT: with what was written before, we could have had many things mapped to a vertex in $V_0$} \n\\item[$(\\mathscr{D}2)$] for all $v \\in V_0$, setting $W_v := N_H(f^{-1}(v))$, we have $W_v \\subseteq X$ and $f(W_v) \\subseteq N_v$;\n\\item[$(\\mathscr{D}3)$] $|f^{-1}(a)| \\leq {\\varepsilon}^{1\/4}m$ for every $a \\in V(R)$;\n\\item[$(\\mathscr{D}4)$] for every edge $uv\\in E(H)$ such that $f(u),f(v) \\notin V_0$, we have $f(u)f(v) \\in E(R)$;\n\\item[$(\\mathscr{D}5)$] for all $y \\in Y$ we have $f(y)=b_{\\chi(y)}$.\n\\end{itemize}\n\\end{lemma}\n\nTo prove Lemma~\\ref{lemmaforH}, we will need an auxiliary result, Lemma~\\ref{findF}, which produces a `framework' $F$ in the reduced graph which we will later use to find $f$. This framework $F$ is a $2r$-trail such that for every $v \\in V_0$ there is a copy \n$T$ of $K_{2r}$ in $F$ such that $V(T)\\subseteq N_v$.\n\n\n\\begin{lemma}\\label{findF}\nLet $0 < 1\/n \\ll 1\/L \\ll {\\varepsilon}  \\ll \\rho \\ll \\eta \\ll d, 1\/r \\leq 1$.\nLet $G$ be an $n$-vertex graph, $R$ an $L$-vertex graph and $\\lbrace b_1,\\ldots,b_r\\rbrace \\subseteq V(R)$ be such that\n\\begin{itemize}\n\\item[$(\\mathscr{G}1)$] $G$ has vertex partition $\\lbrace V_0\\rbrace  \\cup \\lbrace V_a : a \\in V(R)\\rbrace$ where $|V_0| \\leq {\\varepsilon} n$ and $|V_a|=:m$ for all $a \\in V(R)$;\n\\item[$(\\mathscr{G}2)$] each $v \\in V_0$ is equipped with a subset $N_v \\subseteq V(R)$ with $|N_v| \\geq \\eta L$;\n\\item[$(\\mathscr{G}3)$] $R$ is $(\\rho,d)$-dense and $\\delta(R) \\geq (1\/2+\\eta)L$;\n\\item[$(\\mathscr{G}4)$] $R[\\lbrace b_1,\\ldots,b_r\\rbrace] \\cong K_r$ and $\\lbrace b_1,\\ldots,b_r\\rbrace$ lies in a copy of $K_{18 r\/\\eta ^2}$ in $R$.\n\\end{itemize}\n\\COMMENT{deleted a sentence that I thought wasn't used- please check}\nThen there exists an integer $K \\leq L^{2r}$ and a subgraph $F \\subseteq R$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{F}1)$] $F$ is a $2r$-trail with ordering $a_1,\\ldots,a_{t}$ where $t=(8K+1)r$;\n\\item[$(\\mathscr{F}2)$] there is a partition $V_0=V_0^1\\cup \\ldots \\cup V_0^K$ such that $N_v \\supseteq \\lbrace a_{8(i-1)r+1},\\ldots,a_{8(i-1)r+2r}\\rbrace$ for all $v \\in V_0^i$ and $|V_0^i| \\leq \\sqrt{{\\varepsilon}}m\/L^{2r-1}$ for all $i \\in [K]$;\n\\item[$(\\mathscr{F}3)$] $(a_{t-r+1},\\ldots,a_{t})=(b_1,\\ldots,b_r)$;\n\\item[$(\\mathscr{F}4)$] every $a \\in V(R)$ appears at most $L^{2r-1}\/{\\varepsilon}^{1\/12}$ times in the sequence $a_1,\\ldots,a_{t}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nAssume without loss of generality that $V(R)=[L]$.\nWe first prove the following claim.\n\n\\begin{claim}\\label{findKrs}\nThere is a $K \\leq L^{2r}$ and a set $\\mathcal{T} = \\lbrace T_1,\\ldots,T_K\\rbrace$ of $((d\/2)^{2r} \\eta L)$-extendable copies of $K_{2r}$ in $R$ such that there is a partition $V_0 = V_0^1 \\cup \\ldots \\cup V_0^{K}$ with the property that, for all $k \\in [K]$, we have $|V_0^k| \\leq \\sqrt{{\\varepsilon}}m\/L^{2r-1}$ and $T_k \\subseteq R[N_v]$ for all $v \\in V_0^k$.\n\\end{claim}\n\n\\begin{claimproof}\nBy Lemma~\\ref{nbrhd}(i), we see that $R_v := R[N_v]$ is $(\\rho L^2\/|N_v|^2,d)$-dense, and hence $(\\sqrt{\\rho},d)$-dense, where we used $(\\mathscr{G}2)$ and the fact that $\\rho\/\\eta^2 < \\sqrt{\\rho}$.\nLemma~\\ref{nbrhd}(iv) implies that $R_v$ contains at least $(d\/2)^{\\binom{2r+1}{2}} \\eta ^{2r} L^{2r}\/(2r)!$ copies of $K_{2r}$, each of which is $((d\/2)^{2r} \\eta L)$-extendable in $R_v$ (and thus $R$).\n\\COMMENT{We do need these $\\eta$ terms here in the 2 parts of this sentence.... so lots of changes to constants cos of this}\n\nLet $T_1,\\ldots,T_K$ be the set of $((d\/2)^{2r} \\eta L)$-extendable copies of $K_{2r}$ in $R$.\nSo\n\\begin{equation}\\label{Keq99}\nK \\leq \\binom{L}{2r} \\leq L^{2r}.\n\\end{equation}\nThen there is a partition $V_0^1 \\cup \\ldots \\cup V_0^{K}$ of $V_0$ into subsets (some of which may be empty) such that for all $k \\in [K]$ and $v \\in V_0^k$ we have that $R_v \\supseteq T_k$ and\n$$\n|V_0^k| \\leq \\frac{|V_0|}{(d\/2)^{\\binom{2r+1}{2}}\\eta ^{2r} L^{2r}\/(2r)!} \\stackrel{(\\mathscr{G}1)}{\\leq} \\frac{{\\varepsilon} m}{(d\/2)^{\\binom{2r+1}{2}} \\eta ^{2r} (1-{\\varepsilon})L^{2r-1}\/(2r)!} \\leq \\frac{\\sqrt{{\\varepsilon}}m}{L^{2r-1}},\n$$\nas desired.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nLet $\\mathcal{T} := \\lbrace T_i : i \\in [K]\\rbrace$ be obtained from the claim.\nTo complete the proof, we will use the Connecting lemma (Lemma~\\ref{connect}) to join the $K_{2r}$s in $\\mathcal{T}$ into a $2r$-trail.\nIn so doing, we have to be careful not to visit any $a \\in [L]$ too many times so as to ensure  $(\\mathscr{F}4)$ holds.\n\nSuppose, for some $0 \\leq i < K-1$ and all $j \\in [i]$ we have obtained a copy $P_j=x_j^1\\ldots x_j^{6r}$ of $P^{2r}_{6r} \\subseteq R$ such that\n\\begin{itemize}\n\\item[$\\mathscr{P}_1(i)$] $V(T_j)x_j^1\\ldots x_j^{6r}$ induces a copy $P_j'$ of $P^{2r}_{8r}$;\n\\item[$\\mathscr{P}_2(i)$] $x_j^1\\ldots x_j^{6r}V(T_{j+1})$ induces a copy $P_j''$ of $P^{2r}_{8r}$;\n\\item[$\\mathscr{P}_3(i)$] each $a \\in [L]$ lies in at most ${\\varepsilon}^{-1\/12}L^{2r-1}\/2$ of the $2r$-paths $P_1,\\ldots,P_i$.\n\\end{itemize}\nWe would like to find $P_{i+1}$ such that $\\mathscr{P}_1(i+1)$--$\\mathscr{P}_3(i+1)$ hold.\nWe will say that $a \\in [L]$ is \\emph{bad} if it appears in at least ${\\varepsilon}^{-1\/12}L^{2r-1}\/3$ of $P_1,\\ldots,P_i$.\nLet $D$ be the set of bad $a$. Since each $P_j$ contains $6r$ vertices, we have\n$$\n|D| {\\leq} \\frac{6ir}{{\\varepsilon}^{-1\/12}L^{2r-1}\/3} \\leq \\frac{18{\\varepsilon}^{1\/12} Kr}{L^{2r-1}} \\stackrel{(\\ref{Keq99})}{\\leq} 18{\\varepsilon}^{1\/12}rL.\n$$\nRecall from Claim~\\ref{findKrs} that $T_{i+1}$ and $T_{i+2}$ are both $((d\/2)^{2r} \\eta L)$-extendable copies of $K_{2r}$ in $R$.\nSince $\\eta \\ll d,1\/r$, they are $\\eta ^2 L$-extendable copies.\nApply Lemma~\\ref{connect} with $R,V(T_{i+1}),V(T_{i+2}),D,2r,\\eta ^2$ playing the roles of $G,X,Y,W,r,\\eta$ to obtain a copy $P_{i+1}$ of $P^{2r}_{6r} = x_{i+1}^1\\ldots x_{i+1}^{6r}$ which avoids $D$ and such that $V(T_{i+1})x_{i+1}^1\\ldots x_{i+1}^{6r}$ induces a copy $P_{i+1}'$ of $P^{2r}_{8r}$, and $x_{i+1}^1\\ldots x_{i+1}^{6r}V(T_{i+2})$ induces a copy $P_{i+1}''$ of $P^{2r}_{8r}$.\nSo $\\mathscr{P}_1(i+1)$ and $\\mathscr{P}_2(i+1)$ hold.\nNow let $a \\in [L]$. If $a \\notin V(P_{i+1})$, then $a$ lies in at most ${\\varepsilon}^{-1\/12}L^{2r-1}\/2$ of $P_1,\\ldots,P_{i+1}$ by~$\\mathscr{P}_3(i)$.\nOtherwise, since $P_{i+1}$ avoids $D$, $a$ lies in at most ${\\varepsilon}^{-1\/12}L^{2r-1}\/3+1 < {\\varepsilon}^{-1\/12}L^{2r-1}\/2$ of $P_1,\\ldots,P_{i+1}$. So~$\\mathscr{P}_3(i+1)$ holds.\nTherefore we can find $P_1,\\ldots,P_{K-1}$ satisfying $\\mathscr{P}_1(K-1)$--$\\mathscr{P}_3(K-1)$.\n\nNext we want to find a $2r$-path between $T_K$ and $\\lbrace b_1,\\ldots,b_r\\rbrace$.\nLet $\\{b'_1,\\dots, b'_r\\}$ be such that $\\lbrace b_1,\\ldots,b_r,b'_1,\\dots, b'_r\\rbrace$ lies in a copy of $K_{18 r\/\\eta ^2}$ in $R$ (such vertices exist by $(\\mathscr{G}4)$).\nApply Lemma~\\ref{connect} with $R,V(T_K),$ $\\lbrace b_1,\\ldots,b_r, b'_1, \\dots, b'_r\\rbrace,\\emptyset,2r,\\eta^2$ playing the roles of $G,X,Y,W,r,\\eta$ to obtain a copy $P_K$ of $P^{2r}_{6r} = x_{K}^1\\ldots x_{K}^{6r}$ such that $V(T_{K})x_{K}^1\\ldots x_{K}^{6r}$ induces a copy $P_{K}'$ of $P^{2r}_{8r}$, and furthermore\\\\\n$x_{K}^1\\ldots x_{K}^{6r}b_1\\ldots b_r b'_1\\dots b'_r$ induces a copy  of $P^{2r}_{8r}$; thus $x_{K}^1\\ldots x_{K}^{6r}b_1\\ldots b_r$ induces a copy $P_{K}''$ of $P^{2r}_{7r}$.\n(Note that the vertices $b'_1,\\dots, b'_r$ were only introduced so that we could apply Lemma~\\ref{connect}.)\nClearly $\\mathscr{P}_3(K-1)$ implies that each $a \\in [K]$ lies in at most $L^{2r-1}{\\varepsilon}^{-1\/12}\/2+1$ of $P_1,\\ldots,P_K$.\n\nWriting $V(T_i) = \\lbrace y_i^1,\\ldots,y_i^{2r}\\rbrace$ for all $i \\in [K]$, $R$ contains a $2r$-trail $F' := \\bigcup_{i \\in [K]}(P_i' \\cup P_i'')$\nof length $(8K+1)r=t$, with ordering given by\n$$\n(a_1,\\ldots,a_{t}) := (y_1^1,\\ldots y_1^{2r}, x_1^1, \\ldots x_1^{6r}, y_2^1,\\ldots y_2^{2r}, \\ldots, y_K^1,\\ldots, y_K^{2r}, x_K^1,\\ldots, x_K^{6r}, b_1,\\ldots,b_r).\n$$\nBy construction~$(\\mathscr{F}1)$ and~$(\\mathscr{F}3)$ hold. \n\nWe have that $V(T_i) = \\lbrace a_{8(i-1)r+1},\\ldots,a_{8(i-1)r+2r}\\rbrace$ for all $i \\in [K]$, which together with Claim~\\ref{findKrs} implies that~$(\\mathscr{F}2)$ holds.\nNow let $a \\in [L]$. Then~$\\mathscr{P}_3(K-1)$ implies that $a$ plays the role of some $x_i^j$ with $(i,j)\\in[K]\\times[6r]$ at most ${\\varepsilon}^{-1\/12}L^{2r-1}\/2+1$ times.\nSince each $T_i$ with $i \\in [K]$ is a distinct copy of $K_{2r}$ in $R$, we see that $a$ plays the role of some $y_i^j$ with $(i,j)\\in[K]\\times[2r]$ at most $\\binom{L-1}{2r-1} \\leq L^{2r-1}$ times.\nClearly $a$ plays the role of at most one of $b_1,\\ldots,b_r$.\nThus the number of times $a$ appears in the sequence $a_1,\\ldots,a_t$ is at most ${\\varepsilon}^{-1\/12}L^{2r-1}\/2 + L^{2r-1}+2 \\leq {\\varepsilon}^{-1\/12}L^{2r-1}$.\nSo~$(\\mathscr{F}4)$ holds.\n\\end{proof}\n\n\nArmed with Lemma~\\ref{findF}, we can now prove Lemma~\\ref{lemmaforH}.\nThe proof proceeds by splitting $V(H)$ into segments and assigning each one to a copy of $K_r$ in $R$, according to the framework $F$. For example, the first segment of $V(H)$ will be assigned to $\\lbrace a_1,\\ldots,a_r\\rbrace$, and more specifically, those vertices coloured $i$ by $\\chi$ will be mapped to $a_i$.\nIn those special segments assigned to vertex sets of $K_r$s which lie in $N_v$ for $v \\in V_0^i$, we choose $|V_0^i|$ special vertices to be the pre-images of vertices in $V_0^i$.\nThe property~$(\\mathscr{F}4)$ of $F$ will ensure that not too many vertices are mapped to the same cluster of $R$.\n\n\\begin{proof}[Proof of Lemma~\\ref{lemmaforH}]\nLet $G$ and $R$ be as in the statement of the lemma.\nWithout loss of generality we will assume that $V(R)=[L]$.\nApply Lemma~\\ref{findF} to obtain $K \\leq L^{2r}$ and $F \\subseteq R$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{F}1)$] $F$ is a $2r$-trail with ordering $a_1,\\ldots,a_{t}$ where $t=(8K+1)r$;\n\\item[$(\\mathscr{F}2)$] there is a partition $V_0=V_0^1\\cup \\ldots \\cup V_0^K$ such that $N_v \\supseteq \\lbrace a_{8(i-1)r+1},\\ldots,a_{8(i-1)r+2r}\\rbrace$ for all $v \\in V_0^i$ and $|V_0^i| \\leq \\sqrt{{\\varepsilon}}m\/L^{2r-1}$ for all $i \\in [K]$;\n\\item[$(\\mathscr{F}3)$] $(a_{t-r+1},\\ldots,a_{t})=(b_1,\\ldots,b_r)$;\n\\item[$(\\mathscr{F}4)$] every $a \\in [L]$ appears at most $L^{2r-1}\/{\\varepsilon}^{1\/12}$ times in the sequence $a_1,\\ldots,a_{t}$.\n\\end{itemize}\nLet\n$$\ns := 8K{\\varepsilon}^{1\/3}m\/L^{2r-1} \\leq 8L{\\varepsilon}^{1\/3}m \\stackrel{(\\mathscr{G}1)}{\\leq} 8{\\varepsilon}^{1\/3}n \\leq {\\varepsilon}^{1\/4}n.\n$$\nFor all $i \\in [K]$, let\n\\begin{equation}\\label{15sizes}\nu_i := |V_0^i| \\stackrel{(\\mathscr{F}2)}{\\leq} \\sqrt{{\\varepsilon}}m\/L^{2r-1}\\quad\\text{and}\\quad b := {\\varepsilon}^{1\/3}m\/L^{2r-1} > 100\\beta mL \\stackrel{(\\mathscr{G}1)}{>} 99\\beta n.\n\\end{equation}\nLet $H,X,Y$ be as in the statement of the lemma.\nDefine a partition of $X \\cup Y = \\lbrace x_1,\\ldots,x_{s+\\beta n}\\rbrace$ into $8K+1$ intervals\n$$\nB_1^1,B_1^2,\\ldots,B^{8}_1,B^1_2,B_2^2,\\ldots,B_2^{8},B^1_3,\\ldots,B^1_K,B_K^2,\\ldots,B_K^{8},B^1_{K+1}\n$$\nwhere $|B_i^j|=b$ for all $(i,j) \\in [K]\\times[8]$; $|B^1_{K+1}|=\\beta n$; the first $b$ vertices $x_1,\\dots,x_b$ in $X\\cup Y $ form $B_1^1$, the next $b$ vertices in $X\\cup Y$ form  $B_1^2$, and so on.\nIn particular, $B^1_{K+1}=Y$ and\n each interval comes equipped with the ordering inherited from the bandwidth ordering of $H$.\nThe first claim identifies a set $I \\subseteq X$ which will be the pre-image of $V_0$ in our desired mapping.\nRecall that given a graph $J$ and $A \\subseteq V(J)$, we say that $A$ is \\emph{$2$-independent} if every pair of vertices in $A$ are at distance at least $3$ in $J$. In other words, $A$ is an independent set and additionally the neighbourhoods of different vertices in $A$ are disjoint.\n\\medskip\n\\noindent\n\n\\begin{claim}\\label{claim1H}\nFor each $i\\in[K]$, there exists a $2$-independent set $I_i \\subseteq B^1_i$ (with respect to $H$) of size $u_i$ such that $W(i) := \\bigcup_{y \\in I_i}N_H(y) \\subseteq B^1_i$. Further, $I := \\bigcup_{i \\in [K]}I_i$ is a $2$-independent set in $H$.\n\\end{claim}\n\n\\begin{claimproof}\nObtain $A_i$ from $B^1_i$ by removing the first $2\\beta n$ and last $2\\beta n$ elements (which is possible by~(\\ref{15sizes})).\nSuppose we have obtained a $2$-independent set $I^j \\subseteq A_i$ of size $0 \\leq j < u_i$.\nThen for any $y \\in A_i$, the set $I^j \\cup \\lbrace y \\rbrace$ is a $2$-independent set in $H$ of size $j+1$ if $y \\notin I^j \\cup N_H(y') \\cup N_H(N_H(y'))$ for any $y' \\in I^j$.\nThe number of excluded $y$ is at most \n\\begin{eqnarray*}\n|I^j|+\\sum_{x \\in I^j}d_H(x) + \\sum_{x\\in I^j}\\sum_{z \\in N_H(x)}d_H(z) &\\leq& |I^j|(1+\\Delta+\\Delta^2) \\leq 2\\Delta^2 u_i \\stackrel{(\\ref{15sizes})}{\\leq} 2\\Delta^2\\sqrt{{\\varepsilon}}m\/L^{2r-1}\\\\\n&\\stackrel{(\\ref{15sizes})}{<}& b - 4\\beta n = |A_i|.\n\\end{eqnarray*}\nTherefore we can find a $2$-independent set $I_i := I^{u_i}$ of size $u_i$ in $A_i$.\nThis together with the bandwidth property and the definition of $A_i$ implies that $W(i) \\subseteq \\bigcup_{y \\in I_i}(N_H(y) \\cup N_H(N_H(y))) \\subseteq B^1_i$.\nThus there is no edge between $N_H(I_i)$ and $N_H(I_{i'})$ for $i \\neq i'$.\nSo $I = \\bigcup_{i \\in [K]}I_i$ is a $2$-independent set in $H$, proving the claim.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nLet $\\chi : V(H) \\rightarrow [r]$ be the given proper colouring of $H$.\nA second claim finds a suitable homomorphism $\\phi : V(H) \\rightarrow V(F)$ on which $f$ will be based.\n\n\\begin{claim}\\label{claim2H}\nFor each $(i,j) \\in [K]\\times[8] \\cup \\lbrace (K+1,1)\\rbrace$,\nlet\n$$\n\\phi(x) := a_{(8(i-1)+(j-1))r+\\chi(x)}\\quad\\text{if } x \\in B_i^j.\n$$\nThen $\\phi : V(H) \\rightarrow V(F)$ is a graph homomorphism such that \n$|\\phi^{-1}(a)| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$.\n\\end{claim}\n\n\\begin{claimproof}\nNote first that if $a_k$ is in the image of $\\phi$ for some $k\\in \\mathbb{N}$, then, recalling~$(\\mathscr{F}1)$, we have that $k \\in [t]$, so $V(F) \\supseteq \\phi(V(H))$.\nLet us check that $\\phi$ is a homomorphism.\nLet $xy \\in E(H)$.\nLet $(i,j),(i',j')\\in [K]\\times[8] \\cup \\lbrace (K+1,1)\\rbrace$ be such that $x \\in B_i^j$ and $y \\in B_{i'}^{j'}$.\nSince $H$ has bandwidth at most $\\beta n$ and $|B_i^j|,|B_{i'}^{j'}| > \\beta n$, we must have $(i',j')\\in\\lbrace (i,j-1),(i,j),(i,j+1)\\rbrace$, where we let $(i,9) := (i+1,1)$ and $(i,0) := (i-1,8)$.\nSo, writing\n$$\nT_i := \\lbrace a_{(8(i-1)+(j-1))r+p} : p \\in [r]\\rbrace\\quad\\text{and}\\quad T_{i'} := \\lbrace a_{(8(i'-1)r+(j'-1))+p}: p \\in [r]\\rbrace\n$$ \nwe either have $T_i=T_{i'}$, or $T_i$ and $T_{i'}$ are consecutive intervals in $a_1,\\ldots,a_t$ each of length $r$.\nIn both cases we have $\\phi(x) \\neq \\phi(y)$ (in the first case this follows from the fact that $\\chi(x) \\neq \\chi(y)$).\nBut~$(\\mathscr{F}1)$ now implies that $F[T_i \\cup T_{i'}]$ is a clique, so since $\\phi(x) \\in T_i$ and $\\phi(y) \\in T_{i'}$ are distinct, $\\phi(x)\\phi(y) \\in E(F)$, as required.\n\n\nFor the final assertion, each $a \\in V(F)$ appears at most $L^{2r-1}\/{\\varepsilon}^{1\/12}$ times in the sequence $a_1,\\ldots,a_t$ by~$(\\mathscr{F}4)$.\nSo, writing $\\theta : V(H) \\rightarrow [t]$ where $\\phi(x) = a_{\\theta(x)}$, we have\n$$\n|\\phi^{-1}(a)| \\leq \\frac{L^{2r-1}}{{\\varepsilon}^{1\/12}} \\cdot \\max_{k \\in [t]}|\\theta^{-1}(k)| \\leq \\frac{L^{2r-1}b}{{\\varepsilon}^{1\/12}} \\stackrel{(\\ref{15sizes})}{=} {\\varepsilon}^{1\/4}m,\n$$\nas desired.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nNow let $H' := H\\setminus I$ where $I := \\bigcup_{k \\in [K]}I_k$ and $W := \\bigcup_{k \\in [K]}W(k)$ where $W(k)$ is defined in Claim~\\ref{claim1H}.\nNote also that $W \\subseteq V(H')$ since, by Claim~\\ref{claim1H}, $I$ is an independent set.\n\nLet $g : I \\rightarrow V_0$ be a bijection such that $g(I_i) =V_0^i$ for all $i \\in [K]$ (which is clearly possible by Claim~\\ref{claim1H} and~(\\ref{15sizes})).\nSince $I_i$ is a $2$-independent set in $H$, the set of neighbourhoods $N_H(y)$ is pairwise disjoint over all $y \\in I_i$.\nSo for each $w \\in W(i)$, there is a unique $y \\in I_i$ for which $w \\in N_H(y)$. \nClaim~\\ref{claim2H} implies that $|\\phi^{-1}(a)| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$.\n\nWe claim that $f : V(H) \\rightarrow [L] \\cup V_0$ given by\n\\begin{equation}\\label{fdef}\nf(x) = \\begin{cases} \\phi(x) &\\mbox{if } x \\in V(H)\\setminus I \\\\\ng(x) & \\mbox{if } x \\in I \\end{cases} \n\\end{equation}\nis the required mapping.\nNote that $f(V(H)\\setminus I) \\subseteq [L]$ and $f(I) \\subseteq V_0$.\nFor~$(\\mathscr{D}1)$, \nnote that, by Claim~\\ref{claim1H} and~(\\ref{fdef}), $f(I)=g(I)=V_0$, $|I|=|V_0|$ and $I$ is a $2$-independent subset of $X$.\nFor~$(\\mathscr{D}2)$,\nlet $v \\in V_0$ and $W_v := N_H(f^{-1}(v))$.\nLet $k \\in [K]$ be such that $v \\in V_0^k$.\nThen $f^{-1}(v)=g^{-1}(v) \\in I_k$. So $W_v \\subseteq W(i) \\subseteq B^1_i \\subseteq  X$.\nLet $x \\in W_v\\subseteq B^1_i$.\nBy Claim~\\ref{claim2H} and $(\\mathscr{F}2)$ we have that $f(x)=\\phi(x)=a_{8(i-1)+\\chi(x)} \\in N_v$.\nThis completes the proof of~$(\\mathscr{D}2)$.\n\nFor~$(\\mathscr{D}3)$, let $a \\in V(R)$. Then $f^{-1}(a)\\subseteq \\phi^{-1}(a)$ has size at most ${\\varepsilon}^{1\/4}m$ by Claim~\\ref{claim2H}.\nFor~$(\\mathscr{D}4)$, let $uv \\in E(H)$ be such that $f(u),f(v) \\notin V_0$. So $u,v \\in V(H)\\setminus I$ and $f(u)=\\phi(u)$ and $f(v)=\\phi(v)$.\nBy Claim~\\ref{claim2H}, $\\phi : V(H)\\rightarrow V(F)$ is a homomorphism, so $f(u)f(v) \\in E(F) \\subseteq E(R)$.\nFinally, for~$(\\mathscr{D}5)$, we have that $Y \\cap I = \\emptyset$ by Claim~\\ref{claim1H}, so for any $y \\in Y$ we have $f(y)=\\phi(y)=a_{t-r+\\chi(y)}=b_{\\chi(y)}$ by~$(\\mathscr{F}3)$.\n\\end{proof}\n\n\n\n\n\n\\begin{comment}\n\\begin{lemma}\\label{embedV0}\nLet $n,L,m,r,q\\in\\mathbb{N}$ and suppose that $0<1\/n \\ll \\beta \\ll 1\/L \\ll {\\varepsilon} \\ll c \\ll \\delta \\ll 1\/r,1\/\\Delta,1\/q$ and let $K \\leq L^r$ be an integer.\nSuppose that $G$ is a graph on $n$ vertices with vertex partition $V_0 \\cup \\ldots \\cup V_L$ where $|V_0| \\leq {\\varepsilon} n$ and $|V_i|=m$ for all $i \\in [L]$, and $V_0^1\\cup \\ldots \\cup V_0^K$ is a partition of $V_0$ in which $|V_0^k| \\leq \\sqrt{{\\varepsilon}}m\/L^{r-1}$ for all $k \\in [K]$.\nSuppose that $F$ is a graph with $V(F) \\subseteq [L]$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{T}1)$] $F$ is a $2r$-trail with ordering $a_1,\\ldots,a_{t}$ where $t:=(qK+1)r$;\n\\item[$(\\mathscr{T}2)$] for all $i \\in [K]$ and $v \\in V_0^i$, we have $d_G(v,V_{a_j}) \\geq c m$ for all $j \\in \\lbrace q(i-1)r+1,\\ldots,q(i-1)r+r\\rbrace$;\n\\item[$(\\mathscr{T}3)$] every $a \\in [L]$ appears at most $L^{r-1}\/{\\varepsilon}^{1\/12}$ times in the sequence $a_1,\\ldots,a_{t}$.\n\\item[$(\\mathscr{G}'4)$] $G[V_i,V_j]$ is $({\\varepsilon},\\delta)$-regular for all $ij \\in E(F)$.\n\\end{itemize}\nLet \n$\ns = qK{\\varepsilon}^{1\/3}m\/L^{r-1}+\\beta n\n$\nand\nlet $H$ be an $(s+\\beta n)$-vertex graph with $\\Delta(H) \\leq \\Delta$ and $\\chi(H) \\leq r$. Suppose that $H$ has bandwidth at most $\\beta n$, with bandwidth order $\\lbrace x_1,\\ldots,x_{s+\\beta n}\\rbrace$ and let $X := \\lbrace x_1,\\ldots,x_s\\rbrace$ and $Y := \\lbrace x_{s+1},\\ldots,x_{s+\\beta n}\\rbrace$.\nLet $\\chi:V(H)\\rightarrow[r]$ be a proper colouring of $H$.\nThen there exists an embedding $f : X \\rightarrow V(G)$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{H}1)$] $V_0 \\subseteq f(X)$;\n\\item[$(\\mathscr{H}2)$] $|f^{-1}(V_{a})| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$;\n\\item[$(\\mathscr{H}3)$] for all $y \\in Y$, there exists $C_y \\subseteq V_{a_{t-r+\\chi(y)}}\\setminus f(X)$ such that $C_y \\subseteq N_G(f(x))$ for all $x \\in N_{H}(y) \\cap X$ and $|C_y| \\geq cm$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFor all $i \\in [K]$, let\n\\begin{equation}\\label{15sizes}\nu_i := |V_0^i| \\leq \\sqrt{{\\varepsilon}}m\/L^{r-1}\\quad\\text{and}\\quad b := {\\varepsilon}^{1\/3}m\/L^{r-1}.\n\\end{equation}\nDefine a partition of $X \\cup Y = \\lbrace x_1,\\ldots,x_{s+\\beta n}\\rbrace$ into $qK+1$ intervals\n$$\nB_1^1,B_1^2,\\ldots,B^q_1,B^1_2,B_2^2,\\ldots,B_2^q,B^1_3,\\ldots,B^1_K,B_K^2,\\ldots,B_K^q,B^1_{K+1}\n$$\nwhere $|B_i^j|=b$ for all $(i,j) \\in [K]\\times[q]$ and $|B^1_{K+1}|=\\beta n$.\nSo each interval comes equipped with the ordering inherited from the bandwidth ordering of $H$.\nThe first claim identifies a set $I \\subseteq X$ which will be the pre-image of $V_0$ in our desired embedding.\nRecall that given a graph $J$ and $A \\subseteq V(J)$, we say that $J$ is \\emph{$2$-independent} if every pair of vertices in $A$ are distance at least $3$ in $J$. In other words, the neighbourhoods of different vertices in $A$ are disjoint.\n\\medskip\n\\noindent\n\n\\begin{claim}\\label{claim1}\nFor each $i\\in[K]$, there exists a $2$-independent set $I_i \\subseteq B^1_i$ of size $u_i$ such that $W_i := \\bigcup_{y \\in I_i}N_H(y) \\subseteq B^1_i$. Further, $I := \\bigcup_{i \\in [K]}I_i$ is a $2$-independent set.\n\\end{claim}\n\n\\begin{claimproof}\nObtain $A_i$ from $B^1_i$ by removing the first $2\\beta n$ and last $2\\beta n$ elements.\nSuppose we have obtained a $2$-independent set $I^j \\subseteq A_i$ of size $0 \\leq j < u_i$.\nThen for any $y \\in A_i$, the set $I^j \\cup \\lbrace y \\rbrace$ is a $2$-independent of size $j+1$ if $y \\notin I^j \\cup N_H(y') \\cup N_H(N_H(y'))$ for any $y' \\in I^j$.\nThe number of excluded $y$ is at most \n\\begin{eqnarray*}\n|I^j|+\\sum_{x \\in I^j}d_H(x) + \\sum_{x\\in I^j}\\sum_{z \\in N_H(x)}d_H(z) &\\leq& |I^j|(1+\\Delta+\\Delta^2) \\leq 2\\Delta^2 u_i \\stackrel{(\\ref{15sizes})}{\\leq} 2\\Delta^2\\sqrt{{\\varepsilon}}m\/L^{r-1}\\\\\n&\\stackrel{(\\ref{15sizes})}{<}& b - 4\\beta n = |A_i|.\n\\end{eqnarray*}\nTherefore we can find a $2$-independent set $I_i := I^{u_i}$ of size $u_i$ in $A_i$.\nThis together with the bandwidth property and the definition of $A_i$ implies that $W_i \\subseteq \\bigcup_{y \\in I_i}(N_H(y) \\cup N_H(N_H(y))) \\subseteq B^1_i$.\nThus there is no edge between $N_H(I_i)$ and $N_H(I_{i'})$ for $i \\neq i'$.\nSo $I = \\bigcup_{i \\in [K]}I_i$ is a $2$-independent set, proving the claim.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nLet $\\chi : V(H) \\rightarrow [r]$ be the given proper colouring of $H$.\nA second claim finds a suitable homomorphism $\\phi : V(H) \\rightarrow V(F)$ which will be used in applying the embedding lemma, Lemma~\\ref{target}.\n\n\\begin{claim}\\label{claim2}\nFor each $(i,j) \\in [K]\\times[q] \\cup \\lbrace (K+1,0)\\rbrace$,\nlet\n$$\n\\phi(x) := a_{(q(i-1)+(j-1))r+\\chi(x)}\\quad\\text{if } x \\in B_i^j.\n$$\nThen $\\phi : V(H) \\rightarrow V(F)$ is a graph homomorphism such that \n$|\\phi^{-1}(a)| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$.\n\\end{claim}\n\n\\begin{claimproof}\nNote first that if $a_k$ is in the image of $\\phi$, then, recalling~$(\\mathscr{T}2)$, we have that $k \\in [t]$, so $V(F) \\subseteq \\phi(V(H))$.\nLet us check that $\\phi$ is a homomorphism.\nLet $xy \\in E(H)$.\nLet $(i,j),(i',j')\\in [K]\\times[q] \\cup \\lbrace (K+1,1)\\rbrace$ be such that $x \\in B_i^j$ and $y \\in B_{i'}^{j'}$.\nSince $H$ has bandwidth at most $\\beta n$, we must have $(i',j')\\in\\lbrace (i,j-1),(i,j),(i,j+1)\\rbrace$, where we let $(i,q+1) := (i+1,1)$ and $(i,0) := (i-1,q)$.\nSo, writing\n$$\nT_i := \\lbrace a_{(q(i-1)+(j-1))r+p} : p \\in [r]\\rbrace\\quad\\text{and}\\quad T_{i'} := \\lbrace a_{(q(i'-1)r+(j'-1))+p}: p \\in [r]\\rbrace\n$$ \nwe either have $T_i=T_{i'}$, or $T_i$ and $T_{i'}$ are consecutive intervals in $a_1,\\ldots,a_t$ each of length $r$.\nIn both cases we have $\\phi(x) \\neq \\phi(y)$ (in the first case this follows from the fact that $\\chi(x) \\neq \\chi(y)$).\nBut~$(\\mathscr{T}1)$ now implies that $F[T_i \\cup T_{i'}]$ is a clique, so since $\\phi(x) \\in T_i$ and $\\phi(y) \\in T_{i'}$ are distinct, $\\phi(x)\\phi(y) \\in E(F)$, as required.\n\n\nFor the final assertion, each $a \\in V(F)$ appears at most $L^{r-1}\/{\\varepsilon}^{1\/12}$ times in the sequence $a_1,\\ldots,a_t$ by~$(\\mathscr{T}3)$.\nSo, writing $\\theta : V(H) \\rightarrow [t]$ where $\\phi(x) = a_{\\theta(x)}$, we have.\n$$\n|\\phi^{-1}(a)| \\leq \\frac{L^{r-1}}{{\\varepsilon}^{1\/12}} \\cdot \\max_{k \\in [t]}|\\theta^{-1}(k)| \\leq \\frac{L^{r-1}b}{{\\varepsilon}^{1\/12}} \\stackrel{(\\ref{15sizes})}{=} {\\varepsilon}^{1\/4}m,\n$$\nas desired.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nNow let $H' := H\\setminus I$ and $W := \\bigcup_{i \\in [K]}W_i$.\nThe restriction $\\phi'$ of $\\phi$ to $H'$ is still a graph homomorphism.\nNote also that $W \\subseteq V(H')$ since, by Claim~\\ref{claim1}, $W \\subseteq X$ and $I$ is an independent set.\n\nLet $g : I \\rightarrow V_0$ be a bijection such that $g(I_i) =V_0^i$ for all $i \\in [K]$ (which is clearly possible by Claim~\\ref{claim1} and~(\\ref{15sizes})).\nRecall that $W_i = \\bigcup_{y \\in I_i}N_H(y)$.\nSince $I_i$ is a $2$-independent set in $H$, the set of neighbourhoods $N_H(y)$ is pairwise disjoint over all $y \\in I_i$.\nSo for each $w \\in W_i$, there is a unique $y \\in I_i$ for which $w \\in N_H(y)$. Let\n\\begin{equation}\\label{Sw}\nS_w := N_G(g(y),V_{\\phi(w)}) \\subseteq V_{a_{q(i-1)r+\\chi(w)}}\n\\end{equation}\n(here we used the fact that $W_i \\subseteq B_i^1$ from Claim~\\ref{claim1}).\nThen~$(\\mathscr{T}2)$ implies that $|S_w| \\geq c m$.\nClaim~\\ref{claim2} implies that $|\\phi^{-1}(a)| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$.\nIn particular, $W(a) := \\phi^{-1}(a) \\cap W$ is such that $|W(a)| \\leq {\\varepsilon}^{1\/4}m$ for all $a \\in [L]$.\n\nApply Lemma~\\ref{target} with $G,F,H',X\\setminus I,Y,\\phi',{\\varepsilon}^{1\/4}\/2,W(a),S_w$ playing the roles of $G,R,H,X,Y,\\phi,{\\varepsilon},W_i,S_w$ respectively to obtain an embedding $f'$ of $H'[X]$ into $G$ such that\n\\begin{itemize}\n\\item[(i)] $f'(x) \\in V_{\\phi'(x)}$ for all $x \\in V(H)$;\n\\item[(ii)] $f'(w) \\in S_w$ for all $w \\in W$;\n\\item[(iii)] for all $y \\in Y$ there exists $C_y \\subseteq V_{\\phi'(y)}\\setminus f'(X)$ such that $C_y \\subseteq N_G(f'(x))$ for all $x \\in N_H(y) \\cap X$, and $|C_y| \\geq cm$.\n\\end{itemize}\nWe claim that $f : X \\rightarrow V(G)$ given by\n\\begin{equation}\\label{fdef}\nf(x) = \\begin{cases} f'(x) &\\mbox{if } x \\in X\\setminus I \\\\\ng(x) & \\mbox{if } x \\in I \\end{cases} \n\\end{equation}\nis the required embedding.\nSince $f'$ and $g$ are both injective with disjoint domains, $f$ is clearly injective.\nTo show that $f$ is an embedding, we now just need to check that it is a graph homomorphism. So let $xy \\in E(H[X])$. If $x,y \\in X\\setminus I$ then $f(x)f(y)=f'(x)f'(y) \\in E(G)$; and we cannot have $x,y \\in I$ since $I$ is independent.\nThus we may assume that $x \\in X\\setminus I$ and $y \\in I$.\nBut then $x \\in W$ by definition, so there is a unique $v \\in I$ for which $x \\in N_H(v)$. But $y \\in I$ and $x \\in N_H(y)$ so we must have $v=y$.\nTherefore\n$$\nf(x) \\stackrel{(\\ref{fdef})}{=} f'(x) \\stackrel{(ii)}{\\in} S_x \\stackrel{(\\ref{Sw})}{\\subseteq} N_G(g(y)) \\stackrel{(\\ref{fdef})}{=} N_G(f(y)),\n$$\ni.e.~$f(x)f(y) \\in E(G)$, as required.\nSo $f$ is an embedding of $H$ into $G$.\nFurther, $V_0 = g(I) \\subseteq g(X)$, proving~$(\\mathscr{H}1)$.\nFor~$(\\mathscr{H}2)$, we have for all $a \\in [L]$ that\n$$\nf^{-1}(V_a)\\stackrel{(\\ref{fdef})}{=}(f')^{-1}(V_a) \\stackrel{(i)}{=} (\\phi')^{-1}(a) \\subseteq \\phi^{-1}(a)\n$$\nwhich has size at most ${\\varepsilon}^{1\/4}m$ by Claim~\\ref{claim2}.\n\nIt remains to prove~$(\\mathscr{H}3)$.\nFor this, fix $y \\in Y$.\nSince $Y = B_{K+1}^1$, Claim~\\ref{claim2} and the definition of $t$ from~$(\\mathscr{T}1)$ imply that $\\phi'(y) = \\phi(y) = a_{t-r+\\chi(y)}$.\nSo~(iii) implies that $C_y \\subseteq V_{a_{t-r+\\chi(y)}} \\setminus f'(X)$. But by~(\\ref{fdef}), $V_a \\cap f(X) = V_a \\cap f'(X)$, so in fact $C_y \\subseteq V_{a_{t-r+\\chi(y)}} \\setminus f(X)$, as required.\nLet $x \\in N_H(y) \\cap X$ be arbitrary. Now, $y \\notin \\bigcup_{i \\in [K]}B_i^1$, so Claim~\\ref{claim1} implies that $y \\notin W$. Then $x \\notin I$. So $f(x) = f'(x)$, and~(iii) implies that $C_y \\subseteq N_G(f(x))$, as required.\nFinally, $|C_y| \\geq cm$ again by~(iii).\nThis completes the proof of~$(\\mathscr{H}3)$ and hence of the lemma.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The lemma for $G$: adjusting cluster sizes}\\label{sec7}\n\nRecall the definition of $Z^r_\\ell$ from Section~\\ref{notation} and in particular that it contains a $K_r$-factor.\nOur goal in this section is to prove Lemma~\\ref{lemmaforG}. Roughly speaking, it supposes that the reduced graph $R$ of $G$ contains a spanning copy of $Z^{2r}_\\ell$, its clusters $V_1,\\dots,V_L$ are equally sized, and pairs of clusters corresponding to the $K_{2r}$-factor $\\ell\\cdot K_{2r}$ in $Z^{2r}_\\ell$ are superregular.\nThen we can adjust $V_1,\\ldots,V_L$ slightly by reallocating a small number of vertices so that they have given sizes, at the expense of now having superregular pairs corresponding to a $K_r$-factor $2\\ell \\cdot K_r$.\n\nTo formalise the structural properties we need from $G$, we make the following definition (very similar to Definition~8.1 in~\\cite{st1}).\n\n\n\n\\begin{definition}[$r$-Cycle structure]\\label{cyclestructure}\nGiven integers $n,\\ell,r$, a graph $G$ on $n$ vertices, and constants ${\\varepsilon},\\delta>0$, we say that $G$ has an \\emph{$(R,\\ell,r,\\mathcal{V},{\\varepsilon},\\delta)$-cycle structure} $\\mathcal{C}$ if the following hold:\n\\begin{itemize}\n\\item[$(\\mathscr{C}1)$] $\\mathcal{V} = \\lbrace V_0\\rbrace \\cup \\lbrace V_{i,j}:(i,j) \\in [\\ell]\\times[r]\\rbrace$ is a partition of $V(G)$, where $|V_0| \\leq {\\varepsilon} n$.\n\\item[$(\\mathscr{C}2)$] $R$ has vertex set $[\\ell]\\times[r]$ and $R \\supseteq Z^{r}_\\ell$ and $G[V_{i,j},V_{i',j'}]$ is $({\\varepsilon},\\delta)$-regular whenever $(i,j)(i',j') \\in E(R)$;\n\\item[$(\\mathscr{C}3)$] $G[V_{i,j},V_{i,j'}]$ is $({\\varepsilon},\\delta)$-superregular whenever $i \\in [\\ell]$ and $1 \\leq j < j' \\leq r$.\n\\end{itemize}\nWe say that $\\mathcal{V}$ \\emph{induces} $\\mathcal{C}$. If $V_0 = \\emptyset$ we say that $\\mathcal{C}$ is \\emph{spanning}.\n\\end{definition}\n\n\nThe next definition concerns a convenient relabelling of the vertex set of a graph, which we will use for the reduced graph $R$.\n\n\\begin{definition}[Bijection $\\phi^{2r}_\\ell$]\\label{bijdef}\nGiven integers $r,\\ell$, define $\\phi^{2r}_{\\ell} : [\\ell]\\times[2r]\\rightarrow[2\\ell]\\times[r]$ by setting\n\\begin{equation}\\label{phi2rldef}\n\\phi^{2r}_{\\ell}(i,j)=\\left((2i-1)+\\left\\lfloor\\frac{j}{r}\\right\\rfloor,j-\\left(\\left\\lceil \\frac{j}{r}\\right\\rceil-1\\right) r\\right),\\quad\\text{for all }(i,j)\\in[\\ell]\\times[2r].\n\\end{equation}\nIt is easy to check that $\\phi^{2r}_{\\ell}$ is a bijection and\n\\begin{alignat*}{6}\n\\phi^{2r}_{\\ell}&(1,1)\\ldots\\phi^{2r}_{\\ell}&&(1,r)\\phi^{2r}_{\\ell}&&(1,r+1)\\ldots\\phi^{2r}_{\\ell}&&(1,2r)\\ldots\\phi^{2r}_{\\ell}&&(\\ell,r+1)\\ldots\\phi^{2r}_{\\ell}&&(\\ell,2r)\\\\\n= &(1,1)\\ldots&&(1,r)&&(2,1)\\ldots&&(2,r)\\ldots&&(2\\ell,1)\\ldots&&(2\\ell,r).\n\\end{alignat*}\nThis implies that for all $a \\in [2\\ell]$ and distinct $b,b'\\in[r]$, there are $i \\in [\\ell]$ and $(j,j')\\in[[2r]]^2$ such that $(\\phi^{2r}_{\\ell}(i,j),\\phi^{2r}_{\\ell}(i,j'))=((a,b),(a,b'))$.\n\nGiven a graph $R$ and a bijection $\\phi : V(R) \\rightarrow V$ to some set $V$, we write $\\phi(R)$ for the graph with vertex set $\\lbrace \\phi(x) : x \\in V(R)\\rbrace$ and edge set $\\lbrace \\phi(x)\\phi(y) : xy \\in E(R)\\rbrace$. So $\\phi(R) \\cong R$.\n\\end{definition}\n\n\nIn the language of Definition~\\ref{cyclestructure}, the main result of this section states that, given a graph with a (spanning) $2r$-cycle structure, we can obtain from it an $r$-cycle structure which is almost balanced, but the exact deviation from perfect balancedness can be controlled.\n\n\n\n\n\\begin{lemma}[Lemma for $G$]\\label{lemmaforG}\nLet $n,\\ell,m,r \\in \\mathbb{N}$ and $0 < 1\/n \\ll \\xi \\ll 1\/\\ell \\ll {\\varepsilon} \\ll \\delta < 1\/r$.\nSuppose that $G$ is a graph on $n$ vertices with a spanning $(R,\\ell,2r,\\mathcal{V},{\\varepsilon},\\delta)$-cycle structure, where $\\mathcal{V} = \\lbrace V_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ and $|V_{i,j}| = m$ for all $(i,j)\\in[\\ell]\\times[2r]$.\nLet $\\lbrace \\tau_{a,b} \\in \\mathbb Z : (a,b)\\in[2\\ell]\\times[r]\\rbrace$ be such that $0 \\leq \\tau_{a,b}\\leq {\\varepsilon} m$ for all $(a,b)\\in[2\\ell]\\times[r]$.\nThen there exist positive integers $\\lbrace m_{a,b} : (a,b)\\in[2\\ell]\\times[r]\\rbrace$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{L}1)$] $\\sum_{(a,b)\\in[2\\ell]\\times[r]}(m_{a,b}+\\tau_{a,b})=n$ and $m_{a,b} \\geq (1-\\sqrt{{\\varepsilon}})m$ and $|m_{a,b}-m_{a,b'}| \\leq 1$ for all $a \\in [2\\ell]$ and $b,b' \\in [r]$;\n\\item[$(\\mathscr{L}2)$] given any $\\lbrace n_{a,b}\\in \\mathbb N : (a,b)  \\in [2\\ell]\\times[r]\\rbrace$ with $\\sum_{(a,b)\\in[2\\ell]\\times[r]}(n_{a,b}+\\tau_{a,b})=n$ and $|m_{a,b}- n_{a,b}| \\leq \\xi n$, there is a partition $\\mathcal{X} = \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[r] \\rbrace$ of $V(G)$ with $|X_{a,b}| = n_{a,b}+\\tau_{a,b}$ and $|X_{a,b} \\bigtriangleup V_{(\\phi^{2r}_\\ell)^{-1}(a,b)}| \\leq \\sqrt{{\\varepsilon}}m$ for all $(a,b)\\in[2\\ell]\\times[r]$ such that\n$G$ has a spanning $(\\phi^{2r}_\\ell(R),2\\ell,r,\\mathcal{X},{\\varepsilon}^{1\/3},\\delta\/2)$-cycle structure.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nNote that\n\\begin{equation}\\label{msize}\n2r\\ell m=n.\n\\end{equation}\nFor each $(i,j)\\in[\\ell]\\times[2r]$,\nchoose $A_{i,j} \\subseteq V_{i,j}$ satisfying\n\\begin{equation}\\label{Aij}\n|A_{i,j}|=\\tau_{\\phi^{2r}_\\ell(i,j)}\n\\end{equation}\nand let\n\\begin{equation}\\label{Yij}\nY_{i,j} := V_{i,j}\\setminus A_{i,j},\\quad\\text{so}\\quad (1-{\\varepsilon})m \\leq |Y_{i,j}| \\leq m.\n\\end{equation}\nLet $\\mathcal{Y} :=\\lbrace Y_0 \\rbrace \\cup \\lbrace Y_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ where\n$$\nY_0 := V(G)\\setminus \\bigcup_{(i,j)\\in[\\ell]\\times[2r]}Y_{i,j} = \\bigcup_{(i,j)\\in[\\ell]\\times[2r]}A_{i,j}.\n$$\n\n\nGiven a vertex $v \\in V(G)$ and $(i,j)\\in[\\ell]\\times[2r]$, we will say that $v \\rightarrow Y_{i,j}$ is \\emph{valid} if\n\\begin{itemize}\n\\item $j\\in[r]$ and $d_G(v,Y_{i,j'}) \\geq (\\delta-2{\\varepsilon})m$ for all $j' \\in [r]\\setminus\\lbrace j\\rbrace$; or\n\\item $j\\in[2r]\\setminus[r]$ and $d_G(v,Y_{i,j'}) \\geq (\\delta-2{\\varepsilon})m$ for all $j' \\in ([2r]\\setminus [r])\\setminus \\lbrace j\\rbrace$.\n\\end{itemize}\nThe first claim furnishes us with many pairs $(v,Y_{i',j'})$ such that $v \\in Y_{i,j}$ and $v \\rightarrow Y_{i',j'}$ is valid.\n\n\\begin{claim}\\label{goodvx}\nLet $i \\in [\\ell]$\nand suppose that\n$1 \\leq j \\leq r < t \\leq 2r$ or $1 \\leq t \\leq r < j \\leq 2r$.\nThen every vertex $v \\in Y_{i,j}$ is such that $v\\rightarrow Y_{i,j}, Y_{i,t}$ is valid, and at least $(1-\\sqrt{{\\varepsilon}})m$ are such that $v \\rightarrow Y_{i+1,j}, Y_{i+1,t}$ are also valid.\n(Here e.g. $Y_{\\ell+1,j}:=Y_{1,j}$.)\n\\end{claim}\n\n\\begin{claimproof}\nLet $t,j$ be as in the statement. Since, by~$(\\mathscr{C}3)$, $G[V_{i,j},V_{i,j'}]$ is $({\\varepsilon},\\delta)$-superregular for all $j' \\in [2r]\\setminus \\lbrace j \\rbrace$, we have that every vertex $v \\in V_{i,j}$ has at least $\\delta|V_{i,j'}|$ neighbours in $V_{i,j'}$.\nThus every vertex $v \\in Y_{i,j} \\subseteq V_{i,j}$ has at least $\\delta m-{\\varepsilon} m \\geq (\\delta-2{\\varepsilon})m$ neighbours in $Y_{i,j'}$.\nIn particular, $v \\rightarrow V_{i,j},V_{i,t}$ is valid.\n\n\nFrom the definition of regularity, one can see the following. If $G[A,B]$ is an $({\\varepsilon},\\delta)$-regular graph, then there are less than ${\\varepsilon}|A|$ vertices with less than $(\\delta-{\\varepsilon})|B|$ neighbours in $B$.\nThus, if $S_{i,j}$ is a subset of $N_{i,j} := \\lbrace (i',j')\\in V(R) : G[V_{i,j},V_{i',j'}] \\text{ is } ({\\varepsilon},\\delta)\\text{-regular}\\rbrace$, we see that there are at least $(1-{\\varepsilon}|S_{i,j}|)|V_{i,j}|$ vertices in $V_{i,j}$ with at least $(\\delta-{\\varepsilon})m $ neighbours in $V_{i',j'}$ for all $(i',j')\\in S_{i,j}$, and hence at least $(\\delta-2{\\varepsilon})m$ neighbours in $Y_{i',j'}$.\n\nRecall that, since $Z^{2r}_\\ell \\subseteq R$ by~$(\\mathscr{C}2)$, we have that $N_{i,j} \\supseteq \\lbrace (i,j'), (i+1,j') : j' \\in [2r]\\setminus\\lbrace j \\rbrace \\rbrace$.\nThus the second assertion of the claim follows by taking $S_{i,j} := \\lbrace (i+1,j') : j'\\in[2r]\\setminus\\lbrace j,t\\rbrace\\rbrace$ and using the fact that $(1-|S_{i,j}|{\\varepsilon})|V_{i,j}|-|A_{i,j}| \\geq (1-(2r-2){\\varepsilon})m - {\\varepsilon} m \\geq (1-\\sqrt{{\\varepsilon}})m$. \n\\end{claimproof}\n\n\n\n\\medskip\n\\noindent\nNext we prove the following claim, which will give us a `balanced' partition.\n\n\\begin{claim}\\label{Uclaim}\n$V(G)$ has a partition $\\lbrace Y_0 \\rbrace \\cup \\lbrace U_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ such that the following hold for all $i \\in [\\ell]$:\n\\begin{itemize}\n\\item[($\\mathscr{U}$1)] $||U_{i,j}|-|U_{i,j'}|| \\leq 1$ for all $(j,j') \\in [[2r]]^2$;\n\\item[($\\mathscr{U}$2)] $|Y_{i,j}\\bigtriangleup U_{i,j}| \\leq r{\\varepsilon} m$ for all $j\\in [2r]$;\n\\item[($\\mathscr{U}$3)] if $j \\in [r]$ then $U_{i,j}\\setminus Y_{i,j} \\subseteq \\bigcup_{k \\in [2r]\\setminus[r]}Y_{i,k}$ and if $k \\in [2r]\\setminus[r]$ then $U_{i,k}\\subseteq Y_{i,k}$.\n\\end{itemize}\n\\end{claim}\n\n\n\\begin{claimproof}\nFix an $i \\in [\\ell]$, and, to simplify notation, let $A_j := Y_{i,j}$, $a_j := |A_j|$, $B_j := Y_{i,r+j}$ and $b_j := |B_j|$ for all $j \\in [r]$.\nSuppose without loss of generality that $a_1 \\geq \\ldots \\geq a_r$ and $b_1 \\geq \\ldots \\geq b_r$.\nLet\n\\begin{equation}\\label{S}\nS := \\max\\left\\lbrace \\sum_{j \\in [r]}(a_1-a_j), \\sum_{j\\in[r]}(b_j-b_r)\\right\\rbrace \\stackrel{(\\ref{Yij})}{\\leq} r{\\varepsilon} m.\n\\end{equation}\n\nNow let $A_j(0):=A_j$ and $B_j(0) := B_j$, and $a_j(0) := |A_j(0)|$ and $b_j(0) := |B_j(0)|$ for all $j \\in [r]$.\nDo the following for each $0 \\leq s < S$.\nFix $t^-,t^+ \\in [r]$ such that $a_{t^-}(s) \\leq a_j(s)$ and $b_{t^+}(s) \\geq b_j(s)$ for all $j \\in [r]$. Choose $x \\in B_{t^+} \\cap B_{t^+}(s)$ and let\n$$\nA_{j}(s+1) := \\begin{cases} A_{j}(s)\\cup\\lbrace x \\rbrace &\\mbox{if } j=t^- \\\\\nA_{j}(s) & \\mbox{if } j\\in[r]\\setminus\\lbrace t^- \\rbrace; \\end{cases}\n$$\n$$\nB_{j}(s+1) := \\begin{cases} B_{j}(s)\\setminus\\lbrace x \\rbrace &\\mbox{if } j=t^+ \\\\\nB_{j}(s) & \\mbox{if } j\\in[r]\\setminus\\lbrace t^+ \\rbrace. \\end{cases}\n$$\nLet $a_j(s+1):= |A_j(s+1)|$ and $b_j(s+1) := |B_j(s+1)|$ for all $j\\in[r]$.\nThe following properties are clear:\n\\begin{itemize}\n\\item[(i)] for all $0 \\leq s < S$ and $j \\in [r]$ we have $A_j(s) \\supseteq A_j$ and $A_j(s)\\setminus A_j \\subseteq \\bigcup_{k\\in [r]}B_k$, and $B_j(s) \\subseteq B_j$. Furthermore, for all $j \\in [r]$ we have $\\sum_{j\\in[r]}|A_j(s)\\setminus A_j| = \\sum_{k\\in[r]}|B_k\\setminus B_k(s)| = s$;\n\\item[(ii)] letting $s_1 := \\sum_{j\\in[r]}(a_1-a_j)$, we have that $a_1(s_1)=\\ldots = a_r(s_1)=a_1$; and for each $s>s_1$ we have $|a_j(s)-a_{j'}(s)|\\leq 1$;\n\\item[(iii)] letting $s_2 := \\sum_{j \\in [r]}(b_j-b_r)$, we have that $b_1(s_2)=\\ldots = b_r(s_2)=b_r$; and for each $s>s_2$ we have $|b_j(s)-b_{j'}(s)|\\leq 1$.\n\\end{itemize}\nNow let $U_{i,j} := A_j(S)$ if $j\\in[r]$ and $U_{i,j} := B_{j-r}(S)$ if $j\\in[2r]\\setminus[r]$.\nFor ($\\mathscr{U}$1), the fact that $S=\\max\\lbrace s_1,s_2\\rbrace$ together with~(ii) and~(iii) implies that $|a_j(S)-a_{j'}(S)| \\leq 1$ and $|b_j(S)-b_{j'}(S)| \\leq 1$ for all $j,j'\\in[r]$. So~$(\\mathscr{U}1)$ holds.\nFor~($\\mathscr{U}$2), we have by~(i) that\n\\begin{align*}\n|U_{i,j} \\bigtriangleup Y_{i,j}| &= |U_{i,j}\\setminus Y_{i,j}| = |A_j(S)\\setminus A_j| \\leq S \\stackrel{(\\ref{S})}{\\leq} r{\\varepsilon} m\\quad&&\\text{if }j \\in [r],\\quad\\text{and}\\\\\n|U_{i,j} \\bigtriangleup Y_{i,j}| &= |Y_{i,j}\\setminus U_{i,j}| = |B_{j-r}\\setminus B_{j-r}(S)| \\leq S \\stackrel{(\\ref{S})}{\\leq} r{\\varepsilon} m\\quad&&\\text{if }j \\in [2r]\\setminus[r].\n\\end{align*}\nFinally, ($\\mathscr{U}$3) follows immediately from~(i).\n\\end{claimproof}\n\n\n\\medskip\n\\noindent\nThe next claim shows that we can modify $\\lbrace U_{i,j}\\rbrace$ further to obtain a new partition with clusters of given sizes (each of which does not differ much from $|U_{i,j}|$).\n\n\n\\begin{claim}\\label{finalclaim}\nLet $\\lbrace Y_0 \\rbrace \\cup \\lbrace U_{i,j}:(i,j)\\in[\\ell]\\times[2r]\\rbrace$ be any partition of $V(G)$ satisfying $(\\mathscr{U}1)$--$(\\mathscr{U}3)$.\nLet $\\lbrace n'_{i,j} : (i,j) \\in [\\ell]\\times[2r]\\rbrace $ be such that $\\sum_{(i,j)\\in[\\ell]\\times[2r]}n'_{i,j}= \\sum_{(i,j)\\in[\\ell]\\times[2r]}|U_{i,j}| $ and $||U_{i,j}| - n'_{i,j}| \\leq \\xi n$ for all $(i,j)\\in[\\ell]\\times[2r]$. Then\n$V(G)$ has a partition $\\lbrace Y_0 \\rbrace \\cup \\lbrace W_{i,j} : (i,j) \\in [\\ell]\\times[2r]\\rbrace$ such that the following hold for all $(i,j)\\in[\\ell]\\times[2r]$:\n\\begin{itemize}\n\\item[($\\mathscr{W}$1)] $|W_{i,j}| = n'_{i,j}$;\n\\item[($\\mathscr{W}$2)] $|W_{i,j}\\bigtriangleup U_{i,j}| \\leq {\\varepsilon} m$;\n\\item[($\\mathscr{W}$3)] for every $v \\in W_{i,j}$ we have that $v \\rightarrow Y_{i,j}$ is valid. \n\\end{itemize}\n\\end{claim}\n\n\\begin{claimproof}\nLet\n\\begin{equation}\\label{Keq}\nK := 2r\\ell\\xi n \\stackrel{(\\ref{msize})}{=} 4r^2\\ell^2\\xi m \\leq \\frac{{\\varepsilon} m}{2}.\n\\end{equation}\nSuppose, for some $0 \\leq k < K\/2$, we have found for each $(i,j) \\in [\\ell]\\times[2r]$ subsets $U^k_{i,j} \\subseteq V(G)$ such that the following hold:\n\\begin{itemize}\n\\item[$\\mathscr{A}_1(k)$] $\\lbrace Y_0\\rbrace \\cup \\lbrace U^k_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ is a partition of $V(G)$;\n\\item[$\\mathscr{A}_2(k)$] for all $v \\in U^k_{i,j}$ we have that $v \\rightarrow Y_{i,j}$ is valid;\n\\item[$\\mathscr{A}_3(k)$] for all $(i,j)\\in[\\ell]\\times[2r]$ we have $|U^k_{i,j}\\bigtriangleup U_{i,j}| \\leq 2k$;\n\\item[$\\mathscr{A}_4(k)$] $\\sum_{(i,j)\\in[\\ell]\\times[2r]}||U^k_{i,j}|-n'_{i,j}| \\leq 2(r\\ell \\xi n-k)$.\n\\end{itemize}\n\nWe claim that we can set $U^0_{i,j}:=U_{i,j}$ for all $(i,j)\\in[\\ell]\\times[2r]$.\nIndeed, $\\mathscr{A}_1(0)$ holds by Claim~\\ref{Uclaim}.\nFor $\\mathscr{A}_2(0)$, let $(i,j)\\in[\\ell]\\times[2r]$ and let $v \\in U_{i,j}$. If $v \\in Y_{i,j}$, then $v \\rightarrow Y_{i,j}$ is valid by~Claim~\\ref{goodvx}. Otherwise, $v \\in U_{i,j}\\setminus Y_{i,j}$.\nNote that by ($\\mathscr{U}$3) this implies $j \\in [r]$ and further\n $v \\in \\bigcup_{k \\in [2r]\\setminus[r]}Y_{i,k}$. So $v\\rightarrow Y_{i,j}$ is valid by Claim~\\ref{goodvx}.\nProperty $\\mathscr{A}_3(0)$ vacuously holds and $\\mathscr{A}_4(0)$ holds since $||U_{i,j}|-n'_{i,j}| \\leq \\xi n$ for all $(i,j)\\in[\\ell]\\times[2r]$.\n\nIf $|U^k_{i,j}|=n'_{i,j}$ for all $(i,j)\\in[\\ell]\\times[2r]$, then we stop.\nOtherwise, we will obtain sets $U^{k+1}_{i,j}$ from $U^k_{i,j}$.\nThere must exist $(i^-,j^-),(i^+,j^+) \\in [\\ell]\\times[2r]$ for which $|U^k_{i^-,j^-}| \\leq n'_{i^-,j^-}-1$ and $|U^k_{i^+,j^+}| \\geq n'_{i^+,j^+}+1$.\n\nWe will say that $(i_1,j_1)\\rightarrow (i_2,j_2) \\rightarrow \\ldots \\rightarrow (i_s,j_s)$ is a \\emph{good chain (of length $s$)} if for all $p \\in [s-1]$ there exist at least $(1-\\sqrt{{\\varepsilon}})m$ vertices $v \\in Y_{i_p,j_p}$ such that $v\\rightarrow Y_{i_{p+1},j_{p+1}}$ is valid.\nClaim~\\ref{goodvx} implies that the following are good chains of length 3 (where here and for the remainder of the proof of Claim~\\ref{finalclaim} addition is modulo $\\ell$):\n\\begin{align*}\n&(i^+,j^+) \\rightarrow (i^+,j^-+r) \\rightarrow (i^++1,j^-) &&\\text{ if } j^+,j^- \\in [r]\\\\\n&(i^+,j^+) \\rightarrow (i^+,j^--r) \\rightarrow (i^++1,j^-) &&\\text{ if } j^+,j^- \\in [2r]\\setminus[r]\\\\\n&(i^+,j^+) \\rightarrow (i^+,j^-) \\rightarrow (i^++1,j^-) &&\\text{ otherwise,}\n\\end{align*}\nand further, in all cases and for all $t \\geq 0$, the chain $(i^++t,j^-)\\rightarrow (i^++t+1,j^-)$ of length 2 is good.\nTogether this implies that in all cases there is a good chain\n$$\n(i^+,j^+) =: (i_1,j_1)\\rightarrow \\ldots \\rightarrow (i_{S},j_{S}) := (i^-,j^-)\n$$\nof some length $S$, where we choose the shortest such chain. As a crude estimate, we have, say, $S \\leq 2\\ell$, and $(i_s,j_s)\\neq(i_{s'},j_{s'})$ for any distinct $s,s' \\in [S]$ (or we could find a shorter chain).\n\nWe will exchange vertices between successive clusters according to this chain.\nFor each $s \\in [S]$, there are by definition at least $(1-\\sqrt{{\\varepsilon}}) m$ vertices $v \\in Y_{i_s,j_s}$ such that $v \\rightarrow Y_{i_{s+1},j_{s+1}}$ is valid.\nThe number of these vertices which additionally lie in $U^k_{i_s,j_s}$ is by $(\\mathscr{U}2)$,~$\\mathscr{A}_3(0)$ and~(\\ref{Keq}) at least\n$\n(1-\\sqrt{{\\varepsilon}})m - 2k-r{\\varepsilon} m  >  m\/2$.\nSo we can find $x_s \\in U^k_{i_s,j_s}$ such that $x_s \\rightarrow Y_{i_{s+1},j_{s+1}}$ is valid.\nFor each $(i,j)\\in[\\ell]\\times[2r]$, set\n$$\nU^{k+1}_{i,j} = \\begin{cases} U^k_{i,j}\\setminus \\lbrace x_1\\rbrace &\\mbox{if } (i,j)=(i_1,j_1) \\\\\nU^k_{i,j} \\cup \\lbrace x_{s-1}\\rbrace \\setminus \\lbrace x_s\\rbrace & \\mbox{if } (i,j)=(i_s,j_s) \\text{ for some }2 \\leq s < S\\\\\nU^k_{i,j} \\cup \\lbrace x_{S-1}\\rbrace & \\mbox{if } (i,j)=(i_S,j_S)\\\\\nU^k_{i,j} & \\mbox{otherwise.}\\end{cases} \n$$\nProperty~$\\mathscr{A}_1(k+1)$ holds by~$\\mathscr{A}_1(k)$, the definition of $U^{k+1}_{i,j}$ and the fact that each pair in the chain is distinct.\nProperty~$\\mathscr{A}_2(k)$ and the choice of $x_s$ imply that $\\mathscr{A}_2(k+1)$ holds.\nWe have\n$$\n|U^{k+1}_{i,j}\\bigtriangleup Y_{i,j}| \\leq |U^{k+1}_{i,j}\\bigtriangleup U^k_{i,j}| + |U^k_{i,j}\\bigtriangleup Y_{i,j}| \\stackrel{\\mathscr{A}_3(k)}{\\leq} 2(k+1),\n$$\nproving~$\\mathscr{A}_3(k+1)$ (note here we are again using the fact that each pair in our chain is distinct).\nFinally, observe that $||U^{k+1}_{i^\\pm,j^\\pm}|-n'_{i^\\pm,j^\\pm}| = ||U^{k}_{i^\\pm,j^\\pm}|-n'_{i^\\pm,j^\\pm}| -1$ and $|U^{k+1}_{i,j}|=|U^k_{i,j}|$ for all other pairs $(i,j)$.\nTherefore\n$$\n\\sum_{(i,j)\\in[\\ell]\\times[2r]} ||U^{k+1}_{i,j}|-n'_{i,j}| = \\sum_{(i,j)\\in[\\ell]\\times[2r]}||U^k_{i,j}|-n'_{i,j}| - 2 \\stackrel{\\mathscr{A}_4(k)}{\\leq} 2(r\\ell\\xi n - (k + 1)),\n$$\nproving~$\\mathscr{A}_4(k+1)$.\nSo, for each $0 \\leq k \\leq K\/2$, either the procedure has terminated, or we are able to proceed to step $k+1$.\nTherefore there is some $p \\leq K\/2$ such that $\\sum_{(i,j)\\in[\\ell]\\times[2r]}||U^p_{i,j}|-n'_{i,j}|=0$.\nNote that, by $\\mathscr{A}_3(p)$, we have\n$$\n|U^p_{i,j} \\bigtriangleup U_{i,j}| \\leq 2p \\leq K \\stackrel{(\\ref{Keq})}{\\leq} {\\varepsilon} m\n$$\nfor all $(i,j)\\in[\\ell]\\times[2r]$.\nThus setting $W_{i,j} := U^p_{i,j}$ yields the required partition.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nApply Claim~\\ref{Uclaim} to obtain $\\lbrace U_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ satisfying~$(\\mathscr{U}1)$--$(\\mathscr{U}3)$.\n\nLet $\\phi := \\phi^{2r}_\\ell$ as in Definition~\\ref{bijdef}. \nLet\n$$\nU'_{a,b} := U_{\\phi^{-1}(a,b)}\\quad\\text{and}\\quad m_{a,b} := |U'_{a,b}|\\quad\\text{for all}\\quad(a,b) \\in [2\\ell]\\times[r].\n$$\nWe claim that $\\lbrace m_{a,b}\\rbrace$ satisfies~$(\\mathscr{L}1)$.\nIndeed,~$(\\mathscr{U}1)$ implies that $|m_{a,b}-m_{a,b'}|\\leq 1$ for all $a \\in [2\\ell]$ and $b,b'\\in[r]$, and further, writing $\\phi^{-1}(a,b) =: (i,j)$,\n$$\nm_{a,b} = |U_{i,j}| \\stackrel{(\\mathscr{U}2)}{\\geq} |Y_{i,j}|-r{\\varepsilon} m \\stackrel{(\\ref{Aij})}{\\geq} |V_{i,j}| - (r+1){\\varepsilon} m = (1-(r+1){\\varepsilon})m \\geq (1-\\sqrt{{\\varepsilon}})m.\n$$\nFinally,\n$$\n\\sum_{(a,b)\\in[2\\ell]\\times[r]} m_{a,b} = \\sum_{(i,j)\\in[\\ell]\\times[2r]}|U_{i,j}| = n-|Y_0| = n-\\sum_{(a,b)\\in[2\\ell]\\times[r]}\\tau_{a,b},\n$$\nso~$(\\mathscr{L}1)$ holds.\n\nNow let $\\lbrace n_{a,b} \\in \\mathbb N: (a,b)\\in[2\\ell]\\times[r]\\rbrace$ satisfy $\\sum_{(a,b)\\in[2\\ell]\\times[r]}(n_{a,b}+\\tau_{a,b}) =n$ and $|m_{a,b}- n_{a,b}| \\leq\\xi n$.\nLet\n\\begin{equation}\\label{neq}\nn_{i,j}' := n_{\\phi(i,j)}\\quad\\text{for all }(i,j)\\in[\\ell]\\times[2r].\n\\end{equation}\nApply Claim~\\ref{finalclaim} with input partition $\\lbrace Y_0\\rbrace \\cup \\lbrace U_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ and input sizes $\\lbrace n'_{i,j} : (i,j)\\in[\\ell]\\times[2r] \\rbrace$ to obtain a partition $\\lbrace Y_0 \\rbrace \\cup \\lbrace W_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ satisfying $(\\mathscr{W}1)$--$(\\mathscr{W}3)$. Let\n\\begin{equation}\\label{Weq}\nX_{a,b} := W_{\\phi^{-1}(a,b)} \\cup A_{\\phi^{-1}(a,b)} \\quad\\text{for all }(a,b)\\in[2\\ell]\\times[r].\n\\end{equation}\nWe claim that $\\mathcal{X} := \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[r]\\rbrace$ is the required partition for~$(\\mathscr{L}2)$.\nFor all $(a,b)\\in[2\\ell]\\times[r]$ we have\n$$\n|X_{a,b}| \\stackrel{(\\ref{Weq})}{=} |W_{\\phi^{-1}(a,b)}|+|A_{\\phi^{-1}(a,b)}| \\stackrel{(\\ref{Aij}),(\\mathscr{W}1)}{=} n'_{\\phi^{-1}(a,b)}+\\tau_{a,b} \\stackrel{(\\ref{neq})}{=} n_{a,b}+\\tau_{a,b},\n$$\nas required. Also, writing $(i,j):= \\phi^{-1}(a,b) \\in [\\ell]\\times[2r]$ and recalling that $A_{i,j}\\subseteq V_{i,j}$, we have\n\\begin{eqnarray}\n\\label{Xtriangle}|X_{a,b} \\bigtriangleup V_{\\phi^{-1}(a,b)}| &\\stackrel{(\\ref{Weq})}{=}& |W_{i,j}\\bigtriangleup V_{i,j}|\n\\leq |W_{i,j}\\bigtriangleup U_{i,j}| + |U_{i,j}\\bigtriangleup V_{i,j}|\\\\\n\\nonumber &\\stackrel{(\\mathscr{U}2),(\\mathscr{W}2)}{\\leq}& 2 r{\\varepsilon} m \\leq 3r{\\varepsilon}|X_{a,b}| \\leq \\sqrt{{\\varepsilon}}m.\n\\end{eqnarray}\n\nLastly, we need to check that $\\mathcal{X}$\ninduces a $(\\phi(R),2\\ell,r,\\mathcal{X},{\\varepsilon}^{1\/3},\\delta\/2)$-cycle structure.\nThat is, we need to check that $(\\mathscr{C}1)$--$(\\mathscr{C}3)$ hold.\nProperty~$(\\mathscr{W}1)$ implies that $\\mathcal{X} = \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[r]\\rbrace = \\lbrace W_{i,j}\\cup A_{i,j} : (i,j)\\in[\\ell]\\times[2r]\\rbrace$ is a partition of $V(G)$.\nSo $(\\mathscr{C}1)$ holds.\nNow, by~(\\ref{CZ}) and Definition~\\ref{bijdef}, we see that $\\phi(R)$ has vertex set $[2\\ell]\\times[r]$ and, since $Z^{2r}_\\ell \\subseteq R$, we have $Z^r_{2\\ell} \\subseteq \\phi(R)$ (with the correct labelling).\nLet $(a,b),(a',b')\\in E(\\phi(R))$ and write $(i,j):=\\phi^{-1}(a,b)$ and $(i',j')=\\phi^{-1}(a',b')$.\nThen $(i,j)(i',j')\\in E(R)$, so $G[V_{i,j},V_{i',j'}]$ is $({\\varepsilon},\\delta)$-regular by~$(\\mathscr{C}2)$ for $\\mathcal{V}$.\nThen~(\\ref{Xtriangle}) implies that we can apply Proposition~\\ref{newsuperslice} with $\\alpha := 3r{\\varepsilon}$ and ${\\varepsilon} ' := {\\varepsilon}^{1\/3} \\geq {\\varepsilon} + 6\\sqrt{\\alpha}$ to see that $G[X_{a,b},X_{a',b'}]$ is $({\\varepsilon}^{1\/3},\\delta\/2)$-regular.\nSo~$(\\mathscr{C}2)$ holds.\n\nFor~($\\mathscr{C}3)$, fix $a \\in [2\\ell]$ and let $b,b' \\in [r]$ be distinct.\nLet $(i,j):=\\phi^{-1}(a,b)$.\nDefinition~\\ref{bijdef} implies that there exists $j'$ such that $(j,j')\\in[[2r]]^2$ and $\\phi^{-1}(a,b')=(i,j')$.\nLet $x \\in X_{a,b}\\setminus V_{\\phi^{-1}(a,b)}  = W_{i,j}\\setminus Y_{i,j}$.\nThen~$(\\mathscr{W}3)$ implies that $x \\rightarrow Y_{i,j}$ is valid.\nSince $Y_{i,j}\\subseteq V_{i,j}$, this means that $d_G(x,V_{i,j^*}) \\geq (\\delta-2{\\varepsilon})m$ for all $j^*$ such that $(j,j^*)\\in[[2r]]^2$.\nSo $d_G(x,V_{\\phi^{-1}(a,b')}) \\geq (\\delta-2{\\varepsilon})m$, and hence  (\\ref{Xtriangle}) implies $d_G(x,X_{a,b'}) \\geq (\\delta-2{\\varepsilon})m - 2r{\\varepsilon} m \\geq \\delta|X_{a,b'}|\/2$.\nSimilarly, every $y \\in X_{a,b'} \\setminus V_{\\phi^{-1}(a,b')}$ satisfies $d_G(y,X_{a,b}) \\geq \\delta|X_{a,b}|\/2$.\nMoreover,~($\\mathscr{C}3)$ for $\\mathcal V$  and (\\ref{Xtriangle}) implies that $d_G(x,X_{a,b'})  \\geq \\delta|X_{a,b'}|\/2$ for every $x \\in V_{\\phi^{-1}(a,b)}$ and  $d_G(y,X_{a,b}) \\geq \\delta|X_{a,b}|\/2$ for every $y \\in V_{\\phi^{-1}(a,b')}$.\nSo Proposition~\\ref{newsuperslice} applied with $\\alpha := 3r{\\varepsilon}$ and ${\\varepsilon}' := {\\varepsilon}^{1\/3}$ implies that $G[X_{a,b},X_{a,b'}]$ is $({\\varepsilon}^{1\/3},\\delta\/2)$-superregular.\nSo~$(\\mathscr{C}3)$ holds.\nThis completes the proof of~$(\\mathscr{L}2)$ and hence of the lemma.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The proof of Theorem~\\ref{main}}\\label{sec8}\nFirst note that it suffices to prove the theorem under the additional assumption that $\\eta \\ll d,1\/\\Delta $.\nLet $n_0,\\beta,\\rho,{\\varepsilon},c,\\delta,\\rho ', L'>0$ satisfy\n\\begin{equation}\\label{hierarchy}\n0 < 1\/n_0 \\ll \\beta \\ll 1\/L'\\ll \\rho \\ll {\\varepsilon} \\ll c \\ll \\delta \\ll \\rho '  \\ll \\eta \\ll d,1\/\\Delta.\n\\end{equation}\n\\COMMENT{AT: delete all less than $1\/3$ as we want to be dealing with $r=2$ too!}\n\n\n\nLet $G$ be a $(\\rho,d)$-dense graph on $n \\geq n_0$ vertices with $\\delta(G) \\geq (1\/2+\\eta)n$.\nLet $H$ be a graph on $n$ vertices with  $\\Delta(H) \\leq \\Delta$ and bandwidth at most $\\beta n$.\nWrite $r:=\\chi (H)$; so as $\\eta \\ll 1\/\\Delta$, certainly $\\eta \\ll 1\/r$.\n\nApply  the Regularity lemma (Lemma~\\ref{reg}) with parameters ${\\varepsilon},(4r+1)L'$ to obtain $L^* \\in \\mathbb{N}$.\nWe may assume that $\\beta \\ll 1\/L^*$.\n\n\\COMMENT{AT: go back and check that when I talk about $r^*$, $4r$ etc earlier on, we use the correct term}\n\\begin{claim}\\label{claim1}\nThere exists $L' \\leq \\ell \\leq L^*$, a partition $\\mathcal{V} = \\lbrace V_0\\rbrace \\cup \\lbrace V_{i,j} : (i,j) \\in [\\ell]\\times[4r]\\rbrace$ of $V(G)$ with $|V_{i,j}|=:m$ for all $(i,j)\\in[\\ell]\\times[4r]$, a graph $R$ on vertex set $[\\ell]\\times[4r]$ and a spanning subgraph $G'$ of $G$ such that\n\\begin{itemize}\n\\item[(i)] $R$ is $(\\rho ',d)$-dense;\n\\item[(ii)] $\\delta(R) \\geq (1\/2+\\eta\/3)|R|$;\n\\item[(iii)] $G'$ has an $(R,\\ell,4r,\\mathcal{V},7{\\varepsilon}^{1\/4},\\delta\/2)$-cycle structure $\\mathcal{C}$ and $|V_0| \\leq 2{\\varepsilon}^{1\/2}n$;\n\\item[(iv)] $R[\\lbrace(1,1),\\ldots,(1,4r)\\rbrace] \\cong K_{4r}$ and $\\lbrace (1,1),\\ldots,(1,4r)\\rbrace$ lies in a copy of $K_{324r\/\\eta ^2}$ in $R$.\n\\end{itemize}\n\\end{claim}\n\n\\begin{claimproof}\nApply Lemma~\\ref{reg} to $G$ with parameters ${\\varepsilon},\\delta,(4r+1)L'$ to obtain clusters $V_1,\\ldots,V_{L}$ of size $m'$, an exceptional set $V_0'$, a pure graph $G'$ and a reduced graph $R'$. So\n\\begin{equation}\\label{L''m'}\nLm' \\leq n \\leq Lm' + {\\varepsilon} n,\n\\end{equation}\nand $|R'|=L$ where\n\\begin{equation}\\label{Ldef}\n(4r+1)L' \\leq L \\leq L^*\n\\end{equation}\nand $|V_0'| \\leq {\\varepsilon} n$,\n\\begin{equation}\\label{G'}\n\\delta(G') \\geq (1\/2+\\eta)n - (\\delta+{\\varepsilon})n \\geq (1\/2+\\eta\/2)n\n\\end{equation}\nand $G'[V_i,V_j]$ is $({\\varepsilon},\\delta)$-regular whenever $ij \\in E(R')$.\nLemma~\\ref{inherit} implies that $R'$ is $(3\\delta,d)$-dense and $\\delta(R') \\geq (1\/2+\\eta\/2)L$.\n\nLet $r^* := 324r\/\\eta ^2$.\nApply Theorem~\\ref{rcycle} with $R',L,r^*-1,4r,3\\delta,d,\\eta \/2$ playing the roles of $G,n,r,s,\\rho,d,\\eta$ to obtain \nan $(r^*-1)$-cycle $C \\cong C^{r^*-1}_{4r\\ell} \\subseteq R'$ of order $4r \\ell$ where\n\\begin{equation}\\label{leq}\n(1-{\\varepsilon}) L \\leq L -4r \\leq 4r\\ell \\leq L.\n\\end{equation}\nRelabel those clusters of $R'$ corresponding to vertices of $C$\nso that they are now $\\lbrace V_{i,j}' : (i,j) \\in [\\ell]\\times[4r]\\rbrace$, and\n\\begin{equation}\\label{ordering}\n(1,1)(1,2)\\ldots (1,4r)(2,1)\\ldots (2,4r)\\ldots (\\ell,1)\\ldots (\\ell,4r) = C^{r^*-1}_{4r\\ell} {\\supseteq} Z^{4r}_{\\ell}.\n\\end{equation}\n\\COMMENT{AT: this last inclusion doesn't follow from (1) -rather the definitions - so have removed the (1) here}\nLet $R := R'[V(C)]$. So $V(R)=[\\ell]\\times[4r]$.\nObserve that $\\lbrace (1,1),\\ldots,(1,4r)\\rbrace$ lies in a copy of $K_{r^*}$ in $R$.\nFor all $i \\in [\\ell]$ let\n\\begin{equation*}\\label{Ti}\nT(i) := \\textstyle{R\\left[\\bigcup_{j \\in [4r]}(i,j)\\right]} \\stackrel{(\\ref{ordering})}{\\cong} K_{4r}.\n\\end{equation*}\nApply Lemma~\\ref{superslice} with $G'[\\bigcup_{j \\in [4r]}V'_{i,j}],T(i),4r-1,4r,V'_{i,1},\\ldots,V'_{i,4r},m',{\\varepsilon},\\delta$ playing the roles of $G,R, \\Delta,$ $L,V_1,\\ldots,V_L,m,{\\varepsilon},d$ to obtain for each $j \\in [4r]$ a subset $V_{i,j} \\subseteq V'_{i,j}$ of size\n\\begin{equation}\\label{m}\n|V_{i,j}| = m := (1-\\sqrt{{\\varepsilon}})m'\n\\end{equation}\nsuch that for every distinct $j,j'\\in[4r]$ the graph $G'[V_{i,j},V_{i,j'}]$ is $(4\\sqrt{{\\varepsilon}},\\delta\/2)$-superregular.\nLet $V_0 := V(G)\\setminus \\bigcup_{(i,j)\\in[\\ell]\\times[4r]}V_{i,j}$ and\n\\begin{equation}\\label{Vpart}\n\\mathcal{V} := \\lbrace V_0 \\rbrace \\cup \\lbrace V_{i,j} : (i,j)\\in[\\ell]\\times[4r]\\rbrace.\n\\end{equation}\nWe have\n\\begin{align}\\label{4rell}\nn &\\geq 4r\\ell m \\stackrel{(\\ref{m})}{=} 4r\\ell(1-\\sqrt{{\\varepsilon}})m' \\stackrel{(\\ref{leq})}{\\geq} (1-\\sqrt{{\\varepsilon}})(1-{\\varepsilon})Lm' \\stackrel{(\\ref{L''m'})}{\\geq} (1-\\sqrt{{\\varepsilon}})(1-{\\varepsilon})^2 n\\\\\n\\nonumber&\\geq (1-2{\\varepsilon}^{1\/2})n. \n\\end{align}\nSince we will often compare $m$ and $\\beta n$ in calculations, let us note here that\n\\begin{equation}\\label{beta}\n\\beta n \\stackrel{(\\ref{L''m'})}{\\leq} \\frac{\\beta Lm'}{1-{\\varepsilon}} \\stackrel{(\\ref{m})}{=} \\frac{\\beta L m}{(1-\\sqrt{{\\varepsilon}})(1-{\\varepsilon})} \\stackrel{(\\ref{hierarchy}),(\\ref{Ldef})}{\\leq} 2\\beta L^* \\cdot m \\leq \\frac{{\\varepsilon}^2 m}{L^*}.\n\\end{equation}\n\nWe will now show that $\\ell$, $R$ and $\\mathcal{V}$ satisfy~Claim~\\ref{claim1}(i)--(iv).\nWe have that\n\\begin{equation*}\\label{eqL}\n4rL' \\stackrel{(\\ref{hierarchy})}{\\leq} (1-{\\varepsilon})(4r+1)L'  \\stackrel{(\\ref{Ldef})}{\\leq} (1-{\\varepsilon})L \\stackrel{(\\ref{leq})}{\\leq} 4r\\ell \\leq L \\stackrel{(\\ref{Ldef})}{\\leq} L^*.\n\\end{equation*}\nSo $L' \\leq \\ell \\leq L^*$, as required.\nNote that (i) follows from \nLemma~\\ref{nbrhd}(i) since $\\rho ' \\gg \\delta$.\nFurther, $\\delta(R) \\geq \\delta(R')-4r \\geq (1\/2+\\eta\/3)L$ so~(ii) holds.\n\nFor~(iii), we need to show that $\\mathcal{V}$ (see~(\\ref{Vpart})) induces the required cycle structure $\\mathcal{C}$.\nThat is, we need to check that $(\\mathscr{C}1)$--$(\\mathscr{C}3)$ hold with the desired parameters.\nThe sets $V_{i,j}$ are pairwise-disjoint since the same is true for $V'_{i,j}$, so by the definition of $V_0$ we have that $\\mathcal{V}$ is a partition of $V(G')$. Moreover,\n\\begin{eqnarray*}\n|V_0| = n - 4r\\ell m &\\stackrel{(\\ref{4rell})}{\\leq}& 2{\\varepsilon}^{1\/2}n < 7{\\varepsilon}^{1\/4}n,\n\\end{eqnarray*}\nso~$(\\mathscr{C}1)$ holds.\nCertainly $V(R)=[\\ell]\\times[4r]$ and, by~(\\ref{ordering}), $R \\supseteq Z^{4r}_{\\ell}$. Let $(i,j)(i',j')\\in E(R)$.\nThen $(i,j)(i',j')$ has a corresponding edge in $R' \\supseteq R$, so $G'[V'_{i,j},V'_{i',j'}]$ is $({\\varepsilon},\\delta)$-regular.\nNote that ${\\varepsilon} + 6\\sqrt{{\\varepsilon}^{1\/2}} \\leq 7{\\varepsilon}^{1\/4}$ and $\\delta-4{\\varepsilon}^{1\/2} \\geq \\delta\/2$.\nThus Lemma~\\ref{newsuperslice} applied with $V'_{i,j},V_{i,j},V'_{i',j'},V_{i',j'},{\\varepsilon}^{1\/2}$ playing the roles of $A,A',B,B',\\alpha$ implies that $G'[V_{i,j},V_{i',j'}]$ is $(7{\\varepsilon}^{1\/4},\\delta\/2)$-regular, so~$(\\mathscr{C}2)$ holds.\nWe have already seen, for every $i \\in [\\ell]$ and distinct $j,j'\\in[4r]$, that $G'[V_{i,j},V_{i,j'}]$ is $(4\\sqrt{{\\varepsilon}},\\delta\/2)$-superregular. Thus it is $(7{\\varepsilon}^{1\/4},\\delta\/2)$-superregular. So~$(\\mathscr{C}3)$ holds.\nThus~(iii) holds. We saw when we defined $R$ that~(iv) holds.\nThis completes the proof of the claim.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nRecall the definition of the bijection $\\phi^{4r}_{\\ell} : [\\ell]\\times[4r]\\rightarrow[2\\ell]\\times[2r]$ given by\n$$\n\\phi^{4r}_{\\ell}(i,j)=\\left((2i-1)+\\left\\lfloor\\frac{j}{2r}\\right\\rfloor,j-\\left(\\left\\lceil \\frac{j}{2r}\\right\\rceil-1\\right) 2r\\right),\\quad\\text{for all }(i,j)\\in[\\ell]\\times[4r].\n$$\nRecall also that $\\phi^{4r}_\\ell(R)$ is the graph with vertex set $\\phi^{4r}_\\ell(V(R)) = [2\\ell]\\times[2r]$ and edge set $E(\\phi^{4r}_\\ell(R)) = \\lbrace \\phi^{4r}_\\ell(x)\\phi^{4r}_\\ell(y) : xy \\in E(R)\\rbrace$.\nFor ease of notation, we will write\n\\begin{eqnarray}\n\\label{phiprop} \\phi &:=& \\phi^{4r}_{\\ell},\\quad\\text{so}\\quad \\phi(1,b)=(1,b)\\text{ for all }b \\in [2r],\\quad\\text{and}\\\\\n\\nonumber R^* &:=& \\phi(R),\\quad\\text{so }R^* \\cong R,\\quad V(R^*)=[2\\ell]\\times[2r],\n\\end{eqnarray}\nand $V(G')$ has partition $\\mathcal{V} = \\lbrace V_0\\rbrace \\cup \\lbrace V_{\\phi^{-1}(a,b)} : (a,b)\\in[2\\ell]\\times[2r]\\rbrace$.\n\n\n\n\n\n\n\n\n\n\n\n\\begin{claim}\\label{claim2}\nThere exists a partition $\\mathcal{X} = \\lbrace V_0\\rbrace \\cup \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[2r]\\rbrace$ of $V(G')$ and a surjective\\COMMENT{To ensure all of $V_0$ covered} mapping $\\psi : V(H) \\rightarrow ([2\\ell]\\times[2r])\\cup V_0$ such that the following hold:\n\\begin{itemize}\n\\item[(i)]  $|\\psi^{-1}(a,b)|=|X_{a,b}| \\geq (1-{\\varepsilon}^{1\/19})m$ for all $(a,b)\\in[2\\ell]\\times[2r]$;\n\\item[(ii)] $G'$ has an $(R^*,2\\ell,2r,\\mathcal{X},{\\varepsilon}^{1\/27},\\delta\/4)$-cycle structure $\\mathcal{C}'$;\n\\item[(iii)] $I := \\psi^{-1}(V_0) $ is an independent set in $H$ of size $|V_0|$ and for all $w \\in W := \\bigcup_{x \\in I}N_H(x)$, there is a unique $u \\in I$ such that $uw \\in E(H)$, and $d_{G'}(\\psi(u),X_{\\psi(w)}) \\geq cm\/2$; \n\\item[(iv)] $\\psi|_{V(H)\\setminus I} : V(H\\setminus I) \\rightarrow V(R^*)$ is a graph homomorphism;\n\\item[(v)] there exists $X' \\subseteq V(H)\\setminus I$ with $W \\subseteq X'$ and $|\\psi^{-1}(a,b)\\cap X'| \\leq {\\varepsilon}^{1\/10}m$ for all $(a,b)\\in[2\\ell]\\times[2r]$ such that, whenever $uv \\in E(H)$ and $u,v \\notin X' \\cup I$, writing $\\psi(u)=:(a,b)$ and $\\psi(v)=:(a',b')$, we have $a=a'$ and $b \\neq b'$. Moreover, writing\n$$\nN := \\textstyle{\\left(\\bigcup_{x \\in X'}N_H(x)\\right)\\setminus (X' \\cup I)},\n$$\nwe have $|N| \\leq {\\varepsilon} m$.\n\\end{itemize}\n\\end{claim}\n\n\\begin{claimproof}\nFor all $v \\in V(G')$, write\n\\begin{equation}\\label{Nalpha}\nN_{R^*}^c(v) := \\lbrace (a,b)\\in[2\\ell]\\times[2r] : d_{G'}(v,V_{\\phi^{-1}(a,b)})\\geq c m\\rbrace\n\\end{equation}\nand $d_{R^*}^c(v) := |N_{R^*}^c(v)|$. Then\n$$\n(1\/2+\\eta) n - (\\delta+{\\varepsilon})n \\stackrel{(\\ref{G'})}{\\leq} d_{G'}(v) \\leq d_{R^*}^c(v)m + (4r\\ell-d_{R^*}^c(v))c m + |V_0|.\n$$\nClaim~\\ref{claim1}(iii)  implies that\n\\begin{equation*}\\label{meq2}\n4r\\ell m \\leq n \\leq 4r\\ell m + |V_0| \\leq 4r\\ell m+ 2{\\varepsilon}^{1\/2} n.\n\\end{equation*}\nThus\n\\begin{equation}\\label{dalpha}\nd_{R^*}^c(v) \\geq \\frac{(1\/2+\\eta-\\delta-{\\varepsilon}) n - 4r\\ell cm - |V_0|}{(1-c)m} \\stackrel{(\\ref{hierarchy})}{\\geq} \\frac{1\/2+ \\eta\/2}{1-c}\\cdot 4r\\ell \\geq \\frac{|R^*|}{2}.\n\\end{equation}\n\nWe would like to apply Lemma~\\ref{lemmaforH} (Special Lemma for $H$) to obtain an integer $s$, with $G',R^*,4r\\ell,$ $2r,\\eta\/3,2{\\varepsilon}^{1\/2},\\rho ',d,n,m,N_{R^*}^c(v),(1,i)$ playing the roles of $G,R,L,r,\\eta,{\\varepsilon},\\rho,d,n,m,N_v,b_i$.\nFor this, we need to check that~$(\\mathscr{G}1)$--$(\\mathscr{G}4)$ hold.\nFor~$(\\mathscr{G}1)$, we know that $G'$ has vertex partition $\\lbrace V_0 \\rbrace \\cup \\lbrace V_{i,j} : (i,j)\\in[\\ell]\\times[4r]\\rbrace = \\lbrace V_0\\rbrace\\cup \\lbrace V_{\\phi^{-1}(a,b)} : (a,b)\\in[2\\ell]\\times[2r]\\rbrace$, and $|V_0| \\leq {\\varepsilon}^{1\/2}n$ and $|V_p|=m$ for all $p \\in V(R^*)$.\nThat~$(\\mathscr{G}2)$ holds follows from~(\\ref{dalpha}).\nProperty~$(\\mathscr{G}3)$ follows from~Claim~\\ref{claim1}(i) and~(ii) and the fact that $R^* \\cong R$.\nFinally,~$(\\mathscr{G}4)$ follows from~(iv), noting that $324r\/\\eta ^2= 18 \\cdot (2r) \\cdot 1\/(\\eta\/3)^2$, and the fact that $\\phi(1,b)=(1,b)$ for all $b \\in [2r]$ from~(\\ref{phiprop}).\nTherefore we can apply Lemma~\\ref{lemmaforH} with the above parameters to obtain an integer\n\\begin{equation}\\label{seq}\ns \\leq (2{\\varepsilon}^{1\/2})^{1\/4}n \\leq {\\varepsilon}^{1\/9}n.\n\\end{equation}\n\nLet $\\chi : V(H)\\rightarrow[r]$ be a proper colouring of $H$, let $x_1,\\ldots,x_n$ be an ordering of $V(H)$ with bandwidth at most $\\beta n$, and\nlet\n\\begin{align}\n\\label{XYZdef} X &:= \\lbrace x_1,\\ldots,x_{s}\\rbrace,\\quad Y := \\lbrace x_{s+1},\\ldots,x_{s+\\beta n}\\rbrace \\subseteq Z := \\lbrace x_{s+ 1},\\ldots,x_n\\rbrace,\\\\\n\\label{H'def} H' &:= H[X \\cup Y]\\quad\\text{and}\\quad H'' := H[Z].\n\\end{align}\nApply Lemma~\\ref{lemmaforH} (Special Lemma for $H$) with the above parameters and with $s,\\beta,\\Delta,H',X,Y,\\chi$ playing the roles of $s,\\beta,\\Delta,H,X,Y,\\chi$ to obtain a mapping\n\\begin{equation*}\\label{ffinaldef}\nf:X \\cup Y\\rightarrow ([2\\ell]\\times[2r]) \\cup V_0\n\\end{equation*}\nwith the following properties:\n\\begin{itemize}\n\\item[$(\\mathscr{D}1)$]  \nsetting $I := f^{-1}(V_0)$, we have that $I$ is a subset of $X$ which is $2$-independent in $H'$, and each  vertex in $V_0$ is mapped onto from a unique vertex in $H$ (so $|I|=|V_0|$);\n\\item[$(\\mathscr{D}2)$] for all $v \\in V_0$, setting $W_v := N_H(f^{-1}(v))$, we have $W_v \\subseteq X$ and $f(W_v) \\subseteq N_{R^*}^c(v)$;\n\\item[$(\\mathscr{D}3)$] $|f^{-1}(a,b)| \\leq {\\varepsilon}^{1\/9}m$ for every $(a,b) \\in [2\\ell]\\times[2r]$;\n\\item[$(\\mathscr{D}4)$] for every edge $uv\\in E(H)$ such that $f(u),f(v) \\notin V_0$, we have $f(u)f(v) \\in E(R^*)$;\n\\item[$(\\mathscr{D}5)$] for all $y \\in Y$ we have $f(y)=(1,\\chi(y))$.\n\\end{itemize}\n\nLet\n\\begin{equation}\\label{taudef}\n\\tau_{a,b} := |(f|_{X\\setminus I})^{-1}(a,b)| = |(f|_X)^{-1}(a,b)| \\quad\\text{for all }(a,b)\\in[2\\ell]\\times[2r].\n\\end{equation}\nThen $0 \\leq \\tau_{a,b} \\leq {\\varepsilon}^{1\/9}m$ for all $(a,b)\\in[2\\ell]\\times[2r]$ by~$(\\mathscr{D}3)$. \n\n\\smallskip\n\nApply Lemma~\\ref{lemmaforG} (the Lemma for $G$) with $n-|V_0|,\\ell,m,2r,11\\beta,{\\varepsilon}^{1\/9},\\delta\/2,G'\\setminus V_0,R,\\mathcal{V}\\setminus \\lbrace V_0\\rbrace$ playing the roles of $n,\\ell,m,r,\\xi,{\\varepsilon},\\delta,G,R,\\mathcal{V}$ to obtain\npositive integers $\\lbrace m_{a,b} : (a,b)\\in[2\\ell]\\times[2r]\\rbrace$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{L}1)$] $\\sum_{(a,b)\\in[2\\ell]\\times[2r]}(m_{a,b}+\\tau_{a,b})=n-|V_0|$ and $m_{a,b} \\geq (1-{\\varepsilon}^{1\/18})m$ and $|m_{a,b}-m_{a,b'}| \\leq 1$ for all $a \\in [2\\ell]$ and $b,b' \\in [2r]$;\n\\item[$(\\mathscr{L}2)$] given any $\\lbrace n_{a,b} : (a,b) \\in [2\\ell]\\times[2r]\\rbrace$ with $\\sum_{(a,b)\\in[2\\ell]\\times[2r]}(n_{a,b}+\\tau_{a,b})=n-|V_0|$ and $|m_{a,b}- n_{a,b}| \\leq 11\\beta (n-|V_0|)$, there is a partition $\\mathcal{X} = \\lbrace V_0 \\rbrace \\cup \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[2r] \\rbrace$ of $V(G')$ with $|X_{a,b}| = n_{a,b}+\\tau_{a,b}$ and $|X_{a,b}\\bigtriangleup V_{\\phi^{-1}(a,b)}| \\leq {\\varepsilon}^{1\/18}m$ for all $(a,b)\\in[2\\ell]\\times[2r]$ such that\n$G'$ has an $(R^*,2\\ell,2r,\\mathcal{X},{\\varepsilon}^{1\/27},\\delta\/4)$-cycle structure.\n\\end{itemize}\nNote that Lemma~\\ref{lemmaforG} yields a partition of $G'\\setminus V_0$ into clusters, and the partition of $V(G')$ specified in~$(\\mathscr{L}2)$ is simply this partition together with $V_0$.\n\n\\smallskip\n\nThe next step is to apply Lemma~\\ref{lemmaforH1} (Basic Lemma for $H$) to $H'' = H[Z]$ (which overlaps with $H'$ in $Y$). Note that the number of vertices in $H''$ is $n-s \\geq (1-{\\varepsilon}^{1\/9})n$. Further,\n\\begin{equation}\\label{mab}\n\\sum_{(a,b)\\in[2\\ell]\\times[2r]}m_{a,b} \\stackrel{(\\mathscr{L}1)}{=} n - |V_0| - \\sum_{(a,b)\\in[2\\ell]\\times[2r]}\\tau_{a,b} \\stackrel{(\\ref{taudef})}{=} n- |V_0| - |X\\setminus I| \\stackrel{(\\mathscr{D}1)}{=} n-|X| \\stackrel{(\\ref{XYZdef})}{=}|Z|\n\\end{equation}\nand $m_{a,b} \\geq (1-{\\varepsilon}^{1\/18})m \\geq 10\\beta(n-s)$ by~(\\ref{hierarchy}).\nThus we can apply Lemma~\\ref{lemmaforH1} with $n-s,r,2\\ell,\\Delta,\\beta,H'',(x_{s+1},\\ldots,x_n),\\chi,\\lbrace m_{a,b} : (a,b)\\in[2\\ell]\\times[2r]\\rbrace$ playing the roles of\n$\nn,r,\\ell,\\Delta,\\beta,H,$ $(x_1,\\ldots,x_n),\\chi,\\lbrace m_{a,b}:(a,b)\\in[2\\ell]\\times[2r]\\rbrace\n$\nto obtain a mapping\n\\begin{equation}\\label{kdef}\nk : Z \\rightarrow [2\\ell]\\times[2r]\n\\end{equation}\nand $B \\subseteq Z$ with the following properties: \n\\begin{itemize}\n\\item[$(\\mathscr{B}1)$] $B \\subseteq Z\\setminus Y$ and $|B| \\leq 2\\ell\\beta n$;\n\\item[$(\\mathscr{B}2)$] $\\left| |k^{-1}(a,b)| - m_{a,b}\\right| \\leq 10\\beta n$ for every $(a,b) \\in [2\\ell]\\times[2r]$;\n\\item[$(\\mathscr{B}3)$] for every edge $uv\\in E(H'')$, writing $k(u)=:(a,b)$ and $k(v)=:(a',b')$, we have $|a-a'|\\leq 1$ and $b \\neq b'$. If additionally $u,v \\notin B$, then $a=a'$;\n\\item[$(\\mathscr{B}4)$] for all $y \\in Y$ we have $k(y)=(1,\\chi(y))$.\n\\end{itemize}\nLet\n\\begin{equation}\\label{nabdef}\nn_{a,b} := |k^{-1}(a,b)|\\quad\\text{for all }(a,b)\\in[2\\ell]\\times[2r].\n\\end{equation}\nThen\n$$\n\\sum_{(a,b)\\in[2\\ell]\\times[2r]}n_{a,b} \\stackrel{(\\ref{kdef}),(\\ref{nabdef})}{=} |Z| \\stackrel{(\\ref{mab})}{=} \\sum_{(a,b)\\in[2\\ell]\\times[2r]}m_{a,b} \\stackrel{(\\mathscr{L}2)}{=} n - |V_0| - \\sum_{(a,b)\\in[2\\ell]\\times[2r]}\\tau_{a,b}\n$$\nand for all $(a,b)\\in[2\\ell]\\times[2r]$,\n$$\n|n_{a,b}-m_{a,b}|\\stackrel{(\\ref{nabdef})}{=}||k^{-1}(a,b)|-m_{a,b}| \\stackrel{(\\mathscr{B}2)}{\\leq} 10\\beta n \\leq 11\\beta(n-|V_0|).\n$$\nThus~$(\\mathscr{L}2)$ implies that there is a partition $\\mathcal{X} = \\lbrace V_0 \\rbrace \\cup \\lbrace X_{a,b} : (a,b)\\in[2\\ell]\\times[2r] \\rbrace$ of $V(G')$ with, for all $(a,b)\\in[2\\ell]\\times[2r]$,\n\\begin{equation}\\label{Xprops}\n|X_{a,b}| = |k^{-1}(a,b)|+|(f|_{X\\setminus I})^{-1}(a,b)|\\quad\\text{and}\\quad|X_{a,b}\\bigtriangleup V_{\\phi^{-1}(a,b)}| \\leq {\\varepsilon}^{1\/18}m\n\\end{equation}\nsuch that\n$G'$ has an $(R^*,2\\ell,2r,\\mathcal{X},{\\varepsilon}^{1\/27},\\delta\/4)$-cycle structure.\n\nDefine a mapping $\\psi : V(H) \\rightarrow ([2\\ell]\\times[2r]) \\cup V_0$ by\nsetting\n\\begin{equation}\\label{psidef99}\n\\psi(x) = \\begin{cases} f(x) &\\mbox{if } x\\in X \\\\\nk(x) & \\mbox{if } x \\in Z.\\end{cases} \n\\end{equation}\nFinally, let $X' := (X\\setminus I) \\cup B$.\n\nWe need to check that $\\mathcal{X}$, $\\psi$ and $X'$ satisfy Claim~\\ref{claim2}(i)--(v).\nFor~(i), we have\n\\begin{eqnarray*}\n|\\psi^{-1}(a,b)| &\\stackrel{(\\ref{psidef99})}{=}& |k^{-1}(a,b)|+|(f|_{X\\setminus I})^{-1}(a,b)| \\stackrel{(\\ref{Xprops})}{=} |X_{a,b}|  \\stackrel{(\\mathscr{B}2)}{\\geq} m_{a,b}-10\\beta n\\\\\n&\\stackrel{(\\mathscr{L}1)}{\\geq}& (1-{\\varepsilon}^{1\/18})m-10\\beta n \\stackrel{(\\ref{beta})}{\\geq} (1-{\\varepsilon}^{1\/19})m.\n\\end{eqnarray*}\nFurther, we have already seen that~(ii) holds.\n\nNote that $I =f^{-1}(V_0)=\\psi^{-1}(V_0)$ has size $|V_0|$ and is a $2$-independent subset of $X$ in $H'$ by~$(\\mathscr{D}1)$. Let $w \\in W := \\bigcup_{x \\in I}N_H(x)$.\nSince $I$ is $2$-independent, there is a unique $u \\in I \\subseteq X$ such that $uw \\in E(H)$. So $w \\in W_{f(u)}=W_{\\psi(u)} \\subseteq X$ in the notation of~$(\\mathscr{D}2)$.\nSo $\\psi(w) = f(w) \\in N_{R^*}^c(f(u))=N_{R^*}^c(\\psi(u))$.\nThus\n$$\nd_{G'}(\\psi(u),X_{\\psi(w)}) \\stackrel{(\\ref{Xprops})}{\\geq} d_{G'}(\\psi(u),V_{\\phi^{-1}(\\psi(w))}) - {\\varepsilon}^{1\/18}m \\stackrel{(\\ref{Nalpha})}{\\geq} cm\/2,\n$$\nso~(iii) holds.\n\nFor~(iv), note that $k(y)=(1,\\chi(y))  = f(y)=\\psi(y)$ for all $y \\in Y$ by~$(\\mathscr{D}5)$ and~$(\\mathscr{B}4)$.\nObserve that $\\psi' := \\psi|_{V(H)\\setminus I}$ is a map into $V(R^*)=[2\\ell]\\times[2r]$. Let $xy \\in E(H)$ where $x,y \\notin I$.\nSuppose first that $x,y \\in X \\cup Y$. Then $\\psi(x)=f(x)$ and $\\psi(y)=f(y)$.\nThen $f(x),f(y) \\notin V_0$, so~$(\\mathscr{D}4)$ implies that $f(x)f(y) \\in E(R^*)$.\nSuppose now that $x,y \\in Z$.\nWrite $\\psi(x)=k(x)=(a,b)$ and $\\psi(y)=k(y)=(a',b')$ where $(a,b),(a',b')\\in[2\\ell]\\times[2r]$.\nThen~$(\\mathscr{B}3)$ implies that $|a-a'| \\leq 1$ and $b \\neq b'$. Thus $\\psi(x)\\psi(y) \\in E(Z^{2r}_{2\\ell}) \\subseteq E(R^*)$, as required.\nThe only other possibility is that one of $x,y$ is in $X$ and the other is in $Z\\setminus Y$.\nBut then the distance between them in the bandwidth ordering of $H$ is more than $|Y|=\\beta n$, a contradiction to $xy \\in E(H)$.\nThus $\\psi' : V(H\\setminus I) \\rightarrow V(R^*)$ is a graph homomorphism.\nSo~(iv) holds.\n\nFor~(v), \nnote that $B \\subseteq Z$ so $X' \\cap I = \\emptyset$, and $W = \\bigcup_{v \\in V_0}W_v \\subseteq X$, and $W \\cap I = \\emptyset$ since $I$ is $2$-independent in $H'$. So $W \\subseteq X'$.\nNow\nlet $(a,b)\\in[2\\ell]\\times[2r]$. We have\n$$\n|\\psi^{-1}(a,b) \\cap X'| \\leq |\\psi^{-1}(a,b) \\cap X|+|B| \\leq |f^{-1}(a,b)| + |B| \\stackrel{(\\mathscr{D}3),(\\mathscr{B}1)}{\\leq} {\\varepsilon}^{1\/9}m + 2\\ell\\beta n \\stackrel{(\\ref{beta})}{\\leq} {\\varepsilon}^{1\/10}m.\n$$\nNow let $uv \\in E(H)$ where $u,v \\notin X' \\cup I $. So $u,v \\in Z \\setminus B$. Write $\\psi(u)=(a,b)$ and $\\psi(v)=(a',b')$.\nThen $(a,b)=k(u)$ and $(a',b')=k(v)$, and~$(\\mathscr{B}3)$ implies that $a=a'$ and $b \\neq b'$, as required.\n\nFinally, define $N$ as in~(v).\nIf $y \\in N$, then either $y \\in \\bigcup_{x \\in X}N_H(x)\\setminus X \\subseteq Y$; or $y \\in \\bigcup_{x \\in B}N_H(x)$ (or both).\nSo~(\\ref{XYZdef}) and the fact that $\\Delta(H) \\leq \\Delta$ implies that\n\\begin{equation}\\label{N}\n|N| \\leq |Y|+\\Delta|B| \\leq (2\\Delta\\ell+1)\\beta n \\stackrel{(\\ref{hierarchy}),(\\ref{beta})}{\\leq} {\\varepsilon} m.\n\\end{equation}\nThis completes the proof of~(v) and hence of the claim.\n\\end{claimproof}\n\n\\medskip\n\\noindent\nIn the final part of the proof, we will use the cycle structure $\\mathcal{C}'$, mapping $\\psi$ and special set $X'$ obtained in Claim~\\ref{claim2} to find an embedding $g$ of $H$ into $G'\\subseteq G$. We will do this in three stages: (1) first define an embedding $g_1$ of $I$ into $V_0$, according to $\\psi$; (2) find an embedding $g_2$ of $X'$ using $\\psi$ as a framework, such that there are large candidate sets for the neighbouring vertices $N$ of $X'$; (3) find an embedding $g_3$ of the remainder of $H$ using the Blow-up lemma, using the candidate sets obtained in (2) to ensure that $g_2$ is compatible with $g_3$.\nThen set $g$ to be the union of $g_1,g_2,g_3$.\n\nStage (1) is easy; we simply define\n$$\ng_1 : I \\rightarrow V_0\\quad\\text{where}\\quad g_1(x) := \\psi(x)\\quad\\text{for all }x \\in I.\n$$\nSince by Claim~\\ref{claim2}(iii), $I$ is an independent set in $H$ of size $V_0$, we trivially have that $g_1$ is an embedding of $H[I]$ into $V(G')$.\n\nFor Stage~(2), we will apply Lemma~\\ref{target} (Embedding lemma with target sets) to embed vertices in $X'$.\nIndeed, let $\\psi^* := \\psi|_{X' \\cup N}$.\nGiven $w \\in W$, let $u$ be the unique element of $I$ such that $uw \\in E(H)$, as guaranteed by Claim~\\ref{claim2}(iii). Let\n\\begin{equation}\\label{Sw99}\nS_w := N_{G'}(\\psi(u),X_{\\psi(w)}).\n\\end{equation}\nWe will apply Lemma~\\ref{target} with\n$\nG'\\setminus V_0,R^*,H[X' \\cup N],n-|V_0|,4r\\ell,{\\varepsilon}^{1\/27},c\/2,\\delta\/4,\\Delta,$ $ \\lbrace X_{a,b}\\rbrace,(1-{\\varepsilon} ^{19})m,\\psi^*,X',$ $N,W,S_w\n$\nplaying the roles of\n$G,R,H,\nn,L,{\\varepsilon},c,\\delta,\\Delta,\\lbrace V_a : a \\in V(R)\\rbrace,m,\\phi,X,Y,W,S_w\n$.\nTo see why this is possible, note that, by Claim~\\ref{claim1}(iii),\n$G'\\setminus V_0$ has $n-|V_0| \\geq (1-2{\\varepsilon}^{1\/2})n$ vertices and Claim~\\ref{claim2}(ii) (specifically~$(\\mathscr{C}2)$) implies that it has an $({\\varepsilon}^{1\/27},\\delta\/4)$-regular partition $\\lbrace X_{a,b} : (a,b)\\in V(R^*)\\rbrace$.\nClearly, as a restriction of $\\psi$, the function $\\psi^*$ is a suitable graph homomorphism, and by Claim~\\ref{claim2}(v) and~(\\ref{N}), we have\n\\begin{equation}\\label{psistar}\n|(\\psi^*)^{-1}(a,b)| \\leq |\\psi^{-1}(a,b) \\cap X'| + |N| \\leq {\\varepsilon}^{1\/10}m + {\\varepsilon} m \\leq {\\varepsilon}^{1\/12}m.\n\\end{equation}\nFinally, $W \\subseteq X'$ by Claim~\\ref{claim2}(v), and $|S_w| \\geq cm\/2$ by Claim~\\ref{claim2}(iii).\nSo the above are suitable parameters for the application of Lemma~\\ref{target}.\n\nThus there is a mapping\n$$\ng_2 : X' \\rightarrow V(G')\\setminus V_0\n$$\nwhich is an embedding of $H[X']$ into $G'$ such that\n\\begin{itemize}\n\\item[$(\\mathscr{T}1)$] $g_2(x) \\in X_{\\psi^*(x)}$ for all $x \\in X'$;\n\\item[$(\\mathscr{T}2)$] $g_2(w) \\in S_w$ for all $w \\in W$;\n\\item[$(\\mathscr{T}3)$] for all $y \\in N$ there exists $C_y \\subseteq X_{\\psi^*(y)}\\setminus g_2(X')$ such that $C_y \\subseteq N_{G'}(g_2(x))$ for all $x \\in N_H(y) \\cap (X')$, and $|C_y| \\geq cm\/2$.\n\\end{itemize}\n\nFor Stage~(3), we will do the following for each $a \\in [2\\ell]$.\nLet $U_{a,b} := X_{a,b} \\setminus g_2(X')$ for all $b \\in [2r]$.\nWe want to show that $U_{a,b}$ has exactly the right size to embed the remaining vertices of $H$ whose image under $\\psi$ is $(a,b)$.\nIndeed, let $\\psi' := \\psi|_{H\\setminus (X' \\cup I)}$.\nThen Claim~\\ref{claim2}(i) implies that\n$$\n|U_{a,b}| = |X_{a,b}| - |g_2(X') \\cap X_{a,b}| \\stackrel{(\\mathscr{T}1)}{=} |\\psi^{-1}(a,b)| - |(\\psi^*)^{-1}(a,b) \\cap X'| = |(\\psi')^{-1}(a,b)|\n$$\nwhere we used the fact that $\\psi^{-1}(a,b) \\cap I=\\emptyset$.\nThis together with~(\\ref{psistar}) implies that\n$|U_{a,b}\\bigtriangleup X_{a,b}| = |(\\psi^*)^{-1}(a,b) \\cap X'| \\leq {\\varepsilon}^{1\/10}m \\leq 2{\\varepsilon}^{1\/10}|U_{a,b}|$.\nLet $b,b' \\in [2r]$ be distinct.\nSo $|U_{a,b}| \\geq (1-{\\varepsilon}^{1\/20})m$ by Claim~\\ref{claim2}(i).\nRecall from Claim~\\ref{claim2}(ii) (specifically~$(\\mathscr{C}3)$) that $G'[X_{a,b},X_{a,b'}]$ is $({\\varepsilon}^{1\/27},\\delta\/4)$-superregular.\nSo given any $x \\in U_{a,b}$, Claim~\\ref{claim2}(i) implies that \n$$\nd_{G'}(x,U_{a,b'}) \\geq \\delta|X_{a,b}|\/4-{\\varepsilon}^{1\/10}m \\geq (\\delta\/4-\\delta{\\varepsilon}^{1\/19}-{\\varepsilon}^{1\/10})m \\geq \\delta|U_{a,b}|\/5.\n$$\nThus Proposition~\\ref{newsuperslice} with $G',X_{a,b},U_{a,b},X_{a,b'},U_{a,b'},{\\varepsilon}^{1\/27},\\delta\/4,{\\varepsilon}^{1\/10}$ playing the roles of $G,A,A',B,B',$ ${\\varepsilon},\\delta,\\alpha$ implies that\n $G'[U_{a,b},U_{a,b'}]$ is $(2{\\varepsilon}^{1\/27},\\delta\/5)$-superregular for all distinct $b,b'\\in[2r]$.\nThe set $N$ has size at most ${\\varepsilon} m \\leq 2{\\varepsilon}|U_{a,b}|$ for any $b \\in [2r]$, and for each $y \\in N \\cap (\\psi')^{-1}(a,b)$,~$(\\mathscr{T}3)$ guarantees a corresponding set $C_y \\subseteq X_{\\psi^*(y)}\\setminus g_2(X') = U_{\\psi'(y)} = U_{a,b}$ which has size at least $cm\/2 \\geq c|U_{a,b}|\/3$.\nLet $H_a$ denote the subgraph of $H$ induced by the set of all $x \\in V(H) \\setminus (X' \\cup I)$ such that $\\psi '(x)=(a,b)$ for some $b \\in [2r]$. \\COMMENT{AT: note all the changes here}\nNow apply, for each $a \\in [2\\ell]$, Lemma~\\ref{blowlem} (Blow-up Lemma) with \n$G'[\\bigcup_{b \\in [2r]}U_{a,b}]$ and $H_a$ playing the roles of $G$ and $H$ and\n$\n2{\\varepsilon}^{1\/27},2{\\varepsilon},\\delta\/5,c\/3,\\Delta,2r,\\lbrace U_{a,b}: b \\in [2r]\\rbrace,\\psi',C_y\n$\nplaying the roles of\n$\n{\\varepsilon},\\alpha,\\delta,c,\\Delta,k,\\lbrace V_a:a \\in [k]\\rbrace,\\phi,S_y.\n$\nAltogether this yields a mapping\n$$\ng_3 : V(H)\\setminus (X' \\cup I) \\rightarrow V(G')\\setminus (V_0 \\cup g_2(X'))\n$$\nwhich is an embedding of $H\\setminus(X' \\cup I)$ into $V(G')$ such that every $y \\in N$ is mapped to a vertex in $C_y$.\n\nWe claim that the mapping $g$ given by\n\\begin{equation}\\label{psidef}\ng(x) = \\begin{cases} g_1(x) &\\mbox{if } x\\in I \\\\\ng_2(x) &\\mbox{if } x\\in X' \\\\\ng_3(x) & \\mbox{otherwise}.\\end{cases} \n\\end{equation}\nis an embedding of $H$ into $G'$ (and hence into $G$).\n\nFirstly, $g$ is an injective map from $V(H)$ to $V(G')$ by the definitions of $g_1,g_2,g_3$.\nSo we just need to check that it is a graph homomorphism.\nAlso by their definitions, each of $g_1,g_2,g_3$ is an embedding of $H$ induced on their respective domains into $G'$.\nSo it suffices to check that whenever $xy \\in E(H)$ and $x,y$ are not both in $I$ or in $X'$ or in $V(H)\\setminus (X' \\cup I)$, that $g(x)g(y)\\in E(G')$.\n\n\nSuppose first that $x \\in I$ and $y \\in V(H)\\setminus I$. Then $g(x)=g_1(x)=\\psi(x)$ and $y \\in W \\subseteq X'$ (here we used Claim~\\ref{claim2}(v)). So $g(y)=g_2(y)$.\nClaim~\\ref{claim2}(iii) implies that $x$ is the only vertex in $I$ which is a neighbour of $y$.\nThen\n$$\ng(y)\\stackrel{(\\ref{psidef})}{=}g_2(y) \\stackrel{(\\mathscr{T}2)}{\\in} S_y \\stackrel{(\\ref{Sw99})}{=} N_{G'}(\\psi(x),X_{\\psi(y)}) = N_{G'}(g(x),X_{\\psi(y)}).\n$$\nSo $g(x)g(y)\\in E(G')$, as required.\n\nTherefore we may assume that $x \\in X'$ and $y \\in V(H)\\setminus (X' \\cup I)$.\nThen $g(x)=g_2(x)$, $y \\in N$ and $g(y)=g_3(y) \\in C_y$, where $C_y$ was defined in~$(\\mathscr{T}3)$, which guarantees that $C_y \\subseteq N_{G'}(g_2(x)) = N_{G'}(g(x))$.\nSo $g(x)g(y)\\in E(G')$, as required.\nThis completes the proof of Theorem~\\ref{main}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding remarks}\\label{sec9}\nIn this paper we  prove a version of the Bandwidth theorem for locally dense graphs. As mentioned in the introduction, it is also of interest to seek minimum degree conditions that force a given spanning structure in a graph with\nsublinear independence number. In particular, it would be very interesting to obtain an analogue of the Bandwidth theorem in this setting.\n\nIn a step in this direction,\nBalogh, Molla and  Sharifzadeh~\\cite{bms} proved the following result on triangle factors.\n\\begin{theorem}[Balogh, Molla and  Sharifzadeh~\\cite{bms}]\\label{bmsthm}\nFor every ${\\varepsilon} >0$, there exists $\\gamma > 0$ and $n_0 \\in \\mathbb N$ such that the following holds.\nFor every $n$-vertex graph $G$ with $n \\geq n_0$ divisible by $3$, if $\\delta (G) \\geq (1\/2 + {\\varepsilon})n$ and $G$ has independence number $\\alpha(G) \\leq \\gamma n$, then $G$ has\na $K_3$-factor.\n\\end{theorem}\nPerhaps the next natural step is to ascertain whether the conclusion of Theorem~\\ref{bmsthm} can be strengthened to ensure the square of a Hamilton cycle.\n\\begin{conjecture}\nFor every ${\\varepsilon} >0$, there exists $\\gamma > 0$ and $n_0 \\in \\mathbb N$ such that the following holds.\nFor every $n$-vertex graph $G$ with $n \\geq n_0$, if $\\delta (G) \\geq (1\/2 + {\\varepsilon})n$ and  $\\alpha(G) \\leq \\gamma n$, then $G$ \ncontains the square of a Hamilton cycle.\n\\end{conjecture}\n\nIt is also natural to seek a version of Theorem~\\ref{main} where now one replaces the condition of locally dense with a more restrictive \\emph{uniformly dense} condition:\ngiven $\\rho,d >0$, we say that an $n$-vertex graph $G$ is \\emph{$(\\rho,d)$-uniformly-dense} \nif every $X ,Y\\subseteq V(G)$  satisfies $e_G(X,Y) \\geq d|X||Y|-\\rho n^2$.\nIf one restricts to uniformly dense graphs, then one can substantially reduce the minimum degree condition in Theorem~\\ref{main}, as well as remove the bandwidth condition on $H$.\n\\begin{theorem}\\label{udthm}\nFor all $\\Delta \\in \\mathbb{N}$ and $d,\\eta>0$, there exist constants $\\rho,n_0>0$ such that for every $n \\geq n_0$, the following holds.\nLet $H$ be an $n$-vertex graph  with $\\Delta(H) \\leq \\Delta$.\nThen any $(\\rho,d)$-uniformly-dense graph $G$ on $n$ vertices with $\\delta(G)\\geq \\eta n$ contains a copy of $H$.\n\\end{theorem}\nTheorem~\\ref{udthm} can be proven by a  simple application of the Blow-up lemma; a more general `rainbow' version of Theorem~\\ref{udthm} is given in~\\cite[Corollary 1.3]{gj}.\n\n\n\n\\section*{Acknowledgements}\nWe would like to thank Maryam Sharifzadeh for many helpful conversations at the start of the project. The second author is grateful to Stefan Glock and Felix Joos for a conversation on~\\cite{gj} that brought to light the\nversion of the Bandwidth theorem for uniformly dense graphs.\n\nThe authors are also grateful to the referees for their helpful and careful reviews.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\nAlthough the Standard Model (SM) of particle physics agrees very well with various experimental results, some problems and unsatisfactory points have also been pointed out.\nOne of them is the lack of a guiding principle ruling the flavor structure of fermions.\nIn this regard, the introduction of flavor symmetries is one of the most conceivable extensions of the SM \nand also well motivated from a viewpoint of string theories \\cite{stringy}.\n\nSince, as concluded in Ref.~\\cite{no-go}, most of flavor symmetries need to be broken either spontaneously or explicitly at energy scales much above the electroweak scale, it is usually not easy to decide which symmetries are suitable and how they should be imposed.\nTo this end, one needs to find out remnants of such broken symmetries in a low energy Lagrangian, and small parameters could be important for this purpose.\nIf symmetry breaking was slight, its effects should correspondingly be small, yielding small breaking terms in a low energy Lagrangian.\nMoreover, even if they were grossly broken, the breaking effects might be realized as effective non-renormalizable operators suppressed by their breaking scales.\nFor instance, it is well known that a Majorana neutrino mass term can be constructed with only the SM particles at mass-dimension five \\cite{dim5}:\n\\begin{eqnarray}\n{\\cal L}_{\\rm eff}=\\frac{f_{ij}}{\\Lambda_\\nu}L_iL_jHH,\n\\label{eq:wein}\n\\end{eqnarray}\nwhere $L$ and $H$ represent the left-handed lepton and the SM Higgs doublets, respectively, and that the lepton number conservation is\nviolated with this term\\footnote{In Ref. \\cite{chen}, authors relate the tiny neutrino masses with the Peccei-Quinn symmetry.}.\nIn this case, this term can be regarded as a breaking term of the lepton number symmetry broken at the high energy scale $\\Lambda_\\nu$.\nHence, it might be said that small parameters in a low energy theory are the manifestation of new symmetries in high energies; zero limits of the small parameters may correspond to the unbroken limits of the associated symmetries.\n\nIn the case of flavor symmetries, their remnants should appear in the fermion mass terms and\/or the flavor mixing sectors. \nFor example, one can line up the following candidates of small parameters:\n\\begin{enumerate}\n\\item $\\theta_{13}^{\\rm PMNS}\\ll\\theta_{12}^{\\rm PMNS}, ~\\theta_{23}^{\\rm PMNS}$ ~~~({\\rm or}~~$|V^{\\rm PMNS}_{e3}|\\ll {\\rm the~others}$),\n\\item $|\\theta_{23}^{\\rm PMNS}-45^\\circ|\\ll \\theta_{23}^{\\rm PMNS}$~~~({\\rm or}~~$||V^{\\rm PMNS}_{\\mu 3}|-1\/\\sqrt{2}|\\ll |V^{\\rm PMNS}_{\\mu 3}|$),\n\\item $\\Delta m^2_{12}=(m_2^\\nu)^2 - (m_1^\\nu)^2 \\ll \\Delta m^2_{23} =|(m_3^\\nu)^2 - (m_2^\\nu)^2|$,\n\\item $m_1^{u,d,\\ell},~m_2^{u,d,\\ell} \\ll m_3^{u,d,\\ell}$,\n\\item $\\theta_{ij}^{\\rm CKM} \\ll \\theta_{ij}^{\\rm PMNS}$~~~({\\rm or}~~$|V^{\\rm CKM}_{ij}| \\ll |V^{\\rm PMNS}_{ij}|$),\n\\end{enumerate}\nwhere $\\theta_{ij}^{\\rm PMNS}$ and $\\theta_{ij}^{\\rm CKM}$ stand for the mixing angles of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS), $V^{\\rm PMNS}$, and the Cabibbo-Kobayashi-Maskawa (CKM), $V^{\\rm CKM}$, mixing matrices\\footnote{In this paper, we adopt the standard parametrization \\cite{PDG}.}, respectively;\n$m_{i}^{f}$ with $f=u,d,\\ell,\\nu$ and $i=1,2,3$ denotes masses of the up-type quarks, down-type quarks, charged leptons and neutrinos.\nThe first two items have attracted a lot of attention over the years as they predict the $\\mu\\hif\\tau$ permutation symmetry \\cite{mutau} or non-Abelian discrete flavor symmetries \\cite{nADFS} in the limit of $\\theta_{13}^{\\rm PMNS}=0^\\circ$ with $\\theta_{23}^{\\rm PMNS}=45^\\circ$.\nIn fact a number of models based on these symmetries have been proposed \\cite{models}.\nThe recent reactor \\cite{rct} and long-baseline \\cite{acl} neutrino oscillation experiments, however, disfavor the vanishing $\\theta_{13}^{\\rm PMNS}$: the DAYA-BAY experiment \\cite{8sigma} give us $7.7\\sigma$ deviations from $\\theta_{13}^{\\rm PMNS}=0^\\circ$ with a rather large central value of $\\theta_{13}^{\\rm PMNS}\\simeq 8.7^\\circ$.\nFurthermore, considerable deviations of $\\theta_{23}^{\\rm PMNS}$ from $45^\\circ$ have also been found in the global analysis of the neutrino oscillation experiments \\cite{valle, fogli, garc}, yet its confidence level is not high enough to conclude.\nHaving these facts in mind, we would turn our attention to the remaining three items and discuss their consequences in the present study.\n\nFirst, the third item means quasi mass degeneracy between the first and second generation neutrinos unless $m_1^\\nu$ is much smaller than $m_2^\\nu$ in the case of normal mass ordering.\nLet us focus on this partially quasi-degenerated region (roughly $m_1^\\nu = 0.05 \\sim 0.1~\\ev$ and $m_3^\\nu=0 \\sim 0.1~\\ev$ for the normal and inverted ordering cases, respectively) and consider the effective Majorana neutrino mass operator given in Eq. (\\ref{eq:wein}).\nIn the limit of $\\Delta m_{12}^2=0$, the Majorana neutrino mass matrix comes to respect a two-dimensional rotation symmetry \\cite{o2,general}\\footnote{\nThe mass degeneracy is entangled with the mixing in Ref. \\cite{vonDyck}.}, such as $O(2)$, $SO(2)$ and $D_N^{}$:\n\\begin{eqnarray}\n&&\nR^T\n\\bmx{ccc}\n m_1^\\nu & 0 & 0 \\\\\n 0 & m_1^\\nu & 0 \\\\\n 0 & 0 & m_3^\\nu\n\\emx\nR\n=\n\\bmx{ccc}\n m_1^\\nu & 0 & 0 \\\\\n 0 & m_1^\\nu & 0 \\\\\n 0 & 0 & m_3^\\nu\n\\emx\n~~~{\\rm with}~~~\nR=\n\\bmx{ccc}\n \\cos\\theta & \\sin\\theta & 0 \\\\\n-\\sin\\theta & \\cos\\theta & 0 \\\\\n 0 & 0 & 1\n\\emx.\n\\label{eq:Mnu0}\n\\end{eqnarray}\nIn other words, $L_1$ and $L_2$ belong to a doublet representation, e.g. ${\\bf 2}_n$, of the symmetry, while $L_3$ behaves as a singlet representation, e.g. ${\\bf 1}$. \n(Notations of $O(2)$ are given in Appendix.)\nThen, the observed slight mass splitting between $m_1^\\nu$ and $m_2^\\nu$ could be interpreted as slight breaking of the rotation symmetry.\nLike this, the observed neutrino mass spectrum could be explained by starting from a limit of $m_1^\\nu = m_2^\\nu$.\n\nSecondly, the idea of the partial mass degeneracy ($m_1^f=m_2^f$), however, seems to conflict with the fourth item because the charged lepton masses are strongly hierarchical.\nNevertheless, it is possible to realize a hierarchical mass spectrum in the case of Dirac fermions by assigning a different doublet representation, ${\\bf 2}_{m\\neq n}$, or a singlet representation ${\\bf 1}$ to the right-handed charged leptons, resulting in\n\\begin{eqnarray}\nM^{\\ell}=\n\\bmx{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & m_{33}^\\ell\n\\emx\n~~~~{\\rm or}~~~~\nM^\\ell =\n\\bmx{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\nm_{31}^{\\ell} & m_{32}^{\\ell} & m_{33}^{\\ell}\n\\emx ,\\label{eq:Me0}\n\\end{eqnarray}\nrespectively.\nIt can be readily observed that the electron and muon masses are vanishing and thus degenerate in both cases.\nIn this sense, the idea of the partial mass degeneracy may be applicable to not only the charged lepton sector but also the quark sectors.\nRather, that appears reasonable since one can relate the smallness of the light charged fermion masses with symmetry breaking.\nAlso, it may enable us to naturally understand some phenomenological relations among elements of the CKM matrix \\cite{xing}.\n\nLastly, given the above conjectures, the fifth item may naturally be explained at the same time.\nBreaking of the rotation symmetry triggers flavor mixing as well. \nOn one hand, in the quark and charged lepton sectors, small flavor mixings are expected because breaking terms are supposed to be small; thus the observed small CKM mixing would be derived\\footnote{Note that mixing between the first and second generations is not necessarily small because of the mass degeneracy. This might enable us to understand why $\\theta_{12}^{\\rm CKM}$ is a little larger than the others.}.\nOn the other hand, in the neutrino sector, its flavor mixing can be large since the leading-order neutrino mass matrix in Eq. (\\ref{eq:Mnu0}) is almost proportional to the unit matrix in the neutrino mass regions under consideration.\nWe here stress that the mysterious differences between the CKM and PMNS matrices stem from the nature of fermions, that is to say Dirac or Majorana, in this scenario.\n\nTo summarize, the limit of the partial mass degeneracy seems to fit the observed fermion mass spectra and mixing patterns and suggests the existence of a two-dimensional rotation symmetry.\nYet another motivation to consider the partial mass degeneracy is that the effective mass, $\\langle m_\\nu^{} \\rangle$, of the neutrinoless double beta decay is not vanishing even in the case of normal mass ordering\\footnote{We would like to thank E. Takasugi for making us aware of this point.}.\nIn Fig. \\ref{fig:0nbb}, we depict the allowed regions of $\\langle m_\\nu^{} \\rangle$ as a function of the lightest neutrino mass in the standard $3\\nu$ framework, where 3$\\sigma$ constraints of the neutrino oscillation parameters from \\cite{fogli} are imposed while varying CP phases from $0$ to $2\\pi$.\nFor example, if $m_1^\\nu > 0.05~\\ev$, then $\\langle m_\\nu^{} \\rangle > 0.01~\\ev$, which would be accessible by the next generation EXO and KamLAND-Zen experiments \\cite{neut12}.\nIn addition, the sensitivity of cosmic microwave background (CMB) observations on the neutrino mass has just started to enter this mass region \\cite{planck}.\nThus, it should be worthwhile to carry out theoretical studies on this mass region.\n\nIn Sect. II, we show a simple model for the lepton sector by means of a $D_N^{}$ flavor symmetry and demonstrate that its $Z_2^{}$ subgroup forbids the electron mass and $\\theta_{13}^{\\rm PMNS}$.\nThe particle content is enriched with SM-gauge-singlet real scalars.\nIn order to break the $Z_2^{}$ symmetry, in Sect. III, we promote the singlet scalars to complex ones with complex vacuum expectation values.\nAfter mentioning testability of the models in Sect. IV, we summarize our results and discuss what to do next in Sect. V.\nThe group theories of $D_N^{}$ and $O(2)$ are briefly summarized in Appendix A and B, respectively.\n\n\n\\section{Model with real scalars}\n\\begin{table}[ht]\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n & $L_I$ & $L_3$ & $\\ell_{i}$ & $H$ & $S_I$ \\\\ \\hline\n$D_N$ & ${\\bf 2}_2$ & ${\\bf 1}$ & ${\\bf 1}$ & ${\\bf 1}$ & ${\\bf 2}_1$ \\\\ \\hline\n\\end{tabular}\n\\caption{A particle content and charge assignment of the model, where $I=1,2$ and $i=1\\hif 3$ denote the indices of generations; \n$L$ and $\\ell$ represent the left- and right-handed SM leptons, respectively; $H$ and $S_{1,2}^{}$ are the SM Higgs and gauge-singlet real scalars, respectively.}\n\\label{tab:1}\n\\end{table}\n\nTypical examples of a flavor symmetry realizing the partial mass degeneracy are presumably $O(2)$, $SO(2)$ and $D_N^{}$.\nAs we shall explain later, $SO(2)$ may be excluded from the list since it does not include a $Z_2$ parity, whereas both $O(2)$ and $D_N^{}$ would be useful for our purpose.\nWe here adopt $D_N^{}$ as our flavor symmetry just to avoid dangerous massless Nambu-Goldstone bosons or gauge anomalies.\nWe concentrate on the case of $N={\\rm odd}$ and postulate that $N$ is sufficiently large.\nIn this case, one may be able to disregard the tensor product Eq. (\\ref{eq:cg3}) in Appendix A unless higher order terms with huge suppression factors are concerned, and the theory would be governed by only Eqs. (\\ref{eq:cg1}) and (\\ref{eq:cg2}) which are the same as those of $O(2)$.\nIn this sense, the following models would work for $O(2)$, too.\nWe will make a further comment on differences between $D_N^{}$ and $O(2)$ models later in this section.\n\nWe embed $L_1^{}$ and $L_2^{}$ into the doublet representation ${\\bf 2}_2$ of $D_N^{}$ and assign the trivial singlet representation ${\\bf 1}$ to the other leptons and the SM Higgs\\footnote{Alternatively, one can embed $\\ell_1$ and $\\ell_2$ into a doublet representation too as mentioned in Section I. In this case, however, masses of the electron and muon tend to be degenerate within the given particle contents. Such charge assignments are adopted in Ref. \\cite{old-models} with different particle contents.}, resulting in $m_e^{}=m_\\mu^{}=0$ and $m_1^\\nu = m_2^\\nu$.\nIn order to lift the mass degeneracies, the $D_N$ flavor symmetry must be broken by a doublet representation, so that we introduce a set of SM-gauge-singlet real scalars $S^{}_{1,2}$ belonging to ${\\bf 2}_1$.\nA particle content and charge assignment of the model are summarized in Table \\ref{tab:1}.\nWe consider the effective Majorana neutrino mass operator given in Eq. (\\ref{eq:wein}) so as to keep our discussions as general as possible.\nUnder the $D_N$ flavor symmetry, the charged lepton Yukawa and Majorana neutrino mass terms are written by\n\\begin{eqnarray}\n{\\cal L}\n&=& \n  y_{i}^{0} ~\\overline{L}_3^{} H \\ell_i^{}\n+ \\frac{y^{}_{i}}{\\Lambda_F^2} \\overline{L}_I^{} H \\ell_{i}^{} (S_{}^2)_I^{}\n+ \\frac{f_\\nu}{\\Lambda_\\nu}L_I L_I H H\n+ \\frac{f_\\nu^\\prime}{\\Lambda_\\nu}L_3 L_3 H H\n\\nonumber \\\\\n&&\n+ \\frac{g_\\nu}{\\Lambda_\\nu \\Lambda_F^2}L_3 L_I H H (S^2)_I\n+ \\frac{h_\\nu}{\\Lambda_\\nu \\Lambda_F^4}(L_J L_K)_I H H (S^4)_I\\,,\n\\end{eqnarray}\nwhere $y_i^0$, $y_i^{}$, $f_\\nu^{}$, $f_\\nu^\\prime$, $g_\\nu^{}$ and $h_\\nu^{}$ are dimensionless complex couplings and supposed to be ${\\cal O}(1)$, and $\\Lambda_F$ describes a breaking scale of the $D_N$ symmetry.\nNote that the term proportional to ${\\cal O}(1\/\\Lambda_F^{4})$ is omitted in the charged lepton sector since it is absorbed by $(y_i^{}\/\\Lambda_F^2) ~\\overline{L}_I^{} H \\ell_{i}^{} (S_{}^2)_I^{}$.\nWe denote the vacuum expectation values (VEVs) of the scalars as\n\\begin{eqnarray}\n\\langle H \\rangle=v,~~\n\\langle S_I \\rangle=(s_1~~s_2)^T, \\label{eq:vev_real}\n\\end{eqnarray}\nwhich yield the following mass matrices for the charged leptons and neutrinos:\n\\begin{eqnarray}\n\\frac{1}{v} M^\\ell \\eqn{=} \n\\begin{pmatrix}\n0 &0 &0\\\\\n0 &0 &0\\\\\ny_1^0 &y_2^0 &y_3^0\n\\end{pmatrix}\n+ \n\\frac{1}{\\Lambda_F^2}\n\\begin{pmatrix}\ny_1 \\delta_1 &y_2 \\delta_1 &y_3 \\delta_1 \\\\\ny_1 \\delta_2 &y_2 \\delta_2 &y_3 \\delta_2 \\\\\n0 &0 &0\n\\end{pmatrix}\\,,\n\\label{eq:Me1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{\\Lambda_\\nu}{v^2} M^\\nu \n= \n\\begin{pmatrix}\nf_\\nu &0 &0\\\\\n0 &f_\\nu &0\\\\\n0 &0 &f_\\nu'\n\\end{pmatrix}\n+\n\\frac{1}{\\Lambda_F^2}\n\\begin{pmatrix}\n0 &0 &g_\\nu \\delta_1 \\\\\n0 &0 &g_\\nu \\delta_2 \\\\\ng_\\nu \\delta_1 &g_\\nu \\delta_2 &0\n\\end{pmatrix} \n+ \n\\frac{1}{\\Lambda_F^4}\n\\begin{pmatrix}\n h_\\nu (\\delta_1^2 - \\delta_2^2) & \n h_\\nu 2\\delta_1^{}\\delta_2^{} & 0\\\\\n h_\\nu 2\\delta_1^{}\\delta_2^{} &\n-h_\\nu (\\delta_1^2 - \\delta_2^2) & 0\\\\\n0 &0 &0\n\\end{pmatrix} \\,.\n\\label{eq:Mn1}\n\\end{eqnarray}\nHere and hereafter we use the following abbreviations\n\\begin{eqnarray}\n\\delta_1^{} = s_1^2 - s_2^2\\,,\n\\eqn{} \\delta_2^{} =2 s_1^{} s_2^{}\\,,~\n\\delta_s^{}=s_1^2+s_2^2\\,.\n\\end{eqnarray}\nThe charged lepton mass matrix is diagonalized by the unitary transformation\n\\begin{eqnarray}\nV^\\ell_{} \n= \n\\frac{1}{\\delta_s^{}} \n\\begin{pmatrix}\n-\\delta_2 &\\delta_1 &0\\\\\n\\delta_1 &\\delta_2 &0\\\\\n0 &0 & \\delta_s^{}\n\\end{pmatrix}\n\\begin{pmatrix}\n1 &0 &0\\\\\n0 &\\cos \\theta &-\\sin \\theta ~e^{i\\rho}\\\\\n0 &\\sin \\theta ~e^{-i\\rho} &\\cos \\theta\n\\end{pmatrix}\\,,\n\\label{eq:Ve1}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\tan 2 \\theta = \\frac{2 Y' \\delta_s^{} \\Lambda_F^2}{Y \\delta_s^2  - Y^0 \\Lambda_F^4}\n\\sim {\\cal O}\\left(\\frac{1}{\\Lambda_F^2}\\right)\\,,\n\\end{eqnarray}\nwhere $Y^0 = |y_1^0|^2 + |y_2^0|^2 + |y_3^0|^2$, $Y = |y_1|^2 + |y_2|^2 + |y_3|^2$, $Y' = |y_1^* y_1^0 + y_2^* y_2^0 + y_3^* y_3^0|$ and $\\rho={\\rm Arg}[y_1^* y_1^0 + y_2^* y_2^0 + y_3^* y_3^0]$.\nThe eigenvalues are approximately derived as\n\\begin{eqnarray}\n\\frac{1}{v^2} (V^\\ell_{})^\\dag M^\\ell (M^{\\ell})^\\dagger V^\\ell \\simeq\n\\begin{pmatrix}\n0 &0 &0\\\\\n0 &\\frac{Y Y^0 - Y'^2}{Y^0} \\frac{\\delta_s^2}{\\Lambda_F^4} &0\\\\\n0 &0 &Y^0\\\\\n\\end{pmatrix}\\,.\\label{eq:Diag_Ml_R}\n\\end{eqnarray}\nOne immediately observes that the electron remains massless and that $s_i^{}\/\\Lambda_F^{} \\sim \\sqrt{m_\\mu^{}\/m_\\tau^{}}$.\nOn the other hand, the unitary transformation affects the neutrino mass matrix in such a way that\n\\begin{eqnarray}\n\\frac{\\Lambda_\\nu}{v^2} \n(V^\\ell_{})^T M^\\nu V^\\ell_{}\n\\eqn{=} \n\\begin{pmatrix}\n\\bar{M}^\\nu_{11} &0 &0\\\\\n0 & \\bar{M}^\\nu_{22} & \\bar{M}^\\nu_{23}\\\\\n0 & \\bar{M}^\\nu_{32} & \\bar{M}^\\nu_{33} \n\\end{pmatrix}\\,,\n\\label{eq:Mnd}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\bar{M}^\\nu_{11} \\eqn{=} \nf_\\nu - h_\\nu \\frac{\\delta_s^2}{\\Lambda_F^4}\\,,\\\\\n\\bar{M}^\\nu_{22} \\eqn{\\sim} \nf_\\nu + h_\\nu \\frac{\\delta_s^2}{\\Lambda_F^4} \n+ \\frac{Y' ( Y' f_\\nu' - 2 Y^0 g_\\nu)}{(Y^0)^2} \\frac{\\delta_s^2}{\\Lambda_F^4}\\,,\\\\\n\\bar{M}^\\nu_{23} \\eqn{=}\n\\bar{M}^\\nu_{32} \\sim \n\\frac{Y^0 g_\\nu + (f_\\nu - f_\\nu') Y'}{(Y^0)^2} \\frac{\\delta_s^{} }{\\Lambda_F^2}\\,,\\\\\n\\bar{M}^\\nu_{33} \\eqn{\\sim} \nf_\\nu' \n+ \\frac{Y' ( Y' f_\\nu + 2 Y^0 g_\\nu)}{(Y^0)^2} \\frac{\\delta_s^2}{\\Lambda_F^4}\\,,\n\\end{eqnarray}\nand it can be seen that $\\theta_{12}^{\\rm PMNS}$ and $\\theta_{13}^{\\rm PMNS}$ are vanishing as well.\n\nThe vanishing electron mass, $\\theta_{12}^{\\rm PMNS}$ and $\\theta_{13}^{\\rm PMNS}$ are not accidental.\nIn order to explain this, let us first consider the case of $O(2)$.\nSince $O(2)$ is a continuous symmetry, \nthere always exists an $O(2)$ transformation which keeps the VEV configuration Eq. (\\ref{eq:vev_real}) invariant, such as                                                                                                                                                                                                                                                                                                                                     \n\\begin{eqnarray}\n\\bmx{cc}\n \\cos\\theta & \\sin\\theta \\\\\n \\sin\\theta &-\\cos\\theta\n\\emx\n=\\frac{1}{s_1^2 + s_2^2}\n\\bmx{cc}\n s_1^2 - s_2^2 & 2s_1^{}s_2^{} \\\\\n 2s_1^{}s_2^{} & -(s_1^2 - s_2^2) \n\\emx\\,.\n\\label{eq:z2}\n\\end{eqnarray}\nAfter the symmetry breaking, this invariance ends up an unbroken $Z_2^{}$ symmetry under which the left-handed leptons\ntransform as\n\\begin{eqnarray}\nL_i^{} \\rightarrow\n\\frac{1}{\\delta_s^2}\n\\bmx{ccc}\n \\delta_1^2 - \\delta_2^2 & 2\\delta_1^{}\\delta_2^{} & 0 \\\\\n 2\\delta_1^{}\\delta_2^{} & -(\\delta_1^2 - \\delta_2^2) & 0 \\\\\n 0 & 0 & \\delta_s^2\n\\emx_{ij}^{} L_j^{} ,\n\\label{eq:z2f}\n\\end{eqnarray}\nor in the diagonal basis of the charged lepton mass matrix, it becomes\n\\begin{eqnarray}\nL_i^{} \\rightarrow\n\\bmx{ccc}\n -1 & 0 & 0 \\\\\n 0 & 1 & 0 \\\\\n 0 & 0 & 1\n\\emx_{ij}^{} L_j^{} \\,.\n\\label{eq:z2m}\n\\end{eqnarray}\nFrom Eq. (\\ref{eq:z2m}), it is clear that the electron mass, $\\theta_{12}^{\\rm PMNS}$ and $\\theta_{13}^{\\rm PMNS}$ are forbidden by this $Z_2^{}$ symmetry.\nAlso, looking back at Eqs. (\\ref{eq:Me1}) and (\\ref{eq:Mn1}), it can be found that they take the most-general $Z_2$-invariant forms.\nIn the case of $D_N$, the $Z_2$ symmetry may not be an exact one because $D_N$ is a discrete group.\nCorrections stemming from this violation, however, are negligibly small as long as the order of $D_N$ is sufficiently large.\nIn the case of $N=9$, for instance, effects of the $Z_2$ breaking appear in $(x\/\\Lambda_F^7) ~\\overline{L}_I^{} H \\ell_{i}^{} (S_{}^7)_I^{}$ for the first time.\nA $D_9^{}$ invariant term is obtained by constructing ${\\bf 2}_2$ from $(S^7)_I$ via ${\\bf 2}_3 \\otimes {\\bf 2}_4$ with the tensor product Eq. (\\ref{eq:cg3}).\nAfter $S_{1,2}^{}$ develop the VEVs, the resultant term violates the $Z_2^{}$ symmetry and induces a non-zero but negligibly small electron mass.\nSimilarly, the breaking effects in the neutrino sector are ignorable.\nIf $N \\le 7$, it may be possible to reproduce a realistic electron mass.\nAlthough this may be an interesting idea, we will discuss a different breaking mechanism of $Z_2^{}$ in the next section while assuming $N \\ge 9$.\nNote that the $Z_2^{}$ symmetry originates in the parity included in $D_N^{}$ and $O(2)$, whereas $SO(2)$ does not include it.\n\nWe stress that the $Z_2^{}$ symmetry could provide us with a natural explanation for the smallness of the electron mass and $\\theta_{13}^{\\rm PMNS}$: their smallness could be explained by approximate conservation of the $Z_2^{}$ symmetry.\nMoreover, the electron mass and $\\theta_{13}^{\\rm PMNS}$ are possibly correlated with each other through a mechanism of the $Z_2^{}$ symmetry breaking.\nIn contrast, $\\theta_{12}^{\\rm PMNS}$ can easily be large due to the mass degeneracy.\nIn the next section, we will show it is indeed the case.\n\n\n\\section{Model with complex scalars}\n\\subsection{Model}\nOne way to break the $Z_2^{}$ symmetry is to promote $S_{I}^{}$ to complex scalars with complex VEVs. \nIn the current and next subsections, just for simplicity, we assume that all the dimensionless couplings are real and invoke spontaneous CP violation (SCPV).\nThe availability of SCPV will be discussed in the next subsection, and here we simply rewrite the VEVs of the scalars as\n\\begin{eqnarray}\n\\langle H \\rangle=v,~~\n\\langle S_I \\rangle=(s_1 e^{i\\phi_1}~~s_2 e^{i\\phi_2})^T\\,.\\label{eq:vev}\n\\end{eqnarray}\nAs can be seen from Eq. (\\ref{eq:z2}), an unbroken limit of the $Z_2^{}$ symmetry corresponds to $\\phi_1^{} = \\phi_2^{}$, which keeps the right-hand side real, and thus a slight splitting between them is expected to trigger a non-zero electron mass, $\\theta_{12}^{\\rm PMNS}$ and $\\theta_{13}^{\\rm PMNS}$.\nThe Lagrangian is augmented by the following new terms:\n\\begin{eqnarray}\n{\\cal L}_{\\rm new}^{}\n&=&\\frac{y^{'}_{i}}{\\Lambda_F^2} \\overline{L}_I^{} H \\ell_{i}^{} (S_{}^{*2})_I\n+ \\frac{y''_{i}}{\\Lambda_F^2} \\overline{L}_I^{} H \\ell_{i}^{} (|S_{}|^2)_I \n\\nonumber \\\\\n&&\n+ \\frac{g_\\nu^\\prime}{\\Lambda_\\nu \\Lambda_F^2}L_3 L_I H H (S^{*2})_I\n+ \\frac{g_\\nu''}{\\Lambda_\\nu \\Lambda_F^2}L_3 L_I H H (|S|^2)_I \n\\nonumber \\\\\n&&\n+ \\frac{h_\\nu^\\prime}{\\Lambda_\\nu \\Lambda_F^4}(L_J L_K)_I H H (S^{*4})_I\n+ \\frac{h_\\nu''}{\\Lambda_\\nu \\Lambda_F^4}(L_J L_K)_I H H (|S|^4)_I \n\\nonumber \\\\\n&&\n+ \\frac{h_\\nu'''}{\\Lambda_\\nu \\Lambda_F^4}(L_J L_K)_I H H (S^2|S|^2)_I\n+ \\frac{h_\\nu''''}{\\Lambda_\\nu \\Lambda_F^4}(L_J L_K)_I H H (S^{*2}|S|^2)_I\\,.\n\\end{eqnarray}\nSuppose $\\phi_2 = \\phi_1 + \\delta \\phi$ and $\\delta\\phi \\ll 1$, the charged lepton mass matrix is approximated as\n\\begin{eqnarray}\n\\frac{1}{v}M^\\ell \n\\eqn{\\simeq} \n\\bmx{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\ny_1^0 & y_2^0 & y_3^0\n\\emx \n+\n\\frac{1}{\\Lambda_F^2}\n\\bmx{ccc}\n\\tilde{Y}_1^{} \\delta_1^{}  & \\tilde{Y}_2^{} \\delta_1^{} & \\tilde{Y}_3^{} \\delta_1^{} \\\\\n\\tilde{Y}_1^{} \\delta_2^{} & \\tilde{Y}_2^{} \\delta_2^{} & \\tilde{Y}_3^{} \\delta_2^{}\\\\\n0 & 0 & 0\n\\emx\n-\n\\frac{i\\delta\\phi}{\\Lambda_F^2}\n\\bmx{ccc}\n \\tilde{Y}_1^\\prime \\delta_s^{} & \\tilde{Y}_2^\\prime \\delta_s^{} & \\tilde{Y}_3^\\prime \\delta_s^{} \\\\\n 0 & 0 & 0 \\\\\n0 & 0 & 0\n\\emx\\, \n\\label{eq:Me2}\n\\end{eqnarray}\nupto the first order of $\\delta\\phi$, where\n\\begin{equation}\n\\begin{split}\n\\tilde{Y}_i \n&= y_i e^{2i\\phi_1}(1+i\\delta \\phi) + y_i^\\prime e^{-2i\\phi_1}(1-i\\delta \\phi) + y_i''\\,, \\\\\n\\tilde{Y}_i^\\prime \n&= y_i e^{2i\\phi_1} - y_i^\\prime e^{-2i\\phi_1}\\,.\n\\end{split}\n\\end{equation}\nSimilarly, the neutrino mass matrix is\n\\begin{eqnarray}\n\\frac{\\Lambda_\\nu}{v^2_{}} M^\\nu \n&\\simeq &\n\\begin{pmatrix}\nf_\\nu &0 &0\\\\\n0 &f_\\nu &0\\\\\n0 &0 &f_\\nu'\n\\end{pmatrix}\n+ \n\\frac{1}{\\Lambda_F^2}\n\\begin{pmatrix}\n0 &0 &\\tilde{G}_\\nu^{} \\delta_1^{} \\\\\n0 &0 &\\tilde{G}_\\nu^{} \\delta_2^{} \\\\\n\\tilde{G}_\\nu^{} \\delta_1^{} & \\tilde{G}_\\nu^{} \\delta_2^{} &0\n\\end{pmatrix}\n-\n\\frac{i\\delta\\phi}{\\Lambda_F^2}\n\\begin{pmatrix}\n0 &0 & \\tilde{G}_\\nu^\\prime \\delta_s^{}\\\\\n0 &0 & 0 \\\\\n\\tilde{G}_\\nu^\\prime \\delta_s^{} & 0 &0\n\\end{pmatrix} \n\\nonumber \\\\\n&&+ \n\\frac{1}{\\Lambda_F^4}\n\\begin{pmatrix}\n \\tilde{H}_\\nu^{} (\\delta_1^2 - \\delta_2^2) & \n \\tilde{H}_\\nu^{} 2\\delta_1^{}\\delta_2^{} & 0\\\\\n \\tilde{H}_\\nu^{} 2\\delta_1^{}\\delta_2^{} &\n-\\tilde{H}_\\nu^{} (\\delta_1^2 - \\delta_2^2) & 0\\\\\n0 &0 &0\n\\end{pmatrix}\n-\n\\frac{i\\delta\\phi}{\\Lambda_F^4}\n\\begin{pmatrix}\n \\tilde{H}_\\nu^\\prime \\delta_1^{}\\delta_s^{} & \n \\tilde{H}_\\nu^\\prime \\delta_2^{}\\delta_s^{} & 0\\\\\n \\tilde{H}_\\nu^\\prime \\delta_2^{}\\delta_s^{} &\n-\\tilde{H}_\\nu^\\prime \\delta_1^{}\\delta_s^{} & 0\\\\\n0 &0 &0\n\\end{pmatrix} ,\n\\label{eq:Mn2}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\begin{split}\n\\tilde{G}_\\nu^{} \n&= g_\\nu^{} e^{2i\\phi_1}(1+i\\delta \\phi) + g_\\nu^\\prime  e^{-2i\\phi_1}(1-i\\delta \\phi) \n+ g_\\nu'' \\,,\\\\\n\\tilde{G}_\\nu^\\prime \n&= g_\\nu^{} e^{2i\\phi_1} - g_\\nu^\\prime e^{-2i\\phi_1}\\,,\\\\\n\\tilde{H}_\\nu^{}\n&= [h_\\nu^{}e^{4i\\phi_1^{}}(1+2i\\delta\\phi) \n+ h_\\nu^\\prime e^{-4i\\phi_1^{}}(1-2i\\delta\\phi) + h_\\nu''\\\\\n&\\hspace{4cm}\n+ h_\\nu'''e^{2i\\phi_1^{}}(1+i\\delta\\phi) + h_\\nu'''' e^{-2i\\phi_1^{}}(1-i\\delta\\phi)] \\,,\\\\\n\\tilde{H}_\\nu^\\prime \n&= [2h_\\nu^{}e^{4i\\phi_1^{}} - 2h_\\nu' e^{-4i\\phi_1^{}} \n+ h_\\nu''' e^{2i\\phi_1^{}} - h_\\nu'''' e^{-2i\\phi_1^{}}]\\,.\n\\end{split}\n\\end{equation}\nThe terms proportional to $\\delta\\phi$ violate the $Z_2^{}$ symmetry, which indicates that the electron mass and $\\theta_{13}^{\\rm PMNS}$ are proportional to $\\delta\\phi$.\nIn order to demonstrate this, we here simplify the diagonalization of the charged lepton mass matrix by requiring $\\sum_i y_i^0 \\tilde{Y}_i^* = \\sum_i y_i^0 (\\tilde{Y}_i^\\prime)^* = \\sum_i \\tilde{Y}_i^{}(\\tilde{Y}_i^\\prime)^* = 0$.\nWith this simplification, we regard the third term of Eq. (\\ref{eq:Me2}) as small perturbations.\nThen, the electron mass is approximately obtained as\n\\begin{eqnarray}\nm_e^2 \\simeq (\\delta\\phi)^2 \\sum_i |Y_i^\\prime|^2 \\frac{\\delta_2^2}{\\Lambda_F^4} v_{}^2\\,,\n\\end{eqnarray}\nwhile the $12$ and the $13$ element of the neutrino mass matrix gain\n\\begin{equation}\n\\begin{split}\n\\frac{\\Lambda_\\nu}{v^2_{}} \\bar{M}^\\nu_{12} \n&\\simeq i\\delta\\phi\\frac{\\delta_2^{}\\delta_s^{}}{\\Lambda_F^4} \\tilde{H}_\\nu^\\prime \n+ {\\cal O}\\left( \\frac{(\\delta\\phi)^2}{\\Lambda_F^4} \\right)\\,,\\\\\n\\frac{\\Lambda_\\nu}{v^2_{}} \\bar{M}^\\nu_{13} \n&\\simeq i\\delta\\phi\\frac{\\delta_2^{}}{\\Lambda_F^2} \\tilde{G}_\\nu^\\prime \n+ {\\cal O}\\left( \\frac{(\\delta\\phi)^2}{\\Lambda_F^2} \\right)\\,.\n\\end{split}\n\\end{equation}\nin the diagonal basis of the charged lepton mass matrix.\nThe other elements are almost the same as Eq. (\\ref{eq:Mnd}).\nNow, it is obvious that $\\theta_{13}^{\\rm PMNS}$ as well as the electron mass are proportional to and suppressed by $\\delta\\phi$.\nThe expression of $\\theta_{13}^{\\rm PMNS}$ can be derived from them, but it is rather complicated.\nHence, we refrain from showing it.\n\nThe model contains a sufficient number of parameters to fit the experimental data.\nNevertheless, we would like to emphasize that the model can reproduce experimental data without manipulating the dimensionless couplings hierarchical by hand.\nFor instance, we find the following parameter spaces ($\\sum_i y_i^0 \\tilde{Y}_i^* = \\sum_i y_i^0 (\\tilde{Y}_i^\\prime)^* = \\sum_i \\tilde{Y}_i^{}(\\tilde{Y}_i^\\prime)^* = 0$ are not placed):\n\\begin{equation}\n\\begin{split}\n&y^0_1 = y^0_2 = 1.2,~y^0_3 = 1.0,~\ny_1 = -y_2 = y_3  = y_1^\\prime = y_2^\\prime = -y_3^\\prime = 0.8,\\\\\n&y_1'' = -y_2'' = -y_3'' = 0.85\\sim 0.90,\\\\\n&f_\\nu^\\prime=1.0,~f_\\nu=0.93\\sim 0.95,~\ng_\\nu = g_\\nu'' =0.9,~g_\\nu^\\prime = 0.9\\sim 1.3,\\\\\n&h_\\nu = h_\\nu^\\prime = h_\\nu'' = -h_\\nu''' = -h_\\nu'''' = 0.8 \\sim 1.3,\\\\\n&\\frac{s_{1,2}}{\\Lambda_F} = 0.17\\sim 0.27,~\n|\\phi_1| = 1.2\\sim 2.0,~ |\\delta\\phi|= 0.08\\sim 0.10\\,,\n\\end{split}\n\\end{equation}\nfor the normal mass ordering, and\n\\begin{equation}\n\\begin{split}\n&y^0_1 = y^0_2 = 1.2,~y^0_3 = 1.0,~\ny_1 = -y_2 = y_3  = y_1^\\prime = y_2^\\prime = -y_3^\\prime = 0.8,\\\\\n&y_1'' = -y_2'' = -y_3'' = 0.84\\sim 0.88,\\\\\n&f_\\nu^\\prime=1.0,~f_\\nu=1.05\\sim 1.08,~\ng_\\nu = g_\\nu'' =-0.9,~g_\\nu^\\prime = 1.1 \\sim 1.5,\\\\\n&-h_\\nu = -h_\\nu^\\prime = -h_\\nu'' = h_\\nu''' = h_\\nu'''' = 1.3 \\sim 1.8,\\\\\n&\\frac{s_{1,2}}{\\Lambda_F} = 0.18\\sim 0.28,~\n|\\phi_1| = 1.2\\sim 2.0,~ |\\delta\\phi|= 0.08\\sim 0.10\\,,\n\\end{split}\n\\end{equation}\nfor the inverted mass ordering.\nThe parameter spaces are required to reproduce the charged lepton mass ratios at the $Z$-boson mass scale \\cite{rge}:\n\\begin{eqnarray}\n\\frac{m_e}{m_\\mu}=4.74\\times 10^{-3},~~\n\\frac{m_\\mu}{m_\\tau}=5.88\\times 10^{-2}\\,,\n\\end{eqnarray}\nand to satisfy $1\\sigma$ constraints of the oscillation parameters:\n\\begin{eqnarray}\n\\frac{\\Delta m_{12}^2}{\\Delta m_{23}^2}\\eqn{=}\n\\left\\{\\begin{array}{l}\n(2.94\\sim 3.35)\\times 10^{-2} \\hspace{1cm}{\\rm for~Normal} \\\\\n(2.97\\sim 3.30)\\times 10^{-2} \\hspace{1cm}{\\rm for~Inverted}\n\\end{array}\\right. , \\\\\n\\sin^2\\theta_{12}^{\\rm PMNS}\\eqn{=}\n\\left\\{\\begin{array}{l}\n0.291\\sim 0.325 \\hspace{1cm}{\\rm for~Normal} \\\\\n0.303\\sim 0.336 \\hspace{1cm}{\\rm for~Inverted} \n\\end{array}\\right. ,\\\\\n\\sin^2\\theta_{13}^{\\rm PMNS}\\eqn{=}\n\\left\\{\\begin{array}{l}\n(2.16\\sim 2.66)\\times 10^{-2} \\hspace{1cm}{\\rm for~Normal} \\\\\n(2.23\\sim 2.76)\\times 10^{-2} \\hspace{1cm}{\\rm for~Inverted}\n\\end{array}\\right. , \\\\\n\\sin^2\\theta_{23}^{\\rm PMNS}\\eqn{=}\n\\left\\{\\begin{array}{l}\n0.365\\sim 0.410 \\hspace{1cm}{\\rm for~Normal} \\\\\n0.569\\sim 0.626 \\hspace{1cm}{\\rm for~Inverted}\n\\end{array}\\right.\\,,\n\\end{eqnarray}\nfrom Refs. \\cite{fogli} (for Normal) and \\cite{valle} (for Inverted).\nNote that $\\theta_{12}^{\\rm PMNS}$ can be large owing to the mass degeneracy, but at least $10 \\%$ tuning is necessary between $f_\\nu$ and $f_\\nu^\\prime$ in order to reproduce a large $\\theta_{23}^{\\rm PMNS}$.\nNote also that, as discussed below Eq. (\\ref{eq:Diag_Ml_R}), the scales of $s_i^{}\/\\Lambda_F^{}$ are indeed close to that of $\\sqrt{m_\\mu\/m_\\tau}$.\n\nFurthermore, in those parameter spaces, the mass of the second generation neutrino and $\\langle m_\\nu^{} \\rangle$ are computed as\n\\begin{eqnarray}\nm_2^\\nu = 0.07 \\sim 0.08~\\ev\\,,~~\\langle m_\\nu^{} \\rangle = 0.07 \\sim 0.08~\\ev\\,,\n\\end{eqnarray}\nfor both the normal and inverted ordering cases.\n\n\n\\subsection{Scalar potential and spontaneous CP violation}\nWe discuss the possibility for SCPV in our model.\nWe expand the singlet scalar fields with its VEVs as\n\\begin{eqnarray}\nS \\eqn{=} \n\\begin{pmatrix}\nS_1\\\\\nS_2\n\\end{pmatrix}\n\\rightarrow\n\\begin{pmatrix}\ns_1 e^{i\\phi_1} + S_1\\\\\ns_2 e^{i\\phi_2} + S_2\n\\end{pmatrix}\\,.\n\\end{eqnarray}\nWe write the full scalar potential up to renormalizable level:\n\\begin{eqnarray}\nV \\eqn{=} V_{H} + V_S + V_{H S}\\,, \\\\\n&&V_{H} =\n  \\alpha \\left| H \\right|^2 \n+ \\beta \\left| H \\right|^4 \\,,\\label{eq:Higgs_1}\\\\\n&&V_S = \n  \\alpha_S \\left| S \\right|^2 \n+ \\alpha_S' {\\rm Re} \\left[ S^2 \\right] \n+ \\beta_S^a \\left| S \\right|^2_{\\bf 1} \\left| S \\right|^2_{\\bf 1} \n+ \\beta_S^b \\left| S \\right|^2_{\\bf 1'} \\left| S \\right|^2_{\\bf 1'} \n+ \\beta_S^c \\left| S \\right|^2_{\\bf 2} \\left| S \\right|^2_{\\bf 2} \\notag\\\\\n\\eqn{}~~~~~~+ \\beta_S' {\\rm Re} \\left[ S^4 \\right] \n+ \\gamma_S \\left| S \\right|^2 {\\rm Re} \\left[ S^2 \\right]\\,, \\\\\n&&V_{H S} = \n  \\lambda \\left| H \\right|^2 \\left| S \\right|^2 \n+ \\lambda' \\left| H \\right|^2 {\\rm Re} \\left[ S^2 \\right]\\,,\n\\end{eqnarray}\nwhere the couplings $\\beta_S^A \\,(A=a,b,c)$ distinguish different combinations of $S_{1,2}$ under the $D_N$ tensor product rules, and all of the couplings are supposed to be real. \nSubstituting the VEVs into the potential, we obtain\n\\begin{eqnarray}\nV_{H} \\eqn{=} \\alpha v^2 + \\beta v^4 \\,,\\\\\nV_S \\eqn{=} \n\\alpha_S ( s_1^2 + s_2^2 ) \n+ \\alpha_S' ( s_1^2 \\cos 2 \\phi_1 + s_2^2 \\cos 2 \\phi_2 ) \n+ \\beta_S^a ( s_1^2 + s_2^2 )^2 \\notag\\\\\n\\eqn{} \n- 4 \\beta_S^b s_1^2 s_2^2 \\sin^2 ( \\phi_1 - \\phi_2 ) \n+ \\beta_S^c \\{ ( s_1^2 - s_2^2 )^2 + 4 s_1^2 s_2^2 \\cos^2 ( \\phi_1 - \\phi_2 ) \\} \\notag\\\\\n\\eqn{}\n+ \\beta_S' \\left[ s_1^4 \\cos 4 \\phi_1 + s_2^4 \\cos 4 \\phi_2 + 2 s_1^2 s_2^2 \\cos [ 2 ( \\phi_1 + \\phi_2 ) ] \\right] \\notag\\\\\n\\eqn{}\n+ \\gamma_S ( s_1^2 + s_2^2 ) ( s_1^2 \\cos 2 \\phi_1 + s_2^2 \\cos 2 \\phi_2 ) \\,, \\\\\nV_{H S} \\eqn{=} \n\\lambda v^2 ( s_1^2 + s_2^2 ) \n+ \\lambda' v^2 ( s_1^2 \\cos 2 \\phi_1 + s_2^2 \\cos 2 \\phi_2 )\\,.\n\\end{eqnarray}\nHereafter, we presume that the singlet scalar fields were completely decoupled from the theory at a high energy scale and investigate only the potential $V_S$.\nThen, the minimization conditions are calculated as \n\\begin{eqnarray}\n\\frac{\\partial V_S}{\\partial \\phi_1} \\eqn{=} \n  -2s_1^2\\left[~\\{ \\alpha_S' + \\gamma_S \\left( s_1^2 + s_2^2 \\right) \\} \\sin 2 \\phi_1 \n+ 2 ( \\beta_S^b + \\beta_S^c ) s_2^2 \\sin \\left[ 2 \\left( \\phi_1 - \\phi_2 \\right) \\right] \\right.\\notag\\\\\n\\eqn{}\\left. + 2 \\beta_S' \\left( s_1^2 \\sin 4 \\phi_1 + s_2^2 \\sin \\left[ 2 \\left( \\phi_1 + \\phi_2 \\right) \\right] \\right)~\\right]=0\\,,\\label{cond_al_1}\\\\\n\\frac{\\partial V_S}{\\partial \\phi_2} \\eqn{=} \n  -2s_2^2\\left[~\\{ \\alpha_S' + \\gamma_S \\left( s_1^2 + s_2^2 \\right) \\} \\sin 2 \\phi_2 \n- 2 ( \\beta_S^b + \\beta_S^c ) s_1^2 \\sin \\left[ 2 \\left( \\phi_1 - \\phi_2 \\right) \\right] \\right]\\notag\\\\\n\\eqn{}\\left.+ 2 \\beta_S' \\left( s_2^2 \\sin 4 \\phi_2 + s_1^2 \\sin \\left[ 2 \\left( \\phi_1 + \\phi_2 \\right) \\right] \\right)~\\right]=0 \\,,\\label{cond_al_2}\\\\\n\\frac{\\partial V_S}{\\partial s_1} \\eqn{=} \n2s_1\\left[~\\alpha_S + \\alpha_S' \\cos 2 \\phi_1 \n+ 2 \\beta_S^a ( s_1^2 + s_2^2 ) - 4 \\beta_S^b s_2^2 \\sin^2 ( \\phi_1 - \\phi_2) \\right.\\notag\\\\\n\\eqn{} \n+ 2 \\beta_S^c \\{ s_1^2 - s_2^2 + 2 s_2^2 \\cos^2 ( \\phi_1 - \\phi_2 ) \\} \n+ 2 \\beta_S' ( s_1^2 \\cos 4 \\phi_1 + s_2^2 \\cos \\left[ 2 ( \\phi_1 + \\phi_2 ) \\right] ) \\notag\\\\\n\\eqn{}\\left. \n+ \\gamma_S \\{s_2^2 \\cos 2 \\phi_2 + ( 2s_1^2 + s_2^2 ) \\cos 2 \\phi_1 \\}~\\right]=0\n\\,,\\label{cond_d1}\\\\\n\\frac{\\partial V_S}{\\partial s_2} \\eqn{=} \n2s_2\\left[~\\alpha_S + \\alpha_S' \\cos 2 \\phi_2 \n+ 2 \\beta_S^a ( s_1^2 + s_2^2 ) - 4 \\beta_S^b s_1^2 \\sin^2 ( \\phi_1 - \\phi_2) \\right.\\notag\\\\\n\\eqn{} \n- 2 \\beta_S^c \\{ s_1^2 - s_2^2 - 2 s_1^2 \\cos^2 ( \\phi_1 - \\phi_2 ) \\} \n+ 2 \\beta_S' ( s_2^2 \\cos 4 \\phi_2 + s_1^2 \\cos \\left[ 2 ( \\phi_1 + \\phi_2 ) \\right] ) \\notag\\\\\n\\eqn{} \\left.\n+ \\gamma_S \\{ s_1^2 \\cos 2 \\phi_1 + ( s_1^2 + 2s_2^2 ) \\cos 2 \\phi_2 \\}~\\right]=0\n\\,.\\label{cond_d2}\n\\end{eqnarray}\nLet us set $\\alpha_S'$, $\\beta_S'$ and $\\gamma_S$ to zero just for simplicity, then the first two conditions become\n\\begin{eqnarray}\n2(\\beta_S^b + \\beta_S^c)s_2^2\\sin[2(\\phi_1 - \\phi_2)]=0,~~\n2(\\beta_S^b + \\beta_S^c)s_1^2\\sin[2(\\phi_1 - \\phi_2)]=0\\,.\n\\end{eqnarray}\nSuppose $\\phi_1 \\simeq \\phi_2$, which is preferred from a model building point of view, the third and fourth conditions give us\n\\begin{eqnarray}\ns_1^2 + s_2^2 \\eqn{=} -\\frac{\\alpha_S}{2 ( \\beta_S^a + \\beta_S^c )}\\,.\n\\end{eqnarray}\n\n\n\\section{Neutrinoless double beta decay}\nWe here discuss testability of the models.\nIn order to keep generality, we introduce the most-general $Z_2$-breaking terms to Eq. (\\ref{eq:Mnd}) by hand and re-parametrize it as\n\\begin{eqnarray}\n\\frac{\\Lambda_\\nu}{v^2} \n(V^\\ell_{})^T M^\\nu V^\\ell_{}\n\\eqn{=} \n\\begin{pmatrix}\nf_\\nu^{} & 0 &0 \\\\\n0 & f_\\nu^{} & \\bar{M}^\\nu_{23} \\\\\n0 & \\bar{M}^\\nu_{23} & f_\\nu^\\prime \n\\end{pmatrix}\n+\n\\begin{pmatrix}\n\\varepsilon_{11}^{} &\\varepsilon_{12}^{} &\\varepsilon_{13}^{}\\\\\n\\varepsilon_{12}^{} & 0 & 0\\\\\n\\varepsilon_{13}^{} & 0 & 0 \n\\end{pmatrix},\n\\label{eq:pMn}\n\\end{eqnarray}\nwhere all the parameters are complex, the second term violates the $Z_2$ symmetry, and the terms proportional to $s_i^4\/\\Lambda_F^4$ are neglected in the first term. \nHere, $\\bar{M}^\\nu_{23}$ is suppressed with $s_i^2\/\\Lambda_F^2$, and \nas mentioned just below Eq. (\\ref{eq:Diag_Ml_R}) the scales of $s_i^{}\/\\Lambda_F^{}$ can be estimated from the charged lepton mass ratio as $s_i^{}\/\\Lambda_F^{} \\sim \\sqrt{m_\\mu^{}\/m_\\tau^{}}$.\nTherefore, one can infer $|\\bar{M}^\\nu_{23}\/f_\\nu^\\prime| \\simeq 0.06$.\nThe scales of $\\varepsilon_{ij}^{}$ are unknown, but they may be at least smaller than $\\bar{M}^\\nu_{23}$ since $\\varepsilon_{ij}^{}$ are responsible for a small $\\theta_{13}^{\\rm PMNS}$.\n\nWe evaluate the effective mass of the neutrinoless double beta decay:\n\\begin{eqnarray}\n\\langle m_\\nu^{} \\rangle =\n\\left| c_{12}^2c_{13}^2m_1^\\nu e^{i\\gamma_1^{}} + s_{12}^2c_{13}^2m_2^\\nu e^{i\\gamma_2^{}} + s_{13}^2e^{-2i\\delta}m_3^\\nu e^{i\\gamma_3^{}} \\right|,\n\\nonumber\n\\end{eqnarray}\nwhere $s_{ij}^{}$ ($c_{ij}^{}$) is $\\sin\\theta_{ij}^{\\rm PMNS}$ ($\\cos\\theta_{ij}^{\\rm PMNS}$), and $\\gamma_i^{}$ denotes the Majorana CP violating phases.\nIn Eq. (\\ref{eq:pMn}), phases of $f_\\nu^{}$ and $f_\\nu^\\prime$ can be absorbed into the right-handed charged leptons as we are considering the diagonal basis of the charged lepton mass matrix.\nIn view of $f_\\nu^{},f_\\nu^\\prime \\gg \\bar{M}^\\nu_{23},\\varepsilon_{ij}^{}$, the eigenvalues should be dominated by $f_\\nu^{}$ and $f_\\nu^\\prime$.\nAs a result, it is conjectured that the Majorana phases are almost vanishing.\nMoreover, as long as we focus on the neutrino mass regions where $m_1^\\nu \\simeq m_2^\\nu$ holds, the third term can be dropped because of $s_{13}^2 \\ll 1$.\nGiven these facts, $\\langle m_\\nu^{} \\rangle$ is approximately given by\n\\begin{eqnarray}\n\\langle m_\\nu^{} \\rangle \\simeq m_1^\\nu \\simeq m_2^\\nu.\n\\end{eqnarray}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics*[width=0.7\\textwidth]{mass-mee.eps}\n\\end{center}\n\\vspace{-0.8cm}\n\\caption{\\footnotesize \nThe effective mass, $\\langle m_\\nu^{} \\rangle$, of the neutrinoless double beta decay as a function of the lightest neutrino mass, $m_1^\\nu$ ($m_3^\\nu$) for the normal (inverted) ordering case.\nThe region surrounded by the solid (dotted) curves corresponds to the normal (inverted) ordering case in the standard $3\\nu$ framework, where 3$\\sigma$ constraints of the neutrino oscillation parameters from \\cite{fogli} are imposed while varying CP phases from $0$ to $2\\pi$.\nThe red regions are favored by the model for $|\\bar{M}^\\nu_{23}\/f_\\nu^\\prime| > |\\varepsilon_{ij}^{}\/f_\\nu^\\prime|$, and the grey region corresponds to the case of $0.5 > |\\varepsilon_{ij}^{}\/f_\\nu^\\prime|$ (see text).\nThe horizontal-dashed lines display the strongest upper bound on $\\langle m_\\nu^{} \\rangle$ from the combined analysis of the EXO \\cite{exo} and KamLAND-Zen \\cite{kam} experiments, and their expected future bound \\cite{neut12}.\nThe vertical-dashed line represents the $95\\%$ C.L. upper bound on the neutrino mass from the Planck data \\cite{planck} in combination with a WMAP polarization low-multipole likelihood (WP), the high-resolution CMB data (highL), and constraints from baryon acoustic oscillation (BAO) surveys.\n} \n\\label{fig:0nbb}\n\\end{figure}\n\nIn Fig. \\ref{fig:0nbb}, we numerically calculate $\\langle m_\\nu^{} \\rangle$ while assuming $|\\bar{M}^\\nu_{23}\/f_\\nu^\\prime| = 0.06\\pm 0.04$ and $|\\bar{M}^\\nu_{23}\/f_\\nu^\\prime| > |\\varepsilon_{ij}^{}\/f_\\nu^\\prime|$ in Eq. (\\ref{eq:pMn}).\nAs can be seen, the model favors very narrow regions, and thus it can easily be confirmed or excluded once the lower bounds on $\\langle m_\\nu^{} \\rangle$ and the neutrino mass are available.\nEspecially, the experimental sensitivity on $\\langle m_\\nu^{} \\rangle$ would reach these regions in the near future.\nAs we noticed in Sect. III-A, at least $10\\%$ tuning is inevitable between $f_\\nu^{}$ and $f_\\nu^\\prime$ in order to reproduce a large $\\theta_{23}^{\\rm PMNS}$.\nBecause of this tuning, the model cannot cover the neutrino mass regions in which $m_3^\\nu$ is much far from $m_1^\\nu \\simeq m_2^\\nu$.\n\nThe above results and conclusions are based on the requirement $f_\\nu^{},f_\\nu^\\prime \\gg \\bar{M}^\\nu_{23},\\varepsilon_{ij}^{}$.\nFor instance, if one relaxes $|\\bar{M}^\\nu_{23}\/f_\\nu^\\prime| > |\\varepsilon_{ij}^{}\/f_\\nu^\\prime|$ into $0.5 > |\\varepsilon_{ij}^{}\/f_\\nu^\\prime|$, the favored regions start to broaden as depicted by the grey region in the case of normal mass ordering.\nWe also note that Fig. \\ref{fig:0nbb} is not a prediction since the number of parameters in the mass matrix is enough to reproduce any data.\nOur claim is that $\\langle m_\\nu^{} \\rangle$ would be found in such regions soon in the case of $f_\\nu^{},f_\\nu^\\prime \\gg \\bar{M}^\\nu_{23},\\varepsilon_{ij}^{}$, which is suggested by the model-building of the partial mass degeneracy.\n\n\n\\section{Summary and Future Works}\nInspired by $\\Delta m_{12}^2 \\ll \\Delta m_{23}^2$, we focus on the neutrino mass regions in which the first two generation neutrinos are quasi degenerate in mass.\nIn the limit of $\\Delta m_{12}^2 = 0$, a Majorana neutrino mass matrix respects a two-dimensional rotation symmetry: the first two generations constitute a doublet representation while the third one acts as a singlet representation.\nIn the charged fermion sectors, the symmetry results in zero masses for the first two generations, which may be a reasonable first-order approximation to the hierarchical mass spectra of the charged fermions.\nMoreover, the small CKM and large PMNS mixings can naturally be understood as a consequence of the Dirac and Majorana natures of fermions, respectively.\nWe propose a simple model for the lepton sector by means of a $D_N$ flavor symmetry and find that the smallness of the electron mass and $\\theta_{13}^{}$ could be explained by approximate conservation of its $Z_2^{}$ subgroup.\nWe also point out that the model would be tested by the future neutrinoless-double-beta-decay experiments and cosmological observations.\nIn order for the $Z_2^{}$ symmetry to be slightly broken, we extend the scalar sector so as to acquire complex VEVs.\nConsequently, the electron mass and $\\theta_{13}^{\\rm PMNS}$ turn out to be related with each other via CP violating phases.\nThe extended model can successfully reproduce the experimental data without imposing unnatural hierarchies among dimensionless couplings.\nHowever, at least $10\\%$ tuning between $f_\\nu$ and $f_\\nu^\\prime$ is necessary in order to generate a large atmospheric mixing.\n\nIn the present work, we have adopted $D_N^{}$ as our flavor symmetry in order to concentrate of the flavor puzzles of fermions.\nIt may be challenging to enlarge the symmetry to $O(2)$ while including an associated new gauge boson and gauge anomalies.\nThe quark sectors should also be included, and we need to check whether the CP phases responsible for the electron mass can simultaneously explain the up- and the down-quark mass with the Dirac phase in the CKM matrix.\nFurthermore, we plan to implement the Leptogenesis mechanism \nwithin a specific neutrino mass generation framework.\nThese issues will be studied elsewhere.\n\n\n\\section*{ACKNOWLEDGMENTS}\nThe authors would thank to J. Kubo and J. Heeck for useful discussions and comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:hajimeni}\n\nIn a cloud platform, various types of software, such as VMs, containers and UNIX processes, run and work closely together to form a scalable software development and execution platform. \\emph{Workload}~\\cite{SPIFFE} is a concept for managing such various types of software comprehensively.\nVMs and containers are examples of the workload.\n\nVarious packet processing should be applied at network devices like routers and switches to secure the workload, such as packet filtering for discarding packets from unintended workloads, and bandwidth and priority control for surely transmitting important packets.\nConventionally, network based identifiers such as IP addresses and port numbers are used to classify packets in network devices for identifying who is communicating, because a prefix of IP addresses represents a group of similar workloads, and port numbers are known.\nHowever, for inter-workload communications in a cloud platform, it is difficult to widely use network based identifiers for this purpose due to the following two reasons.\n\nFirst, modern system design on cloud, such as microservice architecture\\cite{Microservices}, requires a large number and frequent updates of entries for classifying packets in network devices.\nWorkloads of a system are frequently created and deleted due to autoscaling, deploying new software, etc, and each workload is created on servers where computing resources are available.\nUse of containers accelerates the speed of creation and deletion of workloads.\nIn addition, IP addresses on each server are assigned based on the location of the server\\cite{FBBGP}\\cite{LINEDC} on the network, and some of them will be reassigned to the workloads on the server, so the prefix of IP addresses does not represent a group of similar workloads.\nFurthermore, more and more types of workloads will be deployed due to the nature of microservice architecture, which divides a large application into multiple loosely coupled applications that communicate with each other.\n\nSecond, granularity of network based identifiers is sometimes too low to identify workloads at network devices.\nSmall workloads such as UNIX processes or containers may share one IP address assigned to VM and use random source port numbers, thus workloads working as clients cannot be identified by IP addresses and port numbers.\n\nAt the same time, we have to consider the fact that many legacy systems, which are not orchestrated by the cloud controller, are sometimes essential for systems on the cloud when the legacy systems cannot be easily replaced.\nThis means network devices have to classify packets to\/from many legacy systems and handle packets properly.\n\nOverlay networks enable to assign IP addresses in the same prefix to multiple similar workloads.\nHowever, overlay networks incur additional overhead on the design of the system, e.g., network devices to process packets by access control lists (ACLs) must be placed on the overlay networks.\nThis may hinder flexible system configuration changes, which is one of the benefits of the cloud.\nService Mesh~\\cite{ServiceMesh} represented by Istio~\\cite{Istio} is a middleware that performs various processing such as access control to application level messages.\nThis is achieved by intercepting and modifying packets at a proxy running along with each application, but Istio cannot prevent unintended packets processed at the proxy itself or OS.\n\nThis paper proposes Acila, a system to directly mark packets with the necessary information extracted from the identity of workloads, which mainly comes from orchestrators or controllers in the cloud, so that network devices can classify packets based on the identity of a workload.\nWe assign a \\emph{Service} to workloads whose packets are processed in the same way in the network, which is derived from the identity of the workloads.\nThen, we add an identifier of the service (called SACL ID) to packets, and network devices classify packets based on SACL IDs for ACLs, bandwidth and priority control, etc.\nIn this way, the number of entries can be decreased and there is no need to make any changes to entries even if a set of workloads associated with the same Service is changed.\n\nWe design and implement Acila, and we show that Acila can be applied to various use cases with two examples, packet filtering and priority control.\nThe evaluation shows that mechanisms with Acila greatly reduce the burden of managing entries for classifying packets with little overhead on packet processing performance.\n\nThe contribution of this paper is summarized as follows.\n\n\\begin{itemize}\n\\item We design Acila, a system for Service-based packet classification by marking packets with information extracted from the identity of workloads that send and receive packets.\nBy adding both Services of sender and receivers to packets, Acila can support not only containers and virtual machines, but also legacy systems through gateways.\n\n\\item We demonstrate Acila has various applications with two examples, access control lists whose processing should be applied at somewhere on the path to the destination, and priority control whose processing should be applied at everywhere to the destination.\n\n\\item We show that Acila greatly reduces the burden of managing entries for classifying packets by calculating necessary entries to be installed on the network devices, and comparing with the conventional network based identifier approach. The performance overhead caused by introducing Acila is small.\n\n\\end{itemize}\n\n\\section{Related Works}\n\n\\textbf{Protocols} There are many proposals to separate host identifiers and location identifiers to identify hosts regardless of locations where hosts are, such as HIP~\\cite{HIP}, Mobile IP~\\cite{MobileIP} and LISP~\\cite{LISP}.\nIn our scenario, identifiers in each workload level are too detail for packet processing like packet filtering, bandwidth control and priority control, and the number of entries and update frequencies at network devices are still high.\n\nWe can build an overlay network using protocols like IPIP~\\cite{IPIP} and VXLAN~\\cite{VXLAN}, so that IP addresses with the same prefix are assigned to similar workloads, and entries can be aggregated by the prefixes in a cloud platform.\nOn the other hand, entry management by prefixes may hinder flexible system configuration changes, which is one of the benefits of cloud platforms.\n\n\nGuichard et al.~\\cite{SRHCLS} published a concept which inserts source and destination `Classes' which is similar to our `Services' into SRH~\\cite{SRv6}, a packet format for segment routing in IPv6, so that network devices such as firewalls can identify Classes and reduce the number of entries.\nTheir purpose is similar to us, but the published concept is still in the early stage, and there remain many undiscussed topics, such as detailed packet format, methods to assign Classes, and evaluation. Also, this protocol is built on SRH so segment routing needs to be deployed to the network.\n\nTools for data plane programmability like P4~\\cite{P4} allows operators to specify how programmable network devices like Tofino~\\cite{Tofino} process packets.\nThanks to this, it is now more realistic to process packets having our own packet format at line rate.\nWe can utilize this tool for real deployment of the proposed method.\n\n\n\\textbf{Access control for microservices on cloud}\nSPIFFE~\\cite{SPIFFE} is a framework that manages the identities of workloads in a cloud platform.\nThis authenticates various types of workloads, such as UNIX processes and containers, and gives a certificate of the corresponding identity to the workloads.\nSPIFFE assumes that workloads perform mutual authentication with endpoints by mTLS using its certificates, and Istio~\\cite{Istio} implements access control in the same idea.\nIn this approach, other than payload within mTLS, such as attacks to the kernel and codes to handle mTLS itself, is not protected.\n\n\nCilium~\\cite{Cilium} executes packet filtering and tracing based on the identity of workloads.\nCilium adds identifier to packets about the identity of source workload at the source node, and applies actions to packets at the destination node.\neZTrust~\\cite{eZTrust} attaches similar identifiers to packets.\nThe identifiers can be mapped to more information than Cilium's one.\nSuch information includes process ID, UID and others captured by in-kernel tracers, and is stored in the centralized context map.\nCilium and eZTrust attaches some identifiers related to the source workload only.\nHowever, for network devices to apply ACLs, priority and bandwidth control, etc, information related to the destination is also required to identify the type and importance of the traffic.\nWe propose to add information about the identity of both workloads, at the source and the destination, to packets so that we can perform such processing within the network as well as end hosts.\n\n\\section{A Premise of Cloud Platform}\n\\subsection{Workloads and Labels}\n\n\\label{sec:workload}\n\nA piece of running software that provides a single function is called a workload~\\cite{SPIFFE}.\nFor example, in IaaS (Infrastructure as a Service), a VM provided to users from the cloud administrator or a UNIX process or a container in the VM can be a workload.\nIn CaaS (Container as a Service), a pod in Kubernetes or a container can be a workload.\n\nAs shown in Figure~\\ref{fig:register_workloads}, a workload can be identified from outside of the workload by the IP address, the listening port number, UNIX process information (e.g.~UID), etc., and a combination of these.\n\nAdministrators of workloads (or cloud users) assign labels to the workloads for management purposes.\nLabels contain metadata about the workload, e.g., customer, service name, role and version, and each label consists of a key and a value.\nTypical cloud platforms already have such kind of labels in their cloud orchestrators as metadata of the workload, assigned by cloud administrators or the users.\nFor example, Kubernetes~\\cite{Kubernetes} has metadata of workloads in the form of a set of key and its values.\nWhen such kind of information is not available or too coarse, we assume that further information is provided directly from the cloud administrators and the users.\n\n\\subsection{Trust Model in a Cloud Platform}\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/cloud.pdf}\n\t\t\\caption{Administrators in a cloud. Blue indicates administrative privileges is in the user while others in the administrator of the cloud.}\n\t\t\\label{fig:cloud}\n\t\\end{center}\n\\end{figure}\n\nAs shown in Section~\\ref{sec:workload} and Figure~\\ref{fig:cloud}, workloads are categorized to several types according to administration privileges that the cloud administrator allows a user to have.\nIn this paper, entities managed by the cloud administrator are treated as trustworthy, while those managed by cloud users are not treated as completely trustworthy.\n\nFor example, in IaaS, the cloud administrator deploys a VM and provides it to cloud users, and the users manage the VM.\nThe cloud administrator can confirm that packets sent to and received from the VM are truly done by the VM, but payloads of the packets cannot be trusted because the cloud user can change the payloads of the packets from the VM\\@.\nTherefore, there is no reliable way for the cloud administrator to identify workloads inside VMs such as UNIX processes and containers, and they have to rely on the users to identify them.\nIn CaaS, the cloud administrator can reliably identify workloads running as containers because the infrastructure that runs containers are managed by the cloud administrators.\n\n\\subsection{Target Packet Processing}\n\\label{sec:target_types}\n\nThe proposed method targets packet processing that requires more metadata about workloads than network based identifiers to configure it.\nSuch metadata includes owners of the workloads and roles in the system.\nThe administrators use metadata to express policies indicating how to handle packets to\/from workloads.\nIn conventional network devices, administrators translate such policies into rules using network based identifiers, and install such rules and intended actions to be applied to packets into the devices.\n\nFor example when an administrator tries to configure that database servers are accessible only from application servers, the administrator translates the database servers and the application servers into a set of IP addresses, and installs packet filtering rules using IP addresses into network devices.\nOther examples include prioritizing traffic related to workloads whose response time is sensitive to the overall system performance, limiting bandwidth consumed by workloads that handle background jobs, etc.\n\nThe proposed method tries to remove the use of network based identifiers from such rules and to provide alternative identifiers, given the nature of the workloads on the cloud where usage of resources is elastically changed.\n\n\\section{Acila: Assign Workload Identity Based Identifiers to Packets}\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\linewidth]{images\/acila-overview.pdf}\n\t\t\\caption{Acila overview}\n\t\t\\label{fig:acila_overview}\n\t\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:acila_overview} shows an overview of the proposed system, Acila.\nAcila Controller receives workload information and policies (if any) as input from cloud orchestrators and administrators.\nAcila Controller creates Services and SACL ID corresponding to each Service, and translates policies into rules using SACL ID.\nThen, the controller distributes workload information, SACL IDs and rules to SACL Gateways in network devices and servers, which act as a data plane handling packets with SACL ID.\n\nThis section explains handling SACL IDs including attaching them to packets.\nPolicy is covered at Section~\\ref{sec:policy}.\n\n\\label{sec:assign_sevice_to_w}\n\n\\subsection{Service}\n\nThe identity of a workload is the information that characterizes the workload, which includes not only network based identifiers but also application types, application characteristics, versions, servers deployed on, etc., and such information is expressed as labels.\nThis information varies according to workloads, but not all information is used to express policies by the administrators.\nRather, the policies use a subset of labels to select workloads whose packets are processed in the same way.\n\\emph{Service} represents a subset of labels required to classify packets that the same actions are applied.\nThen we associate each workload with a Service based on its labels.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/workload-services.pdf}\n\t\t\\caption{Workloads, Labels and Services}\n\t\t\\label{fig:register_workloads}\n\t\\end{center}\n\\end{figure}\n\nFor example in Figure~\\ref{fig:register_workloads}, given the same actions should be applied to the workloads whose labels are \\texttt{\\{user:~bob, group:~2\\}}, we create Service $\\beta$ that has the same labels, and then assign workload 2 and workload 3 to Service $\\beta$.\n\nNetwork devices have to identify Services corresponding to the source and destination workloads.\nTo do this, we add an identifier of the Service called \\emph{SACL ID} to packets. SACL ID is unique among all Services.\n\nFor simplicity, we assume that the labels associated with a workload and those associated with a Service have a one-to-one correspondence.\nWorkloads with the same labels belong to the same Service.\nAn example of the correspondence is shown in Figure~\\ref{fig:register_workloads}.\n\nThe creation and assignments of Services are done in Acila Controller, which is an application that manages various data used in Acila.\nOnce a workload is created or updated, such event is sent to Acila Controller with its labels.\nAt this time, if a Service with the same labels already exists, the Service is assigned to the workload.\nOtherwise, Acila Controller creates a new Service corresponding to the labels and assigns it to the workload.\n\n\\subsection{Adding SACL ID to Packets}\n\\label{sec:add_saclid}\n\nAn overview of adding SACL ID to packets sent from workloads is shown in Figure~\\ref{fig:how-to-attach-sacl-id}.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/how-to-attach-sacl-id.pdf}\n\t\t\\caption{Overview of adding SACL ID to packets}\n\t\t\\label{fig:how-to-attach-sacl-id}\n\t\\end{center}\n\\end{figure}\n\nA workload is divided into a Client Workload (on the client-side) and a Server Workload (on the server-side) since the method of identifying workloads and adding SACL IDs greatly differs.\nCorresponding Services are called a Client Service and a Server Service.\nAcila Controller regards all workloads as Client Workloads.\nWhen any listen port number is specified in the condition for identification of workload, the controller recognizes that Server Workloads are also registered, assuming that it also acts as a server.\nIn this case, the listen port number is excluded in the condition for identification.\n\n\\emph{SACL Gateway} is an entity that adds and removes SACL IDs to\/from packets.\nThe SACL Gateway is logically in the middle of Client\/Server Workloads and the rest of the network (called Client\/Server Workloads are under the SACL Gateway), and has information of the Client Workloads and the Server Workloads that is necessary for adding SACL IDs to packets.\nFor example, in the case of packet filtering, SACL Gateway holds the information of the Server Workloads that are allowed to communicate from the Client Workloads under the SACL Gateway.\n\nThe processing of SACL Gateway is attached to the interface where each workload is connected and the cloud administrator manages.\nFor example, when a VM is a workload, SACL Gateway will intercept and modify packets on the TAP interface through which the hypervisor host forward packets to\/from the VM or the physical interface of the hypervisor host.\n\nThe reason is available information to identify workloads and trustworthiness of SACL IDs in packets.\nWhen attaching the SACL Gateway on an interface near the workload, rich information that can only be known near the Client Workload, such as the Service IP in Kubernetes, is available to identify the Server Workload from the destination of a packet.\nThis leads to precise identification of the Server Workload with a smaller number of entries in the SACL Gateway.\nOn the other hand, when the SACL Gateway is installed in the area managed by the user, such as inside VMs or containers, the user may forge the SACL ID in packets, and the cloud administrator cannot trust the SACL ID in packets and additional mechanism to verify the SACL ID is necessary.\n\n\\label{sec:sacl_id_validate}\n\nFor devices that cannot implement SACL Gateway in it or external networks, network devices that support the extension of the SACL ID (e.g.~programmable switches) should be installed and run as SACL Gateway.\n\n\\textbf{SACL Gateway Processing} When packets are sent from a Client Workload to a Server Workload, the Client SACL ID is added by identifying the Client Workload based on the source IP address, and the Server SACL ID is added by identifying the Server Workload based on the destination IP address and port number.\n\nFor the reverse direction, from a Server Workload to a Client Workload, it is difficult to identify the Client Workload from the packet because the port number used by the client is not known in advance. To solve the problem, the server-side SACL Gateway performs Connection Tracking (Conntrack)~\\cite{Conntrack} to find the Client SACL ID from network based identifiers in packets. Conntrack remembers correspondences between the network based identifiers and the SACL ID of active connections between workloads.\n\nSACL Gateway removes SACL ID from arriving packets and forwards them to the workloads.\n\n\\textbf{LID} Some Client Workloads, such UNIX processes in a VM managed by a user, cannot be identified only by the information that the cloud administrator has.\nThe port number is not sufficient since the client uses random source port numbers.\nA connection is not stored in Conntrack when the workload starts to send packets.\n\nTo identify such workloads, we have to use internal information in the computing resource managed by the user.\nFor example, when a UNIX process in a VM managed by the user is defined as a workload, only the VM can identify which process sends packets.\nHowever, we do not want to put SACL Gateway into the VM because the SACL ID will become easily forged and untrustworthy.\n\nFor the above reason, we design a mechanism to partially delegate the trust of a SACL ID associated with a workload to the cloud user who manages the workload, in a way that only affects the user.\nThis mechanism conveys the information for identifying the workload from the computing resource managed by the user to the SACL Gateway.\nThe cloud administrator adds the SACL ID at the SACL Gateway using this information.\n\nThe computing resource that can identify the workload between the workload and the SACL Gateway, such as VM in the previous example, adds a temporary identifier of workload to the packets and conveys it to the SACL Gateway.\nThe identifier is called \\emph{LID} (Local ID) and \\emph{LID Marker} is installed into the computing resources to add the LID to the packets.\nSACL Gateway prevents a malicious LID Marker from impersonating a workload that is not under the LID Marker by mapping a pair of LID Marker and LID to a Client Workload.\n\nThis mechanism does not prevent the LID Marker from impersonating another Client Workload under the same LID Marker.\nUsers can use LID Marker for finer-grained workload identification that cannot be done by the cloud administrator alone.\nInstead, the management of the LID Marker and the trustworthiness of LID are left to the user.\n\n\\subsubsection{Packet Format}\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/packet-format.png}\n\t\t\\caption{Packet format for transmitting SACL ID}\n\t\t\\label{fig:packet_format}\n\t\\end{center}\n\\end{figure}\n\nWe use TLV (Type-length-value) in Hop-by-Hop Options of IPv6~\\cite{IPV6} as the packet format for transmitting SACL IDs. This is because IPv6 is being adopted in large-scale cloud data center networks, due to limited private IPv4 address space, and the following two reasons. Figure~\\ref{fig:packet_format} shows the packet format whose SACL ID is 64 bit long.\n\n\\textbf{Flexibility} A field with a fixed length such as IPv6 flow labels~\\cite{IPV6} is not suitable since the bit length of SACL ID will be limited by such fields. TLV has a variable length so we can design SACL IDs with sufficient length according to the scale of a cloud platform and make future extensions easy.\n\n\\textbf{Compatibility}\nIt is possible to reuse header fields defined by existing protocols such as VXLAN~\\cite{VXLAN}, Geneve~\\cite{Geneve} and MPLS~\\cite{MPLS} for adding SACL IDs to packets.\nHowever, network devices may perform unexpected interpretation and processing by values in header fields reused for different purposes.\n\nSince IPv6 Hop-by-Hop Option is standardized as a part of IPv6, network devices can treat it as a normal IPv6 packet.\nAlso, in the Hop-by-Hop Option (TLV), we can specify the action of the devices when the devices do not support the Type, with the first 2 bits of the Type.\nIf the value is \\texttt{00}, the device will `skip over this option and continue processing the header~\\cite{IPV6}.\nSo even if the device is unaware of SACL ID, the device just ignores the SACL ID and continues processing, and the packet will not be discarded as an unknown protocol.\n\n\\section{Realizing Target Packet Processing Types with Acila}\n\\label{sec:policy}\n\nThis section explains how we can apply the proposed method to targeted packet processing types in Section~\\ref{sec:target_types}, packet filtering and priority control as examples.\n\nAn entry in a network device is in the form of a set of \\texttt{(Client SACL ID, Server SACL ID, Value (Option))}.\nValue indicates additional information such as priority.\n\n\\subsection{Policy Management and Distribution}\n\n\\emph{Policy} is a set of rules to select a group of Client Workloads and a group of Server Workloads based on labels, and to specify actions or Values for actions applied to packets.\nPolicy is basically registered and managed by cloud users.\nABAC (Attribute-Based Access Control)~\\cite{ABAC} is used for this selection because ABAC has a high affinity with labels.\n\nThe users register a policy to select workloads based on their labels without being aware of the Services.\nAcila Controller interprets the policy to select appropriate Services, which is generated from the labels of the workloads.\n\nIn the following, we will describe the process inside Acila Controller. An overview and examples are shown in Figure~\\ref{fig:policy}.\n\n\\begin{figure}[!ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/apply-policy.pdf}\n\t\t\\caption{Policy management and distribution}\n\t\t\\label{fig:policy}\n\t\\end{center}\n\\end{figure}\n\nSelectors are used to select Services whose labels meet the conditions specified by the Selectors.\nEach Selector consists of: Key, which is a key of label to be searched for, Operator which determines whether the value corresponding to the Key should exist (\\texttt{in}) or not (\\texttt{not\\_in}), Values, which is a set of values that should or should not exist.\nIf there is more than one Selector, they are ANDed.\n\nA Policy has two Selectors, one for the client-side and the other for the server-side.\nWhen a Policy or a Service is registered or updated, Acila Controller finds a group of Services for both sides that match the Policy respectively.\nThen it generates pairs of \\texttt{(Client Service, Server Service, Value)} called \\emph{Rule}, and distribute entries based on the Rules to the appropriate network devices.\n\nPolicies are also used to select the Server Workload required for the client-side SACL Gateway to put Server SACL IDs into packets.\nAcila Controller distributes the information of Server Workloads so that each SACL Gateway identify Server Workloads where the Client Workloads under the SACL Gateway may send packets.\n\n\\subsection{Packet Filtering}\n\nWhen a Client Workload initiates a session, we enforce policies at the SACL Gateway on the server-side based on the SACL ID in the packet.\nValue is not used since Policy only needs to represent the client\/server Services to be allowed to communicate.\n\nSACL Gateways on the server-side (`SACL Gateway (server-side)' in Figure~\\ref{fig:how-to-attach-sacl-id}) filter out the packets from the Client Workload to the Server Workload.\nIf this is done only on the client-side, all SACL Gateways on the client-side (`SACL Gateway (client-side)' in Figure~\\ref{fig:how-to-attach-sacl-id}) need to filter out packets correctly in order to prevent undesired access to one Server Workload.\nThis may cause a delay until the policy is updated in the network.\nIn the proposed method, only SACL Gateways that take care of the Server Workload, which is expected to be smaller than the number of client-side SACL Gateways, have to update the rules so the delay is expected to be shorter.\n\nPackets in the reverse direction, from the Server Workload to the Client Workload, are handled by Conntrack as in Section~\\ref{sec:add_saclid}.\nWhen the session has been already recorded in the Conntrack, the packet is allowed.\n\nIn addition, network devices on the network perform packet filtering.\nIn this case, the devices that performs packet filtering based on the SACL ID, such as a programmable switch, needs to be installed between the SACL Gateway on the client-side and the server-side.\nThis method is mainly for supporting connections to legacy systems that is not managed by the cloud orchestrators.\n\n\\subsection{Priority Control}\n\nPriority control assigns a priority to packets based on the Services of both sides, and prioritizes packet processing on network devices.\nIn this case, Policy consists of Selectors for the client\/server Services to set the priority and a Value which represents the priority to be set.\nAcila Controller distributes entries consisting of \\texttt{(Client SACL ID, Server SACL ID, Value (Priority))} to all devices that enforce priority control based on SACL ID\\@.\n\n\\section{Implementation}\n\n\\label{sec:implementation}\n\nWe implemented a prototype system for packet filtering using the proposed method, including Service assignment to workloads, the addition of SACL IDs to packets. An overview is shown in Figure~\\ref{fig:acl_implementation}.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/acl-implementation.pdf}\n\t\t\\caption{Overview of the data plane in the prototype system}\n\t\t\\label{fig:acl_implementation}\n\t\\end{center}\n\\end{figure}\n\nWe add SACL ID to packets between VMs and perform packet filtering.\nClient Workload is a VM or a UNIX process in a VM and the Server Workload is a UNIX process in a VM\\@.\nWhen the Client Workload is a UNIX process in a VM, the LID Marker is installed in the VM\\@.\nSACL Gateway is installed in the hypervisor host of each VM\\@.\nSACL ID is 64 bits long.\n\nEach VM has an IPv6 address and is connected to each other including the hypervisor hosts via Linux Bridge.\n\nSACL Gateway is implemented on the hypervisor host using eBPF~\\cite{eBPF}.\nThe eBPF program adds Client\/Server SACL ID to packets from VMs and filters out the packets to VMs based on the SACL IDs.\nIt is attached to traffic control (tc) on the TAP interface connected to each VM\\@.\nAlso, we implement a simple Conntrack mechanism in the server-side SACL Gateway to store correspondence between network based identifiers and a SACL ID\\@.\n\n\nLID is set in the Hop Limit field of the IPv6 header~\\cite{IPV6} in the form of $LID \\%128+ 100$.\nHop Limit is reset to the default in the SACL Gateway on the hypervisor host.\nLID Marker uses the UID of UNIX process to identify the Client Workload.\nLID Marker is implemented with eBPF, configured in tc on the interface which is the default gateway in the VM\\@.\n\nAcila Controller is implemented as a web server. For simplicity, we assume that the registering workload is divided into Client Workload and Server Workload in advance.\n\n\\section{Evaluation}\n\nWe evaluate the proposed method in terms of the number of entries by simulation and the performance of SACL Gateway.\nWe pick up priority control because, in order to perform priority control effectively, rules should be applied to all the network devices that packets pass through, and a larger number of entries are required compared to packet filtering and bandwidth control that can be applied only at SACL Gateway or network devices at the end.\n\n\\subsection{The Number and Update Amount of Entries}\n\\label{sec:eval_entries}\nThis section evaluates how much the number of entries and their update amount can be reduced by introducing the proposed method.\n\n\\textbf{Cloud Platform Assumption} We assume a 3-tier CLOS network~\\cite{CLOS1}~\\cite{CLOS2} (Leaf-Spine architecture~\\cite{CLOS3}) as the network topology in the cloud platform.\nThis has been adopted in recent data center networks due to its ease of scale-out.\nEach physical server in a server rack is connected to a ToR (Top of Rack) switch in each server rack.\nEach ToR switch is connected to a leaf switch.\nAll leaf switches are connected to all spine switches, and vice versa.\nWe simulate the entries required for spine switches in addition to SACL Gateways since spine switches have entries covering all workloads or Services.\n\nHere, each rack and physical server is denoted by $r_i$ and $m_i$, respectively.\nA set of racks, a set of servers in the network, and a set of servers in rack $r_i$ are denoted as $R$, $M$, and $M_{r_i}$, respectively.\n\nMultiple VMs running in a physical server, and multiple workloads are on a VM\\@.\nWe denote each VM and each workload as $v_i$ and $w_i$ respectively.\nA set of all workloads, a set of VMs in physical server $m_i$ and a set of workloads on VM $v_i$ are represented as $W$, $V_{m_i}$ and $W_{v_i}$, respectively.\nThe number of all workloads $|W|$ is calculated as\n\\begin{equation}\n\t|W| = \\sum_{r_i \\in R} \\sum_{m_j \\in M_{r_i}} \\sum_{v_k \\in V_{m_j}} |W_{v_k}| \\nonumber\n\\end{equation}\n\nWe assume that each workload is assigned an independent network based identifiers since we want to compare the number of entries in the case where both the conventional method using network based identifiers and the proposed method achieve the same priority control.\n\nEach Service is denoted by $s_i$, and a set of all Services is denoted by $S$.\nThe set of Server Services whose connections from a Client Service $s_i$ have configured a priority is denoted by $\\mathit{SS}_{s_i}$. \nThe set of Client Services where a Server Service $s_i$ accept connections with priority-controlled packets is denoted as $\\mathit{CS}_{s_i}$.\n\nA workload belongs to one Service.\n$W_{s_i}$ represents a set of workloads that belong to Service $s_i$.\n$c_{w_i,w_j}$ shows the number of active connections between Client Workload $w_i$ and Server Workload $w_j$.\n\n\\subsubsection{Conventional Network-identifier Based Approach} \n\nWith conventional approach, an additional Conntrack table will be required in order to differentiate whether a packet is from a client to a server or vice versa.\nTo avoid this differentiation, $\\mathit{CS}_{s_i}$ and $\\mathit{SS}_{s_i}$ are treated as the destination and source Services of packets instead of the Client and Server Services, respectively.\nAn entry for priority control should be configured for each workload, and consists of the network based identifiers of the source workload and the destination workload, and priority value.\n\nSince each spine switch is connected to all leaf switches, the number of workloads under a spine switch is equal to $|W|$.\nThis value is the same among all spine switches.\n\nWe then discuss the number of entries required for a spine switch defined as $\\mathit{el}$.\nSince we cannot distinguish the direction of packets, from client to server or vice versa, we assume that a priority is set to packets from the destination Service to the source Service when a priority is set to packets from the source Service to the destination Service, in order to align the conditions with the proposed method.\nIn that case, a prioritized inter-workload communication is counted in both $\\mathit{CS}_{s_i}$ and $\\mathit{SS}_{s_i}$.\nThe number of entries required for a workload is the total number of workloads belonging to each destination Service that the workload makes connections to with priority configuration.\nTherefore, $\\mathit{el}$ is as follows.\n\\begin{equation}\n\t\\mathit{el} =  \\sum_{w_i \\in W} \\sum_{s_k \\in \\mathit{SS}_{s_j} \\atop (W_{s_j} \\ni w_i)} |W_{s_k}| \\nonumber\n\\end{equation}\n\nNext, we consider the number of entries to be updated in a spine switch when there is a change in workloads, Services, or the priority between Services. Note that `update' here includes the addition and removal of workloads and Services.\n\nWhen a workload $w_i$ is created or deleted, the entries for the cases where the workload is at the source and at the destination are added or removed from a spine switch.\nTherefore, the number of entries to be updated at a spine switch, $\\mathit{elu}_{w_i}$, is\n\\begin{equation}\n\t\\mathit{elu}_{w_i} = \\sum_{s_k \\in \\mathit{SS}_{s_j}} |W_{s_k}| + \\sum_{s_l \\in \\mathit{CS}_{s_j}} |W_{s_l}| \\quad (W_{s_j} \\ni w_i) \\nonumber\n\\end{equation}\n\nCreation or deletion of a Service is always accompanied by creation or deletion of workloads, and there is no addition or removal in entries caused by the Service.\nTherefore, the number of updated entries per spine switch when Service $s_i$ is created or deleted, $\\mathit{elu}_{s_i}$, is as follows.\n\\begin{eqnarray}\n\t\\mathit{elu}_{s_i} & =& \\sum_{w_i \\in W_{s_i}} \\mathit{elu}_{w_i} \\nonumber\n\\end{eqnarray}\n\nWhen one priority between Services is updated, priorities in both directions should be set.\nFor each direction, the number of updated entries per spine switch when the priority from Service $s_i$ to Service $s_j$ is updated,$\\mathit{elu}_{{s_i}{s_j}}$, is as follows.\n\\begin{eqnarray}\n\t\\mathit{elu}_{{s_i}{s_j}} &=& |W_{s_i}| * |W_{s_j}| \\nonumber\n\\end{eqnarray}\n\n\n\\subsubsection{With Proposed Approach}\n\nWe assume that SACL Gateways are installed on physical servers and spine switches apply priority controls using SACL ID\\@.\nAn entry for priority control on spine switches is in the form of (Client Service, Destination Service, Priority) and is required for each Service.\n\nA Client Service $s_i$ requires entries for each Server Service ($\\mathit{SS}_{s_i}$), so the number of entries required for a spine switch ($\\mathit{es}$) is as follows.\n\\begin{eqnarray}\n\t\\mathit{es} &=&  \\sum_{s_i \\in S} |\\mathit{SS}_{s_i}| \\nonumber\n\\end{eqnarray}\n\nSuppose that a SACL Gateway $g_i$ is installed on a server $m_i$.\nThe number of elements in the set of workloads under SACL Gateway $g_i$, $W_{g_i}$, is as follows:\n\\begin{eqnarray}\n\t|W_{g_i}| = \\sum_{v_j \\in V_{m_i}} |W_{v_j}| \\nonumber\n\\end{eqnarray}\n\nSACL Gateway $g_i$ has the following three types of entries.\n\n\\textbf{Entries to add Client SACL ID} They are pairs of (IP address and LID (if any), SACL ID) for all Client Workloads under the SACL Gateway.\nThe number of entries, $\\mathit{escc}_{g_i}$, is as follows.\n\\begin{eqnarray*}\n      \\mathit{escc}_{g_i} &=& |W_{g_i}| \n\\end{eqnarray*}\n\n\\textbf{Entries to add Server SACL ID} They are pairs of the network based identifiers and the SACL ID of the Server Workload with which all Client Workloads under the SACL Gateway may communicate. Considering the worst-case in which all workloads in a physical server belong to different Services and all workloads of the Server Services do not overlap, the number of entries, $\\mathit{escs}_{g_i}$, is as follows.\n\\begin{eqnarray*}\n      \\mathit{escs}_{g_i} &=& \\sum_{w_j \\in W_{g_i}} \\sum_{s_l \\in \\mathit{SS}_{s_k} \\atop (W_{s_k} \\ni w_j)} |W_{s_l}| \n\\end{eqnarray*}\n\n\\textbf{Entries to add SACL ID by Conntrack} An entry consists of two values: one is a pair of (Network based identifier of the Client Workload, Network based identifier of the Server Workload) and the other is a pair of both SACL IDs.\nLet $c_{w_i}$ be the number of active connections of the Server Workload $w_i$ to the Client Workloads. $c_{w_i}$ is calculated as follows.\n\\begin{equation}\n\tc_{w_i} = \\sum_{s_k \\in \\mathit{CS}_{s_j} \\atop (w_i \\in W_{s_j})} \\sum_{w_l \\in W_{s_k}} c_{{w_l},{w_i}} \\nonumber\n\\end{equation}\nTherefore, the number of entries of Conntrack, defined as $\\mathit{ess}_{g_i}$, is as follows.\n\\begin{eqnarray*}\n\t\\mathit{ess}_{g_i} &= &\\sum_{w_j \\in W_{g_i}} c_{w_j} \\nonumber \n\\end{eqnarray*}\n\nThe total number of entries in SACL Gateway $\\mathit{es}_{g_i}$ is $\\mathit{es}_{g_i} = \\mathit{escc}_{g_i} + \\mathit{escs}_{g_i} + ess_{g_i}$.\n\nWhen one workload $w_i$ is created or deleted, there is no need to change the entries.\nSo, the number of updated entries in a spine switch in this case, $\\mathit{esu}_{w_i}$, is as follows.\n\\begin{eqnarray}\n\t\\mathit{esu}_{w_i} & =& 0 \\nonumber\n\\end{eqnarray}\n\nCreation or deletion of a Service $s_i$ requires adding or deleting both entries for the cases where the Service is Client Service and Server Service.\nThe number of updated entries in a spine switch in this case, $\\mathit{esu}_{s_i}$, is as follows.\n\\begin{eqnarray}\n\t\\mathit{esu}_{s_i} & =& |\\mathit{SS}_{s_i}| + |\\mathit{CS}_{s_i}| \\nonumber\n\\end{eqnarray}\n\nWhen one priority between Services is updated, only the entries related to the updated Services need to be changed.\nSo, let $s_i$ be the Client Service and $s_j$ be the Server Service whose priority is updated, and the number of updated entries per spine switch, $\\mathit{esu}_{{s_i}{s_j}}$, is\n\\begin{eqnarray}\n\t\\mathit{esu}_{{s_i}{s_j}} & =& 1 \\nonumber\n\\end{eqnarray}\n\n\\subsubsection{Comparison}\n\n\nWe can confirm the number of entries in a spine switch is greatly reduced from the conventional approach $\\mathit{el}$ to the proposed approach, $\\mathit{es}$ as follows.\n\\begin{eqnarray*}\n\t\\mathit{es} &=&  \\sum_{s_i \\in S} |\\mathit{SS}_{s_i}| \\\\\n\t&=& \\sum_{s_i \\in S} \\frac{|W_{s_i}|}{|W_{s_i}|} \\sum_{s_k \\in \\mathit{SS}_{s_i}} \\frac{|W_{s_k}|}{|W_{s_k}|} \\\\\n\t&=& \\sum_{w_i \\in W} \\sum_{s_k \\in \\mathit{SS}_{s_j} \\atop (W_{s_j} \\ni w_i)}   \\frac{|W_{s_k}|}{|W_{s_k}| |W_{s_j}|} < el\n\\end{eqnarray*}\nThis is because we can represent the combination of Client Workload $w_{l} \\in W_{s_j}$ and Server Workload $w_{l} \\in W_{s_j}$ by one entry.\nInstead, servers need to maintain a certain number of entries.\nThis issue is discussed in Section~\\ref{sec:saclgateway_performance}.\n\nWe can also see that the number of updated entries when a workload or a Service is added or deleted, or when a priority is changed because multiple workloads are usually in one Service.\n\n\\subsection{Performance of SACL Gateway}\n\n\\label{sec:saclgateway_performance}\n\nSACL Gateway adds SACL IDs to packets in order to process packets based on SACL ID.\nThis means SACL Gateway has to perform additional processing that is not needed in conventional approach.\n\nTo evaluate the performance impact caused by SACL Gateway, we run a version of SACL Gateway that removed the packet filtering mechanism from the implementation explained in Section \\ref{sec:implementation}.\nWe measure the change in RTT and throughput by the number of entries in SACL Gateway.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/experiment.pdf}\n\t\t\\caption{Overview of SACL Gateway performance evaluation}\n\t\t\\label{fig:experiment}\n\t\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:experiment} shows the overview of the environment conducting this evaluation.\nTwo servers, Server A and Server B, are connected via an Ethernet switch by 100GbE\\@. SACL Gateway is configured on tc of the physical interface on both servers.\nWe set Server A as the client and Server B as the server.\nWe have measured the number of packets per second (pps, (1) in Figure~\\ref{fig:experiment}) when massive UDP packets have sent from Server A or Server B, and the RTT when TCP connections have been established between Server A and Server B ((2) in Figure~\\ref{fig:experiment}).\n\nEach server has AMD EPYC 7282 (16 core, 32 thread), 128GB RAM, and Arch Linux (Linux 5.9.9-arch1-1) is installed.\nMellanox ConnectX-5 is installed as 100GbE NIC, and queue length for both tx and rx is set to 8192.\n\nWe have used Ethr~\\cite{Ethr} 1.0.0 as a measurement program.\nThe number of simultaneous streams has been set to 16 and the measurement time to 60 seconds.\nServer A has run Ethr as a client and vice versa.\n\nFor pps measurement, UDP packets with 1 byte of payloads were generated.\nThe measurement has been performed in two cases independently, when the SACL Gateway is on the client-side (Server A) and when it is on the server-side (Server B).\nFor RTT measurement, TCP connections has been used.\n\nThe SACL Gateway on Server A\/B has been implemented with eBPF as described in Section \\ref{sec:implementation}.\nWe have made several changes from the original implementation for the measurement.\n\n\\textbf{Removal of the packet filtering mechanism}\nIn this experiment, we have removed the packet filtering mechanism since we want to see the overhead of attaching and removing SACL IDs, not filtering packets.\n\n\\textbf{Randomized entry operations}\nIn Ethr, the number of clients is the same as the number of connections, the default value is 16, and the number of servers is one.\nSince entries stored in the SACL Gateway are limited to those connections, it is insufficient for the measurement of a large number of entries that will be needed in cloud platforms.\nTherefore, in this experiment, we reproduced the cost of searching\/updating a large number of entries by randomizing the keys to search\/update as follows.\n\nGiven that the number of entries in the evaluation scenario is $X$, we insert $X$ entries at the beginning so that the key of the $n$th entry is an unsigned integer $n$ and the value a randomized dummy SACL ID\\@.\nWhen the SACL Gateway is searching or updating entries (entry operations), the key is randomly selected: $(rand \\% X)$ is used as the key, where $rand$ is a random 32-bit unsigned integer.\nThis lookup always hit.\nAlthough randomized keys are gathered in bit space differently from the actual environment, there is no gap in the performance during entry operation because all the entry tables used in the eBPF program are a hash map.\nThe hashed keys should be distributed since the type of the hash maps is \\texttt{BPF\\_MAP\\_TYPE\\_HASH} which uses Jenkins hash~\\cite{Jenkins}.\n\nOn the client-side SACL Gateway (Server A), different prefixes of the key in Conntrack are used among for initial data insertion, search and update so that any packet from the client to the server never hit Conntrack to simulate the cost of searching and updating entries.\nNote that normally Conntrack is hit after the first packet in a connection, but not used in the experiment.\n This does not increase performance but may decrease compared to the actual environment.\n\nDespite that it is common for the same key from the same network based identifiers to be used in one connection, the key is changed every packets in the experiment.\nThis drawback may decrease performance but not increase it because it leads to an increase in the cache miss rate and the number of newly inserted entries to Conntrack.\n\nWhile a random value is used for the key, the same operations are performed for the IPv6 header such as parsing in order to measure the correct performance impacts.\n\nThe number of entries used in the experiment is based on a simple assumption about the values.\nThe assumption is that: all servers have 8 VMs, each VM has 16 workloads, every Service include 15 workloads, a Service communicates with 2 Services with priority, 1 connection is established at any time between workloads.\nAs the scale of the cloud grow, more and more entries will be installed, so we introduce a positive coefficient $\\alpha$.\nAs a result, the number of entries for SACL Gateway is estimated as follows.\n\\begin{eqnarray*}\n\t\\mathit{escc}_g &=& 1.3 * 10^2 \\alpha  \\\\\n\t\\mathit{escs}_g &=& 3.8 * 10^3 \\alpha \\\\\n\t\\mathit{ess}_g &=& 3.8 * 10^3  \\alpha \\\\\n\\end{eqnarray*}\nNote that the numbers introduced by the simple assumption can be ignored when $\\alpha$ is large, and LID is not assumed here and the table for LID is not used.\n\nThe measurements have been made when $\\alpha$ was 0.1, 0.5, 1, 2, 4, 8, 16 and 32 respectively and when no SACL Gateway has been installed as the baseline.\n\n\\subsubsection{Result}\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/pps.pdf}\n\t\t\\caption{pps for different coefficient $\\alpha$ for the number of entries, from a client to a server}\n\t\t\\label{fig:pps}\n\t\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:pps} shows the average pps for 60 seconds when packets are from the client to the server.\n\nIn the case where $\\alpha = 1$,\nthe pps keeps $86 \\%$ of the baseline.\nThe pps is decreasing when $\\alpha$ becomes large.\nEven when the number of entries is very large, $\\alpha = 32$, the pps keeps $76 \\%$ of the baseline.\nOn the other hand, even when the number of entries is small ($\\alpha = 0.1$), the pps is reduced to $86 \\%$ of the case without SACL Gateway.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/reverse_pps.pdf}\n\t\t\\caption{pps for different coefficient $\\alpha$ for the number of entries, from a server to a client}\n\t\t\\label{fig:reverse-pps}\n\t\\end{center}\n\\end{figure}\n\nThe average pps for 60 seconds when packets are transmitted from the server to the client is shown in Figure~\\ref{fig:reverse-pps}.\n\nIn the case where $\\alpha = 1$, the pps keeps $88 \\%$ of the baseline.\nThe pps is decreasing when $\\alpha$ becomes large, but not as much as the case from the client to the server.\nEven when the number of entries is very large, $\\alpha = 32$, the pps also keeps $82 \\%$ of the baseline.\nOn the other hand, even when the number of entries is small ($\\alpha = 0.1$), the pps is also reduced to $88 \\%$ of the case without SACL Gateway.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{images\/rtt.pdf}\n\t\t\\caption{RTT between Server A and Server B for coefficient $\\alpha$ for the number of entries}\n\t\t\\label{fig:rtt}\n\t\\end{center}\n\\end{figure}\n\nThe average RTT over 60 seconds for TCP connections from Server A to Server B is shown in Figure~\\ref{fig:rtt}.\nFor $\\alpha = 1$, the RTT is almost the same with the baseline.\nWe can also see that the RTT is almost stable when $\\alpha$ becomes large.\n\n\\subsubsection{Discussion}\n\nThe performance degradation due to the introduction of the SACL Gateway with the expected number of entries was less than $15 \\%$.\nIt was also found that the impact of the number of entries to pps is small.\nTherefore, the performance does not decrease significantly even if the number of entries increases by an increase of workloads or introducing packet filtering.\n\nThe decrease in pps for the number of entries is smaller for packets from the client to the server than for packets from the server to the client is thought to be that the former requires three table searches (Conntrack, Client Workload, and Server Workload) while the latter requires only one (Conntrack).\n\nWhen the number of entries is small, $\\alpha = 0.1$, pps is $86 \\%$ and $88 \\%$ of each baseline, even though the search cost is considered to be low.\nThis should be due to the cost of parsing and expanding packets to insert SACL IDs at the SACL Gateway.\nThis may be improved by inserting SACL IDs during the packet assembly process using a kernel module, etc., instead of parsing and expanding the packet by eBPF after the packet has been assembled.\n\nAs for the RTT, there is no change in the presence of SACL Gateway or the number of entries, and the processing time of the SACL Gateway is so small that it can be ignored.\nMoreover, in many cloud platforms, packets usually pass through more devices such as ToR switches and leaf\/spine switches than in the experimental environment.\nIn such a platform, the baseline of RTT is expected to be higher.\nTherefore, the impact of the processing time of the SACL Gateway to the RTT is almost negligible.\n\n\\section{Discussion}\n\n\\label{sec:kousatsu}\n\n\\subsection{Application to Larger Scale Cloud Platforms}\n\nIn very large cloud platforms having large scale networks, the following issues will appear.\n\n\\textbf{Increase in the Number of Entries} When a spine switch maintains the rules between all Services as it does now, the number of workloads and Services will increase, and the number of entries ($\\mathit{es}$) will increase.\nAs a result, the number of entries may exceed the capacity of the switch.\nIn response to this, the following solutions can be considered.\n\n\\textit{Optimized Service generation}\nIn this paper, labels of a Service have a one-to-one correspondence with labels of a workload, but the labels may contain unnecessary information such as a serial number, which may increase the number of Services and entries.\nIt is possible to specify and limit the key of the label used to associating with the Service, or generate a Service with the appropriate keys to classify packets in all use cases introduced in the network.\n\n\\textit{Structured SACL ID}\nStructuring SACL IDs is also a promising approach so that network devices can decide the actions to be performed by looking at a part of the SACL ID\\@.\nThere are various ways to do this; assigning a common upper bit determined by a subset of labels for a Service, similar to entries of routing tables.\nWith structured SACL ID, the number of entries in a network device can be reduced by handling a group of Services together instead of entries for each Service.\n\n\\textbf{Increasing the Load of Acila}\nCurrently, all the information used is centrally managed by a single Acila Controller.\nIt may become difficult to manage them in a single instance, due to the increased load on the DB for example.\nWe can consider operating multiple Acila Controller instances such as on each data center.\nThere are some concerns about operating multiple Acila Controller instances across the data centers as follows.\n\n\\textit{Support for Policies} When Services are managed in a distributed manner, an Acila Controller instance cannot know which Service defined in another instance matches a policy.\nThere are some solutions, one is to centrally manage Services and their associated labels in one place and each Acila Controller instance converts them into rules by querying them to the centralized management entity.\nAnother one is to develop a mechanism that allows Acila Controllers to query each other for Services that match a policy.\nThe latter method has better scalability in terms of the load of Acila Controller, but the presence of many Acila Controllers may lead to degrade the performance by locking the database when updating a Policy locking problems, and so on.\n\n\\textit{Issuance of a common SACL ID} The same SACL ID must be used for the same Service in multiple Acila Controllers.\nThis can be done by managing the issuance of SACL IDs in one place as described above.\nAnother solution is to generate a common SACL ID in a distributed manner.\nWe can use the hash value from labels as SACL ID with a hash function having sufficient collision resistance.\nHowever, there is a small possibility of collision, and we cannot simply use the hash function when SACL ID is structured.\n\n\\textbf{Exhaustion of SACL ID in Bit Space} While this research does not specify the length of SACL ID in the packet format, when the length is small, there is a possibility that the SACL ID bit space will run out as the scale of the cloud platform grows and the number of Services increases.\nOn the other hand, if the length is unnecessarily large, the packet length will become longer and the goodput will be decreased.\n\n\\subsection{Precise Workload-Service Association}\nAcila relies on network based identifiers to associate a workload with a Service, and SACL Gateways must have a large number of entries to do this.\nSACL Gateways should be richer information for this association, such as an interface where a workload is connected, and use of such information may reduce the number of entries.\nApplications running on workloads have more data, and such data is also useful for the association.\nFor example, designing a new socket option so that applications pass some clues to identify a Service, and the kernel or other modules use the clues to associate with a Service.\n\n\\subsection{Protocol}\n\nThe protocol for adding SACL ID uses IPv6 Hop-by-Hop Options, this means that IPv4 packets are not currently supported and needs to be encapsulated in IPv6 packets.\nHeader fields in the packet format itself is independent of IPv6 addresses, so we can easily design the same options in IPv4 packets, but the compatibility with network devices that do not support SACL ID should be carefully assessed.\n\n\\section{Conclusion}\n\nUse of network based identifiers, IP addresses and port numbers, are inappropriate for classify source and destination hosts of packets in the cloud platforms when identity of such hosts are required to configure the classification rules.\nThis is due to the fact that the cloud elastically creates and deletes workloads in servers whose resources are available according to the load on each system, and that network based identifiers are assigned sorely by the location of the servers.\nTherefore, the number of entries for such classification in network devices is large, and the entries should be frequently updated.\nTo solve these problems, we proposed Acila, a system that extracts the information related to workloads, and reduces the number and update frequency of the entries by handling multiple workloads together as Services.\nThe system assigns identifier of Services (SACL ID) to packets between workloads so that network devices can identify Services that the source and destination of packets belong to.\nWe also proposed applications to packet filtering and priority control using Acila.\nThrough evaluation, we confirmed the effectiveness of Acila by evaluating the number of entries required for simulated priority control.\nWe also benchmarked the performance of our SACL Gateway implementation and confirmed that there was no significant overhead.\n\nFuture works include extending Acila to support more large scale cloud platforms, which hosts tens of thousands of server racks by structuring SACL IDs and optimizing Service generation methods.\n\n\\nocite{*}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMesh-free (also called meshless) methods for solving partial differential equations (PDEs)\narose in 1970s and are still an active topic of research in applied mathematics today.\nIn mesh-free methods the computational domain is represented by a could of\npoints instead of a mesh of elements, as is typical for mesh-based methods.\nThe weak form mesh-free methods are most often analogous to the well established Finite Element\nMethod (FEM), while strong form methods are most often generalization of the Finite Difference\nMethods (FDM).\n\nMany strong form methods have been proposed throughout the years, starting from Smooth Particle\nHydrodynamics (SPH)~\\cite{benz1990smooth}, followed by generalizations of FDM with\nthe Finite Point method (FPM)~\\cite{onate2001finite}, the Generalized Finite Differences\nmethod~\\cite{gavete2003improvements}, and Radial basis function-generated Finite Differences\n(RBF-FD)~\\cite{tolstykh2003using} to name a few.\nA significant\ndevelopment in RBF-FD has been a recently reported by using polyharmonic RBFs\naugmented with monomials~\\cite{bayona2017augment} to avoid stagnation errors\nand allow control over the rate of convergence. Substantial development has also been\nreported in the stabilization of the method in convection dominated\nregimes~\\cite{shankar2018hyperviscosity}, in adaptive\nsolution of elliptic problems~\\cite{oanh2017adaptive}, in methods for\npositioning computational nodes~\\cite{slak2019generation}\nand in surface meshless methods~\\cite{petras2018rbf,suchde2019meshfree}.\n\nA number of mature software implementations exist for FEM, such as\ndeal.II~\\cite{dealii}, DOLFIN (part of the FEniCS\nProject)~\\cite{logg2010dolfin} and FreeFem++~\\cite{hecht2012new}.\nSuch a diverse ecosystem of general purpose implementations has not yet been developed for the\nfield of strong-form meshless methods. There are implementations consisting of\nMatlab scripts and domain specific applications,\nsuch as MFDMtool~\\cite{milewski2013selected}, GEC\\_RBFFD~\\cite{bayona20153},\nMFree2D~\\cite{liu2002mesh}, RBFFD\\_GPU~\\cite{bollig2014radial}, and even a\nreview paper by Nguyen et al.~\\cite{nguyen2008review} that specifically deals with\ncomputer implementation, includes its own set of Matlab scripts.\n\nExtensible, tested, documented and published general-purpose libraries for mesh-free methods\nwhich would facilitate further research and practical applications of the field are scarce.\nFor older and established methods, such as SPH, high quality software packages are available,\nwith DualSPHyiscs~\\cite{crespo2015dualsphysics} being one example.\nAnother such package for particle-based methods is the Aboria library~\\cite{robinson2017particle}.\nTwo commercial mesh-free implementations are known to the authors. One is the Midas\nMeshFree~\\cite{midas} package, which uses the Implicit Boundary Method and a background integration\ngrid to perform simulations, and claims to perform ``finite element analysis''. The other is the\nMESHFREE software~\\cite{meshfree}, which implements the Finite Pointset\nMethod~\\cite{tiwari2003finite} and has an impressive suite of examples. However, it focuses on\napplications and not on the development of strong form mesh-free methods in general.\nIn 2014, Hsieh and Pan published ESFM: An essential software framework for meshfree\nmethods~\\cite{hsieh2014esfm} which is an object-oriented C++ framework for computations using\nweak-form meshless methods and claims to be the first of its kind. However, it is not publicly\navailable and the authors only shows examples of linear elasticity problems in the paper.\nAnother package to note is the RBF python package~\\cite{hines2015rbf} (although not present in the\nstandard Python Package Index), which implements RBF interpolation and RBF-based PDE solution\ntechniques.\n\nMany packages for PDE solving such as deal.II, DOLFIN,\nFreeFem++, DualSPHyiscs, Aboria and ESFM libraries use the C++ programming\nlanguage. FreeFem++ implements its own extended language on top of C++ core,\nwhile FEniCS offers Python bindings. Nonetheless, C++ seems to be the language of choice\nfor many such applications.\nNo open-source C++ library for dealing with strong form meshless methods is\nknown to authors. Therefore, to help further research and development in the\nfield of strong form meshless methods,\nwe present an open source C++ library Medusa (\\url{http:\/\/e6.ijs.si\/medusa}).\n\nOur team started the development of Medusa library in 2015 to support our\nresearch in the field~\\cite{kosec2019weak, kosec2018local} and to ease\nimplementation of applied solutions~\\cite{maksic2019cooling}. Over time, the interface\ngrew and matured, putting emphasis on modularity, extensibility and reusability. Similarly\nto listed FEM libraries, it relies heavily on the C++ template system and allows the programs\nto be written independently of the number of spatial dimensions with negligible run-time and\nmemory overhead. Special care is also taken to increase expressiveness and to be able to\nexplicitly translate mathematical notation into program source code. However, source code is\nstill standard compliant C++, which allows the user to use entirety of the C++ ecosystem.\n\nThe rest of the paper is organized as follows: a brief overview of strong-form meshless methods\nis presented in section~\\ref{sec:sfmm}, where the most common part of strong form meshless methods\nare identified and described. This is followed by the presentation of the library in\nsection~\\ref{sec:soft}, which also includes the relevant abstractions and\nrationale behind some design decisions. Two more interesting computational examples are\npresented in section~\\ref{sec:applications} with measurements of execution time presented\nalong with comparison to FreeFem++ presented in section~\\ref{sec:bench}.\n\n\\section{Strong from mesh-free methods}\n\\label{sec:sfmm}\nSimilarly to many other methods, the general parts of the solution procedure\nfor strong form mesh-free methods are:\n\\begin{enumerate}\n  \\item \\emph{Domain discretization}: the geometry of the spatial domain is discretized,\n  by placing computational nodes and finding their stencils. This part is described in\n  more detail in section~\\ref{sec:dd}.\n  \\item \\emph{Differential operator discretization}: the spatial partial differential operators\n  are discretized using method specific techniques. This part is described in\n  more detail in section~\\ref{sec:od}.\n  \\item \\emph{PDE discretization:} The remaining time-dependent part of the PDE is discretized and\n  then solved either implicitly or explicitly, with time iteration, or by only solving the implicit\n  sparse system once, for elliptic problems. This part is described in\n  more detail in section~\\ref{sec:pded}.\n\\end{enumerate}\n\nEven if the overall problem solution procedure is more complicated, and involves\ncoupled equations, additional physical models or non-linearities, such as in computational\nfluid dynamics, the above three parts represent the core of the solution\nprocedure. From our experience, these parts and their components are the elements worthy of\nabstraction and general implementation.\n\nA more detailed description of the three parts is given in the following subsections,\nwith their respective implementations presented in sections~\\ref{sec:domain-impl},\n\\ref{sec:approx-impl} and~\\ref{sec:op-impl}.\n\n\\subsection{Domain discretization}\n\\label{sec:dd}\nA discretization of a bounded domain $\\Omega \\subset \\mathbb{R}^d$ consists of $N$ nodes\n$\\mathcal{X} = \\{p_0, p_2, \\ldots, p_{N-1}\\}$ placed in the interior and on the boundary of\nthe domain. Each node is assigned a stencil (also called neighborhood or \\emph{support})\nconsisting of some nodes near it. We will denote the size of the stencil of $i$-th\nnode with $n_i$ and the indices of stencil nodes with\n$\\mathcal{I}(i) = (I_{i,1}, I_{i,2}, \\ldots, I_{i,n_i})$. The stencil of the $i$-th node\n$\\mathcal{N}(i)$ is the $n_i$-tuple \\begin{equation}\n  \\mathcal{N}(i) = (p_{I_{i,1}}, p_{I_{i,2}}, \\ldots, p_{I_{i,n_i}}).\n\\end{equation}\nEach node should be in its own stencil, and for simplicity we assume that\nit is the first one, i.e.\\ $I_{i, 1} = i$ holds for all $i = 1, \\ldots, N$.\nBoundary nodes are assigned outer unit normals $\\vec{n}_i$.\n\nGeneration of nodal distributions has often been considered\nas an easy and not too relevant first step. This is partly due to the fact that\nexisting mesh generators could be used to generate a suitable mesh and the user\ncan simply discard the connectivity information~\\cite{liu2002mesh}.\nBesides being conceptually flawed, such approach is also computationally wasteful and does not\neasily generalize to higher dimensions. Some authors even reported having difficulties to obtain\nnode distributions of sufficient quality~\\cite{shankar2018robust}.\nAs a response, there are currently two known algorithm for variable density node generation\nin irregular domains in arbitrary dimensions with our original algorithm~\\cite{slak2019generation}\npublished in 2019 and another described in an arXiv preprint~\\cite{van2019fast}.\nBoth of these algorithms are implemented in Medusa. In addition,\nMedusa provides classic discretizations of basic geometric shapes, support for\ngridded nodes and an ability to easily define custom node generation schemes (e.g.\\ hexagonal).\nMedusa also offers support for adding so-called ``ghost nodes'' to the boundary.\n\nThe remaining part of the discretization is to define the stencils, which is fully automated\nand considered part of the solution procedure in nearly all meshless methods.\nThe most widely used type of stencils consist of some number of closest\nneighbors. Besides those, balanced stencils can be used in adaptive\nsolutions~\\cite{oanh2017adaptive}. Both approaches are implemented in Medusa,\nalong with the ability to only restrict the stencils to certain node types.\nIt is also simple to define custom stencil selection algorithms, which are not included by default,\nfor example visibility-based stencils~\\cite{nguyen2008review}.\n\n\\subsection{Differential operator discretization}\n\\label{sec:od}\n\nMost strong-form meshless approximations approximate a partial differential operator\n$\\L$ at a point $p$ with a linear functional $\\b{w}_{\\L, p}^\\mathsf{T}$, using an approximation of the form\n\\begin{equation} \\label{eq:lapprox}\n  (\\L u)(p) \\approx \\sum_{j \\in \\mathcal{I}(p)} (\\b{w}_{\\L, p})_j u(p_j) = \\b{w}_{\\L, p}^\\mathsf{T} \\b u,\n\\end{equation}\nwhere point $p$ is not necessarily one of the computational nodes.\nHowever, the stencil indices $\\mathcal{I}(p)$ and stencil nodes $\\mathcal{N}(p)$ represent computational nodes.\nThe values $\\b{w}_{\\L, p}$ are called \\emph{stencil weights} or sometimes shape functions,\nas a legacy terminology originating from weak-form methods.\nOther approximations such as of Hermite type collocation~\\cite{li2013meshless} are\nalso possible, but less common.\n\nWe will describe two possibilities to obtain the stencil weights $\\b{w}_{\\L, p}$ which cover many\nmeshless formulations and are also included in Medusa by default.\nThe first is the generalized weighted least squares (GWLS) method,\nwhich includes many commonly used meshless approximations, such as\nSPH approximations~\\cite{benz1990smooth}, Finite Point Method~\\cite{onate2001finite},\nGeneralized Finite Difference method~\\cite{gavete2003improvements}, radial basis\nfunctions-generated finite differences (RBF-FD)~\\cite{tolstykh2003using}, meshless local\nstrong-form method~\\cite{slak2018refined}, Finite Pointset\nMethod~\\cite{tiwari2003finite}, diffuse approximate methods~\\cite{wang2012new}\nand many more.\n\nThe second is a more specific radial basis functions-generated finite differences (RBF-FD)\napproximation with monomial augmentation, which also offers some speed improvements.\nOther custom approximation schemes can be implemented and used, such as schemes that\nput additional constraints on the center weights to achieve diagonal dominance in\ndifferentiation matrices~\\cite{suchde2019meshfree}.\n\n\\subsubsection{Generalized weighted least squares}\nAn approximation of function $u\\colon \\mathbb{R}^d \\to \\mathbb{R}$ around $p^\\ast$ is sought in the form\n\\begin{equation}\n  \\label{eq:uhat}\n  \\hat{u}(p) = \\sum_{i=1}^m \\alpha_i b_i\\left(\\frac{p-p^\\ast}{s}\\right) =\n  \\b{b}\\left(\\frac{p-p^\\ast}{s}\\right)^\\mathsf{T} \\b{\\alpha}\n\\end{equation}\nwhere $\\b{b} = (b_i)_{i=1}^m$ is a set of \\emph{basis functions}, $b_i\\colon \\mathbb{R}^d \\to\\mathbb{R}$,\n$\\b{\\alpha} = (\\alpha_i)_{i=1}^m$ are the unknown coefficients and $s$ is a positive scaling\nfactor. For simplicity we will assume that $I(p) = (1, \\ldots, n)$ and $N(p) = (p_1, \\ldots, p_n)$.\nNote that if monomials are chosen for $b_i$, we obtain the same setup as for the standard\nmoving\/weighted least squares (MLS\/WLS) formulation~\\cite{levin1998approximation}.\n\nUsing the known values $u_i$ in nearby nodes $p_i$, the error\n\\begin{equation}\n    e_i = \\hat{u}(p_i) - u_i = \\b{b}\\left(\\frac{p_i-p^\\ast}{s}\\right)^\\mathsf{T} \\b{\\alpha} - u_i\n\\end{equation}\ncan be computed. A weighted norm of the error vector $\\b{e} = (e_i)_{i=1}^n$ is then minimized.\nIt can be expressed as\n\\begin{equation}\n   \\|\\b{e}\\|_{2,w}^2 = \\sum_{i=1}^n (w_i e_i)^2 = \\|W \\b{e}\\|_2^2 = \\|W(B \\b{\\alpha} - \\b u)\\|_2^2,\n\\end{equation}\nwhere $B$ is a rectangular matrix of dimensions $n \\times m$ with rows containing basis function\nevaluated at points $p_i$:\n\\begin{equation}\nB =\n\\begin{bmatrix}\nb_1\\left(\\frac{p_1-p^\\ast}{s}\\right) & \\ldots & b_m\\left(\\frac{p_1-p^\\ast}{s}\\right) \\\\\n\\vdots & \\ddots & \\vdots \\\\\nb_1\\left(\\frac{p_n-p^\\ast}{s}\\right) & \\ldots & b_m\\left(\\frac{p_n-p^\\ast}{s}\\right)\n\\end{bmatrix} =\n\\left[b_j\\left(\\frac{p_i-p^\\ast}{s}\\right)\\right]_{j=1,i=1}^{m,n} =\n\\left[\\b{b}\\left(\\frac{p_i-p^\\ast}{s}\\right)^\\mathsf{T}\\right]_{i=1}^n,\n\\end{equation}\nand $W$ is a diagonal matrix of weights, $W_{ii} = \\omega((p_i-p^\\ast)\/s)$,\nwhere $\\omega\\colon\\mathbb{R}^d \\to (0, \\infty)$ is a \\emph{weight function}.\nChoosing $\\omega \\equiv 1$ gives the unweighted version. The arguments of $b_j$\nare shifted and scaled to ensure better conditioning of matrix $B$~\\cite{nguyen2008review}.\n\nIf we wanted to construct an approximant from known values of $u_i$, we could just compute\ncoefficients $\\b \\alpha$ with standard methods for solving\nleast square problems, such as normal equations with Cholesky decomposition, QR\ndecomposition or SVD decomposition.\nHowever, to obtain an approximation of $\\L|_p$, we express $\\b\\alpha$ in closed form\nusing Moore-Penrose pseudoinverse as\n\\begin{equation}\n  \\alpha = (WB)^+W\\b u\n\\end{equation}\nand substitute it in the definition~\\eqref{eq:uhat} of $\\hat{u}$ which\nbecomes \\begin{equation}\n  \\hat{u}(p) = \\b{b}\\left(\\frac{p-p^\\ast}{s}\\right)^\\mathsf{T}  (WB)^+W\\b u.\n\\end{equation}\nThe value $(\\L u)(p)$ can be approximated by applying operator $\\L$ to $\\hat{u}$ which\ngives\n\\begin{equation}\n  (\\L u)(p) \\approx (\\L \\hat{u})(p) = (\\L\\b{b})\\left(\\frac{p-p^\\ast}{s}\\right)^\\mathsf{T} (WB)^+W\\b u =\n  \\b{w}_{\\L, p}^\\mathsf{T} \\b u,\n\\end{equation}\nwhere the weights $\\b{w}_{\\L, p}^\\mathsf{T}$ are computed as\n\\begin{equation}\n  \\b{w}_{\\L, p}^\\mathsf{T} = (\\L\\b{b})\\left(\\frac{p-p^\\ast}{s}\\right)^\\mathsf{T} (WB)^+W.\n\\end{equation}\nNote that the computation of Moore-Penrose pseudoinverse is not really necessary,\nsince $\\b{w}_{\\L, p}^\\mathsf{T}$ can be computed by first solving the (possibly) underdetermined system\n\\begin{equation} \\label{eq:wbsys}\n  (WB)^\\mathsf{T} y = (\\L\\b{b})((p-p^\\ast)\/s)\n\\end{equation}\nfor $y$ and then computing $\\b{w}_{\\L, p} = Wy$. System~\\eqref{eq:wbsys} can be solved using\nQR, SVD or any other appropriate decomposition, however, depending on $m$, $n$ and properties\nof $b_j$, it can even be square and positive definite, making it possible to use\nCholesky, LDL$^\\mathsf{T}$ or LU decompositions.\n\n\\subsubsection{Radial basis function-generated finite differences with monomial augmentation}\nWe again consider a partial differential operator $\\L$\nat a point $p$ of form\n\\begin{equation} \\label{eq:lapprox-simple}\n(\\L u)(p) \\approx \\sum_{j = 1}^n (\\b{w}_{\\L, p})_j u(p_j) = \\b{w}_{\\L, p}^\\mathsf{T} \\b u,\n\\end{equation}\nwhere $p_i$ are the neighboring nodes to $p$. The unknown weights in\napproximation~\\eqref{eq:lapprox-simple}\ncan be computed by enforcing equality for $n$ basis functions.\nA natural choice are monomials, which\nare also used in FDM, resulting in the Finite Point Method \\cite{onate2001finite}.\n\nIn the RBF-FD discretization the equality is satisfied for radial basis functions $\\phi_j$,\nwhich are functions\n\\begin{equation}\n  \\left\\{ \\phi_j(p) =\n  \\phi\\Bigg(\\left\\|\\frac{p-p^\\ast}{s} - \\frac{p_j-p^\\ast}{s} \\right\\|\\Bigg) =\n  \\phi\\Bigg(\\frac{\\left\\|p-p_j\\right\\|}{s}\\Bigg), \\; j = 1, \\ldots, n \\right\\},\n\\end{equation}\ngenerated by a radial function $\\phi\\colon[0, \\infty) \\to \\mathbb{R}$ and defined over the set of\nnearby centers $p_j$. The center of the coordinate system is once again shifted to $p^\\ast$ and\ndistances are scaled by $s > 0$ to improve conditioning.\n\nEach $\\phi_j$, for $j = 1, \\ldots, n$ gives rise to one linear equation\n\\begin{equation}\n\\sum_{i=1}^{n} w_i \\phi_j (p_i) = (\\L \\phi_j)\\left(\\frac{p-p^\\ast}{s}\\right)\n\\end{equation}\nfor unknowns $w_i$ obtained by substituting $\\phi_j$ for $u$ in~\\eqref{eq:lapprox}.\nThese equation form the following linear system:\n\\begin{equation} \\label{eq:rbf-system}\n\\begin{bmatrix}\n \\phi\\Big(\\frac{\\left\\|p_1-p_1\\right\\|}{s}\\Big) &\\cdots &\n \\phi\\Big(\\frac{\\left\\|p_n-p_1\\right\\|}{s}\\Big)  \\\\\n\\vdots & \\ddots & \\vdots    \\\\\n \\phi\\Big(\\frac{\\left\\|p_1-p_n\\right\\|}{s}\\Big) &\\cdots &\n \\phi\\Big(\\frac{\\left\\|p_n-p_n\\right\\|}{s}\\Big)\n\\end{bmatrix}\n\\begin{bmatrix}\nw_1 \\\\ \\vdots \\\\ w_n\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n(\\L\\phi_1)\\Big(\\frac{p-p^\\ast}{s}\\Big) \\\\\n\\vdots \\\\\n(\\L\\phi_n)\\Big(\\frac{p-p^\\ast}{s}\\Big) \\\\\n\\end{bmatrix},\n\\end{equation}\nwhere $\\phi_j$ have been expanded for clarity.\nThe above system can be written more compactly as\n\\begin{equation} \\label{eq:rbf-system-c}\nA \\b{w} = \\b \\ell_\\phi.\n\\end{equation}\nThe matrix $A$ is symmetric, and for some $\\phi$ even positive\ndefinite. Other approximation properties are also well studied~\\cite{wendland2004scattered}.\nAdditionally, the computation up to now is the same as using GWLS with $n=m$ and $b_j = \\phi_j$.\n\n\nTo ensure consistency up to a certain order, the computation can be augmented with monomials.\nLet $q_1, \\ldots, q_l$ be polynomials forming the basis of the space of $d$-dimensional\nmultivariate polynomials up to and including total degree $m$, with $l = \\binom{m+d}{d}$.\n\nAdditional constraints are enforced by extending~\\eqref{eq:rbf-system-c}\nas\n\\begin{equation} \\label{eq:rbf-system-aug}\n\\begin{bmatrix}\nA & Q \\\\\nQ^\\mathsf{T} & 0\n\\end{bmatrix}\n\\begin{bmatrix}\n\\b{w} \\\\ \\b \\lambda\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\b \\ell_{\\phi} \\\\ \\b \\ell_q\n\\end{bmatrix}, \\quad\nQ = \\begin{bmatrix}\nq_1(p_1) & \\cdots & q_l(p_1) \\\\\n\\vdots & \\ddots & \\vdots \\\\\nq_1(p_n) & \\cdots & q_l(p_n) \\\\\n\\end{bmatrix}, \\quad\n\\b \\ell_q = \\begin{bmatrix}\n(\\L q_1)(p^\\ast) \\\\\n\\vdots \\\\\n(\\L q_l)(p^\\ast) \\\\\n\\end{bmatrix}\n\\end{equation}\nwhere $Q$ is a $n \\times l$ matrix of polynomials evaluated at nodes $p_i$ and\n$\\b \\ell_q$ is the vector of values assembled by applying considered operator $\\L$ to\nthe polynomials at $p^\\ast$.\n\n\nWeights obtained by solving~\\eqref{eq:rbf-system-aug} are taken as values for $\\b{w}_{\\L, p}$, while\nvalues $\\b \\lambda$ are discarded.\n\n\\subsection{PDE discretization}\n\\label{sec:pded}\nWith stencil weights $\\b{w}_{\\L, p}$ computed, they are mostly used in two main patterns.\nThe first is to explicitly approximate $(\\L u)(p)$, with the field $u$ being known, such as in\nexplicit time iteration, during linearization of nonlinear PDEs, or simply to\nobtain a derivative of the field.\nThe second is in implicit form, when we wish to obtain a field $u$, such that the field values\nsatisfy a set of linear equations.\nThis usually happens when solving elliptic problems or during time iteration with\nat least partially implicit methods, such as Crank-Nicholson and implicit Euler's method.\n\nBoth usage patterns are described on typical examples in low-level detail in the following sections.\nWe judged that these patterns of spatial approximation are common enough that\nsuitable abstractions abstractions are offered in Medusa (see~\\ref{sec:op-impl})\nto avoid error-prone handling of indices, code repetition and poor readability.\n\n\\subsubsection{Explicit evaluation}\n\\label{sec:pded-exp}\nConsider a sample time-dependent initial value problem on domain $\\Omega$\n\\begin{align} \\label{eq:expp-start}\n  \\hspace{4cm}  \\dpar{u}{t}(p, t) &= (\\L u)(p, t) && \\text{ in } \\Omega, \\hspace{4cm} \\\\\n  \\hspace{4cm} u(p, t) &= f(p, t) && \\text{ at } t=0, \\hspace{4cm} \\\\\n  \\hspace{4cm}  u(p, t) &= g_d(p, t)  && \\text{ on } \\Gamma_d, \\hspace{4cm} \\\\\n  \\hspace{4cm}  \\dpar{u}{\\vec n}(p, t) &= g_n(p, t)   && \\text{ on } \\Gamma_n, \\hspace{4cm}\n  \\label{eq:expp-end}\n\\end{align}\nwhere $\\Gamma_d$ and $\\Gamma_n$ are Dirichlet and Neumann boundaries, respectively, and\n$f$, $g_d$ and $g_n$ are known functions. Using explicit Euler scheme in time, starting at $t=0$\nwith time step $\\Delta t$, we define $u^k_i = u(p_i, k \\Delta t)$. Time iteration using\nstrong form meshless approximations is performed as follows:\n\\begin{align} \\label{eq:exp-start}\nu^0_i &= f(p_i), \\\\\nu^{k+1}_i &= u^k_i + \\Delta t \\left( \\b{w}_{\\L, p_i}^\\mathsf{T} u_{I(i)}^k \\right), \\ \\text{ for internal\nnodes\n} p_i, \\\\\nu^{k+1}_i &= g_d(p_i, (k+1)\\Delta t), \\ \\text{ for Dirichlet nodes\n} p_i, \\\\\nu^{k+1}_i &= \\frac{g_n(p_i, (k+1)\\Delta t) - \\sum_{j=2}^{n_i} u_{I_{i,j}}^k \\sum_{\\ell=1}^d n_\\ell\n(\\b{w}_{\\partial_\\ell, p_i})_j}{\\sum_{\\ell=1}^d n_\\ell (\\b{w}_{\\partial_\\ell, p_i})_1}, \\text{ for\nNeumann nodes } \\ p_i, \\label{eq:exp-end}\n\\end{align}\nwhere Neumann boundary conditions are obtained by equating the discretized version~\\eqref{eq:neu}\nto $g_n$ and expressing $u_i$.\nExplicit discretization of Neumann boundary conditions is obtained by\napproximating coordinate partial derivatives with their discrete versions\n\\begin{align}\n \\dpar{u}{\\vec n}(p_i, t) &= \\sum_{\\ell=1}^d n_\\ell (\\partial_\\ell u)(p_i) \\approx\n   \\sum_{\\ell=1}^d n_\\ell \\b{w}_{\\partial_\\ell, p_i}^\\mathsf{T} u_{I(i)}\n = \\sum_{\\ell=1}^d n_\\ell \\sum_{j=1}^{n_i} (\\b{w}_{\\partial_\\ell, p_i})_j u_{I_{i,j}} \\\\\n  &=  \\sum_{\\ell=1}^d n_\\ell \\sum_{j=1}^{n_i} (\\b{w}_{\\partial_\\ell, p_i})_j u_{I_{i,j}}\n  =  \\sum_{j=1}^{n_i} u_{I_{i,j}} \\sum_{\\ell=1}^d n_\\ell (\\b{w}_{\\partial_\\ell, p_i})_j = \\\\\n  &= u_i \\sum_{\\ell=1}^d n_\\ell (\\b{w}_{\\partial_\\ell, p_i})_1 + \\sum_{j=2}^{n_i} u_{I_{i,j}}\n  \\sum_{\\ell=1}^d n_\\ell (\\b{w}_{\\partial_\\ell, p_i})_j, \\label{eq:neu}\n\\end{align}\nwhere we used $I_{i, 1} = i$ and $u_{I(i)}$ is the vector of function values in stencil nodes\n$u_{I(i)} = \\left(u(p_j)\\right)_{j \\in I(i)}$.\n\nThe equations~(\\ref{eq:exp-start}--\\ref{eq:exp-end}) contain explicit evaluations of meshless\ndiscretizations on known fields.\nSimilar expressions, containing the same explicit evaluations can be obtained for other time\ndiscretizations or for vector functions $u$.\n\n\\subsubsection{Implicit solution}\n\\label{sec:implicit}\nConsider a boundary value problem\n\\begin{align} \\label{eq:impp-start}\n\\hspace{4cm}  \\L u &= f && \\text{ in } \\Omega, \\hspace{4cm} \\\\\n\\hspace{4cm}  u &= g_d  && \\text{ on } \\Gamma_d, \\hspace{4cm} \\\\\n\\hspace{4cm}  \\dpar{u}{\\vec n} &= g_n  && \\text{ on } \\Gamma_n, \\hspace{4cm}\n\\label{eq:impp-end}\n\\end{align}\nwhere $\\Gamma_d$ and $\\Gamma_n$ are Dirichlet and Neumann boundaries, respectively, and\n$f$, $g_d$ and $g_n$ are known functions. Each of the above equations is\napproximated by a linear equation in corresponding computational nodes. The system of linear\nequations can be written as $Mu = r$, where $i$-th row of the system corresponds to the\nequation that holds in node $p_i$. Formally, the matrix $M$ and right-hand side $r$ are given by\n\\begin{align} \\label{eq:implicit-start}\n  M_{i, I_{i, j}} &= (\\b{w}_{\\L, p_i})_j, \\ \\text{ for } j = 1, \\ldots, n_i,\n  & r_i &= f(p_i), && \\text{ for internal nodes } p_i, \\\\\n  M_{i, i} &= 1, & r_i &= g_d(p_i), && \\text{ for Dirichlet nodes } p_i, \\\\\n  M_{i, I_{i, j}} &= \\sum_{\\ell=1}^d n_\\ell (\\b{w}_{\\partial_\\ell, p_i})_j,\n  \\ \\text{ for } j = 1, \\ldots, n_i, & r_i &= g_n(p_i), && \\text{ for Neumann nodes }\n  p_i.\\label{eq:implicit-end}\n\\end{align}\nMatrix $M$ is a sparse matrix with at most $\\sum_{i=1}^N n_i$ nonzero entries.\nSolution of the system $Mu=r$ is the numerical approximation of $u$.\n\nThe equations~(\\ref{eq:implicit-start}--\\ref{eq:implicit-end}) define the unknown field $u$\nimplicitly by using stencil weights.\nSimilar approximations can be obtained for vector equations, or in implicit\ntime stepping schemes.\n\n\\section{Software description}\n\\label{sec:soft}\nLooking at existing finite element software packages and based on our\nexperience with implementing\nstrong-form meshless PDE solution procedures, we isolated a set of\nimplementation requirements:\n\\begin{itemize}\n  \\item \\emph{Modularity}. Ability to change approximation, node generation, stencil selection,\n  and other algorithms is of crucial importance for fast prototyping that is\n  needed in research. The goal of Medusa is that different reported meshless\n  methods can be rapidly constructed by using different combinations of\n  provided classes.\n  \\item \\emph{Dimension independence.} The mathematical PDE formulation is independent of the\n  dimension of the problem, and we strive to conserve this property in the\n  implementation as well.\n  Implemented approximations, node placing algorithms and operators can be used\n  in any domain dimensionality simply by changing a template parameter, e.g.\n  there is virtually no difference between code for solution of problem in 2D\n  or 3D, or any other dimensionality.\n  \\item \\emph{Extensibility.} Allowing users to define their own shapes,\n  approximations and operators enables wide applicability, e.g.\\ implementing\n  additional stabilizations such as upwind or hyperviscosity is\n  straightforward.\n  \\item \\emph{Readability.} A clear mapping from mathematical notation to code helps reduce\n  errors in the code. Additionally, dealing with objects representing abstract concepts such as\n  operators, vector fields and domains directly instead of matrices and lists of indices\n  also helps avoid bugs.\n  \\item \\emph{Small overhead due to the abstraction}: the run-time has\n      small and often negligible overheads in comparison with ``bare-bones'' implementations.\n  \\item \\emph{Parallelization.} When possible, parallelization can be handled internally,\n  so that the program can remain relatively unchanged if the user decides for parallel execution.\n  \\item \\emph{Ease of use.} This involves easy import and export of common file formats,\n  access to examples and technical documentation.\n\\end{itemize}\n\nWe designed the Medusa library with above requirements in mind. The library is written\nin C++ using object oriented approach and C++'s strong template system to achieve\nmodularity, extensibility and dimension independence. The library has no requirements,\napart from the C++ standard library and optionally the HDF5 C library~\\cite{hdf5} for\nreading and writing binary HDF5 files. However, we include four open-source\nthird-party\nlibraries, namely the Eigen~\\cite{eigenweb} library for linear algebra,\nnanoflann~\\cite{nanoflann} library for spatial-search structures,\ntinyformat~\\cite{tinyformat} library for simple formatting and\nand RapidXML~\\cite{rapidxml} for XML file processing. These four libraries have been\npackaged together with Medusa source code for simplicity. An external version\nof Eigen can be easily used as well.\n\nMedusa is licensed under MIT license, but the included libraries\nEigen, nanoflann, tinyformat and RapidXML are licensed under\nMozilla Public License (v.\\ 2.0), BSD license, Boost Software License and\ndual Boost Software license \/ MIT license, respectively.\nThe repository also includes the Google test library which\nis licensed under BSD 3-Clause ``New'' or ``Revised'' License, but\nis used for unit testing purposes and not necessary for core functionality.\n\nThe official website of the library is \\url{http:\/\/e6.ijs.si\/medusa}. The library is\ndeveloped using the \\texttt{git} versioning system and the development is ongoing on\nGitLab~\\url{https:\/\/gitlab.com\/e62Lab\/medusa}. The library uses \\texttt{cmake} build system\nand can be used as a \\texttt{cmake} submodule or as a standard standalone static C++ library.\nLong compile times associated with large amounts of C++ templates are somewhat mitigated by\nseparating declarations from template definitions into \\cpp{Medusa_fwd.hpp} and other included\nfiles, explicitly instantiating most common class instances and linking them. If other instances are\ndesired, they can be explicitly instantiated or full template definitions available in\n\\cpp{Medusa.hpp} can be included.\n\nQuality of implementation is ensured through continuous integration, which build the library and\nruns its test suite, documentation generation tools, linters and compiles and runs\nall examples. This aims to minimize the risk of regressions, stale documentation or examples and\nensures code validity, uniform code style and validity of system dependencies. The library also\nincludes numerous assertions, which can be disabled at compile time, that help catch errors earlier\nin the debugging phase. We use Google test testing framework to develop and run over 300 tests. The\nde-facto standard documentation generation tool Doxygen is used to generate the technical\ndocumentation, which is available at \\url{http:\/\/e6.ijs.si\/medusa\/docs}. The \\texttt{cpplint} style\nand code checker is used. Additionally, our wiki page is available at\n\\url{http:\/\/e6.ijs.si\/medusa\/wiki}, where more detailed explanations of examples, the\ntheory behind the methods, practical applications and further information about development and\npotential building issues can be found.\n\nThe following section describe main modules of Medusa, dealing with domains, approximations and PDE\ndiscretization. Almost all core classes are templated using a \\cpp{vec_t} type, which contains two\nessential pieces of information, the dimension of the computational domain (\\cpp{vec_t::dim})\nand the scalar type used for numerical computations (\\cpp{vec_t::scalar_t}),\ne.g.\\ \\cpp{float} or \\cpp{complex<double>}.\n\n\\subsection{Domains}\n\\label{sec:domain-impl}\nThe main class representing domain discretizations is the \\cpp{template <class vec_t>\nDomainDiscretization} class, which closely resembles the description of domains discretizations\ngiven in section~\\ref{sec:dd}. It includes a list of $d$-dimensional points $p_i$,\neach one has an associated \\emph{type} $\\tau_i$, with positive $\\tau_i$ for internal nodes\nand negative $\\tau_i$ for boundary nodes. The boundary nodes also have their outer unit\nnormals $\\vec n_i$ stored. Additionally, stencil indices $I(p_i)$ are stored for each point.\nStencils of varying sizes are supported.\n\nDomain discretizations can be constructing by discretizing one of the predefined shapes,\nincluding $d$-dimensional spheres, cubes, 2d polygons, 3d polyhedra (given by STL files),\nas well as their unions, differences, translations and rotations.\nMost of them support discretization of boundaries with arbitrary spacing function $h$.\nFor discretizations of domain interiors, two dimension independent variable density\nnode generation algorithms are implemented, \\cpp{GeneralFill} and \\cpp{GrainDropFill}, based\non~\\cite{slak2019generation} and~\\cite{van2019fast}, respectively.\nOther node generation algorithms, such as grid-based fills and surface filling algorithms are also\navailable.\n\nTwo stencil selection algorithms are also available, \\cpp{FindClosest}, which\nconstructs stencils using the indices of defined number of closest nodes, and\n\\cpp{FindBalancedSupport}, which also ensures that stencils are balanced around\nthe central node.\n\nListing~\\ref{lst:domains} demonstrates some of the capabilities for creating and\nhandling domains. Figure~\\ref{fig:domains} shows the domains produced by the source code in\nlisting~\\ref{lst:domains}. The left part shows a 2D domain with relatively coarse\nvariable density discretization, with interior and boundary nodes and also shows stencils\nfor a few selected nodes. The right part shows a uniform discretization of a 3D\nmodel, obtained from a STL file.\n\n\\begin{listing}[H]\n\\inputminted[firstline=13,lastline=28,gobble=4]{cpp}{analyses\/src\/domain.cpp}\n\\caption{Construction and discretization of domains.}\n\\label{lst:domains}\n\\end{listing}\n\n\\begin{figure}[H]\n  \\includegraphics[width=0.44\\linewidth]{images\/domain1}\n  \\includegraphics[width=0.55\\linewidth]{images\/domain2}\n  \\caption{Domain discretizations produced by listing~\\ref{lst:domains}. A few selected nodes\n  are shown along with their support nodes in the left figure. The right figure shows a denser\n  discretization of a STL model.}\n  \\label{fig:domains}\n\\end{figure}\n\n\\subsection{Approximations}\n\\label{sec:approx-impl}\nThe library currently includes two approximation engines for computing\nwhich implement the procedures described in section~\\ref{sec:od}. These are\n\\cpp{template <class basis_t, class weight_t, class scale_t, class solver_t> class WLS},\n\\cpp{template <class rbf_t, class vec_t, class scale_t, class solver_t> class RBFFD},\nwith reasonable defaults for last few parameters.\nTemplate parameters allow for various combinations of basis functions $b_j$, RBFs $\\phi$,\nweight functions $\\omega$, scaling function $s$, and solvers to be used. By default,\nthe library includes monomial and RBF bases, Gaussian, Multiquadric, Inverse multiquadric and\nPolyharmonic RBFs, three scaling functions, various weights,\nand a variety of solvers included with Eigen. It is also easy for users to add their own RBFs,\nweights and bases. Since templates offer a (static) version of duck typing, any class with the\ninterface conforming to the e.g.\\ RBF concept as described in the documentation, can be used.\n\nThe power of this generality is shown in Figure~\\ref{fig:approx},\nwhere errors of various approximation setups are shown.\nThe Laplacian operator was approximated on a regular grid $G_h$ of points with\nspacing $h$ covering the unit square $[0, 1]^2$.\nThe error of the approximation was computed as \\[\n  e_h = \\max_{p_i \\in G_h}\\left|\\b{w}_{\\nabla^2\\!, p_i}^\\mathsf{T} \\b u_{I(i)} - (\\nabla^2 u)(p_i)\\right|.\n\\]\nThe test function was chosen to be $u(x,y) = \\sin(\\pi x) \\sin(\\pi y)$.\nFive different approximation setups were tested:\n\\begin{enumerate}\n  \\item RBF-FD with Gaussian RBFs $b_j(p) = \\exp(\\|p - p_j\\|^2\/\\sigma^2)$, using stencil\n  of $n=9$ closest nodes with no monomial augmentation, $\\sigma=100$ and\n  with scaling $s$ equal to the distance to the nearest neighbor.\n  LU decomposition was used to solve the system for stencil weights. \\label{itm:rbfgs}\n  \\item Like~\\eqref{itm:rbfgs}, but with $\\sigma=5$ and without scaling ($s=1$). \\label{itm:rbfnslu}\n  \\item Like~\\eqref{itm:rbfnslu}, but with SVD decomposition. \\label{itm:rbfnssvd}\n  \\item GWLS with $m=5$ monomial basis functions up to order 2, $n=9$ closest nodes,\n  Gaussian weight with $\\sigma=1$, scaling to closest node and SVD decomposition. \\label{itm:fpm}\n  \\item RBF-FD with polyharmonic splines $\\phi(r) = r^5$ and monomial augmentation of order\n  $m=2$ with $n=12$ closest nodes. \\label{itm:phs}\n\\end{enumerate}\nThe definition of these setups in Medusa is shown in listing~\\ref{lst:approx}.\nStencil sizes are not included, as their computation was already shown in listing~\\ref{lst:domains}.\n\n\\begin{listing}[H]\n  \\inputminted[firstline=9,lastline=13,gobble=4]{cpp}{analyses\/src\/approx_sample.cpp}\n  \\caption{Definition of various important approximations.}\n  \\label{lst:approx}\n\\end{listing}\n\nThese setups present some of the problems and answers in meshless strong form methods in recent\nyears. The question of choice of the shape parameter for RBFs is a long standing one, since the\nshape parameter often presents a trade-off between accuracy and the condition number of the\nmatrix~$A$~\\cite{wendland2004scattered}.\nCase~\\eqref{itm:rbfnslu} exhibits the expected behavior that Gaussian approximations converge\nuntil the condition number is too high, and numerical errors become predominant. Jagged behavior\ncan be smoothed by using SVD decomposition, however the overall outcome is the same.\nA simple remedy for this instability is to scale the shape parameter (or the space) to keep the\ncondition number constant. This solves the problems with numerical instability, but causes the\napproximation to diverge in a characteristic fashion with two local minimums~\\cite{bayona2010rbf}.\nThis lack of convergence is also often called divergence due to ``stagnation errors''.\nTwo more convergent cases are included, one is the Finite point method (Case~\\eqref{itm:fpm}),\nwhich achieves similar\nbehavior and accuracy to FDM~\\cite{onate2001finite} and another is RBF-FD using PHS augmented with\nmonomials (Case~\\eqref{itm:phs})~\\cite{bayona2017augment}, where accuracy and convergence order can\nbe easily controlled through augmentation.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{images\/approx_behaviour.pdf}\n  \\caption{Error of approximating the Laplacian with different approximation setups.\n  Less than 1 minute of computing time was needed to produce the data for this plot.}\n  \\label{fig:approx}\n\\end{figure}\n\n\\subsection{Operators}\n\\label{sec:op-impl}\nThis module defines one of the core functions of the library, which takes\na domain discretization with nodes $p_i$, an approximation engine and a list of operators\n$(\\L_1, \\ldots, \\L_\\ell)$\nand computes and stores stencil weights $(\\b{w}_{\\L_j, p_i})_{i=1, j=1}^{N, \\ell}$,\nfor all operators and all computational nodes in the domain.\nThese weights are stored in a \\cpp{ShapeStorage} class.\n\nThe library supports computing shapes for first and second derivatives, as well as for the\nLaplacian operator. Note that this allows for construction of arbitrary\nsecond order operators as\n\\begin{align}\n \\b{w}_{\\L, p} = \\sum_{1 \\leq |\\alpha| \\leq 2} a_\\alpha(p) \\b{w}_{\\partial_\\alpha, p}, \\text{ for }\n \\L = \\sum_{1 \\leq |\\alpha| \\leq 2} a_\\alpha(p) \\dpar{}{x^\\alpha},\n \\label{eq:gen-op}\n\\end{align}\nwhere $|\\alpha| = \\sum_{i=1}^d \\alpha_i$ and $ \\dpar{}{x^\\alpha} = \\dpar{^{|\\alpha|}}{x^\\alpha_1\n\\cdots x^{\\alpha_d}}$ are the standard multiindex notations.\nThis would also cover the Laplacian operator, however, equation~\\eqref{eq:gen-op} is not necessarily\nthe most efficient nor the most numerically stable way of computing the Laplacian for certain basis\nfunctions. User defined operators are supported as well, with the only requirement being that\nthe user implements application of the operator for a class of basis functions that is used in their\ncode. Our examples include solving the biharmonic equation to demonstrate this extensibility.\n\nThe \\cpp{ShapeStorage} class stores the computed weights for a chosen set of operators.\nThese shapes can be used to implicitly express or explicitly compute $\\L u$, as described in\nsections~\\ref{sec:pded} and its subsections.\n\nFor given scalar or vector field $u$, we directly support most common scalar\nand vector operators, such as coordinate derivatives of first and second order,\nLaplacian, gradient, divergence, gradient of divergence, directional derivatives, as\nwell as any user defined operators.\n\nTwo examples of PDE solutions will be given in this section, to illustrate the\nfunctionality of the library for explicit and implicit solving. Special effort\nwas put into readability of the solution procedures, to give the user\na direct mapping from the mathematical solution procedure to the source code.\n\n\\subsubsection{Explicit operators}\nConsider the problem of type~(\\ref{eq:expp-start}--\\ref{eq:expp-end}):\n\\begin{align} \\label{eq:heat-start}\n  \\hspace{4cm} \\dpar{u}{t}(x, y, t) &= \\nabla^2 u + 5 && \\text{ in } \\Omega, \\hspace{4cm} \\\\\n  \\hspace{4cm} u(x, y, t) &= 0 && \\text{ at } t=0, \\hspace{4cm} \\\\\n  \\hspace{4cm} u(x, y, t) &= x  && \\text{ on } \\Gamma_d, \\hspace{4cm} \\\\\n  \\hspace{4cm} \\dpar{u}{\\vec n}(x, y, t) &= 0  && \\text{ on } \\Gamma_n, \\hspace{4cm}\n  \\label{eq:heat-end}\n\\end{align}\non the 2D domain $\\Omega$ constructed in listing~\\ref{lst:domains},\nwhere $\\Gamma_n$ is the inner circle boundary and $\\Gamma_d$ the outer boundary.\nThe problem is solved in listing~\\ref{lst:demo2d} and the solution procedure\nfollows~(\\ref{eq:exp-start}--\\ref{eq:exp-end}). The solution is shown on the left side of\nFigure~\\ref{fig:pde-examples}.\n\nListing~\\ref{lst:demo2d} begins after the domain has been constructed and the sets of indices\n\\cpp{interior}, \\cpp{boundary} and \\cpp{circle}, corresponding to the\ninterior, outer boundary in inner boundary nodes, respectively, have been defined.\nThe \\cpp{computeShapes} method computes shapes for Laplacian and first derivatives,\nwhich are then stored. The explicit operators \\cpp{op} are a collection of\nmethods, that implement the spatial parts of the formulas~(\\ref{eq:exp-start}--\\ref{eq:exp-end})\nfrom section~\\ref{sec:pded-exp} and greatly help with the readability of the solution procedure.\n\n\\begin{listing}[H]\n  \\inputminted[firstline=27,lastline=51,gobble=4]{cpp}{analyses\/src\/demo2d.cpp}\n  \\caption{Solving the heat equation~(\\ref{eq:heat-start}--\\ref{eq:heat-end}) explicitly.}\n  \\label{lst:demo2d}\n\\end{listing}\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.44\\linewidth]{images\/demo2d.pdf}\n  \\includegraphics[width=0.55\\linewidth]{images\/demo3d.png}\n  \\caption{Solution of the heat\n  equation~(\\ref{eq:heat-start}--\\ref{eq:heat-end}) on the left\n           and convection-diffusion problem~\\eqref{eq:pois} on the right.}\n  \\label{fig:pde-examples}\n\\end{figure}\n\n\\subsubsection{Implicit operators}\nConsider a boundary value problem of type~(\\ref{eq:impp-start}--\\ref{eq:impp-end}):\n\\begin{equation} \\label{eq:pois}\n  -2 \\nabla^2 u + 8\\;(2, 1, -1) \\cdot \\nabla u = 1 \\text{ in } \\Omega,\n   \\quad u = 0 \\text{ on }\\partial \\Omega,\n\\end{equation}\nwhere $\\Omega$ is the right domain in Figure~\\ref{fig:domains}.\nThe listing~\\ref{lst:demo3d} shows the source code needed to solve the problem implicitly, as\ndescribed in section~\\ref{sec:implicit}.\n\n\\begin{listing}[H]\n  \\inputminted[firstline=12,lastline=29,gobble=4]{cpp}{analyses\/src\/demo3d.cpp}\n  \\caption{Solving convection-diffusion equation~\\eqref{eq:pois} implicitly.}\n  \\label{lst:demo3d}\n\\end{listing}\n\nAfter computing the weights, the appropriately allocated sparse matrix and right side are assembled.\nThe implicit operators \\cpp{op.lap} hold a reference to the matrix and the right hand side\nand them with the appropriate weights, taken from \\cpp{storage}, implementing\nformulas~(\\ref{eq:implicit-start}--\\ref{eq:implicit-end}). This is done to improve readability;\nnote the similarity between the line of the source code, which\ndefines the equation in the interior, and the equation~\\eqref{eq:pois}.\nThe implicit system can also be amended manually, if desired.\nThe intuitive mathematical syntax supports expressions of form\n$\\sum_\\ell \\alpha_\\ell \\L_\\ell u = r$, where 0th, 1st and 2nd derivatives are supported\nfor $\\L_\\ell$, as well as directional derivatives, gradients of divergence, Laplacian,\nand any user defined operators.\nAnother benefit of this system is that (in \\cpp{DEBUG} mode) checks are performed that\nthe operators added together always write to the same matrix row, to avoid indexing errors.\n\nOverall, the abstractions for implicit and explicit operators are in our opinion one of the best\nfeatures of Medusa library. They allow the user to think in terms of\nfield and operators, instead in terms of arrays and indices, which are\nmuch more error prone. This shift has been present in FEM implementations for a while,\nwith FreeFem++, Fenics and deal.II implementing these types of abstractions,\nhowever it has not been noted in the strong form community and\nfinite difference codes are often riddled with poorly readable\ndiscretization code clouding the overall problem solution procedure.\nMunthe-Kaas and Haveraaen in 1996 introduced the concept of coordinate free\nnumerics~\\cite{munthe1996coordinate}, which encompasses this idea, and Medusa\nhas been investigated in this direction as well~\\cite{slak2018mipro}.\n\n\\subsection{Miscellaneous}\nThere are a few additional modules in the library that simplify its usage or offer\noften needed utilities. The ``types'' module implement nicer interfaces\nand additional functionality to types used to represent (physical) vectors, scalar fields,\nvector fields and containers, while retaining full compatibility with Eigen.\nInput and output capabilities from and to CSV, XML and HDF file formats are supported.\nSome basic integrators for solving ODEs, such as RK4, are also included.\n\n\n\\section{Examples}\n\\label{sec:applications}\nPlenty of examples are included in the project's repository and a tutorial\nfor solving the Poisson equation is available on the website.\nThe examples include many different setups for\nsolving Poisson boundary value problems, which are used to demonstrate different features.\nOther examples include solving problems from electromagnetic scattering, which includes support for\ncomplex numbers, Navier-Stokes equations for fluid simulation, problems from linear elasticity and\nsimulation of wave propagation. The instruction on compiling and running these examples are\navailable on the wiki and from the README in the examples folder.\n\nIn this paper, we include examples from linear elasticity and fluid mechanics.\n\n\\subsection{Linear elasticity}\n\nSmall displacements in an isotropic homogeneous linearly elastic material under stress are\ndescribed by Cauchy-Navier equations\n\\begin{equation}\n(\\lambda + \\mu) \\nabla (\\nabla \\cdot \\vec{u}) + \\mu \\nabla^2 \\vec{u} = \\vec{f},\n\\end{equation}\nwhere $\\vec{u}$ are unknown displacements, $\\vec{f}$ is the loading\nbody force, and $\\lambda$ and $\\mu$ are material constants, called\nLam\\'{e} parameters. The stress tensor $\\sigma$ is computed as\n\\begin{equation}\n\\sigma = \\lambda \\operatorname{tr}(\\varepsilon) I + 2 \\mu \\varepsilon,\n \\quad \\varepsilon = \\frac{\\nabla \\vec{u} + (\\nabla \\vec{u})^\\mathsf{T}}{2},\n\\end{equation}\nwhere $I$ is the identity tensor.\n\nWe consider a beam of dimensions $L \\times W$ in 2D and $L \\times W \\times T$ in 3D,\noccupying the area $[0, L] \\times [0, W]$ in 2D and $[0, L] \\times [0, W] \\times [0, T]$ in 3D.\nThe beam is fixed on the side with the first coordinate equal to 0, experiences\na downwards traction of size $F$ on the side with the fist coordinate equal to $W$ and\nzero traction elsewhere.\n\nNote that this is not the classical Timoshenko beam, although the library was also tested against\nthat problem~\\cite{slak2018refined}. Additionally, some cavities (also with no traction boundary\nconditions) were added to the domain. The problems were solved for $L = 15$, $W = 5$, $T = 2$,\nwith $E = 72.1 \\cdot 10^9$, $\\nu = 0.33$ and $F = 1000$. Polyharmonic radial basis function on 25\nnearest nodes with monomial augmentation od 2nd order were used both in 2D and 3D.\nThe results are shown in Figure~\\ref{fig:beams}, colored according to von Mises stress.\n\n\\begin{figure}[h!]\n  \\includegraphics[width=0.45\\linewidth]{images\/beam2d.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/beam2d_holes.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/beam3d.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/beam3d_holes.png}\n  \\caption{Cantilever beams with and without cavities in 2D and 3D.\n     Displacements are multiplied by a factor $10^5$ in 2D and $5\\cdot 10^4$ in 3D.}\n  \\label{fig:beams}\n\\end{figure}\n\n\\subsection{Simulation of natural convection}\nThe natural convection problem is governed by coupled Navier-Stokes, mass\ncontinuity and heat transfer equations\n\\begin{align}\n\\dpar{\\b v}{t} + (\\b v \\cdot \\nabla) \\b v &= -\\frac{1}{\\rho} \\nabla p +\n\\frac{\\mu}{\\rho} \\nabla^2\n\\b v + \\frac{1}{\\rho} \\b b, \\\\\n\\nabla \\cdot \\b v &= 0, \\\\\n\\b b &= \\rho (1-\\beta(T - T_{\\text{ref}})) \\b g, \\\\\n\\dpar{T}{t} + \\b v \\cdot \\nabla T  &= \\frac{\\lambda}{\\rho c_p} \\nabla^2 T,\n\\end{align}\nwhere $\\b v(u,v,w)$, $p$, $T$, $\\mu$, $\\lambda$, $c_p$, $\\rho$, $\\b g$,\n$\\beta$, $T_{\\text{ref}}$ and\n$\\b b$ stand for velocity, pressure, temperature, viscosity, thermal\nconductivity,\nspecific heat, density, gravitational acceleration, coefficient of thermal\nexpansion,\nreference temperature for Boussinesq approximation, and body force,\nrespectively. The problem is defined on a unit square domain with vertical\nwalls kept at constant different temperatures, while\nhorizontal walls are adiabatic. In generalization to 3D front\nand back walls are also assumed to be adiabatic~\\cite{wang2017numerical}. The\nproblem is solved with implicit time stepping and projection method for pressure-velocity\ncoupling~\\cite{slak2019generation}. Results in terms of velocity and temperature contour plots\nare presented in Figure~\\ref{fig:thermo} for Prandtl number $0.71$ and Rayleigh\nnumbers $10^8$ in 2D and $10^6$ in 3D, respectively.\nMore details about the solution procedure and results\ncan be found in~\\cite{slak2019generation}.\n\n\\begin{figure}[h!]\n  \\includegraphics[width=0.45\\linewidth]{images\/DVD_2D.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/DVD_3D.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/DVD_2D_irreg.png}\n  \\includegraphics[width=0.45\\linewidth]{images\/DVD_3D_irreg.png}\n  \\caption{Solution of natural convection problem for Ra=$10^8$ in 2D (top\n  left), Ra=$10^6$ in 3D (top right), and on irregular 2D and 3D domains\n  (bottom row).}\n  \\label{fig:thermo}\n\\end{figure}\n\n\\section{Benchmarks}\n\\label{sec:bench}\nWhile the design of Medusa is mainly focused on modularity and extensibility,\nwe still take care that the implementation is reasonably efficient.\nTo this end, we compare the performance of Medusa with the mature FreeFem++ library for\nsolving PDEs. Note that we will be comparing two different methods for solving\nPDEs, which by themselves have different complexity,\nand it is not the purpose of this measurements to compare the methods, nor the quality of\nimplementations. We simply wish to establish that Medusa execution times are in the same\nballpark as the FreeFem++ ones for the same problem.\n\nThe comparison is done on the Poisson boundary value problem\n\\begin{equation} \\label{eq:time-prob}\n  -\\nabla u = f \\text{ in } \\Omega, \\; u = u_0 \\text{ on } \\partial\\Omega,\n\\end{equation}\nfor $u_0(x) = \\prod_{i=1}^d \\sin(\\pi x_i)$ and $f = -\\nabla^2 u_0$ on\n$\\Omega = B(0, 1) \\setminus B(0, 1\/2)$ in 2D and 3D.\nMedusa implementation uses RBF-FD with PHS on $n=9$ and $n=35$ closest nodes in 2D and 3D,\nrespectively. FreeFem++ implementation solves the corresponding variational formulation using P1\nelements. The problem itself and the FreeFem++ code were taken from FreeFem++'s own example suite.\n\nFreeFem++ and its dependencies were compiled from source, as was Medusa.\nBoth implementations were run single-threaded on a laptop computer with\n\\texttt{Intel(R) Core(TM) i7-7700HQ CPU @ 2.80GHz} processor with\n16 GB of DDR4 RAM. Each time measurement was repeated 9 times and the median\nvalues are shown, with error bars showing standard deviation of the measurements.\n\n\\begin{figure}\n  \\includegraphics[width=0.49\\linewidth]{images\/time2d_err.pdf}\n  \\includegraphics[width=0.49\\linewidth]{images\/time2d_cpu.pdf}\n  \\includegraphics[width=0.49\\linewidth]{images\/time3d_err.pdf}\n  \\includegraphics[width=0.49\\linewidth]{images\/time3d_cpu.pdf}\n  \\caption{Errors and execution times of FreeFem++ and Medusa when solving~\\eqref{eq:time-prob} in\n  2D and 3D. Each time measurement was repeated 9 times. The median value is shown with error\n  bars representing the standard deviation.}\n  \\label{fig:times}\n\\end{figure}\n\nBoth methods attain expected convergence rate $N^{-2\/d}$ and similar accuracy, with\nRBF-FD performing slightly worse.\nThe difference in execution times is almost exclusively due to node placing in Medusa\nbeing faster than meshing in FreeFem++. The execution time is also highly dependent on the\nnumber of stencil nodes, which can be lowered or increased, and on the choice of\nsparse linear solver and its parameters. The Conjugate Gradient solver was chosen in FreeFem\nbecause it performed best, and BiCGStab with ILUT$(5, 10^{-2})$  preconditioner was\nchosen for Medusa. The solvers took approximately the same amount of time.\n\nParts of the Medusa solution procedure were also timed separately: namely\ndomain discretization, stencil selection, stencil weight computation, matrix assembly,\npreconditioner computation, iterative solution and post-process error computation.\nFigure~\\ref{fig:times-parts} shows these times with respect to the number of nodes\nand a ratio of time spent on each part of the solution procedure.\nThese measurements also show the scaling behavior of different parts of the solution procedure.\nComputational time complexity of most parts is linear or log linear, with the\nexception of the linear solver. For most problems with explicit time iteration, the iteration\nitself is so time consuming that domain discretization and weight computation are negligible, since\nthey are only performed once at the beginning of the iteration.\n\n\\begin{figure}\n  \\includegraphics[width=0.49\\linewidth]{images\/time_distr.pdf}\n  \\includegraphics[width=0.49\\linewidth]{images\/time_distr_ratio.pdf}\n  \\caption{Errors and execution times of FreeFem++ and Medusa when solving~\\eqref{eq:time-prob} in\n    2D and 3D.}\n  \\label{fig:times-parts}\n\\end{figure}\n\nExecution time ratio can vary significantly in different setups. For 2D problems with 2nd order\nmethods, construction of domain discretization can take more than 50\\% of the total time.\nFor high order methods with large support sizes and augmentation orders, weight computation\ncan severely dominate, even as high as 80\\%. For more complicated problems and larger $N$,\nlinear solver can take up almost 90\\% of the time.\n\nThese separate time measurements also serve as a guideline for optimization and parallelization.\nWeight computation is trivially parallelizable and is already included in Medusa for shared memory\narchitectures, using OpenMP. Support for parallel sparse solvers is also included in Eigen,\nand other parallelization efforts are ongoing.\n\nAdditionally, we also reviewed the cost of abstractions in performance critical sections by\ncomparing execution time with a ``bare-bones'' implementation~\\cite{slak2018mipro} and\nby analyzing assembly instructions with Compiler Explorer~\\cite{compilerExplorer}, until\nwe were satisfied with incurred costs, which are now small to negligible.\n\n\\section{Conclusions and outlook}\n\\label{sec:conc}\nIn this paper we presented an overview of abstractions and implementation\nof Medusa, a general purpose C++ library for solving PDEs with strong-form methods.\nThe library provides core elements of meshless solution procedures as standalone\nblocks that can be pieced together or swapped to ease research, development and testing of\nmeshless methods, all in a dimension independent manner.\nIt allows to define custom node generation and stencil selection procedures,\nbasis functions, weights functions, RBFs, approximation schemes, and linear operators,\nrelying heavily on C++ templating system and most commonly used classes\nare explicitly instantiated to avoid long compile times.\nWe have demonstrated this modularity and extensibility by constructing several\nreported mesh-free methods and many more examples are available in the documentation.\nSpecial attention is also paid to readability of the resulting code, which closely\nresembles the mathematical description of the problem and allows the user to think in\nterms of operators and fields instead of arrays and indices.\nThe library is also tested for correctness with a suite of unit tests and offers technical\ndocumentation and other informal discussions on its website. A basic comparison of Medusa with\nFreeFem++ on a Poisson problem showed it is comparable in execution time for similar accuracy.\n\nAlthough Medusa is primarily intended as a research platform\nfor mesh-free community, it offers enough features for\nsolving 3D coupled problems, such as illustrated thermo-fluid transport problem\nin an irregular 3D domain. Other problems, such as linear elasticity,\ncomplex-valued electromagnetic scattering and wave propagation are also included in the examples.\n\nThe ongoing and future development of Medusa is aimed in several directions.\nOne is to increase the geometric capabilities of Medusa, by\nadding a module for discretization of parametric surfaces,\nand potentially extending it to handle Computer-Aided Design objects,\npushing Medusa a step closer to the engineering simulation software.\n\nAnother important directions is parallelism, since at the moment only\nnaive shared memory parallelization of modules that are trivial to execute in\nparallel is offered. We are developing a parallel version of node positioning algorithms\nas well as a domain decomposition module required for distributed parallel execution.\n\nThroughout all other development we will also (albeit conservatively) extend the set of\napproximations, bases, node generation algorithms and other elements offered by default,\nwith useful developments from ongoing research in core meshless areas. Potential future\nadditions include better support for adaptivity and coupled problems.\n\n\n\\begin{acks}\nThe authors would like to acknowledge other contributors to the Medusa library\n(and its previous unpublished versions), listed in alphabetical order:\nUrban Duh,\nMitja Jan\u010di\u010d,\nMaks Kolman,\nJure Lapajne,\nJure Mo\u010dnik - Berljavac,\nAnja Petkovi\u0107,\nAnja Pirnat,\nIvan Pribec,\nTja\u017e Silov\u0161ek and\nBlaz Stojanovi\u010d.\n\nThe authors would also like to acknowledge the financial support of the Slovenian Research\nAgency (ARRS) research core funding No.\\ P2-0095 and the Young Researcher program PR-08346.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Absence of polytopes in dimensions $\\ge$9}\n\\label{new}\n\n\nSuppose that there exists a compact hyperbolic Coxeter $d$-polytope\n$P$ with $d+4$ facets. Denote by $\\Sigma$ its Coxeter diagram.\n\nWe will refer to the classification of compact hyperbolic Coxeter \n$d$-polytopes with $d+k$ facets for $k\\le 3$ \n(see~\\cite{L},~\\cite{Ess_eng},~\\cite{K} and~\\cite{n3}).\nIn particular, we remind that $P_8$ is a unique Coxeter 8-polytope with 11\nfacets, and $\\Sigma(P_8)$ is its Coxeter diagram (see Fig.~\\ref{p8}).\n \n\nFrom now on by polytope we mean a compact hyperbolic Coxeter polytope.\nBy ``a diagram of a polytope'' we  mean  a Coxeter diagram of the polytope.\n\n\n\\begin{lemma}\n\\label{nok}\nSuppose that $d\\ge 9$. Then \\\\\n(1) any node of $\\Sigma$ is incident to at most one non-simple edge;\\\\\n(2) $\\Sigma$ contains no Lann\\'er diagram of order $3$;\\\\\n(3) $\\Sigma$ contains no edge of multiplicity $\\ge 4$.\n\n\\end{lemma}\n\n\\begin{proof}\nSuppose the lemma is broken. Then there exists either a node incident\nto two dotted edges, or a subdiagram\n$S_0\\subset \\Sigma$ of the type $G_2^m$ for some $m\\ge 4$ having at least\none bad neighbor. \n\nSuppose that  there exists a node $v$ incident\nto two dotted edges.\nThen $v$ corresponds to a\n$(d-1)$-dimensional (non-Coxeter) face $f$ with at most \n$d+1=(d-1)+2$ facets,\nso $f$ is either a simplex or a product of two simplices.\nIf $f$ is a simplex then $P$\nhas a large missing face, which is impossible by \nCor.~\\ref{simplex}. If $f$ is a product of two simplices and it has no\nlarge missing faces then $f$  is a product of two 4-dimensional\nsimplices and $D(f)=\\left\\lfloor 5,5 \\right\\rfloor_1$ (since $d\\ge 9$). \nBy Prop.~\\ref{spletayushaya} \nthe diagram $\\left\\lfloor 5,5 \\right\\rfloor_1$ has no liftings, so we contradict   \nLemma~\\ref{lift}.\n\n\nSuppose that $\\Sigma$ contains\n a subdiagram\n$S_0\\subset \\Sigma$ of the type $G_2^m$ for some $m\\ge 4$ having at least\none bad neighbor.\nThen $P(S_0)$ is a Coxeter $(d-2)$-polytope with\nat most $(d-2)+3$ facets, so $(d-2)$ is either equal to 8 or less\nthan 7 (see~\\cite{n3}). If $d-2<7$ we obtain that $d\\le 8$. \nIf $d-2=8$ we have $d=10$, and $P(S_0)$ is a unique Coxeter 8-polytope \nwith 11 facets (see Fig.~\\ref{p8}).\nThe diagram $\\Sigma_{S_0}$ \ncontains a subdiagram of the type $H_4$ with 2\nneighbors (the neighbors are attached to this diagram by simple edges). \nCor.~\\ref{dif2} implies that this diagram corresponds to\na subdiagram  \n$S_1\\subset \\Sigma$ of the type $H_4$ with 2 neighbors.\nSo, $P(S_1)$ is a\nCoxeter  6-polytope with at most 8 facets, which is impossible\n(see~\\cite{Ess_eng},~\\cite{K}).     \n\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{no5-9}\nSuppose that $d\\ge 9$. Then $\\Sigma$ does not contain any Lann\\'er diagram\nof order $5$. \n\n\\end{lemma}\n\n\\begin{proof}\nSuppose there exists a diagram $L_0\\subset \\Sigma$ of order 5. \nClearly, it\ncontains a subdiagram $S_0\\subset L_0$, such that $S_0=H_4$ or\n$F_4$. Then $P(S_0)$ is a $(d-4)$-dimensional Coxeter polytope, and its \nCoxeter diagram $\\Sigma_{S_0}$ is a subdiagram of $\\Sigma$ \n(Cor.~\\ref{dif}).\nThe diagram $S_0$ has at least one neighbor, the node\n$L_0\\setminus S_0$. Thus, $P(S_0)$ has either $d-3$, or $d-2$, or $d-1$\nfacets. \n\n\\bigskip\n\n\\noindent\n{\\bf Case 1:} $P(S_0)$ has $d-3$ facets. Then  $P(S_0)$ is a \nCoxeter simplex, and we obtain $d-4\\le 4$, i.e. $d\\le 8$.\n\n\\bigskip\n\n\\noindent\n{\\bf Case 2:} $P(S_0)$ has $d-2$ facets. Then $P(S_0)$ is a \nCoxeter $(d-4)$-polytope with $(d-4)+2$\nfacets. Results of~\\cite{Ess2} and~\\cite{K} imply that $d-4\\le 5$,\ni.e. $d\\le 9$. If $d=9$, $P(S_0)$ is a 5-dimensional prism.\nSince $\\Sigma_{S_0}=\\overline S_0$,\ncondition (1)  of Lemma~\\ref{nok} and~\\cite{K} imply that there are only\nfour possibilities for $\\Sigma_{S_0}$: all of\nthem are presented on Fig.~\\ref{prisms}.\n\n\\begin{figure}[!h] \n\\begin{center}\n\\psfrag{a}{a)}\n\\psfrag{b}{b)}\n\\psfrag{c}{c)}\n\\psfrag{d}{d)}\n\\psfrag{u}{$u$}\n\\psfrag{u1}{$u_1$}\n\\psfrag{u2}{$u_2$}\n\\psfrag{u3}{$u_3$}\n\\psfrag{u4}{$u_4$}\n\\epsfig{file=.\/pic_new\/prisms.eps,width=0.92\\linewidth}\n\\caption{5-prisms satisfying condition (1) of\nLemma~\\ref{nok}.}\n\\label{prisms}  \n\\end{center}\n\\end{figure}\n\nLet $L_1$ be a unique  Lann\\'er subdiagram of\n$\\overline S_0$ of order 5, and denote by $S_1$ a unique subdiagram of\n$L_1$  \nof the type $H_4$ or $F_4$.\nDenote $u=L_1\\setminus S_1$.\nThe diagram consists of the following parts:\n\\begin{center}\n\\psfrag{L1}{$L_1$}\n\\psfrag{S1}{$S_1$ {\\small = $H_4$ or $F_4$}}\n\\psfrag{S0}{$S_0$ {\\small = $H_4$ or $F_4$}}\n\\psfrag{S'}{$\\overline S_0=\\Sigma_{S_0}$ \n{\\small = one of the 5-prisms from Fig.~\\ref{prisms}}}\n\\psfrag{u1}{$u_1$}\n\\psfrag{u2}{$u_2$}\n\\psfrag{u}{$u$}\n\\psfrag{v}{$a_1$}\n\\psfrag{w}{$a_2$}\n\\psfrag{a}{$a$}\n\\psfrag{b}{$b$}\n\\epsfig{file=.\/pic_new\/diagr1.eps,width=0.9\\linewidth}\n\\end{center}\nwhere $a_1$ and $a_2$ attach to $S_0$, and $S_0$ is not connected to\n$\\overline S_0$. \n Since any two indefinite subdiagrams of\n$\\Sigma$ are joined in $\\Sigma$ by at \nleast one edge, each of $a_1$ and $a_2$ attaches to $L_1$. If they both\nattach to $S_1$, then $S_1$ has at least 3 neighbors and we refer to\nCase~1. So, one of $a_1$ and $a_2$, say $a_1$, attaches to $u=L_1\\setminus\nS_1$  (see Fig.~\\ref{prisms}) and does not attach to $S_1$.\n\n\nConsider the diagrams (a) and (b) with nodes indexed as shown in   \nFig.~\\ref{prisms}.  Let $S_2=\\overline {S_0}\\setminus\\[u_2,u_4\\]$. \n$S_2$ is a diagram of type $B_5$ with at\nleast 3 bad neighbors ($u_4$, $u_2$ and $a_1$).  Thus, $a_2$ does not\nattach to $S_2$. Considering a subdiagram\n$S_3=\\overline S_0\\setminus\\[u_2,u_3\\]$ instead of $S_2$, we obtain\nthat  $a_2$\ndoes not attach to $u_4$ either. Thus, $a_2$ does not attach to $L_1$, and\nan indefinite subdiagram $\\[S_0,a_2\\]$ is not joined with $L_1$. This\nmeans that $\\Sigma$ is superhyperbolic, which is impossible.\n\nNow let $\\overline S_0$ be one of the diagrams (c) and (d) with nodes\nindexed as shown in Fig.~\\ref{prisms}. \nWe have two possibilities: $a_2$ either attaches to $S_1$ or not.\nSuppose that $a_2$ does not attach to $S_1$. Then $P(S_1)$ is a\nCoxeter 5-polytope with 8 facets. However, the Coxeter diagrams of\nsuch polytopes do not satisfy Lemma~\\ref{nok} (see~\\cite{n3}).\nThus, $a_2$ is joined with $S_1$ and the diagram $S_1$ has exactly two\nneighbors (otherwise we refer to Case~1), so the polytope $P(S_1)$ is\na Coxeter 5-prism. \nBy the reasoning of the previous paragraph \nwe may assume that the Coxeter diagram $\\Sigma_{S_1}$ of $P(S_1)$ is of the\ntype (c) or (d) (see Fig.~\\ref{prisms}). Notice that the diagram\n$\\overline S_1=\\Sigma_{S_1}$ contains $S_0$. Therefore, $S_0$ is of the type $H_4$.\n\nFurther, the diagram $\\overline S_1$ contains $u_1$, $u_2$ and $a_1$.\nSince $P(S_1)$ is a 5-prism and $\\overline {S_1}=\\[S_0,u_1,u_2,a_1\\]$, \n$\\overline S_1$\ncontains a Lann\\'er diagram of order 5. The nodes $u_1$ and $u_2$ do\nnot attach to $S_0$, so $\\[S_0,a_1\\]$ is a Lann\\'er\ndiagram. It follows that it should be joined with $L_1$, and by\nLemma~\\ref{2lan} $a_1$ attaches to $u$  by a dotted edge. Therefore, the\ndiagram $\\[S_0,a_1,u, S_1\\]$ is\n\\smallskip\n\\begin{center}\n\\psfrag{u}{$u$}\n\\psfrag{v}{$a_1$}\n\\psfrag{S0}{$S_0$}\n\\psfrag{S1}{$S_1$}\n\\epsfig{file=.\/pic_new\/l1l2.eps,width=0.4\\linewidth}\n\\end{center}\nRecall that the nodes $u_1$ and $u_2$ are joined by a dotted\nedge. Since $u_1,u_2\\in \\overline S_0$ and  $u_1,u_2\\in \\overline S_1$,\nthey may be \njoined in $\\Sigma\\setminus a_2$ with $u$ and $a_1$ only. We \nobtain the following four possibilities for the diagram \n$\\Sigma\\setminus a_2$:  \n\\begin{center}\n\\medskip\n\\psfrag{p1}{$v_1$}\n\\psfrag{p2}{$v_2$}\n\\psfrag{p3}{$v_3$}\n\\psfrag{p4}{$v_4$}\n\\psfrag{p5}{$a_1$}\n\\psfrag{p6}{$u$}\n\\psfrag{p7}{$v_5$}\n\\psfrag{p8}{$v_6$}\n\\psfrag{p9}{$v_7$}\n\\psfrag{p10}{$v_8$}\n\\psfrag{p11}{$u_1$}\n\\psfrag{p12}{$u_2$}\n\\epsfig{file=.\/pic_new\/prismshh.eps,width=0.92\\linewidth}\n\\end{center}\n\n\n\nBy the assumption, the node $a_2$ attaches to  $S_1=\\[v_5,v_6,v_7,v_8\\]$.\nSince $a_2\\notin \\Sigma_{S_0}$, $a_2$ also attaches to   \n$S_0=\\[v_1,v_2,v_3,v_4\\]$. Further, $a_2$ attaches to\n$\\[u_{1},u_{2}\\]$, otherwise an indefinite diagram\n$\\[S_0,a_2\\]$ is not joined with the Lann\\'er diagram $\\[u_{1},u_{2}\\]$.\n Since the diagrams $\\[v_2,v_3,v_4,a_1,u_1\\]$,\n$\\[v_7,v_6,v_5,u,u_{1}\\]$ and\n$\\[v_7,v_6,v_5,u,u_{2}\\]$\nalready have 3 bad neighbors each, no dotted edge ends in\n$a_2$ (see (1) of Lemma~\\ref{nok}). \nCarefully examining possible \nmultiplicities of edges, it is easy\nto see that we always obtain either a Lann\\'er subdiagram of order 3, or\na parabolic subdiagram, or a subdiagram of the type $H_4$ with at\nleast 3 neighbors, and we refer to Case 1.   \n\n\\bigskip\n\n\\noindent\n{\\bf Case 3:} $P(S_0)$ has $d-1$ facets. Then  $P(S_0)$ is a Coxeter\n$(d-4)$-polytope with $(d-4)+3$ \nfacets. Since $d\\ge9$, we have $d-4\\ge 5$. According to~\\cite{n3},\nCoxeter diagrams of \nall such polytopes either do not satisfy the conditions of\nLemma~\\ref{nok}, or contain a subdiagram of the type $H_4$ with at\nleast two neighbors, so we refer to the previous cases.     \n\n\\end{proof}\n\n\\begin{lemma}\n\\label{no4-9}\nSuppose that one of the following holds:\\\\\n1) $d\\ge 9$, or\\\\ \n2) $d=8$, $\\Sigma$ satisfies conditions (1)--(3) of Lemma~\\ref{nok}\nand $\\Sigma$ \ndoes not contain any Lann\\'er diagram of order $5$.\\\\\nThen $\\Sigma$ does not contain any Lann\\'er diagram\nof order $4$. \n\n\\end{lemma}\n\n\\begin{proof}\nSuppose the contrary. \nLet $L_0$ be a Lann\\'er subdiagram  of $\\Sigma$ of order 4. \nDenote by $S_0$ a subdiagram of $L_0$ of the type $H_3$ (if any) or\n$B_3$ (otherwise). Then $P(S_0)$ is a Coxeter \n$(d-3)$-polytope with at most $(d-3)+3$ facets, and its Coxeter\ndiagram $\\Sigma_{S_0}$ does not contain\nLann\\'er diagrams of order 5 (Lemmas~\\ref{no5-9} and~\\ref{lift}). \nIf $P(S_0)$ is a simplex or a product of two\nsimplices, then $d-3\\le 3$ or  $d-3\\le 4$ respectively, so in this\ncase $d\\le 7$. \n\nSuppose that $P(S_0)$ has $(d-3)+3$ facets. According to~\\cite{n3}, for\n$d\\ge 8$ the Coxeter diagrams of almost all Coxeter \n$(d-3)$-polytopes with $d$\nfacets contain Lann\\'er diagrams of order 5. There is only one\nexclusion: a 5-polytope with 8 facets having the following Coxeter\ndiagram $\\Sigma_{S_0}$: \n\\medskip\n\\begin{center}\n\\psfrag{e1}{\\small $e_1$}\n\\psfrag{e2}{\\small $e_2$}\n\\psfrag{b1}{\\small $b_1$}\n\\psfrag{b2}{\\small $b_2$}\n\\psfrag{c1}{\\small $c_1$}\n\\psfrag{c2}{\\small $c_2$}\n\\psfrag{d1}{\\small $d_1$}\n\\psfrag{d2}{\\small $d_2$}\n\\epsfig{file=.\/pic_d7\/multy1.eps,width=0.2\\linewidth}\n\\end{center}\n\nBy Cor.~\\ref{dif2}, no node of $\\overline S_0\\setminus \\[b_1,b_2\\]$\nis a good neighbor of $S_0$.\nThe node $b_1$ is a good neighbor of $S_0$,\notherwise $\\Sigma$ contains an order 3 Lann\\'er diagram  \n$\\[b_1,c_1,d_1\\]$, which contradicts  the assumption. \nThus, the edge  $b_1c_1$ is a simple edge of $\\Sigma$, \nand the subdiagram \n$\\[d_1,c_1,b_1,x_1,x_2\\]\\subset \\Sigma$ is a Lann\\'er diagram of order 5\n(where $x_1$ and $x_2$ are the ends of the simple edge of $S_0$).\nThe contradiction with the assumption of the lemma completes the proof.\n\n\n\\end{proof}\n\n\\begin{theorem}\nThere is no compact hyperbolic Coxeter $d$-polytope with $d+4$ facets\nfor $d\\ge 9$.\n\n\\end{theorem}\n\n\\begin{proof}\nLemmas~\\ref{nok},~\\ref{no5-9} and~\\ref{no4-9} imply that all Lann\\'er\ndiagrams in $\\Sigma$ have order $2$. By~\\cite[Satz~6.9]{Ess2}, in this case\n$d\\le 4$.  \n\n\\end{proof}\n\n\n\\section{Absence of polytopes in dimension 8}\n\\label{d8,n+4}\n\nIn this section, we prove that no compact Coxeter polytope with 12\nfacets exists in ${\\mathbb H}^8$. \nSuppose there exists an 8-polytope $P$ with 12 facets. \nFirst, we show some properties of its Coxeter diagram $\\Sigma$. Most\nof them repeat ones that  \nhold for polytopes in large dimensions. However, the proofs in\neight-dimensional case are much more complicated.\n\n\\begin{lemma}[~\\cite{nodots}, Lemma~1]\n\\label{same}\nLet $\\Sigma(P)$ be a Coxeter diagram of a Coxeter $d$-polytope\n$P$. Then no proper subdiagram of $\\Sigma(P)$ is a Coxeter \ndiagram of a finite volume Coxeter $d$-polytope.\n\n\\end{lemma}\n\nIn particular,\n$\\Sigma$ does not contain $\\Sigma(P_8)$ as a proper subdiagram.\n\n\n\\begin{lemma}\n\\label{nok8-2}\nSuppose that nodes $v_1$ and $v_2$ of $\\Sigma$ are joined by an\nedge of multiplicity $k\\ge 2$. Then\\\\\n(1) the subdiagram $\\[v_1,v_2\\]$ has at most one bad neighbor;\\\\\n(2) $k\\le 3$;\\\\\n(3) if the subdiagram $\\[v_1,v_2\\]$ has a bad neighbor, then it has\nat least one good neighbor, too. \n\n\\end{lemma} \n\n\\begin{proof}\nDenote by $S_0$ the diagram $\\[v_1,v_2\\]$. If $S_0$ has at least two\nbad neighbors, $P(S_0)$ is a Coxeter 6-polytope with at most 8\nfacets, which is impossible~\\cite{K},~\\cite{Ess2}.\n\nSuppose $k\\ge 4$ or $S_0$ has a unique neighbor which is bad. \nIn particular, all neighbors of $S_0$ are bad. In this case $P(S_0)$\nis a Coxeter 6-polytope with \nat most 9 facets, and its Coxeter diagram $\\Sigma_{S_0}$ is a subdiagram\nof $\\Sigma$. \nBy~\\cite{n3}, there are only three such polytopes, their diagrams are \nshown in Fig.~\\ref{d6n9}.\nA Coxeter diagram of each of these three polytopes\ncontains a multiple edge with at least two bad neighbors\n(see Fig.~\\ref{d6n9}).\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\psfrag{1}{$P_6^1$}\n\\psfrag{2}{$P_6^2$}\n\\psfrag{3}{$P_6^3$}\n\\psfrag{10}{\\scriptsize $10$}\n\\epsfig{file=.\/pic_new\/d6n9.eps,width=0.75\\linewidth}\n\\caption{Compact hyperbolic Coxeter 6-polytopes with 9 facets. }\n\\label{d6n9}\n\\end{center}\n\\end{figure}\n\n\n\\end{proof}\n\n\\begin{lemma}\n\\label{no3-8}\n(1) $\\Sigma$ contains no Lann\\'er diagram of order $3$.\\\\\n(2) No end of a dotted edge is incident to a triple edge.\\\\\n(3) If $\\Sigma$ contains a unique dotted edge, then no end of that\nedge is incident to a double edge.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the lemma is broken. Denote by $L_0$ a Lann\\'er diagram\nof order 3 or a subdiagram consisting of adjoint dotted and multiple\nedges. $L_0$ contains an edge of multiplicity 2 or 3. We consider the\nfollowing two cases:  \n\n\\bigskip\n\n\\noindent\n{\\bf Case 1:}  $L_0$ contains an edge of multiplicity 3.\\\\\nDenote by $S_0$ the subdiagram consisting of this edge. $P(S_0)$ is a\nCoxeter\n6-polytope with at most 9 facets, so we may assume that $P(S_0)$ has\nexactly 9 facets. \nThen $\\Sigma_{S_0}$ is one of the diagrams shown in\nFig.~\\ref{d6n9}. By Cor.~\\ref{dif}, the diagram $\\overline\nS_0\\subset\\Sigma$ can be obtained\nfrom $\\Sigma_{S_0}$ by replacing some dotted edges by ordinary edges,\nand some edges labeled by 10 by simple edges. Applying (or not) such a\nprocedure to the Coxeter diagrams of the polytopes $P_6^1$ and $P_6^3$\nwe see that the resulting diagrams contain either a parabolic\nsubdiagram or a multiple edge with at least two bad neighbors. \n\n\nNow suppose that $P(S_0)=P_6^2$. Denote by $u$ a unique bad neighbor\nof $S_0$ (it is unique by Lemma~\\ref{nok8-2}). \nBy Cor.~\\ref{dif}, $\\Sigma_{S_0}$   can by transformed to $\\overline S_0$ \nby the change of some edges labeled by 10 into simple edges and some \ndotted edges into ordinary edges.\nBy Cor.~\\ref{dif2}, the dotted edge of $\\Sigma_{S_0}$ remains dotted in\n$\\overline S_0$.\nBy Lemma~\\ref{nok8-2}, the edge of $\\Sigma_{S_0}$ labeled by 10\nis a simple edge of $\\overline S_0$. Therefore, the leaf\n$a\\in \\Sigma_{S_0}$ that is incident to the edge labeled by 10\nis a good neighbor of \n$S_0$ in $\\Sigma$. The remaining nodes of $\\Sigma_{S_0}$\ncannot be good neighbors of $S_0$ in view of Cor.~\\ref{dif2}.\nSo, the subdiagram $\\Sigma\\setminus u$ is a Coxeter diagram \n$\\Sigma(P_8)$, which is impossible by Lemma~\\ref{same}.\n\n\\bigskip\n\n\n\\noindent\n{\\bf Case 2:}  $L_0$ contains no edge of multiplicity 3.\\\\ \nDenote by $S_0$ the subdiagram consisting of a double edge contained\nin $L_0$. $P(S_0)$ is again a 6-polytope \nwith 9 facets. By Prop.~\\ref{al}, the diagram $\\overline\nS_0\\subset\\Sigma$  can be obtained from  \n$\\Sigma_{S_0}$ by replacing some dotted edges by ordinary (or empty) edges,\nand some double edges by simple or empty edges. Applying (or not) such a\nprocedure to the Coxeter diagrams of the polytopes $P_6^3$ and $P_6^2$\nwe see that the resulting diagrams contain a multiple edge (a triple one or\nlabeled by 10 respectively)  with at least two bad neighbors. \n\nSuppose that $P(S_0)=P_6^1$. Both double edges from $\\Sigma_{S_0}$ become\nsimple or empty edges in $\\overline S_0$, otherwise \n$\\Sigma$ contains a Lann\\'er diagram with a triple edge and\nwe refer to Case 1. \nFurthermore, by Cor.~\\ref{dif2}, only one end of each double edge can\nbe a good neighbor of $S_0$. Hence, by Prop.~\\ref{al} \nboth double edges of $\\Sigma_{S_0}$ \nare simple edges in $\\overline S_0$.\nThis implies that a\nunique dotted edge in $\\Sigma_{S_0}$ remains a dotted edge \nin $\\overline S_0$ \n(otherwise the diagram $\\overline S_0$ is superhyperbolic). \nSo, in this case we may assume that $L_0$ contains no dotted edge. By\nProp.~\\ref{al}, there exist only two good neighbors of $S_0$ --- the\nends of the dotted edge. They attach to $S_0$ by simple edges. \nConsider the diagram $S_1$ of the type $H_4$ containing a triple edge, a\nsimple edge that was double in $\\Sigma_{S_0}$ and a simple edge joining\n$S_0$ with its good neighbor.\nSince\n$L_0$ is a Lann\\'er diagram without edges of multiplicity greater than\ntwo, $L_0$ contains 3 edges, and we obtain that the  diagram \n$S_1$ of the type $H_4$ has at least 4 bad neighbors (two neighbors in\n$\\overline S_0$ and two ones in $S_0$). This contradicts lemma~\\ref{bad}, which\ncompletes the proof of the lemma. \n  \n\n\n\\end{proof}\n\nThe following lemma is a corollary of the main result of~\\cite{nodots}. \n\n\\begin{lemma}[\\cite{nodots}, cor. of Th.~A]\n\\label{e2}\nIf $\\Sigma(P)$ is a Coxeter diagram of a compact Coxeter\npolytope $P\\subset {\\mathbb H}^d$, where $d>4$, then\n$\\Sigma(P)$ contains a dotted edge.\n\n\\end{lemma}\n\n\\begin{lemma}\n\\label{order3}\nLet $S\\subset \\Sigma$ be an elliptic subdiagram of order $3$\ncontaining no component of the type $A_n$. Then $S$ has at most $2$\nbad neighbors. \n\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $S$ has $3$ or more bad neighbors.\nThen $\\Sigma_S$ is a Coxeter $5$-polytope with at most $6$ facets,\nwhich is impossible.\n\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{5-e2-3}\nSuppose that $\\Sigma$ contains a Lann\\'er subdiagram $L_0$ of order\n$5$ and that\nthe dotted edge is unique in $\\Sigma$. Then $\\Sigma$ contains a subdiagram\n$S$ of the type $F_4$ or $H_4$ such that $P(S)$ is not a simplex.   \n \n\\end{lemma}\n\n\\begin{proof} \nSuppose that the lemma is broken. Then we may assume that for any diagram\n$S'\\subset \\Sigma$ of the type $H_4$ or $F_4$ the face $P(S')$ is a Coxeter\nsimplex.\n\nDenote by $S_0$ a subdiagram of $L_0$ of the type $H_4$ or $F_4$. In\nparticular, $P(S_0)$ is a Coxeter 4-simplex.   \nThe Coxeter diagram $\\Sigma_{S_0}$ of $P(S_0)$ is a Lann\\'er diagram\nof order 5. Denote it by $L_1$. Denote by $S_1$ a subdiagram of $L_1$\nof the type $H_4$ or $F_4$. $P(S_1)$ is a Coxeter \nsimplex, and its Coxeter diagram\n$\\overline S_1= \\Sigma_{S_1}$ is a Lann\\'er diagram of order 5. \nDenote it by $L_2$. Notice\nthat $S_0\\subset L_2$, so let $S_2=S_0$. Let $v_i=L_i\\setminus S_i$,\n$i=0,1,2$. \n\nSince $S_2=S_0$ does not attach to $L_1$, $v_2$ is joined with $L_1$. On\nthe other hand, $L_2$ does not attach to $S_1$. Thus, the subdiagram\n$\\[L_1, L_2\\]$ consists of two Lann\\'er diagrams $L_1$ and $L_2$ joined\nby a unique edge $v_1v_2$, such that\n$L_i\\setminus v_i$ are of the type $H_4$ or $F_4$. By\nLemma~\\ref{2lan}, $v_1v_2$ is a dotted edge.  \n\nNow suppose the dotted edge $v_1v_2$ is unique.\nDenote by $a$ and $b$ two nodes not contained in $L_1$ and $L_2$. Note\nthat if $v_2\\ne v_0$ then either $a$ or $b$ coincides with $v_0$. \nBoth $a$ and $b$ are neighbors of $S_1$ and $S_2$. The diagram\n$\\Sigma$ consists of the following parts:\n\n\\begin{center}\n\\psfrag{L1}{$L_1$}\n\\psfrag{L2}{$L_2$}\n\\psfrag{S1}{$S_1$}\n\\psfrag{S2}{$S_2=S_0$}\n\\psfrag{v1}{$v_1$}\n\\psfrag{v2}{$v_2$}\n\\psfrag{a}{$a$}\n\\psfrag{b}{$b$}\n\\epsfig{file=.\/pic_new\/diagr2.eps,width=0.55\\linewidth}\n\\end{center}\nwhere a wave line means that the vertex is joined with the subdiagram\nby some non-dotted edges, and some additional non-dotted edges may\nappear in the subdiagram $\\[a,b,v_1,v_2\\]$.\n  \n\nBy Lemma~\\ref{no3-8}, vertices $v_1$ and $v_2$ attach to $S_1$ and\n$S_2$ respectively by simple edges only. Thus, $L_1$ (as well as\n$L_2$) is one of the diagrams ${\\cal L }_1^5$, ${\\cal L }_4^5$ or ${\\cal L }_5^5$ \n(see Table~\\ref{Lan}).\n\nSuppose that $L_1={\\cal L }^5_5$. Let $w_1\\in S_1$ \nbe a vertex attached to $a$. If $w_1a$ is a triple edge, we obtain\neither a Lann\\'er diagram of order 3 or a diagram of the type $H_3$\nwith at least three bad neighbors, which is impossible by\nCor.~\\ref{simplex}. If $w_1a$ is a double edge or a\nsimple edge, we obtain either a parabolic diagram or a Lann\\'er\ndiagram of order 3. So, the multiplicity of the edge $w_1a$ should be\ngreater than 3, which contradicts  Lemma~\\ref{nok8-2}. \n\nThus, both $S_1$ and $S_2$ are of the type $H_4$. \nFor each of 4 possible pairs of diagrams $L_1$ and $L_2$ there exists\na unique label of the dotted edge $v_1v_2$ such that the determinant\nof the diagram $\\Sigma\\setminus\\[a,b\\]$ vanishes. The label equals\n$(1+\\sqrt{5})\/2$ for $L_1=L_2={\\cal L }^5_1$, \\ $4+2\\sqrt{5}$ for\n$L_1=L_2={\\cal L }^5_4$, and $(3+\\sqrt{5})\/\\sqrt{2}$ for $L_1={\\cal L }^5_1$,\n$L_2={\\cal L }^5_4$ (or $L_2={\\cal L }^5_1$, $L_1={\\cal L }^5_4$).\n\nDenote by $u_1\\in\nS_1$ and $u_2\\in S_2$ the leaves of $\\Sigma\\setminus\\[a,b\\]$ incident\nto the triple edges. If we assume that each of $a$ and $b$ attaches to\n$S_1\\setminus u_1$ and to $S_2\\setminus u_2$, all edges joining\nthem with $S_i\\setminus u_i$ are  simple,\nand $a$ is not joined with $b$,\n then the diagram $\\Sigma\\setminus\\[v_1,v_2\\]$ contains \n a parabolic  subdiagram of the type $\\widetilde A_m$ for some $m$,\n$2\\le m \\le 7$.\nIf in addition there are some multiple edges joining $a$ and $b$ with\n$S_1\\setminus u_1$ and to $S_2\\setminus u_2$, or $a$ is joined with\n$b$, then $\\Sigma\\setminus v_1$ or  $\\Sigma\\setminus v_2$\ncontains a subdiagram of the type\n$H_3$ with at least 3 bad neighbors, or a Lann\\'er diagram of order 3,\nor a parabolic diagram of the type $\\widetilde B_3$. \nTherefore, at least one of $a$ and $b$, say\n$a$, can attach to one of $S_i\\setminus u_i$, say $S_1\\setminus u_1$,\nby multiple edges only. \nThis means that we have got two cases: $a$\neither attaches to $S_1\\setminus u_1$ by a unique multiple edge\n(Lemma~\\ref{no3-8})\nor does not attach to $S_1\\setminus u_1$ (in the latter case $a$\nattaches to $u_1$). \n\n\\bigskip\n\n\\noindent\n{\\bf Case 1:} $a$ attaches to $S_1\\setminus u_1$ by a unique multiple\nedge $au$. It is easy to see that in this case $L_1={\\cal L }^5_4$, $u$ is\nthe leaf of $L_1$ different from $v_1$, and $a$ does not attach \nto $v_1$ \n(otherwise we obtain either a diagram of the type $H_3$ with at least\n3 bad neighbors, or\na Lann\\'er diagram of order 3, or a\nparabolic subdiagram). If $au$ is a double edge then \n$\\[a,L_1\\setminus u_1\\]$ is a parabolic diagram\n$\\widetilde B_4$. Suppose that $au$ is a triple edge.  The vertex $a$\nis not joined with $u_1$, otherwise the diagram $\\[u,a,u_1\\]$ of the\ntype $H_3$ has at least 3 bad neighbors, which contradicts \nCor.~\\ref{simplex}. Now consider the diagram $\\Sigma_1=\\[L_1,a,v_2\\]$. \nThe vertex $a$ may attach (by simple edge only, see Lemma~\\ref{no3-8}) \nto $v_2$ only. An elementary computation shows that $\\Sigma_1$ is\nsuperhyperbolic whenever $a$ attaches to $v_2$ or not. \n\n\\bigskip\n\n\\noindent\n{\\bf Case 2:} $a$ does not attach to $L_1\\setminus u_1$. Therefore,  \n$a$ attaches to $u_1$ by a simple edge. Again, consider the diagram\n$\\Sigma_1=\\[L_1,a,v_2\\]$. The vertex $a$ may attach (by\nsimple edge only) to  $v_1$ and $v_2$ only. For each of 4 pairs of diagrams\n$L_1$ and $L_2$ the diagram $\\Sigma_1$ is superhyperbolic whenever \n$a$ attaches to $v_1$ and (or) $v_2$ or not. \n\n\\end{proof}\n\n\n\n\n\n\n\\begin{lemma}\n\\label{nok8-1}\nAny node of $\\Sigma$ is incident to at most one dotted edge.\n\n\\end{lemma}\n\n\\begin{proof}\nSuppose the contrary. Denote by $u$ a node of $\\Sigma$ incident to at least\ntwo dotted edges. Since the number of bad neighbors of $u$ cannot\nexceed 3, $u$ is incident to either two or three dotted edges. If\nthere are three, the polytope $P(u)$ is a simplicial\n7-face of $P$, which contradicts  Cor.~\\ref{simplex} of Lemma~\\ref{lift}.\n\nThus, $u$ is incident to exactly two dotted edges, and $P(u)$ is a \n7-polytope with 9 facets, i.e. a polytope whose  diagram of the missing\nfaces is $\\left\\lfloor k,9-k \\right\\rfloor_1$ for some $k<9$.\nSince the missing faces\ndiagram of such a polytope contains no large missing faces,\nit is of the type $\\left\\lfloor 4,5 \\right\\rfloor_1$.\nBy Lemma~\\ref{lift54}, this\ndiagram of missing faces  \nhas a unique lifting, namely the diagram $\\Theta_1$ \nshown in Fig.~\\ref{10only5}. \nDenote the nodes of $\\Theta_1$ as shown in Fig.~\\ref{lift-v},\ndenote\nby $v$ and $w$ the remaining two nodes of $\\Sigma$. Each of them \nattaches to the node $u$ by a dotted edge and attaches to the\nsubdiagram $\\[v_5,\\dots,v_9\\]\\subset \\Sigma$, where the nodes are indexed\nas shown in Fig.~\\ref{lift-v}.\n\\begin{figure}[htb]\n\\begin{center}\n\\smallskip\n\\psfrag{v}{$u$}\n\\psfrag{v1}{$v_1$}\n\\psfrag{v2}{$v_2$}\n\\psfrag{v3}{$v_3$}\n\\psfrag{v4}{$v_4$}\n\\psfrag{v5}{$v_5$}\n\\psfrag{v6}{$v_6$}\n\\psfrag{v7}{$v_7$}\n\\psfrag{v8}{$v_8$}\n\\psfrag{v9}{$v_9$}\n\\epsfig{file=.\/pic_new\/lift-v.eps,width=0.32\\linewidth}\n\\caption{A unique lifting of $[4,5]_1$. }\n\\label{lift-v}\n\\end{center} \n\\end{figure}\n\n\n\n\n\\noindent\n{\\bf Case 1:} no dotted edge joins $\\[ v,w\\]$ with $\\Sigma\\setminus\n\\[v,w,u\\]=\\[v_1,\\dots,v_9\\]$.   \n\nLet $S_0=\\[v_2,\\dots,v_8\\]$. The face $P(S_0)$ is one-dimensional, so\n$S_0$ is contained in exactly two elliptic diagrams of order 8. This\nmeans that one of $v$ and $w$, say $v$, either is not joined with $S_0$\nor is a good neighbor of $S_0$. In particular, any double edge from\n$v$ to $S_0$ may end in $v_2$ or $v_8$ only. Now Lemma~\\ref{no3-8}\nimplies that no double edge joins $v$  with $\\Sigma\\setminus\n\\[v,w,u\\]$. Thus, $v$ may be joined by \nsimple edges only with $v_1$, $v_9$ and one node of $S_0$. A\nstraightforward calculation shows that none of obtained subdiagrams\n$\\Sigma\\setminus \\[w,u\\]$ is admissible with positive inertia index at\nmost 8.    \n\n\n\n\\bigskip\n\n\n\\noindent\n{\\bf Case 2:} at least one of $v$ and $w$, say $v$, is joined by a dotted edge\nwith some node $v_x\\in \\Sigma\\setminus \\[v,w,u\\]$.\n\nIn this case $P(v)$ is a 7-face of $P$ with 9 facets, \ni.e. a face with a diagram of missing faces of the type $\\left\\lfloor 5,4 \\right\\rfloor_1$.\nA unique lifting of its missing faces is the diagram $\\Theta_1$ \n(Lemma~\\ref{lift54}). In other words, \ndiagram $\\Sigma \\setminus\\[u,v_x\\]$ looks as shown in \nFig.~\\ref{lift-v}, where\n$v$ takes place of $u$, $w$ takes place of $v_x$,\nand, possibly, the node $v$ is not joined with $v_4$, but joined with \neither $v_6$ (if $v_x\\ne v_6$) or with $w$ (if $v_x=v_6$).\n\n\nIf $v_x$ and $w$ are not joined by a dotted edge, we obtain a\nparabolic diagram $\\[v_{x-1},v_x,v_{x+1},w\\]$ (if $v_x\\ne v_1$, $v_9$)\nor a Lann\\'er diagram  of order 3 (otherwise). Thus, $v_x$ and\n$w$ are  joined by a dotted edge. Now consider $P(v_x)$. It is\nagain a 7-polytope with 9 facets, but in the lifting of its missing faces \ndiagram the node $v_x$ is joined with at least two other nodes, which\ncontradicts Lemma~\\ref{lift}.   \n\n\\end{proof}\n\n\\begin{lemma}\n\\label{2dot}\n$\\Sigma$ contains at least two dotted edges.\n\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma~\\ref{e2}, there exists at least one dotted\nedge. Suppose it is unique. \nThen, by Lemmas~\\ref{nok8-2} and \\ref{no3-8},\n$\\Sigma$ satisfies  conditions (1)--(3) of Lemma~\\ref{nok}. \nBelow we show that this implies that no\nLann\\'er diagram of order 5 is contained in $\\Sigma$. In view of\nLemmas~\\ref{no4-9},~\\ref{no3-8} and~\\ref{disjoint}, this proves the lemma.\n\nWe assume that there exists a Lann\\'er diagram $L_0\\subset \\Sigma$ of order\n5. Denote by $S_0$ a subdiagram of $L_0$ of the type $H_4$ or $F_4$.\nThe Coxeter diagrams of all Coxeter 4-polytopes with 7 facets\ncontain either a Lann\\'er diagram of order 3 or at least two dotted\nedges. The Coxeter diagrams of all Coxeter 4-polytopes with \n6 facets that are not prisms contain Lann\\'er diagrams of order 3. Thus,\n$P(S_0)$ is either a Coxeter 4-prism or a Coxeter\n4-simplex. \n\nBy Lemma~\\ref{5-e2-3}, we can assume that $P(S_0)$ is a prism.\nConsider two cases.\n\n\\bigskip\n\n\\noindent\n{\\bf Case 1:}\n$P(S_0)$ is a prism whose Coxeter diagram $\\Sigma_{S_0}$ is different from \n\\smallskip\n\\begin{center}\n\\psfrag{2,3}{\\scriptsize $2,3$}\n\\psfrag{v1}{}\n\\psfrag{v2}{}\n\\psfrag{v3}{}\n\\psfrag{v4}{}\n\\psfrag{v5}{}\n\\psfrag{v6}{}\n\\epsfig{file=.\/pic_new\/prism4-i.eps,width=0.26\\linewidth}\n\\end{center} \nwhere the label ``$2,3$'' means that the nodes are either not joined\nor joined by a simple edge.\n\nLooking at the list of Coxeter diagrams of 4-prisms~\\cite{K}, we\nsee that $\\Sigma_{S_0}=\\overline S_0$ has a subdiagram \n$S_1$ of the type $H_4$ or $F_4$\nwith a dotted edge attached to $S_1$. \nSince $\\Sigma$ contains a unique dotted edge,\nthe Coxeter diagram $\\Sigma_{S_1}=\\overline S_1$ of $P(S_1)$ \ndoes not contain dotted\nedges, so $P(S_1)$ \nis a simplex. Notice that $S_0\\subset \\overline S_1$.\nDenote by $u$ and $v$ the neighbors of $S_0$, denote by $a$\nthe end of the dotted edge not contained in $S_1$, and by $b$ the remaining\nnode of $\\Sigma$ not contained in $\\[S_0,S_1\\]$.\nSo, the diagram $\\Sigma$ consists of the following parts:\n\n\\begin{center}\n\\psfrag{S0}{$S_0=H_4$ or $F_4$}\n\\psfrag{S1}{$S_1=H_4$ or $F_4$}\n\\psfrag{s0}{} \n\\psfrag{s1}{$\\overline S_0=\\Sigma_{S_0}=$ {\\small a 4-prism}}\n\\psfrag{u}{$u$}\n\\psfrag{v}{$v$}\n\\psfrag{a}{$a$}\n\\psfrag{b}{$b$}\n\\epsfig{file=.\/pic_new\/diagr3.eps,width=0.85\\linewidth}\n\\end{center}\n\n\\noindent\nwhere $u$ and $v$ attach to $S_0$, and $S_0$ does not attach to\n$\\overline S_0$. \nSince  $P(S_1)$\nis a simplex, $b$ and one of $u$ and $v$, say $v$, attach to $S_1$,\nand $\\[S_0,u\\]=\\overline S_1$ is a Lann\\'er diagram. \n\n\nConsider the diagram $\\Sigma\\setminus \\[v,a\\]$. It consists of a Lann\\'er\ndiagram  $\\[S_0,u\\]=\\overline S_1$ and an indefinite diagram \n$\\[S_1,b\\]$. \nThese two subdiagrams may be joined by a unique non-dotted\nedge $ub$ only. It is easy to see that the diagram \n$\\Sigma \\setminus\\[v,a\\]$ is superhyperbolic.\n\n\\bigskip\n\n\\noindent\n{\\bf Case 2.}\n$P(S_0)$ is a prism whose Coxeter diagram $\\Sigma_{S_0}$ is  \n\\begin{center}\n\\smallskip\n\\psfrag{2,3}{\\scriptsize $2, 3$}\n\\psfrag{v1}{$v_1$}\n\\psfrag{v2}{$v_2$}\n\\psfrag{v3}{$v_3$}\n\\psfrag{v4}{$v_4$}\n\\psfrag{v5}{$v_5$}\n\\psfrag{v6}{$v_6$}\n\\epsfig{file=.\/pic_new\/prism4-i.eps,width=0.26\\linewidth}\n\\end{center} \nDenote by $S_1$ a subdiagram of $\\Sigma_{S_0}$ of the type $B_4$, say\n$S_1=\\[v_1,v_2,v_3,v_5\\]$. \nSo, $\\Sigma$ consists of the following parts:\n\\begin{center}\n\\psfrag{S0}{$S_0=H_4$ or $F_4$}\n\\psfrag{S1}{$S_1$}\n\\psfrag{s0}{} \n\\psfrag{s1}{$\\overline S_0$}\n\\psfrag{u}{$u$}\n\\psfrag{v}{$v$}\n\\psfrag{v1}{$v_1$}\n\\psfrag{v2}{$v_2$}\n\\psfrag{v3}{$v_3$}\n\\psfrag{v4}{$v_4$}\n\\psfrag{v5}{$v_5$}\n\\psfrag{v6}{$v_6$}\n\\psfrag{2,3}{\\scriptsize $2, 3$}\n\\psfrag{a}{$a$}\n\\psfrag{b}{$b$}\n\\epsfig{file=.\/pic_new\/diagr4.eps,width=0.8\\linewidth}\n\\end{center}\nwhere $u$ and $v$ attach to $S_0$, and $S_0$ does not attach to\n$\\overline S_0$. Notice that\n$S_1$ has at least two bad neighbors,\n$v_4$ and $v_6$, so $P(S_1)$ is either an Esselmann polytope,\nor a prism, or a simplex.\nConsider these three cases.\n\n\\smallskip\n\n\\noindent\n{\\bf Case 2.1:} $P(S_1)$ is an Esselmann polytope.\\\\\nThen $\\Sigma_{S_1}$ contains two disjoint Lann\\'er diagrams of order 3,\nwhile $\\overline S_1$ contains no subdiagram of order 3. Therefore,\n  $\\Sigma_{S_1}$ contains two nodes that are good neighbors of \nthe diagram $S_1$ in $\\Sigma$. This is impossible\nby Cor.~\\ref{dif2} (we use the list of Esselmann\ndiagrams~\\cite{Ess_eng} and the fact that\n$S_1$ is a diagram of the type $B_4$).\n\n\n\\noindent\n{\\bf Case 2.1:} $P(S_1)$ is a prism.\\\\\nAs above, $u$ and $v$ are  neighbors of $S_0$. \nBy the assumption, $\\Sigma$ contains a unique dotted edge,\nhence \nat least one of $u$ and $v$,\nsay $v$, should be a good neighbor of $S_1$. Then $\\[S_1,v\\]$ is\nof the type $B_5$. If $u$ is also a good neighbor of $S_1$, we\nobtain either a parabolic subdiagram of $\\Sigma$ or a Lann\\'er diagram of\norder 3. Thus, $v$ is a unique good neighbor of $S_1$. \n\n\nConsider the diagram $\\overline S_1=\\[S_0,u,v\\]$. \nThe only node of $\\overline S_1$ that attaches to $S_1$\nis $v$. Thus, \n$\\overline S_1$ may differ from $\\Sigma_{S_1}$ by multiplicities of edges incident to $v$\nonly. By Prop.~\\ref{al}, any simple edge of $\\overline S_1$ incident to $v$ becomes\na double edge in $\\Sigma_{S_1}$, and any other edge of $\\overline S_1$\nincident to $v$ becomes a\ndotted edge. Since  $P(S_1)$ is a prism, $\\Sigma_{S_1}$ contains a \nunique dotted\nedge. At the same time, no Coxeter diagram of a Coxeter 4-prism contains a\nnode incident to a dotted edge and to a multiple edge\nsimultaneously. Thus, $v$ is \njoined with a unique node of $\\overline S_1 \\setminus v=\n\\[S_0,u\\]$, say $w$, and\n$\\overline S_1$ may be obtained from $\\Sigma_{S_1}$ by replacing  \nthe dotted edge by\na double or a triple edge.   \n  \nRecall that $v$ is a unique neighbor of $S_1$ contained in $\\overline\nS_1$, and it is\njoined with $v_5$ by a simple edge. Thus, we obtain either a parabolic\nsubdiagram $\\[v_1,v_2,v_3,v_5,v,w\\]$ or a subdiagram $\\[v_3,v_5,v,w\\]$\nof the type $H_4$ with at least 4 neighbors.\n\n\\smallskip\n\n\n\\noindent\n{\\bf Case 2.3:} $P(S_1)$ is a simplex.\\\\\nWe preserve the notation from Case 1. The difference is that now\none of $u$ and $v$, say $v$, is a bad neighbor of $S_1$. The only way\nto attach a bad neighbor to $S_1$ such that no parabolic or Lann\\'er\ndiagram of order 3 appears is to join $v$ by a triple edge with\n$v_5$. But in this case we obtain again a subdiagram\n$\\[v_2,v_3,v_5,v\\]$ of the type $H_4$ with at least 4 neighbors.\n\n\\bigskip\n\nHence, no Lann\\'er subdiagram of $\\Sigma$ of order five exists, and the\nlemma is proved.     \n   \n\\end{proof}\n\n\\begin{theorem}\n\\label{no8}\nThere are no compact hyperbolic Coxeter $8$-polytopes with $12$ facets.\n\n\\end{theorem}\n\n\\begin{proof}\nSuppose there exists a polytope $P$ with a Coxeter diagram $\\Sigma$. \nBy Lemma~\\ref{e2}, $\\Sigma$ contains a dotted edge $vw$.\nLemma~\\ref{nok8-1} implies that $P(v)$ is a 7-polytope with\n10 facets. By Lemma~\\ref{2dot}, $P(v)$ has a pair of disjoint\nfacets. \n\nThere are only four two-dimensional Gale diagrams of order 10\ncontaining a missing face of order 2 and not containing large\nmissing faces, all of them are listed below. \n\\smallskip\n\\begin{center}\n\\psfrag{1}{\\small $1$}\n\\psfrag{2}{\\small $2$}\n\\psfrag{3}{\\small $3$}\n\\psfrag{4}{\\small $4$}\n\\psfrag{5}{\\small $5$}\n\\psfrag{532}{}\n\\psfrag{442}{}\n\\psfrag{14132}{}\n\\psfrag{32311}{}\n\\epsfig{file=.\/pic_new\/with2.eps,width=0.8\\linewidth}\n\\end{center} \n\n\nBy Lemma~\\ref{lift}, $\\Sigma$ contains either 0- or 1-lifting of one \nof these diagrams. By Lemma~\\ref{no3-8}, $\\Sigma$ contains no Lann\\'er\ndiagram of order 3. Since $P$ is an 8-dimensional polytope, the\npositive inertia index of $\\Sigma$ is at most 8.\nHence, by Lemmas~\\ref{442},~\\ref{14131},~\\ref{32311} and~\\ref{1lift53},\n$\\Sigma$ contains $\\Sigma(P_8)$,\nwhich contradicts Lemma~\\ref{same}.\n\n\\end{proof}\n\n\\input{dim7_1.txt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nCopyright \u00a9 2016 by General Mills, Minneapolis, Minnesota. All rights reserved.\n\nFor information about permission to reproduce selections from this book, write to trade.permissions@hmhco.com or Permissions, Houghton Mifflin Harcourt Publishing Company, 3 Park Avenue, New York, New York 10016.\n\nwww.hmhco.com\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nISBN 978-0-544-64892-0 (hbk); ISBN 978-0-544-81102-7 (ebk)\n\nv1.1016\n\n## General Mills\n\nOwned Media and Publishing Director: Amy Halford\n\nOwned Media and Publishing Manager: Audra Carson\n\nSenior Editors: Grace Wells and Cathy Swanson\n\nFood Editor: Lori Fox\n\nRecipe Development and Testing: Betty Crocker Kitchens\n\nPhotography: General Mills Photography Studios and Image Library\n\nPhotographers: Val Bourassa, Mike Jensen, Chuck Nields, Kayla Pieper, Maja Sahlberg\n\nPhoto Assistants: Dusty Hoskovec, Olga Ivanova, Morgan Marks, Kayla Pieper, Brie Rinkel, Kevin Santavy, Kacey Wyrick\n\nFood Stylists: Junita Bognanni, Sue Brosious, Carol Grones, Sharon Harding, Nancy Johnson, Stacey LeNeave, Karen Linden, Cindy Lund, Alynna Malson, Betsy Nelson, Amy Peterson, Beth Rosendahl, Cindy Syme\n\nFood Styling Assistants: Susan Barrientos Hevey, Sara Bartus, Jerry David Dudycha, Patty Gabbert, Phyllis Kral, Alynna Malson, Kari Setterholm\n\n## Houghton Mifflin Harcourt\n\nEditorial Director: Cindy Kitchel\n\nExecutive Editor: Anne Ficklen\n\nSenior Editor: Adam Kowit\n\nEditorial Assistant: Melissa Fisch\n\nEditorial Associate: Molly Aronica\n\nManaging Editor: Marina Padakis Lowry\n\nAssociate Production Editor: Helen Seachrist\n\nCover Design: Tai Blanche\n\nInterior Design and Layout: Tai Blanche\n\nDirector of Production Technologies: David Futato\n\nSenior Production Coordinator: Kimberly Kiefer\n\nEbook design and Production: Rebecca Springer\n\nThe Betty Crocker Kitchens seal guarantees success in your kitchen. Every recipe has been tested in America's Most Trusted Kitchens\u2122 to meet our high standards of reliability, easy preparation and great taste.\n\nFind more great ideas at\n\n# CONTENTS\n\n1\u2022Getting Started\n\n2\u2022Appetizers\n\n3\u2022Beverages\n\n4\u2022Breakfast & Brunch\n\n5\u2022Salads & Salad Dressings\n\n6\u2022Soups, Stews & Chilies\n\n7\u2022Pasta & Pizza\n\n8\u2022Whole Grains & Rice\n\n9\u2022Vegetables, Beans & Legumes\n\n10\u2022One-Dish Dinners\n\n11\u2022Beef, Pork & Lamb\n\n12\u2022Chicken & Turkey\n\n13\u2022Fish & Shellfish\n\n14\u2022Vegetarian\n\n15\u2022Grilling & Smoking\n\n16\u2022Do It Yourself\n\n17\u2022Sauces, Seasonings & Marinades\n\n18\u2022Breads\n\n19\u2022Cookies, Bars & Candies\n\n20\u2022Cakes & Cupcakes\n\n21\u2022Pies & Tarts\n\n22\u2022Desserts & Fruits\n\n Helpful Nutrition and Cooking Information\n\n Index \nDear Friends,\n\nWhether you are just learning to cook or know your way around the kitchen, you'll find _Betty Crocker Cookbook_ to be an indispensable resource you'll consistently reach for. It's like having a trusted cook nearby whenever you have a cooking question!\n\nBetty Crocker still provides the expertise you need to make cooking approachable and rewarding\u2014everything you'd expect from America's most-trusted test kitchens. This is the 12th edition of the popular cookbook that's been helping cooks for more than 75 years\u2014but it isn't your grandmother's version. Loaded with over 1500 recipes, this edition has hand-selected, frequently requested traditional favorites plus brand new, taste-tempting recipes utilizing fresh ingredients and the wide variety of flavors available today.\n\nIn addition, you'll love the breadth and depth of cooking information, provided by means of the many features woven throughout. Our \"Quick Technique\" photos and descriptions will show you how to easily master cooking skills. Each \"Make Ahead\" feature highlights one recipe and several simple ways to use it. \"Betty's Staples\" shows how to use ingredients typically found in your pantry, in easy ways that are sure to please.\n\nWhether you cook with others or for others, food brings people together; conversations are richer, laughter erupts and guests linger. Each time in the kitchen can be a fun, rewarding and satisfying experience! Let us help you make delicious meals and memories with your family and friends.\n\nSincerely,\n\n## Getting Started\n\nA Place in Your Kitchen Called Home\n\nKeeping Your Kitchen Sanity\n\nMeal Planning\n\nShop Smart\n\nSmart and Creative Storage\n\nSmart Shopping Strategies\n\nSupermarket Skills\n\n## Ingredients\n\nBaking\n\nCondiments, Sauces and Seasonings\n\nDairy\/Refrigerated\n\nNuts and Seeds\n\nPantry\n\n## EQUIPMENT\n\nCooking Tools and Handy Gadgets\n\nCookware\n\nCutting Boards\n\nBakeware\n\nKnives and Related Tools\n\nMeasuring Utensils\n\nOther Ovenware\n\nThermometers\n\nSmall Electric Appliances\n\n## Entertaining\n\nCheese Types\n\nCreating a Cheese Tray\n\nEasy Entertaining Strategies\n\n## Additional Help\n\nCooking at Higher Altitudes\n\nCooking Terms\n\nFood Safety\n\nFood Storage\n\nHealth and Nutrition\n\nMicrowave Cooking\n\nMicrowave Cooking and Heating Chart\n\nQuick Techniques\n\nBasting\n\nBlanching\n\nBoiling and Simmering\n\nBroiling\n\nCoring Fruit\n\nCrushing\n\nCutting\n\nDeep Frying\n\nDrizzling\n\nHulling\n\nMaking Soft and Stiff Peaks\n\nMeasuring Correctly\n\nPeeling Fruit\n\nStraining\n\nUsing Kitchen Scissors\n\nUsing Silicon Mats\n\nToasting Coconut\n\nToasting Nuts and Sesame Seed\n\nRefrigerator and Freezer Food Storage Chart\n\n# Getting Started\n\nWhether you are just learning to cook or to want to make cooking more inspirational, a little organization can be just what you need.\n\n## A Place in Your Kitchen Called Home\n\n  * Store items where you use them\u2014pans by the stove, spices near where you mix, tableware near where you serve.\n  * Keep items you use frequently in easy-to-grab spaces and store things you rarely use in a different spot.\n  * Use cabinet organizers so everything can be located quickly.\n  * Install roll-out shelves if you don't already have them\u2014they make it easier to see and grab items.\n  * Keep your counters clutter-free so they are open for cooking.\n\n## Keeping Your Kitchen Sanity\n\nAre you tired of looking for an item you know you have but can't locate it? Keeping things in order lets you cook more easily and effortlessly:\n\n  * Use Prime Real Estate Well: Reserve the front and center parts of any space for the ingredients you reach for most often. Group items by how often your family uses items. Use the space around (higher and lower shelves) for those items you use occasionally.\n  * Label Liberally: Leave no question as to what things are and where they are stored. Mark foods with name and date.\n  * Tame Clutter: Use containers to keep foods together easily and neatly, such as sauce and seasoning packets or bags of pasta.\n  * Use Trays: Place raw meats on trays or in containers with sides to catch any meat juice drips in the refrigerator. Store these items separately from produce to prevent bacterial contamination.\n  * First In, First Out: Avoid food waste and missed expiration dates by placing newer, unopened packages behind older ones. What you buy first should be used first.\n\n## Smart and Creative Storage\n\nHere are some of our favorite unexpected household storage ideas:\n\n  * Store snacks or bulk spices in leftover food containers or baby food jars.\n  * Use canning jars to store for rice, grains, pasta and other dry goods.\n  * Use plastic shoe boxes to store sauce and seasoning packets, as well as herbs and spices.\n  * Use magazine holders with holes to store potatoes and onions.\n  * Use an over-the-door clear plastic shoe organizer to store individual snacks.\n\n## Smart Shopping Strategies\n\nSet aside time for weekly meal planning to avoid last-minute dashes to the store or take-out splurges. Also, be strategic about grocery shopping, to save money and cut down on food waste. Shopping will be more efficient, helping you stick to your list and budget. It will also be easier, so you'll have the energy and time to enjoy cooking and eating at home.\n\n### Meal Planning\n\n  * Start Fresh: Before you shop, do a quick cleanup and inventory of your fridge, freezer and pantry so you know what you have, as well as what you need to use soon.\n  * Keep a Running List: Keep a dry-erase board inside your pantry or cupboard door. As you finish an item, add it to the board. Each time you go shopping, start a shopping list from this list.\n  * Create a Chart: Map out a chart of the week's needed meals, snacks and brown-bag lunches. Consider your schedule and the time you can spend to make each meal.\n  * Strategize Your List: Plan meals that help you make use of foods you already have\u2014especially those items you noted need to be used soon.\n  * A Meal and Recipe at a Time: As you consider each item on your chart, add any needed ingredients to your shopping list.\n\n### Shop Smart\n\n  * If possible, shop alone so that you aren't distracted.\n  * Bring your own bags\u2014many stores offer a reusable bag discount.\n  * Clip coupons ahead of time and read store ads, both print and online.\n  * Know your budget and stick to it.\n  * Always shop with a list.\n\n### Supermarket Skills\n\n  * Shop and cook seasonally to get the freshest food at the best prices.\n  * Shop the perimeter of the store for the most healthful items (dairy, produce, meats).\n  * Check out the discount shelves for bargains.\n  * Buy in bulk for those items you use often.\n  * Check the **unit price per ounce** to compare package sizes or brands for the best price. Many stores offer this information on the shelf labels, right below the item. If not, you can determine the unit price with this equation: **price\/number of units (such as ounces)** = **cost per unit**. Use the calculator feature on your phone to do the math for you.\n\n## Ingredients\n\n### Baking Ingredients\n\nBaking Powder: A leavening mixture made from baking soda and a moisture absorber. When mixed with liquid, carbon dioxide gas bubbles are released, causing baked goods to rise. Not interchangeable with baking soda.\n\nBaking Soda: A leavening ingredient that must be mixed with an acid ingredient, such as lemon juice or buttermilk, in order for it to cause baked goods to rise.\n\nButter: Available salted or unsalted and must contain at least 80 percent butterfat. Recipes in this book were tested with salted butter unless unsalted butter is noted. Butter (not margarine) was tested for recipes in this book for the best taste, texture, appearance and performance. See Cake Basics; Cookie and Bar Basics; Quick Bread Basics.\n\nChocolate: Cocoa beans are made into a thick paste called chocolate liquor, and then processed to make the various forms of chocolate:\n\n  * **Baking Cocoa:** Made from dried chocolate liquor with cocoa butter removed; unsweetened.\n  * **Bittersweet, Sweet (German), Milk and Semisweet Chocolate:** Available in bars and chips (German chocolate in bars only) for baking and eating.\n  * **Dark Chocolate:** Contains a higher percentage of cocoa solids (more than 60 percent) and little or no added sugar. A rich, intense flavor; available as baking chocolate, chocolate bars and candies.\n  * **Unsweetened Chocolate:** Bitter in flavor, so use just for baking; available in bars.\n  * **White Chocolate:** Contains no chocolate liquor and so it is not truly chocolate. Labeled as white baking chips or vanilla baking bar.\n\n**Photo (above):** **1** Brown Sugar **2** Granulated Sugar **3** Egg **4** Yeast **5** White Baking Chips **6** Coconut **7** Sweet Baking (German) Chocolate **8** Baking Cocoa **9** Chocolate Chips **10** Butter **11** Whole Wheat Flour **12** All-Purpose Flour **13** Salt **14** Ground Cinnamon **15** Baking Powder\n\nCoconut: Made from the fruit of the coconut palm tree; available sweetened or unsweetened. **Shredded Coconut** is moist and therefore best for baking. **Dried (desiccated) Flaked Coconut** is used mostly for decorating.\n\nCorn Syrup: A sweetener made from corn sugar processed with acid or enzymes. **Dark Corn Syrup** has a caramel-like flavor and dark color; it can be used interchangeably with **Light Corn Syrup** , although flavor and color of the finished dish may be affected.\n\nCornstarch: A thickener derived from corn; good for making clear (rather than opaque) sauces.\n\nCream of Tartar: An acid ingredient that adds stability and volume when beating eggs.\n\nEggs: Adds richness, moisture and structure to baked goods.\n\nFlour: A primary baking ingredient; comes in many varieties. Below are the varieties of wheat flour. (See Quick Bread Basics, Yeast Bread Basics, and Cookie and Bar Basics.) In addition, many other non-wheat flour choices are either gluten-free or low in gluten. Follow recipes specifically designed for these products, as they are not directly interchangeable with wheat flour.\n\n  * **All-Purpose:** The most common flour; available bleached and unbleached.\n  * **Bread Flour:** Higher in protein so it gives more structure to bread.\n  * **Cake Flour:** Made from soft wheat; makes tender, fine-textured cakes.\n  * **Quick-Mixing Flour:** Processed to blend easily; used for sauces and gravies.\n  * **Self-Rising Flour:** Flour combined with baking powder and salt. Not interchangeable with other flours.\n  * **Whole Wheat Flour:** Made from the whole grain; gives foods a nutty flavor and heartier texture. **White Whole Wheat Flour** is whole-grain flour made with white wheat.\n\nMolasses: Made from the sugar-refining process; available in light and dark varieties, which are interchangeable unless specified otherwise.\n\nSalt: Adds flavor, controls the growth of yeast in breads.\n\nShortening: Sold in sticks and cans, this solid all-vegetable product is available in regular and butter-flavored varieties. Used to grease pans, make pie pastry and sometimes as all or part of the fat in cookies.\n\nSpices: Adds flavor to baked goods. See Spices\/Seasonings.\n\nSugar: Made from sugar beets or sugarcane.\n\n  * **Brown Sugar:** Made from granulated sugar plus molasses. Light or dark brown in color; use interchangeably unless one is specified in a recipe.\n  * **Coarse, Decorating or Pearl Sugar:** Large grained; use for decorating.\n  * **Colored Sugar:** Use for decorating.\n  * **Granulated Sugar:** Available in boxes or bags, cubes or packets. **Superfine Sugar** dissolves quickly and is great for beverages, meringues and frostings.\n  * **Powdered Sugar:** Granulated sugar that has been processed to a fine powder and contains a very small amount of cornstarch to keep it from clumping.\n  * **Raw or Turbinado Sugar:** This golden brown\u2013colored sugar is coarser than granulated sugar. Use to sprinkle on cookies and pies before baking.\n\nYeast: A leavening ingredient used to make breads rise. There are several varieties available:\n\n  * **Bread Machine Yeast:** Finely granulated yeast for bread machine recipes.\n  * **Fast-Acting Dry Yeast:** Dehydrated yeast that allows bread to rise in less time than regular yeast.\n  * **Regular Active Dry Yeast:** Dehydrated yeast that can be used in most bread recipes.\n\n### Dairy\/Refrigerated Ingredients\n\nButter: See Butter.\n\nCheese: See Cheese Types.\n\nCream: Smooth, rich dairy product made from milk; churned to make butter and buttermilk; pasteurized and processed into several forms:\n\n  * **Half-and-Half:** Contains 10 to 20 percent butterfat and does not whip. To save on fat and calories, look for fat-free half-and-half: although not recommended for baking, it is great for soups and beverages.\n  * **Sour Cream:** Regular sour cream has 18 to 20 percent butterfat. Low-fat and fat-free varieties are available and can often be substituted for regular.\n  * **Whipping Cream:** Available in light and heavy varieties. Contains 36 to 40 percent butterfat and doubles in volume when whipped. **Ultra-Pasteurized Cream** has a longer shelf life than regular cream.\n\nDairy Ingredients\n\nEggs: See Egg Basics.\n\nMilk: Pasteurized dairy milk is available in many types. Today there are also many nondairy options to choose from. See There Is More to Milk. Read labels to compare protein, calcium and calories. Common forms of milk:\n\n  * **Buttermilk:** Fat-free or low-fat product adds a tangy taste to baked goods. See Betty's Staples: Buttermilk.\n  * **Dulce de Leche:** Sweetened milk that's slowly heated to caramelize the sugar, changing the flavor and thickness to be more like caramel sauce. Common in Mexican, Central and South American cooking; meaning \"sweet milk\" or \"milk candy.\"\n  * **Evaporated Milk:** Milk with at least half of the water removed. Available in cans, typically in the baking aisle; whole, low-fat and skim varieties.\n  * **Fat-Free (Skim) Milk:** Contains little or no fat.\n  * **Low-Fat Milk:** Available as 1% milk (99% of milk fat removed) and 2% milk (98% of milk fat removed). Recipes in this book were tested with 2% milk.\n  * **Sweetened Condensed Milk:** A highly sweetened milk product used mostly for baking.\n  * **Whole Milk:** Contains at least 3.5% milk fat.\n\nYogurt: Product made from milk heated and combined with healthful bacteria. A variety of yogurts are available with a range of fat content, sugar and possibly added fruit for flavor. Look for yogurt with \"live and active cultures\" to be sure you are getting the most from your yogurt.\n\nThick, creamy Greek yogurt is regular yogurt with the whey strained off, resulting in yogurt with generally higher protein content and less sugar than regular yogurt. See Betty's Staples: Yogurt; to make your own, see page 444.\n\n### Pantry Ingredients\n\nBouillon\/Broth\/Stock: **Bouillon** is available in cubes or granules. **Broth** and **Stock** are available in cans and larger-quantity boxes. **Soup Base** is a highly concentrated stock available near the broth in supermarkets.\n\nCoffee: Available as whole beans or ground and instant. See Beverage Basics.\n\nCooking Oil: The healthiest oils are those that are high in monounsaturated and polyunsaturated fats, such as **Vegetable Oil** and **Olive Oil**. When cooking, not all oils are equal. Choose the right oil, depending on what you will use it for:\n\n  * **For High-Heat Cooking:** For frying and stir-frying, choose Corn, Soybean, Peanut or Sesame Oil.\n  * **For Moderate-Heat Cooking:** For saut\u00e9ing, choose Olive, Canola or Grapeseed Oil.\n  * **Other Oils:** Some oils don't perform well with heat, such as Flaxseed or Walnut. They can be used when no heat is required, such as in making dressings or dips.\n\nPantry Ingredients\n\nGelatin: Colorless, tasteless thickening agent used to thicken foods when dissolved in liquid and refrigerated. Also available in sweetened fruit flavors.\n\nHerbs and Spices: See Herbs, and Spices\/Seasonings.\n\nHoney: A sweetener produced by bees. Store it at room temperature.\n\nLegumes: See Bean Basics, and Betty's Staples: Canned Beans.\n\nMaple Syrup: Made from maple tree sap. Maple-flavored syrup and pancake syrup are made from corn syrup and other sweeteners with some maple syrup or maple flavor added.\n\nPasta: See Pasta Basics.\n\nRice: See Grains and Rice Basics.\n\nSun-Dried Tomatoes: Available dehydrated and in jars packed in oil; use the type specified in the recipe.\n\nTea: Available in a variety of loose tea, tea bags and instant products; both regular and decaffeinated. See Beverage Basics.\n\nTortillas: Corn and flour tortillas are often used for sandwich wraps and in recipes.\n\n### Condiments, Sauces and Seasonings\n\nCapers: Unopened buds of a Mediterranean plant with a sharp, tangy flavor; packed in vinegar brine. Use in appetizers and cooking.\n\nKetchup: Thick, spicy sauce made from tomatoes, vinegar and seasonings. Use it as sauce or in cooking. To make your own, see page 446. Sriracha is the hot, spicy Thai \"ketchup\"; see Asian Flavors. To make your own, see page 463.\n\n**Photo (above):** **1** Red Pepper Sauce **2** Worcestershire Sauce **3** Balsamic Vinegar **4** Red Wine Vinegar **5** Mayonnaise **6** and **7** Mustards **8** Salsa **9** Roasted Bell Peppers **10** Pesto **11** Capers\n\nMayonnaise\/Salad Dressing: Creamy sauce used in cooking and for salads and sandwiches. Salad dressing is usually sweeter than mayonnaise. To make your own, see page 460.\n\nMustard: A range of flavors available from yellow to darker, more highly flavored varieties; adds a sharp flavor to dishes.\n\nPesto: Italian sauce traditionally made from basil, Parmesan cheese, olive oil and pine nuts. Make your own, page 456, or look for it in the refrigerated area and in jars near the spaghetti sauce.\n\nRed Pepper Sauce: Spicy sauce made from hot peppers.\n\nRoasted Bell Peppers: Both red and yellow bell peppers are available packed in jars for salads, sandwiches, snacking and cooking.\n\nSalsa: Sauce made from tomatoes and a variety of other ingredients. Make your own, page 458, or look for varieties ranging from mild to spicy.\n\nSoy Sauce: Asian sauce made from fermented soybeans.\n\nVinegar: Many varieties with different flavors that can balance other ingredients in salad dressings and other recipes. Varieties include: apple cider, wine (both red and white), balsamic (both dark and white), rice and rice wine.\n\nWorcestershire Sauce: Highly flavored sauce containing a mixture of garlic, soy sauce, onions, molasses and vinegar; for stew and meat dishes.\n\n### Nuts and Seeds\n\nNuts and seeds are great sources of protein, and also contain vitamins, minerals and other components that may contribute to good health. Perfect for eating out of hand or adding to recipes for flavor and crunch. They naturally contain oil, so they may spoil easily; store at room temperature 1 month or see Refrigerator and Freezer Food Storage Chart, for storage times.\n\nAlmonds: Available whole in shell or shelled, blanched, sliced or slivered. Also ground into almond flour and meal or made into almond milk.\n\nBrazil Nuts: Actually seeds. Shells are extremely hard; kernels are rich, oily. Eat raw or use in trail mixes.\n\nCashews: Rich, buttery flavor. Also made into cashew milk.\n\nChestnuts: Available fresh, canned (whole or pieces) or as puree. For fresh, choose firm, plump, blemish-free nuts. Store unshelled canned chestnuts in a cool, dry place. Refrigerate shelled nuts in an airtight container.\n\nHazelnuts (Filberts): Available whole or chopped. Use in a variety of desserts, salads and main dishes.\n\nMacadamia Nuts: Available shelled, either roasted or raw; rich, buttery, sweet flavor.\n\nPecans: Available shelled (halves or chopped) or unshelled. Unshelled should be blemish- and crack-free and the kernels shouldn't rattle when shaken.\n\nPine Nuts: From the pinecones of several varieties of pine trees. Commonly used in pesto.\n\nPistachios: Available raw, roasted, salted or unsalted. Available shelled as well as unshelled (partially open) in tan, red (dyed) or white (blanched).\n\nPumpkin Seeds (Pepitas): Available with or without hulls, raw, roasted or salted. Delicious, delicate flavor; common in Mexican cooking.\n\nSunflower Seeds: Available shelled or unshelled, dried, roasted plain or salted. Good for snacking, or in salads, sandwiches and baked goods.\n\nWalnuts: English (most common) or black varieties, shelled or unshelled. Unshelled walnuts should have shells free from cracks or holes; shelled walnuts should be crisp and meaty (not shriveled).\n\n**Photo (above):** **1** Colanders and Strainers **2** Pancake Turner **3** Graters **4** Spoons **5** Timer **6** Kitchen Towel **7** Salad Spinner **8** Cooling Racks **9** Zester **10** Offset Metal Spatula **11** Basting Brushes **12** Whisks **13** Rolling Pin **14** Pastry Blender **15** Jar Opener **16** Ruler **17** Citrus Reamer **18** Scrubbing Brush **19** Silicone Mat **20** Rubber Spatulas **21** Bowls **22** Meat Mallet **23** Potato Masher **24** Silicone Pot Holder\n\n## The Right Tool for the Job\n\nJust imagine trying to cut a lawn with a pair of scissors\u2014it wouldn't be easy, quick or enjoyable, right? Similarly, having the right tools in the kitchen can help make cooking easier, quicker and more a lot more fun! Don't worry\u2014you won't necessarily need everything listed below. Start by stocking your kitchen with items you'll use often or that can slash prep time. As you expand your cooking abilities, you can continue to add to your kitchen tool collection.\n\n### Cooking Tools and Handy Gadgets\n\nBaster: Use to baste meat and poultry when roasting to keep it moist. The bulb draws up liquid into the stem. Squeeze the bulb again over the meat to release the liquid.\n\nBowls (Large, Medium and Small): Use for combining ingredients.\n\nBrushes: A wide variety of sizes and materials are available. Keep brushes used for pastry separate from those used to baste meat.\n\nCan Opener: Available in handheld and electric versions.\n\nCheese Servers: Sets of **Cheese Knives** are available; use a **Spreader** for soft cheese, a **Curved Blade Knife** for hard cheese, a **Cheese Plane** or **Wide Blade Knife** for semi-hard cheese and a **Thin Blade Knife** for semi-soft cheeses.\n\nCitrus Reamer and Juicer: Both remove juice easily from citrus fruits; juicers also keep seeds from getting into the juice.\n\nColanders and Strainers: Use colanders for draining pasta and vegetables. Use strainers to drain liquid from canned goods.\n\nCooling Racks: Use to cool baked goods so that air can get underneath hot pans or foods.\n\nCutting Boards: See note below.\n\n## Cutting Board Smarts\n\n  * Have at least two boards\u2014one for raw meats and poultry and one for vegetables and fruit.\n  * Hard plastic or glass cutting boards are the best choices for raw meat, poultry and seafood as they are less porous than other types.\n  * Avoid cutting cooked foods with the same knife and board used to cut raw foods.\n\nFat Separator: Made with a special spout so that as fat rises to the top, juices can be poured off, leaving the fat behind.\n\nGraters: Many styles with a variety of hole sizes for grating\/shredding. A handheld **Plane Grater** is sharp with many very small holes.\n\nJar Opener: Aids in opening jars and bottles.\n\nKitchen Towels: Dish and hand towels should be absorbent; tea towels are made for drying stemware and cutlery without scratching or leaving lint.\n\nMeat Mallet: Breaks down tough meat fibers for more tender meat, among other tasks.\n\nOven Mitt\/Pot Holder: Use to remove hot pans from oven or stovetop. **Barbecue Mitts** cover more of the forearm for grilling.\n\nPastry Blender: Handy for making pastry, but you can substitute a potato masher or fork.\n\nPotato Masher: Handheld tool for mashing potatoes, cooked vegetables and apples (for applesauce); also used for making pie pastry.\n\nRolling Pin: Use for rolling cookies, pastry and bread dough. Look for one with sturdy, easy-to-hold handles.\n\nRuler: Use to measure dough and in other recipes requiring food to be shaped to a certain size; also use for determining correct pan sizes.\n\nSalad Spinner: Quickly removes the water from lettuce and leafy herbs after washing.\n\nScrubbing Brush: Aids in pan\/dish washing to remove stuck-on food while keeping hands dry.\n\nSilicone Mat: Use to line cookie sheets when baking cookies or pastries, as an alternate to waxed paper or as a floured surface for rolling out pie pastry.\n\nRubber Spatulas or Scrapers: A must-have for folding, stirring and scraping food from bowls, jars or saucepans. Look for those that are heat resistant.\n\nSpatula or Pancake Turner: Use to turn foods such as burgers or pancakes, remove cookies from baking sheets or serve desserts. Offset Metal Spatulas frost baked goods while keeping fingers from marking the frosted surface.\n\nSpoons: Use to stir thick batters and dough. Use wooden spoons for nonstick cookware and for hot items on the stove, such as soups and sauces. Slotted Spoons allow liquid to drain away from food.\n\nTimer: Alerts you when food is finished cooking or baking and for timed directions in recipes.\n\nTongs: Use to handle raw cuts of meat without touching, for turning foods when cooking and serving or for tossing and serving green salads. (Do not let tongs that have touched raw meat contact other foods until thoroughly washed.)\n\nVegetable Peeler: Used to peel any vegetable or fruit, and to make vegetable ribbons and chocolate curls.\n\nWhisk: Cuts the time to beat eggs, sauces and dressings, and to smooth lumpy batters.\n\nZester: Removes the peel from citrus fruit, leaving the bitter pith below it.\n\n### Knives and Related Tools\n\nAn integral part of cooking, a good set of knives is a worthwhile investment. Because you'll use them daily, invest in the best quality that you can afford. Wash them by hand; keep them sharp and they will last many years. Choose knives that feel good in your hand. Look for stainless steel blades with sturdy, durable handles. If you can't afford an entire set of knives, these three knives would get you by: a chef's knife, a paring knife and a serrated knife. As you can afford more, check out the variety of choices and select those that fit your needs.\n\nCarving Knife: The long, thin blade of this knife makes it the perfect choice for carving meat and poultry or slicing cooked meats.\n\nChef's Knife: Use for chopping, slicing and dicing foods. Choose an 8- or 10-inch blade for your first chef's knife; later, add shorter-blade versions for other specific tasks.\n\nKitchen Scissors: Great for snipping herbs, cutting up chicken, cutting fat from chicken pieces and other tasks.\n\nParing Knife: The short, small blade of this knife is great for peeling vegetables and fruits, removing the seeds of fruit or cutting small food items such as grapes.\n\nSantoku Knife: Similar in size to a chef's knife but with a different-shaped blade for a slightly different cutting action. Typically has a thinner, sharper blade, which gives more precise control when slicing dense food such as carrots.\n\nSerrated (Bread) Knife: Use a longer-blade serrated knife to cut bread or angel food cake without tearing or compressing it.\n\nSharpening Steel: A quick way to occasionally sharpen knives by gently scraping blade back and forth several times across it; rinse and wipe blade before using.\n\nSteak Knife: Use to cut meat into bite-size pieces while dining.\n\nUtility Knife: Smaller than a chef's knife, larger than a paring knife; use for many kitchen tasks when these other knives aren't the right size.\n\n**Photo (above):** **1** Sharpening Steel **2** Chef's Knife **3** Carving or Utility Knife **4** Kitchen Scissors **5** Serrated Knives **6** Paring Knives **7** Steak Knives\n\n### Cutting Techniques\n\n**Chop:** Gather food into close cluster. Hold down top of knife with fingers from one hand while holding knife handle in other hand. Manipulate knife (holding tip of knife on board) up and down over food, until evenly chopped.\n\n**Slice:** Hold food with fingers of one hand tucking fingers under. Slice with knife held in the other hand.\n\n**Julienne:** Slice long, thin, uniform strips of food, using slicing method.\n\n**Dice or Cube:** Hold food with fingers of one hand, tucking fingers under. With knife in other hand, cut into strips; rotate, then cut into small squares.\n\n### Cookware\n\nA variety of cookware is essential. Often, you are able to purchase sets of cookware with an assortment of pieces to provide for basic cooking needs.\n\nThe best heat conductors are copper and aluminum, so buying pans made of these materials ensures better cooking results. Avoid purchasing uncoated aluminum pans, as the aluminum can react with acidic foods, causing off flavors in the food as well as discoloring the pans. Also avoid pans made entirely of stainless steel, as they can get hot spots when heated, causing food to cook unevenly.\n\nDutch Oven: A large pot with a tight-fitting lid used mostly for moist, slow-cooking methods such as braising and stewing or a large batch of chili or soup, if you don't have a stockpot. Sides are shorter than a stockpot.\n\nGriddle: Large, heavy, flat pan with very short sides; great for pancakes, bacon, sandwiches and fried potatoes. Regular pans are used on the stovetop; electric versions are also available.\n\nGrill Pan: Heavy pan with shallow sides and ridged cooking surface that allows fat to drain away from food as it cooks and leaves grill marks on the food. Regular grill pans are used on the stovetop; electric versions are also available.\n\nSaucepan: Comes in a range of sizes from 1- to 4-quart and should have tight-fitting lid. Having a variety of sizes will ensure you have you have the right size for cooking or reheating food. Using the wrong size can be a reason a recipe isn't successful.\n\nSaut\u00e9 Pan: Similar to skillets, saut\u00e9 pans generally have sides that are a little higher. The sides can be straight or slightly sloped. In addition to a long handle, these pans typically have a loop handle on the side opposite the long handle so the pan can be lifted easily. Use to brown and cook meats or almost any other kind of food on the stovetop.\n\nSkillet: Usually available in 8-, 10- and 12-inch sizes. Skillets generally have low, gently sloping sides that allow steam to escape. Use to pan-fry or cook almost any kind of food on the stovetop.\n\nStockpot: Large pan with tall sides; perfect for making stocks and soups or cooking pasta.\n\nWok: A rounded, flat-bottomed pan with high, sloping sides; popular for Asian cooking, for stir-frying, steaming and deep frying. Electric woks are also available.\n\n### Bakeware\n\nBakeware is available in many materials, including aluminum, glass, ceramic and terra-cotta (clay). For the recipes in this book, _pan_ refers to a shiny metal pan and _baking dish_ refers to a heat-resistant glass or ceramic dish.\n\nBaking Dish: Made of heat-resistant glass or ceramic, usually round, square or rectangular. Use for desserts, egg bakes, casseroles and main dishes.\n\nBaking Pans: Metal baking pans are available in a variety of sizes\u2014round, square, rectangular and loaf shapes. Use for cakes, breads and desserts. See below for specific types of cake pans.\n\nCake Pans:\n\n  * **Angel Food Cake Pan (Tube Pan):** Round metal 10-inch pan with a hollow tube in the middle. The bottom is usually removable, making it easy to remove the cake from the pan. Use for angel food, chiffon and sponge cakes.\n  * **Fluted Tube Cake Pan:** Round, fluted pan; typically metal pan with center tube. Use for cakes and coffee cakes.\n  * **Jelly Roll Pan:** The 15x10-inch pan has 1-inch sides and is technically used for baking thin, rectangular cakes (called jelly rolls when filled and rolled up) or bars. Some refer to this pan as a _sheet pan, baking sheet_ or _cookie sheet._ While these pans can be used for baking cookies, the sides can cause cookies near the edges to brown more quickly and the edges of the pan can make removing cookies without damaging them more difficult.\n\nCasserole: Glass or ceramic cookware for baking and serving food; may come with a matching cover.\n\nCookie Sheet: Flat, rectangular aluminum sheets of various sizes with very short sides on one or more edges. The open sides allow for good air circulation while baking cookies, biscuits, scones, shortcakes or bread.\n\nCustard Cup or Ramekin: Custard Cups are small, deep, glass or ceramic heatproof individual dishes (6- and 8-ounce) with flat bottoms. Ramekins are similar but come in a variety of shapes and sizes\u2014and can be short or tall and deep, like custard cups\u2014and are made of a variety of materials. Use for baking individual custards or other desserts.\n\nMuffin Pan: Pan with 6 or 12 individual cups for baking muffins or cupcakes. Cups can range in size from miniature to jumbo.\n\nPie and Tart Pans:\n\n  * **Pie Plate or Pan:** A pie plate is glass with a flared side, designed for baking pies with flaky (not soggy) crusts. A pie pan is the metal version.\n  * **Tart Pan:** Available in a variety of shapes and sizes, typically metal with a removable bottom.\n\nPizza Pan: Round metal pan with no sides or low sides (deep-dish) for baking pizza. Some are perforated to help crisp the crust. A **Pizza Stone** is typically flat and should be preheated in the oven; the unbaked pizza is slid onto the hot stone to help crisp pizza crust while it bakes.\n\nPopover Pan: Pan with 6 or 12 individual extra-deep cups especially designed for baking popovers.\n\nSouffl\u00e9 Dish: Round, open dish with high sides, especially designed for making souffl\u00e9s.\n\nSpringform Pan: Round, deep pan with a removable side; available in various sizes. Use for cheesecakes and desserts that can't be turned upside down to remove from pan.\n\n**Photo (above):** **1** Baking Dish **2** Baking Pans **3** Roasting Pan and Rack **4** Muffin Pan **5** Fluted Tube Cake Pan **6** Baking Dish **7** Cookie Sheet **8** Broiler Pan and Rack **9** Pie Plate and Pie Pan **10** Tart Pan **11** Jelly Roll Pan\n\n### Other Ovenware\n\nBroiler Pan and Rack: Rack has slits to allow fat to drip away from food as it cooks, falling into the pan below. For easier cleanup, line pan and rack separately with foil; cut through foil over broiler rack slits with knife.\n\nMicrowavable Cooking Dishes: Specifically made and designated to be microwavable. Use only dishes labeled \"microwave-safe\" in the microwave.\n\nRoasting Pan: Large, rectangular pan with short sides so that the oven heat can reach as much of the food as possible. Typically comes with a roasting rack to elevate the food and allow the fat to drain away from the food as it cooks. Use to roast whole chicken, turkey, pork and tender cuts of meat with little or no liquid.\n\n### Measuring Utensils\n\nUsing the correct measuring equipment can make a difference in how a recipe turns out:\n\n  * **Glass Measuring Cups:** Use to measure liquids. To measure accurately, pour liquid to desired mark on cup; set on counter and read at eye level.\n  * **Plastic and Metal Measuring Cups:** Use to measure dry ingredients. Spoon dry ingredient into cup; level off with metal spatula or flat side of a knife.\n  * **Measuring Spoons:** Use to measure small amounts of liquid, such as extracts or food color, and dry ingredients, such as salt, baking soda and baking powder. For dry ingredients, fill and level off. For liquid ingredients, fill to rim.\n\n### Measuring Correctly\n\nSpoon in dry ingredients, then level off top using a flat-edged utensil such as a knife or metal spatula.\n\nSpoon brown sugar into measuring cup; firmly pack with back of spoon.\n\nCheck amount of liquid by looking at it at eye level while cup sits steady on counter.\n\nDip measuring spoon into food; level off (if dry) or fill to rim (if liquid).\n\n### Thermometers\n\nLook for a variety of thermometers in domestic retail stores and kitchen specialty stores.\n\nCandy Thermometer: Use to check temperature of candy while it cooks and to check liquid temperature for bread making and deep frying.\n\nDigital Oven Probe Thermometer: Temperature readout goes on your counter or oven door and has a long cord with a probe that allows the thermometer to measure temperature of food without opening the oven door. You can set the temperature you wish to cook your food to and an alarm will sound when the food has reached temperature.\n\nInstant-Read Thermometer: Gives an accurate reading of food's temperature in seconds. These thermometers are not heat-safe: Don't leave in oven or on grill while cooking.\n\nOven Thermometer: Ovens can run hot or cool over time\u2014the wrong temperature can affect the success of a recipe. Use to ensure oven is at the proper temperature; nudge the heat up or down as needed to get the temperature to where it needs to be.\n\nOvenproof Meat Thermometer: Heat-safe so they can be left in the food while in the oven.\n\n**Photo (above):** **1** Oven Thermometers **2** Regular and Digital Candy\/Deep Fry Thermometers **3** Ovenproof Meat Thermometers **4** Digital Thermometer **5** Refrigerator\/Freezer Thermometer **6** Digital Oven Probe Thermometer **7** Instant-Read Thermometer\n\n## Is My Thermometer Accurate?\n\nThermometers can lose accuracy over time. To check whether your food thermometer is reading accurately, heat 2 cups water in 1-quart saucepan to boiling. Immerse the stem of the thermometer 2 inches into the boiling water; it should read 212\u00b0F after 30 seconds. If the thermometer is off a few degrees, calibrate it according to the manufacturer's directions or allow for this adjustment when measuring the temperature of food. Check thermometers once or twice a year for accuracy.\n\n### Small Electric Appliances\n\nSome appliances are necessary if you want to make a recipe\u2014making waffles is impossible if you don't have a waffle iron, for example. Many times small appliances aren't essential, but they certainly can make your time in the kitchen easier, quicker and a lot more enjoyable. As you think about equipment for your kitchen, consider purchasing these items as you need them.\n\nBlender: Use to blend, liquefy and puree. An **Immersion Blender** is a handheld appliance used to puree soups and sauces right in the pan.\n\nBread Machine: Mixes, kneads, rises and bakes bread or can be used to make dough to shape for other foods such as rolls and pizza crust to be baked in your oven.\n\nCoffee Grinder: Quickly grinds whole beans for making coffee.\n\nCoffee Maker: Makes coffee automatically; available in models ranging from making coffee a cup at a time to a full pot, or the option of both.\n\nFood Processor: Handles a host of cooking tasks quickly: chopping, slicing, mixing, shredding, pureeing and kneading.\n\nHand Mixer: Use to whip and mix liquid foods or thin batters quickly. Not good for heavy batters or dough.\n\nMini Chopper: Much smaller than a food processor. Use to chop small amounts of food or grind whole dried spices quickly.\n\nPressure Cooker: Meals cook in much less time than if prepared on the stovetop or in the oven. Moisture from the food builds up pressure inside the cooker, causing the temperature to rise above what it could if cooked conventionally, so it can cook faster.\n\nSlow Cooker: An easy way to make meals with little preparation, as the food cooks for long periods with little attention or electricity, and without heating up your kitchen. Great for less tender cuts of meat.\n\nStand Mixer: Preferred by professional chefs over hand mixers for ease of use (hands-free) and power (heavy-duty and quicker).Handles thicker, stiffer mixtures, such as bread dough.\n\nToaster or Toaster Oven: For toasting bread, bagels and frozen waffles, use a **Toaster.** A **Toaster Oven** can toast as well as heat up, broil or toast foods (such as garlic bread and frozen snacks) more quickly and with less energy than if done in the regular oven.\n\nWaffle Maker: Bakes waffles and is available in a variety of shapes and sizes.\n\n**Photo (above):** **1** Electric Hand Mixer **2** Food Processor **3** Stand Mixer **4** Waffle Maker **5** Mini Chopper **6** Blender\n\n### Quick Techniques\n\nWe've gathered the best ways to tackle the most-used cooking skills you'll need to know when cooking. Refer back to this feature any time you come across a technique that's unfamiliar. It's an easy way to build your cooking prowess. Many techniques have more in-depth instructions elsewhere in this book. See Index for specific techniques.\n\n### Basting\n\nBrushing foods with liquid adds flavor to the food while also keeping it moist during cooking.\n\n### Blanching\n\nImmerse food into boiling water to loosen skins, partially cook or brighten color.\n\nImmediately immerse food into ice water to stop the cooking process.\n\n### Boiling and Simmering\n\nBoil liquid or cook food in liquid at a temperature that causes bubbles to rise continuously and break on surface.\n\nSimmer food in liquid at a temperature just below boiling point. Bubbles rise slowly and break just below surface.\n\n### Broiling\n\nBroil food directly under heated broiler element at a specified distance to keep from burning.\n\n### Coring Fruit\n\nPierce fruit at stem end with corer; twist into fruit and pull out to remove core.\n\nWith paring knife, cut fruit into quarters; cut out core with knife.\n\n### Crushing\n\nPlace food to be crushed into resealable plastic bag; seal. Roll over bag with rolling pin until food is in small, even pieces.\n\n### Deep-Frying\n\nHeat oil to desired temperature.\n\nDrop food carefully into oil; do not crowd.\n\nDrain fried food on paper towel\u2013lined plate.\n\n### Drizzling\n\nWith spoon, drop glaze in a thin stream while moving over food.\n\nCut tiny (\u215b-inch) corner from bag of glaze. Squeeze glaze over food in thin stream.\n\n### Hulling\n\nGently press huller into fruit. Squeeze while twisting and pulling hull away from fruit.\n\n### Making Soft and Stiff Peaks\n\n**Soft Peaks:** Beat just until peaks form but curl over.\n\n**Stiff Peaks:** Continue beating until peaks stand upright.\n\n### Straining\n\nPlace food in sieve; with back of spoon, press liquid through.\n\n### Peeling Fruit\n\nCut into top of fruit just below the skin. Turn the fruit, carefully peeling off skin in strip(s).\n\n### Using Kitchen Scissors\n\n**Snipping:** Place small items such as herbs into small cup. Snip with scissors into uniform size.\n\n**Cutting:** Use scissors to cut small pieces of food into even smaller, similar-shaped pieces.\n\n## Cooking Terms\n\nAl Dente: Used to describe pasta or other food cooked just enough\u2014neither soft nor underdone.\n\nBake: To cook food in the oven with dry heat. Bake uncovered for a dry, crisp top or covered to keep food moist.\n\nBaste: To add liquid or fat over surface of food during cooking to keep food moist. Use a spoon, pastry brush or bulb baster to baste.\n\nBatter: Mixture of flour, eggs, liquid and other ingredients that is thin enough to be spooned or poured.\n\nBeat: To combine ingredients vigorously with a spoon, fork, whisk or electric mixer. When electric mixer is specified, mixer speed is included. See Learn to Beat, Whip and Fold.\n\nBlanch: To place food in boiling water briefly and then into ice water to stop the cooking process as a way to preserve color and texture, or to remove skin. See Learn to Blanch.\n\nBlend: To combine ingredients using a spoon, whisk or rubber spatula, or using a blender or food processor.\n\nBoil: To heat liquid or cook food at a temperature that causes bubbles to rise continuously and break the surface.\n\nBraise: Less tender cuts of meat are seared and then cooked in a large amount of liquid to make them tender. See Learn to Braise.\n\nBread or Coat: To cover food with a coating by dipping or brushing with a liquid (like beaten egg or milk), then dipping or rolling in bread or cracker crumbs or cornmeal before frying or baking.\n\nBroil: To cook food from a measured distance directly under the heat source in the oven. See Learn to Broil.\n\nBrown: To cook quickly, usually over high heat, causing surface of food to turn brown and adding color and flavor to the finished dish.\n\nCaramelize: To melt sugar slowly over low heat until golden brown and syrupy. Sugar can be caramelized on top of food with a kitchen torch or by placing it under the broiler.\n\nChill: To place food in the refrigerator until thoroughly cold.\n\nChop: To cut food into coarse or fine pieces of irregular shapes, using a knife, food processor or chopper.\n\nCore: To remove the center of a fruit (apple, pear, pineapple).\n\nCrisp-Tender: Describes doneness of cooked vegetables when they are between completely soft and hard and crunchy.\n\nCrush: To smash into small pieces using the side of a knife blade (garlic), a meat mallet, mortar and pestle (dried herbs and seeds) or rolling pin.\n\nCube: To cut food with a knife into uniform squares, \u00bd inch or larger.\n\nCut In: To work butter or shortening into dry ingredients. Use a pastry blender, potato masher or fork, lifting up and down with a rocking motion, until particles are of desired size.\n\nCut Up: To cut food into small pieces of irregular sizes, using a knife or kitchen scissors. Or to cut a large food, like whole chicken, into smaller pieces.\n\nDash: Refers to less than \u215b teaspoon of an ingredient.\n\nDeep-Fry: Cooking in hot fat that's deep enough to cover and float the food being fried. See also Fry and Panfry, and Learn to Deep-Fry.\n\nDeglaze: After food has been fried, a small amount of liquid is added to a hot pan to loosen the brown bits, which are full of flavor. See Learn to Deglaze.\n\nDice: To cut food with a knife into uniform \u00bc-inch squares.\n\nDip: To moisten or coat food by submerging into liquid mixture to cover completely.\n\nDissolve: To stir a dry ingredient, like gelatin, into a liquid, like boiling water, until dry ingredient disappears.\n\nDot: To drop small pieces of an ingredient, like butter, randomly over another food.\n\nDough: A stiff but pliable mixture of flour, liquid and other ingredients that can be dropped from a spoon, rolled or kneaded.\n\n## Using Silicone Mats\n\nSilicone mats prevent food from sticking to pans (and no greasing is necessary). Placed in a baking pan on a lower racks in oven, they catch juices bubbling out of pies or casseroles to prevent burning on the bottom of the oven. They're also great for rolling dough, since the dough will release from the mat easily.\n\nDrain: To pour off liquid by putting food into a colander or strainer. To drain fat from meat, place strainer over a disposable container.\n\nDrizzle: To pour a thin stream over food from a spoon, a bag, a squeeze bottle with a tip or a liquid measuring cup.\n\nDust: To sprinkle lightly with flour, granulated sugar, powdered sugar or baking cocoa.\n\nFlake: Using a fork to break off pieces or layers of food.\n\nFlute: Using fingers to squeeze pastry, making a decorative edge.\n\nFold: To combine ingredients without losing volume. See Learn to Beat, Whip and Fold.\n\nFry: To cook in hot fat over medium to high heat. See also Deep-Fry and Panfry.\n\nGarnish: A decorative, edible accessory added when serving a dish to enhance its visual appeal with color, flavor or texture. Garnishes may be placed under, around or on top of finished dishes.\n\nGlaze: To spread, drizzle or brush an ingredient on hot or cold food, adding a thin, glossy coating.\n\nGrate: To rub a hard-textured food across the holes of a grater to make tiny particles.\n\nGrease: To coat the bottom and sides of a pan with shortening, using a pastry brush or paper towel, to prevent food from sticking. Cooking spray can often be used. Don't use butter unless specified in a recipe, as it may make foods stick.\n\nGrease and Flour: After greasing pan, sprinkle with a small amount of flour and shake pan to distribute evenly on bottom and sides. Turn pan upside down over sink; tap bottom to remove excess flour. See Flouring Pan.\n\nGrill: See Grilling Basics.\n\nHull: To remove stems and leaves from strawberries with a paring knife or huller.\n\nHusk: To remove leaves and silk from fresh ears of corn.\n\nJuice: To extract the juice from fruit and vegetables. See Learn to Juice.\n\nJulienne: To cut into long, thin slices. Stack slices; cut into match-like sticks.\n\nKnead: See Learn to Knead.\n\nMarinate: To soak food in a flavorful liquid (called a marinade), which infuses flavor into and\/or tenderizes food. Meat, fish or vegetables can be marinated. See Marinade Success.\n\nMelt: To turn a solid into a liquid or semiliquid by heating.\n\nMicrowave: To cook, reheat or thaw food in a microwave oven. See Microwave Cooking.\n\nMince: To cut food with a knife into very fine pieces that are smaller than chopped but bigger than crushed.\n\nMix: To combine ingredients to distribute evenly.\n\nPanfry: To crisp or brown meat or other food in an uncovered skillet, using less fat than with deep frying (if fat is necessary). See also Deep-Fry and Fry.\n\nPeel: To cut off an outer covering with a paring knife, vegetable peeler or citrus zester, or to remove peel with your fingers. Also refers to the outside colored layer of citrus fruit that contains aromatic oils and flavor.\n\nPoach: To cook in simmering liquid just below the boiling point.\n\nProcess: To use a blender, food processor or mini chopper to liquefy, blend, chop, grind or knead food.\n\nPuree: See Learn to Puree.\n\nReduce: To boil liquid uncovered in order to reduce volume as a way to thicken it and\/or intensify the flavor.\n\nRoast: To cook meat or vegetables in the oven in a shallow, uncovered pan to achieve a brown exterior and moist interior. A dry-heat cooking method, using little to no liquid, it is great for more tender cuts of meat, poultry, potatoes and vegetables. See Learn to Roast.\n\nRoll: To use a rolling pin to flatten dough into a thin, even layer. Also means to shape food into balls or to turn over in all directions to coat food.\n\nRoll Up: To roll a flat food that's spread with filling or with filling placed at one end. Begin at one end and turning food over and over, until food is log-shaped.\n\nSaut\u00e9: To cook food over medium-high heat in a small amount of fat, frequently tossing or turning.\n\nScore: To cut shallow lines, about \u00bc inch deep, through surface of food such as meat or bread to decorate, tenderize or let fat drain away as food cooks or bakes.\n\nSear: See Learn to Sear.\n\nSeason: To add flavor with salt, pepper, herbs, spices or seasoning mixes.\n\nShred: To cut food into thin strips with a knife or with a food processor fitted with a shredding disk. Or to pull apart very tender cooked meat using two forks.\n\nSimmer: See Learn to Simmer.\n\nSkim: To remove fat or foam from soup, broth, stock or jam, using a spoon, ladle or skimmer (a flat utensil with small holes).\n\nSlice: To cut into flat pieces of about the same size.\n\nSmoke: See Smoking Basics.\n\nSnip: To cut into very small pieces with kitchen scissors.\n\nSoft Peaks: Refers to egg whites or whipping cream beaten until peaks curl over when beaters are lifted from bowl. See Learn to Beat, Whip and Fold.\n\nSteam: To cook food by placing it in a steamer basket or steamer pan insert over a small amount of boiling or simmering water in a covered pan.\n\nStew: To cook slowly in a covered pot, pan or casserole in a small amount of liquid to tenderize less tender cuts of meat and\/or to blend flavors.\n\nStiff Peaks: Refers to egg whites or whipping cream beaten until peaks stand up straight when beaters are lifted from the bowl. See also Soft Peaks. See Learn to Beat, Whip and Fold.\n\nStir: To combine ingredients with a circular or figure-eight motion until thoroughly blended.\n\nStir-Fry: Method of cooking small, similar-size pieces of food in a small amount of hot oil in a wok or skillet over high heat while stirring constantly.\n\nStrain: To pour a mixture or liquid through a fine-mesh strainer or cheesecloth to remove unwanted larger particles.\n\nTear: To break into pieces with your fingers.\n\nToast: To brown lightly in a toaster, oven, broiler or skillet. See below.\n\n## Toasting Coconut\n\nTo toast coconut, heat the oven to 350\u00b0F. Spread the coconut in an ungreased shallow pan. Bake uncovered 5 to 7 minutes, stirring occasionally, until golden brown. Or, sprinkle in an ungreased skillet. Cook over medium-low heat 6 to 14 minutes, stirring frequently until coconut begins to brown, then stirring constantly until golden brown.\n\n## Toasting Nuts and Sesame Seed\n\nTo toast nuts, heat the oven to 350\u00b0F. Spread the nuts in an ungreased shallow pan. Bake uncovered 6 to 10 minutes, stirring occasionally, until light brown. Or, sprinkle in an ungreased skillet. Cook over medium heat 5 to 7 minutes, stirring frequently until nuts begin to brown, then stirring constantly until light brown.\n\nTo toast sesame seed, sprinkle in an ungreased skillet. Cook over medium-low heat 5 to 7 minutes, stirring frequently until seed begins to brown, then stirring constantly until golden brown.\n\nToss: To gently combine ingredients by lifting and dropping using hands or utensils.\n\nWhip: To add air by beating ingredients; whipping increases the volume until ingredients are light and fluffy. See Learn to Beat, Whip and Fold.\n\n## COOKING AT HIGHER ALTITUDES\n\nFor elevations of 3,500 feet or higher, there are unique cooking challenges. Air pressure is lower, so water has a lower boiling point and liquids evaporate faster. That means recipes for conventional and microwave cooking may need to be adjusted.\n\nUnfortunately, no set of rules applies to all recipes; sometimes the only way to make improvements is through trial and error. Here are some guidelines to help you with high-altitude cooking challenges:\n\n  * **Boiling foods** such as pasta, rice and vegetables will take longer.\n  * **Microwave cooking** may require more liquid and foods may need to be microwaved longer; the type and amount of food, the water content of the food and the elevation may affect how much.\n  * **Meat and poultry cooking** when braising (see page 330) or boiling takes longer\u2014possibly 50 to 100 percent longer. Cooking large meat cuts such as roasts and turkeys in the oven also take longer.\n  * **Grilling foods** will take longer.\n  * **Baked goods** made with baking powder or baking soda (but not yeast) can be improved with one or more of these changes: \n    * Increase oven temperature by 25\u00b0F\n    * Increase liquid\n    * Decrease baking powder or baking soda\n    * Decrease sugar and\/or use larger pan\n  * **High-fat baked** goods such as pound cakes will turn out better if you decrease the fat. Quick breads and cookies usually don't require as many adjustments.\n  * **Yeast bread dough** may need less flour, as flour dries out more quickly at high altitudes. Use the minimum amount called for or decrease the amount by \u00bc to \u00bd cup. Dough rises faster at high altitudes and can easily overrise: let dough rise just until doubled in size.\n\n## Specific High Altitude Cooking Adjustments\n\n**Slow Cooking:** Meats may take twice as long as specified to get tender. To shorten meat cooking times, use High heat setting. Cut vegetable into smaller pieces than the recipe specifies so that they cook more quickly.\n\n**Deep-Fried Foods:** May brown too quickly on the outside before the centers are cooked. Reduce oil temperature by 3\u00b0F for every 1,000 feet of elevation and increase fry time, if necessary.\n\n**Cooked Sugar Mixtures:** Any cooked sugar mixtures, such as **boiled candy** or **cooked frostings** , will concentrate faster because water evaporates quicker. Watch the recipe closely during cooking so that it doesn't scorch. You can try reducing the recipe temperature by 2\u00b0F for every 1,000 feet of elevation or use the cold water test for candy (see Testing Candy Temperatures).\n\n## More Cooking Help\n\nIf you're new to high-altitude cooking, go to the **U.S. Department of Agriculture website** for more information: <http:\/\/www.fsis.usda.gov\/wps\/portal\/fsis\/topics\/food-safety-education\/get-answers\/food-safety-fact-sheets\/safe-food-handling\/high-altitude-cooking-and-food-safety\/ct_index>.\n\nOr visit the **Colorado State University website** : <http:\/\/www.extension.colostate.edu\/SEA\/High%20Altitude%20Cooking.pdf>\n\n## Easy Entertaining Strategies\n\nGreat food and good friends are the perfect ingredients for a memorable event. Relationships are formed and deepened and conversations flow when people can relax and be themselves. But the secret to pulling off a successful gathering starts with a relaxed host or hostess. Use these strategies to entertain with minimal fuss so you can enjoy the party, too.\n\n  * Sometimes there isn't time to make everything. Select a few \"wow\" recipes to prepare that will make a delicious statement, then fill in the rest with go-to prepared foods.\n  * Consider local delis, specialty bakeries, gourmet shops, ethnic grocers and organic food stores as your behind-the-scenes food providers. Scope them out ahead of time, making sure any unusual items you are interested in will be available when you need them (not just on certain days or for certain occasions).\n  * Consider asking guests to bring a food item to the party. It's a great way to slash the cost of entertaining if you are on a budget. People will be happy to bring something because you are doing the hosting. If ask them to bring something specifically, it will be easy to round out the menu.\n\n## Creating a Cheese Tray\n\nFrom simple and inexpensive to impressive showstoppers, cheese trays offer limitless variety and are a perfectly-easy choice for an appetizer or dessert.\n\n  * Start with at least 3 types of cheese on a tray. For larger groups or buffets, add more varieties and\/or larger quantities (plan on at least 2 ounces per person).\n  * Aim for a variety of flavors and textures (see Cheese Types). Many specialty grocery stores offer countless types and can provide samples for you to try as well as pairing suggestions.\n  * Cheese is easier to cut when cold but tastes better at room temperature. Remove from the refrigerator and unwrap 30 to 60 minutes before eating.\n  * Provide appropriate servers or butter knives appropriate for each type of cheese (see Cheese Servers).\n  * Offer crackers or bread to go with the cheese. Consider also adding other foods for more color, interest and flavors, such as dried or fresh fruit, nuts, olives, preserves, mustard, chutney, honey, fig spread or quince paste.\n\nCheese tray arrangement with a variety of items\n\n## Cheese Types\n\n### Very Hard (Grating)\n\nAsiago: Mild when young; nutty and semisharp when aged. Melts best when grated. Use with pasta, potatoes, rice salads, vegetables.\n\nCotija: Firm, dry, crumbly. Salty, sharp flavor. **Cotija A\u00f1ejo** (aged) is firmer and harder than regular cotija. Both will soften but do not change shape when heated. Crumble over salads or Mexican dishes such as tacos or refried beans.\n\nParmesan: Sharp, savory, salty. Flavor intensifies with age. Melts best when grated. Use in salads, cooked dishes or casseroles, or sprinkle on pizza.\n\nRomano: Sharp, robust; similar to but richer than Parmesan. **Pecorino Romano** is made from sheep's milk. Melts best when grated. Use in pasta, soups, salads, egg dishes and stuffings, or sprinkle on pizza.\n\n### Hard\n\nCheddar: Rich, nutty flavor ranging from mild to full-bodied (sharp Cheddar also has a bite). **Smoked Cheddar** contains smoked hardwood flavor. Other ingredients may be added.\n\nGouda: Mellow, rich caramel flavor. **Smoked Gouda** contains smoked hardwood flavor. Melts best when shredded. Use in sandwiches, soups, salads and snacks.\n\nGruy\u00e8re: Mellow, buttery, nutty. Melts well. Use in fondues, baked dishes and onion soup.\n\nJarlsberg: Similar to Swiss. Slightly sweet, nutty. Use in baked dishes, sandwiches and snacks.\n\nManchego: Creamy, mildly nutty. Melts well. Use as any meal course or in sandwiches, salads, baked dishes and snacks.\n\n### Semisoft\n\nBlue: Rich, robust, salty with a lingering tang. Melts best when crumbled. Use in vegetable, green, fruit and pasta salads, spreads, dressings, dips and with grilled meats.\n\nFeta: Very sharp, salty flavor. Melts best in sauces. Use in salads, Greek or egg dishes and hot or cold pasta dishes.\n\nFontina: Delicate, nutty, with a hint of honey flavor. Melts well. Use in sandwiches, baked dishes and snacks.\n\nGorgonzola: Italian blue cheese; spicy flavor becomes more pronounced as it ages. Melts best when crumbled. Use in salads, pasta, souffl\u00e9s, spreads, dressings and dips.\n\nPaneer: Indian cheese, similar to queso blanco. Mild, fresh, mozzarella-like flavor. Unaged, nonmelting, it's customarily diced and used in a variety of dishes including vegetables and salads.\n\nQueso Blanco: Fresh (unripened), with a crumbly, curdy texture. Commonly fried. Softens but holds its shape when heated. Use in soups, salads, stuffings and snacks.\n\nQueso Fresco: Spongy, fresh (unripened), with a grainy texture and slightly acidic flavor. Softens but holds its shape when heated. Crumble over enchiladas or refried beans.\n\nRoquefort: French blue cheese made from sheep's milk. Similar in flavor to blue cheese with a melt-in-your-mouth texture. See Blue cheese for uses.\n\n### Soft\n\nBrie: Rich, buttery. Softens and flows when heated. Use as an appetizer and in sandwiches or snacks.\n\nCh\u00e8vre (Goat): Creamy, fresh, mildly tangy. Made with goat's milk. Softens but holds its shape when heated. Use in appetizers, salads, baked dishes and sauces.\n\nFresh Mozzarella: Creamy, soft texture with a delicate, milky flavor. Use in salads, sandwiches, as an appetizer or as a pizza topping.\n\nMascarpone: Very soft, sweet, mild with a buttery texture. Melts best in sauces. Use in fillings, toppings, dips, spreads and sauces.\n\nRicotta: Mild, slightly sweet. Use in fillings, stuffings, spreads and cheesecakes.\n\n## Health and Nutrition\n\nWhat is healthy eating? It means eating foods that are good for you and also enjoying them. Trendy diets come and go, as does frequently changing nutrition research, leaving a haze of questions. Bottom line: Common sense and balance are the pillars of good health, and good food choices can have a positive effect on our health.\n\n### Small but Mighty Changes\n\nStart with small changes that build positive habits over time. Try these easy ways based on www.choosemyplate.gov to up your healthy eating game:\n\n  * **Don't Skip Meals:** When meals are missed, it's harder to maintain blood glucose levels, making it easy to overeat at the next meal or overindulge in anything just to take away the hunger. Stick to your food plan, but if you don't have time for a meal, have satisfying snacks on hand.\n  * **Plan Meals and Snacks:** Planning what you eat might seem a little over the top at first, but in time, you'll become aware of what foods keep you satisfied and when your hunger tends to hit, so you can be armed with healthy foods that keep you on track. Plan to grocery shop when you aren't hungry, with a list of the healthy foods you need, so you won't be tempted to purchase foods that aren't part of your plan.\n  * **Make Half Your Plate Fruits and Vegetables:** Focus on whole fruits (rather than juice). Satisfy your sweet tooth with fresh, frozen, canned or dried fruits instead of cookies or candies; they also make a healthy dessert choice. Vary your veggies to include greens as well as other colorful choices like red and orange. Add veggies to salads, side dishes and main dishes as a way to increase your consumption.\n  * **Reduce Sugar, Saturated Fat and Sodium:** The USDA Dietary Guidelines recommend consuming: less than 10 percent of calories per day from added sugar, less than 10 percent of calories per day from saturated fats and less than 2,300 milligrams (mg) per day of sodium. Guidelines are based on gender and age; to find out more information visit **http:\/\/health.gov\/dietaryguidelines\/**\n  * One way to help keep your sodium, saturated fat and added sugars in check is to prepare your veggies without sauces or glazes.\n  * **Make Half Your Grains Whole Grains:** Choose whole-grain foods, such as oats, whole wheat flour and popcorn, over refined grains for at least half of the grains you eat each day. Read Nutrition Facts labels and ingredient lists on packages to be sure foods are whole grain.\n  * **Move to Low-Fat and Fat-Free Dairy:** Choose low-fat or fat-free milk and yogurt and low-fat varieties of cheese more often than full-fat varieties.\n  * **Vary Your Protein Routine:** Mix up your protein choices to include seafood, beans, nuts, seeds, soy, eggs, lean meats and poultry. Eat fish and seafood twice a week. Add beans or peas, unsalted nuts or seeds and soy to main dishes and snacks to add protein.\n\n## Microwave Cooking\n\nConvenient to use and great for many tasks, the microwave oven is an indispensable, time-saving appliance for any kitchen. It's perfect to use for reheating but it can be used to cook many food items with little time or fuss. Look here to get great microwave tips and a chart for cooking a variety of foods.\n\n### Microwaving Tips\n\n  * Increase the cooking time stated in the chart on the next two pages if you increase the amount of food.\n  * Check food at the minimum time to avoid overcooking. Microwave longer if necessary.\n  * Stir food from the outer edge to the center so food cooks evenly.\n  * If food cannot be stirred (or the microwave doesn't have a turntable), rotate dish one-quarter to one-half turn during cooking.\n  * Foods that contain high amounts of sugar, fat or moisture may cook more quickly.\n  * Small pieces of food cook faster than larger pieces.\n  * For even cooking, arrange food in a circle with thickest parts to the outside of the dish.\n  * Use standing time to help finish cooking and distribute heat through food.\n  * For microwaving vegetables, see Fresh Vegetable Cooking Chart.\n\n### Microwave Testing for this Book\n\nRecipes in this book that are cooked or heated in a microwave oven were tested in consumer ovens with 700 to 800 watts of power. Find the wattage of your microwave inside the door or unit, on the back near the UL tag or in your Use and Care Manual. Adjust your cooking times if the wattage of your oven is more or less than ours.\n\n### Microwave Cooking and Heating Chart\n\n---\n\nHere are at-a-glance times for cooking, warming and softening many favorite foods. Be sure to use microwavable dishes and microwave-safe paper towels or plastic wrap. Follow any directions that follow microwave times for stirring or standing.\n\nFood, Utensil and Tips | Power Level | Amount | Time\n\nBacon (cook)\n\nPlace on plate or bacon rack lined with paper towels. Place paper towels between layers; cover with paper towel. Microwave until crisp. | High | 1 slice | 30 seconds to 1\u00bd minutes\n\n2 slices | 1 to 2 minutes\n\n4 slices | 2 to 3 minutes\n\n6 slices | 3 to 5 minutes\n\n8 slices | 4 to 6 minutes\n\nBrown Sugar (soften)\n\nPlace in glass bowl; cover with damp paper towel, then plastic wrap. | High | 1 to 3 cups | 1 minute\n\nLet stand 2 minutes, until softened. Repeat heating once or twice.\n\nButter (melt)\n\nRemove wrapper. Place in glass bowl; cover with paper towel. | High | 1 to 8 tablespoons | 10 to 50 seconds\n\n\u00bd to 1 cup | 60 to 75 seconds\n\nButter (soften)\n\nRemove wrapper. Place in glass bowl, uncovered. | Low (30%) | 1 to 8 tablespoons | 10 to 40 seconds\n\n\u00bd to 1 cup | 30 seconds to 1 minute\n\nCaramels (melt)\n\nRemove wrappers. Place in a 4-cup glass measuring cup, uncovered. | High | 1 bag (11 ounces) plus 2 to 4 tablespoons milk | 2 to 3 minutes, stirring once or twice\n\nBaking Chocolate (melt)\n\nRemove wrappers. Place in glass dish, covered. | Medium (50%) | 1 to 3 ounces | 1\u00bd to 2\u00bd minutes, until can be stirred smooth.\n\nChocolate Chips (melt)\n\nPlace in glass bowl, uncovered. | Medium (50%) | \u00bd to 1 cup | 2 to 3 minutes, until can be stirred smooth.\n\nCoconut (toast)\n\nPlace in pie plate, uncovered. | High | \u00bc to \u00bd cup | 1\u00bd to 2 minutes\n\n1 cup | 2 to 3 minutes\n\nStir every 30 seconds.\n\nCream Cheese (soften)\n\nRemove wrapper or tub. Place in glass bowl, uncovered. | Medium (50%) | 8-oz package | 1 to 1\u00bd minutes\n\n8-oz tub | 45 seconds to 1 minute\n\nFruit, Dried (soften)\n\nPlace in 2-cup glass measuring cup; add \u00bd teaspoon water for each \u00bd cup fruit. Cover with plastic wrap, turning back a corner or \u00bc-inch edge to vent steam. | High | \u00bc to \u00bd cup | 30 to 45 seconds\n\n\u00bd to 1 cup | 45 seconds to 1 minute\n\nLet stand 2 minutes.\n\nFruit, Frozen (thaw)\n\nPlace in glass bowl. | Medium (50%) | 16-oz bag | 3 to 5 minutes\n\nThaw until most of the ice is gone, stirring or rearranging twice.\n\nFruit, Refrigerated (warm)\n\nPlace on turntable or floor of microwave. | High | 1 medium | 15 seconds\n\n2 medium | 20 to 30 seconds\n\nLet stand 2 minutes.\n\nHoney (dissolve crystals)\n\nLeave in jar with lid removed, uncovered. | High | \u00bd to 1 cup | 45 seconds to 1\u00bd minutes, stirring every 20 to 30 seconds or until crystals dissolve.\n\nIce Cream (soften)\n\nLeave in original container; remove any foil or plastic covering and loosen the lid (whether cardboard or plastic). | Low (10%) | \u00bd gallon | 2 to 3 minutes\n\nLet stand 2 to 3 minutes.\n\nMuffins or Rolls, Small to Medium (heat)\n\nPlace on plate, napkin or napkin-lined non-metal basket, uncovered. | High | 1 | 5 to 10 seconds\n\n2 | 10 to 15 seconds\n\n3 | 12 to 20 seconds\n\n4 | 20 to 30 seconds\n\nMuffins, Large to Jumbo (heat)\n\nPlace on plate or napkin, uncovered. | High | 1 | 10 to 20 seconds\n\n2 | 20 to 30 seconds\n\n3 | 30 to 40 seconds\n\n4 | 40 to 50 seconds\n\nLet stand 1 minute.\n\nNuts, Chopped (toasted)\n\nPlace in glass measuring cup, uncovered; add \u00bc teaspoon vegetable oil for each \u00bc cup nuts. | High | \u00bc to \u00bd cup | 2\u00bd to 3\u00bd minutes\n\n\u00bd to 1 cup | 3 to 4 minutes\n\nStir every 30 seconds, until light brown.\n\nSnack, Popcorn, Pretzels, Corn or Potato Chips (crisp)\n\nPlace in paper towel\u2013lined non-metal basket, uncovered. | High | 2 cups | 20 to 40 seconds\n\n4 cups | 40 seconds to 1 minute\n\nSyrup (heat)\n\nPlace in glass measuring cup or pitcher, uncovered. | High | \u00bd cup | 30 to 45 seconds\n\n1 cup | 45 seconds to 1 minute\n\nStir every 30 seconds.\n\nWater (boil)\n\nPlace in glass measuring cup, uncovered. | High | 1 cup | 2 to 3 minutes\n\n## Food Safety\n\nAmerica's food supply is one of the safest in the world. Farmers, ranchers, food processors, supermarkets and restaurants must follow strict rules and regulations while getting food to you. Once food leaves the grocery store, care must be taken to keep it safe.\n\nWhy worry about food safety? Because most illnesses reported from \"bad food\" are caused by bacterial contamination. Nearly all cases can be linked to improper food handling and could have been prevented.\n\nMicroorganisms are always with us: They are on us and on animals, in the air and water and on raw food. Some bacteria are useful, like those used to make cheese or ferment beer. But other bacteria cause foods to spoil or cause food poisoning.\n\nBacteria that can cause food to spoil can grow at refrigerator temperatures (below 40\u00b0F). They usually make the food look or smell bad, raising a red flag that it should be thrown out. Food poisoning bacteria grow at temperatures between 40\u00b0F and 140\u00b0F\u2014those temperatures between hot-from-the-oven and chilled. These are pathogens that you can't see, smell or taste, which if eaten may lead to illness, disease or even death.\n\nTo prevent this type of bacteria from becoming harmful, they must be stopped from multiplying. If contaminated food is eaten, people most often get sick within 4 to 48 hours. It's not always easy to determine if the problem is the flu or food poisoning. Call a doctor or go to a hospital immediately if symptoms are severe.\n\n### Basic Food Safety Rules\n\n#### Wash Your Hands\n\nProper hand washing could eliminate nearly half of all causes of food-borne illness. Always wash hands:\n\n  * Before handling food and utensils and serving and eating food.\n  * After handling food\u2014especially raw meat, poultry, fish, shellfish and eggs.\n  * Between jobs like cutting up raw chicken and making a salad.\n  * After using the bathroom, blowing your nose, changing diapers, touching pets, or handling garbage, dirty dishes, hair, dirty laundry and phones.\n  * If you have any kind of skin cut or infection on your hands; and wear protective plastic or rubber gloves.\n\n#### Keep It Clean\n\n  * Wash all utensils and surfaces with hot, soapy water after contact with raw meat, poultry, fish or seafood. Use paper towels when working with these foods.\n  * Bleach and commercial kitchen-cleaning agents are the most effective at killing bacteria. Hot water and soap or hot water alone may not kill all strains of bacteria that are present.\n  * Keep dishcloths and sponges clean and dry because when wet, these materials harbor bacteria and may promote their growth. Wash them often in hot water in the washing machine and, for added safety, sanitize them three times a week by soaking them in a mixture of \u00be cup bleach to 1 gallon water.\n  * Clean refrigerator surfaces inside and out. Sort through perishable foods each week, tossing out anything past its prime.\n  * Keep pets out the kitchen and away from food.\n\n#### Don't Cross-Contaminate\n\nCross-contamination happens when cooked or ready-to-eat foods pick up bacteria from other foods, hands, cutting boards and utensils. Raw meat, poultry, shellfish, fish and eggs require special handling to keep them from contaminating other foods:\n\n  * Keep them separate from cooked or ready-to-eat foods in your grocery cart.\n  * Prevent leaks by repacking leaky packages and thawing foods in the refrigerator on a tray with sides to catch any juices.\n  * It's best to have at least two cutting boards\u2014one for raw meats and so forth and one for vegetables and fruit. Don't cut raw meat on wooden boards, as the raw juices can get into the wood, where bacteria can grow.\n  * Don't put cooked food on an unwashed plate that was used for raw meats.\n  * Wash sinks and sink mats with hot, soapy water if they've come in contact with raw meat.\n\n#### Keep Hot Food Hot\n\n  * Hot foods can't be left at room temperature for more than 2 hours, including preparation time. Keeping foods hot means keeping them at 140\u00b0F or higher.\n  * Don't partially cook perishable foods and then set them aside or refrigerate to finish them later. It's okay to cook them with one appliance if they are immediately completed with another\u2014grilling recipes will frequently use this technique.\n  * Keep cooked food hot, or refrigerate until ready to serve. This includes take-out foods, too.\n  * Reheat leftovers, stirring often, until steaming hot (165\u00b0F), using a cover to get the food hot in the center and retain moisture. Heat soups, sauces and gravies to a rolling boil, then boil 1 minute, stirring frequently, before serving.\n\n#### Keep Cold Food Cold\n\n  * Cold foods can't be left at room temperature for more than 2 hours, including preparation time. Keeping foods cold means keeping them at 40\u00b0F or lower.\n  * When shopping, put perishable foods (eggs, milk, seafood, fish, meat and poultry or items that contain them) in your cart last. Make sure they are cold to the touch when you select them. Put packages that can leak into plastic bags to prevent dripping on other foods in your cart.\n  * Perishable foods should be taken home and refrigerated immediately. If the trip is longer than 30 minutes or it is a hot day, bring freezer shopping bags or a cooler with freezer packs for these items to prevent them from reaching unsafe temperatures.\n  * Purchase frozen foods that are free from ice crystals, because ice crystals may mean the food had been partially thawed and refrozen.\n  * Foods will chill faster if stored in smaller, shallow containers and there is space between them in the refrigerator or freezer.\n  * Never thaw foods at room temperature\u2014thaw only in the refrigerator or microwave following manufacturer's directions. If food is thawed in the microwave, cook it immediately.\n\n## Safe Buffets\n\n  * Serve food in family-size dishes rather than larger, catering-size ones. Put out new dishes of food as they become empty.\n  * Keep foods hot (at least 140\u00b0F) by using slow cookers, chafing dishes, fondue pots or a warming tray. Use warming units heated by electricity or canned cooking fuel\u2014not candles, as they don't get hot enough to keep foods safe.\n  * Chill both salads made with seafood, poultry, meat or dairy products and the dish before serving; they will stay cold longer.\n  * Hot or cold foods shouldn't stand at room temperature for more than 2 hours. Toss it if you don't know how long it's been out.\n  * At restaurants or potlucks, salad bars and buffets should look clean. Make sure cold foods are cold and hot foods are steaming before putting them on your plate.\n\n## Safe Picnics and Packed Lunches\n\n  * Pack lunches in insulated lunch bags or in a small cooler with a freezer pack, frozen juice box or frozen small plastic bottle of water to keep food cold. Keep out of the sun. Perishable foods packed in an uninsulated bag should be refrigerated.\n  * Wash thermal containers and rinse well with boiling water after each use. Be sure hot foods are boiling when poured into these containers.\n  * Wash fruits and vegetables before packing.\n  * Chill picnic food before packing it in freezer shopping bags or an ice-filled cooler. Because beverage coolers will be opened more frequently, use one cooler for beverages and one for perishable food.\n  * Tightly wrap raw meat, poultry, fish and seafood or pack them in a separate cooler to keep them from dripping on other foods. Bring along hand sanitizer for washing hands after handling raw poultry, meat, fish or seafood.\n\n## Storing Food in the Refrigerator and Freezer\n\n### Refrigerator and Freezer Food Storage Chart\n\n---\n\nFoods | Refrigerator (35\u00b0F to 40\u00b0F) | Freezer (0\u00b0F or below)\n\nBaked Products\n\nBreads, coffee cakes, muffins, quick and yeast breads\n\nCool completely; wrap tightly after frostings have set at room temperature. Freeze muffins uncovered before packaging together. | 5 to 7 days\n\nRefrigerate bread during hot, humid weather or if it contains perishable ingredients like meat or cheese. | 2 to 3 months\n\nFreeze tightly wrapped in freezer plastic bags. Thaw by loosening wrap; let stand 2 to 3 hours.\n\nCakes (unfrosted or frosted)\n\nCool completely; wrap tightly.\n\nDo not freeze cakes with fruit filling.\n\nThaw by loosening wrap; thaw unfrosted cakes at room temperature. For frosted cakes, place in refrigerator overnight. | 3 to 5 days\n\nRefrigerate cakes that contain dairy products in the filling (like custard filling) or frosting. | 3 to 4 months (unfrosted)\n\n2 to 3 months (frosted)\n\nCakes filled and frosted with plain sweetened whipped cream can be frozen. Freeze unwrapped on a plate. When frozen, wrap and freeze.\n\nCheesecakes (baked) | 3 to 5 days | 4 to 5 months\n\nThaw wrapped in refrigerator 4 to 6 hours.\n\nCookies (baked)\n\nPlace delicate or frosted cookies between layers of waxed paper.\n\nDo not freeze meringue, custard-filled or cream-filled cookies. | Only if stated in recipe | Up to 12 months (unfrosted)\n\nUp to 3 months (frosted)\n\nThaw most cookies, covered, in container at room temperature 1 to 2 hours. Crisp cookies: Remove from container to thaw.\n\nPies (unbaked or baked fruit pies, baked pecan and baked pumpkin pies)\n\nThaw frozen baked fruit and pecan pies unwrapped at room temperature or unwrap and thaw at room temperature 1 hour, then heat in 375\u00b0F oven 35 to 40 minutes or until warm.\n\nThaw frozen baked pecan pie unwrapped 3 to 4 hours in refrigerator. | 3 to 5 days (baked pumpkin pie)\n\n2-3 days (baked fruit pie)\n\n1-2 days (unbaked fruit pie)\n\n3-5 days (pecan pie) | Up to 4 months (unbaked and baked fruit pies, baked pecan pie)\n\n1 month (baked pumpkin pie)\n\nTo thaw unbaked fruit pies, unwrap and cut slits in top crust. Bake at 425\u00b0F for 15 minutes. Reduce oven to 375\u00b0F; bake 30 to 45 minutes longer or until juices begin to bubble through slits.\n\nDairy Products\n\nCottage cheese and creamy ricotta | Up to 10 days | Not recommended\n\nHard cheese\n\nIf hard cheese is moldy, trim \u00bd inch from the affected area and rewrap tightly. | 3 to 4 weeks | 6 to 8 weeks\n\nThaw wrapped cheeses in refrigerator. Texture will be crumbly; use in baked goods.\n\nButtermilk, cream and milk | Up to 5 days | Not recommended\n\n1 day (sweetened whipped cream). | Unsweetened and sweetened whipped cream can be frozen by dropping small mounds onto waxed paper; freeze and store in airtight container 1 to 2 weeks. To thaw, let stand at room temperature about 15 minutes.\n\nSour cream and yogurt | Up to 1 week | Not recommended\n\nEggs\n\nFresh eggs in shell | 2 weeks | Not recommended\n\nCooked eggs in shell | 1 week | Not recommended\n\nFats and Oils\n\nButter | No longer than 2 weeks | No longer than 2 months\n\nMargarine and spreads | No longer than 1 month | No longer than 2 months\n\nMayonnaise and salad dressing | No longer than 6 months | Not recommended\n\nMeats\n\nIf wrapped in white butcher paper, rewrap tightly in plastic wrap, foil or plastic freezer bags.\n\nChops (uncooked) | 3 to 5 days | 4 to 6 months\n\nGround (uncooked) | 1 to 2 days | 3 to 4 months\n\nRoasts and steaks (uncooked) | 3 to 5 days | 6 to 12 months\n\nCooked | 3 to 4 days | 2 to 3 months\n\nCold cuts | 3 to 5 days (opened) | Not recommended\n\n2 weeks (unopened) | Not recommended\n\nBacon, cured | 5 to 7 days | No longer than 1 month\n\nHot dogs | 1 week (opened) | 1 to 2 months\n\n2 weeks (unopened) | 1 to 2 months\n\nHam, whole or half (cooked) | 5 to 7 days | 1 to 2 months\n\nHam slices (cooked) | 3 to 4 days | 1 to 2 months\n\nNuts\n\nStore nuts in airtight container\n\n|\n\n6 months | 1 year\n\nPoultry\n\nWhole, including game birds, ducks and geese (uncooked) | 1 to 2 days | No longer than 12 months\n\nCut up (cooked) | 5 to 7 days | No longer than 9 months\n\nCooked | 3 to 4 days | 1 to 2 months\n\nSeafood\n\nFish (uncooked) | 1 to 2 days | 3 to 6 months\n\nFish (breaded, cooked) | Store in freezer | 2 to 3 months\n\nShellfish (uncooked) | Up to 1 to 2 days | 3 to 4 months\n\nShellfish (cooked) | 3 to 4 days | 1 to 2 months\n\n# Chapter 2: Appetizers\n\n## Appetizer Basics\n\nChoosing Appetizers\n\nServing with Style\n\n## Features\n\nBetty's Staples: Yogurt\n\nGlobal Flavors: Charcuterie\n\nHeirloom Recipe and New Twist\n\nLearn to Deep-Fry\n\nMake-Ahead: Meatballs\n\n## Dips and Spreads\n\nBaba Ghanoush Calorie Smart\n\nChipotle Guacamole\n\nCotija Guacamole\n\nCumin Hummus\n\nFig-Hazelnut Tapenade Calorie Smart \u2022 Fast\n\nGreek Layered Dip Calorie Smart \u2022 Fast\n\nGuacamole Calorie Smart \u2022 Fast\n\nHot Artichoke Dip Calorie Smart \u2022 Easy\n\nHot Artichoke-Spinach Dip\n\nHummus Calorie Smart \u2022 Fast\n\nOlive Tapenade Calorie Smart \u2022 Fast\n\nPeanut Butter\u2013Yogurt Dip\n\nPersonal Taco Dip\n\nRoasted Garlic Calorie Smart \u2022 Easy\n\nRoasted Garlic Hummus\n\nRoasted Vegetable Dip with Baked Pita Crisps Calorie Smart\n\nSpicy Vegetable Dip\n\nSun-Dried Tomato Hummus\n\nTzatziki Dip Calorie Smart \u2022 Easy\n\n## Finger Foods\n\nBacon-Cheddar Deviled Eggs\n\nBlue Cheese Deviled Eggs\n\nChipotle Deviled Eggs\n\nCinnamon-Cardamom Mixed Nuts\n\nCinnamon-Sugared Nuts\n\nDeviled Eggs Calorie Smart \u2022 Fast\n\nProvolone-Onion Crostini\n\nReuben Deviled Eggs\n\nShrimp Cocktail Shooters\n\nSouthwestern Spiced Cashews\n\nSouthwestern Spiced Party Nuts Fast\n\nSteamed Edamame in the Shell with Dipping Sauces Calorie Smart \u2022 Fast\n\nTomato-Basil Crostini Calorie Smart \u2022 Fast\n\n## Hot Appetizers\n\nBasil-Cheese Triangles Calorie Smart\n\nBeef or Chicken Nachos\n\nBlack Bean Sliders Calorie Smart\n\nBrie in Puff Pastry with Cranberry Sauce\n\nBrie in Puff Pastry with Double-Orange Cranberry Sauce\n\nBuffalo Chicken Wings Calorie Smart\n\nCheesy Potato Skins\n\nChicken Satay with Peanut Sauce\n\nClassic Crab Cakes Fast\n\nEasy Square Meatballs\n\nHoney-Mustard Chicken Wings\n\nHot and Saucy Cocktail Meatballs Calorie Smart\n\nMeatballs Calorie Smart\n\nNachos Fast\n\nPork Carnitas Taco Bites\n\nSalmon Cakes\n\nSalsa Chicken Wings\n\nShrimp Egg Rolls\n\nSweet Chili Kabobs\n\nSweet-Spicy Chicken Wings\n\nTeriyaki Sliders\n\n## Cold Appetizers\n\nBell Pepper Cheese Ball\n\nCheese Ball Calorie Smart\n\nClassic Shrimp Cocktail Calorie Smart \u2022 Fast\n\nDill Havarti Cheese Ball\n\nMini French Cheese Balls Calorie Smart\n\nMini Indian Cheese Balls\n\nMini Korean Cheese Balls\n\nMini Mexican Cheese Balls\n\nPecan-Coated Cheese Ball\n\nPepper Jack Cheese Ball\n\nPickled Red Onions Calorie Smart\n\nShrimp Cocktail Shooters\n\nVietnamese Beef Lettuce Wraps Calorie Smart\n\n## Bonus Recipes\n\nApple-Yogurt Snacks\n\nBlue Cheese Yogurt Dressing\n\nCheeseburger Hoagies\n\nFrozen Yogurt Pops\n\nGranola-Blueberry Parfaits\n\nMeatball Pizza\n\nOrange-Pineapple Smoothies\n\nPesto Pasta and Meatballs\n\nSalsa Tacos\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Appetizer Basics\n\nWhether you call them appetizers or hors d'oeuvres, variety is the name of the game when these tasty morsels are part of the occasion. And it doesn't need to be a special occasion to enjoy their amazing versatility.\n\nThe word \"appetizers\" simply refers to any food served before a meal to entice the appetite. Often we think of appetizers as being small plates served at the table and hors d'oeuvres as small finger foods that are served separately from the meal\u2014a subtle difference. But the terms are used interchangeably and include all snack-type foods, whether they are served as a meal itself, or simply to awaken taste buds.\n\nThe array of appetizers ranges from dips and spreads to elegant choices such as tapas and fancy canap\u00e9s. But whether you are serving one or two or expecting a crowd, planning is key.\n\n## choosing Appetizers\n\n  * Depending on the occasion, you might want to make a meal of appetizers or just snack on one or two tasty morsels before the meal.\n  * Select a menu with an eye toward different foods, flavors and textures. For larger groups, choose both hot and cold appetizers for variety and ease of serving. Be mindful of oven space when planning hot choices.\n  * Allow for 2 or 3 pieces of each appetizer per person. Many recipes can be doubled or tripled easily.\n  * Map out how many guests will attend, what you want to serve and when and where you will serve food before deciding what to make.\n  * Make recipes or parts of recipes ahead as much as possible.\n\n## Serving with Style\n\n  * Choose a place for the food that guests can access easily. For a larger gathering, set up several areas with different appetizers so guests can help themselves at their leisure.\n  * Consider your theme and who will attend, then decorate and set the table or buffet area ahead of time. Think about serving\u2014will the event require equipment such as a fondue pot, slow cooker or hot plate?\n  * Make cutlery bundles wrapped in paper or cloth napkins for foods that require utensils. Add small trays for guests to gather appetizers\u2014often beverages will fit on these trays, too.\n  * Unless it is a sit-down occasion, choose appetizers that are easy to pick up.\n  * Serve foods in interesting containers\u2014pretty plates\/bowls, hollowed-out loaves of bread, fun baskets.\n\nTomato-Basil Crostini (page 59)\nGlobal Flavors\n\n## Charcuterie\n\nIf you have never assembled a charcuterie board, a wonderful experience awaits you. This crowd-pleasing appetizer is simply a tasty combination of cured meats and other delicious foods to round out the display. Like a cheese platter, a charcuterie board is low maintenance and can serve a crowd easily.\n\nMake your display stand out by including a few other foods tucked in around the meats, adding visual interest, flavors and textures to the board. You'll find good ideas both in this chapter and the Do It Yourself chapter. Most can be made ahead and then simply spooned into serving dishes.\n\nStart by adding bread to the board. Choose toasted French bread, breadsticks, small cocktail breads or even hearty crackers. Add a spicy food, such as spiced nuts or spicy pickles. Offer a sweet sauce, like Sweet Tomato Jam (page 438) or chutney. Sour, crunchy foods like Pickled Red Onions (page 38) or other pickled vegetables add delightful contrast to the rich meats. Or check out gourmet food shops for unique finds when you're short on time. Cheeses can be added, too, for variety. (See page 27 for cheese types.)\n\nCutting boards of many shapes and sizes are available, a large flat tray, marble or other smooth surfaces make visually interesting and practical choices for serving charcuterie.\n\nFor the meats, check out a good deli, ethnic market or butcher shop. Be sure to ask for samples so that you can try any meats that are unfamiliar. Here are some popular choices, but there are many more options available.\n\nBacon: Smoked pork belly\n\nBresaola: Cured beef tenderloin that is air-dried and salted\n\nCapicola (Capocollo): Dry-cured rolled pork shoulder\n\nCesina: Spanish version of bresaola\n\nChorizo: Cured hard sausage with a spicy garlicky flavor\n\nFinocchiona: Cured Italian salami flavored with fennel seed\n\nLomo de Cerdo: Cured pork tenderloin\n\nMortadella: Large Italian hard sausage made with pork\n\nPancetta: Italian bacon made from cured pork belly\n\nP\u00e2t\u00e9 or Terrine: Cooked ground meat and fat minced into a spreadable paste\u2014mixture in a terrine will be more coarsely chopped\n\nPepperoni: Variety of salami, usually made with pork and beef\n\nProsciutto: Dry-cured ham\n\nRillettes: Rustic p\u00e2t\u00e9 made from meat that has been poached in its own fat, then shredded and combined with some of the fat.\n\nSalami: Cured sausage made from fermented and air-dried beef or pork\n\nSerrano Ham: Spanish dry-cured ham similar to prosciutto\n\nSmoked Country Ham: Salty cured ham\n\nSopressata: Cured hard spicy sausage\n\nSpeck and Lardo: German and Italian types of salted and seasoned lard\n\n## Pickled Red Onions Calorie Smart\n\nPrep 20 min Total 8 hr 20 min \u25cf 16 servings\n\n  * 3 medium red onions, thinly sliced\n  * 2 tablespoons vegetable oil\n  * 2 cloves garlic, finely chopped\n  * 1 cup cider vinegar or \u2153 cup fresh lime juice\n  * 4 black peppercorns\n  * 1 teaspoon salt\n  * \u215b teaspoon dried oregano leaves\n  * \u215b teaspoon dried thyme leaves\n  * \u215b teaspoon dried marjoram leaves\n  * 3 dried bay leaves\n\n1 In large bowl, place onions. Cover with boiling water; let stand about 10 minutes. Drain well.\n\n2 In 12-inch skillet, heat oil over medium-high heat. Add onions and garlic; cook 5 to 8 minutes, stirring frequently, until onions are crisp-tender. Transfer onions to medium bowl. Stir in remaining ingredients until blended.\n\n3 Cover tightly; refrigerate at least 8 hours. Remove bay leaves before serving. Store in refrigerator up to 3 days.\n\n1 Serving: Calories 25 (Calories from Fat 15); Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 2g); Protein 0g Exchanges: \u00bd Vegetable Carbohydrate Choices: 0\n\nRoasted Garlic\n\n## Roasted Garlic Calorie Smart \u2022 Easy\n\nRoasting garlic turns it into a sweet and delicious spread. Garlic bulbs (heads of garlic) are made up of as many as 15 sections, called cloves.\n\nPrep 10 min Total 1 hr \u25cf 2 to 8 tablespoons\n\n  * 1 to 4 bulbs garlic\n  * 2 teaspoons olive or vegetable oil for each bulb garlic\n  * Salt and pepper to taste\n  * Sliced French bread, if desired\n\n1 Heat oven to 350\u00b0F. Carefully peel paperlike skin from around each bulb of garlic, leaving just enough to hold cloves together. Cut \u00bc- to \u00bd-inch slice from top of each bulb to expose cloves. Place each bulb, cut side up, on 12-inch square of foil.\n\n2 Drizzle 2 teaspoons oil over each bulb. Sprinkle with salt and pepper. Wrap securely in foil. Place in pie plate or shallow baking pan.\n\n3 Bake 45 to 50 minutes or until garlic is tender when pierced with toothpick or fork. Let stand until cool enough to handle. To serve, gently squeeze soft garlic out of cloves. Spread garlic on bread.\n\n1 Tablespoon: Calories 70; Total Fat 4.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: 1 Vegetable, 1 Fat Carbohydrate Choices: \u00bd\n\nSouthwestern Spiced Party Nuts\n\n## Southwestern Spiced Party Nuts Fast\n\nPrep 10 min Total 20 min \u25cf 2\u00bc cups nuts\n\n  * 1 can (9.5 to 11.5 oz) salted mixed nuts\n  * 1 tablespoon butter, melted\n  * 2 teaspoons chili powder\n  * \u00bd teaspoon garlic powder\n  * \u00bd teaspoon onion powder\n  * \u00bc teaspoon ground cinnamon\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 2 tablespoons sugar\n\n1 Heat oven to 300\u00b0F. In medium bowl, mix nuts and butter until nuts are coated. In small bowl, mix remaining ingredients except sugar; sprinkle over nuts. Stir until nuts are completely coated. In ungreased 15x10x1-inch pan, spread nuts in single layer.\n\n2 Bake uncovered about 10 minutes or until nuts are toasted. Return to medium bowl. While nuts are still hot, sprinkle with sugar and toss to coat. Serve warm, or cool completely, about 1 hour. Store in airtight container at room temperature up to 3 weeks.\n\n\u00bc Cup: Calories 250; Total Fat 21g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 240mg; Total Carbohydrate 11g (Dietary Fiber 3g, Sugars 4g); Protein 6g Exchanges: 1 Starch, \u00bd High-Fat Meat, 3 Fat Carbohydrate Choices: 1\n\nSouthwestern Spiced Cashews Substitute salted whole cashews for the mixed nuts.\n\n## Cinnamon-Sugared Nuts\n\nPrep 10 min Total 40 min \u25cf 2 cups nuts\n\n  * 1 tablespoon slightly beaten egg white\n  * 2 cups pecan halves, unblanched whole almonds or walnut halves\n  * \u00bc cup sugar\n  * 2 teaspoons ground cinnamon\n  * \u00bc teaspoon ground nutmeg\n  * \u00bc teaspoon ground cloves\n\n1 Heat oven to 300\u00b0F. In medium bowl, stir egg white and nuts until nuts are coated and sticky.\n\n2 In small bowl, stir remaining ingredients until well mixed; sprinkle over nuts. Stir until nuts are coated. In ungreased 15x10x1-inch pan, spread nuts in single layer.\n\n3 Bake uncovered about 30 minutes or until toasted (nut coating becomes crunchy during baking and cooling). Serve warm, or cool completely, about 1 hour. Store in airtight container at room temperature up to 3 weeks.\n\n\u00bc Cup: Calories 210; Total Fat 18g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 10g (Dietary Fiber 3g, Sugars 7g); Protein 3g Exchanges: \u00bd Other Carbohydrate, \u00bd High-Fat Meat, 3 Fat Carbohydrate Choices: \u00bd\n\nCinnamon-Cardamom Mixed Nuts Substitute lightly salted mixed nuts for the pecans and ground cardamom for the nutmeg. Omit cloves.\n\nGuacamole\n\n## Guacamole Calorie Smart \u2022 Fast\n\nRipe avocados yield to gentle pressure. To ripen, let stand at room temperature in a closed paper bag until ripe.\n\nPrep 15 min Total 15 min \u25cf 2 cups dip\n\n  * 1 small jalape\u00f1o chile*\n  * 2 large ripe avocados\n  * 2 tablespoons fresh lime or lemon juice\n  * 2 tablespoons finely chopped fresh cilantro\n  * \u00bc to \u00bd teaspoon salt\n  * Dash pepper\n  * 1 small clove garlic, finely chopped\n  * 1 small tomato, seeded, chopped (\u00bd cup)\n  * 2 tablespoons finely chopped red onion\n  * Tortilla chips, if desired\n\n1 Remove stems, seeds and ribs from chile; chop chile. Cut avocados lengthwise in half; remove pit and peel. In medium glass or plastic bowl, mash avocados with fork. Gently stir in chile and remaining ingredients except tortilla chips until mixed.\n\n2 Serve with tortilla chips. Press and smooth plastic wrap directly on surface of leftover guacamole to help prevent discoloration.\n\n*Or substitute 1 tablespoon canned chopped green chiles for the jalape\u00f1o chile.\n\n1 Tablespoon: Calories 15; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 15mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nCotija Guacamole Stir in \u00bd cup crumbled Cotija cheese with remaining ingredients in Step 1.\n\nChipotle Guacamole Omit the jalape\u00f1o and add 1 chopped chipotle chile in adobo sauce instead.\n\n### Cutting an Avocado\n\nCut avocado lengthwise through skin around pit.\n\nWith hands, slowly twist both sides to separate.\n\nSlide flatware tablespoon under pit to remove.\n\nMake cuts through flesh; remove with spoon.\n\n## Hummus Calorie Smart \u2022 Fast\n\nTahini, a thick paste made from ground sesame seed, gives this dip its distinctive flavor. Look for it in the condiment, ethnic foods or pickle aisle.\n\nPrep 5 min Total 5 min \u25cf 2 cups dip\n\n  * 1 can (15 to 16 oz) chickpeas, drained, \u00bc cup liquid reserved\n  * \u00bd cup tahini*\n  * \u00bc cup olive oil\n  * 2 cloves garlic, finely chopped\n  * 3 tablespoons lemon juice\n  * 1 teaspoon salt\n  * \u215b teaspoon pepper\n  * Chopped fresh parsley\n  * Pita bread wedges, crackers or fresh vegetables, if desired\n\n1 In blender or food processor, place beans, reserved bean liquid, tahini, oil, garlic, lemon juice, salt and pepper. Cover and blend on high speed, stopping occasionally to scrape side, until smooth.\n\n2 Spoon into serving dish. Garnish with parsley. Serve with pita bread wedges.\n\n*Or substitute \u00bd cup sesame seed for the tahini, but the hummus will not be smooth.\n\n2 Tablespoons: Calories 60; Total Fat 3g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 85mg; Total Carbohydrate 6g (Dietary Fiber 1g, Sugars 0g); Protein 2g Exchanges: 1 Vegetable, \u00bd Fat Carbohydrate Choices: \u00bd\n\nCumin Hummus Add \u00bc teaspoon ground cumin with the salt and pepper. Omit parsley. Sprinkle with cumin seed, if desired.\n\nRoasted Garlic Hummus Stir 1 to 2 tablespoons Roasted Garlic (page 40) into blended mixture.\n\nSun-Dried Tomato Hummus Stir \u2153 cup chopped drained sun-dried tomatoes (packed in oil) into blended mixture.\n\n## Baba Ghanoush Calorie Smart\n\nThis heathly vegan dip comes together easily and is a flavorful way to enjoy eggplant \u2014 even for those who don't like eggplant!\n\nPrep 10 min Total 1 hr 35 min \u25cf 1\u00bd cups dip\n\n  * 1 large eggplant (about 1\u00bd lb)*\n  * 1 tablespoon olive oil\n  * 2 tablespoons sesame tahini paste\n  * 1 tablespoon fresh lemon juice\n  * 1 clove garlic, minced\n  * \u00be teaspoon salt\n  * \u00bd teaspoon ground cumin\n  * \u215b teaspoon pepper\n  * Pomegranate seeds and chopped fresh parsley, if desired\n  * Toasted baguette slices or pita wedges, if desired\n\n1 Heat oven to 450\u00b0F. Line 15x10x1-inch pan with foil. Pierce eggplant in several places with tip of sharp knife; place on pan. Roast uncovered 45 to 55 minutes, turning once, until skin is blackened and inside begins to soften. Let stand until cool enough to handle, about 30 minutes.\n\n2 Cut eggplant lengthwise in half. Scoop out flesh and place on several layers of paper towel to absorb excess moisture. In food processor, place eggplant, oil, tahini, lemon juice, garlic, salt, cumin and pepper. Cover and process until smooth.\n\n3 Spoon into serving bowl. Sprinkle with pomegranate seeds and parsley. Serve with toasted baguette slices.\n\n*Choose dark purple, smooth, glossy, taut-skinned eggplant that is free from blemishes and rust spots. Caps and stems should be intact and free of mold.\n\n2 Tablespoons: Calories 45; Total Fat 2.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 4g (Dietary Fiber 2g, Sugars 2g); Protein 1g Exchanges: 1 Vegetable, \u00bd Fat Carbohydrate Choices: 0\n\n### Charring Eggplant\n\nRoast eggplant until skin is blackened and insides begin to soften.\n\nScoop out flesh and place on several layers of paper towel to absorb excess moisture.\n\nTzatziki Dip, Baba Ghanoush, Hummus\n\n## Tzatziki Dip Calorie Smart \u2022 Easy\n\nA classic Greek yogurt dip, tzatziki typically includes cucumber and mint.\n\nPrep 10 min Total 2 hr 10 min \u25cf 2 cups dip\n\n  * 1\u00bd cups plain Greek yogurt (to make your own, see page 445)\n  * \u2154 cup finely chopped peeled, seeded English (hothouse) cucumber\n  * 3 tablespoons finely chopped fresh mint leaves\n  * 1\u00bd teaspoons fresh lemon juice\n  * 1 teaspoon olive oil\n  * \u00bd teaspoon salt\n  * Additional fresh mint leaves, if desired\n  * Pita fold breads or pita (pocket) breads, cut into wedges, if desired\n  * Fresh vegetables, if desired\n\n1 In medium bowl, gently mix yogurt, cucumber, chopped mint, lemon juice, oil and salt until blended. Cover and refrigerate 2 hours to blend flavors.\n\n2 Garnish with mint leaves. Serve with pita wedges and vegetables.\n\n2 Tablespoons: Calories 20; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 85mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 1g); Protein 2g Exchanges: Free Carbohydrate Choices: 0\n\nGreek Layered Dip\n\n## Greek Layered Dip Calorie Smart \u2022 Fast\n\nPrep 20 min Total 20 min \u25cf 12 servings\n\n  * 2 whole wheat pita (pocket) breads (6 inch)\n  * Cooking spray\n  * 1 container (8 oz) plain hummus (to make your own, see page 42)\n  * 1 container (6 oz) plain Greek yogurt (to make your own, see page 445)\n  * 1 tablespoon chopped fresh parsley\n  * 1 teaspoon lemon juice\n  * \u215b teaspoon pepper\n  * 1 medium plum (Roma) tomato, seeded, chopped\n  * \u2153 cup pitted kalamata or ripe olives, quartered\n  * \u2153 cup finely chopped seeded cucumber\n  * 1 cup crumbled feta cheese (4 oz)\n  * 4 medium green onions, chopped (\u00bc cup)\n  * 1 tablespoon olive oil\n  * \u00bd medium cucumber, sliced\n  * 1 medium green or red bell pepper, cut into strips\n\n1 Heat oven to 350\u00b0F. Split each pita bread horizontally to make 2 rounds. Cut each round into 6 wedges. Place bread, rough surface up, on ungreased cookie sheet. Spray with cooking spray. Bake 8 to 10 minutes or until golden brown and crisp; cool.\n\n2 Meanwhile, spread hummus on shallow serving platter or in pie plate. In small bowl, mix yogurt, parsley, lemon juice and pepper; spread evenly over hummus. Top with tomato, olives, chopped cucumber, cheese and onions. Drizzle with oil. Serve with pita chips, sliced cucumber and bell pepper strips.\n\n1 Serving (\u00bc Cup Dip, 2 Chips and 4 Veggies): Calories 100; Total Fat 4.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 5mg; Sodium 270mg; Total Carbohydrate 12g (Dietary Fiber 2g, Sugars 3g); Protein 4g Exchanges: \u00bd Starch, \u00bd Skim Milk, \u00bd Fat Carbohydrate Choices: 1\n\nLighter Directions To reduce the fat and calories in this recipe, use fat-free plain Greek yogurt. Reduce feta cheese to \u00bd cup and olive oil to 1 teaspoon.\n\nSteamed Edamame in Shell with Dipping Sauces\n\n## Steamed Edamame in the Shell with Dipping Sauces Calorie Smart \u2022 Fast\n\nPrep 15 min Total 15 min \u25cf 10 servings\n\nGinger-Soy Dipping Sauce\n\n  * \u2154 cup reduced-sodium or regular soy sauce\n  * 2 tablespoons honey\n  * 2 teaspoons finely chopped gingerroot\n  * \u00bd teaspoon crushed red pepper flakes\n  * 2 green onions with green tops, thinly sliced\n\nCreamy Wasabi Dipping Sauce\n\n  * \u00bd cup mayonnaise\n  * \u2153 cup sour cream\n  * 2 teaspoons wasabi powder*\n  * \u00bc teaspoon salt\n  * Toasted sesame seed and black sesame seed, if desired\n\nEdamame\n\n  * 1 bag (12 oz) frozen edamame soybeans (in the pod)**\n\n1 In small bowl, mix soy sauce, honey, gingerroot and pepper flakes until well blended; stir in onions. In another small bowl, mix mayonnaise, sour cream, wasabi powder and salt until smooth; sprinkle with sesame seed. Set sauces aside.\n\n2 In 4-quart saucepan, place steamer basket. Add 1 cup water; heat to boiling. Add edamame to basket; cover and cook 3 to 6 minutes or until crisp-tender. Remove from basket to serving bowl; let stand until cool enough to handle. Serve warm or room temperature with dipping sauces.\n\n*Wasabi, also called Japanese horseradish, has a sharp, pungent, fiery flavor. It comes in both powder and paste form and is available in the Asian-foods aisle of large supermarkets. If you can't find it, regular prepared horseradish makes a great substitute.\n\n**Fresh edamame (green soybeans), found in the produce section, can be substituted for frozen edamame. Increase steaming time to 20 minutes.\n\n1 Serving: (10 Edamame and 1 Tablespoon of Each Sauce): Calories 150; Total Fat 11g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 710mg; Total Carbohydrate 8g (Dietary Fiber 1g, Sugars 5g); Protein 3g Exchanges: \u00bd Other Carbohydrate, \u00bd Vegetable, 2 Fat Carbohydrate Choices: \u00bd\n\n## Olive Tapenade Calorie Smart \u2022 Fast\n\nTapenade is a French condiment often served with vegetables, fish and meat. The flavor is tangy, rich and strong. Fresh rosemary is delicious in this version, but you could use parsley instead.\n\nPrep 10 min Total 10 min \u25cf 1\u00be cups dip\n\n  * 1\u00bd cups pitted kalamata or ripe olives\n  * \u00bc cup chopped walnuts\n  * 3 tablespoons olive oil\n  * 3 tablespoons capers, drained\n  * 1\u00bd teaspoons fresh rosemary leaves\n  * 1 teaspoon Italian seasoning\n  * 2 cloves garlic, peeled\n  * Chopped red bell pepper, if desired\n  * Assorted crackers, if desired\n\n1 In food processor or blender, place all ingredients except bell pepper and crackers. Cover; process, using quick on-and-off motions, until slightly coarse.\n\n2 Spoon into serving dish. Sprinkle with bell pepper. Serve with crackers.\n\n2 Tablespoons: Calories 60; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 190mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1 Fat Carbohydrate Choices: 0\n\n# Heirloom Recipe and New Twist\n\nThis classic cheese ball has been a favorite in the Betty Crocker Kitchens for many years. It is ideal for any occasion and is even easy to tote if you are asked to bring along an appetizer. Often it is the first item on a buffet to be gone. The new twist provides ideas for a variety of mini cheese balls that can be customized for a theme party. Enjoy!\n\nCheese Ball\n\nHeirloom\n\n## Cheese Ball Calorie Smart\n\nPrep 15 min Total 10 hr 45 min \u25cf 16 servings\n\n  * 2 packages (8 oz each) cream cheese\n  * 1 cup shredded sharp Cheddar cheese (4 oz)\n  * \u00be cup crumbled blue, Gorgonzola or feta cheese (3 oz)\n  * 1 small onion, finely chopped (\u00bc cup)\n  * 1 tablespoon Worcestershire sauce\n  * \u00bd cup chopped fresh parsley\n  * Assorted crackers, if desired\n\n1 Place cheeses in medium bowl; let stand at room temperature about 30 minutes or until softened. Add onion and Worcestershire sauce. Beat with electric mixer on low speed until mixed. Beat on medium speed 1 to 2 minutes, scraping bowl frequently, until fluffy. Cover and refrigerate at least 8 hours or until firm enough to shape into ball.\n\n2 Shape cheese mixture into 1 large ball. Roll in parsley; place on serving plate. Cover and refrigerate about 2 hours or until firm. Serve with crackers.\n\n1 Serving: (2 Tablespoons): Calories 160; Total Fat 14g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 45mg; Sodium 240mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 6g Exchanges: 1 High-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\nLighter Directions For 3 grams of fat and 65 calories per serving, use fat-free cream cheese and reduced-fat Cheddar cheese.\n\nPepper Jack Cheese Ball Substitute \u00be cup shredded pepper Jack cheese for the blue cheese. Omit Worcestershire sauce. Substitute cilantro for the parsley.\n\nPecan-Coated Cheese Ball Omit the parsley and roll the cheese ball in \u00bc cup finely chopped pecans.\n\nBell Pepper Cheese Ball Omit the onion and stir in \u00bc cup finely chopped red bell pepper instead.\n\nDill Havarti Cheese Ball Substitute dill Havarti cheese for the blue cheese and roll the cheese ball in chopped fresh dill weed instead of parsley.\nNew Twist\n\n## Mini French Cheese Balls Calorie Smart\n\nPrep 1 hr Total 2 hr \u25cf 75 cheese balls\n\nBasic Cheese Ball\n\n  * 2 packages (8 oz each) cream cheese, softened\n  * 1 package (10 to 11 oz) ch\u00e8vre (goat) cheese, softened\n  * 1 teaspoon fresh lemon juice\n  * \u00bd teaspoon Worcestershire sauce\n  * \u00bc teaspoon kosher (coarse) salt\n  * \u00bc teaspoon freshly ground black pepper\n\nFlavor Stir-Ins\n\n  * 3 cups crumbled Roquefort cheese (12 oz)\n  * 2 tablespoons honey\n  * 3 tablespoons finely chopped shallot\n  * \u00bc cup chopped fresh parsley\n\nCoating\n\n  * \u00be cup chopped fresh parsley\n  * \u00bc cup finely chopped Marcona almonds\n\nServe-With, if desired\n\n  * Toasted baguette slices\n\n1 Line large cookie sheet with waxed paper or cooking parchment paper. In large bowl, beat all basic cheese ball ingredients with electric mixer on medium speed until combined. Stir in Roquefort cheese, honey, shallot and \u00bc cup parsley.\n\n2 With moistened hands, shape mixture into 1-inch balls. (If mixture is too soft to shape, refrigerate 30 minutes.) Place cheese balls on cookie sheet. Refrigerate until firm, about 1 hour.\n\n3 In small bowl, stir together \u00be cup parsley and the almonds. Roll cheese balls in coating before serving. Serve with baguette slices.\n\n1 Cheese Ball: Calories 60; Total Fat 5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 130mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 0\n\nMake-Ahead  Directions Make cheese balls as directed through Step 2. Cover and refrigerate up to 3 days or freeze up to 1 month. If frozen, allow cheese balls to thaw in the refrigerator before rolling in coating and serving.\n\nMini Indian Cheese Balls Omit flavor stir-ins, coating and serve-with. Beat cheese ball ingredients as directed in Step 1. Stir in 3 cups finely shredded paneer cheese (12 ounces), 1 tablespoon each grated lime peel and curry powder and \u00bc cup chopped fresh cilantro. Continue as directed in Step 2. In small bowl, stir together \u00be cup chopped fresh cilantro and \u00be cup finely chopped roasted salted cashews. Before serving, roll cheese balls in cashew mixture. Serve with naan bread if desired.\n\nMini Korean Cheese Balls Omit flavor stir-ins, coating and serve-with. Beat cheese ball ingredients as directed in Step 1. Stir in 3 cups shredded mozzarella cheese (12 ounces), 1\u00bd teaspoons chili paste, \u00be cup finely chopped mild kimchi, 6 tablespoons finely chopped green onions (6 medium) and 1 tablespoon grated lime peel. Continue as directed in Step 2. Before serving, roll cheese balls in 2\u00bc cups black sesame seed. Serve with rice crackers if desired.\n\nMini Mexican Cheese Balls Omit flavor stir-ins, coating and serve-with. Beat cheese ball ingredients as directed in Step 1. Stir in 3 cups crumbled queso fresco cheese, 1\u00bd teaspoons ancho chile powder, 2 tablespoons chopped chipotle chiles in adobo sauce (from 7-ounce can) and 6 tablespoons chopped green onions (6 medium). Continue as directed in Step 2. Before serving, roll cheese balls in 1\u00bd cups finely chopped pepitas (pumpkin seeds). Serve with tortilla chips if desired.\n\nMini Korean Cheese Balls\n\n## Hot Artichoke Dip Calorie Smart \u2022 Easy\n\nPrep 10 min Total 35 min \u25cf About 1\u00bd cups\n\n  * \u00bd cup mayonnaise or salad dressing\n  * \u00bd cup grated Parmesan cheese\n  * 4 medium green onions, chopped (\u00bc cup)\n  * 1 can (14 oz) artichoke hearts, drained, coarsely chopped\n  * Crackers or cocktail rye bread, if desired\n\n1 Heat oven to 350\u00b0F. In small bowl, stir mayonnaise and cheese until well mixed. Stir in onions and artichoke hearts. Spoon into ungreased 1-quart casserole.\n\n2 Cover and bake 20 to 25 minutes or until hot. Serve warm with crackers.\n\n1 Tablespoon: Calories 50; Total Fat 4.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 115mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Fat Carbohydrate Choices: 0\n\nHot Artichoke-Spinach Dip Increase mayonnaise and Parmesan cheese to 1 cup each. Stir in 1 box (9 ounces) frozen chopped spinach, thawed (squeeze to drain; spread on paper towels and pat dry).\n\n## Roasted Vegetable Dip with Baked Pita Crisps Calorie Smart\n\nPrep 20 min Total 1 hr \u25cf 6 servings\n\nBaked Pita Crisps\n\n  * 2 pita (pocket) breads (6 inch)\n  * 1 tablespoon olive oil or melted butter\n  * 1\u00bc teaspoons dried basil leaves\n  * 3 tablespoons grated Parmesan cheese\n\nRoasted Vegetable Dip\n\n  * 1 medium zucchini, sliced (2 cups)\n  * 1 medium yellow summer squash, sliced (1\u00bd cups)\n  * 1 medium red bell pepper, sliced\n  * 1 medium red onion, thinly sliced\n  * 2 cloves garlic, peeled\n  * Cooking spray\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon ground red pepper (cayenne)\n\n1 Heat oven to 375\u00b0F. Split each pita bread horizontally to make 2 rounds. Cut each round into 6 wedges. Lightly brush oil over smooth surface of bread; sprinkle with basil and cheese. Place bread, cheese side up, on ungreased cookie sheet. Bake 6 to 8 minutes or until golden brown and crisp; cool.\n\n2 Increase oven temperature to 400\u00b0F. In 15x10x1-inch pan, spread zucchini, yellow squash, bell pepper, onion and garlic in single layer. Spray vegetables with cooking spray. Sprinkle with salt and red pepper.\n\n3 Roast about 30 minutes, turning vegetables once, until tender and light brown.\n\n4 Place roasted vegetables in blender or food processor. Cover and blend on high speed about 1 minute, stopping occasionally to scrape side, until smooth. Spoon dip into serving bowl. Serve warm with pita crisps, or refrigerate at least 2 hours or until chilled.\n\n1 Serving (\u00bc Cup Dip and 4 Chips): Calories 25; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 170mg; Total Carbohydrate 5g (Dietary Fiber 1g, Sugars 2g); Protein 1g Exchanges: 1 Vegetable Carbohydrate Choices: \u00bd\n\n## Fig-Hazelnut Tapenade Calorie Smart \u2022 Fast\n\nPrep 15 min Total 15 min \u25cf 16 servings\n\n  * 1 cup dried Mission figs\n  * \u2153 cup water\n  * \u00bd cup pitted kalamata olives, chopped\n  * \u00bd cup roasted red bell peppers (from a jar), drained, chopped\n  * 2 tablespoons olive oil\n  * \u00bc cup chopped hazelnuts (filberts)\n  * 1 tablespoon chopped fresh thyme leaves\n  * 1 container (8 oz) mascarpone cheese or 1 package (8 oz) cream cheese, softened\n  * Crisp wheat crackers\n\n1 In small saucepan, heat figs and water to boiling over medium-high heat. Reduce heat; simmer 5 minutes, stirring occasionally, until liquid is absorbed and figs are soft. Cool; chop.\n\n2 In medium bowl, stir together figs, olives, roasted peppers, oil and nuts until well combined. Sprinkle with thyme. To serve, spread cheese on crackers; spoon fig mixture over cheese.\n\n1 Serving: Calories 190; Total Fat 11g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 10mg; Sodium 210mg; Total Carbohydrate 20g (Dietary Fiber 2g, Sugars 7g); Protein 2g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 1\n\n## Betty's Staples  \nYogurt\n\nA favorite food for many, yogurt is wonderful all by itself as part of a breakfast or lunch, or just simply as a snack. But this versatile food is also useful to have on hand for impromptu quick recipes. These ideas suggest a type or flavor and sometimes an amount, but you can certainly vary the recipe to suit your taste or what you might have on hand.\n\n  1. 1. **Apple-Yogurt Snacks:** Place plain Greek yogurt in small bowl; top with apple slices and a drizzle of caramel sauce.\n  2. 2. **Blue Cheese Yogurt Dressing:** In small bowl, mix \u00bc cup each plain Greek yogurt, mayonnaise and crumbled blue cheese until blended. Add 1 to 2 tablespoons milk to thin if desired. Serve on tossed green salads or use as dip for vegetables.\n  3. 3. **Frozen Yogurt Pops:** Place any flavor yogurt in small (5-ounce) paper cups. Insert wooden craft stick into center of yogurt; freeze until firm. To serve, remove paper cup.\n  4. 4. **Granola-Blueberry Parfaits:** Layer Greek yogurt with granola cereal and fresh blueberries in parfait glasses or small bowls.\n  5. 5. **Orange-Pineapple Smoothies:** To make 2 smoothies, place 1 cup orange juice, 1 cup orange or vanilla yogurt, 1 small banana and \u00bd cup fresh or canned pineapple chunks in blender; blend until smooth.\n  6. 6. **Peanut Butter\u2013Yogurt Dip:** In medium serving bowl, mix 1 cup vanilla yogurt with 2 tablespoons creamy or crunchy peanut butter. Sprinkle with chopped peanuts and serve with sliced pears, celery pieces and baby-cut carrots for dipping.\n  7. 7. **Personal Taco Dip:** Layer equal amounts of plain Greek yogurt, guacamole, chunky salsa, shredded lettuce and just a sprinkle of shredded Cheddar cheese in individual custard cups or small glasses. Serve with tortilla chips.\n  8. 8. **Spicy Vegetable Dip:** In medium serving bowl, mix 1 cup plain Greek or regular yogurt, 2 tablespoons chili sauce and 1 tablespoon prepared horseradish until well blended. Serve with cut-up fresh vegetables.\n\nGranola-Blueberry Parfaits, Spicy Vegetable Dip, Apple-Yogurt Snacks\n\n## Learn to Deep-Fry\n\nCooking food in enough hot oil to cover it is called deep-frying. You want to be sure that your food is very dry before frying, as water can cause the oil to splatter. Start with fresh oil\u2014a neutral-flavored oil such as canola, peanut, safflower or sunflower works best so you only taste the flavor of the food and not the oil. These oils also have a high smoke point, which means that the oil can be kept at a fairly high temperature without smoking. A deep-fry thermometer is a must for monitoring the temperature of the cooking oil. Follow recipe directions carefully to bring the oil to the proper temperature for the recipe.\n\nUse a deep fryer or a large pot with a heavy bottom for frying. The vessel should be tall enough to have at least 2 inches between the top of the oil and the top of the pan, as the oil will rise and bubble as the food cooks. Follow these basic steps to crispy-cooked food:\n\n  * Heat oil to desired temperature and maintain it using deep-fry thermometer.\n  * Be sure food to be fried is dry; pat with paper towels if necessary.\n  * Carefully place food into hot oil using metal slotted spoon or tongs; do not crowd.\n  * Remove food from oil with slotted spoon to drain on paper towel\u2013lined plate.\n\nWhen done at the correct temperature, deep-frying produces a deliciously crispy exterior without the food being greasy.\n\n## Shrimp Egg Rolls\n\nShredded cooked chicken or pork can be substituted for the shrimp. Thaw frozen egg roll skins at room temperature.\n\nPrep 1 hr 15 min Total 1 hr 15 min \u25cf 8 egg rolls\n\n  * 1 teaspoon cornstarch\n  * \u00bc teaspoon five-spice powder\n  * 4 teaspoons soy sauce\n  * 1 tablespoon vegetable oil\n  * 4 cups coleslaw mix (from 1-lb bag)\n  * 1 cup fresh bean sprouts\n  * 2 tablespoons sliced green onions\n  * \u00bd teaspoon grated gingerroot\n  * \u00bd cup finely chopped cooked shrimp\n  * Vegetable oil for frying\n  * 8 egg roll skins (from 1-lb package)\n  * 1 egg, beaten\n  * Sweet-and-sour sauce\n\n1 In small bowl, mix cornstarch, five-spice powder and soy sauce until blended.\n\n2 In large skillet or wok, heat 1 tablespoon oil over medium-high heat until hot. Add coleslaw mix, bean sprouts, onions and gingerroot; cook 3 to 4 minutes, stirring constantly, until tender. Add shrimp and cornstarch mixture; cook 1 to 2 minutes, stirring constantly, until mixture is thoroughly coated. Remove from skillet; cool to room temperature. In deep fryer or 4-quart Dutch oven, heat 3 to 4 inches oil to 350\u00b0F.\n\n3 Meanwhile, place 1 egg roll skin on work surface with 1 corner facing you. (Cover remaining skins with damp paper towel to prevent drying out.) Place \u00bc cup coleslaw mixture slightly below center of egg roll skin. Fold corner of egg roll skin closest to filling over filling, tucking point under. Fold in and overlap right and left corners. Brush remaining corner with egg; gently roll egg roll toward remaining corner and press to seal. (Cover filled egg roll with damp paper towel to prevent drying out.) Repeat with remaining egg roll skins and coleslaw mixture.\n\n4 Fry egg rolls, a few at a time, in hot oil 4 to 6 minutes, turning once, until golden brown. Drain on paper towels. Serve with sweet-and-sour sauce.\n\n1 Egg Roll: Calories 220; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 60mg; Sodium 220mg; Total Carbohydrate 22g (Dietary Fiber 2g, Sugars 2g); Protein 8g Exchanges: 1\u00bd Starch, \u00bd Very Lean Meat, 2 Fat Carbohydrate Choices: 1\u00bd\n\nShrimp Egg Rolls\n\n### Forming Egg Rolls\n\nFold corner of egg roll skin over filling, tucking in corner.\n\nFold in and overlap left and right corners.\n\nBrush remaining corner with egg; gently roll egg roll toward remaining corner and press to seal.\n\n## Cheesy Potato Skins\n\nPrep 15 min Total 2 hr \u25cf 8 servings\n\n  * 4 large russet potatoes (about 2 lb)\n  * 2 tablespoons butter, melted\n  * 1 cup shredded Cheddar or Colby\u2013Monterey Jack cheese blend (4 oz)\n  * 4 slices bacon, crisply cooked, crumbled (\u00bc cup)\n  * \u00bd cup sour cream\n  * 8 medium green onions, sliced (\u00bd cup)\n\n1 Heat oven to 375\u00b0F. Prick potatoes with fork. Bake potatoes 1 hour to 1 hour 15 minutes or until tender. Let stand until cool enough to handle. Cut potatoes lengthwise into quarters; carefully scoop out pulp, leaving \u00bc-inch shells. Refrigerate potato pulp for another use.\n\n2 Set oven control to broil. Place potato shells, skin sides down, on rack in broiler pan. Brush with butter. Broil with tops 4 to 5 inches from heat 8 to 10 minutes or until crisp and brown. Sprinkle cheese and bacon over potato shells. Broil about 30 seconds longer or until cheese is melted. Serve hot with sour cream and onions.\n\n1 Serving: Calories 280; Total Fat 12g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 35mg; Sodium 230mg; Total Carbohydrate 33g (Dietary Fiber 3g, Sugars 3g); Protein 9g Exchanges: 2 Starch, \u00bd High-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 2\n\nLighter Directions For 5 grams of fat and 115 calories per serving, decrease cheese to \u00bd cup; use fat-free sour cream.\n\n## Nachos Fast\n\nPrep 5 min Total 10 min \u25cf 4 servings\n\n  * 28 tortilla chips\n  * 1 cup shredded Monterey Jack or Cheddar cheese (4 oz)\n  * \u00bc cup canned chopped green chiles, if desired\n  * 2 tablespoons sliced ripe olives, well drained\n  * \u2153 cup sour cream\n  * \u00bc cup salsa\n\n1 Heat oven to 400\u00b0F. Line cookie sheet with foil. Place tortilla chips on cookie sheet. Sprinkle with cheese, chiles and olives.\n\n2 Bake about 4 minutes or until cheese is melted. Top with sour cream and salsa. Serve hot.\n\n1 Serving: Calories 260; Total Fat 18g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 35mg; Sodium 370mg; Total Carbohydrate 15g (Dietary Fiber 1g, Sugars 2g); Protein 9g Exchanges: 1 Starch, 1 High-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\nBeef or Chicken Nachos Sprinkle 1 cup shredded cooked beef or chicken over tortilla chips. Continue as directed.\n\n## Hot and Saucy Cocktail Meatballs Calorie Smart\n\nPrep 30 min Total 1 hr 20 min \u25cf 72 meatballs\n\n  * 2 lb lean (at least 80%) ground beef\n  * 1 cup dry bread crumbs (any flavor)\n  * \u2154 cup finely chopped onion\n  * \u00bd cup milk\n  * 2 tablespoons chopped fresh parsley\n  * 2 teaspoons salt\n  * 1 teaspoon Worcestershire sauce\n  * \u215b teaspoon pepper\n  * 2 eggs\n  * 2 bottles (12 oz each) chili sauce\n  * 2\u00bd cups grape jelly\n\n1 Heat oven to 400\u00b0F. In medium bowl, mix all ingredients except chili sauce and jelly. Shape into 1-inch meatballs. Place in ungreased 13x9-inch pan or on rack in broiler pan.\n\n2 Bake uncovered about 20 minutes or until thoroughly cooked and no longer pink in center.\n\n3 In 5-quart Dutch oven or stockpot, heat chili sauce and jelly over medium heat, stirring constantly, until jelly is melted. Stir in meatballs until coated. Simmer uncovered 30 minutes. Serve hot with toothpicks.\n\n1 Meatball: Calories 60; Total Fat 1.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 210mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 5g); Protein 3g Exchanges: \u00bd Other Carbohydrate, \u00bd Lean Meat Carbohydrate Choices: \u00bd\n\n## Pork Carnitas Taco Bites\n\nCarnitas, which translates as \"little meats,\" have long been a popular taco filling but recently gained fame as a food truck favorite. Our tasty version follows the tradition of cooking the pork until it gets brown and crispy.\n\nPrep 45 min Total 1 hr 45 min \u25cf 20 servings\n\nCarnitas\n\n  * 2 lb boneless pork shoulder or butt or country-style ribs, cut into 1-inch pieces*\n  * 1 medium onion, chopped (\u00bd cup)\n  * 3 cloves garlic, finely chopped\n  * 1 teaspoon dried marjoram leaves\n  * 1 teaspoon dried thyme leaves\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n\nGreen Salsa Crema\n\n  * \u2154 cup crema (Mexican-style cream) or sour cream\n  * \u2154 cup green tomatillo salsa\n\nTacos\n\n  * 15 flour tortillas (10 to 12 inch)\n  * 3 cups shredded iceberg lettuce\n  * 2 medium avocados, peeled, pitted and each cut into 30 slices\n  * 1 jar (24 oz) roasted red bell peppers, drained, cut into \u00bcx\u00bd-inch strips\n  * 1\u00bc cups crumbled cotija or queso fresco cheese\n\n1 In 12-inch skillet, place meat in single layer. Add cold water until meat is almost but not completely covered (about 1\u00bd cups). Add remaining carnitas ingredients. Heat to boiling; reduce heat to medium. Cook uncovered until liquid has evaporated and meat begins to brown, 35 to 45 minutes.\n\n2 In small bowl, mix crema and salsa; refrigerate until serving time. Using 4-inch round cookie or biscuit cutter, cut out 4 mini tortillas from each tortilla. Cover with plastic wrap to prevent drying out; set aside.\n\n3 Heat oven to 425\u00b0F. Into 15x10x1-inch pan, spoon meat mixture, pan juices and any browned bits from skillet. Bake uncovered 10 to 15 minutes or until pork is brown and edges are crisp. Drain fat if necessary; scrape any browned bits from pan and reserve. Cut meat into \u00bc-inch pieces; place in medium bowl. Add browned bits and mix together.\n\n4 To assemble, spoon 1 tablespoon pork down center of each mini tortilla; top with 1 tablespoon lettuce, 1 avocado slice, about 3 roasted pepper strips and 1 teaspoon cheese. Drizzle each with about 1 teaspoon green salsa crema.\n\n*If using the ribs, they may be too lean to render enough fat to get brown and crisp. If you don't see any fat in the skillet after the water has evaporated in Step 1, add 1 tablespoon vegetable oil with the onion. Continue as directed.\n\n1 Serving (3 Tacos): Calories 310; Total Fat 16g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 40mg; Sodium 690mg; Total Carbohydrate 27g (Dietary Fiber 2g, Sugars 4g); Protein 14g Exchanges: 1 Starch, 1 Other Carbohydrate, 1\u00bd Lean Meat, 2 Fat Carbohydrate Choices: 2\n\n## Make-Ahead Meatballs\n\nCalorie Smart\n\nHomemade meatballs are a versatile snack that can be served in a variety of ways. They are ideal for appetizers or added to soups, but can be served as a main dish too.\n\nPrep 20 min Total 40 min \u25cf 60 meatballs\n\n  * 1 lb lean (at least 80%) ground beef\n  * \u00bd lb ground pork\n  * 1 medium onion, finely chopped (\u00bd cup)\n  * \u00bc cup unseasoned dry bread crumbs\n  * \u00bd teaspoon ground mustard\n  * \u00bd teaspoon seasoned salt\n  * \u215b teaspoon pepper\n  * 1 egg\n\n1 Heat oven to 375\u00b0F. Spray 15x10x1-inch pan with cooking spray. In large bowl, mix all ingredients. Shape into 1-inch meatballs. Place in pan.\n\n2 Bake 15 to 20 minutes or until no longer pink in center and meat thermometer inserted in center of meatball reads 160\u00b0F. Cool completely. Cover tightly and refrigerate 3 to 4 days or freeze 2 to 3 months. Reheat in desired sauce.\n\n1 Meatball: Calories 25; Total Fat 1.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 20mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd Fat Carbohydrate Choices: 0\n\n### Making Meatballs\n\nShape mixture into 1-inch balls, place in pan.\n\nBake until no longer pink in center and thermometer reads 160\u00b0F.\n\n## Easy Square Meatballs\n\nHere is a quick way to make meatballs without making balls. Make the meat mixture. In a 13x9-inch pan, shape the meat mixture into a rectangle, 10x6 inches. With a knife dipped in water, cut the mixture into 60 squares (6 squares x 10 squares) but do not separate. Bake 20 to 25 minutes or until no longer pink. Cool slightly and separate meatballs by cutting apart.\n\n## Ways to Use Meatballs\n\nMake up a batch or two of these easy meatballs and store in the refrigerator for up to 3 days or in the freezer for up to 3 months. The meatballs are easy to heat and can be served as they are with a little ketchup or chili sauce, or try the ideas below.\n\n  1. 1. **Cheeseburger Hoagies:** Fill individual small hoagie buns with 3 or 4 warm meatballs, 2 to 3 tablespoons barbecue sauce or ketchup and 3 or 4 dill pickle slices. Sprinkle with 2 to 3 tablespoons shredded Cheddar cheese. Microwave each sandwich on plate for 30 to 60 seconds or until cheese is melted.\n  2. 2. **Meatball Pizza:** Top a purchased pizza crust (or make your own, page 192) with 1 cup each canned diced tomatoes with basil, garlic and oregano, halved meatballs and shredded mozzarella cheese. Bake as directed on pizza package.\n  3. 3. **Pesto Pasta and Meatballs:** For 4 servings, toss 6 ounces hot cooked linguine with \u00bd cup refrigerated pesto; toss with about 20 warm meatballs.\n  4. 4. **Salsa Tacos:** Fill individual taco shells with 1 or 2 meatballs, \u00bc cup shredded lettuce, 2 tablespoons chunky-style salsa and 2 tablespoons shredded Monterey Jack cheese.\n  5. 5. **Sweet Chili Kabobs:** Thread meatballs on skewers with strips of any color bell pepper and whole fresh mushrooms. Brush with sweet chili sauce. Place kabobs on a cookie sheet. Bake 15 to 20 minutes at 350\u00b0F or until hot and vegetables are crisp-tender.\n  6. 6. **Teriyaki Sliders:** Fill individual slider buns with 1 or 2 warm meatballs, 1 tablespoon teriyaki baste and glaze, 1 slice drained canned pineapple and 1 to 2 tablespoons sliced roasted red bell pepper.\n\nMake-Ahead Meatballs, Sweet Chili Kabobs\n\n## Vietnamese Beef Lettuce Wraps Calorie Smart\n\nMake restaurant food at home with these easy and tasty lettuce wraps. They are great on their own, but also good served with Ginger-Soy Dipping Sauce (page 45), Peanut Sauce (page 462) or your favorite purchased sauce.\n\nPrep 40 min Total 40 min \u25cf 25 servings\n\n  * 1 lb lean (at least 80%) ground beef*\n  * 4 medium green onions, sliced diagonally (\u00bc cup)\n  * 2 teaspoons finely chopped lemongrass**\n  * 1 teaspoon finely chopped gingerroot\n  * 1 teaspoon finely chopped seeded red or green Thai, serrano or jalape\u00f1o chile\n  * 1 clove garlic, finely chopped\n  * 1 tablespoon sugar\n  * 2 tablespoons soy sauce or fish sauce\n  * 2 tablespoons lime juice\n  * 1 cup shredded carrots\n  * 1 can (8 oz) sliced water chestnuts, drained, finely chopped\n  * 25 small (about 3-inch) Bibb lettuce leaves (about 2 heads), breaking larger leaves into small size if necessary\n  * \u00bd cup finely chopped lightly salted dry-roasted peanuts\n  * Fresh cilantro leaves, if desired\n  * Lime slices, if desired\n\n1 In 12-inch skillet, cook beef, onions, lemongrass, gingerroot, chile and garlic over medium heat 8 to 10 minutes, stirring occasionally, until beef is thoroughly cooked; drain. Stir in sugar, soy sauce and lime juice just until sugar dissolves. Stir in carrots and water chestnuts. Heat until mixture is hot, stirring frequently.\n\n2 To assemble, spoon 2 tablespoons beef mixture onto each lettuce leaf. Sprinkle with peanuts and cilantro. Arrange on serving platter; garnish with lime slices. Serve immediately.\n\n*Ground chicken, pork or turkey can be substituted for the beef.\n\n**Once a mysterious ingredient, lemongrass is now available in many supermarkets. Resembling a green reed, it's an ancient herb of Southeast Asia. The flavor is reminiscent of citrus, bay leaves and pepper.\n\n1 Serving: Calories 60; Total Fat 3.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 105mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 0g); Protein 4g Exchanges: \u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\n## Chicken Satay with Peanut Sauce\n\nPrep 15 min Total 2 hr 25 min \u25cf 8 servings\n\n  * 3 tablespoons lime juice\n  * 1 teaspoon curry powder\n  * 2 teaspoons honey\n  * \u00bd teaspoon ground coriander\n  * \u00bd teaspoon ground cumin\n  * \u215b teaspoon salt\n  * 2 cloves garlic, finely chopped\n  * 1 lb boneless skinless chicken breasts, cut into 1-inch cubes\n  * 1 medium red bell pepper, cut into 1\u00bc-inch pieces\n  * 4 medium green onions, cut into 2-inch pieces\n  * Peanut Sauce (page 462) or purchased spicy peanut sauce\n\n1 In small bowl, stir lime juice, curry powder, honey, coriander, cumin, salt and garlic until well mixed. Place chicken in resealable food-storage plastic bag or shallow glass or plastic dish. Add lime juice mixture; seal bag tightly. Shake or stir to coat chicken with juice mixture. Cover and refrigerate 2 hours, turning bag or stirring occasionally.\n\n2 Set oven control to broil. Spray rack in broiler pan with cooking spray. Remove chicken from marinade; reserve marinade. On each of 8 (8-inch) metal or bamboo* skewers, thread chicken, bell pepper and onion pieces, leaving space between each piece. Place skewers on rack in broiler pan.\n\n3 Broil with tops about 3 inches from heat 4 minutes. Turn; brush with reserved marinade. Discard any remaining marinade. Broil chicken 4 to 5 minutes longer or until chicken is no longer pink in center. Serve chicken with peanut sauce.\n\n*Soak bamboo skewers in water at least 30 minutes to prevent burning.\n\n1 Serving (1 Skewer and 1 Tablespoon Sauce): Calories 330; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 70mg; Sodium 500mg; Total Carbohydrate 17g (Dietary Fiber 2g, Sugars 12g); Protein 32g Exchanges: \u00bd Other Carbohydrate, 5 Lean Meat, 3\u00bd Fat Carbohydrate Choices: 1\n\n## Buffalo Chicken Wings Calorie Smart\n\nIn our testing, we found that coating the wings with flour really helped the sauce to cling to the wings.\n\nPrep 20 min Total 55 min \u25cf 24 appetizers\n\n  * 12 chicken wings (about 2 lb)\n  * 2 tablespoons butter\n  * \u00bd cup all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cup barbecue sauce (for homemade sauce, see page 411)\n  * 1 tablespoon red pepper sauce\n  * \u00bd teaspoon Cajun seasoning\n  * \u00bc teaspoon ground cumin\n  * 1 bottle (8 oz) blue cheese dressing (to make your own, see page 138)\n  * Celery, carrot and zucchini sticks, if desired\n\n1 Heat oven to 425\u00b0F. Cut each chicken wing at joints to make 3 pieces; discard tip. Cut off and discard excess skin.\n\n2 In 13x9-inch pan, melt butter in oven. In resealable heavy-duty food-storage plastic bag, mix flour, salt and pepper. Add chicken; seal bag tightly. Shake until chicken is completely coated with flour mixture. Place chicken in pan.\n\n3 Bake uncovered 20 minutes. In small bowl, stir barbecue sauce, pepper sauce, Cajun seasoning and cumin until well mixed. Turn chicken. Pour sauce mixture over chicken; toss until evenly coated with sauce.\n\n4 Bake uncovered 10 to 12 minutes longer or until light golden brown on outside and juice of chicken is clear when thickest part is cut to bone (165\u00b0F). Serve with dressing and vegetable sticks.\n\n1 Appetizer: Calories 80; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 180mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 3g); Protein 5g Exchanges: 1 Medium-Fat Meat Carbohydrate Choices: \u00bd\n\nHoney-Mustard Chicken Wings Substitute honey-mustard barbecue sauce for barbecue sauce. Omit red pepper sauce. Substitute barbecue seasoning for the Cajun seasoning and omit ground cumin. Stir 1 tablespoon mustard seed into sauce mixture.\n\nSalsa Chicken Wings Substitute salsa for the barbecue sauce and chili powder for the Cajun seasoning.\n\nSweet-Spicy Chicken Wings Substitute sweet-spicy French dressing for the barbecue sauce and Montreal chicken grill seasoning for the Cajun seasoning; omit ground cumin.\n\n## Classic Crab Cakes Fast\n\nPrep 25 min Total 25 min \u25cf 6 servings\n\n  * \u2153 cup mayonnaise or salad dressing\n  * 1 egg\n  * 1\u00bc cups soft white bread crumbs (about 2 slices bread)\n  * 1 teaspoon ground mustard\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon ground red pepper (cayenne), if desired\n  * \u215b teaspoon black pepper\n  * 2 medium green onions, chopped (2 tablespoons)\n  * 1 can (1 lb) or 2 refrigerated containers (8 oz each) pasteurized lump crabmeat, well drained, cartilage removed*\n  * \u00bc cup unseasoned dry bread crumbs\n  * 2 tablespoons vegetable oil\n\n1 In medium bowl, mix mayonnaise and egg with whisk. Gently stir in remaining ingredients except dry bread crumbs and oil so that crabmeat doesn't break apart into small pieces. Shape mixture into 6 patties about 3 inches in diameter (mixture will be moist). Coat each patty with dry bread crumbs.\n\n2 In 12-inch nonstick skillet, heat oil over medium heat. Cook patties in oil about 10 minutes, gently turning once, until golden brown and hot in center. Reduce heat if crab cakes become brown too quickly.\n\n*Or substitute 3 cans (6 ounces each) lump crabmeat for the 1-pound can or the refrigerated crabmeat.\n\n1 Serving: Calories 330; Total Fat 18g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 120mg; Sodium 690mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 2g); Protein 22g Exchanges: 1 Starch, 3 Lean Meat, 2 Fat Carbohydrate Choices: 1\n\nSalmon Cakes Substitute 1 can (15 ounces) red salmon, drained, for the lump crabmeat.\n\n## Classic Shrimp Cocktail Calorie Smart \u2022 Fast\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\nShrimp\n\n  * 4 cups water\n  * 1\u00bd lb uncooked large shrimp in shells, thawed if frozen, peeled (tail shells left on) and deveined\n\nCocktail Sauce\n\n  * 1 cup ketchup\n  * 4 to 6 teaspoons prepared horseradish*\n  * 1 teaspoon Worcestershire sauce\n  * 2 or 3 drops red pepper sauce\n\nGarnish, if desired\n\n  * Lemon wedges\n\n1 In 3-quart saucepan, heat water to boiling. Add shrimp. Cover and return to boiling; reduce heat. Simmer uncovered 3 to 5 minutes or until shrimp are pink; drain.\n\n2 Meanwhile, in small bowl, mix sauce ingredients. Serve shrimp with sauce and lemon wedges.\n\n*Stir in 1 to 2 teaspoons additional horseradish if you like a hotter cocktail sauce.\n\n1 Serving: Calories 100; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 105mg; Sodium 770mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 9g); Protein 12g Exchanges: \u00bd Starch, 1\u00bd Very Lean Meat Carbohydrate Choices: 1\n\nShrimp Cocktail Shooters Serve the shrimp in decorative shot glasses. Place about 1 tablespoon of the cocktail sauce in each shot glass, and top with 1 or 2 cooked shrimp. Garnish with a little chopped fresh parsley or celery leaves.\n\n## Basil-Cheese Triangles Calorie Smart\n\nPrep 30 min Total 45 min \u25cf 72 appetizers\n\n  * 1 lb feta cheese*\n  * 2 eggs, slightly beaten\n  * \u00bc cup finely chopped fresh or 1 tablespoon dried basil leaves**\n  * \u00bc teaspoon white or black pepper\n  * 1 package (16 oz) frozen phyllo (filo) sheets (18x14 inch), thawed\n  * \u2153 cup butter, melted\n\n1 Heat oven to 400\u00b0F. Grease cookie sheets with shortening or cooking spray. Crumble cheese into medium bowl; mash with fork. Stir in eggs, basil and pepper until well mixed.\n\n2 Cut phyllo sheets lengthwise into 2-inch strips. (Cover with plastic wrap, then with damp towel to keep them from drying out.) Place 1 level teaspoon cheese mixture on end of 1 strip. Fold strip over cheese mixture, end over end in triangular shape, to opposite end. Place on cookie sheet. Repeat with remaining strips and cheese mixture. Brush triangles lightly with butter.\n\n3 Bake 12 to 15 minutes or until puffed and golden brown. Serve warm.\n\n*Finely shredded Monterey Jack cheese can be substituted for the feta cheese.\n\n**Chopped fresh chives or 1 tablespoon freeze-dried chives can be substituted for the basil.\n\n1 Appetizer: Calories 45; Total Fat 2.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 105mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: 0\n\nMake-Ahead  Directions Cover and refrigerate unbaked triangles up to 24 hours before baking; bake as directed. Or freeze tightly covered up to 2 months; increase bake time by 5 minutes.\n\n### Making Phyllo Triangles\n\nCut phyllo sheets lengthwise into 2-inch strips. Place 1 teaspoon cheese mixture on end of strip.\n\nFold strip over cheese mixture, end over end in triangle shape to opposite end.\n\n## Tomato-Basil Crostini Calorie Smart \u2022 Fast\n\nPrep 15 min Total 25 min \u25cf 12 appetizers\n\n  * 12 slices (\u00bd inch thick) Italian bread\n  * \u00bc cup olive or vegetable oil\n  * 1 large tomato, seeded, chopped (1 cup)\n  * 3 tablespoons chopped fresh basil leaves\n  * 1 tablespoon large capers or chopped pitted ripe olives\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon coarsely ground pepper\n  * 12 slices (1 oz each) regular or fresh mozzarella cheese\n\n1 Heat oven to 375\u00b0F. On ungreased cookie sheet, place bread slices in single layer. Drizzle 1 teaspoon oil over each slice.\n\n2 In small bowl, mix tomato, basil, capers, salt and pepper. Spread half of the tomato mixture over bread slices; top each with a cheese slice. Spread remaining tomato mixture over cheese.\n\n3 Bake about 8 minutes or until bread is hot and cheese is melted. Serve hot.\n\n1 Appetizer: Calories 180; Total Fat 11g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 390mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 0g); Protein 9g Exchanges: 1 Starch, 1 High-Fat Meat Carbohydrate Choices: 1\n\nProvolone-Onion Crostini Substitute a chopped green onion for the capers and provolone cheese for the mozzarella.\n\n## Deviled Eggs Calorie Smart \u2022 Fast\n\nPrep 15 min Total 30 min \u25cf 12 appetizers\n\n  * 6 eggs\n  * 3 tablespoons mayonnaise, salad dressing or half-and-half\n  * 1 teaspoon prepared yellow, Dijon or spicy brown mustard*\n  * \u215b teaspoon pepper\n\n1 In 2-quart saucepan, place eggs in single layer. Cover with cold water at least 1 inch above eggs. Cover saucepan; heat to boiling.\n\n2 Immediately remove from heat; let stand covered 15 minutes for large eggs (12 minutes for medium eggs and 18 minutes for extra-large eggs).\n\n3 Drain. Immediately place eggs in cold water with ice cubes or run cold water over eggs until completely cooled.\n\n4 To peel, gently tap egg on countertop until entire shell is finely crackled. Roll gently between hands to loosen shell. Starting at large end, peel egg under cold running water to help remove shell.\n\n5 Cut lengthwise in half. Slip out yolks into small bowl; mash with fork. To prevent eggs from tipping on serving plate, cut a thin slice from bottom of each egg white half before filling. Or, to stand eggs upright, cut crosswise slice about two-thirds from top of narrow end of egg to remove yolk; cut thin slice from bottom of wide end of egg white to rest on serving platter.\n\n6 Stir mayonnaise, mustard and pepper into yolks. Fill whites with egg yolk mixture, heaping lightly. Cover and refrigerate up to 24 hours.\n\n*Any flavored prepared mustard can be used, or \u00bd teaspoon ground mustard can be substituted for yellow mustard.\n\n1 Appetizer: Calories 60; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 110mg; Sodium 75mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 3g Exchanges: \u00bd Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 0\n\nLighter Directions For 1 gram of fat and 25 calories per serving, mash only 6 yolk halves in Step 5 (reserve remaining yolks for another use or discard). Use fat-free mayonnaise. Stir in \u2153 cup finely chopped zucchini.\n\nBacon-Cheddar Deviled Eggs Mix 2 slices crisply cooked crumbled bacon and 2 tablespoons finely shredded Cheddar cheese into yolk mixture. Garnish with additional crumbled bacon or chopped fresh chives or parsley.\n\nBlue Cheese Deviled Eggs Omit mustard. Mix \u00bc cup crumbled blue cheese into yolk mixture. Garnish with coarse ground black pepper and small celery leaves.\n\nChipotle Deviled Eggs Omit mustard. Mix 1\u00bd to 2 teaspoons finely chopped chipotle chiles in adobo sauce (from 7-ounce can), drained, and 1 thinly sliced green onion into yolk mixture. Garnish with whole or chopped fresh cilantro leaves.\n\nReuben Deviled Eggs Omit mustard. Substitute Thousand Island dressing for the mayonnaise. Stir in 2 tablespoons each finely chopped thinly sliced deli corned beef and finely chopped sauerkraut (squeezed in paper towel to drain). Garnish with shredded Swiss cheese and caraway seed.\n\n### Making Deviled Eggs\n\nCut a thin slice from bottom of egg half so that egg sits on a platter.\n\nSpoon prepared filling into egg halves.\n\n## Brie in Puff Pastry with Cranberry Sauce\n\nPrep 30 min Total 1 hr 25 min \u25cf 12 servings\n\nCranberry Sauce*\n\n  * 1 cup fresh cranberries\n  * 6 tablespoons packed brown sugar\n  * 1 tablespoon orange juice\n  * \u00bd teaspoon grated orange peel\n\nBrie in Pastry\n\n  * 1 tablespoon butter\n  * \u2153 cup sliced almonds\n  * 1 frozen (thawed) puff pastry sheet (from a 17.3-oz package)\n  * 1 round (14 to 15 oz) Brie cheese\n  * 1 egg, beaten\n  * Assorted crackers or sliced fresh fruit, if desired\n\n1 In 1-quart saucepan, stir cranberries, brown sugar and orange juice until well mixed. Heat to boiling, stirring frequently; reduce heat. Simmer uncovered 15 to 20 minutes, stirring frequently, until mixture thickens and cranberries are tender. Stir in orange peel; remove from heat and set aside.\n\n2 In 8-inch skillet, melt butter over medium heat. Cook almonds in butter, stirring frequently, until golden brown; remove from heat and set aside.\n\n3 Heat oven to 400\u00b0F. Spray cookie sheet with cooking spray. On lightly floured surface, roll pastry into 16x9-inch rectangle; cut out 2 circles, 1 (8\u00bd-inch) round and 1 (7-inch) round.\n\n4 Remove paper from cheese; leave rind on. Place cheese round on center of 8\u00bd-inch pastry round. Spoon cranberry sauce and almonds over cheese. Bring pastry up and press around side of cheese. Brush top edge of pastry with egg. Place 7-inch pastry round on top, pressing around edge to seal. Brush top and side of pastry with egg. Cut decorations from remaining pastry and arrange on top; brush with egg. Place on cookie sheet.\n\n5 Bake 20 to 25 minutes or until golden brown. Cool on cookie sheet on cooling rack 30 minutes before serving. Serve with crackers.\n\n*Or substitute 1 cup purchased whole berry cranberry sauce for the cranberry sauce.\n\n1 Serving: Calories 270; Total Fat 19g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 75mg; Sodium 270mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 8g); Protein 9g Exchanges: 1 Starch, 1 High-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\nBrie in Puff Pastry with Double-Orange Cranberry Sauce Add 1 tablespoon orange-flavored liqueur or chai-flavored liqueur to the cranberry mixture before cooking.\n\n### Wrapping Brie in Puff Pastry\n\nCenter cheese on puff pastry. Spoon sauce and almonds over cheese, and wrap pastry around sides of cheese.\n\nPlace 7-inch round on top, pressing around edge to seal.\n\nBrush top and sides of pastry with egg.\n\nCut garnish from pastry scraps; place on top, and press lightly to secure.\n\n## Black Bean Sliders Calorie Smart\n\nPrep 45 min Total 2 hr 45 min \u25cf 20 sandwiches\n\nBuns\n\n  * 5 frozen stone-ground 100% whole wheat Texas rolls (from 3-lb bag)\n  * 2 tablespoons butter, melted\n\nCorn Salsa\n\n  * 1 can (7 oz) whole kernel corn with red and green peppers, drained\n  * \u2154 cup chunky-style medium salsa\n\nBean Patties\n\n  * 5 teaspoons olive oil\n  * \u00bc cup finely chopped red or yellow onion\n  * 1 can (15 oz) black beans, drained, rinsed*\n  * 1 egg, beaten\n  * \u00bd cup unseasoned dry bread crumbs\n  * 2 cloves garlic, finely chopped\n  * 1 teaspoon ground cumin\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon freshly ground pepper\n  * 20 small pieces Bibb or leaf lettuce\n  * 5 slices (\u00be oz each) Cheddar cheese, cut into quarters\n\n1 Thaw rolls at room temperature until soft but still cold, about 10 minutes. Spray cookie sheet with cooking spray. Cut each roll into quarters; shape dough pieces into balls. Place on cookie sheet. Brush tops of dough balls with butter. Cover with plastic wrap sprayed with cooking spray. Let rise in warm place 2 hours to 2 hours 30 minutes or until doubled in size.\n\n2 Heat oven to 350\u00b0F. Remove plastic wrap. Bake buns 12 to 15 minutes or until golden brown. Remove from pan to cooling rack.\n\n3 Meanwhile, in medium bowl, mix corn salsa ingredients. Cover and refrigerate until serving time.\n\n4 In 10-inch nonstick skillet, heat 1 teaspoon of the oil over medium heat. Add onion; cook 3 to 4 minutes, stirring frequently, until softened. Remove from heat; cool slightly.\n\n5 In medium bowl, mash beans with potato masher, leaving a few beans whole. Stir in egg, bread crumbs, garlic, cumin, salt and pepper. Add cooled onion; mix until thoroughly combined. Shape mixture into 20 (1-inch) balls, using slightly less than 1 tablespoon per ball.\n\n6 In same skillet, heat 2 teaspoons oil over medium heat. Place half of the bean balls in skillet; with back of spatula, flatten balls into patties, about 1\u00bd inches in diameter. Cook 4 to 6 minutes, turning once, until browned. Remove from skillet; cover to keep warm. Repeat with remaining 2 teaspoons oil and bean balls.\n\n7 Split buns in half. On bottom half of each bun, place lettuce piece, bean patty, cheese quarter and 1 heaping tablespoon corn salsa. Cover with top halves of buns.\n\n*Cannellini beans or chickpeas (garbanzo beans) can be substituted for the black beans.\n\n1 Sandwich: Calories 130; Total Fat 5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 20mg; Sodium 270mg; Total Carbohydrate 17g (Dietary Fiber 3g, Sugars 2g); Protein 5g Exchanges: 1 Starch, \u00bd Vegetable, 1 Fat Carbohydrate Choices: 1\n\n### Making Black Bean Sliders\n\nSlightly flatten bean balls to about 1\u00bd inches in diameter.\n\nTop each slider with prepared corn salsa.\n\n# Chapter 3: Beverages\n\nMartini Cocktails\n\n## Beverage Basics\n\nCoffee Anytime\n\nTea for Relaxation\n\nThere Is More to Milk\n\n## Features\n\nBetty's Staples: Buttermilk\n\nHeirloom Recipe and New Twist\n\nLearn to Juice\n\nMake-Ahead: Lemonade Concentrate\n\nSimple Syrup\n\n## Coffee and Tea\n\nApple-Mint Iced Green Tea Calorie Smart \u2022 Fast \u2022 Easy\n\nBerry Tea\n\nCappuccino Calorie Smart \u2022 Fast \u2022 Easy\n\nCaramel Latte\n\nChai Calorie Smart \u2022 Fast\n\nDill Tea\n\nEasy Iced Tea\n\nGinger Tea\n\nHazelnut Iced Coffee\n\nIced Coffee Calorie Smart \u2022 Easy\n\nIced Vanilla Soy Latte Calorie Smart \u2022 Fast \u2022 Easy\n\nLemonade Sweet Tea\n\nRaspberry-Ginger Tea Calorie Smart \u2022 Fast \u2022 Easy\n\nSweetened Iced Coffee\n\nVietnamese Iced Coffee\n\n## Hot Beverages\n\nHot Buttered Rum\u2013Spiced Cider\n\nHot Spiced Cider Calorie Smart \u2022 Fast \u2022 Easy\n\n## Milk Beverages\n\nChocolate Milk\n\nDouble-Chocolate Hot Chocolate\n\nEggnog\n\nFresh Strawberry Milk Fast \u2022 Easy\n\nHot Chocolate Fast\n\nMarshmallow-Topped Hot Chocolate\n\nNew York Egg Cream\n\nRoot Beer Milk\n\n## Fruit and Vegetable Beverages\n\nBlueberry Smoothies\n\nCitrus Splash\n\nDill Simple Syrup\n\nEasy-Being-Green Smoothies Fast \u2022 Easy\n\nGinger Lemonade\n\nGinger Simple Syrup\n\nLemonade Ice Cubes\n\nLemonade Concentrate Calorie Smart \u2022 Fast \u2022 Easy\n\nLemonade from Concentrate\n\nLemonade Smoothies\n\nLemonade Sweet Tea\n\nLimeade Concentrate\n\nOrange Smoothies\n\nPeach Smoothies\n\nRaspberry Lemonade\n\nRaspberry Smoothies\n\nRaspberry-Lime Juice Fast \u2022 Easy\n\nRefreshing Green Juice Calorie Smart \u2022 Fast \u2022 Easy\n\nSparkling Lemonade\n\nStrawberry Smoothies Calorie Smart \u2022 Fast \u2022 Easy\n\nStrawberry-Banana Smoothies\n\nStrawberry-Lemonade Slush\n\nTriple-Berry Simple Syrup\n\n## Drinks Basics\n\nTools for the Bar\n\nGlassware\n\nCocktail Mixology\n\nAbout Beer\n\nWine Primer\n\n## Drinks\n\nBeergaritas Fast \u2022 Easy\n\nBerry Sangria\n\nBourbon-Maple Eggnog\n\nChocolate Martini Cocktails\n\nClassic Bloody Mary Calorie Smart \u2022 Fast\n\nDill Martini\n\nFrozen Strawberry Margaritas\n\nGreen Tea Martini Cocktails\n\nLemon Sangria\n\nMango Martini Cocktails\n\nMango Mimosas Calorie Smart \u2022 Fast \u2022 Easy\n\nManhattan Cocktails Calorie Smart \u2022 Fast \u2022 Easy\n\nMargaritas Fast\n\nMartini Cocktails Calorie Smart \u2022 Fast \u2022 Easy\n\nMojitos\n\nMulled Wine Calorie Smart\n\nPineapple Mimosas\n\nPomegranate-Mango Mimosas\n\nPomegranate Martini Cocktails\n\nSangria Calorie Smart \u2022 Fast\n\nSweet Martini Cocktails\n\nWhite Sangria\n\n## Bonus Recipes\n\nChopped Salad with Tangy Dressing\n\nStrawberry Pops\n\nTangy Coleslaw\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Beverage Basics\n\nHaving coffee in the morning, tea in the afternoon or even milk with a meal are all traditional choices for beverages. Why not shake things up a bit? Break out of your routine with flavored coffee, tangy chai tea or almond milk. Or you might even want to try a creamy smoothie or tall glass of lemonade for a refreshing change of pace.\n\n## Coffee Anytime\n\nCoffee is grown in more than 50 countries around the world in tropical climates at high altitudes and in rich soil. _Arabica_ is the most common variety, as well as the highest quality, and is a descendent of the original coffee beans grown in Ethiopia hundreds of years ago. _Robusta_ is another coffee variety that is becoming more common, particularly in blends and instant coffees. This coffee has a higher caffeine level than Arabica.\n\nThere are many methods for brewing coffee and a large selection of coffeemakers to choose from. You might choose an automatic drip coffeemaker, a French press, cone and filter, percolator or even a cappuccino machine. However you make coffee, here some tips:\n\n  * Purchase coffee in small amounts that can be used within one to two weeks.\n  * French or dark roast yields a dark, rich cup of coffee, ideal for espresso. Lighter roasts will be better for a mild morning cup of coffee.\n  * Store unopened packages of coffee in the freezer for up to 1 month.\n  * If you grind whole beans, do it as close to brew time as possible.\n  * Be conscious of the grind you are using in relation to how you are making coffee. If coffee is bitter, it may be ground too finely. If it is flat tasting, the grind may be too coarse. Check with professionals where you purchase beans for the correct grind. Here are some general guidelines:\n\n  * **Automatic drip:** medium grind\n  * **Espresso maker:** fine grind\n  * **Percolator:** coarse grind\n  * **French press:** coarse grind\n\n  * Use good water for best flavor\u2014filtered or bottled water are good choices.\n  * For regular-strength coffee, use 1 level tablespoon coffee for 6 ounces water. If you like stronger coffee, use 2 tablespoons coffee for 6 ounces water.\n\n## Tea for Relaxation\n\nThis simple beverage is the most-commonly consumed drink in the world because it is delicious, healthful, easy to make and just plain relaxing. The thousands of different varieties of tea vary by where they are grown, the time of year they are picked and how they are processed. Black and green teas contain a good amount of antioxidants. Some common varieties of tea include:\n\n  * **Black tea:** This tea contains the most caffeine of all tea, 50 to 65 percent of the amount in coffee. It is processed by crushing and fermenting the leaves. Common black teas include Darjeeling and English Breakfast.\n  * **Green tea:** Pale green with a light, fresh flavor, green tea is favored in Asia. The leaves are steamed and dried but not fermented. Varieties include Gunpowder and Tencha.\n  * **White tea:** Pale in color with a light, sweet flavor, white tea is the purest and least processed of all teas, and contains very little caffeine. Familiar varieties include Silver Needle and White Peony.\n  * **Oolong tea:** Partially fermented and a cross between black and green teas, it is also known as Chinese restaurant tea. Oolong is full bodied with a sweet aroma. Imperial oolong is prized for honey flavor and Formosa oolong tastes a bit like peaches.\n  * **Herbal teas:** Because these teas do not contain any leaves from the Camellia sinensis plant, they are referred to as tisanes. These blends consist of herbs, flowers and fruits.\n\nFor brewing tea, some equipment you might want on hand includes:\n\n  * Tea kettle for boiling water.\n  * Teapot for brewing.\n  * Infuser for holding tea leaves in a teapot.\n  * Strainer for pouring if there is no infuser.\n\nTo brew tea:\n\n  * Start with cold water in a tea kettle. For black or oolong tea, bring water to a boil. For green or white tea, water should reach between 170\u00b0F and 190\u00b0F.\n  * Warm a teapot by filling with hot water and letting it stand for a few minutes; drain.\n  * Place desired tea in teapot, using about 1 teaspoon loose tea or 1 teabag for each 6 ounces of water. Pour boiling or hot water over tea; steep 3 to 5 minutes.\n  * Pour tea into cups; use a strainer to catch loose tea leaves.\n\n## There Is More to Milk\n\nMilk has come a long way, with many varieties to choose from. Look for organic, lactose-free (for those who cannot drink regular milk) and milk with added calcium or omega-3 fatty acids. Goat's milk is easy to digest, has less lactose than cow's milk and is less allergenic.\n\nThere are many tasty nondairy choices, which are delicious and healthful:\n\n  * **Soymilk:** Made from soybeans and available unsweetened, sweetened and flavored. Often organic, it has a nutrition profile and protein content similar to dairy milk \n  * **Almond and cashew milks:** Although these have less protein than dairy milk, they contain about \u2153 fewer calories than 2% milk. They are made with ground almonds or cashews, water and often a sweetener.\n  * **Rice milk:** Made from boiled rice, brown rice syrup and brown rice starch, this alternative is low in protein compared to dairy milk. Look for rice milk that is calcium fortified.\n  * **Coconut milk:** This alternative is the closest to dairy milk in texture but is high in fat content. It is a good choice for those with multiple food allergies and does not contain soy or gluten. It has fewer calories, less protein and less calcium than dairy milk.\n  * **Hemp milk:** Sometimes available, this alternative is another choice for those with multiple allergies. It is made from hemp seeds, water and sweeteners and is often free of soy, nuts and gluten.\n\n## Iced Coffee Calorie Smart \u2022 Easy\n\nThe coffee concentrate base in this recipe uses ground coffee and water. In addition to making iced coffee, you can use it for ice cubes so that cold coffee-flavored drinks are not diluted by regular ice cubes.\n\nPrep 5 min Total 12 hr 35 min \u25cf 4 cups\n\nCoffee Concentrate\n\n  * 1 package (12 oz) medium or coarsely ground medium to dark roasted coffee beans*\n  * 8 cups cold water\n\nIced Coffee\n\n  * Ice cubes\n  * \u00bd cup Coffee Concentrate\n  * \u00bd cup cold whole milk\n\n1 In large bowl, stir coffee grounds and water until coffee grounds are completely moistened. Cover and let stand at room temperature 12 to 15 hours. The longer it stands, the stronger the flavor will be.\n\n2 Line fine-mesh strainer with paper coffee filter or several layers of cheesecloth; place over 2-quart pitcher. Pour coffee mixture through strainer, in batches, allowing all the liquid to drain through without pressing on the grounds (this may take up to 30 minutes; replace cheesecloth or coffee filter as needed). Pressing on the grounds can cause cloudy coffee. Discard grounds.\n\n3 Cover pitcher and store coffee concentrate in refrigerator up to 2 weeks. Stir concentrate before each use. To make iced coffee, fill a glass with ice. Mix \u00bd cup coffee concentrate with \u00bd cup milk, or if desired, water.\n\n*Medium or coarsely ground coffee is recommended because the concentrate will be clearer. Using finely ground coffee will result in a cloudy concentrate.\n\n\u00bd Cup: Calories 80; Total Fat 4g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 70mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 6g); Protein 4g Exchanges: \u00bd Milk Carbohydrate Choices: \u00bd\n\nSweetened Iced Coffee In coffee mug or glass, mix \u00be cup coffee concentrate with \u00bc cup water and 1 to 2 tablespoons Simple Syrup (page 81). Stir well and add ice cubes.\n\nHazelnut Iced Coffee In coffee mug or glass, mix \u00be cup Coffee Concentrate with \u00bc to \u2153 cup hazelnut-flavored liquid nondairy creamer. Add ice cubes. If you don't prefer hazelnut, use your favorite flavor of liquid coffee creamer.\n\nVietnamese Iced Coffee In coffee mug or glass, mix \u00be cup Coffee Concentrate with 2 to 3 tablespoons sweetened condensed milk. Add ice cubes.\n\n### Straining Iced Coffee\n\nLine fine-mesh strainer with several layers of cheesecloth. Pour coffee through cheesecloth to catch all coffee bean particles.\n\n## Cappuccino Calorie Smart \u2022 Fast \u2022 Easy\n\nCappuccino is easy to make at home without special equipment. A microwave or stove and an electric mixer work just fine to heat the milk and create a foamy top. Make sure to look for French roast coffee\u2014its bold, strong flavor is preferred for espresso-based coffee drinks.\n\nPrep 10 min Total 10 min \u25cf 2 servings\n\n  * \u00bd cup ground French roast or espresso coffee\n  * 1\u00bd cups water\n  * \u00be cup whole milk\n  * Ground cinnamon, if desired\n\n1 Brew coffee according to manufacturer's directions, using ground coffee and water.\n\n2 Meanwhile, in medium microwavable bowl, microwave milk on High 45 seconds to 1 minute or until hot; do not boil. (Or heat milk in 1-quart saucepan over medium heat.) With electric mixer, beat milk on low speed, gradually increasing speed until milk begins to thicken and form foam. Continue beating until desired amount of foam forms. Hot milk will settle to the bottom and foam will float on top; do not stir.\n\n3 To serve, pour brewed coffee into 2 mugs or cups. Add hot milk (pour by holding table knife against edge of bowl or saucepan to hold back foam). Spoon foam on top. Sprinkle with cinnamon. Serve immediately.\n\n1 Serving: Calories 60; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 45mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 5g); Protein 3g Exchanges: \u00bd Milk Carbohydrate Choices: \u00bd\n\nCaramel Latte Make recipe as directed\u2014except increase milk to 1 cup and heat in microwavable measuring cup or saucepan; do not beat milk with mixer. For each serving, spoon 3 tablespoons caramel topping into each cup before adding \u00bd cup each coffee and milk; stir to blend. Microwave 40 to 50 seconds or until hot. Top with whipped cream; drizzle with additional caramel topping. Sprinkle with cinnamon.\n\n### Making Foam\n\nBeat hot milk, starting on low speed and gradually increasing speed until milk thickens and foams.\n\nCarefully spoon foam on top of coffee and milk.\n\n## Iced Vanilla Soy Latte Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 10 min Total 10 min \u25cf 2 servings\n\n  * \u00bd cup ground espresso or French roast coffee\n  * 1\u00bd cups water\n  * 2 cups vanilla soymilk\n  * 2 teaspoons caramel or chocolate fat-free topping\n  * Ice cubes\n  * Sugar, if desired\n\n1 Brew coffee according to manufacturer's directions, using ground coffee and water.\n\n2 In heatproof pitcher, stir together coffee and soymilk.\n\n3 Drizzle topping over insides of 2 large glasses. Fill glasses with ice cubes. Pour soymilk mixture over ice. Sweeten to taste with sugar.\n\n1 Serving: Calories 120; Total Fat 3g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 210mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 13g); Protein 7g Exchanges: 1 Low-Fat Milk Carbohydrate Choices: 1\n\nHot Spiced Cider, Chai, Hot Chocolate\n\n## Chai Calorie Smart \u2022 Fast\n\nPrep 5 min Total 10 min \u25cf 4 servings\n\n  * 2 cups water\n  * \u00bc cup loose Darjeeling tea leaves or 5 tea bags black tea\n  * 2 cups milk\n  * \u215b teaspoon ground cardamom\n  * 2 whole cloves, crushed\n  * 2 to 4 black peppercorns, crushed\n  * Pinch of ground cinnamon\n  * \u00bc cup sweetened condensed milk or 4 teaspoons sugar\n\n1 In 2-quart saucepan, heat water to a rapid boil over medium-high heat; reduce heat to low. Add tea leaves; simmer 2 to 4 minutes to blend flavors. (If using tea bags, remove and discard.)\n\n2Stir in remaining ingredients except condensed milk. Heat just to boiling, but do not let boil over. Remove from heat; stir in condensed milk. Strain tea through strainer into cups. Sprinkle with additional ground cinnamon, if desired.\n\n1 Serving: Calories 140; Total Fat 6g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 85mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 16g); Protein 6g Exchanges: 1 Low-Fat Milk, 1 Fat Carbohydrate Choices: 1\n\n## Hot Spiced Cider Calorie Smart \u2022 Fast \u2022 Easy\n\nA slow cooker is a great way to keep this cider warm after it's made.\n\nPrep 5 min Total 15 min \u25cf 6 servings\n\n  * 6 cups apple cider\n  * \u00bd teaspoon whole cloves\n  * \u00bc teaspoon ground nutmeg\n  * 3 sticks cinnamon\n\n1 In 3-quart saucepan, heat all ingredients to boiling over medium-high heat; reduce heat to low. Simmer uncovered 10 minutes.\n\n2 Strain cider mixture through strainer into hot-beverage carafe or pitcher to remove cloves and cinnamon. Serve hot with additional cinnamon sticks, if desired.\n\n1 Serving: Calories 120; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 29g (Dietary Fiber 0g, Sugars 24g); Protein 0g Exchanges: 2 Fruit Carbohydrate Choices: 2\n\nHot Buttered Rum\u2013Spiced Cider Make as directed. For each serving, place 1 tablespoon butter (do not use margarine or vegetable oil spreads), 1 tablespoon packed brown sugar and 2 tablespoons rum in mug. Fill with hot cider.\n\n## Hot Chocolate Fast\n\nPrep 10 min Total 10 min \u25cf 4 servings\n\n  * 2 tablespoons sugar\n  * 2 tablespoons unsweetened baking cocoa\n  * 2\u00bd cups milk\n  * \u00bd cup half-and-half\n  * \u00bc cup semisweet or dark chocolate chips\n  * \u00bd teaspoon vanilla\n\n1 In 2-quart saucepan, mix sugar and cocoa. Using whisk, gradually stir in milk and half-and-half until well blended. Cook and stir over medium heat until thoroughly heated (do not boil). Remove from heat.\n\n2 Add chocolate chips; stir constantly with whisk until chips are melted and mixture is smooth. Stir in vanilla. Pour into cups.\n\n1 Serving: Calories 230; Total Fat 12g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 25mg; Sodium 75mg; Total Carbohydrate 23g (Dietary Fiber 1g, Sugars 20g); Protein 6g Exchanges: \u00bd Other Carbohydrate, 1 Milk, 1 Fat Carbohydrate Choices: 1\u00bd\n\nLighter Directions For 4 grams of fat and 160 calories per serving, substitute fat-free (skim) milk and fat-free half-and-half.\n\nDouble-Chocolate Hot Chocolate Substitute chocolate milk for regular milk.\n\nMarshmallow-Topped Hot Chocolate Top each serving with 2 to 4 tablespoons miniature marshmallows or one large homemade marshmallow (page 550).\n\nNew York Egg Cream, Fresh Strawberry Milk\n\n## Fresh Strawberry Milk Fast \u2022 Easy\n\nBecause fresh strawberries are used to make this beverage, the flavor is light, fresh and sweet.\n\nPrep 5 min Total 5 min \u25cf 4 servings\n\n  * 4 cups milk\n  * 4 cups sliced fresh strawberries\n  * 1 cup powdered sugar\n  * Additional sliced strawberries, if desired\n\nIn blender, place milk, 4 cups strawberries and the powdered sugar. Cover and blend on low speed 30 seconds or until smooth. Pour through fine-mesh strainer, if desired. Pour into 4 glasses. Garnish rim of each glass with strawberry slice.\n\n1 Serving: Calories 300; Total Fat 5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 20mg; Sodium 115mg; Total Carbohydrate 54g (Dietary Fiber 3g, Sugars 50g); Protein 9g Exchanges: 1 Fruit, 2 Other Carbohydrate, 1 Low-Fat Milk Carbohydrate Choices: 3\u00bd\n\nChocolate Milk In small microwavable bowl, mix \u00bd cup milk and \u00bc cup unsweetened baking cocoa. Microwave on High 10 to 15 seconds or until warm. Pour into blender; add 3\u00bd cups milk, \u2153 cup powdered sugar and \u00bd teaspoon vanilla. Cover and blend on low speed about 30 seconds or until well blended. Pour into 4 glasses.\n\nNew York Egg Cream In each of 4 (10-ounce) glasses, place 2 tablespoons chocolate-flavored syrup. Add \u00bd cup cold whole milk or half-and-half to each glass. While beating vigorously, slowly add \u00bd cup cold club soda to each glass.\n\nRoot Beer Milk In blender, place 2 cups cold root beer and 2 cups cold milk. Cover and blend on low speed about 30 seconds or until well blended. Pour into 4 glasses.\n\n## Betty's Staples Buttermilk\n\nIn years past, buttermilk was the portion of milk remaining after it was churned to make butter. Today buttermilk is milk \"cultured\" with lactic acid bacteria. Cultured buttermilk is low in fat and calories but still has the traditional tangy flavor and creamy texture. Store it in the refrigerator and shake the carton before using, as it can separate. Many people drink buttermilk just as it is, but it is also wonderful in a variety of recipes. Here are some ways to use it (and don't miss the Buttermilk Ranch Dressing, page 138).\n\n  1. **Chopped Salad with Tangy Dressing:** In medium bowl, with wire whisk, beat \u00bc cup buttermilk and \u00bc cup lime juice until blended. Beat in \u2153 cup buttermilk until mixture is smooth. Add 1 cup cooked shelled edamame, 1 cup chopped romaine, \u00bd cup chopped red bell pepper and \u00bd cup cooked corn. Toss to mix. Add salt and pepper to taste.\n  2. **Strawberry Pops:** Place 2 cups fresh strawberries in blender and blend until smooth. Add 1 cup buttermilk and \u00bc cup sugar and blend until smooth. Pour into small paper cups and freeze until slightly firm, about 1 hour. Place wooden craft stick into center of each pop and freeze until firm.\n  3. **Tangy Coleslaw:** In medium bowl, mix \u00bc cup buttermilk, \u00bc cup mayonnaise and 1 tablespoon sugar until blended. Stir in 2 cups coleslaw mix or shredded cabbage and carrot mixture. Season to taste with salt and pepper.\n  4. **Orange Smoothies:** In blender, place 1 cup buttermilk, \u00bd cup orange juice and 1 cup orange sherbet. Blend until smooth; pour into glasses.\n\nStrawberry Pops\n\nApple-Mint Iced Green Tea, Raspberry-Ginger Tea\n\n## Apple-Mint Iced Green Tea Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 30 min \u25cf 6 servings\n\n  * 6 tea bags green tea with mint\n  * 4 cups boiling water\n  * 2 cups apple juice\n  * Mint sprigs, if desired\n\n1 In large heatproof bowl or pitcher, place tea bags. Pour boiling water over tea bags. Cover and let steep 10 minutes. Remove tea bags. Cool tea 15 minutes.\n\n2 Stir apple juice into tea. Pour over ice. Garnish with mint.\n\n1 Serving: Calories 40; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 8g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\n## Easy Iced Tea\n\nFor iced tea, brew a pot of tea, using double the amount of tea bags. Remove the tea bags and pour into a pitcher. Cool the tea before refrigerating so it does not get cloudy. Refrigerate until very cold. To sweeten iced tea, add Simple Syrup (page 81), or use honey, agave nectar or sugar.\n\n## Raspberry-Ginger Tea Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 15 min \u25cf 6 servings\n\n  * 3 cups brewed tea\n  * 3 cups cranberry-raspberry or raspberry juice\n  * 3 slices (\u00bc inch thick) gingerroot\n  * Fresh raspberries, if desired\n  * Mint sprigs, if desired\n\n1 In 3-quart saucepan, heat tea, juice and gingerroot to boiling over medium-high heat; reduce heat to low.\n\n2 Simmer uncovered 10 minutes, stirring occasionally. Remove gingerroot. Serve warm. Garnish with raspberries and mint.\n\n1 Serving: Calories 80; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 21g (Dietary Fiber 0g, Sugars 17g); Protein 0g Exchanges: 1\u00bd Other Carbohydrate Carbohydrate Choices: 1\u00bd\n\n## Make-Ahead Lemonade Concentrate\n\nCalorie Smart \u2022 Fast \u2022 Easy\n\nPrep 10 min Total 10 min \u25cf 5 cups\n\n  * 2 cups sugar\n  * 2 cups water\n  * 2 cups lemon juice (about 8 medium lemons, at room temperature)\n  * 2 teaspoons grated lemon peel, if desired\n\n1 In 2-quart saucepan, stir together sugar and water. Heat over medium heat, stirring frequently, until sugar is dissolved. Remove from heat; stir in lemon juice and lemon peel. Store covered in refrigerator up to 1 month or in freezer up to 3 months.\n\n\u00bc Cup: Calories 90; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 22g (Dietary Fiber 0g, Sugars 21g); Protein 0g Exchanges: 1\u00bd Other Carbohydrate Carbohydrate Choices: 1\u00bd\n\nLemonade from Concentrate In a pitcher, mix together 1 cup concentrate and 3 cups cold water.\n\nLimeade Concentrate Substitute lime juice for the lemon juice and lime peel for the lemon peel. Continue as directed. Make limeade as directed for lemonade in step 2.\n\n## Ways to Use Lemonade Concentrate\n\nIt is so easy to make delicious lemonade when you have this concentrate on hand in the refrigerator. But that's not all you can make with the mixture. Try some of these tasty ideas, including a slush, a sangria and a delicious tea. It is such a versatile concentrate that you might decide that one batch is not enough! Be sure to experiment with your own ideas, too.\n\n  1. 1. **Citrus Splash:** Mix 1 cup lemonade concentrate and 2 cups orange juice and pour over ice. Garnish with lemon and orange wedges. \n  2. 2. **Sparkling Lemonade:** In pitcher filled with 2 cups ice cubes, mix 1 cup concentrate with 3 cups club soda until well blended.\n  3. 3. **Raspberry Lemonade:** In pitcher, mix \u00bd cup concentrate, 2 cups raspberry juice and 3 cups cold water until blended.\n  4. 4. **Lemonade Sweet Tea:** In kettle or saucepan, heat 4 cups water to boiling. In large heatproof container, pour hot water over 6 tea bags of black tea. Stir in 3 cups cold water and 1 cup concentrate. Refrigerate until chilled.\n  5. 5. **Strawberry-Lemonade Slush:** Microwave 1 (10-ounce) package frozen strawberries 1 to 2 minutes on High or until partially thawed in syrup. In blender, blend strawberries and syrup with \u00be cup concentrate until smooth. Stir in \u00bd cup vodka and 1 cup water. Freeze until firm (will not totally freeze because of vodka). To serve, spoon slush mixture into pitcher and add 3 to 4 cups ginger ale, club soda or lemon-lime carbonated beverage.\n  6. 6. **Lemon Sangria:** In pitcher, mix 1 cup concentrate, \u00bd cup orange juice and 1 bottle (750 ml) dry red wine. Stir until mixed.\n  7. 7. **Lemonade Smoothies:** In blender, place 1 cup vanilla yogurt, 1 cup orange juice, 1 medium cut-up banana and \u00bc cup concentrate. Blend until smooth.\n  8. 8. **Lemonade** **Ice Cubes:** Pour concentrate into ice cube trays and freeze until firm. Place ice cubes in lemonade or tea when serving.\n\n### Grating Lemon Zest\n\n**Handheld plane grater:** Use to grate tiny pieces of citrus peel or whole nutmeg.\n\nUsing a sharp paring knife or zester, pull across peel pressing lightly to remove peel, but not the bitter white pith below.\n\n### Juicing Lemons\n\nBefore squeezing, roll lemon back and forth on counter with palm of hand, using firm pressure, to help burst the cells holding the juice.\n\nCut lemon in half crosswise. Using citrus reamer, press into center of cut side of lemon and twist to squeeze out juice. Strain juice through mesh strainer.\n\nLemonade Sweet Tea\n\n## Learn to Juice\n\nJuicing is an easy way to incorporate more fresh produce into your diet. And a diet that is rich in produce can make you feel fabulous, with an added plus that it is fun to blend your own flavors together.\n\n### Types of Juicers\n\nThere are two main types of juicers:\n\n  * **Slow juicer:** Also known as an upright juicer, this type uses a single auger to crush and squeeze produce and extract juice. This juicer takes just a few minutes longer than the other type. Juice from a slow juicer will be fairly thick and a pretty color, with a small amount of foam.\n  * **Centrifugal juicer:** Sometimes called a high-speed juicer, this juicer separates juice from pulp with high-speed spinning, removing more pulp and making more foam. This appliance is great for making juice in larger quantities.\n\n### Prepping Produce\n\nChoose vibrant, fresh produce\u2014juice will only be as good as the produce that it is made from. Here are some tips for produce preparation.\n\n  * Wash produce before cutting.\n  * Cut produce into pieces that will fit in the feed tube. Most fruits and veggies can be juiced with the peel, stems and leaves left on, if they are scrubbed and have not been chemically treated. Peels can add color and boost nutrients.\n  * Roll leaves such as lettuce and kale into tight balls before adding.\n  * Remove pits from foods and rind from melon, pineapple and citrus.\n  * Wrap herbs around other produce before adding so that they are evenly incorporated.\n  * Have all produce ready before you start juicing so that it is easy to add produce to the juicer.\n  * Alternate adding different produce pieces to the juicer. For produce stuck in the juicer, add a little juice that has already been made to help unstick the juicer.\n\n## Tips for Success\n\n  * If you are new to juicing, start with fruit-based combinations and slowly work into using more vegetables.\n  * For the best flavor, add fruit to vegetable mixtures. Apples and pears can play a great supporting role in vegetable juices.\n  * Think about the colors of the produce, as eye appeal makes a difference.\n\n## Refreshing Green Juice Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 10 min Total 10 min \u25cf 2 servings\n\n  * 8 large leaves fresh green kale (6 oz), \u00bd inch cut off stem ends\n  * 1 medium Bartlett pear (9 oz)\n  * \u00bc bunch fresh cilantro (1 oz)\n  * \u00bd medium lemon, peeled\n\n1 Turn juicer on. Alternate adding ingredients, a little at a time, to juicer, until fruit and vegetables are processed into juice.\n\n2 Stir juice well; pour into 2 glasses. Serve immediately. Discard pulp.\n\n1 Serving: Calories 140; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 29g (Dietary Fiber 6g, Sugars 13g); Protein 3g Exchanges: 1\u00bd Fruit, 1\u00bd Vegetable Carbohydrate Choices: 2\n\nRefreshing Green Juice\n\n## Raspberry-Lime Juice Fast \u2022 Easy\n\nApples that work well in this juice include Regent, Gala and Fuji. Leave the skin on so it adds to the red color of the juice.\n\nPrep 10 min Total 10 min \u25cf 2 servings\n\n  * 2 unpeeled sweet red apples (about \u00bd lb each)\n  * 1 package (6 oz) fresh raspberries (1\u2153 cups)\n  * \u00bd medium lime, peeled\n\n1 Turn juicer on. Alternate adding ingredients, a little at a time, to juicer, until fruit is processed into juice.\n\n2 Stir juice well; pour into 2 glasses. Serve immediately. Discard pulp.\n\n1 Serving: Calories 170; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 40g (Dietary Fiber 10g, Sugars 25g); Protein 1g Exchanges: 2\u00bd Fruit Carbohydrate Choices: 2\u00bd\n\nPeach, Strawberry, and Blueberry Smoothies\n\n## Strawberry Smoothies Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 5 min \u25cf 4 servings\n\n  * 2 cups fresh strawberries\n  * 1 cup milk\n  * 2 containers (6 oz each) strawberry yogurt (1\u2153 cups)\n\nReserve 4 strawberries for garnish. Cut out the hull, or \"cap,\" from remaining strawberries. In blender, place strawberries, milk and yogurt. Cover and blend on high speed about 30 seconds or until smooth. Pour into 4 glasses. Garnish each with reserved strawberry.\n\n1 Serving: Calories 140; Total Fat 2.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 80mg; Total Carbohydrate 24g (Dietary Fiber 2g, Sugars 21g); Protein 6g Exchanges: 1 Fruit, 1 Skim Milk Carbohydrate Choices: 1\u00bd\n\nBlueberry Smoothies Substitute 2 cups fresh or frozen blueberries for the strawberries and blueberry yogurt for the strawberry yogurt. Sweeten to taste with honey or agave nectar.\n\nPeach Smoothies Substitute frozen sliced peaches for the strawberries and peach yogurt for the strawberry yogurt.\n\nRaspberry Smoothies Substitute fresh or frozen raspberries for the strawberries and raspberry yogurt for the strawberry yogurt. Sweeten to taste with honey or agave nectar.\n\nStrawberry-Banana Smoothies Substitute 1 medium banana, cut into chunks, for 1 cup of the strawberries.\n\n## Easy-Being-Green Smoothies Fast \u2022 Easy\n\nPrep 10 min Total 10 min \u25cf 2 servings\n\n  * 1 box (9 oz) frozen chopped spinach\n  * 1 container (6 oz) Key lime pie low-fat yogurt\n  * 2 medium kiwifruit, peeled, quartered\n  * \u00bd cup ice cubes\n  * \u2153 cup apple juice\n  * Kiwi slices, if desired\n\n1 Microwave spinach as directed on box. Rinse with cold water until cooled. Drain, squeezing out as much liquid as possible.\n\n2 In blender, place \u2153 cup of the cooked spinach and remaining ingredients. (Cover and refrigerate remaining spinach for another use.) Cover and blend on high speed about 30 seconds or until smooth. Pour into 2 glasses. Garnish with kiwi slices. Serve immediately.\n\n1 Serving: Calories 160; Total Fat 1.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 60mg; Total Carbohydrate 32g (Dietary Fiber 3g, Sugars 24g); Protein 4g Exchanges: 1\u00bd Fruit, \u00bd Skim Milk, 1 Vegetable Carbohydrate Choices: 2\n\n## Smoothie Success\n\nFor the coolest smoothies, start with refrigerated juice and fruit. Be sure to cool cooked vegetables before adding to smoothies. If you like frosty smoothies, chill the glasses in the freezer about 15 minutes ahead of time.\n\n# Drinks Basics\n\nWhether you are a cocktail lover, wine drinker or beer aficionado, having a good selection of bar ingredients at hand for gatherings is the secret to being a great host. Start with a sampling of liquors, wines and beers plus a variety of mixers and nonalcoholic beverages and build from there. Also, having the right bar equipment makes it a cinch to whip up drinks quickly and with confidence.\n\n## Tools for the Bar\n\n  * Blender\n  * Bottle opener\n  * Cocktail shaker\n  * Corkscrew\n  * Ice bucket and tongs\n  * Pitcher\n  * Mini measure or shot glass\n  * Muddler\n  * Stir sticks and toothpicks\n  * Coasters and cocktail napkins\n\n## Glassware\n\n  * Highball tumblers (10 to 14 ounces): used for gin and tonics, Tom Collins, coolers, spritzers, iced tea and soft drinks\n  * Old-fashioned, lowball or rocks glasses (6 to 10 ounces): used for Scotch or other on-the-rocks drinks and old-fashioned cocktails\n  * Martini\/cocktail glasses: used for martinis, Manhattans, some iced and frozen drinks\n  * Wine glasses (stemmed or not stemmed): used for white or red wine\n  * Margarita glasses: used for margaritas, daiquiris and some frozen or slushy drinks\n  * Champagne flutes\n  * Beer mugs\n  * Shot glasses\n\n## Cocktail Mixology\n\nMixology is all about making good drinks, understanding what goes into drinks and using quality ingredients. Most cocktails follow formulas that are fairly classic, and the difference in flavor will be due to the brand of liquor, the mixer or even the garnish used.\n\n### Spirits to Stock\n\n  * Bourbon and\/or whiskey\n  * Brandy or cognac\n  * Gin\n  * Liqueurs as desired\n  * Scotch\n  * Tequila\n  * Vermouth\n  * Vodka\n\n### Mixers to Stock\n\n  * Club soda\n  * Cola\n  * Ginger ale\n  * Lemon-lime soda\n  * Drink mixes (Bloody Mary, margarita, daiquiri)\n  * Lime and lemon juices\n  * Bitters\n\n### Garnishes to Stock\n\nMost cocktail recipes will suggest a certain garnish, but you can mix it up a bit, too\u2014use decorative toothpicks for spearing cherries, olives and fruit. Small containers are a fun way to showcase garnishes.\n\n  * Stuffed cocktail olives\n  * Maraschino cherries\n  * Cocktail onions\n  * Fresh fruit (lemons, limes, oranges, berries)\n  * Whole hazelnuts\n  * Coarse salt\n  * Granulated, coarse and powdered sugar\n  * Black pepper\n  * Celery sticks\n  * Fresh mint or basil\n\n### Shaken Cocktails\n\nThe best shaken cocktails are always served refreshingly cold, with a very thin layer of ice floating on the surface. Knowing a few tips will make cocktails an easy task and just might make you legendary.\n\n  * Chill glasses in the freezer.\n  * Use correct amounts of the chosen cocktail ingredients.\n  * Have garnishes ready to add quickly.\n  * Use a cocktail shaker with a strainer insert so that the drink mixture moves around the ice cubes, chilling it well.\n  * Strain the drink immediately after mixing so that ice chills the drink but doesn't dilute it.\n  * Serve shaken drinks immediately.\n\n## About Beer\n\nAlthough the major ingredients of beer are simple (grain, hops and yeast), the explosion of different flavors of craft beer has made this simple beverage an art, with local breweries sprouting up virtually everywhere. It is truly remarkable that the United States now has more beer styles and brands than any other place in the world!\n\nHere are the basic categories of beer with some varieties (or flavors) you might like to sample:\n\n### Ales\n\nOften higher in alcohol than lagers, ales tend to be sweet, tasty beers. Varieties include:\n\n  * India Pale Ale (IPA): general category of pale ales with hoppy flavor\n  * Brown Ale: generally mellow but flavorful, with a caramel taste\n  * Saison\/Farmhouse Ale: refreshing fruity ale with mild to moderate tartness\n  * Belgian Pale Ale: copper colored with a malty, somewhat spicy flavor\n  * Berliner Weisse: wheat based and very pale. but refreshingly sour flavor\n  * Lambic: wheat based, with a complex sour flavor\n  * Oatmeal Stout: dark ale that is full bodied and malty, with a slight oatmeal flavor\n\n### Lagers\n\nMost of the mass-produced beers are lagers, but a large variety of styles are available, including:\n\n  * American Pale Lagers: light color with delicate sweetness; serve very cold\n  * Bock Beer: dark, somewhat strong, with an intense malty flavor\n  * Doppelbock: dark, with a malty, rich flavor and high alcohol content\n  * Pilsner: very aromatic, with a crisp, refreshingly bitter flavor\n\n### Specialty\n\nThese beers do not fit in the other categories. They are often quite memorable and will have distinct flavors from fruits, herbs and spices to make them unusual but interesting and delicious.\n\n  * Fruit Beers: usually light- to medium-bodied lagers or ales that have fruit extract or real fruit added, resulting in sweet-flavored beers\n  * Smoked Beers: flavor and aroma are derived from added smoky flavoring\n  * Herb and Spice Beers: beer flavor resembles the ingredients that are added\n\n## Wine Primer\n\nThree things determine the flavor of wine: the type of grapes, where the grapes were grown and how the grapes are processed for the wine. Whether you are serving wine with a meal or with appetizers, the choice of wine is mostly personal; there are no fixed rules. Wine stores can be very helpful and provide guidance. Experimenting with different wines will be helpful, too, as in the end, it is about what you like and what tastes good!\n\n\"Varietal\" refers to a particular variety of grape. Some wines are made from a single varietal, and others are blends; here are some general flavor guidelines:\n\n### White Grape Varietals\n\n  * Chardonnay: rich and creamy, with a buttery, full flavor that often has vanilla overtones\n  * Chenin Blanc: light and fruity, this wine often has a spicy peach flavor\n  * Pinot Gris or Pinot Grigio: refreshing wine with a little spice and fruit overtones\n  * Riesling: crisp and fresh with a delightful fruity flavor and notes of peach, apple or apricot\n  * Sauvignon Blanc or Fum\u00e9 Blanc: often slightly acidic with citrus and herb overtones\n\n### Red Grape Varietals\n\n  * Cabernet Sauvignon: bold with a velvety texture and rich flavor often reminiscent of chocolate, coffee, blackberries and plums\n  * Malbec: dark red, with rich flavor\n  * Merlot: fruity flavor with a medium body and overtones of blackberry and plum\n  * Pinot Noir: light-bodied wine that is often spicy with the flavors of cherries, blackberries and currants\n  * Syrah (Shiraz): dark red, bold and spicy, with overtones of dark fruit like blackberries and plums\n  * Zinfandel: spicy red with earthy overtones of berries and peppers\n\n### Other Wines\n\n  * Sparkling wines include Champagne, Prosecco, and Cava.\n  * Fortified wines have added liquor and include Madeira, Marsala, port, and sherry.\n  * Aromatic wines such as vermouth are flavored with herbs and spices.\n\n### Storing and Serving Wine\n\n  * Store wine in a cool, dry location away from light. Lay bottles on their sides, not upright, so corks do not dry out and shrink, which can allow air to enter and spoil the wine.\n  * Serve white and sparkling wines between 45\u00b0F and 55\u00b0F. Reds should be a bit warmer at 55\u00b0F to 60\u00b0F for the best flavor.\n  * Wineglasses are designed to maximize aromas and flavors of wine. A set of all-purpose glasses for both red and white wines is a good place to start.\n  * If you have wine left over, place the cork back into the bottle or seal with a wine plug. Refrigerate and use within 3 days.\n\n## Manhattan Cocktails Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 5 min \u25cf 2 servings\n\n  * 2 maraschino cherries\n  * \u00bc cup small ice cubes or shaved ice\n  * \u2153 cup bourbon or whiskey\n  * \u2153 cup sweet vermouth\n  * 2 dashes aromatic bitters\n\n1 Chill 2 (3-oz) stemmed glasses in freezer.\n\n2 Place cherry in each chilled glass. Place ice in martini shaker or pitcher. Add bourbon, vermouth and bitters; shake or stir to blend and chill. Pour into glasses, straining out ice.\n\n1 Serving: Calories 130; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 3g); Protein 0g Carbohydrate Choices: \u00bd\n\n## Martini Cocktails Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 5 min \u25cf 2 servings\n\n  * 2 pimiento-stuffed green olives or lemon twists\n  * \u00bc cup small ice cubes or shaved ice\n  * \u2153 cup gin\n  * \u2153 cup dry vermouth\n  * 2 dashes aromatic bitters\n\n1 Chill 2 (3-oz) stemmed glasses in freezer.\n\n2 Place olive in each chilled glass. Place ice in martini shaker or pitcher. Add gin, vermouth and bitters; shake or stir to blend and chill. Pour into glasses, straining out ice.\n\n1 Serving: Calories 150; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 0g); Protein 0g Carbohydrate Choices: \u00bd\n\nSweet Martini Cocktails Substitute sweet vermouth for the dry vermouth.\n\nChocolate Martini Cocktails Omit all ingredients except ice. Swirl about 1 tablespoon chocolate syrup on inside of each chilled glass. To serve, add \u2153 cup chocolate-flavored liqueur, \u2153 cup Irish cream liqueur, \u2153 cup vanilla-flavored vodka and \u2153 cup cr\u00e8me de cacao to ice in martini shaker; shake or stir and pour over swirled chocolate in each glass.\n\nMango Martini Cocktails Omit all ingredients except ice. Puree 5 slices mango and 5 tablespoons syrup from one jar (24 ounces) refrigerated mango slices in extra-light syrup to make about \u00bd cup puree. Add \u00bd cup mango puree, \u00bd cup peach-flavored vodka, \u00bc cup orange-flavored liqueur and 2 tablespoons fresh lime juice to ice in martini shaker; shake or stir and pour as directed. Garnish each cocktail with a slice of mango and slice of lime, if desired.\n\nPomegranate Martini Cocktails Omit all ingredients except ice. Wet rim of each glass by rubbing with lime wedge; dip into colored sugar. Add 1 cup pomegranate juice, 3 tablespoons citrus-flavored vodka, 2 tablespoons orange-flavored liqueur and 1 tablespoon lime juice to ice in martini shaker; shake or stir and pour as directed.\n\nGreen Tea Martini Cocktails Omit all ingredients except ice. Add \u00bd cup citrus-flavored vodka, \u00bd cup chilled brewed green tea and \u2153 cup sweet tea\u2013flavored vodka or regular vodka to ice in martini shaker; shake or stir and pour as directed.\n\n### Shaking Cocktails\n\nPlace ice in shaker. Add liquor; cover and shake vigorously to blend and chill.\n\nRemove shaker cover; place strainer over top of shaker. Pour liquor into glass, straining out ice.\n\n## Mojitos\n\nPrep 15 min Total 40 min \u25cf 4 servings\n\nMint Syrup\n\n  * \u00bd cup sugar\n  * \u00be cup water\n  * 1 cup packed fresh mint leaves, coarsely chopped\n\nMojitos\n\n  * 1 cup rum\n  * \u00bc cup fresh lime juice (2 medium limes)\n  * 1\u00bd cups chilled club soda\n  * Ice cubes or crushed ice\n  * 12 thin lime slices (from 2 limes), cut in half\n  * 4 mint sprigs\n\n1 In 1-quart saucepan, mix syrup ingredients. Heat to boiling over high heat, stirring occasionally. Reduce heat to low; simmer 5 minutes or until sugar is dissolved. Remove from heat; let stand at room temperature about 20 minutes.\n\n2 Strain mixture into pitcher; do not push on leaves or syrup will turn a dark green color.\n\n3 Stir in rum, lime juice and club soda. Fill 4 lowball glasses with ice. Place lime slices in glasses, pushing down alongside of glass, allowing ice cubes to hold them in place. Pour mint mixture over ice. Garnish with mint sprigs.\n\n1 Serving: Calories 250; Total Fat 0 (Saturated Fat 0, Trans Fat 0); Cholesterol 0; Sodium 25; Total Carbohydrate 30 (Dietary Fiber 1, Sugars 26); Protein 0 Carbohydrate Choices: 2\n\nMake-Ahead Directions Make a big batch of the Mint Syrup and store it in your refrigerator for up to 1 week. If you make it beforehand, the mojitos will take only a few minutes to make. For more mint flavor, place 1 tablespoon chopped mint leaves in glasses before adding the ice.\n\n### Making a Citrus Twist\n\nWith vegetable peeler or small knife, cut strip of orange or lemon peel; twist as desired.\n\n## Simple Syrup\n\nUse this syrup to sweeten drinks instead of granulated sugar\u2014it works especially well in cold drinks. Use 1 teaspoon per drink or more to taste.\n\nTo make simple syrup: In 2-quart saucepan, stir together 2 cups sugar and 2 cups water. Heat over medium heat, stirring frequently, until sugar is dissolved. Store covered in refrigerator up to 1 month. Makes 3 cups.\n\nDill Simple Syrup: Make as directed\u2014except add 1\u00bd cups lightly packed fresh dill weed. After removing from heat, let stand 30 minutes. Strain through fine-mesh strainer; discard dill.\n\n  * **Dill Martini:** Mix up a Martini (page 80) and shake it up with 1 teaspoon syrup. For more dill flavor, try adding a little more of the syrup.\n  * **Dill Tea:** Make hot black or green tea and stir 1 teaspoon syrup into each serving.\n\nGinger Simple Syrup: Make as directed\u2014except add \u00bd pound fresh gingerroot, peeled, cut into \u00bc-inch slices. After removing from heat, let stand 30 minutes. Strain through fine-mesh strainer; discard gingerroot, or reserve for another use.\n\n  * **Ginger Tea:** Make Easy Iced Tea (recipe) and stir in 1 to 2 teaspoons syrup.\n  * **Ginger Lemonade:** Make Lemonade (page 72) and stir in enough syrup (1 to 2 teaspoons) for desired flavor.\n\nTriple-Berry Simple Syrup: Make as directed\u2014except add 3 cups frozen (thawed) mixed berries. After removing from heat, let stand 30 minutes. Strain through fine-mesh strainer; discard berries or reserve for another use.\n\n  * **Berry Tea:** Make Easy Iced Tea (recipe) and stir in 1 to 2 teaspoons syrup.\n  * **Berry Sangria:** Stir 2 to 3 teaspoons syrup into a pitcher of Sangria (page 83).\n\n## Margaritas Fast\n\nPrep 15 min Total 15 min \u25cf 8 servings\n\n  * Margarita salt or kosher (coarse) salt\n  * 2 cups cracked ice\n  * 2\u00bd cups sweet-and-sour drink mix or nonalcoholic margarita drink mix\n  * 1\u2153 cups tequila\n  * 1\u2153 cups fresh lime juice (about 9 limes)*\n  * \u00be cup orange-flavored liqueur (such as Cointreau or triple sec)\n\nRub rims of 8 stemmed glasses with water; dip rims of glasses into salt. Place ice in pitcher. Add drink mix, tequila, lime juice and liqueur; shake or stir until blended and chilled. Pour into glasses, straining out ice.\n\n*Or substitute \u00be cup frozen (thawed) limeade concentrate for the lime juice.\n\n1 Serving: Calories 180; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 8g); Protein 0g Carbohydrate Choices: 3\n\nFrozen Strawberry Margaritas In blender, place 6 ounces (half of 12-ounce can) frozen (thawed) limeade concentrate and 1 box (10 ounces) frozen strawberries. Cover and blend on high speed until smooth. Add 3 cups water and \u00be cup tequila; blend until smooth. Pour into freezer container. Freeze until slushy, at least 24 hours. To serve, stir and spoon \u2154 cup mixture into each glass and fill with \u2153 cup lemon-lime carbonated beverage.\n\n### Coating the Edge of a Glass\n\nWet rim of glass with citrus wedge or dip into lemon or lime juice or water on small plate; shake off excess.\n\nDip wet rim into sugar, coarse (kosher) salt, decorator sugar or chocolate candy sprinkles on small plate.\n\n## Classic Bloody Mary Calorie Smart \u2022 Fast\n\nBloody Marys are very easy to make at home. As with any cocktail, it helps to start with chilled ingredients. Store your vodka in the freezer; it won't freeze but will be icy cold!\n\nPrep 10 min Total 10 min \u25cf 4 servings\n\n  * 1 cup vodka\n  * \u00be cup chilled tomato juice\n  * \u00bc cup lemon juice\n  * \u00bc teaspoon celery salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon Worcestershire sauce\n  * Hot pepper sauce, if desired\n  * Ice cubes\n\nGarnishes, if desired\n\n  * Celery stalk\n  * Pickle spear\n  * Cucumber spear\n  * Beef jerky stick\n  * Lime or lemon wedge\n\n1 In half-gallon pitcher, mix vodka, tomato juice, lemon juice, celery salt, pepper and Worcestershire sauce until well blended. Add pepper sauce to taste.\n\n2 Fill 4 glasses with ice; add tomato juice mixture. Garnish as desired.\n\n1 Serving: Calories 140; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 2g); Protein 0g Carbohydrate Choices: 0\n\n## Mango Mimosas Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 5 min \u25cf 12 servings\n\n  * 2 cups chilled orange juice\n  * 2 cups chilled mango nectar*\n  * 1 bottle (750 ml) regular or nonalcoholic dry champagne or sparkling white wine, chilled\n\nIn 1\u00bd-quart pitcher, mix orange juice and mango nectar. Pour champagne into glasses until half full. Fill glasses with juice mixture.\n\n*Apricot or peach nectar can be substituted for the mango nectar.\n\n1 Serving: Calories 100; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 9g); Protein 0g Carbohydrate Choices: 1\n\nPineapple Mimosas Substitute pineapple juice for the mango nectar. Garnish with a pineapple wedge.\n\nPomegranate-Mango Mimosas Substitute pomegranate juice for the orange juice. Place a few fresh pomegranate seeds in each glass for a fun garnish.\n\n## Sangria Calorie Smart \u2022 Fast\n\nSangria is also lovely made with white or ros\u00e9 (blush) wine. Try adding fresh peach or plum slices instead of the lemon and orange.\n\nPrep 10 min Total 10 min \u25cf 8 servings\n\n  * \u2154 cup lemon juice\n  * \u2153 cup orange juice\n  * \u00bc cup sugar\n  * 1 lemon, cut into thin slices\n  * 1 orange, cut into thin slices\n  * 1 bottle (750 ml) dry red wine or nonalcoholic red wine\n  * Ice cubes, if desired\n\nIn half-gallon glass pitcher, mix juices and sugar until sugar is dissolved. Add lemon and orange slices. Stir in wine. Add ice, if desired.\n\n1 Serving: Calories 100; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 8g); Protein 0g Carbohydrate Choices: \u00bd\n\nWhite Sangria Substitute dry white wine, such as Chardonnay or Sauvignon Blanc, for the dry red wine.\n\n# Heirloom Recipe and New Twist\n\nThis classic eggnog recipe is easy to make ahead of time so is ideal for gatherings. It is a favorite among many on our staff. The new twist is made with bacon and bourbon\u2014our taste panel raved about this fun combination. Serve it at your next holiday gathering.\n\nHeirloom\n\n## Eggnog\n\nFor food safety, our eggnog is made with a cooked egg custard instead of raw eggs.\n\nPrep 35 min Total 2 hr 35 min \u25cf 10 servings\n\nSoft Custard\n\n  * 3 eggs, slightly beaten\n  * \u2153 cup granulated sugar\n  * Dash salt\n  * 2\u00bd cups milk\n  * 1 teaspoon vanilla\n\nEggnog\n\n  * 1 cup whipping cream\n  * 2 tablespoons powdered sugar\n  * \u00bd teaspoon vanilla\n  * \u00bd cup light rum, brandy or whiskey*\n  * 1 to 2 drops yellow food color, if desired\n  * Ground nutmeg\n\n1 In heavy 2-quart saucepan, stir eggs, granulated sugar and salt until well mixed. Gradually stir in milk. Cook over medium heat 10 to 15 minutes, stirring constantly, until mixture just coats a metal spoon; remove from heat. Stir in 1 teaspoon vanilla. Place saucepan in cold water until custard is cool. (If custard curdles, beat vigorously with hand beater until smooth.) Cover and refrigerate at least 2 hours but no longer than 24 hours.\n\n2Just before serving, in chilled medium bowl, beat whipping cream, powdered sugar and \u00bd teaspoon vanilla with electric mixer on high speed until stiff. Gently stir 1 cup of the whipped cream, the rum and the food color into custard.\n\n3 Pour custard mixture into small punch bowl. Drop remaining whipped cream in mounds onto custard mixture. Sprinkle with nutmeg. Serve immediately. Store covered in refrigerator up to 2 days.\n\n*Or substitute 2 tablespoons rum extract and \u2153 cup milk for the rum.\n\n1 Serving: Calories 160; Total Fat 10g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 95mg; Sodium 70mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 12g); Protein 4g Carbohydrate Choices: 1\nNew Twist\n\n## Bourbon-Maple Eggnog\n\nPrep 35 min Total 2 hr 35 min \u25cf 10 servings\n\nCustard\n\n  * 3 eggs, slightly beaten\n  * \u00bd cup real maple syrup\n  * Dash salt\n  * 2\u00bd cups milk\n  * 1 teaspoon vanilla\n\nEggnog\n\n  * 1 cup whipping cream\n  * 2 tablespoons powdered sugar\n  * 2 tablespoons real maple syrup\n  * \u00bd teaspoon vanilla\n  * \u00bd cup bourbon\n  * Crisply cooked bacon, finely chopped, if desired\n\n1 In 2-quart saucepan, stir eggs, \u00bd cup syrup and the salt until well mixed. Gradually stir in milk. Cook over medium heat 10 to 15 minutes, stirring constantly, until mixture just coats a metal spoon; remove from heat. Stir in 1 teas-poon vanilla. Place saucepan in cold water until custard is cool. (If custard curdles, beat vigorously with hand beater until smooth.) Cover and refrigerate at least 2 hours but no longer than 24 hours.\n\n2 Just before serving, in chilled medium bowl, beat whipping cream, powdered sugar, 2 tablespoons syrup and \u00bd teaspoon vanilla with electric mixer on high speed until stiff. Gently stir 1 cup of the whipped cream and the bourbon into custard.\n\n3 Pour custard mixture into small punch bowl. Drop remaining whipped cream in mounds onto custard mixture. Sprinkle each serving with bacon. Serve immediately. Store covered in refrigerator up to 2 days.\n\n1 Serving: Calories 220; Total Fat 12g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 95mg; Sodium 90mg; Total Carbohydrate 19g (Dietary Fiber 0g, Sugars 17g); Protein 4g Carbohydrate Choices: 1\n\nEggnog\n\nBourbon-Maple Eggnog\n\n## Beergaritas Fast \u2022 Easy\n\nThis takeoff on the classic margarita is a fun alternative to plain beer. For an authentic touch, salt the rims of the glasses. Pour \u00bd cup margarita salt or coarse salt onto a small flat plate. Rub a lime wedge around the edge of each glass and then dip into salt.\n\nPrep 10 min Total 10 min \u25cf 8 servings\n\n  * 1 can (12 oz) frozen (thawed) limeade concentrate\n  * 1 cup tequila\n  * \u00bc cup orange-flavored liqueur\n  * 2 bottles (12 oz each) light-colored beer\n  * Crushed ice\n  * 8 lime slices\n\n1 In pitcher, mix limeade concentrate, tequila and liqueur until well blended. Pour beer into pitcher; stir.\n\n2 Fill 8 lowball or margarita glasses with crushed ice. Pour beer mixture over ice. Garnish each with lime slice.\n\n1 Serving: Calories 210; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 31g (Dietary Fiber 0g, Sugars 20g); Protein 0g Carbohydrate Choices: 2\n\n## Mulled Wine Calorie Smart\n\nPrep 5 min Total 1 hr 5 min \u25cf 12 servings\n\n  * 2 bottles (750 ml each) dry red wine\n  * \u00bd cup brandy\n  * 1 cup honey\n  * 2 oranges, thinly sliced\n  * 4 cinnamon sticks\n  * 8 whole cloves\n  * \u00bd teaspoon whole allspice\n  * Cinnamon sticks, if desired\n  * Orange slices, if desired\n\n1 In 3-quart stainless steel or nonstick saucepan (not aluminum), mix all ingredients. Heat to simmering over low heat. Simmer uncovered 1 hour, being careful not to boil.\n\n2 Pour mixture through strainer into heatproof pitcher or punch bowl; serve warm. Garnish with additional cinnamon sticks and orange slices.\n\n1 Serving: Calories 113; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 26g (Dietary Fiber 0g, Sugars 0g); Protein 0g Carbohydrate Choices: 1\u00bd\n\n# Chapter 4: Breakfast and Brunch\n\nGranola, Greek Yogurt\n\n## Breakfast & Brunch Basics\n\nFor Morning Rush Breakfasts\n\nFor a Leisurely Brunch\n\nEgg Basics\n\n## Features\n\nBetty's Staples: Cooked Pork Sausage\n\nGlobal Flavors: Middle Eastern\n\nHeirloom Recipe and New Twist\n\nLearn with Betty: Scrambled Eggs\n\nMake-Ahead: Granola\n\n## Basic Eggs\n\nBaked Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\nBasic Omelet Calorie Smart\n\nFried Eggs, Sunny Side Up Calorie Smart \u2022 Fast\n\nHard-Cooked Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\nPoached Eggs Calorie Smart \u2022 Fast\n\nScrambled Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\n## Cooked Eggs\n\nAvocado, Bacon and Cheese Omelet\n\nBacon, Kale and Egg Cups Calorie Smart\n\nBreakfast Burritos\n\nBreakfast Roll-Up\n\nCheese Omelet\n\nChiles Rellenos Bake Calorie Smart\n\nChorizo, Potato and Egg Scramble\n\nClassic Shakshuka Calorie Smart\n\nDenver Omelet\n\nEggs Benedict Fast\n\nHam 'n Egg Breakfast Sandwich\n\nHam and Cheese Omelet\n\nHam and Cheddar Strata Calorie Smart\n\nHuevos Rancheros\n\nPotato, Bacon and Egg Scramble Calorie Smart \u2022 Fast\n\nPuffy Omelet Calorie Smart \u2022 Fast\n\nQuiche Lorraine\n\nRoasted Poblano Rellenos Bake\n\nSausage 'n Eggs\n\nSeafood Quiche\n\nSkillet Eggs with Summer Squash Hash Calorie Smart\n\nSpinach and Cheddar Quiche Calorie Smart\n\nSpinach Quiche\n\nSun-Dried Tomato Omelet\n\nTex-Mex Egg Bake\n\n## Breakfast Meats\n\nBalsamic and Brown Sugar Bacon\n\nHoney-Mustard Bacon\n\nOven-Baked Bacon Calorie Smart \u2022 Easy\n\nPeppered Brown Sugar Bacon\n\nPraline Bacon\n\n## Pancakes, French Toast and Waffles\n\nApple Oven Pancake\n\nApricot Stuffed French Toast Calorie Smart\n\nBerry Pancakes\n\nBuckwheat Pancakes Calorie Smart \u2022 Fast\n\nButtermilk Pancakes\n\nCornbread Pancakes Calorie Smart \u2022 Fast\n\nCrepes Calorie Smart\n\nDouble Oven Pancake\n\nFrench Toast Calorie Smart \u2022 Fast\n\nFrench Toast Sticks\n\nGranola Pancakes\n\nOrange-Cranberry Pancakes\n\nOven French Toast\n\nPancakes Calorie Smart \u2022 Fast\n\nPuffy Oven Pancake Calorie Smart \u2022 Easy\n\nSausage Waffles\n\nUpside-Down Banana-Walnut French Toast\n\nWaffles Calorie Smart\n\nWhole Wheat Waffles with Honey\u2013Peanut Butter Drizzle Calorie Smart\n\n## Cereal, Oatmeal and Other Grains\n\nBacon and Cheese Grits\n\nBacon and Pecan Wild Rice Porridge\n\nBaked Apple Oatmeal Calorie Smart\n\nBreakfast Quinoa Calorie Smart \u2022 Fast \u2022 Easy\n\nCheese Grits Calorie Smart\n\nCranberry\u2013Wild Rice Porridge\n\nGranola\n\nGranola Bars\n\nGranola Cereal\n\nGranola Fruit Salad\n\nSteel-Cut Oatmeal\n\nTriple-Berry Oatmeal-Flax Muesli Calorie Smart\n\n## Syrups\n\nButter-Pecan Syrup\n\nSpicy Cider Syrup\n\n## Bonus Recipes\n\nEasy Sausage-Broccoli Pizza\n\nGranola Cookies\n\nIce Cream and Granola\n\nYogurt Parfaits\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Breakfast & Brunch Basics\n\nMaking breakfast is about more than nutrition\u2014it's a good thing that you can do for yourself and those in your home. Starting the day with a wholesome, satisfying breakfast just might make everyone's day better.\n\nWhile ready-to-go options like cereal, yogurt and toast are wholesome time-savers for busy days, it never hurts to branch out. Make homemade Granola, page 108, to have ready for eating and using all sorts of ways, or you could even whip up a Basic Omelet, page 94\u2014it just takes a few minutes. On leisurely mornings, indulge in brunch with Eggs Benedict, page 93, or Upside-Down Banana-Walnut French Toast, page 107\u2014two choices to wow guests. Be sure to check other chapters, too, for beverages, breads, sides and even main dishes that you might want to serve for breakfast.\n\n## For Morning Rush Breakfasts\n\n  * Make a list before you grocery shop. Half the battle is remembering to stock your kitchen with quick and easy breakfast options and ingredients for recipes your family enjoys.\n  * Everyone tends to be short on time in the morning, so take a few minutes the night before to pull together breakfast. Hard-Cooked Eggs, page 91, and Granola, page 108, for topping yogurt, are both good choices. Or wrap up a Ham 'n Egg Breakfast Sandwich, page 93, for quick heating in the microwave in the morning.\n\n## For a Leisurely Brunch\n\n  * Plan brunch for any time of the morning that works for you. The foods you make might vary with the time of day. If it is an early brunch, try a make-the-night-before item like Bacon, Kale and Egg Cups, page 97, with a bowl of mixed fruit. For a later morning brunch, you might choose Quiche Lorraine, page 95, with a tossed green salad.\n  * Choose a variety of foods that include one or two main dishes, fresh fruit and\/or vegetable salads and sweet or savory breads. Beverages can be as simple as coffee, tea and juices, or why not add Mango Mimosas, page 83?\n  * Plan for some foods that can be served at room temperature or chilled, if possible. Often it is just the main dish that needs to be served warm.\n\n## Let the Sun Shine!\n\nWhen you sit down for breakfast or even if you are getting food ready to run out the door, open the curtains and enjoy some sunlight. It will make you feel alert and brighten the morning. Add some relaxing music while you eat, and you're on the way to a beautiful day!\n\n## Egg Basics\n\nEggs are the perfect combination of nutrition, versatility and good taste! Packed inside each egg is a great supply of protein, vitamins and minerals. Although eggs do contain some fat and cholesterol, they are low in sodium. And 1 large egg has just 80 calories. Here are a few basic methods for cooking eggs. Choose your favorite or look for more egg recipes in the pages that follow.\n\n## Scrambled Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 10 min Total 10 min \u25cf 4 servings\n\n  * 6 eggs\n  * \u2153 cup water, milk or half-and-half\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper, if desired\n  * 1 tablespoon butter\n\n1 In medium bowl, beat eggs, water, salt and pepper thoroughly with fork or whisk until well mixed.\n\n2 In 10-inch skillet, heat butter over medium heat just until butter begins to sizzle. Pour egg mixture into skillet.\n\n3 As mixture begins to set at bottom and side, gently lift cooked portions with metal spatula so that thin, uncooked portion can flow to bottom. Avoid constant stirring. Cook 3 to 4 minutes or until eggs are thickened throughout but still moist.\n\n1 Serving: Calories 140; Total Fat 11g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 325mg; Sodium 260mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 9g Exchanges: 1 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\n### Learn with Betty Scrambled Eggs\n\n**Perfectly Scrambled Eggs:** These eggs are fluffy and moist with large curds and no visible liquid egg remaining.\n\n**Undercooked Scrambled Eggs:** These eggs haven't been cooked long enough, as liquid egg remains.\n\n**Overcooked Scrambled Eggs:** These eggs are dry with small curds. Brown spots will affect the flavor.\n\n## Fried Eggs, Sunny Side Up Calorie Smart \u2022 Fast\n\nPrep 15 min Total 15 min \u25cf 4 servings\n\n  * 2 tablespoons butter\n  * 4 eggs\n\n1 In 10-inch skillet, melt butter over medium-high heat. Break 1 egg into custard cup or small glass bowl. Carefully slide egg from cup into skillet. Repeat with remaining 3 eggs. Immediately reduce heat to low.\n\n2 Cook uncovered 5 to 7 minutes, frequently spooning butter from skillet over eggs, until film forms over top and whites and yolks are firm, not runny.\n\n1 Serving: Calories 130; Total Fat 11g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 225mg; Sodium 105mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 6g Exchanges: 1 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\nLighter Directions Omit butter and use nonstick skillet. Add eggs to skillet as directed. Cook over low heat about 1 minute or until edges turn white. Add 2 teaspoons water for each egg. Cover and cook about 5 minutes longer or until film forms over top and whites and yolks are firm, not runny.\n\n### Making Fried Eggs\n\nCarefully slide each egg from custard cup into pan.\n\nCook until film forms over top and whites and yolks are firm, not runny.\n\n## Poached Eggs Calorie Smart \u2022 Fast\n\nBe sure to use a large enough pan so the eggs do not touch during cooking.\n\nPrep 15 min Total 15 min \u25cf 4 servings\n\n  * 4 eggs\n\n1 In 10-inch skillet or 2-quart saucepan, heat 2 to 3 inches water to boiling. Reduce heat so water is simmering (bubbles rise slowly and break just below surface).\n\n2 Break 1 egg into custard cup or small glass bowl. Holding cup close to water's surface, carefully slide egg into water. Repeat with remaining 3 eggs.\n\n3 Cook uncovered 3 to 5 minutes or until whites and yolks are firm, not runny. Carefully remove eggs with slotted spoon.\n\n1 Serving: Calories 80; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 210mg; Sodium 60mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 6g Exchanges: 1 Medium-Fat Meat Carbohydrate Choices: 0\n\n### Making Poached Eggs\n\nCarefully slide egg into water.\n\nCook until whites and yolks are firm, not runny.\n\n## Baked Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 5 min Total 25 min \u25cf 2 servings\n\n  * 2 eggs\n  * \u215b teaspoon salt\n  * Dash pepper\n  * 2 tablespoons milk or half-and-half\n  * 2 teaspoons butter\n\n1 Heat oven to 325\u00b0F. Grease 2 (6-oz) custard cups with butter. Break 1 egg into each cup; sprinkle with salt and pepper. Top each with 1 tablespoon milk and 1 teaspoon butter.\n\n2 Bake 15 to 18 minutes or until whites and yolks are firm, not runny.\n\n1 Egg: Calories 120; Total Fat 9g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 200mg; Sodium 250mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 1g); Protein 7g Exchanges: 1 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\n## Hard-Cooked Eggs Calorie Smart \u2022 Fast \u2022 Easy\n\nPrep 10 min Total 30 min \u25cf 6 eggs\n\n  * 6 eggs\n\n1 In 2-quart saucepan, place eggs in single layer; cover with cold water at least 1 inch above eggs. Cover and heat to boiling. Immediately remove from heat; let stand covered 15 minutes for large eggs (12 minutes for medium or 18 minutes for extra-large).\n\n2 Drain. Immediately place eggs in cold water with ice cubes or run cold water over eggs until completely cooled.\n\n3 To peel, gently tap egg on countertop until entire shell is cracked. Roll gently between hands to loosen shell. Starting at large end, peel egg under cold running water to help remove shell.\n\n1 Egg: Calories 80; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 185mg; Sodium 60mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 6g Exchanges: 1 Medium-Fat Meat Carbohydrate Choices: 0\n\n### Making Hard-Cooked Eggs\n\nPlace eggs in single layer, covering with at least 1 inch of water above eggs.\n\nGently tap egg until entire shell is finely cracked.\n\nStarting a large end, peel under water to help remove shell.\n\n## Potato, Bacon and Egg Scramble Calorie Smart \u2022 Fast\n\nPrecooking the potatoes in boiling water helps them cook faster, turning this scramble into a 20-minute meal.\n\nPrep 10 min Total 20 min \u25cf 5 servings\n\n  * 1 lb small red potatoes (6 or 7), cubed\n  * 6 eggs\n  * \u2153 cup milk\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * 2 tablespoons butter\n  * 4 medium green onions, sliced (\u00bc cup)\n  * 5 slices bacon, crisply cooked, crumbled\n\n1 In 2-quart saucepan, heat 1 inch water to boiling. Add potatoes. Heat to boiling; reduce heat to medium-low. Cover and cook 6 to 8 minutes or until potatoes are tender; drain.\n\n2 In medium bowl, beat eggs, milk, salt and pepper with fork or whisk until well mixed; set aside.\n\n3 In 10-inch skillet, melt butter over medium-high heat. Cook potatoes in butter 3 to 5 minutes, turning occasionally, until light brown. Stir in onions. Cook 1 minute, stirring constantly.\n\n4 Pour egg mixture into skillet with potatoes and onions. As mixture begins to set at bottom and side, gently lift cooked portions with spatula so that thin, uncooked portion can flow to bottom. Avoid constant stirring. Cook 3 to 4 minutes or until eggs are thickened throughout but still moist. Sprinkle with bacon.\n\n1 Serving: Calories 290; Total Fat 15g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 275mg; Sodium 370mg; Total Carbohydrate 25g (Dietary Fiber 3g, Sugars 3g); Protein 13g Exchanges: 1 Starch, 1 Vegetable, 1 High-Fat Meat, 1 Fat Carbohydrate Choices: 1\u00bd\n\nChorizo, Potato and Egg Scramble Omit bacon. In 8-inch skillet, crumble \u00bc pound bulk chorizo sausage; add \u00bd cup chopped green bell pepper. Cook and stir until sausage is no longer pink; drain. Stir into eggs at end of cook time. Garnish each serving with 1 to 2 tablespoons salsa.\n\n## Huevos Rancheros\n\n_Huevos rancheros_ is Spanish for \"rancher's eggs.\" When buying tortillas, check for freshness by making sure they are soft and pliable when you bend the package, and check that the edges are not dry or cracked.\n\nPrep 45 min Total 45 min \u25cf 6 servings\n\n  * \u00bd lb bulk chorizo or pork sausage\n  * Vegetable oil\n  * 6 soft corn tortillas (6 to 7 inch)\n  * 1\u00bc cups salsa (for homemade salsa, see page 458)\n  * 6 Fried Eggs (page 90)\n  * 1\u00bd cups shredded Cheddar cheese (6 oz)\n\n1 In 8-inch skillet, cook sausage over medium heat 8 to 10 minutes, stirring occasionally, until no longer pink; drain. Remove to bowl; cover to keep warm.\n\n2 In same skillet, heat \u215b inch oil over medium heat just until hot. Cook 1 tortilla at a time in oil about 1 minute, turning once, until crisp; drain on paper towels.\n\n3 In 1-quart saucepan, heat salsa, stirring occasionally, until hot.\n\n4 Spread 1 tablespoon salsa over each tortilla to soften. Place 1 fried egg on each tortilla. Top with sausage, cheese and remaining salsa.\n\n1 Serving: Calories 460; Total Fat 33g (Saturated Fat 14g, Trans Fat 1g); Cholesterol 275mg; Sodium 1100mg; Total Carbohydrate 16g (Dietary Fiber 2g, Sugars 3g); Protein 25g Exchanges: 1 Starch, 3 High-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\n## Skillet Eggs with Summer Squash Hash Calorie Smart\n\nPrep 35 min Total 1 hr 15 min \u25cf 4 servings\n\n  * 2 medium yellow summer squash (1 lb), shredded\n  * 2 medium zucchini (1 lb), shredded\n  * 1 teaspoon salt\n  * 1 tablespoon olive oil\n  * \u00bc cup chopped red onion\n  * 1 small tomato, chopped (\u00bd cup)\n  * \u00bd cup diced cooked 95% fat-free ham\n  * 1 tablespoon chopped fresh dill weed\n  * 1 teaspoon grated lemon peel\n  * 4 eggs\n  * \u215b teaspoon pepper\n\n1 In large colander, place yellow squash and zucchini. Sprinkle with salt; stir. Let stand in sink 30 minutes to drain.\n\n2 Gently rinse squash to remove excess salt; squeeze to remove as much liquid as possible. Set aside.\n\n3 In 12-inch nonstick skillet, heat oil over medium-high heat. Add onion; cook 2 minutes, stirring frequently. Stir in squash mixture. Cook 8 minutes, stirring occasionally, until vegetables are crisp-tender. Add tomato, ham, dill and lemon peel. Cook and stir 1 minute longer. Reduce heat to medium.\n\n4 Spread squash mixture evenly in skillet. Make 4 (2\u00bd-inch-wide) indentations in mixture with back of spoon. Break eggs, 1 at a time, into custard cup or small glass bowl; carefully slide into indentations. Sprinkle eggs with pepper. Cover and cook 8 to 10 minutes or until whites and yolks are firm, not runny.\n\n1 Serving: Calories 260; Total Fat 13g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 225mg; Sodium 1550mg; Total Carbohydrate 10g (Dietary Fiber 2g, Sugars 7g); Protein 24g Exchanges: 2 Vegetable, 2 Very Lean Meat, 1 Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: \u00bd\n\n## Ham 'n Egg Breakfast Sandwich\n\nMake biscuits as directed on page 488 using 3-inch cutter (yield will be 6 biscuits). Split biscuits in half while still warm. Spread with mayonnaise and mustard if desired. For each sandwich, place \u00bd slice American cheese, 1 Fried Egg (page 90) or Poached Egg (page 91) and 1 thin slice deli ham on bottom half of biscuit. Top each with tops of biscuits. Serve immediately.\n\nOr, cool sandwiches and wrap individually with plastic wrap and refrigerate up to 1 week. To reheat, unwrap sandwich and rewrap with a paper towel. Microwave on High 50 seconds. Let stand 1 minute in microwave before eating.\n\n## Eggs Benedict Fast\n\nAs one story goes, Delmonico's restaurant in New York City was the birthplace of this classic brunch dish. In the late 1800s, when regular patrons Mr. and Mrs. LeGrand Benedict complained of nothing new on the lunch menu, the chef created this now-famous dish and named it after them.\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\n  * Hollandaise Sauce (page 461)\n  * 3 English muffins, split, toasted\n  * 7 teaspoons butter, softened\n  * 6 thin slices Canadian bacon or cooked ham\n  * 6 Poached Eggs (page 91)\n  * Paprika, if desired\n\n1 Keep hollandaise sauce warm.\n\n2 Spread each toasted muffin half with 1 teaspoon butter; keep warm.\n\n3 In 10-inch skillet, melt remaining 1 teaspoon butter over medium heat. Cook bacon in butter until light brown on both sides.\n\n4 Place 1 slice bacon on each muffin half; top with 1 poached egg. Spoon warm hollandaise sauce over eggs. Sprinkle with paprika.\n\n1 Serving: Calories 410; Total Fat 33g (Saturated Fat 14g, Trans Fat 1.5g); Cholesterol 400mg; Sodium 670mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 5g); Protein 15g Exchanges: 1 Starch, 1\u00bd High-Fat Meat, 4 Fat Carbohydrate Choices: 1\n\nDenver Omelet\n\n## Basic Omelet Calorie Smart\n\nOmelets are a great way to start your day or an easy go-to for a quick dinner. Make them just with eggs or try one of the variations.\n\nPrep 40 min Total 40 min \u25cf 4 omelets\n\n  * 8 eggs\n  * Salt and pepper, if desired\n  * 8 teaspoons butter\n\n1 For each omelet, in small bowl, beat 2 of the eggs until fluffy. Add salt and pepper to taste. In 8-inch nonstick omelet pan or skillet, heat 2 teaspoons of the butter over medium-high heat just until butter is hot and sizzling. As butter melts, tilt pan to coat bottom.\n\n2 Quickly pour eggs into pan. While rapidly sliding pan back and forth over heat, quickly and continuously stir with spatula to spread eggs over bottom of pan as they thicken. Let stand over heat a few seconds to lightly brown bottom of omelet. Do not overcook; omelet will continue to cook after folding. If desired, add filling before folding.\n\n3 To remove from pan, first run spatula under one edge of omelet, folding about one-third of it to the center. Transfer to plate by tilting pan, letting flat unfolded edge of omelet slide out onto plate. Using edge of pan as a guide, flip folded edge of omelet over the flat portion on the plate.\n\n4 Repeat with remaining butter and eggs. If desired, omelets can be kept warm on platter in 200\u00b0F oven while preparing remaining omelets.\n\n1 Omelet: Calories 220; Total Fat 18g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 445mg; Sodium 170mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 1g); Protein 13g Exchanges: 2 Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 0\n\nLighter Directions For 8 grams of fat and 130 calories per omelet, substitute \u00bd cup fat-free egg product for the 2 eggs.\n\nCheese Omelet Before folding omelet, sprinkle with \u00bc cup shredded Cheddar, Monterey Jack or Swiss cheese or \u00bc cup crumbled blue cheese.\n\nDenver Omelet Before adding eggs to pan, cook 2 tablespoons chopped cooked ham, 1 tablespoon finely chopped bell pepper and 1 tablespoon finely chopped onion in butter about 2 minutes, stirring frequently. Continue as directed in Step 2.\n\nHam and Cheese Omelet Before folding omelet, sprinkle with 2 tablespoons shredded Cheddar, Monterey Jack or Swiss cheese and 2 tablespoons finely chopped cooked ham.\n\nAvocado, Bacon and Cheese Omelet Before folding omelet, sprinkle with 2 tablespoons shredded Cheddar cheese and top with 1 or 2 strips cooked chopped bacon and 2 or 3 avocado slices.\n\nSun-Dried Tomato Omelet Before folding omelet, sprinkle with 2 tablespoons chopped sun-dried tomatoes in oil and 1 to 2 tablespoons crumbled goat cheese. After folding, top omelet with 1 tablespoon shredded fresh basil.\n\n### Making an Omelet\n\nQuickly and continuously stir with spatula.\n\nFold about one-third of omelet over other side.\n\nUse edge of pan to flip folded edge of omelet onto plate.\n\n## Puffy Omelet Calorie Smart \u2022 Fast\n\nPrep 15 min Total 30 min \u25cf 2 servings\n\n  * 4 eggs, separated\n  * \u00bc cup water\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * 1 tablespoon butter\n  * Cheese Sauce (page 459), if desired\n\n1 Heat oven to 325\u00b0F. In medium bowl, beat egg whites, water and salt with electric mixer on high speed until stiff but not dry. In small bowl, beat egg yolks and pepper on high speed about 3 minutes or until very thick and lemon colored. Fold egg yolks into egg whites.\n\n2 In 10-inch ovenproof skillet, melt butter over medium heat. As butter melts, tilt skillet to coat bottom. Pour egg mixture into skillet. Gently level surface with spatula; reduce heat to low. Cook about 5 minutes or until puffy and bottom is light brown. Carefully lift omelet at edge to see color.\n\n3 Bake uncovered 12 to 15 minutes or until knife inserted in center comes out clean. Tilt skillet and slip pancake turner or metal spatula under omelet to loosen. Carefully fold omelet in half. Slip onto warm serving plate. Serve with cheese sauce.\n\n1 Serving: Calories 200; Total Fat 16g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 440mg; Sodium 460mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 1g); Protein 13g Exchanges: 2 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\n## Quiche Lorraine\n\nPrep 25 min Total 1 hr 40 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586)\n  * 8 slices bacon, chopped\n  * \u2153 cup finely chopped onion\n  * 4 eggs\n  * 2 cups whipping cream or half-and-half\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon black pepper\n  * \u215b teaspoon ground red pepper (cayenne)\n  * 1 cup shredded Swiss or Gruy\u00e8re cheese (4 oz)\n\n1 Heat oven to 425\u00b0F. After folding pastry into fourths, place in 9-inch quiche dish or pie plate. Unfold and ease into dish, pressing firmly against bottom and side. Carefully line pastry with double thickness of foil, gently pressing foil to bottom and side of pastry. Let foil extend over edge to prevent excessive browning. Bake 10 minutes. Carefully remove foil; bake 2 to 4 minutes longer or until pastry just begins to brown and is set. If crust bubbles, gently push bubbles down with back of spoon.\n\n2 Meanwhile, in 10-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Remove bacon with slotted spoon; drain on paper towels. Drain all but 2 teaspoons bacon drippings from skillet. Cook onion in bacon drippings over medium heat 3 to 5 minutes, stirring occasionally, until light golden brown. Set aside.\n\n3 In large bowl, beat eggs slightly with fork or whisk. Beat in whipping cream, salt, black pepper and red pepper. Sprinkle bacon, onion and cheese into warm pie crust. Gently pour egg mixture into crust.\n\n4 Reduce oven to 325\u00b0F. Bake 45 to 50 minutes or until knife inserted in center comes out clean. Let stand 10 minutes before cutting.\n\n1 Serving: Calories 620; Total Fat 52g (Saturated Fat 25g, Trans Fat 3g); Cholesterol 255mg; Sodium 580mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 3g); Protein 17g Exchanges: 1 Starch, 1 Vegetable, 1\u00bd High-Fat Meat, 8 Fat Carbohydrate Choices: 1\n\nSeafood Quiche Substitute 1 cup chopped cooked crabmeat (patted dry), imitation crabmeat, cooked shrimp or cooked salmon for the bacon. Use \u2153 cup finely chopped green onions; increase salt to \u00bd teaspoon.\n\nSpinach Quiche Substitute 1 box (9 ounces) frozen chopped spinach, thawed and squeezed to drain, for the bacon. Sprinkle over onion and cheese.\n\nSpinach Quiche\n\n## Spinach and Cheddar Quiche Calorie Smart\n\nPrep 25 min Total 1 hr 10 min \u25cf 6 servings\n\nCrust\n\n  * \u00bd cup all-purpose flour\n  * \u00bd cup whole wheat flour\n  * \u00bc teaspoon salt\n  * \u00bc cup butter\n  * 3 to 4 tablespoons cold water\n\nFilling\n\n  * \u00bd cup shredded reduced-fat sharp Cheddar cheese (2 oz)\n  * 1 tablespoon all-purpose flour\n  * 4 eggs, slightly beaten\n  * 1\u00bd cups milk\n  * \u00bd teaspoon salt\n  * 1 cup firmly packed fresh spinach leaves, coarsely chopped\n  * 1 small red bell pepper, chopped (\u00bd cup)\n\n1 Heat oven to 375\u00b0F. In medium bowl, mix \u00bd cup all-purpose flour, the whole wheat flour and \u00bc teaspoon salt. Cut in butter, using pastry blender or fork, until mixture looks like small crumbs. Sprinkle with water, 1 tablespoon at a time, tossing with fork just until flour is moistened. Shape into a ball; flatten slightly.\n\n2 On floured surface, roll dough into 12-inch round. Fold dough into fourths; place in 9-inch glass pie plate. Unfold and ease into pie plate, pressing firmly against bottom and side; crimp edge. Trim overhanging edge of pastry 1 inch from rim of pie plate. Fold and roll pastry under, even with plate; flute.\n\n3 In large bowl, mix cheese and 1 tablespoon all-purpose flour. Stir in eggs, milk and \u00bd teaspoon salt until well blended. Stir in spinach and bell pepper. Pour into pie crust.\n\n4 Bake 15 minutes. Cover edge of crust with pie crust shield or strip of foil to prevent excessive browning. Bake 15 to 20 minutes longer or until knife inserted in center comes out clean. Let stand 10 minutes before cutting.\n\n1 Serving: Calories 250; Total Fat 13g (Saturated Fat 7g, Trans Fat 0.5g); Cholesterol 170mg; Sodium 510mg; Total Carbohydrate 21g (Dietary Fiber 2g, Sugars 4g); Protein 11g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 1\u00bd\n\n## Ham and Cheddar Strata Calorie Smart\n\nPrep 15 min Total 1 hr 35 min \u25cf 8 servings\n\n  * 12 slices bread\n  * 2 cups cut-up cooked smoked ham (about 10 oz)\n  * 2 cups shredded Cheddar cheese (8 oz)\n  * 8 medium green onions, sliced (\u00bd cup)\n  * 6 eggs\n  * 2 cups milk\n  * 1 teaspoon ground mustard\n  * \u00bc teaspoon red pepper sauce\n  * Paprika, if desired\n\n1 Heat oven to 300\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 Trim crusts from bread. Arrange 6 of the slices in baking dish. Layer ham, cheese and onions on bread in dish. Cut remaining 6 bread slices diagonally in half; arrange on onions.\n\n3 In medium bowl, beat eggs, milk, mustard and pepper sauce with fork or whisk; pour evenly over bread. Sprinkle with paprika.\n\n4 Bake uncovered 1 hour to 1 hour 10 minutes or until center is set and bread is golden brown. Let stand 10 minutes before cutting.\n\n1 Serving: Calories 350; Total Fat 19g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 215mg; Sodium 920mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 5g); Protein 24g Exchanges: 1 Starch, 3 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 1\n\nMake-Ahead Directions Prepare recipe as directed through Step 3; cover and refrigerate up to 24 hours. Increase bake time by 5 to 10 minutes if necessary.\n\n## Breakfast Burritos\n\nTo make 2 burritos, scramble 4 eggs as directed on page 90. On each of 2 (10-inch) flour tortillas, spread \u00bc cup refried beans and 2 tablespoons salsa. Top each with \u00bc cup shredded Cheddar cheese. Microwave each on microwavable plate 30 to 45 seconds or until cheese is melted. Top each with scrambled eggs and roll up.\n\n## Chiles Rellenos Bake Calorie Smart\n\nPrep 10 min Total 55 min \u25cf 8 servings\n\n  * 8 eggs\n  * 1 container (8 oz) sour cream\n  * \u00bc teaspoon salt\n  * 2 drops red pepper sauce\n  * 2 cups shredded Monterey Jack cheese (8 oz)\n  * 2 cups shredded Cheddar cheese (8 oz)\n  * 2 cans (4.5 oz each) chopped green chiles, undrained\n  * 2 cups salsa (for homemade salsa, see page 458)\n  * 2 tablespoons chopped fresh cilantro\n  * \u00bd cup black beans (from 15-oz can), drained, rinsed\n  * \u00bd cup frozen (thawed) or canned (drained) whole kernel corn, cooked as directed on package; drained and cooled\n\n1 Heat oven to 350\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 In large bowl, beat eggs, sour cream, salt and pepper sauce with whisk. Stir in cheeses and chiles. Pour into baking dish.\n\n3 Bake uncovered 40 to 45 minutes or until golden brown and set in center.\n\n4 In small bowl, mix 1 cup of the salsa and the cilantro. In another small bowl, mix remaining 1 cup salsa, the beans and corn. Serve casserole with salsa mixtures.\n\n1 Serving: Calories 400; Total Fat 29g (Saturated Fat 17g, Trans Fat 0.5g); Cholesterol 285mg; Sodium 870mg; Total Carbohydrate 12g (Dietary Fiber 2g, Sugars 5g); Protein 23g Exchanges: 1 Starch, 3 High-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\n\nRoasted Poblanos Rellenos Bake Set oven control to broil. Place 2 fresh poblano chiles (4\u00bdx3-inch) on cookie sheet. Broil with tops 2 inches from heat about 10 minutes, turning frequently with tongs, until all sides are blackened and blistered. Place chiles in paper bag; seal bag. Let chiles steam 15 minutes. Wearing food-safe plastic gloves, peel blackened skin from chiles. Cut open chiles and remove stems, seeds and membranes; chop. Substitute for the canned chiles.\n\n## Bacon, Kale and Egg Cups Calorie Smart\n\nPrep 30 min Total 1 hr \u25cf 12 servings\n\n  * 3 large kale leaves\n  * 6 slices packaged precooked bacon (from 2.1-oz package)\n  * \u00bc cup finely chopped red bell pepper\n  * 1 green onion, chopped (1 tablespoon)\n  * 3 tablespoons grated Parmesan cheese\n  * 12 eggs\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * Additional grated Parmesan cheese, if desired\n\n1 Heat oven to 350\u00b0F. Grease 12 regular-size muffin cups with shortening.\n\n2 Cut kale into 2- to 3-inch pieces. In 2-quart saucepan, heat \u00bd inch water to boiling. Add kale pieces; cover and cook about 5 minutes or until bright green and just tender.\n\n3 Line bottom and side of each muffin cup with several small leaves of steamed kale (leaves should stick up around edge of cup). Cut each bacon slice crosswise in half; place half slice in bottom of each cup or around edge over kale.\n\n4 In small bowl, stir together bell pepper and onion; reserve 1 tablespoon. Stir cheese into remaining bell pepper and onion. Place about 1 teaspoon vegetable-cheese mixture into each muffin cup. Press mixture down slightly with back of spoon. Break eggs 1 at a time into small custard cup; slide into muffin cup over cheese mixture. Top with reserved bell pepper and onion. Sprinkle with salt, pepper and additional cheese.\n\n5 Bake 25 to 30 minutes or until egg yolks are set. Serve immediately.\n\n1 Serving: Calories 110; Total Fat 7g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 190mg; Sodium 230mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 8g Exchanges: 1 Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 0\n\n### Filling Egg Cups\n\nLine muffin cups with several leaves of softened kale. Place bacon in cup over kale.\nGlobal Flavors\n\n## Middle Eastern\n\nTake a trip to the other side of the world without even leaving your kitchen! Explore the variety of flavors and diversity of Middle Eastern cuisine.\n\nDates The fruit of the date palm. Very sweet; contain a long, narrow seed. Available dried, or fresh in late summer through mid-fall.\n\nFigs Fresh figs can be found May to December, depending on the variety, which can range in color and flavor. Dried figs are available year round.\n\nFlatbread There are many varieties that vary slightly in ingredients and technique. Flatbread is used at nearly every meal to scoop up meat, vegetables and rice and to sop up sauces and dips. **Pita** rounds can be split horizontally to form a pocket that is used to stuff ingredients for sandwiches. **Naan** is traditionally baked in a tandoor oven to achieve its characteristic smoky flavor. **Sabaayed** is a grilled Somali flatbread that has a crispy outside and flaky interior.\n\nHarissa A spicy-hot sauce, both green and red; use as accompaniment for couscous and to flavor soups, stews and other dishes. Available canned or jarred. To make your own, see page 446.\n\nPomegranates Inside the leathery skin of this fruit are hundreds of juicy, tart seeds packed in sections separated by membranes. Typically available August through December; also look for fresh pomegranate seeds, which eliminates peeling and separating the fruit.\n\nPomegranate Molasses Pomegranate juice reduced to thick, tart-sweet syrup, used as a marinade for grilled meats or in everything from dips to stews.\n\nPreserved Lemons A staple in Moroccan cooking, these lemons have been preserved in salty lemon juice brine\u2014possibly with cinnamon, cloves or other spices\u2014giving them a silky texture and unique flavor.\n\nSumac Ground dried berries found in Middle Eastern markets, this brick- to dark-purple spice adds fruity, astringent flavor to meat, fish and vegetables.\n\nTahini A MIddle Eastern staple, this thick paste is made from ground sesame seed. Available in many grocery stores near the international ingredients.\n\nZa'atar Both a native spice of the Middle East with the flavors reminiscent of marjoram, oregano and thyme and also the name of a popular dry spice blend containing toasted sesame seed, marjoram and sumac. Mix with olive oil and salt and drizzle over warm bread or use to season meat and vegetables.\n\n## Classic Shakshuka Calorie Smart\n\nHere is our version of a Middle Eastern favorite.\n\nPrep 25 min Total 50 min \u25cf 6 servings\n\n  * 2 tablespoons olive oil\n  * 1 large onion, chopped (1 cup)\n  * 1 medium red bell pepper, chopped\n  * 3 cloves garlic, finely chopped\n  * 2 teaspoons ground cumin\n  * 1 teaspoon paprika\n  * 1 can (28 oz) crushed tomatoes, undrained\n  * 1 teaspoon sugar\n  * 1\u00bc teaspoons salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon crushed red pepper flakes\n  * 6 pasteurized eggs (see page 177)\n  * \u00be cup crumbled feta cheese (3 oz)\n  * Chopped fresh cilantro or chives, if desired\n  * Crusty bread, if desired\n\n1 Heat oven to 350\u00b0F. In 12-inch ovenproof skillet, heat oil over medium heat. Cook onion, bell pepper and garlic in oil about 8 minutes, stirring occasionally, until tender. Add cumin and paprika; cook 1 minute, stirring once. Stir in tomatoes, sugar, salt, pepper and pepper flakes.\n\n2 Heat to simmer; remove from heat. With back of spoon, make 6 (3-inch) indentations in tomato mixture. Break eggs, 1 at a time, into custard cup; carefully slide egg into each indentation. Place skillet in oven.\n\n3 Bake uncovered 20 to 25 minutes or until whites are firm but yolks are still slightly runny. Bake longer if desired for firm yolks. Sprinkle with cheese and cilantro. Serve with bread.\n\n1 Serving: Calories 220; Total Fat 14g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 205mg; Sodium 960mg; Total Carbohydrate 13g (Dietary Fiber 2g, Sugars 9g); Protein 10g Exchanges: 2 Vegetable, 1 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\n## Betty's Staples Cooked Pork Sausage\n\nTo cook bulk pork sausage, crumble the sausage and cook in a skillet over medium heat 10 to 15 minutes or until well browned and thoroughly cooked, stirring frequently. Drain in a colander or on a paper towel and store tightly covered in the refrigerator for up to 3 days or freeze for up to 2 months.\n\n  1. 1. **Breakfast Roll-Up:** For each roll-up, heat \u00bc cup cooked sausage in the microwave for about 20 seconds. Spoon onto center of flour tortilla. Top with 1 sliced hard-cooked egg and \u00bd cup each shredded lettuce and chopped tomato. Drizzle with taco sauce and roll up.\n  2. 2. **Sausage Waffles:** Make Waffles, page 104. For each waffle, pour batter onto waffle maker as directed; sprinkle with \u00bc cup cooked sausage. Bake waffle as directed.\n  3. 3. **Sausage 'n Eggs:** Make Scrambled Eggs mixture as directed on page 90. Stir in \u00bd to \u00be cup cooked sausage. Cook egg and sausage mixture as directed in recipe.\n  4. 4. **Easy Sausage-Broccoli Pizza:** Sprinkle purchased Italian pizza crust with 1 cup cooked sausage, 1 cup chopped cooked broccoli and 1\u00bd cups shredded Italian cheese blend. Bake as directed on pizza crust package.\n  5. 5. **Tex-Mex Egg Bake:** Spread 5 cups frozen diced hash brown potatoes in greased 13x9-inch (3-quart) glass baking dish. Top with 1 cup cooked sausage, 1 can (4.5 oz) chopped green chiles and 1\u00bd cups shredded Colby\u2013Monterey Jack cheese. In medium bowl, beat 6 eggs with 1\u00bd cups milk; pour over mixture. Sprinkle with 1\u00bd cups additional cheese. Bake at 350\u00b0F for 50 to 60 minutes or until knife inserted in center comes out clean. Serve with salsa.\n\nOven-Baked Bacon, Peppered Brown Sugar Bacon, Praline Bacon\n\n## Oven-Baked Bacon Calorie Smart \u2022 Easy\n\nPrep 5 min Total 40 min \u25cf 12 slices\n\n  * 12 slices bacon\n\n1 Heat oven to 350\u00b0F. Line 15x10x1-inch pan with foil. Place metal cooling rack on foil, making sure rack fits within interior of pan so drippings don't fall to bottom of oven. Arrange bacon in single layer on rack.\n\n2 Bake 20 minutes. Turn bacon over. Bake 10 to 15 minutes longer or until crisp and golden brown. Remove from rack to paper towels. Serve warm.\n\n1 Slice: Calories 60; Total Fat 3g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 135mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 5g); Protein 2g Exchanges: \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: \u00bd\n\nBalsamic and Brown Sugar Bacon In small bowl, mix \u00bc cup packed brown sugar and 2 tablespoons balsamic vinegar. Make bacon as directed\u2014except after turning bacon over, brush slices with balsamic mixture.\n\nHoney-Mustard Bacon Brush both sides of bacon with \u2153 cup honey mustard. Arrange bacon in single layer on rack. Bake as directed.\n\nPeppered Brown Sugar Bacon In large resealable food-storage plastic bag, mix \u00bd cup packed brown sugar, 1 teaspoon coarse ground black pepper and \u00bc teaspoon ground red pepper (cayenne). Separate bacon slices and place in bag; seal and shake to coat. Arrange bacon in single layer on rack. Bake as directed.\n\nPraline Bacon In small bowl, mix \u00bc cup packed brown sugar and \u00bc cup finely chopped pecans. Make bacon as directed\u2014except after turning bacon over, sprinkle evenly with pecan mixture.\n\n## Pancakes Calorie Smart \u2022 Fast\n\nThis classic pancake recipe has appeared in every Betty Crocker cookbook since 1950.\n\nPrep 25 min Total 25 min \u25cf 9 pancakes\n\n  * 1 egg\n  * 1 cup all-purpose or whole wheat flour\n  * 1 tablespoon sugar\n  * 3 teaspoons baking powder\n  * \u00bc teaspoon salt\n  * \u00be cup milk\n  * 2 tablespoons vegetable oil or melted butter\n\n1 In medium bowl, beat egg with whisk until fluffy. Stir in remaining ingredients just until flour is moistened (batter will be slightly lumpy); do not overmix or pancakes will be tough. For thinner pancakes, stir in 1 to 2 tablespoons additional milk.\n\n2 Heat griddle or skillet over medium-high heat (375\u00b0F). (To test griddle, sprinkle with a few drops of water. If bubbles jump around, heat is just right.) Brush with vegetable oil if necessary (or spray with cooking spray before heating).\n\n3 For each pancake, pour slightly less than \u00bc cup batter onto hot griddle. Cook 2 to 3 minutes or until bubbly on top and dry around edges. Turn; cook other side until golden brown.\n\n1 Pancake: Calories 110; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 250mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 3g); Protein 3g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nBerry Pancakes Stir \u00bd cup fresh or frozen (thawed and well drained) sliced strawberries, blackberries, blueberries or raspberries into batter. Top pancakes with vanilla yogurt and additional berries.\n\nButtermilk Pancakes Substitute 1 cup buttermilk for the \u00be cup milk. Reduce baking powder to 1 teaspoon. Add \u00bd teaspoon baking soda.\n\nOrange-Cranberry Pancakes Substitute orange juice for the milk. Stir in \u00bc cup dried cranberries and \u00bc cup chopped dried apples after mixing ingredients in Step 1.\n\n### Turning Pancakes\n\nPancakes are ready to turn when they are puffed and bubbles form on top.\n\n## Cornbread Pancakes Calorie Smart \u2022 Fast\n\nMake the syrup first, then the pancakes, so both are warm and ready at the same time.\n\nPrep 30 min Total 30 min \u25cf 8 servings\n\n  * 1\u00bc cups all-purpose flour\n  * \u00be cup cornmeal\n  * \u00bc cup sugar\n  * 2 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 1\u2153 cups milk\n  * \u00bc cup vegetable oil\n  * 1 egg, beaten\n  * Butter-Pecan Syrup (recipe)\n\n1 Heat griddle or skillet over medium-high heat (375\u00b0F). Brush with vegetable oil if necessary (or spray with cooking spray before heating).\n\n2 In large bowl, mix flour, cornmeal, sugar, baking powder and salt. Stir in milk, oil and egg just until blended.\n\n3 For each pancake, pour about \u00bc cup batter onto hot griddle. Cook 2 to 3 minutes or until bubbly on top and dry around edges. Turn; cook other side until golden brown. Top each serving (2 pancakes) with 2 tablespoons syrup.\n\n1 Serving: Calories 380; Total Fat 15g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 35mg; Sodium 350mg; Total Carbohydrate 57g (Dietary Fiber 2g, Sugars 20g); Protein 6g Exchanges: 1 Starch, 3 Other Carbohydrate, 3 Fat Carbohydrate Choices: 4\n\n## Puffy Oven Pancake Calorie Smart \u2022 Easy\n\nMore like a popover than a pancake, this oven pancake puffs up high around the edges when it's done. Serve it quickly, before it sinks.\n\nPrep 10 min Total 40 min \u25cf 4 servings\n\n  * 2 tablespoons butter\n  * 2 eggs\n  * \u00bd cup all-purpose flour\n  * \u00bc teaspoon salt\n  * \u00bd cup milk\n  * Lemon juice and powdered sugar or cut-up fruit, if desired\n\n1 Heat oven to 400\u00b0F. In 9-inch glass pie plate or cast-iron skillet, melt butter in oven; brush butter over bottom and side of pie plate.\n\n2 In medium bowl, beat eggs slightly with whisk. Stir in flour, salt and milk just until flour is moistened (do not overbeat or pancake may not puff). Pour into pie plate.\n\n3 Bake 25 to 30 minutes or until puffy and deep golden brown. Serve immediately sprinkled with lemon juice and powdered sugar or topped with fruit.\n\n1 Serving: Calories 160; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 125mg; Sodium 230mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 2g); Protein 5g Exchanges: 1 Starch, 2 Fat Carbohydrate Choices: 1\n\nApple Oven Pancake Make pancake as directed\u2014except sprinkle 2 tablespoons packed brown sugar and \u00bc teaspoon ground cinnamon evenly over melted butter in pie plate. Arrange 1 cup thinly sliced peeled baking apple (1 medium) over sugar. Pour batter over apple. Bake 30 to 35 minutes. Immediately loosen edge of pancake and turn upside down onto heatproof serving plate.\n\nDouble Oven Pancake In 13x9-inch pan, melt \u2153 cup butter. Use 4 eggs, 1 cup all-purpose flour, \u00bc teaspoon salt and 1 cup milk. Bake 30 to 35 minutes.\n\n## Cast-Iron Skillets\n\nIt's worth digging out your heavy cast-iron skillet from the recesses of your cupboard or picking one up at a garage sale (if you are lucky enough to find one). What makes cast iron so special? It holds the heat like no other pan. So it's a perfect choice for the Puffy Oven Pancake because the heat of the pan helps to make the pancake puff up and gives it a delicious crispy exterior.\n\nA cast-iron skillet is also great for meats that need to get a nice sear (see page 302) before being braised, delivering a great deep brown color without getting burnt bits in the bottom of the pan. Also oven compatible, it makes delicious roasted veggies with perfectly golden-brown crispy exteriors. And it's the secret of many for gorgeous fried chicken, as the heat of the pan helps keep the oil very hot, producing crispy chicken while the deep side cuts down on oil splatters.\n\n## Buckwheat Pancakes Calorie Smart \u2022 Fast\n\nStore buckwheat tightly covered for up to 1 month in the pantry or in the freezer for up to 2 months.\n\nPrep 30 min Total 30 min \u25cf 5 servings\n\n  * 1 egg\n  * \u00bd cup buckwheat flour\n  * \u00bd cup whole wheat flour\n  * 1 cup milk\n  * 1 tablespoon sugar\n  * 2 tablespoons vegetable oil\n  * 1 tablespoon baking powder\n  * \u00bd teaspoon salt\n  * Whole bran or wheat germ, if desired\n  * Butter-Pecan Syrup (below)\n\n1 Heat griddle or skillet over medium-high heat (375\u00b0F). Grease with vegetable oil if necessary (or spray with cooking spray before heating).\n\n2 In medium bowl, beat egg with whisk until fluffy. Beat in remaining pancake ingredients except bran just until smooth.\n\n3 For each pancake, pour about 3 tablespoons batter onto hot griddle. Cook pancakes until puffed and dry around edges. Sprinkle each pancake with 1 teaspoon bran. Turn; cook other side until golden brown.\n\n4 Top each serving (2 pancakes) with about 2 tablespoons syrup.\n\n1 Serving: Calories 340; Total Fat 13g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 50mg; Sodium 620mg; Total Carbohydrate 48g (Dietary Fiber 3g, Sugars 17g); Protein 6g Exchanges: 2 Starch, 1 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 3\n\n## Butter-Pecan Syrup\n\nIn 1-quart saucepan, melt 2 tablespoons butter over medium heat. Add \u2153 cup pecans; cook 2 to 3 minutes, stirring frequently, until browned. Stir in \u00be cup real maple syrup; heat until hot. Serve over pancakes.\n\nApple Oven Pancake\n\n## Crepes Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 12 crepes\n\n  * 1\u00bd cups all-purpose flour\n  * 1 tablespoon granulated sugar\n  * \u00bd teaspoon baking powder\n  * \u00bd teaspoon salt\n  * 2 cups milk\n  * 2 tablespoons butter, melted\n  * \u00bd teaspoon vanilla\n  * 2 eggs\n  * Applesauce, sweetened berries, jelly or jam, if desired\n  * Powdered sugar, if desired\n\n1 In medium bowl, mix flour, granulated sugar, baking powder and salt. Add milk, 2 tablespoons butter, the vanilla and eggs; beat with whisk just until smooth.\n\n2 Lightly butter 6- to 8-inch skillet or crepe pan. Heat over medium heat until bubbly. For each crepe, pour slightly less than \u00bc cup batter into hot skillet. Immediately tilt and rotate skillet so thin layer of batter covers bottom. Cook until light brown. Run wide spatula around edge to loosen; turn and cook other side until light brown. Repeat with remaining batter, buttering skillet as needed.\n\n3 Stack crepes, placing waxed paper between each; keep covered. Spread applesauce, sweetened berries, jelly or jam thinly over each warm crepe; roll up. (Fill crepes so when rolled the more attractive side is on the outside.) Sprinkle with powdered sugar.\n\n1 Crepe: Calories 140; Total Fat 8g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 55mg; Sodium 190mg; Total Carbohydrate 15g (Dietary Fiber 0g, Sugars 3g); Protein 4g Exchanges: 1 Starch, 1\u00bd Fat Carbohydrate Choices: 1\n\nMake-Ahead Directions Make crepes as directed and cool completely; do not fill. Stack crepes with plastic wrap between each; wrap tightly and freeze up to 2 months.\n\n## Waffles Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 6 (7-inch) round waffles\n\n  * 2 eggs\n  * 2 cups all-purpose or whole wheat flour\n  * 1 tablespoon sugar\n  * 4 teaspoons baking powder\n  * \u00bc teaspoon salt\n  * 1\u00be cups milk\n  * \u00bd cup vegetable oil or melted butter\n  * Syrup or fresh berries, if desired\n\n1 Heat waffle maker. (Waffle makers without a nonstick coating may need to be brushed with vegetable oil or sprayed with cooking spray before batter for each waffle is added.)\n\n2 In large bowl, beat eggs with whisk until fluffy. Beat in flour, sugar, baking powder, salt, milk and oil just until smooth.\n\n3 Pour slightly less than \u00be cup batter onto center of hot waffle maker. (Check manufacturer's directions for recommended amount of batter.) Close lid of waffle maker.\n\n4 Bake about 5 minutes or until steaming stops. Carefully remove waffle. Serve immediately with syrup. Repeat with remaining batter.\n\n1 Waffle: Calories 380; Total Fat 22g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 75mg; Sodium 480mg; Total Carbohydrate 38g (Dietary Fiber 1g, Sugars 6g); Protein 9g Exchanges: 2\u00bd Starch, 4 Fat Carbohydrate Choices: 2\u00bd\n\nLighter Directions For 7 grams of fat and 255 calories per waffle, substitute \u00bd cup fat-free egg product for the eggs, use fat-free (skim) milk and reduce oil to 3 tablespoons.\n\n## Whole Wheat Waffles with Honey\u2013Peanut Butter Drizzle Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 8 servings\n\nWaffles\n\n  * 2 eggs\n  * 1 cup whole wheat flour\n  * 1 cup all-purpose flour\n  * 2 cups buttermilk\n  * 1 tablespoon sugar\n  * 3 tablespoons vegetable oil\n  * 2 teaspoons baking powder\n  * \u00bc teaspoon salt\n\nDrizzle and topping\n\n  * \u00bd cup honey\n  * \u00bc cup creamy peanut butter\n  * \u00bd cup low-fat granola\n\n1 Heat waffle maker. (Waffle makers without a nonstick coating may need to be brushed with vegetable oil or sprayed with cooking spray before batter for each waffle is added.)\n\n2 In medium bowl, beat eggs with whisk until foamy. Beat in remaining waffle ingredients just until smooth.\n\n3 Pour about 1 cup of batter onto the center of hot waffle maker. (Check manufacturer's directions for recommended amount of batter.) Close lid of waffle maker.\n\n4 Bake about 5 minutes or until steaming stops. Carefully remove waffle. Repeat with remaining batter.\n\n5 Meanwhile, in small microwavable bowl, mix honey and peanut butter. Microwave uncovered on High 40 to 60 seconds or until warm; stir until smooth.\n\n6 Top each serving (2 waffle squares) with 4\u00bd teaspoons honey mixture and 1 tablespoon granola.\n\n1 Serving: Calories 350; Total Fat 12g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 55mg; Sodium 330mg; Total Carbohydrate 52g (Dietary Fiber 3g, Sugars 25g); Protein 10g Exchanges: 1\u00bd Starch, 2 Other Carbohydrate, \u00bd High-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 3\u00bd\n\n## Apricot Stuffed French Toast Calorie Smart\n\nPrep 20 min Total 1 hr 15 min \u25cf 6 servings\n\n  * 1 loaf (\u00bd-lb size) or \u00bd loaf (1-lb size) day-old French bread\n  * 3 oz (from 8-oz package) cream cheese, softened\n  * 3 tablespoons apricot preserves\n  * \u00bc teaspoon grated lemon peel\n  * 3 eggs\n  * \u00be cup half-and-half or milk\n  * 2 tablespoons granulated sugar\n  * 1 teaspoon vanilla\n  * \u215b teaspoon salt\n  * \u215b teaspoon ground nutmeg, if desired\n  * 2 tablespoons butter, softened\n  * Powdered sugar, if desired\n\n1 Spray 13x9-inch pan with cooking spray. Cut bread crosswise into 12 (1-inch) slices. Cut a horizontal slit in the side of each bread slice, cutting to but not through the other side.\n\n2 In medium bowl, beat cream cheese, preserves and lemon peel with electric mixer on medium speed about 1 minute or until well mixed. Spread 2 teaspoons cream cheese mixture into slit in each bread slice. Place stuffed bread slices in pan.\n\n3 In medium bowl, beat eggs, half-and-half, granulated sugar, vanilla, salt and nutmeg with fork or whisk until mixed. Pour egg mixture over bread slices in pan; turn slices carefully to coat. Cover and refrigerate at least 30 minutes but no longer than 24 hours.\n\n4 Heat oven to 425\u00b0F. Uncover French toast. Bake 20 to 25 minutes or until golden brown. Top with butter and sprinkle with powdered sugar.\n\n1 Serving (2 Slices): Calories 310; Total Fat 16g (Saturated Fat 8g, Trans Fat 1g); Cholesterol 145mg; Sodium 380mg; Total Carbohydrate 32g (Dietary Fiber 1g, Sugars 11g); Protein 9g Exchanges: 1 Starch, 1 Other Carbohydrate, 1 High-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 2\n\n# Heirloom Recipe and New Twist\n\nFrench toast has been a breakfast favorite for generations, and this recipe has been in our collection for many years. Although there are subtle differences in recipes, this classic fried bread dish is typically crispy on the outside and tender on the inside\u2014and is made and cooked right before serving. The secrets to success are beating the eggs well and having the griddle at the correct temperature. The new twist is ideal for make-ahead occasions. It has wonderful flavor and looks impressive.\n\nHeirloom\n\n## French Toast Calorie Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 8 slices\n\n  * 3 eggs\n  * \u00be cup milk\n  * 1 tablespoon sugar\n  * \u00bc teaspoon vanilla\n  * \u215b teaspoon salt\n  * 8 slices firm-textured sandwich bread, Texas toast or 1-inch-thick French bread\n\n1 In medium bowl, beat eggs, milk, sugar, vanilla and salt with whisk until well mixed. Pour into shallow bowl.\n\n2 Heat griddle or skillet over medium-high heat (375\u00b0F). (To test griddle, sprinkle with a few drops of water. If bubbles jump around, heat is just right.) Grease griddle with vegetable oil if necessary (or spray with cooking spray before heating).\n\n3 Dip bread into egg mixture. Place on hot griddle. Cook about 4 minutes on each side or until golden brown.\n\n1 Slice: Calories 230; Total Fat 3.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 80mg; Sodium 490mg; Total Carbohydrate 39g (Dietary Fiber 1g, Sugars 8g); Protein 10g Exchanges: 2 Starch, \u00bd Other Carbohydrate, \u00bd Medium-Fat Meat Carbohydrate Choices: 2\u00bd\n\nLighter Directions For 95 calories and 2 grams of fat per serving, use 1 whole egg and 2 egg whites for the 3 eggs, use \u2154 cup fat-free (skim) milk and increase vanilla to \u00bd teaspoon.\n\nOven French Toast Heat oven to 450\u00b0F. Generously butter 15x10x1-inch pan. Heat pan in oven 1 minute; remove from oven. Arrange dipped bread in hot pan. Drizzle any remaining egg mixture over bread. Bake 5 to 8 minutes or until bottoms are golden brown; turn bread. Bake 2 to 4 minutes longer or until golden brown.\n\nFrench Toast Sticks Use 8 slices firm-textured whole-grain sandwich bread. Cut each piece into 3 strips. Dip bread strips into batter. Cook on griddle about 4 minutes on each side or until golden brown. Serve with Spicy Cider Syrup, below, or real maple syrup for dipping.\n\n## Spicy Cider Syrup\n\nIn 2-quart saucepan, mix 1 cup sugar, 3 tablespoons all-purpose flour, \u00bc teaspoon each ground cinnamon and nutmeg, 2 cups apple cider, and 2 tablespoons lemon juice. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute; remove from heat. Stir in 4 tablespoons butter, cut into several pieces, until melted.\n\nFrench Toast Sticks\nNew Twist\n\n## Upside-Down Banana-Walnut French Toast\n\nPrep 15 min Total 2 hr 10 min \u25cf 10 servings\n\n  * 1\u00bd cups packed brown sugar\n  * \u00bd cup butter, melted\n  * \u00bc cup light corn syrup\n  * \u00bd cup chopped walnuts\n  * 3 medium bananas, sliced\n  * 1 loaf (about 1 lb) sliced unfrosted firm cinnamon bread\n  * 6 eggs\n  * 1\u00bd cups milk\n  * 1 teaspoon vanilla\n  * Powdered sugar, if desired\n\n1 Spray 13x9-inch (3-quart) glass baking dish with cooking spray. In large bowl, mix brown sugar, butter, corn syrup and walnuts. Gently stir in bananas. Spoon mixture into baking dish.\n\n2 Reserve ends of bread loaf for another use; they do not soak up egg mixture very well. Arrange 2 layers of bread on banana mixture, tearing bread to fit if needed.\n\n3 In medium bowl, beat eggs, milk and vanilla with whisk until well blended. Pour over bread. Cover tightly and refrigerate at least 1 hour but no longer than 24 hours.\n\n4 Heat oven to 325\u00b0F. Uncover baking dish; bake 45 to 55 minutes or until knife inserted in center comes out clean. Serve portions upside down, spooning sauce from bottom of dish over each serving. Sprinkle with powdered sugar.\n\n1 Serving: Calories 490; Total Fat 19g (Saturated Fat 7g, Trans Fat 1g); Cholesterol 155mg; Sodium 380mg; Total Carbohydrate 72g (Dietary Fiber 2g, Sugars 42g); Protein 10g Exchanges: 3 Starch, 2 Other Carbohydrate, 3\u00bd Fat Carbohydrate Choices: 5\n\n## Triple-Berry Oatmeal-Flax Muesli Calorie Smart\n\nPrep 10 min Total 45 min \u25cf 6 servings\n\n  * 2\u00be cups old-fashioned oats or rolled barley\n  * \u00bd cup sliced almonds\n  * 2 containers (6 oz each) banana cr\u00e8me or French vanilla fat-free yogurt\n  * 1\u00bd cups fat-free (skim) milk\n  * \u00bc cup ground flaxseed or flaxseed meal\n  * \u00bd cup fresh blueberries\n  * \u00bd cup fresh raspberries\n  * \u00bd cup sliced fresh strawberries\n\n1 Heat oven to 350\u00b0F. In ungreased 15x10x1-inch pan, spread oats and almonds.\n\n2 Bake 18 to 20 minutes, stirring occasionally, until light golden brown. Cool 15 minutes.\n\n3 In large bowl, mix yogurt and milk until well blended. Stir in flaxseed and toasted oats and almonds. Top each serving with berries.\n\n1 Serving: Calories 290; Total Fat 8g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 55mg; Total Carbohydrate 42g (Dietary Fiber 7g, Sugars 11g); Protein 11g Exchanges: 2\u00bd Starch, \u00bd Skim Milk, 1 Fat Carbohydrate Choices: 3\n\n## Is Muesli Really Granola?\n\nThink of muesli as granola's fresher cousin. While both foods contain oats (traditionally), fruit, seeds and nuts, they are not the same food. Muesli combines toasted grains with other fresh ingredients. Granola is baked. Muesli is usually less sweet\u2014not adding any extra sugar or oil. Granola adds sweeteners and oil to make it more clumpy\u2014perfect for eating out of your hand, anytime. Muesli is a loose mixture\u2014meant for eating with milk, yogurt or water, either hot or cold.\n\n## Make-Ahead Granola\n\nPrep 10 min Total 1 hr \u25cf 8 servings\n\n  * 3 cups old-fashioned or quick-cooking oats\n  * 1 cup whole almonds\n  * 1 teaspoon ground cinnamon\n  * \u2153 cup real maple syrup or honey\n  * \u00bd cup dried cherries\n  * \u00bd cup dried blueberries\n\n1 Heat oven to 300\u00b0F. In ungreased 15x10x1-inch pan, spread oats and almonds. Sprinkle evenly with cinnamon; drizzle with syrup. Toss to coat; spread mixture evenly in pan.\n\n2 Bake 15 minutes. Remove from oven. Stir well to mix so granola toasts evenly. Spread mixture evenly in pan. Bake 15 minutes longer or until mixture is dry and lightly toasted. Cool completely.\n\n3 Place granola in container with tight-fitting lid or resealable food-storage plastic bag. Add dried fruit; stir to distribute. Cover tightly and store at room temperature up to 1 month.\n\n1 Serving (\u00bd Cup): Calories 330; Total Fat 11g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 50g (Dietary Fiber 7g, Sugars 22g); Protein 8g Exchanges: 2 Starch, 1\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\n\n## Ways to Use Granola\n\n  1. 1. **Granola Cereal:** Pour soymilk or almond milk over granola and top with sliced fresh strawberries.\n  2. 2. **Granola Pancakes:** Make Pancake batter, page 101, as directed. Stir in \u00be cup granola before spooning pancakes onto griddle.\n  3. 3. **Granola Cookies:** Make Oatmeal-Raisin Cookies, page 522. Substitute 1 cup granola for the raisins, nuts and chocolate. Bake as directed.\n  4. 4. **Ice Cream and Granola:** For each serving, spoon ice cream or frozen yogurt (any flavor) into bowl. Sprinkle with 2 to 4 tablespoons granola.\n  5. 5. **Yogurt Parfaits:** In parfait glasses or dessert dishes, layer vanilla yogurt with granola and sliced bananas.\n  6. 6. **Granola Bars:** In medium bowl, mix 1 cup all-purpose or whole wheat flour, 1 teaspoon baking powder, 1 cup packed brown sugar, \u2153 cup vegetable oil and 2 eggs until well blended. Stir in 2 cups granola. Spread mixture in lightly greased 9-inch square pan. Bake at 350\u00b0F for 30 minutes or until set. Cool and cut into squares.\n  7. 7. **Granola Fruit Salad:** In medium bowl, mix 1 cup sliced strawberries, 1 cup raspberries, 1 sliced banana, 1 or 2 sectioned oranges and 1 cup granola. Toss with \u00bc cup honey-mustard salad dressing.\n\nGranola Bars, Granola\n\nBaked Apple Oatmeal\n\n## Baked Apple Oatmeal Calorie Smart\n\nSteel-cut oats, also known as Scottish or Irish oats, are kernels of the oat sliced lengthwise.\n\nPrep 10 min Total 1 hr \u25cf 8 servings\n\n  * 2\u2154 cups old-fashioned or steel-cut oats\n  * \u00bd cup raisins, sweetened dried cranberries or dried cherries\n  * \u2153 cup packed brown sugar\n  * 1 teaspoon ground cinnamon\n  * \u00bc teaspoon salt\n  * 4 cups milk\n  * 2 medium apples or pears, chopped (2 cups)\n  * \u00bd cup chopped walnuts, toasted (page 23) if desired\n  * Additional milk, if desired\n\n1 Heat oven to 350\u00b0F. In ungreased 2-quart casserole, mix oats, raisins, brown sugar, cinnamon, salt, 4 cups milk and the apples.\n\n2 Bake uncovered 40 to 50 minutes or until most of the milk is absorbed (a small amount of milk will remain in the center); do not overbake (oats will look chewy). Sprinkle walnuts over top. Serve with additional milk.\n\n1 Serving: Calories 300; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 130mg; Total Carbohydrate 45g (Dietary Fiber 5g, Sugars 24g); Protein 10g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, \u00bd Low-Fat Milk, 1 Fat Carbohydrate Choices: 3\n\n## Steel-Cut Oatmeal\n\nFor about 4 servings of oatmeal, stir 1 cup steel-cut oats into 4 cups boiling water or milk. Reduce heat to low and cook uncovered for 20 to 30 minutes or until thickened and oats are tender, stirring occasionally. For firmer oats, use a little less water.\n\n## Breakfast Quinoa Calorie Smart \u2022 Fast \u2022 Easy\n\nQuinoa makes a quick and hearty base for this new spin on hot cereal. Experiment with your favorite dried or fresh fruits and nuts. Use more liquid for a creamier cereal, or serve additional milk or cream on the side.\n\nPrep 5 min Total 20 min \u25cf 4 servings\n\n  * 1 cup uncooked quinoa\n  * 2 cups water, milk or soymilk\n  * \u00bc teaspoon ground cinnamon\n  * 1 cup fresh or frozen (thawed) blueberries\n  * \u00bc cup pecans or walnuts, toasted (page 23) if desired\n  * Real maple syrup, if desired\n\n1 Rinse quinoa thoroughly by placing in fine-mesh strainer and holding under cold running water until water runs clear; drain well.\n\n2 In 1-quart saucepan, heat water and quinoa to boiling. Reduce heat to low; cover and simmer 15 minutes or until liquid is absorbed (mixture will still be moist).\n\n3 Stir in cinnamon. Spoon quinoa into bowls. Top with blueberries and pecans. Sweeten to taste with a drizzle of syrup.\n\n1 Serving (\u00bd Cup): Calories 230; Total Fat 7g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 34g (Dietary Fiber 4g, Sugars 7g); Protein 7g Exchanges: 2\u00bd Starch, 1 Fat Carbohydrate Choices: 2\n\n## What Is Quinoa?\n\nTiny and bead shaped, quinoa cooks quickly and expands to four times its volume. The flavor is delicate and similar to couscous. Look for this grain at most supermarkets and health food stores.\n\n## Cheese Grits Calorie Smart\n\nCoarsely ground dried corn kernels called hominy grits are a treasured tradition in many families. A bowl of steaming-hot grits, topped with a pat of butter, is a breakfast favorite. The cheese can be left out if you prefer.\n\nPrep 20 min Total 1 hr 10 min \u25cf 8 servings\n\n  * 2 cups milk\n  * 2 cups water\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cup uncooked quick-cooking corn grits\n  * 1\u00bd cups shredded Cheddar cheese (6 oz)\n  * 2 medium green onions, sliced (2 tablespoons)\n  * 2 eggs, slightly beaten\n  * 1 tablespoon butter, cut into small pieces\n  * \u00bc teaspoon paprika\n\n1 Heat oven to 350\u00b0F. Spray 1\u00bd-quart casserole with cooking spray.\n\n2 In 2-quart saucepan, heat milk, water, salt and pepper to boiling. Gradually add grits, stirring constantly; reduce heat. Simmer uncovered about 5 minutes, stirring frequently, until thickened. Stir in cheese and onions.\n\n3 Stir 1 cup of the grits mixture into eggs, then stir back into remaining grits in saucepan. Pour into casserole. Sprinkle with butter and paprika.\n\n4 Bake uncovered 35 to 40 minutes or until set. Let stand 10 minutes before serving.\n\n1 Serving: Calories 220; Total Fat 11g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 85mg; Sodium 340mg; Total Carbohydrate 19g (Dietary Fiber 0g, Sugars 4g); Protein 11g Exchanges: 1 Starch, 1 High-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\n\nLighter Directions For 1 gram of fat and 125 calories per serving, use fat-free (skim) milk and 1 cup reduced-fat Cheddar cheese; substitute \u00bd cup fat-free egg product for the eggs. Omit butter.\n\nBacon and Cheese Grits Stir in 4 slices crumbled cooked bacon just before pouring into casserole. Bake as directed.\n\n## Cranberry\u2013Wild Rice Porridge\n\nIn this recipe, wild rice turns out a toothsome and hearty bowl of \"porridge\" that is a delicious breakfast or brunch treat. Be sure to check out the savory bacon variation, too.\n\nPrep 15 min Total 1 hr 15 min \u25cf 4 servings\n\n  * 4 cups water\n  * 1 cup uncooked wild rice, rinsed, drained\n  * \u00bd cup dried cherries or golden raisins\n  * \u00bd cup dried blueberries\n  * \u2153 cup real maple syrup or honey\n  * 1\u00bc cups whipping cream\n  * \u00bd cup chopped pecans, toasted (page 23)\n\n1 In 2-quart saucepan, heat water and wild rice to boiling; reduce heat to low. Cover and simmer 45 to 55 minutes or until kernels are opened and tender. Drain well in colander or strainer.\n\n2 Return rice to saucepan; add cherries, blueberries and syrup. Cook over medium-high heat about 5 minutes, stirring frequently, until hot. Cook 3 minutes longer, stirring frequently. Stir in whipping cream. Heat to boiling; reduce heat to medium. Simmer 4 to 6 minutes or until mixture is slightly thickened and creamy. Let stand uncovered 5 minutes. Sprinkle with pecans.\n\n1 Serving (About 1 Cup): Calories 740; Total Fat 38g (Saturated Fat 18g, Trans Fat 1g); Cholesterol 100mg; Sodium 45mg; Total Carbohydrate 89g (Dietary Fiber 7g, Sugars 46g); Protein 10g Exchanges: 3 Starch, 1 Fruit, 2 Other Carbohydrate, 7\u00bd Fat Carbohydrate Choices: 6\n\nBacon and Pecan Wild Rice Porridge Make recipe as directed\u2014except omit dried cherries and blueberries. While porridge simmers in Step 2, cook 4 slices bacon until crisp; crumble. Top each serving with crumbled bacon and toasted pecans. For a twist, top with a fried or poached egg.\n\n# Chapter 5: Salads & Salad Dressings\n\nHeirloom Tomato Caprese Salad\n\n## Salad Basics\n\nAbout Greens\n\nBold Greens and Cabbage\n\nMild Greens\n\n## Features\n\nBetty's Staples: Canned Tuna or Salmon\n\nBetty's Staples: Seeds and Nuts\n\nHeirloom Recipe and New Twist\n\nLearn to Blanch\n\nMake-Ahead: Fresh Herb Vinaigrette\n\nOther Crispy Toppers\n\n## Lettuce\/Greens Salads\n\nAsian Salad with Wonton Crisps Fast\n\nBeets and Baby Greens Salad\n\nCaesar Salad Calorie Smart \u2022 Fast\n\nCherry-Walnut Kale Salad Calorie Smart \u2022 Fast\n\nCreamy Coleslaw Calorie Smart\n\nFennel-Asparagus Seven-Layer Salad\n\nGreek Salad Calorie Smart \u2022 Fast\n\nGreens with Prosciutto and Gorgonzola Calorie Smart \u2022 Fast\n\nGrilled Ceasar Salad\n\nKale Caesar Salad\n\nMandarin Salad with Sugared Almonds Calorie Smart \u2022 Fast\n\nPanzanella Salad Calorie Smart\n\nSanta Fe Salad with Tortilla Straws\n\nSpinach-Bacon Salad with Hard-Cooked Eggs Calorie Smart \u2022 Fast\n\nSunny Sunflower Salad\n\nSweet-and-Sour Coleslaw Calorie Smart\n\nTropical Creamy Coleslaw\n\n## Fruit and Vegetable Salads\n\nAsparagus-Almond Salad\n\nAsparagus-Cashew Salad\n\nAsparagus-Orange Salad\n\nAsparagus-Strawberry Salad Calorie Smart \u2022 Fast\n\nCranberry-Orange Mold Calorie Smart \u2022 Easy\n\nCreamy Potato Salad\n\nDilled Potato Salad\n\nGreen and Yellow Bean Salad Calorie Smart\n\nHeirloom Tomato Caprese Salad Calorie Smart \u2022 Fast \u2022 Easy\n\nHot German Potato Salad Calorie Smart\n\nMango-Avocado Salad\n\nRatatouille Salad Calorie Smart\n\nRoasted Beet Salad\n\nRoasted Potato Salad\n\nSeven-Layer Salad\n\n## Main Dish Salads\n\nBlue Cheese Chicken Salad\n\nChicken and Bacon Chopped Salad\n\nChicken Caesar Salad\n\nChicken Chopped Salad\n\nChopped Summer Salad\n\nCinnamon-Maple Glazed Salmon Salad\n\nCobb Salad\n\nCrunchy Chicken-Mandarin Salad\n\nItalian Chopped Salad Fast\n\nMediterranean Tuna Salad\n\nNi\u00e7oise Salmon Salad\n\nNorthern Italian White Bean Salad Calorie Smart\n\nRatatouille Salad Calorie Smart\n\nSalad Lyonnaise Calorie Smart \u2022 Fast\n\nShrimp Caesar Salad\n\nTurkey-Cheese Chopped Salad\n\n## Pasta Salads\n\nChicken-Thyme Penne Salad\n\nSesame Noodle Salad Calorie Smart \u2022 Fast\n\nTurkey-Basil Penne Salad\n\n## Salad Toppings\n\nBacon Crumbles\n\nCandied Nuts\n\nCroutons Calorie Smart \u2022 Easy\n\nEdamame\n\nGarlic Croutons\n\nHerbed Croutons\n\nPumpernickel or Rye Croutons\n\nSweet Peppery Pecans\n\nTortilla Straws\n\n## Dressing Basics\n\nAbout Oil\n\nAbout Vinegar\n\n## Creamy Dressings\n\nBlue Cheese Dressing Calorie Smart \u2022 Easy\n\nButtermilk Ranch Dressing Calorie Smart \u2022 Easy\n\nButtermilk Ranch-Parmesan Dressing\n\nCreamy Fresh Herb Vinaigrette\n\nCreamy Italian Dressing\n\nRussian Dressing\n\nThousand Island Dressing Calorie Smart \u2022 Fast\n\n## Oil Dressings\n\nAsian Dressing\n\nBlue Cheese Herb Vinaigrette\n\nFrench Dressing\n\nFresh Herb Vinaigrette Calorie Smart \u2022 Fast\n\nHoney Miso Dressing\n\nHoney-Mustard Dressing Calorie Smart \u2022 Fast \u2022 Easy\n\nHoney\u2013Poppy Seed Dressing\n\nItalian Dressing Calorie Smart \u2022 Fast\n\nMiso Dressing Calorie Smart \u2022 Fast\n\nRaspberry Vinaigrette Calorie Smart \u2022 Fast \u2022 Easy\n\nSriracha Miso Dressing\n\n## Bonus Recipes\n\nBaked Potatoes with Tuna Topping\n\nCheddar Tuna Melts\n\nHerbed Marinated Chicken\n\nItalian Beef Sandwiches\n\nTuna Quesadillas\n\nTuna-Lettuce Wraps\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Salad Basics\n\nWe love our salads! From mixed greens and fresh fruit to tender pasta and crisp vegetables, salad ingredients span a delicious range of flavors and textures. In this collection of recipes, there are many side salads that will complement other recipes on your menu, as well as salads that can stand alone as a main course.\n\nLooking for ways to break out of your salad rut? Explore some of the newer greens that are available. From mild greens like mesclun, leaf or Bibb lettuce to bolder varieties like kale or radicchio, the many choices let you vary the taste, texture and appearance of salads easily!\n\n## About Greens\n\nWithin the two categories of lettuce (delicate and assertive-flavored), there are four lettuce types: crisphead (iceberg), butterhead (Bibb or Boston), loose leaf (oak leaf) and long leaf (romaine).\n\n### Selecting and Handling Greens\n\n  * Choose the freshest greens possible\u2014there should be no bruising, discoloration or wilting.\n  * Keep greens chilled\u2014store unwashed in the refrigerator, wrapped in a paper towel and placed in a plastic bag or tightly covered container for up to 5 days.\n  * Wash greens well just before using\u2014rinse well in cold water, removing any dirt.\n  * Dry greens well\u2014use a salad spinner or pat dry with paper towels.\n  * Trim greens\u2014remove stems and roots as necessary and discard. Serve small leaves whole and cut up larger pieces.\n\n### Drying Fresh Greens\n\nSalad Spinner: Place washed leaves in basket of spinner; cover spinner and spin until lettuce is dry.\n\nTowel Dry: Place washed leaves on clean kitchen towel or in several layers of paper towels and pat dry.\n\n#### Mild Greens\n\nButterhead Lettuce (Bibb, Boston): Small rounded heads of soft, tender, buttery leaves; delicate, mild flavor.\n\nCrisphead Lettuce (Iceberg): Solid, compact heads with leaves ranging in color from medium green outer leaves to pale green inner ones; very crisp, mild flavor.\n\nLong Leaf Lettuce (Romaine, Cos): Narrow, elongated, dark green, crisp leaves with tips sometimes tinged with red. The broad white center rib is especially crunchy.\n\nLoose Leaf Lettuce (Green, Red, Oak): Tender, crisp leaves in loose heads. Mildly flavored but stronger than iceberg lettuce.\n\nM\u00e2che (Lamb's Lettuce, Corn Salad): Spoon-shaped medium to dark green leaves with velvety texture; mild, subtly sweet and nutty.\n\nMesclun (Field or Wild Greens): A mixture of young, tender, small greens often including arugula, chervil, chickweed, dandelion, fris\u00e9e, mizuna and oak leaf lettuce.\n\nMixed Salad Greens (Prepackaged): These prewashed greens come in many varieties, some with dressing, croutons or cheese.\n\n#### Bold Greens and Cabbage\n\nArugula (Rocket): Small, slender, dark green leaves similar to radish leaves; slightly bitter, peppery mustard flavor. Pick smaller leaves for milder flavor.\n\nBelgian Endive (French): Narrow, cupped, cream-colored leaves tinged with green or red; slightly bitter flavor.\n\nCabbage: Comes in several distinctly flavored varieties. Green and red cabbage are most common; look for compact heads of waxy, tightly wrapped leaves. Savoy cabbage has crinkled leaves and Chinese (or napa) cabbage has long, crisp leaves.\n\nCurly Endive: Frilly, narrow, slightly prickly leaves; slightly bitter taste.\n\nEscarole: Broad, wavy, medium-green leaves; slightly bitter flavor but milder than Belgian or curly endive.\n\nFris\u00e9e: Slender, curly leaves ranging in color from yellow-white to yellow-green; slightly bitter flavor.\n\nGreens (Beet, Chard, Collard, Dandelion, Mustard): All have a strong, sharp flavor. Young greens are milder and more tender for tossed salads; older greens are more bitter and should be cooked for best flavor.\n\nKale: Firm, dark green leaves with frilly edges tinged with shades of blue and purple; mild cabbage taste. Pick young, small leaves for best flavor.\n\nRadicchio: Looks like a small, loose-leaf cabbage with smooth, tender leaves; slightly bitter flavor. The two most common radicchios in the United States are a ruby red variety with broad white veins and one with leaves speckled in shades of pink, red and green.\n\nSorrel (Sour Grass): Resembles spinach, but with smaller leaves; sharp, lemony flavor.\n\nSpinach: Smooth, tapered, dark green leaves, sometimes with crumpled edges; slightly bitter flavor. Baby spinach leaves are smaller and milder in flavor than regular spinach leaves.\n\nWatercress: Small, crisp, dark green, coin-size leaves; strong, peppery flavor.\n\nCaesar Salad\n\n## Caesar Salad Calorie Smart \u2022 Fast\n\nThis classic salad is the perfect go-to recipe to accompany most any meal. It's not hard to whip up, but knowing how to make a great dressing and crisp croutons turns a ho-hum Caesar into a specialty salad that folks will ask you to make again and again.\n\nPREP 15 min TOTAL 15 min \u25cf 6 servings\n\n  * 1 clove garlic, cut in half\n  * 8 anchovy fillets, cut up*\n  * \u2153 cup olive oil\n  * 3 tablespoons fresh lemon juice\n  * 1 pasteurized egg yolk**\n  * 1 teaspoon Worcestershire sauce\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon ground mustard\n  * Freshly ground pepper to taste\n  * 1 large or 2 small bunches romaine lettuce, torn into bite-size pieces (10 cups)\n  * 1 cup Garlic Croutons (page 117)\n  * \u2153 cup freshly grated Parmesan cheese\n\n1 Rub large salad bowl with cut clove of garlic for extra flavor, if desired. Chop garlic and place in bowl.\n\n2 In salad bowl, mix anchovies, oil, lemon juice, egg yolk, Worcestershire sauce, salt, mustard and pepper with whisk.\n\n3 Add lettuce; toss until coated. Sprinkle with croutons and cheese; toss. Serve immediately.\n\n*2 teaspoons anchovy paste can be substituted for the anchovies. If you prefer, anchovies can be omitted.\n\n**Use only pasteurized eggs in this recipe to prevent the risk of food poisoning that can result from eating (raw or undercooked) eggs.\n\n1 Serving (1\u00be Cups): Calories 190; Total Fat 16g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 10mg; Sodium 500mg; Total Carbohydrate 7g (Dietary Fiber 2g, Sugars 2g); Protein 6g Exchanges: 1\u00bd Vegetable, \u00bd High-Fat Meat, 2\u00bd Fat Carbohydrate Choices: \u00bd\n\nChicken Caesar Salad Broil 6 boneless skinless chicken breasts (page 351); slice diagonally and arrange on salads. Serve chicken warm or chilled.\n\nGrilled Caesar Salad Heat gas or charcoal grill. Make dressing as directed in small bowl. Omit halved garlic clove; do not tear lettuce. Carefully cut head of romaine lengthwise in half through stem. Brush each cut side with 1 tablespoon of dressing. Place on grill cut side down; cook 2 to 4 minutes over medium-high heat or until lightly browned. Brush with dressing; turn and cook 2 to 4 minutes or until lightly browned. Place romaine on serving platter. Drizzle with desired amount of remaining dressing; sprinkle with croutons and cheese.\n\nKale Caesar Salad Substitute baby kale leaves for romaine lettuce.\n\nShrimp Caesar Salad Broil or grill 1 pound deveined peeled large shrimp (page 370); arrange on salads. Serve shrimp warm or chilled.\n\n### Making Caesar Salad\n\nRub bowl with cut side of garlic.\n\nMix ingredients with whisk.\n\nToss lettuce until coated with dressing.\n\n## Croutons Calorie Smart \u2022 EASY\n\nHomemade croutons taste so much better than purchased ones and are super easy to make. We've suggested cutting the bread into \u00bd-inch cubes, but by all means, if you love big croutons, cut them larger (they will take longer to get golden and crisp).\n\nPREP 10 min TOTAL 55 min \u25cf 16 servings\n\n  * 10 slices (\u00bd inch thick) firm white or whole-grain bread\n  * \u00bd cup butter, melted*\n\n1 Heat oven to 300\u00b0F. Cut bread slices into \u00bd-inch cubes. In ungreased 15x10x1-inch pan, spread bread cubes in single layer. Drizzle butter evenly over bread cubes; toss until coated.\n\n2 Bake 30 to 35 minutes, stirring occasionally, until golden brown, dry and crisp. Cool completely. Store tightly covered at room temperature up to 2 days.\n\n*Olive oil can be substituted for the melted butter. After drizzling bread cubes with oil, sprinkle evenly with \u00bd teaspoon coarse (kosher or sea) salt; toss until coated.\n\n1 Serving: Calories 90; Total Fat 6g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 15mg; Sodium 150mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: \u00bd Starch, 1 Fat Carbohydrate Choices: \u00bd\n\nGarlic Croutons Add 3 cloves garlic, very finely chopped, or \u00bc teaspoon garlic powder to butter before drizzling over bread cubes.\n\nHerbed Croutons Drizzle bread cubes with olive oil instead of butter. Sprinkle with 2 teaspoons Italian seasoning and \u00bd teaspoon salt; toss until coated.\n\nPumpernickel or Rye Croutons Substitute pumpernickel or rye bread. Bake as directed or until pumpernickel cubes are dry and crisp or rye cubes are light golden brown, dry and crisp.\n\n## Greek Salad Calorie Smart \u2022 Fast\n\nPREP 20 min TOTAL 20 min \u25cf 8 servings\n\nSalad\n\n  * 5 cups fresh baby spinach leaves\n  * 1 head Boston lettuce, torn into bite-size pieces (4 cups)\n  * 1 cup crumbled feta cheese (4 oz)\n  * 24 pitted kalamata or Greek olives*\n  * 4 medium green onions, sliced (\u00bc cup)**\n  * 3 medium tomatoes, cut into wedges\n  * 1 medium cucumber, sliced\n\nLemon Dressing\n\n  * \u00bc cup olive oil\n  * 2 tablespoons fresh lemon juice\n  * 1\u00bd teaspoons Dijon mustard\n  * \u00bd teaspoon sugar\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n\nIn large bowl, mix salad ingredients. In tightly covered container, shake dressing ingredients. Pour dressing over salad; toss until coated. Serve immediately.\n\n*Extra-large pitted ripe olives can be substituted for the kalamata or Greek olives.\n\n**Chopped red onion can be substituted for the green onions.\n\n1 Serving (1\u00be Cups): Calories 140; Total Fat 11g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 680mg; Total Carbohydrate 6g (Dietary Fiber 2g, Sugars 2g); Protein 4g Exchanges: 1 Vegetable, 2\u00bd Fat Carbohydrate Choices: \u00bd\n\n## Other Crispy Toppers\n\nIdeas for other toppings that add crunch and flavor to salads can be as simple as chopped cashews or peanuts, thin or thick chow mein noodles, french-fried crispy onions or shoestring potatoes, or opt for shredded carrots, radishes or cucumber. Here are a couple of other ideas to try:\n\n**Tortilla Straws:** Brush flour tortillas with vegetable oil and sprinkle each with \u00bc teaspoon each chili powder and salt. Cut into \u00bc-inch-wide strips. Place on cookie sheet and bake at 350\u00b0F for 10 to 12 minutes or until crisp.\n\n**Bacon Crumbles:** Chop bacon and cook in skillet over medium heat until crisp. Drain on paper towels.\n\n**Edamame:** Cook frozen shelled edamame until crisp-tender as directed on package. Chill in cold water and drain.\n\n**Sweet Peppery Pecans:** Cook \u00bc cup pecan halves and 1 tablespoon brown sugar in skillet over low heat until sugar begins to melt; sprinkle with coarsely ground pepper. Cook and stir until sugar is melted. Remove pecans to plate to cool before using.\n\n### Making Tortilla Straws\n\nCut tortilla in half; cut each half crosswise into \u00bc-inch strips.\n\nPlace on cookie sheet; bake until crisp and golden brown.\n\nGreens with Prosciutto and Gorgonzola\n\n## Greens with Prosciutto and Gorgonzola Calorie Smart \u2022 Fast\n\nArugula is a bitter, aromatic salad green with a peppery mustard flavor that looks like radish leaves and is sold in bunches with the roots attached. It's very perishable, so wrap it tightly in a resealable food-storage plastic bag and refrigerate no longer than 2 days. Wash it thoroughly before using to remove any grit in the leaves.\n\nPREP 25 min TOTAL 25 min \u25cf 6 servings\n\n  * 4 oz prosciutto (8 to 10 slices), cut into \u215b-inch strips*\n  * 4 cups bite-size pieces mixed salad greens\n  * 1 cup bite-size pieces arugula\n  * 1 small head radicchio, cut into thin strips (1 cup)\n  * \u2153 cup Red Wine Vinaigrette Dressing (page 119)\n  * \u00bd cup crumbled Gorgonzola or feta cheese (2 oz)\n  * 6 pepperoncini peppers (bottled Italian peppers), drained, sliced if desired\n\n1 In 10-inch nonstick skillet, cook prosciutto over medium-high heat 5 minutes, stirring occasionally. Reduce heat to medium. Cook 5 to 10 minutes longer, stirring frequently, until prosciutto becomes mostly crisp. Drain on paper towels.\n\n2 In large bowl, toss mixed greens, arugula, radicchio and dressing. Sprinkle with prosciutto and cheese. Garnish with pepperoncini peppers.\n\n*Very thinly sliced deli ham can be substituted for the prosciutto. Pat ham dry with paper towels before cooking. The ham may not become as crisp and may remain in larger pieces. Another substitution is 6 slices bacon, crisply cooked and crumbled.\n\n1 Serving: Calories 110; Total Fat 8g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 20mg; Sodium 500mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 3g); Protein 5g Exchanges: 1 Vegetable, \u00bd High-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\n## Italian Chopped Salad Fast\n\nGenoa and cotto salamis are the best-known Italian salamis. Genoa is rich, fatty and studded with white peppercorns; cotto is studded with black peppercorns.\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\nSalad\n\n  * 6 cups chopped romaine lettuce\n  * 1 cup chopped fresh basil leaves\n  * 1 cup chopped cooked chicken\n  * 2 large tomatoes, chopped (2 cups)\n  * 2 medium cucumbers, chopped (1\u00bd cups)\n  * 1 package (3 oz) Italian salami, chopped\n  * 1 can (15 to 19 oz) cannellini beans, drained, rinsed\n\nBasil Vinaigrette Dressing\n\n  * \u2153 cup olive oil\n  * \u00bc cup red wine vinegar\n  * 2 tablespoons chopped fresh or 2 teaspoons dried basil leaves\n  * 1 teaspoon sugar\n  * \u00bc teaspoon salt\n\nIn large bowl, mix salad ingredients. In tightly covered container, shake dressing ingredients. Pour dressing over salad; toss until coated. Serve immediately.\n\n1 Serving: Calories 560; Total Fat 31g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 45mg; Sodium 740mg; Total Carbohydrate 41g (Dietary Fiber 11g, Sugars 7g); Protein 27g Exchanges: 2 Starch, 2 Vegetable, 2 Lean Meat, 4 Fat Carbohydrate Choices: 3\n\nChicken and Bacon Chopped Salad Substitute 4 slices crumbled cooked bacon for the salami and navy beans for the cannellini beans.\n\nTurkey-Cheese Chopped Salad Substitute turkey for the chicken and 1 cup shredded pepper Jack cheese for the salami. Use red kidney beans instead of cannellini.\n\n## Quick Chop\n\nTo chop fresh basil leaves quickly, lay the leaves on top of one another and then roll up lengthwise to chop them quickly.\n\nSpinach-Bacon Salad with Hard-Cooked Eggs\n\n## Spinach-Bacon Salad with Hard-Cooked Eggs Calorie Smart \u2022 Fast\n\nA little bit sweet, a little bit salty\u2014this classic salad is delicious. Look for bags of prewashed spinach to get this on the table faster.\n\nPREP 30 min TOTAL 30 min \u25cf 6 servings\n\n  * 4 slices bacon, cut into \u00bd-inch pieces\n  * 3 tablespoons vegetable oil\n  * 5 medium green onions, chopped (\u2153 cup)\n  * 2 teaspoons sugar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 tablespoons white or cider vinegar\n  * 8 oz fresh spinach leaves (9 cups)\n  * 2 Hard-Cooked Eggs (page 91), sliced\n\n1 In 10-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Remove bacon with slotted spoon; drain on paper towels. Drain all but 3 tablespoons bacon drippings from skillet (if fewer than 3 tablespoons remain, add enough vegetable oil to drippings to equal 3 tablespoons).\n\n2 Add oil, onions, sugar, salt and pepper to drippings in skillet. Cook over medium heat 2 to 3 minutes, stirring occasionally, until onions are slightly softened. Stir in vinegar.\n\n3 Place spinach in very large bowl. Pour warm dressing over spinach; toss until coated. Arrange egg slices on top. Crumble bacon and sprinkle on top. Serve immediately.\n\n1 Serving (1\u00bd Cups): Calories 130; Total Fat 11g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 340mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 2g); Protein 5g Exchanges: 1 Vegetable, \u00bd Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 0\n\nSalad Lyonnaise\n\n## Salad Lyonnaise Calorie Smart \u2022 Fast\n\nThis timeless salad from France is made with fresh fris\u00e9e or other crisp greens and always has a poached or fried egg on top.\n\nPREP 15 min TOTAL 15 min \u25cf 4 servings\n\nRed Wine Vinaigrette Dressing\n\n  * 3 tablespoons olive oil\n  * 1 tablespoon red wine vinegar\n  * 2 teaspoons Dijon mustard\n  * 1 teaspoon finely chopped shallot\n  * \u00bd teaspoon sugar\n  * \u215b teaspoon coarse ground black pepper\n\nSalad\n\n  * 6 cups torn fris\u00e9e or escarole\n  * 4 Poached or Fried Eggs (pages  and )\n  * 4 slices bacon, crisply cooked, crumbled\n  * \u00be cup Croutons (page 117)\n  * Additional coarse ground black pepper, if desired\n  * Paprika, if desired\n\n1 In small bowl, mix dressing ingredients with whisk until smooth.\n\n2 In large bowl, toss fris\u00e9e and dressing until coated. Divide among 4 shallow serving bowls or plates. Top each with egg, bacon and croutons. Sprinkle egg with pepper and paprika.\n\n1 Serving: Calories 230; Total Fat 20g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 195mg; Sodium 290mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 1g); Protein 10g Exchanges: 1 Vegetable, 1 Medium-Fat Meat, 3 Fat Carbohydrate Choices: 0\n\nCherry-Walnut Kale Salad\n\n## Cherry-Walnut Kale Salad Calorie Smart \u2022 Fast\n\nMassaging kale means to rub the leaves with a bit of oil, lemon juice and salt. This makes the fibrous green more tender and palatable. Wear plastic gloves to do this if you wish; or instead, toss the leaves with oil, lemon juice and salt and refrigerate overnight. If you like, substitute baby kale for part of the kale and add with the vinaigrette.\n\nPREP 20 min TOTAL 20 min \u25cf 4 servings\n\n  * 1 bunch (8 oz) fresh kale, ribs removed, leaves thinly sliced (6 cups)\n  * 3 tablespoons lemon juice\n  * 1 tablespoon olive oil\n  * \u00bd teaspoon salt\n  * 1 tablespoon white balsamic or white wine vinegar\n  * 1 tablespoon honey\n  * \u00bd teaspoon pepper\n  * 2 tablespoons dried tart cherries\n  * 2 tablespoons chopped walnuts, toasted (page 23), if desired\n  * 2 tablespoons crumbled blue cheese\n\n1 In large bowl, mix kale, lemon juice, 1 teaspoon of the oil and the salt. Massage kale with your hands 3 to 5 minutes or until it starts to wilt; set aside.\n\n2 In small bowl, beat vinegar, honey and pepper with whisk. Slowly pour in remaining 2 teaspoons oil while continuing to beat. Pour vinaigrette over kale mixture; toss to coat.\n\n3 Top individual servings with cherries, walnuts and cheese.\n\n1 Serving: Calories 120; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 380mg; Total Carbohydrate 16g (Dietary Fiber 1g, Sugars 9g); Protein 3g Exchanges: \u00bd Starch, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\nMandarin Salad with Sugared Almonds\n\n## Mandarin Salad with Sugared Almonds Calorie Smart \u2022 Fast\n\nPREP 30 min TOTAL 30 min \u25cf 6 servings\n\nSugared Almonds\n\n  * \u00bc cup sliced almonds\n  * 4 teaspoons sugar\n\nSweet-Sour Dressing\n\n  * \u00bc cup vegetable oil\n  * 2 tablespoons sugar\n  * 2 tablespoons white or cider vinegar\n  * 1 tablespoon chopped fresh parsley\n  * \u00bd teaspoon salt\n  * Dash pepper\n  * Dash red pepper sauce\n\nSalad\n\n  * 3 cups bite-size pieces mixed salad greens or iceberg lettuce\n  * 3 cups bite-size pieces romaine lettuce\n  * 2 medium stalks celery, chopped (1 cup)\n  * 2 medium green onions, thinly sliced (2 tablespoons)\n  * 1 can (11 oz) mandarin orange segments, drained\n\n1 In 1-quart saucepan, cook almonds and 4 teaspoons sugar over low heat about 10 minutes, stirring constantly, until sugar is melted and almonds are coated. Cool; break apart.\n\n2 Meanwhile, in tightly covered container, shake dressing ingredients. Refrigerate until serving time.\n\n3 In large bowl, mix salad ingredients. Shake dressing; pour over salad and toss until coated. Sprinkle with sugared almonds. Serve immediately.\n\n1 Serving (1\u2153 Cups): Calories 170; Total Fat 12g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 15g (Dietary Fiber 2g, Sugars 12g); Protein 2g Exchanges: \u00bd Fruit, 2 Vegetable, 2 Fat Carbohydrate Choices: 1\n\nCrunchy Chicken-Mandarin Salad Top each serving with a grilled boneless skinless chicken breast, sliced (warm or cold) and sprinkle with 2 tablespoons chow mein noodles.\n\n### Making Sugared Almonds\n\nCook almonds and sugar over low heat about 10 minutes, stirring constantly.\n\nCook until sugar is melted and almonds are coated; cool and break apart.\n\n## Roasted Beet Salad\n\nPREP 15 min TOTAL 1 hr 25 min \u25cf 4 servings\n\n  * 1\u00bd lb small beets (1\u00bd to 2 inches in diameter)\n  * 1 tablespoon olive or vegetable oil\n  * 4 cups bite-size pieces mixed salad greens\n  * 1 medium orange, peeled, sliced\n  * \u00bd cup walnut halves, toasted (page 23), coarsely chopped\n  * \u00bc cup crumbled ch\u00e8vre (goat) cheese (1 oz)\n  * \u00bd cup Fresh Herb Vinaigrette (page 136)\n\n1 Heat oven to 425\u00b0F. Remove greens from beets, leaving about \u00bd inch of stem. Wash beets well; leave whole with root ends attached. Place beets in ungreased 13x9-inch pan; drizzle with oil. Bake uncovered about 40 minutes or until tender.\n\n2 Remove skins from beets under running water. Let beets cool until easy to handle, about 30 minutes. Cut off root ends and cut beets into slices (cut slices in half if large).\n\n3 On 4 salad plates, arrange salad greens. Top with beets, orange slices, toasted walnuts and cheese. Serve with dressing.\n\n1 Serving: Calories 410; Total Fat 33g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 5mg; Sodium 440mg; Total Carbohydrate 19g (Dietary Fiber 6g, Sugars 11g); Protein 7g Exchanges: \u00bd Other Carbohydrate, 3 Vegetable, 6\u00bd Fat Carbohydrate Choices: 1\n\n## Betty's Staples Seeds and Nuts\n\nA touch of something crunchy in a salad can be a wonderful surprise. This can come from crisp vegetables or croutons (page 117), but nuts or seeds are a fun way to add both crunch and some protein. Walnuts, pecans, almonds, cashews, pistachios, sunflower seeds and pumpkin seeds (pepitas)\u2014any of these little treasures can be added plain, toasted (see page 23) or for a special treat, candied, which works well with pecan or walnut halves, whole or slivered almonds and whole cashews.\n\n**Candied Nuts:** In medium bowl, mix 1 cup nuts, 2 tablespoons corn syrup, 1 tablespoon sugar and \u00bc teaspoon salt (unless the nuts are salted). If you want a spicy mixture, include \u215b teaspoon ground red (cayenne) pepper too. Spread the nuts on a shallow baking pan lined with parchment paper and bake at 350\u00b0F for 10 to 15 minutes or until nuts are golden brown and sugar mixture is bubbly, stirring once or twice during baking. Cool completely and store tightly covered.\n\n**Sunny Sunflower Salad:** Toss kale leaves with torn butter lettuce leaves, small cauliflower florets, sliced roasted red bell pepper pieces (page 247), sliced yellow summer squash, toasted sunflower seeds and Honey-Mustard Dressing (page 137).\n\n**Asparagus-Almond Salad:** Blanch asparagus spears (see page 130). Arrange spears on individual salad plates. Drizzle with Fresh Herb Vinaigrette (page 136) and sprinkle with chopped toasted sliced almonds.\n\n**Beets and Baby Greens Salad:** In medium bowl, toss mixed baby greens with chopped cooked beets and crumbled goat cheese. Drizzle with vinaigrette and sprinkle with toasted walnuts. Toss lightly to mix.\n\n**Mango-Avocado Salad:** On individual salad plates, arrange sliced mango with sliced avocado and fresh raspberries. Sprinkle with toasted slivered almonds and drizzle with Raspberry Vinaigrette (page 137).\n\n**Chicken Chopped Salad:** In medium bowl, mix chopped romaine, chopped cooked chicken, crumbled bacon, chopped fresh tomato, chopped cauliflower and chopped pistachios. Drizzle with Italian Dressing (page 137) and toss to mix.\n\nBeets and Baby Greens Salad\n\n## Ratatouille Salad Calorie Smart\n\nPREP 30 min TOTAL 4 hr 40 min \u25cf 8 servings\n\n  * 1 cup water\n  * \u00be teaspoon salt\n  * 1 small eggplant (1 lb), cut into \u00bd-inch cubes\n  * \u2153 cup olive or vegetable oil\n  * 2 tablespoons lemon juice\n  * 2 teaspoons chopped fresh or \u00bd teaspoon dried basil leaves\n  * \u00bd teaspoon ground mustard\n  * \u215b teaspoon pepper\n  * 2 medium tomatoes, chopped (1\u00bd cups)\n  * 1 medium zucchini, thinly sliced (2 cups)\n  * 1 small onion, sliced, separated into rings\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * \u2153 cup chopped fresh parsley\n\n1 In 3-quart saucepan, heat water and \u00bc teaspoon of the salt to boiling. Add eggplant. Return to boiling; reduce heat. Simmer uncovered 5 to 8 minutes or until tender; drain.\n\n2 In large glass or plastic bowl, mix the oil, lemon juice, basil, mustard, pepper and remaining \u00bd teaspoon salt. Add eggplant and remaining ingredients; toss. Cover and refrigerate about 4 hours or until chilled.\n\n1 Serving: Calories 120; Total Fat 9g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 230mg; Total Carbohydrate 7g (Dietary Fiber 2g, Sugars 4g); Protein 1g Exchanges: 1\u00bd Vegetable, 2 Fat Carbohydrate Choices: \u00bd\n\nPanzanella\n\n## Panzanella Calorie Smart\n\nPanzanella, or bread salad, is an Italian classic.\n\nPREP 20 min TOTAL 1 hr 20 min \u25cf 6 servings\n\n  * 4 cups cubed (1 inch) day-old Italian or other firm-textured bread\n  * 2 medium tomatoes, cut into bite-size pieces (2 cups)\n  * 2 cloves garlic, finely chopped\n  * 1 medium green bell pepper, chopped (1 cup)\n  * \u2153 cup chopped fresh basil leaves\n  * 2 tablespoons chopped fresh parsley\n  * \u2153 cup olive oil\n  * 2 tablespoons red wine vinegar\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n\n1 In large glass or plastic bowl, mix bread, tomatoes, garlic, bell pepper, basil and parsley.\n\n2 In tightly covered container, shake remaining ingredients. Pour over bread mixture; toss gently until bread is evenly coated.\n\n3 Cover and refrigerate at least 1 hour, but no longer than 8 hours, until bread is softened and flavors are blended. Toss before serving.\n\n1 Serving: Calories 190; Total Fat 13g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 340mg; Total Carbohydrate 15g (Dietary Fiber 2g, Sugars 1g); Protein 3g Exchanges: 1 Starch, 2 Fat Carbohydrate Choices: 1\n\n## Heirloom Tomato Caprese Salad Calorie Smart \u2022 Fast \u2022 EASY\n\nChoose heirloom tomatoes in a variety of colors for a stunning presentation. If you want to splurge with ingredients, buy balsamic vinegar aged over 20 years. It has a syrupy consistency and a heavenly sweet and tangy flavor.\n\nPREP 10 min TOTAL 10 min \u25cf 8 servings\n\n  * 2 containers (8 oz each) bocconcini (small fresh mozzarella cheese balls), drained\n  * 4 medium to large heirloom or other tomatoes, cut into wedges or slices\n  * 2 tablespoons balsamic vinegar\n  * 1 tablespoon olive oil\n  * Coarse sea salt, flaked salt or colored salt, if desired\n  * Freshly ground pepper, if desired\n  * 6 large fresh basil leaves, thinly sliced*\n\nOn serving platter, arrange cheese and tomatoes. Drizzle with vinegar and oil; sprinkle with salt and pepper. Sprinkle with basil. Serve immediately.\n\n*Basil leaves cut into thin strips is called \"chiffonade.\" To cut, stack the leaves on top of each other, then roll up. Slice across the roll to form long shreds.\n\n1 Serving: Calories 170; Total Fat 12g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 40mg; Sodium 170mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 2g); Protein 10g Exchanges: \u00bd Vegetable, 1 Lean Meat, 2 Fat Carbohydrate Choices: \u00bd\n\nHeirloom Tomato Caprese Salad\n\n# Heirloom Recipe and New Twist\n\nIdeal for picnics and gatherings, this Seven-Layer Salad is a classic! The new twist is also layered and shows how versatile this salad style can be.\n\nHeirloom\n\nSeven-Layer Salad\n\n## Seven-Layer Salad\n\nPREP 25 min TOTAL 2 hr 25 min \u25cf 6 servings\n\n  * 6 cups bite-size pieces mixed salad greens\n  * 2 medium stalks celery, thinly sliced (1 cup)\n  * 1 cup thinly sliced radishes\n  * 8 medium green onions, sliced (\u00bd cup)\n  * 12 slices bacon, crisply cooked, crumbled (\u00be cup)\n  * 1 cup frozen sweet peas, cooked, drained\n  * 1\u00bd cups mayonnaise or salad dressing (for homemade mayonnaise, see page 460)\n  * \u00bd cup grated Parmesan or shredded Cheddar cheese (2 oz)\n\n1 In large glass bowl, layer salad greens, celery, radishes, onions, bacon and peas. Spread mayonnaise over peas, covering top completely all the way to edge of bowl. Sprinkle with cheese.\n\n2 Cover and refrigerate at least 2 hours, but no longer than 12 hours, to blend flavors. Toss before serving, if desired.\n\n1 Serving (1\u00bc Cups): Calories 580; Total Fat 54g (Saturated Fat 11g, Trans Fat 0g); Cholesterol 55mg; Sodium 790mg; Total Carbohydrate 11g (Dietary Fiber 3g, Sugars 5g); Protein 12g Exchanges: \u00bd Starch, 1 Vegetable, 1 High-Fat Meat, 9 Fat Carbohydrate Choices: 1\n\nLighter Directions For 12 grams of fat and 200 calories per serving, substitute \u00bd cup reduced-fat mayonnaise and 1 cup plain fat-free yogurt for the 1\u00bd cups mayonnaise. Reduce bacon to 6 slices and cheese to \u00bc cup.\nNew Twist\n\nFennel-Asparagus Seven-Layer Salad\n\n## Fennel-Asparagus Seven-Layer Salad\n\nPREP 25 min TOTAL 2 hr 25 min \u25cf 6 servings\n\n  * 6 cups mesclun (field or wild) greens\n  * 1 cup thinly sliced fennel bulb\n  * \u00bd cup thinly sliced radishes\n  * \u00bd cup shredded carrot\n  * \u00bd cup thinly sliced sweet onion\n  * 6 slices bacon, crisply cooked, crumbled\n  * 1 cup (\u00bd-inch pieces) fresh asparagus\n  * 1\u00bd cups reduced-fat mayonnaise or salad dressing\n  * \u00bd cup grated Parmesan cheese\n  * \u00bc cup shredded aged Gouda or Swiss cheese (2 oz)\n\n1 In large glass bowl, layer mesclun, fennel, radishes, carrot, onion, bacon and asparagus.\n\n2 In medium bowl, mix mayonnaise and Parmesan cheese. Spread mayonnaise mixture over asparagus, covering top completely and sealing to edge of bowl. Sprinkle with Gouda cheese.\n\n3 Cover and refrigerate at least 2 hours, but no longer than 12 hours, to blend flavors. Toss before serving, if desired.\n\n1 Serving (About 1\u00bc Cups): Calories 330; Total Fat 27g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 40mg; Sodium 810mg; Total Carbohydrate 11g (Dietary Fiber 2g, Sugars 5g); Protein 9g Exchanges: 1\u00bd Vegetable, 1 Very Lean Meat, 5\u00bd Fat Carbohydrate Choices: 1\n\nCobb Salad\n\n## Cobb Salad\n\nThe original Hollywood Brown Derby restaurant, where owner Robert H. Cobb created the Cobb salad, is gone, but this hearty dish remains popular.\n\nPREP 25 min TOTAL 1 hr 25 min \u25cf 4 servings\n\nLemon Vinaigrette Dressing\n\n  * \u00bd cup vegetable oil\n  * \u00bc cup fresh lemon juice\n  * 1 tablespoon red wine vinegar\n  * 1 clove garlic, finely chopped\n  * 2 teaspoons sugar\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground mustard\n  * \u00bd teaspoon Worcestershire sauce\n  * \u00bc teaspoon pepper\n\nSalad\n\n  * 1 medium head iceberg lettuce (about 1 lb), shredded (6 cups)\n  * 2 cups cut-up cooked chicken\n  * 3 Hard-Cooked Eggs (page 91), chopped\n  * 2 medium tomatoes, chopped (1\u00bd cups)\n  * 1 medium avocado, pitted, peeled and chopped\n  * \u00bc cup crumbled blue cheese (1 oz)\n  * 4 slices bacon, crisply cooked, crumbled (\u00bc cup)\n\n1 In tightly covered container, shake all dressing ingredients. Refrigerate at least 1 hour to blend flavors.\n\n2 Divide lettuce among 4 salad plates or shallow bowls. Arrange remaining salad ingredients in rows on lettuce. Serve with dressing.\n\n1 Serving: Calories 610; Total Fat 49g (Saturated Fat 10g, Trans Fat 0g); Cholesterol 230mg; Sodium 650mg; Total Carbohydrate 12g (Dietary Fiber 4g, Sugars 6g); Protein 30g Exchanges: 2 Vegetable, 4 High-Fat Meat, 3 Fat Carbohydrate Choices: 1\n\nSanta Fe Salad with Tortilla Straws\n\n## Santa Fe Salad with Tortilla Straws\n\nPREP 25 min TOTAL 40 min \u25cf 6 servings\n\nTortilla Straws\n\n  * 1 flour tortilla (7- or 8-inch)\n  * Cooking spray\n  * \u00bc teaspoon coarse (kosher or sea) salt\n  * \u00bc teaspoon chili powder\n\nCreamy Avocado Dressing\n\n  * \u00bd cup ranch dressing\n  * 1 medium ripe avocado, pitted, peeled and cut into chunks\n\nSalad\n\n  * 2 cups bite-size pieces iceberg lettuce\n  * 2 cups bite-size pieces romaine lettuce\n  * 1 large tomato, chopped (1 cup)\n  * 1 cup diced peeled jicama\n  * \u00be cup shredded Cheddar cheese (3 oz)\n  * \u00bc cup sliced pitted ripe olives\n\n1 Heat oven to 350\u00b0F. Spray one side of tortilla with cooking spray; sprinkle with salt and chili powder. Cut tortilla in half, then cut each half crosswise into \u00bc-inch strips. Place on ungreased cookie sheet. Bake 10 to 12 minutes or until crisp and golden brown.\n\n2 In blender, place ranch dressing and avocado. Cover and blend on medium speed until smooth and creamy. Transfer to small bowl.\n\n3 Place lettuces on shallow serving platter or in large bowl. Top with tomato, jicama, cheese, olives and tortilla straws. Serve with dressing.\n\n1 Serving (1 Cup): Calories 240; Total Fat 20g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 20mg; Sodium 450mg; Total Carbohydrate 11g (Dietary Fiber 3g, Sugars 1g); Protein 5g Exchanges: \u00bd Starch, 1 Vegetable, 4 Fat Carbohydrate Choices: 1\n\nMake-Ahead Directions Make Tortilla Straws up to 2 days ahead of time. Cool completely, then store in a resealable food-storage plastic bag. Dressing can be made up to 2 hours ahead of time; cover and refrigerate.\n\nAsparagus-Strawberry Salad\n\n## Asparagus-Strawberry Salad Calorie Smart \u2022 Fast\n\nPREP 15 min TOTAL 15 min \u25cf 4 servings\n\n  * 1 lb fresh asparagus spears, trimmed, cut into 1-inch pieces (2 cups)\n  * \u00bc cup fresh lemon juice\n  * 2 tablespoons vegetable oil\n  * 2 tablespoons honey\n  * 2 cups sliced fresh strawberries\n\n1 In 2-quart saucepan, heat 1 inch water to boiling. Add asparagus. Return to boiling; boil uncovered 2 to 4 minutes or until crisp-tender. Remove asparagus from boiling water; immediately plunge into ice water until cold to stop cooking. Drain.\n\n2 In tightly covered container, shake lemon juice, oil and honey.\n\n3 On salad plates or in bowls, arrange asparagus and strawberries; drizzle with dressing.\n\n1 Serving: Calories 150; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 19g (Dietary Fiber 3g, Sugars 14g); Protein 2g Exchanges: 1 Other Carbohydrate, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 1\n\nAsparagus-Orange Salad Substitute orange juice for the lemon juice and 2 cups sliced peeled fresh oranges for the strawberries.\n\nAsparagus-Cashew Salad Substitute \u00bc cup sliced green onions and 1\u00bd cups small cauliflower florets for the strawberries. Arrange vegetables on salad plates; drizzle with dressing. Sprinkle with \u00bd cup cashew pieces.\n\nCranberry-Orange Mold\n\n## Cranberry-Orange Mold Calorie Smart \u2022 Easy\n\nPREP 10 min TOTAL 4 hr 10 min \u25cf 8 servings\n\n  * 1 can (11 oz) mandarin orange segments, drained, juice reserved\n  * 1 box (4-serving size) orange-flavored gelatin\n  * 1 can (14 oz) whole berry cranberry sauce\n  * Salad greens, if desired\n\n1 Spray 4-cup mold with cooking spray. Add enough water to reserved mandarin orange juice to measure 1\u00bc cups. In 1-quart saucepan, heat juice mixture to boiling.\n\n2 In medium bowl, pour boiling mixture on gelatin; stir until gelatin is dissolved. Stir in cranberry sauce until sauce is melted. Stir in orange segments. Pour into mold.\n\n3 Refrigerate about 4 hours or until firm. Unmold; serve on salad greens.\n\n1 Serving: Calories 150; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 35g (Dietary Fiber 1g, Sugars 33g); Protein 1g Exchanges: 2\u00bd Fruit Carbohydrate Choices: 2\n\n### Unmolding Gelatin Salads\n\nDip mold into warm (not hot) water for several seconds. Dip tip of knife in warm water; carefully run knife around top edge of mold to loosen.\n\nPlace chilled serving plate on top of mold. Holding both mold and plate firmly, turn upside down onto plate and gently shake to release gelatin. Carefully lift mold from gelatin.\n\nCinnamon-Maple Glazed Salmon Salad\n\n## Cinnamon-Maple Glazed Salmon Salad\n\nPREP 20 min TOTAL 35 min \u25cf 4 servings\n\nDressing\n\n  * \u2153 cup real maple or maple-flavored syrup\n  * \u2153 cup vegetable oil\n  * 2 tablespoons plus 1 teaspoon balsamic vinegar\n  * \u00bd teaspoon ground cinnamon\n\nSalad\n\n  * 1 salmon fillet (1\u00bc lb), cut into 4 serving pieces\n  * 6 cups mixed spring greens\n  * \u2154 cup crumbled feta cheese\n  * \u2153 cup dried cherries\n  * \u2153 cup coarsely chopped pecans, toasted (page 23)\n\n1 Heat oven to 400\u00b0F. Line 15x10x1-inch pan with foil. In small bowl, beat dressing ingredients with whisk until blended. Place 3 tablespoons dressing in small bowl; reserve for brushing on salmon. Set remaining dressing aside for salad. Place salmon pieces, skin side down, in pan.\n\n2 Bake 10 to 14 minutes, brushing with reserved 3 tablespoons dressing twice during last 5 minutes of baking, until salmon flakes easily with fork. Discard any remaining dressing used for salmon.\n\n3 In large bowl, toss greens with reserved dressing for salad. Divide greens among 4 individual plates; sprinkle evenly with cheese, cherries and toasted pecans. With metal spatula, lift pieces of salmon from skin; place 1 salmon piece on each salad. Discard skin and foil.\n\n1 Serving: Calories 630; Total Fat 39g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 100mg; Sodium 400mg; Total Carbohydrate 35g (Dietary Fiber 3g, Sugars 29g); Protein 34g Exchanges: 2 Starch, 1 Vegetable, 3\u00bd Lean Meat, 5\u00bd Fat Carbohydrate Choices: 2\n\nRoasted Potato Salad\n\n## Roasted Potato Salad\n\nPREP 15 min TOTAL 50 min \u25cf 6 servings\n\nPotatoes\n\n  * 6 medium unpeeled Yukon Gold potatoes, each cut into 8 wedges*\n  * 2 tablespoons olive oil\n  * \u00bd teaspoon salt\n  * Fresh rosemary leaves for garnish, if desired\n\nHerb Vinaigrette Dressing\n\n  * \u00bc cup olive oil\n  * 1 tablespoon white balsamic or cider vinegar\n  * 2 teaspoons Dijon mustard\n  * 2 tablespoons finely chopped shallots\n  * 1 tablespoon chopped fresh chives\n  * 1 teaspoon fresh thyme leaves\n  * 1 teaspoon chopped fresh rosemary leaves\n  * \u00bd teaspoon sugar\n  * \u00bc teaspoon salt\n  * 1 clove garlic, finely chopped\n\n1 Heat oven to 400\u00b0F. Place potatoes in center of 15x10x1-inch pan. Drizzle with 2 tablespoons oil; sprinkle with \u00bd teaspoon salt. Toss to coat. Spread in single layer in pan.\n\n2 Roast uncovered 30 to 35 minutes or until potatoes are light brown and tender, turning occasionally.\n\n3 In small bowl, mix dressing ingredients with whisk. Transfer potatoes to serving bowl; drizzle with dressing. Toss gently to coat. Garnish with rosemary leaves. Serve immediately or let stand 30 minutes at room temperature.\n\n*Red potatoes can be substituted for the Yukon Gold potatoes.\n\n1 Serving: Calories 300; Total Fat 14g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 350mg; Total Carbohydrate 38g (Dietary Fiber 4g, Sugars 3g); Protein 4g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nCreamy Potato Salad\n\n## Creamy Potato Salad\n\nThis classic potato salad is always a favorite for picnics and potlucks. Make it your own by adding a variety of flavors, such as bacon, capers, herbs or fresh vegetables.\n\nPREP 20 min TOTAL 5 hr \u25cf 10 servings\n\n  * 6 medium unpeeled round red or Yukon Gold potatoes (2 lb)\n  * 1\u00bd cups mayonnaise or salad dressing (for homemade Mayonnaise, see page 460)\n  * 1 tablespoon white or cider vinegar\n  * 1 tablespoon yellow mustard\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 medium stalks celery, chopped (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 4 Hard-Cooked Eggs (page 91), chopped\n  * Paprika, if desired\n\n1 In 3-quart saucepan, place potatoes and just enough water to cover potatoes. Cover and heat to boiling; reduce heat to low.\n\n2 Cook 30 to 35 minutes or until tender; drain. Let stand until cool enough to handle. Peel potatoes; cut into cubes.\n\n3 In large glass or plastic bowl, mix mayonnaise, vinegar, mustard, salt and pepper. Add potatoes, celery and onion; toss. Stir in eggs. Sprinkle with paprika. Cover and refrigerate at least 4 hours to chill and blend flavors.\n\n1 Serving (\u00be Cup): Calories 350; Total Fat 28g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 105mg; Sodium 480mg; Total Carbohydrate 19g (Dietary Fiber 2g, Sugars 2g); Protein 5g Exchanges: 1\u00bd Starch, 5\u00bd Fat Carbohydrate Choices: 1\n\nLighter Directions For 13 grams of fat and 210 calories per serving, use reduced-fat mayonnaise and 2 eggs.\n\nHot German Potato Salad\n\n## Hot German Potato Salad Calorie Smart\n\nThe tangy sweet-sour dressing, which soaks into the velvety cooked potatoes, and the smoky bacon are what give this salad its characteristic flavor.\n\nPREP 30 min TOTAL 1 hr 5 min \u25cf 6 servings\n\n  * 4 medium unpeeled red or white potatoes (1\u00bd lb)\n  * 3 slices bacon, cut into 1-inch pieces\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 tablespoon all-purpose flour\n  * 1 tablespoon sugar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon celery seed\n  * Dash pepper\n  * \u00bd cup water\n  * \u00bc cup white or cider vinegar\n\n1 In 3-quart saucepan, place potatoes and just enough water to cover potatoes. Cover and heat to boiling; reduce heat to low. Cook 30 to 35 minutes or until tender; drain. Let stand until cool enough to handle. Cut potatoes into \u00bc-inch slices.\n\n2 In 10-inch skillet, cook bacon over medium heat 8 to 10 minutes, stirring occasionally, until crisp. Remove bacon with slotted spoon; drain on paper towels.\n\n3 Cook onion in bacon drippings over medium heat, stirring occasionally, until tender. Stir in flour, sugar, salt, celery seed and pepper. Reduce heat to low. Cook and stir until mixture is bubbly; remove from heat.\n\n4 Stir in water and vinegar. Heat to boiling, stirring constantly. Boil and stir 1 minute; remove from heat. Stir in potatoes and bacon. Heat over medium heat, stirring gently to coat potato slices, until hot and bubbly. Serve warm.\n\n1 Serving (\u2154 Cup): Calories 120; Total Fat 2g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 270mg; Total Carbohydrate 22g (Dietary Fiber 3g, Sugars 5g); Protein 3g Exchanges: 1 Starch, 1 Vegetable, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\nSlow-Cooker Directions Cut uncooked potatoes into \u00bc-inch slices. Cook and drain bacon; refrigerate. Increase flour to 2 tablespoons. Decrease water to \u2153 cup. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except bacon. Cover; cook on Low heat setting 8 to 10 hours or until potatoes are tender. Stir in bacon.\n\n## Creamy Coleslaw Calorie Smart\n\nThis is a delicious coleslaw to make for any occasion. For variety, you could add \u00bc to \u00bd cup chopped red bell pepper. To serve, garnish with a lemon twist and a little fresh parsley.\n\nPREP 15 min TOTAL 1 hr 15 min \u25cf 8 servings\n\n  * \u00bd cup mayonnaise or salad dressing (for homemade Mayonnaise, see page 460)\n  * \u00bc cup sour cream\n  * 1 tablespoon sugar\n  * 2 teaspoons fresh lemon juice\n  * 2 teaspoons Dijon mustard\n  * \u00bd teaspoon celery seed, if desired\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bd medium head green cabbage, thinly sliced or chopped (4 cups)\n  * 1 medium carrot, shredded (\u00bd cup)\n  * 1 small onion, finely chopped (\u2153 cup)\n\n1 In large glass or plastic bowl, mix all ingredients except cabbage, carrot and onion. Add cabbage, carrot and onion; toss until evenly coated.\n\n2 Cover and refrigerate at least 1 hour to blend flavors. Stir before serving.\n\n1 Serving (\u2154 Cup): Calories 140; Total Fat 13g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 200mg; Total Carbohydrate 6g (Dietary Fiber 1g, Sugars 4g); Protein 1g Exchanges: \u00bd Other Carbohydrate, \u00bd Vegetable, 2\u00bd Fat Carbohydrate Choices: \u00bd\n\nLighter Directions For 6 grams of fat and 85 calories per serving, use reduced-fat mayonnaise and sour cream.\n\nTropical Creamy Coleslaw Add 1 can (8 ounces) pineapple tidbits or chunks, drained, with the cabbage, carrot and onion. Just before serving, sprinkle with \u00bd cup chopped macadamia nuts.\n\nSweet-and-Sour Coleslaw\n\n## Sweet-and-Sour Coleslaw Calorie Smart\n\nPREP 15 min TOTAL 3 hr 15 min \u25cf 8 servings\n\n  * \u00bd medium head green cabbage, thinly sliced or chopped (4 cups)\n  * 1 large carrot, finely shredded (1 cup)\n  * 1 medium green bell pepper, chopped (1 cup)\n  * 4 medium green onions, thinly sliced (\u00bc cup)\n  * \u00bd cup sugar\n  * \u00bd cup white wine vinegar, white vinegar or cider vinegar\n  * \u00bc cup vegetable oil\n  * 1 teaspoon ground mustard\n  * \u00bd teaspoon celery seed\n  * \u00bd teaspoon salt\n\n1 In large glass or plastic bowl, toss cabbage, carrot, bell pepper and onions.\n\n2 In tightly covered container, shake remaining ingredients. Pour over vegetables; toss until evenly coated.\n\n3 Cover and refrigerate at least 3 hours, stirring several times, to blend flavors and chill. Stir before serving; serve with slotted spoon.\n\n1 Serving: Calories 140; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 170mg; Total Carbohydrate 17g (Dietary Fiber 1g, Sugars 15g); Protein 1g Exchanges: 1 Other Carbohydrate, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 1\n\nQuicker Sweet-and-Sour Coleslaw Substitute 6\u00bd cups coleslaw mix (from 1-pound bag) for the cabbage and carrot; omit bell pepper.\n\n### Thinly Slicing Cabbage\n\nTo thinly slice cabbage, place a flat side of \u00bc head of cabbage on cutting board. Cut into thin slices with a large sharp knife. Cut slices several times to make smaller pieces.\n\n## Learn to Blanch\n\nThe simple definition of blanching is to place food in boiling water for a brief time to slightly cook it. Blanch foods (typically vegetables and fruits) when you want to preserve their color, crisp texture and nutritional value or as a way to remove skin. Once you realize how simple it is to blanch, it will be easy to see the many ways you can use this technique.\n\nBlanching is most often done in boiling water on the stove top. Although steaming can also be used to blanch a few vegetables, this method does take longer and is not as effective. Microwave blanching is not recommended.\n\nPrepare the vegetables or fruits just before blanching to prevent discoloration. If you are blanching to remove skins (peaches, nectarines, tomatoes), leave whole and wash just before blanching to remove any visible dirt. Some vegetables can be blanched whole (string or wax beans) and some should be cut (carrots). Cut vegetables into uniform sizes.\n\n  1. 1. Fill a large pot with water and bring it to a boil. A vegetable blancher will have a basket for holding the vegetables and immersing in the water. Use 1 gallon of water per pound of vegetables (about 2 cups). Salt can be added (1 teaspoon) to help maintain color and flavor but is not necessary.\n  2. 2. Place the vegetables or fruit in a wire basket or the blancher basket and immerse into the boiling water. (If you don't have a basket, plunge the food into the boiling water and remove with a slotted spoon or small colander.) Bring the water back to a boil. Boil uncovered 2 to 4 minutes for most vegetables or until color brightens. (Exact time will depend on vegetables and size. Watch carefully, as over-blanching can cause loss of flavor and color.) Skin on tomatoes and peaches may start to wrinkle slightly or crack.\n  3. 3. Remove items from the boiling water and immediately plunge into ice water to cool quickly and stop the cooking process.\n  4. 4. Drain well when cooled. Use in recipes or refrigerate or freeze as desired. For tomatoes and peaches, use fingers and a small knife to pull skins from fruit.\n\n## Asian Salad with Wonton Crisps Fast\n\nThis very pretty and fresh-tasting arranged salad would pair great with any grilled meat, chicken, fish or seafood.\n\nPREP 30 min TOTAL 30 min \u25cf 4 servings\n\nWonton Crisps\n\n  * 4 wonton skins (about 3\u00bc-inch square), cut into \u00bc-inch strips\n\nPeanutty Soy Dressing\n\n  * \u00bc cup vegetable oil\n  * 2 tablespoons lemon juice\n  * 4 teaspoons sugar\n  * 4 teaspoons soy sauce\n  * 2 teaspoons Dijon mustard\n  * \u00bc cup chopped dry-roasted peanuts\n\nSalad\n\n  * 1 cup small fresh broccoli florets\n  * 1 cup fresh snow pea pods (4 oz), strings removed\n  * 4 cups bite-size pieces mixed salad greens\n  * \u00bd cup shredded red cabbage\n  * \u00bd cup shredded radishes, patted dry\n\n1 Heat oven to 350\u00b0F. On ungreased cookie sheet, arrange wonton strips in single layer. Bake 5 to 6 minutes or until lightly browned.\n\n2 Meanwhile, in tightly covered container, shake all dressing ingredients; set aside.\n\n3 In 2-quart saucepan, heat 2 cups water to boiling. Add broccoli and pea pods; boil uncovered 1 to 2 minutes or just until bright green. Remove vegetables from boiling water; immediately place in ice water until cold. Drain well.\n\n4 Divide salad greens among 4 plates. Top evenly with broccoli, pea pods, cabbage and radishes. Drizzle with dressing; top with wonton strips.\n\n1 Serving: Calories 270; Total Fat 19g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 0mg; Sodium 500mg; Total Carbohydrate 18g (Dietary Fiber 4g, Sugars 7g); Protein 6g Exchanges: \u00bd Other Carbohydrate, 1\u00bd Vegetable, \u00bd High-Fat Meat, 3 Fat Carbohydrate Choices: 1\n\nGreen and Yellow Bean Salad\n\n## Green and Yellow Bean Salad Calorie Smart\n\nThree-bean salad gets a vibrant take in this makeover of the classic\u2014with two colors of fresh beans, cannellini beans, grape tomatoes and fresh basil. Garnish with shaved Parmesan cheese, if desired.\n\nPREP 20 min TOTAL 1 hr 20 min \u25cf 10 servings\n\nSalad\n\n  * 8 oz fresh green beans, trimmed\n  * 8 oz fresh yellow wax beans, trimmed\n  * 1 cup grape tomatoes, cut in half\n  * 1 can (15 oz) cannellini beans, drained, rinsed\n  * \u00bc cup lightly packed fresh basil leaves, torn\n\nSherry Vinaigrette Dressing\n\n  * \u00bc cup olive oil\n  * 2 tablespoons sherry vinegar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon coarse ground black pepper\n\n1 In 2-quart saucepan, place green and wax beans in 1 inch water. Heat to boiling; reduce heat. Simmer uncovered 8 to 10 minutes or until beans are crisp-tender. Rinse with cold water; drain.\n\n2 In medium bowl, stir together cooked beans, tomatoes, cannellini beans and basil.\n\n3 In jar with tight-fitting lid, shake dressing ingredients. Pour over salad; toss to coat. Refrigerate at least 1 hour to blend flavors. Stir before serving.\n\n1 Serving: Calories 110; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 12g (Dietary Fiber 3g, Sugars 2g); Protein 4g Exchanges: \u00bd Other Carbohydrate, \u00bd Vegetable, \u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\n## Northern Italian White Bean Salad Calorie Smart\n\nPREP 15 min TOTAL 2 hr 15 min \u25cf 6 servings\n\n  * 2 cans (15 or 19 oz each) cannellini beans, drained, rinsed\n  * 1 large tomato, seeded, coarsely chopped (1 cup)\n  * 1 small red bell pepper, chopped (\u00bd cup)\n  * \u00bd cup chopped red onion\n  * \u00bc cup chopped fresh parsley\n  * \u00bc cup olive or vegetable oil\n  * 2 tablespoons chopped fresh or 2 teaspoons dried basil leaves\n  * 2 tablespoons red wine vinegar\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n  * 12 leaves Bibb lettuce\n\n1 In large glass or plastic bowl, gently mix all ingredients except lettuce.\n\n2 Cover and refrigerate at least 2 hours to blend flavors. Use slotted spoon to spoon onto lettuce.\n\n1 Serving: Calories 360; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 210mg; Total Carbohydrate 49g (Dietary Fiber 12g, Sugars 3g); Protein 18g Exchanges: 3 Starch, 1 Vegetable, 1 Very Lean Meat Carbohydrate Choices: 3\n\n## Betty's Staples   \nCanned Tuna or Salmon\n\nIt's easy to see why cans and pouches of tuna and salmon are pantry staples in many homes. They are versatile, inexpensive and convenient. Because the pouches do not contain extra water or oil, they may cost a little more. We all know about making tuna salad and sandwiches, but there are other creative ways to use tuna or salmon that you might like to try.\n\n  1. 1. **Mediterranean Tuna Salad:** On individual salad plates, arrange 1 cup mixed salad greens, \u00bd cup drained canned cannellini beans, \u00bd cup chopped tomato, torn romaine, \u00bd cup sliced ripe olives and \u00bd can (3 ounces) drained tuna or salmon. Drizzle with Italian Dressing (page 137) and sprinkle with chopped fresh parsley.\n  2. 2. **Cheddar Tuna Melts:** In small bowl, mix 1 can (6 ounces) drained tuna or salmon, \u00bc cup chopped fresh chives and 1 to 2 tablespoons mayonnaise. Spread on 4 slices toasted hearty whole grain bread. Top each sandwich with 1 or 2 slices tomato and 1 or 2 slices sharp Cheddar cheese. Broil 2 to 3 minutes or until cheese is melted.\n  3. 3. **Chopped Summer Salad:** In large bowl, toss 4 cups chopped lettuce, 1 can (6 ounces) drained tuna or salmon, 1 cup halved red grapes, 1 cup chopped carrots, 1 cup chopped honey-roasted peanuts and \u00bd cup Honey-Mustard Dressing (page 137). Sprinkle with 2 tablespoons sliced green onions.\n  4. 4. **Tuna Quesadillas:** Toss 1 can (6 ounces) tuna or salmon with 2 tablespoons ranch dressing. Spread evenly on 2 (8-inch) flour tortillas. Sprinkle each with \u00bd cup shredded Monterey Jack cheese and \u00bc cup corn salsa. Top each with another tortilla. Cook each in 1 tablespoon oil in skillet for 3 to 5 minutes or until golden brown, turning once.\n  5. 5. **Baked Potatoes with Tuna Topping:** Bake potatoes and cut in half lengthwise. Mash in shell slightly with fork. Top each with 2 tablespoons chive and sour cream potato topper, \u00bc cup drained canned tuna or salmon, \u00bc cup cooked small broccoli florets and \u00bd cup shredded Cheddar cheese. Microwave on High 1 to 3 minutes or until cheese is melted.\n  6. 6. **Tuna-Lettuce Wraps:** In medium bowl, mix 1 can (6 ounces) drained tuna or salmon, \u00bc cup shredded carrots, \u00bd cup chopped cashews, 2 tablespoons chopped green or red onion and 2 tablespoons Asian Dressing (page 138). Spoon into Bibb lettuce leaves; roll up each one.\n\nMediterranean Tuna Salad, Tuna-Lettuce Wraps, Chopped Summer Salad\n\n## Chicken-Thyme Penne Salad\n\nYou'll need about 2 pounds of uncooked boneless skinless chicken breasts, which equals 4 cups cubed cooked chicken, to make this dish. Cook them however you want, or buy deli rotisserie chicken.\n\nPREP 25 min TOTAL 4 hr 25 min \u25cf 8 servings\n\n  * 3 cups uncooked penne pasta (9 oz)\n  * 4 cups cubed cooked chicken\n  * 2 cups seedless red grapes, halved\n  * 2 medium stalks celery, sliced (1 cup)\n  * 1 small onion, chopped (\u2153 cup)\n  * 3 tablespoons olive or vegetable oil\n  * 2 tablespoons chopped fresh or 2 teaspoons dried thyme leaves, crushed\n  * 1\u00bc cups mayonnaise or salad dressing (for homemade Mayonnaise, see page 460)\n  * 1 tablespoon milk\n  * 1 tablespoon honey\n  * 1 tablespoon coarse-grained mustard\n  * 1 teaspoon salt\n  * 1 cup chopped walnuts, toasted (page 23)\n\n1 Cook and drain pasta as directed on package. Rinse with cold water to cool; drain.\n\n2 In 4-quart bowl, mix pasta, chicken, grapes, celery and onion. In small bowl, mix oil and 1 tablespoon of the fresh thyme (or 1 teaspoon of the dried thyme). Pour over chicken mixture; toss until coated.\n\n3 In small bowl, mix mayonnaise, milk, honey, mustard, salt and remaining thyme. Cover chicken mixture and mayonnaise mixture separately; refrigerate at least 4 hours but no longer than 24 hours.\n\n4 Up to 2 hours before serving, toss chicken mixture and mayonnaise mixture. Cover and refrigerate. Just before serving, stir in \u00be cup of the toasted walnuts. Sprinkle salad with remaining \u00bc cup walnuts.\n\n1 Serving: Calories 690; Total Fat 46g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 80mg; Sodium 600mg; Total Carbohydrate 41g (Dietary Fiber 4g, Sugars 10g); Protein 29g Exchanges: 2 Starch, 1 Fruit, 3 Medium-Fat Meat, 5\u00bd Fat Carbohydrate Choices: 3\n\nTurkey-Basil Penne Salad Substitute cooked turkey for the chicken and fresh basil for the thyme.\n\nSesame Noodle Salad\n\n## Sesame Noodle Salad Calorie Smart \u2022 Fast\n\nLight-colored sesame oil has a delicate, nutty flavor, while dark sesame oil has a stronger, robust flavor.\n\nPREP 15 min TOTAL 25 min \u25cf 12 servings\n\nSesame Dressing\n\n  * 3 tablespoons vegetable oil\n  * 3 tablespoons soy sauce\n  * 1 tablespoon balsamic or rice vinegar\n  * 1 tablespoon dark sesame oil\n  * 4\u00bd teaspoons sugar\n  * 1\u00bd teaspoons grated gingerroot\n  * \u00bd teaspoon garlic powder\n  * \u215b teaspoon ground red pepper (cayenne)\n\nSalad\n\n  * 8 oz uncooked soba (buckwheat) noodles or angel hair pasta\n  * 2 medium carrots, shredded (1 cup)\n  * 1 small cucumber, cut lengthwise in half, then cut crosswise into thin slices\n  * 2 medium green onions, thinly sliced (2 tablespoons)\n  * 1 tablespoon sesame seed, toasted (page 23)\n\n1 In small bowl, stir all dressing ingredients until well mixed; set aside.\n\n2 In 4-quart Dutch oven or saucepan, heat 2 quarts water to boiling. Cook noodles in boiling water 6 to 8 minutes or until tender (if using angel hair pasta, cook as directed on package); drain. Rinse with cold water to cool; drain.\n\n3 In large bowl, toss noodles, remaining salad ingredients and dressing. Serve or cover and refrigerate.\n\n1 Serving: Calories 120; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 230mg; Total Carbohydrate 16g (Dietary Fiber 2g, Sugars 2g); Protein 3g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\n# Dressing Basics\n\nDressings can vary in ingredients, but usually some type of oil plus vinegar or another acidic ingredient make up the base. Oils, depending on the type used, provide body and may add flavor to the dressing or simply carry the flavor of the other ingredients. Traditionally, vinaigrettes are 3 parts oil to 1 part vinegar.\n\n## About Oil\n\nOils range in flavor from earthy and nutty to neutral or fruity and mildly sweet. Recipes often call for a specific oil designed to complement the ingredients, but feel free to experiment with a different oil or combination of oils.\n\nType of Oil | Description\n\n---|---\n\nVegetable Oils\n\n(often generically labeled as vegetable oil; check label for oil source)\n\nCanola, Corn, Soybean, Sunflower | Delicate\/neutral\n\nPeanut | Mildly buttery\n\nSafflower | Mildly nutty\n\nOlive Oils\n\nOlive Oil and Light Olive Oil | Delicate\/neutral; golden color and mild flavor\n\nExtra-Virgin Olive Oil | Fruity olive flavor that is stronger than regular olive oil; golden to green or bright green color\n\nNut Oils\n\n(highly flavorful; start with a small amount, adding more as needed)\n\nAlmond | Nutty, mildly sweet\n\nAvocado | Nutty, sharp\n\nHazelnut | Nutty, rich\n\nWalnut | Mildly nutty\n\nSesame Oil\n\n(highly flavorful; start with a small amount, adding more as needed)\n\nLight Sesame Oil | Nutty, light color\n\nDark Sesame Oil (also called Asian) | Strong toasted sesame seed flavor; amber color\n\n## About Vinegar\n\nThere are many kinds of vinegar on the market, and they all add lively zest to salad dressings. Most recipes call for one type of vinegar, but you can try a little mixing and matching.\n\nType of Vinegar | Description\n\n---|---\n\nBalsamic (white grapes) | Rich, sweet, dark brown or white\n\nCider (fermented apple cider) | Mild apple flavor, golden brown\n\nFruit (blueberries, raspberries or strawberries steeped in cider vinegar or white wine vinegar) | Flavor and color of berry used; mildly sweet\n\nHerb (basil, chives, dill weed or tarragon steeped in white wine vinegar or cider vinegar) | Flavor of herb used\n\nRice (fermented rice) | Plain and sweetened forms; delicate, sweet flavor\n\nWhite (distilled grain alcohol) | Strong, pungent and clear\n\nWine (champagne, sherry, red or white wine) | Flavor of wine used\n\nUse a wire whisk to mix salad dressings.\n\n## Make-Ahead Fresh Herb Vinaigrette\n\n## Calorie Smart \u2022 Fast\n\nIt's always nice to have vinaigrette on hand: It's super simple to make and not just for salads. Choose whatever herb or herbs you like best or have on hand. Vinaigrette stores well in the refrigerator, too, so make a double batch.\n\nPREP 10 min TOTAL 10 min \u25cf About \u00be cup dressing\n\n  * \u00bd cup olive or vegetable oil\n  * 3 tablespoons red or white wine vinegar or your favorite vinegar\n  * 1 tablespoon chopped fresh herb leaves (basil, marjoram, oregano, rosemary, tarragon or thyme)\n  * 1 tablespoon chopped fresh parsley\n  * 1 tablespoon finely chopped shallot or green onion\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n\nIn tightly covered container, shake all ingredients. Shake before serving. Store tightly covered in refrigerator up to 1 week.\n\n1 Tablespoon: Calories 80; Total Fat 9g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nBlue Cheese Herb Vinaigrette Stir in \u2153 cup crumbled blue cheese.\n\nCreamy Fresh Herb Vinaigrette In small bowl, add all vinaigrette ingredients and \u2153 cup mayonnaise. Beat with whisk until well blended. Stir before serving.\n\n## Ways to Use Vinaigrette\n\n  1. 1. **Dilled Potato Salad:** Boil 2 pounds cubed small red potatoes 8 to 10 minutes or until fork-tender. Drain and rinse with cold water to cool. In medium bowl, stir potatoes with \u00bd cup vinaigrette made with fresh dill until coated.\n  2. 2. **Blue Cheese Chicken Salad:** Toss 2 cups mixed baby greens, 1 cup shredded cooked chicken and \u00bd cup grape tomatoes in medium bowl. Drizzle with \u00bc cup vinaigrette; sprinkle with \u00bd cup crumbled blue cheese.\n  3. 3. **Herbed Marinated Chicken:** In shallow bowl, pour vinaigrette over 4 boneless skinless chicken breasts. Let stand in refrigerator 30 to 60 minutes. Grill or broil 11 to 15 minutes or until chicken is no longer pink in center, brushing with marinade occasionally. Discard marinating vinaigrette.\n  4. 4. **Ni\u00e7oise Salmon Salad:** On individual serving plates, arrange torn romaine, sliced hard-cooked egg, sliced tomato and cooked or smoked salmon. Drizzle with vinaigrette and sprinkle with garlic croutons.\n  5. 5. **Italian Beef Sandwiches:** For each sandwich, spread Dijon mustard on crusty slice of bread. Layer with sliced cooked roast beef, provolone cheese, sliced roasted red bell pepper and shredded lettuce. Drizzle with 1 to 2 tablespoons vinaigrette and top with bread.\n\nFresh Herb Vinaigrette, Ni\u00e7oise Salmon Salad\n\n## Raspberry Vinaigrette Calorie Smart \u2022 Fast \u2022 Easy\n\nPREP 5 min TOTAL 5 min \u25cf 1 cup dressing\n\n  * \u00bd cup red wine vinegar\n  * \u2153 cup seedless red raspberry jam\n  * \u00bc cup olive or vegetable oil\n  * \u00bc teaspoon salt\n\nIn small glass or plastic bowl, beat all ingredients with whisk until well blended. Store tightly covered in refrigerator up to 1 week. Stir before serving.\n\n1 Tablespoon: Calories 50; Total Fat 3.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 40mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 4g); Protein 0g Exchanges: \u00bd Fruit, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## Thousand Island Dressing Calorie Smart \u2022 Fast\n\nPREP 10 min TOTAL 10 min \u25cf 1 cup dressing\n\n  * 1 cup mayonnaise or salad dressing (for homemade Mayonnaise, see page 460)\n  * 2 tablespoons chopped pimiento-stuffed green olives or sweet pickle relish\n  * 2 tablespoons chili sauce or ketchup\n  * 1 tablespoon chopped fresh parsley\n  * 1 teaspoon finely chopped onion\n  * 1 Hard-Cooked Egg (page 91), finely chopped\n  * \u00bd teaspoon paprika\n\nIn small bowl, mix all ingredients. Store tightly covered in refrigerator up to 5 days. Stir before serving.\n\n1 Tablespoon: Calories 110; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 20mg; Sodium 130mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nLighter Directions For 5 grams of fat and 55 calories per serving, use reduced-fat mayonnaise; substitute 2 hard-cooked egg whites for the hard-cooked whole egg.\n\nRussian Dressing Omit olives, parsley and egg. Increase chili sauce to \u00bc cup. Add 1 teaspoon prepared horseradish.\n\n## Thousand Island Dressing\n\nThis mayonnaise-based salad dressing, made with chili sauce and finely chopped stuffed green olives and a hard-cooked egg, also makes a yummy sandwich spread.\n\n## Italian Dressing Calorie Smart \u2022 Fast\n\nPREP 10 min TOTAL 10 min \u25cf 1\u00bc cups dressing\n\n  * 1 cup olive or vegetable oil\n  * \u00bc cup white or cider vinegar\n  * 2 tablespoons finely chopped onion\n  * 1 tablespoon chopped fresh or 1 teaspoon dried basil leaves\n  * 1 teaspoon sugar\n  * 1 teaspoon ground mustard\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon dried oregano leaves\n  * \u00bc teaspoon pepper\n  * 2 cloves garlic, finely chopped\n\nIn tightly covered container, shake all ingredients. Store tightly covered in refrigerator up to 1 week. Shake before serving.\n\n1 Tablespoon: Calories 100; Total Fat 11g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 60mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nLighter Directions For 5 grams of fat and 50 calories per serving, substitute \u00bd cup apple juice for \u00bd cup of the oil.\n\nCreamy Italian Dressing In small bowl, beat \u00bd cup Italian Dressing and \u00bd cup mayonnaise or salad dressing with whisk until smooth.\n\nFrench Dressing Omit onion, sugar, oregano and garlic. Add \u00bc cup fresh lemon juice and \u00bd teaspoon paprika; reduce mustard to \u00bd teaspoon.\n\n## Honey-Mustard Dressing Calorie Smart \u2022 Fast \u2022 Easy\n\nPREP 5 min TOTAL 5 min \u25cf 1 cup dressing\n\n  * \u00bd cup vegetable oil\n  * \u2153 cup honey or real maple syrup\n  * \u00bc cup fresh lemon juice\n  * 1 tablespoon Dijon mustard\n\nIn tightly covered container, shake all ingredients. Store tightly covered in refrigerator up to 1 week. Shake before serving.\n\n1 Tablespoon: Calories 90; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 25mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 6g); Protein 0g Exchanges: \u00bd Fruit, 1 Fat Carbohydrate Choices: \u00bd\n\nHoney\u2013Poppy Seed Dressing Omit mustard. Add 1 tablespoon poppy seed.\n\n## Blue Cheese Dressing Calorie Smart \u2022 Easy\n\nPREP 10 min TOTAL 3 hr 10 min \u25cf 1\u2154 cups dressing\n\n  * \u00be cup crumbled blue or Gorgonzola cheese (3 oz)\n  * 3 oz (from 8-oz package) cream cheese, softened\n  * \u00bd cup mayonnaise or salad dressing (for homemade mayonnaise, see page 460)\n  * \u2153 cup half-and-half\n  * 1 teaspoon cider vinegar\n  * \u215b teaspoon pepper\n\n1 Reserve \u2153 cup of the blue cheese. In small bowl, mix remaining blue cheese and the cream cheese until well blended. Stir in mayonnaise, half-and-half, vinegar and pepper until creamy. Stir in reserved \u2153 cup blue cheese.\n\n2 Cover and refrigerate at least 3 hours to blend flavors. Stir before serving. Store tightly covered in refrigerator up to 5 days.\n\n1 Tablespoon: Calories 50; Total Fat 5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 80mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: 1 Fat Carbohydrate Choices: 0\n\nLighter Directions For 3 grams of fat and 35 calories per serving, reduce blue cheese to \u00bd cup and use reduced-fat mayonnaise. Use 4 oz \u2153-less-fat cream cheese and \u00bc cup fat-free (skim) milk.\n\n## Buttermilk Ranch Dressing Calorie Smart \u2022 Easy\n\nPREP 10 min TOTAL 2 hr 10 min \u25cf 1\u00bc cups dressing\n\n  * \u00be cup mayonnaise or salad dressing (for homemade Mayonnaise, see page 460)\n  * 1 clove garlic, finely chopped\n  * \u00bd cup buttermilk\n  * 1 teaspoon parsley flakes\n  * \u00bd teaspoon dried minced onion\n  * \u00bd teaspoon salt\n  * Dash ground pepper\n\nIn small bowl, mix all ingredients. Cover and refrigerate at least 2 hours to blend flavors. Stir before serving. Store tightly covered in refrigerator up to 5 days.\n\n1 Tablespoon: Calories 60; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 110mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1\u00bd Fat Carbohydrate Choices: 0\n\nLighter Directions For 4 grams of fat and 45 calories per serving, use reduced-fat mayonnaise.\n\nButtermilk Ranch\u2013Parmesan Dressing Add \u2153 cup grated Parmesan cheese and \u00bd teaspoon paprika.\n\nMiso Dressing\n\n## Miso Dressing Calorie Smart \u2022 Fast\n\nMiso, also called bean paste, is a Japanese staple available in many flavors and colors. This salty, fermented soybean paste can be found in the refrigerated section of the grocery store with other international foods, often near the tofu, or in Asian food markets.\n\nPREP 5 min TOTAL 5 min \u25cf \u00be cup dressing\n\n  * \u00bc cup rice vinegar or cider vinegar\n  * \u00bc cup vegetable oil\n  * \u00bc cup white or yellow miso\n  * 1 teaspoon dark sesame oil\n  * 1 teaspoon grated gingerroot or \u00bc teaspoon ground ginger\n  * \u00bc teaspoon salt\n  * 1 clove garlic, finely chopped\n\nIn small bowl, beat all ingredients with whisk until well blended. Store tightly covered in refrigerator up to 1 week. Stir before serving.\n\n1 Tablespoon: Calories 60; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 260mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1 Fat Carbohydrate Choices: 0\n\nHoney Miso Dressing Beat in 1 tablespoon honey.\n\nSriracha Miso Dressing Beat in 2 to 3 teaspoons Sriracha sauce.\n\nAsian Dressing Omit miso, sesame oil and garlic. Add 2 tablespoons each dry sherry and soy sauce and 1 teaspoon sugar.\n\n# Chapter 6: Soups, Stews & Chilies\n\nChicken Noodle Soup\n\n## Soups, Stews & Chilies Basics\n\nStorage\n\n## Features\n\nBetty's Staples: Canned Beans\n\nHeirloom Recipe and New Twist\n\nLearn to Puree\n\nMake-Ahead: Cream of Mushroom Soup\n\n## BROTH\n\nBeef and Broth Calorie Smart\n\nChicken and Broth Calorie Smart\n\nFish Broth Calorie Smart\n\nVegetable Broth Calorie Smart\n\n## CHICKEN AND TURKEY\n\nChicken and Wild Rice Soup\n\nChicken Meatball Soup Calorie Smart\n\nChicken Noodle Soup Calorie Smart\n\nChicken Rice Soup\n\nChicken Tortilla Soup Calorie Smart\n\nChicken-Mushroom Soup\n\nConfetti Chicken Noodle Soup\n\nCreamy Herbed Chicken Stew Calorie Smart \u2022 Slow Cooker\n\nHomestyle Chicken Noodle Soup\n\n## LEGUME\n\nChunky Tomato-Bean Soup\n\nCuban Black Bean Soup Calorie Smart\n\nFamily Favorite Chili Calorie Smart \u2022 Slow Cooker\n\nFamily Favorite Turkey Chili with Brown Rice\n\nIndian Lentil Stew Calorie Smart\n\nItalian Chicken-Lentil Soup Calorie Smart \u2022 Slow Cooker\n\nMeatless Minestrone\n\nMinestrone with Italian Sausage Calorie Smart\n\nSplit Pea Soup Calorie Smart\n\nWhite Chicken Chili Calorie Smart\n\n## BEEF, PORK AND LAMB\n\nBeef Pho Calorie Smart\n\nBeef Stew Calorie Smart\n\nBorscht Calorie Smart\n\nCheesy Potato-Bacon Soup Slow Cooker\n\nHearty Pork Stew Calorie Smart \u2022 Slow Cooker\n\nLamb Dijon Calorie Smart \u2022 Slow Cooker\n\nMoroccan Lamb Stew Calorie Smart\n\nSpicy Thai Pork Stew\n\nVegetable Beef Soup Calorie Smart\n\n## VEGETABLE\n\nButternut Squash Soup\n\nCream of Broccoli Soup Calorie Smart\n\nCream of Cauliflower Soup\n\nCream of Mushroom Soup\n\nFrench Onion Soup Calorie Smart\n\nGazpacho Calorie Smart\n\nGolden Onion Soup\n\nGreen Gazpacho\n\nRoasted Red Pepper Soup with Mozzarella Calorie Smart\n\nSriracha Gazpacho\n\nTomato-Basil Soup Calorie Smart\n\nTortilla Green Chili Calorie Smart \u2022 Fast\n\nWild Rice Soup Calorie Smart\n\n## FISH\n\nManhattan Clam Chowder Calorie Smart\n\nNew England Clam Chowder Calorie Smart\n\nOyster Stew Calorie Smart \u2022 Fast \u2022 Easy\n\nProven\u00e7al Fish Soup Calorie Smart \u2022 Fast\n\n## Bonus Recipes\n\nBean and Avocado Roll-Ups\n\nChopped Salad with Beans\n\nGreen Bean Casserole\n\nPork with Mushroom Sauce\n\nQuick Refried Beans\n\nTomato and Black Bean Salsa\n\nTwo-Bean Salad\n\nVegetables with Mushroom Sauce\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Soups, Stews & Chilies Basics\n\nSoups, stews and chilies are the original one-pot meals. While the difference between them is often not clear, all three are simple and often flexible dishes\u2014and are especially convenient if you have a slow cooker handy.\n\n  * Soups are combinations of vegetables, noodles or rice and often meat with a liquid base. Soups can be thick or thin and can be served hot or cold. Chowders fall in the soup category even though they can be very thick.\n  * Stews are made by cooking ingredients slowly in some liquid for a period of time until everything is very tender. Stews are often thicker than their soup cousins.\n  * Chilies are stew-like soups traditionally made with chili powder, chiles, meat and depending on what part of the country you are from, beans.\n\n## Storage\n\n  * Refrigerate soups, stews and chilies in shallow containers so they cool rapidly.\n  * Store in the refrigerator, covered tightly, for up to 3 days. Anything made with fish or shellfish should be stored no longer than 24 hours.\n  * To store in the freezer, pour into freezer containers, leaving \u00bd to 1 inch at the top of the container for expansion. Often freezer food storage bags will work too.\n  * Broth- and tomato-based soups, stews and chilies freeze well.\n  * Recipes made with cream may separate after freezing, so add the cream when reheating.\n  * Potatoes can get mushy when frozen, so add them when reheating.\n  * Thaw frozen soups, stews and chilies overnight in the refrigerator. Reheat over medium heat until hot. Those made with cream should be reheated over low heat, stirring frequently; do not boil, as these mixtures may separate.\n  * Soups and stews can become thicker during storage. While reheating, add a little broth, water or milk to thin a bit if necessary.\n\n## Tips for Great Soups, Stews and Chilies\n\n  * Use the pan size specified in the recipe. If the pan is too small, the mixture may boil over or heat too slowly, resulting in overcooked ingredients.\n  * To ensure even cooking, cut vegetables so they are similar in size.\n  * Rice and pasta absorb liquid, so if you are cooking ahead, cook them separately and add just before serving.\n  * Many recipes work well in the slow cooker\u2014no need to baby-sit a pan on the stove.\n\n# Heirloom Recipe and New Twist\n\nIt's hard to beat chicken noodle soup any time\u2014it's warm, comforting and tasty and it fills the house with a wonderful aroma while it's cooking. This recipe has been a favorite of the test kitchens for many years and is truly a classic. The new twist, with a definite Italian flair, is hearty with cheesy chicken meatballs and orzo pasta.\n\nHeirloom\n\nChicken Noodle Soup\n\n## Chicken Noodle Soup Calorie Smart\n\nPrep 15 min Total 40 min \u25cf 6 servings\n\n  * Chicken and Broth (page 143)\n  * 4 medium carrots, sliced (2 cups)\n  * 4 medium stalks celery, sliced (2 cups)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 cup uncooked medium egg noodles (2 oz)\n  * Chopped fresh parsley, if desired\n\n1 Refrigerate cut-up cooked chicken. Add enough water to broth to measure 5 cups.\n\n2 In 4-quart Dutch oven or stockpot, heat broth, carrots, celery and onion to boiling; reduce heat. Cover and simmer about 15 minutes or until carrots are tender.\n\n3 Stir in noodles and chicken. Heat to boiling; reduce heat. Simmer uncovered 7 to 10 minutes or until noodles are tender. Sprinkle with parsley.\n\n1 Serving (1 Cup): Calories 240; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 90mg; Sodium 990mg; Total Carbohydrate 14g (Dietary Fiber 3g, Sugars 5g); Protein 29g Exchanges: \u00bd Starch, 1 Vegetable, 3\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\nChicken Rice Soup Substitute \u00bd cup uncooked regular long-grain white rice for the noodles. Stir in rice with the vegetables. Cover and simmer about 15 minutes or until rice is tender. Stir in chicken; heat until chicken is hot.\n\nConfetti Chicken Noodle Soup Add \u00bd cup each frozen sweet peas and corn with the noodles and chicken. Continue as directed.\n\nHomestyle Chicken Noodle Soup Substitute 1\u00bd cups frozen homestyle egg noodles (from a 12-oz bag) for the uncooked egg noodles. Add frozen noodles with the onion.\nNew Twist\n\nChicken Meatball Soup\n\n## Chicken Meatball Soup Calorie Smart\n\nPrep 35 min Total 45 min \u25cf 6 servings\n\nMeatballs\n\n  * 1 lb ground chicken\n  * \u00bd cup seasoned dry bread crumbs\n  * 1 egg\n  * 3 oz fontina cheese, cut in 18 (\u00bd-inch) cubes\n\nSoup\n\n  * 2 tablespoons olive oil\n  * 2 cloves garlic, finely chopped\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 medium carrot, chopped (\u00bd cup)\n  * \u00bd cup uncooked orzo or rosamarina pasta\n  * \u2153 cup chopped fresh basil leaves\n  * 1 can (14 oz) diced tomatoes with basil, oregano and garlic, undrained\n  * 1 carton (32 oz) chicken broth or 4 cups Chicken and Broth (page 143)\n  * \u00bc teaspoon pepper\n\n1 Heat oven to 400\u00b0F. Line 13x9-inch pan with foil. In large bowl, mix chicken, bread crumbs and egg. Shape mixture into 18 (1\u00bd-inch) balls. Push 1 piece of cheese into center of each, pressing mixture around cheese and reshaping meatball. Place in pan.\n\n2 Bake uncovered 12 to 15 minutes or until firm and no longer pink in center next to cheese.\n\n3 Meanwhile, in 4-quart saucepan, heat oil over medium-low heat. Stir in garlic, onion and carrot. Cover and cook 10 minutes, stirring occasionally. Stir in remaining soup ingredients. Heat to boiling; reduce heat.\n\n4 Add meatballs. Cover and simmer about 10 minutes or until pasta is tender. To serve, place 3 meatballs in each serving bowl; spoon 1 cup soup over meatballs.\n\n1 Serving: Calories 300; Total Fat 16g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 90mg; Sodium 900mg; Total Carbohydrate 21g (Dietary Fiber 2g, Sugars 2g); Protein 18g Exchanges: 1\u00bd Other Carbohydrate, 2 Very Lean Meat, \u00bd Medium-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 1\u00bd\n\n## Chicken and Broth Calorie Smart\n\nPrep 25 min Total 1 hr 25 min \u25cf 4 cups broth and 2\u00bd to 3 cups cooked chicken\n\n  * 1 cut-up chicken or chicken parts (3 to 3\u00bd lb)\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * 1 medium stalk celery with leaves, cut up\n  * 1 medium carrot, cut up\n  * 1 small onion, cut up\n  * 1 sprig fresh parsley\n  * 4\u00bd cups cold water\n\n1 In 4-quart Dutch oven or pot, place chicken and all remaining ingredients; heat to boiling. Skim foam from broth; reduce heat. Cover and simmer 40 to 45 minutes or until juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F).\n\n2 Carefully remove chicken from broth with tongs. Cool chicken about 10 minutes or just until cool enough to handle. Strain broth through fine-mesh strainer; discard vegetables.\n\n3 Remove skin and bones from chicken. Cut chicken into \u00bd-inch pieces. Skim fat from broth. Use broth and chicken immediately, or cover and refrigerate broth and chicken in separate containers up to 24 hours or freeze up to 6 months.\n\n1 Serving (1 Cup Broth and about \u00bd Cup Chicken): Calories 180; Total Fat 7g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 85mg; Sodium 490mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 27g Exchanges: 4 Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\n### Skimming Fat\n\n**With Fat Separator:** Pour or ladle warm broth into fat separator; let stand a few minutes for fat to rise to surface. Pour out broth carefully, stopping when you get to the layer of fat.\n\n**With Spoon:** Refrigerate broth 6 to 8 hours or overnight, until fat hardens on surface. Carefully scoop fat from surface with spoon.\n\nChicken Tortilla Soup\n\n## Chicken Tortilla Soup Calorie Smart\n\nPrep 15 min Total 35 min \u25cf 4 servings\n\n  * 3 teaspoons vegetable oil\n  * 4 soft corn tortillas (5- to 6-inch), cut into 2x\u00bd-inch strips\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 cans (14 oz each) chicken broth (for homemade broth, see page 143)\n  * 1 can (10 oz) diced tomatoes with green chiles, undrained\n  * 1\u00bd cups shredded cooked chicken\n  * 1 tablespoon lime juice\n  * 1 tablespoon chopped fresh cilantro or parsley\n\n1 In 2-quart nonstick saucepan, heat 2 teaspoons of the oil over medium-high heat. Add tortilla strips; cook 30 to 60 seconds, stirring occasionally, until crisp and light golden brown. Drain on paper towels.\n\n2 In same saucepan, cook onion in remaining 1 teaspoon oil over medium-high heat, stirring occasionally, until tender. Stir in broth and tomatoes. Heat to boiling; reduce heat. Simmer uncovered 20 minutes. Stir in chicken; heat until hot.\n\n3 Stir in lime juice. Sprinkle tortilla strips on individual servings of soup or spoon soup over tortilla strips. Sprinkle with cilantro.\n\n1 Serving: Calories 230; Total Fat 8g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 45mg; Sodium 1100mg; Total Carbohydrate 17g (Dietary Fiber 3g, Sugars 4g); Protein 22g Exchanges: 1 Starch, 2\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\nItalian Chicken-Lentil Soup\n\n## Italian Chicken-Lentil Soup Calorie Smart \u2022 Slow Cooker\n\nPrep 15 min Total 5 hr 30 min \u25cf 6 servings\n\n  * 4 medium carrots, sliced (2 cups)\n  * 1 medium zucchini, chopped (2 cups)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 cup dried lentils (8 oz), sorted, rinsed\n  * 4\u00bd cups chicken broth (for homemade broth, see page 143)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 lb boneless skinless chicken thighs, trimmed of excess fat\n  * 1 can (28 oz) diced tomatoes, undrained\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * \u00bc cup chopped fresh or 1 tablespoon dried basil leaves\n  * Shredded Parmesan cheese, if desired\n\n1 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix carrots, zucchini, onion, lentils, broth, salt and pepper; top with chicken. Cover; cook on Low heat setting 5 to 6 hours.\n\n2 Remove chicken from slow cooker to cutting board or plate. Use 2 forks to pull chicken into shreds. Return chicken to slow cooker. Stir in tomatoes and mushrooms. Cover; cook about 15 minutes longer or until thoroughly heated. Sprinkle individual servings with basil. Serve with cheese.\n\n1 Serving: Calories 280; Total Fat 4g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 30mg; Sodium 1200mg; Total Carbohydrate 35g (Dietary Fiber 10g, Sugars 7g); Protein 25g Exchanges: 2 Starch, 1 Vegetable, 2\u00bd Very Lean Meat Carbohydrate Choices: 2\n\nWhite Chicken Chili\n\n## White Chicken Chili Calorie Smart\n\nTop servings of this tasty chili with shredded cheese, tortilla chips, chopped green onions, diced tomatoes, chopped fresh cilantro, sliced avocado or sour cream.\n\nPrep 20 min Total 45 min \u25cf 6 servings\n\n  * 1 tablespoon vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 3 cups chicken broth (for homemade broth, see page 143)\n  * 1 can (11 oz) white shoepeg or whole kernel sweet corn, drained\n  * 1 can (15.5 oz) great northern beans, drained\n  * 1 can (15.5 oz) butter beans, drained\n  * 2 tablespoons chopped fresh cilantro\n  * 2 tablespoons lime juice\n  * 1 teaspoon ground cumin\n  * \u00bd teaspoon dried oregano leaves\n  * \u00bc teaspoon red pepper sauce\n  * \u00bc teaspoon salt\n  * 2 cups chopped cooked chicken breast\n\n1 In 4-quart Dutch oven or stockpot, heat oil over medium heat. Cook onion and garlic in oil 4 to 6 minutes, stirring occasionally, until onion is tender.\n\n2 Stir in remaining ingredients except chicken. Heat to boiling; reduce heat. Simmer uncovered 20 minutes. Stir in chicken; simmer about 5 minutes or until hot.\n\n1 Serving (1\u2153 Cups): Calories 360; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 40mg; Sodium 920mg; Total Carbohydrate 46g (Dietary Fiber 11g, Sugars 4g); Protein 31g Exchanges: 2 Starch, 3 Vegetable, 2 Lean Meat Carbohydrate Choices: 3\n\nCreamy Herbed Chicken Stew\n\n## Creamy Herbed Chicken Stew Calorie Smart \u2022 Slow Cooker\n\nPrep 30 min Total 7 hr 40 min \u25cf 12 servings\n\n  * 4 cups ready-to-eat baby-cut carrots\n  * 4 medium Yukon Gold potatoes, cut into 1\u00bd-inch pieces\n  * 1 large onion, chopped (1 cup)\n  * 2 medium stalks celery, sliced (1 cup)\n  * 2 teaspoons dried thyme leaves\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon pepper\n  * 2 lb boneless skinless chicken thighs\n  * 3 cups chicken broth (for homemade broth, see page 143)\n  * 2 cups fresh sugar snap peas\n  * 1 cup whipping cream\n  * \u00bd cup all-purpose flour\n\n1 Spray 5- to 6-quart slow cooker with cooking spray. In slow cooker, place carrots, potatoes, onion and celery. Sprinkle with 1 teaspoon of the thyme, the salt and pepper. Top with chicken and broth.\n\n2 Cover; cook on Low heat setting 7 to 8 hours, adding peas for last 5 to 10 minutes of cooking.\n\n3 Using slotted spoon, remove chicken and vegetables from slow cooker; cover to keep warm. Increase heat setting to High. In small bowl, mix whipping cream, flour and remaining 1 teaspoon thyme; stir into cooking liquid. Cover; cook about 10 minutes or until thickened. Pour sauce over chicken and vegetables.\n\n1 Serving (1\u00bd Cups): Calories 270; Total Fat 13g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 70mg; Sodium 430mg; Total Carbohydrate 19g (Dietary Fiber 3g, Sugars 4g); Protein 20g Exchanges: 1 Starch, 1 Vegetable, 2 Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\n\n## Beef and Broth Calorie Smart\n\nPrep 30 min Total 3 hr 45 min \u25cf 6 cups broth and 1 cup cooked beef\n\n  * 2 lb beef shank cross cuts or soup bones\n  * 2 tablespoons vegetable oil, if desired\n  * 6 cups cold water\n  * 1 medium carrot, chopped (\u00bd cup)\n  * 1 medium stalk celery with leaves, chopped (\u00bd cup)\n  * 1 small onion, chopped (\u2153 cup)\n  * 1 teaspoon salt\n  * \u00bc teaspoon dried thyme leaves\n  * 5 black peppercorns\n  * 3 whole cloves\n  * 3 sprigs fresh parsley\n  * 1 dried bay leaf\n\n1 Remove marrow from centers of bones. In 4-quart Dutch oven or stockpot, melt marrow over medium heat until hot, or heat 2 tablespoons oil. Cook beef shanks in marrow or oil until brown on both sides.\n\n2 Add water; heat to boiling. Skim foam from broth. Stir in remaining ingredients; heat to boiling. Skim foam from broth; reduce heat. Cover and simmer 3 hours.\n\n3 Remove beef from broth. Cool beef about 10 minutes or just until cool enough to handle. Strain broth through fine-mesh strainer; discard vegetables and seasonings.\n\n4 Remove beef from bones. Cut beef into \u00bd-inch pieces. Skim fat from broth. Use broth and beef immediately, or cover and refrigerate broth and beef in separate containers up to 24 hours or freeze up to 6 months.\n\n1 Cup: Calories 120; Total Fat 3g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 50mg; Sodium 440mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 20g Exchanges: \u00bd Vegetable, 2\u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\nSlow-Cooker Directions Decrease water to 5 cups. Increase salt to 1\u00bc teaspoons. In 10-inch skillet, heat marrow or 2 tablespoons oil over medium heat. Cook beef in marrow or oil until brown on both sides. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix remaining ingredients; add beef. Cover; cook on Low heat setting 8 to 10 hours. Continue as directed in Step 3.\n\nBorscht\n\n## Borscht Calorie Smart\n\nThis recipe for a traditional favorite from Russia and Poland calls for mixed pickling spice, which is available at most grocery stores. It usually includes a fragrant combination of coarse pieces of allspice, bay leaves, cardamom, cinnamon, cloves, coriander, ginger, mustard seed and peppercorns.\n\nPrep 30 min Total 4 hr 10 min \u25cf 6 servings\n\n  * \u00be lb boneless beef chuck, tip or round roast, cut into \u00bd-inch cubes\n  * 1 smoked pork hock\n  * 4 cups water\n  * 1 can (10.5 oz) condensed beef broth\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 4 medium beets, cooked (page 234), or 1 can (15 oz) sliced beets, drained\n  * 3 cups shredded green cabbage\n  * 2 medium potatoes, peeled, cut into cubes (2 cups)\n  * 1 large onion, sliced\n  * 2 cloves garlic, finely chopped\n  * 1 tablespoon mixed pickling spice\n  * 2 teaspoons dill seed or 1 sprig fresh dill weed\n  * \u00bc cup red wine vinegar\n  * \u00be cup sour cream\n  * Chopped fresh dill weed, if desired\n\n1 In 4-quart Dutch oven or stockpot, place beef, pork hock, water, broth, salt and pepper. Heat to boiling; reduce heat. Cover and simmer 1 hour to 1 hour 30 minutes or until beef is tender.\n\n2 Shred beets, or cut into \u00bc-inch strips. Remove pork hock from soup; let stand until cool enough to handle. Remove pork from bone; cut pork into bite-size pieces.\n\n3 Add pork, beets, cabbage, potatoes, onion and garlic to soup; stir. Tie pickling spice and dill seed in cheesecloth bag or place in tea ball; add to soup. Cover and simmer 2 hours.\n\n4 Stir in vinegar. Simmer uncovered 10 minutes. Remove spice bag. Serve soup with sour cream. Sprinkle with fresh dill.\n\n1 Serving: Calories 270; Total Fat 14g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 60mg; Sodium 810mg; Total Carbohydrate 18g (Dietary Fiber 3g, Sugars 7g); Protein 18g Exchanges: 4 Vegetable, 2 Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\n\nFrench Onion Soup\n\n## French Onion Soup Calorie Smart\n\nThe long, slow cooking of the onions gives this soup its rich flavor and color. Gruy\u00e8re cheese is a rich, nutty, buttery-tasting form of Swiss cheese with fewer and smaller holes.\n\nPrep 35 min Total 1 hr 30 min \u25cf 4 servings\n\n  * 2 tablespoons butter\n  * 4 medium onions, sliced\n  * 2 cans (10\u00bd oz each) condensed beef broth\n  * 1\u00bd cups water\n  * \u215b teaspoon pepper\n  * \u215b teaspoon dried thyme leaves\n  * 1 dried bay leaf\n  * 4 slices (\u00be to 1 inch thick) French bread, toasted\n  * 1 cup shredded Gruy\u00e8re, Swiss or mozzarella cheese (4 oz)\n  * \u00bc cup grated Parmesan cheese\n\n1 In 4-quart nonstick Dutch oven or stockpot, melt butter over medium-high heat. Stir in onions to coat with butter. Cook uncovered 10 minutes, stirring every 3 to 4 minutes. Reduce heat to medium-low. Cook 35 to 40 minutes longer, stirring well every 5 minutes, until onions are deep golden brown (onions will shrink during cooking).\n\n2 Stir in broth, water, pepper, thyme and bay leaf. Heat to boiling; reduce heat. Cover and simmer 15 minutes. Remove bay leaf.\n\n3 Set oven control to broil. Place toasted bread in 4 ovenproof bowls or individual ceramic casseroles.* Spoon soup over bread; sprinkle with cheeses. Place bowls on cookie sheet or in pan with shallow sides.\n\n4 Broil with tops about 5 inches from heat 1 to 2 minutes or just until cheese is melted and golden brown. (Watch carefully so cheese does not burn.) Serve with additional French bread, if desired.\n\n*Do not use glass containers; they cannot withstand the heat from the broiler and may break.\n\n1 Serving: Calories 360; Total Fat 18g (Saturated Fat 10g, Trans Fat 1g); Cholesterol 50mg; Sodium 1210mg; Total Carbohydrate 28g (Dietary Fiber 3g, Sugars 7g); Protein 22g Exchanges: 2 Starch, 2 High-Fat Meat Carbohydrate Choices: 2\n\nGolden Onion Soup Omit French bread and cheeses; do not broil.\n\n### Caramelizing Onions\n\nCook onions 10 minutes, stirring every 3 to 4 minutes.\n\nContinue cooking onions, stirring well every 5 minutes, until deep golden brown.\n\n## Beef Pho Calorie Smart\n\nPho is a Vietnamese soup that features noodles, spicy broth and thinly sliced beef or chicken.\n\nPrep 45 min Total 1 hr 30 min \u25cf 6 servings\n\nBeef and Noodles\n\n  * 2 medium yellow onions, cut in half\n  * 1 piece (5 inches) gingerroot, peeled, cut lengthwise in half\n  * 6 whole cloves\n  * 5 whole star anise\n  * 1 cinnamon stick (3-inch)\n  * 1 carton (32 oz) beef broth (4 cups; for homemade broth, see page 146)\n  * 1 lb beef sirloin steak, 1 inch thick\n  * 7 oz Thai stir-fry rice noodles (from 14-oz box)\n  * 1 cup water\n  * \u00bc cup fish sauce\n  * 2 tablespoons palm sugar or brown sugar\n\nToppings, if desired\n\n  * Sliced green onions\n  * Sliced jalape\u00f1o chiles\n  * Lime wedges\n  * Thinly sliced fresh Thai basil leaves\n  * Fresh cilantro leaves\n  * Hoisin sauce\n  * Sriracha sauce\n\n1 Set oven control to broil. Spray broiler pan rack with cooking spray. Place onions and gingerroot on rack in pan. Broil with tops 4 to 6 inches from heat 10 to 15 minutes or until onions begin to char. Turn; broil 10 to 15 minutes longer or until onions begin to char. Remove from oven; set aside.\n\n2 Heat 4-quart Dutch oven or stockpot over medium heat. Add cloves, star anise and cinnamon stick. Cook over medium heat 2 to 3 minutes, stirring frequently, until toasted. Add 2 cups of the broth, the onions and gingerroot. Heat to boiling; reduce heat to low. Cover and simmer 30 minutes, stirring occasionally.\n\n3 Meanwhile, cut beef crosswise in half; place one half on plate. Freeze until partially frozen, 20 to 30 minutes (do not freeze solid). Cut remaining beef into 1-inch cubes; cover and refrigerate. Cook noodles as directed on package; keep warm. If noodles become sticky, rinse with hot water just before serving.\n\n4 Strain simmered broth through fine-mesh strainer into bowl; discard spices, onions and gingerroot. Return broth to Dutch oven. Add remaining 2 cups broth, the water, fish sauce, sugar and cubed beef. Heat to boiling; reduce heat. Simmer 10 to 15 minutes or until beef is tender.\n\n5 To serve, divide noodles among individual bowls. Cut partially frozen beef into paper-thin slices; place on noodles in each bowl. Immediately pour hot soup over beef to cover. Add desired toppings.\n\n1 Serving: Calories 260; Total Fat 3.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 50mg; Sodium 1690mg; Total Carbohydrate 33g (Dietary Fiber 1g, Sugars 5g); Protein 23g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 2\u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 2\n\n## Vegetable Beef Soup Calorie Smart\n\nThe prep time is long for this recipe, but if you already have the beef broth ready, you are almost there!\n\nPrep 20 min Total 1 hr 5 min \u25cf 7 servings\n\n  * Beef and Broth (page 146)\n  * 1 ear fresh sweet corn or \u00bd cup frozen whole kernel corn\n  * 2 medium potatoes, cut into pieces (2 cups)\n  * 2 medium tomatoes, chopped (1\u00bd cups)\n  * 2 medium carrots, thinly sliced (1 cup)\n  * 2 medium stalks celery, sliced (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 cup 1-inch pieces fresh green beans or frozen cut green beans\n  * 1 cup shelled green peas or frozen sweet peas\n  * \u00bc teaspoon pepper\n\n1 In 4-quart Dutch oven or stockpot, place strained beef and broth.\n\n2 Cut kernels from ear of corn. Stir corn and remaining ingredients into broth. Heat to boiling; reduce heat. Cover and simmer about 30 minutes or until vegetables are tender.\n\n1 Serving (1\u00bd Cups): Calories 240; Total Fat 9g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 45mg; Sodium 640mg; Total Carbohydrate 19g (Dietary Fiber 4g, Sugars 4g); Protein 20g Exchanges: 1 Starch, 1 Vegetable, 2 Lean Meat, \u00bd Fat Carbohydrate Choices: 1\n\nBeef Pho\n\nFamily-Favorite Chili\n\n## Family-Favorite Chili Calorie Smart \u2022 Slow Cooker\n\nPrep 20 min Total 6 hr 35 min \u25cf 8 servings\n\n  * 2 lb lean (at least 80%) ground beef*\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 1 can (28 oz) diced tomatoes, undrained\n  * 1 can (15 oz) tomato sauce\n  * 2 tablespoons chili powder\n  * 1\u00bd teaspoons ground cumin\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon pepper\n  * 1 can (15 to 16 oz) kidney or pinto beans, drained, rinsed\n  * Shredded Cheddar cheese, if desired\n\n1 In 12-inch skillet, cook beef over medium heat 8 to 10 minutes, stirring occasionally, until thoroughly cooked; drain.\n\n2 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix beef and remaining ingredients except beans and cheese. Cover; cook on Low heat setting 6 to 8 hours (or on High heat setting 3 to 4 hours).\n\n3 Stir in beans. If using Low heat setting, increase to High. Cover; cook 15 to 20 minutes longer or until slightly thickened. Sprinkle individual servings with cheese.\n\n*Always cook ground meat before adding to a slow cooker. Getting cold, uncooked ground beef to a safe temperature in a slow cooker takes too long to be safe.\n\n1 Serving: Calories 300; Total Fat 12g (Saturated Fat 4.5g, Trans Fat 0.5g); Cholesterol 70mg; Sodium 800mg; Total Carbohydrate 22g (Dietary Fiber 6g, Sugars 6g); Protein 27g Exchanges: 1 Starch, 1 Vegetable, 3\u00bd Lean Meat Carbohydrate Choices: 1\u00bd\n\nFamily-Favorite Turkey Chili with Brown Rice Substitute ground turkey breast for the ground beef, cooking until no longer pink in Step 1. Serve chili over cooked brown rice. Serve with reduced-fat Cheddar cheese.\n\nBeef Stew\n\n## Beef Stew Calorie Smart\n\nLook for beef stew meat, already trimmed and cut into \u00bd- to 1-inch pieces, in the meat department.\n\nPrep 15 min Total 3 hr 45 min \u25cf 8 servings\n\n  * 1 lb beef stew meat, cut into \u00bd-inch pieces\n  * 1 medium onion, cut into 8 wedges\n  * 1 bag (8 oz) ready-to-eat baby-cut carrots (about 30)\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 can (10.5 oz) condensed beef broth\n  * 1 can (8 oz) tomato sauce\n  * \u2153 cup all-purpose flour\n  * 1 tablespoon Worcestershire sauce\n  * 1 teaspoon salt\n  * 1 teaspoon sugar\n  * 1 teaspoon dried marjoram leaves\n  * \u00bc teaspoon pepper\n  * 12 small red potatoes (1\u00bd lb), cut into quarters\n  * 2 cups sliced fresh mushrooms (about 5 oz) or 1 package (about 3.5 oz) fresh shiitake mushrooms, sliced\n\n1 Heat oven to 325\u00b0F. In 4-quart ovenproof Dutch oven, mix all ingredients except potatoes and mushrooms. Cover and bake 2 hours, stirring once.\n\n2 Stir in potatoes and mushrooms. Cover and bake 1 hour to 1 hour 30 minutes longer or until beef and vegetables are tender.\n\n1 Serving: Calories 310; Total Fat 7g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 35mg; Sodium 820mg; Total Carbohydrate 43g (Dietary Fiber 6g, Sugars 7g); Protein 18g Exchanges: 2 Starch, 2 Vegetable, 1 Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 3\n\nSlow-Cooker Directions Chop onion (\u00bd cup). Omit tomato sauce. Increase flour to \u00bd cup. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except beef. Add beef (do not stir). Cover; cook on Low heat setting 8 to 9 hours. Stir well before serving.\n\nCheesy Potato-Bacon Soup\n\n## Cheesy Potato-Bacon Soup SLOW Cooker\n\nLook for bacon that is already cooked at the grocery store\u2014it's a great time-saver. Or pick up a package of microwavable bacon instead.\n\nPrep 15 min Total 6 hr 35 min \u25cf 6 servings\n\n  * 1 bag (32 oz) frozen Southern-style diced hash brown potatoes, thawed\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 medium stalk celery, diced (\u00bd cup)\n  * 3\u00bd cups chicken broth (for homemade broth, see page 143)\n  * 1 cup water\n  * 3 tablespoons all-purpose flour\n  * 1 cup half-and-half\n  * 2 cups shredded Cheddar cheese (8 oz)\n  * 12 slices bacon, crisply cooked, crumbled\n  * 4 medium green onions, thinly sliced (\u00bc cup)\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix potatoes, onion, celery, broth and water. Cover; cook on Low heat setting 6 to 8 hours.\n\n2 In small bowl, mix flour and half-and-half; stir into potato mixture. Increase heat setting to High. Cover; cook 20 to 30 minutes or until mixture thickens.\n\n3 Stir in cheese until melted; stir in half of the bacon. Sprinkle individual servings with remaining bacon and the green onions.\n\n1 Serving (1\u00bd Cups): Calories 410; Total Fat 15g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 45mg; Sodium 1210mg; Total Carbohydrate 50g (Dietary Fiber 5g, Sugars 5g); Protein 19g Exchanges: 3\u00bd Starch, 1 High-Fat Meat, 1 Fat Carbohydrate Choices: 3\n\n## Minestrone with Italian Sausage Calorie Smart\n\nPrep 45 min Total 45 min \u25cf 7 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 1 lb bulk sweet Italian pork sausage\n  * 2 medium carrots, coarsely chopped (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 teaspoons dried basil leaves\n  * 2 teaspoons finely chopped garlic\n  * 5\u00bc cups beef broth (for homemade broth, see page 146)\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 can (15.5 oz) great northern beans, drained, rinsed\n  * 1 cup uncooked small elbow macaroni (4 oz)\n  * 1 medium zucchini, cut in half lengthwise, then cut into \u00bc-inch slices (1 cup)\n  * 1 cup frozen cut green beans\n\n1 In 5-quart Dutch oven or stockpot, heat oil over medium-high heat. Add sausage, carrots, onion, basil and garlic. Cook 5 to 7 minutes, stirring frequently, until sausage is no longer pink; drain.\n\n2 Stir broth, tomatoes and great northern beans into sausage mixture. Heat to boiling; reduce heat to medium-low. Cover and cook 7 to 8 minutes, stirring occasionally.\n\n3 Stir in macaroni, zucchini and green beans. Heat to boiling over medium-high heat. Cook 5 to 6 minutes, stirring occasionally, until vegetables are hot and macaroni is tender.\n\n1 Serving (1\u00bd Cups): Calories 380; Total Fat 16g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 25mg; Sodium 1400mg; Total Carbohydrate 38g (Dietary Fiber 6g, Sugars 5g); Protein 20g Exchanges: 2 Starch, 1 Vegetable, 1\u00bd Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nMeatless Minestrone Use 1 additional can (15.5 oz) great northern beans (or your favorite canned beans) for the sausage, and substitute vegetable broth for the beef broth.\n\nHearty Pork Stew\n\n## Hearty Pork Stew Calorie Smart \u2022 Slow Cooker\n\nPrep 25 min Total 6 hr 55 min \u25cf 6 servings\n\n  * 1\u00bd lb boneless pork loin, cut into 1-inch cubes\n  * 2 cups diced (\u00bd-inch) peeled parsnips (3 medium)\n  * 1\u00bd cups cubed (1-inch) peeled butternut squash\n  * 3 medium carrots, cut into \u00bc-inch slices (1\u00bd cups)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 carton (32 oz) chicken broth (4 cups; for homemade broth, see page 143)\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon pepper\n  * 3 tablespoons all-purpose flour\n  * 3 tablespoons butter, softened\n\n1 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except flour and butter. Cover; cook on Low heat setting 6 to 7 hours (or on High heat setting 3 to 4 hours) or until pork is no longer pink and vegetables are tender.\n\n2 In small bowl, mix flour and butter. Gently stir into pork mixture, 1 spoonful at a time, until blended. Increase heat setting to High. Cover; cook 30 to 45 minutes, stirring occasionally, until thickened.\n\n1 Serving: Calories 290; Total Fat 11g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 65mg; Sodium 960mg; Total Carbohydrate 19g (Dietary Fiber 4g, Sugars 6g); Protein 27g Exchanges: 1 Starch, 1 Vegetable, 3 Lean Meat, \u00bd Fat Carbohydrate Choices: 1\n\nSpicy Thai Pork Stew\n\n## Spicy Thai Pork Stew\n\nPrep 30 min Total 1 hr 40 min \u25cf 6 servings\n\n  * \u2153 cup all-purpose flour\n  * \u00bd teaspoon garlic-pepper blend\n  * 4 boneless pork loin chops (1\u00bd lb), cut into 1-inch cubes\n  * 3 tablespoons vegetable oil\n  * 1 bottle (11.5 oz) peanut sauce\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1 teaspoon crushed red pepper flakes\n  * 2 cups cubed (1 inch) peeled butternut squash (about \u00be lb)\n  * 1 medium red bell pepper, cut into 1-inch pieces (1\u00bc cups)\n  * 4 oz fresh snow pea pods (1 cup), strings removed, cut diagonally in half\n  * Hot cooked white rice, if desired\n  * Chopped dry-roasted peanuts or fresh cilantro, if desired\n\n1 In large resealable food-storage plastic bag, mix flour and garlic-pepper blend. Add pork; seal bag and shake until pork is evenly coated. In 5-quart Dutch oven or stockpot, heat oil over medium heat. Cook pork in oil, stirring occasionally, until evenly golden brown.\n\n2 Stir in peanut sauce, broth and pepper flakes. Heat to boiling. Boil 1 minute, scraping browned bits from bottom of pan; reduce heat to low. Cover and simmer 30 minutes.\n\n3 Add squash. Cover and simmer 30 minutes or until pork and squash are tender.\n\n4 Stir in bell pepper and pea pods. Simmer uncovered 5 to 7 minutes or until bell pepper is crisp-tender. Serve over rice; garnish with peanuts or cilantro.\n\n1 Serving: Calories 480; Total Fat 30g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 70mg; Sodium 350mg; Total Carbohydrate 19g (Dietary Fiber 3g, Sugars 6g); Protein 34g Exchanges: 1 Starch, \u00bd Vegetable, 4 Lean Meat, 3\u00bd Fat Carbohydrate Choices: 1\n\nMoroccan Lamb Stew\n\n## Moroccan Lamb Stew Calorie Smart\n\nPrep 25 min Total 1 hr 10 min \u25cf 6 servings\n\n  * 1 lb boneless lamb shoulder, trimmed of excess fat, cut into cubes\n  * 1\u00bd cups chicken broth (for homemade broth, see page 143)\n  * 1 large onion, chopped (1 cup)\n  * 3 cloves garlic, finely chopped\n  * 1 tablespoon all-purpose flour\n  * 1 cup dry red wine or chicken broth\n  * 1 cup dried apricots\n  * \u00bc cup pitted dried plums\n  * \u00bc cup raisins\n  * 1 teaspoon paprika\n  * 1 can (14.5 oz) whole peeled tomatoes, undrained\n  * \u2153 cup chopped fresh parsley\n  * 3 cups hot cooked couscous\n\n1 Spray 4-quart Dutch oven or stockpot with cooking spray; heat over medium-high heat. Add lamb; cook, stirring frequently, until brown. Remove lamb; set aside.\n\n2 Add \u00bd cup of the broth to Dutch oven; heat to boiling. Stir in onion and garlic. Cook 3 min-utes, stirring frequently. Stir in flour. Cook 1 minute, stirring constantly.\n\n3 Stir in the lamb, remaining 1 cup broth, the wine, apricots, dried plums, raisins, paprika and tomatoes, breaking up tomatoes. Heat to boiling; reduce heat. Cover and simmer 45 minutes. Stir in parsley. Serve over couscous.\n\n1 Serving: Calories 350; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 50mg; Sodium 480mg; Total Carbohydrate 49g (Dietary Fiber 5g, Sugars 20g); Protein 23g Exchanges: 2 Starch, 1\u00bd Other Carbohydrate, 2 Very Lean Meat, \u00bd Lean Meat, \u00bd Fat Carbohydrate Choices: 3\n\nLamb Dijon\n\n## Lamb Dijon Calorie Smart \u2022 Slow Cooker\n\nPrep 25 min Total 8 hr 35 min \u25cf 6 servings\n\n  * \u00bc cup all-purpose flour\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 lb lamb stew meat, cut into 1-inch cubes\n  * 2 tablespoons olive or vegetable oil\n  * 6 small red potatoes (1\u00bc lb), cut into cubes\n  * \u00bc cup Dijon mustard\n  * \u00bd teaspoon grated lemon peel\n  * 1 tablespoon lemon juice\n  * 2 teaspoons chopped fresh or \u00bd teaspoon dried rosemary leaves\n  * 2 cloves garlic, finely chopped\n  * 1\u00be cups beef broth (for homemade broth, see page 146)\n  * 1 bag (12 oz) frozen sweet peas, thawed\n\n1 In large resealable food-storage plastic bag, mix flour, salt and pepper. Add lamb; seal bag and shake until lamb is evenly coated. In 12-inch skillet, heat oil over medium-high heat. Cook lamb in oil about 20 minutes, stirring occasionally, until brown; drain.\n\n2 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix lamb and remaining ingredients except peas. Cover; cook on Low heat setting 8 to 10 hours or until lamb is tender.\n\n3 Skim fat from juices in slow cooker. Stir peas into lamb mixture. Increase heat setting to High. Cover; cook 10 to 15 minutes or until peas are hot.\n\n1 Serving: Calories 360; Total Fat 14g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 85mg; Sodium 1040mg; Total Carbohydrate 28g (Dietary Fiber 4g, Sugars 3g); Protein 31g Exchanges: 2 Starch, 3\u00bd Lean Meat, \u00bd Fat Carbohydrate Choices: 2\n\n## Fish Broth Calorie Smart\n\nCelery leaves add great flavor to broth. Try adding chopped celery leaves along with the celery stalks called for in soup recipes.\n\nPrep 20 min Total 1 hr \u25cf 6 cups broth\n\n  * 1\u00bd lb fish bones and trimmings\n  * 4 cups cold water\n  * 2 cups dry white wine or clam juice\n  * 1 large stalk celery with leaves, chopped\n  * 1 small onion, sliced\n  * 3 medium fresh mushrooms, chopped\n  * 1 tablespoon lemon juice\n  * 1 teaspoon salt\n  * \u00bd teaspoon dried thyme leaves\n  * 3 sprigs fresh parsley\n  * 1 dried bay leaf\n\n1 Rinse fish bones and trimmings with cold water; drain. In 4-quart Dutch oven or stockpot, mix bones, trimmings, 4 cups water and remaining ingredients; heat to boiling. Skim foam from broth; reduce heat. Cover and simmer 30 minutes.\n\n2 Cool about 10 minutes. Strain broth through fine-mesh strainer; discard skin, bones, vegetables and seasonings. Use broth immediately, or cover and refrigerate up to 24 hours or freeze up to 6 months.\n\n1 Cup: Calories 15; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 400mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nProven\u00e7al Fish Soup\n\n## Proven\u00e7al Fish Soup Calorie Smart \u2022 Fast\n\n_Proven\u00e7al_ refers to dishes prepared in Provence, France, where ingredients like olive oil, onions, garlic and tomatoes are commonly used.\n\nPrep 25 min Total 25 min \u25cf 8 servings\n\n  * 1 tablespoon olive oil\n  * 1 medium onion, chopped (\u00bd cup)\n  * 4 cups fish or chicken broth (for homemade broth, see page 154 or page 143)\n  * 2 cans (14.5 oz each) diced tomatoes with basil, garlic and oregano, undrained\n  * \u00bd cup dry white wine or dry vermouth\n  * \u00bc teaspoon pepper\n  * 1 lb halibut fillet, skinned, cut into 1-inch cubes*\n  * Toasted baguette slices, shaved Parmesan cheese and chopped fresh parsley, if desired\n\n1 In 3-quart saucepan, heat oil over medium heat. Cook onion in oil 2 to 3 minutes, stirring occasionally, until crisp-tender.\n\n2 Add broth, tomatoes, wine and pepper. Heat to boiling; reduce heat. Simmer uncovered 10 minutes, stirring occasionally.\n\n3 Stir in fish. Heat to boiling; reduce heat. Simmer 5 minutes or until fish flakes easily with fork. Serve in shallow bowls topped with baguette slices and cheese; sprinkle with parsley.\n\n*Cod or grouper can be substituted for the halibut.\n\n1 Serving (1 Cup): Calories 100; Total Fat 2.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 25mg; Sodium 640mg; Total Carbohydrate 6g (Dietary Fiber 1g, Sugars 0g); Protein 11g Exchanges: \u00bd Other Carbohydrate, 1\u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## New England Clam Chowder Calorie Smart\n\nPrep 20 min Total 35 min \u25cf 4 servings\n\n  * 4 slices bacon, cut into \u00bd-inch pieces\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 medium stalk celery, sliced (\u00bd cup)\n  * 2 cans (6.5 oz each) minced or chopped clams, drained, \u00bc cup liquid reserved*\n  * 2\u00be cups milk or half-and-half\n  * 2 medium potatoes, peeled, diced (2 cups)\n  * \u00bc teaspoon salt\n  * Dash pepper\n  * \u00bc cup all-purpose flour\n  * Chopped fresh parsley, if desired\n\n1 In 3-quart saucepan, cook bacon, onion and celery over medium heat, stirring occasionally, until bacon is crisp and onion is tender; drain.\n\n2 Stir in clams, reserved clam liquid, \u00be cup of the milk, the potatoes, salt and pepper. Heat to boiling; reduce heat. Cover and simmer about 15 minutes or until potatoes are tender.\n\n3 In medium bowl, mix remaining 2 cups milk and the flour with whisk until smooth and blended. Stir into clam mixture. Heat to boiling, stirring frequently. Boil and stir 1 minute or until thickened. Garnish with parsley.\n\n*Or substitute 1 pint shucked fresh clams with their liquid for the canned clams. Chop the clams and stir in with the potatoes in Step 2.\n\n1 Serving (1\u00bc Cups): Calories 240; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 490mg; Total Carbohydrate 19g (Dietary Fiber 1g, Sugars 8g); Protein 29g Exchanges: \u00bd Starch, \u00bd Low-Fat Milk, 1 Vegetable, 3 Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\n## Manhattan Clam Chowder Calorie Smart\n\nCooks in Rhode Island in the late 1800s liked to throw tomatoes into clam chowder. Sometime in the mid-20th century, their creation became known as Manhattan clam chowder. It resembles New England clam chowder but is made with the tomatoes instead of milk or cream.\n\nPrep 20 min Total 35 min \u25cf 4 servings\n\n  * \u00bc cup chopped bacon or salt pork\n  * 1 small onion, finely chopped (\u2153 cup)\n  * 2 cans (6.5 oz each) minced or chopped clams, undrained*\n  * 2 medium potatoes, peeled, diced (2 cups)\n  * \u2153 cup chopped celery\n  * 1 cup water\n  * 1 can (14.5 oz) whole tomatoes, undrained\n  * 2 teaspoons chopped fresh parsley\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried thyme leaves\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n\n1 In 2-quart saucepan, cook bacon and onion over medium heat 8 to 10 minutes, stirring occasionally, until bacon is crisp and onion is tender; drain.\n\n2 Stir in clams, potatoes, celery and water. Heat to boiling; reduce heat. Cover and simmer about 15 minutes or until potatoes are tender.\n\n3 Stir in remaining ingredients, breaking up tomatoes. Heat until hot, stirring occasionally.\n\n*Or substitute 1 pint shucked fresh clams with their liquid for the canned clams. Chop the clams and stir in with the potatoes in Step 2.\n\n1 Serving: Calories 230; Total Fat 3g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 450mg; Total Carbohydrate 23g (Dietary Fiber 3g, Sugars 4g); Protein 26g Exchanges: 1 Starch, 2 Vegetable, 3 Very Lean Meat Carbohydrate Choices: 1\u00bd\n\n## Oyster Stew Calorie Smart \u2022 Fast \u2022 EASY\n\nPrep 15 min Total 15 min \u25cf 4 servings\n\n  * \u00bc cup butter\n  * 1 pint shucked oysters, undrained, or fresh shucked oysters in their liquid\n  * 2 cups milk\n  * \u00bd cup half-and-half\n  * \u00bd teaspoon salt\n  * Dash pepper\n  * Oyster crackers, if desired\n\n1 In 1\u00bd-quart saucepan, melt butter over low heat. Stir in oysters. Cook 2 to 4 minutes, stirring occasionally, just until edges curl.\n\n2 In 2-quart saucepan, heat milk and half-and-half over medium-low heat until hot. Stir in salt, pepper and oyster mixture; heat until hot. Serve with oyster crackers.\n\n1 Serving (1 Cup): Calories 290; Total Fat 20g (Saturated Fat 10g, Trans Fat 1g); Cholesterol 115mg; Sodium 700mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 8g); Protein 14g Exchanges: 1 Low-Fat Milk, 1 Medium-Fat Meat, 3 Fat Carbohydrate Choices: 1\n\n## Vegetable Broth Calorie Smart\n\nMild vegetables are preferred for a well-balanced neutral flavor. Broccoli, rutabaga, red cabbage or other strong-flavored vegetables would overpower the broth or give it a muddy, objectionable color.\n\nPrep 20 min Total 1 hr 30 min \u25cf 8 cups broth\n\n  * 8 cups cold water\n  * 6 cups coarsely chopped mild vegetables (bell peppers, carrots, celery, leeks, mushrooms,* potatoes, spinach or zucchini)\n  * 1 medium onion, coarsely chopped (\u00bd cup)\n  * 4 cloves garlic, finely chopped\n  * \u00bd cup fresh parsley sprigs\n  * 2 tablespoons chopped fresh or 2 teaspoons dried basil leaves\n  * 2 tablespoons chopped fresh or 2 teaspoons dried thyme leaves\n  * 1 teaspoon salt\n  * \u00bc teaspoon cracked black pepper\n  * 2 dried bay leaves\n\n1 In 4-quart Dutch oven or stockpot, mix all ingredients. Heat to boiling; reduce heat. Cover and simmer 1 hour, stirring occasionally.\n\n2 Cool about 10 minutes. Strain broth through fine-mesh strainer; discard vegetables and seasonings. Use broth immediately, or cover and refrigerate up to 24 hours or freeze up to 6 months. Stir before measuring.\n\n*Some mushrooms have woody stems that cannot be eaten, but they can be used to flavor the broth and then discarded.\n\n1 Cup: Calories 0; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Storing Vegetable Broth\n\nFreeze broth in ice-cube trays. Once frozen, put broth into freezer plastic bags and store in freezer.\n\nGazpacho\n\n## Gazpacho Calorie Smart\n\nPrep 15 min Total 1 hr 15 min \u25cf 8 servings\n\n  * 1 can (28 oz) whole tomatoes, undrained\n  * 1 medium green bell pepper, finely chopped (1 cup)\n  * 1 cup finely chopped cucumber\n  * 1 cup croutons\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 tablespoons dry white wine or chicken broth\n  * 2 tablespoons olive or vegetable oil\n  * 1 tablespoon ground cumin\n  * 1 tablespoon white vinegar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 In blender or food processor, place tomatoes, \u00bd cup of the bell pepper, \u00bd cup of the cucumber, \u00bd cup of the croutons, \u00bc cup of the onion and the remaining ingredients. Cover and blend on medium speed 30 to 60 seconds until smooth. Pour into large bowl.\n\n2 Cover and refrigerate at least 1 hour. Serve remaining bell pepper, cucumber, onion and croutons as accompaniments.\n\n1 Serving (\u00bd Cup): Calories 80; Total Fat 4g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 320mg; Total Carbohydrate 10g (Dietary Fiber 2g, Sugars 4g); Protein 2g Exchanges: 2 Vegetable, \u00bd Fat Carbohydrate Choices: \u00bd\n\nSriracha Gazpacho Add 1 to 2 teaspoons Sriracha sauce to the soup mixture. Just before serving, sprinkle with chopped fresh cilantro.\n\nGreen Gazpacho Use fresh green heirloom tomatoes (such as Green Zebra) instead of the canned tomatoes. Cut up to make about 4 cups. Add 1 avocado, pitted and peeled, to tomato mixture.\n\n## Tomato-Basil Soup Calorie Smart\n\nFully ripe, juicy tomatoes provide the best flavor for this soup. If your tomatoes aren't completely ripe, you may need to increase the salt and sugar just a bit.\n\nPrep 30 min Total 1 hr \u25cf 4 servings\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 medium carrot, finely chopped (\u00bd cup)\n  * 1 medium onion, finely chopped (\u00bd cup)\n  * 1 clove garlic, finely chopped\n  * 6 large tomatoes, peeled, seeded and chopped (6 cups)*\n  * 1 can (8 oz) tomato sauce\n  * \u00bc cup thinly sliced fresh or 2 teaspoons dried basil leaves\n  * \u00bd teaspoon sugar\n  * \u00bc teaspoon salt\n  * Dash pepper\n\n1 In 3-quart saucepan, heat oil over medium heat. Cook carrot, onion and garlic in oil about 10 minutes, stirring occasionally, until tender but not browned.\n\n2 Stir in tomatoes. Cook uncovered about 10 minutes, stirring occasionally, until thoroughly heated.\n\n3 Stir in remaining ingredients. Cook uncovered about 10 minutes, stirring occasionally, until hot.\n\n*Or substitute 3 cans (14.5 oz each) diced tomatoes, undrained, for the fresh tomatoes. Omit salt; increase sugar to 1 teaspoon.\n\n1 Serving: Calories 170; Total Fat 8g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 550mg; Total Carbohydrate 22g (Dietary Fiber 6g, Sugars 10g); Protein 4g Exchanges: 4 Vegetable, 1 Fat Carbohydrate Choices: 1\u00bd\n\nRoasted Red Pepper Soup with Mozzarella\n\n## Roasted Red Pepper Soup with Mozzarella Calorie Smart\n\nPrep 40 min Total 1 hr 10 min \u25cf 4 servings\n\n  * 4 red bell peppers\n  * 1 tablespoon olive oil\n  * 2 large onions, chopped (2 cups)\n  * 3 cloves garlic, sliced\n  * 2 cups vegetable broth or reduced-sodium chicken broth (for homemade broth, see page 156)\n  * 1 cup water\n  * \u00bc teaspoon cracked black pepper\n  * 1 cup thinly sliced fresh basil leaves\n  * \u00bd yellow bell pepper, diced\n  * 8 bocconcini (small fresh mozzarella cheese balls), cut into quarters\n  * 4 slices crusty multigrain or whole wheat bread\n\n1 Set oven control to broil. On rack in broiler pan, place red bell peppers. Broil with tops about 5 inches from heat, turning occasionally, until skin is blistered and evenly browned. Place roasted peppers in large bowl; cover with plastic wrap. Let stand 15 minutes.\n\n2 In 4-quart saucepan, heat oil over medium-low heat. Cook onions and garlic in oil 7 to 9 minutes, stirring occasionally, until onions begin to turn brown; remove from heat.\n\n3 Remove skin, stems, seeds and membranes from roasted peppers; cut peppers into strips. Stir bell pepper strips, broth, water and black pepper into onion mixture. Heat to boiling; reduce heat. Simmer uncovered 10 minutes, stirring occasionally. Stir in \u00bd cup of the basil. Remove from heat; cool slightly.\n\n4 In blender or food processor, place about one-third of the soup mixture. Cover and blend on high speed until smooth, stopping blender to scrape side if necessary. Pour into large bowl. Repeat two times more with remaining soup mixture.\n\n5 Top individual servings with yellow bell pepper and mozzarella; sprinkle evenly with remaining \u00bd cup basil. Serve with bread.\n\n1 Serving (about 1\u00bd Cups): Calories 200; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 620mg; Total Carbohydrate 29g (Dietary Fiber 6g, Sugars 13g); Protein 7g Exchanges: 1\u00bd Starch, 1\u00bd Vegetable, 1 Fat Carbohydrate Choices: 2\n\n## Learn to Puree\n\nThe quick definition of puree is to mash or blend food until smooth, using a blender or food processor or by forcing food through a sieve. But there is more to this process than one might think. Pureed soups are likely the most famous foods prepared using this method. But you can also puree foods to feed babies, or foods like sweet potatoes can be pureed to serve as part of a meal.\n\nEach appliance for pureeing works a little differently.\n\n  1. 1. **Blender:** This provides the smoothest puree. Blend in small batches for the best results.\n  2. 2. **Food Processor:** Because they tend to chop, not blend, food processors do not provide the smoothest mixture. The soup or vegetable mixture is often gritty with small bits of food left behind.\n  3. 3. **Immersion Blender:** This tool will provide a smooth puree with the least work or cleanup. Immersion blenders are made to go right into pans of hot mixture, eliminating the need to puree soup in batches in a blender or food processor. To avoid damaging the motor, submerge the blender just to the top of the stem. Follow the manufacturer's directions before using or washing.\n\n### Tips for Great Results\n\n  * Food should be fully cooked before pureeing. If it is not, it is more difficult to get the puree really smooth. For the best flavor, don't overcook foods that will be pureed.\n  * Blend in a blender or process in a food processor in batches. Having too much food in either appliance will make it difficult to create a smooth puree. Adding the food with a ladle works well.\n  * Hot foods can release a lot of steam, so allow the soup to cool slightly before pureeing. Also, pureeing small amounts at a time allows for room in the appliance for the steam. Trapped steam can push the lid off, so for safety, leave the lid slightly cracked with a kitchen towel over the edge. If the lid has a vent, leave it open.\n  * If you have the option of more than one speed, start slow and increase the speed as you puree.\n  * For the smoothest soups, after pureeing, pass the soup through a strainer into the soup pot. The food pieces left behind can be re-pureed.\n  * For baby food, cook the food in a small amount of water. Drain the vegetables, but save the water. Add small amounts of the water while you process or blend the food until you have the consistency you like. Foods such as avocado or watermelon do not need to be cooked before pureeing.\n\n## Butternut Squash Soup\n\nSprinkle bowls of soup with pumpkin seeds or toasted walnuts.\n\nPrep 30 min Total 45 min \u25cf 6 servings\n\n  * 2 tablespoons butter\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 butternut squash (2 lb), peeled, seeded and cut into pieces (page 237)*\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * \u00bd teaspoon dried marjoram leaves\n  * \u00bc teaspoon black pepper\n  * \u215b teaspoon ground red pepper (cayenne)\n  * 1 package (8 oz) cream cheese, cut into cubes\n  * Additional chicken broth or water, if desired\n\n1 In 3-quart saucepan, melt butter over medium heat. Cook onion in butter, stirring occasionally, until crisp-tender.\n\n2 Add squash, the 2 cups broth, the marjoram, black pepper and red pepper. Heat to boiling; reduce heat. Cover and simmer 12 to 15 minutes or until squash is tender.\n\n3 In blender or food processor, place one-third each of the soup mixture and cream cheese. Cover and blend on high speed until smooth, scraping down side of blender if needed. Repeat twice with remaining soup mixture and cream cheese. Return mixture to saucepan. Heat over medium heat, stirring with whisk, until blended and hot (do not boil). Add additional broth if thinner consistency is desired.\n\n*Squash will be easier to peel if you microwave it first. Pierce whole squash with knife in several places to allow steam to escape. Place on microwavable paper towel; microwave on High 4 to 6 minutes or until squash is hot and peel is firm but easy to cut. Cool slightly before peeling.\n\n1 Serving: Calories 240; Total Fat 17g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 50mg; Sodium 1220mg; Total Carbohydrate 15g (Dietary Fiber 2g, Sugars 6g); Protein 5g Exchanges: 1 Starch, 3\u00bd Fat Carbohydrate Choices: 1\n\nButternut Squash Soup\n\n## Cream of Broccoli Soup Calorie Smart\n\nPrep 35 min Total 45 min \u25cf 8 servings\n\n  * 1\u00bd lb fresh broccoli\n  * 2 cups water\n  * 1 large stalk celery, chopped (\u00be cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 tablespoons butter\n  * 2 tablespoons all-purpose flour\n  * 2\u00bd cups chicken broth (for homemade broth, see page 143)\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n  * Dash ground nutmeg\n  * \u00bd cup whipping cream\n  * Shredded Cheddar cheese, if desired\n\n1 Remove florets from broccoli. Cut stalks into 1-inch pieces, discarding any leaves.\n\n2 In 3-quart saucepan, heat water to boiling. Add broccoli florets and stalk pieces, celery and onion. Heat to boiling; reduce heat. Cover and simmer 7 to 10 minutes or until broccoli is tender (do not drain).\n\n3 In blender, carefully place broccoli mixture. Cover and blend on medium speed 30 to 60 seconds or until smooth. Set aside.\n\n4 In same saucepan, melt butter over medium heat. Stir in flour. Cook, stirring constantly, until mixture is smooth and bubbly; remove from heat. Stir in broth. Heat to boiling, stirring constantly. Boil and stir 1 minute.\n\n5 Stir in broccoli mixture, salt, pepper and nutmeg; heat just to boiling. Stir in whipping cream; heat just until hot (do not boil or soup may curdle). Top individual servings with cheese.\n\n1 Serving (1 Cup): Calories 110; Total Fat 8g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 510mg; Total Carbohydrate 6g (Dietary Fiber 2g, Sugars 2g); Protein 4g Exchanges: 2 Vegetable, 1 Fat Carbohydrate Choices: \u00bd\n\nCream of Cauliflower Soup Substitute 1 medium head cauliflower (about 2 pounds), separated into florets, for the broccoli. Add 1-tablespoon lemon juice with the vegetables in Step 2.\n\nWild Rice Soup\n\n## Wild Rice Soup Calorie Smart\n\nPrep 35 min Total 1 hr 10 min \u25cf 7 servings\n\n  * \u00bc cup butter\n  * 4 medium stalks celery, sliced (2 cups)\n  * 2 medium carrots, coarsely shredded (2 cups)\n  * 1 large onion, chopped (1 cup)\n  * 1 medium green bell pepper, chopped (1 cup)\n  * \u00bc cup plus 2 tablespoons all-purpose flour\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * 3 cups cooked wild rice (page 215)\n  * 2 cups water\n  * 2 cans (10.5 oz each) condensed chicken broth\n  * 3 cups half-and-half\n  * \u2154 cup slivered almonds, toasted if desired (page 23)\n  * \u00bd cup chopped fresh parsley\n\n1 In 4-quart Dutch oven or stockpot, melt butter over medium-high heat. Cook celery, carrots, onion and bell pepper in butter about 10 minutes, stirring frequently, until crisp-tender.\n\n2 Stir in flour, salt and pepper. Stir in wild rice, water and broth. Heat to boiling; reduce heat. Cover and simmer 15 minutes, stirring occasionally.\n\n3 Stir in half-and-half, almonds and parsley. Heat just until hot (do not boil or soup may curdle).\n\n1 Serving: Calories 360; Total Fat 20g (Saturated Fat 11g, Trans Fat 1g); Cholesterol 55mg; Sodium 1240mg; Total Carbohydrate 33g (Dietary Fiber 4g, Sugars 9g); Protein 12g Exchanges: 2 Starch, 1 Vegetable, \u00bd Very Lean Meat, 3\u00bd Fat Carbohydrate Choices: 2\n\nChicken and Wild Rice Soup Stir in 4 cups cubed cooked chicken or turkey in Step 3.\n\n## Make-Ahead Cream of Mushroom Soup\n\nThe flavor of homemade cream of mushroom soup is amazing; it takes less than 30 minutes and has so many uses. Why not try it?\n\nPREP 15 min TOTAL 30 min \u25cf 4 servings\n\n  * 1 lb fresh mushrooms, any type\n  * \u00bc cup butter\n  * 3 tablespoons all-purpose flour\n  * \u00bc teaspoon salt\n  * 1 cup whipping cream\n  * 1 can (14 oz) chicken broth (1\u2154 cups)\n  * 1 tablespoon dry sherry, if desired\n  * Freshly ground pepper, if desired\n\n1 Slice only enough mushrooms to measure 1 cup. Coarsely chop remaining mushrooms.\n\n2 In 3-quart saucepan, melt butter over medium heat. Cook sliced and chopped mushrooms in butter about 10 minutes, stirring occasionally, until golden brown. Sprinkle with flour and salt, stirring until well mixed.\n\n3 Gradually stir in whipping cream and broth; heat to boiling. Cook 2 to 3 minutes, until thickened, stirring constantly. Stir in sherry. Sprinkle each serving with pepper.\n\n1 Serving: Calories 350; Total Fat 31g (Saturated Fat 18g, Trans Fat 1g); Cholesterol 95mg; Sodium 840mg; Total Carbohydrate 11g (Dietary Fiber 2g, Sugars 3g); Protein 7g Exchanges: 3 Vegetable, 6 Fat Carbohydrate Choices: 1\n\nLighter Directions For 7 grams of fat and 145 calories per serving, decrease butter to 2 tablespoons and substitute fat-free half-and-half for the whipping cream.\n\n## Ways to Use Cream of Mushroom Soup\n\n  1. 1. **Green Bean Casserole:** Cook 2 lbs fresh beans, cut into 2-inch pieces, until crisp-tender (see page 234); drain. Mix beans with 1\u00bd cups of the soup and \u00bc teaspoon pepper. Spread in 2-quart baking dish; sprinkle with 1 can (2.8 oz) French-fried onions. Bake at 350\u00b0F 30 to 40 minutes or until heated through.\n  2. 2. **Chicken-Mushroom Soup:** Make soup as directed. Stir in 1 cup chopped cooked chicken and \u00bd cup cooked peas. Heat over low heat until hot, stirring frequently.\n  3. 3. **Pork with Mushroom Sauce:** Broil or grill pork chops. In small saucepan, heat soup (\u00bd cup per serving) over low heat. Serve over pork chops; sprinkle with chopped fresh chives.\n  4. 4. **Vegetables with Mushroom Sauce:** Cook 4 cups cauliflower florets and baby carrots until fork-tender. Heat 1 cup mushroom soup over low heat, stirring frequently. Spoon soup over vegetables. Sprinkle with chopped fresh chives.\n\nGreen Bean Casserole, Cream of Mushroom Soup\n\n## Split Pea Soup Calorie Smart\n\nPrep 20 min Total 2 hr 20 min \u25cf 8 servings\n\n  * 1 bag (16 oz) dried green or yellow split peas (2 cups), sorted, rinsed\n  * 8 cups water\n  * \u00bc teaspoon pepper\n  * 1 large onion, chopped (1 cup)\n  * 2 medium stalks celery, finely chopped (1 cup)\n  * 1 ham bone or 2 lb ham shanks or smoked pork hocks\n  * 3 medium carrots, cut into \u00bc-inch slices (1\u00bd cups)\n\n1 In 4-quart Dutch oven or stockpot, mix all ingredients except carrots. Heat to boiling; reduce heat. Cover and simmer 1 hour to 1 hour 30 minutes.\n\n2 Remove ham bone; let stand until cool enough to handle. Remove ham from bone. Remove excess fat from ham; cut ham into \u00bd-inch pieces.\n\n3 Stir ham and carrots into soup. Heat to boiling; reduce heat. Cover and simmer about 30 minutes or until carrots are tender and soup is desired thickness.\n\n1 Serving: Calories 250; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 15mg; Sodium 250mg; Total Carbohydrate 32g (Dietary Fiber 16g, Sugars 3g); Protein 16g Exchanges: 2 Starch, 1 Vegetable, 1 Very Lean Meat Carbohydrate Choices: 2\n\n## Indian Lentil Stew Calorie Smart\n\nPrep 20 min Total 1 hr \u25cf 4 servings\n\n  * 2 tablespoons butter\n  * 1 large onion, chopped (1 cup)\n  * 1 tablespoon curry powder\n  * 2 tablespoons all-purpose flour\n  * 1 can (14 oz) vegetable broth (for homemade broth, see page 156)\n  * \u00be cup dried lentils (6 oz), sorted, rinsed\n  * \u00bd teaspoon salt\n  * \u00bd cup apple juice\n  * 3 cups 1-inch pieces peeled dark-orange sweet potatoes\n  * 1 cup frozen sweet peas\n  * Sour cream or plain yogurt, if desired\n  * Chutney, if desired\n\n1 In 3-quart saucepan, melt butter over medium-high heat. Cook onion and curry powder in butter 2 minutes, stirring occasionally. Stir in flour. Gradually add broth, stirring constantly, until thickened.\n\n2 Stir in lentils and salt. Reduce heat to low. Cover and simmer 20 minutes, stirring occasionally.\n\n3 Stir in apple juice, sweet potatoes and peas. Heat to boiling; reduce heat to low. Cover and simmer 15 to 20 minutes, stirring occasionally, until vegetables are tender. Top individual servings with sour cream and chutney.\n\n1 Serving: Calories 360; Total Fat 7g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 790mg; Total Carbohydrate 62g (Dietary Fiber 12g, Sugars 22g); Protein 14g Exchanges: 2\u00bd Starch, 1\u00bd Other Carbohydrate, 1 Very Lean Meat, 1 Fat Carbohydrate Choices: 4\n\n## Cuban Black Bean Soup Calorie Smart\n\nPrep 20 min Total 2 hr 35 min \u25cf 8 servings\n\n  * 2 tablespoons vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 3 cloves garlic, finely chopped\n  * 1 bag (16 oz) dried black beans, sorted, rinsed (2 cups)\n  * 1 cup finely chopped cooked ham\n  * 3 cups beef broth (for homemade broth, see page 146)\n  * 3 cups water\n  * \u00bc cup dark rum or apple cider\n  * 1\u00bd teaspoons ground cumin\n  * 1\u00bd teaspoons dried oregano leaves\n  * 1 medium green bell pepper, chopped (1 cup)\n  * 1 large tomato, chopped (1 cup)\n  * Chopped Hard-Cooked Eggs (page 91), if desired\n  * Additional chopped onion, if desired\n\n1 In 4-quart Dutch oven or stockpot, heat oil over medium heat. Cook the 1 cup chopped onion and the garlic in oil 4 to 6 minutes, stirring occasionally, until onion is tender.\n\n2 Stir in remaining ingredients except eggs and additional onion; heat to boiling. Boil 2 minutes; reduce heat. Cover and simmer about 2 hours or until beans are tender. Top individual servings with eggs and onion.\n\n1 Serving (1\u00bd Cups): Calories 290; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 610mg; Total Carbohydrate 40g (Dietary Fiber 14g, Sugars 6g); Protein 19g Exchanges: 2 Starch, 2 Vegetable, 1 Very Lean Meat Carbohydrate Choices: 2\u00bd\n\n## Betty's Staples Canned Beans\n\nIf you are looking to put healthful meals together quickly, reach for canned beans. Traditionally, cooks soak dried beans overnight before a long cooking. But with canned beans, there's no need for soaking. Plus, beans deliver a good dose of inexpensive protein as well as fiber. Varieties of beans that are readily available include red kidney, cannellini, pinto, black, navy and chickpeas (garbanzo). To remove some of the sodium from canned beans, rinse well in a colander. Beans in all of the following recipes have been drained and rinsed.\n\n  1. 1. **Two-Bean Salad:** In large bowl, mix 1 can each (15 ounces each) red kidney beans and chickpeas with 1 cup chopped red bell pepper, \u00bc cup sliced green onions and \u00bd cup Italian Dressing (page 137). Cover and refrigerate at least 4 hours to marinate.\n  2. 2. **Quick Refried Beans:** Heat 1 tablespoon vegetable oil in 10-inch skillet. Add 2 cans (15 ounces each) pinto beans, 2 teaspoons hot pepper sauce, \u00bc teaspoon salt and \u00bc teaspoon garlic powder. Cook and stir over medium heat 10 minutes, mashing as you stir. Add 1 to 2 tablespoons water if necessary.\n  3. 3. **Chunky Tomato-Bean Soup:** In 4-quart saucepan, cook 1 cup each chopped onion and carrot in 1 tablespoon olive oil over medium heat until tender. Stir in 2 cans (14.5 ounces each) diced tomatoes with basil, garlic and oregano, 2 cans (15 ounces each) cannellini beans, 4 cups chicken broth and \u2153 cup sliced sun-dried tomatoes in oil. Heat to boiling; reduce heat to medium-low and cook 10 to 15 minutes or until hot, stirring occasionally. Top servings with shredded Asiago cheese.\n  4. 4. **Tomato and Black Bean Salsa:** In medium bowl, mix 2 cups chopped fresh tomatoes, 1 can (15 ounces) black beans, 1 finely chopped jalape\u00f1o chile, 2 tablespoons chopped fresh cilantro and 2 tablespoons finely chopped green onions. Serve with tortilla chips.\n  5. 5. **Bean and Avocado Roll-Ups:** For each roll-up, spoon about \u00bd cup black beans and \u00bc cup corn salsa on an 8-inch flour tortilla; drizzle with 1 to 2 tablespoons ranch dressing. Top with 1 cup shredded lettuce and slices of avocado. Roll up.\n  6. 6. **Chopped Salad with Beans:** In large bowl, mix 3 cups chopped romaine, 1 cup chopped tomato, 1 can (15 ounces) red kidney beans, 1 cup cooked corn, \u00bd cup chopped bell pepper, 1 cup chopped Genoa salami and \u00bd cup balsamic vinaigrette or Italian Dressing (page 137).\n\nTomato and Black Bean Salsa, Two-Bean Salad, Chunky Tomato-Bean Soup\n\n## Slow Cooker Success\n\nUse these tips to ensure great results from your slow cooker:\n\n  * Spray the inside of the insert or cooker with cooking spray for easy cleanup.\n  * Slow cookers work most efficiently when two-thirds to three-fourths full.\n  * Always keep the slow cooker covered for the specified cooking time. Each time the lid is removed, it allows heat to escape, adding 15 to 20 minutes to the cooking time. Exception: large cuts of meat should be rotated halfway through cooking for best results.\n  * Root vegetables take longer to cook than other vegetables, so cut them smaller and put them on the bottom of the cooker.\n  * Cover raw potatoes with liquid to prevent them from turning dark.\n\n## Slow Cooker Safety\n\nSlow cookers heat up more slowly and cook at lower temperatures than other appliances. To keep food safe, follow these guidelines.\n\n  * Use ingredients that are completely thawed.\n  * Keep perishable foods refrigerated until you are ready to put them in the cooker.\n  * Always cook and drain ground meat just before placing in the cooker\u2014not ahead of time.\n  * Brown poultry and meats just before placing in the cooker\u2014not ahead of time.\n  * Whole chickens or meat loaf shouldn't be put in a cooker. The centers can't reach a safe temperature quickly enough.\n  * Follow recipes exactly for specific sizes when cutting ingredients and for layering so that the dish cooks properly.\n  * Place leftovers in shallow containers and refrigerate within 2 hours of cooking. Don't use the insert for storage.\n  * Always reheat foods with your microwave, conventional oven or stovetop before placing in the slow cooker to keep warm until serving. Don't use a slow cooker for heating up food.\n\n## Testing Bean Doneness\n\nIf you test just one bean when cooking dried beans, you might be fooled into thinking they are all tender. Choose about six beans from different spots in the pan to pierce with a fork. If they all aren't tender, cook them a bit longer.\n\nTortilla Green Chili\n\n## Tortilla Green Chili Calorie Smart \u2022 Fast\n\nPrep 15 min Total 15 min \u25cf 4 servings\n\n  * 1 can (15 oz) black beans with cumin and chili, undrained\n  * 1 can (14.5 oz) diced tomatoes with green chiles, undrained\n  * 1 bag (12 oz) frozen whole kernel corn\n  * 1 jar (16 oz) mild green tomatillo salsa\n  * \u00bd cup coarsely crushed tortilla chips (about 1 oz)\n  * 1 avocado, pitted, peeled and cut into cubes\n  * 1 lime, sliced, then slices cut into quarters\n  * \u00bc cup sour cream\n  * Additional tortilla chips, if desired.\n\n1 In 4-quart saucepan, stir together beans, tomatoes, corn, salsa and tortilla chips. Heat to boiling; reduce heat to medium.\n\n2 Cover and simmer 5 to 10 minutes, stirring occasionally, until slightly thickened. Top individual servings with avocado, lime, sour cream and if desired, additional tortilla chips.\n\n1 Serving (1\u00bd Cups): Calories 390; Total Fat 13g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 10mg; Sodium 940mg; Total Carbohydrate 56g (Dietary Fiber 14g, Sugars 9g); Protein 12g Exchanges: 2 Starch, \u00bd Fruit, \u00bd Other Carbohydrate, 1\u00bd Vegetable, \u00bd Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 4\n\n# Chapter 7: Pasta & Pizza\n\n## Pasta Basics\n\nCooking Pasta\n\nPasta Yields\n\nPurchasing and Storage\n\n## Features\n\nBetty's Staples: Cooked Macaroni\n\nHeirloom Recipe and New Twist\n\nLearn to Make: Pasta\n\nMake-Ahead: Pizza Crust\n\n## Fresh Pasta\n\nBacon, Chive and Parmesan Ravioli\n\nFour-Cheese Homemade Ravioli Calorie Smart \u2022 Easy\n\nGluten-Free Pasta\n\nHerb Pasta\n\nHomemade Pasta Calorie Smart\n\nPotato Gnocchi\n\nRavioli\n\nSpaetzle Calorie Smart \u2022 Fast\n\nWhole Wheat Pasta\n\n## Noodles\n\nAngel Hair Pasta with Basil, Avocado and Tomatoes Fast\n\nBrowned Butter Orecchiette with Broccoli Rabe Fast\n\nButternut Squash Ravioli\n\nCacio e Pepe Fast\n\nCaramelized Onion Macaroni and Cheese\n\nCheesy Rigatoni with Eggplant Sauce\n\nChipotle-Peanut Noodle Bowls Fast\n\nFettuccine Alfredo Fast\n\nFettuccine Alfredo with Baby Spinach\n\nFettuccine and Vegetables Parmesan Calorie Smart \u2022 Fast\n\nFire-Roasted Tomato Macaroni and Cheese\n\nGarden Vegetable Lasagna\n\nGorgonzola Linguine with Toasted Walnuts Fast\n\nItalian Salad Toss\n\nMacaroni and Cheese\n\nPesto Pasta\n\nPuttanesca\n\nSouthwest Macaroni and Cheese\n\nVermicelli with Fresh Herbs Fast\n\nVermicelli with Fresh Herbs and Cannellini\n\nVermicelli with Fresh Herbs, Olives and Fresh Mozzarella\n\n## Meat and Poultry Pastas\n\nAngel Hair Pasta with Chicken\n\nBacon Mac and Cheese\n\nBacon Macaroni and Cheese\n\nBacon, Chive and Parmesan Ravioli\n\nBacon, Kale and Tomato Macaroni and Cheese\n\nBolognese Calorie Smart\n\nChicken and Spaghetti Carbonara\n\nChicken- and Spinach-Stuffed Shells\n\nChicken Fettuccine Alfredo\n\nChicken Spaghetti\n\nChicken-Macaroni Salad\n\nEasy Italian Sausage Lasagna\n\nIsraeli Couscous Risotto with Caramelized Onions and Sausage\n\nItalian Sausage Lasagna\n\nItalian Sausage with Tomatoes and Penne Calorie Smart \u2022 Fast\n\nManicotti\n\nMeatballs and Sauce\n\nPad Thai with Chicken\n\nPad Thai with Pork\n\nPasta with Prosciutto and Asiago Cheese Calorie Smart \u2022 Fast\n\nShort Rib and Sausage Rag\u00f9\n\nSpaghetti Carbonara Fast\n\nSpaghetti and Meat Sauce\n\nSpaghetti and Meatballs\n\nTurkey Manicotti\n\nVermicelli with Fresh Herbs, Spinach and Bacon\n\n## Seafood Pastas\n\nAngel Hair Pasta with Shrimp\n\nBuccatini with Clam-Mushroom Sauce\n\nLinguine with White Clam Sauce and Basil\n\nLobster Macaroni and Cheese\n\nPad Thai with Shrimp\n\nShrimp Alfredo Primavera Calorie Smart \u2022 Fast\n\nSpaghetti with White Clam Sauce Fast\n\nSpanish Clams, Sausage and Linguine Calorie Smart\n\nVermicelli with Fresh Herbs and Shrimp\n\n## Pasta Sauces\n\nBolognese Calorie Smart\n\nCreamy Basil-Chicken Sauce\n\nCreamy Olive Sauce\n\nCreamy Tomato-Vodka Sauce Calorie Smart \u2022 Fast\n\nItalian Tomato Sauce Calorie Smart\n\nRoasted Pepper Puttanesca\n\n## Pizza\n\nCalzone\n\nCanadian Bacon Pizza\n\nCheese Pizza\n\nCheesy Beef Pizza\n\nChicken Sausage, Spinach and Swiss Pizza Fast\n\nDeluxe Stuffed-Crust Tomato Pizza\n\nFresh Tomato Pizza Calorie Smart\n\nHam and Gorgonzola Pizza\n\nMeat Lover's Pizza\n\nPizza Crust Calorie Smart\n\nStuffed-Crust Pizza Calorie Smart\n\nTart Flamb\u00e9e Calorie Smart\n\nVeggie Lover's Pizza with Cornmeal Crust Calorie Smart\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Pasta Basics\n\nWe love our pasta because it is so versatile, easy to cook and ideal for any occasion. Whether mixed in a comforting casserole or tossed with a sensational sauce, the meal possibilities with dried, fresh, frozen or homemade pasta are endless.\n\n## Purchasing and Storage\n\nDried Pasta: Avoid broken pasta or pasta that looks cracked. It might fall apart during cooking. Label, date and store tightly covered in a cool, dry place up to 1 year.\n\nFresh Pasta: Avoid packages with moisture droplets or liquid inside. Pasta should be smooth and evenly colored, without broken or crumbly pieces. Look for sell-by dates on packages.\n\n  * Refrigerate and use by package expiration date. Store opened uncooked pasta in a tightly covered container up to 3 days.\n\nFrozen Pasta: Avoid packages that are frozen as a solid block and those with ice crystals or freezer burn (dry, white spots).\n\n  * Freeze unopened fresh pasta in the original package up to 9 months. Leftover uncooked fresh pasta can be frozen in a tightly covered container up to 3 months, and homemade fresh pasta up to 1 month.\n\nCooked Pasta: To prevent sticking during storage, toss cooked pasta with 1 to 2 teaspoons olive or vegetable oil after draining. Refrigerate tightly covered up to 5 days or freeze up to 2 months.\n\n## Pasta Yields\n\nPlan on \u00bd to \u00be cup cooked pasta per side dish and 1 to 1\u00bd cups per main-dish serving.\n\nTo easily measure 8 ounces of dried spaghetti, make a circle with your thumb and index finger (about the size of a silver dollar) and fill it with the pasta.\n\nType of Pasta | Uncooked | Cooked | Servings\n\n---|---|---|---\n\nShort Pasta | 6 to 7 ounces | 4 cups | 4 to 6\n\nLong Pasta | 7 to 8 ounces | 4 cups | 4 to 6\n\nEgg Noodles | 8 ounces | 4 to 5 cups | 4 to 6\n\n## Cooking Pasta\n\n  * Use 1 quart (4 cups) water for every 4 ounces pasta. When the water is boiling vigorously, gradually add the pasta. Return to a boil, then reduce the heat just a bit so that the pasta can boil gently and stir frequently to prevent sticking.\n  * For added flavor, add \u00bd teaspoon salt for every 8 ounces pasta, if desired. \n  * Follow package directions for the correct cook times. For baked recipes, slightly undercook pasta because it will continue to cook during baking.\n  * Taste pasta as it approaches doneness to determine when it is ready. Cooked pasta should be al dente, or tender but firm to the bite, without any raw flavor. Overcooked pasta is mushy, waterlogged and bland.\n  * Unless the recipe specifies, do not rinse pasta after draining or sauces will not cling well. Pasta is usually only rinsed for cold salads.\n\n## Learn to Make Pasta\n\nIf you have never made homemade pasta, now is the opportunity to find out how simple it is. The flavor, texture and quality of really fresh pasta are unique and delicious. A pasta machine makes the process go quickly, but the dough can be rolled by hand, too.\n\n## Homemade Pasta Calorie Smart\n\nFor hearty sauces, cut pasta wide like fettuccine. For lighter sauces, cut it into thinner strips.\n\nPREP 30 min TOTAL 1 hr 55 min \u25cf 8 servings\n\n  * 2 cups all-purpose flour\n  * \u00bd teaspoon salt\n  * 2 eggs\n  * \u00bc cup water\n  * 1 tablespoon olive or vegetable oil\n\n1 In medium bowl, mix flour and salt. Make a well in center of flour mixture. Add eggs, water and oil to well; mix thoroughly. (If dough is too dry, mix in enough water to make dough easy to handle. If dough is too sticky, gradually add flour when kneading.)\n\n2 Gather dough into a ball. On lightly floured surface, knead 5 to 10 minutes or until smooth and springy. Cover with plastic wrap or foil; let stand 15 minutes.\n\n3 Divide dough into 4 equal parts. On lightly floured surface, roll one-fourth of dough at a time (keep remaining dough covered) into rectangle 1\/16 to \u215b inch thick (if using pasta machine*, pass dough through machine until 1\/16 inch thick). Loosely fold rectangle crosswise into thirds. Cut crosswise into 2-inch strips for lasagna, \u00bc-inch strips for fettuccine or \u215b-inch strips for linguine. Unfold and gently shake out. Hang pasta on pasta drying rack or arrange in single layer on lightly floured towels; let stand 30 minutes or until dry.\n\n4 In stockpot, heat 4 quarts water (salted if desired) to boiling; add pasta. Boil uncovered 2 to 5 minutes, stirring occasionally, until firm but tender. Begin testing for doneness when pasta rises to surface of water; drain.\n\n1 Serving: Calories 150; Total Fat 3.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 55mg; Sodium 160mg; Total Carbohydrate 24g (Dietary Fiber 0g, Sugars 0g); Protein 5g Exchanges: 1\u00bd Starch, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\n*Our recipes were not developed for or tested in electric extrusion pasta machines. We recommend following manufacturer's directions.\n\nHerb Pasta Add 1 tablespoon chopped fresh herbs to flour mixture before adding eggs.\n\nGluten-Free Pasta Substitute Betty Crocker Gluten Free all-purpose rice flour blend for the all-purpose flour and use \u2153 cup water instead of \u00bc cup water. Make pasta as directed, except in Step 3, divide dough into 6 equal parts instead of 4. Gluten-free dough is more fragile than regular dough and can break easily, so smaller dough portions work best.\n\nRavioli To make homemade ravioli, see Four-Cheese Homemade Ravioli (page 172).\n\nWhole Wheat Pasta Make as directed\u2014except use 3 cups whole wheat or white whole wheat flour and 5 eggs, beaten. Omit water.\n\n### Cutting Homemade Pasta\n\nOn lightly floured surface, roll one-fourth of dough at a time into rectangle 1\/16 to \u215b inch thick (keep remaining dough covered).\n\nLoosely fold rectangle crosswise into thirds.\n\nCut pasta crosswise into 2-inch strips for lasagna, \u00bc-inch strips for fettuccine or \u215b-inch strips for linguine.\n\nUnfold and gently shake out strips. Hang strips on pasta drying rack or arrange in single layer on lightly floured towels.\n\nMake a well in the center of the flour mixture. Add eggs, water and oil to well.\n\nPass the dough through pasta machine until dough is 1\/16 to \u215b inch thick.\n\nTo cut pasta with pasta machine, insert handle in hole for desired pasta shape; slowly pass dough through machine.\n\nArrange pasta in single layer on lightly floured towels, about 30 minutes or until dry.\n\nPasta can be cut into several widths: 2 inches (lasagna), \u00bc inch (fettuccine) or \u215b inch (linguine).\n\n## Tips for Making Fresh Pasta\n\n  * If rolling the dough by hand, use a large wooden board or laminated counter. Avoid cold surfaces like granite, metal or marble because the dough tends to stick.\n  * Pasta dough that is dry will be fragile, so handle carefully. Keep dough covered with plastic wrap until ready to roll and cut.\n  * Be sure water is boiling before adding pasta. Do not rinse pasta after cooking. Drain carefully and toss with 1 to 2 teaspoons olive oil.\n  * Store fresh uncooked pasta, tightly wrapped, up to 2 days in the refrigerator or up to 2 weeks in the freezer.\n\nFour-Cheese Homemade Ravioli\n\n## Four-Cheese Homemade Ravioli Calorie Smart \u2022 Easy\n\nHomemade ravioli is easy and fun to make! Serve with your favorite sauce or one of ours, such as Italian Tomato Sauce (page 174), Bolognese (page 180), Basil Pesto (page 456) or Creamy Tomato-Vodka Sauce (page 174).\n\nPREP 1 hr TOTAL 1 hr 15 min \u25cf 5 servings\n\nPasta\n\n  * Homemade Pasta (page 170)\n\nFilling\n\n  * \u00be cup ricotta cheese (from 15-oz container)\n  * \u00bc cup shredded fontina cheese (1 oz)\n  * \u00bc cup shredded Parmesan cheese (1 oz)\n  * \u00bc cup shredded mozzarella cheese (1 oz)\n\n1 Make pasta as directed through Step 2. Divide dough into 4 equal parts. On lightly floured surface, roll one-fourth of dough at a time (keep remaining dough covered with damp towel) into rectangle 1\/16 to \u215b inch thick (if using hand-crank pasta machine, pass dough through machine until 1\/16 inch thick). Trim dough into 4 (14x4-inch) rectangles. Cover with damp towel until ready to fill.\n\n2 In medium bowl, mix filling ingredients until well blended. On 1 dough rectangle, place 10 mounds of filling (using slightly less than 1 measuring tablespoon for each mound) about 1\u00bd inches apart in 2 rows. Moisten dough lightly around mounds with water; top with second rectangle of dough, lining up edges of sheets. Press gently around edges to seal. Cut between mounds, using pizza cutter, pastry wheel or knife, into 10 squares. Trim any uneven edges and reseal if needed. Repeat with remaining dough and filling.\n\n3 In 8-quart Dutch oven or stockpot, heat 4 quarts water (salted if desired) to boiling; add ravioli. Boil uncovered 7 to 10 minutes, stirring occasionally, until tender (ravioli will rise to the top during first few minutes). Remove ravioli with large slotted spoon to colander to drain; do not rinse. Do not dump pasta and water into colander to drain or ravioli will break open. Serve with your favorite sauce.\n\n1 Serving (4 Ravioli): Calories 340; Total Fat 13g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 100mg; Sodium 480mg; Total Carbohydrate 41g (Dietary Fiber 1g, Sugars 0g); Protein 16g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 1\u00bd Medium-Fat Meat, 1 Fat Carbohydrate Choices: 3\n\nBacon, Chive and Parmesan Ravioli Make pasta as directed. Omit filling ingredients. In medium bowl, mix 1 container (8 oz) chives-and-onion cream cheese, 6 slices bacon, crisply cooked and crumbled, and \u00bc cup grated Parmesan cheese until well blended. Continue as directed.\n\n### Making and Cutting Ravioli\n\nTrim dough into 4 rectangles. Cover with damp towel until ready to fill.\n\nPlace 10 mounds of filling (about 1\u00bd inches apart) in 2 rows on each of two dough rectangles.\n\nTop with second rectangles of dough, lining up edges of sheets. Press gently between mounds and dough edges to seal filling inside each ravioli.\n\nCut between mounds, using a pizza cutter, pastry wheel or knife into 10 squares. Trim any uneven edges and reseal if needed.\n\n## Spaetzle Calorie Smart \u2022 Fast\n\nThe word _spaetzle_ is German for \"little sparrow,\" which is what these tiny noodles or dumplings look like. Serve them as a side dish tossed with a little melted butter\u2014as you would serve potatoes or rice\u2014or topped with a creamy sauce or gravy.\n\nPREP 25 min TOTAL 25 min \u25cf 6 servings\n\n  * 2 eggs, beaten\n  * \u00bc cup milk or water\n  * 1 cup all-purpose flour\n  * \u00bc teaspoon salt\n  * Dash pepper\n  * 1 tablespoon butter\n\n1 Fill 4-quart Dutch oven or saucepan half full with water; heat to boiling. In medium bowl, mix eggs, milk, flour, salt and pepper with fork (batter will be thick).\n\n2 Press a few tablespoons of the batter at a time through colander with \u00bc-inch holes, or spaetzle maker, into boiling water. Stir once or twice to prevent sticking. Cook 2 to 5 minutes or until spaetzle rise to surface and are tender; drain. Toss with butter.\n\n1 Serving: Calories 120; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 135mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 0g); Protein 5g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\nLighter Directions For 2 grams of fat and 100 calories per serving, substitute \u00bd cup fat-free egg product for the eggs.\n\n### Making Spaetzle\n\nPress a few tablespoons batter at a time through spaetzle maker or colander with \u00bc-inch holes into boiling water.\n\nCook until spaetzle rise to surface and are tender.\n\nPotato Gnocchi\n\n## Potato Gnocchi\n\nPotato gnocchi are small dumplings made from potatoes; their flavor and texture are similar to fresh pasta. Serve with your favorite pasta sauce or toss with melted butter.\n\nPREP 20 min TOTAL 1 hr \u25cf 6 servings\n\n  * 2 medium russet potatoes (6 oz each)\n  * 1 teaspoon salt\n  * 2 eggs\n  * 2 to 2\u2153 cups all-purpose flour\n  * 1 tablespoon salt\n\n1In 4-quart Dutch oven or saucepan, place potatoes and enough water to cover. Heat to boiling. Cover and boil about 30 minutes or until tender; drain and cool slightly. Peel potatoes; mash in large bowl until smooth. Cool.\n\n2 Stir 1 teaspoon salt, the eggs and enough of the flour into mashed potatoes to make a stiff dough. Shape into 1-inch oval balls. To make the traditional ridges in gnocchi, use a gnocchi board or fork (see below).\n\n3 In 6- to 8-quart Dutch oven or stockpot, heat about 4 quarts water and 1 tablespoon salt to boiling. Add about one-fourth of the gnocchi. After gnocchi rise to surface, boil uncovered 4 minutes. Remove with slotted spoon; drain. Repeat with remaining gnocchi.\n\n1 Serving: Calories 220; Total Fat 2g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 70mg; Sodium 470mg; Total Carbohydrate 43g (Dietary Fiber 2g, Sugars 0g); Protein 7g Exchanges: 3 Starch Carbohydrate Choices: 3\n\n### Shaping Potato Gnocchi\n\nTo make traditional ridges in gnocchi, roll each 1-inch oval ball over gnocchi board.\n\nOr make ridges by rolling each gnocchi over the tines of the back side of a table fork.\n\nItalian Tomato Sauce\n\n## Italian Tomato Sauce Calorie Smart\n\nMake a batch or two of this sauce and freeze it in portion sizes that work for you.\n\nPREP 15 min TOTAL 1 hr \u25cf 4 cups\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 2 cloves garlic, finely chopped\n  * 1 can (28 oz) whole tomatoes, undrained\n  * 2 cans (8 oz each) tomato sauce\n  * 2 tablespoons chopped fresh or 2 teaspoons dried basil leaves\n  * 1 tablespoon chopped fresh or 1 teaspoon dried oregano leaves\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon fennel seed\n  * \u00bc teaspoon pepper\n\n1 In 3-quart saucepan, heat oil over medium heat. Cook onion, bell pepper and garlic in oil 2 minutes, stirring occasionally.\n\n2 Stir in remaining ingredients, breaking up tomatoes. Heat to boiling; reduce heat. Simmer uncovered 45 minutes.\n\n3 Use sauce immediately, or cover and refrigerate up to 2 weeks or freeze up to 1 year.\n\n\u00bd Cup: Calories 90; Total Fat 3.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 670mg; Total Carbohydrate 11g (Dietary Fiber 2g, Sugars 6g); Protein 2g Exchanges: 2 Vegetable, \u00bd Fat Carbohydrate Choices: 1\n\nSlow-Cooker Directions Use 1 medium onion, chopped (\u00bd cup) and 1 can tomato sauce. Substitute 1 can (28 oz) diced tomatoes, undrained for the whole tomatoes. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients. Cover; cook on Low heat setting 8 to 10 hours.\n\nCreamy Tomato-Vodka Sauce\n\n## Creamy Tomato-Vodka Sauce Calorie Smart \u2022 Fast\n\nThe vodka in this sauce adds complexity and depth to the overall flavor, but it doesn't leave behind a vodka taste.\n\nPREP 10 min TOTAL 30 min \u25cf 3 cups\n\n  * 1 tablespoon olive or vegetable oil\n  * 1 small onion, chopped (\u2153 cup)\n  * 2 cloves garlic, finely chopped\n  * 1 can (28 oz) crushed tomatoes with basil, undrained\n  * \u00bd cup vodka or chicken broth\n  * 1 teaspoon sugar\n  * \u00bc teaspoon coarse (kosher or sea) salt\n  * \u215b teaspoon pepper\n  * \u00bd cup whipping cream\n\n1 In 10-inch skillet, heat oil over medium heat. Cook onion and garlic in oil 3 to 4 minutes, stirring constantly, until crisp-tender.\n\n2 Stir in tomatoes, vodka, sugar, salt and pepper. Heat to boiling; reduce heat. Simmer uncovered 20 minutes, stirring occasionally. Stir in whipping cream. Heat just until hot.\n\n\u00bd Cup: Calories 130; Total Fat 9g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 20mg; Sodium 270mg; Total Carbohydrate 8g (Dietary Fiber 1g, Sugars 5g); Protein 2g Exchanges: 1 Vegetable, 2 Fat Carbohydrate Choices: \u00bd\n\nCreamy Basil-Chicken Sauce Add \u00be to 1 cup shredded cooked chicken with tomatoes in Step 2. Continue as directed. Sprinkle finished sauce with 2 tablespoons chopped fresh basil.\n\nCreamy Olive Sauce Add \u00bd cup sliced ripe olives with tomatoes in Step 2. Continue as directed.\n\n## Puttanesca\n\nAccording to one of many theories, the ladies of the night in Naples, Italy, are credited with creating this spicy pasta dish. It uses just a few ingredients, so it's fast and easy to cook.\n\nPREP 15 min TOTAL 40 min \u25cf 4 servings\n\n  * \u2153 cup olive or vegetable oil\n  * 2 cloves garlic, cut in half\n  * 1 tablespoon capers\n  * 4 flat anchovy fillets in oil, drained, finely chopped\n  * 2 cans (28 oz each) whole tomatoes, drained, chopped\n  * 1 small jalape\u00f1o chile, seeded, finely chopped\n  * 1 package (16 oz) spaghetti\n  * \u00bd cup sliced pitted kalamata or ripe olives\n\n1 In 4-quart Dutch oven or saucepan, heat oil over medium-high heat. Cook garlic in oil, stirring frequently, until golden. Remove garlic and discard.\n\n2 Stir capers, anchovy fillets, tomatoes and chile into oil in Dutch oven. Heat to boiling; reduce heat. Simmer uncovered 15 minutes.\n\n3 Meanwhile, cook and drain spaghetti as directed on package. Stir spaghetti and olives into tomato mixture; cook until thoroughly heated.\n\n1 Serving: Calories 750; Total Fat 24g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 790mg; Total Carbohydrate 112g (Dietary Fiber 9g, Sugars 9g); Protein 22g Exchanges: 6 Starch, \u00bd Other Carbohydrate, 2\u00bd Vegetable, 4 Fat Carbohydrate Choices: 7\u00bd\n\nRoasted Pepper Puttanesca Add \u00bd cup roasted red bell peppers (from a jar) with tomatoes in Step 2. Continue as directed.\n\nVermicelli with Fresh Herbs\n\n## Vermicelli with Fresh Herbs Fast\n\nPREP 20 min TOTAL 20 min \u25cf 6 servings\n\n  * 1 package (16 oz) vermicelli\n  * 1 tablespoon capers\n  * \u00bc cup olive or vegetable oil\n  * 2 tablespoons chopped pine nuts\n  * 1 tablespoon chopped fresh parsley\n  * 2 teaspoons chopped fresh rosemary leaves\n  * 2 teaspoons chopped fresh sage leaves\n  * 1 teaspoon chopped fresh basil leaves\n  * 1 pint (2 cups) cherry or grape tomatoes, cut into quarters\n  * Freshly ground pepper, if desired\n\n1 Cook and drain vermicelli as directed on package.\n\n2 Meanwhile, coarsely chop capers if they are large. In medium bowl, mix capers and remaining ingredients except tomatoes and pepper.\n\n3 In large bowl, toss vermicelli and herb mixture. Stir in tomatoes. Sprinkle with pepper.\n\n1 Serving: Calories 410; Total Fat 12g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 64g (Dietary Fiber 6g, Sugars 2g); Protein 11g Exchanges: 4 Starch, 1 Vegetable, 1 Fat Carbohydrate Choices: 4\n\nVermicelli with Fresh Herbs and Cannellini Stir in 1 can (15 oz) cannellini beans, rinsed with hot water and drained, with vermicelli in Step 3.\n\nVermicelli with Fresh Herbs, Olives and Fresh Mozzarella Add \u00bd cup coarsely chopped pitted kalamata olives and 1 cup bocconcini (small fresh mozzarella cheese balls), cut in half, with vermicelli in Step 3.\n\nVermicelli with Fresh Herbs and Shrimp Stir in \u00be pound cooked medium shrimp with tomatoes in Step 3.\n\nVermicelli with Fresh Herbs, Spinach and Bacon Toss 1 cup firmly packed thinly sliced fresh spinach leaves with vermicelli in Step 3. Sprinkle with 6 slices crisply cooked crumbled bacon, 2 tablespoons shredded Parmesan cheese and pepper.\n\nCacio e Pepe\n\n## Cacio e Pepe Fast\n\n_Cacio e pepe_ translates to \"cheese and pepper\" and is a very simple, zesty pasta dish that can be made quickly.\n\nPREP 25 min TOTAL 25 min \u25cf 6 servings\n\n  * 1 tablespoon salt\n  * 1 package (16 oz) spaghetti\n  * \u00bc cup olive oil\n  * 2 teaspoons coarsely ground black pepper\n  * 1\u00bd cups finely shredded Pecorino Romano cheese (6 oz)\n  * Chopped fresh parsley, if desired\n\n1 In 8-quart Dutch oven or stockpot, heat about 5 quarts water and salt to boiling. Add spaghetti; cook as directed on package. Remove 1 cup of the cooking water; set aside. Drain spaghetti but do not rinse; set aside.\n\n2 In Dutch oven, heat oil over medium heat. Add pepper; cook 1 minute, stirring constantly. Add \u00be cup of the cooking water; heat until simmering. Add spaghetti; sprinkle with cheese. Toss with tongs until mixture clings to spaghetti. If mixture seems dry, add remaining \u00bc cup reserved cooking water (pasta dish will not be saucy).\n\n3 Sprinkle with parsley. Serve with additional cheese.\n\n1 Serving: Calories 520; Total Fat 19g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 30mg; Sodium 640mg; Total Carbohydrate 67g (Dietary Fiber 4g, Sugars 1g); Protein 21g Exchanges: 4 Starch, \u00bd Other Carbohydrate, 1 High-Fat Meat, 2 Fat Carbohydrate Choices: 4\u00bd\n\n## Angel Hair Pasta with Basil, Avocado and Tomatoes Fast\n\nPREP 20 min TOTAL 20 min \u25cf 6 servings\n\n  * 8 oz uncooked angel hair (capellini) pasta\n  * 2 tablespoons olive or vegetable oil\n  * 2 cloves garlic, finely chopped\n  * \u00be cup chopped fresh basil leaves\n  * 1 ripe small avocado, pitted, peeled and diced\n  * 4 medium tomatoes, cut into small cubes\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 Cook and drain pasta as directed on package.\n\n2 Meanwhile, in 3-quart saucepan, heat oil over medium heat. Cook garlic in oil about 5 minutes, stirring occasionally, until tender but not brown. Remove from heat; stir in basil, avocado and tomatoes.\n\n3 In large bowl, gently toss vegetable mixture with pasta. Sprinkle with salt and pepper.\n\n1 Serving: Calories 260; Total Fat 8g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 350mg; Total Carbohydrate 38g (Dietary Fiber 4g, Sugars 3g); Protein 7g Exchanges: 2 Starch, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nAngel Hair Pasta with Chicken Stir 1 cup shredded cooked chicken into mixture with the tomatoes. Cook just until heated.\n\nAngel Hair Pasta with Shrimp Stir \u00bd pound cooked medium shrimp into mixture with the tomatoes. Cook just until heated.\n\nSpaghetti Carbonara\n\n## Spaghetti Carbonara Fast\n\nPREP 25 min TOTAL 25 min \u25cf 6 servings\n\n  * 1 package (16 oz) spaghetti\n  * 1 clove garlic, finely chopped\n  * 6 slices bacon, cut into 1-inch pieces\n  * 3 pasteurized eggs,* beaten, or \u00be cup fat-free egg product\n  * 1 tablespoon olive or vegetable oil\n  * \u00bd cup freshly grated Parmesan cheese\n  * \u00bd cup freshly grated Romano cheese\n  * 2 tablespoons chopped fresh parsley\n  * \u00bc teaspoon pepper\n  * Additional grated Parmesan cheese, if desired\n  * Freshly ground pepper, if desired\n\n1 In 4-quart Dutch oven or saucepan, cook spaghetti as directed on package; drain and return to Dutch oven.\n\n2 Meanwhile, in 10-inch skillet, cook garlic and bacon over medium heat, stirring occasionally, until bacon is crisp; drain.\n\n3 In small bowl, mix eggs, oil, Parmesan and Romano cheeses, parsley and \u00bc teaspoon pepper.\n\n4 Add bacon mixture and egg mixture to hot cooked spaghetti. Cook over low heat, tossing constantly, until egg mixture coats spaghetti. Serve with additional Parmesan cheese and freshly ground pepper.\n\n*Pasteurized eggs are uncooked eggs that have been heat-treated to kill bacteria that can cause food poisoning and gastrointestinal distress. Because the eggs in this recipe are not cooked, be sure to use pasteurized eggs. They can be found in the dairy case at large supermarkets. Fat-free egg product is also pasteurized.\n\n1 Serving: Calories 450; Total Fat 13g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 25mg; Sodium 450mg; Total Carbohydrate 62g (Dietary Fiber 5g, Sugars 2g); Protein 22g Exchanges: 4 Starch, \u00bd Medium-Fat Meat Carbohydrate Choices: 4\n\nChicken and Spaghetti Carbonara Add 2 cups cut-up cooked chicken in Step 4 before heating.\n\n## Fettuccine Alfredo Fast\n\nHere's our version of the famous dish created in the early 1990s by restaurateur Alfredo di Lello in Rome. It is just as rich with butter and cheese as the original was then. The combination of whipping cream and Parmesan cheese is the secret to thickening this sauce.\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 8 oz uncooked fettuccine (for homemade fettuccine, see page 170)\n\nAlfredo sauce\n\n  * \u00bd cup butter, cut into pieces\n  * \u00bd cup whipping cream\n  * \u00be cup grated Parmesan cheese\n  * \u00bd teaspoon salt\n  * Dash pepper\n  * Chopped fresh parsley, if desired\n\n1 Cook and drain fettuccine as directed on package.\n\n2 Meanwhile, in 10-inch skillet, heat butter and whipping cream over medium heat, stirring frequently, until butter is melted and mixture starts to bubble. Reduce heat to low; simmer uncovered 6 minutes, stirring frequently, until slightly thickened. Remove from heat. Stir in cheese, salt and pepper.\n\n3 In large bowl, toss fettuccine with sauce until well coated. Sprinkle with parsley.\n\n1 Serving: Calories 570; Total Fat 40g (Saturated Fat 21g, Trans Fat 2g); Cholesterol 155mg; Sodium 810mg; Total Carbohydrate 38g (Dietary Fiber 2g, Sugars 2g); Protein 15g Exchanges: 2\u00bd Starch, 1 High-Fat Meat, 5\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nLighter Directions For 17 grams of fat and 370 calories per serving, reduce butter to \u00bc cup and Parmesan cheese to \u00bd cup; substitute evaporated milk for the whipping cream.\n\nChicken Fettuccine Alfredo Stir in 1\u00bd cups diced cooked chicken with the cheese, salt and pepper in Step 2.\n\nFettuccine Alfredo with Baby Spinach Omit parsley. Add 1 cup fresh baby spinach when tossing the fettuccine in Step 3. Toss until the spinach wilts.\n\n## Pasta with Prosciutto and Asiago Cheese Calorie Smart \u2022 Fast\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 2 cups uncooked fusilli (corkscrew) pasta (6 oz)\n  * 2 tablespoons olive or vegetable oil\n  * 1 package (8 oz) sliced fresh mushrooms (3 cups)\n  * 6 medium green onions, cut into \u00bd-inch pieces\n  * 1 medium red bell pepper, coarsely chopped (1 cup)\n  * 1 clove garlic, finely chopped\n  * 1 package (3 oz) thinly sliced prosciutto, cut into thin strips*\n  * 1 tablespoon chopped fresh or \u00bd teaspoon dried basil leaves\n  * 2 teaspoons chopped fresh or \u00bc teaspoon dried oregano leaves\n  * \u00bc teaspoon salt\n  * \u00bc cup shredded Asiago cheese (1 oz)\n\n1 Cook and drain pasta as directed on package.\n\n2 Meanwhile, in 10-inch nonstick skillet, heat 1 tablespoon of the oil over medium-high heat. Cook mushrooms, onions, bell pepper and garlic in oil 2 to 3 minutes, stirring occasionally, until vegetables are tender. Stir in prosciutto, basil, oregano and salt.\n\n3 In large bowl, toss pasta, vegetable mixture and remaining 1 tablespoon oil. Sprinkle with  cheese.\n\n*Half of a 6-oz package thinly sliced cooked ham can be substituted for the prosciutto.\n\n1 Serving: Calories 320; Total Fat 12g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 20mg; Sodium 560mg; Total Carbohydrate 39g (Dietary Fiber 4g, Sugars 4g); Protein 15g Exchanges: 2 Starch, 2 Vegetable, 1 Lean Meat, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nIsraeli Couscous Risotto with Caramelized Onions and Sausage\n\n## Israeli Couscous Risotto with Caramelized Onions and Sausage\n\nIsraeli couscous is a larger couscous that tastes delicious when cooked risotto-style. Try this sweet-salty-spicy take on a risotto with a new grain.\n\nPrep 35 min Total 35 min \u25cf 4 servings\n\n  * 1 teaspoon olive oil\n  * 1 large onion, diced (1 cup)\n  * 1 teaspoon sugar\n  * 3 links (4 oz each) lean sweet Italian turkey sausage, casings removed\n  * 1\u00bd cups uncooked Israeli couscous\n  * \u00bc cup white wine or chicken broth\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * 1 cup water\n  * \u00bc teaspoon crushed red pepper flakes\n  * 1 tablespoon freshly grated lemon peel\n  * 2 cups baby spinach leaves, roughly chopped\n  * \u2153 cup shredded Parmesan cheese\n\n1 Heat 4-quart Dutch oven or heavy-bottomed saucepan over medium heat. Add oil and onion. Stir in sugar. Cook 5 to 10 minutes, stirring occasionally, until onion starts to caramelize. Add sausage. Cook, breaking up sausage with spoon, until browned.\n\n2 Add couscous and stir; cook 1 minute. Add wine; cook until wine has been absorbed by couscous. Add 1 cup of the broth; cook until broth is absorbed, stirring occasionally, about 5 minutes. Add remaining 1 cup broth, cooking until liquid is absorbed, stirring occasionally. Add water, cooking until liquid is absorbed, stirring occasionally.\n\n3 Remove from heat. Stir in red pepper flakes, lemon peel, spinach leaves and Parmesan cheese, stirring until spinach wilts. Sprinkle with additional shredded Parmesan cheese, if desired. Serve immediately.\n\n1 Serving: Calories 450 (Calories from Fat 100); Total Fat 12g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 60mg; Sodium 720mg; Total Carbohydrate 57g (Dietary Fiber 4g, Sugars 4g); Protein 27g Exchanges: 3\u00bd Starch, \u00bd Vegetable, 2 Lean Meat, 1 Fat Carbohydrate Choices: 4\n\n## Spaghetti and Meatballs\n\nThis family-style dish is easy to make ahead. Both the sauce and meatballs can be refrigerated or frozen separately for a quick meal any night.\n\nPREP 1 hr 45 min TOTAL 2 hr 15 min \u25cf 6 servings\n\n  * Italian Tomato Sauce (page 174)\n  * Meatballs (page 54)\n  * 4 cups hot cooked spaghetti (page 167)\n  * Grated or shredded Parmesan cheese, if desired\n\n1 Make tomato sauce and meatballs.\n\n2 Stir meatballs into sauce. Simmer uncovered over low heat 30 minutes, stirring occasionally. Serve over spaghetti. Sprinkle with cheese.\n\n1 Serving: Calories 430; Total Fat 16g (Saturated Fat 4.5g, Trans Fat 0.5g); Cholesterol 85mg; Sodium 1320mg; Total Carbohydrate 49g (Dietary Fiber 6g, Sugars 10g); Protein 23g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 2 Vegetable, 2 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 3\n\nSpaghetti and Meat Sauce Omit meatballs. In 10-inch skillet, cook 1 pound lean (at least 80%) ground beef or bulk Italian pork sausage, 1 large onion, chopped (1 cup), and 2 cloves garlic, finely chopped, over medium heat 8 to 10 minutes, stirring occasionally, until meat is thoroughly cooked; drain. Stir meat mixture into sauce. Simmer as directed in Step 2.\n\nChicken Spaghetti Omit meatballs. In 10-inch skillet, cook 1 pound ground chicken, 1 large onion, chopped (1 cup), and 2 cloves garlic, finely chopped, over medium heat 8 to 10 minutes, stirring occasionally, until chicken is no longer pink; drain. Stir chicken mixture into sauce. Simmer as directed in Step 2.\n\n## Betty's Staples Cooked Macaroni\n\nCook up some macaroni\u2014whole grain or regular\u2014and you're ready to put together quick meals at a moment's notice. Drain the cooked pasta and toss with 1 to 2 teaspoons olive oil, then store in the refrigerator for up to 5 days or freeze for up to 2 months. Each idea below uses 4 cups cooked macaroni.\n\n  1. 1. **Bacon Mac and Cheese:** Make Cheese Sauce, page 459, and stir in cooked macaroni and 1 cup crumbled cooked bacon until thoroughly heated.\n  2. 2. **Chicken-Macaroni Salad:** Toss cold cooked macaroni with 1 cup chopped cooked chicken, \u00bd cup sliced celery, \u00bd cup halved red grapes and \u00bc cup mayonnaise. Sprinkle with cashews or almonds.\n  3. 3. **Italian Salad Toss:** Toss cold cooked macaroni with 1 medium chopped fresh tomato, 1 cup diced mozzarella cheese, \u00bc cup sliced green onions, 2 tablespoons chopped fresh basil and \u00bc cup Italian dressing.\n  4. 4. **Meatballs and Sauce:** Heat 1 cup pasta sauce with 8 ounces frozen cooked meatballs until hot. Stir in 2 tablespoons chopped fresh parsley and serve over warm cooked macaroni.\n  5. 5. **Pesto Pasta:** Toss warm cooked macaroni with \u00bc cup pesto, \u00bc cup roasted bell peppers, 2 tablespoons sliced ripe olives and 1 cup diced cooked chicken.\n\nItalian Salad Toss\n\n# Heirloom Recipe and New Twist\n\nA staple of northern Italy, Bolognese is a hearty, thick and creamy tomato and meat sauce made with canned tomatoes, wine and milk or cream. In contrast, the Short Rib and Sausage Rag\u00f9 is a very rich meat sauce that has very little tomato and no milk or cream. Make either recipe 1 to 2 days ahead and reheat before serving for the best flavor.\n\nHeirloom\n\n## Bolognese Calorie Smart\n\nPREP 25 min TOTAL 1 hr 10 min \u25cf 6 cups\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 tablespoon butter\n  * 2 medium carrots, finely chopped (1 cup)\n  * 1 medium stalk celery, finely chopped (\u00bd cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 cloves garlic, finely chopped\n  * 1 lb lean (at least 80%) ground beef\n  * \u00bc cup chopped pancetta or bacon\n  * \u00bd cup dry red wine, nonalcoholic red wine or beef broth\n  * 3 cans (28 oz each) whole tomatoes, drained, chopped\n  * 1 teaspoon dried oregano leaves\n  * \u00bd teaspoon pepper\n  * \u00bd cup milk or whipping cream\n\n1In 12-inch skillet, heat oil and butter over medium-high heat. Add carrots, celery, onion and garlic; cook, stirring frequently, until crisp-tender. Stir in beef and pancetta. Cook 8 to 10 minutes, stirring occasionally, until beef is thoroughly cooked; drain.\n\n2 Stir in wine. Heat to boiling; reduce heat to low. Simmer uncovered until wine has evaporated. Stir in tomatoes, oregano and pepper. Heat to boiling; reduce heat to low. Cover and simmer 45 minutes, stirring occasionally. Remove from heat; stir in milk.\n\n3 Use sauce immediately, or cover and refrigerate up to 48 hours or freeze up to 2 months.\n\n\u00bd Cup: Calories 100; Total Fat 4g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 25mg; Sodium 360mg; Total Carbohydrate 8g (Dietary Fiber 2g, Sugars 4g); Protein 9g Exchanges: 1\u00bd Vegetable, 1 Medium-Fat Meat Carbohydrate Choices: \u00bd\n\nBolognese\nNew Twist\n\nShort Rib and Sausage Rag\u00f9\n\n## Short Rib and Sausage Rag\u00f9\n\nPREP 45 min TOTAL 3 hr 45 min \u25cf 8 servings\n\n  * 3 lb boneless beef short ribs\n  * 1 teaspoon salt\n  * 2 tablespoons olive oil\n  * \u00bc cup finely chopped carrot\n  * \u00bc cup finely chopped celery\n  * \u00bc cup finely chopped onion\n  * 2 cloves garlic, finely chopped\n  * \u00bc lb mild bulk Italian sausage\n  * \u00bd cup dry red wine\n  * 2 tablespoons tomato paste\n  * 1 tablespoon all-purpose flour\n  * 3\u00bd cups reduced-sodium beef broth (for homemade broth, see page 146)\n  * \u00bc teaspoon pepper\n  * 2 slices bacon, crisply cooked, crumbled\n  * 2 sprigs (5 inch) fresh thyme\n  * 2 sprigs (5 inch) fresh rosemary\n  * Hot cooked pappardelle or rigatoni pasta, if desired\n  * Shredded Parmesan cheese, if desired\n\n1 Rub all sides of ribs with salt. In 6-quart ovenproof Dutch oven or stockpot, heat oil over medium-high heat. Cook ribs in oil 6 to 8 minutes, turning frequently, until brown on all sides (brown in batches if necessary). Remove ribs; set aside.\n\n2 Add carrot, celery, onion, garlic and sausage to Dutch oven. Cook over medium heat 6 to 8 minutes, stirring frequently and scraping up any browned bits, until sausage is brown; drain if necessary. Stir in wine. Heat to boiling; reduce heat. Simmer uncovered about 6 minutes or until wine is almost evaporated.\n\n3 Heat oven to 350\u00b0F. Add tomato paste and flour to mixture in Dutch oven; stir until well blended. Gradually stir in 2\u00bd cups of the broth. Add pepper and bacon. Tie herb sprigs together with kitchen string; place in broth. Place ribs in broth, turning to coat all sides (ribs may not fit in single layer).\n\n4 Cover and bake 2 hours 30 minutes to 3 hours or until ribs begin to fall apart when tested with fork. Remove ribs. Skim fat from mixture in Dutch oven. Cut excess fat from ribs and discard. Pull ribs into 1-inch pieces; return to Dutch oven. (This type of rag\u00f9 isn't meant to be saucy, but if desired, gently stir desired amount of remaining 1 cup broth into mixture to moisten without making it soupy, being careful not to break up meat pieces.) Heat until hot. Serve over pasta; sprinkle with cheese.\n\n1 Serving: Calories 440; Total Fat 31g (Saturated Fat 11g, Trans Fat 1g); Cholesterol 125mg; Sodium 750mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 1g); Protein 35g Exchanges: 5 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\n## Italian Sausage with Tomatoes and Penne Calorie Smart \u2022 Fast\n\nPREP 30 min TOTAL 30 min \u25cf 4 servings\n\n  * 3 cups uncooked penne pasta (9 oz)\n  * 1 lb Italian sausage links, cut crosswise into \u00bc-inch slices\n  * \u00bd cup beef broth (for homemade broth, see page 146)\n  * 1 medium yellow summer squash, cut in half lengthwise, then cut crosswise into \u00bc-inch slices\n  * 1 pint (2 cups) grape or cherry tomatoes, cut in half lengthwise\n  * \u00bc cup chopped fresh or 1 tablespoon dried basil leaves\n  * 6 green onions, cut into \u00bd-inch pieces\n  * 2 tablespoons olive oil\n  * Shaved Parmesan cheese, if desired\n\n1 Cook and drain pasta as directed on package.\n\n2 Meanwhile, spray 12-inch skillet with cooking spray; heat over medium-high heat. Add sausage; cook 4 to 6 minutes, stirring frequently, until brown. Stir in broth; reduce heat to medium. Cover and cook 5 minutes.\n\n3 Stir in squash, tomatoes and 2 tablespoons of the basil. Heat to boiling; reduce heat to low. Cover and simmer 5 minutes, stirring occasionally. Stir in onions. Simmer uncovered 1 minute.\n\n4 In large bowl, toss pasta, oil and remaining 2 tablespoons basil. Divide pasta among individual bowls; top evenly with sausage mixture. Serve with cheese.\n\n1 Serving: Calories 490; Total Fat 31g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 70mg; Sodium 1140mg; Total Carbohydrate 30g (Dietary Fiber 3g, Sugars 5g); Protein 21g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2 Vegetable, 2 High-Fat Meat, 3 Fat Carbohydrate Choices: 2\n\nSpaghetti with White Clam Sauce\n\n## Spaghetti with White Clam Sauce Fast\n\nPREP 20 min TOTAL 20 min \u25cf 4 servings\n\n  * 1 package (7 oz) spaghetti\n  * \u00bc cup butter or olive oil\n  * 2 cloves garlic, finely chopped\n  * 2 tablespoons chopped fresh parsley\n  * 2 cans (6.5 oz each) minced clams, undrained\n  * \u00bd cup grated Parmesan cheese\n  * Additional chopped fresh parsley\n\n1 Cook and drain spaghetti as directed on package.\n\n2 Meanwhile, in 1\u00bd-quart saucepan, melt butter over medium heat. Cook garlic in butter about 3 minutes, stirring occasionally, until light golden. Stir in 2 tablespoons parsley and the clams. Heat to boiling; reduce heat. Simmer uncovered 3 to 5 minutes.\n\n3 In large bowl, toss spaghetti and clam sauce. Sprinkle with cheese and additional parsley.\n\n1 Serving: Calories 480; Total Fat 18g (Saturated Fat 8g, Trans Fat 1g); Cholesterol 100mg; Sodium 410mg; Total Carbohydrate 45g (Dietary Fiber 3g, Sugars 0g); Protein 36g Exchanges: 3 Starch, 3 Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 3\n\nLinguine with White Clam Sauce and Basil Substitute linguine for the spaghetti and chopped basil for the parsley.\n\nBucatini with Clam-Mushroom Sauce Substitute bucatini or any other long-shaped pasta for the spaghetti. Cook 1 cup sliced mushrooms in butter 2 minutes before adding garlic. Stir in \u00bd teaspoon fresh thyme leaves just before serving.\n\n## Spanish Clams, Sausage and Linguine Calorie Smart\n\nPREP 25 min TOTAL 45 min \u25cf 6 servings\n\n  * \u00bd lb bulk spicy Italian pork sausage\n  * 3 medium onions, chopped (1\u00bd cups)\n  * 2 medium carrots, chopped (1 cup)\n  * 3 cloves garlic, finely chopped\n  * 1 can (28 oz) crushed tomatoes, undrained\n  * 2 cans (14.5 oz each) diced tomatoes with roasted garlic, undrained\n  * \u00bd cup dry red wine or clam juice\n  * 1\u00bd teaspoons ground cumin\n  * 1 teaspoon dried rosemary leaves, crushed\n  * \u00bd cup small pimiento-stuffed olives, coarsely chopped\n  * 12 oz uncooked linguine\n  * 24 littleneck clams (about 2 lb), scrubbed*\n  * Grated Pecorino Romano or Parmesan cheese, if desired\n\n1 In 4-quart Dutch oven, cook sausage, onions, carrots and garlic over medium-high heat 5 to 7 minutes, stirring frequently, until sausage is no longer pink; drain.\n\n2 Stir in remaining ingredients except linguine, clams, and cheese. Heat to boiling; reduce heat to medium-low. Partially cover and simmer 10 minutes. Meanwhile, cook and drain linguine as directed on package.\n\n3 Add clams to sausage mixture. Cover and cook over medium-high heat 5 to 7 minutes or until clams open. Discard any unopened clams. Serve clam sauce over linguine, sprinkled with cheese.\n\n*Or substitute 3 cans (10 oz each) drained whole clams for the fresh clams.\n\n1 Serving: Calories 440; Total Fat 11g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 35mg; Sodium 870mg; Total Carbohydrate 65g (Dietary Fiber 8g, Sugars 10g); Protein 21g Exchanges: 3 Starch, 4 Vegetable, \u00bd Medium-Fat Meat, 1 Fat Carbohydrate Choices: 4\n\n## Littleneck Clams\n\nLittleneck clams are the smallest hardshell clam found on the East Coast. Their shell diameter is less than 2 inches. They are also known by their Indian name, quahog. Pacific littlenecks are usually slightly larger, with shells about 2\u00bd inches in diameter.\n\nSpanish Clams, Sausage and Linguine\n\nPad Thai with Shrimp\n\n## Pad Thai with Shrimp\n\nPad Thai is Thailand's most well-known noodle dish, which you'll find on almost all Thai restaurant menus in America. Look for the traditional ingredients in the ethnic foods aisle.\n\nPREP 35 min TOTAL 40 min \u25cf 4 servings\n\n  * 1 package (6 to 8 oz) linguine-style stir-fry rice noodles*\n  * 3 tablespoons lime juice\n  * 3 tablespoons packed brown sugar\n  * 2 tablespoons fish sauce or soy sauce\n  * 2 tablespoons soy sauce\n  * 1 tablespoon rice vinegar or white vinegar\n  * \u00be teaspoon ground red pepper (cayenne)\n  * 3 tablespoons vegetable oil\n  * 3 cloves garlic, finely chopped\n  * 1 medium shallot, finely chopped, or \u00bc cup finely chopped onion\n  * 2 eggs, beaten\n  * \u00be lb cooked deveined peeled medium shrimp, thawed if frozen\n  * \u00bc cup finely chopped dry-roasted peanuts\n  * 3 cups fresh bean sprouts\n  * 4 medium green onions, thinly sliced (\u00bc cup)\n  * \u00bc cup firmly packed fresh cilantro leaves\n  * Additional chopped dry-roasted peanuts, if desired\n  * Additional sliced green onions, if desired\n\n1 In 3-quart saucepan, heat 4 cups water to boiling. Remove from heat; add noodles (push noodles into water with back of spoon to cover completely with water if necessary). Soak 3 to 5 minutes or until noodles are soft but firm. Drain; rinse noodles with cold water.\n\n2 In small bowl, stir lime juice, brown sugar, fish sauce, soy sauce, vinegar, red pepper and 1 tablespoon of the oil until well mixed; set aside.\n\n3 In nonstick wok or 12-inch skillet, heat remaining 2 tablespoons oil over medium heat. Cook garlic and shallot in oil about 30 seconds, stirring constantly, until starting to brown. Add eggs. Cook about 2 minutes, stirring gently and constantly, until scrambled but still moist.\n\n4 Stir in noodles and lime juice mixture. Increase heat to high. Cook about 1 minute, tossing constantly with 2 wooden spoons, until sauce begins to thicken. Add remaining ingredients except cilantro. Cook 2 to 3 minutes, tossing with 2 wooden spoons, until noodles are tender and bean sprouts are thoroughly cooked and no longer crisp.\n\n5 Spoon noodle mixture onto serving platter. Sprinkle with cilantro. Garnish with additional chopped peanuts and green onions.\n\n*Thin or thick rice stick noodles can be substituted for the linguine-style stir-fry rice noodles.\n\n1 Serving: Calories 560; Total Fat 23g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 240mg; Sodium 1310mg; Total Carbohydrate 59g (Dietary Fiber 4g, Sugars 15g); Protein 29g Exchanges: 3 Starch, \u00bd Other Carbohydrate, 1 Vegetable, 2\u00bd Lean Meat, 2\u00bd Fat Carbohydrate Choices: 4\n\nPad Thai with Chicken Substitute 2 cups chopped cooked chicken for the shrimp.\n\nPad Thai with Pork Substitute 2 cups shredded cooked pork for the shrimp.\n\n### Soaking Rice Noodles\n\nPush noodles into water with back of spoon to cover completely. Soak 3 to 5 minutes or until noodles are soft but firm.\n\n## Shrimp Alfredo Primavera Calorie Smart \u2022 Fast\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 3 cups uncooked bow-tie (farfalle) pasta (8 oz)\n  * 2 slices bacon, cut into \u00bd-inch pieces\n  * 1\u00bd cups frozen sweet peas\n  * \u00bc cup water\n  * 1 lb peeled deveined medium shrimp, thawed if frozen\n  * \u00be cup Alfredo sauce (from 10-oz container; for homemade sauce, see page 177)\n  * 2 tablespoons chopped fresh chives\n\n1 Cook and drain pasta as directed on package.\n\n2 Meanwhile, in 12-inch nonstick skillet, cook bacon over medium heat 4 to 5 minutes, stirring occasionally, until crisp. Stir in peas; cook 2 minutes, stirring occasionally. Add water; cover and cook 3 to 5 minutes or until peas are tender and water has evaporated. Add shrimp; cook 2 to 3 minutes, stirring occasionally, until shrimp are pink.\n\n3 Stir in Alfredo sauce and pasta. Cook over medium-low heat, stirring occasionally, until thoroughly heated. Sprinkle with chives.\n\n1 Serving (1\u00bd Cups): Calories 510; Total Fat 20g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 215mg; Sodium 720mg; Total Carbohydrate 54g (Dietary Fiber 5g, Sugars 4g); Protein 32g Exchanges: 3 Starch, 1 Vegetable, 3 Very Lean Meat, 3\u00bd Fat Carbohydrate Choices: 3\n\n## Chipotle-Peanut Noodle Bowls Fast\n\nPREP 30 min TOTAL 30 min \u25cf 4 servings\n\n  * \u00bd cup creamy peanut butter\n  * \u00bd cup apple juice\n  * 2 tablespoons soy sauce\n  * 2 chipotle chiles in adobo sauce (from 7-oz can), seeded, chopped\n  * 1 teaspoon adobo sauce from can of chipotle chiles\n  * \u00bc cup chopped fresh cilantro\n  * 2 medium carrots, cut into julienne strips\n  * 1 medium red bell pepper, cut into julienne strips\n  * 1 package (8 to 10 oz) Chinese curly noodles\n  * 2 tablespoons chopped peanuts\n\n1 In large bowl, mix peanut butter, apple juice, soy sauce, chiles and adobo sauce until smooth. Stir in cilantro; set aside.\n\n2 In 2-quart saucepan, heat 4 cups water to boiling. Add carrots and bell pepper; cook 1 minute. Remove from water with slotted spoon; set aside. Add noodles to water; cook and drain as directed on package.\n\n3 Add noodles to peanut butter mixture; toss. Divide noodles among 4 bowls; top with carrots and bell pepper. Sprinkle with peanuts.\n\n1 Serving: Calories 590; Total Fat 36g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 0mg; Sodium 980mg; Total Carbohydrate 51g (Dietary Fiber 7g, Sugars 11g); Protein 15g Exchanges: 3 Starch, 1 Vegetable, \u00bd High-Fat Meat, 6 Fat Carbohydrate Choices: 3\u00bd\n\n## Gorgonzola Linguine with Toasted Walnuts Fast\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 8 oz uncooked linguine\n  * 1 tablespoon butter\n  * 1 clove garlic, finely chopped\n  * 1\u00bd cups whipping cream\n  * \u00bc cup dry white wine or chicken broth\n  * \u00bc teaspoon salt\n  * \u00bd cup crumbled Gorgonzola cheese (2 oz)\n  * \u00bc cup coarsely chopped walnuts, toasted (page 23)\n  * Freshly ground pepper, if desired\n  * Chopped fresh parsley, if desired\n\n1 Cook and drain linguine as directed on package.\n\n2 Meanwhile, in 10-inch skillet, melt butter over low heat. Cook garlic in butter, stirring occasionally, until golden. Stir in whipping cream, wine and salt. Simmer about 6 minutes, stirring constantly, until slightly thickened; remove from heat. Stir in cheese until melted.\n\n3 In large bowl, toss linguine and sauce until well coated. Sprinkle with toasted walnuts, pepper and parsley.\n\n1 Serving: Calories 620; Total Fat 41g (Saturated Fat 22g, Trans Fat 1g); Cholesterol 120mg; Sodium 620mg; Total Carbohydrate 50g (Dietary Fiber 4g, Sugars 4g); Protein 14g Exchanges: 3 Starch, \u00bd High-Fat Meat, 7 Fat Carbohydrate Choices: 3\n\nLighter Directions For 11 grams of fat and 400 calories per serving, substitute evaporated fat-free milk for the whipping cream; use 2 tablespoons finely chopped walnuts.\n\nBrowned Butter Orecchiette with Broccoli Rabe\n\n## Browned Butter Orecchiette with Broccoli Rabe Fast\n\nBroccoli rabe (rapini) is related to broccoli, but isn't the same as regular or baby broccoli. Broccoli rabe has slender stems, is mostly leaves and has a pleasantly bitter flavor.\n\nPREP 30 min TOTAL 30 min \u25cf 4 servings\n\n  * 1 lb broccoli rabe (rapini)\n  * \u2153 cup butter\n  * 1 tablespoon fresh lemon juice\n  * \u00bd teaspoon salt\n  * 1 package (8.8 oz) orecchiette (tiny disk) pasta (about 6\u00be cups)\n  * Toasted pine nuts or chopped walnuts, if desired\n  * Shredded Parmesan cheese, if desired\n\n1 Cut off tough stem ends (about 1 inch) from broccoli rabe and discard. Cut remaining stems into 1-inch pieces. Cut leaves crosswise into 2-inch pieces (about 10 loosely packed cups). Cut any florets into bite-size pieces if necessary. Set aside.\n\n2 In 1-quart saucepan, heat butter over medium heat, stirring constantly, until medium golden brown. (Watch carefully because butter can brown and then burn quickly.) Remove from heat; stir in lemon juice and salt.\n\n3 Cook and drain pasta as directed on package, except add reserved broccoli rabe stems during last 5 minutes of cooking.\n\n4 Meanwhile, in 12-inch nonstick skillet, heat 1 tablespoon of the browned butter over medium heat. Add reserved broccoli rabe leaves and florets; cook 4 to 6 minutes, stirring frequently, until leaves are wilted. Add pasta mixture and remaining browned butter; toss to coat. Heat until hot. Sprinkle with nuts and cheese.\n\n1 Serving (1\u00bc Cups): Calories 450; Total Fat 17g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 40mg; Sodium 710mg; Total Carbohydrate 61g (Dietary Fiber 6g, Sugars 3g); Protein 13g Exchanges: 3\u00bd Starch, 1 Vegetable, 3 Fat Carbohydrate Choices: 4\n\n### Cutting Broccoli Rabe\n\nCut off tough stem ends (about 1 inch) and discard. Cut remaining stems into 1-inch pieces and leaves crosswise into 2-inch pieces.\n\n## Fettuccine and Vegetables Parmesan Calorie Smart \u2022 Fast\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 8 oz uncooked fettuccine or linguine\n  * 1 cup fresh broccoli florets\n  * 1 cup fresh cauliflower florets\n  * 1 cup frozen sweet peas\n  * 2 medium carrots, thinly sliced (1 cup)\n  * 1 small onion, chopped (\u00bc cup)\n  * 1 tablespoon butter\n  * \u00be cup evaporated fat-free milk (from 12-oz can)\n  * \u2153 cup grated Parmesan cheese\n  * \u00bd teaspoon garlic salt\n  * \u215b teaspoon ground nutmeg, if desired\n  * Dash pepper\n\n1 Cook fettuccine as directed on package, omitting salt and adding broccoli, cauliflower, peas, carrots and onion for last 3 minutes of cooking. Drain and return to saucepan; cover to keep warm.\n\n2 Meanwhile, in 10-inch skillet, heat butter and milk over medium heat, stirring frequently, until butter is melted and mixture starts to bubble. Reduce heat to low. Simmer uncovered 3 to 4 minutes, stirring frequently, until slightly thickened. Remove from heat. Stir in cheese, garlic salt, nutmeg and pepper. Stir cheese mixture into pasta mixture.\n\n1 Serving (1\u00bd Cups): Calories 370; Total Fat 9g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 60mg; Sodium 410mg; Total Carbohydrate 55g (Dietary Fiber 5g, Sugars 11g); Protein 17g Exchanges: 2 Starch, 1 Other Carbohydrate, 2 Vegetable, 1 Lean Meat, 1 Fat Carbohydrate Choices: 3\u00bd\n\nButternut Squash Ravioli\n\n## Butternut Squash Ravioli\n\nPREP 1 hr TOTAL 1 hr 30 min \u25cf 6 servings\n\nRavioli\n\n  * 1 medium butternut squash (about 1\u00bd lb), cut in half, seeds removed\n  * 1 tablespoon olive oil\n  * \u00bd cup reduced-fat ricotta cheese\n  * \u00bc cup shredded Parmesan cheese (1 oz)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon ground nutmeg\n  * 36 wonton skins (about 3\u00bc-inch square)\n  * 1 egg\n  * 2 tablespoons water\n\nBrowned Butter\u2013Sage Sauce\n\n  * \u00bd cup butter\n  * 12 to 15 fresh sage leaves\n  * 1 clove garlic, finely chopped\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n\nToppings\n\n  * 6 tablespoons shredded Parmesan cheese\n  * 6 fresh sage leaves\n\n1 Heat oven to 400\u00b0F. Place squash halves, cut side up, on ungreased cookie sheet. Brush with oil. Roast 30 minutes or until squash is very tender. Cool slightly.\n\n2 When squash is cool enough to handle, use spoon to remove pulp; place in medium bowl. Stir in ricotta cheese, Parmesan cheese, salt, pepper and nutmeg.\n\n3 Spoon 1 tablespoon squash mixture onto center of each wonton skin. In small bowl, beat egg and water. Brush edges of wonton skins with egg mixture. Fold skins tightly over filling, pressing gently to remove any air trapped inside, forming triangles and sealing filling tightly inside skins. (If filled wonton skins have large air pockets, they can float to surface when boiled, causing them to cook unevenly.)\n\n4 For sauce, in 1-quart saucepan, heat butter over medium heat about 3 minutes until melted. Add sage leaves; cook 3 to 4 minutes, watching carefully and stirring frequently, until butter is golden and sage wilts and then crisps. Add garlic; cook 1 minute or until butter is amber in color and garlic is tender. Stir in salt and pepper. Set aside; keep warm.\n\n5 In 6- to 8-quart Dutch oven or stockpot, heat about 4 quarts water to boiling. Slip filled wonton skins into boiling water in batches so as not to crowd them. Boil 3 minutes or until tender (do not overcook). Use slotted spoon to lift ravioli from boiling water; place on serving plates.\n\n6 Drizzle butter mixture evenly over ravioli. Immediately sprinkle each serving with 1 tablespoon Parmesan cheese; garnish each serving with 1 sage leaf.\n\n1 Serving (6 Ravioli and about 1 Tablespoon Sauce): Calories 430; Total Fat 24g (Saturated Fat 13g, Trans Fat 0.5g); Cholesterol 95mg; Sodium 910mg; Total Carbohydrate 40g (Dietary Fiber 2g, Sugars 5g); Protein 13g Exchanges: 2\u00bd Starch, 1 Medium-Fat Meat, 3\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nMake-Ahead Directions Arrange uncooked filled ravioli in single layer on cookie sheet; freeze 1 to 2 hours or until firm. Transfer to freezer plastic bag; seal bag and freeze up to 3 months. Do not thaw before cooking; just increase the cooking time about 1 minute.\n\n### Using Wonton Skins to Make Ravioli\n\nSpoon 1 tablespoon squash mixture onto center of each wonton skin. Brush edges of wonton skins with egg mixture.\n\nFold skins tightly over filling to form triangles, pressing gently to remove any air trapped inside; seal tightly.\n\nChicken- and Spinach-Stuffed Shells\n\n## Chicken- and Spinach-Stuffed Shells\n\nPREP 30 min TOTAL 1 hr 10 min \u25cf 6 servings\n\n  * 18 uncooked jumbo pasta shells\n  * 1 container (15 oz) whole-milk ricotta cheese\n  * 1 egg, slightly beaten\n  * \u00bc cup grated Parmesan cheese\n  * 2 cups frozen cut-leaf spinach, thawed, squeezed to drain\n  * 1 cup chopped cooked chicken\n  * 1 jar (26 oz) tomato pasta sauce (for homemade sauce, see page 174)\n  * 2 cups shredded Italian cheese blend (8 oz)\n\n1 Heat oven to 350\u00b0F. Cook and drain pasta shells as directed on package, using minimum cook time. Rinse with cold water to cool; drain.\n\n2 Meanwhile, in medium bowl, mix ricotta cheese, egg, Parmesan cheese, spinach and chicken.\n\n3 Spread 1 cup of the pasta sauce in bottom of ungreased 13x9-inch (3-quart) glass baking dish. Spoon about 2 tablespoons ricotta mixture into each pasta shell. Arrange shells, filled sides up, on sauce in baking dish. Spoon remaining sauce over stuffed shells.\n\n4 Cover and bake 30 minutes. Uncover; sprinkle with Italian cheese blend. Bake 5 to 10 minutes longer or until cheese is melted.\n\n1 Serving: Calories 570; Total Fat 28g (Saturated Fat 15g, Trans Fat 0.5g); Cholesterol 120mg; Sodium 1330mg; Total Carbohydrate 48g (Dietary Fiber 4g, Sugars 12g); Protein 33g Exchanges: 3 Starch, 1 Vegetable, 3 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 3\n\nMake-Ahead Directions Make as directed through Step 3. Cover tightly and refrigerate up to 24 hours. Add 5 to 10 minutes to the first bake time before topping with cheese.\n\nCheesy Rigatoni with Eggplant Sauce\n\n## Cheesy Rigatoni with Eggplant Sauce\n\nPREP 20 min TOTAL 50 min \u25cf 4 servings\n\n  * 2\u00bd cups uncooked rigatoni pasta (7\u00bd oz)\n  * 2 tablespoons olive oil\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 small unpeeled eggplant, cut into \u00bd-inch cubes (3 cups)\n  * 1 medium zucchini, halved lengthwise, cut into \u00bc-inch slices (1\u00bd cups)\n  * 1 can (14.5 oz) diced tomatoes with basil, garlic and oregano, undrained\n  * 1 can (8 oz) tomato sauce\n  * 1\u00bd cups shredded mozzarella cheese (6 oz)\n\n1 Heat oven to 350\u00b0F. Spray 12x8-inch (2-quart) glass baking dish with cooking spray. Cook and drain pasta as directed on package, using minimum cook time.\n\n2 Meanwhile, in 12-inch nonstick skillet, heat oil over medium-high heat. Add onion, eggplant and zucchini; cook 5 to 7 minutes, stirring frequently, until crisp-tender. Stir in tomatoes and tomato sauce.\n\n3 Spoon cooked pasta into baking dish. Spoon vegetable sauce over pasta.\n\n4 Cover and bake 20 minutes. Uncover; sprinkle with mozzarella cheese. Bake 5 to 7 minutes longer or until cheese is melted.\n\n1 Serving (2 Cups): Calories 540; Total Fat 17g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 25mg; Sodium 1000mg; Total Carbohydrate 71g (Dietary Fiber 8g, Sugars 11g); Protein 25g Exchanges: 4 Starch, 2 Vegetable, 1 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 5\n\nMake-Ahead Directions Make as directed through Step 3. Cover and refrigerate up to 24 hours. Bake about 10 minutes longer before topping with the cheese.\n\n## Macaroni and Cheese\n\nOnce you've tasted the depth of flavor in our homemade mac and cheese, you'll see why it's the perfect choice for the entire family. Don't even try to compare it to the box variety\u2014it's in a league of its own.\n\nPREP 25 min TOTAL 50 min \u25cf 4 servings\n\n  * 2\u00bd cups elbow macaroni (7 oz)\n  * \u00bc cup butter\n  * \u00bc cup all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon ground mustard\n  * \u00bc teaspoon Worcestershire sauce\n  * 2 cups milk or half-and-half\n  * 2 cups shredded sharp Cheddar cheese (8 oz)\n\n1 Heat oven to 350\u00b0F. Cook and drain macaroni as directed on package, using minimum cook time.\n\n2 Meanwhile, in 3-quart saucepan, melt butter over low heat. Stir in flour, salt, pepper, mustard and Worcestershire sauce. Cook over low heat, stirring constantly, until mixture is smooth and bubbly; remove from heat.\n\n3 Stir in milk. Heat to boiling, stirring constantly. Boil and stir 1 minute; remove from heat. Stir in cheese until melted. Gently stir macaroni into cheese sauce.\n\n4 Pour into ungreased 2-quart casserole. Bake uncovered 20 to 25 minutes or until bubbly.\n\n1 Serving (1 Cup): Calories 610; Total Fat 34g (Saturated Fat 19g, Trans Fat 1g); Cholesterol 100mg; Sodium 980mg; Total Carbohydrate 51g (Dietary Fiber 3g, Sugars 8g); Protein 26g Exchanges: 3 Starch, \u00bd Low-Fat Milk, 2 High-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 3\u00bd\n\nStove-Top Directions Cook and drain macaroni as directed on package. Continue as directed in Steps 2 and 3; omit Step 4.\n\nLighter Directions For 10 grams of fat and 390 calories per serving, reduce butter to 2 tablespoons. Use fat-free (skim) milk and 1\u00bd cups reduced-fat Cheddar cheese (6 oz).\n\nBacon Macaroni and Cheese Stir in \u2154 cup crumbled crisply cooked bacon with macaroni in Step 3.\n\nCaramelized Onion Macaroni and Cheese Stir in 1 cup Caramelized Onions (page 246) with macaroni in Step 3.\n\nFire-Roasted Tomato Macaroni and Cheese Stir in \u00bd teaspoon smoked paprika with flour in Step 2. Add 1 can (14.5 oz) fire-roasted diced tomatoes, drained, with macaroni in Step 3.\n\nBacon, Kale and Tomato Macaroni and Cheese In Step 3, stir in 4 slices cooked crumbled bacon, 1 cup finely chopped fresh kale (stems removed) and 1 can (14.5 oz) drained fire-roasted tomatoes with macaroni. Bake 15 minutes; stir. Mix 1 tablespoon melted butter with \u00bd cup panko bread crumbs; sprinkle over top. Bake 10 to 15 minutes longer or until topping is golden brown.\n\nLobster Macaroni and Cheese Add \u215b teaspoon each ground red pepper (cayenne) and ground nutmeg with the flour in Step 2. Substitute \u00be cup each shredded fontina and Gruy\u00e8re cheese for 1\u00bd cups of the Cheddar. Stir in 2 cups cooked lobster pieces with macaroni in Step 3.\n\nSouthwest Macaroni and Cheese Omit ground mustard and Worcestershire sauce. Add 2 teaspoons ground ancho chile pepper or chili powder, \u00bd teaspoon ground cumin and \u00bc teaspoon garlic powder with the flour in Step 2. Stir in 1 can (4.5 oz) diced green chiles, \u00bc cup thinly sliced green onions, \u00bc cup diced bell pepper and \u00bc cup chopped fresh cilantro with macaroni in Step 3. Top with coarsely crushed corn chips before serving.\n\nBacon, Kale, and Tomato Macaroni and Cheese\n\nItalian Sausage Lasagna\n\n## Italian Sausage Lasagna\n\nPREP 1 hr TOTAL 2 hr \u25cf 8 servings\n\n  * 1 lb bulk Italian pork sausage*\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 clove garlic, finely chopped\n  * 3 tablespoons chopped fresh parsley\n  * 1 tablespoon chopped fresh or 1 teaspoon dried basil leaves\n  * 1 teaspoon sugar\n  * 1 can (15 oz) tomato sauce\n  * 1 can (14.5 oz) whole tomatoes, undrained\n  * 8 uncooked lasagna noodles\n  * 1 container (15 to 16 oz) ricotta cheese or small-curd cottage cheese\n  * \u00bd cup grated Parmesan cheese\n  * 1 tablespoon chopped fresh or 1\u00bd teaspoons dried oregano leaves\n  * 2 cups shredded mozzarella cheese (8 oz)\n\n1 In 10-inch skillet, cook sausage, onion and garlic over medium heat 8 to 10 minutes, stirring occasionally, until sausage is no longer pink; drain.\n\n2 Stir in 2 tablespoons of the parsley, the basil, sugar, tomato sauce and tomatoes, breaking up tomatoes. Heat to boiling, stirring occasionally; reduce heat. Simmer uncovered about 45 minutes or until slightly thickened.\n\n3 Heat oven to 350\u00b0F. Cook and drain noodles as directed on package, using minimum cook time. Meanwhile, in small bowl, mix ricotta cheese, \u00bc cup of the Parmesan cheese, the oregano and remaining 1 tablespoon parsley.\n\n4 In ungreased 13x9-inch (3-quart) glass baking dish, spread half of the sausage mixture (about 2 cups). Top with 4 noodles. Spread half of the ricotta cheese mixture (about 1 cup) over noodles. Sprinkle with half of the mozzarella cheese. Repeat layers, ending with mozzarella. Sprinkle with remaining \u00bc cup Parmesan cheese.\n\n5 Cover and bake 30 minutes. Uncover; bake about 15 minutes longer or until hot and bubbly. Let stand 15 minutes before cutting.\n\n*Or substitute 1 pound lean (at least 90%) ground beef for the sausage.\n\n1 Serving: Calories 430; Total Fat 23g (Saturated Fat 11g, Trans Fat 0g); Cholesterol 70mg; Sodium 1110mg; Total Carbohydrate 28g (Dietary Fiber 3g, Sugars 6g); Protein 28g Exchanges: 2 Starch, 3 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 2\n\nMake-Ahead Directions Cover unbaked lasagna with foil; refrigerate no longer than 24 hours or freeze up to 2 months. Bake covered 45 minutes, then bake uncovered 15 to 20 minutes longer (35 to 45 minutes if frozen). Check the center and bake a little longer if necessary, until hot and bubbly.\n\nEasy Italian Sausage Lasagna Substitute 4 cups (from two 26- to 28-oz jars) tomato pasta sauce with meat for the first 8 ingredients. Omit Steps 1 and 2.\n\nGarden Vegetable Lasagna Substitute 1 jar (26 oz) chunky vegetable tomato pasta sauce for the first 8 ingredients. Heat 1 tablespoon olive oil in 10-inch skillet. Add 3 cups frozen broccoli cuts, 1\u00bd cups sliced fresh mushrooms, 1 cup chopped bell pepper and \u00bd teaspoon garlic powder. Cook and stir 3 to 4 minutes, or until vegetables are crisp-tender. Stir in pasta sauce. Layer noodles, sauce, ricotta mixture and mozzarella cheese as directed in Step 4. Bake as directed.\n\n## Lasagna Noodles\n\nThere are a variety of brands of lasagna noodles. Cook the noodles in boiling water, following the directions on the package, just until tender or al dente. If the noodles are overcooked, they might fall apart while you are layering the lasagna. After cooking the noodles, place them in cold water until you're ready to layer. This keeps them from sticking together.\n\nManicotti\n\n## Manicotti\n\nPREP 40 min TOTAL 1 hr 35 min \u25cf 7 servings\n\n  * 14 uncooked manicotti pasta shells\n  * 1 lb lean (at least 80%) ground beef\n  * 1 cup sliced fresh mushrooms (3 oz)*\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 1 jar (26 to 28 oz) tomato pasta sauce (for homemade sauce, see page 174)\n  * 2 boxes (9 oz each) frozen chopped spinach, thawed\n  * 2 cups small-curd cottage cheese\n  * \u2153 cup grated Parmesan cheese\n  * \u00bc teaspoon ground nutmeg\n  * \u00bc teaspoon pepper\n  * 2 cups shredded mozzarella cheese (8 oz)\n  * 2 tablespoons grated Parmesan cheese\n\n1 Cook and drain pasta shells as directed on package, using minimum cook time.\n\n2 Meanwhile, in 10-inch skillet, cook beef, mushrooms, onion and garlic over medium heat 8 to 10 minutes, stirring occasionally, until beef is thoroughly cooked; drain. Stir in pasta sauce.\n\n3 Heat oven to 350\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n4 Squeeze thawed spinach to drain; spread on paper towels and pat dry. In medium bowl, mix spinach, cottage cheese, \u2153 cup Parmesan cheese, the nutmeg and pepper.\n\n5 In baking dish, spread 1 cup of the beef mixture. Fill pasta shells with spinach mixture; place on beef mixture in dish. Pour remaining beef mixture evenly over shells, covering completely. Sprinkle with mozzarella cheese and 2 tablespoons Parmesan cheese.\n\n6 Cover and bake 30 minutes. Uncover; bake 20 to 25 minutes longer or until hot and bubbly.\n\n*Or substitute 1 can (4 oz) mushroom pieces and stems, drained, for the fresh mushrooms.\n\n1 Serving: Calories 570; Total Fat 21g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 70mg; Sodium 1360mg; Total Carbohydrate 55g (Dietary Fiber 6g, Sugars 12g); Protein 39g Exchanges: 2\u00bd Starch, 1 Other Carbohydrate, 4\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 3\u00bd\n\nTurkey Manicotti Substitute ground turkey for the ground beef and substitute chopped red or green bell pepper for the mushrooms.\n\n### Filling Manicotti Shells\n\nSpoon filling into resealable food-storage plastic bag; seal bag.\n\nWith scissors, make \u00bd-inch cut from one corner of bag.\n\nInsert tip of bag into each manicotti shell; gently squeeze filling into each shell.\n\nOr use teaspoon to carefully fill shells.\n\n## Make-Ahead PIZZA CRUST\n\nCalorie Smart\n\nIf your family likes pizza, why not make your own and top it how you like? Once you make the dough and roll it into balls, it can be stored in the refrigerator or freezer to use later. This crust recipe is a favorite in our test kitchens, and we have had many gatherings where this crust has been made and topped with numerous ingredients. Some of us use the sugar in the dough and others like to substitute honey\u2014whichever you prefer is up to you. If you choose the honey, add it with the oil and warm water.\n\nPREP 45 min TOTAL 1 hr 35 min \u25cf 2 pizza crusts (8 slices each)\n\n  * 2\u00bd to 3 cups all-purpose, bread or whole wheat flour\n  * 1 tablespoon sugar\n  * 1 teaspoon salt\n  * 1 package (2\u00bc teaspoons) regular active or fast-acting dry yeast\n  * 3 tablespoons olive or vegetable oil\n  * 1 cup very warm water (120\u00b0F to 130\u00b0F)\n\n1 In large bowl, mix 1 cup of the flour, the sugar, salt and yeast. Add oil and warm water. Beat with electric mixer on medium speed 3 minutes, scraping bowl frequently. Stir in enough of the remaining flour until dough is soft and leaves side of bowl. Place dough on lightly floured surface; knead 5 to 8 minutes or until dough is smooth and springy.\n\n2 Place dough in greased bowl; cover with plastic wrap. Refrigerate up to 24 hours. Remove from refrigerator 1 hour before shaping crusts. Shape crusts as directed in the photos. Partially bake as directed, right, for thin or thick crusts.\n\n3 Bake topped thin-crust pizzas at 425\u00b0F for 8 to 10 minutes, thick-crust pizzas at 375\u00b0F for 18 to 20 minutes, or until cheese is melted. Cut into wedges to serve.\n\n1 Slice: Calories 100; Total Fat 3g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nTo use dough more quickly, knead as directed. Cover dough loosely with plastic wrap; let rest 30 minutes. Pre-bake, top and bake as directed.\n\nTo freeze dough, remove dough from bowl after rising in Step 2; shape into 2 balls. Wrap each ball tightly in plastic wrap. Freeze up to 2 months. To use, remove from freezer and place in refrigerator until thawed, 4 to 6 hours. Shape and bake as directed in recipe.\n\nFor Thin Crusts Heat oven to 425\u00b0F. Grease 2 cookie sheets or 12-inch pizza pans with oil. Sprinkle with cornmeal. Divide dough in half. Pat each half into 12-inch round on cookie sheet using floured fingers. Partially bake 7 to 8 minutes or until crust just begins to brown. Top as desired; bake as directed in Step 3.\n\nFor Thick Crusts Grease 2 (8-inch square or 9-inch round) baking pans with oil. Sprinkle with cornmeal. Divide dough in half. Pat each half in bottom of pan using floured fingers. Cover loosely with plastic wrap; let rise in warm place 30 to 45 minutes or until almost doubled in size. Move oven rack to lowest position. Heat oven to 375\u00b0F. Partially bake 20 to 22 minutes or until crust just begins to brown. Top as desired; bake as directed in Step 3.\n\nCheesy Beef Pizza Partially bake desired crust as directed. Spread 1 can (8 oz) pizza sauce over crust. In 10-inch skillet, cook 1 lb lean ground beef with 1 cup chopped onion, 1 teaspoon Italian seasoning and \u00bd teaspoon garlic powder until beef is thoroughly cooked. Spoon over pizza sauce. Top with 2 cups each sliced fresh mushrooms and shredded mozzarella cheese. Bake thin crust for 8 to 10 minutes or thick crust for 18 to 20 minutes or until cheese is melted.\n\nMeat Lover's Pizza Make Cheesy Beef Pizza as directed using Italian sausage instead of the ground beef. Add 1 cup diced or sliced pepperoni with the sausage.\n\nCheese Pizza Partially bake desired crust as directed. Sprinkle with 3 cups shredded mozzarella cheese or Italian cheese blend. Bake thin crust for 8 to 10 minutes or thick crust for 18 to 20 minutes or until cheese is melted.\n\nHam and Gorgonzola Pizza Partially bake desired crust as directed. Spread \u2153 cup refrigerated Alfredo sauce over crust. Top with 1 cup diced cooked ham, \u00bd cup crumbled Gorgonzola cheese, \u00bc cup sliced green onions, 1 cup shredded mozzarella cheese and \u00bd teaspoon dried oregano leaves. Bake thin crust pizza for 8 to 10 minutes or thick crust for 18 to 20 minutes or until cheese is melted.\n\nCanadian Bacon Pizza Partially bake desired crust as directed. Spread 1 can (8 oz) pizza sauce over crust. Sprinkle with 1 package (6 oz) sliced Canadian bacon, cut into fourths, 2 cups shredded mozzarella and Parmesan cheese blend and \u00bd cup chopped green bell pepper. Bake thin crust pizza for 8 to 10 minutes or thick crust for 18 to 20 minutes or until cheese is melted.\n\n### Shaping Pizza Dough\n\nFor thin crust, pat each half of dough into 12-inch round on cookie sheet or pizza pan greased with oil and sprinkled with cornmeal.\n\nFor thick crust, pat each half of dough in bottom of square or round pan greased with oil and sprinkled with cornmeal. Cover and let rise until almost doubled in size.\n\n## Fresh Tomato Pizza Calorie Smart\n\nPREP 35 min TOTAL 3 hr 15 min \u25cf 1 pizza (8 slices)\n\nItalian-Style Crust\n\n  * 1 package (2\u00bc teaspoons) regular active or fast-acting dry yeast\n  * \u00bd cup warm water (105\u00b0F to 115\u00b0F)\n  * 1\u00bc to 1\u00bd cups all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon sugar\n  * 1 teaspoon olive oil\n  * Cornmeal\n\nToppings\n\n  * 4 oz fresh mozzarella cheese, well drained, cut into \u00bc-inch-thick slices\n  * 2 plum (Roma) tomatoes, thinly sliced\n  * \u00bc teaspoon salt\n  * Freshly ground pepper to taste\n  * \u00bc cup thin strips fresh basil leaves\n  * 1 tablespoon chopped fresh oregano leaves\n  * 1 tablespoon small capers, if desired\n  * 1 tablespoon olive oil\n\n1 For crust, in large bowl, dissolve yeast in warm water. Stir in \u00be cup of the flour, the salt, sugar and oil. Stir in enough of the remaining flour to make dough easy to handle. Place dough on lightly floured surface; knead about 10 minutes or until smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover and let rise in warm place 20 minutes. Gently push fist into dough to deflate. Cover and refrigerate at least 2 hours but no longer than 48 hours. (If dough should double in size during refrigeration, gently push fist into dough to deflate.)\n\n2 Move oven rack to lowest position. Heat oven to 425\u00b0F. Grease cookie sheet or 12-inch pizza pan with oil. Sprinkle with cornmeal. Pat dough into 12-inch round on cookie sheet or pat in pizza pan using floured fingers. Press dough from center to edge so edge is slightly thicker than center.\n\n3 Place cheese slices on dough to within \u00bd inch of edge. Arrange tomatoes on cheese. Sprinkle with salt, pepper, 2 tablespoons of the basil, the oregano and capers. Drizzle with oil.\n\n4 Bake about 20 minutes or until crust is golden brown and cheese is melted. Sprinkle with remaining 2 tablespoons basil. Cut into slices to serve.\n\n1 Slice: Calories 140; Total Fat 5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 300mg; Total Carbohydrate 17g (Dietary Fiber 1g, Sugars 0g); Protein 6g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\n## Calzone\n\nPREP 45 min TOTAL 1 hr 40 min \u25cf 6 servings\n\n  * Pizza Crust (page 192)\n  * 2 cups shredded mozzarella cheese (8 oz)\n  * \u00bc lb salami, cut into thin strips\n  * \u00bd cup ricotta cheese\n  * \u00bc cup chopped fresh basil leaves\n  * 2 plum (Roma) tomatoes, chopped\n  * Freshly ground pepper to taste\n  * 1 egg, slightly beaten\n\n1 Let pizza crust dough rest 30 minutes.\n\n2 Heat oven to 375\u00b0F. Grease 2 cookie sheets with shortening or cooking spray.\n\n3 Divide dough into 6 equal parts. On lightly floured surface, roll each part into 7-inch circle with floured rolling pin.\n\n4 On half of each dough circle, place equal portions mozzarella cheese, salami, ricotta cheese, basil and tomatoes to within 1 inch of edge. Sprinkle with pepper. Carefully fold dough over filling; pinch edges or press with fork to seal securely.\n\n5 Place calzones on cookie sheets. Brush with egg. Bake about 25 minutes or until golden brown.\n\n1 Serving: Calories 490; Total Fat 24g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 75mg; Sodium 980mg; Total Carbohydrate 46g (Dietary Fiber 2g, Sugars 4g); Protein 24g Exchanges: 3 Starch, 2 High-Fat Meat, \u00bd Fat Carbohydrate Choices: 3\n\nLighter Directions For 8 grams of fat and 350 calories per serving, use reduced-fat mozzarella cheese and fat-free ricotta cheese; substitute cooked chicken for the salami.\n\nDeluxe Stuffed-Crust Tomato Pizza\n\n## Stuffed-Crust Pizza Calorie Smart\n\nPREP 25 min TOTAL 1 hr 45 min \u25cf 1 pizza (6 slices)\n\n  * 1 recipe Honey\u2013Whole Wheat Bread (page 504) or 1 loaf (1 lb) frozen 100% whole-wheat bread dough, thawed\n  * 1 tablespoon yellow cornmeal\n  * 4 sticks (1 oz each) string cheese, cut lengthwise in half\n  * \u00bc cup Italian-style tomato paste (from 6-oz can)\n  * 1 small onion, cut lengthwise in half, then thinly sliced\n  * 1 medium bell pepper, thinly sliced\n  * 1 can (4 oz) mushrooms pieces and stems, drained\n  * 1 oz sliced pepperoni, coarsely chopped (\u00bc cup)\n  * 12 pitted kalamata or Greek olives, coarsely chopped (\u2153 cup)\n  * 2 cups shredded mozzarella cheese (8 oz)\n\n1 Prepare Honey\u2013Whole Wheat Bread as directed through Step 2, using honey.\n\n2 Gently push fist into dough to deflate. Divide dough in half; save one half for another use.*\n\n3 Heat oven to 400\u00b0F. Grease cookie sheet with shortening or cooking spray. Sprinkle cornmeal over cookie sheet. On cookie sheet, press or roll remaining dough half into 13-inch round. Arrange string cheese in circle around edge of dough. Carefully roll edge of dough up over string cheese; seal well.\n\n4 Spread tomato paste evenly over dough. Top with onion, bell pepper, mushrooms, pepperoni and olives. Sprinkle with shredded cheese.\n\n5 Bake 15 to 17 minutes or until crust is golden brown and cheese is melted. Cut into wedges to serve.\n\n*Reserved dough can be shaped and baked as directed in Honey\u2013Whole Wheat Bread recipe or used to make another pizza crust.\n\n1 Serving: Calories 530; Total Fat 20g (Saturated Fat 9g, Trans Fat 0.5g); Cholesterol 40mg; Sodium 1270mg; Total Carbohydrate 64g (Dietary Fiber 6g, Sugars 12g); Protein 24g Exchanges: 3 Starch, 1 Other Carbohydrate, \u00bd Vegetable, 2 Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 4\n\nDeluxe Stuffed-Crust Tomato Pizza Omit mushrooms. Thinly slice 2 plum (Roma) tomatoes. Layer tomatoes over olives. Sprinkle with \u00bd teaspoon dried basil leaves. Continue as directed.\n\n### Making Stuffed Crust Pizza\n\nCarefully roll edge of dough up over cheese; seal well.\n\n## Chicken Sausage, Spinach and Swiss Pizza Fast\n\nPREP 15 min TOTAL 25 min \u25cf 4 pizzas\n\n  * 4 baked flatbreads (from 11.2-oz package)\n  * 2 tablespoons olive oil\n  * 1 bag (6 oz) fresh baby spinach leaves\n  * \u00bd teaspoon garlic-pepper blend\n  * 3 fully cooked chicken sausage links (from 12-oz package), thinly sliced\n  * \u00bc to \u00bd teaspoon crushed red pepper flakes\n  * \u00bd cup sliced green onions (8 medium)\n  * 2 cups shredded Gruy\u00e8re or Swiss cheese (8 oz)\n\n1 Heat oven to 400\u00b0F. Place flatbreads on 2 large ungreased cookie sheets. Brush flatbreads with 1 tablespoon of the oil. Bake 5 minutes.\n\n2 Meanwhile, in large bowl, toss spinach with remaining 1 tablespoon oil and the garlic-pepper blend.\n\n3 Top partially baked flatbreads evenly with spinach and sausage. Sprinkle with pepper flakes, onions and cheese. Bake 7 to 9 minutes or until cheese is melted.\n\n1 Pizza: Calories 500; Total Fat 29g (Saturated Fat 12g, Trans Fat 0g); Cholesterol 95mg; Sodium 860mg; Total Carbohydrate 26g (Dietary Fiber 9g, Sugars 6g); Protein 33g Exchanges: 1\u00bd Starch, \u00bd Vegetable, 4 Lean Meat, 3 Fat Carbohydrate Choices: 2\n\nTart Flamb\u00e9e\n\n## Tart Flamb\u00e9e Calorie Smart\n\nThough not traditional, we think our optional spin of adding fresh arugula and thyme to Tart Flamb\u00e9e after baking is delicious.\n\nPrep 55 min Total 1 hour 10 min \u25cf 2 pizzas, 8 slices each\n\nCrust\n\n  * Pizza Crust (page 192)\n\nTopping\n\n  * 12 slices thick-cut bacon, cut into 1-inch pieces\n  * 1 large white onion, cut in half, sliced (2 cups)\n  * \u00be cup cr\u00e8me fra\u00eeche or sour cream (7 oz)\n  * 3 oz cream cheese, softened\n  * 1 cup shredded Gruy\u00e8re or Parmesan cheese (4 oz)\n  * Arugula, if desired\n  * Fresh thyme leaves, if desired\n\n1 Heat oven to 425\u00b0F. Place clean pizza stone (baking stone) in oven to heat.* Make pizza crust as directed for Thin Crust but press each dough half into 14x10-inch rectangle for rectangular stone or 12-inch round for round stone on separate pieces of foil. Place one crust (on foil) on hot pizza stone. Partially bake 8 to 10 minutes or until light golden brown. Remove crust to work surface. Repeat with second crust. Keep pizza stone hot in oven.\n\n2 Meanwhile, in 12-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Remove bacon with slotted spoon; drain on paper towels. Drain all but 1 tablespoon bacon drippings from skillet. Add onions. Cook 5 to 6 minutes, stirring occasionally, until tender. Remove from heat; set aside.\n\n3 In small bowl, whisk together cr\u00e8me fra\u00eeche and cream cheese until smooth; divide mixture between crusts, spreading evenly over each to within \u00bd inch of edges. Divide onions and bacon evenly over crusts; sprinkle each with \u00bd cup cheese.\n\n4 Bake for 10 to 12 minutes or until cheese is melted and crust is golden brown. Top each pizza with arugula and thyme.\n\n*If desired, pizza crust can be made using 2 cookie sheets or two 12-inch round pizza pans. Heat the oven after the dough has risen.\n\n1 Slice: Calories 220; Total Fat 13g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 30mg; Sodium 350mg; Total Carbohydrate 17g (Dietary Fiber 1g, Sugars 2g); Protein 8g Exchanges: 1 Starch, 1 High-Fat Meat, 1 Fat Carbohydrate Choices: 1\n\n## Veggie Lover's Pizza with Cornmeal Crust Calorie Smart\n\nPREP 20 min TOTAL 55 min \u25cf 1 pizza (6 slices)\n\nCrust\n\n  * 1\u00bc cups all-purpose flour\n  * \u00be cup white or yellow cornmeal\n  * 1 teaspoon baking powder\n  * 1 tablespoon salt\n  * \u2154 cup milk\n  * \u00bc cup oil\n\nTopping\n\n  * 1 can (8-oz.) pizza sauce\n  * 1 medium green bell pepper, cut into thin rings\n  * 1 medium onion, thinly sliced, separated into rings\n  * 1 medium zucchini, thinly sliced\n  * 8 oz (2 cups) shredded mozzarella cheese\n  * \u00bc cup grated Parmesan cheese\n\n1 Heat oven to 425\u00b0F. Grease 14-inch pizza pan or 15x10x1-inch pan. In medium bowl, combine flour, cornmeal, baking powder and salt; mix well. Add milk and oil; stir until mixture forms a ball. Let stand 3 minutes. With floured fingers, press dough into pan, shaping edge to form rim.\n\n2 Bake 11 to 15 minutes or until edges are very light golden brown.\n\n3 Spread pizza sauce evenly over partially baked crust. Top with vegetables and cheeses.\n\n4 Bake an additional 15 to 20 minutes or until cheese is melted and edges are golden brown.\n\n1 Slice: Calories 430; Total Fat 20g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 30mg; Sodium 890mg; Total Carbohydrate 47g (Dietary Fiber 3g, Sugars 6g); Protein 17g Exchanges: \u00bd Starch, 1\u00bd Vegetable, 1\u00bd Medium-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 3\n\n# Chapter 8: Whole Grains & Rice\n\n## Grain and Rice Basics\n\nCooking Hints\n\nJust What Is a Whole Grain?\n\nRice\n\nStorage\n\nWhole Grains\n\n## Features\n\nBetty's Staples: Oats\n\nHeirloom Recipe and New Twist\n\nLearn to Simmer\n\nLearn with Betty: Cooking Polenta\n\nLearn with Betty: Cooking Risotto\n\nMake-Ahead: Brown Rice\n\nTimetable for Cooking Common Grains\n\nTimetable for Cooking Rice\n\nTypes of Barley\n\n## Other Grains\n\nApple-Rosemary Pork and Barley Calorie Smart \u2022 Fast\n\nBalsamic Pork Chops with Quinoa Calorie Smart\n\nBarley with Mixed Vegetables Calorie Smart\n\nBarley, Cauliflower and Red Lentil Tabbouleh Calorie Smart\n\nBeef and Kasha Mexicana Calorie Smart \u2022 Fast\n\nBeef and Kasha Tortillas\n\nBulgur Pilaf Calorie Smart \u2022 Fast\n\nCherry-Pecan Quinoa Calorie Smart \u2022 Fast \u2022 Easy\n\nClassic Tabbouleh Calorie Smart\n\nCranberry-Almond Rye Berry Salad\n\nFaster Fried Polenta\n\nFried Polenta\n\nFruited Tabbouleh with Walnuts and Feta Calorie Smart\n\nGarbanzo Bean Tabbouleh\n\nIndian-Spiced Farro Salad Calorie Smart\n\nKasha and Bow-Tie Pilaf Calorie Smart\n\nKibbeh Patties Calorie Smart\n\nMediterranean Bulgur and Lentils Calorie Smart \u2022 Slow Cooker\n\nMinty Tabbouleh\n\nParmesan Polenta\n\nPolenta Calorie Smart \u2022 Fast\n\nRich and Creamy Polenta\n\nSouthwestern Tabbouleh\n\nSummer Quinoa-Tomato Salad Calorie Smart\n\nThree-Grain Medley Calorie Smart \u2022 Slow Cooker\n\nVegetable Polenta\n\nWheat Berry Salad Calorie Smart\n\n## Rice\n\nAlmond\u2013Brown Rice Pilaf\n\nBrown Rice Pilaf\n\nCashew Pilaf\n\nCherry Pistachio Rice Pilaf\n\nChicken Fried Rice\n\nChicken\u2013Brown Rice Salad\n\nClassic Risotto\n\nClassic Risotto with Butternut Squash\n\nClassic Risotto with Peas\n\nClassic Risotto with Shrimp\n\nCurried Veggie Brown Rice Fast\n\nCurry Pilaf\n\nFried Brown Rice\n\nMushroom Pilaf\n\nOrange and Cranberry Pilaf\n\nPine Nut and Green Onion Pilaf\n\nPork Fried Rice Calorie Smart \u2022 Fast\n\nProsciutto, Parmesan and Pea Pilaf\n\nRice Pilaf Fast \u2022 Easy\n\nShrimp Fried Rice\n\nSpanish Rice Calorie Smart\n\nSpanish Rice with Bacon\n\nSticky Rice with Mango Calorie Smart\n\nWild Rice, Vegetable and Bacon Skillet Calorie Smart\n\n## Bonus Recipes\n\nEasy Rice Pudding\n\nGolden Harvest Streusel Muffins\n\nNo-Bake Chocolate-Oatmeal Drops\n\nOat Pie Crust\n\nQuick Lettuce Wraps\n\nSalsa\u2013Brown Rice Burritos\n\nWhole-Grain Pancakes\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Grain and Rice Basics\n\nCultivated for centuries, grains are eaten by every culture in a variety of dishes. Whole grains are amazing foods. Not only are they budget-friendly, but they also have many nutritional benefits. In general, they are low in fat, have little or no cholesterol and are high in fiber.\n\n## Just What Is a Whole Grain?\n\n  * It is the entire seed of the plant containing all parts of the kernel, including the fiber-rich outer coating of bran, the energy-dense middle layer called the endosperm and the nutrient-packed germ. If any part of the grain is removed, it's not considered whole. Whole grains contain essential vitamins, minerals, antioxidants and phytonutrients plus healthy fats. Whole grains include barley, quinoa and brown rice.\n  * In contrast, a refined grain has been milled, which removes the bran and germ, giving the grain a finer texture, but also removing important fiber and vitamins. Most refined grains are labeled \"enriched,\" meaning some nutrients have been added back in after processing. Refined grains include white rice and white flour.\n\n## Storage\n\n  * Because whole grains contain healthy fats, careful attention to storage of both whole grain and whole grain products is important; they won't keep as long as their refined counterparts. Whole grain flours can be kept 1 to 3 months in airtight containers on a cool, dry shelf or can be kept in the freezer 2 to 6 months.\n\n## Cooking Hints\n\n  * In general, heat the water and grain to a boil and add the grain. Then reduce the heat. Water should be barely simmering when cooking grain so that it cooks slowly.\n  * Keep it covered\u2014For most grains, simmer with the lid on and check occasionally to be sure the liquid is at a low simmer. \n  * Let it rest\u2014When cooking is finished, remove pan from heat and let the grain stand covered for 5 to 10 minutes. Fluff with a fork just before serving.\n\n  * Cook extra\u2014Make a little extra for other uses like stirring into soups, stews or salads. Store cooked grains tightly covered in the refrigerator up to 5 days or freeze for up to 3 months.\n\n## Whole Grains vs. Fiber\n\nWhole grains naturally contain fiber in the outer coating of the grain kernel or bran. So fiber is a component of whole grains, but there are many other important nutrients and natural compounds found in whole grains that work together for health benefits. Eating whole grains can provide fiber, but not all grains are high in fiber.\n\nSummer Quinoa-Tomato Salad (page 206)\n\n## Whole Grains\n\nLook for these grains in your grocery or health-food store:\n\nAmaranth: High in protein and gluten-free, it's used in baked goods.\n\nBarley: See Types of Barley.\n\nBuckwheat: High in protein and gluten-free, ideal for people who struggle with wheat allergies and can't tolerate gluten.\n\nBulgur: Boiled, dried and cracked wheat kernels.\n\nCornmeal: Ground whole-grain, whole corn kernels. Available in white, yellow, and blue forms. For cooking directions, see Polenta, page 207.\n\nFlaxseed: While not actually a grain, its nutritional profile is similar to that of whole grains. Add to baked goods.\n\nKamut\u00ae: Trademarked name of organically grown grain resembling wheat kernels.\n\nKasha: Generic for any crushed grain, commonly buckwheat kernels.\n\nMillet: Mild in flavor, mixes well with other foods and is delicious toasted.\n\nOats: Oats are a healthful grain with several types available:\n\nQuick-Cooking and Old-Fashioned Oats: Both are whole grain with bran and germ still intact.\n\nOat Groats: Whole oats (before being steamed and rolled), so they have the highest nutritional value of all oat products.\n\nSteel-Cut Oats: Groats cut into smaller pieces; they are chewier and nuttier than quick-cooking or old-fashioned oats.\n\nQuinoa: Quick-cooking grain perfect for a light, fluffy side dish or to add to soups and salads.\n\nRye: Fibrous grain with a deep, rich flavor. Substitute rolled rye flakes for old-fashioned oats.\n\nSpelt: Nutty-flavored, higher-protein variety of wheat.\n\nTeff: A whole grain with a sweet, molasses-like flavor typically used to make Ethiopia's spongy flatbread, injera.\n\nWheat Berries: Chewy, nutty, whole wheat kernels for main or side dishes or for adding to salads.\n\n### Timetable For Cooking Common Grains\n\nCheck packages of products for additional information.\n\nType of Grain (1 cup) | Amount of Water in Cups | Cooking Directions | Yield in Cups\n\n---|---|---|---\n\nBarley, Hulled or Hulless | 3\u00bd | Simmer 40 minutes. Drain if needed. For larger, softer barley, simmer an additional 40 minutes, stirring occasionally. | 3\n\nBarley, Pearled | 4 | Simmer covered for 45 to 50 minutes. | 4\n\nBarley, Quick Cooking | 2 | Simmer covered for 10 to 12 minutes. Remove from heat. Let stand 5 minutes. | 3\n\nBulgur | 2 | Bring water to boil, then add bulgur; cover and remove from the heat. Let stand 15 minutes or until all water is absorbed. Or cook as directed on package. | 2\u00bd\n\nBulgur, Light (Golden) | 2 | Mix bulgur and cold water in a saucepan; cover and heat to boiling. Reduce heat; simmer 12 to 15 minutes or until all water is absorbed. | 2 to 2\u00bd\n\nKasha | 2 | Bring water to boil, then add kasha; cover and remove from heat. Let stand 10 to 15 minutes. Drain if needed. Or cook as directed on package. | 4\n\nMillet | 2\u00bd | Simmer covered for 15 to 20 minutes. | 4\n\nOats, Steel Cut | 4 | Bring water to boil, then add oats. Simmer uncovered 25 to 30 minutes. | 1\n\nQuinoa | 2 | Simmer covered for 15 minutes. | 3 to 4\n\nWheat Berries | 2\u00bd | Simmer covered for 50 to 60 minutes. | 2\u00be to 3\n\n# Heirloom Recipe and New Twist\n\nTabbouleh, a traditional Middle Eastern whole-grain salad, has also become a favorite in the United States and is easy to make and serve. It typically uses bulgur, parsley, mint, olive oil, lemon and tomatoes in varying amounts and is always served chilled or at room temperature. The new twist features barley instead of bulgur, among other new flavors, but still uses plenty of the fresh herbs that are a signature of any tabbouleh. In fact, we love tabbouleh so much we've included a third featuring fresh fruit. Be sure to try them all!\n\nHeirloom\n\nClassic Tabbouleh\n\n## Classic Tabbouleh Calorie Smart\n\nServe this traditional version of tabbouleh with crisp bread.\n\nPrep 10 min Total 1 hr 40 min \u25cf 6 servings\n\n  * \u00be cup uncooked bulgur\n  * 1\u00bd cups chopped fresh parsley\n  * 3 medium tomatoes, chopped (2\u00bc cups)\n  * 5 medium green onions, thinly sliced (\u2153 cup)\n  * 2 tablespoons chopped fresh or 2 teaspoons crushed dried mint leaves\n  * \u00bc cup olive or vegetable oil\n  * \u00bc cup fresh lemon juice\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n  * Whole ripe olives, if desired\n\n1 In small bowl, cover bulgur with cold water. Let stand 30 minutes. Drain; press out as much water as possible.\n\n2 In medium glass or plastic bowl, stir together bulgur, parsley, tomatoes, onions and mint.\n\n3 In tightly covered container, shake oil, lemon juice, salt and pepper. Pour over bulgur mixture; toss until coated. Cover and refrigerate at least 1 hour to blend flavors. Garnish with olives.\n\n1 Serving (\u00be Cup): Calories 170; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 320mg; Total Carbohydrate 19g (Dietary Fiber 5g, Sugars 2g); Protein 3g Exchanges: 1 Starch, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\nGarbanzo Bean Tabbouleh Add 1 can (15 to 16 oz) garbanzo beans, drained and rinsed, and 1 cup chopped green bell pepper with the bulgur in Step 2.\n\nMinty Tabbouleh Use \u00be cup each chopped fresh mint and parsley for the 1\u00bd cups chopped fresh parsley. Omit the additional 2 tablespoons mint.\n\nSouthwestern Tabbouleh Substitute 1 cup chopped fresh cilantro for the parsley. Reduce lemon juice to 2 tablespoons; increase salt to 1 teaspoon. Add 2 teaspoons ground cumin with the salt and pepper in Step 3.\nNew Twist\n\nBarley, Cauliflower and Red Lentil Tabbouleh\n\n## Barley, Cauliflower and Red Lentil Tabbouleh Calorie Smart\n\nPrep 10 min Total 2 hr 10 min \u25cf 15 servings\n\nSalad\n\n  * \u00bd cup uncooked hulled (not pearl) barley (not quick-cooking)\n  * 2\u00bc cups water\n  * 1 cup dried red lentils, sorted, rinsed\n  * 2 cups fresh cauliflower florets (1 inch), separated into mini florets (about \u00bc inch)\n  * 1 cup chopped fresh parsley\n  * \u00bd cup chopped fresh mint leaves\n  * 1 jar (12 oz) roasted red bell peppers, drained, patted dry and chopped\n  * 1 cup crumbled feta cheese (4 oz)\n\nDressing\n\n  * \u00bc cup olive oil\n  * 2 tablespoons fresh lemon juice\n  * 1\u00bc teaspoons Greek seasoning or salt\n\n1 In 2-quart saucepan, heat barley and water to boiling. Reduce heat; cover and simmer 40 minutes. Add lentils; cook 10 to 15 minutes longer or until barley is chewy but tender and lentils are just tender. (Do not overcook or lentils will lose their shape and become mushy.) Let stand 5 minutes or until liquid is absorbed. Place in strainer; rinse with cold water. Drain well.\n\n2 In medium glass or plastic bowl, place barley, lentils and remaining salad ingredients. In tightly covered container, shake dressing ingredients. Pour over barley mixture; gently toss to combine. Cover and refrigerate at least 1 hour to blend flavors.\n\n1 Serving (\u00bd Cup): Calories 130; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 170mg; Total Carbohydrate 15g (Dietary Fiber 4g, Sugars 2g); Protein 5g Exchanges: 1 Other Carbohydrate, \u00bd Vegetable, \u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\n## Types of Barley\n\n**Hulled barley** is whole grain barley. Hulless barley is a variety of barley grown to lose its hulls during harvesting. **Pearl barley** has been polished to remove part or all of the outer bran layer, along with the hull. This is not a whole grain, but some of the bran may still be intact. **Quick-cooking barley** is made from pearl barley that's been partially cooked and dried. **Barley grits** are barley kernels cut into pieces.\n\nFruited Tabbouleh with Walnuts and Feta\n\n## Fruited Tabbouleh with Walnuts and Feta Calorie Smart\n\nPrep 20 min Total 3 hr 20 min \u25cf 10 servings\n\n  * 1 cup uncooked bulgur\n  * 1 cup boiling water\n  * \u00bc cup orange juice\n  * \u00bc cup olive oil\n  * \u00bd medium unpeeled cucumber, seeded, chopped (about 1 cup)\n  * \u00bd cup chopped red onion\n  * \u00bd cup sweetened dried cranberries\n  * \u2153 cup loosely packed fresh parsley, finely chopped\n  * \u2153 cup loosely packed fresh mint leaves, finely chopped\n  * 1 tablespoon grated orange peel\n  * \u00bd teaspoon salt\n  * 1 orange, peeled, divided into sections and chopped\n  * \u00bd cup chopped walnuts, toasted (page 23)\n  * \u00bd cup crumbled feta cheese (2 oz)*\n\n1 Place bulgur in large heatproof bowl. Pour boiling water over bulgur; stir. Let stand about 1 hour or until water is absorbed.\n\n2 Stir in orange juice, oil, cucumber, onion, cranberries, parsley, mint, orange peel and salt; toss well. Cover and refrigerate 2 to 3 hours or until well chilled.\n\n3 Just before serving, stir in chopped orange; sprinkle with walnuts and feta cheese.\n\n*Crumbled ch\u00e8vre (goat) cheese can be substituted for the feta cheese.\n\n1 Serving (\u00bd Cup): Calories 200; Total Fat 11g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 210mg; Total Carbohydrate 21g (Dietary Fiber 4g, Sugars 7g); Protein 4g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 1\u00bd\n\nBulgur Pilaf\n\n## Bulgur Pilaf Calorie Smart \u2022 Fast\n\nBulgur comes from wheat berries (wheat kernels) that have been steamed, dried and crushed. With its mild flavor and chewy texture, bulgur is a great stand-in for rice in nearly any recipe.\n\nPrep 15 min Total 30 min \u25cf 6 servings\n\n  * 2 tablespoons butter\n  * \u00bd cup slivered almonds\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 medium carrot, chopped (\u00bd cup)\n  * 1\u00be cups chicken broth (for homemade broth, see page 143)\n  * 1 cup uncooked bulgur\n  * \u00bc teaspoon lemon-pepper seasoning or black pepper\n  * \u00bc cup chopped fresh parsley\n\n1 In 12-inch skillet, melt 1 tablespoon of the butter over medium-high heat. Cook almonds in butter 2 to 3 minutes, stirring constantly, until golden brown. Remove almonds from skillet; set aside.\n\n2 Add remaining 1 tablespoon butter, the onion and carrot to skillet. Cook about 3 minutes, stirring occasionally, until vegetables are crisp-tender.\n\n3 Stir in broth, bulgur and lemon-pepper seasoning. Heat to boiling; reduce heat. Cover and simmer about 15 minutes or until liquid is absorbed and bulgur is tender. Stir in parsley and toasted almonds.\n\n1 Serving: Calories 200; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 330mg; Total Carbohydrate 23g (Dietary Fiber 6g, Sugars 2g); Protein 7g Exchanges: 1 Starch, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 1\u00bd\n\nMediterranean Bulgur and Lentils\n\n## Mediterranean Bulgur and Lentils Calorie Smart \u2022 Slow Cooker\n\nPrep 15 min Total 3 hr 30 min \u25cf 8 servings\n\n  * 1 cup uncooked bulgur or cracked wheat\n  * \u00bd cup dried lentils, sorted, rinsed\n  * 1 teaspoon ground cumin\n  * \u00bc teaspoon salt\n  * 3 cloves garlic, finely chopped\n  * 1 can (15.25 oz) whole kernel corn, drained\n  * 3\u00bd cups vegetable or chicken broth (for homemade broth, see page 156 or )\n  * 2 medium tomatoes, chopped (1\u00bd cups)\n  * \u00bd cup drained pitted kalamata olives\n  * 1 cup crumbled reduced-fat feta cheese (4 oz)\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix bulgur, lentils, cumin, salt, garlic, corn and broth.\n\n2 Cover; cook on Low heat setting 3 to 4 hours or until lentils are tender.\n\n3 Stir in tomatoes and olives. Increase heat setting to High. Cover; cook 15 minutes. Top individual servings with cheese.\n\n1 Serving: Calories 210; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 880mg; Total Carbohydrate 33g (Dietary Fiber 7g, Sugars 4g); Protein 10g Exchanges: 2 Starch, \u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 2\n\n## Complete Proteins\n\nIf you are eating meatless and are counting on vegetable sources for your protein requirements, it's important to combine them in a way that makes them \"complete.\" What are the best combos to make complete proteins?\n\n  * Whole grains and legumes\n  * Legumes with nuts and\/or seeds\n  * Animal dairy products (eggs, milk and other products) with any vegetable protein\n\nKibbeh Patties with Tzatziki Dip\n\n## Kibbeh Patties Calorie Smart\n\nBulgur, partially processed whole wheat, is well known in Middle Eastern cooking, especially in tabbouleh and kibbeh. Kibbeh is claimed to be the national dish by many Middle Eastern countries. It's often torpedo-shaped and usually deep-fried, but we've streamlined the process to make delicious pan-fried patties. Try them as is, or serve in pita bread or on a bun for a new burger flavor.\n\nPrep 25 min Total 55 min \u25cf 6 servings\n\nPatties\n\n  * \u00be cup uncooked bulgur\n  * 1 cup cold water\n  * 1 lb ground lamb\n  * 1 medium onion, finely chopped (\u00bd cup)\n  * \u00bc cup finely chopped fresh mint leaves\n  * 1 teaspoon dried marjoram leaves\n  * \u00be teaspoon salt\n  * \u00bd teaspoon ground allspice\n  * \u00bd teaspoon ground cinnamon\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 2 tablespoons olive oil\n\nToppings, if desired\n\n  * Tzatziki Dip (page 44) or Hummus (page 42)\n  * Cucumber and tomato slices\n\n1 In medium bowl, cover bulgur with cold water. Let stand 30 minutes. Drain; press out as much water as possible.\n\n2 Add lamb, onion, mint, marjoram, salt, allspice, cinnamon, red pepper and 1 tablespoon of the oil to bulgur; mix just until blended. Divide mixture into 6 equal portions; shape into patties, about \u00bd inch thick.\n\n3 In 12-inch nonstick skillet, heat remaining 1 tablespoon oil over medium heat. Cook patties in oil 9 to 12 minutes, turning once, until deep golden brown and meat thermometer inserted in center of patties reads 160\u00b0F. Serve with tzatziki dip or hummus, cucumber and tomato.\n\n1 Serving: Calories 260; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 50mg; Sodium 340mg; Total Carbohydrate 16g (Dietary Fiber 4g, Sugars 0g); Protein 14g Exchanges: 1 Starch, 1\u00bd Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\n### Making Kibbeh Patties\n\nCover bulgur with cold water. Let stand 30 minutes.\n\nDivide mixture into 6 equal portions; shape into patties about \u00bd inch thick.\n\n## Cherry-Pecan Quinoa Calorie Smart \u2022 Fast \u2022 EASY\n\nPrep 10 min Total 25 min \u25cf 8 servings\n\n  * 1 cup uncooked quinoa\n  * 2 cups water\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon coarsely ground black pepper\n  * \u00bd cup chopped pecans\n  * \u00bd cup dried cherries, coarsely chopped\n  * \u00bc cup finely chopped fresh cilantro or basil leaves\n  * 1 tablespoon grated orange peel\n\n1 Rinse quinoa thoroughly by placing in a fine-mesh strainer and holding under cold running water until water runs clear; drain well.\n\n2 In 2-quart saucepan, heat quinoa, water, salt and pepper to boiling; reduce heat. Cover and simmer 12 to 15 minutes or until water is absorbed and quinoa is tender.\n\n3 Stir in remaining ingredients. Serve warm.\n\n1 Serving (\u00bd Cup): Calories 160; Total Fat 6g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 23g (Dietary Fiber 2g, Sugars 9g); Protein 4g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\u00bd\n\n## Summer Quinoa-Tomato Salad Calorie Smart\n\nPrep 15 min Total 35 min \u25cf 8 servings\n\n  * \u00be cup uncooked quinoa\n  * 1\u00bd cups water\n  * 2 large tomatoes, quartered, cut into chunks\n  * \u00bc medium red onion, chopped\n  * 2 cloves garlic, finely chopped\n  * 3 tablespoons chopped fresh basil leaves\n  * 3 tablespoons chopped fresh parsley\n  * 4\u00bd teaspoons olive oil\n  * 3 tablespoons balsamic vinegar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon freshly ground pepper\n  * 2 teaspoons sugar\n  * \u00bc cup grated Parmesan cheese\n\n1 Rinse quinoa thoroughly by placing in a fine-mesh strainer and holding under cold running water until water runs clear; drain well.\n\n2 In 2-quart saucepan, heat quinoa and water to boiling; reduce heat to low. Cover and simmer 15 to 20 minutes or until water is absorbed and quinoa is tender. Cool slightly.\n\n3 In medium bowl, toss tomatoes, onion, garlic, basil, parsley, oil, vinegar, salt, pepper and sugar.\n\n4 Spread quinoa in large serving bowl or on platter; spoon tomato mixture over top. Serve or cover and refrigerate up to 24 hours. Sprinkle with cheese just before serving.\n\n1 Serving: Calories 110; Total Fat 4.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 210mg; Total Carbohydrate 14g (Dietary Fiber 1g, Sugars 4g); Protein 4g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\nBalsamic Pork Chops with Quinoa\n\n## Balsamic Pork Chops with Quinoa Calorie Smart\n\nPrep 25 min Total 45 min \u25cf 4 servings\n\n  * \u00bd cup uncooked quinoa\n  * 5 teaspoons vegetable oil\n  * 1 medium onion, cut into wedges\n  * 4 boneless pork loin chops, \u00bd to \u00be inch thick (6 oz each), trimmed of fat\n  * \u00be teaspoon seasoned salt\n  * \u00bc teaspoon dried thyme leaves\n  * \u00bc teaspoon pepper\n  * \u00be cup water\n  * \u00bc cup white balsamic vinegar\n  * 2 tablespoons chopped fresh parsley\n  * 2 cups small fresh broccoli florets\n  * 1\u00bd cups frozen peach slices\n\n1 Rinse quinoa thoroughly by placing in a fine-mesh strainer and holding under cold running water until water runs clear; drain well. Set aside.\n\n2 In 12-inch nonstick skillet, heat 2 teaspoons of the oil over medium-high heat. Cook onion in oil 5 minutes, stirring occasionally, until it begins to brown. Remove from skillet; set aside.\n\n3 Sprinkle both sides of pork chops with \u00bc teaspoon of the seasoned salt, the thyme and pepper. In skillet, heat remaining 3 teaspoons oil over medium heat. Add pork; cook 4 to 6 minutes, turning once, until browned. Remove from skillet to plate.\n\n4 Add quinoa, onion, water, vinegar, parsley and remaining \u00bd teaspoon seasoned salt to skillet. Heat to boiling; reduce heat. Place pork and any juices on top of quinoa mixture. Cover and simmer 10 minutes.\n\n5 Add broccoli and peaches. Cover and simmer about 10 minutes longer or until quinoa is tender and meat thermometer inserted in center of pork reads 145\u00b0F.\n\n1 Serving: Calories 470; Total Fat 21g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 115mg; Sodium 350mg; Total Carbohydrate 27g (Dietary Fiber 4g, Sugars 10g); Protein 44g Exchanges: 1 Starch, \u00bd Fruit, 1 Vegetable, 5\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 2\n\nFried Polenta\n\n## Polenta Calorie Smart \u2022 Fast\n\nPolenta is a staple in northern Italy and Eastern European countries, but we know it as cornmeal mush. It's very versatile, showing up for breakfast with maple syrup, nestled next to a grilled pork chop or topped with pasta sauce.\n\nPrep 20 min Total 20 min \u25cf 6 servings\n\n  * 1 cup yellow cornmeal\n  * 4 cups water\n  * 1\u00bd teaspoons salt\n  * 1 tablespoon butter or olive oil\n\n1 In 2-quart saucepan, mix cornmeal and \u00be cup of the water (this helps to prevent clumping). Stir in remaining 3\u00bc cups water and the salt. Cook over medium heat 8 to 10 minutes, stirring constantly, until mixture thickens and boils; reduce heat.\n\n2 Cover and simmer about 10 minutes, stirring occasionally, until very thick. Remove from heat. Add butter; stir until polenta is smooth.\n\n1 Serving (\u00be Cup): Calories 110; Total Fat 2.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 610mg; Total Carbohydrate 21g (Dietary Fiber 1g, Sugars 0g); Protein 2g Exchanges: \u00bd Starch, 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\nFried Polenta Spray 9x5-inch loaf pan with cooking spray. After simmering polenta 10 minutes in Step 2, spread in loaf pan. Cover and refrigerate at least 12 hours or until firm. Turn pan upside down to unmold. Cut into \u00bd-inch slices. Coat slices with flour. In 10-inch skillet, melt 2 tablespoons butter over low heat. Cook slices in butter about 5 minutes on each side or until brown.\n\nFaster Fried Polenta Use 13x9-in pan instead of loaf pan; refrigerate polenta uncovered about 3 hours or until firm. Cut into 6 squares. (Cut squares diagonally into triangles, if desired.) Cook in butter as directed for Fried Polenta.\n\nParmesan Polenta Make Polenta or Rich and Creamy Polenta as directed\u2014except stir in \u00bd cup shredded Parmesan cheese with the butter in Step 2.\n\nRich and Creamy Polenta Substitute half-and-half for 1\u00bd cups of the water. Continue as directed.\n\nVegetable Polenta Simmer 5 minutes in Step 2, then stir in 1\u00bd cups frozen mixed vegetables; simmer 5 minutes longer or until polenta is thick and vegetables are tender.\n\n### Learn with Betty Cooking Polenta\n\n**Perfect Polenta:** This polenta is smooth and creamy.\n\n**Undercooked Polenta:** This polenta mixture was not cooked long enough and all of the water was added at once.\n\n**Overcooked Polenta:** This polenta was cooked too long.\n\n## Learn to Simmer\n\nSimmering is one of the most useful techniques in cooking. Poached eggs, soups, stews and sauces will be a cinch when you understand this method. Simmering is a way to gently cook ingredients and allow flavors to blend and become more concentrated. But why simmer when you could just boil the food to get it done faster?\n\nBoiling displays large bubbles constantly and vigorously appearing on the surface, while simmering will show smaller bubbles slowly rising and breaking just below the surface. A rolling boil causes ingredients to bump around a lot, often cooking the outside before the inside is done. For foods like cream sauces and custards, high heat can cause curdling, and in some instances, food will burn at the bottom.\n\nAt a simmer, the food is jostled just enough to move it around and mix flavors. Simmering keeps foods from cooking too quickly, so they cook more evenly and tenderize; it also helps prevent curdling. Maintaining a simmer can be a little tricky because it requires careful regulation of the temperature and you may need to adjust the heat during cooking. Here are a few ways to reach a simmer:\n\n  * Bring the liquid to a boil over medium-high or high heat; reduce the heat until the liquid reaches a simmer. This method is often used for soups, stews and vegetables cooked in water.\n  * Gradually increase the heat of the liquid until it reaches a simmer. This method might be used for foods that shouldn't boil, such as milk or cream sauces, eggs and custards. Recipes in this category might indicate \"do not boil.\"\n\nSimmering can be divided further into the following simmer levels. If a recipe does not specify, use the middle simmer level.\n\n  * **Slow Simmer:** Usually low heat is used and very little movement of the liquid. You might see a bubble or two, but the idea is to cook very slowly. Most often used for stocks or grains that will soak up liquid.\n  * **Simmer:** Usually over medium-low heat with gentle bubbling. Use this level for soups, sauces and vegetables cooked in water.\n  * **Rapid Simmer:** Medium heat is typically used with more aggressive bubbling in the pot, but bubbles will still be fairly small. This level is most often used for sauce reduction.\n\nIn the Barley with Mixed Vegetables recipe (page 208), use the slow simmer level so that the barley cooks slowly and is tender throughout. Note that there are two simmer times in this recipe to allow the barley to completely cook but keep the vegetables from overcooking.\n\n## Barley with Mixed Vegetables Calorie Smart\n\nPrep 20 min Total 1 hr 20 min \u25cf 6 servings\n\n  * \u00bd cup uncooked hulled (not pearl) barley\n  * 1\u00be cups chicken broth (for homemade broth, see page 143)\n  * 2 large onions, chopped (2 cups)\n  * 2 packages (8 oz each) sliced fresh mushrooms (6 cups)\n  * 4 medium carrots, cut into julienne strips (2 cups)\n  * 1 large red bell pepper, coarsely chopped (1\u00bd cups)\n  * 2 tablespoons chopped fresh or 1 tablespoon dried dill weed\n  * \u00bd teaspoon pepper\n  * 4 medium green onions, chopped (\u00bc cup)\n\n1 Spray 12-inch skillet with cooking spray; heat over medium heat. Add barley; cook 6 to 8 minutes, stirring frequently until barley begins to brown, then stirring constantly until golden brown. Reduce heat to low; stir in broth. Cover and simmer 40 minutes.\n\n2 Stir in remaining ingredients except green onions. Heat to boiling over high heat; reduce heat to low. Cover and simmer 20 minutes longer or until vegetables are tender. Sprinkle with green onions.\n\n1 Serving: Calories 140; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 320mg; Total Carbohydrate 26g (Dietary Fiber 6g, Sugars 6g); Protein 6g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Vegetable Carbohydrate Choices: 2\n\nSimmering Barley\n\n## Apple-Rosemary Pork and Barley Calorie Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1\u00bd cups apple juice\n  * \u00be cup uncooked quick-cooking barley\n  * 2 tablespoons chopped fresh or 2 teaspoons dried rosemary leaves, crushed\n  * 2 teaspoons vegetable oil\n  * 1 pork tenderloin (\u00be lb), cut into \u00bc-inch slices\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 clove garlic, finely chopped\n  * \u00bc cup apple jelly\n  * 1 large unpeeled red cooking apple, sliced (1\u00bd cups)\n\n1 In 2-quart saucepan, heat apple juice to boiling. Stir in barley and 1 tablespoon of the rosemary; reduce heat to low. Cover and simmer 10 to 12 minutes or until liquid is absorbed and barley is tender.\n\n2 Meanwhile, in 10-inch nonstick skillet, heat oil over medium-high heat. Add pork, onion, garlic and remaining 1 tablespoon rosemary; cook about 5 minutes, stirring frequently, until pork is no longer pink in center. Stir in apple jelly and apple slices; cook until hot. Serve pork mixture over barley.\n\n1 Serving: Calories 400; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 55mg; Sodium 50mg; Total Carbohydrate 63g (Dietary Fiber 8g, Sugars 26g); Protein 23g Exchanges: 2 Starch, 2 Fruit, 2\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 4\n\n## Three-Grain Medley Calorie Smart \u2022 Slow Cooker\n\nPrep 10 min Total 4 hr 10 min \u25cf 8 servings\n\n  * \u2154 cup uncooked wheat berries\n  * \u00bd cup uncooked hulled or pearl barley\n  * \u00bd cup uncooked wild rice\n  * \u00bc cup chopped fresh parsley\n  * \u00bc cup butter, melted\n  * 2 teaspoons finely shredded lemon peel\n  * 6 medium green onions, thinly sliced (6 tablespoons)\n  * 2 cloves garlic, finely chopped\n  * 3\u00bd cups vegetable or chicken broth (for homemade broth, see page 156 or )\n  * 1 jar (2 oz) diced pimientos, undrained\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix all ingredients.\n\n2 Cover; cook on Low heat setting 4 to 6 hours or until liquid is absorbed. Stir before serving.\n\n1 Serving: Calories 200; Total Fat 6g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 190mg; Total Carbohydrate 30g (Dietary Fiber 5g, Sugars 0g); Protein 5g Exchanges: 2 Starch, 1 Fat Carbohydrate Choices: 2\n\n## Beef and Kasha Mexicana Calorie Smart \u2022 FAst\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\n  * 1 lb extra-lean (at least 90%) ground beef\n  * 1 small onion, chopped (\u2153 cup)\n  * 1 cup uncooked buckwheat kernels or groats (kasha)\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 can (4.5 oz) chopped green chiles, undrained\n  * 1 package (1 oz) 40% less-sodium taco seasoning mix\n  * 2 cups frozen whole kernel corn, thawed\n  * 1\u00bd cups water\n  * 1 cup shredded reduced-fat Cheddar cheese (4 oz)\n  * 2 tablespoons chopped fresh cilantro, if desired\n  * 2 tablespoons sliced pitted ripe olives, if desired\n\n1 In 12-inch skillet, cook beef and onion over medium-high heat 5 to 7 minutes, stirring occasionally, until beef is thoroughly cooked; drain. Stir in kasha until kernels are moistened.\n\n2 Stir in tomatoes, chiles, taco seasoning mix, corn and water. Heat to boiling; reduce heat to low. Cover and simmer 5 to 7 minutes, stirring occasionally, until kasha is tender.\n\n3 Sprinkle cheese over kasha mixture. Cover and cook 2 to 3 minutes or until cheese is melted. Sprinkle with cilantro and olives.\n\n1 Serving (1\u2153 Cups): Calories 300; Total Fat 8g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 50mg; Sodium 720mg; Total Carbohydrate 33g (Dietary Fiber 4g, Sugars 5g); Protein 23g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 2\u00bd Lean Meat Carbohydrate Choices: 2\n\nBeef and Kasha Tortillas Spoon mixture onto soft corn or flour tortillas; roll up.\n\nKasha and Bow-Tie Pilaf\n\n## Kasha and Bow-Tie Pilaf Calorie Smart\n\nLook for kasha, sometimes called roasted buckwheat kernels or groats, in the health-food, cereal or kosher-food area.\n\nPrep 20 min Total 35 min \u25cf 12 servings\n\n  * 3 tablespoons butter\n  * 1 large onion, coarsely chopped (1 cup)\n  * 1 medium red bell pepper, coarsely chopped\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * 1 cup uncooked buckwheat kernels or groats (kasha)\n  * 1 egg, beaten\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cup uncooked bow-tie (farfalle) pasta (2 oz)\n  * \u00bd cup chopped fresh parsley\n\n1 In 12-inch nonstick skillet, melt butter over medium heat. Cook onion, bell pepper and mushrooms in butter 3 to 4 minutes, stirring occasionally, until tender. Remove from skillet; set aside.\n\n2 In small bowl, stir kasha and egg, coating well. In same skillet, cook kasha over medium heat about 3 minutes, stirring constantly, until browned and dry.\n\n3 Return vegetables to skillet; stir in broth, salt and pepper. Heat to boiling; reduce heat to low. Cover and simmer 10 to 15 minutes or until broth is absorbed and kasha is tender.\n\n4 Meanwhile, cook and drain pasta as directed on package. Stir pasta and parsley into kasha mixture.\n\n1 Serving: Calories 100; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 300mg; Total Carbohydrate 13g (Dietary Fiber 2g, Sugars 2g); Protein 4g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nIndian-Spiced Farro Salad\n\n## Indian-Spiced Farro Salad Calorie Smart\n\nFarro is an ancient Tuscan grain similar in taste and texture to pearl barley. Look for it in the grain aisle, specialty stores or Italian markets.\n\nPrep 15 min Total 1 hr 45 min \u25cf 12 servings\n\nFarro\n\n  * 1\u00bd cups uncooked farro*\n  * 4 cups water\n\nSalad\n\n  * 1 medium red bell pepper, chopped\n  * \u00bd cup finely chopped red onion\n  * \u2153 cup finely chopped fresh cilantro leaves\n  * \u2153 cup finely chopped fresh mint leaves\n  * 2 tablespoons finely chopped seeded jalape\u00f1o chile\n  * \u2153 cup coarsely chopped dry-roasted peanuts\n\nDressing\n\n  * 3 tablespoons vegetable oil\n  * 2 tablespoons fresh lime juice\n  * 2\u00bd teaspoons ground cumin\n  * 1 teaspoon grated lime peel\n  * 1\u00bc teaspoons salt\n\n1 In 2-quart saucepan, heat farro and water to boiling; reduce heat to low. Cover and simmer 25 to 30 minutes or until tender. Rinse with cold water; drain well.\n\n2 In large bowl, mix all salad ingredients except peanuts; stir in farro. In small bowl, beat dressing ingredients with whisk until blended. Pour over salad; toss to coat. Cover and refrigerate 1 hour to blend flavors. Just before serving, stir in peanuts. Garnish with additional peanuts, cilantro and mint, if desired.\n\n*Pearl barley can be substituted for the farro. Cook 45 to 50 minutes.\n\n1 Serving (\u00bd Cup): Calories 150; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 19g (Dietary Fiber 2g, Sugars 1g); Protein 4g Exchanges: 1\u00bd Starch, 1 Fat Carbohydrate Choices: 1\n\nWheat Berry Salad\n\n## Wheat Berry Salad Calorie Smart\n\nWheat berries are whole, unprocessed kernels of wheat. Look for them in the cereal, bulk-bins or natural-food section of your supermarket.\n\nPrep 15 min Total 2 hr 45 min \u25cf 10 servings\n\n  * \u00be cup uncooked wheat berries\n  * 3 cups water\n\nCreamy Vinaigrette Dressing\n\n  * \u2153 cup vegetable oil\n  * 2 tablespoons mayonnaise or salad dressing\n  * 2 tablespoons red wine vinegar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon garlic powder\n  * \u215b teaspoon pepper\n\nSalad\n\n  * 1 cup small fresh broccoli florets\n  * 1 cup small fresh cauliflower florets\n  * 1 cup cherry tomatoes, cut in half\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 4 medium green onions, sliced (\u00bc cup)\n\n1 In 3-quart saucepan, soak wheat berries in water 30 minutes. Heat to boiling over high heat; reduce heat to low. Partially cover and simmer 55 to 60 minutes or until wheat berries are tender. Drain; rinse with cold water.\n\n2 In small bowl, stir all dressing ingredients until well mixed. In large serving bowl, toss wheat berries, salad ingredients and dressing. Cover and refrigerate at least 1 hour.\n\n1 Serving: Calories 170; Total Fat 11g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 230mg; Total Carbohydrate 13g (Dietary Fiber 3g, Sugars 2g); Protein 3g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 1\n\nCranberry-Almond Rye Berry Salad\n\n## Cranberry-Almond Rye Berry Salad\n\nRye berries can often be found in the self-serve bins in large supermarkets, or in co-ops or natural-food stores or online. If you love grains, this chewy, flavorful side dish will soon become a new favorite; it sure disappeared fast in our test kitchens.\n\nPrep 10 min Total 2 hr 40 min \u25cf 8 servings\n\nRye Berries\n\n  * 1 cup uncooked rye berries or wheat berries\n  * 4 cups water\n\nDressing\n\n  * \u00bc cup vegetable oil\n  * \u00bc cup honey\n  * 2 tablespoons fresh lemon juice\n  * \u00bd teaspoon salt\n\nSalad\n\n  * 1 cup diagonally sliced celery\n  * \u00bd cup dried cranberries\n  * 4 medium green onions, sliced (\u00bc cup)\n  * \u00bc cup chopped fresh parsley\n  * \u00bd cup slivered almonds, toasted (page 23)\n\n1 In 2-quart saucepan, soak rye berries in water 30 minutes. Heat to boiling; reduce heat to low. Partially cover and simmer 55 to 60 minutes or until tender. Drain; rinse with cold water.\n\n2 In small bowl, beat all dressing ingredients with whisk until blended.\n\n3 In medium bowl, toss rye berries, celery, cranberries, onions, parsley and dressing. Cover and refrigerate 1 hour to blend flavors. Stir in almonds just before serving.\n\n1 Serving (\u00bd Cup): Calories 250; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 160mg; Total Carbohydrate 34g (Dietary Fiber 4g, Sugars 14g); Protein 4g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\n\n## Betty's Staples Oats\n\nOats can be used for so much more than just oatmeal! Use this whole-grain powerhouse to make recipes even more nutritious and help keep you fuller longer. And don't miss the Granola recipe on page 108.\n\n  1. 1. **Golden Harvest Streusel Muffins:** Make Golden Harvest Muffins (page 484) as directed, except mix \u00bd cup oats, \u00bc cup each all-purpose flour and packed brown sugar, \u00bc teaspoon ground cinnamon and 2 tablespoons cold butter, cut into small pieces, until crumbly; stir in remaining \u00bc cup almonds. Sprinkle over batter in cups before baking.\n  2. 2. **Oat Pie Crust:** In medium bowl, mix \u00bd cup melted butter, \u00be cup all-purpose flour, \u00bd cup each quick-cooking oats and chopped nuts and 2 tablespoons sugar. Press in bottom and up side of 9-inch pie plate. Bake at 400\u00b0F 11 to 13 minutes or until edges are golden brown. Cool; fill with your favorite filling or ice cream.\n  3. 3. **Whole-Grain Pancakes:** Whisk \u00be cup each quick-cooking oats and whole wheat flour, 1 cup milk, 1 tablespoon sugar, 2 tablespoons vegetable oil, 2 teaspoons baking powder, \u00bc teaspoon salt and 1 egg until blended. Cook as for Pancakes (page 101) until golden brown.\n  4. 4. **No-Bake Chocolate-Oatmeal Drops:** Mix 2 cups oats, \u2154 cup peanut butter, 1 can (3\u00bd ounces) flaked coconut, \u00bc cup unsweetened baking cocoa and 1 teaspoon vanilla. In 1-quart saucepan heat 2 cups sugar, \u00bd cup milk and \u00bc cup butter to boiling; boil 1 minute, stirring constantly. Pour over oat mixture; quickly mix. Drop by heaping teaspoonfuls onto waxed paper\u2013lined cookie sheets. Cool completely.\n\nGolden Harvest Struesel Muffins\n\nWild Rice, Vegetable and Bacon Skillet\n\n## Wild Rice, Vegetable and Bacon Skillet Calorie Smart\n\nTo quickly determine if you've reduced the liquid enough, tip the pan toward you, and the liquid will gather in one spot\u2014simply eyeball the amount.\n\nPrep 20 min Total 1 hr 20 min \u25cf 8 servings\n\n  * 6 slices bacon, cut into 1-inch pieces\n  * 1 tablespoon butter\n  * 4 medium stalks celery, sliced (1 cup)\n  * 4 medium carrots, sliced (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 3 cups water\n  * 1 cup uncooked wild rice, rinsed, drained\n  * 1 teaspoon Italian seasoning\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * Toasted pecan halves or cashews, if desired\n\n1 In 10-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Remove bacon with slotted spoon to paper towels; set aside. Drain, reserving 1 tablespoon drippings in skillet.\n\n2 Add butter to bacon drippings; heat until melted. Add celery, carrots and onion. Cook 3 to 4 minutes, stirring occasionally, until crisp-tender. Remove from skillet; set aside. Add water, wild rice, Italian seasoning, salt and pepper to skillet. Heat to boiling; reduce heat. Cover and cook 45 minutes or until water is reduced to about \u00bd cup.\n\n3 Add mushrooms and reserved vegetable mixture to wild rice; stir to mix. Cover and cook 10 minutes. Uncover; cook 5 minutes or until water is absorbed. Sprinkle with bacon and pecans.\n\n1 Serving: Calories 160; Total Fat 6g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 310mg; Total Carbohydrate 22g (Dietary Fiber 3g, Sugars 3g); Protein 6g Exchanges: 1\u00bd Starch, 1 Fat Carbohydrate Choices: 1\u00bd\n\n## Rice\n\nRice is a food staple for almost half of the world's population. It is incredibly versatile and can add flavor and texture to meals. White rice comes in many forms: **White rice** is available in long, medium and short varieties. **Converted rice** is partially cooked and takes a little less time to cook. When you are really in a hurry, **instant rice** is the fastest, cooking in about 5 minutes. **Instant or precooked brown rice** is also quick to make but takes just a bit longer, about 10 minutes. It's also a great last-minute choice.\n\nTo cook rice, bring rice and water to a boil; reduce heat to low, cover and follow the specific directions for each kind of rice.\n\nArborio: Contains a high proportion of starch that gives risotto its characteristic creamy texture.\n\nBamboo: White rice infused with fresh bamboo juice, which imparts a pale jade color when cooked and a fragrance similar to jasmine tea. A very moist, sticky rice.\n\nBasmati: Long grains stay separate (not sticky) when cooked. Aromatic with subtle a flavor.\n\nBrown: White rice without the outer brown hull removed, so it's considered a whole grain. It contains more nutrients and fiber than white rice.\n\nJasmine: Fragrant white rice with a subtle nutty flavor that differs from basmati. Less sticky than other forms of rice.\n\nOrganic Sweet Brown: Sweeter and more glutinous than regular brown rice, with the benefits of whole grain.\n\nRed: Russet-colored whole-grain rice. It has a complex, nutty, flavor and soft texture that pairs well with assertive flavors like duck, pork and wild mushrooms. Great steamed plain, or used in pilafs and risotto.\n\nSushi: The right proportion of starches allows the grains to stick together and therefore help hold sushi together.\n\nWhite: White rice includes three main types, with different characteristics:\n\n> Long-Grain: Grains stay separate and fluffier when cooked; good for side dishes.\n> \n> Medium-Grain: Plumper and shorter; good for paella or as a substitute for Arborio in risotto.\n> \n> Short-Grain: The shortest and most moist; good for puddings and molded salads.\n\nWild: Not actually rice, but seeds from a semiaquatic grass with the benefits of whole grain. Somewhat chewy with a nutty flavor, wild rice is significantly higher in protein than regular white rice and is a good source of fiber.\n\n### Timetable for Cooking Rice\n\nTo cook rice, bring rice and water to a boil, reduce heat to low, cover and follow the specific directions for each kind of rice.\n\nType of Rice (1 Cup) | Amount of Water in Cups | Cooking Directions | Yield in Cups\n\n---|---|---|---\n\nBasmati White | 1\u00bd | Simmer 15 to 20 minutes. | 3\n\nJasmine | 1\u00be | Simmer 15 to 20 minutes. | 3\n\nLong-Grain White | 2 | Simmer 20 to 25 minutes | 3\n\nParboiled (Converted) White | 2\u00bd | Simmer 20 minutes. Let stand 5 minutes. | 3 to 4\n\nPrecooked (Instant) White | 1 | After stirring in rice, cover and remove from heat; let stand 5 minutes. | 2\n\nLong-Grain Brown (Regular and Basmati) | 2\u00be | Simmer 45 to 50 minutes. | 4\n\nPrecooked (Instant) Brown | 1\u00bd | Simmer 10 minutes. | 2\n\nWild | 2\u00bd | Simmer 40 to 50 minutes. | 3\n\nCherry Pistachio Rice Pilaf\n\n## Rice Pilaf Fast \u2022 EASY\n\nRice pilaf can be a new dish each time you make it just by adding different ingredients. The secret to the flavor is browning the rice in butter until it is golden brown before adding the liquid to the pan.\n\nPrep 10 min Total 30 min \u25cf 4 servings\n\n  * 2 tablespoons butter\n  * 1 small onion, chopped (\u2153 cup)\n  * 1 cup uncooked regular long-grain white rice\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * \u00bc teaspoon salt\n\n1 In 3-quart saucepan, melt butter over medium heat. Cook onion in butter about 3 minutes, stirring occasionally, until tender.\n\n2 Stir in rice. Cook 5 minutes, stirring frequently, until rice is golden brown. Stir in broth and salt.\n\n3 Heat to boiling, stirring once or twice; reduce heat to low. Cover and simmer 15 minutes (do not lift cover or stir). Remove from heat; let stand covered 5 minutes.\n\n1 Serving: Calories 260; Total Fat 7g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 700mg; Total Carbohydrate 42g (Dietary Fiber 0g, Sugars 0g); Protein 7g Exchanges: 2 Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 3\n\nBrown Rice Pilaf Substitute uncooked regular brown rice for the white rice. Make as directed\u2014except in Step 3, increase simmer time to 45 to 50 minutes.\n\nCashew Pilaf Stir \u00bc cup coarsely chopped cashews and 2 tablespoons chopped fresh parsley into the cooked rice pilaf.\n\nCurry Pilaf Stir in \u00bd cup diced dried fruit and raisin mixture, \u00bc teaspoon ground allspice, \u00bc teaspoon ground turmeric and \u00bc teaspoon curry powder with the broth and salt in Step 2.\n\nMushroom Pilaf Cook 1 cup sliced fresh mushrooms or 1 can (4 oz) mushrooms pieces and stems, drained, with the onion in Step 1.\n\nOrange and Cranberry Pilaf Add \u00bd teaspoon grated orange peel with the broth and salt in Step 2. Stir \u00bc cup sweetened dried cranberries into the cooked rice pilaf.\n\nPine Nut and Green Onion Pilaf Cook \u00bd cup sliced green onions and \u00bc cup pine nuts with the rice in Step 2.\n\nProsciutto, Parmesan and Pea Pilaf Cook 3 ounces prosciutto, chopped, with the rice in Step 2. Stir \u00bc cup frozen sweet peas, thawed, and \u00bc cup shredded Parmesan cheese into the cooked rice pilaf.\n\nCherry Pistachio Rice Pilaf Substitute uncooked wild and whole-grain brown rice blend for the white rice. Make as directed, except in Step 3, increase simmer time to 45 to 50 minutes. Stir \u00bc cup each dried cherries and chopped pistachio nuts into the cooked rice pilaf.\n\nSpanish Rice\n\n## Spanish Rice Calorie Smart\n\nPrep 15 min Total 40 min \u25cf 10 servings\n\n  * 2 tablespoons vegetable oil\n  * 1 cup uncooked regular long-grain white rice\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2\u00bd cups water\n  * 1\u00bd teaspoons salt\n  * \u00be teaspoon chili powder\n  * \u215b teaspoon garlic powder\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 1 can (8 oz) tomato sauce\n  * 1 large tomato, seeded, chopped (1 cup)\n\n1 In 10-inch skillet, heat oil over medium heat. Cook rice and onion in oil about 5 minutes, stirring frequently, until rice is golden brown and onion is crisp-tender.\n\n2 Stir in remaining ingredients except tomato. Heat to boiling; reduce heat. Cover and simmer 20 minutes, stirring occasionally. Stir in tomato. Cover and simmer about 5 minutes longer or until rice is tender.\n\n1 Serving (\u00bd Cup): Calories 120; Total Fat 3g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 500mg; Total Carbohydrate 20g (Dietary Fiber 1g, Sugars 2g); Protein 2g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nSpanish Rice with Bacon Omit oil. Cut 6 slices bacon into 1-inch pieces. In 10-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Remove from skillet with slotted spoon and drain on paper towels. Drain all but 2 tablespoons drippings from skillet. Cook rice and onion in bacon drippings over medium heat about 5 minutes, stirring frequently, until rice is golden brown and onion is crisp-tender. Continue as directed in Step 2. Stir in bacon just before serving.\n\nPork Fried Rice\n\n## Pork Fried Rice Calorie Smart \u2022 Fast\n\nUsing cold cooked rice is best so the grains stay separated during frying.\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 cup fresh bean sprouts\n  * 2 tablespoons vegetable oil\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * 3 cups cold cooked regular long-grain white rice\n  * 1 cup cut-up cooked pork\n  * 2 medium green onions, sliced (2 tablespoons)\n  * 2 eggs, slightly beaten\n  * 3 tablespoons soy sauce\n  * Dash white pepper\n\n1 Rinse bean sprouts with cold water; drain and set aside.\n\n2 In 10-inch skillet, heat 1 tablespoon of the oil over medium heat; rotate skillet until oil covers bottom. Cook mushrooms in oil about 1 minute, stirring frequently, until coated.\n\n3 Add bean sprouts, rice, pork and onions to skillet. Cook over medium heat about 5 minutes, stirring and breaking up rice, until sprouts are thoroughly cooked and no longer crisp and mixture is hot.\n\n4 Move rice mixture to side of skillet. Add remaining 1 tablespoon oil to other side of skillet. Cook eggs in oil over medium heat, stirring constantly, until eggs are thickened throughout but still moist. Stir eggs into rice mixture. Stir in soy sauce and pepper.\n\n1 Serving: Calories 360; Total Fat 14g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 135mg; Sodium 1190mg; Total Carbohydrate 37g (Dietary Fiber 1g, Sugars 1g); Protein 20g Exchanges: 2\u00bd Starch, 2 Lean Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\nLighter Directions For 3 grams of fat and 240 calories per serving, reduce pork to \u00bd cup and finely chop. Use a nonstick skillet and omit oil in Step 4. Substitute \u00bd cup fat-free egg product for the eggs.\n\nChicken Fried Rice Substitute 1 cup shredded cooked chicken for the pork and add 1 cup frozen sweet peas, cooked, in Step 3.\n\nShrimp Fried Rice Substitute 1 cup frozen cooked salad shrimp, thawed, for the pork.\n\n## Make-Ahead Brown Rice\n\nMild, nutty brown rice is a wonderful addition to any meal. It's delicious, healthful and easy to cook and store. But because brown rice is a whole grain and not refined, it does take a bit longer to cook than white rice. So why not make a double batch to have on hand for a variety of uses?\n\nTo make 4 cups cooked brown rice, bring 2\u00bc cups water to a boil in a 2-quart saucepan. Stir in 1 cup uncooked brown rice; reduce heat to low and cover. Cook 45 to 50 minutes or until rice is tender and water is absorbed.\n\nCool and store covered in the refrigerator for up to 5 days or freeze for up to 3 months.\n\n## Ways to Use Brown Rice\n\n  1. 1. **Chicken\u2013Brown Rice Salad:** In medium bowl, toss 1 cup cooked brown rice with 1 cup shredded cooked chicken, \u00bd cup small red grapes, \u00bc cup sliced green onions or celery and 2 to 4 tablespoons honey-mustard dressing. Serve on torn leaf lettuce.\n  2. 2. **Almond\u2013Brown Rice Pilaf:** In 10-inch skillet, heat 1 tablespoon olive oil or butter over medium heat. Stir in \u00bd cup slivered almonds and \u00bd cup chopped carrot. Cook and stir 2 to 4 minutes or until carrot is tender. Stir in \u00bc cup dried currants and 1\u00bd cups cooked brown rice. Cook and stir just until heated. Sprinkle with chopped fresh parsley.\n  3. 3. **Quick Lettuce Wraps:** In 10-inch skillet, cook 1 pound ground turkey, \u00bc cup finely chopped onion and \u00bc teaspoon ground ginger until turkey is no longer pink. Stir in 2 tablespoons stir-fry sauce, \u00bd cup chopped red bell pepper and 1 cup cooked brown rice. Spoon mixture into butterhead lettuce leaves; roll up each. Serve with sweet-and-sour sauce.\n  4. 4. **Salsa\u2013Brown Rice Burritos:** In 2-quart saucepan, heat 1 cup chunky salsa just until hot. Stir in 1 teaspoon chili powder, 1 cup cooked or canned corn, 1 can (15 ounces) drained, rinsed black beans and 1 cup cooked brown rice. Spoon mixture down center of flour tortillas; roll up.\n  5. 5. **Fried Brown Rice:** In 10-inch skillet, heat 1 tablespoon olive oil or butter over medium heat. Stir in \u00bd cup slivered almonds and \u00bd cup chopped carrot. Cook and stir 2 to 4 minutes or until carrot is tender. Stir in \u00bc cup dried currants and 1\u00bd cups cooked brown rice. Cook and stir just until heated. Sprinkle with chopped fresh parsley.\n  6. 6. **Easy Rice Pudding:** In 2-quart saucepan, combine 2 cups cooked brown rice, 2 cups milk, \u00bc cup dried cranberries and \u00bc cup sugar. Bring to a boil. Reduce the heat to low and cook 15 to 20 minutes or until thickened, stirring occasionally. Stir in 1 teaspoon vanilla.\n\nQuick Lettuce Wraps, Almond\u2013Brown Rice Pilaf, Brown Rice\n\nClassic Risotto with Peas\n\n## Classic Risotto\n\nPrep 35 min Total 35 min \u25cf 4 servings\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 tablespoon butter\n  * 1 small onion, thinly sliced\n  * 1 tablespoon chopped fresh parsley\n  * 1 cup uncooked Arborio rice or regular long-grain white rice\n  * \u00bd cup dry white wine or chicken broth\n  * 3 cups chicken broth (for homemade broth, see page 143), warmed\n  * \u00bd cup freshly grated Parmesan cheese\n  * \u00bc teaspoon coarsely ground black pepper\n\n1 In 10-inch nonstick skillet or 3-quart saucepan, heat oil and butter over medium-high heat until butter is melted. Add onion and parsley; cook about 5 minutes, stirring frequently, until onion is tender.\n\n2 Stir in rice. Cook, stirring occasionally, until edges of kernels are translucent. Stir in wine. Cook about 3 minutes, stirring constantly, until wine is absorbed.\n\n3 Reduce heat to medium. Stir in 1 cup of the warm broth. Cook uncovered about 5 minutes, stirring frequently, until broth is absorbed. Repeat, adding another 1 cup of broth. Stir in remaining 1 cup broth. Cook about 8 minutes, stirring frequently, until rice is just tender and mixture is creamy. Stir in cheese and pepper.\n\n1 Serving: Calories 360; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 15mg; Sodium 1020mg; Total Carbohydrate 43g (Dietary Fiber 0g, Sugars 2g); Protein 13g Exchanges: 3 Starch, 2\u00bd Fat Carbohydrate Choices: 3\n\nClassic Risotto with Butternut Squash Stir in 1 cup mashed cooked butternut squash or sweet potato with the first cup of broth in Step 3. Continue as directed.\n\nClassic Risotto with Peas Omit parsley. Stir in 1 box (9 ounces) frozen baby sweet peas, cooked and drained.\n\nClassic Risotto with Shrimp Stir in 8 ounces cooked medium shrimp during last 3 to 5 minutes of cooking.\n\n### Learn with Betty Cooking Risotto\n\n**Perfect Risotto:** This risotto is thickened and creamy.\n\n**Undercooked Risotto:** This risotto was not cooked long enough and is soupy or watery instead of creamy.\n\n**Overcooked Risotto:** This risotto was cooked too long and is dried out and no longer creamy.\n\n### Cooking Risotto\n\nStir rice into the onion and parsley, cooking until the edges of the grains are translucent.\n\nStir warmed broth, 1 cup at a time, into the rice, stirring until each broth addition is absorbed.\n\nSticky Rice with Mango\n\n## Sticky Rice with Mango Calorie Smart\n\nGlutinous rice, also known as sweet or sticky rice, gets its sticky character from a high starch content. Garnish this classic Thai dessert with fresh, flaked or shredded coconut, either toasted or right out of the package.\n\nPrep 15 min Total 2 hr 5 min \u25cf 10 servings\n\nSticky Rice\n\n  * 1\u00bd cups sweet rice (sticky or glutinous rice)*\n  * 1 can (13.66 oz) coconut milk (not reduced-fat and not cream of coconut)\n  * \u00bd cup sugar\n  * \u00bc teaspoon salt\n  * 1 large mango, peeled, pitted and cut into \u00bc-inch slices\n\nCoconut Sauce\n\n  * 2 tablespoons sugar\n  * \u00bd teaspoon cornstarch\n  * Toasted flake coconut or sesame seed, if desired\n\n1 Soak rice in cold water 1 hour; drain. Cook rice in rice cooker using the Sweet Rice setting according to manufacturer's directions. Or spray 2-quart saucepan with cooking spray. Add rice and 2 cups water. Heat to boiling; reduce heat to low. Cover and simmer 15 to 20 minutes or until water is absorbed. Spoon rice into large bowl and gently even top; set aside.\n\n2 Meanwhile, in 2-quart saucepan, heat 1 cup of the coconut milk (reserve remaining coconut milk for sauce), \u00bd cup sugar and the salt over medium heat until hot. Simmer, stirring frequently, just until sugar is dissolved. Pour over rice; gently fold into rice. Let stand 30 to 35 minutes at room temperature or until almost all of milk mixture is absorbed.\n\n3 In 1-quart saucepan, mix the reserved coconut milk, 2 tablespoons sugar and the cornstarch until cornstarch is dissolved. Heat to boiling; boil 1 minute, stirring constantly. Remove from heat. Spoon rice into individual bowls. Garnish with mango slices; drizzle each serving with 2 teaspoons coconut sauce. Sprinkle with sesame seed.\n\n*One package (12 oz) Arborio rice can be substituted for the sweet rice. If making in rice cooker, follow manufacturer's directions. If making on stove top, make as directed on package for plain or basic white Arborio rice (not risotto), except omit any fat or salt called for on package. Continue as directed in Step 2.\n\n1 Serving (\u00bd Cup): Calories 190; Total Fat 7g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 30g (Dietary Fiber 1g, Sugars 19g); Protein 1g Exchanges: \u00bd Starch, 1\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\n### Making Sticky Rice\n\nPour cooked coconut mixture over cooked rice.\n\nDrizzle each serving with coconut sauce.\n\n## Curried Veggie Brown Rice Fast\n\nPrep 5 min Total 20 min \u25cf 8 servings\n\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * 1\u00bd teaspoons curry powder\n  * \u00bd teaspoon salt\n  * 2 cups uncooked instant brown rice\n  * 1\u00bd cups frozen mixed vegetables\n  * \u2153 cup raisins\n  * 2 tablespoons chopped fresh parsley\n\n1 In 2-quart saucepan, heat broth, curry powder and salt to boiling. Stir in rice; reduce heat to medium-low. Cover and cook 3 minutes.\n\n2 Add vegetables. Cover and cook 4 to 6 minutes longer or until rice and vegetables are tender.\n\n3 Remove from heat; let stand 3 to 4 minutes or until liquid is absorbed. Fluff rice with fork. Stir in raisins and parsley.\n\n1 Serving (\u00bd Cup): Calories 230; Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 430mg; Total Carbohydrate 47g (Dietary Fiber 6g, Sugars 6g); Protein 6g Exchanges: 2 Starch, 1 Other Carbohydrate, \u00bd Vegetable Carbohydrate Choices: 3\n\n# Chapter 9: Vegetables, Beans & Legumes\n\n## Vegetable Basics\n\nBuying Vegetables\n\nCabbage\n\nChiles\n\nFresh Legumes\n\nFresh Vegetable Cooking Chart\n\nHealth Benefits\n\nMushrooms\n\nNightshade Vegetables\n\nOnions\n\nOther Vegetables\n\nPotatoes\n\nRoot Vegetables\n\nSquash\n\nStoring Vegetables\n\n## Features\n\nBetty's Staples: Edamame\n\nGlobal Flavors: India\n\nHeirloom Recipe and New Twist\n\nLearn to Roast\n\nLearn with Betty: Caramelized Onions\n\n## Vegetables\n\nArtichokes with Rosemary Butter Easy\n\nAsparagus with Goat Cheese\n\nAsparagus with Parmesan Calorie Smart \u2022 Fast\n\nBroccoli with Roasted Red Peppers and Hazelnuts Calorie Smart \u2022 Fast\n\nCaramelized Onions Calorie Smart\n\nCheesy Bacon Broccoli\n\nCheesy Bacon Brussels Sprouts Calorie Smart\n\nCheesy Bacon Green Beans\n\nCollard Greens with Black-Eyed Peas\n\nCranberry Caramelized Onions\n\nCranberry-Pistachio Brussels Sprouts Calorie Smart \u2022 Fast\n\nCumin-Chili Roasted Carrots Calorie Smart\n\nFried Green Tomatoes Calorie Smart\n\nGlazed Carrots Calorie Smart \u2022 Fast\n\nGreen Beans with Glazed Shallots Calorie Smart \u2022 Fast\n\nGreen Beans with Parmesan\n\nHoney-Glazed Lemon Carrots\n\nOkra, Corn and Tomatoes Calorie Smart \u2022 Fast\n\nRoasted Beets Calorie Smart \u2022 Easy\n\nRoasted Bell Peppers Calorie Smart \u2022 Easy\n\nRoasted Root Vegetables\n\nRoasted Vegetables Calorie Smart\n\nSaut\u00e9ed Cauliflower with Browned Bread Crumbs Fast\n\nSpicy Collard Greens with Bacon Calorie Smart\n\nSpiral Summer Squash Calorie Smart \u2022 Fast \u2022 Easy\n\nSweet-Sour Red Cabbage Calorie Smart\n\nWilted Spinach Calorie Smart \u2022 Fast\n\nWilted Spinach and Kale\n\n## Potatoes\n\nAu Gratin Potatoes\n\nBacon and Pepper Twice-Baked Potatoes\n\nButtermilk Mashed Potatoes\n\nCandied Sweet Potatoes Easy\n\nFried Potatoes Calorie Smart \u2022 Fast\n\nGarlic Mashed Potatoes\n\nHash Brown Potatoes Calorie Smart\n\nHerbed Mashed Potatoes\n\nHorseradish Mashed Potatoes\n\nItalian Twice-Baked Potatoes\n\nMashed Potatoes Calorie Smart \u2022 Easy\n\nPotato Pancakes Calorie Smart\n\nScalloped Potatoes\n\nTwice-Baked Potatoes Calorie Smart\n\nUltimate Smashed Potatoes Calorie Smart \u2022 Slow Cooker\n\nBeans and Legumes\n\n## Bean Basics\n\nCooking Dried Beans\n\nDried Beans and Lentils\n\nTimetable for Cooking Dried Beans\n\n## Bean Dishes\n\nBacon-Edamame Toss\n\nCannellini with Tomatoes and Sage Calorie Smart \u2022 Fast\n\nCaribbean Black Beans\n\nEdamame and Vegetable Salad\n\nEdamame Boiled in the Pods\n\nGarbanzo Beans with Vegetables Slow Cooker\n\nHomemade Baked Beans\n\nHoppin' John\n\nIndian Lentils and Rice Calorie Smart\n\nPasta with Edamame and Pesto\n\nRed Beans and Rice Calorie Smart\n\nSpicy Black-Eyed Peas Calorie Smart \u2022 Slow Cooker \u2022 Easy\n\nSpicy Refried Black Beans Calorie Smart \u2022 Fast\n\nStir-Fried Edamame with Corn and Peppers\n\nWhite Beans with Sun-Dried Tomatoes Slow Cooker \u2022 Easy\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Vegetable Basics\n\nWhether part of the main dish or served as a side, vegetables add texture, color, flavor and a bounty of healthful nutrients to any meal. Browse any farmers' market or grocery store, and you will see the array of choices available, from standards like carrots and broccoli to more exotic vegetables like golden beets and carnival squash. Among the varieties shown on the following pages, you'll find many treasures to brighten your table.\n\n## Health Benefits\n\nEating vegetables provides health benefits. People who eat more vegetables as part of an overall healthy diet are likely to have a reduced risk of some chronic diseases. Vegetables provide nutrients vital for general wellness.\n\n  * Most vegetables are naturally low in fat and calories, and none have any cholesterol.\n  * Vegetables are important sources of many nutrients, including potassium, dietary fiber, folic acid, vitamin A and vitamin C.\n  * Dietary fiber from vegetables as part of an overall healthy diet can help reduce blood cholesterol levels and may lower risk of heart disease.\n  * To gain the most value from vegetables, choose a variety and plan for several servings every day. For specific amounts to eat daily, refer to the USDA's ChooseMyPlate.gov.\n\n## Buying Vegetables\n\nLook for the freshest vegetables possible\u2014appearance is usually the best indication with good, bright color at the top of the list. Avoid any vegetable that looks withered or has soft spots. Leafy tops and vegetable greens should be crisp and fresh looking.\n\nOrganic vegetables are grown without synthetic fertilizers or pesticides. Look for organic produce at most grocery stores and specialty food stores.\n\n## Storing Vegetables\n\nStorage varies for vegetables but in general, fresh vegetables should be used within a few days of purchase, as they do start to lose flavor quality and nutritional value if stored too long. See the vegetable listings, starting on page 225, for recommendations on individual vegetables.\n\n## Nightshade Vegetables\n\nThis group also includes potatoes, page 231.\n\nEggplant: Choose firm, even-colored, unblemished eggplants heavy for their size. Caps and stems should be intact with no mold. Refrigerate in plastic bag up to 5 days.\n\n> Asian (Chinese, Japanese, Thai): Colors include purple, green, white or striated. Longer, smaller than common eggplant.\n\nSweet Peppers: Choose shiny, brightly colored, unblemished peppers with firm sides. Refrigerate in plastic bag up to 5 days.\n\n> Banana: Long, yellow, banana-shaped.\n> \n> Bell: Available in green, red, yellow, orange, purple, brown.\n\nTomatillos: Choose firm, unblemished fruit with tight-fitting husks. Store in paper bag in refrigerator up to 1 month. Remove husks; wash fruit before using.\n\nTomatoes: Choose firm, unblemished fruit heavy for their size. Store at room temperature (refrigerating will cause flavor loss); use within a few days.\n\n> Beefsteak: Rich flavor, meaty texture. Good fresh or cooked.\n> \n> Cherry: Red or yellow; bite-size, great flavor. Use in salads, as garnish or cooked.\n> \n> Heirloom: Seeds are passed down through generations. Have distinct attributes like exceptional flavor or unusual coloring.\n> \n> Plum (Roma): Great flavor; fleshy, making them great for sauces and salads, as there is less juice than in a round tomato. **Grape** tomatoes are baby plum tomatoes.\n\n## Chiles\n\nChoose firm chiles; avoid shriveled or soft spots. Store fresh chiles in refrigerator drawer; store dried ones in airtight container in a cool, dark place. Typically, the larger the chile, the milder the flavor.\n\nAnaheim: Fresh, canned. Mild; frequently stuffed.\n\nFresno: Fresh. Medium-size. Similar in flavor to jalape\u00f1o. Usually red.\n\nHabanero: Fresh or dried. Range from light green to bright orange. Distinctively hot.\n\nJalape\u00f1o: Fresh or canned. Dark green; hot to very hot. **Chipotle** is a dried, smoked jalape\u00f1o.\n\nPoblano: Fresh or canned. Green: flavor ranges from mild to zippy. Reddish brown: sweeter than green. **Ancho** is a dried poblano.\n\nSerrano: Fresh. Medium-size. Very hot and green at first, ripening to red or orange.\n\nThai: Fresh. Small; several colors. Very hot.\n\n## Onions\n\nThe onion group of vegetables is often called the Lily family and includes not only onions but also garlic, leeks, shallots and asparagus. They all grow tubers, bulbs or rhizomes.\n\nAsparagus: Spears are green, white or purple. Choose firm, brightly colored stalks with tight tips. Refrigerate in plastic bag up to 3 days.\n\nGarlic: Made of papery-coated sections called cloves. Choose fresh, firm heads of garlic that have no green sprouts.\n\nLeeks: Milder than onions. Look for crisp leaves and unblemished white bulb ends; avoid withered, yellow leaves. Refrigerate in a plastic bag up to 5 days. Cut lengthwise; wash thoroughly to remove dirt trapped between layers.\n\nOnions: Choose firm, blemish-free onions with thin skins and no sign of mold or sprouting. Do not store near potatoes\u2014onions will absorb moisture.\n\n> Green (Scallions): Look for bright green, fresh tops, white bulb ends. Refrigerate in plastic bag up to 1 week.\n> \n> Red, Yellow, White: Store whole in cool (45\u00b0F to 60\u00b0F), dry, dark place with good ventilation up to 2 months. Once cut, cover with plastic wrap; refrigerate up to 4 days.\n> \n> Sweet: Very juicy, mild sugary taste without the \"burn\" associated with other onions. Refrigerate whole in single layer 4 to 6 weeks.\n> \n> Shallots: Papery skin; more like garlic than onions. Store in cool, dry place up to 2 weeks.\n\n## Mushrooms\n\nStore unwashed, wrapped in paper towel in a plastic or paper bag. Refrigerate up to 4 days. For fresh, look for firm caps with good color and no soft spots or decay. Wash just before using.\n\nChanterelle: Fresh, dried. Delicate, nutty flavor; chewy texture. Cook as side dish, or add to soups, sauces, stir-fries.\n\nDried (Various): More concentrated flavor than fresh. Store in cool, dry place up to 1 year. Rehydrate as directed on package. One ounce dried (rehydrated) = 4 ounces fresh.\n\nMorel: Fresh, dried. Smoky, earthy, nutty flavor. Saut\u00e9 in butter.\n\nPorcini: Often dried. Earthy flavor; firm texture. Popular in French and Italian cooking.\n\nPortabella: Very large, with dense, meaty texture. Grill for sandwiches, saut\u00e9s, slice for salads. **Cremini** is a young portabella.\n\nShiitake: Meaty texture; strong flavor. Choose plump ones with edges that curl under.\n\nWhite (Button): Mild, earthy flavor. Small to quite large (for stuffing). Choose firm, white ones with no sign of decay.\n\n## Cabbage\n\nThese are all cruciferous vegetables, part of the cabbage family.\n\nBok Choy: Mild. Crunchy stalks; tender leaves. Use in salads and stir-fries; as side dish.\n\nBroccoli: Florets and stalks edible. Use raw in salads, or cook. Florets should be tight and not flowering.\n\nBroccolini: Cross between broccoli and Chinese kale. Mildly sweet, slightly peppery; crunchy. Refrigerate up to 10 days. Use raw in salads, crudit\u00e9s; cook crisp-tender as side dish.\n\nBrussels Sprouts: Strong flavor similar to cabbage. Small sprouts will be milder than large ones. Look for those with compact heads. Cook for side dishes.\n\nCabbage: Choose cabbage with fresh, crisp leaves (firmly packed, if head variety). Refrigerate, tightly wrapped, up to 1 week.\n\nCauliflower: Mild and usually white. Florets should be firm and compact. Largest, toughest stalks usually not eaten. Use raw in salads, or cook.\n\nKale: Mild cabbage flavor; many varieties and colors. Store in coldest part of refrigerator 2 to 3 days (any longer, and flavor gets very strong). Remove and discard tough center stalk.\n\nKohlrabi (Cabbage Turnip): Bulbs have mild broccoli\u2013celery root flavor. Leaves have flavor similar to collard greens. Use raw; add to soups, stews, stir-fries. Refrigerate leaves up to 4 days, bulbs up to 10 days.\n\n## Other Vegetables\n\nArtichoke, Globe: Available fresh, canned (hearts and bottoms) in various sizes: jumbo for stuffing and steaming, baby in marinated salads. Or use saut\u00e9ed or roasted. Refrigerate in plastic bag up to 1 week.\n\nArtichoke, Jerusalem (Sunchoke): Available year-round (season peaks October to March). Choose hard artichokes with smooth skins and no soft spots. Use raw in salads, or boiled, steamed, fried. Refrigerate in plastic bag up to 1 week.\n\nCelery: Crisp, mild and stringy stalks. Use raw in salads, or cook.\n\nCorn: White, yellow, bicolored. Choose bright green, tight-fitting husks; fresh-looking silk; plump, consistently sized kernels. Wrap unhusked in damp paper towels; refrigerate up to 2 days.\n\nEnglish Cucumber: Available year-round (peak is May to August). Virtually seedless. Choose firm cucumbers with smooth skins; avoid shriveled or soft spots. Refrigerate in plastic bag up to 10 days.\n\nOkra: Available fresh year-round in the South (May to October in rest of United States), canned, frozen. Choose tender, unblemished, bright green pods less than 4 inches long. Refrigerate in plastic bag up to 3 days.\n\n## Root Vegetables\n\nBeets: Colors include reddish-purple, golden, rainbow and white. Choose brightly colored, firm beets with smooth skins. Remove greens as soon as possible; store separately. Greens and roots should be cooked. Refrigerate in plastic bag up to 1 week.\n\nCarrots and Parsnips: Mild sweet flavor. Choose those with no signs of decay or cracking. Refrigerate in plastic bag up to 2 weeks.\n\nCassava (Yuca): Bland, starchy; absorbs other flavors. Two types: sweet and bitter (which must be cooked before eating). Choose hard, evenly shaped yuca; avoid blemishes, cracks, mold, stickiness. Store in cool, dark place up to 3 days. Use as side dish or in soups.\n\nFennel: Mild licorice flavor. Bulbs and greens can be eaten raw or cooked. Choose clean, firm bulbs with fresh-looking greens. Refrigerate in plastic bag up to 1 week.\n\nJicama: A sweet vegetable that looks like a turnip and can be eaten raw; use in salads.\n\nGingerroot: Peppery, slightly sweet. Choose firm, smooth roots. Refrigerate fresh, unpeeled in plastic bag up to 3 weeks; freeze up to 6 months.\n\nRadishes: Mild to peppery flavor. Great in salads, as snacks. Choose firm radishes with bright colors. **Daikon** are much larger; good in salads, stir-fries. Refrigerate either variety in plastic bag up to 1 week.\n\nSweet Potatoes: Available in varieties ranging from pale skin with light yellow flesh to dark orange or red skin and flesh. Often mistakenly called yams but true yams are not sweet potatoes. Choose nicely shaped, smooth, firm potatoes with evenly colored skins. Store in cool (45\u00b0F to 60\u00b0F), dry, dark, well-ventilated place up to 2 weeks.\n\nTurnips, Rutabagas, Celeriac: Good roasted, used in stews, baked dishes. Choose firm vegetables with no signs of decay or bruising. Refrigerate in plastic bag up to 2 weeks.\n\n## Potatoes\n\nStore potatoes in a cool (45\u00b0F to 60\u00b0F) dry, dark, well-ventilated place as directed below.\n\nAll-Purpose Potatoes: **Purple** and **Yukon Gold** can be used with any cooking technique. Most of the color of purple potatoes will fade during cooking. Store 2 to 3 weeks.\n\nStarchy Potatoes: Both **Russet** and **Idaho** are great for baking or frying. Store 3 to 5 weeks.\n\nWaxy Potatoes: **Fingerling** , **New** and **Red** are great in soups or salads, as they hold their shape after cooking; also great for roasting and boiling. Store 2 to 3 weeks.\n\n## Squash\n\nAcorn: Green skin; acorn-shaped. Bake halves with filling or remove rind, cut up and roast.\n\nButternut: Yellow or gold; peanut-shaped. Cook as for acorn squash.\n\nCarnival: Cream, orange or green. Flavor similar to sweet potatoes or butternut squash. Bake or steam.\n\nChayote: Mild flavor; requires bold seasoning. Refrigerate in plastic bag up to 1 month. Use raw in salads, or stuff and bake.\n\nDelicata: Pale yellow skin with green stripes. Sweet, buttered-corn flavor. Seed cavity is small; lots of edible flesh. Bake or steam.\n\nPattypan: These small, flying saucer-shaped squash have thin, smooth to slightly bumpy skin that typically isn't removed for cooking or eating.\n\nSummer: Thin, edible skin; soft, edible seeds. Choose firm squash with shiny, unblemished skin. Refrigerate in plastic bag up to 5 days.\n\nSweet Dumpling: Green, white stripes. Sweet; good stuffed with rice or stuffing.\n\nWinter: Thick, hard skin, seeds. Choose squash heavy for its size and with dull-colored hard rind (no soft spots). Whole: store in cool (45\u00b0F to 60\u00b0F), dry, dark place with good ventilation up to 2 months. Cut and peeled: refrigerate in plastic wrap up to 5 days.\n\n## Fresh Legumes\n\nLegumes are seed pods that split along both sides when ripe. Choose bright, smooth, crisp pods. Refrigerate in plastic bag up to 4 days.\n\nEdamame (Fresh Soybeans): Often sold in their fuzzy green pods (avoid blemishes) or frozen. Serve as snack; add to rice, salads, scrambled eggs; mash into guacamole.\n\nLong Beans (Asparagus Bean, Chinese Long Bean, Yard-Long Bean): Same family as black-eyed peas. Milder, less sweet than green beans. Usually cut in half or smaller pieces, then saut\u00e9ed or stir-fried. Mushy if overcooked.\n\nMung Bean Sprouts: Tender; great for salads or stir-fries. Refrigerate in plastic bag up to 3 days.\n\nPea Shoots: The best leaves and tendrils from pea plants. Choose fresh, crisp, young, tender shoots with top leaves. Remove coarse stems. Use raw in salads; steam as a side dish with lemon or sprinkled with ginger and sugar. Use within 2 days.\n\nPeas (English, Sugar Snap, Snow): English peas must be removed from pods before cooking; sugar snaps and snow peas can be eaten in the pod. All can be eaten raw or cooked; great in salads or side dishes.\n\n### Fresh Vegetable Cooking Chart\n\nStore vegetables in plastic bag in refrigerator or at room temperature if indicated in chart. Wash veg-etables before using. Use fresh vegetables within a few days. For grilling direction, see page 422.\n\n**To Boil:** In saucepan, heat 1 inch water to boiling. Add vegetables. Heat to boiling; reduce heat to low. Cook as directed and drain.\n\n**To Steam:** In saucepan or skillet, place steamer basket in \u00bd inch water (water should not touch bottom of basket). Place vegetables in basket. Cover; heat to boiling; reduce heat to low. Steam as directed.\n\n**To Saut\u00e9:** In skillet, cook in butter or oil over medium-high heat as directed.\n\n**To Bake:** Heat oven to 350\u00b0F. Place vegetables in oven and bake as directed.\n\n**To Roast:** Heat oven to 425\u00b0F. Toss cut (unless stated otherwise) vegetables with about 1 tablespoon olive oil and season as desired. Place vegetables in baking pan and roast as directed.\n\n**To Microwave:** Use microwavable dish with cover or use plastic wrap to cover. Paper towels or plastic wrap used in the microwave should be microwave-safe. Add 2 tablespoons water to dish with vegetables. Microwave on High, unless stated otherwise, as directed; drain. Stir or rearrange vegetables once or twice during cooking. Let stand covered for 1 to 2 minutes to finish cooking.*\n\n###\n\nVegetable with Amount for 4 Servings | Preparation | Conventional Directions | Microwave Directions*\n\n---|---|---|---\n\nArtichokes, Globe (4 medium) | Remove discolored leaves; trim stem even with base. Cut 1 inch off top. Snip tips off leaves. To retain color, dip in cold water mixed with lemon juice. | **Boil:** Covered with water, adding 2 tablespoons lemon juice to water, 20 to 30 minutes or until leaves pull out easily and bottom is tender when pierced with knife. | Place 1 or 2 artichokes in dish; add \u00bc cup water. Microwave 5 to 7 minutes or until leaves pull out easily and bottom is tender when pierced with knife.\n\nArtichokes, Jerusalem (1 lb) | Leave whole, or cut. To retain color, toss with cold water mixed with lemon juice. | **Boil:** Covered 7 to 9 minutes or until crisp-tender.\n\n**Steam:** 15 to 20 minutes or until crisp-tender. | Place in dish. Microwave 5 to 7 minutes or until crisp-tender.\n\nAsparagus (1\u00bd lb) | Break off ends of spears. Tie spears in bundles with string, or hold together with band of foil (do not use foil if microwaving). Or cut into 1-inch pieces. | **Boil:** Uncovered 6 to 8 minutes or until crisp-tender.\n\n**Steam:** 6 to 8 minutes or until crisp-tender.\n\n**Roast whole:** 10 to 12 minutes. | Place in dish. Microwave 4 to 6 minutes or until crisp-tender.\n\nBeans, Green, Purple Wax and Yellow Wax (1 lb) | Remove ends. Leave beans whole, or cut into 1-inch pieces. | **Boil:** Uncovered 6 to 8 minutes or until crisp-tender.\n\n**Steam:** 10 to 12 minutes or until crisp-tender.\n\n**Roast (whole):** 6 to 8 minutes. | Place in dish. Microwave 8 to 10 minutes or until crisp-tender.\n\nBeans, Lima (3 lb unshelled; 3 cups shelled) | To shell beans, remove thin outer edge of pod with sharp knife or scissors. Slip out beans. | **Boil:** Covered 15 to 20 minutes or until tender. | Place in dish; add \u00bd cup water. Microwave on High 4 to 5 minutes or until boiling; then Medium-Low 20 to 25 minutes or until tender.\n\nBeets (5 medium) | Cut off all but 2 inches of tops. Leave whole with root ends attached. Peel after cooking. | **Boil:** Add water to cover and 1 tablespoon vinegar. Boil, covered, 20 to 30 minutes.\n\n**Steam:** 45 to 50 minutes or until tender.\n\n**Roast:** 35 to 40 minutes. | Place in dish with 2 tablespoons water. Microwave 12 to 16 minutes or until tender.\n\nBroccoli (1\u00bd lb) | Trim off large leaves; remove tough ends of stems. Cut as desired. | **Boil:** Uncovered 4 to 6 minutes or until crisp-tender.\n\n**Steam:** 10 to 11 minutes or until crisp-tender. | Place in dish. Microwave 6 to 8 minutes or until crisp-tender.\n\nBrussels Sprouts (1 lb) | Remove discolored leaves; cut off stem ends. Cut large sprouts in half. | **Boil:** Uncovered 8 to 12 minutes or until tender.\n\n**Steam:** 8 to 12 minutes or until tender.\n\n**Roast:** 12 to 15 minutes. | Place in dish. Microwave 5 to 6 minutes or until tender.\n\nCarrots (6 or 7 medium) | Peel; cut off ends. Leave ready-to-eat baby-cut carrots whole or cut as desired. | **Boil:** Covered 7 to 10 minutes or until tender.\n\n**Steam:** 8 to 10 minutes or until tender.\n\n**Roast:** 25 to 30 minutes. | Place in dish. Microwave 5 to 9 minutes or until crisp-tender.\n\nCauliflower (1 medium head) | Remove outer leaves and stalk; cut off any discoloration. Leave whole or separate into florets. | **Boil:** Uncovered 8 to 12 minutes or until tender.\n\n**Steam:** 8 to 12 minutes or until tender.\n\n**Roast:** 15 to 20 minutes. | Place in dish. Microwave 8 to 10 minutes or until tender.\n\nCorn (4 ears) | Husk ears and remove silk just before cooking. | **Boil:** Add water to cover and 1 tablespoon sugar. Boil, uncovered, 5 to 7 minutes.\n\n**Steam:** 5 to 7 minutes. | Wrap ears in waxed paper or place in dish.\n\n1 ear: Microwave 2 to 3 minutes.\n\n2 ears: Microwave 3 to 4 minutes.\n\nEggplant (1 medium) | Remove stems. Peel can be left on, or peel if desired. Cut as desired. | **Boil:** Covered 5 to 8 minutes or until tender.\n\n**Steam:** 5 to 7 minutes or until tender.\n\n**Saut\u00e9:** With 2 tablespoons butter, 5 to 10 minutes or until tender. | Place in dish. Microwave 7 to 9 minutes or until tender.\n\nFennel (3 to 4 medium) | Remove feathery tops and tough or discolored outer ribs; trim base. Cut bulbs into fourths. | **Boil:** Covered 8 to 11 minutes or until tender.\n\n**Steam:** 12 to 15 minutes or until tender.\n\n**Roast:** 20 to 25 minutes. | Place in dish. Microwave 4 to 5 minutes or until tender.\n\nGreens: Beet, Chicory, Collards, Escarole, Kale, Mustard, Spinach, Swiss Chard and Turnip (1 lb) | Remove root ends and imperfect leaves. | **Steam:** 5 to 8 minutes or until tender. | Place in dish. Microwave beet, chicory or escarole 8 to 10 minutes or until tender.\n\nMicrowave collards, kale, mustard, spinach, Swiss chard or turnips 4 to 6 minutes or until tender.\n\nKohlrabi (4 medium) | Cut off root ends and tops. Cut as desired. | **Boil:** Covered 15 to 20 minutes or until tender.\n\n**Steam:** 8 to 12 minutes or until tender. | Place in dish. Microwave 3 to 5 minutes or until tender.\n\nLeeks (6 medium) | Remove green tops to within 2 inches of white part. Peel outside layer of bulbs. Cut as desired. | **Boil:** Covered 10 to 12 minutes or until tender.\n\n**Steam:** 10 to 12 minutes or until tender.\n\n**Roast:** 12 to 15 minutes. | Place in dish. Microwave 4 to 5 minutes or until tender.\n\nMushrooms (1 lb) | Wipe each mushroom with damp paper towel to remove dirt. Trim off stem. Leave whole, or cut as desired. | **Saut\u00e9:** With 1 tablespoon butter, 4 to 6 minutes, stirring frequently, until tender.\n\n**Roast:** 5 to 10 minutes. | Place in dish; add 1 tablespoon butter. Microwave 3 to 4 minutes or until tender.\n\nOkra (1 lb) | Remove ends. Leave whole or cut into slices. | **Boil:** Uncovered 8 to 10 minutes or until tender.\n\n**Steam:** 6 to 8 minutes or until tender. | Place in dish; add \u00bc cup water. Microwave 5 to 6 minutes or until tender.\n\nOnions, White, Yellow and Red (8 to 10 small) | Peel onions in cold water to prevent eyes from watering. Cut as desired. | **Boil:** Covered 15 to 20 minutes.\n\n**Steam:** 15 to 20 minutes.\n\n**Saut\u00e9:** With 2 tablespoons butter, 6 to 9 minutes or until tender.\n\n**Roast:** 30 to 40 minutes. | Place in dish; add \u00bc cup water. Microwave 7 to 9 minutes or until tender.\n\nParsnips (6 to 8 medium) | Peel; cut off ends. Leave whole or cut as desired. | **Boil:** Covered 9 to 15 minutes or until tender.\n\n**Steam:** 9 to 15 minutes or until tender.\n\n**Roast:** 25 to 30 minutes. | Place in dish. Microwave 5 to 6 minutes or until tender.\n\nPea Pods, Snow or Chinese (1 lb) | Remove tips and strings. | **Boil:** Uncovered 2 to 3 minutes or until crisp-tender.\n\n**Steam:** 3 to 5 minutes or until crisp-tender. | Place in dish. Microwave 6 to 7 minutes or until crisp-tender.\n\nPeas, Sweet (2 lb) | Shell just before cooking. | **Boil:** Uncovered 5 to 10 minutes or until tender.\n\n**Steam:** 8 to 10 minutes or until tender. | Place in dish. Microwave 4 to 6 minutes or until tender.\n\nPeas, Sugar Snap (1 lb) | Snip off stem ends and remove strings. | **Boil:** Uncovered 4 to 5 minutes or until crisp-tender.\n\n**Steam:** 6 to 7 minutes or until crisp-tender. | Place in dish. Microwave 6 to 7 minutes or until crisp-tender.\n\nPeppers, Bell (2 medium) | Remove stems, seeds and membranes. Leave whole to stuff and bake, or cut as desired. | **Steam:** 4 to 6 minutes or until crisp-tender.\n\n**Saut\u00e9:** With 1 tablespoon butter, 3 to 5 minutes or until crisp-tender.\n\n**Roast:** 15 to 20 minutes. | Place in dish. Microwave 3 to 4 minutes or until crisp-tender.\n\nPotatoes,\n\nFingerling (10 to 12)\n\nSmall, Red and White (10 to 12) | Leave whole, or cut as desired. | **Boil:** Add water to cover. Boil, uncovered, 15 to 20 minutes or until tender.\n\n**Steam:** 18 to 22 minutes or until tender.\n\n**Roast:** 25 to 30 minutes. | Place in dish; add \u00bc cup water. Microwave 9 to 11 minutes or until tender.\n\nPotatoes,\n\nRed and White (6 medium)\n\nRusset (4 medium)\n\nYukon Gold (6 medium) | Leave whole, or peel and cut as desired. | **Boil:** Add water to cover. Boil, uncovered, 15 to 20 minutes or until tender.\n\n**Steam:** 15 to 20 minutes or until tender.\n\n**Bake:** Uncovered 1 hour or until tender.\n\n**Roast:** 40 to 45 minutes. | Pierce whole potatoes to allow steam to escape. Place on paper towel.\n\n1 or 2 potatoes: Microwave 4 to 6 minutes or until tender. Cover; let stand 5 minutes.\n\n3 or 4 potatoes: Microwave 8 to 12 minutes or until tender. Cover; let stand 5 minutes.\n\nPotatoes, Sweet (4 medium) | Leave whole, or peel and cut as desired. | **Boil:** Add water to cover. Boil, uncovered, 10 to 15 minutes or until tender.\n\n**Steam:** 15 to 20 minutes or until tender.\n\n**Bake:** Uncovered 1 hour or until tender.\n\n**Roast:** 40 to 45 minutes. | Pierce whole potatoes to allow steam to escape. Place on paper towel. Microwave 9 to 11 minutes, or until tender. Cover; let stand 5 minutes.\n\nRutabagas (2 medium) | Peel; cut as desired. | **Boil:** Covered 20 to 25 minutes or until tender.\n\n**Steam:** 20 to 25 minutes or until tender.\n\n**Roast:** 40 to 45 minutes. | Place in dish; add \u00bc cup water. Microwave 13 to 15 minutes or until tender.\n\nSquash, Summer: Chayote, Crookneck, Zucchini, Pattypan, Straightneck (1\u00bd lb) | Remove stem and blossom ends, but do not peel. Cut as desired. | **Boil:** Uncovered 5 to 10 minutes.\n\n**Steam:** 5 to 7 minutes or until tender.\n\n**Saut\u00e9:** With 1 tablespoon olive oil, 5 to 10 minutes or until tender. | Place in dish. Microwave 4 to 6 minutes or until almost tender.\n\nSquash, Winter: Acorn, Buttercup, Butternut, Pumpkin, Spaghetti (2 lb) | Cook in halves or pieces with seeds removed. | **Boil:** Peeled and cut up, covered, 10 to 15 minutes or until tender.\n\n**Steam:** 10 to 15 minutes.\n\n**Bake:** Place squash halves cut side up in baking dish. Cover and bake 40 minutes or until tender. | Whole squash except spaghetti: Pierce with knife in several places to allow steam to escape. Place on paper towel. Microwave uncovered 5 minutes or until squash feels warm to the touch. Cut in half; remove seeds. Arrange halves in dish. Microwave 5 to 8 minutes or until tender.\n\nWhole spaghetti squash: Pierce with knife in several places to allow steam to escape. Place on paper towel. Microwave uncovered 18 to 23 minutes or until tender. Cut in half; remove seeds and fibers.\n\nTurnips (4 medium) | Cut off tops. Leave whole, or cut as desired. | **Boil:** Covered 20 to 25 minutes.\n\n**Steam:** 15 to 20 minutes or until tender.\n\n**Roast:** 30 to 35 minutes. | Place in dish. Microwave 6 to 8 minutes or until tender.\n\n*Vegetables continue to cook for a short time after being microwaved. Most vegetables require a stand time after cooking, which completes the cooking and equalizes the temperature throughout the food.\n\n## Artichokes with Rosemary Butter EASY\n\nTo eat a fresh artichoke, remove a leaf, dip it in the butter mixture and draw it between your teeth to scrape off the meaty part. Continue removing leaves until you see the center cone of light green leaves. Cut the leaves from the cone, then remove and discard the fuzzy choke to reveal the meaty heart in the center.\n\nPREP 10 min TOTAL 45 min \u25cf 4 servings\n\n  * 4 medium artichokes\n  * \u00bd cup butter\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried rosemary leaves, crushed\n  * 1 teaspoon lemon juice\n\n1 Remove any discolored leaves and the small leaves at base of artichokes. Trim stems even with base of artichokes. Cutting straight across, slice 1 inch off tops and discard tops. Cut off points of remaining leaves with scissors. Rinse artichokes with cold water.\n\n2 Place steamer basket in \u00bd inch water in 4-quart Dutch oven (water should not touch bottom of basket). Place artichokes in basket. Cover tightly. Heat to boiling; reduce heat. Steam 20 to 25 minutes or until bottoms are tender when pierced with knife.\n\n3 In small microwavable bowl, microwave butter until melted; stir in rosemary and lemon juice. Pluck out artichoke leaves one at a time. Dip base of leaf into rosemary butter.\n\n1 Serving: Calories 280; Total Fat 23g (Saturated Fat 15g, Trans Fat 1.5g); Cholesterol 60mg; Sodium 280mg; Total Carbohydrate 14g (Dietary Fiber 7g, Sugars 1g); Protein 4g Exchanges: 2 Vegetable, 5 Fat Carbohydrate Choices: 1\n\n### Preparing Artichokes\n\nRemove discolored leaves, leaves with thorns and small leaves at base of artichoke.\n\nTrim stem even with base of artichoke; cut 1 inch from top of artichoke and discard top.\n\nUsing scissors, cut points from remaining leaves; rinse artichokes with cold water.\n\n## Asparagus with Parmesan CALORIE Smart \u2022 Fast\n\nPREP 10 min TOTAL 20 min \u25cf 4 servings\n\n  * 1\u00bd lb fresh asparagus\n  * 1 tablespoon olive oil\n  * 1 tablespoon butter\n  * 1 medium green onion, sliced (1 tablespoon)\n  * 1 clove garlic, finely chopped\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon freshly ground black pepper\n  * \u00bc cup freshly grated or shredded Parmesan cheese\n\n1 Heat oven to 375\u00b0F. Snap off tough bottom ends of asparagus. In 3-quart saucepan, heat 1 inch water. Add asparagus. Return to boiling; reduce heat. Simmer uncovered 4 minutes.\n\n2 Meanwhile, in ungreased 8-inch square pan, place oil, butter, onion and garlic. Heat uncovered in oven 5 minutes.\n\n3 Drain asparagus; spread in oil mixture in pan. Sprinkle with salt, pepper and cheese. Bake uncovered about 10 minutes or until cheese is melted.\n\n1 Serving: Calories 110; Total Fat 8g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 430mg; Total Carbohydrate 4g (Dietary Fiber 2g, Sugars 2g); Protein 4g Exchanges: 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 0\n\nGreen Beans with Parmesan Substitute fresh whole green beans for the asparagus.\n\nAsparagus with Goat Cheese Omit green onion. Substitute crumbled plain goat cheese. Sprinkle 2 tablespoons chopped walnuts and 1 teaspoon grated lemon peel with the goat cheese. Bake about 10 minutes or until cheese is warm.\n\n### Snapping Ends from Asparagus\n\nSnap off tough ends of asparagus where they break easily.\n\nGreen Beans with Glazed Shallots\n\n## Green Beans with Glazed Shallots Calorie Smart \u2022 Fast\n\nPREP 15 min TOTAL 15 min \u25cf 6 servings\n\n  * 1 lb fresh green beans, trimmed\n  * 2 tablespoons butter\n  * 2 shallots, finely chopped\n  * \u00bd teaspoon sugar\n  * 1 teaspoon lemon juice\n  * 1 tablespoon chopped fresh dill weed\n  * \u00bc teaspoon salt\n\n1 In 3-quart saucepan, place green beans in 1 inch water. Heat to boiling; boil uncovered 6 to 8 minutes or until crisp-tender. Drain; return to saucepan.\n\n2 Meanwhile, in 10-inch skillet, melt butter over medium heat. Cook shallots in butter 2 to 3 minutes, stirring occasionally, until crisp-tender. Stir in sugar. Cook 2 to 3 minutes longer, stirring occasionally, until shallots are glazed and brown. Stir in lemon juice, dill weed and salt.\n\n3 Add shallot mixture to green beans; toss to coat.\n\n1 Serving: Calories 60; Total Fat 4g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 130mg; Total Carbohydrate 5g (Dietary Fiber 2g, Sugars 2g); Protein 1g Exchanges: 1 Vegetable, 1 Fat Carbohydrate Choices: \u00bd\n\nRoasted Beets\n\n## Roasted Beets CALORIE Smart \u2022 Easy\n\nBeets are usually available year-round. If given a choice, pick smaller, young beets for the best flavor.\n\nPREP 10 min TOTAL 1 hr 20 min \u25cf 6 servings\n\n  * 2 lb small beets (1\u00bd to 2 inch)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon coarsely ground black pepper\n  * 2 tablespoons olive oil\n  * 2 tablespoons chopped fresh basil leaves\n  * 1 tablespoon balsamic vinegar\n\n1 Heat oven to 425\u00b0F. Cut off all but 2 inches of beet tops. Wash beets; leave whole with root ends attached. Place beets in ungreased 13x9-inch pan. Sprinkle with salt and pepper. Drizzle with oil.\n\n2 Roast uncovered about 40 minutes or until beets are tender. Let beets cool until easy to handle, about 30 minutes. Peel beets and cut off root ends; cut beets into \u00bd-inch slices.\n\n3 In medium bowl, toss beets, basil and vinegar. Serve warm or at room temperature.\n\n1 Serving: Calories 90; Total Fat 4.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 270mg; Total Carbohydrate 10g (Dietary Fiber 2g, Sugars 9g); Protein 2g Exchanges: 2 Vegetable, 1 Fat Carbohydrate Choices: \u00bd\n\n### Preparing Beets\n\nBefore cooking, cut off all but 2 inches of stems. Wash beets; leave whole with root ends attached.\n\nAfter cooking, peel beets with paring knife under cold water, removing stems, skins and roots.\n\n## Cooking Beet Greens\n\nIf you are lucky enough to have fresh beet greens, here's how to cook them. Wash them well and cut the greens and stems into pieces. Heat 1 tablespoon olive oil in a skillet and cook 1 or 2 cloves chopped garlic until tender. Add the cooked greens; sprinkle with salt and pepper. Cook and stir 5 to 8 minutes or until wilted.\n\n## Learn to Roast\n\nRoasting is a simple dry-heat cooking technique that is easy to master. \"Dry heat\" simply means that no liquid is added and it is the heat of the oven that cooks the food, rather than moisture being added to the food. Foods that are roasted are cooked uncovered. The primary difference between roasting and baking is based on the foods being cooked and the temperature used to cook the food. Crispy on the outside and tender on the inside, roasting makes meats and veggies look and taste fantastic. Sugars in vegetables are caramelized; heightening and sweetening their flavors. Meats develop a delicious, brown crust when roasted.\n\nTo roast, the oven temperature will usually be higher than for baked foods\u2014above 400\u00b0F. It is important to use the correct oven temperature for roasting so that foods have a browned, slightly caramelized exterior and a tender interior.\n\nMany vegetables, such as onions, carrots, potatoes and bell peppers, benefit from roasting. Each vegetable might cook at a different rate. For instance, if you want to add zucchini to the mix, you might add it after potatoes and carrots are partially cooked. Here are some guidelines for times for vegetables that can easily be roasted. For most vegetables, roast at 425\u00b0F unless a recipe indicates a different temperature. For roasting meats, see chart on page 314 for times.\n\n  * Cut vegetables into similar-size pieces so that they cook evenly. \n  * Use a large enough pan so that the vegetables can be in a single layer.\n  * Use a short-sided pan, such as a 15x10x1-inch or shallow roasting pan, to evaporate moisture from the foods, so they roast rather than steam.\n  * Toss vegetables with a small amount of olive oil and season as desired. \n  * Individual recipes will provide specific directions, but in general, cook until just slightly firm when pierced with a fork. \n  * Try roasting a single vegetable, then experiment with different combinations of vegetables that you like.\n  * Check the food pieces for color on the sides that touch the pan, as these sides will brown more quickly than the tops. Stir or turn pieces so that all sides brown.\n\n## Vegetable Roasting\n\nWhile times differ based on factors such as how hot your oven runs or the size of the vegetables, here are some basic time guidelines for perfectly raosted vegetables:\n\n  * **Carrots, parsnips, potatoes, sweet potatoes, Brussels sprouts, onions and fennel:** 35 to 45 minutes\n  * **Bell peppers, cauliflower, green beans, butternut squash:** 20 to 25 minutes\n  * **Zucchini, asparagus, leeks, pattypan squash, eggplant:** 15 to 20 minutes\n\n## Roasted Vegetables calorie smart\n\nPick baby-cut carrots that are all about the same size so they cook evenly.\n\nPREP 15 min TOTAL 40 min \u25cf 10 servings\n\n  * 3 tablespoons olive or vegetable oil\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n  * 1 clove garlic, finely chopped\n  * 1 cup ready-to-eat baby-cut carrots\n  * 6 small red potatoes, cut into quarters\n  * 2 small onions, cut into \u00bd-inch wedges\n  * 1 small red bell pepper, cut into 1-inch pieces\n  * 1 medium zucchini, cut lengthwise in half, then cut crosswise into 1-inch slices\n  * 1 cup grape or cherry tomatoes\n\n1 Heat oven to 425\u00b0F. In small bowl, stir oil, salt, pepper and garlic until well mixed. In ungreased 15x10x1-inch pan, toss carrots, potatoes, onions, bell pepper and zucchini with oil mixture until coated.\n\n2 Roast uncovered 25 minutes, stirring once. Stir in tomatoes. Roast 5 to 10 minutes longer or until vegetables are tender and starting to brown.\n\n1 Serving: Calories 110; Total Fat 4.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 130mg; Total Carbohydrate 16g (Dietary Fiber 3g, Sugars 3g); Protein 2g Exchanges: \u00bd Starch, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\nRoasted Root Vegetables Omit bell pepper, zucchini and tomatoes. Add 1\u00bd cups each peeled, cubed parsnips and peeled, cubed turnips. Roast 35 to 40 minutes or until vegetables are fork-tender.\n\n## Broccoli with Roasted Red Peppers and Hazelnuts CALORIE Smart \u2022 Fast\n\nPREP 20 min TOTAL 20 min \u25cf 8 servings\n\n  * 5 cups fresh broccoli florets\n  * 1 tablespoon olive or vegetable oil\n  * 1 clove garlic, finely chopped\n  * \u00bc cup chopped hazelnuts (filberts)\n  * \u00bd cup chopped drained roasted red bell peppers (from 7-oz jar) (to roast your own, see page 247)\n  * \u00bc teaspoon salt\n\n1 In 2-quart saucepan, heat 1 cup water to boiling. Add broccoli. Cover; cook about 1 minute or just until crisp. Drain and immediately place broccoli in ice water.\n\n2 In 12-inch skillet, heat oil over medium heat. Cook garlic and hazelnuts in oil 1 to 2 minutes, stirring frequently, until nuts are lightly toasted.\n\n3 Drain broccoli. Stir broccoli, roasted peppers and salt into nut mixture. Cook about 3 minutes, stirring occasionally, until broccoli is crisp-tender.\n\n1 Serving (\u00bd Cup): Calories 70; Total Fat 4g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 90mg; Total Carbohydrate 5g (Dietary Fiber 2g, Sugars 2g); Protein 2g Exchanges: 1 Vegetable, 1 Fat Carbohydrate Choices: \u00bd\n\nRoasted Red Bell Peppers (page 247)\n\nCheesy Bacon Brussels Sprouts\n\n## Cheesy Bacon Brussels Sprouts CALORIE Smart\n\nBrussels sprouts have gotten a bad rap over the years because they are often boiled to mushiness, which doesn't show them at their best. These spectacular sprouts, combined with bacon and cheese, may just convert even sworn Brussels sprouts haters.\n\nPREP 35 min TOTAL 1 hr 15 min \u25cf 10 servings\n\n  * 1\u00bc lb fresh Brussels sprouts (1 to 1\u00bd inch), cut in half (5 cups)\n  * 6 slices bacon, cut into 1-inch pieces\n  * 1 tablespoon butter\n  * 1 large onion, finely chopped (1 cup)\n  * 1 tablespoon all-purpose flour\n  * \u00bc teaspoon dried thyme leaves\n  * \u00bc teaspoon pepper\n  * \u00be cup half-and-half\n  * 2 teaspoons chicken bouillon granules\n  * \u2153 cup shredded Parmesan cheese\n  * \u00bd cup shredded Cheddar cheese (2 oz)\n\n1 Heat oven to 350\u00b0F. In 3-quart saucepan, place Brussels sprouts and enough water just to cover. Heat to boiling; boil 6 to 8 minutes or until crisp-tender. Drain and return to saucepan; set aside.\n\n2 In 10-inch skillet, cook bacon over medium heat 6 to 8 minutes, stirring occasionally, until crisp. Drain on paper towels; set aside. Reserve 1 tablespoon drippings in skillet.\n\n3 Add butter to bacon drippings. Heat over medium-high heat until butter is melted. Add onion; cook 2 to 3 minutes, stirring frequently, until crisp-tender. Add flour, thyme and pepper; cook and stir until well blended. Gradually stir in half-and-half and bouillon granules. Heat to boiling, stirring constantly; boil and stir 1 minute.\n\n4 Remove from heat; stir in Parmesan cheese. Pour sauce over Brussels sprouts in saucepan; mix gently. Spoon into ungreased 2-quart casserole.\n\n5 Bake uncovered 15 minutes. Sprinkle with bacon and Cheddar cheese. Bake 10 to 15 minutes longer or until bubbly and cheese is melted.\n\n1 Serving (\u00bd Cup): Calories 130; Total Fat 8g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 20mg; Sodium 330mg; Total Carbohydrate 8g (Dietary Fiber 3g, Sugars 3g); Protein 6g Exchanges: 1 Vegetable, \u00bd High-Fat Meat, 1 Fat Carbohydrate Choices: \u00bd\n\nCheesy Bacon Broccoli Substitute 1\u00bc lb fresh broccoli, cut into 1\u00bd-inch pieces, for the Brussels sprouts.\n\nCheesy Bacon Green Beans Substitute 1 bag (1 lb) frozen cut green beans for the Brussels sprouts. In 2-quart microwavable casserole, place beans and 2 tablespoons water. Cover and microwave on High 5 minutes; drain. Continue as directed in Steps 2 through 5.\n\n## Brussels Sprouts\n\nPart of the cruciferous vegetable family, Brussels sprouts look like tiny cabbages but deliver big on flavor. Choose fresh-looking sprouts and remove any discolored outer leaves before cooking.\n\nCranberry-Pistachio Brussels Sprouts\n\n## Cranberry-Pistachio Brussels Sprouts CALORIE Smart \u2022 Fast\n\nPREP 25 min TOTAL 25 min \u25cf 6 servings\n\n  * 2 slices thick-sliced bacon\n  * 1 lb Brussels sprouts, trimmed, cut in half\n  * \u00bc cup chicken broth\n  * \u2153 cup sweetened dried cranberries\n  * \u00bc cup salted roasted pistachio nuts\n  * 1 tablespoon honey\n  * \u00bc teaspoon cracked black pepper\n\n1 In 10-inch skillet, cook bacon over medium-high heat until crisp. Drain on paper towels. Crumble bacon; set aside. Reserve 1 tablespoon drippings in skillet.\n\n2 Cook Brussels sprouts in bacon drippings over medium heat 3 to 4 minutes, stirring frequently, until they just begin to brown. Add broth; reduce heat to medium-low. Cover and cook 5 to 6 minutes longer or until broth is evaporated and Brussels sprouts are tender.\n\n3 Add cranberries, nuts, honey, pepper and bacon; toss to coat.\n\n1 Serving: Calories 110; Total Fat 4.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 170mg; Total Carbohydrate 14g (Dietary Fiber 2g, Sugars 9g); Protein 4g Exchanges: \u00bd Starch, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\n## Sweet-Sour Red Cabbage CALORIE Smart\n\nSome like this classic cabbage dish sweet, and some prefer it sour. For a sweeter touch, add another tablespoon of brown sugar; for a more sour taste, add an extra tablespoon of vinegar.\n\nPREP 35 min TOTAL 50 min \u25cf 8 servings\n\n  * 1 medium head red cabbage (1\u00bd lb), shredded\n  * 4 slices bacon, chopped\n  * 1 small onion, sliced\n  * \u00bc cup packed brown sugar\n  * 2 tablespoons all-purpose flour\n  * \u00bc cup water\n  * 3 tablespoons white vinegar\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n\n1 In 10-inch skillet, heat 1 inch water to boiling. Add cabbage; return to boiling. Boil uncovered about 15 minutes, stirring occasionally, until tender; drain and set aside. Wipe out skillet and dry with paper towel.\n\n2 In same skillet, cook bacon over medium heat 4 minutes, stirring occasionally. Add onion. Cook 2 to 4 minutes, stirring occasionally, until bacon is crisp. Remove bacon and onion with slotted spoon; drain on paper towels. Reserve 1 tablespoon drippings in skillet.\n\n3 Stir brown sugar and flour into bacon drippings. Stir in water, vinegar, salt and pepper until well mixed. Stir in cabbage, bacon and onion. Cook over medium heat 1 to 2 minutes, stirring occasionally, until hot.\n\n1 Serving: Calories 100; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 160mg; Total Carbohydrate 13g (Dietary Fiber 2g, Sugars 10g); Protein 3g Exchanges: \u00bd Other Carbohydrate, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\n# Heirloom Recipe and New Twist\n\nA perennial favorite, Glazed Carrots are easy to make, colorful and delicious to eat. This recipe has been part of our collection for generations and always brings rave reviews. For the new twist, cumin, chili powder and honey are combined for a slightly spicy-sweet flavor that is heightened with roasting. Both recipes are winners and a great addition to any meal.\n\nHeirloom\n\nGlazed Carrots\n\n## Glazed Carrots CALORIE Smart \u2022 Fast\n\nPREP 15 min TOTAL 25 min \u25cf 6 servings\n\n  * 1\u00bd lb fresh carrots, cut into julienne strips (page 12)*\n  * \u2153 cup packed brown sugar\n  * 2 tablespoons butter\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon grated orange peel\n\n1 In 2-quart saucepan, heat 1 inch water to boiling. Add carrots. Return to boiling; reduce heat. Simmer uncovered 6 to 9 minutes or until crisp-tender; drain.\n\n2 In 12-inch skillet, cook remaining ingredients over medium heat, stirring constantly, until bubbly. Stir in carrots. Cook over low heat about 5 minutes, stirring occasionally, until carrots are glazed and hot.\n\n*Or substitute 1\u00bd bags (16-oz size) frozen sliced carrots, cooked as directed on bag, or 1\u00bd lb ready-to-eat baby-cut carrots for the julienned carrots.\n\n1 Serving: Calories 130; Total Fat 4g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 260mg; Total Carbohydrate 22g (Dietary Fiber 3g, Sugars 17g); Protein 1g Exchanges: 1 Other Carbohydrate, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\u00bd\n\nHoney-Glazed Lemon Carrots Substitute 2 tablespoons honey for the brown sugar and lemon peel for the orange peel.\nNew Twist\n\nCumin-Chili Roasted Carrots\n\n## Cumin-Chili Roasted Carrots Calorie Smart\n\nThese snazzy, savory carrots go really well with roasted or grilled pork or chicken. If you've never tried the pairing of cumin and carrots, you'll discover it's a happy union of flavors.\n\nPREP 10 min TOTAL 35 min \u25cf 6 servings\n\n  * 1 bag (2 lb) fresh carrots, cut into \u00bd-inch-wide strips*\n  * 2 tablespoons vegetable oil\n  * 1 tablespoon honey or real maple syrup\n  * 2 teaspoons ground cumin\n  * 2 teaspoons chili powder\n  * 1 teaspoon salt\n\n1 Heat oven to 425\u00b0F. Line 15x10x1-inch pan with foil; spray foil with cooking spray.\n\n2 In large bowl, place carrots. Drizzle with oil and honey; toss to coat. Sprinkle with cumin, chili powder and salt; toss to coat. Spread carrots in single layer in pan.\n\n3 Roast 20 to 25 minutes, turning once, until tender.\n\n*Or substitute 2 lb parsnips, peeled, for the carrots; or use 1 lb of each.\n\n1 Serving: Calories 130; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 510mg; Total Carbohydrate 18g (Dietary Fiber 4g, Sugars 10g); Protein 1g Exchanges: 1 Other Carbohydrate, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\nSaut\u00e9ed Cauliflower with Browned Bread Crumbs\n\n## Saut\u00e9ed Cauliflower with Browned Bread Crumbs FAST\n\nIf anyone at your table typically turns up their nose at cauliflower, this recipe might tempt them into giving it another try. The buttery, flavorful topping is our version of fairy dust.\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 1 medium head cauliflower (2 lb), separated into florets\n  * 4 tablespoons butter\n  * 1 tablespoon Dijon mustard\n  * 2 cups soft bread crumbs (about 3 slices bread)\n  * 2 tablespoons olive oil\n\n1 In 10-inch skillet, heat 1 inch water (salted if desired) to boiling. Add cauliflower; return to boiling. Boil uncovered 3 minutes or until almost tender; drain. Immediately rinse under cold water to stop cooking; set aside.\n\n2 In same skillet, melt 3 tablespoons of the butter over medium heat. Stir mustard into butter; add bread crumbs. Cook 8 to 10 minutes, stirring frequently, until crumbs are golden brown. Remove crumbs to plate.\n\n3 Wipe out skillet with paper towel. Add remaining 1 tablespoon butter and the oil; heat over medium-high heat. Stir in cauliflower. Cook about 5 minutes, stirring occasionally, until lightly browned and tender. Sprinkle crumbs over top.\n\n1 Serving (\u00be Cup): Calories 420; Total Fat 22g (Saturated Fat 7g, Trans Fat 0.5g); Cholesterol 30mg; Sodium 1270mg; Total Carbohydrate 47g (Dietary Fiber 5g, Sugars 7g); Protein 10g Exchanges: 2\u00bd Starch, \u00bd Other Carbohydrate, 1 Vegetable, 4 Fat Carbohydrate Choices: 3\n\n## Spicy Collard Greens with Bacon CALORIE Smart\n\nCollard greens are part of the cruciferous family of vegetables and are similar to cabbage, but instead of forming a head, they stay in loose-leaf form.\n\nPREP 25 min TOTAL 1 hr 30 min \u25cf 4 servings\n\n  * 2 lb collard greens, ribs and stems removed, leaves coarsely chopped (about 8 cups)\n  * 6 slices bacon, chopped\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 jalape\u00f1o chile, seeded, finely chopped\n  * \u00bd teaspoon dried thyme leaves\n  * \u00bd teaspoon seasoned salt\n  * \u00bd teaspoon pepper\n\n1 In 4-quart Dutch oven or saucepan, heat 6 cups water to boiling. Add collard greens; return to boiling. Boil 30 minutes.\n\n2 Meanwhile, in 12-inch skillet, cook bacon over medium-high heat, stirring occasionally, until crisp. Drain on paper towels; set aside. Reserve 1 tablespoon drippings in skillet.\n\n3 Heat bacon drippings over medium heat. Add remaining ingredients; cook 5 minutes, stirring frequently.\n\n4 Drain collard greens. Stir greens and bacon into skillet. Reduce heat to low. Cover and cook about 15 minutes, stirring occasionally, until greens are very tender.\n\n1 Serving: Calories 130; Total Fat 10g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 380mg; Total Carbohydrate 6g (Dietary Fiber 3g, Sugars 2g); Protein 6g Exchanges: 1 Vegetable, \u00bd High-Fat Meat, 1 Fat Carbohydrate Choices: \u00bd\n\nCollard Greens with Black-Eyed Peas During last 5 minutes of cooking, stir in 1 can (15 to 16 oz) black-eyed peas, drained, rinsed.\n\n### Preparing Collard Greens\n\nRemove thick ribs from centers of collard leaves by holding each leaf with hand and stripping away rib with other hand.\n\nLayer several leaves in stack; roll up and cut into \u00bd- to 1-inch-thick slices. Coarsely chop slices.\n\n## Caramelized Onions CALORIE Smart\n\nSweet onions have more sugar than other onions, and it's the sugar that caramelizes, giving a deep golden brown color and rich flavor. If you can't find sweet onions, regular yellow onions can be used\u2014just sprinkle about 1 tablespoon of brown sugar over the onions and cook as directed.\n\nPREP 55 min TOTAL 55 min \u25cf 7 servings\n\n  * 2 tablespoons butter*\n  * 3 large sweet onions (Bermuda, Maui, Spanish or Walla Walla), sliced (8 cups)\n  * \u00bc teaspoon salt\n\n1 In 12-inch nonstick skillet, melt butter over medium-high heat. Stir in onions to coat with butter. Cook uncovered 10 minutes, stirring every 3 to 4 minutes.\n\n2 Reduce heat to medium-low. Sprinkle salt over onions. Cook 35 to 40 minutes longer, stirring well every 5 minutes, until onions are deep golden brown (onions will shrink during cooking).\n\n*Do not use margarine or vegetable oil spreads.\n\n1 Serving: Calories 60; Total Fat 3.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 110mg; Total Carbohydrate 6g (Dietary Fiber 1g, Sugars 3g); Protein 0g Exchanges: 1 Vegetable, \u00bd Fat Carbohydrate Choices: \u00bd\n\nCranberry Caramelized Onions Stir in \u00bd cup dried cranberries during last 5 to 10 minutes of cooking. Serve on turkey or chicken sandwiches.\n\n### Learn with Betty Caramelized Onions\n\n**Perfect Onions:** These onions are deep golden brown in color and will taste caramelized.\n\n**Undercooked Onions:** These onions are too light in color and will not taste caramelized.\n\n**Overcooked Onions:** These onions are too dark with charred edges and will taste burned.\n\n## Okra, Corn and Tomatoes CALORIE Smart \u2022 Fast\n\nPREP 30 min TOTAL 30 min \u25cf 8 servings\n\n  * 4 slices bacon\n  * 1 cup coarsely chopped onion (1 large)\n  * 2 cups diced tomatoes (from 28-oz can), undrained\n  * 2 cans (15.25 oz each) whole kernel corn, drained\n  * 2\u00bd cups frozen cut okra (from 1-lb bag)\n  * 1 teaspoon seasoned salt\n  * \u00bc teaspoon garlic powder\n  * \u00bd teaspoon red pepper sauce\n\n1 In 12-inch skillet, cook bacon over low heat 8 to 10 minutes, turning occasionally, until crisp. Drain on paper towels. Crumble bacon; set aside. Reserve 1 tablespoon bacon drippings in skillet.\n\n2 Cook onion in bacon drippings over medium heat 2 to 3 minutes, stirring occasionally, until tender. Stir in remaining ingredients except bacon. Heat to boiling; reduce heat to low. Cover and simmer 6 to 8 minutes, stirring occasionally, until okra is tender. Sprinkle with bacon.\n\n1 Serving (\u00be Cup): Calories 170; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 550mg; Total Carbohydrate 26g (Dietary Fiber 4g, Sugars 7g); Protein 6g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Vegetable, \u00bd Fat Carbohydrate Choices: 2\n\n## Roasted Bell Peppers CALORIE Smart \u2022 Easy\n\nThere is no reason to buy store-bought roasted peppers when making them at home is so, so easy! Give them a try.\n\nPREP 10 min TOTAL 1 hr 20 min \u25cf 12 servings\n\n  * 6 large red, yellow or orange bell peppers (do not cut)\n  * 1 tablespoon olive oil\n\n1 Heat oven to 400\u00b0F. Line 15x10x1-inch pan with foil. Arrange peppers in pan; brush with oil.\n\n2 Roast 20 minutes; turn peppers over with tongs. Roast 20 minutes longer or until peppers are deep golden brown and charred in spots and slightly collapsed. Remove from oven. Wrap peppers in foil; let stand 15 minutes to steam and allow skins to loosen. Remove foil; let stand 15 minutes or until cool enough to handle. Reserve pan juices for another use, if desired.*\n\n3 Remove stems. Cut peppers open; place, skin side down, on cutting board. Scrape off seeds with spoon. Turn peppers over; peel off skins, scraping off with edge of knife if necessary. Dice peppers or cut into strips. Cover and refrigerate up to 1 week or freeze up to 3 months.\n\n*Use the reserved pan juices in soups, salad dressings or marinades, or brush on grilled chicken or add to a Bloody Mary.\n\n1 Serving (\u00bc Cup): Calories 30; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 2g); Protein 0g Exchanges: 1 Vegetable, \u00bd Fat Carbohydrate Choices: 0\n\n### Preparing Roasted Peppers\n\nWrap peppers in foil; let stand 15 minutes to steam and allow skins to loosen.\n\nPlace cut peppers, skin side down, on cutting board. Scrape off seeds with spoon.\n\n## Mashed Potatoes CALORIE Smart \u2022 Easy\n\nDot the mashed potatoes with small pieces of butter or sprinkle with paprika, chopped fresh parsley or chives, if you like.\n\nPREP 10 min TOTAL 40 min \u25cf 6 servings\n\n  * 6 medium round red or white potatoes, Yukon Gold potatoes or russet potatoes (2 lb), peeled if desired, cut into quarters\n  * \u2153 to \u00bd cup milk, warmed\n  * \u00bc cup butter, softened\n  * \u00bd teaspoon salt\n  * Dash pepper\n\n1 In 2-quart saucepan, place potatoes and enough water just to cover potatoes. Heat to boiling; reduce heat. Cover and simmer 20 to 30 minutes or until potatoes are tender when pierced with fork; drain. Shake pan with potatoes over low heat to dry (this will help mashed potatoes be fluffier).\n\n2 Mash potatoes in pan with potato masher until no lumps remain. Add milk in small amounts, mashing after each addition (amount of milk needed to make potatoes smooth and fluffy depends on type of potatoes used).\n\n3 Add butter, salt and pepper. Mash vigorously until potatoes are light and fluffy and desired consistency.\n\n1 Serving: Calories 200; Total Fat 8g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 20mg; Sodium 260mg; Total Carbohydrate 28g (Dietary Fiber 3g, Sugars 1g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nLighter Directions For 4 grams of fat and 145 calories per serving, use fat-free (skim) milk and reduce butter to 2 tablespoons.\n\nButtermilk Mashed Potatoes Substitute buttermilk for the milk.\n\nGarlic Mashed Potatoes Cook 6 cloves garlic, peeled, with the potatoes. Mash garlic with potatoes.\n\nHerbed Mashed Potatoes Stir in 2 tablespoons chopped fresh chives and \u00bd teaspoon each chopped fresh thyme leaves and rosemary leaves with the butter, salt and pepper in Step 3.\n\nHorseradish Mashed Potatoes Add 2 tablespoons prepared mild or hot horseradish with the butter, salt and pepper in Step 3.\n\n## Ultimate Smashed Potatoes CALORIE Smart \u2022 Slow Cooker\n\nThis is the perfect recipe for your big holiday meal; it cooks as you work on the rest of the meal\u2014leaving your oven free for other dishes.\n\nPREP 30 min TOTAL 5 hr \u25cf 20 servings\n\n  * 5 lb russet or Idaho potatoes, peeled, cut into 1-inch chunks\n  * 1\u2153 cups chicken broth (for homemade broth, see page 143)\n  * \u00bc cup butter or margarine, cut into pieces\n  * 1 cup sour cream or plain yogurt\n  * 1 teaspoon garlic powder\n  * 1 teaspoon onion powder\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bd to 1 cup milk, warmed\n\n1 Spray 4- to 5-quart slow cooker with cooking spray. In slow cooker, place potatoes, broth and butter.\n\n2 Cover; cook with High heat setting 4 hours to 4 hours 30 minutes (or Low heat setting 8 hours) or until potatoes are tender.\n\n3 Add remaining ingredients except milk. Gently smash potatoes using potato masher or fork or beat with electric mixer on low speed until well blended (do not overmix). Stir in enough milk for desired cream consistency. Cover; keep warm on Low or Warm heat setting until serving time, up to 2 hours. Stir before serving.\n\n1 Serving: Calories 150; Total Fat 5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 160mg; Total Carbohydrate 23g (Dietary Fiber 2g, Sugars 2g); Protein 2g Exchanges: 1\u00bd Starch, 1 Fat Carbohydrate Choices: 1\u00bd\n\nTwice-Baked Potatoes\n\n## Twice-Baked Potatoes  CALORIE Smart\n\nPREP 15 min TOTAL 1 hr 50 min \u25cf 8 servings\n\n  * 4 large russet potatoes (8 to 10 oz each)\n  * \u00bc to \u00bd cup milk\n  * \u00bc cup butter, softened\n  * \u00bc teaspoon salt\n  * Dash pepper\n  * 1 cup shredded Cheddar cheese (4 oz)\n  * 1 tablespoon chopped fresh chives\n\n1 Heat oven to 375\u00b0F. Scrub potatoes, but do not peel. Pierce potatoes several times with a fork to allow steam to escape. Bake 1 hour to 1 hour 15 minutes or until potatoes are tender when pierced in center with a fork.\n\n2 When potatoes are cool enough to handle, cut in half lengthwise; scoop out inside, leaving a \u00bc-inch shell. In medium bowl, mash potatoes with potato masher or electric mixer on low speed until no lumps remain. Add milk in small amounts, beating after each addition (amount of milk needed depends on type of potato used).\n\n3 Add butter, salt and pepper; beat vigorously until potatoes are light and fluffy. Stir in cheese and chives. Fill shells with potato mixture.\n\n4 Increase oven temperature to 400\u00b0F. Place potatoes on ungreased cookie sheet. Bake about 20 minutes or until hot.\n\n1 Serving: Calories 200; Total Fat 11g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 30mg; Sodium 210mg; Total Carbohydrate 20g (Dietary Fiber 2g, Sugars 2g); Protein 6g Exchanges: 1 Starch, \u00bd High-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\nMake-Ahead Directions Prepare recipe as directed through Step 3. Wrap stuffed potatoes tightly; refrigerate up to 24 hours or freeze up to 1 month. Bake as directed in Step 4. Bake refrigerated potatoes 30 minutes, frozen potatoes about 40 minutes.\n\nBacon and Pepper Twice-Baked Potatoes Stir in \u00bd cup crumbled crisply cooked bacon (8 slices) and \u00bd cup diced red bell pepper with the cheese and chives in Step 3.\n\nItalian Twice-Baked Potatoes Substitute 1 cup shredded Italian cheese blend for the Cheddar cheese and add \u00bd teaspoon dried Italian seasoning.\n\n## Fried Potatoes CALORIE Smart \u2022 Fast\n\nPREP 30 min TOTAL 30 min \u25cf 4 servings\n\n  * 2 tablespoons vegetable oil\n  * 4 medium potatoes, peeled if desired, thinly sliced (4 cups)\n  * 1 medium onion, thinly sliced, if desired\n  * 1\u00bd teaspoons salt\n  * \u00bc teaspoon pepper\n\n1 In 10-inch nonstick skillet, heat oil over medium heat. Add potatoes and onion; sprinkle with salt and pepper.\n\n2 Cover and cook over medium heat 10 minutes, stirring occasionally. Uncover; cook 10 minutes longer, stirring occasionally, until potatoes are brown and tender.\n\n1 Serving: Calories 180; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 890mg; Total Carbohydrate 27g (Dietary Fiber 2g, Sugars 1g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\n## Hash Brown Potatoes CALORIE Smart\n\nPREP 45 min Total 45 min \u25cf 4 servings\n\n  * 4 medium Idaho or russet baking potatoes (1\u00bd lb), peeled if desired\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n  * 2 tablespoons vegetable oil\n\n1 Shred enough potatoes to measure 4 cups. Rinse well; drain and pat dry.\n\n2 In large bowl, mix potatoes, salt and pepper. In 10-inch nonstick skillet, heat 1 tablespoon of the oil over medium heat. Pack potato mixture firmly in skillet, leaving \u00bd-inch space around edge.\n\n3 Reduce heat to medium-low. Cook about 15 minutes or until bottom is brown. Drizzle remaining 1 tablespoon oil evenly over potatoes. Cut potato mixture into quarters; turn over. Cook about 12 minutes longer or until bottom is brown.\n\n1 Serving: Calories 160; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 23g (Dietary Fiber 2g, Sugars 1g); Protein 2g Exchanges:1 Starch, \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\u00bd\n\n## Scalloped Potatoes\n\nPREP 20 min TOTAL 2 hr 10 min \u25cf 6 servings\n\n  * 4 tablespoons butter\n  * 1 small onion, finely chopped (\u2153 cup)\n  * 3 tablespoons all-purpose flour\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2\u00bd cups milk\n  * 6 medium potatoes, peeled if desired, thinly sliced (6 cups)\n\n1 Heat oven to 350\u00b0F. Grease 2-quart casserole with shortening or cooking spray.\n\n2 In 2-quart saucepan, melt 3 tablespoons of the butter over medium heat. Cook onion in butter about 2 minutes, stirring occasionally, until tender. Stir in flour, salt and pepper. Cook and stir until smooth and bubbly; remove from heat. Stir in milk. Heat to boiling, stirring constantly. Boil and stir 1 minute.\n\n3 Spread potatoes in casserole. Pour sauce over potatoes. Cut remaining 1 tablespoon butter into small pieces; sprinkle over potatoes.\n\n4 Cover and bake 30 minutes. Uncover; bake 1 hour to 1 hour 10 minutes longer or until potatoes are tender. Let stand 5 to 10 minutes before serving (sauce thickens as it stands).\n\n1 Serving: Calories 260; Total Fat 10g (Saturated Fat 5g, Trans Fat 0.5g); Cholesterol 30mg; Sodium 500mg; Total Carbohydrate 36g (Dietary Fiber 2g, Sugars 7g); Protein 6g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nAu Gratin Potatoes After stirring in milk and boiling 1 minute in Step 2, stir in 1 cup shredded sharp Cheddar cheese (4 oz). Cook and stir until cheese is melted. Continue as directed.\n\n## Potato Pancakes CALORIE Smart\n\nPotato pancakes, or _latkes,_ are eaten traditionally during Hanukkah, but they are great to serve any time. Serve them with sour cream or applesauce.\n\nPREP 35 min TOTAL 35 min \u25cf 16 pancakes\n\n  * 4 medium russet potatoes (1\u00bd lb), peeled\n  * 4 eggs, beaten\n  * 1 small onion, finely chopped (\u2153 cup), if desired\n  * \u00bc cup all-purpose flour\n  * 1 teaspoon salt\n  * \u00bc cup vegetable oil\n\n1 Shred enough potatoes to measure 4 cups. Rinse well; drain and pat dry.\n\n2 In large bowl, mix potatoes, eggs, onion, flour and salt. In 12-inch skillet, heat 2 tablespoons of the oil over medium heat. Using \u00bc cup potato mixture for each pancake, place 4 mounds into skillet. Flatten each with spatula to about 4 inches in diameter.\n\n3 Cook pancakes about 4 minutes, turning once, or until golden brown. Remove from skillet; cover to keep warm. Repeat with remaining potato mixture; as mixture stands, liquid and potatoes will separate, so stir to mix as necessary. Add remaining oil to skillet as needed to prevent sticking.\n\n1 Pancake: Calories 80; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 55mg; Sodium 160mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd Starch, 1 Fat Carbohydrate Choices: \u00bd\n\nCandied Sweet Potatoes\n\n## Candied Sweet Potatoes EASY\n\nPREP 10 min TOTAL 35 min \u25cf 6 servings\n\n  * 6 medium unpeeled sweet potatoes (2 lb)*\n  * \u2153 cup packed brown sugar\n  * 3 tablespoons butter\n  * 3 tablespoons water\n  * \u00bd teaspoon salt\n\n1 In 2-quart saucepan, place sweet potatoes and enough water just to cover sweet potatoes. Heat to boiling; reduce heat. Cover and simmer 20 to 25 minutes or until potatoes are tender when pierced with fork; drain. When potatoes are cool enough to handle, peel and cut into \u00bc-inch slices.\n\n2 In 10-inch skillet, heat remaining ingredients over medium heat, stirring constantly, until smooth and bubbly. Add potatoes. Gently stir until glazed and hot.\n\n*Or substitute 1 can (23 oz) sweet potatoes, drained and cut into \u00bd-inch slices, for the fresh sweet potatoes; omit Step 1.\n\n1 Serving: Calories 220; Total Fat 6g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 250mg; Total Carbohydrate 41g (Dietary Fiber 4g, Sugars 29g); Protein 2g Exchanges: \u00bd Starch, 2 Other Carbohydrate, 1 Fat Carbohydrate Choices: 3\n\n## Wilted Spinach CALORIE Smart \u2022 Fast\n\nPREP 25 min TOTAL 25 min \u25cf 4 servings\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 slice bacon, chopped\n  * 1 clove garlic, finely chopped\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon ground nutmeg\n  * 1 lb fresh spinach, stems removed\n  * 2 tablespoons fresh lime juice\n\n1 In 4-quart Dutch oven or saucepan, heat oil over medium heat. Cook onion, bacon and garlic in oil 8 to 10 minutes, stirring occasionally, until bacon is crisp; reduce heat to low.\n\n2 Stir in salt, pepper and nutmeg. Gradually add spinach. Toss just until spinach is wilted. Drizzle with lime juice.\n\n1 Serving (\u00bd Cup): Calories 110; Total Fat 8g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 390mg; Total Carbohydrate 6g (Dietary Fiber 3g, Sugars 2g); Protein 3g Exchanges: 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: \u00bd\n\nWilted Spinach and Kale Use \u00bd lb each fresh spinach and kale. Substitute \u00bd teaspoon dried oregano for the nutmeg.\n\nSpiral Summer Squash\n\n## Spiral Summer Squash CALORIE Smart \u2022 Fast \u2022 Easy\n\nIf you love gadgets, try a vegetable spiralizer! Straight, even vegetables work best when using spiralizers. If you don't have a spiralizer, use a vegetable peeler. Pull the peeler down the sides of the squash, making long strips. The baking time may be shorter.\n\nPrep 10 min Total 20 min \u25cf 4 servings\n\n  * 1 zucchini (6 inches)\n  * 1 yellow summer squash (6 inches)\n  * 2 teaspoons olive oil\n  * \u00bc teaspoon salt\n  * Freshly ground pepper, chopped parsley and grated Parmesan, if desired\n\n1 Heat oven to 400\u00b0F. Line 15x10x1-inch pan with foil.\n\n2 Cut off ends of squash. Cut squash with spiralizer according to manufacturer's directions. Place in pan. Drizzle with oil; sprinkle with salt. Toss to coat; spread squash in single layer.\n\n3 Bake 8 to 10 minutes or just until tender. Sprinkle with pepper, parsley and Parmesan.\n\n1 Serving: Calories 40; Total Fat 2.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 3g (Dietary Fiber 1g, Sugars 2g); Protein 1g Exchanges: \u00bd Vegetable, \u00bd Fat Carbohydrate Choices: 0\n\n### Making Spiral Summer Squash\n\nCreate veggie noodles by running firm, fresh veggies through spiralizer.\n\nPlace veggie spirals in pan; toss with oil and salt.\n\nFried Green Tomatoes\n\n## Fried Green Tomatoes CALORIE Smart\n\nPREP 20 min TOTAL 40 min \u25cf 4 servings\n\n  * 2 medium firm green tomatoes, cut into \u00bc-inch slices\n  * 1 teaspoon salt\n  * \u00bd cup buttermilk\n  * \u00bd cup all-purpose flour\n  * \u215b teaspoon pepper\n  * 2 cups vegetable oil\n  * Mayonnaise or Tartar Sauce (to make your own, see page 460 or )\n\n1 Place tomato slices in colander; sprinkle with \u00bd teaspoon of the salt. Let stand 30 minutes. Pat tomatoes dry with paper towels.\n\n2 Pour buttermilk into shallow bowl. In separate shallow bowl, mix flour and remaining salt and pepper. Dip tomatoes into buttermilk, coating both sides. Dip tomatoes into flour mixture, coating both sides; gently shake off excess.\n\n3 In deep fryer or 10-inch skillet, heat \u00bd inch oil to 375\u00b0F. Fry 4 or 5 tomato slices at a time (do not let them touch) in oil 4 to 6 minutes, turning once, until golden brown. Drain on paper towels. Serve warm with Mayonnaise or Tartar Sauce.\n\n1 Serving: Calories 130; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 630mg; Total Carbohydrate 17g (Dietary Fiber 1g, Sugars 4g); Protein 3g Exchanges: 1 Other Carbohydrate, 1 Vegetable, 1 Fat Carbohydrate Choices: 1\n\n### Making Fried Green Tomatoes\n\nDip tomatoes in buttermilk, coating both sides; shake off excess.\n\nFry tomatoes in oil, turning once, until golden brown.\n\n# Bean Basics\n\nAvailable dried, canned and frozen, beans are extremely versatile and pair well with countless seasonings. They can also be an excellent source of protein instead of meat. Beans, along with peanuts, lentils, soybeans and peas, are considered legumes. Pulses are the edible seeds of legume plants.\n\n## Cooking Dried Beans\n\nBefore cooking dried beans, sort through to remove any shriveled, small or damaged beans. Rinse and drain well. With the exception of black-eyed peas, lentils and split peas, dried legumes need to be soaked before cooking to soften and plump them. Because they rehydrate to triple their dried size, choose a pot that's large enough for them to cook with plenty of room to expand. There are two methods for soaking beans.\n\n  * Quick-Soak Method: Place dried beans in a large saucepan and add enough water to cover. Heat to boiling and boil 2 minutes. Remove from heat; cover and let stand for at least 1 hour. Drain, then cook in fresh, cold water.\n  * Long-Soak Method: Place dried beans in a large saucepan or bowl and add enough water to cover. Let stand 8 to 24 hours. Drain, then cook in fresh, cold water.\n\nPlace soaked beans in a 4-quart saucepan and add fresh, cold water to cover. Heat to boiling and boil uncovered for 2 minutes. Reduce heat to low. Cover and simmer (do not boil or beans will burst), stirring occasionally, for time indicated in cooking chart, right, or until tender.\n\n## Dried Beans and Lentils\n\nThere are many types of beans and lentils available, and here is some information on the more unusual varieties.\n\nAdzuki Beans: Mild, nutty flavor, similar to kidney beans and interchangeable with them.\n\nFava Beans: Large, flat beans; earthy flavor. Appear brown and wrinkled when dried. **Habas** are dried favas with skins removed. Used in Mediterranean and Middle Eastern dishes, such as falafel.\n\nFrench Green Lentils: Considered to be the most flavorful lentils. They don't get mushy as fast as other varieties, making them a good choice in recipes requiring firmer lentils.\n\nMayocoba (Peruano) Beans: Light, buttery taste; soft, creamy texture. Common in Latin American and northern Mexican cooking.\n\nMung Beans: Sweet flavor; popular in India and China. Interchangeable with lentils or peas in recipes. Called **Gram** , or when hulled, **Moong** **Dal**.\n\nPeas (Green, Yellow): Field peas grown for drying. Available whole, hulled and split. Yellow peas are milder than green; split peas are sweeter, less starchy than whole peas.\n\nPink Beans: Interchangeable with pinto beans. Popular in Southwestern cooking.\n\n### Timetable for Cooking Dried Beans\n\nPlace beans in a large saucepan; add enough water to cover and soak using one of the methods described on previous page. Bring to a boil; reduce heat to low; then cover and simmer, stirring occasionally, for the amount of time specified below.\n\nType of Bean (2\u20443 Cup) | Simmer Time | Yield in Cups\n\n---|---|---\n\nAdzuki Beans Lentils | 30 to 45 minutes | 2 to 3\n\nMung Beans Split Peas | 45 to 60 minutes | 2 to 2\u00bc\n\nBlack-Eyed Peas, Butter Beans, Cannellini Beans, Great Northern Beans, Lima Beans, Navy Beans, Pinto Beans | 1 to 1\u00bd hours | 2 to 2\u00bd\n\nAnasazi Beans, Black Beans, Fava Beans, Kidney Beans | 1 to 2 hours | 2\n\nChickpeas (Garbanzo Beans) | 2 to 2\u00bd hours | 2\n\nSoybeans | 3 to 4 hours | 2\n\n## Betty's Staples Edamame\n\nIf you are already a fan of edamame, you know how tasty this vegetable is. The word \"edamame\" means \"beans on branches,\" as these immature soybeans grow in clusters on bushy branches. The pods are not edible, but the beans inside are tender and green. You may find them fresh, but most often they will be parboiled and frozen. To cook them, follow the directions on the package.\n\n  1. 1. **Boiled in the Pods:** Cook the beans in the pods and drain. Sprinkle with sea salt and teriyaki sauce. Use your fingers to squeeze the beans directly from the pods into your mouth.\n  2. 2. **Stir-Fried with Corn and Peppers:** Cook the shelled edamame with frozen corn and chopped bell pepper in a small amount of oil until tender, stirring frequently. Sprinkle with salt, pepper and garlic powder to taste.\n  3. 3. **Edamame and Vegetable Salad:** In large bowl, toss chopped romaine lettuce, cooked shelled edamame beans, chopped fresh tomatoes, chopped roasted bell peppers, chopped avocado and balsamic vinaigrette. Sprinkle with crumbled blue or feta cheese.\n  4. 4. **Pasta with Edamame and Pesto:** Cook refrigerated linguini with shelled edamame beans. Drain and toss with refrigerated pesto.\n  5. 5. **Bacon-Edamame Toss:** Cook 2 slices diced bacon in skillet until crisp. Stir in 1\u00bd cups shelled edamame beans and \u00bd cup red bell pepper. Cook and stir 4 minutes. Sprinkle with 2 tablespoons chopped fresh oregano and \u00bc teaspoon salt. Cook and stir 3 to 4 minutes or until vegetables are tender.\n\nPasta with Edamame and Pesto\n\nCannellini with Tomatoes and Sage\n\n## Cannellini with Tomatoes and Sage CALORIE Smart \u2022 Fast\n\nPREP 15 min TOTAL 15 min \u25cf 4 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 2 cloves garlic, finely chopped\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 can (15 or 19 oz) cannellini beans, drained, rinsed\n  * 1 tablespoon finely chopped fresh or \u00bd teaspoon dried sage leaves\n  * \u00bc teaspoon coarse (kosher or sea) salt\n  * \u215b teaspoon freshly ground pepper\n\n1 In 10-inch skillet, heat oil over medium heat. Cook garlic in oil 1 to 2 minutes, stirring constantly, until light golden brown.\n\n2 Stir in tomatoes, beans and sage. Heat to boiling; reduce heat. Simmer uncovered 5 to 7 minutes, stirring occasionally, until most of the liquid has evaporated. Stir in salt and pepper.\n\n1 Serving: Calories 190; Total Fat 4g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 280mg; Total Carbohydrate 28g (Dietary Fiber 7g, Sugars 3g); Protein 10g Exchanges: 1\u00bd Starch, 1 Vegetable, \u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 2\n\n## White Beans with Sun-Dried Tomatoes SLOW Cooker \u2022 Easy\n\nPREP 10 min TOTAL 4 hr 10 min \u25cf 5 servings\n\n  * 2 cups dried great northern beans (16 oz), sorted, rinsed\n  * 2 cloves garlic, crushed\n  * 6 cups water\n  * 1\u00bd teaspoons dried basil leaves\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00be cup finely chopped sun-dried tomatoes in oil\n  * 1 can (2.25 oz) sliced ripe olives, drained\n\n1 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix beans, garlic, water, basil, salt and pepper.\n\n2 Cover; cook on High heat setting 4 to 5 hours. Before serving, stir in tomatoes and olives.\n\n1 Serving: Calories 380; Total Fat 4g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 620mg; Total Carbohydrate 61g (Dietary Fiber 15g, Sugars 3g); Protein 23g Exchanges: 3\u00bd Starch, 1\u00bd Vegetable, 1\u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: 4\n\n## Spicy Refried Black Beans CALORIE Smart \u2022 Fast\n\nFor a tamer version of these refried beans, swap one 4.5-ounce can of diced green chiles for the fresh jalape\u00f1os and use just 1 teaspoon of the chipotle chiles.\n\nPREP 5 min TOTAL 20 min \u25cf 6 servings\n\n  * 2 tablespoons vegetable oil\n  * 1 small onion, chopped (\u00bc cup)\n  * 2 jalape\u00f1o chiles, seeded, finely chopped\n  * 2 cloves garlic, finely chopped\n  * 2 cans (15 oz each) black beans, undrained\n  * 1 chipotle chile in adobo sauce (from 7-oz can), chopped*\n  * 1 teaspoon chili powder\n  * \u00bd teaspoon salt\n  * Sliced green onion and chopped tomato, if desired\n\n1 In 10-inch skillet, heat oil over medium-high heat. Cook onion, jalape\u00f1o chiles and garlic in oil, stirring occasionally, until onion is tender, about 3 minutes. Stir in beans, chipotle chile, chili powder, and salt; mash beans.\n\n2 Reduce heat. Simmer uncovered about 15 minutes, stirring occasionally, until desired consistency. Sprinkle with green onion and tomato.\n\n*Freeze any unused chipotle chiles on cookie sheet lined with waxed paper, parchment paper or foil. Once frozen, place in resealable plastic food storage bag and store in freezer up to 6 months.\n\n1 Serving: Calories 170; Total Fat 5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 690mg; Total Carbohydrate 22g (Dietary Fiber 6g, Sugars 0g); Protein 7g Exchanges: 1\u00bd Starch, \u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 1\u00bd\n\n## Red Beans and Rice CALORIE Smart\n\nPREP 20 min TOTAL 2 hr \u25cf 8 servings\n\n  * 1 cup dried kidney beans (8 oz), sorted, rinsed*\n  * 3 cups water\n  * 2 oz salt pork (with rind) or 3 slices bacon, cut up\n  * 1 medium green bell pepper, chopped (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 cup uncooked regular long-grain white rice\n  * 1 teaspoon salt\n\n1 In 3-quart saucepan, heat beans and water to boiling. Boil uncovered 2 minutes; reduce heat. Cover and simmer 1 hour to 1 hour 15 minutes or until tender (do not boil or beans will fall apart). Drain, reserving liquid.\n\n2 In 10-inch skillet, cook salt pork over medium heat, stirring occasionally, until crisp. Stir in bell pepper and onion. Cook, stirring occasionally, until onion is tender.\n\n3 Add enough water to bean liquid, if necessary, to measure 2 cups. Add bean liquid, salt pork mixture, rice and salt to beans in saucepan. Heat to boiling, stirring once or twice; reduce heat. Cover and simmer 14 minutes (do not lift cover or stir). Remove from heat. Fluff with fork. Cover; let steam 5 to 10 minutes.\n\n*One can (15 to 15.5 oz) red kidney beans, drained and liquid reserved, can be substituted for the dried beans. Omit water and skip Step 1.\n\n1 Serving (\u00be Cup): Calories 210; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 360mg; Total Carbohydrate 35g (Dietary Fiber 4g, Sugars 2g); Protein 7g Exchanges: 2 Starch, 1 Vegetable, \u00bd Fat Carbohydrate Choices: 2\n\nSlow-Cooker Directions Increase water to 3\u00bc cups. Use 1\u2153 cups uncooked instant white rice. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except rice. Cover; cook on High heat setting 3 hours 30 minutes to 4 hours 30 minutes or until beans are tender. Stir in rice. Cover; cook 15 minutes longer. Stir well.\n\nHoppin' John Substitute 1 cup dried black-eyed peas for the kidney beans. Omit bell pepper.\n\nHomemade Baked Beans\n\n## Homemade Baked Beans\n\nPREP 15 min TOTAL 8 hr 30 min \u25cf 10 servings\n\n  * 2 cups dried navy beans (16 oz), sorted, rinsed\n  * 13 cups water\n  * \u00bd cup packed brown sugar\n  * \u00bc cup molasses\n  * 1 teaspoon salt\n  * 6 slices bacon, crisply cooked, crumbled\n  * 1 medium onion, chopped (\u00bd cup)\n\n1 Heat oven to 350\u00b0F. In 4-quart ovenproof Dutch oven, heat beans and 10 cups of the water to boiling. Boil uncovered 2 minutes.\n\n2 Stir in brown sugar, molasses, salt, bacon and onion. Cover and bake 4 hours, stirring occasionally.\n\n3 Stir in remaining 3 cups water. Bake uncovered 2 hours to 2 hours 15 minutes longer, stirring occasionally, until beans are tender and desired consistency.\n\n1 Serving: Calories 240; Total Fat 2.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 340mg; Total Carbohydrate 45g (Dietary Fiber 11g, Sugars 16g); Protein 10g Exchanges: 1 Starch, 2 Other Carbohydrate, 1 Lean Meat Carbohydrate Choices: 3\n\nSlow-Cooker Directions Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, place beans and 5 cups water. Cover; cook on High heat setting 2 hours. Turn off heat; let stand 8 to 24 hours. Stir in brown sugar, molasses, salt, bacon and onion. Cover; cook on Low heat setting 10 to 12 hours or until beans are very tender and most of liquid is absorbed.\nGlobal Flavors\n\n## India\n\nLoved throughout the world for the intoxicating aromas and unique flavors, Indian dishes have gained popularity because of their variety of ingredients and unique flavor combinations.\n\nGhee Clarified butter (slowly melted to separate the milk solids from the golden surface liquid) that's been further simmered until all of the moisture has evaporated and the milk solids begin to brown, giving it a caramel-like, nutty flavor and aroma. Used for frying and saut\u00e9ing.\n\nJaggery A distinctly flavored coarse, dark sugar. Many forms are available; the most common is a soft form that is spread on breads and sweets and a solid form that is typically crushed and used on cereal or for making candy.\n\nLentils A staple in most Indian kitchens, these high-protein pulses are commonly used in place of meat. Many dishes from flat bread to noodles are made with lentils\u2014dal, a spicy East Indian dish, is one of the most popular.\n\nMangosteen Similar in size and form to tangerines. The sweet-tart flavor and soft, creamy colored fruit is juicy and refreshing. The purple-brown skin is hard.\n\nPaneer (Panir) Fresh cheese made from cow or buffalo milk curdled with lemon or lime juice and traditionally pressed. Similar to tofu in texture.\n\nPappadam (Poppadum) Tortilla-like bread made with lentil flour, widely available in India in various flavors. In northern India, it may be seasoned with red or black pepper, garlic or other seasonings. Southern India prefers an unflavored variety. Deep fried, they puff up to double their size, grilling pappadam gives them a smoky flavor.\n\nSpices **Bishop Weed** or **Ajowain** is available ground or as seeds. Its astringent thyme-like flavor is commonly used in curry dishes, chutneys, breads and legumes. A member of the ginger family, aromatic **Cardamom** can be purchased either in the pod or ground. The warm, spicy-sweet flavor is used in stews and curries\u2014a little goes a long way. A dried fruit of a plant from the parsley family, **Cumin** is available both as seeds and ground. Aromatic **Fenugreek** seeds can be purchased whole or ground. Dried leaves are more commonly available than fresh. Typically used in curry powders, spice blends and tea. **Garam Masala** is a blend of up to 12 dry-roasted, ground spices. There are many varieties. Used to add \"warmth\" to a dish, add it towards the end of cooking or sprinkle over a dish before serving. **Nigella Seeds** , with a peppery-nutty flavor, are used to season vegetables, legumes and bread. **Tamarind** is a sweet-sour fruit often called Indian date. When dried, it becomes very sour. Used like lemon juice in full-flavored chutneys, curried dishes, to make a syrup to flavor soft drinks, Worcestershire sauce. Available in jars of concentrated pulp with seeds, canned paste, whole pods dried into bricks or ground. **Turmeric** , also known as **Indian Saffron** , imparts its characteristic deep yellow-orange to many Indian dishes.\n\n## Indian Lentils and Rice CALORIE Smart\n\nPREP 25 min TOTAL 55 min \u25cf 6 servings\n\n  * 8 medium green onions, chopped (\u00bd cup)\n  * 1 tablespoon finely chopped gingerroot\n  * \u215b teaspoon crushed red pepper flakes\n  * 2 cloves garlic, finely chopped\n  * 3 cans (14 oz each) vegetable broth (5\u00bc cups; for homemade broth, see page 156)\n  * 1\u00bd cups dried lentils (12 oz), sorted, rinsed\n  * 1 teaspoon ground turmeric\n  * \u00bd teaspoon salt\n  * 1 large tomato, chopped (1 cup)\n  * \u00bc cup shredded coconut\n  * 2 tablespoons chopped fresh or 2 teaspoons dried mint leaves\n  * 3 cups hot cooked rice\n  * 1\u00bd cups plain fat-free yogurt\n\n1 Spray 3-quart saucepan with cooking spray. Add onions, gingerroot, pepper flakes and garlic; cook over medium heat 3 to 5 minutes, stirring occasionally, until onions are tender.\n\n2 Stir in 5 cups of the broth, the lentils, turmeric and salt. Heat to boiling; reduce heat. Cover and simmer 25 to 30 minutes, adding remaining \u00bc cup broth if needed, until lentils are tender.\n\n3 Stir in tomato, coconut and mint. Serve over rice; top with yogurt.\n\n1 Serving: Calories 330; Total Fat 2g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 1050mg; Total Carbohydrate 61g (Dietary Fiber 9g, Sugars 9g); Protein 17g Exchanges: 4 Starch, \u00bd Very Lean Meat Carbohydrate Choices: 4\n\n## Spicy Black-Eyed Peas CALORIE Smart \u2022 Slow Cooker \u2022 Easy\n\nPREP 5 min TOTAL 3 hr 15 min \u25cf 8 servings\n\n  * 2 cups dried black-eyed peas (16 oz), sorted, rinsed\n  * 1 medium onion, chopped (\u00bd cup)\n  * 6 cups water\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * \u00be cup chunky-style salsa\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except salsa.\n\n2 Cover; cook on High heat setting 3 to 4 hours.\n\n3 Stir in salsa. Cover; cook about 10 minutes longer or until hot.\n\n1 Serving: Calories 200; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 480mg; Total Carbohydrate 36g (Dietary Fiber 10g, Sugars 4g); Protein 12g Exchanges: 2 Starch, 1 Vegetable, \u00bd Very Lean Meat Carbohydrate Choices: 2\u00bd\n\nCaribbean Black Beans\n\n## Caribbean Black Beans\n\nPREP 25 min TOTAL 1 hr 10 min \u25cf 5 servings\n\n  * 4\u00bd cups water\n  * 1\u00bd cups dried black beans (12 oz)*, sorted, rinsed\n  * 2 teaspoons vegetable oil\n  * 1 medium papaya or mango, peeled, seeded and diced (about 1\u00bd cups)\n  * 1 medium red bell pepper, finely chopped\n  * \u00bd cup finely chopped red onion\n  * \u00bd cup orange juice\n  * \u00bc cup lime juice\n  * 2 tablespoons chopped fresh cilantro\n  * \u00bd teaspoon ground red pepper (cayenne)\n  * 2 cloves garlic, finely chopped\n  * 5 cups hot cooked rice (page 215)\n\n1 In 2-quart saucepan, heat water and beans to boiling. Boil uncovered 2 minutes; reduce heat. Cover; simmer 45 minutes, stirring occasionally, until beans are tender; drain.\n\n2 In 10-inch skillet, heat oil over medium heat. Add remaining ingredients except rice; cook 5 minutes, stirring occasionally, until bell pepper is crisp-tender. Stir in beans. Cook 5 minutes or until hot. Serve over rice.\n\n*Or substitute 2 cans (15 oz each) black beans, drained and rinsed, for the dried beans. Omit water and Step 1.\n\n1 Serving: Calories 450; Total Fat 3g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 89g (Dietary Fiber 14g, Sugars 12g); Protein 17g Exchanges: 4 Starch, 2 Fruit Carbohydrate Choices: 6\n\n## Garbanzo Beans with Vegetables SLOW Cooker\n\nPREP 15 min TOTAL 4 hr 30 min \u25cf 8 servings\n\n  * 2 cups dried garbanzo beans (chickpeas; 16 oz), sorted, rinsed\n  * 5\u00bd cups water\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * 2 tablespoons olive or vegetable oil\n  * 2 cups sliced fresh mushrooms (about 5 oz)\n  * 1 cup shredded carrots (1\u00bd medium)\n  * 4 medium green onions, thinly sliced (\u00bc cup)\n  * 2 cloves garlic, finely chopped\n  * 2 tablespoons lemon juice\n  * 1 to 2 tablespoons prepared horseradish\n  * 2 teaspoons yellow mustard\n\n1 Spray 4- to 5-quart slow cooker with cooking spray. In slow cooker, stir together beans, water, salt and pepper. Cover; cook on High heat setting 4 to 5 hours.\n\n2 In 12-inch skillet, heat oil over medium heat. Add mushrooms, carrots, onions and garlic; cook about 5 minutes, stirring occasionally, until vegetables are tender. Stir vegetables, lemon juice, horseradish and mustard into beans. Cover; cook 15 minutes longer to blend flavors.\n\n1 Serving: Calories 250; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 350mg; Total Carbohydrate 37g (Dietary Fiber 10g, Sugars 4g); Protein 11g Exchanges: 2 Starch, 1\u00bd Vegetable, \u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\n# Chapter 10: One-Dish Dinners\n\nCranberry-Chicken Pot Pie\n\n## One-Dish Basics\n\nOne-Dish Appeal\n\nThrow-Together Meals\n\n## Features\n\nBetty's Staples: Shredded Cheeses\n\nHeirloom Recipe and New Twist\n\nLearn to Make: Foil Packets\n\n## Chicken and Turkey\n\nChicken and Barley Risotto with Edamame Calorie Smart \u2022 Slow Cooker\n\nChicken and Pasta with Creamy Basil Sauce Fast\n\nChicken and Vegetables Pasta\n\nChicken with Pan-Roasted Cauliflower and Orzo Calorie Smart \u2022 Fast\n\nCranberry-Chicken Pot Pie Calorie Smart\n\nEasy Chicken with Tomatoes and Spinach Calorie Smart\n\nGreek Chicken with Orzo\n\nMexican Cheesy Chicken and Sausage with Black Beans\n\nMoroccan Chicken Pie\n\nRoasted Chicken Sausage with Potatoes and Cheese Calorie Smart\n\nSkillet Chicken Thighs with Bacon and Spinach Calorie Smart\n\nTurkey and Butternut Squash Ragout Calorie Smart \u2022 Slow Cooker\n\nVegetable Chicken Paella\n\n## Beef and Pork\n\nAsian Beef Roast with Cabbage and Pasta Slow Cooker\n\nBeef with Bow-Tie Pasta Calorie Smart \u2022 Fast\n\nBeef with Bow-Tie Pasta and Feta\n\nClassic Cabbage Rolls Calorie Smart\n\nFiesta Taco Casserole\n\nParmesan Orzo and Meatballs Calorie Smart \u2022 Fast\n\nPasta Bolognese\n\nPizza Quinoa with Sausage, Onion and Pepper Calorie Smart\n\nPork and Pasta with Creamy Basil Sauce\n\nPotato-Topped Meat Loaf Casserole Calorie Smart\n\nQuick Potato-Topped Meat Loaf Casserole\n\nSamosa Pot Pie Calorie Smart\n\nSmoked Pork Chops with Apple and Sweet Potato Calorie Smart\n\nSouthwestern Pork Packets Calorie Smart\n\nSouthwestern Steak with Corn and Chiles Calorie Smart \u2022 Fast\n\n## Vegetable and Fish\n\nDilled Tuna Steak Packets Calorie Smart\n\nLemon and Herb Salmon Packets Calorie Smart\n\nTilapia with Lemon Potatoes\n\nTropical Stuffed Cabbage Rolls Slow Cooker\n\nVegetable Paella Calorie Smart \u2022 Fast\n\nVegetarian Barley Risotto with Edamame\n\nVeggie Rigatoni Bake\n\n## Bonus Recipes\n\nBacon-Cheese Spread\n\nBaked Jalape\u00f1o Poppers\n\nCheesy Eggs\n\nEasy Alfredo Sauce\n\nGarlic-Cheese Crostini\n\nJacked-Up Rice\n\nSimple Cheese Enchilada\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# One-Dish Basics\n\nWe love meals that come together in one dish. They offer the flavor and comfort that your family will love. But today's recipes aren't the same old casseroles! From skillet meals to flaky pot pies, oven-roasted meats to convenient foil packets, one-dish meals are an easy way to showcase fresh ingredients and flavors in new and exciting combinations.\n\n## One-Dish Appeal\n\n  * **No Meal Planning Required:** No juggling of multiple recipes when you're just trying to get dinner on the table. Round out the meal with a salad if you want\u2014dinner is done.\n  * **Easy Cleanup:** Many recipes use just one pot or are baked in foil packets, so you won't end up with a sink full of dishes.\n  * **Serve Once, Eat Twice:** Many of these make enough to give you a lunch for tomorrow that beats any fast food. Or freeze individual portions for those nights when there's only one or two\u2014and you don't feel like cooking.\n\n## THROW-TOGETHER MEALS\n\nWhether you want to use up things from your fridge or feel inspired to \"whip something up,\" use these guidelines for fresh meals in minutes.\n\n### Skillet Meals\n\n  * Pick a meat or chicken, veggies and pasta or rice. Use quick-cooking meats such as ground beef, sausage or cut chicken breast, or precooked meat such as chicken sausage, pepperoni or chopped cooked chicken.\n  * Cook the rice or pasta, adding veggies during the last few minutes of cooking. Undercook the veggies by a minute or two.\n  * Cook (and drain if necessary) or heat precooked meat in a skillet or Dutch oven.\n  * Add the veggies and pasta or rice to the meat; toss with broth and\/or olive oil for a lightly dressed dish or add a sauce for more flavor and heartiness; heat through. Toss or top with grated or shredded cheese, if you like.\n\n### Instant Pot Pies\n\n  * Start with precooked meat and cooked veggies and refrigerated pie crust.\n  * Using Samosa Pot Pie (page 274) as a guide, make a sauce with beef or chicken broth **.** Other sauces great for pot pies are White Sauce or Cheese Sauce (both page 459, Italian Tomato Sauce (page 174), or Alfredo Sauce (page 177). Mix the sauce with cooked meat and veggies; assemble and bake pie as directed in guide recipe.\n\n### Foil Packets\n\n  * See Learn to Make Foil Packets (page 278).\n\nEasy Chicken with Tomatoes and Spinach (page 262)\n\n## Easy Chicken with Tomatoes and Spinach Calorie Smart\n\nPrep 10 min Total 40 min \u25cf 4 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * 1 clove garlic, finely chopped\n  * \u00bd teaspoon dried oregano leaves\n  * \u00bd teaspoon seasoned salt\n  * \u00bc teaspoon pepper\n  * \u00bc cup dry white wine or water\n  * 2 medium plum (Roma) tomatoes, sliced\n  * 1 bag (6 oz) fresh baby spinach leaves\n\n1 In 12-inch nonstick skillet, heat oil over medium heat. Sprinkle chicken with garlic, oregano, seasoned salt and pepper. Add chicken to skillet; cook uncovered 15 to 20 minutes, turning once, until juice of chicken is clear when center of thickest part is cut (at least 165\u00b0F).\n\n2 Stir in wine. Arrange tomato slices on chicken. Cover and cook 2 to 3 minutes or until tomatoes are thoroughly heated.\n\n3 Add spinach to skillet. Cover and cook 2 to 3 minutes longer or until spinach is wilted.\n\n1 Serving: Calories 230; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 90mg; Sodium 270mg; Total Carbohydrate 3g (Dietary Fiber 1g, Sugars 0g); Protein 33g Exchanges: \u00bd Vegetable, 2 Very Lean Meat, 2\u00bd Lean Meat Carbohydrate Choices: 0\n\nChicken and Pasta with Creamy Basil Sauce\n\n## Chicken and Pasta with Creamy Basil Sauce FAST\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\n  * 3 cups uncooked bow-tie (farfalle) pasta (6 oz)\n  * 2 cups fresh broccoli florets\n  * 1 tablespoon vegetable oil\n  * 1 lb boneless skinless chicken breasts, cut into 1-inch pieces\n  * \u00bd teaspoon garlic-pepper blend\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1 tablespoon cornstarch\n  * 1 teaspoon sugar\n  * 1 small red bell pepper, chopped (\u00bd cup)\n  * \u00be cup plain fat-free yogurt\n  * 3 tablespoons chopped fresh or 1 tablespoon dried basil leaves\n\n1 Cook pasta as directed on package, adding broccoli during last 4 minutes of cooking; drain.\n\n2 Meanwhile, in 12-inch nonstick skillet, heat oil over medium-high heat. Sprinkle chicken with garlic-pepper blend. Add chicken to skillet; cook about 3 minutes, stirring occasionally, until no longer pink in center.\n\n3 In small bowl, mix broth, cornstarch and sugar. Add broth mixture and bell pepper to skillet; cook about 3 minutes or until sauce is thickened. Stir in pasta, broccoli, yogurt and basil. Heat over low heat just until hot (do not boil).\n\n1 Serving (1\u00be Cups): Calories 640; Total Fat 10g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 70mg; Sodium 760mg; Total Carbohydrate 93g (Dietary Fiber 6g, Sugars 7g); Protein 45g Exchanges: 5\u00bd Starch, \u00bd Other Carbohydrate, \u00bd Vegetable, 2\u00bd Very Lean Meat, 1 Lean Meat, \u00bd Fat Carbohydrate Choices: 6\n\nChicken and Vegetable Pasta Add \u00bd cup sliced sun-dried tomatoes in oil (drained) with the basil.\n\nPork and Pasta with Creamy Basil Sauce Substitute 1 lb pork tenderloin, cut in half lengthwise, then cut crosswise into thin slices, for the chicken breasts. Cook pork about 6 minutes, stirring occasionally, or until no longer pink in center.\n\n## Mexican Cheesy Chicken and Sausage with Black Beans\n\nPrep 40 min Total 40 min \u25cf 6 servings\n\n  * 2 tablespoons olive oil\n  * \u00bd lb fresh chorizo or bulk pork sausage\n  * 1\u00bc lb boneless skinless chicken thighs, cut into 1-inch cubes\n  * 1 large onion, chopped (1 cup)\n  * \u00bd teaspoon salt\n  * 1 tablespoon taco seasoning mix (from 1-oz package)\n  * 1 can (4.5 oz) chopped green chiles\n  * 1 can (28 oz) whole tomatoes, undrained\n  * 1 can (15 oz) black beans, drained, rinsed\n  * 2 cups shredded Monterey Jack or pepper Jack cheese (8 oz)\n  * 4\u00bd cups hot cooked rice or 6 flour tortillas (6 inch), heated as directed on package, if desired\n  * 2 cups coarsely crushed tortilla chips\n  * \u00bc cup chopped fresh cilantro\n  * 6 lime wedges\n\n1 In 4-quart Dutch oven or stockpot, heat 1 tablespoon of the oil over medium heat. Cook sausage in oil 6 to 10 minutes, stirring occasionally, until no longer pink. Drain. Transfer sausage to plate; set aside.\n\n2 Add chicken to Dutch oven; cook 6 to 10 minutes, stirring occasionally, until no longer pink in center. Transfer to plate with sausage.\n\n3 Add remaining 1 tablespoon oil to Dutch oven. Add onion and salt. Cook 4 to 5 minutes or until onion is softened. Stir in taco seasoning mix and chiles; cook 1 minute longer. Add tomatoes. Heat to simmering. Add sausage, chicken and beans. Return to simmering; cook 5 minutes for flavors to combine. Stir in cheese.\n\n4 Serve over rice or in tortillas. Top with chips and cilantro. Serve with lime wedges.\n\n1 Serving (1\u2153 Cups): Calories 730; Total Fat 42g (Saturated Fat 16g, Trans Fat 0g); Cholesterol 130mg; Sodium 1460mg; Total Carbohydrate 42g (Dietary Fiber 8g, Sugars 2g); Protein 46g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, 1 Vegetable, 3 Lean Meat, 2\u00bd High-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 3\n\nSkillet Chicken Thighs with Bacon and Spinach\n\n## Skillet Chicken Thighs with Bacon and Spinach Calorie Smart\n\nPrep 40 min Total 40 min \u25cf 4 servings\n\n  * 8 boneless skinless chicken thighs (about 1\u00bd lb)\n  * 3 slices bacon, chopped\n  * 2 large carrots, chopped (1\u00bd cups)\n  * 2 small onions, sliced\n  * 3 cloves garlic, finely chopped\n  * \u00bd cup chicken broth (for homemade broth, see page 143)\n  * 1 bag (8 oz) fresh baby spinach leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 tablespoon chopped fresh or \u00bd teaspoon dried sage leaves\n  * 1 tablespoon grated lemon peel\n\n1 In 12-inch skillet, cook chicken and bacon over medium-high heat 5 minutes, turning chicken once. Stir in carrots, onions, garlic and broth. Cook uncovered 15 to 20 minutes, turning chicken and stirring frequently, until juice of chicken is clear when thickest part is cut (at least 165\u00b0F) and vegetables are tender.\n\n2 Remove from heat. Add spinach, salt and pepper; stir about 3 minutes or until spinach is wilted. Stir in sage and lemon peel.\n\n1 Serving: Calories 350; Total Fat 16g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 110mg; Sodium 720mg; Total Carbohydrate 11g (Dietary Fiber 3g, Sugars 4g); Protein 40g Exchanges: 1\u00bd Vegetable, 5\u00bd Lean Meat Carbohydrate Choices: 1\n\n## Chicken with Pan-Roasted Cauliflower and Orzo CALORIE Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 tablespoon olive oil\n  * 2 cups bite-size fresh cauliflower florets\n  * 1\u00bd cups chicken broth (for homemade broth, see page 143)\n  * 1\u00bc lb boneless skinless chicken thighs, cut into bite-size pieces\n  * 1 cup uncooked orzo or rosamarina pasta (6 oz)\n  * \u00bc cup thinly sliced green onions (4 medium)\n  * 1 can (14.5 oz) diced tomatoes with basil, garlic and oregano, drained\n  * 2 cups packed arugula or fresh baby spinach leaves\n  * \u00bd cup shredded Parmesan cheese (2 oz)\n\n1 In 12-inch nonstick skillet, heat oil over medium heat. Cook cauliflower in oil about 5 minutes, stirring occasionally, until lightly browned.\n\n2 Add broth, chicken, pasta, onions and tomatoes. Heat to boiling; reduce heat. Cover and simmer 8 to 10 minutes, stirring occasionally, until chicken is no longer pink in center and pasta is tender.\n\n3 Remove from heat. Stir in arugula; cover and let stand about 1 minute or until arugula is partially wilted. Sprinkle with cheese before serving.\n\n1 Serving (1\u00bd Cups): Calories 380; Total Fat 14g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 55mg; Sodium 780mg; Total Carbohydrate 35g (Dietary Fiber 3g, Sugars 2g); Protein 28g Exchanges: 1\u00bd Starch, 2 Vegetable, \u00bd Lean Meat, 2\u00bd Medium-Fat Meat Carbohydrate Choices: 2\n\nGreek Chicken with Orzo Stir in \u00bc cup sliced ripe olives with the arugula. Substitute crumbled feta cheese for the Parmesan. Sprinkle 2 tablespoons chopped fresh parsley over the cheese.\n\n### Slicing Green Onions\n\nPlace onions on cutting board. Hold firmly with hand and cut into thin slices starting at white (bulb) end.\n\nChicken with Pan-Roasted Cauliflower and Orzo\n\n## Roasted Chicken Sausage with Potatoes and Cheese CALORIE Smart\n\nPrep 15min Total 1 hr 10 min \u25cf 4 servings\n\n  * 3 medium potatoes, peeled and cut into 1-inch pieces\n  * 2 tablespoons vegetable oil\n  * 1 teaspoon salt\n  * \u00bd teaspoon garlic powder\n  * \u00bd teaspoon onion powder\n  * \u00bd teaspoon paprika\n  * \u215b teaspoon pepper\n  * 2 large bell peppers (any color), seeded, coarsely chopped\n  * 1 package (12 oz) cooked chicken sausages (any flavor)*\n  * \u2153 cup finely shredded sharp Cheddar cheese\n\n1 Heat oven to 375\u00b0F. Line 15x10x1-inch pan with heavy-duty foil; spray foil with cooking spray.\n\n2 In pan, toss potatoes with oil. Sprinkle with salt, garlic powder, onion powder, paprika and pepper; toss until evenly coated. Spread evenly in pan. Bake 10 minutes; toss with peppers. Poke tops of sausages with fork; arrange on top of potato mixture.\n\n3 Bake 20 minutes. Stir potato mixture; spread evenly in pan. Rearrange sausages on top. Bake 20 minutes longer or until vegetables are tender and sausages are golden brown on top. Sprinkle with cheese. Bake 1 to 2 minutes longer or until cheese is melted.\n\n*Any smoked sausage, such as cooked kielbasa or bratwurst, can be substituted for the chicken sausage.\n\n1 Serving: Calories 380; Total Fat 16g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 65mg; Sodium 1320mg; Total Carbohydrate 39g (Dietary Fiber 4g, Sugars 3g); Protein 19g Exchanges: 2\u00bd Starch, 1 Vegetable, 1\u00bd High-Fat Meat, \u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## Chicken Sausage\n\nIf your family is tired of the same old chicken, chicken sausage is a yummy addition to many dishes. It comes in a variety of unique flavor combinations that will perk up your taste buds. Some are fully cooked, others are uncooked\u2014be sure to check the label for the style you are looking for. Many products are lower in calories and fat than their pork counterparts and some are also gluten free and all natural.\n\n## Betty's Staples Shredded Cheeses\n\nWhat isn't better with cheese? Shredded cheese is a handy go-to for quick meals and snacks. If you wish, freeze larger amounts to keep it fresh longer. When small amounts are lurking in your fridge, put them to use with these ideas. We've suggested cheeses to use in each dish, but feel free to use whatever you have on hand.\n\n  1. 1. **Cheesy Eggs:** Make Scrambled Eggs (page 90) as directed, except beat \u00bd cup shredded Cheddar cheese into scrambled egg mixture before cooking. Sprinkle tops of scrambled eggs with additional cheese during final few minutes of cooking.\n  2. 2. **Simple Cheese Enchilada:** Spoon about 2 tablespoons spicy chili beans and sauce in center of large flour tortilla; sprinkle with 2 tablespoons shredded Colby-Jack cheese. Roll up; place seam side down on microwave-safe plate. Sprinkle with additional cheese. Microwave 30 to 40 seconds or until hot.\n  3. 3. **Jacked-Up Rice:** Stir \u00bd cup shredded pepper Jack cheese, \u00bc cup chopped pickled jalape\u00f1o peppers and 2 tablespoons of the pickled jalape\u00f1o liquid into 2 cups hot cooked rice; cover and let stand a few minutes to melt cheese.\n  4. In medium bowl, mix 2 tablespoons each shredded Cheddar cheese, chopped cooked bacon and chopped fresh chives with one 8-ounce package softened cream cheese. Serve with crackers.\n  5. Mix 1 tablespoon olive oil and 1 clove finely chopped garlic; brush lightly on eight \u00bd-inch-thick slices baguette. Place on broiler pan; sprinkle with 2 tablespoons shredded Parmesan cheese. Broil with tops 4 to 6 inches from heat 1 to 2 minutes or until edges of bread are golden brown.\n  6. 6. **Easy Alfredo Sauce:** Make White Sauce (page 459) as directed, except after boiling and stirring sauce 1 minute, stir in \u00bd cup shredded Romano cheese and a dash of grated nutmeg until melted.\n  7. 7. **Baked Jalape\u00f1o Poppers:** In small bowl, mix 4 ounces softened cream cheese and 3 tablespoons shredded taco cheese blend. Cut 8 to 10 whole jalape\u00f1o chiles lengthwise and remove seeds. Fill chile halves with cheese mixture. Bake on cookie sheet at 350\u00b0F 10 to 15 minutes or until hot.\n\nGarlic-Cheese Crostini, Jacked-Up Rice\n\nMoroccan Chicken Pie\n\n## Moroccan Chicken Pie\n\n_Phyllo_ means \"leaf\" in Greek. The paper-thin layers of phyllo don't contain much fat so they can dry out quickly. Sheets are usually brushed with melted butter (or in this case, sprayed with cooking oil) and layered to create a crisp, light and airy crust.\n\nPrep 30 min Total 1 hr 20 min \u25cf 6 servings\n\n  * 2 tablespoons butter\n  * 1 large onion, chopped (1 cup)\n  * \u00bd teaspoon ground cinnamon\n  * \u00bd teaspoon ground turmeric\n  * 2 tablespoons all-purpose flour\n  * 1 cup reduced-sodium chicken broth\n  * \u2153 cup raisins\n  * 3 cups shredded deli rotisserie chicken (from 2-lb chicken)\n  * \u00bc cup chopped fresh cilantro\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon freshly ground pepper\n  * 16 sheets frozen phyllo (filo) pastry (14x9 inch), thawed\n  * Cooking spray\n  * \u00bd cup slivered almonds, finely ground\n\n1 Heat oven to 350\u00b0F. Lightly spray 9-inch glass pie plate with cooking spray.\n\n2 In 10-inch nonstick skillet, melt butter over medium heat. Cook onion in butter 5 to 6 minutes, stirring occasionally, until softened and lightly browned. Stir in cinnamon and turmeric. Stir in flour; cook 1 minute, stirring constantly. Slowly stir in broth. Add raisins. Heat to boiling; boil gently 1 minute, stirring occasionally. In large bowl, stir together onion mixture, chicken, cilantro, salt and pepper.\n\n3 Place phyllo sheets on work surface; cover with clean towel to keep from drying out. Place 1 phyllo sheet in bottom and up side of pie plate, allowing excess phyllo to extend over edge. Lightly spray with cooking spray; sprinkle with 1\u00bd teaspoons of the ground almonds. Repeat layers, using 7 more phyllo sheets and alternating position of sheets to cover bottom and side of pie plate. (It does not matter if phyllo splits or creases when placed in the pan; it won't be noticeable after the pie is baked.)\n\n4 Spoon chicken mixture into pastry-lined plate. Using kitchen scissors, cut excess phyllo from edges. Layer with remaining 8 phyllo sheets, spraying each with cooking spray and sprinkling with almonds. Fold excess phyllo under to form rustic crust edge.\n\n5 Bake 30 to 40 minutes or until crust is golden brown. Let stand 10 minutes before serving.\n\n1 Serving: Calories 440; Total Fat 15g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 70mg; Sodium 760mg; Total Carbohydrate 49g (Dietary Fiber 3g, Sugars 6g); Protein 27g Exchanges: 3 Starch, \u00bd Other Carbohydrate, 2\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 3\n\n## Turkey and Butternut Squash Ragout CALORIE Smart \u2022 Slow Cooker\n\nPrep 10 min Total 7 hr 10 min \u25cf 4 servings\n\n  * 1\u00bd lb turkey thighs (about 2 medium), skin removed\n  * 1 small butternut squash (about 2 lb), peeled, seeded and cut into 1\u00bd-inch pieces (3 cups)\n  * 1 medium onion, cut in half, sliced\n  * 1 can (16 oz) baked beans, undrained\n  * 1 can (14.5 oz) diced tomatoes with Italian-style herbs, undrained\n  * 2 tablespoons chopped fresh parsley\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except parsley.\n\n2 Cover; cook on Low heat setting 7 to 8 hours.\n\n3 Remove turkey from slow cooker to cutting board. Remove meat from bones; discard bones. Return turkey to slow cooker. Just before serving, sprinkle with parsley.\n\n1 Serving: Calories 380; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 115mg; Sodium 730mg; Total Carbohydrate 46g (Dietary Fiber 10g, Sugars 16g); Protein 36g Exchanges: 3 Starch, 1 Vegetable, 3 Very Lean Meat Carbohydrate Choices: 3\n\nChicken and Barley Risotto with Edamame\n\n## Chicken and Barley Risotto with Edamame CALORIE Smart \u2022 Slow Cooker\n\nPrep 15 min Total 4 hr 40 min \u25cf 9 servings\n\n  * 1\u00bc lb boneless skinless chicken breasts, cut into \u00be-inch cubes\n  * 3 medium onions, chopped (1\u00bd cups)\n  * 1\u00bc cups uncooked pearl barley (not quick-cooking)\n  * \u00bd cup shredded carrot\n  * 2 cloves garlic, finely chopped\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon dried thyme leaves\n  * 1 carton (32 oz) chicken broth (4 cups; for homemade broth, see page 143)\n  * 1 cup frozen shelled edamame (from 10-oz bag), thawed\n  * \u00bd cup shredded Parmesan cheese (2 oz)\n\n1 Spray 4- to 5-quart slow cooker with cooking spray. In slow cooker, mix chicken, onions, barley, carrot, garlic, salt, thyme and 3 cups of the broth.\n\n2 Cover; cook on Low heat setting 4 to 5 hours.\n\n3 In 2-cup microwavable measuring cup, microwave remaining 1 cup broth uncovered on High 2 to 3 minutes or until boiling. Stir boiling broth and thawed edamame into chicken mixture.\n\n4 Increase heat setting to High. Cover; cook 25 to 30 minutes or until edamame are tender. Stir in cheese.\n\n1 Serving (About 1 Cup): Calories 250; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 45mg; Sodium 690mg; Total Carbohydrate 27g (Dietary Fiber 6g, Sugars 2g); Protein 23g Exchanges: 1\u00bd Starch, 1 Vegetable, 2 Lean Meat Carbohydrate Choices: 2\n\nVegetarian Barley Risotto with Edamame Omit the chicken and use vegetable broth instead of chicken broth.\n\n## Beef with Bow-Tie Pasta CALORIE Smart \u2022 Fast\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\n  * 3 cups uncooked bow-tie (farfalle) pasta (6 oz)\n  * 1\u00bd lb boneless beef sirloin steak, trimmed of fat*\n  * 3 cups 2-inch pieces asparagus (about 1 lb)\n  * 2 medium onions, sliced\n  * 1\u00bd cups beef broth (for homemade broth, see page 146)\n  * 1 cup tomato puree (from 28-oz can)\n  * 3 tablespoons chopped fresh or 1 tablespoon dried basil leaves\n  * 3 tablespoons chopped dry-pack sun-dried tomatoes\n  * \u00bc teaspoon pepper\n  * 2 tablespoons shredded Parmesan cheese\n\n1 Cook and drain pasta as directed on package.\n\n2 Meanwhile, cut beef with grain into 2-inch strips; cut strips across grain into \u215b-inch slices. Spray 12-inch skillet with cooking spray; heat over medium heat. Add asparagus, onions and 1 cup of the broth to skillet. Cook 5 to 7 minutes, stirring occasionally, until liquid has evaporated. Transfer mixture to bowl.\n\n3 Add beef to skillet; cook about 2 minutes over medium heat, stirring frequently, until beef is no longer pink.\n\n4 Return asparagus mixture to skillet. Stir in tomato puree, basil, tomatoes, pepper and remaining \u00bd cup broth. Cook about 2 minutes, stirring frequently, until mixture is hot. Toss beef mixture with cooked pasta. Sprinkle with cheese.\n\n*For easier cutting, partially freeze beef about 1 hour.\n\n1 Serving: Calories 360; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 75mg; Sodium 670mg; Total Carbohydrate 39g (Dietary Fiber 4g, Sugars 6g); Protein 38g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 2 Vegetable, 3 Very Lean Meat, 1 Lean Meat Carbohydrate Choices: 2\u00bd\n\nBeef with Bow-Tie Pasta and Feta Substitute crumbled feta cheese for the Parmesan cheese, and stir in \u00bc cup capers.\n\nParmesan Orzo and Meatballs\n\n## Parmesan Orzo and Meatballs CALORIE Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 medium bell pepper, cut lengthwise into strips\n  * 1 medium onion, cut into thin wedges\n  * 2 tablespoons Italian dressing\n  * 1\u00be cups beef broth (for homemade broth, see page 146)\n  * 1 cup uncooked orzo or rosamarina pasta (6 oz)\n  * 16 Make-Ahead Meatballs (page 54)*\n  * 1 large tomato, chopped (1 cup)\n  * 2 tablespoons chopped fresh parsley\n  * \u00bc cup shredded Parmesan cheese\n\n1 In 12-inch nonstick skillet, cook bell pepper, onion and dressing over medium-high heat 2 minutes; remove from skillet to heatproof bowl. Add broth to skillet; heat to boiling. Stir in pasta and meatballs. Heat to boiling; reduce heat to low. Cover and simmer 10 minutes, stirring occasionally.\n\n2 Stir in tomato and bell pepper mixture. Cover and simmer 3 to 5 minutes longer or until most of the liquid has been absorbed and pasta is tender. Stir in parsley. Sprinkle with cheese.\n\n*16 frozen cooked Italian-style meatballs (from 16-oz bag) can be susbstituted for the Make-Ahead Meatballs, if desired.\n\n1 Serving: Calories 480; Total Fat 23g (Saturated Fat 7g, Trans Fat 0.5g); Cholesterol 30mg; Sodium 1120mg; Total Carbohydrate 45g (Dietary Fiber 4g, Sugars 5g); Protein 23g Exchanges: 2\u00bd Starch, \u00bd Other Carbohydrate, \u00bd Vegetable, 1 Lean Meat, 1 High-Fat Meat, 2 Fat Carbohydrate Choices: 3\n\nFiesta Taco Casserole\n\n## Fiesta Taco Casserole\n\nPrep 15 min Total 45 min \u25cf 4 servings\n\n  * 1 lb lean (at least 80%) ground beef\n  * 1 can (15 to 16 oz) spicy chili beans in sauce, undrained\n  * 1 cup chunky-style salsa\n  * 2 cups coarsely broken tortilla chips\n  * 4 medium green onions, sliced (\u00bc cup)\n  * 1 medium tomato, chopped (\u00be cup)\n  * 1 cup shredded Cheddar or Monterey Jack cheese (4 oz)\n  * Additional tortilla chips, if desired\n  * Shredded lettuce, if desired\n  * Additional chunky-style salsa, if desired\n\n1 Heat oven to 350\u00b0F. In 10-inch skillet, cook beef over medium heat 8 to 10 minutes, stirring occasionally, until thoroughly cooked; drain. Stir in beans and 1 cup salsa. Heat to boiling, stirring occasionally.\n\n2 In ungreased 2-quart casserole, place broken tortilla chips. Spoon beef mixture over chips. Sprinkle with onions, tomato and cheese.\n\n3 Bake uncovered 20 to 30 minutes or until hot and bubbly. Arrange tortilla chips around edge of casserole. Serve with lettuce and additional salsa.\n\n1 Serving: Calories 530; Total Fat 27g (Saturated Fat 11g, Trans Fat 1g); Cholesterol 100mg; Sodium 1540mg; Total Carbohydrate 36g (Dietary Fiber 7g, Sugars 7g); Protein 35g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 4 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\nLighter Directions For 19 grams of fat and 470 calories per serving, use ground turkey breast and reduced-fat Cheddar cheese.\n\nPasta Bolognese\n\n## Pasta Bolognese\n\nPrep 30 min Total 45 min \u25cf 6 servings\n\n  * 2 tablespoons olive oil\n  * 2 large onions, chopped (2 cups)\n  * 2 medium carrots, chopped (about 1 cup)\n  * 1 teaspoon salt\n  * 1 lb lean (at least 80%) ground beef\n  * \u00bc cup tomato paste (from 6-oz can)\n  * 1 can (28 oz) fire-roasted diced tomatoes, undrained\n  * 1 carton (32 oz) beef broth (4 cups; for homemade broth, see page 146)\n  * \u00bd teaspoon crushed red pepper flakes\n  * 2 teaspoons Italian seasoning\n  * 1 lb uncooked spaghetti\n  * \u00bd cup shredded Parmesan cheese (2 oz)\n  * \u00bc cup thinly sliced fresh basil leaves\n\n1 In 4-quart Dutch oven or stockpot, heat oil over medium-high heat. Cook onions, carrots and salt in oil 5 to 8 minutes, stirring occasionally, until softened. Add beef; cook 5 to 8 minutes, stirring frequently, until no longer pink.\n\n2 Stir in tomato paste and tomatoes. Stir in broth, pepper flakes and Italian seasoning; heat to simmering. Break spaghetti in half; rinse under cold water. Tuck spaghetti into simmering liquid, covering completely.\n\n3 Reduce heat to medium-low; cook uncovered 13 to 15 minutes or until spaghetti is tender and sauce is reduced slightly. Top with cheese and basil.\n\n1 Serving (1\u2154 Cups): Calories 630; Total Fat 18g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 55mg; Sodium 1520mg; Total Carbohydrate 86g (Dietary Fiber 7g, Sugars 7g); Protein 32g Exchanges: 2 Starch, 3 Other Carbohydrate, 2 Vegetable, 3 Lean Meat, 2 Fat Carbohydrate Choices: 6\n\n## Potato-Topped Meat Loaf Casserole CALORIE Smart\n\nPrep 30 min Total 1 hour 25 minutes \u25cf 6 servings\n\nTopping\n\n  * Mashed Potatoes (page 247)\n  * 1 egg\n  * 1\u00bd cups frozen chopped broccoli, thawed\n  * \u00bd cup shredded sharp Cheddar cheese (2 oz)\n\nMeat Loaf\n\n  * 1 lb extra-lean (at least 90%) ground beef\n  * 3 tablespoons unseasoned dry bread crumbs\n  * 3 tablespoons steak sauce\n  * 1 tablespoon dried minced onion\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 egg\n\n1 Heat oven to 350\u00b0F. Spray 8-inch square (2-quart) glass baking dish with cooking spray. Prepare Mashed Potatoes. Stir in egg. Stir in broccoli and cheese; set aside.\n\n2 Meanwhile, in medium bowl, mix meat loaf ingredients. Press mixture in bottom and up sides of baking dish to within \u00bd inch of top.\n\nBake 25 to 30 minutes or until meat loaf is thoroughly cooked and meat thermometer inserted in center of loaf reads 160\u00b0F. Let stand 5 minutes before serving; drain liquid along edges.\n\n1 Serving: Calories 330; Total Fat 15g (Saturated Fat 8g, Trans Fat 0.5g); Cholesterol 140mg; Sodium 610mg; Total Carbohydrate 26g (Dietary Fiber 2g, Sugars 4g); Protein 23g Exchanges: 1\u00bd Starch, 2\u00bd Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 2\n\nQuick Potato-Topped Meat Loaf Casserole Use 2\u00bd to 3 cups leftover mashed potatoes. Place in medium microwavable bowl; cover with plastic wrap. Microwave 2 to 4 minutes on High, stirring occasionally, or until warm. Continue as directed.\n\n## Mashed or Smashed?\n\nMashed potatoes are typically prepared with peeled potatoes and mashed or beaten until silky smooth. Smashed potatoes are traditionally left lumpy on purpose and may or may not have the skins left on, for a dish with more texture, color and additional nutritional value. But there are no potato police: Feel free to prepare your potatoes however you like them.\n\n## Southwestern Steak with Corn and Chiles Calorie Smart \u2022 Fast\n\nPumpkin seeds (pepitas) can usually be found near the packaged items in the produce section of your grocery store or near other prepackaged nuts.\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\n  * 2 tablespoons roasted salted hulled pumpkin seeds (pepitas), finely chopped\n  * 2 teaspoons Southwest seasoning\n  * \u00bd teaspoon pepper\n  * 1 boneless beef top sirloin steak, 1 inch thick (1 lb)\n  * 4 ears fresh sweet corn, husks removed, cut in half crosswise, or 8 frozen small pieces corn on the cob\n  * 2 large poblano chiles, cut in half lengthwise, stems and seeds removed\n  * 1 medium onion, cut into 8 wedges\n  * 4 teaspoons olive oil\n  * \u00bc cup roasted salted hulled pumpkin seeds (pepitas)\n\n1 In small bowl, mix chopped pumpkin seeds, 1 teaspoon of the seasoning blend and the pepper. Press mixture into all sides of beef; let stand at room temperature 15 minutes.\n\n2 Meanwhile, set oven control to broil. Line 15x10x1-inch pan with heavy-duty foil; spray foil with cooking spray. Place corn, chiles and onion in pan. In same small bowl, mix oil and remaining 1 teaspoon seasoning blend. Drizzle mixture over vegetables; toss until evenly coated. Place vegetables in single layer on sides of pan. Place beef in center of pan.\n\n3 Broil 4 to 6 inches from heat 10 to 15 minutes for medium doneness (150\u00b0F), turning beef and vegetables halfway through cooking time.\n\n4 Slice beef; place on platter. Add vegetables to platter; sprinkle with \u00bc cup pumpkin seeds.\n\n1 Serving: Calories 410; Total Fat 16g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 80mg; Sodium 170mg; Total Carbohydrate 27g (Dietary Fiber 4g, Sugars 7g); Protein 39g Exchanges: 2 Starch, 4 Lean Meat, \u00bd High-Fat Meat Carbohydrate Choices: 2\n\nMake-Ahead Directions Rub the beef and marinate the vegetables up to 24 hours in advance and refrigerate. Add a few extra minutes to the broiling time.\n\nCranberry-Chicken Pot Pie\n\n## Cranberry-Chicken Pot Pie CALORIE Smart\n\nPrep 20 min Total 55 min \u25cf 6 servings\n\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1\u00bd cups cubed (\u00bd inch) peeled dark orange sweet potatoes (about 9 oz)\n  * 1 cup chicken gravy (from 12-oz jar, or to make you own, see page 346)\n  * 2 cups fresh baby spinach leaves\n  * \u00bd cup sweetened dried cranberries\n  * 3 cups cut-up cooked chicken or turkey\n  * 1 box refrigerated pie crusts, softened as directed on box (or to make your own, see page 586)\n  * 1 teaspoon milk\n\n1 Heat oven to 425\u00b0F. In 3-quart saucepan, heat broth and sweet potato to boiling; reduce heat. Cover and simmer about 6 minutes or until sweet potato is almost tender when pierced with fork. Gently stir in gravy, spinach and cranberries; fold in chicken. Remove from heat; cover and set aside.\n\n2 On lightly floured surface, roll out 1 pie crust until 12 inches in diameter. Place crust in ungreased 9\u00bd- or 10-inch deep-dish glass pie plate or shallow 2-quart casserole; gently press in bottom and up side of pie plate. Do not stretch.\n\n3 Spoon chicken mixture into crust-lined plate. Top with second crust; seal edge and flute. Brush top crust with milk. Cut several slits in top crust for steam to escape.\n\n4 Bake 25 to 30 minutes or until crust is golden brown and filling is bubbly. Let stand 5 minutes before serving.\n\n*Prepare as directed through Step 2.\n\n1 Serving: Calories 500; Total Fat 24g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 70mg; Sodium 740mg; Total Carbohydrate 50g (Dietary Fiber 2g, Sugars 8g); Protein 23g Exchanges: 2 Starch, 1 Other Carbohydrate, 1 Vegetable, 2 Lean Meat, 3\u00bd Fat Carbohydrate Choices: 3\n\n# Heirloom Recipe and New Twist\n\nCabbage rolls have a rich history in many cuisines, and there are many variations that you will find. Our classic cabbage roll recipe is full on flavor but not on calories\u2014you can feel good about eating this hearty, homemade favorite. Our meatless twist gives cabbage rolls a tropical vibe with coconut, black beans and the crunch of cashews. It also has the bonus of being a make-ahead meal for the slow cooker, to be ready for you to enjoy when you walk in the door.\n\nHeirloom\n\nClassic Cabbage Rolls\n\n## Classic Cabbage Rolls CALORIE Smart\n\nPrep 35 min Total 1 hr 20 min \u25cf 4 servings\n\n  * 1 large head green cabbage (about 3 lb)\n  * 1 tablespoon butter\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 clove garlic, finely chopped\n  * 1 lb lean (at least 80%) ground beef\n  * 1 cup cooked long-grain white rice, cooled\n  * 1 can (15 oz) tomato sauce\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon dried thyme leaves\n  * \u00bc teaspoon dried marjoram leaves\n  * 1 teaspoon sugar\n  * \u00bd teaspoon lemon juice\n  * 1 tablespoon cornstarch\n  * 1 tablespoon water\n\n1 Heat oven to 350\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 Remove core from cabbage. Place cabbage, cored side up, under cool running water until outer leaves loosen; carefully peel leaves off from core end of head. As you peel the leaves away, you may need to trim the core occasionally to keep the leaves free from the thicker center area around the core.\n\n3 In large microwavable bowl, place 8 cabbage leaves (refrigerate any remaining cabbage for another use). Add \u00bc cup water; cover with microwavable plastic wrap. Microwave on High 5 minutes; let stand covered for 10 minutes or until leaves are limp.\n\n4 Meanwhile, in 10-inch skillet, melt butter over medium heat. Cook onion and garlic in butter, stirring occasionally, until tender, about 5 minutes. In large bowl, mix beef, onion mixture, rice, \u00bd cup of the tomato sauce, the salt, pepper, thyme and marjoram.\n\n5 Drain cabbage; remove thick rib at base of each leaf by cutting a small triangle shape (this will make it easier to roll). Place about \u2153 cup beef mixture at stem end of each leaf. Roll leaf around beef mixture, tucking in sides. Place rolls, seam sides down, in baking dish.\n\n6 In small bowl, mix remaining tomato sauce, the sugar and lemon juice; pour over cabbage rolls. Cover and bake 40 to 45 minutes or until beef is no longer pink in center and thermometer reads 160\u00b0F when inserted in center of rolls.\n\n7 Remove cabbage rolls with slotted spoon to platter; keep warm. In 1-quart saucepan, mix cornstarch and water until smooth. Stir in liquid from baking dish. Heat to boiling, stirring constantly. Boil and stir 1 minute. Pour sauce over cabbage rolls. Serve immediately.\n\n1 Serving (2 Cabbage Rolls): Calories 340; Total Fat 16g (Saturated Fat 7g, Trans Fat 1g); Cholesterol 80mg; Sodium 1090mg; Total Carbohydrate 27g (Dietary Fiber 3g, Sugars 9g); Protein 23g Exchanges: 1\u00bd Other Carbohydrate, 1 Vegetable, 3 Lean Meat, 1\u00bd Fat Carbohydrate Choices: 2\nNew Twist\n\nTropical Stuffed Cabbage Rolls\n\n## Tropical Stuffed Cabbage Rolls SLOW Cooker\n\nPrep 30 min Total 7 hr 40 min \u25cf 4 servings\n\n  * 8 large cabbage leaves\n  * 1 can (14 oz) reduced-fat (lite) unsweetened coconut milk (not cream of coconut)\n  * \u00bd cup pineapple preserves\n  * 1 cup cooked orzo or rosamarina pasta\n  * 1 medium onion, chopped (\u00bd cup)\n  * \u2153 cup raisins\n  * \u2153 cup cashew pieces*\n  * 2 teaspoons curry powder\n  * \u00bd teaspoon garlic salt\n  * 1 can (15 oz) black beans, drained, rinsed\n  * Sweetened flaked coconut, if desired\n  * Additional cashew pieces, if desired\n\n1 In large bowl, cover cabbage leaves with boiling water. Cover and let stand about 10 minutes or until leaves are limp.\n\n2 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In small bowl, mix coconut milk and preserves. Spread \u00bd cup mixture in bottom of slow cooker; set remaining mixture aside. In medium bowl, mix pasta, onion, raisins, \u2153 cup cashews, the curry powder, garlic salt and beans.\n\n3 Drain cabbage leaves. Place about \u2153 cup pasta mixture at stem end of each leaf. Roll leaf around pasta mixture, tucking in sides. Place as many cabbage rolls, seam side down, as will comfortably fit in bottom of slow cooker. Pour \u2153 cup of the coconut milk mixture over cabbage rolls. Add remaining cabbage rolls to slow cooker. Pour remaining coconut milk mixture over rolls.\n\n4 Cover; cook on Low heat setting 7 to 9 hours (or on High heat setting 3 hours 30 minutes to 4 hours 30 minutes).\n\n5 With spatula, carefully remove cabbage rolls, one at a time, from slow cooker to serving plate. Sprinkle with coconut and additional cashews.\n\n*Chopped macadamia nuts can be substituted for the cashews.\n\n1 Serving (2 Cabbage Rolls): Calories 670; Total Fat 27g (Saturated Fat 19g, Trans Fat 0g); Cholesterol 0mg; Sodium 620mg; Total Carbohydrate 91g (Dietary Fiber 16g, Sugars 36g); Protein 16g Exchanges: 2 Starch, 3\u00bd Other Carbohydrate, \u00bd Milk, 1 Vegetable, \u00bd High-Fat Meat, 3\u00bd Fat Carbohydrate Choices: 6\n\n### Making Cabbage Rolls in a Slow Cooker\n\nCover cabbage leaves completely with boiling water. Let stand until leaves are limp.\n\nPlace about \u2153 cup of pasta mixture at stem end of each leaf. Roll leaf around pasta mixture, tucking in sides.\n\nPlace one layer of cabbage rolls, seam side down, in bottom of slow cooker. Pour \u2153 cup coconut milk over rolls.\n\nAsian Beef Roast with Cabbage and Pasta\n\n## Asian Beef Roast with Cabbage and Pasta SLOW Cooker\n\nPrep 20 min Total 7 hr 30 min \u25cf 6 servings\n\n  * 1 boneless beef chuck roast (3 lb), trimmed of excess fat\n  * 1 teaspoon garlic salt\n  * \u00bd teaspoon pepper\n  * 1 cup water\n  * \u00bd cup hoisin sauce\n  * \u00bc cup reduced-sodium soy sauce\n  * 2 tablespoons finely chopped gingerroot\n  * 1 teaspoon sesame oil\n  * \u00bd teaspoon crushed red pepper flakes\n  * 4 medium carrots, cut into 1-inch pieces\n  * \u00bd head green cabbage, cut into 6 wedges\n  * 6 oz uncooked angel hair pasta\n  * Toasted sesame seed, chopped red bell pepper and sliced green onions, if desired\n\n1 Spray 5- to 6-quart slow cooker with cooking spray. Sprinkle beef with \u00bd teaspoon of the garlic salt and the pepper. Pour water into slow cooker; place beef in center.\n\n2 In small bowl, stir together hoisin sauce, soy sauce, gingerroot, sesame oil, pepper flakes and remaining \u00bd teaspoon garlic salt. Pour over top and sides of beef. Arrange carrots around beef. Arrange cabbage wedges in single layer on top of beef and carrots.\n\n3 Cover; cook on Low heat setting 7 to 8 hours.\n\n4 Increase heat setting to High. Using slotted spoon, remove vegetables from slow cooker to serving bowl; cover with foil. Remove beef from slow cooker to cutting board; cover with foil. Let stand 5 minutes.\n\n5 Meanwhile, break pasta in half; add to sauce in slow cooker, pushing pasta below surface of sauce. Cover; cook 9 to 11 minutes or until pasta is tender.\n\n6 Cut beef across grain into slices. Serve beef, pasta and vegetables with any remaining sauce. Garnish with sesame seed, bell pepper and onions.\n\n1 Serving: Calories 590; Total Fat 26g (Saturated Fat 10g, Trans Fat 1g); Cholesterol 145mg; Sodium 980mg; Total Carbohydrate 40g (Dietary Fiber 3g, Sugars 9g); Protein 48g Exchanges: 2 Starch, 2 Vegetable, 5\u00bd Lean Meat, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## Samosa Pot Pie CALORIE Smart\n\nPrep 40 min Total 1 hr 30 min \u25cf 6 servings\n\nFilling\n\n  * 1\u00bc cups diced red potatoes (about \u00bd lb)\n  * 1 lb lean (at least 80%) ground beef\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 2 tablespoons curry powder\n  * 2 teaspoons ground cumin\n  * 1 teaspoon ground ginger\n  * \u00bc to \u00bd teaspoon crushed red pepper flakes\n  * \u00be cup beef broth (for homemade broth, see page 146)\n  * 2 tablespoons cornstarch\n  * \u00bd teaspoon salt\n  * \u00be cup frozen sweet peas, thawed\n  * \u00bc cup chopped fresh cilantro\n\nCrust\n\n  * 1 box refrigerated pie crusts, softened as directed on box (or to make your own, see page 586)\n\nCucumber-Yogurt Sauce, if desired\n\n  * \u00bd cup plain fat-free yogurt\n  * \u00bc cup finely chopped peeled cucumber\n  * 2 tablespoons chopped fresh mint leaves\n\n1 In 1-quart saucepan, place potatoes and enough water to cover. Heat to boiling over medium-high heat; reduce heat to medium-low. Cook uncovered 10 minutes or until tender. Drain; set aside.\n\n2 Meanwhile, in 10-inch nonstick skillet, cook beef over medium-high heat 5 minutes, stirring occasionally. Reduce heat to medium. Add onion; cook 3 minutes longer, stirring occasionally, until beef is thoroughly cooked and onion is softened. Drain. Stir in garlic, curry powder, cumin, ginger and pepper flakes; cook and stir 30 seconds.\n\n3 In small bowl, mix broth, cornstarch and salt with whisk. Add to beef mixture. Heat to boiling over medium heat; reduce heat. Simmer 1 minute or until broth is slightly thickened. Remove from heat. Stir in potatoes, peas and cilantro; set aside.\n\n4 Heat oven to 400\u00b0F. Place 1 pie crust in 9-inch glass pie plate. Spoon filling into crust-lined plate. Top with second crust; seal edge and flute. Cut several slits in top crust for steam to escape.\n\n5 Bake 20 minutes. Cover crust edge with pie crust shield ring or strips of foil to prevent excessive browning. Bake 15 to 20 minutes longer or until crust is golden brown and filling is bubbly. Let stand 10 minutes before serving.\n\n6 Meanwhile, in small bowl, stir together sauce ingredients. Serve sauce with pot pie.\n\n1 Serving: Calories 550; Total Fat 32g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 45mg; Sodium 750mg; Total Carbohydrate 44g (Dietary Fiber 3g, Sugars 2g); Protein 19g Exchanges: 2\u00bd Starch, 1 Vegetable, 1\u00bd Medium-Fat Meat, 4\u00bd Fat Carbohydrate Choices: 3\n\nPizza Quinoa with Sausage, Onion and Pepper\n\n## Pizza Quinoa with Sausage, Onion and Pepper CALORIE Smart\n\nPrep 40 min Total 1 hr 5 min \u25cf 8 servings\n\n  * 1 lb bulk spicy Italian pork sausage\n  * 2 tablespoons olive oil\n  * 1 red onion, thinly sliced (about 2 cups)\n  * 1 green bell pepper, cut into thin strips (about 1 cup)\n  * 1 package (8 oz) sliced fresh mushrooms (about 3 cups)\n  * 1\u00bd teaspoons salt\n  * 1 teaspoon Italian seasoning\n  * 1 package (12 oz) quinoa, rinsed (2 cups)\n  * 4 cups milk*\n  * 2 cups shredded mozzarella cheese (8 oz)\n  * 2 cups cherry tomatoes, cut into quarters\n  * \u00bd cup thinly sliced fresh basil leaves\n  * Grated or shredded Parmesan cheese, if desired\n\n1 In 5-quart nonstick Dutch oven or stockpot, cook sausage over medium heat 8 to 10 minutes, stirring occasionally, until no longer pink. Drain; transfer to large bowl. Cover with foil; set aside.\n\n2 Wipe out Dutch oven with paper towel. Add 1 tablespoon of the oil; heat over medium heat. Cook onion in oil 3 to 5 minutes, stirring occasionally, until softened. Add bell pepper; cook 3 to 5 minutes longer or until softened. Using slotted spoon, transfer onion and bell pepper to bowl with sausage; cover with foil.\n\n3 Increase heat to high. Add remaining 1 tablespoon oil, the mushrooms, salt and Italian seasoning. Cook 5 to 7 minutes or until mushrooms are browned and liquid is evaporated. Transfer to bowl with sausage and vegetables; cover with foil.\n\n4 Wipe out Dutch oven. Add quinoa; cook over medium heat 1 to 2 minutes, stirring constantly, until fragrant. Slowly stir in milk. Heat to simmering over high heat, stirring occasionally; reduce heat to low. Cover and cook 20 to 25 minutes or until liquid is absorbed and quinoa is thoroughly cooked.\n\n5 Stir in sausage-vegetable mixture. Remove from heat; stir in mozzarella cheese. Top with tomatoes, basil and Parmesan cheese.\n\n*Cooking the quinoa in milk makes it much creamier, but increases the cooking time. If you'd rather use water or broth, reduce the cooking time to 10 to 15 minutes.\n\n1 Serving (1\u00bd Cups): Calories 450; Total Fat 22g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 45mg; Sodium 890mg; Total Carbohydrate 39g (Dietary Fiber 4g, Sugars 12g); Protein 24g Exchanges: 2 Starch, 1 Vegetable, 1 Medium-Fat Meat, 1\u00bd High-Fat Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\nSmoked Pork Chops with Apple and Sweet Potato\n\n## Smoked Pork Chops with Apple and Sweet Potato CALORIE Smart\n\nPrep 15 min Total 55 min \u25cf 4 servings\n\n  * 2 tablespoons vegetable oil\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon ground cumin\n  * \u00bd teaspoon ground coriander\n  * 1 large dark orange sweet potato, peeled, cut into \u00bd-inch slices and slices quartered\n  * 1 large unpeeled Granny Smith apple, cut into \u00bd-inch pieces\n  * \u00bd small onion, cut into thin wedges, wedges separated\n  * 1 tablespoon packed brown sugar\n  * 4 fully cooked smoked boneless pork chops (about \u00be lb)\n\n1 Heat oven to 425\u00b0F. Spray 15x10x1-inch pan with cooking spray. In small bowl, mix oil, cinnamon, cumin and coriander.\n\n2 In large bowl, toss sweet potato, apple and onion with 1 tablespoon of the oil-spice mixture to coat. Spread mixture in pan.\n\n3 Bake 30 minutes. Meanwhile, stir brown sugar into remaining oil-spice mixture (mixture will be thick). Rub mixture over one side of each pork chop.\n\n4 Remove pan from oven. Move vegetable mixture to one side of pan. Place pork, coated side up, in other half of pan. Bake 8 to 10 minutes longer or until sweet potato and apple are tender and pork is thoroughly heated.\n\n1 Serving: Calories 260; Total Fat 12g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 45mg; Sodium 45mg; Total Carbohydrate 22g (Dietary Fiber 3g, Sugars 12g); Protein 16g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2 Lean Meat, 1 Fat Carbohydrate Choices: 1\u00bd\n\n## Southwestern Pork Packets CALORIE Smart\n\nPrep 15 min Total 45 min \u25cf 4 servings\n\n  * 2 cups chicken broth (for homemade broth, see page 143)\n  * 2 cups uncooked instant white rice\n  * 5 teaspoons Mexican seasoning\n  * 1 can (15.25 oz) whole kernel corn, drained\n  * 1 small bell pepper (any color), chopped (\u00bd cup)\n  * 4 medium green onions, sliced (\u00bc cup)\n  * 4 boneless pork rib or loin chops, \u00be to 1 inch thick (about 1\u00bc lb)\n  * Chunky-style salsa, if desired\n\n1 Heat oven to 450\u00b0F. Cut 4 (18x12-inch) sheets of heavy-duty foil; spray foil with cooking spray.\n\n2 In 1\u00bd-quart saucepan, heat broth to boiling; remove from heat. Stir in rice and 3 teaspoons of the Mexican seasoning. Cover and let stand about 5 minutes or until broth is absorbed. Stir in corn, bell pepper and onions.\n\n3 Sprinkle pork chops evenly with remaining 2 teaspoons Mexican seasoning; place 1 chop on one half of each foil sheet. Spoon rice mixture over pork. Bring up other side of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal. Place packets in ungreased 15x10x1-inch pan.\n\n4 Bake 20 to 25 minutes or until pork is no longer pink in center. Cut large X across top of each packet; carefully fold back foil to allow steam to escape. Transfer pork and rice mixture to individual plates. Serve with salsa.\n\n1 Serving: Calories 500; Total Fat 12g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 85mg; Sodium 790mg; Total Carbohydrate 61g (Dietary Fiber 3g, Sugars 6g); Protein 37g Exchanges: 3\u00bd Starch, 2 Vegetable, 3 Lean Meat Carbohydrate Choices: 4\n\nDilled Tuna Steak Packets\n\n## Dilled Tuna Steak Packets CALORIE Smart\n\nPrep 25 min Total 50 min \u25cf 4 servings\n\nCouscous\n\n  * 1 cup water\n  * 2 teaspoons olive oil\n  * \u00bc teaspoon salt\n  * 1 clove garlic, finely chopped\n  * 1 cup uncoooked couscous\n\nTuna\n\n  * 4 tuna steaks (6 oz each)\n  * 2 teaspoons chopped fresh dill weed\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n\nVegetables\n\n  * 1 medium zucchini, cut in half lengthwise, then cut crosswise into \u00bc-inch slices\n  * 1 medium yellow squash, cut in half lengthwise, then cut crosswise into \u00bc-inch slices\n  * 3 plum (Roma) tomatoes, cut into quarters\n  * 4 teaspoons olive oil\n  * 1 tablespoon chopped fresh dill weed\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 Heat oven to 425\u00b0F. In a 2-quart saucepan, heat water, oil, salt and garlic to boiling. Stir couscous in quickly; remove from heat. Cover and let stand 4 to 5 minutes or until water is absorbed. Cut 4 (18x12-inch) sheets of heavy-duty foil. Fluff couscous with fork; divide evenly onto one half of foil sheets (about \u00bd cup each).\n\n2 Place 1 tuna steak on top of couscous for each packet. Sprinkle each tuna steak with \u00bd teaspoon dill, \u00bc teaspoon salt and \u215b teaspoon pepper.\n\n3 In medium bowl, toss all ingredients for vegetables; divide mixture evenly around couscous. Bring up other half of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal. Place packets in ungreased 15x10x1-inch pan.\n\n4 Bake about 20 minutes or until tuna flakes easily with fork and is slightly pink in center and vegetables are tender. Cut large X across top of each packet; carefully fold back foil to allow steam to escape. Transfer tuna, vegetables and couscous to individual plates.\n\n1 Serving: Calories 500; Total Fat 17g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 65mg; Sodium 900mg; Total Carbohydrate 38g (Dietary Fiber 4g, Sugars 6g); Protein 47g Exchanges: 2 Starch, 1\u00bd Vegetable, 5\u00bd Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## Tilapia with Lemon Potatoes\n\nPrep 25 min Total 40 min \u25cf 2 servings\n\n  * 1 large unpeeled potato, cut into \u00bd-inch pieces\n  * 2 teaspoons olive oil\n  * \u00bc teaspoon salt\n  * 2 tablespoons all-purpose flour\n  * 1 egg\n  * \u00be cup panko bread crumbs\n  * \u00be teaspoon seasoned salt\n  * 2 tablespoons butter, melted\n  * 2 tilapia fillets (5 to 6 oz each)\n  * 1 medium zucchini, cut into \u00bd-inch slices\n  * 2 teaspoons chopped fresh thyme leaves\n  * 2 teaspoons grated lemon peel\n\n1 Heat oven to 425\u00b0F. Spray 15x10x1-inch pan with cooking spray. In medium bowl, toss potato, oil and salt to coat. Spread potato in one half of pan. Bake 15 to 20 minutes or until potato is tender.\n\n2 Meanwhile, place flour on plate. In shallow dish, beat egg with fork. In another shallow dish, mix bread crumbs, seasoned salt and butter. Coat fillets with flour. Dip into egg; coat well with bread crumb mixture.\n\n3 Remove pan from oven. Toss zucchini with potato. Place fillets in other half of pan.\n\n4 Bake 10 to 12 minutes or until fish flakes easily with fork and vegetables are tender. Sprinkle fish with thyme. Sprinkle vegetables with lemon peel; toss.\n\n1 Serving: Calories 660; Total Fat 23g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 210mg; Sodium 1360mg; Total Carbohydrate 71g (Dietary Fiber 6g, Sugars 7g); Protein 41g Exchanges: 4 Starch, 1\u00bd Vegetable, 3\u00bd Very Lean Meat, 4 Fat Carbohydrate Choices: 5\n\n## Learn to Make   \nFOIL PACKETS\n\nThis old campfire cooking technique is hands down one of the easiest ways to get dinner on the table quickly, easily and deliciously. With this method of cooking, you can prepare a whole meal or part of a meal in a tightly sealed foil packet, baking it in the oven. Or you can cook these packets over medium heat on your grill\u2014adjusting cooking time as needed. Another benefit of cooking in packets is the easy cleanup!\n\nHere are some tips for success:\n\n  * Use heavy-duty foil for the most secure packets.\n  * Cut the foil large enough that you can easily wrap food tightly with a good fold at top and sides. A general size for individual servings is 18x12 inches, but follow the recipe you're using for specific size.\n  * Spray the foil with cooking spray or use nonstick foil.\n  * Fill with foods that have similar cook times\u2014cut foods that take longer to cook, like fresh carrots, into smaller or thinner pieces, so that they will be done with everything else.\n  * Use boneless meats such as patties, pork chops, chicken breasts, fish fillets or steaks.\n  * Place food centered on one half of foil, usually with meat pieces first, topped with other foods.\n  * Add a sprig of fresh herbs or sprinkle with dried herbs or seasoning.\n  * marinades.\n  * To seal each packet, fold other side of foil over food so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing a little space for heat circulation and expansion. Fold other sides to seal.\n  * Place packets on cookie sheet with sides. Bake at a high temperature to create steam in the packets, 425\u00b0F or 450\u00b0F. Depending on the foods selected, bake time can range from 10 to 25 minutes.\n  * Open one packet to test for doneness. Carefully unwrap one fold (steam will be hot).Check doneness and carefully refold if more cooking time is needed. \n  * When done, remove from oven carefully. Cut large X across top of packets; carefully fold back foil.\n\nFoil packets are easy to make ahead of time. Just prepare and refrigerate until ready to bake. If they are really cold, you may need to bake them a little extra longer.\n\n## Lemon and Herb Salmon Packets CALORIE Smart\n\nPrep 15 min Total 35 min \u25cf 2 servings\n\n  * 1\u00bc cups reduced-sodium chicken broth\n  * 1 cup uncooked instant brown rice\n  * \u00bd cup julienne carrots (see page 12)\n  * \u00bd to \u00be lb salmon fillet, cut into 2 pieces\n  * \u00bd teaspoon lemon-pepper seasoning\n  * 2 tablespoons chopped fresh chives\n  * 2 lemon slices (\u00bc inch thick)\n\n1 Heat oven to 450\u00b0F. In 1-quart saucepan, heat broth to boiling over high heat. Stir in rice; reduce heat to low. Cover and simmer 5 minutes or until most of broth is absorbed. Stir in carrots.\n\n2 Meanwhile, cut 2 (18x12-inch) sheets of heavy-duty foil; spray foil with cooking spray. Place salmon piece on one half of each sheet. Sprinkle with lemon-pepper seasoning; top with chives. Arrange lemon slices over fish.\n\n3 Spoon rice mixture around salmon. Bring up other half of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal. Place packets in ungreased 15x10x1-inch pan.\n\n4 Bake 16 to 20 minutes or until fish flakes easily with fork. Place packets on plates. Cut large X across top of each packet; carefully fold back foil to allow steam to escape.\n\n1 Serving: Calories 380; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 75mg; Sodium 540mg; Total Carbohydrate 47g (Dietary Fiber 3g, Sugars 2g); Protein 31g Exchanges: 3 Starch, 3 Very Lean Meat, 1 Fat Carbohydrate Choices: 3\n\nLemon and Herb Salmon Packets\n\nVeggie Rigatoni Bake\n\n## Veggie Rigatoni Bake\n\nThere's no need to boil noodles for this vegetarian dish. Just mix the pasta ingredients in the dish, pour on the sauce and bake!\n\nPrep 25 min Total 1 hr 30 min \u25cf 6 servings\n\nPasta Mixture\n\n  * 2 cups uncooked rigatoni pasta (6 oz)\n  * 4 cups loosely packed fresh baby kale or spinach leaves\n  * \u2154 cup coarsely chopped drained roasted red bell peppers (from 12-oz jar, or to make your own, see page 247)\n  * 1 can (15 or 15.5 oz) light red or red kidney beans, drained, rinsed\n  * 1 medium yellow summer squash, cut into \u00bd-inch cubes (1\u00bd cups)\n  * 2 cloves garlic, finely chopped\n\nSauce\n\n  * \u00bc cup butter\n  * \u00bc cup all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon crushed red pepper flakes\n  * 2 cups milk\n  * 1 cup vegetable broth (for homemade broth, see page 156)\n  * 2 cups shredded Cheddar cheese (8 oz)\n\nTopping\n\n  * 1 cup shredded Cheddar cheese (4 oz)\n  * \u00bd cup Italian-style panko crispy bread crumbs\n  * Freshly ground black pepper, if desired\n\n1 Heat oven to 350\u00b0F. In ungreased 13x9-inch (3-quart) glass baking dish, stir together all pasta mixture ingredients.\n\n2 In 2-quart saucepan, melt butter over low heat. Stir in flour, salt, pepper and pepper flakes. Cook over medium heat, stirring constantly, until mixture is smooth and bubbly. Remove from heat. Gradually stir in milk and broth. Heat to boiling, stirring constantly; boil and stir 1 minute. Remove from heat; stir in 2 cups cheese until melted. Pour over pasta mixture.\n\n3 Cover with foil. Bake 45 to 55 minutes or until pasta is tender. In small bowl, mix topping ingredients. Sprinkle topping evenly over pasta mixture. Bake uncovered 3 to 5 minutes or until cheese is melted. Let stand 5 minutes before serving. Sprinkle with pepper.\n\n1 Serving: Calories 670; Total Fat 32g (Saturated Fat 18g, Trans Fat 1g); Cholesterol 85mg; Sodium 1220mg; Total Carbohydrate 64g (Dietary Fiber 6g, Sugars 10g); Protein 30g Exchanges: 3 Starch, 1 Other Carbohydrate, 1 Vegetable, \u00bd Very Lean Meat, 2 High-Fat Meat, 3 Fat Carbohydrate Choices: 4\n\n## Vegetable Paella Calorie Smart \u2022 Fast\n\nPrep 15 min Total 20 min \u25cf 4 servings\n\n  * 2 tablespoons olive or vegetable oil\n  * 2 cloves garlic, finely chopped\n  * 1 large red onion, cut into thin wedges\n  * 1 cup uncooked instant brown rice\n  * 1 cup vegetable or chicken broth (for homemade broth, see page 156 or 143)\n  * \u00bd teaspoon saffron threads,* crushed\n  * 1 bag (16 oz) frozen sweet peas, potatoes and carrots\n  * 1 can (14.5 oz) stewed tomatoes, undrained\n\n1 In 12-inch nonstick skillet, heat oil over medium-high heat. Cook garlic and onion in oil, stirring frequently, until onion is tender.\n\n2 Stir in remaining ingredients. Heat to boiling; reduce heat to medium-low. Cover and cook 5 minutes, stirring occasionally. Remove from heat; let stand covered 5 minutes.\n\n*Saffron gives paella its characteristic bright yellow color and distinct flavor, but it can be expensive. You can substitute the same amount of ground turmeric for the saffron.\n\n1 Serving: Calories 320; Total Fat 8g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 290mg; Total Carbohydrate 56g (Dietary Fiber 7g, Sugars 2g); Protein 6g Exchanges: 1\u00bd Starch, 2 Other Carbohydrate, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 4\n\nVegetable Chicken Paella Add 1 cup cubed cooked chicken with the remaining ingredients in Step 2. Continue as directed.\n\n# Chapter 11: Beef, Pork & Lamb\n\nBroiled Boneless Top Sirloin, page 317, with Herb Butter, page 470\n\n## Beef, Pork and Lamb Basics\n\nBraising Meat\n\nBroiling Meat\n\nRetail Beef Cuts\n\nRetail Lamb Cuts\n\nRetail Pork Cuts\n\nRetail Veal Cuts\n\nRoasting Meat\n\nPanfrying Meat\n\nServings per Pound\n\nStoring Meat\n\nThawing Meat\n\nTips for Purchasing\n\nWhat's on the Label\n\n## Features\n\nGlobal Flavors: South America\n\nHeirloom Recipe and New Twist\n\nLearn to Sear and Panfry\n\nMake-Ahead: Oven-Roasted Pulled Pork\n\n## Beef and Veal\n\nBacon-Wrapped Herb-Roasted Beef Tenderloin\n\nBarbeque Beef Sandwiches Calorie Smart \u2022 Slow Cooker\n\nBarbeque Pot Roast\n\nBeef Rib Roast with Oven-Browned Potatoes\n\nBeef Short Ribs\n\nBeef Stroganoff Calorie Smart\n\nBeer-Braised Short Rib Dinner\n\nBrandy-Herb Pan Steaks Calorie Smart \u2022 Fast\n\nCaramelized Onion Pot Roast\n\nChicken-Fried Steak\n\nConfetti Pot Roast\n\nCream Gravy Pot Roast\n\nFrench Dip Sandwiches Calorie Smart \u2022 Slow Cooker\n\nGarlic-Herb Pot Roast\n\nGround Beef Stroganoff\n\nHerb and Garlic\u2013Crusted Beef Rib Roast Calorie Smart \u2022 Easy\n\nHerb-Roasted Beef Tenderloin Calorie Smart\n\nHorseradish Meat Loaf\n\nKorean Barbequed Beef Calorie Smart\n\nMeat Loaf Calorie Smart\n\nMini Meat Loaves\n\nMushroom Brandy-Herb Pan Steaks\n\nOsso Buco\n\nPeppered Flank Steak Calorie Smart\n\nPot Roast\n\nReuben or Rachel Sandwich\n\nRoast Beef with Orange and Thyme Calorie Smart\n\nRopa Vieja Toastadas Calorie Smart \u2022 Slow Cooker\n\nSloppy Joes Calorie Smart \u2022 Fast\n\nSpiced Corned Beef Brisket with Horseradish Sour Cream Calorie Smart \u2022 Slow Cooker\n\nVeal Parmigiana Calorie Smart\n\n## Pork\n\nChopped Salad with Pork\n\nCornbread and Bacon\u2013Stuffed Pork Chops\n\nCountry-Style Pork Ribs\n\nDijon-Herbed Pork Loin Roast\n\nGlazed Baked Ham Calorie Smart\n\nHam with Brown Sugar\u2013Orange Glaze\n\nHerbed Pork Loin Roast Calorie Smart\n\nItalian Pork Pot Roast Calorie Smart\n\nJerk Pulled Pork Sandwiches Slow Cooker \u2022 Easy\n\nLoaded Fries\n\nPanfried Pork Chops with Cider Sauce Calorie Smart \u2022 Fast\n\nPecan-Crusted Pork Medallions Fast\n\nPepper Garlic Pork Loin Roast\n\nPork Burrito Bowls\n\nPork and Coleslaw Sandwiches\n\nPork Chops with Mustard-Chive Cream Sauce\n\nPork Crown Roast with Fruited Stuffing Supreme\n\nPork Spareribs\n\nPork Tenderloins with Roasted Vegetables Calorie Smart\n\nPorketta Pot Roast Calorie Smart \u2022 Slow Cooker\n\nPulled Pork Nachos\n\nPulled Pork Pizza\n\nPulled Pork Quesadillas\n\nPumpkin Seed\u2013Crusted Pork Medallions\n\nSkillet Pork Hash\n\nSpicy Barbeque Pork Loin Ribs\n\nSpicy Southwestern Pork Chops\n\n## Lamb and Other Meats\n\nBalsamic Braised Lamb Shanks Calorie Smart\n\nChicken Parmigiana\n\nGreek Lamb Chops Calorie Smart\n\nHerb and Garlic Roast Leg of Lamb Calorie Smart\n\nVenison with Cranberry-Wine Sauce Calorie Smart\n\n## Sauces\n\nMolasses-Mustard Barbeque Sauce\n\nSweet-Savory Barbeque Sauce\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Beef, Pork & Lamb Basics\n\nChoosing the right cut of meat will be just a little easier with the knowledge you will pick up in this chapter. From photos identifying a variety of cuts to detailed cooking times and delicious recipes, here's all you need to select and prepare meat perfectly.\n\n## Tips for Purchasing\n\n  * Choose packages without any tears, holes or leaks. There should be little or no liquid in the bottom of the tray.\n  * Packages should be cold and feel firm. Avoid packages that are stacked too high in the meat case, as they may not have been kept cold enough.\n  * Pay attention to the sell-by date and use or freeze within 2 days of the date.\n  * Choose meat that has good color. Don't purchase any that is gray or off-color.\n  * Purchasing meat from a butcher can ensure you are getting exactly what you want in terms of cut, preparation and amount. Evaluate the cost vs. prepackaged; it may save you time, money and waste.\n  * Place packages of meat in plastic bags so that the juices will not contaminate other food in your grocery cart.\n  * Refrigerate meat as soon as you get home from shopping. For longer commutes from the grocery store or on hot days, place meat in a small cooler with ice packs.\n\n## What's on the Label?\n\n**Organic** : To be labeled organic, livestock must be raised in compliance with the National Organic Program. Regulations require that animals are able to graze in pastures, are fed with 100 percent organic feed and are not given antibiotics or hormones.\n\n**Grass-Fed vs. Grain-Fed** : Grass-fed livestock feeds from a pasture. Livestock may have been \"finished\" on a diet of grains and supplements to gain weight rapidly. Most beef, lamb and goat is grain-fed, but grass-fed meats are becoming more accessible.\n\n**Natural** : Many people think \"natural\" is the same as \"organic\"\u2014it is not. If a meat package is labeled natural, it means that it doesn't contain artificial ingredients or added color or is minimally processed. Read on\u2014the statement that follows the natural label will indicate what is natural about the product.\n\n## Servings Per Pound\n\nThe number of servings per pound varies depending on the type of meat and the amount of bone and fat present.\n\nType of Meat | Servings Per Pound\n\n---|---\n\nBoneless Cuts (ground, boneless chops, loin, tenderloin) | 3 to 4\n\nBone-In Cuts (rib roasts, pot roasts, country-style ribs) | 2 to 3\n\nVery Bony Cuts (back ribs, spareribs, short ribs, shanks) | 1 to 1\u00bd\n\n## Retail Beef Cuts\n\nA variety of cuts is available in the marketplace. The information and images here should help you choose the right cut for your recipe needs.\n\nDiagram: **1** Chuck **2** Rib **3** Loin **4** Sirloin **5** Round **6** Brisket **7** Shank **8** Plate **9** Flank\n\n## Retail Veal Cuts\n\nHere are some of the more commonly used cuts of veal that you might find called for in recipes.\n\n## Storing Meat\n\n  * Meat wrapped in butcher paper should be repackaged tightly in plastic wrap, foil or freezer plastic bags.\n  * Meat packaged in clear plastic wrap on a plastic or Styrofoam tray doesn't need to be repackaged.\n  * Store meat in the meat compartment or coldest part of your refrigerator, or freeze it as soon as possible. Ground meat is more perishable than other cuts, so use it within 2 days.\n  * Cook or freeze meat within 2 days of the sell-by date.\n  * If meat was purchased frozen or was frozen right after purchasing, keep it in the refrigerator after thawing for the number of days listed in the Timetable for Storing Meat, below. If the meat was refrigerated several days before freezing, use it the same day you thaw it.\n\n### Timetable for Storing Meat\n\nCut of Meat | Refrigerator (36\u00b0F to 40\u00b0F) | Freezer (0\u00b0F or Colder)\n\n---|---|---\n\nGround Meat\n\nBeef | 1 to 2 days | 3 to 4 months\n\nVeal | 1 to 2 days | 2 to 3 months\n\nPork | 1 to 2 days | 1 to 3 months\n\nSteaks and Roasts\n\nBeef | 3 to 4 days | 6 to 12 months\n\nVeal | 1 to 2 days | 6 to 9 months\n\nPork | 2 to 4 days | 3 to 6 months\n\nLamb | 3 to 5 days | 6 to 9 months\n\nCubes and Slices | 2 to 3 days | 6 to 12 months\n\nLeftover Cooked Meats | 3 to 4 days | 2 to 3 months\n\nCured, Smoked and Ready-to-Serve Meat Products\n\nCorned Beef | 1 week | 2 weeks\n\nHot Dogs | 3 to 5 days | 1 month\n\nBacon | 5 to 7 days | 1 month\n\n## Handling Raw Meat\n\nCooking meat to the recommended doneness destroys any bacteria. To avoid any cross-contamination when preparing raw meat for cooking, follow the tips in Don't Cross-Contaminate, page 31.\n\n## Cutting Raw Meat\n\nNeed to cut raw meat into cubes, thin slices or strips? Put it in the freezer first. Leave the meat in the freezer until it's firm but not frozen, 30 to 60 minutes, depending on the size of the piece. It'll be easy to cube or slice\u2014even paper thin.\n\nEasily cut raw meat into cubes, thin slices or strips by partially freezing it first.\n\n## Thawing Meat\n\nThaw meat slowly in the refrigerator in a dish or baking pan with sides or in a resealable food-storage plastic bag to catch any drips during thawing. Don't thaw meat on the countertop because bacteria thrive at room temperature.\n\nAmount of Frozen Meat | Thawing Time in Refrigerator*\n\n---|---\n\nLarge Roast (4 pounds or larger) | 4 to 7 hours per lb\n\nSmall Roast (under 4 pounds) | 3 to 5 hours per lb\n\nSteak or Chops (1 inch thick) | 12 to 14 hours total\n\nGround Beef or Beef Pieces (1- to 1\u00bd-pound package) | 24 hours total\n\nGround Beef Patties (\u00bd to \u00be inch thick) | 12 hours total\n\n*To thaw meat in the microwave, follow manufacturer's directions.\n\nBrandy-Herb Pan Steaks\n\n## Brandy-Herb Pan Steaks Calorie Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 teaspoon cracked or coarse ground black pepper\n  * \u00be teaspoon chopped fresh or \u00bc teaspoon dried basil leaves\n  * \u00be teaspoon chopped fresh or \u00bc teaspoon dried rosemary leaves\n  * \u00bd teaspoon coarse (kosher or sea) salt\n  * 4 boneless beef tenderloin steaks, 1 inch thick (1 to 1\u00bd lb)\n  * 1 tablespoon butter\n  * \u00bc cup beef broth (for homemade broth, see page 146)\n  * 2 tablespoons brandy or beef broth\n\n1 In small bowl, mix pepper, basil, rosemary and salt. Rub mixture into both sides of each steak.\n\n2 In 12-inch skillet, melt butter over medium heat. Cook steaks in butter 6 to 8 minutes, turning once, until medium-rare to medium (145\u00b0F to 160\u00b0F). Remove beef from skillet; cover to keep warm.\n\n3 Add broth and brandy to skillet. Heat to boiling, stirring to loosen browned bits from bottom of skillet; reduce heat. Simmer uncovered 3 to 4 minutes or until slightly thickened. Serve with steaks.\n\n1 Serving: Calories 200; Total Fat 5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 90mg; Sodium 110mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 37g Exchanges: 5 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\nMushroom Brandy-Herb Pan Steaks After removing steaks from skillet, add 1 teaspoon additional butter and 1 cup sliced fresh mushrooms. Cook 2 to 3 minutes, stirring frequently, until tender. Leave mushrooms in skillet and make sauce as directed in Step 3.\n\n### Making a Pan Sauce\n\nRemove cooked meat from skillet; add liquid, stirring to loosen browned bits from bottom of skillet.\n\nSimmer uncovered until mixture is slightly thickened.\n\n## Peppered Flank Steak Calorie Smart\n\nPrep 15 min Total 4 hr 20 min \u25cf 6 servings\n\nMarinade\n\n  * \u00bd cup olive or vegetable oil\n  * \u00bd cup dry red wine\n  * 1 tablespoon honey\n  * \u00bd teaspoon dried thyme leaves\n  * \u00bd teaspoon dried marjoram leaves\n  * 1 clove garlic, finely chopped\n\nSteak\n\n  * 1 beef flank steak (about 1\u00bc lb)\n  * \u00be teaspoon coarse sea salt\n  * 1 tablespoon coarsely ground mixed peppercorns or black peppercorns*\n  * \u00bd cup chopped fresh parsley\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all marinade ingredients. Add beef. Cover dish or seal bag; refrigerate at least 4 hours but no longer than 24 hours, turning occasionally.\n\n2 Set oven control to broil. Spray broiler pan rack with cooking spray. Remove beef from marinade; discard marinade. Sprinkle both sides of beef with salt; rub into surface. Place beef on rack in pan.\n\n3 Broil with top 4 to 6 inches from heat 5 minutes. Turn beef over; sprinkle with pepper. Broil 4 to 5 minutes longer or until desired doneness. Let stand 5 minutes. Cut beef across grain into thin slices. Sprinkle with parsley.\n\n*Look for jars of peppercorns with a grinder top in your supermarket. The mixed peppercorns usually include black, white and green peppercorns.\n\n1 Serving: Calories 190; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 70mg; Sodium 330mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 28g Exchanges: 4 Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nChicken-Fried Steak\n\n## Chicken-Fried Steak\n\nSome say it gets its name from the fact that the steak is cooked in oil that has already been used to fry chicken. Others say it's because it's the same method as frying chicken. Or, it may have evolved from the wiener schnitzel recipes German and Austrian immigrants brought with them from Europe to Texas in the 19th century.\n\nPrep 40 min Total 40 min \u25cf 4 servings\n\nBeef and Coatings\n\n  * 1 beef sirloin steak, about 1 inch thick (1\u00bc lb)*\n  * 1\u00bc teaspoons salt\n  * \u00bc cup all-purpose flour\n  * \u00bd teaspoon black pepper\n  * \u215b teaspoon ground red pepper (cayenne)\n  * 1 egg\n  * \u00bc cup milk\n  * 1 cup panko bread crumbs or regular bread crumbs\n  * \u00bc cup vegetable oil\n\nCreamy Gravy\n\n  * 1 tablespoon vegetable oil\n  * 2 tablespoons all-purpose flour\n  * 1 cup milk\n  * \u00bd cup beef broth\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon black pepper\n  * Mashed potatoes, if desired\n\n1 Heat oven to 225\u00b0F or warm setting. Place beef between 2 sheets of plastic wrap or waxed paper; pound with flat side of meat mallet or rolling pin until \u00bd inch thick. Sprinkle both sides with \u00bc teaspoon of the salt; rub into beef. Cut into 4 serving pieces.\n\n2 In shallow dish, mix \u00bc cup flour, remaining 1 teaspoon salt, \u00bd teaspoon black pepper and the red pepper. In another shallow dish, beat egg and \u00bc cup milk with fork. Place bread crumbs on sheet of waxed paper. Dip beef into flour mixture, coating well; shake off excess. Dip into egg mixture, then coat with bread crumbs.\n\n3 Line 15x10x1-inch pan with foil. In 12-inch nonstick skillet, heat \u00bc cup oil over medium heat. Cook beef in oil 12 to 14 minutes, turning once, until golden brown. Remove from skillet to pan; place in oven to keep warm.\n\n4 Add 1 tablespoon oil and 2 tablespoons flour to skillet, scraping up browned bits. Cook and stir until flour mixture turns light golden brown. Gradually stir in 1 cup milk and the broth, stirring constantly until well blended. Add \u00bc teaspoon salt and \u00bc teaspoon pepper. Heat to boiling, stirring constantly. Cook 1 minute, stirring frequently. (If thinner gravy is desired, add additional beef broth, milk or water.) Serve gravy with steaks and mashed potatoes.\n\n*Four beef cube steaks or eye of round steaks can be substituted for the sirloin. Cube steaks are pre-tenderized and pounded, so do not pound them. Pound the eye of round steaks only if thicker than \u00bd inch.\n\n1 Serving (1 Steak and \u00bc Cup Gravy): Calories 560; Total Fat 28g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 145mg; Sodium 1140mg; Total Carbohydrate 32g (Dietary Fiber 0g, Sugars 4g); Protein 44g Exchanges: \u00bd Starch, 1\u00bd Other Carbohydrate, 6 Lean Meat, 2 Fat Carbohydrate Choices: 2\n\n### Making Chicken-Fried Steak\n\nPlace beef between plastic wrap or waxed paper. Pound with flat side of meat mallet or rolling pin.\n\nCoat beef with flour mixture, then dip into egg mixture and coat with bread crumbs.\n\nVeal Parmigiana\n\n## Veal Parmigiana Calorie Smart\n\nPrep 20 min Total 45 min \u25cf 6 servings\n\n  * 1 egg\n  * 2 tablespoons water\n  * \u2154 cup unseasoned dry bread crumbs\n  * \u2153 cup grated Parmesan cheese\n  * 1\u00bd lb veal leg cutlets, \u00bc inch thick*\n  * \u00bc cup olive or vegetable oil\n  * 1 cup tomato pasta sauce (for homemade Italian tomato pasta sauce, see page 174)\n  * 2 cups shredded mozzarella cheese (8 oz)\n\n1 Heat oven to 350\u00b0F. In small bowl, beat egg and water. In shallow dish, mix bread crumbs and Parmesan cheese. Dip veal into egg mixture, then coat with bread crumb mixture.\n\n2 In 12-inch skillet, heat 2 tablespoons of the oil over medium heat. Cook half of the veal in oil about 5 minutes, turning once, until light brown; drain. Repeat with remaining veal and 2 tablespoons oil.\n\n3 In ungreased 13x9-inch (3-quart) glass baking dish, arrange veal, overlapping slices slightly. Spoon pasta sauce over veal. Sprinkle with mozzarella cheese.\n\n4 Bake uncovered about 25 minutes or until sauce is bubbly and cheese is light brown.\n\n*1\u00bd lb veal round steak can be substituted for the veal leg cutlets. Cut veal into 12 pieces. Place between pieces of plastic wrap or waxed paper. Pound with flat side of meat mallet until \u00bc inch thick.\n\n1 Serving: Calories 340; Total Fat 20g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 120mg; Sodium 500mg; Total Carbohydrate 15g (Dietary Fiber 1g, Sugars 5g); Protein 26g Exchanges: 1 Other Carbohydrate, 4 Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\nChicken Parmigiana Substitute 6 boneless skinless chicken breasts (about 1\u00bd lb) for the veal. Place each chicken breast, smooth side down, between pieces of plastic wrap or waxed paper. Pound with flat side of meat mallet until \u00bc inch thick. In Step 2, cook chicken until no longer pink in center.\n\nKorean Barbecued Beef\n\n## Korean Barbecued Beef Calorie Smart\n\nBulgogi, or marinated beef, is one of the most popular dishes in Korea, with its distinctive sweet-salty flavor. If you're adventurous, add a side of kimchi, a super-spicy cabbage condiment you can find in many supermarkets. (Or, to make your own kimchi, see page 443.)\n\nPrep 10 min Total 40 min \u25cf 4 servings\n\n  * 1 lb boneless beef top loin or sirloin steak\n  * \u00bc cup soy sauce\n  * 3 tablespoons sugar\n  * 2 tablespoons sesame or vegetable oil\n  * \u00bc teaspoon pepper\n  * 3 medium green onions, finely chopped (3 tablespoons)\n  * 2 cloves garlic, chopped\n  * Hot cooked rice (page 215), if desired\n\n1 Cut beef diagonally across grain into \u215b-inch slices (beef is easier to cut if partially frozen). In medium glass or plastic bowl, mix remaining ingredients except rice. Add beef; stir until well coated. Cover and refrigerate 30 minutes.\n\n2 Drain beef; discard marinade. Heat 10-inch skillet over medium heat. Add beef; cook 2 to 3 minutes, stirring frequently, until brown. Serve with rice.\n\n1 Serving: Calories 280; Total Fat 11g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 80mg; Sodium 940mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 10g); Protein 33g Exchanges: 1 Other Carbohydrate, 4\u00bd Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\nHerb and Garlic\u2013Crusted Beef Rib Roast\n\n## Herb and Garlic\u2013Crusted Beef Rib Roast Calorie Smart \u2022 Easy\n\nWhen buying the roast, look for it under several names: beef rib roast, standing rib roast or prime rib roast.\n\nPrep 10 min Total 3 hr \u25cf 8 servings\n\n  * 1 beef rib roast, small end (4 to 6 lb)\n  * 2 teaspoons dried basil leaves\n  * 2 teaspoons dried thyme leaves\n  * 1 teaspoon coarse (kosher or sea) salt\n  * 1 teaspoon garlic powder\n  * \u00bc teaspoon coarse ground black pepper\n\n1 Heat oven to 350\u00b0F. Line shallow roasting pan with foil. Place beef, fat side up, in pan.\n\n2 In small bowl, mix basil, thyme, salt, garlic powder and pepper; press mixture onto all surfaces of beef. Insert ovenproof meat thermometer so tip is in thickest part of beef and does not rest in fat or touch bone.\n\n3 For medium-rare, roast uncovered 1 hour 45 minutes to 2 hours 15 minutes or until thermometer reads 135\u00b0F. Cover loosely with foil; let stand 15 to 20 minutes until thermometer reads 145\u00b0F. For medium, roast 2 hours 15 minutes to 2 hours 45 minutes or until thermometer reads 150\u00b0F. Cover loosely with foil; let stand 15 to 20 minutes or until thermometer reads 160\u00b0F.\n\n4 Remove beef from pan to cutting board; cut into slices. Serve with pan drippings, if desired.\n\n1 Serving: Calories 340; Total Fat 17g (Saturated Fat 7g, Trans Fat 0.5g); Cholesterol 140mg; Sodium 390mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 2g); Protein 32g Exchanges: 1 Starch, 4 Lean Meat, 1 Fat Carbohydrate Choices: 1\n\nBeef Rib Roast with Oven-Browned Potatoes About 1 hour 30 minutes before beef is done, prepare and boil 8 medium potatoes as directed on page 237. For decorative potatoes, make crosswise cuts almost through whole potatoes to make thin slices and reduce boiling time to 10 minutes. Place potatoes in beef drippings in pan, turning to coat completely; or brush potatoes with melted butter and place on rack with beef. Continue cooking about 1 hour 15 minutes, turning potatoes once, until golden brown. Season with salt and pepper, if desired.\n\n## Roast Beef with Orange and Thyme Calorie Smart\n\nPrep 15 min Total 2 hr \u25cf 8 servings\n\n  * 1 boneless beef sirloin tip roast (2 to 2\u00bd lb)\n  * \u00bd cup orange marmalade\n  * 1 tablespoon olive or vegetable oil\n  * 2 teaspoons Sriracha sauce\n  * 1 tablespoon dried thyme leaves\n  * 1\u00bd teaspoons coarse kosher or sea salt\n  * \u00bd teaspoon coarse ground black pepper\n  * 8 small red potatoes (about 1\u00bd lb), cut in half\n  * 1 lb fresh asparagus spears, trimmed\n\n1 Heat oven to 325\u00b0F. Line 15x10x1-inch pan with heavy-duty foil; spray foil with cooking spray. Place beef in pan. Insert ovenproof meat thermometer so tip is in thickest part of beef.\n\n2 In small bowl, mix marmalade, oil and Sriracha sauce; reserve 3 tablespoons. Generously brush some of the remaining marmalade mixture over beef. Roast uncovered 40 minutes, brushing beef occasionally with marmalade mixture.\n\n3 In another small bowl, mix thyme, salt and pepper. Sprinkle top and sides of beef with 2 teaspoons of the thyme mixture. Place potatoes around beef. Generously brush potatoes with marmalade mixture; sprinkle with half of the remaining thyme mixture.\n\n4 Roast 45 to 55 minutes longer or until meat thermometer reads 135\u00b0F for medium-rare (150\u00b0F for medium). Remove beef to cutting board; cover loosely with foil. Let stand about 10 minutes or until thermometer reads 145\u00b0F (160\u00b0F for medium).\n\n5 Increase oven temperature to 425\u00b0F. Arrange asparagus in single layer in pan with potatoes. Brush asparagus with remaining reserved marmalade mixture; sprinkle with remaining thyme mixture. Roast 7 to 10 minutes or until asparagus is crisp-tender.\n\n6 Cut beef across grain into thin slices; arrange on serving platter. Add potatoes and asparagus to platter. Spoon pan drippings over beef.\n\n1 Serving: Calories 310; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 70mg; Sodium 290mg; Total Carbohydrate 31g (Dietary Fiber 3g, Sugars 12g); Protein 32g Exchanges: 1\u00bd Starch, 2 Vegetable, 2 Very Lean Meat, 1\u00bd Lean Meat Carbohydrate Choices: 2\n\n### Carving a Beef Rib Roast\n\nWith roast on cutting board, carefully cut bone portion away from meat.\n\nTurn roast bone side down. Cut slices evenly and toward bone side of roast.\n\nHerb-Roasted Beef Tenderloin\n\n## Herb-Roasted Beef Tenderloin Calorie Smart\n\nPrep 15 min Total 1 hr 20 min \u25cf 6 servings\n\n  * 1 beef tenderloin (about 2\u00bd lb)\n  * 1 tablespoon olive or vegetable oil\n  * \u00bd teaspoon coarse ground black pepper\n  * \u00bd teaspoon dried marjoram leaves\n  * \u00bc teaspoon coarse (kosher or sea) salt\n\n1 Heat oven to 425\u00b0F. Turn small end of beef under about 6 inches. Tie beef with kitchen string at about 1\u00bd-inch intervals. Place in shallow roasting pan. Brush with oil. Sprinkle with pepper, marjoram and salt. Insert ovenproof meat thermometer so tip is in thickest part of beef.\n\n2 For medium-rare, roast 35 to 40 minutes or until thermometer reads 135\u00b0F. Cover loosely with foil; let stand 15 to 20 minutes until thermometer reads 145\u00b0F. For medium, roast uncovered 45 to 50 minutes or until thermometer reads 150\u00b0F. Cover loosely with foil; let stand 10 to 15 minutes until thermometer reads 160\u00b0F. Remove string from beef before slicing.\n\n1 Serving: Calories 300; Total Fat 16g (Saturated Fat 5g, Trans Fat 0.5g); Cholesterol 110mg; Sodium 190mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 41g Exchanges: 5\u00bd Lean Meat Carbohydrate Choices: 0\n\nBacon-Wrapped Herb-Roasted Beef Tenderloin After sprinkling beef with pepper, marjoram and salt, place 6 slices bacon (don't use thick-sliced) over top of beef, tucking ends under bottom. Continue as directed.\n\n### Tying a Whole Beef Tenderloin\n\nTo make tenderloin same thickness, turn small end of beef under about 6 inches.\n\nTie beef with kitchen string at about 1\u00bd-inch intervals.\n\nCut string from cooked beef with kitchen scissors and remove before carving.\n\nBeef Stroganoff\n\n## Beef Stroganoff Calorie Smart\n\nPrep 25 min Total 50 min \u25cf 6 servings\n\n  * 1\u00bd lb beef tenderloin or boneless top loin steak\n  * 2 tablespoons butter\n  * 1\u00bd cups beef broth (for homemade broth, see page 146)\n  * 2 tablespoons ketchup\n  * 1 teaspoon salt\n  * 1 clove garlic, finely chopped\n  * 1 package (8 oz) sliced fresh mushrooms (about 3 cups)\n  * 1 medium onion, chopped (\u00bd cup)\n  * \u00bc cup all-purpose flour\n  * 1 cup sour cream or plain yogurt\n  * Hot cooked noodles (page 167) or rice (page 215), if desired\n\n1 Cut beef across grain into 1\u00bdx\u00bd-inch strips (beef is easier to cut if partially frozen). In 12-inch skillet, melt butter over medium-high heat. Cook beef in butter, stirring occasionally, until brown.\n\n2 Reserve \u2153 cup of the broth. Stir remaining broth, the ketchup, salt and garlic into beef. Heat to boiling; reduce heat. Cover and simmer about 10 minutes or until beef is tender.\n\n3 Stir in mushrooms and onion. Heat to boiling; reduce heat. Cover and simmer about 5 minutes longer or until onion is tender.\n\n4 In tightly covered container, shake reserved \u2153 cup broth and the flour until mixed; grad-ually stir into beef mixture. Heat to boiling, stirring constantly. Boil and stir 1 minute; reduce heat to low. Stir in sour cream; heat until hot. Serve over noodles.\n\n1 Serving: Calories 330; Total Fat 20g (Saturated Fat 10g, Trans Fat 1g); Cholesterol 100mg; Sodium 810mg; Total Carbohydrate 10g (Dietary Fiber 1g, Sugars 4g); Protein 28g Exchanges: 2 Vegetable, 3\u00bd Lean Meat, 1 Fat Carbohydrate Choices: \u00bd\n\nGround Beef Stroganoff Substitute 1 lb lean (at least 90%) ground beef for the tenderloin; omit butter. In 12-inch skillet, cook beef over medium heat, stirring occasionally, until thoroughly cooked; drain. Continue as directed.\n\n## Spiced Corned Beef Brisket with Horseradish Sour Cream Calorie Smart \u2022 Slow Cooker\n\nPrep 10 min Total 8 hr 10 min \u25cf 8 servings\n\n  * 1 corned beef brisket (3 to 3\u00bd lb)\n  * 1 large sweet onion (Bermuda, Maui or Spanish), sliced\n  * \u00be teaspoon crushed red pepper flakes\n  * 1 cup reduced-sodium chicken broth\n  * 1 tablespoon Worcestershire sauce\n  * \u00bd cup sour cream\n  * 1 tablespoon cream-style prepared horseradish\n  * 2 tablespoons chopped fresh parsley\n\n1 Remove beef from package; discard liquid and seasoning packet. Trim fat from beef and thoroughly rinse.\n\n2 Spray 5- to 6-quart slow cooker with cooking spray. Place onion in slow cooker. Top with beef; sprinkle with pepper flakes. In small bowl, mix broth and Worcestershire sauce; pour over beef.\n\n3 Cover; cook on Low heat setting 8 to 9 hours or until beef is tender.\n\n4 In small bowl, stir sour cream, horseradish and parsley until well mixed. Slice beef across the grain; serve with horseradish sour cream.\n\n1 Serving: Calories 340; Total Fat 26g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 125mg; Sodium 1480mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 2g); Protein 23g Exchanges: 3\u00bd High-Fat Meat Carbohydrate Choices: 0\n\n## Reuben or Rachel Sandwich\n\nFor each Reuben sandwich, spread 1 tablespoon Thousand Island dressing on 1 slice marble rye or pumpernickel bread. Top with 1 slice Swiss cheese, \u00bc cup sauerkraut, 2 ounces sliced corned beef and another bread slice. Spread outsides of bread with 1 to 2 tablespoons butter. Cook in covered skillet over low heat 8 to 10 minutes or until toasted and hot. Turn halfway through cooking. For a Rachel sandwich, substitute turkey or pastrami for the corned beef.\n\nBeer-Braised Short Rib Dinner\n\n## Beer-Braised Short Rib Dinner\n\nPrep 15 min Total 8 hr 15 min \u25cf 4 servings\n\n  * 4 beef short ribs (about 3\u00bc lb), cut in half\n  * 2 tablespoons applewood rub for meat\n  * 1 can (6 oz) tomato paste\n  * 1 bottle (12 oz) pale ale\n  * 6 medium carrots\n  * 1 medium leek, finely chopped\n  * 3 cloves garlic, finely chopped\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 3 medium Yukon Gold potatoes (about 1\u00bc lb), cut lengthwise into sixths\n  * Chopped fresh parsley, if desired\n\n1 Line 15x10x1-inch pan with heavy-duty foil; spray foil with cooking spray. Rub ribs on all sides with applewood rub. Place ribs in pan.\n\n2 In medium bowl, stir together tomato paste and ale with whisk until smooth. Finely chop 1 of the carrots. Stir chopped carrot, leek, garlic, salt and pepper into tomato mixture. Pour over ribs; turn to coat. Cover pan tightly with heavy-duty foil. Refrigerate at least 6 hours or up to 24 hours to marinate, turning ribs occasionally.\n\n3 Heat oven to 400\u00b0F. Cut remaining 5 carrots into 2-inch pieces and place in pan with ribs, turning to coat. Arrange ribs, fat sides up, in marinade. Bake uncovered 1 hour.\n\n4 Reduce oven temperature to 350\u00b0F. Turn ribs on side; add potatoes to pan, turning in marinade to coat. Bake 45 to 60 minutes longer, occasionally turning ribs, until ribs and potatoes are tender. (If too much ale mixture evaporates, add a little water to pan.) Sprinkle with parsley.\n\n1 Serving: Calories 1090; Total Fat 78g (Saturated Fat 33g, Trans Fat 3.5g); Cholesterol 170mg; Sodium 1360mg; Total Carbohydrate 51g (Dietary Fiber 7g, Sugars 13g); Protein 46g Exchanges: 3 Starch, 1 Vegetable, 5 High-Fat Meat, 7 Fat Carbohydrate Choices: 3\u00bd\n\nOsso Buco\n\n## Osso Buco\n\nThese braised veal shanks, featuring white wine, beef broth and a touch of lemon peel, are very tender. Instead of serving with mashed potatoes or spaghetti, try Classic Risotto (make 1\u00bd times the recipe, page 219).\n\nPrep 40 min Total 2 hr 40 min \u25cf 6 servings\n\n  * 6 veal or beef shank crosscuts, 2 to 2\u00bd inches thick (3 to 3\u00bd lb)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc cup all-purpose flour\n  * 2 tablespoons olive or vegetable oil\n  * \u2153 cup dry white wine or apple juice\n  * 1 can (10\u00bd oz) condensed beef broth\n  * 1 clove garlic, finely chopped\n  * 1 dried bay leaf\n  * 2 tablespoons chopped fresh parsley\n  * 1 teaspoon grated lemon peel\n  * Shredded Parmesan or Asiago cheese, if desired\n  * Hot cooked mashed potatoes (page 247) or spaghetti (page 167)\n\n1 Sprinkle veal with salt and pepper; coat with flour. In 4-quart Dutch oven, heat oil over medium heat. Cook veal in oil about 20 minutes, turning occasionally, until brown on all sides.\n\n2 Stir in wine, broth, garlic and bay leaf. Heat to boiling; reduce heat. Cover and simmer 1 hour 30 minutes to 2 hours or until veal is tender.\n\n3 Remove veal to serving platter. Skim fat from broth; remove and discard bay leaf. Pour broth over veal; sprinkle with parsley, lemon peel and cheese. Serve with mashed potatoes.\n\n1 Serving: Calories 700; Total Fat 30g (Saturated Fat 9g, Trans Fat 4g); Cholesterol 255mg; Sodium 1090mg; Total Carbohydrate 41g (Dietary Fiber 4g, Sugars 3g); Protein 68g Exchanges: 2\u00bd Starch, 7 Lean Meat, 1 Fat Carbohydrate Choices: 3\nGlobal Flavors\n\n## South America\n\nThe diverse cuisines of South America are as different as the countries and regions found there. Influences from Africa and European countries have created a mash-up of indigenous ingredients paired with external cuisines.\n\nCarne Seca Brazilian salted dried beef, similar to beef jerky.\n\nDried Shrimp\/Prawns Popular in Brazilian cooking and also in other parts of South America, dried shrimp have a strong, fishy taste. Used in small amounts to add flavor to stews, spicy sauces and other dishes; reconstituted whole, chopped or ground. The liquid used to soak them may also be added for flavor.\n\nMat\u00e9 The leaves and young twigs of the mat\u00e9 tree, dried, shredded and aged in cedar containers before being sold to make the popular tea-like, energy-boosting drink.\n\nPalm Oil Distinctively flavored reddish-orange oil, integral to Brazilian cooking. Also known as dende oil, it's extracted from the fruit of the African palm, It's not to be confused with palm kernel oil, which is used to make margarine and cosmetics.\n\nPlantains Large, firm cooking bananas, usually cooked when green and used like potatoes in vegetable side dishes. When green, they have a mild, squash-like flavor; when allowed to ripen and turn yellow, the flavor becomes slightly sweet, with a soft, spongy texture when cooked.\n\nSeasonings **Annatto** is a derivative of achiote seed used to add color and flavor to Latin American dishes. **Aj\u00ed Amarillo Paste** is made from aj\u00ed amarillo hot pepper, a Peruvian staple made almost daily for use in most recipes. **Black Mint Paste (Huacatay)** is made from the pungent herb, adding a unique flavor to Peruvian meals.\n\nYerba Mat\u00e9 A tea-like drink made by steeping the dried, aged leaves and young twigs of the mat\u00e9 tree in hot water. South Americans have been sipping this energy-boosting earthy, herbal beverage for centuries, but its popularity is now spreading globally. Bottled varieties (frequently sweetened) and the dried \"tea\" are available in Latin markets.\n\nYuca\/Yuca Root Essential tuber in South American cuisine. Also called cassava or manioc. Sold with a waxy coating to protect its skin from bruising, yuca is traditionally boiled, roasted or fried in thin chips. It can be ground into flour or tapioca to thicken dishes.\n\n## Ropa Vieja Tostadas Calorie Smart \u2022 Slow Cooker\n\nThis mildly seasoned hearty meat filling is great in tacos, enchiladas, burritos, on taco salads and served over hot cooked rice.\n\nPrep 20 min Total 6 hr 35 min \u25cf 12 servings\n\n  * 1\u00bd lb beef skirt steak\n  * 1 dried bay leaf\n  * 1\u00bd teaspoons ground coriander\n  * 1 teaspoon ground cumin\n  * \u00bd teaspoon dried oregano leaves\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cup chopped red onion\n  * 1 cup chopped green bell pepper\n  * 3 cloves garlic, finely chopped\n  * \u00bd cup pitted green olives, sliced\n  * 1 tablespoon capers\n  * 1 can (14.5 oz) fire-roasted diced tomatoes, undrained\n  * 1 can (8 oz) tomato sauce\n  * 1 box (4.5 oz) tostada shells (12 shells)\n  * 3 ripe avocados, pitted, peeled and sliced\n  * 2 cups crumbled queso fresco cheese (8 oz)\n\n1 Spray 6-quart slow cooker with cooking spray. In slow cooker, place all ingredients except tostada shells, avocados and cheese. Cover; cook on Low heat setting 6 hours or until beef is tender.\n\n2 Remove beef from slow cooker to plate or cutting board; pull into shreds with 2 forks. Remove and discard bay leaf. Return beef to slow cooker; stir to combine with liquid.\n\n3 Heat tostada shells as directed on box. Spoon beef mixture onto shells; top with avocados and cheese.\n\n1 Serving (1 Tostada and \u00bd Cup Meat): Calories 300; Total Fat 19g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 50mg; Sodium 620mg; Total Carbohydrate 15g (Dietary Fiber 4g, Sugars 2g); Protein 16g Exchanges: 1 Starch, 2 Lean Meat, 2\u00bd Fat Carbohydrate Choices: 1\n\n# Heirloom Recipe and New Twist\n\nIf you like a very tender, tasty beef roast, this tried-and-true recipe is the one to use. There are slow cooker directions too, for those days when you are busy but want a great dinner at the end of the day. We wanted something different for the new twist and created the Italian Pork Pot Roast. This is another great recipe with tender meat and long-cooking goodness. Either recipe is elegant enough to serve to family or for entertaining anytime.\n\nHeirloom\n\nPot Roast\n\n## Pot Roast\n\nSpreading a layer of horseradish all over the outside of the meat is the secret to making this pot roast. Contrary to what you might think, the horseradish doesn't add a hot or spicy flavor. Instead, it mellows during cooking, leaving behind a delicious flavor.\n\nPrep 30 min Total 4 hr \u25cf 8 servings\n\n  * 1 boneless beef chuck, arm, shoulder or blade pot roast (4 lb)*\n  * 1 teaspoon salt\n  * 1 teaspoon pepper\n  * 1 jar (8 oz) prepared horseradish\n  * 1 cup water\n  * 8 small potatoes, cut in half\n  * 8 medium carrots or parsnips, peeled, cut into quarters\n  * 8 small whole onions, peeled\n  * \u00bd cup cold water\n  * \u00bc cup all-purpose flour\n\n1 In 4-quart Dutch oven, cook beef over medium heat until brown on all sides; reduce heat to low.\n\n2 Sprinkle beef with salt and pepper. Spread horseradish over all sides of beef. Add 1 cup water to Dutch oven. Heat to boiling; reduce heat. Cover and simmer 2 hours 30 minutes.\n\n3 Add potatoes, carrots and onions to Dutch oven. Cover and simmer about 1 hour longer or until beef and vegetables are tender.\n\n4 Remove beef and vegetables to warm platter; cover to keep warm. Skim excess fat from broth in Dutch oven. Add enough water to broth to measure 2 cups. In tightly covered container, shake \u00bd cup cold water and the flour; gradually stir into broth. Heat to boiling, stirring constantly. Boil and stir 1 minute. Serve gravy with beef and vegetables.\n\n*A 3-lb beef bottom round, rolled rump, tip or chuck eye roast can also be used; reduce salt to \u00be teaspoon.\n\n1 Serving: Calories 590; Total Fat 24g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 145mg; Sodium 560mg; Total Carbohydrate 47g (Dietary Fiber 7g, Sugars 9g); Protein 46g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 1\u00bd Vegetable, 5\u00bd Lean Meat, 1\u00bd Fat Carbohydrate Choices: 3\n\nSlow-Cooker Directions In 12-inch skillet, cook beef over medium heat until brown on all sides. Spray 4- to 6-quart slow cooker with cooking spray. In slow cooker, place potatoes, carrots and onions. Place beef on vegetables. In small bowl, mix horseradish, salt and pepper; spread evenly over beef. Add 1 cup water. Cover; cook on Low heat setting 8 to 10 hours or until beef and vegetables are tender.\n\nBarbecue Pot Roast Reduce pepper to \u00bd teaspoon. Omit horseradish and water. Make Smoky Barbecue Sauce (page 420). After browning the beef in Step 1, pour barbecue sauce over beef. Omit \u00bd cup cold water and the flour. After removing beef and vegetables in Step 4, skim fat from sauce. Serve sauce with beef and vegetables.\n\nCaramelized Onion Pot Roast Omit horseradish, potatoes, carrots, water and flour. Substitute 6 medium onions, sliced, for the small whole onions. Cook beef as directed; sprinkle with salt and pepper. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. Place onions in slow cooker; top with beef. In small bowl, mix 1\u00bd cups beef broth, \u00be cup beer, 2 tablespoons brown sugar, 3 tablespoons Dijon mustard and 2 tablespoons cider vinegar. Cover; cook on Low heat setting 8 to 10 hours or until beef is tender. Remove beef from cooker and slice. Skim fat from juices in cooker; serve juices with beef.\n\nConfetti Pot Roast Omit horseradish and potatoes. Sprinkle beef with \u00bd teaspoon Italian seasoning. Add \u00bd lb fresh green beans and 1 cup frozen corn with the carrots in Step 3. Continue as directed. Serve with mashed potatoes.\n\nCream Gravy Pot Roast Substitute 1 can (10\u00bd oz) condensed beef broth for the 1 cup water. For the gravy, add enough half-and-half or milk, instead of water, to the cooking liquid to measure 2 cups. Substitute \u00bd cup half-and-half or milk for the \u00bd cup cold water.\n\nGarlic-Herb Pot Roast Reduce pepper to \u00bd teaspoon. Omit horseradish. After browning the beef in Step 1, sprinkle with 1 tablespoon chopped fresh or 1 teaspoon dried marjoram leaves, 1 tablespoon chopped fresh or 1 teaspoon dried thyme leaves, 2 teaspoons chopped fresh or \u00bd teaspoon dried oregano leaves and 4 cloves garlic, finely chopped. Substitute 1 can (10\u00bd oz) condensed beef broth for the 1 cup water.\nNew Twist\n\nItalian Pork Pot Roast\n\n## Italian Pork Pot Roast Calorie Smart\n\nPrep 25 min Total 3 hr 30 min \u25cf 8 servings\n\nItalian Seasoning Rub\n\n  * 1 tablespoon fennel seed, crushed*\n  * 1 tablespoon Italian seasoning\n  * 1\u00bd teaspoons salt\n  * 1 teaspoon celery seed\n  * \u00bd teaspoon pepper\n\nPork and Vegetables\n\n  * 4 teaspoons olive oil\n  * 1 boneless pork shoulder roast (2\u00bd to 3 lb)\n  * \u00bd cup dry white wine\n  * 8 small red potatoes, cut in half\n  * 2 medium bulbs fennel, cut into \u00bc-inch slices**\n  * 1 large red bell pepper, cut into 8 pieces\n  * 1 large orange bell pepper, cut into 8 pieces\n  * 1 can (14 oz) artichoke hearts, drained, cut in half\n  * \u00bd cup shredded Parmesan cheese (2 oz)\n\n1 Heat oven to 325\u00b0F. In small bowl, mix all seasoning rub ingredients. Brush 2 teaspoons of the oil over pork. Rub 2 tablespoons of the seasoning rub over all surfaces of pork.\n\n2 Pour wine in shallow roasting pan. Place pork in pan. In medium bowl, toss potatoes and fennel with remaining 2 teaspoons oil; arrange around pork. Sprinkle vegetables with remaining seasoning rub.\n\n3 Cover and bake 2 hours. Arrange peppers and artichokes around pork. Cover and bake 1 hour longer or until pork is fork-tender. Sprinkle vegetables with cheese. Cover with foil; let stand 5 minutes or until cheese is melted. Serve pork and vegetables with pan juices.\n\n*To crush fennel seed, place in small resealable food-storage plastic bag and crush with rolling pin or meat mallet.\n\n**Save dill-like tops of fennel bulbs to use as a garnish.\n\n1 Serving: Calories 440; Total Fat 22g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 95mg; Sodium 740mg; Total Carbohydrate 23g (Dietary Fiber 7g, Sugars 6g); Protein 36g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, 1\u00bd Vegetable, 2\u00bd Very Lean Meat, 2 Lean Meat, 3 Fat Carbohydrate Choices: 1\u00bd\n\nMeat Loaf\n\n## Meat Loaf Calorie Smart\n\nPrep 20 min Total 1 hr 40 min \u25cf 6 servings\n\n  * 1\u00bd lb lean (at least 80%) ground beef\n  * 1 cup milk\n  * 1 tablespoon Worcestershire sauce\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried sage leaves\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground mustard\n  * \u00bc teaspoon pepper\n  * 1 clove garlic, finely chopped, or \u215b teaspoon garlic powder\n  * 1 egg\n  * 3 slices bread, torn into small pieces*\n  * 1 small onion, finely chopped (\u2153 cup)\n  * \u00bd cup ketchup, chili sauce or barbecue sauce\n\n1 Heat oven to 350\u00b0F. In large bowl, mix all ingredients except ketchup. Spread mixture in ungreased 8x4- or 9x5-inch loaf pan, or shape into 9x5-inch loaf in ungreased 13x9-inch pan. Spread ketchup over top.\n\n2 Insert ovenproof meat thermometer so tip is in center of loaf. Bake uncovered 1 hour to 1 hour 15 minutes or until beef is no longer pink in center and thermometer reads 160\u00b0F; drain. Let stand 5 minutes; remove from pan.\n\n*\u00bd cup dry bread crumbs or \u00be cup quick-cooking oats can be substituted for the bread.\n\n1 Serving: Calories 290; Total Fat 15g (Saturated Fat 6g, Trans Fat 1g); Cholesterol 110mg; Sodium 610mg; Total Carbohydrate 15g (Dietary Fiber 0g, Sugars 8g); Protein 24g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, 3 Medium-Fat Meat Carbohydrate Choices: 1\n\nLighter Directions For 7 grams of fat and 240 calories per serving, substitute ground turkey breast for the ground beef and \u00bc cup fat-free egg product for the egg. Use fat-free (skim) milk. Bake as directed until turkey is no longer pink in center and thermometer reads at least 165\u00b0F.\n\nMini Meat Loaves Spray 12 regular-size muffin cups with cooking spray. Divide beef mixture evenly among cups (cups will be very full). Brush tops with about \u00bc cup ketchup. Place muffin pan on cookie sheet. Bake about 30 minutes or until loaves are no longer pink in center and thermometer reads 160\u00b0F when inserted in center of loaves in middle of muffin pan (outer loaves will be done sooner). Immediately remove from cups.\n\nHorseradish Meat Loaf Omit sage. Stir in 1 to 2 tablespoons cream-style prepared horseradish in Step 1.\n\n## Sloppy Joes Calorie Smart \u2022 Fast\n\nPrep 30 min Total 30 min \u25cf 6 sandwiches\n\n  * 1 lb lean (at least 80%) ground beef\n  * 1 medium onion, chopped (\u00bd cup)\n  * \u00bc cup chopped celery\n  * 1 cup ketchup\n  * 1 tablespoon Worcestershire sauce\n  * 1 teaspoon ground mustard\n  * \u215b teaspoon pepper\n  * 6 burger buns, split\n\n1 In 10-inch skillet, cook beef, onion and celery over medium heat 8 to 10 minutes, stirring occasionally, until beef is thoroughly cooked; drain.\n\n2 Stir in ketchup, Worcestershire sauce, mustard and pepper. Heat to boiling; reduce heat. Simmer uncovered 10 to 15 minutes, stirring occasionally, until hot and bubbly. Spoon into buns.\n\n1 Sandwich: Calories 320; Total Fat 11g (Saturated Fat 4g, Trans Fat 1.5g); Cholesterol 50mg; Sodium 630mg; Total Carbohydrate 35g (Dietary Fiber 1g, Sugars 12g); Protein 18g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, 2 Medium-Fat Meat Carbohydrate Choices: 2\n\nLighter Directions For 3 grams of fat and 235 calories per serving, substitute ground turkey breast for the ground beef; spray skillet with cooking spray before heating.\n\nBarbecue Beef Sandwiches\n\n## Barbecue Beef Sandwiches Calorie Smart \u2022 slow Cooker\n\nPrep 15 min Total 7 hr 35 min \u25cf 12 sandwiches\n\n  * 1 boneless beef chuck roast (3 lb), trimmed of excess fat\n  * 1 cup barbecue sauce (for homemade sauce, see page 411)\n  * \u00bd cup apricot or peach preserves\n  * 1 small onion, sliced\n  * \u2153 cup chopped green bell pepper\n  * 1 tablespoon Dijon mustard\n  * 2 teaspoons packed brown sugar\n  * 12 kaiser rolls or burger buns, split\n\n1 Spray 4- to 5-quart slow cooker with cooking spray. Cut beef into 4 pieces; place in slow cooker. In medium bowl, mix remaining ingredients except rolls; pour over beef.\n\n2 Cover; cook on Low heat setting 7 to 8 hours or until beef is tender.\n\n3 Remove beef from slow cooker to cutting board; cut into thin slices. Return to slow cooker. Cover; cook 20 to 30 minutes longer or until hot. Spoon beef mixture into buns. If desired, serve juices in small bowls for dipping.\n\n1 Sandwich: Calories 410; Total Fat 14g (Saturated Fat 5g, Trans Fat 1g); Cholesterol 60mg; Sodium 570mg; Total Carbohydrate 46g (Dietary Fiber 1g, Sugars 15g); Protein 26g Exchanges: 2 Starch, 1 Other Carbohydrate, 2\u00bd Medium-Fat Meat Carbohydrate Choices: 3\n\nFrench Dip Sandwiches\n\n## French Dip Sandwiches Calorie Smart \u2022 Slow Cooker\n\nPrep 10 min Total 7 hr 10 min \u25cf 10 sandwiches\n\n  * 1 boneless beef chuck roast (3 lb)\n  * 1\u00bd cups water\n  * \u2153 cup soy sauce\n  * 1 teaspoon dried rosemary leaves\n  * 1 teaspoon dried thyme leaves\n  * 1 clove garlic, finely chopped\n  * 1 dried bay leaf\n  * 3 or 4 black peppercorns\n  * 2 loaves (1 lb each) French bread\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. Place beef in slow cooker (if roast comes in netting or is tied, do not remove). In medium bowl, mix remaining ingredients except bread; pour over beef.\n\n2 Cover; cook on Low heat setting 7 to 8 hours. Skim fat from surface of juices; discard bay leaf and peppercorns. Remove beef from slow cooker; place on cutting board (remove netting or strings). Cut beef into thin slices.\n\n3 Cut each loaf of bread into 5 pieces, about 4 inches long; cut in half horizontally. Fill bread with beef. Serve cooking juices in small bowls for dipping.\n\n1 Sandwich: Calories 400; Total Fat 13g (Saturated Fat 4.5g, Trans Fat 1g); Cholesterol 50mg; Sodium 1050mg; Total Carbohydrate 46g (Dietary Fiber 3g, Sugars 0g); Protein 25g Exchanges: 3 Starch, 2 Medium-Fat Meat Carbohydrate Choices: 3\n\n## Retail Pork Cuts\n\nHere are many of the pork cuts available at the market. They will have different names in different regions of the country, but this should help you identify what you are purchasing.\n\nDiagram: **1** Shoulder Butt **2** Loin **3** Picnic Shoulder **4** Side **5** Leg\n\n## Retail Lamb Cuts\n\nThese cuts of lamb show what is available at grocery stores. For specific cuts, ask a butcher to provide it for you, because lamb is not as readily available as pork or beef.\n\nDiagram: **1** Shoulder **2** Rib **3** Loin and Sirloin **4** Leg and Hind Shank **5** Breast and Foreshank\n\n## Learn to SEAR and PANFRY\n\nSearing and panfrying have similar techniques but are used for different purposes. Searing caramelizes the outsides of meat, which adds a wonderful color, builds flavor, and seals in juices by quickly cooking it. The meat is then finished cooking in another way\u2014braising, roasting or simmering.\n\nPanfrying involves cooking food in a small amount of oil, first by browning and then cooking it through. Panfrying requires less oil than deep frying. Often, foods are coated with flour or other ingredients before cooking to create a nice crust. Meats and vegetables or even tofu are candidates for this technique.\n\n## Tips for Searing or Panfrying\n\n  1. **Choose the correct pan:** Stainless steel or cast-iron skillets are the best pans for searing. These pans can be brought to a high enough heat to sear the meat evenly and quickly. \n  2. **Prepare the meat:** Searing can be done with small cubes of meat or steaks or chops; or sear larger cuts of meat before roasting. For panfrying, coat meat with seasonings, flour or crumbs just before cooking.\n  3. **Add oil to the skillet:** To sear, use no oil or heat a small amount\u20141 to 3 teaspoons swirled to cover the bottom of pan. To panfry, add \u00bc inch oil to the pan.\n  4. **Heat the skillet:** Heat the pan over medium-high heat\u2014the pan and\/or oil should be hot when you add the meat. \n  5. **Cook the meat:** To sear, let the meat cook\u2014when it's properly browned; it will start to come loose from the bottom of the pan. As it starts to loosen, turn the pieces and sear the other sides. For panfrying, allow meat to brown before turning\u2014ideally, turn only one time to get a good sear on the surface\u2014and continue cooking, uncovered, usually at a lower temperature, as directed in the recipe. \n  6. **Deglaze the pan:** After removing the meat, you can add liquid to create a sauce from the brown bits left in the pan (see page 466).\n\n## Panfried Pork Chops with Cider Sauce Calorie Smart \u2022 FAst\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 4 boneless pork loin chops, \u00bd to \u00be inch thick (1 lb), trimmed of fat\n  * \u00bc teaspoon seasoned salt\n  * \u00bc teaspoon dried thyme leaves\n  * \u215b teaspoon garlic powder\n  * \u215b teaspoon pepper\n  * 1 tablespoon vegetable oil\n  * \u00bd cup apple cider\n  * \u00bc cup brandy or apple cider\n  * 1 tablespoon butter\n\n1 Sprinkle both sides of pork chops with seasoned salt, thyme, garlic powder and pepper. In 12-inch skillet, heat oil over medium-high heat. Add pork; cook about 5 minutes or until bottoms are browned.\n\n2 Turn pork chops; reduce heat to medium-low. Cook uncovered 5 to 10 minutes longer or until pork is no longer pink in center. Remove from skillet to plate; cover to keep warm.\n\n3 Add cider and brandy to skillet. Heat to boiling, stirring to loosen browned bits from bottom of skillet; reduce heat to low. Add butter; simmer 1 to 3 minutes, stirring frequently, until slightly thickened. Serve sauce over pork.\n\n1 Serving: Calories 250; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 75mg; Sodium 150mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 3g); Protein 24g Exchanges: \u00bd Starch, 3 Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nPork Chops with Mustard-Chive Cream Sauce Omit cider, brandy and butter. Prepare recipe through Step 2. Add \u00bc cup dry sherry to skillet. Heat to boiling, stirring to loosen browned bits. Stir in \u00bd cup whipping cream, 1 tablespoon country-style Dijon mustard, 2 teaspoons chopped fresh chives, \u215b teaspoon salt and \u215b teaspoon pepper. Simmer 3 minutes, stirring frequently, until slightly thickened.\n\nSpicy Southwestern Pork Chops Omit all ingredients except pork chops. In small bowl, mix 1 teaspoon chili powder, 1 teaspoon chipotle chile pepper powder, \u00bd teaspoon ground cumin, \u00bc teaspoon salt, \u00bc teaspoon ground red pepper (cayenne), \u00bc teaspoon garlic powder and \u215b teaspoon black pepper. Rub on both sides of pork. Cook as directed through Step 2.\n\nCornbread and Bacon\u2013Stuffed Pork Chops\n\n## Cornbread and Bacon\u2013Stuffed Pork Chops\n\nPrep 30 min Total 1 hr 30 min \u25cf 6 servings\n\n  * 6 bone-in pork rib or loin chops, 1 to 1\u00bc inches thick (about 4 lb)\n  * 4 slices bacon, cut into \u00bd-inch pieces\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 1 cup cornbread stuffing mix\n  * \u00bd cup water\n  * \u00bd cup shredded Cheddar cheese (2 oz)\n  * \u00bd teaspoon seasoned salt\n  * \u00bd teaspoon dried marjoram leaves\n  * \u00bc teaspoon pepper\n\n1 Heat oven to 350\u00b0F. Make a pocket in each pork chop by cutting into side of chop toward the bone.\n\n2 In 12-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Add onion and bell pepper. Cook 2 to 3 minutes, stirring occasionally, until vegetables are crisp-tender; drain. Remove from heat; stir in stuffing mix and water until well blended. Stir in cheese.\n\n3 Sprinkle both sides of pork chops with seasoned salt, marjoram and pepper. Fill each pocket with about \u2153 cup stuffing. In same skillet, cook pork over medium heat until brown on both sides. Place pork in 13x9-inch pan.\n\n4 Cover tightly with foil; bake 45 minutes. Uncover; bake about 15 minutes longer or until pork is no longer pink and meat thermometer inserted in center reads 145\u00b0F.\n\n1 Serving: Calories 600; Total Fat 30g (Saturated Fat 11g, Trans Fat 0g); Cholesterol 210mg; Sodium 540mg; Total Carbohydrate 9g (Dietary Fiber 1g, Sugars 1g); Protein 73g Exchanges: \u00bd Starch, 9\u00bd Very Lean Meat, \u00bd High-Fat Meat, 4 Fat Carbohydrate Choices: \u00bd\n\n### Cutting a Pocket in a Pork Chop\n\nWith sharp knife, make pocket in side of pork chop, cutting toward bone.\n\n## Pecan-Crusted Pork Medallions Fast\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1 egg\n  * \u00bc cup honey\n  * 1 pork tenderloin (1 lb), cut into \u00bd-inch slices\n  * 1 cup chopped pecans\n  * \u00bd cup yellow cornmeal\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * 2 tablespoons vegetable oil\n\n1 In large bowl, mix egg and honey. Add pork slices; toss to coat.\n\n2 In food processor, place pecans, cornmeal, salt and pepper. Cover and process until finely chopped. Place pecan mixture in resealable plastic food-storage bag. Add pork slices; seal bag and shake to coat.\n\n3 In 10-inch nonstick skillet, heat oil over medium-high heat. Cook pork in oil 6 to 8 minutes, turning once, until golden brown on outside and no longer pink in center.\n\n1 Serving: Calories 530; Total Fat 32g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 105mg; Sodium 640mg; Total Carbohydrate 35g (Dietary Fiber 3g, Sugars 19g); Protein 25g Exchanges: 2 Starch, 2 Lean Meat, 4 Fat Carbohydrate Choices: 2\n\nPumpkin Seed\u2013Crusted Pork Medallions Substitute 1 cup pepitas (pumpkin seeds) for the pecans. Add \u00bd teaspoon ancho chile powder with the pepper. Top each serving with 1 tablespoon each chopped fresh cilantro and crumbled cotija (Mexican) cheese.\n\nPork Tenderloin with Roasted Vegetables\n\n## Pork Tenderloin with Roasted Vegetables Calorie Smart\n\nPrep 10 min Total 50 min \u25cf 6 servings\n\n  * 2 pork tenderloins (about \u00be lb each)\n  * 1 bag (16 oz) ready-to-eat baby-cut carrots\n  * 2 lb small red potatoes (16 to 20), cut in half\n  * 1 medium onion, cut into wedges\n  * 6 cloves garlic, peeled\n  * 1 tablespoon olive or vegetable oil\n  * 2 teaspoons dried rosemary leaves, crushed\n  * 1 teaspoon dried sage leaves, crushed\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 Heat oven to 450\u00b0F. Spray shallow roasting pan with cooking spray. Place pork in pan. Insert ovenproof meat thermometer so tip is in thickest part of pork.\n\n2 Place carrots, potatoes, onion and garlic around pork. Drizzle with oil; sprinkle with rosemary, sage, salt and pepper.\n\n3 Roast uncovered 25 to 30 minutes or until thermometer reads 145\u00b0F. Remove pork from pan. Stir vegetables; roast 5 to 10 minutes longer or until tender. Serve pork with vegetables and garlic.\n\n1 Serving: Calories 320; Total Fat 7g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 50mg; Sodium 220mg; Total Carbohydrate 38g (Dietary Fiber 5g, Sugars 6g); Protein 26g Exchanges: 2 Starch, 1\u00bd Vegetable, 2\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\nJerk Pulled Pork Sandwiches\n\n## Jerk Pulled Pork Sandwiches Slow Cooker\n\nPulled pork, a favorite style of Southern and some Southeastern barbecues, refers to the method of preparing the cooked pork by pulling it apart with your fingers or two forks.\n\nPrep 10 min Total 8 hr 40 min \u25cf 8 sandwiches\n\n  * 1 tablespoon Jamaican jerk seasoning (dry)\n  * 1 boneless pork shoulder (2\u00bd lb)\n  * \u00bc teaspoon dried thyme leaves\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 cup cola carbonated beverage\n  * 2 cups barbecue sauce (for homemade sauce, see page 411)\n  * 8 burger buns, split, or flour tortillas (9 to 10 inch)\n  * Creamy coleslaw (from deli; for homemade coleslaw, see page 129), if desired\n\n1 Rub jerk seasoning over pork (if pork comes in netting or is tied, do not remove); sprinkle with thyme. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, place pork and onion; pour cola over pork.\n\n2 Cover; cook on Low heat setting 8 to 10 hours. Remove pork from slow cooker to cutting board or plate (remove netting or strings). Remove juices from slow cooker and reserve.\n\n3 Using 2 forks or your fingers, shred pork into desired-size pieces; return to slow cooker. Stir in barbecue sauce and \u00bd cup reserved juices. Increase heat setting to High. Cover; cook 30 to 45 minutes or until thoroughly heated.\n\n4 In 2-quart saucepan, heat remaining reserved juices to boiling. Spoon pork into buns; top with coleslaw. If desired, serve juices in small bowls for dipping.\n\n1 Sandwich: Calories 420; Total Fat 14g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 900mg; Total Carbohydrate 49g (Dietary Fiber 2g, Sugars 21g); Protein 25g Exchanges: 2 Starch, 1 Other Carbohydrate, 3 Lean Meat, 1 Fat Carbohydrate Choices: 3\n\nPork Crown Roast with Fruited Stuffing Supreme\n\n## Pork Crown Roast with Fruited Stuffing Supreme\n\nPrep 30 min Total 4 hr 25 min \u25cf 16 servings\n\nPork Roast\n\n  * 1 pork crown roast, about 16 to 18 ribs (8 to 10 lb)*\n  * 2 teaspoons salt\n  * 1 teaspoon pepper\n\nStuffing\n\n  * 1 lb bulk pork sausage\n  * \u00bc cup butter\n  * 4 medium stalks celery, chopped (2 cups)\n  * \u00be cup chopped onion (about 1 large)\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1 teaspoon dried sage leaves, crushed\n  * 1 teaspoon poultry seasoning\n  * 6\u00bd cups unseasoned stuffing cubes (16 oz)\n  * 1 can (8 oz) crushed pineapple in juice, undrained\n  * 1 cup applesauce\n  * 1 cup orange marmalade\n\n1 Heat oven to 325\u00b0F. Sprinkle pork with salt and pepper. Place pork, bone ends up, on rack in shallow roasting pan. Wrap bone ends in foil to prevent excessive browning. Insert ovenproof meat thermometer so tip is in thickest part of pork and does not touch bone or rest in fat.\n\n2 Place small heatproof bowl or crumpled foil in crown to hold shape of roast evenly. Do not add water. Roast uncovered 2 hours 40 minutes to 3 hours 20 minutes.\n\n3 In 10-inch skillet, cook sausage over medium heat 8 to 10 minutes, stirring occasionally, until no longer pink; drain well and set aside. In same skillet, melt butter over medium-high heat. Cook celery and onion in butter about 5 minutes, stirring occasionally, until crisp-tender. Stir in broth, sage and poultry seasoning. In very large bowl, mix stuffing cubes, sausage, pineapple, applesauce and marmalade. Add vegetable mixture; stir until well blended.\n\n4 About 1 hour before pork is done, remove bowl and fill center of crown with stuffing. (Remaining stuffing can be baked in 1\u00bd-quart covered casserole.) Cover stuffing with foil for first 30 minutes. Remove foil and finish roasting.\n\n5 Remove pork from oven when thermometer reads 145\u00b0F. Cover loosely with foil; let stand at least 3 minutes. Remove foil from bone ends; place paper frills on bone ends, if desired. To serve, spoon stuffing into bowl and cut pork between ribs.\n\n*A 4-lb boneless pork loin can be substituted for the crown roast. Place in roasting pan. Insert ovenproof meat thermometer so tip is in thickest part of pork and does not rest in fat. Bake uncovered about 1 hour 20 minutes. Remove pork from oven when thermometer reads 145\u00b0F. Cover loosely with foil; let stand at least 3 minutes. Make stuffing as directed and bake in covered casserole.\n\n1 Serving: Calories 580; Total Fat 24g (Saturated Fat 9g, Trans Fat 0.5g); Cholesterol 135mg; Sodium 1020mg; Total Carbohydrate 44g (Dietary Fiber 2g, Sugars 19g); Protein 46g Exchanges: 2 Starch, 5 Lean Meat, 1 Fat Carbohydrate Choices: 3\n\n### Carving a Pork Crown Roast\n\nPlace roast on carving board or serving platter. Insert meat fork into roast to keep it from moving; cut slices between ribs.\n\nHerbed Pork Loin Roast\n\n## Herbed Pork Loin Roast Calorie Smart\n\nPrep 10 min Total 1 hr 15 min \u25cf 6 servings\n\n  * 1 boneless pork loin roast (1\u00bd lb), trimmed of fat\n  * \u00bd teaspoon olive oil\n  * 3 tablespoons chopped fresh sage leaves\n  * 2 tablespoons chopped fresh parsley\n  * 2 tablespoons chopped fresh oregano leaves\n  * 4\u00bd teaspoons chopped fresh chives\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon black pepper\n\n1 Heat oven to 375\u00b0F. Spray broiler pan with cooking spray.\n\n2 Rub pork with oil. In small bowl, mix remaining ingredients. Rub mixture over pork. Place pork on rack in pan. Insert ovenproof meat thermometer so tip is in thickest part of pork and does not rest in fat.\n\n3 Roast uncovered 1 hour or until thermometer reads 145\u00b0F. Cover loosely with foil; let stand at least 3 minutes before slicing.\n\n1 Serving: Calories 130; Total Fat 6g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 250mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 18g Exchanges: 2 Lean Meat Carbohydrate Choices: 0\n\nDijon-Herbed Pork Loin Roast In Step 2, omit olive oil. Mix 1 tablepoon Dijon mustard and 1 finely chopped clove garlic in small bowl; spread over pork. Continue as directed.\n\nPepper Garlic Pork Loin Roast Increase olive oil to 1 teaspoon; omit sage, oregano, chives and pepper. Mix parsley with 4 to 6 finely chopped cloves garlic and 1 teaspoon coarsely ground black pepper. Continue as directed.\n\nPorketta Pot Roast\n\n## Porketta Pot Roast Calorie smart \u2022 Slow Cooker\n\nPrep 20 min Total 9 hr 20 min \u25cf 8 servings\n\n  * 2 teaspoons Italian seasoning\n  * 1\u00bd teaspoons fennel seed, crushed\n  * \u00be teaspoon salt\n  * \u00bd teaspoon celery seed\n  * 1 boneless pork shoulder (3 lb)\n  * 3 medium parsnips, peeled, cut into \u00be-inch pieces (3 cups)\n  * 2 medium sweet potatoes, peeled, cut into \u00be-inch pieces (3 cups)\n  * 12 cloves garlic, peeled, cut in half\n  * 1\u00bd cups water\n\n1 In small bowl, mix Italian seasoning, fennel seed, salt and celery seed. Pat seasoning mixture evenly onto pork (if pork comes in netting or is tied, do not remove). Heat 12-inch nonstick skillet over medium-high heat. Add pork; cook 5 to 10 minutes, turning several times, until brown on all sides.\n\n2 Spray 4- to 5-quart slow cooker with cooking spray. In slow cooker, place parsnips, sweet potatoes and garlic; pour water over vegetables. Place pork on vegetables.\n\n3 Cover; cook on Low heat setting 9 to 10 hours or until pork is tender. Remove pork from slow cooker to cutting board (remove netting or strings). Cut pork across grain into slices; serve with vegetables and cooking juices.\n\n1 Serving: Calories 350; Total Fat 20g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 90mg; Sodium 290mg; Total Carbohydrate 16g (Dietary Fiber 3g, Sugars 6g); Protein 26g Exchanges: 1 Starch, 3 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 1\n\n## Make-Ahead Oven-Roasted Pulled Pork\n\nCalorie smart\n\nThis flavorful, tender pulled pork can be used for sandwiches (try topping with coleslaw!), tacos, burritos, quesadillas, salads or soup.\n\nPrep 10 min Total 5 hr 20 min \u25cf 10 servings\n\n  * 1 boneless pork shoulder blade roast (4 lb)\n  * \u00bc cup packed brown sugar\n  * 2 teaspoons salt\n  * 1 teaspoon garlic powder\n  * 1 teaspoon onion powder\n  * 1 teaspoon paprika\n  * 1 teaspoon smoked paprika\n  * 1 teaspoon pepper\n\n1 Heat oven to 325\u00b0F. In 13x9-inch (3-quart) glass baking dish, place pork, fat side up (if roast has netting, do not remove). In small bowl, mix remaining ingredients. Sprinkle mixture over all pork surfaces; rub and press into meat.\n\n2 Cover with foil. Roast 4 hours to 4 hours 30 minutes or until very tender. Place roast on cutting board; remove excess fat from top of pork and discard. Cover roast loosely with foil; let stand 30 minutes or until cool enough to handle. Pull pork apart into bite-size pieces, discarding any fat or gristle.\n\n3 Pour pan drippings into fat separator, or skim fat from drippings. Pour pan drippings into small bowl, leaving fat in separator. If desired, add some pan drippings to pulled pork. Reheat to use immediately, or cover and refrigerate 3 to 4 days or freeze 2 to 3 months.\n\n1 Serving: Calories 270; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 80mg; Sodium 520mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 5g); Protein 27g Exchanges: \u00bd Other Carbohydrate, 4 Lean Meat, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## The Origins of Pulled Pork\n\nPulled pork goes back to colonial days, when Southerners would roast less tender, fatty cuts of pork over low heat for long periods. The fat would keep the meat moist while the long cook time would make the pork ultra-tender, allowing it to be pulled apart.\n\n## Ways to Use Pulled Pork\n\nSpend a little time on the weekend cooking this flavorful pulled pork, then plan to use it in a variety of ways during the week. The recipe makes about 5 cups of pulled pork to serve in sandwiches or to add to recipes. Check out these easy ideas for using the pork.\n\n  1. 1. **Pork and Coleslaw Sandwiches:** Spread cut sides of buns with spicy mustard. Layer pulled pork and creamy coleslaw in buns.\n  2. 2. **Pulled Pork Pizza:** Top purchased pizza crust with about 1 cup pulled pork, 1 cup chopped tomatoes and 1\u00bd cups shredded Monterey Jack or Cheddar cheese. Bake as directed on pizza package.\n  3. 3. **Chopped Salad with Pork:** In large bowl, mix 3 cups chopped romaine, 1 cup chopped red cabbage, 1 cup chopped red bell pepper, \u00bd cup sliced green onions, 1 cup corn salsa and 1\u00bd cups pulled pork. Toss with \u00bc cup Italian dressing.\n  4. 4. **Pulled Pork Nachos:** On microwavable serving plate, layer 2 to 3 cups tortilla chips, 1 cup chopped tomatoes, 1 cup pulled pork and 1\u00bd cups shredded Monterey Jack or Cheddar cheese. Microwave on High 1 to 2 minutes or until cheese is melted.\n  5. 5. **Pulled Pork Quesadillas:** For each quesadilla, between 2 (8-inch) flour tortillas, layer \u00be cup pulled pork, \u00bc cup chunky salsa, \u00bd cup coleslaw mix and \u00be cup shredded cheese. Brush outsides of tortillas with 1 to 2 teaspoons oil. Cook in skillet over medium heat 3 to 5 minutes or until browned on both sides. Cut into wedges.\n  6. 6. **Pork Burrito Bowls:** For each serving, spoon 1 cup hot cooked rice in bowl. Top with \u00bd cup hot pulled pork, \u00bc cup shredded Mexican blend cheese, \u00bc cup shredded lettuce and 1 tablespoon salsa.\n  7. 7. **Skillet Pork Hash:** In 10-inch skillet, cook \u00bd cup each chopped onion and bell pepper in 1 tablespoon oil over medium heat 3 to 5 minutes or until tender. Stir in 2 cups pulled pork, \u00bd cup cooked corn and 2 to 4 tablespoons chunky salsa. Cook and stir 8 to 10 minutes or until hot and flavors are blended.\n  8. 8. **Loaded Fries:** Place about 1\u00bd pounds prepared white potato or sweet potato fries on an ovenproof platter. Top with 1 to 1\u00bd cups of the pulled pork, 1 cup shredded Colby-Jack cheese and finely chopped green onions. Return to the oven for 2 to 3 minutes or until cheese is melted.\n\nOven-Roasted Pulled Pork, Pulled Pork Quesadillas\n\nGlazed Baked Ham\n\n## Glazed Baked Ham Calorie Smart\n\nPrep 10 min Total 1 hr 50 min \u25cf 10 servings\n\n  * 1 cooked smoked bone-in ham (6 lb)\n  * 1 cup packed brown sugar\n  * 1 tablespoon cornstarch\n  * \u00bc teaspoon salt\n  * 1 can (8 oz) crushed pineapple, undrained\n  * 2 tablespoons lemon juice\n  * 1 tablespoon yellow or Dijon mustard\n\n1 Heat oven to 325\u00b0F. Place ham on rack in shallow roasting pan. Insert ovenproof meat thermometer so tip is in center of thickest part of ham and does not touch bone or rest in fat. Bake uncovered 1 hour 30 minutes.\n\n2 Meanwhile, in 1-quart saucepan, mix brown sugar, cornstarch and salt. Stir in pineapple, lemon juice and mustard. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute. Brush glaze over ham during last 45 minutes of baking.\n\n3 When thermometer reads at least 140\u00b0F, remove ham from oven. Cover loosely with foil; let stand 10 to 15 minutes before slicing.\n\n1 Serving: Calories 140; Total Fat 3.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 35mg; Sodium 830mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 14g); Protein 14g Exchanges: 1 Other Carbohydrate, 4 Lean Meat Carbohydrate Choices: 1\n\nHam with Brown Sugar\u2013Orange Glaze Omit all ingredients except ham. Bake ham as directed. In small bowl, mix \u00bd cup packed brown sugar, 2 tablespoons orange juice and \u00bd teaspoon ground mustard. Brush glaze over ham during last 45 minutes of baking.\n\n### Carving Ham\n\nPlace ham on carving board or platter, fat side up, bone facing you. Cut in half next to bone.\n\nPlace boneless side of ham fat side up; cut slices. Cut slices from bone-in portion, cutting away from bone.\n\n## All About Hams\n\nWhatever kind of ham you choose, they all come from cured pork leg meat. Curing gives ham its distinctive sweet-smoky and salty flavor. Hams are usually either wet or dry cured. Most grocery store hams, including spiral-cut hams, are wet cured, meaning they have been processed with a brine of water, salt, sugar and spices. Brining keeps the meat moist and tender.\n\nDry-cured hams are rubbed with salt, sugar and spices, then aged anywhere from several weeks to more than a year. These hams, referred to as country hams, are often named for the city where they are processed. Dry-cured hams are slightly salty, so follow package directions for using.\n\nSpicy Barbecue Pork Loin Ribs with Molasses-Mustard Barbecue Sauce\n\n## Spicy Barbecue Pork Loin Ribs\n\nPrep 15 min Total 2 hr 45 min \u25cf 6 servings\n\nRibs\n\n  * 4\u00bd lb pork loin back ribs\n  * Salt and pepper to taste\n\nSpicy Barbecue Sauce\n\n  * \u2153 cup butter\n  * 2 tablespoons white or cider vinegar\n  * 2 tablespoons water\n  * 1 teaspoon sugar\n  * \u00bd teaspoon garlic powder\n  * \u00bd teaspoon onion powder\n  * \u00bd teaspoon black pepper\n  * Dash ground red pepper (cayenne)\n\n1 Heat oven to 325\u00b0F. With knife or kitchen scissors, cut ribs into 6 serving pieces. Place ribs, meat side up, on rack in roasting pan or 13x9-inch pan. Sprinkle with salt and pepper. Cover and bake 1 hour 45 minutes.\n\n2 In 1-quart saucepan, heat sauce ingredients over medium heat, stirring frequently, until butter is melted.\n\n3 Brush some of the sauce over ribs. Bake uncovered about 45 minutes longer, brushing frequently with sauce, until meat is tender and no longer pink next to bones. Heat any remaining sauce to boiling; serve with ribs.\n\n1 Serving: Calories 730; Total Fat 60g (Saturated Fat 23g, Trans Fat 0.5g); Cholesterol 225mg; Sodium 220mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 48g Exchanges: 7 High-Fat Meat, 1 Fat Carbohydrate Choices: 0\n\nSlow-Cooker Directions Spray 5- to 6-quart slow cooker with cooking spray. Cut ribs into 2- or 3-rib portions; place in slow cooker. Sprinkle with \u00bd teaspoon salt and \u00bc teaspoon pepper. Pour \u00bd cup water into slow cooker. Cover; cook on Low heat setting 8 to 9 hours. Remove ribs. Drain and discard liquid from slow cooker. Make sauce as directed; pour into bowl. Dip ribs into sauce to coat. Return ribs to slow cooker. Pour any remaining sauce over ribs. Cover; cook 1 hour.\n\nBeef Short Ribs Substitute beef short ribs for the pork ribs. Heat oven to 350\u00b0F. Place ribs in 13x9-inch pan. Pour sauce over ribs. Cover and bake about 2 hours 30 minutes or until meat is tender.\n\nCountry-Style Pork Ribs Substitute 3 lbs country-style pork loin ribs for the back ribs. Heat oven to 350\u00b0F. Place ribs in 13x9-inch pan. Cover and bake about 2 hours; drain. Pour sauce over ribs. Bake uncovered 30 minutes longer or until meat is tender.\n\nPork Spareribs Substitute pork spareribs for the back ribs. Heat oven to 325\u00b0F. Cut ribs into 6 serving pieces. Place ribs, meat side up, on rack in roasting pan or 13x9-inch pan. Cover and bake 1 hour 15 minutes. Brush with sauce. Bake uncovered about 45 minutes longer, brushing frequently with sauce, until meat is tender and no longer pink next to bones.\n\nMolasses-Mustard Barbecue Sauce In small bowl, mix \u00bd cup molasses and \u2153 cup Dijon mustard; stir in \u2153 cup cider or white vinegar.\n\nSweet-Savory Barbecue Sauce In 1-quart saucepan, mix 1 cup chili sauce, \u00be cup grape jelly, 1 tablespoon plus 1\u00bd teaspoons dry red wine or beef broth and 1 teaspoon Dijon mustard. Heat over medium heat, stirring occasionally, until jelly is melted.\n\n## Greek Lamb Chops Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 4 servings\n\n  * 4 lamb sirloin chops or 8 lamb loin or rib chops (1\u00bd lb)\n  * 1 teaspoon dried oregano leaves\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 tablespoon finely chopped garlic\n  * 1 tablespoon olive or vegetable oil\n  * \u00bd cup chicken broth (for homemade broth, see page 143)\n  * 1 tablespoon lemon juice\n  * 1 tablespoon butter\n  * \u00bc cup sliced pitted kalamata or ripe olives\n  * 2 tablespoons chopped fresh parsley\n  * 2 tablespoons crumbled feta cheese\n\n1 Sprinkle both sides of lamb chops with oregano, salt and pepper. Press garlic into lamb. In 12-inch skillet, heat oil over medium-high heat. Cook lamb in oil 4 to 6 minutes, turning once, until brown.\n\n2 Add broth to skillet; reduce heat to medium-low. Cover and cook 8 to 10 minutes, turning once, until lamb is tender. Remove lamb from skillet; cover to keep warm.\n\n3 Heat cooking juices to boiling; boil 1 to 2 minutes or until slightly reduced. Stir in lemon juice and butter. Cook and stir just until slightly thickened. Stir in olives and parsley. Spoon sauce over lamb. Top with cheese.\n\n1 Serving: Calories 270; Total Fat 17g (Saturated Fat 6g, Trans Fat 1g); Cholesterol 95mg; Sodium 490mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 27g Exchanges: 4 Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nBalsamic Braised Lamb Shanks\n\n## Balsamic Braised Lamb Shanks Calorie Smart\n\nThe classic long, slow method of braising in a flavorful marinade is perfect for making lamb shanks fall-off-the-bone tender. Pair with Parmesan Polenta, page 207, for a fabulous meal.\n\nPrep 25 min Total 11 hr 20 min \u25cf 6 servings\n\nLamb Shanks\n\n  * 1 tablespoon olive oil\n  * 6 lamb shanks (\u00be to 1 lb each)\n\nMarinade\n\n  * 1 cup balsamic vinegar\n  * 1 cup beef broth\n  * \u00bc cup soy sauce\n  * \u00bc cup olive oil\n  * \u00bc cup packed brown sugar\n  * 2 teaspoons ground mustard\n  * \u00bd teaspoon coarsely ground black pepper\n  * \u00bc cup chili sauce\n  * Chopped fresh parsley, if desired\n\n1 In 12-inch nonstick skillet, heat 1 tablespoon oil over medium-high heat. Cook lamb in oil 8 to 10 minutes, turning frequently, until brown on all sides (brown in batches if necessary). Remove from heat.\n\n2 In large shallow glass or plastic dish or 2-gallon resealable food-storage plastic bag, mix all marinade ingredients except chili sauce and parsley. Add lamb. Cover dish or seal bag; refrigerate at least 8 hours but no longer than 24 hours, turning occasionally.\n\n3 In 8-quart Dutch oven or stockpot, place lamb and marinade. Heat to boiling; reduce heat to low. Cover and simmer 2 hours 30 minutes to 3 hours, turning lamb once, until very tender. Remove lamb to heatproof serving platter; cover with foil. Keep warm in 225\u00b0F oven.\n\n4 Pour marinade and pan drippings into fat separator, or skim fat from marinade. Return marinade and drippings to Dutch oven, leaving fat in separator. Stir in chili sauce. Heat to boiling; reduce heat to medium. Cook until reduced by half, 20 to 30 minutes. Serve lamb with marinade; sprinkle with parsley.\n\n1 Serving: Calories 440; Total Fat 17g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 165mg; Sodium 950mg; Total Carbohydrate 19g (Dietary Fiber 1g, Sugars 17g); Protein 53g Exchanges: 1 Other Carbohydrate, 7\u00bd Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 1\n\nHerb and Garlic Roast Leg of Lamb\n\n## Herb and Garlic Roast Leg of Lamb Calorie Smart\n\nPrep 10 min Total 2 hr 45 min \u25cf 10 servings\n\n  * 1 boneless leg of lamb (5 to 6 lb)\n  * \u00bc cup finely chopped fresh parsley\n  * 1 tablespoon chopped fresh or 1 teaspoon dried rosemary leaves, crushed\n  * 1 tablespoon chopped fresh or 1 teaspoon dried thyme leaves, crushed\n  * 2 teaspoons coarse kosher salt\n  * \u00bd teaspoon pepper\n  * 2 cloves garlic, finely chopped\n  * 3 tablespoons olive or vegetable oil\n\n1 Heat oven to 325\u00b0F. Place lamb in shallow roasting pan (if lamb comes in netting or is tied, do not remove).\n\n2 In small bowl, stir remaining ingredients until well mixed. Spread mixture over entire surface of lamb. Insert meat thermometer so tip is in thickest part of lamb and does not rest in fat.\n\n3 For medium-rare, roast uncovered 2 hours 5 minutes to 2 hours 15 minutes or until thermometer reads 140\u00b0F. (For medium, roast until thermometer reads 155\u00b0F.) Cover loosely with foil; let stand 15 to 20 minutes or until thermometer reads 145\u00b0F (or 160\u00b0F for medium). Remove netting or string before slicing. Serve lamb with pan juices, if desired.\n\n1 Serving: Calories 370; Total Fat 20g (Saturated Fat 6g, Trans Fat 1g); Cholesterol 155mg; Sodium 590mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 48g Exchanges: 7 Lean Meat Carbohydrate Choices: 0\n\n## Venison with Cranberry-Wine Sauce Calorie Smart\n\nPrep 35 min Total 2 hr 35 min \u25cf 4 servings\n\n  * 4 venison tenderloin steaks, about 1 inch thick (1\u00bc lb)\n  * \u00bd cup dry red wine or nonalcoholic red wine\n  * 1 tablespoon Dijon mustard\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon coarse ground black pepper\n  * 1 tablespoon olive or vegetable oil\n  * \u00bd cup beef broth (for homemade broth, see page 146)\n  * \u00bd cup sweetened dried cranberries\n  * 2 tablespoons currant or apple jelly\n  * 1 tablespoon butter\n  * 2 medium green onions, sliced (2 tablespoons)\n\n1 Place venison in resealable food-storage plastic bag or shallow glass or plastic dish. In small bowl, mix wine and mustard. Pour over venison; turn to coat. Seal bag or cover dish; refrigerate at least 2 hours but no longer than 4 hours to marinate, turning venison occasionally.\n\n2 Remove venison from marinade; reserve marinade. Sprinkle venison with salt and pepper. In 12-inch nonstick skillet, heat oil over medium-high heat. Cook venison in oil about 4 minutes, turning once, until brown.\n\n3 Add broth to skillet; reduce heat to low. Cover and cook about 10 minutes, turning venison once, until venison is tender and desired doneness. (Don't overcook or venison will become tough.)\n\n4 Remove venison from skillet; cover to keep warm. Stir reserved marinade into skillet. Heat to boiling, scraping up any browned bits from bottom of skillet; reduce heat to medium. Cook about 5 minutes or until mixture is slightly reduced. Stir in cranberries, jelly, butter and onions. Cook 1 to 2 minutes, stirring occasionally, until butter is melted and mixture is hot. Serve sauce with venison.\n\n1 Serving: Calories 320; Total Fat 10g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 125mg; Sodium 440mg; Total Carbohydrate 20g (Dietary Fiber 1g, Sugars 15g); Protein 32g Exchanges: \u00bd Fruit, 1 Other Carbohydrate, 4\u00bd Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\n## Roasting Meat\n\nRoasting is a great method for cooking large, tender cuts of meat. Roast meat at 325\u00b0F or above. Lower temperatures can start bacterial growth before cooking is finished. See page 31 for more information.\n\nPlace meat fat side up on a rack in a shallow roasting pan. (Bone-in roasts don't need a rack.) Insert ovenproof meat thermometer so tip is in center of thickest part of meat and does not touch bone or rest in fat. Don't add water or liquid; don't cover. Roast as directed in the chart below for desired final doneness, but removing from oven as directed in fourth column. Remove meat from oven; cover loosely with foil and let stand as directed below before carving.\n\n*Large-end roasts add approximately 15 minutes to cooking time.\n\n*The USDA recommends pork be cooked to 145\u00b0F with a 3-minute resting period. If you prefer pork more well-done, cook to 160\u00b0F.\n\n### Timetable for Roasting and baking Meat\n\nCut of Meat | Weight in Pounds | Approximate Total Roasting Time in Hours (unless noted otherwise) | Remove from Oven When Internal Temperature Reaches | Final Doneness Temperature (after standing time)\n\n---|---|---|---|---\n\nBeef\n\nRib-Eye Roast (small-end)*\n\nRoast at 350\u00b0F. Let stand 15 to 20 minutes. | 3 to 4\n\n4 to 6\n\n6 to 8 | 1\u00bd to 1\u00be\n\n1\u00be to 2\n\n1\u00be to 2\n\n2 to 2\u00bc\n\n2\u00bd to 2\u00be | 135\u00b0F\n\n150\u00b0F\n\n135\u00b0F\n\n150\u00b0F\n\n135\u00b0F\n\n150\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\nRib Roast (bone-in)\n\nRoast at 350\u00b0F. Let stand 15 to 20 minutes. | 4 to 6 (2 ribs)\n\n6 to 8 (2 to 4 ribs)\n\n8 to 10 (4 to 5 ribs) | 1\u00be to 2\u00bc\n\n2\u00bc to 2\u00be\n\n2\u00bc to 2\u00bd\n\n2\u00be to 3\n\n2\u00bd to 3\n\n3 to 3\u00bd | 135\u00b0F\n\n150\u00b0F\n\n135\u00b0F\n\n150\u00b0F\n\n135\u00b0F\n\n150\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\nTenderloin Roast\n\nRoast at 425\u00b0F. Let stand 15 to 20 minutes. | 2 to 3\n\n4 to 5 | 35 to 40 minutes\n\n45 to 50 minutes\n\n50 to 60 minutes\n\n60 to 70 minutes | 135\u00b0F\n\n150\u00b0F\n\n135\u00b0F\n\n150\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\nTri-Tip Roast\n\nRoast at 425\u00b0F. Let stand 15 to 20 minutes. | 1\u00bd to 2 | 30 to 40 minutes\n\n40 to 45 minutes | 135\u00b0F\n\n150\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\nRound Tip Roast\n\nRoast at 325\u00b0F. Let stand 15 to 20 minutes. | 3 to 4\n\n4 to 6\n\n6 to 8 | 1\u00be to 2\n\n2\u00bc to 2\u00bd\n\n2 to 2\u00bd\n\n2\u00bd to 3\n\n2\u00bd to 3\n\n3 to 3\u00bd | 140\u00b0F\n\n155\u00b0F\n\n140\u00b0F\n\n155\u00b0F\n\n140\u00b0F\n\n155\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\n145\u00b0F medium-rare\n\n160\u00b0F medium\n\nRump Roast\n\nRoast at 325\u00b0F. Let stand 15 to 20 minutes. | 3 to 4 | 1\u00bd to 2 | 135\u00b0F | 145\u00b0F medium-rare\n\nBottom Round Roast\n\nRoast at 325\u00b0F. Let stand 15 to 20 minutes. | 3 to 4 | 1\u00bd to 2 | 135\u00b0F | 145\u00b0F medium-rare\n\nEye Round Roast\n\nRoast at 325\u00b0F. Let stand 15 to 20 minutes. | 2 to 3 | 1\u00bd to 1\u00be | 135\u00b0F | 145\u00b0F medium-rare\n\nBrisket, Fresh\n\nBake at 325\u00b0F. Let stand 15 to 20 minutes. | 2\u00bd to 4 | 2\u00bd to 3\n\n(must be cooked with liquid to partially cover) |   | 170\u00b0F well-done\n\nBrisket, Corned\n\nBake at 325\u00b0F. Let stand 15 to 20 minutes. | 2\u00bd to 3\u00bd\n\n3\u00bd to 5 | 2\u00bd to 3\u00bd\n\n3\u00bd to 4\u00bd | Follow package\n\ndirections. | 170\u00b0F well-done; follow package directions\n\nVeal\n\nRump or Shoulder Roast (boneless)\n\nRoast at 325\u00b0F. | 2 to 3 | 25 to 30 minutes per lb | 145\u00b0F | Let stand at least 3 minutes.\n\nRib Roast (bone-in)\n\nRoast at 325\u00b0F. | 2 to 4 | 30 to 34 minutes per lb | 145\u00b0F | Let stand at least 3 minutes.\n\nPork*\n\nLoin Roast, bone-in\n\nboneless\n\nRoast at 325\u00b0F. | 3 to 5\n\n2 to 4 | 20 to 25 minutes per lb\n\n23 to 33 minutes per lb | 145\u00b0F\n\n145\u00b0F | Let stand at least 3 minutes.\n\nCrown Roast\n\nRoast at 325\u00b0F. | 6 to 10 | 20 minutes per lb | 145\u00b0F | Let stand at least 3 minutes.\n\nShoulder Roast\/Butt\n\nRoast at 325\u00b0F. | 3 to 6 | 20 minutes per lb | 145\u00b0F | Let stand at least 3 minutes.\n\nRibs (back, spareribs)\n\nBake at 325\u00b0F. |   | Cover and cook 1 hour 15 minutes, uncover and cook 45 minutes longer | Tender | No standing time\n\nRibs (country-style)\n\nBake at 350\u00b0F. |   | Cover and cook 2 hours, uncover and cook 30 minutes longer. | Tender | No standing time\n\nTenderloin\n\nRoast at 425\u00b0F to450\u00b0F. | \u00bd to 1\u00bd | 20 to 30 minutes | 145\u00b0F | Let stand at least 3 minutes.\n\nHam, Fully Cooked (bone-in or boneless)\n\nRoast at 325\u00b0F. | 3 to 4 (boneless)\n\n7 to 8 (bone-in)\n\n14 to 16 (bone-in) | 18 to 25 minutes per lb | 140\u00b0F | No standing time\n\nFresh Leg\/Uncured Ham, Cook Before Eating (bone-in)\n\nRoast at 325\u00b0F. |   | 15 to 18 minutes per lb | 145\u00b0F | Let stand at least 3 minutes.\n\nLamb\n\nLeg (bone-in)\n\nRoast at 325\u00b0F. | 5 to 7 | 20 to 25 minutes per lb\n\n25 to 30 minutes per lb | 140\u00b0F\n\n155\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\nLeg (rolled, boneless)\n\nRoast at 325\u00b0F. | 4 to 7 | 20 to 25 minutes per lb\n\n30 to 35 minutes | 140\u00b0F\n\n155\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\nShoulder Roast or Shank Leg Half(bone-in)\n\nRoast at 325\u00b0F. | 3 to 4 | 20 to 25 minutes\n\n25 to 30 minutes | 140\u00b0F\n\n155\u00b0F | 145\u00b0F medium-rare\n\n160\u00b0F medium\n\n## Braising Meat\n\nBraising (or pot roasting) is great for large, less tender cuts of meat, such as those coming from the chuck, shank, brisket or round. These cuts are generally less expensive than more tender cuts. The meat is browned first to add depth of flavor, then covered and cooked slowly with a small amount of liquid to make the meat very tender.\n\nSee Learn to Braise, page 330.\n\n## Panfrying Meat\n\nPanfrying (also known as pan broiling) is a great method for cooking tender cuts of meat without liquid and is faster than oven broiling, if you're in a hurry. It's important not to overcook meat with this method, or it can become tough and dry.\n\nUse a nonstick skillet, or lightly coat a regular skillet with vegetable oil. Heat the skillet over medium heat (unless otherwise noted) for 5 minutes. Season meat as desired. Add the meat to the skillet. Don't add water, liquids or fats; don't cover. Cook for the minimum time listed or until the thermometer reaches the final doneness temperature, turning once. If the meat browns too quickly, reduce heat to medium-low.\n\nSee Learn to Sear and Panfry, page 302.\n\n## Broiling Meat\n\nBroiling is a great method for cooking smaller cuts of meat directly under the high heat of your oven quickly and without fat. Broiling enhances the flavor of smaller cuts by caramelizing the surface of the meat, creating a flavorful \"crust\" and sealing in the juices.\n\nSet the oven control to broil; heat 10 minutes. Check the oven's use-and-care manual for whether the oven door should be partially opened or closed during broiling. Place the meat on a rack in the broiler pan; season to taste with salt and pepper. For cuts less than 1 inch thick, broil 2 to 3 inches from the heat. For cuts 1 to 1\u00bd inches thick, broil 3 to 4 inches from the heat unless chart (page 317) gives a different distance.\n\nBroil for the minimum time listed, turning once and adding additional time as needed.\n\n## Cooking Ham\n\nThe most popular ham is fully cooked and ready to eat. Packaged cooked hams should be cooked to at least 140\u00b0F. If you are not sure what kind of ham you have, or the cooked ham has been repackaged, it must be heated to an internal temperature of 165\u00b0F. Raw fresh ham should be cooked to at least 145\u00b0F and allowed to stand or rest at least 3 minutes.\n\n### Timetable for Broiling Meat\n\nCut of Meat | Thickness in Inches | distance from heat | Approximate Cooking Time in Minutes | Final Doneness Temperature\n\n---|---|---|---|---\n\nBeef\n\nSteak: Rib Eye, Porterhouse, T-Bone, Top Loin (Strip) Steak | \u00be\n\n1\n\n1\u00bd | 2 to 3\n\n3 to 4\n\n3 to 4 | 9 tp 13\n\n13 to 20\n\n24 to 32 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nTop Blade Chuck Steak | \u00be\n\n1 | 2 to 3\n\n3 to 4 | 8 to 11\n\n12 to 15 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nTenderloin Steak | 1\n\n1\u00bd | 2 to 3\n\n3 to 4 | 13 to 16\n\n18 to 22 | 145\u00b0F medium-rare to160\u00b0F medium\n\nTop Sirloin Steak\n\n(boneless) | \u00be\n\n1\n\n1\u00bd\n\n2 | 2 to 3\n\n3 to 4\n\n3 to 4\n\n3 to 4 | 9 to 12\n\n16 to 21\n\n26 to 31\n\n34 to 39 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nGround Patties (4-inch diameter)\n\n4 per lb\n\n4 per 1\u00bd lb | \u00bd\n\n\u00be | 2 to 3\n\n3 to 4 | 12 to 13\n\n12 to 14 | 160\u00b0F medium\n\n160\u00b0F medium\n\nVeal\n\nChop, Loin or Rib | 1 | 4 | 14 to 16 | 145\u00b0F medium-rare\n\nPork\n\nChop (bone-in or boneless) | \u00be to 1\n\n1\u00bd | 4 to 5\n\n4 to 5 | 8 to 12\n\n12 to 22 | Let stand at least 3 minutes.\n\n145\u00b0F medium\n\n145\u00b0F medium\n\nGround Patties | \u00bd\n\n|  |\n\n8 to 12 | 160\u00b0F medium\n\nLamb\n\nChop | 1 to 1\u00bc | 4 to 5 | 9 to 12 minutes per lb | Let stand for at least 3 minutes. 145\u00b0F rare to 160\u00b0F medium.\n\n### Timetable for Panfrying Meat\n\nCut of Meat | Thickness in Inches | Approximate Cooking Time in Minutes | Final Doneness Temperature\n\n---|---|---|---\n\nBeef\n\nRib-Eye Steak | \u00be\n\n1 | 9 to 11\n\n12 to 15 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nPorterhouse\/T-Bone Steak | \u00be\n\n1 | 10 to 13\n\n14 to 17 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nStrip Steak (Boneless) | \u00be\n\n1 | 8 to 11\n\n12 to 15 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nRanch Steak (shoulder center) | \u00be\n\n1 | 8 to 11\n\n12 to 15 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nFlatiron Steak (shoulder top blade) | 8 oz each | 11 to 14 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nTenderloin Steak\n\n(use medium-high heat for \u00bd-inch thickness) | \u00bd\n\n\u00be\n\n1 | 3 to 5\n\n7 to 10\n\n10 to 13 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nTop Sirloin Steak (boneless) | \u00be\n\n1 | 12 to 15\n\n15 to 18 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nTop Round Steak (best when marinated before cooking) | \u00be\n\n1 | 12 to 15\n\n15 to 17 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nCubed Steak (use medium-high heat) |   | 3 to 5 | 145\u00b0F medium-rare to 160\u00b0F medium\n\nGround Patties (4-inch diameter)\n\n4 per lb\n\n4 per 1\u00bd lb | \u00bd\n\n\u00be | 10 to 12\n\n14 to 16 | 160\u00b0F medium\n\n160\u00baF medium\n\nVeal\n\nChop, Loin or Rib (use medium-high heat) | \u00be | 10 to 14 | 145\u00b0F medium-rare (let stand 3 minutes)\n\nCutlet (use medium-high heat) | \u215b\n\n\u00bc | 3 to 4\n\n5 to 6 | 145\u00b0F medium-rare (let stand 3 minutes)\n\nGround Patties (use medium-high heat) | \u00bd | 9 to 12 | 160\u00b0F medium\n\nPork*\n\nChop, Loin or Rib (bone-in or boneless) | \u00be to 1 | 8 to 16 | 145\u00b0F medium-rare (let stand 3 minutes)\n\nCutlets (bone-in or boneless) | \u215b\n\n\u00bc | 3 to 4\n\n5 to 6 | 145\u00b0F medium-rare (let stand 3 minutes)\n\nTenderloin Medallions | \u00bc to \u00bd | 4 to 8 | 145\u00b0F (let stand 3 minutes)\n\nGround Patties | \u00bd | 8 to 11 | 160\u00b0F medium\n\nHam Slice, Cooked\n\nHam Steak, Cooked | \u00bd\n\n1 to 1 | 6 to 8\n\n8 to 10 | 140\u00b0F medium\n\n140\u00b0F medium\n\nLamb\n\nChop, Loin or Rib | 1 to 1\u00bc | 9 to 11 | 145\u00b0F medium-rare (let stand 3 minutes)\n\nGround Patties | \u00bd | 9 to 12 | 160\u00b0F medium\n\n*The USDA recommends pork be cooked to 145\u00b0F with a 3-minute resting period. If you prefer pork more well done, cook to 160\u00b0F.\n\nType of Meat | Servings per Pound\n\n---|---\n\nBoneless Cuts (ground, boneless chops, loin, tenderloin) | 3 to 4\n\nBone-In Cuts (rib roasts, pot roasts, country-style ribs) | 2 to 3\n\nVery Bony Cuts (back ribs, spareribs, short ribs, shanks) | 1 to 1\u00bd\n\n# Chapter 12: Chicken & Turkey\n\nSkillet-Fried Chicken\n\n## Chicken and Turkey Basics\n\nBuying\n\nStoring\n\nThawing\n\n## Features and Charts\n\nBetty's Staples: Cooked Wild Rice\n\nBoning Chicken Breasts\n\nCarving a Whole Chicken\n\nCarving Roast Turkey\n\nCooked Poultry Yields\n\nCutting Up a Whole Chicken\n\nGlobal Flavors: Mediterranean\n\nHeirloom Recipe and New Twist\n\nLearn to Braise\n\nMaking Bread Stuffing\n\nMaking Pan Gravy\n\nMaking Roast Turkey\n\nPreparing Chicken for Roasting\n\nSetting a Buffet Table\n\nSetting a Festive Holiday Table\n\nTimetable for Broiling\n\nTimetable for Thawing\n\nTimetable for Roasting\n\n## Chicken\n\nAsian Chicken Rolls-Ups Calorie Smart \u2022 Fast\n\nBaked Chicken Panzanella Calorie Smart\n\nButtermilk Fried Chicken\n\nButtery Rosemary-Honey Roasted Chicken\n\nCaramel Chicken with Pickled Cucumber and Onion\n\nChicken Adobo Calorie Smart\n\nChicken and Dumplings Calorie Smart\n\nChicken Cacciatore Calorie Smart\n\nChicken Divan Pot Pie\n\nChicken Korma Calorie Smart\n\nChicken Marsala Calorie Smart\n\nChicken Piccata Calorie Smart \u2022 Fast\n\nChicken Pot Pie\n\nChicken Salad Sandwiches Fast \u2022 Easy\n\nChicken Tagine Calorie Smart\n\nCoffee Chicken with Quick Mole Sauce Calorie Smart\n\nCoq Au Vin Calorie Smart\n\nCountry French Chicken and Rice Calorie Smart\n\nCrunchy Cornmeal Chicken and Mango-Peach Salsa Fast\n\nCrunchy Oven-Fried Chicken\n\nDijon Chicken Smothered in Mushrooms Calorie Smart \u2022 Fast\n\nHerb-Garlic Butter Spatchcocked Chicken Calorie Smart\n\nHerb-Roasted Chicken and Vegetables Calorie Smart\n\nIndividual Chicken Pot Pies\n\nKorean Fried Chicken Calorie Smart\n\nOven Chicken Cordon Bleu Calorie Smart\n\nOven-Fried Chicken Calorie Smart \u2022 Easy\n\nOven-Fried Chicken Fingers\n\nPan-Fried Chicken with Romesco Sauce Calorie Smart\n\nPortabella Chicken Pot Pie\n\nProven\u00e7al Roasted Chicken\n\nRotisserie Spiced Chicken Calorie Smart \u2022 Slow Cooker\n\nSage Chicken and Potatoes Calorie Smart\n\nSkillet Chicken Nachos Fast\n\nSkillet-Fried Chicken Calorie Smart \u2022 Easy\n\nSpicy Chicken in Peanut Sauce Slow Cooker\n\nThai-Style Coconut Chicken Calorie Smart\n\nTuscan Rosemary Chicken and White Beans Calorie Smart \u2022 Fast\n\nWild Rice and Chicken Salad\n\n## Game Fowl\n\nDuckling with Orange Sauce\n\nOrange-Roasted Pheasants\n\nRoast Goose with Apple Stuffing\n\n## Turkey\n\nApple-Maple Brined Turkey Breast\n\nBrined Whole Turkey\n\nHerbed Turkey and Wild Rice Casserole Calorie Smart \u2022 Slow Cooker\n\nRaspberry Turkey-Rice Salad\n\nRoast Turkey\n\n## Sauces and Stuffings\n\nBread Stuffing\n\nCornbread Stuffing\n\nOyster Stuffing\n\nPan Cream Gravy\n\nPan Gravy Calorie Smart \u2022 Fast \u2022 Easy\n\nSausage Stuffing\n\nThin Gravy\n\nWild Rice Stuffing\n\n## Bonus Recipes\n\nBacon-Roasted Pepper Rice\n\nDijon Pork Smothered in Mushrooms\n\nEgg Salad Sandwiches\n\nEggs with Wild Rice\n\nHam Salad Sandwiches\n\nTuna Salad Sandwiches\n\nSkillet Beef Nachos\n\nWinter Squash with Wild Rice\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Chicken and Turkey Basics\n\nFrom fast-fix skillet meals and classic stews to roasted whole birds, chicken and turkey are ideal for everyday cooking and special occasions. Poultry works with many techniques and flavors, including many global dishes. Here are some useful guidelines for buying and storing poultry.\n\n## Buying Chicken and Turkey\n\n  * Look for tightly wrapped packages without tears or leaks, and with little or no liquid.\n  * Check for a fresh odor; if it does not smell right, don't buy it.\n  * Choose cold packages; those stacked above the meat case might not be cold enough.\n  * Do not purchase any packages if the sell-by or use-by date has already passed.\n  * Whole birds and cut-up pieces should be plump and meaty, with smooth, moist-looking skin and no traces of feathers.\n  * Boneless skinless products should look plump and moist.\n  * Chicken skin color can vary from yellow to white and does not indicate quality. Turkey skin should be cream colored.\n  * Cut ends of bones should be pink to red.\n  * Place packages in plastic bags so juices don't drip and contaminate other foods.\n  * Place all poultry in the refrigerator as soon as you get home. If you're shopping on a hot day or are more than 30 minutes from home, place poultry in an ice-packed cooler until you can refrigerate it.\n\n## Storing Chicken and Turkey\n\n  * Poultry wrapped in butcher paper should be repackaged in plastic wrap, foil or resealable freezer plastic bags.\n  * Store poultry in the meat compartment or coldest part of the refrigerator, or freeze as soon as possible.\n  * Cook or freeze poultry within 2 days of the sell-by date.\n  * If poultry was purchased frozen or was frozen at home and thawed, use within 2 days.\n\n## Thawing Chicken and Turkey\n\n  * Thaw poultry completely before cooking.\n  * Thaw in the refrigerator, not at room temperature.\n  * Remove giblets as soon as possible during thawing, then wrap and refrigerate.\n  * Fully thawed poultry will have no ice crystals and will be soft, with flexible joints.\n\nHerb-Roasted Chicken and Vegetables (page 322)\n\n## Herb-Roasted Chicken and Vegetables Calorie Smart\n\nPrep 20 min Total 2 hr 5 min \u25cf 6 servings\n\n  * \u00bc cup olive or vegetable oil\n  * 2 tablespoons chopped fresh or 1 teaspoon dried thyme leaves\n  * 2 tablespoons chopped fresh or 1 teaspoon dried marjoram leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon coarsely ground black pepper\n  * 1 lemon\n  * 1 whole chicken (4 to 5 lb)\n  * 6 small red potatoes, cut in half\n  * 1 cup ready-to-eat baby-cut carrots\n  * \u00bd lb fresh green beans, trimmed\n\n1 Heat oven to 375\u00b0F. In small bowl, mix oil, thyme, marjoram, salt and pepper. Grate 1 teaspoon peel from lemon; stir peel into oil mixture. Cut lemon into quarters; place in cavity of chicken.\n\n2 Fold wings across back of chicken so tips are touching. Skewer or tie legs together. Place chicken, breast side up, on rack in shallow roasting pan or 13x9-inch pan fitted with rack. Brush some of the oil mixture on chicken. Insert ovenproof meat thermometer so tip is in thickest part of thigh and does not touch bone.\n\n3 Roast uncovered 45 minutes. Arrange potatoes, carrots and green beans around chicken; brush remaining oil mixture on chicken and vegetables. Roast uncovered 30 to 45 minutes longer or until thermometer reads at least 165\u00b0F, legs move easily when lifted or twisted and vegetables are tender. Cover loosely with foil; let stand 15 to 20 minutes.\n\n4 Remove lemon and discard. Place chicken on platter; arrange vegetables around chicken. Serve with pan drippings.\n\n1 Serving: Calories 480; Total Fat 27g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 115mg; Sodium 320mg; Total Carbohydrate 23g (Dietary Fiber 4g, Sugars 3g); Protein 38g Exchanges: 1 Starch, 2 Vegetable, 4\u00bd Lean Meat Carbohydrate Choices: 1\u00bd\n\nButtery Rosemary-Honey Roasted Chicken Omit all ingredients except chicken. Place chicken on rack in roasting pan as directed in Step 2. Arrange 1\u00bd pounds buttercup or acorn squash, seeded, cut into \u00bd-inch rings or slices, then cut in half crosswise, and 2 medium onions, cut into 1-inch wedges, around chicken. In small bowl, mix \u00bd cup melted butter, \u00bc cup lemon juice, 2 tablespoons honey, 2 teaspoons dried rosemary, crushed, and 1 clove garlic, finely chopped. Brush half of mixture on chicken and vegetables. Insert thermometer as directed. Roast uncovered 1 hour. Brush chicken and vegetables with remaining butter mixture. Cover loosely with foil. Roast 45 to 55 minutes longer. Continue as directed.\n\nProven\u00e7al Roasted Chicken Heat oven to 400\u00b0F. Reduce oil to 1 teaspoon. Omit remaining ingredients except chicken, pepper and lemon. Place chicken on rack in roasting pan as directed in Step 2. In small bowl, mix grated peel and juice of the lemon and the oil. Drizzle half of lemon mixture over chicken. Pat pepper and 1 tablespoon herbes de Provence on skin of chicken. Place squeezed lemon halves inside chicken cavity.\n\nIn large bowl, toss 8 small red potatoes (1\u00bd lb), cut into quarters; 2 medium zucchini, cut into 1\u00bd-inch pieces; 1 can (14.5 oz) diced tomatoes with basil, garlic and oregano, drained; \u00bd cup chopped pitted kalamata olives and remaining lemon mixture. Arrange vegetables around chicken in pan. Insert thermometer as directed. Roast 1 hour 45 minutes to 2 hours. Continue as directed.\n\n### Preparing Chicken for Roasting\n\nCut lemon into quarters; place in cavity of chicken.\n\nSkewer or tie legs together.\n\nBrush some of the oil mixture on chicken.\n\nInsert ovenproof meat thermometer so tip is in thickest part of thigh but does not touch bone.\n\nRotisserie Spiced Chicken\n\n## Rotisserie Spiced Chicken CALORIE SMART \u2022 SLOW COOKER\n\nPrep 10 min Total 4 hr 10 min \u25cf 4 servings\n\nSpice Rub\n\n  * 2 teaspoons paprika\n  * 1 teaspoon garlic salt\n  * 1 teaspoon onion powder\n  * 1 teaspoon sugar\n  * 1 teaspoon chili powder\n  * \u00bd teaspoon dried thyme leaves, crushed\n  * \u00bd teaspoon dried marjoram leaves, crushed\n  * \u00bd teaspoon pepper\n\nChicken\n\n  * 1 whole chicken (3\u00bd to 4\u00bd lb)\n\n1 Spray 5- to 6-quart oval slow cooker with cooking spray. In small bowl, mix spice rub ingredients until well blended.\n\n2 Rub spice mixture on all sides of chicken. Do not tie legs. Place chicken in slow cooker, making sure it fits loosely (leave at least 1 inch of space around chicken).\n\n3 Cover; cook on Low heat setting 4 to 5 hours or until instant-read meat thermometer inserted in thickest part of inside thigh muscle and not touching bone reads at least 165\u00b0F and legs move easily when lifted or twisted. Do not remove lid before 4 hours.\n\n1 Serving: Calories 270; Total Fat 11g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 125mg; Sodium 370mg; Total Carbohydrate 3g (Dietary Fiber 1g, Sugars 1g); Protein 40g Exchanges: 5\u00bd Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 0\n\n### Carving a Whole Chicken\n\nPlace chicken, breast up, on cutting board. Remove ties or skewers.\n\nWhen holding drumstick, cut through joint between thigh and body. Separate drumstick and thigh by cutting through connecting joint.\n\nRemove wing from body by cutting through wing joint.\n\nJust to right of breastbone, cut down through meat to remove; slice breast meat.\n\nHerb-Garlic Butter Spatchcocked Chicken\n\n## Herb-Garlic Butter Spatchcocked Chicken CALORIE SMART\n\nSimilar to butterflying, spatchcocking refers to cutting poultry down the backbone and spreading it open like a book. Only poultry can be spatchcocked, but any boneless meat can be butterflied.\n\nPrep 20 min Total 1 hr 25 min \u25cf 4 servings\n\n  * 1 whole chicken (3\u00bd to 4\u00bd lb)\n  * \u2153 cup butter, softened\n  * 3 tablespoons chopped fresh or 1 tablespoon dried herb leaves (basil, chives, oregano, tarragon or thyme or combination)\n  * 5 teaspoons fresh lemon juice\n  * \u00bd teaspoon salt\n  * 2 cloves garlic, finely chopped\n\n1 Heat oven to 375\u00b0F. Line 15x10x1-inch pan with foil.\n\n2 Pat chicken dry with paper towels. Place chicken, breast side down, on cutting board. Using heavy-duty kitchen scissors or poultry shears, cut closely along one side of backbone from thigh end to neck. Repeat on other side. Remove backbone; save for making Chicken and Broth (page 143) or discard. Turn chicken over; flatten breast area by pressing firmly with heel of hand. Place chicken, breast side up, in pan. Tuck wings under breast.\n\n3 In small bowl, beat remaining ingredients with electric mixer on medium speed until light and fluffy. Starting at leg end of chicken, gently separate skin (do not peel back) from chicken breast using fingers, being careful not to tear or puncture skin. Rub half of butter mixture under skin to cover entire breast area; gently replace skin. Rub remaining butter mixture on outside of chicken.\n\n4 Roast uncovered 30 minutes. Brush chicken with pan juices. Roast 30 to 35 minutes longer or until thermometer inserted in thickest part of breast reads at least 165\u00b0F. Serve chicken with pan juices, if desired.\n\n1 Serving: Calories 400; Total Fat 26g (Saturated Fat 13g, Trans Fat 1g); Cholesterol 165mg; Sodium 550mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 40g Exchanges: 5\u00bd Lean Meat, 2 Fat Carbohydrate Choices: 0\n\nMake-Ahead Directions The herb-garlic butter can be made 1 day ahead of time and refrigerated; soften at room temperature before preparing the chicken.\n\n### Spatchcocking a Chicken\n\nWith chicken breast-side down, using heavy-duty kitchen or poultry shears, cut chicken closely along one side of backbone from thigh end to neck.\n\nCut along other side of backbone; remove bone.\n\nTurn chicken over. Flatten breast area by pressing firmly with heel of hand.\n\n### Cutting Up a Whole Chicken\n\nCut off each leg by cutting skin between thigh and body; continue cutting through meat between tail and hip joint, cutting as closely as possible to backbone. Bend leg back until hip joint pops out as shown.\n\nSeparate thigh and drumstick by cutting about \u215b inch from the fat line toward the drumstick as shown. (A thin white fat line runs crosswise at joint between drumstick and thigh.)\n\nRemove each wing from body by cutting into wing joint with sharp knife, rolling knife to let the blade follow through at the curve of joint as shown.\n\nSeparate back from breast by holding body, neck end down, and cutting downward along each side of backbone.\n\nBend breast halves back to pop out the keel bone. Remove keel bone as shown in Boning Chicken Breasts below.\n\n### Boning Chicken Breasts\n\nLoosen keel bone and white cartilage by running tip of index finger around both sides. Pull out bone in one or two pieces.\n\nInsert tip of knife under long rib bone. Resting knife against bones, use steady, even pressure to gradually trim meat away from bones. Cut rib cage away from breast, cutting through shoulder joint to remove entire rib cage. Repeat on other side.\n\nSlip knife under white tendons on either side of breast; loosen and pull out tendons (grasp end of tendons with paper towel if tendons are slippery). Remove skin if desired. Cut breast lengthwise in half.\n\n# Heirloom Recipe and New Twist\n\nClassic fried chicken is such a favorite\u2014delicious, moist, juicy. No wonder it's always welcome at the dinner table! We're sharing this often-requested, time-honored classic so you can make it for your family. The new twist is a slightly spicy Korean version\u2014traditionally fried twice, providing extra flavor and crispness to the chicken.\n\nHeirloom\n\nSkillet-Fried Chicken\n\n## Skillet-Fried Chicken Calorie Smart \u2022 EASY\n\nPrep 10 min Total 40 min \u25cf 6 servings\n\n  * \u00bd cup all-purpose flour\n  * 1 tablespoon paprika\n  * 1\u00bd teaspoons salt\n  * \u00bd teaspoon pepper\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n  * Vegetable oil\n\n1 In shallow dish, mix flour, paprika, salt and pepper. Coat chicken with flour mixture.\n\n2 In 12-inch nonstick skillet, heat \u00bc inch oil over medium-high heat. Add chicken, skin side down; cook about 10 minutes or until light brown on all sides.\n\n3 Reduce heat to low. Turn chicken skin side up. Simmer uncovered about 20 minutes, without turning, until juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F).\n\n1 Serving: Calories 330; Total Fat 20g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 85mg; Sodium 670mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 0g); Protein 28g Exchanges: \u00bd Starch, 4 Medium-Fat Meat Carbohydrate Choices: \u00bd\n\nLighter Directions For 250 calories and 11 grams of fat per serving, remove skin from chicken before cooking. Use 2 tablespoons oil in Step 2.\n\nButtermilk Fried Chicken Increase flour to 1 cup. Dip chicken into 1 cup buttermilk before coating with flour mixture.\nNew Twist\n\nKorean Fried Chicken\n\n## Korean Fried Chicken Calorie Smart\n\nPrep 1 hr 20 min Total 1 hr 20 min \u25cf 10 servings (3 pieces each)\n\nSauce\n\n  * \u00bc cup gochujang (Korean chile pepper paste)*\n  * \u00bc cup honey\n  * 1 tablespoon soy sauce\n  * 1 tablespoon unseasoned rice vinegar\n  * 1 tablespoon toasted sesame oil\n\nChicken\n\n  * Vegetable oil\n  * \u00bd cup cornstarch\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground ginger\n  * \u00bd teaspoon pepper\n  * 1 package (3 lb) chicken wingettes and drumettes, patted dry\n  * Cilantro leaves, if desired\n\n1 Heat oven to 350\u00b0F. In small bowl, mix together all sauce ingredients until well blended and smooth; set aside.\n\n2 In deep fryer or 4-quart Dutch oven, heat 3 to 4 inches oil to 350\u00b0F. In gallon-size resealable food-storage plastic bag, mix together cornstarch, salt, ginger and pepper. Add chicken; seal and toss to thoroughly coat.\n\n3 Fry 4 to 6 pieces of chicken at a time in oil 6 minutes; drain on paper towels. Fry the same pieces again for about 4 to 6 minutes longer or until golden brown and juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F). Drain on paper towels. Repeat with remaining chicken. (To keep first batches of chicken hot while frying remaining chicken, place in single layer on baking pan and place in oven.)\n\n4 Pour sauce over chicken; toss until evenly coated with sauce. Garnish with cilantro leaves.\n\n*Gochujang is a spicy Korean condiment. This dark red paste is usually made from red chile, glutinous rice, fermented soybeans, salt and sometimes a sweetener such as sugar. It is traditionally used in various dishes such as bibimbap or tteokbokki, and also in salads, stews, soups and marinated dishes.\n\n1 Serving (3 Pieces): Calories 180; Total Fat 10g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 25mg; Sodium 410mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 7g); Protein 8g Exchanges: 1 Other Carbohydrate, 1 Lean Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\n## Oven-Fried Chicken Calorie Smart \u2022 EASY\n\nPrep 10 min Total 1 hr \u25cf 6 servings\n\n  * \u00bc cup butter\n  * \u00bd cup all-purpose flour\n  * 1 teaspoon seasoned salt\n  * 1 teaspoon paprika\n  * \u00bd teaspoon garlic powder\n  * \u00bc teaspoon pepper\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n\n1 Heat oven to 425\u00b0F. In 13x9-inch pan, melt butter in oven. In shallow dish, mix flour, seasoned salt, paprika, garlic powder and pepper. Coat chicken with flour mixture. Place chicken, skin side down, in pan.\n\n2 Bake uncovered 30 minutes. Turn chicken skin side up; bake about 20 minutes longer or until juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F).\n\n1 Serving: Calories 330; Total Fat 21g (Saturated Fat 8g, Trans Fat 1g); Cholesterol 105mg; Sodium 330mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 0g); Protein 28g Exchanges: \u00bd Starch, 4 Lean Meat, 1\u00bd Fat Carbohydrate Choices: \u00bd\n\nLighter Directions For 240 calories and 11 grams of fat per serving, remove skin from chicken before cooking. Do not melt butter in pan; spray pan with cooking spray. Reduce butter to 2 tablespoons; drizzle melted butter over chicken after turning in Step 2.\n\nCrunchy Oven-Fried Chicken Remove skin from chicken, if desired. Substitute 1 cup cornflake crumbs or panko bread crumbs for the \u00bd cup flour. Dip chicken into \u00bc cup melted butter before coating with crumb mixture.\n\nOven-Fried Chicken Fingers Substitute 1\u00bd lb boneless skinless chicken breasts, cut crosswise into 1\u00bd-inch strips, for the cut-up whole chicken. Reduce butter to 2 tablespoons. After coating chicken with flour mixture in Step 1, toss with melted butter in pan. Bake uncovered 15 minutes. Turn strips; bake 10 to 15 minutes longer or until no longer pink in center.\n\nChicken and Dumplings\n\n## Chicken and Dumplings Calorie Smart\n\nPrep 20 min Total 1 hr 10 min \u25cf 6 servings\n\nBroth\n\n  * \u00bd cup all-purpose flour\n  * 5 cups chicken broth (for homemade broth, see page 143)\n  * 1 cup frozen sweet peas\n  * 2 medium carrots, thinly sliced (1 cup)\n  * 1 large onion, chopped (1 cup)\n  * \u00bc cup chopped fresh parsley or 1 tablespoon parsley flakes\n  * \u215b teaspoon pepper\n  * 3 cups cubed cooked chicken\n\nDumplings\n\n  * 1\u00bd cups all-purpose flour\n  * 1 tablespoon chopped fresh parsley or dried parsley flakes\n  * 2 teaspoons baking powder\n  * \u00bc teaspoon salt\n  * 3 tablespoons cold butter or shortening\n  * \u00be cup milk\n\n1 In 4-quart Dutch oven, place \u00bd cup flour. Gradually whisk in 1 cup of the broth until smooth. Gradually stir in remaining broth. Add vegetables and pepper; heat to boiling. Reduce heat to low. Stir in chicken. Cook uncovered about 20 minutes or until carrots are tender.\n\n2 In medium bowl, mix 1\u00bd cups flour, parsley, baking powder and salt. Cut in butter, using pastry blender or fork, until mixture looks like fine crumbs. Stir in milk. Divide dough into six portions, about \u2153 cup each. Drop dough by heaping tablespoonsful onto hot chicken mixture. Cook uncovered over low heat 10 minutes. Cover; cook 10 minutes longer or until dumplings are completely cooked in center.\n\n1 Serving: Calories 400; Total Fat 14g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 110mg; Sodium 1190mg; Total Carbohydrate 42g (Dietary Fiber 3g, Sugars 5g); Protein 26g Exchanges: 2 Starch, \u00bd Other Carbohydrate, \u00bd Vegetable, 2\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 3\n\nChicken Cacciatore\n\n## Chicken Cacciatore Calorie Smart\n\nPrep 35 min Total 1 hr 15 min \u25cf 6 servings\n\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n  * \u00bd cup all-purpose flour\n  * \u00bc cup vegetable oil\n  * 2 medium onions\n  * 1 medium green bell pepper\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 can (8 oz) tomato sauce\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * 1\u00bd teaspoons chopped fresh or \u00bd teaspoon dried oregano leaves\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried basil leaves\n  * \u00bd teaspoon salt\n  * 2 cloves garlic, finely chopped\n  * Hot cooked pasta (page 167), if desired\n  * Grated Parmesan cheese, if desired\n\n1 Coat chicken with flour. In 12-inch skillet, heat oil over medium-high heat. Cook chicken in oil 15 to 20 minutes or until brown on all sides; drain.\n\n2 Cut onions and bell pepper in half; remove seeds. Cut each half crosswise into quarters.. Add onions, bell pepper and remaining ingredients except cheese to skillet with chicken; stir.\n\n3 Heat to boiling; reduce heat. Cover and simmer 30 to 40 minutes or until juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F). Serve over pasta with cheese.\n\n1 Serving: Calories 400; Total Fat 23g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 85mg; Sodium 630mg; Total Carbohydrate 19g (Dietary Fiber 3g, Sugars 6g); Protein 30g Exchanges: 3 Vegetable, 3 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\nSlow-Cooker Directions Remove skin from chicken. Reduce flour to \u2153 cup and oil to 2 tablespoons; omit tomato sauce. Substitute 1 jar (4.5 oz) sliced mushrooms, drained, for the fresh mushrooms. Brown chicken as directed. Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, place half of chicken. Mix raw onions, bell pepper and remaining ingredients except cheese; spoon half of mixture over chicken. Repeat with remaining chicken and vegetable mixture. Cover; cook on Low heat setting 4 to 6 hours. Serve with cheese.\n\nCoq au Vin\n\n## Coq au Vin Calorie Smart\n\nA classic French dish of chicken simmered in red wine with mushrooms, onions, bacon or salt pork and various herbs.\n\nPrep 45 min Total 1 hr 20 min \u25cf 6 servings\n\n  * \u00bd cup all-purpose flour\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n  * 8 slices bacon\n  * \u00be cup frozen small whole onions\n  * 1 package (8 oz) sliced fresh mushrooms (about 3 cups)\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1 cup dry red wine or nonalcoholic red wine\n  * 4 medium carrots, cut into 2-inch pieces\n  * 1 clove garlic, finely chopped\n  * \u00bd teaspoon salt\n  * Bouquet garni*\n\n1 In shallow dish, mix flour, 1 teaspoon salt and the pepper. Coat chicken with flour mixture; set aside.\n\n2 In 12-inch skillet, cook bacon over medium heat 8 to 10 minutes, turning once, until crisp. Remove bacon with slotted spoon and drain on paper towels; set aside. Cook chicken in bacon drippings over medium heat about 15 minutes, turning occasionally, until brown on all sides.\n\n3 Move chicken to one side of skillet; add onions and mushrooms to other side. Cook uncovered over medium-high heat about 6 min-utes, stirring occasionally, until mushrooms are tender. Drain drippings from skillet.\n\n4 Crumble bacon. Add bacon, broth, wine, carrots, garlic, \u00bd teaspoon salt and the bouquet garni to skillet; stir. Heat to boiling; reduce heat. Cover and simmer about 35 minutes or until juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F). Remove bouquet garni; skim off excess fat.\n\n*Tie \u00bd teaspoon dried thyme leaves, 2 large sprigs fresh parsley and 1 dried bay leaf in cheesecloth bag or place in tea ball.\n\n1 Serving: Calories 350; Total Fat 19g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 95mg; Sodium 1020mg; Total Carbohydrate 12g (Dietary Fiber 1g, Sugars 2g); Protein 33g Exchanges: 1 Starch, 2 Vegetable, 4 Lean Meat, \u00bd Fat Carbohydrate Choices: 1\n\n## Browning Chicken\n\nUse medium-high heat to brown chicken. For the best browning, the skillet should be hot when you add the chicken. Let the chicken cook on one side without turning. When it is properly browned, it should release from the pan without sticking so the other side can be browned.\n\n## Learn to BRAISE\n\nBraising cuts of meat or poultry involves first browning them in a small amount of oil, then cooking in a little liquid\u2014either on the stovetop or in the oven. The long, slow cook time helps develop rich flavor while tenderizing the meat, making it a good choice for tougher, more economical cuts. Fish and vegetables can also be braised.\n\nMost braising uses broth or water, but other liquids can be used to add flavor and color. Braising with wine or beer mellows the alcohol, but the flavor lingers with the meat. Try apple cider when braising pork, for a slightly sweet, subtle flavor.\n\nFor the best flavor, cook braised dishes the day before you want to serve them, and refrigerate the food overnight so the flavors can develop more fully. Spoon off any congealed fat from the top; reheat slowly and serve.\n\n### Braising Tips\n\n  * Use a pot or skillet with a tight cover. If you are braising in the oven, the pot should be ovenproof.\n  * Season the meat or poultry on all sides. Brown or sear (see page 302) the meat in a small amount of oil. Remove the meat from the pan and place on a plate. \n  * Add liquid to the pan to deglaze (\u00be to 1 cup). Cook and stir, scraping any browned bits from the bottom of the pan.\n  * Return the meat or poultry to the pan with any accumulated liquid. There should be just enough liquid to come one-fourth to two-thirds of the way up the sides of the meat.\n  * Often other ingredients will be added, such as potatoes, carrots or other vegetables. For the best results, add these ingredients as directed in the recipe to prevent overcooking.\n  * Cover and simmer on the stovetop or in the oven until everything is tender.\n\n## Is Braising the Same as Stewing?\n\nThese are similar techniques, using different sizes of meat and amounts of liquid. Braising typically involves larger cuts of meat and less liquid. Stewing typically uses smaller pieces of meat with enough liquid to submerge it.\n\n## Chicken Tagine Calorie Smart\n\nTagines are Moroccan dishes of meat or poultry braised with vegetables, olives, spices and garlic. They are typically served over couscous.\n\nPrep 30 min Total 1 hr \u25cf 6 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n  * 1 medium onion, sliced\n  * 2 cloves garlic, finely chopped\n  * \u00bc cup chopped fresh cilantro\n  * 1 teaspoon ground cumin\n  * 1 teaspoon ground turmeric\n  * 1 teaspoon ground ginger\n  * 1 teaspoon salt\n  * 1 cinnamon stick (2 inch)\n  * 1 cup chicken broth (for homemade broth, see page 143)\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 1 cup pitted dried plums, cut into bite-size pieces\n  * \u00bd cup pitted green olives\n  * 1 small lemon, cut into quarters\n  * Hot cooked couscous or rice (page 215), if desired\n\n1 In 4-quart Dutch oven, heat oil over medium-high heat. Add chicken, skin side down, onion and garlic. Cook uncovered 6 to 10 minutes, turning chicken occasionally, until chicken is brown on all sides.\n\n2 Reduce heat to medium. Sprinkle cilantro, cumin, turmeric, ginger and salt over chicken. Add cinnamon stick; pour broth and tomatoes over chicken. Turn chicken several times to coat evenly. Add dried plums, olives and lemon, pressing into liquid around chicken. Reduce heat to low. Cover and simmer 30 minutes or until juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F).\n\n3 Place chicken on deep serving platter; cover to keep warm. Increase heat to high; boil sauce uncovered about 5 minutes, stirring occasionally, until thickened. Pour sauce over chicken. Garnish with additional chopped fresh cilantro, if desired. Serve over couscous.\n\n1 Serving: Calories 370; Total Fat 18g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 85mg; Sodium 920mg; Total Carbohydrate 23g (Dietary Fiber 4g, Sugars 13g); Protein 29g Exchanges: \u00bd Fruit, 2 Vegetable, 3\u00bd Medium-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\nChicken Adobo\n\n## Chicken Adobo Calorie Smart\n\nPrep 30 min Total 4 hr 50 min \u25cf 4 servings\n\nMarinade\n\n  * \u00bd cup cider vinegar\n  * \u00bd cup soy sauce\n  * \u00bc cup packed brown sugar\n  * 1 teaspoon black peppercorns\n  * \u215b teaspoon crushed red pepper flakes\n  * 4 cloves garlic, crushed\n  * 3 bay leaves\n\nChicken\n\n  * 4 chicken thighs\n  * 4 chicken drumsticks\n  * Hot cooked rice, if desired\n  * Diagonally sliced green onions, if desired\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all marinade ingredients. Add chicken. Cover dish or seal bag; refrigerate at least 4 hours but no longer than 24 hours, turning occasionally.\n\n2 Heat 12-inch nonstick skillet over medium-high heat. Remove chicken from marinade; reserve marinade. Place chicken, skin side down, in skillet. Cook 6 to 8 minutes, turning once, until deep golden brown. Pour marinade over chicken; heat to boiling. Cover and simmer 20 minutes, turning once, until juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F). Remove chicken to plate; cover to keep warm.\n\n3 Heat marinade to boiling; reduce heat to medium. Cook 5 to 7 minutes or until reduced by half or to desired consistency. Pour into fat separator, or skim fat from marinade. Serve chicken and marinade with rice; sprinkle with onions.\n\n1 Serving: Calories 280; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 155mg; Sodium 1880mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 14g); Protein 36g Exchanges: 1 Other Carbohydrate, 5 Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\nSpicy Chicken in Peanut Sauce\n\n## Spicy Chicken in Peanut Sauce SLOW Cooker\n\nPass small bowls of dry-roasted peanuts and chopped fresh cilantro to sprinkle over the chicken, and serve with wedges of warm pita bread.\n\nPrep 15 min Total 7 hr 15 min \u25cf 4 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 8 large bone-in chicken thighs (about 3 lb), skin removed\n  * 1 large onion, chopped (1 cup)\n  * 2 cans (14.5 oz each) diced tomatoes with green chiles, undrained\n  * 1 can (14.5 oz) crushed fire-roasted or regular diced tomatoes, undrained\n  * 2 tablespoons honey\n  * 1\u00bd teaspoons ground cumin\n  * 1 teaspoon ground cinnamon\n  * \u2153 cup creamy peanut butter\n  * 2 cups hot cooked couscous or jasmine or basmati rice (page 215)\n\n1 In 12-inch nonstick skillet, heat oil over medium-high heat. Cook chicken in oil about 4 minutes, turning once, until brown.\n\n2 Spray 4- to 5-quart slow cooker with cooking spray. In slow cooker, mix onion, diced and crushed tomatoes, honey, cumin and cinnamon. Add chicken; spoon tomato mixture over chicken. Cover; cook on Low heat setting 7 to 8 hours.\n\n3 Stir peanut butter into mixture in slow cooker until melted and well blended. Serve chicken and sauce over couscous.\n\n1 Serving: Calories 740; Total Fat 33g (Saturated Fat 8g, Trans Fat 0.5g); Cholesterol 140mg; Sodium 1180mg; Total Carbohydrate 52g (Dietary Fiber 8g, Sugars 24g); Protein 60g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 3 Vegetable, 7\u00bd Lean Meat, 2 Fat Carbohydrate Choices: 3\u00bd\n\nChicken Marsala\n\n## Chicken Marsala Calorie Smart\n\nPrep 25 min Total 35 min \u25cf 4 servings\n\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * \u00bd cup all-purpose flour\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 tablespoons olive or vegetable oil\n  * 2 cloves garlic, finely chopped\n  * 1 cup sliced fresh mushrooms (3 oz)\n  * \u00bc cup chopped fresh parsley or 1 tablespoon parsley flakes\n  * \u00bd cup Marsala wine or chicken broth\n  * Hot cooked pasta (page 167), if desired\n\n1 Flatten chicken as directed below. In shallow dish, mix flour, salt and pepper. Coat chicken with flour mixture.\n\n2 In 10-inch skillet, heat oil over medium-high heat. Cook garlic, mushrooms and parsley in oil 5 minutes, stirring frequently. Add chicken to skillet. Cook about 8 minutes, turning once, until brown.\n\n3 Stir in wine. Cook 8 to 10 minutes longer or until chicken is no longer pink in center. Serve with pasta.\n\n1 Serving: Calories 280; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 85mg; Sodium 230mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 2g); Protein 34g Exchanges: 1 Starch, 4 Very Lean Meat, 1 Fat Carbohydrate Choices: 1\n\n### Flattening Chicken Breasts\n\nPlace chicken breast between pieces of plastic wrap or waxed paper. Using flat side of meat mallet or rolling pin, gently pound chicken breasts until \u00bc inch thick.\n\nChicken Piccata\n\n## Chicken Piccata Calorie Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 6 servings\n\n  * 6 boneless skinless chicken breasts (about 1\u00bd lb)\n  * 1 egg, slightly beaten\n  * 1 tablespoon water\n  * \u00bd cup dry bread crumbs (any flavor)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u215b teaspoon garlic powder\n  * \u00bc cup all-purpose flour\n  * 2 tablespoons butter\n  * 2 tablespoons vegetable oil\n  * 2 tablespoons lemon juice\n  * 2 tablespoons dry white wine or chicken broth\n  * Chopped fresh parsley, if desired\n  * Lemon wedges, if desired\n\n1 Between pieces of plastic wrap or waxed paper, place each chicken breast; gently pound with flat side of meat mallet or rolling pin until about \u00bc inch thick. In small bowl, mix egg and water. In shallow dish, mix bread crumbs, salt, pepper and garlic powder. Coat chicken with flour. Dip into egg mixture; coat with bread crumb mixture.\n\n2 In 12-inch skillet, heat butter and oil over medium heat. Add chicken; cook 8 to 10 minutes, turning once, until chicken is no longer pink in center. Remove chicken from skillet; cover to keep warm.\n\n3 Stir lemon juice and wine into drippings in skillet; heat to boiling. Pour over chicken. Sprinkle with parsley; serve with lemon wedges.\n\n1 Serving: Calories 290; Total Fat 14g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 115mg; Sodium 370mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 0g); Protein 28g Exchanges: \u00bd Starch, 4 Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 1\nGlobal Flavors\n\n## Mediterranean\n\nSimple ingredients used in uncomplicated ways bring out the flavors of Mediterranean cooking.\n\nAnchovies Tiny salt-cured fish fillets canned in oil. Used in tiny amounts to intensify and add depth of flavor to sauces, pasta and other dishes. **Anchovy Paste,** available in tubes near the other condiments, is a convenient substitution and keeps longer than anchovies in the refrigerator.\n\nCapers Small ower buds from a bush native to the Mediterranean. Sun-dried and pickled or salt-cured, they add a pungent bite as garnishes or additions to sauces. **Caperberries** are the larger fruit from the same plant, with a similar but less intense flavor.\n\nChorizo Spicy paprika gives Spanish chorizo its signature ruddy red color and smoky flavor. It adds flavor to stews and other dishes when fried, or can be sliced and served as tapas.\n\nGreek Cheese Cheese is the main protein source in the Greek diet\u2014it's eaten all day long. **Kefalotiri,** one of the most popular hard cheeses, has a sharp, salty flavor, somewhat similar to Gruy\u00e8re or Pecorino Romano. **Manouri** is a typically unsalted soft white cheese similar to cream cheese.\n\nHoneycomb Greeks make and use a lot of honey in both sweet and savory dishes. Honey varies greatly in flavor, depending on where it's made. Both the comb and the honey in it are edible.\n\nMarcona Almonds Spanish almonds long loved for being more tender, sweet and delicate than California almonds. Traditionally used in tapas and turr\u00f3n\u2014a Spanish nougat candy.\n\nPhyllo (Filo) Similar to strudel dough, tissue-thin sheets of pastry used in Greek savory and sweet dishes\u2014most commonly, baklava and spanakopita. Look for it near other frozen dough.\n\nSeasonings **Dukka** Egyptian spice blend sprinkled over meats or vegetables or made into a dip. Typically it contains hazelnuts or chickpeas ground together with toasted nuts, sesame seed, coriander and cumin. **Saffron** Handpicked and dried stigmas from the purple crocus; available in threads or powdered. It adds color, aroma and avor to many Mediterranean dishes. Very expensive\u2014a little goes a long way. **Urfa Biber** Sun-dried Turkish chiles with chocolate and wine flavors, typically ground with a bit of salt. Adds heat, saltiness and acidity to dishes.\n\n## Panfried Chicken with Romesco Sauce Calorie Smart\n\nPrep 25 min Total 45 min \u25cf 6 servings\n\nRomesco Sauce\n\n  * \u00bd cup slivered almonds\n  * 1 slice firm crusty white bread (about 5x5x\u00bd inch)\n  * 1 medium tomato, cut in half, seeded\n  * 2 cloves garlic, peeled\n  * 1 jar (12 oz) roasted red bell peppers, rinsed, drained and patted dry\n  * \u00bc cup olive oil\n  * 1 tablespoon sherry vinegar\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon smoked paprika\n\nChicken\n\n  * 2 tablespoons olive oil\n  * 6 boneless skinless chicken breasts (about 1\u00be lb)\n  * 1 teaspoon garlic salt\n  * Chopped fresh parsley, if desired\n\n1 Heat oven to 400\u00b0F. In 15x10x1-inch pan, place almonds, bread, tomato and garlic in single layer. Bake 5 to 6 minutes or until almonds and bread are lightly toasted. Break bread into bite-size pieces.\n\n2 In food processor, place toasted almonds, bread, tomato and garlic. Cover and process, using quick on-and-off motions, until coarsely chopped. Add remaining sauce ingredients. Cover and process, using quick on-and-off motions, until almost smooth. Transfer sauce to small bowl; set aside. Sauce will thicken as it stands.\n\n3 In 12-inch nonstick skillet, heat 2 tablespoons oil over medium heat. Sprinkle both sides of chicken with garlic salt; add to skillet. Cook 15 to 20 minutes, turning once, until juice of chicken is clear when center of thickest piece is cut (at least 165\u00b0F). Serve chicken with sauce; sprinkle with parsley.\n\n1 Serving: Calories 400; Total Fat 23g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 105mg; Sodium 630mg; Total Carbohydrate 8g (Dietary Fiber 1g, Sugars 3g); Protein 39g Exchanges: \u00bd Other Carbohydrate, 5\u00bd Very Lean Meat, 4 Fat Carbohydrate Choices: \u00bd\n\nMake-Ahead Directions The Romesco Sauce can be made up to 1 day ahead; cover and refrigerate. Let stand at room temperature 30 minutes before serving.\n\nCoffee Chicken with Quick Mole Sauce\n\n## Coffee Chicken with Quick Mole Sauce Calorie Smart\n\nPrep 45 min Total 2 hr 45 min \u25cf 6 servings\n\nChicken and Marinade\n\n  * 4 cups water\n  * \u2153 cup salt\n  * \u2153 cup packed dark brown sugar\n  * 1 tablespoon cumin seed\n  * 1 tablespoon chili powder\n  * 1 lime, thinly sliced\n  * 1 cup strong brewed coffee\n  * 6 boneless skinless chicken breasts (about 1\u00be lb)\n  * 2 tablespoons vegetable oil\n\nMole Sauce\n\n  * 1 medium onion, thinly sliced\n  * 2 cloves garlic, finely chopped\n  * 1 can (28 oz) crushed tomatoes, undrained\n  * 1\u00bc cups chicken broth (for homemade broth, see page 143)\n  * 1 tablespoon creamy peanut butter\n  * 1 teaspoon granulated sugar\n  * \u00bd teaspoon ground cumin\n  * Dash ground cinnamon\n  * 1 oz unsweetened baking chocolate\n  * 2 chipotle chiles in adobo sauce (from 7-oz can), chopped\n  * *Corn tortillas, if desired.\n\n1 In 2-quart saucepan, heat water, salt, brown sugar, cumin seed, chili powder and lime slices over medium heat, stirring occasionally, until salt and brown sugar are dissolved. Remove from heat; stir in coffee. Cool to room temperature.\n\n2 Place chicken in 1-gallon resealable food-storage plastic bag or large bowl; pour cooled coffee mixture over chicken. Seal bag or cover bowl; refrigerate 2 to 3 hours to marinate, turning chicken occasionally.\n\n3 Remove chicken from marinade; discard marinade. Pat chicken dry with paper towels. In 12-inch skillet, heat oil over medium heat. Cook chicken in oil 5 to 6 minutes, turning once, until golden brown. Remove from skillet; set aside.\n\n4 Add onion to skillet; cook over medium heat 5 minutes, stirring occasionally, until soft and lightly browned. Add garlic; cook 30 seconds. Add remaining sauce ingredients, except corn tortillas. Cook uncovered 3 minutes, stirring occasionally, until mixture simmers and peanut butter and chocolate are melted.\n\n5 Reduce heat to medium-low; arrange chicken in sauce. Cook uncovered 15 to 20 minutes, stirring occasionally, until sauce thickens and juice of chicken is clear when center of thickest piece is cut (at least 165\u00b0F). Serve with corn tortillas.\n\n*Warm corn tortillas as directed on package, if desired.\n\n1 Serving: Calories 320; Total Fat 14g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 80mg; Sodium 2130mg; Total Carbohydrate 15g (Dietary Fiber 3g, Sugars 8g); Protein 33g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, 4\u00bd Lean Meat Carbohydrate Choices: 1\n\n## Tuscan Rosemary Chicken and White Beans Calorie Smart \u2022 FAst\n\nPrep 15 min Total 30 min \u25cf 4 servings\n\n  * \u2153 cup Italian dressing\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * \u00bc cup water\n  * 2 medium carrots, sliced (1 cup)\n  * 2 medium stalks celery, sliced (1 cup)\n  * \u00bc cup coarsely chopped drained sun-dried tomatoes in oil\n  * 1 teaspoon dried rosemary leaves, crushed\n  * 1 can (19 oz) cannellini beans, drained, rinsed\n\n1 In 12-inch skillet, heat dressing over medium-high heat. Add chicken; cook 4 to 6 minutes, turning once, until lightly browned.\n\n2 Reduce heat to medium-low. Add water, carrots, celery, tomatoes and rosemary. Cover and simmer about 10 minutes or until carrots are crisp-tender and juice of chicken is clear when center of thickest piece is cut (at least 165\u00b0F).\n\n3 Stir in beans. Cover and cook 5 minutes or until beans are thoroughly heated.\n\n1 Serving: Calories 390; Total Fat 9g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 85mg; Sodium 340mg; Total Carbohydrate 33g (Dietary Fiber 8g, Sugars 5g); Protein 42g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 5\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 2\n\nThai-Style Coconut Chicken\n\n## Thai-Style Coconut Chicken Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 4 servings\n\n  * 1 tablespoon vegetable oil\n  * 1 lb boneless skinless chicken breasts, cut into bite-size pieces\n  * 1 teaspoon grated lime peel\n  * 1 teaspoon grated gingerroot\n  * 1 clove garlic, finely chopped\n  * 2 serrano chiles or 1 jalape\u00f1o chile, seeded, finely chopped\n  * \u00bc cup finely chopped fresh cilantro\n  * 1 can (14 oz) coconut milk (not cream of coconut)\n  * 1 teaspoon packed brown sugar\n  * \u00bd teaspoon salt\n  * 1 tablespoon soy sauce\n  * 1 cup fresh sugar snap peas\n  * 1 medium green bell pepper, cut into 1-inch pieces\n  * 1 medium tomato, seeded, chopped (\u00be cup)\n  * 1 tablespoon chopped fresh basil leaves\n  * Hot cooked jasmine rice (page 215), if desired\n\n1 In nonstick wok or 12-inch skillet, heat oil over medium-high heat. Add chicken; cook 2 to 3 minutes, stirring constantly, until chicken is no longer pink in center. Add lime peel, gingerroot, garlic, chiles and cilantro; cook and stir 1 minute.\n\n2 Pour coconut milk over chicken. Stir in brown sugar, salt, soy sauce, peas and bell pepper. Reduce heat to medium. Simmer uncovered 3 to 5 minutes, stirring occasionally, until vegetables are crisp-tender. Stir in tomato.\n\n3 Spoon into shallow serving bowls; top with basil. Serve with rice.\n\n1 Serving: Calories 430; Total Fat 26g (Saturated Fat 17g, Trans Fat 0g); Cholesterol 85mg; Sodium 650mg; Total Carbohydrate 14g (Dietary Fiber 4g, Sugars 8g); Protein 35g Exchanges: \u00bd Starch, 1 Vegetable, 5 Lean Meat, 2 Fat Carbohydrate Choices: 1\n\nChicken Korma\n\n## Chicken Korma Calorie Smart\n\nPrep 20 min Total 1 hr 30 min \u25cf 4 servings\n\n  * 1 lb boneless skinless chicken breasts, cut crosswise into \u00bd-inch strips\n  * \u00bc cup whipping cream\n  * 2 tablespoons finely chopped gingerroot\n  * 5 cloves garlic, finely chopped\n  * 1 tablespoon finely chopped fresh cilantro\n  * 1 teaspoon coriander seed, ground\n  * \u00bd teaspoon cumin seed, ground\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 2 tablespoons Clarified Butter (page 469) or butter\n  * \u00bd cup tomato sauce\n  * \u00bc cup finely chopped fresh or 2 tablespoons crumbled dried fenugreek leaves (m\u00e9thi)*\n\n1 In medium bowl, mix all ingredients except clarified butter, tomato sauce and fenugreek. Cover and refrigerate at least 1 hour but no longer than 24 hours.\n\n2 In 10-inch skillet, heat clarified butter over medium heat. Add chicken mixture and tomato sauce. Cook about 5 minutes, stirring frequently, until chicken is partially cooked.\n\n3 Stir in fenugreek; reduce heat. Cover and simmer about 10 minutes or until chicken is no longer pink in center. Serve with rice, if desired.\n\n*Watercress leaves or fresh parsley can be substituted for the fenugreek, but the flavor will be milder.\n\n1 Serving: Calories 270; Total Fat 15g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 100mg; Sodium 570mg; Total Carbohydrate 7g (Dietary Fiber 2g, Sugars 2g); Protein 28g Exchanges: \u00bd Starch, 3\u00bd Lean Meat, 1 Fat Carbohydrate Choices: \u00bd\n\nCaramel Chicken with Pickled Cucumber and Onion\n\n## Caramel Chicken with Pickled Cucumber and Onion\n\nPrep 25 min Total 50 min \u25cf 4 servings\n\nPickled Cucumber and Onion\n\n  * \u00bc cup cider vinegar\n  * 1 teaspoon granulated sugar\n  * \u00bc teaspoon crushed red pepper flakes\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * 1 medium cucumber\n  * 1 small red onion, cut in half, sliced (\u00bd cup)\n\nChicken and Sauce\n\n  * \u00bd cup packed brown sugar\n  * \u2154 cup chicken broth (for homemade broth, see page 143)\n  * 3 tablespoons rice vinegar\n  * 2 tablespoons soy sauce\n  * 1 tablespoon vegetable oil\n  * 8 boneless skinless chicken thighs, cut into 1-inch pieces\n  * 1 teaspoon finely chopped gingerroot\n  * 2 cloves garlic, finely chopped\n  * Hot cooked rice (page 215), if desired\n  * Black sesame seed or toasted sesame seed, if desired\n\n1 In medium glass or plastic bowl, mix cider vinegar, granulated sugar, pepper flakes, salt and pepper until sugar is dissolved. Cut cucumber in half lengthwise; remove seeds with spoon. Cut each half crosswise into \u00bc-inch slices. Add cucumber and onion to bowl; toss to coat. Set aside; stir occasionally.\n\n2 In small bowl, mix brown sugar, broth, rice vinegar and soy sauce until sugar is almost dissolved; set aside. In 12-inch nonstick skillet, heat oil over medium-high heat. Add chicken, gingerroot and garlic; cook about 5 minutes, stirring occasionally, until chicken is lightly browned.\n\n3 Add broth mixture to skillet. Heat to boiling; reduce heat to medium-low. Cook uncovered 20 to 25 minutes or until chicken is golden brown and sauce is thickened and reduced by half.\n\n4 Serve chicken and sauce over rice. Top with pickled cucumber and onion; sprinkle with sesame seed.\n\n1 Serving: Calories 430; Total Fat 14g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 120mg; Sodium 860mg; Total Carbohydrate 33g (Dietary Fiber 1g, Sugars 30g); Protein 42g Exchanges: 2 Other Carbohydrate, 6 Very Lean Meat, 2 Fat Carbohydrate Choices: 2\n\n## Skillet Chicken Nachos FAST\n\nPrep 20 min Total 20 min \u25cf 6 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 1\u00bc lb boneless skinless chicken breasts, cut into \u00bc-inch pieces\n  * 1 package (1 oz) taco seasoning mix\n  * 1 can (8 oz) tomato sauce\n  * 1 medium red bell pepper, chopped (1 cup)\n  * 1 can (15 oz) black beans, drained, rinsed\n  * 1 can (7 oz) whole kernel sweet corn, drained\n  * 2 cups shredded Mexican cheese blend (8 oz)\n  * 6 oz tortilla chips (about 42 chips)\n  * \u00bc cup chopped fresh cilantro\n\n1 In 12-inch nonstick skillet, heat oil over medium-high heat. Cook chicken in oil 3 to 5 minutes, stirring occasionally, until no longer pink in center.\n\n2 Stir in taco seasoning mix, tomato sauce, bell pepper, beans, corn and 1 cup of the cheese. Reduce heat to medium; cook 3 to 5 minutes, stirring occasionally, until thoroughly heated and cheese is melted.\n\n3 Divide tortilla chips among 6 plates. Spoon chicken mixture evenly over chips. Sprinkle with remaining 1 cup cheese and the cilantro.\n\n1 Roll-up: Calories 520; Total Fat 24g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 95mg; Sodium 1320mg; Total Carbohydrate 38g (Dietary Fiber 5g, Sugars 4g); Protein 36g Exchanges: 2 Starch, \u00bd Other Carbohydrate, \u00bd Vegetable, 4 Very Lean Meat, 4 Fat Carbohydrate Choices: 2\u00bd\n\nSkillet Beef Nachos Substitute 1\u00bc lb ground beef for the chicken. In Step 1, cook beef 5 to 7 minutes or until thoroughly cooked. Drain and continue as directed.\n\n## Chicken Salad Sandwiches FAST \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 4 sandwiches\n\n  * 1\u00bd cups chopped cooked chicken or turkey\n  * 1 medium stalk celery, chopped (\u00bd cup)\n  * 1 small onion, finely chopped (\u2153 cup)\n  * \u00bd cup mayonnaise or salad dressing\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * 8 slices bread\n\nIn medium bowl, mix chicken, celery, onion, mayonnaise, salt and pepper. Spread mixture on 4 of the bread slices. Top with remaining bread.\n\n1 Sandwich: Calories 430; Total Fat 27g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 60mg; Sodium 630mg; Total Carbohydrate 27g (Dietary Fiber 1g, Sugars 2g); Protein 19g Exchanges: 2 Starch, 2 Lean Meat, 4 Fat Carbohydrate Choices: 2\n\nLighter Directions For 260 calories and 6 grams of fat per serving, use fat-free mayonnaise.\n\nEgg Salad Sandwiches Substitute 6 Hard-Cooked Eggs (page 91), chopped, for the chicken.\n\nHam Salad Sandwiches Substitute 1\u00bd cups chopped cooked ham for the chicken. Omit salt and pepper. Stir in 1 teaspoon yellow mustard.\n\nTuna Salad Sandwiches Substitute 2 cans (5 oz each) tuna in water, drained, for the chicken. Stir in 1 teaspoon lemon juice.\n\n## Asian Chicken Roll-Ups Calorie Smart \u2022 FAst\n\nPrep 15 min Total 15 min \u25cf 4 roll-ups\n\n  * 2 tablespoons creamy peanut butter\n  * 2 tablespoons teriyaki baste and glaze (from 12-oz bottle) or stir-fry sauce\n  * 1 tablespoon packed brown sugar\n  * 1 tablespoon hot water\n  * 1 teaspoon sesame or vegetable oil\n  * 4 flour tortillas (8 to 10 inch)\n  * \u00bd lb thinly sliced roasted chicken or turkey breast (from deli)\n  * 1\u00bd cups shredded iceberg lettuce\n  * 1\u00bd cups shredded carrots\n  * \u00bd cup chopped fresh cilantro\n\n1 In small bowl, beat peanut butter, teriyaki glaze, brown sugar, water and oil with whisk until blended.\n\n2 Spread about 2 tablespoons peanut butter mixture over each tortilla. Top each with one-fourth of the chicken, about \u2153 cup lettuce, about \u2153 cup carrots and 2 tablespoons cilantro. Roll up tortillas.\n\n1 Serving: Calories 290; Total Fat 10g (Saturated Fat 2g, Trans Fat 0.5g); Cholesterol 25mg; Sodium 540mg; Total Carbohydrate 35g (Dietary Fiber 3g, Sugars 7g); Protein 16g Exchanges: 2 Starch, 1 Vegetable, 1 Lean Meat, 1 Fat Carbohydrate Choices: 2\n\nSage Chicken and Potatoes\n\n## Sage Chicken and Potatoes Calorie Smart\n\nPrep 10 min Total 1 hr 10 min \u25cf 4 servings\n\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * 3 medium unpeeled russet potatoes, cut into \u00be-inch pieces (3 cups)\n  * 1\u00bd cups ready-to-eat baby-cut carrots\n  * I\u00bd cups Pan Gravy (page 346) or 1 jar (12 oz) home-style gravy\n  * 2 tablespoons Worcestershire sauce\n  * 1 teaspoon dried sage leaves\n  * \u00bd teaspoon garlic-pepper blend\n\n1 Heat oven to 400\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 In baking dish, arrange chicken, potatoes and carrots. In small bowl, mix remaining ingredients; pour over chicken and vegetables.\n\n3 Spray sheet of foil with cooking spray; place sprayed side down over baking dish. Bake 50 to 60 minutes or until vegetables are tender and juice of chicken is clear when center of thickest piece is cut (at least 165\u00b0F).\n\n1 Serving: Calories 320; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 680mg; Total Carbohydrate 30g (Dietary Fiber 4g, Sugars 5g); Protein 31g Exchanges: 2 Starch, 3 Very Lean Meat, 1 Fat Carbohydrate Choices: 2\n\n## Chicken Pot Pie\n\nPot pie for dinner is possible even if you're busy. Just pick up refrigerated pie crusts instead of making them yourself. You'll cut your preparation time in half.\n\nPrep 40 min Total 1 hr 15 min \u25cf 6 servings\n\n  * \u2153 cup butter\n  * \u2153 cup all-purpose flour\n  * \u2153 cup chopped onion\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1\u00be cups chicken broth (for homemade broth, see page 143)\n  * \u2154 cup milk\n  * 2\u00bd to 3 cups cut-up cooked chicken or turkey\n  * 1 box (10 oz) frozen peas and carrots*\n  * Two-Crust Pastry (page 586)\n\n1 In 2-quart saucepan, melt butter over medium heat. Stir in flour, onion, salt and pepper. Cook and stir until mixture is bubbly; remove from heat. Stir in broth and milk. Heat to boiling, stirring constantly. Boil and stir 1 minute. Stir in chicken and peas and carrots; remove from heat.\n\n2 Heat oven to 425\u00b0F. Roll two-thirds of pastry into 13-inch square. Ease into ungreased 9-inch square (2-quart) glass baking dish. Pour chicken mixture into pastry-lined dish.\n\n3 Roll remaining pastry into 11-inch square. Cut out designs with 1-inch cookie cutter. Place square over chicken mixture. Arrange cutouts on pastry. Turn edges of pastry under and flute (see Pastry Edge Treatments, page 587). Bake about 35 minutes or until golden brown.\n\n*About 1\u00bd cups of any frozen vegetable combination can be substituted. (Smaller vegetable pieces will be distributed more evenly throughout the filling.)\n\n1 Serving: Calories 670; Total Fat 43g (Saturated Fat 14g, Trans Fat 5g); Cholesterol 80mg; Sodium 1050mg; Total Carbohydrate 44g (Dietary Fiber 3g, Sugars 4g); Protein 25g Exchanges: 3 Starch, 2\u00bd Lean Meat, 6\u00bd Fat Carbohydrate Choices: 3\n\nLighter Directions For 22 grams of fat and 425 calories per serving, reduce butter to 2 tablespoons, increase flour to \u00bd cup, reduce broth to 1\u00bd cups and use fat-free (skim) milk. After melting butter in Step 1, stir in milk, onion, salt and pepper; stir in flour. Cook and stir until mixture is bubbly; remove from heat. Stir in broth. Continue as directed, except use One-Crust Pastry (page 586) for top crust. Spray 9-inch square baking dish or pan with cooking spray. Pour chicken mixture into dish (not pastry-lined). Bake 25 to 35 minutes or until golden brown.\n\nChicken Divan Pot Pie Substitute 1 box (9 oz) frozen broccoli cuts for the peas and carrots. Stir 1 teaspoon curry powder and a few drops red pepper sauce into flour mixture before cooking. Continue as directed.\n\nPortabella Chicken Pot Pie Add 1\u00bd cups sliced fresh baby portabella mushrooms to the flour mixture before cooking in Step 1. Continue as directed.\n\nIndividual Chicken Pot Pies\n\n## Individual Chicken Pot Pies\n\nIf you wish, make a double batch of Cream of Mushroom Soup, page 161, and save out 3\u00bd cups for another use. Use the remaining soup in place of the condensed soup, half-and-half, sherry and water below.\n\nPrep 35 min Total 1 hr 15 min \u25cf 8 servings\n\n  * 1 tablespoon olive oil\n  * 1 cup diced unpeeled red potatoes\n  * 1 medium carrot, thinly sliced (\u00bd cup)\n  * 1 medium stalk celery, thinly sliced (\u00bd cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 cloves garlic, finely chopped\n  * 2 cans (10.75 oz each) condensed 98% fat-free cream of chicken soup with 45% less sodium\n  * 1 cup fat-free or regular half-and-half\n  * \u00bd cup dry sherry, nonalcoholic wine or chicken broth\n  * \u00be cup water\n  * 1\u00bd teaspoons parsley flakes\n  * 1 teaspoon poultry seasoning\n  * 2 cups cubed cooked chicken breast\n  * 1 cup frozen sweet peas, thawed\n  * 1 package (17.3 oz) frozen puff pastry sheets, thawed\n\n1 Heat oven to 400\u00b0F. Lightly spray 8 (8-oz) individual baking dishes (ramekins) with cooking spray. Place dishes on cookie sheet.\n\n2 In 3-quart saucepan, heat oil over medium heat. Add potatoes, carrot, celery, onion and garlic; cook about 10 minutes, stirring occasionally, until vegetables are tender. Stir in soup, half-and-half, sherry, water, 1 teaspoon of the parsley flakes, the poultry seasoning, chicken and peas; heat to boiling. Remove from heat.\n\n3 Cut 8 (4-inch) squares from puff pastry. Divide chicken mixture evenly among baking dishes; top each with puff pastry square. Sprinkle evenly with remaining \u00bd teaspoon parsley flakes.\n\n4 Bake 30 to 35 minutes or until pastry is puffed and golden brown. Cool 5 minutes before serving.\n\n1 Serving: Calories 540; Total Fat 29g (Saturated Fat 7g, Trans Fat 6g); Cholesterol 35mg; Sodium 530mg; Total Carbohydrate 48g (Dietary Fiber 2g, Sugars 4g); Protein 18g Exchanges: 3 Starch, \u00bd Vegetable, 1 Very Lean Meat, 5\u00bd Fat Carbohydrate Choices: 3\n\nBaked Chicken Panzanella\n\n## Baked Chicken Panzanella Calorie Smart\n\nPrep 10 min Total 40 min \u25cf 6 servings\n\n  * 2 cups chopped cooked chicken\n  * 1 can (14.5 oz) diced tomatoes with garlic, onion and oregano, drained\n  * \u00bc cup sliced green onions (4 medium)\n  * 1 package (5 oz) Italian-seasoned croutons\n  * \u00bc cup Italian dressing\n  * \u00be cup shredded Parmesan cheese (3 oz)\n  * \u00bc cup sliced fresh basil leaves\n\n1 Heat oven to 350\u00b0F. In ungreased 11x7-inch (2-quart) baking dish, layer chicken, tomatoes, onions and croutons. Drizzle with dressing.\n\n2 Cover with foil; bake 20 minutes. Uncover; top with cheese. Bake about 10 minutes longer or until hot and cheese is melted. Sprinkle with basil.\n\n1 Serving (1\u00bd Cups): Calories 290; Total Fat 14g (Saturated Fat 4.5g, Trans Fat 1.5g); Cholesterol 50mg; Sodium 830mg; Total Carbohydrate 20g (Dietary Fiber 2g, Sugars 4g); Protein 21g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2\u00bd Medium-Fat Meat Carbohydrate Choices: 1\n\nCrunchy Cornmeal Chicken with Mango-Peach Salsa\n\n## Crunchy Cornmeal Chicken with Mango-Peach Salsa FAST\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\nChicken\n\n  * \u00bd cup yellow cornmeal\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * 2 tablespoons vegetable oil\n  * Hot cooked rice (page 215), if desired\n\nMango-Peach Salsa\n\n  * 3 medium peaches, peeled, chopped (1\u00bd cups)*\n  * 1 ripe large mango, peeled, seeded and chopped (1\u00bd cups)**\n  * 1 large tomato, seeded, chopped (1 cup)\n  * \u00bc cup chopped fresh cilantro\n  * 3 tablespoons vegetable oil\n  * 2 tablespoons white vinegar\n  * \u00bc teaspoon salt\n\n1 In shallow dish, mix cornmeal, \u00bd teaspoon salt and the pepper. Coat chicken with cornmeal mixture.\n\n2 In 10-inch skillet, heat 2 tablespoons oil over medium-high heat. Cook chicken in oil 15 to 20 minutes, turning once, until juice of chicken is clear when center of thickest piece is cut (at least 165\u00b0F).\n\n3 Meanwhile, in large bowl, mix all salsa ingredients. Serve chicken with salsa over rice.\n\n*About 1\u00bd cups chopped frozen (thawed) sliced peaches can be substituted for the fresh peaches.\n\n**Refrigerated mango slices (from a jar), well drained and chopped, can be substituted for the fresh mango.\n\n1 Serving: Calories 450; Total Fat 22g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 85mg; Sodium 520mg; Total Carbohydrate 30g (Dietary Fiber 5g, Sugars 12g); Protein 34g Exchanges: 2 Fruit, 5 Lean Meat, 1 Fat Carbohydrate Choices: 2\n\nDijon Chicken Smothered in Mushrooms\n\n## Dijon Chicken Smothered in Mushrooms Calorie Smart \u2022 FAst\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * \u00bc cup all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 tablespoons olive or vegetable oil\n  * \u00bd cup chicken broth (for homemade broth, see page 143)\n  * 1 jar (4.5 oz) sliced mushrooms, drained\n  * 4\u00bd teaspoons Dijon mustard\n  * Fresh thyme sprigs, if desired\n\n1 Between pieces of plastic wrap or waxed paper, place each chicken breast; gently pound with flat side of meat mallet or rolling pin until about \u00bc inch thick. In shallow dish, mix flour, salt and pepper. Coat chicken with flour mixture.\n\n2 In 12-inch nonstick skillet, heat oil over medium-high heat. Cook chicken in oil 6 to 8 minutes, turning once, until no longer pink in center. Remove chicken from skillet to serving plate; cover to keep warm.\n\n3 Stir broth into skillet. Heat to boiling over medium-high heat. Stir in mushrooms and mustard. Cook 2 to 3 minutes, stirring frequently, until slightly thickened. Spoon sauce over chicken. Garnish with thyme.\n\n1 Serving: Calories 240; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 70mg; Sodium 750mg; Total Carbohydrate 8g (Dietary Fiber 1g, Sugars 0g); Protein 27g Exchanges: \u00bd Starch, 3\u00bd Lean Meat Carbohydrate Choices: \u00bd\n\nDijon Pork Smothered in Mushrooms Substitute pork tenderloin for the chicken. Cut into 1-inch slices, and flatten as directed in Step 1. Continue as directed, cooking until pork is no longer pink in center.\n\nOven Chicken Cordon Bleu\n\n## Oven Chicken Cordon Bleu Calorie Smart\n\nPrep 20 min Total 50 min \u25cf 4 servings\n\n  * 4 boneless skinless chicken breasts (about 1\u00bc lb)\n  * 2 teaspoons Dijon mustard\n  * 4 teaspoons chopped fresh chives\n  * 4 very thin slices (about \u00be oz each) lean cooked ham\n  * 4 very thin slices (about \u00be oz each) reduced-fat Swiss cheese\n  * 1 egg white\n  * 1 tablespoon water\n  * \u2153 cup finely crushed cornflakes or bran flakes cereal\n  * \u00bc teaspoon paprika\n\n1 Heat oven to 375\u00b0F. Spray 8-inch square (2-quart) glass baking dish with cooking spray. Between pieces of plastic wrap or waxed paper, place each chicken breast; gently pound with flat side of meat mallet or rolling pin until about \u00bc inch thick.\n\n2 Spread each chicken breast with \u00bd teaspoon mustard; sprinkle with 1 teaspoon chives. Cut ham and cheese slices to fit chicken. Place ham and cheese on chicken; roll up, tucking ends inside.\n\n3 In shallow dish, slightly beat egg white and water. Place cereal crumbs in another shallow dish. Coat chicken rolls with egg white mixture; roll in crumbs. Place in baking dish; sprinkle with paprika.\n\n4 Bake uncovered 25 to 30 minutes or until chicken is no longer pink in center.\n\n1 Serving: Calories 150; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 60mg; Sodium 440mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 25g Exchanges: 3 Lean Meat Carbohydrate Choices: 0\n\nCountry French Chicken and Rice\n\n## Country French Chicken and Rice Calorie Smart\n\nPrep 25 min Total 3 hr 25 min \u25cf 8 servings\n\n  * \u00bc cup chopped sun-dried tomatoes in oil, drained\n  * 2 tablespoons herbes de Provence*\n  * 2 tablespoons olive oil\n  * 2 tablespoons lemon juice\n  * 1 tablespoon finely chopped garlic\n  * 1 teaspoon salt\n  * 8 bone-in chicken thighs (about 2 lb), skin and fat removed\n  * 1\u00bd cups sliced fresh mushrooms (4 oz)\n  * 1 cup uncooked regular long-grain white rice\n  * I medium carrot, shredded (\u00bd cup)\n  * 2 cups boiling water\n  * 1 tablespoon chopped fresh parsley\n  * 2 teaspoons grated lemon peel\n\n1 In 1-gallon resealable food-storage plastic bag, mix tomatoes, herbes de Provence, oil, lemon juice, garlic and \u00bd teaspoon of the salt. Add chicken and mushrooms; seal bag. Turn to coat chicken and mushrooms in marinade. Refrigerate at least 2 hours but no longer than 24 hours.\n\n2 Heat oven to 375\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray. In baking dish, stir together rice, carrot and remaining \u00bd teaspoon salt; stir in boiling water. Pour chicken, mushrooms and marinade evenly over rice mixture.\n\n3 Cover with foil. Bake 50 to 60 minutes or until liquid is absorbed and juice of chicken is clear when thickest piece is cut to bone (at least 165\u00b0F). Sprinkle with parsley and lemon peel.\n\n*Any combination of dried basil, fennel seed, lavender, marjoram, rosemary, sage, summer savory, tarragon or thyme can be substituted for the herbes de Provence.\n\n1 Serving: Calories 260; Total Fat 10g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 45mg; Sodium 360mg; Total Carbohydrate 23g (Dietary Fiber 1g, Sugars 1g); Protein 18g Exchanges: 1\u00bd Starch, 2 Lean Meat, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\nRoast Turkey\n\n## Roast Turkey Calorie Smart\n\nPrep 20 min Total 4 hr 20 min \u25cf 12 to 15 servings\n\n  * 1 whole turkey (12 to 15 lb), thawed if frozen*\n  * 3 tablespoons butter, melted, or vegetable oil\n\n1 Heat oven to 325\u00b0F. Discard giblets and neck or reserve for another use.\n\n2 Fold wings across back of turkey so tips are touching. If turkey doesn't have an ovenproof plastic leg band holding legs together, tuck legs under band of skin at tail (if present), or tie legs together with kitchen string, then tie to tail if desired. Place turkey, breast side up, on rack in large roasting pan. Brush butter over turkey. Insert ovenproof meat thermometer so tip is in thickest part of thigh but does not touch bone. (Do not add water or cover turkey.)\n\n3 Roast uncovered 3 hours to 3 hours 45 minutes. After roasting about 2 hours, remove leg band, cut band of skin or remove string holding legs to allow the inside of the thighs to cook thoroughly and evenly. Place tent of foil loosely over turkey to prevent excessive browning.\n\n4 Turkey is done when thermometer reads at least 165\u00b0F and legs move easily when lifted or twisted. Place turkey on warm platter; cover with foil to keep warm. Reserve drippings if making gravy (see page 346). Let turkey stand 15 to 20 minutes for easiest carving.\n\n*For optimal food safety and even doneness, the USDA recommends cooking stuffing separately. However, if you choose to stuff poultry or game birds, it's necessary to use an accurate food thermometer to make sure the center of the stuffing reaches a safe minimum temperature of 165\u00b0F. Cooking home-stuffed poultry or game birds is riskier than cooking those that are not stuffed. Even if the poultry or game bird itself has reached the safe minimum internal temperature of 165\u00b0F, the stuffing may not have reached same temperature. Bacteria can survive in stuffing that has not reached 165\u00b0F, possibly resulting in food-borne illness. Do not stuff poultry or game birds that will be grilled, smoked, fried or microwaved, because the stuffing will never get hot enough in the center to be safe.\n\n1 Serving: Calories 320; Total Fat 15g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 150mg; Sodium 160mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 46g Exchanges: 6 \u00bd Very Lean Meat, 2 \u00bd Fat Carbohydrate Choices: 0\n\nBrined Whole Turkey In large stainless steel stockpot, mix 2 gallons cold water and 2 cups coarse kosher salt until salt is dissolved. Add turkey (brine should cover bird). Cover and refrigerate 8 to 12 hours. Remove turkey from brine; thoroughly rinse under cool water, gently rubbing inside and outside of turkey to release salt. Pat dry inside and out with paper towels. Follow directions for roast turkey to prepare turkey for roasting. In medium bowl, toss 1 quartered medium onion, 1 coarsely chopped carrot, 1 coarsely chopped stalk celery, 1 teaspoon dried thyme leaves and 1 tablespoon of the melted butter. Spoon into turkey cavity. Place turkey in roasting pan and insert thermometer as directed in recipe. Brush all sides of turkey with remaining melted butter. Roast turkey as directed. Remove turkey from oven. Remove vegetables from cavity.\n\n### Making Roast Turkey\n\nFold wings across back of turkey so tips are touching.\n\nTuck legs under plastic leg band, if present, or tie together with kitchen string.\n\nInsert ovenproof meat thermometer so tip is in thigh but does not touch bone.\n\n### Carving Roast Turkey\n\nPlace turkey, breast up, on cutting board. Remove ties or skewers. While gently pulling leg away from body, cut through joint between thigh and body. Separate drumstick and thigh by cutting through connecting joint.\n\nCarefully cut down through breast meat next to bone, removing one side of breast. Cut turkey breast into slices.\n\nServe drumsticks and thighs whole or carved. To carve, remove meat from drumstick by slicing at an angle, and slice thigh by cutting even slices parallel to the bone.\n\n### Timetable for Thawing Poultry\n\nFor food safety reasons, never thaw poultry at room temperature. Cook poultry the same day it's thawed. _Refrigerator Thawing:_ Place poultry in a pan with sides or in a resealable food-storage plastic bag to catch drips. Allow about 24 hours for every 4 to 5 pounds. _Cold-Water Thawing:_ Submerge the packaged poultry in cold water, changing the water every 30 minutes. Allow 30 minutes per pound.\n\nType of Poultry | Weight in Pounds | Refrigerator Thawing Time (Days) | Cold-Water Thawing Time (Hours)\n\n---|---|---|---\n\nChicken, Whole | 3 to 4 | 1 to 2 | 2 to 2\u00bd\n\nTurkey, Whole | 8 to 12\n\n12 to 16\n\n16 to 20\n\n20 to 24 | 2 to 2\u00bd\n\n2\u00bd to 4\n\n4 to 5\n\n5 to 6 | 4 to 6\n\n6 to 8\n\n8 to 10\n\n10 to 12\n\nCut-Up Pieces | Up to 4 | \u00bd up to 3 | 1\u00bd to 3\n\n## Pan Gravy Calorie Smart \u2022 Fast \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1 cup gravy\n\n  * Drippings from roast turkey or other cooked meat\n  * 2 tablespoons all-purpose flour\n  * 1 cup liquid (turkey or meat juices, broth, water)\n  * Few drops browning sauce, if desired\n  * Salt and pepper to taste\n\n1 After removing turkey from roasting pan, pour drippings (turkey juices and fat) into fat separator or glass measuring cup, leaving browned bits in pan. The fat will rise to the top. With spoon, return 2 tablespoons of the fat to the pan. Pour or spoon off and discard any remaining fat; reserve remaining drippings.\n\n2 Stir flour into fat in roasting pan. Cook over low heat, stirring constantly and scraping up browned bits, until mixture is smooth and bubbly; remove from heat.\n\n3 Gradually stir in reserved drippings plus enough broth or water to equal 1 cup. Heat to boiling, stirring constantly. Boil and stir 1 minute. Stir in browning sauce if a darker color is desired. Stir in salt and pepper.\n\n1 Tablespoon: Calories 20; Total Fat 1.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nPan Cream Gravy Substitute half-and-half or whole milk for the liquid.\n\nThin Gravy Reduce fat from meat drippings and flour to 1 tablespoon each.\n\n### Making Pan Gravy\n\nPour turkey drippings into fat separator or glass measuring cup, leaving browned particles in pan.\n\nStir flour into fat in cooking pan, stirring constantly and scraping up browned bits.\n\nGradually stir in reserved drippings plus enough broth to equal 1 cup.\n\n## Bread Stuffing\n\nPrep 15 min Total 55 min \u25cf 10 servings\n\n  * \u00be cup butter\n  * 2 large stalks celery (with leaves), chopped (1\u00bd cups)\n  * 1 large onion, chopped (1 cup)\n  * 9 cups soft bread cubes (about 15 slices bread)\n  * 1\u00bd teaspoons chopped fresh or \u00bd teaspoon dried thyme leaves\n  * 1\u00bd teaspoons chopped fresh or \u00bd teaspoon dried sage leaves or \u00bc teaspoon ground sage\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 Heat oven to 350\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 In 4-quart Dutch oven or saucepan, melt butter over medium-high heat. Cook celery and onion in butter 4 to 6 minutes, stirring occasionally, until tender. Add remaining ingredients; stir gently to mix well. Spoon into baking dish.\n\n3 Cover with foil; bake 25 minutes. Remove foil; bake 10 to 15 minutes longer or until center is hot and edges are beginning to brown.\n\n1 Serving (\u00bd Cup): Calories 230; Total Fat 15g (Saturated Fat 7g, Trans Fat 1g); Cholesterol 35mg; Sodium 530mg; Total Carbohydrate 19g (Dietary Fiber 1g, Sugars 2g); Protein 3g Exchanges: 1 Starch, 3 Fat Carbohydrate Choices: 1\n\nCornbread Stuffing Substitute cornbread cubes for the soft bread cubes. For Cornbread-Sausage Stuffing, omit salt; add \u00bd lb cooked crumbled bulk pork or chorizo sausage (see Sausage Stuffing below) and 1 cup chopped toasted pecans with the remaining stuffing ingredients.\n\nOyster Stuffing Add 2 cans (8 oz each) whole oysters, drained and chopped, with the remaining stuffing ingredients.\n\nSausage Stuffing Omit salt. In 10-inch skillet, cook 1 lb bulk pork sausage or fresh chorizo sausage over medium heat, stirring occasionally, until no longer pink; drain, reserving drippings. Substitute drippings for part of the butter. Add cooked sausage with the remaining stuffing ingredients.\n\nWild Rice Stuffing In ungreased 2-quart casserole, mix 3 cups cooked wild rice, \u00bc cup melted butter, 1 cup orange juice, 1 medium apple, peeled and cut into chunks, 1 cup dry bread crumbs and \u00bd cup each raisins and walnuts. Cover and bake at 325\u00b0F for 55 to 60 minutes or until apple is tender.\n\n### Making Bread Stuffing\n\nCook celery and onion in butter until tender.\n\nSpoon stuffing into baking dish. Bake until center is hot and edges are beginning to brown.\n\n## Apple-Maple Brined Turkey Breast\n\nPrep 30 min Total 14 hr 30 min \u25cf 8 servings\n\nBrine and Turkey\n\n  * \u00bd gallon (64 oz) apple cider\n  * 1 cup real maple syrup or maple-flavored syrup\n  * \u00bd cup coarse kosher salt\n  * \u00bc cup chopped fresh or 1 tablespoon rubbed sage leaves\n  * 1 tablespoon dried marjoram leaves\n  * 5 cloves garlic, finely chopped\n  * 1 bone-in whole turkey breast (5 to 6 lb), thawed if frozen\n\nBasting Sauce and Gravy\n\n  * \u00bc cup butter, cut into pieces\n  * 1\u00bd teaspoons chopped fresh or \u00bd teaspoon rubbed sage leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon dried marjoram leaves\n  * \u00bc teaspoon coarsely ground black pepper\n  * \u00bc cup all-purpose flour\n\n1 Reserve 1 cup apple cider for basting; cover and refrigerate. In 6-quart nonreactive bowl (glass or stainless steel) or stockpot (stainless steel), stir remaining cider, the syrup, kosher salt, \u00bc cup fresh sage, 1 tablespoon marjoram and the garlic until salt is dissolved. Add turkey breast. Cover and refrigerate 12 hours. Do not brine turkey for more than 12 hours or it will become too salty.\n\n2 Heat oven to 325\u00b0F. Remove turkey from brine; discard brine. Thoroughly rinse turkey under cool running water, gently rubbing outside and inside of turkey to release salt. Pat skin and cavity dry with paper towels.\n\n3 Place turkey, skin side up, on rack in large shallow roasting pan. Insert ovenproof meat thermometer so tip is in thickest part of breast and does not touch bone. Roast uncovered 1 hour.\n\n4 In 1-quart saucepan, heat reserved cider, the butter, 1\u00bd teaspoons fresh sage, \u00bd teaspoon salt, \u00bc teaspoon marjoram and the pepper over medium heat until butter is melted and mixture is hot. Roast turkey about 1 hour longer or until thermometer reads at least 165\u00b0F, basting generously with cider mixture and pan juices every 15 minutes.\n\n5 Remove turkey from oven. Cover loosely with foil; let stand 15 to 20 minutes. Meanwhile, pour pan drippings and browned bits into measuring cup; let stand 5 minutes. Skim 4 tablespoons fat from top of drippings and pour into 2-quart saucepan; skim and discard any remaining fat. Add enough water to remaining drippings to measure 2 cups; set aside.\n\n6 Stir flour into fat in saucepan with whisk. Cook and stir over medium heat until mixture is smooth and bubbly; remove from heat. Gradually stir in reserved 2 cups drippings. Heat to boiling, stirring constantly. Boil and stir about 1 minute or until gravy thickens. Serve with turkey.\n\n1 Serving: Calories 470; Total Fat 21g (Saturated Fat 8g, Trans Fat 0.5g); Cholesterol 165mg; Sodium 2080mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 12g); Protein 55g Exchanges: 1 Other Carbohydrate, 7\u00bd Lean Meat Carbohydrate Choices: 1\n\nHerbed Turkey and Wild Rice Casserole\n\n## Herbed Turkey and Wild Rice Casserole Calorie Smart \u2022 Slow Cooker\n\nPrep 25 min Total 6 hr 55 min \u25cf 6 servings\n\n  * 6 slices bacon, cut into \u00bd-inch pieces\n  * 1 lb turkey breast tenderloins, cut into \u00be-inch pieces\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 medium carrot, sliced (\u00bd cup)\n  * 1 medium stalk celery, sliced (\u00bd cup)\n  * 3\u00bd cups chicken broth (for homemade broth, see page 143)\n  * 1 can (10.75 oz) condensed cream of chicken soup\n  * \u00bc teaspoon dried marjoram leaves\n  * \u215b teaspoon pepper\n  * 1\u00bc cups uncooked wild rice, rinsed, drained\n\n1 In 10-inch skillet, cook bacon over medium heat, stirring occasionally, until crisp. Stir in turkey. Cook 3 to 5 minutes, stirring occasionally, until turkey is brown. Stir in onion, carrot and celery. Cook 2 minutes, stirring occasionally; drain.\n\n2 Spray 3\u00bd- to 6-quart slow cooker with cooking spray. In slow cooker, beat 1\u00be cups of the broth and the soup with whisk until smooth. Stir in remaining 1\u00be cups broth, the marjoram and pepper. Stir in turkey mixture and wild rice.\n\n3 Cover; cook on High heat setting 30 minutes. Reduce heat setting to Low. Cook 6 to 7 hours or until rice is tender and liquid is absorbed.\n\n1 Serving: Calories 340; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 1190mg; Total Carbohydrate 36g (Dietary Fiber 3g, Sugars 2g); Protein 30g Exchanges: 2 Starch, 1 Vegetable, 3 Very Lean Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\n## Betty's Staples Cooked Wild Rice\n\nWild rice takes some time to cook, so it's nice to have some on hand to use in various ways. See page 215 for cooking instructions. Store in the refrigerator up to 5 days or freeze 3 months.\n\n  1. 1. **Wild Rice and Chicken Salad:** In medium bowl, mix 1 cup cooked wild rice, 1 cup shredded or chopped cooked chicken, \u00bd cup dried cranberries and \u00bd cup chopped walnuts. Toss with Honey Mustard Dressing (page 137). Serve on Bibb lettuce leaves.\n  2. 2. **Bacon-Roasted Pepper Rice:** Cook 2 slices chopped bacon in skillet until crisp. Stir in 1\u00bd cups cooked wild rice, \u00bd cup sliced roasted red bell peppers and \u00bd cup cooked corn. Cook and stir about 5 minutes or until heated. Toss with 2 tablespoons chopped fresh parsley.\n  3. 3. **Raspberry Turkey-Rice Salad:** In medium bowl, toss 1 cup cooked wild rice, 1 cup chopped cooked turkey and 2 tablespoons chopped fresh oregano. Toss with \u00bd cup Raspberry Vinaigrette (page 137). Carefully stir in 1 cup fresh raspberries. Serve on red leaf lettuce.\n  4. 4. **Winter Squash with Wild Rice:** Cook halved acorn or buttercup squash as directed on page 237. Spoon about \u00bd cup cooked wild rice into each squash half. In small bowl, mix 1 tablespoon melted butter and 2 tablespoons maple syrup; drizzle over squash and rice. Bake 5 to 10 minutes or until heated through.\n  5. 5. **Eggs with Wild Rice:** Scramble 3 or 4 eggs as directed on page 90. As eggs are cooking, stir in \u00bd cup cooked wild rice and 2 tablespoons chopped fresh basil.\n\nBacon-Roasted Pepper Rice, Wild Rice and Chicken Salad, Winter Squash with Wild Rice\n\nDuckling with Orange Sauce\n\n## Duckling with Orange Sauce\n\nServe this delicious recipe over a rice blend\u2014it will really bring out the flavor of the orange sauce.\n\nPrep 20 min Total 3 hr 5 min \u25cf 4 servings\n\n  * 1 whole duckling (4 to 5 lb), thawed if frozen\n  * 2 teaspoons grated orange peel\n  * \u00bd cup orange juice\n  * \u00bc cup currant jelly\n  * 1 tablespoon lemon juice\n  * \u215b teaspoon ground mustard\n  * \u215b teaspoon salt\n  * 1\u00bd teaspoons cornstarch\n  * 1 tablespoon cold water\n  * 1 medium orange, peeled, sectioned\n  * 1 tablespoon orange-flavored liqueur, if desired\n\n1 Heat oven to 350\u00b0F. Fasten neck skin to back of duckling with skewer. Fold wings across back of duckling so tips are touching. Place duckling, breast side up, on rack in shallow roasting pan.\n\n2 Pierce skin all over with fork, especially at the breast, but don't pierce the flesh. Loosely tie legs to the tail, if desired, to better hold an even shape during cooking. Insert ovenproof meat thermometer so tip is in thickest part of inside thigh and does not touch bone.\n\n3 Roast uncovered about 2 hours 30 minutes or until thermometer reads at least 165\u00b0F and legs move easily when lifted or twisted. If necessary, place tent of foil loosely over breast during last hour to prevent excessive browning. Place duckling on warm platter; cover and let stand 15 to 20 minutes.\n\n4 Meanwhile, in 1-quart saucepan, heat orange peel, orange juice, jelly, lemon juice, mustard and salt to boiling. In small bowl, mix cornstarch and water; stir into sauce. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute. Stir in orange sections and liqueur. Brush duckling with some of the sauce. Serve with remaining sauce.\n\n1 Serving: Calories 490; Total Fat 24g (Saturated Fat 7g, Trans Fat 1g); Cholesterol 155mg; Sodium 230mg; Total Carbohydrate 20g (Dietary Fiber 1g, Sugars 15g); Protein 49g Exchanges: 1 Starch, 6\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 1\n\nOrange-Roasted Pheasants\n\n## Orange-Roasted Pheasants\n\nPrep 20 min Total 1 hr 35 min \u25cf 4 servings\n\n  * \u00bd cup butter, softened\n  * 2 teaspoons grated orange peel\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 tablespoons honey\n  * 4 medium green onions, sliced (\u00bc cup)\n  * 2 pheasants (2\u00bd to 3 lb each), thawed if frozen\n  * \u00bd cup orange-flavored liqueur or orange juice\n  * 1 cup chicken broth (for homemade broth, see page 143)\n\n1 Heat oven to 350\u00b0F. In small bowl, stir butter, orange peel, salt, pepper, honey and onions until well mixed.\n\n2 Place pheasants, breast side up, in shallow roasting pan. Starting at the back opening, gently separate skin from breast of each pheasant, using fingers. Spread butter mixture under skin. Secure skin to flesh with toothpick if necessary. Insert ovenproof meat thermometer so tip is in thickest part of inside thigh and does not touch bone.\n\n3 Roast uncovered 50 to 60 minutes or until thermometer reads at least 165\u00b0F and legs move easily when lifted or twisted. Remove pheasants to a platter; cover loosely with foil.\n\n4 Pour liqueur and broth into roasting pan, scraping up browned bits from pan; pour mixture into 1-quart saucepan. Heat to boiling, stirring occasionally; boil about 15 minutes, stirring occasionally, until liquid is reduced to 1 cup. Serve sauce with pheasants.\n\n1 Serving: Calories 670; Total Fat 45g (Saturated Fat 18g, Trans Fat 2g); Cholesterol 220mg; Sodium 1170mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 9g); Protein 56g Exchanges: \u00bd Starch, 8 Medium-Fat Meat Carbohydrate Choices: \u00bd\n\nRoast Goose with Apple Stuffing\n\n## Roast Goose with Apple Stuffing\n\nPrep 30 min Total 4 hr 20 min \u25cf 8 servings\n\n  * 1 whole goose (8 to 10 lb), thawed if frozen\n  * \u00bc cup butter\n  * 2 medium stalks celery (with leaves), chopped (1 cup)\n  * 1 medium onion, chopped (\u00bd cup)\n  * 6 cups soft bread crumbs (about 9 slices bread)\n  * 3 medium unpeeled tart apples, chopped (3 cups)\n  * 1\u00bd teaspoons chopped fresh sage leaves or \u00bd teaspoon rubbed sage or ground sage\n  * \u00be teaspoon chopped fresh or \u00bc teaspoon dried thyme leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * Pan Gravy (page 346)*, if desired\n\n1 Heat oven to 350\u00b0F. Discard giblets and neck or reserve for another use. Remove excess fat from goose.\n\n2 Place goose, breast side up, on rack in shallow roasting pan. Fasten neck skin to back of goose with skewer. Fold wings across back of goose so tips are touching. Pierce skin all over with fork so fat can drain. Insert ovenproof meat thermometer so tip is in thickest part of inside thigh and does not touch bone. (Do not add water or cover goose.)\n\n3 Roast goose uncovered 3 hours to 3 hours 30 minutes, removing excess fat from pan occasionally. If necessary, place tent of foil loosely over goose during last hour to prevent excessive browning. Meanwhile, spray 2-quart casserole with cooking spray. In 3-quart saucepan, melt butter over medium-high heat. Cook celery and onion in butter, stirring occasionally, until tender; remove from heat. Stir in remaining ingredients except gravy. Spoon into casserole. Cover with foil and refrigerate until baking time.\n\n4 Bake stuffing covered alongside goose for last 35 to 45 minutes of roasting time, removing foil from stuffing during last 10 minutes of baking, until center is hot and edges are beginning to brown. Goose is done when thermometer reads at least 165\u00b0F and legs move easily when lifted or twisted. Reserve drippings if making gravy. Let stand 15 to 20 minutes for easiest carving.\n\n*Pan Gravy recipe makes 1 cup. Increase, if needed, for desired amount.\n\n1 Serving: Calories 820; Total Fat 54g (Saturated Fat 18g, Trans Fat 1.5g); Cholesterol 210mg; Sodium 630mg; Total Carbohydrate 26g (Dietary Fiber 2g, Sugars 7g); Protein 57g Exchanges: 1 Starch, Fruit, 8 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 2\n\n### Timetable for Roasting Poultry\n\nRoasting times are general guidelines. Also check turkey labels for timing recommendations.\n\nBegin checking turkey doneness about 1 hour before end of recommended roasting time. For purchased stuffed turkeys, follow package directions instead of this timetable.\n\nType of Poultry | Weight in Pounds | Oven Temperature | Roasting Time in Hours\n\n---|---|---|---\n\nChicken\n\nWhole Chicken (Not Stuffed)* | 3 to 3\u00bd | 375\u00b0F | 1\u00be to 2\n\nTurkey\n\nWhole Turkey (Not Stuffed)* | 8 to 12\n\n12 to 14\n\n14 to 18\n\n18 to 20\n\n20 to 24 | 325\u00b0F\n\n325\u00b0F\n\n325\u00b0F\n\n325\u00b0F\n\n325\u00b0F | 2\u00be to 3\n\n3 to 3\u00be\n\n3\u00be to 4\u00bc\n\n4\u00bc to 4\u00bd\n\n4\u00bd to 5\n\nWhole Turkey Breast (Bone-In) | 4 to 6\n\n6 to 8 | 325\u00b0F\n\n325\u00b0F | 1\u00bd to 2\u00bc\n\n2\u00bc to 3\u00bc\n\nGame\n\nWhole Duck\n\n(domestic)\n\n(wild)\n\n|  |\n\n375\u00b0F\n\n350\u00b0F | 20 minutes per lb\n\n18 to 20 minutes per lb\n\nWhole Goose | 7 to 9\n\n9 to 11\n\n11 to 13 | 325\u00b0F\n\n325\u00b0F\n\n325\u00b0F | 2\u00bc to 3\u00be\n\n3\u00be to 4\u00bd\n\n4\u00bd to 5\u00bd\n\nWhole Pheasant | 2 to 3 | 350\u00b0F | 1 to 1\u00bd\n\nWhole Rock Cornish Hen | 1\u00bd to 2 | 350\u00b0F | 1\u00bc to 1\u00bd\n\n*For optimal food safety and even doneness, the USDA recommends cooking stuffing separately. However, if you choose to stuff poultry or game birds, it's necessary to use an accurate food thermometer to make sure the center of the stuffing reaches a safe minimum temperature of 165\u00b0F. Cooking home-stuffed poultry or game birds is riskier than cooking those that are not stuffed. Even if the poultry or game bird itself has reached the safe minimum internal temperature of 165\u00b0F, the stuffing may not have reached same temperature. Bacteria can survive in stuffing that has not reached 165\u00b0F, possibly resulting in food-borne illness. Do not stuff poultry or game birds that will be grilled, smoked, fried or microwaved, because the stuffing will never get hot enough in the center to be safe.\n\n### Timetable for Broiling Poultry\n\nCook poultry right from the refrigerator. Set oven control to broil. Check the owner's manual for whether the oven door should be partially opened or closed during broiling. Place poultry on rack in broiler pan. Broil for the time listed, turning once, until thermometer reaches at least 165\u00b0F or until juice is clear when centers of thickest pieces are cut.\n\nCut of Poultry | Weight in Pounds | Broiling Time\n\n---|---|---\n\nChicken\n\nCut-Up | 3 to 3\u00bd | Skin side down 30 minutes; turn. Broil 15 to 25 minutes longer (7 to 9 inches from heat)\n\nBone-In Split Breasts | 2\u00bd to 3 | 25 to 35 minutes (7 to 9 inches from heat)\n\nBoneless Skinless Breasts | 1\u00bc | 15 to 20 minutes (4 to 6 inches from heat)\n\nWings | 2 to 2\u00bd | 10 minutes (5 to 7 inches from heat)\n\nTurkey\n\nTenderloins | 1 to 1\u00bd | 8 to 12 minutes (4 to 6 inches from heat)\n\nBreast Slices | 1 to 1\u00bd | 7 minutes (4 to 6 inches from heat)\n\n### Cooked Poultry Yields\n\nRemove the uncertainty of wondering if you will have enough cooked poultry for your favorite recipe by using this handy chart.\n\nType of Poultry | Weight in Pounds | Yield of Chopped, Cubed or Shredded Cooked Poultry in Cups\n\n---|---|---\n\nChicken\n\nWhole Chicken | 3 to 3\u00bd | 2\u00bd to 3\n\nBone-In Split Breast | 1\u00bd | 2\n\nBoneless Skinless Breasts | 1\u00bd | 3\n\nLegs (Thighs and Drumsticks) | 1\u00bd | 1\u00be\n\nTurkey\n\nWhole Turkey | 6 to 8 | 7 to 10\n\nBone-In Breast | 3 to 9 | 5 to 13\n\nTenderloins | 1\u00bd | 3\n\n## Setting a Festive Holiday Table\n\nThese time-tested guidelines will help you set a beautiful table:\n\n  * Coordinate placemats, napkins, centerpieces and serving pieces; everything doesn't need to match. An eclectic mix can be fun and less reserved than serving on a matched set of china.\n  * Allow enough elbow room for each person and leg room uninhibited by table legs.\n  * Place flatware 1 inch from the edge of the table.\n\n**For a casual place setting:** Place a fork to the left of the plate, a knife to the right (blade toward the plate) and a spoon to the right of the knife.\n\n**For a formal place setting:** Arrange pieces to be used first farthest from the plate, as shown in photo. Place first-course salad or soup plate on top of main-course plate. Dessert flatware can be placed above the dinner plate, or bring it when you serve dessert. Napkins can go on the center of the plate, to the left of the forks or tucked inside empty glasses. Before serving dessert, clear the table of all items that won't be needed.\n\n## Setting a Buffet Table\n\nBuffets are a great option when the gathering is less formal or you are short on table space for platters and serving dishes. See Safe Buffets, page 32, for food safety considerations when hosting a buffet. Use these tips to help in planning a buffet:\n\n  * Anywhere there is ample room for people to move around the serving area is a place a buffet could be set up: a center island or counter, dining room or kitchen table (remove chairs for easy access), sideboard, picnic or folding table. Don't have one large space? Consider breaking up the buffet into stations by food type. It allows guests to move around and avoids congestion at one location.\n  * Arrange buffet items starting with the main course and then side dishes, salad, condiments, bread. End with flatware, napkins (make cutlery bundles for easier carrying) and glassware last so that guests don't have to juggle these items while serving themselves.\n  * If people will be standing to eat, skip paper plates, opting for dishes or plastic plates instead. If you must use paper, make sure they are heavy-duty and avoid serving foods that require cutting to eat.\n\n# Chapter 13: Fish & Shellfish\n\n## Fish Basics\n\nBuying Fish\n\nChoosing Fish by Flavor and Texture\n\nCooking Fish\n\nFish Pound Per Serving\n\nStoring Fish\n\n## Fish Features\n\nHeirloom Recipe and New Twist\n\nLearn to Broil\n\nLearn with Betty: Salmon Doneness\n\n## Fish\n\nAsian Tuna with Wasabi Aioli Calorie Smart\n\nBeer-Batter Fried Fish Calorie Smart \u2022 Fast\n\nBranzino with Sun-Dried Tomato Pesto\n\nBroiled Fish Fillets\n\nBrowned-Butter Panfried Fish\n\nButtery Broiled Fish Steaks Calorie Smart \u2022 Fast\n\nCajun Fish Soft Tacos Fast\n\nCornmeal-Crusted Catfish Calorie Smart \u2022 Fast\n\nCornmeal-Crusted Cod\n\nFresh Branzino with Sweet Pea-Mint Pesto Fast\n\nGarlic-Butter Panfried Fish\n\nHerbed Butter Branzino\n\nHoney-Mustard Glazed Salmon Calorie Smart\n\nLemon- and Parmesan-Crusted Salmon Calorie Smart \u2022 Easy\n\nLemon-Garlic Branzino\n\nOrange Sole Amandine\n\nOven-Fried Fish Calorie Smart \u2022 Fast\n\nPanfried Fish Calorie Smart \u2022 Fast\n\nParmesan, Herb and Lemon Panfried Fish\n\nRoasted Tilapia and Vegetables Calorie Smart\n\nRosemary-Lemon Branzino\n\nSalmon Patties with Sour Cream\u2013Dill Sauce Calorie Smart \u2022 Fast\n\nSnapper with Tomato-Pepper Sauce Calorie Smart \u2022 Fast\n\nSole Amandine Calorie Smart \u2022 Fast\n\nToasted Spice Seared Mackerel Fast\n\n## Shellfish Basics\n\nBuying and Using Shellfish\n\nStoring Shellfish\n\nShellfish Per Serving\n\nSuccessfully Cooking Shellfish\n\nTimetable for Cooking Shellfish\n\nReheating Cooked Crab Legs\n\nUnderstanding Shrimp Count\n\n## SHELLFISH FEATURES\n\nBetty's Staples: Cooked Shrimp\n\nBuying Clams\n\nDetermining Spoiled Mussels\n\nDeveining Shrimp\n\nImitation Seafood Products\n\nOpening Raw Clams\n\nRemoving Cooked Lobster Meat\n\nRemoving Cooked Crab Meat\n\nTypes of Shellfish\n\n## Shellfish\n\nBoiled Hard-Shell Blue Crabs Calorie Smart\n\nBoiled Lobsters Calorie Smart\n\nCitrus Seafood Skillet Calorie Smart \u2022 Fast\n\nDeep-Fried Oysters or Clams\n\nDeep-Fried Sea Scallops\n\nDeep-Fried Shrimp Calorie Smart\n\nScallops with Artichokes and Tomatoes Calorie Smart \u2022 Fast\n\nScrambled Eggs and Shrimp\n\nShrimp and Egg Salad Wraps\n\nShrimp and Melon Salad\n\nShrimp Creole Calorie Smart\n\nShrimp Gumbo Calorie Smart\n\nShrimp Scampi Calorie Smart \u2022 Fast\n\nShrimp Tacos\n\nSteamed Clams Calorie Smart\n\nSteamed Mussels in Wine Sauce Calorie Smart \u2022 Fast\n\nTomato-Basil Shrimp and Pasta\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Fish Basics\n\nFish is delicious, healthful and available in a multitude of varieties, both fresh and freshly frozen. As you find out just how easy it is to select great fish and terrific ways to prepare it, you'll want to have fish more often. Here's what you need to know about buying, storing and cooking fish.\n\n## Buying Fish\n\n  * Flesh should be shiny, firm and spring back when touched. Avoid fish with dark edges or brown or yellowish discoloration.\n  * Contents of the package should smell fresh and mild, not fishy or like ammonia.\n  * Buy and use the fish by the sell-by date.\n  * For whole fish, eyes should be bright, clear and just slightly bulging. Only a few fish, like walleye, have naturally cloudy eyes. \n  * Gills should be bright pink to red and scales should be bright and shiny.\n  * Frozen fish should be tightly wrapped and not have any freezer burn (dry or dark spots).\n\n## Storing Fish\n\n  * Keep fresh or thawed fish in the original packaging in the coldest part of your refrigerator. Use within 2 days.\n  * Fish that is packaged in clear plastic wrap on a tray can be frozen as it is. Other fish should be tightly wrapped in freezer bags or foil. Freeze for up to 6 months. Thaw overnight in the refrigerator.\n\nRed Snapper, Tuna, Tilapia, Salmon\n\n### Choosing Fish by Flavor and Texture\n\nFish can be categorized by flavor and texture, making it easy to substitute one fish for another within a category.\n\nMild Flavor | Moderate Flavor | Full Flavor\n\n---|---|---\n\nDelicate to Medium Texture\n\nAlaskan Pollock | Atlantic Salmon | Herring\/Sardines\n\nBarramundi | Branzino | Smelt\n\nFlounder | Chinook Salmon\n\n|\n\nSole | Walleye\n\n|   \n|\n\nWhitefish\n\n|\n\nMedium-Firm Texture\n\nButterfish\/Sablesfish | Catfish | Bluefish\n\nCod | Char | Mackerel\n\nHaddock | Coho Salmon | Salmon (Alaskan, Sockeye)\n\nLingcod | Drum | Shad\n\nSnapper\/Red Snapper | Hake\/Whiting | Wahoo\/Ono\n\nTilapia | Lake Perch\n\n|\n\nTilefish | Mahi Mahi\n\n|   \n|\n\nPompano\n\n|   \n|\n\nPorgy\/Scup\n\n|   \n|\n\nRainbow Trout\n\n|   \n|\n\nRedfish\n\n|   \n|\n\nRockfish\/Ocean Perch\n\n|   \n|\n\nSea Bass\n\n|\n\nFirm Texture\n\nGrouper | Shark | Cobia\n\nHalibut | Tuna (Albacore, Skipjack, Yellowfin\/Ahi) | Marlin\n\nMonkfish\n\n|  |\n\nSwordfish\n\nPompano\n\n|  |\n\nTuna (Bluefin, Bonito)\n\nStriped Bass\n\n|  |\n\nSturgeon\n\n|  | \n\n### The Best cooking methods for fish\n\nInstead of shopping for a specific fish, scan the seafood counter for what's freshest and on sale. You can decide what to buy based on the cooking method you wish to use or the type of fish that appeals to you most that day. Look to this chart to match the fish to the best cooking methods and you're on your way to cooking fish successfully!\n\ncooking methods | fish type\n\n---|---\n\nPanfrying | Flounder, Halibut, Salmon or Sole (with skin on) or medium-firm fish such as char, perch or snapper\n\nOven-Frying | Medium-firm, mild flavored skinless fish such as cod or haddock\n\nBroiling | Fillets or steaks of any texture fish such as shark or swordfish (thin or delicate-textured fish should not be turned or it can fall apart)\n\nGrilling | Fillets with skins or steaks of medium-firm texture, steaks or whole firm-textured fish such as mahi mahi, red fish or ono\n\nBaking | Delicate or medium-firm fillets or steaks, such as tilapia, rockfish or red snapper. (If fish has skin, it can be removed either before or after cooking.)\n\nSteaming (packets made with parchment or foil*) | Skinless fillets or steaks of any texture fish such as flounder, grouper or tuna\n\nRoasting | Whole fish or large fillets or steaks such as snapper, trout, sea bass\n\nDeep-Frying | Mild-flavored delicate or medium-firm textured skinless fillets such as cod, halibut or catfish or whole fish such as sea ball, a small grouper or red snapper\n\n*See Learn to Make Foil Packets, page 278.\n\n## Cooking Fish\n\nFish is naturally delicate and tender so overcooking makes it dry. Use the \"10-minute rule\" for moist, flaky fish\u2014here's how to do it.\n\n  * Measure the fish at its thickest point. If it will be stuffed or rolled, measure it before stuffing or rolling.\n  * Fish fillets will often have skin on one side. Cook the fish skin side down. The skin helps to hold the fish together and is easy to remove after cooking.\n  * Cook the fish 10 minutes per inch of thickness. Turn over halfway through cooking time only if specified. Fillets less than \u00bd inch thick generally need no turning. Cook frozen fish (not thawed) 20 minutes per inch. Add 5 minutes to the total cooking time if the fish is cooked in foil or in a sauce.\n  * Cook just until the fish flakes easily with a fork. Insert the tines of a fork gently into the thickest part of the fish and twist slightly. The flesh should begin to separate along the natural lines.\n  * Baking, broiling or grilling fish with the skin on helps to hold the delicate flesh together. To remove skin after cooking, carefully insert a metal spatula between the skin and the flesh, starting at the tail end if present (if no tail end, start at any edge). Hold on to a small piece of the skin and slide the fish off.\n\n### Fish Pounds per Serving\n\nThe number of servings per pound varies depending on the form of fish.\n\nType of Fish | Pounds per Serving\n\n---|---\n\nFillets or Steaks | \u2153 to \u00bd\n\nPan-Dressed (often scaled with internal organs, head, tail and fins removed) | \u00bd\n\nDrawn (whole with head and tail; only internal organs removed) | \u00bd to \u00be\n\nWhole (right from the water) | \u00be to 1\n\n## Seafood Sustainability\n\nMany chefs, restaurateurs and consumers ask for sustainable seafood. It's sustainable when the population of a fish species is managed to provide for today's needs without damaging the ability of the species to reproduce. If you buy fish managed under a U.S. fishery management plan, it considers social and economic outcomes for fishing communities; prevents overfishing; rebuilds depleted stocks; minimizes bycatch and interactions with protected species; and identifies and conserves essential fish habitat.\n\nFor more information, go to www.fishwatch.gov.\n\n## Panfried Fish Calorie Smart \u2022 Fast\n\nPrep 20 min Total 20 min \u25cf 6 servings\n\n  * 1\u00bd lb perch, red snapper or other medium-firm fish fillets (\u00bd to \u00be inch thick), skin removed\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 egg\n  * 1 tablespoon water\n  * \u2154 cup all-purpose flour, cornmeal or dry bread crumbs\n  * Vegetable oil or shortening\n\n1 Cut fish into 6 serving pieces. Sprinkle both sides with salt and pepper. In small bowl, beat egg and water with fork or whisk until blended. Place flour in shallow dish. Dip fish into egg, then coat with flour.\n\n2 In 12-inch skillet, heat about \u215b inch oil over medium heat. Fry fish in oil 6 to 8 minutes, turning once, until fish flakes easily with fork and is brown on both sides. (Fish cooks very quickly, especially the thinner tail sections; be careful not to overcook.) Remove with slotted spatula; drain on paper towels.\n\n1 Serving: Calories 200; Total Fat 7g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 95mg; Sodium 230mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 0g); Protein 24g Exchanges: 1 Starch, 3 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 1\n\nBrowned-Butter Panfried Fish Omit oil. In skillet, heat \u00bc cup butter over medium heat 3 to 4 minutes, stirring constantly, until light brown. Coat fish as directed in Step 1; cook in butter as directed in Step 2.\n\nGarlic-Butter Panfried Fish Omit salt. Substitute crushed garlic butter\u2013flavored croutons for the flour. (Place croutons in resealable food-storage plastic bag and crush with rolling pin.) Continue as directed.\n\nParmesan, Herb and Lemon Panfried Fish Omit salt and pepper. Substitute \u2153 cup Italian-style panko bread crumbs for the \u2154 cup flour. In shallow dish, mix bread crumbs, \u2153 cup grated Parmesan cheese and 1 teaspoon grated lemon peel. Continue as directed.\n\n### Panfrying Fish\n\nDip fish in egg.\n\nCoat both sides with flour.\n\nFry fish in oil, turning once.\n\nFish is done when brown on both sides and flakes easily with a fork.\n\n## Oven-Fried Fish Calorie smart \u2022 FAst\n\nPrep 10 min Total 25 min \u25cf 4 servings\n\n  * 1 lb cod, haddock or other medium-firm fish fillets (about \u00be inch thick), skin removed\n  * \u00bc cup cornmeal\n  * \u00bc cup unseasoned dry bread crumbs\n  * \u00be teaspoon chopped fresh or \u00bc teaspoon dried dill weed\n  * \u00bd teaspoon paprika\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * \u00bc cup milk\n  * 3 tablespoons butter, melted\n\n1 Move oven rack to position slightly above middle of oven. Heat oven to 450\u00b0F.\n\n2 Cut fish into 4 serving pieces. In shallow dish, mix cornmeal, bread crumbs, dill, paprika, salt and pepper. Pour milk into another shallow dish. Dip fish into milk, then coat with cornmeal mixture.\n\n3 Place fish in ungreased 13x9-inch pan. Drizzle butter over fish. Bake uncovered about 12 minutes or until fish flakes easily with fork.\n\n1 Serving: Calories 240; Total Fat 11g (Saturated Fat 5g, Trans Fat 0.5g); Cholesterol 85mg; Sodium 360mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 1g); Protein 24g Exchanges: 1 Starch, 3 Lean Meat Carbohydrate Choices: 1\n\n## Learn to BROIL\n\nBroiling is a high-heat method where food is cooked a short distance from a direct heat source above\u2014usually a flame in a gas oven or hot coil in an electric oven. The broiler will be located in the oven or sometimes in a compartment under the oven. Some think of broiling as a cousin to grilling because the surfaces of food are well-browned, creating a wonderful caramelized taste. Advantages of broiling include quickly cooked foods with great flavor, plus minimal cleanup.\n\nTo check the distance between the food and the pan, place the broiler pan in a cold oven. Then measure the distance from where the food will be to the broiler element. Use the distance recommended in the recipe or refer to the oven manufacturer's recommendations. Always watch food carefully while broiling to prevent burning.\n\n### Broiling Tips\n\n  * Turn the broiler on 5 to 10 minutes ahead of time so that it can heat up.\n  * For easy cleanup, line the broiler pan with foil and poke holes to let the juices drain. The bottom pan can also be lined with foil.\n  * Place the food in the center of the pan and place the pan under the broiler at the distance indicated in the recipe (see above).\n  * On most ovens, you can leave the oven door open slightly to allow air to circulate. This also helps to keep steam from forming\u2014too much steam and the food will not crisp. Check the oven manufacturer's guidelines to be sure this is recommended.\n  * Remove the pan from the oven to turn the food as indicated in the recipe directions.\n  * Follow recipe times carefully. If food is not quite cooked, you can place it under the broiler for a bit longer.\n\n### Best Foods for Broiling\n\n  * Beef and Pork: Meats less than 11\/2 inches thick, such as steaks and chops, are good candidates for broiling. Use tender cuts or marinate for tenderness. Ground meat patties and sausages are also great.\n  * Chicken and Turkey: Most pieces work well under the broiler. Items such as chicken quarters, legs and bone-in or boneless breasts are ideal, as are turkey cutlets and tenderloins. Try to broil the same thickness or size of pieces for even cooking.\n  * Fish and Shellfish: Fillets and steaks are ideal to cook under the broiler. They cook quickly and evenly. Shrimp and scallops also perform well under the broiler. \n  * Vegetables: Many vegetables are great under the broiler. Good candidates include asparagus spears, bell pepper halves or strips and onion wedges.\n  * Fruits: Many fruits can be broiled, such as peach or nectarine halves, grapefruit halves, pineapple slices and bananas.\n\n## Buttery Broiled Fish Steaks Calorie smart \u2022 FAst\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 4 salmon, tuna, halibut or other medium-firm to firm fish steaks, about \u00be inch thick (6 oz each)\n  * Salt and pepper to taste\n  * 2 tablespoons butter, melted\n  * Lemon wedges, if desired\n\n1 Set oven control to broil. Sprinkle both sides of fish with salt and pepper. Place fish on rack in broiler pan. Brush with 1 tablespoon of the butter.\n\n2 Broil with tops about 4 inches from heat 5 minutes. Carefully turn fish; brush with remaining 1 tablespoon butter. Broil 4 to 6 minutes longer or until fish flakes easily with fork. Serve with lemon wedges.\n\n1 Serving: Calories 280; Total Fat 15g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 125mg; Sodium 340mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 36g Exchanges: 5 Lean Meat Carbohydrate Choices: 0\n\nBroiled Fish Fillets Substitute 1 lb fish fillets, cut into 4 serving pieces, for the fish steaks. Broil with tops about 4 inches from heat 5 to 6 minutes or until fish flakes easily with fork (do not turn).\n\nButtery Broiled Fish Steaks\n\nSole Amandine\n\n## Sole Amandine Calorie Smart \u2022 Fast\n\nA French term, amandine means \"garnished with almonds\" and is often misspelled as \"almondine.\" If you own a gratin dish (a shallow oval-shaped ovenproof baking dish), use it to bake this classic.\n\nPrep 10 min Total 30 min \u25cf 6 servings\n\n  * 1\u00bd lb sole, orange roughy or other delicate- to medium-texture fish fillets (\u00bd to \u00be inch thick), skin removed\n  * \u00bd cup sliced almonds\n  * \u00bc cup butter, softened\n  * 2 tablespoons grated lemon peel\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon paprika\n  * 2 tablespoons fresh lemon juice\n\n1 Heat oven to 375\u00b0F. Spray 11x7-inch (2-quart) glass baking dish with cooking spray. Cut fish into 6 serving pieces. Place in baking dish, tucking under any thin ends for more even cooking.\n\n2 In small bowl, mix almonds, butter, lemon peel, salt and paprika; spoon evenly over fish. Sprinkle with lemon juice.\n\n3 Bake uncovered 15 to 20 minutes or until fish flakes easily with fork.\n\n1 Serving: Calories 210; Total Fat 13g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 330mg; Total Carbohydrate 2g (Dietary Fiber 1g, Sugars 0g); Protein 21g Exchanges: 3 Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 0\n\nLighter Directions For 140 calories and 6 grams of fat per serving, reduce both the almonds and butter to 2 tablespoons.\n\nOrange Sole Amandine Substitute orange peel for the lemon peel and orange juice for the lemon juice. Garnish with fresh basil leaves if desired.\n\nRoasted Tilapia and Vegetables\n\n## Roasted Tilapia and Vegetables Calorie smart\n\nPrep 15 min Total 40 min \u25cf 4 servings\n\n  * \u00bd lb fresh asparagus spears\n  * 2 small zucchini, cut in half lengthwise, then cut into \u00bd-inch pieces\n  * 1 medium bell pepper, cut into \u00bd-inch strips\n  * 1 large onion, cut into \u00bd-inch wedges, separated\n  * 2 tablespoons olive oil\n  * 2 teaspoons Montreal steak grill seasoning\n  * 4 tilapia fillets (about 1\u00bd lb)\n  * 1 tablespoon butter, melted\n  * \u00bd teaspoon paprika\n\n1 Heat oven to 450\u00b0F. Snap off tough ends of asparagus; cut each spear in half. In large bowl, mix asparagus, zucchini, bell pepper, onion and oil. Sprinkle with 1 teaspoon of the grill seasoning; toss to coat. Spread vegetables in ungreased 15x10x1-inch pan. Roast 5 minutes on lowest oven rack.\n\n2 Meanwhile, spray 13x9-inch (3-quart) glass baking dish with cooking spray. Pat fish dry with paper towels. Brush fish with butter; sprinkle with paprika and remaining 1 teaspoon grill seasoning. Place in baking dish.\n\n3 Place baking dish on middle oven rack. Roast fish and vegetables uncovered 17 to 18 minutes or until fish flakes easily with fork and vegetables are tender.\n\n1 Serving: Calories 290; Total Fat 12g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 100mg; Sodium 520mg; Total Carbohydrate 10g (Dietary Fiber 3g, Sugars 5g); Protein 34g Exchanges: 2 Vegetable, 4 Lean Meat, \u00bd Fat Carbohydrate Choices: \u00bd\n\nSnapper with Tomato-Pepper Sauce\n\n## Snapper with Tomato-Pepper Sauce Calorie Smart \u2022 Fast\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1 lb red snapper, cod or other medium-firm fish fillets (\u00bd inch thick)\n  * 1 large tomato, chopped (1 cup)\n  * 1 small green bell pepper, chopped (\u00bd cup)*\n  * 1 small onion, sliced\n  * 2 tablespoons finely chopped fresh cilantro or parsley\n  * \u00bc teaspoon salt\n  * \u00bc cup dry white wine or chicken broth\n  * 2 cups hot cooked rice (page 215), if desired\n\n1 Heat 12-inch nonstick skillet over medium heat. If fish fillets are large, cut into 4 serving pieces. Arrange fish, skin side down, in single layer in skillet. Cook uncovered 4 to 6 minutes, turning once, until fish flakes easily with fork. Remove fish to warm platter; keep warm.\n\n2 Add tomato, bell pepper, onion, cilantro and salt to skillet. Cook over medium heat 3 to 5 minutes, stirring frequently, until bell pepper and onion are crisp-tender. Stir in wine; cook about 1 minute or until hot. Spoon sauce over fish. Serve with rice.\n\n*For a slightly sweeter-tasting sauce, use a red, yellow or orange bell pepper instead of green.\n\n1 Serving: Calories 120; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 60mg; Sodium 250mg; Total Carbohydrate 5g (Dietary Fiber 1g, Sugars 2g); Protein 22g Exchanges: 1 Vegetable, 3 Very Lean Meat Carbohydrate Choices: \u00bd\n\n## Asian Tuna with Wasabi Aioli Calorie smart\n\nPrep 25 min Total 2 hr 25 min \u25cf 8 servings\n\nTuna\n\n  * 2 lb tuna steaks (\u00be to 1 inch thick)\n  * \u00bd cup vegetable oil\n  * \u2153 cup soy sauce\n  * 2 tablespoons packed brown sugar\n  * 2 teaspoons sesame oil\n  * 2 teaspoons grated gingerroot\n  * 2 cloves garlic, finely chopped\n\nWasabi Aioli\n\n  * \u00bd cup mayonnaise or salad dressing\n  * 1 teaspoon wasabi powder*\n\nGarnish\n\n  * 2 teaspoons sesame seed, toasted (page 23) if desired\n\n1 If tuna steaks are large, cut into 8 serving pieces. In shallow glass or plastic dish or resealable food-storage plastic bag, mix vegetable oil, soy sauce, brown sugar, sesame oil, gingerroot and garlic. Add tuna; turn to coat with marinade. Cover dish or seal bag; refrigerate at least 2 hours but no longer than 4 hours to marinate, turning once.\n\n2 Meanwhile, in small bowl, mix aioli ingredients. Cover and refrigerate until serving time.\n\n3 Set oven control to broil. Remove tuna from marinade; reserve marinade. Place tuna on rack in broiler pan. Broil with tops 4 to 6 inches from heat 10 to 15 minutes, brushing 2 or 3 times with marinade and turning once, until fish flakes easily with fork and is slightly pink in center. Discard any remaining marinade. Sprinkle sesame seed over tuna. Serve with aioli.\n\n*Prepared horseradish can be substituted for the wasabi powder.\n\n1 Serving: Calories 350; Total Fat 27g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 600mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 3g); Protein 22g Exchanges: 3 Medium-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 0\n\n## Common Types of Tuna\n\nAlbacore is also know as \"white tuna\" because of its light flesh. Mild-flavored, albacore is the most expensive canned tuna. Young bluefin tuna are lighter in flesh with a milder flavor than mature bluefin, which has dark-red flesh and a strong flavor.\n\nCornmeal-Crusted Catfish\n\n## Cornmeal-Crusted Catfish Calorie Smart \u2022 Fast\n\nYou can substitute regular ranch dressing for the fat-free variety, if that's what you have on hand. Or feel free to make your own, see page 138.\n\nPrep 10 min Total 25 min \u25cf 4 servings\n\n  * 1 lb catfish fillets (about \u00be inch thick)\n  * \u00bc cup fat-free ranch dressing\n  * \u00bc cup yellow cornmeal\n  * \u00bc cup unseasoned dry bread crumbs\n  * 1 teaspoon chili powder\n  * \u00bd teaspoon paprika\n  * \u00bd teaspoon garlic salt\n  * \u00bc teaspoon pepper\n  * Chopped fresh parsley, if desired\n\n1 Heat oven to 450\u00b0F. Spray rack in broiler pan with cooking spray. Cut fish into 4 serving pieces. Lightly brush dressing on both sides of fish.\n\n2 In shallow dish, mix remaining ingredients except parsley. Coat fish with cornmeal mixture. Place fish on rack in pan.\n\n3 Bake about 15 minutes or until fish flakes easily with fork. Sprinkle with parsley.\n\n1 Serving: Calories 230; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 70mg; Sodium 440mg; Total Carbohydrate 17g (Dietary Fiber 1g, Sugars 1g); Protein 20g Exchanges: 1 Starch, 2\u00bd Lean Meat Carbohydrate Choices: 1\n\nCornmeal-Crusted Cod Substitute 1 pound cod fillets, cut into serving size pieces, for the catfish.\n\nLemon- and Parmesan-Crusted Salmon\n\n## Lemon- and Parmesan-Crusted Salmon Calorie smart\n\nPrep 10 min Total 35 min \u25cf 4 servings\n\n  * 1 salmon fillet (1\u00bc lb)\n  * 2 tablespoons butter, melted\n  * \u00bc teaspoon salt\n  * \u00be cup medium- to firm-textured bread crumbs (about 1 slice bread)*\n  * \u00bc cup grated Parmesan cheese\n  * 2 medium green onions, thinly sliced (2 tablespoons)\n  * 2 teaspoons grated lemon peel\n  * \u00bc teaspoon dried thyme leaves\n\n1 Heat oven to 375\u00b0F. Spray shallow baking pan with cooking spray. Pat salmon dry with paper towels. Place salmon, skin side down, in pan. Brush with 1 tablespoon of the butter. Sprinkle with salt.\n\n2 In small bowl, mix bread crumbs, cheese, onions, lemon peel and thyme. Stir in remaining 1 tablespoon butter. Press bread crumb mixture evenly on salmon.\n\n3 Bake uncovered 15 to 25 minutes or until fish flakes easily with fork. Cut into 4 serving pieces. Serve immediately.\n\n*Soft-textured bread is not recommended because it's too moist and won't create a crisp crumb topping.\n\n1 Serving: Calories 290; Total Fat 16g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 115mg; Sodium 420mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 0g); Protein 33g Exchanges: 5 Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\n## Honey-Mustard Glazed Salmon Calorie Smart\n\nPrep 20 min Total 35 min \u25cf 4 servings\n\n  * 1 salmon fillet (1 lb)\n  * 1 tablespoon packed brown sugar\n  * 1 tablespoon butter, melted\n  * 1 tablespoon olive or vegetable oil\n  * 1 tablespoon honey\n  * 1 tablespoon soy sauce\n  * 1 tablespoon Dijon mustard\n  * 1 clove garlic, finely chopped\n\n1 Place salmon, skin side down, in shallow glass or plastic dish. In small bowl, mix remaining ingredients; pour over salmon. Cover and refrigerate at least 15 minutes but no longer than 1 hour.\n\n2 Set oven control to broil. Remove salmon from marinade; reserve marinade. Place salmon, skin side down, on rack in broiler pan. Broil with top 4 to 6 inches from heat 10 to 15 minutes, brushing 2 or 3 times with marinade, until fish flakes easily with fork. Discard any remaining marinade. Cut into 4 pieces to serve.\n\n1 Serving: Calories 230; Total Fat 11g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 80mg; Sodium 320mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 6g); Protein 24g Exchanges: \u00bd Starch, 3 Lean Meat Carbohydrate Choices: \u00bd\n\n### Salmon Doneness\n\n**Perfectly Done:** Salmon is pink and cloudy-looking and flakes easily with a fork.\n\n**Underdone:** Salmon is translucent and does not flake easily with a fork.\n\n**Overdone:** Salmon is dry and tough.\n\nSalmon Burgers with Sour Cream\u2013Dill Sauce\n\n## Salmon Patties with Sour Cream\u2013Dill Sauce Calorie smart \u2022 FAst\n\nServe these patties as burgers by placing in buns with lettuce leaves.\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\n  * \u2153 cup sour cream\n  * 3 tablespoons mayonnaise or salad dressing\n  * \u00be teaspoon dried dill weed\n  * 1 egg\n  * 2 tablespoons milk\n  * 1 can (14\u00be oz) salmon, drained, skin and bones removed, flaked\n  * 2 medium green onions, chopped (2 tablespoons)\n  * 1 cup soft bread crumbs (about 1\u00bd slices bread)\n  * \u00bc teaspoon salt\n  * 1 tablespoon vegetable oil\n\n1 In small bowl, stir sour cream, mayonnaise and dill until well mixed. Cover and refrigerate until serving time.\n\n2 In medium bowl, beat egg and milk with fork or whisk. Stir in salmon, onions, bread crumbs and salt. Shape mixture into 4 patties about 4 inches in diameter.\n\n3 In 10-inch nonstick skillet, heat oil over medium heat. Cook patties in oil about 8 minutes, turning once, until golden brown. Serve with sauce.\n\n1 Serving: Calories 300; Total Fat 22g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 120mg; Sodium 750mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 2g); Protein 20g Exchanges: \u00bd Starch, 2\u00bd Lean Meat, 3 Fat Carbohydrate Choices: \u00bd\n\n## Fresh Branzino with Sweet Pea\u2013Mint Pesto Fast\n\nFresh branzino is very mild and tender. The term \"pan-dressed\" refers to a whole fish that has been split open and its internal organs removed. If the fish has hard or rough scales, those are often removed too. The head, tail and fins are usually left on small fish such as branzino or trout. You can ask the seafood department to remove those for you, or they can be easily cut off with a chef's knife at home.\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\nbranzino\n\n  * 4 pan-dressed branzino or trout (about \u00bd lb each), heads removed if desired\n  * 1 tablespoon butter, melted\n  * 1 tablespoon lemon juice\n  * \u00bd teaspoon coarse sea salt\n  * 4 slices fresh lemon, cut in half\n  * Kosher salt (coarse), if desired\n  * Fresh mint leaves, if desired (optional)\n\nSweet Pea\u2013Mint Pesto\n\n  * 1 cup frozen sweet peas\n  * \u00bc cup freshly grated Parmesan cheese\n  * 3 tablespoons olive oil\n  * 2 tablespoons finely chopped fresh mint leaves\n  * 1 tablespoon water\n  * 1 clove garlic\n  * \u00bc teaspoon coarse sea salt\n  * \u00bc teaspoon freshly ground pepper\n\n1 Heat oven to 400\u00b0F. Line 15x10x1-inch pan with foil. Rinse fish and pat dry with paper towel. Arrange fish in pan.\n\n2 In small bowl, mix melted butter and lemon juice; brush mixture lightly on inside and outside of fish. Sprinkle inside of fish with \u00bd teaspoon salt; place 2 lemon halves inside each. Bake 15 to 20 minutes or until fish flakes easily with fork.\n\n3 Meanwhile, cook peas as directed on package. Rinse with cold water; drain well. In food processor, place peas and remaining pesto ingredients. Cover and process until smooth, stopping to scrape down side if necessary. Spoon pesto into serving bowl.\n\n4 To serve, use a sharp knife to cut off the head and tail. Starting at head, slice along backbone and remove in one piece. Serve branzino with pesto. Sprinkle with kosher sea salt and fresh mint leaves.\n\n1 Serving: Calories 460; Total Fat 28g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 130mg; Sodium 700mg; Total Carbohydrate 6g (Dietary Fiber 1g, Sugars 2g); Protein 45g Exchanges: \u00bd Starch, 6 Very Lean Meat, 5 Fat Carbohydrate Choices: \u00bd\n\nRosemary-Lemon Branzino Place a sprig of fresh rosemary inside each fish with the lemon slices. Omit Sweet Pea\u2013Mint Pesto.\n\nLemon-Garlic Branzino Place 1 garlic clove, sliced, inside each fish with the lemon slices.\n\nHerbed Butter Branzino Melt 2 tablespoons Herbed Butter (page 469); brush on fish instead of lemon-butter mixture.\n\nBranzino with Sun-Dried Tomato Pesto Omit Sweet Pea\u2013Mint Pesto. Serve with Sun-Dried Tomato Pesto (page 456).\n\n### Removing Fillets from Cooked Fish\n\nRemove any fatty, gelatinous portion from belly of fish with spoon.\n\nCut off head and tail using a sharp knife.\n\nGently slice along backbone, using a table knife, loosening top fillet.\n\nOpen fish flat; pull back bone from fish, starting at head end.\n\nFresh Branzino with Sweet Pea\u2013Mint Pesto\n\n# Heirloom Recipe and New Twist\n\nThis classic Beer-Batter Fried Fish is often requested because the light flavor of the batter gets crispy while the fish inside is tender and flavorful. The secret is in the temperature of the oil and the consistency of the batter. For the Cajun Fish Soft Tacos, we use a flavorful coating so the fish is cooked in a skillet with equally delicious results. If you enjoy a bit of spice, this recipe will soon be one of your favorites too.\n\nHeirloom\n\nBeer-Batter Fried Fish\n\n## Beer-Batter Fried Fish Calorie Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 lb walleye, sole or other delicate- to medium-texture fish fillets (about \u00bd inch thick), skin removed\n  * 3 to 4 tablespoons all-purpose flour\n  * 1 cup all-purpose flour\n  * \u00bd cup regular or nonalcoholic beer\n  * 1 egg\n  * \u00bd teaspoon salt\n  * Vegetable oil for frying\n  * Tartar Sauce (page 461), if desired\n\n1 Cut fish into 8 serving pieces. Lightly coat with 3 to 4 tablespoons flour. In medium bowl, mix 1 cup flour, the beer, egg and salt with wooden spoon until smooth. (If batter is too thick, stir in additional beer, 1 tablespoon at a time, until desired consistency.)\n\n2 In deep fryer or 4-quart Dutch oven, heat oil (1\u00bd inches) to 350\u00b0F. Dip fish into batter, letting excess drip into bowl. Fry fish in batches in oil about 4 minutes, turning once, until golden brown (tail sections may cook more quickly). Remove with slotted spoon; drain on paper towels. Serve hot with tartar sauce.\n\n1 Serving: Calories 390; Total Fat 17g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 115mg; Sodium 410mg; Total Carbohydrate 30g (Dietary Fiber 1g, Sugars 0g); Protein 27g Exchanges: 2 Starch, 3 Very Lean Meat, 3 Fat Carbohydrate Choices: 2\nNew Twist\n\nCajun Fish Soft Tacos\n\n## Cajun Fish Soft Tacos Fast\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\nPickled Cabbage\n\n  * 1\u00bd cups thinly sliced green cabbage\n  * \u00bd cup thinly sliced red cabbage\n  * 2 tablespoons chopped fresh chives\n  * 3 tablespoons cider vinegar\n  * 1 tablespoon sugar\n  * \u00bc teaspoon salt\n\nFish\n\n  * 1 lb tilapia or other medium-firm fish fillets (about \u00bd inch thick), skin removed\n  * 1\u00bd teaspoons Cajun seasoning\n  * 1 egg\n  * 1 tablespoon red pepper sauce\n  * \u00bd cup cornmeal\n  * 2 tablespoons vegetable oil\n  * 8 flour tortillas (6 inch), heated as directed on package\n\nSeasoned Sour Cream\n\n  * \u00bd cup sour cream\n  * 2 teaspoons chopped fresh chives\n  * \u00bc teaspoon Cajun seasoning*\n\nToppings, if desired\n\n  * Chopped seeded tomato\n  * Cubed avocado\n\n1 In medium bowl, mix cabbages and 2 tablespoons chives. In tightly covered container, shake vinegar, sugar and salt until sugar is dissolved. Pour over cabbage mixture; toss to coat. Set aside; stir before serving.\n\n2 Sprinkle fish with 1\u00bd teaspoons Cajun seasoning and press into flesh. In small bowl, beat egg and pepper sauce with fork or whisk until blended. Place cornmeal in shallow dish. Dip fish into egg mixture, then coat with cornmeal.\n\n3 In 12-inch nonstick skillet, heat oil over medium heat. Cook fish in oil 6 to 8 minutes, turning once, until fish flakes easily with fork and is brown on both sides. (Fish cooks very quickly, especially the thinner tail sections; be careful not to overcook.)\n\n4 Meanwhile, in small bowl, mix seasoned sour cream ingredients. Gently break fish into bite-size pieces; divide fish among tortillas, placing 4 or 5 pieces down center of each. Top with pickled cabbage, seasoned sour cream, tomato and avocado.\n\n*Or substitute 2 teaspoons red pepper sauce for the Cajun seasoning.\n\n1 Serving (2 Tacos): Calories 510; Total Fat 20g (Saturated Fat 7g, Trans Fat 1g); Cholesterol 110mg; Sodium 1200mg; Total Carbohydrate 51g (Dietary Fiber 3g, Sugars 8g); Protein 31g Exchanges: 3 Starch, \u00bd Vegetable, 3 Very Lean Meat, 3\u00bd Fat Carbohydrate Choices: 3\u00bd\n\nToasted Spice Seared Mackerel\n\n## Toasted Spice Seared Mackerel Fast\n\nPrep 15 min Total 25 min \u25cf 4 servings\n\nGremolata Mayonnaise\n\n  * \u00bd cup mayonnaise\n  * 2 tablespoons chopped fresh parsley\n  * 1 teaspoon grated orange peel\n  * 1 teaspoon grated lemon peel\n\nToasted Spice Rub\n\n  * 1 tablespoon coriander seed\n  * 2 teaspoons cumin seed\n  * 1 teaspoon fennel seed\n  * 1 tablespoon sugar\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n\nFish\n\n  * 1\u00bc lb mackerel, snapper, tilapia or other medium-firm fish fillets (about \u00bd inch thick)\n  * 2 tablespoons vegetable oil\n\n1 In small bowl, mix gremolata mayonnaise ingredients until well blended. Cover and refrigerate until serving time.\n\n2 Heat 8-inch skillet over medium-high heat. Add coriander, cumin and fennel seed; toast 2 to 3 minutes, stirring constantly, until coriander seed turns golden brown and mixture becomes fragrant. Place toasted spices in spice grinder. Grind until mixture looks like finely ground pepper. Transfer to small bowl; stir in sugar, salt and pepper.\n\n3 Heat oven to 400\u00b0F. Rub tops of fish fillets with spice mixture; let stand 5 minutes. In 12-inch ovenproof skillet, heat oil over medium-high heat. Place fish, skin side down, in skillet; sear 2 minutes or until deep golden brown. Remove from heat. Carefully turn fish fillets over.\n\n4 Bake 2 to 4 minutes or until fish flakes easily with fork. Serve with gremolata mayonnaise.\n\n1 Serving: Calories 560; Total Fat 48g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 95mg; Sodium 860mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 4g); Protein 27g Exchanges: \u00bd Other Carbohydrate, 4 Lean Meat, 7 Fat Carbohydrate Choices: \u00bd\n\n# Shellfish Basics\n\nBecause shellfish are easy to prepare plus taste delicious, their popularity is no surprise. Shellfish are grouped into two main categories: crustaceans and mollusks. Crustaceans have long bodies with soft, jointed shells and include crab, crayfish, lobsters and shrimp. Mollusks have soft bodies, may or may not be covered by a shell and have no spinal column. Types include clams, mussels, scallops, oysters, abalone, octopus and squid (calamari).\n\n## Buying and Using Shellfish\n\n  * Clams, Mussels and Oysters (in the shell): Clam varieties include **hard shell** (cherrystone and littleneck) and **soft-shell** (razor and steamers). Clams, mussels and oysters in shells should be purchased alive. Look for tightly closed shells that are not cracked, chipped or broken. They should have a very mild aroma. Shells may open naturally but will close if lightly tapped, indicating they are still alive. If shells do not close, throw them out. \n  * Shucked Clams, Mussels and Oysters (no shells): These shellfish should be plump and surrounded by a clear, slightly milky or light gray liquid. They should have a mild aroma.\n  * Scallops: Varieties include **sea scallops** (1\u00bd to 2 inches wide) and tiny **bay scallops** (\u00bd inch wide). Scallops should look moist and have a mild, sweet aroma. They should not be standing in liquid or stored on ice. These shellfish are creamy white and may have a light orange, light tan or pink tint.\n  * Crabs and Lobsters: Live crabs and lobsters will show some leg movement and lobsters will curl their tails under when picked up. These shellfish must be cooked live or killed immediately before cooking; throw out any that are dead. Crabmeat is available fresh, and both crab and lobster meat are available canned and frozen.\n  * Shrimp: Varieties include raw (\"green\") with the heads on; raw in the shell without the heads; raw, peeled and deveined; cooked in the shell; or cooked, peeled and deveined. Shrimp should have a clean sea aroma. If they have an ammonia smell, they are spoiled and should be discarded.\n  * Squid (Calamari): Color of squid ranges from cream with reddish brown spots to light pink. Buy fresh squid that's whole with clear eyes and a fresh sea aroma. Cleaned squid is also available in juices and the meat should be firm.\n\n## Imitation Seafood Products\n\nImitation seafood products, shaped into pieces to resemble crab or lobster, are usually made from pollock and are less expensive than shellfish but are similar in taste and texture to the real thing. Because some of these products, often labeled surimi, contain a shellfish extract, check labels carefully if you have a seafood allergy.\n\n## Storing Shellfish\n\n  * Live crabs (hard shell or soft-shell) and lobsters should be cooked the same day they're purchased. Put them on a tray with sides, cover with a damp cloth and refrigerate until ready to cook.\n  * Raw scallops, shrimp, squid and shucked shellfish should be stored in a leak-proof bag or plastic container with a lid. Cook scallops, shrimp and squid within 2 days. Shucked shellfish should be used within 7 days.\n  * Live clams, mussels and oysters should be refrigerated in a container covered with a clean, damp cloth, not with an airtight lid or in a plastic bag, which will cause suffocation. Live clams, mussels and oysters may open their shells even when refrigerated. Give the shells a tap\u2014they will close if alive; if not, throw them out. Cook within 2 days.\n\n### Shellfish Per Serving\n\nThe amount of shellfish you need per person depends on the type you're preparing.\n\nType of Shellfish | Amount per Serving\n\n---|---\n\nClams, Mussels or Oysters\n\nIn shell\n\nShucked | 3 large hard-shell clams\n\n6 small hard-shell clams\n\n18 soft-shell clams\n\n18 mussels\n\n6 oysters\n\n\u00bc lb\n\nCrab (hard shell) or Lobster\n\nCrab (soft-shell) | 1\u00bc lb live or \u00bc lb cooked\n\n2 crabs\n\nShrimp\n\nWith head, unpeeled\n\nWithout head, unpeeled\n\nWithout head, peeled | 1 lb\n\n\u00bd lb\n\n\u00bc lb\n\nSquid\n\nWhole\n\nCleaned | \u00bd lb\n\n\u00bc lb\n\n## Successfully Cooking Shellfish\n\nCooked shellfish should be moist and slightly chewy; overcooking makes it tough and rubbery. Follow these guidelines to determine when shellfish is done:\n\n  * Crabs and lobsters turn bright red.\n  * Scallops turn milky white or opaque and get firm.\n  * Raw shrimp turns pink and gets firm.\n  * Live clams, oysters and mussels open their shells.\n  * Shucked clams, oysters and mussels get plump and opaque. Oyster edges start to curl.\n  * **To cook in a skillet (panfry),** heat about 1 tablespoon oil over medium-high heat. Cook the shellfish as directed below.\n  * **To boil,** bring water to a boil in large saucepan. Add shellfish and cook as directed below.\n  * **To broil,** place shellfish on broiler rack. Broil 7 to 9 inches from heat as directed below.\n\n### Timetable for Cooking Shellfish\n\nShellfish can be cooked in a variety of ways. Keep in mind the cooking time varies depending on the method of preparation.\n\nType of Shellfish | Amount in Pounds | Method and Cook Time\n\n---|---|---\n\nScallops (bay or sea) | 1 | Panfry 3 to 6 minutes\n\nBoil 2 to 3 minutes\n\nBroil 3 to 5 minutes\n\nShrimp (peeled and deveined) | 1 | Panfry 3 to 6 minutes\n\nBoil 2 to 3 minutes\n\nBroil 3 to 5 minutes\n\nShrimp (in shell) | 1 | Boil 4 to 6 minutes\n\nClams or Oysters (in shell) | 4 | Boil or steam 5 to 7 minutes\n\nMussels (in shell) | 2 | Boil or steam 5 to 7 minutes\n\nSquid | 1 | Panfry 2 to 3 minutes\n\n### Reheating Cooked Crab Legs\n\nMost crab legs available for purchase are already cooked. Follow these steps to reheat them before serving.\n\n1. Cut crab legs to fit 8-inch square microwavable dish.\n\n2. Cover with plastic wrap, folding one edge or corner back about \u00bc inch to vent steam.\n\n3. Microwave on Medium-High (70%), rotating dish once if the microwave has no turntable, until hot. Let stand as directed before serving.\n\nType of Shellfish | Approximate Weight in Pounds | Microwave Time in Minutes | Stand Time in Minutes\n\n---|---|---|---\n\nCooked Crab Legs (thawed if frozen) | 2 | 5 to 6 | 5\n\n### Understanding Shrimp Count\n\nFresh and frozen shrimp are sold by a descriptive size name like jumbo or large, and by \"count,\" or number per pound. The larger the shrimp, the lower the count. Size and count vary throughout the United States.\n\nCommon Shrimp Size Market Name | Count (Approximate Number) per Pound\n\n---|---\n\nColossal | Fewer than 10\n\nJumbo | 11 to 15\n\nExtra Large | 16 to 20\n\nLarge | 21 to 30\n\nMedium | 31 to 35\n\nSmall | 36 to 45\n\nMiniature or Tiny | About 100\n\n## Soft-Shell Crabs\n\nSoft-shell crabs are the same species as Atlantic hard-shell blue crabs, except soft-shell crabs have been caught immediately after shedding their old hard shells. They stay soft for only a couple of hours before their shells harden again. Like all live crabs, they should be cooked the same day they're purchased.\n\nNote: For more information about shellfish handling, safety and nutrition, call the FDA's Center for Food Safety and Applied Nutrition, 1-888-SAFEFOOD (723-3366). Recorded information is available 24 hours a day, or you can speak to an information specialist by calling Monday through Friday, 10 a.m. to 4 p.m. Eastern Standard Time. Or check the FDA's website at www.fda.gov.\n\n### Shucking Fresh Oysters\n\nSlip tip of oyster knife between top and bottom shells, next to hinge. Carefully run knife around oyster to other side of hinge, using twisting motion, to pry top and bottom shells apart.\n\nCut muscle attached in middle of oyster to both top and bottom shells to spread apart and separate. Remove top shell from bottom, being careful to not spill oyster juice in bottom shell.\n\nRemove meat from bottom shell by sliding knife inward, under meat, close against surface of bottom shell, to keep meat intact.\n\n### Deveining Shrimp\n\nUsing a small, pointed knife or shrimp deveiner, make a shallow cut along the center back of each shrimp; wash out vein.\n\nDeep-Fried Shrimp\n\n## Deep-Fried Shrimp Calorie Smart\n\nWatch the temperature of the oil carefully during frying. If it starts to smoke, turn down the heat and wait until the temperature returns to 350\u00b0F before putting in the next batch of shrimp.\n\nPrep 40 min Total 40 min \u25cf 4 servings\n\n  * \u00bd cup all-purpose flour\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n  * 2 eggs\n  * \u00be cup dry bread crumbs\n  * 1 lb uncooked medium shrimp, thawed if frozen, peeled (with tail shells left on), deveined\n  * Vegetable oil for frying\n\n1 In shallow dish, mix flour, salt and pepper. In another shallow dish, beat eggs slightly with fork or whisk. In third shallow dish, place bread crumbs. Pat shrimp dry with paper towels. Coat shrimp with flour mixture; dip into egg, then coat with bread crumbs.\n\n2 In deep fryer or 4-quart Dutch oven, heat oil (2 to 3 inches) to 350\u00b0F. Fry 4 or 5 shrimp at a time in oil about 1 minute, turning once, until golden brown. Drain on paper towels.\n\n1 Serving: Calories 300; Total Fat 14g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 215mg; Sodium 920mg; Total Carbohydrate 27g (Dietary Fiber 0g, Sugars 2g); Protein 19g Exchanges: 2 Starch, 2 Lean Meat, 1 Fat Carbohydrate Choices: 2\n\nDeep-Fried Oysters or Clams Substitute \u00be pound shucked oysters or clams, drained, for the shrimp.\n\nDeep-Fried Sea Scallops Substitute \u00be pound sea scallops, drained, for the shrimp. Fry 3 to 4 minutes or until golden brown on outside and white and opaque inside. Bay scallops, which are smaller, will cook more quickly.\n\nShrimp Creole\n\n## Shrimp Creole Calorie smart\n\nThis traditional dish from Louisiana is a slightly spicy mixture of spices with tomatoes and shrimp. It's similar to gumbo but thicker.\n\nPrep 30 min Total 1 hr \u25cf 6 servings\n\n  * \u00bc cup butter\n  * 3 medium onions, chopped (1\u00bd cups)\n  * 2 medium green bell peppers, finely chopped (2 cups)\n  * 2 medium stalks celery, finely chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 1 can (15 oz) tomato sauce\n  * 1 cup water\n  * 2 teaspoons chopped fresh parsley\n  * 1\u00bd teaspoons salt\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 2 dried bay leaves\n  * 2 lb uncooked medium shrimp, thawed if frozen, peeled, deveined\n  * 6 cups hot cooked rice (page 215)\n\n1 In 3-quart saucepan, melt butter over medium heat. Cook onions, bell peppers, celery and garlic in butter about 10 minutes, stirring occasionally, until onions are tender. Stir in remaining ingredients except shrimp and rice. Heat to boiling; reduce heat to low. Simmer uncovered 10 minutes.\n\n2 Stir in shrimp. Heat to boiling; reduce heat to medium. Cover and cook 4 to 6 minutes, stirring occasionally, until shrimp are pink. Remove and discard bay leaves. Serve shrimp mixture over rice.\n\n1 Serving: Calories 400; Total Fat 9g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 160mg; Sodium 1860mg; Total Carbohydrate 57g (Dietary Fiber 3g, Sugars 6g); Protein 22g Exchanges: 3\u00bd Starch, 1 Vegetable, 1\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 4\n\nLighter Directions For 335 calories and 3 grams of fat per serving, reduce butter to 1 tablespoon and use a nonstick saucepan.\n\n## Shrimp Scampi Calorie smart \u2022 FAst\n\nServe these succulent shrimp over a bed of hot angel hair pasta or fettuccine and sprinkle with extra chopped fresh parsley.\n\nPrep 20 min Total 20 min \u25cf 6 servings\n\n  * 2 tablespoons olive or vegetable oil\n  * 1\u00bd lb uncooked medium shrimp, thawed if frozen, peeled, deveined\n  * 2 medium green onions, thinly sliced (2 tablespoons)\n  * 2 tablespoons fresh lemon juice\n  * 1 tablespoon chopped fresh parsley\n  * 2 cloves garlic, finely chopped\n  * \u00bc teaspoon salt\n  * Grated or shredded Parmesan cheese, if desired\n\nIn 10-inch skillet, heat oil over medium heat. Add shrimp, onions, lemon juice, parsley, garlic and salt to skillet; cook 2 to 3 minutes, stirring frequently, until shrimp are pink. Remove from heat; sprinkle with cheese.\n\n1 Serving: Calories 90; Total Fat 4g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 105mg; Sodium 220mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 12g Exchanges: 2 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\n## Shrimp Gumbo Calorie smart\n\nPrep 45 min Total 45 min \u25cf 6 servings\n\n  * \u00bc cup butter, divided\n  * 2 medium onions, sliced\n  * 1 medium green bell pepper, cut into thin strips\n  * 2 cloves garlic, finely chopped\n  * 2 tablespoons all-purpose flour\n  * 3 cups beef broth (for homemade broth, see page 146)\n  * \u00bd teaspoon red pepper sauce\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 dried bay leaf\n  * 1 package (12 oz) frozen cut okra, thawed and drained\n  * 1 can (14.5 oz) whole tomatoes, undrained\n  * 1 can (6 oz) tomato paste\n  * 1 lb uncooked medium shrimp, thawed if frozen, peeled, deveined\n  * 3 cups hot cooked rice (page 215)\n  * \u00bc cup chopped fresh parsley\n\n1 In 4-quart Dutch oven, melt 1 tablespoon of the butter over medium heat. Cook onions, bell pepper and garlic in butter 5 minutes, stirring occasionally. Remove and set aside. Melt remaining butter in Dutch oven over medium heat. Stir in flour. Cook and stir over medium heat until bubbly and medium golden brown, 2 to 3 minutes; remove from heat.\n\n2 Stir in onions, peppers and remaining ingredients except shrimp, rice and parsley, breaking up tomatoes. Heat to boiling; reduce heat. Simmer uncovered 10 minutes, stirring occasionally.\n\n3 Stir in shrimp. Cover and simmer about 5 minutes or until shrimp turn pink. Discard bay leaf. Serve gumbo over rice. Sprinkle with parsley.\n\n1 Serving: Calories 320; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 125mg; Sodium 1310mg; Total Carbohydrate 41g (Dietary Fiber 4g, Sugars 9g); Protein 20g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, \u00bd Vegetable, 2 Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 3\n\n## Scallops with Artichokes and Tomatoes Calorie smart \u2022 FAst\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1 lb bay scallops\n  * 4 medium green onions, sliced (\u00bc cup)\n  * \u00bc teaspoon salt\n  * \u215b teaspoon white pepper\n  * 1 clove garlic, finely chopped\n  * 1 box (9 oz) frozen artichoke hearts, thawed, drained*\n  * 1 cup cherry tomatoes, cut into quarters\n  * 1 cup shredded fresh spinach leaves\n  * 1 tablespoon lemon juice\n  * Hot cooked linguine or angel hair pasta, if desired\n\n1 In 10-inch nonstick skillet, cook scallops, onions, salt, pepper and garlic over medium-high heat 4 minutes, stirring frequently, until scallops are white and opaque.\n\n2 Stir in artichokes, tomatoes and spinach. Cook, stirring occasionally, until tomatoes are hot and spinach is wilted; drain. Sprinkle with lemon juice. Serve over linguine.\n\n*A 14-ounce can of artichoke hearts, drained and cut into quarters, can be substituted for the frozen artichoke hearts.\n\n1 Serving: Calories 110; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 25mg; Sodium 580mg; Total Carbohydrate 12g (Dietary Fiber 5g, Sugars 2g); Protein 14g Exchanges: 2\u00bd Vegetable, 1 Very Lean Meat Carbohydrate Choices: 1\n\n## Betty's Staples Cooked Shrimp\n\nWith cooked shrimp on hand, a quick, delicious meal is minutes away. Buy them precooked or cook them yourself; refrigerate up to 2 days or freeze up to 2 months.\n\n  1. 1. **Shrimp and Melon Salad:** On individual salad plates, arrange 3 or 4 Bibb lettuce leaves and top with 1 cup each cubed cantaloupe and watermelon and \u00bd to \u00be cup cooked shrimp. Drizzle with balsamic vinaigrette.\n  2. 2. **Tomato-Basil Shrimp and Pasta:** In skillet, heat 1 can diced tomatoes with garlic, basil and oregano with 1 chopped bell pepper over medium-low heat until pepper is fork-tender. Stir in 1\u00bd cups cooked shrimp and 2 tablespoons chopped fresh basil; heat through. Serve over pasta.\n  3. 3. **Shrimp Tacos:** For each taco, in taco shell, layer about \u00bc cup each shredded lettuce and chopped fresh tomato, \u00bd cup heated cooked shrimp and a few slices avocado. Drizzle with 1 to 2 tablespoons each salsa and French dressing.\n  4. 4. **Scrambled Eggs and Shrimp:** Make scrambled eggs, page 90. Just before eggs are cooked, stir in 1\u00bd cups cooked shrimp and 2 tablespoons chopped fresh parsley.\n  5. 5. **Shrimp and Egg Salad Wraps:** In medium bowl, mix 4 chopped hard-cooked eggs, 1 cup chopped cooked shrimp, 1 tablespoon chopped fresh dill and 3 tablespoons creamy mustard-mayonnaise sauce. Spoon \u00bc of mixture down center of flour tortilla; top with \u00bd cup shredded lettuce. Roll up tortilla.\n\nShrimp Tacos\n\nCitrus Seafood Skillet\n\n## Citrus Seafood Skillet Fast\n\n\"Fresh and light\" best describes this seafood trio. If you love the flavors of ceviche, you'll find them all here in this quickly cooked dinner.\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1 tablespoon olive oil\n  * 1 orange bell pepper, cut into \u00bc-inch strips\n  * \u00bd medium red onion (halved lengthwise), sliced\n  * 1 tablespoon finely chopped seeded jalape\u00f1o chile\n  * \u00bd lb bay scallops\n  * \u00bd lb uncooked deveined peeled medium shrimp, thawed if frozen\n  * \u00bd lb uncooked squid (calamari) tubes, thawed if frozen and cut into rings\n  * \u00bd teaspoon salt\n  * 3 tablespoons fresh lime juice\n  * 2 teaspoons grated lime peel\n  * 2 tablespoons chopped fresh cilantro\n  * Hot cooked rice, if desired\n\n1 In 12-inch skillet, heat oil over medium-high heat. Cook bell pepper, onion and chile in oil 2 to 3 minutes, stirring occasionally, until crisp-tender.\n\n2 Pat seafood dry with paper towels. Add scallops and shrimp to skillet; sprinkle seafood and vegetables with salt. Cook 3 to 4 minutes, adding calamari during last 1 minute of cooking, until shrimp turn pink and scallops and calamari turn white and opaque. Add lime juice and lime peel; cook and stir just until thoroughly heated. Sprinkle with cilantro. Serve with rice.\n\n1 Serving: Calories 170; Total Fat 4.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 210mg; Sodium 550mg; Total Carbohydrate 7g (Dietary Fiber 1g, Sugars 2g); Protein 24g Exchanges: \u00bd Other Carbohydrate, 3\u00bd Very Lean Meat, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## Steamed Clams Calorie Smart\n\nPrep 20 min Total 50 min \u25cf 4 servings\n\n  * 4 lb cherrystone, littleneck or steamer clams in shells\n  * 6 cups water\n  * \u2153 cup white vinegar\n  * \u00bd cup boiling water, chicken broth or white wine\n  * Melted butter or Clarified Butter (page 469), if desired\n\n1 Discard any broken-shell or open (dead) clams that do not close when tapped. Place remaining clams in large container. Cover with 6 cups water and the vinegar. Let stand 30 minutes; drain. Scrub clams in cold water.\n\n2 In steamer*, place clams and boiling water. Cover and steam 5 to 8 minutes or until clam shells open at least 1 inch, removing clams as they open. Discard any unopened clams. Serve hot clams in shells with butter.\n\n*If steamer is not available, place clams in 6-quart Dutch oven. Add 1 inch boiling water; cover tightly.\n\n1 Serving: Calories 50; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 25mg; Sodium 40mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 9g Exchanges: 1 Lean Meat Carbohydrate Choices: 0\n\n## Buying Clams\n\nIn the fish market, you might find a variety of clams and names for them. Littleneck (topnecks) are the smallest hard-shell clams, with cherrystone (quahogs or chowder clams) being a little bigger. Steamer clams are soft-shell clams but can be cooked in the same way as hard-shell clams.\n\n### Opening Raw Clams\n\nHold clam with hinged side against heavy cloth or oven mitt. Insert oyster knife or blunt-tipped knife between shell halves.\n\nGently twist knife to pry open the shell and release juices. Be sure to work over bowl or plate to catch juices.\n\nHolding clam firmly, move sharp knife around clam, cutting muscle at hinge. Gently twist knife to pry open shell. Cut clam meat from shell.\n\n## Steamed Mussels in Wine Sauce Calorie smart \u2022 FAst\n\nServe the mussels with crusty Italian bread for sopping up the garlic-flavored wine sauce.\n\nPrep 30 min Total 30 min \u25cf 4 servings\n\n  * 24 large fresh mussels in shells (about 2 lb)\n  * 2 tablespoons olive oil\n  * \u00bd cup chopped fresh parsley\n  * 4 cloves garlic, finely chopped\n  * 2 plum (Roma) tomatoes, chopped\n  * 1 cup dry white wine or chicken broth\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon freshly ground pepper\n\n1 Discard any broken-shell or open (dead) mussels that do not close when tapped. Scrub remaining mussels, removing any barnacles with a dull paring knife. If there are beards, remove them by tugging them away from shells.\n\n2 Place mussels in large container. Cover with cool water. Agitate water with hand, then drain and discard water. Repeat several times until water runs clear; drain.\n\n3 In 12-inch skillet, heat oil over medium-high heat. Cook parsley and garlic in oil, stirring frequently, until garlic is lightly golden. Add tomatoes, mussels, wine, salt and pepper. Cover and cook about 10 minutes or until mussel shells open.\n\n4 Discard any unopened mussels. Spoon sauce from skillet over each serving.\n\n1 Serving: Calories 170; Total Fat 8g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 40mg; Sodium 640mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 0g); Protein 16g Exchanges: 1 Vegetable, 2 Lean Meat, \u00bd Fat Carbohydrate Choices: \u00bd\n\n### Determining Spoiled Mussels\n\nAfter cooking mussels, discard those that do not open; they are spoiled.\n\n## Boiled Lobsters Calorie smart\n\nPrep 20 min Total 35 min \u25cf 2 servings\n\n  * 2 to 4 quarts water\n  * 2 live lobsters (about 1 lb each)\n  * Melted butter or Clarified Butter (page 469), if desired\n  * Lemon wedges, if desired\n\n1 Fill 6-quart Dutch oven or stockpot one-third full with water. Heat to boiling. Plunge lobsters headfirst into water. Cover and return to boiling; reduce heat to low. Simmer 10 to 12 minutes or until lobsters turn bright red; drain.\n\n2 To remove meat, follow directions below. Serve with butter and lemon wedges. Serve green tomalley (liver) and coral roe (only in females) if desired.\n\n1 Serving: Calories 100; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 75mg; Sodium 400mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 21g Exchanges: 3 Very Lean Meat Carbohydrate Choices: 0\n\n### Removing Cooked Lobster Meat\n\nSeparate tail from body by breaking shell in half where tail and body meet.\n\nCut away membrane on tail to expose meat. Discard vein that runs through tail and sac near head of lobster.\n\nTwist large claws away from body of lobster. Use nutcracker to break open claws. Remove meat from claws, tail and body.\n\n## Boiled Hard-Shell Blue Crabs Calorie smart\n\nBlue crab is the most familiar hard-shell crab. It is usually 4 to 6 inches in diameter and often has red claw tips.\n\nPrep 20 min Total 1 hr \u25cf 4 servings\n\n  * 4 quarts water\n  * 16 live hard-shell blue crabs\n  * Melted butter or Clarified Butter (page 469), if desired\n  * Cocktail Sauce (page 58), if desired\n\n1 In 6-quart Dutch oven or stockpot, heat water to boiling. Drop 4 crabs at a time into water. Cover and return to boiling; reduce heat to low. Simmer 10 minutes; drain. Repeat with remaining crabs.\n\n2 To remove meat, follow directions below. Serve with butter and cocktail sauce.\n\n1 Serving: Calories 100; Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 105mg; Sodium 290mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 21g Exchanges: 3 Very Lean Meat Carbohydrate Choices: 0\n\n### Removing Cooked Crabmeat\n\nPlace crab on its back. Using thumb, pry up tail flap, twist off and discard. Turn right side up; pry up top shell. Pull it away from body and discard.\n\nUsing small knife (or fingers), cut the gray-white gills from both sides of crab. Discard gills and internal organs.\n\nTwist off claws and legs; use nutcracker to crack shells at joints. Remove meat with cocktail fork or nutpick. Break body in half; remove remaining meat.\n\n# Chapter 14: Vegetarian\n\nPortabella Muffuletta Sandwiches\n\n## Vegetarian Basics\n\nPlant-Based Diets\n\nSoy Foods\n\nSoy Products\n\n## Features\n\nBetty's Staples: Canned Tomatoes\n\nHeirloom Recipe and New Twist\n\nLearn to Prepare Tofu\n\nMake-Ahead: Cooked Veggie Crumbles\n\n## Main Dishes\n\nArtichoke-Spinach Lasagna\n\nAsian Stuffed Portabellas Calorie Smart\n\nBaked Buffalo Frittata Calorie Smart\n\nBarbeque Sandwiches\n\nBarley and Black Bean Pilaf Calorie Smart\n\nBlack Bean Quesadillas\n\nCalifornia Black Bean Burgers Calorie Smart \u2022 Fast\n\nCalifornia Black Bean Burgers with Tomato and Cheese\n\nCheese Enchiladas\n\nCheese Enchiladas Verde\n\nChickpea and Tomato Curry Calorie Smart \u2022 Fast\n\nDouble Black Bean and Avocado Burgers\n\nEasy Pizza\n\nEggplant Parmigiana\n\nEggplant-Olive Parmigiana\n\nEggplant-Ravioli Parmigiana\n\nEggs and Crumbles\n\nFalafel (Fava Bean Rounds) Calorie Smart\n\nGreat Northern Bean and Veggie Sausage Cassoulet Slow Cooker\n\nGrilled Cheese Sandwiches Fast\n\nItalian Tofu-Rice Skillet\n\nLasagna Cupcakes Calorie Smart\n\nMediterranean Lasagna\n\nMexican Tofu-Rice Skillet Calorie Smart \u2022 Fast\n\nMexican Wraps\n\nMoussaka Calorie Smart\n\nPepper and Soybean Stir-Fry Calorie Smart \u2022 Fast\n\nPesto-Parmesan Grilled Cheese Sandwiches\n\nPortabella Muffuletta Sandwiches Calorie Smart \u2022 Fast\n\nRatatouille Polenta Bake Calorie Smart\n\nSmoky Chipotle Soft Tacos Calorie Smart \u2022 Slow Cooker\n\nSpaghetti and Spicy Rice Balls Fast\n\nSweet Potato\u2013Cauliflower Curry Calorie Smart\n\nTaco Salad\n\nTempeh Stir-Fry with Noodles\n\nTempeh Stir-Fry with Yogurt Peanut Sauce Calorie Smart\n\nTofu Scramble Calorie Smart \u2022 Fast\n\nTofu Stir-Fry with Black Bean Sauce Calorie Smart \u2022 Fast\n\nTofu Teriyaki Noodles Calorie Smart\n\nTofu Teriyaki Noodles and Peppers\n\nTomato and Avocado Grilled Cheese Sandwiches\n\nTomatoes with Ravioli\n\nVeggie and Bean \"Meatballs\"\n\nVeggie and Bean Burgers Fast\n\nVeggie Joes Calorie Smart \u2022 Slow Cooker\n\nVeggie Pizza\n\nVeggie-Stuffed Potatoes\n\n## Side Dishes\n\nCajun Tomatoes with Black-Eye Peas\n\nCauliflower Steaks with Olive-Tomato Relish\n\nGarden Ratatouille Calorie Smart \u2022 Fast\n\nIndian-Style Vegetable Casserole Calorie Smart\n\nSpicy Samosa Patties Calorie Smart\n\n## Bonus Recipes\n\nChicken Tortilla Soup\n\nItalian Meatballs with Tomato Sauce\n\nQuick Skillet Pasta Sauce\n\nShrimp with Tomatoes\n\nThyme Tomatoes with Chicken\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Vegetarian Basics\n\nWelcome to the colorful, nutrition-packed, tasty world of vegetarian cooking. Whether you are looking for occasional meatless meals; are increasing the amount of grains, legumes, and vegetables your family eats; or you have health, ethical or environmental reasons for eating meatless, this is the chapter for you.\n\n## Plant-Based Diets\n\nWhat we eat can make a difference in our lives. Plant-based diets, which emphasize fruits, vegetables, grains, soy, beans, legumes and nuts, are generally rich in fiber, vitamins and other important nutrients. If you are new to eating meatless, start with one or two meals a week. Use protein-rich foods containing beans and legumes, tofu, soy or cheese. Add plenty of vegetables and fruits for taste and eye appeal.\n\nMeatless diets vary based on the types of foods allowed. Below are the most common plant-based diets, although there might be some variation within each eating style. | DIET\n\n---|---\n\nFOOD ALLOWED | Vegan | Semi-Vegetarian (Flexitarian) | Lacto-Ovo-Vegetarian | Lacto-Vegetarian | Ovo-Vegetarian\n\nFruit | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nVegetables | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nGrains | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nLegumes | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nNuts & Seeds | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nTofu & Soy | \u2713 | \u2713 | \u2713 | \u2713 | \u2713\n\nEggs\n\n|  |\n\n\u2713 | \u2713\n\n|  |\n\n\u2713\n\nDairy Products\n\n|  |\n\n\u2713 | \u2713 | \u2713\n\n|\n\nFish & Seafood\n\n|  |\n\nmay include occasionally\n\n|  |  |\n\nMeat\n\n|  |\n\nmay include occasionally\n\n|  |  |\n\nPoultry\n\n|  |\n\nmay include occasionally\n\n|  |  |\n\nCalifornia Black Bean Burgers (page 399)\n\n## Soy Foods\n\nAll the foods listed here are made from soybeans, which can be an important part of a healthy diet. High-protein powerhouses, soy-based foods can replace meat and poultry sources of protein, helping to reduce the amount of saturated fat in your diet.\n\nEdamame: see page 233.\n\nMiso: Fermented bean paste\u2014a basic flavoring in Japanese cooking. There are many different colors and flavors, ranging from sweet to salty. Use lighter versions in mild soups or sauces, darker versions in highly flavored dishes. Found in Japanese markets or natural food stores. Refrigerate in airtight container up to 1 year.\n\nSoy Cheese: Available as full-, low- and nonfat versions in many popular cheese flavors. A few may be available at grocery stores, but many are found in health food stores. All are lactose-free, but be sure to check labels for added milk protein if dairy is a problem for you. Best served chilled or at room temperature; may separate if heated.\n\nSoy Flour: Made from roasted soybeans that are ground into a powder. Look for full-fat soy flour and defatted soy flour, which has had the oils removed during processing. Store in the refrigerator or freezer.\n\nSoy Ice Cream: Dairy-free ice cream available in many flavors.\n\nSoy Mayonnaise: Available in flavors such as plain, Dijon, wasabi. Made from soybean oil. Vegan versions are available.\n\nSoy Milk: Available in aseptic (non-refrigerated) cartons or refrigerated, or as soy milk powder. Varieties include regular and light versions in plain, chocolate and vanilla. Once opened, refrigerate all soy milk; use within 5 days. Store soy milk powder in refrigerator or freezer.\n\nSoy Nuts: Roasted soybeans available salted or unsalted and in flavors such as wasabi, barbecue and honey roasted.\n\nSoy Pudding: Dairy-free refrigerated pudding available in many flavors.\n\nSoy Yogurt: Available flavors include plain, vanilla and fruit flavors. Dairy-free.\n\nSoybean Oil: Clean taste. A cooking oil low in saturated fat, high in omega-3 fatty acids. Many oils sold as \"vegetable oil\" are made from soybeans, but not all. Read ingredient list to be sure.\n\nSoynut Butter: Available in plain, unsweetened, honey, chocolate flavors. Use like peanut butter; milder \"nut\" flavor than peanut butter. Some varieties are 30 percent lower in total fat or 50 percent lower in saturated fat than regular peanut butter. Great for those with peanut allergies.\n\nTamari: Similar to but thicker than soy sauce; unique, mellow flavor; often gluten-free.\n\nTempeh: Fermented cooked soybeans, formed into dense, chewy cakes with nutty, yeast flavor; use as a meat substitute. Holds shape when cooked\u2014great for veggie burgers. When crumbled, great in casseroles. Available fresh or frozen. Refrigerate fresh up to 2 weeks; freeze up to 3 months.\n\nTextured Soy Protein: Available dried, granular or in chunks of compressed, processed soy flour (rehydrate before using). Textured soy crumbles (frozen, ready-to-use crumbles) resemble ground beef and are used to replace ground meat in recipes.\n\nTofu: Three main types available both flavored or unflavored. **Extra-Firm** **or Firm Tofu:** Highest in protein, fat and calcium; solid dense form holds its shape in recipes. Use in casseroles, stews, soups, sandwiches, salads, stir-fries and grilling. **Soft Tofu:** Soft, doesn't hold its shape. Best used in dips, dressings, desserts, sauces, smoothies and spreads. **Silken Tofu:** Creamy, smooth, custard-like texture. Use in dips, dressings, desserts, sauces, smoothies and spreads. (See also Learn to Prepare Tofu, page 392.)\n\n## How Tofu Is Made\n\nTo make tofu, soybeans are soaked, cooked, ground and then mixed with a curdling ingredient. The resulting curds are pressed into cakes, which are tofu. Firmness depends on the amount of whey liquid extracted. Tofu is very mild in flavor.\n\n## Cheese Enchiladas\n\nPrep 25 min Total 55 min \u25cf 4 servings\n\n  * 2 cups shredded Monterey Jack cheese (8 oz)\n  * 1 cup shredded Cheddar cheese (4 oz)\n  * \u00bd cup sour cream\n  * 1 medium onion, chopped (\u00bd cup)\n  * 2 tablespoons chopped fresh parsley\n  * \u00bc teaspoon pepper\n  * 1 can (15 oz) tomato sauce\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 1 can (4.5 oz) chopped green chiles, drained\n  * 1 clove garlic, finely chopped\n  * \u2154 cup water\n  * 1 tablespoon chili powder\n  * 1\u00bd teaspoons chopped fresh or \u00bd teaspoon dried oregano leaves\n  * \u00bc teaspoon ground cumin\n  * 8 soft corn tortillas (5 or 6 inch)\n  * Chopped green onions and additional sour cream, if desired\n\n1 Heat oven to 350\u00b0F. In medium bowl, mix cheeses, sour cream, onion, parsley and pepper; set aside.\n\n2 In 2-quart saucepan, heat remaining ingredients except tortillas to boiling, stirring occasionally; reduce heat. Simmer uncovered 5 minutes. Spread \u00bc cup sauce in ungreased 9-inch glass pie plate.\n\nSpoon about \u00bc cup cheese mixture onto each tortilla; roll tortilla around filling. Place seam side down in ungreased 11x7-inch (2-quart) glass baking dish. Pour remaining sauce over top.\n\n4 Bake uncovered 25 to 30 minutes or until bubbly. Garnish with chopped green onions and additional sour cream.\n\n1 Serving (2 Enchiladas): Calories 530; Total Fat 34g (Saturated Fat 20g, Trans Fat 1g); Cholesterol 100mg; Sodium 1310mg; Total Carbohydrate 31g (Dietary Fiber 5g, Sugars 9g); Protein 26g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Vegetable, 3 High-Fat Meat, 2 Fat Carbohydrate Choices: 2\n\n## Cheese Enchiladas Verde\n\nPrep 15 min Total 40 min \u25cf 4 servings\n\n  * 1\u00bd cups shredded Mexican cheese blend (6 oz)\n  * 1 small zucchini, shredded (\u00be cup)\n  * \u00bc cup chopped fresh cilantro\n  * \u2153 cup garden vegetable cream cheese spread\n  * 8 soft corn tortillas (5 or 6 inch)\n  * 2 cups salsa verde (for homemade Salsa Verde, see page 458)\n  * 1 medium avocado, pitted, peeled and diced\n  * 4 medium green onions, sliced (\u00bc cup)\n\n1 Heat oven to 350\u00b0F. Spray 11x7-inch (2-quart) glass baking dish with cooking spray.\n\n2 In medium bowl, mix shredded cheese, zucchini and cilantro. Spread cream cheese on tortillas. Top each with about \u2153 cup shredded cheese mixture to within 1 inch of edge; roll tortilla around filling. Place seam side down in baking dish. Pour salsa over top.\n\n3 Cover and bake about 25 minutes or until hot. Garnish with avocado and onions.\n\n1 Serving (2 Enchiladas): Calories 490; Total Fat 31g (Saturated Fat 15g, Trans Fat 1g); Cholesterol 65mg; Sodium 550mg; Total Carbohydrate 36g (Dietary Fiber 7g, Sugars 7g); Protein 17g Exchanges: \u00bd Starch, \u00bd Fruit, \u00bd Other Carbohydrate, 2\u00bd Vegetable, 1\u00bd Medium-Fat Meat, 4\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## Baked Buffalo Frittata CALORIE Smart\n\nPrep 10 min Total 35 min \u25cf 4 servings\n\n  * 1 tablespoon butter\n  * 6 eggs\n  * 2 tablespoons Buffalo wing sauce\n  * 1 cup shredded Mexican cheese blend (4 oz)\n  * 2 tablespoons crumbled blue cheese\n  * 2 stalks celery, cut into 3-inch pieces\n\n1 Heat oven to 375\u00b0F. Place butter in 8-inch square (2-quart) glass baking dish; place dish in oven until butter is melted.\n\n2 In medium bowl, beat eggs and Buffalo wing sauce until well blended. Stir in cheese blend. Pour egg mixture over melted butter in baking dish.\n\n3 Bake uncovered 18 to 22 minutes or until set. Sprinkle frittata with blue cheese. Serve with celery pieces.\n\n1 Serving: Calories 270; Total Fat 21g (Saturated Fat 11g, Trans Fat 0g); Cholesterol 355mg; Sodium 460mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 2g); Protein 17g Exchanges: 2\u00bd Medium-Fat Meat, 1\u00bd Fat Carbohydrate Choices: 0\n\n## Indian-Style Vegetable Casserole CALORIE Smart\n\nPrep 15 min Total 1 hr 10 min \u25cf 8 servings\n\n  * 1 can (13.5 oz) unsweetened coconut milk (not cream of coconut)\n  * \u00be cup water\n  * 1\u00bd teaspoons salt\n  * 1 cup uncooked basmati rice\n  * 1 bag (1 lb) frozen sliced carrots, thawed, drained (about 4 cups)\n  * 1 bag (1 lb) frozen cauliflower florets, thawed, drained (about 5 cups)\n  * 1 can (15.8 oz) black-eyed peas, drained, rinsed\n  * 1 serrano chile, seeded, finely chopped\n  * 2 tablespoons olive oil\n  * 1\u00bd teaspoons garam masala\n  * Chopped fresh cilantro, if desired\n  * Lime wedges, if desired\n\n1 Heat oven to 375\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 In 2-quart saucepan, stir 1 cup of the coconut milk, the water, \u00bd teaspoon of the salt and the rice. Heat to boiling over high heat. Stir; reduce heat to low. Cover and simmer 15 minutes or until liquid is absorbed. Remove from heat; fluff rice with fork.\n\n3 In large bowl, stir carrots, cauliflower, peas, chile, oil, remaining 1 teaspoon salt and the garam masala. Stir in cooked rice and remaining coconut milk until well blended. Spoon into baking dish.\n\n4 Cover and bake 40 minutes or until thoroughly heated. Garnish with cilantro; squeeze lime wedges over top.\n\n1 Serving (1\u2153 Cups): Calories 320; Total Fat 14g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 0mg; Sodium 630mg; Total Carbohydrate 39g (Dietary Fiber 6g, Sugars 5g); Protein 7g Exchanges: 2 Starch, 1\u00bd Vegetable, 2\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## What is Garam Masala?\n\nTranslated from Hindi, it means \"warm\" or \"hot.\" A blend of dry-roasted ground spices common to the colder climates of northern India, it's used to give a sense of warmth to dishes, both in spirit and in taste. Containing up to 12 spices, including cinnamon, cloves, coriander, cumin and dried chilies, each cook may have their own unqiue variation.\n\nYou can make your own easily, however you'd want to do it in small batches to be used within 6 months to retain its freshness, or you can purchase small jars of it in most grocery stores or Indian markets.\n\nChickpea and Tomato Curry\n\n## Chickpea and Tomato Curry CALORIE Smart \u2022 Fast\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\n  * 1 tablespoon olive or vegetable oil\n  * 1 medium onion, chopped (\u00bd cup)\n  * 3 cloves garlic, finely chopped\n  * 1 tablespoon finely chopped gingerroot\n  * 1 tablespoon curry powder\n  * 2 cans (15 oz each) chickpeas (garbanzo beans), drained, rinsed\n  * 2 cans (14.5 oz each) fire-roasted diced tomatoes, undrained\n  * \u00bd cup chopped fresh cilantro\n  * 1 tablespoon fresh lemon juice\n  * \u00bd teaspoon coarse (kosher or sea) salt\n  * Hot cooked rice, if desired\n  * Plain yogurt, if desired\n\n1 In 3-quart saucepan, heat oil over medium heat. Cook onion, garlic, gingerroot and curry powder in oil about 2 minutes, stirring frequently, until onion is tender.\n\n2 Stir in chickpeas and tomatoes. Heat to boiling; reduce heat. Simmer uncovered 15 minutes, stirring occasionally.\n\n3 Stir in cilantro, lemon juice and salt. Serve over rice; top individual servings with yogurt.\n\n1 Serving (1 Cup): Calories 270; Total Fat 6g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 380mg; Total Carbohydrate 42g (Dietary Fiber 10g, Sugars 5g); Protein 12g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 1 Vegetable, \u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 3\n\n## Artichoke-Spinach Lasagna\n\nPrep 25 min Total 1 hr 35 min \u25cf 8 servings\n\n  * 1 medium onion, chopped (\u00bd cup)\n  * 4 cloves garlic, finely chopped\n  * 1\u00be cups vegetable broth (for homemade broth, see page 156)\n  * 1 tablespoon chopped fresh or 1 teaspoon dried rosemary leaves\n  * 1 can (14 oz) artichoke hearts, drained, coarsely chopped\n  * 1 box (9 oz) frozen chopped spinach, thawed, squeezed to drain\n  * 1 jar (15 to 17 oz) Alfredo sauce (for homemade sauce, see page 177)\n  * 9 uncooked fresh or dried lasagna noodles\n  * 3 cups shredded mozzarella cheese (12 oz)\n  * 1 package (4 oz) crumbled herb-and-garlic or plain feta cheese (1 cup)\n  * Rosemary sprigs, if desired\n  * Lemon wedges, if desired\n\n1 Heat oven to 350\u00b0F. Spray 13x9-inch (3-quart) glass baking dish with cooking spray.\n\n2 Spray 12-inch skillet with cooking spray; heat over medium-high heat. Add onion and garlic; cook about 3 minutes, stirring occasionally, until onion is crisp-tender. Stir in broth and rosemary leaves. Heat to boiling. Stir in artichokes and spinach; reduce heat. Cover and simmer 5 minutes. Stir in Alfredo sauce.\n\n3 Spread one-fourth of the artichoke mixture in baking dish; top with 3 noodles. Sprinkle with \u00be cup of the mozzarella cheese. Repeat layers twice. Spread with remaining artichoke mixture; sprinkle with remaining \u00be cup mozzarella cheese. Sprinkle with feta cheese.\n\n4 Cover and bake 40 minutes. Uncover; bake about 15 minutes longer or until bubbly and noodles are tender. Let stand 10 to 15 minutes before cutting. Garnish with rosemary sprigs and lemon wedges.\n\n1 Serving: Calories 520; Total Fat 31g (Saturated Fat 19g, Trans Fat 1g); Cholesterol 95mg; Sodium 960mg; Total Carbohydrate 35g (Dietary Fiber 7g, Sugars 3g); Protein 24g Exchanges: 2\u00bd Starch, 2 Very Lean Meat, 5\u00bd Fat Carbohydrate Choices: 2\n\nMediterranean Lasagna Stir in \u00bd cup chopped pitted kalamata, Greek or ripe olives with the Alfredo sauce.\n\nLasagna Cupcakes\n\n## Lasagna Cupcakes CALORIE Smart\n\nThese are fantastic little casseroles to freeze and eat for a quick dinner\u2014or to take to work for lunch. Microwave 1 frozen cupcake uncovered on Medium (50%) 5 to 6 minutes or until hot.\n\nPrep 15 min Total 1 hr \u25cf 12 servings\n\n  * 1 cup ricotta cheese\n  * \u00bd cup grated Parmesan cheese\n  * 1 egg\n  * 1 jar (25.5 oz) pasta sauce (any meatless variety; for homemade sauce, see page 174)\n  * \u00bd lb frozen Italian sausage\u2013style soy-protein crumbles (2 cups)\n  * 36 pot sticker (gyoza) wrappers\n  * 1 cup shredded mozzarella cheese (4 oz)\n\n1 Heat oven to 375\u00b0F. Spray 12 regular-size muffin cups with cooking spray. In small bowl, mix ricotta cheese, Parmesan cheese and egg. In another small bowl, mix pasta sauce and soy-protein crumbles.\n\n2 Place 1 pot sticker wrapper in bottom of each muffin cup; top each with 1 heaping tablespoon pasta sauce mixture and 1 tablespoon cheese mixture. Repeat layers twice, ending with pasta sauce mixture. Sprinkle evenly with mozzarella cheese.\n\n3 Spray sheet of foil large enough to cover pan with cooking spray; place foil, sprayed side down, over pan. Bake 15 minutes. Uncover; bake 15 minutes longer. Let stand 15 minutes before serving.\n\n1 Serving: Calories 170; Total Fat 7g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 30mg; Sodium 500mg; Total Carbohydrate 13g (Dietary Fiber 2g, Sugars 6g); Protein 12g Exchanges: 1 Starch, 1 Medium-Fat Meat Carbohydrate Choices: 1\n\nEggplant Parmigiana\n\n## Eggplant Parmigiana\n\nPrep 25 min Total 50 min \u25cf 6 servings\n\n  * 1 egg\n  * 2 tablespoons water\n  * \u2154 cup seasoned dry bread crumbs\n  * \u2153 cup grated Parmesan cheese\n  * 2 small unpeeled eggplants (about 1 lb each), cut into \u00bc-inch slices\n  * \u00bc cup olive or vegetable oil\n  * 2 cups tomato pasta sauce (any meatless variety; for homemade sauce, see page 174)\n  * 2 cups shredded mozzarella cheese (8 oz)\n\n1 Heat oven to 350\u00b0F. In shallow dish, beat egg and water. In another shallow dish, mix bread crumbs and Parmesan cheese. Dip eggplant in egg mixture, then coat with crumb mixture.\n\n2 In 12-inch nonstick skillet, heat 2 tablespoons of the oil over medium heat. Cook half of the eggplant in oil about 5 minutes, turning once, until light golden brown; drain on paper towels. Repeat with remaining 2 tablespoons oil and eggplant, using additional oil if necessary.\n\n3 In ungreased 11x7-inch (2-quart) glass baking dish, place half of the eggplant, overlapping slices slightly. Spoon half of the pasta sauce over eggplant. Sprinkle with 1 cup of the mozzarella cheese. Repeat with remaining eggplant, pasta sauce and mozzarella cheese.\n\n4 Bake uncovered about 25 minutes or until sauce is bubbly, cheese is light brown and eggplant is still slightly firm but can be easily pierced with fork.\n\n1 Serving: Calories 410; Total Fat 23g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 60mg; Sodium 830mg; Total Carbohydrate 35g (Dietary Fiber 5g, Sugars 13g); Protein 17g Exchanges: 1 Starch, 4 Vegetable, 1\u00bd High-Fat Meat, 2 Fat Carbohydrate Choices: 2\n\nEggplant-Olive Parmigiana Prepare as directed, except stir \u00bd cup sliced pitted imported Italian black olives into the pasta sauce before spooning over eggplant in Step 3.\n\nEggplant-Ravioli Parmigiana Cook 1 bag (25 oz) frozen cheese-filled ravioli as directed on package; drain and keep warm. Prepare eggplant as directed, except arrange half of cooked ravioli over eggplant in dish in Step 3. Spoon half of pasta sauce over ravioli. Sprinkle with 1 cup of the mozzarella cheese. Repeat with remaining eggplant, ravioli, sauce and cheese. Bake as directed.\n\n### Preparing Eggplant Parmigiana\n\nDip eggplant slices in egg mixture, coat with crumb mixture.\n\nCook half of the eggplant at a time, turning once, until light golden brown.\n\nPlace half of the eggplant in dish, overlapping slices slightly.\n\n# Heirloom Recipe and New Twist\n\nRatatouille is a classic French dish that simmers eggplant, tomatoes, onions, bell peppers and zucchini in olive oil with garlic and herbs. The vegetables can be altered to suit your tastes, making it an ideal recipe to use when you have a variety of fresh vegetables on hand. Ratatouille Polenta Bake, the new twist for this recipe, pairs ratatouille with the heartiness of polenta for a very satisfying, extraordinary meal.\n\nHeirloom\n\nGarden Ratatouille\n\n## Garden Ratatouille Calorie Smart \u2022 Fast\n\nEnjoy this easy vegetable side dish that's ready in 25 minutes. Perfect if you love French cuisine.\n\nPrep 25 min Total 25 min \u25cf 6 servings\n\n  * 1 medium unpeeled eggplant (1\u00bd cups), cut into \u00bd-inch pieces (3 cups)\n  * 1 medium zucchini, cut into \u00bc-inch slices (1 cup)\n  * 1 small onion, sliced\n  * 1 medium green bell pepper, cut into strips\n  * 2 cloves garlic, finely chopped\n  * 2 tablespoons chopped fresh parsley\n  * 1 tablespoon chopped fresh or \u00bd teaspoon dried basil leaves\n  * 2 tablespoons water\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 medium ripe tomatoes, each cut into eighths\n  * 2 tablespoons olive oil\n  * Additional chopped fresh basil leaves, if desired\n\n1 In 10-inch skillet, cook all ingredients except tomatoes, oil and additional chopped fresh basil leaves over medium heat about 10 minutes, stirring occasionally, until vegetables are tender; remove from heat.\n\n2 Stir in tomatoes and oil. Cover and let stand 2 to 3 minutes or until tomatoes are warm. Sprinkle with additional chopped fresh basil leaves.\n\n1 Serving: Calories 100; Total Fat 5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 210mg; Total Carbohydrate 11g (Dietary Fiber 4g, Sugars 6g); Protein 2g Exchanges:2Vegetable, 1 Fat Carbohydrate Choices: 1\n\n## Ratatouille Anytime\n\nPronounced _ra-tuh-TOO-ee_ , ratatouille can be served at any temperature, either as a side dish or as an appetizer with bread or crackers.\nNew Twist\n\nRatatouille Polenta Bake\n\n## Ratatouille Polenta Bake CALORIE Smart\n\nPrep 25 min Total 1 hr 15 min \u25cf 6 servings\n\n  * 1 medium onion, coarsely chopped (\u00bd cup)\n  * 1 medium green bell pepper, coarsely chopped (1 cup)\n  * 1 small unpeeled eggplant (1 lb), chopped (2 cups)\n  * 1 medium zucchini, chopped (1 cup)\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 can (14.5 oz) Italian-style stewed tomatoes, undrained\n  * 1 roll (1 lb) refrigerated plain or flavored polenta (for homemade polenta, see page 207)\n  * 2 tablespoons shredded Parmesan cheese\n  * \u00be cup finely shredded mozzarella cheese (3 oz)\n  * \u00bc cup chopped fresh parsley\n\n1 Heat oven to 375\u00b0F. Spray 11x7-inch (2-quart) glass baking dish with cooking spray.\n\n2 Spray 12-inch skillet with cooking spray; heat over medium-high heat. Add onion and bell pepper; cook 2 minutes, stirring occasionally. Add eggplant, zucchini, salt and pepper. Cook 3 to 4 minutes, stirring occasionally, until vegetables are tender. Stir in tomatoes, breaking up with spoon. Reduce heat to low. Cook 3 minutes, stirring occasionally. Remove from heat.\n\n3 Cut polenta into \u00bc-inch slices. Arrange slices in bottom of baking dish, overlapping and cutting to fit as necessary. (For homemade polenta, spoon into bottom of dish; spreading evenly.) Sprinkle with Parmesan cheese. Spoon vegetable mixture evenly over top.\n\n4 Cover and bake 30 minutes. Uncover; sprinkle with mozzarella cheese. Bake about 15 minutes longer or until bubbly and cheese is melted. Sprinkle with parsley. Let stand 5 minutes before serving.\n\n1 Serving: Calories 260; Total Fat 7g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 10mg; Sodium 830mg; Total Carbohydrate 45g (Dietary Fiber 6g, Sugars 0g); Protein 10g Exchanges: 2 Starch, 3 Vegetable, \u00bd Fat Carbohydrate Choices: 3\n\n## Sweet Potato\u2013Cauliflower Curry CALORIE Smart\n\nCouscous would also be good with this curry instead of rice. And raisins, coconut or chutney are other good choices for toppings.\n\nPrep 30 min Total 1 hr \u25cf 6 servings\n\nCurry\n\n  * 2 tablespoons vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 2 medium dark-orange sweet potatoes, peeled, cut into 1-inch cubes (4 cups)\n  * 4 cups fresh cauliflower florets\n  * 1 tablespoon finely chopped gingerroot\n  * 4 teaspoons curry powder\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 can (13.5 oz) coconut milk (not cream of coconut)\n  * 1 cup vegetable broth (for homemade broth, see page 156)\n  * 2 teaspoons cornstarch\n  * 2 tablespoons cold water\n\nToppings\n\n  * 1\u00bd cups hot cooked white or brown rice (page 215)\n  * \u2153 cup sliced almonds, toasted (page 23)\n  * \u2153 cup sliced green onions (about 5 medium)\n  * \u2153 cup chopped fresh cilantro\n\n1 In 5-quart Dutch oven or stockpot, heat oil over medium-high heat. Cook onion in oil 5 minutes, stirring frequently, until tender. Add garlic; cook 1 minute. Stir in remaining curry ingredients except cornstarch and water. Reduce heat to low. Cover and cook 30 minutes or until vegetables are tender.\n\n2 Uncover; increase heat to medium-high. In small bowl, blend cornstarch and water until smooth paste forms. Stir cornstarch mixture into curry. Cook uncovered 5 to 10 minutes, stirring frequently, until thickened and bubbly.\n\n3 Serve curry in individual deep bowls. Top each serving with \u00bc cup rice. Sprinkle evenly with almonds, green onions and cilantro.\n\n1 Serving: Calories 400; Total Fat 23g (Saturated Fat 13g, Trans Fat 0g); Cholesterol 0mg; Sodium 440mg; Total Carbohydrate 40g (Dietary Fiber 7g, Sugars 11g); Protein 7g Exchanges: 2 Starch, 1\u00bd Vegetable, 4\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nMoussaka\n\n## Moussaka CALORIE Smart\n\nOur version of moussaka uses soy-protein burgers in the flavorful eggplant \"lasagna.\" You'll love the protein but won't miss the meat!\n\nPrep 30 min Total 1 hr 40 min \u25cf 4 servings\n\n  * 4 frozen soy-protein burgers or soy-protein vegetable burgers\n  * 1 medium unpeeled eggplant (about 1\u00bd lb), cut into \u00bc-inch slices\n  * \u00bc cup all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon pepper\n  * \u00bc teaspoon ground nutmeg\n  * \u00bc teaspoon ground cinnamon\n  * 3 cups milk\n  * 4 oz (half of 8-oz package) cream cheese, cut into cubes\n  * 1 can (15 oz) tomato sauce\n  * 2 eggs, beaten\n\n1 Heat oven to 375\u00b0F. Spray 3-quart casserole with cooking spray. On large microwavable plate, microwave burgers uncovered on High 2 to 3 minutes, turning once, until thawed. Cut into 1-inch pieces; set aside.\n\n2 In 2-quart saucepan, place eggplant and enough water to cover. Heat to boiling; cook 5 to 8 minutes or until tender. Drain in colander; set aside.\n\n3 In same saucepan, mix flour, salt, pepper, nutmeg, cinnamon and milk. Heat to boiling, stirring constantly. Boil 1 minute, stirring constantly. Remove from heat. Stir in cream cheese until melted and smooth.\n\n4 In casserole, layer half of the eggplant, the burger pieces, tomato sauce, 1\u00bd cups of the white sauce and the remaining eggplant. Stir eggs into remaining white sauce; pour over eggplant.\n\n5 Bake uncovered about 1 hour or until firm. Let stand 10 minutes before serving.\n\n1 Serving: Calories 330; Total Fat 6g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 1560mg; Total Carbohydrate 40g (Dietary Fiber 7g, Sugars 21g); Protein 29g Exchanges: 2 Starch, 2 Vegetable, 2\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 2\u00bd\n\n## Asian Stuffed Portabellas CALORIE Smart\n\nPrep 20 min Total 1 hr 25 min \u25cf 4 servings\n\n  * \u00be cup uncooked regular brown rice\n  * 1\u00bd cups water\n  * 4 large portabella mushrooms (4 to 5 inches in diameter), stems removed\n  * 1 tablespoon vegetable oil\n  * 1\u00bd cups coleslaw mix (shredded cabbage and carrots)\n  * \u00bd cup cut-up fresh snow pea pods\n  * \u00bc cup sliced green onions (4 medium)\n  * 1 tablespoon finely chopped gingerroot\n  * 1 clove garlic, finely chopped\n  * \u00bc cup reduced-sodium soy sauce\n  * 2 teaspoons sesame oil\n  * 4 teaspoons sesame seed\n\n1 In 1\u00bd-quart saucepan, heat brown rice and water to boiling; reduce heat. Cover and simmer 45 to 50 minutes or until water is absorbed.\n\n2 Heat oven to 400\u00b0F. Spray 15x10x1-inch pan with cooking spray. Place mushrooms, gill sides down, in pan. Bake 5 minutes or until tender and beginning to brown.\n\n3 Meanwhile, in 10-inch nonstick skillet, heat vegetable oil over medium-high heat. Add coleslaw mix, pea pods, onions, gingerroot and garlic. Cook 2 to 3 minutes, stirring frequently, until coleslaw mix is wilted and vegetables are crisp-tender. Stir in cooked rice; sprinkle with soy sauce and sesame oil. Cook, tossing gently with soy sauce and oil, until thoroughly heated; remove from heat.\n\n4 Turn mushrooms gill sides up. Spoon about \u00be cup rice mixture onto each mushroom; sprinkle each with 1 teaspoon sesame seed. Bake 13 to 15 minutes or until sesame seed begins to turn brown.\n\n1 Serving: Calories 250; Total Fat 9g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 560mg; Total Carbohydrate 35g (Dietary Fiber 6g, Sugars 3g); Protein 7g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 1\u00bd Vegetable, 1\u00bd Fat Carbohydrate Choices: 2\n\n## Betty's Staples Canned Tomatoes\n\nIf you have a can of diced or whole tomatoes in your pantry, an easy meal has a great start. With just a few ingredients and some imagination, it's simple to take that can of tomatoes and turn it into a delicious dinner.\n\n  1. 1. **Quick Skillet Pasta Sauce:** Heat 1 tablespoon olive oil in skillet over medium heat. Add 2 sliced garlic cloves; cook 30 seconds. Add a 28-ounce can tomatoes (breaking any large pieces with a spoon), 1 teaspoon dried basil leaves and \u00bc teaspoon crushed red pepper flakes. Heat to boiling; reduce heat. Simmer uncovered about 10 minutes or until thickened. Stir in \u00bd teaspoon salt and \u00bc teaspoon pepper. Serve over hot cooked pasta.\n  2. 2. **Cajun Tomatoes with Black-Eyed Peas:** Cook chopped onion, garlic and bell pepper in small amount of oil until tender. Stir in a 14-ounce can of tomatoes, a 16-ounce can drained black-eyed peas and sprinkle with a little ground red pepper (cayenne) and thyme. Simmer until hot and desired consistency. If desired, serve with rice.\n  3. 3. **Easy Pizza:** Heat a 14-ounce can of tomatoes in a saucepan until slightly thickened. Spoon onto a purchased pizza crust and top with pepperoni slices and shredded mozzarella cheese. Bake as directed on the pizza package.\n  4. 4. **Shrimp with Tomatoes:** Heat a 14-ounce can of tomatoes (breaking any large pieces with a spoon) in skillet with chopped green bell pepper, a sprinkle of garlic powder, a few drops of sriracha sauce and chopped fresh or dried oregano. Stir in 1 to 2 cups cooked shrimp (any size) and heat until very hot and desired consistency. \n  5. 5. **Italian Meatballs with Tomato Sauce:** Mix 15 to 20 frozen Italian meatballs with a 28-ounce can of tomatoes in a skillet. Cover and cook, stirring occasionally, until meatballs are thawed and mixture is very hot. Serve with mashed potatoes or over pasta.\n  6. 6. **Chicken Tortilla Soup:** Heat tomatoes with 4 cups of chicken broth, about 1 cup shredded rotisserie chicken and a 4-ounce can diced green chiles until boiling. Reduce heat and simmer for about 5 minutes and sprinkle with baked corn chips and chopped fresh cilantro.\n  7. 7. **Thyme Tomatoes with Chicken:** Brown 4 chicken breast halves (seasoned with salt, pepper and garlic powder) in skillet and pour a 14-ounce can of tomatoes over them. Sprinkle with chopped fresh or dried thyme and cover. Simmer 10 to 15 minutes or until chicken is no longer pink in the center and sauce is desired consistency. \n  8. 8. **Tomatoes with Ravioli:** Heat a 20-ounce package of refrigerated ravioli with a 28-ounce can of tomatoes, sliced zucchini and chopped red onion until ravioli are tender. Sprinkle with a little crumbled goat cheese.\n\nCajun Tomatoes with Black-Eyed Peas\n\n## Cauliflower Steaks with Olive-Tomato Relish\n\nCauliflower steaks are created by cutting straight down through a large head of cauliflower to create 1-inch slabs or \"steaks.\" When roasted, cauliflower morphs into a slightly sweet and caramelized version of its former self and is delicious!\n\nPrep 15 min Total 55 min \u25cf 4 servings\n\nCauliflower Steaks\n\n  * 2 large heads cauliflower\n  * 2 tablespoons olive oil\n\nOlive-Tomato Relish\n\n  * \u00bc cup chopped pimiento-stuffed olives\n  * \u00bc cup chopped pitted kalamata olives\n  * 1 plum (Roma) tomato, seeded, chopped (\u00bd cup)\n  * 1 tablespoon chopped fresh parsley\n  * 1 tablespoon olive oil\n  * \u00bc teaspoon dried oregano leaves\n  * \u215b teaspoon pepper\n  * 1 clove garlic, finely chopped\n\n1 Heat oven to 400\u00b0F. Line 15x10x1-inch pan with foil; spray foil with cooking spray.\n\n2 Trim tough leaves off bottom of cauliflower, leaving stem intact. Cut off tough bottom of stem, about \u00bd inch. Carefully cut 2 (1-inch) slabs (\"steaks\") from each cauliflower. Reserve remaining cauliflower for another use. Place cauliflower steaks in pan. Brush tops lightly with 1 tablespoon of the oil.\n\n3 Roast 15 minutes. Carefully turn cauliflower steaks; brush with remaining 1 tablespoon oil. Roast about 15 to 20 minutes longer or until golden brown.\n\n4 Meanwhile, in small bowl, gently mix all relish ingredients until blended. Top each cauliflower steak with \u00bc cup relish. Serve warm.\n\n1 Serving: Calories 250; Total Fat 14g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 320mg; Total Carbohydrate 23g (Dietary Fiber 9g, Sugars 9g); Protein 8g Exchanges: 4 Vegetable, 3 Fat Carbohydrate Choices: 1\u00bd\n\n### Making Cauliflower Steaks\n\nLeaving stem intact, cut each cauliflower vertically into 2 (1-inch) \"steaks.\"\n\nRoast cauliflower 15 minutes. Carefully turn steaks and roast until golden brown.\n\n## Barley and Black Bean Pilaf CALORIE Smart\n\nPrep 40 min Total 1 hr 20 min \u25cf 6 servings (1\u2153 cups each)\n\n  * 1 tablespoon vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 2 cloves garlic, finely chopped\n  * 4 cups vegetable broth (for homemade broth, see page 156)\n  * 1 cup uncooked medium pearled barley\n  * \u00bd teaspoon ground turmeric\n  * \u00bd teaspoon ground cumin\n  * \u00bd teaspoon ground chipotle chile pepper\n  * \u00bd teaspoon salt\n  * 8 pattypan squash, cut into quarters, or 2 medium yellow summer squash, cut in half lengthwise, then crosswise into \u00bd-inch slices\n  * 1 large red bell pepper, coarsely chopped (1\u00bd cups)\n  * 1 can (15 oz) black beans, drained, rinsed\n  * 1 cup frozen sweet peas\n\n1 In 12-inch nonstick skillet, heat oil over medium heat. Add onion and garlic; cook 4 minutes, stirring occasionally.\n\n2 Stir in broth, barley, turmeric, cumin, chile pepper and salt. Heat to boiling. Reduce heat; cover and simmer 40 minutes, stirring occasionally.\n\n3 Stir in squash, bell pepper and beans. Cover; cook 10 to 15 minutes, stirring occasionally, until squash and barley are just tender.\n\n4 Stir in peas. Cover; cook 3 to 5 minutes longer or until peas are cooked. Let stand covered 5 to 10 minutes to let flavors blend before serving.\n\n1 Serving (1\u2153 Cups): Calories 270; Total Fat 3.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 490mg; Total Carbohydrate 50g (Dietary Fiber 13g, Sugars 7 g); Protein 10g Exchanges: 1 \u00bd Starch, 1 \u00bd Other Carbohydrate, 1 Vegetable, 1 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 3\n\nCauliflower Steaks with Olive-Tomato Relish\n\n## Learn to Prepare TOFU\n\nSometimes called bean curd, tofu is made from soybeans and processed into a solid form. This versatile food absorbs flavors easily and can be used as a healthful source of protein in meals. Tofu is readily available in grocery stores\u2014look for it in the refrigerated section, as it needs to be kept cold. Cooking tofu is not difficult, but there are some tips that will help make the process easy and result in great-tasting tofu dishes.\n\n### Tofu Storage\n\n  * Tofu is perishable, so check the sell-by date on the package. If unopened, it will keep in the refrigerator up to 5 days. Once opened, use within 2 days. Store covered with water and change the water daily.\n  * Tofu can be frozen up to 5 months. Freeze in original container or wrap tightly in plastic wrap.\n\n### Cooking Tips\n\nStir-frying and baking are easy ways to cook tofu. Choose the method depending on the texture you want and what you are cooking with it. For use in most cooking, extra-firm and firm tofu will be the best types to use.\n\n  * Most tofu is packaged in water and is like a sponge, so it needs to be drained well before cooking. On a plate or other surface, press the tofu between layers of paper towels or kitchen towels; top with a weight such as a heavy skillet or plate. Let it stand for at least 1 hour.\n  * Determine how to cut the tofu according to what you are making or the recipe used. Cubes will work well in stir-fries, soups and stews; cut into larger pieces or slices for frying, baking or grilling.\n  * To marinate and add flavor, place the tofu pieces in a container and pour the marinade mixture over it. Cover and marinate in the refrigerator at least 1 hour or overnight. The tofu in the marinade can also be frozen. When ready to cook, just thaw in the refrigerator and cook as desired. Tofu becomes a bit firmer after freezing.\n  * **Stir-Frying Tofu:** Results in tofu with dry, firm texture and crispy edges. \n    * Heat a nonstick pan over medium heat and add 1 tablespoon vegetable or olive oil. Season tofu with salt and pepper or other desired seasonings.\n    * Cook 4 to 5 minutes or until light golden brown on both sides, turning once.\n  * **Baking Tofu:** Results in tofu with denser, firmer texture. \n    * Heat oven to 350\u00b0F. Lightly spray a shallow pan with cooking spray or grease lightly.\n    * Place tofu in the pan. It can be seasoned with salt and pepper, soy sauce or other desired seasonings, or it can be marinated.\n    * Bake 30 to 35 minutes or until thoroughly heated and slightly golden brown.\n\nTofu Stir-Fry with Black Bean Sauce\n\n## Tofu Stir-Fry with Black Bean Sauce CALORIE Smart \u2022 Fast\n\nBlack bean sauce is robust in flavor, thin and salty. It is made from fermented black beans, garlic and often star anise. Look for it with other Asian ingredients.\n\nPrep 25 min Total 25 min \u25cf 5 servings\n\n  * 1 tablespoon olive oil\n  * 1 large onion, thinly sliced (1 cup)\n  * 4 cups chopped fresh broccoli\n  * 1 medium red bell pepper, cut into \u00bd-inch slices, then cut in half (1 cup)\n  * \u2153 cup black bean garlic sauce (from 7-oz jar)\n  * 2 tablespoons soy sauce\n  * 1 tablespoon chili-garlic sauce (from 8-oz jar)\n  * 1 package (12 to 14 oz) firm lite tofu, drained if needed, cut into \u00be-inch cubes\n  * \u2153 cup chopped fresh cilantro\n  * 5 cups hot cooked brown basmati rice (page 215)\n  * \u00bc cup honey-roasted or regular sunflower seeds\n\n1 In wok or 12-inch skillet, heat oil over medium-high heat. Cook onion in oil about 2 minutes, stirring occasionally, until crisp-tender. Add broccoli and bell pepper; stir-fry about 2 minutes or until almost crisp-tender.\n\n2 In small bowl, mix black bean sauce, soy sauce and chili-garlic sauce. Stir into vegetable mixture. Add tofu; stir-fry 2 to 3 minutes or until thoroughly heated. Stir in cilantro. Serve over rice. Sprinkle with sunflower seeds.\n\n1 Serving (1 Cup Tofu Mixture and 1 Cup Rice): Calories 340; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 540mg; Total Carbohydrate 48g (Dietary Fiber 9g, Sugars 4g); Protein 14g Exchanges: 3 Starch, 1 Vegetable, \u00bd Medium-Fat Meat, 1 Fat Carbohydrate Choices: 3\n\n### Preparing Tofu\n\nPress tofu between layers of paper towels; top with a weight, such as a heavy skillet or plate, to remove moisture.\n\nTo stir-fry tofu, cook 4 to 5 minutes or until light golden brown on both sides, turning once.\n\nTo bake tofu, place on pan; season and bake until thoroughly heated and slightly golden brown.\n\n## Tofu-cation\n\nIf you're new to the tofus scene, here's the skinny on how to purchase tofu: Look for tofu packed in clear liquid with no gelatinous or discolored areas. If the liquid is cloudy, it may be spoiled. When opened, fresh tofu shouldn't smell or taste sour.\n\n## Tofu Scramble CALORIE Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\n  * 1 package (14 oz) firm tofu, drained\n  * 1 tablespoon olive oil\n  * 1 medium red onion, cut in half, thinly sliced (1 cup)\n  * 1 package (8 oz) sliced fresh mushrooms (about 3 cups)\n  * 1 orange bell pepper, chopped (1 cup)\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bc teaspoon ground turmeric\n  * \u215b teaspoon crushed red pepper flakes\n  * 2 cups fresh baby spinach leaves, coarsely chopped\n  * 2 plum (Roma) tomatoes, seeded, finely chopped\n  * \u00bc cup fresh basil leaves, thinly sliced\n\n1 Crumble tofu; squeeze dry with paper towels. Set aside.\n\n2 Heat 10-inch nonstick skillet over medium heat. Add oil and onion; cook 3 to 4 minutes, stirring occasionally, until onion begins to soften. Add mushrooms and bell pepper. Cook about 5 minutes or until bell pepper begins to soften. Stir in tofu, salt, pepper, turmeric and pepper flakes.\n\n3 Add spinach; cook and stir until spinach begins to wilt. Remove from heat. Top with tomatoes and basil. Serve immediately.\n\n1 Serving (1\u00bc Cups): Calories 160; Total Fat 8g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 620mg; Total Carbohydrate 11g (Dietary Fiber 3g, Sugars 6g); Protein 11g Exchanges: 2 Vegetable, 1 Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 1\n\nTofu Teriyaki Noodles\n\n## Tofu Teriyaki Noodles CALORIE Smart\n\nPrep 20 min Total 40 min \u25cf 4 servings\n\n  * 1 cup hot water\n  * 6 dried black or shiitake mushrooms (\u00bd oz)\n  * 8 oz uncooked soba (buckwheat) noodles or whole wheat spaghetti\n  * 1 tablespoon vegetable oil\n  * 1 large onion, sliced\n  * 1 package (12 to 14 oz) firm or extra-firm tofu, drained if needed, cut into \u00bc-inch cubes\n  * 1 package (8 oz) sliced fresh mushrooms (about 3 cups)\n  * \u00bd lb fresh shiitake, cremini or baby portabella mushrooms, sliced\n  * \u2153 cup teriyaki sauce\n  * \u00bc cup chopped fresh cilantro\n  * 1 tablespoon sesame seed, toasted if desired (page 23)\n\n1 In small bowl, pour hot water over dried mushrooms. Let stand about 20 minutes or until soft; drain. Rinse with warm water; drain. Squeeze out excess moisture from mushrooms. Remove and discard stems; cut caps into \u00bd-inch strips.\n\n2 Meanwhile, cook and drain noodles as directed on package.\n\n3 In 12-inch skillet, heat oil over medium-high heat. Cook onion in oil 3 minutes, stirring frequently. Stir in tofu and all mushrooms; cook 3 minutes, stirring frequently. Stir in teriyaki sauce; reduce heat. Partially cover and simmer about 2 minutes or until vegetables are tender. Stir in noodles, cilantro and sesame seed.\n\n1 Serving (2 Cups): Calories 370; Total Fat 8g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 1170mg; Total Carbohydrate 54g (Dietary Fiber 9g, Sugars 6g); Protein 20g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 3 Vegetable, 1 Very Lean Meat, 1\u00bd Fat Carbohydrate Choices: 3\u00bd\n\nTofu Teriyaki Noodles and Peppers Substitute 3 sliced medium bell peppers (any color) for the fresh mushrooms. Add the peppers with the onions.\n\n## Mexican Tofu-Rice Skillet CALORIE Smart \u2022 Fast\n\nLooking for an icebreaker for introducing tofu into your eating plan? This is it. Familiar Mexican flavors let tofu do its chameleon act of absorbing all the wonderful tastes in the dish.\n\nPrep 25 min Total 30 min \u25cf 4 servings\n\n  * 1 package (12 to 14 oz) extra-firm tofu, drained if needed\n  * 1 cup frozen whole kernel corn\n  * 1 can (14.5 oz) diced tomatoes with green chiles, undrained\n  * 1 package (5.6 oz) Spanish-flavor rice and pasta blend mix in tomato sauce\n  * 1\u00bc cups water\n  * 1 cup shredded Mexican cheese blend (4 oz)\n  * 1\u00bd cups shredded lettuce\n  * 1 large tomato, seeded, chopped (1 cup)\n  * 4 medium green onions, sliced (\u00bc cup)\n\n1 Place tofu between 2 layers of paper towels; press gently to remove as much water as possible. Cut into \u00bd-inch cubes; set aside.\n\n2 In 12-inch nonstick skillet, mix corn, canned tomatoes, contents of rice mix package and water. Gently stir in tofu. Heat to boiling; reduce heat to low. Cover and simmer 12 to 14 minutes, stirring occasionally, until rice is tender.\n\n3 Remove from heat. Sprinkle cheese over rice mixture. Cover and let stand 4 to 5 minutes or until liquid is absorbed and cheese is melted. Top individual servings with lettuce, chopped tomato and onions.\n\n1 Serving: Calories 420; Total Fat 16g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 25mg; Sodium 1060mg; Total Carbohydrate 48g (Dietary Fiber 5g, Sugars 9g); Protein 23g Exchanges: 3 Starch, 1 Vegetable, 1\u00bd High-Fat Meat, \u00bd Fat Carbohydrate Choices: 3\n\nItalian Tofu-Rice Skillet Substitute 1 can (14.5 oz) diced tomatoes with Italian seasoning for the diced tomatoes with green chiles. Use mozzarella cheese instead of the Mexican cheese blend.\n\nMexican Wraps After cooking rice mixture, do not sprinkle with cheese. Cover and let stand 4 to 5 minutes or until liquid is absorbed. Spoon mixture onto center of 4 flatbread wraps. Sprinkle with cheese; top with lettuce, tomato and onion. Fold opposite sides of each tortilla over filling. Beginning at one open end, roll up each tortilla. Cut diagonally in half; if desired, secure with toothpick. Serve immediately.\n\nGreat Northern Bean and Veggie Sausage Cassoulet\n\n## Great Northern Bean and Veggie Sausage Cassoulet SLOW Cooker\n\nPrep 20 min Total 7 hr 50 min \u25cf 6 servings\n\n  * 3 carrots, sliced (1\u00bd cups)\n  * 1 large onion, finely chopped (1 cup)\n  * 3 cans (15.5 oz each) great northern beans, drained, rinsed\n  * \u00be cup dry white wine or vegetable broth\n  * \u00bc cup chopped dry-pack sun-dried tomatoes\n  * 2 teaspoons dried minced garlic\n  * 1 teaspoon dried thyme leaves\n  * 1 dried bay leaf\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 package (14 oz) frozen Italian-style soy-protein sausages with sun-dried tomatoes and basil, each cut into 3 pieces\n  * 1 can (14.5 oz) diced tomatoes, undrained\n  * 2 tablespoons finely chopped fresh Italian (flat-leaf) parsley\n  * 1 tablespoon finely chopped fresh thyme leaves\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, gently mix carrots, onion, beans, wine, sun-dried tomatoes, dried minced garlic, dried thyme, bay leaf, salt and pepper. Arrange sausage pieces on top of mixture. Pour tomatoes and their liquid over top.\n\n2 Cover; cook on Low heat setting 7 hours 30 minutes to 8 hours 30 minutes (or High heat setting 3 hours 30 minutes to 4 hours 30 minutes). Remove and discard bay leaf. Sprinkle individual servings with parsley and fresh thyme.\n\n1 Serving (1\u00bd Cups): Calories 580; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 850mg; Total Carbohydrate 76g (Dietary Fiber 22g, Sugars 9g); Protein 42g Exchanges: 4\u00bd Starch, 2 Vegetable, 3\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 5\n\nSpaghetti and Spicy Rice Balls\n\n## Spaghetti and Spicy Rice Balls FAST\n\nThese rice and oat \"meatballs\" get a light wheat germ coating, giving them a golden-brown color and a bit of nutty-flavored crunch. Wheat germ is high in oil content, so it can turn rancid quickly; store it in the refrigerator or freezer.\n\nPrep 30 min Total 30 min \u25cf 5 servings\n\n  * 2 cups cooked regular long-grain white or brown rice (page 215)*\n  * \u00bd cup quick-cooking oats\n  * 1 medium onion, chopped (\u00bd cup)\n  * \u00bc cup unseasoned dry bread crumbs\n  * \u00bc cup milk\n  * 1 tablespoon chopped fresh or 1 teaspoon dried basil leaves\n  * 2 teaspoons chopped fresh or \u00bd teaspoon dried oregano leaves\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 1 egg, beaten\n  * \u00bd cup wheat germ\n  * 1 tablespoon vegetable oil\n  * 1 package (1 lb) spaghetti\n  * 2 cups tomato pasta sauce (any meatless variety; for homemade sauce, see page 174), heated\n  * Shredded Parmesan cheese, if desired\n\n1 In medium bowl, mix cooked rice, oats, onion, bread crumbs, milk, basil, oregano, red pepper and egg. Shape mixture into 10 balls; roll in wheat germ to coat.\n\n2 In 10-inch skillet, heat oil over medium heat. Cook rice balls in oil about 10 minutes, turning occasionally, until light golden brown. Meanwhile, cook and drain spaghetti as directed on package.\n\n3 Serve rice balls and pasta sauce over spaghetti; sprinkle with cheese.\n\n*Do not use instant rice.\n\n1 Serving: Calories 740; Total Fat 12g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 45mg; Sodium 1180mg; Total Carbohydrate 133g (Dietary Fiber 9g, Sugars 14g); Protein 24g Exchanges: 7\u00bd Starch, 1 Other Carbohydrate, 1 Vegetable, 1\u00bd Fat Carbohydrate Choices: 9\n\n## Pepper and Soybean Stir-Fry CALORIE Smart \u2022 Fast\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1 tablespoon vegetable oil\n  * 1 tablespoon curry powder\n  * 1 bag (1 lb) frozen bell pepper and onion  stir-fry\n  * 1 bag (12 oz) frozen shelled edamame (green soybeans)\n  * 4 cloves garlic, finely chopped\n  * 1 cup canned unsweetened coconut milk (not cream of coconut)\n  * 2 cups hot cooked jasmine rice (page 215)*\n  * \u00bd cup salted roasted cashews\n  * Chopped fresh cilantro or parsley, if desired\n\n1 In wok or 12-inch nonstick skillet, heat oil over medium-high heat. Add curry powder; cook 1 minute, stirring constantly. Add stir-fry vegetables, edamame and garlic. Cook 2 minutes, stirring frequently. Cover and cook about 3 minutes longer or until vegetables are tender.\n\n2 Stir in coconut milk. Reduce heat to medium-low; simmer uncovered 2 minutes, stirring occasionally. Serve over rice. Sprinkle with cashews and cilantro.\n\n*Regular long-grain white or brown rice can be substituted for the jasmine rice.\n\n1 Serving: Calories 540; Total Fat 28g (Saturated Fat 12g, Trans Fat 0g); Cholesterol 0mg; Sodium 370mg; Total Carbohydrate 53g (Dietary Fiber 11g, Sugars 8g); Protein 19g Exchanges: 2\u00bd Starch, \u00bd Other Carbohydrate, 1 Vegetable, 1\u00bd Very Lean Meat, 5 Fat Carbohydrate Choices: 3\u00bd\n\n## Make-Ahead COOKED VEGGIE CRUMBLES\n\nA convenient and tasty ingredient, veggie crumbles, made from soy, are easy to cook and add hot to recipes or simply stir frozen into recipes before cooking.\n\nTo cook veggie crumbles: For about 4 cups of crumbles, place contents of 1 (12-ounce) bag in skillet. Add \u2153 cup water and cover skillet. Cook over medium-high heat for 4 minutes. Reduce heat to medium; cook an additional 4 minutes, stirring occasionally. Uncover and cook 2 to 3 minutes or until hot and most of water has evaporated.\n\nHere are some recipe ideas for using the cooked crumbles.\n\nTaco Salad: In medium bowl, toss about 1\u00bd cups hot cooked veggie crumbles with \u00bc cup chunky salsa, 2 cups shredded lettuce, 1 cup chopped tomatoes, \u00bc cup sliced green onion and 1 medium avocado, diced. Top with \u00bd cup shredded Cheddar cheese and drizzle with \u00bc cup taco sauce.\n\nEggs and Crumbles: In medium skillet, scramble 4 eggs (page 90). As eggs cook, stir in 1 cup hot cooked veggie crumbles and 1 teaspoon dried basil leaves. Sprinkle eggs and crumbles with \u00bd cup shredded mozzarella cheese.\n\nVeggie Pizza: Top purchased Italian pizza crust (10- to 12-inch) with 1 cup pizza sauce or seasoned tomato sauce. Sprinkle with 1 cup frozen veggie crumbles, 1 cup chopped green bell pepper, \u00bd cup sliced ripe olives and 1 cup sliced fresh mushrooms. Sprinkle with 1\u00bd cups shredded Italian cheese blend. Bake as directed on pizza crust package.\n\nBarbecue Sandwiches: In 2-quart saucepan, heat 1\u00bd cups frozen veggie crumbles, \u00bd cup barbecue sauce and 1 cup frozen bell pepper and onion mixture until very hot and bell peppers are fork-tender. Spoon mixture onto sandwich buns.\n\nBlack Bean Quesadillas: For each quesadilla, top 1 flour tortilla with \u00bd cup hot cooked veggie crumbles, \u00bd cup drained canned black beans, \u00bc cup chunky salsa and \u00bd cup shredded Cheddar cheese. Top with another tortilla. Cook in nonstick skillet until golden brown on both sides and cheese is melted.\n\nVeggie-Stuffed Potatoes: Bake potatoes (page 237). Slit top and open carefully. With fork, mash slightly. Top each potato with \u00bd cup hot cooked veggie crumbles, \u00bc cup cooked chopped broccoli, \u00bc cup cooked sliced carrots and 2 tablespoons sour cream. Sprinkle with 2 tablespoons chopped fresh parsley. Add salt and pepper to taste.\n\nVeggie-Stuffed Potatoes\n\nTempeh Stir-Fry with Yogurt Peanut Sauce\n\n## Tempeh Stir-Fry with Yogurt Peanut Sauce CALORIE Smart\n\nTempeh, pronounced TEHM-pay and also spelled tempe, is a fermented, high-protein soybean cake with a chewy texture and slightly nutty flavor. If you can't find it in your supermarket (sometimes it's sold in the freezer case with other vegetarian foods), check a co-op or natural foods store.\n\nPrep 40 min Total 40 min \u25cf 4 servings\n\n  * \u00bc cup creamy peanut butter\n  * \u00bc cup vanilla low-fat yogurt\n  * 3 tablespoons teriyaki marinade (from 12-oz bottle)\n  * 1 tablespoon honey\n  * 2 tablespoons vegetable oil\n  * 1 package (8 oz) tempeh, cut into 2x\u00bcx\u00bc-inch strips\n  * 1 medium onion, cut into thin wedges\n  * 4 medium carrots, cut into 2x\u00bcx\u00bc-inch strips (2 cups)\n  * 12 oz fresh green beans, cut in half crosswise (2 cups)\n  * \u00bc cup water\n  * 1 medium red bell pepper, cut into thin bite-size strips\n  * 2 cups hot cooked white or brown rice (page 215)\n  * \u00bc cup chopped fresh cilantro\n\n1 In small bowl, beat peanut butter, yogurt, teriyaki marinade and honey with whisk until smooth; set aside.\n\n2 In wok or 12-inch skillet, heat 1 tablespoon of the oil over medium heat. Cook tempeh in oil 5 to 6 minutes, turning frequently, until light golden brown. Remove tempeh to plate or bowl; cover to keep warm.\n\n3 In same skillet, heat remaining 1 tablespoon oil over medium heat. Cook onion in oil 1 minute, stirring occasionally. Stir in carrots, green beans and water. Cover and cook 5 minutes. Stir in bell pepper. Cook 2 to 3 minutes, stirring occasionally, until vegetables are crisp-tender.\n\n4 Stir in tempeh and reserved peanut butter mixture until well mixed. Cook 1 to 2 minutes, stirring occasionally, until hot. Serve over rice. Sprinkle with cilantro.\n\n1 Serving: Calories 490; Total Fat 22g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 0mg; Sodium 600mg; Total Carbohydrate 52g (Dietary Fiber 8g, Sugars 17g); Protein 20g Exchanges: 1\u00bd Starch, 1\u00bd Other Carbohydrate, 2 Vegetable, 1\u00bd Medium-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 3\u00bd\n\nTempeh Stir-Fry with Noodles Substitute 8 oz angel hair (capellini) pasta, cooked and drained, for the hot cooked rice. Sprinkle each serving with chopped peanuts.\n\nVeggie and Bean Burgers\n\n## Veggie and Bean Burgers FAST\n\nPrep 25 min Total 25 min \u25cf 4 sandwiches\n\n  * \u00bd cup small fresh broccoli florets\n  * 2 oz fresh mushrooms (about 4 medium)\n  * \u00bd small red bell pepper, cut into large pieces\n  * \u00bd cup cooked instant white or brown rice\n  * 1 can (15 to 16 oz) chickpeas (garbanzo beans), drained, rinsed\n  * 1 egg\n  * 1 clove garlic\n  * \u00bd teaspoon seasoned salt\n  * 1 teaspoon dried chopped onion\n  * \u2153 cup seasoned dry bread crumbs\n  * 3 tablespoons vegetable oil\n  * 4 whole wheat burger buns, split\n  * Toppings (Cheddar cheese slices, lettuce, sliced tomato, sliced onion and mayonnaise), if desired\n\n1 In food processor, place broccoli, mushrooms and bell pepper. Cover and process, using quick on-and-off motions, to finely chop vegetables (do not puree). Remove vegetables to medium bowl; stir in rice.\n\n2 Add chickpeas, egg, garlic and seasoned salt to food processor. Cover and process until smooth. Stir bean mixture, dried chopped onion and bread crumbs into vegetable mixture. Shape mixture into 4 patties, about \u00bd inch thick.\n\n3 In 10-inch nonstick skillet, heat oil over medium-high heat. Cook patties in oil 8 to 10 minutes, turning once, until brown and crisp. Serve in buns with toppings.\n\n1 Sandwich: Calories 460; Total Fat 16g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 55mg; Sodium 540mg; Total Carbohydrate 61g (Dietary Fiber 9g, Sugars 4g); Protein 17g Exchanges: 3\u00bd Starch, 2 Vegetable, 2 Fat Carbohydrate Choices: 4\n\nVeggie and Bean \"Meatballs\" Heat oven to 400\u00b0F. Generously spray 15x10x1-inch pan with cooking spray. Shape vegetable mixture into 16 balls; place in pan. Generously spray tops of balls with cooking spray. Bake about 20 minutes or until crisp. Serve with cheese sauce or tomato pasta sauce (any meatless variety; for homemade sauce, see page 174).\n\n## California Black Bean Burgers CALORIE Smart \u2022 Fast\n\nPrep 25 min Total 25 min \u25cf 6 sandwiches\n\n  * 1 can (15 oz) black beans, undrained\n  * 1 can (4.5 oz) chopped green chiles, undrained\n  * 1 cup unseasoned dry bread crumbs\n  * 1 teaspoon chili powder\n  * 1 egg, beaten\n  * \u00bc cup yellow cornmeal\n  * 2 tablespoons vegetable oil\n  * 6 burger buns, split, toasted\n  * 2 tablespoons mayonnaise or salad dressing\n  * 1\u00bc cups shredded lettuce\n  * 3 tablespoons chunky-style salsa\n\n1 In food processor or blender, place beans. Cover and process on high speed until slightly mashed. Remove beans from food processor to medium bowl. Stir in chiles, bread crumbs, chili powder and egg.\n\n2 Shape mixture into 6 patties, about \u00bd inch thick. Coat each patty with cornmeal.\n\n3 In 10-inch skillet, heat oil over medium heat. Cook patties in oil 5 to 10 minutes, turning once, until crisp and thoroughly cooked on both sides. Spread bun bottoms with mayonnaise. Top with lettuce, patties and salsa. Cover with bun tops.\n\n1 Sandwich: Calories 400; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 35mg; Sodium 1090mg; Total Carbohydrate 60g (Dietary Fiber 7g, Sugars 6g); Protein 14g Exchanges: 4 Starch, 1 Fat Carbohydrate Choices: 4\n\nCalifornia Black Bean Burgers with Tomato and Cheese Cut 1 medium tomato into 6 slices. Add 1 slice to each sandwich along with 1 slice of mozzarella cheese.\n\nDouble Black Bean and Avocado Burgers Substitute pesto for the mayonnaise and prepared black bean and corn salsa for the salsa. (To make your own pesto, see page 456.) Omit the lettuce. Top patties with sliced avocado and fresh sliced jalape\u00f1os, if desired, before topping with salsa and bun tops.\n\nPortabella Muffuletta Sandwiches\n\n## Portabella Muffuletta Sandwiches CALORIE Smart \u2022 Fast\n\nPortabella mushrooms are wonderful large, earthy-flavored flat mushrooms that are perfect for sandwiches. Look for ones with firm, fresh-looking caps.\n\nPrep 20 min Total 20 min \u25cf 2 sandwiches\n\nOlive Salad\n\n  * \u00bc cup chopped celery\n  * 3 tablespoons chopped pimiento-stuffed green olives\n  * 1 tablespoon reduced-fat mayonnaise or salad dressing\n\nSandwiches\n\n  * 6 oz portabella mushrooms, cut into \u00bd-inch-thick slices\n  * 1 to 2 teaspoons olive or vegetable oil\n  * \u215b teaspoon garlic powder\n  * 2 French rolls (6 inch), split\n  * 2 slices (\u00be oz each) provolone or mozzarella cheese\n  * \u00bd medium tomato, thinly sliced\n\n1 In small bowl, mix olive salad ingredients; set aside.\n\n2 Spray broiler pan with cooking spray. Place mushroom slices on pan. In small bowl, mix oil and garlic powder. Brush half of oil mixture on mushrooms. Broil 4 to 6 inches from heat 5 to 6 minutes or until mushrooms are tender, turning and brushing with remaining oil mixture halfway through cooking. Remove mushrooms from pan.\n\n3 Place rolls, cut side up, on broiler pan. Broil 4 to 6 inches from heat 30 to 60 seconds or until golden brown. Place cheese on roll bottoms. Top with mushrooms, olive salad and tomato slices. Cover with roll tops.\n\n1 Sandwich: Calories 330; Total Fat 16g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 15mg; Sodium 780mg; Total Carbohydrate 33g (Dietary Fiber 3g, Sugars 7g); Protein 13g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 2 Vegetable, 1 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 2\n\n## Veggie Joes Calorie Smart \u2022 Slow Cooker\n\nThese vegetarian sloppy joes have all the flavor and texture of the traditional ground beef version. For a new twist, skip the buns and try spooning the mixture over tortilla chips, pasta or rice instead.\n\nPrep 10 min Total 4 hr 10 min \u25cf 16 sandwiches\n\n  * 4 cups frozen soy-protein burger crumbles (from two 12-oz packages)\n  * 1 medium onion, finely chopped (\u00bd cup)\n  * 1\u00bd cups ketchup\n  * \u00bd cup water\n  * 2 tablespoons packed brown sugar\n  * 2 tablespoons white vinegar\n  * 1 tablespoon yellow mustard\n  * \u00bd teaspoon pepper\n  * \u00bc teaspoon salt\n  * 16 burger buns, split\n\n1 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, mix all ingredients except buns.\n\n2 Cover; cook on Low heat setting 4 to 6 hours. (If slow cooker has black liner, do not cook longer than 6 hours or mixture may burn around edge.) Spoon \u2153 cup mixture into each bun.\n\n1 Sandwich: Calories 190; Total Fat 3g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 640mg; Total Carbohydrate 30g (Dietary Fiber 3g, Sugars 10g); Protein 10g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 1 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 2\n\nMake-Ahead Directions Make sloppy joe filling as directed. Freeze desired portions in freezer containers up to 4 months.\n\n## Falafel (Fava Bean Rounds) Calorie Smart\n\nPrep 30 min Total 1 hr 30 min \u25cf 4 servings\n\n  * 2 cans (15 to 16 oz each) fava beans or chickpeas (garbanzo beans), drained, liquid reserved\n  * 3 tablespoons chopped fresh parsley\n  * 2 tablespoons all-purpose flour\n  * 2 teaspoons finely chopped garlic\n  * 1 teaspoon salt\n  * 1 teaspoon ground coriander\n  * \u00be teaspoon ground cumin\n  * \u00bc teaspoon baking powder\n  * \u215b teaspoon ground red pepper (cayenne)\n  * 1 egg\n  * 1 small red onion, finely chopped\n  * Vegetable oil for frying\n\n1 Mash beans with fork; add 2 to 3 tablespoons reserved liquid if necessary. (Do not puree beans in blender or food processor.) Stir in remaining ingredients except oil (mixture should be thick.) Cover and let stand 1 hour.\n\n2 In 3-quart saucepan, heat oil (2 inches) to 375\u00b0F. Pinch off 1-inch pieces of bean mixture; shape into rounds and flatten. Fry 4 or 5 rounds at a time in oil 2 to 3 minutes, turning once, until golden brown. Remove with slotted spoon; drain on paper towels.\n\n1 Serving: Calories 230; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 55mg; Sodium 650mg; Total Carbohydrate 24g (Dietary Fiber 5g, Sugars 2g); Protein 9g Exchanges: 1 Starch, 1\u00bd Vegetable, \u00bd Very Lean Meat, 2 Fat Carbohydrate Choices: 1\u00bd\n\n### Shaping Falafel Patties\n\nMash beans with fork, adding 2 to 3 tablespoons reserved liquid if necessary.\n\nPinch off 1-inch pieces of bean mixture; shape into rounds and flatten.\n\nSpicy Samosa Patties\n\n## Spicy Samosa Patties Calorie Smart\n\nPrep 45 min Total 45 min \u25cf 4 servings\n\n  * 3 medium russet potatoes (1 lb), peeled, cut into 1-inch pieces\n  * 1 tablespoon butter\n  * 1 medium onion, chopped (\u00bd cup)\n  * 1 teaspoon finely chopped gingerroot\n  * 2 serrano chiles, seeded, finely chopped\n  * \u00bd cup frozen sweet peas, thawed\n  * 2 tablespoons chopped cashews\n  * 1 teaspoon garam masala\n  * 1 teaspoon salt\n  * 1 egg white, slightly beaten\n  * 2 tablespoons vegetable oil\n  * \u00bd cup loosely packed fresh mint leaves, finely chopped\n  * \u00bc cup loosely packed fresh cilantro, finely chopped\n  * 3 tablespoons lime juice\n\n1 In 2-quart saucepan, place potatoes and enough water to cover. Heat to boiling; reduce heat. Cover and simmer 20 minutes or just until tender; drain. Mash potatoes slightly with potato masher, leaving potatoes slightly lumpy; set aside.\n\n2 Meanwhile, in 10-inch nonstick skillet, melt butter over medium heat. Add onion, gingerroot and half the chiles; cook about 6 minutes, stirring frequently, until onion is tender. Stir in peas, cashews, garam masala and \u00bd teaspoon of the salt; cook until thoroughly heated.\n\n3 Stir onion mixture into potato mixture until well combined. Stir in egg white. Shape mixture into 4 patties, about \u00be inch thick.\n\n4 Wipe skillet clean with paper towels. In skillet, heat oil over medium heat. Cook patties in oil 10 minutes, turning once, until golden brown on each side.\n\n5 To make relish, in small bowl mix mint, cilantro, remaining chiles, lime juice, and remaining \u00bd teaspoon salt. Serve patties with relish.\n\n1 Serving (1 Patty and 2 Teaspoons Relish): Calories 230; Total Fat 12g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 640mg; Total Carbohydrate 27g (Dietary Fiber 3g, Sugars 4g); Protein 4g Exchanges: 1\u00bd Starch, \u00bd Vegetable, 2 Fat Carbohydrate Choices: 2\n\n## Smoky Chipotle Soft Tacos CALORIE Smart \u2022 Slow Cooker\n\nPrep 20 min Total 4 hr 20 min \u25cf 18 tacos\n\n  * Cooking spray\n  * 1 large onion, chopped (1 cup)\n  * 1 Anaheim chile, chopped (\u2153 cup)\n  * 6 cups frozen soy-protein burger crumbles (from two 12-oz packages)\n  * 1\u00bd cups water\n  * \u00be cup chili sauce\n  * \u00bd cup mole sauce (from 8.25-oz jar)\n  * 1 tablespoon chopped chipotle chiles in adobo sauce (from 7-oz can)\n  * 1 teaspoon ground cumin\n  * \u00be teaspoon salt\n  * 18 flour tortillas (6 inch)\n  * 2 cups shredded Cheddar cheese (8 oz)\n  * 3 medium tomatoes, chopped (2\u00bc cups)\n\n1 Generously spray 8-inch skillet with cooking spray. Add onion and Anaheim chile; spray onion and chile with cooking spray. Cook over medium heat 4 to 5 minutes, stirring occasionally, until onion is crisp-tender.\n\n2 Spray 3\u00bd- to 4-quart slow cooker with cooking spray. In slow cooker, stir together onion mixture and remaining ingredients except tortillas, cheese and tomatoes.\n\n3 Cover; cook on Low heat setting 4 to 5 hours. Spoon \u2153 cup mixture down center of each tortilla; top with 1 heaping tablespoon cheese and 2 tablespoons tomatoes. Roll tortilla around filling.\n\n1 Taco: Calories 250; Total Fat 10g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 730mg; Total Carbohydrate 24g (Dietary Fiber 4g, Sugars 4g); Protein 15g Exchanges: 1\u00bd Starch, 1\u00bd Lean Meat, 1 Fat Carbohydrate Choices: 1\u00bd\n\nGrilled Cheese Sandwiches\n\n## Grilled Cheese Sandwiches FAst\n\nThere's plenty of debate about what makes a great grilled cheese sandwich. Some insist on using American cheese, while others use only natural cheeses. Soft bread that squishes down when you take a bite is favored by many, whereas a substantial, sturdy bread is a must for others. The choice is yours\u2014enjoy!\n\nPrep 20 min Total 20 min \u25cf 4 sandwiches\n\n  * 8 slices white or whole wheat bread\n  * 12 slices American cheese (about 8 oz) or 2 cups shredded Cheddar cheese (8 oz)*\n  * \u2153 cup butter, softened\n\n1 On each of 4 bread slices, place 3 slices cheese or \u00bd cup shredded cheese. Top with remaining bread. Spread 2 teaspoons butter over each top slice of bread.\n\n2 Place sandwiches, buttered side down, on griddle or 12-inch skillet. Spread remaining butter over top slices of bread. Cook uncovered over medium heat about 5 minutes or until bottoms are golden brown. Turn; cook 2 to 3 minutes longer or until bottoms are golden brown and cheese is melted.\n\n*If you're using natural cheese, any type of shredded cheese can be used, such as Monterey Jack, Swiss, Gruy\u00e8re, Jarlsberg or Gouda.\n\n1 Sandwich: Calories 480; Total Fat 35g (Saturated Fat 19g, Trans Fat 1.5g); Cholesterol 95mg; Sodium 1180mg; Total Carbohydrate 26g (Dietary Fiber 0g, Sugars 1g); Protein 17g Exchanges: 2 Starch, 2 High-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 2\n\nTomato and Avocado Grilled Cheese Sandwiches On each of 4 bread slices, place \u00bc cup shredded cheese. Top evenly with 1 small onion, chopped (\u2153 cup), 1 medium tomato, thinly sliced, and 1 medium avocado, pitted, peeled and thinly sliced. Top with an additional \u00bc cup cheese and remaining bread. Spread butter over top slices of bread. Continue as directed in Step 2.\n\nPesto-Parmesan Grilled Cheese Sandwiches Spread Basil Pesto (page 456) or purchased basil pesto lightly over each bread slice before adding cheese in Step 1. Sprinkle butter-topped slices with Parmesan cheese before grilling.\n\n# Chapter 15: Grilling & Smoking\n\nEasy Grilled Steak Kabobs\n\n## Grilling Basics\n\nAll About Coals\n\nControlling the Heat\n\nDirect and Indirect Heat\n\nFood Safety When Grilling\n\nPreventing and Taming Flare-Ups\n\nTypes of Grills\n\n## Features\n\nFresh Vegetable Grilling Chart\n\nGrilling Sausage and Bratwurst\n\nHeirloom Recipe and New Twist\n\nLearn with Betty: Beef Steak Doneness\n\nLearn with Betty: Hamburgers\n\nGrilled Corn on the Cob\n\n## Seasonings and Rubs\n\nJamaican Jerk Seasoning Calorie Smart \u2022 Fast \u2022 Easy\n\nSouthwestern Rub Calorie Smart \u2022 Fast \u2022 Easy\n\n## Chicken and Turkey\n\nBeer Can Chicken Calorie Smart \u2022 Easy\n\nBourbon Chicken with Corn Relish Calorie Smart\n\nChicken Fajitas\n\nEasy Grilled Chicken Kabobs\n\nGrilled Barbecue Chicken Calorie Smart\n\nGrilled Barbecue Chicken Breasts\n\nLemon Chicken with Grilled Fennel and Onion Calorie Smart\n\nMediterranean Chicken Packets Calorie Smart\n\nThree-Herb Chicken\n\nTurkey Breast with Chili-Cumin Rub Calorie Smart\n\n## Beef, Pork and Lamb\n\nBeef and Chorizo Burgers with Roasted Chile Mayonnaise\n\nBeef Fajitas\n\nBlue Cheese Burgers\n\nCaramelized Beer-Onion and Bacon Burgers\n\nCuban Pork Chops Calorie Smart \u2022 Fast\n\nEasy Grilled Steak Kabobs Calorie Smart\n\nFlank Steak with Smoky Honey Mustard Sauce Calorie Smart\n\nGrilled Balsamic- and Roasted Garlic\u2013Marinated Steak Calorie Smart\n\nGrilled Jerk Flank Steak Calorie Smart\n\nGrilled Lemon-Pepper Pork Tenderloin Calorie Smart \u2022 Fast\n\nGrilled Rosemary Lamb Chops Calorie Smart \u2022 Fast\n\nPatty Melts with Smothered Onions Calorie Smart\n\nPork Ribs with Smoky Barbecue Sauce\n\nSirloin Steak with Smoky Honey Mustard Sauce\n\nSouthwest Pork Ribs with Smoky Barbecue Sauce\n\nSweet Onion\u2013Topped Caesar Burgers Fast\n\nTexas T-Bones Calorie Smart\n\n## Fish and Shellfish\n\nGrilled Fish Calorie Smart \u2022 Fast\n\nGrilled Fish Tacos\n\nHalibut and Summer Squash Packets Calorie Smart\n\nHerbed Grilled Fish\n\nHoney-Dijon Grilled Fish\n\nShrimp Fajitas\n\nSpicy Coconut-Curry Grilled Shrimp Calorie Smart\n\n## Vegetables\n\nBuffalo Corn on the Cob\n\nChili-Lime Corn on the Cob\n\nFresh Herb Corn on the Cob\n\nGarlic-Cilantro Corn on the Cob\n\nGrilled Corn with Chile-Lime Butter\n\nGrilled Corn with Herb Butter Fast\n\nGrilled Herbed Red Potatoes Fast\n\nLemon-Dill Corn on the Cob\n\nParmesan-Pepper Corn on the Cob\n\n## BONUS RECIPES\n\nOven-Baked Beef Brisket\n\nSlow Cooker Pork Ribs with Smoky Barbecue Sauce\n\nSmoking\n\n## Smoking Basics\n\nTypes of Smokers\n\nSafety Tips for Smoking\n\nSmoking on a Grill\n\n## Smoked Foods\n\nOven-Baked Beef Brisket\n\nSmoked Beef Brisket Calorie Smart\n\nSmoked Brined Salmon Calorie Smart\n\nSmoked Cajun Chicken Calorie Smart\n\nSmoked Hot and Spicy Ribs\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Grilling Basics\n\nGrilling aficionados will grill no matter what the season because they know it's easy to do\u2014and easy to clean up. Beef, poultry, pork and seafood are all elevated to star status as the smoky flavor and charred edges get mouths watering! Here's everything you need to know to quickly become a grill master.\n\n## TYPES OF GRILLS\n\nFor **outside grilling** , choose between gas, charcoal or electric types. For **indoor grilling** , you can use a countertop electric grill appliance or a grill pan on your stovetop. The type of grill you choose will depend on your personal needs, space and preferences. No matter which type you choose, follow the manufacturer's directions for best results.\n\n**Gas:** An advantage to using a gas grill is that it heats up in 5 to 10 minutes, making it possible to grill on the spur of the moment. It can be more costly than a charcoal grill, but cleanup is quick and easy. A gas grill requires a tank of propane as fuel. You can get your tank refilled or exchange it for a filled one when it is empty.\n\n**Charcoal:** Grilling with charcoal gives foods the iconic flavor and aroma that some people prefer. However, it can take close to half an hour for the coals to get hot, so you need to plan ahead when making your meals. Charcoal grills can be a cost-friendly option.\n\n**Electric:** **Outdoor** electric grills are perfect for people living in an apartment or condominium where gas and charcoal grills are prohibited on small decks. Like gas grills, they are quick to light and easy to clean. **Indoors** , if money and space are at a premium, choose a countertop\n\ngrill appliance. While the flavor on these electric grills won't be the same as food grilled outside, the convenience of being able to grill without stepping outside could make this an appealing option\u2014and you can still make grill marks on your food.\n\n**Grill Pans:** Used on your stovetop, grill pans can't replicate the flavor of cooking over a traditional grill, but they can still cook meats and veggies with crusty, charred edges and grill marks. Many come with presses for panini and grilled cheese sandwiches, which can be another reason to choose this option.\n\n## ALL ABOUT COALS\n\n### Starting Briquettes\n\n  * **Pyramid Style:** Place briquettes in the grill, arranging them into a mound like a pyramid. Squirt lighter fluid over the top and sides of the mound, following the lighter fluid directions. **NEVER USE GASOLINE OR KEROSENE TO START A FIRE.** Briquettes can also be lit with electric starters, fire starter gels or paraffin starters. Or look for instant-lighting briquettes that don't require lighter fluid.\n  * **Charcoal Chimney Starter:** This easy-to-use cylindrical canister with a large handle lights coals quickly and evenly with newspaper rather than lighter fluid. Follow the manufacturer's directions for filling and lighting the newspaper and coals in the chimney. When the coals are ready, remove the chimney and spread out the hot coals.\n\n### When Are Coals Ready?\n\nAfter lighting briquettes with either method, leave them in a mounded shape until they are glowing red (10 to 20 minutes) and mostly covered in gray ash. (After dark, you'll know coals are ready when they have an even red glow.) Spread the coals into a single layer for grilling.\n\nDetermine the temperature of the coals by holding the palm of your hand near the grill rack and timing how long you can comfortably keep it there:\n\n  * 2 seconds = high heat\n  * 3 seconds = medium-high heat\n  * 4 seconds = medium heat\n  * 5 seconds = low heat\n\n## CONTROLLING THE HEAT\n\nBelow are ways adjust the grill temperature\u2014but also consider moving your food to hotter or cooler parts of the grill to get the right heat under the food you are grilling as an alternative method if needed. (Also see Direct and Indirect Heat, right.)\n\n  * Gas\/Electric Grill: If the heat is too hot, use a lower burner setting. If heat is too low, use a higher heat setting.\n  * **Charcoal Grill:** If the coals are too hot, spread them out or close the air vents halfway. If too cool, move the coals closer together or knock off the ashes by tapping them with long-handled tongs or open the air vents.\n\n## PREVENTING AND TAMING FLARE-UPS\n\nFats and sugary sauces dripping through the grill rack can cause flare-ups that burn the food and can cause a fire hazard. Use these tricks to prevent flare-ups:\n\n  * Trim excess fat from meats.\n  * Keep the bottom of the grill uncovered so that the grease can drain easily into the grease catch pan.\n  * Keep the grill bottom and catch pan clean and debris-free.\n  * Brush sugary or tomato-based sauces on food only during the last 10 to 15 minutes to prevent them from burning.\n  * Clean off residue from the grill rack after each use. For a gas grill, turn the heat setting on high for 10 to 15 minutes with the grill cover closed to burn off residue from the grill rack and lava rock or ceramic briquettes. For a charcoal grill, use a brass bristle brush (or crumble a large sheet of aluminum foil and use long-handled tongs) to scrape residue off the grill rack while the grill is still warm.\n\n### When a Flare-Up Happens\n\n  * Move food to a different area of the grill rack; cover the grill to extinguish the flames.\n  * For a charcoal grill, spread the coals farther apart or move food and spritz the flames with water from a spray bottle. When the flames are extinguished, move food back to the desired area.\n  * For a gas or electric grill, turn all burners off. NEVER USE WATER TO EXTINGUISH FLAMES ON A GAS OR ELECTRIC GRILL. When the flames are gone, light the grill again.\n\n## FOOD SAFETY WHEN GRILLING\n\n  * If you've brushed raw meat or poultry with a marinade or sauce, wash the brush with hot, soapy water before using on cooked meat.\n  * Use a clean plate to place the grilled food on, not the same plate that carried the raw food.\n  * Set aside some of the marinade or sauce for serving before any of it is brushed on raw meat. If the marinade has been brushed on raw meat and the brush has been dipped again into the marinade, you can boil the remaining portion for 1 minute before serving.\n\n## DIRECT AND INDIRECT HEAT\n\nFor best results, follow the grill manufacturer's directions for both methods.\n\n  * Direct-Heat Grilling: Food is cooked on the grill rack directly over heat. Best for foods that cook in less than 25 minutes: burgers, chicken pieces, chops, steaks.\n  * Indirect-Heat Grilling: Food is cooked on the grill rack next to, not directly over, the heat source. Best for foods that take longer than 25 minutes: ribs, whole chickens or turkeys, roasts. For best results, place large cuts of meat or whole turkeys or chickens on a roasting rack set inside a disposable heavy-duty foil pan on grill rack. If necessary, add water to foil pan to keep meat drippings from burning.\n\n### Setting Up Grill for Direct Heat\n\nFor charcoal grill, evenly spread hot, white coals over firebox of grill.\n\nFor gas grill, heat all burners on high for 10 to 15 minutes; then reduce heat as needed.\n\n### Setting Up Grill for Indirect Heat\n\nFor charcoal grill, move hot, white coals to edge of firebox, leaving area in middle free of coals; place drip pan in open area. Grill food over drip pan.\n\nFor gas grill, heat all burners on high for 10 to 15 minutes. Turn off center burner. Grill food over center burner. Adjust temperature of ignited burners as needed. If grill only has two burners, turn off one and grill over that burner.\n\n## Southwestern Rub Calorie smart \u2022 FAst \u2022 EASY\n\nRubs are simple ways to season meats\u2014just sprinkle on the seasoning mixture and rub it into the meat with your fingers. Because this rub has oil in it, it's known as a wet rub rather than a dry rub.\n\nPrep 10 min Total 25 min \u25cf 3 tablespoons\n\n  * 1 tablespoon chili powder\n  * 1 teaspoon ground cumin\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon ground red pepper (cayenne)\n  * 1 clove garlic, finely chopped\n  * 1 tablespoon vegetable oil\n\n1 In small bowl, mix all ingredients.\n\n2 Spread rub evenly on 1 pound boneless meat (chicken, pork or beef) or 2\u00bd pounds \u00bd-inch-thick pork chops (about 8). Cover and refrigerate at least 15 minutes but no longer than 24 hours. Cook meat as desired.\n\n1 Teaspoon: Calories 15; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 75mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n## Jamaican Jerk Seasoning Calorie smart \u2022 FAst \u2022 EASY\n\nPrep 10 min Total 25 min \u25cf 3 tablespoons\n\n  * 1 tablespoon dried minced onion\n  * 2 teaspoons dried thyme leaves\n  * 1 teaspoon ground allspice\n  * 1 teaspoon pepper\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground cinnamon\n  * \u00bc teaspoon ground red pepper (cayenne)\n\n1 In small bowl, mix all ingredients.\n\n2 Spread rub evenly on 1 pound boneless meat (chicken, pork or beef). Cover and refrigerate at least 15 minutes but no longer than 24 hours. Cook meat as desired.\n\n1 Teaspoon: Calories 0; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 130mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n## Three-Herb Chicken\n\nPrep 55 min Total 1 hr 15 min \u25cf 4 servings\n\n  * \u00bd cup vegetable oil\n  * \u00bd cup lime juice\n  * 2 tablespoons chopped fresh or 2 teaspoons dried basil leaves\n  * 2 tablespoons chopped fresh or 2 teaspoons dried oregano leaves\n  * 2 tablespoons chopped fresh or 2 teaspoons dried thyme leaves\n  * 1 teaspoon onion powder\n  * \u00bc teaspoon lemon-pepper seasoning\n  * 4 chicken thighs (about 1 lb)\n  * 4 chicken drumsticks (about 1 lb)\n\n1 In shallow glass or plastic dish, mix all ingredients except chicken. Add chicken; turn to coat. Cover and refrigerate at least 30 minutes but no longer than 24 hours, turning occasionally.\n\n2 Heat gas or charcoal grill. Remove chicken from marinade; reserve marinade. Place chicken, skin side down, on grill over medium heat. Cover grill; cook 8 to 10 minutes. Turn chicken; brush with reserved marinade. Cover grill; cook 25 to 35 minutes longer, turning occasionally and brushing with marinade, until juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F). Discard any remaining marinade.\n\n1 Serving: Calories 430; Total Fat 33g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 100mg; Sodium 95mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 31g Exchanges: 4 Medium-Fat Meat, 3 Fat Carbohydrate Choices: 0\n\nLemon Chicken with Grilled Fennel and Onion\n\n## Lemon Chicken with Grilled Fennel and Onion Calorie Smart\n\nPrep 35 min Total 50 min \u25cf 6 servings\n\n  * 6 bone-in chicken breasts (about 3 lb)\n  * \u00bc cup olive or vegetable oil\n  * 1 teaspoon grated lemon peel\n  * \u00bc cup lemon juice\n  * 2 tablespoons chopped fresh or 2 teaspoons dried oregano leaves*\n  * \u00bd teaspoon salt\n  * 2 medium bulbs fennel, cut into \u00bd-inch slices\n  * 1 medium red onion, cut into \u00bd-inch slices\n\n1 Place chicken in shallow glass or plastic dish. In small bowl, mix oil, lemon peel, lemon juice, oregano and salt; pour over chicken. Cover and let stand 15 minutes.\n\n2 Heat gas or charcoal grill. Remove chicken from marinade; reserve marinade. Brush fennel and onion slices with marinade.\n\n3 Place chicken, skin side down, fennel and onion on grill over medium heat. Cover grill; cook 15 to 20 minutes, turning chicken and vegetables once and brushing frequently with marinade, until juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F). Discard any remaining marinade.\n\n*Use whatever fresh herb you have on hand\u2014chopped fresh thyme or rosemary would also be good in this recipe.\n\n1 Serving: Calories 240; Total Fat 11g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 75mg; Sodium 310mg; Total Carbohydrate 8g (Dietary Fiber 3g, Sugars 3g); Protein 28g Exchanges: 1 Vegetable, 4 Lean Meat, 1 Fat Carbohydrate Choices: \u00bd\n\n## Bourbon Chicken with Corn Relish Calorie Smart\n\nPrep 45 min Total 45 min \u25cf 6 servings\n\n  * \u00bc cup soy sauce\n  * \u00bc cup bourbon or apple cider\n  * 5 tablespoons vegetable oil\n  * \u00bc cup packed brown sugar\n  * 1 tablespoon Dijon mustard\n  * 2 teaspoons red pepper sauce\n  * 1 can (15.25 oz) whole kernel corn, drained\n  * 1 jar (2 oz) diced pimientos, drained\n  * 3 medium green onions, sliced (3 tablespoons)\n  * \u00bc cup granulated sugar\n  * 1\u00bc teaspoons salt\n  * \u00bc teaspoon celery seed\n  * \u2153 cup cider vinegar\n  * 6 boneless skinless chicken breasts (about 1\u00be lb)\n  * \u00bc teaspoon pepper\n\n1 Heat gas or charcoal grill. In 1-quart saucepan, mix soy sauce, bourbon, 4 tablespoons of the oil, the brown sugar, mustard and pepper sauce. Heat to boiling over medium-high heat. Cook 7 to 9 minutes or until glaze is reduced to \u00bd cup. Remove from heat; cover to keep warm.\n\n2 In medium bowl, mix corn, pimientos, onions and remaining 1 tablespoon oil. In another 1-quart saucepan, mix granulated sugar, \u00bc teaspoon of the salt, the celery seed and vinegar. Heat to boiling; reduce heat. Simmer 2 minutes or until sugar is dissolved. Pour over corn mixture. Cover and refrigerate until serving time or prepare and refrigerate up to 6 hours in advance.\n\n3 Sprinkle chicken with remaining 1 teaspoon salt and the pepper. Place chicken on grill over medium heat. Cover grill; cook 15 to 20 minutes, turning once, until juice of chicken is clear when center of thickest part is cut (at least 165\u00b0F). Brush with glaze.\n\n4 Drain corn relish. Spoon \u00bc cup relish on each chicken breast. Serve with mixed greens, if desired.\n\n1 Serving: Calories 380; Total Fat 16g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 75mg; Sodium 1430mg; Total Carbohydrate 30g (Dietary Fiber 1g, Sugars 20g); Protein 29g Exchanges: 1\u00bd Other Carbohydrate, 1 Vegetable, 4 Very Lean Meat, 3 Fat Carbohydrate Choices: 2\n\nMediterranean Chicken Packets\n\n## Mediterranean Chicken Packets Calorie Smart\n\nPrep 15 min Total 40 min \u25cf 4 servings\n\n  * 1 package (4 oz) crumbled basil-and-tomato or plain feta cheese\n  * 2 tablespoons grated lemon peel\n  * 1 teaspoon dried oregano leaves\n  * 4 boneless skinless chicken breasts (1\u00bc lb)\n  * 4 plum (Roma) tomatoes, each cut into 3 slices\n  * 1 medium red onion, finely chopped (1 cup)\n  * 20 pitted kalamata olives\n\n1 Heat gas or charcoal grill. In small bowl, mix cheese, lemon peel and oregano.\n\n2 Cut 4 (18x12-inch) sheets of heavy-duty foil. In center of each sheet, place 1 chicken breast, 3 tomato slices, \u00bc cup of the onion and 5 of the olives. Spoon cheese mixture over chicken and vegetables. Bring up 2 sides of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal.\n\n3 Place packets on grill over medium heat. Cover grill; cook 20 to 25 minutes or until juice of chicken is clear when center of thickest part is cut (at least 165\u00b0F). Cut large X across top of each packet; carefully fold back foil to allow steam to escape.\n\n1 Serving: Calories 260; Total Fat 12g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 100mg; Sodium 560mg; Total Carbohydrate 7g (Dietary Fiber 2g, Sugars 3g); Protein 31g Exchanges: 1 Vegetable, 4 Lean Meat Carbohydrate Choices: \u00bd\n\n### Pitting Olives\n\nPlace olive in olive pitter. Press through pitter to remove pit.\n\nBeer Can Chicken\n\n## Beer Can Chicken Calorie Smart \u2022 Easy\n\nThe steam from the beer and the savory rub make this chicken very moist and flavorful.\n\nPrep 10 min Total 1 hr 55 min \u25cf 6 servings\n\nBasic Barbecue Rub\n\n  * 1 tablespoon paprika\n  * 2 teaspoons salt\n  * \u00bd teaspoon garlic powder\n  * \u00bd teaspoon onion powder\n  * \u00bd teaspoon pepper\n\nChicken\n\n  * 1 whole chicken (4 to 4\u00bd lb)\n  * 1 can (12 oz) regular or nonalcoholic beer\n\n1 In small bowl, mix rub ingredients. Fold wings of chicken across back with tips touching. Sprinkle rub inside cavity and all over outside of chicken; rub with fingers. If desired, tie legs together with kitchen string.\n\n2 Pour out \u00bd cup of beer from can for another use. Holding chicken upright with larger opening of body cavity downward; insert beer can into larger cavity. Insert ovenproof meat, grilling or digital thermometer so tip is in thickest part of inside thigh and does not touch bone.\n\n3 Heat gas or charcoal grill for indirect cooking. For two-burner gas grill, heat one burner to medium; place chicken with beer can upright on unheated side. For one-burner gas grill, place chicken on grill over low heat. For charcoal grill, move medium coals to edge of firebox; place chicken with beer can upright on grill rack over drip pan.\n\n4 Cover grill; cook 1 hour 15 minutes to 1 hour 30 minutes or until thermometer reads at least 165\u00b0F and legs move easily when lifted or twisted. Using tongs, carefully lift chicken to 13x9-inch pan, holding large metal spatula under beer can for support. Let stand 15 minutes before carving. Remove beer can; discard.\n\n1 Serving: Calories 300; Total Fat 18g (Saturated Fat 5g, Trans Fat 0.5g); Cholesterol 115mg; Sodium 700mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 36g Exchanges: 5 Lean Meat, 1 Fat Carbohydrate Choices: 0\n\n### Making Beer Can Chicken\n\nSprinkle seasoning inside cavity of chicken and rub all over outside.\n\nInsert open beer can into cavity of chicken.\n\nUsing tongs, carefully lift finished chicken to pan, holding metal spatula under beer can for support.\n\n## Grilled Barbecue Chicken Calorie Smart\n\nPrep 50 min Total 50 min \u25cf 6 servings\n\nBarbecue Sauce\n\n  * 1 medium onion, finely chopped (\u00bd cup)\n  * 1 cup ketchup or 1 can (8 oz) tomato sauce\n  * \u00bd cup hot water\n  * \u2153 cup lemon juice\n  * 1 tablespoon Worcestershire sauce\n  * 2 teaspoons paprika\n  * 1 teaspoon sugar\n  * 1 teaspoon salt\n  * \u00bd teaspoon pepper\n\nChicken\n\n  * 1 teaspoon salt\n  * 1 teaspoon paprika\n  * \u00bc teaspoon pepper\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n\n1 Heat gas or charcoal grill. In 1-quart saucepan, stir together all sauce ingredients. Heat over medium heat until boiling, stirring occasionally. Remove from heat.\n\n2 In small bowl, mix 1 teaspoon salt, 1 teaspoon paprika and \u00bc teaspoon pepper. Sprinkle mixture over both sides of chicken.\n\n3 Carefully brush oil on grill rack. Place chicken, skin side up, on grill over medium heat. Cover grill; cook 15 minutes. Turn chicken; brush with barbecue sauce. Cover grill; cook 20 to 25 minutes longer, turning and brushing with sauce, until juice of chicken is clear when thickest pieces are cut to bone (at least 165\u00b0F).\n\n1 Serving: Calories 380; Total Fat 13g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 155mg; Sodium 1410mg; Total Carbohydrate 14g (Dietary Fiber 1g, Sugars 11g); Protein 51g Exchanges: 1 Other Carbohydrate, 7 Very Lean Meat, 2 Fat Carbohydrate Choices: 1\n\nGrilled Barbecue Chicken Breasts Substitute 6 bone-in chicken breasts for the cut-up whole chicken. Place chicken, skin side up, on grill over medium heat. Cover grill; cook 10 minutes. Turn chicken; brush with barbecue sauce. Cover grill; cook 10 to 15 minutes longer or until juice of chicken is clear when thickest part is cut to bone (at least 165\u00b0F).\n\n## Turkey Breast with Chili-Cumin Rub Calorie Smart\n\nWhole turkey breasts grill beautifully. Because they're available in different sizes, adjust the grilling time as needed and check the meat thermometer for doneness.\n\nPrep 10 min Total 2 hr 50 min \u25cf 8 servings\n\nChili-Cumin Rub\n\n  * 2 tablespoons packed brown sugar\n  * 2 teaspoons paprika\n  * 1 teaspoon chili powder\n  * 1 teaspoon ground cumin\n  * \u00bd teaspoon ground mustard\n  * \u00bd teaspoon ground ginger\n  * \u00bd teaspoon garlic-pepper blend\n  * \u00bd teaspoon salt\n\nTurkey\n\n  * 1 bone-in whole turkey breast (5 to 6 lb), thawed if frozen\n  * 2 tablespoons cold butter\n\n1 In small bowl, mix rub ingredients. Loosen skin on turkey breast in 4 or 5 places. Cut butter into small slices; place randomly under skin of turkey. Sprinkle rub mixture over entire outside of turkey; rub with fingers. Insert ovenproof meat, grilling or digital thermometer so tip is in thickest part of breast and does not touch bone.\n\n2 Heat gas or charcoal grill for indirect cooking. For two-burner gas grill, heat one burner to medium; place turkey, breast side up, on unheated side. For one-burner gas grill, place turkey, breast side up, on grill over low heat. For charcoal grill, move medium coals to edge of firebox; place turkey, breast side up, on grill rack over drip pan filled with \u00bd inch water.\n\n3 Cover grill; cook 30 minutes. Turn turkey. Cover grill; cook 1 hour 30 minutes to 2 hours longer or until thermometer reads at least 165\u00b0F. Remove turkey from grill; cover with foil. Let stand 10 minutes before slicing.\n\n1 Serving: Calories 390; Total Fat 18g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 155mg; Sodium 290mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 3g); Protein 54g Exchanges: 7\u00bd Very Lean Meat, 3 Fat Carbohydrate Choices: 0\n\n## Texas T-Bones Calorie Smart\n\nPrep 25 min Total 55 min \u25cf 8 servings\n\nSteaks\n\n  * 4 beef T-bone steaks, about \u00be inch thick (10 to 12 oz each)\n  * 2 cloves garlic, cut in half\n  * 4 teaspoons black peppercorns, crushed\n  * Salt and pepper to taste, if desired\n\nMustard Butter\n\n  * \u00bc cup butter, softened\n  * 1 tablespoon Dijon mustard\n  * \u00bd teaspoon Worcestershire sauce\n  * \u00bc teaspoon lime juice\n\n1 Heat gas or charcoal grill. Trim fat from steaks to \u00bc-inch thickness. Rub garlic on beef. Press peppercorns into beef. Let stand for 30 minutes to reach room temperature. In small bowl, mix mustard butter ingredients; set aside.\n\n2 Place beef on grill over medium heat. Cover grill; cook 10 to 12 minutes for medium doneness (160\u00b0F), turning once. Sprinkle with salt and pepper. Serve with mustard butter.\n\n1 Serving: Calories 310; Total Fat 12g (Saturated Fat 4.5g, Trans Fat 0.5g); Cholesterol 95mg; Sodium 240mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 10g); Protein 37g Exchanges: 1 Fruit, 5 Lean Meat Carbohydrate Choices: 1\n\n### Preparing Steaks for the Grill\n\nTrim fat from steaks to \u00bc-inch thickness to cut down on flare-ups while grilling.\n\nRub seasonings such as garlic and pepper into beef. Let beef stand for 30 minutes for best flavor and texture when grilled.\n\n## Grilled Balsamic- and Roasted Garlic\u2013Marinated Steak Calorie Smart\n\nPrep 30 min Total 8 hr 30 min \u25cf 6 servings\n\n  * \u00bd cup balsamic vinegar\n  * \u00bc cup chili sauce\n  * 2 tablespoons packed brown sugar\n  * 2 tablespoons olive or vegetable oil\n  * 2 teaspoons chopped roasted garlic (from 4-oz jar)\n  * \u00bd teaspoon Italian seasoning\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon coarsely ground black pepper\n  * 1 boneless beef top round steak, 1 to 1\u00bd inches thick (1\u00bd lb)\n\n1 In shallow glass dish or resealable food-storage plastic bag, mix all ingredients except beef. Add beef; turn to coat. Cover dish or seal bag. Refrigerate 8 hours or overnight, turning beef occasionally.\n\n2 Heat gas or charcoal grill. Remove beef from marinade; reserve marinade.\n\n3 Place beef on grill over medium heat. Cover grill; cook 16 to 18 minutes, turning and brushing with marinade once or twice, until beef is of desired doneness. Discard any remaining marinade. Cut beef across grain into thin slices.\n\n1 Serving: Calories 140; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 60mg; Sodium 105mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 23g Exchanges: 3 Very Lean Meat, \u00bd Fat Carbohydrate Choices: 0\n\n### Learn with Betty Beef Steak Doneness\n\n**Medium-Rare (145\u00b0F):** Steak is very pink in the center and slightly brown toward the exterior.\n\n**Medium (160\u00b0F):** Steak is light pink in the center and brown toward the exterior.\n\n**Well-Done (170\u00b0F):** Steak is uniformly brown throughout.\n\nBeef Fajitas\n\n## Beef Fajitas\n\nPrep 1 hr Total 9 hr \u25cf 6 servings\n\n  * 1 boneless beef top sirloin steak, 1\u00bd inches thick (1\u00bd lb), trimmed of excess fat\n  * Fajita Marinade (page 476)\n  * 2 large onions, sliced\n  * 2 medium green or red bell peppers, cut into \u00bc-inch strips\n  * 2 tablespoons vegetable oil\n  * 12 flour tortillas (9 or 10 inch)\n  * 1 cup picante sauce\n  * 1 cup shredded Cheddar or Monterey Jack cheese (4 oz)\n  * 1\u00bd cups guacamole (for homemade Guacamole, see page 41)\n  * \u00be cup sour cream\n\n1 Pierce beef with fork in several places. Place in resealable food-storage plastic bag or shallow glass or plastic dish. Pour marinade over beef; turn to coat. Cover and refrigerate at least 8 hours but no longer than 24 hours, turning occasionally.\n\n2 Heat gas or charcoal grill. In large bowl, toss onions and bell peppers with oil; place in grill basket (grill \"wok\"). Set aside.\n\n3 Remove beef from marinade; reserve marinade. Place beef on grill over medium heat. Cover grill; cook 22 to 26 minutes for medium-rare (145\u00b0F) to medium doneness (160\u00b0F), turning once and brushing occasionally with reserved marinade. Add vegetables to grill for last 6 to 8 minutes of cooking, shaking basket or stirring vegetables once or twice, until crisp-tender.\n\n4 Meanwhile, heat oven to 325\u00b0F. Wrap tortillas in foil. Heat in oven about 15 minutes or until warm. Remove tortillas from oven; keep wrapped.\n\n5 Discard any remaining marinade. Cut beef across grain into very thin slices. For each fajita, place a few slices of beef, some of the onion mixture, 1 heaping tablespoonful each picante sauce and cheese, 2 tablespoons guacamole and 1 tablespoon sour cream on center of tortilla. Fold 1 end of tortilla up about 1 inch over filling; fold right and left sides over folded end, overlapping. Fold remaining end down.\n\n1 Serving: Calories 710; Total Fat 33g (Saturated Fat 12g, Trans Fat 1.5g); Cholesterol 100mg; Sodium 910mg; Total Carbohydrate 63g (Dietary Fiber 7g, Sugars 8g); Protein 38g Exchanges: 3\u00bd Starch, \u00bd Other Carbohydrate, 4 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 4\n\nLighter Directions For 11 grams of fat and 520 calories per serving, omit oil; spray onions and bell peppers with cooking spray and toss before placing in grill basket. Use fat-free tortillas and reduced-fat cheese and sour cream. Omit guacamole.\n\nChicken Fajitas Substitute 1 pound boneless skinless chicken breasts for the beef. Marinate 2 hours. Cook chicken on grill over medium heat 15 to 20 minutes, turning occasionally, until juice of chicken is clear when center of thickest part is cut (at least 165\u00b0F). Cut chicken into thin slices.\n\nShrimp Fajitas Substitute 1 pound uncooked medium shrimp, peeled and deveined, for the beef. Marinate 1 hour. Cook shrimp in grill basket (grill \"wok\") over medium heat 4 to 6 minutes, until pink, turning occasionally.\n\n# Heirloom Recipe and New Twist\n\nFlank steak is a great recipe to have in your repertoire. With these easy-to-follow directions, it will be a success every time. Use any leftovers to make quick Mexican-inspired meals. For a flavor twist, try our jerk version. You'll be surprised how a few common herbs and spices, combined in a new way, make a great new flavor.\n\nHeirloom\n\nFlank Steak with Smoky Honey-Mustard Sauce\n\n## Flank Steak with Smoky Honey-Mustard Sauce Calorie Smart\n\nPrep 30 min Total 35 min \u25cf 6 servings\n\nHoney-Mustard Sauce\n\n  * \u00bc cup Honey-Mustard Dressing (see page 137)\n  * 1 tablespoon frozen (thawed) orange juice concentrate\n  * 1 tablespoon water\n  * 1 clove garlic, finely chopped\n  * 1 chipotle chile in adobo sauce (from 7-oz can), finely chopped\n\nSteak\n\n  * 1 beef flank steak (about 1\u00bd lb)\n  * 6 flour tortillas (8 to 10 inch), heated as directed on package\n  * Sliced tomato and avocado, if desired\n\n1 Heat gas or charcoal grill. In small bowl, mix sauce ingredients. In both sides of beef, make cuts about \u00bd inch apart and \u215b inch deep in diamond pattern. Brush 2 tablespoons of the sauce on both sides of beef.\n\n2 Place beef on grill over medium heat. Cover grill; cook 17 to 21 minutes, turning once, until beef is of desired doneness. Remove beef from grill to cutting board or plate; cover and let stand 5 minutes. Cut beef across grain into thin slices. Serve with remaining sauce, tortillas, tomato and avocado.\n\n1 Serving: Calories 360; Total Fat 16g (Saturated Fat 4.5g, Trans Fat 1.5g); Cholesterol 55mg; Sodium 460mg; Total Carbohydrate 22g (Dietary Fiber 0g, Sugars 2g); Protein 31g Exchanges: 1\u00bd Starch, 4 Lean Meat, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\nSirloin Steak with Smoky Honey-Mustard Sauce Substitute 1\u00bd-pound boneless beef top sirloin steak (1 inch thick) for the flank steak.\nNew Twist\n\nGrilled Jerk Flank Steak\n\n## Grilled Jerk Flank Steak Calorie Smart\n\nPrep 30 min Total 4 hr 35 min \u25cf 6 servings\n\n  * 3 tablespoons teriyaki marinade and sauce (from 10-oz bottle)\n  * 1 tablespoon vegetable oil\n  * 2 teaspoons pumpkin pie spice\n  * \u00bd teaspoon dried thyme leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 2 cloves garlic, finely chopped\n  * 1 jalape\u00f1o chile, finely chopped (not seeded)\n  * 1 beef flank steak (1\u00bd lb)\n\n1 In shallow glass dish or resealable food-storage plastic bag, mix all ingredients except beef. Add beef; turn to coat. Cover dish or seal bag. Refrigerate at least 4 hours but no longer than 24 hours.\n\n2 Heat gas or charcoal grill. Remove beef from marinade; discard marinade.\n\n3 Place beef on grill over medium heat. Cover grill; cook 17 to 21 minutes, turning once, until beef is of desired doneness. Remove beef from grill to cutting board or plate; cover and let stand 5 minutes. Cut beef across grain into thin slices.\n\n1 Serving: Calories 180; Total Fat 8g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 50mg; Sodium 170mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 26g Exchanges: 3\u00bd Lean Meat Carbohydrate Choices: 0\n\nBroiler Directions Marinate beef as directed in Step 1. Place beef on rack in broiler pan. Broil with top 4 to 5 inches from heat about 10 minutes, turning once, until beef is of desired doneness.\n\n### Slicing Flank Steak\n\nUsing sharp knife, cut steak across grain, cutting at a slight angle.\n\nEasy Grilled Steak Kabobs\n\n## Easy Grilled Steak Kabobs Calorie Smart\n\nPrep 35 min Total 35 min \u25cf 4 servings\n\n  * 1 boneless beef top sirloin steak, 1 inch thick (1 lb)\n  * 1 large bell pepper (any color), cut into 16 pieces\n  * 16 medium fresh mushrooms\n  * 1 tablespoon chopped fresh or 1 teaspoon dried dill weed\n  * 1 tablespoon lemon juice\n  * 1 tablespoon olive or vegetable oil\n  * 1 tablespoon honey mustard\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 Heat gas or charcoal grill. Cut beef into 24 (1-inch) cubes.\n\n2 On each of 8 (10- to 12-inch) metal skewers, alternately thread beef, bell pepper and mushrooms, leaving about \u00bc inch space between each piece. In small bowl, mix remaining ingredients.\n\n3 Place kabobs on grill over medium heat. Cover grill; cook 15 to 18 minutes, turning and brushing 3 or 4 times with oil mixture, until beef is desired doneness and vegetables are tender.\n\n1 Serving: Calories 220; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 75mg; Sodium 200mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 2g); Protein 32g Exchanges: 1 Vegetable, 4 Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nEasy Grilled Chicken Kabobs Substitute boneless skinless chicken breasts or chicken thighs, cut into 1-inch pieces, for the steak. Cook until chicken is no longer pink in center.\n\nCaramelized Beer-Onion and Bacon Burgers\n\n## Caramelized Beer-Onion and Bacon Burgers\n\n1 hour 15 min 1 hour 15 min \u25cf 4 burgers\n\n  * 8 slices bacon, crisply cooked, drippings reserved\n  * 2 large sweet onions, thinly sliced (about 1 \u00bd lb)\n  * 1 bottle (12 oz) pilsner-style beer\n  * 1\u00bd lb lean (at least 80%) ground beef\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n  * 4 slices (about \u00be oz each) Cheddar cheese\n  * 8 leaves red leaf lettuce or red romaine lettuce\n  * 4 slices tomato\n  * 4 kaiser rolls or crusty rolls, split\n\n1 In 10-inch skillet, heat 1 tablespoon bacon drippings over medium-high heat. Add onions; cook about 15 minutes, stirring frequently.\n\n2 Reserve \u00bd cup beer. Stir remaining beer into onions. Heat to boiling; reduce heat to medium and simmer uncovered, stirring occasionally, until onions are golden and liquid has evaporated, about 30 minutes. Cool. Coarsely chop 4 slices of the cooked bacon; set aside.\n\n3 Heat charcoal or gas grill. Chop half of the onion mixture. In medium bowl, stir chopped onion mixture and bacon into beef. Add salt and pepper. Shape beef mixture into 4 large patties, thinner at center and thicker at edges.\n\n4 Place patties on grill over medium heat. Cover grill; cook 11 to 13 minutes, turning once, until meat thermometer inserted in center of patties reads 160\u00b0F. After turning, brush reserved \u00bd cup beer on burgers, then top burgers with cheese. Cover grill; cook about 1 minute longer or until cheese is melted.\n\n5 Place burgers on bun bottoms; top with lettuce leaves, tomato slices, and remaining caramelized onions. Cut remaining 4 slices bacon in half; crisscross 2 pieces on top of onions on each burger, and cover with bun tops.\n\n1 Burger: Calories 620; Total Fat 35g (Saturated Fat 14g, Trans Fat 2g); Cholesterol 145 mg, Sodium 1090mg; Total Carbohydrate 30g (Dietary Fiber 2g, Sugars 5g ); Protein 46g Exchanges: 1 Starch, Other Carbohydrate, 2 Vegetable, 3 Medium-Fat Meat, 2 Fat Carbohydrate Choices: 2\n\n## Grilled Corn on the Cob\n\nGrilled corn is an easy and delicious side dish for any summer meal. Start with four ears of fresh corn (with husks) and butter and add your choice of toppings below. Grill and enjoy!\n\nHeat gas or charcoal grill. In small bowl, mix \u00bc cup softened butter with desired topping ingredients. Remove large outer husks from each ear of corn; gently pull back inner husks and remove silk. Spread each ear of corn with about 2 teaspoons topping; reserve any remaining topping. Pull husks up over ears.\n\nPlace corn on grill. Cook uncovered over medium heat 10 to 15 minutes, turning frequently, until tender. Let stand 5 minutes. Pull back or remove husks; spread corn with remaining topping.\n\n  1. 1. **Buffalo:** 1 ounce crumbled blue cheese, 2 teaspoons red pepper sauce, 2 teaspoons finely chopped green onion.\n  2. 2. **Chili-Lime:** \u00bc teaspoon grated lime peel, 4 teaspoons lime juice, 1 to 2 teaspoons ground red chiles or chili powder.\n  3. 3. **Parmesan-Pepper:** \u00bc cup grated Parmesan cheese, \u00bc teaspoon freshly ground black pepper.\n  4. 4. **Garlic-Cilantro:** 2 teaspoons each finely chopped garlic and fresh cilantro leaves, \u00bc teaspoon salt.\n  5. 5. **Fresh Herb:** 2 tablespoons chopped fresh basil leaves, 1 tablespoon chopped fresh chives, \u00bd teaspoon garlic salt.\n  6. 6. **Lemon-Dill:** 1\u00bd teaspoons grated lemon peel, 1 tablespoon fresh lemon juice, 1 teaspoon chopped fresh dill weed, pinch of salt.\n\nPatty Melts with Smothered Onions\n\n## Patty Melts with Smothered Onions Calorie Smart\n\nAlthough patty melts are usually made with ground beef, you could also make them with ground turkey or a mixture of ground pork and beef. Varying the cheese is also a nice option.\n\nPrep 35 min Total 35 min \u25cf 4 sandwiches\n\n  * 1 lb lean (at least 80%) ground beef\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon dried thyme leaves\n  * \u00bc teaspoon dried oregano leaves\n  * \u215b teaspoon pepper\n  * 2 teaspoons Dijon mustard\n  * 1 clove garlic, finely chopped\n  * 1 teaspoon olive or vegetable oil\n  * 2 medium red onions, thinly sliced, separated into rings\n  * \u215b teaspoon salt\n  * Dash pepper\n  * 8 slices rye bread\n  * 4 slices (\u00be oz each) Swiss cheese\n\n1 Heat gas or charcoal grill. In medium bowl, mix beef, \u00bd teaspoon salt, the thyme, oregano, \u215b teaspoon pepper, the mustard and garlic. Shape mixture into 4 patties, about \u00be inch thick.\n\n2 In 8-inch skillet, heat oil over medium-high heat. Cook onions in oil 8 to 10 minutes, stirring frequently, until tender. Stir in \u215b teaspoon salt and dash pepper; keep warm.\n\n3 Place patties on grill over medium heat. Cover grill; cook 13 to 15 minutes, turning once, until meat thermometer inserted in center of patties reads 160\u00b0F. Add bread slices to side of grill for last 5 minutes of cooking, turning once, until lightly toasted.\n\n4 Top patties with cheese. Cover grill; cook about 1 minute longer or until cheese is melted. Place each patty on 1 bread slice; top with onions and remaining bread.\n\n1 Sandwich: Calories 440; Total Fat 22g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 90mg; Sodium 880mg; Total Carbohydrate 31g (Dietary Fiber 3g, Sugars 5g); Protein 31g Exchanges: 2 Starch, 4 Medium-Fat Meat, 1 Fat Carbohydrate Choices: 2\n\n### Learn with Betty Hamburgers\n\n**Perfect Hamburger:** The center of this hamburger has reached a safe 160\u00b0F and is no longer pink.\n\n**Undercooked Hamburger:** The center of this hamburger has not reached a safe temperature of 160\u00b0F and is too pink.\n\n**Overcooked Hamburger:** This burger has been cooked over 160\u00b0F and is hard and dry.\n\n## Grilling Sausage and Bratwurst\n\nPlace uncooked sausage or bratwurst on heated grill over medium-low heat. Cover grill; cook 15 to 20 minutes, turning occasionally, until no longer pink in center. If flare-ups occur, move links to another part of grill.\n\nOr, to cut down on flare-ups and to prevent overbrowning, cook bratwurst first in cider, beer or water and then brown them on the grill. For 4 bratwurst, heat 1\u00bd cups cider, beer or water to boiling in 2-quart saucepan. Add links; reduce heat to low. Cover and simmer 15 minutes. Drain meat; grill over medium heat 6 minutes, turning once, until brown.\n\nGrill fully cooked bratwurst or sausage over medium-low heat 7 to 10 minutes, turning frequently, until golden brown and heated through.\n\nBeef and Chorizo Burgers with Roasted Chile Mayonnaise\n\n## Beef and Chorizo Burgers with Roasted Chile Mayonnaise\n\nChorizo comes in two forms\u2014fresh and cured. The fresh chorizo used in this recipe is a common ingredient in Mexican dishes, while the cured, which comes completely cooked, is more often found in Spanish cuisine.\n\nPrep 40 min Total 45 min \u25cf 4 burgers\n\nRoasted Chile Mayonnaise\n\n  * \u00bc cup mayonnaise or salad dressing\n  * 2 teaspoons lime juice\n  * 1 clove garlic, finely chopped\n  * 1 tablespoon chopped fresh cilantro\n  * 1 small poblano chile\n\nBurgers\n\n  * 1 lb lean (at least 80%) ground beef\n  * \u00bd lb bulk fresh chorizo sausage or Italian pork sausage\n  * \u00be teaspoon salt\n  * \u00bc teaspoon pepper\n  * 4 slices (1 oz each) Monterey Jack cheese\n  * 4 burger buns, split, or 8 slices bread\n  * 1 medium tomato, coarsely chopped\n\n1 Heat gas or charcoal grill. In small bowl, mix mayonnaise, lime juice, garlic and cilantro. Cover and refrigerate until serving time.\n\n2 Remove stem, seeds and membranes from chile; cut lengthwise into quarters. Place chile, skin side down, on grill over medium heat. Cover grill; cook about 10 minutes or until skin is blackened and blistered. Immediately place chile in bowl. Cover tightly with plastic wrap; cool 5 minutes. Peel blackened skin from chile; rinse chile with water. Finely chop; set aside.\n\n3 In large bowl, mix beef, sausage, salt and pepper. Shape mixture into 4 patties, \u00bd inch thick.\n\n4 Place patties on grill over medium heat. Cover grill; cook 11 to 13 minutes, turning once, until meat thermometer inserted in center of patties reads 160\u00b0F. During last 2 minutes of cooking, place cheese slice on each patty and place buns, cut sides down, on grill. Remove burgers and buns from grill; cover to keep warm.\n\n5 Stir chopped roasted chile into mayonnaise mixture. Spread 1 tablespoon mixture on cut sides of buns. Place burgers on bun bottoms; top with tomato. Cover with bun tops.\n\n1 Burger: Calories 560; Total Fat 35g (Saturated Fat 14g, Trans Fat 1.5g); Cholesterol 125mg; Sodium 1580mg; Total Carbohydrate 27g (Dietary Fiber 1g, Sugars 6g); Protein 36g Exchanges: 1\u00bd Starch, \u00bd Vegetable, 4 High-Fat Meat, \u00bd Fat Carbohydrate Choices: 2\n\n## Sweet Onion\u2013Topped Caesar Burgers Fast\n\nPrep 30 min Total 30 min \u25cf 4 burgers\n\n  * 1 lb lean (at least 80%) ground beef\n  * 2 tablespoons chopped fresh parsley\n  * \u00bd cup Caesar dressing\n  * \u00bd teaspoon peppered seasoned salt\n  * 1 small sweet onion (Bermuda, Maui, Spanish or Walla Walla), cut into \u00bc- to \u00bd-inch slices\n  * 1\u00bd cups shredded romaine lettuce\n  * 2 tablespoons shredded Parmesan cheese\n  * 4 burger buns, split\n\n1 Heat gas or charcoal grill. In medium bowl, mix beef, parsley, 2 tablespoons of the dressing and the peppered seasoned salt. Shape mixture into 4 patties, about \u00bd inch thick.\n\n2 Place patties on grill over medium heat. Cover grill; cook 10 to 12 minutes, turning once, until meat thermometer inserted in center of patties reads 160\u00b0F. Add onion slices to side of grill for last 8 to 10 minutes of cooking, brushing with 2 tablespoons of the dressing and turning once, until crisp-tender.\n\n3 In small bowl, toss lettuce, cheese and remaining \u00bc cup dressing. On bun bottoms, place lettuce mixture, burgers and onion. Cover with bun tops.\n\n1 Burger: Calories 500; Total Fat 33g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 75mg; Sodium 840mg; Total Carbohydrate 25g (Dietary Fiber 2g, Sugars 3g); Protein 26g Exchanges: 1\u00bd Starch, 1 Vegetable, 3 High-Fat Meat, 1 Fat Carbohydrate Choices: 1\u00bd\n\nBlue Cheese Burgers Substitute blue cheese dressing for the Caesar dressing and blue cheese for the Parmesan cheese.\n\nGrilled Lemon-Pepper Pork Tenderloin\n\n## Grilled Lemon-Pepper Pork Tenderloin Calorie Smart \u2022 Fast\n\nPrep 25 min Total 30 min \u25cf 6 servings\n\n  * 1 teaspoon grated lemon peel\n  * \u00bd teaspoon seasoned salt\n  * \u00bd teaspoon coarsely ground black pepper\n  * \u00bd teaspoon paprika\n  * \u00bc teaspoon dried thyme or marjoram leaves\n  * 2 teaspoons olive or vegetable oil\n  * 2 pork tenderloins (about \u00be lb each)\n\n1 Heat gas or charcoal grill. In small bowl, mix lemon peel, seasoned salt, pepper, paprika and thyme. Brush oil over all sides of pork. Rub lemon peel mixture over pork.\n\n2 Place pork on grill over medium-low heat. Cover grill; cook 15 to 20 minutes, turning occasionally, until pork has slight blush of pink in center and meat thermometer inserted in center reads 145\u00b0F.\n\n3 Remove pork from grill to cutting board or plate; cover with foil. Let stand at least 3 minutes before slicing.\n\n1 Serving: Calories 160; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 70mg; Sodium 160mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 26g Exchanges: 3\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\n### Removing Silverskin from Pork Tenderloin\n\nSlide long, narrow knife under silverskin; hold silverskin tightly with one hand while slicing under membrane.\n\n## Cuban Pork Chops Calorie Smart \u2022 Fast\n\nServe these flavor-packed pork chops with corn on the cob and cooked rice tossed with black beans.\n\nPrep 15 min Total 20 min \u25cf 4 servings\n\nCuban Rub\n\n  * 2 tablespoons grated lime peel\n  * 1 tablespoon cracked black pepper\n  * 1 tablespoon cumin seed\n  * 2 tablespoons olive or vegetable oil\n  * \u00bd teaspoon salt\n  * 1 clove garlic, finely chopped\n\nPork\n\n  * 4 boneless pork loin or rib chops, about 1 inch thick (about 2 lb), trimmed of excess fat\n  * Mango slices, if desired\n\n1 Heat gas or charcoal grill. In small bowl, mix rub ingredients. Rub mixture evenly on both sides of pork chops.\n\n2 Place pork chops on grill over medium heat. Cover grill; cook 8 to 10 minutes, turning once, until pork is no longer pink and meat thermometer inserted in center reads 145\u00b0F. Let stand at least 3 minutes before serving. Garnish with mango slices.\n\n1 Serving: Calories 330; Total Fat 16g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 95mg; Sodium 400mg; Total Carbohydrate 2g (Dietary Fiber 1g, Sugars 0g); Protein 45g Exchanges: 6\u00bd Very Lean Meat, 2\u00bd Fat Carbohydrate Choices: 0\n\nPork Ribs with Smoky Barbecue Sauce\n\n## Pork Ribs with Smoky Barbecue Sauce\n\nPrep 30 min Total 1 hr 40 min \u25cf 4 servings\n\nRibs\n\n  * 2 racks (2 lb each) pork back ribs (not cut into serving pieces)\n  * 1 tablespoon vegetable oil\n  * 4 teaspoons chopped fresh or 1\u00bd teaspoons dried thyme leaves\n\nSmoky Barbecue Sauce\n\n  * \u00bd cup ketchup\n  * \u00bc cup water\n  * 3 tablespoons packed brown sugar\n  * 2 tablespoons white vinegar\n  * 2 teaspoons celery seed\n  * \u00bc teaspoon liquid smoke\n  * \u00bc teaspoon red pepper sauce\n\n1 Brush meaty side of ribs with oil; sprinkle with thyme. Heat gas or charcoal grill for indirect cooking. For two-burner gas grill, heat one burner to medium; place ribs, meaty side up, on unheated side. For one-burner gas grill, place ribs, meaty side up, on grill over low heat. For charcoal grill, move medium coals to edge of firebox; place ribs, meaty side up, on grill rack over drip pan.\n\n2 Cover grill; cook 1 hour to 1 hour 10 minutes or until meat is tender and no longer pink next to bones. Meanwhile, in 1-quart saucepan, mix sauce ingredients; heat to boiling. Reduce heat; simmer uncovered 15 minutes, stirring occasionally. Brush sauce over pork 2 or 3 times during last 15 minutes of grilling. Heat any remaining sauce to boiling; boil and stir 1 minute. Cut ribs into 4 serving pieces. Serve with sauce.\n\n1 Serving: Calories 960; Total Fat 70g (Saturated Fat 25g, Trans Fat 0g); Cholesterol 265mg; Sodium 550mg; Total Carbohydrate 18g (Dietary Fiber 0g, Sugars 17g); Protein 64g Exchanges: 1 Other Carbohydrate, 9 Medium-Fat Meat, 5 Fat Carbohydrate Choices: 1\n\nSlow-Cooker Directions Spray 5- to 6-quart slow cooker with cooking spray. Cut ribs into 4 serving pieces. Omit oil and thyme. Place ribs in slow cooker. Cover; cook on Low heat setting 6 hours. Place ribs on grill over medium heat. Brush with sauce. Cover grill; cook about 15 minutes.\n\nSouthwest Pork Ribs with Smoky Barbecue Sauce Omit oil and thyme. In small bowl, mix 2 tablespoons each regular chili powder or ancho chile pepper powder, ground cumin, smoked or regular paprika, garlic powder, salt and packed brown sugar. Rub desired amount of mixture into meaty side of ribs. Continue as directed. Store any remaining rub in airtight container in cool, dark location for up to 6 months.\n\nGrilled Rosemary Lamb Chops\n\n## Grilled Rosemary Lamb Chops Calorie Smart \u2022 Fast\n\nFrench-cut lamb chops are small and very tender. If these chops are not available, the recipe can also be made with lamb loin or sirloin chops. Both of these cuts will be a little meatier, so allow 1 or 2 chops per serving.\n\nPrep 20 min Total 20 min \u25cf 2 servings\n\n  * 1 tablespoon country-style Dijon mustard\n  * 1 tablespoon chopped fresh rosemary leaves\n  * 2 teaspoons honey\n  * 1 clove garlic, finely chopped\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon coarsely ground black pepper\n  * 6 French-cut baby lamb chops (1 to 1\u00bc inches thick)\n\n1 Heat gas or charcoal grill. In small bowl, mix all ingredients except lamb. Spread mixture on one side of each lamb chop.\n\n2 Place lamb, coated side up, on grill over medium heat. Cover grill; cook 12 to 15 minutes for medium doneness (160\u00b0F).\n\n1 Serving (3 Chops): Calories 330; Total Fat 14g (Saturated Fat 5g, Trans Fat 0.5g); Cholesterol 140mg; Sodium 880mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 6g); Protein 43g Exchanges: \u00bd Other Carbohydrate, 6 Very Lean Meat, 2 Fat Carbohydrate Choices: \u00bd\n\n## Grilled Fish Calorie Smart \u2022 Fast\n\nPrep 20 min Total 20 min \u25cf 4 servings\n\n  * 1\u00bd lb fish steaks or 1 lb fish fillets (halibut, lake trout, mahimahi, marlin, red snapper, salmon, swordfish or tuna), about \u00be inch thick\n  * 2 tablespoons butter, melted\n  * 1 teaspoon salt\n  * \u00bc teaspoon pepper\n  * 1 lemon, cut into 4 wedges\n\n1 Heat gas or charcoal grill. Cut fish into 4 serving pieces if necessary. Brush fish with 1 tablespoon of the butter; sprinkle with salt and pepper.\n\n2 Carefully brush oil on grill rack. Place fish on grill over medium heat (if fish fillets have skin, place skin side down). Cover grill; cook 10 to 14 minutes, brushing 2 or 3 times with remaining 1 tablespoon butter, until fish flakes easily with fork. Serve with lemon.\n\n1 Serving: Calories 210; Total Fat 8g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 105mg; Sodium 770mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 32g Exchanges: 4\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nHerbed Grilled Fish Sprinkle fish with 1 teaspoon Italian seasoning with the salt and pepper. Substitute olive oil for butter.\n\nHoney-Dijon Grilled Fish Mix 2 tablespoons each honey and Dijon mustard. Brush mixture on fish before grilling. Omit butter and salt.\n\nGrilled Fish Tacos Wrap 4 flour tortillas (6 inch) in foil; place on grill during last 3 minutes the fish is cooking. Cook tortillas, turning once, until warm. Cut grilled fish into serving-size pieces. Place fish on tortillas. Top each with 2 tablespoons Sweet-and-Sour Coleslaw (page 129) or desired coleslaw and 1 tablespoon chopped fresh mango or desired salsa. Fold tortillas over filling.\n\n### Removing Skin from Cooked Fish\n\nGrab skin from tail end with tongs or fingers and gently peel off.\n\nSpicy Coconut-Curry Grilled Shrimp\n\n## Spicy Coconut-Curry Grilled Shrimp Calorie Smart\n\nPrep 25 min Total 55 min \u25cf 6 servings\n\n  * 18 bamboo skewers (6 inch)\n  * \u00be cup canned coconut milk (not cream of coconut)\n  * 2 teaspoons curry powder\n  * 2 teaspoons cornstarch\n  * 1 teaspoon honey\n  * \u00bc teaspoon salt\n  * 54 uncooked medium shrimp (about 2 lb), thawed if frozen, peeled, deveined\n  * 2 tablespoons olive or vegetable oil\n  * 1 teaspoon red pepper sauce\n\n1 Soak skewers in water at least 30 minutes before using to prevent burning. Meanwhile, heat gas or charcoal grill. In small microwavable bowl, mix coconut milk, curry powder, cornstarch, honey and salt. Microwave uncovered on High about 2 minutes, stirring every 30 seconds, until mixture bubbles and thickens; set aside.\n\n2 In large bowl, place shrimp. Drizzle with oil and pepper sauce; toss to coat. Thread 3 shrimp on each skewer, leaving space between each.\n\n3 Place kabobs on grill over medium heat. Cover grill; cook 4 to 6 minutes, turning once, until shrimp are pink. Serve with coconut-curry sauce.\n\n1 Serving (3 Kabobs and 4 Teaspoons Sauce): Calories 210; Total Fat 11g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 215mg; Sodium 370mg; Total Carbohydrate 5g (Dietary Fiber 1g, Sugars 2g); Protein 24g Exchanges: \u00bd Other Carbohydrate, 3\u00bd Lean Meat Carbohydrate Choices: \u00bd\n\nHalibut and Summer Squash Packets\n\n## Halibut and Summer Squash Packets Calorie Smart\n\nPrep 15 min Total 35 min \u25cf 4 servings\n\n  * 1 lb halibut or other mild-flavored fish fillets, \u00bd to \u00be inch thick\n  * 2 teaspoons dried basil leaves\n  * 1 teaspoon lemon-pepper seasoning\n  * 1 teaspoon seasoned salt\n  * 1 medium zucchini, cut into 2x1-inch strips\n  * 1 medium yellow summer squash, cut into 2x1-inch strips\n  * 1 medium bell pepper (any color), cut into 1-inch pieces\n  * 2 tablespoons olive or vegetable oil\n\n1 Heat gas or charcoal grill. Cut 4 (18x12-inch) sheets of heavy-duty foil; spray foil with cooking spray.\n\n2 Cut halibut into 4 serving pieces if necessary. Place 1 piece on center of each sheet. Sprinkle 1 teaspoon of the basil, \u00bd teaspoon of the lemon-pepper seasoning salt and \u00bd teaspoon of the seasoned salt over halibut pieces. Top with vegetables. Sprinkle evenly with remaining seasonings. Drizzle with oil.\n\n3 Bring up 2 sides of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal.\n\n4 Place packets on grill over medium heat. Cover grill; cook 15 to 20 minutes, rotating packets half turn after 10 minutes, until fish flakes easily with fork and vegetables are tender. Cut large X across top of each packet; carefully fold back foil to allow steam to escape.\n\n1 Serving: Calories 190; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 60mg; Sodium 390mg; Total Carbohydrate 6g (Dietary Fiber 2g, Sugars 3g); Protein 23g Exchanges: 1 Vegetable, 3 Lean Meat Carbohydrate Choices: \u00bd\n\n### Making Foil Packets for Grill\n\nFold top over twice to prevent leakage from packet, leaving room for heat circulation and expansion.\n\n### FRESH VEGETABLE GRILLING CHART\n\nToss vegetables (except corn) with a little olive oil or vegetbale oil, if desired; sprinkle with salt and pepper.Grill in a grill basket (grill \"wok\") over medium heat for time indicated.\n\nVegetable | Form | Minutes\n\n---|---|---\n\nAsparagus Spears | Whole | 8 to 10\n\nBell Peppers | Cut into \u00bd-inch strips | 5 to 10\n\nBroccoli Florets | Cut in half lengthwise | 15 to 20\n\nCarrots, Baby-Cut | Whole | 20 to 25\n\nCauliflower Florets | Cut in half lengthwise | 10 to 15\n\nCherry Tomatoes | Whole | 5 to 10\n\nCorn on the Cob (husked and wrapped in foil) | Whole | 12 to 18\n\nGreen Beans | Whole | 10 to 15\n\nMushrooms, White | Whole with stems removed | 8 to 10\n\nMushrooms, Portabella | Whole with stems removed | 8 to 10\n\nOnions | Cut into \u00bd-inch slices | 10 to 15\n\nPotatoes, Russet or Idaho | Cut into quarters lengthwise | 20 to 30\n\nPotatoes, Small Red | Cut into quarters | 10 to 15\n\nZucchini, Small | Cut in half lengthwise | 10 to 15\n\nGrilled Corn with Herb Butter\n\n## Grilled Corn with Herb Butter Fast\n\nIf you've run out of heavy-duty foil, layer two sheets of standard foil to make your packet.\n\nPrep 30 min Total 30 min \u25cf 6 servings\n\n  * \u00bc cup butter\n  * 2 tablespoons chopped fresh chives\n  * \u00bd teaspoon garlic salt\n  * 6 ears fresh sweet corn, husks removed, cleaned\n  * \u00bc cup grated Parmesan cheese\n\n1 Heat gas or charcoal grill. Cut 30x18-inch sheet of heavy-duty foil. In small microwavable bowl, microwave butter, chives and garlic salt uncovered on High 15 to 20 seconds, until butter is melted.\n\n2 Brush butter mixture over each ear of corn. Place corn on center of foil. Pour any remaining butter mixture over corn. Sprinkle with cheese. Bring up 2 sides of foil so edges meet. Seal edges, making tight \u00bd-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal.\n\n3 Place packet on grill over low heat. Cover grill; cook 12 to 18 minutes, rotating packet half turn after every 6 minutes, until corn is tender. Cut large X across top of packet; carefully fold back foil to allow steam to escape.\n\n1 Serving: Calories 210; Total Fat 10g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 25mg; Sodium 220mg; Total Carbohydrate 25g (Dietary Fiber 4g, Sugars 3g); Protein 5g Exchanges: 1\u00bd Starch, 2 Fat Carbohydrate Choices: 1\u00bd\n\nGrilled Corn with Chile-Lime Butter Omit chives and garlic salt. Increase butter to \u00bd cup. Add \u00bd teaspoon grated lime peel, 1\u00bd tablespoons fresh lime juice, 1 teaspoon chili powder and \u00bc teaspoon salt with butter.\n\nGrilled Herbed Red Potatoes\n\nGrilled Herbed Red Potatoes Fast\n\nPrep 25 min Total 25 min \u25cf 4 servings\n\nSour Cream Sauce\n\n  * \u2153 cup sour cream\n  * 1 tablespoon chopped fresh or 1 teaspoon dried rosemary leaves, crushed\n  * \u00bc teaspoon lemon-pepper seasoning\n  * \u215b teaspoon garlic powder\n\nPotatoes\n\n  * 2 tablespoons olive or vegetable oil\n  * 1 tablespoon chopped fresh parsley\n  * 1 tablespoon chopped fresh or 1 teaspoon dried rosemary leaves, crushed\n  * \u00bd teaspoon lemon-pepper seasoning\n  * \u00bc teaspoon salt\n  * 8 small red potatoes, cut into quarters\n\n1 Heat gas or charcoal grill. In small bowl, mix sauce ingredients. Cover and refrigerate until serving time.\n\n2 In large bowl, mix remaining ingredients except potatoes. Add potatoes; toss to coat. Place potatoes in grill basket (grill \"wok\").\n\n3 Place grill basket on grill over medium heat. Cover grill; cook 10 to 15 minutes, shaking basket or stirring potatoes occasionally, until tender. Serve potatoes with sour cream sauce.\n\n1 Serving: Calories 270; Total Fat 11g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 240mg; Total Carbohydrate 40g (Dietary Fiber 4g, Sugars 3g); Protein 5g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\u00bd\n\n### Using a Grill Basket\n\nSpray basket with cooking spray (away from grill flame to prevent flare-up). Place food in basket; place basket directly on grill. Shake or stir food in basket for even cooking.\n\n# Smoking Basics\n\nTo traditionalists, smoking is the original BBQ. Smoking is an unfussy form of outdoor cooking that uses low heat and hardwoods to impart smoky flavor to meat, poultry and fish. Smoking foods at home is easy with these tips.\n\n## TYPES OF SMOKERS\n\nSmokers come in a variety of shapes and sizes. Regardless of the style, all smokers contain a firebox, water pan and grill racks.\n\nTo smoke, the food is placed on a grill rack, which is high above or next to the heat. A pan of water or other liquid (beer, fruit juice, wine, soda) rests between the heat source and the food. Water-soaked aromatic wood chunks, chips or shreds are added to the smoker. Foods are cooked very slowly in a dense fog created from the damp wood chips, infusing the characteristic flavor, moistness and tenderness smoked meats are known for. Foods best for smoking include ribs, beef brisket, roasts, poultry and fish.\n\n## SAFETY TIPS FOR SMOKING\n\nFollow the manufacturer's directions for specific safety tips for your smoker and use these basic tips to ensure successful results:\n\n  * Keep the smoker away from buildings and from where people will frequently walk by, since the smoky aromas can last for hours.\n  * Use only completely thawed meats, poultry, fish or seafood. The heat inside the smoker is too low to thaw and cook the food safely.\n  * It's best to smoke foods when the outside temperature is 65\u00b0F or higher and there is little or no wind. Below 55\u00b0F, the smoker and food will not get hot enough to cook properly and may result in an additional 2 to 3 hours of cooking.\n  * Use long tongs and barbecue mitts to avoid burns when doing anything in the hot smoker.\n  * It's important throughout the cooking process to monitor the temperature inside the smoker (to ensure that the smoker stays between 225\u00b0F and 300\u00b0F) and of the meat; smoked meat can have a reddish appearance even when fully cooked. A digital oven-probe thermometer can stay in the food in the smoker and sound an alarm when the food has reached the correct temperature. This prevents you from having to open the smoker to check the temperature. If using an ovenproof meat thermometer, you'll have to open the smoker to check it, which can let heat out and increase the cook time. See Thermometers, page 17, for more information.\n\n## Smoking on a Grill\n\nHere's how to get smoked flavor on a grill.\n\n  * Cover 1 to 2 cups wood chips or shreds or 2 to 3 wood chunks with water, and soak at least 30 minutes; drain.\n  * Put soaked wood onto a piece of heavy-duty foil; seal tightly to form a pouch. Poke 6 to 8 slits in top of pouch with a sharp knife.\n  * Put pouch on grill rack or follow manufacturer's directions for adding wood chips. Cover grill and let pouch get hot enough to start smoking, about 10 minutes. Add food, leaving the pouch in the grill during cooking.\n\nWood chips for smoking, clockwise from top left: Hickory chips, mesquite chips, apple chips and cedar chips\n\nSmoked Beef Brisket\n\n## Smoked Beef Brisket Calorie Smart\n\nPrep 10 min Total 5 hr 50 min \u25cf 12 servings\n\n  * 2 cups hickory wood chips\n\nBrisket Rub\n\n  * 1 tablespoon coarse sea salt\n  * 1 tablespoon coarsely ground black pepper\n  * 1 tablespoon packed brown sugar\n  * 2 teaspoons onion powder\n  * 2 teaspoons garlic powder\n  * 1 teaspoon ground mustard\n  * 1 teaspoon smoked paprika\n  * 1 teaspoon chili powder\n\nBrisket\n\n  * 1 fresh beef brisket (not corned beef), 3 lb\n\n1 In large bowl, cover wood chips with water; soak 30 minutes. Drain. Prepare and heat smoker using wood chips and adding water to pan following manufacturer's directions.\n\n2 In small bowl, mix all rub ingredients. Rub mixture onto all surfaces of beef.\n\n3 Carefully brush oil on smoker rack. Place beef on rack in smoker. Cover and smoke 4 to 5 hours or until tender. If smoking stops, add additional wood chips through side door of smoker. Remove beef from smoker to cutting board or plate; cover and let stand 10 minutes. Cut beef across grain into thin slices.\n\n1 Serving: Calories 170; Total Fat 7g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 70mg; Sodium 1210mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 24g Exchanges: 3\u00bd Very Lean Meat, 1 Fat Carbohydrate Choices: 0\n\nOven-Baked Beef Brisket Heat oven to 325\u00b0F. Omit wood chips. Rub mix on beef. Place beef in 4-quart ovenproof Dutch oven or 3-quart casserole. Cover and bake 3 hours, turning beef over halfway through baking, until tender. Let stand 10 minutes before slicing.\n\nSmoked Hot and Spicy Ribs\n\n## Smoked Hot and Spicy Ribs\n\nPrep 15 min Total 5 hr 15 min \u25cf 6 servings\n\n  * 4 cups hickory wood chips\n\nHot and Spicy Rub\n\n  * 1 tablespoon garlic powder\n  * 1 tablespoon paprika\n  * 2 teaspoons ground red pepper (cayenne)\n  * 2 teaspoons dried thyme leaves, crushed\n  * 1 teaspoon salt\n  * 1 teaspoon black pepper\n\nRibs\n\n  * 5 lb pork spareribs (not cut into serving pieces)\n  * Barbecue sauce, if desired (for homemade, see page 411)\n\n1 In large bowl, cover wood chips with water; soak 30 minutes. Drain. Prepare and heat smoker using wood chips and adding water to water pan following manufacturer's directions.\n\n2 In small bowl, mix rub ingredients. Cut rack of ribs in half to fit on smoker rack if necessary. Rub spice mixture into pork.\n\n3 Place pork on rack in smoker. Cover smoker; cook 4 hours 30 minutes to 5 hours or until meat is tender and no longer pink next to bones. If smoking stops, add additional wood chips through side door of smoker. Cut ribs into serving pieces; serve with barbecue sauce.\n\n1 Serving: Calories 580; Total Fat 45g (Saturated Fat 16g, Trans Fat 0g); Cholesterol 180mg; Sodium 530mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 43g Exchanges: 6 Medium-Fat Meat, 3 Fat Carbohydrate Choices: 0\n\n### Removing Silverskin from Pork Ribs\n\nSlide knife under silverskin; lift and loosen until you can grab it with a paper towel. Pull it off in one piece if possible.\n\n## Smoked Cajun Chicken Calorie Smart\n\nPrep 10 min Total 3 hr 10 min \u25cf 6 servings\n\n  * 2 cups hickory wood chips\n\nCajun Spice Rub\n\n  * 1 teaspoon black pepper\n  * \u00bd teaspoon white pepper\n  * \u00bd teaspoon ground red pepper (cayenne)\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground cumin\n  * \u00bd teaspoon ground nutmeg\n\nCHICKEN\n\n  * 1 cut-up whole chicken (3 to 3\u00bd lb)\n  * 1 tablespoon vegetable oil\n\n1 In large bowl, cover wood chips with water; soak 30 minutes. Drain. Prepare and heat smoker using wood chips and adding water to water pan following manufacturer's directions.\n\n2 In small bowl, mix rub ingredients. Brush chicken with oil. Rub spice mixture into all sides of chicken.\n\n3 Place chicken, skin side up, on rack in smoker. Cover smoker; cook 2 hours 30 minutes to 3 hours or until juice of chicken is clear when thickest pieces are cut to bone (at least 165\u00b0F). If smoking stops, add additional wood chips through side door of smoker.\n\n1 Serving: Calories 170; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 70mg; Sodium 270mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 23g Exchanges: 3 Lean Meat Carbohydrate Choices: 0\n\nSmoked Brined Salmon\n\n## Smoked Brined Salmon Calorie Smart\n\nPrep 10 min Total 4 hr 10 min \u25cf 6 servings\n\n  * \u00bd cup sugar\n  * \u00bc cup salt\n  * 1 tablespoon grated orange peel (from 1 small orange)\n  * 1 teaspoon black peppercorns\n  * 1 medium onion, sliced\n  * 4 cups water\n  * 1 skin-on salmon fillet (2\u00bd to 3 lb)\n  * 4 to 6 wood chunks (hickory, mesquite or apple)*\n  * 2 tablespoons vegetable oil\n  * \u00bd teaspoon paprika\n  * \u00bc teaspoon pepper\n  * Orange slices and sliced green onions, if desired.\n\n1 In 1-gallon resealable food-storage plastic bag, make brine by mixing sugar, salt, orange peel, peppercorns, onion and 4 cups water. Seal bag and squeeze to mix until sugar and salt are dissolved. Place salmon in brine; seal bag. Refrigerate at least 3 hours but no longer than 4 hours.\n\n2 In large bowl, cover wood chunks with water; soak 30 minutes. Drain. Prepare and heat smoker using wood chunks and adding water to water pan following manufacturer's directions.\n\n3 Remove salmon from brine; discard brine. Blot salmon dry with paper towels. Brush both sides of salmon with oil.\n\n4 Place salmon, skin side down, on top grill rack in smoker. Sprinkle with paprika and pepper. Cover smoker; cook about 1 hour or until fish flakes easily with fork. If smoking stops, add additional wood chunks through side door of smoker.\n\n5 Place salmon, skin side up, on foil. Peel skin from fish and discard. Use foil to turn fish onto serving platter. Garnish with orange slices and sliced green onion.\n\n*Or substitute 2 cups hickory, mesquite or applewood chips for the wood chunks.\n\n1 Serving: Calories 270; Total Fat 13g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 105mg; Sodium 490mg; Total Carbohydrate 4g (Dietary Fiber 0g, Sugars 4g); Protein 34g Exchanges: 5 Lean Meat Carbohydrate Choices: 0\n\n# Chapter 16: Do It Yourself\n\nStrawberry Freezer Jam\n\n## Do-It-Yourself Basics\n\nChoosing Containers\n\nEating Local\n\n## Freezing and Canning Basics\n\nFreezing Foods\n\nHome Canning\n\nPreparing Jars for Canning\n\nPre-Sterilizing Jars\n\n## Features\n\nBetty's Staples: Fresh Herbs\n\nDrying Fresh Herbs\n\nFreezing Fresh Herbs\n\nHeirloom Recipe and New Twist\n\nLearn to Can: Peach Jalape\u00f1o Jam\n\n## Jams, Jellies and Preserves\n\nApple Butter Calorie Smart\n\nApple-Pepper Jelly Calorie Smart \u2022 Easy\n\nBlueberry Freezer Jam\n\nPeach-Jalape\u00f1o Jam Calorie Smart\n\nPear-Orange Conserve Calorie Smart\n\nRaspberry Freezer Jam\n\nRosemary Peach Jam\n\nSpiced Apricot-Ginger Preserves Calorie Smart \u2022 Easy\n\nSpiced Peach Jam\n\nStrawberry Freezer Jam Calorie Smart \u2022 Easy\n\nStrawberry-Balsamic Freezer Jam\n\nSweet Tomato Jam Calorie Smart\n\nTriple-Berry Pomegranate Freezer Jam Calorie Smart\n\n## Chutney and Relishes\n\nApple-Cranberry Chutney Calorie Smart\n\nCorn Relish Calorie Smart\n\nCranberry-Orange Relish Calorie Smart \u2022 Easy\n\nCranberry-Orange-Ginger Relish\n\nCrunchy Veggie Relish Calorie Smart\n\nMango Relish Calorie Smart \u2022 Easy\n\nPear-Cranberry Chutney\n\n## Pickles\n\nEasy Refrigerator Pickles Calorie Smart\n\nGarlic Dill Pickles Calorie Smart\n\nJalape\u00f1o-Garlic Pickled Green Beans Calorie Smart\n\nKimchi Calorie Smart\n\nLemon-Dill Pickled Green Beans\n\nPickled Beets Calorie Smart\n\nPickled Ginger\n\nPickled Green Beans\n\nPickled Peppers Calorie Smart\n\nPickled Tarragon Baby Carrots Calorie Smart \u2022 Easy\n\nSpicy Pickled Carrots\n\nSpicy Pickled Vegetables Calorie Smart\n\n## Dairy Products\n\nBacon Ranch Greek Yogurt\n\nBlueberry Kefir\n\nCardamom-Scented Yogurt Cheese\n\nGreek Yogurt Calorie Smart \u2022 Easy\n\nKefir Calorie Smart\n\nMango Kefir\n\nStrawberry Kefir\n\nStrawberry-Banana Kefir\n\n## Condiments, Homemade Mixes and Sauces\n\nBerry Vinegar\n\nBrownie Mix\n\nChai Mix Calorie Smart \u2022 Fast\n\nConfetti Cookie Mix Calorie Smart\n\nCrystallized Ginger Calorie Smart\n\nGarlic Vinegar\n\nHarissa Calorie Smart\n\nHerb Vinegar Calorie Smart \u2022 Easy\n\nKetchup Calorie Smart\n\nLemon Vinegar\n\nNatural Homemade Food Color\n\nOven-Dried Tomatoes Calorie Smart \u2022 Easy\n\nSalt-Free Herb Mix Calorie Smart \u2022 Fast\n\nSalt-Free Taco Seasoning Mix Calorie Smart \u2022 Fast\n\nSmoky Salt-Free Taco Seasoning Mix\n\n## Homemade Baby Food\n\nCarrot Baby Food Calorie Smart \u2022 Easy\n\nGreen Bean Baby Food\n\nSweet Pea Baby Food\n\nWinter Squash Baby Food\n\n## Bonus Recipes\n\nEasy Quesadillas\n\nFresh Herb Buttermilk-Ranch Dressing\n\nGarlic-Herb Butter\n\nHoney-Rosemary Simple Syrup\n\nMeat Glaze\n\nOrange\u2013Lemon Balm Sun Tea\n\nPasta with Fried Sage\n\nQuick Spread\n\nSavory Bread Sidekick\n\nSeafood Sauce\n\nSlow Cooker Apple Butter\n\nSlow Cooker Pear-Orange Conserve\n\nSpicy Almond-Cilantro Pesto\n\nSpicy Peach Vinaigrette\n\nStrawberry-Basil Water\n\nSweet-and-Savory Yogurt\n\nZesty BBQ Sauce\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Do-It-Yourself Basics\n\nDIY has come full circle, when you talk about food. Past generations made their own pickles, jellies and jams and canned vegetables because they had to. Today, the make-it-yourself movement in the kitchen has become, \"We want to.\" While there are many options to choose from in our grocery stores, people are making their own versions of these foods for many reasons. Whether it's avoiding preservatives, controlling seasoning and salt levels, taking advantage of fresh, locally grown produce at its peak or making foods to share as gifts from your kitchen\u2014nothing beats homemade.\n\n## Choosing Containers\n\nWhen making your own food, choosing the right container is important so the food will be delicious when you are ready to enjoy it. Follow these general guidelines, or see individual recipes for specific requirements.\n\nFreezer Jam\/Chutney: Choose plastic freezer containers or glass preserving jars without curves under the neck of the jars. Be sure to choose a size that will leave enough room for the food to expand.\n\nFruit Butter: Choose wide-mouth glass preserving jars or plastic containers with tight-fitting lids.\n\nCanned Jelly: Choose glass preserving jars (regular or wide mouth) with lids and bands.\n\nPickles: Choose glass preserving jars (regular or wide mouth) with lids and bands. For whole pickles, choose wide-mouth jars.\n\nPickled Vegetables: Choose nonreactive covered containers.\n\nVinegar: Choose glass bottles with a narrow neck and screw top, cork or two-piece canning lids. Sterilize bottles before using. see page 431.\n\nDried Vegetables: Choose containers with tight-fitting lids.\n\nSeasoning Mixes\/Dried Herbs: Choose small containers with tight-fitting lids.\n\nDessert\/Beverage Mixes: Choose food-safe glass jars with screw-on lids to show off the layers or plastic containers with tight-fitting lids.\n\n## Eating Local\n\nOptions for eating local may be all around you\u2014it's knowing where to look and understanding what the offerings are that can help you make the best choices for yourself..\n\nFarmers' Markets: Farmers from the area all bring their produce to sell at nearby markets.\n\nU-Picks: Farms where you are allowed to come in and pick your own produce.\n\nFarm Stands: Farmers set up stands along the side of the road or in store parking lots where their produce is brought in daily for purchase.\n\nFood Co-ops: Grocery stores owned and controlled by their members, although you typically don't need to be a member to shop there. They usually specialize in natural foods and often offer local and organic foods.\n\nCommunity-Supported Agriculture (CSA):You buy a \"share\" of the local farmer's produce in advance and then receive a weekly portion of fresh produce or other items.\n\nGrocery Stores: Many grocery stores are getting on the fresh bandwagon, offering in-season locally grown produce.\n\n## freezing and Canning Basics\n\nFreezing and canning are both rewarding and smart. Both are great ways to preserve foods to enjoy later. Freezing is a quick method for preserving foods such as fruits and veggies. Canning is a great way to put up vegetables or make your own pickles or jam.\n\n## Freezing Foods\n\nFreezing is a simple way to preserve in-season produce. When it's frozen at the peak of freshness, much of the nutrient value is left in the food, making it a great way to have fresh-tasting, nutritious food all year long. It's a terrific way to take advantage of a bountiful harvest and low prices and to keep food to enjoy later. Both fruits and vegetables can be stored in freezer bags or clean plastic containers with tight-fitting lids. Produce will freeze faster, be easier to store and keep longer if containers are not too large. Most fruits and vegetables can be frozen up to 1 year.\n\n### Vegetables\n\nIn order to freeze most vegetables successfully, you'll want to blanch them first. Blanching stops the enzyme actions that cause loss of flavor, color and texture. It also slows the loss of vitamins, brightens the color and softens vegetables for easier packing in freezer containers.\n\nTo blanch, bring 1 gallon of water per pound of cleaned vegetables to a boil. Add the vegetables and let the water return to boiling.\n\nAfter boiling, immediately remove vegetables from the boiling water and plunge into ice-cold water to cool them quickly.\n\nBlanch using the following guidelines:\n\nVegetable | Blanching time\n\n---|---\n\nGreen peas, turnips or parsnips (cubed) | 2 minutes\n\nAsparagus (medium stalks), broccoli or cauliflower florets, Brussels sprouts (small), bell pepper halves, green or wax beans, okra | 3 minutes\n\nAsparagus (large stalks), Brussels sprouts (large), carrots (baby), edamame pods, corn on the cob (small to medium ears)* | 5 minutes\n\n*Cut corn kernels from cob after boiling and cooling.\n\n### Fruits\n\nTo freeze fruits, sort, wash and drain carefully. Prepare fruits as you would to serve them, removing stems, coring, peeling and slicing as needed. Frozen fruit will retain much of the nutritional value but will be softer after freezing.\n\nSmall whole fruits such as berries and cherries can be frozen individually on a tray, and then placed in bags for use in salads and desserts. Some fruits, such as apples, peaches, nectarines and pears, might need added acid to keep their color during freezing. Look for ascorbic acid products available near the canning supplies, to help fruits maintain their color. Be sure to follow the package directions carefully. Citric acid or lemon juice can also be used by placing \u00bc teaspoon in a quart of cold water. Place the sliced fruit in the mixture for 1 to 2 minutes. Drain and freeze in resealable plastic freezer bags or pack with sugar, water or fruit juice as directed in individual recipes.\n\n## Home Canning\n\nHome canning isn't difficult, but it's important to do it correctly so the food you preserve is safe from bacterial contamination. To destroy spoilage organisms such as molds, yeast and bacteria, food must be processed long enough and at a high enough temperature. The amount of acidity of the food determines which canning method you use:\n\nPressure Canning: Use for low-acid foods such as meat, poultry and vegetables. Most pressure canners today have removable racks, an automatic vent\/cover lock, a vent pipe (steam vent) and a safety fuse. It's important to know your canner is working properly and that you are carefully following both the canner manual instructions and the recipe to be sure you are cooking and cooling your food accurately, to preserve it safely for eating at a later date.\n\nWater Bath Canning: Recommended for all acidic foods, such as fruits, tomatoes with added acid, jams, jellies and pickled vegetables. A boiling water canner consists of a large pot with a lid and a rack to hold canning jars. The open kettle method of canning (which doesn't use a lid) isn't recommended, as it can't get the jars to temperatures hot enough to kill bacteria.\n\nIf you are unsure about the safety of certain home-canned foods, remove it from its container and boil the food for 10 minutes at altitudes below 1,000 feet to destroy these toxins. Boil an additional minute for every 1,000 feet of additional elevation.\n\n### Preparing Jars for Canning\n\nAlways clean jars just prior to filling them when canning; either washing in the dishwasher or by hand, using detergent and rinsing well. Keep them warm until ready to fill by leaving them in the closed dishwasher after the cycle, placing them in your canner as the water is heating or placing them in another pan of hot water that will keep the jars both clean and warm.\n\nCheck the jars for any chips or cracks while washing them\u2014discard or repurpose any with imperfections.\n\nJars need to be sterile to kill any organisms that could contaminate the food being processed in them, so nothing has a chance to grow when the food is being stored. If the processing time for your food is 10 minutes or more or if you are using a pressure canner, the jars will be sterilized at the same time the food is being processed in the canner. If the processing time is less than 10 minutes, you will need to pre-sterilize them before filling.\n\n### Pre-Sterilizing Jars\n\nPlace the cleaned jars right side up on the rack in the canner; fill jars and canner with water to 1 inch above the tops of the jars. Bring the water to a boil and then boil for 10 minutes at altitudes less than 1,000 feet elevation. Add 1 additional minute for each additional 1,000 feet of elevation. To fill the jars, remove one at a time, emptying the water from the jar back into the canner, so the water can be reused for processing the filled jars.\n\n## When in Doubt, Throw It Out!\n\nIf any of these signs are present, do not eat the food. Discard the food in such a way that children or pets won't accidentally eat it either.\n\n  * Lid is loose and is no longer vacuum sealed or lid swells.\n  * Gas bubbles are present inside the jar.\n  * Liquid spurts out when the container is opened.\n  * Mold, cloudiness, yeast growth or fermentation is present.\n  * Container is leaking.\n  * Unpleasant odor is present when container is opened.\n\n## Strawberry Freezer Jam Calorie Smart \u2022 Easy\n\nFreezer jams have become very popular, not only because of their fresh just-picked flavor but also because they are easy to make. Strawberry Freezer Jam is a favorite for us and was first printed in the 1995 edition of this cookbook.\n\nPrep 10 min Total 24 hr 20 min \u25cf 5 half-pints\n\n  * 1 quart (4 cups) strawberries, cut in half\n  * 4 cups sugar\n  * \u00be cup water\n  * 1 package (1\u00be oz) powdered fruit pectin\n\n1 In large bowl, mash strawberries with potato masher (or process in food processor until slightly chunky, not pureed) to make 2 cups crushed strawberries. Add sugar; mix well. Let stand at room temperature 10 minutes, stirring occasionally.\n\n2 In 1-quart saucepan, mix water and pectin until pectin is dissolved. Heat to a full rolling boil that cannot be stirred down, stirring constantly. Boil hard 1 minute, continuing to stir. Pour hot pectin mixture over strawberry mixture; stir constantly until sugar is completely dissolved, about 3 minutes.\n\n3 Immediately pour mixture into hot sterilized jars or freezer containers, leaving \u00bd inch headspace; cover. Let stand at room temperature 24 hours or until set.\n\n4 Store in freezer up to 12 months or in refrigerator up to 3 weeks. Thaw frozen jam in refrigerator and stir before serving.\n\n1 Tablespoon: Calories 45; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 11g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: 1\n\nBlueberry Freezer Jam Substitute 2 pints (4 cups), blueberries, crushed (2\u00bd cups crushed) for the strawberries. Reduce sugar to 3 cups. Add 1 teaspoon grated lemon or orange peel if desired. Reduce water to \u00bd cup.\n\nRaspberry Freezer Jam Substitute 3 pints (6 cups) raspberries, crushed (3 cups crushed) for the strawberries. Increase sugar to 5\u00bc cups.\n\nStrawberry-Balsamic Freezer Jam Make as directed, except\u2014reduce water to \u00bc cup and add \u00bd cup balsamic vinegar in Step 2.\n\n### Dissolving Pectin for Canning\n\nIn saucepan, mix water and pectin until pectin is dissolved. Heat to boiling, stirring constantly.\n\nPour hot pectin mixture over fruit mixture; stir constantly until sugar is completely dissolved.\n\n## Triple-Berry Pomegranate Freezer Jam Calorie Smart\n\nPrep 10 min Total 24 hr 20 min \u25cf 6 half-pints\n\n  * 2 cups strawberries, cut in half\n  * 2 cups raspberries\n  * 1 cup blueberries\n  * 4\u00bc cups sugar\n  * \u00be cup pomegranate juice\n  * 1 package (1\u00be oz) powdered fruit pectin\n\n1 In large bowl, mash berries with potato masher (or process in food processor until slightly chunky, not pureed) to make 2\u00bd cups crushed berries. Add sugar; mix well. Let stand at room temperature 10 minutes, stirring occasionally.\n\n2 In 1-quart saucepan, mix juice and pectin until pectin is dissolved. Heat to a full rolling boil that cannot be stirred down, stirring constantly. Boil hard 1 minute, continuing to stir. Pour hot pectin mixture over berry mixture; stir constantly until sugar is completely dissolved, about 3 minutes.\n\n3 Immediately pour mixture into hot sterilized jars or freezer containers, leaving \u00bd inch headspace; cover. Let stand at room temperature about 24 hours or until set.\n\n4 Store in freezer up to 12 months or in refrigerator up to 3 weeks. Thaw frozen jam in refrigerator and stir before serving.\n\n1 Tablespoon: Calories 40; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 10g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\nPear-Orange Conserve\n\n## Pear-Orange Conserve Calorie Smart\n\nUsually served as a spread for toast, biscuits, scones and bread, conserves differ from single-fruit preserves in that they have two or more fruits and some type of nut. The orange flavor of this conserve makes it a perfect match for grilled meats or holiday turkey or ham.\n\nPrep 30 min Total 2 hr 30 min \u25cf 4 cups\n\n  * 5 medium pears, chopped (6 cups)\n  * 1 medium unpeeled seedless orange, chopped\n  * 1\u00bd cups sugar\n  * \u00bd cup chopped well-drained maraschino cherries or dried cherries\n  * \u00bd cup chopped walnuts or slivered almonds\n\n1 In 2-quart saucepan, mix pears, orange and sugar. Heat to boiling, stirring frequently; reduce heat to low. Cook uncovered 15 minutes, stirring occasionally. Add cherries and nuts; cook 5 to 10 minutes, stirring occasionally.\n\n2 Spoon into glass or plastic containers. Cool 2 hours. Cover and store in refrigerator up to 3 weeks or freeze up to 1 year.\n\n1 Tablespoon: Calories 35; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 7g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\nSlow-Cooker Directions Spray 2- to 3\u00bd-quart slow cooker with cooking spray. In slow cooker, mix pears, orange and sugar. Cover; cook on High heat setting 2 hours. Stir; cook uncovered 1 hour 30 minutes to 2 hours or until very thick. Stir in cherries and nuts. Spoon into glass or plastic containers. Cool 2 hours. Cover and store in refrigerator up to 3 weeks or freezer up to 1 year.\n\n## Spiced Apricot-Ginger Preserves Calorie Smart \u2022 Easy\n\nPrep 10 min Total 2 hr 10 min \u25cf 1 cup\n\n  * 1 cup apricot or peach preserves\n  * 1 tablespoon finely chopped crystallized ginger\n  * 1 tablespoon orange-flavored liqueur or orange juice\n  * \u00bc teaspoon ground nutmeg\n\n1 In 1-quart saucepan, mix all ingredients. Heat to boiling; boil 1 minute, stirring frequently.\n\n2 Spoon into glass or plastic container. Cool 2 hours. Cover and store in refrigerator up to 2 months.\n\n1 Tablespoon: Calories 60; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 15g (Dietary Fiber 0g, Sugars 10g); Protein 0g Exchanges: 1 Other Carbohydrate Carbohydrate Choices: 1\n\n## Apple-Pepper Jelly Calorie Smart \u2022 Easy\n\nPrep 10 min Total 24 hr 10 min \u25cf 4 half-pints\n\n  * 2 cups water\n  * 1 can (6 oz) frozen apple juice concentrate, thawed\n  * 1 package (1\u00be oz) powdered fruit pectin\n  * 3\u00be cups sugar\n  * Few drops red food color, if desired\n  * 1 to 2 tablespoons crushed red pepper flakes\n\n1 In 3-quart saucepan, mix water, apple juice concentrate and pectin until pectin is dissolved. Stir in sugar. Heat to a full rolling boil that cannot be stirred down, stirring constantly. Boil hard 1 minute, continuing to stir. Remove from heat; stir in food color. Quickly skim off foam. Stir in pepper flakes.\n\n2 Immediately pour mixture into hot sterilized jars or freezer containers, leaving \u00bd inch headspace; cover. Let stand at room temperature about 24 hours or until set.\n\n3 Store covered in refrigerator up to 1 month or in freezer up to 2 months. Thaw jelly in refrigerator before serving.\n\n1 Tablespoon: Calories 50; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 13g); Protein 0g Exchanges: 1 Other Carbohydrate Carbohydrate Choices: 1\n\nApple Butter\n\n## Apple Butter Calorie Smart\n\nPrep 20 min Total 4 hr 20 min \u25cf 4 cups\n\n  * 12 medium Granny Smith or other cooking apples (4 lb), peeled, cut into quarters\n  * 1\u00bd cups packed brown sugar\n  * 1 tablespoon ground cinnamon\n  * 1 teaspoon ground allspice\n  * 1 teaspoon ground nutmeg\n  * \u00bd teaspoon ground cloves\n  * 1\u00bc cups apple juice\n  * 1 tablespoon lemon juice\n\n1 In 4-quart Dutch oven or saucepan, mix all ingredients. Heat to boiling, stirring occasionally; reduce heat. Cover and simmer 1 hour.\n\n2 Mash apples with potato masher or large fork. Simmer uncovered about 1 hour longer, stirring occasionally, until mixture is very thick. Cool about 2 hours.\n\n3 Spoon into glass or plastic containers. Cover and store in refrigerator up to 3 weeks.\n\n1 Tablespoon: Calories 40; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 8g); Protein 0g Exchanges: \u00bd Fruit Carbohydrate Choices: \u00bd\n\nSlow-Cooker Directions Reduce apple juice to \u00bd cup. Spray 5- to 6-quart slow cooker with cooking spray. In slow cooker, mix all ingredients. Cover; cook on Low heat setting 8 to 10 hours or until apples are very tender. Mash apples with potato masher or large fork. Cook uncovered 1 to 2 hours longer, stirring occasionally, until very thick.\n\nApple-Cranberry Chutney\n\n## Apple-Cranberry Chutney Calorie Smart\n\nServe the chutney with roast turkey, ham or pork. Or for a quick appetizer, spoon chutney over a block of cream cheese and serve with crackers.\n\nPrep 15 min Total 3 hr \u25cf 2 cups\n\n  * 2 medium cooking apples, chopped (2 cups)\n  * 2 cups fresh or frozen cranberries\n  * 1 medium red bell pepper, chopped (1 cup)\n  * 1 small onion, finely chopped (\u2153 cup)\n  * \u00be cup packed brown sugar\n  * \u00bd cup golden raisins, dried cranberries or dried cherries\n  * \u00bd cup white vinegar\n  * 1\u00bd teaspoons finely chopped gingerroot\n  * 1 clove garlic, finely chopped\n\n1 In 2-quart saucepan, mix all ingredients. Heat to boiling, stirring occasionally; reduce heat. Cover and simmer 30 minutes.\n\n2 Uncover; simmer about 15 minutes longer, stirring occasionally, until mixture has thickened and fruit is tender. Cool about 2 hours.\n\n3 Spoon into glass or plastic containers; cover. Store in refrigerator up to 2 weeks or in freezer up to 2 months.\n\n\u00bc Cup: Calories 150; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 36g (Dietary Fiber 2g, Sugars 31g); Protein 0g Exchanges: 1 Fruit, 1\u00bd Other Carbohydrate Carbohydrate Choices: 2\u00bd\n\nPear-Cranberry Chutney Substitute 2 medium firm, ripe pears, chopped (2 cups), for the apples.\n\n## Peach-Jalape\u00f1o Jam (see page 436) Calorie Smart\n\nPrep 1 hr 10 min Total 25 hr 55 min \u25cf 6 half-pints\n\n  * 6 (\u00bd pint) glass preserving jars with lids and bands\n  * 4 cups finely chopped, peeled peaches (about 3 pounds)\n  * \u00bc cup fresh lemon juice\n  * \u00bd teaspoon butter\n  * 1 package (1\u00be oz) powdered fruit pectin\n  * 3 cups regular granulated sugar\n  * \u2153 cup diced jalape\u00f1o chiles*\n\n1 Fill boiling water canner half full with water; heat until water is simmering. Wash jars as directed in Preparing Jars for Canning, page 431.\n\n2 In 4-quart saucepan, mix peaches, lemon juice, butter (helps prevent foaming) and pectin, stirring until pectin is dissolved. Heat to boiling, stirring frequently. Add sugar all at once; mix well. Stir in chiles. Heat to a full rolling boil that covers the entire surface, not just the edges of the pan, stirring constantly. Boil and stir 1 minute; remove from heat.\n\n3 Quickly skim off any foam. Immediately ladle or pour into jars, leaving \u00bd inch headspace. Drain flat lids well. Wipe rims and threads of jars. Place flat lids on jars and screw bands on tightly to seal. Place jars on elevated rack in canner; lower rack into canner. To process jam safely, water must cover jars by 1 to 2 inches; add additional boiling water if necessary. Cover. Heat to boiling; continue boiling 10 minutes. Turn off heat. Remove canner lid. Let jars stand in canner 5 minutes.\n\n4 Carefully remove jars with canning jar lifter or heavy-duty tongs and place upright on dry towel. Cool completely. After 24 hours, check seal on cooled jars by pressing middle of lids with finger. If the lids spring back, jars are not sealed and must be stored in the refrigerator. Store completely sealed, unopened jars in a cool, dry place up to 1 year. Once opened, store in refrigerator up to 3 weeks.\n\n*If chiles are very hot try using \u00bc cup.\n\n1 Tablespoon: Calories 30; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol: 0mg; Sodium 0mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 7g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\nRosemary Peach Jam Omit jalape\u00f1o chiles. Add three 5-inch sprigs fresh rosemary with sugar in Step 2. Discard rosemary before ladling jam into jars.\n\nSpiced Peach Jam Omit jalape\u00f1o chiles. Add 2 cinnamon sticks and 6 whole cloves with pectin in Step 2. Discard cinnamon sticks and cloves before ladling jam into jars.\n\nSweet Tomato Jam\n\n## Sweet Tomato Jam Calorie Smart\n\nTry this sweet jam made with fresh tomatoes on crostini or grilled cheese, with hummus and pita chips or sandwiched between shortbread cookies.\n\nPrep 20 min Total 1 hr 50 min \u25cf 2 cups\n\n  * 12 medium ripe tomatoes, seeded, chopped (6 cups)\n  * \u00bd cup sugar\n  * \u00bc cup cider vinegar*\n  * \u00bc cup olive oil\n  * 1 teaspoon salt\n  * \u00bd teaspoon coarsely ground black pepper\n\n1 In 12-inch skillet, heat all ingredients to boiling. Reduce heat to medium-low. Simmer uncovered 45 to 60 minutes, stirring occasionally, until thickened and almost all liquid is absorbed. Remove from heat; cool 30 minutes.\n\n2 Spoon into clean jars or nonmetal containers; cover with tight-fitting lids. Refrigerate up to 1 week or freeze up to 3 months.\n\n* Red or white wine vinegar or balsamic vinegar can be substituted for the cider vinegar.\n\n1 Tablespoon: Calories 40; Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 75mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 4g); Protein 0g Exchanges: \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## Learn to Can PEACH-JALAPE\u00d1O JAM\n\nMaking homemade jam isn't difficult, but it does require close attention to directions from trusted recipes, so that the end results are safe to store and eat. Food canning techniques have changed over the years, so it is best be up to date with the latest techniques and recipes, rather than processing Grandma's famous jam using her outdated instructions. If done incorrectly, you can put your family and friends at risk of ingesting dangerous bacteria in the food you processed. Follow the general guidelines on page 431 and specific directions in recipes (see page 435 for Peach-Jalape\u00f1o Jam recipe).\n\n### How to \"Put Up\" Jam Successfully\n\n  * Use the proper canning techniques. see page 431 for more information.\n  * Inspect and wash jars; keep warm until ready to fill. Repurpose jars that have defects.\n  * Skim foam from jam mixture. Foam forms when cooking jam mixture; skim and discard it for the best-looking jam and so it won't affect headspace.\n  * Fill jars as directed in recipe; do not over-fill. Proper amount of headspace is necessary to ensure no mold growth.\n  * Wipe rims and threads. Prevent spoilage during storage by wiping off any jam that might be on the rim of jar or on the threads.\n  * Cover with new lids. New lids have fresh seals, which are necessary for proper processing to get a vacuum seal. Do not reuse lids.\n  * Screw on bands securely. Bands can be reused. Tighten enough to secure lid well but do not over-tighten. Bands may loosen during processing but can be retightened after the processed jam has cooled.\n  * Place jars on rack in canner. Do not put jars directly on bottom of pan. Jars need water boiling all around them to process properly and could potentially break if processed directly on the bottom of the pan.\n  * Have enough water in the canner. There should be enough water to cover the jars 1 inch above the tops. Add additional boiling water, if necessary.\n  * Cover and boil for the full time given. Follow directions exactly, making sure water is fully boiling, pan is covered and jars are being processed the full time called for in the recipe. This kills organisms that could grow in the food and\/or on the jar when stored.\n  * Remove and cool jars carefully. Jars can slip and break if not taken out of boiling water with large canning or heavy-duty tongs. To prevent jars from cracking, let them cool on a towel rather than directly on a cold countertop.\n  * Check seals after 24 hours. Press on center of each lid with finger. If lid springs back, it didn't seal properly. Reprocess any that didn't seal well or refrigerate and use within 1 month. Tighten any bands that have loosened.\n  * Store jars properly. Properly sealed, unopened jars can be stored in a cool, dry place up to 1 year. Once jars are opened, they should be stored in the refrigerator and used within 1 month.\n\n## Jelly, Jam or Preserves?\n\n**Jelly** is the clear, bright condiment made with fruit juice, sugar and possibly pectin to help it gel. It will hold its shape when turned out of its container. **Jam** uses crushed or chopped fruits and is usually less thick than jelly. **Conserves** are jams made with a variety of fruit and nuts. **Preserves** are like jam only usually with larger pieces of fruit left intact. **Marmalade** is preserves with the addition of fruit rind.\n\n### Creative Uses for Peach-Jalape\u00f1o Jam\n\nNow that you've got fresh jam on hand, here are some terrific ways to use it:\n\nMeat Glaze Brush it on chicken, turkey, pork or beef during cooking for a flavorful glaze. Brush it on during the last 15 to 20 minutes of cooking to prevent burning.\n\nQuick Spread Spoon it over a block of softened cream cheese to serve with crackers as a quick appetizer or snack.\n\nZesty BBQ Sauce Heat 1 cup barbecue sauce in a small saucepan over medium-low heat, stirring occasionally. Stir in 3 to 4 tablespoons jam until melted. Add more if desired.\n\nSweet-and-Savory Yogurt Top plain yogurt with a spoonful of jam and roasted pumpkin seeds for a breakfast or snack with kick.\n\nSavory Bread Sidekick Offer the jam with corn muffins or any bread or rolls instead of butter for an unexpected side dish.\n\nSpicy Peach Vinaigrette Prepare Raspberry Vinaigrette (page 137), except use white wine vinegar and Peach-Jalape\u00f1o Jam.\n\nSeafood Sauce Serve with crab cakes or coconut shrimp or in place of cocktail sauce with cold cooked shrimp.\n\nEasy Quesadillas Place 4 (8-inch) tortillas on cookie sheet; spread with about 1 tablespoon jam. Top each with 2 tablespoons shredded Monterey Jack cheese and 1 tablespoon chopped walnuts; top with another tortilla and press lightly. Bake at 375\u00b0F 8 to 10 minutes or until cheese begins to melt and tortillas begin to brown.\n\nAdd sugar and pectin and bring to a full rolling boil.\n\nAfter processing, carefully remove jars from water.\n\nQuickly ladle into sterilized jars.\n\nCheck seals after 24 hours.\n\n## Cranberry-Orange Relish Calorie Smart \u2022 Easy\n\nTry these creative ways to use this condiment beyond your Thanksgiving meal: as a spread on turkey or ham sandwiches, to fill baked mushrooms or phyllo cups for an appetizer or to stir into hot oatmeal.\n\nPrep 10 min Total 24 hr 10 min \u25cf 2\u00bd cups\n\n  * 1 bag (12 oz) fresh or frozen cranberries (3 cups)\n  * 1 unpeeled seedless orange\n  * 1 cup sugar\n\n1 Wash cranberries; remove any stems and discard any blemished berries. Cut orange into 1-inch pieces. (The peel adds a tangy orange flavor; it won't make the relish bitter tasting.)\n\n2 Place sugar in medium bowl. In food processor, place half of the cranberries and orange pieces. Cover and process about 15 seconds, using short on-and-off motions, until evenly chopped. Stir cranberry mixture into sugar. Repeat with remaining cranberries and orange pieces.\n\n3 Cover and refrigerate at least 24 hours to blend flavors before serving, but no longer than 1 week.\n\n2 Tablespoons: Calories 50; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 12g); Protein 0g Exchanges: 1 Fruit Carbohydrate Choices: 1\n\nCranberry-Orange-Ginger Relish Stir 1 tablespoon finely chopped crystallized ginger into sugar in Step 2.\n\n### Making Cranberry-Orange Relish\n\nPlace half of cranberries and orange pieces in food processor.\n\nProcess until evenly chopped; stir into sugar.\n\nCrunchy Veggie Relish\n\n## Crunchy Veggie Relish Calorie Smart\n\nPrep 15 min Total 24 hr 15 min \u25cf 6 cups\n\n  * 6 cups finely chopped green cabbage\n  * 2 medium cucumbers, peeled, shredded (3 cups)\n  * 1 medium red bell pepper, chopped (1 cup)\n  * 1 cup sugar\n  * 1\u00bd teaspoons salt\n  * 1 teaspoon mustard seed\n  * \u00bd teaspoon celery seed\n  * \u00bd cup white or cider vinegar\n\n1 In large bowl, mix all ingredients. Spoon into 1-quart or 1-pint jars or plastic containers.\n\n2 Cover and refrigerate at least 24 hours before serving. Store covered in refrigerator up to 2 weeks.\n\n\u00bc Cup: Calories 45; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 9g); Protein 0g Exchanges: \u00bd Other Carbohydrate, 1 Vegetable Carbohydrate Choices: \u00bd\n\n## Mango Relish Calorie Smart \u2022 Easy\n\nPrep 5 min Total 20 min \u25cf 1\u00bc cups\n\n  * 1 mango, peeled, pitted and chopped (1 cup)\n  * 1 small jalape\u00f1o chile, seeded and finely chopped (2 to 3 teaspoons)\n  * \u00bc cup finely chopped red onion\n  * 2 tablespoons finely chopped fresh mint leaves\n  * 2 tablespoons lime juice\n  * \u215b teaspoon salt\n\nMix all ingredients in glass or plastic bowl. Cover and refrigerate 15 minutes. Serve with grilled chicken breasts or grilled fish.\n\n2 Tablespoons: Calories 10 (Calories from Fat 0); Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Sodium 15mg; Total Carbohydrate 2g (Dietary Fiber 0g); Protein 0g Exchanges: Free Carbohydrate Choices:\n\n## Corn Relish Calorie Smart\n\nTry this relish on hot dogs, brats and burgers, or spoon it on grilled chicken or pork. Black or brown mustard seed can be used instead of the more readily available yellow mustard seed.\n\nPrep 15 min Total 4 hr 15 min \u25cf 2 cups\n\n  * 2 cups frozen whole kernel corn or 1 can (15.25 oz) whole kernel corn, drained\n  * 2 tablespoons chopped green bell pepper\n  * 1 tablespoon finely chopped onion\n  * 1 jar (2 oz) diced pimientos, drained\n  * \u00bd cup sugar\n  * \u00bd teaspoon celery seed\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon mustard seed\n  * \u00bd cup cider or white vinegar\n  * \u00bc teaspoon red pepper sauce\n\n1 If using frozen corn, cook and drain as directed on package. In medium heatproof glass or plastic bowl, mix corn, bell pepper, onion and pimientos.\n\n2 In 1-quart saucepan, heat remaining ingredients to boiling, stirring occasionally. Boil 2 minutes. Pour over corn mixture; stir.\n\n3 Spoon into glass or plastic containers. Cover and refrigerate at least 4 hours before serving. Store covered in refrigerator up to 5 days.\n\n2 Tablespoons: Calories 45; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 40mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 7g); Protein 0g Exchanges: 2 Vegetable Carbohydrate Choices: 1\n\nPickled Peppers\n\n## Pickled Peppers Calorie Smart\n\nThere's no lengthy canning process required to pickle these peppers . . . just the microwave and refrigerator. A jar of this tangy condiment would make a great gift from the kitchen.\n\nPrep 30 min Total 1 week \u25cf 1 quart\n\n  * 1 tablespoon sugar\n  * 1 tablespoon salt\n  * 2 cups water\n  * 1 cup white vinegar\n  * 1 red bell pepper, cut into 6 pieces\n  * 1 yellow bell pepper, cut into 6 pieces\n  * 1 green bell pepper, cut into 6 pieces\n  * 1 small onion, sliced, separated into rings\n  * 2 cloves garlic, cut in half\n  * 1 sprig fresh tarragon or \u00bc teaspoon dried tarragon leaves\n\n1 In 2-quart microwavable bowl, mix sugar, salt, water and vinegar. Microwave uncovered on High 8 to 14 minutes or until mixture boils. Stir to dissolve sugar.\n\n2 In clean, hot 1-quart jar, pack all remaining ingredients. Pour hot vinegar mixture over vegetables; cover with tight-fitting lid. Refrigerate at least 1 week before serving to blend flavors. Store in refrigerator up to 3 weeks.\n\n\u00bc Cup: Calories 10; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 440mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 2g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n## Pickled Ginger\n\nMaking your own pickled ginger is easy and it might just become your new favorite condiment. It's typically served with sushi but is also good in stir-fried greens, on deviled eggs or on sandwiches. It can be stored in the refrigerator for up to 3 months.\n\n  1. In a small saucepan, heat \u00bc cup sugar, 1 cup chicken broth, \u00bd cup seasoned or unseasoned rice vinegar and 2 tablespoons dry sherry or mirin (rice wine) to boiling, stirring frequently until sugar is dissolved. Stir in 4 ounces peeled, very thinly sliced gingerroot. Cool 15 minutes.\n  2. Pour into a nonreactive container. Cover and refrigerate at least 24 hours before serving.\n\n# Heirloom Recipe and New Twist\n\nIt may be strange to call pickles fresh, but try our Easy Refrigerator Pickles, and you'll quickly see why homemade pickles are so special. Our twist recipe is geared for those who love foods with kicked-up flavor\u2014but we didn't stop with pickles. While we were at it, we pickled a variety of veggies just for fun.\n\nHeirloom\n\nEasy Refrigerator Pickles\n\n## Easy Refrigerator Pickles Calorie Smart\n\nWhen choosing cucumbers for pickles, pick out the smaller ones, and pickle them soon after buying or harvesting.\n\nPrep 10 min Total 24 hr 10 min \u25cf 6 cups\n\n  * 6 cups thinly sliced unpeeled unwaxed cucumbers\n  * 2 small onions, sliced\n  * 1 medium carrot, thinly sliced (\u00bd cup)\n  * 1\u00be cups sugar\n  * 2 tablespoons salt\n  * 1 tablespoon chopped fresh or 1 teaspoon dried dill weed\n  * 1 cup white or cider vinegar\n\n1 In 2\u00bd- or 3-quart glass container, layer cucumbers, onions and carrot.\n\n2 In medium bowl, stir remaining ingredients until sugar is dissolved; pour over vegetables. Cover and refrigerate at least 24 hours before serving. Store covered in refrigerator up to 2 weeks.\n\n\u00bc Cup: Calories 70; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 590mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 16g); Protein 0g Exchanges: 1 Other Carbohydrate Carbohydrate Choices: 1\nNew Twist\n\nSpicy Pickled Vegetables\n\n## Spicy Pickled Vegetables Calorie Smart\n\nPrep 15 min Total 24 hr 15 min \u25cf 6 cups\n\n  * 3 cups white wine vinegar\n  * 1\u00bd cups sugar\n  * 2 cloves garlic, finely chopped\n  * 1 medium head cauliflower (1\u00bd lb), separated into small florets (6 cups)\n  * \u00bd lb fresh green beans, trimmed\n  * 2 medium carrots, cut into 3x\u00bcx\u00bc-inch pieces (1\u00bd cups)\n  * 2 jalape\u00f1o chiles, cut lengthwise into quarters, seeded\n\n1 In 1-quart saucepan, heat vinegar, sugar and garlic over medium heat just until mixture begins to simmer and sugar is dissolved. Remove from heat; cool 5 minutes.\n\n2 Meanwhile, in 2 (1-quart) jars or glass containers, layer cauliflower, beans, carrots and chiles, dividing evenly between jars. Pour vinegar mixture over vegetables. Cover and refrigerate at least 24 hours before serving but no longer than 2 weeks.\n\n\u00bc Cup: Calories 70; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 15mg; Total Carbohydrate 16g (Dietary Fiber 1g, Sugars 14g); Protein 1g Exchanges: 1 Other Carbohydrate, \u00bd Vegetable Carbohydrate Choices: 1\n\n## Garlic Dill Pickles Calorie Smart\n\nPrep 1 hr 20 min Total 1 week 2 hr \u25cf 4 quarts\n\n  * 4 (1-quart) canning jars with lids\n  * 7 pints unwaxed pickling cucumbers, 24 to 30 (3-inch) cucumbers\n  * 12 to 16 sprigs fresh dill weed\n  * 12 cloves garlic, peeled\n  * 4 slices (\u00bd inch thick) onion\n  * 4 teaspoons mustard seed\n  * 4 cups water\n  * 2 cups cider or white vinegar\n  * 2 tablespoons pickling or canning salt*\n\n1 Fill boiling water canner half full with water; heat until water is simmering. Wash jars as directed in Preparing Jars for Canning (page 431).\n\n2 Wash cucumbers; cut off stems and blossom ends. Pack cucumbers loosely in jars. To each jar, add 3 or 4 dill weed sprigs, 3 garlic cloves, 1 onion slice and 1 teaspoon mustard seed. In large saucepan, heat water, vinegar and salt to boiling. Pour into jars, leaving \u00bd inch headspace.\n\n3 Drain flat lids well. Wipe rims and threads of jars. Place flat lids on jars and screw bands on tightly to seal. Place jars on elevated rack in canner; lower rack into canner. To process pickles safely, water must cover jars by 1 to 2 inches; add additional boiling water if necessary. Cover. Heat to boiling; continue boiling 10 minutes. Carefully remove jars with canning jar lifter or heavy-duty tongs and place upright on towel. Cool completely.\n\n4 After 24 hours, check seal on cooled jars by pressing middles of lids with finger. If the lids spring back, jars are not sealed and must be stored in refrigerator. Store completely sealed, unopened jars in a cool, dry place up to 1 year (let stand at least 1 week to blend flavors before serving). Once opened, store in refrigerator up to 3 weeks.\n\n* Pickling or canning salt, available in most grocery stores, is preferred for making pickles. Table salt can cause the brine to become cloudy and darken the pickles.\n\n1 Pickle: Calories 5; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Packing Cucumbers\n\nWash cucumbers; cut off stems and blossom ends.\n\nPack cucumbers loosely in jars. Add dill, garlic, onion and mustard seed to jars.\n\n## Why Use Unwaxed Cucumbers?\n\nIt's important to use unwaxed cucumbers because the brine can't get through the wax to pickle the flesh. Waxing cucumbers preserves them for transportation to the grocery store. Unwaxed cucumbers can be found at farmers' markets and roadside stands.\n\n## Pickled Tarragon Baby Carrots Calorie Smart \u2022 Easy\n\nPrep 10 min Total 24 hr 10 min \u25cf 3\u00bd cups\n\n  * 1 bag (1 lb) ready-to-eat baby-cut carrots\n  * \u00bd cup tarragon vinegar\n  * 1 tablespoon chopped fresh or 1 teaspoon dried tarragon leaves\n  * 1 tablespoon olive or vegetable oil\n  * \u00bc teaspoon coarsely ground black pepper\n\n1 In 3-quart saucepan, heat 2 quarts water to boiling. Add carrots; cook 3 minutes. Meanwhile, in 1- to 1\u00bd-quart heatproof glass or plastic container, mix remaining ingredients. Drain carrots; immediately stir into mixture in container.\n\n2 Cover and refrigerate 24 hours to blend flavors, stirring once, before serving. Store covered in refrigerator up to 3 months.\n\n\u00bc Cup: Calories 25; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 25mg; Total Carbohydrate 3g (Dietary Fiber 1g, Sugars 2g); Protein 0g Exchanges: \u00bd Vegetable Carbohydrate Choices: 0\n\nSpicy Pickled Carrots Cook 1 sliced jalape\u00f1o chile with the carrots. Substitute distilled white vinegar for the tarragon vinegar and cilantro for the fresh tarragon.\n\nPickled Green Beans Substitute 1 pound trimmed fresh green beans for the carrots. Add 1 clove finely chopped garlic with the beans. Substitute white wine vinegar for the tarragon vinegar and thyme for the fresh tarragon.\n\nPickled Beets\n\n## Pickled Beets Calorie Smart\n\nClassic pickled beets got a bit of an update with less sugar in the brine. They are still sweetly delicious. Try them on a salad with toasted walnuts and crumbles of a creamy Gorgonzola or blue cheese.\n\nPrep 1 hr 10 min Total 25 hr 10 min \u25cf 4 pints (8 cups)\n\n  * 3 lb small to medium beets (1\u00bd to 2\u00bd inch)\n  * 1\u00bc cups cider vinegar\n  * \u00be cup water\n  * \u00be cup sugar\n  * \u00bd teaspoon whole cloves\n  * \u00bd teaspoon salt\n  * 2 (3-inch) cinnamon sticks\n\n1 Cut off all but 2 inches of beet tops. Wash beets; leave whole with root ends attached. Place in 4-quart saucepan. Add water to cover. Heat to boiling; boil uncovered, 20 to 30 minutes or just until beets are tender. Check smaller beets for doneness first. Let beets cool until easy to handle, about 30 minutes. Peel beets and cut off root ends. Cut beets into wedges, 1\u00bd-inch chunks or \u00bc-inch slices.\n\n2 In 3-quart stainless steel saucepan, heat vinegar, water, sugar, cloves, salt and cinnamon sticks to boiling. Reduce heat; simmer 5 minutes. Add beets; heat to boiling. Remove from heat; discard cinnamon sticks.\n\n3 Spoon hot beets and liquid into 4 (1-pint) glass preserving jars, leaving \u00bd inch headspace. Place flat lids on jars and screw bands on tightly to seal. Let stand at room temperature to cool completely. Refrigerate at least 24 hours or up to 3 months. Flavor will intensify the longer they are refrigerated. Or, to process for canning, see Garlic Dill Pickles (page 441).\n\n\u00bc Cup: Calories 25; Total Fat 0g(Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 5g (Dietary Fiber 1g, Sugars 4g); Protein 0g Exchanges: \u00bd Vegetable Carbohydrate Choices: \u00bd\n\nJalape\u00f1o-Garlic Pickled Green Beans\n\n## Jalape\u00f1o-Garlic Pickled Green Beans Calorie Smart\n\nPrep 30 min Total 24 hr 30 min \u25cf 4 pints (8 cups)\n\n  * 2 lb fresh green or yellow wax beans, trimmed\n  * 2\u00be cups white vinegar\n  * 1\u00bd cups water\n  * \u00bc cup sugar\n  * 1 teaspoon canning and pickling salt\n  * 12 slices (\u00bcinch) red or green jalape\u00f1o chiles\n  * 8 cloves garlic, peeled\n\n1 Cut beans into 4-inch lengths to fit into jars. In 4-quart stainless steel saucepan, heat vinegar, water, sugar and salt to boiling. Add beans, pushing them into liquid; heat to boiling. Boil 1 minute. Place strainer over large glass bowl. Drain beans in strainer over bowl, reserving liquid.\n\n2 Pack beans upright into 4 (1-pint) glass preserving jars. To each jar, add 3 slices jalape\u00f1o and 2 cloves garlic. Pour hot liquid into jars, leaving \u00bd inch headspace. Make sure all beans are submerged in liquid. Place flat lids on jars and screw bands on tightly to seal. Let stand at room temperature to cool completely. Refrigerate at least 24 hours up to 3 months. Flavor will intensify the longer the beans are refrigerated. Or, to process for canning, see Garlic Dill Pickles (page 441).\n\n\u00bc Cup: Calories 15; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 30mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nLemon-Dill Pickled Green Beans Make as directed, except omit jalape\u00f1o and garlic. To each jar, add 2 (\u215b-inch) strips lemon peel and 4 sprigs fresh dill weed. Continue as directed.\n\n## Kimchi Calorie Smart\n\nA daikon radish is a large, long, Asian radish with a fresh, sweet flavor and moist, crunchy texture. The flesh is white; the skin is white or black. Look for firm, unwrinkled radishes.\n\nPrep 20 min Total 5 days \u25cf 3 cups\n\n  * 1 large head Chinese (napa) cabbage (2 lb)\n  * \u00bc cup non-iodized coarse sea salt or kosher salt*\n  * 1 tablespoon red pepper flakes\n  * 1 teaspoon finely chopped gingerroot\n  * 1 teaspoon sugar\n  * 4 cloves garlic, finely chopped\n  * 2 tablespoons fish sauce or water\n  * \u00bd lb daikon radish, peeled, cut into 2x\u00bc-inch strips (2\u00bc cups)\n  * 2 medium carrots, peeled, cut into 2x\u00bc-inch strips (1 cup)\n  * 4 medium green onions, cut into \u00bc-inch slices\n\n1 Cut cabbage lengthwise into quarters; remove core pieces. Cut quarters crosswise into 1-inch strips. Place cabbage in large bowl. Sprinkle with salt. Gently rub salt into cabbage until cabbage starts to soften just a little. Cover cabbage with water. Place plate on top and weight with something heavy so it doesn't float. Let stand 1 hour.\n\n2 Place the cabbage in large colander to drain. Rinse very thoroughly for several minutes, turning the cabbage over so you are rinsing all of the cabbage leaves to remove excess salt. Drain. Rinse out and dry the bowl used for the soaking the cabbage. Place cabbage in bowl; set aside.\n\n3 In small bowl, mix pepper flakes, gingerroot, sugar, garlic and fish sauce. Stir until sugar is dissolved. Working in batches, lift cabbage above bowl and squeeze cabbage over the bowl to catch the juices. Return cabbage to bowl with juices. Add radish, carrots and onions. Add pepper flake mixture; toss to coat, making sure mixture is thoroughly coated (gloved hands may work more easily).\n\n4 Spoon into glass or plastic container with tight-fitting lid, leaving 1 inch headspace. Press the kimchi into the container with the back of a spoon, allowing the brine to rise to the top covering the vegetables. Cover. Let stand at room temperature 3 to 5 days to ferment and develop flavor. You may see bubbles develop inside the container. Check the kimchi every day to make sure the vegetables are submerged in the brine. If not, push the vegetable mixture down with the back of a spoon.\n\n3 months.\n\n* Iodized salt can interfere with fermentation; non-iodized salt is readily available.\n\n1 Tablespoon: Calories 5; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Making Kimchi\n\nPlace a plate with something heavy on top to keep cabbage submerged in water.\n\nUse a spoon to press kimchi ingredients into container with tight-fitting lid, leaving 1 inch of headspace.\n\n## Fermenting Vegetables\n\nUntil a few generations ago, fermenting was a common practice for food preservation. Now both home cooks and small restaurateurs are getting interested in fermenting their own vegetables. Why? Fermented vegetables take on new flavors, adding interest\u2014they are usually tarter in flavor than their raw predecessors. They also are easier to digest than raw and may be helpful for overall digestive health.\n\nJust about any raw vegetable can be safely fermented at home, if done properly. They can actually be safer to eat than raw vegetables because the fermentation process hunts down and kills any harmful bacteria that may be present in the raw vegetables.\n\nWhile there's no veggie you can't ferment, green tomatoes, string beans, cucumbers, cabbage, daikon radishes, parsnips and okra are the most flavorful choices for fermenting. Some vegetables, due to their biological makeup, end up with an undesirable flavor when fermented. Kale, for example, has a high chlorophyll content that gives the fermented version a flavor most people won't enjoy.\n\n## Greek Yogurt Calorie Smart \u2022 Easy\n\nSometimes called yogurt cheese, Greek yogurt is concentrated yogurt with a silky, smooth texture. Do not start with yogurt that contains gelatin. The gelatin prevents the liquid from draining from the yogurt.\n\nPrep 5 min Total 7 hr 5 min \u25cf 8 servings\n\n  * 2 cups regular or fat-free plain yogurt (1 lb)\n\n1 Line medium colander with cheesecloth or muslin. Place colander in bowl. Spoon yogurt into lined colander; drain 1 hour.\n\n2 Loosely cover colander and bowl. Refrigerate 6 to 8 hours or until texture of yogurt is smooth and silky. Transfer yogurt from cheesecloth to medium bowl; set aside. Discard liquid in bowl. Store covered in refrigerator 2 to 3 days.\n\n1 Serving (\u00bc Cup): Calories 35; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 3g); Protein 3g Exchanges: \u00bd Skim Milk Carbohydrate Choices: \u00bd\n\nCardamom-Scented Yogurt Cheese Make yogurt as directed in recipe. Stir \u00bc teaspoon saffron threads into 1 tablespoon hot milk. Let stand 1 to 2 minutes or until milk is a rich, golden yellow-orange color. Stir milk mixture, \u00bc cup chopped unsalted pistachios, \u215b teaspoon ground nutmeg and \u215b teaspoon ground cardamom into yogurt. Cover and refrigerate about 2 hours or until chilled.\n\nBacon-Ranch Greek Yogurt In a medium bowl, stir together 1 cup Greek Yogurt, \u2153 cup shredded cheddar cheese, 2 tablespoon crumbled cooked bacon, 2 tablespoon chopped green onion (with tops) and 2 tablespoons ranch dressing (to make your own, see page 138). Cover and refrigerate 2 hours to blend flavors.\n\n### Straining Yogurt\n\nSpoon yogurt into cheesecloth-lined colander.\n\nDrain yogurt until very thick.\n\nBlueberry, Strawberry and Plain Kefir\n\n## Kefir Calorie Smart\n\nPlain, unflavored kefir tastes like a cross between buttermilk and sour cream, with the benefit of containing probiotics (just like yogurt) and strains of beneficial yeast. Our recipe uses the convenience of shelf-stable freeze-dried starter, but there are mail-order sources for perishable kefir grains.\n\nPrep 10 min Total 25 hr 10 min \u25cf 4 servings\n\n  * 4 cups whole milk*\n  * 1 packet (5 grams) freeze-dried kefir starter (from 1 ounce box)\n\n1 In 3-quart saucepan, heat milk just to boiling, stirring frequently to avoid scorching. Pour into large bowl. Cool milk at room temperature until it reaches 73\u00b0F to 77\u00b0F, 45 to 60 minutes.\n\n2 Remove 1 cup of cooled milk into small bowl; add kefir starter packet. Whisk to mix thoroughly. Add milk with kefir starter back into milk in large bowl; whisk to blend thoroughly. Cover with plastic wrap and let stand at room temperature for 24 hours or until kefir has thickened.\n\n3 Cover and refrigerate 8 hours to stop fermentation process. Stir before each use to liquefy. Store covered in refrigerator up to 2 weeks.\n\n* For food safety, make sure to use pasteurized milk for this recipe. Milk alternatives, such as almond, cashew, coconut, hemp, rice and soy, may not work with kefir starter products. Please check product packages and websites for information.\n\n1 Cup: Calories 150; Total Fat 8g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 125mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 13g); Protein 7g Exchanges: 1 Milk Carbohydrate Choices: 1\n\nBlueberry Kefir In blender container, place \u00bd cup fresh blueberries, 1 cup kefir and 1 to 2 tablespoons honey or agave nectar. Cover and blend on high speed about 30 seconds or until smooth.\n\nMango Kefir In blender container, place 1 cup fresh cubed ripe mango, 1 cup kefir and 1 to 2 tablespoons agave nectar or honey. Cover and blend on high speed about 30 seconds or until smooth.\n\nStrawberry-Banana Kefir In blender container, place \u00bd cup sliced fresh ripe strawberries, \u00bd sliced very ripe banana and 2 to 3 tablespoons agave nectar or honey. Cover and blend on high speed about 30 seconds or until smooth.\n\nStrawberry Kefir In blender container, place 1 cup sliced fresh ripe strawberries, 1 cup kefir and 4 to 5 tablespoons agave nectar or honey. Cover and blend on high speed about 30 seconds or until smooth.\n\n## Harissa Calorie Smart\n\nHarissa is a hot Tunisian chile paste flavored with dried chiles, garlic, coriander, cumin and a bit of caraway. It has a few steps, but it's easy to make. Stir into sauces, hummus, dips or melted butter to add zing, spread on meats before grilling or try it with feta cheese and pita chips.\n\nPrep 35 min Total 1 hr 45 min \u25cf 1\u00be cups\n\n  * 1 large red bell pepper\n  * \u00bd teaspoon olive oil\n  * 2 oz dried ancho or pasilla chiles*\n  * 2 oz dried guajillo or New Mexican chiles\n  * 1\u00bc teaspoons coriander seed\n  * 1\u00bc teaspoons cumin seed\n  * \u00bd teaspoon caraway seed\n  * 2 tablespoons olive oil\n  * 2 cloves garlic, peeled\n  * 1\u00bd teaspoons salt\n  * 1 teaspoon dried mint\n\n1 Heat oven to 400\u00b0F. Line pie plate with foil. Place bell pepper in plate; brush with \u00bd teaspoon oil. Roast 30 to 40 minutes, turning every 10 minutes, until pepper is charred and golden brown in spots and slightly collapsed.\n\n2 Remove from oven. Wrap pepper in foil; let stand 15 minutes to steam and allow skin to loosen. Unwrap; remove stem. Cut pepper open; place, skin side down, on cutting board. Scrape off seeds with spoon. Turn pepper over; peel off skin, scraping off with edge of knife if necessary. Place roasted pepper in food processor.\n\n3 Meanwhile, place dried chiles in large bowl. Cover with boiling water. Place plate or bowl over chiles, weighted if necessary to keep chiles submerged. Let stand 30 minutes to soften completely. Drain. Remove stems. Cut peppers open; place, skin side down, on cutting board. Scrape off seeds with spoon. Add chiles to food processor.\n\n4 Heat 8-inch skillet over medium-high heat. Add coriander, cumin and caraway seed; toast 2 to 3 minutes, stirring constantly, until coriander seed turns golden brown and mixture becomes fragrant.** Place toasted spices in spice grinder. Grind until mixture looks like finely ground pepper. Add to food processor; add 2 tablespoons oil, the garlic, salt and mint. Cover and process until smooth, scraping side of bowl if necessary. Store covered in refrigerator up to 1 week or in the freezer up to 3 months.\n\n*If you can't find the dried chiles in your area, check online sources.\n\n**We found that toasting the spices adds depth of flavor and more complexity to the sauce, but this step can be left out if desired.\n\n1 Teaspoon: Calories 10; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Making Harissa\n\nCover dried chiles with boiling water. Place a plate or bowl over chiles to keep chiles submerged.\n\nPlace toasted spices in spice grinder. Grind until mixture looks like finely ground pepper.\n\n## Ketchup Calorie Smart\n\nChoose fleshy, ripe tomatoes for best flavor and consistency. Beefsteak varieties are an excellent choice for this iconic condiment.\n\nPrep 40 min Total 2 hr 10 min \u25cf 5\u00bc cups\n\n  * 2 tablespoons vegetable oil\n  * 1 large onion, chopped (1 cup)\n  * 3 stalks celery, chopped (1 cup)\n  * 8 lb ripe tomatoes, coarsely chopped\n  * 1 cup cider vinegar\n  * 4 teaspoons salt\n  * 2 teaspoons pepper\n  * 1 teaspoon chili powder\n  * \u00bd teaspoon ground coriander\n  * \u00bd teaspoon ground cloves\n  * \u00bd cup sugar\n\n1 In 8-quart Dutch oven or stockpot, heat oil over medium heat. Cook onion and celery in oil until tender, stirring occasionally. Add remaining ingredients except sugar. Heat to boiling; reduce heat. Cook uncovered until tomatoes are completely softened, stirring occasionally, about 20 minutes.\n\n2 Working in batches, place tomato mixture in blender or food processor. Cover and blend until smooth. Strain through fine-mesh strainer, pressing firmly with back of spoon until only seeds remain. Return each pureed batch to Dutch oven.\n\n3 Add sugar to tomato mixture. Heat to boiling; reduce heat to medium-low. Cook uncovered 1 hour to 1 hour 30 minutes or until desired thickness. Cool to room temperature. Spoon into half-pint jars or plastic containers; cover with lids. Refrigerate up to 6 months or freeze up to 1 year.\n\n1 Tablespoon: Calories 20; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 115mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 2g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Straining Ketchup Mixture\n\nStrain pureed mixture in fine-mesh strainer, pressing with back of spoon until only the seeds remain.\n\n## Drying Fresh Herbs\n\nDrying herbs while in the peak of growing season is a way to save money and enjoy their flavors throughout the year. For best flavor, pick herbs in the morning, after the dew has dried. Dry the herbs until the leaves are crisp enough that they will crumble in your hands. Store dried herbs as whole leaves in small containers with tight-fitting lids in a cool, dry place. For the best flavor, crumble the leaves when you are ready to use them. Dried herbs are stronger in flavor than fresh, so use about half the amount of the fresh herb called for; if recipe calls for a tablespoon of a fresh herb, substitute 1 teaspoon dried.\n\n### Sturdy Herbs\n\nPick herbs on the stems for easy drying. Bundle the stems and secure the bottoms with a rubber band. Hang and let dry indoors for best color and flavor. Remove leaves from stems; discard stems.\n\n### Tender Herbs\n\n**Basil, lemon balm, mint, oregano and tarragon**\n\nThese herbs have a high moisture content and will mold if not dried quickly. Prepare as directed above for sturdy herbs, except put small bunches inside paper lunch bags; tie stems and bag with rubber band. Cut holes in bags to allow air to circulate. Hang in place where there are air currents, so they can dry quickly. The bags will catch any leaves or seeds that fall. In humid areas, dry mint and bay leaves only (remove stems) in a single layer, with no leaves touching each other, on paper towel\u2013lined trays. Cover with another layer of paper towel. To dry a large quantity, stack layers of leaves in this manner with paper towels in between the layers.\n\n## Freezing Fresh Herbs\n\nAnother way to preserve herbs is to freeze them while at their peak freshness to use whenever you like.\n\n### Sturdy Herbs\n\nPlace the chopped leaves of sturdy herbs (see above) in ice cube trays or mini muffin cups. Cover with water, broth or olive or vegetable oil. Cover and freeze until firm. Place frozen cubes in a resealable freezer plastic bag (pressing all the air out) to add to simmering soups, stews or sauces.\n\n### Tender Herbs\n\nFreeze leaves in a single layer on a cookie sheet and then transfer to a freezer bag, as directed above. Alternatively for basil or cilantro leaves, you could make pesto (see page 456) and freeze it in ice cube trays or mini muffin cups, as for sturdy herbs.\n\n## Crystallized Ginger Calorie Smart\n\nHomemade crystallized ginger (candied ginger) is as easy to make as boiling water and is significantly cheaper than store-bought. It's delicious in scones, cookies and cakes, to name a few uses, and is perfect for gift giving.\n\nPrep 20 min Total 25 hr 10 min \u25cf 2\u00bd cups\n\n  * 1\u00bc cups water\n  * 1\u00bc cups sugar\n  * 1 lb gingerroot, peeled, cut into \u215b-inch slices (about 2\u00bd cups)\n  * \u00bd to \u2154 cup sugar\n\n1 In 2-quart saucepan, heat water and 1\u00bc cups sugar to boiling. Stir constantly until sugar is dissolved. Add gingerroot; return to boiling. Reduce heat; simmer uncovered 25 to 30 minutes, stirring occasionally, until gingerroot is tender.\n\n2 Place waxed paper, plastic wrap or foil under cooling rack. With slotted spoon, remove gingerroot from liquid and place in single layer on cooling rack.* Let stand 20 minutes or until completely cool.\n\n3 Line 15x10x1-inch pan with paper towels. Place \u00bd cup sugar in 1-gallon resealable food-storage plastic bag. Using tongs, place ginger pieces in bag. Seal and toss until ginger is thoroughly coated with sugar, adding remaining sugar if needed. Arrange ginger pieces in single layer in pan. Let stand uncovered at room temperature 24 hours to dry. Store in tightly covered container at room temperature up to 2 weeks, or refrigerate up to 3 months or freeze up to 6 months.\n\n* After removing the gingerroot, you can continue to cook the liquid until it thickens and becomes syrupy. Mix the syrup with chilled club soda or seltzer for homemade ginger ale, add to cold or hot tea or drizzle over ice cream.\n\n1 Teaspoon: Calories 10; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Making Crystallized Ginger\n\nPlace sugar mixture\u2013soaked ginger slices on cooling rack until cool.\n\nToss cooled ginger with sugar in sealed bag until ginger is thoroughly coated.\n\n## Oven-Dried Tomatoes Calorie Smart \u2022 Easy\n\nPlum tomatoes work best for drying because they are thick-fleshed with few seeds. Regular tomatoes can be used but may take longer to dry.\n\nPrep 10 min Total 12 hr 10 min \u25cf 12 servings\n\n  * 24 plum (Roma) tomatoes\n\n1 Heat oven to 200\u00b0F. Rinse tomatoes under cold running water and remove stems; drain on paper towels. Cut each tomato lengthwise in half. Using small spoon, scoop out and discard seeds.\n\n2 Place tomato halves, cut sides down, in single layer on broiler rack. Place larger halves on outer edges of rack for even drying.\n\n3 Dry in oven 6 to 12 hours, turning once after 4 hours, or until tomatoes are shriveled, dry to the touch and very chewy in texture. Edges may be crisp, but tomatoes should not be crisp in center; they should be somewhat pliable. Check after 6 hours and remove any smaller tomatoes that have dried. Tomatoes will be dark red and some pieces may have brown or dark spots.\n\n4 Remove from oven and cool completely. Store in tightly covered container in cool, dark location. To rehydrate, place tomatoes in small bowl. Cover with hot or boiling water. Let stand 15 to 30 minutes or until softened.\n\n1 Serving (2 Tomatoes): Calories 20; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 5mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 3g); Protein 1g Exchanges: Free Carbohydrate Choices: 0\n\n### Drying Tomatoes\n\nCut each tomato in half lengthwise; remove seeds with small spoon.\n\nPlace tomato halves, cut side down, on broiler rack.\n\nCarrot Baby Food\n\n## Carrot Baby Food Calorie Smart \u2022 Easy\n\nMaking baby food at home is easy and puts you in charge! This recipe can easily be doubled or tripled to meet your needs.\n\nPrep 10 min Total 50 min \u25cf 4 servings\n\n  * 4 medium carrots, cut into 1-inch chunks\n\n1 In 2-quart saucepan, heat 1 inch of water to boiling. Add carrots; heat to boiling. Reduce heat to low. Simmer covered 7 to 10 minutes or until carrots are tender. Drain.\n\n2 Place carrots in food processor or blender. Cover and process until smooth, adding water as needed for desired consistency. Spoon puree into ice cube trays or freezer-safe jars. Cover and freeze. If using ice cube tray, remove frozen cubes from tray and transfer to resealable freezer plastic bag. Freeze up 3 months.\n\n3 To serve, thaw in the refrigerator several hours. Or, place cube in small microwavable bowl. Microwave uncovered on Defrost, stirring every few seconds, just until thawed and barely warm. Check temperature of baby's food before serving. Discard any uneaten portions.\n\n1 Serving: Calories 50; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 85mg; Total Carbohydrate 12g (Dietary Fiber 3g, Sugars 6g); Protein 1g Exchanges: 2 Vegetable Carbohydrate Choices: 1\n\nSweet Pea Baby Food Cook 1 bag (12 ounces) frozen sweet peas as directed on bag; cool 5 minutes. Continue as directed in Step 2.\n\nGreen Bean Baby Food Cut ends from \u00be pound fresh green beans. Cut beans into 1-inch pieces. Cook beans as directed above, except cook uncovered 6 to 8 minutes.\n\nWinter Squash Baby Food Peel 2 lb acorn squash or pumpkin; cut open and remove seeds and fibers. Cut squash into 1-inch pieces. Cook squash as directed above, except cook 20 to 25 minutes.\n\n## Homemade Baby Food\n\nMaking your own baby food is a great way to know exactly what your baby is eating. It may be more economical, too. You can serve your baby what you are eating for a meal just by cooking vegetables or fruits long enough to make them soft to puree them in your blender or food processor.\n\nHomemade baby food may be more perishable than commercially prepared food. If you won't be using your prepared food in the next day or two, freeze it (see Carrot Baby Food, left) in ice cube trays to take out portions as you need them.\n\n## Herb Vinegar Calorie Smart \u2022 Easy\n\nFlavored vinegars can add distinctive taste to your favorite recipes. Making them yourself not only gives satisfaction, but also can be less costly than buying them.\n\nPrep 5 min Total 10 days \u25cf 2 cups\n\n  * 2 cups white wine vinegar or white vinegar\n  * \u00bd cup firmly packed fresh herb leaves (basil, chives, dill weed, mint, oregano, rosemary or tarragon)\n  * 1 sprig of fresh herb used above, if desired\n\n1 In tightly covered glass jar or bottle, shake vinegar and herb leaves. Let stand in cool, dry place 10 days.\n\n2 Strain vinegar; discard herbs. Place 1 sprig of fresh herb in jar to identify flavor if desired. Store covered at room temperature up to 6 months.\n\n1 Tablespoon: Calories 0; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nBerry Vinegar Substitute 2 cups fresh berries, crushed, for the herbs.\n\nGarlic Vinegar Substitute 6 cloves garlic, peeled and cut in half, for the herbs.\n\nLemon Vinegar Substitute peel from 2 lemons for the herbs.\n\n## Betty's Staples Fresh Herbs\n\nDon't let extra herbs go to waste\u2014there are lots of creative ways to heighten the flavor of food with herbs. Start with these inspirational ideas and soon you'll be finding more creative ways to use them.\n\n  1. 1. **Orange\u2013Lemon Balm Sun Tea:** For 8 servings, place \u00bd cup loosely packed fresh lemon balm leaves and 6 orange tea bags in 2-quart glass jar or pitcher. Fill with 2 quarts water. Cover and set jar in sun 6 to 8 hours. Remove leaves and tea bags from tea. To serve, pour over glasses of ice and slices of fresh orange.\n  2. 2. **Honey-Rosemary Simple Syrup:** In 1-quart saucepan, heat \u00bd cup each honey and water with 1 sprig fresh rosemary. Heat to boiling; reduce heat. Simmer until mixture is reduced by half, about 5 minutes. Remove rosemary sprig; cool completely. Stir 1 to 2 teaspoons into iced tea or Lemonade (page 72). \n  3. 3. **Fresh Herb\u2013Buttermilk Ranch Dressing:** Make Buttermilk Ranch Dressing (page 138) as directed, except omit parsley flakes. Stir 2 tablespoons chopped fresh basil or cilantro leaves or 2 teaspoons chopped fresh dill with mayonnaise. Or stir the same amount of herbs into 1\u00bd cups purchased ranch dressing.\n  4. 4. **Pasta with Fried Sage:** For 4 servings, heat \u00bc cup butter in skillet over medium heat until butter is melted. Cook 12 fresh medium sage leaves in butter in pan, turning once until crisp, about 3 minutes. Remove to plate with slotted spoon. Toss 8 ounces hot cooked pasta, 1 cup arugula leaves and \u00bd cup cherry tomato halves in butter until warm. Arrange pasta on dinner plates; sprinkle with grated Parmesan cheese and black pepper; top with fried sage leaves.\n  5. 5. **Garlic-Herb Butter:** In small food processor, place 1 clove garlic and 1 thin red onion wedge (about 1 tablespoon). Cover and process until finely chopped. Place in small bowl with \u00bd cup softened butter. Process enough basil or cilantro leaves or combination of other herbs, such as thyme, rosemary, and sage leaves, to equal 1 tablespoon. Add to butter; mix evenly. Serve on warm rolls or corn on the cob or stir into hot mashed potatoes.\n  6. 6. **Spicy Almond-Cilantro Pesto:** Make Cilantro Pesto (page 456) as directed, except substitute slivered almonds for the pine nuts and add 1 seeded serrano chile.\n  7. 7. **Strawberry-Basil Water:** For 8 servings, fill 2-quart pitcher with water. Add \u00bd cup whole strawberries and 3 to 5 whole fresh basil leaves, pressing down under the water. Cover and refrigerate 2 hours to blend flavors before serving.\n\n## Natural Homemade Food Color\n\nMaking homemade natural food color is as easy as turning to the produce, juice and vegetable aisle in your grocery store. The colors are more muted and subtle, and it takes a bit of trial and error to get the look you want, but it's a fun experiment along the way!\n\nFresh Berry-Based (mauve, pink, purple)\n\n  * 1 cup blackberries, raspberries or strawberries\n\nFresh Vegetable-Based (green)\n\n  * 1 cup firmly packed fresh curly parsley or small spinach leaves\n  * 3 to 4 tablespoons water\n\nFresh Vegetable-Based (green, mauve, purple)\n\n  * 3 cups shredded or chopped red cabbage\n  * 4 cups water\n  * \u00bd teaspoon baking soda\n\nJuice-Based (orange, pale yellow, peach, pink, purple)\n\n  * Frozen (thawed) grape juice concentrate (use desired amount)\n  * 1 cup carrot juice\n  * 1 cup canned beet juice\n\n1 To use the coloring for frosting, start with 1 teaspoon color for each cup of Creamy Vanilla Frosting (page 579; omit vanilla for best color or substitute clear vanilla). If using canned frosting, colors will be much more muted.\n\n2 To make Fresh Berry-Based color: Place desired type of berry in food processor or blender. Cover and process to puree. Place in fine-mesh strainer over bowl, stirring and pressing through with back of spoon to remove seeds. For muted, lighter color, use uncooked puree. For more intense color, place berry puree in 1-quart saucepan. Heat to boiling. Reduce heat to medium-low. Simmer until puree has been reduced to 1 tablespoon for most intense color, stirring occasionally. Cool completely before use. Blackberries make mauve to purple; raspberries make light to deep pink; strawberries make light to medium pink.\n\n3 To make Fresh Vegetable-Based color (green only): Place parsley or spinach and water in food processor or blender. Cover and process to finely chop. Place in fine-mesh strainer over bowl and press with back of spoon to extract as much green liquid as possible from vegetable pulp. For muted, lighter color, use uncooked liquid. For more intense color, place liquid in 1-quart saucepan. Heat to boiling. Reduce heat to medium-low. Simmer until liquid has been reduced to 1 tablespoon for most intense color, stirring occasionally. Cool completely before use. Uncooked makes pale, light green; cooked makes medium sage green.\n\n4 To make Fresh Vegetable-Based color: In 3-quart saucepan, heat cabbage and water to boiling. Reduce heat; simmer 20 minutes to extract as much color as possible. Remove from heat. Place strainer over large bowl. Drain cabbage in strainer over bowl, reserving liquid. Divide liquid in half in two 2-quart saucepans for two colors; one purple and one that will become sage green. To create the sage green, add the baking soda to the liquid in one of the saucepans. Heat to boiling. Reduce heat to medium-low. Simmer until liquid has been reduced to 1 tablespoon for most intense color, stirring occasionally. Cool completely before use. The two-way cooked cabbage coloring method gives you mauve and purple shades without baking soda and sage green shades with the addition of baking soda.\n\n5 To make Juice-Based color: Use thawed frozen grape juice concentrate as is for mauve to purple colors; it does not need to be cooked and reduced for intense color. For muted, light color, use uncooked beet and carrot juice. For more intense color, place liquid in 1-quart saucepan. Heat to boiling. Reduce heat to medium-low. Simmer until liquid has been reduced to 1 tablespoon for most intense color, stirring occasionally. Cool completely before use. Uncooked makes pale, light yellow to peach and pink; cooked makes orange to deep orange, deep pink, purple.\n\n## Use It or Lose It\n\nIt's best to make and use your natural food colors as soon as you make them or within a day or two, since they don't have any preservatives to keep them longer. If you won't be using them immediately, cover and refrigerate up to 2 days.\n\n## Salt-Free Herb Mix Calorie Smart \u2022 Fast\n\nThis super-versatile salt-free seasoning is great to sprinkle on meats, poultry, fish and vegetables. Or, toss hot cooked pasta with olive oil or melted butter and some of this blend, then sprinkle with shredded cheese.\n\nPrep 5 min Total 5 min \u25cf \u00bc cup\n\n  * 1 tablespoon onion powder\n  * 1 tablespoon garlic powder\n  * 1 tablespoon parsley flakes\n  * 1 teaspoon dried basil leaves\n  * 1 teaspoon dried thyme leaves\n  * 1 teaspoon dried marjoram leaves\n  * 1 teaspoon coarsely ground black pepper\n\nIn small container with tight-fitting lid, mix all ingredients until well blended; cover. Store in cool, dry place up to 6 months. Stir or shake well before each use.\n\n1 Teaspoon: Calories 5; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n## Salt-Free Taco Seasoning Mix Calorie Smart \u2022 Fast\n\nUse this homemade taco seasoning in any recipe calling for the store-bought version. The beauty of this mix is you can add your own salt to taste. Twelve teaspoons of this mix is equal to a 1-ounce package of purchased taco seasoning mix.\n\nPrep 5 min Total 5 min \u25cf \u00bd cup\n\n  * 2 tablespoons chili powder\n  * 1 tablespoon onion powder\n  * 5 teaspoons ground cumin\n  * 5 teaspoons paprika\n  * 2\u00bd teaspoons garlic powder\n  * \u215b to \u00bc teaspoon ground red pepper (cayenne)\n\nIn small container with tight-fitting lid, mix all ingredients until well blended; cover. Store in cool, dry place up to 6 months. Stir or shake well before each use.\n\n1 Teaspoon: Calories 10; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nSmoky Salt-Free Taco Seasoning Mix Substitute 2 teaspoons smoked paprika for 2 teaspoons of the regular paprika.\n\nConfetti Cookie Mix\n\n## Confetti Cookie Mix Calorie Smart\n\nPrep 50 min Total 50 min \u25cf 1 jar mix (3 dozen cookies)\n\nDry Cookie Mix\n\n  * 1 cup sugar\n  * \u00be cup all-purpose flour\n  * \u2153 cup unsweetened baking cocoa\n  * \u00bd teaspoon baking soda\n  * \u00bc teaspoon salt\n  * 1\u00bd cups quick-cooking or old-fashioned oats\n  * 1 cup miniature candy-coated milk chocolate candies\n\nTo Complete Cookies\n\n  * \u00bd cup butter, softened\n  * 2 tablespoons water\n  * \u00bd teaspoon vanilla\n  * 3 eggs\n\n1 In medium bowl, mix sugar, flour, cocoa, baking soda and salt. Transfer mixture to 1-quart jar with tight-fitting lid; tap lightly to pack. Top with oats and candies; cover.\n\n2 To make cookies, heat oven to 350\u00b0F. In large bowl, place contents of jar, butter, water, vanilla and eggs. Stir with spoon 30 seconds or until combined. On ungreased cookie sheets, spoon dough by rounded teaspoonfuls 2 inches apart. Bake 10 to 12 minutes or until edges are set. Cool 5 minutes; remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 100; Total Fat 4.5g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 60mg; Total Carbohydrate 14g (Dietary Fiber 1g, Sugars 9g); Protein 1g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\n\nChai Mix\n\n## Chai Mix Calorie Smart \u2022 Fast\n\n_Chai_ is the Hindi name for tea made with milk and spices such as cardamom, cinnamon, cloves, ginger, nutmeg and pepper. Give a jar of this chai mix with a lovely mug and cinnamon sticks for stirring, and don't forget to include the preparation instructions.\n\nPrep 10 min Total 10 min \u25cf 4 servings\n\n  * 1 cup instant nonfat dry milk\n  * \u00bd cup dry vanilla-flavored nondairy creamer\n  * \u00bc cup plus 2 tablespoons powdered sugar\n  * \u00bd cup dry unsweetened instant tea\n  * 2 teaspoons ground cinnamon\n  * 1 teaspoon ground cardamom\n  * \u00bd teaspoon ground cloves\n\n1 In small bowl, mix dry milk, creamer and powdered sugar. In another small bowl, mix dry tea and spices. Into 1-pint jar with tight fitting-lid, alternately spoon milk and tea mixtures, packing lightly; cover.\n\n2 For each serving, place \u00bc cup mix in cup or mug. Fill with 1 cup very hot water. For creamier chai, use half milk and half water.\n\n1 Serving: Calories 110; Total Fat 3.5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 15g); Protein 4g Exchanges: 1 Other Carbohydrate, \u00bd Skim Milk, \u00bd Fat Carbohydrate Choices: 1\n\n## Brownie Mix\n\nAn impressive layered brownie mix makes the perfect gift from your kitchen to someone you love. Include a card with instructions to complete the brownies.\n\nPrep 15 min Total 1 hr 40 min \u25cf 1 jar mix (16 brownies)\n\nDry Brownie Mix\n\n  * 1 cup semisweet or dark chocolate chips\n  * \u00bd cup plus 2 tablespoons all-purpose flour\n  * \u00bd teaspoon salt\n  * \u00bd cup unsweetened baking cocoa\n  * \u00bd cup all-purpose flour*\n  * \u2154 cup granulated sugar\n  * \u2154 cup packed brown sugar\n  * \u00bd cup white vanilla baking chips\n  * \u00bd cup chopped pecans or walnuts, if desired\n\nTo Complete Brownies\n\n  * \u2154 cup butter, melted\n  * 3 eggs, beaten\n  * 1 teaspoon vanilla\n\n1 In 1-quart jar with tight-fitting lid, layer all brownie mix ingredients in order given; cover.\n\n2 To make brownies, heat oven to 350\u00b0F. Grease 8- or 9-inch square pan with shortening or cooking spray. In large bowl, place contents of jar, butter, eggs and vanilla. (If brown sugar has become slightly firm, break up with fork.) Using spoon, beat about 30 strokes or until well blended. Spoon and spread in pan. Bake 35 to 40 minutes or until set. Cool completely, about 45 minutes. Cut into 4 rows by 4 rows.\n\n* Two separate measurements of flour are listed in order to create an extra layer.\n\n1 Brownie: Calories 290; Total Fat 14g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 60mg; Sodium 160mg; Total Carbohydrate 37g (Dietary Fiber 2g, Sugars 27g); Protein 3g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n## Giftable Jars\n\nFabric-topped jars make cute gifts. To make, use pinking shears to cut squares or circles of fabric that are 2 inches wider than the diameter of your jar lids. Place the fabric on top of the lids before screwing on the bands. For a quilted appearance, cut circles of quilt batting the size of the lids and place on tops of lids before adding the fabric.\n\n# Chapter 17: Sauces, Seasonings & Marinades\n\n## Sauce Basics\n\nFixing Sauces\n\nThickening Sauces\n\nTips for Perfect Sauces\n\n## Features\n\nGlobal Flavors: Asia\n\nHeirloom Recipe and New Twist\n\nLearn to Deglaze\n\nLearn with Betty: Hollandaise Sauce\n\nMake-Ahead: Mayonnaise\n\nMarinade Success\n\nWine\n\n## Sauces\n\nBasil Pesto Calorie Smart \u2022 Fast \u2022 Easy\n\nBasil, Roasted Red Pepper and Walnut Pesto\n\nB\u00e9arnaise Sauce\n\nBeet-Dill Pesto\n\nBlack Bean Salsa\n\nBroccoli-Walnut Pesto Calorie Smart \u2022 Fast\n\nBrussels Sprouts\u2013Cashew Pesto\n\nCheese Sauce\n\nChimichurri Sauce Calorie Smart \u2022 Easy\n\nCilantro Pesto\n\nCurried Mayonnaise\n\nCurry Sauce\n\nCurry-Chutney Tartar Sauce\n\nDill Sauce\n\nDill Tartar Sauce\n\nFresh Peach Salsa Calorie Smart\n\nGarlic Mayonnaise\n\nGravy\n\nHerbed Mayonnaise\n\nHollandaise Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nMayonnaise Calorie Smart \u2022 Fast \u2022 Easy\n\nMole Poblano Calorie Smart\n\nMornay Sauce Calorie Smart \u2022 Fast\n\nMustard Sauce\n\nPeanut Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nPeppery Red Wine Sauce\n\nPort Cranberry Sauce\n\nRoasted Carrot\u2013Almond Pesto\n\nRoux\n\nSalsa Calorie Smart\n\nSalsa Verde Calorie Smart \u2022 Fast\n\nSpinach Pesto\n\nSriracha Dipping Sauce for Fries\n\nSriracha Sauce Calorie Smart\n\nSun-Dried Tomato Pesto\n\nTartar Sauce Calorie Smart \u2022 Easy\n\nThai Red Curry Paste Calorie Smart\n\nThick White Sauce\n\nThin White Sauce\n\nVelout\u00e9 Sauce\n\nWhite Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nWhite Wine Sauce\n\nWhite Wine\u2013Garlic Sauce\n\n## Flavored Butters\n\nAlmond Butter\n\nClarified Butter Fast\n\nFlavored Butters Fast \u2022 Easy\n\nGarlic Butter\n\nGarlic Clarified Butter\n\nHerb Butter\n\nHerbed Browned Butter Sauce\n\nHerbed Butter Calorie Smart \u2022 Fast \u2022 Easy\n\nHoney or Maple Butter\n\nItalian Parmesan Butter\n\nLemon-Pepper Butter\n\nOrange Butter\n\nPecan Butter\n\nRaspberry Butter\n\n## Cream Cheese Spreads\n\nBacon-Chive-Cheddar Cream Cheese Spread\n\nCream Cheese Spreads Calorie Smart \u2022 Fast \u2022 Easy\n\nHoney-Walnut Cream Cheese Spread\n\nMaple-Cinnamon Cream Cheese Spread\n\nOlive-Walnut Cream Cheese Spread\n\nPeanut Butter\u2013Honey Cream Cheese Spread\n\nSeasonings\n\n## Seasoning Basics\n\nWhen to Season\n\nHow Much to Add\n\nHerbs\n\nSpices\/Seasonings\n\n## Marinades\n\nBalsamic Garlic Marinade\n\nFajita Marinade Calorie Smart\n\nGarlic Marinade Calorie Smart\n\nLemon-Herb Marinade Calorie Smart\n\nMaple-Bourbon Marinade\n\nOnion and Garlic Marinade\n\nOrange-Thyme Marinade Calorie Smart \u2022 Easy\n\nSeasoned Brine Calorie Smart \u2022 Easy\n\nTeriyaki Marinade Calorie Smart \u2022 Easy\n\n## Bonus Recipes\n\nMoist Garlic Turkey Burgers\n\nMulled Wine\n\nStew\n\nWine Cubes\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Sauce Basics\n\nWhether a plain white sauce or the flavor punch of a mole poblano, sauces can transform a meal from ordinary to extraordinary. Learn how easy it is to masterfully create delicious sauces, worthy of putting you in the \"great cook\" category!\n\n## Thickening Sauces\n\nSauces can be thickened in different ways, depending on what they are used for and the ingredients used. Here are the four classic thickening methods:\n\nRoux: A mixture of flour and fat (usually butter) is cooked over low to medium heat until smooth and bubbly, cooking the flour to remove the floury taste before liquid is added. Some recipes call for cooking the roux until it turns golden to deep brown, which adds color and flavor. See White Sauce and Roux, page 459.\n\nCornstarch: Often stirred into cold water or other liquid to thoroughly blend before being added to a hot mixture. Sauces thickened with cornstarch become clear and almost shiny.\n\nEmulsion: An egg-based sauce cooked over low heat. If the temperature is too hot, the mixture will overcook and curdle. These sauces must be cooked thoroughly to kill any potential salmonella bacteria. See Hollandaise Sauce, page 461.\n\nReduction: A liquid such as broth, wine or a sauce mixture is boiled or simmered to evaporate some of the water in it until the volume is reduced and the mixture thickens, which concentrates the flavor. To speed this process, use a skillet (which has a large surface area) instead of a saucepan. See Peppery Red Wine Sauce, page 466.\n\n## Tips for Perfect Sauces\n\nIt's easy to make an ultra-smooth, lump-free sauce if you follow these tips. Take your time\u2014if you rush it, the sauce could curdle or separate. If it doesn't turn out the way you wanted, try our kitchen secrets for fixing sauces:\n\n### Making Sauce\n\n  * Gather all of the ingredients before beginning to give full attention to cooking the sauce.\n  * Use the heat specified in the recipe.\n  * Use a whisk to mix sauces and stir constantly to avoid lumps.\n  * Finish savory sauces with a pat of butter to enhance the flavor and give the sauce a velvety texture. Season with salt and pepper to taste before serving.\n\n### Fixing Sauces\n\nLumpy Sauce: Strain through a tightly woven mesh strainer; pushing liquid through the strainer with the back of a spoon. Discard what is left in the strainer.\n\nToo-Thin Sauce: For a gravy or other sauce thickened with flour, stir a little plain-flavored instant mashed potato flakes into the hot sauce until desired consistency. (The sauce will continue to thicken as it cools, so make the consistency slightly thinner than you would like it.)\n\nCurdled Egg-Based Sauce: Strain out the lumps and try whisking the sauce into another gently heated egg yolk in a clean bowl; heat until hot.\n\nCurdled Cream Sauce: Add one-third more of the original cream volume; slowly stir into the hot, curdled sauce, 1 tablespoon at a time.\n\n# Heirloom Recipe and New Twist\n\nBasil pesto is a classic recipe you'll want to make often and have on hand. Few dishes pack as much flavor punch as fresh pesto. It's a natural to make when fresh basil is abundant. See the next page to learn how to freeze small amounts to have on hand whenever you want it. Toss pesto with hot cooked pasta, dollop it on pizza or use it as a sandwich spread. Our twist recipe is a great example of how adaptable pesto can be. With broccoli as the base instead of herbs, the pesto takes on a completely different flavor.\n\nHeirloom\n\nBasil Pesto\n\n## Basil Pesto CALORIE SMART \u2022 FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1\u00bc cups\n\n  * 2 cups firmly packed fresh basil leaves\n  * \u00be cup grated Parmesan cheese\n  * \u00bc cup pine nuts, toasted if desired (page 23)\n  * \u00bd cup olive or vegetable oil\n  * 3 cloves garlic, peeled\n\n1 In food processor or blender, place all ingredients. Cover and process on medium speed about 3 minutes, stopping occasionally to scrape down sides with rubber spatula, until smooth.\n\n2 Use immediately, or cover tightly and refrigerate up to 5 days or freeze up to 1 month (color of pesto will darken as it stands).\n\n1 Tablespoon: Calories 80; Total Fat 8g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 70mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1\u00bd Fat Carbohydrate Choices: 0\n\nBasil, Roasted Red Pepper and Walnut Pesto Use food processor. Reduce basil to 1 cup and oil to \u2153 cup. Add \u00bd cup drained roasted red or yellow bell peppers (from a jar). Substitute walnuts for the pine nuts.\n\nCilantro Pesto Substitute 1\u00bd cups firmly packed fresh cilantro and \u00bd cup firmly packed fresh parsley for the basil.\n\nSpinach Pesto Substitute 2 cups firmly packed fresh spinach leaves for the basil.\n\nSun-Dried Tomato Pesto Use food processor. Omit basil. Reduce oil to \u2153 cup. Add \u00bd cup sun-dried tomatoes in oil, undrained.\n\n### Blending Pesto\n\nPlace all ingredients in food processor or blender.\n\nCover and process, stopping occasionally to scrape down side.\nNew Twist\n\nBroccoli-Walnut Pesto and Beet-Dill Pesto\n\n## Broccoli-Walnut Pesto CALORIE SMART \u2022 FAst\n\nPesto doesn't have to be \"herb-centric.\" In fact, vegetable-based pestos are varied, come in beautiful hues and are absolutely delicious! Try these four amazing beauties with pasta, as a crostini topper, with cheese and crackers or, as we did in the Test Kitchens, eaten right out of the jar with a spoon!\n\nPrep 15 min Total 15 min \u25cf 1\u00bd cups\n\n  * 2 cups small fresh broccoli florets\n  * \u00bd cup lightly packed fresh basil leaves\n  * \u00bd cup chopped walnuts, toasted (page 23)\n  * 1 clove garlic, peeled\n  * \u2153 cup shredded Parmesan cheese\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon pepper\n  * \u00bd cup olive oil\n\n1 In 2-quart saucepan, heat 2 cups water to boiling. Add broccoli; boil uncovered 1 to 2 minutes or just until bright green. Remove broccoli from boiling water; immediately place in ice water until cold. Drain well.\n\n2 In food processor, place broccoli and remaining ingredients. Cover and process until smooth, stopping occasionally to scrape down sides with rubber spatula, until smooth.\n\n3 Use pesto immediately, or press and smooth plastic wrap directly on surface of pesto and refrigerate up to 3 days or freeze up to 1 month.\n\n2 Tablespoons: Calories 130; Total Fat 13g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 105mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd Vegetable, 2\u00bd Fat Carbohydrate Choices: 0\n\nBeet-Dill Pesto Substitute 1 pound fresh red beets (cooked as directed in Roasted Beets, page 239), cooled and peeled, or 1 can (15 ounces) whole beets, well drained, for the broccoli. Substitute 3 tablespoons fresh dill weed for the basil and reduce oil to 2 tablespoons. Continue as directed in Step 2.\n\nBrussels Sprouts\u2013Cashew Pesto Make as directed, except substitute 2 cups halved fresh Brussels sprouts for the broccoli and boil 3 minutes, omit basil, substitute \u00bd cup cashew halves and pieces for the walnuts (not toasted) and add 2 tablespoons water.\n\nRoasted Carrot\u2013Almond Pesto Omit broccoli and basil. Substitute slivered almonds for the walnuts. Heat oven to 350\u00b0F. Spray 15x10x1-inch pan with cooking spray. Spread 1 bag (12 ounces) ready-to-eat petite baby-cut carrots in pan. Drizzle with 1 tablespoon of the oil; toss to coat. Bake uncovered 35 to 45 minutes, stirring occasionally, until tender. Cool completely. Continue as directed in Step 2. Add 1 to 2 tablespoons water if necessary.\n\n### Making Pesto Cubes\n\nFor homemade pesto anytime, freeze pesto in ice cube trays. Once frozen, move pesto cubes to a resealable plastic freezer bag and store in freezer up to 6 months.\n\n## Chimichurri Sauce CALORIE SMART \u2022 EASY\n\nChimichurri is a common condiment in Argentina. Either curly- or flat-leaf (Italian) parsley can be used in this recipe. Flat-leaf parsley has a stronger peppery flavor. Use this sauce to perk up the flavor of grilled meat or poultry.\n\nPrep 10 min Total 2 hr 10 min \u25cf 1 cup\n\n  * 1 bunch fresh parsley, stems removed\n  * 2 cloves garlic, peeled\n  * \u00be cup olive or vegetable oil\n  * \u00bc cup white vinegar\n  * \u00bd teaspoon salt\n  * \u215b teaspoon pepper\n\nIn food processor, place parsley. Cover and process until chopped. Add remaining ingredients. Cover and process until well blended. Transfer to small bowl; cover and refrigerate 2 hours to blend flavors. Store covered in refrigerator up to 1 week.\n\n1 Tablespoon: Calories 90; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 75mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\n## Salsa Calorie Smart\n\nThis amazing classic salsa is a traditional Southwestern-style sauce, full of tomato flavor. Pico de gallo is another name often used for uncooked salsa like this.\n\nPrep 15 min Total 1 hr 15 min \u25cf 3\u00bd cups\n\n  * 3 large tomatoes, seeded, chopped (3 cups)\n  * 1 small green bell pepper, chopped (\u00bd cup)\n  * 8 medium green onions, sliced (\u00bd cup)\n  * 3 cloves garlic, finely chopped\n  * 2 tablespoons chopped fresh cilantro\n  * 1 tablespoon finely chopped seeded jalape\u00f1o chiles\n  * 2 to 3 tablespoons fresh lime juice\n  * \u00bd teaspoon salt\n\nIn medium glass or plastic bowl, mix all ingredients. Cover and refrigerate at least 1 hour to blend flavors. Store covered in refrigerator up to 1 week.\n\n\u00bc Cup: Calories 15; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 90mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 1g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nBlack Bean Salsa Stir in 1 can (15 ounces) black beans, drained and rinsed. Makes about 5 cups.\n\nSalsa Verde\n\n## Salsa Verde Calorie Smart \u2022 FAst\n\nPrep 15 min Total 15 min \u25cf 1\u00bc cups\n\n  * \u00bd lb tomatillos, husks removed, rinsed and cut in half*\n  * 2 canned serrano chiles, rinsed and seeded, or 1 fresh serrano chile, seeded\n  * \u00bc cup chopped red onion\n  * \u00bc cup chopped fresh cilantro or parsley\n  * \u00bc teaspoon salt\n  * Tortilla chips, if desired\n\nIn food processor or blender, place all ingredients. Cover and process until well blended. Serve, or cover and refrigerate up to 1 week. Serve with tortilla chips.\n\n*Small green tomatoes can be substituted for the tomatillos.\n\n\u00bc Cup: Calories 20; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 120mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 2g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n### Preparing Tomatillos\n\nOpen dry husk and pull up toward top of tomatillo. Pull or cut with paring knife to remove husk and stem.\n\nRinse tomatillo; rub gently to remove sticky film. Cut as desired.\n\n## Tomatillos\n\nTomatillos (tohm-uh-TEE-ohs) are from the same family as tomatoes but very different. Tomatillos have a thick skin with a papery covering and a firm texture. They are green when ripe, with a flavor that hints of lemon and apples.\n\n## Fresh Peach Salsa Calorie Smart\n\nThis peach salsa is a tasty topper for poultry, pork or seafood . . . or delicious with tortilla chips for a snack.\n\nPrep 15 min Total 1 hr 15 min \u25cf 2\u00bc cups\n\n  * 2 cups chopped fresh or frozen (thawed) peaches\n  * \u00bc cup finely chopped red bell pepper\n  * 2 to 4 tablespoons grapefruit juice\n  * 4 teaspoons chopped fresh cilantro\n  * 2 medium green onions, chopped (2 tablespoons)\n  * Dash salt\n\nIn medium glass or plastic bowl, mix all ingredients. Cover and refrigerate 1 to 2 hours to blend flavors. Store covered in refrigerator up to 1 week.\n\n\u00bc Cup: Calories 20; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 120mg; Total Carbohydrate 4g (Dietary Fiber 1g, Sugars 2g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n## White Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nAlso known as b\u00e9chamel sauce, white sauce has many uses. If you're not partial to canned cream soups, this is a great alternative as a base sauce. Check out the variations to make it thicker or thinner, or to customize the flavor.\n\nPrep 10 min Total 10 min \u25cf 1 cup\n\n  * 2 tablespoons butter\n  * 2 tablespoons all-purpose flour\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n  * 1 cup milk\n\n1 In 1\u00bd-quart saucepan, melt butter over low heat. Stir in flour, salt and pepper. Cook and stir over medium heat until mixture is smooth and bubbly; remove from heat.\n\n2 Gradually stir in milk. Heat to boiling, stirring constantly; boil and stir 1 minute. Serve immediately.\n\n1 Tablespoon: Calories 25; Total Fat 1.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 5mg; Sodium 55mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: \u00bd Fat Carbohydrate Choices: 0\n\nThick White Sauce Increase butter and flour to \u00bc cup each.\n\nThin White Sauce Reduce butter and flour to 1 tablespoon each.\n\nCheese Sauce Stir in \u00bc teaspoon ground mustard with the flour. After boiling and stirring sauce 1 minute, stir in \u00bd cup shredded Cheddar cheese (2 ounces) until melted. Serve with eggs or vegetables. Makes about 1\u2153 cups.\n\nCurry Sauce Stir in \u00bd teaspoon curry powder with the flour. Serve with chicken, lamb or shrimp.\n\nDill Sauce Stir in 1 teaspoon chopped fresh or \u00bd teaspoon dried dill weed and dash ground nutmeg with the flour. Serve with fish.\n\nMustard Sauce Reduce butter and flour to 1 tablespoon each. After boiling and stirring sauce 1 minute, stir in 3 tablespoons Dijon mustard. Serve with beef, veal, ham or vegetables.\n\nRoux Use to thicken soup or gumbo. Prepare White Sauce as directed in Step 1. Add liquid for soup or gumbo (omit milk); cook as directed in Step 2. To thicken gravy, cook and stir flour mixture longer in Step 1 to develop rich brown color and deeper flavor. Add desired liquid for gravy (omit milk); cook as directed in Step 2.\n\nMornay Sauce\n\n## Mornay Sauce Calorie Smart \u2022 FAst\n\nMornay is the perfect sauce to serve with meat, fish, shellfish, eggs or vegetables.\n\nPrep 10 min Total 10 min \u25cf 1 cup\n\n  * 2 tablespoons butter\n  * 1 tablespoon all-purpose flour\n  * \u00bd cup half-and-half\n  * \u00bd cup chicken broth (for homemade broth, see page 143)\n  * \u00bc cup shredded Parmesan cheese (1 oz)\n  * \u00bc cup shredded Gruy\u00e8re or Swiss cheese (1 oz)\n\n1 In 1\u00bd-quart saucepan, melt butter over low heat. Stir in flour. Cook and stir over low heat until mixture is smooth and bubbly; remove from heat.\n\n2 Gradually stir in half-and-half and broth. Heat to boiling, stirring constantly. Boil and stir 1 minute. Remove from heat; stir in cheeses with whisk until melted and sauce is smooth. Serve immediately.\n\n1 Tablespoon: Calories 40; Total Fat 3.5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 140mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Fat Carbohydrate Choices: 0\n\nVelout\u00e9 Sauce Omit half-and-half and cheeses. Increase broth to 1 cup. Add \u215b teaspoon ground nutmeg and \u215b teaspoon pepper.\n\n### Preparing Roux\n\nCook flour and butter until smooth and bubbly. Cook an additional minute to remove floury taste.\n\n## Make-Ahead Mayonnaise\n\n## Calorie Smart \u2022 Fast \u2022 EASY\n\nHomemade mayonnaise is deceptively easy to make and deliciously lush. You may never go back to store-bought again.\n\nPrep 10 min Total 10 min \u25cf 1\u00be cups\n\n  * 1\u00bc cups vegetable oil\n  * 2 tablespoons fresh lemon juice\n  * 1 tablespoon Dijon mustard\n  * \u00bc teaspoon salt\n  * 1 pasteurized egg*\n\n1 In blender, place \u00bc cup of the oil, the lemon juice, mustard, salt and egg. Cover and blend until smooth.\n\n2 Leave cover on blender but remove center cap. With blender running on medium-high speed, slowly add remaining 1 cup oil in thin stream through opening in cover. Scrape down sides of container as needed. Store in tightly covered container in refrigerator up to 5 days.\n\n*Pasteurized eggs are uncooked eggs that have been heat-treated to kill bacteria that can cause food poisoning and gastrointestinal distress. Because the egg in this recipe is not cooked, be sure to use a pasteurized egg.\n\n1 Tablespoon: Calories 90; Total Fat 10g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 35mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nCurried Mayonnaise Add \u00bd to 1 teaspoon curry powder to egg mixture before blending.\n\nGarlic Mayonnaise Stir 1 clove garlic, finely chopped, into prepared mayonnaise.\n\nHerbed Mayonnaise Stir 1 tablespoon chopped fresh basil, chervil, chives, cilantro, marjoram, oregano or parsley into mayonnaise. If using dried herbs, use only 1 teaspoon.\n\nMayonnaise and Moist Garlic Turkey Burger\n\n## Uses for Mayonnaise\n\nMayonnaise doesn't have to be the boring, unused condiment in your fridge, destined only for a quick sandwich. Did you know it's the base for making moist turkey burgers and chicken, dressings and flavorful sauces? The following ideas give mayo a makeover with fresh new flavors to perk up your meals.\n\n  1. 1. **Sriracha Dipping Sauce for Fries:** In small bowl, mix \u00bd cup mayonnaise, 2 teaspoons sriracha sauce, 2 tablespoons chopped green onions and \u215b teaspoon seasoned salt. Cover and refrigerate 1 hour to blend flavors.\n  2. 2. **Curry-Chutney Tartar Sauce:** Mix 1 cup Curried Mayonnaise (page 461), 2 tablespoons finely chopped prepared chutney and 1 tablespoon chopped fresh cilantro leaves. Cover and refrigerate 1 hour to blend flavors. Use as a sandwich spread or as an accompaniment for seafood.\n  3. 3. **Moist Garlic Turkey Burgers:** In medium bowl, mix 1\u00bc pounds ground turkey, \u2153 cup Garlic Mayonnaise (recipe above), \u00bc cup each garlic-herb bread crumbs and finely chopped onions, \u00bd teaspoon salt and \u00bc teaspoon pepper; shape into patties. Grill burgers over medium heat 13 to 15 minutes, turning once, until meat thermometer inserted in center of patties reads 165\u00b0F.\n\nTartar Sauce\n\n## Tartar Sauce Calorie Smart \u2022 EASY\n\nTartar sauce is a favorite to serve with most any type of fish or seafood. If you make it with salad dressing, it will be sweeter.\n\nPrep 5 min Total 1 hr 5 min \u25cf 1 cup\n\n  * 1 cup mayonnaise or salad dressing\n  * 2 tablespoons pickle relish or finely chopped dill pickles\n  * 1 tablespoon chopped fresh parsley\n  * 2 teaspoons diced pimientos, drained\n  * 1 teaspoon grated onion\n\nIn small bowl, mix all ingredients. Cover and refrigerate about 1 hour or until chilled. Store covered in refrigerator up to 5 days.\n\n1 Tablespoon: Calories 100; Total Fat 11g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 95mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nLighter Directions For 5 grams of fat and 50 calories per serving, use reduced-fat mayonnaise.\n\nDill Tartar Sauce Stir in 1 tablespoon chopped fresh or 1 teaspoon dried dill weed.\n\n## Hollandaise Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nHere's the classic hollandaise\u2014lusciously rich, lemony and smooth, and perfect over vegetables, eggs, fish and seafood.\n\nPrep 10 min Total 10 min \u25cf \u00be cup\n\n  * 3 egg yolks\n  * 1 tablespoon fresh lemon juice\n  * \u00bd cup cold butter, cut into pieces*\n\n1 In 1\u00bd-quart saucepan, vigorously stir egg yolks and lemon juice with whisk. Add \u00bc cup of the butter. Heat over very low heat, stirring constantly with whisk, until butter is melted.\n\n2 Add remaining \u00bc cup butter. Continue stirring vigorously until butter is melted and sauce is thickened. (Be sure butter melts slowly so eggs have time to cook and thicken sauce without curdling.) If the sauce curdles (begins to separate), add about 1 tablespoon boiling water and beat vigorously with whisk until smooth. Serve immediately.\n\n*Do not use margarine or vegetable oil spread.\n\n1 Tablespoon: Calories 80; Total Fat 9g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 55mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2 Fat Carbohydrate Choices: 0\n\nB\u00e9arnaise Sauce Stir in 1 tablespoon dry white wine with the lemon juice. After sauce thickens, stir in 1 tablespoon finely chopped onion, 1\u00bd teaspoons chopped fresh or \u00bd teaspoon dried tarragon leaves and 1\u00bd teaspoons chopped fresh or \u00bc teaspoon dried chervil leaves.\n\n### Learn with Betty Hollandaise Sauce\n\n**Perfect Sauce:** This sauce is just the right consistency, smooth without lumps.\n\n**Undercooked Sauce:** This sauce was not cooked long enough and is too thin.\n\n**Overcooked Sauce:** This sauce was cooked too long, or heat was too high, and it has curdled.\n\nMole Poblano\n\n## Mole Poblano Calorie Smart\n\nThis thick, dark, lush version of mole poblano is accented by chocolate, which is common to this classic Mexican sauce. There are hundreds of moles, depending on the region and the cook. Our version is easy and delicious. It's great to use with chicken, pork and beef.\n\nPrep 45 min Total 1 hr 15 min \u25cf 3 cups\n\n  * 3 oz dried ancho chiles*\n  * Boiling water\n  * \u00bc cup slivered almonds\n  * \u00bc cup pumpkin seeds (pepitas)\n  * 1 tablespoon sesame seed\n  * \u00bd cup chopped white or Spanish onion\n  * 3 tablespoons raisins\n  * 3 tablespoons creamy peanut butter\n  * 3 tablespoons sugar\n  * 1 oz semisweet baking chocolate, finely chopped\n  * \u00bd teaspoon ground cinnamon\n  * \u00bd teaspoon salt\n  * 2\u2153 cups chicken broth (for homemade broth, see page 143)\n  * 1 tablespoon vegetable oil\n\n1 In medium bowl, cover dried chiles with boiling water. Place plate or bowl over chiles, weighted if necessary to keep chiles submerged. Let stand 30 minutes to soften completely. Drain. Remove stems. Cut chiles open; place, skin side down, on cutting board. Scrape off seeds with spoon; set chiles aside.\n\n2 Heat 8-inch skillet over medium heat. Add almonds, pepitas and sesame seed. Cook 5 to 7 minutes, stirring frequently, until nuts and seed begin to brown, then stirring constantly until nuts are golden brown.\n\n3 In food processor, place chiles, almonds, pepitas, sesame seed, onion, raisins, peanut butter, sugar, chocolate, cinnamon, salt and \u00bd cup of the broth. Cover and process until smooth (mixture will be thick).\n\n4 In 10-inch skillet, heat oil over medium heat. Add mole. Cook 3 minutes, stirring frequently. Gradually add remaining broth. Heat to boiling, stirring constantly; reduce heat. Simmer uncovered 15 minutes, stirring frequently. Cool. Transfer to tightly covered container. Refrigerate up to 1 week or freeze up to 3 months.\n\n*Ancho chiles are dried fresh poblanos. If you can't find them in your area, check online sources.\n\n\u00bc Cup: Calories 140; Total Fat 8g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 190mg; Total Carbohydrate 12g (Dietary Fiber 3g, Sugars 7g); Protein 4g Exchanges: 1 Other Carbohydrate, \u00bd High-Fat Meat, 1 Fat Carbohydrate Choices: 1\n\n## Peanut Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nPeanut sauce goes hand in hand with chicken satay, but it also tastes great with grilled poultry or meats or tossed with noodles and vegetables. Try it as a salad dressing for Asian-inspired combinations.\n\nPrep 10 min Total 10 min \u25cf 1 cup\n\n  * \u00bd cup creamy peanut butter\n  * \u00bd cup water\n  * 2 tablespoons fresh lime juice\n  * \u00bd teaspoon ground coriander\n  * \u00bd teaspoon ground cumin\n  * \u215b teaspoon salt\n  * \u215b teaspoon ground red pepper (cayenne), if desired\n  * 2 cloves garlic, finely chopped\n\n1 Place peanut butter in medium bowl. Gradually stir in water and lime juice with whisk until smooth and creamy; stir in remaining ingredients.\n\n2 Use immediately, or to make a warm sauce, place all ingredients in 1-quart saucepan. Heat over medium heat, stirring with whisk frequently, until smooth and warm.\n\n3 Or cover and refrigerate up to 3 days or freeze up to 2 months.\n\n1 Tablespoon: Calories 50; Total Fat 4g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 55mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: \u00bd High-Fat Meat Carbohydrate Choices: 0\n\n## Sriracha Sauce Calorie Smart\n\nSriracha is a fiery fermented sauce that adds a huge punch of flavor. Our homemade version is so easy to make, so flavorful and so beautifully red that you may never get store-bought again! Make sure to look for completely red jalape\u00f1os for best flavor and color.\n\nPrep 25 min Total 8 days \u25cf 1\u00bc cups\n\n  * 1 lb red jalape\u00f1o chiles (12 to 14 medium)\n  * 3 tablespoons sugar\n  * 4 cloves garlic, peeled\n  * 2\u00bd teaspoons salt\n  * \u2153 cup cider vinegar\n\n1 Remove stems from chiles but do not remove seeds; cut chiles in half.* In food processor, place chiles, sugar, garlic and salt. Cover and process about 20 seconds, using short on-and-off motions, until chiles are very finely chopped. Be aware that the chile fumes may be strong.\n\n2 Transfer mixture to glass jar. Cover and let stand at room temperature 5 to 8 days, stirring every day, until mixture begins to ferment when small bubbles form on bottom and sides of jar. Mixture may rise in volume in jar.\n\n3 In blender, place chile mixture and vinegar. Cover and blend until smooth. Place fine-mesh strainer over 2-quart saucepan. Pour chile mixture into strainer, pressing firmly with back of spoon until only seeds remain. Discard seeds. Heat sauce to boiling; reduce heat to low. Simmer uncovered 10 minutes or until sauce thickens and clings to metal spoon. Cool. Transfer to tightly covered container. Store in refrigerator up to 6 months.\n\n*Use plastic or rubber gloves when handling chiles. Wash knives and cutting boards thoroughly. The fumes from fresh chiles can also be irritating to some people.\n\n1 Teaspoon: Calories 5; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 100mg; Total Carbohydrate 1g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\nThai Red Curry Paste\n\n## Thai Red Curry Paste Calorie Smart\n\nPrep 50 min Total 50 min \u25cf \u00bd cup\n\n  * 10 dried chiles de \u00e1rbol or red spur chiles\n  * boiling water\n  * 4 fresh red Fresno or serrano chiles\n  * \u00bd cup coarsely chopped shallots\n  * \u00bc cup coarsely chopped garlic\n  * 2 tablespoons chopped fresh cilantro stems*\n  * 1 tablespoon finely chopped gingerroot\n  * 1 tablespoon ground coriander\n  * 2 teaspoons finely chopped lemongrass**\n  * 1 teaspoon ground cumin\n  * 1 strip fresh lime peel, 1\u00bd x \u00bd inches\n  * \u00bd teaspoon salt\n  * 4 to 6 teaspoons water\n\n1 Soak dried chiles in boiling water 30 minutes (place plate on top and weight with can to keep chiles submerged). Cut tops off fresh chiles. Cut in half lengthwise, remove seeds and coarsely chop.\n\n2 Place all ingredients except water in small food processor or blender. Cover and process until finely chopped, scraping down sides as needed. Add just enough water until a paste forms, scraping down sides as needed. Store in tightly covered jar in refrigerator up to 6 months.\n\n* Only the cilantro stems are used in this Thai curry paste. For the most flavor, avoid the woody portions of the stem and select the tender bright green portions.\n\n** Resembling a green reed, lemongrass is an ancient herb of Southeast Asia. The flavor is reminiscent of citrus, bay leaves and pepper. Peel away the tough outer layers and use only the tender inner core.\n\n1 Tablespoon: Calories 25; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Sodium 150mg; Total Carbohydrate 4g (Dietary Fiber 1g); Protein 0g Exchanges: \u00bd Other Carbohydrate Carbohydrate Choices: 0\nGlobal Flavors\n\n## Asia\n\nThe ingredients of Asian cuisines encompass a large diversity of tastes and regional specialties. Some exotic, some familiar, the enticing array will draw you in with their beckoning aromas and tempting tastes.\n\nAsian Condiments: Also known as **Gochujang** in Korea, **Chile bean paste** or sauce is a popular condiment in Szechuan, Hunan and Korean cooking, containing fermented soybeans, dried chiles, garlic and seasoning. Other bean sauces and pastes, ranging in thickness, are intensely flavored mixtures used to perk up the flavor of many dishes. **Sriracha,** similar in consistency to ketchup (and used as frequently as ketchup in Southeast Asian cuisine) is a spicy mixture of red chiles, vinegar, garlic and salt. **Sambal oelek** is a mixture of chiles, brown sugar and salt usually served with Chinese rice and curried dishes.\n\nAsian Noodles: Noodles can be made from wheat, rice, potato or buckwheat flour, cornstarch and bean, yam or soybean starch. Popular Chinese noodles include **Cellophane** (made from mung-bean starch), **Egg noodles** (usually wheat) and **Rice noodles**. Japan's most-popular noodles include **Harsume** (made with rice, potato or soybean flour), **Ramen** (wheat egg noodles) and **Soba** noodles (made with buckwheat flour). Korea's **Naengmyeon** noodles are like soba noodles but chewier. Many dried noodles are available in supermarkets' both fresh and dried noodles can be found in Asian markets. Noodles can be served hot or cold and cooked in a variety of ways including stir-frying and steaming.\n\nAsian Seasonings: Native to Asia, **Star anise** is the pod from a small evergreen tree. Slightly more bitter than anise, it's commonly used in tea and as a spice. **Chrysanthemum leaves** are very popular in both Chinese and Japanese cuisines. Both the dark green fragrant leaves and tender stems are used in a variety of dishes. **Lemongrass** is an integral flavoring in Thai and Vietnamese cooking, with a sour lemon flavor and fragrance; available both fresh and dried.\n\nThai chiles are popular in many South Asian dishes, are small, very spicy chiles that don't lose their flavor when cooked. They can range in color from green to red when ripe. **Galangal** has a hot, peppery ginger flavor and is often used as a substitute for ginger in Thai cooking. **Greater galangal** is the dried version found in Asian markets, with a slightly more intense flavor than fresh. **Laos** is the powdered form of greater galangal.\n\nDried Seafood: Small amounts of dried anchovies, shrimp and tuna (bonito), with their strong fishy taste, are commonly used to flavor various Asian dishes. They may need to be reconstituted in warm water to be used. Many dishes also use the water the seafood was soaked in for additional flavor.\n\nEdible Seaweed: An important component of Asian cuisine, belonging to the algae family. Different varieties are used for different dishes. Deep green **Wakame** is used fresh or dried as a vegetable in soups and simmered dishes. **Kelp** is edible brown seaweed. **Nori** is seaweed dried into sheets to use in sushi.\n\nKimchi: Korean pickled and fermented vegetables, sometimes made extraordinarily spicy with chiles added. Served at nearly every Korean meal, kimchi recipes can vary widely depending on the produce in season.\n\nMirin: Rice wine made from glutinous rice. An essential Japanese staple used to add sweetness and flavor to a variety of dishes.\n\nMiso: Fermented soybean paste used in many Japanese recipes, including sauces, marinades, dips, dressings and as a condiment. You can find various types and colors available with different flavors for different dishes.\n\nSake: Often called rice wine, although it is made from various types of rice and includes other ingredients. Available in a wide variety of types used both for drinking and, most commonly, in Japanese sauces and marinades.\n\nSesame Oil: An excellent choice for frying, sesame oil is made from sesame seeds and is available in both light and dark varieties. Light has a light nutty flavor and is used for salad dressings and a variety of dishes. Dark has a stronger flavor and fragrance, so it is used sparingly as an accent in Asian dishes.\n\nTofu: Also known as bean curd or soybean curd. Made from ground cooked soybeans, the curds are pressed in a manner similar to the way cheese is made. It's bland, slightly nutty flavor can take on the other flavors in a dish. Used as a plant-based source of protein in many Asian dishes.\n\n## Learn to Deglaze\n\nDeglazing is a simple technique of capturing the flavorful bits from the bottom of a pan by heating a small amount of liquid in it after cooking meat. The bits are loosened and become the start of a very flavorful sauce. The liquid added may be reduced to concentrate the flavors and butter may be added to help flavor the sauce and give it a velvety texture.\n\nHere are some helpful tips for deglazing and making sauces successfully:\n\n  * A flavorful sauce begins with flavorful bits; be sure to season the meat with salt and pepper before cooking in a small amount of liquid.\n  * Remove any burnt bits from the pan before deglazing, as they can make the sauce bitter.\n  * Remove excess fat from the pan before deglazing.\n  * Add liquid such as wine, spirits, broth or apple cider to the pan over medium-high heat.\n  * If you are using alcohol such as wine to deglaze, remove the pan from the heat to prevent flaming.\n  * If savory seasonings weren't already added, now is the time. Stir in finely chopped garlic, shallots or dried herbs.\n  * Scrape the brown bits from the bottom of the pan using a spoon or spatula.\n  * Stir constantly while deglazing.\n  * Turn down the heat when the bits are loosened.\n  * Reduce the liquid (see page 208), if desired or necessary, to concentrate flavors and reduce amount of liquid in pan.\n  * Stir in a pat of butter until melted.\n  * Season the finished sauce with salt and pepper, garlic or onion powder or fresh herbs to taste.\n\nRemove any burnt bits or excess fat before deglazing.\n\nAdd liquid and stir constantly to loosen stuck bits.\n\n## Peppery Red Wine Sauce\n\nPrep 15 min Total 35 min \u25cf 1 cup\n\n  * 6 tablespoons butter\n  * \u00bd cup finely chopped shallots (3 medium)\n  * 1 cup dry red wine (such as Merlot or Zinfandel) or nonalcoholic red wine\n  * 1 cup beef broth (for homemade broth, see page 146)\n  * 1 tablespoon balsamic vinegar\n  * 2 teaspoons Dijon mustard\n  * 1 teaspoon coarsely ground black pepper\n\n1 In 8-inch skillet, melt 2 tablespoons of the butter over medium-high heat. Add shallots; cook 1 minute, stirring frequently. Add wine; heat to boiling over high heat. Reduce heat to medium-high; cook until mixture coats back of metal spoon, 10 to 12 minutes.\n\n2 Stir in broth. Heat to boiling. Reduce heat to medium-high; cook 8 to 10 minutes longer, stirring constantly, until mixture coats back of metal spoon and is slightly reduced.\n\n3 With whisk, beat in remaining 4 tablespoons butter, 1 tablespoon at a time. Reduce heat to low. Beat in vinegar, mustard and pepper.\n\n\u00bc Cup: Calories 200; Total Fat 17g (Saturated Fat 11g, Trans Fat 0.5g); Cholesterol 45mg; Sodium 330mg; Total Carbohydrate 5g (Dietary Fiber 1g, Sugars 2g); Protein 2g Exchanges: 3\u00bd Fat, \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\n### Making Wine Sauce\n\nCook until mixture coats back of metal spoon.\n\nBeat in remaining 4 tablespoons butter, 1 tablespoon at a time.\n\nWhite Wine\u2013Garlic Sauce\n\n## White Wine\u2013Garlic Sauce\n\nPrep 15 min Total 35 min \u25cf 1 cup\n\n  * \u00bc cup olive oil\n  * 4 cloves garlic, finely chopped\n  * 2 cups dry white wine (such as Chardonnay or Sauvignon Blanc) or nonalcoholic white wine\n  * \u00bc cup white wine vinegar\n  * \u00bd cup whipping cream\n  * \u00bc cup chopped fresh parsley\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon pepper\n\n1 In 10-inch skillet, heat oil over medium-high heat. Add garlic; cook 1 minute, stirring constantly. Add wine and vinegar; heat to boiling over high heat. Reduce heat to medium-high; simmer until mixture coats back of metal spoon, 10 to 12 minutes.\n\n2 Stir in whipping cream; simmer until mixture coats back of metal spoon and is slightly reduced, 3 to 4 minutes longer, stirring constantly. Remove from heat. Stir in parsley, salt and pepper.\n\n\u00bc Cup: Calories 260; Total Fat 25g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 40mg; Sodium 310mg; Total Carbohydrate 3g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: \u00bd Other Carbohydrate, 5 Fat Carbohydrate Choices: 0\n\n## Wine\n\nLeftover wine can be used in a variety of ways to heighten the flavor of dishes. When wine is heated, the alcohol is burned off, leaving the essence to add complexity to the flavors of the dish. Use dry white wines for savory poultry dishes, dry red wines for savory beef or pork dishes. Sweet wines can be used in desserts. Try one of these ideas next time you find yourself with an opened bottle:\n\n  1. Gravy: Try substituting dry white wine (up to half) as part of the liquid for turkey or chicken gravy. Or use a dry red wine (in the same manner) in a beef or pork gravy.\n  2. Stew: Use leftover red wine to replace some of the broth (up to half) in your favorite stew recipe.\n  3. Pasta Sauce: When heating pasta sauce, add a little leftover red wine (about \u00bd cup wine to 2 cups pasta sauce). For a thicker sauce, bring it to a boil and reduce heat. Simmer uncovered until desired consistency.\n  4. Mulled Wine: Pour leftover red wine into a freezer container and freeze. Each time you have leftover wine, add to the container. When you have collected approximately 2 bottles worth, empty the frozen wine into a 3-quart saucepan and continue as directed for Mulled Wine (page 468).\n  5. Wine Cubes: Pour leftover wine into an ice cube tray; freeze. Use white wine cubes to keep glasses of white wine chilled on hot days without diluting it, or use white or red cubes for white or red sangria, respectively.\n  6. White Wine Sauce: Prepare White Sauce (page 459), substituting \u00bd cup of dry white wine for \u00bd cup of the milk.\n  7. Port Cranberry Sauce: Prepare Cranberry Sauce (page 641), substituting 1 cup sweet red wine, such as Port, for 1 cup of the water.\n\nHerbed Butter\n\n## Herbed Butter EASY\n\nThis simple sauce is sensational drizzled over fish, seafood, poultry or vegetables or tossed with pasta.\n\nPrep 5 min Total 5 min \u25cf \u00bd cup\n\n  * \u00bd cup butter, cut into pieces*\n  * 2 tablespoons chopped fresh or 2 teaspoons dried herb leaves (basil, chives, oregano, savory, tarragon or thyme or a mix of these varieties)\n  * 2 teaspoons fresh lemon juice\n\nIn 1-quart saucepan or 8-inch skillet, melt butter over medium heat. Stir in herbs and lemon juice. Use immediately.\n\n*Do not use margarine or vegetable oil spread.\n\n1 Tablespoon: Calories 100; Total Fat 12g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 30mg; Sodium 75mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 2\u00bd Fat Carbohydrate Choices: 0\n\nHerbed Browned Butter Sauce In 1-quart saucepan, heat butter over medium heat just until light brown. Watch carefully because butter can brown and then burn quickly. Continue as directed.\n\n## Clarified Butter Fast\n\nThink you've never heard of clarified butter? Many seafood restaurants serve it hot with lobster and crab. A staple in Indian cuisine, it's known as ghee (see page 256). Aside from serving it with seafood, it's great brushed over warm Naan (page 516) or pita bread, or drizzled over hot cooked basmati rice.\n\nPrep 20 min Total 30 min \u25cf \u00be cup\n\n  * 1 lb unsalted butter, cut into 1-inch pieces\n\n1 In 1-quart saucepan, melt butter over low heat. Continue simmering over low heat 15 to 20 minutes, occasionally skimming off milk solids that rise to surface, until butter is clear and deep yellow in color. Do not stir (some of the milk solids will turn brown and sink to bottom of pan).\n\n2 Remove from heat. Cool 5 to 10 minutes; do not stir. Gently pour clarified butter into airtight container, leaving milk solids in saucepan. Discard milk solids. Store covered in refrigerator up to 1 month.\n\n1 Tablespoon: Calories 280; Total Fat 31g (Saturated Fat 19g, Trans Fat 1g); Cholesterol 80mg; Sodium 0mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 6 Fat Carbohydrate Choices: 0\n\nGarlic Clarified Butter Stir 1 or 2 finely chopped cloves garlic into butter after removing from heat. After cooling, use fine-mesh strainer to gently pour clarified butter into container so that garlic pieces will be removed before storage.\n\n### Clarifying Butter\n\nSimmer butter over low heat, occasionally skimming off milk solids that rise to surface.\n\nContinue until butter is clear and deep yellow in color.\n\nCarefully pour warm clarified butter into airtight container, leaving any milk solids in saucepan.\n\n## Flavored Butters Fast \u2022 EASY\n\nThe sky's the limit with creative additions to whipped butter. Here are some of our favorite sweet and savory ideas.\n\nPrep 5 min Total 5 min \u25cf \u00bd cup\n\n  * \u00bd cup butter, softened\n\nIn small bowl, beat butter and ingredients for one of the flavor variations below with electric mixer on medium speed until light and fluffy. Use immediately, or refrigerate about 1 hour to blend flavors. Store covered in refrigerator up to 1 week.\n\n#### Sweet Butters\n\nTop muffins, scones, biscuits and toast.\n\nAlmond Butter Add 1 tablespoon finely chopped almonds, toasted if desired (page 23), and \u00bd teaspoon almond extract.\n\nHoney or Maple Butter Add \u00bd cup honey or real or maple-flavored syrup.\n\nOrange Butter Add 1 teaspoon grated orange peel and 1 tablespoon fresh orange juice.\n\nPecan Butter Add \u00bc cup chopped pecans, toasted if desired (page 23), and 2 tablespoons packed brown sugar.\n\nRaspberry Butter Add \u00bd cup raspberries, crushed, and 1 tablespoon sugar. Or add \u00bc cup seedless raspberry jam.\n\n#### Savory Butters\n\nTop grilled or broiled beef or fish steaks, pork chops or chicken breasts. Toss with hot pasta, rice or vegetables. Spread on bread.\n\nGarlic Butter Add 1 or 2 finely chopped cloves garlic or \u00bc to \u00bd teaspoon garlic powder.\n\nHerb Butter Add 2 tablespoons to \u00bc cup chopped fresh or 2 to 4 teaspoons dried herb leaves (basil, chives, oregano, savory, tarragon or thyme or a mix) and 2 teaspoons fresh lemon juice.\n\nItalian Parmesan Butter Add 2 tablespoons grated Parmesan cheese and \u00bd teaspoon dried Italian seasoning.\n\nLemon-Pepper Butter Add 2 tablespoons finely grated lemon peel, 1 teaspoon fresh lemon juice and \u215b to \u00bc teaspoon pepper.\n\nBacon-Chive-Cheddar and Honey-Walnut Cream Cheese Spreads\n\n## Cream Cheese Spreads Calorie Smart \u2022 FAst \u2022 EASY\n\nThese spreads are all amazing on bagels, but don't stop there! Try them spread on vegetables, as a delicious sandwich condiment or whisk a few tablespoons into beaten eggs for a tasty omelet.\n\nPrep 5 min Total 5 min \u25cf 1 \u00bc cups\n\n  * 1 package (8 oz) cream cheese, softened\n  * \u00bc cup milk\n\nIn small bowl, stir cream cheese and milk until smooth and creamy. Add ingredients for one of the flavor variations below; stir with spoon until well blended. Use immediately, or refrigerate about 1 hour to blend flavors. Store covered in refrigerator up to 1 week.\n\nBacon-Chive-Cheddar Cream Cheese Spread Stir in \u00bd cup shredded Cheddar cheese (2 ounces), \u00bc cup crumbled crisply cooked bacon and 2 tablespoons chopped fresh chives.\n\nHoney-Walnut Cream Cheese Spread Omit milk. Stir in \u00bc cup honey (or real or maple-flavored syrup) and \u00bd cup finely chopped walnuts, toasted if desired (page 23).\n\nMaple-Cinnamon Cream Cheese Spread Omit milk. Stir in \u00bc cup real or maple-flavored syrup and \u00bd teaspoon ground cinnamon.\n\nOlive-Walnut Cream Cheese Spread Stir in \u00bd cup finely chopped walnuts, toasted if desired (page 23), and \u00bc to \u00bd cup finely chopped kalamata or pimiento-stuffed green olives.\n\nPeanut Butter\u2013Honey Cream Cheese Spread Stir in \u00bc to \u00bd cup crunchy peanut butter and 2 tablespoons honey.\n\n# Seasoning Basics\n\nMuch of what makes a dish great is the flavor. Seasonings are added in the right combination and at the right level. With a little practice, you can learn what seasonings enhance your dishes and create your own signature flavor combinations.\n\n## When to Season\n\nSeasoning can be done at the beginning of making a dish, while working on parts of a recipe and also once the dish is finished.\n\n  * Meat dishes have better flavor if the meat is seasoned before it is cooked and then the entire dish is seasoned at the end, so that both the meat and other ingredients have a lot of flavor.\n  * Salt causes moisture to be drawn out of foods. Add it before cooking if you are going to cook an ingredient that you want to soften, such as onions. If you want a food to be crisper, salt after cooking.\n  * Spices and herbs should be used to enhance the natural flavors in a dish, not disguise it. Many spices and herbs go well with a dish and with each other, but too much won't make it better. Avoid using too many at one time.\n  * Whole spices and bay leaves release their flavors over time, so they can be added before cooking. But be sure to remove bay leaves before serving.\n  * Ground spices and herbs release their flavors easily, so it's better to add them toward the end of long-cooking dishes.\n  * For uncooked foods like salad dressings, sauces such as salsa and salads like pasta salads, add the spices and herbs with the other ingredients and then allow enough time for the flavors to blend before eating.\n  * Taste foods at their serving temperature to accurately determine if the level of seasoning is good or not. Also, keep in mind that flavors can build in your mouth, so what can seem like too little seasoning when a small spoonful is eaten could actually be very flavorful when an entire serving is eaten.\n\n## How Much to Add?\n\nEveryone has a different idea of what tastes good. Some are more susceptible to salt levels, others to spiciness or heat of a dish. In general, less is more\u2014particularly if you are serving people you know will be sensitive. It's easier to add more when the dish is served than it is to correct a dish that has too much. You can place the salt, pepper and hot sauce on the table for folks to add more if necessary.\n\nHerbs: Fresh herbs are less potent than dried. In general, use three times as much fresh herbs as what is called for dried. So in general, 1 teaspoon dried herbs equals 3 teaspoons fresh herbs\n\nRed Pepper Flakes: When adding red pepper flakes to a dish, keep in mind that the heat can intensify over time as the flavor is released. What seems to be a mildly spicy dish when first seasoned might actually be spicier when served.\n\n## Herbs\n\nBasil: Sweet and spicy, fragrant. Chop leaves just before using. Use with eggs, meat, pesto, salads, soups, stews, tomato dishes or herb butters.\n\nBay Leaves: Earthy, grassy, slightly piney (fresh leaves are more potent than dried). Use in Greek dishes, soups, stews or under chicken skin or to top fish while cooking. Discard leaves before eating.\n\nChervil: Mild, parsley-like. Use with eggs, fish, in salads, sauces, soups or stuffing.\n\nChives: Mild green onion flavor. Use with potatoes, eggs or fish, in appetizers or as a garnish.\n\nCilantro: Looks similar to Italian parsley but with a lively, pungent smell and taste. Chop leaves; discard stems. Use in salsas and other Mexican, Indian, Indonesian dishes, marinades, pasta salads, soups or stews.\n\nDill Weed: Fresh, peppery, tangy. Chop leaves and tender stems; discard woody stems. Use in bread, dips, fish, pickles, herb butters, salads (sparingly) or sauces.\n\nFennel: Anise flavor. Chop leaves and stems. Use in soups, salads, stuffing or tossed with cooked vegetables. Seeds are used as spice.\n\nHorseradish: Pungent, peppery. Choose firm roots without blemishes; peel and grate before using. Use in sauces or condiments for fish or meat.\n\nItalian Parsley: Also known as flat-leaf parsley, it is more fragrant, less bitter than regular (curly) parsley. Chop leaves; discard stems. Use in Italian, European and Middle Eastern dishes for stronger parsley flavor.\n\nLemongrass: Sour lemon flavor; extremely fibrous and stringy. Cut off and discard bulb; remove tough or outer leaves. Cut yellow (lower section) into thin slices. Bend, bruise pieces to release flavor (discard after cooking) or process in food processor; cook thoroughly to soften enough to eat. Use in Asian dishes or soups.\n\nMarjoram: Sweet, slightly oregano-like yet delicate flavor; add at the end of cooking. Use with vegetables such as carrots, zucchini, mushrooms, peas, spinach and tomatoes, with fish, lamb, poultry or in soups, stews or stuffing.\n\nMint: Strong, cool, fresh and sweet flavor. Many varieties, with a range of flavors. Use in Greek or Italian dishes, in beverages or desserts or with cucumbers, peas, potatoes, carrots or lamb.\n\nOregano: Pungent\u2014use sparingly. Use in Italian dishes, meat, poultry, sauces, soups or vegetables.\n\nParsley: Helps the fresh flavor of a dish to come through. Works well with garlic to temper its flavor. Chop curly leaves; discard stems. Use in sauces, eggs, stuffing, herb butters or with corn or potatoes.\n\nRosemary: Strong flavor with hints of pine and lemon\u2014use sparingly. Remove and chop leaves; discard stems. Use in soups, breads, fish, lamb, salads, seafood, tomato dishes or vegetables.\n\nSage: Slightly bitter with subtle musty mint flavor\u2014use sparingly. Many varieties, with range of colors and flavors. Use in Mediterranean foods, particularly Italian dishes, stuffing, marinades or with peas, eggplant, lima beans, carrots or poultry.\n\nSavory: Mint-thyme flavor; combines well with other herbs. Use in poultry, fish, meat, vegetables, any kind of beans or in herb butters.\n\nTarragon: Delicate, anise-like flavor\u2014use sparingly. Use in salads, salad dressings, egg dishes, sauces or with poultry, fish or cheese.\n\nThyme: Peppery, minty, mild lemon flavor\u2014use sparingly. Many varieties, with range of colors and flavors. Chop leaves and soft stems; discard woody stems. Use in stews, dressings or with vegetables, poultry, fish, pork, beef or lamb.\n\n## Adjusting for Flavor Preferences\n\nOur recipes are tasted by several people with a wide range of taste preferences. You'll find the seasoning levels to be in the \"average\" category, in terms of salt level and spiciness, unless otherwise indicated by the recipe title. Sometimes we call for an ingredient with a range of amount\u2014typically done where hotness of a dish can be varied. Use the lesser amount if you like things milder or more if you like foods with more kick.\n\n## Spices\/Seasonings\n\nMany dried spices and seasonings can be toasted to enhance their flavors before being added to a dish. To toast spices and seasonings, heat in small, heavy skillet over medium heat, stirring constantly, until fragrant. Watch closely\u2014they can go from toasted to burnt very quickly.\n\nAllspice: Available whole, ground. Cinnamon-clove-nutmeg flavor. Use in baked goods, jerk seasoning, pickling, pies or with fruit.\n\nAnise Seed: Available whole, ground. Sweet licorice flavor. Use in cakes, cookies, breads or fruit dishes.\n\nCaraway Seed: Available whole. Delicate, nutty anise flavor. Use in breads, stews, with vegetables or meats.\n\nCardamom: Available as whole or split pods, whole or ground seeds. The flavor has hints of eucalyptus, camphor and lemon. Use in Indian and Middle Eastern dishes and to flavor coffee or tea. Dominant flavor in chai.\n\nCelery Seed: Available whole, ground. Strong flavor\u2014use sparingly. Use in soups, salads, salad dressings, meat dishes or pickling.\n\nCinnamon: Available in sticks, ground. Sweet, woodsy flavor. Use in baked goods, hot drinks, pickling or savory dishes.\n\nCloves: Available whole, ground. Sweet, peppery flavor. Use in baked goods, pickling, sauces, tea or with fruit, ham or other meat.\n\nCoriander: Available whole, ground. Lemon-sage flavor. Seeds of the cilantro plant\u2014but not to be substituted for cilantro, which has a different flavor. Use in soups, stews, sausages, ratatouille, pickling or curries.\n\nCumin: Available whole, ground. Nutty, earthy, with a peppery bite. Use in Mexican, Thai, Mediterranean dishes, dry rubs, chilies, stews or with beans or lentils.\n\nDill Seed: Available whole, ground. Caraway-dill flavor. Not to be substituted for dill weed, which has a different flavor. Use in pickling, breads or salad dressings.\n\nDill Weed (Dried): Dried dill leaves. Not to be substituted for dill seed, which has a different flavor. Use in breads, dips, pickling, salads, sauces or with vegetables or fish.\n\nFennel Seed: Available whole, ground. Sweet anise flavor. Use in breads, cakes or sweet sausages.\n\nGarlic: Available minced, powdered (finely ground), dehydrated, flaked, fresh or as paste or juice. Slightly musty flavor. Use in appetizers, bread, salads, soups or with vegetables, fish, poultry or meat.\n\nGinger: Available ground, crystallized, fresh. Spicy-hot, sweet flavor. Use in beverages, baked goods, sauces, soup, Asian dishes, tea or with fruit, vegetables, fish, poultry or meat.\n\nMustard Seed: Available whole, ground, powdered (finely ground), or as oil. Use seeds in pickling or with vegetables, cooked meats, ground or powdered in sauces, main dishes, pasta salads or salad dressings; oil for stir-fries, salad dressings or marinades.\n\nNutmeg: Available whole, ground (Mace is the covering of the nutmeg seed, so it can be used the same way). Sweet, spicy flavor. Use in baked goods, eggnog and other beverages, custards, sauces or with vegetables.\n\nPaprika: Available ground. Ranges from sweet to hot in flavor; slightly bitter. Use in casseroles, eggs, salads, soups or with vegetables, fish or meat.\n\nPeppercorns: Available in black, white and green varieties; whole, ground, cracked (green sold packed in brine or dried). Slightly hot flavor with a hint of sweetness (black is the strongest, green the mildest). Use in savory foods of all kinds. Pink peppercorns are not true peppercorns but rather dried berries of the Baies rose plant.\n\nRed Pepper (Cayenne): Available ground. Very hot, peppery flavor. Use in savory foods of all kinds.\n\nSaffron: Available in threads, powdered. Distinctive flavor; mildly bitter. Use in Indian and Spanish dishes.\n\nSesame Seed: Seeds available in a variety of colors or as oil. Light, nutty flavor (white seeds milder than black). Use to add flavor and crunch to breads, salads, stir-fries or other Asian dishes.\n\nStar Anise: Available whole, ground. Bitter, more pungent flavor than anise. Key ingredient in five-spice powder. Use in Asian dishes.\n\nTurmeric: Available ground and increasingly available fresh. Fragrant, woodsy flavor. Key ingredient in prepackaged curry powder, food color. Use in eggs, pickling, rice, Indian dishes, poultry or seafood.\n\n## Teriyaki Marinade Calorie Smart \u2022 EASY\n\nPrep 5 min Total 1 hr 5 min \u25cf \u00bd cup\n\n  * \u00bc cup water\n  * 3 tablespoons soy sauce\n  * 1 tablespoon fresh lemon juice\n  * 1 tablespoon vegetable oil\n  * 1 tablespoon packed brown sugar\n  * \u215b teaspoon coarsely ground black pepper\n  * 1 clove garlic, finely chopped\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all ingredients.\n\n2 Add about 1 pound boneless or 2 to 3 pounds bone-in beef, pork or chicken, turning to coat with marinade. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 25; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 340mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 2g); Protein 0g Exchanges: \u00bd Fat Carbohydrate Choices: 0\n\nFajita Marinade\n\n## Fajita Marinade Calorie Smart\n\nPrep 5 min Total 1 hr 5 min \u25cf \u00bd cup\n\n  * \u00bc cup vegetable oil\n  * \u00bc cup red wine vinegar\n  * 1 teaspoon sugar\n  * 1 teaspoon dried oregano leaves\n  * 1 teaspoon chili powder\n  * \u00bd teaspoon garlic powder\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon pepper\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all ingredients.\n\n2 Add about 1 pound boneless or 2 to 3 pounds bone-in beef, pork or chicken, turning to coat with marinade. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 70; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1\u00bd Fat Carbohydrate Choices: 0\n\nSeasoned Brine\n\n## Seasoned Brine Easy\n\nBrining is a great way to impart flavor into meat or poultry but also is a sure-fire way to ensure it will cook up incredibly tender and juicy. You may find that brined meat and poultry takes a little longer to cook, due to the added moisture.\n\nPrep 5 min Total 24 hr 5 min \u25cf 4 cups\n\n  * 4 cups water\n  * \u00bd cup packed brown sugar\n  * 2 tablespoons salt\n  * 2 cloves garlic, crushed\n  * 1 teaspoon cracked black pepper*\n\n1 In medium bowl, mix all ingredients until brown sugar and salt are dissolved.\n\n2 Place up to 5 pounds meat or poultry in 2-gallon heavy-duty resealable food-storage plastic bag. Add brine; seal bag. Refrigerate 24 hours, turning occasionally.\n\n3 Remove meat or poultry from brine; discard brine. Grill or cook as desired.\n\n* If you can't find cracked pepper, use whole peppercorns. To crack peppercorns, place in small resealable food-storage plastic bag. Seal bag; pound with meat mallet or rolling pin just to crack open. Or, use mortar and pestle.\n\n1 Tablespoon: Calories 110; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 3550mg; Total Carbohydrate 28g (Dietary Fiber 0g, Sugars 27g); Protein 0g Exchanges: Free Carbohydrate Choices: 2\n\nCut of Meat or Poultry | Weight in Pounds | Brining Time in Hours\n\n---|---|---\n\nTurkey\n\nWhole | 8 to 12\n\n12 to 14\n\n14 to 20\n\n20 to 24 | 6 to 12\n\n9 to 14\n\n10 to 20\n\n15 to 24\n\nBreast (bone-in) | 5 to 7 | 3\u00bd to 5\u00bd\n\nChicken\n\nWhole | 3 to 3\u00bd | 8 to 24\n\nPieces\n\n|  |\n\n1 to 3\n\nCornish Game Hens\n\n|  |\n\n8 to 24\n\nPork\n\nChops\n\n|  |\n\n3 to 8\n\nTenderloin | \u00be to 1 | 8 to 12\n\nPork Loin Roast (bone-in or boneless) | 2 to 5 | 12 to 24\n\n## Garlic Marinade Calorie Smart\n\nPrep 10 min Total 1 hr 15 min \u25cf \u00be cup\n\n  * \u00bc cup vegetable oil\n  * 4 cloves garlic, finely chopped\n  * 1 tablespoon chopped fresh or 1 teaspoon dried rosemary leaves, crushed\n  * \u00bd teaspoon ground mustard\n  * 2 teaspoons soy sauce\n  * \u00bc cup red or white wine vinegar\n  * \u00bc cup dry sherry or apple juice\n\n1 In 10-inch skillet, heat oil over medium-high heat. Cook garlic in oil, stirring frequently, until golden. Stir in rosemary, mustard and soy sauce; remove from heat. Stir in vinegar and sherry; cool.\n\n2 In shallow glass or plastic dish or resealable food-storage plastic bag, place 2 to 2\u00bd pounds boneless or 3 to 4 pounds bone-in beef, pork or lamb. Pour marinade over meat. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 45; Total Fat 4.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 50mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1 Fat Carbohydrate Choices: 0\n\nBalsamic Garlic Marinade Substitute \u00bd cup balsamic vinegar for the wine vinegar and sherry.\n\nOnion and Garlic Marinade Cook \u00bc cup finely chopped onion with the garlic until tender. Substitute thyme for the rosemary.\n\n## Marinade Success\n\n  * Use glass or plastic containers or food-storage plastic bags to mix and marinate in. They won't react with acidic ingredients like wine, lemon juice or vinegar.\n  * Use \u00bd cup marinade for each 1 pound of meat, poultry, fish or vegetables.\n  * Marinate all foods except fish in the refrigerator up to 24 hours\u2014any longer and food can become mushy. Marinate fish only 15 to 30 minutes.\n\n## Lemon-Herb Marinade Calorie Smart\n\nPrep 10 min Total 1 hr 10 min \u25cf \u2154 cup\n\n  * \u2153 cup vegetable oil\n  * 1 teaspoon grated lemon peel\n  * 2 tablespoons fresh lemon juice\n  * 1 tablespoon dry vermouth, dry white wine or beef broth\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried sage leaves\n  * 1 teaspoon chopped fresh or \u00bc teaspoon dried oregano leaves\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon coarsely ground black pepper\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all ingredients.\n\n2 Add up to 1\u00bd to 2 pounds boneless or 2\u00bd to 3\u00bd pounds bone-in beef, pork, chicken or turkey, turning to coat with marinade. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 60; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 110mg; Total Carbohydrate 0g (Dietary Fiber 0g, Sugars 0g); Protein 0g Exchanges: 1\u00bd Fat Carbohydrate Choices: 0\n\n## Maple-Bourbon Marinade Easy\n\nPrep 5 min Total 1 hour 5 min \u25cf \u00bd cup\n\n  * \u00bc cup maple-flavored syrup\n  * \u00bc cup bourbon or apple juice\n  * 1 tablespoon vegetable oil\n  * \u00bd teaspoon salt\n\n1 In shallow glass or plastic dish or resealable food-storage plastic bag, mix all ingredients.\n\n2 Add 1 to 2 pounds boneless or 1\u00bd to 2\u00bd pounds bone-in beef, pork or chicken, turning to coat with marinade. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 45; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Sodium 150mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 4 g); Protein 0g Exchanges: \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## Orange-Thyme Marinade Easy\n\nPrep 15 min Total 1 hr 15 min \u25cf \u00be cup\n\n  * 1 large orange\n  * 2 tablespoons balsamic or red wine vinegar\n  * 1 tablespoon vegetable oil\n  * 2 medium green onions, finely chopped (2 tablespoons)\n  * 1 tablespoon chopped fresh thyme leaves\n  * \u00bc teaspoon salt\n  * \u215b teaspoon pepper\n\n1 Grate enough orange peel to equal 1 teaspoon; set aside. Squeeze enough juice to equal \u00bd cup. In shallow glass or plastic dish or resealable food-storage plastic bag, mix all ingredients.\n\n2 Add 2 to 2 \u00bd pounds boneless or 3 to 4 pounds bone-in beef, pork or chicken, turning to coat with marinade. Cover dish or seal bag; refrigerate at least 1 hour but no longer than 24 hours, turning meat occasionally.\n\n3 Remove meat from marinade; reserve marinade. Cook meat as desired, brushing occasionally with marinade. Remaining marinade must be boiled to be served as a sauce (if not boiled, discard marinade). In 1-quart saucepan, heat marinade to boiling, stirring constantly; boil and stir 1 minute.\n\n1 Tablespoon: Calories 20 (Calories from Fat 1); Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Sodium 50mg; Total Carbohydrate 2g (Dietary Fiber 0g, Sugars 1g); Protein 0g Exchanges: Free Carbohydrate Choices: 0\n\n# Chapter 18: Breads\n\n## Quick Bread Basics\n\nSelecting Pans\n\nPrepping Ingredients\n\nTips for Perfect Quick Breads\n\n## Features\n\nHeirloom Recipe and New Twist\n\nLearn to Knead Yeast Dough\n\nLearn with Betty: Biscuits\n\nLearn with Betty: Muffins\n\nLearn with Betty: Yeast Bread\n\nRoll Top Treatments\n\n## Muffins and Popovers\n\nApple-Cinnamon Muffins\n\nBanana Muffins\n\nBlueberry Muffins Calorie Smart\n\nBran Muffins Calorie Smart\n\nChocolate Chip Muffins\n\nCranberry-Orange Muffins\n\nDate-Bran Muffins\n\nGolden Harvest Muffins\n\nJumbo Upside-Down Muffins\n\nPopovers Calorie Smart \u2022 Easy\n\nStreusel-Topped Blueberry Muffins\n\nYorkshire Pudding\n\n## Loaves\n\nBanana Bread Calorie Smart\n\nBlueberry-Banana Bread\n\nCorn Muffins\n\nCornbread Calorie Smart\n\nCranberry Bread\n\nCranberry\u2013Sweet Potato Bread\n\nOatmeal Streusel Bread\n\nOrange-Zucchini Bread\n\nPumpkin Bread\n\nZucchini Bread Calorie Smart\n\n## Biscuits and Scones\n\nBaking Powder Biscuits Calorie Smart \u2022 Fast \u2022 Easy\n\nButtermilk Biscuits\n\nCheddar-Chile Cornbread Scones\n\nCherry\u2013Chocolate Chip Scones\n\nChocolate Chip Cream Biscuits\n\nChocolate Chip Scones\n\nCurrant Scones\n\nDrop Biscuits\n\nEasy Cream Biscuits Calorie Smart \u2022 Easy\n\nHam 'n Egg Biscuit Sandwiches\n\nHerbed Cream Biscuits\n\nLemon-Cherry Scones\n\nRaspberry\u2013White Chocolate Scones\n\nScones\n\nSweet Potato\u2013Bacon Biscuits Calorie Smart\n\n## Doughnuts and Cakes\n\nApplesauce Doughnuts\n\nBaked Blueberry-Orange Doughnuts\n\nBeignets with Espresso Sugar Calorie Smart\n\nCake Doughnuts Calorie Smart\n\nGlazed Doughnuts\n\nLemon Curd\u2013Filled Butter Cake\n\nSour Cream Coffee Cake\n\nStrawberry Cream Brunch Cake\n\n## Yeast Bread Basics\n\nYeast Bread Ingredients\n\nStoring and Serving Yeast Breads\n\n## Kneaded and Shaped Breads\n\nArtisan Asiago Bread Calorie Smart\n\nBakery-Style Pumpernickel Loaf\n\nBlack Pepper and Parmesan Grissini Calorie Smart\n\nChallah Bread\n\nCinnamon Swirl Raisin Bread\n\nClassic White Bread Calorie Smart\n\nDried Cherry and Walnut Bread\n\nEasy Naan\n\nEnglish Muffins Calorie Smart\n\nFocaccia Calorie Smart\n\nFocaccia Breadsticks\n\nFrench Bread Calorie Smart\n\nHoney\u2013Whole Wheat Bread Calorie Smart\n\nOnion Focaccia\n\nRich Egg Bread Calorie Smart\n\nSourdough Bread CALORIE SMART\n\nSourdough Starter\n\nSunflower-Herb Whole Wheat Bread\n\nWhole Grain Artisan Bread CALORIE SMART\n\n## No-Knead Breads\n\nFour-Grain Batter Bread CALORIE SMART\n\nNo-Knead Artisan Bread Calorie Smart\n\nWhole Wheat\u2013Raisin Batter Bread\n\n## Rolls and Pretzels\n\nBakery-Style Pumpernickel Rolls Calorie Smart\n\nCaramel Sticky Rolls\n\nCinnamon Rolls\n\nCloverleaf Rolls\n\nCrescent Rolls\n\nDinner Rolls CALORIE SMART\n\nParmesan-Herb Soft Pretzels\n\nSoft Pretzels CALORIE SMART\n\n## Bonus Recipes\n\nBLT Crostini\n\nBread Salad\n\nCheesy Garlic Bread\n\nHazelnut-Stuffed French Toast\n\nRosemary-Parmesan Croutons\n\nSpaghetti & Meatballs Pizza\n\nTurkey-Provolone-Dill Party Sub\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Quick Bread Basics\n\nBreads that use baking powder or baking soda as the leavening to make them rise are called quick breads. Unlike yeast breads, which require rising time, quick breads usually can be baked as soon as they are mixed. Muffins, scones and biscuits are a few common types of quick breads that you can make, no matter what your skill level, as they offer fresh-from-the oven results, easily.\n\n## SELECTING PANS\n\n  * Use only the pan size indicated in the recipe.\n  * For golden-brown color and tender crusts, use shiny pans and cookie sheets rather than pans with dark finishes, to reflect the heat.\n  * If using dark or nonstick pans, reduce oven temperature by 25\u00b0F. These pans absorb heat more easily than shiny pans, causing baked goods to brown more quickly.\n  * Insulated pans often require slightly longer baking times and result in baked goods that may be less brown.\n\n## PREPPING INGREDIENTS\n\n  * Check expiration dates on leavening packages for freshness. Old products may inhibit quick breads from rising properly.\n  * Chop, mash or shred fruits, vegetables and nuts in advance so they are ready to be added to the batter.\n  * Measure flour by spooning into a dry measuring cup and leveling off with the straight edge of a knife or metal spatula.\n  * Use liquid measuring cups to measure liquids by filling and placing on the counter to read amount accurately at eye level.\n  * In recipes calling for butter, it's best to use real butter. If you use margarine, be sure the product has at least 65 percent fat. Don't use reduced-fat butter or whipped products.\n\n## Tips for Perfect Quick Breads\n\n  * Mix most batters for quick breads with a spoon, not an electric mixer, just until the dry ingredients are moistened, as overmixing can result in tough quick breads with peaked tops.\n  * For those recipes that require a mixer, we tested with handheld electric mixers. If you have a stand mixer, follow the manufacturer's directions for speed settings.\n  * To prevent gummy, soggy or heavy loaves, don't use more fruit or vegetables than called for in the recipe.\n  * Grease bottom only of loaf or muffin pans, unless otherwise specified; this prevents a lip and a hard, dry edge from forming.\n  * Allow at least 2 inches of space around pans in the oven for heat circulation.\n  * Quick bread loaves normally form cracks on top; this is due to the leavening action and is perfectly fine.\n  * Cool loaves completely, about 2 hours, to prevent crumbling when sliced.\n  * Cut loaves with a serrated knife, using a light sawing motion. Flavor improves and slicing is easier if loaves have been refrigerated for 24 hours.\n  * Wrap completely cooled loaves tightly in plastic wrap or foil and refrigerate up to 1 week. To freeze, place tightly wrapped loaves in freezer bags and freeze up to 3 months.\n\nCake Doughnuts (page 492)\n\nBlueberry Muffins\n\n## Blueberry Muffins Calorie Smart\n\nHomemade blueberry muffins burst with fresh flavor that store-bought can never duplicate\u2014and it doesn't take rocket science to make them great. Overmixing leads to peaked tops, tough interiors and dry muffins, so follow these simple steps and your muffins will be beautiful and delicious!\n\nPrep 10 min Total 40 min \u25cf 12 muffins\n\n  * \u00be cup milk\n  * \u00bc cup vegetable oil or melted butter\n  * 1 egg\n  * 2 cups all-purpose flour\n  * \u00bd cup granulated sugar\n  * 2 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 1 cup fresh, canned (drained) or frozen (do not thaw) blueberries\n  * 2 tablespoons coarse sugar or additional granulated sugar, if desired\n\n1 Heat oven to 400\u00b0F. Grease bottoms only of 12 regular-size muffin cups with shortening or cooking spray, or place paper baking cup in each muffin cup.\n\n2 In large bowl, beat milk, oil and egg with fork or whisk until well mixed. Stir in flour, granulated sugar, baking powder and salt all at once just until flour is moistened (batter will be lumpy). Fold in blueberries. Divide batter evenly among muffin cups; sprinkle each with \u00bd teaspoon coarse sugar.\n\n3 Bake 20 to 25 minutes or until golden brown and toothpick inserted in center comes out clean. If baked in greased pan, let stand about 5 minutes in pan, then remove from pan to cooling rack; if baked in paper baking cups, immediately remove from pan to cooling rack. Serve warm if desired.\n\n1 Muffin: Calories 170; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 20mg; Sodium 190mg; Total Carbohydrate 27g (Dietary Fiber 0g, Sugars 10g); Protein 3g Exchanges: 1 Starch, 1 Fruit, 1 Fat Carbohydrate Choices: 2\n\nApple-Cinnamon Muffins Omit blueberries. Stir in 1 cup chopped peeled apple (about 1 medium) and \u00bd teaspoon ground cinnamon with the flour. Bake 25 to 30 minutes.\n\nBanana Muffins Omit blueberries. Reduce milk to \u2153 cup. Beat in 1 cup mashed very ripe bananas (2 medium) with the milk. Use packed brown sugar instead of granulated sugar in the muffins.\n\nChocolate Chip Muffins Omit blueberries. Fold 1 cup miniature semisweet chocolate chips into batter.\n\nCranberry-Orange Muffins Omit blueberries. Beat in 1 tablespoon grated orange peel with the milk. Fold 1 cup coarsely chopped sweetened dried cranberries into batter.\n\nStreusel-Topped Blueberry Muffins Omit coarse sugar. Make batter as directed; divide evenly among muffin cups. In medium bowl, mix \u00bc cup all-purpose flour, \u00bc cup packed brown sugar and \u00bc teaspoon ground cinnamon. Cut in 2 tablespoons cold butter, using pastry blender or fork, until crumbly. Sprinkle about 1 tablespoon streusel over batter in each muffin cup. Bake as directed.\n\n### Preparing Muffins\n\nStir dry ingredients in all at once just until flour is moistened.\n\nDivide batter evenly among muffin cups.\n\nBran Muffins\n\n## Bran Muffins Calorie Smart\n\nPrep 15 min Total 50 min \u25cf 12 muffins\n\n  * 1\u00bc cups original bran cereal shreds or 2 cups bran cereal flakes, crushed\n  * 1\u2153 cups milk\n  * \u00bd cup raisins, dried cherries or sweetened dried cranberries, if desired\n  * \u00bd teaspoon vanilla\n  * \u00bc cup vegetable oil\n  * 1 egg\n  * 1\u00bc cups all-purpose or whole wheat flour\n  * \u00bd cup packed brown sugar\n  * 1 tablespoon baking powder\n  * \u00bc teaspoon salt\n  * \u00bc teaspoon ground cinnamon, if desired\n\n1 Heat oven to 400\u00b0F. Grease bottoms only of 12 regular-size muffin cups with shortening or cooking spray, or place paper baking cup in each muffin cup.\n\n2 In medium bowl, stir cereal, milk, raisins and vanilla until well mixed. Let stand about 5 minutes or until cereal has softened. Beat in oil and egg with fork.\n\n3 In another medium bowl, stir remaining ingredients until well mixed; stir into cereal mixture just until moistened. Divide batter evenly among muffin cups.\n\n4 Bake 18 to 25 minutes or until toothpick inserted in center comes out clean. If baked in greased pan, let stand about 5 minutes in pan, then remove from pan to cooling rack; if baked in paper baking cups, immediately remove from pan to cooling rack. Serve warm if desired.\n\n1 Muffin: Calories 170; Total Fat 6g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 20mg; Sodium 220mg; Total Carbohydrate 26g (Dietary Fiber 3g, Sugars 10g); Protein 3g Exchanges: 1\u00bd Starch, 1 Fat Carbohydrate Choices: 2\n\nMake-Ahead Directions Bake muffins as directed; cool completely. Place in freezer containers or resealable freezer plastic bags. Freeze up to 3 months. Reheat in the microwave.\n\nDate-Bran Muffins Stir in 1 cup chopped dates with remaining ingredients in Step 3.\n\nJumbo Upside-Down Muffins Grease bottoms only of 6 jumbo muffin cups with shortening or cooking spray (do not use paper baking cups). Mix 3 tablespoons packed brown sugar, 2 tablespoons melted butter and 1 tablespoon light corn syrup in small bowl until well mixed. Spoon about 2 teaspoons into the bottom of each muffin cup. Prepare batter as directed, except substitute \u00be cup chopped dates for the raisins. Bake 18 to 25 minutes. Run metal spatula around muffins to loosen. Immediately place cookie sheet upside down on muffin pan; turn cookie sheet and pan over to remove muffins.\n\n### Learn with Betty Muffins\n\n**Perfect Muffin:** This muffin is slightly rounded, with a bumpy top.\n\n**Overmixed Muffin:** This muffin has a peaked, smooth top.\n\n**Overbaked Muffin:** This muffin is dry with a rough top and is too brown.\n\n## Golden Harvest Muffins\n\nPrep 20 min Total 50 min \u25cf 18 muffins\n\n  * 2 eggs\n  * \u00be cup vegetable oil\n  * \u00bc cup milk\n  * 2 teaspoons vanilla\n  * 2 cups all-purpose flour\n  * 1 cup packed brown sugar\n  * 2 teaspoons baking soda\n  * 2 teaspoons ground cinnamon\n  * \u00bd teaspoon salt\n  * 1\u00bd cups shredded carrots (2 or 3 medium)\n  * 1 cup shredded peeled apple (1 medium)\n  * \u00bd cup coconut\n  * \u00bd cup raisins\n  * \u00be cup sliced almonds\n\n1 Heat oven to 350\u00b0F. Grease bottoms only of 18 regular-size muffin cups with shortening or cooking spray, or place paper baking cup in each muffin cup.\n\n2 In large bowl, beat eggs, oil, milk and vanilla with whisk until well blended. Add flour, brown sugar, baking soda, cinnamon and salt; stir just until dry ingredients are moistened. Stir in carrots, apple, coconut, raisins and \u00bd cup of the almonds. Divide batter evenly among muffin cups. Sprinkle remaining \u00bc cup almonds over batter.\n\n3 Bake 20 to 25 minutes or until toothpick inserted in center comes out clean. If baked in greased pan, let stand about 5 minutes in pan, then remove from pan to cooling rack; if baked in paper baking cups, immediately remove from pan to cooling rack. Serve warm if desired.\n\n1 Muffin: Calories 250; Total Fat 13g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 230mg; Total Carbohydrate 29g (Dietary Fiber 2g, Sugars 17g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 2\n\nBanana Bread\n\n## Banana Bread Calorie Smart\n\nAlthough the true origin of banana bread is not known, this enduring favorite has been around at least since the 1930s. Our version of banana bread has appeared in all editions of the Betty Crocker Cookbook. It's a great way to use those bananas that are turning brown and soft.\n\nPrep 15 min Total 3 hr 40 min \u25cf 2 loaves (12 slices each)\n\n  * 1\u00bc cups sugar\n  * \u00bd cup butter, softened\n  * 2 eggs\n  * 1\u00bd cups mashed very ripe bananas (3 medium)\n  * \u00bd cup buttermilk\n  * 1 teaspoon vanilla\n  * 2\u00bd cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon salt\n  * 1 cup chopped nuts, if desired\n\n1 Heat oven to 350\u00b0F. Grease bottoms only of 2 (8x4- or 9x5-inch) loaf pans with shortening or cooking spray.\n\n2 In large bowl, stir sugar and butter until well mixed. Stir in eggs until well mixed. Stir in bananas, buttermilk and vanilla; beat with spoon until smooth. Stir in flour, baking soda and salt just until moistened. Stir in nuts. Divide batter evenly between pans.\n\n3 Bake 8-inch loaves about 1 hour, 9-inch loaves about 1 hour 15 minutes, or until toothpick inserted in center comes out clean. Cool 10 minutes in pans on cooling rack.\n\n4 Loosen sides of loaves from pans; remove from pans and place top side up on cooling rack. Cool completely, about 2 hours, before slicing. Wrap tightly and store at room temperature up to 4 days, or refrigerate.\n\n1 Slice: Calories 150; Total Fat 4.5g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 30mg; Sodium 190mg; Total Carbohydrate 24g (Dietary Fiber 0g, Sugars 12g); Protein 2g Exchanges: \u00bd Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\u00bd\n\nBlueberry-Banana Bread Omit nuts. Stir in 1 cup fresh or frozen (do not thaw) blueberries into batter.\n\nCranberry Bread\n\n## Zucchini Bread Calorie Smart\n\nYou don't need to peel the zucchini if it is very fresh and green. Older, larger zucchini may require peeling before shredding. The peel adds little green flecks to the bread.\n\nPrep 20 min Total 3 hr 50 min \u25cf 2 loaves (12 slices each)\n\n  * 3 cups shredded zucchini (2 or 3 medium)\n  * 1\u2154 cups sugar\n  * \u2154 cup vegetable oil\n  * 2 teaspoons vanilla\n  * 4 eggs\n  * 3 cups all-purpose or whole wheat flour\n  * 2 teaspoons baking soda\n  * 1 teaspoon salt\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon baking powder\n  * \u00bd teaspoon ground cloves\n  * \u00bd cup chopped nuts\n  * \u00bd cup raisins, if desired\n\n1 Heat oven to 350\u00b0F. Grease bottoms only of 2 (8x4- or 9x5-inch) loaf pans with shortening or cooking spray.\n\n2 In large bowl, stir zucchini, sugar, oil, vanilla and eggs until well mixed. Stir in flour, baking soda, salt, cinnamon, baking powder and cloves. Stir in nuts and raisins. Divide batter evenly between pans.\n\n3 Bake 8-inch loaves 50 to 60 minutes, 9-inch loaves 1 hour 10 minutes to 1 hour 20 minutes, or until toothpick inserted in center comes out clean. Cool 10 minutes in pans on cooling rack.\n\n4 Loosen sides of loaves from pans; remove from pans and place top side up on cooling rack. Cool completely, about 2 hours, before slicing. Wrap tightly and store at room temperature up to 4 days, or refrigerate up to 10 days.\n\n1 Slice: Calories 200; Total Fat 9g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 35mg; Sodium 230mg; Total Carbohydrate 27g (Dietary Fiber 1g, Sugars 14g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nCranberry Bread Omit zucchini, cinnamon, cloves and raisins. Add \u00bd cup milk and 2 teaspoons grated orange peel with the oil. Stir 3 cups fresh or frozen (thawed and drained) cranberries into batter. Bake 1 hour to 1 hour 10 minutes.\n\nOrange-Zucchini Bread Stir in 2 teaspoons grated orange peel with the flour mixture. When loaf is cooled, drizzle with Orange Glaze (page 581). Let stand until set.\n\nPumpkin Bread Substitute 1 can (15 ounces) pureed pumpkin (not pumpkin pie mix) for the zucchini.\n\nCranberry\u2013Sweet Potato Bread\n\n## Cranberry\u2013Sweet Potato Bread\n\nPrep 25 min Total 3 hr 35 min \u25cf 2 loaves (12 slices each)\n\n  * 2\u2153 cups sugar\n  * \u2154 cup water\n  * \u2154 cup vegetable oil\n  * 1 teaspoon vanilla\n  * 2 cups mashed cooked dark-orange sweet potatoes (about 1\u00bc lb)*\n  * 4 eggs\n  * 3\u2153 cups all-purpose flour\n  * 2 teaspoons baking soda\n  * 1\u00bd teaspoons salt\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon baking powder\n  * \u00bd teaspoon ground nutmeg\n  * 1 cup sweetened dried cranberries\n  * 1 cup chopped pecans, if desired\n\n1 Heat oven to 350\u00b0F. Grease 2 loaf pans (8x4- or 9x5-inch) with shortening or cooking spray; lightly flour.\n\n2 In large bowl, mix sugar, water, oil, vanilla, sweet potatoes and eggs until well blended. In medium bowl, mix flour, baking soda, salt, cinnamon, baking powder and nutmeg. Add to sweet potato mixture; stir just until dry ingredients are moistened. Stir in cranberries and pecans. Divide batter evenly between pans.\n\n3 Bake 8-inch loaves about 1 hour, 9-inch loaves about 1 hour 10 minutes, or until toothpick inserted in center comes out clean. Cool 15 minutes in pans on cooling rack.\n\n4 Loosen sides of loaves from pans; remove from pans and place top side up on cooling rack. Cool completely, about 2 hours, before slicing. Wrap tightly and store at room temperature up to 4 days, or refrigerate.\n\n*Or substitute 1 can (23 ounces) sweet potatoes in syrup, drained and mashed, for the cooked sweet potatoes.\n\n1 Slice: Calories 240; Total Fat 7g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 35mg; Sodium 280mg; Total Carbohydrate 41g (Dietary Fiber 1g, Sugars 25g); Protein 3g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 3\n\nCornbread\n\n## Cornbread Calorie Smart\n\nPrep 10 min Total 35 min \u25cf 12 servings\n\n  * 1 cup milk\n  * \u00bc cup butter, melted\n  * 1 egg\n  * 1\u00bc cups yellow, white or blue cornmeal\n  * 1 cup all-purpose flour\n  * \u00bd cup sugar\n  * 1 tablespoon baking powder\n  * \u00bd teaspoon salt\n\n1 Heat oven to 400\u00b0F. Grease bottom and side of 9-inch round pan, 8-inch square pan or 8-inch cast-iron skillet with shortening.\n\n2 In large bowl, beat milk, butter and egg with whisk. Stir in remaining ingredients all at once just until flour is moistened (batter will be lumpy). Pour into pan.\n\n3 Bake 20 to 25 minutes or until golden brown and toothpick inserted in center comes out clean. Serve warm if desired.\n\n1 Serving: Calories 170; Total Fat 5g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 30mg; Sodium 260mg; Total Carbohydrate 29g (Dietary Fiber 0g, Sugars 10g); Protein 4g Exchanges: 1 Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\nCorn Muffins Grease bottoms only of 12 regular-size muffin cups with shortening or cooking spray. Divide batter evenly among muffin cups. Bake as directed.\n\nOatmeal Streusel Bread\n\n## Oatmeal Streusel Bread\n\nPrep 25 min Total 3 hr 30 min \u25cf 1 loaf (12 slices)\n\nStreusel\n\n  * \u00bc cup packed brown sugar\n  * \u00bc cup chopped walnuts, toasted (page 23)\n  * 2 teaspoons ground cinnamon\n\nBread\n\n  * 1 cup all-purpose flour\n  * \u00bd cup whole wheat flour\n  * \u00bd cup old-fashioned oats\n  * 2 tablespoons ground flaxseed or flaxseed meal\n  * 1 teaspoon baking powder\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon baking soda\n  * \u00be cup packed brown sugar\n  * \u2154 cup vegetable oil\n  * 2 eggs\n  * \u00bc cup sour cream\n  * 2 teaspoons vanilla\n  * \u00bd cup milk\n\nIcing\n\n  * \u00be to 1 cup powdered sugar\n  * 1 tablespoon milk\n  * 2 teaspoons light corn syrup\n\n1 Heat oven to 350\u00b0F. Spray bottom only of 9x5-inch loaf pan with cooking spray. In small bowl, stir together streusel ingredients; set aside.\n\n2 In medium bowl, stir all-purpose flour, whole wheat flour, oats, flaxseed, baking powder, salt and baking soda with whisk until blended.\n\n3 In large bowl, beat brown sugar and oil with electric mixer on medium speed 1 minute or until blended. Add eggs, one at a time, beating well after each addition. Beat in sour cream and vanilla. Add flour mixture alternately with milk, beating on low speed after each addition until smooth.\n\n4 Spoon one-third of batter in pan (about 1 cup); sprinkle with half of the streusel (about \u2153 cup). Top with half of remaining batter (about 1 cup), spreading gently. Sprinkle with remaining streusel. Top with remaining batter. Gently run knife through batter to swirl.\n\n5 Bake 50 to 55 minutes or until toothpick inserted in center comes out clean. Cool on cooling rack 10 minutes. Loosen sides of loaf from pan; remove from pan and place top side up on cooling rack. Cool completely, about 2 hours.\n\n6 In small bowl, beat icing ingredients, adding enough of the powdered sugar for desired drizzling consistency. Drizzle icing over bread. Let stand until set. Wrap tightly and store at room temperature up to 4 days, or refrigerate.\n\n1 Slice: Calories 320; Total Fat 16g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 40mg; Sodium 190mg; Total Carbohydrate 40g (Dietary Fiber 1g, Sugars 27g); Protein 4g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, 3 Fat Carbohydrate Choices: 2\u00bd\n\n# Heirloom Recipe and New Twist\n\nThere's nothing like hot, homemade biscuits when you're craving comfort food. Our heirloom recipe, made with all-purpose flour, makes the classic biscuits the South is known for. The light, fluffy texture is perfect for holding a pat of butter or a drizzle of honey. When made with whole wheat flour, they'll take on a heartier texture and nutty flavor. Our twist recipe amplifies the Southern appeal by using sweet potatoes and bacon.\n\nHeirloom\n\nDrop Biscuits\n\n## Baking Powder Biscuits Calorie Smart \u2022 Fast \u2022 EASY\n\nPrep 10 min Total 25 min \u25cf 12 biscuits\n\n  * 2 cups all-purpose or whole wheat flour\n  * 1 tablespoon sugar\n  * 1 tablespoon baking powder\n  * 1 teaspoon salt\n  * \u00bd cup shortening or cold butter, cut into 8 pieces\n  * \u00be cup milk\n\n1 Heat oven to 450\u00b0F. In medium bowl, mix flour, sugar, baking powder and salt. Cut in shortening, using pastry blender or fork, until mixture looks like fine crumbs. Stir in milk until dough leaves side of bowl (dough will be soft and sticky).\n\n2 Place dough on lightly floured surface. Knead lightly 10 times. Roll or pat until \u00bd inch thick. Cut with floured 2- to 2\u00bc-inch biscuit cutter. On ungreased cookie sheet, place biscuits about 1 inch apart for crusty sides, touching for soft sides.\n\n3 Bake 10 to 12 minutes or until golden brown. Immediately remove from cookie sheet to cooling rack. Serve warm.\n\n1 Biscuit: Calories 160; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 1.5g); Cholesterol 0mg; Sodium 330mg; Total Carbohydrate 18g (Dietary Fiber 0g, Sugars 2g); Protein 3g Exchanges: 1 Starch, 2 Fat Carbohydrate Choices: 1\n\nButtermilk Biscuits Reduce baking powder to 2 teaspoons; add \u00bc teaspoon baking soda with the sugar. Substitute buttermilk for the milk. (If buttermilk is thick, use slightly more than \u00be cup.)\n\nDrop Biscuits Increase milk to 1 cup. Grease cookie sheet with shortening. Drop dough by 12 spoonfuls about 2 inches apart onto cookie sheet.\n\nHam 'n Egg Biscuit Sandwiches Make biscuits as directed\u2014except use 3-inch cutter (yield will be 6 biscuits). Split biscuits in half while still warm. Spread with mayonnaise and mustard if desired. On each biscuit bottom, place 1 slice American cheese, 1 fried or poached egg (see pages 90 and 91) and 1 slice deli ham; cover with biscuit tops. Serve immediately.\n\n### Making Baking Powder Biscuits\n\nCut shortening into flour mixture until the mixture looks like fine crumbs.\n\n### Learn with Betty Biscuits\n\n**Perfect Biscuit:** This biscuit is light golden brown and high, with fairly smooth sides. It will be tender and light.\n\n**Underleavened Biscuit:** This biscuit did not have enough leavening and was mixed too much. It will be tough and not flaky.\n\n**Overbaked Biscuit:** This biscuit was overbaked and the oven was too hot. It will be hard and crumbly.\nNew Twist\n\nSweet Potato\u2013Bacon Biscuits\n\n## Sweet Potato\u2013Bacon Biscuits Calorie Smart\n\nPrep 20 min Total 35 min \u25cf 8 biscuits\n\n  * 1\u00bd cups all-purpose flour\n  * 1 tablespoon baking powder\n  * \u00bc teaspoon salt\n  * 8 slices bacon, crisply cooked, crumbled\n  * \u00bd teaspoon chopped fresh thyme leaves\n  * \u00bc cup butter, frozen\n  * \u2154 cup mashed canned sweet potatoes (from 15-oz can)\n  * 2 tablespoons buttermilk\n  * 2 tablespoons butter, melted\n\n1 Heat oven to 450\u00b0F. In large bowl, mix flour, baking powder and salt; stir in bacon and thyme. Shred frozen butter, using large holes of grater; add to flour mixture and toss to coat. Stir in sweet potatoes and buttermilk just until soft dough forms.\n\n2 Place dough on lightly floured surface; gently roll in flour to coat. Knead lightly 4 or 5 times. Press or roll dough until \u00be inch thick. Cut with floured 2\u00bd-inch biscuit cutter. On ungreased cookie sheet, place biscuits about 1 inch apart.\n\n3 Bake 12 to 15 minutes or until light golden brown. Brush with melted butter. Serve warm.\n\n1 Biscuit: Calories 180; Total Fat 7g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 480mg; Total Carbohydrate 24g (Dietary Fiber 1g, Sugars 4g); Protein 6g Exchanges: 1\u00bd Starch, 1\u00bd Fat Carbohydrate Choices: 1\u00bd\n\n## Easy Cream Biscuits Calorie Smart \u2022 EASY\n\nWhipping cream replaces shortening or butter, making these biscuits rich and super tender.\n\nPrep 10 min Total 35 min \u25cf 12 biscuits\n\n  * 1\u00be cups all-purpose flour\n  * 2\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * About 1\u00bc cups whipping cream\n\n1 Heat oven to 450\u00b0F. In large bowl, mix flour, baking powder and salt. Stir in just enough whipping cream so dough leaves side of bowl and forms a ball. (If dough is too dry, mix in 1 to 2 teaspoons more whipping cream.)\n\n2 Place dough on lightly floured surface; gently roll in flour to coat. Knead lightly 10 times, sprinkling with flour if dough is too sticky. Roll or pat until \u00bd inch thick. Cut with floured 2- to 2\u00bc-inch biscuit cutter. On ungreased cookie sheet, place biscuits about 1 inch apart.\n\n3 Bake 10 to 12 minutes or until golden brown. Immediately remove from cookie sheet to cooling rack. Serve warm.\n\n1 Biscuit: Calories 140; Total Fat 8g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 30mg; Sodium 210mg; Total Carbohydrate 15g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\nChocolate Chip Cream Biscuits Stir in 1 tablespoon sugar and \u00bc cup miniature semisweet chocolate chips with the flour. Drizzle with Chocolate Glaze (page 582) if desired.\n\nHerbed Cream Biscuits Stir in \u00bd teaspoon Italian seasoning with the flour.\n\nCherry\u2013Chocolate Chip Scones\n\n## Scones\n\nPrep 15 min Total 35 min \u25cf 8 scones\n\n  * 1\u00be cups all-purpose flour\n  * 3 tablespoons granulated sugar\n  * 2\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u2153 cup cold butter, cut into 8 pieces\n  * 1 egg, beaten\n  * \u00bd teaspoon vanilla\n  * 4 to 6 tablespoons whipping cream\n  * Additional whipping cream\n  * Coarse sugar or additional granulated sugar\n\n1 Heat oven to 400\u00b0F. In large bowl, mix flour, granulated sugar, baking powder and salt. Cut in butter, using pastry blender or fork, until mixture looks like fine crumbs. Stir in egg, vanilla and just enough of the 4 to 6 tablespoons whipping cream so dough leaves side of bowl.\n\n2 Place dough on lightly floured surface; gently roll in flour to coat. Knead lightly 10 times. On ungreased cookie sheet, roll or pat dough into 8-inch round. Cut into 8 wedges with sharp knife that has been dipped in flour, but do not separate wedges. Brush with additional whipping cream; sprinkle with coarse sugar.\n\n3 Bake 14 to 16 minutes or until light golden brown. Immediately remove from cookie sheet; carefully separate wedges. Serve warm.\n\n1 Scone: Calories 240; Total Fat 11g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 55mg; Sodium 360mg; Total Carbohydrate 31g (Dietary Fiber 0g, Sugars 10g); Protein 4g Exchanges: 1 Starch, 1 Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\n\nCherry\u2013Chocolate Chip Scones Stir in \u00bd cup each dried cherries and miniature semisweet chocolate chips with the egg, vanilla and whipping cream.\n\nChocolate Chip Scones Stir in \u00bd cup miniature semisweet chocolate chips with the egg, vanilla and whipping cream.\n\nCurrant Scones Stir in \u00bd cup dried currants or raisins with the egg, vanilla and whipping cream.\n\nLemon-Cherry Scones Stir in \u00bd cup dried cherries and 2 teaspoons grated lemon peel with the egg, vanilla and whipping cream.\n\nRaspberry-White Chocolate Scones Substitute almond extract for the vanilla; increase whipping cream to \u00bd cup. Stir in \u00be cup frozen unsweetened raspberries (do not thaw) and \u2154 cup white vanilla baking chips with the egg, almond extract and whipping cream. Omit kneading step. On ungreased cookie sheet, pat dough into 8-inch round. Continue as directed, except bake 18 to 23 minutes. Raspberries will color the dough a little bit.\n\n### Making Scones\n\nRoll or pat dough into 8-inch round on ungreased cookie sheet.\n\nCut into 8 wedges with sharp knife or pizza cutter dipped in flour; do not separate.\n\nCheddar-Chile Cornbread Scones\n\n## Cheddar-Chile Cornbread Scones\n\nPrep 20 min Total 45 min \u25cf 8 scones\n\n  * 1\u00bc cups white whole wheat flour\n  * 1 cup cornmeal\n  * 1 tablespoon sugar\n  * 2 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u00bd cup cold butter, cut into 8 pieces\n  * \u00bc cup milk\n  * 1 egg, beaten\n  * \u00be cup shredded Cheddar cheese (3 oz)\n  * 1 can (4.5 oz) chopped green chiles, undrained\n  * \u2154 cup crumbled crisply cooked bacon, if desired\n  * Honey, if desired\n\n1 Heat oven to 425\u00b0F. Grease cookie sheet with shortening or cooking spray. In large bowl, mix flour, cornmeal, sugar, baking powder and salt. Cut in butter, using pastry blender or fork, until mixture looks like coarse crumbs. Stir in milk, egg, cheese, chiles and bacon.\n\n2 Place dough on lightly floured surface; knead lightly 10 times. On cookie sheet, roll or pat dough into 8-inch round. Cut into 8 wedges, but do not separate wedges.\n\n3 Bake 18 to 23 minutes or until golden brown. Immediately remove from cookie sheet; carefully separate wedges. Serve warm with honey.\n\n1 Scone: Calories 310; Total Fat 17g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 65mg; Sodium 510mg; Total Carbohydrate 33g (Dietary Fiber 2g, Sugars 3g); Protein 7g Exchanges: 1 Starch, 1 Other Carbohydrate, \u00bd High-Fat Meat, 2\u00bd Fat Carbohydrate Choices: 2\n\nPopovers\n\n## Popovers Calorie Smart \u2022 EASY\n\nPopovers get their name because they pop up high out of their baking cups in the oven. Serve them hot with butter or try one of the sweet or savory butters on page 470.\n\nPrep 10 min Total 45 min \u25cf 6 popovers\n\n  * 2 eggs\n  * 1 cup all-purpose flour\n  * 1 cup milk\n  * \u00bd teaspoon salt\n\n1 Heat oven to 450\u00b0F. Generously grease 6-cup popover pan with shortening. Heat popover pan in oven 5 minutes.\n\n2 Meanwhile, in medium bowl, beat eggs slightly with fork or whisk. Beat in remaining ingredients just until smooth (do not overbeat or popovers may not puff as high). Fill cups about half full.\n\n3 Bake 20 minutes. Reduce oven temperature to 325\u00b0F. Bake 10 to 15 minutes longer or until deep golden brown. Immediately remove from cups. Serve hot.\n\n1 Popover: Calories 120; Total Fat 3g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 75mg; Sodium 240mg; Total Carbohydrate 18g (Dietary Fiber 0g, Sugars 2g); Protein 6g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\nYorkshire Pudding Heat oven to 350\u00b0F. Place \u00bc cup vegetable oil or meat drippings in 9-inch square pan; place pan in oven and heat until hot. Increase oven temperature to 450\u00b0F. Prepare popover batter; carefully pour into hot oil. Bake 18 to 23 minutes or until puffy and golden brown (pudding will puff during baking but will deflate shortly after being removed from oven). Cut pudding into squares; serve immediately.\n\n## Cake Doughnuts Calorie smart\n\nDon't forget to also fry the holes\u2014no one can resist bite-size treats.\n\nPrep 45 min Total 45 min \u25cf 24 doughnuts\n\n  * Vegetable oil\n  * 3\u2153 cups all-purpose flour\n  * 1 cup sugar\n  * \u00be cup milk\n  * 1 tablespoon baking powder\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground cinnamon\n  * \u00bc teaspoon ground nutmeg\n  * 2 tablespoons shortening\n  * 2 eggs\n  * Additional sugar, if desired\n\n1 In deep fryer or 3-quart saucepan, heat 3 to 4 inches oil to 375\u00b0F.\n\n2 In large bowl, beat 1\u00bd cups of the flour and the remaining ingredients except additional sugar with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on medium speed 2 minutes, scraping bowl occasionally. Stir in remaining flour.\n\n3 On generously floured surface, roll dough lightly to coat. Gently roll to 3\/8-inch thickness. Cut dough with floured 2\u00bd-inch doughnut cutter.\n\n4 Slide doughnuts into hot oil, using wide spatula. Turn doughnuts as they rise to the surface. Fry 2 to 3 minutes or until golden brown on both sides. Remove from oil with long fork; do not prick doughnuts. Drain on paper towels. While warm, roll in powdered or granulated sugar, if desired.\n\n1 Doughnut: Calories 190; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 20mg; Sodium 120mg; Total Carbohydrate 35g (Dietary Fiber 0g, Sugars 21g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\nApplesauce Doughnuts Reduce sugar to \u00be cup. Add 1 cup applesauce. Increase cinnamon to 1 teaspoon; omit nutmeg. Cover and refrigerate about 1 hour or until dough stiffens. Divide dough in half; roll half of the dough; cut doughnuts. Repeat with remaining dough. Sprinkle hot doughnuts with cinnamon-sugar if desired.\n\nGlazed Doughnuts Frost doughnuts with Chocolate or Vanilla Glaze (page  or ). Sprinkle with candy sprinkles, if desired.\n\n## Baked Blueberry-Orange Doughnuts\n\nPrep 25 min Total 40 min \u25cf 12 doughnuts\n\nStreusel\n\n  * 3 tablespoons all-purpose flour\n  * 3 tablespoons sugar\n  * 1 tablespoon butter\n  * 3 tablespoons sliced almonds\n  * \u00bd teaspoon grated orange peel\n\nDoughnuts\n\n  * \u00bd cup butter, softened\n  * \u2154 cup sugar\n  * 2 eggs\n  * \u00bc cup milk\n  * \u00bc cup fresh orange juice\n  * 2 cups all-purpose flour\n  * 2 teaspoons grated orange peel\n  * 1\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u00be cup fresh blueberries\n\n1 Heat oven to 425\u00b0F. Lightly spray 2 regular-size doughnut pans (6 doughnuts per pan) with cooking spray.\n\n2 In small bowl, mix 3 tablespoons flour and 3 tablespoons sugar. Cut in 1 tablespoon butter, using pastry blender or fork, until mixture is crumbly. Stir in almonds and \u00bd teaspoon orange peel; set streusel aside.\n\n3 In medium bowl, beat \u00bd cup butter, \u2154 cup sugar and the eggs with electric mixer on medium speed until smooth. Add milk and orange juice; beat on low speed until well mixed. Stir in 2 cups flour, 2 teaspoons orange peel, the baking powder and salt just until flour is moistened. Fold in blueberries.\n\n4 Spoon batter evenly into doughnut pans, filling wells about \u00bc inch from top of pan; sprinkle evenly with streusel. Bake 6 to 8 minutes or until toothpick inserted in center comes out clean. Cool 5 minutes; remove from pan to cooling rack. Serve warm if desired.\n\n1 Doughnut: Calories 250; Total Fat 11g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 55mg; Sodium 250mg; Total Carbohydrate 34g (Dietary Fiber 1g, Sugars 16g); Protein 4g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\n\nBeignets with Espresso Sugar\n\n## Beignets with Espresso Sugar Calorie Smart\n\nPrep 50 min Total 3 hr 30 min \u25cf 30 beignets\n\nEspresso Sugar\n\n  * \u2154 cup sugar\n  * 1 to 1\u00bc teaspoons espresso coffee powder\n\nBeignets\n\n  * 2\u00be cups all-purpose flour\n  * \u00bc cup sugar\n  * \u00bc teaspoon salt\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * \u00be cup very warm milk (120\u00b0F to 130\u00b0F)\n  * \u00bc cup butter, melted\n  * 1 teaspoon vanilla\n  * 2 eggs\n  * Vegetable oil for frying\n\nGlaze\n\n  * \u2153 cup dark, semisweet or milk chocolate chips\n  * 1 teaspoon vegetable oil\n\n1 In small bowl, stir together \u2154 cup sugar and the espresso powder; set aside. Grease large bowl with shortening or cooking spray; set aside.\n\n2 Insert paddle attachment in electric stand mixer. In another large bowl, mix 2 cups of the flour, \u00bc cup sugar, the salt and yeast. Add warm milk and butter. Beat on low speed 1 minute, scraping bowl and paddle frequently. Remove paddle attachment and insert dough hook. Add vanilla and eggs; beat until smooth. Beat on low speed 1 minute, scraping bowl and dough hook frequently. Stir in enough remaining flour to make dough easy to handle (dough will be slightly sticky).\n\n3 Place dough in greased bowl, turning dough to grease all sides. Cover bowl with plastic wrap and let rise in warm place about 2 hours or until dough has doubled in size.\n\n4 Sprinkle cookie sheet with flour; set aside. Place dough on surface generously sprinkled with flour. Roll dough in flour to coat. With floured rolling pin, roll dough to \u00bd-inch thickness. Using 2-inch round cookie cutter, cut 30 rounds, gently pressing together and rerolling dough scraps as necessary. On lightly floured surface, sprinkle both sides of each round with flour; place about 2 inches apart on cookie sheet. Cover loosely with plastic wrap sprayed with cooking spray; let rise in warm place 30 to 40 minutes or until slightly risen.\n\n5 In deep fryer or 4-quart Dutch oven, heat 2 inches oil to 325\u00b0F over medium heat. Cover cooling rack with paper towels. Fry 4 or 5 beignets at a time in hot oil until golden brown on both sides, about 1 minute on each side. Remove from oil with slotted spoon to cooling rack. Immediately roll in espresso sugar.\n\n6 In small microwavable bowl, microwave chocolate chips on High 30 to 60 seconds, stirring once, until chips are softened and can be stirred smooth. Stir in 1 teaspoon oil until smooth. Spoon into small resealable food-storage plastic bag; seal bag. Cut off tiny corner of bag; squeeze bag to drizzle chocolate over beignets.\n\n1 Beignet: Calories 150; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 40mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 8g); Protein 2g Exchanges: 1 Starch, 1\u00bd Fat Carbohydrate Choices: 1\n\n### Making Beignets\n\nCut rounds using 2-inch cookie cutter, gently pressing together and rerolling dough scraps as necessary.\n\nFry 4 or 5 beignets at a time until golden brown on both sides. Remove from oil; immediately roll in espresso sugar.\n\nStrawberry Cream Brunch Cake\n\n## Strawberry Cream Brunch Cake\n\nPrep 20 min Total 1 hr 35 min \u25cf 16 servings\n\nCream Cheese Filling\n\n  * 1 package (8 oz) cream cheese, softened\n  * \u00bc cup sugar\n  * 2 tablespoons all-purpose flour\n  * 1 egg\n\nCake\n\n  * 2\u00bc cups all-purpose flour\n  * \u00be cup sugar\n  * \u00be cup cold butter\n  * \u00bd teaspoon baking powder\n  * \u00bd teaspoon baking soda\n  * \u00bc teaspoon salt\n  * \u00be cup sour cream\n  * 1 teaspoon almond extract\n  * 1 egg\n  * \u00bd cup strawberry or raspberry preserves\n  * \u00bd cup sliced almonds\n\n1 Heat oven to 350\u00b0F. Grease bottom and side of 10-inch springform pan or 11x7-inch (2-quart) glass baking dish with shortening; lightly flour.\n\n2 In small bowl, mix filling ingredients until smooth; set aside.\n\n3 In large bowl, mix 2\u00bc cups flour and \u00be cup sugar. Cut in butter, using pastry blender or fork, until mixture looks like coarse crumbs. Reserve 1 cup of the crumb mixture. Stir baking powder, baking soda, salt, sour cream, almond extract and 1 egg into remaining crumb mixture. Spread batter over bottom and 2 inches up side (about \u00bc inch thick) of pan.\n\n4 Spoon filling over batter. Carefully spoon preserves evenly over filling. Mix almonds and reserved crumb mixture; sprinkle over preserves.\n\n5 Bake springform pan 50 to 60 minutes, 11x7-inch dish 35 to 45 minutes, or until filling is set and crust is deep golden brown. Cool 15 minutes; remove side of springform pan. Serve warm if desired. Store covered in refrigerator.\n\n1 Serving: Calories 320; Total Fat 18g (Saturated Fat 10g, Trans Fat 0.5g); Cholesterol 70mg; Sodium 220mg; Total Carbohydrate 35g (Dietary Fiber 1g, Sugars 18g); Protein 4g Exchanges: 1\u00bd Starch, 1 Other Carbohydrate, 3\u00bd Fat Carbohydrate Choices: 2\n\nMake-Ahead Directions Bake Strawberry Cream Brunch Cake as directed. Cool cake completely, then wrap tightly and freeze up to 1 month. To thaw, let stand at room temperature several hours before serving.\n\n## Sour Cream Coffee Cake\n\nPrep 30 min Total 2 hr \u25cf 16 servings\n\nBrown Sugar Filling\n\n  * \u00bd cup packed brown sugar\n  * \u00bd cup finely chopped nuts\n  * 1\u00bd teaspoons ground cinnamon\n\nCoffee Cake\n\n  * 3 cups all-purpose or whole wheat flour\n  * 1\u00bd teaspoons baking powder\n  * 1\u00bd teaspoons baking soda\n  * \u00be teaspoon salt\n  * 1\u00bd cups granulated sugar\n  * \u00be cup butter, softened\n  * 1\u00bd teaspoons vanilla\n  * 3 eggs\n  * 1\u00bd cups sour cream\n\nGlaze\n\n  * \u00bd cup powdered sugar\n  * \u00bc teaspoon vanilla\n  * 2 to 3 teaspoons milk\n\n1 Heat oven to 350\u00b0F. Grease bottom, side and tube of 10-inch angel food (tube) cake pan or 12-cup fluted tube cake pan or bottoms and sides of 2 (9x5-inch) loaf pans with shortening or cooking spray.\n\n2 In small bowl, stir filling ingredients until well mixed; set aside. In large bowl, stir flour, baking powder, baking soda and salt until well mixed; set aside.\n\n3 In another large bowl, beat granulated sugar, butter, 1\u00bd teaspoons vanilla and the eggs with electric mixer on medium speed 2 minutes, scraping bowl occasionally. Alternately add flour mixture and sour cream, beating on low speed until blended.\n\n4 For angel food or fluted tube cake pan, spread one-third of the batter (about 2 cups) in pan, then sprinkle with one-third of the filling; repeat twice. For loaf pans, spread one-fourth of the batter (about 1\u00bd cups) in each pan, then sprinkle each with one-fourth of the filling; repeat once.\n\n5 Bake angel food or fluted tube cake pan about 1 hour, loaf pans about 45 minutes, or until toothpick inserted near center comes out clean. Cool 10 minutes; remove cake from pan to cooling rack. Cool 20 minutes.\n\n6 In small bowl, stir glaze ingredients until smooth and thin enough to drizzle. Drizzle glaze over coffee cake. Serve warm if desired.\n\n1 Serving: Calories 360; Total Fat 16g (Saturated Fat 8g, Trans Fat 0.5g); Cholesterol 75mg; Sodium 360mg; Total Carbohydrate 49g (Dietary Fiber 1g, Sugars 30g); Protein 5g Exchanges: 2 Starch, 1 Other Carbohydrate, 3 Fat Carbohydrate Choices: 3\n\nLemon Curd\u2013Filled Butter Cake\n\n## Lemon Curd\u2013Filled Butter Cake\n\nA classic lemon curd is the filling in this buttery-rich dense yellow cake. To save time, you can substitute a jar of purchased lemon curd.\n\nPrep 30 min Total 2 hr 45 min \u25cf 12 servings\n\nLemon Curd\n\n  * \u00bc cup granulated sugar\n  * 2 tablespoons cornstarch\n  * \u00be cup cold water\n  * 3 egg yolks\n  * 1 tablespoon grated lemon peel\n  * 3 tablespoons fresh lemon juice\n\nCake\n\n  * 1 cup butter, softened\n  * 1 cup granulated sugar\n  * 5 eggs\n  * 1\u00be cups all-purpose flour\n  * 2 teaspoons grated lemon peel\n  * 1\u00bd teaspoons baking powder\n  * 1 teaspoon vanilla\n  * \u2153 cup slivered almonds, toasted (page 23)\n  * \u00bd teaspoon powdered sugar, if desired\n\n1 In 1-quart saucepan, mix \u00bc cup granulated sugar and the cornstarch. Stir in water and egg yolks with whisk until well mixed and no lumps remain. Heat to boiling over medium heat, stirring constantly, until mixture begins to thicken. Cook and stir 1 minute; remove from heat. Stir in 1 tablespoon lemon peel and the lemon juice. Refrigerate uncovered 20 minutes, stirring once, until room temperature.\n\n2 Heat oven to 350\u00b0F. Grease bottom and side of 9-inch springform pan with shortening; lightly flour.\n\n3 In large bowl, beat butter and 1 cup granulated sugar with electric mixer on medium speed about 1 minute or until smooth. Beat in eggs, one at a time, just until blended, then continue beating on medium speed 2 minutes, scraping bowl once. On low speed, beat in flour, 2 teaspoons lemon peel, the baking powder and vanilla about 30 seconds or just until blended.\n\n4 Spread half of the batter (about 2 cups) in bottom of pan. Spoon lemon curd evenly onto batter, spreading to \u00bd inch of edge. Drop remaining batter by tablespoonfuls around edge of curd and pan; spread batter evenly and toward center to cover curd. Sprinkle almonds over top.\n\n5 Bake 45 to 55 minutes or until center is set, cake is firm to the touch and top is golden brown. Cool in pan on cooling rack at least 1 hour (center will sink slightly). Run thin knife around side of cake; remove side of pan. Sprinkle with powdered sugar before serving. Store covered in refrigerator.\n\n1 Serving: Calories 360; Total Fat 20g (Saturated Fat 9g, Trans Fat 1g); Cholesterol 180mg; Sodium 190mg; Total Carbohydrate 37g (Dietary Fiber 0g, Sugars 21g); Protein 6g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 4 Fat Carbohydrate Choices: 2\u00bd\n\n# Yeast Bread Basics\n\nMaking yeast breads is a lot easier than you would think when you know all the tips and tricks. And with our kitchen-tested recipes, you can turn out beautiful, delicious breads every time.\n\n## YEAST BREAD INGREDIENTS\n\nFlour: All-purpose and bread flours both have enough gluten-forming protein to develop the gluten necessary to work for yeast bread recipes. (See Learn to Knead Yeast Dough, page 498.)\n\nWhole wheat flour has a high level of protein but won't make the same lofty loaves that all-purpose or bread flour can because the shards of bran in the flour tear the gluten strands, inhibiting its development. Rye flour doesn't have enough gluten-forming protein to be used on its own\u2014it is used in combination with all-purpose or bread flour so it develops enough gluten to make desirable loaves.\n\nYeast: Yeast is temperature sensitive\u2014proper yeast growth is necessary for dough to rise the way it should. If the liquid in the dough recipe is added too hot, it can kill the yeast. If liquid is too cool when added, it can prevent growth. Always check the package expiration date before use.\n\nDry yeast is the most commonly available form and comes in active dry yeast and instant (fast-acting) yeast varieties, which may shorten rising times. Bread machine yeast is a type of instant yeast that can also be used in traditional baking methods. Be sure to use the type of yeast called for in a recipe, as not all yeasts work well in all types of dough-preparation methods.\n\nYeast can be dissolved using the traditional mixing method. Before combining with other ingredients, the yeast is dissolved in warm water with a small amount of sugar; the sugar acts as food for the yeast. The quick-mix method mixes yeast with the other dry ingredients before adding very warm liquid.\n\nLiquid: Water gives bread a crisp crust, while milk results in a softer crust and bread with added nutrients. Use a thermometer to accurately determine the liquid's temperature before adding.\n\nSweetener: Sugar, honey and molasses feed the yeast to help it grow, add flavor and help brown the crust. Don't use artificial sweeteners because they won't feed the yeast.\n\nSalt: Salt adds flavor and keeps yeast growth in check. Never omit salt if specified in recipe.\n\nFat: Butter, shortening and vegetable or olive oil make bread tender and moist and add flavor.\n\nEggs: Eggs add richness and color and structure.\n\n## STORING AND SERVING YEAST BREADS\n\n  * See Refrigerator and Freezer Storage Chart, page 33.\n  * If breads are to be frozen, frost or glaze after they are thawed.\n  * Store soft-crust breads and rolls in resealable food-storage plastic bags or airtight con-tainers in a cool, dry place up to 7 days.\n  * Store unsliced crisp-crust breads unwrapped at room temperature up to 3 days. Once sliced, keep in a closed paper bag or tightly wrapped in foil and use within 1 day or freeze.\n  * To freeze, wrap loaves tightly in plastic wrap and place in freezer bags. Freeze up to 3 months.\n  * To reheat thawed breads or rolls, wrap in foil and bake at 350\u00b0F (10 to 15 minutes for rolls or 15 to 20 minutes for loaves). For a crisper crust, unwrap during the last 5 minutes.\n\n### Bread Machine Tips\n\nFollow these tips for bread machine techniques:\n\n  * Follow the directions in your manual for the order ingredients should be placed in your bread machine.\n  * Carefully measure the ingredients, using dry measuring cups for flour and sugar and liquid measuring cups for liquids and measuring, not serving spoons.\n  * Use bread machine yeast: It has a fine granulation that disperses more evenly during the mixing and kneading cycles.\n  * Don't open the machine during rising or baking cycles, as it can cause the dough to collapse. If you need to check the progress of your dough, only check during mixing or kneading cycles.\n  * When using the delay cycle, be sure the yeast does not come in contact with the liquid. Don't use the delay cycle with recipes using eggs, fresh dairy products (except butter), honey, meats or fresh fruits or vegetables, because bacteria can grow during this cycle.\n  * Keep the area around the bread machine open for good ventilation.\n\n#### Why Use a Bread Machine?\n\nHere are some reasons why a bread machine might be a great tool for your family:\n\n  * It's a quick and easy way to mix, raise and bake homemade bread to have fresh bread any time, saving time and money.\n  * It's a great way to control what goes into your bread, without added preservatives or other ingredients. (But homemade bread won't stay fresh as long as store-bought, for this very reason. So purchase the machine that will make the size of loaf your family can consume in a day or two.)\n  * It's also a super-simple way to get yeast doughs that are all ready to shape and bake. French bread, pizza dough and pretzel and yeast roll dough can be made in the bread machine on the dough setting, skipping a lot of time and attention from making them by hand. Then you take it from there\u2014shaping the dough and baking it.\n  * All the ingredients get dumped into the machine at once; no dissolving yeast separately or using a lot of bowls and utensils.\n  * Since you don't have to knead the dough by hand, it only takes a few minutes to put everything into the machine and let it do its thing.\n\n### Making Bread Dough\n\nAfter the first addition of flour has been beaten in, dough will be very soft and fall in \"sheets\" off rubber spatula.\n\nTo knead, fold dough toward you. Push dough away from you with short rocking motion. Rotate dough a quarter turn; repeat. Dough will feel springy and smooth.\n\nDough should rise until doubled in size. Press fingertips about \u00bd inch into dough. If indentations remain, dough has risen enough.\n\n### Shaping Traditional Bread Loaves\n\nFlatten dough with hands or rolling pin into 18x9-inch rectangle.\n\nTightly roll dough up toward you, beginning at 9-inch side.\n\n### Making Free-Form Loaves\n\nShape dough into smooth ball by stretching surface of dough around to bottom on all four sides; pinch bottom to seal.\n\nPlace on cookie sheet; carefully slash tic-tac-toe pattern on each loaf top with serrated knife.\n\n## Learn to Knead Yeast Dough\n\nProperly kneaded yeast dough helps the dough rise and produces bread with a finer grain and better texture. When flour is mixed with liquid, gluten strands are formed. Kneading makes the gluten stronger and more elastic, allowing the dough to stretch, expand and hold its shape as it rises and then bakes.\n\nDough recipes usually give a range of how much flour to add when making bread. This is because conditions can change, particularly the moisture content of the flour, due to weather\/storage conditions and humidity level in the air when making bread. Don't worry if you don't use all the flour called for. Use just enough to knead the dough until it's smooth and elastic, so it will rise and bake well. The range of flour given includes what you will use to make the dough and to knead it.\n\nFor the best volume and texture of yeast breads, use these tips:\n\n  * Keep any remaining flour (of the maximum amount called for in the recipe) not used to make the dough in a measuring cup on the counter.\n  * Sprinkle counter lightly with some of the flour. Dust hands with a little of the flour.\n  * Place raised dough on counter. Press dough into an oval shape; fold it in half toward you.\n  * Using the floured lower palm and heel of your hand, press down lightly on dough and press it away from you.\n  * Turn dough \u00bc turn; fold dough toward you in half; repeat kneading motion.\n  * Repeat, adding additional flour under dough as needed if counter is sticky, until dough is smooth and elastic.\n  * Check for proper kneading by pressing a floured fingertip lightly into the top of the dough. If it springs back to its original shape, it has been kneaded long enough.\n  * If hands become sticky, sprinkle a little of the flour onto your hands, rubbing hands together to remove stickiness.\n\n## Artisan Asiago Bread Calorie Smart\n\nPrep 25 min Total 4 hr 15 min \u25cf 1 loaf (24 slices)\n\n  * 3\u00bd to 3\u00be cups bread flour\n  * 1 teaspoon sugar\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * 1\u00bc cups very warm water (120\u00b0F to 130\u00b0F)\n  * 2 tablespoons olive or vegetable oil\n  * 2 teaspoons dried rosemary or thyme leaves, if desired\n  * 1 teaspoon salt\n  * 1\u00bc cups diced Asiago cheese*\n\n1 In large bowl, mix 1\u00bd cups of the flour, the sugar and yeast. Add warm water. Beat with wooden spoon or electric mixer on low speed 1 minute, scraping bowl frequently. Cover tightly with plastic wrap and let stand about 1 hour or until bubbly.\n\n2 Stir in oil, rosemary and salt. Stir in enough remaining flour, \u00bd cup at a time, until a soft, smooth dough forms. Let stand 15 minutes.\n\n3 Place dough on lightly floured surface. Knead 5 to 10 minutes or until dough is smooth and springy. Knead in 1 cup of the cheese. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl tightly with plastic wrap and let rise in warm place 45 to 60 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n4 Lightly grease uninsulated cookie sheet with shortening or cooking spray. Place dough on lightly floured surface. Gently shape into football-shaped loaf, about 12 inches long, by stretching sides of dough downward to make a smooth top. Place loaf, smooth side up, on cookie sheet. Coat loaf generously with flour. Cover loosely with plastic wrap and let rise in warm place 45 to 60 minutes or until dough has almost doubled in size.\n\n5 Heat oven to 450\u00b0F. Place 8- or 9-inch square pan on bottom oven rack; add hot water to pan until about \u00bd inch from the top. Spray loaf with cool water; sprinkle with flour. With serrated knife, carefully cut \u00bd-inch-deep slash lengthwise down center of loaf. Sprinkle remaining \u00bc cup cheese into slash.\n\n6 Bake 10 minutes. Reduce oven temperature to 400\u00b0F. Bake 20 to 25 minutes longer or until loaf is deep golden and sounds hollow when tapped. Remove from cookie sheet to cooling rack; cool.\n\n*Swiss or another firm cheese can be substituted for the Asiago.\n\n1 Slice: Calories 110; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 5mg; Sodium 115mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 0g); Protein 4g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nMix batter 1 minute, scraping bowl frequently.\n\nGently shape dough into football-shaped loaf.\n\nSpray loaf with hot water; sprinkle with flour.\n\nCarefully cut \u00bd-inch-deep slash lengthwise down center of loaf.\n\nFocaccia\n\n## Focaccia Calorie Smart\n\nPrep 30 min Total 1 hr 50 min \u25cf 2 loaves (12 slices each)\n\n  * 2\u00bd to 3 cups all-purpose or bread flour\n  * 2 tablespoons chopped fresh or 1 tablespoon dried rosemary leaves, crumbled\n  * 1 tablespoon sugar\n  * 1 teaspoon salt\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * 5 tablespoons olive or vegetable oil\n  * 1 cup very warm water (120\u00b0F to 130\u00b0F)\n  * \u00bc cup grated Parmesan cheese\n\n1 In large bowl, mix 1 cup of the flour, the rosemary, sugar, salt and yeast. Add 3 tablespoons of the oil and the warm water. Beat with electric mixer on medium speed 3 minutes, scraping bowl frequently. Stir in enough remaining flour until dough is soft and leaves side of bowl.\n\n2 Place dough on lightly floured surface. Knead 5 to 8 minutes or until smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 30 minutes or until dough has almost doubled in size. Dough is ready if indentation remains when touched.\n\n3 Grease 2 cookie sheets or 12-inch pizza pans with small amount of oil or spray with cooking spray. Gently push fist into dough to deflate. Divide dough in half. Shape each half into a flattened 10-inch round on cookie sheet. Cover loosely with plastic wrap lightly sprayed with cooking spray and let rise in warm place about 30 minutes or until dough has doubled in size.\n\n4 Heat oven to 400\u00b0F. Gently make \u00bd-inch-deep depressions about 2 inches apart in dough with fingers. Carefully brush with remaining 2 tablespoons oil; sprinkle with cheese. Bake 15 to 20 minutes or until golden brown. Cut into wedges; serve warm if desired.\n\n1 Slice: Calories 80; Total Fat 3.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 120mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch Carbohydrate Choices: 1\n\nFocaccia Breadsticks Make dough as directed. After kneading, cover loosely with plastic wrap; let rest 30 minutes. Grease 2 cookie sheets with shortening or cooking spray; sprinkle with cornmeal. Divide dough into 12 pieces. Roll and shape each piece into 12-inch rope, sprinkling with flour if dough is too sticky. Place \u00bd inch apart on cookie sheets. Brush with oil and sprinkle with Parmesan cheese or coarse (kosher or sea) salt if desired. Cover loosely with plastic wrap and let rise in warm place about 20 minutes or until dough has almost doubled in size. Heat oven to 425\u00b0F. Uncover breadsticks. Bake 10 to 12 minutes or until golden brown. Makes 12 breadsticks.\n\nOnion Focaccia Make dough as directed, except omit rosemary. In 10-inch skillet, heat \u2153 cup olive oil over medium heat. Stir in 4 cups thinly sliced onions and 4 cloves garlic, finely chopped. Cook uncovered 10 minutes, stirring every 3 to 4 minutes. Reduce heat to medium-low. Cook 30 to 40 minutes longer, stirring well every 5 minutes, until onions are light golden brown. Continue and bake as directed, except do not brush dough with oil or sprinkle with Parmesan cheese; after second rising, carefully spread onion mixture over loaves.\n\n### Making Impressions in Focaccia Dough\n\nGently make \u00bd-inch-deep depressions about 2 inches apart with fingertips.\n\nBlack Pepper and Parmesan Grissini\n\n## Black Pepper and Parmesan Grissini Calorie Smart\n\nGrissini is Italian for \"breadsticks\"; they are thin and crisp.\n\nPrep 30 min Total 2 hr 30 min \u25cf 32 grissini\n\nBreadsticks\n\n  * \u00be cup warm water (105\u00b0F to 115\u00b0F)\n  * 1 teaspoon regular active dry yeast\n  * \u00bd teaspoon sugar\n  * 2 tablespoons olive oil\n  * \u00bd cup whole wheat flour\n  * 1\u00bc to 1\u00be cups all-purpose flour\n  * \u2153 cup grated Parmesan cheese\n  * 1 teaspoon salt\n  * \u00bd teaspoon coarsely ground black pepper\n\nTopping\n\n  * 2 to 3 teaspoons olive oil\n  * \u00bc cup grated Parmesan cheese\n  * 1 teaspoon coarsely ground black pepper\n\n1 In large bowl, stir together warm water, yeast and sugar until yeast is dissolved. Add 2 tablespoons oil. Stir in whole wheat flour and \u00bd cup of the all-purpose flour with whisk until mixture is smooth. Cover and let stand in warm place 35 to 45 minutes or until mixture is light and bubbly.\n\n2 Stir in \u2153 cup cheese, the salt and \u00bd teaspoon pepper. Add \u00be cup of the all-purpose flour, slowly adding additional flour as necessary to make dough easy to handle.\n\n3 Place dough on lightly floured surface. Knead 10 minutes or until dough is smooth and springy. Place dough in large bowl greased with oil, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 1 hour or until dough is doubled in size. Dough is ready if indentation remains when touched.\n\n4 Position one oven rack in lower third of oven and second oven rack in upper third of oven. Heat oven to 425\u00b0F. Line 2 cookie sheets with cooking parchment paper.\n\n5 Gently push fist into dough to deflate. On lightly oiled surface, pat dough to 8-inch square. Using pizza cutter or sharp knife, cut square in half. Cut each half crosswise into 16 strips. Roll each strip to 10- to 12-inch length; place on cookie sheet. Brush with 2 to 3 teaspoons oil; sprinkle with \u00bc cup cheese and 1 teaspoon pepper.\n\n6 Bake 12 to 15 minutes or until golden brown and crisp, moving cookie sheets to other rack position halfway through baking. Cool completely on cooling rack.\n\n1 Grissini: Calories 45; Total Fat 2g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 100mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: \u00bd\n\nMake-Ahead Directions After kneading, the bread can rise overnight (covered) in the refrigerator. When ready to bake, bring to room temperature, deflate dough and continue as directed.\n\nFrench Bread\n\n## French Bread Calorie Smart\n\nPrep 25 min Total 8 hr \u25cf 2 loaves (16 slices each)\n\n  * 1\u00bd cups all-purpose flour\n  * 1 package regular active dry yeast (2\u00bc teaspoons)\n  * 1 cup very warm water (120\u00b0F to 130\u00b0F)\n  * 1 teaspoon salt\n  * 1\u2153 to 1\u2154 cups all-purpose or bread flour\n\n1 In large bowl, mix 1\u00bd cups all-purpose flour and the yeast. Add warm water. Beat with whisk or electric mixer on low speed 1 minute, scraping bowl frequently, until batter is very smooth. Cover tightly with plastic wrap and let stand about 1 hour or until bubbly.\n\n2 Stir in salt and enough flour, \u00bd cup at a time, until a soft dough forms. Place dough on lightly floured surface. Knead 5 to 10 minutes or until dough is smooth and springy (dough will be soft). Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place 1 hour to 1 hour 15 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n3 Grease uninsulated cookie sheet with shortening or cooking spray. Place dough on lightly floured surface, forming it into an oval-shaped mound. Sprinkle top of dough with flour. With a straight-edged knife, press straight down on dough lengthwise to divide it into 2 equal parts (the parts will be elongated in shape). Gently shape each part into a narrow loaf, about 16 inches long, stretching the top of the loaf slightly to make it smooth. Place loaves, smooth sides up, about 4 inches apart on cookie sheet.\n\n4 Cover loaves loosely, but airtight, with plastic wrap. (Loaves will expand slightly in the refrigerator.) Refrigerate at least 4 hours but no longer than 24 hours. (This step can be omitted, but refrigerating develops the flavor and texture of the bread.)\n\n5 Uncover loaves and spray with cool water; let rise in warm place about 1 hour or until refrigerated loaves have come to room temperature.\n\n6 Place 8- or 9-inch square pan on bottom oven rack; add hot water to pan until about \u00bd inch from the top. Heat oven to 450\u00b0F.\n\n7 With serrated knife, carefully cut \u00bc-inch-deep slashes diagonally across loaves at 2-inch intervals. Spray loaves with cool water. Place cookie sheet on middle oven rack. Spray loaves again.\n\n8 Bake 18 to 20 minutes or until loaves are deep golden brown with crisp crust and sound hollow when tapped. Remove from cookie sheet to cooling rack; cool.\n\n1 Slice: Calories 40; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 75mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: \u00bd Starch Carbohydrate Choices: \u00bd\n\n### Making French Bread\n\nMix batter until very smooth. Cover batter in bowl tightly with plastic wrap; let stand 1 hour or until bubbly.\n\nAdd flour, \u00bd cup at a time, until soft dough forms. Knead 5 to 10 minutes or until smooth and springy.\n\nPress straight down on oval mound\u2013shaped dough with straight knife to divide into 2 equal parts.\n\nWith serrated knife, carefully cut \u00bc-inch-deep slashes diagonally across loaves at 2-inch intervals.\n\n## Uses for French Bread\n\nFrench bread is a delicious surprise when served with a meal, but if you have leftovers, you know how fast it can get stale. Rather than let that baked goodness go to waste, try some of these ideas next time you have extra French bread on hand.\n\n  1. 1. **Hazelnut-Stuffed French Toast:** Spread 6 slices of bread (\u00bd inch thick) with about 1 tablespoon hazelnut spread with cocoa; top with additional bread slices. Beat 3 eggs, \u00bd cup milk and 2 tablespoons granulated sugar in shallow container. Dip each side of sandwich, soaking up egg mixture. Cook over medium-low heat in large skillet sprayed with cooking spray; cook 2 to 3 minutes on each side or until golden brown. Sprinkle with powdered sugar.\n  2. 2. **Rosemary-Parmesan Croutons:** Prepare Croutons (page 117) as directed, except mix the butter with 2 tablespoons shredded Parmesan cheese and 1 tablespoon chopped fresh rosemary leaves before drizzling over the bread cubes.\n  3. 3. **Turkey-Provolone-Dill Party Sub:** Slice 12-inch piece of French bread horizontally in half. Spread one cut side with 2 tablespoons mayonnaise and the other cut side with 2 tablespoons Dijon mustard. Top loaf bottom with lettuce leaves, thinly sliced deli turkey, provolone cheese slices, thinly sliced tomato and dill pickle chips. Place top of loaf on sandwich; secure with toothpicks. Cut into 1-inch slices; serve immediately.\n  4. 4. **Cheesy Garlic Bread:** Make slashes in loaf on diagonal about 1 inch apart, to about \u00bd inch from bottom of loaf; place loaf on cookie sheet. Fill each opening with 1 ounce thinly sliced mozzarella cheese (cut in half) and 1 tablespoon shredded Cheddar cheese. Mix 1 tablespoon melted butter, 1 teaspoon chopped fresh parsley and \u215b teaspoon garlic powder; brush over top of loaf. Bake at 375\u00b0F 8 to 12 minutes or until cheese is melted.\n  5. 5. **Spaghetti & Meatballs Pizza:** Slice 12-inch piece of French bread horizontally in half; place on cookie sheet, cut sides up. Spread each with \u00bc cup pasta sauce. Top each with about 1 cup cooked spaghetti and 6 to 8 frozen (thawed) meatballs, cut in half. Sprinkle each with \u00bd cup shredded mozzarella and 2 tablespoons grated Parmesan cheese. Sprinkle with chopped fresh parsley. Bake at 450\u00b0F 10 to 13 minutes or until meatballs are 160\u00b0F and cheese is melted.\n  6. 6. **Bread Salad:** For 4 servings, cut enough day-old French bread into 2 cups of 1-inch cubes. In medium bowl, toss bread with 2 cups bite-size pieces of lettuce; 1 cup each chopped tomatoes and cucumber; \u00bc cup each chopped red onion, kalamata or ripe olives and crumbled feta cheese; and \u00be cup of your favorite vinaigrette dressing. Serve immediately.\n  7. 7. **BLT Crostini:** Mix \u00bd cup shredded Swiss cheese (2 ounces), \u00bc cup mayonnaise and 2 sliced green onions. Spread about 1 teaspoon on each of 12 slices French bread (\u00bc inch thick). Place on ungreased cookie sheet; top each with \u00bd slice precooked bacon. Bake at 375\u00b0F until edges are golden brown and cheese is melted. Top with finely shredded lettuce and cherry tomato slices.\n\nHazelnut-Stuffed French Toast\n\n## Classic White Bread Calorie Smart\n\nPREP 35 min TOTAL 3 hr \u25cf 2 loaves (16 slices each)\n\n  * 6 to 7 cups all-purpose or bread flour\n  * 3 tablespoons sugar\n  * 1 tablespoon salt\n  * 2 tablespoons shortening or softened butter\n  * 2 packages regular active or fast-acting dry yeast (4\u00bd teaspoons)\n  * 2\u00bc cups very warm water (120\u00b0F to 130\u00b0F)\n  * 2 tablespoons butter, melted, if desired\n\n1 In large bowl, stir 3\u00bd cups of the flour, the sugar, salt, shortening and yeast until well mixed. Add warm water. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Stir in enough remaining flour, 1 cup at a time, to make dough easy to handle.\n\n2 Place dough on lightly floured surface. Knead about 10 minutes or until dough is smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place 40 to 60 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n3 Grease bottoms and sides of 2 (8x4- or 9x5-inch) loaf pans with shortening or cooking spray. Gently push fist into dough to deflate. Divide dough in half. On lightly floured surface, flatten each half with hands or rolling pin into 18x9-inch rectangle. Roll dough up tightly, beginning at 9-inch side. Press with thumbs to seal after each turn. Pinch edge of dough into roll to seal. Pinch each end of roll to seal. Fold ends under loaf. Place loaves, seam sides down, in pans. Brush loaves lightly with 1 tablespoon of the melted butter. Cover loosely with plastic wrap and let rise in warm place 35 to 50 minutes or until dough has doubled in size.\n\n4 Move oven rack to low position so that tops of pans will be in center of oven. Heat oven to 425\u00b0F. Bake 25 to 30 minutes or until loaves are deep golden brown and sound hollow when tapped. Remove from pans to cooling rack. Brush loaves with remaining 1 tablespoon melted butter. Cool completely.\n\n1 Slice: Calories 100; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 21g (Dietary Fiber 0g, Sugars 1g); Protein 3g Exchanges: 1 Starch Carbohydrate Choices: 1\u00bd\n\n### Learn with Betty Yeast Bread\n\n**Perfect Yeast Bread:** This loaf is high and evenly shaped and golden brown, with an even texture.\n\n**Underrisen Yeast Bread:** This loaf did not rise because yeast got too hot and dough was not kneaded enough.\n\n**Overrisen Yeast Bread:** This loaf was kneaded too much and contained too much flour.\n\n## Honey\u2013Whole Wheat Bread Calorie Smart\n\nPrep 35 min Total 3 hr 10 min \u25cf 2 loaves (16 slices each)\n\n  * 3 cups whole wheat flour\n  * \u2153 cup honey or real maple syrup\n  * \u00bc cup shortening or softened butter\n  * 3 teaspoons salt\n  * 2 packages regular active or fast-acting dry yeast (4\u00bd teaspoons)\n  * 2\u00bc cups very warm water (120\u00b0F to 130\u00b0F)\n  * 3 to 4 cups all-purpose or bread flour\n  * 2 tablespoons butter, melted, if desired\n\n1 In large bowl, beat whole wheat flour, honey, shortening, salt and yeast with electric mixer on low speed until well mixed. Add warm water. Beat on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Stir in enough all-purpose flour, 1 cup at a time, to make dough easy to handle.\n\n2 Place dough on lightly floured surface. Knead about 10 minutes or until dough is smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place 40 to 60 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n3 Gently push fist into dough to deflate. Divide dough in half. On lightly floured surface, flatten each half with hands or rolling pin into 18x9-inch rectangle. Roll dough up tightly, beginning at 9-inch side. Press with thumbs to seal after each turn. Pinch edge of dough into roll to seal. Pinch each end of roll to seal. Fold ends under loaf.\n\n4 Grease bottoms and sides of 2 (8x4- or 9x5-inch) loaf pans with shortening or cooking spray. Place loaves, seam sides down, in pans. Brush loaves with 1 tablespoon of the melted butter. Cover loosely with plastic wrap and let rise in warm place 35 to 50 minutes or until dough has doubled in size.\n\n5 Move oven rack to low position so that tops of pans will be in center of oven. Heat oven to 375\u00b0F. Uncover loaves. Bake 40 to 45 minutes or until loaves are deep golden brown and sound hollow when tapped. Remove from pans to cooling rack. Brush loaves with remaining 1 tablespoon melted butter; cool.\n\n1 Slice: Calories 120; Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 22g (Dietary Fiber 2g, Sugars 3g); Protein 3g Exchanges: 1\u00bd Starch Carbohydrate Choices: 1\u00bd\n\nSunflower-Herb Whole Wheat Bread Add 1 tablespoon dried basil leaves and 2 teaspoons dried thyme leaves with the salt. Stir in 1 cup unsalted sunflower nuts with the all-purpose flour.\n\nFour-Grain Batter Bread\n\n## Four-Grain Batter Bread Calorie Smart\n\nHomemade bread doesn't get much easier than this! It's called batter bread because the dough is soft and doesn't require kneading. Just mix it, put it in the pan, let it rise and bake.\n\nPrep 15 min Total 1 hr 10 min \u25cf 2 loaves (16 slices each)\n\n  * Cornmeal\n  * 4\u00bd to 4\u00be cups all-purpose or bread flour\n  * 2 tablespoons sugar\n  * 1 teaspoon salt\n  * \u00bc teaspoon baking soda\n  * 2 packages regular active or fast-acting dry yeast (4\u00bd teaspoons)\n  * 2 cups milk\n  * \u00bd cup water\n  * \u00bd cup whole wheat flour\n  * \u00bd cup wheat germ\n  * \u00bd cup quick-cooking oats\n\n1 Grease bottoms and sides of 2 (8x4-inch) loaf pans with shortening or cooking spray; sprinkle with cornmeal.\n\n2 In large bowl, mix 3\u00bd cups of the all-purpose flour, the sugar, salt, baking soda and yeast. In 1-quart saucepan, heat milk and water over medium heat, stirring occasionally, until very warm (120\u00b0F to 130\u00b0F). Add milk mixture to flour mixture. Beat with electric mixer on low speed until moistened. Beat on medium speed 3 minutes, scraping bowl occasionally.\n\n3 Stir in whole wheat flour, wheat germ, oats and enough remaining all-purpose flour to make a stiff batter. Divide batter evenly between pans. Round tops of loaves by patting with floured hands. Sprinkle with cornmeal. Cover loosely with plastic wrap and let rise in warm place about 30 minutes or until batter is about 1 inch below tops of pans.\n\n4 Heat oven to 400\u00b0F. Bake about 25 minutes or until tops of loaves are light brown. Remove from pans to cooling rack; cool.\n\n1 Slice: Calories 100; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 90mg; Total Carbohydrate 19g (Dietary Fiber 1g, Sugars 2g); Protein 4g Exchanges: 1 Starch Carbohydrate Choices: 1\n\nWhole Wheat\u2013Raisin Batter Bread Increase whole wheat flour to 2 cups. Omit wheat germ and oats. Stir in 1 cup raisins with the second addition of all-purpose flour.\n\nSourdough Bread\n\n## Sourdough Bread Calorie Smart\n\nPrep 30 min Total 11 hr 45 min \u25cf 2 loaves (16 slices each)\n\n  * 1 cup Sourdough Starter\n  * 2\u00bd cups all-purpose or bread flour\n  * 2 cups warm water (105\u00b0F to 115\u00b0F)\n  * 3\u00be to 4\u00bc cups all-purpose or bread flour\n  * 3 tablespoons sugar\n  * 1 teaspoon salt\n  * 3 tablespoons vegetable oil\n\n1 In 3-quart glass bowl, mix sourdough starter, 2\u00bd cups flour and the warm water with wooden spoon until smooth. Cover and let stand in warm, draft-free place 8 hours.\n\n2 Add 3\u00be cups flour, the sugar, salt and oil to mixture in bowl. Stir with wooden spoon until dough is smooth and flour is completely absorbed. (Dough should be just firm enough to gather into ball. If necessary, add remaining \u00bd cup flour gradually, stirring until all flour is absorbed.)\n\n3 On heavily floured surface, knead dough about 10 minutes or until smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning to grease all sides. Cover and let rise in warm place about 1 hour 30 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n4 Grease large cookie sheet with shortening. Gently push fist into dough several times to remove air bubbles. Divide dough in half. Shape each half into a round, slightly flat loaf. Do not tear dough by pulling. Place loaves on opposite corners on cookie sheet. With sharp serrated knife, make three \u00bc-inch-deep slashes in top of each loaf. Cover and let rise about 45 minutes or until dough has doubled in size.\n\n5 Heat oven to 375\u00b0F. Brush loaves with cold water. Place in middle of oven. Bake 35 to 45 minutes, brushing occasionally with water, until loaves sound hollow when tapped. Remove from cookie sheet to cooling rack. Cool completely, about 1 hour.\n\n1 Slice: Calories 110; Total Fat 2.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 220mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 2g); Protein 3g Exchanges: 1\u00bd Starch Carbohydrate Choices: 1\n\n## Sourdough Starter\n\n  * 1 teaspoon regular active dry yeast\n  * \u00bc cup warm water (105\u00b0F to 115\u00b0F)\n  * \u00be cup milk\n  * 1 cup all-purpose flour\n\n  1. In 3-quart glass bowl, dissolve yeast in warm water. Stir in milk. Gradually stir in flour; beat until smooth. Cover with towel or cheesecloth; let stand in warm, draft-free place (80\u00b0F to 85\u00b0F) about 24 hours or until starter begins to ferment (bubbles will appear on surface of starter). If starter has not begun fermentation after 24 hours, discard and begin again. If fermentation has begun, stir well; cover tightly with plastic wrap and return to warm, draft-free place. Let starter stand 2 to 3 days or until foamy.\n  2. When starter has become foamy, stir well; pour into 1-quart crock or glass jar with tight-fitting cover. Store in refrigerator. Starter is ready to use when a clear liquid has risen to top. Stir before using. Use 1 cup starter in Sourdough Bread recipe; reserve remaining starter. To remaining starter, add \u00be cup milk and \u00be cup flour. Store covered at room temperature about 12 hours or until bubbles appear; refrigerate.\n  3. Use starter regularly, every week or so. If the volume of the breads you bake begins to decrease, dissolve 1 teaspoon active dry yeast in \u00bc cup warm water. Stir in \u00bd cup milk, \u00be cup flour and the remaining starter.\n\nChallah Braid\n\n## Rich Egg Bread Calorie Smart\n\nPrep 30 min Total 3 hr 5 min \u25cf 1 loaf (16 slices)\n\n  * 3 to 3\u00bc cups all-purpose or bread flour\n  * \u00bc cup sugar\n  * 1\u00bd teaspoons salt\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * 1 cup very warm water (120\u00b0F to 130\u00b0F)\n  * 2 tablespoons vegetable oil\n  * 1 egg\n  * Melted butter, if desired\n\n1 In large bowl, mix 1\u00bd cups of the flour, the sugar, salt and yeast. Add warm water and oil. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Add egg; beat until smooth. Stir in enough remaining flour, \u00bc cup at a time, to make dough easy to handle.\n\n2 Place dough on lightly floured surface. Knead about 10 minutes or until dough is smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 1 hour or until dough has doubled in size. (At this point, dough can be refrigerated up to 24 hours.) Dough is ready if indentation remains when touched.\n\n3 Grease bottom and sides of 9x5- or 8x4-inch loaf pan with shortening or cooking spray. Gently push fist into dough to deflate. On lightly floured surface, flatten dough with hands or rolling pin into 18x9-inch rectangle. Roll up tightly, beginning at 9-inch side. Pinch edge of dough into roll to seal. Pinch each end of roll to seal. Fold ends under loaf. Place loaf, seam side down, in pan. Cover loosely with plastic wrap lightly sprayed with cooking spray and let rise in warm place about 1 hour or until dough has doubled in size.\n\n4 Move oven rack to low position so that top of pan will be in center of oven. Heat oven to 375\u00b0F. Bake 30 to 35 minutes or until loaf is deep golden brown and sounds hollow when tapped. Remove from pan to cooling rack. Brush loaf with melted butter. Cool completely.\n\n1 Slice: Calories 120; Total Fat 2.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 15mg; Sodium 230mg; Total Carbohydrate 21g (Dietary Fiber 0g, Sugars 3g); Protein 3g Exchanges: 1\u00bd Starch Carbohydrate Choices: 1\u00bd\n\nChallah Braid Make dough as directed. Lightly grease cookie sheet with shortening or cooking spray. After pushing fist into dough, divide into 3 equal parts. Roll each part into 14-inch rope. Place ropes close together on cookie sheet. Braid ropes gently and loosely, starting in middle; do not stretch. Pinch ends; tuck ends under braid securely. Brush with vegetable oil. Cover loosely with plastic wrap and let rise in warm place 40 to 50 minutes or until dough has doubled in size. Heat oven to 375\u00b0F. In small bowl, beat 1 egg yolk and 2 tablespoons water; brush over braid. Sprinkle with poppy seed. Bake 25 to 30 minutes or until golden brown.\n\nCinnamon Swirl Raisin Bread Knead \u00bd cup raisins into dough after deflating in Step 3. Before rolling up dough, brush with 2 teaspoons softened butter. Mix \u00bc cup sugar with 1\u00bd teaspoons ground cinnamon; sprinkle over butter.\n\n### Braiding Challah\n\nBraid ropes gently and loosely, starting at middle; do not stretch.\n\nWhole Grain Artisan Bread\n\n## Whole Grain Artisan Bread Calorie Smart\n\nPrep 25 min Total 4 hr 35 min \u25cf 1 loaf (12 slices)\n\n  * 1\u00bc cups warm water (105\u00b0F to 115\u00b0F)\n  * 1 teaspoon regular active dry yeast\n  * \u00bd teaspoon sugar\n  * 1\u00bd cups whole wheat flour\n  * \u00bd cup rye flour\n  * 1\u00bd teaspoons salt\n  * 1\u00bd to 1\u00be cups bread flour\n  * 1 teaspoon cornmeal\n\n1 In large bowl, stir together warm water, yeast and sugar until yeast is dissolved. Stir in whole wheat flour and rye flour until mixture is smooth. Cover and let stand in warm place 1 hour or until mixture has doubled in size.\n\n2 Stir in salt and 1\u00bd cups of the bread flour, adding additional bread flour, 1 tablespoon at a time, to make dough easy to handle.\n\n3 Place dough on lightly floured surface; knead 10 minutes or until smooth and springy. Grease large bowl with oil. Place dough in bowl, turning to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 1 hour or until doubled in size. Dough is ready if indentation remains when touched.\n\n4 Line cookie sheet with cooking parchment paper; sprinkle center with cornmeal. Gently push fist into dough to deflate. Shape dough into round ball; place on cookie sheet and flatten slightly. Cover with a plastic wrap that has been sprayed with cooking spray and let rise in warm place 45 minutes or until doubled in size.\n\n5 Heat oven to 425\u00b0F. Using small strainer, sprinkle 1 teaspoon bread flour over top of dough. Using sharp serrated knife, cut 4 (\u00bc-inch-deep) slashes on top of loaf, then make 4 diagonal cuts to create crisscross pattern. Slide dough and parchment paper from cookie sheet onto pizza stone.\n\n6 Bake 25 to 30 minutes or until crisp golden brown and bread sounds hollow when tapped. Cool completely on cooling rack, about 1 hour.\n\n1 Slice: Calories 130; Total Fat 0.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 27g (Dietary Fiber 3g, Sugars 0g); Protein 4g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate Carbohydrate Choices: 2\n\nNo-Knead Artisan Bread\n\n## No-Knead Artisan Bread Calorie Smart\n\nPrep 10 min Total 4 hr \u25cf 2 loaves (12 slices each)\n\n  * 3 to 3\u00bd cups all-purpose flour\n  * 1 tablespoon sugar\n  * 1\u00bd teaspoons salt\n  * 1 package regular active dry yeast (2\u00bc teaspoons)\n  * 1\u00bc cups very warm water (120\u00b0F to 130\u00b0F)\n  * 1 tablespoon olive oil\n  * \u00bd cup well-drained sun-dried tomatoes (from 7-oz jar)\n  * 1 tablespoon fresh or 1 teaspoon dried basil leaves\n  * Cornmeal\n\n1 In large bowl, mix 2 cups of the flour, the sugar, salt and yeast. Stir in warm water and oil until well mixed, about 1 minute. Beat with wooden spoon 2 minutes.\n\n2 Stir in 1 cup flour. Stir in additional flour, 2 tablespoons at a time, until dough leaves side of bowl and dough is not sticky. Cover tightly with plastic wrap and refrigerate at least 2 hours but no longer than 24 hours.\n\n3 Grease large cookie sheet; sprinkle with cornmeal, shaking off excess. Divide dough in half. Sprinkle each dough half equally with the sun-dried tomatoes and basil. With floured hands, fold sun-dried tomatoes and basil into dough. Shape each half of dough into smooth ball by stretching surface of dough around to bottom on all 4 sides; pinch bottom to seal. Place dough balls about 5 inches apart on cookie sheet. Cover loosely with plastic wrap and let rise in warm place about 1 hour 30 minutes or until dough has doubled in size.\n\n4 Heat oven to 375\u00b0F. Place 8- or 9-inch pan on bottom oven rack; add hot water to pan until about \u00bd inch from top. Uncover dough; carefully slash tic-tac-toe pattern on top of each loaf with serrated knife. Place cookie sheet on middle oven rack. Bake 15 to 20 minutes or until loaves are dark golden brown and bread sounds hollow when tapped. Remove from cookie sheet to cooling rack; cool.\n\n1 Slice: Calories 70; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 15g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch Carbohydrate Choices: 1\n\n## Dried Cherry and Walnut Bread\n\nPrep 25 min Total 3 hr 30 min \u25cf 1 loaf (16 slices)\n\n  * 3\u00bc to 3\u00be cups bread flour\n  * 2 tablespoons sugar\n  * 1 package fast-acting dry yeast (2\u00bc teaspoons)\n  * 1\u00bc cups very warm water (120\u00b0F to 130\u00b0F)\n  * 2 tablespoons vegetable oil\n  * 1 cup coarsely chopped walnuts\n  * 1 cup dried cherries\n  * 1 tablespoon grated orange peel\n  * 1 teaspoon salt\n  * 1 tablespoon cornmeal\n  * 1 teaspoon milk\n\n1 In large bowl, mix 1\u00bd cups of the flour, the sugar and yeast. Add warm water and oil. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Stir in 1\u00be cups flour, the walnuts, cherries, orange peel and salt with wooden spoon until dough pulls away from side of bowl and forms a sticky ball.\n\n2 Place dough on lightly floured surface. Knead 2 to 3 minutes or until dough is smooth and springy, using enough of the remaining \u00bd cup flour as necessary to form into smooth ball. Grease large bowl with oil. Place dough in bowl, turning to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place 1 hour or until dough has doubled in size.\n\n3 Lightly grease large cookie sheet with shortening or cooking spray; sprinkle with cornmeal. Gently push fist into dough to deflate. Place dough on lightly floured surface. Gently shape into smooth ball by stretching surface of dough around to bottom on all sides; pinch bottom to seal. Place on cookie sheet. Cover dough loosely with plastic wrap sprayed with cooking spray and let rise in warm place 30 to 40 minutes or until doubled in size.\n\n4 Heat oven to 350\u00b0F. Brush top of loaf with milk. With serrated knife, cut 3 (\u00bc-inch-deep) slashes across top of dough. Bake 35 to 40 minutes or until loaf is deep golden brown and sounds hollow when tapped. Cool completely on cooling rack, about 1 hour.\n\n1 Slice: Calories 210; Total Fat 7g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 32g (Dietary Fiber 2g, Sugars 9g); Protein 5g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nCloverleaf Rolls, Crescent Rolls and Dinner Rolls\n\n## Dinner Rolls Calorie Smart\n\nPrep 30 min Total 2 hr 15 min \u25cf 15 rolls\n\n  * 3\u00bd to 3\u00be cups all-purpose or bread flour\n  * \u00bc cup sugar\n  * \u00bc cup butter, softened\n  * 1 teaspoon salt\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * \u00bd cup very warm water (120\u00b0F to 130\u00b0F)\n  * \u00bd cup very warm milk (120\u00b0F to 130\u00b0F)\n  * 1 egg\n  * Melted butter, if desired\n\n1 In large bowl, stir 2 cups of the flour, the sugar, \u00bc cup butter, the salt and yeast until well mixed. Add warm water, warm milk and egg. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Stir in enough remaining flour, \u00bc cup at a time, to make dough easy to handle.\n\n2 Place dough on lightly floured surface. Knead about 5 minutes or until dough is smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 1 hour or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n3 Grease bottom and sides of 13x9-inch pan with shortening or cooking spray. Gently push fist into dough to deflate. Divide dough into 15 equal pieces. Shape each piece into a ball; place in pan. Brush with melted butter. Cover loosely with plastic wrap and let rise in warm place about 30 minutes or until dough has doubled in size.\n\n4 Heat oven to 375\u00b0F. Uncover rolls. Bake 12 to 15 minutes or until golden brown. Serve warm if desired.\n\n1 Roll: Calories 160; Total Fat 4g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 25mg; Sodium 190mg; Total Carbohydrate 26g (Dietary Fiber 1g, Sugars 4g); Protein 4g Exchanges: 2 Starch Carbohydrate Choices: 2\n\nMake-Ahead Directions After placing rolls in pan, cover tightly with foil and refrigerate 4 to 24 hours. About 2 hours before baking, remove from refrigerator; remove foil and cover loosely with plastic wrap. Let rise in warm place until dough has doubled in size. If some rising has occurred in the refrigerator, rising time may be less than 2 hours. Bake as directed.\n\nCloverleaf Rolls Grease 24 regular-size muffin cups with shortening or cooking spray. Make dough as directed, except after pushing fist into dough, divide dough into 72 equal pieces. (To divide, cut dough in half, then continue cutting pieces in half until there are 72 pieces.) Shape each piece into a ball. Place 3 balls in each muffin cup. Brush with melted butter. Cover loosely with plastic wrap and let rise in warm place about 30 minutes or until dough has doubled in size. Bake as directed. Makes 24 rolls.\n\nCrescent Rolls Grease cookie sheet with shortening or cooking spray. Make dough as directed, except after pushing fist into dough, cut dough in half. Roll each half into 12-inch round on floured surface. Spread with softened butter. Cut each circle into 16 wedges. Roll up each wedge, beginning at rounded edge. Place rolls, with points underneath, on cookie sheet and curve slightly. Brush with melted butter. Cover loosely with plastic wrap and let rise in warm place about 30 minutes or until dough has doubled in size. Bake as directed. Makes 32 rolls.\n\n### Making Dinner Rolls\n\nPlace dough balls slightly apart in pan so they have room to double in size.\n\n### Making Cloverleaf Rolls\n\nPlace 3 dough balls next to each other in each muffin cup.\n\n### Making Crescent Rolls\n\nCut dough circles into wedges. Roll up each wedge, beginning at rounded edge.\n\n## Roll Top Treatments\n\nBrush unbaked rolls with melted butter (about 1 tablespoon for each 12 rolls). Then sprinkle with one of the following before baking:\n\nRosemary Sea Salt: Mix 3 tablespoons coarsely chopped fresh rosemary leaves and 1 teaspoon sea salt.\n\nGarlic-Parmesan: Mix 2 tablespoons grated Parmesan and a dash of garlic powder.\n\nTwo-Seed: Mix 2 tablespoons each sesame seed and poppy seed.\n\n## Soft Pretzels Calorie Smart\n\nPrep 45 min Total 2 hr \u25cf 16 pretzels\n\n  * 3\u00be to 4\u00bc cups all-purpose flour\n  * 1 tablespoon sugar\n  * 1\u00bd teaspoons salt\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * 1\u00bd cups warm water (120\u00b0F to 130\u00b0F)\n  * 2 tablespoons vegetable oil\n  * 1 cup water\n  * 2 teaspoons baking soda\n  * 2 teaspoons coarse (kosher or sea) salt\n\n1 In large bowl, stir 2 cups of the flour, the sugar, table salt and yeast with wooden spoon until well mixed. Add warm water and oil. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. With wooden spoon, stir in enough of the remaining flour, about \u00bd cup at time, until dough is soft, leaves side of bowl and is easy to handle (dough may be slightly sticky).\n\n2 Place dough on lightly floured surface. Knead 5 to 10 minutes or until dough is smooth and springy. Cover loosely with plastic wrap sprayed with cooking spray and let rest 10 minutes.\n\n3 Heat oven to 425\u00b0F. Spray cookie sheets with cooking spray. In shallow bowl, stir 1 cup water and baking soda to make pretzel \"wash.\"\n\n4 Divide dough into 16 equal pieces. Roll each piece into 24-inch rope (dip hands in pretzel wash to make rolling dough easier). To make pretzel shape, form rope into a circle, crossing ends at top. Fold dough so crossed ends rest on bottom of circle. Stir pretzel wash; brush over both sides of pretzel, using pastry brush. Place pretzel on cookie sheet. Repeat with remaining dough. Reserve remaining pretzel wash.\n\n5 Cover pretzels loosely with plastic wrap. To make thin pretzels, let rest about 5 minutes or until very slightly puffed. To make thicker pretzels, let rise in warm place 15 to 20 minutes or until puffed.\n\n6 Just before baking, brush pretzels with reserved wash; sprinkle with coarse salt. Bake 1 cookie sheet at a time 10 to 13 minutes or until golden brown. Remove from cookie sheets to cooling rack; cool at least 15 minutes before serving.\n\n1 Pretzel: Calories 120; Total Fat 2g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 670mg; Total Carbohydrate 23g (Dietary Fiber 0g, Sugars 0g); Protein 3g Exchanges: 1 Starch, \u00bd Other Carbohydrate Carbohydrate Choices: 1\u00bd\n\nParmesan-Herb Soft Pretzels In small bowl, mix 2 tablespoons grated Parmesan cheese, \u00bd teaspoon dried basil leaves and \u00bc teaspoon garlic powder. Brush hot baked pretzels with melted butter; sprinkle with cheese mixture.\n\n### Shaping Soft Pretzels\n\nRoll each dough piece into 24-inch rope.\n\nForm rope into a circle, crossing ends at top. Fold dough so crossed ends rest on bottom of circle.\n\n## Bakery-Style Pumpernickel Rolls Calorie Smart\n\nPrep 30 min Total 4 hr 30 min \u25cf 12 rolls\n\n  * 2\u00bd cups all-purpose flour\n  * \u00be cup rye flour\n  * 1 tablespoon regular active dry yeast\n  * 2 tablespoons unsweetened baking cocoa\n  * 1\u00bd teaspoons coarse (kosher or sea) salt\n  * 1\u2153 cups warm water (105\u00b0F to 115\u00b0F)\n  * 3 tablespoons full-flavored (dark) molasses\n  * 1 tablespoon cornmeal\n  * 1 egg\n  * 1 tablespoon water\n  * 1 teaspoon caraway seed, if desired\n\n1 In large bowl, mix all-purpose flour, rye flour, yeast, cocoa and salt. In small bowl, mix warm water and molasses. Add water mixture to flour mixture; stir with wooden spoon until dough is thoroughly mixed and no dry spots remain, 1 to 2 minutes. Cover loosely with plastic wrap. Let rise at room temperature 2 hours; refrigerate at least 1 hour but no longer than 24 hours.\n\n2 Line large cookie sheet with cooking parchment paper; sprinkle with cornmeal. Place dough on generously floured surface; turn dough to lightly coat with flour. Shape dough into a ball; cut in half. Cut each half into 6 equal pieces, coating with flour as needed to prevent sticking. Shape each piece into a ball by stretching surface of dough around to bottom on all sides; pinch bottom to seal. Place about 2 inches apart on cookie sheet. Cover loosely with plastic wrap sprayed with cooking spray; let rise in warm place 40 minutes or until doubled in size.\n\n3 Place empty broiler tray or cast-iron skillet on bottom oven rack. Heat oven to 400\u00b0F. In small bowl, beat egg and 1 tablespoon water. Brush buns with egg mixture; sprinkle with caraway seed. Place cookie sheet on middle oven rack. Gently pull out bottom rack and pour 1 cup hot water into broiler tray; gently slide rack back and quickly shut oven door.\n\n4 Bake 12 to 15 minutes or until buns begin to slightly brown and sound hollow when tapped. Cool 5 minutes on cooling rack. Serve warm or at room temperature.\n\n1 Roll: Calories 150; Total Fat 1g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 20mg; Sodium 300mg; Total Carbohydrate 30g (Dietary Fiber 2g, Sugars 3g); Protein 4g Exchanges: 1\u00bd Starch, \u00bd Other Carbohydrate Carbohydrate Choices: 2\n\nBakery-Style Pumpernickel Loaf Shape dough into 7x5-inch oval and place on cookie sheet. Cover loosely with plastic wrap sprayed with cooking spray; let rise in warm place 1 hour or until doubled in size. Continue as directed in Steps 3 and 4, brushing top and sides of loaf with egg mixture; sprinkle top with caraway seed. Bake 25 to 30 minutes or until loaf sounds hollow when tapped. Cool completely.\n\n## English Muffins Calorie Smart\n\nPrep 35 min Total 3 hr \u25cf 9 muffins\n\n  * 2 to 2\u00bd cups all-purpose flour\n  * 1 tablespoon sugar\n  * 1 teaspoon salt\n  * 2 tablespoons shortening\n  * 1 package regular active or fast-acting dry yeast (2\u00bc teaspoons)\n  * 1 cup very warm milk (120\u00b0F to 130\u00b0F)\n  * Cornmeal\n\n1 In large bowl, mix 2 cups of the flour, the sugar, salt, shortening and yeast. Add milk; stir until dough forms. If necessary, stir in enough remaining flour, 2 tablespoons at a time, until dough is no longer sticky. Dough should be soft.\n\n2 Place dough on lightly floured surface. Knead 4 to 5 minutes or until smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place 40 minutes to 1 hour or until doubled in size. Dough is ready if indentation remains when touched.\n\n3 Gently push fist into dough to deflate. On lightly floured surface, roll dough into 13-inch round \u00bc inch thick. Cut dough into 9 (3\u00bd-inch) rounds, rerolling dough as necessary. Generously sprinkle ungreased cookie sheet with cornmeal. Carefully transfer rounds to cookie sheet, placing about 1 inch apart. Sprinkle tops of rounds with cornmeal. Cover and let rise about 1 hour or until almost doubled in size.\n\n4 Heat oven to 350\u00b0F. Heat nonstick griddle or 12-inch skillet over medium heat (350\u00b0F). Carefully transfer rounds to hot griddle. Cook 6 to 8 minutes, turning once, until deep golden brown. Return to cookie sheet. Bake 5 to 10 minutes or until center is completely cooked. Cool 10 to 15 minutes. To serve, split with fork.\n\n1 Muffin: Calories 150; Total Fat 3.5(Saturated Fat 1, Trans Fat 0); Cholesterol 0mg; Sodium 280mg; Total Carbohydrate 24g(Dietary Fiber 1g, Sugars 3g); Protein 4g Exchanges: 1 Starch, \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\u00bd\n\n### Making English Muffins\n\nCook rounds 6 to 8 minutes, turning once, until deep golden brown.\n\nBake rounds 5 to 10 minutes or until center is completely cooked.\n\n## Caramel Sticky Rolls\n\nPrep 40 min Total 3 hr 20 min \u25cf 15 rolls\n\nRolls\n\n  * 3\u00bd to 4 cups all-purpose or bread flour\n  * \u2153 cup granulated sugar\n  * 1 teaspoon salt\n  * 2 packages regular active or fast-acting dry yeast (4\u00bd teaspoons)\n  * 1 cup very warm milk (120\u00b0F to 130\u00b0F)\n  * \u00bc cup butter, softened\n  * 1 egg\n\nCaramel Topping\n\n  * 1 cup packed brown sugar\n  * \u00bd cup butter, softened\n  * \u00bc cup light corn syrup\n  * 1 cup pecan halves, if desired\n\nFilling\n\n  * \u00bd cup chopped pecans or raisins, if desired\n  * \u00bc cup granulated sugar or packed brown sugar\n  * 1 teaspoon ground cinnamon\n  * 2 tablespoons butter, softened\n\n1 In large bowl, mix 2 cups of the flour, \u2153 cup granulated sugar, the salt and yeast. Add warm milk, \u00bc cup butter and the egg. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Beat on medium speed 1 minute, scraping bowl frequently. Stir in enough remaining flour, \u00bd cup at a time, to make dough easy to handle.\n\n2 Place dough on lightly floured surface. Knead about 5 minutes or until dough is smooth and springy. Grease large bowl with shortening. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap and let rise in warm place about 1 hour 30 minutes or until dough has doubled in size. Dough is ready if indentation remains when touched.\n\n3 In 2-quart saucepan, heat brown sugar and \u00bd cup butter to boiling, stirring constantly; remove from heat. Stir in corn syrup. Pour into 13x9-inchpan. Sprinkle with pecan halves.\n\n4 In small bowl, mix filling ingredients except 2 tablespoons butter.\n\n5 Gently push fist into dough to deflate. On lightly floured surface, flatten dough with hands or rolling pin into 15x10-inch rectangle. Spread with 2 tablespoons butter; sprinkle with filling. Roll rectangle up tightly, beginning at 15-inch side. Pinch edge of dough into roll to seal. With fingers, shape until even. With dental floss or serrated knife, cut roll into 15 (1-inch) slices.\n\n6 Place slices slightly apart in pan. Cover loosely with plastic wrap and let rise in warm place about 30 minutes or until dough has doubled in size.\n\n7 Heat oven to 350\u00b0F. Uncover rolls. Bake 30 to 35 minutes or until golden brown. Let stand 2 to 3 minutes. Place heatproof plate upside down over pan; turn plate and pan over. Let stand 1 minute so caramel can drizzle over rolls; remove pan. Serve warm.\n\n1 Roll: Calories 320; Total Fat 12g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 45mg; Sodium 250mg; Total Carbohydrate 50g (Dietary Fiber 1g, Sugars 25g); Protein 4g Exchanges: 1 Starch, 2\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\n\nLighter Directions For 4 grams of fat and 255 calories per serving, make recipe as directed, except omit caramel topping and pecan halves. Line pan with foil; spray foil with cooking spray. Drizzle 1 cup caramel ice cream topping over foil (heat topping slightly if it is stiff). Continue as directed, except omit chopped pecans from filling.\n\nMake-Ahead Directions After placing slices in pan, cover tightly with plastic wrap or foil; refrigerate 4 to 24 hours. About 2 hours before baking, remove from refrigerator; remove plastic wrap or foil and cover loosely with plastic wrap. Let rise in warm place until dough has doubled in size. If some rising has occurred in the refrigerator, rising time may be less than 2 hours. Bake as directed.\n\nCinnamon Rolls Omit caramel topping and pecan halves. Grease bottom and sides of 13x9-inch pan with shortening or cooking spray. Place dough slices in pan. Let rise and bake as directed in Steps 6 and 7, except do not turn pan upside down. Remove rolls from pan to cooling rack. Cool 10 minutes. Drizzle rolls with Vanilla Glaze (page 581) if desired.\n\nCaramel Sticky Rolls\n\n### Placing Pecans\n\nSprinkle pecans over brown sugar mixture in pan.\n\n### Rolling and Cutting Rolls\n\nRoll rectangle up tightly, beginning at 15-inch side.\n\nUsing dental floss or a serrated knife, cut rolls into slices.\n\n### Placing Slices\n\nPlace slices slightly apart in pan so they have room to double in size.\n\nEasy Naan\n\n## Easy Naan\n\nIf you have never seen the way naan breads are made, take a peek the next time you are at an Indian restaurant: The chef slaps a piece of dough between his hands, stretching it into that familiar teardrop shape. Working quickly, he reaches into the tandoor (clay-lined oven) and slaps the dough against its inner wall. Within seconds, the dough puffs up and forms brown spots. He peels the bread away from the wall with a flat-edged skewer and brushes it with clarified butter.\n\nPrep 20 min Total 1 hr 20 min \u25cf 4 breads\n\n  * 3 cups all-purpose flour\n  * 1 tablespoon sugar\n  * 1 teaspoon salt\n  * \u00bd cup milk\n  * 2 tablespoons vegetable oil\n  * \u00bd cup plain yogurt\n  * \u00bd to \u00be cup water\n  * 1 tablespoon butter, melted*\n\n1 In large bowl, mix together flour, sugar and salt. Stir in milk and oil. Stir in yogurt. Stir in water, 2 tablespoons at a time, until dough forms. The dough should not be sticky or dry.\n\n2 Knead dough in bowl or on lightly floured surface 2 to 3 minutes or until dough becomes smooth and elastic. Brush dough lightly with some of the melted butter and cover with plastic wrap; set aside 30 minutes. (At this point, dough can be covered and refrigerated up to 24 hours. When ready to roll, let dough stand at room temperature 30 minutes so it becomes soft and easy to handle.)\n\n3 Heat oven to 450\u00b0F. Place pizza stone in oven to heat. Divide dough into 4 equal pieces. Shape each into a ball; brush each lightly with some of the butter.\n\n4 Roll one piece of dough into 8-inch round or teardrop shape, about \u00bc inch thick, on lightly floured surface, taking care not to tear dough. Repeat with remaining dough.\n\n5 Place one piece of dough on hot pizza stone. Bake 3 to 4 minutes or until brown spots form and bubbles start to appear on surface. Turn; bake 7 to 8 minutes longer. Remove bread with spatula; brush lightly with melted butter. Wrap bread in foil to keep warm while baking remaining dough. Cut breads in half to serve.\n\n*Clarified butter (page 469) can be substituted for the melted butter.\n\n1 Serving (\u00bd Bread): Calories 240; Total Fat 7g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 520mg; Total Carbohydrate 38g (Dietary Fiber 1g, Sugars 2g); Protein 5g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\u00bd\n\n### Making Naan\n\nRoll 1 piece of dough at a time into circle or teardrop shape about \u00bc inch thick.\n\nBake dough on hot pizza stone until brown spots or bubbles begin to appear on surface. Turn; bake other side.\n\n## Delicious Uses for Naan\n\nTraditionally, Indian food is eaten with your right hand, not with utensils. Pieces of naan would be ripped off and used to scoop up food to be eaten or used to dip into saucy foods to enjoy the sauce.\n\nToday, in addition to being a tasty scoop, naan is being used in creative new ways. Any way flatbread is used, naan can be used as well. Turn it into a pizza by topping it with your favorite pizza toppings before baking. Spoon your favorite sandwich fillings on it and roll it up, like a wrap. Or serve it with soups and stews in place of bread or with dips, in place of chips.\n\n# Chapter 19: Cookies, Bars & Candies\n\nAlmond Biscotti, page 539, dipped in chocolate, page 533\n\n## Cookies and Bar Basics\n\nBaking Pans for Bars\n\nChoosing the Right Ingredients\n\nCookie Sheets\n\nMixing\n\nStoring Cookies and Bars\n\nSuccessful Baking on Any Type of Cookie Sheet\n\nTips for Perfect Cookies and Bars\n\n## Features\n\nHeirloom Recipe and New Twist\n\nLearn to Decorate\n\nLearn with Betty: Chocolate Chip Cookies\n\nLearn with Betty: Melting Chocolate\n\nDried Fruit\n\n## Drop Cookies\n\nCandy Cookies\n\nChocolate Chip Cookies CALORIE SMART\n\nChocolate Chip Whoopie Pies\n\nChocolate Crinkles CALORIE SMART\n\nChocolate-Caramel Compost Cookies\n\nDulce de Leche Chocolate Chip Cookies\n\nJumbo Chocolate Chip Cookies\n\nOatmeal\u2013Chocolate Chip Cookies\n\nOatmeal-Cranberry Cookies\n\nOatmeal-Raisin Cookies CALORIE SMART\n\nPink Peppermint Whoopie Pies\n\nPumpkin Cookies with Browned Butter Frosting CALORIE SMART\n\nSpicy Pumpkin Cookies with Browned Butter Frosting\n\nToffee Whoopie Pies\n\nWhoopie Pies\n\n## Shaped and Rolled Cookies\n\nAlmond Biscotti\n\nBrown Sugar Refrigerator Cookies CALORIE SMART\n\nCherry Shortbread Cookies\n\nChocolate Buttery Spritz Cookies\n\nChocolate Chip, Cherry and Pistachio Soft No-Roll Sugar Cookies\n\nChocolate-Dipped Shortbread Cookies\n\nChocolate-Filled Orange-Rosemary Butter Balls CALORIE SMART\n\nChocolate-Hazelnut Sandwiches CALORIE SMART\n\nCitrus\u2013Brown Sugar Refrigerator Cookies\n\nColored Sugar\n\nFrench Macarons with Bittersweet Chocolate Ganache CALORIE SMART\n\nFrosted and Decorated Sugar Cookies\n\nGinger Shortbread Cookies\n\nGingerbread Cookies CALORIE SMART\n\nGingersnaps CALORIE SMART\n\nGlazed Soft No-Roll Sugar Cookies\n\nHazelnut Biscotti CALORIE SMART\n\nHazelnut-Chocolate Thumbprint Cookies\n\nLemon-Almond Thumbprint Cookies\n\nLemon Macarons\n\nMaple\u2013Brown Sugar Refrigerator Cookies\n\nMint Macarons\n\nOatmeal-Date Quinoa Thumbprints CALORIE SMART\n\nOrange Macarons\n\nPaintbrush Sugar Cookies\n\nPeanut Butter Cookies CALORIE SMART\n\nPeanut-Fudge Thumbprint Cookies\n\nRich Peanut Butter Chip Cookies\n\nShortbread Cookie Wedges\n\nShortbread Cookies Calorie Smart\n\nShortbread Nut Cookies\n\nSnickerdoodles CALORIE SMART\n\nSoft No-Roll Sugar Cookies CALORIE SMART\n\nSpritz Cookies CALORIE SMART\n\nSugar Cookies CALORIE SMART\n\nThumbprint Cookies CALORIE SMART\n\nToasted Coconut\u2013Brown Sugar Refrigerator Cookies\n\nToffee Shortbread Cookies\n\nWhole Wheat Shortbread Cookies\n\n## Brownies and Bars\n\nBlondies with Brown Sugar Frosting CALORIE SMART\n\nChocolate Brownie Pie\n\nChocolate Brownies\n\nChocolate Chip Bars\n\nChocolate\u2013Peanut Butter Brownies\n\nDate Bars CALORIE SMART\n\nDouble-Chocolate Brownies\n\nFig Bars\n\nFruitcake-Inspired Brownies\n\nLemon Bars Calorie Smart\n\nOutrageous Caramel-Fudge Brownies\n\nTriple-Nut Bars\n\n## Candy Basics\n\nSelecting and Preparing Pans\n\nMixing, Cooking and Cooling\n\n## Candy\n\nAlmond or Cashew Brittle\n\nCaramels Calorie Smart\n\nCherry-Pistachio Fudge\n\nChocolate Caramels\n\nChocolate-Topped Sea Salt Caramels\n\nCocoa Marshmallows\n\nCoconut Truffles\n\nLuscious Chocolate Truffles CALORIE SMART\n\nMarshmallows CALORIE SMART\n\nPeanut Brittle CALORIE SMART\n\nPeanut Butter Candy Fudge\n\nPeppermint Marshmallows\n\nPistachio-Cranberry Divinity CALORIE SMART\n\nPralines CALORIE SMART\n\nToffee Truffles\n\nTriple-Chocolate and Candy Fudge\n\nTriple-Chocolate Fudge CALORIE SMART\n\nWhite Chocolate\u2013Chocolate Truffles\n\n## Bonus Recipes\n\nFruit 'n Nuts Salad\n\nFruity Couscous\n\nLemon-Fig Muffins\n\nOn-the-Go Mix\n\nPomegranate Oatmeal\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Cookie and Bar Basics\n\nFreshly baked homemade cookies are simply irresistible . . . seriously, who can say no to one of these sweet treats? Cookies are easy to make and a great place for beginner bakers to start. No matter what your baking experience, knowing all of the secrets to perfectly baked cookies will up your baking game.\n\n## COOKIE SHEETS\n\n  * Choose cookie sheets that are at least 2 inches smaller than the inside of your oven to allow heat to circulate.\n  * Cookie sheets may be open on one to three sides. Jelly roll pans (15x10x1 inch) have four sides; if used for cookies, the cookies may not brown as evenly.\n  * Owning at least two cookie sheets is helpful. When one batch is baking, you can be preparing the second batch for the oven.\n\n### Successful Baking on Any Type of Cookie Sheet\n\nShiny Aluminum: The best cookie sheets are shiny aluminum with a smooth surface. They reflect heat, letting cookies bake evenly and brown properly. The recipes in this book were tested using shiny aluminum cookie sheets.\n\nInsulated: These cookie sheets help prevent cookies from becoming too dark on the bottom. Cookies baked on these sheets may take longer to bake; the bottoms will be light colored and cookies may not brown as much overall. Cookies may be difficult to remove from these cookie sheets because the bottoms of the cookies are more tender.\n\nNonstick and Dark Surface: Cookies baked on these cookie sheets may be smaller in diameter and more rounded. The tops and especially the bottoms will be more browned, and the bottoms will be hard. To avoid over-browning and burning, check cookies at the minimum bake time and then add time as necessary. Follow manufacturer's directions; some recommend reducing the oven temperature by 25\u00b0F.\n\n## BAKING PANS FOR BARS\n\nUse the exact size of pan called for in a recipe when baking bars. Bars made in pans that are too big become hard and overcooked, and those made in pans that are too small can be doughy and raw in the center and hard on the edges.\n\nShiny metal pans are recommended for baking bars. They reflect heat and prevent the bottom of the bars from getting too brown and hard. Follow the same guidelines as for nonstick and dark-surface cookie sheets.\n\n## Greasing Pans\n\n  * Grease pans only when a recipe calls for it. Grease cookie sheets with shortening or cooking spray. Unless the recipe directs, do not use butter, margarine or oil because the area between the cookies can burn during baking. Grease pans for bars as directed in the recipe.\n  * Don't grease nonstick cookie sheets, as the cookies can spread too much.\n  * As an alternative to greasing, line pans or sheets with parchment paper; sheets can also be lined with a silicone baking mat.\n\nOatmeal-Date Quinoa Thumbprints (page 523)\n\n## CHOOSING THE RIGHT INGREDIENTS\n\nFlour: For cookies and bars, all-purpose flour is recommended. Whole wheat flour can also be used, but only substitute it for one-third to one-half the amount of all-purpose flour to keep cookies from becoming too dry. White whole wheat flour is also available. It is has the same nutritional benefits as whole wheat flour but with a lighter color and milder flavor than regular whole wheat. Experiment with substituting some or all whole wheat flour for all-purpose flour in your cookie and bar recipes. For more information on flour and how to measure it correctly, see Flour, page 6, and Measuring Correctly, page 16.\n\nSweeteners: In addition to adding sweetness, sugar also helps to brown and add tenderness to baked goods.\n\nLeavenings: Cookies usually call for baking powder or baking soda, which are not interchangeable. For more information, see Baking Ingredients, page 4.\n\nFats and Oils: Fats add tenderness and flavor to cookies and bars, but they are not all equal. For best results, use butter or, if the recipe calls for it, shortening. If you choose to use margarine, use only products with at least 65 percent fat. Any other spreads or reduced-fat products contain more water, resulting in cookies that are soft, tough and puffy.\n\nEggs: Eggs give richness, moisture and structure to cookies and bars. All the recipes in this book have been tested with large eggs. Egg product substitutes, made of egg whites, can be substituted for whole eggs, but the cookies and bars may come out drier.\n\nLiquids: Water, fruit juice, cream or milk tends to make cookies crisper by causing them to spread more. Add only as much liquid as the recipe calls for.\n\nOats: Quick-cooking and old-fashioned oats are interchangeable unless a recipe calls for a specific type. Instant oatmeal products are not the same as quick-cooking oats and should not be used for baking.\n\nNuts, Peanuts and Almond Brickle Chips: When a recipe calls for nuts, feel free to substitute any variety of nut or peanuts. Nuts and almond brickle chips can easily become rancid, giving them an unpleasant, strong flavor that can ruin the taste of cookies. To prevent rancidity, store nuts and peanuts tightly covered in the refrigerator or freezer up to 2 years. Do not freeze cashews, as they will become soggy. Store almond brickle chips in the refrigerator or freezer up to 6 months. Always taste items that tend to turn rancid before adding them to a recipe. If they don't taste fresh, compost or throw them out.\n\n## Bake a Test Cookie\n\nIf you're new to cookie baking or are trying a new recipe, bake just one cookie as directed in the recipe to see how it performs. This is a great way to make adjustments before baking an entire batch. If the cookie spreads too much, add 1 to 2 tablespoons flour to the dough, or refrigerate the dough 1 to 2 hours before baking. If the cookie is too round or hard, add 1 to 2 tablespoons milk to the dough.\n\n## Softening Butter\n\nLet butter soften at room temperature for 30 to 40 minutes. You can also soften it in the microwave; see Microwave Cooking and Heating chart, pages 29 and 30. If the butter is too soft, the dough will be too soft, causing the cookies to spread too much.\n\n## Mixing\n\nFor most recipes in this book, you can use an electric mixer or a spoon when mixing cookie or bar dough. The sugars, fats and liquids are usually beaten together first until well mixed. Flour and other ingredients are almost always stirred in by hand to avoid overmixing the dough, which can result in tough cookies.\n\n### Beating Butter and Sugar\n\nButter and sugar have been beaten long enough when mixture is light and fluffy and light in color.\n\n## TIPS FOR PERFECT COOKIES AND BARS\n\n  * Use completely cooled cookie sheets, as cookies will spread too much if placed and baked on a warm or hot cookie sheet.\n  * Make cookies all the same size so they bake evenly. Small ice cream\u2013type scoops measure the same amount of cookie dough. The spring-loaded handles release the dough easily onto the cookie sheet. See Dropping Dough onto Pan, page 524. In general, a #16 scoop yields \u00bc cup and a #70 scoop equals 1 level tablespoon, but scoops can vary by manufacturer. To ensure the scoop is the size you want to use, measure the volume by filling the scoop with water and measuring the amount of water.\n  * Bake pans of cookies and bars in the middle of the oven. For even baking, bake one sheet at a time. If you do bake two sheets at once, position oven racks as close to the middle as possible and rotate the position of the sheets halfway through baking.\n  * Check cookies and bars at the minimum bake time, baking longer if needed.\n  * Cool cookies as directed. Use a flat, thin metal spatula to remove them from the baking sheet to a cooling rack.\n  * Cool bars and brownies in the pan on a cooling rack.\n  * Use a plastic knife to cut brownies and soft, sticky bars such as Lemon Bars, page 542. It cuts bars with the nicest edges and also prevents pans from getting scratched from a metal knife.\n\n## STORING COOKIES AND BARS\n\n  * Store crisp cookies at room temperature in loosely covered containers.\n  * Keep chewy and soft cookies at room temperature in resealable food-storage plastic bags or tightly covered containers.\n  * Let frosted or decorated cookies set or harden before storing; then store by placing them between layers of parchment or waxed paper, plastic wrap or foil.\n  * Store only one type of cookie in a container to keep the texture and flavor intact. If different types of cookies are stored in the same container, they can transfer moisture to one another, making crisp cookies soft, and they can also pick up flavors from each other.\n  * Most bars can be stored tightly covered, but follow specific recipe directions, as some may need to be stored loosely covered or may need to be refrigerated.\n  * For general directions on freezing and thawing cookies and bars, see Food Storage, page 33, and Refrigerator and Freezer Food Storage Chart, page 33. Do not freeze meringue, custard-filled or cream-filled cookies.\n\n## Removing Stuck Cookies\n\nIf cookies are difficult to remove from the cookie sheet because they were left to cool too long, put the cookies back into the oven for 1 to 2 minutes, and then remove them from the sheet. They should then come off easily.\n\n## Quick Pan Cleanup\n\nLine baking pans for bars with foil for super-easy cleanup and ease in cutting. Turn pan upside down. Tear off a piece of foil longer than the pan. Smooth the foil over the bottom and sides of pan; then remove foil. Flip the pan over and gently fit the shaped foil into the pan. When the bars are cool, lift them out of the pan using the edges of the foil as handles. Peel back the foil and cut bars as directed.\n\n## Baking Cocoa\n\nWhen a recipe calls for cocoa, it means unsweetened baking cocoa, not hot chocolate products, which are sweetened and have additional ingredients in them. Three types of baking cocoa are available: Regular (nonalkalized) is most commonly available and used for most recipes calling for cocoa. Dutch or European (alkalized) goes through a \"Dutching\" process to neutralize the natural acids found in cocoa, resulting in a darker cocoa with a more mellow chocolate flavor. Regular cocoa can be substituted for Dutch, but don't substitute Dutch in recipes calling for regular cocoa because the leavening in the recipe may not work well with it, affecting rising. Dark is a blend of both regular and Dutch cocoas. It can be substituted for regular baking cocoa in recipes, but the baked good will have a darker color and more intense flavor than if made with regular baking cocoa.\n\n# Heirloom Recipe and New Twist\n\nOatmeal-raisin cookies have been a cookie jar favorite for generations. Their chewy texture and sweet taste never go out of style. If you're an oatmeal-raisin fan, give our new twist a try. We've done a mash-up of oatmeal-raisin cookies with thumbprint cookies, plus the addition of quinoa both inside and outside the cookie. Yum!\n\nHeirloom\n\nOatmeal-Raisin Cookies\n\n## Oatmeal-Raisin Cookies Calorie smart\n\nQuick-cooking and old-fashioned rolled oats are interchangeable unless a recipe calls for a specific type. Instant oatmeal products are not the same and should not be used for baking\u2014you will have mushy cookies.\n\nPrep 45 min Total 45 min \u25cf 3 dozen cookies\n\n  * \u2154 cup granulated sugar\n  * \u2154 cup packed brown sugar\n  * \u00bd cup butter, softened\n  * \u00bd cup shortening\n  * 1 teaspoon vanilla\n  * 2 eggs\n  * 1 teaspoon baking soda\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon baking powder\n  * \u00bd teaspoon salt\n  * 3 cups quick-cooking or old-fashioned oats\n  * 1 cup all-purpose flour\n  * 1 cup raisins\n  * \u00bd cup chopped nuts, if desired\n\n1 Heat oven to 375\u00b0F. In large bowl, beat granulated sugar, brown sugar, butter, shortening, vanilla and eggs with electric mixer on medium speed, or mix with spoon, until well blended. Beat or stir in baking soda, cinnamon, baking powder and salt. Stir in oats, flour, raisins and nuts.\n\n2 Onto ungreased cookie sheet, drop dough by rounded tablespoonfuls about 2 inches apart.\n\n3 Bake 9 to 11 minutes or until light brown. Immediately remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 140; Total Fat 6g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 105mg; Total Carbohydrate 18g (Dietary Fiber 1g, Sugars 10g); Protein 1g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\nLighter Directions For 3 grams of fat and 95 calories per serving, substitute unsweetened applesauce for the shortening and \u00bd cup fat-free egg product for the eggs. Increase cinnamon and vanilla to 1\u00bd teaspoons each.\n\nOatmeal\u2013Chocolate Chip Cookies Substitute 1 cup semisweet chocolate chips for the raisins.\n\nOatmeal-Cranberry Cookies Substitute 1 cup dried cranberries for the raisins.\nNew Twist\n\nOatmeal-Date Quinoa Thumbprints\n\n## Oatmeal-Date Quinoa Thumbprints Calorie smart\n\nThe classic oatmeal-raisin cookie goes rogue with our new thumbprint style. The twist gets dates instead of raisins and a double hit of grainy texture with quinoa\u2014cooked quinoa in the dough and crispy quinoa on the outside! Fill them with jelly or jam, or try apple butter or lemon curd.\n\nPrep 45 min Total 2 hr 25 min \u25cf 6\u00bd dozen cookies\n\nCrispy Quinoa\n\n  * 3 cups water\n  * 1\u00bd cups uncooked white quinoa\n  * 2 teaspoons vegetable oil\n\nDough\n\n  * \u2154 cup granulated sugar\n  * \u2154 cup packed brown sugar\n  * 1 teaspoon baking soda\n  * \u00bd teaspoon baking powder\n  * \u00bc teaspoon salt\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla\n  * 2 eggs\n  * 3 cups quick-cooking or old-fashioned oats\n  * 1\u2153 cups all-purpose flour\n  * 1 cup chopped dates or raisins\n  * About \u00be cup jelly or jam (any flavor)\n\n1 Heat oven to 400\u00b0F. In 2-quart saucepan, heat water and quinoa to boiling. Reduce heat; cover and simmer 15 minutes or until water is absorbed. Remove 1 cup cooked quinoa to shallow dish; let cool about 10 minutes (this will be used in the dough). Set aside.\n\n2 Drizzle oil over remaining cooked quinoa in saucepan; toss to coat. Spread evenly in 15x10x1-inch pan. Bake 50 minutes, stirring every 10 minutes, until quinoa is light golden brown and crisp. Cool completely, about 20 minutes. Place in shallow dish (dough balls will be rolled in this crispy quinoa); set aside.\n\n3 Reduce heat to 375\u00b0F. Line large cookie sheets with cooking parchment paper; spray paper with cooking spray. In large bowl, beat granulated sugar, brown sugar, baking soda, baking powder, salt, butter, vanilla and eggs with electric mixer on medium speed until well blended, or mix with spoon. Stir in oats, flour, reserved 1 cup cooked quinoa and the dates.\n\n4 Shape dough into 1-inch balls or use #70 (1 level tablespoon) cookie scoop (dough will be soft). Roll balls in crispy quinoa. On cookie sheets, place balls about 2 inches apart. Press thumb into center of each cookie to make indentation, but do not press all the way to cookie sheet.\n\n5 Bake 8 to 10 minutes or until light brown. Quickly remake indentations with end of wooden spoon. Cool 2 minutes. Remove from cookie sheets to cooling racks; cool completely. Fill each with about \u00bd teaspoon jelly.\n\n1 Cookie: Calories 80; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 50mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 6g); Protein 1g Exchanges: 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\n## Chocolate Chip Cookies Calorie smart\n\nThe secret to great chocolate chip cookies is the combination of fresh ingredients, correct measuring and a good baking technique.\n\nPrep 1 hr Total 1 hr \u25cf 4 dozen cookies\n\n  * \u00be cup granulated sugar\n  * \u00be cup packed brown sugar\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla\n  * 1 egg\n  * 2\u00bc cups all-purpose flour\n  * 1 teaspoon baking soda\n  * \u00bd teaspoon salt\n  * 2 cups semisweet or dark chocolate chips (about 12 oz)\n  * 1 cup coarsely chopped nuts, if desired\n\n1 Heat oven to 375\u00b0F. In large bowl, beat granulated sugar, brown sugar, butter, vanilla and egg with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda and salt (dough will be stiff). Stir in chocolate chips and nuts.\n\n2 Onto ungreased cookie sheets, drop dough by rounded tablespoonfuls about 2 inches apart. For perfectly sized and shaped cookies, use a #70 (1 level tablespoon) cookie scoop.\n\n3 Bake 8 to 10 minutes or until light brown (centers will be soft). Cool 1 to 2 minutes; remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 140; Total Fat 8g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 80mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 10g); Protein 1g Exchanges: 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\nLighter Directions For 5 grams of fat and 90 calories per serving, reduce butter to \u00be cup and omit nuts. Substitute 1 cup miniature semisweet chocolate chips for the chocolate chips.\n\nCandy Cookies Substitute 2 cups candy-coated chocolate candies for the chocolate chips.\n\nChocolate Chip Bars Press dough in ungreased 13x9-inch pan. Bake 15 to 20 minutes or until golden brown. Cool in pan on cooling rack. Cut into 8 rows by 6 rows. Makes 48 bars.\n\nJumbo Chocolate Chip Cookies Onto ungreased cookie sheets, drop dough by \u00bc cupfuls or #16 cookie scoop about 3 inches apart. Bake 12 to 15 minutes or until edges are set (centers will be soft). Cool 1 to 2 minutes; remove from cookie sheets to cooling racks. Makes 1\u00bd dozen cookies.\n\n### Learn with Betty Chocolate Chip Cookies\n\n**Perfect Cookie:** It is slightly rounded on top and evenly golden brown.\n\n**Flat Cookie** : The butter was too soft or partially melted, the flour was under-measured or the cookie sheet was too hot.\n\n**Round or Hard Cookie:** Too much flour was used and cookie was overbaked.\n\n### Dropping Dough onto Pan\n\nScoop dough with one spoon and use another spoon to push dough onto cookie sheet.\n\nOr use a spring-handle cookie scoop to quickly make evenly sized and shaped cookies.\n\nFor jumbo cookies, use a measuring cup or a large spring-handle ice cream scoop.\n\n## Dulce de Leche Chocolate Chip Cookies\n\nPrep 1 hr Total 1 hr 45 min \u25cf 3 dozen cookies\n\n  * 1 can (13.4 oz) dulce de leche (caramelized sweetened condensed milk)*\n  * 1\u00bd cups butter, softened\n  * 1\u00bc cups granulated sugar\n  * 1\u00bc cups packed brown sugar\n  * 1 tablespoon vanilla\n  * 2 eggs\n  * 4 cups all-purpose flour\n  * 2 teaspoons baking soda\n  * 1 teaspoon salt\n  * 4 cups semisweet chocolate chips (about 24 oz)\n  * 1 cup chopped pecans, toasted (page 23)\n\n1 Line cookie sheet with waxed paper or foil. Spoon 36 measuring teaspoonfuls dulce de leche onto cookie sheet. Freeze 1 hour or until slightly firm (dollops will not freeze solid).\n\n2 Heat oven to 350\u00baF. Line cookie sheets with cooking parchment paper. In large bowl, beat butter, granulated sugar, brown sugar, vanilla and eggs with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda and salt. Using heavy-duty wooden spoon, stir in chocolate chips and pecans; dough will be stiff and chunky (you may need to use your hands).\n\n3 Onto cookie sheets, drop dough by \u00bc cupfuls or #16 ice cream scoop about 3 inches apart. Press thumb into center of each cookie to make deep indentation, but do not press all the way to cookie sheet. Place 1 dollop dulce de leche into center of each cookie, forming dough around dollop to enclose. (If dulce de leche dollops become too warm to work with, return to freezer.)\n\n4 Bake 12 to 15 minutes or until light brown (centers will be soft). Cool 1 to 2 minutes. Remove from cookie sheets to cooling racks.\n\n*see page 7. If you can't find dulce de leche, substitute 42 round chewy caramels in milk chocolate (from 12-ounce bag), unwrapped. No need to freeze these, just unwrap from the gold foil and insert in the indentation as directed in Step 3. Continue as directed.\n\n1 Cookie: Calories 340; Total Fat 17g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 35mg; Sodium 220mg Total Carbohydrate 43g (Dietary Fiber 2g, Sugars 30g); Protein 3g Exchanges:1 Starch, 2 Other Carbohydrate 3\u00bd Fat Carbohydrate Choices: 3\n\nChocolate Crinkles\n\n## Chocolate Crinkles Calorie smart\n\nPrep 1 hr Total 4 hr \u25cf 6 dozen cookies\n\n  * 2 cups granulated sugar\n  * \u00bd cup vegetable oil\n  * 2 teaspoons vanilla\n  * 4 oz unsweetened baking chocolate, melted, cooled\n  * 4 eggs\n  * 2 cups all-purpose flour\n  * 2 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 1 cup powdered sugar\n\n1 In large bowl, stir granulated sugar, oil, vanilla and chocolate until well mixed. Stir in eggs, one at a time. Stir in flour, baking powder and salt. Cover and refrigerate at least 3 hours.\n\n2 Heat oven to 350\u00b0F. Grease cookie sheets with shortening or cooking spray, or line with cooking parchment paper or silicone baking mats.\n\n3 In small bowl, place powdered sugar. Drop dough by teaspoonfuls into powdered sugar; roll around to coat. Shape into balls. On cookie sheets, place balls about 2 inches apart.\n\n4 Bake 10 to 12 minutes or until almost no indentation remains when touched in center. Immediately remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 70; Total Fat 2.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 10mg; Sodium 35mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 7g); Protein 0g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: \u00bd\n\nChocolate-Caramel Compost Cookies\n\n## Chocolate-Caramel Compost Cookies\n\nPrep 45 min Total 45 min \u25cf 2 dozen cookies\n\n  * \u00be cup packed brown sugar\n  * \u00bd cup granulated sugar\n  * \u00bd cup butter, softened\n  * \u00bd cup shortening\n  * 2 tablespoons coffee grounds from brewed coffee\n  * 1\u00bd teaspoons vanilla\n  * 1 egg\n  * 2 cups all-purpose flour\n  * 1 teaspoon baking soda\n  * \u00bd teaspoon salt\n  * 1 cup semisweet chocolate chips\n  * 1\u00bd cups slightly crushed rippled potato chips\n  * 1\u00bd cups frosted cocoa corn puff cereal\n  * 30 round chewy caramels in milk chocolate (from 12-oz bag), unwrapped, cut into quarters\n\n1 Heat oven to 350\u00b0F. Spray cookie sheets with cooking spray.\n\n2 In large bowl, beat brown sugar, granulated sugar, butter and shortening with electric mixer on medium speed until light and fluffy. Add coffee grounds, vanilla and egg; beat on low speed until blended. Add flour, baking soda and salt; beat until soft dough forms. Stir in remaining ingredients. Onto cookie sheets, drop dough by \u00bc cupfuls 2 inches apart.\n\n3 Bake 11 to 14 minutes or until light golden brown. Cool 4 minutes; remove from cookie sheets to cooling racks. Store cooled cookies in tightly covered container.\n\n1 Cookie: Calories 260; Total Fat 13g (Saturated Fat 6g, Trans Fat 1g); Cholesterol 20mg; Sodium 190mg; Total Carbohydrate 33g (Dietary Fiber 1g, Sugars 20g); Protein 2g Exchanges: 1 Starch, 1 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 2\n\nPeanut Butter Cookies\n\n## Peanut Butter Cookies Calorie smart\n\nAlthough we call for creamy peanut butter, you could use the chunky variety instead. Or, add \u00bd to \u00be cup chopped honey-roasted peanuts after the flour is mixed in.\n\nPrep 45 min Total 45 min \u25cf 2\u00bd dozen cookies\n\n  * \u00bd cup granulated sugar\n  * \u00bd cup packed brown sugar\n  * \u00bd cup creamy peanut butter\n  * \u00bd cup butter, softened\n  * 1 egg\n  * 1\u00bc cups all-purpose flour\n  * \u00be teaspoon baking soda\n  * \u00bd teaspoon baking powder\n  * \u00bc teaspoon salt\n  * Additional granulated sugar\n\n1 Heat oven to 375\u00b0F. In large bowl, beat \u00bd cup granulated sugar, the brown sugar, peanut butter, butter and egg with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda, baking powder and salt.\n\n2 Shape dough into 1\u00bc-inch balls. On ungreased cookie sheets, place balls about 3 inches apart. Flatten in crisscross pattern with fork dipped in additional granulated sugar.\n\n3 Bake 9 to 10 minutes or until light brown. Cool 5 minutes; remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 110; Total Fat 6g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 95mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 8g); Protein 2g Exchanges: 1 Starch, 1 Fat Carbohydrate Choices: 1\n\nRich Peanut Butter Chip Cookies Omit \u00bd cup granulated sugar; increase brown sugar to 1 cup. After stirring in flour, baking soda, baking powder and salt, stir in 1 cup peanut butter chips. Shape dough into balls as directed. Dip tops of balls into granulated sugar, but do not flatten. Bake as directed.\n\n### Making Peanut Butter Cookies\n\nShape dough into 1\u00bc-inch balls; place on cookie sheet about 3 inches apart.\n\nFlatten in crisscross pattern with fork dipped in sugar.\n\nPumpkin Cookies with Browned Butter Frosting\n\n## Pumpkin Cookies with Browned Butter Frosting Calorie smart\n\nPrep 55 min Total 1 hr 40 min \u25cf 2\u00bd dozen cookies\n\nCookies\n\n  * \u2154 cup granulated sugar\n  * \u2154 cup packed brown sugar\n  * \u00be cup butter, softened\n  * 1 teaspoon vanilla\n  * \u00bd cup canned pureed pumpkin (not pumpkin pie mix)\n  * 2 eggs\n  * 2\u00bc cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon salt\n\nBrowned Butter Frosting\n\n  * 3 cups powdered sugar\n  * 1 teaspoon vanilla\n  * 3 to 4 tablespoons milk\n  * \u2153 cup butter*\n\n1 Heat oven to 375\u00b0F. In large bowl, beat granulated sugar, brown sugar, \u00be cup butter and 1 teaspoon vanilla with electric mixer on medium speed, scraping bowl occasionally, until well blended. Beat in pumpkin and eggs until well mixed. On low speed, beat in flour, baking soda, cinnamon and salt.\n\n2 Onto ungreased cookie sheets, drop dough by heaping tablespoonfuls about 2 inches apart.\n\n3 Bake 10 to 12 minutes or until almost no indentation remains when touched in center. Immediately remove from cookie sheets to cooling racks. Cool completely, about 45 minutes.\n\n4 In medium bowl, place powdered sugar, 1 teaspoon vanilla and 3 tablespoons milk. In 1-quart saucepan, heat \u2153 cup butter over medium heat, stirring constantly, just until light brown.\n\n5 Pour browned butter over powdered sugar mixture. Beat on low speed about 1 minute or until smooth. Gradually add just enough of the remaining 1 tablespoon milk to make frosting creamy and spreadable. Generously frost cookies. Store in tightly covered container.\n\n* Do not use margarine or vegetable oil spreads in the frosting; it will burn.\n\n1 Cookie: Calories 190; Total Fat 7g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 30mg; Sodium 135mg; Total Carbohydrate 29g (Dietary Fiber 0g, Sugars 21g); Protein 2g Exchanges: \u00bd Starch, 1\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nSpicy Pumpkin Cookies with Browned Butter Frosting Add \u215b teaspoon each ground cloves and ground ginger with the flour.\n\nChocolate-Filled Orange-Rosemary Butter Balls\n\n## Chocolate-Filled Orange-Rosemary Butter Balls Calorie smart\n\nPrep 1 hr Total 1 hr 30 min \u25cf 30 sandwich cookies\n\nCookies\n\n  * 1 cup butter, softened\n  * \u00bd cup powdered sugar\n  * 1 tablespoon grated orange peel\n  * 1 tablespoon finely chopped fresh rosemary leaves\n  * \u00bc teaspoon salt\n  * 1 teaspoon vanilla\n  * 2 cups all-purpose flour\n  * \u00bd teaspoon baking powder\n  * \u00bc cup white decorator sugar crystals, if desired\n\nFilling\n\n  * \u00bd cup dark chocolate chips\n  * 2 tablespoons whipping cream\n\n1 Heat oven to 400\u00b0F. In large bowl, beat butter, powdered sugar, orange peel, rosemary, salt and vanilla with electric mixer on medium speed until creamy. On low speed, beat in flour and baking powder until dough forms.\n\n2 In small bowl, place sugar crystals. Shape dough into 1-inch balls. Roll in sugar. On ungreased cookie sheets, place balls 2 inches apart. Press lightly with tines of fork to flatten slightly.\n\n3 Bake 6 to 8 minutes or until set. Remove from cookie sheets to cooling racks. Cool completely, about 30 minutes.\n\n4 In small microwavable bowl, microwave chocolate chips and whipping cream uncovered on High in 15-second intervals until chips can be stirred smooth. Refrigerate until cooled and mixture starts to set, about 10 minutes.\n\n5 For each sandwich cookie, spread \u00bd teaspoon filling on bottom of 1 cookie. Top with second cookie, bottom side down; gently press together. Let stand until set.\n\n1 Sandwich Cookie: Calories 110; Total Fat 7g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 20mg; Sodium 75mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 3g); Protein 1g Exchanges: \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: \u00bd\n\nGingersnaps\n\n## Gingersnaps Calorie smart\n\nPrep 45 min Total 45 min \u25cf 4 dozen cookies\n\n  * 1 cup packed brown sugar\n  * \u00be cup shortening\n  * \u00bc cup molasses\n  * 1 egg\n  * 2\u00bc cups all-purpose flour\n  * 2 teaspoons baking soda\n  * 1 teaspoon ground cinnamon\n  * 1 teaspoon ground ginger\n  * \u00bd teaspoon ground cloves\n  * \u00bc teaspoon salt\n  * Granulated sugar\n\n1 Heat oven to 375\u00b0F. Lightly grease cookie sheets with shortening or cooking spray, or line with cooking parchment paper or silicone baking mats.\n\n2 In large bowl, beat brown sugar, shortening, molasses and egg with electric mixer on medium speed, or mix with spoon, until well blended. Stir in remaining ingredients except granulated sugar.\n\n3 Shape dough by rounded teaspoonfuls into balls. Dip tops into granulated sugar. On cookie sheets, place balls, sugared sides up, about 3 inches apart.\n\n4 Bake 10 to 12 minutes or just until set. Immediately remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 80; Total Fat 3.5g (Saturated Fat 1g, Trans Fat 0.5g); Cholesterol 0mg; Sodium 70mg; Total Carbohydrate 11g (Dietary Fiber 0g, Sugars 6g); Protein 0g Exchanges: 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\nSoft No-Roll Sugar Cookies\n\n## Soft No-Roll Sugar Cookies Calorie smart\n\nPrep 50 min Total 2 hr 50 min \u25cf 3\u00bd dozen cookies\n\n  * 1\u00bd cups granulated sugar\n  * 1 cup powdered sugar\n  * 1 cup butter, softened\n  * \u00be cup vegetable oil\n  * 2 tablespoons milk\n  * 1 tablespoon vanilla\n  * 2 eggs\n  * 4\u00bc cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon cream of tartar\n  * \u00bd teaspoon salt\n\n1 In large bowl, beat 1 cup of the granulated sugar, the powdered sugar, butter, oil, milk, vanilla and eggs with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda, cream of tartar and salt. Cover and refrigerate about 2 hours or until firm.\n\n2 Heat oven to 350\u00b0F. Shape dough into 1\u00bd-inch balls. In small bowl, place remaining \u00bd cup granulated sugar. Roll balls in sugar. On ungreased cookie sheets, place balls about 3 inches apart. Press bottom of drinking glass on each ball until about \u00bc inch thick. Sprinkle each cookie with a little additional granulated sugar.\n\n3 Bake 13 to 15 minutes or until set and edges just begin to turn brown. Immediately remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 160; Total Fat 9g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 20mg; Sodium 90mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 10g); Protein 2g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\nChocolate Chip, Cherry and Pistachio Soft No-Roll Sugar Cookies Stir in 1 cup semisweet chocolate chips, \u00bd cup dried cherries and \u00bd cup pistachio nuts before refrigerating.\n\nGlazed Soft No-Roll Sugar Cookies After cooling, drizzle cookies with Chocolate Glaze (page 582). Let stand until set.\n\nSnickerdoodles\n\n## Snickerdoodles Calorie smart\n\nThis favorite, whimsically named cookie originated in New England in the 1800s. It's traditionally rolled in cinnamon-sugar before baking.\n\nPrep 40 min Total 40 min \u25cf 4 dozen cookies\n\n  * 1\u00be cups sugar\n  * \u00bd cup butter, softened\n  * \u00bd cup shortening\n  * 2 eggs\n  * 2\u00be cups all-purpose flour\n  * 2 teaspoons cream of tartar\n  * 1 teaspoon baking soda\n  * \u00bc teaspoon salt\n  * 1 tablespoon ground cinnamon\n\n1 Heat oven to 400\u00b0F. In large bowl, beat 1\u00bd cups of the sugar, the butter, shortening and eggs with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, cream of tartar, baking soda and salt.\n\n2 Shape dough into 1\u00bc-inch balls. In small bowl, mix remaining \u00bc cup sugar and the cinnamon. Roll balls in cinnamon-sugar mixture. On ungreased cookie sheets, place balls 2 inches apart.\n\n3 Bake 8 to 10 minutes or just until set. Immediately remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 90; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 55mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 7g); Protein 1g Exchanges: 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\n\n## Learn to DECORATE\n\nThe options for decorating cookies can range from ridiculously easy to over-the-top elaborate. How you wish to decorate your cookies might depend on the time you have, the purpose for the cookies and your skill level. Look here for some basic tips to help you create beautiful cookies that will entice people to grab them and take a bite.\n\nCookies can be decorated before baking or after they are baked and cooled. See Storing Cookies and Bars, page 521, and Refrigerator and Freezer Storage Chart, page 33, for how to store them.\n\n### Decorate Before Baking\n\n  * Sprinkle unbaked cookies on cookie sheet with coarse clear or colored sugar. To make your own, see Colored Sugar, right. To keep the sugar on top of the cookie, place the cookie cutter used to cut the cookie back over it on the cookie sheet before sprinkling. Or roll unbaked balls of cookie dough in the sugar before placing on the cookie sheet.\n  * Make an Egg Yolk Paint to brush on sugar cookies before baking (see Paintbrush Sugar Cookies, right).\n  * Make a Baked-On Decorating Mixture: Mix \u00bd cup softened butter, \u00bd cup all-purpose flour and 3 teaspoons milk in a small bowl until well mixed. Divide mixture in half; tint each half with 4 drops food color. Place the mixtures in decorating bags fitted with a small writing tip. Pipe colors onto unbaked sugar cookies.\n\n### Decorating Baked Cookies\n\n  * Always cool cookies completely before frosting.\n  * Spread cookies with Vanilla Glaze (page 581) or Chocolate Glaze (page 582) on a cooling rack so that drips can fall away from the cookies and the glaze can dry. Place a jelly roll pan or piece of parchment or waxed paper under the cooling rack to catch the drips for easy cleanup.\n  * Sprinkle the glaze while wet with finely chopped nuts, colored sugars or candy sprinkles.\n  * Once glaze is dry, frost or drizzle dots of colored frosting or gel; pull through the dots with a toothpick to make designs.\n  * Pipe designs onto cookies using icing in a decorating bag with various tips or by placing the icing into small, resealable food-storage plastic bags (cutting a tiny corner off one end) or food-safe plastic squeeze bottles.\n\n### Rolling Cookies\n\nOn lightly floured surface, shape refrigerated dough (half at a time) into flattened round without cracks.\n\nUsing ruler or sticks as guide, roll dough \u00bc inch thick.\n\nUsing cookie cutters dipped in flour, cut desired shapes.\n\n### Using a Decorating Bag\n\nInsert coupler inside disposable pastry bag. (Cut end if necessary so coupler will reach tip of bag.)\n\nAttach tip with ring; open bag, pulling opening far over hand, and fill with frosting.\n\nSqueeze and twist with even pressure to pipe frosting onto cookies.\n\nSugar Cookies and variations\n\n## Sugar Cookies Calorie smart\n\nPrep 1 hr Total 3 hr \u25cf 5 dozen cookies\n\n  * 1\u00bd cups powdered sugar\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla\n  * \u00bd teaspoon almond extract\n  * 1 egg\n  * 2\u00bd cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon cream of tartar\n  * Granulated sugar or Colored Sugar (follows)\n\n1 In large bowl, beat powdered sugar, butter, vanilla, almond extract and egg with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda and cream of tartar. Cover and refrigerate at least 2 hours.\n\n2 Heat oven to 375\u00b0F. Lightly grease cookie sheets with shortening or cooking spray, or line with cooking parchment paper or silicone baking mats.\n\n3 Divide dough in half. Roll each half on lightly floured surface until \u00bc inch thick. Cut with floured 2-inch cookie cutters. Sprinkle with granulated sugar. On cookie sheets, place cutouts about 2 inches apart.\n\n4 Bake 7 to 8 minutes or until edges are light brown. Remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 60; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 45mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 4g); Protein 0g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: \u00bd\n\nColored Sugar Place \u00bd cup granulated sugar in resealable food-storage plastic bag. Add liquid food color to tint as desired. Seal bag. Squeeze and rub sugar in bag until it becomes evenly colored. Homemade colored sugar might clump; if it does, just break apart until clumps are gone.\n\nFrosted and Decorated Sugar Cookies Omit granulated sugar. Frost cooled cookies with Creamy Vanilla Frosting (page 579), tinted with food color if desired. Decorate with colored sugar, small candies, candied fruit or nuts if desired.\n\nPaintbrush Sugar Cookies Omit granulated sugar. Cut rolled dough into desired shapes with cookie cutters. (Cut no more than 12 cookies at a time to keep dough from drying out.) Mix 1 egg yolk and \u00bc teaspoon water. Divide mixture among several custard cups. Tint each with different food color to make bright colors. (If paint thickens while standing, stir in a few drops water.) Paint designs on cookies with small new paintbrushes. Bake as directed.\n\n### Decorating Cookies\n\nColored Sugar: Sprinkle cookies with decorating sugar or decors before baking.\n\nFrosted: Frost cookies and top with nuts, decorating sugar or sprinkles. Or frost and top with dots of colored frosting or gel food color; use a toothpick to make designs.\n\nPainted: Make egg yolk paints and paint cookies before baking.\n\nGingerbread Cookies\n\n## Gingerbread Cookies Calorie smart\n\nPrep 1 hr 40 min Total 3 hr 40 min \u25cf 5 dozen cookies\n\nCookies\n\n  * \u00bd cup packed brown sugar\n  * \u00bd cup butter, softened\n  * \u00bd cup molasses\n  * \u2153 cup cold water\n  * 3\u00bd cups all-purpose flour\n  * 2 teaspoons baking soda\n  * 2 teaspoons ground ginger\n  * \u00bd teaspoon ground allspice\n  * \u00bd teaspoon ground cinnamon\n  * \u00bc teaspoon ground cloves\n  * \u00bc teaspoon salt\n\nDecorator's Frosting\n\n  * 2 cups powdered sugar\n  * 2 tablespoons water or milk\n  * \u00bd teaspoon vanilla\n  * Food color, colored sugars and small candies, if desired\n\n1 In large bowl, beat brown sugar, butter, molasses and water with electric mixer on medium speed, or mix with spoon, until well blended. Stir in remaining cookie ingredients until mixed. Cover and refrigerate at least 2 hours.\n\n2 Heat oven to 350\u00b0F. Grease cookie sheets with shortening or cooking spray, or line with cooking parchment paper or silicone baking mats.\n\n3 Divide dough in half. Roll each half on lightly floured surface until \u00bc inch thick. Cut with floured 2\u00bd-inch gingerbread boy or girl cookie cutter. On cookie sheets, place cutouts about 2 inches apart. After cutting as many cookies as possible, lightly press scraps of dough together; reroll dough and cut additional cookies.\n\n4 Bake 10 to 12 minutes or until no indentation remains when touched in center. Immediately remove from cookie sheets to cooling racks. Cool completely, about 30 minutes. Cool cookie sheets 10 minutes between batches.\n\n5 Meanwhile, in medium bowl, mix powdered sugar, water and vanilla with spoon until smooth and spreadable. Stir in liquid food color, one drop at a time, until frosting is desired color. (For intense, vivid color, use paste food color. You would have to use too much liquid food color to get a vivid color, and the frosting will begin to separate and look curdled.) Cover and set aside.\n\n6 When ready to decorate, spoon frosting into small resealable food-storage plastic bag. Seal bag; push frosting down in one corner. Cut off tiny corner of bag. Squeeze bag to pipe frosting onto cookies and make desired design. Or spread over cookies with small metal spatula. Decorate with colored sugars and candies.\n\n1 Cookie: Calories 60; Total Fat 1.5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 65mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 3g); Protein 0g Exchanges: \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: \u00bd\n\n### Learn with Betty Melting Chocolate\n\n**Perfectly Melted:** This chocolate was melted perfectly; it is smooth and creamy.\n\n**Too Melted or Seized:** This chocolate has either been melted over too high a heat or has come in contact with a small amount of liquid, causing it to seize.\n\n**Correcting Seized Chocolate:** To make chocolate smooth again, use a whisk to beat in at least 1 tablespoon melted butter, vegetable oil or melted vegetable shortening. Heat over very low heat, if necessary, stirring constantly.\n\nShortbread Cookies and variations\n\n## Shortbread Cookies Calorie smart\n\nPrep 40 min Total 40 min \u25cf 2 dozen cookies\n\n  * \u00be cup butter, softened\n  * 5 tablespoons sugar\n  * 2 cups all-purpose flour\n\n1 Heat oven to 350\u00b0F. In large bowl, stir butter and 4 tablespoons of the sugar until well mixed. Stir in flour. (If dough is crumbly, mix in 1 to 2 tablespoons more softened butter.)\n\n2 Roll dough on lightly floured surface until \u00bd inch thick. Cut with floured 1\u00bd-inch cookie cutters. On ungreased cookie sheet, place cutouts \u00bd inch apart. Sprinkle evenly with remaining 1 tablespoon sugar.\n\n3 Bake 15 to 20 minutes or until set. Immediately remove from cookie sheet to cooling rack.\n\n1 Cookie: Calories 100; Total Fat 6g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 40mg; Total Carbohydrate 10g (Dietary Fiber 0g, Sugars 2g); Protein 1g Exchanges: \u00bd Starch, 1 Fat Carbohydrate Choices: \u00bd\n\nShortbread Cookie Wedges Make dough as directed. On ungreased cookie sheet, pat dough into 8-inch round. Sprinkle with 1 tablespoon sugar. Bake 20 to 24 minutes or until set. Cool 10 minutes on cookie sheet on cooling rack. Using sharp knife, cut into 24 wedges.\n\nCherry Shortbread Cookies Stir in \u00bd cup chopped dried cherries, cranberries or blueberries with the flour.\n\nChocolate-Dipped Shortbread Cookies Make and bake cookies as directed; cool completely. In small microwavable bowl, microwave 1 cup chocolate chips and 1 teaspoon vegetable oil on High 30 to 60 seconds or until mixture can be stirred smooth. Dip cookies in chocolate; wipe excess on edge of bowl. Place on waxed paper to set.\n\nGinger Shortbread Cookies Stir in 3 tablespoons chopped crystallized ginger with the flour.\n\nShortbread Nut Cookies Stir in \u00bd cup finely chopped hazelnuts (filberts), macadamia nuts, pecans, pistachio nuts or slivered almonds, toasted if desired, with the flour.\n\nToffee Shortbread Cookies Stir in \u00bd cup toffee bits with the flour.\n\nWhole Wheat Shortbread Cookies Substitute 1\u00bc cups whole wheat flour for the all-purpose flour. Continue as directed.\n\n### Dipping Cookies in Chocolate\n\nHeat 1 cup chocolate chips and 1 teaspoon oil until mixture can be stirred smooth.\n\nDip cookies; wipe excess on edge of bowl. Place on waxed paper to set.\n\nBrown Sugar Refrigerator Cookies and variations\n\n## Brown Sugar Refrigerator Cookies Calorie smart\n\nPrep 50 min Total 2 hr 50 min \u25cf 6 dozen cookies\n\n  * 1 cup packed brown sugar\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla\n  * 1 egg\n  * 3 cups all-purpose flour\n  * 1\u00bd teaspoons ground cinnamon\n  * \u00bd teaspoon baking soda\n  * \u00bd teaspoon salt\n  * \u2153 cup finely chopped nuts\n\n1 In large bowl, beat brown sugar, butter, vanilla and egg with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, cinnamon, baking soda and salt. Stir in nuts.\n\n2 On plastic wrap, shape dough into 10x3-inch rectangle. Wrap and refrigerate about 2 hours or until firm but no longer than 24 hours.\n\n3 Heat oven to 375\u00b0F. Unwrap dough; cut into \u215b-inch slices. On ungreased cookie sheets, place slices 2 inches apart.\n\n4 Bake 6 to 8 minutes or until light brown. Cool 1 to 2 minutes; remove from cookie sheets to cooling racks.\n\n1 Cookie: Calories 60; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 45mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 3g); Protein 0g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: \u00bd\n\nCitrus\u2013Brown Sugar Refrigerator Cookies Add 1 tablespoon grated lemon or orange peel with the flour. Frost cookies with **Vanilla Glaze**. For glaze, mix 2 cups powdered sugar, \u2153 cup melted butter and 1 teaspoon vanilla. Stir in 2 to 4 tablespoons hot water, 1 tablespoon at a time, until glaze is smooth and has the consistency of thick syrup. Drizzle over cookies, sprinkle with grated lemon peel.\n\nMaple\u2013Brown Sugar Refrigerator Cookies Substitute 2 teaspoons maple flavor for the vanilla.\n\nToasted Coconut\u2013Brown Sugar Refrigerator Cookies Add 1 cup toasted coconut (page 23) with the flour. Frost cookies with Vanilla Glaze (page 581) and sprinkle with additional toasted coconut.\n\n### Making Refrigerator Cookies\n\nBeat cookie mixture with electric mixer or with spoon; stir in nuts.\n\nShape dough on plastic wrap into 10x3-inch rectangle.\n\nWrap; refrigerate about 2 hours or until firm.\n\nUnwrap; cut rectangle into \u215b-inch slices.\n\nLeft to right: Lemon-Almond Thumbprint Cookies, Thumbprint Cookies, Hazelnut Thumbprint Cookies\n\n## Thumbprint Cookies Calorie smart\n\nPrep 50 min Total 50 min \u25cf 3 dozen cookies\n\n  * \u00bc cup packed brown sugar\n  * \u00bc cup shortening\n  * \u00bc cup butter, softened\n  * \u00bd teaspoon vanilla\n  * 1 egg, separated\n  * 1 cup all-purpose flour\n  * \u00bc teaspoon salt\n  * 1 cup finely chopped nuts\n  * About 6 tablespoons jelly or jam (any flavor)\n\n1 Heat oven to 350\u00b0F. In medium bowl, beat brown sugar, shortening, butter, vanilla and egg yolk with electric mixer on medium speed, or mix with spoon. Stir in flour and salt.\n\n2 Shape dough into 1-inch balls. In small bowl, beat egg white slightly with fork. Place nuts in another small bowl. Dip each ball into egg white, then roll in nuts. On ungreased cookie sheets, place balls about 1 inch apart. Press thumb into center of each cookie to make indentation, but do not press all the way to the cookie sheet.\n\n3 Bake about 10 minutes or until light brown. Quickly remake indentations with end of wooden spoon handle if necessary. Immediately remove from cookie sheets to cooling racks. Fill each with about \u00bd teaspoon jelly.\n\n1 Cookie: Calories 80; Total Fat 5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 55mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 3g); Protein 1g Exchanges: \u00bd Starch, 1 Fat Carbohydrate Choices: \u00bd\n\nHazelnut-Chocolate Thumbprint Cookies Roll balls of dough in finely chopped hazelnuts (filberts). Instead of jelly, fill indentations with hazelnut spread with cocoa.\n\nLemon-Almond Thumbprint Cookies Roll balls of dough in finely chopped slivered almonds. Fill indentations with lemon curd.\n\nPeanut-Fudge Thumbprint Cookies Roll balls of dough in finely chopped dry-roasted peanuts. Instead of jelly, fill indentations with hot fudge topping (look for the thick topping, not pourable or syrup varieties).\n\nSpritz Cookies\n\n## Spritz Cookies Calorie smart\n\nPrep 45 min Total 45 min \u25cf 5 dozen cookies\n\n  * 1 cup butter, softened*\n  * \u00bd cup sugar\n  * 2\u00bc cups all-purpose flour\n  * 1 egg\n  * 1 teaspoon almond extract or vanilla\n  * \u00bd teaspoon salt\n  * Few drops red or green food color, if desired\n  * Sugar, colored sugar or multi-colored candy sprinkles, if desired\n\n1 Heat oven to 400\u00b0F. In large bowl, beat butter and \u00bd sugar with electric mixer on medium speed, or mix with spoon, until well blended. Stir in remaining ingredients.\n\n2 Place dough in cookie press. On ungreased cookie sheets, form desired shapes. Sprinkle with sugar.\n\n3 Bake 6 to 9 minutes or until set but not brown. Immediately remove from cookie sheets to cooling racks.\n\n*Do not use margarine or vegetable oil spreads.\n\n1 Cookie: Calories 50; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 40mg; Total Carbohydrate 5g (Dietary Fiber 0g, Sugars 2g); Protein 0g Exchanges: \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: \u00bd\n\nChocolate Buttery Spritz Cookies Stir 2 ounces unsweetened baking chocolate, melted and cooled, into butter-sugar mixture.\n\n### Making Spritz Cookies\n\nPlace desired design disk in cookie press. Place dough in press; press dough through press onto cookie sheet.\n\n## Whoopie Pies\n\nPrep 45 min Total 1 hr 25 min \u25cf 1\u00bd dozen sandwich cookies\n\nCookies\n\n  * 1 cup granulated sugar\n  * \u00bd cup butter, softened\n  * \u00bd cup buttermilk\n  * 2 teaspoons vanilla\n  * 1 egg\n  * 2 oz unsweetened baking chocolate, melted, cooled\n  * 1\u00be cups all-purpose flour\n  * \u00bd teaspoon baking soda\n  * \u00bd teaspoon salt\n\nCreamy Marshmallow Filling\n\n  * 3 cups powdered sugar\n  * 1 jar (7 oz) marshmallow cr\u00e8me\n  * \u00be cup butter, softened\n  * 6 to 7 teaspoons milk\n\n1 Heat oven to 400\u00b0F. Grease cookie sheets with shortening or cooking spray, or line with cooking parchment paper or silicone baking mats.\n\n2 In large bowl, beat granulated sugar, \u00bd cup butter, the buttermilk, vanilla, egg and chocolate with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking soda and salt. Onto cookie sheets, drop dough by rounded tablespoonfuls about 2 inches apart.\n\n3 Bake 8 to 10 minutes or until almost no indentation remains when touched in center. Immediately remove from cookie sheets to cooling racks. Cool completely, about 30 minutes.\n\n4 In large bowl, beat all filling ingredients on medium speed about 2 minutes or until light and fluffy. For each whoopie pie, spread slightly less than 3 tablespoons filling on bottom of 1 cookie; top with second cookie, bottom side down; gently press together. Store in tightly covered container.\n\n1 Sandwich Cookie: Calories 350; Total Fat 15g (Saturated Fat 9g, Trans Fat 0.5g); Cholesterol 45mg; Sodium 210mg; Total Carbohydrate 50g (Dietary Fiber 1g, Sugars 38g); Protein 2g Exchanges: 1 Starch, 2\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 3\n\nChocolate Chip Whoopie Pies Fold \u00bd cup miniature semisweet chocolate chips into filling.\n\nPink Peppermint Whoopie Pies Add 6 drops red food color to filling ingredients. Once whoopie pies are assembled, sprinkle edges of filling with crushed peppermint candies or candy canes.\n\nToffee Whoopie Pies Fold \u00bd cup chocolate-covered toffee bits into filling.\n\nWhoopie Pies and variations\n\n## Chocolate-Hazelnut Sandwiches Calorie smart\n\nPrep 55 min Total 3hr10min \u25cf 3 dozen sandwich cookies\n\n  * 1 cup butter, softened\n  * 1 cup sugar\n  * 4 oz bittersweet chocolate, melted, cooled\n  * 1 egg\n  * 1 teaspoon vanilla\n  * 2 cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bc teaspoon salt\n  * \u00be cup chopped hazelnuts (filberts), toasted\n  * \u00be cup hazelnut spread with cocoa\n\n1 In large bowl, beat butter and sugar with electric mixer on high speed 3 to 5 minutes or until creamy. Beat in chocolate until well blended. Add egg and vanilla, beating until well blended. On low speed, beat in flour, baking powder and salt just until moistened. Stir in hazelnuts.\n\n2 Divide dough in half. On 2 sheets of plastic wrap, shape each half into 14-inch log 1\u00bd inches in diameter. Wrap tightly; refrigerate at least 2 hours.\n\n3 Heat oven to 325\u00b0F. Line cookie sheets with cooking parchment paper. Cut each log into 36 (about \u00bc-inch) slices. On cookie sheets, place slices 1 inch apart.\n\n4 Bake 12 to 13 minutes or until set. Cool 1 minute; remove from cookie sheets to cooling racks. Cool completely.\n\n5 For each sandwich cookie, spread 1 teaspoon hazelnut spread on bottom of 1 cookie; top with second cookie, bottom side down.\n\n1 Sandwich Cookie: Calories 158 ; Total Fat 10g (Saturated Fat 6g, Trans Fat 0g); Sodium 78mg; Total Carbohydrate 17g (Dietary Fiber 1g); Protein 2g Exchanges: 1 Other Carbohydrate, 2 Fat Carbohydrate Choices: 1\n\nFrench Macarons with Bittersweet Chocolate Ganache\n\n## French Macarons with Bittersweet Chocolate Ganache Calorie smart\n\nIn France, these popular cookies are sold in patisseries in a wide variety of colors and flavors.\n\nPrep 55 min Total 3 hr 10 min \u25cf 20 sandwich cookies\n\nCookies\n\n  * 1\u00bc cups blanched whole almonds (about 7 oz)\n  * 1\u00bd cups powdered sugar\n  * 3 egg whites, at room temperature*\n  * \u00bc teaspoon salt\n  * \u00bd teaspoon red paste or gel food color\n  * 3 tablespoons granulated sugar\n\nGanache\n\n  * \u00bc cup whipping cream\n  * 1 teaspoon light corn syrup\n  * 2 oz bittersweet or semisweet baking chocolate, finely chopped\n  * 1 tablespoon butter, softened\n\n1 In food processor or blender, place almonds and \u00bd cup of the powdered sugar. Cover and process, scraping sides occasionally, until almonds are very finely ground. Place strainer over medium bowl. Using rubber spatula, press almond mixture through strainer into bowl. Return any almonds left in strainer to blender; add remaining 1 cup powdered sugar. Cover and process, scraping sides occasionally, until well blended.\n\n2 In large bowl, beat egg whites, salt and food color with electric mixer on high speed just until foamy. Gradually add granulated sugar, 1 tablespoon at a time, beating on high speed 1 to 2 minutes or just until stiff peaks form. Using rubber spatula, fold about half of the almond mixture into egg white mixture until completely incorporated. Fold in remaining almond mixture.\n\n3 Line 2 large cookie sheets with cooking parchment paper or silicone baking mats. Spoon half of the batter into decorating bag fitted with \u00bd-inch round tip. Onto one cookie sheet, pipe batter in 20 (1\u00bd-inch) rounds about 1\u00bd inches apart. Repeat with remaining batter and second cookie sheet. If tops have a peak, wet fingertips lightly on damp paper towel and press down to flatten. Tap bottom of cookie sheet on counter a few times to flatten cookies. Let stand uncovered at room temperature 30 minutes to allow a light crust to form on tops.\n\n4 Heat oven to 300\u00b0F. Bake one cookie sheet at a time 17 to 18 minutes or until tops look set. Cool 10 minutes. With metal pancake turner, remove cookies from cookie sheets to cooling racks; cool completely.\n\n5 Meanwhile, in 2-cup microwavable measuring cup, microwave whipping cream and corn syrup uncovered on High 1 minute. Stir in chocolate and butter until chocolate is melted. Let stand about 30 minutes. Cover and refrigerate about 30 minutes or until spreading consistency.\n\n6 For each sandwich cookie, spread slightly less than 1 teaspoon ganache on bottom of 1 cookie. Top with second cookie, bottom side down; gently press together. Store in tightly covered container.\n\n*To quickly bring the whites to room temperature, soak whole eggs in a bowl of hot water for about 10 minutes before cracking and separating.\n\n1 Sandwich Cookie: Calories 140; Total Fat 7g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 45mg; Total Carbohydrate 15g (Dietary Fiber 1g, Sugars 13g); Protein 3g Exchanges: 1 Other Carbohydrate, \u00bd High-Fat Meat, \u00bd Fat Carbohydrate Choices: 1\n\nLemon Macarons Tint egg white mixture with desired amount of yellow food color in Step 2. Omit ganache. Fill with Lemon Frosting (page 579) or Lemon Curd (page 581).\n\nMint Macarons Tint egg white mixture with desired amount of green food color in Step 2. In Step 5, add 1 tablespoon finely chopped fresh mint leaves to heated cream mixture; let stand 30 minutes. Place fine-mesh strainer over medium bowl. Using back of wooden spoon, press mint mixture through strainer into bowl; discard mint. Return cream mixture to measuring cup. Microwave on High 1 minute. Continue with ganache as directed.\n\nOrange Macarons Tint egg white mixture orange with desired food color in Step 2. Use ganache if desired, or fill with Orange Frosting (page 579).\n\n### Making Macarons\n\nGradually add sugar to egg whites, 1 tablespoon at a time, beating on high speed just until stiff peaks form.\n\nPipe batter in 20 (1\u00bd-inch) rounds about 1\u00bd inches apart on parchment-lined cookie sheet.\n\nSpread ganache on bottom of 1 cookie. Top with second cookie, bottom side down; gently press together.\n\n## Hazelnut Biscotti Calorie smart\n\nPrep 25 min Total 1 hr 30 min \u25cf 40 cookies\n\n  * 1 cup hazelnuts (filberts), coarsely chopped\n  * 1 cup sugar\n  * \u00bd cup butter, softened\n  * 1 teaspoon almond extract\n  * 1 teaspoon vanilla\n  * 2 eggs\n  * 3\u00bd cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bd teaspoon baking soda\n\n1 Heat oven to 350\u00b0F. Spread hazelnuts in ungreased shallow pan. Bake uncovered about 10 minutes, stirring occasionally, until golden brown; cool.\n\n2 In large bowl, beat sugar, butter, almond extract, vanilla and eggs with electric mixer on medium speed, or mix with spoon, until well blended. Stir in flour, baking powder and baking soda. Stir in hazelnuts.\n\n3 Place dough on lightly floured surface. Gently knead 2 to 3 minutes or until dough holds together and hazelnuts are evenly distributed. Divide dough in half. On large ungreased cookie sheet, shape each half into 10x3-inch rectangle, rounding edges slightly.\n\n4 Bake about 25 minutes or until center is firm to the touch. Cool on cookie sheet 15 minutes; move to cutting board. Using sharp knife, cut each rectangle crosswise into \u00bd-inch slices.\n\n5 Place slices, cut sides down, on ungreased cookie sheet. Bake about 15 minutes or until crisp and light brown. Immediately remove from cookie sheet to cooling rack.\n\n1 Cookie: Calories 100; Total Fat 4.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 45mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 5g); Protein 2g Exchanges: 1 Starch, \u00bd Fat Carbohydrate Choices: 1\n\nAlmond Biscotti Substitute 1 cup slivered almonds for the hazelnuts.\n\n### Making Biscotti\n\nOn cookie sheet, shape each dough half into 10x3-inch rectangle, rounding edges slightly.\n\nCool baked rectangles 15 minutes; cut into \u00bd-inch slices. Place cut side down on same cookie sheet. Bake until crisp and golden brown.\n\n## Chocolate Brownies\n\nIf you like chewy, fudgy chocolate brownies, this is your recipe. We really like the walnuts in the recipe, but you could use pecans or cashews instead, or make the brownies without nuts.\n\nPrep 20 min Total 3 hr 10 min \u25cf 16 brownies\n\n  * \u2154 cup butter\n  * 5 oz unsweetened baking chocolate, chopped\n  * 1\u00be cups sugar\n  * 2 teaspoons vanilla\n  * 3 eggs\n  * 1 cup all-purpose flour\n  * \u00bd cup chopped walnuts, toasted if desired (page 23)\n  * Creamy Chocolate Frosting or Mocha Frosting (one-half recipe, page 579), if desired\n\n1 Heat oven to 350\u00b0F. Grease bottom and sides of 9-inch square pan with shortening.\n\n2 In 1-quart saucepan, melt butter and chocolate over low heat, stirring constantly. Cool 5 minutes.\n\n3 In medium bowl, beat sugar, vanilla and eggs with electric mixer on high speed 5 minutes. Beat in chocolate mixture on low speed, scraping bowl occasionally. Beat in flour just until blended, scraping bowl occasionally. Stir in walnuts. Spread batter in pan.\n\n4 Bake 40 to 45 minutes or just until brownies begin to pull away from sides of pan. Cool completely in pan on cooling rack, about 2 hours. Spread frosting over brownies. Cut into 4 rows by 4 rows.\n\n1 Brownie: Calories 310; Total Fat 18g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 60mg; Sodium 65mg; Total Carbohydrate 31g (Dietary Fiber 2g, Sugars 22g); Protein 4g Exchanges: 1 Starch, 1 Other Carbohydrate, 3\u00bd Fat Carbohydrate Choices: 2\n\nChocolate Brownie Pie Grease bottom and side of 10-inch glass pie plate with shortening. Spread batter in pie plate. Bake 35 to 40 minutes or until center is set. Cool completely on cooling rack. Cut into wedges. Omit frosting. Serve with ice cream and Hot Fudge Sauce (page 642) if desired. Makes 12 servings.\n\nChocolate\u2013Peanut Butter Brownies Substitute \u2153 cup crunchy peanut butter for \u2153 cup of the butter. Omit walnuts. Before baking, arrange 16 miniature chocolate-covered peanut butter cup candies, unwrapped, on top of batter. Press into batter so tops of cups are even with top of batter.\n\nDouble-Chocolate Brownies Stir in \u00bd cup semisweet or dark chocolate chips after adding the flour.\n\n### Determining Brownie Doneness\n\nBrownies will start to pull away from sides of pan when done.\n\nBlondies with Brown Sugar Frosting\n\n## Blondies with Brown Sugar Frosting Calorie smart\n\nPrep 25 min Total 1 hr 50 min \u25cf 36 brownies\n\nBrownies\n\n  * 1 cup granulated sugar\n  * \u00bd cup packed brown sugar\n  * \u00bd cup butter, softened\n  * 1 teaspoon vanilla\n  * 2 eggs\n  * 1\u00bd cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bd teaspoon salt\n\nFrosting\n\n  * \u2153 cup butter*\n  * \u2154 cup packed brown sugar\n  * 3 tablespoons milk\n  * 2 cups powdered sugar\n  * \u00bd teaspoon vanilla\n  * \u00bd cup chopped pecans, toasted if desired (page 23)\n\n1 Heat oven to 350\u00b0F. In large bowl, beat granulated sugar, \u00bd cup brown sugar, \u00bd cup butter, 1 teaspoon vanilla and the eggs with electric mixer on medium speed, or mix with spoon, until light and fluffy. Stir in flour, baking powder and salt. In ungreased 13x9-inch pan, spread batter evenly.\n\n2 Bake 20 to 23 minutes or until golden brown and toothpick inserted in center comes out clean. Cool completely in pan on cooling rack, about 1 hour.\n\n3 In 2-quart saucepan, melt \u2153 cup butter over low heat. Stir in \u2154 cup brown sugar; cook over low heat 2 minutes, stirring constantly. Stir in milk; cook until mixture comes to a rolling boil. Remove from heat. Gradually stir in powdered sugar and vanilla, mixing well with spoon after each addition, until smooth and spreadable. If necessary, add more milk, a few drops at a time. Spread frosting over brownies. Immediately sprinkle with pecans. Cut into 6 rows by 6 rows.\n\n*Do not use margarine or vegetable oil spreads in the frosting.\n\n1 Brownie: Calories 150; Total Fat 6g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 80mg; Total Carbohydrate 23g (Dietary Fiber 0g, Sugars 19g); Protein 1g Exchanges: \u00bd Starch, 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\u00bd\n\nOutrageous Caramel Fudge Brownies\n\n## Outrageous Caramel-Fudge Brownies\n\nPrep 30 min Total 3 hr \u25cf 24 brownies\n\n  * 1 bag (14 oz) caramels, unwrapped\n  * \u00bd cup evaporated milk\n  * 1 cup butter\n  * 2 cups sugar\n  * 2 teaspoons vanilla\n  * 4 eggs, slightly beaten\n  * 1\u00bc cups all-purpose flour\n  * \u00be cup unsweetened baking cocoa\n  * \u00bc teaspoon salt\n  * 2 cups semisweet chocolate chunks (about 12 oz)\n  * 1\u00bd cups chopped pecans\n  * 1 teaspoon vegetable oil\n\n1 Heat oven to 350\u00b0F. Grease bottom and sides of 13x9-inch pan with shortening or cooking spray. In 3-quart saucepan, heat caramels and milk over low heat, stirring frequently, until caramels are melted and smooth.\n\n2 In 2-quart saucepan, melt butter over low heat; remove from heat. Stir in sugar, vanilla and eggs until well blended. Stir in flour, cocoa and salt. Stir in 1\u00bd cups of the chocolate chunks and 1 cup of the pecans. Spread batter in pan.\n\n3 Gently and evenly drizzle melted caramel over batter, preventing large pockets of caramel and preventing caramel from reaching bottom of brownies. (Caramel can cover entire surface of batter.) Bake 35 to 40 minutes or until set.\n\n4 In 1-quart saucepan, heat remaining \u00bd cup chocolate chunks and the oil over low heat, stirring frequently, until smooth. Drizzle over warm brownies. Sprinkle with remaining \u00bd cup pecans; press in lightly. Cool 20 minutes. Refrigerate about 1 hour 30 minutes or until chocolate is set. Cut into 6 rows by 4 rows. If refrigerated longer, let stand at room temperature 20 minutes before serving.\n\n1 Brownie: Calories 380; Total Fat 20g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 60mg; Sodium 140mg; Total Carbohydrate 46g (Dietary Fiber 2g, Sugars 33g); Protein 4g Exchanges: 1 Starch, 2 Other Carbohydrate, 4 Fat Carbohydrate Choices: 3\n\nLemon Bars\n\n## Lemon Bars Calorie smart\n\nPrep 10 min Total 2 hr \u25cf 25 bars\n\n  * 1 cup all-purpose flour\n  * \u00bd cup butter, softened\n  * \u00bc cup powdered sugar\n  * 2 eggs\n  * 1 cup granulated sugar\n  * 2 teaspoons grated lemon peel\n  * 2 tablespoons fresh lemon juice\n  * \u00bd teaspoon baking powder\n  * \u00bc teaspoon salt\n  * Additional powdered sugar\n\n1 Heat oven to 350\u00b0F. In medium bowl, mix flour, butter and \u00bc cup powdered sugar with spoon until well mixed. Press in ungreased 8- or 9-inch square pan, building up \u00bd-inch edges.\n\n2 Bake crust 20 minutes; remove from oven. In medium bowl, beat remaining ingredients except additional powdered sugar with electric mixer on high speed about 3 minutes or until light and fluffy. Pour over hot crust.\n\n3 Bake 25 to 30 minutes or until no indentation remains when touched lightly in center. Cool completely in pan on cooling rack, about 1 hour. Sprinkle with powdered sugar. Cut into 5 rows by 5 rows.\n\n1 Bar: Calories 100; Total Fat 4g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 25mg; Sodium 65mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 10g); Protein 1g Exchanges: 1 Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\n\n### Pressing Crust in Pan\n\nUsing fingertips, press dough in pan; build up \u00bd-inch edges.\n\nTriple-Nut Bars\n\n## Triple-Nut Bars\n\nPrep 35 min Total 1 hr 50 min \u25cf 36 bars\n\nCrust\n\n  * \u00be cup butter, softened\n  * \u00bd cup packed brown sugar\n  * \u00bd teaspoon almond extract\n  * 1\u00bd cups all-purpose flour\n\nTopping\n\n  * 1 cup butter\n  * 1\u00bd cups packed brown sugar\n  * \u00bc cup honey\n  * \u00bc cup light corn syrup\n  * \u00bd teaspoon vanilla\n  * 1 cup walnut pieces\n  * 1 cup unblanched or blanched whole almonds\n  * 1 cup pecan halves\n\n1 Heat oven to 350\u00b0F. In medium bowl, beat \u00be cup butter, \u00bd cup brown sugar and the almond extract with electric mixer on medium-low speed until blended. On low speed, beat in flour until soft dough forms. Press dough in bottom of ungreased 13x9-inch pan. Bake 17 to 20 minutes or until golden brown.\n\n2 Meanwhile, in 2-quart saucepan, cook all topping ingredients except nuts over medium-high heat 12 to 15 minutes, stirring frequently, until mixture comes to a full rolling boil. Boil 1 to 2 minutes, stirring frequently. Remove from heat.\n\n3 Sprinkle walnuts, almonds and pecans over crust. Pour brown sugar mixture over nuts. Bake 13 to 15 minutes or until top of mixture is bubbly. Cool completely in pan on cooling rack, about 1 hour. Cut into 6 rows by 6 rows.\n\n1 Bar: Calories 230; Total Fat 15g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 25mg; Sodium 70mg; Total Carbohydrate 21g (Dietary Fiber 1g, Sugars 15g); Protein 2g Exchanges: 1\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 1\u00bd\n\n## Dried Fruit\n\nDoes dried fruit get passed over in your cupboard? If you are looking for ways to use dried fruit other than eating it out of your hand, try the ideas below. If you don't have the exact dried fruit called for, feel free to improvise with what you do have.\n\n  1. 1. **On-the-Go Mix:** In medium bowl, mix 1 cup bite-size pretzels, \u00bd cup each toasted oat cereal and dried fruit (any variety) and \u00bc cup each dry-roasted whole almonds or peanuts and roasted pumpkin seeds. Place in small resealable plastic bags or plastic containers.\n  2. 2. **Fruit 'n Nuts Salad:** For each serving, top 1 cup baby spinach leaves with 2 tablespoons each dried cherries, crumbled feta cheese and toasted pecan halves. Drizzle with your favorite vinaigrette dressing.\n  3. 3. **Pomegranate Oatmeal:** Make oatmeal as directed on package, except\u2014stir in 2 tablespoons dried pomegranate seeds with the water.\n  4. 4. **Fruity Couscous:** Make 10-ounce box of couscous as directed on package, except\u2014add \u00bc cup each dried cranberries and chopped dried apricots with the water. \n  5. 5. **Lemon-Fig Muffins:** Prepare Blueberry Muffins (page 482) as directed, except\u2014stir in 2 teaspoons grated lemon peel with the milk and substitute chopped dried figs for the blueberries.\n  6. 6. **Fruitcake-Inspired Brownies:** Toss \u00bc cup each chopped dried apricots, dates and cherries with 1 tablespoon all-purpose flour; set aside. Prepare Chocolate Brownies (page 540) as directed, except\u2014add apricots, dates and cherries with the walnuts.\n\nFruitcake-Inspired Brownies\n\nDate Bars\n\n## Date Bars Calorie smart\n\nPrep 20 min Total 1 hr \u25cf 36 bars\n\nFilling\n\n  * 3 cups chopped pitted dates (1 lb)\n  * 1\u00bd cups water\n  * \u00bc cup granulated sugar\n\nBars\n\n  * 1 cup packed brown sugar\n  * 1 cup butter, softened\n  * 1\u00be cups all-purpose or whole wheat flour\n  * 1\u00bd cups quick-cooking oats\n  * \u00bd teaspoon baking soda\n  * \u00bd teaspoon salt\n\n1 In 2-quart saucepan, cook filling ingredients over low heat about 10 minutes, stirring constantly, until thickened. Cool 5 minutes.\n\n2 Heat oven to 400\u00b0F. Grease bottom and sides of 13x9-inch pan with shortening.\n\n3 In large bowl, stir brown sugar and butter until well mixed. Stir in flour, oats, baking soda and salt until crumbly. Press half of the crumb mixture evenly in bottom of pan. Spread with filling. Top with remaining crumb mixture; press lightly.\n\n4 Bake 25 to 30 minutes or until light brown. Cool 5 minutes in pan on cooling rack. Cut into 6 rows by 6 rows while warm.\n\n1 Bar: Calories 160; Total Fat 5g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 85mg; Total Carbohydrate 25g (Dietary Fiber 2g, Sugars 17g); Protein 2g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\u00bd\n\nFig Bars Substitute 3 cups chopped dried figs for the dates.\n\n# Candy Basics\n\nCandy making is a rewarding way to spend time in your kitchen, whether alone or as a bonding experience with family or friends. If you're new at making candy, don't be intimidated! With the right tools and tips, you can make no-fuss treats or traditional candy recipes with ease.\n\n## SELECTING AND PREPARING PANS\n\n  * Use the exact size saucepan called for in the recipe or cooking time will be affected.\n  * Use the size metal pan or glass dish called for in the recipe so the candy will have the right thickness, set up properly and have the desired texture.\n  * Grease the pan with butter or line it with foil so the candy will be easily removed. See Quick Pan Cleanup, page 521, for how to line a pan with foil.\n\n## MIXING, COOKING AND COOLING\n\n  * For best results, use butter. If you use margarine, it needs to be at least 65 percent fat. Do not use vegetable oil spreads or reduced-calorie, reduced-fat or tub products.\n  * If directed, be sure to butter the side of the saucepan, as it prevents crystals from forming while cooking the candy mixture. Crystals can result in a grainy candy.\n  * Don't double a candy recipe. Increasing the amount of ingredients changes cooking time, so the resulting candy may not set up properly. Make two batches instead.\n  * A cool, dry day is best for making candy. Some candies, like pralines, won't set up properly on a humid day.\n  * Unless otherwise directed, do not stir the candy mixture while cooling. Stirring during cooling can cause the mixture to crystallize, resulting in a grainy candy. Follow recipe exactly, stirring only when indicated.\n\n## Using a Candy Thermometer\n\nHere are a few things to know when using a candy thermometer:\n\n  * Always check the accuracy of your thermometer before making candy. Put the thermometer into a pan of water and bring the water to a boil. The thermometer should read 212\u00b0F. If it doesn't, note how much higher or lower the temperature reads than 212\u00b0F and make that adjustment to the temperature needed in the recipe you are making.\n  * If you live in a high-altitude area (above 3,500 feet), refer to an altitude table to find out the boiling point and adjust the cooking time if necessary.\n  * To get an accurate reading, the thermometer should stand upright in the candy mixture. The bulb or tip of the thermometer should be submerged in the candy mixture but shouldn't rest on the bottom of the pan. Read the thermometer at eye level. Watch the temperature closely\u2014after 200\u00b0F, it goes up very quickly.\n  * If you don't have a candy thermometer, use the cold-water test, in which a small amount of candy mixture is dropped into a cupful of very cold water (see Testing Candy Temperatures at right).\n\nCaramels\n\n## Triple-Chocolate Fudge Calorie smart\n\nPrep 30 min Total 2 hr \u25cf 120 squares\n\n  * 4\u00bd cups sugar\n  * \u00bd cup butter\n  * 1 can (12 oz) evaporated milk\n  * 4\u00bd cups miniature marshmallows\n  * 2 cups semisweet chocolate chips (about 12 oz)\n  * 12 oz sweet baking chocolate, cut into pieces\n  * 2 oz unsweetened baking chocolate, cut into pieces\n  * 2 teaspoons vanilla\n  * \u00bc teaspoon almond extract\n  * 1 cup chopped walnuts or pecans\n  * Colored sugar, if desired\n\n1 Line 15x10x1-inch pan with foil, extending foil over sides of pan; grease foil with shortening or butter. In 5- to 6-quart saucepan, cook sugar, butter and milk over medium heat, stirring constantly, until sugar is dissolved. Heat to full boil, stirring constantly. Boil uncovered over medium heat without stirring 5 minutes.\n\n2 Remove from heat. Add marshmallows; stir until melted. Add chocolate chips and baking chocolates, stirring constantly until all chocolate is melted and mixture is smooth. Stir in vanilla, almond extract and walnuts. Quickly spread mixture in pan. Sprinkle with colored sugar. Cool completely, about 1 hour 30 minutes.\n\n3 Remove fudge from pan by lifting foil; remove foil from sides of fudge. With long knife, cut fudge into 12 rows by 10 rows.\n\n1 Square: Calories 90; Total Fat 3.5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 12g); Protein 0g Exchanges: 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\nCherry-Pistachio Fudge Omit walnuts or pecans and colored sugar. Stir in 1\u00bd cups sweetened dried cherries and \u00bd cup chopped pistachio nuts. After spreading fudge in pan, sprinkle top with \u00bd cup chopped pistachio nuts; press gently into fudge.\n\nCoconut Fudge Substitute 1 cup flake sweetened coconut for the colored sugar; press lightly into fudge.\n\nPeanut Butter Candy Fudge Omit almond extract and walnuts or pecans. Stir in 1 cup candy-coated peanut butter pieces and 1 cup chopped peanuts.\n\nTriple-Chocolate and Candy Fudge Omit walnuts or pecans. Stir in 1\u00bd cups miniature candy-coated chocolate candies.\n\n### Testing Candy Temperatures\n\n**Thread stage** (230\u00b0F to 234\u00b0F): Fine, thin, 2-inch thread falls off spoon when removed from hot mixture.\n\n**Soft-ball stage** (234\u00b0F to 240\u00b0F): When dropped into very cold water, forms a soft ball that flattens between fingers.\n\n**Firm-ball stage** (242\u00b0F to 248\u00b0F): When dropped into very cold water, forms a firm ball that holds its shape until pressed.\n\n**Hard-ball stage** (250\u00b0F to 265\u00b0F): When dropped into very cold water, forms a hard ball that holds its shape but is still pliable.\n\n**Soft-crack stage** (270\u00b0F to 290\u00b0F): When dropped into very cold water, separates into hard but pliable threads.\n\n**Hard-crack stage** (300\u00b0F to 310\u00b0F): When dropped into very cold water, separates into hard, brittle threads that break easily.\n\nLuscious Chocolate Truffles\n\n## Luscious Chocolate Truffles Calorie smart\n\nPrep 35 min Total 1 hr 15 min \u25cf 24 truffles\n\n  * 2 cups semisweet, dark or milk chocolate chips (about 12 oz)\n  * 2 tablespoons butter, softened\n  * \u00bc cup whipping cream\n  * 2 tablespoons liqueur (almond, cherry, coffee, hazelnut, Irish cream, orange, raspberry or other flavor), if desired\n  * 1 tablespoon shortening\n  * Finely chopped nuts, if desired\n  * \u00bc cup powdered sugar, if desired\n  * \u00bd teaspoon milk, if desired\n\n1 In 2-quart saucepan, melt 1 cup of the chocolate chips over low heat, stirring constantly; remove from heat. Stir in butter. Stir in whipping cream and liqueur. Refrigerate 10 to 15 minutes, stirring frequently, just until thick enough to hold a shape.\n\n2 Line cookie sheet with foil. Onto cookie sheet, drop chocolate mixture by teaspoonfuls; shape into balls. (If mixture is too sticky, refrigerate until firm enough to shape.) Freeze 30 minutes.\n\n3 In 1-quart saucepan, heat shortening and remaining 1 cup chocolate chips over low heat, stirring constantly, until chocolate is melted and mixture is smooth; remove from heat. Using fork, dip truffles, one at a time, into chocolate, tapping fork on side of pan to remove excess chocolate . Return to cookie sheet. Immediately sprinkle some of the truffles with nuts. Refrigerate about 10 minutes or until coating is set.\n\n4 In small bowl, stir powdered sugar and milk until smooth; drizzle over some of the truffles. Refrigerate just until set. Store in tightly covered container in refrigerator. Remove from refrigerator about 30 minutes before serving.\n\n1 Truffle: Calories 160; Total Fat 10g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 10mg; Sodium 15mg; Total Carbohydrate 14g (Dietary Fiber 1g, Sugars 12g); Protein 1g Exchanges: \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\nToffee Truffles Stir in 3 tablespoons chopped toffee bits with the whipping cream.\n\nWhite Chocolate\u2013Chocolate Truffles Stir in 3 tablespoons chopped white chocolate with the whipping cream.\n\n### Making Truffles\n\nRefrigerate truffle mixture just until thick enough to hold a shape.\n\nUsing fork, dip truffles into chocolate, covering completely; tap off excess\n\nPralines\n\n## Pralines Calorie smart\n\nPronounced prah-LEEN or PRAY-leen, this confection originated in Louisiana, where brown sugar and pecans are abundant.\n\nPrep 25 min Total 1 hr 15 min \u25cf 36 candies\n\n  * 1\u00bd cups granulated sugar\n  * 1\u00bd cups packed light brown sugar\n  * 1 cup half-and-half\n  * 2 tablespoons light corn syrup\n  * \u215b teaspoon salt\n  * 2 tablespoons butter\n  * 1 teaspoon vanilla\n  * 1\u00be cups pecan halves\n\n1 Check the accuracy of your candy thermometer before starting (see Using a Candy Thermometer, page 544). In 3-quart saucepan, mix granulated sugar, brown sugar, half-and-half, corn syrup and salt. Heat to boiling, stirring constantly. Reduce heat to medium. Cook uncovered about 6 minutes, without stirring, to 236\u00b0F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a soft ball that flattens when removed from water; remove from heat. Drop butter into hot mixture (do not stir in). Cool uncovered 10 minutes without stirring.\n\n2 Add vanilla and pecans. Beat with spoon about 2 minutes or until mixture is thickened and just begins to lose its gloss. On waxed paper, quickly drop mixture by heaping tablespoonfuls, dividing pecans equally; spread slightly. Let stand uncovered 30 minutes or until candies are firm.\n\n3 Wrap candies individually in waxed paper or plastic wrap. Store in tightly covered container.\n\n1 Candy: Calories 120; Total Fat 5g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 20mg; Total Carbohydrate 19g (Dietary Fiber 0g, Sugars 18g); Protein 0g Exchanges: 1\u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\n\nPistachio-Cranberry Divinity\n\n## Pistachio-Cranberry Divinity Calorie smart\n\nPrep 30 min Total 1 hr \u25cf 20 candies\n\n  * 1 egg white\n  * \u00bc cup water\n  * \u00bc cup light corn syrup\n  * 1\u00bd cups sugar\n  * \u00bd teaspoon vanilla\n  * \u00bd cup chopped pistachio nuts\n  * \u00bc cup sweetened dried cranberries, coarsely chopped\n\n1 Check the accuracy of your candy thermometer before starting (see Using a Candy Thermometer, page 544). Line large cookie sheet with waxed paper. In medium bowl, beat egg white with electric mixer on high speed until soft peaks form; set aside.\n\n2 In 2-quart saucepan, mix water, corn syrup and sugar. Cook over medium heat, stirring constantly, until sugar is dissolved. Without stirring, cook over medium heat 8 to 10 minutes or until syrup reaches 250\u00b0F on candy thermometer.\n\n3 When syrup is 250\u00b0F, continue beating egg white on high speed while slowly pouring syrup into egg white. Beat 2 to 3 minutes or until mixture holds a soft peak and does not flatten when dropped from a spoon.\n\n4 Fold in vanilla, nuts and cranberries. Quickly spoon mixture by rounded teaspoonfuls onto cookie sheet. Let stand about 30 minutes or until completely set.\n\n1 Candy: Calories 100; Total Fat 1.5g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 20mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 17g); Protein 0g Exchanges: 1\u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\nCaramels and variations\n\n## Caramels Calorie smart\n\nCut little rectangles of waxed paper ahead of time, so you're ready to wrap when it's time. Here's another hint\u2014cutting the caramels with kitchen scissors is quicker and easier than using a knife.\n\nPrep 40 min Total 2 hr 40 min \u25cf 64 candies\n\n  * 2 cups sugar\n  * \u00bd cup butter, cut into pieces\n  * 2 cups whipping cream\n  * \u00be cup light corn syrup\n\n1 Check the accuracy of your candy thermometer before starting (see Using a Candy Thermometer, page 544). Line 8- or 9-inch square pan with foil, leaving 1 inch of foil hanging over 2 opposite sides of pan; grease foil with butter.\n\n2 In 3-quart saucepan, heat all ingredients to boiling over medium heat, stirring constantly. Boil uncovered about 35 minutes, stirring frequently, to 245\u00b0F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a firm ball that holds its shape until pressed. Immediately spread in pan. Cool completely, about 2 hours.\n\n3 Using foil, lift caramels out of pan; place on cutting board. Peel foil from caramels. Cut into squares, 8 rows by 8 rows. Wrap individually in waxed paper or plastic wrap; store in tightly covered container at room temperature.\n\n1 Candy: Calories 70; Total Fat 4g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 15mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 8g); Protein 0g Exchanges: \u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: \u00bd\n\nChocolate Caramels Heat 2 ounces unsweetened baking chocolate, chopped, with the sugar mixture.\n\nChocolate-Topped Sea Salt Caramels After spreading caramel mixture into pan in Step 2, refrigerate about 1 hour or until completely cooled. In small microwavable bowl, microwave \u00bd cup semisweet chocolate chips and \u00bd teaspoon vegetable oil on High 30 to 45 seconds, stirring every 10 seconds, until chocolate is melted. Using spatula, spread chocolate evenly over caramel layer. Sprinkle with 1 teaspoon coarse sea salt. Refrigerate about 30 minutes or until chocolate is set. Continue as directed in Step 3.\n\n### Making Caramels\n\nLift caramels from pan; peel foil from caramels. Cut into squares.\n\nPeanut Brittle\n\n## Peanut Brittle Calorie smart\n\nPrep 55 min Total 1 hr 55 min \u25cf 72 pieces\n\n  * 1\u00bd teaspoons baking soda\n  * 1 teaspoon water\n  * 1 teaspoon vanilla\n  * 1\u00bd cups sugar\n  * 1 cup water\n  * 1 cup light corn syrup\n  * 3 tablespoons butter, cut into pieces\n  * 1 lb unsalted raw Spanish peanuts (3 cups)\n\n1 Check the accuracy of your candy thermometer before starting (see Using a Candy Thermometer, page 544). Heat oven to 200\u00b0F. Grease 2 (15x10x1-inch) pans with butter; keep warm in oven. (Keeping the pans warm allows the candy to be spread \u00bc inch thick without it setting up.) Grease long metal spatula with butter; set aside.\n\n2 In small bowl, mix baking soda, 1 teaspoon water and the vanilla; set aside. In 3-quart saucepan, mix sugar, 1 cup water and the corn syrup. Cook over medium heat about 25 minutes, stirring occasionally, to 240\u00b0F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a soft ball that flattens when removed from water.\n\n3 Stir in butter and peanuts. Cook over medium heat about 13 minutes, stirring constantly, to 300\u00b0F or until small amount of mixture dropped into cup of very cold water separates into hard, brittle threads. (Watch carefully so mixture does not burn.) Immediately remove from heat. Quickly stir in baking soda mixture until light and foamy.\n\n4 Pour half of the mixture onto each cookie sheet and quickly spread about \u00bc inch thick with buttered spatula. Cool completely, at least 1 hour. Break into pieces. Store in tightly covered container up to 2 weeks.\n\n1 Piece: Calories 70; Total Fat 3.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 35mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 6g); Protein 2g Exchanges: \u00bd Starch, 1 Fat Carbohydrate Choices: \u00bd\n\nMicrowave Directions Prepare pans and spatula as directed in Step 1. Omit all water. In 8-cup microwavable measuring cup, mix sugar, corn syrup and peanuts. Microwave uncovered on High 10 to 12 minutes, stirring every 5 minutes, until peanuts are light brown. Stir in vanilla and butter. Microwave uncovered on High 4 to 6 minutes to 300\u00b0F on microwave candy thermometer or until small amount of mixture dropped into cup of very cold water separates into hard, brittle threads (see Testing Candy Temperatures, page 545). Quickly stir in baking soda until mixture is light and foamy. Continue as directed in Step 4.\n\nAlmond or Cashew Brittle Substitute unsalted almonds or cashews for the peanuts.\n\nPeppermint Marshmallows\n\n## Marshmallows Calorie smart\n\nPrep 55 min Total 9 hr \u25cf 77 marshmallows\n\n  * \u2153 cup powdered sugar\n  * 2\u00bd tablespoons unflavored gelatin\n  * \u00bd cup cold water\n  * 1\u00bd cups granulated sugar\n  * 1 cup corn syrup\n  * \u00bc teaspoon salt\n  * \u00bd cup water\n\n1 Check the accuracy of your candy thermometer before starting (see Using a Candy Thermometer, page 544). Generously grease bottom and sides of 11x7-inch (2-quart) glass baking dish with butter; sprinkle with 1 tablespoon of the powdered sugar. In bowl of stand electric mixer, sprinkle gelatin over \u00bd cup cold water to soften; set aside.\n\n2 In 2-quart saucepan, heat granulated sugar, corn syrup, salt and \u00bd cup water over medium heat, stirring constantly, until sugar is dissolved. Heat to boiling; cook uncovered about 25 minutes, without stirring, to 240\u00b0F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a ball that flattens between fingers (see Testing Candy Temperatures, page 545).\n\n3 Remove from heat; slowly pour syrup into softened gelatin while beating on low speed. Increase speed to high; beat 6 to 8 minutes or until mixture is white and has almost tripled in volume. Pour into baking dish, patting lightly with wet hands. Let stand uncovered at least 8 hours or overnight.\n\n4 Sprinkle cutting board with about 1 tablespoon powdered sugar. Place remaining powdered sugar in small bowl. To remove marshmallow mixture, loosen sides from dish and gently lift in one piece onto cutting board. Using sharp knife greased with butter, cut into 1-inch squares (11 rows by 7 rows). Dip bottom and sides of each marshmallow into bowl of powdered sugar. Store in tightly covered container at room temperature up to 3 weeks.\n\n1 Marshmallow: Calories 35; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 8g (Dietary Fiber 0g, Sugars 6g); Protein 0gExchanges: \u00bd Other Carbohydrate Carbohydrate Choices: \u00bd\n\nCocoa Marshmallows In small bowl, mix \u00bc cup powdered sugar and 1 tablespoon unsweetened baking cocoa. Sprinkle 1 tablespoon of the mixture in greased baking dish; set remaining powdered sugar mixture aside. In Step 3, after increasing mixer speed to high, gradually add \u00bc cup unsweetened baking cocoa and 1 teaspoon vanilla, beating 6 to 8 minutes or until mixture is light brown in color and has almost tripled in volume. Continue as directed, using remaining powdered sugar mixture for sprinkling on the cutting board and dipping the marshmallows.\n\nPeppermint Marshmallows In Step 3, after beating mixture 6 to 8 minutes, add 1 teaspoon peppermint extract; beat 1 minute longer. Pour into baking dish, patting lightly with wet hands. Drop 8 to 10 drops red food color randomly onto top of marshmallow mixture. Pull table knife through food color to create swirl pattern over top. Continue as directed.\n\n### Making Marshmallows\n\nBeat syrup-gelatin mixture until almost tripled in volume; beat in peppermint extract.\n\nDrop food color randomly onto marshmallow mixture; pull table knife through to swirl pattern.\n\n# Chapter 20: Cakes & Cupcakes\n\nChocolate Cupcakes, page 574, with Fudge Frosting, page 578\n\n## Cake Basics\n\nAbout Butter Cakes\n\nAbout Foam Cakes\n\nStoring Cakes\n\nTips for Perfect Cakes\n\n## Features\n\nCupcake and Frosting Basics\n\nHeirloom Recipe and New Twist\n\nLearn to Beat, Whip & Fold\n\nLearn with Betty: Molten Chocolate Cake\n\nLearn with Betty: Browned Butter\n\nLearn with Betty: Butter Cakes\n\nLearn with Betty: Foam Cakes\n\nWays to Use Pound Cake\n\n## Chocolate Cakes\n\nChocolate Cake Roll\n\nChocolate Confetti Angel Food Cake\n\nChocolate-Fig Cake\n\nChocolate Layer Cake\n\nChocolate Sheet Cake\n\nDouble-Chocolate Hot Fudge Sundae Cake\n\nGerman Chocolate Cake\n\nMolten Chocolate Cakes\n\nRed Velvet Cake\n\nTurtle Layer Cake\n\n## White and Yellow Cakes\n\nAngel Food Cake\n\nBoston Cream Pie\n\nBoston Cream Shooters\n\nCaramel Snickerdoodle Cake\n\nChocolate Chip Cake\n\nCookies 'n Creme Cake\n\nDessert Kabobs\n\nEspresso Angel Food Cake\n\nEspresso Cake with Mocha Streusel Ribbon\n\nItalian Cream Cake\n\nMarble Cake\n\nPound Cake\n\nSilver White Cake\n\nSour Cream Spice Cake\n\nStarlight Yellow Cake\n\nToasted Almond Pound Cake\n\nTres Leches Cake\n\nTropical Tres Leches Cake\n\n## Fruit and Vegetable Cakes\n\nBanana Split\u2013Chocolate Mug Cake\n\nCarrot Cake\n\nCherry Angel Food Cake\n\nChocolate-Cherry Angel Food Cake\n\nCoconut Cake\n\nCranberry-Orange Pound Cake\n\nJelly Roll\n\nLemon Curd Jelly Roll\n\nLemon\u2013Poppy Seed Pound Cake\n\nOrange-Coconut Pound Cake\n\nPear-Ginger Upside-Down Cake\n\nRaspberry-Topped Pound Cake\n\nSpiced Apple Cake with Maple Glaze\n\nStrawberry Cake\n\nZucchini Cake\n\n## Cupcakes\n\nAngel Food Cupcakes\n\nCarrot Cupcakes\n\nChocolate Cupcakes Calorie Smart\n\nGinger and Peach Cupcakes\n\nLemon Meringue Cupcakes\n\nMaple\u2013Cornmeal Cupcakes with Maple-Butter Frosting\n\nPound Cake Cupcakes\n\nRed Velvet Cupcakes\n\nWhite Cupcakes Calorie Smart\n\nYellow Cupcakes Calorie Smart\n\n## Frostings\n\nBrowned Butter Frosting\n\nButterscotch Frosting\n\nCaramel Frosting Easy\n\nCherry-Nut Frosting\n\nChocolate Cream Cheese Frosting\n\nCinnamon Cream Cheese Frosting\n\nCream Cheese Frosting Fast \u2022 Easy\n\nCreamy Chocolate Frosting Fast \u2022 Easy\n\nCreamy Cocoa Frosting\n\nCreamy Vanilla Frosting Fast \u2022 Easy\n\nFluffy White Frosting Calorie Smart\n\nFudge Frosting\n\nLemon Frosting\n\nMaple-Nut Frosting\n\nMocha Frosting\n\nOrange Frosting\n\nPeanut Butter Frosting\n\nPeppermint Frosting\n\nSalty Caramel Frosting\n\nWhite Chocolate Frosting\n\n## Glazes\n\nBittersweet Chocolate Ganache\n\nBrowned Butter Glaze\n\nChocolate Ganache Fast \u2022 Easy\n\nChocolate Glaze Fast \u2022 Easy\n\nDark Chocolate Glaze\n\nLemon Glaze\n\nMilk Chocolate Glaze\n\nMint Chocolate Glaze\n\nOrange Glaze\n\nVanilla Glaze Fast \u2022 Easy\n\nWhite Chocolate Glaze\n\n## Bonus Recipes\n\nGlazed Pound Cake Doughnut Holes\n\nSnickerdoodle French Toast\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Cake Basics\n\nWith our foolproof, kitchen-tested recipes, you'll be sweetly surprised at just how easy it is to make a delicious cake from scratch. Once you take a bite of a tender, moist, delicious scratch cake, you'll quickly see what makes them so special. Straight from the Betty Crocker Kitchens, you'll find all the tips and techniques you need to become a \"Betty,\" too!\n\n## TIPS FOR PERFECT CAKES\n\nUnlike cooking, baking requires accuracy for success.\n\n  * Heat the oven to the correct temperature. Invest in an oven thermometer and to be sure your oven is at the correct temperature by placing it in the center of your oven.\n  * Be sure to measure ingredients accurately, using the right measuring equipment for the ingredient being measured (see Measuring Correctly, page 16), Add ingredients in the order they are called for.\n  * Use butter for best results and flavor. If you choose to substitute margarine, use those with at least 65 percent fat. Do not use reduced-fat butter or whipped products.\n  * For testing, we used handheld electric mixers. If you have a stand mixer, follow the manufacturer's directions for speed settings.\n  * Don't judge cake doneness just by the time given. Bake for the minimum amount of time listed, checking for doneness by using the doneness indicator given in the recipe (usually the toothpick test or by touching lightly with a finger). Bake longer if needed.\n  * Follow directions for cake cooling and pan removal. If a cake is left in the pan too long and sticks, try reheating it in the oven for 1 minute, and then remove.\n  * Cool cakes on cooling racks to allow for air circulation.\n\n## STORING CAKES\n\n  * See Refrigerator and Freezer Food Storage Chart, page 33, for information on wrapping, refrigerating, freezing and thawing cakes.\n  * Store layer or tube cakes on a plate under a cake cover or inverted mixing bowl.\n  * Serve cake with fluffy frosting as soon as possible, as the cake will tend to absorb the frosting. Store under a cake cover, but slip a knife under the edge so it isn't airtight.\n\n## Picking Cake Pans\n\n  * Use the pan size called for in the recipe. If the size isn't printed on the bottom of the pan, measure the length and width from inside edge to inside edge. Too small a pan may allow batter to overflow and bake up with the center still raw; too large a pan may result in a short cake with an overbaked, dry texture.\n  * Shiny pans reflect heat, baking cakes that are tender and light. Dark pans or pans with nonstick coating absorb heat faster, causing cakes to brown too quickly.\n  * Unless otherwise specified, fill cake pans half-full with batter.\n\nOrange-Coconut Pound Cake, page 564\n\n## About Butter Cakes\n\nSometimes called shortening cakes, these are cakes made with butter or shortening, flour, eggs, liquid and baking powder or soda.\n\n  * Avoid overmixing batter, which can cause tunnels or a sunken center.\n  * Bake cakes on center oven rack, arranging pans so that there is at least 1 inch of space between the pans and the sides of the oven.\n  * If not all pans will fit on one rack, refrigerate one pan of batter until the others are baked, and then bake remaining layer separately.\n  * Grease pans with solid shortening, not butter or margarine. Use cooking spray if recipe calls for it.\n  * To remove cake from pan, insert knife between cake and pan, then slide around edge to loosen it. Place wire rack upside down over cake. Invert carefully so cake is on rack and remove the pan. Invert onto another rack to flip cake right side up.\n  * Cut cake with thin, sharp knife in gentle sawing motion.\n\n### Removing Cakes from Pans\n\nPlace wire rack upside down over loosened cake. Invert cake carefully onto rack; remove pan.\n\nInvert cake and rack onto another rack and flip cake right side up. Remove first rack.\n\n## About Foam Cakes\n\nAngel food, sponge and chiffon cakes are types of foam cakes, which depend on beaten egg whites for their light and airy texture.\n\n**Angel food cakes** contain no added leavening, fat or egg yolks. They have a high proportion of beaten egg whites to flour.\n\n**Sponge cakes** use both egg whites and yolks and sometimes a little leavening, but do not contain other added fat.\n\n**Chiffon cakes** are a cross between foam and butter cakes, made with some leavening, vegetable oil or shortening and egg yolks, as well as beaten egg whites.\n\n  * Use a clean, dry bowl and beaters to beat egg whites so they will whip properly. Even a speck of fat from an egg yolk can keep whites from whipping up properly.\n  * Do not grease and flour pans unless directed in a recipe. During baking, batter needs to cling and climb up the sides of the pans.\n  * For any tube pan cake, move oven rack to the lowest position in oven so the cake will bake completely without browning too much.\n\n## Foolproof Foam Cakes\n\nThe best foam cakes are high, golden brown with cracks in the surface, soft, moist and delicate. Here are some things that can happen, with solutions to help.\n\n  * Low and compact\u2014underbeaten, overmixed or overfolded batter, incorrect cooling or underbaking.\n  * Coarse and not tender\u2014underbeaten egg whites or underfolded batter.\n\n### Learn with Betty Foam Cakes\n\n**Perfect Cake:** This cake is high and golden brown, with feathery, light texture.\n\n**Underrisen Cake:** This cake is compact because of underbeaten or very overbeaten egg whites, overfolded batter or underbaking.\n\n**Overbaked Cake:** This cake is too brown and crumbly from batter not being properly folded or underbeaten egg whites and overbaking.\n\nGerman Chocolate Cake\n\n## German Chocolate Cake\n\nSamuel German, an employee of the Baker's Chocolate Company, developed a sweet chocolate in 1852. Over a hundred years later, in 1957, a reader of a Dallas newspaper submitted her recipe for this now-famous three-tiered cake with coconut-pecan frosting. Sales of sweet chocolate soared, and the rest is delicious history!\n\nPrep 30 min Total 2 hr 20 min \u25cf 12 servings\n\nCake\n\n  * 4 oz sweet baking chocolate, chopped\n  * \u00bd cup water\n  * 2\u00bc cups all-purpose flour or 2\u00bd cups cake flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon salt\n  * 2 cups granulated sugar\n  * 1 cup butter, softened\n  * 4 whole eggs, separated\n  * 1 teaspoon vanilla\n  * 1 cup buttermilk\n\nCoconut-Pecan Frosting\n\n  * 1 cup granulated sugar or packed brown sugar\n  * \u00bd cup butter, cut into pieces\n  * 1 cup evaporated milk or half-and-half\n  * 1 teaspoon vanilla\n  * 3 egg yolks\n  * 1\u2153 cups coconut\n  * 1 cup chopped pecans, toasted if desired (page 23)\n\n1 Heat oven to 350\u00b0F. Grease bottoms and sides of 3 (8- or 9-inch) round cake pans with shortening. Line pan bottoms with waxed paper or cooking parchment paper; grease paper.\n\n2 In 1-quart saucepan, heat chocolate and water over low heat, stirring frequently, until chocolate is completely melted; cool.\n\n3 In medium bowl, mix flour, baking soda and salt; set aside. In another medium bowl, beat 2 cups granulated sugar and 1 cup butter with electric mixer on high speed until light and fluffy. Beat in 4 egg yolks, one at a time. Beat in chocolate and 1 teaspoon vanilla on low speed. Add flour mixture alternately with buttermilk, beating on low speed after each addition just until smooth.\n\n4 Wash and dry mixer beaters. In small bowl, beat 4 egg whites on high speed until stiff; fold into batter. Pour into pans. If all pans won't fit in oven at one time, refrigerate batter in third pan and bake separately.\n\n5 Bake 8-inch pans 35 to 40 minutes, 9-inch pans 30 to 35 minutes, or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling racks. Remove waxed paper. Cool completely, about 1 hour.\n\n6 Meanwhile, in 2-quart saucepan, stir together 1 cup granulated sugar, \u00bd cup butter, the milk, 1 teaspoon vanilla and 3 egg yolks until well mixed. Cook over medium heat about 12 minutes, stirring frequently, until thickened and bubbly. Stir in coconut and pecans. Cool about 30 minutes, beating occasionally with spoon, until spreadable.\n\n7 Spread frosting between layers and on top of cake, leaving side of cake unfrosted. Store covered in refrigerator.\n\n1 Serving: Calories 670; Total Fat 33g (Saturated Fat 21g, Trans Fat 1g); Cholesterol 175mg; Sodium 600mg; Total Carbohydrate 82g (Dietary Fiber 2g, Sugars 63g); Protein 8g Exchanges: 2\u00bd Starch, 3 Other Carbohydrate, 6\u00bd Fat Carbohydrate Choices: 5\u00bd\n\nChocolate Layer Cake\n\n## Chocolate Layer Cake\n\nHomemade chocolate cake is impressive but certainly not hard to achieve. To make great cakes, use the right pans, measure ingredients accurately and follow directions carefully.\n\nPrep 20 min Total 2 hr 5 min \u25cf 12 servings\n\n  * 2\u00bc cups all-purpose flour or 2\u00bd cups cake flour\n  * 1\u2154 cups sugar\n  * \u2154 cup unsweetened baking cocoa\n  * 1\u00bc teaspoons baking soda\n  * 1 teaspoon salt\n  * \u00bc teaspoon baking powder\n  * 1\u00bc cups water\n  * \u00be cup butter, softened\n  * 2 eggs\n  * 1 teaspoon vanilla\n  * Fudge Frosting (page 578) or Fluffy White Frosting (page 581), if desired\n\n1 Heat oven to 350\u00b0F. Grease bottoms and sides of 2 (9-inch) or 3 (8-inch) round cake pans with shortening; lightly flour.\n\n2 In large bowl, beat all ingredients except frosting with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on high speed 3 minutes, scraping bowl occasionally. Pour batter into pans.\n\n3 Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling racks. Cool completely, about 1 hour.\n\n4 Fill and frost layers with frosting.\n\n1 Serving: Calories 330; Total Fat 13g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 65mg; Sodium 430mg; Total Carbohydrate 48g (Dietary Fiber 2g, Sugars 28g); Protein 5g Exchanges: 2 Starch, 1 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 3\n\nTurtle Layer Cake Substitute Caramel Frosting (page 580) for Fudge Frosting. After frosting first layer, press \u00bd cup coarsely chopped pecans gently into frosting on top of cake before topping with second layer. Sprinkle top of frosted cake with an additional \u00bd cup coarsely chopped pecans; press lightly into frosting.\n\nChocolate Sheet Cake Grease bottom and sides of 13x9-inch pan with shortening; lightly flour. Make batter as directed; pour into pan. Bake 40 to 45 minutes or until toothpick inserted in center comes out clean. Cool completely in pan on cooling rack. Frost top of cake with frosting. Decorate cake, if desired. See Easy Chocolate Dessert Garnishes, page 611.\n\n### Frosting a Layer Cake\n\nBrush any loose crumbs from cooled cake layer. Place 4 strips of waxed paper around edge of plate. Place layer, rounded side down, on plate.\n\nSpread \u2153 to \u00bd cup frosting over top of first layer to within about \u00bc inch of edge.\n\nPlace second cake layer, rounded side up, on frosted first layer. Coat side of cake with a very thin layer of frosting to seal in crumbs.\n\nFrost side of cake in swirls, making a rim about \u00bc inch high above top of cake. Spread remaining frosting on top, just to the built-up rim. Carefully remove waxed paper strips.\n\nRed Velvet Cake\n\n## Red Velvet Cake\n\nPrep 25 min Total 2 hr \u25cf 12 servings\n\nCake\n\n  * 2\u00bd cups all-purpose flour\n  * 1\u00bd cups sugar\n  * 2 tablespoons unsweetened baking cocoa\n  * 1 tablespoon baking powder\n  * 1 teaspoon salt\n  * 1\u00bd cups vegetable oil\n  * 1 cup buttermilk\n  * 1 teaspoon vanilla\n  * 1 bottle (1 oz) red food color\n  * 2 eggs\n\nFrosting\n\n  * \u00bd cup all-purpose flour\n  * 1\u00bd cups milk\n  * 1\u00bd cups sugar\n  * 1\u00bd cups butter, softened\n  * 1 tablespoon vanilla\n\n1 Heat oven to 350\u00b0F. Grease bottoms and sides of 3 (8- or 9-inch) round cake pans with shortening; lightly flour.\n\n2 In large bowl, beat all cake ingredients with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat 2 minutes on medium speed, scraping bowl occasionally. Pour batter into pans.\n\n3 Bake 25 to 35 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling rack. Cool completely, about 1 hour.\n\n4 Meanwhile, in 2-quart saucepan, mix \u00bd cup flour and 1\u00bd cups milk with whisk until smooth. Cook over medium heat until mixture is very thick, stirring constantly. Remove from heat; cool 10 minutes. In large bowl, beat 1\u00bd cups sugar and the butter with electric mixer on medium speed until light and fluffy. Gradually add flour mixture by tablespoonfuls, beating on high speed until smooth. Beat in 1 tablespoon vanilla.\n\n5 Fill and frost cake, using 1 cup frosting between layers. Store covered in refrigerator.\n\n1 Serving: Calories 800; Total Fat 52g (Saturated Fat 20g, Trans Fat 1g); Cholesterol 100mg; Sodium 510mg; Total Carbohydrate 76g (Dietary Fiber 1g, Sugars 52g); Protein 5g Exchanges: 1\u00bd Starch, 3\u00bd Other Carbohydrate, 10\u00bd Fat Carbohydrate Choices: 5\n\nRed Velvet Cupcakes Place paper baking cup in each of 24 regular-size muffin cups. Heat oven and make cake batter as directed in recipe. Divide batter among muffin cups, filling each about two-thirds full. Bake 20 to 22 minutes or until toothpick inserted in center comes out clean. Remove cupcakes from pan to cooling rack. Cool completely, about 30 minutes. Frost with red velvet cake frosting (above) or Fluffy White Frosting (page 581).\n\n### Flouring Pan\n\nPlace 1 to 2 tablespoons flour in greased pan. Rotate pan until all sides and bottom are coated; tap pan and shake out excess flour.\n\n### Scraping Bowl\n\nBeat cake ingredients on low speed 30 seconds, scraping bowl constantly. Beat on high speed 3 minutes, scraping bowl occasionally.\n\n### Making Red Velvet Cake Frosting\n\nCook frosting in saucepan over medium heat, stirring constantly, until very thick.\n\nGradually add flour mixture by tablespoonfuls, beating on high speed until smooth and thick.\n\nChocolate-Fig Cake\n\n## Chocolate-Fig Cake\n\nPrep 30 min Total 3 hr 30 min \u25cf 16 servings\n\nCake\n\n  * 6 oz dried Calimyrna figs, stems removed, chopped (1 cup)\n  * 1 cup dry red wine (such as Merlot or Cabernet Sauvignon) or water\n  * 1 cup butter, softened\n  * 1 cup granulated sugar\n  * 2 eggs\n  * 1 teaspoon vanilla\n  * 1\u00be cups all-purpose flour\n  * 2 tablespoons unsweetened regular baking cocoa\n  * 1 teaspoon baking soda\n  * 4 oz semisweet baking chocolate, melted, cooled\n\nFrosting and Garnish\n\n  * 1 container (8 oz) mascarpone cheese\n  * 2\u00bd cups powdered sugar\n  * \u00bd teaspoon vanilla\n  * 1 bar (1.55 oz) milk chocolate, if desired\n\n1 Heat oven to 350\u00b0F. Grease bottom and side of 9- or 10-inch springform pan with shortening; lightly flour. In 1\u00bd-quart saucepan, heat figs and wine to boiling over medium-high heat. Remove from heat; cool.\n\n2 In large bowl, beat butter and granulated sugar with electric mixer on medium speed until creamy. Add eggs and 1 teaspoon vanilla. Beat until thick and creamy, scraping bowl occasionally. Add flour, cocoa and baking soda. Beat on low speed until moistened, scraping bowl occasionally. Add melted chocolate and fig mixture. Beat until blended. Spread batter in pan.\n\n3 Bake 55 to 60 minutes or until toothpick inserted in center comes out clean. Cool completely in pan on cooling rack, about 2 hours. Remove side of pan.\n\n4 In medium bowl, beat cheese, powdered sugar and \u00bd teaspoon vanilla with electric mixer on low speed until blended. Beat on medium speed until thick and creamy. Frost side and top of cake.\n\n5 To make garnish, see Chocolate Curls, page 611. Arrange curls on cake. Garnish cake with chocolate curls. Store in refrigerator.\n\n1 Serving: Calories 380; Total Fat 19g (Saturated Fat 12g, Trans Fat 0g); Cholesterol 65mg; Sodium 190mg; Total Carbohydrate 47g (Dietary Fiber 2g, Sugars 32g); Protein 3g Exchanges: 3 Other Carbohydrate, \u00bd Medium-Fat Meat, 3\u00bd Fat Carbohydrate Choices: 3\n\n## Double-Chocolate Hot Fudge Sundae Cake\n\nPrep 15 min Total 1 hr 5 min \u25cf 9 servings\n\n  * 1 cup all-purpose flour\n  * \u00be cup granulated sugar\n  * 2 tablespoons unsweetened baking cocoa\n  * 2 teaspoons baking powder\n  * \u00bc teaspoon salt\n  * \u00bd cup chocolate or regular milk\n  * 2 tablespoons butter, melted, or vegetable oil\n  * 1 teaspoon vanilla\n  * \u00bd cup semisweet chocolate chips\n  * \u00bd cup chopped walnuts or pecans, toasted (page 23), if desired\n  * 1 cup packed brown sugar\n  * \u00bc cup unsweetened baking cocoa\n  * 1\u00be cups hot strong brewed coffee or hot water\n  * Ice cream, if desired\n\n1 Heat oven to 350\u00b0F. In ungreased 9-inch square pan, mix flour, granulated sugar, 2 tablespoons cocoa, the baking powder and salt. Mix in milk, butter and vanilla with fork until smooth. Stir in chocolate chips and nuts.\n\n2 Spread batter in pan. Sprinkle with brown sugar and \u00bc cup cocoa. Pour hot coffee evenly over batter.\n\n3 Bake about 40 minutes or until top is dry. Cool 10 minutes. Spoon warm cake into dessert dishes. Top with ice cream. Spoon sauce from pan onto each serving.\n\n1 Serving: Calories 260; Total Fat 4g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 0mg; Sodium 190mg; Total Carbohydrate 54g (Dietary Fiber 2g, Sugars 41g); Protein 3g Exchanges: 2 Starch, 1 Fat Carbohydrate Choices: 3\u00bd\n\nStarlight Yellow Cake\n\n## Starlight Yellow Cake\n\nPrep 20 min Total 2 hr 10 min \u25cf 12 servings\n\n  * 2\u00bc cups all-purpose flour\n  * 1\u00bd cups sugar\n  * 3\u00bd teaspoons baking powder\n  * 1 teaspoon salt\n  * \u00bd cup butter, softened\n  * 1\u00bc cups milk\n  * 1 teaspoon vanilla\n  * 3 eggs\n  * Creamy Chocolate Frosting (page 579) or Peanut Butter Frosting (page 579), if desired\n\n1 Heat oven to 350\u00b0F. Grease bottoms and sides of 2 (9-inch) round cake pans, 3 (8-inch) round cake pans or 13x9-inch pan with shortening; lightly flour.\n\n2 In large bowl, beat all ingredients except frosting with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on high speed 3 minutes, scraping bowl occasionally. Pour batter into pan(s).\n\n3 Bake 9-inch pans 25 to 30 minutes, 8-inch pans 30 to 35 minutes, 13x9-inch pan 35 to 40 minutes, or until toothpick inserted in center comes out clean or cake springs back when touched lightly in center. Cool rounds 10 minutes; remove from pans to cooling racks. Cool 13x9-inch cake in pan on cooling rack. Cool completely, about 1 hour.\n\n4 Fill and frost round layers or frost top of 13x9-inch cake with frosting.\n\n1 Serving: Calories 290; Total Fat 10g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 75mg; Sodium 420mg; Total Carbohydrate 45g (Dietary Fiber 0g, Sugars 27g); Protein 5g Exchanges: 2 Starch, 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 3\n\nBoston Cream Pie While cake layers are baking, in 2-quart heavy saucepan, stir 1\u00bc cups whole milk, \u00bc cup sugar, 2 egg yolks, 2 tablespoons cornstarch and \u00bc teaspoon salt. Heat to boiling over medium heat, stirring constantly; boil 1 minute or until thickened and mixture coats back of spoon. Remove from heat; stir in 1 tablespoon butter and 1 teaspoon vanilla. Cover surface of custard with plastic wrap. Refrigerate until set, about 2 hours. Omit Creamy Chocolate Frosting. Fill cooled cake layers with custard. Top cake with Chocolate Glaze (page 582).\n\n### Learn with Betty Butter Cakes\n\n**Perfect Cake:** This cake is high and golden brown, with a slightly rounded top. The texture is soft, velvety and tender.\n\n**Underrisen Cake:** This cake did not rise and is pale in color because of too much liquid or fat, not enough leavening or pan was too large.\n\n**Overbaked Cake:** This cake is too brown and sunk in the middle because of too much leavening, too much liquid or too long in the oven.\n\n### Testing Cakes for Doneness\n\nInsert wooden cake tester or toothpick into center of cake.\n\nWhen you remove cake tester, it should be clean, maybe with some dry crumbs on it. If there is wet batter, the cake is not done.\n\nMarble Cake with Fluffy White Frosting (page 581)\n\n## Silver White Cake\n\nPrep 20 min Total 2 hr 15 min \u25cf 12 servings\n\n  * 2\u00bc cups all-purpose flour\n  * 1\u2154 cups sugar\n  * \u2154 cup shortening\n  * 1\u00bc cups milk\n  * 3\u00bd teaspoons baking powder\n  * 1 teaspoon salt\n  * 1 teaspoon vanilla or almond extract\n  * 5 egg whites\n  * Fluffy White Frosting (page 581) or Creamy Chocolate Frosting (page 579), if desired\n\n1 Heat oven to 350\u00baF. Grease 2 (9-inch) round cake pans, 3 (8-inch) round cake pans or 13x9-inch pan with shortening; lightly flour.\n\n2 In large bowl, beat all ingredients except egg whites and frosting with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on high speed 2 minutes, scraping bowl occasionally. Beat in egg whites on high speed 2 minutes, scraping bowl occasionally. Pour batter into pan(s).\n\n3 Bake 9-inch rounds 30 to 35 minutes, 8-inch rounds 23 to 28 minutes, 13x9-inch pan 40 to 45 minutes, or until toothpick inserted in center comes out clean or cake springs back when touched lightly in center. Cool rounds 10 minutes; remove from pans to cooling racks. Cool 13x9-inch cake in pan on cooling rack. Cool completely, about 1 hour.\n\n4 Fill and frost round layers or frost top of 13x9-inch cake with frosting.\n\n1 Serving: Calories 320; Total Fat 12g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 0mg; Sodium 370mg; Total Carbohydrate 47g (Dietary Fiber 0g, Sugars 29g); Protein 4g Exchanges: 1 Starch, 2 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 3\n\nChocolate Chip Cake Fold \u00bd cup miniature or finely chopped regular semisweet chocolate chips into batter after beating in egg whites.\n\nCookies 'n Creme Cake Stir 1 cup crushed creme-filled chocolate sandwich cookies into batter after beating in egg whites. Bake and cool as directed. Frost with Fluffy White Frosting; garnish with whole or coarsely crushed creme-filled chocolate sandwich cookies.\n\nMarble Cake Reserve 1\u00be cups of the batter. Pour remaining batter into pan(s). Stir 3 tablespoons unsweetened baking cocoa and \u215b teaspoon baking soda into reserved batter. Drop chocolate batter by tablespoonfuls randomly onto white batter. Cut through batters with knife for marbled design. Bake as directed.\n\nStrawberry Cake Use 9-inch round cake pans and Fluffy White Frosting. After frosting first layer, top with 1 cup sliced fresh strawberries, overlapping as needed. Add second layer; frost cake. Garnish cake with whole or sliced strawberries.\n\n### Marbling Batter\n\nDrop chocolate batter by tablespoonfuls randomly over white batter. Cut through batters with knife.\n\nCoconut Cake\n\n## Coconut Cake\n\nPrep 50 min Total 2 hr 20 min \u25cf 16 servings\n\nCake\n\n  * 2\u00be cups all-purpose flour\n  * 2 teaspoons baking powder\n  * 1 teaspoon salt\n  * 1 cup butter, softened\n  * 2 cups sugar\n  * 4 whole eggs\n  * 1\u00bd teaspoons vanilla\n  * 1\u00bd teaspoons almond extract\n  * 1 cup milk\n\nBoiled Frosting\n\n  * 1\u00bd cups sugar\n  * \u00bd cup water\n  * 4 egg whites\n  * \u00bd teaspoon cream of tartar\n  * \u215b teaspoon salt\n  * 6 large marshmallows, cut into small pieces\n\nFilling\n\n  * 2 tablespoons sugar\n  * \u00bc cup reduced-fat (lite) coconut milk (not cream of coconut)\n  * 2 to 3 cups sweetened flaked coconut\n\n1 Heat oven to 350\u00b0F. Grease 3 (9-inch) round cake pans with shortening; lightly flour. In medium bowl, mix flour, baking powder and 1 teaspoon salt; set aside.\n\n2 In large bowl, beat butter with electric mixer on medium speed 30 seconds. Gradually add 2 cups sugar, \u00bc cup at a time, beating well after each addition. Beat 2 minutes longer. Add whole eggs, one at a time, beating well after each addition. Beat in vanilla and almond extract. Add flour mixture alternately with milk, beating on low speed just until blended. Pour batter into pans.\n\n3 Bake 18 to 20 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling racks. Cool completely, about 1 hour.\n\n4 In 2-quart saucepan, mix 1\u00bd cups sugar and the water. Cook over medium heat, stirring constantly, until mixture is clear. Cook, without stirring, to 240\u00b0F on candy thermometer, about 10 minutes. Meanwhile, in large bowl, beat egg whites with electric mixer on low speed until foamy. Add cream of tartar and \u215b teaspoon salt; beat on medium speed until soft peaks form. Increase speed to high; pour hot syrup into egg white mixture. Add marshmallows, a few pieces at a time, beating until stiff peaks form and frosting is thick enough to spread.\n\n5 In small microwavable bowl, microwave 2 tablespoons sugar and the coconut milk on High 1 minute; stir until sugar is dissolved. Brush half of the coconut milk mixture over one cake layer to within \u00bd inch of edge. Frost with 1 cup of the frosting; sprinkle with \u00bd cup of the coconut. Top with second cake layer; brush with remaining coconut milk mixture. Frost with 1 cup frosting; sprinkle with \u00bd cup coconut. Top with remaining cake layer. Spread remaining frosting on top and side of cake; sprinkle with remaining coconut.\n\n1 Serving: Calories 460; Total Fat 17g (Saturated Fat 11g, Trans Fat 0g); Cholesterol 80mg; Sodium 400mg; Total Carbohydrate 70g (Dietary Fiber 1g, Sugars 52g); Protein 5g Exchanges: \u00bd Starch, 4 Other Carbohydrate, \u00bd Lean Meat, 3 Fat Carbohydrate Choices: 4\u00bd\n\nTres Leches Cake\n\n## Tres Leches Cake\n\nIn Spanish, tres leches means \"three milks.\"\n\nPrep 25 min Total 4 hr 15 min \u25cf 15 servings\n\n  * Batter for Starlight Yellow Cake (page 559) or Silver White Cake (page 560)\n  * 2 cups whipping cream\n  * 1 cup whole milk\n  * 1 can (14 oz) sweetened condensed milk (not evaporated)\n  * \u2153 cup rum or 1 tablespoon rum extract plus enough water to measure \u2153 cup\n  * 2 tablespoons rum or 1 teaspoon rum extract\n  * \u00bd teaspoon vanilla\n  * \u00bd cup chopped pecans, toasted (page 23)\n\n1 Heat oven to 350\u00b0F. Grease bottom only of 13x9-inch pan with shortening. Make batter as directed in recipe; pour into pan. Bake 35 to 40 minutes for yellow cake, 40 to 45 minutes for white cake, or until toothpick inserted in center comes out clean or cake springs back when touched lightly in center. Let stand 5 minutes.\n\n2 Pierce top of hot cake every \u00bd inch with long-tined fork, wiping fork occasionally to reduce sticking. In large bowl, stir 1 cup of the whipping cream, the whole milk, condensed milk and \u2153 cup rum until well mixed. Carefully pour milk mixture evenly over top of cake. Cover and refrigerate about 3 hours or until chilled and most of milk mixture has been absorbed into cake (when cutting cake to serve, you may notice some of the milk mixture on the bottom of the pan).\n\n3 In chilled large deep bowl, beat remaining 1 cup whipping cream, 2 tablespoons rum and the vanilla with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat just until soft peaks form, lifting beaters occasionally to check thickness. Frost cake with whipped cream mixture. Sprinkle with pecans. Store covered in refrigerator.\n\n1 Serving: Calories 480; Total Fat 24g (Saturated Fat 12g, Trans Fat 1g); Cholesterol 110mg; Sodium 400mg; Total Carbohydrate 57g (Dietary Fiber 0g, Sugars 42g); Protein 8g Exchanges: 2 Starch, 2 Other Carbohydrate, 4\u00bd Fat Carbohydrate Choices: 4\n\nTropical Tres Leches Cake Make cake through Step 3, except omit pecans. Sprinkle with 1 cup coconut, toasted (page 23), and \u00bd cup chopped macadamia nuts, toasted (page 23).\n\nCarrot Cake\n\n## Carrot Cake\n\nPrep 20 min Total 2 hr 5 min \u25cf 12 servings\n\n  * 1\u00bd cups sugar\n  * 1 cup vegetable oil\n  * 3 eggs\n  * 2 cups all-purpose flour\n  * 2 teaspoons ground cinnamon\n  * 1 teaspoon baking soda\n  * \u00bd teaspoon salt\n  * 1 teaspoon vanilla\n  * 3 cups shredded carrots (4 medium)\n  * 1 can (8 oz) crushed pineapple in juice, drained (\u00bd cup)\n  * 1 cup chopped nuts\n  * \u00bd cup coconut\n  * Cream Cheese Frosting (page 580), if desired\n\n1 Heat oven to 350\u00b0F. Grease bottom and sides of 13x9-inch pan or 2 (8- or 9-inch) round cake pans with shortening; lightly flour.\n\n2 In large bowl, beat sugar, oil and eggs with electric mixer on low speed about 30 seconds or until blended. Add flour, cinnamon, baking soda, salt and vanilla; beat on medium speed 1 minute. Stir in carrots, pineapple, nuts and coconut (batter will be thick). Pour into pan(s).\n\n3 Bake 13x9-inch pan 40 to 45 minutes, round pans 30 to 35 minutes, or until toothpick inserted in center comes out clean. Cool 13x9-inch cake in pan on cooling rack. Cool rounds 10 minutes; remove from pans to cooling racks. Cool completely, about 1 hour.\n\n4 Frost top of 13x9-inch cake or fill and frost round layers with frosting. Store covered in refrigerator.\n\n1 Serving: Calories 470; Total Fat 28g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 55mg; Sodium 240mg; Total Carbohydrate 47g (Dietary Fiber 3g, Sugars 28g); Protein 5g Exchanges: 1 Starch, 2 Other Carbohydrate, \u00bd Vegetable, 5\u00bd Fat Carbohydrate Choices: 3\n\nLighter Directions For 10 grams of fat and 280 calories per serving, substitute \u00bd cup unsweetened applesauce for \u00bd cup of the oil, and 1 egg plus 4 egg whites for the eggs. Omit nuts.\n\nCarrot Cupcakes Place paper baking cup in each of 24 regular-size muffin cups. Divide batter evenly among muffin cups. Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling racks. Cool completely, 30 minutes.\n\nZucchini Cake Substitute 3 cups shredded zucchini (2 or 3 medium) for the carrots.\n\nPear-Ginger Upside-Down Cake\n\n## Pear-Ginger Upside-Down Cake\n\nPrep 30 min Total 1 hr 50 min \u25cf 8 servings\n\nTopping\n\n  * \u00bc cup butter\n  * \u2154 cup packed brown sugar\n  * \u00bd teaspoon ground ginger\n  * 3 pears, peeled, cut into \u00bd-inch wedges\n  * \u00bc cup finely chopped crystallized ginger\n\nCake\n\n  * 1 tablespoon all-purpose flour\n  * \u00bc cup finely chopped crystallized ginger\n  * 1\u2153 cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bc teaspoon salt\n  * 1 cup packed brown sugar\n  * 6 tablespoons butter, softened\n  * 2 eggs\n  * \u00bd teaspoon vanilla\n  * \u00bc cup milk\n\nGinger Whipped Cream\n\n  * 1 cup whipping cream\n  * 2 tablespoons granulated sugar\n  * \u00bc teaspoon ground ginger\n  * 1 tablespoon chopped crystallized ginger (to make your own, see page 447)\n\n1 Heat oven to 325\u00b0F. Grease bottom and sides of 8- or 9-inch square pan with shortening.\n\n2 In 1-quart saucepan, melt \u00bc cup butter over medium heat, stirring occasionally. Stir in \u2154 cup brown sugar. Heat to boiling; remove from heat. Stir in \u00bd teaspoon ground ginger. Pour into pan; spread evenly. Arrange pear wedges on sugar mixture, overlapping tightly and making 2 layers if necessary. Sprinkle \u00bc cup crystallized ginger over pears.\n\n3 In small bowl, toss 1 tablespoon flour and \u00bc cup crystallized ginger; set aside. In another small bowl, mix 1\u2153 cups flour, the baking powder and salt; set aside.\n\n4 In large bowl, beat 1 cup brown sugar and 6 tablespoons butter with electric mixer on medium speed, scraping bowl occasionally, until fluffy. Beat in eggs, one at a time, until smooth. Add vanilla. Alternately add flour mixture with milk, beating on low speed after each addition until smooth. Stir in ginger-flour mixture. Spread batter over pears in pan.\n\n5 Bake 55 to 65 minutes or until toothpick inserted in center of cake comes out clean. Cool 15 minutes on cooling rack.\n\n6 Meanwhile, in chilled medium bowl, beat whipping cream on high speed until it begins to thicken. Gradually add granulated sugar and \u00bc teaspoon ground ginger, beating until soft peaks form. Sprinkle with crystallized ginger. Loosen edges of cake with small knife. Place serving plate upside down over pan; turn plate and pan over. Serve cake warm with ginger whipped cream.\n\n1 Serving: Calories 600; Total Fat 27g (Saturated Fat 15g, Trans Fat 1g); Cholesterol 135mg; Sodium 290mg; Total Carbohydrate 84g (Dietary Fiber 3g, Sugars 62g); Protein 5g Exchanges: 1\u00bd Starch, 4 Other Carbohydrate, 5\u00bd Fat Carbohydrate Choices: 5\u00bd\n\n## Pound Cake\n\nIn 1895, Sperry Flour Company (a firm that later joined General Mills) produced a recipe booklet called \"Easy Cooking for Little Cooks.\" Included in this booklet was a recipe for pound cake, given its name because the primary ingredients were 1 pound each of butter, sugar and flour. This recipe has been a favorite ever since.\n\nPrep 15 min Total 3 hr 55 min \u25cf 24 servings\n\n  * 3 cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bc teaspoon salt\n  * 2\u00bd cups granulated sugar\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla or almond extract\n  * 5 eggs\n  * 1 cup milk or evaporated milk\n  * Powdered sugar, if desired\n\n1 Heat oven to 350\u00b0F. Generously grease bottom, side and tube of 10-inch angel food (tube) cake pan or 12-cup fluted tube cake pan or bottoms and sides of 2 (9x5-inch) loaf pans with shortening; lightly flour.\n\n2 In medium bowl, mix flour, baking powder and salt. In large bowl, beat granulated sugar, butter, vanilla and eggs with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on high speed 5 minutes, scraping bowl occasionally. Add flour mixture alternately with milk, beating on low speed just until smooth after each addition. Pour batter into pan(s).\n\n3 Bake angel food or fluted tube cake pan 1 hour 10 minutes to 1 hour 20 minutes, loaf pans 55 to 60 minutes, or until toothpick inserted in center comes out clean. Cool 20 minutes; remove from pan(s) to cooling rack. Cool completely, about 2 hours. Sprinkle with powdered sugar.\n\n1 Serving: Calories 230; Total Fat 9g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 115mg; Total Carbohydrate 33g (Dietary Fiber 0g, Sugars 22g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\n\nCranberry-Orange Pound Cake Stir \u00bd cup coarsely chopped fresh cranberries or dried cranberries and 1 teaspoon grated orange peel into batter.\n\nLemon\u2013Poppy Seed Pound Cake Substitute 1 teaspoon lemon extract for the vanilla. Fold 1 tablespoon grated lemon peel and \u00bc cup poppy seed into batter. Drizzle with Lemon Glaze (page 581) if desired.\n\nOrange-Coconut Pound Cake Fold 1\u2153 cups coconut and 2 tablespoons grated orange peel into batter. Drizzle with Orange Glaze (page 581) if desired.\n\nPound Cake Cupcakes Heat oven to 350\u00b0F. Place paper baking cup in each of 30 regular-size muffin cups. Make batter as directed. Spoon \u00bc cup batter into each muffin cup. Bake 20 to 23 minutes or until toothpick inserted in center comes out clean. Cool 15 minutes; remove from pans to cooling racks. Sprinkle with powdered sugar or frost if desired.\n\nToasted Almond Pound Cake Substitute almond extract for the vanilla. Fold 1\u00bd cups slivered almonds, toasted (page 23), into batter. Drizzle with Chocolate Glaze (page 582) or Vanilla Glaze (page 581) if desired.\n\nRaspberry-Topped Pound Cake Make Raspberry Sauce (page 641). Spoon sauce over slices of pound cake. Top with Sweetened Whipped Cream (page 622).\n\n## Ways to Use Pound Cake\n\nTry these delicious ways to use pound cake:\n\n  1. 1. **Boston Cream Shooters:** For each shooter, layer pieces of pound cake, vanilla pudding and fudge ice cream topping in 3x2-inch shot glass. Top with whipped cream.\n  2. 2. **Snickerdoodle French Toast:** Make French Toast (page 106), except use slices of pound cake instead of bread. Sprinkle cooked French toast with cinnamon-sugar.\n  3. 3. **Glazed Pound Cake Doughnut Holes:** Cut 1-inch-thick slices of pound cake with 1-inch round cutter. Make Vanilla Glaze (page 581). Dip 2 or 3 doughnut holes at a time into glaze on slotted spatula; shake off excess. Place on cooling rack. Immediately sprinkle with desired sprinkles; let dry.\n  4. 4. **Dessert Kabobs:** Cut pound cake into 12 (1\u00bd-inch) cubes. Make Chocolate Ganache (page 582). Place 1 cup coconut in small bowl. Dip each cake cube in ganache on fork; shake off excess. Place in bowl with coconut; spoon over top and sides of cake cube. Place on cooling rack to set up. Thread 3 cake cubes on skewers alternately with whole small strawberries.\n\nPound Cake, Glazed Pound Cake Doughnut Holes\n\n# Heirloom Recipe and New Twist\n\nJust the right combination of sweet, spicy and tangy, our heirloom recipe conjures up visions of a can't-wait-to-sink-my-teeth-into-it cake showcased on grandmother's sideboard or in a bakery display. Our twist version combines spice cake with apples and then adds a lighter, fresh maple glaze. Either way, you can't go wrong and aren't likely to see leftovers!\n\nHeirloom\n\nSour Cream Spice Cake\n\n## Sour Cream Spice Cake\n\nPrep 30 min Total 2 hr 25 min \u25cf 16 servings\n\nCake\n\n  * 2\u00bc cups all-purpose flour\n  * 1\u00bd cups packed brown sugar\n  * 1 cup raisins, chopped\n  * 1 cup sour cream\n  * \u00bd cup chopped walnuts\n  * \u00bc cup butter, softened\n  * \u00bc cup shortening\n  * \u00bd cup water\n  * 2 teaspoons ground cinnamon\n  * 1\u00bc teaspoons baking soda\n  * 1 teaspoon baking powder\n  * \u00be teaspoon ground cloves\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground nutmeg\n  * 2 eggs\n\nBrowned Butter Cream Frosting\n\n  * \u2153 cup butter*\n  * 3 cups powdered sugar\n  * 1\u00bd teaspoons vanilla\n  * 1 to 2 tablespoons milk\n  * Walnut halves, broken, if desired\n\n1 Heat oven to 350\u00b0F. Grease bottom and sides of 13x9-inch pan or 2 (8- or 9-inch) round cake pans with shortening; lightly flour.\n\n2 In large bowl, beat all cake ingredients with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on high speed 3 minutes, scraping bowl occasionally. Pour batter into pan(s).\n\n3 Bake 13x9-inch pan 40 to 45 minutes, round pans 30 to 35 minutes, or until toothpick inserted in center comes out clean. Cool 13x9-inch cake in pan on cooling rack. Cool rounds 10 minutes; remove from pans to cooling racks. Cool completely, about 1 hour.\n\n4 Meanwhile, in 1-quart saucepan, heat \u2153 cup butter over medium heat just until light brown, stirring constantly. Watch carefully, because butter can brown and then burn quickly. Remove from heat; cool. In medium bowl, mix powdered sugar and browned butter until well blended. Add vanilla and milk; beat until frosting is smooth and spreadable. Frost top of 13x9-inch cake or fill and frost round layers with frosting. Garnish with broken walnut halves.\n\n* Do not use margarine or vegetable oil spreads in the frosting.\n\n1 Serving: Calories 420; Total Fat 16g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 50 mg; Sodium 290mg; Total Carbohydrate 65g (Dietary Fiber 1g, Sugars 48g); Protein 4g Exchanges: 1 Starch, 3\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 1 Serving:\nNew Twist\n\nSpiced Apple Cake with Maple Glaze\n\n## Spiced Apple Cake with Maple Glaze\n\nPrep 30 min Total 2 hr 35 min \u25cf 16 servings\n\nCake\n\n  * 3 cups chopped peeled apples (3 medium)\n  * 3 cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bd teaspoon baking soda\n  * \u00bd teaspoon salt\n  * 2 teaspoons ground cinnamon\n  * \u00bd teaspoon ground nutmeg\n  * \u00bc teaspoon ground cloves\n  * 2 cups packed brown sugar\n  * 1 cup vegetable oil\n  * 4 eggs\n  * 1 teaspoon vanilla\n\nGlaze\n\n  * 1\u00bd cups powdered sugar\n  * 3 to 4 tablespoons whipping cream\n  * \u00bc teaspoon maple flavor\n\n1 Heat oven to 350\u00b0F. Generously grease 12-cup fluted tube cake pan with shortening; lightly flour.\n\n2 In medium bowl, toss apples with 2 tablespoons of the flour. In another medium bowl, mix baking powder, baking soda, salt, cinnamon, nutmeg, cloves and remaining flour. In large bowl, beat brown sugar, oil, eggs and vanilla with electric mixer on medium speed until mixed. On low speed, beat in flour mixture just until blended. Stir in apples. Spoon batter into pan.\n\n3 Bake 55 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pan to cooling rack. Cool completely, about 1 hour.\n\n4 In medium bowl, mix glaze ingredients until smooth. Drizzle glaze over cake.\n\n1 Serving: Calories 400; Total Fat 16g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 50mg; Sodium 170mg; Total Carbohydrate 60g (Dietary Fiber 1g, Sugars 40g); Protein 4g Exchanges: 1\u00bd Starch, 2\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 4\n\nCaramel Snickerdoodle Cake\n\n## Caramel Snickerdoodle Cake\n\nPrep 20 min Total 2 hr 40 min \u25cf 16 servings\n\n  * 2 tablespoons sugar\n  * 2 teaspoons ground cinnamon\n  * 2\u00bd cups all-purpose flour\n  * 1\u00be cups sugar\n  * 2 teaspoons baking soda\n  * 1 teaspoon salt\n  * 1 can (5 oz) evaporated milk\n  * 1 cup sour cream\n  * \u00bd cup butter, melted\n  * 1 teaspoon vanilla\n  * 2 eggs, beaten\n  * 10 caramels, unwrapped\n\n1 Heat oven to 350\u00b0F. Grease 12-cup fluted tube cake pan with shortening. In small bowl, mix 2 tablespoons sugar and 1 teaspoon of the cinnamon. Sprinkle mixture over inside of pan, turning to evenly coat. Shake out any excess.\n\n2 In large bowl, mix flour, 1\u00be cups sugar, the baking soda, salt and remaining 1 teaspoon cinnamon. Reserve 1 tablespoon of the milk for topping. Add remaining milk, the sour cream, butter, vanilla and eggs to dry ingredients; stir until well blended. Pour batter into pan.\n\n3 Bake 40 to 50 minutes or until toothpick inserted in center comes out clean. Cool 30 minutes; remove from pan to cooling rack. Cool completely, about 1 hour.\n\n4 In small microwavable bowl, microwave caramels with reserved milk uncovered on High 1 to 2 minutes, stirring every 30 seconds, until caramels are melted and mixture is smooth. Drizzle over cake.\n\n1 Serving: Calories 340; Total Fat 11g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 50mg; Sodium 420mg; Total Carbohydrate 54g (Dietary Fiber 0g, Sugars 33g); Protein 4g Exchanges: 1\u00bd Starch, 2 Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\u00bd\n\nMolten Chocolate Cakes\n\n## Molten Chocolate Cakes\n\nMolten cakes or lava cakes, made popular by restaurants, are easy to make at home. The batter can be made a day ahead of time\u2014perfect for entertaining. Be sure to follow the recipe steps exactly and bake at the correct oven temperature. If the centers are too cake-like in texture, bake a few minutes less the next time; if they're too soft, bake a minute or two longer.\n\nPrep 20 min Total 40 min \u25cf 6 servings\n\n  * Unsweetened baking cocoa\n  * 6 oz semisweet baking chocolate, chopped\n  * \u00bd cup plus 2 tablespoons butter\n  * 3 whole eggs\n  * 3 egg yolks\n  * 1\u00bd cups powdered sugar\n  * \u00bd cup all-purpose flour\n  * Additional powdered sugar, if desired\n\n1 Heat oven to 450\u00b0F. Grease bottoms and sides of 6 (6-oz) custard cups with shortening; sprinkle with cocoa.\n\n2 In 2-quart saucepan, melt chocolate and butter over low heat, stirring frequently. Cool slightly.\n\n3 In large bowl, beat whole eggs and egg yolks with whisk or hand beater until well blended. Beat in 1\u00bd cups powdered sugar. Beat in melted chocolate mixture and flour. Divide batter evenly among custard cups. Place cups on cookie sheet with sides.\n\n4 Bake 12 to 14 minutes or until sides are set and centers are still soft (tops will be puffed and cracked). Let stand 3 minutes. Run small knife or metal spatula around side of each cake to loosen. Immediately place heatproof serving plate upside down over each cup; turn plate and cup over and remove cup. Sprinkle cakes with additional powdered sugar. Serve warm.\n\n1 Serving: Calories 550; Total Fat 33g (Saturated Fat 16g, Trans Fat 1g); Cholesterol 265mg; Sodium 170mg; Total Carbohydrate 56g (Dietary Fiber 2g, Sugars 44g); Protein 7g Exchanges: 2 Starch, 2 Other Carbohydrate, 6 Fat Carbohydrate Choices: 4\n\nMake-Ahead Directions After pouring batter into custard cups, cover tightly with plastic wrap and refrigerate up to 24 hours. Remove plastic wrap before baking. You may need to bake the cakes 1 to 2 minutes longer.\n\n### Learn with Betty Molten Chocolate Cake\n\n**Perfect Cake:** This cake has a flowing chocolate center. There are no problems.\n\n**Underbaked Cake:** This cake is too runny because it was not baked long enough.\n\n**Overbaked Cake:** This cake is completely cooked through because it was baked too long.\n\nEspresso Cake with Mocha Streusel Ribbon\n\n## Espresso Cake with Mocha Streusel Ribbon\n\nPrep 30 min Total 4 hr 30 min \u25cf 16 servings\n\nStreusel\n\n  * 2 tablespoons all-purpose flour\n  * 2 tablespoons packed brown sugar\n  * 1 tablespoon instant espresso coffee powder or granules\n  * 1 tablespoon cold butter\n  * \u00bd cup miniature semisweet chocolate chips\n  * \u00bc cup chopped pecans\n\nCake\n\n  * 3 cups all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bc teaspoon salt\n  * 1 cup milk\n  * \u00bc cup instant espresso coffee powder or granules\n  * 2\u00bd cups granulated sugar\n  * 1 cup butter, softened\n  * 1 teaspoon vanilla\n  * 5 eggs\n\nGlaze\n\n  * 3 tablespoons whipping cream\n  * \u2154 cup miniature semisweet chocolate chips\n  * 2 teaspoons instant espresso coffee powder or granules\n\n1 Heat oven to 325\u00b0F. Grease 12-cup fluted tube cake pan with shortening; lightly flour. In small bowl, mix 2 tablespoons flour, the brown sugar and 1 tablespoon espresso powder. Cut in 1 tablespoon butter, using pastry blender or fork, until crumbly. Stir in \u00bd cup chocolate chips and the pecans; set aside.\n\n2 In medium bowl, mix 3 cups flour, the baking powder and salt; set aside. In 1-cup glass measuring cup, stir milk and \u00bc cup espresso powder; set aside.\n\n3 In large bowl, beat granulated sugar, 1 cup butter, the vanilla and eggs with electric mixer on low speed 30 seconds. Beat on high speed 5 minutes, scraping bowl occasionally. Add flour mixture alternately with milk mixture, beating on low speed just until smooth after each addition. Spread 3 cups batter into pan. Sprinkle streusel over batter. Carefully spread remaining batter over streusel.\n\n4 Bake 1 hour 30 minutes to 1 hour 40 minutes or until toothpick inserted in center comes out clean and crack at the center top of cake no longer looks moist. Cool 20 minutes; remove from pan to cooling rack. Cool completely, about 2 hours.\n\n5 In small microwavable bowl, mix glaze ingredients. Microwave uncovered on High 30 seconds; stir until smooth. Spoon over cake, allowing some to drip down side of cake.\n\nCalories 450; Total Fat 20g (Saturated Fat 11g, Trans Fat 0.5g); Cholesterol 95mg; Sodium 210mg; Total Carbohydrate 61g (Dietary Fiber 2g, Sugars 41g); Protein 6g Exchanges: 2 Starch, 2 Other Carbohydrate, 4 Fat Carbohydrate Choices: 4\n\nItalian Cream Cake\n\n## Italian Cream Cake\n\nPrep 40 min Total 2 hr \u25cf 16 servings\n\nCake\n\n  * \u00bd cup butter, softened\n  * 2 cups granulated sugar\n  * 5 eggs\n  * 2 cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 cup buttermilk\n  * \u00bd cup vegetable oil\n  * 1 teaspoon vanilla\n  * 1 cup sweetened coconut\n\nFrosting and Garnish\n\n  * 12 oz cream cheese, softened\n  * \u00bc cup butter, softened\n  * 1 teaspoon vanilla\n  * 5 cups powdered sugar\n  * 1 to 2 tablespoons milk\n  * \u00bd cup chopped pecans\n  * 16 whole pecans\n\n1 Heat oven to 350\u00b0F. Grease bottoms and sides of 3 (8- or 9-inch) round cake pans with shortening. Line pan bottoms with waxed paper or cooking parchment paper; grease paper and lightly flour.\n\n2 In large bowl, beat \u00bd cup butter and the granulated sugar with electric mixer on low speed until creamy. Add eggs; beat on medium speed until creamy. Add flour and baking soda; beat on low speed until combined. Beat on medium speed until smooth. Add buttermilk, oil and 1 teaspoon vanilla; beat on low speed until combined. Beat on medium speed 2 minutes or until smooth. Fold in coconut. Pour batter into pans.\n\n3 Bake 20 to 25 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes; remove from pans to cooling racks. Remove waxed paper. Cool completely, about 45 minutes.\n\n4 In large bowl, beat cream cheese, \u00bc cup butter, 1 teaspoon vanilla, the powdered sugar and 1 tablespoon of the milk with electric mixer on low speed until blended. Gradually beat in enough remaining milk to make frosting smooth and spreadable. Reserve \u00be cup of the frosting for top of cake.\n\n5 On cake plate, place one cake layer, top side up. Spread with \u2154 cup frosting. Top with another layer, top side down; spread with \u2154 cup frosting. Top with remaining layer, top side up. Spread side and top of cake with remaining frosting. Press chopped pecans into side of cake. Arrange pecan halves on top of cake. Store in refrigerator.\n\n1 Serving: Calories 630; Total Fat 31g (Saturated Fat 13g, Trans Fat 0.5g); Cholesterol 105mg; Sodium 270mg; Total Carbohydrate 80g (Dietary Fiber 1g, Sugars 65g); Protein 6g Exchanges: 2 Starch, 3\u00bd Other Carbohydrate, 6 Fat Carbohydrate Choices: 5\n\nJelly Roll\n\n## Jelly Roll\n\nIf you tried to roll a regular layer cake, it would crack. This special kind of cake, called a sponge cake, is soft and flexible so it can be rolled up.\n\nPrep 30 min Total 1 hr 15 min \u25cf 10 servings\n\n  * 3 eggs\n  * 1 cup granulated sugar\n  * \u2153 cup water\n  * 1 teaspoon vanilla\n  * \u00be cup all-purpose flour\n  * 1 teaspoon baking powder\n  * \u00bc teaspoon salt\n  * Powdered sugar\n  * About \u2154 cup jelly or jam (any flavor)\n\n1 Heat oven to 375\u00b0F. Line 15x10x1-inch pan with waxed paper, foil or cooking parchment paper; generously grease paper or foil with shortening.\n\n2 In medium bowl, beat eggs with electric mixer on high speed about 5 minutes or until very thick and lemon colored. Gradually beat in granulated sugar. Beat in water and vanilla on low speed. Gradually add flour, baking powder and salt, beating on low speed just until smooth. Pour batter into pan, spreading to corners.\n\n3 Bake 12 to 15 minutes or until toothpick inserted in center comes out clean. Immediately loosen cake from sides of pan and turn upside down onto towel generously sprinkled with powdered sugar. Carefully remove paper. Trim off stiff edges of cake if necessary. While cake is hot, carefully roll cake and towel from narrow end. Cool on cooling rack at least 30 minutes.\n\n4 Unroll cake and remove towel. Beat jelly slightly with fork to soften; spread over cake. Roll up cake again; place seam side down on serving plate. Sprinkle with powdered sugar.\n\n1 Serving: Calories 200; Total Fat 1.5g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 65mg; Sodium 135mg; Total Carbohydrate 42g (Dietary Fiber 0g, Sugars 31g); Protein 3g Exchanges: 1 Starch, 2 Other Carbohydrate Carbohydrate Choices: 3\n\nChocolate Cake Roll Increase eggs to 4. Beat in \u00bc cup unsweetened baking cocoa with the flour. If desired, fill cake with ice cream instead of jelly or jam. Spread 1 to 1\u00bd pints (2 to 3 cups) slightly softened ice cream over cooled cake. Roll up cake; wrap in plastic wrap. Freeze about 4 hours or until firm. Serve slices drizzled with hot fudge sauce.\n\nLemon Curd Jelly Roll Make cake as directed, except add 2 teaspoons grated lemon peel with the flour. Omit jelly; spread cake with \u2154 cup Lemon Curd (page 602) or purchased lemon curd. Roll up as directed. Store covered in refrigerator.\n\n### Making Jelly Roll\n\nPour batter into 15x10x1-inch pan, spreading batter to the corners.\n\nWhile cake is hot, carefully roll up with towel from narrow end.\n\nUnroll cake, remove towel and spread jelly over the cake.\n\n## Learn to BEAT, WHIP & FOLD\n\nYou'll need to use different techniques in baking cakes to achieve different results:\n\nBEAT Stirring in a rapid, circular motion to incorporate air into the mixture to help lighten the texture of baked goods. Can be done with a spoon or rubber spatula or an electric mixer; 100 strokes by hand equals approximately 1 minute with a mixer. If you are beating ingredients by hand and they don't seem to be combining easily, try a whisk\u2014the wires of a whisk can break up ingredients quickly.\n\nWHIP Beating ingredients (such as egg whites or cream) to incorporate a lot of air; making them light and fluffy with loads of volume. Egg whites are beaten this way for foam cakes, supplying the structure instead of leavening ingredients. Cream of tartar is commonly added to egg whites while whipping to improve stability and increase volume.\n\nFOLD Gently blending a light, whipped mixture (such as egg whites) with a heavier mixture (such as remaining cake ingredients) so that the combined batter is also light and fluffy. When this technique is used for cake, the resulting texture will be light and fluffy. See Making Angel Food Cake (page 573).\n\nAngel Food Cake with Chocolate Glaze\n\n## Angel Food Cake Calorie smart\n\nWhat to do with the leftover egg yolks from this recipe? One idea would be to make some Lemon Curd (page 602).\n\nPrep 20 min Total 3 hr 25 min \u25cf 12 servings\n\n  * 1\u00bd cups egg whites (about 12)\n  * 1\u00bd cups sugar\n  * 1 cup cake flour\n  * 1\u00bd teaspoons cream of tartar\n  * 1\u00bd teaspoons vanilla\n  * \u00bd teaspoon almond extract\n  * \u00bc teaspoon salt\n  * Chocolate Glaze (recipe) or Vanilla Glaze (recipe), if desired\n\n1 Let egg whites stand at room temperature for 30 minutes. Room temperature egg whites will have more volume when beaten than cold egg whites. Move oven rack to lowest position. Heat oven to 375\u00b0F.\n\n2 In medium bowl, mix 3\/4 cup of the sugar and the flour; set aside. In large, clean, dry bowl, beat egg whites and cream of tartar with electric mixer on medium speed until foamy. Beat in remaining 3\/4 cup sugar, 2 tablespoons at a time, on high speed, adding vanilla, almond extract and salt with the last addition of sugar. Continue beating until stiff and glossy. Do not underbeat.\n\n3 Sprinkle flour mixture, \u00bc cup at a time, over egg white mixture, folding in with rubber spatula just until flour mixture disappears. Spoon and spread batter into ungreased 10-inch angel food (tube) cake pan. Cut gently through batter with metal spatula or knife to break air pockets.\n\n4 Bake 30 to 35 minutes or until cracks feel dry and top springs back when touched lightly. Immediately turn pan upside down onto heatproof bottle or funnel. Let hang about 2 hours or until cake is completely cool.\n\n5 Loosen side of cake with knife or long metal spatula; remove from pan. Drizzle glaze over top of cake, allowing some of glaze to drip down side.\n\n1 Serving: Calories 160; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 100mg; Total Carbohydrate 34g (Dietary Fiber 0g, Sugars 25g); Protein 4g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate Carbohydrate Choices: 2\n\nAngel Food Cupcakes Place paper baking cup in each of 24 regular-size muffin cups. Spoon \u2153 cup batter into each cup. Bake 17 to 23 minutes or until cracks in cupcakes feel dry and tops spring back when touched lightly. Cool 10 minutes; remove from pans to cooling racks. Cool completely. Glaze if desired or serve with whipped cream and fresh fruit.\n\nCherry Angel Food Cake Gently fold \u2153 cup chopped, very well drained maraschino cherries into batter in Step 3.\n\nChocolate-Cherry Angel Food Cake Stir 2 ounces semisweet baking chocolate, grated, into sugar and flour in Step 2. Continue as directed. Gently fold \u2153 cup chopped, very well drained maraschino cherries into batter in Step 3.\n\nChocolate Confetti Angel Food Cake Stir 2 ounces semisweet baking chocolate, grated, into sugar and flour in Step 2.\n\nEspresso Angel Food Cake Stir 2 tablespoons instant espresso coffee powder or granules into sugar and flour in Step 2.\n\nBeat batter on high until stiff and glossy.\n\nSprinkle flour mixture over egg-white mixture, folding in with spatula.\n\nCut gently through batter to break air pockets.\n\n# Cupcake and Frosting Basics\n\nFrosted cupcakes are sure to bring smiles, so choose from the Frosting recipes, starting on page 578, to top your creations. If you like heavily frosted, bakery-style cupcakes, consider doubling the recipe you choose.\n\n## Mini Cupcakes\n\nTo make mini cupcakes, place mini paper baking cup in each of 24 mini muffin cups. Make batter as directed in recipe. Fill each cup until about two-thirds full. (Cover and refrigerate remaining batter until ready to bake; cool pan before reusing.) Bake 12 to 16 minutes or until toothpick inserted in center comes out clean. Cool in pan 5 minutes. Remove cupcakes from pans; place on cooling racks to cool. Repeat with remaining batter to make an additional 48 mini cupcakes. Frost with desired frosting. Makes 72 mini cupcakes.\n\nChocolate, Yellow and White Cupcakes with Creamy Chocolate Frosting (page 579)\n\n## Chocolate Cupcakes calorie smart\n\nPrep 20 min Total 1 hr 15 min \u25cf 24 cupcakes\n\n  * 2 cups all-purpose flour\n  * 1\u00bc teaspoons baking soda\n  * 1 teaspoon salt\n  * \u00bc teaspoon baking powder\n  * 1 cup hot water\n  * \u2154 cup unsweetened baking cocoa\n  * \u00be cup shortening\n  * 1\u00bd cups sugar\n  * 2 eggs\n  * 1 teaspoon vanilla\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 24 regular-size muffin cups.\n\n2 In medium bowl, mix flour, baking soda, salt and baking powder; set aside. In small bowl, mix hot water and cocoa until dissolved; set aside.\n\n3 In large bowl, beat shortening with electric mixer on medium speed 30 seconds. Gradually add sugar, about \u00bc cup at a time, beating well after each addition and scraping bowl occasionally. Beat 2 minutes longer. Add eggs, one at a time, beating well after each addition. Beat in vanilla. On low speed, alternately add flour mixture, about one-third of mixture at a time, and cocoa mixture, about one-half at a time, beating just until blended.\n\n4 Divide batter evenly among muffin cups, filling each about two-thirds full.\n\n5 Bake 20 to 25 minutes or until toothpick inserted in center comes out clean. Cool in pans 5 minutes. Remove cupcakes from pans; place on cooling racks to cool. Frost with desired frosting.\n\n1 Cupcake: Calories 160; Total Fat 7g (Saturated Fat 2g, Trans Fat 1g); Cholesterol 20mg; Sodium 180mg; Total Carbohydrate 22g (Dietary Fiber 1g, Sugars 13g); Protein 2g Exchanges: \u00bd Starch, 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\u00bd\n\n## Yellow Cupcakes calorie smart\n\nPrep 15 min Total 1 hr 15 min \u25cf 24 cupcakes\n\n  * 2\u2153 cups all-purpose flour\n  * 2\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 1 cup butter or margarine, softened\n  * 1\u00bc cups sugar\n  * 3 eggs\n  * 1 teaspoon vanilla\n  * \u2154 cup milk\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 24 regular-size muffin cups. Grease and flour muffin cups, or spray with baking spray with flour.\n\n2 In medium bowl, mix flour, baking powder and salt; set aside.\n\n3 In large bowl, beat butter with electric mixer on medium speed 30 seconds. Gradually add sugar, about \u00bc cup at a time, beating well after each addition and scraping bowl occasionally. Beat 2 minutes longer. Add eggs, one at a time, beating well after each addition. Beat in vanilla. On low speed, alternately add flour mixture, about one-third of mixture at a time, and milk, about one-half at a time, beating just until blended.\n\n4 Divide batter evenly among muffin cups, filling each with about 3 tablespoons batter or until two-thirds full.\n\n5 Bake 20 to 25 minutes or until golden brown and toothpick inserted in center comes out clean. Cool in pans 5 minutes. Remove cupcakes from pans; place on cooling racks to cool. Frost with desired frosting.\n\n1 Cupcake: Calories 170; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 45mg; Sodium 190mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 11g); Protein 2g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\n## White Cupcakes calorie smart\n\nPrep 15 min Total 1 hr 15 min \u25cf 24 cupcakes\n\n  * 2\u00be cups all-purpose flour\n  * 3 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u00be cup shortening or butter\n  * 1\u2154 cups sugar\n  * 5 egg whites\n  * 2\u00bd teaspoons vanilla\n  * 1\u00bc cups milk\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 24 regular-size muffin cups.\n\n2 In medium bowl, mix flour, baking powder and salt; set aside.\n\n3 In large bowl, beat shortening with electric mixer on medium speed 30 seconds. Gradually add sugar, about \u2153 cup at a time, beating well after each addition and scraping bowl occasionally. Beat 2 minutes longer. Add egg whites, one at a time, beating well after each addition. Beat in vanilla. On low speed, alternately add flour mixture, about one-third of mixture at a time, and milk, about one-half at a time, beating just until blended.\n\n4 Divide batter evenly among muffin cups, filling each about two-thirds full.\n\n5 Bake 18 to 20 minutes or until toothpick inserted in center comes out clean. Cool in pans 5 minutes. Remove cupcakes from pans; place on cooling racks to cool. Frost with desired frosting.\n\n1 Cupcake: Calories 180; Total Fat 7g (Saturated Fat 2g, Trans Fat 1g); Cholesterol 0mg; Sodium 125mg; Total Carbohydrate 26g (Dietary Fiber 0g, Sugars 15g); Protein 2g Exchanges: \u00bd Starch, 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nMaple-Cornmeal Cupcakes with Maple-Butter Frosting\n\n## Maple-Cornmeal Cupcakes with Maple-Butter Frosting\n\nPrep 45 min Total 1 hr 45 min \u25cf 18 cupcakes\n\nCupcakes\n\n  * 1\u00bd cups all-purpose flour\n  * \u2153 cup yellow cornmeal\n  * 2 teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u00bd cup butter, softened\n  * 1 cup granulated sugar\n  * 1 teaspoon maple flavor\n  * 2 eggs\n  * \u00be cup milk\n\nFrosting\n\n  * 4 cups powdered sugar\n  * 2 tablespoons butter, softened\n  * 2 teaspoons maple flavor\n  * 3 to 4 tablespoons milk\n\nGarnish\n\n  * 18 maple leaf candies or pecan halves, if desired\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 18 regular-size muffin cups. In medium bowl, mix flour, cornmeal, baking powder and salt; set aside.\n\n2 In large bowl, beat \u00bd cup butter with electric mixer on medium speed 30 seconds. Gradually add granulated sugar, about \u00bc cup at a time, beating well after each addition. Beat 2 minutes longer. Beat in 1 teaspoon maple flavor and the eggs. Add flour mixture alternately with \u00be cup milk, beating on low speed just until blended. Divide batter evenly among muffin cups, filling each about two-thirds full.\n\n3 Bake 20 to 25 minutes or until golden brown and toothpick inserted in center comes out clean. Cool 5 minutes; remove from pans to cooling racks. Cool completely, about 30 minutes.\n\n4 In medium bowl, beat frosting ingredients, adding enough milk, until frosting is smooth and spreadable. Frost cupcakes. Garnish each with candy or pecan half.\n\n1 Cupcake: Calories 280; Total Fat 7g (Saturated Fat 4.5g, Trans Fat 0g); Cholesterol 40mg; Sodium 180mg; Total Carbohydrate 50g (Dietary Fiber 0g, Sugars 38g); Protein 2g Exchanges: 1\u00bd Starch, 2 Other Carbohydrate, 1 Fat Carbohydrate Choices: 3\n\n## Banana Split\u2013Chocolate Mug Cake\n\nPrep 10 min Total 15 min \u25cf 1 Serving:\n\n  * \u00bc cup all-purpose flour\n  * 2 tablespoons sugar\n  * 2 tablespoons unsweetened baking cocoa\n  * \u00bd teaspoon baking powder\n  * \u215b teaspoon salt\n  * \u2153 cup mashed banana (about 1 small)\n  * 3 tablespoons milk\n  * 1 tablespoon butter, melted\n  * \u2153 cup vanilla or favorite flavor ice cream\n  * 1 tablespoon hot fudge topping\n  * 1 fresh strawberry\n\n1 In small bowl, mix flour, sugar, cocoa, baking powder and salt with wooden spoon until well blended. Beat in banana, milk and butter with whisk until smooth. Pour into 10-ounce microwavable mug.\n\n2 Microwave uncovered on High 1 to 2 minutes or until set and cake rises (center will have a slightly wet appearance). Cool 5 minutes. Top with ice cream, hot fudge and strawberry.\n\n1 Serving: Calories 620; Total Fat 21g (Saturated Fat 13g, Trans Fat 0.5g); Cholesterol 55mg; Sodium 770mg; Total Carbohydrate 98g (Dietary Fiber 7g, Sugars 56g); Protein 10g Exchanges: \u00bd Starch, \u00bd Fruit, 4\u00bd Other Carbohydrate, 1 Milk, 2\u00bd Fat Carbohydrate Choices: 6\u00bd\n\n### Making Chocolate Mug Cake\n\nMix cake ingredients as directed in recipe; pour into 10-ounce microwavable mug.\n\nMicrowave on High 1 to 2 minutes or until set and cake rises (center will have a slightly wet appearance).\n\nLemon Meringue Cupcakes\n\n## Lemon Meringue Cupcakes\n\nPrep 1 hr Total 2 hr \u25cf 24 cupcakes\n\nCupcakes\n\n  * 2\u2153 cups all-purpose flour\n  * 2\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 1 cup butter, softened\n  * 1\u00bc cups sugar\n  * 3 whole eggs\n  * 2 tablespoons grated lemon peel\n  * 1 teaspoon vanilla\n  * \u2154 cup milk\n  * 1 jar (10 to 12 oz) lemon curd\n\nMeringue\n\n  * 4 egg whites\n  * \u00bc teaspoon cream of tartar\n  * 1\u00bd teaspoons vanilla\n  * \u2154 cup sugar\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 24 regular-size muffin cups; spray paper cups with baking spray with flour. In medium bowl, mix flour, baking powder and salt; set aside.\n\n2 In large bowl, beat butter with electric mixer on medium speed 30 seconds. Gradually add 1\u00bc cups sugar, about \u00bc cup at a time, beating well after each addition. Beat 2 minutes longer. Add eggs, one at a time, beating well after each addition. Beat in lemon peel and 1 teaspoon vanilla. Alternately add flour mixture and milk, beating on low speed just until blended. Divide batter evenly among muffin cups, filling each about two-thirds full.\n\n3 Bake 20 to 25 minutes or until toothpick inserted in center comes out clean. Cool 5 minutes; remove from pans to cooling racks. Cool completely, about 30 minutes.\n\n4 By slowly spinning end of round handle of wooden spoon back and forth, make deep, \u00be-inch-wide indentation in center of top of each cupcake, not quite to bottom. (Wiggle end of spoon in cupcake to make opening large enough.) Spoon lemon curd into small resealable food-storage plastic bag; seal bag. Cut 3\/8-inch tip off bottom corner of bag. Insert tip of bag into opening in each cupcake; squeeze bag to fill opening.\n\n5 Increase oven temperature to 450\u00b0F. In medium bowl, beat egg whites, cream of tartar and 1\u00bd teaspoons vanilla with electric mixer on high speed until soft peaks form. Gradually add \u2154 cup sugar, 1 tablespoon at a time, beating until stiff peaks form and mixture is glossy. Frost cupcakes with meringue; place on cookie sheet. Bake 2 to 3 minutes or until lightly browned.\n\n1 Cupcake: Calories 230; Total Fat 9g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 60mg; Sodium 210mg; Total Carbohydrate 34g (Dietary Fiber 0g, Sugars 24g); Protein 3g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nGinger and Peach Cupcakes\n\n## Ginger and Peach Cupcakes\n\nPrep 45 min Total 1 hr 45 min \u25cf 24 cupcakes\n\nCupcakes\n\n  * 2\u2153 cups all-purpose flour\n  * 2\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * \u00bc teaspoon ground ginger\n  * 1 cup butter, softened\n  * 1\u00bc cups granulated sugar\n  * 3 eggs\n  * 1 teaspoon vanilla\n  * 1 container (6 oz) peach low-fat yogurt\n  * 1 large peach, peeled, chopped (about 1 cup)*\n  * 1 tablespoon all-purpose flour\n\nFrosting\n\n  * 4\u00bd cups powdered sugar\n  * 1 container (6 oz) peach low-fat yogurt\n\nGarnish\n\n  * \u2153 cup Crystallized Ginger (page 447) or 1 jar (2 oz) crystallized ginger, chopped\n\n1 Heat oven to 350\u00b0F. Place paper baking cup in each of 24 regular-size muffin cups. In medium bowl, mix 2\u2153 cups flour, the baking powder, salt and ground ginger; set aside.\n\n2 In large bowl, beat butter with electric mixer on medium speed 30 seconds. Gradually add granulated sugar, about \u00bc cup at a time, beating well after each addition. Beat 2 minutes longer. Add eggs, one at a time, beating well after each addition. Beat in vanilla. Alternately add flour mixture and 1 container yogurt, beating on low speed just until blended.\n\n3 In small bowl, toss peach with 1 tablespoon flour; fold into batter. Divide batter evenly among muffin cups, filling each about two-thirds full.\n\n4 Bake 20 to 25 minutes or until golden brown and toothpick inserted in center comes out clean. Cool 5 minutes; remove from pans to cooling racks. Cool completely, about 30 minutes.\n\n5 In small bowl, mix powdered sugar and 1 container yogurt until smooth and spreadable. Frost cupcakes. Sprinkle each with about \u00bd teaspoon crystallized ginger.\n\n*Or substitute 1 cup frozen peach slices, thawed, well drained and chopped, for the fresh peach.\n\n1 Cupcake: Calories 280; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 45mg; Sodium 190mg; Total Carbohydrate 48g (Dietary Fiber 0g, Sugars 35g); Protein 2g Exchanges: 1\u00bd Starch, 1\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 3\n\n## Fudge Frosting\n\nPrep 10 min Total 55 min \u25cf 3\u00bd cups\n\n  * 2 cups granulated sugar\n  * 1 cup unsweetened baking cocoa\n  * 1 cup milk\n  * \u00bd cup butter, cut into pieces\n  * \u00bc cup light corn syrup\n  * \u00bc teaspoon salt\n  * 2 teaspoons vanilla\n  * 2\u00bd to 3 cups powdered sugar\n\n1 In 3-quart saucepan, mix granulated sugar and cocoa. Stir in milk, butter, corn syrup and salt. Heat to boiling, stirring frequently. Boil 3 minutes, stirring occasionally. Cool 45 minutes.\n\n2 Beat in vanilla and enough powdered sugar for spreading consistency.\n\n3 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake. Leftover frosting can be tightly covered and refrigerated up to 5 days or frozen up to 1 month. Let stand 30 minutes at room temperature to soften; stir before using.\n\nAbout \u00bc Cup: Calories 170; Total Fat 4.5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 10mg; Sodium 35mg; Total Carbohydrate 32g (Dietary Fiber 1g, Sugars 28g); Protein 1g Exchanges: 2 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\n## Creamy Vanilla Frosting FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1\u00be cups\n\n  * 3 cups powdered sugar\n  * \u2153 cup butter, softened\n  * 1\u00bd teaspoons vanilla\n  * 1 to 2 tablespoons milk\n\n1 In large bowl, mix powdered sugar and butter with spoon or electric mixer on low speed until blended. Stir in vanilla and 1 tablespoon of the milk.\n\n2 Gradually beat in just enough remaining milk to make frosting smooth and spreadable. If frosting is too thick, beat in more milk, a few drops at a time. If frosting becomes too thin, beat in a small amount of powdered sugar.\n\n3 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake.* Leftover frosting can be tightly covered and refrigerated up to 5 days or frozen up to 1 month. Let stand 30 minutes at room temperature to soften; stir before using.\n\n*To fill and frost 8- or 9-inch three-layer cake, use 4\u00bd cups powdered sugar, \u00bd cup butter, 2 teaspoons vanilla and about 3 tablespoons milk.\n\nAbout 2 Tablespoons: Calories 170; Total Fat 5g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 35mg; Total Carbohydrate 30g (Dietary Fiber 0g, Sugars 29g); Protein 0g Exchanges: 2 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\nBrowned Butter Frosting In 1-quart saucepan, heat \u2153 cup butter (do not use margarine or vegetable oil spreads) over medium heat just until light brown, stirring constantly. Watch carefully, because butter can brown and then burn quickly. Cool. Substitute browned butter for the softened butter.\n\nLemon Frosting Omit vanilla. Substitute lemon juice for the milk. Stir in 1 teaspoon grated lemon peel.\n\nMaple-Nut Frosting Omit vanilla. Substitute 1 to 2 tablespoons real maple or maple-flavored syrup for the milk. Stir in \u00bc cup finely chopped nuts.\n\nOrange Frosting Omit vanilla. Substitute orange juice for the milk. Stir in 1 teaspoon grated orange peel.\n\nPeanut Butter Frosting Substitute creamy peanut butter for the butter. Increase milk to about \u00bc cup, adding more if necessary, a few drops at a time.\n\n## Creamy Chocolate Frosting FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 2 cups\n\n  * \u2153 cup butter, softened\n  * 3 oz unsweetened baking chocolate, melted, cooled at least 5 minutes\n  * 3 cups powdered sugar\n  * 2 teaspoons vanilla\n  * 3 to 4 tablespoons milk\n\n1In large bowl, beat butter and chocolate with electric mixer on low speed until blended. Gradually beat in powdered sugar until blended.\n\n2 Gradually beat in vanilla and just enough milk to make frosting smooth and spreadable. If frosting is too thick, beat in more milk, a few drops at a time. If frosting becomes too thin, beat in a small amount of powdered sugar.\n\n3 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake.* Leftover frosting can be tightly covered and refrigerated up to 5 days or frozen up to 1 month. Let stand 30 minutes at room temperature to soften; stir before using.\n\n*To fill and frost 8- or 9-inch three-layer cake, use \u00bd cup butter, 4 ounces chocolate, 4\u00bd cups powdered sugar, 1 tablespoon vanilla and about \u00bc cup milk.\n\nAbout 3 Tablespoons: Calories 210; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 15mg; Sodium 35mg; Total Carbohydrate 32g (Dietary Fiber 1g, Sugars 29g); Protein 0g Exchanges: \u00bd Starch, 2 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\n\nCreamy Cocoa Frosting Substitute \u2153 cup unsweetened baking cocoa for the chocolate.\n\nMocha Frosting Add 2\u00bd teaspoons instant coffee powder or granules with the powdered sugar.\n\nWhite Chocolate Frosting Substitute 2 ounces white chocolate baking squares or bars, melted and cooled at least 5 minutes, for the chocolate. Do not use white vanilla baking chips, because they will add a grainy texture.\n\n## Clever Cupcake Toppers\n\n  * Candy sprinkles, decors or colored sugar\n  * Shredded coconut (toasted if desired, page 23)\n  * Chopped nuts\n  * Finely chopped candy\n  * Finely chopped crystallized ginger (page 447) \n  * Tiny chocolate candies or cookies\n  * Grated orange, lemon or lime peel\n\n## Cream Cheese Frosting FAst \u2022 EASY\n\nThis is the perfect frosting for carrot cake and spice or applesauce cakes. Be sure to refrigerate the frosted cake, since cream cheese is perishable.\n\nPrep 10 min Total 10 min \u25cf 2\u00bd cups\n\n  * 1 package (8 oz) cream cheese, softened\n  * \u00bc cup butter, softened\n  * 2 to 3 teaspoons milk\n  * 1 teaspoon vanilla\n  * 4 cups powdered sugar\n\n1 In large bowl, beat cream cheese, butter, milk and vanilla with electric mixer on low speed until smooth.\n\n2 Gradually beat in powdered sugar, 1 cup at a time, on low speed until frosting is smooth and spreadable.\n\n3 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake. Leftover frosting can be tightly covered and refrigerated up to 5 days or frozen up to 1 month. Let stand 30 minutes at room temperature to soften; stir before using.\n\nAbout 3 Tablespoons: Calories 260; Total Fat 10g (Saturated Fat 6g, Trans Fat 0g); Cholesterol 30mg; Sodium 80mg; Total Carbohydrate 40g (Dietary Fiber 0g, Sugars 39g); Protein 2g Exchanges: 1 Starch, 1\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 2\u00bd\n\nChocolate Cream Cheese Frosting Add 2 ounces unsweetened baking chocolate, melted and cooled 10 minutes, with the butter.\n\nCinnamon Cream Cheese Frosting Stir in \u00bd teaspoon ground cinnamon with the vanilla.\n\n## Caramel Frosting EASY\n\nFor a richer frosting, use whole milk or half-and-half.\n\nPrep 10 min Total 40 min \u25cf 2 cups\n\n  * \u00bd cup butter\n  * 1 cup packed brown sugar\n  * \u00bc cup milk\n  * 2 cups powdered sugar\n\n1 In 2-quart saucepan, melt butter over medium heat. Stir in brown sugar. Heat to boiling, stirring constantly; reduce heat to low. Boil and stir 2 minutes. Stir in milk. Heat to boiling; remove from heat. Cool to lukewarm, about 30 minutes.\n\n2 Gradually stir in powdered sugar. Place saucepan of frosting in bowl of cold water. Beat with spoon until frosting is smooth and spreadable. If frosting becomes too stiff, stir in additional milk, 1 teaspoon at a time, or heat over low heat, stirring constantly.\n\n3 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake. Leftover frosting can be tightly covered and refrigerated up to 5 days or frozen up to 1 month. Let stand 30 minutes at room temperature to soften; stir before using.\n\nAbout 3 Tablespoons: Calories 220; Total Fat 8g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 20mg; Sodium 60mg; Total Carbohydrate 38g (Dietary Fiber 0g, Sugars 37g); Protein 0g Exchanges: 2\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 2\u00bd\n\nSalty Caramel Frosting Sprinkle \u00bd teaspoon kosher (coarse) salt over frosted cupcakes or cake.\n\n### Learn with Betty Browned Butter\n\n**Perfectly browned butter:** This butter is golden brown.\n\n**Underbrowned butter:** This butter is not browned enough.\n\n**Overbrowned butter:** This butter is scorched and too brown.\n\nFluffy White Frosting\n\n## Fluffy White Frosting calorie smart\n\nThis frosting got its name because its large white peaks hold up long after it's beaten. It's sometimes called White Mountain Frosting.\n\nPrep 25 min Total 55 min \u25cf 3 cups\n\n  * 2 egg whites\n  * \u00bd cup sugar\n  * \u00bc cup light corn syrup\n  * 2 tablespoons water\n  * 1 teaspoon vanilla\n\n1 Let egg whites stand at room temperature for 30 minutes. Room temperature egg whites will have more volume when beaten than cold egg whites. In medium bowl, beat egg whites with electric mixer on high speed just until stiff peaks form.\n\n2 In 1-quart saucepan, stir sugar, corn syrup and water until well mixed. Cover and heat to rolling boil over medium heat. Uncover and boil 4 to 8 minutes, without stirring, to 242\u00b0F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a firm ball that holds its shape until pressed (see Testing Candy Temperatures, page 545). For an accurate temperature reading, tilt the saucepan slightly so mixture is deep enough for thermometer.\n\n3 Pour hot syrup very slowly in thin stream into egg whites, beating constantly on medium speed. Add vanilla. Beat on high speed about 10 minutes or until stiff peaks form.\n\n4 Frost 13x9-inch cake, or fill and frost 8- or 9-inch two-layer cake. Leftover frosting can be tightly covered and refrigerated up to 2 days; do not freeze. Let stand 30 minutes at room temperature to soften; do not stir.\n\n\u00bc Cup: Calories 60; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 15mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 11g); Protein 0g Exchanges: 1 Other Carbohydrate Carbohydrate Choices: 1\n\nButterscotch Frosting Substitute packed brown sugar for the sugar. Reduce vanilla to \u00bd teaspoon.\n\nCherry-Nut Frosting Stir in \u00bc cup chopped candied cherries, \u00bc cup chopped nuts and, if desired, 6 to 8 drops red food color.\n\nPeppermint Frosting Stir in \u2153 cup coarsely crushed hard peppermint candies or \u00bd teaspoon peppermint extract.\n\n## Vanilla Glaze FAst \u2022 EASY\n\nPrep 5 min Total 5 min \u25cf 1 cup\n\n  * \u2153 cup butter\n  * 2 cups powdered sugar\n  * 1\u00bd teaspoons vanilla\n  * 2 to 4 tablespoons hot water\n\n1 In 1\u00bd-quart saucepan, melt butter over low heat; remove from heat. Stir in powdered sugar and vanilla.\n\n2 Stir in hot water, 1 tablespoon at a time, until glaze is smooth and has the consistency of thick syrup.\n\n3 Glaze 12-cup fluted tube cake, 10-inch angel food or chiffon cake or top of 8- or 9-inch layer cake.\n\n4 Teaspoons: Calories 130; Total Fat 5g (Saturated Fat 3g, Trans Fat 0g); Cholesterol 15mg; Sodium 35mg; Total Carbohydrate 20g (Dietary Fiber 0g, Sugars 19g); Protein 0g Exchanges: 1\u00bd Other Carbohydrate, 1 Fat Carbohydrate Choices: 1\n\nBrowned Butter Glaze Brown the butter as directed in Browned Butter Frosting (page 579; see Learn with Betty on left page). Continue as directed.\n\nLemon Glaze Stir 1 teaspoon grated lemon peel into melted butter. Omit vanilla. Substitute lemon juice, heated, for the hot water.\n\nOrange Glaze Stir 1 teaspoon grated orange peel into melted butter. Omit vanilla. Substitute orange juice, heated, for the hot water.\n\n## Chocolate Glaze FAst \u2022 EASY\n\nThe corn syrup in this glaze not only adds sweetness but also gives it a glossy sheen.\n\nPrep 5 min Total 15 min \u25cf 1 cup\n\n  * 1 cup semisweet chocolate chips\n  * \u00bc cup butter\n  * \u00bc cup light corn syrup\n  * 2 to 4 teaspoons hot water\n\n1 In 1-quart saucepan, heat chocolate chips, butter and corn syrup over low heat, stirring frequently, until chocolate chips are melted. Cool about 10 minutes.\n\n2 Stir in hot water, 1 teaspoon at a time, until glaze is smooth and has the consistency of thick syrup.\n\n3 Glaze 12-cup fluted tube cake, 10-inch angel food or chiffon cake or top of 8- or 9-inch layer cake.\n\n4 Teaspoons: Calories 130; Total Fat 8g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 10mg; Sodium 40mg; Total Carbohydrate 14g (Dietary Fiber 0g, Sugars 10g); Protein 0g Exchanges: 1 Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\n\nDark Chocolate Glaze Substitute dark chocolate chips for the semisweet chips.\n\nMilk Chocolate Glaze Substitute milk chocolate chips for the semisweet chips.\n\nMint Chocolate Glaze  Substitute mint-flavored chocolate chips for the semisweet chips.\n\nWhite Chocolate Glaze Substitute white vanilla baking chips for the semisweet chocolate chips.\n\n### Making and Drizzling Glaze\n\nGlaze should be consistency of thick syrup.\n\nWith spoon, drizzle glaze over top of cake.\n\n## Chocolate Ganache FAst \u2022 EASY\n\nGanache is a very rich chocolate glaze. If you glaze the cake on a cooling rack with waxed paper underneath the rack, the extra glaze will fall onto the paper. When the ganache hardens, just slide the cake\u2014easily and neatly\u2014onto your serving plate.\n\nPrep 5 min Total 10 min \u25cf 1\u00bc cups\n\n  * \u2154 cup whipping cream\n  * 6 oz semisweet baking chocolate, chopped\n\n1 In 1-quart saucepan, heat whipping cream over low heat until hot but not boiling; remove from heat.\n\n2 Stir in chocolate until melted. Let stand about 5 minutes. Ganache is ready to use when it mounds slightly when dropped from a spoon. It will become firmer the longer it cools.\n\n3 Glaze 13x9-inch cake or top and side of 8- or 9-inch two-layer cake. Pour ganache carefully onto top center of cake; spread with large spatula so it flows evenly over top and down to cover side of cake. Leftover glaze can be tightly covered and refrigerated up to 5 days; do not freeze. Let stand 30 minutes at room temperature to soften; stir before using.\n\nAbout 2 Tablespoons: Calories 120; Total Fat 8g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 15mg; Sodium 5mg; Total Carbohydrate 9g (Dietary Fiber 0g, Sugars 8g); Protein 0g Exchanges: \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: \u00bd\n\nBittersweet Chocolate Ganache Substitute bittersweet chocolate for the semisweet chocolate.\n\n### Using Ganache\n\nGanache is ready to use when it is fairly thick and mounds slightly when dropped from a spoon.\n\nPour ganache carefully onto center of cake; spread to cover top and down side of cake.\n\n# Chapter 21: Pies & Tarts\n\nBlueberry Pie\n\n## Pie Basics\n\nBaking Methods for Pie Crust\n\nFreezing Filled Pies\n\nFreezing Pie Pastry\n\nPicking Pie Pans\n\n## Features\n\nHeirloom Recipe and New Twist\n\nLearn to Make: Lattice Crust\n\nStone Fruit\n\nPastry Edge Treatments\n\n## Pastry and Crusts\n\nBaked One-Crust Pie\n\nBaked Tart Shells and Individual Pie Crusts\n\nButter Pastry\n\nCookie Crumb Crust\n\nEasy Buttermilk Pastry Fast \u2022 Easy\n\nEasy Nut Crust Fast \u2022 Easy\n\nGraham Cracker Crust Fast \u2022 Easy\n\nPartially Baked One-Crust Pie\n\nPie Pastry\n\nPress-in-the-Pan Oil Pastry Fast \u2022 Easy\n\nPress-in-the-Pan Tart Pastry Fast \u2022 Easy\n\n## Fruit Pies\n\nApple Slab Pie CALORIE SMART\n\nBlackberry, Boysenberry, Loganberry or Raspberry Pie\n\nBlueberry Hand Pies\n\nBlueberry Pie\n\nBlueberry-Peach Pie\n\nCherry Pie\n\nClassic Apple Pie\n\nClassic Peach Pie\n\nElegant Pear Custard Pie\n\nFrench Apple Pie\n\nGinger-Lemon Curd Petite Pies CALORIE SMART\n\nNectarine-Plum Crostata\n\nQuick Cherry Pie\n\nQuick Rhubarb Pie\n\nRaspberry-Almond Pie\n\nRhubarb Pie\n\nStrawberry-Rhubarb Pie\n\n## Cream, Custard and Pecan Pies\n\nBanana Cream Pie\n\nBourbon-Chocolate-Pecan Mini Pies\n\nButterscotch Cream Pie\n\nChocolate Cream Pie\n\nChocolate-Banana Cream Pie\n\nClassic French Silk Pie\n\nCoconut Cream Pie\n\nKentucky Pecan Pie\n\nKey Lime Pie\n\nLemon Meringue Pie\n\nMeringue for 9-Inch Pie\n\nMocha French Silk Pie\n\nPecan Pie\n\nPraline Pumpkin Pie\n\nPumpkin Pie\n\nRoasted Sweet Potato Meringue Pie\n\nRoasted Sweet Potato Pie\n\n## Tarts\n\nEasy Apple Tart\n\nLemon Mascarpone Tart\n\nMixed Berry Crumble Tart\n\nSalted Cashew\u2013Bittersweet Chocolate Tart\n\nStrawberries and Cream Tart\n\nTiny Caramel Tarts\n\nWhite and Dark Chocolate Raspberry Tart\n\n## Bonus Recipes\n\nBerry-Plum Pops\n\nCherry BBQ Sauce\n\nEasy Bellini Cocktails\n\nGrilled Pluot Dessert\n\nPeachy Lemonade\n\nSour Cream\u2013Nectarine Pancakes\n\nStrawberry-Peach Salsa\n\nSummery Salad\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Pie Basics\n\nWhat makes a pie stand out from the crowd? It's the crust. When you bite into a really flaky, tender crust, you immediately know that the pie is special. When you know how to work with it, it's not hard to turn out a beautiful pie. We'll show you all the tricks and tips you need to become a legendary pie maker.\n\n## PICKING PIE PANS\n\nThe best pans to use for making pies are actually heat-resistant glass pie plates. The next best alternative would be a dull aluminum pie pan. Shiny or disposable pie pans reflect the heat and prevent crusts from browning. Dark pans absorb heat, causing overbrowning. Nonstick pans can cause a filled crust that isn't well anchored to the edge of the pan or an unfilled pie crust to shrink excessively. Use the size pie plate or pan called for in the recipe. Because pastry is high in fat, there is no need to grease the pie plate.\n\n## FREEZING PIE PASTRY\n\nYou can make pastry ahead and freeze it to have on hand for when you are ready to make a pie. Better yet, freeze the pastry in a freezer-to-oven glass pie plate so you are even closer to homemade pie. Unbaked and baked pie pastry (without filling) can be frozen up to 2 months.\n\n  * For rounds of pastry dough (see Pie Pastry, page 586, for how to make pastry rounds), wrap tightly in plastic wrap and freeze. Thaw in refrigerator before rolling and filling.\n  * For unbaked pastry crust in a pie plate, wrap tightly in foil or place in a freezer plastic bag. Fill crust while frozen and bake so that the crust won't be soggy.\n  * For baked pastry crust in a pie plate, wrap tightly (when completely cooled) in foil or place in a freezer plastic bag. Thaw before using.\n\n## FREEZING FILLED PIES\n\nSee and Refrigerator and Freezer Food Storage Chart, page 33, for freezing pies.\n\n  * Completely cool baked pies before freezing.\n  * Freeze pies with room around them so that the pastry edge won't get broken or misshapen during freezing. Once they are frozen, other items can touch up against the pies.\n  * Do not freeze cream, custard and meringue-topped pies. The filling and the meringue will break down and become watery.\n  * Fruit pies can be frozen baked or unbaked. Pecan and pumpkin pies need to be baked before freezing.\n\n## Tips for Perfect Pie Pastry\n\n  * Using a pastry blender is the fastest way to get the pastry ingredients to pea-size crumbs. If you don't have one, using a potato masher is a great alternative. Finish mixing with a fork if the crumbs are larger than pea size. Alternately, a fork can be used for all the mixing, but it will take longer to do.\n  * Overworking the pastry dough will make it tough\u2014handle it as little as possible.\n  * Another easy method for rolling the pastry is to do so between two sheets of waxed paper (anchor the bottom sheet to the counter by sprinkling counter with a few drops of water first). Flip pastry with both sheets of waxed paper over; carefully peel off the top piece of waxed paper. Flip pastry so remaining piece of waxed paper is on top; center over the pie plate and set on pie plate. Gently peel off waxed paper and ease pastry into plate.\n\nBaked One-Crust Pie (page 587)\n\n## Pie Pastry\n\nThis pastry is perfect for making any pie or tart. Follow individual recipes for which amount of pastry to make. Make your pies extra special; see Pastry Edge Treatments (page 587) and Learn to Make a Lattice Crust (page 596).\n\nPrep 20 min Total 1 hr 5 min \u25cf 8 servings\n\nOne-Crust Pastry\n\n  * 1 cup plus 1 tablespoon all-purpose flour\n  * \u00bd teaspoon salt\n  * \u2153 cup cold shortening\n  * 3 to 5 tablespoons ice-cold water\n\nTwo-Crust Pastry\n\n  * 2 cups plus 2 tablespoons all-purpose flour\n  * 1 teaspoon salt\n  * \u2154 cup cold shortening\n  * 6 to 8 tablespoons ice-cold water\n\n1 In medium bowl, mix flour and salt. Cut in shortening, using pastry blender or fork, until mixture forms coarse crumbs the size of small peas. Sprinkle with water, 1 tablespoon at a time, tossing with fork until all flour is moistened and pastry almost leaves side of bowl (1 to 2 teaspoons more water can be added if necessary).\n\n2 Gather pastry into a ball. For one-crust pastry, shape into flattened round on lightly floured surface. For two-crust pastry, divide pastry in half and shape into 2 rounds on lightly floured surface. Wrap flattened round or rounds in plastic wrap and refrigerate 45 minutes or until dough is firm and cold, yet pliable. This allows the shortening to become slightly firm, which helps make the baked pastry flaky. If refrigerated longer, let pastry soften slightly at room temperature before rolling.\n\n3 Using floured rolling pin, roll one round of pastry on lightly floured surface (or pastry board with floured pastry cloth) into round 2 inches larger than upside-down 9-inch glass pie plate or 3 inches larger than 10- or 11-inch tart pan. Fold pastry into fourths and place in pie plate or tart pan, or roll pastry loosely around rolling pin and transfer to pie plate or tart pan. Unfold or unroll pastry and ease into plate or pan, pressing firmly against bottom and side and being careful not to stretch pastry, which will cause it to shrink when baked.\n\n4 For one-crust pie, trim overhanging edge of pastry 1 inch from rim of pie plate. Fold edge under to form standing rim; flute edges (see Pastry Edge Treatments, page 587). For tart, trim overhanging edge of pastry even with top of tart pan. Fill and bake as directed in pie or tart recipe.\n\n5 For two-crust pie, roll out second pastry round. Fold into fourths and place over filling, or roll loosely around rolling pin and place over filling. Unfold or unroll pastry over filling. Cut slits in pastry so steam can escape. Trim overhanging edge of top pastry 1 inch from rim of plate. Fold edge of top crust under bottom crust, forming a stand-up rim of pastry that is even thickness on edge of pie plate, pressing on rim to seal; flute edges (see Pastry Edge Treatments, page 587). Bake as directed in pie recipe.\n\n1 Serving One-Crust Pastry: Calories 140; Total Fat 9g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 0g); Protein 1g Exchanges: 1 Starch, 1\u00bd Fat Carbohydrate Choices: 1\n\n1 Serving Two-Crust Pastry: Calories 290; Total Fat 20g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 25g (Dietary Fiber 1g, Sugars 0g); Protein 3g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 4 Fat Carbohydrate Choices: 1\u00bd\n\n### Making Pastry\n\nCut in shortening, using a pastry blender, until mixture forms coarse, pea-sized crumbs.\n\nAlternatively, use a fork to cut in the shortening.\n\nRoll pastry loosely around rolling pin and transfer to pie plate or tart pan.\n\nEase pastry into pie plate, pressing firmly against bottom and side. Trim overhanging edge 1 inch from rim of pie plate.\n\n## Baking Methods for Pie Crust\n\nFor many pies, such as apple or pecan pie, the filling is added to the unbaked pastry so that the crust and filling bake together at the same time. For other pies, such as pumpkin or banana cream pie, the crust is either partially baked or completely baked before the filling is added. This method is used to prevent the crust from becoming soggy from liquid fillings or when the filling doesn't need to be baked.\n\n## Partially Baked One-Crust Pie\n\nUse this when indicated in recipe to prevent bottom crust from becoming soggy.\n\n1 Heat oven to 425\u00b0F. Prepare One-Crust Pastry (page 586). Carefully line pastry with a double thickness of foil, gently pressing foil to bottom of and side of pastry. Let foil extend over edge to prevent excessive browning.\n\n2 Bake 10 minutes; carefully remove foil and bake 2 to 4 minutes longer or until pasty just begins to brown and has become set. If crust bubbles, gently push bubbles down with back of spoon.\n\n3 Fill and bake as directed in pie or tart recipe, changing oven temperature if necessary.\n\n### Partially Baked One-Crust Pie\n\nCarefully line pastry with foil, gently pressing foil to bottom and side of pastry and bake in oven as directed.\n\nFill partially baked pastry; bake pie as directed.\n\n## Baked One-Crust Pie\n\nUse this for one-crust pies and tarts baked completely before filling is added, such as banana cream or lemon meringue pie.\n\n1 Heat oven to 475\u00b0F. Prepare One-Crust Pastry (page 586), except for pie, trim overhanging edge of pastry 1 inch from rim of pie plate. For tart, trim overhanging edge of pastry even with top of tart pan. Prick bottom and side of pastry thoroughly with fork to prevent puffing.\n\n2 Bake 8 to 10 minutes or until light brown; cool on cooling rack.\n\n3 Fill cooled crust as directed in pie or tart recipe.\n\n## Decorative Cutouts\n\nFor an extra-pretty pie, make decorative cutouts on the top of a two-crust pie. Use a small cookie cutter or paring knife and cut shapes from the top crust before placing it on the filling. Once top crust is on filling, attach cutouts on top of crust with a little cold water. Sprinkle with coarse sugar.\n\n## Pastry Edge Treatments\n\nIt's important that the bottom and top crusts are well sealed to prevent the filling from escaping through it while baking. Seal edge as directed for Two-Crust Pastry (page 586). Then make the edge attractive by using one of these treatments:\n\nScalloped Edge\n\nPlace thumb and index finger about 1 inch apart on outside of raised edge. With other index finger, push pastry toward outside, continuing around edge, repeating motion, to form scalloped edge.\n\nRope or Pinched Edge\n\nPlace side of thumb on pastry rim at angle. Pinch pastry by pressing knuckle of index finger down into pastry toward thumb. Continue around edge, repeating motion, to form rope edge.\n\nFork or Herringbone Edge\n\nDip fork tines in flour; press fork diagonally onto edge without pressing through pastry. Rotate tines 90 degrees and press again next to first set of marks. Continue around edge of pastry, rotating tines back and forth.\n\n## Baked Tart Shells and Individual Pie Crusts\n\n1 Prepare One-Crust Pastry (page 586), except roll pastry into 13-inch round. Cut into 8 (4\u00bd-inch) rounds, rerolling pastry scraps if necessary.\n\n2 Heat oven to 475\u00b0F. Fit rounds over backs of regular-size muffin cups or 6-oz custard cups, making pleats so pastry will fit snugly against the cups. (If using individual pie pans or tart pans, cut pastry rounds 1 inch larger than the pans when upside down; then flip pans right side up and fit into pans. Prick pastry thoroughly with fork to prevent puffing. Place on cookie sheet.\n\n3 Bake 8 to 10 minutes or until light brown; cool before removing from cups. Fill each shell with \u2153 to \u00bd cup of your favorite filling, pudding, fresh fruit or ice cream.\n\n### Baked One-Crust Pie\n\nBake pastry, which has been pricked with a fork, until light brown; cool on cooling rack.\n\nBanana Cream Pie (page 605) with Easy Buttermilk Pastry\n\n## Easy Buttermilk Pastry FAST \u2022 EASY\n\nThis pastry is a dream to work with because it's extra easy to roll and handle, and the baked crust is very flaky.\n\nPrep 15 min Total 15 min \u25cf 2 (9-inch) crusts (8 servings each)\n\n  * 2 cups all-purpose flour\n  * 1 teaspoon salt\n  * \u2154 cup cold shortening\n  * 3 tablespoons cold butter\n  * \u2153 cup buttermilk\n  * 2 teaspoons vegetable oil\n\n1 In medium bowl, mix flour and salt. Cut in shortening and butter, using pastry blender or fork, until mixture forms coarse crumbs the size of small peas.\n\n2 Mix in buttermilk and oil with fork until all flour is moistened and pastry leaves side of bowl. Divide in half; shape each half into a ball. If making one-crust pie, wrap second ball of pastry and freeze for later use.\n\n3 Roll pastry as directed in Step 3 of One-Crust Pastry (page 586). Fill and bake as directed in pie recipe. Or, to bake before filling is added, heat oven to 450\u00b0F. Prick bottom and side of pastry thoroughly with fork. Bake 8 to 10 minutes or until light brown; cool on cooling rack.\n\n1 Serving: Calories 160; Total Fat 11g (Saturated Fat 3.5g, Trans Fat 1.5g); Cholesterol 5mg; Sodium 170mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch, 2 Fat Carbohydrate Choices: 1\n\n## Press-in-the-Pan Oil Pastry FAST \u2022 EASY\n\nNo rolling is needed for this crust! Use it for pies that have only a bottom crust. Try it for Pumpkin Pie (page 604), as well as for pie crusts that are baked before being filled.\n\nPrep 10 min Total 10 min \u25cf 1 (9-inch) crust (8 servings)\n\n  * 1\u2153 cups all-purpose flour\n  * \u00bd teaspoon salt\n  * \u2153 cup vegetable oil\n  * 2 tablespoons ice-cold water\n\n1 In medium bowl, stir flour, salt and oil until all flour is moistened. Sprinkle with cold water, 1 tablespoon at a time, tossing with fork until all water is absorbed. Gather pastry into a ball. Press firmly and evenly against bottom and up side of 9-inch glass pie plate; flute (see Pastry Edge Treatments, page 587).\n\n2 Fill and bake as directed in pie recipe. Or, to bake before filling is added, heat oven to 450\u00b0F. Prick bottom and side of pastry thoroughly with fork. Bake 10 to 12 minutes or until light brown; cool on cooling rack.\n\n1 Serving: Calories 150; Total Fat 9g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 150mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch, 1\u00bd Fat Carbohydrate Choices: 1\n\n## Press-in-the-Pan Tart Pastry FAST \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1 (10- or 11-inch) crust (8 servings)\n\n  * 1\u00bc cups all-purpose flour\n  * \u00bd cup butter, softened\n  * 2 tablespoons packed brown sugar\n  * 1 egg\n\n1 In medium bowl, stir all ingredients until soft dough forms. Using lightly floured fingers, press firmly and evenly against bottom and side of ungreased 9-, 10- or 11-inch tart pan with removable bottom.\n\n2 Fill and bake as directed in tart recipe. Or, to bake before filling is added, heat oven to 450\u00b0F. Bake 8 to 10 minutes or until light brown; cool on cooling rack.\n\n1 Serving: Calories 200; Total Fat 12g (Saturated Fat 6g, Trans Fat 0.5g); Cholesterol 55mg; Sodium 85mg; Total Carbohydrate 18g (Dietary Fiber 0g, Sugars 3g); Protein 3g Exchanges: 1 Starch, 2\u00bd Fat Carbohydrate Choices: 1\n\n## Easy Nut Crust FAST \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1 (9-inch) crust (8 servings)\n\n  * 1 cup all-purpose flour\n  * \u00bd cup butter, softened\n  * \u00bc cup finely chopped nuts\n\n1 In medium bowl, mix all ingredients until soft dough forms. Using lightly floured fingers, press firmly and evenly against bottom and side of 9-inch glass pie plate.\n\n2 Fill and bake as directed in pie recipe. Or, to bake before filling is added, heat oven to 450\u00b0F. Bake 7 to 8 minutes or until light brown; cool on cooling rack.\n\n1 Serving: Calories 190; Total Fat 14g (Saturated Fat 8g, Trans Fat 0g); Cholesterol 30mg; Sodium 80mg; Total Carbohydrate 12g (Dietary Fiber 0g, Sugars 0g); Protein 2g Exchanges: 1 Starch, 2\u00bd Fat Carbohydrate Choices: 1\n\nKey Lime Pie (recipe) with Graham Cracker Crust\n\n## Graham Cracker Crust FAST \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1 (9-inch) crust (8 servings)\n\n  * 1\u00bd cups finely crushed regular or cinnamon graham crackers (24 squares)\n  * \u2153 cup butter, melted\n  * 3 tablespoons sugar\n\n1 In medium bowl, stir all ingredients until well mixed. Reserve 3 tablespoons crumb mixture for garnishing top of pie before serving, if desired. Press remaining mixture firmly and evenly against bottom and side of 9-inch glass pie plate.\n\n2 Fill and bake as directed in pie recipe. Or, to bake before filling is added, heat oven to 350\u00b0F. Bake about 10 minutes or until light brown; cool on cooling rack.\n\n1 Serving: Calories 150; Total Fat 9g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 20mg; Sodium 140mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 9g); Protein 1g Exchanges: 1 Other Carbohydrate, 2 Fat Carbohydrate Choices: 1\n\nCookie Crumb Crust Substitute 1\u00bd cups finely crushed chocolate wafer cookies (24 cookies), vanilla wafer cookies (35 cookies) or gingersnaps (30 cookies) for the graham crackers. Reduce butter to \u00bc cup; omit sugar.\n\n### Making Graham Cracker Crust\n\nMix crumbs until butter is combined and mixture is crumbly.\n\nPress crumb mixture firmly against side and bottom of pie plate.\n\n# Heirloom Recipe and New Twist\n\nWhat makes homemade apple pie so special? You can use local apples for the freshest, juiciest pie\u2014mix two or three varieties for best taste and texture. Sweetly wrapped in a tender, flaky crust, it's drool-worthy. Our twist Apple Slab Pie makes it easy to serve pie to a crowd. This one-crust version gets topped with a powdered sugar glaze . . . mmm!\n\nHeirloom\n\nClassic Apple Pie\n\n## Classic Apple Pie\n\nPrep 30 min Total 3 hr 20 min \u25cf 8 servings\n\n  * Two-Crust Pastry (page 586)\n  * \u00bd cup sugar\n  * \u00bc cup all-purpose flour\n  * \u00be teaspoon ground cinnamon\n  * \u00bc teaspoon ground nutmeg\n  * Dash salt\n  * 6 cups thinly sliced (\u215b inch thick) peeled tart apples (6 medium)\n  * 2 tablespoons cold butter, if desired\n  * 2 teaspoons water\n  * 1 tablespoon sugar\n\n1 Heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate.\n\n2 In large bowl, mix \u00bd cup sugar, the flour, cinnamon, nutmeg and salt. Stir in apples. Spoon into pastry-lined pie plate. Cut butter into small pieces; sprinkle over apples. Cover with top pastry; cut slits in pastry. Seal and flute (see Pastry Edge Treatments, page 587).\n\n3 Brush top crust with 2 teaspoons water; sprinkle with 1 tablespoon sugar. Cover edge with 2- to 3-inch-wide strip of foil to prevent excessive browning; remove foil during last 15 minutes of baking.\n\n4 Bake 40 to 50 minutes or until crust is golden brown and juice begins to bubble through slits in crust. Cool on cooling rack at least 2 hours.\n\n1 Serving: Calories 420; Total Fat 21g (Saturated Fat 7g, Trans Fat 2g); Cholesterol 10mg; Sodium 330mg; Total Carbohydrate 53g (Dietary Fiber 3g, Sugars 23g); Protein 4g Exchanges: 1\u00bd Starch, 2 Fruit, 4 Fat Carbohydrate Choices: 3\u00bd\n\nFrench Apple Pie Heat oven to 400\u00b0F. Make One-Crust Pastry (page 586). Spoon apple mixture into pastry-lined pie plate. Omit butter, 2 teaspoons water and 1 tablespoon sugar. In small bowl, mix 1 cup all-purpose flour and \u00bd cup packed brown sugar. Cut in \u00bd cup cold butter with fork until crumbly. Sprinkle over apple mixture. Bake 35 to 40 minutes or until golden brown. Cover top with foil during last 10 to 15 minutes of baking, if necessary, to prevent excessive browning. Serve warm.\n\nClassic Peach Pie Prepare Classic Apple Pie as directed, except use \u2154 cup sugar and \u2153 cup flour. Decrease cinnamon to \u00bc teaspoon and omit nutmeg. Substitute sliced peeled fresh peaches (6 to 8) for the sliced apples. Stir in 1 teaspoon lemon juice with the peaches. Decrease butter to 1 tablespoon. Bake pie about 45 minutes.\n\n### Making Classic Apple Pie\n\nSpoon apple mixture into pastry. Cut butter into small pieces; sprinkle over apples.\n\nCover pastry edge with pie crust shield to prevent excess browning.\n\nOr cover pastry edge with 2- to 3-inch-wide strip of foil to prevent excess browning.\n\nCool baked pie on cooling rack at least 2 hours.\nNew Twist\n\nApple Slab Pie\n\n## Apple Slab Pie Calorie Smart\n\nPrep 30 min Total 2 hr 25 min \u25cf 24 servings\n\n  * Two-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * 1 cup granulated sugar\n  * 3 tablespoons all-purpose flour\n  * 1 teaspoon ground cinnamon\n  * \u00bc teaspoon ground nutmeg\n  * \u00bc teaspoon salt\n  * 4\u00bd teaspoons lemon juice\n  * 9 cups thinly sliced (\u215b inch thick) peeled apples (9 medium)\n  * 1 cup powdered sugar\n  * 2 tablespoons milk\n\n1 Heat oven to 450\u00b0F. Stack pastry rounds on top of each other on lightly floured surface. Roll to 17x12-inch rectangle. Fit crust into 15x10x1-inch pan, pressing into corners. Fold extra pastry crust under, even with edges of pan; seal edges.\n\n2 In large bowl, mix granulated sugar, flour, cinnamon, nutmeg, salt and lemon juice. Add apples; toss to coat. Spoon into pastry-lined pan.\n\n3 Bake 33 to 38 minutes or until crust is golden brown and filling is bubbly. Cool on cooling rack 45 minutes.\n\n4 In small bowl, mix powdered sugar and milk until well blended. Drizzle over pie. Allow glaze to set before serving, about 30 minutes.\n\n1 Serving: Calories 150; Total Fat 4g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 110mg; Total Carbohydrate 28g (Dietary Fiber 0g, Sugars 17g); Protein 0g Exchanges: 2 Other Carbohydrate, 1 Fat Carbohydrate Choices: 2\n\n### Making Apple Slab Pie\n\nStack pastry rounds on top of each other on lightly floured surface. Roll into rectangle.\n\nFit crust into pan, pressing into corners. Fold extra pastry crust under, even with edges of pan; seal edges.\n\nSpoon apple mixture into pastry-lined pan.\n\nIn small bowl, mix powdered sugar and milk. Drizzle over baked pie. Allow glaze to set before serving.\n\nBlueberry Pie\n\n## Blueberry Pie\n\nPrep 30 min Total 3 hr 15 min \u25cf 8 servings\n\n  * Two-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * 1\u00bc cups sugar\n  * \u00bd cup cornstarch\n  * \u00bd teaspoon ground cinnamon, if desired\n  * 6 cups fresh blueberries\n  * 1 tablespoon lemon juice\n  * 1 tablespoon cold butter, if desired\n\n1 Heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate.\n\n2 In large bowl, mix sugar, cornstarch and cinnamon. Stir in blueberries. Spoon into pastry-lined pie plate. Sprinkle with lemon juice. Cut butter into small pieces; sprinkle over blueberries.\n\n3 Make Classic Lattice Top (page 596) or cover with top pastry; cut slits in pastry. Seal and flute (see Pastry Edge Treatments, page 587). Cover edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning; remove shield or foil during last 15 minutes of baking.\n\n4 Bake 35 to 45 minutes or until crust is golden brown and juice begins to bubble through slits in crust. Cool on cooling rack at least 2 hours.\n\n1 Serving: Calories 410; Total Fat 19g (Saturated Fat 4.5g, Trans Fat 3g); Cholesterol 0mg; Sodium 270mg; Total Carbohydrate 57g (Dietary Fiber 4g, Sugars 24g); Protein 4g Exchanges: 1 Starch, 1 Fruit, 2 Other Carbohydrate, 3\u00bd Fat Carbohydrate Choices: 4\n\nBlackberry, Boysenberry, Loganberry or Raspberry Pie Substitute any of these fresh berries for the blueberries; omit lemon juice.\n\nBlueberry-Peach Pie Substitute 2\u00bd cups sliced peeled fresh peaches for 4 cups of the blueberries; omit lemon juice. Place blueberries in pastry-lined pie plate; sprinkle with half of the sugar mixture. Top with peaches; sprinkle with remaining sugar mixture. Continue as directed in Step 3.\n\n## Blueberry Hand Pies\n\nIf you like, drizzle these little pies with a sweet glaze. Mix 1 cup powdered sugar with about 1 tablespoon of milk or enough milk to make desired consistency. Drizzle over cooled pies. Let stand 1 hour for glaze to dry.\n\nPrep 1 hr Total 1 hr 55 min \u25cf 8 pies\n\nCrust\n\n  * Two-Crust Pastry (page 586)\n\nFilling\n\n  * 1\u00bd cups fresh or frozen (thawed and drained) blueberries\n  * \u00bd cup granulated sugar\n  * 2 tablespoons cornstarch\n  * \u00bc teaspoon ground cinnamon\n\nTopping\n\n  * 1 egg, beaten\n  * 2 teaspoons white decorator sugar crystals or granulated sugar\n\n1 Heat oven to 375\u00b0F. Line large cookie sheet with cooking parchment paper. Make pastry as directed, rolling into 2 (11-inch) rounds. With 4-inch round cutter, cut 8 rounds from each crust, rerolling scraps if necessary.\n\n2 In medium bowl, mix filling ingredients. Spoon filling evenly in center of 8 pastry rounds to within \u00bc inch of edges. Brush edges of pastry with water. Place remaining pastry rounds on top, stretching dough gently to fit if necessary; press edges firmly with fork to seal. Cut slit in top pastry.\n\n3 Place pies on cookie sheet. Brush egg over tops of pies; sprinkle with decorator sugar.\n\n4 Bake 22 to 25 minutes or until light golden brown. Remove from cookie sheet to cooling rack; cool at least 30 minutes.\n\n1 Pie: Calories 380; Total Fat 20g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 25mg; Sodium 300mg; Total Carbohydrate 45g (Dietary Fiber 2g, Sugars 16g); Protein 4g Exchanges: 1\u00bd Starch, \u00bd Fruit, 1 Other Carbohydrate, 4 Fat Carbohydrate Choices: 3\n\nBlueberry Hand Pies\n\nEasy Apple Tart\n\n## Easy Apple Tart\n\nThere's no pie plate required for this recipe. The filling is partially wrapped in a pastry crust and baked on a cookie sheet.\n\nPrep 25 min Total 2 hr \u25cf 8 servings\n\n  * One-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * \u2154 cup packed brown sugar\n  * \u2153 cup all-purpose flour\n  * 4 cups thinly sliced (\u215b inch thick) peeled tart apples (4 medium)\n  * 1 tablespoon cold butter\n  * Granulated sugar, if desired\n\n1 Heat oven to 425\u00b0F. Make pastry as directed\u2014except roll into 13-inch round. Place on ungreased cookie sheet. Cover with plastic wrap to keep pastry moist while making filling.\n\n2 In large bowl, mix brown sugar and flour. Stir in apples. Mound apple mixture on center of pastry to within 3 inches of edge. Cut butter into small pieces; sprinkle over apples. Fold edge of pastry over apples, making pleats so it lays flat on apples (pastry will not cover apples in center). Sprinkle pastry with granulated sugar.\n\n3 Bake 30 to 35 minutes or until crust is light golden brown. To prevent excessive browning, cover center of pie with 5-inch square of foil during last 10 to 15 minutes of baking. Cool on cookie sheet on cooling rack 1 hour, or serve warm if desired.\n\n1 Serving: Calories 290; Total Fat 12g (Saturated Fat 3.5g, Trans Fat 2g); Cholesterol 0mg; Sodium 160mg; Total Carbohydrate 43g (Dietary Fiber 2g, Sugars 25g); Protein 2g Exchanges: 1 Starch, 1 Fruit, 1 Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\n\nRhubarb Pie\n\n## Rhubarb Pie\n\nRhubarb is very tart, so there's a lot of sugar in this pie. Use the lower amount of sugar for young, thin rhubarb stalks.\n\nPrep 30 min Total 3 hr 20 min \u25cf 8 servings\n\n  * Two-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * 2 to 2\u2153 cups sugar\n  * \u2154 cup all-purpose flour\n  * 1 teaspoon grated orange peel, if desired\n  * 6 cups chopped (\u00bd-inch pieces) fresh rhubarb\n  * 1 tablespoon cold butter, if desired\n\n1 Heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate.\n\n2 In large bowl, mix sugar, flour and orange peel. Stir in rhubarb. Spoon into pastry-lined pie plate. Cut butter into small pieces; sprinkle over rhubarb.\n\n3 Cover with top pastry; cut slits in pastry. Seal and flute (see Pastry Edge Treatments, page 587). Cover edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning; remove shield or foil during last 15 minutes of baking.\n\n4 Bake 50 to 55 minutes or until crust is golden brown and juice begins to bubble through slits in crust. Cool on cooling rack at least 2 hours.\n\n1 Serving: Calories 540; Total Fat 21g (Saturated Fat 5g, Trans Fat 3.5g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 84g (Dietary Fiber 3g, Sugars 51g); Protein 5g Exchanges: 2 Starch, 1 Fruit, 2\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 5\u00bd\n\nQuick Rhubarb Pie Substitute 1 box refrigerated pie crusts, softened as directed on box, for the Two-Crust Pastry or Easy Buttermilk Pastry. Substitute 2 bags (1 pound each) frozen unsweetened rhubarb, thawed and drained, for the fresh rhubarb.\n\nStrawberry-Rhubarb Pie Substitute 3 cups sliced fresh strawberries for 3 cups of the rhubarb. Use 2 cups sugar.\n\nNectarine-Plum Crostata\n\n## Nectarine-Plum Crostata\n\nPrep 40 min Total 2 hr 50 min \u25cf 8 servings\n\nCrust\n\n  * 1\u00bd cups all-purpose flour\n  * 1 teaspoon sugar\n  * \u00bc teaspoon salt\n  * \u00bd cup cold butter\n  * 1 egg yolk\n  * 4 to 5 tablespoons cold water\n\nFilling\n\n  * \u00bd cup sugar\n  * 3 tablespoons all-purpose flour\n  * \u00bc teaspoon ground cinnamon\n  * 3 cups sliced nectarines\n  * 2 cups sliced plums\n  * 1 tablespoon lemon juice\n  * 2 tablespoons cold butter\n  * 1 tablespoon sugar\n\n1 In medium bowl, mix 1\u00bd cups flour, 1 teaspoon sugar and the salt. Cut in \u00bd cup butter, using pastry blender or fork, until crumbly. Stir in egg yolk with fork. Sprinkle with water, 1 tablespoon at a time, tossing until ball of dough forms. Flatten ball to \u00bd-inch thickness. Wrap in plastic wrap; refrigerate 30 minutes.\n\n2 Heat oven to 425\u00b0F. On lightly floured surface, roll pastry into 13-inch round, about \u215b inch thick. Place on ungreased large cookie sheet.\n\n3 In large bowl, mix \u00bd cup sugar, 3 tablespoons flour and the cinnamon. Stir in nectarines and plums until coated. Sprinkle with lemon juice; mix. Spoon fruit mixture onto center of pastry, spreading to within 3 inches of edge. Cut 2 tablespoons butter into small pieces; sprinkle over filling. Fold edge of pastry up and over fruit mixture, pleating crust slightly as necessary. Brush edge of pastry with small amount of water; sprinkle with 1 tablespoon sugar.\n\n4 Bake 30 to 40 minutes or until crust is golden brown and fruit is tender. Cool on cooling rack at least 1 hour. Store at room temperature up to 2 days, then store loosely covered in refrigerator up to 2 days longer.\n\n1 Serving: Calories 340; Total Fat 15g (Saturated Fat 9g, Trans Fat 0.5g); Cholesterol 65mg; Sodium 180mg; Total Carbohydrate 45g (Dietary Fiber 2g, Sugars 23g); Protein 4g Exchanges: 1 Starch, 1 Fruit, 1 Other Carbohydrate, 3 Fat Carbohydrate Choices: 3\n\n## Learn to Make LATTICE CRUST\n\nA lattice top crust is a gorgeous way to decorate a two-crust pie, allowing the filling to show through for an easy \"wow.\" Make Two-Crust Pastry (page 586), except trim overhanging edge of bottom crust 1 inch from rim of plate. Place filling in crust. After rolling pastry for top crust, cut into \u00bd- to 1-inch wide strips using a pastry wheel or pizza cutter with a straight edge, such as a ruler, to even strips.\n\n## Classic Lattice Top\n\nPlace half of the strips equal distance apart on filling, using shorter strips toward edge of pie. Weave remaining strips over and under first strips. Trim strips evenly with edge of overhanging bottom crust. Fold edge up, forming high stand-up rim; flute as desired (see Pastry Edge Treatments, page 587).\n\n### Making a Lattice Top\n\nPlace strips about \u00bd inch apart across the pie vertically, then weave strips horizontally over and under the strips.\n\n## Making Pastry in the Food Processor\n\nUse One-Crust Pastry or Two-Crust Pastry recipe on page 586, except make these changes when using your food processor to mix the dough:\n\n  1. Measure 2 tablespoons ice-cold water for One-Crust Pastry or 4 tablespoons ice-cold water for Two-Crust Pastry into a liquid measuring cup; set aside.\n  2. Place flour, salt and shortening in food processor. Cover and process, using quick on-and-off motions, until particles are the size of small peas.\n  3. With food processor running, pour water all at once through feed tube just until dough leaves side of bowl (dough should not form a ball). Continue as directed in Step 2 of pastry recipe.\n\n## Cherry Pie\n\nSour cherries\u2014also called pie cherries, tart cherries or tart red cherries\u2014make wonderful pies. In fact, this is the pie featured on the cover. Take advantage of their short season by making a homemade pie. If fresh aren't available, use our variation below, which uses bags of frozen pitted sweet cherries.\n\nPrep 30 min Total 3 hr 15 min \u25cf 8 servings\n\n  * Two-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * 1\u2153 cups sugar\n  * \u00bd cup all-purpose flour\n  * 6 cups fresh sour cherries, pitted\n  * 2 tablespoons cold butter, if desired\n\n1 Heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate.\n\n2 In large bowl, mix sugar and flour. Stir in cherries. Spoon into pastry-lined pie plate. Cut butter into small pieces; sprinkle over cherries.\n\n3 Make Classic Lattice Top (page 596) or cover with top pastry; cut slits in pastry if covering with top pastry. Seal and flute (see Pastry Edge Treatments, page 587). Cover edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning; remove shield or foil during last 15 minutes of baking.\n\n4 Bake 35 to 45 minutes or until crust is golden brown and juice begins to bubble through slits in crust. Cool on cooling rack at least 2 hours.\n\n1 Serving: Calories 540; Total Fat 22g (Saturated Fat 5g, Trans Fat 3.5g); Cholesterol 0mg; Sodium 300mg; Total Carbohydrate 81g (Dietary Fiber 4g, Sugars 49g); Protein 5g Exchanges: 2 Starch, 1 Fruit, 2 Other Carbohydrate, 4 Fat Carbohydrate Choices: 5\u00bd\n\nQuick Cherry Pie Decrease sugar to \u2154 cup and substitute 2 bags (1 pound each) frozen (thawed) pitted dark sweet cherries for the fresh sour cherries.\n\n## Butter Pastry\n\nWe prefer pies made with shortening because of the tender, ultra-flaky texture and the way the flavor (or lack of it) lets the flavors of the fillings shine. If you're a fan of butter pastry, use the Pie Pastry recipe on page 586, substituting butter (cut into \u00bd-inch pieces) for half of the shortening.\n\nCherry Pie\n\n## Stone Fruit\n\nStone fruit are eaten plain or can be used in so many types of dishes. Blanch them about 30 seconds and the peels will pull right off (see Blanching, page 19).\n\n  1. 1. **Easy Bellini Cocktails:** Puree sliced peeled ripe nectarines or peaches in food processor or blender until smooth. For each Bellini, spoon 2 to 3 tablespoons puree into a champagne flute. Fill glass with chilled sparkling white wine. Stir and serve immediately.\n  2. 2. **Grilled Pluot Dessert:** Place 4 ripe pluot or peach halves cut sides down on hot, oiled grill over medium heat. Grill 3 minutes. Turn cut sides up; brush with 1 tablespoon butter; sprinkle with mixture of 1\u00bd teaspoons sugar and \u215b teaspoon each ground cinnamon and ground ginger. Grill 3 minutes longer or until softened. Serve with ice cream.\n  3. 3. **Cherry BBQ Sauce:** Stir \u00bc cup pureed or finely chopped pitted ripe cherries into \u2154 cup prepared barbecue sauce. Brush on chicken or pork while grilling.\n  4. 4. **Peachy Lemonade:** Make Lemonade (page 72), except\u2014mix 1 cup water, 2 medium peeled, sliced peaches or nectarines and 1 cup sugar in 2-quart saucepan. Heat to boiling; reduce heat. Simmer uncovered 5 to 7 minutes or until peaches are fork-tender. Remove from heat; cool. Place mixture in blender container. Cover and blend on medium speed until smooth; pour into large pitcher. Stir in remaining water and lemon juice. Serve over ice with slices of fresh peaches.\n  5. 5. **Summery Salad:** For 4 servings, toss 4 cups desired salad greens with purchased balsamic vinaigrette. Arrange on salad plates. Top each with sliced (peeled if desired) fresh peaches, blue cheese crumbles and Cinnamon-Sugared Nuts (page 41) or purchased candied pecans.\n  6. 6. **Strawberry-Peach Salsa:** Mix \u00bc cup lime juice, 1 tablespoon honey and \u00bc teaspoon salt in medium bowl. Toss with 1 cup chopped fresh strawberries, 2 cups chopped fresh peaches (peeled if desired), \u00bc cup chopped fresh cilantro and 1 jalape\u00f1o chile, seeded and chopped. Cover and refrigerate 1 hour to blend flavors. Serve with chicken or fish.\n  7. 7. **Berry-Plum Pops:** Place 1 cup almond milk (any variety), 3 medium peeled, sliced fresh plums, 1 container (6 ounces) berry yogurt (any variety) and 2 tablespoons agave or honey in blender; blend until smooth. Pour about \u00bc cup into 3-ounce paper cups. Insert flat wooden sticks with rounded ends in cups. Freeze until firm, about 1 hour 30 minutes. Remove cups to serve.\n  8. 8. **Sour Cream\u2013Nectarine Pancakes:** Heat skillet over medium-high heat. Beat 2 cups original baking mix, 1 cup sour cream, \u00bd cup milk, 2 eggs and 2 teaspoons poppy seed in large bowl with whisk until smooth. For each pancake, pour about 3 tablespoons batter onto hot skillet. Immediately press 3 or 4 thin slices nectarine into batter. Cook about 3 minutes or until edges are dry. Turn; cook about 3 minutes or until golden brown.\n\nStrawberry-Peach Salsa\n\nRaspberry-Almond Pie\n\n## Raspberry-Almond Pie\n\nPrep 50 min Total 4 hr 45 min \u25cf 8 servings\n\n  * Two-Crust Pastry (page 586)\n  * \u00bd cup whole blanched almonds\n  * 2 tablespoons butter\n  * \u00bc cup granulated sugar\n  * 2 teaspoons all-purpose flour\n  * 1 egg\n  * \u00bd teaspoon almond extract\n  * 1 cup granulated sugar\n  * 5 tablespoons cornstarch\n  * 1 tablespoon grated orange peel\n  * 5 cups fresh or frozen (thawed) raspberries\n  * 1 egg\n  * 1 tablespoon whipping cream\n  * 2 tablespoons coarse sugar or sanding sugar\n\n1 Heat oven to 400\u00b0F. Place 1 pastry round in 9-inch glass pie plate; flute edges. On lightly floured surface, roll second pastry to 13-inch round. With 1\u00bd-inch star cookie cutter, cut out 45 stars; set aside.\n\n2 Place almonds in food processor. Cover and process until finely chopped. Add butter, \u00bc cup granulated sugar and the flour; process until blended. Add 1 egg and the almond extract; process until blended. Spread mixture evenly in pastry-lined pie plate.\n\n3 In large bowl, mix 1 cup granulated sugar, the cornstarch and orange peel. Add raspberries; toss gently. Spoon berry mixture evenly over almond layer. Arrange stars on top of filling, covering entire surface. In small bowl, beat 1 egg and the whipping cream. Brush mixture over stars and edge of crust; sprinkle with coarse sugar.\n\n4 Place cookie sheet on rack below pie in case of spillover. Bake 10 minutes. Cover crust edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning. Bake 40 to 45 minutes longer or until crust is golden brown and juices are bubbly. (If crust becomes too brown, cover entire crust loosely with foil.) Cool on cooling rack at least 3 hours.\n\n1 Serving: Calories 580; Total Fat 28g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 55mg; Sodium 340mg; Total Carbohydrate 76g (Dietary Fiber 7g, Sugars 38g); Protein 8g Exchanges: 2\u00bd Starch, 1 Fruit, 1\u00bd Other Carbohydrate, 5\u00bd Fat Carbohydrate Choices: 5\n\nMixed Berry Crumble Tart\n\n## Mixed Berry Crumble Tart\n\nPrep 30 min Total 1 hr 15 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586)\n  * 1\u00bd cups sliced fresh strawberries\n  * 1\u00bd cups fresh blueberries\n  * 1 cup fresh raspberries\n  * \u2154 cup sugar\n  * 2 tablespoons cornstarch\n  * \u00be cup all-purpose flour\n  * \u00bd cup sugar\n  * 1 teaspoon grated orange peel\n  * \u2153 cup butter, melted\n\n1 Heat oven to 425\u00b0F. Make pastry as directed, except roll into 13-inch round. Place pastry in 10- or 11-inch tart pan with removable bottom; press against bottom and side of pan. Trim overhanging edge of pastry even with top of pan.\n\n2 In large bowl, gently toss berries with \u2154 cup sugar and the cornstarch. Spoon into pastry-lined pan.\n\n3 In small bowl, stir flour, \u00bd cup sugar, the orange peel and butter with fork until crumbly. Sprinkle evenly over berries.\n\n4 Bake 35 to 45 minutes or until fruit bubbles in center. Remove tart from side of pan. Serve warm.\n\n1 Serving: Calories 420; Total Fat 18g (Saturated Fat 6g, Trans Fat 2g); Cholesterol 20mg; Sodium 200mg; Total Carbohydrate 60g (Dietary Fiber 3g, Sugars 35g); Protein 3g Exchanges: 1 Starch, 3 Fruit, 3\u00bd Fat Carbohydrate Choices: 4\n\nStrawberries and Cream Tart\n\n## Strawberries and Cream Tart\n\nPrep 25 min Total 1 hr 40 min \u25cf 10 servings\n\nCrust\n\n  * One-Crust Pastry (page 586) or Press-in-the-Pan Tart Pastry (page 589)\n\nFilling\n\n  * 1 package (8 oz) cream cheese, softened\n  * \u2153 cup sugar\n  * \u00bc to \u00bd teaspoon almond extract\n  * 1 cup whipping cream, whipped\n\nTopping\n\n  * 4 cups fresh strawberries, cut in half\n  * \u00bd cup semisweet chocolate chips\n  * 1 tablespoon shortening\n\n1 Heat oven to 450\u00b0F. Place pastry in 10-inch tart pan with removable bottom. Bake as directed for Baked One-Crust Pie (page 587). Cool completely.\n\n2 In large bowl, beat cream cheese with electric mixer on medium speed until fluffy. Gradually add sugar and almond extract; beat well. Fold in whipped cream. Spread filling into cooled baked crust. Arrange strawberry halves over filling. Refrigerate while making drizzle.\n\n3 In 1-quart saucepan, melt chocolate chips and shortening over low heat, stirring constantly, until smooth. Drizzle over strawberries and filling. Refrigerate until set, about 30 minutes. Remove tart from side of pan. Cover and refrigerate any remaining tart.\n\n1 Serving: Calories 340; Total Fat 24g (Saturated Fat 13g, Trans Fat 0.5g); Cholesterol 55mg; Sodium 190mg; Total Carbohydrate 28g (Dietary Fiber 1g, Sugars 16g); Protein 3g Exchanges: 1 Fruit, 1 Other Carbohydrate, \u00bd High-Fat Meat, 4 Fat Carbohydrate Choices: 2\n\n## White and Dark Chocolate Raspberry Tart\n\nFor a pretty garnish, make white chocolate curls (see Chocolate Curls, page 611).\n\nPrep 1 hr Total 4 hr 50 min \u25cf 10 servings\n\n  * One-Crust Pastry (page 586) or Press-in-the-Pan Tart Pastry (page 589)\n  * 2 tablespoons orange juice\n  * 1 teaspoon unflavored gelatin\n  * 1\u00bd cups whipping cream\n  * 1 package (6 oz) white chocolate baking bars, chopped\n  * 1 bag (12 oz) frozen raspberries, thawed, drained and juice reserved\n  * 1 tablespoon cornstarch\n  * 1 tablespoon sugar\n  * 1 cup fresh raspberries, if desired\n  * 2 oz semisweet baking chocolate, cut into pieces\n  * 2 tablespoons butter\n\n1 Heat oven to 450\u00b0F. Place pastry in 10-inch tart pan with removable bottom or 10-inch springform pan, pressing 1 inch up side of pan. Bake as directed for Baked One-Crust Pie (page 587). Cool completely.\n\n2 Pour orange juice into 2-quart saucepan. Sprinkle gelatin over juice; let stand 5 minutes to soften. Stir in \u00be cup of the whipping cream; heat over low heat, stirring frequently, until gelatin is dissolved. Stir in white chocolate until melted and smooth. Transfer to medium bowl; refrigerate about 30 minutes, stirring occasionally, until cool but not set.\n\n3 In blender or food processor, place thawed raspberries and reserved juice. Cover and blend until pureed. Set strainer over 2-cup measuring cup. Press puree with back of spoon through strainer to remove seeds. If necessary, add water to raspberry puree to measure \u00bd cup. In 1-quart saucepan, mix cornstarch and sugar. Gradually add raspberry puree. Cook and stir over low heat until thickened. Fold in fresh raspberries. Spread into cooled baked crust. Refrigerate 15 minutes.\n\n4 Meanwhile, in chilled small, deep bowl, beat remaining \u00be cup whipping cream with electric mixer on high speed until stiff peaks form. Fold whipped cream into white chocolate mixture. Spoon and spread over raspberry layer. Refrigerate 1 hour or until filling is set.\n\n5 In 1-quart saucepan, melt semisweet chocolate and butter over low heat, stirring frequently; carefully pour and spread over white chocolate layer. Refrigerate at least 2 hours or until set. Remove from refrigerator 30 minutes before serving to soften chocolate layers. Remove tart from side of pan (if using springform pan, remove side of pan). Cover and refrigerate any remaining tart.\n\n1 Serving: Calories 390; Total Fat 27g (Saturated Fat 16g, Trans Fat 0g); Cholesterol 50mg; Sodium 135mg; Total Carbohydrate 33g (Dietary Fiber 2g, Sugars 19g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 5\u00bd Fat Carbohydrate Choices: 2\n\n## Tiny Caramel Tarts\n\nThese rich and creamy caramel custard bite-size tarts are a perfect choice for a crowd.\n\nPrep 25 min Total 4 hr 25 min \u25cf 75 tarts\n\n  * 2 cups sugar\n  * \u00bd cup cold butter, cut into pieces\n  * 6 tablespoons all-purpose flour\n  * 4 egg yolks\n  * 2 cups milk\n  * 5 boxes (1.9 oz each) frozen mini phyllo (filo) shells (75 shells)\n  * 4 cups Sweetened Whipped Cream (page 622)\n  * Grated chocolate, if desired\n\n1 In 10-inch heavy skillet, heat 1 cup of the sugar over medium heat 6 to 8 minutes, stirring constantly, until sugar is melted and golden brown. Stir in butter until melted. Remove from heat.\n\n2 In 3-quart heavy saucepan, stir together flour, egg yolks, milk and remaining 1 cup sugar with wire whisk. Heat to simmering over low heat, stirring constantly. Add melted sugar mixture; cook 1 to 2 minutes, stirring constantly, until thickened. Spoon mixture into large bowl. Cover; refrigerate 4 hours.\n\n3 To serve, spoon caramel mixture into phyllo shells; top with each with about 2 rounded teaspoons whipped cream and a sprinkle of grated chocolate.\n\n1 Tart: Calories 270; Total Fat 17g (Saturated Fat 3.5g, Trans Fat 0 g); Sodium 115mg; Total Carbohydrate 25g (Dietary Fiber 0g, Sugars 0g); Protein 3g Exchanges: 1 Starch, \u00bd Other Carbohydrate: 3\u00bd Fat Carbohydrate Choices: 1 \u00bd\n\nSalted Cashew\u2013Bittersweet Chocolate Tart\n\n## Salted Cashew\u2013Bittersweet Chocolate Tart\n\nPrep 25 min Total 2 hr 40 min \u25cf 16 servings\n\n  * One-Crust Pastry (page 586) or Press-in-the-Pan Tart Pastry (page 589)\n  * 1\u00bc cups whipping cream\n  * 12 oz bittersweet baking chocolate, chopped*\n  * \u00bd cup cashew halves and pieces, coarsely chopped\n  * 1 can (13.4 oz) dulce de leche (caramelized sweetened condensed milk)\n  * 2 tablespoons cashew halves and pieces\n  * \u00bc teaspoon coarse (kosher or sea) salt\n\n1 Heat oven to 450\u00b0F. Place pastry in 9-inch tart pan with removable bottom. Bake as directed for Baked One-Crust Pie (page 587), except do not prick crust. Cool completely.\n\n2 Meanwhile, in 2-quart saucepan, heat whipping cream over medium-high heat just to boiling. Remove from heat; stir in chocolate with whisk until smooth. Stir in \u00bd cup chopped cashews. Pour into cooled baked crust. Refrigerate 2 hours or until firm.\n\n3 Spread dulce de leche over chocolate layer. Sprinkle 2 tablespoons cashews and the salt evenly over top. Remove tart from side of pan. For easier cutting, use a thin, sharp knife, wiping it clean with paper towels and running it under hot water between cutting each slice.\n\n*Semisweet baking chocolate can be substituted for the bittersweet chocolate.\n\n1 Serving: Calories 370; Total Fat 26g (Saturated Fat 14g, Trans Fat 0g); Cholesterol 30mg; Sodium 150mg; Total Carbohydrate 28g (Dietary Fiber 4g, Sugars 13g); Protein 6g Exchanges: 1 Starch, 1 Other Carbohydrate, \u00bd Medium-Fat Meat, 4\u00bd Fat Carbohydrate Choices: 2\n\nGinger\u2013Lemon Curd Petite Pies\n\n## Ginger\u2013Lemon Curd Petite Pies Calorie Smart\n\nPrep 40 min Total 1 hr 10 min \u25cf 16 pies\n\nLemon Curd\n\n  * \u00bd cup granulated sugar\n  * 1 tablespoon cornstarch\n  * 2 teaspoons grated lemon peel\n  * \u00bd cup fresh lemon juice\n  * 2 eggs, slightly beaten\n  * 1 tablespoon finely chopped Crystallized Ginger (page 447) or crystallized ginger from 2-oz jar\n\nCrust\n\n  * Two-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n\nGarnish\n\n  * Powdered sugar\n\n1 In 1-quart saucepan, mix granulated sugar and cornstarch. Stir in lemon peel, lemon juice, eggs and ginger with whisk until blended. Cook over medium heat about 2 minutes, stirring constantly, until mixture thickens and boils. Remove from heat; transfer to bowl. Cover with plastic wrap and refrigerate 30 minutes or until completely cool.\n\n2 Meanwhile, heat oven to 375\u00b0F. Line 2 cookie sheets with cooking parchment paper. On lightly floured surface, roll 1 pastry round to \u215b-inch thickness. With 3-inch round cookie cutter, cut out 16 rounds (you may need to reroll scraps to get 16 rounds). Place on cookie sheet. With fork, poke holes all over rounds. Repeat with second pastry\u2014except don't poke with fork. With 1\u00bd-inch cookie cutter, cut out design from center of second batch of rounds. Place on cookie sheet.\n\n3 Bake 8 to 10 minutes or until golden brown. Remove from cookie sheets to cooling racks; cool completely.\n\n4 To serve, place plain pastry rounds in jumbo paper baking cups or on individual dessert plates. Top each with about 1 tablespoon lemon curd. Sprinkle powdered sugar over rounds with cutouts; place over lemon curd.\n\n1 Pie: Calories 180; Total Fat 9g (Saturated Fat 2.5g, Trans Fat 0g); Cholesterol 25mg; Sodium 160mg; Total Carbohydrate 21g (Dietary Fiber 0g, Sugars 7g); Protein 2g Exchanges: 1 Starch, \u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 1\u00bd\n\nLemon Meringue Pie\n\n## Lemon Meringue Pie\n\nWhen you separate the eggs, be sure to save the egg whites to make the meringue. To speed up the beating, let the egg whites warm up to room temperature (about 30 minutes).\n\nPrep 50 min Total 3 hr 5 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586), Easy Buttermilk Pastry (page 588) or Press-in-the-Pan Oil Pastry (page 588)\n  * Meringue for 9-Inch Pie (below)\n  * 3 egg yolks (reserve whites for meringue)\n  * 1\u00bd cups sugar\n  * \u2153 cup plus 1 tablespoon cornstarch\n  * 1\u00bd cups water\n  * 3 tablespoons butter\n  * 2 teaspoons grated lemon peel\n  * \u00bd cup fresh lemon juice\n  * 2 drops yellow food color, if desired\n\n1 Heat oven to 450\u00b0F. Place pastry in 9-inch glass pie plate. Bake as directed for Baked One-Crust Pie (page 587). Cool completely.\n\n2 Make sugar mixture for meringue (recipe below). While sugar mixture is cooling, in small bowl, beat egg yolks with fork; set aside. In 2-quart saucepan, mix sugar and cornstarch. Gradually stir in water. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute.\n\n3 Immediately stir at least half of the hot mixture gradually into egg yolks, then stir back into hot mixture in saucepan. Boil and stir 2 minutes or until very thick. (Do not boil less than 2 minutes or filling may stay too soft or become runny.) Remove from heat. Stir in butter, lemon peel, lemon juice and food color. Press plastic wrap on filling to prevent tough layer from forming on top.\n\n4 Heat oven to 350\u00b0F. Continue making meringue. Pour hot lemon filling into cooled baked crust. Spoon meringue onto hot lemon filling. Spread over filling, carefully sealing meringue to edge of crust to prevent shrinking or weeping.\n\n5 Bake 12 to 15 minutes or until meringue is light brown. Cool away from drafts 2 hours. Store covered in refrigerator.\n\n1 Serving: Calories 450; Total Fat 17g (Saturated Fat 6g, Trans Fat 1.5g); Cholesterol 95mg; Sodium 250mg; Total Carbohydrate 70g (Dietary Fiber 0g, Sugars 51g); Protein 5g Exchanges: 2 Starch, 2\u00bd Other Carbohydrate, 3 Fat Carbohydrate Choices: 4\u00bd\n\nMeringue for 9-Inch Pie In 1-quart saucepan, mix \u00bd cup sugar and 4 teaspoons cornstarch. Stir in \u00bd cup cold water. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute; remove from heat. Cool completely while making filling for pie recipe. (To cool more quickly, place in freezer about 10 minutes.) In large bowl, beat 4 egg whites and \u215b teaspoon salt with electric mixer on high speed just until soft peaks begin to form. Very gradually, beat in sugar mixture until stiff peaks form.\n\n### Spreading Meringue over Pie\n\nSpread meringue evenly over hot filling, sealing to edge of crust to help keep meringue from shrinking.\n\n## Lemon Mascarpone Tart\n\nPrep 30 min Total 3 hr 35 min \u25cf 12 servings\n\n  * Easy Nut Crust (page 589)\n  * \u00be cup mascarpone cheese (from 8-oz container)\n  * 1 cup sugar\n  * 2 tablespoons grated lemon peel\n  * \u2153 cup fresh lemon juice\n  * 2 tablespoons whipping cream\n  * 1 teaspoon vanilla\n  * 2 whole eggs\n  * 1 egg yolk\n  * 1 cup whipping cream\n  * 3 tablespoons mascarpone cheese (from 8-oz container)\n  * 2 tablespoons honey\n  * \u00bd teaspoon vanilla\n  * Grated lemon peel or lemon peel twists\n\n1 Make crust as directed, using pistachio nuts. Press crust evenly in bottom and up side of 10-inch tart pan with removable bottom. Refrigerate 30 minutes.\n\n2Heat oven to 350\u00b0F. In medium bowl, beat \u00be cup mascarpone cheese, the sugar and 2 tablespoons lemon peel with electric mixer on low speed until blended. Add lemon juice, 2 tablespoons whipping cream and 1 teaspoon vanilla; beat until well blended. Add eggs and egg yolk, beating on low speed just until combined. Spread filling evenly in chilled crust. Place cookie sheet on rack below tart in case of spillover.\n\n3 Bake about 35 minutes or until filling is set but center still jiggles slightly when moved. Cool on cooling rack 1 hour. Refrigerate at least 1 hour.\n\n4 In chilled large, deep bowl, beat 1 cup whipping cream and 3 tablespoons mascarpone cheese with electric mixer on high speed until soft peaks form. Stir in honey and \u00bd teaspoon vanilla. Spread over top of tart. Garnish with lemon peel. Remove tart from side of pan.\n\n1 Serving: Calories 340; Total Fat 23g (Saturated Fat 13g, Trans Fat 0.5g); Cholesterol 110mg; Sodium 90mg; Total Carbohydrate 29g (Dietary Fiber 0g, Sugars 21g); Protein 3g Exchanges: 1 Starch, 1 Other Carbohydrate, 4\u00bd Fat Carbohydrate Choices: 2\n\nPumpkin Pie\n\n## Pumpkin Pie\n\nBe sure to use canned pumpkin puree, not pumpkin pie mix, in this recipe. The mix has sugar and spices already in it, so if you have purchased the pumpkin pie mix, follow the directions on that label. Or, if you like, use 1\u00bd cups cooked, mashed fresh pumpkin.\n\nPrep 30 min Total 3 hr 45 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586) or Press-in-the-Pan Oil Pastry (page 588)\n  * 2 eggs\n  * \u00bd cup sugar\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon salt\n  * \u00bd teaspoon ground ginger\n  * \u215b teaspoon ground cloves\n  * 1 can (15 oz) pureed pumpkin (not pumpkin pie mix)\n  * 1 can (12 oz) evaporated milk\n  * Sweetened Whipped Cream (page 622), if desired\n\n1 Heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate. After fluting edge, carefully line pastry with double thickness of foil, gently pressing foil to bottom and side of pastry. Let foil extend over edge to prevent excessive browning. Bake 10 minutes; carefully remove foil. Bake 2 to 4 minutes longer or until pastry just begins to brown and has become set. If crust bubbles, gently push bubbles down with back of spoon.\n\n2 In medium bowl, beat eggs slightly with whisk. Beat in remaining ingredients except whipped cream.\n\n3 Pour filling into hot crust. (To prevent spilling, place pie plate on oven rack before adding filling.) Cover edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning; remove shield or foil during last 15 minutes of baking.\n\n4 Bake 15 minutes. Reduce oven temperature to 350\u00b0F. Bake about 45 minutes longer or until knife inserted in center comes out clean. Cool on cooling rack 1 hour; refrigerate uncovered until completely chilled. Store loosely covered in refrigerator. Serve with whipped cream.\n\n1 Serving: Calories 300; Total Fat 15g (Saturated Fat 5g, Trans Fat 2g); Cholesterol 65mg; Sodium 360mg; Total Carbohydrate 33g (Dietary Fiber 2g, Sugars 19g); Protein 7g Exchanges: 1 Starch, 1 Fruit, 3 Fat Carbohydrate Choices: 2\n\nPraline Pumpkin Pie Make pie as directed, except decrease second bake time to 35 minutes. Mix \u2153 cup packed brown sugar, \u2153 cup chopped pecans and 1 tablespoon softened butter. Sprinkle over pie. Bake about 10 minutes longer or until knife inserted in center comes out clean.\n\n### Testing Pumpkin Pie for Doneness\n\nBake until knife inserted in center comes out clean.\n\n## Banana Cream Pie\n\nPrep 30 min Total 2 hr 30 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586), Easy Buttermilk Pastry (page 588) or Press-in-the-Pan Oil Pastry (page 588)\n  * 4 egg yolks\n  * \u2154 cup sugar\n  * \u00bc cup cornstarch\n  * \u00bd teaspoon salt\n  * 3 cups milk\n  * 2 tablespoons butter, softened\n  * 2 teaspoons vanilla\n  * 2 large ripe but firm bananas\n  * 1 cup Sweetened Whipped Cream (page 622)\n\n1 Heat oven to 450\u00b0F. Place pastry in 9-inch glass pie plate. Bake as directed for Baked One-Crust Pie (page 587). Cool completely.\n\n2Meanwhile, in medium bowl, beat egg yolks with fork; set aside. In 2-quart saucepan, mix sugar, cornstarch and salt. Gradually stir in milk. Cook over medium heat, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute. Immediately stir at least half of the hot mixture gradually into egg yolks, then stir back into hot mixture in saucepan. Boil and stir 1 minute; remove from heat. Stir in butter and vanilla; cool filling slightly.\n\n3 Slice bananas evenly into cooled baked crust; pour warm filling over bananas. Press plastic wrap on filling to prevent a tough layer from forming on top. Refrigerate at least 2 hours or until set. Remove plastic wrap. Top pie with whipped cream. Store covered in refrigerator.\n\n1 Serving: Calories 410; Total Fat 22g (Saturated Fat 8g, Trans Fat 2g); Cholesterol 135mg; Sodium 370mg; Total Carbohydrate 46g (Dietary Fiber 1g, Sugars 27g); Protein 7g Exchanges: 1 Starch, 1 Fruit, \u00bd Other Carbohydrate, \u00bd Low-Fat Milk, 4\u00bd Fat Carbohydrate Choices: 3\n\nButterscotch Cream Pie Substitute packed brown sugar for the granulated sugar. Omit bananas.\n\nChocolate-Banana Cream Pie Make Chocolate Cream Pie filling (below). Cool slightly. Slice 2 large ripe but firm bananas evenly into cooled baked crust; pour warm filling over bananas. Continue as directed.\n\nChocolate Cream Pie Increase sugar to 1\u00bd cups and cornstarch to \u2153 cup; omit butter and bananas. Stir in 2 ounces chopped unsweetened baking chocolate after stirring in milk.\n\nCoconut Cream Pie Increase cornstarch to \u2153 cup. Substitute 1 can (14 ounces) coconut milk (not cream of coconut) and milk to equal 3 cups for the milk. Stir in \u00be cup toasted coconut with the butter. Omit bananas. Refrigerate pie 3 hours or until set. Top with whipped cream; sprinkle with \u00bc cup toasted coconut.\n\nPecan Pie\n\n## Pecan Pie\n\nPrep 20 min Total 1 hr 10 min \u25cf 8 servings\n\n  * One-Crust Pastry (page 586), Easy Buttermilk Pastry (page 588) or Press-in-the-Pan Oil Pastry (page 588)\n  * \u2154 cup sugar*\n  * \u2153 cup butter, melted\n  * 1 cup light or dark corn syrup\n  * 3 eggs\n  * 1 cup pecan halves or broken pecans\n\n1 Heat oven to 375\u00b0F. Place pastry in 9-inch glass pie plate.\n\n2 In medium bowl, beat sugar, butter, corn syrup and eggs with whisk until well blended. Stir in pecans. Pour into pastry-lined pie plate.\n\n3 Bake 40 to 50 minutes or until center is set. Serve warm or chilled.\n\n*Or use \u2153 cup packed brown sugar for half of the sugar.\n\n1 Serving: Calories 530; Total Fat 29g (Saturated Fat 8g, Trans Fat 2g); Cholesterol 100mg; Sodium 420mg; Total Carbohydrate 62g (Dietary Fiber 2g, Sugars 33g); Protein 5g Exchanges: 2 Starch, 2 Other Carbohydrate, 6 Fat Carbohydrate Choices: 4\n\nKentucky Pecan Pie Add 2 tablespoons bourbon or 1 teaspoon brandy extract or vanilla with the corn syrup. Stir in 1 cup semisweet chocolate chips with the pecans.\n\nRoasted Sweet Potato Pie\n\n## Roasted Sweet Potato Pie\n\nPrep 30 min Total 6 hr 20 min \u25cf 8 servings\n\n  * 1\u00bd lb dark-orange sweet potatoes (about 2 medium-large)\n  * One-Crust Pastry (page 586)\n  * \u00bd cup butter, softened\n  * \u00bd cup packed brown sugar\n  * \u00bd cup granulated sugar\n  * \u00bd cup whipping cream\n  * 2 tablespoons bourbon or 1 teaspoon bourbon extract, brandy extract or vanilla\n  * \u00bd teaspoon ground cinnamon\n  * \u00bd teaspoon ground nutmeg\n  * \u00bc teaspoon salt\n  * 2 eggs\n\n1 Heat oven to 400\u00b0F. Line cookie sheet with foil. Pierce sweet potatoes with fork; place on cookie sheet. Roast 1 hour or until tender. Cut potatoes in half. Scoop out pulp into medium bowl; discard skins. Reduce oven temperature to 350\u00b0F.\n\n2 Place pastry in 9\u00bd-inch glass deep-dish pie plate. Beat sweet potato pulp with electric mixer on medium speed until creamy. Add remaining ingredients; beat 1 minute or until well blended. Pour filling into pastry-lined pie plate.\n\n3 Bake 45 to 50 minutes or until knife inserted near center comes out clean. Cover edge with pie crust shield ring or 2- to 3-inch-wide strip of foil during last 15 minutes of baking if necessary to prevent excessive browning. Cool completely on cooling rack, about 4 hours. Store in refrigerator.\n\n1 Serving: Calories 440; Total Fat 24g (Saturated Fat 14g, Trans Fat 0.5g); Cholesterol 100mg; Sodium 360mg; Total Carbohydrate 51g (Dietary Fiber 2g, Sugars 29g); Protein 3g Exchanges: 3 Other Carbohydrate, 1 Vegetable, 5 Fat Carbohydrate Choices: 3\u00bd\n\nRoasted Sweet Potato Meringue Pie While pie is baking, make Meringue for 9-Inch Pie (page 603). Remove from oven; spread meringue over hot filling, carefully sealing meringue to edge of crust to prevent shrinking or weeping. Bake 12 to 15 minutes or until meringue is light brown. Cool as directed.\n\n## Key Lime Pie\n\nPrep 15 min Total 2 hr 50 min \u25cf 8 servings\n\n  * Graham Cracker Crust (page 589)\n  * 4 egg yolks\n  * 2 cans (14 oz each) sweetened condensed milk (not evaporated)\n  * \u00be cup bottled or fresh Key lime juice or regular lime juice\n  * 1 or 2 drops green food color, if desired\n  * 1\u00bd cups Sweetened Whipped Cream (page 622)\n\n1 Heat oven to 375\u00b0F. Make crust as directed but do not bake.\n\n2 In medium bowl, beat egg yolks, condensed milk, lime juice and food color with electric mixer on medium speed about 1 minute or until well blended. Pour into unbaked crust.\n\n3 Bake 14 to 16 minutes or until center is set. Cool on cooling rack 15 minutes. Cover and refrigerate until chilled, at least 2 hours. Spread with whipped cream. Store covered in refrigerator up to 3 days.\n\n1 Serving: Calories 430; Total Fat 23g (Saturated Fat 13g, Trans Fat 1g); Cholesterol 165mg; Sodium 230mg; Total Carbohydrate 48g (Dietary Fiber 0g, Sugars 38g); Protein 7g Exchanges: \u00bd Starch, 2\u00bd Other Carbohydrate, 1 High-Fat Meat, 3 Fat Carbohydrate Choices: 3\n\nElegant Pear Custard Pie\n\n## Elegant Pear Custard Pie\n\nPrep 1 hr Total 6 hr 10 min \u25cf 12 servings\n\nPoached Pears\n\n  * 1 cup sugar\n  * 1 cup water\n  * 1 cup pear vodka*\n  * \u00bd cup elderflower liqueur*\n  * 1 vanilla bean\n  * 3 lb Bosc or Bartlett pears, peeled, cut in half\n\nCrust\n\n  * One-Crust Pastry (page 586)\n\nCustard\n\n  * \u00bd cup whipping cream\n  * \u00bc cup reserved poaching liquid\n  * \u00bc cup sugar\n  * Dash salt\n  * 4\u00bd teaspoons cornstarch\n  * 1 whole egg\n  * 1 egg yolk\n  * \u00bd teaspoon vanilla\n\nGlaze\n\n  * 1 tablespoon reserved poaching liquid\n\n1 In 4-quart saucepan, stir together 1 cup sugar, the water, vodka and liqueur. Cut slit down center of vanilla bean; scrape out seeds. Discard bean; add seeds to mixture in saucepan. Heat to boiling over medium-high heat; add pears. Reduce heat to low. Cover loosely and simmer 25 to 45 minutes or until pears are tender. (Poaching time will vary depending on ripeness of pears.) Remove from heat. Let pears cool in liquid 1 hour.\n\n2 Meanwhile, heat oven to 425\u00b0F. Place pastry in 9-inch glass pie plate. Bake as directed for Partially Baked One-Crust Pie (page 587). Cool crust 10 minutes. Reduce oven temperature to 350\u00b0F.\n\n3 Remove pears from liquid to cutting board. Strain poaching liquid through fine-mesh strainer; set liquid aside. Thinly slice pears; arrange in layers in decorative pattern in partially baked crust.\n\n4 In large bowl, beat all custard ingredients with whisk just until combined. Slowly pour over pears. Carefully place pie in oven (pie plate will be full). Place cookie sheet on rack below pie in case of spillover. Cover crust edge with pie crust shield ring or 2- to 3-inch-wide strip of foil to prevent excessive browning; remove shield or foil during last 15 minutes of baking.\n\n5 Bake 1 hour 10 minutes or until knife inserted in center comes out clean. Brush top of pie with 1 tablespoon reserved poaching liquid. Cool on cooling rack 1 hour. Refrigerate 2 hours or until completely cool.\n\n*Pear nectar can be substituted for both the vodka and liqueur.\n\n1 Serving: Calories 310; Total Fat 10g (Saturated Fat 4g, Trans Fat 0g); Cholesterol 45mg; Sodium 125mg; Total Carbohydrate 48g (Dietary Fiber 3g, Sugars 32g); Protein 2g Exchanges: 1 Starch, \u00bd Fruit, 1\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\n\n## Bourbon-Chocolate-Pecan Mini Pies\n\nPrep 45 min Total 1 hr 40 min \u25cf 12 pies\n\nCrust\n\n  * 1\u2153 cups all-purpose flour\n  * \u00bc cup unsweetened baking cocoa\n  * 2 tablespoons granulated sugar\n  * \u00be teaspoon salt\n  * \u00bd cup shortening\n  * \u00bc cup ice-cold water\n\nFilling\n\n  * 1 egg\n  * \u00bc cup packed dark brown sugar\n  * 2 tablespoons butter, melted\n  * 2 tablespoons light or dark corn syrup\n  * 2 tablespoons real maple syrup\n  * 1 tablespoon bourbon\n  * \u00bd teaspoon vanilla\n  * \u00be cup whole pecan halves, toasted (page 23)\n  * 4 oz bittersweet baking chocolate, coarsely chopped\n\nTopping\n\n  * \u00be cup whipping cream\n  * 2 tablespoons granulated or powdered sugar\n  * 2 tablespoons shaved bittersweet chocolate\n\n1 Heat oven to 375\u00b0F. In food processor, place flour, cocoa, 2 tablespoons granulated sugar and the salt. Cover and process, using quick on-and-off motions, until blended. Add shortening. Cover and process, using quick on-and-off motions, until particles are size of small peas. With food processor running, pour water all at once through feed tube just until dough leaves side of bowl. Gather dough into a ball.\n\n2 Divide dough into 12 pieces. Place 1 piece in each of 12 ungreased regular-size muffin cups, pressing dough in bottoms and up sides of cups. Sprinkle pecan halves and chopped chocolate into crust-lined cups.\n\n3 In medium bowl, beat egg, brown sugar, butter, corn syrup, maple syrup, bourbon and vanilla with whisk. Spoon mixture evenly into muffin cups, about 4 teaspoons per cup.\n\n4 Bake 20 to 25 minutes or until filling is golden and slightly puffed in center. Cool 30 minutes; remove from pan to cooling rack.\n\n5 In chilled small, deep bowl, beat whipping cream and 2 tablespoons sugar with electric mixer on high speed until soft peaks form. Spoon about 2 tablespoons whipped cream onto each pie. Sprinkle with shaved chocolate.\n\n1 Pie: Calories 370; Total Fat 26g (Saturated Fat 10g, Trans Fat 0g); Cholesterol 35mg; Sodium 180mg; Total Carbohydrate 30g (Dietary Fiber 3g, Sugars 12g); Protein 4g Exchanges: 1 Starch, 1 Other Carbohydrate, 5 Fat Carbohydrate Choices: 2\n\n## Classic French Silk Pie\n\nUse only pasteurized eggs in this recipe to prevent the risk of food poisoning that can result from eating raw eggs.\n\nPrep 30 min Total 4 hr 30 min \u25cf 10 servings\n\n  * One-Crust Pastry (page 586) or Easy Buttermilk Pastry (page 588)\n  * 1 cup butter, softened*\n  * 1\u00bd cups sugar\n  * 1 tablespoon vanilla\n  * 3 oz unsweetened baking chocolate, melted, cooled\n  * 5 pasteurized eggs\n  * 1\u00bd cups Sweetened Whipped Cream (page 622)\n  * Chocolate Curls (page 611), if desired\n\n1 Heat oven to 450\u00b0F. Place pastry in 9-inch glass pie plate. Bake as directed for Baked One-Crust Pie (page 587). Cool completely.\n\n2 In medium bowl, beat butter with electric mixer on medium speed until creamy. Gradually beat in sugar until light and fluffy and sugar begins to dissolve, 3 to 5 minutes. Beat in vanilla and chocolate. Add eggs, one at a time, beating well on high speed after each addition. Beat until light and fluffy and sugar is completely dissolved, about 3 minutes. Pour into cooled baked crust. Refrigerate until set, at least 4 hours.\n\n3 Spread with whipped cream. Garnish with chocolate curls. Store covered in refrigerator.\n\n*Do not use margarine or vegetable oil spreads because filling will be curdled instead of smooth and creamy.\n\n1 Serving: Calories 420; Total Fat 29g (Saturated Fat 13g, Trans Fat 2g); Cholesterol 35mg; Sodium 250mg; Total Carbohydrate 35g (Dietary Fiber 2g, Sugars 22g); Protein 4g Exchanges: 1 Starch, 1 Other Carbohydrate, 6 Fat Carbohydrate Choices: 2\n\nMocha French Silk Pie Beat in 1\u00bd teaspoons instant coffee powder or granules with the baking chocolate.\n\n# Chapter 22: Desserts & Fruits\n\n## Dessert and Fruit Basics\n\nApples\n\nBerries\n\nBuying and Storing Fresh Fruit\n\nCitrus\n\nEasy Chocolate Dessert Garnishes\n\nPears\n\nStone Fruits\n\nTropical Fruits\n\n## Features\n\nLearn to Make: Cr\u00e8me Br\u00fbl\u00e9e\n\nFresh Berries\n\n## Fruit Desserts\n\nAlmond-Cherry Rice Pudding\n\nApple Cobbler\n\nApple Crisp\n\nApple Dumplings\n\nApple-Fig Bread Pudding Cupcakes with Maple Sauce\n\nApplesauce Calorie Smart \u2022 Fast \u2022 Easy\n\nBaked Apples Calorie Smart \u2022 Easy\n\nBlueberry Crisp\n\nBroiled Grapefruit\n\nBroiled Pineapple Calorie Smart \u2022 Fast\n\nCaramel Apple Crisp\n\nCaramel-Maple Pears Slow Cooker \u2022 Easy\n\nDrop Shortcakes\n\nFresh Berry Crisp\n\nFresh Blueberry Cobbler\n\nFresh Cherry Cobbler\n\nPeach Cobbler\n\nRaspberry-Peach Cobbler\n\nRhubarb Crisp\n\nStrawberry Shortcakes\n\n## Cheesecakes and Meringues\n\nCaramel-Cappuccino Cheesecake\n\nChocolate Chip New York Cheesecake\n\nChocolate Meringue Shell\n\nGingerbread\n\nIndividual Meringues\n\nMeringue Shell Easy\n\nNew York Cheesecake\n\n## Puddings and Custards\n\nBaked Custard Calorie Smart \u2022 Easy\n\nBread Pudding with Lemon Curd\n\nBread Pudding with Whiskey Sauce\n\nButterscotch Pudding\n\nCaramel Custard (Cr\u00e8me Caramel)\n\nChocolate Mousse Easy\n\nChocolate Pudding\n\nChocolate\u2013Sour Cream Fondue Fast\n\nCream Puffs\n\nCr\u00e8me Br\u00fbl\u00e9e\n\nDouble-Chocolate Puffs\n\n\u00c9clairs\n\nEggnog Puffs\n\nEnglish Trifle\n\nIce Cream Puffs\n\nIndividual English Trifles\n\nLemon Cr\u00e8me Br\u00fbl\u00e9e\n\nPanna Cotta Easy\n\nProfiteroles\n\nPumpkin Pudding Cake\n\nRice Pudding CALORIE SMART\n\nTiramisu\n\nVanilla Pudding Easy\n\nWhipped Cream Puffs\n\n## Frozen Desserts\n\nCaramel Ice Cream Easy\n\nChocolate Ice Cream\n\nCranberry-Raspberry Sorbet\n\nEggnog Ice Cream\n\nFresh Blueberry Ice Cream\n\nFresh Peach Ice Cream\n\nFresh Raspberry Ice Cream\n\nFresh Strawberry Ice Cream\n\nLemonade Sorbet Calorie Smart \u2022 Easy\n\nLimeade Sorbet\n\nOrange Sorbet\n\nPineapple Sorbet\n\nVanilla Ice Cream Easy\n\nVery Berry Frozen Yogurt Calorie Smart \u2022 Easy\n\nWatermelon Granita Calorie Smart\n\n## Sauces\n\nCaramel Sauce Calorie Smart \u2022 Easy\n\nChocolate Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nCranberry Sauce Calorie Smart \u2022 Easy\n\nHot Fudge Sauce Calorie Smart \u2022 Easy\n\nLemon Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nRaspberry Sauce Calorie Smart \u2022 Fast \u2022 Easy\n\nRhubarb Sauce Calorie Smart \u2022 Fast\n\nStrawberry Sauce\n\nStrawberry-Rhubarb Sauce\n\nSweetened Whipped Cream\n\n## Bonus Recipes\n\nBerry-Chia Breakfast Pudding\n\nBlackberry-Thyme Cocktail\n\nBlueberry-Ginger Scones\n\nChocolate Curls\n\nChocolate Swirls and Filigree\n\nChocolate-Covered Confections\n\nCreamy Strawberry-Poppy Dressing\n\nCucumber-Avocado-Berry Salad\n\nFresh Berries and Greens Smoothie\n\nRaspberry Sangria Pops\n\nCalorie Smart = See Helpful Nutrition and Cooking Information\n\nFast = Ready in 30 minutes or less\n\nEasy = Prep in 10 minutes or less **plus** five ingredients or less\n\nLighter = 25% fewer calories or grams of fat\n\nMake Ahead = Make-ahead directions\n\nSlow Cooker = Slow cooker directions\n\n# Dessert and Fruit Basics\n\nFruit can be the dessert or can be showcased in delicious dessert recipes\u2014either way, it's a perfect choice when you're looking for a lighter ending to your meal. Look here for how to buy and store fresh fruit so it will be at the peak of ripeness when you are ready to enjoy or use it in dessert recipes. Want to make your desserts just a bit more polished? Try one of our easy chocolate decorations.\n\n## BUYING AND STORING FRESH FRUIT\n\nPurchase fruit that is bright in color, heavy for its size (indicating juiciness) and firm to the touch for the variety. Fresh fruit should not show any sign of decay or have soft spots. For berries, check for mold, which can occur in closed containers.\n\nFruits such as berries, pineapples, watermelon, cherries and apples will not ripen after purchase, so look for ripe fruit.\n\nFruit such as bananas, pears, peaches, cantaloupe and kiwifruit will ripen after purchase.\n\nTo ripen fruits after purchase, place them in a paper bag at room temperature 1 to 2 days. Don't use a plastic bag, as mold can form.\n\n## EASY CHOCOLATE DESSERT GARNISHES\n\nNext time you want your dessert to look extra special, try one of these garnishes:\n\nChocolate-Covered Confections: Melt chocolate (follows). Dip dried apricots, fresh strawberries, cherries with stems, pretzels, marshmallows or whole nuts three-fourths of the way into chocolate; shake or gently wipe off excess. If desired, sprinkle or roll confections in candy sprinkles, finely chopped nuts or shredded coconut. Place on cookie sheet lined with waxed paper. Refrigerate 10 minutes to set; chill until serving.\n\nChocolate Curls: Let milk chocolate bar stand at room temperature about 15 minutes. (Semisweet chocolate can also be used, but curls will be smaller.) Pull swivel vegetable peeler or thin sharp knife across block, using long, thin strokes. (Make chocolate shavings by using shorter strokes.) Use a toothpick to lift curls to prevent breakage.\n\nChocolate Swirls and Filigree: Melt chocolate (below). Place chocolate in resealable plastic food-storage bag. Cut off tiny corner of bag. Squeeze bag to pipe swirls inside stemware to use for shakes or ice cream drinks or as a clever way to serve mousse or pudding. Refrigerate 10 minutes to set; chill until serving. For filigree, pipe designs (such as stars, trees, letters, numbers or random patterns) onto cookie sheet lined with waxed paper. (Print out or draw designs on paper and place underneath waxed paper to trace with the chocolate.) Refrigerate until chocolate is set. Gently remove designs from paper; chill. Garnish desserts with filigree just before serving.\n\n## Melting Chocolate\n\nIn 1-quart saucepan, heat 1 cup semisweet or dark chocolate chips with 1 teaspoon shortening over low heat, stirring constantly until melted; remove from heat.\n\n## Berries\n\nChoose firm, plump berries; avoid moldy, mushy or soft spots. Remove any moldy or damaged berries before refrigerating unwashed. Very perishable; use within a few days. Available year-round but best when in season locally for superior flavor.\n\nBlackberries: Largest wild berry. Best used immediately but can be refrigerated up to 2 days.\n\nBlueberries: Juicy and sweet; range from tiny wild blueberries to large cultivated ones.\n\nCranberries: Available fresh September to December (avoid discolored, shriveled ones); also frozen, dried. Fresh berries are very tart. Dried are sweetened; use like raisins.\n\nGooseberries: Varieties include green, white, yellow or red with smooth or fuzzy skins. Limited season (March to June) depending on region; also canned. Great in jams, jellies, desserts.\n\nRaspberries (Red, Golden, Black): Most intensely flavored berry. Red are available year-round; yellow from June to September; black from mid-June to mid-August.\n\nRed Currants: Related to gooseberries. Available June to August in red, black, white. Black typically used for preserves, syrups, liqueurs. Use red or white as snacks or for preserves.\n\nStrawberries: Choose berries that are all red, brightly colored, with bright green tops. Great in salads, desserts, snacks.\n\n## Stone Fruits\n\nChoose plump stone fruits; free of bruises, soft or shriveled spots; not excessively hard. Avoid any greening (in lighter-colored fruit). Refrigerate 3 to 5 days.\n\nCherries: **Sweet cherries** are available fresh (choose firm\u2014not hard\u2014cherries), dried, frozen. Stemmed cherries last longer. Eat out of hand or cooked. **Sour cherries** , also known as **tart cherries** , are smaller and typically too tart to eat fresh. They are, however, delicious in main dishes, jams, jellies and pies, such Cherry Pie, page 596, the recipe featured on the cover of this book. Fresh sour cherries have a very short harvesting season and are typically found only at farmer's markets and roadside stands. Dried, frozen and canned tart cherries are available year around.\n\nNectarines: Sweeter, richer, firmer flesh than peaches. Available mid-spring to late September; peak season is July to August. Choose fragrant fruit that gives to gentle pressure.\n\nPeaches: Fuzzy-skinned varieties range from lightly shaded pink\/creamy white to red-shaded yellow. Flesh ranges from pink-tinged white to yellow-gold. U.S. varieties available May to October in most regions (or imports available during off-season). Choose intensely aromatic fruit that gives to gentle pressure.\n\nPlums: Japanese varieties: larger, rounder; juicy; soft flesh; good for eating. European varieties (including prune plums or sugar plums): smaller; less juicy; firmer flesh; good for baking and drying. **Plumcots** and **Pluots** are varieties of plums crossed with apricots.\n\n## Tropical Fruits\n\nExotic tropical fruits are available at different times of year. Look for new and interesting varieties that are firm, with great color and a fresh aroma.\n\nCherimoya: Creamy white flesh; sweet custard taste (discard large black seeds). Ripen unwashed at room temperature until fruit yields to gentle pressure. For more intense flavor, chill before serving.\n\nDragon Fruit: Pink skin (white or red flesh) or yellow skin (white flesh). Kiwi-pineapple flavor; edible seeds. Choose evenly colored skin with few blotches; avoid very dry, brown stem or brown-tipped \"leaves.\" Fruit should yield to gentle pressure. Eat plain, in salads or smoothies.\n\nKiwano (Horned Melon): Bright, lime-green pulp; banana-cucumber taste; jellylike texture; edible seeds. Golden-orange at peak ripeness. Store at room temperature up to 2 weeks.\n\nMango: Golden-orange flesh with large center seed; very sweet flavor. Chill before serving.\n\nPineapple: Firm yellow flesh; very fragrant with sweet\/tart flavor. Choose firm fruit with a crisp green top and no sign of decay.\n\nPlantain: \"Cooking banana.\" Less sweet; more firm than bananas. Popular in Latin American and some Asian and African cuisines. Avoid those with bruises. Green: mild, squash-like flavor. Ripened: soft, spongy texture. Use both types in side dishes. Store at room temperature.\n\n## Citrus\n\nChoose fruit heavy for size, with smooth-textured skin. Store unwashed at room temperature or refrigerate up to 2 weeks.\n\nBlood Orange (Moro Orange): Deep red; sweet, juicy flesh. Peel is smooth or pitted with red blush. Peak season: December to May. Refrigerate up to 2 weeks.\n\nKumquat: Entirely edible; skin is sweet, pulp is tart. Peak season: December to May. Store unwashed at room temperature up to 2 days or refrigerate up to 2 weeks.\n\nMeyer Lemon: Mild, juicy. Sweeter than common lemons; rind isn't bitter. Peak season: November to May. Refrigerate in plastic bag up to 2 weeks.\n\nTangelo: Cross between grapefruit and tangerine. Range of sizes. Yellow-orange to deep orange; smooth skin. Sweet-tart; juicy; few seeds. Minneola is most common variety. Peak season: November to March. Refrigerate up to 2 weeks.\n\n## Apples\n\nChoose firm, smooth-skinned apples without bruises, blemishes or musty smell. Choose apples based on purpose\u2014eating, baking or cooking. Store refrigerated in plastic bag up to 1 month.\n\nSweet: Crispin\/Mutsu (crisp), Fuji (crisp), Gala (crisp), Ginger Gold (crisp), Golden Delicious (crisp), Honeycrisp (crisp), Honey Gold (slightly crisp), Red Delicious (slightly crisp), Regent (crisp)\n\nSlightly Sweet: Fireside (slightly crisp), Prairie Spy (crisp)\n\nSweet-Tart: Braeburn (crisp), Jonagold (crisp), Jonamac (tender), McIntosh (tender)\n\nSlightly Tart: Cortland (slightly crisp), Ginger Gold (crisp), Ida Red (slightly crisp), Jonathan (crisp), Newtown Pippin (crisp), Northern Spy (crisp), Paula Red (slightly crisp), Rome (crisp), Winesap (crisp)\n\nTart: Granny Smith (crisp), Haralson (crisp)\n\n## Pears\n\nChoose blemish-free pears. Ripen at room temperature; store in refrigerator 1 month.\n\nCooking\/Baking Pears: Anjou (green, red when slightly ripe), Bosc, French Butter, Seckel\n\nEating\/Salad Pears: Anjou (green, red when ripe), Asian, Bartlett, Cornice, French Butter\n\n## Baked Apples Calorie Smart \u2022 EASY\n\nOften overlooked, baked apples make a perfect last-minute dessert. Use the larger amount of brown sugar for very tart apples.\n\nPrep 10 min Total 50 min \u25cf 4 servings\n\n  * 4 large unpeeled tart cooking apples (Granny Smith or Rome)\n  * 2 to 4 tablespoons packed brown sugar\n  * 4 teaspoons butter\n  * \u00bd teaspoon ground cinnamon\n\n1 Heat oven to 375\u00b0F. Core apples to within \u00bd inch of bottom. Remove 1-inch strip of peel from around middle of each apple, or peel upper half of each apple to prevent splitting. Place apples in ungreased 8-inch square (2-quart) glass baking dish.\n\n2 In center of each apple, place 1\u00bd teaspoons to 1 tablespoon sugar, 1 teaspoon butter and \u215b teaspoon cinnamon. Sprinkle with additional cinnamon if desired. Pour water into baking dish until \u00bc inch deep.\n\n3 Bake 30 to 40 minutes or until apples are tender when pierced with fork. (Time will vary depending on size and variety of apple.) Spoon syrup in dish over apples several times during baking if desired.\n\n1 Serving: Calories 200; Total Fat 4.5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 10mg; Sodium 25mg; Total Carbohydrate 39g (Dietary Fiber 5g, Sugars 28g); Protein 0g Exchanges: 2 Fruit, \u00bd Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 2\u00bd\n\nMICROWAVE Directions Prepare apples as directed. Place each apple in 10-ounce custard cup or individual microwavable casserole or bowl. Do not add water. Microwave uncovered on High 5 to 10 minutes, rotating cups half turn after 3 minutes, until apples are tender when pierced with fork.\n\n## Applesauce Calorie Smart \u2022 FAst \u2022 EASY\n\nPrep 20 min Total 20 min \u25cf 3 cups\n\n  * 4 medium apples (1\u00bd lb), peeled, quartered\n  * \u00bd cup water\n  * \u00bc cup packed brown sugar or 3 to 4 tablespoons granulated sugar\n  * \u00bc teaspoon ground cinnamon\n  * \u215b teaspoon ground nutmeg\n\n1 In 2-quart saucepan, heat apples and water to boiling over medium heat, stirring occasionally; reduce heat. Simmer uncovered 5 to 10 minutes, stirring occasionally to break up apples, until tender.\n\n2 Stir in remaining ingredients. Heat to boiling; boil and stir 1 minute. Serve warm or chilled. Store covered in refrigerator.\n\n\u00bd Cup: Calories 90; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 22g (Dietary Fiber 2g, Sugars 18g); Protein 0g Exchanges: 1 Fruit, \u00bd Other Carbohydrate Carbohydrate Choices: 1\u00bd\n\nCaramel-Maple Pears\n\n## Caramel-Maple Pears SLOW Cooker \u2022 EASY\n\nPrep 15 min Total 2 hr 15 min \u25cf 6 servings\n\n  * 6 yellow- or red-skinned pears, cored from bottom leaving stems attached\n  * 1 cup caramel ice cream topping\n  * \u2153 cup real maple syrup\n  * Grated lemon peel\n\n1 Spray 5- to 6-quart slow cooker with cooking spray. Place pears upright in cooker. (If all pears do not fit in bottom of cooker, place remaining pears suspended between 2 others in cooker.)\n\n2 In small bowl, mix brown ice cream topping and syrup; pour over pears.\n\n3 Cover; cook on High heat setting 2 hours to 2 hours 30 minutes to 3 hours 30 minutes.\n\n4 Remove pears from slow cooker; place upright in individual dessert dishes. Spoon about \u00bc cup topping mixture over each pear; sprinkle with lemon peel.\n\n1 Serving: Calories 310; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 190mg; Total Carbohydrate 76g (Dietary Fiber 6g, Sugars 55g); Protein 1g Exchanges: 1 Starch, \u00bd Fruit, 3\u00bd Other Carbohydrate Carbohydrate Choices: 5\n\nApple Dumplings\n\n## Apple Dumplings\n\nThe recipe for apple dumplings was available free in Gold Medal\u2122 flour bags, as noted in a 1938 advertisement, and has also been included in many of the company's cookbooks over the years, beginning with the 1904 Christmas edition of _Gold Medal Flour Cook Book_.\n\nPrep 55 min Total 1 hr 35 min \u25cf 6 servings\n\n  * Two-Crust Pastry (page 586)\n  * 6 small cooking apples (Golden Delicious, Braeburn or Rome), about 3 inches in diameter*\n  * 3 tablespoons raisins, dried cranberries or dried cherries, if desired\n  * 3 tablespoons chopped nuts, if desired\n  * 2\u00bd cups packed brown sugar\n  * 1\u2153 cups water\n  * 2 tablespoons butter, softened\n  * \u00bc teaspoon ground cinnamon\n  * Cream or Sweetened Whipped Cream (page 622), if desired\n\n1 Heat oven to 425\u00b0F. Make pastry as directed, except roll two-thirds of the pastry into 14-inch square; cut into 4 (7-inch) squares. Roll remaining pastry into 14x7-inch rectangle; cut into 2 (7-inch) squares.\n\n2 Peel and core apples; place 1 apple on each pastry square. Mix raisins and nuts; fill apples with mixture. Moisten corners of pastry squares. Bring 2 opposite corners up over apple and pinch together. Repeat with remaining corners, and pinch edges of pastry to seal.\n\n3 Place dumplings in ungreased 13x9-inch (3-quart) glass baking dish. In 2-quart saucepan, heat brown sugar, water, butter and cinnamon to boiling over high heat, stirring frequently. Carefully pour syrup around dumplings.\n\n4 Bake about 40 minutes, spooning syrup over dumplings two or three times, until crust is golden and apples are tender when pierced with small knife or toothpick. Serve warm or cool with cream.\n\n*Dough squares will not be large enough to seal larger apples.\n\n1 Serving: Calories 680; Total Fat 32g (Saturated Fat 9g, Trans Fat 5g); Cholesterol 10mg; Sodium 450mg; Total Carbohydrate 94g (Dietary Fiber 4g, Sugars 45g); Protein 5g Exchanges: 2 Starch, 4 Other Carbohydrate, 6 Fat Carbohydrate Choices: 6\n\n### Making Apple Dumplings\n\nMoisten corners of pastry squares. Bring 2 opposite corners up over apple and pinch together.\n\nRepeat with remaining corners; pinch edges of pastry to seal.\n\nApple Crisp\n\n## Apple Crisp\n\nPrep 20 min Total 1 hr \u25cf 6 servings\n\n  * 6 medium tart cooking apples (Granny Smith or Rome), peeled, sliced (about 6 cups)\n  * \u00be cup packed brown sugar\n  * \u00bd cup all-purpose flour\n  * \u00bd cup quick-cooking or old-fashioned oats\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon ground nutmeg\n  * \u2153 cup cold butter\n  * Cream or ice cream, if desired\n\n1 Heat oven to 375\u00b0F. Spray bottom and sides of 8-inch square (2-quart) glass baking dish with cooking spray.\n\n2 Spread apples in baking dish. In medium bowl, mix brown sugar, flour, oats, cinnamon and nutmeg. Cut in butter, using pastry blender or fork, until mixture is crumbly. Sprinkle evenly over apples.\n\n3 Bake 35 to 40 minutes or until topping is golden brown and apples are tender when pierced with fork. Serve warm with cream.\n\n1 Serving: Calories 330; Total Fat 11g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 25mg; Sodium 85mg; Total Carbohydrate 55g (Dietary Fiber 4g, Sugars 38g); Protein 3g Exchanges: 1 Starch, 1 Fruit, 1\u00bd Other Carbohydrate, 2 Fat Carbohydrate Choices: 3\u00bd\n\nBlueberry Crisp Substitute 6 cups fresh or frozen (thawed and drained) blueberries for the apples.\n\nCaramel Apple Crisp Toss apples with \u00bd cup butterscotch caramel topping before spreading in pan.\n\nRhubarb Crisp Substitute 6 cups chopped fresh or frozen (thawed and drained) rhubarb for the apples. Sprinkle \u00bd cup granulated sugar over rhubarb; stir to combine. Continue as directed in Step 2.\n\nFresh Berry Crisp\n\n## Fresh Berry Crisp\n\nPrep 15 min Total 45 min \u25cf 6 servings\n\n  * 3 cups fresh strawberries, sliced\n  * 3 tablespoons cornstarch\n  * 2 tablespoons granulated sugar\n  * 1 pint (2 cups) fresh blueberries\n  * 1 pint (2 cups) fresh raspberries\n  * \u2154 cup packed brown sugar\n  * \u00bd cup whole wheat flour\n  * \u00bd cup old-fashioned or quick-cooking oats\n  * \u00bd teaspoon ground cinnamon\n  * \u00bc teaspoon salt\n  * \u2153 cup butter, softened\n  * Ice cream or Sweetened Whipped Cream (page 622), if desired\n\n1 Heat oven to 350\u00b0F. In 2-quart saucepan, mash 2 cups of the strawberries. Stir in cornstarch and granulated sugar. Cook over medium heat, stirring constantly, until mixture boils. Boil and stir 1 minute.\n\n2 Carefully stir in blueberries, raspberries and remaining 1 cup strawberries. Pour mixture into ungreased 8-inch square (2-quart) glass baking dish or 9-inch glass pie plate.\n\n3 In small bowl, mix brown sugar, flour, oats, cinnamon, salt and butter with pastry blender or fork until crumbly; sprinkle over berry mixture.\n\n4 Bake about 30 minutes or until topping is golden brown and filling is bubbly. Serve warm with ice cream.\n\n1 Serving: Calories 370; Total Fat 12g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 25mg; Sodium 180mg; Total Carbohydrate 62g (Dietary Fiber 7g, Sugars 39g); Protein 3g Exchanges: 1 Starch, 1 Fruit, 2 Other Carbohydrate, 2\u00bd Fat Carbohydrate Choices: 4\n\n## Fresh Berries\n\n  1. In medium bowl, whisk together 1 cup raspberry sherbet, 1 single-serve container raspberry yogurt, and \u00bd cup each fresh raspberries and sangria. Divide between 4 (5-ounce) paper cups. Cover cups with foil; insert flat craft sticks with rounded ends through foil into center of pops. Freeze 4 to 6 hours or until firm. Remove cups to serve.\n  2. 1. **Creamy Strawberry-Poppy Dressing:** For 4 servings, in small bowl, whisk together \u2153 cup canola oil, 2 tablespoons raspberry vinegar, 3 tablespoons sugar, 1 tablespoon sour cream, 1\u00bd teaspoons Dijon mustard and \u00bc teaspoon poppy seed. Fold in \u2153 cup finely chopped fresh strawberries. To serve, drizzle over salad greens and sprinkle with chopped walnuts and sliced strawberries.\n  3. 2. **Blueberry-Ginger Scones:** Make Scones (page 490) as directed, except\u2014increase whipping cream to \u00bd cup. Stir in \u00be cup fresh blueberries with egg. Omit kneading step. Sprinkle 8-inch round with 1 tablespoon coarse sugar mixed with 2 teaspoons finely chopped crystallized ginger; press lightly into dough.\n  4. 3. **Blackberry-Thyme Cocktail:** For 3 cocktails, place thyme leaves from 1 fresh stem in food processor. Add \u00bd cup fresh blackberries. Process until almost smooth; place in pitcher; stir in 3 shots blackberry vodka, 1 tablespoon fresh lemon juice and 2 cups tonic water. Refrigerate at least 2 hours to blend flavors. Pour into sugar-rimmed glasses; garnish with additional fresh blackberries and thyme stem if desired.\n  5. 4. **Fresh Berries and Greens Smoothie:** For 2 smoothies, place \u00bd cup orange juice in blender. Add 1 cup kale or spinach leaves, 1 tablespoon flaxseed, 1 small banana, 1 cup fresh berries and \u00bd teaspoon coconut oil. Add 1 to 2 cups water; blend until smooth.\n  6. 5. **Cucumber-Avocado-Berry Salad:** For 4 to 6 servings, in medium bowl, mix \u00bc cup olive oil, 4 teaspoons red wine vinegar, \u00bc teaspoon salt and \u215b teaspoon pepper; toss with 1 chopped avocado. Add 1 sliced English cucumber, 2 cups bite-size pieces fresh berries and \u00bd cup crumbled queso fresco or feta cheese; toss. Serve immediately.\n  7. 6. **Berry-Chia Breakfast Pudding:** For 2 servings, mix 1 cup almond milk, \u00bc cup chia seeds and 2 tablespoons honey. Cover and refrigerate overnight to soften seeds. To serve, spoon into bowls; top each with \u00bd cup fresh berries and \u00bd teaspoon grated lime peel.\n\nBlueberry-Ginger Scones\n\nPeach Cobbler\n\n## Peach Cobbler\n\nPrep 25 min Total 55 min \u25cf 6 servings\n\n  * \u00bd cup sugar\n  * 1 tablespoon cornstarch\n  * \u00bc teaspoon ground cinnamon\n  * 4 cups sliced peeled fresh peaches (6 medium)*\n  * 1 teaspoon lemon juice\n  * 1 cup all-purpose flour\n  * 1 tablespoon sugar\n  * 1\u00bd teaspoons baking powder\n  * \u00bd teaspoon salt\n  * 3 tablespoons cold butter\n  * \u00bd cup milk\n  * 2 tablespoons sugar, if desired\n  * Cream, if desired\n\n1 Heat oven to 400\u00b0F. In 2-quart saucepan, mix \u00bd cup sugar, the cornstarch and cinnamon. Stir in peaches and lemon juice. Cook over medium-high heat 4 to 5 minutes, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute. Pour into ungreased 2-quart casserole; keep peach mixture hot in oven.\n\n2 In medium bowl, mix flour, 1 tablespoon sugar, the baking powder and salt. Cut in butter, using pastry blender or fork, until mixture looks like fine crumbs. Stir in milk. Drop dough by 6 spoonfuls onto hot peach mixture. Sprinkle 2 tablespoons sugar over dough.\n\n3 Bake 25 to 30 minutes or until topping is golden brown. Serve warm with cream.\n\n*Or substitute 4 cups frozen sliced peaches (from two 1-pound bags), thawed and drained, for the fresh peaches.\n\n1 Serving: Calories 300; Total Fat 13g (Saturated Fat 7g, Trans Fat 0.5g); Cholesterol 40mg; Sodium 240mg; Total Carbohydrate 44g (Dietary Fiber 4g, Sugars 31g); Protein 3g Exchanges: 1\u00bd Starch, 1\u00bd Fruit, 1 Fat Carbohydrate Choices: 3\n\nApple Cobbler Substitute 4 cups sliced peeled tart cooking apples (Granny Smith or Rome) for the peaches.\n\nFresh Blueberry Cobbler Substitute 4 cups fresh blueberries for the peaches. Omit cinnamon.\n\nFresh Cherry Cobbler Substitute 4 cups pitted tart red or sweet fresh cherries for the peaches. Increase sugar in cherry mixture to 1\u00bc cups for tart cherries and 1 cup for sweet cherries, and cornstarch to 3 tablespoons. Substitute \u00bc teaspoon almond extract for the lemon juice.\n\nRaspberry-Peach Cobbler Substitute 2 cups fresh raspberries for 1 cup of the peaches.\n\n## Strawberry Shortcakes\n\nPrep 15 min Total 2 hr \u25cf 6 servings\n\n  * 1 quart (4 cups) fresh strawberries, sliced\n  * 1 cup sugar\n  * 2 cups all-purpose flour\n  * 1 tablespoon baking powder\n  * \u00bd teaspoon salt\n  * \u00bd cup cold butter\n  * \u2154 cup milk\n  * 1 egg, slightly beaten\n  * Sweetened Whipped Cream (follows), if desired\n\n1 In large bowl, stir strawberries and \u00bd cup of the sugar until well mixed. Let stand about 1 hour so juice forms.\n\n2 Heat oven to 375\u00b0F. Grease bottom and side of 8- or 9-inch round cake pan with shortening; lightly flour.\n\n3 In medium bowl, mix flour, remaining \u00bd cup sugar, the baking powder and salt. Cut in butter, using pastry blender or fork, until mixture looks like coarse crumbs. Stir in milk and egg just until blended. Spoon and spread batter in pan.\n\n4 Bake 30 to 35 minutes or until toothpick inserted in center comes out clean. Cool 10 minutes.\n\n5 Place cooling rack upside down over pan; turn rack and pan over and remove pan. Carefully invert shortcake onto serving plate. Cut shortcake into 6 wedges. If desired, split each wedge horizontally in half. Fill and top with strawberries and whipped cream.\n\n1 Serving: Calories 490; Total Fat 18g (Saturated Fat 8g, Trans Fat 1g); Cholesterol 80mg; Sodium 570mg; Total Carbohydrate 75g (Dietary Fiber 4g, Sugars 41g); Protein 7g Exchanges: 2 Starch, 1 Fruit, 2 Other Carbohydrate, 3\u00bd Fat Carbohydrate Choices: 5\n\nDrop Shortcakes Heat oven to 425\u00b0F. Make dough as directed. Onto ungreased cookie sheet, drop dough in 6 mounds about 2 inches apart. Bake 12 to 14 minutes or until golden brown.\n\nSweetened Whipped Cream Chill bowl and beaters in freezer or refrigerator 10 to 20 minutes. In chilled medium, deep bowl, beat \u00bd cup whipping cream, 1 tablespoon powdered or granulated sugar and \u00bd teaspoon vanilla with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat just until soft peaks form. Do not overbeat or mixture will curdle. Makes 1 cup. Double ingredients to make 2 cups.\n\n### Cutting Shortcake\n\nCut shortcake into 6 wedges.\n\nIf desired, split each wedge horizontally in half.\n\nStrawberry Shortcake\n\n## Broiled Pineapple Calorie Smart \u2022 FAST\n\nPrep 15 min Total 15 min \u25cf 4 servings\n\n  * 1 can (20 oz) sliced pineapple in juice, drained, or 12 slices (\u00bd inch thick) fresh pineapple\n  * 1 tablespoon packed brown sugar\n  * 2 tablespoons lime juice\n  * 2 tablespoons honey\n  * \u00bd cup vanilla fat-free yogurt\n  * 1 teaspoon honey\n  * \u00bd teaspoon grated lime peel\n\n1 Set oven control to broil. Place pineapple slices on ungreased broiler pan. In small bowl, mix brown sugar, lime juice and 2 tablespoons honey; drizzle over pineapple. Broil with tops of pineapple 4 inches from heat 6 to 8 minutes, turning once, until light brown.\n\n2 In another small bowl, mix yogurt, 1 teaspoon honey and the lime peel.\n\n3 For each serving, top 3 pineapple slices with 2 tablespoons yogurt mixture.\n\n1 Serving: Calories 150; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 20mg; Total Carbohydrate 35g (Dietary Fiber 1g, Sugars 34g); Protein 2g Exchanges: 1 Fruit, 1\u00bd Other Carbohydrate Carbohydrate Choices: 2\n\nBroiled Grapefruit Omit lime. Cut 2 medium grapefruits in half. Place halves, cut sides up, on broiler pan. Top each grapefruit half with brown sugar mixture. Serve with yogurt mixture.\n\nGingerbread\n\n## Gingerbread\n\nPrep 10 min Total 1 hr 5 min \u25cf 9 servings\n\n  * 2\u2153 cups all-purpose flour\n  * 1 teaspoon baking soda\n  * 1 teaspoon ground ginger\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon salt\n  * \u00bd cup butter, softened, or shortening\n  * \u2153 cup sugar\n  * 1 cup molasses*\n  * \u00be cup hot water\n  * 1 egg\n  * Lemon Sauce (page 641), if desired\n  * Sweetened Whipped Cream (page 622), if desired\n\n1 Heat oven to 325\u00b0F. Grease bottom and sides of 9-inch square pan with shortening; lightly flour.\n\n2 In large bowl, beat all ingredients except lemon sauce and whipped cream with electric mixer on low speed 30 seconds, scraping bowl constantly. Beat on medium speed 3 minutes, scraping bowl occasionally. Pour batter into pan.\n\n3 Bake 50 to 55 minutes or until toothpick inserted in center comes out clean. Serve warm with lemon sauce and whipped cream.\n\n*Full-flavored (dark) molasses will result in a bold and robust flavor. For milder flavor, use mild-flavored (light) molasses. For extra-mild molasses flavor, use \u00bd cup light molasses and \u00bd cup real maple syrup.\n\n1 Serving: Calories 370; Total Fat 12g (Saturated Fat 3g, Trans Fat 2g); Cholesterol 25mg; Sodium 360mg; Total Carbohydrate 60g (Dietary Fiber 1g, Sugars 28g); Protein 4g Exchanges: 1 Starch, 3 Other Carbohydrate, 2 Fat Carbohydrate Choices: 4\n\nPumpkin Pudding Cake\n\n## Pumpkin Pudding Cake\n\nPrep 15 min Total 1 hr 15 min \u25cf 9 servings\n\n  * 1\u00bd cups all-purpose flour\n  * \u00be cup granulated sugar\n  * 1 teaspoon baking powder\n  * \u00bd teaspoon baking soda\n  * \u00bd teaspoon salt\n  * 1\u00bd teaspoons ground cinnamon\n  * \u00bd cup buttermilk\n  * \u00bd cup canned pureed pumpkin (not pumpkin pie mix)\n  * \u00bc cup butter, melted\n  * \u2153 cup chopped pecans\n  * 1\u00bd cups water\n  * \u00be cup packed brown sugar\n\n1 Heat oven to 350\u00b0F. In large bowl, mix flour, granulated sugar, baking powder, baking soda, salt and cinnamon. Add buttermilk, pumpkin and butter; mix well with spoon. Spread in ungreased 8- or 9-inch square pan. Sprinkle with pecans, pressing into top.\n\n2 In small saucepan, mix water and brown sugar. Heat over medium heat until simmering, stirring until sugar is dissolved. Carefully pour over batter. (Nuts may float, but will settle as cake bakes.)\n\n3 Bake 35 to 45 minutes or until toothpick inserted in center comes out clean. Cool 15 minutes. Serve warm, spooning sauce from bottom of pan over cake.\n\n1 Serving: Calories 300; Total Fat 9g (Saturated Fat 3.5g, Trans Fat 0g); Cholesterol 15mg; Sodium 320mg; Total Carbohydrate 53g (Dietary Fiber 1g, Sugars 36g); Protein 3g Exchanges: 1 Starch, 2\u00bd Other Carbohydrate, 1\u00bd Fat Carbohydrate Choices: 3\u00bd\n\nIndividual English Trifles\n\n## English Trifle\n\nPrep 30 min Total 3 hr 30 min \u25cf 10 servings\n\n  * \u00bd cup sugar\n  * 3 tablespoons cornstarch\n  * \u00bc teaspoon salt\n  * 3 cups milk\n  * \u00bd cup dry sherry or other dry white wine or white grape juice\n  * 3 egg yolks, beaten\n  * 3 tablespoons butter, softened\n  * 1 tablespoon vanilla\n  * 2 packages (3 oz each) ladyfingers (24 ladyfingers)*\n  * \u00bd cup strawberry preserves\n  * 2 cups fresh strawberries, sliced, or 1 box (10 oz) frozen sliced strawberries, thawed, drained\n  * 1 cup whipping cream\n  * 2 tablespoons sugar\n  * 2 tablespoons slivered almonds, toasted (page 23)\n\n1 In 3-quart saucepan, mix \u00bd cup sugar, the cornstarch and salt. Gradually stir in milk and sherry. Heat to boiling over medium heat, stirring constantly. Boil and stir 1 minute.\n\n2 Gradually stir at least half of the hot mixture into egg yolks, then stir back into hot mixture in saucepan. Boil and stir 1 minute; remove from heat. Stir in butter and vanilla. Press plastic wrap on surface of pudding to prevent tough layer from forming on top. Refrigerate about 3 hours or until chilled.\n\n3 Split ladyfingers horizontally in half. Spread cut sides with preserves. In 2-quart trifle bowl or glass serving bowl, layer one-fourth of the ladyfingers, cut sides up, half of the strawberries and half of the pudding; repeat layers. Arrange remaining ladyfingers around edge of bowl in upright position with cut sides toward center. (It may be necessary to gently ease ladyfingers down into pudding about 1 inch so they remain upright.) Cover and refrigerate until serving time.\n\n4 In chilled large, deep bowl, beat whipping cream and 2 tablespoons sugar with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat until stiff peaks form. Spread over trifle. Sprinkle with almonds. Cover and refrigerate any remaining trifle.\n\n*Or substitute 2 packages (10.75 ounces each) frozen (thawed) pound cake for the ladyfingers. Place each pound cake on its side; carefully cut each cake lengthwise into 3 slices. Lay slices cut-side down; cut each slice crosswise into 4 slices (to total 24 slices). Spread one side of each slice with preserves and layer as directed.\n\n1 Serving: Calories 360; Total Fat 16g (Saturated Fat 8g, Trans Fat 1g); Cholesterol 110mg; Sodium 180mg; Total Carbohydrate 46g (Dietary Fiber 1g, Sugars 35g); Protein 6g Exchanges: 2 Starch, 1 Other Carbohydrate, 3 Fat Carbohydrate Choices: 3\n\nIndividual English Trifles Prepare as directed through step 2. Do not split ladyfingers; omit preserves. Cut ladyfingers in half crosswise; layer 3 halves in the bottom of 8 (10 oz) glasses. Divide 1 cup of the strawberries among glasses; top each with about 3 tablespoons of the pudding. Repeat layers, except reserve 8 ladyfinger halves for garnish, if desired (using only 2 ladyfinger halves for second layer). Divide whipped cream and almonds among glasses. Garnish each glass with a sliced strawberry and reserved ladyfinger half.\n\n## New York Cheesecake\n\nCheesecake seems so special to your guests, they'll think you went all out. But don't say a word. We've worked hard to find the secrets to a perfectly baked crust and a creamy, rich filling\u2014now you will know them, too.\n\nPrep 45 min Total 15 hr 45 min \u25cf 16 servings\n\nCrust\n\n  * 1 cup all-purpose flour\n  * \u00bd cup butter, softened\n  * \u00bc cup sugar\n  * 1 egg yolk\n\nCheesecake\n\n  * 5 packages (8 oz each) cream cheese, softened\n  * 1\u00be cups sugar\n  * 3 tablespoons all-purpose flour\n  * 1 tablespoon grated orange peel, if desired\n  * 1 tablespoon grated lemon peel, if desired\n  * \u00bc teaspoon salt\n  * \u00bc cup whipping cream\n  * 5 whole eggs\n  * 2 egg yolks\n  * \u00be cup whipping cream\n  * \u2153 cup slivered almonds, toasted (page 23), or fresh fruit, if desired\n\n1 Heat oven to 400\u00b0F. Spray bottom and side of 9-inch springform pan with cooking spray; remove bottom. In medium bowl, mix crust ingredients, using pastry blender or fork, until dough forms; gather into a ball. Press one-third of the dough on bottom of pan. Place on cookie sheet. Bake 8 to 10 minutes or until light golden brown; cool 10 minutes.\n\n2 Assemble bottom and side of pan; secure side. Press remaining dough 2 inches up side of pan. Wrap outside bottom and side of pan with heavy-duty foil to prevent leaking.\n\n3 Increase oven temperature to 450\u00b0F. In large bowl, beat cream cheese, 1\u00be cups sugar, 3 tablespoons flour, the orange peel, lemon peel and salt with electric mixer on medium speed about 1 minute or until smooth. On low speed, beat in \u00bc cup whipping cream, the eggs and egg yolks just until blended. Pour into crust.\n\n4 Bake 15 minutes. Reduce oven temperature to 200\u00b0F. Bake 1 hour 25 minutes to 1 hour 40 minutes longer or until cheesecake is set at least 2 inches from edge of pan but center of cheesecake still jiggles slightly when moved and surface is golden brown. (Cheesecake will become firm during refrigeration.) Do not insert knife into cheesecake because the hole may cause cheesecake to crack. Turn off oven; leave cheesecake in oven 30 minutes longer. Remove from oven and cool in pan on cooling rack away from drafts 30 minutes.\n\n5 Without releasing side of pan, run knife around side of pan carefully to loosen cheesecake. Refrigerate uncovered about 3 hours or until chilled; cover and continue refrigerating at least 9 hours or overnight before serving.\n\n6 Run knife around side of pan to loosen cheesecake again; remove side of pan. Leave cheesecake on pan bottom to serve. In chilled medium, deep bowl, beat \u00be cup whipping cream with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat until stiff peaks form. Spread whipped cream over top of cheesecake. Sprinkle with almonds. Store covered in refrigerator. To cut cheesecake, use long, thin-bladed, non-serrated knife. Dip knife into hot water after each cut, cleaning it off with paper towel.\n\n1 Serving: Calories 520; Total Fat 38g (Saturated Fat 22g, Trans Fat 1g); Cholesterol 215mg; Sodium 310mg; Total Carbohydrate 35g (Dietary Fiber 0g, Sugars 27g); Protein 9g Exchanges: 2\u00bd Other Carbohydrate, 1 High-Fat Meat, 6 Fat Carbohydrate Choices: 2\n\nLighter Directions For 19 grams of fat and 330 calories per serving, omit crust. Move oven rack to lowest position. Heat oven to 425\u00b0F. Lightly grease side only of 9-inch springform pan with shortening. Mix \u00be cup graham cracker crumbs, 2 tablespoons melted butter and 2 tablespoons sugar; press evenly in bottom of pan. Use reduced-fat cream cheese (Neufch\u00e2tel); increase flour to \u00bc cup. Omit \u00bc cup whipping cream. Substitute 1\u00bc cups fat-free egg product for the 5 eggs. Continue as directed in Step 3. Omit \u00be cup whipping cream and the almonds. Serve with fruit if desired.\n\nChocolate Chip New York Cheesecake Fold 1 cup miniature semisweet chocolate chips into cream cheese mixture before pouring into crust.\n\nNew York Cheesecake\n\n### Making Cheesecake\n\nPress one-third of the dough evenly on bottom of springform pan. Place on cookie sheet. Bake 8 to 10 minutes.\n\nAssemble bottom and side of pan; secure side. Press remaining dough 2 inches up side of pan.\n\nWrap outside and bottom of pan with foil to prevent leaking during baking.\n\nWithout releasing side of pan, run knife around side of pan to loosen baked cheesecake.\n\nSpread whipped cream over top of cheesecake. Sprinkle with almonds.\n\nCut with long, thin-bladed, non-serrated knife. After each cut, dip knife into hot water and clean it off with towel.\n\n## Caramel-Cappuccino Cheesecake\n\nPrep 30 min Total 8 hr 50 min \u25cf 16 servings\n\nCrust\n\n  * 1\u00bc cups chocolate cookie crumbs (from 15-oz box)\n  * \u00bc cup butter, melted\n\nFilling\n\n  * 2 tablespoons instant espresso coffee powder or granules\n  * 2 teaspoons vanilla\n  * 4 packages (8 oz each) cream cheese, softened\n  * 1\u00bd cups granulated sugar\n  * 4 eggs\n  * 1 teaspoon ground cinnamon\n  * \u00bc cup caramel topping\n\nTopping\n\n  * 1 cup whipping cream\n  * 2 tablespoons powdered sugar or granulated sugar\n  * \u00bc cup caramel topping\n\n1 Heat oven to 300\u00b0F. Wrap outside of 10-inch springform pan with foil. To minimize cracking, place shallow pan half full of hot water on bottom oven rack.\n\n2 In small bowl, mix cookie crumbs and butter with fork. Press mixture evenly over bottom of pan. Refrigerate crust while preparing filling. In another small bowl, stir espresso powder and vanilla until dissolved; set aside.\n\n3 In large bowl, beat cream cheese and granulated sugar with electric mixer on medium speed until light and fluffy. Beat in eggs, one at a time, just until blended. Add espresso mixture, cinnamon and \u00bc cup caramel topping; beat about 30 seconds or until mixture is well blended. Pour over crust.\n\n4 Bake 1 hour 10 minutes to 1 hour 20 minutes or until cheesecake is set at least 2 inches from edge of pan but center of cheesecake still jiggles slightly when moved. Turn oven off; open oven door at least 4 inches. Let cheesecake remain in oven 30 minutes. Run small metal spatula around edge of pan to loosen cheesecake. Cool in pan on cooling rack 30 minutes. Refrigerate at least 6 hours or overnight.\n\n5 Just before serving, run small metal spatula around edge of pan; carefully remove side of pan. In chilled medium, deep bowl, beat 1 cup whipping cream and powdered sugar with electric mixer on high speed until soft peaks form. Spread whipped cream over top of cheesecake; drizzle with \u00bc cup caramel topping. Cover and refrigerate any remaining cheesecake.\n\n1 Serving: Calories 440; Total Fat 30g (Saturated Fat 18g, Trans Fat 1g); Cholesterol 140mg; Sodium 300mg; Total Carbohydrate 35g (Dietary Fiber 0g, Sugars 29g); Protein 7g Exchanges: \u00bd Starch, 2 Other Carbohydrate, 1 High-Fat Meat, 4 Fat Carbohydrate Choices: 2\n\nProfiterole and Cream Puff\n\n## Cream Puffs\n\nPrep 30 min Total 2 hr 40 min \u25cf 12 cream puffs\n\nPuffs\n\n  * \u00bd cup butter, cut into pieces\n  * 1 cup water\n  * 1 cup all-purpose flour\n  * 4 whole eggs\n\nFilling\n\n  * \u2153 cup granulated sugar\n  * 2 tablespoons cornstarch\n  * 2 cups milk\n  * 2 egg yolks, slightly beaten\n  * 2 tablespoons butter, softened\n  * 2 teaspoons vanilla\n  * Powdered sugar, if desired\n\n1 Heat oven to 400\u00b0F. In 2-quart saucepan, heat \u00bd cup butter and the water over high heat, stirring occasionally, until boiling rapidly. Stir in flour; reduce heat to low. With wooden spoon, beat vigorously over low heat about 1 minute or until mixture forms a ball; remove from heat.\n\n2 Add eggs, one at a time, beating vigorously with spoon after each addition until mixture is smooth and glossy.* Onto ungreased cookie sheet, drop dough by slightly less than \u00bc cupfuls in mounds about 3 inches apart.\n\n3 Bake 35 to 40 minutes or until puffed and golden. Remove from cookie sheet to cooling rack; prick side of each puff with tip of sharp knife to release steam. Cool away from drafts 30 minutes.\n\n4 Meanwhile, in 2-quart saucepan, stir granulated sugar and cornstarch with whisk; gradually stir in milk. Cook and stir over medium heat until mixture thickens and boils. Boil and stir 1 minute. Gradually stir at least half of the hot mixture into egg yolks, then stir back into hot mixture in saucepan. Boil and stir 1 minute; remove from heat. Stir in 2 tablespoons butter and the vanilla. Press plastic wrap on surface of filling to prevent tough layer from forming on top. Refrigerate at least 1 hour or until cool.\n\n5 Using sharp knife, cut off top third of each puff; reserve tops. Pull out any strands of soft dough from puffs. Just before serving, fill bottom of each puff with about 2 rounded tablespoons filling; replace tops. Sprinkle with powdered sugar. Store cream puffs covered in refrigerator.\n\n* Or beat in eggs with electric mixer on medium speed. Beat 1 minute after adding each egg.\n\n1 Cream Puff: Calories 210; Total Fat 13g (Saturated Fat 7g, Trans Fat 0g); Cholesterol 135mg; Sodium 140mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 8g); Protein 6g Exchanges: 1 Starch, \u00bd Medium-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\n\u00c9clairs After dropping dough onto cookie sheet, shape each mound into finger shape, 4\u00bdx1\u00bd inch, with metal spatula. Bake and cool as directed. Fill \u00e9clairs with filling. Drizzle with Vanilla Glaze (page 581).\n\nIce Cream Puffs Omit filling. Fill cream puffs with your favorite flavor of ice cream (page 638). Cover and freeze until serving. Serve with Chocolate Sauce (page 642), Hot Fudge Sauce (page 642) or Caramel Sauce (page 642).\n\nProfiteroles Fill cream puffs with filling or ice cream (page 638). Drizzle with Chocolate Sauce (page 642).\n\nWhipped Cream Puffs Omit filling. Make Sweetened Whipped Cream (page 622), using 2 cups whipping cream and \u00bc cup granulated or powdered sugar and a large, deep bowl. Fill puffs with cream.\n\nDouble-Chocolate Puffs Prepare Cream Puffs as directed, except reduce flour to \u00be cup plus 2 tablespoons. Mix flour with 2 tablespoons unsweetened baking cocoa and 1 tablespoon sugar. Continue as directed, baking puffs 35 to 40 minutes or until puffed and darker brown on top. Substitute about 2\u00bc cups Chocolate Mousse (page 632) for the filling.\n\nEggnog Puffs Prepare Cream Puffs as directed, except reduce vanilla to 1 teaspoon. Add 1 teaspoon rum extract, \u00bd teaspoon ground nutmeg and \u215b teaspoon ground ginger with the vanilla.\n\n### Making Cream Puffs and Eclairs\n\nBeat vigorously with wooden spoon after adding the flour until mixture forms a ball.\n\nTo shape cream puffs, drop dough by slightly less than \u00bc cupfuls in mounds about 3 inches apart.\n\nTo shape \u00e9clairs, shape each mound into finger shape, 4\u00bdx1\u00bd inch, with metal spatula.\n\nCut off top third of each cooked puff; pull out any strands of soft dough.\n\nTiramisu\n\n## Tiramisu\n\nMascarpone is a rich double- or triple-cream cheese that originated in the Lombardy region of Italy. It has a delicate, buttery flavor with just a hint of sweetness. Look for it in the cheese case in large supermarkets, specialty cheese shops or gourmet food stores.\n\nPrep 35 min Total 5 hr 35 min \u25cf 8 servings\n\n  * 6 egg yolks\n  * \u00be cup sugar\n  * \u2154 cup milk\n  * 2 containers (8 oz each) mascarpone cheese or 2 packages (8 oz each) cream cheese, softened\n  * 1\u00bc cups whipping cream\n  * \u00bd teaspoon vanilla\n  * \u00bc cup cold brewed espresso or very strong coffee\n  * 2 tablespoons rum*\n  * 2 packages (3 oz each) ladyfingers (24 ladyfingers)**\n  * 1\u00bd teaspoons unsweetened baking cocoa\n\n1 In 2-quart saucepan, beat egg yolks and sugar with whisk until well mixed. Beat in milk. Heat to boiling over medium heat, stirring constantly; reduce heat to low. Boil and stir 1 minute; remove from heat. Pour into medium bowl; press plastic wrap on surface of custard mixture. Refrigerate about 1 hour or until chilled.\n\n2 Add cheese to custard mixture. Beat with electric mixer on medium speed until smooth; set aside. Wash and dry beaters. In chilled large, deep bowl, beat whipping cream and vanilla with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat until stiff peaks form; set aside.\n\n3 In small bowl, mix espresso and rum. Split ladyfingers horizontally in half.\n\n4 In ungreased 11x7-inch (2-quart) glass baking dish, arrange half of the ladyfingers in single layer. Brush with espresso mixture (do not soak). Spread half of the cheese mixture over ladyfingers; spread with half of the whipped cream. Repeat layers with remaining ladyfingers, espresso mixture, cheese mixture and whipped cream. Sprinkle with cocoa. Cover and refrigerate at least 4 hours to develop flavors but no longer than 24 hours. Store covered in refrigerator.\n\n* Or substitute \u215b teaspoon rum extract mixed with 2 tablespoons water for the rum.\n\n** Or substitute 2 packages (10.75 ounces each) frozen (thawed) pound cake for the ladyfingers. Place each pound cake on its side; carefully cut each cake lengthwise into 3 slices. Lay slices cut-side down; cut each slice crosswise into 4 slices (to total 24 slices). Brush each slice with espresso mixture. Arrange 3 slices crosswise in baking dish. Spread half of the cheese mixture over cake; spread with half of the whipped cream. Repeat layers with remaining pound cake, espresso mixture, cheese mixture and whipped cream.\n\n1 Serving: Calories 540; Total Fat 38g (Saturated Fat 22g, Trans Fat 1.5g); Cholesterol 270mg; Sodium 260mg; Total Carbohydrate 39g (Dietary Fiber 0g, Sugars 32g); Protein 9g Exchanges: 2 Starch, \u00bd Other Carbohydrate, 7\u00bd Fat Carbohydrate Choices: 2\u00bd\n\n### Making Tiramisu\n\nSeparate ladyfingers horizontally; brush with espresso mixture.\n\nSpread half of cheese mixture over ladyfingers; spread with half of whipped cream.\n\nButterscotch, Chocolate and Vanilla Pudding\n\n## Vanilla Pudding\n\nPrep 10 min Total 1 hr 10 min \u25cf 4 servings\n\n  * \u2153 cup sugar\n  * 2 tablespoons cornstarch\n  * \u215b teaspoon salt\n  * 2 cups milk\n  * 2 egg yolks, slightly beaten\n  * 2 tablespoons butter, softened\n  * 2 teaspoons vanilla\n\n1 In 2-quart saucepan, mix sugar, cornstarch and salt. Gradually stir in milk. Cook over medium heat, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute.\n\n2 Gradually stir at least half of the hot mixture into egg yolks, then stir back into hot mixture in saucepan. Boil and stir 1 minute; remove from heat. Stir in butter and vanilla.\n\n3 Pour pudding into dessert dishes. Cover and refrigerate about 1 hour or until chilled. Store covered in refrigerator.\n\n1 Serving: Calories 220; Total Fat 11g (Saturated Fat 5g, Trans Fat 0g); Cholesterol 130mg; Sodium 170mg; Total Carbohydrate 26g (Dietary Fiber 0g, Sugars 23g); Protein 5g Exchanges: 1\u00bd Other Carbohydrate, 1 Low-Fat Milk, 2 Fat Carbohydrate Choices: 2\n\nButterscotch Pudding Substitute \u2154 cup packed dark or light brown sugar for the granulated sugar; reduce vanilla to 1 teaspoon.\n\nChocolate Pudding Increase sugar to \u00bd cup; stir \u2153 cup unsweetened baking cocoa into sugar mixture. Omit butter.\n\nPanna Cotta\n\n## Panna Cotta EASY\n\nPrep 10 min Total 4 hr 20 min \u25cf 4 servings\n\n  * 2 cups cold half-and-half\n  * 1\u00bd teaspoons unflavored gelatin\n  * \u00bc cup sugar\n  * 1 teaspoon vanilla\n  * Dash salt\n  * Raspberry Sauce (page 641), if desired\n\n1 Pour half-and-half into 1\u00bd-quart saucepan. Sprinkle gelatin evenly over half-and-half. Let stand 10 minutes.\n\n2 Heat mixture over medium-high heat 5 to 7 minutes, stirring constantly, until gelatin is dissolved and mixture is just beginning to simmer (do not boil). Remove from heat. Stir in sugar, vanilla and salt until sugar is dissolved.\n\n3 Pour mixture into 4 (6-oz) ramekins or custard cups.* Cover tightly with plastic wrap; refrigerate about 4 hours or until set.\n\n4 When ready to serve, run thin knife around edge of each panna cotta. Dip bottom of each ramekin into bowl of very hot water for 5 seconds. Immediately place serving plate upside down over each ramekin; turn plate and ramekin over and remove ramekin. Serve with raspberry sauce.\n\n* Or, instead of using ramekins and having to unmold them, pour the custard into 4 parfait glasses, making sure to leave room at the top for the raspberry sauce.\n\n1 Serving: Calories 210; Total Fat 14g (Saturated Fat 9g, Trans Fat 0g); Cholesterol 45mg; Sodium 125mg; Total Carbohydrate 18g (Dietary Fiber 0g, Sugars 18g); Protein 4g Exchanges: 1 Other Carbohydrate, \u00bd High-Fat Meat, 2 Fat Carbohydrate Choices: 1\n\nChocolate Mousse\n\n## Chocolate Mousse\n\nPrep 20 min Total 2 hr 20 min \u25cf 8 servings\n\n  * 4 egg yolks\n  * \u00bc cup sugar\n  * 2\u00bd cups whipping cream\n  * 8 oz semisweet or dark baking chocolate, chopped\n\n1 In small bowl, beat egg yolks with electric mixer on high speed 3 minutes or until thick and lemon colored. Slowly beat in sugar.\n\n2 In 2-quart saucepan, heat 1 cup of the whipping cream over medium heat just until hot. Gradually stir at least half of the hot cream into egg yolk mixture, then stir back into hot cream in saucepan. Cook over low heat about 5 minutes, stirring constantly, until mixture thickens (do not boil).\n\n3 Stir in chocolate until melted. Cover and refrigerate about 2 hours, stirring occasionally, just until chilled.\n\n4 In chilled large, deep bowl, beat remaining 1\u00bd cups whipping cream with electric mixer on low speed until mixture begins to thicken. Gradually increase speed to high and beat until stiff peaks form. Fold chocolate mixture into whipped cream.\n\n5 Pipe or spoon mixture into 8 dessert dishes or stemmed glasses. Refrigerate until serving. Store covered in refrigerator.\n\n1 Serving: Calories 460; Total Fat 41g (Saturated Fat 24g, Trans Fat 0.5g); Cholesterol 190mg; Sodium 35mg; Total Carbohydrate 17g (Dietary Fiber 4g, Sugars 9g); Protein 6g Exchanges: \u00bd Starch, \u00bd Other Carbohydrate, \u00bd Medium-Fat Meat, 7\u00bd Fat Carbohydrate Choices: 1\n\nLighter Directions For 14 grams of fat and 255 calories per serving, substitute 2 whole eggs for the 4 egg yolks, and half-and-half for the 1 cup whipping cream in Step 2. Substitute 3 cups frozen (thawed) fat-free whipped topping for the 1\u00bd cups whipping cream.\n\n## Rice Pudding Calorie Smart\n\nLeftover cooked rice can be used in this recipe. Use 1\u00bd cups cooked rice and increase the bake time by about 5 minutes because the rice will be cold.\n\nPrep 15 min Total 1 hr 30 min \u25cf 8 servings\n\n  * \u00bd cup uncooked regular long-grain white rice\n  * 1 cup water\n  * 2 whole eggs or 4 egg yolks\n  * \u00bd cup sugar\n  * \u00bd cup raisins or chopped dried apricots\n  * 2\u00bd cups milk\n  * 1 teaspoon vanilla\n  * \u00bc teaspoon salt\n  * Ground cinnamon or nutmeg\n  * Raspberry Sauce (page 641), if desired\n  * Sweetened Whipped Cream (page 622), if desired\n\n1 In 1\u00bd-quart saucepan, heat rice and water to boiling, stirring once or twice; reduce heat to low. Cover and simmer 14 minutes (do not lift cover or stir). All water should be absorbed; if not, drain excess water.\n\n2 Heat oven to 325\u00b0F. In ungreased 1\u00bd-quart casserole, beat eggs with whisk or fork. Stir in sugar, raisins, milk, vanilla, salt and hot rice. Sprinkle with cinnamon.\n\n3 Bake uncovered 45 minutes, stirring every 15 minutes. Top of pudding will be very wet and not set (overbaking may cause pudding to curdle).\n\n4 Stir well; let stand 15 minutes. Enough liquid will be absorbed while standing to make pudding creamy. Serve warm, or cover and refrigerate about 3 hours or until chilled. Serve with raspberry sauce and whipped cream. Store covered in refrigerator.\n\n1 Serving: Calories 180; Total Fat 3g (Saturated Fat 1g, Trans Fat 0g); Cholesterol 60mg; Sodium 125mg; Total Carbohydrate 33g (Dietary Fiber 0g, Sugars 22g); Protein 5g Exchanges: 2 Starch, \u00bd Fat Carbohydrate Choices: 2\n\nAlmond-Cherry Rice Pudding Substitute dried cherries for the raisins and \u00bd teaspoon almond extract for the 1 teaspoon vanilla. Omit raspberry sauce.\n\nCaramel Custard (Cr\u00e8me Caramel)\n\n## Baked Custard EASY\n\nPrep 10 min Total 1 hr 25 min \u25cf 6 servings\n\n  * 3 eggs, slightly beaten\n  * \u2153 cup sugar\n  * 1 teaspoon vanilla\n  * Dash salt\n  * 2\u00bd cups very warm milk or half-and-half (120\u00b0F to 130\u00b0F)\n  * Ground nutmeg\n\n1 Heat oven to 350\u00b0F. In medium bowl, beat eggs, sugar, vanilla and salt with whisk or fork. Gradually stir in milk.\n\n2 Pour mixture into 6 (6-oz) ramekins or custard cups. Sprinkle with nutmeg. Place ramekins in 13x9-inch pan; place pan in oven. Carefully pour very hot water into pan to within \u00bd inch of tops of ramekins.\n\n3 Bake about 45 minutes or until knife inserted halfway between center and edge comes out clean. Carefully remove ramekins from water using tongs or grasping tops of ramekins with pot holder. Cool about 30 minutes. Serve warm or cool. If desired, unmold custard onto individual plates. Store covered in refrigerator.\n\n1 Serving: Calories 130; Total Fat 5g (Saturated Fat 2g, Trans Fat 0g); Cholesterol 115mg; Sodium 120mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 17g); Protein 7g Exchanges: 1 Low-Fat Milk, 1 Fat Carbohydrate Choices: 1\n\nCaramel Custard (Cr\u00e8me Caramel) Before making custard, in heavy 1-quart saucepan, heat \u00bd cup sugar over low heat 10 to 15 minutes, stirring constantly with wooden spoon, until sugar is melted and golden brown (sugar becomes very hot and could melt a plastic spoon). Immediately divide syrup among 6 (6-oz) custard cups before it hardens in saucepan; carefully tilt cups to coat bottoms (syrup will be extremely hot). Let syrup harden in cups about 10 minutes. Make custard as directed in Step 1; pour over syrup in cups. Bake as directed in Steps 2 and 3. Cool completely. Cover and refrigerate until serving or up to 48 hours.\n\nTo unmold, carefully loosen side of custard with knife or small spatula. Place dessert dish on top of cup and, holding tightly, turn dish and cup upside down. Shake cup gently to loosen custard. Caramel syrup will drizzle over custard, forming a sauce.\n\n### Unmolding Custard\n\nRun thin knife around edge of each baked custard; dip bottom of each ramekin into hot water for about 5 seconds.\n\nImmediately place serving plate upside down onto each ramekin; hold both ramekin and plate firmly. Turn ramekin upside down onto plate; remove ramekin.\n\n## Learn to Make Cr\u00e8me Br\u00fbl\u00e9e\n\nThis elegant dessert, loved for its contrast in textures\u2014the creaminess of the custard with the brittle caramelized sugar topping\u2014is a popular star on restaurant menus. It's a great dessert to make ahead and dazzle your guests with because it's actually easy to make when you know the secrets to success.\n\n### Tips for Making Cr\u00e8me Br\u00fbl\u00e9e\n\n  * Use whipping cream (not half-and-half or milk) to correctly set the custard and achieve the velvety smooth texture and creamy richness cr\u00e8me br\u00fbl\u00e9e is known for.\n  * Use ceramic ramekins, not glass custard cups or glass pie plates, which may not be able to withstand the heat from the kitchen torch or broiler and break.\n  * Slightly beat the eggs with a whisk to break them up before adding to the cream mixture. Whisk eggs with cream mixture until well blended so the custard will be smooth when baked.\n  * Add enough boiling water to the pan until it comes up to two-thirds of the height of the ramekins in order to protect the custard from the direct heat of the oven during baking.\n  * Don't allow any boiling water to splash into the ramekins, which could affect the custard texture.\n  * Blot any condensation from the tops of the chilled custards before sprinkling with sugar.\n  * Use a kitchen torch (see right for directions) or broiler to caramelize sugar. To broil topping, sprinkle 2 teaspoons brown sugar over each chilled custard. (Brown sugar melts much more evenly under the broiler.) Place ramekins in 15x10x1-inch pan or on cookie sheet with sides. Broil with tops 4 to 6 inches from heat 5 to 6 minutes.\n  * Caramelize the sugar on top of the custard until sugar is melted and light golden brown, watching carefully so the sugar doesn't burn and forms a glaze.\n\n## Cr\u00e8me Br\u00fbl\u00e9e\n\nPrep 20 min Total 7 hr \u25cf 4 servings\n\n  * 2 cups whipping cream\n  * \u2153 cup sugar\n  * 1 teaspoon vanilla\n  * 6 egg yolks, slightly beaten\n  * Boiling water\n  * 8 teaspoons sugar\n\n1 Heat oven to 350\u00b0F. In large bowl, stir whipping cream, \u2153 cup sugar and the vanilla until well mixed. Add egg yolks; beat with whisk until evenly colored and well blended.\n\n2 In 13x9-inch pan, place 4 (6-oz) ceramic ramekins. Pour cream mixture evenly into ramekins. Carefully place pan in oven. Carefully pour boiling water into pan to within \u00bd inch of tops of ramekins.\n\n3 Bake 30 to 40 minutes or until tops are light golden brown and sides are set (centers will be jiggly). Carefully transfer ramekins to cooling rack, using tongs or grasping tops of ramekins with pot holder. Cool 2 hours or until room temperature. Cover tightly with plastic wrap; refrigerate until chilled, at least 4 hours but no longer than 2 days.\n\n4 Uncover ramekins; gently blot any condensation on custards with paper towel. Sprinkle 2 teaspoons sugar over each custard. Holding kitchen torch 3 to 4 inches from custard, caramelize sugar by heating with torch about 2 minutes, moving flame continuously over sugar in circular motion, until sugar is melted and light golden brown. (Or caramelize under broiler; see left.) Serve immediately, or refrigerate up to 8 hours before serving. If served immediately, custard will be softer.\n\n1 Serving: Calories 540; Total Fat 45g (Saturated Fat 25g, Trans Fat 1g); Cholesterol 450mg; Sodium 50mg; Total Carbohydrate 29g (Dietary Fiber 0g, Sugars 29g); Protein 7g Exchanges: 2 Other Carbohydrate, 1 High-Fat Meat, 7\u00bd Fat Carbohydrate Choices: 2\n\nLemon Cr\u00e8me Br\u00fbl\u00e9e Reduce sugar in custard to \u2153 cup. Add 1 tablespoon grated lemon peel with the vanilla. After caramelizing sugar, top each ramekin with 1 tablespoon fresh raspberries, if desired.\n\nApple-Fig Bread Pudding Cupcakes with Maple Sauce\n\n## Apple-Fig Bread Pudding Cupcakes with Maple Sauce\n\nPrep 30 min Total 1 hr 5 min \u25cf 6 servings\n\nCupcakes\n\n  * 7 cups cubes (1 inch) day-old French or Italian bread (from 1-lb loaf)\n  * 1 large tart cooking apple (Braeburn, Cortland or Granny Smith), peeled, chopped (1\u00bd cups)\n  * \u00bd cup chopped dried figs\n  * 1 cup packed brown sugar\n  * 1 cup milk\n  * \u00bc cup butter, cut into pieces\n  * 1 teaspoon ground cinnamon\n  * \u00bd teaspoon vanilla\n  * 2 eggs, beaten\n\nSauce\n\n  * \u2153 cup granulated sugar\n  * \u2153 cup packed brown sugar\n  * \u2153 cup whipping cream\n  * \u2153 cup butter, cut into pieces\n  * \u00bd teaspoon maple flavor\n  * Maple candies, if desired\n\n1 Heat oven to 350\u00b0F. Grease 6 jumbo muffin cups with shortening.\n\n2 In large bowl, mix bread cubes, apple and figs. In small saucepan, cook 1 cup brown sugar, the milk and \u00bc cup butter over medium heat until butter is melted. Remove from heat; stir in cinnamon and vanilla. Pour over bread mixture in bowl. Add eggs; toss to coat. Spoon mixture evenly into muffin cups, filling to tops of cups.\n\n3 Bake 30 to 34 minutes or until center is set and apples are tender. Cool while making sauce.\n\n4 In 1-quart saucepan, stir granulated sugar, \u2153 cup brown sugar, the whipping cream and \u2153 cup butter. Heat to boiling over medium heat, stirring occasionally. Remove from heat; stir in maple flavor.\n\n5 Remove warm cupcakes from pan to individual serving plates. Spoon warm sauce over cupcakes. Garnish with maple candies.\n\n1 Serving: Calories 660; Total Fat 26g (Saturated Fat 16g, Trans Fat 1g); Cholesterol 140mg; Sodium 450mg; Total Carbohydrate 98g (Dietary Fiber 3g, Sugars 74g); Protein 9g Exchanges: 2\u00bd Starch, \u00bd Fruit, 3\u00bd Other Carbohydrate, 5 Fat Carbohydrate Choices: 6\u00bd\n\nBread Pudding with Whiskey Sauce\n\n## Bread Pudding with Whiskey Sauce\n\nPrep 25 min Total 2 hr 20 min \u25cf 12 servings\n\nBread Pudding\n\n  * 4 whole eggs\n  * 1 egg yolk\n  * \u00be cup sugar\n  * 2\u00bd cups milk\n  * 2\u00bd cups whipping cream\n  * 1 tablespoon vanilla\n  * 1\u00bd teaspoons ground cinnamon\n  * 12 oz French or other firm bread, cut into \u00bd-inch slices, then cut into 1\u00bd-inch pieces (10 cups)\n  * \u00bd cup raisins, if desired\n  * 2 tablespoons sugar\n  * 2 tablespoons butter, melted\n\nWhiskey Sauce\n\n  * \u00bd cup butter, cut into pieces\n  * 2 tablespoons water\n  * 1 whole egg\n  * 1 cup sugar\n  * 2 tablespoons whiskey or bourbon or 1 teaspoon brandy extract or vanilla\n\n1 Heat oven to 325\u00b0F. Grease bottom and sides of 13x9-inch (3-quart) glass baking dish with shortening or cooking spray.\n\n2 In large bowl, beat 4 eggs, 1 egg yolk and \u00be cup sugar with whisk until well blended. Beat in milk, whipping cream, vanilla and 1 teaspoon of the cinnamon until well blended. Stir in 7 cups of the bread pieces and the raisins. Let stand 20 minutes. Pour into baking dish. Lightly press remaining 3 cups bread pieces on top of mixture.\n\n3 In small bowl, mix 2 tablespoons sugar and remaining \u00bd teaspoon cinnamon. Brush top of bread mixture with melted butter; sprinkle with cinnamon-sugar. Bake uncovered 55 to 65 minutes or until top is puffed and light golden brown (center will jiggle slightly). Cool 30 minutes.\n\n4 Meanwhile, in 1-quart saucepan, melt \u00bd cup butter over low heat (do not allow to simmer). Remove from heat; cool 10 minutes. In small bowl, mix water and 1 egg; stir into butter until blended. Stir in 1 cup sugar. Cook and stir over medium-low heat until sugar is dissolved and mixture begins to boil; remove from heat. Stir in whiskey. Cool at least 10 minutes before serving.\n\n5 Serve sauce over warm bread pudding. Store bread pudding and sauce covered in refrigerator.\n\n1 Serving: Calories 500; Total Fat 30g (Saturated Fat 16g, Trans Fat 1.5g); Cholesterol 190mg; Sodium 300mg; Total Carbohydrate 50g (Dietary Fiber 0g, Sugars 36g); Protein 8g Exchanges: 2 Starch, 1\u00bd Other Carbohydrate, 5\u00bd Fat Carbohydrate Choices: 3\n\nLighter Directions For 7 grams of fat and 300 calories per serving, substitute 2 eggs and \u00bd cup fat-free egg product for the 4 eggs and 1 egg yolk. Substitute 4 cups fat-free (skim) milk and 1 cup half-and-half for the milk and whipping cream. Instead of whiskey sauce, stir 1 tablespoon bourbon or 1 teaspoon brandy extract into 1 cup fat-free caramel topping; heat if desired.\n\nBread Pudding with Lemon Curd Make bread pudding as directed, except\u2014omit whiskey sauce. Serve Lemon Curd (page 602) or purchased lemon curd over warm bread pudding.\n\nChocolate\u2013Sour Cream Fondue\n\n## Chocolate\u2013Sour Cream Fondue FAST\n\nPrep 15 min Total 15 min \u25cf 12 servings\n\n  * 4 cups milk chocolate or semisweet chocolate chips (about 24 oz) or 6 packages (4 oz each) sweet baking chocolate, chopped\n  * 1 cup sour cream\n  * \u00bd cup half-and-half\n  * 1 tablespoon instant coffee powder or granules or \u00bd teaspoon ground cinnamon\n  * 4 cups cut-up fresh fruit\n  * 4 cups cubed angel food cake (to make your own, see page 572) or pound cake (to make your own, see page 564)\n\n1 In 2-quart saucepan, heat chocolate, sour cream, half-and-half and coffee powder over low heat, stirring constantly, until chocolate is melted and mixture is smooth; remove from heat.\n\n2 Pour chocolate mixture into fondue or chafing dish to keep warm. Spear fruit and cake with fondue forks; dip and swirl in fondue with stirring motion. If mixture becomes too thick, stir in small amount of half-and-half.\n\n1 Serving: Calories 410; Total Fat 22g (Saturated Fat 13g, Trans Fat 0g); Cholesterol 25mg; Sodium 180mg; Total Carbohydrate 48g (Dietary Fiber 3g, Sugars 42g); Protein 6g Exchanges: 1 Starch, 2 Other Carbohydrate, 4\u00bd Fat Carbohydrate Choices: 3\n\n## Funtastic Dippers\n\nHere are some other fun foods to dunk in chocolate fondue:\n\n  * Mini cupcakes (page 574)\n  * Doughnuts (page 492)\n  * Marshmallows (page 550)\n  * Cooked bacon slices\n  * Pretzels\n  * Mini chocolate chip cookies\n  * Mini cereal-marshmallow bars\n\nIndividual Meringues filled with Lemon Curd (page 602)\n\n## Meringue Shell EASY\n\nA meringue shell bakes up crisp yet melts in your mouth when you bite into it. Just before serving, fill with fresh fruit, ice cream, Chocolate Mousse (page 632), Lemon Curd (page 602) or Vanilla Pudding (page 631).\n\nPrep 10 min Total 4 hr 40 min \u25cf 8 servings\n\n  * 3 egg whites\n  * \u00bc teaspoon cream of tartar\n  * \u00be cup sugar\n\n1 Heat oven to 275\u00b0F. Line cookie sheet with cooking parchment paper.\n\n2 In medium bowl, beat egg whites and cream of tartar with electric mixer on high speed until foamy. Beat in sugar, 1 tablespoon at a time; continue beating until stiff peaks form and mixture is glossy. Do not underbeat. On cookie sheet, shape meringue into 9-inch round with back of spoon, building up side.\n\n3 Bake 1 hour 30 minutes. Turn off oven; leave meringue in oven with door closed 1 hour. Finish cooling at room temperature, about 2 hours.\n\n1 Serving: Calories 80; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 20mg; Total Carbohydrate 19g (Dietary Fiber 0g, Sugars 19g); Protein 1g Exchanges: 1 Starch Carbohydrate Choices: 1\n\nIndividual Meringues Drop meringue by \u2153 cupfuls onto cookie sheet. Shape into rounds, building up sides. Bake 1 hour. Turn off oven; leave meringues in oven with door closed 1 hour. Finish cooling at room temperature, about 2 hours. Makes 8 to 10 meringues.\n\nChocolate Meringue Shell After beating egg white mixture, sprinkle 2 tablespoons unsweetened baking cocoa through strainer over top; fold cocoa into mixture. Bake as directed.\n\n### Shaping Individual Meringue Shells\n\nShape each meringue with back of spoon into desired shape, building up side.\n\n## Vanilla Ice Cream EASY\n\nPrep 10 min Total 2 hr 40 min \u25cf 1 quart\n\n  * 3 egg yolks, slightly beaten\n  * \u00bd cup sugar\n  * 1 cup milk\n  * \u00bc teaspoon salt\n  * 2 cups whipping cream\n  * 1 tablespoon vanilla\n\n1 In 2-quart saucepan, stir egg yolks, sugar, milk and salt until well mixed. Heat just to boiling over medium heat, stirring constantly (do not allow mixture to continue to boil or it will curdle). Immediately remove from heat.\n\n2 Pour milk mixture into chilled bowl. Refrigerate uncovered 2 to 3 hours, stirring occasionally, until room temperature. At this point, mixture can be refrigerated up to 24 hours before completing recipe if desired.\n\n3 Stir whipping cream and vanilla into milk mixture. Pour into 1-quart ice cream freezer and freeze according to manufacturer's directions.\n\n\u00bd Cup: Calories 270; Total Fat 21g (Saturated Fat 12g, Trans Fat 0.5g); Cholesterol 150mg; Sodium 110mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 16g); Protein 3g Exchanges: 1 Starch, 4 Fat Carbohydrate Choices: 1\n\nChocolate Ice Cream Increase sugar to 1 cup. Beat 2 ounces unsweetened baking chocolate, melted and cooled, into milk mixture before cooking. Reduce vanilla to 1 teaspoon.\n\nEggnog Ice Cream Add \u00bc teaspoon ground nutmeg with the sugar. Add 1 teaspoon rum extract with the vanilla.\n\nFresh Blueberry Ice Cream Reduce vanilla to 1 teaspoon. Mash 2 cups fresh blueberries and an additional \u2153 cup sugar with potato masher or process in food processor until slightly chunky (not pureed); stir into milk mixture after adding vanilla.\n\nFresh Peach Ice Cream Reduce vanilla to 1 teaspoon. Mash 4 or 5 peeled fresh peaches with potato masher or process in food processor until slightly chunky (not pureed) to make 2 cups; stir into milk mixture after adding vanilla.\n\nFresh Raspberry Ice Cream Substitute \u00bd teaspoon almond extract for the vanilla. Mash 2 cups fresh raspberries with potato masher or process in food processor until slightly chunky (not pureed); stir into milk mixture after adding almond extract.\n\nFresh Strawberry Ice Cream Reduce vanilla to 1 teaspoon. Mash 2 cups fresh strawberries and an additional \u00bd cup sugar with potato masher or process in food processor until slightly chunky (not pureed); stir into milk mixture after adding vanilla. Stir in a few drops red food color if desired.\n\nCaramel Ice Cream\n\n## Caramel Ice Cream\n\nPrep 15 min Total 2 hr 45 min \u25cf 1 quart\n\n  * \u00bd cup sugar\n  * 2 cups whipping cream\n  * 1 cup milk\n  * 3 egg yolks, slightly beaten\n  * \u00bc teaspoon salt\n  * 2 teaspoons vanilla\n\n1 In 2-quart saucepan, heat sugar over medium heat until sugar begins to melt. Stir until sugar is completely dissolved and becomes deep golden brown to amber in color. Remove from heat; carefully and gradually stir in 1 cup of the whipping cream (mixture will boil and splatter at first). Heat over low heat, stirring frequently, until any hardened sugar bits dissolve; remove from heat.\n\n2 With whisk, stir in remaining 1 cup whipping cream, the milk, egg yolks, salt and vanilla until well mixed. Heat just to boiling over medium heat, stirring constantly (do not allow mixture to continue to boil or it will curdle). Immediately remove from heat.\n\n3 Pour milk mixture into chilled bowl. Refrigerate uncovered 2 to 3 hours, stirring occasionally, until room temperature. At this point, mixture can be refrigerated up to 24 hours before completing recipe if desired.\n\n4 Pour mixture into 1-quart ice cream freezer and freeze according to manufacturer's directions.\n\n\u00bd Cup: Calories 300; Total Fat 25g (Saturated Fat 15g, Trans Fat 1g); Cholesterol 160mg; Sodium 110mg; Total Carbohydrate 16g (Dietary Fiber 0g, Sugars 16g); Protein 3g Exchanges: 1 Other Carbohydrate, \u00bd Medium-Fat Meat, 4\u00bd Fat Carbohydrate Choices: 1\n\n### Caramelizing Sugar for Caramel Ice Cream\n\nHeat sugar in saucepan over medium heat until sugar begins to melt.\n\nStir until sugar is completely dissolved and is deep golden-brown to amber colored.\n\nGradually stir in whipping cream.\n\nHeat over low heat, stirring frequently, until hardened sugar bits dissolve.\n\n## Very Berry Frozen Yogurt Calorie Smart \u2022 EASY\n\nPrep 5 min Total 40 min \u25cf 12 servings\n\n  * 2 cups fresh strawberries or raspberries\n  * \u2153 cup sugar\n  * 4 cups vanilla low-fat yogurt\n\n1 In large bowl, mash berries with sugar. Stir in yogurt.\n\n2 Pour mixture into 2-quart ice cream freezer. Freeze according to manufacturer's directions. Scoop into serving dishes or ice cream cones.\n\n1 Serving: Calories 100; Total Fat 1g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 55mg; Total Carbohydrate 19g (Dietary Fiber 0g, Sugars 18g); Protein 4g Exchanges: \u00bd Fruit , \u00bd Other Carbohydrate, \u00bd Low-Fat Milk Carbohydrate Choices: 1\n\nWatermelon Granita\n\n## Watermelon Granita Calorie Smart\n\nGranita is the Italian term for an \"ice\" or frozen dessert made with fruit or fruit juice. The texture will be grainier than a sorbet\u2014very similar to a slush.\n\nPrep 20 min Total 4 hr 25 min \u25cf 10 servings\n\n  * 3 tablespoons lime juice (2 to 3 limes)\n  * 1 cup water\n  * \u00bd cup sugar\n  * 6 cups 1-inch cubes seeded watermelon\n\n1 In 1-quart saucepan, mix lime juice, water and sugar. Cook over low heat about 5 minutes, stirring occasionally, until sugar is dissolved. Cool slightly, about 5 minutes.\n\n2 In blender or food processor, place watermelon. Cover and blend on high speed about 2 minutes or until smooth. Add lime juice mixture. Cover and blend until well mixed. Pour into 13x9-inch (3-quart) glass baking dish. Cover and freeze 1 hour.\n\n3 Scrape with fork to distribute ice crystals evenly. Freeze at least 3 hours longer, repeating scraping procedure every 30 minutes, until mixture is consistency of fine ice crystals. To serve, scoop into chilled dessert cups.\n\n1 Serving: Calories 70; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 16g); Protein 0g Exchanges: \u00bd Fruit, \u00bd Other Carbohydrate Carbohydrate Choices: 1\n\nLemonade Sorbet\n\n## Lemonade Sorbet Calorie Smart \u2022 EASY\n\nPrep 5 min Total 4 hr 5 min \u25cf 4 servings\n\n  * 1\u00bd cups cold water\n  * 1 cup lemonade concentrate (to make your own, see page 72, or thawed from 12-oz can)\n  * 3 tablespoons honey\n  * Lemon slices, if desired\n  * Fresh blueberries, if desired\n\n1 In blender or food processor, place water, lemonade concentrate and honey. Cover and blend on low speed about 30 seconds or until smooth. Pour into 8-inch square (2-quart) glass baking dish.\n\n2 Cover and freeze about 4 hours or until firm, stirring several times to keep mixture smooth. Garnish with lemon slices and blueberries.\n\n1 Serving: Calories 190; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 47g (Dietary Fiber 0g, Sugars 43g); Protein 0g Exchanges: 1 Fruit, 2 Other Carbohydrate Carbohydrate Choices: 3\n\nCranberry-Raspberry Sorbet Substitute cranberry-raspberry concentrate for the lemonade concentrate.\n\nLimeade Sorbet Substitute limeade concentrate for the lemonade concentrate.\n\nOrange Sorbet Substitute orange juice concentrate for the lemonade concentrate.\n\nPineapple Sorbet Substitute pineapple juice concentrate for the lemonade concentrate.\n\n## Lemon Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1\u00bc cups\n\n  * \u00bd cup sugar\n  * 2 tablespoons cornstarch\n  * \u00be cup water\n  * 1 tablespoon grated lemon peel\n  * \u00bc cup fresh lemon juice\n  * 2 tablespoons butter, softened\n\n1 In 1-quart saucepan, mix sugar and cornstarch. Gradually stir in water. Cook over medium heat, stirring constantly, until mixture thickens and boils. Boil and stir 1 minute; remove from heat.\n\n2 Stir in remaining ingredients. Serve warm or cool. Store covered in refrigerator up to 10 days.\n\n1 Tablespoon: Calories 35; Total Fat 1g (Saturated Fat 0.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 6g (Dietary Fiber 0g, Sugars 5g); Protein 0g Exchanges: \u00bd Fruit Carbohydrate Choices: \u00bd\n\nRaspberry Sauce\n\n## Raspberry Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1 cup\n\n  * 3 tablespoons sugar\n  * 2 teaspoons cornstarch\n  * \u2153 cup water\n  * 1 box (10 oz) frozen raspberries in syrup, thawed, undrained\n\n1 In 1-quart saucepan, mix sugar and cornstarch. Stir in water and raspberries with syrup. Cook over medium heat, stirring constantly, until mixture thickens and boils (mixture will become translucent during cooking). Boil and stir 1 minute.\n\n2 Strain sauce through fine-mesh strainer into bowl to remove seeds if desired. Serve sauce warm or cool. Store covered in refrigerator up to 10 days.\n\n1 Tablespoon: Calories 30; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 6g); Protein 0g Exchanges: \u00bd Fruit Carbohydrate Choices: \u00bd\n\nStrawberry Sauce Substitute frozen strawberries in syrup for the raspberries.\n\n### Making Raspberry Sauce\n\nStir water and raspberries into sugar and cornstarch.\n\nCook, stirring constantly, until mixture thickens and boils and becomes translucent.\n\n## Cranberry Sauce Calorie Smart \u2022 EASY\n\nThe natural pectin in the cranberries thickens this easy homemade sauce. Be sure to cook the cranberries until they pop so the pectin is released and the sauce gels as it cools.\n\nPrep 15 min Total 3 hr 15 min \u25cf 4 cups\n\n  * 4 cups fresh or frozen cranberries\n  * 2 cups sugar\n  * 2 cups water\n  * 1 tablespoon grated orange peel, if desired\n\n1 Wash cranberries; remove any stems and discard any blemished berries.\n\n2 In 3-quart saucepan, heat sugar and water to boiling over medium heat, stirring occasionally. Boil 5 minutes.\n\n3 Stir in cranberries. Heat to boiling; boil about 5 minutes or until cranberries pop. Stir in orange peel. Cover and refrigerate about 3 hours or until chilled. Store covered in refrigerator up to 1 week.\n\n\u00bc Cup: Calories 110; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 28g (Dietary Fiber 1g, Sugars 26g); Protein 0g Exchanges: 2 Fruit Carbohydrate Choices: 2\n\n## Rhubarb Sauce Calorie Smart \u2022 FAst\n\nRhubarb varies in sweetness, so add sugar to taste. Serve it for dessert, either by itself or over pound cake or ice cream. Or serve as a sweet-tart accompaniment to pork or chicken.\n\nPrep 20 min Total 20 min \u25cf 3 cups\n\n  * \u00bd to \u00be cup sugar\n  * \u00bd cup water\n  * 1 lb fresh or frozen rhubarb, cut into 1-inch pieces (3 cups)\n  * Ground cinnamon, if desired\n\n1 In 2-quart saucepan, heat \u00bd cup sugar and the water to boiling over medium heat, stirring occasionally. Stir in rhubarb; reduce heat to low. Simmer uncovered about 10 minutes, stirring occasionally, until rhubarb is tender and slightly transparent, adding up to \u00bc cup more sugar, if desired.\n\n2 Stir in cinnamon. Serve sauce warm or chilled. Store covered in refrigerator up to 1 week.\n\n\u00bd Cup: Calories 80; Total Fat 0g (Saturated Fat 0g, Trans Fat 0g); Cholesterol 0mg; Sodium 0mg; Total Carbohydrate 20g (Dietary Fiber 1g, Sugars 17g); Protein 0g Exchanges: 1 \u00bd Other Carbohydrate Carbohydrate Choices: 1\n\nStrawberry-Rhubarb Sauce Substitute 1 cup fresh strawberries, cut in half, for 1 cup of the rhubarb. After simmering rhubarb, stir in strawberries; heat just to boiling.\n\n## Hot Fudge Sauce Calorie Smart \u2022 EASY\n\nPrep 10 min Total 40 min \u25cf 3 cups\n\n  * 1 can (12 oz) evaporated milk\n  * 2 cups semisweet or dark chocolate chips (about 12 oz)\n  * \u00bd cup sugar\n  * 1 tablespoon butter, softened\n  * 1 teaspoon vanilla\n\n1 In 2-quart saucepan, heat milk, chocolate chips and sugar to boiling over medium heat, stirring constantly; remove from heat. Stir in butter and vanilla until mixture is smooth and creamy.\n\n2 Cool about 30 minutes or until sauce begins to thicken. Serve warm. Store sauce covered in refrigerator up to 4 weeks. Sauce becomes firm when refrigerated; heat slightly before serving (sauce will become thin if overheated).\n\n1 Tablespoon: Calories 60; Total Fat 2.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 10mg; Total Carbohydrate 7g (Dietary Fiber 0g, Sugars 7g); Protein 0g Exchanges: \u00bd Starch, \u00bd Fat Carbohydrate Choices: \u00bd\n\n## Chocolate Sauce Calorie Smart \u2022 FAst \u2022 EASY\n\nPrep 10 min Total 10 min \u25cf 1\u00bd cups\n\n  * 1\u00bd cups light corn syrup\n  * 3 oz unsweetened baking chocolate, chopped\n  * 1 tablespoon butter, softened\n  * \u00be teaspoon vanilla\n\n1 In 1-quart saucepan, heat corn syrup and chocolate over low heat, stirring frequently, until chocolate is melted; remove from heat. Stir in butter and vanilla.\n\n2 Serve warm or cool. Store covered in refrigerator up to 10 days. Reheat slightly before serving if desired.\n\n1 Tablespoon: Calories 90; Total Fat 2.5g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 0mg; Sodium 30mg; Total Carbohydrate 17g (Dietary Fiber 0g, Sugars 8g); Protein 0g Exchanges: 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\n## Caramel Sauce Calorie Smart \u2022 EASY\n\nWhat else besides ice cream is caramel sauce great on? Drizzle it over cheesecake or sliced bananas and sprinkle with chopped pecans. Or you can use it as a dip for apple slices or cubes of pound cake.\n\nPrep 10 min Total 40 min \u25cf 2\u00bd cups\n\n  * 1 cup light corn syrup\n  * 1\u00bc cups packed brown sugar\n  * \u00bc cup butter, cut into pieces\n  * 1 cup whipping cream\n\n1 In 2-quart saucepan, heat corn syrup, brown sugar and butter to boiling over low heat, stirring constantly. Boil 5 minutes, stirring occasionally.\n\n2 Stir in whipping cream; heat to boiling, stirring constantly. Cool about 30 minutes. Serve warm. Store covered in refrigerator up to 10 days. Reheat slightly before serving if desired.\n\n1 Tablespoon: Calories 80; Total Fat 3g (Saturated Fat 1.5g, Trans Fat 0g); Cholesterol 10mg; Sodium 20mg; Total Carbohydrate 13g (Dietary Fiber 0g, Sugars 10g); Protein 0g Exchanges: 1 Other Carbohydrate, \u00bd Fat Carbohydrate Choices: 1\n\n# Helpful Nutrition and Cooking Information\n\nWe provide nutrition information for each recipe that includes calories, fat, cholesterol, sodium, carbohydrate, fiber and protein. Individual food choices can be based on this information.\n\n### Criteria Used for Calculating Nutrition Information\n\n  * The first ingredient was used wherever a choice is given (such as \u2153 cup sour cream or plain yogurt).\n  * The first ingredient amount was used wherever a range is given (such as 3- to 3\u00bd-pound cut-up whole chicken).\n  * The first serving number was used wherever a range is given (such as 4 to 6 servings).\n  * \"If desired\" ingredients and recipe variations were not included (such as sprinkle with brown sugar, if desired).\n  * Only the amount of a marinade or frying oil that is estimated to be absorbed by the food during preparation or cooking was calculated.\n\n### Criteria Used for Calculating Lower-Calorie Recipes\n\nRecipes designated as Lower Calorie were determined using the following guidelines:\n\n  * Main dishes: 350 calories or less\n  * Chilies, stews, plain meats: 250 calories or less\n  * Soups, breads, side dishes, appetizers: 175 calories or less\n  * Desserts: 200 calories or less\n  * Condiments, beverages, dips, sauces, dressings: 80 calories or less\n\n### Ingredients Used in Recipe Testing and Nutrition Calculations\n\n  * Ingredients used for testing represent those that the majority of consumers use in their homes: large eggs, 2% milk, 80%-lean ground beef and canned ready-to-use chicken broth.\n  * Fat-free, low-fat or low-sodium products were not used, unless otherwise indicated.\n  * Solid vegetable shortening was used to grease pans, unless otherwise indicated.\n\n### Equipment Used in Recipe Testing\n\nWe use equipment for testing that the majority of consumers use in their homes. If a specific piece of equipment (such as a wire whisk) is necessary for recipe success, it is listed in the recipe.\n\n  * Cookware and bakeware without nonstick coatings were used, unless otherwise indicated.\n  * No dark-colored, black or insulated bakeware was used.\n  * When a pan is specified in a recipe, a metal pan was used; a baking dish or pie plate means ovenproof glass was used.\n  * An electric hand mixer was used for mixing only when mixer speeds are specified in the recipe directions. If a stand mixer is necessary, the recipe will call for that appliance. When a mixer speed is not given, a spoon or fork was used.\n\n# Index\n\n## A\n\nAcorn squash,\n\nAdzuki beans,\n\nAioli, Wasabi, Asian Tuna with,  CALORIE SMART\n\nAj\u00ed amarillo paste,\n\nAjowain (bishop weed),\n\nAle,\n\nAlfredo Sauce,\n\nEasy,\n\nShrimp, Primavera,  CALORIE SMART \u2022 Fast\n\nAllspice,\n\nAlmond(s),\n\nBiscotti,\n\nBrittle,\n\n\u2013Brown Rice Pilaf,\n\nButter,\n\n-Cranberry Rye Berry Salad,\n\nMacarons, French, with Bittersweet Chocolate Ganache, 538\u2013539 CALORIE SMART\n\nMarcona,\n\nPesto, Carrot-, Roasted,\n\nPesto, -Cilantro, Spicy,\n\nPound Cake, Toasted,\n\n-Raspberry Pie,\n\nRice Pudding, -Cherry,\n\nRomesco Sauce,\n\nSole Amandine,  CALORIE SMART \u2022 Fast\n\nSole Amandine, Orange,\n\nSugared,\n\nThumbprint Cookies, -Lemon,  CALORIE SMART\n\nAlmond milk,\n\nAmaranth,\n\nAnaheim chiles,\n\nAnchovies,\n\nAnchovy paste,\n\nAngel Food Cake, ,  CALORIE SMART\n\nCherry,\n\nChocolate-Cherry,\n\nChocolate Confetti,\n\nEspresso,\n\nAngel Food Cupcakes,\n\nAngel Hair Pasta\n\nwith Basil, Avocado and Tomatoes,  Fast\n\nwith Chicken,\n\nwith Shrimp,\n\nAnise seed,\n\nAnnatto,\n\nAppetizers. _See also_ Dip(s); Spreads\n\nBlack Bean Sliders,  CALORIE SMART\n\nBrie in Puff Pastry with Cranberry Sauce,\n\nBrie in Puff Pastry with Cranberry Sauce, Double-Orange,\n\nCashews, Southwestern Spiced,  Fast\n\nCharcuterie,\n\ncheese tray,\n\nChicken Satay with Peanut Sauce, 56\u201357\n\nChicken Wings\n\nBuffalo,  CALORIE SMART\n\nHoney-Mustard,\n\nSalsa,\n\nSweet-Spicy,\n\nCrab Cakes, Classic,  Fast\n\nCrostini, Provolone-Onion,\n\nCrostini, Tomato-Basil,  CALORIE SMART \u2022 Fast\n\nDeviled Eggs,  CALORIE SMART \u2022 Fast\n\nBacon-Cheddar,\n\nBlue Cheese,\n\nChipotle,\n\nReuben,\n\nEdamame in the Shell with Dipping Sauces, Steamed,  CALORIE SMART \u2022 Fast\n\nEgg Rolls, Shrimp,\n\nGarlic, Roasted,  CALORIE SMART \u2022 EASY\n\nJalape\u00f1o Poppers, Baked,\n\nLettuce Wraps, Beef, Vietnamese,  CALORIE SMART\n\nMeatballs, Hot and Saucy Cocktail,  CALORIE SMART\n\nMeatballs, Make-Ahead,  CALORIE SMART\n\nNachos,  Fast\n\nNachos, Beef or Chicken,\n\nNuts, Cinnamon-Cardamom Mixed,\n\nNuts, Cinnamon-Sugared,\n\nNuts, Southwestern Spiced Party,  Fast\n\nPotato Skins, Cheesy,\n\nRed Onions, Pickled,  CALORIE SMART\n\nSalmon Cakes,\n\nselecting\/serving,\n\nShrimp Cocktail, Classic,  CALORIE SMART \u2022 Fast\n\nShrimp Cocktail Shooters,\n\nTaco Bites, Pork Carnitas,\n\nTriangles, Basil-Cheese,  CALORIE SMART\n\nApple(s)\n\nBaked,  CALORIE SMART \u2022 EASY\n\nBread Pudding Cupcakes, -Fig, with Maple Sauce,\n\nButter,  CALORIE SMART\n\nCake, Spiced, with Maple Glaze,\n\nChutney, -Cranberry,  CALORIE SMART\n\nCobbler,\n\nCrisp,\n\nCrisp, Caramel,\n\nDumplings,\n\nGreen Tea, Iced, -Mint,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nJelly, -Pepper,  CALORIE SMART \u2022 EASY\n\nMuffins, -Cinnamon,\n\nOatmeal, Baked,  CALORIE SMART\n\nPancake, Oven,\n\nPie, Classic,\n\nPie, French,\n\nPie, Slab,  CALORIE SMART\n\nPork and Barley, -Rosemary,  CALORIE SMART \u2022 Fast\n\nPork Chops, Smoked, with Sweet Potato and,  CALORIE SMART\n\nStuffing, Roast Goose with,\n\nTart, Easy,\n\nTurkey Breast, -Maple Brined,\n\nvarieties of,\n\n-Yogurt Snacks,\n\nApple Cider. _See_ Cider\n\nApplesauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nDoughnuts,\n\nApricot-Ginger Preserves, Spiced,  CALORIE SMART \u2022 EASY\n\nApricot-Stuffed French Toast,  CALORIE SMART\n\nArborio rice,\n\nArtichoke(s),\n\nDip, Hot,  CALORIE SMART \u2022 EASY\n\nLasagna, -Spinach,\n\npreparing\/cooking,\n\nwith Rosemary Butter,  EASY\n\nScallops with Tomatoes and,  CALORIE SMART \u2022 Fast\n\n-Spinach Dip, Hot,\n\nArtisan Asiago Bread, 498\u2013499 CALORIE SMART\n\nArugula,  _15_\n\nAsiago Bread, Artisan, 498\u2013499 CALORIE SMART\n\nAsiago Cheese, Pasta with Prosciutto and,  CALORIE SMART \u2022 Fast\n\nAsian\n\nBeef Roast with Cabbage and Pasta,  SLOW COOKER\n\nChicken Roll-Ups,  CALORIE SMART \u2022 Fast\n\nDressing,\n\nEggplant,\n\ningredients\/seasonings,\n\nPortabellas, Stuffed,  CALORIE SMART\n\nSalad with Wonton Crisps,  Fast\n\nTuna with Wasabi Aioli,  CALORIE SMART\n\nAsparagus,\n\n-Almond Salad,\n\n-Cashew Salad,\n\n-Fennel Seven-Layer Salad,\n\nwith Goat Cheese,\n\ngrilling,\n\n-Orange Salad,\n\nwith Parmesan,  CALORIE SMART \u2022 Fast\n\npreparing\/cooking,\n\nsnapping ends,\n\n-Strawberry Salad,  CALORIE SMART \u2022 Fast\n\nAu Gratin Potatoes,\n\nAvocado\n\nBacon and Cheese Omelet,\n\nand Bean Roll-Ups,\n\n-Cucumber-Berry Salad,\n\ncutting, __\n\nDressing, Creamy,\n\nGrilled Cheese Sandwiches, Tomato and,\n\nGuacamole,  CALORIE SMART \u2022 Fast\n\nGuacamole, Chipotle,\n\nGuacamole, Cotija,\n\n-Mango Salad,\n\n## B\n\nBaba Ghanoush, , _43_ CALORIE SMART\n\nBaby Food\n\nCarrot,  CALORIE SMART \u2022 EASY\n\nGreen Bean,\n\nhomemade,\n\nSweet Pea,\n\nWinter Squash,\n\nBacon\n\nAvocado and Cheese Omelet,\n\nBalsamic and Brown Sugar,\n\nBeef Tenderloin, -Wrapped Herb-Roasted,\n\nBroccoli, Cheesy,\n\nBrussels Sprouts, Cheesy, 242\u2013243 CALORIE SMART\n\nBurgers, Caramelized Beer-Onion and,\n\nCanadian, Pizza,\n\nand Cheese Grits,\n\n-Cheese Spread,\n\nChicken Thighs with Spinach and,  CALORIE SMART\n\nChopped Salad, Chicken and,\n\nCollard Greens with, Spicy,  CALORIE SMART\n\n-and Cornbread-Stuffed Pork Chops,\n\nCream Cheese Spread, Chive-Cheddar-,\n\nCrostini, BLT,\n\nCrumbles,\n\nDeviled Eggs, -Cheddar,\n\n-Edamame Toss,\n\nin German Potato Salad, Hot,  CALORIE SMART\n\nGreen Beans, Cheesy,\n\nHoney-Mustard,\n\nKale and Egg Cups,  CALORIE SMART\n\nMacaroni and Cheese, ,\n\nMacaroni and Cheese, Kale, Tomato and,\n\nmicrowave cooking,\n\nOven-Baked,  CALORIE SMART \u2022 EASY\n\nPeppered Brown Sugar,\n\nPotato and Egg Scramble,  CALORIE SMART \u2022 Fast\n\n-Potato Soup, Cheesy,  SLOW COOKER\n\nPraline,\n\nQuiche Lorraine,\n\n-Ranch Yogurt,\n\nRavioli, Chive, Parmesan and,\n\nSpaghetti Carbonara,  Fast\n\nSpaghetti Carbonara, Chicken and,\n\nSpanish Rice with,\n\n-Spinach Salad, with Hard-Cooked Eggs,  CALORIE SMART \u2022 Fast\n\n\u2013Sweet Potato Biscuits,  CALORIE SMART\n\nTart Flamb\u00e9e,  CALORIE SMART\n\nTwice-Baked Potatoes, and Pepper,\n\nVermicelli with Fresh Herbs, Spinach and,\n\nWild Rice, Pepper, -Roasted,\n\nWild Rice Porridge, and Pecan,\n\nWild Rice and Vegetable Skillet,  CALORIE SMART\n\nBacteria,\n\nBaked Apple Oatmeal,  CALORIE SMART\n\nBaked Apples,  CALORIE SMART \u2022 EASY\n\nBaked Blueberry-Orange Doughnuts,\n\nBaked Buffalo Frittata,  CALORIE SMART\n\nBaked Chicken Panzanella,  CALORIE SMART\n\nBaked Custard,  CALORIE SMART \u2022 EASY\n\nBaked Eggs,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nBaked goods, ingredients, 4\u20136,\n\nBaked Jalape\u00f1o Poppers,\n\nBaked One-Crust Pie,\n\nBaked Pita Crisps,\n\nBaked Potatoes with Tuna Topping,\n\nBaked Tart Shells and Individual Pie Crusts,\n\nBakery-Style Pumpernickel Loaf,\n\nBakery-Style Pumpernickel Rolls, 512\u2013513 CALORIE SMART\n\nBaking dish,\n\nBaking pans,\n\nbars,\n\ncakes, ,\n\ncleanup,\n\nmuffins,\n\npies and tarts, ,\n\nquick breads,\n\nBaking Powder,\n\nBiscuits,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nBaking soda,\n\nBalsamic,\n\nand Brown Sugar Bacon,\n\nGarlic Marinade,\n\nLamb Shanks, -Braised,  CALORIE SMART\n\nPork Chops with Quinoa,  CALORIE SMART\n\nSteak, -and Roasted Garlic\u2013Marinated, Grilled,  CALORIE SMART\n\n-Strawberry Freezer Jam,\n\nBamboo rice,\n\nBanana\n\nBread,  CALORIE SMART\n\nBread, Blueberry-,\n\nChocolate Mug Cake, Banana Split-, 576\u2013577\n\nCream Pie,\n\nCream Pie, -Chocolate,\n\nFrench Toast, Walnut-, Upside-Down,\n\nKefir, -Strawberry,\n\nMuffins,\n\nSmoothies, -Strawberry,\n\nBanana peppers,\n\nBarbecue(d)\n\nBeef, Korean,  CALORIE SMART\n\nBeef Sandwiches, 298\u2013299 CALORIE SMART \u2022 SLOW COOKER\n\nChicken Breasts, Grilled,\n\nChicken, Grilled,  CALORIE SMART\n\nPork Loin Ribs, Spicy, 310\u2013311\n\nBarbecue(d) _(cont.)_\n\nPot Roast,\n\nRub, Basic,\n\nSandwiches,\n\nBarbecue Sauce,\n\nCherry BBQ,\n\nMolasses-Mustard,\n\nSmoky,\n\nSpicy, 310\u2013311\n\nSweet-Savory,\n\nZesty BBQ,\n\nBarley,\n\nand Black Bean Pilaf,  CALORIE SMART\n\nCauliflower and Red Lentil Tabbouleh,  CALORIE SMART\n\nand Chicken Risotto with Edamame,  CALORIE SMART \u2022 SLOW COOKER\n\nPork and, Apple-Rosemary,  CALORIE SMART \u2022 Fast\n\nRisotto with Edamame, Vegetarian,\n\nsimmering,\n\nThree-Grain Medley,  CALORIE SMART \u2022 SLOW COOKER\n\nwith Vegetables, Mixed,  CALORIE SMART\n\nBars. _See also_ Brownie(s), Chocolate\n\nbaking pans for,\n\nBlondies with Brown Sugar Frosting,  CALORIE SMART\n\nChocolate Chip,\n\nDate,  CALORIE SMART\n\nFig,\n\nGranola,\n\ningredients,\n\nLemon,  CALORIE SMART\n\nmixing,\n\nNut, Triple-,\n\nstoring,\n\ntips for,\n\nBasic Barbecue Rub,\n\nBasic Omelet,  CALORIE SMART\n\nBasil,\n\nAngel Hair Pasta with Avocado, Tomatoes and,  Fast\n\nCaprese Salad, Heirloom Tomato,  CALORIE SMART \u2022 Fast \u2022 EASY\n\n-Cheese Triangles,  CALORIE SMART\n\n-Chicken Sauce, Creamy,\n\nchopping,\n\nCrostini, Tomato-,  CALORIE SMART \u2022 Fast\n\nLinguine with White Clam Sauce and,\n\nPesto,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nPesto, Roasted Red Pepper, Walnut and,\n\nSauce, Creamy, Chicken and Pasta with,  Fast\n\nSauce, Creamy, Pork and Pasta with,\n\nShrimp and Pasta, Tomato-,\n\n-Strawberry Water,\n\n-Tomato Soup,  CALORIE SMART\n\n-Turkey Penne Salad,\n\nVinaigrette Dressing,\n\nBasmati rice,\n\nBaster,\n\nBasting method,\n\nBay leaves,\n\nBean(s). _See also_ Black Bean(s); Black-Eyed Peas; Edamame; Garbanzo Bean(s); Green Bean(s)\n\nand Avocado Roll-Ups,\n\nBaked, Homemade,\n\nBurgers, Veggie and,  Fast\n\ncanned, as staple,\n\nCannellini with Tomatoes and Sage,  CALORIE SMART \u2022 Fast\n\nCannellini, Vermicelli with Fresh Herbs and,\n\nCassoulet, Great Northern Bean and Veggie Sausage,  SLOW COOKER\n\nChicken and White Beans, Tuscan Rosemary,  CALORIE SMART \u2022 Fast\n\nin Chili, Family Favorite,  CALORIE SMART \u2022 SLOW COOKER\n\nin Chili, White Chicken,  CALORIE SMART\n\ndried\n\ncooking timetable,\n\ndoneness,\n\nsoaking,\n\nvarieties of,\n\nFava, Rounds (Falafel),  CALORIE SMART\n\nfresh, preparing\/cooking,\n\nlong beans,\n\n\"Meatballs\", Veggie and,\n\nin Minestrone with Italian Sausage,  CALORIE SMART\n\nin Minestrone, Meatless,\n\nRed, and Rice,  CALORIE SMART\n\nRefried, Quick,\n\nSalad, Two-Bean,\n\nSoup, Tomato and, Chunky,\n\nWhite Bean Salad, Northern Italian,  CALORIE SMART\n\nWhite, with Sun-Dried Tomatoes,  SLOW COOKER \u2022 EASY\n\nBean paste,\n\nB\u00e9arnaise Sauce,\n\nBeef. _See also_ Burgers, Beef; Meat(s); Steak(s)\n\nBarbecued, Korean,  CALORIE SMART\n\nBolognese,  CALORIE SMART\n\nBolognese, Pasta,\n\nBorscht, 146\u2013147 CALORIE SMART\n\nwith Bow-Tie Pasta,  CALORIE SMART \u2022 Fast\n\nwith Bow-Tie Pasta and Feta,\n\nBrisket, Oven-Baked,\n\nBrisket, Smoked,  CALORIE SMART\n\nbroiling, ,\n\nand Broth,  CALORIE SMART\n\nCabbage Rolls, Classic,  CALORIE SMART\n\nin Chili, Family Favorite,  CALORIE SMART \u2022 SLOW COOKER\n\nCorned Beef Brisket, Spiced, with Horseradish Sour Cream,  CALORIE SMART \u2022 SLOW COOKER\n\ncuts of, 284\u2013285\n\nFajitas,\n\nand Kasha Mexicana,  CALORIE SMART \u2022 Fast\n\nand Kasha Tortillas,\n\nLettuce Wraps, Vietnamese,  CALORIE SMART\n\nManicotti,\n\nMeatballs, Hot and Saucy Cocktail,  CALORIE SMART\n\nMeatballs, Make-Ahead,  CALORIE SMART\n\nMeat Loaf,  CALORIE SMART\n\nMeat Loaf Casserole, Potato-Topped,  CALORIE SMART\n\nMeat Loaf Casserole, Quick Potato-Topped,\n\nMeat Loaf, Horseradish,\n\nMeat Loaves, Mini,\n\nNachos,  Fast\n\nNachos, Skillet,\n\nOsso Buco,\n\npanfrying, 317\u2013318\n\nPho,  CALORIE SMART\n\nPizza, Cheesy, 192\u2013193\n\nPot Pie, Samosa, 274\u2013275 CALORIE SMART\n\nPot Roast,\n\nBarbecue,\n\nCaramelized Onion,\n\nConfetti,\n\nCream Gravy,\n\nGarlic-Herb,\n\nItalian Pork,  CALORIE SMART\n\nRib Roast\n\ncarving\n\nHerb- and Garlic-Crusted,  CALORIE SMART \u2022 EASY\n\nwith Oven-Browned Potatoes,\n\nroasting timetable,\n\nRoast with Cabbage and Pasta, Asian,  SLOW COOKER\n\nroasting, 314\u2013315\n\nRoast, with Orange and Thyme,  CALORIE SMART\n\nRopa Vieja Tostadas,  CALORIE SMART \u2022 SLOW COOKER\n\nSandwiches\n\nBarbecue, 298\u2013299 CALORIE SMART \u2022 SLOW COOKER\n\nFrench Dip,  CALORIE SMART \u2022 SLOW COOKER\n\nItalian,\n\nReuben or Rachel,\n\nShort Rib Dinner, Beer-Braised,\n\nShort Ribs,\n\nShort Ribs and Sausage Rag\u00f9,\n\nSloppy Joes,  CALORIE SMART \u2022 Fast\n\nStew, 150\u2013151 CALORIE SMART\n\nStroganoff,  CALORIE SMART\n\nTaco Casserole, Fiesta,\n\nTenderloin, Bacon-Wrapped Herb-Roasted,\n\nTenderloin, Herb-Roasted,  CALORIE SMART\n\ntenderloin, whole, tying\n\nVegetable Soup,  CALORIE SMART\n\nBeer, 78\u201379\n\n-Batter Fried Fish,  CALORIE SMART \u2022 Fast\n\nBeergaritas,  Fast \u2022 EASY\n\n-Braised Short Rib Dinner,\n\n-Caramelized Onion and Bacon Burgers,\n\nvarieties of, 78\u201379\n\nBeer Can Chicken,  CALORIE SMART \u2022 EASY\n\nBeergaritas,  Fast \u2022 EASY\n\nBeet(s),\n\nand Baby Greens Salad,\n\nBorscht, 146\u2013147 CALORIE SMART\n\nPesto, -Dill,\n\nPickled,  CALORIE SMART\n\npreparing\/cooking,\n\nRoasted,  CALORIE SMART \u2022 EASY\n\nSalad, Roasted,\n\nBeet greens, ,\n\nBeignets with Espresso Sugar,  CALORIE SMART\n\nBelgian Endive,\n\nBellini Cocktails, Easy,\n\nBell Pepper(s),\n\nCheese Ball,  CALORIE SMART\n\nEdamame, Stir-Fried with Corn and,\n\nFajitas, Beef,\n\ngrilling,\n\nPickled,  CALORIE SMART\n\npreparing\/cooking,\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRatatouille Salad,  CALORIE SMART\n\nred pepper sauce,\n\nRoasted,  CALORIE SMART \u2022 EASY\n\nBacon-, Rice,\n\ncondiment,\n\nPuttanesca,\n\nRed, Basil and Walnut Pesto,\n\nRed, Broccoli with Hazelnuts and,  CALORIE SMART \u2022 Fast\n\nRed, Soup with Mozzarella, , CALORIE SMART\n\nand Soybean Stir-Fry,  CALORIE SMART \u2022 Fast\n\nTofu Teriyaki Noodles and,\n\n-Tomato Sauce, Snapper with,  CALORIE SMART \u2022 Fast\n\nTwice-Baked Potatoes, Bacon and,\n\nBerry(ies). _See also specific berries_\n\n-Chia Breakfast Pudding,\n\nCrisp, Fresh,\n\n-Cucumber-Avocado Salad,\n\nFood Color, Natural Homemade,\n\nand Greens Smoothie, Fresh,\n\nhulling\n\nJam, Freezer, Triple-Berry Pomegranate,  CALORIE SMART\n\nOatmeal-Flax Muesli, Triple,  CALORIE SMART\n\nPancakes,\n\nPops, -Plum,\n\nSangria,\n\nSimple Syrup, Triple-,\n\nTart, Crumble, Mixed,\n\nTea,\n\nvarieties of,\n\nVinegar,\n\nYogurt, Very Berry Frozen,  CALORIE SMART \u2022 EASY\n\nBeverages. _See also_ Cocktails; Coffee; Kefir; Lemonade; Tea; Smoothies\n\nbar, stocking, 77\u201378\n\nbeers, 78\u201379\n\nChocolate Milk,\n\nCider, Hot Buttered Rum-Spiced,\n\nCider, Hot Spiced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCitrus Splash,\n\nEgg Cream, New York,\n\nEggnog,\n\nEggnog, Bourbon-Maple,\n\nGreen Juice, Refreshing,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHot Chocolate,  Fast\n\nHot Chocolate, Double-Chocolate,\n\nHot Chocolate, Marshmallow-Topped,\n\nRaspberry-Lime Juice,  Fast \u2022 EASY\n\nStrawberry-Basil Water,\n\nStrawberry Milk, Fresh,  Fast \u2022 EASY\n\nwine varietals,\n\nBiscotti, Almond,\n\nBiscotti, Hazelnut,  CALORIE SMART\n\nBiscuits\n\nBaking Powder,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nButtermilk,\n\nCream, Chocolate Chip,\n\nCream, Easy,  CALORIE SMART \u2022 EASY\n\nCream, Herbed,\n\nDrop,\n\nHam 'n Egg Biscuit Sandwiches,\n\nproblems with\n\nSweet Potato\u2013Bacon,  CALORIE SMART\n\nBittersweet Chocolate Ganache,\n\nBlack Bean(s)\n\nand Barley Pilaf,  CALORIE SMART\n\nBurgers, California,  CALORIE SMART \u2022 Fast\n\nBurgers with Tomato and Cheese, California,\n\nCaribbean,\n\nChicken and Sausage with, Cheesy, Mexican,\n\nQuesadillas,\n\nRefried, Spicy,  CALORIE SMART \u2022 Fast\n\nSalsa,  CALORIE SMART\n\nSauce, Stir-Fry Tofu with, 392\u2013393 CALORIE SMART \u2022 Fast\n\nSliders,  CALORIE SMART\n\nSoup, Cuban,  CALORIE SMART\n\nand Tomato Salsa,\n\nBlackberry(ies),\n\nPie,\n\n-Thyme Cocktail,\n\nBlack-Eyed Peas\n\nCajun Tomatoes with,\n\nCollard Greens with,\n\nHoppin' John,\n\nSpicy,  CALORIE SMART \u2022 SLOW COOKER \u2022 EASY\n\nBlack mint paste,\n\nBlack Pepper and Parmesan Grissini,  CALORIE SMART\n\nBlack tea,\n\nBlanching method, ,\n\nBlender, ,\n\nBlondies with Brown Sugar Frosting,  CALORIE SMART\n\nBlood Orange,\n\nBloody Mary, Classic,  CALORIE SMART \u2022 Fast\n\nBLT Crostini,\n\nBlueberry(ies),\n\n-Banana Bread,\n\nCobbler, Fresh,\n\nCrisp,\n\nDoughnuts, -Orange, Baked,\n\n-Granola Parfaits,\n\nIce Cream, Fresh,\n\nJam, Freezer,\n\nKefir,\n\nMuffins,  CALORIE SMART\n\nMuffins, Streusel-Topped,\n\nPie,\n\nPie, -Peach,\n\nPies, Hand,\n\nScones, -Ginger,\n\nSmoothies,\n\nBlue Cheese,\n\nBurgers,\n\nChicken Salad,\n\nDeviled Eggs,\n\nDressing,  CALORIE SMART \u2022 EASY\n\nDressing, Yogurt,\n\nVinaigrette, Herb,\n\nBoiled Frosting,\n\nBoiled Hard-Shell Blue Crabs,  CALORIE SMART\n\nBoiled Lobsters,  CALORIE SMART\n\nBoiling method,\n\nBok Choy,\n\nBolognese _,_,  CALORIE SMART\n\nPasta,\n\nBorscht, _146,_ 146\u2013147 CALORIE SMART\n\nBoston Cream Pie,\n\nBoston Cream Shooters,\n\nBouillon,\n\nBourbon\n\nChicken with Corn Relish,  CALORIE SMART\n\n-Chocolate-Pecan Mini Pies,\n\nManhattan Cocktails,  CALORIE SMART \u2022 Fast \u2022 EASY\n\n-Maple Eggnog,\n\n-Maple Marinade,  CALORIE SMART \u2022 EASY\n\nBow-Tie Pasta\n\nBeef with Feta and,\n\nBeef with,  CALORIE SMART \u2022 Fast\n\nChicken and Pasta with Creamy Basil Sauce,  Fast\n\nand Kasha Pilaf,  CALORIE SMART\n\nShrimp Alfredo Primavera,  CALORIE SMART \u2022 Fast\n\nBoysenberry Pie,\n\nBraising poultry,\n\nBrandy-Herb Pan Steaks, _287,_ 287 CALORIE SMART \u2022 Fast\n\nMushroom,\n\nBran Muffins,  CALORIE SMART\n\nDate-,\n\nBranzino\n\nHerbed Butter,\n\nLemon-Garlic,\n\nRosemary-Lemon,\n\nwith Sun-Dried Tomato Pesto,\n\nwith Sweet Pea\u2013Mint Pesto, Fresh,  Fast\n\nBrazil nuts,\n\nBread. _See also_ Crostini; Croutons; French Toast\n\nCrumbs, Browned, Saut\u00e9ed Cauliflower with,  Fast\n\nFrench, uses for,\n\nGarlic, Cheesy,\n\nPanzanella,  CALORIE SMART\n\nStuffing, 346\u2013347\n\nApple, Roast Goose with,\n\nCornbread,\n\nOyster,\n\nSausage,\n\nSalad,\n\nBread(s), Quick. _See also_ Biscuits; Doughnuts; Muffins; Scones\n\nBanana,  CALORIE SMART\n\nBanana, Blueberry-,\n\nBeignets with Espresso Sugar,  CALORIE SMART\n\nBrunch Cake, Strawberry Cream,\n\nButter Cake, Lemon Curd\u2013Filled,\n\nCoffee Cake, Sour Cream, 494\u2013495\n\nCornbread,  CALORIE SMART\n\nCranberry,\n\nCranberry\u2013Sweet Potato,\n\nOatmeal Streusel,\n\npans for,\n\nPopovers,  CALORIE SMART \u2022 EASY\n\nPumpkin,\n\nstoring,\n\ntips for,\n\nYorkshire Pudding,\n\nZucchini,  CALORIE SMART\n\nZucchini, Orange-,\n\nBread(s), Yeast. _See also_ Pizza Crust; Rolls\n\nAsiago, Artisan, 498\u2013499 CALORIE SMART\n\nbread machine tips, 496\u2013497\n\nChallah Braid,\n\nCherry, Dried, and Walnut,\n\nCinnamon Swirl Raisin,\n\nEgg, Rich,  CALORIE SMART\n\nEnglish Muffins,  CALORIE SMART\n\nFocaccia,  CALORIE SMART\n\nFocaccia Breadsticks,\n\nFocaccia, Onion,\n\nFour-Grain Batter,  CALORIE SMART\n\nFrench,  CALORIE SMART\n\nGrissini, Black Pepper and Parmesan,  CALORIE SMART\n\ningredients for,\n\nkneading dough,\n\nmaking\/shaping dough\n\nNaan, Easy,\n\nNo-Knead Artisan,  CALORIE SMART\n\nPretzels, Soft,  CALORIE SMART\n\nPretzels, Soft, Parmesan-Herb,\n\nproblems with\n\nPumpernickel Loaf, Bakery-Style,\n\nSourdough,  CALORIE SMART\n\nstoring, ,\n\nWhite, Classic,  CALORIE SMART\n\nWhole Grain Artisan,  CALORIE SMART\n\nWhole Wheat, Honey\u2013, 504\u2013505 CALORIE SMART\n\nWhole Wheat, Sunflower-Herb,\n\nBread machine, , 496\u2013497\n\nBread Pudding\n\nCupcakes, Apple-Fig, with Maple Sauce,\n\nwith Lemon Curd,\n\nwith Whiskey Sauce, 636\u2013637\n\nBreadsticks, Focaccia,\n\nBreadsticks, Grissini, Black Pepper and Parmesan,  CALORIE SMART\n\nBreakfast Burritos,\n\nBreakfast Quinoa,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nBreakfast Roll-Up,\n\nBreakfasts, planning,\n\nBrie,\n\nin Puff Pastry, Double-Orange, with Cranberry Sauce,\n\nin Puff Pastry with Cranberry Sauce,\n\nBrine(d)\n\nSalmon, Smoked,  CALORIE SMART\n\nSeasoned,  CALORIE SMART \u2022 EASY\n\ntimetable, brining,\n\nTurkey, Whole,\n\nTurkey Breast, Apple-Maple,\n\nBrisket\n\nCorned Beef, Spiced, with Horseradish Sour Cream,  CALORIE SMART \u2022 SLOW COOKER\n\nOven-Baked Beef,\n\nRub,\n\nSmoked Beef,  CALORIE SMART\n\nBrittle\n\nAlmond,\n\nCashew,\n\nPeanut,  CALORIE SMART\n\nBroccoli,\n\nin Asian Salad with Wonton Crisps,  Fast\n\nCheesy Bacon,\n\nin Chicken Divan Pot Pie,\n\ngrilling,\n\npreparing\/cooking,\n\nwith Red Peppers, Roasted, and Hazelnuts,  CALORIE SMART \u2022 Fast\n\n-Sausage Pizza, Easy,\n\nSoup, Cream of,  CALORIE SMART\n\nBroccolini,\n\nBroccoli Rabe with Orecchiette, Browned Butter,  Fast\n\nBroiled Fish Fillets,\n\nBroiled Grapefruit,\n\nBroiled Pineapple,  CALORIE SMART \u2022 Fast\n\nBroiling, ,\n\nbest foods for,\n\nmeat, 316\u2013317\n\npoultry,\n\ntips for,\n\nBroth,\n\nBeef and,  CALORIE SMART\n\nChicken and,  CALORIE SMART\n\nFish,  CALORIE SMART\n\nVegetable,  CALORIE SMART\n\nvegetable, storing,\n\nBrowned Butter\n\nFish, Panfried,\n\nFrosting, ,\n\nFrosting, Cream,\n\nGlaze,\n\nOrecchiette with Broccoli Rabe,  Fast\n\nproblems with\n\nSauce, Herbed,\n\nSauce, \u2013Sage,\n\nBrownie(s), Chocolate,\n\nCaramel-Fudge, Outrageous,\n\ndoneness,\n\nDouble-Chocolate,\n\nFruitcake-Inspired,\n\nMix,\n\n\u2013Peanut Butter,\n\nPie,\n\nBrown Rice, ,\n\nBurritos, Salsa\u2013,\n\n\u2013Chicken Salad,\n\nCurried Veggie,  Fast\n\nFried,\n\nLettuce Wraps, Quick,\n\nPaella, Vegetable,  CALORIE SMART \u2022 Fast\n\nPaella, Vegetable Chicken,\n\nPilaf,\n\nPilaf, Almond\u2013,\n\nPudding, Easy,\n\nBrown Sugar,\n\nBacon, Peppered,\n\nBacon, and Balsamic,\n\nBacon, Praline,\n\nFrosting,\n\n\u2013Orange Glaze, Ham with,\n\nRefrigerator Cookies,  CALORIE SMART\n\nCitrus\u2013,\n\nCoconut, Toasted\u2013,\n\nMaple\u2013,\n\nsoftening in microwave,\n\nBrunch, planning,\n\nBrushes,\n\nBrussels Sprouts,\n\n\u2013Cashew Pesto,\n\nCheesy Bacon, 242\u2013243 CALORIE SMART\n\nCranberry-Pistachio,  CALORIE SMART \u2022 Fast\n\npreparing\/cooking,\n\nBucatini with Clam-Mushroom Sauce,\n\nBuckwheat, . _See also_ Kasha\n\nPancakes,  CALORIE SMART \u2022 Fast\n\nBuffalo\n\nChicken Wings,  CALORIE SMART\n\nCorn on the Cob,\n\nFrittata, Baked,  CALORIE SMART\n\nBuffet service, ,\n\nBulgur,\n\ncooking timetable,\n\nKibbeh Patties,  CALORIE SMART\n\nand Lentils, Mediterranean,  CALORIE SMART \u2022 SLOW COOKER\n\nPilaf,  CALORIE SMART \u2022 Fast\n\nTabbouleh, Classic,  CALORIE SMART\n\nTabbouleh, Minty,\n\nTabbouleh, Southwestern,\n\nBurgers\n\nBlack Bean, California,  CALORIE SMART \u2022 Fast\n\nBlack Bean, with Tomato and Cheese, California,\n\nBeef and Chorizo, with Roasted Chile Mayonnaise,\n\nTurkey, Garlic, Moist,\n\nVeggie and Bean,  Fast\n\nBurgers, Beef\n\nBlue Cheese,\n\nCaesar, Sweet Onion-Topped,  Fast\n\nCaramelized Beer-Onion and Bacon,\n\ndoneness,\n\nPatty Melts with Smothered Onions,  CALORIE SMART\n\nBurrito(s)\n\nBowls, Pork,\n\nBreakfast,\n\nSalsa\u2013Brown Rice,\n\nButter. _See also_Browned Butter\n\nAlmond,\n\nin baked goods,\n\nChile-Lime,\n\nClarified,  Fast\n\nClarified, Garlic,\n\nFlavored,  Fast \u2022 EASY\n\nGarlic,\n\nGarlic-Herb,\n\nGarlic-, Panfried Fish,\n\nghee,\n\nHerb,\n\nHerbed, Branzino,\n\nHerbed,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHerb-Garlic, Spatchcocked Chicken,  CALORIE SMART\n\nHerb, Grilled Corn with,  Fast\n\nHoney,\n\nLemon-Pepper,\n\nMaple,\n\nMaple-, Frosting,\n\nmelting in microwave,\n\nMustard,\n\nOrange,\n\nParmesan, Italian,\n\npastry,\n\nPecan,\n\n-Pecan Syrup,\n\nRaspberry,\n\nRosemary, Artichokes with,  EASY\n\nsoftening,\n\nsoynut,\n\nstoring,\n\nButter Balls, Orange-Rosemary, Chocolate-Filled,  CALORIE SMART\n\nButter Cake,\n\nLemon Curd\u2013Filled,\n\nButterhead lettuce,\n\nButtermilk, ,\n\nBiscuits,\n\nChicken, Fried,\n\nColeslaw, Tangy,\n\nin Cream Cake, Italian,\n\nDressing, Tangy, Chopped Salad with,\n\nMashed Potatoes,\n\nPancakes,\n\nButtermilk _(cont.)_\n\nPastry, Easy,  Fast \u2022 EASY\n\nRanch Dressing,  CALORIE SMART \u2022 EASY\n\nRanch Dressing, Fresh Herb,\n\nRanch Dressing, -Parmesan,\n\nin Red Velvet Cake,\n\nSmoothies, Orange,\n\nstoring,\n\nStrawberry Pops,\n\nWaffles, Whole Wheat, with Honey\u2013Peanut Butter Drizzle,\n\nButternut Squash,\n\nRavioli,\n\nRisotto with, Classic,\n\nSoup,\n\nand Turkey Ragout,  CALORIE SMART \u2022 SLOW COOKER\n\nButterscotch\n\nCream Pie,\n\nFrosting,\n\nPudding,\n\nButtery Broiled Fish Steaks,  CALORIE SMART \u2022 Fast\n\nButtery Rosemary-Honey Roasted Chicken,\n\n## C\n\nCabbage, , . _See also_ Coleslaw\n\nBeef Roast with Pasta and, Asian,  SLOW COOKER\n\nBorscht, 146\u2013147 CALORIE SMART\n\nKimchi,  CALORIE SMART\n\nPickled, 366\u2013367\n\nRed, Sweet-Sour,  CALORIE SMART\n\nRolls, Classic,  CALORIE SMART\n\nRolls, Tropical Stuffed,  SLOW COOKER\n\nthinly slicing\n\nVeggie Relish, Crunchy,  CALORIE SMART\n\nCacio e Pepe,  Fast\n\nCaesar Burgers, Sweet Onion-Topped,  Fast\n\nCaesar Salad,  CALORIE SMART \u2022 Fast\n\nChicken,\n\nGrilled,\n\nKale,\n\nShrimp,\n\nCajun\n\nChicken, Smoked,  CALORIE SMART\n\nFish Tacos, Soft, 366\u2013367 Fast\n\nSpice Rub,\n\nTomatoes with Black-Eyed Peas,\n\nCake(s). _See also_ Cheesecake; Cupcakes; Frosting; Pound Cake\n\nAngel Food, ,  CALORIE SMART\n\nCherry,\n\nChocolate-Cherry,\n\nChocolate Confetti,\n\nEspresso,\n\nApple, Spiced, with Maple Glaze,\n\nbeating\/whipping\/folding,\n\nBoston Cream Pie,\n\nbutter, about, ,\n\nButter, Lemon Curd\u2013Filled,\n\nCarrot, 562\u2013563\n\nChocolate Chip,\n\nChocolate-Fig,\n\nChocolate, German,\n\nChocolate Layer,\n\nChocolate, Molten,\n\nChocolate Mug, Banana Split-, 576\u2013577\n\nChocolate Sheet,\n\nCoconut,\n\nCoffee, Sour Cream, 494\u2013495\n\nCookies 'n Creme,\n\nCream, Italian,\n\ndoneness,\n\nEspresso, with Mocha Streusel Ribbon,\n\nfoam cakes,\n\nGingerbread,\n\nHot Fudge Sundae, Double-Chocolate,\n\nJelly Roll,\n\nJelly Roll, Lemon Curd,\n\nMarble,\n\nmarbling\n\nPear-Ginger Upside-Down,\n\nPumpkin Pudding,\n\nRed Velvet,\n\nremoving from pan\n\nShortcakes, Strawberry,\n\nShortcakes, Drop,\n\nSilver White,\n\nSnickerdoodle, Caramel,\n\nSpice, Sour Cream,\n\nStarlight Yellow,\n\nstoring, ,\n\nStrawberry,\n\nStrawberry Cream Brunch,\n\ntips for,\n\nTres Leches,\n\nTres Leches, Tropical,\n\nTurtle Layer,\n\nZucchini,\n\nCake Doughnuts,  CALORIE SMART\n\nCake pans, ,\n\nCalifornia Black Bean Burgers,  CALORIE SMART \u2022 Fast\n\nCalifornia Black Bean Burgers with Tomato and Cheese,\n\nCalzone,\n\nCanadian Bacon Pizza,\n\nCandied Nuts,\n\nCandied Sweet Potatoes,  EASY\n\nCandy(ies)\n\nBrittle, Almond,\n\nBrittle, Cashew,\n\nBrittle, Peanut,  Calorie Smart\n\nCaramels,  CALORIE SMART\n\nCaramels, Chocolate,\n\nCaramels, Sea Salt, Chocolate-Topped,\n\nCookies,\n\nDivinity, Pistachio-Cranberry,  CALORIE SMART\n\nequipment and tools,\n\nFudge\n\nCherry-Pistachio,\n\nCoconut,\n\nPeanut Butter Candy,\n\nTriple Chocolate,  CALORIE SMART\n\nTriple-Chocolate and Candy,\n\nmaking,\n\nPralines,  CALORIE SMART\n\nTruffles, Chocolate, Luscious,  CALORIE SMART\n\nTruffles, Toffee,\n\nTruffles, White-Chocolate-Chocolate,\n\nCandy thermometer, ,\n\nCannellini with Tomatoes and Sage,  CALORIE SMART \u2022 Fast\n\nCannellini, Vermicelli with Fresh Herbs and,\n\nCanning and preserving\n\ncontainers for,\n\nequipment,\n\nfood safety in,\n\njars, preparing,\n\npectin, dissolving\n\nCan opener,\n\nCaperberries,\n\nCapers, ,\n\nCappuccino, _68,_ 68 CALORIE SMART \u2022 Fast \u2022 EASY\n\n-Caramel Cheesecake,\n\nCaprese Salad, Heirloom Tomato,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCaramel(s),  CALORIE SMART\n\nApple Crisp,\n\n-Cappuccino Cheesecake,\n\nChicken with Pickled Cucumber and Onion,\n\nChocolate,\n\n-Chocolate Compost Cookies,\n\nChocolate-Topped Sea Salt,\n\nCustard (Cr\u00e8me Caramel),\n\nFrosting,  EASY\n\nFrosting, Salty,\n\n-Fudge Brownies, Outrageous,\n\nIce Cream,\n\nLatte,\n\nmelting in microwave,\n\nPears, -Maple,  SLOW COOKER \u2022 EASY\n\nSauce,  CALORIE SMART \u2022 EASY\n\nSnickerdoodle Cake,\n\nSticky Rolls,\n\nTarts, Tiny,\n\nCaramelized Onion(s). _See_ Onion(s), Caramelized\n\nCaramelizing,\n\nonions\n\nsugar,\n\nCaraway,\n\nCarbonara, Spaghetti,  Fast\n\nChicken and,\n\nCardamom, ,\n\n-Cinnamon Mixed Nuts,\n\nYogurt Cheese, -Scented,\n\nCaribbean Black Beans,\n\nCarne seca,\n\nCarnitas, Pork, Taco Bites,\n\nCarnival squash,\n\nCarrot(s),\n\nBaby Food,  CALORIE SMART \u2022 EASY\n\nCake, 562\u2013563\n\nCumin-Chile Roasted,  CALORIE SMART\n\nCupcakes,\n\nGlazed,  CALORIE SMART \u2022 Fast\n\ngrilling,\n\nHoney-Glazed Lemon,\n\nin Kimchi,  CALORIE SMART\n\nMuffins, Golden Harvest,\n\nPickled, Tarragon Baby,  CALORIE SMART \u2022 EASY\n\nPickled, Spicy,\n\npreparing\/cooking,\n\nRoasted, -Almond Pesto,\n\nCashew(s),\n\n-Asparagus Salad,\n\nBrittle,\n\n\u2013Brussels Sprouts Pesto,\n\nPilaf,\n\nSalted, -Bittersweet Chocolate Tart,\n\nSouthwestern Spiced Party,  Fast\n\nCashew milk,\n\nCassava,\n\nCasserole(s)\n\nGreen Bean,\n\nMeat Loaf, Potato-Topped,  CALORIE SMART\n\nMeat Loaf, Potato-Topped, Quick,\n\nTaco, Fiesta,\n\nTurkey and Wild Rice, Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\nVegetable, Indian-Style,  CALORIE SMART\n\nCasserole pan,\n\nCassoulet, Great Northern Bean and Veggie Sausage,  SLOW COOKER\n\nCatfish, Cornmeal-Crusted,  CALORIE SMART \u2022 Fast\n\nCauliflower,\n\nBarley and Red Lentil Tabbouleh,  CALORIE SMART\n\ngrilling,\n\nPan-Roasted, Chicken with Orzo and,  CALORIE SMART \u2022 Fast\n\npreparing\/cooking,\n\nSaut\u00e9ed, with Browned Bread Crumbs,  Fast\n\nSoup, Cream of,\n\nSteaks with Olive-Tomato Relish,\n\n\u2013Sweet Potato Curry,  CALORIE SMART\n\nCayenne (red pepper),\n\nCelery,\n\nCelery seed,\n\nCereal\n\nGranola,\n\nMuesli, Triple-Berry Oatmeal-Flax,  CALORIE SMART\n\nOatmeal, Apple, Baked,  CALORIE SMART\n\nPorridge, Bacon and Pecan Wild Rice,\n\nPorridge, Cranberry\u2013Wild Rice,\n\nQuinoa, Breakfast,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChai, _69,_ 69 CALORIE SMART \u2022 Fast\n\nChai Mix,  CALORIE SMART \u2022 Fast\n\nChallah Braid,\n\nChanterelle mushrooms,\n\nCharcoal grills, . _See also_ Grilling\n\nCharcuterie, , _photo_\n\nChard,\n\nCheddar, . _See also_ Macaroni, and Cheese\n\nCornbread Scones, -Chile,\n\nCream Cheese Spread, -Bacon-Chive,\n\nDeviled Eggs, -Bacon,\n\nQuiche, and Spinach,  CALORIE SMART\n\nStrata, and Ham, 96\u201397 CALORIE SMART\n\nTuna Melts,\n\nCheese. _See also_ Cheese Balls; Cheesy; _specific cheeses_\n\n-Bacon Spread,\n\nBlack Bean Burgers with Tomato and, California,\n\nCalzone,\n\nChicken Sausage with Potatoes and, Roasted,  CALORIE SMART\n\nChopped Salad, Turkey-,\n\nCrostini, Garlic-,\n\nEnchiladas,\n\nEnchilada, Simple,\n\nEnchiladas Verde,\n\nGrilled Cheese Sandwiches,  Fast\n\nPesto Parmesan,\n\nTomato and Avocado,\n\nGrits,  CALORIE SMART\n\nGrits, Bacon and,\n\nin Lasagna, Italian Sausage,\n\nMacaroni and. _See_ Macaroni, and Cheese\n\nin Manicotti,\n\nOmelet,\n\nOmelet, Avocado, Bacon and,\n\nOmelet, Ham and,\n\nPatty Melts with Smothered Onions,  CALORIE SMART\n\nPizza,\n\nRavioli, Four-Cheese Homemade,  CALORIE SMART \u2022 EASY\n\nSauce,\n\nshredded,\n\nsoy,\n\nstoring,\n\ntray, creating,\n\nTriangles, Basil-,  CALORIE SMART\n\nvarieties of,\n\nYogurt, Cardamom-Scented,\n\nCheese Ball(s), _46,_ 46 CALORIE SMART\n\nBell Pepper,\n\nDill Havarti,\n\nMini\n\nFrench,  CALORIE SMART\n\nIndian,\n\nKorean,\n\nMexican,\n\nPecan-Coated,\n\nPepper Jack,\n\nCheeseburger Hoagies,\n\nCheesecake(s)\n\nCaramel-Cappuccino,\n\nChocolate Chip New York,\n\nmaking\n\nNew York,\n\nstoring,\n\nCheese knives\/spreaders,\n\nCheesy\n\nBeef Pizza, 192\u2013193\n\nBroccoli, Bacon,\n\nBrussels Sprouts, Bacon, 242\u2013243 CALORIE SMART\n\nChicken and Sausage with Black Beans, Mexican,\n\nEggs,\n\nGarlic Bread,\n\nGreen Beans, Bacon,\n\nPotato-Bacon Soup,  SLOW COOKER\n\nPotato Skins,\n\nRigatoni with Eggplant Sauce,\n\nCherimoya,\n\nCherry(ies),\n\nAngel Food Cake,\n\nAngel Food Cake, Chocolate-,\n\nBBQ Sauce,\n\nCobbler, Fresh,\n\nDried, and Walnut Bread,\n\nFrosting, -Nut,\n\nFudge, -Pistachio,\n\nKale Salad, -Walnut,  CALORIE SMART \u2022 Fast\n\nPie,\n\nPie, Quick,\n\nQuinoa, -Pecan,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRice Pilaf, Pistachio,\n\nRice Pudding, Almond-,\n\nScones, \u2013Chocolate Chip,\n\nScones, Lemon-,\n\nShortbread Cookies,\n\nsour\/sweet,\n\nSugar Cookies, Chocolate Chip, Pistachio and, Soft No-Roll,\n\nChervil,\n\nChestnuts,\n\nCh\u00e8vre. _See_ Goat Cheese\n\nChia-Berry Breakfast Pudding,\n\nChicken. _See also_ Poultry\n\nAdobo,  CALORIE SMART\n\n-and Shells, Spinach-Stuffed,\n\nAngel Hair Pasta with,\n\nand Barley Risotto with Edamame,  CALORIE SMART \u2022 SLOW COOKER\n\n-Basil Sauce, Creamy,\n\nbreasts, boning\n\nbreasts, flattening,\n\nbrining time,\n\nbroiling, ,\n\nand Broth,  CALORIE SMART\n\nbrowning,\n\n\u2013Brown Rice Salad,\n\nCacciatore,  CALORIE SMART\n\nCaesar Salad,\n\nCaramel, with Pickled Cucumber and Onion,\n\nwith Cauliflower, Pan-Roasted, and Orzo,  CALORIE SMART \u2022 Fast\n\nChili, White,  CALORIE SMART\n\nChopped Salad,\n\nChopped Salad, and Bacon,\n\nin Cobb Salad,\n\nCoconut, Thai-Style,  CALORIE SMART\n\nCoffee, with Quick Mole Sauce,  CALORIE SMART\n\nCoq au Vin,  CALORIE SMART\n\nCordon Bleu, Oven,  CALORIE SMART\n\nCornmeal, Crunchy, with Mango-Peach Salsa,  Fast\n\nDijon, Smothered in Mushrooms,  CALORIE SMART \u2022 Fast\n\nand Dumplings,  CALORIE SMART\n\nFettuccine Alfredo,\n\nFingers, Oven-Fried,\n\nFried, Buttermilk,\n\nFried, Korean,  CALORIE SMART\n\nFried, Skillet-,  CALORIE SMART \u2022 EASY\n\nGrilled\n\nBarbecue,  CALORIE SMART\n\nBeer Can,  CALORIE SMART \u2022 EASY\n\nBourbon, with Corn Relish,  CALORIE SMART\n\nBreasts, Barbecue,\n\nFajitas,\n\nHerb, Three-,\n\nKabobs, Easy,\n\nLemon, with Grilled Fennel and Onion,  CALORIE SMART\n\nPackets, Mediterranean,  CALORIE SMART\n\nHerbed Marinated,\n\nKorma,  CALORIE SMART\n\n-Macaroni Salad,\n\n-Mandarin Salad, Crunchy,\n\nMarsala,  CALORIE SMART\n\nMeatballs,\n\n-Mushroom Soup,\n\nNachos,\n\nNachos, Skillet,  Fast\n\nwith Orzo, Greek,\n\nOven-Fried,  CALORIE SMART \u2022 EASY\n\nOven-Fried, Crunchy,\n\nPad Thai with,\n\nPaella, Vegetable,\n\nPan-Fried, with Romesco Sauce,  CALORIE SMART\n\nPanzanella, Baked,  CALORIE SMART\n\nParmigiana,\n\nand Pasta with Creamy Basil Sauce,  Fast\n\nin Peanut Sauce, Spicy,  SLOW COOKER\n\nPenne Salad, -Thyme,\n\nPiccata,  CALORIE SMART \u2022 Fast\n\nPie, Moroccan,\n\nand Potatoes, Sage,  CALORIE SMART\n\nPot Pie(s),\n\nCranberry-,  CALORIE SMART\n\nDivan,\n\nIndividual,\n\nPortabella,\n\nand Rice, Country French,  CALORIE SMART\n\nRice, Fried,\n\nRoasted\n\nButtery Rosemary-Honey,\n\ncarving\n\nHerb-, and Vegetables,  CALORIE SMART\n\nProven\u00e7al,\n\nroasting,\n\nRoll-Ups, Asian,  CALORIE SMART \u2022 Fast\n\nwith Romesco Sauce, Pan-Fried,  CALORIE SMART\n\nRotisserie Spiced,  CALORIE SMART \u2022 SLOW COOKER\n\nSalad, Blue Cheese,\n\nSalad Sandwiches,  Fast \u2022 EASY\n\nSatay with Peanut Sauce, 56\u201357\n\nSausage,\n\nwith Potatoes and Cheese, Roasted,  CALORIE SMART\n\nSpinach and Swiss Pizza,  Fast\n\nand Sausage with Black Beans, Mexican Cheesy,\n\nSmoked Cajun,  CALORIE SMART\n\nSoup\n\n-Lentil, Italian,  CALORIE SMART \u2022 SLOW COOKER\n\nMeatball,  CALORIE SMART\n\nNoodle,  CALORIE SMART\n\nNoodle, Confetti,\n\nRice,\n\nTortilla, ,  CALORIE SMART\n\nSpaghetti,\n\nand Spaghetti Carbonara,\n\nSpatchcocked, Herb-Garlic Butter,  CALORIE SMART\n\nspatchcocking\n\nStew, Creamy Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\nTagine,  CALORIE SMART\n\nThighs with Bacon and Spinach, Skillet,  CALORIE SMART\n\nwith Tomatoes and Spinach, Easy,  CALORIE SMART\n\nTomatoes with, Thyme,\n\nand Vegetables Pasta,\n\nand White Beans, Tuscan Rosemary,  CALORIE SMART \u2022 Fast\n\nwhole, cutting up\n\nand Wild Rice Salad,\n\nand Wild Rice Soup,\n\nWings\n\nBuffalo,  CALORIE SMART\n\nHoney-Mustard,\n\nSalsa,\n\nSweet-Spicy,\n\nyields,\n\nChicken-Fried Steak,\n\nChickpeas. _See_ Garbanzo Bean(s)\n\nChiffon cakes,\n\nChile(s). _See also_ Chipotle; Jalape\u00f1o; Sriracha\n\nCornbread Scones, Cheddar-,\n\n-Cumin Roasted Carrots,  CALORIE SMART\n\n-Cumin Rub,\n\nHarissa, ,  CALORIE SMART\n\nKabobs, Sweet,\n\n-Lime Butter, Grilled Corn with,\n\n-Lime Corn on the Cob,\n\nMayonnaise, Roasted,\n\nMole Poblano,  CALORIE SMART\n\nPoblanos Rellenos Bake, Roasted,\n\nRed Curry Paste, Thai,  CALORIE SMART\n\nRellenos Bake,  CALORIE SMART\n\nSalsa Verde,  CALORIE SMART \u2022 Fast\n\nSteak with Corn and, Southwestern,  CALORIE SMART \u2022 Fast\n\nThai, ,\n\nvarieties of,\n\nChile bean paste,\n\nChili\n\nChicken, White,  CALORIE SMART\n\nFamily Favorite,  CALORIE SMART \u2022 SLOW COOKER\n\ntips for,\n\nTortilla Green,  CALORIE SMART \u2022 Fast\n\nChimichurri Sauce,  CALORIE SMART \u2022 EASY\n\nChipotle\n\nDeviled Eggs,\n\nGuacamole,\n\n-Peanut Noodle Bowls,  Fast\n\nTacos, Soft, Smoky,  CALORIE SMART \u2022 SLOW COOKER\n\nChive(s),\n\n-Bacon-Cheddar Cream Cheese Spread,\n\nBacon and Parmesan Ravioli,\n\n-Mustard Cream Sauce, Pork Chops with,\n\nChocolate. _See also_ Brownie(s), Chocolate; Cocoa; Fudge; White Chocolate\n\nCake. _See_ Chocolate Cake(s)\n\n-Caramel Compost Cookies,\n\nCaramels,\n\nCaramels, Sea Salt, -Topped,\n\n-covered confections,\n\nCream Pie,\n\nCream Pie, -Banana,\n\nCrinkles,  CALORIE SMART\n\nCupcakes,  CALORIE SMART\n\ncurls,\n\ndipping cookies in\n\n-Filled Butter Balls, Orange-Rosemary,  CALORIE SMART\n\nFondue, -Sour Cream,  Fast\n\nFrench Silk Pie, Classic,\n\nFrosting, Cream Cheese,\n\nFrosting, Creamy,  Fast \u2022 EASY\n\nGanache,  Fast \u2022 EASY\n\nGanache, Bittersweet,\n\nGanache, Bittersweet, French Macarons with, 538\u2013539 CALORIE SMART\n\nGlaze,  Fast \u2022 EASY\n\nDark Chocolate,\n\nMilk Chocolate,\n\nMint Chocolate,\n\nWhite Chocolate,\n\n-Hazelnut Sandwiches,  CALORIE SMART\n\n-Hazelnut Thumbprint Cookies,\n\nHot,  Fast\n\nHot, Double-Chocolate,\n\nHot, Marshmallow-Topped,\n\nIce Cream,\n\nMartini Cocktails,\n\nmelting, ,\n\nMeringue Shell,\n\nMilk,\n\nMole Poblano,  CALORIE SMART\n\nMousse,\n\n-Oatmeal Drops, No-Bake,\n\nPies, Mini, Bourbon-Pecan-,\n\nPudding,\n\nPuffs, Double-,\n\nSauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nShortbread Cookies, Dipped-,\n\nSpritz Cookies, Buttery,\n\nswirls and filigree,\n\nTart, Bittersweet, Salted Cashew-,\n\nTart, Raspberry, White and Dark Chocolate, 600\u2013601\n\nTruffles\n\nLuscious,  CALORIE SMART\n\nToffee,\n\nWhite-Chocolate-Chocolate,\n\nvarieties of, ,\n\nChocolate Cake(s)\n\nAngel Food, -Cherry,\n\nAngel Food, Confetti,\n\n-Fig,\n\nGerman,\n\nHot Fudge Sundae, Double-Chocolate,\n\nLayer,\n\nMolten,\n\nMug, Banana Split-, 576\u2013577\n\nRoll,\n\nSheet,\n\nChocolate Chip(s)\n\nBars,\n\nCake,\n\nCheesecake, New York,\n\nCream Biscuits,\n\nmelting in microwave,\n\nMuffins,\n\nScones,\n\nScones, Cherry\u2013,\n\nSugar Cookies, Cherry, Pistachio and, Soft No-Roll,\n\nWhoopie Pies,\n\nChocolate Chip Cookies,  CALORIE SMART\n\nDulce de Leche,\n\nJumbo,\n\nOatmeal\u2013,\n\nproblems with\n\nChopped Salad. _See_ Salad(s), Chopped\n\nChorizo,\n\nand Beef Burgers with Roasted Chile Mayonnaise,\n\nPotato and Egg Scramble,\n\nChowder, Manhattan Clam,  CALORIE SMART\n\nChowder, New England Clam,  CALORIE SMART\n\nChrysanthemum leaves,\n\nChunky Tomato-Bean Soup, _163,_ 163\n\nChutney\n\nApple-Cranberry,  CALORIE SMART\n\ncontainers for,\n\nPear-Cranberry,\n\nTartar Sauce, Curry-,\n\nCider\n\nHot Buttered Rum-Spiced,\n\nHot Spiced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSauce, Panfried Pork Chops with,  CALORIE SMART \u2022 Fast\n\nSyrup, Spicy,\n\nCider vinegar,\n\nCilantro,\n\n-Almond Pesto, Spicy,\n\n-Garlic Corn on the Cob,\n\nPesto,\n\nCinnamon,\n\n-Apple Muffins,\n\n-Cardamom Nuts, Mixed,\n\n\u2013Cream Cheese Frosting,\n\nCream Cheese Spread, -Maple,\n\nRolls,\n\nSalmon Salad, -Maple Glazed,\n\n-Sugared Nuts,\n\nSwirl Raisin Bread,\n\nCitrus, . _See also specific fruits_\n\n\u2013Brown Sugar Refrigerator Cookies,\n\nSeafood Skillet,  CALORIE SMART \u2022 Fast\n\nSplash,\n\ntwist, making\n\nCitrus reamer\/juicer,\n\nClam(s). _See also_ Shellfish\n\nbuying, ,\n\nChowder, Manhattan,  CALORIE SMART\n\nChowder, New England,  CALORIE SMART\n\nDeep-Fried,\n\nlittleneck,\n\nraw, opening\n\nSauce, Bucatini with,\n\nSausage and Linguine, Spanish,  CALORIE SMART\n\nSteamed,  CALORIE SMART\n\nvarieties of,\n\nWhite Clam Sauce, Linguine with Basil and,\n\nWhite Clam Sauce, Spaghetti with,  Fast\n\nClarified Butter,  Fast\n\nGarlic,\n\nClassic Apple Pie,\n\nClassic Bloody Mary,  CALORIE SMART \u2022 Fast\n\nClassic Cabbage Rolls,  CALORIE SMART\n\nClassic Crab Cakes,  Fast\n\nClassic French Silk Pie,\n\nClassic Peach Pie,\n\nClassic Risotto. _See_ Risotto, Classic\n\nClassic Shakshuka,  CALORIE SMART\n\nClassic Shrimp Cocktail,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nClassic Tabbouleh,  CALORIE SMART\n\nClassic White Bread,  CALORIE SMART\n\nCleaning\/sanitizing,\n\nCloverleaf Rolls,\n\nCloves,\n\nCobbler(s)\n\nApple,\n\nBlueberry, Fresh,\n\nCherry, Fresh,\n\nPeach,\n\nRaspberry-Peach,\n\nCobb Salad, _125,_ 125\n\nCocktails. _See also_Martini Cocktails; Mimosas; Sangria\n\nbar, stocking, 77\u201378\n\nBeergaritas,  Fast \u2022 EASY\n\nBellini, Easy,\n\nBlackberry-Thyme,\n\nBloody Mary, Classic,  CALORIE SMART \u2022 Fast\n\ncitrus twist for\n\nManhattan,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nMargaritas,  Fast\n\nMargaritas, Strawberry, Frozen,\n\nMojitos,\n\nshaken,\n\nCocktail Sauce,\n\nCocoa\n\nfor baking, ,\n\nFrosting, Creamy,\n\nMarshmallows,\n\nCoconut,\n\nCake,\n\nChicken, Thai-Style,  CALORIE SMART\n\nCream Pie,\n\n-Curry, Spicy, Grilled Shrimp,  CALORIE SMART\n\nFudge,\n\n-Pecan Frosting,\n\nPound Cake, Orange-,\n\nSauce,\n\nToasted, \u2013Brown Sugar Refrigerator Cookies,\n\ntoasting, ,\n\nCoconut milk,\n\nCod, Cornmeal-Crusted,\n\nCoffee, . _See also_ Espresso; Mocha\n\nappliances for,\n\nbrewing,\n\nCappuccino,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCappuccino-Caramel Cheesecake,\n\nChicken with Quick Mole Sauce,  CALORIE SMART\n\nIced\n\n CALORIE SMART \u2022 EASY\n\nHazelnut,\n\nSoy Latte, Vanilla,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSweetened,\n\nVietnamese,\n\nLatte, Caramel,\n\nTiramisu,\n\nCoffee Cake, Sour Cream, 494\u2013495\n\nColanders\/strainers,\n\nCold foods, safe handling of,\n\nColeslaw\n\nCreamy,  CALORIE SMART\n\nCreamy, Tropical,\n\nand Pork Sandwiches,\n\nSweet-and-Sour,  CALORIE SMART\n\nTangy,\n\nCollard Greens,\n\nwith Bacon, Spicy,  CALORIE SMART\n\nwith Black-Eyed Peas,\n\npreparing\n\nCondensed milk, sweetened,\n\nCondiments, 8\u20139\n\nConfetti Chicken Noodle Soup,\n\nConfetti Cookie Mix,  CALORIE SMART\n\nConfetti Pot Roast,\n\nConserve,\n\nPear-Orange,  CALORIE SMART\n\nContainers for preserving,\n\nCookie Crumb Crust,\n\nCookies. _See also_ Bars; Chocolate Chip Cookie(s); Shortbread Cookies\n\nBiscotti, Almond,\n\nBiscotti, Hazelnut,  CALORIE SMART\n\nBrown Sugar Refrigerator\n\n CALORIE SMART\n\nCitrus\u2013,\n\nCoconut, Toasted\u2013,\n\nMaple\u2013,\n\nButter Balls, Chocolate-Filled Orange-Rosemary,  CALORIE SMART\n\nCandy,\n\nChocolate-Caramel Compost,\n\nChocolate Crinkles,  CALORIE SMART\n\nChocolate-Hazelnut Sandwiches,  CALORIE SMART\n\nChocolate-Oatmeal Drops, No-Bake,\n\ndecorating,\n\ndipping in chocolate\n\nGingerbread, 532\u2013533 CALORIE SMART\n\nGingersnaps,  CALORIE SMART\n\nGranola,\n\ningredients,\n\nMacarons\n\nFrench, with Bittersweet Chocolate Ganache, 538\u2013539 CALORIE SMART\n\nLemon,\n\nMint,\n\nOrange,\n\nMix, Confetti Cookie,  CALORIE SMART\n\nmixing,\n\nOatmeal\u2013Chocolate Chip,\n\nOatmeal-Cranberry,\n\nOatmeal-Raisin,  CALORIE SMART\n\nPeanut Butter,  CALORIE SMART\n\nPeanut Butter Chip, Rich,\n\nPumpkin, with Browned Butter Frosting,\n\nPumpkin, Spicy, with Browned Butter Frosting,\n\nrolling\n\nSnickerdoodles,  CALORIE SMART\n\nSpritz,  CALORIE SMART\n\nSpritz, Chocolate Buttery,\n\nSugar. _See_ Sugar Cookies\n\nstoring, ,\n\ntest cookie, baking,\n\nThumbprint(s)\n\n534\u2013535 CALORIE SMART\n\nHazelnut-Chocolate,\n\nLemon-Almond,\n\nOatmeal-Date Quinoa,  CALORIE SMART\n\nPeanut-Fudge,\n\ntips for,\n\nWhoopie Pies,\n\nChocolate Chip,\n\nPeppermint, Pink,\n\nToffee,\n\nCookie sheets, ,\n\nCookies 'n Creme Cake,\n\nCooking. _See also_Microwave cooking\/heating; _specific foods and methods_\n\nquick techniques, 19\u201320\n\nslow cooker,\n\nterms, 21\u201323\n\ntools\/gadgets, 11\u201312\n\nCookware, ,\n\nCooling racks,\n\nCoq au Vin,  CALORIE SMART\n\nCordon Bleu, Oven Chicken,  CALORIE SMART\n\nCoriander,\n\nCoring,\n\nCorn,\n\nwith Chile-Lime Butter,\n\non the Cob, Grilled,\n\nEdamame, Stir-Fried with Peppers and,\n\ngrilling,\n\nwith Herb Butter,  Fast\n\nOkra and Tomatoes,  CALORIE SMART \u2022 Fast\n\npreparing\/cooking,\n\nRelish, ,  CALORIE SMART\n\nSalsa,\n\nSteak with Chiles and, Southwestern,  CALORIE SMART \u2022 Fast\n\nCornbread,  CALORIE SMART\n\n-and Bacon-Stuffed Pork Chops,\n\nPancakes,  CALORIE SMART \u2022 Fast\n\nScones, Cheddar-Chile,\n\nStuffing,\n\nCorned Beef\n\nBrisket, Spiced, with Horseradish Sour Cream,  CALORIE SMART \u2022 SLOW COOKER\n\nin Reuben sandwich,\n\nroasting,\n\nCornish hen, roasting,\n\nCornmeal, . _See also_ Cornbread; Polenta\n\nCatfish, -Crusted,  CALORIE SMART \u2022 Fast\n\nChicken, Crunchy, with Mango-Peach Salsa,  Fast\n\nCod, -Crusted,\n\nCorn Muffins,\n\nCupcakes, Maple-, with Maple-Butter Frosting,\n\nCorn Muffins,\n\nCornstarch, ,\n\nCorn Syrup,\n\nCotija,\n\nGuacamole,\n\nCountry French Chicken and Rice,  CALORIE SMART\n\nCountry-Style Pork Ribs,\n\nCouscous,\n\nFruity,\n\nCrab(s). _See also_ Shellfish\n\nBoiled Hard-Shell Blue,  CALORIE SMART\n\nbuying,\n\nCakes, Classic,  Fast\n\ncooked meat, removing\n\nlegs, reheating,\n\nsoft-shell,\n\nCranberry(ies),\n\n-Apple Chutney,  CALORIE SMART\n\nBread,\n\nBrussels Sprouts, -Pistachio,  CALORIE SMART \u2022 Fast\n\n-Chicken Pot Pie,  CALORIE SMART\n\nDivinity, Pistachio-,  CALORIE SMART\n\n-Oatmeal Cookies,\n\nOnions, Caramelized,\n\n-Orange Mold,  CALORIE SMART \u2022 EASY\n\n-Orange Muffins,\n\n-Orange Pancakes,\n\nand Orange Pilaf,\n\n-Orange Pound Cake,\n\n-Orange Relish,  CALORIE SMART \u2022 EASY\n\n-Orange Relish, -Ginger,\n\nOrange Sauce, Double-, Brie in Puff Pastry with,\n\n-Pear Chutney,\n\nPort Sauce,\n\nRye Berry Salad, -Almond,\n\nSauce,  CALORIE SMART \u2022 EASY\n\nSauce, Brie in Puff Pastry with,\n\nSorbet, -Raspberry,\n\n\u2013Sweet Potato Bread,\n\n\u2013Wild Rice Porridge,\n\n-Wine Sauce, Venison with,  CALORIE SMART\n\nCream. _See also_ Creamy\n\nof Broccoli Soup,  CALORIE SMART\n\nof Cauliflower Soup,\n\nGravy Pot Roast,\n\nof Mushroom Soup,\n\nPan Gravy,\n\nSauce, Mustard-Chive, Pork Chops with,\n\nCream Biscuits. _See_ Biscuits, Cream\n\nCream Cake, Italian,\n\nCream Cheese. _See also_ Cheesecake\n\nCheese Ball,  CALORIE SMART\n\nCheese Balls, Mini French,  CALORIE SMART\n\nFrosting, ,  Fast \u2022 EASY\n\nFrosting, Chocolate,\n\nin Moussaka,  CALORIE SMART\n\nsoftening,\n\nSpread(s)\n\nBacon-Chive-Cheddar,\n\nHoney-Walnut,\n\nMaple-Cinnamon,\n\nOlive-Walnut,\n\nPeanut Butter\u2013Honey,\n\nStrawberries and Cream Tart,\n\nStrawberry Cream Brunch Cake,\n\nCream Pie. _See_ Pie(s), Cream\n\nCream Puffs, 628\u2013629\n\nChocolate, -Double,\n\n\u00c9clairs,\n\nEggnog,\n\nIce Cream,\n\nCream Puffs _(cont.)_\n\nProfiteroles,\n\nWhipped Cream,\n\nCream of Tartar,\n\nCreamy\n\nAvocado Dressing,\n\nBasil-Chicken Sauce,\n\nBasil Sauce, Chicken and Pasta with,  Fast\n\nBasil Sauce, Pork and Pasta with,\n\nChicken Stew, Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\nChocolate Frosting,  Fast \u2022 EASY\n\nColeslaw,  CALORIE SMART\n\nGravy,\n\nItalian Dressing,\n\nPotato Salad,\n\nStrawberry-Poppy Dressing,\n\nTomato-Vodka Sauce,  CALORIE SMART \u2022 Fast\n\nVanilla Frosting,  Fast \u2022 EASY\n\nVinaigrette Dressing,\n\nVinaigrette Dressing, Fresh Herb,\n\nWasabi Dipping Sauce,\n\nCr\u00e8me Br\u00fbl\u00e9e,\n\nLemon,\n\ntips for,\n\nCr\u00e8me Caramel,\n\nCrepes,  CALORIE SMART\n\nCrescent Rolls,\n\nCrisp(s)\n\nApple,\n\nApple, Caramel,\n\nBerry, Fresh,\n\nBlueberry,\n\nRhubarb,\n\nCrisps, Pita, Baked,\n\nCrisps, Wonton,  Fast\n\nCross-contamination,\n\nCrostata, Nectarine-Plum,\n\nCrostini\n\nBLT,\n\nGarlic-Cheese,\n\nProvolone-Onion,\n\nTomato-Basil,  CALORIE SMART \u2022 Fast\n\nCroutons,  CALORIE SMART \u2022 EASY\n\nChicken Panzanella, Baked,  CALORIE SMART\n\nGarlic,\n\nPumpernickel\/Rye,\n\nRosemary-Parmesan,\n\nCrunchy Chicken-Mandarin Salad,\n\nCrunchy Cornmeal Chicken with Mango-Peach Salsa,  Fast\n\nCrunchy Oven-Fried Chicken,\n\nCrunchy Veggie Relish,  CALORIE SMART\n\nCrushing foods,\n\nCrust. _See_ Pie Crust; Pizza Crust\n\nCrystallized Ginger,  CALORIE SMART\n\nCuban\n\nBlack Bean Soup,  CALORIE SMART\n\nPork Chops,  CALORIE SMART \u2022 Fast\n\nRub,\n\nCucumber, . _See also_ Pickles\n\n-Avocado-Berry Salad,\n\nand Onion, Pickled,\n\nTzatziki,  CALORIE SMART \u2022 EASY\n\nVeggie Relish, Crunchy,  CALORIE SMART\n\n-Yogurt Sauce,\n\nCumin, ,\n\n-Chile Roasted Carrots,  CALORIE SMART\n\n-Chile Rub,\n\nHummus,\n\nCupcakes\n\nAngel Food,\n\nApple-Fig Bread Pudding, with Maple Sauce,\n\nCarrot,\n\nChocolate,  CALORIE SMART\n\nGinger and Peach,\n\nLemon Meringue,\n\nMaple-Cornmeal, with Maple-Butter Frosting,\n\nmini,\n\nPound Cake,\n\nRed Velvet,\n\nWhite,  CALORIE SMART\n\nYellow, 574\u2013575 CALORIE SMART\n\nCurd. _See_Lemon(s), Curd\n\nCurly endive,\n\nCurrant Scones,\n\nCurry(ied)\n\nBrown Rice, Veggie,  Fast\n\nChickpea and Tomato,  CALORIE SMART \u2022 Fast\n\n-Coconut, Spicy, Grilled Shrimp,  CALORIE SMART\n\nMayonnaise,\n\nPilaf,\n\nRed Curry Paste, Thai,  CALORIE SMART\n\nSauce,\n\nSweet Potato\u2013Cauliflower,  CALORIE SMART\n\nTartar Sauce, -Chutney,\n\nCustard\n\nBaked,  CALORIE SMART \u2022 EASY\n\nCaramel (Cr\u00e8me Caramel),\n\nCr\u00e8me Br\u00fbl\u00e9e,\n\nCr\u00e8me Br\u00fbl\u00e9e, Lemon,\n\ncr\u00e8me br\u00fbl\u00e9e, tips for,\n\nPanna Cotta,  EASY\n\nPear Custard Pie, Elegant,\n\nTiramisu,\n\nunmolding\n\nCustard cup\/ramekin,\n\nCutting boards, ,\n\nCutting techniques,\n\n## D\n\nDaikon Radish,\n\nin Kimchi,  CALORIE SMART\n\nDairy products, , 33\u201334\n\nDandelion greens,\n\nDark Chocolate Glaze,\n\nDate(s),\n\nBars,  CALORIE SMART\n\n-Bran Muffins,\n\n-Oatmeal Quinoa Thumbprints,  CALORIE SMART\n\nDecorating bag\n\nDecorating cookies,\n\nDecorator's Frosting,\n\nDeep-Fried Clams,\n\nDeep-Fried Oysters,\n\nDeep-Fried Sea Scallops,\n\nDeep-Fried Shrimp,  CALORIE SMART\n\nDeep frying method, ,\n\nDeglazing sauces, ,\n\nDelicata squash,\n\nDeluxe Stuffed-Crust Tomato Pizza,\n\nDenver Omelet,\n\nDessert Sauce\n\nCaramel,  CALORIE SMART \u2022 EASY\n\nChocolate,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCranberry,  CALORIE SMART \u2022 EASY\n\nHot Fudge,  CALORIE SMART \u2022 EASY\n\nLemon,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nMaple,\n\nRaspberry,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRhubarb,  CALORIE SMART \u2022 Fast\n\nStrawberry,\n\nStrawberry-Rhubarb,\n\nWhiskey,\n\nDesserts. _See_ Bars; Cake(s); Cookies; Cream Puffs; Custard; Frozen Desserts; Fruit Desserts; Pie(s); Pudding(s); Tart(s)\n\nDeviled Eggs,  CALORIE SMART \u2022 Fast\n\nBacon-Cheddar,\n\nBlue Cheese,\n\nChipotle,\n\nReuben,\n\nDijon\n\nChicken Smothered in Mushrooms,  CALORIE SMART \u2022 Fast\n\nFish, Grilled, Honey-,\n\nLamb,  CALORIE SMART \u2022 SLOW COOKER\n\nPork Loin Roast, -Herbed,\n\nPork Smothered in Mushrooms,\n\nDill(ed),\n\n-Beet Pesto,\n\nGreen Beans, Pickled, Lemon-,\n\nHavarti Cheese Ball,\n\n-Lemon Corn on the Cob,\n\nMartini,\n\nPickles, Garlic,  CALORIE SMART\n\nPotato Salad,\n\nSauce,\n\nseed,\n\nSimple Syrup,\n\n\u2013Sour Cream Sauce,  CALORIE SMART \u2022 Fast\n\nTea,\n\nTuna Steak Packets,  CALORIE SMART\n\nTurkey-Provolone Party Sub,\n\nDinner Rolls,  CALORIE SMART\n\nDip(s). _See also_ Spread(s)\n\nArtichoke, Hot,  CALORIE SMART \u2022 EASY\n\nArtichoke-Spinach, Hot,\n\nBaba Ghanoush,  CALORIE SMART\n\nBlue Cheese Yogurt Dressing,\n\nGuacamole,  CALORIE SMART \u2022 Fast\n\nChipotle,\n\nCotija,\n\nHummus,  CALORIE SMART \u2022 Fast\n\nGarlic, Roasted,\n\nSun-Dried Tomato,\n\nLayered, Greek,  CALORIE SMART \u2022 Fast\n\nPeanut Butter\u2013Yogurt,\n\nTaco, Personal,\n\nTzatziki,  CALORIE SMART \u2022 EASY\n\nVegetable, Roasted, with Baked Pita Crisps,  CALORIE SMART\n\nVegetable, Spicy,\n\nDipping Sauce\n\nGinger-Soy,\n\nSriracha, for Fries,\n\nWasabi, Creamy,\n\nDirect-heat grilling,\n\nDivinity, Pistachio-Cranberry,  CALORIE SMART\n\nDouble-Chocolate Brownies,\n\nDouble-Chocolate Hot Chocolate,\n\nDouble-Chocolate Hot Fudge Sundae Cake,\n\nDouble-Chocolate Puffs,\n\nDouble Oven Pancake,\n\nDoughnut Holes, Glazed Pound Cake,\n\nDoughnuts\n\nApplesauce,\n\nBlueberry-Orange, Baked,\n\nCake,  CALORIE SMART\n\nGlazed,\n\nDragon fruit,\n\nDressing. _See_ Salad Dressing\n\nDried Cherry and Walnut Bread,\n\nDrizzling\n\nDrop Biscuits,\n\nDrop Shortcakes,\n\nDuckling with Orange Sauce,\n\nDuck, roasting,\n\nDukka,\n\nDulce de Leche,\n\nChocolate Chip Cookies,\n\nDumplings\n\nApple,\n\nChicken and,  CALORIE SMART\n\nGnocchi, Potato,\n\nSpaetzle,  CALORIE SMART \u2022 Fast\n\nDutch oven,\n\n## E\n\nEasy Alfredo Sauce,\n\nEasy Apple Tart,\n\nEasy-Being-Green Smoothies,  Fast \u2022 EASY\n\nEasy Buttermilk Pastry,  Fast \u2022 EASY\n\nEasy Chicken with Tomatoes and Spinach,  CALORIE SMART\n\nEasy Cream Biscuits,  CALORIE SMART \u2022 EASY\n\nEasy Grilled Chicken Kabobs,\n\nEasy Grilled Steak Kabobs,  CALORIE SMART\n\nEasy Italian Sausage Lasagna,\n\nEasy Naan,\n\nEasy Nut Crust,  Fast \u2022 EASY\n\nEasy Pizza,\n\nEasy Quesadillas,\n\nEasy Refrigerator Pickles,  CALORIE SMART\n\nEasy Rice Pudding,\n\nEasy Sausage-Broccoli Pizza,\n\n\u00c9clairs,\n\nEdamame,\n\n-Bacon Toss,\n\nBarley Risotto with, Vegetarian,\n\nBoiled in the Pods,\n\nChicken and Barley Risotto with,  CALORIE SMART \u2022 SLOW COOKER\n\nPasta with Pesto and,\n\nas salad topper,\n\nin the Shell with Dipping Sauces,  CALORIE SMART \u2022 Fast\n\nStir-Fried with Corn and Peppers,\n\nStir-Fry, Pepper and Soybean,  CALORIE SMART \u2022 Fast\n\nand Vegetable Salad,\n\nEgg(s), . _See also_ Custard; French Toast; Meringue; Omelet(s); Quiche\n\nBacon, and Kale Cups,  CALORIE SMART\n\nBake, Tex-Mex,\n\nBaked,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nin baked goods,\n\nBenedict,  Fast\n\nBurritos, Breakfast,\n\nCheesy,\n\nChiles Rellenos Bake,  CALORIE SMART\n\nChorizo and Potato Scramble,\n\nand Crumbles,\n\nDeviled. _See_ Deviled Eggs\n\nFried, Sunny Side Up,  CALORIE SMART \u2022 Fast\n\nFrittata, Baked Buffalo,  CALORIE SMART\n\nHard-Cooked,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHard-Cooked, in Cobb Salad,\n\nHard-Cooked, Spinach-Bacon Salad with,  CALORIE SMART \u2022 Fast\n\nHuevos Rancheros,\n\nPoached,  CALORIE SMART \u2022 Fast\n\nPoblanos Rellenos Bake, Roasted,\n\nSalad Wraps, Shrimp and,\n\nSandwich, Breakfast, 'n Ham,\n\nSandwiches, Biscuit, 'n Ham,\n\nSandwiches, Egg Salad,\n\nSausage 'n,\n\nScrambled,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nScrambled, and Shrimp,\n\nScramble, Potato and Bacon,  CALORIE SMART \u2022 Fast\n\nShakshuka, Classic,  CALORIE SMART\n\nSkillet, with Summer Squash Hash,  CALORIE SMART\n\nstoring,\n\nStrata, Ham and Cheddar, 96\u201397 CALORIE SMART\n\nwith Wild Rice,\n\nEgg Bread, Rich,  CALORIE SMART\n\nEgg Cream, New York,\n\nEggnog,\n\nBourbon-Maple,\n\nIce Cream,\n\nPuffs,\n\nEggplant,\n\nAsian,\n\nBaba Ghanoush,  CALORIE SMART\n\ncharring, __\n\nMoussaka,  CALORIE SMART\n\nParmigiana,\n\nParmigiana, -Olive,\n\nParmigiana, -Ravioli,\n\npreparing\/cooking,\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRatatouille Salad,  CALORIE SMART\n\nSauce, Cheesy Rigatoni with,\n\nEgg Rolls\n\nforming\n\nShrimp,\n\nElectric appliances,\n\nElectric grills,\n\nElegant Pear Custard Pie,\n\nEmulsion,\n\nEnchilada(s)\n\nCheese,\n\nCheese, Simple,\n\nVerde, Cheese,\n\nEnglish Muffins,  CALORIE SMART\n\nEnglish Trifle,\n\nEnglish Trifles, Individual,\n\nEntertaining\n\nbuffets,\n\nholiday,\n\nstrategies for,\n\ntable settings,\n\nEquipment and tools. _See also_ Baking pans\n\nbar tending,\n\nbread machine, , 496\u2013497\n\ncandy making,\n\ncanning\/preserving,\n\ncleaning\/sanitizing of,\n\ncoffee making,\n\ncooking utensils, 11\u201312\n\ncookware, ,\n\nelectric appliances (small),\n\njuicers, ,\n\nknives\/cutting,\n\nmeasuring,\n\nfor pureeing,\n\nskillets, cast-iron,\n\ntea brewing,\n\nthermometers, ,\n\nEscarole,\n\nEspresso\n\nAngel Food Cake,\n\nCake with Mocha Streusel Ribbon,\n\nSugar,\n\nEvaporated milk,\n\n## F\n\nFajita(s)\n\nBeef,\n\nChicken,\n\nMarinade,  CALORIE SMART\n\nShrimp,\n\nFalafel (Fava Bean Rounds),  CALORIE SMART\n\nFamily-Favorite Chili,  CALORIE SMART \u2022 SLOW COOKER\n\nFarfalle, _See_ Bow-Tie Pasta\n\nFarro Salad, Indian-Spiced,  CALORIE SMART\n\nFaster Fried Polenta,\n\nFats, ,\n\nFat separator,\n\nFava Bean Rounds (Falafel),  CALORIE SMART\n\nFava beans,\n\nFennel, ,\n\n-Asparagus Seven-Layer Salad,\n\nand Onion, Grilled, Lemon Chicken with,  CALORIE SMART\n\npreparing\/cooking,\n\nseed,\n\nFenugreek,\n\nFermented vegetables,\n\nKimchi,  CALORIE SMART\n\nFeta,\n\nBow-Tie Pasta with Beef and,\n\nChicken Packets, Mediterranean,  CALORIE SMART\n\nin Greek Salad,  CALORIE SMART \u2022 Fast\n\nin Lasagna, Artichoke-Spinach,\n\nLayered Dip, Greek,  CALORIE SMART \u2022 Fast\n\nTabbouleh, Fruited, with Walnuts and,  CALORIE SMART\n\nTriangles, Basil-Cheese,  CALORIE SMART\n\nFettuccine\n\nAlfredo,  Fast\n\nAlfredo, Chicken,\n\nAlfredo, with Spinach, Baby,\n\nand Vegetables Parmesan,  CALORIE SMART \u2022 Fast\n\nFiber, in whole grains,\n\nFiesta Taco Casserole,\n\nFig(s),\n\nBars,\n\nBread Pudding Cupcakes, -Apple, with Maple Sauce,\n\n-Chocolate Cake,\n\nMuffins, Lemon-,\n\nTapenade, -Hazelnut,  CALORIE SMART \u2022 Fast\n\nFingerling potatoes, ,\n\nFire-Roasted Tomato Macaroni and Cheese,\n\nFirm tofu,\n\nFish. _See also_ Branzino; Salmon; Tuna\n\nBeer-Batter Fried,  CALORIE SMART \u2022 Fast\n\nbroiling,\n\nBroth,  CALORIE SMART\n\nCatfish, Cornmeal-Crusted,  CALORIE SMART \u2022 Fast\n\nCod, Cornmeal-Crusted,\n\ncooked, removing skin,\n\ncooking methods,\n\nfilleting\n\nFillets, Broiled,\n\nGrilled,  CALORIE SMART \u2022 Fast\n\nHerbed,\n\nHoney-Dijon,\n\nTacos,\n\nHalibut and Summer Squash Packets,  CALORIE SMART\n\nimitation seafood products,\n\nMackerel, Toasted Spice Seared,  Fast\n\nOven-Fried,  CALORIE SMART \u2022 Fast\n\nPanfried,  CALORIE SMART \u2022 Fast\n\nBrowned-Butter,\n\nGarlic-Butter,\n\nParmesan, Herb and Lemon,\n\npanfrying method\n\nQuiche, Seafood,\n\nselecting,\n\nservings per pound,\n\nSnapper with Tomato-Pepper Sauce,\n\nSole Amandine,  CALORIE SMART \u2022 Fast\n\nSole Amandine, Orange,\n\nSteaks, Buttery Broiled,  CALORIE SMART \u2022 Fast\n\nstoring, ,\n\nsustainable,\n\nTacos, Grilled,\n\nTacos, Soft, Cajun, 366\u2013367 Fast\n\nTilapia with Lemon Potatoes,\n\nTilapia and Vegetables, Roasted,  CALORIE SMART\n\nFlank Steak\n\nGrilled Jerk,  CALORIE SMART\n\nPeppered,  CALORIE SMART\n\nslicing\n\nwith Smoky Honey-Mustard Sauce,  CALORIE SMART\n\nFlare-ups, grilling,\n\nFlatbread,\n\nPizza, Chicken Sausage, Spinach and Swiss,  Fast\n\nFlax-Oatmeal Triple-Berry Muesli, _107,_ 107 CALORIE SMART\n\nFlaxseed,\n\nFlexitarian diet,\n\nFlour, ,\n\nsoy,\n\nFluffy White Frosting,  CALORIE SMART\n\nFoam cakes,\n\nFocaccia,  CALORIE SMART\n\nBreadsticks,\n\nOnion,\n\nFoil Packets,\n\nChicken, Mediterranean,  CALORIE SMART\n\nHalibut and Summer Squash,  CALORIE SMART\n\nmaking,\n\nPork, Southwestern,  CALORIE SMART\n\nSalmon, Lemon and Herb,  CALORIE SMART\n\nTuna Steak, Dilled,  CALORIE SMART\n\nFondue, Chocolate\u2013Sour Cream,  Fast\n\nFontina,\n\nFood Color, Natural Homemade,\n\nFood labels,\n\nFood poisoning,\n\nFood processor, ,\n\nFood safety\n\nbasic rules for, 31\u201332\n\nin buffets,\n\nin canning\/preserving,\n\ncleaning\/sanitizing,\n\nin grilling,\n\nhot\/cold foods,\n\nin picnics\/packed lunches,\n\nin refrigeration\/freezing, 33\u201334\n\nwith slow cooker,\n\nin smoking,\n\nFour-Cheese Homemade Ravioli,  CALORIE SMART \u2022 EASY\n\nFour-Grain Batter Bread,  CALORIE SMART\n\nFreezing\/thawing. _See also_ Jam, Freezer\n\nfood safety in,\n\nfood storage chart, 33\u201334\n\nfruits, 430\u2013431\n\nherbs,\n\nmeats,\n\npies\/pie crust,\n\npoultry, , ,\n\nvegetables,\n\nyeast breads,\n\nFrench Apple Pie,\n\nFrench Bread,  CALORIE SMART\n\nFrench Cheese Balls, Mini,  CALORIE SMART\n\nFrench Chicken and Rice, Country,  CALORIE SMART\n\nFrench Dip Sandwiches,  CALORIE SMART \u2022 SLOW COOKER\n\nFrench Dressing,\n\nFrench green lentils,\n\nFrench Macarons with Bittersweet Chocolate Ganache, 538\u2013539 CALORIE SMART\n\nFrench Onion Soup,  CALORIE SMART\n\nFrench Silk Pie, Classic,\n\nFrench Silk Pie, Mocha,\n\nFrench Toast,  CALORIE SMART \u2022 Fast\n\nApricot Stuffed,  CALORIE SMART\n\nBanana-Walnut, Upside-Down,\n\nHazelnut-Stuffed,\n\nOven,\n\nSnickerdoodle,\n\nSticks,\n\nFresh Berries and Greens Smoothie,\n\nFresh Blueberry Cobbler,\n\nFresh Blueberry Ice Cream,\n\nFresh Branzino with Sweet Pea\u2013Mint Pesto,  Fast\n\nFresh Cherry Cobbler,\n\nFresh Herb Vinaigrette,  CALORIE SMART \u2022 Fast\n\nFresh Peach Ice Cream,\n\nFresh Peach Salsa,  CALORIE SMART\n\nFresh Raspberry Ice Cream,\n\nFresh Strawberry Ice Cream,\n\nFresh Strawberry Milk,  Fast \u2022 EASY\n\nFresh Tomato Pizza,  CALORIE SMART\n\nFresno chiles,\n\nFried Eggs, Sunny Side Up,  CALORIE SMART \u2022 Fast\n\nFried Green Tomatoes,  CALORIE SMART\n\nFried Polenta,\n\nFried Potatoes,  CALORIE SMART \u2022 Fast\n\nFries, Loaded,\n\nFris\u00e9e,\n\nFrittata, Baked Buffalo,  CALORIE SMART\n\nFrosted and Decorated Sugar Cookies,\n\nFrosting\n\nBoiled,\n\nBrowned Butter, ,\n\nBrowned Butter Cream,\n\nBrown Sugar,\n\nButter Cream, Browned,\n\nButterscotch,\n\nCaramel,  EASY\n\nCaramel, Salty,\n\nCherry-Nut,\n\nChocolate, Creamy,  Fast \u2022 EASY\n\nCocoa, Creamy,\n\nCoconut-Pecan,\n\nCream Cheese, ,  Fast \u2022 EASY\n\nCream Cheese, Chocolate,\n\nCream Cheese, Cinnamon,\n\nDecorator's,\n\nFluffy White,  CALORIE SMART\n\nfood color for,\n\nFudge,\n\nlayer cakes\n\nLemon,\n\nMaple-Butter,\n\nMaple-Nut,\n\nMocha,\n\nOrange,\n\nPeanut Butter,\n\nPeppermint,\n\nRed Velvet Cake,\n\nVanilla, Creamy,  Fast \u2022 EASY\n\nWhite Chocolate,\n\nFrozen Desserts. _See also_ Ice Cream; Pops; Sorbet\n\nGranita, Watermelon,  CALORIE SMART\n\nYogurt, Very Berry Frozen,  CALORIE SMART \u2022 EASY\n\nFrozen Strawberry Margaritas,\n\nFrozen Yogurt Pops,\n\nFruit(s). _See also_ Citrus; Berry(ies); _specific fruits_\n\nbroiling,\n\nButter, Apple,  CALORIE SMART\n\nbutters, containers for,\n\nchocolate-covered,\n\ncoring,\n\nCouscous, Fruity,\n\ndried, using,\n\nfreezing\/thawing, 430\u2013431\n\njuicing,\n\nlocal sources for,\n\nin microwave,\n\nin Middle Eastern cuisine,\n\n'n Nuts Salad,\n\nOn-the-Go Mix,\n\npeeling\n\nripening,\n\nSalad, Granola,\n\nselecting,\n\nstone fruits,\n\nStuffing Supreme, Fruited, Pork Crown Roast with,\n\nTabbouleh, Fruited, with Walnuts and Feta,  CALORIE SMART\n\ntropical fruits,\n\nvinegars,\n\nFruitcake-Inspired Brownies,\n\nFruit Desserts. _See also_ Cobbler(s); Crisp(s); Pie(s); Tart(s)\n\nApple Dumplings,\n\nApplesauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nFruit Desserts _(cont.)_\n\nApples, Baked,  CALORIE SMART \u2022 EASY\n\nGrapefruit, Broiled,\n\nPears, Caramel-Maple,  SLOW COOKER \u2022 EASY\n\nPineapple, Broiled,  CALORIE SMART \u2022 Fast\n\nPluot, Grilled,\n\nStrawberry Shortcakes,\n\nFruited Tabbouleh with Walnuts and Feta,  CALORIE SMART\n\nFudge\n\n-Caramel Brownies, Outrageous,\n\nCherry-Pistachio,\n\nCoconut,\n\nFrosting,\n\nHot Fudge Sauce,  CALORIE SMART \u2022 EASY\n\nHot Fudge Sundae Cake, Double-Chocolate,\n\nPeanut Butter Candy,\n\n-Peanut Thumbprint Cookies,\n\nTriple-Chocolate,  CALORIE SMART\n\nTriple-Chocolate and Candy,\n\n## G\n\nGalangal,\n\nGame Fowl\n\nbrining time,\n\nDuckling with Orange Sauce,\n\nGoose, Roast, with Apple Stuffing,\n\nPheasants, Orange-Roasted,\n\nroasting,\n\nGanache, Chocolate,  Fast \u2022 EASY\n\nBittersweet,\n\nBittersweet, French Macarons with, 538\u2013539 CALORIE SMART\n\nGaram Masala, ,\n\nGarbanzo Bean(s). _See also_ Hummus\n\nBurgers, Veggie and,  Fast\n\nFalafel,  CALORIE SMART\n\nTabbouleh,\n\nand Tomato Curry,  CALORIE SMART \u2022 Fast\n\nwith Vegetables,  SLOW COOKER\n\nGarden Ratatouille,  CALORIE SMART \u2022 Fast\n\nGarden Vegetable Lasagna,\n\nGarlic,\n\nBranzino, Lemon-,\n\nBread, Cheesy,\n\nButter,\n\nButter, -Herb, Spatchcocked Chicken,  CALORIE SMART\n\n-Butter Panfried Fish,\n\n-Cilantro Grilled Corn on the Cob,\n\nClarified Butter,\n\nCrostini, -Cheese,\n\nCroutons,\n\nDill Pickles,  CALORIE SMART\n\nforms of,\n\nHummus, Roasted,\n\n-Jalape\u00f1o Pickled Green Beans,  CALORIE SMART\n\nLeg of Lamb, Roast, Herb and,  CALORIE SMART\n\nMarinade,  CALORIE SMART\n\nMarinade, Balsamic,\n\nMarinade, Onion and,\n\nMashed Potatoes,\n\nMayonnaise,\n\nPork Loin Roast, Peppered-,\n\nPot Roast, -Herb,\n\nRib Roast, -and Herb-Crusted,  CALORIE SMART \u2022 EASY\n\nRoasted,  CALORIE SMART \u2022 EASY\n\nRoasted, -and Balsamic-Marinated Steak, Grilled,  CALORIE SMART\n\nTurkey Burgers, Moist,\n\nVinegar,\n\n\u2013White Wine Sauce,\n\nGarnishes,\n\nchocolate dessert,\n\ncocktail,\n\nGas grills, . _See also_ Grilling\n\nGazpacho,  CALORIE SMART\n\nGreen,\n\nSriracha,\n\nGelatin,\n\nCranberry-Orange Mold,  CALORIE SMART \u2022 EASY\n\nsalads, unmolding\n\nGerman Chocolate Cake,\n\nGerman Potato Salad, Hot,  CALORIE SMART\n\nGhee,\n\nGin, in Martini Cocktails,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nGinger,\n\n-Apricot Preserves, Spiced,  CALORIE SMART \u2022 EASY\n\nBlueberry Scones,\n\nCranberry-Orange Relish,\n\nCrystallized,  CALORIE SMART\n\nLemonade,\n\n\u2013Lemon Curd Petite Pies,  CALORIE SMART\n\nand Peach Cupcakes,\n\n-Pear Upside-Down Cake,\n\nPickled,\n\n-Raspberry Tea,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nShortbread Cookies,\n\nSimple Syrup,\n\n-Soy Dipping Sauce,\n\nTea,\n\nWhipped Cream,\n\nGingerbread,\n\nGingerbread Cookies, 532\u2013533 CALORIE SMART\n\nGingerroot,\n\nGingersnaps,  CALORIE SMART\n\nGlassware,\n\ncoating edge of\n\nGlaze. _See also_ Chocolate, Glaze\n\nBrowned Butter,\n\ndrizzling\n\nLemon,\n\nMaple, Spiced Apple Cake with,\n\nmeat,\n\nOrange,\n\nVanilla,  Fast \u2022 EASY\n\nGlazed\n\nCarrots,  CALORIE SMART \u2022 Fast\n\nCarrots, Lemon, Honey-,\n\nDoughnuts,\n\nGreen Beans with Shallots,  CALORIE SMART \u2022 Fast\n\nHam, Baked,  CALORIE SMART\n\nSalmon, Honey-Mustard,  CALORIE SMART\n\nSalmon Salad, Cinnamon-Maple,\n\nSugar Cookies, Soft No-Roll,\n\nGluten-Free Pasta, Homemade,\n\nGnocchi, Potato,\n\nGoat Cheese,\n\nAsparagus with,\n\nGolden Harvest Muffins,\n\nStreusel,\n\nGolden Onion Soup,\n\nGooseberries,\n\nGoose, Roast, with Apple Stuffing,\n\nGoose, roasting,\n\nGorgonzola,\n\nGreens with Prosciutto and,  CALORIE SMART \u2022 Fast\n\nand Ham Pizza,\n\nLinguine with Toasted Walnuts,  Fast\n\nGouda,\n\nGraham Cracker Crust,  Fast \u2022 EASY\n\nGrains. _See also specific grains_\n\ncooking hints,\n\ncooking timetable,\n\nstoring,\n\nThree-Grain Medley,  CALORIE SMART \u2022 SLOW COOKER\n\nvarieties of,\n\nwhole, defined,\n\nGranita, Watermelon,  CALORIE SMART\n\nGranola,\n\nBars,\n\nCereal,\n\nCookies,\n\nFruit Salad,\n\nIce Cream and,\n\nPancakes,\n\nYogurt Parfaits,\n\nYogurt Parfaits, -Blueberry,\n\nGrapefruit, Broiled,\n\nGraters,\n\nGravy\n\nCream, Pot Roast,\n\nCreamy,\n\nPan,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nPan Cream,\n\nwine in,\n\nGreat Northern Bean and Veggie Sausage Cassoulet,  SLOW COOKER\n\nGreek\n\ncheese,\n\nChicken with Orzo,\n\nLamb Chops,  CALORIE SMART\n\nLayered Dip,  CALORIE SMART \u2022 Fast\n\nSalad,  CALORIE SMART \u2022 Fast\n\nGreek Yogurt,  CALORIE SMART \u2022 EASY\n\nBacon-Ranch,\n\nLayered Dip, Greek, , CALORIE SMART \u2022 Fast\n\nGreen Bean(s)\n\nBaby Food,\n\nCasserole,\n\nCheesy Bacon,\n\ngrilling,\n\nwith Parmesan,\n\nPickled,\n\nPickled, Jalape\u00f1o-Garlic,  CALORIE SMART\n\nPickled, Lemon-Dill,\n\npreparing\/cooking,\n\nwith Shallots, Glazed,  CALORIE SMART \u2022 Fast\n\nand Yellow Bean Salad,  CALORIE SMART\n\nGreen Gazpacho,\n\nGreen Juice, Refreshing,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nGreen Onion(s),\n\nand Pine Nut Pilaf,\n\nslicing\n\nGreens _,_, . _See also specific greens_\n\nBaby, and Beets Salad,\n\nand Berries Smoothie, Fresh,\n\npreparing\/cooking,\n\nwith Prosciutto and Gorgonzola,  CALORIE SMART \u2022 Fast\n\nsalad, 113\u2013115\n\nGreen Salsa Crema,\n\nGreen Smoothies, Easy-Being-,  Fast \u2022 EASY\n\nGreen Tea,\n\nApple-Mint Iced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nMartini Cocktails,\n\nGreen Tomatoes, Fried,  CALORIE SMART\n\nGremolata Mayonnaise,\n\nGriddle,\n\nGrilled Balsamic- and Roasted Garlic\u2013Marinated Steak,  CALORIE SMART\n\nGrilled Caesar Salad,\n\nGrilled Cheese Sandwiches,  Fast\n\nGrilled Corn with Herb Butter,  Fast\n\nGrilled dishes. _See also_ Burgers, Beef; Grilling\n\nBeef Fajitas,\n\nChicken, Barbecue,  CALORIE SMART\n\nChicken, Beer Can,  CALORIE SMART \u2022 EASY\n\nChicken, Bourbon, with Corn Relish,  CALORIE SMART\n\nChicken Breasts, Barbecue,\n\nChicken Fajitas,\n\nChicken, Herb, Three-,\n\nChicken Kabobs, Easy,\n\nChicken, Lemon, with Grilled Fennel and Onion,  CALORIE SMART\n\nChicken Packets, Mediterranean,  CALORIE SMART\n\nCorn with Chile-Lime Butter,\n\nCorn on the Cob,\n\nCorn with Herb Butter,  Fast\n\nFish,  CALORIE SMART \u2022 Fast\n\nFish, Herbed,\n\nFish, Honey-Dijon,\n\nFish Tacos,\n\nFlank Steak, Jerk,  CALORIE SMART\n\nFlank Steak with Smoky Honey-Mustard Sauce,  CALORIE SMART\n\nHalibut and Summer Squash Packets,  CALORIE SMART\n\nLamb Chops, Rosemary,  CALORIE SMART \u2022 Fast\n\nPatty Melts with Smothered Onions,  CALORIE SMART\n\nPork Chops, Cuban,  CALORIE SMART \u2022 Fast\n\nPork Ribs with Smoky Barbecue Sauce,\n\nPork Ribs, Southwest, with Smoky Barbecue Sauce,\n\nPork Tenderloin, Lemon-Pepper,  CALORIE SMART \u2022 Fast\n\nRed Potatoes, Herbed,  Fast\n\nShrimp, Coconut-Curry, Spicy,  CALORIE SMART\n\nShrimp Fajitas,\n\nSirloin Steak with Smoky Honey-Mustard Sauce,\n\nSteak, Balsamic- and Roasted Garlic\u2013Marinated,  CALORIE SMART\n\nSteak Kabobs, Easy,  CALORIE SMART\n\nT-Bones, Texas,  CALORIE SMART\n\nTurkey Breast with Chile-Cumin Rub,  CALORIE SMART\n\nGrilled Fish,  CALORIE SMART \u2022 Fast\n\nGrilled Jerk Flank Steak,  CALORIE SMART\n\nGrilled Lemon-Pepper Pork Tenderloin,  CALORIE SMART \u2022 Fast\n\nGrilled Pluot Dessert,\n\nGrilled Rosemary Lamb Chops,  CALORIE SMART \u2022 Fast\n\nGrilling\n\ncharcoal, starting, 405\u2013406\n\ndirect\/indirect heat,\n\nflare-ups, preventing,\n\nfoil packets\n\nfood safety in,\n\nin grill basket\n\ngrill types,\n\nsausage\/bratwurst,\n\nsmoked flavor in,\n\ntemperature control in,\n\nvegetables,\n\nGrill pans, ,\n\nGrissini, Black Pepper and Parmesan,  CALORIE SMART\n\nGrits, Cheese,  CALORIE SMART\n\nBacon and,\n\nGround Beef Stroganoff,\n\nGruy\u00e8re,\n\nGuacamole,  CALORIE SMART \u2022 Fast\n\nChipotle,\n\nCotija,\n\nGumbo, Shrimp,  CALORIE SMART\n\n## H\n\nHabanero chiles,\n\nHalf-and-half,\n\nHalibut and Summer Squash Packets,  CALORIE SMART\n\nHam\n\nwith Brown Sugar\u2013Orange Glaze,\n\ncarving\n\nHam _(cont.)_\n\nand Cheddar Strata, 96\u201397 CALORIE SMART\n\nand Cheese Omelet,\n\nin Chicken Cordon Bleu, Oven,  CALORIE SMART\n\ncuring,\n\nDenver Omelet,\n\n'n Egg Breakfast Sandwich,\n\n'n Egg Biscuit Sandwiches,\n\nGlazed Baked,  CALORIE SMART\n\nand Gorgonzola Pizza,\n\nroasting timetable,\n\nSalad Sandwiches,\n\nHand washing,\n\nHard-Cooked Eggs,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHarissa, ,  CALORIE SMART\n\nHash Brown Potatoes,  CALORIE SMART\n\nHash, Pork, Skillet,\n\nHash, Summer Squash, Skillet Eggs with,  CALORIE SMART\n\nHavarti Dill Cheese Ball,\n\nHazelnut(s),\n\nBiscotti,  CALORIE SMART\n\nBroccoli with Roasted Red Peppers and,  CALORIE SMART \u2022 Fast\n\n-Chocolate Sandwiches,  CALORIE SMART\n\n-Chocolate Thumbprint Cookies,\n\n-Fig Tapenade,  CALORIE SMART \u2022 Fast\n\nFrench Toast, -Stuffed,\n\nIced Coffee,\n\nHearty Pork Stew,  CALORIE SMART \u2022 SLOW COOKER\n\nHeirloom Tomato Caprese Salad,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHemp milk,\n\nHerbs, Herbed. _See also specific herbs_\n\nAsian,\n\nBeef Tenderloin, Bacon-Wrapped, -Roasted,\n\nBeef Tenderloin, -Roasted,  CALORIE SMART\n\nBrowned Butter Sauce,\n\nButter, ,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nButter Branzino,\n\nButter, Grilled Corn with,  Fast\n\nChicken, Marinated,\n\nChicken Stew, Creamy,  CALORIE SMART \u2022 SLOW COOKER\n\nChicken, Three-,\n\nChicken and Vegetables, -Roasted,  CALORIE SMART\n\nCorn on the Cob, Fresh, Grilled,\n\nCream Biscuits,\n\ndrying,\n\nFish, Grilled,\n\nfreezing,\n\nfresh and dried, ,\n\n-Garlic Butter,\n\n-Garlic Butter, Spatchcocked Chicken,  CALORIE SMART\n\nLeg of Lamb, Roast, and Garlic,  CALORIE SMART\n\n-Lemon Marinade,  CALORIE SMART\n\nMashed Potatoes,\n\nMayonnaise,\n\nMix, Salt-Free,  CALORIE SMART \u2022 Fast\n\nPan Steaks, -Brandy,  CALORIE SMART \u2022 Fast\n\nPan Steaks, Mushroom Brandy-,\n\nPasta, Homemade,\n\nPork Loin Roast,  CALORIE SMART\n\nPork Loin Roast, Dijon-,\n\nPot Roast, Garlic-,\n\nRed Potatoes, Grilled,  Fast\n\nRib Roast, -and Garlic-Crusted,  CALORIE SMART \u2022 EASY\n\nSalmon Packets, and Lemon,  CALORIE SMART\n\nseasoning with,\n\n-Sunflower Whole Wheat Bread,\n\nteas, herbal,\n\nTurkey and Wild Rice Casserole,  CALORIE SMART \u2022 SLOW COOKER\n\nvarieties of,\n\nVermicelli with Fresh Herbs,  Fast\n\nand Cannellini,\n\nOlives and Fresh Mozzarella,\n\nand Shrimp,\n\nSpinach and Bacon,\n\nVinaigrette,\n\nBlue Cheese,\n\nCreamy Fresh,\n\nFresh Herb,  CALORIE SMART \u2022 Fast\n\nVinegar,  CALORIE SMART \u2022 EASY\n\nvinegars,\n\nHoagies, Cheeseburger,\n\nHollandaise Sauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHomemade Baked Beans,\n\nHomemade Pasta, 170\u2013171 CALORIE SMART\n\nHoney\n\nButter,\n\nChicken, -Rosemary Roasted,\n\ncomb,\n\nCream Cheese Spread, Peanut Butter\u2013,\n\nCream Cheese Spread, -Walnut,\n\nLemon Carrots, -Glazed,\n\nin microwave,\n\nMiso Dressing,\n\n\u2013Peanut Butter Drizzle, Whole Wheat Waffles with,  CALORIE SMART\n\n\u2013Poppy Seed Dressing,\n\nSimple Syrup, -Rosemary,\n\n\u2013Whole Wheat Bread, 504\u2013505 CALORIE SMART\n\nHoney-Mustard\n\nBacon,\n\nChicken Wings,\n\nDressing,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nFish Grilled, -Dijon,\n\nSalmon, Glazed,  CALORIE SMART\n\nSauce, Smoky,\n\nHoppin' John,\n\nHors d'oeuvres. _See_ Appetizers\n\nHorseradish,\n\nMashed Potatoes,\n\nMeat Loaf,\n\nSour Cream, Spiced Corned Beef Brisket with,  CALORIE SMART \u2022 SLOW COOKER\n\nHot Artichoke Dip,  CALORIE SMART \u2022 EASY\n\nHot Artichoke-Spinach Dip,\n\nHot Buttered Rum-Spiced Cider,\n\nHot Chocolate,  Fast\n\nHot foods, safe handling of,\n\nHot Fudge Sauce,  CALORIE SMART \u2022 EASY\n\nHot German Potato Salad,  CALORIE SMART\n\nHot and Saucy Cocktail Meatballs,  CALORIE SMART\n\nHot Spiced Cider, , CALORIE SMART \u2022 Fast \u2022 EASY\n\nHot and Spicy Rub,\n\nHuevos Rancheros,\n\nHulling berries\n\nHummus,  CALORIE SMART \u2022 Fast\n\nCumin,\n\nGarlic, Roasted,\n\nLayered Dip, Greek,  CALORIE SMART \u2022 Fast\n\nSun-Dried Tomato,\n\n## I\n\nIceberg lettuce,\n\nIce Cream\n\nBlueberry, Fresh,\n\nCaramel,\n\nChocolate,\n\nEggnog,\n\nand Granola,\n\nHot Fudge Sundae Cake, Double-Chocolate,\n\nPeach, Fresh,\n\nPuffs,\n\nRaspberry, Fresh,\n\nsoftening in microwave,\n\nsoy,\n\nStrawberry, Fresh,\n\nVanilla,  EASY\n\nIce Cubes\n\nLemonade,\n\nPesto,\n\nWine,\n\nIced Coffee,  CALORIE SMART \u2022 EASY\n\nImmersion blender, ,\n\nIndian(-Style)\n\nCheese Balls, Mini,\n\nFarro Salad, -Spiced,  CALORIE SMART\n\ningredients\/flavorings,\n\nLentils and Rice,  CALORIE SMART\n\nLentil Stew,  CALORIE SMART\n\nVegetable Casserole,  CALORIE SMART\n\nIndirect-heat grilling,\n\nIndividual Chicken Pot Pies,\n\nIndividual English Trifles,\n\nIndividual Meringues,\n\nIngredients\n\nAsian,\n\nbaked goods, ,\n\nbreadmaking,\n\nfor cocktails, 77\u201378\n\ndairy,\n\nIndian cuisine,\n\nMediterranean cuisine,\n\nMiddle Eastern cuisine,\n\npantry staples,\n\nshopping strategies,\n\nSouth American cuisine,\n\nIsraeli Couscous Risotto with Caramelized Onions and Sausage,\n\nItalian(-Style)\n\nBeef Sandwiches,\n\nChicken-Lentil Soup,  CALORIE SMART \u2022 SLOW COOKER\n\nChopped Salad,  Fast\n\nCream Cake,\n\nDressing,  CALORIE SMART \u2022 Fast\n\nDressing, Creamy,\n\nMeatballs with Tomato Sauce,\n\nParmesan Butter,\n\nPizza Crust,\n\nPork Pot Roast,  CALORIE SMART\n\nSalad Toss,\n\nSeasoning Rub,\n\nTofu-Rice Skillet,\n\nTomato Sauce,  Calorie Smart\n\nTwice-Baked Potatoes, 248\u2013249\n\nWhite Bean Salad, Northern,  CALORIE SMART\n\nItalian Sausage\n\nLasagna,\n\nLasagna, Easy,\n\nMinestrone with,  CALORIE SMART\n\nPizza, Meat Lover's,\n\nwith Tomatoes and Penne,  CALORIE SMART \u2022 Fast\n\n## J\n\nJacked-Up Rice,\n\nJaggery,\n\nJalape\u00f1o,\n\nGreen Beans, -Garlic Pickled,  CALORIE SMART\n\n-Peach Jam,  CALORIE SMART\n\nmaking,\n\nuses for, 436\u2013437\n\nPoppers, Baked,\n\nSriracha Sauce,  CALORIE SMART\n\nJam\n\nFreezer\n\nBerry, Triple-, Pomegranate,  CALORIE SMART\n\nBlueberry,\n\ncontainers for,\n\nRaspberry,\n\nStrawberry,  CALORIE SMART \u2022 EASY\n\nStrawberry-Balsamic,\n\nmaking,\n\nPeach-Jalape\u00f1o,  CALORIE SMART\n\nmaking,\n\nuses for, 436\u2013437\n\nPeach Rosemary,\n\nPeach, Spiced,\n\nPear-Orange Conserve,  CALORIE SMART\n\nTomato, Sweet,  CALORIE SMART\n\nJamaican Jerk Seasoning,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nJarlsberg,\n\nJar opener,\n\nJars, decorating as gifts,\n\nJars, preparing for canning,\n\nJasmine rice,\n\nJelly,\n\nApple-Pepper,  CALORIE SMART \u2022 EASY\n\nJelly Roll,\n\nLemon Curd,\n\nJelly roll pan,\n\nJerk\n\nFlank Steak, Grilled,  CALORIE SMART\n\nPulled Pork Sandwiches,  SLOW COOKER\n\nSeasoning, Jamaican,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nJerusalem artichokes,\n\npreparing\/cooking,\n\nJ\u00edcama,\n\nJuice(s)\n\nFood Color, Natural Homemade,\n\nGreen, Refreshing,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nprepping produce for,\n\nRaspberry-Lime,  Fast \u2022 EASY\n\nJuicers, ,\n\nJumbo Chocolate Chip Cookies,\n\nJumbo Upside-Down Muffins,\n\n## K\n\nKabobs\n\nChicken, Easy Grilled,\n\nDessert,\n\nSteak, Easy Grilled,  CALORIE SMART\n\nSweet Chile,\n\nKale, ,\n\nBacon and Egg Cups,  CALORIE SMART\n\nBacon and Tomato Macaroni and Cheese,\n\nCaesar Salad,\n\nGreen Juice, Refreshing,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSalad, Cherry-Walnut,  CALORIE SMART \u2022 Fast\n\nand Spinach, Wilted,\n\nKamut,\n\nKasha,\n\nand Beef Mexicana,  CALORIE SMART \u2022 Fast\n\nand Beef Tortillas,\n\nand Bow-Tie Pilaf,  CALORIE SMART\n\ncooking timetable,\n\nKefalotiri,\n\nKefir,  CALORIE SMART\n\nBlueberry,\n\nMango,\n\nStrawberry,\n\nStrawberry-Banana,\n\nKentucky Pecan Pie,\n\nKetchup, ,  CALORIE SMART\n\nKey Lime Pie,\n\nKibbeh Patties,  CALORIE SMART\n\nKimchi,  CALORIE SMART\n\nabout,\n\nKiwano (Horned Melon),\n\nKiwi Smoothies, Easy-Being-Green,  Fast \u2022 EASY\n\nKnives,\n\nKohlrabi,\n\npreparing\/cooking,\n\nKorean\n\nBarbecued Beef,  CALORIE SMART\n\nCheese Balls, Mini,\n\nChicken, Fried,  CALORIE SMART\n\nKumquat,\n\n## L\n\nLacto-ovo vegetarian diet,\n\nLacto-vegetarian diet,\n\nLagers, 78\u201379\n\nLamb\n\nbroiling,\n\nChops, Greek,  CALORIE SMART\n\nChops, Grilled Rosemary,  CALORIE SMART \u2022 Fast\n\ncuts\n\nDijon,  CALORIE SMART \u2022 SLOW COOKER\n\nKibbeh Patties,  CALORIE SMART\n\nLeg of, Herb and Garlic Roast,  CALORIE SMART\n\npanfrying,\n\nroasting,\n\nShanks, Balsamic-Braised,  CALORIE SMART\n\nStew, Moroccan,  CALORIE SMART\n\nLasagna\n\nArtichoke-Spinach,\n\nCupcakes,  CALORIE SMART\n\nGarden Vegetable,\n\nItalian Sausage,\n\nItalian Sausage, Easy,\n\nMediterranean,\n\nLatte, Caramel,\n\nLatte, Iced Vanilla Soy,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nLattice top crust,\n\nLayer Cake Chocolate,\n\nfrosting\n\nTurtle,\n\nLayered Dip, Greek,  CALORIE SMART \u2022 EASY\n\nLeaf lettuce,\n\nLeeks,\n\npreparing\/cooking,\n\nLeftovers, reheating,\n\nLegumes. _See_ Bean(s); Lentil(s)\n\nLegumes, fresh,\n\nLemon(s), . _See also_ Citrus\n\nBars,  CALORIE SMART\n\nBranzino, -Garlic,\n\nBranzino, -Rosemary,\n\nCarrots, Honey-Glazed,\n\nChicken, with Grilled Fennel and Onion,  CALORIE SMART\n\nCr\u00e8me Br\u00fbl\u00e9e,\n\nCurd,\n\nBread Pudding with,\n\nButter Cake, \u2013Filled,\n\n\u2013Ginger Petite Pies,  CALORIE SMART\n\nJelly Roll,\n\n-Dill Corn on the Cob, Grilled,\n\n-Dill Pickled Green Beans,\n\nDressing,\n\nFrosting,\n\nGlaze,\n\njuicing,\n\nMacarons,\n\nMarinade, -Herb,  CALORIE SMART\n\nMascarpone Tart,\n\nMeringue Cupcakes,\n\nMeringue Pie, 602\u2013603\n\nMuffins, -Fig,\n\n-Pepper Butter,\n\n-Pepper Pork Tenderloin, Grilled,  CALORIE SMART \u2022 Fast\n\nPotatoes, Tilapia with,\n\nPound Cake,\u2013Poppy Seed,\n\npreserved,\n\nSalmon, -and Parmesan-Crusted,  CALORIE SMART\n\nSalmon Packets, Herb and,  CALORIE SMART\n\nSangria,\n\nSauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nScones, -Cherry,\n\nThumbprint Cookies, -Almond,\n\nVinaigrette Dressing,\n\nVinegar,\n\nzest, grating\n\nLemonade\n\nfrom Concentrate,\n\nConcentrate, Make-Ahead,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nGinger,\n\nIce Cubes,\n\nPeachy,\n\nRaspberry,\n\nSmoothies,\n\nSorbet,  CALORIE SMART \u2022 EASY\n\nSparkling,\n\n-Strawberry Slush,\n\nSweet Tea,\n\nLemon Balm-Orange Sun Tea,\n\nLemongrass, ,\n\nLentil(s), ,\n\nand Bulgur, Mediterranean,  CALORIE SMART \u2022 SLOW COOKER\n\n-Chicken Soup, Italian,  CALORIE SMART \u2022 SLOW COOKER\n\nRed, Barley and Cauliflower Tabbouleh,  CALORIE SMART\n\nand Rice, Indian,  CALORIE SMART\n\nStew, Indian,  CALORIE SMART\n\nLettuce\n\ndrying\n\nselecting\/handling,\n\nvarieties of, 113\u2013115\n\nLettuce Wraps\n\nBeef, Vietnamese,  CALORIE SMART\n\nQuick,\n\nTuna-,\n\nLime\n\n-Chile Butter, Grilled Corn with,\n\n-Chile Corn on the Cob,\n\nKey Lime Pie,\n\n-Raspberry Juice,  Fast \u2022 EASY\n\nLimeade Concentrate,\n\nLimeade Sorbet,\n\nLinguine\n\nClams and Sausage, Spanish,  CALORIE SMART\n\nGorgonzola, with Toasted Walnuts,  Fast\n\nwith White Clam Sauce and Basil,\n\nLoaded Fries,\n\nLobster(s)\n\nBoiled,  CALORIE SMART\n\nbuying,\n\ncooked meat, removing\n\nMacaroni and Cheese,\n\nLoganberry Pie,\n\nLuscious Chocolate Truffles,  CALORIE SMART\n\nLyonnaise Salad,  CALORIE SMART \u2022 Fast\n\n## M\n\nMacadamia,\n\nMacaroni\n\nand Cheese,\n\nBacon, ,\n\nBacon, Kale and Tomato,\n\nLobster,\n\nOnion, Caramelized,\n\nSouthwest,\n\nTomato, Fire-Roasted,\n\n-Chicken Salad,\n\nMeatballs and Sauce,\n\nPesto Pasta,\n\nSalad Toss, Italian,\n\nMacarons\n\nFrench, with Bittersweet Chocolate Ganache, 538\u2013539 CALORIE SMART\n\nLemon,\n\nMint,\n\nOrange,\n\nM\u00e2che,\n\nMackerel, Toasted Spice Seared,  Fast\n\nMake-Ahead Meatballs, , _55_ CALORIE SMART\n\nManchego,\n\nMandarin Salad, Crunchy Chicken-,\n\nMandarin Salad with Sugared Almonds, 120\u2013121 CALORIE SMART \u2022 Fast\n\nMango,\n\n-Avocado Salad,\n\nKefir,\n\nMartini Cocktails,\n\nMimosas,  CALORIE SMART \u2022 Fast \u2022 EASY\n\n-Peach Salsa,\n\n-Pomegranate Mimosas,\n\nRelish,  CALORIE SMART \u2022 EASY\n\nSticky Rice with,  CALORIE SMART\n\nMangosteen,\n\nManhattan Clam Chowder,  CALORIE SMART\n\nManhattan Cocktails,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nManicotti,\n\nTurkey,\n\nManouri,\n\nMaple\n\n-Bourbon Eggnog,\n\n-Bourbon Marinade,  CALORIE SMART \u2022 EASY\n\n\u2013Brown Sugar Refrigerator Cookies,\n\nButter,\n\nCaramel Pears,  SLOW COOKER \u2022 EASY\n\n-Cornmeal Cupcakes with Maple-Butter Frosting,\n\nCream Cheese Spread, -Cinnamon,\n\nFrosting, -Butter,\n\nFrosting, -Nut,\n\nGlazed Salmon Salad, -Cinnamon,\n\nGlaze, Spiced Apple Cake with,\n\nSauce,\n\nTurkey Breast, -Apple Brined,\n\nMaple syrup,\n\nMarble Cake,\n\nMargaritas,  Fast\n\nStrawberry, Frozen,\n\nMarinade(s)\n\nFajita,  CALORIE SMART\n\nGarlic,  CALORIE SMART\n\nGarlic, Balsamic,\n\nGarlic and Onion,\n\nLemon-Herb,  CALORIE SMART\n\nMaple-Bourbon,  CALORIE SMART \u2022 EASY\n\nOrange-Thyme,  CALORIE SMART \u2022 EASY\n\nTeriyaki,  CALORIE SMART \u2022 EASY\n\nMarjoram,\n\nMarmalade,\n\nMarshmallow(s),  CALORIE SMART\n\nCocoa,\n\nHot Chocolate, -Topped,\n\nPeppermint,\n\nin Whoopie Pies,\n\nMartini Cocktails,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChocolate,\n\nDill,\n\nGreen Tea,\n\nMango,\n\nPomegranate,\n\nSweet,\n\nMascarpone,\n\nLemon Tart,\n\nTiramisu,\n\nMashed Potatoes,  CALORIE SMART \u2022 EASY\n\nMat\u00e9,\n\nMayocoba beans,\n\nMayonnaise, ,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChile, Roasted,\n\nCurried,\n\nGarlic,\n\nGremolata,\n\nHerbed,\n\nsoy,\n\nuses for,\n\nMeal planning,\n\nMeasuring\/measuring utensils,\n\nMeat(s). _See also_ Beef; Lamb; Pork; Veal\n\nbraising,\n\nbroiling, 316\u2013317\n\nCharcuterie, , _photo_\n\ncuts, 284\u2013285, 300\u2013301\n\nglaze,\n\nlabel information,\n\npanfrying,\n\nroasting, 314\u2013316\n\nsafe handling of,\n\nSauce, Spaghetti and,\n\nsearing,\n\nselecting,\n\nservings per pound,\n\nstoring, ,\n\nVenison with Cranberry-Wine Sauce,  CALORIE SMART\n\nMeatball(s)\n\nCheeseburger Hoagies,\n\nChicken, Soup,  CALORIE SMART\n\nChile Kabobs, Sweet,\n\nHot and Saucy Cocktail,  CALORIE SMART\n\nItalian, with Tomato Sauce,\n\nKabobs, Sweet Chile,\n\nMake-Ahead,  CALORIE SMART\n\nand Orzo, Parmesan,  CALORIE SMART \u2022 Fast\n\nPesto Pasta and,\n\nPizza,\n\nand Sauce,\n\nSpaghetti and,\n\nSpaghetti and, Pizza,\n\nTacos, Salsa,\n\nTeriyaki Sliders,\n\n\"Meatballs\", Veggie and Bean,\n\nMeatless Minestrone,\n\nMeat Loaf,  CALORIE SMART\n\nCasserole, Potato-Topped,  CALORIE SMART\n\nCasserole, Potato-Topped, Quick,\n\nHorseradish,\n\nMini Meat Loaves,\n\nMeat Lover's Pizza,\n\nMeat mallet,\n\nMeat thermometer,\n\nMediterranean\n\nBulgur and Lentils,  CALORIE SMART \u2022 SLOW COOKER\n\nChicken Packets,  CALORIE SMART\n\ningredients\/spices,\n\nLasagna,\n\nTuna Salad,\n\nMelon and Shrimp Salad,\n\nMeringue(s)\n\nIndividual,\n\nLemon Meringue Cupcakes,\n\nLemon Meringue Pie, 602\u2013603\n\nfor 9\u2013inch pie,\n\nShell,  EASY\n\nShell, Chocolate,\n\nspreading over pie\n\nSweet Potato Meringue Pie, Roasted,\n\nMesclun,\n\nMexican\n\nCheese Balls, Mini,\n\nChicken and Sausage with Black Beans, Cheesy,\n\nTofu-Rice Skillet,  CALORIE SMART \u2022 Fast\n\nWraps,\n\nMeyer Lemon,\n\nMicrowave cooking\/heating\n\nApples, Baked,\n\nchart for favorite foods, 29\u201330\n\nchocolate mug cake, 576\u2013577\n\ndishes for,\n\nPeanut Brittle,\n\nvegetables, 234\u2013237\n\nMiddle Eastern ingredients\/spices,\n\nMilk. _See also_ Buttermilk; Soymilk\n\nChocolate,\n\nEgg Cream, New York,\n\nLatte, Caramel,\n\nLatte, Soy, Iced Vanilla,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRoot Beer,\n\nStrawberry, Fresh,  Fast \u2022 EASY\n\nvarieties of, ,\n\nMilk Chocolate Glaze,\n\nMillet,\n\ncooking timetable,\n\nMimosas\n\nMango,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nPineapple,\n\nPomegranate-Mango,\n\nMinestrone with Italian Sausage,  CALORIE SMART\n\nMinestrone, Meatless,\n\nMini French Cheese Balls,  CALORIE SMART\n\nMini Indian Cheese Balls,\n\nMini Korean Cheese Balls,\n\nMini Meat Loaves,\n\nMini Mexican Cheese Balls,\n\nMint,\n\n-Apple Green Tea, Iced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChocolate Glaze,\n\nMacarons,\n\n\u2013Sweet Pea Pesto,\n\nSyrup,\n\nTabbouleh, Minty,\n\nTzatziki Dip,  CALORIE SMART \u2022 EASY\n\nMirin,\n\nMiso, ,\n\nDressing,  CALORIE SMART \u2022 Fast\n\nHoney Dressing,\n\nSriracha Dressing,\n\nMixed Berry Crumble Tart,\n\nMixers, electric,\n\nMixing bowls,\n\nMocha\n\nFrench Silk Pie,\n\nFrosting,\n\nStreusel Ribbon, Espresso Cake with,\n\nMojitos,\n\nMolasses,\n\n-Mustard Barbecue Sauce,\n\nMole Poblano,  CALORIE SMART\n\nMole Sauce, Quick, Coffee Chicken with,  CALORIE SMART\n\nMolten Chocolate Cakes,\n\nMorel mushrooms,\n\nMornay Sauce,  CALORIE SMART \u2022 Fast\n\nMoroccan\n\nChicken Pie,\n\nChicken Tagine,  CALORIE SMART\n\nLamb Stew,  CALORIE SMART\n\nMoussaka,  CALORIE SMART\n\nMousse, Chocolate,\n\nMozzarella\n\nCalzone,\n\nCaprese Salad, Heirloom Tomato,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCupcakes,  CALORIE SMART\n\nEggplant Parmigiana,\n\nfresh,\n\nFresh, Vermicelli with Fresh Herbs, Olives and,\n\nLasagna, Artichoke-Spinach,\n\nLasagna, Italian Sausage,\n\nLasagna Cupcakes,  CALORIE SMART\n\nManicotti,\n\nPizza, Stuffed-Crust,  CALORIE SMART\n\nPizza, Veggie Lover's, with Cornmeal Crust,  CALORIE SMART\n\nwith Red Pepper Soup, Roasted, , CALORIE SMART\n\nRigatoni, Cheesy, with Eggplant Sauce,\n\nMuesli, Triple-Berry Oatmeal-Flax,  CALORIE SMART\n\nMuffins\n\nApple-Cinnamon,\n\nBanana,\n\nBlueberry,  CALORIE SMART\n\nBlueberry, Streusel-Topped,\n\nBran,  CALORIE SMART\n\nBran, Date-,\n\nChocolate Chip,\n\nCorn,\n\nCranberry-Orange,\n\nGolden Harvest,\n\nGolden Harvest, Streusel,\n\nJumbo Upside-Down,\n\nLemon-Fig,\n\npan for,\n\npreparing\n\nproblems with\n\nreheating in microwave,\n\nMuffuletta Sandwiches, Portabella,  CALORIE SMART \u2022 Fast\n\nMug Cake, Banana Split-Chocolate, 576\u2013577\n\nMulled Wine, ,  CALORIE SMART\n\nMung beans,\n\nMung bean sprouts,\n\nMushroom(s)\n\nChicken, Dijon, Smothered in,  CALORIE SMART \u2022 Fast\n\ngrilling,\n\nPan Steaks, Brandy-Herb,\n\nPilaf,\n\nPork, Dijon, Smothered in,\n\nPortabella Chicken Pot Pie,\n\nPortabella Muffuletta Sandwiches,  CALORIE SMART \u2022 Fast\n\nPortabellas, Asian Stuffed,  CALORIE SMART\n\npreparing\/cooking,\n\nSauce, Pork with,\n\nSauce, Vegetables with,\n\nSoup, Chicken-,\n\nSoup, Cream of,\n\nin Tofu Teriyaki Noodles,  CALORIE SMART\n\nvarieties of,\n\nMussels\n\nbuying,\n\nspoiled\n\nSteamed, in Wine Sauce,  CALORIE SMART \u2022 Fast\n\nMustard, . _See also_ Dijon; Honey-Mustard\n\nButter,\n\n-Chive Cream Sauce, Pork Chops with,\n\n-Molasses Barbecue Sauce,\n\nSauce,\n\nseed,\n\nMustard greens,\n\n## N\n\nNaan, Easy,\n\nNachos,  Fast\n\nBeef or Chicken,\n\nBeef, Skillet,\n\nChicken, Skillet,  Fast\n\nNectarine(s),\n\nPancakes, \u2013Sour Cream,\n\n-Plum Crostata,\n\nNew England Clam Chowder,  CALORIE SMART\n\nNew York Cheesecake,\n\nNew York Egg Cream,\n\nNi\u00e7oise Salmon Salad,\n\nNigella seed,\n\nNo-Bake Chocolate-Oatmeal Drops,\n\nNo-Knead Artisan Bread,  CALORIE SMART\n\nNoodle(s)\n\nAsian,\n\nin Beef Pho,  CALORIE SMART\n\nBowls, Chipotle-Peanut,  Fast\n\nChicken Soup,  CALORIE SMART\n\nChicken Soup, Confetti,\n\nPad Thai with Chicken,\n\nPad Thai with Pork,\n\nPad Thai with Shrimp,\n\nrice, soaking\n\nSesame Salad,  CALORIE SMART \u2022 Fast\n\nSpaetzle,  CALORIE SMART \u2022 Fast\n\nTempeh Stir-Fry with,\n\nTofu Teriyaki,  CALORIE SMART\n\nTofu Teriyaki, and Peppers,\n\nNorthern Italian White Bean Salad,  CALORIE SMART\n\nNut(s). _See also specific nuts_\n\nin baked goods,\n\nBars, Triple-,\n\nCandied,\n\n-Cherry Frosting,\n\nchocolate-covered,\n\nCinnamon-Cardamom Mixed,\n\nCinnamon-Sugared,\n\nCrust, Easy,  Fast \u2022 EASY\n\n-Maple Frosting,\n\nOn-the-Go Mix,\n\nShortbread Cookies,\n\nSouthwestern Spiced Party,  Fast\n\nsoy,\n\nstoring,\n\ntoasting, ,\n\nvarieties of,\n\nNutmeg,\n\nNut oils,\n\n## O\n\nOatmeal\n\nApple, Baked,  CALORIE SMART\n\nin baked goods,\n\n-Chocolate Drops, No-Bake,\n\nCookies, \u2013Chocolate Chip,\n\nCookies, -Cranberry,\n\nCookies, -Raisin,  CALORIE SMART\n\ncooking timetable,\n\n-Flax Muesli, Triple-Berry,\n\nin Granola,\n\nMuffins, Streusel, Golden Harvest,\n\nPancakes, Whole-Grain,\n\nPie Crust,\n\nPomegranate,\n\nQuinoa Thumbprints, -Date,  CALORIE SMART\n\nsteel-cut, ,\n\nStreusel Bread,\n\ntypes of,\n\nOil Pastry, Press-in-the-Pan,  Fast \u2022 EASY\n\nOils\n\nin baked goods,\n\ncooking, 7\u20138\n\nfor deep frying,\n\nfor salad dressing,\n\nsesame,\n\nsoybean,\n\nstoring,\n\ntypes of,\n\nOkra,\n\nCorn and Tomatoes,  CALORIE SMART \u2022 Fast\n\npreparing\/cooking,\n\nOlive(s)\n\nChicken Packets, Mediterranean,  CALORIE SMART\n\nCream Cheese Spread, -Walnut,\n\n-Eggplant Parmigiana,\n\nLayered Dip, Greek,  CALORIE SMART \u2022 Fast\n\npitting\n\nSalad,\n\nSauce, Creamy,\n\nTapenade,  CALORIE SMART \u2022 Fast\n\n-Tomato Relish,\n\nVermicelli with Fresh Herbs, Fresh Mozzarella and,\n\nOlive oil, ,\n\nOmelet(s)\n\nAvocado, Bacon and Cheese,\n\nBasic,  CALORIE SMART\n\nCheese,\n\nDenver,\n\nHam and Cheese,\n\nPuffy,  CALORIE SMART \u2022 Fast\n\nSun-Dried Tomato,\n\nOnion(s)\n\nCaramelized,  CALORIE SMART\n\nBeer-, and Bacon Burgers,\n\nCranberry,\n\nIsraeli Couscous Risotto with Sausage and,\n\nMacaroni and Cheese,\n\nPot Roast,\n\ncaramelizing method\n\nand Cucumber, Pickled,\n\nFocaccia,\n\nand Garlic Marinade,\n\ngrilling,\n\npreparing\/cooking,\n\n-Provolone Crostini,\n\nRed, Pickled,  CALORIE SMART\n\nSmothered, Patty Melts with,  CALORIE SMART\n\nSoup, French,  CALORIE SMART\n\nSoup, Golden,\n\nvarieties of,\n\nOolong tea,\n\nOrange(s). _615_ , . _See also_ Citrus\n\n-Asparagus Salad,\n\nBeef, Roast, with Thyme and,  CALORIE SMART\n\n-Blueberry Doughnuts, Baked,\n\n\u2013Brown Sugar Glaze, Ham with,\n\nButter,\n\nButter Balls, Chocolate-Filled, -Rosemary,  CALORIE SMART\n\nCitrus Splash,\n\n-Cranberry Mold,  CALORIE SMART \u2022 EASY\n\n-Cranberry Muffins,\n\n-Cranberry Pancakes,\n\nand Cranberry Pilaf,\n\n-Cranberry Relish,  CALORIE SMART \u2022 EASY\n\n-Cranberry Relish, -Ginger,\n\nCranberry Sauce, Double-, Brie in Puff Pastry with,\n\nFrosting,\n\nGlaze,\n\n\u2013Lemon Balm Sun Tea,\n\nMacarons,\n\nMandarin Salad, Crunchy Chicken-,\n\nMandarin Salad with Sugared Almonds, 120\u2013121 CALORIE SMART \u2022 Fast\n\nMarinade, -Thyme,  CALORIE SMART \u2022 EASY\n\n-Pear Conserve,  CALORIE SMART\n\nPheasants, -Roasted,\n\nPound Cake, -Coconut,\n\nPound Cake, Cranberry-,\n\nSauce, Duckling with,\n\nSmoothies,\n\nSmoothies, -Pineapple,\n\nSole Amandine,\n\nSorbet,\n\n-Zucchini Bread,\n\nOrecchiette, Browned Butter, with Broccoli Rabe,  Fast\n\nOregano,\n\nOrzo\n\nChicken with Pan-Roasted Cauliflower and,  CALORIE SMART \u2022 Fast\n\nChicken with, Greek,\n\nand Meatballs, Parmesan,  CALORIE SMART \u2022 Fast\n\nOsso Buco,\n\nOutrageous Caramel-Fudge Brownies,\n\nOven-Baked Bacon,  CALORIE SMART \u2022 EASY\n\nOven-Baked Beef Brisket,\n\nOven Chicken Cordon Bleu,  CALORIE SMART\n\nOven-Dried Tomatoes,  CALORIE SMART \u2022 EASY\n\nOven French Toast,\n\nOven-Fried Chicken,  CALORIE SMART \u2022 EASY\n\nOven-Fried Chicken Fingers,\n\nOven-Fried Fish,  CALORIE SMART \u2022 Fast\n\nOven-Roasted Pulled Pork,  CALORIE SMART\n\nOvo vegetarian diet,\n\nOysters. _See also_Shellfish\n\nDeep-Fried,\n\nshucking\n\nStew,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nStuffing,\n\n## P\n\nPackets. _See_ Foil Packets\n\nPad Thai\n\nwith Chicken,\n\nwith Pork,\n\nwith Shrimp,\n\nPaella, Vegetable,  CALORIE SMART \u2022 Fast\n\nPaella, Vegetable Chicken,\n\nPaintbrush Sugar Cookies,\n\nPalm Oil,\n\nPancake(s),  CALORIE SMART \u2022 Fast\n\nBerry,\n\nBuckwheat,  CALORIE SMART \u2022 Fast\n\nButtermilk,\n\nCornbread,  CALORIE SMART \u2022 Fast\n\nGranola,\n\nOrange-Cranberry,\n\nOven, Apple,\n\nOven, Double,\n\nOven, Puffy,  CALORIE SMART \u2022 EASY\n\nPotato,  CALORIE SMART\n\nSour Cream\u2013Nectarine,\n\nturning\n\nWhole-Grain,\n\nPaneer,\n\nPan-Fried Chicken with Romesco Sauce,  CALORIE SMART\n\nPanfried Fish,  CALORIE SMART \u2022 Fast\n\nPanfried Pork Chops with Cider Sauce,  CALORIE SMART \u2022 Fast\n\nPanfrying fish\n\nPanfrying meat, , 317\u2013318\n\nPan Gravy,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCream,\n\nPanna Cotta,  EASY\n\nPan sauce, making\n\nPantry ingredients, -9\n\nPanzanella,  CALORIE SMART\n\nChicken, Baked,  CALORIE SMART\n\nPappadam,\n\nPaprika,\n\nParfaits, Blueberry-Granola,\n\nParfaits, Yogurt,\n\nParmesan,\n\n-and Lemon-Crusted Salmon,  CALORIE SMART\n\nAsparagus with,  CALORIE SMART \u2022 Fast\n\nButter, Italian,\n\n-Buttermilk Ranch Dressing,\n\nChicken Parmigiana,\n\nCroutons, Rosemary-,\n\nEggplant Parmigiana,\n\nEggplant Parmigiana, -Olive,\n\nEggplant Parmigiana, -Ravioli,\n\nFettuccine and Vegetables,  CALORIE SMART \u2022 Fast\n\nFish, Panfried, Herb, Lemon and,\n\nGreen Beans with,\n\nGrissini, Black Pepper and,  CALORIE SMART\n\nOrzo and Meatballs,  CALORIE SMART \u2022 Fast\n\n-Pepper Corn on the Cob,\n\n-Pesto Grilled Cheese Sandwiches,\n\nPolenta,\n\nPretzels, Soft, -Herb,\n\nProsciutto and Pea Pilaf,\n\nRavioli, Bacon, Chive and,\n\nVeal Parmigiana,  CALORIE SMART\n\nParsley,\n\nin Chimichurri Sauce,  CALORIE SMART \u2022 EASY\n\nParsnips, ,\n\nPartially Baked One-Crust Pie,\n\nPasta. _See also_ Bow-Tie Pasta; Fettuccine; Lasagna; Noodle(s); Penne; Spaghetti; Vermicelli\n\nAngel Hair, with Basil, Avocado and Tomato,  Fast\n\nAngel Hair, with Chicken,\n\nAngel Hair, with Shrimp,\n\nBeef Roast with Cabbage and, Asian,  SLOW COOKER\n\nBucatini with Clam Sauce,\n\nCacio e Pepe,  Fast\n\nChicken and, with Creamy Basil Sauce,  Fast\n\nChicken and Vegetables,\n\ncooking,\n\nwith Edamame and Pesto,\n\nHomemade, 170\u2013171 CALORIE SMART\n\ncutting\n\nGluten-Free,\n\nHerb,\n\nRavioli,\n\nRavioli, Four-Cheese,  CALORIE SMART \u2022 EASY\n\ntips for,\n\nWhole Wheat,\n\nIsraeli Couscous Risotto with Caramelized Onions and Sausage,\n\nLinguine, Clams and Sausage, Spanish,  CALORIE SMART\n\nLinguine, Gorgonzola, with Toasted Walnuts,  Fast\n\nLinguine with White Clam Sauce and Basil,\n\nManicotti,\n\nManicotti, Turkey,\n\nOrecchiette, Browned Butter, with Broccoli Rabe,  Fast\n\nOrzo, Chicken with, Greek,\n\nOrzo, Chicken with Pan-Roasted Cauliflower and,  CALORIE SMART \u2022 Fast\n\nOrzo and Meatballs, Parmesan,  CALORIE SMART \u2022 Fast\n\nPesto, and Meatballs,\n\nPork and, with Creamy Basil Sauce,\n\nwith Prosciutto and Asiago Cheese,  CALORIE SMART \u2022 Fast\n\nPuttanesca,\n\nPuttanesca, Pepper, Roasted,\n\nRavioli, Butternut Squash,\n\nRigatoni Bake, Veggie,\n\nRigatoni, Cheesy, with Eggplant Sauce,\n\nwith Sage, Fried,\n\nselecting\/storing,\n\nShells, Chicken- and Spinach-Stuffed,\n\nand Shrimp, Tomato-Basil,\n\nvarieties of, 168\u2013169\n\nyields,\n\nPasta Sauce,\n\nAlfredo,\n\nAlfredo, Easy,\n\nBolognese,  CALORIE SMART\n\nBolognese, Pasta,\n\nBrowned Butter, \u2013Sage,\n\nClam-Mushroom, Bucatini with,\n\nClam, White, Spaghetti with,  Fast\n\nClam, White, Linguine with Basil and,\n\nEggplant, Cheesy Rigatoni with,\n\nMeat, Spaghetti and,\n\nShort Rib and Sausage Rag\u00f9,\n\nSkillet, Quick,\n\nTomato, Italian,  CALORIE SMART\n\nTomato-Vodka, Creamy,  CALORIE SMART \u2022 Fast\n\nTomato-Vodka, Creamy Basil-Chicken,\n\nTomato-Vodka, Creamy Olive,\n\nPastry(ies). _See also_ Cream Puffs; Phyllo; Pie Crust\n\nPuff, Brie in, with Cranberry Sauce,\n\nPuff, Brie in, with Cranberry Sauce, Double-Orange,\n\nTart, Press-in-the Pan,  Fast \u2022 EASY\n\nPastry blender,\n\nPatties. _See also_ Burgers\n\nFalafel (Fava Bean Rounds),  CALORIE SMART\n\nKibbeh,  CALORIE SMART\n\nSalmon, with Sour Cream\u2013Dill Sauce,  CALORIE SMART \u2022 Fast\n\nSamosa, Spicy,  CALORIE SMART\n\nPatty Melts with Smothered Onions,  CALORIE SMART\n\nPattypan squash,\n\nPea(s),\n\nBaby Food, Sweet Pea,\n\ndried,\n\nPesto, Sweet Pea\u2013Mint,\n\npreparing\/cooking,\n\nProsciutto and Parmesan Pilaf,\n\nRisotto with, Classic,\n\nSnow, in Asian Salad with Wonton Crisps,  Fast\n\nSplit Pea Soup,  CALORIE SMART\n\nPeach(es),\n\nCobbler,\n\nCobbler, Raspberry-,\n\nCupcakes, Ginger and,\n\nIce Cream, Fresh,\n\nJam, -Jalape\u00f1o,  CALORIE SMART\n\nJam, -Jalape\u00f1o, making,\n\nJam, -Jalape\u00f1o, uses for, 436\u2013437\n\nJam, Rosemary,  CALORIE SMART\n\nJam, Spiced,  CALORIE SMART\n\nLemonade, Peachy,\n\nPie, Blueberry-,\n\nPie, Classic,\n\nSalad, Summery,\n\nSalsa, Fresh,  CALORIE SMART\n\nSalsa, -Mango,\n\nSalsa, -Strawberry,\n\nSmoothies,\n\nVinaigrette, Spicy,\n\nPeachy Lemonade,\n\nPeaks, soft\/stiff,\n\nPeanut(s)\n\nin baked goods,\n\nBrittle,  CALORIE SMART\n\n-Chipotle Noodle Bowls,  Fast\n\n-Fudge Thumbprint Cookies,\n\nSoy Peanutty Dressing,\n\nPeanut Butter\n\nCandy Fudge,\n\n\u2013Chocolate Brownies,\n\nCookies,  CALORIE SMART\n\nCookies, Rich,\n\nFrosting,\n\n\u2013Honey Cream Cheese Spread,\n\n\u2013Honey Drizzle, Whole Wheat Waffles with,  CALORIE SMART\n\n\u2013Yogurt Dip,\n\nPeanut Sauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChicken Satay with, 56\u201357\n\nChicken, Spicy, in,  SLOW COOKER\n\nYogurt, Tempeh Stir-Fry with,  CALORIE SMART\n\nPear(s),\n\nCaramel-Maple,  SLOW COOKER \u2022 EASY\n\n-Cranberry Chutney,\n\nCustard Pie, Elegant,\n\n-Orange Conserve,  CALORIE SMART\n\nUpside-Down Cake, -Ginger,\n\nPea shoots,\n\nPecan(s),\n\nButter,\n\nButter-Pecan Syrup,\n\nCheese Ball, -Coated,\n\n-Coconut Frosting,\n\nPie,\n\nPie, Kentucky,\n\nPies, Mini, Bourbon-Chocolate-,\n\nPork Medallions, -Crusted,  Fast\n\nPraline, Bacon,\n\nPraline Pumpkin Pie,\n\nPralines,  CALORIE SMART\n\nQuinoa, -Cherry,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSticky Rolls, Caramel,\n\nSweet Peppery,\n\nTurtle Layer Cake,\n\nWild Rice Porridge, Bacon and,\n\nPenne\n\nChicken-Thyme,\n\nItalian Sausage with Tomatoes and,  CALORIE SMART \u2022 Fast\n\nTurkey-Basil,\n\nPepper(ed)\n\n-Apple Jelly,  CALORIE SMART \u2022 EASY\n\nBrown Sugar Bacon,\n\nCacio e Pepe,  Fast\n\nFlank Steak,  CALORIE SMART\n\nGrissini, Black Pepper and Parmesan,  CALORIE SMART\n\nJack Cheese Ball,\n\n-Lemon Butter,\n\n-Lemon Pork Tenderloin, Grilled,  CALORIE SMART \u2022 Fast\n\n-Parmesan Corn on the Cob,\n\nPork Loin Roast, Garlic,\n\nred pepper flakes,\n\nPepper(s). _See_ Bell Pepper(s); Chile(s)\n\nPeppercorns,\n\nPeppermint\n\nFrosting,\n\nMarshmallows,\n\nWhoopie Pies, Pink,\n\nPeppery Red Wine Sauce,\n\nPersonal Taco Dip,\n\nPesto,\n\nAlmond-Cilantro, Spicy,\n\nBasil,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nBasil, Roasted Red Pepper and Walnut,\n\nBeet-Dill,\n\nblending\n\nBroccoli-Walnut,  CALORIE SMART \u2022 Fast\n\nBrussels Sprouts\u2013Cashew,\n\nCarrot, Roasted, -Almond,\n\nCilantro,\n\nice cubes\n\n-Parmesan Grilled Cheese Sandwiches,\n\nPasta,\n\nPasta with Edamame and,\n\nPasta and Meatballs,\n\nSpinach,\n\nSun-Dried Tomato,\n\nSun-Dried Tomato, Branzino with,\n\nSweet Pea\u2013Mint,\n\nPheasant, roasting,\n\nPheasants, Orange-Roasted,\n\nPho, Beef,  CALORIE SMART\n\nPhyllo,\n\nCaramel Tarts, Tiny,\n\nChicken Pie, Moroccan,\n\nTriangles, Basil-Cheese,  CALORIE SMART\n\nPickled\n\nBeets,  CALORIE SMART\n\nCabbage, 366\u2013367\n\nCarrots, Spicy,\n\nCarrots, Tarragon Baby,  CALORIE SMART \u2022 EASY\n\nCucumber and Onion,\n\nGinger,\n\nGreen Beans,\n\nGreen Beans, Jalape\u00f1o-Garlic,  CALORIE SMART\n\nGreen Beans, Lemon-Dill,\n\nPeppers,  CALORIE SMART\n\nRed Onions,  CALORIE SMART\n\nVegetables, Spicy,  CALORIE SMART\n\nPickles\n\ncontainers for,\n\nDill, Garlic,  CALORIE SMART\n\npacking cucumbers\n\nPickles _(cont.)_\n\nRefrigerator, Easy,  CALORIE SMART\n\nPicnics, food safety in\n\nPie(s). _See also_ Pie Crust; Pizza; Pot Pie(s); Quiche; Tart(s)\n\nApple, Classic,\n\nApple, French,\n\nApple, Slab,  CALORIE SMART\n\nBlackberry,\n\nBlueberry,\n\nBlueberry Hand Pies,\n\nBlueberry, -Peach,\n\nBourbon-Chocolate-Pecan Mini,\n\nBoysenberry,\n\nCherry,\n\nCherry, Quick,\n\nChicken, Moroccan,\n\nChocolate Brownie,\n\nCream\n\nBanana,\n\nButterscotch,\n\nChocolate,\n\nChocolate-Banana,\n\nCoconut,\n\nFrench Silk, Classic,\n\nFrench Silk, Mocha,\n\nKey Lime,\n\nLemon Curd, Ginger\u2013, Petite,  CALORIE SMART\n\nLemon Meringue, 602\u2013603\n\nLoganberry,\n\nNectarine-Plum Crostata,\n\nPeach, Classic,\n\nPear Custard, Elegant,\n\nPecan,\n\nPecan, Kentucky,\n\nPumpkin,\n\nPumpkin, Praline,\n\nRaspberry,\n\nRaspberry-Almond,\n\nRhubarb,\n\nRhubarb, Quick,\n\nstoring, ,\n\nStrawberry-Rhubarb,\n\nSweet Potato Meringue, Roasted,\n\nSweet Potato, Roasted,\n\nPie Crust\n\nBaked Individual,\n\nbaking methods,\n\nButtermilk Pastry, Easy,  Fast \u2022 EASY\n\nbutter pastry,\n\nCookie Crumb,\n\nedge treatments\n\nin food processor,\n\nfreezing,\n\nGraham Cracker,  Fast \u2022 EASY\n\nlattice top,\n\nmaking\n\nNut, Easy,  Fast \u2022 EASY\n\nOat,\n\nOil Pastry, Press-in-the-Pan,  Fast \u2022 EASY\n\nOne-Crust, Baked,\n\nOne-Crust, Partially Baked\n\nOne-Crust Pastry,\n\ntips for,\n\nTwo-Crust Pastry,\n\nPie pans, ,\n\nPilaf. _See also_ Rice, Pilaf\n\nBarley, Black Bean and,  CALORIE SMART\n\nBulgur,  CALORIE SMART \u2022 Fast\n\nKasha and Bow-Tie,  CALORIE SMART\n\nPineapple,\n\nBroiled,  CALORIE SMART \u2022 Fast\n\nMimosas,\n\n-Orange Smoothies,\n\nSorbet,\n\nPine Nut and Green Onion Pilaf,\n\nPink beans,\n\nPistachio\n\nBrussels Sprouts, -Cranberry,  CALORIE SMART \u2022 Fast\n\nDivinity, -Cranberry,  CALORIE SMART\n\nFudge, -Cherry,\n\nRice Pilaf, Cherry,\n\nSugar Cookies, Chocolate Chip, Cherry and, Soft No-Roll,\n\nPita Crisps, Baked,\n\nPizza\n\nBeef, Cheesy, 192\u2013193\n\nCalzone,\n\nCanadian Bacon,\n\nCheese,\n\nChicken Sausage, Spinach and Swiss,  Fast\n\nEasy,\n\nHam and Gorgonzola,\n\nMeatball,\n\nMeat Lover's,\n\nPulled Pork,\n\nSausage-Broccoli, Easy,\n\nSpaghetti and Meatballs,\n\nStuffed-Crust,  CALORIE SMART\n\nStuffed-Crust Tomato, Deluxe,\n\nTart Flamb\u00e9e,  CALORIE SMART\n\nTomato, Fresh,  CALORIE SMART\n\nVeggie,\n\nVeggie Lover's, with Cornmeal Crust,  CALORIE SMART\n\nPizza Crust,  CALORIE SMART\n\nCornmeal,\n\nItalian-Style,\n\nshaping dough\n\nstuffed\n\nThick,\n\nThin,\n\nWhole Wheat,\n\nPizza pan\/stone,\n\nPizza Quinoa with Sausage, Onion and Pepper,  CALORIE SMART\n\nPlantain, ,\n\nPlum(s),\n\n-Berry Pops,\n\n-Nectarine Crostata,\n\nPlumcots,\n\nPluot(s),\n\nDessert, Grilled,\n\nPoached Eggs,  CALORIE SMART \u2022 Fast\n\nPoblano(s),\n\nRellenos Bake, Roasted,\n\nPolenta,  CALORIE SMART \u2022 Fast\n\nFaster Fried,\n\nFried,\n\nmaking\n\nParmesan,\n\nRatatouille Bake,  CALORIE SMART\n\nRich and Creamy,\n\nVegetable,\n\nPomegranate(s),\n\nFreezer Jam, Triple-Berry,  CALORIE SMART\n\n-Mango Mimosas,\n\nMartini Cocktails,\n\nOatmeal,\n\nPomegranate Molasses,\n\nPopovers,  CALORIE SMART \u2022 EASY\n\nPoppy Seed\n\n\u2013Honey Dressing,\n\n\u2013Lemon Pound Cake,\n\n\u2013Strawberry Dressing, Creamy,\n\nPops\n\nBerry-Plum,\n\nRaspberry Sangria,\n\nStrawberry,\n\nYogurt, Frozen,\n\nPorcini mushrooms,\n\nPork. _See also_ Bacon; Ham; Ribs, Pork; Sausage, Pork\n\nand Barley, Apple-Rosemary,  CALORIE SMART \u2022 Fast\n\nbrining time,\n\nbroiling, ,\n\nCarnitas Taco Bites,\n\nChops\n\nBalsamic, with Quinoa,  CALORIE SMART\n\nCornbread- and Bacon-Stuffed,\n\nCuban,  CALORIE SMART \u2022 Fast\n\ncutting pocket\n\nwith Mustard-Chive Cream Sauce,\n\nPanfried, with Cider Sauce,  CALORIE SMART \u2022 Fast\n\nSmoked, with Apple and Sweet Potato,  CALORIE SMART\n\nSpicy Southwestern,\n\nCrown Roast with Fruited Stuffing Supreme,\n\ncuts\n\nDijon, Smothered in Mushrooms,\n\nLoin Roast, Dijon-Herbed,\n\nLoin Roast, Herbed,  CALORIE SMART\n\nLoin Roast, Peppered Garlic,\n\nMeatballs, Make-Ahead,\n\nMedallions, Pecan-Crusted,  Fast\n\nMedallions, Pumpkin Seed\u2013Crusted,\n\nwith Mushroom Sauce,\n\nPackets, Southwestern,  CALORIE SMART\n\nPad Thai with,\n\npanfrying,\n\nand Pasta with Creamy Basil Sauce,\n\nPot Roast, Italian,  CALORIE SMART\n\nPot Roast, Porketta,  CALORIE SMART \u2022 SLOW COOKER\n\nPulled\n\nBurrito Bowls,\n\nChopped Salad with,\n\nand Coleslaw Sandwiches,\n\nFries, Loaded,\n\nHash, Skillet,\n\nOven-Roasted,  CALORIE SMART\n\nPizza,\n\nQuesadillas,\n\nSandwiches, Jerk,  SLOW COOKER\n\nRice, Fried,  CALORIE SMART \u2022 Fast\n\nroasting timetable,\n\nStew, Hearty,  CALORIE SMART \u2022 SLOW COOKER\n\nStew, Thai, Spicy,\n\ntenderloin, removing silverskin\n\nTenderloin, Grilled Lemon-Pepper,  CALORIE SMART \u2022 Fast\n\nTenderloin, with Roasted Vegetables,  CALORIE SMART\n\nPorketta Pot Roast,  CALORIE SMART \u2022 SLOW COOKER\n\nPorridge, Bacon and Pecan Wild Rice,\n\nPorridge, Cranberry\u2013Wild Rice,\n\nPortabella(s),\n\nAsian Stuffed,  CALORIE SMART\n\nChicken Pot Pie,\n\nMuffuletta Sandwiches,  CALORIE SMART \u2022 Fast\n\nPort Cranberry Sauce,\n\nPotato(es)\n\nAu Gratin,\n\nBacon and Egg Scramble,  CALORIE SMART \u2022 Fast\n\nBaked, with Tuna Topping,\n\nChicken and, Sage,  CALORIE SMART\n\nChicken Sausage with Cheese and, Roasted,  CALORIE SMART\n\nChorizo and Egg Scramble,\n\nEgg Bake, Tex-Mex,\n\nFried,  CALORIE SMART \u2022 Fast\n\nGnocchi,\n\ngrilling,\n\nHash Brown,  CALORIE SMART\n\nLemon, Tilapia with,\n\nMashed,  CALORIE SMART \u2022 EASY\n\nButtermilk,\n\nGarlic,\n\nHerbed,\n\nHorseradish,\n\nMeat Loaf Casserole, -Topped,  CALORIE SMART\n\nMeat Loaf Casserole, -Topped, Quick,\n\nOven-Browned, Beef Rib Roast with,\n\nPancakes,  CALORIE SMART\n\npreparing\/cooking, 236\u2013237\n\nRed, Grilled Herbed,  Fast\n\nSalad\n\nCreamy,\n\nDilled,\n\nGerman, Hot,  CALORIE SMART\n\nRoasted,\n\nSamosa Patties, Spicy,  CALORIE SMART\n\nScalloped,\n\nSkins, Cheesy,\n\nsmashed,\n\nSmashed, Ultimate,  CALORIE SMART \u2022 SLOW COOKER\n\nSoup, -Bacon, Cheesy,  SLOW COOKER\n\nTwice-Baked,  CALORIE SMART\n\nBacon and Pepper,\n\nItalian, 248\u2013249\n\nvarieties of,\n\nVeggie-Stuffed,\n\nPotato masher,\n\nPot holder,\n\nPot Pie(s),\n\nChicken,\n\nChicken, Cranberry-,  CALORIE SMART\n\nChicken Divan,\n\nChicken, Individual,\n\nChicken, Portabella,\n\nSamosa, 274\u2013275 CALORIE SMART\n\nPot Roast,\n\nBarbecue,\n\nCaramelized Onion,\n\nConfetti,\n\nCream Gravy,\n\nGarlic-Herb,\n\nPork, Italian,  CALORIE SMART\n\nPorketta,  CALORIE SMART \u2022 SLOW COOKER\n\nSlow Cooker,\n\nPoultry. _See also_ Chicken; Game Fowl; Turkey\n\nbraising,\n\nbroiling,\n\nbuying tips,\n\nfreezing\/thawing, , ,\n\nroasting,\n\nstoring,\n\nyields,\n\nPound Cake,\n\nAlmond, Toasted,\n\nCranberry-Orange,\n\nCupcakes,\n\nDoughnut Holes, Glazed,\n\nLemon\u2013Poppy Seed,\n\nOrange-Coconut,\n\nRaspberry-Topped,\n\nuses for,\n\nPraline(s),  CALORIE SMART\n\nBacon,\n\nPumpkin Pie,\n\nPreserves, . _See also_ Chutney; Jam; Jelly; Relish\n\nApricot-Ginger, Spiced,  CALORIE SMART \u2022 EASY\n\nPreserving. _See_ Canning and preserving\n\nPress-in-the-Pan Oil Pastry,  Fast \u2022 EASY\n\nPress-in-the Pan Tart Pastry,  Fast \u2022 EASY\n\nPressure canning,\n\nPressure cooker,\n\nPretzels, Soft,  CALORIE SMART\n\nParmesan-Herb,\n\nProfiteroles,\n\nProsciutto\n\nGreens with Gorgonzola and,  CALORIE SMART \u2022 Fast\n\nParmesan and Pea Pilaf,\n\nPasta with Asiago Cheese and,  CALORIE SMART \u2022 Fast\n\nProtein, complete,\n\nProven\u00e7al Fish Soup,  CALORIE SMART \u2022 Fast\n\nProven\u00e7al Roasted Chicken,\n\nProvolone-Onion Crostini,\n\nProvolone-Turkey-Dill Party Sub,\n\nPudding(s). _See also_ Bread Pudding; Custard\n\nBerry-Chia Breakfast,\n\nButterscotch,\n\nChocolate,\n\nChocolate Mousse,  EASY\n\nRice, 632\u2013633\n\nRice, Almond-Cherry,\n\nRice, Easy,\n\nsoy,\n\nTrifle, English,\n\nTrifles, Individual, English,\n\nVanilla,  EASY\n\nYorkshire,\n\nPudding Cake, Pumpkin,\n\nPuff Pastry, Brie in, with Cranberry Sauce,\n\nDouble-Orange,\n\nPuffy Omelet,  CALORIE SMART \u2022 Fast\n\nPuffy Oven Pancake,  CALORIE SMART \u2022 EASY\n\nPulled Pork. _See_ Pork, Pulled\n\nPumpernickel\n\nCroutons,\n\nLoaf, Bakery-Style,\n\nRolls, Bakery-Style, 512\u2013513 CALORIE SMART\n\nPumpkin\n\nBread,\n\nCookies with Browned Butter Frosting,\n\nCookies, Spicy, with Browned Butter Frosting,\n\nPie,\n\nPie, Praline,\n\nPudding Cake,\n\nPumpkin Seed\u2013Crusted Pork Medallions,\n\nPureeing, equipment\/tips for,\n\nPuttanesca,\n\nPepper, Roasted,\n\n## Q\n\nQuesadillas\n\nBlack Bean,\n\nEasy,\n\nPulled Pork,\n\nTuna,\n\nQueso Blanco,\n\nQueso Fresco,\n\nQuiche\n\nLorraine,\n\nSeafood,\n\nSpinach,\n\nSpinach and Cheddar,  CALORIE SMART\n\nQuick Cherry Pie,\n\nQuick Lettuce Wraps,\n\nQuick Potato-Topped Meat Loaf Casserole,\n\nQuick Refried Beans,\n\nQuick Rhubarb Pie,\n\nQuick Skillet Pasta Sauce,\n\nQuinoa,\n\nBreakfast,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCherry-Pecan,  CALORIE SMART \u2022 Fast \u2022 EASY\n\ncooking timetable,\n\nOatmeal Thumbprints, -Date,  CALORIE SMART\n\nPizza, with Sausage, Onion and Pepper,  CALORIE SMART\n\nPork Chops with, Balsamic,  CALORIE SMART\n\n-Tomato Salad, Summer,  CALORIE SMART\n\n## R\n\nRachel sandwich,\n\nRadicchio,\n\nRadish,\n\nRaisin\n\nBatter Bread, Whole Wheat\u2013,\n\nBread, Cinnamon Swirl,\n\n-Oatmeal Cookies,  CALORIE SMART\n\nRanch Dressing, Buttermilk,  CALORIE SMART \u2022 EASY\n\nHerb, Fresh,\n\n-Parmesan,\n\nRaspberry(ies),\n\nButter,\n\nChocolate Tart, White and Dark Chocolate, 600\u2013601\n\n-Ginger Tea,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nIce Cream, Fresh,\n\nJam, Freezer,\n\nLemonade,\n\n-Lime Juice,  Fast \u2022 EASY\n\n-Peach Cobbler,\n\nPie,\n\nPie, -Almond,\n\nPops, Sangria,\n\nPound Cake, -Topped,\n\nSauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nScones, \u2013White Chocolate,\n\nSmoothies,\n\nSorbet, Cranberry-,\n\nTurkey\u2013Wild Rice Salad,\n\nVinaigrette,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRatatouille\n\nGarden,  CALORIE SMART \u2022 Fast\n\nPolenta Bake,  CALORIE SMART\n\nSalad,  CALORIE SMART\n\nRavioli\n\nButternut Squash,\n\n-Eggplant Parmigiana,\n\nHomemade,\n\nBacon, Chive and Parmesan,\n\nFour-Cheese,  CALORIE SMART \u2022 EASY\n\nmaking\/cutting,\n\nTomatoes with,\n\nwith wonton skins\n\nRed Beans and Rice,  CALORIE SMART\n\nRed Cabbage, Sweet-Sour,  CALORIE SMART\n\nRed Currants,\n\nRed Curry Paste, Thai,  CALORIE SMART\n\nRed Lentil, Barley and Cauliflower Tabbouleh,  CALORIE SMART\n\nRed Onions,\n\nPickled,  CALORIE SMART\n\nRed pepper (cayenne),\n\nRed Pepper(s). _See also_ Bell Pepper(s)\n\nRoasted, Basil and Walnut Pesto,\n\nRoasted, Broccoli with Hazelnuts and,  CALORIE SMART \u2022 Fast\n\nsauce (condiment),\n\nSoup, Roasted, with Mozzarella, , CALORIE SMART\n\nRed pepper flakes,\n\nRed Potatoes, Grilled Herbed,  Fast\n\nRed rice,\n\nReduction, Sauce,\n\nRed Velvet Cake,\n\nRed Velvet Cupcakes,\n\nRed Wine\n\nMulled,  CALORIE SMART\n\nSangria,  CALORIE SMART \u2022 Fast\n\nSauce, Peppery,\n\nvarietals,\n\nVinaigrette Dressing,\n\nRefreshing Green Juice,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRefried Beans, Quick,\n\nRefried Black Beans, Spicy,  CALORIE SMART \u2022 Fast\n\nRefrigeration\n\ncleaning\/sanitizing,\n\nand food safety, ,\n\nfood storage chart, 33\u201334\n\nRefrigerator Cookies. _See_ Brown Sugar, Refrigerator Cookies\n\nRelish\n\nCorn, ,  CALORIE SMART\n\nCranberry-Orange,  CALORIE SMART \u2022 EASY\n\nCranberry-Orange-Ginger,\n\nMango,  CALORIE SMART \u2022 EASY\n\nOlive-Tomato,\n\nVeggie, Crunchy,  CALORIE SMART\n\nReuben Deviled Eggs,\n\nReuben sandwich,\n\nRhubarb\n\nCrisp,\n\nPie,\n\nPie, Quick, 594\u2013595\n\nSauce,  CALORIE SMART \u2022 Fast\n\n-Strawberry Pie,\n\n-Strawberry Sauce,\n\nRib Roast, Beef\n\ncarving,\n\nHerb- and Garlic-Crusted,  CALORIE SMART \u2022 EASY\n\nwith Potatoes, Oven-Browned,\n\nroasting,\n\nRibs, Pork. _See also_ Short Rib(s), Beef\n\nCountry-Style,\n\nLoin, Spicy Barbecue, 310\u2013311\n\nremoving silverskin from\n\nSmoked Hot and Spicy,\n\nwith Smoky Barbecue Sauce,\n\nSouthwest, with Smoky Barbecue Sauce,\n\nSpareribs,\n\nRice. _See also_ Brown Rice; Risotto, Classic; Wild Rice\n\nBalls, Spicy, Spaghetti and,  Fast\n\nChicken and, Country French,  CALORIE SMART\n\nChicken Soup,\n\ncooking timetable,\n\nFried\n\nBrown Rice,\n\nChicken,\n\nPork,  CALORIE SMART \u2022 Fast\n\nShrimp,\n\nJacked-Up,\n\nLentils and, Indian,  CALORIE SMART\n\nPilaf,  Fast \u2022 EASY\n\nAlmond\u2013Brown Rice,\n\nBrown Rice,\n\nCashew,\n\nCherry Pistachio,\n\nCurry,\n\nMushroom,\n\nOrange and Cranberry,\n\nPine Nut and Green Onion,\n\nProsciutto, Parmesan and Pea,\n\nPudding, 632\u2013633\n\nPudding, Almond-Cherry,\n\nPudding, Easy,\n\nRed Beans and,  CALORIE SMART\n\nSpanish,  CALORIE SMART\n\nSpanish, with Bacon,\n\nSticky, with Mango,  CALORIE SMART\n\n-Tofu Skillet, Italian,\n\n-Tofu Skillet, Mexican,  CALORIE SMART \u2022 Fast\n\nvarieties of,\n\nRice milk,\n\nRice vinegar,\n\nRich and Creamy Polenta,\n\nRich Egg Bread,  CALORIE SMART\n\nRich Peanut Butter Cookies,\n\nRicotta,\n\nLasagna Cupcakes,  CALORIE SMART\n\nLasagna, Italian Sausage,\n\nShells, Chicken- and Spinach-Stuffed,\n\nRigatoni Bake, Veggie,\n\nRigatoni, Cheesy, with Eggplant Sauce,\n\nRisotto\n\nBarley, with Edamame, Vegetarian,\n\nChicken and Barley, with Edamame,  CALORIE SMART \u2022 SLOW COOKER\n\nIsraeli Couscous, with Caramelized Onions and Sausage,\n\nRice. _See_ Risotto, Classic\n\nRisotto, Classic,\n\nwith Butternut Squash,\n\nwith Peas,\n\nwith Shrimp,\n\nRoasted Beets,  CALORIE SMART \u2022 EASY\n\nRoasted Beet Salad,\n\nRoasted Bell Peppers,  CALORIE SMART \u2022 EASY\n\nRoasted Carrot\u2013Almond Pesto,\n\nRoasted Chicken Sausage with Potatoes and Cheese,  CALORIE SMART\n\nRoasted Chile Mayonnaise,\n\nRoasted Garlic,  CALORIE SMART \u2022 EASY\n\nRoasted Garlic Hummus,\n\nRoasted Pepper Puttanesca,\n\nRoasted Poblanos Rellenos Bake,\n\nRoasted Potato Salad,\n\nRoasted Red Pepper Soup with Mozzarella, , CALORIE SMART\n\nRoasted Root Vegetable,\n\nRoasted Sweet Potato Meringue Pie,\n\nRoasted Sweet Potato Pie,\n\nRoasted Tilapia and Vegetables,  CALORIE SMART\n\nRoasted Vegetable Dip with Baked Pita Crisps,  CALORIE SMART\n\nRoasted Vegetables,  CALORIE SMART\n\nRoast Goose with Apple Stuffing,\n\nRoasting\n\nmeats, 314\u2013315\n\npoultry,\n\nvegetables,\n\nRoasting pan,\n\nRoast Turkey,  CALORIE SMART\n\nRolling pin,\n\nRolls\n\nCaramel Sticky,\n\nCinnamon,\n\nCloverleaf,\n\nCrescent,\n\nDinner,  CALORIE SMART\n\nPumpernickel, Bakery-Style, 512\u2013513 CALORIE SMART\n\ntoppings for,\n\nRoll-Up(s)\n\nBean and Avocado,\n\nChicken, Asian,  CALORIE SMART \u2022 Fast\n\nSausage, Breakfast,\n\nRomaine (cos) lettuce,\n\nRomano,\n\nRomesco Sauce,\n\nRoot Beer Milk,\n\nRoot vegetables,\n\nRopa Vieja Tostadas,  CALORIE SMART \u2022 SLOW COOKER\n\nRoquefort,\n\nRosemary,\n\n-Apple Pork and Barley,  CALORIE SMART \u2022 Fast\n\nButter, Artichokes with,  EASY\n\nChicken and White Beans, Tuscan,  CALORIE SMART \u2022 Fast\n\n-Honey Roasted Chicken,\n\n-Honey Simple Syrup,\n\nLamb Chops, Grilled,  CALORIE SMART \u2022 Fast\n\n-Lemon Branzino,\n\n-Orange Butter Balls, Chocolate-Filled,  CALORIE SMART\n\n-Parmesan Croutons,\n\nPeach Jam,\n\nRotisserie Spiced Chicken, Slow-Cooker,  CALORIE SMART\n\nRoux, ,\n\nRub(s)\n\nBrisket,\n\nCajun Spice,\n\nChile-Cumin,\n\nCuban,\n\nHot and Spicy,\n\nItalian Seasoning,\n\nSouthwestern,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSpice,\n\nSpice, Toasted,\n\nRuler,\n\nRum\n\nHot Buttered, -Spiced Cider,\n\nMojitos,\n\nTres Leches Cake,\n\nRussian Dressing,\n\nRutabagas, ,\n\nRye, ,\n\nCroutons,\n\nRye Berry Salad, Cranberry-Almond,\n\n## S\n\nSafety. _See_ Food safety\n\nSaffron, ,\n\nSage,\n\n\u2013Browned Butter Sauce,\n\nCannellini with Tomatoes and,  CALORIE SMART \u2022 Fast\n\nChicken and Potatoes, Sage,  CALORIE SMART\n\nFried, Pasta with,\n\nSake,\n\nSalad(s). _See also_ Caesar Salad; Coleslaw; Tabbouleh\n\nAsian, with Wonton Crisps,  Fast\n\nAsparagus-Cashew, ,\n\nAsparagus-Orange,\n\nAsparagus-Strawberry,  CALORIE SMART \u2022 Fast\n\nBean, Green and Yellow,  CALORIE SMART\n\nBean, Two-,\n\nBeet, Roasted,\n\nBeets and Baby Greens,\n\nBread,\n\nCaprese, Heirloom Tomato,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nChicken, Blue Cheese,\n\nChicken\u2013Brown Rice,\n\nChicken-Macaroni,\n\nChicken-Thyme Penne,\n\nChopped\n\nwith Beans,\n\nChicken,\n\nChicken and Bacon,\n\nItalian,  Fast\n\nPulled Pork with,\n\nSummer,\n\nwith Tangy Dressing,\n\nTurkey-Cheese,\n\nCobb,\n\nCranberry-Orange Mold,  CALORIE SMART \u2022 EASY\n\nCucumber-Avocado-Berry,\n\nEdamame and Vegetable,\n\nFarro, Indian-Spiced,  CALORIE SMART\n\nFruit, Granola,\n\nFruit 'n Nuts,\n\ngelatin, unmolding\n\nGreek,  CALORIE SMART \u2022 Fast\n\nGreens with Prosciutto and Gorgonzola,  CALORIE SMART \u2022 Fast\n\ngreens, varieties of, 113\u2013115\n\nItalian Salad Toss,\n\nKale, Cherry-Walnut,  CALORIE SMART \u2022 Fast\n\nLyonnaise,  CALORIE SMART \u2022 Fast\n\nMandarin, Crunchy Chicken-,\n\nMandarin, with Sugared Almonds, 120\u2013121 CALORIE SMART \u2022 Fast\n\nMango-Avocado,\n\nOlive,\n\nPanzanella,  CALORIE SMART\n\nPotato\n\nCreamy,\n\nDilled,\n\nGerman, Hot,  CALORIE SMART\n\nRoasted,\n\nQuinoa-Tomato, Summer,  CALORIE SMART\n\nRatatouille,  CALORIE SMART\n\nRye Berry, Cranberry-Almond,\n\nSalmon, Cinnamon-Maple Glazed,\n\nSalmon, Ni\u00e7oise,\n\nSanta Fe, with Tortilla Straws,\n\nSesame Noodle,  CALORIE SMART \u2022 Fast\n\nSeven-Layer,\n\nSeven-Layer, Fennel-Asparagus,\n\nShrimp and Melon,\n\nSpinach-Bacon, with Hard-Cooked Eggs,  CALORIE SMART \u2022 Fast\n\nSummery,\n\nSunflower, Sunny,\n\nTaco,\n\ntoppers, crispy, . _See also_ Croutons\n\nTuna, Mediterranean,\n\nTurkey-Basil Penne,\n\nWheat Berry,  CALORIE SMART\n\nWhite Bean, Northern Italian,  CALORIE SMART\n\nWild Rice and Chicken,\n\nSalad Dressing, . _See also_ Vinaigrette Dressing\n\nAsian,\n\nAvocado, Creamy,\n\nBlue Cheese,  CALORIE SMART \u2022 EASY\n\nBlue Cheese Yogurt,\n\nButtermilk Ranch,  CALORIE SMART \u2022 EASY\n\nButtermilk Ranch, Fresh Herb,\n\nButtermilk Ranch\u2013Parmesan,\n\nFrench,\n\nHoney-Mustard,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHoney\u2013Poppy Seed,\n\nItalian,  CALORIE SMART \u2022 Fast\n\nItalian, Creamy,\n\nLemon,\n\nMiso,  CALORIE SMART \u2022 Fast\n\nMiso, Honey,\n\nMiso, Sriracha,\n\noil\/vinegar for,\n\nPeanutty Soy,\n\nRussian,\n\nSesame,\n\nStrawberry-Poppy, Creamy,\n\nSweet-Sour,\n\nTangy, Chopped Salad with,\n\nThousand Island,  CALORIE SMART \u2022 Fast\n\nSalad spinner,\n\nSalmon\n\nCakes,\n\nChopped Salad, Summer,\n\ndoneness of,\n\nHoney Mustard\u2013Glazed,  CALORIE SMART\n\nLemon- and Parmesan-Crusted,  CALORIE SMART\n\nPackets, Lemon and Herb,  CALORIE SMART\n\nPatties with Sour Cream\u2013Dill Sauce,  CALORIE SMART \u2022 Fast\n\nSalad, Cinnamon-Maple Glazed,\n\nSalad, Ni\u00e7oise,\n\nSmoked Brined,  CALORIE SMART\n\nSalsa, ,  CALORIE SMART\n\nBlack Bean,\n\n\u2013Brown Rice Burritos,\n\nChicken Wings,\n\nCorn,\n\nGreen, Crema,\n\nMango-Peach,\n\nPeach, Fresh,  CALORIE SMART\n\nStrawberry-Peach,\n\nTacos,\n\nTomato and Black Bean,\n\nVerde,  CALORIE SMART \u2022 Fast\n\nSalt,\n\nSalted Cashew-Bittersweet Chocolate Tart,\n\nSalt-Free Herb Mix,\n\nSalt-Free Taco Seasoning Mix,\n\nSalty Caramel Frosting,\n\nSambal oelek,\n\nSamosa Patties, Spicy,  CALORIE SMART\n\nSamosa Pot Pie, 274\u2013275 CALORIE SMART\n\nSandwich(es). _See also_ Burgers; Burgers, Beef; Wraps\n\nBarbecue,\n\nBeef, Barbecue, 298\u2013299 CALORIE SMART \u2022 SLOW COOKER\n\nBeef, Italian,\n\nBeef Lettuce Wraps, Vietnamese,  CALORIE SMART\n\nBlack Bean Sliders,  CALORIE SMART\n\nCheeseburger Hoagies,\n\nCheese, Grilled,  Fast\n\nCheese, Grilled, Pesto-Parmesan,\n\nCheese, Grilled, Tomato and Avocado,\n\nChicken Salad,  Fast \u2022 EASY\n\nEgg Salad,\n\nFrench Dip,  CALORIE SMART \u2022 SLOW COOKER\n\nHam 'n Egg Biscuit,\n\nHam 'n Egg Breakfast,\n\nHam Salad,\n\nPatty Melts with Smothered Onions,  CALORIE SMART\n\nPork and Coleslaw,\n\nPortabella Muffuletta,  CALORIE SMART \u2022 Fast\n\nPulled Jerk Pork,  SLOW COOKER\n\nReuben and Rachel,\n\nSloppy Joes,  CALORIE SMART \u2022 Fast\n\nTeriyaki Sliders,\n\nTuna Melts, Cheddar,\n\nTuna Salad,\n\nTurkey-Provolone-Dill Party Sub,\n\nVeggie Joes,  CALORIE SMART \u2022 SLOW COOKER\n\nSangria,  CALORIE SMART \u2022 Fast\n\nBerry,\n\nLemon,\n\nRaspberry Pops,\n\nWhite,\n\nSanta Fe Salad with Tortilla Straws,\n\nSatay, Chicken, with Peanut Sauce, 56\u201357\n\nSauce(s). _See also_ Barbecue Sauce; Gravy; Mayonnaise; Pasta Sauce; Pesto; Salsa\n\nAioli, Wasabi, Asian Tuna with,  CALORIE SMART\n\nBasil, Creamy, Chicken and Pasta with,  Fast\n\nBasil, Creamy, Pork and Pasta with,\n\nB\u00e9arnaise,\n\nBlack Bean, Stir-Fry Tofu with, 392\u2013393 CALORIE SMART \u2022 Fast\n\nBrowned Butter, Herbed,\n\nBrowned Butter, \u2013Sage,\n\nCheese,\n\nChimichurri,  EASY\n\nCider, Panfried Pork Chops with,  CALORIE SMART \u2022 Fast\n\nCocktail,\n\nCoconut,\n\ncondiments, 8\u20139\n\nCranberry, Brie in Puff Pastry with,\n\nCranberry, Double-Orange, Brie in Puff Pastry with,\n\nCranberry-Wine, Venison with,  CALORIE SMART\n\nCucumber-Yogurt,\n\nCurry,\n\ndeglazing technique,\n\nDessert. _See_ Dessert Sauce\n\nDill,\n\nDipping, Ginger-Soy,\n\nDipping, Sriracha, for Fries,\n\nDipping, Wasabi, Creamy,\n\nHollandaise,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nHoney-Mustard,\n\nMole Poblano,  CALORIE SMART\n\nMole, Quick, Coffee Chicken with,  CALORIE SMART\n\nMornay,  CALORIE SMART \u2022 Fast\n\nMushroom, Pork with,\n\nMushroom, Vegetables with,\n\nMustard,\n\nMustard-Chive Cream, Pork Chops with,\n\nOrange, Duckling with,\n\npan, making,\n\nPeanut\n\n462 CALORIE SMART \u2022 Fast \u2022 EASY\n\nChicken, Spicy, in,  SLOW COOKER\n\nChicken Satay with, 56\u201357\n\nYogurt, Tempeh Stir-Fry with,  CALORIE SMART\n\nRed Wine, Peppery,\n\nRomesco,\n\nSour Cream,\n\nSour Cream\u2013Dill,  CALORIE SMART \u2022 Fast\n\nSriracha,  CALORIE SMART\n\nTartar,  CALORIE SMART \u2022 EASY\n\nTartar, Curry-Chutney,\n\nTartar, Dill,\n\nthickeners,\n\ntips for,\n\nTomato. _See_ Tomato Sauce\n\nTomato-Pepper, Snapper with,\n\nVelout\u00e9,\n\nWhite,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nThick,\n\nThin,\n\nWhite Wine,\n\nWhite Wine\u2013Garlic,\n\nWine, Steamed Mussels in,  CALORIE SMART \u2022 Fast\n\nSaucepan,\n\nSausage. _See also_ Sausage, Pork\n\nChicken,\n\nChicken, with Potatoes and Cheese, Roasted,  CALORIE SMART\n\nChicken, Spinach and Swiss Pizza, 195\u2013196 Fast\n\nVeggie, Great Northern Bean Cassoulet and,\n\nSausage, Pork. _See also_ Chorizo\n\n-Broccoli Pizza, Easy,\n\nand Chicken with Black Beans, Mexican Cheesy,\n\nChorizo and Beef Burgers with Roasted Chile Mayonnaise,\n\nClams and Linguine, Spanish,  CALORIE SMART\n\nEgg Bake, Tex-Mex,\n\n'n Eggs,\n\ngrilling,\n\nIsraeli Couscous Risotto with Caramelized Onions and,\n\nItalian\n\nLasagna,\n\nLasagna, Easy,\n\nMinestrone with,  CALORIE SMART\n\nPizza, Meat Lover's,\n\nwith Tomatoes and Penne,  CALORIE SMART \u2022 Fast\n\nQuinoa Pizza, with Onion, Pepper and,  CALORIE SMART\n\nRoll-Up, Breakfast,\n\nand Short Rib Rag\u00f9,\n\nStuffing,\n\nWaffles,\n\nSaut\u00e9ed Cauliflower with Browned Bread Crumbs,  Fast\n\nSaut\u00e9 pan,\n\nSavory,\n\nScalloped Potatoes,\n\nScallops\n\nwith Artichokes and Tomatoes,  CALORIE SMART \u2022 Fast\n\nCitrus Seafood Skillet,  CALORIE SMART \u2022 Fast\n\nDeep-Fried Sea,\n\nvarieties of,\n\nScissors, kitchen\n\nScones,\n\nBlueberry-Ginger,\n\nCherry\u2013Chocolate Chip,\n\nChocolate Chip,\n\nCornbread, Cheddar-Chile,\n\nCurrant,\n\nLemon-Cherry,\n\nRaspberry-White Chocolate,\n\nScrambled Eggs, ,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nand Shrimp,\n\nScrubbing brush,\n\nSeafood. _See also_ Fish; Shellfish\n\ndried,\n\nSauce,\n\nSearing method,\n\nSeasoned Brine,  CALORIE SMART \u2022 EASY\n\nSeasoning Mix\n\ncontainers for,\n\nGaram Masala,\n\nHerb, Salt-Free,\n\nJamaican Jerk,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nTaco, Salt-Free,\n\nTaco, Smoky, Salt-Free,\n\nSeasonings. _See_ Herb(s), Herbed; Rub(s); Seasoning Mix; Spice(s)\n\nSeaweed,\n\nSeeds, . _See also specific seeds_\n\nSerrano chiles,\n\nSesame (seeds),\n\nDressing,\n\nNoodle Salad,  CALORIE SMART \u2022 Fast\n\ntoasting,\n\nSesame oil, ,\n\nSeven-Layer Salad, _124,_ 124\n\nSeven-Layer Salad, Fennel-Asparagus,\n\nShakshuka, Classic,  CALORIE SMART\n\nShallots,\n\nGreen Beans with, Glazed,  CALORIE SMART \u2022 Fast\n\nShellfish. See also Clam(s); Crab(s); Lobster(s); Oyster(s); Scallops; Shrimp\n\nbroiling,\n\nbuying tips, ,\n\nCitrus Seafood Skillet,  CALORIE SMART \u2022 Fast\n\ncooking timetable,\n\ndoneness,\n\nimitation products,\n\nMussels, Steamed, in Wine Sauce,  CALORIE SMART \u2022 Fast\n\nQuiche, Seafood,\n\nserving size,\n\nstoring, ,\n\nsustainable,\n\nvarieties of,\n\nShells, Chicken-and Spinach-Stuffed,\n\nSherry Vinaigrette Dressing,\n\nShiitake mushrooms,\n\nShooters, Boston Cream,\n\nShopping strategies,\n\nShortbread Cookie(s),  CALORIE SMART\n\nCherry,\n\nChocolate-Dipped,\n\nGinger,\n\nNut,\n\nToffee,\n\nWedges,\n\nWhole Wheat,\n\nShortcakes, Drop,\n\nShortcakes, Strawberry,\n\nShortening,\n\nShort Rib(s), Beef,\n\nBeer-Braised, Dinner,\n\nand Sausage Rag\u00f9,\n\nShrimp. _See also_ Shellfish\n\nAlfredo Primavera,  CALORIE SMART \u2022 Fast\n\nAngel Hair Pasta with,\n\nCaesar Salad,\n\nCitrus Seafood Skillet,  CALORIE SMART \u2022 Fast\n\nCocktail, Classic,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCocktail Shooters,\n\nCoconut-Curry Grilled, Spicy,  CALORIE SMART\n\nCreole,  CALORIE SMART\n\nDeep-Fried,  CALORIE SMART\n\ndeveining,\n\ndried,\n\nEgg Rolls,\n\nand Egg Salad Wraps,\n\nFajitas,\n\nGumbo,  CALORIE SMART\n\nand Melon Salad,\n\nPad Thai with,\n\nand Pasta, Tomato-Basil,\n\nRice, Fried,\n\nRisotto with, Classic,\n\nScampi,  CALORIE SMART \u2022 Fast\n\nand Scrambled Eggs,\n\nsize and count,\n\nTacos,\n\nwith Tomatoes,\n\nvarieties of,\n\nVermicelli with Fresh Herbs and,\n\nSilicon mats, ,\n\nSilken tofu,\n\nSilver White Cake,\n\nSimmering foods,\n\nSimple Cheese Enchilada,\n\nSimple Syrup,\n\nBerry, Triple-,\n\nDill,\n\nGinger,\n\nHoney-Rosemary,\n\nMint,\n\nSirloin Steak with Smoky Honey Mustard Sauce,\n\nSkillet Beef Nachos,\n\nSkillet Chicken Nachos,  Fast\n\nSkillet Chicken Thighs with Bacon and Spinach,  CALORIE SMART\n\nSkillet Eggs with Summer Squash Hash,  CALORIE SMART\n\nSkillet-Fried Chicken,  CALORIE SMART \u2022 EASY\n\nSkillet Pork Hash,\n\nSkillets,\n\ncast-iron,\n\nSliders, Black Bean,  CALORIE SMART\n\nSliders, Teriyaki,\n\nSloppy Joes,  CALORIE SMART \u2022 Fast\n\nSlow cooker,\n\nfood safety,\n\ntips for,\n\nSlow cooker dishes\n\nApple Butter,  CALORIE SMART\n\nBarbecue Beef Sandwiches, 298\u2013299 CALORIE SMART\n\nBeans, Baked, Homemade,\n\nBeef and Broth,  CALORIE SMART\n\nBeef Roast with Cabbage and Pasta, Asian,\n\nBeef Stew,  CALORIE SMART\n\nBlack-Eyed Peas, Spicy,  CALORIE SMART \u2022 EASY\n\nBulgur and Lentils, Mediterranean,  CALORIE SMART\n\nCabbage Rolls, Tropical Stuffed,\n\nCassoulet, Great Northern Bean and Veggie Sausage,\n\nChicken and Barley Risotto with Edamame,  CALORIE SMART\n\nChicken Cacciatore,  CALORIE SMART\n\nChicken-Lentil Soup, Italian,  CALORIE SMART\n\nChicken in Peanut Sauce, Spicy,\n\nChicken, Rotisserie Spiced,  CALORIE SMART\n\nChicken Stew, Creamy Herbed,  CALORIE SMART\n\nChili, Family Favorite,  CALORIE SMART\n\nCorned Beef Brisket, Spiced, with Horseradish Sour Cream,  CALORIE SMART\n\nFrench Dip Sandwiches,  CALORIE SMART\n\nGarbanzo Beans with Vegetables,\n\nGerman Potato Salad, Hot,  CALORIE SMART\n\nGrain Medley, Three-,  CALORIE SMART\n\nLamb Dijon,  CALORIE SMART\n\nPear-Orange Conserve,  CALORIE SMART\n\nPears, Caramel-Maple,  EASY\n\nPorketta Pot Roast,  CALORIE SMART\n\nPork Loin Ribs, Spicy Barbecue, 310\u2013311\n\nPork Ribs with Smoky Barbecue Sauce,\n\nPork Stew, Hearty,  CALORIE SMART\n\nPotato-Bacon Soup,\n\nPotatoes, Smashed, Ultimate,  CALORIE SMART\n\nPot Roast,\n\nPulled Jerk Pork Sandwiches,\n\nRed Beans and Rice,  CALORIE SMART\n\nRopa Vieja Tostadas,  CALORIE SMART\n\nTacos, Smoky Chipotle Soft,  CALORIE SMART\n\nTomato Sauce, Italian,  CALORIE SMART\n\nTurkey and Wild Rice Casserole, Herbed,  CALORIE SMART\n\nVeggie Joes,  CALORIE SMART\n\nWhite Beans with Sun-Dried Tomatoes,  EASY\n\nSlush, Strawberry-Lemonade,\n\nSmoked dishes\n\nBeef Brisket,  CALORIE SMART\n\nChicken, Cajun,  CALORIE SMART\n\nPork Chops with Apple and Sweet Potato,  CALORIE SMART\n\nRibs, Hot and Spicy,\n\nSalmon, Brined,  CALORIE SMART\n\nSmoking\n\non grill,\n\nsafety tips for,\n\ntypes of smokers,\n\nSmoky Barbecue Sauce,\n\nSmoky Chipotle Soft Tacos,  CALORIE SMART \u2022 SLOW COOKER\n\nSmoky Salt-Free Taco Seasoning Mix,\n\nSmoothies\n\nBerries and Greens, Fresh,\n\nBlueberry,\n\nGreen, Easy-Being-,  Fast \u2022 EASY\n\nLemonade,\n\nOrange,\n\nOrange-Pineapple,\n\nPeach,\n\nRaspberry,\n\nStrawberry,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nStrawberry-Banana,\n\nSnapper with Tomato-Pepper Sauce,\n\nSnickerdoodle(s), _529,_ 529\n\nCake, Caramel,\n\nFrench Toast,\n\nSoft No-Roll Sugar Cookies,  CALORIE SMART\n\nSoft Pretzels,  CALORIE SMART\n\nSole Amandine,  CALORIE SMART \u2022 Fast\n\nOrange,\n\nSorbet\n\nCranberry-Raspberry,\n\nLemonade,  CALORIE SMART \u2022 EASY\n\nLimeade,\n\nOrange,\n\nPineapple,\n\nSorrel,\n\nSouffl\u00e9 dish,\n\nSoup(s). _See also_ Broth; Chili; Chowder\n\nBeef Pho,  CALORIE SMART\n\nBlack Bean, Cuban,  CALORIE SMART\n\nBorscht, 146\u2013147 CALORIE SMART\n\nBroccoli, Cream of,  CALORIE SMART\n\nButternut Squash,\n\nCauliflower, Cream of,\n\nChicken\n\n-Lentil, Italian,  CALORIE SMART \u2022 SLOW COOKER\n\nNoodle,  CALORIE SMART\n\nNoodle, Confetti,\n\nRice,\n\nTortilla, ,  CALORIE SMART\n\nfat, skimming\n\nFish, Proven\u00e7al,  CALORIE SMART \u2022 Fast\n\nGazpacho,  CALORIE SMART\n\nGazpacho, Green,\n\nGazpacho, Sriracha,\n\nMinestrone with Italian Sausage,  CALORIE SMART\n\nMinestrone, Meatless,\n\nMushroom, Chicken-,\n\nMushroom, Cream of,\n\nOnion, French,  CALORIE SMART\n\nOnion, Golden,\n\nPotato-Bacon, Cheesy,  SLOW COOKER\n\nRed Pepper, Roasted, with Mozzarella, , CALORIE SMART\n\nSplit Pea,  CALORIE SMART\n\ntips for,\n\nTomato, -Basil,  CALORIE SMART\n\nTomato-Bean, Chunky,\n\nVegetable Beef,  CALORIE SMART\n\nWild Rice,  CALORIE SMART\n\nWild Rice, Chicken and,\n\nSour Cream,\n\n\u2013Chocolate Fondue,  Fast\n\nCoffee Cake, 494\u2013495\n\n\u2013Dill Sauce,  CALORIE SMART \u2022 Fast\n\nGreen Salsa Crema,\n\nHorseradish, Spiced Corned Beef Brisket with,  CALORIE SMART \u2022 SLOW COOKER\n\n\u2013Nectarine Pancakes,\n\nSauce,\n\nSeasoned,\n\nSnickerdoodle Cake, Caramel,\n\nSpice Cake,\n\nstoring,\n\nTart Flamb\u00e9e,  CALORIE SMART\n\nWasabi Dipping Sauce, Creamy,\n\nSourdough Bread,  CALORIE SMART\n\nSourdough starter,\n\nSouth American cuisine,\n\nSouthwest(ern)\n\nCashews, Spiced,  Fast\n\nMacaroni and Cheese,\n\nParty Nuts, Spiced,  Fast\n\nPork Chops, Spicy,\n\nPork Packets,  CALORIE SMART\n\nPork Ribs with Smoky Barbecue Sauce,\n\nRub,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSteak with Corn and Chiles,  CALORIE SMART \u2022 Fast\n\nTabbouleh,\n\nSoybean oil,\n\nSoybeans. _See_ Edamame; Tofu\n\nSoymilk, ,\n\nLatte, Iced Vanilla,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSoy Peanutty Dressing,\n\nSoy products, . _See also_ Tofu\n\nSoy protein, textured, . _See also_ Veggie Crumbles\n\nin Lasagna Cupcakes,  CALORIE SMART\n\nin Moussaka,  CALORIE SMART\n\nSausage, Veggie, and Great Northern Bean Cassoulet,  SLOW COOKER\n\nSoy sauce,\n\nSpaetzle,\n\nSpaghetti\n\nBolognese, Pasta,\n\nCacio e Pepe,  Fast\n\nCarbonara,  Fast\n\nCarbonara, Chicken and,\n\nChicken,\n\nand Meatball Pizza,\n\nand Meatballs,\n\nand Meat Sauce,\n\nPuttanesca,\n\nPuttanesca, Pepper, Roasted,\n\nand Rice Balls, Spicy,  Fast\n\nwith White Clam Sauce,  Fast\n\nSpanish Clams, Sausage and Linguine,  CALORIE SMART\n\nSpanish Rice,  CALORIE SMART\n\nwith Bacon,\n\nSpareribs, Pork,\n\nSparkling Lemonade,\n\nSpatchcocked Chicken, Herb-Garlic Butter,  CALORIE SMART\n\nSpatulas\/scrapers,\n\nSpelt,\n\nSpice(s), . _See also_ Rub(s); Seasoning Mix\n\nAsian,\n\nCake, Sour Cream,\n\ngaram masala,\n\nIndian,\n\nMediterranean,\n\nMiddle Eastern,\n\nseasoning with,\n\nSouth American, ,\n\nvarieties of,\n\nSpiced Apple Cake with Maple Glaze,\n\nSpiced Apricot-Ginger Preserves,  CALORIE SMART \u2022 EASY\n\nSpiced Corned Beef Brisket with Horseradish Sour Cream,  CALORIE SMART \u2022 SLOW COOKER\n\nSpiced Peach Jam,\n\nSpicy Almond-Cilantro Pesto,\n\nSpicy Barbecue Pork Loin Ribs, 310\u2013311\n\nSpicy Black-Eyed Peas,  CALORIE SMART \u2022 SLOW COOKER *EASY\n\nSpicy Chicken in Peanut Sauce,  SLOW COOKER\n\nSpicy Cider Syrup,\n\nSpicy Coconut-Curry Grilled Shrimp,  CALORIE SMART\n\nSpicy Collard Greens with Bacon,  CALORIE SMART\n\nSpicy Pickled Carrots,\n\nSpicy Pickled Vegetables,  CALORIE SMART\n\nSpicy Pumpkin Cookies with Browned Butter Frosting,\n\nSpicy Refried Black Beans,  CALORIE SMART \u2022 Fast\n\nSpicy Samosa Patties,  CALORIE SMART\n\nSpicy Southwestern Pork Chops,\n\nSpicy Thai Pork Stew,\n\nSpicy Vegetable Dip,\n\nSpinach,\n\n-Artichoke Dip, Hot,\n\nChicken Sausage and Swiss Pizza, 195\u2013196 Fast\n\nChicken Thighs with Bacon and,  CALORIE SMART\n\nChicken with Tomatoes and, Easy,  CALORIE SMART\n\nand Chicken-Stuffed Shells,\n\nEggs Benedict,\n\nFettuccine Alfredo with Baby Spinach,\n\nand Kale, Wilted,\n\nLasagna, -Artichoke,\n\nManicotti,\n\nPesto,\n\nQuiche,\n\nQuiche, and Cheddar,  CALORIE SMART\n\nSalad, -Bacon, with Hard-Cooked Eggs,  CALORIE SMART \u2022 Fast\n\nSmoothies, Easy-Being-Green,  Fast \u2022 EASY\n\nVermicelli with Fresh Herbs, Bacon and,\n\nWilted,  CALORIE SMART \u2022 Fast\n\nSpiral Summer Squash,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSpirits, cocktail,\n\nSplit Pea Soup,  CALORIE SMART\n\nSponge cakes,\n\nSpoons,\n\nSpread(s). _See also_ Cheese Ball(s); Cream Cheese, Spread(s); Dip(s)\n\nBacon-Cheese,\n\nTapenade, Fig-Hazelnut,  CALORIE SMART \u2022 Fast\n\nTapenade, Olive,  CALORIE SMART \u2022 Fast\n\nSpringform pan,\n\nSpritz Cookies,  CALORIE SMART\n\nChocolate Buttery,\n\nSquash, Summer. _See_ Summer Squash; Zucchini\n\nSquash, Winter. _See_ Butternut Squash; Winter Squash\n\nSquid,\n\nCitrus Seafood Skillet,  CALORIE SMART \u2022 Fast\n\nSriracha,\n\nDipping Sauce for Fries,\n\nGazpacho,\n\nMiso Dressing,\n\nSauce,  CALORIE SMART\n\nStar anise, ,\n\nStarlight Yellow Cake,\n\nSteak(s)\n\nBalsamic- and Roasted Garlic\u2013Marinated, Grilled,  CALORIE SMART\n\nChicken-Fried,\n\nwith Corn and Chiles, Southwestern,  CALORIE SMART \u2022 Fast\n\ndoneness,\n\nFlank, Grilled Jerk,  CALORIE SMART\n\nFlank, Peppered,  CALORIE SMART\n\nFlank, with Smoky Honey Mustard Sauce,  CALORIE SMART\n\nKabobs, Easy Grilled,  CALORIE SMART\n\nPan, Brandy-Herb,  CALORIE SMART \u2022 Fast\n\nPan, Mushroom Brandy-Herb,\n\npreparing for grill\n\nSirloin, with Smoky Honey Mustard Sauce,\n\nT-Bones, Texas,  CALORIE SMART\n\nSteamed Clams,  CALORIE SMART\n\nSteamed Edamame in the Shell with Dipping Sauces,  CALORIE SMART \u2022 Fast\n\nSteamed Mussels in Wine Sauce,  CALORIE SMART \u2022 Fast\n\nStew(s)\n\nBeef, 150\u2013151 CALORIE SMART\n\nChicken, Creamy Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\nLamb Dijon,  CALORIE SMART \u2022 SLOW COOKER\n\nLamb, Moroccan,  CALORIE SMART\n\nLentil, Indian,  CALORIE SMART\n\nOyster,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nPork, Hearty,  CALORIE SMART \u2022 SLOW COOKER\n\nPork, Thai, Spicy,\n\ntips for,\n\nTurkey and Butternut Squash Ragout,  CALORIE SMART \u2022 SLOW COOKER\n\nSticky Rice with Mango,  CALORIE SMART\n\nSticky Rolls, Caramel,\n\nStir-Fry(ied)\n\nEdamame with Corn and Peppers,\n\nPepper and Soybean,\n\nTempeh, with Noodles,\n\nTempeh, with Yogurt Peanut Sauce,  CALORIE SMART\n\ntofu,\n\nTofu with Black Bean Sauce, 392\u2013393 CALORIE SMART \u2022 Fast\n\nStock,\n\nStorage, kitchen,\n\nStrainers\/straining,\n\nStrata, Ham and Cheddar, 96\u201397 CALORIE SMART\n\nStrawberry(ies),\n\n-Asparagus Salad,  CALORIE SMART \u2022 Fast\n\n-Basil Water,\n\nCake,\n\nCream Brunch Cake,\n\nand Cream Tart,\n\nIce Cream, Fresh,\n\nJam, Freezer,  CALORIE SMART \u2022 EASY\n\nJam, Freezer, -Balsamic,\n\nKefir,\n\nKefir, -Banana,\n\n-Lemonade Slush,\n\nMargaritas, Frozen,\n\nMilk, Fresh,  Fast \u2022 EASY\n\n-Peach Salsa,\n\n-Poppy Dressing, Creamy,\n\nPops,\n\n-Rhubarb Pie,\n\n-Rhubarb Sauce,\n\nSauce,\n\nShortcakes,\n\nSmoothies,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSmoothies, -Banana,\n\nStraws, Tortilla,\n\nSanta Fe Salad with,\n\nStreusel\n\nDoughnuts, Blueberry-Orange, Baked,\n\nMocha Ribbon, Espresso Cake with,\n\nMuffins, Blueberry, -Topped,\n\nMuffins, Golden Harvest,\n\nOatmeal Bread,\n\nStroganoff, Beef,  CALORIE SMART\n\nStroganoff, Ground Beef,\n\nStuffed-Crust Pizza,  CALORIE SMART\n\nStuffing\n\nApple, Roast Goose with,\n\nBread, 346\u2013347\n\nCornbread,\n\nCornbread-and Bacon-Stuffed Pork Chops,\n\nFruited Supreme, Pork Crown Roast with,\n\nOyster,\n\nSausage,\n\nWild Rice,\n\nSugar, , . _See also_ Brown Sugar\n\ncaramelizing,\n\nEspresso,\n\nSugar Cookies,  CALORIE SMART\n\nFrosted and Decorated,\n\nPaintbrush,\n\nSoft No-Roll,  CALORIE SMART\n\nSoft No-Roll, Chocolate Chip, Cherry and Pistachio,\n\nSoft No-Roll, Glazed,\n\nSugared Almonds,\n\nSumac,\n\nSummer Quinoa-Tomato Salad,  CALORIE SMART\n\nSummer Squash, . _See also_ Zucchini\n\nand Halibut Packets,  CALORIE SMART\n\nHash, Skillet Eggs with,  CALORIE SMART\n\npreparing\/cooking,\n\nSpiral,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSummery Salad,\n\nSun-Dried Tomato(es),\n\nHummus,\n\nOmelet,\n\nPesto,\n\nPesto, Branzino with,\n\nWhite Beans with, Slow-Cooker,  EASY\n\nSunflower-Herb Whole Wheat Bread,\n\nSunflower Salad, Sunny,\n\nSunny Sunflower Salad,\n\nSushi rice,\n\nSweet-and-Savory Yogurt,\n\nSweet Chile Kabobs,\n\nSweetened Ice Coffee,\n\nSweetened Whipped Cream,\n\nSweet Martini Cocktails,\n\nSweet Onion\u2013Topped Caesar Burgers,  Fast\n\nSweet Pea\u2013Mint Pesto,\n\nSweet Pepper(s), . _See also_ Bell Pepper(s)\n\nSweet Peppery Pecans,\n\nSweet Potato(es),\n\nBiscuits, -Bacon,  CALORIE SMART\n\nCandied,  EASY\n\n\u2013Cauliflower Curry,  CALORIE SMART\n\n\u2013Cranberry Bread,\n\nFries, Loaded,\n\nMeringue Pie, Roasted,\n\nPie, Roasted,\n\nPork Chops, Smoked, with Apple and Sweet Potato,  CALORIE SMART\n\npreparing\/cooking,\n\nSweet-Savory Barbecue Sauce,\n\nSweet-Sour\n\nColeslaw,  CALORIE SMART\n\nDressing,\n\nRed Cabbage,  CALORIE SMART\n\nSweet-Spicy Chicken Wings,\n\nSweet Tomato Jam,  CALORIE SMART\n\nSwiss, Chicken Sausage and Spinach Pizza, 195\u2013196 Fast\n\nSyrup. _See also_ Simple Syrup\n\nButter-Pecan,\n\nCider, Spicy,\n\nmaple,\n\nin microwave,\n\n## T\n\nTabbouleh\n\nBarley, Cauliflower and Red Lentil,  CALORIE SMART\n\nClassic,  CALORIE SMART\n\nFruited, with Walnuts and Feta,  CALORIE SMART\n\nGarbanzo Bean,\n\nMinty,\n\nSouthwestern,\n\nTable settings, ,\n\nTaco(s)\n\nCasserole, Fiesta,\n\nChipotle, Smoky, Soft,\n\nDip, Personal,\n\nFish, Cajun, Soft, 366\u2013367 Fast\n\nFish, Grilled,\n\nPork Carnitas Taco Bites,\n\nSalad,\n\nSalsa,\n\nSeasoning Mix, Salt-Free,\n\nSeasoning Mix, Smoky, Salt-Free,\n\nShrimp,\n\nTagine, Chicken,  CALORIE SMART\n\nTahini,\n\nTamari,\n\nTamarind,\n\nTangelo,\n\nTangy Coleslaw,\n\nTapenade, Fig-Hazelnut,  CALORIE SMART \u2022 Fast\n\nTapenade, Olive,  CALORIE SMART \u2022 Fast\n\nTarragon,\n\nCarrots, Pickled Baby,  CALORIE SMART \u2022 EASY\n\nTart(s)\n\nApple, Easy,\n\nTart(s) _(cont.)_\n\nBerry Crumble, Mixed,\n\nCaramel, Tiny,\n\nChocolate, Bittersweet, \u2013Salted Cashew,\n\nChocolate Raspberry, White and Dark Chocolate, 600\u2013601\n\nFlamb\u00e9e,  CALORIE SMART\n\nLemon Mascarpone,\n\nPastry, Press-in-the Pan,  Fast \u2022 EASY\n\nShells, Baked,\n\nStrawberries and Cream,\n\nTartar Sauce,  CALORIE SMART \u2022 EASY\n\nCurry-Chutney,\n\nDill,\n\nTart pan,\n\nT-Bones, Texas,  CALORIE SMART\n\nTea,\n\nBerry,\n\nbrewing,\n\nChai,  CALORIE SMART \u2022 Fast\n\nChai Mix,  CALORIE SMART \u2022 Fast\n\nDill,\n\nGinger,\n\nGreen, Apple-Mint Iced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nGreen, Martini Cocktails,\n\niced, easy,\n\nLemonade Sweet,\n\nRaspberry-Ginger,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nSun, Orange\u2013Lemon Balm,\n\nvarieties of,\n\nTeff,\n\nTempeh,\n\nStir-Fry with Noodles,\n\nStir-Fry with Yogurt Peanut Sauce,  CALORIE SMART\n\nTequila\n\nin Beergaritas,  Fast \u2022 EASY\n\nin Margaritas,  Fast\n\nTeriyaki\n\nMarinade,\n\nSliders,\n\nTofu Noodles,  CALORIE SMART\n\nTofu Noodles, and Peppers,\n\nTexas T-Bones,  CALORIE SMART\n\nTex-Mex Egg Bake,\n\nTex-Mex Scramble,  CALORIE SMART \u2022 Fast\n\nTextured soy protein. _See_ Soy protein, textured,\n\nThai(-Style)\n\nCoconut Chicken,  CALORIE SMART\n\nPork Stew, Spicy,\n\nRed Curry Paste,  CALORIE SMART\n\nThai chiles, ,\n\nThermometers,\n\nThickeners, sauce,\n\nThousand Island Dressing,  CALORIE SMART \u2022 Fast\n\nThree-Grain Medley,  CALORIE SMART \u2022 SLOW COOKER\n\nThumbprint Cookies. _See_ Cookies, Thumbprint(s)\n\nThyme,\n\nBeef, Roast, with Orange and,  CALORIE SMART\n\n-Blackberry Cocktail,\n\n-Chicken Penne,\n\nOrange Marinade,  CALORIE SMART \u2022 EASY\n\nTomatoes with Chicken,\n\nTilapia with Lemon Potatoes,\n\nTilapia and Vegetables, Roasted,  CALORIE SMART\n\nTimer,\n\nTiny Caramel Tarts,\n\nTiramisu,\n\nToasted Almond Pound Cake,\n\nToasted Coconut\u2013Brown Sugar Refrigerator Cookies,\n\nToaster\/toaster oven,\n\nToasts. _See also_ Crostini; Croutons; French Toast\n\nPita Crisps, Baked,\n\nWonton Crisps,\n\nToffee\n\nShortbread Cookies,\n\nTruffles,\n\nWhoopie Pies,\n\nTofu\n\nbaking,\n\nmaking,\n\npreparing,\n\n-Rice Skillet, Italian,\n\n-Rice Skillet, Mexican,  CALORIE SMART \u2022 Fast\n\nStir-Fry with Black Bean Sauce, 392\u2013393 CALORIE SMART \u2022 Fast\n\nstir-frying,\n\nstoring,\n\nTeriyaki Noodles,  CALORIE SMART\n\nTeriyaki Noodles and Peppers,\n\nvarieties of,\n\nWraps, Mexican,\n\nTomatillos, ,\n\nSalsa Verde,  CALORIE SMART \u2022 Fast\n\nTomato(es), . _See also_ Sun-Dried Tomato(es); Tomato Sauce\n\nAngel Hair Pasta with Basil, Avocado and,  Fast\n\nBlack Bean Burgers with Cheese and, California,\n\nBloody Mary, Classic,  CALORIE SMART \u2022 Fast\n\nCajun, with Black-Eyed Peas,\n\ncanned, using,\n\nCannellini with Sage and,  CALORIE SMART \u2022 Fast\n\nCaprese Salad, Heirloom,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nCheese Sandwiches, Grilled, and Avocado,\n\nwith Chicken, Thyme,\n\nChicken with Spinach and, Easy,  CALORIE SMART\n\nand Chickpea Curry,  CALORIE SMART \u2022 Fast\n\nCrostini, -Basil,  CALORIE SMART \u2022 Fast\n\nFried Green,  CALORIE SMART\n\nGazpacho,  CALORIE SMART\n\nGazpacho, Green,\n\nGazpacho, Sriracha,\n\nItalian Sausage with Penne and,  CALORIE SMART \u2022 Fast\n\nJam, Sweet,  CALORIE SMART\n\nKetchup,  CALORIE SMART\n\nMacaroni and Cheese, Fire-Roasted,\n\nMacaroni and Cheese, Kale, Bacon and,\n\nOkra and Corn,  CALORIE SMART \u2022 Fast\n\n-Olive Relish,\n\nOven-Dried,  CALORIE SMART \u2022 EASY\n\nPizza, Fresh,  CALORIE SMART\n\nPizza, Stuffed-Crust Deluxe,\n\n-Quinoa Salad, Summer,  CALORIE SMART\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRatatouille Salad,  CALORIE SMART\n\nwith Ravioli,\n\nSalsa,  CALORIE SMART\n\nSalsa, and Black Bean,\n\nSalsa, Black Bean,\n\nScallops with Artichokes and,  CALORIE SMART \u2022 Fast\n\nShrimp and Pasta, -Basil,\n\nShrimp with,\n\nSoup, -Basil,  CALORIE SMART\n\nSoup, -Bean, Chunky,\n\nTomato Sauce\n\nBolognese,  CALORIE SMART\n\nBolognese Pasta,\n\nItalian,  CALORIE SMART\n\nwith Meatballs, Italian,\n\n-Pepper, Snapper with,\n\nQuick Skillet Pasta,\n\n-Vodka, Creamy,  CALORIE SMART \u2022 Fast\n\n-Vodka, Creamy Basil-Chicken,\n\n-Vodka, Creamy Olive,\n\nTongs,\n\nTools. _See_ Equipment and tools\n\nTortilla(s), . _See also_ Nachos; Quesadillas; Taco(s)\n\nBeef and Kasha,\n\nBurritos, Breakfast,\n\nBurritos, Salsa\u2013Brown Rice,\n\nChicken Soup, ,  CALORIE SMART\n\nEnchilada, Cheese, Simple,\n\nEnchiladas, Cheese,\n\nEnchiladas Verde, Cheese,\n\nFajitas, Beef,\n\nFajitas, Chicken,\n\nFajitas, Shrimp,\n\nGreen Chili,  CALORIE SMART \u2022 Fast\n\nHuevos Rancheros,\n\nNachos,  Fast\n\nRoll-Up(s)\n\nBean and Avocado,\n\nChicken, Asian,  CALORIE SMART \u2022 Fast\n\nSausage, Breakfast,\n\nStraws,\n\nStraws, Santa Fe Salad with,\n\nWraps, Shrimp and Egg Salad,\n\nTostadas, Ropa Vieja,  CALORIE SMART \u2022 SLOW COOKER\n\nTowels, kitchen,\n\nTres Leches Cake,\n\nTropical,\n\nTriangles, Basil-Cheese, _59,_ 59 CALORIE SMART\n\nTrifle(s), English,\n\nIndividual,\n\nTriple-Berry Oatmeal-Flax Muesli,  CALORIE SMART\n\nTriple-Berry Pomegranate Freezer Jam,  CALORIE SMART\n\nTriple-Chocolate and Candy Fudge,\n\nTriple-Chocolate Fudge,\n\nTriple-Nut Bars,\n\nTropical Creamy Coleslaw,\n\nTropical fruits,\n\nTropical Stuffed Cabbage Rolls,  SLOW COOKER\n\nTropical Tres Leches Cake,\n\nTruffles\n\nChocolate, Luscious,  CALORIE SMART\n\nToffee,\n\nWhite Chocolate\u2013Chocolate,\n\nTuna\n\nChopped Salad, Summer,\n\n-Lettuce Wraps,\n\nMelts, Cheddar,\n\nQuesadillas,\n\nSalad, Mediterranean,\n\nSalad Sandwiches,\n\nSteak Packets, Dilled,  CALORIE SMART\n\nTopping, Baked Potatoes with,\n\ntypes of,\n\nwith Wasabi Aioli, Asian,  CALORIE SMART\n\nTurkey. _See also_ Poultry\n\nBreast with Chile-Cumin Rub,  CALORIE SMART\n\nBrined, Breast, Apple-Maple,\n\nBrined Whole,\n\nbrining time,\n\nbroiling, ,\n\nBurgers, Garlic, Moist,\n\nand Butternut Squash Ragout,  CALORIE SMART \u2022 SLOW COOKER\n\nChopped Salad, -Cheese,\n\nLettuce Wraps, Quick,\n\nManicotti,\n\nPenne, -Basil,\n\n-Provolone-Dill Party Sub,\n\nRoast,  CALORIE SMART\n\nroasting,\n\nand Wild Rice Casserole, Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\n\u2013Wild Rice Salad, Raspberry,\n\nyields,\n\nTurmeric, ,\n\nTurnips, ,\n\nTurtle Layer Cake,\n\nTuscan Rosemary Chicken and White Beans,  CALORIE SMART \u2022 Fast\n\nTwice-Baked Potatoes,  CALORIE SMART\n\nTwo-Bean Salad,\n\nTzatziki,  CALORIE SMART \u2022 EASY\n\n## U\n\nUltimate Smashed Potatoes,  CALORIE SMART \u2022 SLOW COOKER\n\nUpside-Down Banana-Walnut French Toast,\n\nUpside-Down Cake, Pear-Ginger,\n\nUrfa Biber,\n\n## V\n\nVanilla\n\nFrosting, Creamy,  Fast \u2022 EASY\n\nGlaze,  Fast \u2022 EASY\n\nIce Cream,  EASY\n\nPudding,  EASY\n\nSoy Latte, Iced,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nVeal\n\nbroiling,\n\ncuts of\n\nOsso Buco,\n\npanfrying,\n\nParmigiana,  CALORIE SMART\n\nroasting,\n\nVegan diet,\n\nVegetable(s). _See also specific vegetables_\n\nBarley with Mixed Vegetables,  CALORIE SMART\n\nblanching,\n\nbroiling,\n\nBroth,  CALORIE SMART\n\nbroth, freezing,\n\nBrown Rice, Curried Veggie,  Fast\n\nbuying,\n\nCasserole, Indian-Style,  CALORIE SMART\n\nand Chicken, Herb-Roasted,  CALORIE SMART\n\ncooking chart, 234\u2013237\n\nDip, Roasted, with Baked Pita Crisps,  CALORIE SMART\n\nDip, Spicy,\n\nand Edamame Salad,\n\nfermenting,\n\nand Fettuccine Parmesan,  CALORIE SMART \u2022 Fast\n\nFood Color, Natural Homemade,\n\nfreezing\/thawing,\n\nGarbanzo Beans with,  SLOW COOKER\n\ngrilling,\n\nhealth benefits of,\n\njuicing,\n\nLasagna, Garden,\n\nlocal sources for,\n\nMinestrone with Italian Sausage,  CALORIE SMART\n\nMinestrone, Meatless,\n\nMushroom Sauce with,\n\nPaella,  CALORIE SMART \u2022 Fast\n\nPaella, Chicken,\n\nPickled, Spicy,  CALORIE SMART\n\nPizza, Veggie Lover's, with Cornmeal Crust,  CALORIE SMART\n\nPolenta,\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRatatouille Salad,  CALORIE SMART\n\nRelish, Crunchy Veggie,  CALORIE SMART\n\nVegetable(s) _(cont.)_\n\nRoasted,  CALORIE SMART\n\nRoasted, Pork Tenderloin with,  CALORIE SMART\n\nRoasted Root,\n\nroasting,\n\nroot vegetables,\n\nSoup, Beef,  CALORIE SMART\n\nstoring,\n\nTilapia and, Roasted,  CALORIE SMART\n\nvarieties of, 224\u2013233\n\nWild Rice and Bacon Skillet,  CALORIE SMART\n\nVegetable oil, ,\n\nVegetable peeler,\n\nVegetarian diet,\n\nVegetarian dishes. _See also_ Soy protein, textured; Tofu; Veggie Crumbles\n\nBurgers, Black Bean, California,  CALORIE SMART \u2022 Fast\n\nBurgers, Black Bean, with Tomato and Cheese, California,\n\nBurgers, Veggie and Bean,  Fast\n\nCassoulet, Great Northern Bean and Veggie Sausage,  SLOW COOKER\n\nCauliflower Steaks with Olive-Tomato Relish,\n\nCheese Sandwiches, Grilled,  Fast\n\nCheese Sandwiches, Grilled, Pesto-Parmesan,\n\nCheese Sandwiches, Grilled, Tomato and Avocado,\n\nCurry, Chickpea and Tomato,  CALORIE SMART \u2022 Fast\n\nCurry, Sweet Potato\u2013Cauliflower,  CALORIE SMART\n\nEggplant Parmigiana,\n\nEggplant Parmigiana, -Olive,\n\nEggplant Parmigiana, -Ravioli,\n\nEnchiladas, Cheese,\n\nEnchiladas Verde, Cheese,\n\nFalafel (Fava Bean Rounds),  CALORIE SMART\n\nFrittata, Baked Buffalo,  CALORIE SMART\n\nLasagna, Artichoke-Spinach,\n\nLasagna Cupcakes,  CALORIE SMART\n\nLasagna, Mediterranean,\n\nMoussaka,  CALORIE SMART\n\nPepper and Soybean Stir-Fry,\n\nPilaf, Barley and Black Bean,  CALORIE SMART\n\nPortabella Muffuletta Sandwiches,  CALORIE SMART \u2022 Fast\n\nPortabellas, Asian Stuffed,  CALORIE SMART\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRigatoni Bake, Veggie,\n\nRisotto, Barley, with Edamame,\n\nSamosa Patties, Spicy,  CALORIE SMART\n\nSpaghetti and Spicy Rice Balls,  Fast\n\nTacos, Chipotle, Smoky, Soft,  CALORIE SMART \u2022 SLOW COOKER\n\nTempeh Stir-Fry with Noodles,\n\nTempeh Stir-Fry with Yogurt Peanut Sauce,  CALORIE SMART\n\nVegetable Casserole, Indian-Style,  CALORIE SMART\n\nVeggie Joes,  CALORIE SMART \u2022 SLOW COOKER\n\nVeggie and Bean Burgers,  Fast\n\nVeggie and Bean \"Meatballs\",\n\nVeggie Crumbles\n\nBarbecue Sandwiches,\n\ncooking,\n\nEggs and,\n\nPizza, Veggie,\n\nQuesadillas, Black Bean,\n\nTaco Salad,\n\nVeggie Joes,  CALORIE SMART \u2022 SLOW COOKER\n\nVeggie Joes,  CALORIE SMART \u2022 SLOW COOKER\n\nVeggie Lover's Pizza with Cornmeal Crust,  CALORIE SMART\n\nVeggie Rigatoni Bake,\n\nVeggie-Stuffed Potatoes,\n\nVelout\u00e9 Sauce,\n\nVenison with Cranberry-Wine Sauce,  CALORIE SMART\n\nVermicelli with Fresh Herbs,  Fast\n\nand Cannellini,\n\nOlives and Fresh Mozzarella,\n\nand Shrimp,\n\nSpinach and Bacon,\n\nVery Berry Frozen Yogurt,  CALORIE SMART \u2022 EASY\n\nVietnamese Beef Lettuce Wraps, , _56_ CALORIE SMART\n\nVietnamese Iced Coffee,\n\nVinaigrette Dressing\n\nBasil,\n\nBlue Cheese Herb,\n\nCreamy,\n\nHerb,\n\nHerb, Fresh,  CALORIE SMART \u2022 Fast\n\nHerb, Fresh, Creamy,\n\nLemon,\n\nPeach, Spicy,\n\nRaspberry,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nRed Wine,\n\nSherry,\n\nVinegar, . _See also_ Balsamic\n\nBerry,\n\ncontainers for,\n\nGarlic,\n\nHerb,  CALORIE SMART \u2022 EASY\n\nLemon,\n\ntypes of,\n\nVodka\n\nBlackberry-Thyme Cocktail,\n\nBloody Mary, Classic,  CALORIE SMART \u2022 Fast\n\nMartini Cocktails\n\nChocolate,\n\nGreen Tea,\n\nMango,\n\nPomegranate,\n\n-Tomato Sauce, Creamy,  CALORIE SMART \u2022 Fast\n\n## W\n\nWaffles, 104\u2013105 CALORIE SMART\n\nSausage,\n\nWhole Wheat, with Honey\u2013Peanut Butter Drizzle,\n\nWalnut(s)\n\nBread, Dried Cherry and,\n\nCream Cheese Spread, -Honey,\n\nCream Cheese Spread, -Olive,\n\nFrench Toast, Banana-, Upside-Down,\n\nKale Salad, -Cherry,  CALORIE SMART \u2022 Fast\n\nPesto, Basil, Roasted Red Pepper and,\n\nTabbouleh, Fruited, with Feta and,  CALORIE SMART\n\nToasted, Gorgonzola Linguine with,  Fast\n\nWasabi Aioli, Asian Tuna with,\n\nWasabi Dipping Sauce, Creamy,\n\nWater bath canning,\n\nWatercress,\n\nWatermelon Granita,  CALORIE SMART\n\nWheat Berry(ies),\n\ncooking timetable,\n\nSalad,  CALORIE SMART\n\nThree-Grain Medley,  CALORIE SMART \u2022 SLOW COOKER\n\nWhipped Cream\n\nGinger,\n\nJelly Roll,\n\nPuffs,\n\nSweetened,\n\nWhipping cream,\n\nWhisk,\n\nWhiskey, in Manhattan Cocktails,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nWhiskey Sauce,\n\nWhite Bean(s)\n\nRosemary Chicken and, Tuscan,  CALORIE SMART \u2022 Fast\n\nSalad, Northern Italian,  CALORIE SMART\n\nwith Sun-Dried Tomatoes, Slow-Cooker,\n\nWhite Bread, Classic,  CALORIE SMART\n\nWhite Chicken Chili,  CALORIE SMART\n\nWhite Chocolate,\n\nChocolate Raspberry Tart, White and Dark Chocolate,\n\n\u2013Chocolate Truffles,\n\nFrosting,\n\nGlaze,\n\nScones, Raspberry-,\n\nWhite Clam Sauce, Linguine with Basil and,\n\nWhite Clam Sauce, Spaghetti with,  Fast\n\nWhite Cupcakes,\n\nWhite Sauce,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nThick,\n\nThin,\n\nWhite tea,\n\nWhite Wine. _See also_ Mimosas\n\nBellini Cocktails, Easy,\n\n\u2013Garlic Sauce,\n\nSangria, White,\n\nSauce,\n\nSauce, Steamed Mussels in,  CALORIE SMART \u2022 Fast\n\nvarietals,\n\nWhole Grain Artisan Bread,  CALORIE SMART\n\nWhole-Grain Pancakes,\n\nWhole grains,\n\nWhole Wheat\n\nBatter Bread, \u2013Raisin,\n\nBread, Honey\u2013, 504\u2013505 CALORIE SMART\n\nBread, Sunflower-Herb,\n\nflour,\n\nPasta,\n\nPizza Crust,\n\nShortbread Cookies,\n\nWaffles with Honey\u2013Peanut Butter Drizzle,\n\nWhoopie Pies,\n\nChocolate Chip,\n\nPeppermint, Pink,\n\nToffee,\n\nWild Rice,\n\nBacon\u2013Roasted Pepper,\n\nand Chicken Salad,\n\nEggs with,\n\nPorridge, Bacon and Pecan,\n\nPorridge, Cranberry\u2013,\n\nSoup,  CALORIE SMART\n\nSoup, Chicken and,\n\nStuffing,\n\nThree-Grain Medley,  CALORIE SMART \u2022 SLOW COOKER\n\nand Turkey Casserole, Herbed,  CALORIE SMART \u2022 SLOW COOKER\n\n\u2013Turkey Salad, Raspberry,\n\nVegetable and Bacon Skillet,  CALORIE SMART\n\nWinter Squash with,\n\nWilted Spinach,  CALORIE SMART \u2022 Fast\n\nWilted Spinach and Kale,\n\nWine. _See also_ Mimosas; Sangria\n\nBellini Cocktails, Easy,\n\nChicken Marsala,  CALORIE SMART\n\nCoq au Vin,  CALORIE SMART\n\n-Cranberry Sauce, Venison with,  CALORIE SMART\n\nculinary uses of,\n\ndeglazing with, ,\n\nice cubes,\n\nMulled, ,  CALORIE SMART\n\nRed Wine Sauce, Peppery,\n\nRed Wine Vinaigrette Dressing,\n\nSauce, Steamed Mussels in,  CALORIE SMART \u2022 Fast\n\nsauce, making,\n\nstoring\/serving,\n\nvarietals,\n\nWhite Wine\u2013Garlic Sauce,\n\nWine vinegar,\n\nWinter Squash. _See also_ Butternut Squash\n\nBaby Food,\n\npreparing\/cooking,\n\nvarieties of,\n\nwith Wild Rice,\n\nWonton Crisps, Asian Salad with,  Fast\n\nWonton skins, in Ravioli\n\nWorcestershire sauce,\n\nWraps\n\nLettuce, Beef, Vietnamese,\n\nLettuce, Tuna-,\n\nMexican,\n\nShrimp and Egg Salad,\n\n## Y\n\nYeast, ,\n\nYeast Bread(s). _See_ Bread(s), Yeast\n\nYellow Cake, Starlight,\n\nYellow Cupcakes, 574\u2013575 CALORIE SMART\n\nYellow and Green Bean Salad,  CALORIE SMART\n\nYellow Squash. _See_ Summer Squash\n\nYerba Mat\u00e9,\n\nYogurt,\n\n-Apple Snacks,\n\nBacon-Ranch,\n\nBlue Cheese Dressing,\n\nCheese, Cardamom-Scented,\n\n-Cucumber Sauce,\n\nFrozen, Very Berry,  CALORIE SMART \u2022 EASY\n\nGreek,  CALORIE SMART \u2022 EASY\n\nLayered Dip, Greek,  CALORIE SMART \u2022 Fast\n\nParfaits, Granola,\n\nParfaits, Granola-Blueberry,\n\n\u2013Peanut Butter Dip,\n\nPeanut Sauce,  CALORIE SMART\n\nPops, Frozen,\n\nSmoothies\n\nBlueberry,\n\nEasy-Being-Green,  Fast \u2022 EASY\n\nOrange-Pineapple,\n\nPeach,\n\nRaspberry,\n\nStrawberry,  CALORIE SMART \u2022 Fast \u2022 EASY\n\nStrawberry-Banana,\n\nsoy,\n\nstoring,\n\nstraining,\n\nSweet-and-Savory,\n\nTaco Dip, Personal,\n\nTzatziki,  CALORIE SMART \u2022 EASY\n\nVegetable Dip, Spicy,\n\nYorkshire Pudding,\n\nYuca, ,\n\nYukon Gold potatoes, ,\n\n## Z\n\nZa'atar,\n\nZester,\n\nZucchini\n\nBread, Orange-,\n\nCake,\n\nEggs, Skillet, with Summer Squash Hash,  CALORIE SMART\n\ngrilling,\n\nPackets, Summer Squash and Halibut,  CALORIE SMART\n\nRatatouille, Garden,  CALORIE SMART \u2022 Fast\n\nRatatouille Polenta Bake,  CALORIE SMART\n\nRatatouille Salad,  CALORIE SMART\n  1. Cover\n  2. Title Page\n  3. Copyright Page\n  4. Contents\n  5. Dear Friends...\n  6. Getting Started\n  7. Appetizers\n  8. Beverages\n  9. Breakfast & Brunch\n  10. Salads & Salad Dressings \n  11. Soups, Stews & Chilies\n  12. Pasta & Pizza\n  13. Whole Grains & Rice\n  14. Vegetables, Beans & Legumes\n  15. One-Dish Dinners\n  16. Beef, Pork & Lamb\n  17. Chicken & Turkey\n  18. Fish & Shellfish\n  19. Vegetarian\n  20. Grilling & Smoking\n  21. Do It Yourself\n  22. Sauces, Seasonings & Marinades\n  23. Breads\n  24. Cookies, Bars & Candies\n  25. Cakes & Cupcakes\n  26. Pies & Tarts\n  27. Desserts & Fruits\n  28. Helpful Nutrition and Cooking Information\n  29. Index\n\n  1. Title Page\n  2. Table of Contents\n  3. Index\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nGeneral Disclaimer:\n\nSpecific Disclaimer: Alternative Investment Strategies (Chapter 6)\n\nAcknowledgements\n\nForeword\n\nTAKING FULL ADVANTAGE OF A WEALTH OF OPPORTUNITIES\n\nAPPRECIATING WHAT A DIFFERENCE A DECADE MAKES\n\nLEVERAGING HOLISTIC WEALTH MANAGEMENT TO MEET THE DEMANDS OF THE MASERATI ...\n\nRECOGNIZING \"ASSET ALLOCATION\" AS SIMPLY CODE FOR \"RISK MANAGEMENT\"\n\nPREPARING FOR INCREASED DEPENDENCE ON ALTERNATIVE INVESTMENTS\n\nWITNESSING THE RISE OF INTERNATIONAL INVESTING\n\nEXPLORING THE EMERGENCE OF GENERATIONAL WEALTH ISSUES\n\nCHANGING THE RULES AND ROLES OF PHILANTHROPY\n\nLOOKING TO THE FUTURE OF WEALTH MANAGEMENT\n\nCHAPTER 1 - The Evolution of Wealth Management\n\nPREPARING THE NEXT GENERATION\n\nTRANSITIONING FROM A TRANSACTION-BASED INDUSTRY TO WEALTH MANAGEMENT\n\nADVOCATING THE TRIPLE-A WEALTH MANAGEMENT STRATEGY\n\nPLACING A RENEWED FOCUS ON THE WEALTHY\n\nTHE CHANGING ROLE OF THE FINANCIAL ADVISOR\n\nIMPROVING THE ADVISOR-CLIENT DIALOGUE\n\nUNDERSTANDING THE HNWI ADVANTAGE: HOW THE WEALTHY MANAGE TO ENJOY ROBUST ...\n\nDESCRIBING THE STATE OF THE WORLD'S WEALTH\n\nPRIME DRIVERS OF HNWI WEALTH\n\nLOOKING FORWARD\n\nCHAPTER 2 - Global Trends and What They Mean for International Investing\n\nCOINING THE TERM \"EMERGING MARKETS\"\n\nGLOBALIZATION'S GROWING IMPACT ON HIGH-NET-WORTH INDIVIDUALS\n\nCHINA: AN EMERGING MARKET FOR FOREIGN INVESTMENT\n\nINDIA: A GROWING MAJOR ECONOMIC FORCE\n\nTHE EUROPEAN UNION: THE WORLD'S LARGEST INTEGRATED ECONOMY\n\nUNITED STATES: LOST LUSTER\n\nLATIN AMERICA: A GROWING REGIONAL ECONOMY\n\nGLOBALIZATION PROMOTES LOCAL INVESTING\n\nRADICAL CHANGES IMPACT INVESTMENT OPPORTUNITIES\n\nINTERNATIONAL MARKETS' SUCCESS IS FORCING U.S. INVESTORS TO RETHINK THEIR ...\n\nEMERGING MARKETS BECOME MAINSTREAM\n\nBEYOND EMERGING MARKETS LIES THE FINAL FRONTIER\n\nCHAPTER 3 - Technology's Critical Role in Wealth Management\n\nDAWNING OF THE DIGITAL AGE\n\nDISPELLING FIVE MYTHS OF ONLINE PRIVATE BANKING\n\nEXAMINING THE THREE FACES OF INFORMATION TECHNOLOGY\n\nASSESSING THE ROLE OF TECHNOLOGY IN THE ADVISOR-CLIENT RELATIONSHIP: A ...\n\nOUTSOURCING FUNCTIONS: THE JOYS AND PERILS\n\nMELDING INFORMATION TECHNOLOGY AND CULTURE\n\nCHAPTER 4 - Holistic Wealth Management\n\nMOVING BEYOND INVESTMENTS TO A BROADER RANGE OF SERVICES\n\nEXPECTING MORE EXTENSIVE FINANCIAL ADVICE\n\nUSING DEBT TO BUILD WEALTH\n\nDEFINING THE ROLE OF FAMILY OFFICES WITHIN WEALTH MANAGEMENT\n\nEXPECTING CLEAR REPORTING\n\nDEMANDING PRIVACY\n\nCHAPTER 5 - The Doctrine of Asset Allocation\n\nENHANCING RETURNS WITH MODERN PORTFOLIO THEORY (MPT)\n\nUNCOVERING THE PREFERRED ASSET ALLOCATIONS OF THE WEALTHY AND ULTRA-WEALTHY\n\nMOVING BEYOND MODERN PORTFOLIO THEORY\n\nCOMMENTING EXPERTLY ON ASSET ALLOCATION\n\nCHAPTER 6 - Alternative Investment Strategies\n\nTHE APPEAL OF ALTERNATIVE INVESTMENTS\n\nSTRUCTURED PRODUCTS FOR INDIVIDUALS\n\nDEVELOPING NEW PRODUCTS BY LISTENING TO INVESTORS\n\nGRABBING HEADLINES: HEDGE FUNDS AND PRIVATE EQUITY\n\nFACING THE CHALLENGES AHEAD\n\nCHAPTER 7 - Family Matters\n\nFOSTERING FINANCIAL LITERACY, HNW STYLE\n\nPARTAKING IN VALUES RETREATS\n\nENABLING THE NEXT GENERATION OF WEALTHY THROUGH PEER NETWORKING\n\nPREPARING FOR GREAT WEALTH TRANSFER\n\nCREATING A WEALTH TRANSFER PLAN\n\nRECOGNIZING REGIONAL DIVERSITY\n\nAPPRECIATING GENDER DIFFERENCES\n\nPRESERVING THE FAMILY LEGACY\n\nCHAPTER 8 - The New Philanthropy: Proactive Involvement\n\nGIVING WITH PURPOSE\n\nPHILANTHROPY LESSONS FROM ONE OF GREAT BRITAIN'S WEALTHIEST MEN\n\nCOMMITTING FUNDS AND DEMANDING ACCOUNTABILITY\n\nCOMPARING REGIONAL DIFFERENCES\n\nGLOBAL TREND 1: THE RISE OF VENTURE PHILANTHROPY\n\nGLOBAL TREND 2: GIVING WHILE LIVING\n\nTAKING TAX CONSEQUENCES INTO ACCOUNT\n\nMAKING A DIFFERENCE WITH FAMILY FOUNDATIONS AND DONOR-ADVISED FUNDS\n\nWITNESSING A DECLINE IN GIVING?\n\nEXPRESSING OPTIMISM FOR HNWIS' GRANDCHILDREN\n\nCHAPTER 9 - The Future of Holistic Wealth Management\n\nCREATING CONDITIONS TO BUILD AND PRESERVE WEALTH FOR THE FUTURE\u2014THE MICHAEL ...\n\nIDENTIFYING FUTURE TRENDS TO TAKE INTO CONSIDERATION\n\nINCREASING WEALTH ACROSS THE GLOBE\n\nEASING THE STRESS OF INTERGENERATIONAL WEALTH TRANSFER\n\nTRENDING TOWARD OPEN ARCHITECTURE AND UNBIASED ADVICE\n\nINDEX\n\n**Copyright \u00a9 2008 by Merrill Lynch, Capgemini**\n\nAll rights reserved. No part of this work covered by the copyright herein may be reproduced or used in any form or by any means\u2014graphic, electronic or mechanical without the prior written permission of the publisher. Any request for photocopying, recording, taping or information storage and retrieval systems of any part of this book shall be directed in writing to The Canadian Copyright Licensing Agency (Access Copyright). For an Access Copyright license, visit www.accesscopyright.ca or call toll free 1-800-893-5777.\n\nCare has been taken to trace ownership of copyright material contained in this book. The publisher will gladly receive any information that will enable them to rectify any reference or credit line in subsequent editions.\n\n_**Library and Archives Canada Cataloguing in Publication Data**_\n\nWealth : how the world's high-net-worth grow, sustain and manage their fortunes \/ by associates of Merrill Lynch and Capgemini.\n\nIncludes index.\n\neISBN : 978-0-470-67511-3\n\n1. Finance, Personal. 2. Wealth. 3. Investments. 4. Rich people\u2014Finance, Personal. I. Merrill Lynch & Co. (1973- ) II. Capgemini (Firm)\n\nHG179.W.024'01 C2008-900607-0\n\n**Production Credits**   \nCover design: Michael Freeland   \nInterior text design: Adrian So   \nPrinter: Friesens\n\nJohn Wiley & Sons Canada, Ltd.   \n6045 Freemont Blvd.   \nMississauga, Ontario   \nL5R 4J3\n\nFP \nTo Our Clients and Our Colleagues\n\n#  **General Disclaimer:**\n\nThe information contained herein was obtained from various sources; Capgemini and Merrill Lynch do not guarantee its accuracy or completeness nor the accuracy or completeness of the analysis relating thereto. This book is for general circulation and is provided for general information only. Any party relying on the contents hereof does so at its own risk.\n\nThe information and case studies in this book are intended to be a general introduction to wealth management. They are not intended to be either a specific offer by any Merrill Lynch entity or person to sell or provide, or a specific invitation to apply for, any particular product or service. Merrill Lynch and its affiliates offer a broad range of brokerage, investment advisory (including financial planning), banking, trust, mortgage and other financial services and products. The nature and degree of advice and assistance provided, the fees charged, and client rights and Merrill Lynch's obligations will differ among these services. Neither Merrill Lynch nor its Financial Advisors provide individual tax advice. Clients should review any planned financial transactions and strategies that may have tax implications with their personal tax advisors.\n\n#  **Specific Disclaimer: Alternative Investment Strategies (Chapter 6)**\n\nMost alternative investments are sold in private placements and may be offered only to individuals who are both Qualified Purchasers and Accredited Investors. No assurance can be given that any product's investment objectives will be achieved. Many alternative investment products are sold pursuant to exemptions from regulation and, for example, may not be subject to the same regulatory requirements as mutual funds. Each product will be subject to its own specific risks, including strategy and market risk.\n\nHedge funds are speculative and involve a high degree of risk. An investor could lose all or a substantial amount of his or her investment. There is no secondary market nor is one expected to develop for investments in hedge funds and there may be restrictions on transferring fund investments. Hedge funds may be leveraged and performance may be volatile. Hedge funds have high fees and expenses that reduce returns.\n\nA private equity investment involves significant risks and is illiquid on a long-term basis. Interests may not be transferred, assigned or otherwise disposed of without the prior written consent of the manager. Investors may lose their entire investment. Private equity funds are subject to significant fees and expenses, including management fees and, typically, a 20% carried interest in the net profits generated by the fund paid to the manager. Private equity funds may make a limited number of investments, and such investments generally will involve a high degree of risk, such as start-up ventures with little or no operating histories, or companies that may utilize significant leverage.\n**ACKNOWLEDGMENTS**\n\nWhile the idea for _WEALTH_ was inspired by the tenth anniversary of the _World Wealth Report_ , it quickly took on a life of its own. This deeply collaborative effort has evolved over the past two years to, we hope, provide a richer view of the world of wealth.\n\nFirst and foremost, we would like to thank the management of Merrill Lynch and Capgemini for their support in agreeing to support this effort amidst multiple priorities, as well as the many thought leaders who graciously agreed to share their views, insights and lessons learned about wealth and its preservation. Of these, we are particularly grateful for the expertise and guidance of Bob McCann, Vice Chairman and President of Global Wealth Management at Merrill Lynch, and Bertrand Lavayssi\u00e8re, Managing Director of Global Financial Services for the Capgemini Group. If not for their steadfast support and deep industry knowledge, the storyline could not have matured.\n\nSecondly, we would like to thank the management of Merrill Lynch and Capgemini for their support in agreeing to support this effort amidst multiple priorities, as well as the many executives at both firms who invested considerable time and effort in reflecting and commenting on the points of view expressed in this book. The ability of both organizations to reach out to a global network of economists, academics and experts in the field for validation and support was a key factor in broadening the scope of the work.\n\nThirdly, Capgemini and Merrill Lynch would like to thank Michael Sisk of Hamilton, Bridges Media Inc., for the numerous interviews he conducted across the globe, and for his skilled organization of the many stories and ensuing research into a commendable first draft. In the process of guiding our manuscript to completion, Stephen Fenichell of Merrill Lynch's Executive Communications group provided color, texture and a narrative framework, in addition to conducting further interviews and research.\n\nWe would also like to extend our sincere appreciation to Petrina Dolby of SECOR Consulting for her insight, creativity and tenacity in overseeing this complex and collaborative process.\n\nOn the Capgemini team, we would like to acknowledge the key contributions of William Sullivan, Manager of Capgemini's Global Wealth Management Center of Excellence, for his leadership in the book's development; Scott Becchi, Karen Cohen, Chris Gant, Andres Guibert, Bob McGraw, Patrick Neuwirth, Eric Rajendra, Stephen Ray, Ileana van der Linde, Liz Witt, and Alan Young from Capgemini's Financial Services practice for providing industry insights and expertise; and Capgemini's Strategic Research Group for providing research and in-depth market analysis. We would also like to thank Capgemini's Group management for supporting this endeavor.\n\nAt Merrill Lynch, Erik Hendrickson, Michael O'Looney and Tricia Nestfield provided industry perspective and oversight of the editorial and research process.\n\nWe would additionally like to recognize the contributions of a number of senior executives at Merrill Lynch, including: Stacy Allred, Sameer Aurora, John Barrett, Nancy Bello, Richard Bernstein, Stephen Bodurtha, Candace Browning, Darcie Burk, August Cenname, Daniel Dunn, John Hogarty, Richard Jones, Karen Klein, Joseph Lam, Albert Lee, Rahul Malhotra, Patricia McLaughlin, Marcus Mitchell, Alyssa Moeder, former CEO Stan O'Neal, Paula Polito, David Ratcliffe, Diane Schueneman, Raj Sharma, Dan Sontag, Ed Specter, Michael Sullivan, Tom Sweeney, Victor Tan, John Thiel, Kevin Waldron, and Ken Winch.\n\nWe would like to express our appreciation to our colleagues at Wiley for adeptly steering _WEALTH_ through the publishing process and for their collective commitment to editorial excellence. A special thank you goes to Karen Milner, Elizabeth McCurdy, Pamela Vokey, Peter Knapp and Lucas Wilk.\n\nFinally, we have benefited enormously from the many industry authorities and clients who have been so generous with their time in granting extensive interviews for this book. Namely: Antoine van Agtmael of Emerging Markets Management, LLC, Patricia Angus of Shelterwood Financial Services, Jacques Attali of PlaNet Finance, Edward Bernard and James A.C. Kennedy of T. Rowe Price, Dr. James Canton, Jon Carroll of Family Office Metrics, Charles Collier of Harvard University, Larry Fink and Ralph Schlosstein of BlackRock, Doug Freeman and Lee Hausner of IFF Advsiors, Gerald Cavendish Grosvenor, The Duke of Westminster, Gregory Johnson of Franklin Templeton Investments, Michael Lee-Chin of AIC Investments, James Rothenberg of Capital Research and Management, Michael Saadie of ANZ Private Bank, Dr. Lee Shau Kee of Henderson Land Development, Charlie Simonyi, Domhnal Slattery of International Aviation Management Group, Dr. Paul Tiffany of The Wharton School of Business of the University of Pennsylvania and of the Haas School of Business of the University of California, Berkeley, and Dr. Cheng Yu-Tung of New World Development.\n**FOREWORD**\n\n**Our Eight Convictions about the Future of Wealth Management**\n\nBy Bertrand Lavayssi\u00e8re, Managing Director of Global   \nFinancial Services for the Capgemini Group and Bob   \nMcCann, Vice Chairman and President of Global Wealth   \nManagement at Merrill Lynch & Co.\n\nThe world is witnessing the greatest period of wealth accumulation in history. Never before have so many people from so many different regions of the earth become so wealthy in so short a period of time. And never before have so many opportunities existed to create new wealth, both as the natural outcome of new ideas and the product of existing capital appropriately and prudently leveraged. Due to the downfall of the Berlin Wall in 1989, coupled with the concomitant rise of global capitalism and globally integrated capital markets, the twenty-first century presents an unprecedented opportunity for innovative and entrepreneurial spirits to profit from these long-term secular trends. This will continue to fuel the concentration and consolidation of wealth at the unprecedented velocity witnessed over the past decade. Although economic disruptions and volatility can by no means be relegated to history, we firmly believe that the future of global capitalism is bright. In support of this belief, we have documented and validated eight convictions, which we are convinced will continue to shape the world of wealth as it evolves in the twenty-first century.\n\n**Eight Firmly Held Convictions Regarding the Future of Wealth Management**\n\nBased on our collective experience, extensive research and dozens of interviews conducted specifically for this book, we have developed eight firmly held convictions regarding the future of wealth management. We will illustrate all eight throughout the book by using detailed examples and case studies. We will demonstrate the myriad ways by which wealthy investors and their financial advisors, as well as the broader community of aggressive and informed investors worldwide, can materially benefit from the wide universe of wealth creation opportunities before us.\n\n1. The global economy and investment focus for high-net-worth individuals is evolving from a two-pole world dominated by Europe and North America to a multi-polar world including Asia and the Middle East.\n\n2. This multi-polar evolution will accelerate as an unprecedented wave of wealth transfer occurs, putting assets into the hands of a younger, more internationally focused, technologically savvy generation.\n\n3. Diversification of investments across different investment vehicles, asset classes and geographies will be the key to consistent and, at times, superior performance.\n\n4. New investment products will continue to arrive at a rapid pace, accompanied by rapid obsolescence and product cycles. As in the past, early adopters may benefit substantially before a new product is commoditized and returns inevitably revert to the mean.\n\n5. Investor sophistication\u2014the ability to understand these new products, to invest in them and to compare them\u2014will depend on a mixture of cutting-edge technology, information and expertise.\n\n6. Investors will have an increasing responsibility to stay informed and educated. They must be ready to engage and question their financial advisors about wealth management strategies. They must participate in the dialogue.\n\n7. Family education around financial issues will become increasingly important to demystify money, create family unity and prepare children to be good stewards of capital\u2014whether through investments or philanthropy.\n\n8. Business ownership will remain the main driver of wealth creation. Business owners will increasingly demand that advisors manage their wealth as professionally as they manage their own businesses, prompting advisors to offer an ever-broader and deeper range of financial products and services to the wealthy and ultra-wealthy.\n\n#  **TAKING FULL ADVANTAGE OF A WEALTH OF OPPORTUNITIES**\n\nThe recent growth of global wealth prompts an obvious question: How can the wealthy (and those of us who aspire to join them) and their families maximize their returns, commensurate with their risk appetite, in a rational and measured way that permits them to sleep at night while achieving their increasingly complex financial and personal goals? The purpose of this book is to provide some insightful working responses to this question. Not only are we convinced that the current era promises an unprecedented opportunity for abundance, but that all of us can share in the seemingly limitless prosperity of our era.\n\nWhile the various topics treated herein may be of keenest interest to present and aspiring possessors of wealth, the rapidly increasing population of financial advisors and their associates will find much to provoke their interest and stimulate discussion of the ever-shifting landscape of wealth management. While we can by no means guarantee that we will address every question posed by professionals, we can offer assurances that most of the broader dimensions of this holistic wealth management process have been covered.\n\nWhen Capgemini and Merrill Lynch published their first _World Wealth Report_ in 1997, the goal of the project was modest and simple: to compile and distribute a discrete set of data points, conclusions and insights describing an explosively expanding class of individuals whom we dubbed HNWIs (\"high-net-worth individuals\"),  individuals with more than US$1 million in financial assets\u2014and about whom the financial services industry of the time knew astonishingly little.\n\nWe could not have known then that this project would inadvertently give birth to the now commonly used acronym, HNWI (pronounced by the cognoscenti as \"HUNWEE\") which would be destined to take pride of its place in the popular parlance of the burgeoning wealth management industry. As we began more closely to observe, track and interpret the distinctive needs and behaviors of these individuals, we identified an even more rarefied class of investors with financial assets in excess of US$30 million, which we subsequently called ultra-HNWIs (ultra-high-net-worth individuals) who exhibited behaviors more institutional-like than the \"average\" HNWI.\n\n#  **APPRECIATING WHAT A DIFFERENCE A DECADE MAKES**\n\nThe world of wealth described in that first seven-page report was positively quaint by today's standards. In 1996, the number of dollar millionaires had just topped six million; ten years later the ranks of the wealthy had swelled to nearly ten million. In 1996, the pool of assets controlled by the wealthy stood at approximately US$17 trillion. By 2007, that number had surged to US$37 trillion.\n\nThe approximately twenty trillion dollars in newly accrued assets accumulated by the wealthy and ultra-wealthy over the past decade represents not only a trend toward ever-greater wealth consolidation and concentration, but also its accelerating globalization. In 1996, the most dynamic regional wealth generators worldwide were in Asia and Latin America, although all the emerging markets were shortly to be significantly if briefly disturbed by the onset of economic contagion from Asia and its fallout. This included the disruptive default on Russian sovereign debt, which first raised its ugly head in the summer of 1997.\n\nWe also documented a trend by which many of the world's wealthiest individuals and families were abandoning traditional private banking havens in Switzerland in favor of offshore centers in the Caribbean. Additionally, we identified a behavioral pattern that has since accelerated, as the wealthy and ultra-wealthy diversified their portfolios to embrace a number of then-radical and exotic investment strategies, loosely grouped under the generic term \"alternative investments.\"\n\nWhether those alternative investments included taking the plunge into frontier emerging markets, making a proprietary private equity placement, or perhaps purchasing one or more holdings from a rapidly expanding menu of comparatively exotic structured financial instruments and vehicles, the penchant of the wealthy and ultra-wealthy to seek enhanced performance closely mirrored an increased appetite for measured and structured risk\u2014i.e., alternative investments\u2014on the part of institutional investors. By late 2007 and 2008, however, the global credit crunch threatens to cast many of the rosy scenarios underlying these pooled assets into question.\n\nA brief history and synopsis of the ten-year trends can be found in Figure 0.1.\n\n**Figure 0.1** \\- _World Wealth Report_ Spotlight Sections, 1996-2007\n\nFrom 1997 to 2007, the number of HNWIs grew at an annual rate of 7.6 percent, from nearly six million to nearly ten million. At the same time, their financial assets surged at an annual rate of 8 percent, to US$37.2 trillion from US$16.6 trillion. In 2006, for the first time, the Forbes 400 was composed strictly of billionaires. HNWIs, who tend to be older than the general population and hover somewhere on the cusp of traditional retirement age, have been saving and spending in their lifetimes according to patterns and strategies quite different from the wealth accumulation and preservation strategies employed by their parents. What is more, as these wealthy individuals pass their assets on to their children, they will in all likelihood usher in a new generation, armed with a different mindset and approach, which is likely to give rise to yet newer forms of investment methods and strategies. The profound social and economic implications of what is predicted to be the largest intergenerational transfer of wealth in history will be discussed at length in later chapters. This growth over the past decade is broken down into an annual timeline in Figure 0.2 below.\n\n**Figure 0.2** **-** _World Wealth Report_ Scorecard, 1996-2006\n\n_Source_ : Capgemini Lorenz Curve Model; Capgemini\/Merrill Lynch 2007 _World Wealth Report_\n\nRunning concurrently with these long-term economic and demographic trends has been the overarching phenomenon of globalization, coupled with the advent of sophisticated technologies that have integrated the world's capital markets while fostering the creation of a steady stream of complex financial instruments. If two decades ago the financial services industry resembled a mom-and-pop sidewalk food stand offering a mere handful of plain-vanilla entrees on its menu\u2014stocks, bonds and mutual funds\u2014today's financial marketplace more closely evokes a full-service restaurant located inside a five-star hotel, replete with smiling and eminently capable ma\u00eetres d'h\u00f4tel and a concierge prepared to meet\u2014and ideally anticipate\u2014every conceivable need, articulated or not, of the discerning investor.\n\nThe popularity of structured instruments\u2014including collateralized debt obligations and other forms of asset-backed securities, some of which courted controversy and even outright incredulity during the credit crunch of 2007\u2014is clearly responsible for a significant proportion of the new wealth spawned within the first decade of the twenty-first century. Yet broader access to these high-performance and often high-risk vehicles (the Porsches and Maseratis of the financial world) has not come without cost or anxiety. The sheer complexity of such instruments demands a level of expertise to fully comprehend and navigate through their exotic universe and has only increased the value of expert guidance and advice to baffled yet justifiably intrigued clients. Many of today's structured products require advanced math degrees not only to create, but even to comprehend. The global nature of these investments also means that numerous tax and regulatory jurisdictions may come into play, further enhancing the chances that expert guidance in the field will be required by clients and advisors alike when engaging with a wide variety of products.\n\nJust as the _World Wealth Report_ colorfully accomplished over the past decade, this book will trace the evolution of the financial services industry, from its comparatively humble beginnings as a straightforward purveyor of stocks, bonds, mutual funds and garden-variety investment advice to the affluent masses, to a globally distributed group of high-level custom-shops or boutiques, frequently contained within and supported by global financial powerhouses, and offering a staggering array of products and services precisely targeted at the wealthiest of wealthy individuals and households. We will also provide guidance on the strategies and insights developed by many HNWIs to take full advantage of the plethora of often exotic products and services on the menu.\n\nAs we noted a decade ago, the traditional world of the obsequiously discreet and low-key private banker was even then fading fast into a dusty past. \"Private banks can no longer expect to receive handsome fees for providing old-fashioned trust and investment management services merely to protect assets from inflation or domestic taxation.\" Clients then, as now, \"are becoming increasingly sophisticated and demanding...and expect not only outstanding service but consistent strong performance, active wealth management, and a multi-market investment strategy.\"\n\nFamily offices and multi-family offices that provide tailored advice to an exclusive and limited network of families, relatives and friends have continued to proliferate in the HNW universe. Financial services boutiques (many located within larger institutions) that strive to maintain the high-touch intimacy of the traditional private bank have continued to thrive. Several of the largest financial institutions have been quite aggressive in preserving the best of the private-banking tradition by carving out private banks within their larger wealth management divisions, while supplementing the skill sets of their high-production private bankers with expertise drawn from the broad array of activities conducted by the firm.\n\n#  **LEVERAGING HOLISTIC WEALTH MANAGEMENT TO MEET THE DEMANDS OF THE MASERATI INVESTOR CLASS**\n\nWealthy and ultra-wealthy investors' perennially urgent demand for convenient and comprehensive access to markets and information has given rise to a dazzlingly broad range of bespoke products and asset classes. The most avid consumers of these new asset classes consist primarily of a new client class we have labeled \"active performance-driven investors.\" While once upon a time the average wealthy investor might have sought out a private banker for his or her high-end investment advice\u2014probably the same private banking house that served his or her parents or grandparents\u2014today's more versatile, knowledgeable and empowered HNWIs are inclined to take a broader view of their entire financial picture. This broader view has in turn ushered in an era of financial service in which a comparatively small number of firms with a global footprint are prepared to combine the intimacy of a financial boutique with the intellectual capital of a financial powerhouse.\n\nActive performance-driven investors require returns backdated into the past and thrusting far into the future, encompassing asset allocation, risk management, credit, tax strategies, insurance, estate planning, college planning, financial education and philanthropy\u2014not to mention the occasional provision of a sought-after ballet ticket, entree to an intimate lecture by a world-class financial expert, or possibly even a financial boot camp for their children so they are more fully grounded in legacy issues and the ins and outs of intra-generational wealth transfer.\n\nWhen a HNWI turns to a wealth manager in our era, he or she invariably demands to know about and perhaps even fully understand the identity of the newest exotic investment vehicle being offered, much as the same discerning consumer might prefer to acquire a custom-made designer suit as opposed to a mass-production item pulled straight off the rack. Whether it is the most secure risk management tactic, the latest estate tax strategy, or the best way to structure a charitable foundation in accordance with local tax law, the inherent complexity of managing wealth in this day and age demands a genuinely holistic approach.\n\nParticularly for many of the more recently minted HNWIs, as well as for entrepreneurs so often busy building their businesses that they lack the time or inclination to dwell on wealth management issues, it is critical to grasp the fact that a more holistic wealth management view offers the surest path to wealth for the broadest class of discerning investors. For advisors, it pays to know that HNWIs will increasingly expect this breadth of service. For large banks, with complex global structures, it has become all the more urgent to break down silos and integrate across lines of business and geographies to provide the holistic service HNWIs need and will expect.\n\nIn our chapter on investor sophistication, we examine the remarkable strides in financial literacy among HNWIs, arguably driven by three factors:\n\n1. The enormous amount of information available via the Internet, which allows HNWIs to educate themselves more easily than ever before;\n\n2. Globalization, which has forcibly broadened HNWIs' investment views;\n\n3. Steadily increasing household liquidity. Through initial public offerings and private equity buyouts, entrepreneurial HNWIs, or those with a family business, may suddenly find themselves with tens or even hundreds of millions of dollars on hand, which they must wisely allocate across asset classes and geographies.\n\nA key finding here is that as HNWIs become more engaged than ever before in their investments, they are less and less self-directed.\n\nHNWIs will continue to require and value the high-level expertise and sophistication that a good financial advisor can bring to the table. That may be good news for advisors, but it also means that HNWIs are relentlessly raising the bar of expectations for them. They fully expect their financial advisor to understand and explain how products fit into the overall portfolio. Or, depending upon the product or instrument in question, they expect their advisor to be in a position to call upon the right expert to add their specialized knowledge to the relationship\u2014a synergy that private banks cradled within global financial institutions and offering a wide array of products and services excel at delivering.\n\nTo ensure the highest level of service, HNWIs need to be active participants in a dynamic dialogue between client and advisor. Investors cannot be passive bystanders to the management of their wealth. Advisors, for their part, need to know how to encourage and stimulate this dialogue so that goals may be clearly understood and expectations managed on both sides. The wealth of opportunities available today mean that one's net worth should be viewed as a means to achieve life goals, in addition to being a potential revenue stream.\n\n#  **RECOGNIZING \"ASSET ALLOCATION\" AS SIMPLY CODE FOR \"RISK MANAGEMENT\"**\n\n\"Asset allocation\" is, in fact, code for \"risk management.\" HNWIs and their advisors should use asset allocation as a tool to achieve acceptable risk levels. If conducted correctly, unexpectedly high returns are as much a warning sign that risk levels are out of sync as low returns. While concerns about achieving high returns are perfectly understandable, rest assured that the rules of investing are suspended for no one and that unexpectedly high returns will very likely over time lead to unexpected volatility and losses. According to the \"wealth allocation framework\" designed by Ashvin Chhabra, former head of wealth management strategies and analytics at Merrill Lynch\u2014now at the Institute of Advanced Studies at Princeton\u2014HNWIs and their advisors should move beyond modern portfolio theory, today the basis of virtually all investment strategies, and put _risk allocation_ ahead of _asset allocation_.\n\n#  **PREPARING FOR INCREASED DEPENDENCE ON ALTERNATIVE INVESTMENTS**\n\nWe delve deeply into the heady issue of alternative asset classes, including hedge funds, private equity, derivatives of all stripes and even investments of passion such as art or yachts. All are playing an increasingly critical role in the ongoing quest for diversification of assets and concomitant mitigation of risk. Collectively, alternative assets have shifted from a minor component of the average HNWI investment portfolio ten years ago to roughly 25 percent today; in addition, this multifaceted asset class has become a veritable wellspring of financial innovation and high returns for clients and advisors alike. While financial advisors must ensure that their clients have been provided with access to the best thinking around these innovative products, attaining diversity through an appropriate distribution of risk across multiple alternative asset classes remains the key to consistent and even superior performance. HNWIs who fail to demand educated access to these asset classes are, by definition, failing to optimize their returns in the current environment.\n\n#  **WITNESSING THE RISE OF INTERNATIONAL INVESTING**\n\nThe rise of international investing is the natural outgrowth of globalization and the free flow of capital it encourages. This international flavor in investing has been adopted by a large percentage of investors globally. An exception has been North American investors, who have not taken full advantage of the growth and benefits international diversification has to offer. They have historically invested the least overseas, certainly by comparison to their European and Asian counterparts, thanks to the deep and stable U.S. market. This provincial strategy has until recently served them fairly well, but as exotic frontier markets including Vietnam, Dubai and Brazil continue to mature, such reluctance to invest overseas will limit potential returns. Overall, international investing will continue to grow and play a more significant role in investor portfolios, including those in North America as has been highlighted in the _WWRs_ over the past three years.\n\nIn the meantime, emerging markets are attracting local investment on a scale never seen before. In Asia and the Middle East, where local wealth frequently flowed without question to Europe and North America, more capital is staying onshore. We see local\n\nHNWIs as canaries in the coal mine, with their increasing participation in their own markets and economies a powerful validation of the growing internationalization of capital (and of returns), which every wealthy individual or household worldwide should strongly consider emulating.\n\n#  **EXPLORING THE EMERGENCE OF GENERATIONAL WEALTH ISSUES**\n\nWe zero in on the trends around educating children and multiple generations about wealth. Not long ago, parents could, if they chose, keep most of the details of their fortune hidden from children. Many did this not for the purpose of deception but out of a not unreasonable fear that if the full scope of their fortunes were known they would demotivate their children and create passive \"trust funders\"\u2014children who sponge off their inherited wealth and never develop into fully formed adults who contribute to society.\n\nToday's parents no longer have the luxury of nondisclosure\u2014even if they think they still do. Information age tools, such as the Internet, and greater mandated disclosure by players in increasingly transparent financial markets mean that children are able to find out plenty about their family fortunes in the absence of parental consent. According to the most sophisticated advisors in this area, it is generally advisable to talk directly with these children about wealth, the responsibility it brings, and commonly held expectations about what they must accomplish with their lives than to leave offspring to a guessing game that might help to foster precisely the problems that most parents would prefer to avoid.\n\nAt educational seminars and \"financial boot camps\" sponsored by advisors such as Merrill Lynch, psychologists specializing in money and family dynamics increasingly play an invaluable role in helping wealthy families face the complex issues around great wealth\u2014as will be discussed in greater detail in Chapters 1 and 3. These formal events have become critical for families engaged in preserving, protecting and passing on their wealth to future generations since, upon each generational transfer, HNW families must work harder to keep the family's portfolio from splintering into insignificance.\n\n#  **CHANGING THE RULES AND ROLES OF PHILANTHROPY**\n\nWe identify and explore two major trends on the philanthropic scene: \"venture philanthropy\" and \"giving while living.\" Today's millionaires\u2014many with strong entrepreneurial bents\u2014are working to change the rules of philanthropy by bringing corporate world discipline to philanthropic pursuits: they set specific goals and timelines, track and measure results, tie future funding to meeting certain milestones and, whenever possible, aim for profitability and self-sustainability. Today's entrepreneurially inclined philanthropists like to encourage entrepreneurship in their grant recipients, not hand-outs. In 2006, eBay co-founder Pierre Omidyar donated US$100 million to his alma mater, Tufts University, while stipulating that the funds be used exclusively to make micro loans to people in poor countries looking to start a business.\n\n\"Giving while living\" has been provided a recent boost by Bill Gates, who in 2006 stepped down from his day-to-day duties at Microsoft to devote more time to the Bill and Melinda Gates Foundation, which focuses on global health and education. Just a few weeks after Gates made his announcement Warren Buffett declared that he would donate US$31 billion to the Foundation. The carefully considered decision on the part of the two wealthiest men in the world to tie their fortunes together\u2014even posthumously\u2014propelled philanthropy and its latest trends to the front pages of media outlets worldwide. For the less well-heeled, \"giving while living\" is being reinforced by the rise of donor-advised funds, which firms can set up with an initial contribution of no more than a few thousand dollars. Not only do donors enjoy the satisfaction of seeing their donations at work while still alive, but they can also keep tabs on their charities.\n\n#  **LOOKING TO THE FUTURE OF WEALTH MANAGEMENT**\n\nWe wrap up with a look into the future. And we predict that the same drivers at work today\u2014demographics, globalization and technology\u2014will continue to propel and shape wealth management for decades to come. And they will do so in ways not necessarily obvious. For instance, the aging of HNWIs is generally expected to mean a wave of retirement and the need to implement \"decumulation\" or distribution strategies to manage cash flow over a thirty-year span of leisure. But how many HNWIs will truly retire? And how late might they push the age of that retirement? Dr. James Canton, CEO of the Institute for Global Futures and author of _The Extreme Future_ , contends that many of today's HNWIs will continue to accumulate wealth well beyond the age that past generations retired.\n\nThis is just one example of the twists and turns wealth management will take over the coming decades. For instance, what is the future of globalization? While it is likely to remain a powerful force, it is not without its vulnerabilities. Dr. Paul Tiffany, an adjunct associate professor of management at the Wharton School of the University of Pennsylvania, has observed the rise of more anti-trade legislation across the globe over the last four years than the previous twenty-five. The perceived inequalities created by China's managed currency will need to be corrected, he argues, or developed countries may pull back on free trade. Meanwhile, technology innovation and development continues apace in this space. Will we ever witness the rise of a truly integrated global electronic stock market? The political and regula-tory hurdles are certainly not insignificant or even fully understood at this stage, but such a development lies certainly within the realm of imagination, not to mention technological feasibility.\n\nDespite the fact that this book plainly focuses on wealth management strategies employed by the wealthy, we firmly believe the messages here are deeply relevant for a broader community of investors. Our reasoning is simple: the wealthy are trendsetters. By watching how they put their money to use, one gets a valuable insight and feeling for what investments will become popular and more widely available in the future. A case in point is how hedge funds and private equity investments have steadily moved downstream from HNWIs into the mainstream of strategies open to the mass affluent. Such forward-looking insight tends to be particularly valuable since HNW investment strategies tend to outperform those routinely employed by other categories of investors. The rich do, on average, get richer\u2014and do so faster than other investors\u2014because their purchasing power and their access to the best thinking and innovative products gives them an advantage, an advantage that we hope to confer through the knowledge embodied in this book.\n\nAs you can see in Figure 0.3, the higher the net worth, the faster wealth accumulates, with UHNWI's accounting for 1 percent of HNWI population, yet holding 35 percent of HNWI wealth.\n\nFor the broader investment community, the most persuasive argument for studying how HNWIs manage their wealth is to spot early trends and best practices that can help _everyone_ with investable assets achieve superior results and returns\u2014manifestly a pressing concern for many of us in an era when defined benefit plans are fast disappearing and even average-income investors must manage their retirement investments more aggressively. For HNWIs, learning best practices from each other has long been a proven and valuable strategy.\n\n**Figure 0.3** \\- Number of HNWIs and Growth, 2004-2006\n\n_Source_ : Capgemini Lorenz Curve Model; Capgemini\/Merrill Lynch 2007 _World Wealth Report_\n\nThis traditionally spontaneous practice\u2014of which the most common variant is most likely the swapping of hot stock tips within the confines of private club locker rooms\u2014has been increasingly systematized and reinforced in recent years by the establishment of peer networks such as TIGER 21. Co-founded by wealth investors Michael Sonnenfeldt, Richard Lavin and Tommy Gallagher, the New York-based TIGER 21 ( _T_ he _I_ nvestment _G_ roup for _E_ nhanced _R_ eturns in the _21_ st Century) requires members to possess in excess of US$10 million in investible assets and relies upon candid peer review and sharing to enhance members' already sophisticated personal wealth management tools and strategies. Other similar groups, including the Institute for Private Investors and the Met Circle, offer comparable peer counseling to wealthy investors.\n\nThe highly touted success of such groups should not, however, lead us to assume that all innovation comes from the top. In some cases, valuable acumen and advice can \"trickle up\" to HNWIs from the entrepreneurial middle class. For example, much of the impetus and trial-and-error innovation around real-time, on-line trading of currencies and securities, and even the push for e-IPOs over the past decade, has come from middle-income \"mass-affluent\" investors who enthusiastically capitalized on decreasing transaction costs in the rapidly evolving open electronic marketplaces. Their ability to seize control of hitherto esoteric investing techniques and break into new markets without the direct use of any financial advisor made many HNWIs sit up and take notice, and led them to eschew the occasionally conventional guidance they were passively receiving from their advisors.\n\nIt is important for all of us who are intrigued and possibly even fascinated by the fast-changing face of wealth management to keep up with the latest trends whether generated from above or below. That said, it is also necessary to bear in mind that the nature, timing or origination of valuable new ideas is not easy to predict. Smart investors will always keep their ears to the ground and be constantly prepared to act with prudence and judgment. Our overriding objective is to provide you with some of the insights and guidance that the wealthiest individuals and most talented investment advisors currently enjoy so that no matter what your current net worth, you will find yourself in a position to take charge of and prudently safeguard your own financial future.\n\n\u2014Bertrand and Bob\n**CHAPTER 1**\n\n**The Evolution of Wealth Management**\n\n#  **PREPARING THE NEXT GENERATION**\n\nJudged strictly by outward appearances, the twenty-odd twenty- somethings gathered at the Steinberg Conference Center at the Wharton School are indistinguishable from the grad students passionately debating the pros and cons of conducting business in Africa in Conference Room A, or collectively salivating over the bounties bestowed by the judicious deployment of Investment Logarithms in Conference Room B. Among the nineteen males and seven females (median age: twenty-five) dutifully taking notes at the afternoon session are the generationally and demographically predictable percentage of skin piercings in locations other than the ear (4), hair dyed in primary colors other than blonde (3), two-ply V-necked cashmere sweaters in a dazzling array of designer colors (6) and non-native accents (2). In fact, the only commonly shared trait separating this group from any other attending conferences at Wharton is rather discrete: the approximate average net worth of the participants' parents was easily north of US$100 million. Or as the folks at MasterCard put it, \"Priceless.\"\n\nWelcome to Merrill Lynch's Financial Boot Camp, a high-touch, high-focus financial literacy program offered exclusively to the offspring of ultra-high-net-worth (UHNW) clients around the globe by the firm's Global Wealth Management division \"designed to give the next generation a higher level of understanding and comfort with money,\" according to its mission statement. \"The agenda not only targets the hard financial concepts but also addresses the softer issues associated with family wealth.\" This is a fine example of how the wealth management industry has developed into one of the most sought after segments in financial services\u2014it is no longer simply based on transactions with well-established clients, but has begun to focus on investing in relationships that will be of value in the future.\n\nExpertly honing in on a few of the softer issues, Beverly Hills psychologist Lee Hausner (author of _Children of Paradise_ , \"a comprehensive parenting guide for financially advantaged families [that] offers a clear nine-step program for affluent parents to improve their skills and inspire healthy values in their children\") kicks off the session by breezily drilling her raw recruits on the broader implications for human relations of the uneasy conversations on the ever-emotionally-charged subject of money that the wealthy frequently have with their non-wealthy peers.\n\nNow a partner at IFF Advisors, a California-based consulting firm \"created to enhance the human, intellectual, social and financial capitals\" of its clients, \"empowering them to do the right things by doing things right\" with their money, Dr. Hausner candidly confronts such hot-button topics as that hardy perennial of the celebrity tabloids, the prenuptial agreement, followed by the awkwardness associated with requests for repayment of casually extended personal loans to non-wealthy \"friends,\" and the bizarre hesitation so many people (wealthy and non-wealthy alike) seem to feel about challenging glaring overcharges on restaurant checks, for fear of being made to look cheap. But by far the keenest source point of tension, which drew the highest volume of knowing nods and winces from the crowd, is Dr. Hausner's scalding treatment of the topic \"Speaking with Significant Others When Fiscal Unequals.\" The gist of Dr. Hausner's advice is that when the wealthy enter into relationships (God forbid marriage) with those from less privileged backgrounds, difficulties may well ensue that could make that staple of late-night talk show humor, the prenup, seem like the soundest investment the family could make since it bought big into Google at US$120.\n\nIn many wealthy families, particularly those in which the family fortune has been earned not inherited, some degree of confusion may reign regarding the question of how best to talk to the offspring about the money they're likely to inherit as opposed to earn...because every self-made entrepreneur or corporate executive who goes on to reproduce is destined to rear not fellow scrappy pull-oneself-up-by-the-bootstraps types but children of privilege\u2014or as Dr. Hausner would prefer to have it, \"paradise.\" With the notable caveat, of course, that there's often trouble in paradise.\n\n\"In less affluent families,\" Hausner cautions, \"money is often more about context than subtext. Kids constantly hear their parents argue about money. They're told that the family can't afford this or that, whether it's a trip to Walt Disney World or a college education. In high-net-worth (HNW) families, the kids might see the glossy brochures on the dining room table describing the next high-end vacation, or see the grand estate with the ten cars parked in the garage. But despite the visibility of that wealth, those same families often prefer to treat money as if it's taboo\u2014a topic off-limits to honest discussion.\"\n\nAs Bertrand and Bob observe in the Foreword, the pace of global wealth creation has been remarkable\u2014if not downright startling\u2014over the past decade, which goes a long way toward explaining why wealth management issues have become so top of mind from New York to London to Mumbai to Beijing. Naturally, these established and newly minted millionaires want to know how to stay ahead\u2014making wealth management so topical. But the mass affluent\u2014emerging everywhere from the United States, Europe and increasingly the far-flung corners of the world\u2014are paying attention to what works and what doesn't. They wisely want to learn from other successes and mistakes. They want a shot at joining the club. And they want a shot at doing what the wealthy have done since time immemorial\u2014preserving and protecting their wealth for the next generation and even generations beyond. For that to take place, the next generation has to be educated about the perils and minefields of money\u2014hence the financial boot camps conducted at Wharton and elsewhere globally.\n\nIn the here and now of the twenty-first century, wealth management firms of any size and scope are aggressively stepping into the breach, spying an opportunity to strengthen the advisor-client bond with the family by offering financial education for offspring as a natural adjunct to estate planning, philanthropic advice, alternative investments or exclusive access to an event starring a noted author, an invitation to a museum gallery on a closed day, tickets to the theater or a concert, a fly fishing outing on a noble estate in Scotland, or a round of golf at St. Andrews or Pebble Beach.\n\nAnd if that all sounds like a far cry from Merrill Lynch's humble origins, well, welcome to the brave new world dominated by the wealthy and ultra-wealthy\u2014the natural outcome of the stunning success of global capitalism in concentrating and consolidating family, personal and household wealth worldwide. In an era of galloping financial surplus, many firms have been obliged to adopt an outlook and skill set more commonly associated with private bankers domiciled in Switzerland, Luxembourg or other tax havens. They must transform into firms intensely focused on meeting the needs of the wealthy and ultra-wealthy in all their variability and complexity. As Merrill Lynch's Private Banking and Investment Group promisingly proclaims, \"Substantial wealth is different. It deserves substantially different care.\"\n\nClearly such services are not typically available to the HNWI or the average investor, yet many of the lessons learned and applied herein can be adapted to multiple environments. For example, when anticipating the advent of a pending intergenerational wealth transfer, clearly communicate your expectations for the next generation, and seek to ensure that the next generation grasps and internalizes what tools and skills will be required to meet those expectations. Getting the next generation more deeply involved in actively thinking about and managing whatever level of wealth they aspire to or already possess can only be a positive development for all concerned.\n\n**Taking It from Wall Street to Main Street to Rodeo Drive**\n\nThe securities industry in the United States in the early decades of the twentieth century was so scandalously under-regulated that most Americans, not without cause, saw little reason to distinguish between the purchase of corporate securities\u2014stocks or bonds\u2014and spending a long boozy night at a casino.\n\nOne of Wall Street's most rigidly held traditional precepts at the time was that it was ungentlemanly for brokers to advertise. Charlie Merrill, the son of a prosperous doctor who had suffered an irreversible financial setback after being badly beaten during a holdup near his home in Jacksonville, Florida, was one of the first to break from tradition by garnering his first clients by sending out a soberly written letter to a select list of New York physicians, offering them the chance to consider the purchase of a few promising shares in firms whose attractiveness he outlined in a low-key fashion. Merrill's conviction that a core value would be the provision of accurate and timely information to clients\u2014a common practice fifty years later\u2014was decades ahead of its time.\n\nIn 1915, Merrill and his partner, Edward Lynch, established a retail organization distinguished by the embrace of a pervasive philosophy that today would go by the name of \"transparency.\" Merrill personally coached every new recruit to avoid overselling clients on specific securities, particularly if they offered a rapid speculative route to profit. \"A salesman deserves no credit for any sale made on the strength of exaggerated statements,\" Merrill insisted in a widely circulated internal memo.\n\nNot long before the 1929 stock market crash that he predicted, Merrill placed a US$5 million bet on the long-term future of the securities industry by purchasing a controlling share in E.A. Pierce & Co., the largest \"wire-house\" in the country. The overarching vision was to create a nationwide chain of brokerage houses, linked by telephone and telegraph wire, that would serve the financial needs of America's burgeoning middle class in the same way that the Safeway supermarket chain\u2014Merrill's largest single personal investment\u2014met its biological requirements.\n\nAt a two-day managerial conference held at New York's Waldorf Astoria hotel in April 1940, Merrill outraged his audience by suggesting that the lofty financial services industry had much to learn from the lowly food industry. \"When a customer comes into a Safeway,\" he insisted, \"she is entitled to buy with confidence, [knowing] that she will get full value at the lowest possible price.\" Brokerage or investment banking customers of the _new_ Merrill Lynch should expect nothing more and nothing less. \"We must bring Wall Street to Main Street,\" Merrill charged persuasively, \"and we must use the efficient, mass merchandising methods of the chain store to do it!\"\n\nFinancial writer Martin Mayer would eulogize Merrill's crowning achievement as having coaxed the broad public into the financial markets \"not as lambs to be fleeced but as partners in the benefits.\"\n\n#  **TRANSITIONING FROM A TRANSACTION-BASED INDUSTRY TO WEALTH MANAGEMENT**\n\nFrom its inception in 1915 by Charles Merrill and Edward Lynch, the firm of Merrill Lynch pioneered the transformation of the financial services industry from an enterprise based on the processing of individual transactions to a fee-based wealth management enterprise. A key figure in this evolution was Donald Regan, an ex-Marine World War II veteran and Harvard graduate who became CEO of Merrill Lynch in 1971 (and would go on to serve as Treasury Secretary under President Reagan).\n\nAlong with a number of his senior executives, Regan pushed for the scrapping of the regulated commission rates that for decades had kept Wall Street profits artificially high while providing a nearly insurmountable barrier to entry for \"average investors.\" He further challenged the traditional ways of the Street by taking the firm public in 1972, insisting that Wall Street firms that championed the sales of securities by publicly owned firms should practice what they preached and go public themselves.\n\nBut the chief insight that crowned the Regan era was his realization that trading and brokerage not just on Wall Street but around the world would soon be extensively automated. The long-term impact of technology on the brokerage business, he forecast, would be the erosion of the traditional source of profit for retail financial firms: high commissions on individual and block trades. In light of this realization, Regan launched Merrill Lynch Asset Management, a pioneering attempt to redefine the fundamental purpose of a traditional brokerage firm into an organization specializing in wealth management.\n\nIn the wake of wide-ranging deregulation at the national level, the overarching question posed by Regan's grand strategy was, what was the future of the financial services industry? Regan's key contribution was to promote the continuing conversion of the industry from a gentleman's club catering to the needs of the old money elite into a convenience-based service model dedicated to vastly improving the lives and financial security of the mass affluent. What would become industry practice within a few decades was\u2014like Merrill's own innovations\u2014rather radical for its time.\n\nParamount among the mass-market tools that Merrill Lynch developed to engineer this pronounced and profound shift of emphasis was the CMA (Cash Management Account), a first in the industry. The product of a creative collaboration between Merrill Lynch's wealth management business and a team of academics at Stanford's Research Institute, the CMA combined a checking account with a money market account, a brokerage account and a credit card account\u2014a potent combination providing an extraordinarily convenient streamlining of services in an era before ATMs, PCs, and even handheld calculators. Having a CMA meant that for the first time, brokerage customers intending to buy and sell stocks no longer had to physically write checks and ensure that they were physically delivered to their brokers, who would then deposit the checks in their brokerage houses. The brokers would in turn track purchases of individual securities by actual stock certificates and check stubs, which clients were obliged to physically store in safety deposit boxes at their banks or brokerage houses.\n\nCritics hotly denounced the CMA as a sneak attack on the Glass-Steagall Act, the Depression Era U.S. law that maintained a strict separation between the banking and securities industries. But the CMA had been scrupulously designed to pass muster with the relevant regulatory authorities and by 1977 the Federal Reserve Board, the Justice Department and the SEC had all given the green light for its rollout. As for the death of Glass-Steagall, although it survived for another two decades, the critics were spot on. When Citibank CEO Walter Wriston was asked in 1979 to describe the financial institution of the future, he replied without hesitation: \"Don Regan already runs it\u2014and it 's called Merrill Lynch.\"\n\n#  **ADVOCATING THE TRIPLE-A WEALTH MANAGEMENT STRATEGY**\n\nFrom the point of view of both firm and consumer, the CMA was a highly effective tool for managing the purchase and tracking of securities transactions. But its larger implication for the industry (and for its clients) was that it spearheaded a broad-based philosophical shift on the part of the entire industry away from a transaction-based to a fee-based foundation. John L. \"Launny\" Steffens, Merrill's Consumer Markets division chief at the time, publicly proclaimed that \"profitability\u2014both for our clients and for the firm\u2014can no longer hinge on day-to-day transactions.\" He passionately advocated a \"Triple-A\" strategic approach to managing the new broker-client relationship, which was comprised of a tripod supported by the following three legs:\n\n1. Asset gathering;\n\n2. Asset allocation;\n\n3. Asset management.\n\n\"We believe long-term relationship building is best achieved by consultative selling,\" Steffens contended, describing the ideal broker-client relationship as one in which the financial firm would assist its clients across a broad spectrum of financial activities, including setting aside funds for education and retirement, establishing trusts, long-term health care financing and even estate planning services\u2014a collaborative process that would bring in expertise provided by attorneys and certified public accountants. The mechanics of the CMA, coupled with the Triple-A approach to investment counseling, ushered in the era of holistic wealth management (HWM).\n\nAsset gathering provided the foundation of this new approach. It emphasized placing the entirety of a client's assets under one roof as the only practical means to tackle the multiple challenges faced by clients attempting to get a grip on their financial lives in their totality. The newly enshrined doctrine of asset allocation\u2014the rational distribution of a client's total pool of investible assets, to be treated in greater depth in a later chapter\u2014could only be effectively implemented through comprehensive client profiling and extensive consultation.\n\nAsset allocation called for a client's assets to be rationally allocated among a broad range of savings, investment and insurance instruments, including liabilities and debts such as mortgages and other loans. On the investment side, an appropriate mix of equities, fixed income securities, credit derivatives and mutual funds was understood to compose the ideally balanced portfolio.\n\nThe third leg of the Triple-A tripod, asset management, was frequently farmed out under the new dispensation to a select group of outside money managers deemed capable of providing the highly specialized expertise needed to match an investment strategy with a carefully prepared client risk profile, which Merrill Lynch dubbed \"The Financial Foundation.\"\n\n#  **PLACING A RENEWED FOCUS ON THE WEALTHY**\n\nBy the dawn of the new millennium, an estimated seven million people worldwide held liquid financial assets exceeding US$1 million, according to the 2000 _WWR_. More than a million new people had joined that exclusive club in 1999, and their financial assets worldwide had increased in that single year by 18 percent to over US$25.5 trillion. Global stock markets expanded by 37 percent in 1999, with Asian stock markets shooting up an astonishing 70 percent, boosting the wealth of high-net-worth households in Asia.\n\nAt the close of the twentieth century, the United States was leading the world \"in the creation of younger active HNWIs (high-net-worth individuals) likely to have created their wealth in the technology sector as the digital economy takes hold,\" the 2000 _WWR_ advised. Unbeknownst to the report's authors, the tech boom and the world's stock markets were destined to suffer a sharp decline toward the end of the following year. Yet the long-term trends of wealth creation and consolidation set off by the abrupt collapse of socialism, the advent of global capitalism and the liberalization of capital markets in virtually every nation and continent on earth were bound to continue and even accelerate unabated into the new century.\n\nGlobal, full-service financial services firms began to systematically segment the mass affluent and the affluent from the wealthy and the ultra-wealthy. Merrill Lynch's dividing lines slightly diverged from those adopted by the _WWR_. \"Mass-affluent\" clients with less than US$250,000 of investible assets would find the majority of their needs efficiently served by calling a newly established Financial Advisory Center. At the upper demographic ranges, the firm's private banking services were expanded and reinvigorated by the establishment of a dedicated Private Banking and Investment Group (PBIG).\n\nTo those inclined to decry this client segmentation strategy as representing the abandonment of Merrill's historic mission of taking \"Wall Street to Main Street,\" supporters of the new system responded that the world had changed over the eighty-five years since the firm's establishment in 1915. By the dawn of the twenty-first century, the HNWIs (\"Hunwees\"), whose customs and aspirations were being avidly documented in the _WWR_ , were transforming the financial services industry as decisively as the rise of the mass-affluent class had changed the game during the twentieth century.\n\nThe world's financial industry was evolving and adapting to this changing landscape by becoming an industry in which the most basic transactions were increasingly frictionless and cost-free, and where the true differentiating factors and mark of excellence would be the degree to which firms were able to engage with their wealthiest and most sophisticated clients at the broadest, deepest and even most profound and emotionally intimate levels. The leading edge of the global financial services industry had become, for better or worse, the high-net-worth wealth management industry.\n\n#  **THE CHANGING ROLE OF THE FINANCIAL ADVISOR**\n\nIt didn't take long for independently minded financial advisors (FAs) to be faced with discount brokers like Fidelity and Schwab offering block trades at US$29 a pop on the Internet, calling the entire proposition of the traditional full-service brokerage firm into question.\n\nThe overriding question became whether firms with bloated brokerage arms could survive a hitherto unimaginable degree of automation. At a time when so many of the business's traditional functions were rapidly becoming commoditized, what value-add was being provided?\n\nWhat differentiated full-service financial firms like Merrill Lynch from the low-cost competition were several things: the quality of that advisor-client relationship combined (particularly at the higher end of the scale) with the ability of a huge firm with a global footprint to deliver to clients access to expertise in a variety of investment instruments, vehicles, strategies, methods and philosophies that few of the boutique firms could ever hope to match.\n\nIn a 2000 speech to the Securities Industry Association, Stan O'Neal, who headed Merrill Lynch's Global Private Client Group at the time and who would go on to become CEO from 2002 to 2007, observed that the industry was \"in a period of wealth creation such as the world has never seen.\" In light of that development, he argued that financial services firms were obliged to grow their share of the high-net-worth market or see the fattest piece of the pie go to their competitors.\n\nMost shocking of all, especially to the healthy number of traditionally minded brokers sitting in the audience, was O'Neal's blunt statement that \"we do not even consider ourselves to be in the retail securities business anymore. We're in the wealth management business.\"\n\nBut rather than congratulate the other leaders of Merrill Lynch on being the \"Christopher Columbuses\" of the financial services industry, O'Neal frankly acknowledged that \"we're all engaged in the process of trying to figure out what it is that will distinguish us in the eyes of the wealthy investor.\" The firm's wealthiest clients were being underserved\u2014not because their financial advisors were doing something wrong, but because the firm as a whole had not developed a rational system to segment clients according not just to their net worth but their actual needs, goals, objectives and aspirations as stated in their client risk profiles, which were subject to frequent rebalancing, typically on an annual basis. The real job of an FA must be to do the best possible job of identifying and meeting these increasingly complex and sophisticated client needs by utilizing the entire intellectual capital of the firm. This was an extraordinary transition for not only FAs, but also for HNWIs.\n\n**Bullish Times for Financial Advisors**\n\nIn early 2007, _Money_ magazine named \"financial advisor\" the third best job in America. \"As company pensions die out and Americans increasingly have to manage their own retirement savings, financial planning is no longer only for the rich. With Gen X-ers entering their peak earning years and boomers nearing retirement, business will get better still.\" As individuals receiving these services, we need to also keep in mind that the advisors we work with should be able to instill in us the confidence that they enjoy their jobs, value their clients and are able to protect us from some na\u00efve decisions we might otherwise make.\n\nIn Singapore, which is rapidly becoming a hub of wealth management, there are currently about two thousand financial advisors. But according to a survey by the Singapore-based Calamander Group, the immediate demand is for twenty-five hundred with the need expected to rise to five thousand to six thousand in five years. As a result, the government and local bankers are putting on a full court press to recruit and train people. This indicates that in Asia, individuals are realizing the benefit and are reaching out to professionals for advice.\n\nMichael Saadie, managing director for Asia and former head of the private bank at ANZ Bank (also known as The Australia and New Zealand Banking Group and the fourth largest bank in Australia) observes that identifying, attracting and retaining talent is one of his top priorities\u2014not to mention _the_ paramount issue that keeps him and his colleagues awake at night. Growth in wealth management in his country and the rest of Asia is putting enormous personnel strains on firms across the region. The Singapore government is going so far as training people from other professions to become private bankers. Through the recently created Singapore Wealth Management Institute, the government offers year-long masters programs and a two-month course to retrain anyone, from curtain salesmen to pianists and to sewage engineers, to become financial advisors and planners.\n\nOf course, becoming a productive FA takes more than a good mind for numbers. It takes an understanding of etiquette, cultural taboos, attention to client details, and an ability to make connections with people. Joseph Lam, a wealth advisor in Merrill Lynch's Hong Kong office, vividly recalls his entry-level job at Goldman Sachs. One of his first big assignments? Get everyone's lunch orders\u2014and don't get any of them wrong, or else.\n\nAt first it might seem like a way to serve a bit of humble pie to a new-comer (and perhaps it was) but Mr. Lam saw it differently. He got to speak to everyone on the floor, including the most senior people. And even if it was just to take a lunch order, he made a small, first connection.\n\nConnections, insists Mr. Lam, are vital to being a successful financial advisor. As FA to two of the most highly respected billionaires in Hong Kong, Dr. Lee Shau-Kee and Dr. Cheng Yu-Tung, Mr. Lam encourages young FAs to get out from behind their desks, take a walk, get a cup of coffee, clear their minds, find ways to expand their social horizons and work closely with senior managers. In fact, our research for this book has led us to believe that a general rule for all clients is that they need to feel that they know their FAs, and be confident that they are not simply a voice on the phone. They need to feel that they are a personality with specific needs. If this is not the case, you (the client) do not have the right FA for long-term success, regardless of the economic climate.\n\nAccording to Lam, the most important thing is for clients to like you, \"because people like to do business with people they like.\" This, he insists, cannot be faked. \"An advisor must truly care about the client to win their friendship. Don't pitch to them every time you see them. Sometimes just have some soup and listen to what is on their mind, financial issues or not.\"\n\nThat sentiment is echoed by Victor Tan, Market Director of Merrill Lynch's global wealth management division in Hong Kong. \"Be genuine. Be who you are.\" This is essential, he says, because personal authenticity is the best way to truly differentiate oneself. \"It can be surprisingly easy to get a first meeting with even the wealthiest prospective clients because they are always on the lookout for new talent. They welcome new faces. They want to talk to people,\" Mr. Tan says. The hard part \"is establishing your identity in that meeting and getting a second [meeting].\" As a client on the receiving end of these interactions, we also need to ensure that we listen and realize that the individuals we are dealing with as advisors need to understand us as best they can to enable them to help us most effectively.\n\n#  **IMPROVING THE ADVISOR-CLIENT DIALOGUE**\n\nThe defining benchmark for determining the quality of any wealth management service is the quality of the dialogue between wealth manager and HNWI, contends Bertrand Lavayssi\u00e8re, Capgemini's managing director for Global Financial Services. The construction of a productive and often subtle dialogue between a client and his or her financial advisor is the name of the game. \"When it's your money, you\u2014the client\u2014 _must_ have an opinion. _You need to be part of the dialogue.\"_ According to Lavayssi\u00e8re, \"If wealth is managed passively and with limited or no dialogue with advisors or specialists who possess in-depth knowledge of your situation, many opportunities are missed and assets are often placed at greater risk.\"\n\nAt Merrill Lynch, Private Wealth Advisors (PWAs) are required to trim client rosters to a few dozen choice individuals and households in order to maintain their PWA designation. Those individuals and households would preferably be those possessing the highest net worth and the most complex financial needs, those prone to participate and collaborate with their advisors in establishing a financial framework and clearly articulating their goals and objectives, so that their FA can devise strategies tailor-made to achieve them.\n\nThe primary value-add of the FA-client relationship is inevitably defined by the nature and substance of the dialogue between them, coupled with the support services that a firm can provide to take the resulting profile and design a bespoke portfolio to consistently meet those objectives. Prior to entering into a relationship with a financial advisor, a few well proven success strategies can enhance virtually any client's selection process.\n\n**Finding Your Holistic Wealth Advisor: A Punch List**\n\nBecause establishing an open dialogue is the key to forging a productive relationship between a financial advisor and a client, consider the introductory meeting as the onset of that dialogue. What should you ask? How do you find an advisor that best fits your personal profile?\n\nMany of us are sometimes overwhelmed when we meet with financial advisors and discuss our needs\u2014we may not always be as inquisitive as we should be when selecting who should manage our portfolios, possibly because we are concerned that we will not understand the answer, we don't know what questions to ask, or we don't know how to measure their performance. We certainly behave differently when selecting a realtor, who\u2014like a financial advisor\u2014may be responsible for helping us to make some of the most important investments decisions of our life.\n\nHere are a few key questions to ask and areas to probe when making decisions related to financial advisors. And here is a list of ten areas that individuals and families should consider to not merely evaluate a potential FA, but also to become more actively involved in the dialogue regarding the management of their assets.\n\n1. Source of FA's Name: HNWIs typically learn about financial advisors through personal referrals. Lawyers, accountants, colleagues and other service professionals tend to be in a position to recommend an individual, team or firm. Referrals can also come through friends and family members. The advantage of a personal referral is that the referrer already knows something about your needs and interests and also about the firm they are referring you to, and you have common ground to start a conversation.\n\n2. Background check\u2014training, licensing and certification: Most, if not all, financial advisors have undergone an extensive training program. Some may style themselves as financial planners, others as financial advisors, still others as wealth managers or private bankers. Most established professionals in the majority of countries, including the United States, are accredited by a variety of reputable professional organizations, including the National Association of Personal Financial Advisors (NAPFA), the Financial Planning Association (FPA) or the American Institute of Certified Public Accountants (AICPA). Similarly, in the U.K. clients should look for firms they deal with to be FSA (Financial Services Authority) authorized. If the advisor is not registered with a government agency, that should raise a red flag.\n\n3. Established investment philosophy and strategy: Whether looking for aggressive growth or to protect your assets, you should establish a set of personal goals and objectives beforehand and compare this to the philosophy of the financial advisor you are interviewing. Good financial advisors will work closely with you to develop a strategy appropriate to achieving your personal lifestyle and financial goals. Similarly, a financial advisor's interests should be aligned to your personal interests: How aligned are the firm's interests to yours as a client? Are the firm's financial advisors rewarded in a way that will encourage them to work more in your interests than those of the firm? What is their total remuneration based on? Which metrics have the most weight in determining their remuneration\u2014the money they bring into the bank, investment performance, or maybe customer satisfaction\/retention? Asking this last question will tell you a lot about how the firm sees its clients. The way FAs are measured for remuneration tends to drive their behavior.\n\n4. Range of activities undertaken and products and services offered: You should also check what other activities or services are conducted by the institution you are entrusting your money to. Is private wealth management a large part of what they do? Are they committed to it (and to you) for the long term? What is the turnover rate of advisors in the firm? Will your FA keep changing, leading to discontinuity in your relationship with the institution? Some attention should be given when assessing the products and services offered. As this book will highlight, the investment planning and financial advice field is crowded and fragmented, running the gamut from single-proprietor consultants and planners to discount brokerage houses that conduct the bulk of their interaction with clients online. HNWIs typically demand and enjoy a high level of service, with the most rarefied model being a family office that attends to clients' any and every need, from travel arrangements to speaking engagements to estate planning and other legacy issues.\n\nThe distinguishing factor between firms is the advisor's capacity to determine the optimal mix of products and services that will fit your particular goals and objectives and then deliver on that strategy. Does the firm or individual offer a truly holistic approach? If so, can they explain this in greater detail? Additionally, as the wealth management industry becomes more global and more complex, you should ensure your advisor is aware of and able to navigate global markets or have access to this information in a form that can be readily and easily shared and understood.\n\n5. Fee structure and other charges: What are the fees you will be charged? How are these calculated? What percentage is the FA prepared to put at risk against investment performance? How do the fees reduce as the sum you entrust to your FA increases? Typically, you will be quoted an annual fee that is levied as a percentage of your entire portfolio. Ask your FA to explain what the fee includes. Also ask what it does not cover and what any additional charges might be. If you are investing for the long term, a difference of a few basis points in fees for a similar product\/service could materially affect the long-term value of your investments.\n\n6. Support in creating a legacy: Find out if a prospective financial advisor is willing to discuss how he or she intends to assist you in transferring not just your assets but also your moral framework and values to the next generation and beyond.\n\n7. Frequency and discipline of the review process, timeliness of reporting and sophistication of technology tools: How does your prospective financial advisor intend to help you reset your strategy and objectives in light of changing circumstances? Almost as important is how timely and accurate the individual's, team's or firm's reporting process claims to be. How sophisticated is the technological platform used by the advisor and how seamlessly is a client able to access that network?\n\n8. Access to leading money managers: Is the FA going to handle the details of asset allocation him or herself or will this activity be handled by outside investment advisors? Where will your money be invested\u2014in the firm's own funds or in third party funds?\n\n9. Target client profile: What is the FA's typical client profile? Do you and your family or household fit it? What is the tenure of these clients?\n\n10. Communication of performance: What performance benchmarks will the advisor employ to measure progress against meeting goals and objectives? How will these be communicated to you? Most professionals shy away from promising specific performance objectives. A common caveat is that if a prospective advisor starts laying out specific parameters or expected rates of return, regard such promises with the degree of skepticism they deserve.\n\n#  **UNDERSTANDING THE HNWI ADVANTAGE: HOW THE WEALTHY MANAGE TO ENJOY ROBUST RETURNS DURING GOOD TIMES AND BAD**\n\nThe main reasons that the industry has shifted to servicing the needs and demands of wealthy and ultra-wealthy clients are extensively documented in the _WWR_ s. The overarching fact is that even during down periods when market returns tend to be dismal, returns enjoyed by the wealthy are generally quite robust. The wealthy tend to be beneficiaries of a virtuous cycle in which access to superior information, expertise and service from FAs and money managers further fuels the process of wealth aggregation and accumulation that produced their wealth in the first place. Even during the sharp slowdown of 2004-2005, when rising oil prices, inflationary pressures and monetary tightening took their toll on the world economy, the ranks of the ultra-wealthy continued to climb by 6.1 percent, while their total financial assets grew 8.5 percent.\n\n**Figure 1.1** \\- Primary Source of Wealth, Regional Breakdown, 2006\n\n_Source_ : Capgemini\/Merrill Lynch Relationship Manager Survey, 2007.\n\nWhen studying wealth, it pays to consider where the wealth comes from. According to the 2007 _WWR_ , 36 percent of the world's millionaires derive their wealth from the ownership or sale of a business, 22 percent from income, 20 percent from inheritance, 11 percent from investments, 9 percent from restricted stock and stock options and 2 percent from elsewhere.\n\nTake a good look at those numbers, particularly the 11 percent figure for investments. This figure tells us that HNWIs are typically talented at _not_ relying on global economic prosperity to expand their financial holdings through dividends and capital appreciation\u2014which helps in turn to explain why HNWIs' returns consistently beat market averages.\n\nAcross the industry, investment advisors have discovered that investment challenges and demands vary dramatically according to how a client's wealth was originally generated. In the case of inherited wealth, it's often found that much of the client's total assets have been invested for the long term, in stocks or real estate. Those clients shifting out of legacy investments could, as a result, suffer significant tax consequences from certain transactions. For those with inherited wealth, flexibility of style and strategy is in some cases reduced. Wealth earners, by contrast, tend to enjoy considerably greater flexibility and tend to be more creative and open to investment mandates.\n\nEntrepreneurs, particularly those who have enjoyed a huge wind-fall from the sale of a business, often require an investment strategy designed to put a lump sum of funds to work at a single point in time, projecting needs five and ten years out from the present. Income earners, on the other hand, enjoy the luxury of constructing portfolios over time through stable cash flows, which means their time horizons may prove flexible. For all three major sources of wealth\u2014income, business and inheritance\u2014the destination is typically the same, yet the route varies.\n\n#  **DESCRIBING THE STATE OF THE WORLD'S WEALTH**\n\nWhile it's a comparatively simple matter to identify global trends in wealth _accumulation_ , there remain strong regional differences in _sources_ of wealth origination. In Europe, business ownership, or the sale of a business, was the number one source of wealth, at 45 percent, while income, which is much more heavily taxed than in the United States, accounts for a mere 26 percent.\n\nIn North America, income accounts for 27 percent of wealth\u2014higher than anywhere else in the world. In the United States in 2005, the number of HNWIs and the value of their assets grew at twice the rate of the domestic economy\u2014driven largely by substantial gains in real income that have far outpaced gains at lower income levels. According to the Center on Budget and Policy Priorities, the poorest fifth of US taxpayers have seen a 6 percent growth in income ($900) over the 25 years prior to 2005, adjusted for inflation. Meanwhile, the top 1 percent saw their incomes jump 228 percent over the same period.\n\nIn the United States, where GDP and stock market returns have proven remarkably steady but hardly dramatic, this growing income gap and concentration of wealth is magnified by a long-term trend of scaling back the progressive income tax. As recently as 1980, the progressive income tax imposed a 70 percent tax rate on income above a certain level. That rate has been halved. Also, new compensation strategies\u2014such as restricted stock and stock options, which are not widely available to the population at large\u2014have helped HNWIs accumulate greater wealth at a faster pace than the economy at large.\n\nA study recently conducted by the Massachusetts Institute of Technology's Sloan School of Management found that 75 percent of chief executives culled from a sample of one hundred publicly traded companies enjoyed an average net worth of US$25 million in 2004, mainly from stock and options in their companies. That was up from 31 percent in 1989, adjusted for inflation. In 2005, CEOs in the United States earned 231 times what an average hourly worker at their company earned, up from forty-two times in 1980.\n\nAll over the world, earned wealth is expanding more rapidly than inherited wealth. Income plays the largest role in wealth accumulation in North America, but in every other region of the world, save the Middle East, business ownership creates the majority of wealth. As recently as 2001, according to the 2007 _WWR_ , 21 percent of North American wealth was inherited, compared to 17 percent in 2006, and 37 percent of European wealth was inherited, compared to 20 percent in 2006.\n\n**Figure 1.2** \\- Percentage Breakdown in HNWI Assets by Region, 2005-2008F\n\n_Source_ : Capgemini\/Merrill Lynch Ralationship Manager Survey, 2007.\n\nBecause of rapidly growing income among HNWIs in the United States, and a flurry of entrepreneurship in Asia, Russia and Eastern Europe as those societies embrace free markets, the global trend is toward earned wealth in general. (The diminished role of inheritance in wealth creation is due in no small part to the fact that there is no wealth to be inherited in former communist countries, where all the businesses were state owned.) If this trend continues, and all evidence suggests that it will, we can expect to see an ever-growing number of wealth earners joining the ranks of the world's HNWIs.\n\nIf the way wealth is accumulated varies in each region, it's no wonder that the pace of HNWI growth and asset accumulation also varies by region. North America and Europe are home to most of the world's HNWIs, about 6.1 million of the world's 9.5 million. But these regions grew the slowest in 2006. The Middle East, Latin America and Asia-Pacific all outpaced the more developed nations in minting new millionaires. Africa, albeit working off a tiny base, saw the biggest jump\u2014a 12.5 percent increase. And the same trends hold true for the accumulation of financial assets, which occurred most rapidly in Africa, the Middle East and Latin America, while wealth grew the slowest in Europe.\n\n**Figure 1.3** \\- HNWIs' Wealth by Region, in Dollars and Percentage Growth, 2004-2006\n\n_Source_ : Capgemini\/Merrill Lynch 2007 _World Wealth Report._\n\n#  **PRIME DRIVERS OF HNWI WEALTH**\n\nGains in local GDP and regional stock exchanges accounted for much of the disparity in HNWIs' wealth growth. In 2006, real GDP grew worldwide to 5.4 percent, compared with 5.0 percent gains posted in 2005. The accelerating growth of real GDP was partially driven by consistent performance of the world's most mature economies. While U.S. GDP growth in 2006 was relatively constant at 3.3 percent in 2006, compared to 3.2 percent in 2005, European economies saw slightly accelerated growth\u2014Germany (from 0.9 percent to 2.3 percent), U.K. (from 1.9 percent to 2.7 percent) and France (from 1.2 percent to 2.0 percent).\n\nHowever, emerging markets continued to outperform the rest of the world. In general, the reason for the disparity is that emerging markets continue to play a moderate game of \"catch up\" with major markets. Financial gains were particularly strong in Latin America, Eastern Europe, Asia-Pacific, Africa and the Middle East, where certain countries enjoyed real GDP growth rates that outperformed the global average of 5.4 percent. This trend was particularly noticeable in Brazil, Russia, India and China, four countries that are often grouped together because of their vast sizes and referred to as the \"BRIC\" nations. In 2006, China and India, for example, sustained real GDP growth rates of 10.5 percent and 8.8 percent, respectively, among the highest of any economy in the world.\n\n**Figure 1.4** \\- Real GDP Growth, 2004-2006\n\n_Source_ : Capgemini\/Merrill Lynch 2007 _World Wealth Report._\n\nMeanwhile, the other macro driver of wealth accumulation\u2014market capitalization\u2014handsomely rewarded those invested in regional stock exchanges. The S&P 500 in the United States bounced back with a 13.6 percent return in 2006, compared to 3.0 percent in 2005. Markets in Europe, Asia-Pacific and Latin America delivered even stronger returns, after record gains in these geographies in 2005 enticed foreign investors to invest even more heavily in these markets in 2006.\n\nIn Europe in 2006, markets rose in response to corporate restructurings, technology investments and a range of cost-cutting moves. In Germany, the DAX gained 22.0 percent for the year, albeit a decline from the 27.1 percent gains posted in 2005. France's CAC 40 and the London FTSE 100, too, were up by 17.5 percent and 10.7 percent respectively for the year. In Russia, the RTS Index saw a 70.7 percent return in 2006.\n\n**Figure 1.5** \\- Stock Market Returns, 2006\n\n_Source_ : Capgemini\/Merrill Lynch 2007 _World Wealth Report._\n\nAsia-Pacific markets\u2014with the exception of Japan\u2014remained particularly strong in 2006. The Shanghai\/Shenzhen market capitalization grew by 220.6 percent in 2006, mainly on the strength of Initial Public Offerings (IPOs). Also, smaller Asian markets, including Indonesia and the Philippines, did well in 2006, where Dow Jones Indexes returned 62.0 percent and 49.5 percent, respectively, on the back of strong economic fundamentals. Overall, we see HNWIs being interested in opportunities around the globe and moving quickly to take advantage of them.\n\nSuch outsized returns not only mint new local millionaires, they are drawing HNWIs' investment from other parts of the world. More than ever before, HNWIs are moving their investments across international borders and the reason is clear\u2014for HNWIs to stay ahead, they must have an international outlook. Of course, these outsized gains cannot continue indefinitely, but there's a sense among HNWIs and their advisors that many emerging markets have turned a corner, and confidence is growing that these far-flung economies are stable.\n\nA hallmark of today's global economic expansion is the way emerging markets are reinforcing each other's expansion\u2014a phenomenon most evident in China and in other commodity-rich emerging markets around the globe. China, with its abundance of cheap skilled labor, has rapidly become the factory to the world, and some experts expect its economy to surpass that of the United States by 2030. With the cash it's generating from manufacturing, it has the power to invest overseas\u2014and it is doing just that. Some investment is flowing into the real estate markets of Hong Kong and Singapore, but China's main focus has been to secure the raw materials necessary to keep its voracious economic engine humming. It 's doing this on many fronts, everything from, literally, building bridges and other infrastructure in Thailand and Myanmar to forging close relationships with Chile and Brazil to lining up allies in Africa.\n\nMeanwhile, the more stable and democratic countries in Latin America are only too happy to have another large trading partner to counterbalance the sway that the United States has long had in the region. For them, China represents a gigantic hedge against the United States, and may at long last permit the region to escape the boom-and-bust cycle that has dogged it for decades. This stability helps in the accumulation of wealth in this region. Brazil, Mexico and Venezuela, in particular, all profited from the worldwide surge in oil prices, buttressing Latin American growth in 2005.\n\nCommodities, particularly energy, have been the primary drivers of the Russian economy. The RTS Index rode to historic highs on the backs of its two primary exports\u2014oil and natural gas. The number of HNWIs in the country jumped by 15.5 percent and the accumulation of assets was even higher. These kinds of results lend credence to a report earlier this decade by Goldman Sachs that projects that by 2040, the BRIC nations will leapfrog the G7 nations in terms of total output. In fact, the BRIC nations grew faster than expected from 2003 to 2005. Morgan Stanley's emerging markets index is up 171 percent since the end of 2002, while its BRIC index has surged 262 percent.\n\nIf China is the factory of the world, India has become the outsourcing capital of the world. It too has a growing thirst for energy and other commodities. Its educational system\u2014though not uniformly excellent\u2014churns out hundreds of thousands of skilled workers every year; what's more, the widespread use of English, its status as a relatively stable democracy, its familiarity with capitalism (though not a wholly free market), the comparatively low wages, even the time difference with developed nations (India works while North America and Europe sleeps) are factors in India's success. None of this would have been possible, of course, had India not consciously decided ten years ago to begin dismantling the protectionist economic policies in place since its independence.\n\nElsewhere in the world the emerging market firmament is the Middle East, which accounts for nearly 65 percent of the world's oil reserves. It continues to benefit enormously from the large industrialized nations' ongoing and deepening dependence on fossil fuel\u2014not to mention the growing thirst of India and China, the latter of which has rapidly become the number two consumer of oil after the United States; rising prices continued to drive oil export revenues skyward in the Middle East and one need look no further to understand its economic success.\n\nIn emerging markets where exchange rates are allowed to freely float\u2014China's yuan being the notable exception though progress is being made\u2014the local currencies generally gained strength during the past decade of relative stability since the destabilizing Asian Contagion of 1997-98. This has been the case in commodity-rich nations such as Mexico, Brazil and Canada. The increased value of the local currencies boosts the wealth of local HNWIs, and it has another salubrious effect: a strong currency engenders confidence in the local economy, increasing reinvestment by local HNWIs, which in turn helps reinforce the economy. In a post-oil boom, for instance, oil revenue generated in the Middle East from sales to developed countries made a round trip. With no place to confidently invest at home, Middle Eastern HNWIs sent their investments overseas\u2014mostly back to Europe and North America.\n\n**The Cost of Living Extremely Well**\n\nHow much one earns is only half the picture when framing someone's financial well-being; the other half is how much one spends. It's no easy matter to accumulate wealth, no matter how impressive the financial returns, if your cost of living is spiraling upward. Quite simply, the \"admission and maintenance charges\" to a life of privilege cannot be overlooked when discussing the state of wealth. The exception is, of course, those who are the richest in the world, such as Oracle's co-founder, Lawrence Ellison, according to economist Christopher Carroll at Johns Hopkins University. In his article \"Why Do the Rich Save So Much?\" Carroll estimates that with Ellison's net worth in 2006 at US$16 billion and at a 10 percent annual rate of return, he would need to spend more than US$30 million a week just to stop accumulating more money than he already has, as well as eat into the US$16 billion.\n\nTo get a handle on costs, _Forbes_ has been tracking the price of a basket of luxury goods since 1976 with its Cost of Living Extremely Well Index (CLEWI). While the old saying \"if you have to ask, you can't afford it\" might still be true, this index is instructive nonetheless. It turns out that HNWIs' financial returns\u2014which generally beat the market average\u2014had better continue to outperform. HNWIs' cost of living is escalating much faster than the average consumer's. Since inception, the value of the _Forbes_ basket of forty-odd luxury goods\u2014which includes everything from a sable coat to a yacht to a catered dinner for forty\u2014has grown more than 700 percent, roughly twice the rate of the Consumer Price Index (CPI).\n\nFor a period in the nineteen nineties, the CLEWI and CPI rose at about the same rate, but since then the CLEWI has consistently risen faster than the CPI and 2006 was no different, with the average cost of such items as caviar, a thoroughbred yearling and a psychiatrist increasing 7 percent compared to the CPI's 4 percent rise. The histor y of the CLEWI does show a cyclical pattern to these costs, accelerating and decelerating over stretches of time, but it certainly never dips below the CPI.\n\nFor a more detailed look at HNWIs' costs, the _WWR_ began tracking prices on a regional basis to ascertain relative purchasing power in Asia-Pacific, Europe and North America. If you were a HNWI, where would you get the best and worst deals? Like _Forbes_ , the _WWR_ also uses a basket of luxury items, including five-star hotel rooms, spa visits and boarding school tuitions. Turns out the Europeans bore the highest cost of maintaining a wealthy lifestyle. Europeans are lagging the rest of the world in their population growth and asset accumulation, and now European HNWIs are shelling out more to live the same lifestyle as other HNWIs! HNWIs in Asia-Pacific, meanwhile, have the lowest cost of living of the HNWIs.\n\n#  **LOOKING FORWARD**\n\nAll of the above would certainly appear to add up to a healthy global environment for growing wealth in the coming decade. Only during one year out of the last decade\u20142000\u2014has the number of HNWIs and their assets actually declined. This success has not, as in the past, been largely limited to the developed nations. A hallmark of the past decade is that wealth has grown faster in the emerging markets than the developed ones, creating a more globally dispersed population of HNWIs.\n\nIn both emerging and developed markets, the rules of investing are the same. In order to stay ahead of the game, the wealthy and those who aspire to join their ranks must bear in mind the four pillars of wealth management:\n\n1. Time horizon;\n\n2. Risk tolerance;\n\n3. Liquidity needs;\n\n4. Performance expectations.\n\nPracticing disciplined wealth management means, first and foremost, protecting and growing the wealth of an individual or family. Although it might sound simple, regardless of strategy, maintaining an effective long-term focus on all four pillars of wealth management requires negotiating cascading layers of complexity. Wealth management is inherently multidimensional, taking into account everything from the health care costs of a grandparent, to tax strategies for the head of the household, to college savings for the children. Disciplined wealth management also must account for the fact all four of these pillars continuously change over time, requiring the frequent updating and refinement of even the best-laid wealth management plans. This remains the irrefutable reality for all wealthy clients, prospective wealthy clients and aspiring wealthy clients\u2014and of course their financial advisors, from Bar Harbor to Singapore.\n**CHAPTER 2**\n\n**Global Trends and What They Mean for International Investing**\n\n#  **COINING THE TERM \"EMERGING MARKETS\"**\n\nOne unseasonably warm day in September 1981, a young, Dutch-born banker employed by the International Finance Corporation (IFC, the private sector arm of The World Bank) stood sweating at a lectern in an auditorium at the Salomon Brothers headquarters in New York. He was preparing to pitch a rather zany idea he and a few colleagues had recently dreamed up for what they were calling a \"Third World Equity Fund.\"\n\nDuring a stint at Banker's Trust in the mid-seventies, Antoine van Agtmael, a graduate of the Netherlands School of Economics with a master's degree in Russian and Eastern European Studies from Yale, had fallen madly in love with the frequently turbulent and often lackluster economies of what was then referred to as the \"Third World.\" Many years later, he vividly recalled the indignant reaction he induced from a senior executive at that major money-center bank when he asked for his opinion of the prospect of recycling petrodollars overseas. \"There are _no_ markets outside the U.S.!\" the senior banker had thundered while pulling wrathfully on his bright crimson suspenders.\n\nAlthough van Agtmael realized that the senior executive was not literally correct in his assessment, the man's provincialism accurately mirrored the investment landscape of the seventies and early eighties, when not just North Americans but just about every investor on earth displayed a strong propensity for investing in their own local markets, if for no other reason that for even the wealthiest individuals to invest in markets beyond their own jurisdiction was logistically and practically next to impossible, not to mention prohibitively costly.\n\nThe young banker had every reason to anxiously reflect on that unsettling moment of six years before as he leaned into his lectern to launch his presentation to prospective investors, a thesis premised on the decidedly unconventional proposition that\u2014as he later put it\u2014\"our data demonstrated a real possibility of making real money in [what would come to be called] emerging markets, despite their admitted volatility.\"\n\n\"Developing countries, we argued, enjoyed higher economic growth rates and boasted a rich set of hitherto-ignored promising companies. We persuasively demonstrated that investing in a basket of companies and countries would provide the diversification required to mitigate the risk of investing in individual stocks and countries.\"\n\nJudging from the faces in the crowd of thirty-odd fund managers staring up at him as he spoke, van Agtmael sensed \"that some were clearly intrigued, others were skeptical, and that just possibly we might win over one or two confirmed skeptics to our cause.\" At the conclusion of van Agtmael's presentation, a well-respected senior investment banker, Francis Finlay of JPMorgan, helpfully observed, \"This is a very interesting idea you've got there, young man, but you will _never_ sell it using the name 'Third World Equity Fund'!\"\n\nBriefly crestfallen, van Agtmael nevertheless recognized that the man had a point. The term \"Third World,\" with its vaguely derisive tone that strongly evoked \"flimsy polyester, cheap toys, rampant corruption, Soviet-style tractors, and flooded rice paddies,\" was hardly likely to induce any great sense of confidence in the traditionally conservative community of international bankers and brokers.\n\nOver the course of the following weekend, the aspiring international fund manager racked his brain for a more uplifting alternative label. Just before his self-imposed deadline of Monday morning, he conceived of a term that struck him as \"more positive and invigorating\" and that was destined to become the preferred nomenclature for this fledging field for the next several decades: _emerging markets._\n\nHaving experienced a classic eureka moment, van Agtmael sat down at his desk on Monday morning and dashed off a memo mandating that he and his colleagues henceforth refer to their forthcoming vehicle as The Emerging Markets Fund. The first index they were working on that tracked the performance of \"Third World\" firms would be promoted as the IFC Emerging Markets Index. Although it took a few years of hard work and artful persuasion to pull the deal off, IFC did in fact introduce its pioneering Emerging Markets Growth Fund (managed by Capital Investment, Inc.) in the mid-eighties. Fortunately not just for van Agtmael but for the world at large, this first fund of its kind was healthily subscribed to by an intrepid group of institutional investors from around the world who over time did very well by their initial investments\u2014thank you.\n\nNot long thereafter, another pioneering international investor, the British-born Sir John Templeton, hired Mark Mobius, a Long Island-born, Hong Kong-based pioneer expert in Asian and Latin American investing, to head up the Templeton Emerging Markets Fund, later subsumed into the family of funds offered by Franklin Templeton. Both van Agtmael's own firm, established with a group of colleagues from The World Bank, and Mobius's Templeton Emerging Markets Fund began to solicit subscribers shortly before the stock market crash of October 1987\u2014anything but an auspicious time to entice new investors with exotic strategies.\n\nYet the most significant geopolitical phenomenon of the latter part of the twentieth century\u2014the downfall of the Berlin Wall and concomitant collapse of communism two years later in 1989\u2014led to a radical revision and liberation of capital markets, not just in Eastern Europe and the former Soviet Bloc but also in Asia and Latin America. The resulting unleashing of capitalistic energy and vitality throughout the nineties, spurred by often controversial and in some cases corrupt privatizations of government-controlled firms, became a driving force behind the most powerful economic trend the world had yet seen: globalization.\n\nDespite a series of traumatic international crises in the nineties, including the Mexican peso devaluation of 1994 and the Asian Contagion of 1997 (which started in Thailand with the devaluation of the _bhat_ but quickly spread to all the emerging markets, ultimately leading to Russia's default on its sovereign debt), the fundamental premise of emerging-markets investing\u2014that \"developing countries...enjoyed higher economic growth rates and boasted a rich set of promising companies,\" as van Agtmael opined\u2014was a rich promise destined to be fulfilled exponentially over the coming decades.\n\n\"In 1988,\" van Agtmael recalled, \"when we launched our first fund, there were just twenty companies in emerging markets with sales of over US$1 billion... By 2005, there were thirty-eight companies in the emerging markets with sales of over US$10 billion, and 270 with sales of over US$1 billion.\" By midway through the first decade of the twenty-first century, a significant number of emerging-markets-based multinationals\u2014including Samsung Electronics and Hyundai of South Korea, iron-ore producer CVRD of Brazil, energy producer Gazprom of Russia and synthetic fuel producer Sasol of South Africa, to name just a few\u2014had attained positions of global dominance as the leading producers in their industries.\n\nWhile the stunning rise to economic prominence and prowess of India and China (following the equally remarkable growth of the Asian tigers, including Taiwan, Korea, Thailand and Singapore) easily garnered the highest volume of anxious headlines in the North American and northern European press, the reality is that the advent of the \"emerging-markets century\" had fundamentally altered the investment equation for mass-affluent and high-net-worth individuals and households worldwide.\n\nAnother defining moment for the newly globalized century came about in 2007, when Carlos Slim Helu, who built a massive empire out of the privatization of Mexican telecom giant Telmex, was hailed as the world's richest man, moving up from the number three position he'd held on the _Forbes_ list and smartly surpassing both Buffett and Gates.\n\n#  **GLOBALIZATION'S GROWING IMPACT ON HIGH-NET-WORTH INDIVIDUALS**\n\nAs Larry Fink, chairman and CEO of asset management giant BlackRock\u2014now owned in part by Merrill Lynch\u2014puts it: \"Globalization has been _the_ transformational theme of the past six years in investing.\" The International Monetary Fund (IMF) defines globalization as \"the growing economic interdependence of countries worldwide through increasing volume and variety of cross-border transactions in goods and services, free international capital flows, and more rapid and widespread diffusion of technology.\" Typically, this broad phenomenon is broken down into a cluster of discrete facets, just about every one of which powerfully influences the investment complexities, decisions and opportunities facing wealthy individuals today.\n\n\u2022 Industrial globalization\u2014the rise and expansion of multinational enterprises;\n\n\u2022 Financial globalization\u2014the emergence of worldwide financial markets and better access to external financing for corporate, national and sub-national borrowers;\n\n\u2022 Political globalization\u2014the spread of political spheres of interest to regions and countries outside the neighborhood of purely political (state and non-state) actors;\n\n\u2022 Informational globalization\u2014the increase in information flows between geographically remote locations;\n\n\u2022 Cultural globalization\u2014the growth of cross-cultural contacts.\n\nAs Bertrand Lavayssi\u00e8re and Bob McCann pointed out in the foreword to this book, the world is irreversibly evolving from a two-pole into a multi-polar world, with Asia, Latin America and the Middle East emerging alongside North America and Europe. This broad-based phenomenon, in which nations formerly below the radar of capital markets are now drawing serious investment from developed countries, is largely due to the fact that so much of the incalculable wealth being produced and accumulated in Asia, Latin America and Eastern Europe is now being reinvested in the exploding emerging markets, reversing a longstanding trend in those countries to experience the dwindling of assets that goes along with capital flight.\n\nNot only patriotic or nationalistic impulses prompt so many wealthy emerging-markets-based investors to keep their assets at home. It's also the fact that a gradual \"decoupling\" of the Asian and Latin American economies from their counterparts in North America and Western Europe has substantially reduced the risk of investing in Asia and other emerging markets, while brightening the unprecedented opportunity of attaining growth rates unparalleled by any other region on earth.\n\nAccording to Mr. van Agtmael, one of the primary reasons that emerging markets have soared so dramatically in recent years is that wealthy investors located inside those markets altered their perception of the risk of investing closer to home. \"Traditionally,\" he points out, \"you saw significant volumes of capital inflow into emerging-markets countries based on workers' remittances sent back by immigrants home to their families. Historically, you also saw emerging markets' entrepreneurs and wealthy households\u2014many of whom owned their own businesses\u2014investing in the more developed markets out of an interest in diversification, because they and their financial advisors understood that so much of their assets were tied up in their own businesses that it made sense to invest some significant portion of their liquid capital offshore.\"\n\nEmerging markets' entrepreneurs from Asia and Latin America were also enthusiastic depositors in the famously discreet tax havens of Switzerland, Luxembourg, the Isle of Man and other low-tax jurisdictions. \"The common phenomenon of capital flight was nothing but simply rational economic behavior in its day,\" van Agtmael contends. \"But at some point, as the risk premium shifted, all that began to change.\"\n\nEven in the wake of the integration of global capital markets, a \"certain lingering prejudice prevented many wealthy individuals and investors in emerging markets from grasping the fact that their own economies were doing so well.\" It took a while for the lesson to sink in even among the savviest investors that, for example, infrastructural spending by emerging markets' governments was spurring a wave of internal investment leading to rapidly expanding economies. Transparency on the part of corporations aspiring to tap the international capital markets improved the investment profile of many firms that had hitherto operated according to the loosey-goosey rules of \"crony capitalism.\"\n\nSeptember 11th and its aftermath, which prompted a rising xenophobia among some American investors and the American public, has played a role, van Agtmael says, in influencing emerging markets' investors to keep their money at home. \"Particularly in the Middle East, many wealthy individuals and investors feel that their funds and assets are no longer welcome in the U.S. Nowadays, they're more inclined than ever to invest in their own regions, which they regard as in many ways safer, more solid and offering a higher rate of return than the former safe haven of the U.S.\" By 2008, declining asset prices in the U.S. touched off a wave of foreign investment in the U.S., with newly emergent Sovereign Wealth Products (SWPs) leading the way in infusing capital into the world's largest economy at a rate unrivaled in modern memory.\n\nThe global trend toward economic liberation has been intertwined with advances in information technology (documented in Chapter 3), enhanced by the greater ease of transportation of goods, funds, people and, most importantly, ideas since the downfall of the Berlin Wall in 1989. It is a classic positive feedback loop, producing the net effect of a world that is ever more tightly knit.\n\nNot all leaders and people, of course, enthusiastically embrace globalization. While its proponents credit the phenomenon with being an engine of commerce that brings an improved standard of living to developing countries and additional wealth to developed nations alike, its detractors argue that globalization is a latter-day form of corporate imperialism that tramples human rights in developing countries, destroys their local ecologies and steamrolls indigenous traditions through cultural assimilation. At the same time, foes of globalization maintain that it erodes the living standards of blue-collar workers in developed countries, whose jobs are shipped overseas, thus exacerbating the already widening gap between the rich and the poor. What proponents consider commerce and prosperity, detractors regard as plundering and profiteering.\n\nJames Rothenberg, chairman and principal executive officer of Capital Research and Management Co. (a principal subsidiary of the Capital Group Companies, and an investment adviser to the American Funds Group), observes that \"globalization is a powerful force. But capitalism, while it may lift all boats, also creates enormous disparities.\" Mr. Rothenberg, who has held his current position since 1994, points to \"the rise of populism and socialism associated with leaders like Venezuela's Hugo Chavez. Will you see actual rebellion or revolts driven by populism? You do have to worry about that. Globalization may not be reversible, but it will have bumps along the way. I don't want anyone to believe that the trend is a 45 degree uninterrupted up line.\"\n\nIt should not go unnoticed that more than 87,000 _officially acknowledged_ riots and demonstrations broke out in China in 2005, nearly all of them connected to the state's seizure of farmland for development. Protectionist forces in the United States and European Union (EU) are acting more assertively against cheap Chinese imports, demanding trade barriers and basing their actions on an argument that China has unfairly exploited its natural competitive advantages by means of currency manipulation. Ultimately, globalization benefits from the participation of as many as possible, so addressing local concerns is not merely an act of goodwill, it's a prudent strategy to avoid a globalization backlash.\n\nJacques Attali, an academically trained economist, former French presidential advisor, the first president of the London-based European Bank for Reconstruction and Development and today the president of PlaNet Finance, plays his own private part in promoting globalization (and preventing the possibly forthcoming populist backlash against it) by offering financial assistance to micro-lending programs around the world. Muhammad Yunus of Bangladesh, awarded the Nobel Peace Prize in 2006 for his pioneering microfinance work at Grameen Bank, is co-president of the PlaNet Finance international advisory board. Attali points out that the last great wave of globalization concluded with the onset of World War I in August 1914. The year before, foreign trade accounted for 13 percent of the world's GDP, but from there foreign trade went into a prolonged reversal, from which it failed to fully recover for nearly eighty years. Not until 1991 did global trade attain its pre-World War I peak of 13 percent of GDP.\n\nThe reason? Two world wars and a cold war, which cordoned off much of the globe, as well as the fading sway of the world's nineteenth century superpower, England. Before World War I, Britain, with its global empire and naval prowess, kept world trade routes open. But as its might waned after World War I, countries began enacting trade barriers. By the thirties, many developing countries taxed imports by as much as 75 percent, virtually wiping out world trade.\n\n\"There is no reason to believe this kind of reversal cannot be repeated,\" Attali warns. \"The risk is that there will be globalization of markets without a globalization of democracy and regulations.\" This, he contends, would further disenfranchise large swaths of the earth's population while discouraging more widespread participation in the global economy, further debilitating an economic system reliant on connectivity and participation. As mentioned earlier, globalization, like the Internet itself, derives its power from participation. The greater the number of people engaged in globalization, the more everyone involved benefits. As Attali observes, \"What's the value of the Internet if you're the only one on it?\"\n\nYet nearly all economic observers would likely agree that the prevailing momentum today is propelling globalization irreversibly onward and upward. Most economies of the world are making an effort to open their borders and capture increased foreign and domestic investment in their economies. Still, the integration of capital markets is not without risk\u2014risk on a global scale hitherto unimaginable.\n\n\"Thanks to globalization, systemic risks are everywhere,\" insists Edward Bernard, a vice chairman at T. Rowe Price. He cites \"interest rate increases in Japan, depreciation of the U.S. dollar, an industrial accident in China, [and] a hedge fund meltdown\" as all potential examples of systemic risk that might pose a shock to the international financial system\u2014shocks against which, he contends, the vast majority of enthusiastic international investors have failed to prepare. One way to prepare is to retain exposure to international markets while limiting risk, by investing in global blue-chip companies that sell their products in emerging markets. Another would be to maintain a prudent asset allocation that limits exposure to these more volatile markets to a set percentage of the portfolio, in much the same way that institutional investors practice portfolio risk management.\n\nLarry Fink of BlackRock strikes a similarly cautionary note regarding the twin dangers of global systemic risk and investor complacency. \"The biggest worry I have\u2014and many of my contemporaries have\u2014is that the market is not pricing in liquidity risk.\" Liquidity, generally defined as the ability of an entity to sell an asset quickly without substantially affecting its price, certainly seized up during the international credit crisis of 2007 and 2008. A wholesale \"breakage in liquidity,\" according to Fink, was presaged by the fact that in early 2007 the market was pricing liquidity risk for emerging-market stocks as aggressively as it priced liquidity risk for Internet stocks in 1999, which was to say that there was virtually no risk.\n\nNot surprisingly, Fink's concerns regarding liquidity risk proved well grounded by the summer of that turbulent year. Investors alarmed by subprime fallout and reversals in other mainly developed markets could point to a number of danger signs erupting circa 2006, including the collapse of the US$9.2 billion U.S.-based hedge fund Amaranth Advisors, which lost US$6.5 billion in less than a month and had to shut down because its sophisticated algorithms and risk modeling did not predict a crash in natural gas prices entering the fall heating season.\n\nThe Thai government rattled markets later in that year by announcing capital controls on foreign investments in stocks, bonds and commercial paper, which, while eventually reversed, shook the confidence of international investors and evoked memories of the Asian Contagion of ten years before. Russia, for its part, which had defaulted on its debt in 1997, flexed its newfound energy-rich muscles by strong-arming Royal Dutch Shell into turning over a majority stake in a US$20 billion natural gas project in the Far East. As for China and India, while both economies continued to expand at dizzying rates, many observers cautioned that such rates were unsustainable and would at some point decline\u2014perhaps precipitously.\n\nCaveats notwithstanding, the fact is that HNWIs from both developing and developed markets have little choice but to invest in emerging markets to maintain earnings that meet today's enhanced expectations. The aging populations in Europe and North America, in particular, have plenty of money to invest, and they and their advisors must seek above-average returns to fund decades-long retirements.\n\n#  **CHINA: AN EMERGING MARKET FOR FOREIGN INVESTMENT**\n\nJust twenty-five years or so ago, Chinese leader Deng Xiaoping famously said\u2014perhaps apocryphally\u2014\"to get rich is glorious,\" unleashing market forces that challenged socialist orthodoxy and ultimately turned it on its head in the most populous nation on earth. According to _Forbes_ , it took fewer than twenty years for China to mint its first billionaires\u2014the Liu brothers, founders of the Hope Group, who made the famous _Forbes_ list in 2001. The fact that Deng's declaration occurred just before the rise of globalization is no coincidence; China's headlong participation in the world economy has been a primary driver of the phenomenon.\n\nSince the nineties, the government of China has almost exclusively focused on foreign trade and exports as its chosen vehicle for attaining economic growth, gradually opening its market to foreign investment by creating special economic zones where investment laws are relaxed in order to attract foreign capital. The result has been a six-fold increase in GDP since 1978, with an average annual GDP rate of 9.4 percent for the past 25 years. By the end of 2005, China had become the world's fourth largest economy by exchange rate and the second largest in the world after the United States by purchasing power parity.\n\nAll this has propelled China, the former economic basket case, from an agrarian, inward-looking society to one enthusiastically embracing technology and all that the twenty-first century promises. Consider this: In 1995, there were virtually no cell phone users among the country's population. By 2005, there were 400 million subscribers and China had become the number one consumer market for mobile phones in the world.\n\nThe Chinese economy is still far from completely open and transparent, yet even its staunchest critics are inclined to concede that considerable progress is being made. China is attempting to harmonize the system of taxes and duties it imposes on enterprises, domestic and foreign alike; in many industries where the old system in which personal \"connections\" were necessary to conduct business, crony capitalism is diminishing. On the downside, bankruptcy rules, intellectual property rights and ownership rules remain murky or nonexistent. The rules governing foreign ownership of businesses, for instance, vary from industry to industry. In the retail industry, firms such as Wal-Mart, Best Buy and Tesco have been permitted to purchase significant stakes in Chinese companies since the start of 2005, but the foreign banks are more hamstrung by government fear and fiat.\n\nThe Chinese fixed exchange rate has also been a sore point with trading partners. Here, too, a decision in 2005 to move to a floating peg, allowing the Chinese renminbi to move against the U.S. dollar in a narrow \"trading band,\" represents a form of progress. But Merrill Lynch's CEO in Taiwan, Albert Lee, seriously doubts that China will permit a fully floating currency any sooner than 2010, because to do so would endanger many exports by making them more expensive. Not only do exporters account for a third of the economy, but they employ ninety million people and operate on razor-thin margins. A drop in sales could force millions out of work. And in a country where millions of people are leaving their farms each year to work in the cities and factories, that could spell disaster. \"They need to keep a tight lid on it; they can't just let it go,\" Lee insists.\n\nOne major policy change that Lee does expect to see by 2010 is the granting of permission by the Chinese government to the country's citizens to invest directly offshore, something they can't do today. China tends to prefer a gradual approach in its economic reforms. According to Lee, the Chinese government is likely to go this route when it comes to loosening capital controls. He looks to Taiwan as a possible model. In 1985, Taiwan first allowed citizens to move investments overseas. The amount started at US$100,000 and has gradually ratcheted up until today a Taiwanese citizen can move US$5 million out of the country. Given the vast wealth creation occurring in China, the prospect that some capital will start moving offshore is a tantalizing one, both for the HNWIs who live in China as well as for financial institutions, which could offer these HNWIs a whole new array of products and services. It's also an encouraging prospect for globalization itself, since the free flow of capital is one of its cornerstones.\n\nThe expectation that the Chinese market will one day be a huge market for wealth management services added some degree of urgency to a recent competition among more than one hundred financial firms to win a mandate from the Chinese government to invest money on behalf of its US$28.5 billion social security fund. The deal is relatively small in terms of money management, so the ten firms eventually chosen are clearly eyeing future opportunities. China is a society of savers, with an estimated US$2 trillion to US$3 trillion in bank accounts, while the mutual fund industry remains small, roughly the size of the U.S. mutual fund industry in the late seventies.\n\n**A New Approach to Wealth Management: Private Banking in Asia**\n\nBorn and educated in Taiwan, Albert Lee obtained a law degree there and attended a graduate program in finance at the University of Illinois at Urbana-Champaign before practicing law with a prominent firm in Taipei. After joining Citibank's Taiwan office as a management trainee, he rose to department head and branch manager of the institutional finance division before joining Merrill Lynch in 1991 as a private banker in Taiwan.\n\n\"Merrill's banking license at that time was strictly advisory,\" Lee recalls. \"This was the early stage of wealth management in this virgin territor y. The Taiwanese government was concerned about granting too much infl u-ence to foreign wealth managers. In Taiwan and elsewhere in Asia, there was virtually no concept of private banking.\" As the regulatory environment gradually loosened up in Taiwan, by 2001 Lee was managing three Merrill Lynch branches there and had over a hundred FAs in Taiwan and Singapore reporting to him.\n\nA year later, Lee transferred to Merrill's Hong Kong office as chief of its Global Private Client division for the entire Pacific Rim. \"Wealth management as a separate entity scarcely existed at that time,\" Lee contends brightly. \"I witnessed\"\u2014and personally participated in\u2014\"the development of private banking in Asia.\"\n\nAs a discipline, wealth management was still immature compared to the more developed markets in Europe and the United States, according to Lee. \"There remained a casino mentality in some of the stock markets in Asia, which remain highly volatile.\" By Lee's account, when he started out in the business in the early nineties, Merrill's major competitor in private wealth management in the Pacific Rim was Citibank, with JPMorgan running a distant second. \"Merrill was still largely perceived by many high-net-worth individuals,\" he modestly says, \"as mainly staffed by stock jockeys.\"\n\nLee made it his mission to \"raise the image of Merrill in Asia.\" This was not only a matter of attracting top clients; it was a matter of attracting top talent. \"In Hong Kong we were successful at building up our image even in an immature market. Yet for some years, it was difficult to recruit FAs from Citi, JPMorgan or Morgan Stanley because so many expressed a disinterest in joining what they regarded as a 'stock brokerage firm' as opposed to a 'private wealth manager.'\"\n\nUnder Lee all that was destined to change. \"Client behavior evolved dramatically during the final decade of the twentieth century. Alternative investments, including hedge funds, private equity, other direct investment opportunities and a variety of more sophisticated synthetic, structured and derivative products, really took off during that time. The economic development of China created a dramatic wealth effect throughout the region.\"\n\nOn the private wealth management side, \"If a client was still stock picking, we would attempt to convert the client to the gospel of asset allocation. Some clients would respond, 'Don't tell me about asset allocation. I've got some money with you that I want you to play with and give me a high return.' And as long as the client understands the risk and rewards involved, we will adapt to that client's personal style. Yet others were intrigued by this asset allocation idea, and determined to check out this new mantra.\"\n\nAt the same time, during the tech boom of the late nineties, Lee observed significant changes in client behavior. \"I noticed in Taiwan, Hong Kong, Singapore and the Philippines a sharply increased appetite for derivative products.\" Today, Lee\u2014who was recently appointed CEO for Merrill Lynch in Taiwan\u2014is keen on the fact that \"at last count we have 280 billionaires in our region. Every financial firm is pitching product to these clients. Clients are no longer satisfied with our just picking stocks. In general clients are getting smarter, which is another way of saying that the FAs are doing a better job of educating them about new opportunities in the markets.\"\n\nIn the Pacific Rim, Merrill's latest thrust is its Private Banking and Investment Group, a joint venture between its Global Wealth Management (GWM) retail division and its Global Markets and Investment Banking Group (GMI). As Lee describes it, \"FAs draw on the specialized knowledge and product specialists within GMI while GMI gains access to the deeper and more personal relationships enjoyed by GPC and the FAs.\"\n\n#  **INDIA: A GROWING MAJOR ECONOMIC FORCE**\n\nChina's giant neighbor, India, is another growing power in Asia, and the liberalization of its capital markets has also played a significant role in the pace of the region's globalization. With a GDP of US$719.8 billion in 2005, India has become the world's twelfth-largest economy and the second-fastest growing major economy in the world, with a GDP growth rate of 8.8 percent as of 2006. It boasts the world's third-largest GDP of US$4.042 trillion as measured by purchasing power parity (PPP).\n\nSince its independence in 1947, India has mostly followed a socialist-inspired approach to its economy, with strict government control over private sector participation, foreign trade and foreign direct investment. But since the early nineties, India has gradually opened up its markets through economic reforms by reducing government controls on foreign trade and investment. The privatization of publicly owned industries and the opening up of certain sectors to private and foreign interests has proceeded slowly amid the kind of political debate impossible in China. But as was the case with China, globalization has gathered momentum from its participation.\n\n**India and China: Two Powerful Trading Partners**\n\nEconomic relations between India and China were strained for decades as the two jockeyed for economic advantage and tried to prove which of their nations was better equipped to sustain the high growth rates of the past several years. But lately the two are driving growth through stronger ties, and this is expected to continue.\n\nAt the heart of this historical tension are complex social and political differences. India is the world's largest democracy, with a mixed history of entrepreneurialism and state control. China is the world's largest communist state and now a cauldron of capitalist experimentation. There are large swaths of disputed territories in India's northeast, and while not as prone to violence as the India\/Pakistan tussle over Kashmir, the feud does simmer.\n\nBut there are growing signs that the differences that have hindered trade between the two nations for many years are acting as trade stimulants, with each side poised to take advantage of its distinctive strengths and, ultimately, complementary differences\u2014in China's case, a genius for manufacturing and computer hardware, and in India's, for services and software.\n\nFrom 2001 to 2005, trading volume between India and China grew by more than 50 percent annually, reaching US$18.7 billion in 2005. By contrast, during the same period, India-U.S. trade grew by 18.7 percent annually. A further sign of recent progress: in July 2006, the Nathu La Pass\u2014the former silk trade route between China and India\u2014reopened after a forty-four-year closure, creating new overland trade opportunities to complement a trading relationship that has existed primarily by sea. Predictions that China will become India's largest trading partner in the near future seem highly credible.\n\nThe economic power inherent in this burgeoning relationship is likely to be a welcome hedge against some of the risks associated with being too reliant on the United States and EU, whose growth rates are expected to slow over the next several years. If India and China succeed in supporting each other's growth, the Asia-Pacific region may one day outgrow its dependence on the economic health of western countries and become self-sustaining.\n\nToday, India's economy is increasingly diverse and encompasses agriculture, textiles, manufacturing and a multitude of services, including high-tech outsourcing. Although two-thirds of the Indian workforce still earns their livelihood directly or indirectly through agriculture, service is a growing sector and is playing an increasingly important role in India's economy. The advent of the digital age, and the large number of young and educated populace fluent in English, is gradually transforming India. India is a major exporter of highly skilled workers in software, financial services and engineering. And the country has quickly emerged as an important \"back office\" destination for global companies as they look to cut costs by outsourcing customer services and technical support. India continues to push the envelope, moving higher and higher up the ladder in the provision of more strategic and front office services.\n\nIndustrial policy reforms have substantially reduced licensing requirements, removed restrictions on expansion and facilitated easy access to foreign technology and foreign direct investment. India liberalized its foreign direct investment policies in 2005, allowing a 100 percent FDI (Foreign Direct Investment) stake in some ventures, such as the construction business, and reforms in the retail sectors are expected shortly.\n\nNot only do these reforms pave the way for non-domestic HNWIs' investment in the Indian economy, they lay the groundwork for those in India's huge middle class, 300 million strong, to move up into the ranks of the wealthy. Reform significantly impacts a class of investor unique to India: Non-Resident Indians (NRIs). NRIs have in the past sent home more than US$20 billion a year to support family members. But due to a lack of good investment opportunities and irksome capital controls, NRIs tended not to invest directly in the Indian economy. Now that's changing as investment options burgeon and capital controls are expected to gradually lift. This new \"financial connection\" that will accompany the long-nurtured \"cultural connection\" is sure to create even more wealth on the subcontinent in the coming decades.\n\n**Non-Resident Indians Play a Unique Role in Wealth Accumulation in India**\n\nNon-Resident Indians (NRIs) are people of Indian origin or descent working or living outside of India\u2014and their influence on the future wealth accumulation within India is likely to be significant. Even NRIs that left India more than a generation earlier have maintained strong cultural and economic links with India. In 2005, NRI deposits in the banking system totaled US$33 billion and remittances amounted to US$21.7 billion, representing approximately 3 percent of India's GDP and four times its foreign direct investment. From 2002 to 2005, NRIs acquired more than US$3.3 billion of shares on Indian stock exchanges.\n\nMany NRIs are successful business owners in the textile, gem and jewelry, hospitality and trading industries, and are integral to the business communities in Hong Kong, Singapore, Indonesia, Thailand and Malaysia. Additionally, the NRI community contributes many skilled workers to the financial services, information technology and shipping industries. Because of their strong local business ties in their adopted region and the maintenance of ties with India, NRIs are particularly global in their outlook\u2014both in terms of their business interests and personal investment opportunities.\n\nAccording to the 2007 _WWR_ , HNWIs in the Asia-Pacific region\u2014numbering about 2.6 million\u2014trade actively in both the local and international financial markets. As one Hong Kong NRI financial advisor commented, \"Some of my NRIs are more knowledgeable about Hong Kong stock markets than [are] the local residents!\" And due to the close-knit nature of the NRI community, NRIs openly share investment information with each other and provide excellent referrals for the wealth advisors lucky enough to work with them. The professional NRI client segment offers some distinctive opportunities for specialist financial advisors. Because of currency convertibility restrictions, it makes sense for these Indians to retain some wealth offshore in foreign currency. Many hold Indian ADRs (American Depositary Receipts) or mutual funds, and some have gone through the process of registering for the Indian government's NRI Portfolio Investment Scheme, enabling them to trade in Indian domestic stocks from overseas. Business owners are inclined toward high-volume transactions, say advisors, and make bold opportunistic moves, such as margin trading.\n\nAt the same time, NRIs are highly demanding clients who are notorious for spreading their investments and comparing pricing across numerous wealth management advisors. From the standpoint of the financial advisor, therefore, NRI business owners make for difficult, but rewarding, clients. To this end, nearly all of the international private banks, along with many of the domestic Indian banks, have set up teams of financial advisors specializing in servicing NRIs.\n\nIn early 2007, a Goldman Sachs report forecast that India could overtake Britain and boast the world's fifth-largest economy within a decade, as the country's growth accelerates apace. Moreover, contrary to the demographic profiles of many developed countries, India's workforce will remain young and robust; the massive retirement waves likely to stall growth in other regions of the world will not figure in the near futures of most Asia-Pacific markets, with the possible exception of Japan and China. As a result, India's HNWI population\u2014along with the opportunities for accumulating wealth\u2014is likely to expand significantly in the years to come. Investors might be well served to watch and determine how much of their portfolio they can allocate to similar vehicles based on the experience of India.\n\n**Merrill's Passage to India**\n\nFor the first time, India has become the largest revenue contributor among Asia's emerging economies for Merrill Lynch, highlighting the country's increasing importance for global investment banks. A record flow of share offerings and significant mergers and acquisitions, coupled with increasing principal investment by Merrill Lynch, meant that \"India has finally become relevant to Merrill Lynch in revenue terms, not just regionally but globally,\" Patricia McLaughlin, Merrill's vice-chairman and managing director for investment banking in India proudly informed _The Financial Times_ in August, 2007.\n\nAs global investment banks rushed in to create beachheads in India during the past few years to take advantage of a rapidly growing economy, mergers and acquisitions volume in India rose in 2006 to an unprecedented US$63.9 billion, more than double that of the same period the previous year, according to data firm Dealogic. A rise in new share issuance was responsible for investment banks underwriting a record US$23.3 billion of share sales in India within the first six months of 2006. Just five years earlier, by sharp contrast, Indian companies sold a comparatively paltry US$603 million worth of shares. In 2007, Merrill plunged even further into India's rapidly expanding real estate market, paying US$377 million for a 49 percent share in a portfolio of residential projects managed by DLF, the country's largest listed developer, in one of the largest deals of its type ever struck on the subcontinent.\n\n#  **THE EUROPEAN UNION: THE WORLD'S LARGEST INTEGRATED ECONOMY**\n\nThe rise of China and India has grabbed so many headlines of late that it can be easy to gloss over the pivotal role that Europe has played in the evolution of globalization. The Maastricht Treaty, defining the guidelines for the EU in 1992 and following soon after the fall of the Berlin Wall, took a continent of more than two dozen countries and created a single market. In fact, it created the world's largest single market.\n\nThe EU consists of a customs union, a single currency managed by the European Central Bank (in 2008, fifteen of the twenty-seven member states are using the euro), and a set of policies to create a single market of more than 350 million customers. The Schengen Agreement abolished passport control for some member states, and customs checks were also abolished at many of the EU's internal borders, creating a single space of mobility where EU citizens can live, travel, work and invest.\n\nThe EU remains the world's largest integrated economy, with a 2005 GDP of US$13.5 trillion compared to US$12.5 trillion in the United States, according to the IMF. Enhanced trade and higher growth from the newest EU members of the former Eastern Bloc are helping to drive EU growth. Indeed, private banks are now grappling with how to service the emerging millionaires of Eastern Europe, a group that is growing very quickly, but whose private banking preferences are still largely unknown.\n\nFor all the strides that the EU has made during the past fifteen years, it faces significant long-term challenges. As briefly mentioned earlier, low birth rate and aging demographics are likely to weigh on economic growth, but so will EU policies that may make the continent a bit less competitive on the world stage. The EU is a highly regulated, highly taxed jurisdiction. Even the tax and privacy haven of Switzerland, which has not joined the EU, must pay some heed. Switzerland has agreed to levy a withholding tax (initially at 15 percent) to return on savings paid to the citizens of EU member states, which means that EU citizens cannot sidestep taxes as they once could.\n\nIn fact, EU rules may eventually topple Switzerland's place at the pinnacle of offshore banking; Singapore is poised to capture that honor, positioning itself as a tax-friendly alternative and aggressively courting emerging Asian HNWIs and HNWIs from the developed markets. That said, one should not overstate the plight of Switzerland, currently the third-largest center of wealth in the world. Even if overtaken by Singapore, Swiss banking is in no danger of disappearing. It continues to be a haven for Latin America and the Middle Eastern HNWIs, who remain unaffected by the tax rules imposed by the EU on its citizens.\n\nSwitzerland's dilemma in Europe represents the flip side of globalization for the developed world. If developing economies worry they will be plundered and assimilated by globalization, developed countries worry their standards (living\/environmental\/human rights) will be degraded and their leadership roles eroded. Liberalized markets are wonderful for financial markets and the creation of wealth, but the financial herd is merciless and unsentimental. HNWIs and their advisors will always look to move their assets to the most advantageous markets\u2014as well they should. Taxes too high in Switzerland? Move money to Singapore. Wages too high in Germany? Move jobs to China. Regulatory and legal costs too high in the United States? Move operations to India.\n\n#  **UNITED STATES: LOST LUSTER**\n\nFor another example of how globalization can put established powers back on their heels, look no further than New York City. In 2000, as the Internet boom peaked, 90 percent of foreign IPOs were issued in the United States. Six years later, those numbers were flipped; just 10 percent of foreign IPOs chose to list in the United States. The total number of foreign IPOs has fallen 75 percent, and more and more non-U.S. companies (and U.S. companies for that matter) are delisting through private equity and management buyouts.\n\nWhy? Two reasons are commonly cited. The first is linked to regulations enacted after the corporate scandals of Enron and WorldCom. Sarbanes-Oxley (SOX) added investor protections, but some say it went too far with overly burdensome audit requirements. Large companies spend millions per year on these rules; meanwhile, the attention of boards of directors is less on strategy and more on regulatory compliance. But SOX is not the only regulatory thorn in the side of the United States. In the U.S. there are at least ten federal, state and regulatory bodies governing the capital markets, with overlapping and sometimes competing jurisdictions. (Rather ironic since Americans often chide emerging markets for their labyrinthian bureaucracies and inefficient business environments.) The second reason often cited for the falloff in U.S. listings by foreign companies is U.S. tort law. In 2005, class action lawsuits cost American corporations US$9.6 billion, up from US$150 million in 1997. In fact, comparing the cost of litigation to GDP shows that the United States is clearly the most litigious society.\n\nTo the credit of the Securities and Exchange Commission, it's been willing to consider that some of these rules and regulations may have overshot the mark. In late 2006, for instance, it proposed softening the costly oversight rule for small companies. That's a good thing, because the message from global entrepreneurs and investors is clear: the United States must have a regulatory and legal regime that's in step with other financial centers, or it will lose even more market share.\n\nCompanies from developing countries, as they grow richer and their financial markets become better governed, have plenty of options besides the United States to raise money. It also doesn't help U.S. market share that investment banking fees attached to IPOs are higher in the States than either Asia or Europe. Russian firms can turn to the U.K., and China's and India's entrepreneurs can look to Hong Kong. Globalization of financial markets, to put it bluntly, makes a New York listing seem less necessary or brag worthy. Retail investors themselves, especially given that most hold stocks via mutual funds, care even less where a company is listed. Once again, this is the flip side of globalization. Once financial advisors outside the United States fully understand the U.S. business models and can compete with respect to service, regulatory issues, etc., competition for HNWIs could go global as long as the perception remains that service delivery was local.\n\nForeign investors' interest in North America has decreased in recent years to 43 percent from about 46 percent and is forecast to drop even further. In 2004, enthusiasm for North American investing dimmed due to the gains witnessed in other markets, and also another important development: HNWIs' confidence in the U.S. dollar began to crack. Huge deficits, the mounting cost of the war in Iraq (not to mention Afghanistan), and entitlements in the form of Social Security and Medicare that will eat up a greater percentage of the budget as baby boomers retire are all starting to weigh on the dollar. The dollar's further losses in 2006 added to the urgency with which some HNWIs began investigating other markets and considering how they will hedge their dollar exposure. BlackRock cofounder Ralph Schlosstein is forthright in his belief that HNWIs are more aware than ever that a fall in the dollar could impact their long-term standard of living.\n\nIn 2005, the Asia-Pacific region surpassed Europe as the second most popular destination for HNWI investments, accounting for 23 percent of total assets. A number of emerging economies with large growth prospects\u2014such as China and India\u2014combined with strong performance in the region's more mature markets\u2014such as Singapore and Hong Kong\u2014is likely to keep international interest focused on this part of the world for some time to come. Such numbers clearly buttress Mr. Lavayssi\u00e8re's observation that the world is evolving from a two-pole world to a multi-polar world. Asia-Pacific is that third pole; this triumvirate will drive the economic and political narrative of the twenty-first century.\n\n#  **LATIN AMERICA: A GROWING REGIONAL ECONOMY**\n\nAlthough Latin American markets have not enjoyed the explosive growth that Asian markets have experienced over the past ten years, the tangible effects of globalization remain strong nonetheless. The regional economy enjoyed a 5.3 percent growth rate in 2006, the third strong year in a row, and although countries such as Venezuela and Bolivia seem to be sliding toward socialist policies, many Central and South American countries are embracing globalization.\n\nTake Brazil, which saw its HNWI population grow 10.1 percent in 2006, and its main stock exchange, the S\u00e3o Paulo Bovespa, jump 32.9 percent. Thanks to smart fiscal policies and a marked improvement in Brazil's current account, foreign exchange reserves, and balance of trade, the country has attracted foreign investment and grown the economy to the world's ninth-largest by purchasing power parity and eleventh largest\u2014both according to early 2007 market exchange rates. It possesses large and well-developed agricultural, mining, manufacturing and service sectors, as well as a large labor pool.\n\nWith the increased economic stability provided by the Plano Real, Brazilian and multinational businesses have invested heavily in new equipment and technology, a large proportion of which has been purchased from North American enterprises. Many of its companies are regional and, increasingly, global powerhouses, such as Gerdau and CSN.\n\nBrazil has not been without its ups and downs over the past decade, but it has successfully moved from a fixed to a floating real and, ahead of time in 2006, paid back IMF loans related to its currency devaluation. This has occurred while the country and others in the region, particularly Chile, have forged closer ties with China for its abundant raw materials.\n\nDarcie Burk, head of Merrill Lynch's Wealth Management Latin American division and a twenty-year veteran at the firm, can distinctly recall a time not so long ago when traders \"physically clipped the coupons on bonds\" and \"international investing was regarded as kind of a gray area\u2014there was the U.S. and everywhere else, which was just this great empty white space on the map.\"\n\nIt was the pervasive influence of the doctrine of asset allocation within the firm that finally propelled international investing to its present prominence, Ms. Burk contends. \"We started getting into providing hedge funds for our private clients,\" she recalls, \"many of which were either headquartered overseas or heavily involved in international investments of various kinds, some of them rather exotic. A lot of hedge funds were also investing in sovereign debt immediately prior to the 1997 crash, so that taught all of us a number of lessons about risk reduction and client risk profiles.\"\n\nThe twin principles of asset allocation and diversification created a virtual mandate that every FA had to know something about international investing, which inevitably involved an engagement in emerging markets because those generally offered favorable rates of return. In recent years, the most dramatic change on the international investing front, according to Burk, is that \"it's no longer about wealthy entrepreneurs parking their money safely offshore.\" These markets have spawned their own indigenous derivative products, which successfully mimic their originals in developed market economies.\n\n\"All of these compelling secular trends offer strong implications for the Latin American economy,\" Burk notes. \"If the U.S. suffers a crash, China offers a little bit of a hedge due to decoupling. The entire region is stronger thanks to globalization.\"\n\nMexico's mortgage market offers another example of what 's happening in Latin America today, and how fiscal policies have created enormous wealth and opportunities for domestic and non-domestic HNWIs. Ten years ago there was no mortgage market in Mexico to speak of. As a result, a person had to pay 100 percent for their home, which, not surprisingly, kept home ownership low. But by reining in inflation and successfully issuing sovereign debt, the country has developed one of the deepest local debt markets in Latin America. In 2003, it issued its first twenty-year peso-denominated bond, and in 2006 issued the first thirty-year bond. In 2005, it was estimated that non-Mexicans, in search of long-term, relatively safe, high-yielding investments, owned about 75 percent of the twenty-year notes.\n\nA deep debt market has allowed the mortgage market to develop, since banks now have the confidence to lend over such long time horizons. People can now finance home purchases, and with low interest rates and President Vicente Fox's push for home ownership the Mexican housing market has heated up. That's good for the middle class in Mexico, which can now more readily access a more liquid real estate market.\n\n#  **GLOBALIZATION PROMOTES LOCAL INVESTING**\n\nDespite improvements brought by globalization, emerging markets clearly have political and financial risks greater than do developed markets. In the Middle East, for instance, regional business practices do not have the same standards of transparency and accounting as exist in the West, according to Hawkamah, a corporate governance institute. On the whole, however, these risks have lessened. That's partly because developing markets have improved monetary and legal policies, but also because the financial community is better at assessing financial risk and political risk, lessening the chance of a surprise.\n\nThe capability of global credit rating agencies to more reliably quantify risk\u2014even high risk\u2014has proven critical to stabilizing these markets and soothing investors; instead of panicking at the first sign of trouble and stampeding for the exits, investors who knew the risks going in to an investment are more likely to ride out trouble. Investors hate surprises, and the fewer surprises there are, thanks to rigorous due diligence, the more confident investors are and the more stable the whole system is. This should be the responsibility and role of the financial advisor, and individuals should demand clarity from their advisors on what the risks will be with these types of investments.\n\nOne of the interesting results of globalization and stability of markets has been the degree to which it has encouraged local HNWIs to invest in their local economies. By keeping inflation under control through smart monetary and fiscal policies, creating a legal system that guarantees rights of ownership and promoting free and transparent capital markets, developing countries not only attract investment from overseas investors, they convince homegrown HNWIs to invest in the local economy, helping to further the economic transformation.\n\nThis is already a well-established trend in the Middle East, where oil revenue that once automatically returned to Europe and North America is being invested locally (though much of the oil wealth is still parked in safe government bonds issued by Western governments). Stock markets in Kuwait and Saudi Arabia have become viable, and Dubai is building a financial center\u2014really a city within a city\u2014with zero tax on income and profits, 100 percent foreign ownership rules, no foreign exchanges or repatriation of profits, and a regulatory regime akin to the U.K. and Australia.\n\nHNWIs' growing confidence in their countries and regions is not limited to the oil-rich Middle East. Advisors note that business ownership and income accounts for 57 percent of wealth in Asia and 63 percent in Latin America, and that it's within Asia and Latin America that the lion's share of new wealth is being created. In places like South Korea, Taiwan and Singapore, the standard method of operation is to create wealth onshore, and then at certain points move wealth offshore to developing markets to preserve wealth and guard against the political and financial uncertainties that still exist in their markets. In India, a booming economy and galloping stock market (the Bombay Sensex returned 46.7 percent in 2006) give local HNWIs confidence to keep their money working in the Indian economy and not stealthily move money offshore. What's more, high domestic returns and a well-run financial market mean that once capital controls are lifted, as is expected over the next several years, HNWIs are less likely to suddenly shift huge amounts of capital out of the country.\n\nThis marks a radical change from ten years ago, when HNWIs in developing markets possessed few investment options in their home markets. In those days they coped with inflation, uncertain taxes, the threat of expropriation and poor wealth management service in their home market. They moved their money to developed markets for safety and to build their wealth. Those bad old days may be over in some developing markets, but not in all of them. Many HNWIs in Africa as well as certain Latin American and Asian countries still consider it wise to stash most assets offshore\u2014and will continue to do so for years to come. Therefore, the ability to research and gather information on global markets can certainly help individual portfolios.\n\n#  **RADICAL CHANGES IMPACT INVESTMENT OPPORTUNITIES**\n\nGlobalization and the mobility of capital it implies are having a radical impact on how HNWIs across the globe are investing. Free-moving capital means that investors are empowered to continually cross borders to find the best returns in an ever-increasing number of well-run, transparent financial markets. It also means they are free to depart those markets when financial discipline breaks down, or when another jurisdiction becomes comparatively more attractive. And, ironically, it also means that the best returns might be had by _not_ sending investment overseas.\n\nIf HNWIs wish to use their wealth to accrue more wealth (and we assume that virtually all of them do), investment opportunities\u2014no matter where they are\u2014cannot be overlooked. HNWIs must open their eyes to the entire globe, shed the antiquated Cold War mindset of political and financial borders, and move their investments adroitly to pockets of growth around the world. This acumen will be necessary to build wealth and to maintain it; what's more, HNWIs and their advisors must work to hone their investment skills quickly. Speed is required to stay ahead of the next wave of investors, which will surely be moving more of their investments overseas soon.\n\nIndeed, an aging population in the West concurrent with the decline of defined benefit pension plans portends a ferocious competition for investments that generate high returns and can fund long-term retirements. It's not a race anyone will care to lose\u2014and settling for lesser returns in more mature economies will not sustain wealth and build a family legacy. HNWIs, if they are to remain HNWIs, must remain in tune with the changes and opportunities of globalization.\n\nWhile there are certainly risks to globalization and investing in the economies new to free markets, the momentum is clearly toward globalization. As more and more markets around the world open themselves to capital, there will be a series of waves of new investment opportunities. For aspiring HNWIs, the best bet is to keep an eye on how and where the established HNWIs are allocating their capital and follow suit when possible, closing the gap and capturing as many of those early, often higher, returns as possible.\n\nUnderstandably, some wealthy individuals and households will remain skittish about investing in overseas markets, particularly developing ones; but they should take comfort in the fact that the HNWIs from the emerging market themselves are increasingly confident about investing in their own economies thanks to reforms and opportunities brought about by globalization\u2014and they are the ones in the best position to know what they're getting into. Foreign investors would be wise to scrutinize their investment and sentiment. This means that despite the increasing globalization and complexity, the basic fundamentals will remain:\n\n\u2022 Understand your risk profile and appetite;\n\n\u2022 Determine your \"disposable income\" level;\n\n\u2022 Set floors and ceilings on exposure, upsides and downsides;\n\n\u2022 Agree to set review and reallocation periods;\n\n\u2022 Aggressively adhere to your plan.\n\nThis will become more and more important. Being aware of this and having the discipline will allow individuals to raise critical issues with their advisors.\n\nAs the international situation evolves, the challenge for advisors is to ensure that they are aware of and able to navigate global markets, or have access to this information and can readily and easily understand and share it. Many HNWIs who were interviewed stated that for them, this is now a key financial advisor selection criteria. This means that advisors need to be a steady hand in this tumultuous investment environment, and wealth management firms must also resist using globalization as a cover to treat all HNWIs the same. For instance, a firm cannot hope to succeed in India if it does not set up a truly local operation there, one that operates in the local time zone and designs services and products that fit the local culture. A financial institution operating at arm's length could miss the important nuances of how to manage the wealth of non-resident Indians and their families still in India.\n\nTo be successful, the financial advisor needs to fully comprehend the individual style of the investor, based on where the money originated (income versus business versus inheritance), and then overlay that with appropriate products and services. Concurrently, HNWIs need to be prepared to provide that information, which will then allow the advisor to build the most appropriate and compelling investment strategy for their investment needs.\n\n#  **INTERNATIONAL MARKETS' SUCCESS IS FORCING U.S. INVESTORS TO RETHINK THEIR DOMESTIC FOCUS**\n\nThe increasing interest and allocation to international investments is occurring because of many of the same reasons as those driving alternative investments such as hedge funds and private equity: investors' growing familiarity with these markets, a hunt for yield and diversification. The dilemma for the HNWI is the same as well: greater risk. The following must be considered:\n\n\u2022 How liquid will these investments be in a time of crisis?\n\n\u2022 How well does the financial community understand the risks\u2014political and economical\u2014in all of these remote corners of the world where investments are now possible, often for the first time?\n\n\u2022 And, even if the risks are fairly well understood, are HNWIs being properly rewarded for the risk they are taking, or is the rush of capital into new markets driving up prices and discounting the risks too much?\n\n\u2022 Therefore: Should the fifty stocks on the Vietnamese stock exchange really be trading at a higher price-to-earnings multiple than the S&P 500\u2014as was the case in early 2007?\n\nMake no mistake, however, that while these questions must be asked and precautions taken, they cannot be used as a pretext for inaction. Smart investors continue to diversify their holdings as their confidence in less familiar markets grows, and this international approach is playing a larger role in their wealth creation strategy as markets outside North America and Europe rack up double-digit gains year after year. To keep up, other investors will be forced to follow suit and fundamentally shift the composition of their portfolios. The so-called \"wheel of fortune\" below highlights some of the rationale behind the drive toward an increasingly international view.\n\nFor HNWIs tempted to wait on the sidelines to see if international investing will fall out of favor, money managers stress that this new global investing approach is no flash in the pan. The smart money will continue to flow into international investments. \"This is part of a permanent trend. The world has changed and the world view of investors is different today. Globalization is the biggest shift in history in terms of economic development; countries are participating that never have before,\" says Gregory Johnson, CEO of Franklin Templeton Investments, founded by his grandfather in 1947 and which today boasts US$465 billion of assets under management. Mr. Johnson's father, Charles Bartlett Johnson, serves as co-chairman of Franklin Resources, has an estimated net worth of US$4.3 billion and was ranked in 2006 by _Forbes_ as the _147th richest person in the world_.\n\n**Figure 2.1** \\- Wealth Creation in Global Economies\n\n_Source_ : Capgemini analysis.\n\nMarket capitalizations and benchmark index performance have grown rapidly in Eastern European, Asian-Pacific and Latin American markets, driven by ongoing foreign investment and strong corporate profits. HNWIs have paid close attention and taken advantage. In 2006, emerging-market mutual funds worldwide saw a record US$22 billion of net new money, up from US$20 billion in 2005, according to Emerging Portfolio Fund Research. The Dow Jones World Stock Index\u2014excluding U.S. equities\u2014rose 88 percent from 2002 to 2005 and then tacked on another 23 percent in 2006.\n\nIn the Asia-Pacific region, stock market growth has remained consistently strong. The MSCI AC Asia-Pacific Index returned 21.0 percent in 2005, up from 14.1 percent in 2004. Similarly, commodity-producing nations such as Brazil, Mexico and Canada benefited from surging prices, as well as from a boost in the value of their currencies. Some smaller, less developed financial markets such as those of the United Arab Emirates, Romania and Singapore were also winners for HNWIs in 2005.\n\nThe year 2006 saw a continuation of many of these trends, with substantial returns in China, Russia and India, among others, thanks to a combination of growing domestic demand, overseas trade and commodity prices. In Russia, the benchmark RTS Index jumped 70.7 percent. India's Bombay Sensex rose 46.7 percent. Meanwhile, the Dow Jones China 88, a measure of China's largest and most traded stocks, scored a 97 percent rise.\n\nTo some degree, success begets success, not just within the same country, but in other countries in the region. Mr. Johnson argues that leaders of Vietnam see what's occurring in China and have decided to take part, paving the way for a wave of Western investment. The main Ho Chi Min Stock Index more than doubled in value in 2006.\n\n#  **EMERGING MARKETS BECOME MAINSTREAM**\n\nThe overarching narrative in terms of HNWI international allocation is that investments in North America and Europe are expected to continue to decline over the next few years as HNWIs redeploy assets to the Asia-Pacific region and other emerging markets that outperform North America and Europe. This is virtually inevitable if HNWIs want to position their portfolios for the highest growth. In 2006, for instance, China notched another year of 10-percent-plus growth and is on track to surpass Germany in 2008 as the world's third-largest economy.\n\nHNWIs' heightened interest in international investments, combined with their growing exposure to equities and alternative investments, are clear signs that the world's wealthiest individuals are not only becoming more sophisticated investors; they have also become somewhat more aggressive than in the past and seem even more determined to achieve returns that surpass market averages. The lesson for investors is that the global economy is not only altering real-time communications, but also the ways that money is managed and leveraged.\n\nIndeed, the strong results in emerging and international markets\u2014coupled with the capital flowing into them\u2014have prompted some observers to argue that the emerging markets are \"decoupling\" from the United States, whose performance remained less robust\u2014albeit respectable\u2014in 2006 and became more severely challenged by the subprime mortgage crisis that dominated the final half of 2007 and continued to roil global markets well into 2008. Many formerly Third World countries are in better financial shape than in the past, with lower deficits, increased reserves and reduced dependence on exports to the United States. The rise of China and, increasingly India, has helped dilute the dependence of many emerging markets on the United States, particularly those that rely on exporting commodities.\n\nHow resilient these emerging economies would really be if U. S. consumer demand dried up, however, remains to be seen. This is one of the great unanswered questions going forward as the world is transformed by globalization.\n\n#  **BEYOND EMERGING MARKETS LIES THE FINAL FRONTIER**\n\nWith so much money chasing investments in places like China and Russia, the more established emerging markets are starting to look mainstream. As investments become more widely available to the mass of investors and demand increases, returns often decline, which leads aggressive investors casting about for the next big thing. This formula has played out with hedge funds and other alternative investments, and now it looks like it's starting to occur with emerging markets as well. Indeed, some HNWIs and their advisors say returns in many emerging markets are declining at too fast a rate, leading them to worry that investors are getting lax about risk.\n\nIn pursuit of the highest yields, some HNWIs are looking beyond mere emerging markets to what are known as \"frontier markets,\" where the potential risks and returns are both a magnitude higher than those of emerging markets. Frontier markets are tiny. With a total market capitalization of about US$40 billion, the whole sector is worth less than 20 percent of the current value of China's ICBC bank. They include countries such as Bangladesh, C\u00f4te d'Ivoire, Jamaica, Slovenia and many other countries in Africa, South Asia, Eastern Europe and the Caribbean.\n\nThe International Finance Corporation (IFC) first launched the S&P\/IFC Frontier Index in the mid-nineties. The index measures the performance of stocks of twenty-two countries and has returned an average of 37 percent over the past five years, compared with an average 25 percent for the Morgan Stanley Emerging Markets Index.\n\nAmong the countries there are some big winners and big losers. Since March 2003, in dollar terms, the Ukraine has risen by 570 percent, Kenya by 279 percent and Bangladesh by 79 percent. In 2006, Namibia was the top performer by posting a 126 percent return, while Tunisia declined 8 percent. And Saudi Arabia's market plummeted a whopping 50 percent in 2006.\n\nThe Vietnam Stock Exchange, which was added to the S&P\/IFC Frontier Index in 2006, doubled in value, due to commodity exports and a young, cheap workforce. Foreign investors spent US$123 million for Vietnamese stocks and by the end of 2006 owned about 31 percent of the market. Investors have raised US$1.8 billion to invest in Vietnam ahead of a privatization program next year, which will include eight US$1 billion listings, according to Nicholas Vardy, editor of _The Global Guru_.\n\nFor the past several years, investing in these countries has been the province of only the most sophisticated HNWIs but, right on cue, new \"frontier funds\" are now cropping up to take advantage of such attractive returns, giving more investors a chance to take part. Investment firms Hamilton Bradshaw and AKD Securities recently launched the first fund focused solely on Pakistan, while Investec is pushing the case for Africa and managers are talking up the prospects they're finding in Vietnam.\n\nMany of the opportunities, for now, are open only to institutional investors, including a fund opened in December 2006 by Acadian Asset Management, which quickly raised US$200 million for a frontier markets fund based on the S&P\/IFC Frontier Index. Meanwhile, Imara Asset Management's Imara African Opportunities Fund focuses primarily on Africa, excluding South Africa, attempting to tap Africa's wealth of natural resources and growing trade links with booming economies like China. But the minimum investment in the fund is US$100,000. A few mutual fund companies, including Franklin Templeton, Eaton Vance, T. Rowe Price and Fidelity Investments, offer limited frontier exposure through emerging-market mutual funds.\n\nAccording to Merrill Lynch global emerging-markets equity strategist Michael Hartnett, frontier markets have outperformed both emerging and developed equity markets for several years, earning annualized returns\u2014as measured by the S&P\/IFC Frontier Index of 23.75 percent between 2000, when data first became available, and September 2007. Compared with a 12.42 percent rise for emerging markets, as measured by the MSCI Emerging Markets Index, and a 1.18 percent increase for the S&P 500 Index in the United States, both over the same period, frontier markets may display a compelling case for savvy and risk-tolerant investors.\n\nThe fundamental driver of their growth potential, Hartnett observes, is the global boom in basic commodities. In 2007, Dubai's stock market rose more than 37 percent, while Abu Dhabi's was up over 40 percent and Qatar's 35 percent. Even sub-Saharan Africa has been experiencing its strongest growth and lowest inflation in more than thirty years, according to the IMF.\n\nOn the downside, the challenges faced by these markets are fairly obvious and include not insignificant political uprisings, human rights and environmental issues, hyperinflation, and poor infrastructure that can make power, transportation and information intermittent. Some countries, like Bangladesh, are prone to natural disasters. Laws and regulations can be nonexistent or subject to the whims of local bureaucracies, making investor rights iffy at best.\n\nOn the plus side, frontier markets tend to be not closely correlated to the economic winds in the developed countries, which mean they offer a degree of diversification to the intrepid investor with the risk appetite to go where others justifiably fear to tread. In fact, intrepid may be too subdued a word to describe the attitude required to plunge into these markets without losing several years' worth of sleep. Many of these stocks are highly illiquid, which makes them impractical for big mutual funds to participate in and more suitable for private pools of capital that can commit to these investments for the long term and need not worry about redemptions. Also, the markets themselves might gyrate radically, and the ultimate goal\u2014that a frontier market becomes an emerging market and eventually a developed market\u2014may in fact never occur. Despite such hurdles, the S&P\/IFC Frontier Index includes Bulgaria, Estonia, Latvia, Lithuania, Romania, Slovakia and Slovenia, some of which are looking decidedly more advanced than their frontier status might suggest.\n\nAn additional challenge with frontier markets is that they may be bid way up as investors rush in, even though experienced investment pros say frontier markets are really only worthwhile if they are cheap and overlooked. A case in point: Vietnam, where shares on the Vietnam Stock Index traded at approximately twenty-six times estimated 2006 earnings, above the S&P 500's trailing price\/earnings ratio of eighteen. As a result, HNWIs and their advisors need to be extraordinarily disciplined when dipping their toes into frontier markets and not be lured into taking on too much risk, chasing possibly phantom returns. Such lessons need to be given serious consideration by everyone\u2014clients and advisors alike.\n\nMore than a quarter century after Antoine van Agtmael stood at that podium at the Salomon Brothers headquarters in New York, pitching his offbeat idea of an emerging-markets equities investment fund, emerging-market stocks still represent only about 10 percent of market capitalization worldwide. Yet the boundless potential of emerging markets' companies and economies is neatly encapsulated by the fact that taken together, the top one hundred companies from the world's fastest-growing economies are growing ten times faster than their U.S. counterparts, twenty-four times faster than Japan's and thirty-four times faster than Germany's, according to The Boston Consulting Group.\n\nFew investors, regardless of risk tolerance or time horizon, can afford to ignore the fact that the total shareholder return of the MSCI Emerging Markets Index jumped 150 percent during the first six years of the twenty-first century, while the Standard & Poor's 500 Index declined over the same period. \"It's astounding that so many people continue to remain oblivious to this new reality,\" van Agtmael contends.\n\nThis new reality is also one more accurately defined by the emergence of dozens of world-class companies, as opposed to world-class economies, according to Mr. van Agtmael. Too many Western investors continue to think about pulling off \"country plays\" while they should be focused\u2014as they are with companies based in the more developed economies\u2014on individual corporate performance.\n\nAs the world's capital markets become ever more closely linked because of the blessings of advancing technology, the global economy is becoming increasingly dominated by forward-looking firms, many based in the developing economies, which possess not just a global footprint but a global outlook. World-class firms like Taiwan's Hon Hai, the world's largest electronics contractor, which produces iPods for Apple; Mexico's Grupo Modelo (Corona beer); Samsung Electronics; or Mexico's Cemex, the largest producer of cement in the United States, are not just the blue chips of the future but the blue chips of today.\n\n**Five Tips for International Investing**\n\n1. **Go where the growth is:** With the growth of the HNWI population strongest in emerging markets (up 21 percent in Singapore, 20 percent in India, and more than 15 percent in Indonesia, Russia and the UAE), much of the market action in those economies appears to be heavily influenced by higher levels of domestic investment, with inflows into emerging-markets funds from developed countries remaining relatively stable.\n\n2. **Seek proper asset allocation and diversification:** Emerging-markets investors, wherever they may be domiciled, are gaining exposure to economies not traditionally tightly correlated with developed economies. Fans of the \"decoupling theme\" saw their pet theory stress-tested during the global turbulence caused by the credit collapse in the third quarter of 2007.\n\n3. **Focus on world-class companies, not world-class economies:** India and China may be the national heroes, but maintaining one's focus on individual corporate success stories offers higher rewards than a more scattershot top-down investment approach.\n\n4. **Don't forget frontier markets:** While highly volatile, frontier markets may be worth investigating provided that these targeted investments are undertaken in accordance with a professionally constructed, risk-balanced portfolio strategy.\n\n5. **Embrace globalization; it's no flash in the pan:** To ignore the rewards of international investing would be comparable to refusing to invest in technology, media networking, or other significant social and geopolitical trends. Maintaining a provincial approach that avoids the perceived risk of \"foreign\" investments is no longer a practical option for investors seeking to maximize opportunities in the twenty-first century.\n**CHAPTER 3**\n\n**Technology's Critical Role in Wealth Management**\n\n#  **DAWNING OF THE DIGITAL AGE**\n\nIn the 2006 _World Wealth Report_ , two-thirds of wealth advisors responded to a survey question on technology that the quality and quantity of information available to clients was the most transformative trend in wealth management over the past decade. Forty percent contended that technology would continue to be the most powerful trend for the next ten years. Already, the velocity, volume and scope of information reaching today's investors have made them more sophisticated and demanding than their predecessors. Just as technology empowers wealthy clients and enables advisors to offer them superior service, technology increases the pressure on advisors to keep up with the furious pace of innovation and rapidly ratchets up investor expectations.\n\nIt would be difficult to overstate the impact of technology on the art and science of wealth management. Information technology puts sophisticated financial tools in the hands of the wealthy and their advisors. It provides an efficient and effective way for clients to communicate with their advisors as well as each other. The advent of information technology, in tandem with financial innovation in general, has been the primary driver of globalization, binding the world's markets closer together and enabling the nearly instantaneous transfer of information, ideas and liquid assets around the globe at a velocity that is historically unprecedented.\n\nAccess to aggregated account information (a single balance sheet across business lines and companies), is not merely desired but demanded by clients, not least by HNWIs. These now conventional expectations nonetheless sorely challenge the technology infrastructure of many a wealth advisory firm that has failed to invest the resources in cutting-edge IT. Yet it's a challenge they have no choice but to tackle. Today's U.S. and Europe-based financial firms must compete head-on with those based in Asia\u2014and vice versa.\n\nFor clients, the primary benefits bestowed by technology are that they can simultaneously have access to the same up-to-date and sophisticated information as do their advisors and (presumably) the most informed market-makers. As information technology has steadily become more user-friendly, cost-effective and intuitive, the information gap between advisor and client has dramatically contracted. The closing of the information gap vastly improves the quality of the dialogue between client and advisor that lies at the heart of every productive investment relationship.\n\nCharles Simonyi, former head of Microsoft's application software group (which oversaw the creation of Microsoft Word and Excel) and who now heads his own company, Intentional Software, observes that \"because financial markets are entirely information-based, technology will drive the future of financial markets.\" Proprietary databases and algorithms will one day become as valuable as capital and liquidity itself, he predicts. He cites as an example of the cutting edge of this trend the stunning success of Renaissance Technologies (Rentec), the quantitative hedge fund management company founded by former MIT math professor James Simons, whose flagship US$5 billion Medallion Fund has averaged 35 percent annual returns, after fees, from 1989 through 2007.\n\n\"The algorithms at their disposal have such incredible value that they truly take the place of capital,\" Simonyi contends. The Medallion Fund's remarkable yield allows it to charge clients a 5 percent management fee and a 44 percent incentive fee\u2014generous even by hedge fund standards. The often daunting task of incorporating new technologies into the discipline of wealth management becomes even more pressing because many technologies considered novel today will not be considered worth talking about to the children of today's wealthy clients. Rapid technological obsolescence is simply a fact of modern corporate life. As younger generations become HNWIs in their own right, they are unlikely to display much patience for firms that have failed to seamlessly weave the most advanced technologies into the very fabric of their operations. \"Technology is something invented after you were born,\" observes James A.C. Kennedy, CEO of financial firm T. Rowe Price. \"For kids the Internet isn't a technology, it 's an appliance. They couldn't imagine life without it because they've never had to.\"\n\n#  **DISPELLING FIVE MYTHS OF ONLINE PRIVATE BANKING**\n\nDespite the technology-fueled revolution around investor education and sophistication, it's hardly surprising that it's still possible to find advisors who believe that technology's place in wealth management is limited. The rationale seems to be that since private banking has thrive so long so successfully, it must be immune to the IT advances all around it. A corollary to this erroneous belief is that HNWIs aren't interested in technology, or that deploying technology will somehow put the wealth management firm itself at some kind of competitive disadvantage. This sentiment is particularly stubborn when it comes to shunning the Internet, which is quite simply a no-growth strategy for both the firm and, potentially, HNWIs. While it's true that any new and potentially costly IT strategy or deployment should not be undertaken lightly\u2014especially when it touches the client as ubiquitously as the Internet does\u2014too many advisors suffer from an abundance of caution when contemplating technology upgrades. Too many are fooled or seduced by what Capgemini has identified as the five myths of online private banking:\n\n##  _**Myth #1: Most private banking clients are not sufficiently Internet-savvy for online banking.**_\n\nWe have extensively documented a widespread belief that wealthy investors are not sufficiently familiar with the Internet to move private banking online. Indeed, there is some basis for this misperception. Most wealthy clients in North America and Europe are in their late fifties, an age group that as a rule uses the Internet less frequently than younger demographics. (As recently as 2004, only 20 percent of Europeans between fifty-five and seventy-four had logged on within twelve months.) But the conclusion that many draw from this fact\u2014that wealthy individuals are not good candidates for Internet banking services\u2014is simply erroneous.\n\nConclusions based on crude demographics frequently miss critical details. Retail brokerage clients as a group are frequent and committed users of online communication. The typical advised client in 2005 spent nearly fourteen hours per week online, and more than half had a broadband connection. In fact, the wealthier the clients, the more active they tend to be online. In 2005, an affluent European was 73 percent more likely than a non-affluent counterpart to bank online at least weekly.\n\nNearly half of the affluent U.S. households conduct online stock research, with one in three HNWIs visiting financial provider sites and one in five trading stocks, mutual funds, options and bonds online. Indeed, for clients with actively managed portfolios, managing finances was the second most frequently performed online activity after emailing. HNWIs still demand ample telephone and face time with their advisors, but 56 percent require full Internet access to their portfolios and investment accounts as a prerequisite for client satisfaction\u2014a figure that is surely to grow.\n\n**Figure 3.1** \\- Share of Investors Who Are Active Online, per Level of Wealth, United States, 2005\n\n_Sources_ : _Private Banker International_ , May 31, 2005; _Private Banker International_ , Apr. 27, 2004; Forrester research, \"The affluent are the most active online investors,\" Sept. 2005; Forrester research, \"How to boost online banking in Italy,\" Sept. 2004; Forrester research, \"Europe's affluent rely on online finance,\" Feb. 2004.\n\nFurthermore, the wealthier the client, the more they tend to factor websites into their overall satisfaction with their advisor. Forrester Research found that 58 percent of brokerage website visitors in the U.S. consider the quality of their brokerage firm's website extremely important to their satisfaction with the company as a whole. Remarkably, the second-strongest predictor of high client satisfaction with their brokerage firm was how they viewed the firm's use of technology. And of course, when it comes to future generations, since younger people are by far the heaviest users of the Internet, their numbers will continue to grow as they age and turn into the next generation of HNWIs.\n\nTechnology, in short, is critical to serving clients properly\u2014and clients know it. In a fiercely competitive environment, where companies are looking for every conceivable advantage to retain and attract wealthy clients, the online channel must be embraced. Clients, in turn, would be wise to make the best use of the online channel as a means of optimizing communication with their advisor.\n\n##  _**Myth #2: An advisor-based business like private banking can only be distributed through high-touch channels.**_\n\nPrivate banking is quite rightly viewed as a people-based relationship business. But that description leads some observers to falsely assume that there is little to gain from providing clients with access to electronic channels. Frequent arguments against electronic channels and Web access include:\n\n1. The high level of customization and personal service HNWIs expect can't be delivered through the Web;\n\n2. Clients' complex financial situations require advanced services and products that cannot be delivered online;\n\n3. The low frequency of transactions (compared to retail banking) makes an Internet channel too costly to maintain.\n\nYet as has been previously noted and well documented, HNWIs are increasingly intent on taking a more active role in the dialogue about their wealth. Whether this means they will or won't want to speak to their advisors on a daily basis is unknown, but it does mean that they require instantaneous real-time access to timely financial information. Ideally, a firm should be able to deliver this information and increase the touch points between the advisor and the clients to \"extend the conversation,\" further proving the advisor's value as a trusted and accessible counselor. For example, a firm could design customized online investment alerts for deployment following every investor-advisor telephone conversation.\n\nOf course, there are always going to be some tech-averse clients temperamentally inclined to avoid the online channel, preferring the intimacy of face-to-face meetings and phone calls. Personal preferences will perennially vary. Not all channels are suitable to all clients. But the fact is that by offering the broadest possible menu of channels, a bank can customize the mix to each client's specific needs and preferences, which will allow the bank to more accurately target services.\n\nThere will always be a significant difference between processing a stock purchase and providing estate planning services. To squeeze the maximum value from its distribution channels, private banks need to make low-value, low-touch distribution channels available for low-value and commoditized products and services such as stock purchases, while maintaining a high degree of flexibility to each investor's preferences in the event that they may want to perform, say, some estate planning online.\n\n**Figure 3.2** \\- Product Complexity and Value of Distribution Channel\n\n_Source_ : Capgemini analysis.\n\nNot only can the \"low-touch\" Web channels improve client satisfaction by offering more information and additional routes of communication with their advisor, they can materially assist banks to boost profits by increasing revenue while lowering costs. Revenue growth can be ascribed to higher client satisfaction and retention; shorter revenue cycles and increased cross-selling thanks to faster communication; and stronger client acquisition and lead generation. On the cost side, improved coordination between client servicing team members and processes yields cost efficiencies; and the cost of standard transactions is lower online than through any other channel.\n\n##  _**Myth #3: Because of concerns about security and confidentiality, private clients will never fully adopt the Internet channel.**_\n\nAs the number of high-profile credit card security breaches in recent years highlight, security issues surrounding identify theft and online fraud must be adequately addressed if online banking stands any chance of truly attaining critical mass, particularly among the highest net-worth demographic. It's certainly clear that credit card fraud is the most sophisticated and frequent, but also that the behavior modeling and risk mitigation strategies are the most advanced in this area, as those of us who have been contacted by our service providers to validate purchases, location of sale, etc. know all too well.\n\nRecently conducted primary research indicates that more HNWIs are using the Internet channel more frequently for banking and managing their wealth in conjunction with a conversation with their advisors. However, those who wish to keep financial transactions as \"recordless\" as possible will clearly look to avoid any type of electronic transaction that will leave an auditable trace. The ultra-rich clients of Swiss private banks, for instance, often don't even trust regular mail, relying instead on face-to-face real-time communication.\n\nRegardless, the broad masses of wealthy investors require functional and reliable online channels; in fact, they should demand them. To meet their needs, providers are taking three broad measures to address security concerns: technological, educational\/informational and legal.\n\nOn the technology front, private banks are developing more advanced functionality to tackle online security threats. In fact, in the United States, regulators mandate the use of two-factor authentication to provide a highly secure form of online access. (An ATM card with a chip and PIN that must be used in conjunction with the card is an example of two-factor authentication.) Elsewhere, firms are introducing the DigiTAG device, which generates a unique eight-digit security code to be used with a standard user ID and password to authorize access to online sites. Some banks offer their clients free downloads of anti-virus and anti-spam software, as well as free security checks for their personal computers. New initiatives are being announced regularly.\n\nOn the education\/information front, private banks are educating their clients about security measures in order to boost client confidence in the Web channel. The challenge for private banks and advisors is to discuss security measures with clients without scaring them off or making the security measures appear too bothersome. Many firms are using visual clues to communicate security and confidentiality by using an email security zone with the retail banking client's first and last name as well as the last four digits of an ATM\/debit card. This encourages users to look for personal clues to verify the information.\n\nOn the legal front, banks are educating customers on security best practices and explaining clearly what the bank's policies are in the event of fraud. Best practice examples include firms putting their \"peace of mind\" guarantee prominently on every Internet page, promising to compensate customers in the event of fraud.\n\n##  _**Myth #4: Financial advisors don't benefit from the increased transparency that online private banking provides clients.**_\n\nTo avoid taking the leap into online banking, advisors often tell themselves that clients don't want full transparency and, even if they did, clients would not stand to benefit from such \"uncontrolled\" access to information. Aside from being self-serving and faulty, such reasoning blinds advisors to how increased transparency technology can help enhance their business.\n\nYes, pressures on advisors do increase with online banking and transparency. With constant real-time client access and with clients viewing and acting upon Web pages that provide them with details of their full financial portfolio, the advisor needs to provide more education about market conditions and the strategy and tactics of the investments, and be prepared to explain their decisions to the client. Clients should also be prepared to ask questions of their advisors versus solely accepting their advice.\n\nAdvisors also benefit. According to a 2006 Forrester Research study, advisors who proactively encouraged usage of online channels had better financial results compared to advisors who didn't encourage online usage. They had 11 percent more clients, 50 percent more assets under management and 38 percent more annual revenue. Perhaps this is because such openness also reflects and promotes a level of trust between advisor and clients. Forrester further found that advisors who don't encourage site use are twice as likely to say that their clients don't trust them.\n\n##  _**Myth #5: Online channel implementation requires major IT investments by the private banks with little return and little ability to pass costs on to clients.**_\n\nThe overriding concern at many firms is that systems designed to support online trading transactions are excessively complicated and costly due to compliance issues, fraud potential, and complex system integration between front-office and back-office systems. Certainly, no IT investment or upgrade is without cost, but firms may find that those costs are less than originally feared and may in some cases generate quicker returns than anticipated.\n\nThrough the use of middleware and service-oriented architecture (SOA; systems that enable software to work together and data to be shared), firms do not necessarily need to replace their entire existing infrastructure to implement additional functionalities. By leveraging existing technologies, private banks may increase the return from their technology investments. Many packages and platforms available today do not require as intense a customization as was necessary in the past to meet the needs of high-net-worth clients. For example, to enhance and enable superior online collaboration between clients and advisors, some firms are deploying a content management system to run their client portals and use financial planning tools or workstations for their advisors.\n\n#  **EXAMINING THE THREE FACES OF INFORMATION TECHNOLOGY**\n\nThe universe of investing technology can be divided into three broad categories:\n\n1. _Client_ facing;\n\n2. _Provider_ facing;\n\n3. _Advisor_ facing.\n\nWhile some technologies overlap, and some might not perfectly fit any of these categories, for the wealthy and ultra-wealthy it has become critical to know not only what tools a financial advisor is using, but also what tools are available that he or she may not be using, or at least not using to their fullest capacity. It's highly unlikely, however, that an advisor will volunteer that his firm is weighed down by legacy systems and is thus _not_ adopting new technology, or is _not_ providing the support he needs to do a better job. Wealthy and aspiring HNWIs need to educate themselves as to the gold standards of IT and press their advisors to either provide the very best or consider taking their business elsewhere. To improve their dialogue and their relationship, it's critical for HNWIs to talk with their advisors about the power of technology. If that subject remains in any way off-limits or taboo, the broader wealth management dialogue is likely to be stunted.\n\n##  **Client Facing**\n\nMany wealthy clients have financial terminals in their homes that provide them with access to a host of real-time, professional-grade data. They expect and frequently have instantaneous access not just to garden-variety financial news but to sophisticated portfolio modeling and rebalancing software as well. They typically demand, as a matter of course, an aggregated view of their accounts. And they assume that they will be able to peruse at their convenience these \"family balance sheets\" in the comfort of their own home offices so that they can pull up a private asset statement alongside a business's profit and loss statement. HNWIs feel that they have the right to an informed dialogue with their advisors about their investments and that if they don't have the ability to share data online with their advisors that their dialogue is not sufficiently well supported by the firm.\n\nIn order to ensure the highest level of service, clients expecting a technological capacity appropriate to their level need to do the following:\n\n1. Ask their advisors if the firm is capable of offering the benefits of open architecture, which enables systems to access data from a variety of internal and external sources, as opposed to closed or proprietary architecture, which limits the offering to data provided by the firm's own analysts and experts.\n\n2. Ask their advisors if the firm's technology platform is set up to enable continuous online reporting on every aspect of their portfolio.\n\n3. Ask their advisors how the workflow of the practitioner or team is enabled by the firm's technological platform. How much does the advisor need to accomplish manually as opposed to being handled automatically by the system?\n\n4. Get a clear sense of how sensitive the advisor is to their technological questions and concerns. This provides an indication of the degree to which the advisor comprehends and is available to leverage the available technology.\n\n##  **Provider Facing**\n\nTo make client-facing tools live up to today's high expectations, wealth management providers need to invest heavily in their back-office technologies to integrate client information across channels in real time. Such \"provider-facing\" technologies allow an institution to communicate internally about each client. High-net-worth individuals typically don't display any great appetite for sifting through fifty roughly cobbled-together pages outlining their financial position. They quite reasonably expect, or should expect, software capable of telling them three primary things:\n\n1. How am I doing?\n\n2. Are there any deviations occurring from my financial plan?\n\n3. If so, where, why and what's the strategy to realign back to my original plan?\n\nAn institution must seamlessly integrate its channels across product lines and geographies in order to retain any hope of providing precise answers to those deceptively simple questions. This goal, not surprisingly, creates huge operational and cultural challenges for some large institutions. Operationally, most legacy systems provide an _account_ and not a _client_ view, making the provision of a truly holistic \"family balance sheet\" difficult, unwieldy and challenging. Back-end data must be fully integrated. Turf wars must be tamped down and coordination spurred across every silo in the enterprise; to most capably serve the highest-end clients, a channel must be opened between the investment banking, private client, hedge fund and private equity operations. On the flip side, these challenges may provide a comparative competitive advantage to small and nimble firms capable of implementing new technologies more quickly and smoothly across the organization than their juggernaut brethren.\n\n##  **Advisor Facing**\n\nThe most capable advisors are supported by the best technological tools. Institutions need to ensure that their advisors are provided with \"advisor-facing technologies\" designed to help them sort out and sift through the growing complexity of client accounts without exponentially increasing the administrative burden of servicing them. To give their advisors the ability to offer the best financial services to HNWIs, institutions must carefully evaluate the workstation needs for their advisors and the entire wealth management process. Some needs include:\n\n\u2022 Intelligent alerts with dashboards that provide advisors with reminders \/reasons to call their clients, data mining of relevant market portfolios and lifestyle alerts.\n\n\u2022 Family and client insight with a comprehensive relationship view that provides an advisor with sufficient specific information on every relationship so that he or she can immediately answer common client questions such as How am I doing?\n\n\u2022 Integrated wealth management that includes seamless integration of advice tools with client profiles and presentation builders.\n\n\u2022 A product marketing catalog that matches clients' needs with relevant offers and delivers actionable product ideas to advisors.\n\n\u2022 Client-review capabilities that deliver consolidated relationship-based reporting and reduce report-generation time, allowing advisors to conduct more frequent and enriched client assessments.\n\nNot surprisingly, the root cause of many of the challenges besetting financial firms' IT departments is the back-end data at most institutions, which, as mentioned, typically offers an _account_ but not a _client_ view. Most global firms hoping to capture any reasonable percentage of the high-net-worth clientele have acknowledged the need to create more elegant solutions to the problem of creating the integrated \"family balance sheet\" views and reports that their wealthiest clients increasingly demand\u2014and which most family offices (organizations of investment professionals that work exclusively for a single family or related cluster of families) are already equipped to provide.\n\nA problem faced by many global institutions is that as new technologies continue to roll out, boutique shops and nimble start-ups, unburdened by the legacy issues affecting some large institutions, can adopt these newest technologies at comparatively low cost, to their competitive advantage.\n\n#  **ASSESSING THE ROLE OF TECHNOLOGY IN THE ADVISOR-CLIENT RELATIONSHIP: A CAUTIONARY TALE**\n\nAt the beginning of the twenty-first century, few savvy business people could honestly disagree with the assessment offered by Microsoft CEO Bill Gates that \"the Internet changes everything.\"  As Gates pointed out, \"Now that customers can deal directly with manufacturers and service providers, there is little value added in simply transferring goods or information.\" As the phenomenon of \"disintermediation\"\u2014i.e., the elimination of the middleman\u2014threatened to profoundly disrupt just about every service industry on the planet, E*TRADE Securities, which pioneered Internet stock brokerage in 1992, touched off a tidal wave that swept through the financial services universe. Meanwhile, Forrester Research accurately estimated that by the turn of the twenty-first century, fourteen million U.S. brokerage customers would be buying, selling and trading stocks, bonds, mutual funds and other securities online, accounting for as much as 20 percent of individual brokerage transactions. Gates astutely analyzed the dilemma facing traditional financial services providers at the dawn of the twenty-first century:\n\n_\"These firms faced a fundamental strategy decision: Do you use technology to play the same game that e-traders play? And if you do that, how do you differentiate yourself from them? Or do you use technology to play to your traditional strengths-highly trained staff accustomed to managing long-term customer relationships? If you adopt the latter strategy, how can you use technology to be more efficient, and how can you turn the Internet's popularity to your advantage? 35_\n\nPerhaps not surprisingly, Merrill Lynch spied an opportunity to play to its traditional strength by using technology to support its \"highly trained staff accustomed to managing long-term customer relationships\" and created a highly-touted \"FA-centric\" system capable of managing the data flow required to develop, implement and monitor comprehensive financial plans for clients. The result was a desktop-PC-based proprietary system, Trusted Global Advisor (TGA), unveiled to great fanfare at an approximate cost of US$850 million.\n\nUnfortunately, not long after TGA's rollout, more than a few frustrated FAs were complaining bitterly to management that the new platform lacked any number of essential features they regarded as essential to compete in the brave, new digital age, particularly when going up against a new crop of upstart firms for whom the provision of state-of-the-art technology\u2014including every conceivable bell and whistle\u2014was a critical component of their value proposition.\n\nDespite its technological sophistication, TGA failed to meet the needs of _its_ clients\u2014the FAs\u2014who as a result did not feel supported to meet the needs of _their_ clients. Neither was TGA particularly adept at integrating the FAs' workflow with that of the Client Associates (CAs), assistants to the FAs who handled the bulk of routine communications with clients.\n\nThe key to providing top-notch software and hardware support is to make sure that the user never has to tell a client, \"I'll get back to you on that.\" FAs did not have online access to customers' monthly account statements or client correspondence. The paradigm shift that had to be executed\u2014and quickly\u2014was changing from a \"product-centric\" system to a \"client-centric\" system.\n\nMerrill's second-generation, Internet-based Wealth Management Work Station is an example of the now-dominant IT paradigm of the wealth management industry: open architecture. Diane Schueneman, who developed the system as chief of Global Infrastructure Solutions, defines the term as follows: \"As an end user, I'm going to take information from any source in an integrated fashion, in such a way that I really don't know or care whether the information is sourced internally or externally. That distinction should be invisible and immaterial to the end-user.\"\n\nAccording to Capgemini's Bertrand Lavayssi\u00e8re, \"It may sound surprising, but even today you will find long-established firms that either won't or _can't_ perform the tasks necessary to achieve truly open architecture. Many firms continue to be reliant on an outmoded model that has the financial advisor pick stocks for you, and think about what is best for you, on his or her own terms, using proprietary systems and software.\"\n\nThe primary problem with closed architecture, as Merrill Lynch and its peers have discovered, is that FAs don't like it. The principal reason they don't like it is that it puts them at a strong disadvantage to their competitors empowered by open architecture. As Lavayssi\u00e8re asserts, \"The key here is transforming the business from a traditional product-management business model to a relationship-based model. In such a business, customer interactions are handled holistically across the business. Ideal solutions assess the client's current position, help define the customer experience, determine the technical architectures required and how they can best be implemented across all viable channels to market.\"\n\nWhen asked to describe how the advent of technology had transformed the advisor-client relationship, Lavayssi\u00e8re responds, \"The first thing to keep in mind about technology is that the critical component is the nature of the person using it\u2014that is, _the person facing the screen_. Many of the more sophisticated clients today will have their own planning and simulation tools on their desktops at home. This enables them and empowers them to develop a set of pretty strong opinions when they meet with their FAs. In some cases, they might test the FA's knowledge and capabilities against their own knowledge and capabilities. Some self-made wealthy individuals prefer, in many cases, to act as their own FAs.\"\n\nThe second thing to keep in mind, Lavayssi\u00e8re insists, is that \"too much information _kills_ information. Technology is only useful if it helps the advisor to perform his or her primary task, which is to cull through the universe of information to select what is most meaningful to meeting the needs of the client.\" One example of technology's enhanced role is in the realm of reporting: \"What are we _actually_ doing with those assets, not just today but yesterday and tomorrow? With all these new vehicles of increasing complexity, both advisors and clients need an accurate real-time snapshot of what the portfolio is _actually_ doing. Even with the most up-to-date technology, it can be difficult to maintain a comprehensive view of the present, let alone the future.\"\n\nScott Becchi, vice president in Capgemini's North America Financial Services Businesses Unit, observes that \"the ultimate goal is turning technology into an information system. I call it 'the personalization of pertinent facts.' This challenge increases exponentially when you have more demanding and knowledgeable clients who understand that their needs will best be met by 'uptiering' to the next category.\" By \"uptiering,\" Becchi is referring to a phenomenon observed by Capgemini consultants by which clients in one demographic segment aspire to a level of service they know (or strongly suspect) is being offered to the segment above them. This is, of course, only human nature.\n\nChris Gant, head of Capgemini's Wealth Management practice in the U.K., contends that as the global phenomenon of wealth consolidation continues apace, an industry rooted in the discreet guidance traditionally provided to wealthy families by a small private bank in Luxembourg or Switzerland remains highly fragmented, with an estimated six thousand firms holding themselves forth as possessed of the expertise to run money for high-net-worth individuals and households. Of those, he maintains, only the relatively small handful committed to combining the capabilities of major players with a large global footprint with the intimacy and level of service provided by the traditional private bank will attain leadership positions in this fiercely competitive field.\n\nThe resulting polarization, Gant says, is largely driven by variations in technological capabilities, with regard to accuracy, timeliness of reporting and the provision of user-friendly aggregated data and actionable knowledge. Since the underlying purpose of technology is to enhance and support the dialogue between advisor and client, savvy clients should be on the lookout for firms willing to invest the substantial resources required to develop and maintain the most capable tech platforms\u2014while educating end-users to keep in mind that technology should facilitate the human dialogue as opposed to restrain or replace it.\n\n#  **OUTSOURCING FUNCTIONS: THE JOYS AND PERILS**\n\nIn the face of competitive pressures to both retain clients and maintain high profit margins, firms must avoid becoming or being perceived as a distributor of routine commoditized products. Wealth advisors must provide value-added advice along with operational excellence, cutting-edge technologies, and broad product lines at competitive prices. For many firms this involves smart outsourcing of certain functions that will allow them to focus on delivering the products and services they do best and that differentiate them in the market.\n\nProviders are taking a hard look at which IT activities to conduct themselves and which to outsource. Pure-play administrative specialists offer providers a way to outsource costly back-office tasks to organizations boasting the necessary global scale to keep costs down. This, in turn, permits advisors to offer wealthy clients competitively priced global products and services while freeing up their own time to concentrate on providing clients with the relationship management they deserve.\n\nVirtual Service Networks and Service-Oriented Architecture are two examples of technology that enable multiple service suppliers to work closely and seamlessly with HNWIs.\n\n**Virtual Service Networks**\n\nThe ongoing surge in global wealth reinforces the importance of serving the needs of \"mid-tier millionaires\"\u2014corporate executive and small to mid-sized business owners with anywhere between US$5 million and US$30 million in investible assets. While such clients have every right to expect a level of service commensurate with their asset base, dedicated family investment offices are luxuries typically out of reach of any but the ultra-affluent US$100-million-plus households. For the rest of the HNW population, the industry has developed Virtual Service Networks (VSNs), which are virtual networks of financial, legal and accounting experts who collaborate via technology to deliver a level of service to clients roughly commensurate with family offices, at a fraction of the cost. We estimate that as many as 8 percent of the total HNWIs could potentially afford to tap into VSNs to obtain a cost-effective wealth advisory solution located somewhere between the more general private banking function and the full-scale family office.\n\nJon Carroll, co-founder of Family Office Metrics, a consultancy that provides business advice to family offices around business organization, management, operations and technology applications, observes that \"a VSN allows the family to operate like a board of directors, and to maintain control over strategy and who they work with.\" Despite the advantages, Mr. Carroll warns that even with a VSN, multi-family offices can find it costly and difficult to offer the kind of carefully personalized service seen in the single family office setting. This is a major issue, he says, since \"relatively\" lower bands of wealth always want the same service as those above them\u2014a phenomenon we have described as \"up-tiering.\"\n\n**Service-Oriented Architecture**\n\nThe latest innovation in this field\u2014a development in some respects more ambitious than the VSN\u2014is Service-Oriented Architecture (SOA), which permits firms to establish a framework that supports multi-channel advisor and client-service strategies. OASIS (the Organization for the Advancement of Structured Information Standards) defines SOA as:\n\n_\"Aparadigm for organizing and utilizing distributed capabilities that may be under the control of different ownership domains. It provides a uniform means to offer, discover, interact with and use capabilities to produce desired effects consistent with measurable preconditions and expectations._ 37\"\n\nOne should not, for example, have to provide redundantly the same personal information to open an online checking, savings or IRA account, and further, the interfaces one interacts with should have the same look and feel and use the same level and type of input data validation. Building all applications from the same pool of services makes achieving this goal much easier and more deployable to affiliate companies. An example of this might be interacting with a rental car company's reservation system even though you are doing so from an airline's reservation system.\n\n**Figure 3.3** \\- Wealth Management Service-Oriented Architecture\n\n_Source_ : Capgemini analysis.\n\n#  **MELDING INFORMATION TECHNOLOGY AND CULTURE**\n\nBuilding the ultimate adaptive private banking capability calls for more than the right strategies, business processes, technologies and infrastructures. It also requires adaptable people and a culture that supports change. Wealth management providers must establish and nurture cultures that foster experimentation, set clear targets at all levels and reward employees who are truly motivated to put clients first. In the absence of such a backdrop, the potential value of all IT investments can never be fully realized.\n\nIn today's hyper-competitive marketplace, private banks must develop online channels to better serve their clients and increase their own top and bottom lines. But banks should not settle for commoditized functions such as online trading; they need to push themselves to focus on innovative tools centered on collaboration between advisor and client. The online channel is a great source of potential value for private banks\u2014but they need the foresight and fortitude to tap this resource. Furthermore, banks must confront the indisputable fact that future generations of high-net-worth individuals and households will demand the best IT services available, or quite justifiably take their assets and liabilities elsewhere. And as we will see in the next chapter, providing clients with appropriate levels of conveniently accessed user-friendly technology is just one facet of a larger challenge: the development of a full-scale and comprehensive holistic wealth management model.\n**CHAPTER 4**\n\n**Holistic Wealth Management**\n\n#  **MOVING BEYOND INVESTMENTS TO A BROADER RANGE OF SERVICES**\n\nRooted in the industry-wide transition from stock brokering to wealth preservation, the rise of Holistic Wealth Management (HWM) dials up the medley of services offered by advisors to include virtually every aspect of an individual or household's financial life. Investments, risk management, insurance, tax, trust and estate planning, retirement planning, credit and cash management\u2014all fall loosely under the broad umbrella of HWM. And all, not surprisingly, are rendered exponentially more complex when the individual or household fits into the HNWI or UHNWI category.\n\nA gentleman exceedingly well versed in all aspects of HWM is D\u00f3mhnal Slattery, co-founder of International Aviation Management Group (IAMG)\u2014a firm he sold in 2001 to the Royal Bank of Scotland for what we can safely assume was a price tag sufficient to make all future labor strictly voluntary. Yet he remains an exceptionally busy person at this stage of his life. Through Dublin-based Claret Capital, he manages his own family fortune along with the assets of three other families, and has participated in some of the world's largest private equity transactions.\n\nWhile Slattery is a professed fan of the concept of HWM and regards it as the coming thing in the field, he cautions that it's critical to draw a firm distinction between HWM and schmoozing (or, for that matter, the unctuous provision of what some in the field refer to\u2014frequently in hushed tones\u2014as \"concierge services\"). Financial advisors, he strongly cautions, should not fancy themselves glorified butlers.\n\n\"There's this belief that ultra-high-net-worth individuals need to be looked after and loved,\" Slattery indignantly states. \"There's this belief that high-net-worth individuals are always demanding free opera tickets from their financial advisors. But in my experience, it's simply not true. Most of my colleagues consider such events a chore and a bore. If I want to go to the opera, I'll go with my wife!\"\n\nMerrill Lynch Managing Director Michael Sullivan, co-head of a newly minted Cross Organizational Client Coverage Initiative that aims to drive greater synergies between the investment banking and wealth management side of the business, heartily concurs with Slattery. \"Concierge services?\" he asks. His team's clients, he insists, \"want their money run the way they run their companies. They want spreadsheets, they want goals and objectives, they want projections, they want rigor and accountability from their FAs just as they demand it from their senior executives. If they're looking for tickets to a Knicks game, they'll go to Ticketmaster.\" A lesson for aspiring HNWIs is that although these tickets and services appear to be icing on the cake, it is probably more likely that the cost of these services is bundled elsewhere.\n\nSullivan, similarly to Joseph Lam, insists that what HNWIs must want and deserve is to feel valued by their financial advisors, with their needs understood and acted upon. And they quite rightly demand and require a broader and deeper range of service than the traditional private banks have historically offered. Meeting these demands is where the opportunity for new entrants, such as brokerage houses and banks building capabilities for the HNW, lies. Today, many HNWIs prefer to draw on the intellectual resources of a large institution with a global footprint combined with the intimacy of a boutique.\n\nAccording to Sullivan, the four things all high-net-worth clients reasonably demand from their advisors, at the end of the day, are simply expressed.\n\n1. Performance;\n\n2. Service;\n\n3. Proactive solutions;\n\n4. Trust.\n\n\"They want performance against their game plan and performance against the markets. They want the best possible financial guidance and advice\u2014bar none. They want the best information we can provide on a proactive basis, with solutions based on an open architecture platform. But what they want above all is trust\u2014because without that, all the rest fades into insignificance.\"\n\nConcierge services aside, HNWIs who can clearly afford tickets to cultural events may be eager to seek advice on tax, family wealth, estate and philanthropic issues, along with a whole host of other ancillary services for clients and households with over US$10 million in investible assets, including, for example, the aforementioned financial boot camps for their kids.\n\nHNWIs want advisors to arrange access to some of the world's largest and most well-respected investment experts. They want ideas, strategies and products that are special and novel to the marketplace. In short, they want what Merrill Lynch's Private Banking and Investment Group (PBIG) aims to provide\u2014high-end, high-touch services\u2014by constantly endeavoring to deliver the \"whole firm\" to clients. If that means referring a client to another division within the firm, John Thiel, head of PBIG, insists that it's all about sharing knowledge among Merrill's various businesses.\n\nEarly on in his career as an FA, Thiel was often shocked to run into other FAs \"who didn't regard it as somehow appropriate to ask a client how much money they had. I used to ask them why not. They'd get these befuddled expressions on their faces and say they just didn't think it was any of their business.\" In those early days, Thiel recalls, \"I met a lot of good people, but the business was all about pushing product and not about determining clients' unmet needs, aspirations and long-term financial goals. If the number one rule of salesmanship is 'know your customer,' the simple answer was that we didn't.\" Conversely, clients should be willing to be open in sharing information with their financial advisors, given the FA has built the trust and credibility mentioned above. This information can often assist the FA in better understanding options and preparing proposals for how to best manage assets, either internally or externally.\n\nStarting out in the nineties, the stirrings of HWM emerged at Merrill Lynch under John L. \"Launny\" Steffens, who insisted that financial advisors deliver Financial Foundation reports\u2014comprehensive financial plans based on extensive personal surveys expressly designed to elicit and define clients' long-range financial goals and objectives\u2014for all of their clients or pay the consequences in reduced compensation. Although that initially onerous mandate made Steffens one of the least popular people at the firm for a number of years\u2014because in its early phases the shift from a transaction-based account to an asset-and-fee-based account can generate less profit\u2014the advent of the Financial Foundation provided was the underpinning of what would come to be dubbed HWM.\n\nAs Steffens repeatedly stressed, a critical facet of HWM is the high value placed on a client's willingness to entrust all or at least most of their total assets _and_ liabilities (loans, mortgages and the like) to the care of a single financial advisor or team domiciled at a single firm. Since it is not uncommon for high-net-worth individuals and households to spread their assets among a number of advisors and firms, the theory of HWM is that not unlike romance, fidelity is the key to enduring success.\n\nBy the mid-nineties, the rise of the Internet placed the full-service financial industry under an ominous cloud of perceived obsolescence. The whole notion of the trusted advisor was in genuine jeopardy. Why not just go buy an index fund? What was the point of managed money? Since managers didn't do any better, on average, than the _Wall Street Journal'_ s notorious dartboard, why bother? The critics weren't totally off base, because it can be hard, statistically speaking, for even the best money managers to consistently beat the market.\n\nThe answer for both firm and client turned out to be a wholesale adoption of a broader outlook encompassing the provision of a full range of legal, accounting, insurance and financial advisory services, rolled up into a collaborative effort potentially drawing on the expertise of outside attorneys, independent money managers, possibly even a family office, and possibly a team of private equity specialists, derivatives designers or investment bankers.\n\nUltra-HNWIs might seek referrals from their financial advisor to commission such services as family security or even property management. In the best-case scenario, the client would have a primary and, ideally, sole point of contact at the firm capable of arranging and brokering these ancillary services with third-party providers. The financial advisor would perform like a financial maestro, orchestrating the variety of instruments required to play the client's concert. Ultimately, the role of the wealth advisor would be to simplify a wealthy client's financial life; in an era of growing complexity, streamlining can command a steep premium.\n\nIf a HNWI desires information and insight into foreign markets and exotic asset classes, a global institution might provide specialist wealth strategy teams spanning the globe and equipped with multiple specialties. A select number of wealth management providers employ this business model on a micro level, providing product specialists serving a narrow geographic range. Companies such as Northern Trust employ an informal network to service clients' requirements regardless of source. In order to be truly effective, the specialist team approach must be accompanied by rigorous training for the relationship manager, who needs to be on familiar terms with all of the firm's global offerings or know whom to call upon elsewhere in the firm to tap the expertise required.\n\nThis team-based approach is typically most effective in advisory areas apart from investing. Some relationship managers may leverage groups of specialists prepared to advise clients on wealth transfer and other family and legacy issues, philanthropy and business ownership matters. Teams might commission attorneys and business analysts for tax and estate planning, in addition to ancillary services such as business evaluation.\n\nSome clients may place a high value on the advice a financial advisor provides on the perennially anxiety-provoking topic of adequately planning for long-term health care expenses. Frequently, high-net-worth clients are concerned more about their aging parents than their own financial futures. Some clients are asking for\u2014and receiving from financial advisors\u2014dedicated health care financing services, including the evaluation of a range of health care options, and\/or putting their clients in touch with relevant outside experts in the field.\n\n**M &A Reshapes Private Banking**\n\nThe private banking business is beginning to display a barbell appearance\u2014on the one end, large institutions with a global footprint and deep pockets, and on the other end, boutique shops with fewer resources and an emphasis on high-touch customer service. The middle ground is thinning out rapidly as mergers reshape the industry. For HNWIs, the decision of where to place their money between the two poles can pose some tough choices.\n\nThis dilemma was thrown into sharp relief in late 2006 when Bank of America (BofA) announced its acquisition of U.S. Trust, the private banking arm of Charles Schwab, for US$3.3 billion. Then in early 2007, Merrill Lynch announced it would buy San Francisco-based First Republic Bank for US$1.8 billion in cash and stock. With assets of US$10.7 billion, First Republic provides investment services including trust banking and luxury home lending through over forty-three branch offices in the United States.\n\nBofA's rationale is to join the top tier of private banking\u2014and asset-wise the deal will accomplish precisely that. JPMorgan has about US$230 billion in client accounts and Citi Private Bank has approximately US$220 billion, while the new private banking arm of BofA will have roughly US$260 billion. BofA officials insist that they are intent on maintaining the kind of top-notch service that U.S. Trust clients are accustomed to, but as with all acquisitions and ensuing integrations, how well the ambition and anticipated value add to clients and shareholders is implemented remains to be seen. Based on a SECOR Consulting analysis of over 2000 deals globally, included in a submission to the Canadian Federal Department of Finance on Shareholder Value Created and Destroyed in Financial Services Mergers, it was validated that acquisitions across all industries are better served through a related diversification strategy (i.e., bank-investment bank), rather than a consolidation (i.e., bank-bank) strategy. What is true in general holds true for Financial Services. Only 34 percent of the bank-bank mergers in the US and Europe from 1990 have been successful in adding shareholder value. However, related diversification deals have performed better, with bank-investment bank deals being successful for the acquirer 55 percent of the time.\n\nIf lessons from this research and other post merger integrations hold true, the rationale for the BofA\/US Trust deal stands. The two cultures\u2014a mass-market retail culture on the BofA side and an exclusive, catering mindset on the U.S. Trust side\u2014will not be so easy to fold together, and will require care, attention and time. Indeed, immediately after the announcement, BofA defended its planned cost efficiencies by assuring analysts that cuts would not target front-line customer services. Subsequently, a few months after the announcement, Peter Scaturro (CEO of US Trust) departed the company despite previous statements regarding his necessity to the integration.\n\nThe big private banking players argue, not surprisingly, that they can and do deliver top-notch customer service. Plus, they say, their strong balance sheets support loan underwriting, while they can bring the global resources and clout to bear to get HNWIs into coveted hedge and private equity funds. Boutiques, for their part, maintain that they provide superior personal service, and that big banks tend to push their own products, while boutiques are more likely to design best-of-breed solutions tailored to HNWIs' needs. As for breadth of products and services, boutiques are improving thanks to an ironic twist: they are outsourcing some products and services, such as loans, to the big banks.\n\nMeanwhile, the middle ground, which some HNWIs argue has the best of both worlds\u2014an institutional focus on wealth management, solid resources and personalized service\u2014is occupied by fewer and fewer players. In December of 2006, Bank of New York (BoNY) announced its intention to buy Mellon Financial for US$16.5 billion. The union will unite BoNY's US$60 billion wealth management group with Mellon's US$92 billion group, combining two mid-tier players. By the end of 2006, the remaining mid-tier firms were a lonely group. Bessemer Trust, with US$47 billion in client accounts, and Northern Trust Corp., with approximately US$128 billion, both still profess an intention to remain independent. But the dynamics of the industry may overtake them.\n\nWhen a HNWI's wealth management provider is undergoing a merger, it may take some extra vocalizing to ensure that he or she continues to receive the attention they deserve. A merger is a time of transition for the employees, who may shift jobs and firms, as well as for the clients, who must adjust to the new corporate culture being forged. HNWIs may not need to search for a new wealth provider in the event of a merger or sale, but that said, it may not be the time for HNWIs to put their wealth management on autopilot either.\n\n#  **EXPECTING MORE EXTENSIVE FINANCIAL ADVICE**\n\nYet for all the emphasis placed in some reports on ancillary services and the wide variety of other fringe benefits increasingly provided by financial firms seeking to broaden their share of the HNWI market, the fact remains that the service most HNWIs seek from their financial advisor is precisely what their professional name implies: financial advice.\n\nClients will always demand that the advice provided by their advisors be objective, candid and solution-oriented as opposed to product-oriented. The most sophisticated clients are not impressed by products and services being \"sold\" to them, particularly if the prime motivation might be to generate a steady stream of fees for the firm. In this day and age, if a HNWI senses any untoward bias towards a particular family of products, it's likely the relationship with the financial institution will conclude fast. \"The business model for most global banks hasn't changed much from just selling the product of the week,\" D\u00f3mhnal Slattery avers. Entrepreneurs like him have a sixth sense when they are being sold, and a marked aversion to being sold to.\n\nWhen HNWIs are interested in buying in-house products it's often because the products are innovative vehicles specifically structured in-house and tailored to meet their needs. Ultra-HNWIs don't stay ahead of the curve by reverting to the mean; they demand products not available to the average Joe. HNWIs typically demand value added that's outside the mainstream\u2014very likely an investment that is complex and illiquid. Which brings us to the greatest conundrum of holistic wealth management: _wealth advisors are being asked to add complexity to the high-net-worth investment portfolio at the same time that they're being asked to manage its growing complexities._\n\nYet as previously noted, a holistic approach amounts to much more than traditional financial advice\u2014no matter how brilliantly conceived. One key differentiator is the quality of the advice and strategies offered in support of non-financial assets. Increasingly, HNWIs are demanding that art, collectibles and other \"investments of passion\" be treated similarly to more common and liquid financial investments. Globalization has impacted philanthropy, with affluent philanthropists allocating a substantial portion of their assets abroad, slowly bringing philanthropy to the global stage. Lifestyle, legacy and philanthropic guidance are not traditionally core offerings for providers but, as is the case with financial investments, such issues require careful asset allocation and portfolio attention\u2014subjects all treated in greater depth elsewhere in the book.\n\n\"The advice needs to be complete,\" contends Paula Polito, head of strategic brand marketing and management for Merrill Lynch's Wealth Management division. \"High-end clients require a high order of benefits, and they demand a personal relationship. They want someone who knows how many children and grandchildren they're sending to private school.\"\n\n\"Money is always personal,\" Polito observes. \"Financial institutions would do well to acknowledge that fact as a critical component of any holistic strategy. High-net-worth individuals have a right to, and should expect a partner they can trust\u2014a desire that, not surprisingly, transcends age and geographic borders.\"\n\nOf course, for the majority of HNWIs, money may be _so_ personal that it can be challenging to view one's own needs, goals and requirements with an impartial or critical eye. One approach employed by some savvy FAs to overcome perfectly natural human biases is to encourage HNWIs to view their wealth no differently than how they might view their businesses or jobs. Few businesspeople feel comfortable making a significant decision regarding one part of their business without considering its possible impact on other aspects of their business. A similar approach may be effective when mulling specific asset allocations, charitable donations, estate planning or other investment strategies. Applying simple business management techniques to wealth management often helps HNWIs view certain personal decisions a bit more dispassionately and with a clearer head.\n\nTo bring even more context to the HNWIs' financial pictures\u2014and to encourage them to be even more holistic in their approach\u2014some advisors are pushing into actual business owner services. Business ownership has become the leading source of HNWIs' wealth across the globe. Often, business owners and the operations of those businesses are intricately entwined, so there is a growing need for increased convergence between commercial, private and investment banking. At most firms the traditional wealth manager is not equipped to serve at the cross-road between personal and business opportunity. This disconnect is a prime example of how wealth advisors are hamstrung by old technologies and business practices that divide their institutional view into accounts instead of clients\u2014even as they profess to offer holistic services.\n\nClients understandably view themselves as having one relationship and one master account with each organization; hence, they would like to see different cuts, slices and groupings based on their distinctive needs. There should be one risk profile and one strategy governing an entire relationship, but unfortunately this has historically not been the case at many large global institutions. This is a frustration voiced by many clients who maintain multiple points of contact across departments of a financial services provider. Clients who are looking to grow wealth want to have access to multiple products regardless of the manufacturer. Some of the leading financial services providers are responding and are working to overcome this gap. For business owners, this is clearly something to consider when selecting your primary financial advisor.\n\n#  **USING DEBT TO BUILD WEALTH**\n\nOne of the foundations of HWM is that for financial advice to be comprehensive, it needs to take a client's entire range of needs into account. In addition to asset management, skilled liability management can be critical to a client's broader financial health and well-being. And as investor sophistication increases, advisors and clients alike realize that a comprehensive approach demands leveraging liabilities as a way of preserving and even gaining wealth.\n\nThe key concept here is opportunity cost\u2014for many sophisticated investors, taking on a liability in one area of the portfolio produces an asset in another. This is why, according to a Federal Reserve Board survey of Consumer Finance, the nation's richest 1 percent took on US$342 billion in new debt between 1998 and 2004. That 1 percent now holds 7 percent of the nation's debt, with a total of US$650 billion, up from 5 percent in 1998. In other words, debt for this top group grew 150 percent in those six years, while debt for Americans in the fiftieth to ninetieth percentile range grew 100 percent in those years.\n\nSo does this growth mean that HNWIs have lost their financial discipline? Not by a long shot. For many HNWIs, debt is a financial tool, a way to allocate resources to the most high-yielding investments. Which is better, paying cash for a US$10 million ranch, or taking out a 6.5 percent mortgage and plowing that US$10 million into a private equity fund that's expected to yield 20 percent? The math is really pretty simple.\n\nNon-HNWIs, on the other hand, typically don't have that kind of cash to redirect, and thus aren't targeting investments to offset the interest payments they are taking on. Interest payments on home equity lines of credit and credit cards detract from their wealth, while HNWIs take on debt specifically to _expand_ their wealth. What's more, the high dollar amount of debt being taken on by HNWIs can be a bit deceiving. True, HNWI debt is increasing, but debt for the top 1 percent still represents only 3.7 percent of their total wealth. For those in the fiftieth to ninetieth percentile, debt represents 24 percent of their total wealth.\n\nMarcus Mitchell, managing director for Americas Lending in Merrill Lynch's Global Bank Group, sees four broad ways that HNWIs are using debt to manage their portfolios. The first is leveraging their investment to diversify or enhance returns. He observes this often with private equity and hedge fund managers who are sitting on large, long-term positions tied to their own funds. By pledging their limited partner or general partner interest in the funds, they can borrow money from the bank and invest elsewhere.\n\nYet another frequent use of debt is for asset acquisition. Sometimes this takes the form of financing aircraft and boats; sometimes it's more of an equity injection into a new business funded by borrowing against a large block of stock. Other times, borrowing is performed to create a liquidity cushion available to purchase other assets or in case there's a sudden cash crunch.\n\nFinally, borrowing is frequently employed in conjunction with an estate plan or generational-transfer strategy. Life insurance may be purchased by HNWIs to help heirs pay estate taxes; the dilemma then becomes how to pay for the hefty premiums. The answer: by pledging marketable securities along with the cash value of the life insurance policy, the HNWI can borrow against the policy to fund the premiums, eliminating out-of-pocket expenses.\n\n\"Why do wealthy people borrow when they've got the money?\" Mitchell rhetorically asks before replying as follows: \"Let's say I want to finance an aircraft or diversify my assets. It's all about the power of leverage\u2014a concept that has transformed investment banking in recent decades. Just as companies seek to strike a balance between debt and equity, so do wealthy individuals, many of whom are used to thinking this way from running businesses, so it doesn't scare them to assume the risk as long as it's accompanied by a reward.\"\n\nAs an example, the senior partners in a hedge fund may be required to invest their own money in a deal, but their funds are already tied up in other deals. They borrow that money as an alternative to liquidating other assets, fully confident that they can repay the loan out of the proceeds of the deal. Of course, as Mitchell cautions, only individuals and households deemed qualified and capable of shouldering such leverage need apply; it takes investment savvy and sophistication to benefit from a prudent program of enhanced liabilities.\n\nFor advisors, HNWIs' attitude toward debt presents a number of significant business opportunities. Not only can advisors offer counsel on the wisest uses of debt and leverage for their clients, but they can also make attractive loans available to their clients through the financial institution\u2014a product and service that in combination help to deepen the HNWI relationship.\n\n#  **DEFINING THE ROLE OF FAMILY OFFICES WITHIN WEALTH MANAGEMENT**\n\nIndividual family offices, where a wealthy family (with at least US$100 million in assets) employs a full-time staff of financial advisors and other experts to see to it that every conceivable planning need will be met, obviously provide the ultimate in holistic service. Multiple family offices, serving a comparatively small number of often affiliated families, represent the second tier. Often unfettered by legacy systems, such organizations may more effectively manage the holistic total view and maintain personal relationships. However, their access to global resources in an era when such access is regarded as increasingly critical tends to be inferior to the major financial institutions.\n\nThe majority of family offices have four primary functions:\n\n1. Centralization of records (all financial statements, bill paying, tax planning);\n\n2. Business management (purchase and sale of business holdings, distribution of financial information, administration of tangible assets such as real estate and yachts);\n\n3. Family management (family governance, education, philanthropy, wealth transfer planning);\n\n4. Investment-related functions (investment policy setting, asset allocation, monitoring).\n\n**Figure 4.1** \\- Traditional Financial Services Institution Model Approach vs. Family Office Model Approach\n\n_Source_ : John Carrol, \"The Functions of a Family Office,\" _The Journal of Wealth Management_ , 4, no.2 (Fall 2001).\n\nD\u00f3mhnal Slattery notes that typically even large, well-run and well-equipped family offices do not operate in a vacuum. They want the services and resources of big global banks if only those global banks could break down silos\u2014or at least build bridges between them\u2014and give family offices access to the services of the firm. Mr. Slattery recalls a business opportunity he recently had that required aggressive financing. His private bankers balked and the opportunity passed. The very next month, while meeting with that same bank's investment bankers, he was told: \"Of course we could have done that\u2014why didn't you bring it to us?\"\n\nDisheartening as this occurrence was for Mr. Slattery, he remains optimistic that large global banks will improve their holistic offerings to family offices, if for no other reason than that the hybrid model of his family office\u2014part individual, part institutional, highly aggressive, with assets north of US$200 million\u2014is a huge growth area. \"There will be lots and lots of these opportunities in the next decade. Banks will want to find the next Blackstone and the next MSD Capital (Michael Dell's family office) and grow with them.\"\n\nSlattery predicts that ten years from now, global firms prepared to make a strong commitment to these kinds of clients and this kind of service will attain and maintain a dominant position with regard to this incalculably valuable client group. To create loyalty among these hybrids (or \"instividuals\"\u2014institutions providing individual service) he believes banks should allow clients to invest alongside the global bank. \"Co-investing with the firm is the sweet spot, in my opinion. That engenders a lot of loyalty. That alignment of interest is the key to building a bond.\"\n\n#  **EXPECTING CLEAR REPORTING**\n\nThere is no single aspect of holistic wealth management more critical to long-term investment success than clear, comprehensive and timely reporting of the ever-changing position of client portfolios. The higher net worth the individual or household, the more emphasis tends to be paid to providing a broad view of the portfolio position that can\u2014if the client desires\u2014be easily drilled down into to achieve ever greater levels of detail\u2014all rendered seamlessly by the latest portfolio management software.\n\nThis is the essence of the wealth management platform, and the place where the rubber meets the road when it comes to client service. Institutions that fail to provide clear reporting can never compensate for that failure by offering top-notch personal service or financial advice. One high-profile issue is that some global institutions, as well as regional banks and brokerages, may be constrained by the limitations of legacy back-end and account-based processes, as noted in the previous chapter on technology. Historically, many providers lack visibility from one account to another due to disordered back-ends. Accounts held in different countries add to the disarray. A full 78 percent of relationship managers surveyed for the 2006 _WWR_ said reports have become more complex even as HNWIs increasingly clamor for greater simplicity and clarity.\n\nUntil fairly recently, HNWIs have grudgingly tolerated awkward reporting, but the impact of globalization on client portfolio complexity is triggering a call to action. In fact, HNWIs' inability to view asset holdings coherently is one of the biggest reasons cited by HNWIs for wanting to change firms or add independent advisors. According to the relationship manager survey for the 2006 _WWR_ , only 50 percent of HNWIs are satisfied with current reporting. In addition, investors stated that clear and aggregated reporting was a leading reason why they would potentially switch financial service providers. That's a pretty frightening statistic for financial services providers and a powerful incentive to supply simplified one-stop reporting.\n\nIn any wealth management relationship, four pillars of reporting must be addressed.\n\n1. Quality: Diversified investing requires reporting that is accurate and complete. As HNWIs shift out of their traditional comfort zones and invest in more complex instruments, many denominated in different currencies or linked to businesses operating in different time zones, they demand greater transparency. Providing timely tracking of shifts in country and multi-currency breakdowns, for example, is critical to meeting HNWIs' goal of maintaining tight control of global positions.\n\n2. Customization: Not all HNWIs want the same reporting formats. The ability to provide desired detail and to include performance reports according to preferences lies at the very heart of maintaining a successful relationship. HNWIs expect recommendations to be positioned in the context of their total net worth and overall asset allocation, and when reviewing their ongoing relationship with their advisor they will consider how well this was done.\n\n3. Simplification: Providing accurate positions, balances and information in line with a client's preferences or requests does not necessarily bring satisfaction. Finding a way to represent this information in an easily understandable format that aligns back to their overall financial plan and wealth strategies will become a higher priority as HNWIs diversify into new products and regions.\n\n4. Cost containment and clarity: In the past, fees have been layered into services and products in ways that have made the costs of those products and services obtuse and difficult to examine individually or compare. This is no longer acceptable practice. Today, HNWIs demand reports that show exactly how much they are paying and for what. Firms should not fear this disclosure. Far from it. The _WWR_ surveys show that while HNWIs are willing to pay a premium for good service, they want to see what they are paying. Clarity and transparency promote trust and in the end provide another way to strengthen the bond between advisor and HNWI.\n\nThe good news for advisors is that by working toward a more holistic approach, by reducing complexities, by offering comprehensive products and services, and by clearly reporting costs, a firm can establish and ensure its role as the family's trusted financial advisor. Indeed, as portfolios become increasingly global in scope, holistic wealth management will become increasingly in demand by HNWIs. Firms that can deliver will be in an excellent position to establish deep, long-term relationships. The more indispensable a firm makes itself to a high-net-worth family, the more likely it will be to retain those assets in the wake of a generational transfer.\n\n#  **DEMANDING PRIVACY**\n\nMany HNWIs experience a degree of tension between the seemingly conflicting needs for _comprehensive reporting_ and _privacy_. In a perfect world, not only HNWIs but everyone would prefer a single, easy-to-read report encompassing all assets held by all providers _\u2014_ domestically and internationally. In real life, however, any of a number of issues can hamper this goal.\n\nOn the more prosaic side, some HNWIs understandably hesitate to share all their financial information with any one entity out of fear that their privacy will be compromised. For aggregated reporting a client must pick a trusted provider to manage information from all of their outside providers and third parties, and some HNWIs worry that that is too much information for any one advisor to have. Such information might, for instance, entice an FA to focus on luring those assets to his or her firm instead of focusing on advice and performance.\n\nSuch reluctance to share information for fear of being oversold is merely the tip of the iceberg. In reality, people may have multiple reasons not to want to pull all their financials together to provide a single, unified, easy-to-read view. A nasty divorce, estrangement between relatives, or murky business arrangements may require what some might regard as a dubious level of discretion. How people want financial information combined varies enormously from person to person and household to household; even if a bank already maintains many different accounts under an individual's name, in some jurisdictions it may be actually illegal to combine these accounts into a single report in the absence of the client's consent.\n\nIn some areas of the world, privacy needs to be guarded to avoid confiscation of assets by governments, or even the prospect of actual physical harm. In describing visits to clients in Latin America, relationship managers recall being told not to bring documents, a laptop or even business cards, out of fear that their family's security might be compromised if the level of their wealth were to be exposed. Many HNWIs in insecure regions or countries have very real reason to fear increased prospects of them and\/or their family members being kidnapped the more people know about their total net worth.\n\nIn Africa, the Middle East and some parts of Asia, the wealthy are known for their obsession with discretion. There, too, it's not uncommon for all forms of bank-initiated correspondence to be discouraged. A commonly used term, HAM (hold all mail), is often used. Regulatory reasons dictate that some correspondence needs to be sent. In Switzerland, meanwhile, it remains culturally acceptable to fax information. But financial institutions dare not try this in many developing countries or they run the risk of losing a client due to confidentiality issues.\n\nAny act that may be construed by relevant regulatory and law enforcement authorities as an attempt to conceal assets from their government tends to create complicated legal trails\u2014and travails\u2014for wealthy individuals. A favorite for Latin American HNWIs are private investments corporations (PICs) and trusts, frequently set up in the United States to shield their names behind seemingly unaffiliated legal entities. These are perfectly legal strategies, and can make it a challenge to follow the legal trail. But they also make it very hard to have a full view of the customer and provide holistic wealth management.\n\nFortunately, most HNWIs need not go to such lengths to protect their assets and loved ones, but these issues do exist for a sizable population of HNWIs. The lesson here is that global financial institutions must remain flexible when catering to their international clientele and never assume to impose any single approach, even an apparently advantageous holistic one.\n\nClearly, HWM is a multi-faceted discipline, incorporating asset and liability management, framed within exceptional service and alternative service methods and assurance of privacy. Applying the doctrine of asset allocation requires paying exquisite attention to cultivating a portfolio's overall balance, risk-adjusted for the long term. An appropriate analogy may be to artistic gardening\u2014a similarly sophisticated discipline, part art and part science, in which the most talented landscape architects grasp at a deep level how best to mix hardy perennials with annuals to achieve a climatically balanced long-term effect. All of which rests firmly on a theoretical and conceptual foundation known as Modern Portfolio Theory (MPT).\n**CHAPTER 5**\n\n**The Doctrine of Asset Allocation**\n\n#  **ENHANCING RETURNS WITH MODERN PORTFOLIO THEORY (MPT)**\n\nOne day in 1951, Harry Markowitz, a graduate student in mathematics at the University of Chicago, was sitting outside his thesis advisor's office waiting to discuss the subject of his doctoral dissertation. His field of research was linear programming, a discipline that employs complex mathematical models to maximize output for a given level of cost or to minimize cost for a given level of output. One of the most common applications of linear programming is a mathematical equation by which auto manufacturers seek to determine how many cars should optimally be built when constrained by specific amounts of materials and worker hours.\n\nFortunately for the future development of the fine art of risk management, the mathematics professor was not immediately free to see Markowitz. While cooling his heels in the waiting room, Markowitz struck up a conversation with a stockbroker also waiting to see one of the professors in the department. Strictly to kill time, the broker asked Markowitz to describe his field of research. After listening intently for a few minutes, the broker casually observed that a promising application of linear programming might be for a young mathematician to tackle the challenges inherent in forecasting and tracking fluctuations in financial portfolios.\n\nAnd so, the discipline of asset allocation\u2014the most critical conceptual component of the now-widely-accepted doctrine known as Modern Portfolio Theory (MPT)\u2014was born. Intellectually stimulated by the notion of applying his arcane tool kit to such a hitherto virgin field of research, Markowitz began by drawing what turned out to be a stunningly fruitful analogy between desired output as a manufacturer might define the term and desired output as an investor might define it. For investors, the desired output of a well-managed portfolio would be an above-average rate of return over time. In his linear analysis, Markowitz defined costs as the level of volatility to which any particular portfolio might be exposed in order to generate an above-average rate of return. With regard to risk, Markowitz worked forward from the not unreasonable proposition that most investors' primary goal would be to reduce it or\u2014if practically possible\u2014to avoid it whenever possible.\n\nThe key to MPT, which Markowitz formally unveiled in the _Journal of Finance_ 42 less than two years after his seminal and random conversation with the stockbroker, was the notion that optimal portfolio diversification could be obtained by drawing precise relationships between three forms of data: 1) the expected _return_ of each component in the portfolio; 2) the expected _volatility_ of each component's return; and 3) the expected _correlation_ of each component with each of the other components.\n\nA well-balanced and diversified portfolio, Markowitz's theory strongly suggested, is one that combines a number of assets whose market fluctuations are not naturally well correlated with each other. For example, if the phenomenon noted by economists that emerging-markets economies appear to be \"decoupling\" from those in developed markets continues to hold true, a degree of diversification could be derived by constructing a portfolio containing a number of \"non-correlated\" assets drawn from developed and developing economies.\n\nWhat made Markowitz's MPT so revolutionary was that it turned the traditional advice doled out to investors\u2014to construct a portfolio containing as many securities as possible that could reasonably be expected to increase in value over time\u2014virtually on its head. After Markowitz, investors and advisors who subscribed to the power of \"asset allocation\" favored portfolios assembled on the premise that a key predictor of long-term investment potential was the degree to which any particular basket of assets contained a sufficient number of components whose anticipated returns could be presumed to be non-correlated with each other. The point was not just to pick good stocks, but rather to pick the right _combination_ of stocks that promises optimal diversification. Investors, Markowitz argued, reasonably expect and deserve to be compensated for taking on any level of risk above that of a Treasury Bill\u2014a level of compensation formally referred to as the \"risk premium\" attached to an asset.\n\nBy 1964, William Sharpe (destined to share the Nobel Prize in Economics with Merton Miller and Markowitz for their joint development of MPT) improved upon Markowitz's methods of evaluating assets, and continued to further refine the methods of risk assessment and management employed by the majority of asset managers today. Although it took a few decades to embed itself thoroughly into the investment universe and to gain widespread acceptance among the intellectually conservative brokerage community, by the eighties many more (although by no means a majority) of financial advisors had embraced the notion that traditional stock picking was not as effective a means of portfolio construction as one in which investments are evaluated on a statistical basis in terms of their long-term rate of return potential and expected short-term volatility. By some estimates, up to 90 percent of a portfolio's long-term positive performance could be ascribed to the quality of the asset allocation strategy employed, and less than 10 percent to the accuracy of the forecasts related to individual stocks in the portfolio.\n\nThe effects of MPT on wealth and asset management best practices were nothing less than profound. If stock picking\u2014the traditional pursuit of the portfolio manager\u2014was less critical to a portfolio's long-term performance than asset allocation and diversification, such an assumption only reinforced the industry's gathering trend toward redefining the role of an FA as a _relationship manager_ : one who comprehends the entirety and complexity of a client's financial picture and aspirations, as opposed to one who can claim to have a pretty decent idea of which securities are likely to rise or fall in value over time.\n\nMPT also strongly reinforced the notion that for an advisor to achieve the optimal asset allocation of a client's portfolio, a comprehensive approach to balancing the client's long-term goals and objectives was virtually a prerequisite. For financial services providers and asset managers, the overwhelming internal logic of MPT reinforced the argument that an asset allocation strategy was only effective if a majority\u2014if not a totality\u2014of client assets were maintained under one roof at a single financial institution. Otherwise, it would be impossible for any single money manager to see the entire household portfolio comprehensively.\n\nSo even if most HNWIs are not particularly comfortable computing Sharpe ratios, standard deviations and other arcane calculi of risk, today's widespread embrace of MPT reflects the nearly universal appreciation of the fact that asset diversification and allocation are critical components of any financial plan. Most investors, of course, intuitively grasp the notion that it is important not to put all of one's eggs in one basket. It's not unreasonable to conclude that the essence of Markowitz and Sharpe's achievement has been to mathematically codify and refine what really amounts to a modicum of common sense.\n\n#  **UNCOVERING THE PREFERRED ASSET ALLOCATIONS OF THE WEALTHY AND ULTRA-WEALTHY**\n\nThe basic building blocks\u2014otherwise known as asset classes\u2014of any portfolio are:\n\n\u2022 Equities (stocks in companies);\n\n\u2022 Fixed income (bonds issued by companies or government agencies);\n\n\u2022 Cash\/Deposits;\n\n\u2022 Real estate (either direct investment or through real estate investment trusts, known as REITs);\n\n\u2022 Alternative investments, a growing category consisting of an eclectic product mix, that may include structured products, hedge funds, foreign currency, commodities and private equity to be dealt with in greater detail in the next chapter.\n\nThe asset mix of any portfolio will typically include a percentage of some or all of the above, although alternative investments remain primarily the province of HNWIs. As history and _WWR_ surveys have repeatedly demonstrated, the wealthy display a heightened sensitivity to the economic environment and tend to reallocate portfolios in accordance with current and impending economic conditions. In 2002, for example, HNWIs adopted defensive allocations of their assets as a means of weathering a recession and a period of poor market performance. A market recovery a year later prompted these same wealthy individuals to reallocate assets to equities and alternative investments, which offered higher return potential.\n\n**Figure 5.1** \\- HNWI Assets by Investment Class, 2004-2008F\n\n_Source_ : Capgemini\/Merrill Lynch Relationship Manager Survey, 2007.\n\nIn 2004, following exceptionally high returns in 2003, HNWIs adopted a conservative and diversified approach to their portfolios, increasing their allocations to fixed income and cash\/deposits to combat market volatility. In 2005, HNWIs adopted even more aggressive strategies than in 2004. They increased their allocations to equities and alternative investments and, anticipating interest rate hikes, shifted funds away from fixed income. In 2006, HNWIs shifted away from alternative investments and toward real estate (as illustrated in Figure 5.1).\n\nWhile clearly everyone would love to have an \"ideal\" portfolio\u2014a Holy Grail typically defined as one promising a high potential return on investment unburdened by anything other than reasonable risk\u2014in reality, investors understand that they must accept a certain level of risk to attain an expected level of reward; just like everything else in life, there are inevitably going to be trade-offs between the two. And how much risk the investor will be comfortably willing to take involves a medley of factors, including age, goals, time horizon and liquidity requirements.\n\n**Ultra-HNWIs Take the Lead**\n\nOver the years, the _WWR_ has consistently shown that ultra-HNWIs\u2014individuals with more than US$30 million in net financial assets\u2014tend to make investing decisions ahead of market trends. Ultra-HNWIs account for 1 percent of the global HNWI population and about a third of its financial wealth, and tend to be more sophisticated and better informed than other HNWI investors when it comes to managing their assets. In many ways, ultra-HNWIs are thought leaders whose behaviors and attitudes about wealth management hold important lessons for HNWIs and other investors alike.\n\nUltra-wealthy portfolios tend not only to be more diversified than those of the mass affluent, they are also more aggressively invested than those of individuals in lower wealth bands. Overall, ultra-HNWIs allocate a greater percentage of their assets to alternative investments than do average HNWIs\u201424 percent compared to 20 percent, respectively, according to the 2006 _WWR_ survey. The use of alternative investments demonstrates ultra-HNWIs' \"tax intelligence,\" as many of these investments allow them to minimize or defer taxes. While ultra-HNWIs allocate a smaller portion of their portfolios to equities than do HNWIs, they exhibit a taste for more complex investment products such as hedge funds, private equity\/venture capital, structured products and investments in their own businesses. Furthermore, ultra-HNWIs hold a significantly lower share of their assets in fixed income and cash, while allocating more of their wealth to real estate, than the average HNWI. The rationale is that they have more wealth to distribute, and history shows that real estate over the long term may be a relatively safer investment.\n\nUltra-HNWIs also tend to be more aware of how wealth is managed internationally, according to the professionals who help manage their portfolios. They tend to tolerate greater exposure to international markets and have more geographically diversified portfolios than their HNWI counterparts. They allocate a lower percentage of assets to North America in favor of regions with a higher number of emerging markets, such as Asia-Pacific and Latin America. In fact, in the survey, ultra-HNWIs say they believe that this movement of investment funds out of mature economies, such as the United States, and into attractive growth economies in other parts of the world is likely to pick up steam.\n\nSince a large percentage of ultra-HNWIs are creating physical and financial presences in international locations, it's hardly surprising that more than 25 percent of HNWIs maintain both homes and relationship managers in countries other than their primary residence. For ultra-HNWIs, such personal and household globalization has become even more pronounced in recent years. More than half maintain residences and financial accounts located outside the nation in which they are officially domiciled, while 45 percent have wealth managers located beyond the borders of the nation they claim as a primary residence.\n\nOverall, ultra-HNWIs' sophisticated asset allocation and risk management strategies help to explain how they have achieved and maintained their elite status on the world stage. Given their proven power to move markets, we anticipate that ultra-HNWIs' behavior will prompt HNWIs to consider embracing more sophisticated and diverse investment vehicles in developed and emerging economies when suitable.\n\nUnexpectedly low returns, or unexpectedly high returns, are evidence that an investor's asset allocation may be unbalanced. The investment mix might be too conservative or too risky, but truly unexpected returns would typically reflect the fact that the original risk parameters are not well defined. Of course, human nature being what it is, when returns are surprisingly strong it can be particularly difficult to question risk\/return objectives\u2014but the problem with achieving unexpectedly high returns during one point in time is that it tends to indicate that the level of risk being taken offers the promise of unexpectedly lower returns at some point in the future. Figure 5.2 lays out a number of risk factors to be taken into consideration.\n\nWhile it may be tempting to bask in the heady illusion that the rules of investing have been suspended for one fortunate individual or period, or that a manager has simply and brilliantly outmaneuvered the market, the latter is sadly hardly ever the case while the former has been proven well near impossible. An obvious example of this fact is the rise and fall of the Internet bubble in the late nineties, when dot-com companies were viewed by many investors as a risk-free way to earn enormous returns; ultimately, of course, these investments proved risky indeed and cost many their life savings.\n\n**Figure 5.2** \\- Typical Risk Factors that a Wealthy Family Should Be Aware Of\n\nLarry Fink, founder and CEO of asset management firm BlackRock, cites the recent example of commodities hedge fund Amaranth, based in Greenwich, Conn., a self-styled multi-strategy \"market neutral\" hedge fund that advertised itself as being able to maintain a perpetually low risk profile by quickly shifting in and out of different markets as opportunities wax and wane. Amaranth's wrong-way bets in the energy market in 2006 triggered roughly US$6.4 billion in losses, or 70 percent of its assets, forcing it to liquidate its investments to pay investors. Nine months before Amaranth ran aground, BlackRock began to question the risks the fund was taking because the return for 2005, about 30 percent, seemed too good. \"When you see aberrant success, you have to ask, 'Did I take too much risk?'\" Larry Fink asks. BlackRock began probing Amaranth to understand how it achieved those returns if it was still hewing to its stated investment strategy. After receiving unsatisfactory explanations, BlackRock decided Amaranth must be deviating from its multi-strategy in some way and taking on more risk than BlackRock wanted to be exposed to. Ultimately, it paid a 3 percent early redemption fee to exit the Amaranth fund, and when the fund abruptly disintegrated a few months later he was happy to have done so.\n\nFink's lesson is a valuable one for advisors and HNWIs constantly searching for new financial products that will generate the potential for high yields. The search for the next hot thing is hardly unreasonable. After all, the first investors into an effective investment strategy\u2014whether it's a hedge fund or private equity\u2014are almost always the ones who make the largest returns. As other investors see the results, more money floods in, driving up demand and reducing potential returns. HNWIs and their FAs must then move on once more if they want to stand any chance of consistently beating the market. In other words, the investment universe is populated by a continuous search for a higher, greener pasture in which to graze before the rest of the herd arrives. Yet being always in the vanguard carries risks that may be difficult to measure precisely, because they are new.\n\n#  **MOVING BEYOND MODERN PORTFOLIO THEORY**\n\nOne practical problem with MPT and its lessons regarding the value of asset allocation and diversification is that most HNWIs, particularly self-made ones, have accumulated their wealth by defying precisely what MPT preaches. They tend to be exceedingly concentrated in one investment\u2014typically their own business. The old belief that you get rich through concentration and stay rich through diversification may be pretty well understood, but it can be a pretty tough pill to swallow for HNWIs who for a variety of reasons are not well diversified and like it that way.\n\nDiversification typically means reversion to average returns; HNWIs have not gotten rich by accepting average returns as the norm. \"Most have become wealthy thanks to one or two success stories\u2014not asset allocation,\" says Mr. Rothenberg of Capital Research and Management. \"But once you have a certain amount of wealth you need to stop worrying about enhancing it, and start worrying about protecting it.\"\n\nHNWIs cite emotional ties to their businesses as what makes the transition from concentration to diversification a difficult one. There's also the fact that these HNWIs really do know more about a business that has made them wealthy than any other investment they might make. As one HNWI says, \"I'm confident I can grow my business by 15 percent. There's nothing out there I know of that can do as well. I know I'm unbelievably undiversified, but why should I sell to buy the S&P 500?\"\n\nThere are still other reasons it can be difficult and not always advisable for HNWIs to follow conventional advice and diversify. For top executives at a public company who may have a huge amount of wealth tied up in company stock, selling a large block of stock can be a politically sensitive issue. Investors don't like to see management sell their stakes; they consider it a lack of confidence in the future of the business; no end of explanations about personal portfolio rebalancing and asset allocation is likely to fully assuage their concerns.\n\nAlso, there may be tax implications associated with liquidating a long-held position, which makes adopting a standard strategy of diversification difficult or ill-advised. Most investors do not want to liquidate positions to reinvest elsewhere unless they are compelled to do so. Of course, if you're lucky to be tapped for government service, then you can have your entire tax bill forgiven. Robert Rubin and Henry Paulson were both tapped to be Treasury secretaries while leading Goldman Sachs. In Paulson's case, he owned 3.23 million Goldman shares worth about US$485 million today, according to regulatory filings. He agreed to sell other investments, including a stake in China's largest bank, for a reported US$800 million. At a 15 percent capital gains rate, that's a tax savings on the Goldman shares of at least US$73 million and total savings of about US$120 million. Most other HNWIs, however, will have to find other ways to pay tax efficiently and diversify their assets.\n\nIn \"Beyond Markowitz: A Comprehensive Wealth Allocation Framework (WAF) for Individual Investors,\" a paper published in _The Journal of Wealth Management_  by Ashvin Chhabra, Merrill Lynch's former chief of Wealth Management Strategies and Analytics, a number of HNWIs are profiled who have profited enormously by defying MPT's asset allocation provisions. Examples of their profitable defiance include:\n\n1. Investors who took out substantial mortgages on real estate that doubled or tripled in value in just a few years, racking up above-average returns on their initial investment, which was the down payment on the property.\n\n2. Corporate executives who hung on to their original stock options and grants in firms whose equity value doubled or tripled, far above the market.\n\n3. Entrepreneurs who invested substantial amounts of financial and intellectual capital in their own business, whose value rose steeply over a comparatively short period of time.\n\nThe primary question posed by the paper: were these individuals taking foolhardy or unnecessary risk, or behaving in some way irrationally? The answer is clearly not the former but rather the latter. Certain individuals, particularly those blessed with special insight or knowledge of the value of certain assets (such as an entrepreneur's insight into the likely growth rate of their own businesses) might well achieve above-average returns based on a risk-reward profile that is, if not impossible, certainly difficult to quantify.\n\nYet another aspect of risk that Chhabra's Wealth Allocation Framework addresses is the notion of _aspirational_ risk. In general, one requires a lot of money (relative to what you have) in order to increase your wealth percentile. For example, for a household to move from the 40th to the 60th wealth percentile, its net worth would need to approximately _triple_. In order to move up in the wealth pantheon, one must acquire substantially, as opposed to incrementally, greater wealth. One cannot meet such an aspirational goal without taking on substantial risk\u2014hence the term \"aspirational risk.\" On the surface, taking such a degree of risk would seem to be at odds with the imperative to protect oneself and one's family from falling into a lower wealth percentile. It would seem reasonable to assume that most investors in this situation would pass up the opportunity for improvement in favor of the certainty of the status quo. However, human behavior frequently defies this assumption\u2014particularly when we're speaking about the behavior of HNWIs and those who aspire to be like them and to invest like them.\n\nAt virtually all economic levels, aspirational risk-taking is a common phenomenon. Examples range from the frequent buying of lottery tickets to more mainstream financial investments, such as the substantial purchase of a single stock, investment in real estate, or small business startup. Every time a member of the peer group succeeds in his aspirational risk-taking, the difficulty of remaining in the same peer group is raised for those who passed on the opportunity. Consider how hard it is to improve or even maintain one's position on the Forbes 400 list. The net worth required to be the 400th richest American has increased by a factor of ten since the early eighties, to US$1 billion from US$100 million.\n\nThe conventional MPT framework lacks the flexibility to accommodate the dual demands of safety and aspiration, Chhabra contends. The WAF represents an important step toward incorporating these aspects of safety and aspiration while building upon the foundation of MPT. A HNWI's portfolio should allow him or her the opportunity to achieve the following:\n\n\u2022 Protection from anxiety and poverty;\n\n\u2022 Ability to maintain his or her living standards;\n\n\u2022 Provide an opportunity to increase wealth substantially or meet aspirational goals.\n\n**Figure 5.3** \\- Personal Risk, Market Risk and Aspirational Risk\n\nFor all the strengths of MPT, its essential weakness is that it only recognizes market risk and seeks to minimize it through optimal asset allocation. In contrast, the WAF identifies three very different risk dimensions and optimizes all three of them simultaneously. The fundamental idea is to create a financial strategy that organizes and allocates all of a HNWI's assets and liabilities into 1) a protective (low-risk) bucket, 2) a market (market-level risk) bucket and 3) a riskier aspirational bucket. According to Chhabra, successful HNWIs pay as much attention to _risk allocation_ as they do to _asset allocation._ According to Chhabra, these are not\u2014as some MPT theorists would have it\u2014\"deviations from rational investing\" but investment decisions whose logic lies outside and beyond the rather limited confines of MPT as traditionally conceived.\n\n#  **COMMENTING EXPERTLY ON ASSET ALLOCATION**\n\nMerrill Lynch Chief Investment Strategist Richard Bernstein points out that the true test of a good asset allocation is the degree to which a client's portfolio is appropriately matched not just to a client's assets but also to his or her liabilities. \"What even sophisticated investors tend to forget when exclusively focused on rate of return is that returns must be adjusted and expressed in the context of a client's _liabilities_ ,\" Bernstein observes _._ \"Let's say a client has three kids heading off to college and he wants to retire while they're still in school.\" Bernstein points out that \"In such a case, the liability side is going to be paramount.\"\n\nFully comprehending the liability side is the really tough trick, Bernstein contends. In fact, it is what separates the wheat from the chaff in the investment advisory business. \"You can download a pretty decent asset allocation model from the Internet for free,\" he observes, emphasizing yet another area in which investment advisory services have been commoditized. \"That model will do an OK job of telling you about sectors and weightings and even factor in a comparatively unsophisticated risk profile.\"\n\nBut the area in which a truly talented FA will outshine the computer model (in a way reminiscent of world chess champion Gary Kasparov beating IBM's Deep Blue) is by presenting a client with the full menu of possibilities and products designed to meet, mitigate and offset the needs on both the asset and liability side\u2014an act of empathy that typically takes a human perspective to comprehend. According to Bernstein, a client's risk tolerance level is never simply a function of the size of his or her portfolio. \"Someone with four billion dollars might rather die than lose one billion dollars, even if they still have three billion left over...a twenty-three-year-old unmarried male with two hundred thousand dollars might have a far more aggressive style than a billionaire, because he\u2014in both senses of the term\u2014has considerably less to lose.\"\n\nAccurately defining and where appropriate _extending_ a client's time horizon, Bernstein maintains, is the key to long-term superior performance. \"We conducted a study that persuasively demonstrated that the [farther away] a client sets their time horizon, the higher [the] probability exists of attaining higher long-term returns.\" The reason day traders are so rarely successful over the long term, he suggests, is that by shortening their time horizon to a matter of hours and days, they reduce their probability of making money to practically zero. On the flip side, a client willing to take a much longer view\u2014possibly extending out decades\u2014will vastly improve his or her chances of achieving superior performance over time.\n\n**Asset Allocations Vary Regionally**\n\nAsset allocation for specific HNWIs is affected by the maturity and pace of development of their local markets. The question is not just what a HNWI should invest in, but what is literally available from market to market. For instance, the Merrill Lynch Capgemini 2007 _Asia-Pacific Wealth Report_ found that in 2006, Asia-Pacific's allocations to cash\/deposits, a traditionally \"safe\" asset class, were significantly higher (24 percent) than the global average (14 percent), driven by wealthy investors in places like South Korea, Japan and China. The high cash allocation in South Korea and China reflected both a lack of alternative products and a lack of sophistication in the WM market. Meanwhile, Japanese HNWIs also had a limited availability of international product but, more significantly, were the most focused on wealth preservation, as opposed to wealth accumulation, which led them to allocate much of their assets to cash\/deposits.\n\nIn addition to their cash\/deposits, South Korean HNWIs allocated a quarter of their assets to fixed income, much higher than elsewhere in the region\u2014again because a narrow range of products limited their investment options. HNWIs in Indonesia, on the other hand, had among the smallest allocation (about 13 percent) to fixed income, chiefly because they had limited access to fixed income products. In fact, Indonesia's local-currency bond market is among the smallest in the region: it is less than one-tenth the size of South Korea's local-currency bond market. (To spur its bond market, the Indonesian government in 2006 was considering refashioning its bond market's trading system along the lines of South Korea's.)\n\nObserving these differences closely can be valuable for HNWIs and their advisors alike. On the one hand, HNWIs can see investment patterns in more developed markets and thus try to position themselves better. Advisors, for their part, get a hint of what new products and services clients might yearn for as the WM market develops. Clearly, for instance, there is a strong correlation between a high allocation to alternative investments and the maturity of the WM market.\n\nThe 2007 _Asia-Pacific Wealth Report_ also studied Asia-Pacific HNWIs' allocations to alternative investments, and found that the allocations were highest in Singapore (10 percent). Considering that market experts view Singapore as one of the most mature and sophisticated WM markets in the region, it was not surprising to find that HNWIs there had the highest allocation to alternative investments.\n\nHNWIs in China (one of the least mature markets in the region) also tended to allocate one of the larger percentages toward alternative investments (9 percent). Why? The most likely explanation is that Chinese high-net-worth individuals have a strong interest in personal business and private equity, a trend largely ascribable to the fact that the vast majority of Chinese HNWIs are self-made entrepreneurs. In contrast, HNWIs in South Korea, where the WM market is relatively immature, had the lowest exposure to alternative investments.\n\nAs WM in Asia's other developing markets matures and becomes more sophisticated, one can expect HNWIs outside today's leading WM markets to adopt the more diversified and aggressive investment strategies now found in places like Singapore and Taiwan. Alternative investments will gain favor and real estate will grow even more popular across the region as securitization of properties becomes more established. Savvy HNWIs and advisors have already begun to position themselves and their clients for this evolution.\n\nEven with the modifications, questions, revisions and refinements suggested by theorists from Chhabra to Bernstein that improve on the original theories proposed by Markowitz, Sharpe and the other pioneering Modern Portfolio proponents, the greatest and most enduring accomplishment of the doctrine of asset allocation is that it has allowed individual investors to become more sophisticated, because it has encouraged them\u2014via the ongoing imperative to diversify\u2014to look beyond investments that were deemed safe strictly because they were familiar. Redefining risk has encouraged investors to seek a more diversified portfolio by investing in such ostensibly non-correlated assets as emerging-markets securities and alternative investments dependent on more flexible mandates than the traditional \"long-only\" strategies employed by garden-variety mutual funds. By stressing the link between volatility and reward, wealthy investors have developed a more sophisticated risk appetite, displaying a taste for adventurously moving beyond the financial equivalent of macaroni and cheese or a ham sandwich to the global grazing associated with eclectic nouvelle cuisine.\n**CHAPTER 6**\n\n**Alternative Investment Strategies**\n\n#  **THE APPEAL OF ALTERNATIVE INVESTMENTS**\n\nSo many different kinds of investments tend to be lumped together under the \"alternative\" tag\u2014including structured products, derivatives, hedge funds, private equity, foreign exchange, commodities and even so-called investments of passion\u2014that the term threatens to be rendered meaningless. Just the same, items in the last category, including fine art, classic cars and sports memorabilia, require a significant cash outlay, and the wealthy are taking an increasingly sophisticated approach to vetting these purchases. In response, a number of high-end advisors are offering consulting services around art acquisition, prompting Bijan Khezri, CEO of the London-based Artist Pension Trust, to proclaim fine art \"a new asset class\" in the op-ed pages of the _Wall Street Journal_.\n\n**Art**\n\nThe art market is a notoriously boom-and-bust affair, frequently jumping during strong economic conditions and declining when the economic winds turn less favorable. Even so, interest in art as an investment has maintained its high pitch even in the face of a turbulent economy. During the last Gilded Age, fine art was primarily purchased by HNWIs as status symbols and as a sign that the collector had arrived, not that he or she was looking to make money on their acquisitions. Collectors have always hoped that their purchases had lasting value, of course, but today's focus on future returns when considering an art purchase is unprecedented.\n\nApart from the vagaries of the economy at large, the art market is (not surprisingly) strongly affected by changing tastes, as artists fall in and out of favor, insuring that pricing information is zealously guarded by auction houses and that liquidity (the ability to sell art for what it was bought) can, at times, be as murky as some collectibles' provenance. Thanks to the Internet, the art market is possibly more transparent than ever, yet it remains a market that is positively opaque by virtually any standards of investing.\n\nNevertheless, more and more investors continue to rush in. In 2006, Sotheby's Holdings sold US$3.6 billion of art, up a third from 2005. In the fall of 2006 alone, Sotheby's and Christie's International, the other major auction house, sold US$1 billion of Impressionist, modern and contemporary art. Christie's reports that the 2006 sales of Impressionist art were up 81 percent, while postwar and contemporary art were up 58 percent, from 2005. Sales of contemporary Asian art, most of it Chinese, zoomed to US$190 million in 2006 from US$22 million in 2004.\n\nA number of factors would appear to be fueling this surge. First is the amount of wealth still sloshing around the globe despite the depredations of the 2007-08 credit crisis, turning the trend truly international. In 2006, a Hong Kong investor paid US$17.3 million for Andy Warhol's \"Mao,\" a record for the artist. In the United States, at least up until the fall of 2007, hedge fund managers were shelling out huge sums to hastily assemble prominent collections, while many were becoming significant art patrons and board members of art institutions.\n\nAnother driver behind the surge in fine art is scarcity value. Despite the high volume of art on the market, big-name artists and indisputable fine work typically remains in short supply, making the search for second-tier and new artists more intense. All this has recently led to respectable returns for art and opportunities for huge returns. Michael Ovitz, co-founder of Creative Artists Agency, bought a painting by the relatively unknown Barnaby Furnas for about US$20,000 in 2003 and sold it for US$520,000 in 2006.\n\nAs art prices have surged, a more concerted effort has ensued on the part of academics to more accurately measure returns. New York University business professors Jianping Mei and Michael Moses compile an index that compares repeat auction sales of the same art objects over time\u2014the Mei Moses All Art Index. Their index of postwar and contemporary art shows a compound annual return of 12.6 percent for the past decade, compared with a return of 9 percent for the Standard & Poor's 500 stock index. Meanwhile, to bring greater pricing transparency to the realm of art investing, Artnet tallies fine art auction sales involving 180,000 international artists. The database represents auction records from more than 500 international auction houses since 1985 and covers more than 2.9 million auction sales.\n\nYet the reliability of such measurements, given the tight-lipped nature of art sellers and buyers, can be difficult to gauge. Sotheby's and Christie's International are the biggest auction houses and the most obvious source of pricing information. But they often do not disclose terms of sales or even the identity of sellers or buyers. It's even more difficult, if not virtually impossible, to obtain accurate and timely information on art gallery sales. The art market is, as mentioned earlier, an opaque one, even with the advent of Internet auction sites such as eBay. It is also an unregulated market.\n\nStill, art is more than just a numbers game\u2014and always will be. Great returns tend to accrue to those with a great eye\u2014the art investor's equivalent of having a sharp nose for business evaluation. Good, old-fashioned art appreciation (the non-financial kind) will help steer collectors. Former Microsoft executive and Russian space tourist Charles Simonyi vows that he will never sell his collection of works by pop artist Roy Lichtenstein or op art pioneer Victor Vasarely, but he also ascribes his success as an art investor to self-education, by which he has enhanced his personal understanding of the context of art both within the scope of art history and within the personal development of the artist. (It's also helpful, according to Simonyi, to become friendly with artists, both established and aspiring, and, if possible, to become something of a fixture on the art scene, as would be the case with just about any investing landscape.)\n\nCombine an interest in art collecting, the hefty sums involved, and the opaque nature of the market, and we've got a corner of WM that's a tailor-made opportunity for highly specialized wealth advisors. Some financial institutions already offer loans to make art purchases and some provide advice and market research on fine art. Whether the current art market is peaking is hard to say, but art collecting itself is timeless, a passion shared from the Caesars to the Medicis to the hedge funds managers of today.\n\nJust how dramatic has the recent rise in alternative investments been? From 2002 to 2006, HNWIs more than doubled their allocations to alternative investments, boosting the share of the average HNW portfolio to 22 percent from 10, mainly at the expense of cash and bonds. By 2006, however, the wealthy and ultra-wealthy, possibly collectively sensing a negative shift in the class' risk profile, had slightly shifted their allocations out of traditional alternatives and into real estate.\n\nOne reason for this shift may be that alternative investments op+portunistically capitalize on highly volatile markets; the 2007 _WWR_ suggests that in 2006, a sharp reduction in the Volatility Index (VIX) could be presumed responsible for a discernible bias away from hedge funds toward more tangible assets\u2014including investments of passion such as high-end collectibles.\n\nYet this shift in allocation is viewed as likely to be short-lived, because \"a majority of investors simply don't find the expected return versus risk in the public stock and bond markets compelling,\" maintains Ralph Schlosstein, the recently retired co-founder of BlackRock. \"They see alternative investments as a way to preserve wealth through absolute returns\u2014which means seeking positive returns in both bull and bear markets while not being tied down to traditional performance benchmarks like the S&P 500 Index.\"\n\nAccording to Daniel Sontag, head of Merrill Lynch's Global Wealth Management division for the Americas, the remarkable rise in interest in alternative investments on the part of high-net-worth clients has conferred at least one additional and little-noted benefit to portfolio managers: many alternative asset classes have historically been negatively or non-correlated with traditional asset classes, providing a classic MPT allocation balance. Particularly during periods in which stocks and bonds\u2014which have historically moved in opposite directions\u2014fluctuate roughly in tandem, a basket of non-correlated assets including alternative investments typically imparts added value while enhancing portfolio principal protection.\n\n**Real Estate**\n\nReal estate would appear to straddle the line between an alternative investment and its own discrete asset class. HNWIs can be invested in real estate in numerous ways: primary residence, vacation homes domestically and abroad, direct investment in residential and commercial real estate, or investment through REITs or other partnerships. And certain kinds of real estate\u2014such as industrial warehouses that often have long leases\u2014tend to hold up well in a down economy.\n\nNotably, despite the headline grabbing nature of the sector, in which every interest rate move and inventory report seems to make the front page, HNWIs had been consistent with their allocation to real estate, holding steady at 15 or 16 percent for most of this decade.\n\nSuch faith has by and large been amply rewarded, although risks remain. The subprime mortgage meltdown in the United States beginning in early 2007 through 2008 battered the stocks of many mortgage companies and REITs. Residential real estate as an asset class suffered severely in many parts of the U.S. in 2008 albeit with some compelling exceptions, particularly at the ultra-high-end and in the most desirable neighborhoods.\n\nIt's important to bear in mind that the real estate market includes various types of assets, and that the major real estate categories do not necessarily move in lockstep, which can help to diversify a portfolio. The net result is that while the real estate market tends to be referred to in aggregate, sweeping generalizations about the direction of the market are difficult given the range of real estate and the idiosyncrasies of individual markets.\n\nOne indication of just how bipolar the market has become is that in early 2007, as single-home prices continued to decline in much of the United States and inventories increased, a pitched battle broke out between two prominent private equity investment consortia to purchase Sam Zell's Equity Office Properties Trust, a commercial REIT that owns directly or through partnerships about 600 office buildings in twenty-five markets, making it the country's largest office landlord.\n\nJames Marrelli, National Association of Realtors vice president of commercial real estate, notes that in 2006, a record amount of capital flowed into commercial real estate. \"Institutional investors, pension funds and foreign investors have focused on commercial grade properties to diversify portfolio assets, with expectations of solid long-term gains.\" Outside of the hotel sector, more than US$236 billion in commercial real estate transaction volume was recorded in the first ten months of 2006, up from US$231.9 billion in the same period of 2005, not including properties valued at less than US$5 million.\n\nAs the scope of real estate investing grows, so does the opportunity for both HNWIs and their advisors. More than ever, HNWIs can take on a view of real estate markets around the world, cashing in on a reliable wealth-creating asset. Increasingly, investors in commercial real estate are looking globally, beyond the United States and Europe to Russia, Turkey, China, India and Japan. For advisors, perceived expertise in the more exotic markets is bound to matter more than ever in this age of increased investor sophistication and connoisseurship. If the question is whether to invest in a shopping center in Russia (or more likely a fund that specializes in such investments), advisors will need to deeply understand and be able to explain in plain language the pros and cons of such an investment, or have the resources to obtain such expertise. Every different corner of the real estate market has its own particular drivers; the health of each type of real estate varies greatly from city to city, much less country to country.\n\n#  **STRUCTURED PRODUCTS FOR INDIVIDUALS**\n\nStephen Bodurtha, a senior vice president and vice chairman of Merrill Lynch's Global Wealth Management division, has led teams that create innovative investment solutions tailored to the tastes of high-net-worth individuals. Bodurtha, who has self-described roots as \"a financial products engineer,\" and his team try to \"take some of the innovation sweeping the financial markets and adapt it prudently to the private client universe.\"\n\nThe critical differences between individual and institutional clients in the structured products world, Bodurtha explains, are the limits that clients place on risk tolerance and time horizon. \"Individual investors typically have a hard time taking comfort in the fact that, if their portfolio value drops 20 percent in a year, sticking to the right asset allocation strategy will have a good chance of swinging the portfolio into the positive in the next cycle.\"\n\nSo how do advisors address the crying need on the part of people who have acquired great wealth to gain access to some of the more compelling opportunities available to investors in the marketplace without threatening a loss of principal? Starting out in the mid-nineties, Market Index Target-Term Securities (MITTs) were among the first protected equity-linked notes (PENs)\u2014the private-client version of proprietary products employed by institutional clients as investment vehicles to manage risk by guaranteeing a protection of principal. The twist was that the issuer of these notes, in this case Merrill Lynch, was able to structure the products so that the client traded off a set amount of the notes' upside profit potential in exchange for minimizing downside risk. For comparatively risk-averse clients whose primary interest was principal protection, these notes became attractive components of their overall asset allocation and risk management strategy.\n\nYet another twist that further increased client demand for products that offered principal protection combined with a growth potential somewhat better than a Treasury note or a savings account was that the ongoing boom in the equity markets was producing ever-greater equity-related wealth. Much of this wealth was being captured in vehicles ranging from Employee Stock Option Plans (ESOPs) to the proceeds from Initial Public Offerings (IPOs). \"A number of senior and highly compensated executives,\" Bodurtha observes, \"would suddenly find themselves the proud possessors of material amounts of wealth concentrated in a very small number of stock positions.\"\n\nWhat those clients needed to do was diversify their holdings. The essence of the wealth protection strategy achieved by the next generation of proprietary products (known as \"collars\") was to hedge the client's highly concentrated portfolio by permitting the client to 1) hang onto the stock while 2) permitting the client to shed some of the stock's downside risk by transferring some of the upside opportunity to a third party. \"From a risk management standpoint,\" Bodurtha explains, \"what these collars did was to insure the portfolio against the risk of a drop in a single stock that would unduly affect the value of the entire portfolio.\"\n\nThese products amounted to the first wave of a trend that Bodurtha characterizes as \"a customization of risk,\" or what might be better termed \"investor-centric innovation.\" \"One of the most critical challenges when creating these products and tailoring them to individual client needs,\" Bodurtha stresses, \"is persuading investors to clearly articulate their investment and risk parameters. Then it becomes our challenge to convey to them the fact that these goals\u2014some of which might appear to be in conflict\u2014might actually not be irreconcilable. That's where the innovation and ingenuity come in.\"\n\nAs Bodurtha points out, twenty years ago who would have believed that exchange traded funds (ETFs), introduced by The American Stock Exchange in the nineties, could be a viable product for private clients? But now these ETFs allow investors to own whole sectors, countries or asset classes with a single purchase. Another output of innovative financial engineering includes products permitting HNWIs to take a view on real estate markets in individual cities, thanks to the 2006 launch by the CME of future and option contracts on the housing markets in ten different U.S. cities.\n\nKevin Waldron, an FA with Merrill Lynch in Bala Cynwyd, Pennsylvania, has been helping clients plan for retirement using structured \"bear notes\" designed to protect clients with concentrated positions in certain sectors. Recently, such customized products have enabled clients to hedge against expected shocks from prospective housing market declines. \"That market was so out-of-whack, we wanted to turn to [notes],\" Waldron recently observed, recalling that Merrill provided similar portfolio protection to clients in the technology industry before the tech meltdown in the spring of 2000.\n\nMany of Waldron's HNW clients have accumulated substantial wealth through leverage or concentrated positions in certain industries, and viscerally resist \"walking away from the way that they made their wealth.\" But many are finding an appealing hedge in housing-oriented structured bear notes, tied to the Philadelphia Housing Index, which are engineered to benefit from the negative environment in the housing industry. Some notes are structured so that even if the housing index rises, the client doesn't risk losing principal.\n\nThe list of innovative products migrating to the public markets goes on and on. For a quick lesson on the advantages and risks of a new alternative investment, consider the wave of catastrophe bonds issued since Hurricane Katrina. In 2006, hedge funds, pension funds and other money managers invested more than US$9.2 billion\u2014more than double from 2004\u2014in disaster-related investments. If a hurricane or some other disaster hits, the investors are on the hook, but if no damage occurs the gains can be substantial.\n\nAdvisors have been busy tailoring all sorts of tweaks, such as \"wind risk,\" to these bonds, opening more opportunities for investors to make money and for sellers to obviate risk. One of the great advantages of these new investments is they are not correlated with other investments. But, as with any new, successful strategy, the window to enjoy big gains can close quickly as other investors rush in. In the case of hurricane risk, for instance, recent yields of 10 percent may start to attract too many followers, inevitably pushing up prices, lowering returns and, perhaps, encouraging investors to take on excessive risk.\n\n**A Tycoon's Tale**\n\nJoseph Lam is in the billionaire business. Born and raised in Hong Kong and educated first as a nuclear engineer at Texas A&M and then in business at the University of Chicago, the former Goldman Sachs private banker who decamped to Merrill Lynch in 2000 did so because \"Merrill was very strong in the mass-affluent category but not yet strong in wealth management.\" Offered an opportunity to build up Merrill Lynch's HNW practice in Hong Kong, Lam leapt (with his team of nine FAs, three senior partners, analysts, associates and support staff) to the burgeoning Pac-Rim operation, bringing with him approximately a billion dollars in assets under management (comprised of about twenty household and client relationships) and a longstanding focus, as he modestly puts it, \"on serving best-of-class, highly sophisticated investment products to tycoons.\"\n\n\"Tycoons have no interest in plain-vanilla products,\" Joseph Lam states, commenting that for big-league clients of his, including major Hong Kong property developer Dr. Lee Shau Kee, \"mutual funds are just not [their] bowl of soup. That's a commodity business,\" he states bluntly, \"because anyone can do that\u2014today, you don't even need an FA to do it.\"\n\nWhat Lam's billionaire clients\u2014he handles just a small handful, but that is enough\u2014want from him and his fellow members of Merrill Lynch's newly established Pac-Rim Private Investment Banking Group is to \"become partners with Merrill on a wide range of proprietary investments and private equity arrangements.\"\n\n\"When one of my clients partners with us, it becomes a much more significant and deeper relationship than that of a passive investor. If we go into business together on whatever it might be\u2014a shopping mall in China or a major proprietary deal\u2014and come out together both making money, that's a classic win-win for both sides.\"\n\nAs Lam describes it, his client Dr. Lee is chairman of so many things it's nearly impossible to keep track of them, but probably the largest and best known of his entities is the Hong Kong-based Henderson Land Development. When Joseph Lam first took on managing wealth in 1994 for Dr. Lee, he was worth\u2014give or take a few hundred million due to market fluctuations\u2014something like US$16 billion, with a fortune based almost exclusively on the value of his extensive properties in Hong Kong.\n\nBut then, the nearly eighty-year-old Dr. Lee\u2014\"who still doesn't have to wear glasses,\" Lam approvingly observes\u2014began reading up on Warren Buffett, and conceived of a late-life ambition to beat Mr. Buffett at his own game: value investing, except Dr. Lee was more interested in expanding his wealth in the comparative short term than in the long term.\n\nDr. Lee was at the time a classic candidate for asset allocation, as nearly all of his wealth was tied up in one asset category, which Dr. Lee frankly stated, might one day run out of steam for the simple reason that his traditional stomping ground, Hong Kong, was rapidly running out of developable land.\n\nWhile sharply stepping up real estate development activities on the mainland (sometimes in private partnership with Merrill Lynch), Dr. Lee took on almost as a hobby the pursuit of Warren Buffett as a world-class investor. As Dr. Lee later put it, according to Lam, \"he started out at a distinct advantage to Buffett, because the tax structure in the U.S. is much less favorable to wealth creation and preservation than it is in Hong Kong. So as Dr. Lee often told me, 'for every dollar Buffett makes he keeps fifty cents, whereas for every dollar I make, I get to keep eighty-five.'\"\n\nDr. Lee resolved to more than double his fortune in two and a half years, aiming for a compound annualized rate of return well in excess of 30 percent. By 2007\u2014not without the advice of Lam and his team\u2014Lee had handily met his goal. He achieved this astonishing investment success by perfecting taking a logical approach to investing that reflected his knowledge and personal style.\n\nHe began by looking at China, Japan, the United States, Taiwan, Singapore and all of the countries he knew intimately and employing a macro-economic, analytic, \"top-down\" approach to choosing the countries he wanted to be in. He made his selection based on an impartial and dispassionate analysis of those countries' projected economic growth in the future.\n\nHe decided to invest the bulk of his funds in mainland China. Having done so, he analyzed the various industrial and commercial sectors and made a similar analysis of which sectors were most likely to prosper over the coming decade of massive economic expansion. He decided that if he was going to invest in China, he needed to rely on the best local expertise available.\n\nHe then bought a basket of three stocks culled from his personal macro-economic analysis\u2014stocks that, as Joseph Lam puts it, \"he already likes very much...he likes the country, he likes the sector, he likes the industry, and most importantly, he likes the companies.\"\n\nHe then purchased a note, custom-tailored by Lam and his team, keyed to a value set at two years in the future. At the end of his first year, he and Lam checked out the performance of the three selected stocks, and pruned his portfolio by selling the worst performers if their value declined to an agreed-upon percentage of their price at the time of purchase.\n\nBy employing this refined variant of Buffett-style value investing, Dr. Lee met his goal and remains not just one of the world's wealthiest men, but one of a handful of the mega-wealthy.\n\n#  **DEVELOPING NEW PRODUCTS BY LISTENING TO INVESTORS**\n\nAccording to Tom Sweeney, a managing director in Merrill Lynch's Private Banking and Investment group, a significant source of inspiration is often HNWIs themselves. Some of Merrill's best alternative investment ideas come from HNWIs through what he calls \"reverse inquiry.\" Sweeney is seeing more and more reverse inquiry\u2014a process by which a client need is so specific that the products virtually design themselves\u2014around credit default swaps and foreign exchange, which lead them to customized solutions for HNWIs.\n\nCredit default swaps (CDSs)\u2014the most widely used credit derivatives\u2014are designed to transfer the credit exposure of fixed income products between parties. Consider an ultra-HNWI who owns US$10 million worth of a five-year bond issued by a high-risk corporation. In order to manage the risk of losing money if the company defaults on its debt, the ultra-HNWI enters into a CDS. Though the protection payments reduce investment returns, the risk of loss in a default scenario is eliminated.\n\nIn the case of foreign exchange (\"forex\") investments, Sweeney has been getting an increasing number of requests from ultra-HNWIs with substantial overseas property and other assets about designing customized investment products to protect them against declines in the U.S. dollar. \"There's an unending opportunity for creative solutions,\" Sweeney says.\n\nIdeally, these new ideas for ultra-HNWIs can eventually be applied to lower bands of wealth. Less wealthy investors always want the products and services being offered to those just above them, and advisors have a vested interest in perfecting these new products and services and finding ways to bring them downstream in a cost-effective manner.\n\nMichael Saadie of Australia's ANZ Private Bank notes that he's seen numerous instances of innovative financial products designed for the bank's ultra-HNW clients drifting down to the HNW band. A good example is a protected equity product originally designed to leverage a single stock exposure held by someone in the firm's top one hundred families. ANZ was able to take that structure and adapt it for the executive segment, whose members often have large equity positions in the companies for which they work. \"We can roll out specific strategies for each segment and build specific product opportunities for each.\"\n\n**Developing Bespoke Products**\n\nJohn Barrett and Nancy Bello of a Merrill Lynch team based in New York recount the story of a wealthy client family that approached them for a solution to a problem many of us would like to have. They were holders of a US$50 million \"long bond\" that paid a comparatively low rate of interest. \"It was a completely illiquid piece of paper,\" Barrett recalls, \"which carried interest but whose value at any particular point in time was not necessarily easy to determine.\" Working with Merrill's Equity Derivatives Desk, Barrett and Bello began designing a completely customized derivative product with an option structure that Bello describes as \"permitting the family to gain equity exposure without forgoing the security of the bond.\" The product combined a 60 percent link to the returns of the S&P, a 20 percent link to a basket of Asian equities, and a 20 percent link to European equities\u2014a truly global synthetic product.\n\n#  **GRABBING HEADLINES: HEDGE FUNDS AND PRIVATE EQUITY**\n\nThe two alternative investments that have grabbed the most headlines over the past decade have been 1) hedge funds and 2) private equity. Performance of these investments is listed in Figure 6.1.\n\n**Figure 6.1** \\- Greenwich-Van U.S. Hedge Fund Index vs. Thomson Financials' U.S. Private Equity Performance Index, 2003-2005\n\n##  **Hedge Funds**\n\nAccording to a report on the hedge fund industry recently issued by The Milken Institute, the global hedge fund industry has experienced continuous and rapid growth (with the exception of a fleeting decline in the number of funds experienced during the first six months of 2006). From 1981 to June 2006, funds grew in number worldwide at an average annual rate of 30 percent. In a sure sign that the capitalist phenomenon of creative destruction is alive and well in the volatile hedge fund universe, as of June 2006, 100 funds with US$257 billion in total assets had exited the industry. Yet 6,445 funds remained in operation, commanding US$969 billion in total assets.\n\nNet asset growth has expanded even more rapidly than the number of funds in absolute terms: hedge funds' average size has increased by more than 2,000 percent from 1981 to June 2006, from US$7 million to US$150 million. While funds with US$1 billion or more in assets account for only 3 percent of the total number of funds, they account for 35 percent of total assets. Conversely, those with US$100 million or less in total assets under management account for 70 percent of the number of funds but just 12 percent of total assets.\n\nOf the 6,445 funds as of June 2006 slightly more than 50 percent, or 3,354, are domiciled in the United States. Of these, about half have their assets in the United States, with nearly all the remaining assets invested globally. Europe is second to the United States in terms of the number of domiciled funds, accounting for approximately 33 percent, or 2,153, of the total number of funds; of these only about 20 percent have their assets invested in Europe. The majority of the funds, 64 percent, have invested their assets globally. However, 76 percent of the funds with 75 percent of total fund assets still have their investments in U.S. dollar-denominated assets, regardless of where the assets are located.\n\nIn the United States total hedge fund asset levels increased by 8 percent in the second quarter of 2007 to US$2.593 trillion, up from $2.401 trillion in the first quarter of 2007, according to the 2007 Hedge Fund Industry Asset Flow Report. New allocations to the hedge fund industry increased total asset levels by an estimated US$339 billion through the first three quarters of 2007, but the reduction in assets from liquidations outpaced the increase from new fund launches in the second and third quarters.\n\nConcern likewise abounded by year-end 2007 regarding the trendiness of certain hedge funds, a fashion only exacerbated by a tendency on the part of some managers to open funds simply to focus on a hot sector and sop up investment dollars. For instance, total assets in emerging market hedge funds reached an estimated US$299 billion in Q3 2007, an increase of US$48 billion through new allocations in the first nine months of the year. Emerging market equity focused fund assets are estimated at US$106.1 billion, widening the gap over EM debt fund assets estimated at US$77.3 billion.\n\nAmong other hot trends, China-focused hedge funds were the fastest growing sector of the hedge fund world in 2007, with total assets invested in hedge funds investing primarily in China reaching an estimated US$22.8 billion, up 89 percent over the previous year. Continuing on the hot theme, hedge funds investing in commodities and other natural resources grew to US$170 billion in assets under management in 2007, led by energy focused funds with US$133.8 billion. Energy sector hedge funds investing in equities returned 15.3 percent through September, but those investing directly in physical commodities did best during the volatile third quarter.\n\nA key attraction of investing in distressed hedge funds for investors and advisors pursuing asset allocation and portfolio balancing strategies is that some\u2014yet not all\u2014hedge funds have remained historically uncorrelated to broad equity markets. As a result of this attractive non-correlation, distressed funds experienced double digit growth rates in 2007, attracting nearly US$10 billion in new allocations, bringing their grand total to just under US$200 billion.\n\nJust as the hedge fund environment has changed, so too has the profile of the typical hedge fund investor: HNWI's have been largely supplanted by an influx of institutional investors. Some estimates suggest that today as much as 60 percent of hedge fund assets are institutional pension funds, endowments and other investment pools\u2014which may over time alter hedge fund strategy, as these institutions tend to be more risk-averse and demand greater transparency.\n\nAt the same time, certain hedge fund strategies have become available even to registered investment companies. For instance, several dozen mutual funds now mimic certain hedge fund strategies although they remain subject to limitations not applicable to private funds. Essentially, these \"long\/short\" mutual funds can bet either that a stock will rise or fall, while traditional funds only invest if managers believe a stock will rise.\n\nSuch funds are, however, limited in the amount of leverage they are permitted to employ and the degree to which they can be net short. In the first half of 2007, the broadest of all hedge fund strategies, long\/short equity funds, attracted an estimated US$15.3 billion in new allocations. However, the majority of the total asset level rise came from fund performance, which added an additional US$39.3 billion, the second largest increase on record.\n\n###  _**Funds of Funds**_\n\nA popular hedge fund vehicle tailored to less wealthy investors is the \"fund of funds,\" which invests directly in traditional hedge funds rather than investing in individual securities. Total fund of fund assets, which are not double counted in the total industry figure, increased 9.4 percent to an estimated US$1.250 trillion, including US$74.5 billion in new assets, the largest quarterly new allocation on record.\n\nSuch entities held about US$500 billion at the end of September 2007, up 27 percent from a year earlier, according to Chicago-based Hedge Fund Research. Some investors who would be unable to invest in a hedge fund directly may be able to purchase shares of registered funds of hedge funds. In 2007, the hedge fund industry attracted a record US$194.5 billion in new capital, to US$1.87 trillion, according to HFR. Funds of funds saw net new inflows of US$59.2 billion for the year sending total assets invested in the category globally to US$498 billion.\n\n##  **Private Equity**\n\nAlthough private equity (PE) is best known in its modern form pioneered by Kohlberg Kravis Roberts (KKR) in the early eighties, private equity's origins date back to 1901, after J.P. Morgan purchased the Carnegie Steel Company from Andrew Carnegie and Henry Phipps for US$480 million. With his share of the proceeds Phipps established the Bessemer Trust\u2014the first \"family office\" in the U.S. A century later Bessemer continues to be a player in the private equity industry under Phipps' great grandson Stuart Janney. Throughout much of its history, PE has largely been available only to institutional clients. PE firms\u2014also known as \"financial sponsors\"\u2014typically raise large pools of capital that they deploy to gain control of firms that they believe to be undervalued. PE investors typically control the firms they take over in hopes of significantly boosting their value\u2014whether by installing superior management or aggressively cutting costs, including by performing controversial lay-offs and downsizing.\n\nFinancial sponsors do assume significant risks when they engineer such takeovers, but much of that risk is typically borne by the investors willing to place funds in their capital pools. With such significant sums involved, the game has historically been a preserve of large institutional clients. Yet many ultra-HNW individuals, eager to share in the often outsized profits of PE, are willing and eager participants in PE deals\u2014obviously willing to shoulder the not-insignificant risk in the hope of gaining a not-insignificant reward.\n\nAccording to Private Equity Intelligence, a London-based company that does research on the industry, private equity funds raised a record US$406 billion in 2006. More than 170 funds each holding US$1 billion or more in assets conducted US$475 billion in deals in 2006. While U.S. companies are leading the industry's continued expansion, Europe-based PE funds raised some \u00a390 billion in 2006, up 25 percent from the prior year. In yet another globalization trend, private equity firms from both continents are presently partnering to fund large trans-Atlantic deals.\n\nThe year 2007 also saw a number of prominent private equity firms moving to raise capital\u2014somewhat ironically\u2014in the public markets, beginning with the trend-setting US$4 billion IPO of Blackstone Group, followed by Carlyle Group's decision to sell a 7.5 percent share of itself to an investment group owned by the government of Abu Dhabi.\n\nIn 2006, PE fundraising reached new record levels, with data from Private Equity Intelligence showing that a total of 684 funds worldwide closed on US$432 billion in commitments. In the United States alone, PE firms raised US$215.4 billion, a record amount. This growth is best explained by the superior returns that these funds delivered in 2005. Indeed, the U.S. Private Equity Performance Index returned 22.6 percent in 2005, compared to 16.4 percent in 2004.\n\nWhile in 2006, leveraged buyout firms, a subset of PE focused on buying mature companies, raised about US$150 billion, half of that raised by only eight firms, including industry leaders Bain Capital, First Reserve Corp. and KKR. Yet, shortly thereafter, Blackstone Group announced that it was raising a US$20 billion fund, making it the largest PE fund ever. Blackstone won a bidding war to purchase Equity Office Properties Trust in a leveraged buyout for a record US$39 billion\u2014a record soon topped by the proposed buyout of Dallas-based power producer TXU Corp. for US$45 billion. Yet by the second half of 2007, liquidity dried up and the psychology of the PE market abruptly shifted into reverse gear.\n\nIn fact, according to a late 2007 survey of more than 700 industry practitioners by _Financial News,_ 55 \"Private-equity firms expect markedly lower returns in 2008, with large buyout funds likely to suffer most,\" with three quarters of respondents reporting that they expected returns from large buyouts to be \"somewhat lower\" or \"significantly lower\" than last year, and with nearly a third of respondents expressing more pessimistic outlooks on their industry than at virtually any time during the first decade of the twenty-first century.\n\nBecause of the high investment minimums in the $5 million dollar range, PE, like hedge funds, has historically been available only to HNWIs. Unlike hedge funds, which may be subject to annual or quarterly redemptions, PE investors have typically been obliged to leave their money tied up for years in PE deals, making these extremely illiquid investments. BlackRock co-founder Ralph Schlosstein describes this illiquidity as the magic key to higher returns: \"Giving up liquidity has been, at least up until recently, the most intelligent way to get good returns.\"\n\nSchlosstein is convinced that PE's vaunted advantage is sustainable given the long-term nature of the investment and the growing costs of being a public company (thanks to legal fees and regulatory demands tied to Sarbanes-Oxley). According to Thomson Financial, the average annual returns on PE have topped 13 percent over the past two decades. Therefore, this vehicle should appeal to those with a long-term view and who are willing to trade liquidity for potential long-term gains.\n\nYet another key difference between PE and hedge funds is that hedge funds started with ultra-HNWIs and moved up to institutions and downstream to HNWIs. PE started with institutions and migrated down to ultra-HNWIs and now to HNWIs, with minimum investments in PE funds drifting from the millions down to US$250,000.\n\nHow far downstream PE may go remains to be seen. But yet another way that the mass affluent may be able to cash in on the PE wave is by purchasing shares in publicly owned PE companies.\n\nIn early 2007, Fortress Investment Group, which managed US$30 billion at the time, became the first PE and hedge fund manager to sell shares on U.S. markets. Indeed, while one of the hottest IPOs in years (closing the day up 68 percent from its initial price and making all five partners billionaires), by January 2008 Fortress shares dropped to record lows while the credit crunch continued to bite. By April 2007, Blackstone had also gone public like Fortress, but by January 2008, as private equity firms experienced unforseen challenges in obtaining financing and closing deals, Blackstone shares sank below US$20.\n\nDespite such obstacles, Wall Street continues to flirt with methods of opening up the PE market to the less well heeled. PowerShares Listed Private Equity Portfolio, an exchange-traded fund launched in the fall of 2006, buys companies whose main business is investing in or lending money to private companies. There are also publicly traded \"special purpose acquisition companies,\" or SPACs, that raise money specifically to purchase private companies. In 2006, SPAC IPOs raised US$2.6 billion, up from IPOs worth US$2 billion in 2005, according to Dealogic.\n\nAnother PE vehicle that might be of particular interest to HNWIs is a PE fund of funds, a format offered by several big Wall Street outfits including Lehman Brothers and Morgan Stanley. These funds pool money from HNWIs to invest in private equity funds. At about US$100,000, the investment minimum is lower than a typical single manager fund. Like investors in single PE manager funds, fund of funds investors should be prepared to hold onto their investment for years. That's because the PE investment life cycle\u2014buy an ailing or undervalued company, turn it around, sell it\u2014is rarely a rapid recipe.\n\nOne of the largest downsides of PE's outsized success is that the PE business has begun to draw the attention of regulators in the United States and Europe. A similar phenomenon occurred with hedge funds, much to the chagrin of the industry. In the U.S., the SEC at first tried to impose some rudimentary rules on hedge funds, such as requiring certain hedge funds with a specified number of investors to register with the Federal Securities Agency, a rule that was ultimately overturned by the courts. The SEC also voted to tighten anti-fraud rules regarding hedge funds.\n\n#  **FACING THE CHALLENGES AHEAD**\n\nAlternative assets provide a myriad of challenges and opportunities for HNWIs and their advisors. For HNWIs, certain alternative assets may provide a way to gain returns superior to the traditional stock and bond markets or to reduce a portfolio's overall volatility. They may also offer risk-distribution and mitigation characteristics, as many of these investments have historically demonstrated various levels of non-correlation both with other and with more conventional investments. This diversification has gained added importance during a period when stocks and bonds are more likely to rise and fall in unison.\n\nBut real risks remain. In a ceaseless search for the next new innovative product and higher returns, HNWIs must be prudent and vigilant against falling prey to the latest investment fad. They must bear in mind that past performance does not guarantee future returns and, as Mr. Fink of BlackRock warns, should be just as suspicious of unexpectedly high returns as they are dissatisfied with poor results. They must be willing but discriminating about tying up capital in an illiquid investment for several years, and never take management talent for granted. As sectors like hedge funds and private equity heat up and capital becomes plentiful, they should expect some marginally talented managers to jump into the fray. HNWIs, although tempted, may need to take a cautionary approach. They shouldn't jump in; rather, they should ensure that their risk appetite and time horizon can accommodate the anticipated returns.\n\nTo provide best-in-class advice to their clients about this complex and competitive investment category, wealth managers must be well-trained and highly knowledgeable as to the benefits and risks inherent in alternative investments and how to clearly explain their characteristics to clients; they need to offer a wide range of well-conceived alternative investment products, working seamlessly with their asset management and investment banking divisions and\u2014if working in an open architecture or multimanager platform\u2014with outside investment managers. Incentives must be well conceived and fully disclosed; and, finally, providers must improve transparency with HNWIs. As previously mentioned, consolidated reporting will become increasingly critical as HNWIs invest in a wider range of complex instruments manufactured by different parties. Responsiveness, communication and coordination are essential for alternative investments and form the basis of a good checklist for HNWIs to work from.\n**CHAPTER 7**\n\n**Family Matters**\n\n#  **FOSTERING FINANCIAL LITERACY, HNW STYLE**\n\n\"At midnight,\" Johnson & Johnson heir Jamie Johnson observes in a bittersweet voice-over narration that kicks off his 2003 documentary _Born Rich_ , \"I'm going to inherit more money than most people can earn or spend in a lifetime. I've been waiting for this night for as long as I can remember. But the thing is, now that it's here, I'm not really sure what to make of it all.\"\n\nFor better or worse, he's not alone.\n\nAnd almost surely for the better, help is on the way.\n\nFor parents and children alike, the challenges of preserving family wealth and passing it on to the next generation can be daunting, baffling and occasionally demoralizing. Self-made millionaires and billionaires worry (not without reason) that children born into great wealth will either personally lack, or fail to appreciate, the spunky entrepreneurial spirit that propelled their parents to their current position of status. Since their own self-esteem is often intimately connected with the cultivation of such a can-do spirit, it can be doubly difficult for them to empathize with their children's very different struggles with the issue.\n\nOf course, if the offspring of every self-made entrepreneur were as entrepreneurial as the first in the line, multigenerational family wealth preservation would be a pretty straightforward affair. A case in point is that of the self-made millionaire Cornelius Vanderbilt, a one-time Staten Island ferry boat captain who upon his death in 1877 was worth more than US$100 million, making him the richest man of his generation in America. His son William, evidently something of a chip off the old block, doubled that fortune by the time of his own death in 1885. Yet less than a century later, when a family reunion at Vanderbilt University attracted more than one hundred and twenty descendents, not a single millionaire could be found among them. Obviously, somewhere after William the entrepreneurial spirit died out and was rarely, if ever, revived.\n\nThough to be fair to the Vanderbilt family, a few wealthy Vanderbilts failed to attend that reunion, their saga of familial financial diminution nonetheless remains a dramatic example of how rapidly even the vastest family fortune can dissipate in a century, or over the span of four short generations. Of course, since family fortunes are typically divided among children, grandchildren and so on down the line, only the active presence of a major-league wealth creator among the immediate descendents is likely to keep the family fortune intact.\n\nYet math and time are only two reasons that family fortunes are so fragile; interpersonal dynamics also play a critical role. Apart from all the obvious causes\u2014heirs feeling a sense of entitlement, or lacking the skills necessary to preserve the wealth they inherited or to create wealth of their own\u2014the WM industry has traditionally offered a menu of solutions to these perennial dilemmas. One that appears to be rapidly falling out of favor is nevertheless adopted by many HNWIs by default, who with the very best of intentions attempt to shield their children as long as possible\u2014often fruitlessly\u2014from knowledge of the full extent of the family's wealth.\n\nSuch a strategy\u2014akin to trying to keep Adam from taking a bite of the apple\u2014can give rise to awkward scenes like one recounted by Jamie Johnson in _Born Rich_ , in which he recalls a friend finding his father's name on the _Forbes_ list of the wealthiest Americans and reading the citation aloud to his entire fifth grade class, much to his chagrin and embarrassment. Despite the fact that the information was obviously in the public domain, the ten-year-old Johnson heir experienced an odd sense of having committed a personal violation, as if he'd stumbled upon some awful skeleton in the family closet: \"I felt I was learning a secret that I wasn't supposed to know.\"\n\nYet to wealthy parents hoping to raise \"normal\" kids, such discretion may seem like the better part of valor, as full disclosure might compromise their offspring's budding sense of self-worth or undermine any latent desire they might harbor to make it on their own in their own chosen field. On the other hand, trying to teach kids \"the value of a buck\" by avoiding facts often in plain sight may backfire if it leaves them woefully unprepared for the moment when they do in fact come into their handsome inheritance.\n\nStories abound of adults in their thirties and forties abruptly inheriting several million dollars and\u2014after crying \"Hallelujah\"\u2014quitting their jobs, making poor investments, or otherwise squandering the money in short order. It is widely assumed within the WM industry that such behaviors cannot necessarily be ascribed to lack of intelligence or character, but rather to the fact that the kids were ill-prepared, both by parents and society, to take on the responsibility of managing their own wealth for the future as opposed to the present.\n\n\"The real challenge,\" contends James Rothenberg, chairman and principal executive officer of Capital Research and Management, \"is to try to educate children without their getting so focused on the amount of money that it obliterates their incentives. You need to talk to them and say, 'You must make your own way in the world and set this money aside.' You have lots of conversations like that.\" \"If wealth is going to make them less passionate, that's a problem,\" insists BlackRock founder Larry Fink. \"Personally, I don't care what they do, as long as they do it with passion.\" But the myriad difficulties surrounding open discussions of wealth between parents and children are often compounded by the fact that since every child is inherently different, it can be just about impossible for parents to adopt a unified approach to education, inheritance and other legacy issues. \"It's not something you can read about in a book,\" Mr. Fink adds. \"Each child interprets what I say differently. Each has different concerns and each has different probable financial outcomes based on the evolution of their careers.\"\n\nThe methods and techniques a family adopts to advance their own wealth education as well as that of their children may vary dramatically depending upon circumstances. The greatest variance is perhaps produced by differences in the way the original family fortune was made. One HNWI, who for personal reasons prefers to remain anonymous, sympathetically observes that both of his immigrant parents worked hard all their lives and that his bartender father never made more than US$10,000 in any given year of his life, yet he contends that that apparent disadvantage is actually what encouraged his entrepreneurial spirit. \"Each family's way of dealing with money is colored by the upbringing and childhood of the parents,\" he observes.\n\nThis particular HNWI remains anxious about the issue of motivating his children to be wealth creators as opposed to wealth dissipaters. He openly (if anonymously) agonizes over the possibility that if his children lack the motivational spur of poverty that he had as a child, they will never know the satisfaction of self-made success. In short, while many a HNWI wrestles with the possibility that their offspring will be irresponsible and squander their wealth, many more HNW parents display congruent angst about their kids squandering their lives and failing to live up to their full potential.\n\nPatricia Angus, the managing director and head of Wealth Advisory Services at Shelterwood Financial Services (a multi-family office advisory firm providing investment management, family office, and business consulting services to high-net-worth families) observes that \"even as recently as five or ten years ago, the common attitude was 'he's got to make a living.' But now many parents are starting to realize that maybe they don't; that the second generation's experience is just completely different.\"\n\nYet another post-modern twist has been added to the centuries-old dilemma surrounding family financial privacy. Heightened transparency and greater disclosure of financial information required of companies with regard to compensation for senior executives has combined with a circus-like atmosphere of media attention paid to the lifestyles of the rich and famous, all of which has in some cases fatally compromised many a wealthy family's determined efforts to keep their financial lives out of the public domain. For example, keeping salaries and the value of stock options secret from today's tech savvy kids has become increasingly difficult\u2014if not impossible. As T. Rowe Price CEO James Kennedy observes with disarming frankness, \"All they've got to do is Google me.\"\n\nYet the psychological upside of the financial information revolution is that it has forced a new level of frankness, openness and honesty about the complex issues surrounding wealth and its disconnects between generations. This occasionally means that, like it or not, some parents feel increasingly unfettered about being the bearers of (possibly) bad news. Some parents want children to understand in no uncertain terms that just because they are worth US$500 million, that doesn't mean the children are going to get it all\u2014so they shouldn't plan on it. Warren Buffett famously summed up the view of many HNWIs when he said that he wanted to leave his children \"enough to do anything, but not enough to do nothing.\"\n\nAlso, by loudly broadcasting the dysfunction found within some wealthy families, today's media may be doing at least a few wealthy families a real service by forcing some of the less salubrious aspects of fortune and fame out into the open, confirming Chief Justice Louis Brandeis's celebrated admonition that \"sunlight makes the best disinfectant.\" When the family feud between the son and grandson of New York society matriarch Brooke Astor broke out into the open in 2006, the whole sorry saga at least served as a cautionary tale regarding the potential divisiveness of money a bit more up-to-date than John Steinbeck's _The Pearl_. \"So much more is being written than ten years ago, so there's an enormous amount of interest in working to prevent these problems and sustaining a healthy, productive family,\" notes Dr. Lee Hausner, a partner in IFF Advisors **,** a consulting firm created to enhance the human, intellectual, social and financial capitals of individual, family, foundation and non-profit clients.\n\nDoug Freeman, a prominent California-based tax attorney and Dr. Hausner's partner in IFF Advisors, advises the participants at the Merrill Lynch Financial Boot Camp at Wharton\u2014and indirectly, all HNWI offspring. \"Some people take pride in not knowing how much they're worth, believing they have more important things to think about than money. Others like to think that like celebrities and highly paid athletes, they can afford to hire business managers to keep track of all that stuff. But you can't plan where you want to go in your financial life if you don't know where you are. And you can't get there without a strategy. You're like a ship without a rudder.\"\n\nFreeman drills his charges on the fundamentals: understanding how net worth\u2014\"the basic scorecard for financial condition\"\u2014is calculated, and \"how by tracking it over time, you can measure the results of your financial decisions.\" He explains how the wealthy calculate and track their net worth, how to develop a cash flow statement, how to create a spending plan, and, in general, how to take control of their financial life. Maintaining a sense of control over these precious assets, Freeman insists, is not merely critical to preserving the wealth itself, but also their marriages, and above all, their sanity. Possessing great wealth in just about any form\u2014but the inherited version perhaps most of all\u2014can be inherently psychologically destabilizing.\n\nApart from information age wealth effects, there have been significant strides over the past quarter century around the science of behavioral finance. It is far more widely assumed today than in the past that many otherwise intelligent and rational people do not necessarily make rational decisions about and around money. In fact, deep-seated issues can lead people to make poor decisions over and over again. Danny Dunn, a Private Wealth Advisor with Merrill Lynch who is also a trained clinical psychologist, observes that \"some investors are addicted to market timing and tend to chase returns to their detriment. I've had conversations with people whom I've told, 'Three times in a row you've missed a market turnaround,' which forces me to have to re-educate them once again about the dubious returns typically generated when trying to time the market.\"\n\nDunn also points out that on the less damaging end of the scale, a wealthy individual might become overly emotionally attached to a particular stock because their parents gave it to them, or because it was received from the sale of the family business. They might insist on not selling it even though for diversification purposes it's the correct course of action.\n\nOn the more damaging end of the scale is what Dunn describes as a \"gap of thought\"\u2014or willful blindness to the emotionality of their decisions. The good news, however, is that as the concept of behavioral financial science has become better understood and accepted, the WM industry has reacted by adopting more subtle and nuanced positions regarding the emotional content of investment decisions.\n\nNot surprisingly, Merrill Lynch, Citigroup, and JPMorgan Chase all provide versions of financial training for the children of HNWIs. Merrill Lynch's Financial Boot Camps provide lessons in portfolio management, asset allocation, transferring wealth to the next generation, estate planning and philanthropy management. Jeremy Arnold, head of Wealth Advisory at Barclays Wealth Management, advises that the best way to \"work out your priorities...may be to sit down and write a family constitution. That requires a disciplined approach and might take six to nine months to do.\" At the family office level, special consultants may be hired to lead family meetings and educational seminars. In all cases, the range of education is usually broad, encompassing investing basics as well as information on trusts, philanthropy and estate planning.\n\nFranklin Templeton CEO Greg Johnson notes that marrying trusts, values and education can be a significant challenge. For him, the primary goal underlying his decision to set up trusts for his children was to find ways to encourage positive behavior over the long term. Tax considerations are often a lesser issue than parenting concerns. \"How do you pass along values through the trust? That's where the wealth advisors can add huge value.\"\n\n#  **PARTAKING IN VALUES RETREATS**\n\nOverwhelmingly, the greatest concern we hear from parents is \"How can we prepare our children for wealth, and what impact will the inheritance have on my child?\" says Stacy Allred, a vice president in Merrill Lynch's PBIG. Among the questions she typically fields from wealthy clients:\n\n\u2022 How should I balance my assets between retaining what I need and sharing with my family and community?\n\n\u2022 How can we make sure our children are informed about the family wealth, yet not give away too many details?\n\n\u2022 How can we communicate with multiple generations while ensuring that each age group finds out just what it needs to know\u2014and on their individual wavelengths?\n\n\u2022 How can we set up a transfer of wealth so that future generations can benefit but also maintain the values so important to us?\n\nAccording to Karen Klein, a director in Merrill Lynch's Client Relationship Group, the first step toward achieving clarity around these legacy issues is to focus on creating a values-based multigeneration financial plan. PBIG assists clients through this process by inviting them to six-hour-long \"values retreats\" designed to help them get to the heart of the values that parents hope to pass on to their children, and possibly even to the generations to come.\n\nValues retreats begin with a questionnaire that focuses on the emotional challenges and opportunities of possessing wealth. \"It's like peeling back the layers of an onion to get to the core values of the family and then building a framework where they can use their shared values around earning, spending, saving and sharing of money to make financial decisions,\" observes Ms. Allred. At the end of the retreat, families are able to clarify their values and start the process of clearly communicating them to their children.\n\nSuch careful planning can help parents solve the often thorny riddle of when to begin transferring assets to heirs\u2014and how much to give. If having motivated and productive children is a key goal, the family may collectively decide that delaying the transfer can be a powerful motivating tool. \"When significant amounts are transferred during career-building years,\" comments Dr. Lee Hausner, \"career building often stops. If adults in their twenties can make as much interest on the distribution from trusts as they can earn in their first job, there may be a temptation to choose not to become fully dedicated to building a career and to 'play' instead.\"\n\nSome families are even turning to that tried-and-true staple of corporate life, the mission statement, for help in clarifying their financial-moral legacy goals. Charles Collier, Senior Philanthropic Adviser at Harvard University, observes that \"the best result of this process is a sense of togetherness around a shared dream.\"\n\nAs an example of one family's mission statement, Collier offers the following partial citation:\n\n_As members of this family we honor and respect our family's civic, social and corporate heritage by supporting our communities each in our own way, guided by principles of generosity, honesty and hard work to improve our lives and the lives of others. We shall act with care and integrity to inspire trust within our family and within our communities._ 64\n\n#  **ENABLING THE NEXT GENERATION OF WEALTHY THROUGH PEER NETWORKING**\n\nPeer networking may also provide useful knowledge. Simply talking with people in similar circumstances and sharing ideas can be invaluable. Some peer networks, such as TIGER 21, are focused on sharing investment insights, while others have a broader mission and are more geared to bringing HNW families together to discuss a variety of financial and non-financial topics. CCC Alliance is a group of successful families and individuals that collaborate regularly on wealth management and family office matters, drawing on a deep network to invite leading experts to speak on wealth management topics, including investments, philanthropy and family governance. Private investor George Russell's Threshold Group asks applicants to fill out detailed questionnaires on topics including family values. New York-based HNW peer network Synergos focuses on philanthropy. The Chicago-based Family Office Exchange runs a network of 350 families across twenty-two countries who meet annually. According to _Newsweek_ , \"the group's most popular service is its Listserv, where families query each other on things like estate planning and buying private jets.\"\n\nNot surprisingly, members of the younger generation are often the more likely to join these peer groups, as HNW children often feel specific frustrations around their or their parents' wealth but may feel reluctant to express them among their non-wealthy peers, who might feel more resentful than sympathetic. As one advisor dryly comments in somewhat snide reference to the rising popularity of these peer networks. \"The offspring can feel free to whine because everyone else at the table is in the same situation.\"\n\nAlbert Lee, Merrill Lynch's CEO for Taiwan and a veteran advisor to the HNW crowd of Asia, observes that he has seen peer networks spring up of their own accord from the educational seminars that Merrill Lynch has held for Asian HNWIs in Hong Kong. \"These can serve as excellent networks of business contacts for young HNWIs more interested in creating their own wealth than simply living off their parents' accumulated assets.\"\n\nUltimately, every wealthy family must use education to develop its own intellectual wealth capital and partner it with financial capital. It has become more widely recognized that the WM industry needs to view wealth education in a more holistic way, and that its primary purpose is to enhance the well-being of the family members at every level\u2014beyond just helping everyone to benefit from better or more stable financial returns.\n\nThe lesson for HNWIs is that educating their children around wealth is essential\u2014and the result is more likely to be a family fortune that could endure for decades and perhaps even centuries. Ultimately, fears that knowledge of wealth will de-motivate children must be balanced with the need to prepare them for that wealth. What's more, hiding the truth of a family's circumstance rarely works for the long term, and can be counterproductive if children grow frustrated or resentful in the belief they are being kept in the dark.\n\n#  **PREPARING FOR GREAT WEALTH TRANSFER**\n\nIn the coming decades, baby boomers in the United States are expected to both receive and bequeath huge sums of money, a transfer of assets that by 2053 is conservatively estimated to be US$41 trillion in the United States alone. In some regions of the world, shrinking populations will concentrate wealth. In China, it will not be uncommon for a set of four grandparents to share just one grandchild.\n\nMake no mistake\u2014the bulk of this wealth transfer is likely to go to family members. Even with the recent headlines surrounding the rise of philanthropy and the high-profile examples of Bill Gates and Warren Buffett, the tendency on the part of parents to leave their children the greater part of their assets is so well established in most cultures, if not in human nature at large, that prevailing patterns are unlikely to change dramatically within the foreseeable future. The 2006 _WWR_ found that 84 percent of HNWIs are married, 83 percent of HNWIs have children, and that more than three-quarters of HNWIs' wealth is anticipated to go directly to their children.\n\nWealth transfer and estate planning, in other words, will be in high demand for the foreseeable future and there is much work for wealth managers to do. According to a survey of relationship managers for the 2007 _WWR_ , 56 percent of HNWIs are considered inadequately prepared, while only 2 percent of HNWIs are believed to be completely prepared. Therefore, as individuals are planning their future, putting FAs to work on these topics now will begin to address future issues.\n\n#  **CREATING A WEALTH TRANSFER PLAN**\n\nAccording to IFF Financial Advisors, a number of key questions need to be answered by every wealthy family preparing for generational wealth transfer:\n\n1. What are my goals for family members, and what are the resources available or needed?\n\n2. What will be the impact of the planning I do?\n\n3. How should funds be distributed while my child is still a minor or in school?\n\n4. How should funds be distributed once my child is an adult and completes (or quits) school at a given age?\n\n5. When should I start distributing principal outright to the child?\n\n6. Who should be responsible for managing the trust assets of my children? (Parents? Siblings? Business associate? Friend? Bank? Financial advisor?)\n\n7. Should I permit my child to become a trustee of his or her own trust?\n\n8. Should I distribute my estate equally amongst my children?\n\n9. How do I insulate my estate from risk\u2014legal, financial, death, illness, incapacity, disasters and accidents?\n\n10. Do I create a wealth management or distribution plan (either through a will or trust)?\n\nPreparing to seize this opportunity, FAs must recognize that they are usually not the top choice among HNWIs for wealth transfer guidance. Trust and estate attorneys, tax attorneys and accountants have traditionally been the most sought-after wealth transfer specialists.\n\nMerrill Lynch has developed dedicated awareness workshops on the subject of wealth transfer that have been implemented in all major regions\u2014Latin America, North America, Europe and Asia-Pacific. Such seminars typically serve a dual purpose: 1) to build trust between HNWIs and FAs on the issues of wealth transfer and 2) to help advisors build rapport with the next generation, a critical step to retaining the family relationship after the transfer occurs.\n\nRetention of inheritor accounts requires pulling off the hat trick of precisely balancing investment needs and motivations of both the current HNWI and the benefactor\u2014not to mention the financial realities of the bank, which cannot afford to extend golden glove service to every heir. Michael Saadie's ANZ Bank segments clients into five categories:\n\n\u2022 The top one hundred families;\n\n\u2022 Business owners;\n\n\u2022 Large investors\/retirees\n\n\u2022 Executives of the top one hundred listed companies;\n\n\u2022 Professionals (partners of legal and accounting firms and medical).\n\nIn each case, the children of those clients are handled differently. If a child is a member of the firm's top one hundred families, all financial issues are handled by the private bank, no questions asked. In the case of an executive's son graduating from college, on the other hand, private bankers will arrange an introduction for the son with other financial professionals at ANZ outside the private bank.\n\nTo retain inheritors' business, advisors must prove to young HNWIs that they are not wedded to an \"old school\" investment philosophy and WM strategy. Financial advisors must realize that many younger HNWIs will approach investing differently and therefore have WM needs that may be dramatically different from those of their parents.\n\nFor example, the next generation of HNWIs will have a very international outlook, far more so than the previous generation. Advances in communication and international travel have become easier, spurring this trend. Family members, even those involved in the same business, live and work further away from one another, perhaps with residences in multiple jurisdictions. There's been a sort of HNWI diaspora that requires HNWIs and their advisors to manage wealth not just in relation to where they own their primary residence, but everywhere they do business, own homes, have family, or travel.\n\nThe next generation is already more technologically savvy and plugged in than their parents. This contributes to a more sophisticated outlook but also a greater degree of impatience with the level of service they are currently accorded by their advisors.\n\nThe next generation is less likely to be bound by tradition or custom to their parents' financial firms\u2014whether a traditional private bank, a boutique, or a private-banking boutique incorporated within a firm with global reach.\n\nEdward Bernard, a vice chairman at T. Rowe Price who joined the firm in 1988 and has been the director of the investment services division since 2006, asserts that the children of today's HNWIs are \"growing up in a global context.\" He predicts that in the United States, more young professionals will begin to spend some of their career overseas, as has been the case for years among the British and Australians.\n\nThe Internet also de-emphasizes national boundaries, Mr. Bernard says, or at least makes them seem much less relevant when it comes to information gathering and investing. \"A URL is a URL no matter what country the company is in. Over time, investment behavior will reflect this. If, in your daily life, you think globally, then your investment behavior will be global.\"\n\n#  **RECOGNIZING REGIONAL DIVERSITY**\n\nThe 2006 _WWR_ found that 83 percent of wealth managers said inheritors will want increased exposure to international investments, and 76 percent said that inheritors will require the ability to be served in multiple geographies\u2014a big potential advantage for global institutions. Underscoring this international trend is that 19 percent of HNWIs have children living abroad, making the international transfer and management of assets a certainty for a substantial number of future HNWIs.\n\nThese younger, more risk tolerant, globe-trotting HNWIs will force advisors and the institutions for which they work to have a global, integrated view, capable of fluid communication and coordination. The ability to tap several markets with a consolidated service model, rather than with multiple relationship managers, will be vital. It also seems likely these younger HNWIs will be more engaged in the investment process, pushing advisors to expand and innovate around international product offerings.\n\nVictor Tan, market director of wealth management for Merrill Lynch in Hong Kong, notes that in Asia he's observed a growing trend: scions of wealthy businesspeople taking positions outside the family firm to gain a broader experience and perspective before returning home to roost. This career path gives the children a chance to prove themselves outside the safe haven of the family business and can boost confidence and decision-making skills. The net effect is that the children of wealth entrepreneurs are, in general, no longer having a difficult time assuming control of the family business; they have become more sophisticated and prepared, and these challenges may become opportunities, altering their investment approach to adopt a more fundamentally global perspective.\n\nDarcie Burk, the Miami-based head of wealth management for Latin America at Merrill Lynch, says that wealth managers are closely observing the changes occurring in the southern hemisphere and are getting a hint of what's to come. The generational transfer is already under way there, with younger HNWIs, many of whom have received MBAs in North America and Europe and have returned with a far more aggressive approach to investing than their parents, starting to actively manage the family wealth. They're more comfortable with equities\u2014an asset class shunned by their parents\u2014and are generally more sophisticated and open-minded when it comes to innovative products.\n\nTen years ago, Ms. Burk notes, most Latin American HNWIs focused on preserving capital and were willing to accept lower returns in exchange for that guarantee of principal. That's changing fast as the next generation seeks to tap into the equity markets and structured products that contain some risk to principal but offer much more potential upside. Wealth _accumulation_ , as opposed to _preservation_ , has become their explicit objective.\n\n#  **APPRECIATING GENDER DIFFERENCES**\n\nTo a significant degree, the gender differential trends shaping WM today reflect the specific priorities of wealthy women, from holistic management to an emphasis on education. \"Education is absolutely critical,\" says Alyssa Moeder, a private wealth advisor at Merrill Lynch. \"Women are willing to invest more time up front to understand what the financial advisor is doing, and once they get that comfort level they're usually willing to follow the advice pretty closely.\"\n\nThe profound influence of women on WM as a whole should not come as a huge surprise. According to TrendSight Group, by 2010, U.S. women will control 60 percent of the nation's wealth, or a whopping US$22 trillion. Meanwhile, women are starting new businesses at twice the rate of men (10 million business owners in the United States today are women), and 90 percent of women will have sole charge of their assets sometime in their lifetime. Little wonder their priorities are becoming the priorities of the industry.\n\nWhile the percentages of wealthy women are highest in North America and Europe, their numbers are growing in Asia. According to the _2007 Asia-Pacific Wealth Report_ , women account for 43 percent of HNWIs in Taiwan, 38 percent in China, 34 percent in Hong Kong, 27 percent in Japan, 22 percent in Singapore, 22 percent in Indonesia, 19 percent in South Korea, 15 percent in Australia and 12 percent in India.\n\nMs. Moeder notes that women are strong drivers behind the HWM trend. They tend to view wealth as a mechanism to achieve broader objectives and want their advisors to adopt a broad view of their entire personal, family and household circumstances. It's not so much that they want different products and services than men; they want a different service model. First, they want their lives made easier and so, in general, they are more open to a single, primary financial relationship than men; and, second, they value networking and finding various sorts of expertise through that primary relationship.\n\nWhile women do not necessarily want different products and services than men do, it is true that their challenges are slightly different. The average age of widowhood in the United States is fifty-five, and women live an average of 5.4 years longer than men. So not only must women plan wisely so their investments can support their longer lives, they must also be prepared to manage their finances on their own and, more and more often, manage those of their parents, as well.\n\nIn light of the intensity of their drive to become more financially sophisticated, advisors may initially spend more time with a female HNWI than a male HNWI to get the relationship off the ground. The time spent may be well worth it, however, if it generates a more prolonged and stable advisor-client relationship. As is the case with many relationships, as a rule \"women are less fickle than men, and are less concerned with the next hot thing,\" Ms. Moeder observes at the risk of stereotyping behavior. \"It's a stereotype,\" she says, \"but it's a correct stereotype. Women are more relationship-oriented; they value communication more.\"\n\nWomen also have a tendency to gather expertise from as many sources as possible. They want access to other individuals and institutions. In the past, the universe of advisors to a wealthy woman might be a financial advisor, an accountant and an attorney. But today women often seek out art experts, philanthropy advisors, consultants to help with their children's educations and entrance into colleges, even psychotherapists. \"Women love to share best practices. They like to hear what others are experiencing.\"\n\n#  **PRESERVING THE FAMILY LEGACY**\n\nIn general, wealthy families are increasingly conscious that the HWM concept permeates everything that they do relating to the issue of family and intergenerational wealth transfer. If wealthy families are doing a better job of educating their children as to the long-term impact of wealth on their lives\u2014or are turning the job over to surrogates who are imparting such knowledge\u2014clients are also gaining a new degree of sophistication regarding the rights, responsibilities, blessings and challenges enjoyed and suffered by the world's burgeoning HNW and ultra-HNW population. In no area of wealth management are these trends so pronounced as in the field of philanthropy, where issues of values, legacy, morality, social service and self-interest all come into play in a way that bears profound implications for the long-term evolution of our shared society.\n**CHAPTER 8**\n\n**The New Philanthropy: Proactive Involvement**\n\n#  **GIVING WITH PURPOSE**\n\nIn 2007, the _WWR_ detailed philanthropic donations for the first time, documenting a record US$285 billion donated to charities by high-net-worth individuals\u2014an impressive sum, to be sure, yet one totaling a mere 1 percent of HNWIs' total US$37.2 trillion net worth. Amidst this comparative charitable cornucopia, the evidence was overwhelming that a new breed of high-net-worth benefactors, the vast majority of whom were self-made entrepreneurs, were eager to make their mark on philanthropy while demanding an unprecedented degree of accountability from grant recipients. Assets managed in donor-advised funds\u2014which permit wealthy individuals to direct tax-free donations to good causes\u2014rose 50 percent to US $4.9 billion from 2003 to 2005. Of this, a significant factor was a marked increase in the percentage of people worth more than US$30 million (17 percent) who donated more than 10 percent of their total wealth. (This trend was further confirmed by the _Chronicle of Philanthropy_ , which noted that twenty-one Americans gave away at least $100 million last year, compared with eleven in 2005.) By 2006, this wave of contemporary philanthropists was giving record amounts to charities while insisting upon seeing \"their money used wisely.\"\n\n#  **PHILANTHROPY LESSONS FROM ONE OF GREAT BRITAIN'S WEALTHIEST MEN**\n\nA leading light amongst this new class of \"activist-entrepreneurs,\" the Duke of Westminster (who is not only one of Britain's wealthiest men but a lifelong and fiercely devoted philanthropist) comments approvingly on the relatively recent phenomenon that \"donors on the whole are displaying a far greater interest in truly comprehending the uses to which their time, money and human as well as financial capital are being put.\"\n\nSpeaking from a modest office off a high-ceilinged corridor at the Eaton Estate Office, hub of an 11,000-acre estate near Chester in England that has been in the Grosvenor family since the late fifteenth century, the Duke is adamant about being characterized not as a \"philanthropist\" but as an individual privately yet intensely devoted to forging new and productive links between private industry and public social improvement.\n\nThe term that probably best describes his role as inspiration and motivator among this new breed of activist-entrepreneur-philanthropists is \"social entrepreneur,\" although he is far too modest to apply the term to himself. His abiding interest in combining the best aspects of business and philanthropy is reflected in his description of his own efforts as \"individual Corporate Social Responsibility.\" Today's heightened emphasis on accountability and efficiency of aid delivery can only be to the good, the Duke asserts, since this more energetic and engaged philanthropic style tends to make social work a more useful and productive force for the good of the entire society.\n\nThe origins of the vast and rapidly expanding business interests that form the Grosvenor Estate today are widely believed to be founded on the legacy of Gilbert Le Grosveneur\u2014a name that loosely translates as \"Chief Huntsman\"\u2014who loyally served William the Conqueror as Master of the Hunt. The Eaton Estate came into the family when Ralph Grosvenor married Joan, heiress to the estate, during the reign of Henry VI. But the bulk of the family fortune is the happy result of the Grosvenor ancestors' fortuitous ownership, timely development and long-term management of 300 acres of land it owned in London, which it smartly transformed into and maintains today as the fashionable neighborhoods of Mayfair and Belgravia.\n\nGerald Cavendish Grosvenor, the present-day and sixth Duke of Westminster, has famously followed in the philanthropic foot-steps of his illustrious ancestor the first Duke of Westminister. In 1874, Hugh Lupus Grosvenor was elevated by Queen Victoria from Marquess of Westminster to Duke (the pinnacle of the peerage) in royal recognition of his outstanding career as one of his nation's non-profit pioneers. The First Duke had spent a great deal of time improving the lives of the less fortunate.\n\nToday's Duke, following precedents set by previous generations of his lineage, artfully combines a tripartite role by being 1) one of the richest men in England, 2) a loyal soldier to the Crown (he has served as Assistant Chief of Defence Staff [Reserves and Cadets]) and 3) a devoted philanthropist. Unlike his ancestors, however, this twenty-first century aristocrat combines a deep personal commitment to philanthropy with an equally deep personal commitment to commerce. He not only serves as chairman of the Grosvenor Trustees, the shareholders of an extensive real estate investment and development portfolio in Britain, Ireland, Canada, the United States and Asia, but he also serves on the boards of dozens of charities, which he actively encourages to govern themselves along business lines.\n\nWhile exuding a sense of relief at this trend toward greater accountability and rationality in the non-profit sector, the Duke ascribes the recent rise of philanthropy across the globe to the chronic inability on the part of municipal, regional and national governments (as well as multilateral institutions like the UN and the World Bank) to solve the world's often daunting social problems. Among the long litany of social woes on which he has personally expended significant amounts of social, human and personal financial capital are service charities (that benefit those who have served in Her Majesty's armed forces and their families), deeply embedded urban and rural poverty, violent crime, domestic child abuse, drug addition and substance abuse. The Duke personally applauds the recent entry into philanthropy of a number of self-made billionaire and millionaire philanthropists, because only when philanthropy is openly challenged by business to be more productive and more transparent does it stand a chance of evolving itself for the twenty-first century.\n\nThe most positive aspect of this development, Westminster believes, is that a welcome infusion of business ideas and ideals has revolutionized the hidebound world of philanthropy, leaving it far better prepared than it might otherwise have been to tackle tough problems in a new century that promises to be infinitely more complex than any before it. \"Today's philanthropists,\" the Duke approvingly notes, \"are able to take a long-term view, which governments either will not or cannot take. They have been able to both plan and act with an efficiency, agility and attention to issues of accountability that the State has either overlooked or been unwilling or incapable of addressing.\"\n\nAs recently as thirty years ago, he recalls, \"too many philanthropists were content to simply write a check and be done with it.\" Not today, when taking a hands-off approach simply won't wash, in part because it does a disservice to the cause of charity and philanthropy itself. A prime example of this trend is the tough-love approach he took when asked to support a U.K.-based children's charity. Before providing any financial assistance of his own, he joined a chorus of voices insisting that the charity sell off its valuable headquarters property in London, thereby releasing considerable pent-up value that could be devoted to the needs of the children served by the charity.\n\nA related and equally positive development, the Duke avers, is that he and a fair number of his fellow philanthropists, many with years of organizational training and experience under their belts, are offering\u2014occasionally in lieu of but more often in addition to financial support\u2014management and marketing expertise.\n\nAs for tendering advice to prospective philanthropists, the Duke suggests that above all, \"people should support charities that mean something to them personally.\" Although, as a rule, he is not big on medical charities, he supports a national arthritis charity because his mother, a talented pianist, was personally afflicted by the disease. The Duke is a strong believer in private firms supporting their employees' philanthropic pursuits, approvingly citing as an example that of a property surveyor within the Grosvenor property group, who looks after the St. John Ambulance charity's property portfolio in his spare time because the charity could not afford to hire its own land surveyor. \"Businesses need to take the lead in encouraging charitable work by their employees,\" Grosvenor insists, contending that HR departments should be made aware of what employees are doing in the philanthropic arenas and taking concrete steps to recognize such activities' value to the firm.\n\n#  **COMMITTING FUNDS AND DEMANDING ACCOUNTABILITY**\n\nThe wealthiest of individuals, some of whom have never before shared their wealth with charitable organizations, are starting to do so in the current Age of Wealth. Many are demanding, contrary to custom, a more direct say in how the funds they contribute are allocated.\n\nIn 2007, even Carlos Slim Helu of Mexico, whose estimated US$68 billion telecom fortune is believed to have made him the world's richest man, reversed a previously poor philanthropic track record by announcing an admittedly belated commitment to set up a philanthropic foundation.\n\nAs one account notes:\n\n_In Europe, Jean Pierre Cuoni, chairman of private bank EFG International, has become a prominent supporter of The Right to Play, an organization that promotes sport and play programs for impoverished children in the emerging world. Belgian industrialist Baron Guy Ullens has been selling part of his art collection to fund humanitarian work, including an orphanage in Katmandu. Anil Agarwal, chairman of Vedanta Resources is putting US$1bn into an Indian university modeled on Harvard. Activist-investor] Chris Hohn this month confirmed that his hedge fund, The Children's Investment Fund, donated US$460m to charity last year. Peter Cruddas, founder of finance group CMC Markets, plans to give US$200m to charities helping young people._[ 68\n\nMany observers have noted that the trend hailed by Westminster and others of the billionaire band recalls the Victorian era of the present Duke's ancestor, the First Duke of Westminster, when vast sums generated by industry and trade were concentrated in the hands of a select few and state support for the disadvantaged was limited to nonexistent. This new breed of activist-philanthropists is ensuring that all funds dispensed for virtually any cause come with strings attached\u2014and that these strings are, if not stronger than steel, at least as binding as polyester or nylon.\n\nThe Indianapolis-based Center for Excellence in Higher Education, an initiative established by a number of major philanthropists (including Home Depot co-founder Bernard Marcus and legendary investor Sir John Templeton), is dedicated to insuring that donors can attach legally enforceable conditions to their gifts, sharply curbing educational institutions' discretion in spending donors' contributions.\n\n#  **COMPARING REGIONAL DIFFERENCES**\n\nAccording to a 2006 study by the Johns Hopkins Comparative Non-profit Sector Project, the United States led all countries by giving 1.85 percent of total GDP from 1995 to 2002. By comparison, the U.K. gave 0.84 percent, Brazil 0.29 percent, Japan 0.22 percent, South Korea 0.18 percent and Germany 0.13 percent. In 2006, according to the 2007 _WWR_ , North American HNWIs donated 7.6 percent of their portfolios, a more than 20 percent increase from 2005 levels, a boost largely ascribable to \"a heightened sense of social responsibility among North American HNWIs.\" By contrast, philanthropists in the Asia-Pacific region and the Middle East donated approximately 11.8 percent and 7.7 percent, respectively of their portfolios to charity. European HNWIs scored next to lowest on the regional list, allocating a mere 4.6 percent of their wealth to charity, while Latin America formed the philanthropic caboose, with HNWIs from the region giving approximately 3 percent to charitable causes.\n\nIn the United States, two big philanthropic themes have emerged: one is so-called \"venture philanthropy\" and the other is \"giving while living.\" While clearly distinct trends, both share a number of key drivers, among them the fact that so many HNWIs are comparatively young by historical standards, and have ample money to give at a relatively tender age. Other self-made philanthropists are applying some of the principles that made them successful in business to their philanthropic pursuits, and have the time and energy to help actively drive\u2014and in some cases define or refine\u2014their charities' missions.\n\n#  **GLOBAL TREND 1: THE RISE OF VENTURE PHILANTHROPY**\n\nIn 2006, the evolving concept of \"venture philanthropy\" was given a powerful boost when Bangladesh-based Muhammad Yunus was awarded the Nobel Peace Prize for his innovative method of alleviating poverty among the world's most desperately poor people. Through his Grameen Bank, Mr. Yunus has issued seven million \"micro-loans\" since 1983 to start small village businesses. The loans are usually for $100 or less, often to women, and there is a 99 percent repayment rate. What can $100 do? A lot, it turns out, in countries where many people live on less than $1 a day.\n\nAlthough Yunus chafes at the term \"venture philanthropy,\" he is willing to acknowledge its stated goal of eradicating poverty and evolving a more equitable society by advancing micro-loans mainly to poor rural women as a means of creating a culture of accountability and sustainability in regions of the world and swathes of society where such notions had never taken root.\n\nSome observers of the foundation scene have gone so far as to claim that Yunus has thereby given rise to an entirely new institutional category\u2014that of a \"for-profit charity,\" a new hybrid entity forged by the marriage (or blurring of boundaries between) three traditional institutional categories: 1) government, 2) the for-profit sector, and 3) the not-for-profit sector.\n\nYet another comparatively recent trend in philanthropy is for non-profit organizations frequently founded by self-made entrepreneurs to blur the traditional boundaries between charity and commerce. In 2004, eBay co-founder Pierre Omidyar founded the Omidyar Network, which makes investments in both for-profit and non-profit ventures. He has, for instance, provided US$4 million to the Grameen Foundation, and has supported Unitus, which develops microfinance institutions, and the International Development Law Institute, which develops law to support the growth of the microfinance industry. Last year he donated US$100 million to his alma mater, Tufts University, stipulating that the funds be used exclusively to make micro-loans.\n\nYet another example of the increasingly blurry boundary between charity and commerce is Google.org, a for-profit entity founded last year to conduct charitable operations for Google. By forgoing non-profit status, Google can donate to what it believes are worthy causes and not worry about whether IRS considers it a charity. Richard Branson, for his part, is investing US$3 billion of his personal fortune in the clean energy industry through what he calls his \"hybrid venture capital unit,\" while also looking to tackle big issues like poverty and climate change. In an interview with the _New York Times_ , Mr. Branson characterized the hybrid concept this way: \"It may be giving, it may not be giving, depending on how things turn out...Although it's very risky capital, hopefully it won't be wasted.\"\n\nIn a speech delivered at the Tech Museum of Innovation in San Jose in early 2007, Microsoft chairman Bill Gates ventured that, \"No foundation alone can solve the health problems of the developing world. We need businesses and governments as partners. That means we need to get these issues on the political agenda, and we need to tap into market forces to get the private sector involved. It means we all need to embrace a broader definition of responsibility. We must be willing to look at the failure of collective action and see how we can change it. Because these problems are so complex, government has to be involved in solving them.\"\n\nAnother prominent case in point is that of James Simons, the billionaire founder of Stony Brook, Long Island-based hedge fund Renaissance Technologies, whose Simons Foundation has committed US$38 million to autism research and has announced plans to spend another US$100 million in what is fast becoming the largest private investment in autism research in the world. While the outright sums are dwarfed by the billions donated to medical research by Bill Gates and Warren Buffett, the extraordinary degree of control that Simons personally asserts over where and how his money is spent makes his a case of venture philanthropy taken to the next level.\n\nSimons, whose daughter suffers from autism, has provided DNA from his own family for research purposes, provided assistance in helping solve key mathematical research problems, and responded positively to a request from MIT (on whose board Simons serves) for brain research funding by expressly stipulating that the project focus exclusively on autism and that it put scientists of his selection on the payroll. And while some not-for-profit officials positively bristle at the idea of relinquishing even a degree of programmatic control to activist-donors, most realists are inclined to acknowledge that more emotionally engaged donors are likely to remain more deeply committed over the long haul.\n\nSo what does all this mean? The main driver behind this growing trend\u2014as confirmed by the Duke of Westminster and others of his wealth and class, as well as self-made entrepreneurs who feel that the old ways of giving just aren't working\u2014is talk about venture philanthropy and for-profit charity as a new form of sustainability, aimed at insuring that once the original infusion of capital is spent, a project can continue and remain self-sustaining.\n\nExamples of this trend are the Acumen Fund, a non-profit charity that invests in companies that tackle global poverty and aims to provide the high-level management and entrepreneurial expertise more commonly found at venture capital funds. It has renounced making conventional grants in favor of loans and equity investments, finding that these create greater clarity of expectations and instill financial discipline. A recent success story is the Fund's substantial investment in International Development Enterprises in India, which manufactures drip irrigation systems that it sells to poor farmers for less than US$30. Acumen and others like it are non-profit organizations that engage in profit-making activities in order to decrease reliance on traditional, passive donors. Trappist monasteries in Belgium and elsewhere in Europe, which actively market their sought-after, original micro-brews and other spirits, represent the originals from whom these present-day models of fund-generating philanthropies derive.\n\nAccording to Jacques Attali of PlaNet Finance, this more muscular and activist approach is especially well suited for today's young and wealthy entrepreneurs, because at a fundamental level, entrepreneur \/philanthropists find the idea of funding fellow aspiring entrepreneurs around the globe to get themselves off the ground emotionally appealing, as to do so bestows a sense of connection, almost a kinship, or \"an intertwining of interests.\"\n\n#  **GLOBAL TREND 2: GIVING WHILE LIVING**\n\nBoth John D. Rockefeller Sr. and Andrew Carnegie famously gave away substantial proportions of their vast fortunes during their lifetimes. But Warren Buffett's 2006 decision to donate some US$37 billion of his fortune\u2014US$31 billion to the Bill and Melinda Gates Foundation, and another US$6 billion to four family foundations\u2014renewed interest in this approach among many HNWIs who in the past might have done most of their charitable giving through bequests at their death.\n\nNot only tycoons and titans but other celebrities have gotten into the act. Of course, U2's Bono is probably the most widely known and admired entertainer-philanthropist, but actress Angelina Jolie is quickly becoming one of the most high-profile humanitarians of our time, putting her money and persona where her mouth is. Not only is she an ambassador for the United Nations, and an adoptive mother to children from underdeveloped nations, she is also responsible for building schools and improving communities. She takes her show on the road, both overseas and to Washington, where she regularly uses her star power to lobby for her favorite charities. \"As much as I would love to never have to visit Washington, that's the way to move the ball,\" was how she put it to _Forbes_. Since giving birth to Shiloh, her third child, in May 2006\u2014she adopted son Maddox in 2002 from Cambodia and daughter Zahara in 2005 from Ethiopia\u2014she has adopted a child from Vietnam and promises more cameos on the Hill. \"The more children I have, the more I feel it's my duty.\"\n\nIn early 2007, amidst predictable fanfare and the beneficent presence of Nelson Mandela, UHNWI Oprah Winfrey opened the Oprah Winfrey Leadership Academy for poor but talented girls in South Africa. A crowd of other celebrities joined Winfrey and Mandela at the opening day celebrations, including singers Tina Turner, Mary J. Blige and Mariah Carey, actors Sidney Poitier and Chris Tucker, and film director Spike Lee. Her decision to do her own thing was based in part, she maintained, on a sense of weariness with conventional philanthropy, passive donation and a desire to feel a more intimate connection to the people she was hoping to help.\n\nWhile Oprah's rationale strikes a strong chord with many of her fellow HNWIs (for whom the most appealing aspect of donating money is enjoying the sense of well-being it bestows while still on the mortal plane), the satisfaction of managing one's philanthropic donations as actively as one might manage other assets is a deeply pragmatic emotion. Of course, other rationales may impinge on a number of these investment decisions, including the well-recognized tax advantages of philanthropy and the social engagement such activities bring.\n\nDavid Ratcliffe, head of the Merrill Lynch Center for Philanthropy and Non-profit Management, insists that HNWIs are actively researching charities and keeping tabs on them through new online tools, such as GuideStar.org, CharityNavigator.org and Charity-Watch. org, to make sure organizations are efficient and that gifts are being used correctly. \"People have become more focusd on the measurement of success,\" Ratcliffe remarks. \"They want the non-profit to be more accountable, to report back on how money has been used and how it's been effective.\" It's not unusual, he says, for a donor to tell a charity it must demonstrate progress, using certain metrics, within a certain time frame, or the HNWI will consider redirecting their charitable contributions to other organizations.\n\nSome of this insistence upon greater accountability has simply been a heightened cynicism regarding the probity of large organizations in the wake of the Enron, WorldCom and Tyco corporate scandals of the late nineties. But non-profits also suffered some tarnishing of their own image when the American Red Cross raised US$564 million for the Liberty Fund, set up in response to 9\/11, and then told donors it would use the majority of that money for other undefined purposes. Oral Suer, former chief executive of the United Way of the National Capital Area in Washington, was sentenced to more than two years in jail in 2004 for stealing about US$500,000 from the organization during his twenty-seven-year tenure.\n\nA survey in 2006 by the Center on Philanthropy at Indiana University found that 75 percent of HNWIs would give more if less money was spent on administration, and nearly 60 percent said they would give more if they could determine the impact of their gifts.\n\n\"These are highly competitive and performance-oriented people, and if they give money away they want it to be put to the best use,\" notes BlackRock co-founder Ralph Schlosstein. \"These people worked hard to make their money, and they want as much impact as possible. I do believe there's a greater level of accountability of the non-profit sector than has historically been the case.\"\n\nYet Mr. Schlosstein also emphasizes that while he sees a tremendous drive among HNWIs to more precisely measure results, it comes from a tremendous desire to give. \"The vast majority of people in my business believe they are working to benefit those more in need, because everything they make now is going to charity. There's an extremely strong sensation that we're incredibly fortunate and we need to give back to something we care about.\"\n\nHis comments were echoed by James Kennedy of T. Rowe Price, who cited his Jesuit upbringing as providing him with a sense of \"obligation to turn around and help the one behind you.\" James Rothenberg of Capital Research concurs: \"I've got far more wealth than I will ever need or my family will ever need under any reasonable circumstance, so I look to philanthropy as a way not just to write a check but [to be] actively involved and engaged with helping people and helping an organization get better.\"\n\n#  **TAKING TAX CONSEQUENCES INTO ACCOUNT**\n\nA study recently conducted by the University of Indiana found that 52 percent of respondents would keep their giving levels the same even if they received no income-tax deduction whatsoever. Thirty-eight percent said their donations would probably decrease somewhat, while only 7 percent said they would decrease substantially.\n\nYet former Microsoft executive Charles Simonyi, who has already given away some US$85 million (most of it to a foundation he created to fund the arts and sciences) cautions that it's easy to underestimate the degree to which the U.S. tax code\u2014in stark contrast to Europe's\u2014encourages charitable giving. \"Essentially, whenever you make a donation you are making a challenge grant to Uncle Sam, and every time he steps up to the plate. In Western Europe the idea is that it's the government's duty, and I think this will accelerate.\"\n\nOne salutary effect of the U.S. tax code is that there are a multitude of tax rules to take into account when giving, depending on the type of assets being donated and the type of charity. HNWIs are educating themselves on the possibilities so they can make the best decisions. David Ratcliffe notes that a general increase in investor sophistication is having an effect on HNWIs' approach to philanthropy. \"Donors are much more strategic in the way they think about philanthropy. As financial services advisors, we spend a lot of time educating them about effective and efficient planning to meet their objectives, and that's spilling over into philanthropy.\"\n\nCharitable gift annuities, charitable remainder trusts and pooled income funds all allow donors to receive a cash flow even after making the gift. Oftentimes such vehicles are funded with appreciated stock, real estate, the proceeds from the sale of a business or other assets that otherwise would be heavily taxed. Among the most prominent of these vehicles are:\n\n\u2022 Charitable remainder trusts: The donor transfers assets to an irrevocable trust and receives a regular payment stream, which is taxable. At the end of the trust's term, what's left goes to a charity. The donor receives an income-tax deduction for the amount estimated to end up with the charity.\n\n\u2022 Charitable gift annuities: The donor gives money to a charity, which promises to pay him or her a fixed amount regularly for life. Although part of the annuity payments is taxable, the donor will also receive an upfront tax deduction for the amount estimated to end up with the charity.\n\n\u2022 Pooled income funds: The donor gives money to a charity, which invests the money in a special fund and pays the donor his or her share of the fund's earnings. When the donor dies, whatever's left of the investments goes to the charity.\n\nIn addition, certain techniques, like Charitable Lead Trusts, provide donors with the ability to make current distributions to the charities of their choice and ultimately transfer the assets to other family members. These techniques allow for charitable contributions as well as \"leveraged\" transfer of assets to children or grandchildren with reduced gift and\/or estate tax ramifications.\n\nConsider the case of an American HNWI who had accumulated US$1 million of company stock via an employee stock-option purchase program. He wants that to generate an income stream of at least US$5,000 per month to help support his retirement costs. If he had sold his company stock outright, he would have had an upfront tax hit on all of the appreciation over the past twenty years. The capital gains tax rate at the time the HNWI was mulling his options was still 20 percent and the shares had appreciated by US$833,660, so the tax hit would have been US$166,732.\n\nTo avoid those taxes, the HNWI funds a charitable gift annuity contract in exchange for US$1 million in appreciated shares of stock. The Internal Revenue Service then applies a 53.73 percent charitable bargain-sale ratio to his gift, which effectively reduces the reportable capital gain to US$537,300 from US$833,660 and prorates the reporting over the donor's life expectancy of 13.2 years. Based on his current age, the donor qualifies for a 7.7 percent annuity rate, which pays him US$77,000 per year, or monthly payments of US$6,417 in retirement.\n\n#  **MAKING A DIFFERENCE WITH FAMILY FOUNDATIONS AND DONOR-ADVISED FUNDS**\n\nFamily foundations represent yet another preferred route to charitable giving that has grown greatly in popularity and numbers over the years\u2014from 23,770 foundations in 1982 to 64,843 in 2002, according to Monitor Group. But there are signs that this explosive growth may be slowing thanks to increasingly complex taxes and regulations governing such foundations.\n\nNot only do family foundations provide an effective means to pursue philanthropic goals, they frequently serve to unite families around a common mission and set of goals. For a family foundation to function properly, however, fundamental principles, policies and practices must be set in place. This is especially important as the foundations created over the last twenty-five years mature and the original family founders die. To handle this transition to the next generation, the governance of the foundation should be clear, or risk the intrusion of potentially dysfunctional family dynamics.\n\nFor some, a family foundation arguably begins to lose its luster; however, another grants management charitable structure is gaining momentum: the donor-advised fund. In 2005, the country's largest donor-advised funds held assets of US$15.5 billion, up 22 percent from the year before, according to the _Chronicle of Philanthropy_. Here's how they work: An individual contributes at least US$5,000 in cash, stocks or other assets and gets an immediate tax deduction for the contribution and retains the right to recommend how the money should be distributed to charitable organizations. The donor can postpone the actual distribution of the money for years and in the meantime choose from a menu of investment options to grow the pool of money.\n\nSuch funds are typically offered by charities set up by major financial institutions as well as community foundations, and provide a donor with many options in recommending disbursements of the funds. Technically, the fund managers are not bound by the donor's wishes since the donor's funds are owned by the charity, yet donors retain the right to name the charity and make disbursement \"recommendations,\" and in practice those suggestions are typically honored once properly vetted by the managing charity. Donor-advised funds can be set up quickly and inexpensively, as opposed to family foundations, which can be cumbersome to set up and manage from a tax and regulatory standpoint.\n\nApart from their obvious tax, cost and management benefits, David Ratcliffe of Merrill Lynch observes that donor-advised funds, which permit HNWIs to recommend multiple grants over many years, provide investors in them both the capability and the right to carefully monitor the charities they choose to recommend and advise and redirect funding to other charities when they deem appropriate.\n\nDonor-advised funds offer something else in great demand by today's HNWIs: _portability_. People are much more mobile than in the past. When living in, perhaps, the Northeast and raising their children, HNWIs may want to donate to local causes; but once their children are grown, they may spend more time in Palm Springs, California, or Naples, Florida, and wish to redirect charitable dollars to those areas. Donor-advised funds can move with the HNWIs, and help fund their changing priorities throughout their lives.\n\nDue to the low threshold to open donor-advised funds\u2014often just $5,000 to $10,000\u2014such funds are increasingly being created by HNWIs who nominate their children or grandchildren as grant advisors. It's a way to help young adults engage a network of philanthropic interests, learn how to research their own charities and develop the skills necessary to be responsible stewards of capital. Thus, donor-advised funds dovetail nicely with HNWIs' use of philanthropy to develop their children's moral compass and to shape a life of purpose as well as privilege.\n\nFor wealth advisors, an additional benefit of donor-advised funds is to deepen an advisor's existing relationship with the HNWI while possibly establishing a rapport with the next generation. Some advisors argue that working closely with HNWIs on their philanthropy is the very best way to deepen a relationship since an advisor gains insight into HNWIs' personal priorities. This broader, more holistic view of the client can help the advisor tailor value-added, differentiated advice.\n\nWhile the United States may lead the world in charitable giving, and Europe may continue to lag, HNWIs in other parts of the world are starting to give more openly and frequently than ever before. Michael Saadie of ANZ has seen a marked increase in the desire of HNWIs in Australia to give back to their society\u2014mostly, he says, for the sheer joy of sharing their wealth and affecting change, but also to create a legacy and perhaps to assuage a touch of guilt. \"The newer money is setting up more and more charitable trusts, many more than seven years ago.\"\n\nA similar charitable awakening is also occurring elsewhere in Asia, where in the past donations have been made sparingly and with little publicity, in part due to a now-outdated reverence for the Confucian values of self-reliance and individual responsibility. In 2006, Li Ka-shing, the chairman of Hong Kong conglomerate Hutchison Whampoa Ltd. (and Asia's richest man, with a fortune estimated at US$30 billion plus) roiled the virtually nonexistent world of Mainland Chinese philanthropy by announcing plans to give a third of his fortune\u2014a pledge estimated at more than US$10 billion\u2014to his family Li Ka-shing Foundation.\n\nAccording to the _Wall Street Journal_ 71 Mr. Li's pioneering social entrepreneurial lead is being enthusiastically followed by Yang Lan, a prominent talk-show host frequently referred to as the Chinese Oprah Winfrey; Yu Pengnian, the eighty-five-year-old head of the Chinese luxury-hotel empire Shenzhen Pengnian Hotels, who has donated roughly 80 percent of his net worth to fund cataract operations for thousands in rural China; and Niu Gensheng, chief executive of Mengniu Dairy Group, who has donated shares in his firm valued at nearly US$600 million to a foundation devoted to agriculture, education and medical endeavors.\n\n\"In Asia, our traditional values encourage and even demand that wealth and means pass through lineage as an imperative duty,\" Mr. Li told the _Wall Street Journal_ , exhorting his fellow Asians to \"transcend this traditional belief.\" They, for the most part, are eagerly awaiting the announcement of a comprehensive law on charitable giving for China, an event predicted by Liu Youping, chief editor of _China Philanthropy Times_ , published by China's Ministry of Civil Affairs.\n\nFor his part, Mr. Li has personally underwritten the construction of Shantou University, a regular hotbed of corporate social responsibility, where Starbucks Coffee Co. founder Howard Schultz has lectured on business ethics, former CNN correspondent Peter Arnett teaches journalism, and the pedagogical philosophy openly espouses the traditionally Western values of open discussion, transparency, creativity and freedom. As West and East meet in the world of philanthropy, trends toward a fusion of styles bode well for the evolution of an active non-profit sector in China.\n\nIn Hong Kong, Dr. Lee Shau Kee of Henderson Land Development has grown concerned with the growing disparity in wealth and prosperity between urban and rural China. He intends to train one million children of farmers to work in the city, where they will earn higher wages and thus be able to support their families back on the farm. \"I want to help people to help themselves,\" Dr. Lee notes, \"which is much more valuable for both sides than just making a donation. By training the child of a farmer you compound the value of that dollar spent on training since that child will now help their parents. This is the multiplier affect. There's no point in just feeding them or just giving them money.\"\n\nDr. Cheng Yu-Tung of New World Development\u2014another client of Joseph Lam's\u2014has interests in property, infrastructure, telecom, jewelry stores and casinos all around Asia. In the eighties, Dr. Cheng embarked on a program to train Chinese doctors in the United States. The problem was that the student doctors, once they landed in San Francisco, usually did not come back\u2014in fact only about 20 percent did, a trend that worsened after Tiananmen Square. In response, Dr. Cheng began having students trained in Hong Kong where they couldn't seek political asylum. This is a good example of prudent business acumen being applied to philanthropy, and how corporations can bring some of their best practices to bear.\n\n#  **WITNESSING A DECLINE IN GIVING?**\n\nWhile philanthropy is unquestionably a force for good, in a world where the gap between the rich and poor is growing steadily, one unsettling trend cannot be ignored: in the United States at least, HNWIs as a group are giving less as a percentage of their net worth than ever before, while non-HNWIs are giving more. According to Internal Revenue Service data published by the _New York Times_ , the richest individuals, worth US$20 million or more, left 20.8 percent of their estates to charity in 2004, down from 25.3 percent in 1995. Among taxpayers with US$1 million or more of income, the percentage of their income they gave away dropped to 3.6 percent in 2003 from 4.1 percent in 1995. Meanwhile, those with incomes of less than US$1 million became more generous, upping their donations to 3.5 percent of income from 2.8 percent in the same time frame.\n\nWhether such numbers may be an anomaly, Jeff Sachs, special advisor to the United Nations secretary-general, told the _Financial Times_ in early 2007 that wealthy philanthropists could do more to lift Africa out of poverty than could the Group of Eight leading nations. He exhorted the wealthy to follow the examples of Gates and Buffett and contribute to a new private sector foundation to help speed the elimination of diseases and tackle specific challenges.\n\nAs Sachs put it to the _Financial Times,_ if the 950 billionaires whose wealth is estimated at US$3.5 trillion were to contribute an annual 5 percent \"foundation\" payout of their total net worth, that would provide US$175 billion per year to charitable causes. \"We don't need the G8, but we do need the 950 people on the _Forbes_ list...Maybe private philanthropists will champion solutions to individual problems rather than the G8.\"\n\n#  **EXPRESSING OPTIMISM FOR HNWIS' GRANDCHILDREN**\n\nIn 1930, in the wake of the Roaring Twenties and at the outset of the worst economic depression the world had yet seen, British economist John Maynard Keynes published an upbeat essay entitled \"Economic Possibilities for Our Grandchildren\" in which he defied his disciples who predicted a semi-permanent depression and forecast instead a shining new world of a century hence, when steady but gradual technological progress accompanied by gradually compounding capital accumulation would produce\u2014as he optimistically put it\u2014\"ever larger and larger classes and groups of people for whom problems of economic necessity have been practically removed.\"\n\nAlthough Keynes failed to live to see the coinage of the term HNWI (perhaps fortunately for him), he accurately forecast a period and even precisely enumerated the drivers of a new Gilded Age in which wealth would be far more greatly dispersed than at any other time in human history. This period would be one in which goods and services would become abundant and nearly cost-free, interest rates on capital would plummet to near zero, and the greatest challenges facing the wealthy class would be how to live a life of unlimited leisure that would also be filled with purpose and meaning.\n\nWhile Keynes' financial future of nearly universally distributed wealth above and beyond the means of necessity has yet to come to pass, by the twenty-first century (not quite a hundred years from 1930) the world had indeed fulfilled Keynes' prophecy of producing a \"standard of living in progressive countries one hundred years hence that will be between four and eight times as high as to-day.\"\n\nIn \"Creating a Moral Biography of Wealth,\" Paul Schervish, the director of the Center on Wealth and Philanthropy at Boston College, ponders the social and even existential implications of the possibility that Keynes' era of nearly universal prosperity may one day soon be reality as opposed to fantasy. Among today's younger HNWIs, he has observed a propensity to search for a sense of ultimate meaning and higher purpose to life\u2014that hitherto unimaginable luxury for many that the possession of great wealth confers.\n\nOne natural extension of portfolio management envisioned by Schervish is the creation of a \"moral biography\" of personal wealth in which its possessor employs total assets under management as tools to embark upon \"a spiritual process of self-examination\" that goes well beyond and far afield of any conventional process of portfolio analysis. It is the sort of adventure that Charles Simonyi embarked upon when, after accumulating great wealth, he decided to give nearly US$100 million of it away while gaining fame as one of the planet's pioneering space tourists. It is the sort of adventure that philanthropists like Bill Gates, Warren Buffett, Gerald Grosvenor, Dr. Lee and Dr. Cheng have embarked upon as they seek fulfillment in improving the lives of others, using the same analytical tools and skills that helped them first improve their own.\n\nAs more and more HNWIs and non-HNWIs begin to look beyond mere wealth, and the period of acquisition and accumulation that produced it, to the next phase of their lives, they will\u2014as John Maynard Keynes precisely predicted\u2014\"for the first time since creation...be faced with [the] real [and] permanent problem of how to use this freedom from pressing economic cares, how to occupy the leisure that science and compound interest will have won...to live wisely and agreeably and well.\"\n\nThis book represents one modest attempt to lead us to the place where a few of us will be prepared to ask\u2014and even answer\u2014such questions. As Keynes presciently wrote, \"it will be those who can keep alive and cultivate into a fuller perfection the art of life itself and do not sell themselves for the means of life, who will be able to enjoy the [age of] abundance when it comes.\"\n\nCyclical credit crunches, market volatility and upheavals aside, the age of abundance is already here for many of us and a fair percentage seem to be striving to \"live wisely and agreeably and well.\" As Winston Churchill, who certainly knew something about living wisely, agreeably and well, once said, \"We make a _living_ by what we get, but we make a _life_ by what we give.\"\n**CHAPTER 9**\n\n**The Future of Holistic Wealth Management**\n\n#  **CREATING CONDITIONS TO BUILD AND PRESERVE WEALTH FOR THE FUTURE\u2014THE MICHAEL LEE-CHIN WAY**\n\nThe Daniel Libeskind-designed, aluminum-and-glass, 175,000-square-foot Michael Lee-Chin Crystal juts jarringly out over bustling Bloor Street West, a stylish and starkly contrasting modern addition to the neo-Byzantine fa\u00e7ade of Toronto's Royal Ontario Museum (ROM). Yet striking as the structure is architecturally, perhaps its least likely aspect is that it is the fruit of a C$32 million donation from prominent investor and financial celebrity Michael Lee-Chin, a Jamaican-born immigrant to Canada who in 2007 ranked number 618 on the _Forbes_ list of the world's wealthiest individuals, with a net worth of US$1.6 billion. It's least likely not because Mr. Lee-Chin isn't philanthropically minded\u2014he's famously so\u2014but because he's the son of a grocery-store-owning father in the rural backwater of Port Antonio, Jamaica, and of an entrepreneurially minded mother who sold Avon products door to door and worked long hours as a bookkeeper to make ends meet. He is also the author of a unique framework and philosophy of diversification and wealth accumulation and preservation that was proven highly successful for many long-term investors in AIC, the mutual fund company he established more than two decades ago.\n\nSo how on earth, Lee-Chin rhetorically asks while swiveling in a leather wing chair in the formally furnished boardroom of AIC, the Canadian mutual fund giant he organically grew from a tiny entity he acquired in 1987, could someone from his ostensibly disadvantaged background have made it on the annual _Forbes_ list of the world's wealthiest individuals for six years running? The answer came to him in 2002 while sitting in the office of the Jamaican Minister of Finance in Kingston, writing a check to purchase a controlling 75 percent share of the National Commercial Bank of Jamaica.\n\n\"For me, buying the National Commercial Bank\"\u2014Jamaica's largest consumer bank\u2014\"would be tantamount to an American buying Citibank,\" he explains. \"So as I was writing out that check, I thought to myself, 'How is this possible? How is it possible for the son of an orphan born of a teenaged mother to be writing a check to buy the National Commercial Bank?' Just before I wrote out the check, the question came to mind. Just _after_ I wrote it, the answer came to mind.\"\n\nFor starters, he decided that he owed a fair part of his remarkable good fortune to the fact that he was born in a country and at a time when a poor boy (whose grandfathers were Hakka Chinese immigrants to Jamaica and whose grandmothers were Afro-Caribbean Jamaicans) was free to attend a good local high school and an excellent college, McMaster University in Hamilton, Ontario, where he studied civil engineering on a Jamaican government scholarship. After returning to Jamaica to work on a national highway project, he decided that civil engineering was not his cup of tea and made his way back to McMaster to pursue a graduate degree in business.\n\nBut he also would realize, some thirty years hence, that sheer happenstance had conspired with character and conditioning to propel him to the next stop on the road to riches. While working after hours as a bouncer in the campus pub, he encountered a former college classmate happily celebrating his $200-a-day take-home pay from selling mutual funds for Investors Group. With that pay scale comparing favorably to the $2.50 an hour he was taking home as a bouncer, Lee-Chin's otherwise neutral outlook on the financial services industry abruptly brightened from \"hold\" to \"buy.\"\n\nDespite failing the psychological test for prospective trainees, he talked his way into the sales training program at his friend's firm. Days after completing the course he found himself waiting at the Toronto airport for his mother's delayed flight from Miami. \"That day,\" he recalled, \"was a Sunday. I was feeling stressed out because\n\nI knew nothing about the financial world. My entire work experience had been as an engineer combined with a few months as a bar bouncer. I was twenty-six and all of my friends were broke. How on earth was I going to get clients to see Monday morning?\"\n\n\"So that Sunday afternoon after I heard my mum's flight was delayed I decided to get in my car and drive around the nearest neighborhood to get my thoughts together. As I drove around this very nice neighborhood I saw people outside tending their gardens. So I said to myself, 'Okay, Mike, here they are, they _really_ want to see you.' I told myself that the next person I saw, I was just going to close my eyes, grit my teeth, put my foot on the brake, jump out of the car, and talk to my first real live prospect.\"\n\nTo every one of those real live prospects he posed precisely the same question. \"Sir\u2014or Madam\u2014I'm here working with members of this community, showing them how to minimize their taxes and taking their tax receipts to create a substantial asset base. What time would be good for you to see me? Early next week or later on in the week?\" Out of the six prospects he approached, he snagged five appointments and ended signing up his first clients that Monday morning.\n\nIt didn't take long for the consummate salesman to build up an enviable track record as a pitcher of mutual funds to mass-affluent clients. But only after stumbling across a brief biography of Warren Buffett did he experience the eureka moment that set him on the path to true prominence as an investor. \"Warren's methodology to me made total sense. I felt I could replicate it. I didn't think I had the ability to prognosticate where markets were going, or prognosticate where interest rates were going, or where the economy was going. But I did think I could train myself to identify great businesses, which is the secret of Warren's success.\"\n\nLee-Chin's devotion to \"The Buffett Way\" extends to the AIC slogan\/mission statement printed on his business card: \"Buy. Hold. Prosper.\" Inspired by Buffett, he distilled his own investment philosophy to five cardinal rules:\n\n1. Invest in a handful\u2014\"less than the fingers on your hand\"\u2014of high-quality firms;\n\n2. Ensure that those firms are domiciled in long-term growth industries;\n\n3. Understand those firms and industries comprehensively;\n\n4. Invest for the long term and with discipline, focus and commitment, ignoring market fluctuations and pressure to sell in down cycles\u2014also to maintain a high rate of return _after taxes_ ;\n\n5. Use other people's money to leverage those investments.\n\nIn applying his five principles, Lee-Chin was well aware that he was single-mindedly defying one of the cardinal rules of his own industry: the gospel of diversification and broad-based asset allocation. Too many of the mutual funds he was advising clients to buy into, he decided, owned too many stocks for the fund managers to truly understand the firms and the industries at the deepest level. Relying on analysts to conduct due diligence didn't strike him as a sound strategy. Instead, he decided to move aggressively \"beyond Markowitz\" and put all of his eggs into one (he believed) promising basket.\n\nIn 1983, he used his diversified portfolio of stocks as collateral to borrow C$500,000\u2014considerably more than his net worth at the time\u2014from the Continental Bank of Canada to buy 500,000 shares of financial services firm Mackenzie Group at C$1.00 a share. He arrived at this momentous and obviously risky decision by using the following thought process: \"I asked myself the question, which industry do I best understand? Remember that I'm a financial advisor. So [for] which industry do I best understand the regulatory environment? The competitive nature of the landscape? Who the competitors are? Margins pertaining to the different products? The long-term nature of the industry? I answered that question in the following way: 'The industry I'm in\u2014the wealth management industry!'\"\n\nAs he plunged more deeply into his analysis of the long-term future of the WM industry, he found that it met his long-term growth-and-value criteria like a glove. \"In the early eighties, the wealth management industry wasn't nearly as developed as it is today. So I worked my way down the list one by one. 1) If I had to narrow down my investment choices to a handful of industries, what would one be? 2) Is it an industry I understand? 3) Is it predicated on other people's money? 4) Is it domiciled in a strong, stable, long-term growth industry? 5) Will it stay strong and stable over the long run?\"\n\nIn five years, Lee-Chin's faith in the future of the financial services industry in general and in Mackenzie in particular was validated as his bucket of shares soared in value to C$3.5 million. Not only was his theory vindicated, but the run-up provided him with sufficient personal credit with his banker in 1987 to enable him to borrow more funds to finance the purchase of a small, local firm of financial advisors. At the time, AIC had a mere C$800,000 under management. Within the decade, AIC had blossomed into a mutual fund behemoth, with over C$14 billion under management throughout the period its flagship Advantage Fund, which had become a darling of the Canadian mutual fund industry, rigidly adhered to the Michael Lee-Chin philosophy by holding fewer than twenty stocks in its bulging portfolio.\n\nYet over the years, and particularly during periods of heightened speculation and bubble mania, AIC has suffered its fair share of redemptions as Lee-Chin's steadfast commitment to \"Buy. Hold. Prosper.\" has been sorely tested by his firm refusal to ride the roller-coaster into tech stocks or plunge into other fashionable investments. Regardless of what's going on in his business life, though, Lee-Chin has remained both a UHNWI _par excellence_ and a proverbially generous man. With his personal wealth, Lee-Chin has underwritten business institutes in and around Toronto, donated heavily to the ROM, and aggressively invested in his native Jamaica and elsewhere in the Caribbean\u2014places and sectors he both understands and to whose often fragile economies he wants to \"give back.\" Using Portland Holdings, his privately held investment vehicle, he has been snapping up land, real estate holdings including office buildings, telecom firms and, of course, banks and other financial firms.\n\nAsked about the future trends of high-net-worth investment, Lee-Chin emphasizes that no advances in technology, social changes or innovative products are likely to undermine the fundamentals of value investing. That said, he is a firm believer in\u2014and a product of\u2014the continuation of the relentless globalization that will continue to drive WM across borders to service increasingly globe-trotting HNW clients. He firmly believes that access to markets will become more transparent and that global consolidation of industries is likely to drive global diversification of assets. The HNWI population shifting funds in and out of alternative investments such as hedge funds, derivatives or structured products into real estate, or demanding concierge services or plunging into \"investments of passion\" will not and cannot, in his opinion, fundamentally alter the competitive landscape of the WM industry or shift the essential dynamics of long-term investment.\n\nIn this view, he concurs with fellow Buffett disciple the Hong Kong billionaire Dr. Lee Shau Kee that at the end of the day all consistently successful long-term investors follow precisely the same rules. In the timeless world of the long-term value investor, the future is likely to look very much like the past.\n\n#  **IDENTIFYING FUTURE TRENDS TO TAKE INTO CONSIDERATION**\n\nAs we continue with our conclusion of _Wealth_ , and taking Michael's philosophy into consideration, we believe there are a number of trends that will continue to shape the Wealth Management Industry moving forward. Some of them are evident today and others are still emerging. Regardless, individuals and their FAs will do well to consider the wide-ranging impacts of these trends when investing their considerable assets.\n\n1. Products designed for the HNWI market will continue to become more complex, sophisticated and international in nature and scope supported by the continuing globalization of services;\n\n2. Technological advances will drive down the WM barriers to entry, enabling one-stop shopping. This will further cement the fact that the ultimate differentiating factor will be the quality of the relationship between client and advisor. The quality of this relationship will be driven by the client, versus the advisor;\n\n3. The Asia-Pacific region will continue to become the dominant source of wealth creation;\n\n4. Intergenerational wealth transfer will drive a focus back to capital growth as a source of wealth creation, complementing the current dominant source of wealth creation from business ownership to capital growth;\n\n5. Industries that will remain attractive and dominant for the long-term include financial services and agriculture;\n\n6. The devaluation of the U.S. dollar will further drive global investment to Europe and Asia, as well as increase foreign investment in the United States.\n\nWhat remains clear is that the Wealth Management Industry will remain buoyant and attractive, regardless of economic ups and downs and the inherently cyclical nature of financial services.\n\n#  **INCREASING WEALTH ACROSS THE GLOBE**\n\nDespite the unsettling credit crunch of 2007-08\u2014and barring the development of unforeseen economic reversals in the form of a global recession or depression\u2014the most influential and seemingly resilient secular trend affecting the burgeoning population of HNWIs of the world is that the ongoing spread of global capitalism is producing pools of ever-more consolidated and concentrated wealth. In absolute terms, according to the 2007 _WWR_ , the worldwide number of HNWIs with portfolios of more than US$1 million rose by 8 percent in 2006, while the total assets of the wealthy grew by 11 percent.\n\nBolstered by buoyant equity and real estate markets, the ranks of the wealthy worldwide have swelled by significant percentages throughout the first decade of the twenty-first century, nearly doubling in less than a decade to 9.6 million from 5.2 million. The ongoing globalization of wealth means that regionally, while wealth continues to be unevenly concentrated in North America and Europe, in Latin America, Asia and the Middle East (regions long home to the world's greatest inequalities) the growth in wealth has been nothing short of breathtaking. According to the 2007 _WWR_ :\n\n\u2022 Of the 9.6 million HNWIs in 2006, 49 percent lived in North America;\n\n\u2022 Latin America boasts 14 percent of the world's wealth, while harboring just 5 percent of HNWIs;\n\n\u2022 Asia is home to roughly a quarter of the world's HNWIs, of which approximately one-fifth are mainland Chinese;\n\n\u2022 Mainland China is home to 320,000 HNWIs, who collectively possess US$1.59 trillion of assets and boast an average net worth of roughly US$5 million. Approximately 4,540 ultra-HNWIs in China control assets upwards of US$30 million;\n\n\u2022 The HNWIs of Hong Kong are collectively worth around US$460 billion, an astounding example of wealth concentration in a city of roughly seven million people;\n\n\u2022 Out of a population of nearly 1.3 billion, India's HNWIs, by contrast, control some US$350 billion (India's 150,000 expatriate millionaires contribute to the explosive economy);\n\n\u2022 Japan, the world's second-largest economy, is home to nearly half of Asia's millionaires, yet its longstanding lead as a haven of the wealthy is rapidly giving ground to China;\n\n\u2022 Singapore, with its venerable tradition of independence and legal and political stability\u2014and strong legal protections of financial privacy\u2014has become a haven of offshore and private banking for Asia's mega-affluent;\n\n\u2022 While the total net worth of Asia's HNWIs rose 10.5 percent to US$8.4 trillion in 2006, private bankers in Hong Kong and Singapore\u2014where the bulk of Asia's private banking industry is concentrated\u2014managed a total of US$600 billion;\n\n\u2022 Taiwan, Thailand, Indonesia and Malaysia are home to rapidly growing HNWI populations;\n\n\u2022 In the Middle East, high oil prices and widespread financial reforms\u2014particularly in the financial centers of Abu Dhabi and the UAE\u2014have contributed to a culture of wealth creation and preservation that contrasts with the free-spending seventies in that more indigenous wealth is being invested within the region.\n\n\u2022 Africa, once the world's economic basket case, is enjoying a wave of wealth creation unseen on the continent since the late Middle Ages. According to the IMF, since 2001 African economies have grown on average by 5 percent annually, a pace that the IMF predicts will quicken over the next five years to a robust 5.6 percent (compared to 4.8 percent elsewhere).\n\nThe seemingly irreversible globalization of wealth means that in order to not merely survive but thrive in the twenty-first century, the WM industry must go where the money is\u2014which it is, with a predictable vengeance. In 2007, the Bank of China embarked on a potentially profitable partnership with the Royal Bank of Scotland (RBS) to provide financial-planning expertise to the local HNWI population. The Bank of China now boasts only one hundred to two hundred private-banking clients, with average investible assets of US$655,000, an amount one private banker estimates represents at best a tiny fraction of their individual and household net worth.\n\nYet opportunities for growth remain considerable, if for no other reason than as the _Financial Times_ reports, less than half of all wealth managers today hold more than 40 percent of their clients' investible wealth. \"Over the next three years, this proportion is expected to increase dramatically so that 80 per cent of wealth managers hold 40 per cent of a client's wealth.\"\n\nWhat this worldwide wave of wealth means for the WM industry is that if it is successful in gaining ever-greater advisory influence over ever-greater proportions of HNWIs' assets, the already fierce competition among a comparatively small number of brand-name providers for high-quality service will only increase at every conceivable level, from breadth, range and quality of products to quality of human relationships to quality of technology.\n\nAmong the surging ranks of the ultra-affluent, levels of sophisticated service, guidance and advice hitherto reserved for the fortunate few will need to be provided to the comparatively many. Stand-alone private banks, independent wealth advisory services, and private banking organizations cradled within larger organizations will all be competing with the _ne plus ultra_ of mega-wealth preservation, the family office.\n\nAt the same time, as the wealth of the world is created and ideally preserved by an historically unprecedented number of households and individuals in absolute terms\u2014yet proportionally far fewer in percentage terms\u2014the level of client sophistication and expectations will only relentlessly rise over time. As access to vast flows of undiluted information is enhanced by ever-more nuanced and powerful technology, the degree of importance accorded by clients to the human factors (trust, empathy, comprehension of the \"whole picture\") will also climb exponentially.\n\nThis has only increased the tendency for competition in the industry to give rise to consolidation. Brokerages, banks, fund companies, and independent money advisors are falling over themselves seeking an ever-greater share of the HNWI pie by offering an ever-broadening array of advisory services, ranging from the mere management of money (clearly number one on a relative scale of importance yet largely viewed as a given by the average discerning HNWI) to the more discretionary and therefore differentiating factors of handling tax and estate issues, guiding philanthropic endeavors, arranging private equity participation deals, providing access to exotic instruments and hedge funds, and even planning exotic vacations down to the most minor detail or counseling wayward children on behavioral issues. The more they compete, the more likely they are to realize that by joining forces they can better serve their clients, and it's likely that HNWIs are well aware of this fact.\n\nAccording to _Forbes,_ the intensity of the competition gives rise, at some points, to a level of concierge service that may strike some as faintly laughable. Citigroup offers \"equestrian management\" services for clients who own horses. JPMorgan Chase counters with art advisory services. Still, according to Sara Hamilton, founder and chief executive of Family Office Exchange (as quoted in _Forbes_ ), a resource network based in Chicago, 30 percent of the work conducted by private bankers serving the HNWI population is \"devoted to accounting, filing taxes and keeping records in compliance, another 18 percent to 20 percent is devoted to investment management, and 12 percent to 15 percent to finance, tax and estate planning. Just 7 percent of the time is devoted to managing a wealthy client's 'lifestyle demands.'\"\n\nAnd while the establishment of family offices\u2014replete with their own staff of lawyers, accountants and investment planners employed full time by the office\u2014remains the _ne plus ultra_ of WM, such high-maintenance entities are only deemed worthwhile and affordable for ultra-HNWI households, families or affiliations of families controlling liquid assets in excess of US$100 million. For the ever-increasing number of families that fall below that threshold yet command significant wealth, multi-family offices that combine the resources while divvying up the costs are seen as a coming trend.\n\n#  **EASING THE STRESS OF INTERGENERATIONAL WEALTH TRANSFER**\n\nIt is widely agreed that the pending transfer of assets from the boomer generation to their children and grandchildren will collectively amount to the greatest intergenerational legacy the world has ever seen. A 2006 study conducted by Paul Schervish and John Havens of Boston College's Center on Wealth and Philanthropy estimates that between 2001 and 2055 some US$41 trillion will have changed hands in the United States alone.\n\nWhat is also becoming clear is that while offspring may be in a position to inherit more wealth than at any point in history, parental attitudes toward wealth transfer may also be evolving as parents sort through a range of often conflicting concerns: their desire to support their offspring financially, and their legitimate anxieties regarding the depletion of motivation and a potential for the diminution of their personal happiness as their wealth saps satisfaction.\n\nNick Tucker, a managing director in the Global Wealth Management division of Merrill Lynch in the U.K., has observed an increase in the number of clients who express anxiety and discomfort about their children receiving large sums of money\u2014particularly among self-made entrepreneurs. \"Often entrepreneurs and wealth creators want their children to be like them,\" Tucker recently reported to the _Financial Times._ 78 \"They want to give their children the same start they had and do feel that, if they give more than that, then they are depriving the children from doing what they have done.\"\n\nThe most promising and effective way for parents to resolve these issues, in Tucker's experience, is for them to more deeply engage their children in open discussions regarding family values with values-based retreats, promote their financial literacy, foster open dialogues around often sensitive issues through programs like boot camps, and in general boost transparency and candor between the generations regarding what is reasonable and equitable to expect from the impending transfer and what is not.\n\n#  **TRENDING TOWARD OPEN ARCHITECTURE AND UNBIASED ADVICE**\n\nThe frequency and severity of equity market fluctuations since the dot-com crash of 2001 has directed the HNWI population worldwide toward a preference for alternative investments and assets classes, ranging from investments in real estate and other commercial property, private equity and hedge funds, and structured and derivative products, as investors and advisors alike seek to protect portfolios from sudden and unsettling equity market corrections.\n\nSophisticated investors' increasing appetite for alternative investments has only fueled the shift toward \"open architecture,\" as the majority of WM firms recognize that it is nearly impossible to maintain all expertise inside the firm. The most client-centric approach will always be to maintain a continual search for \"best in class\" and \"best of breed\" fund managers and funds outside their own firm.\n\nSome private banks, including Merrill Lynch, make a point of not allocating private investor money to their own funds under discretionary mandates. Sophisticated investors are also increasingly aware that a shift from a transaction-based to a fee-based model (in which the advisor does not gain financially from selling a product) is the only guarantor of true independence and impartiality.\n\nThis trend has further fueled an ongoing shift toward providing a wider range of ancillary services and the holistic approach, because advisors who might once have eschewed the provision of tax, estate, retirement, pension, and mortgage and debt planning now compete with each other to offer services that mimic or emulate institutional collaborations between divisions of the whole firm, as has been the case with the shift toward a systematic drawing on expertise from across the firm at some of the larger institutions like Merrill Lynch.\n\nIn the end, the aspirational phenomenon identified by Capgemini as \"uptiering\"\u2014in which clients at virtually every rung on the net-worth spectrum aspire to the level of service provided to the tier above\u2014combined with the inevitable march of technology strongly suggests that the preferences, objectives and expectations currently enjoyed by the wealthy and the ultra-wealthy will, over time, trickle down to an ever-greater share of society. At the same time, the seemingly irreversible tendency for essential financial services to be commoditized means that the high-net-worth-centric share of the business will provide leadership for some time to come.\n\nSo we conclude, appropriately enough, with the final paragraph of the introduction to our first (1997) _WWR_ :\n\n_\"HNWIs not only demand a broader range of products, but also expect their financial advisors to be well-versed in these products and the global markets in which they are offered. Most of all, perhaps, they are seeking advisors with the wisdom to recommend investments that precisely meet their particular needs and objectives.\"_\n\nA similarly timeless and universal sentiment was expressed in 1911 by the twenty-six-year-old Charles Edward Merrill in _Leslie's Illustrated Weekly_ :\n\n_\"To select intelligently the investment securities best suited to the needs of each individual requires of the investment banker long experience, special training, and a thorough knowledge not only of the intrinsic merit of the securities that he recommends, but also of the investment situation of that particular individual...To make effective this personal service, the cooperation of the client is imperative.\"_\n**INDEX**\n\n##  **A**\n\nAbu Dhabi\n\nAcadian Asset Management\n\naccountability, in philanthropy\n\nactive performance-driven investors\n\nactivist-philanthropists\n\nAcumen Fund\n\nAdvantage Fund\n\nadvertising\n\nadvisor-client relationships. _See_ client-advisor relationships\n\nadvisor facing technology\n\nAfrica\n\nfrontier market\n\nGDP\n\nHNWIs in\n\nlocal HNWI investment\n\nwealth creation in\n\nAgarwal, Anil\n\naggregated account information\n\naging populations\n\nAIC\n\nAKD Securities\n\nAllred, Stacy\n\nalternative investments\n\nappeal of\n\nart\n\nincreased dependence on\n\nreal estate\n\nregional differences\n\nrisks\n\nstructured products\n\nAmaranth Advisors\n\nAmerican Institute of Certified Public Accountants (AICPA)\n\nAmerican Red Cross\n\nThe American Stock Exchange\n\nAngus, Patricia\n\nanti-trade legislation\n\nANZ Bank\n\nArnett, Peter\n\nArnold, Jeremy\n\nart\n\nArtist Pension Trust\n\nArtnet\n\nAsian Contagion (1997)\n\nAsia-Pacific\n\nemerging markets in\n\nentrepreneurs\n\nGDP\n\nHNWIs in\n\nlocal investment\n\npeer networks\n\nphilanthropy\n\nprivate banking\n\nstock market growth\n\n_Asia-Pacific Wealth Report (2007)_\n\naspirational risk\n\nasset acquisition\n\nasset allocation\n\ndefying\n\nmeeting clients' needs\n\nregional variations\n\n_see also_ Modern Portfolio Theory (MPT)\n\nasset-backed securities (ABS)\n\nasset classes\n\nasset gathering\n\nasset management\n\nAstor, Brooke\n\nAttali, Jacques\n\nauction houses\n\nAustralia, HNWIs in\n\nThe Australia and New Zealand Banking Group (ANZ Bank). _See_ ANZ Bank\n\n##  **B**\n\nBain Capital\n\nBangladesh\n\nBank of America (BofA)\n\nBank of China\n\nBank of New York (BoNY)\n\nBarrett, John\n\nbear notes\n\nBecchi, Scott\n\nbehavioral finance\n\nBello, Nancy\n\nBerlin Wall, collapse of\n\nBernard, Edward\n\nBernstein, Richard\n\nbespoke products\n\nBessemer Trust\n\n\"Beyond Markowitz\" (Chhabra)\n\nBill and Melinda Gates Foundation\n\nBlackRock\n\nBlackstone Group\n\nBodurtha, Stephen\n\nBombay Sensex\n\nBono\n\n_Born Rich_ (documentary)\n\nboutiques (financial services)\n\nBrandeis, Louis\n\nBranson, Richard\n\nBrazil\n\nBRIC nations\n\nBritain\n\nBuffett, Warren\n\ninfluence on Michael Lee-Chin\n\nBulgaria\n\nBurk, Darcie\n\nbusiness ownership\n\n##  **C**\n\nCAC 40 (France)\n\nCalamander Group\n\nCanada\n\nCanton, James\n\nCapgemini\n\nCapital Investment, Inc.\n\ncapitalism\n\nCapital Research and Management Co.\n\nCaribbean private banks\n\nCarlyle Group\n\nCarnegie, Andrew\n\nCarnegie Steel Company\n\nCarroll, Christopher\n\nCarroll, Jon\n\ncash\/deposits\n\ncatastrophe bonds\n\nCCC Alliance\n\nCemex (Mexico)\n\nCenter for Excellence in Higher Education\n\nCenter of Philanthropy, Indiana University\n\nCentral Registration Depository (CRD)\n\ncertified financial planners (CFPs)\n\ncertified public accountants\/personal financial specialists (CPAs\/ PFSs)\n\ncharitable gift annuities\n\nCharitable Lead Trusts\n\ncharitable reminder trusts\n\nchartered financial consultants (ChFCs)\n\nChavez, Hugo\n\nCheng Yu-Tang\n\nChhabra, Ashvin\n\n_Children of Paradise_ (Hausner)\n\nchildren of wealthy people\n\ncomfort with equities\n\nfinancial education\n\nglobal context of\n\nkeeping wealth a secret from\n\nand philanthropy\n\npreparing to inherit wealth\n\ntrusts\n\nChina\n\ncell phone users\n\nfactory to the world\n\nfixed exchange rate\n\nforeign trade and exports\n\nGDP\n\nHNWIs in\n\nand India\n\nand Latin America\n\nmanaged currency of\n\noffshore investments\n\npolicy changes\n\npolitical unrest in\n\ntrading partners\n\nwealth management in\n\nThe Children's Investment Fund\n\nChristie's International\n\n_Chronicle of Philanthropy_\n\nChurchill, Winston\n\nCitigroup\n\nCitibank\n\nCiti Private Bank\n\nclient-advisor relationships\n\ntechnology and\n\nwomen and\n\nClient Associates (CAs)\n\nclient facing technology\n\nCMA (Cash Management Account)\n\ncollars (proprietary products)\n\ncollateralized debt obligations (CDOs)\n\nCollier, Charles\n\nCommittee on the Global Financial System (CGFS)\n\ncommodities\n\ncommunism, collapse of\n\nconcierge services\n\nconfidentiality\n\n_see also_ privacy\n\nconsumer price index (CPI)\n\ncontent management systems\n\nCost of Living Extremely Well Index\n\nC\u00f4te d'Ivoire\n\nCreative Artists Agency\n\ncredit card fraud\n\ncredit crisis (2007-2008)\n\ncredit default swaps (CDSs)\n\ncrony capitalism\n\nCruddas, Peter\n\ncultural globalization\n\nCuoni, Jean Pierre\n\nCVRD (Brazil)\n\n##  **D**\n\nDAX (Germany)\n\nday traders\n\ndebt, as a financial tool\n\ndecoupling\n\ndecumulation\n\nDeng Xiaoping\n\nderegulation\n\nderivatives\n\ndeveloping countries\n\n_see also_ emerging markets\n\nDigiTAG device\n\ndisaster-related investments\n\ndisclosure\n\ndiscount brokers\n\ndisintermediation\n\ndistressed hedge funds\n\ndiversification\n\nchallenges of\n\nDLF (India)\n\ndonor-advised funds\n\ndot-com crash of\n\nDow Jones Indexes\n\nChina\n\nWorld Stock Index\n\nDubai\n\nDunn, Danny\n\n##  **E**\n\nE.A. Pierce & Co.\n\nEaton Estate\n\neBay\n\neconomic slowdown (2004-2005)\n\ne-IPOs\n\nEllison, Lawrence\n\nemerging markets\n\nbecoming mainstream\n\ncoining the term\n\ndeclining returns\n\nGDP growth in\n\nhedge funds\n\nHNWI\/ultra-HNWI investment in\n\nmutual funds\n\nEmerging Markets Fund\/Growth Fund\n\nEmerging Portfolio Fund Research\n\nemotionality of wealth\n\nEmployee Stock Option Plans (ESOPs)\n\nenergy sector hedge funds\n\nEnron\n\nentrepreneurs\n\nin emerging markets\n\ninvested in own businesses\n\nequities\n\nEquity Office Properties Trust\n\nEstonia\n\nE*TRADE Securities\n\nEurope\n\naging population\/low birth rate\n\nEastern\n\nGDP\n\nand globalization\n\nHNWIs in\n\nwealthy lifestyles in\n\nEuropean Central Bank\n\nEuropean Union\n\nexchange traded funds (ETFs)\n\n_Extreme Future, The_ (Canton)\n\n##  **F**\n\nfamily balance sheets\n\nfamily education\n\nfamily foundations\n\nFamily Office Exchange\n\nFamily Office Metrics\n\nfamily offices\n\nprimary functions of\n\nFDI (Foreign Direct Investment) in India\n\nFederal Reserve Board\n\nsurvey of Consumer Finance\n\nFederal Securities Agency\n\nFidelity Investments\n\nfinancial advisors\n\naccreditation and designations\n\nin Asia\n\nchanging role of\n\nchildren of HNWIs\n\nexpectations of\n\nfee structure\n\nand female HNWIs\n\nfinding\n\nholistic approach to services\n\nmeeting client needs\n\nperformance benchmarks\n\nphilosophy and interests\n\nrecruiting and training\n\nreferrals to\n\nrelationships with clients\n\nremuneration\n\nservices offered\n\ntechnology for\n\nand wealth transfers\n\nfinancial boot camps\n\nfinancial education\n\nboot camps\n\nand security\n\nfinancial globalization\n\nFinancial Industry Regulation Authority (FINRA)\n\nfinancial literacy\n\nFinancial Planning Association (FPA)\n\nFinancial Services Authority (FSA)\n\nfinancial services industry\n\nadvertising\n\nand art purchases\n\nautomation\n\nboutiques\n\nbusiness ownership services\n\nclient satisfaction\n\ncommission\n\ncompetition\n\nconfidentiality\n\nevolution of\n\nmergers and acquisitions\n\nregulation\n\nservices for HNWIs and ultra-HNWIs\n\nand technology\n\ntraditional approach\n\ntransitioning from transaction-based to wealth management\n\nfinancial sponsors\n\n_see also_ private equity (PE)\n\nFink, Larry\n\nFinlay, Francis\n\nFirst Republic Bank\n\nFirst Reserve Corp\n\nfixed income\n\nForbes 400\n\nforeign currency\n\nforeign exchange (\"forex\") investments\n\nfor-profit charity\n\n_see also_ venture philanthropy\n\nForrester Research\n\nFortress Investment Group\n\nFox, Vincent\n\nFrance\n\nFranklin Resources\n\nFranklin Templeton Investments\n\nFreeman, Doug\n\nfrontier markets\n\nfunds of funds\n\nhedge funds\n\nprivate equity\n\n##  **G**\n\nGallagher, Tommy\n\nGant, Chris\n\ngap of thought\n\nGates, Bill\n\nGazprom (Russia)\n\ngender, and wealth management\n\nGen X-ers\n\nGermany\n\ngiving while living\n\nGlass-Steagall Act\n\nglobal capitalism _see also_ globalization\n\nglobal economies, wealth creation in\n\nglobalization\n\nchallenges for United States\n\ndetractors of\n\nfacets of\n\nfuture of\n\nimpact on HNWIs\n\nand investment opportunities\n\nand local investing\n\nand philanthropy\n\nand private equity\n\nproblems with\n\nand reporting\n\nrise in international investing\n\nand technology\n\nand ultra-HNWIs\n\nand wealth management\n\nGoldman Sachs\n\nGoogle.org\n\nGrameen Bank\n\nGrameen Foundation\n\ngross domestic product (GDP)\n\nGrosvenor, Gerald Cavendishh Duke of Westminster\n\nGrosvenor, Hugh Lupust Duke of Westminster\n\nGrosvenor, Ralph\n\nGrupo Modelo (Mexico)\n\n##  **H**\n\nHAM (hold all mail)\n\nHamilton Bradshaw\n\nHamilton, Sara\n\nHartnett, Michael\n\nHausner, Lee\n\nHavens, John\n\nHawkamah\n\nhealth care expenses\n\nHedge Fund Industry Asset Flow Report (2007)\n\nhedge funds\n\ndistressed\n\nemerging markets\n\nenergy sector\n\nfunds of funds\n\ngrowth of\n\ninstitutional investors\n\nand liquidity\n\nregulations\n\nHenderson Land Development\n\nhigh-net-worth individuals (HNWIs)\n\nbreakdown of regional assets\n\nchildren living abroad\n\nexpectations for service\n\ngender\n\nand globalization\n\ngrowth of wealth\n\nin insecure regions\n\ninternational investing\n\nlifestyles of\n\nlocal investments\n\nnext generation of\n\nnumber of\n\npreferred asset allocations\n\nprime driver of wealth\n\nregional differences in wealth\n\nrequirement for financial advisors\n\nretirement\n\nsources of wealth\n\ntalking to children about wealth\n\nHNWIs. _See_ high-net-worth individuals (HNWIs)\n\nHo Chi Min Stock Index\n\nHohn, Chris\n\nHolistic Wealth Management (HWM)\n\nclear reporting\n\nfinding an advisor\n\norigins of\n\nand specialist teams\n\nusing debt to build wealth\n\nHong Kong\n\nHon Hai (Taiwan)\n\nHope Group\n\nHurricane Katrina\n\nhurricane risk\n\nHutchison Whampoa Ltd.\n\nHyundai (South Korea)\n\n##  **I**\n\nIFC Emerging Markets Index\n\nIFF Advisors\n\nImara African Opportunities Fund\n\nImara Asset Management\n\nincome earners\n\nincome gap\n\nincome taxes\n\nIndia\n\nADRs (American Depositary Receipts)\n\nand China\n\nforeign direct investment policies\n\nglobalization\n\nGDP\n\nHNWIs in\n\nlocal investment\n\nNon-Resident Indians (NRIs)\n\noutsourcing to\n\nIndonesia\n\nindustrial globalization\n\ninformational globalization\n\ninformation gap\n\ninformation technology (IT)\n\n_see also_ technology\n\ninherited wealth\n\ninheritor accounts, retaining\n\nInitial Public Offerings (IPOs)\n\nInstitute for Private Investors\n\ninstitutional investors and hedge funds\n\n\"instividuals\"\n\nintergenerational wealth transfer\n\nanxieties about\n\nbaby boomers'\n\nhow wealth was originally made\n\nand peer networking\n\nregional diversity of\n\nvalues-based plan\n\nwealth transfer plan\n\nwhen to transfer assets\n\nInternational Aviation Management Group (IAMG)\n\nInternational Development Enterprises\n\nInternational Development Law Institute\n\nInternational Finance Corporation (IFC)\n\ninternational investing\n\nInternational Monetary Fund (IMF)\n\nInternet\n\nbanking services\n\ninformation available on\n\nuse by clients\n\nInternet bubble (1990s)\n\nInvestec\n\ninvesting\n\nfundamentals of\n\nLee-Chin's rules of\n\ninvestments of passion\n\ninvestor-centric innovation\n\ninvestor sophistication\n\nand philanthropy\n\nIsle of Man\n\n##  **J**\n\nJamaica\n\nJanney, Stuart\n\nJapan\n\nHNWIs in\n\nmarkets\n\nJianping Mei\n\nJohnson, Charles Bartlett\n\nJohnson, Gregory\n\nJohnson, Jamie\n\nJolie, Angelina\n\nJPMorganChase\n\n##  **K**\n\nKennedy, James A.C.\n\nKenya\n\nKeynes, John Maynard\n\nKhezri, Bijan\n\nKlein, Karen\n\nKohlberg Kravis Roberts (KKR)\n\nKuwait\n\n##  **L**\n\nLam, Joseph\n\nLatin America\n\nentrepreneurs\n\nGDP\n\nhedge funds\n\nHNWIs in\n\nlocal investment\n\ntrading with China\n\nLatvia\n\nLavayssi\u00e8re, Bertrand\n\nLavin, Richard\n\nLee, Albert\n\nLee-Chin, Michael\n\nrules of investing\n\nWarren Buffett's influence on\n\nLee Shau-Kee\n\nlegacy systems\n\nLe Grosveneur, Gilbert\n\nleveraging investments\n\nliabilities\n\nLiberty Fund\n\nlife insurance\n\nLi Ka-Shing\n\nLi Ka-Shing Foundation\n\nlinear programming\n\nliquidity\n\nliquidity risk\n\nLithuania\n\nLiu Youping\n\nlocal currencies\n\nLondon FTSE\n\nLuxembourg\n\nluxury goods\n\nLynch, Edward\n\n##  **M**\n\nMaastricht Treaty\n\nMackenzie Group\n\nMalaysia\n\nMarcus, Bernard\n\nmarket capitalization\n\nMarket Index Target-Term Securities (MITTs)\n\nmarket risk\n\nMarkowitz, Harry\n\nMarrelli, James\n\n\"mass-affluent\" investors\n\nmature economies\n\nMayer, Martin\n\nMcCann, Bob\n\nMcLaughlin, Patricia\n\nMedallion Fund\n\nMei Moses All Art Index\n\nMellon Financial\n\nmergers and acquisitions\n\nMerrill, Charles Edward\n\nMerrill Lynch\n\nacquisition of First Republic Bank\n\nin Asia\n\nFinancial Boot Camp\n\n\"The Financial Foundation\"\n\nHolistic Wealth Management (HWM)\n\nin India\n\ninnovations by\n\nin Latin America\n\nPrivate Banking and Investment Group (PBIG)\n\nPrivate Wealth Advisors (PWAs)\n\nservices for HNWIs and ultra-HNWIs\n\nand technology\n\nTrusted Global Advisor (TGA)\n\nwealth transfer workshops\n\nWealth Management Work Station\n\nMet Circle\n\nMexico\n\nmortgage market\n\npeso devaluation (1994)\n\nMichael Lee-Chin Crystal, Royal Ontario Museum\n\nmicro-lending programs\n\nMiddle East\n\nemerging markets in\n\nGDP\n\nHNWIs in\n\nand investing in developed markets\n\nlocal investing in\n\noil reserves\n\nmiddleware\n\nThe Milken Institute\n\nMiller, Merton\n\nmission statements, for families\n\nMitchell, Marcus\n\nMobius, Mark\n\nModern Portfolio Theory (MPT)\n\neffects on wealth management\n\nmoving beyond\n\nMoeder, Alyssa\n\nMorgan, J.P.\n\nMorgan Stanley\n\nEmerging Markets Index\n\nmortgages\n\nmortgage meltdown (2007-2008)\n\nMoses, Michael\n\nMSCI AC Asia-Pacific Index\n\nMSCI Emerging Markets Index\n\nmulti-family offices _see also_ family offices\n\nmutual funds\n\n##  **N**\n\nNamibia\n\nNathu La Pass\n\nNational Association for Personal Financial Advisors (NAPFA)\n\nnatural gas\n\nNiu Gensheng\n\nnon-profits, and scandals\n\nNon-Resident Indians (NRIs)\n\nNorth America\n\nforeign investment in\n\nHNWIs in\n\nand international investing\n\nsources of wealth\n\nNorthern Trust Corp.\n\nNRI Portfolio Investment Scheme\n\n##  **O**\n\nOASIS (Organization for the Advancement of Structured Information Standards)\n\noffspring of wealthy people. _See_ children of wealthy people\n\noil\n\nOmidyar Network\n\nOmidyar, Pierre\n\nO'Neal, Stan\n\nonline private banking\n\nonline security threats\n\nonline trading\n\nopen architecture\n\nOprah Winfrey Leadership Academy\n\noutsourcing technology functions\n\nOvitz, Michael\n\n##  **P**\n\nPakistan\n\nPaulson, Henry\n\npeer networking\n\npersonal risk\n\nphilanthropy\n\naccountability\n\ncelebrities\n\ndecline in\n\nencouraging in employees\n\nand globalization\n\ninfusion of business ideas into\n\nmeasurable results\n\nonline tools\n\nregional differences\n\nrise in\n\ntax implications\n\nPhiladelphia Housing Index\n\nPhilippines\n\nPhipps, Henry\n\nPlaNet Finance\n\nPlano Real\n\npolitical globalization\n\nPolito, Paula\n\npooled income funds\n\npopulism\n\nportfolios, ideal\n\nPortland Holdings\n\nPowerShares Listed Private Equity Portfolio\n\nprenuptial agreements\n\nprivacy\n\nprivate banks\n\nAsia\n\nhigh-touch versus technology\n\nmergers and acquisitions\n\nMerrill Lynch\n\nonline\n\nSwitzerland\n\ntraditional\n\nPrivate Equity Intelligence\n\nprivate equity (PE)\n\nfunds of funds\n\nreturns on\n\nprivate investment corporations (PICs)\n\nPrivate Wealth Advisors (PWAs)\n\nproducts\n\nnew\n\npurchasing or selling\n\nprofessional organizations\n\nproprietary technology\n\nprotected equity-linked notes (PENs)\n\nprovider facing technology\n\npure-play administrative specialists\n\n##  **Q**\n\nQatar\n\n##  **R**\n\nRatcliffe, David\n\nreal estate\n\nRegan, Donald\n\nREITs (real estate investment trusts)\n\nRenaissance Technologies (Rentec)\n\nreporting\n\nfour pillars of\n\nretirement\n\nreverse inquiries\n\nThe Right to Play\n\nrisk\n\nand alternative investments\n\nand diversification\n\nglobal\n\nand ideal portfolio\n\nliquidity\n\nprivate equity and\n\nand structured products\n\ntolerance for\n\ntypical factors\n\nunexpectedly high\/low returns\n\nrisk allocation\n\nrisk management _see also_ asset allocation\n\nrisk premium\n\nRockefeller, John D., Sr.\n\nRomania\n\nRothenberg, James\n\nRoyal Bank of Scotland (RBS)\n\nRoyal Dutch Shell\n\nRoyal Ontario Museum\n\nMichael Lee-Chin Crystal\n\nRTS Index (Russia)\n\nRubin, Robert\n\nRussell, George\n\nRussia\n\ncommodities\n\nGDP\n\nsovereign debt default\n\n##  **S**\n\nSaadie, Michael\n\nSachs, Jeff\n\nSamsung Electronics (South Korea)\n\nS\u00e3o Paulo Bovespa\n\nSarbanes-Oxley (SOX)\n\nSasol (South Africa)\n\nSaudi Arabia\n\nscandals, corporate\n\nScaturro, Peter\n\nSchengen Agreement\n\nSchervish, Paul\n\nSchlosstein, Ralph\n\nSchueneman, Diane\n\nSchultz, Howard\n\nSecurities and Exchange Commission (SEC)\n\nsecurity (technology)\n\nSeptember\n\nService-Oriented Architecture\n\nShanhai\/Shenzhen market\n\nSharpe ratios\n\nSharpe, William\n\nShelterwood Financial Services\n\nShenzhen Pengnian Hotels\n\nSimons Foundation\n\nSimons, James\n\nSimonyi, Charles\n\nSingapore\n\nfinancial advisors in\n\nHNWIs in\n\noffshore banking in\n\nSingapore Wealth Management Institute\n\nSlattery\u00f3mhnal\n\nSlim Helu, Carlos\n\nSlovakia\n\nSlovenia\n\nsocial entrepreneurs\n\nsocialism\n\nSonnenfeldt, Michael\n\nSontag, Daniel\n\nSotheby's Holdings\n\nSouth Korea\n\nSovereign Wealth Products (SWPs)\n\nspecialist teams\n\nspecial purpose acquisition companies (SPACs)\n\nspending\n\nStandard & Poor's 500 Index\n\nStandard & Poor's\/IFC Frontier Index\n\nSteffens, John L. \"Launny\"\n\nstock exchanges, regional\n\nstocks and bonds\n\nStony Brook\n\nstructured products\n\nbear notes\n\nindividual versus institutional clients\n\nsubprime mortgage meltdown (2007- 2008)\n\nSuer, Oral\n\nSullivan, Michael\n\nSweeney, Tom\n\nSwitzerland\n\nSynergos\n\n##  **T**\n\nTaiwan\n\nTan, Victor\n\ntaxes\n\nincome\n\nand philanthropy\n\nwithholding tax (Switzerland)\n\ntax havens\n\ntechnology\n\nbenefits for clients\n\nclient familiarity with\n\nclient satisfaction\n\nand competition\n\ncost of investment\n\nand culture\n\nand globalization\n\noutsourcing\n\nand philanthropy\n\nrevenue growth\n\nsecurity\n\nand transparency\n\nwealth made in\n\nand workflow\n\nTempleton Emerging Markets Fund\n\nTempleton, Sir John\n\nThailand\n\nThiel, John\n\nThird World Equity Fund\n\nThreshold Group\n\nTiffany, Paul\n\nTIGER 21\n\ntime horizon\n\ntransparency\n\nTrappist monasteries\n\nTrendSight Group\n\nTriple A strategic approach to wealth management\n\nTrusted Global Advisor (TGA)\n\ntrusts\n\nTucker, Nick\n\nTXU Corp.\n\nTyco\n\n##  **U**\n\nUkraine\n\nUllens, Baron Guy\n\nultra-high-net-worth individuals (ultra-HNWIs)\n\nand asset allocation\n\ndemands of\n\neducation of offspring\n\nand emerging markets\n\nand globalization\n\nand Holistic Wealth Management (HWM)\n\nand new products\n\nultra-HNWIs. _See_ ultra-high-net-worth individuals (ultra-HNWIs)\n\nUnited Arab Emirates\n\nUnited States\n\nfalling dollar\n\nGDP\n\nand globalization\n\nincome taxes\n\nphilanthropy\n\nregulations\n\nUnitus\n\nuptiering\n\nU.S. Private Equity Performance Index\n\nU.S. Trust\n\n##  **V**\n\nvalues retreats\n\nvan Agtmael, Antoine\n\nVanderbilt, Cornelius\n\nVanderbilt, William\n\nVardy, Nicholas\n\nventure philanthropy\n\nVietnam\n\nVietnam Stock Exchange\n\nVirtual Service Networks\n\nVolatility Index (VIX)\n\n##  **W**\n\nWaldron, Kevin\n\nwealth\n\nearned\n\ninherited\n\nregional differences\n\nspending\n\nWealth Allocation Framework (WAF)\n\nwealth management\n\neight convictions about\n\nfour pillars of\n\nfuture of\n\nand globalization\n\nholistic approach to\n\nimpact of technology on\n\nService-Oriented Architecture\n\nwealth management firms\n\n_see also_ financial services industry\n\nwealth management framework\n\nwealthy families\n\ndysfunction\n\nemotionality of financial decisions\n\nmission statements\n\ntracking net worth\n\nWestminster, Duke of\n\nwind risk\n\nWinfrey, Oprah\n\nwithholding tax (Switzerland)\n\nwomen, and wealth\n\nWorldCom\n\nWorld War 1\n\n_World Wealth Report_\n\n_1997_\n\n_2000_\n\n_2006_\n\n_2007_\n\nWriston, Walter\n\n##  **Y**\n\nYang Lan\n\nYunus, Muhammad\n\nYu Pengnian\n\n##  **Z**\n\nZell, Sam\n\nHNWIs are defined as individuals with US$1 million in investible assets, excluding their primary residence.\n\nUltra-HNWIs are defined as individuals with US$30 million in investible assets, excluding their primary residence.\n\nThe Committee on the Global Financial System (CGFS) provided the following definition of structured finance in its January 2005 report, _The Role of Ratings in Structured Finance: Issues and Implications_ : \"Structured finance instruments can be defined through three key characteristics: (1) pooling of assets (either cash-based or synthetically created); (2) tranching of liabilities that are backed by the asset pool (this property differentiates structured finance from traditional \"pass-through\" securitisations); (3) de-linking of the credit risk of the collateral asset pool from the credit risk of the originator, usually through use of a finite-lived, standalone special purpose vehicle (SPV).\"\n\nAmong the most commonly formulated structured finance instruments:\n\n\u2022 _Asset-backed securities_ (ABS) are bonds or notes based on pools of assets, or collateralized by the cash flows from a specified pool of underlying assets.\n\n\u2022 _Mortgage-backed securities_ (MBS) are asset-backed securities whose cash flows are backed by the principal and interest payments of a set of mortgage loans.\n\n\u2022 _Collateralized debt obligations_ (CDOs) consolidate a group of fixed income assets such as high-yield debt or asset-backed securities into a pool, which is then divided into various tranches.\n\n\u2022 _Collateralized mortgage obligations_ (CMOs) are CDOs backed primarily by mortgages.\n\n\u2022 _Collateralized bond obligations_ (CBOs) are CDOs backed primarily by corporate bonds.\n\n\u2022 _Collateralized loan obligations_ (CLOs) are CDOs backed primarily by leveraged bank loans.\n\n\u2022 _Credit derivatives_ are contracts to transfer the risk of the total return on a credit asset falling below an agreed level, without transfer of the underlying asset.\n\n1997 _World Wealth Report_ ; Capgemini\/Merrill Lynch.\n\nInterview with the authors.\n\nInterview with the authors.\n\nMartin Mayer, _Wall Street: Men and Money_ (New York: Harper, 1955).\n\n\"Don Regan: For Rhyme and Reason,\" William R. Doerner, _Time_ , January 21, 1985.\n\nSince renamed the Securities Industry and Financial Markets Association (SIFMA).\n\n\"Singapore Makes a Pitch to Draw the Wealthy,\" Wayne Arnold, _The New York Times_ , April 26, 2007.\n\nAll financial advisors and planners in the United States are obliged to register with either the Securities and Exchange Commission or the securities agency in the state of their primary place of business. Depending upon jurisdiction, advisors may be designated certified financial planners (CFPs), chartered financial consultant (ChFCc), NAPFA-registered financial advisors, or certified public accountants\/personal financial specialists (CPAs\/PFSs). The Central Registration Depository (CRD) maintains a database that contains information on brokers, their representatives and the firms they work for. Background information is also available from the Financial Industry Regulation Authority (FINRA) and\/or state securities regulators.\n\n\"Income Inequality Hits Record Levels,\" Arloc Sherman, Center on Budget and Policy Priorities, December 14, 2007.\n\nCarroll, Christopher \"Why Do the Rich Save So Much?\"; published in _Does Atlas Shrug: The Economic Consequences of Rating the Rich_. Harvard University Press, 2000; ed. Joel B. Slemrod.\n\nAntoine van Agtmael, _The Emerging Markets Century: How a New Breed of World-Class Companies Is Overtaking the World_ (Free Press, 2007).\n\nIbid, 31.\n\nAlthough subject to change in accordance with share price fluctuations, Merrill Lynch in 2008 owned just under 50 percent or less than controlling interest in BlackRock.\n\nProponents of \"decoupling\" observed the credit crunch of 2007-2008 in the hopes of confirmation of their theory, but as of press time the global jury was still out.\n\nInterview with the authors.\n\nVan Agtmael interview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\n\"India's Rising Growth Potential,\" Global Economics Paper #152, economic research from the GS Institutional Portal; <http:\/\/portal.gs.com>.\n\nInterview with the authors.\n\nMerrill Lynch and Capgemini, 2007 _World Wealth Report._\n\nInterview with the authors.\n\nThomas Landon, Jr.,\"Pack Mentality Among Hedge Funds Fuels Market Volatilty,\" _New York Times_ , August 13, 2007.\n\nInterview with the authors.\n\n\"How Mass Affluent Investors Use the Web,\" Tom Watson with Bill Doyle, Forrester Research, August 31, 2004.\n\n\"Full-Service Brokerages: Stop Neglecting the Net,\" _Outdated Web Sites Make You Increasingly Vulnerable to Direct Firms_ \"Affluent Investors Series\" Forrester Research, September 13, 2006.\n\nBill Gates, _Business @ the Speed of Thought: Succeeding in the Digital Economy_ (Time Warner, 2000), 72.\n\nIbid., 79.\n\nIbid., 80.\n\nInterview with the authors.\n\n\"Service Oriented Architecture (SOA) and Specialized Messaging Patterns,\" Technical White Paper, Duane Nickul, ed. www.oasis.org.\n\nSECOR Consulting Submission to the Canadian Federal Department of Finance on Shareholder Value Created and Destroyed in Financial Services Mergers: Dec 22, 2003.\n\n\"Recent Changes in U.S. Family Finances,\" Brian K. Bucks, Author B. Kennickell, and Kevin B. Moore, _Federal Reserve Bulletin_ , Vol. 92 (Fall 2006), pp. A1-A38.\n\nStory derived from _Against the Gods: The Remarkable Story of Risk_ by Peter L. Bernstein ( Wiley, 1996), 250.\n\nMarkowitz's definition of costs as a function of volatility has since generated some controversy, as critics have argued that a more accurate measure of the costs related to managing a portfolio would be some quantifiable level of risk.\n\nHarry Markowitz, \"Portfolio Selection,\" _Journal of Finance_ (March 1952).\n\nThe Brandes Institute, a division of Brandes Investment Partners, \"The Past, The Future, and Modern Portfolio Theory,\" 1998-2004, www.brandes.com.\n\nAshvin B. Chhabra, \"Beyond Markowitz,\" _The Journal of Wealth Management_ 7, no. 4 (Spring 2005).\n\nKhezri, Bijan, \"The New Art of Art Finance,\" _Wall Street Journal,_ September 19, 2007, D10.\n\nInterview with the authors.\n\nElizabeth Wine, \"Revamping Retirement,\" _On Wall Street_ , September 2007, <http:\/\/www.onwallstreet.com>.\n\nJames R. Barth, Tong Li, Triphon Phumiwasana and Glenn Yago, \"Hedge Funds: Risks and Returns in Global Capital Markets,\" December 2006, The Milken Institute.\n\nInstitutional Investor News and HedgeFund.net, \"2008 Hedge Fund Asset Flows & Trends Report,\" November 29, 2007.\n\nUnless registered, individuals must be deemed qualified purchasers to invest in a fund of funds.\n\nIbid., James R. Barth, Tong Li, Triphon Phumiwasana and Glenn Yago, \"Hedge Funds: Risks and Returns in Global Capital Markets,\" December 2006, The Milken Institute.\n\n_Hedgeweek_ , January 16, 2008.\n\nwww.bessamer.com\n\nEuropean Private Equity and Venture Capital Association (EVCA), www.evca.com.\n\n\"How Psychology Is Private Equity's Biggest Risk in 2008\" _Financial News\/Deal Journal,_ December 17, 2007.\n\nInterview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\nInterview with the authors.\n\nCharles Batchelor, \"Family Dynasties,\" _Financial Times,_ July 8, 2007.\n\nInterview with the authors.\n\nMerrill Lynch Private Banking and Investment Group, \"Building Your Legacy: From One Generation to the Next, Families Can Pass On Wealth\u2014and Values\u2014as a Means of Inspiration\"; The White Papers: Quarterly Intelligence for Informed Investors, Winter 2006.\n\nInterview with the authors.\n\nEmily Vencat and Ginanne Brownell, \"Ah, the Secluded Life; How the Superrich Are Finding New Ways to Set Themselves Apart,\" _Newsweek_. October 12, 2007.\n\nInterview with the authors.\n\nMark Cobley and Mike Foster, \"Philanthropy Becomes an Investment Class,\" _Financial News,_ July 16, 2007.\n\nIbid.\n\nJohn Hechinger, \"Big-Money Donors Move to Curb Colleges' Discretion to Spend Gifts,\" _Wall Street Journal_ , September 18, 2007.\n\nNorthern Trust, \"Wealth In America 2007,\" January 2007; Capgemini Analysis.\n\nKate Linebaugh and Jane Spencer, \"The Revolution of Chairman Li\u2014China's Richest Man Leads Others to Give, Bucking Nation's Taboos,\" _Wall Street Journal_ , November 2, 2007.\n\nJohn Maynard Keynes, \"Economic Possibilities for Our Grandchildren,\" _Essays in Persuasion_ (New York: Harcourt, Brace & Co., 1932), 358-373.\n\n\"Creating a Moral Biography of Wealth: A Conversation with Paul G. Schervish,\" The Whitepapers: Quarterly Intelligence for Informed Investors, Winter 2006, a publication of Merrill Lynch's Private Bank and Investment Group (PBIG).\n\nInterview with the authors.\n\nJason Leow, \"China's Rich Get Catering\u2014Private Bankers Tap a Rising Wealth Pool,\" _Wall Street Journal_ , June 5, 2007.\n\nJane Croft, \"Private Banks Set for Boom Period,\" _Financial Times,_ June 27, 2007.\n\nLiz Moyer, \"Where the Rich Bank Their Money,\" _Forbes_ , October 11, 2007, <http:\/\/www.forbes.com>.\n\nSharlene Goff, \"Parents Opting for Tough Love with the Family Fortune,\" _Financial Times,_ October 27, 2007.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## Praise\n\n_'You're Not Proper is a real insight into communities more often talked about than listened to. But its not just an important book - it's full of heart and a cracking good read as well. Highly recommen_ ded!' **Melvin Burgess**\n\n_'Contemporary and hard-hitting! High on impact and highly engaging, it truly is a story of our age'_ **Jake Hope** , critic, librarian and coordinator of the Lancashire Children's Book of the Year Award\n\n# **YOU'RE NOT PROPER**\n\n_Tariq Mehmood_\n\n## **Kiran**\n\nI live in Boarhead West. And on the other side, in Boarhead East, live the scarfies, turbans and beards. In between us, there's a great big graveyard. There used to be a textile mill where the graveyard ended. My granddad worked there. The mill's gone now. The graveyard took it over. It's where the Muslims are buried. In the middle of the graveyard there is a roofless church, with a huge weeping willow tree near it. That's where the Willow Tree Mob, the WTM, hang out.\n\nIn this Northern English town of mine, especially during the long summer days like now, when the sun shone well into the night, I was happy. I belonged. I had my gang, and nobody bothered me. But then I woke up and couldn't work out who I was.\n\nIt all started a few weeks after my fourteenth birthday. I was hiding from Mum in my bedroom, listening to Lady Gaga on my headphones. I had a poster of her on my wall, in snakeskin, with high-heeled, snakeskin shoes. A great, big, green snake with black stripes, almost as thick as her waist, crossed her legs and went under her back. An orange snake with black patches curled around her neck and slithered across the green one towards her waist, looping around her neck. The poster covered half the wall opposite my bed. It was huge. It was awesome. It was perfect. It stopped my thoughts from flying out of my bedroom and banging on Donna's head and asking her, 'What did I do to you?'\n\nI turned away from Lady Gaga and pushed my head into my pillow. I had washed my face so many times, trying to clean off the cross that Donna had drawn on my forehead; I could smell the soap from my pillow.\n\nI turned over onto my back. The crack in the ceiling that ran from one end of my bedroom to the other jeered down at me. I heard Mum come out of her room and walk down the stairs. She called out to me when she got to the bottom. I didn't answer her. I could still see an image of Jake, standing under the willow tree, watching, just watching as Donna drew the cross on my forehead. The words she hissed rang in my ears, 'Now you are a Christian.' It hurt when she started but the pain stopped when the other girls laughed. I begged them to stop. But they just laughed and laughed. I wanted to scream but instead I laughed as well.\n\n'Leave me alone,' I said, tossing over again, hoping to chase the memories out of my head. 'You'll feel better in the morning, girl,' I assured myself.\n\nLittle did I know how wrong I would be.\n\nIt was nearly midday when I got my head out from under my quilt. Mum had knocked on the door a few times, and I had grunted in reply and gone back to sleep. Yesterday felt like a bad dream. I had forgotten to draw the curtains last night. The sun lit up my room. A ray of light shone up on Lady Gaga.\n\nI rubbed my forehead. I could still feel Donna's pen going up and down and across. The memory of yesterday came flooding back to me. I had wanted to get away from the heavy silence of _it_. Mum and Dad had stopped talking to each other and I had gone to see my gang. The quickest way to get to the old church was through the broken railings of the Muslim side of the graveyard. I could have walked over the railings but didn't. I took a running jump instead, stumbled and fell. As I was getting off the ground, I heard them laugh.\n\nShamshad Ali, a big, busty scarfie, who goes to the same school as me and who hates my guts, was pointing at me. Laila Khan was sitting next to her on a bench not far from where I had fallen and Aisha Sadiq, wearing a black tracksuit top and bottom was doing stretches touching the ground and standing up again.\n\nMy backside was up in the air. I stood up, brushed my skirt and wanted to die. A sharp pain ran down my right leg. I tried not to limp but couldn't help it.\n\nAfter a bit of jogging on the spot, Aisha said something to Shamshad and then ran towards me, she shouldered me as she ran past and out of the graveyard.\n\nTo get to the old church I had to go on the path that ran right past them. As I got closer, Shamshad stood up, blocked my way, and said, 'What's with you here?'\n\nI looked at her ugly face and wanted to say, 'Does it belong to you?' Instead, I smiled sheepishly, and said, 'Nothing!'\n\nI slowed right down thinking, 'How could you just say, _nothing_? Why didn't you tell her, \"What do you think? You think you own everything! You think you're better because you're Muslim?\"'\n\n' _Kali Gori_ ,' Shamshad said, sizing me up.\n\nI stopped.\n\n'You're white inside aren't you?' Shamshad tutted pointing to her arm, 'but brown like me.'\n\nI kept quiet.\n\nShamshad came towards me saying, 'Oreo.'\n\n'I like Oreos,' I knew I shouldn't have said this even as the words came out of my mouth.\n\nShamshad looked at Laila and the two of them laughed. I didn't find anything funny but I laughed as well. Just then, I caught a glimpse of Donna and the gang. I waved at them. Shamshad stepped aside and I ran past her.\n\nNo one in the gang greeted me when I got to them. Jake stood on his own, under the flowing branches of the tree, kicking the trunk gently. Megan and Chloe stood either side of Donna. Megan had her arms folded across her chest and Chloe played with a twirl of her blond hair. Donna glared at me like I was dirt.\n\n'Alright, Don?' I asked.\n\nMegan scratched her back and gaped at Donna.\n\nDonna ignored me and nodded for Megan and Chloe to follow her and walked towards a hedge close to me. I went up to Jake and asked, 'What's with everyone?'\n\nHe grabbed hold of a small branch in his fist and stripped the leaves off it.\n\n'Ouch, that must have hurt, Jake,' I said.\n\n'It's our Dex,' Jake said tossing the leaves to the ground.\n\nBefore he could say anything else, Donna pushed Laila through the hedge and came out holding Shamshad by the wrist. Laila stumbled and fell. Ripping the hijab off Shamshad's head, Donna said, 'Spying on us, eh?'\n\n'Oh please don't,' Shamshad cried, trying to get the hijab back. 'Me Dad'll kill me!'\n\n'Me Dad'll kill me!' Chloe mimicked, snatching the hijab from Donna and waving it just out of Shamshad's reach.\n\nDonna's impersonation of Shamshad, especially with her Pakistani accent, was so good I couldn't help but laugh. I laughed all the louder remembering what Shamshad had said to me earlier.\n\n'Stop it, Donna,' Jake said, coming out from under the tree.\n\nDonna ignored Jake and kept waving the hijab in front of Shamshad. 'I said give it to her,' Jake said, stepping towards Donna.\n\nDonna stared at Jake for a moment and then crumpled the hijab and threw it into a bush. As Shamshad retrieved her hijab, Laila brushed her clothes with her hands saying, 'You're just a coward at heart, aren't you?'\n\nDonna clenched her fists and turned towards Laila. I had seen her batter people when she was like this. I quickly stepped in between her and Laila, and said, 'That's enough, Donna.'\n\nDonna came up to me, grabbed me by the shoulders, and whispered into my ear, 'Whose side are you on?'\n\nAs I stepped away from Donna, I saw Shamshad and Laila run down the path away from us.\n\nWhen they were out of sight, Donna held up a pretend gun and pointed it towards me, saying, 'Bang! Bang! Bang!'\n\n'What's up with everyone?' I asked.\n\nDonna put her hand into her trouser pocket, pulled out a photograph and squatted onto the ground. She kissed the photograph and started crying.\n\nMegan went up to Donna, took the photograph from her hand, came up to me and held it close to my face. It was of Donna's boyfriend, Jake's brother Dex, in army uniform posing with a gun.\n\nDonna stood up, pointed the pretend gun at me and said, 'You Taliban have got my Dex and if anything happens to him, you're dead. Paki.'\n\n'Let up, Don,' I said.\n\n'You're Muslamic, aren't you, _Karen_?' Megan sneered.\n\n'I'm one of us,' I said, turning to Jake, hoping he would take my side. He just looked at me, a look I had not seen before. Like he was looking at someone he didn't know. Turning to Donna I said, 'And it's Islamic, like, or Muslim, you know.'\n\n'It's all the same, innit?' Donna glared at me and asked, 'You're really one of them aren't you?' My stomach knotted.\n\n'Com' on, say it then,' Megan said poking me in the chest with her finger.\n\n'I'm not a Muslim,' I said and then I laughed falsely. I wanted to say, none of us in WTM believe in God any road, but instead, I said, 'And unlike you, I go to church, at least sometimes.'\n\nHolding a small silver crucifix around her neck, Donna stood up, pointed at me with her fat finger and asked, 'Where's yours?'\n\nEveryone just stared at me. I was frightened and as I turned to leave, Chloe blocked my way saying, 'You're not proper, not like us.' Her freckly, pink face a burning red.\n\n'I am. I'm proper, just like you,' I said choking on my words.\n\n'I'll make you proper,' Donna grabbed my wrist. She towered over me. Megan held me by the shoulders and Chloe pushed my arm up my back. 'Jake!' I called out.\n\nHe ignored me and went back under the tree.\n\nI felt Donna's pen digging into my forehead. I don't know how long she rubbed it into my head but she stopped when I noticed Shamshad and a load of hijab-wearing girls, coming towards us. Donna, Megan and Chloe scampered.\n\nThe sound of a song from prehistoric times chased yesterday's nightmare out of my mind. Someone was playing _Lovely Day_.\n\nThrowing the quilt off me, I sat up in bed and looked around my bedroom. My school jacket was hanging on the back of my door, not on the floor where I had dumped it. My skirt and socks were in the wash basket and not by the side of my bed. The music outside got louder and I said to Lady Gaga's poster, 'It's not your type.'\n\nMy mobile buzzed. I tried to remember where I had put it. I saw it flashing in my jacket pocket. By the time I got to it, it stopped. I had four missed calls from Jake and one text. I read the text: _Sorry for what they did to you._\n\nClenching the mobile tightly in my hand, I felt like throwing it against the wall. Instead, I replied to Jake: _U?_ Jake replied before I got back into bed: _I did nthng._\n\nMe: _Yeh nthng._\n\nI waited for Jake's reply and then I wrote: _Lol._\n\nAnd then I sent another message: _A Paki eh. Lol._\n\nMy eyes began to burn and blur as I sent another: _Cross me forehead+hope 2 die. Lol._\n\nI dropped the mobile on the floor and pulled the quilt over my head. The telephone started buzzing. I picked it up, pulled the quilt over my head and read it. It was Jake. I rejected the call.\n\nHe rang again and I answered this time, 'What you do want?'\n\n'You know our Dex is missing out there don't you...'\n\n'I know now. It's writ on me for'ead, isn't it. Beside, I didn't ask him to go did I...'\n\n'And you know what I think. I didn't want him to go. I hate this war. You know I hate it. I told him \"I don't want you to go...\"' I'd had enough of Jake and disconnected the call.\n\nHe immediately sent a message: _Sorry._\n\nI replied: _Lol._\n\nHe replied: _C u @school_.\n\nI replied: _H8 u._\n\nI sent another: _All of you,_ before turning the mobile off.\n\n' _Karen_ , you're really one of the gang now, aren't you, girl?' I said aloud, taking my head out from under the quilt. 'Islamic, Muslamic,' I sniggered as I thought about how I had laughed at the way the gang made fun of the hijab-wearing women. I felt a knot in my stomach as I remembered how I had laughed when Jake had gone up to a man with a long beard and a turban, and said, 'Run for your wives.'\n\nThe knot tightened and I felt the pain coming back in my forehead. I knew now, I never did belong to the WTM, my gang. They never saw me as I saw them. I never saw me as they saw me. I thought I was just me. And who was I? Mixed-race? Oreo? Christian? Muslim?\n\nThe image of Donna and the gang scampering at the sight of Shamshad and the hijab-wearing girls flashed through my mind. I felt a pang of jealousy. They knew who they were. They belonged. They believed. They didn't need to pretend to be anything, they just were. And me, what was I? I certainly wasn't what I thought I was. What was I anyway? What a messed up family I had. A Muslim Dad who loved beer and bacon and a Christian Mum who didn't believe in God, but went to church.\n\nI lay in bed thinking back to how Mum used to take me to church on Sundays. How beautiful she looked in her flowery red dresses, with her blonde hair falling on her shoulders and her thin nose. She wore a silver nose stud, which she only put on when going to church, one Dad had bought her when they had first met and the story of which she always mentioned on these days with a sly remark, 'If Man U had not won, I am sure he would never have bought this for me.'\n\nSometimes, getting ready for church, I would stare at my own nose in the mirror. It was thin, but no matter if I looked at it a thousand times and told myself it looked like hers, it just didn't. And whenever I asked her about this, she would go silent for a moment, as though I had asked her the most difficult question in the world and then squeeze my nose saying, 'It doesn't matter dear. You have the loveliest nose in the whole wide world.'\n\nMum's chin is beautiful and perfectly round and I hate mine. It's pointy and too long. Mum's eyebrows are so perfect, she never needs to have them plucked and mine grow so fast, one day I'll need a hedge trimmer.\n\nAt church she would smile at this person and laugh with that one, her green eyes lighting up each time, like she had not seen them for ages. She once told me she got her eyes from her Dad and blonde hair from her Mum. But I never did see these grandparents of mine. They died before I was born, Mum said. They were upset with my Mum 'cause they didn't want her to marry my Dad. They didn't think much of Pakistanis. But Mum loved my Dad. Even though he's a slob my Dad, Mum still loves him.\n\nThe last time I went to church with Mum, Dad was fidgeting about in the living room. As we were about to leave he sniggered, 'And say hello to _Him_ and his son.'\n\nMum cleared her throat, letting out a disapproving, 'Ahm.'\n\n'Stop filling me daughter's head with all this rubbish,' Dad said, as we were about leave the house.\n\nMum looked at me, grinned, flicked her eyes, shook her head, and then shouted back to Dad, 'Make sure you keep an eye on the chicken, its simmering.'\n\n'Not going to lay an egg now is it?' Dad replied.\n\nIn church we stood in our usual place, in the last aisle near the door, holding our hymn books, but as soon as we started singing the first hymn, 'Onward, Christian Soldiers', Mum grabbed my hand and whispered into my ear, 'I hate this one,' and led me out of church. I was glad.\n\nWhen we got home Dad was stomping about in the kitchen.\n\n'I bet he burnt the chicken,' Mum said to me.\n\n'I turned it off, before you ask,' Dad said coming out of kitchen, rubbing his head in both his hands.\n\nHe had curry stains on his white vest, which he had clearly tried to wipe, smudging it all the more.\n\nMum folded her arms in front of her, smiling one of her _herehegoesagain_ sort of smiles and nudged me in the ribs.\n\nAs Dad was about to turn into the living room, Mum said, 'Well, Lucky, aren't you going to ask how it was?'\n\n'How was it?' Dad asked, opening the door to the living room. The television was on. And what else would he be watching but football?\n\n'Aren't you a bundle of laughs to come back to,' Mum smirked, following him.\n\nI loved moments like this, Mum and Dad. They were such kids.\n\n'Aren't you going to ask your daughter how it was for her?'\n\n'How was it, Karen?' Dad asked, placing his elbows on the coffee table, his chin in his hands and his eyes fixed on the television screen.\n\n'Alright, we left just when the first hymn started,' I said.\n\n'Go on sweetheart, sing it for your Dad, you know it by heart.'\n\n'No.'\n\n'Go on dear, you know how much I love seeing him like this,' Mum whispered into my ear and then gave me one of those great big smiles, which meant that if I did what she asked, then there were a lot of brownie points for me.\n\nI started singing: 'Onward, Christian soldiers, marching as to war, With the cross of Jesus going on before.'\n\n'Oh God, why this,' Dad snapped and stood up. He turned around. He was white with rage. His fist clenched.\n\nMum pushed me behind her saying, 'Lucky, don't you bloody well dare.'\n\nI thought Dad was going to hit Mum. I had never seen him like this. He stood there shaking. After a moment, he pointed at the television screen and said, 'City are at the top of the league.'\n\nI slid out of bed, went into the bathroom and looked at myself in the mirror. I was a flat-chested, brown girl with a thin nose. I screamed, thinking, 'Who are you?'\n\nThe girl looking back at me had shoulder-length, curly, black hair. She had black pupils and thick eyebrows. Her arms were like her face, brown with black hair on them. This wasn't me. I must be dreaming, I thought, and slapped myself on the face. I felt numb.\n\nI screamed again, and this time I couldn't stop.\n\nMum came rushing upstairs, soapsuds dripping off her yellow rubber gloves, 'What's up, dear?' she asked.\n\nI heard Dad jumping out of bed.\n\nI placed my arm next to Mum's white-skinned arm and cried, 'I didn't turn out like _you_!' I cried, 'I'm not white, Mum!'\n\nMum gave me a hug, kissed me on the head, saying, 'It doesn't matter Karen, dear. I love you all the same.'\n\n'My name's not Karen, is it Mum?'\n\n'Did someone hurt you, Karey?' Mum asked.\n\n'No,' I said. 'It's not Karey, is it Mum?'\n\nTouching my forehead Mum asked, 'What happened here?' 'It's not Karey, is it?' I asked again.\n\nMum didn't say anything back.\n\n'What happened, Karen?' Dad asked, rubbing his eyes.\n\nI saw Dad's reflection in the mirror. He was wearing a white vest and tartan boxer shorts. The vest was crumpled up over his big belly. His greying chest hair stuck out of the vest towards his chin and stomach. He had really hairy arms and hands. His eyes were bloodshot.\n\n'My name is Kiran, Dad. It's Kiran and not Karen, isn't it?'\n\nDad looked at my Mum. They exchanged a _whatssheonabout_ look. 'Your daughter says she's not white,' Mum said to Dad. I could tell they were having a hard time keeping a straight face.\n\n'Neither am I, sweety,' Dad said. 'You look just like me.' 'Oh god Dad, I don't want to look like _you_ ,' I said.\n\nMum ran her hand across my face, wiping my tears. Dad went back to his bedroom holding his forehead.\n\nMum gave me a _don'tyouworry_ type of a hug. Dad snored. Mum and I looked at each other and laughed. She kissed me on the head and went downstairs. I went to my bedroom and tore the poster of Lady Gaga off the wall; tore it into pieces and shoved it into the bin next to my wardrobe.\n\nDad snored even louder.\n\nYep, they're right, I thought. I've got a sort of Muslim Dad.\n\n'Well Dad, you can drink and snore your way out of your religion, but I can't get you out of me.' I said aloud. 'Help me find _me_ Dad, I need you...' Just then, Dad let out the loudest snore I had ever heard from him.\n\nI looked at the back of my hand. It was brown. I turned it over and my palm was a lighter brown, with little splotches and three lines curving down the middle. I saw the faces of my gang flash in front of me: Jake looking at me with his blue eyes, Donna glaring at me, her double chin wobbling, the airhead Megan nodding and Chloe shaking her hair out of her face. They didn't see me. I was buried somewhere in a mosque behind my brown skin. Well gang, I thought, I am Kiran Malik and I am what I am. That is what I am and there is no getting away from it for me. 'And what are you?' I asked aloud and I answered myself, 'I am going to find out where I belong.' And I thought of Shamshad. I felt pride thinking about her gang, even though they hated me. They were right to hate me. I hated myself.\n\nI grabbed my CDs and flung them towards the bin, one by one. Some hit their target; others hit the wall. 'What's music got to do with it?' I said to myself. 'And besides, _Karen_ spent a lot of her pocket money there.'\n\nI picked up the CDs. Some of the cases had cracked. Putting the CDs back made me feel better. When they were all back in their positions, I pointed at them and said, 'When I'm ready, I'll get rid of some of you, so make sure you behave!'\n\nI touched my forehead. It still hurt, a strange sort of hurt that went deep down, somewhere, somewhere, where I was hiding from myself. Well, Kiran, I thought, whoever is hiding behind this skin is not a Christian.\n\nI don't know how long I faffed about in my bedroom, when I went down, Mum was wiping invisible dust off our immaculately clean kitchen worktop.\n\nMy Mum is not like normal Mums who tell you to tidy up after yourself. Mine has a place for everything and everything has its place, especially on one of those days when the East Boarhead Curry Club come round. That's Mum's new hobby. She teaches our neighbours how to make authentic curry. In the curry club, there's old George from next door, and snooty Elizabeth from number 31, and one or two other dinosaurs.\n\n'Are you feeling better dear?' Mum asked, wiping the inside of the sink.\n\n'Mum, can I talk to you? There's a lot I want to say.'\n\nMum rinsed the cloth and replied, wiping the worktop, 'Course you can love.'\n\nMum stopped, let out a sigh, rinsed the cloth and said, hanging it on a tap to dry, 'Why did you take me to church, Mum?' I cried wiping tears with the back of my hand.\n\n'I didn't want you to turn out like me or your Dad, dear,' Mum said letting out a deep sigh. 'If I had faith, this pain inside me, maybe it wouldn't hurt so much, maybe it wouldn't hurt. Sometimes it hurts so bad, I just don't want to live, just can't take it anymore. Oh my love, I should have held you tighter. How could I have let you go?'\n\n'What pain, Mum?' I wanted to ask. 'Held who Mum? Please Mum, tell me what is this secret that you and Dad are keeping from me, this thing that comes like a terrifying shadow of silence over our house?' She would never tell me and I was always too scared to ask. Whatever it was I had to find out, without this I couldn't find me. I didn't know then that my search to find me would destroy my family and my world.\n\nShe looked at me with eyes that hid behind a coldness that came on when _it_ took hold of her, but today, her eyes were rolling, as if she was fighting to stave _it_ off, now winning, now losing. Her face went pale, ghost-like and then her colour returned.\n\n'I never want to go to church again, Mum. I don't feel Christian, I just don't. I don't know what I want but I don't want to be what I am. I hate my gang now, Mum. I don't want to be like them. I'm something else. I want to have faith like you used to say. And I've decided I want to be a Muslim. That's how I feel, Mum.'\n\nMum untied her stripy blue apron, took it off her and said, 'That's nice dear.' I suddenly felt a rush of hatred for Mum and said choking on my words, 'Don't you care what happens to me, Mum?' 'Of course I do, I love you.'\n\n'Is that it? That's all you have to say?'\n\nHanging the apron on a hook on the back door, Mum turned around and gave me a look, which said, _you'rejustastupidkid_. 'If you want to try Islam for a while, that's OK, sweetheart.'\n\n'Oh Mum, it's not a game. I'm really, really serious,' I cried.\n\nMum snapped out of her look, kissed me on the cheeks, held my face in her hands and said, 'I love you because I love you for being you.' Hugging Mum I thought of me walking into school wearing a hijab. I knew they were going to laugh at me when they first saw me at school wearing a headscarf. But I didn't care. I was going to wear it. That was how it was going to be.\n\nPulling away from me, Mum said, 'I think you should talk to your Dad.'\n\n'I know, Mum.' I said, 'Love you.'\n\nDad was in his usual place in the living room.\n\n'How do I become a Muslim?' I asked Dad walking into the living room. 'Yes pet, you can. Take a fiver from me wallet,' Dad was glued to a football match he was watching on television. 'You don't know what I need Dad,' I said. 'Alright, pet...'\n\n'I'm not your pet, Dad,' I said, gritting my teeth.\n\n'Alright, you can have a tenner,' he said, waving his hand in the air.\n\nMum stood behind me. I stomped over and stood in between the telly and Dad, and said, 'How do I become a Muslim?'\n\nHe looked at me in bewilderment, like I had just asked him the most difficult question on earth. He looked at my Mum and they exchanged a _whatnow_ type of look.\n\n'You don't know, do you Dad?' I said.\n\nDad shuffled his bum on the settee. 'You don't do you?' I repeated.\n\n'He doesn't,' Mum said. 'His real God's football, really. And if Mohammed the Prophet played for Manchester United, then you'd have known everything, wouldn't you, Lucky?' Mum jibed.\n\n'Less of the tongue, Sharon. And since when did you start believing in all that Holy Mother stuff?' Dad said. 'Apart from Guinness, the only decent thing you Catholics have come up with is Man U.'\n\n'You really don't know a thing, do you Dad?'\n\nHe turned the television off, rubbed his hand through his thick curly hair, and said, 'Course I do.'\n\nPointing to the sofa opposite him, he said, 'Sit there.'\n\nI did. Mum came and sat next to me. I wanted Mum to say something, to say no, she had tried to turn me into a Christian. Then I would tell her what was really, really on my mind but she just stayed silent. I said to her, 'You're not bothered what I become are you?'\n\nMum and Dad looked at each coldly, like they were about to fight. Like _it_ was going to start right now and Mum would turn into a ghost. I didn't care. Mum took a deep breath and sighed, 'I just want you to be happy, Kiran.'\n\n'You're lying,' I said. 'You've always lied to me.' 'All you have to do is believe in Allah,' Dad said.\n\n'That's just God.'\n\n'Yeh,' said Dad.\n\n'I can teach you more later.'\n\nMum put her hand in front of her mouth. I couldn't tell if she was laughing or crying.\n\n'But what do I have to do to become a Muslim?'\n\n'You have to recite the _Kalma_ s,' Dad said, 'as a start, like.'\n\nDad recited the _Kalma_ , ' _La ilaha, illallah, Muhammad ur Rasoolullah_.' He looked a bit unsure of himself, and said, 'It means: There is no God but Allah, and Mohammad is his prophet.'\n\n'Are you sure, Dad?'\n\n'Yeh, pretty much. It's something like that.' Mum left the room.\n\nI tried to repeat what Dad said but kept forgetting.\n\nDad went back to the television, and I left to buy my first hijab. As I was leaving the house, Mum stood by the front door and looked at me.\n\n'What is it Mum?'\n\nShe just looked at me with eyes that seemed to flicker between different worlds, one where she saw me and the other where I couldn't see her. Her blonde hair was held back by a black ribbon. It was the first time I noticed streaks of grey in her hair. The lines on her forehead twitched. She didn't look like the strong woman she was, always knowing what had to be done and when.\n\n'Dinner'll be ready when you get back, Kiran,' she said.\n\nIt was bright and sunny outside, with a slight chill in the air. A few drops of rain came down out of a cloudless sky. Our neighbour, old George sat in his front garden, with Bruno his dog, near him. I waved to him and George nodded to me.\n\nGetting out of our gate, I said a loud 'Hi' to Elizabeth, our other neighbour. She was coming towards me, holding the leash of her poodle, which was a few steps behind her, lifting its leg near a broken street lamp. She lifted her chin to make sure I knew she had been to the hairdresser and smiled back at me. I stepped onto the road. There were so many unfilled holes in it nowadays, ever since the last freaky cold winter a couple of years back, when, after the snow melted, the tarmac had cracked, so everyone drove really carefully.\n\nI turned left and crossed the road close to where a car had been burnt. Some older boys were sitting in the carcass, taking turns smoking a large, hand-rolled cigarette and eyeing me. I felt like sticking my finger up at them, and telling them to go get a job, instead of doing this all day. But there weren't any jobs round here.\n\nThey were still eyeing me up so I quickened my pace. I didn't really know where I was going to get my hijab from, but there were plenty of shops in East Boarhead, I was sure to find one. The quickest way to get there was through the graveyard, but I didn't want to bump into anyone and went the long way round, along St. Enoch's Road, past boarded-up shops and derelict houses with broken windows. The road snaked around the cemetery and changed into Market Road as it got closer to the city centre.\n\nAt the junction where the new road began there were a group of shops I hadn't noticed before. I stopped outside the window of a shop called 'Cluck-Cluck'. A shop worker, a small white woman, was dressing a female mannequin in blue lacy underwear. The dummy was wearing a shiny, white hijab. I was giggling at the sight of the mannequin when a tall woman dressed in flowing black robes stopped close to me and looked at the same mannequin. Apart from her eyes, everything else about her was covered. She even wore black gloves. She walked into the shop and I went in after her.\n\nInside the shop, she went up to the mannequin that was being dressed and felt the underwear. As she was doing this, the shop worker pointed to a stand and said, 'They're over there.'\n\nThe woman in black went over to the stand, picked some underwear and went to the counter to pay.\n\nI went to the back of the store, chose a couple of hijabs from a pile in a wooden basket and stood behind the woman in black to pay. She turned round to me and said, 'And less of those dirty looks, young lady.'\n\nWhen I got home with my hijabs, the East Boarhead Curry Club, pans in hand, were walking into our house. Old George, with his violin case dangling off his shoulder was at the back. Once they were in, I carefully closed the door and tried to sneak upstairs, unnoticed. I avoided the first step, as it squeaked, and had just put my foot on the next one when Mum called, 'Kiran, sweetheart, can you come here?'\n\nI pretended not to hear her and took another step.\n\n'Come on love, I know you heard me,' Mum called again. I really dreaded moments like this.\n\n'Mum, please, can't Dad do it this time,' I yelled back. 'No one trusts his judgement,' Mum replied.\n\nI stomped into the kitchen, folded my arms in front of me, my shopping bag still in hand and hissed, 'What?!'\n\nOld George as ever was in his white trainers, yellow shorts and a matching, sleeveless sweatshirt. He had a great, big, proud smile across his ugly, bony face. He tapped on his creation, which was in an oblong, silver pan. Next to George stood Elizabeth, her ridiculous brown hat with small plastic flowers still on her head and whatever she was cooking in a reddish, brown baking bowl, whose lid was in the shape of a flower. And next to Elizabeth was a flat, round woman, in a fading, black skirt and a fading, black blouse. I didn't know her name but everyone called her the 'bulldog' and kept away from her. Her pink, chubby face was all blotchy. Each time she moved, the flab under her chin wobbled. She stood behind a tall, stainless steel pan. Mum was leaning against the sink. Someone's cat was perched outside in the garden behind Mum.\n\n'Please Mum, not again,' I pleaded, with my stomach turning at the thought of what was about to befall me.\n\n'Your Mum's such a good curry cooker, she is,' George said rubbing his tongue over what few teeth he had left. He touched the lid of his pan and added, 'I've made a special doo peeza, I have.'\n\nI trembled.\n\n'You know that dish don't you?' George asked me and then he answered himself, 'course you do.'\n\n'It's do peeaaza, George,' Mum corrected him.\n\n'I've forgotten exactly what it means, like,' George said scratching his shiny, bald head.\n\n'Two onions,' Mum said.\n\n'Two onions!' George said, raising his silver eyebrows, his clean-shaven face twitching. 'Oh, dear.'\n\n'Double onion, you idiots,' Dad shouted from the living room. 'Double Onion!' George repeated, a look of bewilderment on his face. 'I hate onions, Mum, you know I hate them,' I protested.\n\n'Oh my,' George sighed, lifting the lid off his pan.\n\nI could have just died at the sight of his creation. It was something sickly reddish yellow, with lumps in it.\n\n'I thought it meant double vindaloo,' George said meekly. 'What's that?' I snapped.\n\nMum put her hand to her mouth, her eyes laughing.\n\n'Cornflake do peeaaza, that's what it's meant to be, like,' he said. 'You daft git,' Elizabeth said lifting the lid of her pan, 'you can't make Cornflake curry. I made chicken tikka masala.'\n\nThe only way you could tell that Elizabeth's creation was once a chicken was from the burnt drumsticks, which were almost completely black, floating in a gooey, brown sauce.\n\nI dropped my arms to my sides and yelped. I wanted to leg it out of the kitchen when Mrs Bulldog stirred and lifted the lid of her pan. A rich aroma of freshly chopped coriander mixed with the smell of spices and lentils and filled the kitchen. She smiled and said in a soft voice, 'I used real ghee and lots of fresh garlic for the garnish. I think you should taste mine first, love.'\n\n'It all looks delicious, you know,' I said throwing Mum a pleading look.\n\nMum flicked her eyebrows and nodded, freeing me from my ordeal. Just then, Mrs Bulldog put her nose up at George and said, 'I bet you wouldn't dare feed it to your dog, George.'\n\n'I already have and he loved it,' George interjected.\n\nThey all burst out laughing and I bolted out of the kitchen, thanking my lucky stars.\n\nI had hardly got into my bedroom when I heard Mum running upstairs. I dived onto my bed and quickly got under the quilt, but Mum hadn't come for me and went into her own room. Just then, George starting playing a jig on the violin downstairs. I cringed and shouted from under the quilt, 'Oh, God save me!'\n\n'You coming, love?' Mum said, entering into my room.\n\nI kept my head under the quilt and didn't answer. Downstairs, the violin was now being accompanied by clapping and the muffled curses of Dad.\n\n'Come on, love,' Mum said.\n\nI lifted the quilt off my head and there was Mum, wearing a green dress with black stockings and her black dancing shoes; her hair was neatly brushed and was held in place by a green hairband across her forehead. 'What's up, Mum?'\n\n'It's such a lovely day...'\n\n'But what's with the dancing?' I interrupted.\n\nHolding her hand straight by her sides, Mum started dancing, her feet beating a perfect rhythm on my bedroom floor.\n\nMum looked like a little girl skipping. She danced out of my room and down the stairs. I followed, I just had to see what Dad made of all this. It was a match day.\n\nHe was standing by the door of the living room tapping his feet.\n\nElizabeth was twirling round and the flab on Mrs Bulldog was doing a frenzied dance to the tune of George's jig.\n\n'Can't you play anything other than Morrison's jig, George?' Dad asked.\n\nGeorge flicked his eyebrows and carried on playing. He suddenly looked years younger.\n\nIn the morning, Mum left for work at Asda. She did the early shift on Mondays. I lay in bed a bit longer than I should have. I kept thinking about what everyone would say at school. I jumped up, got dressed, put my headscarf on, ran downstairs, wolfed down some Cornflakes and bolted out of the door.\n\nOn the bus to school, a few girls gave me the evil eye, but I ignored them. This was the first time I wasn't worried about meeting Shamshad. I was going to tell her I knew the _Kalma_ and I was sorry for everything that had happened in the past. This was the new me and I'd like us to be friends.\n\nI got off the bus reciting the _Kalma_ to myself when she jumped out of the old warehouse. I was startled, but that was nothing compared to what I felt when I realised what I had done.\n\nI looked down at my legs and realised what Shamshad was looking at: me, wearing a black hijab, a white shirt, a school tie, a black blazer, a short, black skirt, and white socks. I turned round to run back home and ran into Aisha Sadiq. Laila was close by.\n\n'Push her to me, Laila,' Shamshad said.\n\nLaila didn't touch me.\n\n'Whose side are you on, Laila?' Shamshad asked, grabbing me.\n\nShamshad called everyone towards her, and I was trapped in the middle of a circle. Their mouths were wide open. I dropped my arms by my side. They were really heavy. She spun me round. I kept thinking, 'You deserve everything you get, Kiran.' The whole world was laughing at me. And so it should. Shamshad was pointing her mobile phone at me. Mixed up rubbish, that's all I was, trash. You're right Shamshad, and you're right Laila, and every one of you. How could I do this to you?\n\nI don't know what happened. All of a sudden everyone just vanished. I turned around and walked, not knowing where I was going.\n\n## **Shamshad**\n\nI had gone to the graveyard with my friend Laila to see the new headstone on my granddad's grave. May the Almighty grant him a place in Heaven. It was made of royal white marble, with the _Kalma_ carved into it, running around its flowery edge. Someone had written 'EDL' on the last one. We were sitting on a bench under one of the new CCTV cameras, which covered our graveyard, when Karen Malik came flying over the fence and landed on her bum not far from us.\n\nWhat else can you do but laugh at someone whose behind is stuck up in the air and whose knickers have gone up their bum?\n\n'I can't stand people like her,' I said loudly to Laila. 'Especially when she's sucking up to her _gora_ gang.'\n\n'Oh, Shami,' Laila protested, squeezing my hand, 'let's not let her spoil it.'\n\n'Look what the cat's brought in,' Aisha said as she was about to jog. I nodded, letting out a sigh of relief, but still couldn't help thinking about how her gang had made me feel so bad, so often; how they had humiliated me because I'm Muslim, especially that Donna.\n\n'You know, Laila, once when I went past her gang, Donna started singing, \"God made little, brown people. He made 'em in the night. He made 'em in a hurry and forgot to paint 'em white.\"'\n\n'It's stupid,' Laila laughed.\n\n'It is a bit, but then I didn't think it was, funny, you know. But what really got under my nose was the way Karen laughed with them.'\n\n'What a cow,' Laila said.\n\n_'Qasmein_ , I swear. Isn't she just.'\n\nAfter Karen went to see her gang, Laila and I strolled up the path to see what she was going to do.\n\nI didn't see them coming. They jumped up from behind us. Donna pushed Laila through the hedge. She held my wrists so tightly in her fat hand it hurt. I wish I had said nothing to her, just punched her in the face when she ripped off my hijab, but ended up saying the most stupid thing in the world, 'Me Dad'll kill me!'\n\nI don't know why, but when that Chloe was waving my hijab in front of my face, pretending somehow I talked like that, it made me think about her seventh birthday party. She was so excited she couldn't blow the candles out, so I blew them out for her and she ran to me and hugged me.\n\nWhat made me really, really mad was the way Karen joined in laughing at me. It's one thing leaving Islam for her Mum's religion, but it's another ganging up with her WTM posse and insulting mine. It's just as well they ran off when I came back with the girls. On the way back home, I said to Laila, 'I'm going to teach that _Karen_ a lesson tomorrow, she'll never ever forget.'\n\nI prayed inside my head, _'Ya Allah_ , give me strength to get my own back on her.'\n\nI couldn't even keep Karen out of my dreams that night, when I slept. I dreamed I was going to a special assembly at school. Everyone was there. There was only one chair left empty. It was in the front of the hall. Everyone was looking at me. Karen was standing next to me, her eye on the chair. I ran for the chair. Everyone cheered. She ran for the same chair. I grabbed her hand, to pull her back but she snatched it free and beat me to the chair. The teachers were praising her. Everyone was clapping for her and laughing at me. My mother stood at the back of the hall. Stone-faced, as ever. I ran to her, crying. I held my arms out for Mum to hug me. She folded her arms. I ran past her and just as I got outside the assembly hall, there was Karen and her mother. Her white mother, combing Karen's hair.\n\nThe next morning, I put a pair of scissors in my bag and caught the bus to school early to wait for Karen.\n\nAfter getting off the bus at our school stop, Karen usually went to Gilani's to get a drink. The shop was next to an ivy-covered abandoned warehouse at the end of a street full of 'For Sale' signs. I was going to grab Karen, pull her into the warehouse, and give her a hiding she would never forget. I was going to cut her hair and rub muck in her face.\n\nI saw her bus coming, grabbed some muck and hid. She was muttering something to herself. I held my breath and then jumped in front of her. She yelled like she had seen a ghost. I grabbed her by the throat and pulled her towards me. I shoved some of the dirt right into her mouth and rubbed it into her face. As I pulled the scissors out of my bag, I saw Laila coming towards us. I let go of Karen and laughed and laughed at the sight of her. Not at how dirty her face looked, though that was hilarious, but at what she was wearing.\n\n'Look at you!' I cried pointing to her legs.\n\nKaren looked down and went white. She turned around to leave, but Aisha was right behind her.\n\nAs I was leaving school, I stared at a poster of our Halloween party, a copy of which was stuck on either side of the exit door. It said: _Cum On Yea Ghouls, Vamps and Bats._ Against a yellow background were black outlines of witches flying about on broomsticks and bats of all shapes and sizes. Underneath these in thick gothic letters was written #freakyfriday.\n\nI left school that day, flushed with happiness, made all the better by a bright October day. Walking out of the doors, I thanked _Allahjee_ for giving me a chance to get my own back on Karen.\n\nI just couldn't get comfortable in the bus seat on the way home; I held a prickly pain in my bum and kept moving about. I could feel a spot or a lump. It hurt each time I pressed down on it.\n\nAs soon as I got off the bus, I rushed back home to help Mum. Dad was having one of his parties. They always took place on a Monday. I hated these parties. It was usually the same old men, saying the same old things and leaving their same old smells behind.\n\nThere are no pictures on any walls in my house. Dad has forbidden them. Un-Islamic, they are. There is no music. No television. Not allowed. I am allowed a computer, though. Dad got it for my fourteenth birthday. It turned out to be more of a gift for my mother than me!\n\nThough she can't read or write, she's learned to turn the computer on and use Skype. She's always online, gabbing away with her family in Pakistan. She shuts the door if she knows I'm around. Usually, she looks all shrivelled up, but talking to someone in Pakistan on Skype, she seems to grow taller, pointing at the screen and shaking her head.\n\nWhen I got home, Mum was looking at a darkish, grainy figure of a woman on the computer screen. She didn't realise I was back. 'How many goats did I give you to look after?' Mum asked. 'Ten,' the dark woman replied.\n\n'How many have you got now? 'Nine.'\n\n'What happened to my goat?\n\nThe dark woman went quiet for a moment and then said, 'The Khyber Express went over her.'\n\nI dropped my bag in the hallway. Mum shut the door and continued. 'What were you doing?' Mum was furious. She swore.\n\n'I was bringing my animals back from the jungle,' she said. 'I have twenty six goats and four cows.'\n\n'Which one of my goats died?\n\n'The soil-coloured one with white patches.'\n\n'Didn't that give birth only three months ago?' Mum asked. 'And it was pregnant again.'\n\n'What happened to the offspring?'\n\n'One of them was born without eyes, and a wolf took the other one.' 'How is it that it's always my goats that the wolf takes? It's only my goats that go under the Khyber Express.'\n\nI could hear people in the room in Pakistan with her. They laughed. Just then the Skype connection went. 'Load-shedding. Always load-shedding when I want to talk,' Mum cursed the electricity people in Pakistan.\n\nShe came out of the front room, nodded to me and disappeared towards the kitchen at the back of the house.\n\nI put my school bag on the table near the stairs, took off my shoes, placed them on the shoe rack and went upstairs.\n\nI ran the bath and watched steam from the water rising. Wiping our big bathroom mirror, for a moment I felt a shock run through my body. I thought I saw Karen's long, boney face in my reflection in the mirror.\n\n'Get out of my head, you cow,' I cursed, looking closely at my own reflection. Was this chubby face with thick eyebrows really me?\n\nI could hear Mum and Dad talking about something outside the bathroom door, but couldn't work out what they were saying to each other.\n\nI got hold of Dad's round shaving mirror and looked at what was causing me so much pain. It was a ginormous boil and without thinking, I shouted, 'Dad, I've got a boil on my bum.'\n\n'That's OK. I've got one on my bum as well.'\n\nMum and Dad laughed out loud, a rare thing in our house. When they finished laughing, and they laughed for a long time, Dad said, 'I'm popping out to the shops, do you want anything Shamshad?'\n\nI didn't reply. In our house you never talked to someone in the bathroom, let alone what I had just said, and to Dad at that. I felt so ashamed. I got into the bath, lowered my head and watched my hair spreading in the water.\n\nAfter having my bath and getting changed, I ran downstairs and went straight into the kitchen. Mum was chopping onions. I called out for my cat and listened for the patter of her feet coming down the stairs. She loved curling up in a basket in my bedroom.\n\n'She's not been in all day,' Mum said, putting the chopped onions into a plastic bowl.\n\n'All day!' I exclaimed. 'She must be really hungry.'\n\nMum didn't say anything at first. As I bent down to pick up a half-empty packet of cat food she said, 'Hungry animals find food. Or they die trying to find it.'\n\nI didn't say anything to her. It was my cat. I looked after her. I opened the back door and shook the cat food bag a few times. I called to her, 'Rani! Rani! Food! Come on, Rani!' I couldn't see her and shook the bag again. I was about to step out into the back garden when I felt her curling herself around my legs.\n\nAfter feeding the cat, I went to my bedroom, put my blazer on a coat hanger and hung it on the back of the door. I had just put my dirty clothes in the laundry basket when Rani popped her head round my door. I was about to pick her up when I heard the front door slamming shut. Dad was home.\n\n'You'll have to wait,' I said to Rani, running my hand over her back. She closed her eyes, arched her back, and curled up her tail.\n\nBy the time I got to the kitchen, Mum was frying onions in a big pan. Steam from the pan was being sucked noisily into the extractor.\n\n'How many are coming?' I asked. 'Four or five,' she said.\n\nI took some more garlic out of the basket that hung on the wall opposite the window and started peeling.\n\n'Bring my _Kala Kola_ ,' Dad popped his head into the kitchen and asked me in English, touching his hair. He always spoke to me in English.\n\n'OK, Dad,' I replied, putting a peeled garlic clove in an empty plate Mum had put in front of me. I washed and dried my hands and took a small box of hair dye out of the cupboard. I looked at the box, _Quick-drying Kala Kola._ I took the bottle out, crushed the box in my hand, threw it into a bin and went upstairs to the bathroom.\n\nThe door to the bathroom was open. Dad was wiping his face with a towel. Placing the towel in the sink, he held out his hand and I gave him the bottle. He glared back at me and I realised I hadn't opened the bottle for him. I snatched it back, opened the bottle and gave it back to him.\n\nDad nodded pouring some hair colour into a saucer. Dipping a toothbrush into the saucer he said, 'Too many cars.' 'Yes, Dad.' I said turning to leave.\n\n'Getting stolen,' Dad said, dyeing his hair with the toothbrush. 'No one has a proper job round here. When I was young, we went where work was. This lot just want to make quick money.'\n\n'Yes, Dad.' I said.\n\n'And I spend my life phoning the council. Our streetlights don't work. Nothing works round here anymore. I leave a message on a machine and no one bothers to get back to me.'\n\n'Yes, Dad.' I said.\n\nI went back to the kitchen and helped Mum cook. After the main dishes were cooked, I made some salad and then cleaned up. Rani came in a few times, curled around me leg and went out again.\n\nI cringed when I heard Dad welcoming someone. 'The old farts are here,' I thought.\n\n'What was that?' Mum asked.\n\n'I said nothing, Mum,' I said. 'Was that your father, I mean?'\n\n'Yes Mum, he was welcoming our guests.'\n\nI washed my hands, placed some orange and apple juice cartons on a tray along with some glasses, adjusted my hijab, and took the drinks into the living room. The room was bursting with grey beards and bloated bellies. I gave my _salaam_ to everyone, put the drinks on the table and came out.\n\n'Daughters grow up so quickly,' Dad said as I turned around to leave. 'Such a good girl, your Shamshad, Amjad Saab,' Baba Alam, with his ever-bulging belly said, clearing his throat.\n\nI stopped outside the door and listened.\n\n'Never a bad rumour about her,' the croaky, old voice of Baba Bagha agreed. That's what he always does: agree.\n\n'Amjad Saab, it is official now. Victoria Baths are going to have a women's only day, and our women will end up going. What a shame on us.' 'Yes!' I whispered, clenching my fist.\n\n'Such a shame,' Baba Bagha added.\n\n'And can you believe it, there are seven days in a week and they chose Friday for women's-only swimming. Friday! They're rubbing salt into our wounds,' Baba Alam said.\n\n'How could they do it on a Friday?' Baba Bagha asked.\n\n'There was a petition signed by lots of women from East Boarhead,' Dad said in his matter-of-fact voice. 'And some of the women, even the illiterate ones from round here, put their thumbprints on the petition.\n\nThe other old men went silent.\n\nMum tapped on the side of the pan with her spoon. I rushed back to the kitchen and blurted out, 'There's going to be women's only swimming in Boarhead Mum. Isn't that great?'\n\nMum pursed her lips, shook her head and carried on with the chores. 'Hurry up.' Dad popped into the kitchen.\n\nMum took the lid off a pan with one hand and stirred the keema, the mince meat with the other, over and over again. She had a special way of cooking keema. Whilst everyone else added the spice at the beginning and then fried the onions, garlic and ginger, Mum always quickly tossed the meat in a hot wok without any oil and turned it over and over again until it released its water and began to shine. Only then did she add some hot oil, onions and the other ingredients. There was nothing in the world like Mum's keema.\n\n'Sorry Dad,' I said taking my eyes off Mum.\n\nMum stopped stirring the food once Dad left the kitchen and looked at me in a strange un-Mum like way. I couldn't tell if she was smirking, smiling or smouldering.\n\n'What?' I asked Mum.\n\nMum took hold of my hand, pressed it in both of hers and said, 'Nothing.'\n\nI picked up a tray and went to collect the empty glasses. Dad was sitting on his favourite sofa, set into the alcove of a window, dressed as ever in his grey shirt, flowery tie and striped suit. The old men were discussing the same old things, same as ever.\n\n'Amjad Saab, you are a very influential man. There are many Muslims in this town now and, god willing, there will be many more in years to come.' This was the permanently red-bearded Abdullah Khan from number 127. He was really, really prehistoric and like my Dad, dyed his hair and eyebrows jet black. Mister Khan looked comical as no matter how hard he tried, you could see that the roots of the hair on his head and brows were white. Ever since I could remember, he had driven a taxi for On-time Cars and still drove them now. His sons, like most of the men around here also worked on the taxis or worked in take-aways. Every one of them who came to talk in our house was depressed about not making ends meet. Mister Khan's favourite subject was that he wanted to get the name of our town changed. For as long as I could remember, he had been going on about it. 'I am always ashamed to tell people I live in this city. I have to say, B.O.A.R.head. I can never pronounce its name, only spell it. How can we name our mosque B.O.A.R head Central Mosque.'\n\n'How can we?' the rest of the old farts agreed.\n\n'One day, _Inshahallah_ , one day, god willing,' my Dad replied as he always does when he doesn't know what else he should say.\n\nI went back into the kitchen, placed the empties in the sink and the half-full cartons of juice in the fridge. After wiping the tray, I put some hot kebabs in a serving bowl and placed it on the tray. As I put spoons and side plates on the tray, I remembered it was time for prayers.\n\n'Mum, its getting late for _Isha_ prayers.'\n\nShe nodded, picking the tray up off the table. I followed her out of the room. She knocked on the front room door and waited. Dad popped out, took the tray and went back into the room. I went upstairs to do my _wudhu_ , to wash myself properly before the prayers. I came out of the bathroom, went into my bedroom and spread the prayer mat on the floor. Rani was curled up on my bed, her tail across her eyes. As I touched the floor with my head, at a moment when I should have only been reciting my prayers, I thought of Karen and laughed. I got off the mat. I brushed Rani off the bed and cursed myself. 'Please forgive me, Almighty, for not concentrating on my prayers,' I pleaded, and went back and started my prayers again. This time Karen didn't come into my mind. But as soon as I finished, I jumped on my bed, pulled my mobile out of my bag and watched the recording of Karen that I had made that morning. Then it occurred to me!\n\n'Yes, you're going to be famous, Karen,' I said, watching her with her arms by her side, her face all white with shame. I was going to put her on YouTube as soon as I got near the computer and then send the link to everyone.\n\n## **Kiran**\n\n'Well, Kiran, you really are just dirt,' I thought aloud and walked on. I don't know how long I walked, or in which direction I went. I didn't stop at any road junctions, just crossed. Sometimes cars screeched to a halt, other times they beeped their horns. I just kept going. My ears burnt. My head ached. I felt so cold. My stomach was tight and my throat became dry. I sat down and my breakfast came out all over the pavement. I vomited a few more times and then stood up and walked again.\n\nMy mobile rang. It was Mum. I ignored her. My Dad rang. I ignored him.\n\nMum texted me. I didn't bother opening it and carried on walking. I kept seeing monstrous faces going round and round in front of my eyes. Shamshad laughing, jeering. Holding her mobile phone in front of me.\n\n'Why, Shamshad?' I called out to her. 'What did I do to you?'\n\nShamshad didn't answer. The monstrous mouths were open, but they were not saying anything. The mobile vibrated in my hand.\n\nA car went swerving past. A red-faced driver, his mouth wide open, saying something to me. But there was only silence.\n\nA cold wind came from somewhere. My head began to throb. It felt as if someone had put a metal clamp on it and was tightening it. Tighter and tighter! I walked on and on.\n\nThe clouds thickened above me, but there was no rain.\n\nMy hands were wet. My shirt was wet. I felt so hot. I sat down and vomited again.\n\nThere were only birds around. I was alone somewhere. And then, out of the silence, I heard Mum's voice. She was calling me. I tried to answer, but my voice was gone. Then I saw her running towards me.\n\n'What are you doing here, you silly girl?' She asked. 'Look at the fool I am, Mum,' I hugged her and cried.\n\nWhen I stopped crying, I realised I was in the graveyard, sitting under the weeping willow.\n\n'Thank goodness you're alright,' Mum said, wiping my face. 'Love you Mum,' I said.\n\nShe held my hand and we went home. Mum put me to bed.\n\nThe first time I woke up, there was a glass of water on the desk near my bed. I was in my nightclothes. The door was open. Mum and Dad were talking downstairs. They were arguing.\n\nMum screamed, 'It was because of you.' It was a scream I had not heard her scream for a long, long time. I was terrified. _It_ would be back in our house.\n\n'It was an accident,' Dad said. 'I wish I gave you a boy.'\n\n'I said I didn't want to,' Mum said, 'didn't I?' 'Let it go, Sharon.' 'You. You. You.'\n\n'I know you wanted a boy, Mum,' I said lifting my head off the pillow. It was too heavy, I slumped back down. 'Please don't fight over me. I'm sorry for being an accident. Don't fight. Please.'\n\nI forced myself to sit up in bed. I was going to go downstairs and beg them to stop fighting over me. The front door opened and slammed shut. My throat was dry. It hurt. I drank the water and slumped back on the bed. Behind my closed eyes, open mouths jeered at me. Big, bulging eyes stared at me. Everything was hurled around by a raging wind. The voices, they were the ones that really frightened me. They jeered and asked me so many questions: They wanted a boy, not you. What are you? Who are you? Where are you from? You thought you were white? White? Black? Coloured? Asian? Pakistani? Christian? Muslim? What are you? I kept waking up. I was telling Mum about Shamshad. How she tormented me. How she hated me. Mum just stared back at me. Silently. Sadly.\n\nI heard someone calling my name and woke. Mum was sitting beside my bed. She smiled a _thanksgoodnessyou'realright_ kind of smile. Dad was standing in the doorway, one of his lovely, hairy arms dangling by his side, the other behind his back. His loose, white vest rolled over his bulging stomach.\n\n'Nice to have you back,' he smiled, 'Karen, Kiran or Karey? Whoever you are today.'\n\nI sat up in my bed. You're really one for words Dad, I thought. 'It's Kiran Dad,' I said. 'It'll always be Kiran.'\n\nMum threw a daggerish look at Dad. She kissed me on the forehead and left the room.\n\n'And you really are one,' Dad said, looking at a ray of sunlight coming through the window.\n\n'What?' I leaned up and looked out of the window.\n\nTwigs, leaves and broken branches littered the pavement. Some dustbins were on their sides. Mr and Mrs Mason from next door were surveying the damage from the storm. George was clearing rubbish from his garden, his dog watching him from a short distance.\n\nDad cleared his throat and said, nodding to the light, 'Kiran. It means a ray of light,' he said.\n\nSuddenly I was filled with rage and snapped at him, 'You've taught me nothing, Dad, nothing! I want to be a proper Muslim,' I cried, 'Proper something!'\n\nDad kept quiet. He put one of his feet on top of the other. He was in deep thought.\n\n'Can you teach me, Dad?' 'Yes,' he said. 'I'll teach you...'\n\nMum came back with a glass of hot milk and gave it to me. I held the glass in both my hands and asked, 'Dad, do you actually know anything, about being a Muslim?'\n\n'I do.'\n\nMum coughed a _pulltheotherone_ type of a cough.\n\n'I know a lot more than you think I know,' he said to Mum, but avoided eye contact with her.\n\n'Like fasting?' Mum asked.\n\nDad curled his big, fat lips, held them up to his nose, and breathed out. 'And praying?' Mum said.\n\nDad interrupted Mum with a protesting sigh, and said to me, 'I can teach you what you need to know, whenever you are ready.'\n\nI blew into the milk and drank a big mouthful, and then said, 'I'm ready now, Dad.'\n\nHe smiled, and said as he was leaving, 'OK, but get better first.'\n\nI finished off the milk and handed Mum the empty cup. She said, taking it from me, 'Let's talk about it if you want.'\n\nI nodded. I wasn't going to let her talk me out of it.\n\nAfter Mum left, I went back to sleep. It was a deep, deep sleep, without monsters. I woke up in the afternoon. I was feeling sticky but really good with myself. I was going to be a new me. Gone was Karey. Gone was Karen. The WTM were dead. Here's to Kiran, I thought!\n\nI was going to surprise them all. The first thing that Kiran did, which Karen would never have done and Karey wouldn't have thought about, was to change the sheets and pillow cases on her bed. After putting the dirty ones in the wash, I had a shower and went downstairs.\n\nAs I was going down, I heard the hiss of a beer can opening. Well, it is after 6 p.m. I thought.\n\nI checked my mobile. There was a text from Mum: _If you want anything from the shops, ring me on my mobile._\n\nI went to the kitchen, poured some orange juice into a large glass and walked towards the front room. The door was ajar. Dad was sitting close to the television. The sound was low. I sneaked into the room and peeped at him from behind a large shelf. He was watching an Islam channel. An Imam was giving a talk to some youngsters on the proper way of praying. Dad was puffing away on a cigarette.\n\nMum, who usually brought the shopping home from Asda on her way back from work, was taking a long time at the shops. I texted her: _Where are you Mum?_\n\nMum replied almost instantly: _Done in 15 mins. Are you OK?_\n\nMe: _Fine. Dad glued 2 TV._\n\nMum: _What's new? Tell him to pick me up._\n\nAs I was about to send my reply to Mum, Dad shouted, 'Kiran, I'm going to pick your Mum up. Text her. Tell her I'll be a bit late.'\n\nHe was out of the door before I had a chance to reply.\n\nI sent a text to Mum, _Ur taxi'll b a bit l8. Lol,_ and went into the living room.\n\nThe living room stank of cigarettes and beer. I opened the windows. A cold wind lapped up the staleness. I sat on the sofa and flicked on the television. The news was on. A helicopter was broadcasting pictures of a mansion surrounded by woods. Breaking News: Islamic Terror Attack Foiled. I turned the television off. All of a sudden, I felt exhausted and dozed off. I was in an empty, white room. Without doors. Without windows. 'Karey?' someone called. And then, 'Karen?' And then, 'Kiran?'\n\nI woke up with saliva dribbling from my mouth. Mum and Dad were back and by the time I got off the sofa they had already taken the shopping into the kitchen and Mum was putting it away.\n\nI sleepily walked in, wiping my mouth with a tissue. Mum gave me a great, big, _you'llbealright_ kind of a look, and Dad shoved tins into a cupboard. He was in a hurry, and I guessed it was getting nearer to the match. A tin rolled out and fell on his toe. He yelped.\n\n'Serves you right, Lucky,' Mum laughed, looking at Dad who was holding his foot. 'Put them in properly.'\n\nDad didn't bother saying anything back and blew his nose into his handkerchief. His weekly lottery ticket fell out of his pocket. I bent down, picked it up and put it on the table, thinking, 'There's nothing lucky about my Dad.'\n\nMum picked up the lottery ticket and said, 'Lucky, stop wasting money.'\n\n'You won't say that when I win the jackpot,' Dad said, making a face of exaggerated pain.\n\n'You're the only person I know who has never even won a tenner,' Mum said.\n\nYep. 'There's nothing lucky about my Dad,' I thought again, and asked, 'shall I get you a cold compress, Dad?'\n\n'He needs an ambulance,' Mum said, as Dad put his foot back down and adjusted the tins inside the cupboard, 'don't you Lucky?'\n\nI suddenly felt really angry with Mum and said, 'His name's not Lucky, is it? It's Liaqat.'\n\nMum gave me an icy stare. Dad took a deep breath, and said, 'No, Kiran, it's not.'\n\nMum wiped the kitchen table, went over to the sink, and washed the dishes, which had already been washed \u2013 every now and again looking at me in the reflection in the window.\n\nWhen she had finished, she looked around the kitchen to make sure everything was where it should be. 'Don't hate me Mum,' I said. I was determined not to cry. 'But I can't go to church with you anymore. Your Jesus is white. Your Holy Mother is white, and The Father has to be white as well. Look at me! Just look at me!'\n\nMum whispered, 'It's not like that.'\n\n'I know you hate me,' I said, holding back my tears. 'It's not like that,' Mum repeated, walking up to me.\n\n'You can be what you like, Kiran,' Dad stood up and put his hand on my head.\n\nMum and Dad exchanged looks that sent shivers down my spine.\n\nI sat down on a chair, put my head in my hands and wept. Mum got me a glass of water. I drank it.\n\nMum leaned over to an unpacked bag, picked it up and put it on the table. Pushing it towards me, she said, 'Here I bought these for you.'\n\nDad picked something out from under a pile of empty bags and hid it behind his back. A big grin ran across his face.\n\nThe bag Mum gave me had a green hijab and a long-sleeved, white _kurta_ along with some books and a DVD in it. There were small books called _Qaida_ s and there was the Holy Quran. There was a book on the Prophet, a book on Islamic history, and a book about fasting. The DVD was called _The Messenger_. I felt so ashamed of myself that I stood up, kissed Mum on the cheeks and hugged her.\n\n'What've you got for her, Lucky?' Mum asked, in a knowing kind of a tone.\n\nDad walked out of the kitchen backwards, hiding something behind his back, 'Come on, Dad, what you got for me, eh?'\n\n'I'll give it to you tomorrow,' he said, trying to get away.\n\n'Let's see it, Dad.'\n\n'Show her, Lucky,' Mum said.\n\n'Liaqat, Mum!'\n\n'He'll always be Lucky to me,' Mum replied.\n\nDad brought his hand round to the front slowly. He was holding a book as well: _Islam for Dummies._\n\nWe had a really good laugh, and then I said, 'Dad, you can teach me how to pray, can't you?'\n\n'Course I can' he said, Mum sniggered.\n\n'Can you teach me to read the Quran, Dad?'\n\n'Yep. I read it all, twice, in Pakistan when I was a boy,' Dad said, 'but don't expect me to start believing in all that stuff.'\n\nMum went silent. Her eyes drifted off for a moment into another world, but then she came back all of a sudden, and said, 'Lucky, go into the living room with Kiran and start teaching her what you know. Let me get the dinner on.'\n\n'I can teach her, you know,' Dad said. 'Go on then,' Mum said.\n\nDad rubbed his hand on his stomach and then on his head, and said, 'I will.' Turning to me, he said, 'Come on Kiran, let's start right now. We'll start with the _Qaida_ , and you can learn it just like I did. And then I'll teach you how to pray.'\n\nMum sniggered again. I went to the living room with Dad carrying the books. He cleared the coffee table and sat on the sofa. I sat down next to him.\n\n'Do you mind if I just check on the score before we start?' Dad asked, holding the TV remote.\n\nI smiled and shook my head. He flicked through a few screens and then turned the television off. He opened the first page of the _Qaida_ , pointed to the first letter, and said, ' _Alif_.'\n\n' _Alif_ ,' I repeated.\n\n' _Bey_ ,' he said, pointing to the next letter.\n\nI repeated as he went along. I was so happy to be sitting with my Dad learning Arabic letters that I must have been the happiest girl in the world. I couldn't remember any other time we had sat together for so long, me and my Dad. As we were going over the letters for a second or third time, Mum came into the room and moaned, 'Lucky! Don't smoke a cigarette next to Kiran, and not while you're trying to teach her to read the Quran!'\n\nDad quickly stubbed it out, blowing the smoke away from me. 'I think you've done enough for today on the reading,' Dad said. 'You're going to teach me how to pray?'\n\n'Course I am,' he said, walking out of the room. Mum smiled at me a sort of _I'dliketosee_ kind of a look and followed Dad out. I put the _Qaida_ away and went to the kitchen to help Mum. I heard Dad stomping down the stairs calling me back into the living room. Mum followed me.\n\nDad was looking all around the room a bit puzzled. He said, 'I just can't remember which way the _Kabah_ is.'\n\n'Mum nodded towards the television corner. Dad looked up at the ceiling for a moment, and said, 'Yes. You're absolutely right.'\n\n'I bet you'd have said that whatever Mum said,' I thought. 'You haven't got a prayer mat,' Mum said.\n\n'I have,' he said, spreading a big, red towel in the space in front of the television. It was a Manchester United towel, with a picture of Alex Ferguson on it.\n\n'I'm not praying on that...'\n\nDad picked the towel up, turned it over and spread it out on the floor. It had a picture of Rooney on the other side.\n\n'Oh Dad!' He stepped onto the towel, raised his hands to his ears, and said, 'First you do this, and you say \" _Allah O Akbar_ ,\" bending down towards his knees, he said, 'and then you do this and,' he stood upright, and said, 'then you do this...'\n\nMum interrupted, 'For god's sake, Lucky, you're drinking beer and teaching Kiran how to pray!'\n\nHanding a half-drunk can of beer he was holding in his hand to Mum, he said, 'You're right about that.'\n\n'You'd better take this girl to the mosque or somewhere, Lucky,' Mum said, 'She wants to learn it properly.'\n\nAren't Mums just full of surprises? I had got myself into a bit of sulk. I really wanted to dress up but couldn't decide what to do. I was going to ask Mum to help me dress up for the Halloween party at school, but the way she had gone all quiet ever since I told her I wanted to be a Muslim, like _it_ was going to hit our house again, made me think it better not to bother.\n\n'Kiran sweetheart...?'\n\n'What've I done now?' I grunted turning my face away from her. 'Haven't you forgotten something?' Mum said coming closer to me. 'What?'\n\n'What's tonight, dear?'\n\n'Tonight's, tonight. What's it with you?' 'You're angry aren't you?'\n\n'No.'\n\n'You _are_ ,' Mum said stroking my head. 'Not,' I replied brushing her hand off. Don't you just hate it when Mums do this? 'It's Halloween, Kiran?'\n\n'So.'\n\n'I thought you wanted to go to your party at school.'\n\nI turned around. She was all smiles. She had a shopping bag in her hand. Putting the bag on my bed she asked, 'What do you want to dress up as?'\n\nI didn't reply. I heard Dad stomping up the stairs.\n\n'What about a big, hairy monster?' Mum said pulling a big mask out of the bag with long, straggly hair, a hairy face with red vampire teeth.\n\nI burst out laughing. Dad popped his head into my room and then came in. He stared at Mum's hideous thing, running his hand over his big belly.\n\n'What's so funny?' Mum asked. 'It's a Halloween costume.'\n\nDad started laughing and Mum asked him all serious like, 'What's with you, Lucky?'\n\n'I'm just laughing 'cause she's laughing,' and then turning to me he asked, 'What's the joke, Kiran?' 'It's not a joke, Dad. If I put this thing on at school they'll say I look like you,' I said.\n\nMum laughed. Dad waved his hand dismissively in the air and went to the bathroom. After he left, I said to Mum, 'Seriously though, Mum, I want to be something special not just a mask from a shop.'\n\n'Well I won't bother showing you the other one then,' Mum said shoving the hairy thing back into the bag. 'I got a load of paints. What shall we do?'\n\nI turned my computer on and searched for some ideas. And then I saw one, the perfect one for me. I could just picture myself in it. One side of my face painted in sharp angles of blood red. The other side white, with shining red lipstick, with my messed up hair falling on my face.\n\nDad flushed the toilet, came out of the bathroom, popped his head into my room again and said, 'Mum and daughter moment is it?'\n\nHe waited for us to respond but we ignored him and he added, 'Halloween this week, sports week next week. Don't you lot study any more, just party?'\n\nWe ignored him again and he went back downstairs.\n\nI pointed to the one I liked and Mum said, 'Perfect.'\n\nShe rummaged about in the bag and brought some paints out. I sat on a chair in front of my desk and looked at myself in my big mirror that sat on top of the desk. Mum ran her fingers through my hair saying, 'It's so soft, your curly hair, so soft.'\n\n'I was thinking Mum, I'd like it all messy but what do you think of it being spiky?'\n\n'Spiky!' Mum giggled like a little girl. 'Yeh, spiky, really spiky.'\n\n'Crikey!' Mum said. 'OK. I got all the stuff for it.' We both laughed.\n\nMum went downstairs to get a tray so she could squeeze the paint onto it and I went through a few more images of ghouls and things like that. I thought maybe I would look good as a hook-nosed witch, but then rejected the idea with the thought that half the gormless girls at school would turn up as witches with stupid broomsticks. Then there was this Dracula-type with big, bloody fangs, but I didn't like this one either, as half the barmy boys fancied themselves as vampires. By the time Mum came back, I was convinced I had made the wrong choice but then decided I could see what I looked like and if I didn't like it, I'd go for a witch.\n\nSqueezing some red paint into a silver, stainless steel tray, Mum said, 'You've always been like this you know.' 'Like what, Mum, a monster?'\n\n'No, silly! Once you make your mind up, there's no changing it,' Mum said dipping a brush into the red paint. 'If only,' I thought.\n\n'You know, Kiran, if you have spiky hair, maybe blood red lightening streaks might be better than splotches.\n\n'OK,' I said, and thought, 'You're wicked, Mum. Perfect.'\n\nAs she drew the first lightning shape on the left side of my face, Mum said, 'Do you remember the mp3 player we bought you for your tenth birthday?'\n\nI smiled. I loved Mum telling me things about when I was younger. 'You'd taken it to school with you and you're not allowed to do that are you?'\n\nI shook my head. Mum's brush slipped.\n\n'Don't move!' Mum said, putting the brush down in the tray. She picked up a tissue and cleaned my face a little, saying, 'I had to go to school and get it back for you. All the way home in the car, you kept saying, \"I didn't know it was in the bag.\" And, \"So and so brings one, and nothing happens to them, and it's so unfair.\" When we got home, your Dad was snoring on the sofa. I went into the kitchen and you came in after me, with your new headphones plugged into your ears, holding your new toy. The music was so loud I could hear it even as I washed. I could hear your Dad shouting for something. And then he barged into the kitchen. You were so startled you dropped the mp3 player on the floor and it broke.' Mum dipped the brush into some black paint and asked, 'Do you know what you said to your Dad?'\n\n'No!' I lied.\n\nMum picked up a clean brush and said, swirling it around in the red paint, 'You turned around towards your Dad and shouted, \"You broke my mp3 player.\" He was so shocked, he nearly jumped out of his skin and he said, \"You dropped it, not me.\" And you said, \"It's because of you and Mum that I was born, and if I hadn't been born, I wouldn't have dropped it and it wouldn't have broken and I hate you.\"'\n\n'What did Dad say?' I asked Mum, hoping this time I might get the bit of this story that never seemed to come out.\n\nMum went quiet and continued swirling the brush in the red paint. She took a deep breath and said, carefully painting the outline for the teeth on my face, 'He didn't.'\n\nI wanted to say to Mum, I know he went quiet, just like you go all quiet, but instead I let out a false laugh and Mum continued painting my face.\n\nMum was about to leave to get some spray for my hair but I stopped her.\n\nPointing to the mirror I said, 'Mum, this is stupid.'\n\nDad was coming back up the stairs. Mum nodded towards Dad's footsteps and said, 'He's on the beer again.'\n\nIgnoring Mum, I said, 'I look ridiculous,' and rushed past her. I wanted to get to the bathroom before Dad went in. He was already at the top of the stairs. He saw me and screamed.\n\n'Dad!' I hissed back, ran into the bathroom and quickly washed everything off.\n\nI came back into my bedroom still drying my face with a towel. Mum and Dad were looking at each other. Mum had her hands on her hips and Dad was scratching his arm.\n\n'I'm not going,' I said.\n\nMum smiled, opened the bag on the bed and pulled out another mask. It was of Guy Fawkes, his white plastic face and his perfect black eyebrows, with his moustache going up towards his puffed, red cheeks and his sliver of a beard running down his chin from the middle of his bottom lip.\n\n'Mum, you are the best,' I said. It was perfect. I could just put it on and take it off.\n\n'Guy Fawkes, Sharon!' Dad said, rubbing his head with both his hands. 'What's wrong with that?' Mum asked in disbelief at Dad.\n\n'Yeh, what's wrong with that, Dad?' I asked, throwing the towel onto my bed.\n\nDad ignored me and said to Mum, 'He was a Catholic, Guy Fawkes was.' 'And?' Mum said.\n\n'You're one. Have you no shame, all this burning poor Catholic Guy every fifth of November?' Dad looked so serious.\n\n'It's not the fifth of November, if you haven't worked it out,' Mum said. She was having difficulty stopping herself from laughing.\n\n'He's the only man who has ever wanted to go to the Houses of Parliament for the right reason,' Dad said.\n\nMum burst out laughing. She pushed Dad towards the door saying, 'You're outrageous. Outrageous. Just go to the bog, you've had too much beer.'\n\nAs Dad walked out of my room, I tried to work out what was so funny, but couldn't and then I understood and shouted after Dad, 'You are just mad, Dad. Mad.'\n\nAs he shut the door of the bathroom Mum shouted at him, 'And just so you don't forget, Kiran is in the 100 metre finals at school next Wednesday. And you're going to cheer her on. I'm working late.'\n\nI put the mask on my face and said to Mum, 'He won't remember.' 'He'll be there,' Mum said as Dad went past.\n\n'He won't,' I said.\n\nDad looked at me, shook his head and said, 'He will. And if you win, he'll take you on a treat.'\n\n'You will take her on a treat, even if she doesn't,' Mum said. 'It's about time you spent some quality time with your daughter, and less with the lager cans.'\n\n'Bitter,' Dad corrected Mum. Mum dropped me off at the school party.\n\nThe assembly hall looked really spooky. There were dark, painted lanterns all around. Black drapes hanging down the walls. Large candles burned on the stage. The hall was full of Draculas; Spider-Men with fangs; witches and skeletons galore and to my surprise there was only one other Guy Fawkes. This other Guy Fawkes saw me and swaggered toward me followed by four broom-wielding witches.\n\nWhen Guy Fawkes got close to me, I said, 'Hey Guy, great minds think alike, eh.'\n\nGuy Fawkes's hands went up, as if in disbelief. The witches looked at each other and giggled.\n\nJust then, Guy Fawkes grabbed a broomstick off one of the witches and prodded me hard in the chest. It hurt.\n\n'How dare you!' Guy Fawkes said. It was a voice I knew well and dreaded.\n\nAs soon as I heard Shamshad's voice from behind the Guy Fawkes mask, my blood froze. She prodded the broomstick into me again, harder this time. I think it was a Year 8 who stopped me from falling. I hated my mask, but I was glad I was wearing it. Shamshad and her witches wouldn't be able to see my stupid, ugly face. I know I should have said something to her, told her what a bitch she was, but that was me, the good little girl, who never did anything wrong, just always wanted to please everybody.\n\nI moved away from her and her cronies as fast as I could. A Dracula with one of his canine teeth missing said to me as I rushed past him, 'Penny for the Guy.' A Frankenstein laughed. But I found nothing funny. I kept turning around, trying to see if Shamshad was looking for me. I waited for a little while and when I was sure she was no where near the main door, I bolted out of the hall and telephoned Mum to pick me up. I was so relieved when she said she was already in the school car park.\n\n'How was it dear?' Mum asked as I got into the car. 'Great.'\n\n'Didn't you want to stay longer and have fun?' Mum said. 'I was happy to sit and read the newspaper, done that for years.'\n\n'Yes, Mum,' I said, as Mum reversed the car out of the car park. 'I bet there were lots of Guy Fawkes?' Mum said.\n\n'Just one,' I said.\n\n'That's good then, you were unique.' 'No, there was one other.'\n\n'Who was that?' 'Shamshad,' I said coldly.\n\nMum looked away from me and went quiet. Just as well. I didn't want to talk to her, besides even if I did, she was now locked behind her silence. I was happy to think about what a mess my life was. What was it with Shamshad? She just never let up putting me down. She could be so bad and have so many friends and I was alone, always just being nice. I was good at home, I was good at school and I thought I was good with my gang but look at me. A laughing stock at school, and at home I never knew what I said that was so bad that _it_ was always lurking in some dark corner.\n\nSitting next to Mum, I watched the world outside the car flash by like a bad dream. I saw myself at school the next day, at the final, Shamshad's friends cheering for her. She was bigger than me, but clumsy. She always got a better time than me, but then she had everything I didn't.\n\n'Well, Kiran,' I said to myself inside my head, 'stop being like yourself for once; give her a piece of her own game.'\n\nUnlike me, Shamshad had to win and be the best at everything. I didn't care if I didn't win, I don't even know how I made it to the finals, but she always had to be first. A sly smile came across my face as I saw myself running next to her. She'd be really, really focused. I was going to make sure she didn't win. I would wind her up just by being close to her. And then, just as the race started I'd fall and trip her, and watch her roll on the floor. Perfect I thought. And then I thought, no, they would just start the race again, it was only 100 metres. Maybe I could go up to her and say some things that would really upset her, it was easy for me, I just had to open my mouth in her company.\n\nAs Mum pulled into our street, I had another idea. Why not beat her fair and square? Well, that is, after winding her up at the beginning.\n\nWhen we got home, Mum said to Dad, in an icy sort of a way, 'It's Kiran's 100 metre final tomorrow.'\n\n'OK,' Dad said from the living room. 'You don't have to come, Dad, its fine.' 'OK,' he replied.\n\nI flicked off a shoe; it landed in the middle of the hallway. I flicked the other; it went past the first and hit the kitchen door. Mum, who was halfway up the stairs, glared down at me. I glared back at her. No, I'm not going to pick it up, and I don't care what you think. Besides, you're in one of your moods, where you're just going to lock yourself away anyway.\n\nMum turned around without saying anything and went up into her bedroom.\n\nAs Mum slammed her bedroom door shut, Dad called out, 'Kiran, dear, can you get us a can from the fridge?'\n\n'No,' I shouted back to Dad.\n\n'Good girl, make sure you get one of the cold ones from the back,' Dad said.\n\n'Dad!' I hissed under my breath, thinking, 'What planet are you on?' I was half way up the stairs when I felt really guilty. Dad was just a big softy who didn't know what time of day it was, unless it was a match time. I stopped, turned round and was about to go and get him his beer when I felt a sudden surge of anger. No, he can get it himself, I thought, and went upstairs.\n\n## **Shamshad**\n\nLaila and Aisha turned up at the Halloween party as witches. Aisha, with her ridiculously long wobbly nose and rubbery chin, looked a sight. And Leila was wearing a long black burka, all torn into strips that dangled about her.\n\nI didn't see the other Guy Fawkes at first, it was Laila who did. She nudged me in the side and teased, 'Shamshad, look what Jake's turned up as.' 'He's too small and skinny,' I said.\n\nThe witches made some sort of spooky noise.\n\nLaila flicked her broomstick at me, reciting some things by way of a spell. The broom bit of her broomstick was just a bit of cardboard cut-out that had been hurriedly painted and sellotaped to a stick. It broke off and poked her in the eye. She yelped. We laughed. I laughed all the more at the way she was dressed in her stupid, little witch outfit. The only thing witchy about it was the black cloak; she could easily have passed for a fairy, with a bit of red paint on her face.\n\nAt first, when Laila pointed Guy Fawkes out to me, I was dead chuffed; at least I wasn't the only one. But when this Guy said in Karen's voice, 'Hey Guy, great minds think alike, eh,' I just flipped and prodded her as hard as I could. If it hadn't been for the stupid, red-headed, hairy monster of a clown behind her, she would have fallen flat on her backside.\n\nI was about to go up to her and give her a piece of my mind when Laila tugged my arm and said, 'Let up, girl, not worth it.'\n\nAs always, she was right. Me and my anger, we were never so bad until Karen came onto the scene. I took a deep breath, went up to Karen and said in her ears 'You pathetic copycat.'\n\nIf she didn't have her mask on, I would have been able to see her dirty little face for what it was, a no good, dirty little half-breed face.\n\nWhen she got her balance back, she turned her back on me and walked off into the party.\n\nAfter Karen left, Laila said, 'You know she's in the 100 metre finals.' 'No,' I lied. Of course I knew. Everyone knew, and ever since she had got into the same final as me, I had been running twice a day, just to make sure I beat her.\n\n'There's your chance to get one on her in front of the whole school,' Aisha Sadiq said, holding her stick in one hand and the broom head in the other.\n\n'She always wants to be the best,' one of the witches next to Laila said.\n\n'Yeh, Laila, why don't you put another spell on her,' I said. 'What, with a broken broom?' She replied.\n\nThe others laughed and I joined in. I was scouring the party, trying to see where Karen had disappeared to. Though I didn't see Karen leaving, one of me witches said Guy Fawkes ran out of the hall. Then she said, with broomstick between her legs, her bum wiggling about, 'Like it was bonfire night.' It was such a laugh.\n\nI got to school well early for the races, tied my hair up tightly in a thin, white cloth, got into my black leggings and a black and golden top, put on my trainers and warmed up in the changing rooms. I was still warming up when Laila and my gang came in. We did a few high fives and I went out into the fields. As I was walking out of the main entrance, I saw such a pathetic sight: Karen and her Dad, Lucky, coming towards me. They were late. Karen walked fast. Her Dad was panting behind her. He stopped a short distance away from the gate. Karen came straight towards me, bumped into me and said, 'Get out of my way, you fat cow.'\n\nI was stunned. Before I could say anything else, she was gone. My gang looked at me in disbelief.\n\n'That's it,' I said, clenching my fist. I turned around and was about go for Karen when the bald-headed Mr Armstrong, our P.E. teacher, came out. He was wearing his normal blue sports suit with white stripes. He had a stopwatch dangling in front of his chest.\n\n'Come on girls, why are you hanging about here?' he said, walking past.\n\nHe stopped, did a little jog on the spot, cracked his neck adding, 'And you need to get warmed up, Shamshad.'\n\n'Yes, sir,' I said, still reeling from what Karen had just said.\n\n'And good luck,' he said walking away, moving his shoulders up and down.\n\nKaren's Dad followed Mr Armstrong to the parents and teachers section of the playing fields.\n\nLaila held my hand and said, 'Come on.'\n\nWalking to the fields, I chewed my teeth and promised myself that I would get Karen for what she had just done.\n\nEveryone who was in the final had a parent or two at school. Walking away from my friends, how I wished mine would come to something at school, just once. I shook my head, trying to chase this silly thought away. Dad would never come, he never had the time and Mum didn't even know where my school was.\n\nKaren came out and ran to her Dad, gave him a hug and came towards us in the line-up. She smiled at the other kids in the final and walked straight up to me and said, 'Your leggings are sticking up your big, fat arse.' She looked me straight in the eye, nodded to my chest and added, 'And you should have worn something looser, you show-off.'\n\nBlood rushed to my ears. 'Oh Allah, stop me from flattening her right here,' I thought. I looked round to see if anyone else had heard what she said. No one else was close by. Laila waved at me. I turned around and Karen was in her place at the starting line, next to me, swinging her arms about.\n\n'Wait till this is over,' I hissed at her. She looked at me and tapped her behind.\n\n'Finalists, get ready,' Mr Armstrong's voice boomed out of the loudspeakers.\n\nThe kids let out a roar. I took a deep breath and tried to forget Karen. Then the starting gun went off. Just as we started, the gun went off again. A false start. A teacher went up to Karen and gave her a warning.\n\nAfter a few moments, Mr Armstrong's voice boomed out again, 'Finalists ready.'\n\nGo on Karen, I thought, do it again.\n\nJust then, the gun went and we were off. Karen was away so fast, but I quickly caught up with her. We were all neck and neck at first. I remembered Mr Armstrong's words, 'Hold yourself together for the first 20 metres and then let go.' I was blind to the others in the race. I could see Karen giving it all she had. She was a little in front of me. And then I blotted her out of my mind. I blotted everything out of my mind. I was going to win this. Twenty metres went by in a flash. Karen was a bit further ahead of me by then, but I gave it all I had and caught up with her. As I went past I saw her in the corner of my eyes, panting madly. I pushed on and on. She kept catching up. I heard someone cheering me on. I pushed myself even harder. Karen was a hair's breadth behind. I saw the finishing line. Mr Armstrong was there, slightly bent forward, holding the stopwatch in his hand. There were two other teachers with stopwatches. I ran and ran and ran as fast as I could and crossed the line. I raised my hands to the Almighty for granting me my wish.\n\nI looked for Karen. She was on her back, on the track, not far from me. I waved at Laila. My gang were jumping up and down with joy. Mr Armstrong wrote something down on his scorecard and conferred with the other teachers, nodded, pulled a cordless microphone out of his pocket and tapped on it saying, 'What an absolutely amazing race that was.' He paused, waved his hand towards the kids and added, 'Don't you all agree?'\n\nA deafening cheer went up.\n\n'And I am pleased to announce,' Mr Armstrong continued when the noise went down, 'In the third place, Jonathan West at 12.35, in the second, Susanne Williams.' He paused again.\n\n'You piece of trash,' I cursed Karen. 'Didn't make it, did you?'\n\nShe turned away from me. Mr Armstrong continued, 'Shamshad Khan, 11.29, what an absolutely amazing result. One of the best ever,' the head who was standing close by Mr Armstrong tapped him on the shoulder. He turned to her and the two of them looked through a piece of paper. Mr Armstrong's face lit up and he continued, 'I've just been told, that this is our new school record for the girls 100 metres.' A deafening roar went up. He waited for the noise to go down and said, 'And Kiran Malik, 11.29, congratulations to the joint winners.'\n\nSurely I've misheard, I thought, it can't be. I looked up into the sky. A big, rolling white cloud was merging into a collapsing face. Just then, I became enraged. I didn't care that everyone was watching. I clenched my fist and went for Karen.\n\n## **Kiran**\n\nI was flat on my back looking up at a big, white twisting and turning cloud, sinking into a sad face. From the corner of my eye, I saw Shamshad come towards me, grunting. I got up quickly and went towards Mr Armstrong. Dad was already there. He was out of breath. He hugged me and said, 'Well done, Kiran, you came second. Well done.' Mr Armstrong heard Dad and said, 'No, Mr Malik, She didn't.'\n\n'Well done anyway, Kiran,' Dad said.\n\n'She came first, along with Shamshad Khan,' Mr Armstrong said going back towards the starting line, where the next race was lining up.\n\nDad went silent.\n\nI looked around to see where Shamshad was. I didn't want to meet her in the changing rooms. She was on the other side of the track, surrounded by her cronies.\n\n'I don't feel well Dad, can I go home now, please?' I said. He looked right through me.\n\n'Dad!' I said, 'I don't feel well, I want to go home.' He nodded.\n\n'Can you please come with me to reception, and tell them.' He nodded.\n\nI looked around once again at Shamshad. She was staring at me. She said something to her mates and walked away from them. I knew she was coming for me.\n\n'Dad, please hurry up,' I said, walking back to school.\n\nHe quickened his pace. I ran past reception, bolted to the changing rooms, grabbed my clothes and rushed back. Shamshad was already in the hallway. Dad was talking to the receptionist.\n\nAs I went past Shamshad, she hissed, 'Just wait and see.'\n\nDad had snapped out of wherever he'd gone and said, walking out of the reception, 'It doesn't matter. I'm still proud of you.'\n\nI didn't reply. I was stung by the word 'still'. _Still,_ I thought, you mean because I came joint first or because I came joint first with Shamshad.\n\nGetting into the car, Dad said, 'And on Saturday, me and you are going out to celebrate, just the two of us.'\n\nI threw my stuff on the back seat and sat in the back of the car. 'D and D,' Dad chuckled. 'Daughter and Dad.'\n\n'Oh, yeh,' I thought to myself, 'D and D, that's a new one. As if you'll remember this on a match day.'\n\nOn the way home, I thought about the look on Shamshad's face when I said what I did to her before the start of the race. It was worth everything she was going to do to me. At last, for once, me, little miss goody two shoes, me, the little doormat, had slapped her in the face. Dad said something about what I wanted to do on Saturday. I ignored him. He started humming a Pakistani song. I was rummaging about in my bag looking for my mobile. I had a text from Mum: _How did you do?_\n\nI text her back: _Ask Dad._\n\nA moment or so later, Dad's mobile buzzed. He got a text. Still humming, he looked at his mobile and stopped humming.\n\nI was thinking about D and D when I got home. Dad had never said anything like that. I tried to work out why he'd said this. It wasn't just Dad, but Mum had also been different towards me ever since I wanted to be a Muslim. It certainly wasn't because of the race. Maybe they were trying to get me back on track, whatever that was. And the way Mum had sat with me, for Halloween, the way she'd stroked my hair and spoken, it wasn't like her usual, 'Yes dear, fine dear, love you dear, blah, blah, blah.' No, for once I really felt she was there, for me, with me, just for me. And now, I had a special Saturday coming up with Dad...\n\nI was taking a bag of rubbish out when Mum got back from work, but she didn't say anything to me. No hug. No, 'Hello, I'm back!' No 'Well done.' Nothing. She went straight into the living room, and yes, Dad was up to his usual, glued to the telly, beer in hand. She turned the television off. Dad didn't protest. They looked at each other, in a strange sort of way. Mum saw me standing by the door, walked past Dad and shut the door.\n\nI did what I do, when _it_ takes over our house and ran to my bedroom, jumped on the bed and said aloud, 'Well, Saturday, you're just a hoax.'\n\nBut Saturday did come. And before it came, Mum and Dad talked to each other, in heavy, sad voices. Waiting for Saturday, I tried to catch their words, but they were just words, everyday words, about shopping, and work and just normal everyday chores, but wrapped in something dark, something unmentionable.\n\nSaturday did come and by the time it came, _it_ was gone. And on this Saturday, Mum came and woke me up with a kiss. And this Saturday, Mum was wearing a beautiful dress. At first, I thought I was waking up in a dream, but then I heard Dad calling me, 'Breakfast ready in five.'\n\nI looked at Mum. She looked at me. I touched my ear, waiting for the smoke alarm to go off. Me and my Mum burst out laughing.\n\nAnd this Saturday brought a beautiful, sunny, blue sky. And Dad didn't watch the match. I floated about during the morning trying to work out the big secret that Dad had lined up for me.\n\nAll throughout the breakfast of burnt toast and greasy eggs, I kept trying to work out my surprise. Monday was a bank holiday and then we had a teacher-training day on Tuesday, so maybe D and D meant Dad was going to take me away on holiday, and if this was the case, we were off to Italy, I reckoned. After Man U, Dad loved A.C. Milan. Or maybe we would go to Bodrum, in Turkey. Dad's mates always went there. He often talked about going there, sitting on a beach, getting a boat and going out into the sea, fishing. Or maybe, just maybe, he would take me into town and get me a new laptop.\n\nFinally, we finished our breakfast and Dad said to me, 'I'm taking you somewhere really special.'\n\n'Where?' I asked.\n\n'Dad and his lovely daughter are going for a boat ride...'\n\n'Boat ride!' I said. I was filled with pride for having worked out he was going to take me to Turkey.\n\n'We are going rowing. I've booked a boat in Boarhead Park.' 'Boarhead Park. Rowing,' I thought and said, 'Great, Dad.'\n\nAnd a-rowing we went, Dad and me, in the lake in the middle of Boarhead Park. I sat stiffly in the back of our car, while Dad went on about how wonderful it was for the two of us to spend time together, just me and him.\n\nAt one end of the lake, there is a small caf\u00e9 and next to this is the office for hiring boats. Dad bought me an ice-cream and whilst I sat there stabbing it with a small white plastic spoon he went off to sort out the boat. There are two islands in the middle of the 's' shaped lake, one in each of the curves of the 's', where ducks and geese and other birds breed. Along the banks of the lake there are big oak trees, some bent towards the lake, others so big and tall their shadows fall across the waters like large monsters, moving over the ripples from the oars of passing boats and the swimming ducks and geese.\n\nBy the time Dad came out, my ice-cream had been reduced to a gooey mess.\n\n'Ready?' Dad asked. I nodded.\n\n'Like the ice-cream?' He asked.\n\n'Yeh, Dad.'\n\nHe went so confidently towards the boat, like he had been rowing all his life and stepped into the boat with such force that it almost capsized. A boat assistant ran up and held the boat whilst Dad got his balance back.\n\nThe boat had two places to sit. One next to the oars and the other in front of that. I sat in front of Dad. The oars were placed on either side of the boat. Dad picked them up, put them into the metal holders and rowed. As he did this, he stood up, took some keys out of his back pocket and sat down again. The boat shook from side to side.\n\n'Careful, Dad.' I laughed. 'Mum says you swim like a brick.' 'Less of that!' Dad said rowing, 'there'll be no need for that.'\n\nI was surprised. He was so good at rowing! He used both oars together as we pulled out and then used one to turn and the other to move forward and make sure we missed another boat, full of loud-mouthed teenage boys, which was coming at us with full speed. As we went into one of the turns, Dad's rowing became smooth and methodical, the oars going into the water, pushing the boat through it, coming out, dripping, and dropping back again, sounding almost musical.\n\n'He's not such a clumsy oaf,' your Dad, I thought, looking at his big smile on his big, beautiful, fat face. His big, beautiful belly was popping out of his shirt as some of the buttons had become undone. He closed his eyes for a moment and started singing in his own language, a song I had heard him hum many times but never heard him sing.\n\nI wished I could understand, but even without this, it felt so good, listening to him singing.\n\n'What does it mean, Dad?'\n\nHe smiled and said, 'Do you remember when I went to Pakistan, when my friend Aziz died.'\n\nI tried to think. I remembered him going, but not much more than that. I put my hands between my knees and nodded.\n\n'I used to go to the same school as him when I was young. We were like brothers, we were. He was a poor man, who worked in the hills behind my village, for a stone contractor. He used to load trucks with rocks. One day, when he was loading a truck, there was a landslide and he was crushed to death.'\n\nDad went silent like he was back there. Letting out a deep sigh, he continued, 'It took them two days to move the rocks and get his body out. I got to Pakistan as his body was brought to his house and helped to give him his final _ghusal_ , his final wash. When the time came to carry him to the graveyard for his funeral, his wife came and insisted on helping to lift his body out of her house. That stupid old Imam Butta said women weren't allowed to do this, but she ignored him. I let her hold her husband's bed and she helped to lift it up and placed it on my shoulder. She started walking with the men and Imam Butta told her to stop. Women aren't allowed to do this. She ignored him and walked with us. I thought she would stop where the other women stop, just where the last house of the village is and then they come and stand outside the graveyard until the men finish reading the funeral. But she didn't stop. She kept walking. When everyone recited religious chants, she kept quiet. Not crying. But silent. Along the way a woman tried to hold her hand to stop her going, she snatched it free and kept on walking with us.\n\n'Just as we got past the last house, she sang in a voice so loud and so clear it tore right through me.'\n\nDad looked at me with sad, sad eyes and sang: _'Baghe ander hik bulbul alarnaan paee see banandhi Ajay na charya toor Mohammed, ud gaee ay kurlandhi.'_\n\nMy Dad sang these lines so beautifully, in a voice I didn't recognise, a voice that came out from some depth within him, a place where he didn't go very often.\n\nBefore I could ask him what it meant, he translated: 'A nightingale was building her nest in the garden, Oh Mohammed, it had yet to be finished and she flew away screaming.'\n\nDad looked behind and rowed away from some branches that were hanging into the lake. Pulling out into a sunny bit of the lake, he looked at me and said, 'I love you, Kiran.'\n\n'Love you too, Dad, now, can you sing it again?' He did.\n\nAnd then I asked him to sing it again and he did.\n\nI hadn't noticed we had done a full round of the lake and were going round again.\n\nAs we went past the first curve, I saw Shamshad standing close to the bank, near some bushes. She bent down, picked something up and held her hand out as though to lob it at us. Just then I saw a startled bird fly, screaming out of a tree above her.\n\n## **Shamshad**\n\nOn the way back home from the park, each time I saw a bird, I thought I saw the one that had screamed out from above me, and each time I heard it, it was as though it was the first time I'd heard it. Had I not heard such a frightful cry of a bird above me, a cry that cut deep into me, I would've lobbed the stone at Karen.\n\nThat night I kept falling in and out of sleep, waking up all sweaty. In my dreams, I kept seeing a thick, rolling cloud, changing into faces, mocking faces, faces I should know, but to which I could put no name.\n\nAfter putting on my bedside lamp, I opened the bottom draw of my chest of draws, carefully so as not to wake anyone else. It is a deep draw, with a secret section, which I made myself by placing a bit of matching plywood at the bottom. In this, I keep my secret, secret things, especially my drawings.\n\nOn a blank A4 piece of paper, I sketched the outer lines of the cloud.\n\nThe lines got thicker and deeper, as though my hand had a mind of its own. A face began to form, a disfigured, horrible face, with accusing eyes, fiery eyes, sorrowful eyes and then just a dark smudge.\n\nI felt cleansed after I finished this drawing. I took out my notebooks from the bottom drawer, lifted the plywood false bottom, placed the drawing on top of the others and went back to sleep: a deep, peaceful sleep.\n\nI woke up late the next morning, glad I wasn't going to see _her_ ugly face for a few days. I didn't know she would end up taking my best friend from me.\n\nOur central mosque is in the middle of Boarhead East. Surrounded by boarded up mills and pubs that have long since closed, their yards full of discarded furniture and rusty fridges. Our mosque used to be a church. _Alhamdulillah_ , praise be to God, it is a mosque now. To get inside, you have to climb many old steps and walk in through tall, wooden doors. A balcony runs around the main hall. On the sides of the balcony are smaller rooms in which we have lessons. There is a smaller hall for women at the back of the main hall.\n\nIt was a bright, sunny day and I went to the mosque early to help clean up after Friday prayers. Laila was with me. She didn't usually come here on Fridays. She went somewhere else, a small place near the college where Muslim students prayed and discussed things. We were waiting for the caretaker to come and open up the study rooms, so we sat on a bench on the balcony, looking out as the mosque began to fill up. It was the wrong time of month for me and Laila, so we weren't going to pray anyway. I didn't notice her at first. She was sitting on her own not far from us. Her face was turned away, so I couldn't tell who it was. There was something oddly familiar about her. The strange thing was that she was wearing the same green-coloured hijab as me and a long-sleeved, white shirt, exactly the same as mine; even her black trousers were the same as mine. I prodded Laila and nodded towards her. Laila shrugged her shoulders. I coughed loudly, hoping to get her to turn this way, but she sat stiffly, looking down into the main prayer hall. I coughed again. This time loudly. A few men looked up at me from the hall. I pulled back out of their sight.\n\n'Stop it,' Laila whispered.\n\nThe girl didn't move. I stood up.\n\n'What you doing?' Laila asked.\n\nI nodded for her to follow me towards the girl. She did. I took a few steps towards the girl. She took her hands off the balcony and pressed them together.\n\n'It's her,' I whispered to Laila. 'Karen the flashing _hijaban_.'\n\nLaila didn't reply. I sat down next to Karen. Laila next to me.\n\nI pressed my elbow into Karen's ribs and asked her, 'What's it with you, Karen? What do you want from us?'\n\n'Please Shamshad, I just want to start again and be friends,' she said.\n\n'You don't belong here,' I said. Even the sound of her voice made me go into a rage. 'And what's with you dressing up like, eh?'\n\n'I wasn't, I swear, I wasn't...'\n\nLaila interrupted, 'Leave her alone, Shami.' Turning to me, Laila said, 'Everyone is welcome in the house of Allah.'\n\n'I'm with my Dad, and I am going to come here and I'm going to learn, and my name is Kiran.' Karen said.\n\nJust then the prayers started. It was just as well. I was so vexed with Karen, or Kiran, as she wants to call herself for now, I could have smacked her right there. What did she think it was? A fashion item! You are Karen one day and Kiran the next? You sneer at us, call us scarfies one day, and become one the next? You go to church one day, claiming God had a son, and the next you come here and you can say, 'No, God was not a man, he had no son.' You hang around with your big, ugly gang, you neck them, and they rip our hijabs off our heads, but now you come here all innocent, like?\n\nI looked down into the hall and recognised her father. He came to our street once a week, every Thursday without fail. He was called Lucky Saab on our street. He was in the fourth row down from the front, towards my end of the balcony. It was hard not to miss him, with his long arms and his big stomach. He was doing his _sajda_ , touching the floor with his forehead. His shirt had rolled out of his trousers, exposing his builder's bottom. Like daughter, like father, I thought. Shameless! As Mr Lucky Saab raised his head out of the _sajda,_ he touched the side of his trouser leg and then raised his fist in the air, looked around, then quickly went down into the _sajda_ again with the rest of the line. 'How dare your father do this in the mosque,' I hissed into Karen's ear. 'Disgusting, all of you!'\n\nKaren didn't answer. As the prayers were about to finish, Laila and I left.\n\nThe room in which we read the Quran has long, stained glass windows that curve around the side of the building. We squat on the floor beneath the windows, leaning against the wall, with the Holy Quran in front of us. Girls and children sit in this room. Older boys read next door, in another room like this. I have read the Holy Quran three times and I am on my way to memorising the third _spirah_. _Insha'Allah_ , god willing, I am going to learn the holy book by heart.\n\nOnce a week I teach younger children how to read the Holy Quran. Laila had gone to the bathroom and I was laying out stands for the children's _Qaida_ s when my Dad came in and said to me, 'Someone new is joining today.'\n\nI looked up and put my hand to my mouth. I just managed to stop a squeal turning into a loud ' _OMG_.' Karen was standing next to him. Dad looked at me accusingly. I wanted to stand up and say, 'Dad she's not my friend. I hate her. I had nothing to do with her coming here.' He adjusted the white-laced prayer cap on his head and left, tapping a small cane in his hand, which he always carried in class.\n\nI looked at Karen and thought, you're not getting away with whatever you're up to, I'll get you! She turned her face away from me, looked up at the windows, then down towards the row of stands, and then towards the storeroom in the corner. She had the same headscarf on as she did when she came to school in her miniskirt, but today she was wearing a white-coloured, full-sleeved top with loose-fitting, black trousers.\n\nLaila came back, and I said to her, 'Look what the cat brought in.'\n\n'Hi, Karen,' Laila said, stepping into the room past her.\n\nKaren nodded, and said, 'It's Kiran.' She locked her hands in front of her and tapped her right foot.\n\nLaila smiled, tapped her own head, and said a bit louder, 'Hi Kiran.' I was upset with Laila for putting me down in front of Karen, and said, 'You will always be Karen to me.'\n\nI wanted to smack Karen right there, on her nose, and throw her out of the mosque.\n\n'Where do you think you are, in a disco?' I said to her, nodding towards her foot.\n\nShe looked down at the ground and started tapping her right foot. 'Come on in Kiran...'\n\nI gave Laila such a stiff stare she stopped mid-sentence, but then she looked away from me and said, 'I'll teach you today.'\n\n'You've only been here a minute, Karen,' I thought, 'and you've already taken my friend from me.'\n\n'It's my class and I'll teach _her,_ ' I said to Laila. Turning to Karen, I pointed to the far side of the room, and said, 'Over there, with the babies.'\n\nAs Karen went over to where I had told her to sit, the children came running noisily up the stairs and went to their places. Laila placed a _siparah_ in front of her. Kiran sat down and crossed her legs. 'Start your _sabaq_ , your lesson,' I said to the class, and the room burst into the beautiful sound of Arabic from the mouths of the children.\n\n'Keep still, Karen!' I said, sitting down in front of her.\n\nShe looked round the room, but didn't say anything back to me. 'Open the _Qaida_ ,' I said.\n\nShe did, and before I had given her the first lesson, she started reading, ' _Alif zabr Aa - Alif Zair Ae; Bey zabr Ba..._ '\n\n'Stop! Stop! Stop!' I shouted.\n\nEveryone in the room went silent. Some of the children giggled. Turning to the kids, I asked, 'Who told you lot to stop?'\n\nThey burst into rhythmic recitations again. Seeing my Dad standing at the door, the children increased their volume.\n\nPointing to the letters in front of Karen, I said to her, 'That's not proper, _Karen_. It's _Aa_ , _Ba_...'\n\n'That's not how my Dad taught me,' Karen said.\n\n'What's that rubbish collector know? 'I whispered to her, 'He's just trash.'\n\n'Don't insult my Dad,' Karen said, grinding her teeth. 'Trash! Trash! Trash!' I taunted.\n\nMy Dad came up to us. I moved back a little. He held the cane in his hand behind his back. He asked, 'How's Lucky Saab's daughter getting on in her first _sabaq_?'\n\n'She refuses to learn the proper way to pronounce the Arabic letters,' I said,....'\n\n...'Pindu villagers,' Dad said to himself, 'And England did nothing for them. 'That's how my Dad taught me, sir,' Karen whimpered.\n\n'Imam Lucky Saab did, did he?' Dad laughed. 'Dad, you're proper, brilliant,' I thought.\n\nThe children lowered their voices. Dad looked around the room and they started reciting loudly again.\n\n'My granddad once taught me like this,' Karen said.\n\n'Yes', I thought to myself, clenching my fist, 'go on, say more, _Karen_.'\n\nDad brought the cane out into his right hand. The children recited even louder. 'Go on, Dad', I thought, 'smack her one.' Dad pushed the cane towards Karen's chest. He moved the cane around her chest and slowly took it up to her chin, forcing her to lift her head up. He said, 'I know that _kafir_ very well...'\n\nI caught a glimpse of Laila. With her hand in front of her mouth, she was staring at Dad, shaking her head.\n\n'Don't you call my granddad a _kafir_!' Karen shouted and jumped up. She was crying.\n\nThe children stopped. Dad stepped back from her. She lowered her head and ran out of the room. Laila went after her.\n\n## **Kiran**\n\nLaila walked out with me. We went quickly down the bare wooden stairs, the noise of the children reciting their lessons fading slowly away, into the echoes of our feet. I felt so angry with myself for going to the mosque and for not pushing the cane away. I felt dirty inside.\n\nIt felt as if the walls of the mosque were caving in on me. Someone laughed somewhere. It bounced around me. Laila tried to hold my hand. I snatched it away.\n\n'It was disgusting what he did with the cane,' Laila said, 'disgusting.' I cringed. A shiver ran down my spine.\n\n'I'm alright, you know,' I said to Laila, barging out of the doors of the mosque. Stepping into a gusty wind, I said, 'You don't have to come with me.' Patchy, dark clouds were blotting out the sun. There was a slight chill in the air.\n\n'You're so brave you know, and I never did thank you for standing up for me the other day.'\n\n'I didn't,' I said, 'I just hate Donna.'\n\nLaila pressed on my hand, and we walked on. Dad was leaning against the wall at the bottom of the steps, glued to his mobile, oblivious to the litter flying around him. Clenching his fist, smiling, frowning. He had an earphone in his ear, which was plugged into the mobile.\n\nWhen I got close to him, I pulled the earphone out of the mobile. He was listening to a football match commentary. He looked at me in disbelief. His unshaven face twitched. It was like seeing the face of a little boy who'd had his candy snatched from him.\n\nLaila looked at me and we burst out laughing.\n\n'What's so funny?' Dad asked. 'It's a girly thing, Uncle,' Laila said.\n\nWe laughed more. And then I gave Laila a big hug and burst into tears.\n\n'What's the matter, Kiran?' Dad asked.\n\nGetting myself together, I pulled away from Laila, wiped my face with the corner of the hijab, and said to Dad, 'Like Laila said, Dad.'\n\nWe burst out laughing again. Dad plugged his earphone back. A moment or so later he pulled it out and threw it on the ground, then bent down, picked it up and put it in his pocket, saying, 'I can't believe it. I just can't.'\n\n'He always says this when Man U lose,' I whispered into Laila's ears, 'He's going to clench his fist and say exactly the same thing again.'\n\nHe did.\n\nWhen he calmed down, I pointed back to the mosque, and asked, 'What were you doing in there, Dad?'\n\n'Praying, of course.'\n\n'What was that with the fist?' He leaned closer to us, lowered his voice, shrugged his shoulders like a child, and said, 'I'd set the mobile to receive a message when Man U scored. How was I to know we would lose 6\u20131 to City? I can't believe it, I just can't!'\n\n'Dad, you're terrible!' I said.\n\n'I didn't mean to raise my fist,' Dad blushed. Well, the closest to blushing I have ever seen him. 'I felt really silly when everyone else went into _sajda._ '\n\n'Didn't anyone say anything to you, Dad?'\n\n'When praying, we are meant to be so deep in prayer, we're supposed forget about the world around us,' Laila answered.\n\nDad nodded, sheepishly.\n\n'Your builder's bottom was showing, Dad.'\n\n'Maybe Uncle, you could get braces like some of the old men,' Laila said, with a giggle.\n\n'Hey, less of the _old_ , young lady!' Dad said, pushing himself off the step. 'I'm not coming back here again, Dad.'\n\n'You mean the Almighty got a free prayer out of me,' Dad smirked.\n\n'Uncle, you're unbelievable!' Laila said. Just then, the sun came out.\n\nLaila put her arms around me, and said, 'You can come with me and learn.'\n\n'Dad, you go home and I'll stay with Laila for a bit.'\n\nDad was clearly relieved. He put his hand in his pocket, pulled out a five-pound note, and said handing it to me, 'Don't be late.' 'I won't,' I said.\n\nAs he turned around to leave, Laila said, 'You can come along to the lessons as well, Uncle.'\n\nHe waved his hand dismissively in the air and walked on.\n\nWe crossed the road and as we stepped onto the pavement, Laila pulled her headscarf off, and shook her head. She had shoulder-length, sharply cut hair. A green ribbon was tied across her forehead. Her star-shaped, silver earrings flickered in the bright sunlight.\n\n'What are you doing?' I asked.\n\nFolding the headscarf into a small ball, she said, 'Sometimes I feel like being covered, sometimes I don't. Right now I don't.' She put the headscarf into her pocket, and said, 'You can take yours off if you like.'\n\n'No, I feel better with it on,' I said.\n\n'I know what you mean,' Laila said. After looking round to see if anyone was watching, she adjusted her bra, and said, 'Sometimes when I wear my hijab, I make them stick out.'\n\nPointing to my chest, I laughed, 'Not much I can do with these.' 'Men can't keep their eyes off women in hijabs,' Laila said, 'especially _goray,_ you know.' We stopped talking as a Boarhead Operational Services road-cleaning van went past us, its brushes sweeping up everything from the sides of the road. When its noise faded, I told Laila how, when I'd been little, Dad told me he worked for the local council. I was so proud of him. I thought he had the most important job in the world. He told me he worked for Operational Services, which made sure that everything was kept clean so that there were no diseases like the plague. Then when he got promoted, he told me he was now like a pilot of a plane. When I got older, I realised that he was really a dustbin man. And now he drives the big collection truck. And he loves being called Lucky. Laila and I laughed and laughed till our sides ached.\n\n'I feel like you're me best friend,' I said after we got our breath back, 'and I hardly know you.'\n\n'Me Mum and Dad got married and hadn't seen each before that,' Laila said.\n\nWe broke into laughter. When we stopped again, Laila sucked her teeth and said, 'Shamshad told me she hates you 'cause she says you're half-caste. 'Did she really say \"half-caste\"?'\n\n'And she hates you for leaving Islam...' 'But I am trying now, aren't I, Laila?'\n\n'She's just a messed-up kid. But tell me, what's it with your family and Shamshad's?'\n\nI thought about it and tried to work it out so it would make sense, 'I don't know where to start.'\n\n'Just say it,' Laila said.\n\nI took a deep breath and said, 'Well, my Dad's family and her Dad's family hate each other.'\n\n'Well, who doesn't know that,' Laila said. 'I heard your Dad and her Dad were once best mates.'\n\n'I heard that too, but it's one of those things we don't talk about at home. You know how Mums and Dads can be when they don't want you to talk about something...'\n\n'I know what you mean, in my house', Laila interrupted, 'if we're watching telly and someone kisses on the screen, everyone looks away and the men start talking about cars and the women about cooking. And as soon as the kissing stops, they start watching again and pretend as though nothing happened.'\n\n'Seriously though', I continued, 'I don't really know what happened between our families, but I think someone from their side in Pakistan fancied someone from our side. Their someone got engaged to someone from our side against everyone's wishes, and our someone charged in on the wedding night and took away their someone. There was a shootout, like, and people got killed.'\n\n'Oh my god,' Laila said.\n\n'It gets worse,' I said, 'You know Shamshad's lot have a lot of their clan here.'\n\nLaila nodded.\n\n'Well that wedding thing happened yonks ago...'\n\nLaila interrupted me, 'But here's here and there's there, and it was a long time back so what's it with all the agro' between your families now?' 'Well, things never let up back there and a few years ago, I'm not sure exactly what happened but someone from Shamshad's side there and three of her cousins from here went and killed someone from our side.'\n\n'It was on the news here, wasn't it?!' Laila exclaimed, 'I didn't know it was your lot.'\n\n'So you know why she hates me, then,' I said. 'Bet your lot aren't as bad as ours.'\n\n'Our lot, if they have a big meal together, can't digest it without a good scrap afterwards,' Laila said.\n\n'But they don't kill a few people...'\n\n'If something like this happened back there, in our part, they wouldn't kill a few people like your lot,' Laila paused and then laughed, 'They'd wipe the whole tribe out.'\n\nWe both laughed and laughed like we'd lost it. When we stopped laughing, I said, 'But then I heard another story, Laila. You see they own a lot of land in the village and we worked for them. When my granddad or my great granddad refused to work for them, their side came and took some girls from our side and someone got killed, maybe at a wedding...'\n\n'When did all this happen?' Laila asked.\n\n'About a zillion years ago,' I said, and we laughed a bit more.\n\nThen Laila asked, 'Have you ever asked your Dad what really happened?' 'I did once or twice, but he said what he usually does,' I said, doing the best impersonation of Dad I could, 'It's all water under the bridge.'\n\n'You know, Kiran,' Laila said flicking her hand in the air. 'None of it all matters, really.'\n\n'What doesn't?' I was trying to find a way of saying something I really, really wanted to tell Laila.\n\n'All this stuff. I mean, what does it really matter?'\n\n'No, there's something else, you don't know Laila.' 'Tell. Tell.'\n\n'I don't know how to tell you.'\n\n'We're mates, you can tell me anything,' Laila said, placing her hand on mine.\n\n'It's not that I want to keep a secret from you,' I said. 'There's something else between us and them, I mean between my family and Shamshad's. You know when old folk talk, well, you know how they all talk about each other.'\n\nLaila let out a _don'tIknowit_ sort of a laugh.\n\n'But if I am ever around and someone mentions Shamshad's family, a strange frightening silence comes into the room. They all go quiet and don't look at each other. It's like there's something they really don't want me to know.'\n\n'They're all like that,' Laila said.\n\n'Who are?'\n\n'Grown ups. They're like kids. They think they can keep a secret when everyone knows about it.'\n\n'But I want to know what it is!'\n\n'That's what I mean,' Laila said, 'You want to be like the grown ups.'\n\n'How?'\n\n'You wanna be a kid.'\n\nWe were giggling away in our own worlds when I noticed Donna and Chloe up the road, coming towards us. There were a couple of boys and another girl with them who I didn't recognise. I elbowed Laila and said, 'We've got trouble.'\n\n'It's your gang again,' she replied. 'Yeh, my gang,' I replied.\n\nDonna and her crew stopped before they got to us. They started discussing something among themselves and then walked menacingly towards us.\n\nI felt a strange tingling on my neck. Like I was being watched. I looked back at the mosque. Someone was looking at us. I blinked and there was no one there.\n\nThe mosque's doors opened and a load of men came out.\n\n## **Shamshad**\n\nWhen I saw Karen and Laila on the other side of the road from the mosque, I hated Karen more than anything in the world. She had stolen my best friend. I prayed that Karen would be given a hiding by her old gang. I recognised Donna even from where I was. But it was not to be. Too many people came out of the mosque at the same time.\n\nI went back into the class and started snapping at the kids as they tried to recite the Holy Quran. Part of me wanted to go out, right there and then and give Karen a piece of my mind and part of me wanted to let Laila know what I thought of _her_ now.\n\nThe day dragged on and on, and no matter how hard I tried to forget about what had happened, I just couldn't do it.\n\nOn the way home, I sat in the back seat of the car thinking about what Dad did to her with the cane. Why didn't he just hit her with it? Why did he lift her face up, as if there was anything in it to see?\n\nLooking at me through the rear mirror, Dad said, 'You know you are not allowed to have anything to do with any member of Lucky's family.'\n\nHow many times I had been told this. The reasons were all jumbled up inside my head. We were _Jats_ , landowners, with lots of land in Pakistan. Three of my cousins are in jail over there. Someone in Lucky's family had dishonoured someone in ours. Somebody died in Pakistan. We hated them. They hated us. And none of them could be trusted. But there was something else, something unspoken about us and them. Everyone knew my Dad and her Dad were once friends, but this was never talked about, in what little was talked about in my house. Even the mention of their family name sent the silence of my house into a deeper well. I wanted to know what this was, but I knew I was meant to keep well away from Karen.\n\n'I hate her Dad, I hate her!' I snapped. 'I don't know why she came.'\n\nDad released me from his stare and I looked out at a woman pushing a baby in a pram. The baby was crying. The woman gently lifted it out, kissed it and hugged it. I wanted to be in a pram with Mum pushing me.\n\nDad parked the car outside our house, leaned over and looked at me. I was expecting him to tell me off, like he'd read my thoughts, but he had a warm look in his usually cold, black eyes. His thick, raised brows showed the grey roots of the blackened brows.\n\n'I've been meaning to buy you this,' Dad said leaning back to the space in front of the front passenger's seat.\n\n'You've finally got me a new laptop, Dad,' I thought, 'you're so transparent.'\n\n'I know you might not believe me, but I am so proud of the fact that you came top in the school chess tournament,' Dad said putting his hand into the bag. 'That was last year Dad,' I thought with a sinking heart.\n\n'We invented _shatrange_ , chess, and not the _goray_.' Dad paused to look at me for a reaction.\n\nI gave him the best false smile I could muster thinking, 'Yes Dad, you've told me this a thousand times.'\n\nDad continued, 'Chess is such a great way to learn about strategy and planning. And I still remember how you taught me that the quickest check, what do you call it, is foolish mate...'\n\n'Fools mate, Dad.'\n\n'Yeh, fools mate, that's in four moves.' 'Two, Dad.'\n\n'Yes, the old mates of mine are getting into a bit of chess and I want to show them how it can be done in four.'\n\n'It's in two, not four, Dad.' 'No, that's impossible.'\n\n'Come on, hurry up,' I thought. I think I know what's in that bag. Handing me a chess set, Dad said, 'These pieces were hand-carved in Pakistan.'\n\n'That's really great, Dad,' I said taking the chess set from him. 'Just what I wanted.'\n\n'Really! I knew, I just knew,' Dad said. 'Will you show me the moves?'\n\n'White pawn to F3. Black pawn to E6. White pawn to G4. Black Queen to H4. Check mate!' I said quickly. Dad smiled, a stupid _Ireallydontunderstand_ type of a smile.\n\n'The great thing about playing chess is that you have to plan out what you want to do. Life is like that. It requires hard skills. We have to set our goals and follow them. Do you understand, Shamshad?'\n\n'Yes Dad, I understand.'\n\n'Love is a soft thing, and to survive in this world we need hard things, strength, thinking tactics. Love is beautiful, but it is not enough. Though I do love you, Shamshad, I want you to be strong.'\n\n'I understand, Dad. And did you just say you love me?' Dad smiled. 'I did.' He looked as astonished as me.\n\nJust then, Dad's mobile rang. I went inside. As I opened the door, I saw Mum walking out of the living room, heading for the kitchen. Skype, I thought. If it was just me and she noticed I had come in, she would have shut the door and carried on. But when Dad came back, she always turned the computer off and entered into the kitchen.\n\nDad walked in. He was still talking on the mobile, his face beaming.\n\nGiving me a twenty-pound note, he said quickly, 'Second sale in two days. Things are picking up for Medina Estates.'\n\nI took the money and stepped aside as he walked into the living room. My Dad's company, Medina Estates, was the oldest estate agency in Boarhead East. It wasn't just an estate agency. We did money transfers to Pakistan and sold airline tickets, rented property and sold insurance.\n\nHe was in such a good mood today, I saw my chance. Mum was bringing in his tea. I took the tea from Mum and gave it to Dad. He was still on the mobile. Putting the tea on the coffee table in front of him, I prayed inside my head, 'Please God make it another sale.'\n\nI sat down in front of him, fidgeting with the twenty-pound note. When he finished, I asked, 'Another sale, Dad?'\n\n'No, silly,' he said. 'Baba Zaman just had a baby son.'\n\nEveryone knew Baba Zaman. He was the first one to move to Boarhead from Pakistan. He walked with a bent back. He coughed all the time, spitting phlegm on the floor. He was on wife number three or four; the others were all buried in the graveyard.\n\n'How's the baby?' I asked.\n\n'Alright, I think,' Dad said sadly, sipping his tea. 'It's a girl.' He quickly added, 'Such is Allah's will.'\n\n'And the new Mum?' I asked.\n\n'She's young, she'll be alright,' Dad said. 'You're asking a lot of questions today.'\n\nI don't know where I got the courage from, but I said, 'I'm fourteen now Dad, not young any more...'\n\n'An old woman, eh,' Dad interrupted with a laugh.\n\n'It's not important, Dad, but it's been bothering me, like. Were you and Karen's Dad once best mates? I asked Mum and she said to ask you.'\n\nDad called Mum, 'Sakina, in here now!' I stood up and stepped toward the door.\n\nMum came running in. Dad took his shoe off and held it in his hand, shouting, 'What have I said about talking about that family?'\n\n'She's growing up. She keeps asking questions. I always tell her to ask her father, you know best,' Mum said.\n\n'How dare you disobey me!'\n\nDad stepped towards Mum. He raised the hand with the shoe. She lowered her head.\n\n'It's my fault Dad, honest. I couldn't help it. I'll never ask again,' I said. He looked at me with raging eyes.\n\nI ran upstairs, plugged my headphones in and listened to Lady Gaga.\n\n## **Kiran**\n\nEven though Shamshad was always horrible to me, I still felt sorry for her. When I told this to Laila, she said, 'Maybe she was found in a dustbin?'\n\n'Maybe,' I said thinking about how my Mum could just flip from one mood to another.\n\n'Mums,' Laila laughed, 'can't live with 'em.' 'Can't live without 'em,' I added.\n\nLaila and I chatted all the way to Boarhead College. I told her what Donna had done to me. Laila shook her head in disbelief. It was like we'd been best friends all our lives. I had never been to the college but Laila knew her way round. We went down a flight of stairs, along a corridor into the women's toilets and Laila taught me how to do my _voozu_ , to clean myself up before prayers. After that, we went into a large room. There were kids from East Boarhead, people from Africa and even some white people. Most of the women wore hijabs. The men had beards. Some of them wore long skirts; some wore _shalwars_ that ended above their ankles. Someone tapped on a microphone, and a white man, in his early twenties stood up on the platform. He had light-brown, unkempt hair and a small, stubbly beard. His thin face reminded me of the pictures of Jesus. Everyone went silent. He gave the call to prayer, the _azaan_. I had heard this at my grandfather's and often on the television, but never as beautiful as this. It was like he was singing something that was coming from deep inside me. After the _azaan_ in Arabic, he did it in English. By the time the _azaan_ finished, the men had placed mats on the floor. Some people sat at the back but most stood in neat lines on the mats. Women on one side, men on the other. I stood next to Laila and we all prayed.\n\nAfter the prayers, we all squatted down and the white man who had given the _azaan_ began to speak: 'Brothers and sisters, may peace be upon you all. How many of you heard the _azaan_ in English for the first time?'\n\nAlong with a few others, I put my hand up. He looked around, stroked his beard, and said, ' _Allah O Akbar_ , Allah is great. It means no one is greater than Allah. This means we are all equal in the eyes of Allah. When we prostrate, we do so as equals. In this equality, we seek peace for us and justice for mankind. And why do we get together five times a day to pray? This is so that five times a day we have a chance to get organised and find peace.'\n\nHe paused and pointed at the posters around the room. The words, Kashmir, Chechnya, Iraq, Afghanistan, Somalia, Lebanon and Palestine were written in large letters. Underneath them were pictures of destruction and horrible injuries.\n\nHe continued, 'And we pray for an end to suffering. For those of you who have come here for the first time, I would like you to note there is no picture of God, nor of our Holy Prophet, may peace be upon him, nor of Jesus, may peace be upon him.'\n\nHe paused again, and said with a sly laugh, 'And he was an Arab. Not a blond-haired, blue-eyed boy from up the road. And don't any of you dare to think I look like him! 'Someone laughed. I froze thinking he was talking about me.\n\nHe continued, 'And Jesus, may peace be upon him, felt it his duty to stand up against Rome...'\n\nA mans voice from the audience interrupted, 'Didn't Jesus say, \"Render unto Caesar that which is Caesar's?\"\n\nThe speaker brushed a lock of hair off his forehead, smiled at the questioner and nodding said, 'It's true brother. But Jesus also said, \"And unto God, that which is God's. And what is God's? Life is God's. Everything is His.\" But what did Rome think? That he was a threat. Yes it did. And did he not stand up for God? Yes he did. Did he not stand up against the West? Yes he did. Today, the West is sending its crusading armies to invade Muslim lands. Is it not our duty to follow in the tradition of our prophets?'\n\nEveryone clapped. Waving a leaflet in his hand, he said, 'The crusader, this country's prime minister is coming to Manchester next week. We call upon all believers to go and protest against him.'\n\nAs we left, Laila took some leaflets and said we could give them out at school. On the way home, I asked her, 'I'm not sure about all this crusader stuff.'\n\n'They get a bit hot headed, this lot, and don't make sense,' she replied, 'and sometimes they're the only ones that make sense.'\n\nWhen I got home, I said a loud _himumhidadI'mhome_ and without waiting for a reply, I ran upstairs and slumped onto my bed and without thinking, I started listening to Lady Gaga.\n\nI heard the front door slamming shut and smiled, that had to be Dad. By the time I got downstairs, he was already watching the television, beer can in hand.\n\n'Dad, can I go to Manchester for the day on Saturday?'\n\n'Where were you all day today?' Dad asked.\n\n'I went to a meeting and learned how to pray properly, and learned what the _azaan_ means.'\n\nHe went into a deep silence.\n\n'Can I go or not?' I asked.\n\nHe rubbed his hand across his face and nodded. I turned around to leave, and he said, 'Do you want another fiver?'\n\n'I still haven't spent the one you gave me.'\n\n'You've become so honest with all this hijab stuff,' he laughed. I picked up a cushion and chucked it at him.\n\nOn the way to dropping me off at Boarhead station, Dad said, 'Just be careful in Manchester,' 'I will Dad,' I said. 'I will.'\n\nAs the train set off, I thought how Boarhead was a world of its own. To get here by car, you come along a motorway, which runs along the side of a hill. Boarhead is hidden behind the hill. If you come by train, the rail track is on the other side of another hill, an even bigger one than the motorway one.\n\nI arrived early and went for a wander down Market Street. I got a text from Laila: _Running 15 mins late._ I was about to reply when I saw them. Jake and his mates were coming towards me. Shamshad was with him, her arm in his. She was wearing a low-cut top and a short flowery skirt. She had a black bag on her shoulder. She kept shaking her head to push her hair off her face. They stopped by an ice-cream van and ordered. Shamshad giggled each time Jake said anything. She had bright lipstick and thick eyeliner on. I was about to turn around and leave when she saw me looking at her.\n\nJake whispered something in her ear. She smiled without taking her eyes off me, and then Jake yanked her arm and they went in the opposite direction. Shamshad kept turning around to look at me as they disappeared into the crowd of shoppers.\n\n## **Shamshad**\n\nWhen Karen saw me with Jake, I could have just died. I kept looking over my shoulder, all the time. I was convinced Karen would follow me. Jake kept asking me what the matter was, but I kept quiet. After a while, Jake got fed up with me and went off with his mates. I changed back into my normal clothes at Piccadilly Station and went back home. Before catching the train, I threw my skirt and top into a bin.\n\nDad was out. Mum was on Skype. She shut the door to the living room when I went in. I went into the bathroom. I cleaned my teeth over and over again, trying to chase the smell of vodka away. I scrubbed my teeth until my gums bled, trying to chase the stench of cigarettes all the while begging the Almighty to forgive me. 'I have been bad Almighty,' I prayed. 'You are merciful. All forgiving. Forgive me. Forgive me.' I did my _wudhu_ , cleaned myself, and readied myself to pray. I went to my bedroom, spread my prayer mat on the floor of my bedroom and prayed. Rani came into the room as I prayed and curled up next to me. After I had finished, I picked her up and stroked her from her head all the way to her tail. She closed her eyes and pushed her head up against mine, arching her back as my hand went over it. As I stroked Rani, I heard Mum laughing downstairs. It's a sound I rarely hear and made me think about the one time she really, really made me happy. It was on a Friday when I saw a side of Mum I would never have dreamt existed. I'd just come home from school and was expecting her to be, as ever, glued to the computer screen, yapping away to some goat-herder in Pakistan on Skype, but she was standing by the window, looking out on the street and waved at me as soon as she saw me.\n\nShe opened the front door and said, taking my school bag from me, 'Come on Shamshad, we're going.'\n\nThere was a decisiveness in her voice I had not heard before. For a moment, I thought maybe she'd had a big fight with Dad and was leaving the house, but then I looked in her eyes. They looked like the eyes of a woman much younger then my Mum. Mischievous eyes. Determined eyes. But one about to leave home? No.\n\nMum quickly hung my school bag onto a hook, picked up a large brown bag that was behind the door, and said in English, something she rarely did, 'Swimming.'\n\n'Swimming!' I almost jumped out of my skin. My Mum and swimming! She grabbed my hand, giggled and slammed the door shut behind us.\n\nBefore I could get myself together, a white van, full of _buddies_ \u2013 old women \u2013 from our neighbourhood pulled up outside our house and Mum shoved me inside and then got in herself. The amount of noise these buddies made on the way to the baths would put an army of hormonal teenagers to shame. And the words they used, about each other and about the men they saw on the way? May the Almighty forgive them, especially my Mum.\n\nThe baths were even noisier then the van. They were heaving with happy women and kids. I have never seen so many bums, bigger then coffee tables, tree trunk legs wobblier than jelly jumping into water, ignoring the whistles of the irate lifeguards.\n\nMum handed me a plastic bag with my towel and a new swimsuit in it, kissed me on the forehead and rushed off to change, along the way waving to a big, fat woman who was standing on the edge of the pool, slightly bent, with one foot touching the top of the water. The woman was wearing a purple swimsuit, leggings and a matching top that came down from a tight hood from her head to her waist.\n\nThis was the first time I had been here since the Victoria Baths had been modernised. The old swimming pool in the middle with changing cubicles all around was still here, but now two larger pools, one with diving boards as well as slides and water features for children, had been built. The old ceiling, with its long beams, had been replaced by a much taller glass one.\n\nMum had bought me a normal, dark blue swimming suit with a matching cap. I got changed as quickly as I could and when I came out, Mum was waiting for me outside my cubicle. I put my hand to my mouth as soon as I saw her. I was expecting her to be dressed like her purple-suited friend, but Mum was wearing a swimsuit like mine, with one of Dad's large white t-shirts on top. She burst out laughing when she saw me looking at her.\n\nMum was like an excited girl and I felt like the adult. She hopped over to the deep pool and as she was about to jump in, a lifeguard dressed in a bright yellow top and red shorts whistled at us. The guard pointed at Mum and waved her index finger telling us not to do what Mum was about to do. I was a bit puzzled and tried to work out what it was we had done, when Mum took off the t-shirt, and said, nodding to the lifeguard, 'This one, she doesn't let us go into the water with normal t-shirts on.'\n\n'You've been here before, Mum?!' I asked incredulously.\n\n'But not with you,' she said rolling up the t-shirt and lobbing it towards a bench on the side, behind the lifeguard.\n\nAs she jumped into the water I shouted, 'It's the deep end, Mum.'\n\nShe went down into the water and then when her head popped out she said, 'Race you to the other end.'\n\n'You've no chance,' I said jumping in next to her.\n\nI was wrong. Mum swam like a fish, doing her doggy paddle, panting and splashing all the way to the other end of the pool. She beat me, but she was really out of breath.\n\nWe did a couple of slow lengths, me showing off my breaststroke, front and back crawls and butterfly. Mum stuck to her doggy paddle.\n\nWhen we stopped for a breather, Mum pointed at the tall diving board in the pool behind us and said, 'Lets jump from that one.' 'It's much higher than it looks from here, Mum.' She didn't wait for my reply and got out of the water.\n\nAs we walked towards the queue for the jump, a lifeguard stopped us. 'Why won't she let me go?' Mum asked me.\n\nI asked the lifeguard what the matter was and she replied, 'Everyone has to swim a width before they are allowed to jump.'\n\nI translated for Mum and she lowered herself gently into the water and swam across to the other end. I followed her.\n\nWaiting in the queue, I began to feel goose pimples. I hate heights, but didn't want to let Mum down.\n\n'It's all right if you don't want to jump,' Mum said walking up the steps. 'It's not that...'\n\n'I might not be able to do it when I get to the top, so you wait here and get ready to pull me out of the water if I do manage to do it.' 'OK, Mum.'\n\n'And don't laugh at me if I can't.' 'OK, Mum,' I laughed.\n\nMum looked at a woman who was waiting at the edge of the high jump. She was a skinny, blonde-haired woman who kept looking down, then moving away towards the steps only to go back again. She did this a few times and came down the stairs, crying.\n\nWhen Mum's turn came, she went up the stairs, walked onto the platform and jumped straight off, without waiting or thinking. She came down making a lot of noise. It was difficult to tell whether she was screaming in fear or joy. She landed in the water with a great big splash. I was relieved when her head popped out of the water. She swam towards some steps close to me.\n\nComing out of the water, she said, 'I did it.'\n\nShe looked round the pools and said, 'Do you want a go?' I shook my head.\n\n'Then I don't either,' Mum said.\n\nI was so deep in thought, remembering my one special time with Mum, I didn't hear Mum calling me at first.\n\n'Yes, Mum,' I replied loudly.\n\n'You left the bathroom sink dirty,' she said. 'Sorry, Mum.'\n\nI felt all dirty again; I held my hands open in prayer, and prayed, 'Almighty, forgive me for what I have done. I will never do it again.'\n\nKaren's face flashed through my mind. I slammed my hand on the side of the table. Rani jumped out of my lap.\n\n## **Kiran**\n\n'What's Shamshad like?' I kept thinking, waiting for Laila. As soon as I saw her, I told her about Shamshad and Jake. She said, 'Nothing surprises me about her.'\n\nThough neither of us admitted it to the other, we didn't want to bump into Shamshad. We messed around for a little while and went home.\n\nThe next morning I put on a clean headscarf and left for school. Shamshad was waiting at the top of our street, on the other side of the road. At first, I didn't believe my eyes and looked at her again. She stood on the other side of the road, unconcerned by all the eyes staring at her. Her arms were folded across her chest.\n\nI quickened my pace. She did the same. I started running. She did the same. I ran as fast as I could and stopped by Mrs Johnston, the lollipop lady, at the top of our road. Shamshad was beside me. Panting. I moved closer to a woman with two children and crossed the road. I could feel Shamshad's breath on the back of my neck. I went along the main road instead of the short cut to the bus stop. Shamshad followed me all the way.\n\nAt the bus stop there were lots of kids, many of them going to our school. I bought my weekly bus pass from the driver and went to sit down. I felt relieved as most of the seats had someone sitting on them. I sat next to an old lady. As I was sitting down, she handed me a bag and said, 'Can you help me off with this, love.' I took her bag and moved back whilst she got up off the seat.\n\nShamshad sat down next to me, stuck her elbow into my ribs, and hissed 'Keep that little trap of yours shut!' I kept quiet.\n\n'Is she wearing her mini-skirt again?' Aisha glared her braced teeth at me.\n\nDigging her elbow into me even harder, Shamshad said, 'She's better now. I'm teaching her to be proper.'\n\nI squealed. The cronies laughed. Aisha laughed the loudest. Shamshad stepped on my foot as hard as she could before getting up to leave.\n\nI went to the toilets and washed my face before going to class. Everyone was staring at me when I entered. But for a slight clicking noise, it was really quiet, like everyone was holding their breath for a big bang. I wanted to yell, \"what have I done to any of you\", but I couldn't say anything. My feet felt really heavy. My side ached and I felt the weight of all those eyes glaring at me. Tears began to burn in my eyes. 'Don't cry in front of this lot, Kiran,' I said to myself. 'Don't you cry, girl. They're just ignorant. They're just making you the fool.'\n\nWhen I got to the desk, I saw what the joke was. Everyone started laughing. A small toy monkey, wearing a hijab over a mini-skirt, was dancing on my desk.\n\n'Very funny,' I said as I picked it up. I gave Aisha a dirty look. She stopped laughing. I turned off the monkey and opened my bag. I realised my pencil case was still in my locker. I looked at the clock. There were still five minutes to go and I rushed out. Someone clapped, then everyone clapped. When I got to the door, Shamshad was standing there, peeping in. She grabbed me by the collar, pulled me away from the door and slapped me hard across my face, hissing, 'How dare you spread lies about me.'\n\n'I haven't told anyone, I swear,' I said.\n\nShe was about to hit me again, when I heard Mr Mayflower's shoes clicking on the tiles. He walked with a limp. The sound was unmistakably his. Shamshad let go of me.\n\n'Shouldn't you girls be in class?' Mr Mayflower said. 'I just came to borrow something, sir,' Shamshad said.\n\n'Very good. Very good,' Mr Mayflower replied in his usual manner. 'You look a bit sore this morning Karen,' he said to me.\n\n'I left my pencil case in my locker, Mr Mayflower,' I said.\n\n'Very good. Very good. Chop chop, now. Go get it.' Mr Mayflower said. 'And you can carry on chatting at break.'\n\n'Yes, _Karen_ , I'll see you at break,' Shamshad said, running towards her class, the echo of her feet fading down the corridor.\n\nI didn't pay any attention to Mr Mayflower's lesson. He didn't notice these things. He just looked out from his thick glasses, brushed his wild hair off his head when it fell forward and carried on with whatever he was going to do. I tried not to think about the coming break by thinking about the stories that went around about Mr Mayflower. You could make any excuse to him and he would believe you. He was a legend in our school for being the dopiest of teachers. Mr Mayflower couldn't stop break from coming round. I had only told Laila, and she had promised not to say a word to anyone.\n\n'I didn't say a word to anyone,' Laila said when I saw her at break. She was carrying a plastic bag in her hand. Then she smirked, 'Except for mentioning it on my Facebook wall...'\n\n'How could you, Laila?' I protested, looking round for Shamshad. 'Don't worry about her for a bit,' Laila said, 'her and Jake have gone to Mrs Seaver'. Mrs Seaver is the Head. I was about to ask why when Laila grabbed my hand and led me towards a covered area, and said, 'That's why.'\n\nSomeone had written: _Shamshad fancies Jake. '_ Ouch,' I said.\n\nLaila laughed, 'Sorry, cross my heart and hope to die.' 'Cow!' I swore.\n\n'Moo,' Laila said.\n\n'Come on, are you going to help me give 'em out?' Laila asked. 'Just me and you, Laila,' I asked.\n\n'There's a few others,' Laila said. 'Are we allowed?'\n\n'No one will know who did it,' Laila said, 'We'll do it quickly.'\n\n'Well, nothing much's going right for me,' I said. 'Besides, you're really good at keeping secrets.'\n\nPutting some leaflets into my hand, Laila said, 'Moo.'\n\nPast the covered area, our school grounds are dotted with trees, which go all the way down towards a farm, where cows often sit, staring at us, chewing their cud. Kids from West Boarhead hang out under the birches, and the East Boarhead lot at the opposite end, by the poplars. We had hardly given any leaflets out when Donna, Chloe and Megan came towards us. Jake was a few steps behind them, coming from the direction of the covered area. He was talking to Shamshad. Behind Jake, kids from West Boarhead were standing around close to each other. Fluff from the birch tree was blowing in the wind like snowballs.\n\nI was handing leaflets to some Muslim boys from Year 8, when Donna came up to me, grabbed the leaflets out of my hand and swore. I tried to snatch the leaflets back; she pushed me in the chest. I steadied myself. Donna's chubby cheeks were red. Wiping her blond hair off her face, she stared at me with raging eyes and said, 'Showing your colours, eh.'\n\n'Get lost, Donna,' I said putting my arm around Laila, 'Just go away.' The crucifix drawn on my forehead began to sting.\n\nGirls and boys from West Boarhead ran down towards us.\n\nChloe pointed at us with her index finger and then rubbed it across her neck as if it was a knife hissing, 'Talibans.'\n\nDonna crumpled the leaflets she had taken from me, threw them to the ground and turned towards Laila saying, 'You should be supporting our lads in Afghan.'\n\nLaila's face reddened. She said, 'British soldiers went to my granddad's house in our village in Kandahar. Shot him in front of everyone. One of them put his boot on my granddad's head and had his photo taken. Did he come to Boarhead and bomb it?'\n\n'If you don't like it here, why don't you...'\n\nMuslim boys and girls from East Boarhead had come round as well. Laila and I were between two heaving lines.\n\n'Terrorists,' someone shouted, from the East Boarhead line.\n\n'Up yours, Osama,' Megan pointed to a turbaned Sikh from Year 8. Donna turned around and shouted over to Jake, who was still talking to Shamshad, 'Jake, come here, right now.' Jake ran over. Shamshad went back into school.\n\nAbove us, the wind stopped. The air filled with a seething anger. Some boys pushed each other, swearing. I looked at the West Boarhead line. It was thicker than ours, and getting bigger. Jake came and stood next to Donna. She pointed her fat finger at our line, and shouted, 'If you lot don't like it here, go back to Shariaistatan or wherever you come from.' The angry silence from above came down and engulfed us. Donna's eyes filled with tears and she said in broken words, 'My Dex is out there in Afghanistan, getting shot at by youse lot. And some of you know Dex, don't you?' She held Jake's hand, pointed to me and said, 'And you know him well don't you? I don't know if he's alive or dead now, do I?'\n\nThere was a gasp from the western line. Jake said, 'Our Dex is missing. They think he might be dead or captured and taken to Pakistan...'\n\nDonna's eyes narrowed. She pointed at me and said, 'And if something happens to Dex, I'll have one of you.'\n\nJake shook his head, 'I didn't want our kid to join and I didn't want him to go.'\n\n'What've you become?' Donna stepped away from Jake, looked him in the face and shouted, 'He's your brother and was just doing his job...' 'Doing what?' Jake interrupted.\n\nDonna gritted her teeth.\n\nMegan put her arm in Donna's, spat at Jake and, 'You Paki-loving traitor.'\n\n'Donna, I love our Dex and you know that and I know you love him...' 'He should be with you here, Donna,' Laila said.\n\n'He should,' Donna cried. 'But he might be dead.' 'He's alive, I know he is,' Jake said.\n\nThe Head came charging up towards us with an army of teachers behind her. Everyone scampered.\n\nI followed Laila and we went down past the poplars, and hid behind a bush. A large brown and white cow was sitting a few feet away from us on the other side of our grounds, staring at us with its big eyes.\n\nAfter getting my breath back, I said to Laila, 'I'm sorry about your granddad. I didn't know.'\n\n'How would you?' she said, pulling out a blade of grass. Putting it in her mouth, she whispered, 'It was my _dada_ , me granddad. My Dad couldn't even get to his funeral.'\n\nI pressed on her hand, and asked, 'When?''\n\n'Exactly thirteen months and four days ago,' she said, spitting on the ground.\n\nWe sat in silence for a while. Laila had picked up a twig and was digging into the ground with it. I looked into the vacant eyes of the cow. A small bird had perched itself on the cows head. The cow carried on chewing. A fly buzzed above it. It flicked its tail. The fly continued to buzz. The cow shook its head and the bird flew off.\n\nEven though I was feeling calm on the outside, inside I was really, really mad. I was mad at Shamshad for tormenting me. I was mad at myself for not standing up for myself. Laila had just stood up to that nut case. Why couldn't I? I was really angry with Mum for taking me to church. And my Dad, I thought, 'How could you be you?' And I was angry for Laila.\n\nJust when I was thinking of her, she said, 'He was going to give water to a dying soldier, someone who had come from so far away and who had caused so much suffering to his village. He was going to give one of them water, and they killed him and took photographs.'\n\nI didn't say anything. I searched for the right words, but all I could find was rage.\n\n'That's what my Dad said happened,' Laila said. 'He doesn't say much any more, my Dad. He wants to go home, but there's no one there now. They brought bulldozers and flattened the village, they did. And everyone just went somewhere else. Some of us now live in Pakistan. That's what my Dad said they did.'\n\nWe walked silently back to our classroom. I took the bag with the leaflets from Laila and hid it under my jacket. After shoving the plastic bag into my locker, I went to my classroom and floated through English.\n\nThe whole school was called in for a special assembly that afternoon. Our assembly hall was a long, tall building. All along the wall were engraved names of past Heads and Head Boys and Girls. When empty, it echoed each time you took a step. Your voice, even in whispers, bounced around the room. The Head, a police officer and Mr Mayflower were sitting on the stage. Everyone filed in and sat down, class by class, looking stiffly towards the front. Nobody spoke as they entered. The teachers sat on the sides. Year 9s and 10s stood behind the teachers at the ends of the rows of chairs.\n\nMrs Seaver stood up and tapped on the microphone, three times. The noise from her fingers subdued the echoes. 'I think you know why you are here. I have a few words to say to you. As always in my assemblies, if you have a question to ask, you may do so. If you have something important to say, which we should all hear, you may say it. But first, we have a special guest here.' She nodded towards the police officer and continued, 'Chief Constable Sharon Brittle has come here, and has a few words to say to you.'\n\nMrs Seaver moved back from the microphone stand and gave it to the Chief Constable, who placed a folder in front of her, and said, 'I won't take up too much of your time, school, I know you have a lot of learning to do and are eager to get back to your classes.' She paused. A giggle rippled through the assembly. She continued, 'Yes, that's how I felt when I was sitting where you are now.' The ripple rose a bit louder. Even Mrs Seaver smiled. The Chief Constable continued, 'I am not here to teach you. Nor to preach to you. I am here to tell you simply about the law. All this week, I have been going to schools across the northwest, giving the same message. In this country, we have laws; it is my job to enforce them. In many schools, troublemakers from outside, with obvious support from inside, have been giving leaflets out about the Prime Minister's visit to Manchester. I know emotions are running very high about the war, especially around West and East Boarhead, from where many soldiers have gone to fight. They are doing their jobs. Whether they like it or not. I am doing mine. Whether I like or not. You are doing yours.' She paused, and said with a smile, 'And I know you like it.' Everyone laughed. When the laughter subsided, she said, holding one of the leaflets we were handing out in the playground, 'We are lucky to live in a country that enjoys freedom of speech. Some of you have seen this. Some of you have been giving this out in this school. My job is to tell you that any child not at school during the forthcoming protests about the PM, and without a valid note from their school or their parents, will be treated as a lawbreaker, and will be dealt with accordingly.'\n\nSomeone clapped. I looked around. All the teachers were clapping. As were most students from East Boarhead. I raised my hands involuntarily, but stopped before I clapped. I looked over at Shamshad. She was sitting with her fists clenched.\n\n'Let me read you a sentence from this leaflet,' the Chief Constable said, looking down at the leaflet, \"Stop the Crusade of the Western Armies.\"' She cleared her throat, looked across at us, and said, 'You are young and cannot understand the subliminal message embedded in this sentence. It is the soft edge of terrorism. It is the breeding ground of extremism, and we don't want any more 7\/7s. This is where it hides. It is ideas like this from which it grows. At the back of the room, as Mrs Seaver said, I have left some literature that will help you to identify extremism. For those of you who want to make your point to the Prime Minister, let me assure you that when I meet him in Manchester, I will let him know the strength of feelings about the government's policies. And, finally, I want to warn you again. Anyone breaking the law will be held accountable.' She turned to Mrs Seaver and thanked her, and then walked off the stage. Her footsteps echoed all round the hall as she walked down the centre and out of the back.\n\nWhen the police officer had left, Mrs Seaver said, 'In all my years, I have never called an assembly like this.' She stopped and looked down at us. Her gaze cut through the fading echoes of her cold voice. Someone coughed. Mrs Seaver continued, 'We have children from 85 countries in this school. We have 15 languages. Yes, this is a Christian school. We pride ourselves on this. There are many of you who are Muslims in our school and we welcome you. We celebrate Eid. And there are Hindus and Sikhs. And we celebrate Diwali and Vaisakhi. We try our best to accommodate everyone's faith here. We do this because we _are_ a Christian School. And Christ taught us to love. To love each other. To love peace. And even on the Cross, he forgave. That is what we have tried to teach all of you in this school.' She paused and looked down towards us. I felt she was looking at me. Mrs Seaver continued, her voice softened, 'Now children, it is good to have different opinions, but we want to be able to discuss these in a healthy atmosphere. It makes all of us stronger. Especially in the spirit of Christ. It is not about West and East. I want us all to be one happy family. Now, as always, I believe when passions are high we should let them vent. Those of you who feel they have an important question to raise or something to say, why don't you say it right now?' she stopped.\n\nThe assembly sank into a deep silence. I looked across at Laila. She was crying. I don't know why I did it, but I stuck my hand up. Mrs Seaver raised her head back, as though startled. She thought for a moment, and said, 'Yes, Karen, isn't it?'\n\n'It's Kiran, Mrs Seaver,' I said. A few people laughed. Shamshad coughed.\n\n'Stand up,' Mrs Seaver ordered.\n\nMy legs turned to jelly. Sweat tricked down my spine. Holding on to the chair in front of me, I pushed myself up.\n\n'The Romans were westerners, yeah, Mrs Seaver? And did Jesus not stand up to them?'\n\nFrom the corner of my eye, I saw Laila clap. A few others from East Boarhead joined in. I saw Jake. He clapped, and then suddenly stopped.\n\nI was trembling. Sweat poured down my face. My shirt was stuck to my back. The echoes had run away. Everyone was staring at me.\n\n'Well, Jesus certainly taught us to love one another and to forgive,' Mrs Seaver said as I sat down. 'But Jesus would not allow for unauthorised absence from my school.'\n\nEveryone laughed. I joined in as well. Mrs Seaver always managed to find a way of saying things at the end of an assembly that were funny.\n\nI hurried out of school that day. Everyone was staring at me in a strange sort of way. I really wanted to walk out with Laila, but her Dad was picking her up and I was scared of Shamshad. She had many reasons now for getting at me. I saw Jake coming out of the door. He waved at me to stop, but I pretended I hadn't seen him, lowered my head and carried on. Two white boys blocked my path just as I was leaving the school gates. One of them was taller than the other. The tall one had curly red hair and a stubbly face. The shorter one was stocky, with short, cropped hair. They were Year 9s or 10s. I didn't know their names. As I stepped off the pavement to get past them, one of them, the short haired boy, stepped aside and said, 'You don't need to do that.'\n\nI stopped.\n\nThe taller one said, 'I wish I was as brave as you.'\n\nI was trying to digest the compliment when the shorter one scratched his head, and said, 'Tony Blair lied to us. Me Dad went to Iraq and never came back.'\n\n'Year 10 are going on the demo,' the tall one said.\n\nI stepped forward and hugged him.\n\nThe short one laughed, 'I didn't know you did that.' 'What do you mean?' I asked pulling away. 'Scarfies and hugging... scarfies...'\n\n'Oh, get lost,' I said, wiping away tears of relief.\n\nAs the two boys left, Jake put his hand on my shoulder. I turned around. Behind him, still in the school grounds, Shamshad was giving me the evil eye.\n\n## **Shamshad**\n\nI couldn't take my eyes off Karen, with Jake's hand on her shoulder. I thought back to what had happened. The whole world knew. It was all over Facebook. When I saw Karen in Manchester, I kept looking back at her. I didn't know how long she had been spying on me, but I was sure she had taken photographs of me with Jake on her mobile. I was going to get them deleted before they went up as well.\n\nI hid in the toilets at the start of the dinner break. I cried, and swore to really, really teach her a lesson. I was sitting in the cubical when I heard some girls talking about a big fight coming up in the playground. I wouldn't have cared, but someone mentioned Laila. I waited until they had left and went into the playground through the back entrance of the main hall. Everyone was walking down towards the other end of the grounds. It looked like East Boarhead was already there and West Boarhead was coming from all over. I was behind West Boarhead. I stopped and thought about turning around. Two long lines had formed. I was about to turn around when Jake came up behind me. I felt so ashamed when I saw him, and lowered my eyes and stared down at the patches of chewing gum splattered onto the paved ground.\n\n'I need to ask you something,' Jake said.\n\nThere was a lot of shouting and swearing coming from the lines in front of us.\n\n'I swear I didn't do anything, Jake!'\n\nHe looked at me all confused. Maybe he doesn't know about all the stuff on Facebook. His hair flew off his face. His earring glittered in the sunlight.\n\n'You don't know, do you, Jake?' I asked.\n\n'Why did you write that stuff in the covered area?' Jake asked. I just died.\n\n'It wasn't me, Jake,' I cried. 'It was her, that Karen. She saw us together in Manchester and she's jealous of us.' Jake's face turned red. He looked away from me. Someone was calling him.\n\nTurning to me, he said, 'There is no _us,_ Shamshad. You came out to let your hair down and mess about with me mates. That's all.'\n\n'How could I have said what I just did?' I thought. Jake held my hand and said, 'We're friends aren't we?' I nodded.\n\n'I need your help.'\n\n'Anything,' I said.\n\nHe took a deep breath, and said, 'It's about our Dex. I know your Dad's an important man. Can I come to your house and speak to him about our kid?'\n\n'Oh no, please don't do that. Don't come to my house. You don't know my Dad. He'll kill me if he finds out about us.' Jake held me by the shoulders, and said, 'Look, there's no _us._ I've told you. I want some help to find my brother.'\n\nSomeone called Jake again. He let go of me and went away. I stood where I was. Ashamed! I had made such a fool of myself.\n\nWhen I saw her going home that day after the afternoon assembly, I could have killed her on the spot. Everyone was looking at her as if she was something really special. And now Jake was chasing after her as well. I promised myself, there and then, that I was going to kill Karen even if it meant spending the rest of my life in jail. I was going to kill her the next day. But I couldn't wait. When I saw her cooing up to Jake outside the school gates, I knew she could see me and was rubbing my nose in it. I got so angry. I took out a pair of scissors from my bag.\n\n## **Kiran**\n\n'Sorry, Kiran,' Jake said, pressing my shoulder.\n\nHe was standing under a crab apple tree. I had gone past this tree so many times but only today noticed its bent, leafless branches dangling out over the pavement, shedding its walnut-sized, red apples onto the pavement.\n\nI shrugged my shoulder, forcing him to take his hand off me. His hand slid down my arm and I felt the tips of his fingers on the tips of mine. I moved my hand away and stepped back from him, crushing some of the fallen crab apples under my feet. Jake put his hands in his pockets and kicked some of the fallen leaves. Shamshad was making her way towards us, fidgeting with something in her bag.\n\nI looked Jake in the face. He seemed to have grown older. He turned his eyes away from me and looked at Shamshad. She knelt down and started tying her laces. A brown leaf twisted and turned in the air and landed on Jake's head, perching itself like a feather. He took his hands out of his pockets, brushed his hair, and said, 'You're welcome back, you know. I mean we're still us. And I gave Donna a piece of mind, for what she did. I swear I did. I don't know why I didn't do anything. I just don't know. Sorry, Kiran.'\n\nI suddenly got enraged with Jake. Oh yeah, I thought to myself I can just see it, Jake. Kiran the Muslim, one of the gang again, like nothing's happened, eh? I'm not who you think I am any more, Jake. Kiran was never WTM. And Karen is dead!\n\n'I know, Jake,' I said. 'Thanks though.' Jake nodded.\n\n'You really know how to say what you feel, girl,' I thought.\n\n'I don't know what to do, Kiran,' Jake said. 'You know I'm not for all this war, but he's me brother.' His eyes filled with tears, and he said, 'Please help me.'\n\nShamshad was still tying her laces. Her eyes fixed on me. I wasn't going to look away from her. I'm tired of your games, Shamshad, I wanted to shout, really, really tired. Come and do whatever you want. Jake turned towards Shamshad, pointed at her, and said loudly, 'There's nothing going on between her and me.'\n\nShamshad stood up and ran towards me, screaming.\n\nI was trying to work out what to do when Jake grabbed her hand. She stumbled. The scissors dropped onto the pavement. She fell, banging her head against a wall. Her hand twisted behind her back. Her books spread out across the pavement. Her hijab stuck in a privet above her. Jake picked up the scissors, and said to Shamshad, 'What you doing with these?'\n\nShamshad pulled her knees up to her chest, placing her head on them; she hissed and then started to cry.\n\n'Serves you right,' I thought. 'I ought to sock you one right in the gob.'\n\nI stepped towards her, untangled her hijab and gave it to her. She snatched it out of my hands and put it on her head. 'You could have hurt Kiran with these,' Jake said, waving the scissors in the air.\n\nI held out my hand for her. She took it, and pulling herself up said, 'Thanks, _Karen._ '\n\nI should have let her fall, but I didn't. She stuffed her books into her bag and left.\n\nThere were piles of dried leaves huddled together here and there. A bird twittered in a tree close to me. I looked at a nest, but I couldn't see the bird. It was still singing somewhere close by. I smiled at the thought that I hadn't noticed the clouds clearing. The sun winked down through the leafless branches of the trees that lined the path of School Lane, which led from our school gates down to the main road.\n\nThe wind suddenly changed direction. It was a cold, biting wind. It pushed a pile of brown leaves towards us, along the pavement. They were twisting and turning as they came. More leaves fell off the trees all along the lane. Some small, hardly dead. Others, large, larger than the size of my hand. A dog barked somewhere in the gardens, and I thought of Shamshad and shook my head. 'Oh God, free me from her. Is this too much to ask? This is the first and only thing I have really asked of you. Is it too much?'\n\n'Come on, Kiran.' Jake brought me back down to earth.\n\nHe held my hand and pulled me towards a huge pile of leaves that had collected around a discarded pram. He let go of me and started kicking the leaves. He bent down, picked up a pile of leaves, and tossed them into the air. I ran, stepped into the pile of leaves and started kicking them, shouting with joy. Jake stopped, and asked, 'What's the matter with you?'\n\n'Me prayers just got answered,' I said.\n\n'Barmy,' Jake said, joining me.\n\nWhen we stopped kicking the leaves, I noticed Mrs _Getoffmystreet_. That's what we called her. She was always there, by her gate, each day, before the start of school and at the end of the day, staring out at us, with her arms folded across her chest like a wrestler, with a look that said, _getofmystreet._ I looked her in the eyes, stuck my tongue out at her, and followed Jake as he ran down the lane. Shamshad was standing on her own at the bus stop, on the other side of the pelican crossing. I stopped.\n\n'I'm going to give her a piece of my mind,' Jake said, stepping into the road.\n\nI pulled on his hand, and said, 'Lay off, Jake.'\n\nHe stepped back onto the pavement and looked me in the face. 'Just lay off,' I said.\n\nI pressed the pelican-crossing button.\n\n'What is it with you girls? Don't you know what's good for you?' A bus pulled up on the other side of the road.\n\n'Girls are girls,' I said. 'Yeah,' he said. 'Yeah,' I said.\n\nThe bus pulled away. Shamshad was on it.\n\nThe pelican lights turned green. We didn't cross the road. 'You coming to the willow tree later?' Jake asked.\n\n'So the mob can make fun of me, eh?'\n\n'There's no mob anymore.'\n\nI pressed the pelican lights button again. Jake said, 'We've got a lot of things to sort out and there's...'\n\nThe lights changed and crossing the road, I said, 'No!' Jake followed me, and said, 'Please. It's not what you think.'\n\nGetting onto the pavement on the other side of the road, I quickened my pace. Another bus was coming. Stepping onto the bus, I said, 'No!'\n\nThere was hardly anyone on the bus. Jake sat on a seat in front of me. I touched the window with my forehead. The glass was cold. It started raining. At first just a few drops. They hit the windows and slid down past my face. Moments later, the glass misted. I wiped the glass and saw Jake's reflection. He was looking at me, pleading with those blue eyes.\n\nWhen his stop came, he stood up, looked at me, sat down again, and asked, 'Do you mind if I walk home with you?'\n\n'Why?'\n\n'Don't want to see Dad. He drinks even more since Dex went missing.'\n\n## **Shamshad**\n\n' _Allahjee_ , dear God, how can you let Karen humiliate me?' I thought, 'Why don't you throw a bolt of lightning at her and get rid of her right now?' I didn't try to hide myself from her. I just stood there. I knew Karen knew I was watching her. I tried to cheer myself up with thoughts of cutting her throat with a knife, or putting my scissors into her stomach, pushing her under a bus, or running her over with a car. I laughed when I remembered I couldn't drive.\n\nI was so angry I was trembling, especially when she jumped about kicking leaves with Jake. And what did the stupid, little cow think she was doing sticking her tongue out at that old lady?\n\nI couldn't get Karen out of my mind all the way home on the bus. Killing you would be too easy, _Karen_ , I resolved. I had to find a way that would make you feel like I do. I don't know how you've done it so quickly, taking all my friends away from me, but I won't let you get away with it!\n\nI missed my stop and ended up getting off at the top end of Boarhead East High Street. Our house was at the other end, past the restaurants and takeaways that lined both sides of the High Street. My Dad's office was in the centre of the High Street.\n\nMumtaz Ali Khan, the old man who lived in a house behind ours, Baba Khanu as everyone called him, nodded when he saw me walking past the window of Lala's Kebab House. I flashed a false smile back at him. He was a short, fat man, who had worked at Lala's before I was born. He waved at me with a hand covered in kebab mix and went back to spiking meat onto skewers and placing them into a refrigerated display. The door to Lala's was open. As I walked past it, I heard the meat sizzling on the grills. The scent of the roasting spicy meat made my mouth water. I slowed down.\n\nA police helicopter came out of nowhere and started hovering above. A moment or so later I saw them. Men with masks on their faces, carrying English and Israeli flags, came charging down the middle of the High Street, shouting, 'Muslim Terrorists out of Boarhead!'\n\n'Come inside, daughter, quickly,' Baba Khanu said, stepping out of the shop. Before I could say anything, he grabbed hold of me and yanked me inside the shop.\n\nFour policemen on horses followed the masked men towards the town centre. After they were out of sight, Baba Khanu said, walking back behind the counter, 'Come, let me make you a big, fat kebab.'\n\n'I'm not hungry, _Babaji_ ,'\n\n'And if you're worried about your father, I'll phone him right now.'\n\nBaba Khanu picked up a phone and spoke to my Dad. 'And he said, \"As long as you don't charge me.\"'\n\nPushing a plate of kebabs and a can of coke towards me he said, 'It is so good to see you youngsters turning to Islam.'\n\n'I didn't know I'd left it, _Babaji_ ,' I laughed, opening the Coke can. A bit of Coke sprayed onto the window. I took a sip. The fizz, the spice, the meat and the sugar: I was in heaven!\n\n'I wasn't thinking of you,' Baba Khanu said, settling back again. 'I was thinking about Ghazanfer's granddaughter.'\n\nBaba Ghazanfer's granddaughter, the name I had heard, but no face fitted it. A few drops of rain crashed against the window. I looked at the raindrops sliding down the glass and tried to work out who she was.\n\n'You know, Liaqat Malik's Dad,' Baba Khanu said.\n\nI couldn't work out who this was either. Suddenly it started pouring it down.\n\n'You know, he calls himself \"Lucky\". It's so good to see her leaving the church and coming back to our religion.' Baba Khanu said.\n\nI put my hand to my mouth. I nearly choked. Karen's face flashed in front of my eyes.\n\nOh god, why can't I get away from her, I thought. I stared at Baba Khanu's reflection in the window. He moved slowly, his hands shaking a little. 'And yes, it's Kiran,' he told me. 'She won't let anyone call her Karen any more, like she used to.'\n\nI was about to get up to leave, when Baba Khanu said, 'And Ghazanfer was here only yesterday. He has already been asked for her hand. From that handsome boy, Jani...'\n\n'What?' I stammered. 'Jani, my neighbour!'\n\n'The Almighty works in strange ways indeed,' Baba Khanu said. 'Ghazanfer and Jani's father are going round to their house tonight.'\n\nI settled back into my seat. I was hungry again. The food looked delicious and the beautiful rain was washing the world clean outside. I couldn't wait to finish my food. I wolfed it down. Karen's marrying Jani \u2013 I could see it written on the walls of the covered area at school. I was going to Photoshop her: in her bridal dress, head bowed, a photo of Jake in one hand and in the other, a photo of her, in a headscarf, with a miniskirt, sitting next to her husband, Jani, dripping in his _sehra_ of golden tinsels. And isn't the bride-to-be going to love seeing Jake at her door! After squeezing the last drops of Coke out of the can, I thanked Baba Khanu and walked, humming, out into the rain.\n\n## **Kiran**\n\nThe rain poured down the windows of the bus. Shamshad's face formed in the water. I heard the hiss of the doors opening, but was so engrossed in watching Shamshad's face being cut apart that I didn't realise it was my stop.\n\n'What you planning to do, go all the way to Manchester?' Laila asked, stepping into the bus. She looked at Jake, all puzzled. She looked at me and then back at Jake. I swear Jake blushed, but he turned his face away quickly and looked out of the window.\n\nI grabbed my bag, and asked Jake, 'Aren't you coming, then?'\n\n'No,' he said without turning around.\n\nI didn't have time to work out what Jake was up to, and said sorry to the driver and got off the bus.\n\nBefore I could say anything to Laila, she went into the bus shelter, stepping around a broken beer bottle, and said, 'Before you ask, you know who, texted me.'\n\nI laughed, stepping into the shelter. An old lady, with curlers clearly visible under her flowery hat, gave us a _whatyoudoinghere_ kind of a look and stepped away from us, swinging her umbrella. I gave her a _whatyougonnadoaboutit_ kind of a look back and then blocked her from my mind.\n\n'It's not what you think,' I said to Laila.\n\n'About what, Kiran?'\n\n'You know what?'\n\n'About me and Jake, there's nothing, so stop thinking what you're thinking.'\n\n'I _don't_ think,' Laila giggled. 'Cow!' I said.\n\n'Moo,' she replied We laughed. The old lady tutted, opened her umbrella and stepped out into the rain. 'What're you doing here, Laila?' I asked, getting out of the shelter and dodging the water dripping down from it.\n\n'You were awesome in assembly. Awesome!' Laila said. 'You didn't come to tell me that.'\n\n'Where did you get it all from?' Laila asked.\n\n'I don't know,' I said. 'Maybe, I've just had enough, or it's the meetings you've been taking me to.'\n\n'Wicked,' Laila said.\n\n'Lailoo, I got to get home. Mum'll be worried.'\n\n'I told me Dad. He's going to pick me up from your house later,' Laila said. 'And you still got the miniskirt?'\n\nI elbowed Laila in the ribs as hard as I could and ran. She ran after me, laughing and cursing. We stopped at the top of White Haven Road, which circled our estate. The road was lined with oak trees. Every now and again, overgrown branches of the trees would crack against the upper decks of buses and tall vans. On the opposite side of the road where I lived, ran the broken railings of East Boarhead Park.\n\nThree boys were kicking a can to each other in the middle of St. George's Street.\n\nThe boys stared at us. Laila put her arm in mine. The boys started kicking the can to each other again. I knew them by face, but not by name. A dog barked somewhere close by. One of the boys said something that I didn't quite catch.\n\nLaila squared up to him. He was a short, stocky, grey-haired boy, still in the uniform of St. George's Primary School. He had one foot on the can and both his hands were on his waist. He stepped back and kicked the can at us. It missed us and hit a parked car.\n\nA man swore from one of the houses and the boys scampered. The dog stopped barking.\n\nLaila got a text. 'It's Jake,' Laila said. 'Texting you?'\n\n'Cow!' Laila sniggered.\n\n'Moo,' I laughed.\n\nSeriously, though,' Laila said, showing me Jake's text.\n\n_Kiran's mobile's off,_ Jake's text said.\n\nI wrote back on Laila's mobile: _She's home._\n\n_Liar,_ Jake texted back.\n\n_Where RU?_ I texted. _Across the road,_ Jake wrote. And he was.\n\nLaila waved at him to come over. He did, blushing. Jake looked at us suspiciously, 'Were you talking about me?' 'Nah,' Laila answered, and then dug her nails into my shoulders and whispered into my ear, 'Jake, eh?'\n\n'I'm not proper like you,' I sniggered into her ear. 'Remember.' ' _Khothee_ ,' Laila said, moving away from me.\n\nI pulled her back and brayed loudly like a donkey.\n\n'What's that mean?' Jake asked. 'Donkey,' I said.\n\n'What does it mean, though?' Jake asked.\n\nLaila and I looked at each other. Jake answered himself, 'I'm a donkey, then, eh?'\n\nLaila and I laughed.'Will you help me find our Dex, Laila?'\n\n'I don't know what happens in my house, how can I help you find Dex in Afghanistan?'\n\n'Someone must be able to help,' Jake said.\n\n'Shamshad's Dad knows everyone in East Boarhead, you know,' I said. 'You should go to her house and ask her Dad.'\n\nLeila stamped on my foot as Jake nodded and walked past us up the street.\n\nIt was only a short walk to our house, but we took ages to get there. When we got to the butcher's shop, I waved at Mr Mason, Elizabeth's husband. He's a tall, red-faced man with a big, bulging stomach. He was sharpening a knife and ignored me. Laila pulled a face at the half carcass of a pig dangling in the window. At first, I was puzzled, but then I remembered and did the same.\n\nPointing to my neighbour, I said, 'That's George. Sometimes he sits there like a stiff, other times he blasts out music for all the street to hear.'\n\nOur house is in the middle of Willow Bank behind a row of manicured privet hedges, courtesy of George. He made sure the hedges were perfectly squared-off at the top, low enough for him to see out onto the street. No one at home minded, as it was one less job to nag Dad to do.\n\n'Hi Elizabeth,' I shouted above the noise of George's music, closing the gate after Laila.\n\nMrs Elizabeth Mason was in her garden, hanging clothes out on the line. She looked across at us and then turned around. Since I put the hijab on, she has stopped smiling at me, which was mostly what she did when we met.\n\nI waved at George, who carried on staring at whatever he was staring at. His dog was sitting in its favourite position, its head on its front paws, its tired eyes focused on its master.\n\n'Hi Bruno,' I called out to the dog.\n\nThe dog lifted its head a little turned it a little towards me, moved its ear slightly and went back to its usual position.\n\n'Since I put on my headscarf,' I said to Laila as we got inside my house, 'so many of the neighbours have stopped talking to me.' 'It's a white thing,' Laila said.\n\nMum had left me a note on the kitchen table: _Gone to pick some things up for tonight._\n\nLaila went straight for the fridge and took a can of Dad's beer out, touched it to her face and laughed, 'It happens to us he-ja-bees.'\n\n'You're not going to have that are you, Laila?'\n\nShe looked up at the clock, and said, 'Don't know. Got some time before Dad comes to pick me up.'\n\n'He'll smell it on you, stupid.'\n\n'He won't mind,' Laila said, turning the beer can in her hand. 'Won't he!' I said, incredulously.\n\n'Nah. Maybe just kill me,' Laila said putting the can back in the fridge.\n\nWe laughed about how, when we got on the buses, Asian women clocked us, and how we checked them back. We nattered about how some white men were scared of us little girls wearing a bit of cloth on our heads and how wonderful it would be to get away from Boarhead, especially during the coming Christmas holidays.\n\n'I'm going to meet my family in Pakistan this year,' Laila laughed. 'No _Abbaji_ Christmas for me.'\n\n'Lots of _Abbaji_ Christmases for me,' I said. 'Why don't you come as well?' She asked.\n\n'Everyone's here,' I said. 'My Dad never mentions going to Pakistan.' 'Come with us.' Laila said. 'No Rudolf...'\n\n'I'll ask,' I laughed.\n\nWe chatted about how Shamshad had been cut down to size. We roared and slapped our legs talking about how Jake thought he could get off with any girl.\n\nLaila's father came too quickly. She had hardly got into the car when Jake texted me: _What u doing now?_\n\nMe: _Who wants 2 know?_ Jake: _Me_ Me: _Who?_ Jake: _Jake_ He is so simple, I thought.\n\nI texted back: _Busy_ Jake: _Old place_ Me: _No_ Jake: _Plz_ Me: _No_ Jake: _Plz. Plz_ Me: _No_\n\nI went to see Jake. I reckoned all he wanted to do was to talk about Dex, and I had to get it into his thick skull that there was nothing I could do to help him. He was waiting for me at the top of our street.\n\n'I thought you said by the tree,' I said.\n\nJake was lost deep in thought and we walked quietly up the St. George's towards White Haven Road. Leaves were raining off the trees. There was hardly any traffic on the road. Just then, a red car came towards us. It slowed down as it approached. Someone who looked like Dex was sitting in the front passenger seat. He put his head out of the window and said something. I saw the driver's teeth.\n\nAs the car went past us, the Dex look-alike shouted at me, 'Go back home, you terrorist.'\n\nA boy from the back of the car hurled a can of beer at me. It came flying towards me, leaving a trail of beer behind. It missed me and fell on the ground. Jake picked it up and ran after the car.\n\n'Jake, stop!'\n\nHe kept running. The car slowed down to let another car turn. Jake threw the can at the car. It hit its target and rolled under the wheels of an oncoming bus. 'It doesn't matter,' I said, when I got close to him.\n\n'It does,' Jake said.\n\n'How many are you going to throw that at?' I asked. He ignored my question and we crossed the road. 'Don't ask about Dex,' I said. 'I know nothing.'\n\n'I know,' Jake said.\n\n'If Dex was held by a Man U fan, then me Dad would probably know who he was,' I said.\n\n'He might just be,' Jake laughed. I laughed.\n\n'Man U supporters are everywhere,' Jake said.\n\nWe walked in silence to our favourite place. Jake pulled a twig out of the ground and said, twisting it in his fingers, 'I told him not to sign up for the army, but you know our kid, he's not one for listening. I didn't hear much from him after he went off to Afghanistan, not that I expected him to write anyway. But he never came back from a patrol, that's what I heard. I keep asking around, I know I must sound stupid, but I do and I don't know where Dex is still. Yesterday, some men came to our house. Army men. Dad was his usual, drunk self. I was waiting for the letter. They bring it round. I've seen what it's like. Lots of them have been given out round here. I guess you know that, don't you? But they didn't give me a letter. They said they knew I was going round asking for help to find our Dex, and told me to stay out of things I didn't understand.' Throwing the twig as far as he could, Jake said, 'What should I do now, Kiran?'\n\nHe went quiet for a bit.\n\n'I didn't know what to say.\n\nHe looked at me, and asked, 'What would you do if you were in my shoes?'\n\nWe stood under the willow tree for a while, and then Jake walked me back home. As I was leaving him, I said, 'Even though I hate Dex, I wish I could help you.'\n\n'You know Taliban are Man U supporters...'\n\n'Don't you dare come to my house again,' I said, slamming the door on Jake. I was mad with him. He was like a broken record. I ran upstairs, slumped on the bed and after a little while dozed off.\n\nA heavy, booming voice racing through the house woke me up. This was the unmistakeable roar of my granddad. It was dark outside. My room was all clean. My shoes were gone. The dirty towel was gone. My dirty underwear was off the floor. My desk was all sorted, and everything that should be hanging on the hook behind the door was hanging there and not lying on the floor.\n\nEach time Granddad met my Dad, it was like he hadn't seen him for years. Hugging and kissing, complaining to my Dad that so-and-so's-father's cousin died in Pakistan so he should go to their house in Sheffield to give them condolences; my father would always spend ages pretending to work out who this so-and-so was. A moment or so later, there was sure to be a heated argument between father and son.\n\nGranddad saw me as I went past the door. There was someone else in the room, but I didn't see who it was. Granddad raised his hand, but I shot into the kitchen before he could call me. The air in the kitchen was rich with the scent of fried samosas and pakoras. Below the noisy extractor, steam was coming out of three pans. Mum was putting some cups on a tray, next to a plate of freshly made kebabs.\n\n'Wow, Mum,' I said, taking a samosa and biting into it. I asked, 'Why didn't you wake me up, and who's the special visitor?'\n\n'You were out cold, you were, and don't talk with your mouth full,' Mum said. 'You're not a child now.' She picked up the tray with the tea, nodded to the plates, and said, 'Bring those in.'\n\nThe samosas were so delicious, I bit into another one. To Mum's disapproving eyes, I left the half eaten one on the table and followed her into the living room. Jani from Year 11 was sat down furiously texting on his phone. His father, who sat next to him, rocked his head continuously from side to side; I used to think this meant 'no' but it actually means 'yes'. My granddad and Dad were arguing as they always did.\n\n'If it wasn't for the Americans, where would your team be, lad?' Granddad asked, flicking his thick eyebrows at me, giving me that _hellosweetiejustkeepquiet_ kind of a look.\n\nDad slapped the side of the chair and said, 'And look at your team, Dad, where would your lot be if wasn't for Arab money?'\n\nMum put the tray on the coffee table and pulled it into the room, in between Granddad and Dad. I put the plates of food on the table, giving Jani a _whatthefyoudoinghere_ stare. He looked away from me and sent his text. 'And how is my luv'ly granddaughter?' Granddad asked. 'And how grown-up you are now.'\n\nDad didn't give me a chance to reply and said, 'At least people across the world care for us. We're not merely have-beens, losers, acting-big-on-oil-money-from-Arab-fat-cats.'\n\n'You know what I think about football?' I said, plonking the samosa plate on the table. 'How can anyone get excited about a load of big, sweaty men running around after a ball?'\n\nGranddad laughed. Dad said, 'Man U is not just a football team, it's a legend...'\n\nGranddad whispered into Mum's ear. Mum pulled away, held me by the hand and led me out of the room. As I was leaving, Jani's Dad said, 'Manchester United or Manchester City, we're all Muslims.'\n\nIn the kitchen, Mum said, 'Granddad wants you to dress properly for the occasion.'\n\n'What occasion, Mum?'\n\nMum hesitated, and said, 'You know, dear, your Granddad has come with his friend and their son.'\n\n'That loser, Jani, yeah,' I said. I didn't like where this was going.\n\n'Well, you know.' 'Know what, Mum?'\n\nMum lifted the lid off a pan, waited for the steam to escape, stirred the food, and said, 'Your granddad wants to know what you think about, about him arranging, you know, your marriage...'\n\n'Mum!'\n\nMarriage, I thought. They've got to be kidding. My parents have gone mad. I don't even know if they ever really got married themselves, or just pretended they were to keep everyone happy. An arranged marriage!\n\n'You know what Dad calls this sort of a thing?' I asked. 'A lucky bag job.' 'It's not my idea,' Mum said, putting the lid back on.\n\n'Did you ever get married, Mum?'\n\n'Oh, darling, how can you say such a horrible thing,'\n\nMum grinned. 'Of course we did. I still have my dress.'\n\n'You want me to wear it, do you?' She didn't answer.\n\n'I'm only fourteen.'\n\n'That's what I said,' Mum said, taking plates out of the cupboard. 'Your granddad said it's just a thought for the future, not now.'\n\n'To that idiot!' I laughed. 'You know what girls say...' 'I don't need to know...'\n\n'Why didn't you tell me before, at least I could have dressed _properly_?' I asked.\n\n'Your Dad only told me last night...'\n\nLast night, I hissed inside my head. I could throw the food away. No, spit in it. Chuck it at them. Then an idea came to me. 'How could _you_ keep it from me, Mum?' I said, turning to leave.\n\n'Where you going, luv?' 'To dress properly,' 'I'm sorry.'\n\n'It's no big deal, Mum, honest,'\n\n'Oh, Karey...'\n\n'It's Kiran, Mum.'\n\nI got changed as quickly as I could and came down to help Mum, wearing the shortest skirt I could find.\n\n'That's my girl,' Mum said, her eyes almost popping out of her head.\n\nMum had cooked the chicken and was making fresh rotis. I chopped some salad and spread it out on a plate. Mum filled a large serving bowl with chicken, placed it on a tray with the rotis, put the plate of salad on top of the rotis and asked me to fetch a jug of water.\n\nI took the empty plates, walked up to the front room and peeped in through the half-open door.\n\nMy Dad and granddad were still arguing about Manchester United and City. Jani's Dad was smiling stupidly, stroking his hennaed beard. My granddad stopped talking, closed his eyes and breathed in the aroma from the food. He opened his eyes, took off his glasses, took a handkerchief out of his top pocket and started cleaning them. Holding the glasses in his left hand, he pulled the table closer to him, and said, 'I could eat a horse and chase the jockey, I could.' Then he laughed loudly. He always said this when we put food in front of him in our house. 'I bet you could,' I thought, 'and probably take a bite out of the spectators as well.'\n\nI walked in and said the loudest _As-salam alaykum_ I had ever said. Jani's Dad's mouth dropped open when he saw me. My Dad looked at me and started wiping the table clean with a piece of tissue. Granddad put on his glasses, took them off and started cleaning them again. Jani was still texting.\n\nCome on girl, I thought to myself, fill a glass of water and chuck it at him. No, better to throw something hot so it'll burn. Mum brought in a steaming bowl of chicken. I imagined pouring it down his trousers. Jani screaming down the road. Him, turning round to look at me. Me, sticking my middle finger up at him. I slumped down next to Dad, folded my arms and glared at Granddad. He looked at me, winked at my Dad, and said, 'Young people. They have their ways, I suppose. Why don't us men go for a walk before we eat dinner?'\n\n'Perfect,' I thought. 'This is the right time. As soon as Mum leaves, it all goes down his pants.' Jani didn't notice the men leaving, or me glaring at him. I waited for a bit, and said, 'So, what's this then, you sack of farts?'\n\n'What's what?' Jani said, without taking his eyes off his mobile phone. 'Put that thing down and talk to me now or...' I looked at the chicken bowl and laughed, 'you're going to need a lot of ice.' Jani put the phone in his pocket, looked sheepishly at me, and said, 'It's not what you think.'\n\n'What's not what I think?' I asked. 'I don't want to marry you,' Jani said.\n\nMum stood by the door pretending not to be listening. 'Who the hell wants to marry _you?'_ I asked.\n\n'I mean, I didn't come here to marry you,' Jani said.\n\n'Why did you come at all you, idiot?' I said. 'You know I hate you.' 'Me Dad said he would buy me a new bike if I just came with him.'\n\nMum burst out laughing. I went up to Jani and whispered things no Mum should hear. His freckly face went bright red.\n\nWhen the men returned, Mum and I looked at each other and burst out laughing. Granddad looked shrivelled up and sunken. I could tell he felt really let down and humiliated in front of Jani's Dad. When he saw how happy we all were, he straightened and nudged my Dad, with a _seeItoldyouso_ type of a nudge. My Dad frowned and rubbed his hand over his head not knowing which way to look. Jani's Dad twiddled with some prayer beads and said aloud, 'AllahOSomething'.\n\nThe men sat down. Mum and I walked out of the living room. They ate quietly. Mum and I collected a few things from the kitchen and came back. As we got to the front room, Jani's father said, 'Well, son, are you happy with the decision?'\n\nMum and I stopped where we were. There was a pause, and Jani's father said a bit louder, 'Put that thing away. I'm talking to you.' 'What's up, Dad?' Jani stammered.\n\n'Are you happy with our choice?' Jani's Dad asked.\n\n'Yeah, but make sure I get an extended guarantee,' Jani said. 'Guarantee!' Jani's Dad said.\n\n'I want to make sure it's as good uphill as it is down,' Jani said. 'Up. Down!' Jani's father exclaimed.\n\n'Don't want me gears knackering up Da...' Jani stopped mid-sentence with a gasp.\n\nMum stayed where she was but I walked back into the room, and said, 'Uncle, he's only here for a bicycle. It's cheaper than a bride.' And then I walked back out again, grabbed Mum's hand and we went into the kitchen. We shut the door after us and had a really good _mumdaughter_ laugh.\n\nWe were wiping our eyes when the front door slammed shut. We looked at each other and burst out laughing again. Dad came into the kitchen. He had the biggest grin on him I have ever seen. He walked past us, opened the fridge, pulled out a can of beer and went back into the living room, shaking his head incredulously.\n\nAfter everyone had left, Mum asked me if I wanted to go shopping with her and Dad. I slumped on the settee instead and didn't wake up again until they came back home. I opened the door. Mum walked right past me as though I wasn't there. Dad was singing, a newspaper under his armpit. It fell on the floor. It was the Boarhead Evening News. The front page had a smiling Alex Ferguson, the Man U manager, on it. Now was my time, I thought.\n\nI picked up the newspaper, hugged Dad, and asked, 'Can we all please go to Pakistan for Christmas this year?'\n\nMum gave me a look that frightened me. She went like this once a year. Near Christmas.\n\n'I don't have any time this year,' Dad said, avoiding eye contact with me. 'My friend Laila's going with her family, I could go with her and visit your village and see the relatives I have never seen.' 'No,' Dad stood up and looked out of the window.\n\n'Why can't I go, Dad?' I asked. 'I was born there, why can't I go for a visit?'\n\nHe didn't turn around. 'Mum?'\n\nShe remained silent. She was ghostly white.\n\n'If it's money, I'll pay you back when I grow up,' I said.\n\nI waited for either of them to say something back. They just looked through each other. I left them to their stares and went up to my bedroom.\n\nBy the time I got to my bedroom, Mum and Dad were shouting at each other. _It_ was back. But this time it was different. I shut my door, but I could still hear them arguing. I couldn't work out what they were saying to each other, only that my name kept coming up.\n\nMum and Dad were always arguing since I asked if I could go to Pakistan. If they knew I was around, they would go silent and stare at each other as though they were strangers. I could see my family falling apart. I wanted to ask them why they were destroying each other. I had rights, didn't I? I was a part of this family, wasn't I? I had a say about us staying together, didn't I? I loved them and I knew, deep down, they loved each other. What was so bad that they were ripping each other apart? Why couldn't they just kiss and make up, like they used to? But I was not a part of this grown-up world and they made sure I was kept out.\n\nMum and Dad were not fighting like Mums and Dads fight: about shopping, cleaning, going out or watching television. Mine were fighting about something they did not want me to know. It was to do with me. Maybe because I was a Muslim now? I asked Mum this very question when she was sitting alone in the kitchen dicing the onions, over and over again. A pot of daal was boiling on the cooker behind her.\n\nShe put the knife down on the table, wiped her hands, leaned over and kissed me on the forehead. She stood up, lifted the lid off the pan, stirred the daal and switched the extractor on. It burst noisily into life, sucking the steam up into its innards.\n\nMum's kiss was a dry, empty kiss. Her lips had hardly touched me. It fell off my forehead as soon as she turned around.\n\nAfter putting the lid back on the pan, she picked up a bulb of garlic, sat back down, placed the garlic next to her and started chopping the diced onions again.\n\n'You know I love you and Dad, Mum,' I said. 'Can't you two just make it up like you used to?'\n\nShe didn't answer. I didn't think she would. I started peeling the garlic.\n\nPlacing the peeled garlic in front of her, I said, 'It doesn't matter, I won't go to Pakistan. I just asked. It's not that important.'\n\nMum stopped chopping and looked at me. She turned ghostly white. The knife still in her hand. Its blade up. Her eyes suddenly filled with tears. They trembled on the edges of her eyelids. One of them slid down her face. She flicked it off her cheek.\n\n'Mum, please,' I called out.\n\nShe wiped her eyes with the back of her hand. She pursed her lips and shook her head.\n\n'Is it because of Jake?' I asked. 'I need to know what's going on Mum. He's not my boyfriend, Mum. Honest. I'm not like that. He's my mate. You know he's from WTM. Besides he's not my type.'\n\nI laughed falsely and waited for Mum to smile. She just looked through me and I said, 'I wouldn't do anything to hurt you. Honest, Mum. I love you.'\n\nHer face tightened. She looked at me with eyes that could have ripped me apart. Her grip tightened on the knife. Her lips quivered. The front door slammed shut. My Dad was back.\n\nHe came straight into the kitchen. 'Hi, Dad,' I said.\n\nHe kissed me on the head and put his sandwich box on the table next to me. Mum and Dad didn't look at each other. Mum scraped the onions onto a plate and chopped up the garlic I'd peeled. Dad took a can of beer out of the fridge. He stirred the daal and stood there drinking his beer. I looked at him and tried to find where the lovely cuddly bear of a Dad of mine had gone.\n\n'Daal, again,' Dad said.\n\nMum dug the knife into the board, and said, 'at least you can eat...' 'Will you never let go?' Dad asked.\n\nI ran upstairs and slumped on my bed. After a while, I got up and opened the window. A cold wind came rushing in. I stuck my head outside. It was a bright day. George's dog was sitting in its usual place, staring at a magpie, sitting on the dance mat. George was looking through some CDs.\n\nDownstairs they were shouting at each other. I heard the muffled words.\n\n_Pakistan._\n\n_Why?_\n\nGeorge put a CD in his ghetto blaster. The noise of the music drowned Mum and Dad out. I left my collapsing world downstairs and thought back. I could not remember a time they went out together, other than to the supermarket. Maybe I could give them some time together. They could go out for a bit. Mum was always wandering through the house, like a ghost, looking for something and Dad was always glued to the television, avoiding the ghost. Why hadn't I seen it before, I cursed myself. They were both spending all their time together running away from each other.\n\nI lost track of time. I came back to my world with a jolt when George turned the music off.\n\nDad was walking out of the gate.\n\nJust then, I saw Shamshad coming towards our house. There was a knock on our door. I ran down to tell her to get lost, but Mum got the door. Mum had a glass of juice in her hand.\n\n'I just wanted to tell you that that half-caste daughter of yours is going out with Jake Smith,' Shamshad said.\n\nShe sniggered and left.\n\n'She's lying, Mum. Lying. And that's gone now as well,' I said. And then I shouted after Shamshad, 'I hope you die!'\n\nMum said in a strangely calm way, 'You shouldn't say things like that about her...'\n\n'She's just evil,' I hissed. 'Evil!'\n\nMum said coldly, 'You should not wish evil things onto others. There is much you don't understand. You are young. What she did might be bad but she was not born bad. No one is...'\n\n'What you taking her side for?' I interrupted rudely. 'Who do you think you are?'\n\n'I will not be spoken to like this,' Mum replied sternly. 'Go to your room, right now and think about what you just did.' 'What did I do?'\n\n'Go upstairs!'\n\n'You're on her side against me,' I said pointing to Shamshad, who was standing on the pavement enjoying every moment of my misery. 'What is it Mum, didn't you want me? Did you want a Boy?'\n\nMy angry Mum became a ghost of a Mum. I slammed the door shut. I don't know what I had said that was so bad. _It_ was back. She took deep slow breaths. The sides of her mouth were twitching like she was chewing on something. She began to sweat. She raised her hand to hit me and then she looked at her open palm. She had never hit me before. I screamed, 'Mum, you're scaring me.'\n\nHer hand froze in the air. She turned her head towards me, slowly, with a blank look in her eyes. She turned and walked up the stairs, mumbling something under her breath. I went after her. By the time I got to the top of the stairs, she had already shut the door and locked herself in her room.\n\nWhy do you have to say things that you know will hurt her, I cursed myself in my head.\n\nI banged on the door, begging her to let me in. I wanted to tell her how sorry I was and that I would never do it again. What she did today, she had never done before and I knew it was no good knocking; she would never open the door when she was in one of her freaky silences. Not for me, not for my Dad, no one could enter into this world of hers. Sometimes she went into this world, leaving the door ajar. I would walk past her bedroom, as quietly as I could. She just sat there, on the floor, next to her bed, clutching her little box, which I had seen her hiding under the floorboards. Sometimes she would let out a scream. A terrifying scream that came from deep inside her, in a voice that did not sound like hers. She would scream and scream until she went hoarse. She would just as quickly stop screaming and start swaying to and fro with her box. Her face drained of colour. Her hair covering her face. Her eyes so empty. I always thought that if I ever witnessed an animal being killed, it would make a sound like my Mum, moments before its throat was slit.\n\nI was about to leave when I heard her walking around in her bedroom. I heard her unlock the door. I stopped. The door creaked open. I heard her walking in the room again. I turned around and she was sitting exactly where she always did when she went into one of these moods. She took a deep breath, and said without looking at me, 'I have cooked for you.'\n\n'I'm not hungry, Mum.' 'I have cooked for you.'\n\n'I told you, I'm not hungry.' 'I have cooked for you.'\n\nThe front door slammed shut. Dad was back.\n\n'Dad, come upstairs now. It's Mum!' I called out to him. He didn't answer.\n\n'Mum, please, you're frightening me.'\n\nI went into the room. She turned towards me. I brushed the hair off her face. Her eyes were bloodshot.\n\nI ran downstairs. Dad was flicking through the TV channels. 'There's something wrong with Mum, Dad.' He pursed his lips and nodded. 'Dad!' I shouted.\n\nHe turned away from me, clenched his fist, cracking his fingers. There was a strange wall between us. His mobile phone buzzed into life. 'Answer me, Dad,' I said, grabbing the phone off the sideboard. Switching the mobile off, I asked, 'What's wrong with Mum?'\n\n'Nothing,' he sighed.\n\n'I am not a child any more, Dad. Tell me now.'\n\n'Come upstairs and look at her, and then tell me there's nothing wrong with her.'\n\nDad kept quiet.\n\n'There's nothing proper about this house, is there?' I waited for him to answer. He sat down. I cried, 'Please Dad, call a doctor or something. Mum needs help.'\n\n'She'll be alright, sweetheart.' His voice was heavy. 'Come upstairs with me, please Dad.'\n\nHe looked blankly out of the window.\n\nHe never did go to Mum when she was like this. It was like they didn't know each other. I ran back upstairs to Mum. She was hiding the box.\n\nI waited for her to finish hiding it before going in. 'Mum, I'm really, really scared.'\n\n'It will have gone cold now,' she said, tying her hair.\n\nI went into my room and thought aloud, 'Well Kiran, you're going to find out what's in that little box.'\n\nThey were arguing again. This time they were not shouting. They were in the living room. I went in. They were sitting in front of each other, across the coffee table: the television was off, the curtains shut. They looked at me as I walked in and it was a look I didn't want.\n\n'Sit down, Kiran,' Dad said.\n\nI stood there, looking at them. Their eyes were bloodshot, their faces sunken. Mum's hair was all curled up and the tip of her nose was red. Dad sat on the edge of the settee, elbows on the table, hands locked together, chin on his hands, eyes fixed on me.\n\n'We need to talk to you, Kiran,' Dad said.\n\n'I don't want to hear what you're about to say to me,' I replied. A shiver ran down my neck. He went quiet. I looked at Mum, and asked, 'Mum?'\n\nA tear fell out of her eye. It landed on the back of her bony hand. She rubbed it into her hand with her other hand.\n\n'You love each other, right, Dad?' I asked, 'Right, Mum?'\n\n'Sometimes things are just not meant to be, Kiran,' Dad said. 'Even when two people love each other, they can lose what they had.'\n\nMy legs were giving way. I sat on a chair at the edge of the coffee table, 'I just want my family,' I said.\n\n'Is it because I put this on?' I said, ripping off my hijab. I threw it towards the television.\n\nMum stood up, picked up my hijab and put it back on my head with shaking hands.\n\n'At least tell me why?' I pleaded.\n\nNo one spoke for what seemed like an eternity. And then Dad said, 'Some things can't be explained, Kiran.'\n\n'Are you really breaking up? My family? You two? Really, really breaking it all up?' I stood up and went into the kitchen. Two suitcases were packed and a few boxes were stacked on top of each other. I kicked the suitcases, went back into the living room, and shouted at Dad, 'You're really, really leaving, aren't you?'\n\n'It's for the best, Kiran,' Mum said. 'And what's going to happen to me?'\n\n'You're coming with me,' Dad said. 'We'll live at your granddad's for a while and then get our own place. Everything will work out just fine. You'll see.'\n\n'Just fine,' I thought. Yep. I'll be just fine. All my dreams destroyed.\n\nEverything shattered. Just fine! That's what you call it. Yep.\n\n'Do you think I'm going to just leave Mum on her own?' I cried. 'Do you?'\n\nMum stood up and hugged me. She was trembling. She whispered into my ear, with a faded voice, 'You're staying with me, love, always staying with me.'\n\n'Your Mum's going to be off work for a while,' Dad said, standing up, 'I'll send you some money every week and don't worry about any of the bills...'\n\nI stepped forward and tried to punch Dad in the chest, saying, 'How can you be so calm? How can you?' He held my wrists in his strong, rough hands, hugged me and cried. He pulled away from me, kissed me on the cheeks and stepped out of the living room.\n\nAt first, I didn't believe Mum and Dad would really break up. Even when he had left, I thought he was going to come back and slobber in front of the television. I thought Mum would just snap out of her silences. But Dad didn't come back and Mum went more and more silent. Dad came round each Friday evening and handed me an envelope, standing on the front doorstep. We talked about things that didn't matter, and he left for a week. When he came round, Mum hid somewhere in the house. When he left, Mum didn't really come out, she just changed rooms. Most of the time, she was in her room, mumbling to herself, words that I didn't understand, in a broken family, whose reasons for breaking up didn't make sense.\n\nI gave the envelopes with the money to Mum. She put them on top of each other in a cupboard in the kitchen.\n\nDad took care of the bills, and I did the shopping, cooking, and cleaning. Mum withered away.\n\nI spent endless hours walking around the house trying to work out why my family had broken up. Thinking about ways I could get it back together again. I thought back over all the arguments I had heard Mum and Dad having. I cursed myself for not listening to what they were saying to each other. I recalled the fragments of the loaded words that shattered my family: _Pakistan_\n\nWhatever it was, it had to do with me. That much was clear. I was born in Pakistan, in my Dad's village, when Mum and Dad went out there in 1996. That much I knew. I came to England when I was just a few months old. That much I knew. But I never went back again. Maybe we just couldn't afford it? I don't know. Maybe I was just an accident, not what they wanted. Maybe, they wanted a boy and I turned up and I was horrible. But Mum and Dad loved me. I never felt unwanted in this family. The day of the breakup had been the first birthday Dad forgot about. He never forgot my birthday even though he usually forgot his own.\n\nWhat was there to move on from? Was this what they meant all along?\n\nBreak up the family. And those terrible cries of Mum? Calling out to God. The screams that pierced through the house when they argued. I thought back to when they argued, it was always close to Christmas. They shouted at each other until Mum hid in her room. Dad drank and watched football. I used to put my headphones on and blot them out. That's what grown-ups do, I thought. It was all normal. A happy family arguing with itself.\n\nI cursed myself over and over again. Why didn't I see there was something terribly wrong with my family? Maybe Mum or Dad was having an affair with someone else? I couldn't believe this. They never went out other than for shopping. And in all their arguments, I didn't once hear a word that might have meant this could possibly be the end of my family. One thing I was certain about though, whatever it was, it involved me. And if it involved me, I was going to find out what it was and get Mum and Dad back together again.\n\nI had to find out what it was that Mum kept hidden under the floorboards. Whatever it might be, I would find an answer in there. Mum might kill me, I didn't care. I had to know. But Mum never left the house when I was in.\n\nAt school, I avoided everyone but Laila. Jake came up to me a few times, looking all apologetic and wanting to talk. I gave him the cold shoulder and he kept his distance, but always looked at me in a way that he wanted to say something to me. It was difficult avoiding Shamshad, though. She often blocked my path after school, and said, 'Half-caste families, they're not proper. They always break up.' She didn't frighten me any more, though. She just hurt me. I always kept my head high and walked past her, and then forgot her. I needed to be home. To make sure Mum was OK.\n\nTime went by so quickly. I did the shopping, cooking and cleaning. Laila came round at the weekends. Mum would come out of her bedroom when Laila was around, nod to her, and then go back into her room again.\n\nWhen she came round, she not only helped me to clean up but also helped me to read the Quran. I looked forward to Laila's visits. In my new mad world, she made me feel I belonged to something. We prayed together and then chatted about her new favourite subjects: The _X Factor_ and boys.\n\nA week before the Christmas holidays, Laila went to Pakistan.\n\nThe first snow of the winter fell the week after Laila left. I saw it falling through my bedroom window. I watched it fall on the rooftops, watched it fall on children throwing snowballs at each other, and watched it cover the cars and hedges. I remembered how Dad used to make me a snowman. He built the best snowman on the street. Even if the snowman was much thinner than he was, and even if it was the only one in East Boarhead with a cucumber for a nose, it was still the best. I fell asleep dreaming of my snowman.\n\n## **Shamshad**\n\nIt was bin day, and Karen's Dad was going to pass our street. I had my camera ready. It was charged and waiting on the kitchen table. I'd been meaning to take a photograph of his big, fat ugly face driving the rubbish truck. Our bins are at the back of our house. We place them outside our back gate, in a small alley, and they are taken from there. It was collection day for the black bins. But as no one could remember which day was for the green, the brown or the black, everyone followed whoever got theirs out first. Sometimes it meant most people got it wrong, and if this happened with the green ones, into which the food was meant to go, it didn't half smell, especially in the summer.\n\nThe two-legged _majja,_ the buffaloes, were out. The short round Mrs Khan, our neighbour, with her big, gold earrings was nattering to two others. The _paan_ -chewing Janat Choudry, with her hennaed hair and bell-shaped earrings, bigger than those of Mrs Khan, was shaking her head at the third buffalo: Asmat Jaan. She had a moustache bigger then her husband's, and was always sneaking out for a smoke.\n\nIt was really cold. I had my mittens on, but it made little difference. The buffaloes were standing close to each other. This meant there was some really juicy gossip going round otherwise they would be talking to each other from over their fences. Pulling my bin behind me, I got close enough to hear.\n\n'Left that _goree_ , his white woman, didn't he?' Mrs Khan said, nodding to the bleeping of the rubbish truck, which was somewhere on the other side, next to the houses that backed onto ours.\n\n'Who?' Janat Choudry asked, with a puzzled wrinkly look.\n\n'Ghanzanfer's son, who lives with that white woman in West Boarhead,'\n\nAsmat Jaan adjusted her _dupatta_ across her breasts, and said, 'They never stay with their white women.'\n\n'Nah,' Mrs Khan said.\n\n'But they can't keep their hands off them either, can they?' Janat Choudry said, lifting the lid off her bin. She spat into the bin, and added, ' _Goreyaan_!'\n\n'He's going to Pakistan,' Mrs Khan said.\n\n'Who?' Janat Choudry asked, putting another _paan_ into her mouth. 'Ghanzanfer's son,' Asmat Jaan said, lighting a cigarette. 'He's coming back next week with another woman.'\n\n'They always do,' Mrs Khan said, untangling an earring, which had got stuck in her hair.\n\n'When their _goreyaan_ , their white women, kick them out,' Janat Khan said, chewing her _paan_.\n\nBlowing smoke out of her nostrils, Asmat Jaan said, 'They have the luck of fate.' 'The _goreyaan_ ,' Asmat Jaan laughed. 'They do. That they do.' 'And we get stuck with ours,' Mrs Khan said.\n\nThe buffaloes laughed.\n\nI asked loudly, 'Any news you can share with me, aunties?'\n\nAsmat Jaan let her cigarette drop to the ground, and said, 'It's going to be cold this year.'\n\n'You should wrap yourself up, young girl,' Mrs Khan said. 'Look after yourself properly,' Janat Choudry said.\n\n'And you're out without even your jumper,' I said.\n\n'Nothing'll happen to us old buffaloes,' Asmat Jaan said.\n\nThe old women laughed as I walked back into the house, clenching my fist with joy. I washed my hands and could see the words flying off on my Facebook: _Karen's Dad is off to Pakistan to get another wife. Hey Karen, Your Dad really is Lucky!! He's got a new Mum for you in Pakistan. He's bringing her back next week. Wanna see pictures. Lol._\n\nAfter I put these words up on my Facebook status, I waited for Likes. I waited and waited, but not one Liked my post. I expected lots of LOLs and hahahas. But nothing. Eventually Aisha wrote: omg. Sad.\n\nI don't know why Aisha's word _sad_ so upset me. But I suddenly felt what I had done was really, really bad. I was bad. Always showing off. A big bad bully. It was me who was the sad case.\n\n_'Allahjee_ , help.' I cried, 'Please God forgive me.'\n\nI went back and started writing another status. I wanted to say sorry, but I just couldn't do it. I would look even more stupid then I was.\n\n## **Kiran**\n\nIt was past 11 o'clock in the morning when I woke up the next day. I could hear Mum in the bathroom. I put on my dressing gown and went downstairs. There was a letter on the front door mat. It was from Dad. It said: _Going to Pakistan tomorrow. Will tell you all about it when I come back. My number is on the back of this paper._\n\nI opened the front door. Footprints in the snow led from the door to the garden. The cold wrapped itself around me. I slipped into my shoes and stepped out. There was a snowman in the garden. Just like Dad used to make. With tomato eyes and a cucumber nose. I snatched my eyes off the snowman; it was already making me feel sad and angry.\n\nI shut the door and pinned the letter from Dad onto a board near the cloak rack.\n\n'Come upstairs now,' Mum called. 'Dad's gone to Pakistan,' I shouted back.\n\nMum went silent for a moment and then said, 'And bring lots of bin liners with you.'\n\nI went into the kitchen. It was a mess. The sink was full of unwashed dishes. Two pots, half-filled of food with their lids off were on the cooker. An overflowing ashtray, with half-smoked cigarettes, was on the worktop next to a dirty serving spoon.\n\nI opened the drawer in which the bin liners were kept and took the whole bundle with me and went up to Mum. The room stank of cigarettes. It was damp and cold. The curtains were drawn. Her bedside lamp was on. She was standing in front of her open wardrobe, staring at her clothes.\n\nShe unfolded a bin bag, fanned it open, and gave it to me to hold. And then one after the other, she took her dresses out of the wardrobe and dumped them into the bag.\n\n'Going to buy some new ones at last, Mum?' I asked.\n\nShe ignored me. When it was full, she said, 'Take it downstairs.'\n\nI did. When I came back, she had filled another one and handed it to me. I took that down as well. When I went back up, she was on the phone 'George, the curry club is dead. Finished. Tell the others.'\n\nI left Mum, went into the living room, half expecting to see Dad sitting there, glued to the television. The silence of the room boomed around my ears.\n\nMum came downstairs with a bin liner in her hand and walked out of the house. She came back a few moments later, picked up another two and as she was leaving, I asked, 'What you going to do with them, Mum?'\n\nWithout stopping to look at me, she mumbled, 'Someone else may like them,' and left, leaving the door open after her. I got up and stood in the doorway, watching her throw the bags into the back of the car and drive away.\n\nI don't know where the next few days went. She just smoked and got rid of most of her things.\n\nI couldn't remember the last time I had seen her eat anything so I went into the kitchen and made her a cheese and cucumber sandwich. She was sitting on the top of the stairs, looking down at me, with empty eyes.\n\nWalking up the stairs I held the sandwich up to her, I said, 'Here Mum, please eat this.'\n\nShe ignored me. Putting the sandwich next to her I said, 'Please, Mum talk to me.'\n\n'The bins need putting out,' Mum said.\n\n'Damn you, Mum,' I thought getting up. How long's this going to go on for?\n\nI ran downstairs and went into the living room, slamming the door shut after me.\n\nJust then I got a text from Laila: _Is it true?_ I messaged her back: _What?_\n\n_Go to FB,_ she texted.\n\nI rushed to my bedroom; Mum was still sitting on the stairs. I logged onto Facebook.\n\nI went to Shamshad's FB profile and read what she had written. 'Damn you Shamshad,' I cursed. I stomped downstairs and picked up the landline. I got the piece of paper with Dad's number from the noticeboard and dialled. Waiting for the line to connect, I looked at the time on the telephone. It was 8pm. I tried to work out what time it would be in Pakistan but gave up. I didn't care if he was asleep or awake. After a little while, a long tone started ringing. When he answered, I said, 'Dad?'\n\n'Kiran, sweetie, is everything alright?' he said, sleepily.\n\n'You tell me.' Dad went silent.\n\n'You couldn't wait, eh. You're done with the _goree_ , have you? Not good enough for you now, eh. Got yourself a proper one, eh.'\n\n'Kiran, Kiran. Listen,' Dad interrupted.\n\nI didn't let him finish, 'So you left your half-caste kid, eh. Not proper, am I. eh?'\n\n'It's not like that, I love...'\n\n'And that white woman you walked out on. Every day a bit of her dies. The phone was stuck to my hand. I was sweating all over. I didn't hear Mum coming. She sat on the last step close to me. 'What's the matter?' Mum asked.\n\n'It's Mr Lucky. I'm talking to him in Pakistan.'\n\nMum went quiet. I said to Dad, 'Go on then, what've you got to tell me.'\n\n'It's not what you think, Kiran. You need to know everything. And when I get back I promise you'll know. I'm doing this for you, Kiran.'\n\nThat just did it. I let rip down the telephone, 'I hate you and never want to see you again!' I said, slamming the receiver down.\n\nTurning to Mum, I said, 'He's got another woman!'\n\nShe threw her shoulders up and asked, 'What else did he say?'\n\n'It's not what you think, Kiran. You need to know everything. And when I get back I promise you'll know,' I said mimicking Dad. 'I'm doing it for you, _Kiran._ '\n\n'You had to find out someday,' Mum said.\n\nShe went back to her bedroom. Her words were banging against the inside of my head. What did I have to know? Is this how it happens? Mums and Dads break up, just like that. Dads go off to get married again. Just like that.\n\nThe telephone rang. It was Dad.\n\n'There's a lot you need to know. I'm back next week. I promise I'll tell you everything. Everything! No more secrets,' he said. After a pause, he asked, 'Can I speak to your Mum?'\n\n'You've really got a nerve, haven't you? Don't you dare come to our house! Do you hear?' I said, slamming the telephone down.\n\n## **Shamshad**\n\nWhat should have been another boring day, with me coming home from mosque school, Mum oblivious of my arriving as she nattered on Skype, cursing some villager about her goats or the condition of her fields, became a day when my world just collapsed. Everything was a lie.\n\nWhen I walked in, Mum was on Skype, but she was not talking, just staring at the screen. This was a first. She stared at my reflection in a mirror that sat on top of a chest of drawers in the room. Her eyes moved from the computer screen to me, and then back to the computer again.\n\n'Is everything OK, Mum?' I asked, stepping into the room, waiting for her to do one of her outbursts or shut the door in my face. But she just flicked another look at me and went back to staring at the computer.\n\nWhen I got a bit closer, I felt a jolt go through my body. There was no mistaking who she was looking at. Karen's father, Lucky, was standing next to Mum's goat herder, a hard-faced, dark woman. They were looking into the camera. When I got a bit closer, Karen's Dad's face tilted towards the woman. It looked like the screen was stuck. The woman put her hand on her mouth.\n\n'What's up Mum?' I asked.\n\nThe woman standing behind Karen's Dad in Pakistan let out a wail. The connection was bad, cutting her voice.\n\nA tear rolled down Mum's face.\n\nThe front door opened and shut with a loud bang. The glass chandelier hanging in the middle of the room jingled. Dad was home. And Mum hadn't turned the computer off. I didn't want all hell breaking loose.\n\n'What're you doing, Mum?' I whispered, moving my hand forward to turn the computer off. 'He's home.'\n\nMum grabbed my hand and stopped me from turning the computer off.\n\nShe was looking at me in the mirror. There was no fear in her eyes.\n\nIn the mirror, I saw the door opening behind me. Dad was taken aback when he saw us together, standing in front of the computer. When he realised who we were looking at his eyes narrowed. His face tightened and he hissed through his nose. I stepped away from Mum, and stood pressing my back against the wall, underneath a picture frame with the word 'Allah' written in golden Arabic lettering.\n\nDad looked at me and pointed to the door. I turned to leave, but Mum grabbed me by the wrist, holding me tightly. Dad's eyes burned down on Mum's hand.\n\nHe took off his black winter coat and threw it on the sofa, showering the carpet with snow. He unplugged the computer from the mains.\n\n'She has to know,' Mum said, loosening her grip.\n\n'You dare defy me,' Dad said, taking off his shoe. Mum just stood there. Her head unbowed. 'Oh, Dad, no!' I pleaded.\n\n'Lower your head, woman,' Dad said, stepping towards us, the shoe in his hand raised.\n\nMum let go of my hand. 'Dad, please, it's all my fault.'\n\n'What have we done?' Mum said, looking Dad in the face.\n\nDad brought his hand down to hit Mum. I just snapped and ran at him screaming, 'You will never do that again. It's a crime against Allah and it's a crime against Mum and it's a crime against me.'\n\nHis arm froze mid-air, shoe in hand.\n\n'Never as long as I live, Dad, never,' I screamed.\n\nMum stepped forward, put her arm around me and said, 'We have lied to Shamshad. We have lied to God.'\n\nDad dropped the shoe. He was so shocked he stumbled backwards. Regaining his posture, he let out a stream of obscenities. He no longer looked terrifying. Just pathetic. I put my hands on my ears and ran out of the room as their words lashed each other. 'You will never hit me again!'\n\n'It is all your fault!' 'You're seedless!'\n\n'Leave this house now, woman!' 'It is mine!'\n\nTheir words chased me all the way up the stairs. I shut my bedroom door. I could still hear them, but the words were muffled now. I wanted to go back and ask them to tell me what I had done to bring this down on us. I waited for a moment and then opened my door to go down.\n\nMum was screaming at Dad, 'Is there a bigger crime than what we have committed?'\n\nI turned around and went back into my room. The snow had stopped. The world was frozen. I picked up my headphones and put them on my head. With my shoes still on, I jumped onto my bed, pressed the play button on my iPod and sang loudly along with Lady Gaga.\n\nWhen the song finished, I took off the headphones and threw them against the door. The war downstairs was still on. I turned my head and buried my face into my pillow, cried and fell asleep.\n\nI woke sometime later. My curtains were drawn. It was past 2 a.m. A cat was wailing somewhere, like a lost baby crying for its mother. Popping my head behind the curtains, I half expected the cat to be sat outside my window. There was no cat, only a world drowning in falling snow.\n\nMy stomach rumbled a hungry rumble. I put on my bedside lamp. There was a glass of milk next to it, along with some rotis wrapped in a cloth. I woke up late the next morning, and stayed in bed a bit longer, just to make sure Dad would have left for work, all the while trying to make sense of the words Mum and Dad had hurled at each other.\n\nWhen I opened my bedroom door, the house was filled with the smell of roasting parathas and buttered rotis. I didn't bother cleaning my teeth or washing my face and went down.\n\nDad was still at home. He was sitting on his own, in the living room, staring at the wall in front of him. The door was wide open. His coat was still where he had left it. His shoe was where it had fallen the night before.\n\nMum called me from the kitchen, 'Shamshad, come here.'\n\n'I'm late for the mosque, Mum.' 'Come here,' Mum said.\n\nI went into the kitchen, and said, 'I'm not hungry Mum, honest.'\n\nMy stomach rumbled in protest as I walked into the kitchen, swallowing a hard lump in my throat. Mum smiled at me. She was holding an egg in her hand. She cracked it on the side of the frying pan and the contents of the egg sizzled in the hot oil.\n\nPutting the fried egg onto a small plate, Mum placed it next to the paratha and a glass of orange juice. She stood there looking at me. I pulled the food towards me, broke a piece of paratha and dipped it into the yoke. The yoke broke and dribbled into the white. I ate in silence, drank my juice and walked out of the kitchen.\n\nDad stole a look at me as I went past the front living room and out of the door.\n\nIn the mosque, I didn't help with teaching the younger ones like I usually did. I squatted against a wall and stared down at the open pages of the Quran. I tried to remember the sounds in my head, the sounds of a language I didn't understand, but which was being recited by little children all around the room.\n\nI kept hearing Mum and Dad's words going round and round in my head: _Is there a bigger crime than what we have committed? Barren. Seedless._\n\nWhat had they done? Whatever else it is, 'Girl,' I thought, 'you are not wanted. And you're not a kid. You know what they're talking about.'\n\n'Stop now', I said to my thoughts inside my head. They were hurting me. I didn't want to know. I chased my thoughts away by reciting the Quran, but they clung on.\n\nI would have stayed at the mosque much longer, but had to leave before the start of the boys' session. I walked home as slowly as I could, dragging my feet through the slush, oblivious to the falling snow. A snow flake went into my eye. I was burning up inside.\n\nI thought about running away from home. But where would I go? I was not only hated at home, but I knew, deep down, I was hated by everyone, and even those who said they were my friends were just frightened of me. My mobile rang a few times. It was Mum. It had to be her. I didn't reply. When I got on our road, Mum was standing outside on the street, looking in my direction. She was wearing her white cardigan, which she had knitted herself. She also had on her black shawl, which was draped across her shoulders. It was peppered with white dots.\n\nWhen I was close enough to hear, she smiled and said, 'You are a little late, my daughter.'\n\nVapour came out of Mum's mouth as she spoke.\n\nWhen she said this, I stopped in my tracks. I just melted inside. I wanted to scream, 'Why have you never called me daughter before?' but instead I kept quiet and glared at her.\n\nA cracked line of red ran through the whites of her eyes.\n\nSomething rumbled on the roof of our house. Suddenly an avalanche of snow crashed to the ground by the side of our house.\n\nShe stepped forward, wiped the snow off my hijab and said, 'I was just worried.'\n\nI said, 'Sorry!'\n\nMum's hand slid down my arm and brushed against mine. I began to feel really shaky inside. I thought she was going to hug me. I wanted so much for her to hug me and tell me what was going on. To make me feel we were a proper family, just for once, rather than what we really were, two people living in different prisons in the same house.\n\nMum held my hand, and said, 'I've cooked your favourite, _koftay_.'\n\n'I don't want _koftay_ ,' I thought, 'I just want a hug.'\n\nSnatching my hand from her, I went into the house. There was a school photograph of me from the infant's school. I was taken aback. A photograph in my house and of me!\n\nDad was sitting on his own in the living room, staring at the wall.\n\nMum shut the door behind me, and said softly, 'And I have lived behind hidden feelings for long enough.'\n\nI walked straight into the kitchen, brushed the snow off my coat and hung it on the back of a chair. There were more, newer photographs on the walls. There was one of me and Mum, standing outside our house. Steam from a pot on the cooker was being devoured by the extractor, and the air was filled with the scent of cooked daal. Mum came in after me, lifted the lid off the pan and turned the gas off. She chopped some fresh _dhania_ , some coriander, and sprinkled it over the daal in the pan. Putting her nose close to the rising steam, she inhaled, and said, 'The scent of the daal just after the _dhania_ goes in always reminds me of the fall of the rain on the dry summer soil of my village.' I gritted my teeth, unsure of what to say. If I could have, I would have said, 'Damn your village.' But I said nothing.\n\nNow that I was inside the warmth of our house, I felt cold. After putting the lid back on the pan, Mum took a rolling pin out of a drawer. She rolled a ball out of some freshly made dough, dipped her hand into a bowl of flour, sprinkled it on the worktop, and rolling a roti out said, 'Make some salad, daughter.'\n\nHer voice was soft. A voice I had not heard before. 'Just tell me what's going on, Mum,' I thought. 'Or whoever you are, just tell me.' But I kept quiet. I got up, stood on the other side of the sink, close to where she was and started preparing the salad.\n\nThe floorboards of Dad's bedroom, above the kitchen, announced his arrival into his bedroom. Mum looked up, looked at me and let out a little laugh while placing the rolled roti onto the hot _tava_ , a hotplate, which had been placed on the cooker. She flipped it over after a few moments, then lifted the _tava_ in her left hand, and with the other hand she placed the roti onto the naked flame. It began to puff up and up. When it was fully inflated, she tapped it on the top and quickly withdrew her hand as steam burst out of the roti. She picked it up and placed it on a cloth.\n\nI made a bed of lettuce leaves. Then chopped the cucumbers and tomatoes and put them onto the lettuce.\n\nDad was moving about in his room.\n\nAfter putting the salad plate on the table, I sat back down and watched Mum as she made more rotis. She washed her hands after she had finished the final one. Letting out a deep sigh, she looked at me. Her face was sad again. The wrinkles were back. She was looking through me with her dark eyes. She let out another sigh. A smile had flown in from somewhere and landed on her face. Her eyes were sullen, but they were no longer the ones that were glazing over me, not the ones I had grown up with. These were calm. Her gaze was so intense, as if she had not seen me for a long time. The wrinkles on her brown face were gone again.\n\n'What is it, Mum?' I asked, unwrapping the rotis from the cloth.\n\nMum snatched her eyes away, put some daal into two bowls, put one in front of me and one in front of her. She then placed a plate of _koftay,_ meat-balls, in front of me and said sitting down, 'You've grown up so quickly.'\n\nThe scent of fresh coriander rose up in the steam from the daal on my plate.\n\n'I'm fourteen, Mum,' I said, dipping a piece of roti into the daal and putting it into my mouth. I didn't feel like eating.\n\nRunning a finger over the lettuce in the salad plate, Mum said, 'I don't know where it went.'\n\n'What?' I asked. 'Time.'\n\n'I'm not that old,' I said. 'You're all grown up, a woman.'\n\nNodding to the ceiling, I let out my first little laugh in days, 'He's not thinking about a goat herder for me is he?'\n\nI knew I shouldn't have said what I said, even as I said it. But it was too late.\n\nMum's cheeks sank. She looked right through me. The pupils in her eyes shrank. Colour left her face. And with another sigh she was back, 'He will never hurt us again.'\n\nMy throat dried. The roti in my mouth refused to go down.\n\nDad dropped something upstairs. He took a few heavy steps, and then took a few more and slumped onto his bed.\n\nI felt cold all of a sudden, a chill that went deep into my bones.\n\nMum stood up, came round the table and kissed me on each eye. She whispered, 'No one will ever hurt my angel. No one.'\n\nShe sat down next to me, broke a piece of roti, dipped it in my daal and put it in my mouth, saying, 'Shush, now. Eat.'\n\nI was trembling.\n\nShe stroked my cheek. The lump in my throat became bigger. Drier. I chewed slowly. Scared of the volcano rising inside of me.\n\nAfter a few moments in which the clock ticked, in which a blaring police siren from somewhere close by invaded the room, in which someone laughed in the back alley, in which music came in through the walls of the house next door, Mum took my hands in hers, held them tightly, and said, 'You need to know everything. Soon. But I've not been a good mother, have I?'\n\n'It's alright, Mum.'\n\n'I wish I was...'\n\n'You're my Mum, and that's all that matters.' 'I wish I was.'\n\n## **Kiran**\n\nThat night after talking to Dad in Pakistan I kept tossing and turning in bed, falling in and out of sleep, cursing him in my dreams, 'Lucky, I hope you die.' In the morning when Mum came downstairs, she was dressed in her long, black winter coat. Her hair was tucked into a woolly hat and she was wearing her red gloves. She opened the door and left.\n\nShe didn't tell me where she was going or when she would be back, and I didn't ask her.\n\nI watched her leave the yard and turn left.\n\nI was going to find out what was under her bed. I didn't really care even if she came back early and caught me in the act; but when I got to her door, I got scared. A chill ran down my back. This small room suddenly felt so big, like a huge, terrifying cave. I could feel her presence. I could smell her.\n\nThe quilt on Mum's bed was folded back at the corner from where she'd got out of bed. The rest of it looked as if it hadn't been touched. The main light bulb was fused. I put my hand up her lacy bedside lamp and turned it on. As the light came on, shadows from the moving tassels bounced off a picture of Mum and Dad. It was a picture of when they were young. They were standing together yet seemed so far apart. Mum had that lost look in her green eyes. Dad had a _saycheese_ type of a smile. Turning away from the picture, I sat down and looked under the bed.\n\nApart from a few coins, there was nothing else under the bed. There was no place anything could be hidden. Maybe she lifts the bed up and gets under the floorboards that way? I thought. I tried to lift the bed up. It didn't budge. I thought of calling Jake, then I looked a bit closer. The carpet was covered in a layer of dust and the corner next to the leg of the bed had finger marks on it. I touched the carpet there. It moved. It was cut at the point where the leg of the bed sat. I peeled it back. The underlay was also cut. I peeled that back too. One of the floorboards had a long, thin cut in it. At one end, there was a small hole. I put my finger into the hole and lifted the wood. Blood pumped around my ears. A damp, musky smell came out from under the floorboards. There was nothing there. I put my hand inside and rummaged about under the floorboard. I felt the box and took it out.\n\nI took it over to the lamp and looked at it. It was a carved brown wooden box similar to the one we had in the kitchen for the tissues. Everyone in Dad's family had one of these. This was like the jewellery boxes but unlike the tissue boxes, it had a lid on it, with a small brass hook.\n\nI brought the box downstairs. My heart was pounding. Why would she hide it? What was in it? For a moment, I felt guilty. This was her secret, secret world and I was rummaging through it. I felt bad. But then I flushed with anger. How dare she hide it from me! I put the box down on the kitchen table and stared at the flowers carved into it. I hoped it's little bronze lock would just flip off and the lid open by itself. I waited and waited for the lid to open by itself but it didn't. 'Please open now,' I cried. But it would not.\n\nI wished Laila was here and not in Pakistan. I really, really wanted her now.\n\n'What shall I do, Laila?' I asked aloud.\n\nI saw her face flash through my mind. She was smiling.\n\n'Yes, I'll call him, Laila,' I said to the smiling face in my mind and called Jake.\n\nWaiting for Jake, I sat down on a chair in front of the box and didn't take my eyes off it, half wanting to throw it onto the ground and smash it into pieces and half in terror of what might jump out at me from inside it.\n\nI lost track of time and came back out of my thoughts when I heard Jake talking outside my house.\n\nMum was back, I thought, who else could he be talking to?\n\nI opened the door. Jake was alone, recording a message into his mobile, 'I'm outside her house now. Bye Laila.'\n\nBefore I could ask what he was doing messaging Laila he said, 'I was driving around and we were just chatting when you called.'\n\nI was gob-smacked by the 'we'. But what the hell.\n\n'I borrowed me Dad's car,' Jake said dangling a set of keys in front of me. 'Bet you didn't know I could drive,' he said nervously.\n\nOf course I know you can drive, jerk, I thought. I'd seen you nick cars.\n\nFor a moment, I forgot about the box. But only for a moment.\n\nI went into the kitchen, Jake followed me.\n\n'That,' I said pointing to the box. 'Whatever is in that box...' 'It's just a box,' he said.\n\n'Mum kept it hidden under her bed.'\n\n'All Mums and Dads have something hidden under their bed. Mine's got Whiskey...'\n\n'It's serious, this, Jake.'\n\n'Sorry, Kiran, I didn't mean it like that.'\n\nI looked at the box for one last time and quickly opened it. Inside the box was a dummy, a baby grower and a Hi8 camera tape. The dummy was white and the baby grower, a faded blue. I sniffed the baby suit. It smelt of mould. I took the tape out and looked at the word Hi8. I hadn't seen anything like it before. Jake looked at the contents of the box for a while and said, touching the baby grower, 'Its blue. It's a boy's.'\n\nI felt numb inside. I picked up the tape, and asked 'What would play this?'\n\n'Let's get it transferred onto a disk,' Jake said. 'Andy's Videos'll do it.'\n\nNo, I thought. I don't want to know. I don't want to see whatever was on it.\n\n'When's your Mum back?' Jake asked I shrugged my shoulders.\n\n'Come on,' Jake said. I followed him out of the door.\n\nAndy's Videos was at the bottom of St. George's, a few minutes walk from home. I floated dreamlike to the shop. Jake gave the tape in and an eternity later, we walked out with the tape and a CD. Back home, Jake handed me the CD and the tape. I put the CD into the DVD player.\n\nHolding the remote in my hand, I said, 'I can't do this.' 'Up to you, Kiran.'\n\nI pressed play.\n\nThere was Mum showing off her bulging belly, smiling proudly, laughing. She was so beautiful with her golden hair flowing across her face. The screen went blank and there was Mum again. With a baby. Very small. Very pink. Dad was laughing.\n\nAfter a few minutes of the baby in a basket, the screen went dead again.\n\nWhen the picture came back, there was Dad with a can of beer in his hand. Mum was filming. He was standing close to the baby. The baby cried.\n\n'Pick the baby up,' Mum laughed, her voice booming loudly.\n\nDad took a swig. The screen went blank again and came back. The baby was sitting up on the ground. The screen went blank and remained blank.\n\n'She had a baby before me, Jake,' I said, taking the CD out of the player. My stomach knotted. I felt my chest caving in. My ears were burning. So I have a baby brother, I thought. Mum's face flashed through me mind. How dare you keep this from me! How dare you! Where is my brother? What happened to him?\n\nMy throat dried. I felt dizzy. Though I didn't want to admit it to myself, deep down I knew. I slumped into a settee.\n\nNo one said anything for what felt like an eternity. The front door opened and shut. Mum was back.\n\nShe walked into the kitchen. I heard the sound of breaking glass. She came into the living room a few moments later. Box in hand. She picked up the Hi8 off the coffee table, opened the lid of the box, put the tape into it, shut the lid and turned around to leave.\n\n'I know what's on the tape, Mum.'\n\nMum turned round and without saying anything started to walk out of the room.\n\nAs she was leaving I asked, 'Is that it? You've got nothing to say to me? Nothing? You liar! I hope you die and go to hell!'\n\nAfter she went upstairs, I said to Jake, 'Let's go.'\n\n'Where?' he asked. 'Away from here,' I said.\n\nWe went out of the house and walked silently, but quickly. We got into Jake's car. He drove us to the willow tree. He broke a branch off the tree and hit its trunk with it all the while looking at me. Maybe this tree and this graveyard, with its broken outer wall and its decaying walkways is where I belonged. At least the dead didn't lie. Then I randomly began to think about my grandparents on my Mum's side who I don't know. They were buried round here, Henry and Ada, but I never went to their graves. Why didn't you tell me anything more about them than their names, Mum? I asked inside.\n\nWe stood around for a while, and then I said, 'I'm going home. I want to be alone.' Jake didn't say anything to me and walked with me to the gate of the graveyard.\n\n'Are you alright, Kiran?'\n\nI just kept walking. When we got to the gate, I started running. 'Let me drive you back, Kiran,' He shouted after me, 'please.'\n\nI ignored him and kept running, oblivious to the traffic. I was hungry all of a sudden. I searched my pockets, found a five-pound note and went to the chippy to get some food. Standing in the queue, I ignored someone asking me to let them see what I had under my hijab, and got myself some fish and chips and a drink and left the shop. I sat on a bench on the way home and ate, and as I ate, I sent a text to Dad: _I know the secret under the bed._ And shoving chips into my mouth I sent a text to Laila in Pakistan: _My Dad in Pakistan, got another woman_.\n\nI shoved more chips into my mouth, thinking about how I knew nothing about my Dad. How that smiling lump of hairy flab was a cold-hearted monster, going off to Pakistan just like that and marrying another woman. And then I hated Mum, for not telling me anything all my life. What happened to that baby, Mum? I thought, And why didn't you tell me? If you had told me what had happened, what difference would it have made, Mum? Where is that baby now, Mum? Was he told a lie as well? Is that why I was invisible to you sometimes, or was it you didn't want me, or was it because I didn't turn out white?'\n\nI suddenly began to feel really, really sorry for Mum. 'Everything's out now, Mum,' I thought aloud, wrapping up what was left my food, everything was out now. I made up my mind. I was going to go home and give Mum a great big hug, tell her I loved her, and tell her I understood, and give her all my love and make her come back to me. She was still my Mum and she was all I had left.\n\nBy the time I got home, it was dark. As I stepped through the front gate, I got a text from Dad: _Back today._\n\nA moment later, I got one from Laila: _On the plane with your Dad!_ The front door was slightly open. 'I must have forgotten to shut it,' I thought. The kitchen light was on, so I went into the kitchen. There was an empty bottle of vodka on the kitchen table. I didn't care if she was drunk; she was going to answer my questions. I called her as loudly as I could. She didn't answer.\n\nWalking out of the kitchen, I saw a blood stain on the floor. I called her again and followed the bloodstains. I rushed into the living room. She was lying on the sofa, her hand over the side of the arm. A knife on the floor covered in blood.\n\nI screamed. I touched her head. She opened her eyes for a moment. I ran to the telephone and dialled 999. And then I called Jake. I held her bleeding arm up. I pressed on her cut. That's what she'd always said, press on the cut. Stop it bleeding, but it didn't. I held her arm up above her head, like she had told me; hold your nose up if it bleeds, press on it. I pressed on her cut. She opened her eyes again. She turned her head towards the coffee table and closed her eyes again. There was a letter with my name on it.\n\nThe ambulance arrived before Jake. An ambulance woman took Mum's arm out of my hand, saying, 'Good girl.'\n\n'Will she die?' I asked the ambulance woman.\n\nI don't know what the ambulance woman said in reply as just then old George, my neighbour, came into the room. The ambulance woman said something to George. He went out again and came back with a wet cloth from the kitchen.\n\n'It's in God's hands, love,' George said, wiping my hands clean with the towel.\n\nMum was picked up, put on a stretcher and taken out of the house. I grabbed the letter Mum had written to me and followed her out.\n\n'I'll lock up, love,' George said, following me out of the house. His dog sitting in the yard stared at me, wagging its tail as I went past. The Browns looked out of their window and then drew their curtains. I got into the ambulance with Mum. We wailed off under the gaze of the eyes of our street. As we were leaving, I saw Jake running towards our house.\n\n## **Shamshad**\n\n_Ya Allah_ , how could you let us sit there looking through each other for a couple of eternities. That is what we did. Just looked through each other. The wall clock hammered inside my head, in between the words of what Mum had just said. _I wish I was._ That's what she said. She wished she was my mother.\n\n'Shamshad,' Mum said, 'It's not easy to say this.'\n\nAnd you think it is easy to listen to what you have to say to me, I thought. I tried to blot out the ticking of the clock. I searched inside my head for a prayer to take the pain away. But what prayer was I going to find for the pain of my mother telling me she wished she was my mother?\n\nI waited for Mum to tell me, whatever it was, but I really, really didn't want to know and yet still wanted her to quickly tell me, everything. She broke a piece of roti, dipped it in the daal and stirred it round and round and round, making a little whirlpool in her bowl.\n\nAfter a lifetime, she dropped the roti into the daal and nodded up towards Dad's bedroom. She opened her mouth to speak. I put my hands on my ears. The clock ticked louder and her words slashed into me. I could hear them clearly but they made no sense.\n\nI went cold inside. I put my hands under the table, locked my fingers and bit my lower lip until it hurt.\n\n'And you didn't want me,' I thought cracking my fingers, I'm not stupid.\n\nShe put her face in her hands and looked down at the table. Her _dupatta_ slid off her head, went down the parting of her greying hair and fell onto her shoulders.\n\nShe said, looking up, 'He was once a very close friend of Liaqat, Kiran's Dad, Ghanzanfer's son. A long time ago, we went to Pakistan together. Me,' she nodded up towards Dad's bedroom, 'and him and Liaqat and his white wife. She had a baby with her, about a year old. A boy.' I put my hands on the table. Flat, palms down.\n\n## **Kiran**\n\nI was left on my own in the hospital's reception room, whilst Mum was whisked into the emergency room. I texted Dad: _Mum in hospital. Tried to kill herself. Nearly dead. Hope you have a nice life with your new wife._\n\nI sent it.\n\nI texted Laila: _Mum's tried to kill herself. Slashed her wrist. Nearly dead. In hospital..._\n\nI sent it.\n\nI wrote a text to Allah: _Give me back my Mum, Almighty. You have so many souls. You don't need a Mum. I need her_.\n\nI searched for His name in my address book and kept looking for it until a policeman tapped me on the shoulder and asked me some questions about who I was and what had happened. He was still asking me questions when Jake came in. I moved away from the policeman.\n\n'She's going to die,' I cried on Jake's shoulders.\n\nHe held me tightly. I felt his tears falling on my shoulders.\n\nThe ambulance woman who had brought us here came back from the emergency room. 'Will she die?' I asked her.\n\n'Everything's being done that can be done for her, love,' the ambulance woman said, walking past me.\n\n'Did she leave a letter?' The policeman asked.\n\nI pulled it out of my jacket pocket. He put his hand forward to take it from me.\n\n'I haven't read it yet,' I said.\n\nI sat down on a cold plastic chair with Jake next to me and opened Mum's letter. Jake leaned over and read it with me. It said, in her beautifully neat handwriting: _My beloved Kiran. Don't hate me. I love you. I have always loved you and whatever else I have done, I have never stopped loving you. I am not angry with you for opening my box. You had to know some day. But the pain, the pain, I can't keep it down any more Don't ever feel guilty. It's not your fault._\n\n_'Yes, I had a baby before you. I guess you know that anyway. His name was Ajmal. I had two miscarriages before I had him. When I was carrying him, I didn't tell anyone, not even Lucky, until it was obvious._\n\n_'And don't hate Liaqat. He has raised you as a father should. He loves you. Always call him Dad. Always._\n\n_'We went to Pakistan when Ajmal was just one years old. We were in the city near your Dad's village. He wanted to sit on a tanga, a horse carriage. We were going to his uncle's village. I said we should take a taxi. But he wouldn't listen. We sat in the back, with Ajmal in my lap. I don't know why it happened, maybe it was the road, it was all broken and full of holes. I don't know how but one of the wheels of the carriage came off. I was thrown off. Ajmal slipped out of my hands and fell onto the road on his head. He never woke up. He never cried. He just died. Right there. In my hands. The funeral was so fast, too fast. I wanted to bring him back with me but couldn't cope with the idea of him coming home, dead._\n\n_I will never forgive myself for taking him out there and leaving him there. I should not have buried him. He should have buried me. Just the thought of him kills me inside._\n\n_I want him back in my arms. I want to smell his breath and his voice and be next to me when I wake up in the middle of the night._\n\n_Years have gone by but a part of me still can't stop searching for him. I only have little pieces of him under my bed. I am not angry with you for what you did. I'm angry at myself for not knowing how to let go._\n\n_God knows how I've wanted to, so I can be there for you, but I just don't know how._\n\nJake put his trembling hand on mine.\n\nJust then, I got a text from Dad: _Just arrived._\n\nA moment later I got a text from Laila: _Landed. Coming straight to you_.\n\n## **Shamshad**\n\nMum put a hand forward towards me. I clenched my fists as her fingertips touched me.\n\nShe said, 'There was a terrible accident and the _goree_ lost her baby. She sat by the baby's grave and cried and screamed day after day.' She paused, took a deep breath and continued, letting out a deep sigh, 'And there was a woman in the village, Khatija, a poor woman who worked in the house of the Choudry's, the big landlord of the village. He had made her pregnant. She was about to give birth and her offspring would be killed. Murdered. Everyone knew that.' Mum paused.\n\n'Oh God, kill her now,' I thought, 'right in front of my eyes, I don't want to hear the rest.'\n\nI tried to get out of the chair but couldn't.\n\nMum continued, 'The white woman was mad with grief. She wasn't going to live without her baby. On the night Khatija gave birth, all four of us were together, Liaqat and your Dad and me and her. And we stopped the murder.'\n\nMum stopped again. I didn't see her getting up. She was now sitting next to me. She had her arm around me, holding me tightly. 'Khatija gave birth to twins. Two girls...'\n\n'I hate you,' I cried. 'And don't tell me, me mother's your goat herder.' Wiping my tears, Mum said, 'Shush, child, let me tell you everything.\n\nAnd the Choudry, who fathered you, was shot and killed by his own son, Asmat Choudry, for what he had done. And I took you, and called you Shamshad. And your sister is called Kiran.'\n\nI pushed her away and ran upstairs to my bedroom. She followed me. I slammed the door shut and locked it from inside. Rani was sleeping on my bed. She stood up. Her body all erect. She jumped off and went under the bed. Mum knocked on the door, pleading for me to open it. I ignored her. I pulled a pair of scissors out of the drawer and looked at myself in the mirror. My dark skinned face was hideous. I pulled my hijab off and ripped it up. I loosened my hair. Curly black snakes dangled over my shoulders.\n\n'Open the door, Shamshad,' Mum called again and again.\n\nIgnoring her, I cut the snakes and slashed my prayer mat.\n\nI looked at my hand. My veins were throbbing. I put the blade of the scissors on my wrist. The door burst open. They were both there, _my Mum and my Dad._\n\nI clenched my fist and held the scissors in my hand like a dagger.\n\nShe stepped forward, nodding, 'Stab me now. It is your right.'\n\nAnd he stepped forward, fell to the ground and kissed my feet, saying, 'Do as you will to me.' I raised my scissor-knife. My mobile rang. It was Laila. I ignored it. It rang and rang. I ignored it and clutched the scissors tighter. There was a text. I stepped away and read it: _Kiran's Mum in hospital. Dying. Just landed. Will be there. I know everything._\n\nDad stood up.\n\nI put the scissors on the desk, showed him the text from Laila, and said, 'Take me to the hospital right now!'\n\n## **Kiran**\n\nI got another text from Dad: _In taxi._\n\nI read more of Mum's letter. _All I could do was to live and relive the minutes, trying to wind the clock back, to get my Ajmal back, to leave this place and go back home with my baby. Your father's village is a cruel village. In the village, there was a woman, called Khatija, who had been made pregnant by a big, powerful man. Khatija gave birth to twins, two girls. They were going to be killed at birth, but we didn't let that happen. Liaqat and I took you and loved you. We named you Kiran, our light. And the other one, your sister, is called Shamshad, a shade for her family. I don't understand fully why, but in that village a vow was taken, never ever to mention this. I don't know why I went along with it for all those years. I should have stood up to them. Forgive me, my daughter._\n\n'Could I please have the letter, young lady?' the policeman asked me.\n\nI ignored the policeman and read on. _'And some times when I used to cry, it was not just for my baby. I cried for you not knowing, for me not knowing how to tell you, but also for the mother who lost two, from whom we took her babies. I know you must be hurting now and I ask you to forgive me. And tell your father I have always loved him and even as I write my last word, I still love him, but I can't live with all this pain any more. I love you and have always loved you. Mum.'_\n\nI stood up and handed the policeman the letter.\n\nA nurse called my name. Jake grabbed my hand and we went into the emergency corridor. I searched the nurse's face to see if Mum had died. The nurse opened a door and Jake and I walked in.\n\n'Is she dead?' I asked.\n\nThe nurse looked away from me, and said, 'The doctor is coming to see you,' and left.\n\nA few moments later a doctor came in, his lips pursed.\n\n'She has lost a lot of blood,' the doctor said. 'But she should be OK.' 'Should be or will be?' I asked, 'Can I see her, please?'\n\n'She's asleep now. When she wakes up, we'll call you. Please stay in the waiting room.'\n\nI prayed inside my head, 'Oh God save her, or I will never believe in you.'\n\nAt first, I couldn't believe my eyes when I saw her walking in through the main door of the hospital. Her hair was a mess, all chopped up. One side of it above her ears, the other on her shoulder. But it was Shamshad all right. When she saw me, she ran towards me, her arms open. Her Mum and Dad behind her.\n\nWe hugged each other and cried. 'Kiran, I'm so sorry,' Shamshad said. 'So do you know?' I asked.\n\n'I know,' she said. 'It hurts.'\n\n'I know,' I said.\n\n'How's your Mum?' Shamshad laughed. 'They said she'll be fine. She's sleeping now.' ' _Insha'Allah_ , god willing,' Shamshad said. ' _Insha'Allah_ ,' I said.\n\nShamshad stepped back from me a little, wiped my face with her hand, and said, 'I hope so, Kiran, I really hope so.' Jake gave Shamshad and me each a tissue. Shamshad closed her blood-shot eyes and wiped them. I took off my hijab and tried to put it on her head. She backed off, shaking her head. I forced it on her head, asking, 'Who's older, you or me?'\n\n'Me,' Shamshad smiled, adjusting her headscarf, 'probably.'\n\n'No, you're just a big bully,' I laughed.\n\nShamshad's smile disappeared. She said, 'I said, \"I'm sorry.\"' 'Just joking, sis.'\n\nI got a text from Laila: _How's your Mum?_\n\nI showed it to Shamshad, and then replied: _She'll pull through._ Laila replied: _B there in 2 mins._\n\nThen I got another one Laila: _Love your Mum._ Then I got another one from Laila: _Mums!_\n\nI showed the messages to Shamshad and we burst out laughing. 'What's so funny?' Jake asked, leaning towards us, trying to read the texts.\n\nI showed him my mobile. He grinned.\n\nShamshad whispered in my ear, 'What's he doing here?' 'Shall we adopt him as a brother?' I whispered back. Shamshad looked Jake up and down and said, 'Him, a _gora_!' 'You know!' I laughed.\n\n'What's up with your two?' Jake asked.\n\nWe ignored him and giggled.\n\nLaila came in through the big swing doors of the entrance of the hospital. The Pakistan International Airlines tab still on her handbag.\n\nThe three of us hugged.\n\nLooking at Jake standing on his own not far from us, Laila exclaimed, 'Jake!'\n\nShamshad and I looked at each, and then said together, 'Our brother!' Jake blushed. He was about to say something when Laila interrupted, 'And Jake, I heard a story about an English soldier who got captured and converted...'\n\n'Converted!' Jake said, 'Dex? My brother! You must be mad.' 'I didn't say it was Dex,' Laila said.\n\n'It can't be Dex, can it?' Jake asked.\n\nLaila shrugged her shoulders, and said, 'I said, I just heard a story.' Dad came in.\n\nHugging Laila, I saw the other woman with Dad. She was thin and dark.\n\nIs that what you left Mum for? I thought.\n\n## **Shamshad**\n\nA few moments after Laila came, Kiran's Dad walked in with the dark woman I had often seen on Skype. It was our mother. She was wearing a blue _shalwar kameez_ suit and a thick brown jumper. Her cheeks were sunken. She had deep dark patches under her eyes. She had her arms pressed into her side. She stood a few yards away from Kiran and me. Staring at us. She joined her hands and put them up to her face, staring at us, trembling. Kiran's Dad came up to me and put his hand on my head and stepped towards Kiran to hug her.\n\nPushing her Dad away from her, Kiran said, 'So you brought your wife with you to hospital, eh?'\n\n## **Kiran**\n\nI thought Dad would get angry with me for what I had said to him, but he didn't. He looked at the dark woman, then at me and said, 'Kiran, this is your real mother.'\n\nI felt faint. Shamshad put her arm around me and walked me towards mother. Mother stepped back a little. She seemed to shrink, like a scared bird. ' _Ameejee_ ,' Shamshad cried, 'Mother.'\n\nMother kissed her hand and touched it to my face. Then she stepped forwards, stroked her hand down my face and then Shamshad's. She said something in Punjabi.\n\n'She wants to know if we understand her language.' Shamshad said to me.\n\nShamshad said a lot of words back to her. Mother shed heavier tears with each word. Turning to me, Shamshad said, 'I told her you can't.'\n\nSo many eyes were staring at us. A white man, with one arm in plaster and a bandage round his head, came out of the emergency room, and said, 'Does your whole tribe need to turn up if one of youse has got a headache?' Jake turned to him so fast I thought he was going to hit him. But before Jake could say anything, Elizabeth from the East Boarhead Curry Club stepped in between the man and Jake. George and Mrs Bulldog were there as well. Elizabeth wasn't wearing her usual fancy clothes but George as ever was in his usual attire. Pointing a finger straight at the face of the white man with his arm in plaster, Elizabeth said, 'Beat it you twat, before I gob you one.' Nodding to Mrs Bulldog, Elizabeth added quickly, 'And then you'll 'ave Hilda to deal with.'\n\n'Blimey, Liz, never thought you had it in yer to say sommet like this,' Hilda said.\n\n'Leave it, it doesn't matter,' I said. 'It does,' George said.\n\nThe man with the injured arm lowered his head and walked out of the hospital.\n\nHilda went up to some people who were sitting in a corner and said something to them. They moved to other chairs, leaving four seats empty in a corner of the waiting room. Hilda came up to me and gave me a great big hug. She was still in her blue work clothes. I could smell the scent of baking biscuits on her. Shamshad, Mother, Laila and I sat down in the corner. Dad was talking to a nurse. After he finished talking to the nurse, he came up to me and said, 'Love you Kiran. Let me see Sharon on my own for a bit, first.'\n\nI nodded and he left.\n\nLaila took out a cream-coloured scarf from her purse and gave it to me, saying, 'I bought this for you at Islamabad airport.'\n\n'It's beautiful,' I said.\n\nMother looked us up and down, and she said something to Shamshad. 'Is she angry with me?' I asked Shamshad.\n\n'Mother wants to know if you are angry with her?' Shamshad said. I shook my head.\n\nMother's face lit up with a massive smile. She spoke slowly. Her words falling out of her mouth like flowers.\n\nShamshad translated as she spoke, 'There was never a day in which I didn't think of you. There was never a night when I didn't dream of you. There was never a ray of light, in which I didn't see my Kiran, and there was never a tree whose shade didn't remind me of my Shamshad. I knew your names. That's all I knew. And yes, I was angry, but with God, for giving me children they way He did. And then I was grateful to him for keeping you safe. And then I was angry with him for not letting me see you, even your faces, once, before today, not even a photograph. And now, I am the happiest mother on this earth, in front of whom shine the most beautiful jewels of this world.'\n\nDad came back and said to me, 'She's awake, you can see her.'\n\nI looked at Mother. 'Go see her, Kiran, and give her my _salaam_. Tell her I prayed for her all the way through the flight.'\n\nShamshad translated for me.\n\n'How does she know what Dad said?'\n\nShamshad asked mother. Mother smiled and replied to Shamshad. 'She said the look on your face told her that the Almighty had listened to her prayers.'\n\nGetting up to see Mum, I took hold of Dad's hand, and said to Shamshad, 'We're a _right_ family, aren't we!'\n\nThrowing a look at her Mum and Dad, who had stood at the side of the entrance to the hospital all this time, Shamshad replied, 'Aren't we just!'\n\nI went with Dad to see Mum, the East Boarhead Curry Club came along. As we went into a corridor, a nurse stopped us saying, 'Two at a time please.'\n\n'Give it a rest nurse,' Hilda replied to the nurse, 'We need to see our lass.'\n\nThe nurse stepped back and we went down a corridor and into a room. Mum had a tube in her nose. She was plugged into a machine, with all sorts of readings on small screens. Her eyes were shut. When Dad touched her hand, she opened her eyes and said in a weak voice, 'Liaqat.'\n\n'It's Lucky, you daft git, and aren't I just, to get you back.'\n\nMum nodded to her curry club and nodded to me to come close to her. I did. She stroked the back of my hand with a finger and then closed her eyes again.\n\nThe way she closed her eyes filled me with terror. I thought she was dead, but the machines all kept on as they were.\n\nDad must have known what I was thinking. He said, 'Don't worry, she's just tired. They said she'll be home in a couple of days.' We stood looking at Mum for a while. All the time I kept thinking, how lucky I was to have a family now with no secrets hidden under the bed and to have discovered a brother, a sister and, then I smiled at the thought, two Mums.\n\n## **Shamshad**\n\nAfter Kiran and her Dad went to see her Mum, we stood up and walked towards my Mum and Dad. When she got close to Mum, Mother sat on the floor and touched and tried to kiss my Dad's feet.\n\nHe quickly bent down and stopped her.\n\nMum helped Mother to her feet, and said, 'You must never do that again, never.'\n\n'Forgive us?' Dad asked.\n\n'Can you?' Mum asked.\n\nTurning to me, Dad asked, 'Can you forgive us, Shamshad?'\n\nMother flicked her shining eyes upwards, held her hands out and prayed, 'He is _Gafoor-ur-rahim_ , the merciful, the forgiver. He has forgiven, who am I not to?'\n\nMum and Dad looked at me. I smiled back at them, thinking, 'Shamshad girl, you've got a lot of making up to do.' And then I too held out my hands and repeated Mother's words, 'He is _Gafoor-ur-rahim_ , the merciful, the forgiver. Ameen. He has forgiven, who am I not to?'\n\n## About the Author\n\nTariq Mehmood is an award winning writer and film-maker. His books include _Hand On the Sun_ (Penguin), _While There Is Light_ (Comma). He co-directed the award-winning documentary _Injustice_. He currently teaches at the American University of Beirut, in Lebanon and lives in Beirut and Manchester.\n\n## By the Same Author\n\nHand On The Sun\n\nWhile There Is light\n\nCourageous Ali And The Heartless King\n\nHalf Brother\/Pahi Adha\nHopeRoad Publishing Ltd\n\nP O Box 55544\n\nExhibition Road\n\nLondon SW7 2DB\n\nwww.hoperoadpublishing.com\n\nFirst published by HopeRoad 2015\n\nThe right of Tariq Mehmood to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988\n\nCopyright \u00a9 2015 Tariq Mehmood\n\nA CIP catalogue for this book is available from the British Library\n\nAll rights reserved. No part of this book 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 permission of the publisher.\n\nISBN 978-1-908446-30-5\n\neISBN 978-1-908446-37-4\n\nPrinted and bound by\n\nLightning Source\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## Table of Contents\n\nTitle\n\nCopyright\n\nDedication\n\nAcknowledgments\n\nTwenty-One Years After Escape from Chapterhouse\n\nChapter 01\n\nChapter 02\n\nChapter 03\n\nChapter 04\n\nChapter 05\n\nChapter 06\n\nChapter 07\n\nChapter 08\n\nChapter 09\n\nChapter 10\n\nChapter 11\n\nTwenty-Two Years After Escape from Chapterhouse\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\nTwenty-Three Years After Escape from Chapterhouse\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nTwenty-Four Years After Escape from Chapterhouse\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nChapter 51\n\nChapter 52\n\nChapter 53\n\nChapter 54\n\nChapter 55\n\nChapter 56\n\nChapter 57\n\nChapter 58\n\nChapter 59\n\nChapter 60\n\nChapter 61\n\nChapter 62\n\nChapter 63\n\nChapter 64\n\nChapter 65\n\nChapter 66\n\nChapter 67\n\nChapter 68\n\nChapter 69\n\nChapter 70\n\nChapter 71\n\nChapter 72\n\nChapter 73\n\nChapter 74\n\nChapter 75\n\nChapter 76\n\nChapter 77\n\nChapter 78\n\nChapter 79\n\nChapter 80\n\nChapter 81\n\nChapter 82\n\nChapter 83\n\nChapter 84\n\nChapter 85\n\nChapter 86\n\nChapter 87\n\nChapter 88\n\nChapter 89\n\nChapter 90\n\nChapter 91\n\nChapter 92\n\nEpilogue\nSANDWORMS OF  \nDUNE\nTHE DUNE SERIES\n\nBY FRANK HERBERT\n\n_Dune  \nDune Messiah  \nChildren of Dune  \nGod Emperor of Dune  \nHeretics of Dune  \nChapterhouse: Dune_\n\nBY FRANK HERBERT, BRIAN HERBERT,  \nAND KEVIN J. ANDERSON\n\n_The Road to Dune_ (includes the original short novel _Spice Planet_ )\n\nBY BRIAN HERBERT AND KEVIN J. ANDERSON\n\n_Dune: House Atreides  \nDune: House Harkonnen  \nDune: House Corrino_\n\n_Dune: The Butlerian Jihad  \nDune: The Machine Crusade  \nDune: The Battle of Corrin_\n\n_Hunters of Dune  \nSandworms of Dune  \nPaul of Dune_ (forthcoming)\n\nBY BRIAN HERBERT\n\n_Dreamer of Dune_\n\n(biography of Frank Herbert)\n\n# SANDWORMS OF  \nDUNE\n\n_Brian Herbert  \nand  \nKevin J. Anderson_\n\nBased on an outline by Frank Herbert\n\nA TOM DOHERTY ASSOCIATES BOOK  \nNEW YORK\nThis is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the authors' imaginations or are used fictitiously.\n\nSANDWORMS OF DUNE\n\nCopyright \u00a9 2007 by Herbert Properties LLC\n\nAll rights reserved, including the right to reproduce this book, or portions thereof, in any form.\n\nA Tor Book  \nPublished by Tom Doherty Associates, LLC  \n175 Fifth Avenue  \nNew York, NY 10010\n\nwww.tor.com\n\nwww.dunenovels.com\n\nTor\u00ae is a registered trademark of Tom Doherty Associates, LLC.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nHerbert, Brian.\n\nSandworms of Dune \/ Brian Herbert and Kevin J. Anderson.\u20141st ed.  \np. cm.\n\n\"A Tom Doherty Associates Book.\"\n\nISBN-13: 978-0-7653-1293-8\n\nISBN-10: 0-7653-1293-X\n\n1. Dune (Imaginary place)\u2014Fiction. 2. Life on other planets\u2014Fiction. 3. Robots\u2014Fiction. I. Anderson, Kevin J., 1962\u2013 II. Title.\n\nPS3558.E617S26 2007\n\n813'.54\u2014dc22\n\n2006102742\n\nFirst Edition: August 2007\n\nPrinted in the United States of America\n\n0 9 8 7 6 5 4 3 2 1\n\n_We can never overstate the appreciation we owe to the genius  \nwho created this incredible series. Once again, this book is for  \nFrank Herbert, a man of wondrous and important ideas, who has  \nbeen our mentor as we continue to write new stories in his  \nfantastic Dune universe_. Sandworms of Dune _is the chronological  \ngrand finale that he envisioned, and we are pleased to finally  \nbring it to his millions of loyal fans_.\n\n## ACKNOWLEDGMENTS\n\nAs with all of our previous Dune novels, we have depended on the efforts of a great many people to make the manuscript as good as possible. We would like to thank Pat LoBrutto, Tom Doherty, and Paul Stevens at Tor Books; Carolyn Caughey at Hodder & Stoughton; Catherine Sidor, Louis Moesta, and Diane Jones at WordFire, Inc; Penny Merritt, Kim Herbert, and Byron Merritt at Herbert Properties LLC; and Mike Anderson at the dunenovels.com Web site, as well as Dr. Attila Torkos, who worked on fact-checking and consistency.\n\nIn addition, we have had many supporters of the new Dune novels, including John Silbersack, Robert Gottlieb, and Claire Roberts at Trident Media Group; Richard Rubinstein, Mike Messina, John Harrison, and Emily Austin-Bruns at New Amsterdam Entertainment; Ron Merritt, David Merritt, Julie Herbert, Robert Merritt, Margaux Herbert, and Theresa Shackelford at Herbert Properties LLC.\n\nAnd as always, these books would not exist without the unending help and support from our wives, Janet Herbert and Rebecca Moesta Anderson.\n> _Soon after the Honored Matres careened into the Old Empire, the Bene Gesserit Sisterhood learned to hate and fear them. The intruders used their terrible Obliterator weapons to destroy Bene Gesserit and Tleilaxu planets, Richese with its vast industries and weapon shops, even Rakis itself_.\n> \n> _But in order to survive the even greater Enemy that pursued them, the Honored Matres desperately needed knowledge that only the Sisterhood possessed. To obtain it, they struck like angry vipers, lashing out with extreme violence_.\n> \n> _After the Battle of Junction, the two opposing groups were forcibly united into a New Sisterhood, but the factions continued to wrestle for control and dominance. Such a waste of time, talent, and blood! The real threat came from outside, but we continued to fight the wrong enemy_.\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> address to the New Sisterhood\n> \n> _Two people drift in a lifeboat on an uncharted sea. One says, \"There! I see an island. Our best chance is to go ashore, build a shelter, and await rescue.\" The other says, \"No, we must go farther out to sea and hope to find the shipping lanes_. That _is our best chance.\" Unable to agree, the two fight, the lifeboat capsizes, and they drown_.\n> \n> _This is the nature of humanity. Even if only two people are left in the entire universe, they will come to represent opposing factions_.\n> \n> \u2014The Bene Gesserit Acolytes' Handbook\n> \n> _In re-creating particular gholas, we reweave the fabric of history. Once more, Paul Muad'Dib walks among us, with his beloved Chani, his mother the Lady Jessica, and his son Leto II, the God Emperor of Dune. The presence of Suk doctor Wellington Yueh, whose treachery brought a great house to its knees, is at once disturbing and comforting. Also with us are the warrior-MentatThufir Hawat, the Fremen Naib Stilgar, and the great planetologist Liet-Kynes. Consider the possibilities!_\n> \n> _Such genius constitutes a formidable army. We will need that brilliance, because we face an opponent more terrible than we ever imagined_.\n> \n> \u2014DUNCAN IDAHO,  \n>  _Memories of More Than a Mentat_\n> \n> _I have waited and planned and built my strength for fifteen thousand years. I have evolved. It is time._\n> \n> \u2014OMNIUS\nSANDWORMS OF  \nDUNE\n\n## TWENTY-ONE YEARS AFTER\n\nESCAPE FROM CHAPTERHOUSE\n\n> _So many people I knew in the past are not yet reborn. I still miss them, even though I do not remember them. The axlotl tanks will soon remedy that_.\n> \n> \u2014LADY JESSICA,  \n> the ghola\n\nAboard the wandering no-ship _Ithaca_ , Jessica witnessed the birth of her daughter, but only as an observer. Just fourteen years old, she and many others crowded the medical center, while two Bene Gesserit Suk doctors in the adjacent creche prepared to extract the tiny girl child from an axlotl tank.\n\n\"Alia,\" one of the female doctors murmured.\n\nThis was not truly Jessica's daughter, but a ghola grown from preserved cells. None of the young gholas on the no-ship were \"themselves\" yet. They had regained none of their memories, none of their pasts.\n\nSomething tried to surface at the back of her mind, and though she worried at it like a loose tooth, Jessica could not remember the first time Alia had been born. In the archives, she had read and reread the legendary accounts generated by Muad'Dib's biographers. But she couldn't _remember_.\n\nAll she had were images from her studies: _A dry and dusty sietch on Arrakis, surrounded by Fremen. Jessica and her son Paul had been on the run, taken in by the desert tribe. Duke Leto was dead, murdered by Harkonnens.Pregnant, Jessica had drunk the Water of Life, forever changing the fetus inside her_. From the moment of her birth, the original Alia had been different from all other babies, filled with ancient wisdom and madness, able to tap into Other Memory without having gone through the Spice Agony. Abomination!\n\n_That had been another Alia. Another time and another way._\n\nNow Jessica stood beside her ghola \"son\" Paul, who was chronologically a year older than she. Paul waited with his beloved Fremen mate Chani and the nine-year-old ghola of a boy who had in turn been _their_ son, Leto II. In a prior shuffle of lives, this had been Jessica's family.\n\nThe Bene Gesserit order had resurrected these figures from history to help fight against the terrible Outside Enemy that hunted them. They had Thufir Hawat, the planetologist Liet-Kynes, the Fremen leader Stilgar, and even the notorious Dr. Yueh. Now, after almost a decade of hiatus in the ghola program, Alia had joined the group. Others would come soon; the three remaining axlotl tanks were already pregnant with new children: Gurney Halleck, Serena Butler, Xavier Harkonnen.\n\nDuncan Idaho gave Jessica a quizzical look. Eternal Duncan, with all of his memories restored from all of his prior lives . . . She wondered what he thought of this new ghola baby, a bubble of the past rising up to the present. Long ago, the first ghola of Duncan had been Alia's consort. . . .\n\nCONCEALING HIS AGE well, Duncan was a full-grown man with dark wiry hair. He looked exactly like the hero shown in so many archival records, from the time of Muad'Dib, through the God Emperor's thirty-five-century reign, to now, another fifteen centuries later.\n\nBreathless and late, the old Rabbi bustled into the birthing chamber accompanied by twelve-year-old Wellington Yueh. Young Yueh's forehead did not bear the diamond tattoo of the famous Suk School. The bearded Rabbi seemed to think he could save the gangly young man from repeating the terrible crimes he had committed in his prior life.\n\nAt the moment the Rabbi looked angry, as he invariably did whenever he came near the axlotl tanks. Since the Bene Gesserit doctors ignored him, the old man vented his displeasure on Sheeana. \"After years of sanity, you have done it again! When will you learn to stop taunting God?\"\n\nAfter receiving an ominous prescient dream, Sheeana had declared a temporary moratorium on the ghola project that had been her passion from its inception. But their recent ordeal on the planet of the Handlers and their near capture by the Enemy hunters had forced Sheeana to reassess that decision. The wealth of historical and tactical experience the reawakened gholas could offer might be the greatest weapon the no-ship possessed. Sheeana had decided to take the risk.\n\n_Perhaps we will be saved by Alia one day_ , Jessica thought. _Or by one of the other gholas . . ._\n\nTempting fate, Sheeana had performed an experiment on this unborn ghola in an effort to make it more like _the_ Alia. Estimating the point in the pregnancy when the original Jessica had consumed the Water of Life, Sheeana had instructed Bene Gesserit Suk doctors to flood the axlotl tank with a near-fatal spice overdose. Saturating the fetus. _Trying_ to re-create an Abomination.\n\nJessica had been horrified to learn of it\u2014too late, when she could do nothing about it. How would the spice affect that innocent baby? A melange overdose was different from undergoing the Agony.\n\nOne of the Suk doctors told the Rabbi to stay out of the birthing creche. Scowling, the old man held up a trembling hand, as if making a blessing on the pale flesh of the axlotl tank. \"You witches think these tanks are no longer women, no longer human\u2014but this is still Rebecca. She remains a child of my flock.\"\n\n\"Rebecca fulfilled a vital need.\" Sheeana said. \"All of the volunteers knew exactly what they were doing. She accepted her responsibility. Why can't you?\"\n\nThe Rabbi turned in exasperation toward the young man at his side. \"Speak to them, Yueh. Maybe they will listen to you.\"\n\nJessica thought the sallow young ghola seemed more intrigued than incensed about the tanks. \"As a Suk doctor,\" he said, \"I delivered many children. But never like this. At least I don't think so. With my ghola memories still locked away, I get confused sometimes.\"\n\n\"And Rebecca is human\u2014not just some biological machine to produce melange and a brood of gholas. You have to see that.\" The Rabbi's voice grew in volume.\n\nYueh shrugged. \"Because I was born in the same fashion, I cannot be entirely objective. If my memories were restored, maybe I'd agree with you.\"\n\n\"You don't need original memories to think! You can _think_ , can't you?\"\n\n\"The baby is ready,\" one of the doctors interrupted. \"We must decant it now.\" She turned impatiently to the Rabbi. \"Let us do our work\u2014or the tank could be harmed as well.\"\n\nWith a sound of disgust, the Rabbi shouldered his way from the birthing creche. Yueh remained behind, continuing to watch.\n\nOne of the Suk women tied off the umbilical cord from the fleshy tank. Her shorter colleague cut the purplish-red whip; then she wiped off the slick infant and lifted little Alia into the air. The child let out a loud and immediate cry, as if she had been impatient to be born. Jessica sighed in relief at the healthy sound, which told her the girl was not an Abomination this time. The original newborn Alia had purportedly looked upon the world with the eyes and intelligence of a full adult. This baby's crying sounded normal. But it stopped abruptly.\n\nWhile one doctor tended the now-sagging axlotl tank, the other dried the infant and wrapped her in a blanket. Unable to help feeling a tug at her heart, Jessica wanted to reach out and hold the baby, but resisted the urge. Would Alia suddenly start speaking, uttering voices from Other Memory? Instead, the baby looked around the medical center, without seeming to focus.\n\nOthers would care for Alia, not unlike the way Bene Gesserit sisters took baby girls under their collective wing. The first Jessica, born under the close scrutiny of breeding mistresses, had never known a mother in the traditional sense. Nor would this Jessica, nor Alia, nor any of the other experimental ghola babies. The new daughter would be raised communally in an improvised society, more an object of scientific curiosity than love.\n\n\"What an odd family we all are,\" Jessica whispered.\n\n> _Humans are never capable of complete accuracy. Despite all the knowledge and experiences we have absorbed from countless Face Dancer \"ambassadors,\" we are left with a confused picture. Nonetheless, the flawed accounts of human history provide amusing insights into the delusions of mankind_.\n> \n> \u2014ERASMUS,  \n> Records and Analyses, Backup #242\n\nIn spite of a decades-long effort, the thinking machines had not yet captured the no-ship and its precious cargo. That did not, however, stop the computer evermind from launching his vast extermination fleet against the rest of humanity.\n\nDuncan Idaho continued to elude Omnius and Erasmus, who repeatedly cast their sparkling tachyon net into the nothingness, searching for their quarry. The no-ship's veiling capability normally prevented it from being seen, but from time to time the pursuers caught glimpses, as of something concealed behind shrubbery. At first the hunt had been a challenge, but now the evermind was growing frustrated.\n\n\"You have lost the ship again,\" Omnius boomed through wall speakers in the central, cathedral-like chamber in the technological metropolis of Synchrony.\n\n\"Inaccurate. I must first find it before I can lose it.\" Erasmus tried to sound carefree as he shifted his flowmetal skin, reverting from his guise as a kindly old woman to the more familiar appearance of a platinum-surfaced robot.\n\nLike overarching tree trunks, metal spires towered above Erasmus to form a vaulted dome within the machine cathedral. Photons glittered from the activated skins of the pillars, bathing his new laboratory in light. He had even installed a glowing fountain that bubbled with lava\u2014a useless decoration, but the robot often indulged his carefully cultivated artistic sensibilities. \"Do not be impatient. Remember the mathematical projections. Everything is nicely predetermined.\"\n\n\"Your mathematical projections could be myths, like any prophecy. How do I know they are correct?\"\n\n\"Because I have said they are correct.\"\n\nWith the launch of the machine fleet, the long-foretold Kralizec had begun, at last. Kralizec . . . Armageddon . . . the Battle at the End of the Universe . . . Ragnarok . . . Azrafel . . . the End Times . . . the Cloud Darkness. It was a time of fundamental change, of the entire universe shifting on its cosmic axis. Human legends had predicted such a cataclysmic event since the dawn of civilization. Indeed, they had already been through several iterations of similar cataclysms: the Butlerian Jihad itself, the jihad of Paul Muad'Dib, the reign of the Tyrant Leto II. By manipulating computer projections, and thus creating expectations in the mind of Omnius, Erasmus had succeeded in initiating the events that would bring about another fundamental shift. Prophecy and reality\u2014the order of things really didn't matter.\n\nLike an arrow, all of Erasmus's infinitely complex calculations, running trillions of data points through the most sophisticated routines, pointed to one result: The final Kwisatz Haderach\u2014whoever that was\u2014would determine the course of events at the end of Kralizec. The projection also revealed that the Kwisatz Haderach was on the no-ship, so Omnius naturally wanted such a force fighting on his side. Ergo, the thinking machines needed to capture that ship. The first to exert control over the final Kwisatz Haderach would win.\n\nErasmus didn't fully understand exactly what the superhuman might do when he was located and seized. Though the robot was a longtime student of mankind, he was still a thinking machine, while the Kwisatz Haderach was not. The new Face Dancers, who had long infiltrated humanity and brought vital information back to the Synchronized Empire, fell somewhere in between, like hybrid biological machines. He and Omnius had both absorbed so many of the lives stolen by the Face Dancers that sometimes they forgot who they were. The original Tleilaxu Masters had not foreseen the significance of what they had helped create.\n\nThe independent robot knew he still had to keep Omnius under control, though. \"We have time. You have a galaxy to conquer before we need the Kwisatz Haderach aboard that ship.\"\n\n\"I am glad I did not wait for you to succeed.\"\n\nFor centuries Omnius had been building his invincible force. Using traditional but supremely efficient lightspeed engines, the millions and millions of machine vessels now swept forward and spread out, conquering one star system at a time. The evermind could have made use of the surrogate mathematical navigation systems, which his Face Dancers had \"given\" to the Spacing Guild, but one element of the Holtzman technology simply remained too incomprehensible. Something indefinably human was required to travel through foldspace, an intangible \"leap of faith.\" The evermind would never admit that the bizarre technology actually made him . . . nervous.\n\nFollowing a flurry of test skirmishes, the wall of robotic battleships had encountered and swiftly destroyed fringe outpost worlds settled by humans. Vanguard drones mapped out the planets ahead and distributed deadly biological plagues that Erasmus had developed; by the time the actual machine fleet arrived at a target world, military action was often unnecessary against a dying population. Each combat engagement, even clashes with isolated groups of Honored Matres, was equally decisive.\n\nTo keep himself occupied, the independent robot reviewed the streams of data sent back to him. This was the part he enjoyed best. A buzzing watcheye flitted in front of him, and he brushed it away. \"If you allow me to concentrate, Omnius, I may find some way to speed up our progress against the humans.\"\n\n\"How do I know you will not make another mistake?\"\n\n\"Because you have confidence in my abilities.\"\n\nThe watcheye flitted away.\n\nWhile the machine fleet crushed one human planet after another, Erasmus issued additional instructions for the invader robots. As the infected humans lay writhing, vomiting, and bleeding from their pores, machine scouts casually ransacked databases, halls of records, libraries, and other sources. This was different from the information to be winnowed from the random lives that Face Dancers had assimilated.\n\nWith all the fresh data flowing in, Erasmus had the luxury of becoming a scientist again, as he had been long ago. The pursuit of scientific truth had always been his true reason for existence. Now the flood was greater than ever before. Glad to possess so much new information, so much undigested data, he gorged his elaborate mind on raw facts and histories.\n\nAfter the supposed destruction of thinking machines more than fifteen millennia earlier, the fecund humans had spread, building civilizations and destroying them. Erasmus was intrigued by how, after the Battle of Corrin, the Butler family had founded an empire and ruled it under the name of Corrino for ten thousand years, with a few gaps and interregnums, only to be overthrown by a fanatical leader named Muad'Dib.\n\n_Paul Atreides_. The first Kwisatz Haderach.\n\nA more fundamental change, however, had come from his son Leto II, called the God Emperor or Tyrant. Another Kwisatz Haderach\u2014a unique hybrid of man and sandworm that had imposed a draconian rule for thirty-five hundred years. After his assassination, human civilization fragmented. Fleeing to the far reaches of the galaxy in the Scattering, people became hardened by their privations until the worst sort of humans\u2014Honored Matres\u2014had blundered into the burgeoning machine empire. . . .\n\nAnother flitting watcheye scanned the same records Erasmus was reading. Omnius spoke through resonating plates in the walls. \"I find their contradictions\u2014posed as fact\u2014to be unsettling.\"\n\n\"Unsettling perhaps, but fascinating.\" Erasmus disengaged himself from the stacks of historical files. \"Their histories show how they view themselves and the universe around them. Obviously, these humans need someone to take firm control again.\"\n\n> _Why is religion important? Because logic alone does not compel a person to make great sacrifices. Given sufficient religious fervor, however, people will throw themselves against impossible odds and consider themselves blessed for doing so_.\n> \n> \u2014MISSIONARIA PROTECTIVA,  \n> First Primer\n\nTwo male workers appeared at the door of Murbella's coldly ostentatious council chambers during a tense meeting. Using suspensor clamps, they hauled a large, motionless robot between them. \"Mother Commander? You asked for this to be delivered here.\"\n\nThe combat machine was built from blue and black metal, reinforced with struts and overlapping armor. Its conical head contained a suite of sensors and targeting arrays, and four engine-driven arms were wrapped with cables and augmented with weapons. Damaged during a recent skirmish, the fighting robot had dark smears across its bulky torso where high-energy blasts had knocked out its internal processors. The robotic thing was shut down, dead, defeated. But even deactivated, it was cause for nightmares.\n\nMurbella's advisors, startled out of their discussions and arguments, stared at the big machine. All of the gathered women wore the plain black unitard of the New Sisterhood, following a code of homogenized dress that allowed no indication of their origins as either Bene Gesserit or Honored Matre.\n\nMurbella gestured to the intimidated-looking workers. \"Bring that thing inside where we can see it every time we talk about the Enemy. It will do us good to be reminded of the adversary we're up against.\"\n\nEven with the suspensor clamps, the men sweated as they wrestled the machine into the room. Murbella strode to the bulky combat robot and stared with defiance up into its dull optic sensors. She glanced proudly at her daughter. \"Bashar Idaho brought this specimen back from the battle at Duvalle.\"\n\n\"It should be sent to the scrap heap. Or shot into space,\" said Kiria, a hard-edged former Honored Matre. \"What if it still has passive spy programming?\"\n\n\"It's been thoroughly purged,\" said Janess Idaho. As the newly appointed commandant of the Sisterhood's military forces, she had become a very pragmatic young woman.\n\n\"A trophy, Mother Commander?\" asked Laera, a dark-skinned Reverend Mother who often quietly supported Murbella. \"Or a prisoner of war?\"\n\n\"This is the only one our armies found intact. We blew up four machine ships before we retreated and let them destroy the planet behind us. They had already turned their plagues loose on Ronto and Pital, leaving no survivors. Total population losses number in the billions.\"\n\nDuvalle, Ronto, and Pital were just the latest casualties as the machine army continued its forward march through the outlying systems. Because of the distances involved and the sheer might of the attacking ships, reports were sketchy and often outdated. Refugees and couriers surged away from battle zones, heading inward from the fringes of the Scattering.\n\nMurbella turned her back on the deactivated robot and faced the Sisters. \"Knowing that a tempest approaches, we have the option of simply evacuating\u2014abandoning everything we have. That is the Honored Matre way.\"\n\nSome of the Sisters flinched at the comment. Long ago, Honored Matres had chosen to run from the Enemy, pillaging on their way out, hoping to stay one step ahead of the storm. To them, the Old Empire had been no more than a crude barricade to be thrown up against the Enemy; they had simply hoped it would last long enough for them to get away.\n\n\" _Or_ , we can board up our windows, strengthen our walls, and ride it out. And hope we survive.\"\n\n\"This is no mere storm, Mother Commander,\" said Laera. \"The repercussions are already being felt. Refugees fleeing the battlefront are overwhelming the support systems of second-wave worlds, all of which are preparing for evacuation as well. The people won't stand and fight.\"\n\n\"Like waterlogged rats crowding to the corner of a sinking raft,\" Kiria muttered.\n\n\"Says the Honored Matre, who did exactly the same,\" Janess said from the end of the table, then tried to cover her comment by loudly sipping her spice coffee. Kiria glared at her.\n\n\"A shadow deep in our Honored Matre past,\" Murbella said. \"Through hubris, and a violent predisposition to strike first and understand later, the whores caused all these problems.\" By digging deep into her mind and history, she had been the first to remember how her long-dead sisters had stupidly provoked the thinking machines.\n\nKiria was indignant, clearly still associating herself with the Honored Matres. Murbella found it disturbing. \"You yourself revealed why the Honored Matres are what they are, Mother Commander. Descended from tortured Tleilaxu females, rogue Reverend Mothers, and a few Fish Speakers. They had every right to be vengeful.\"\n\n\"They had no right to be stupid!\" Murbella snapped. \"A painful past did not give them the right to lash out against anything they encountered. They couldn't salve their conscience by pretending they knew what they were doing when they attacked a machine outpost and stole weapons they didn't understand.\" She smiled slightly. \"If anything, I can relate to\u2014though not approve of\u2014their revenge against the Tleilaxu worlds. In Other Memory I know what the Tleilaxu did to my ancestors . . . I _remember_ being one of their vile axlotl tanks. But make no mistake, that kind of provocative and poorly planned violence has caused immeasurable trouble for the human race. And now look what we face!\"\n\n\"How can we strengthen ourselves against this storm, Mother Commander?\" The question came from ancient Accadia, a Reverend Mother who lived in the Chapterhouse Archives. Accadia hardly ever slept and rarely allowed sunlight to touch her parchment skin. \"What defenses do we have?\" The hulking combat robot seemed to mock them from the corner of the room, where the men had left it.\n\n\"We have the weapon of religion. Especially _Sheeana_.\"\n\n\"Sheeana is of no use to us!\" Janess said. \"Her followers believe she died on Rakis decades ago.\"\n\nThe priests on Rakis had once made much of the girl who could command sandworms. The Bene Gesserit had created a grassroots religion around Sheeana, and the annihilation of Dune had only served the Sisterhood's greater purpose. After her supposed death, the rescued girl was isolated on Chapterhouse, so that one day she might \"return from the grave\" to great fanfare. But the real Sheeana had escaped with Duncan on the no-ship more than twenty years ago.\n\n\"It's not necessary for us to have _her_ , specifically. Simply find Sisters who resemble her and apply any necessary makeup and facial modifications.\" Murbella tapped fingers against her lips. \"Yes, we shall begin with twelve new Sheeanas. Disperse them to the refugee worlds, since the displaced survivors will be our most impressionable recruits. The resurrected Sheeana will seem to appear everywhere at once\u2014a messiah, a visionary, a leader.\"\n\nLaera spoke in an eminently reasonable voice. \"Genetic tests will prove that these impostors are not Sheeana. Your plan will backfire once people see we have tried to trick them.\"\n\nKiria had already thought of the obvious solution. \"We can have Bene Gesserit doctors\u2014Suk doctors\u2014perform the tests . . . and lie for us.\"\n\n\"Also, don't underestimate the greatest advantage we have.\" Murbella held out her hand like a mendicant asking for alms. \"The people _want_ to believe. For thousands of years, our Missionaria Protectiva wove religious beliefs among populations. Now we must use those techniques not just for our own protection, but as a functional weapon, a means of influencing armies. No longer passive and protective, but an active force. A Missionaria _Aggressiva_.\"\n\nThe other women, especially Kiria, seemed to like the idea. Accadia scowled down at her Ridulian crystal sheets, as if trying to find profound answers written in the dense characters.\n\nMurbella flashed a defiant look at the combat robot. \"The twelve Sheeanas will carry spice from our stockpiles. Each will distribute extravagant amounts of melange as she makes her pronouncements. She will say that Shaitan told her in a dream that spice would flow again soon. Though Rakis was burned as lifeless as Sodom and Gomorrah, many new Dunes will appear elsewhere. Sheeana will promise them this.\" Years ago, groups of Reverend Mothers had been sent out on a secret Scattering, taking ships and all-important sandtrout to seed additional planets and create more desert worlds for the sandworms.\n\n\"False prophets and sightings of the messiah. It's been done before.\" Kiria sounded bored. \"Explain how this will benefit us.\"\n\nMurbella shot her a calculated smile. \"We take advantage of the superstitions that will run rampant. People believe they must endure a time of tribulation, a cycle as old as the most ancient religions, long before the First Great Movement or the Zensunni Hajj. So, we tailor that belief to our own uses. The thinking machines are the great evil we have to defeat before humanity can reap its reward.\"\n\nTurning to the aged mistress of the Archives, she said, \"Accadia, read everything you can find about the Butlerian Jihad and how Serena Butler led her forces. The same for Paul Muad'Dib. We could even say that the Tyrant began to prepare us for this. Study his writings and take sections out of context to support our message, so the people will be convinced that this final universal conflict has been foretold all along: _Kralizec_. If they believe in the prophecies, they'll continue to fight long after any rational hope should be dashed.\"\n\nShe motioned for the women to go about their tasks. \"In the meantime, I have set up a meeting with the Ixians and the Guild. Since Richese is destroyed, I'll demand that they devote their manufacturing capabilities to our war effort. We need every scrap of resistance the human race can muster.\"\n\nAs she was leaving, Accadia asked, \"And what if those old prophecies prove to be correct? What if these truly are the End Times?\"\n\n\"Then our efforts are all the more justified. And we still fight. It's all we can do.\" Facing the robot, Murbella spoke to it as if the machine mind could still hear her. \"And that's how we will defeat you.\"\n\n> _I am the keeper of private knowledge and uncounted secrets. You will never know what I know! I would pity you, if you were not an infidel_.\n> \n> \u2014 _Mirage in the Shariat Road_ ,  \n> an apocryphal Tleilaxu writing\n\nIn the enormous Guild Heighliner, no passengers ever guessed what the Navigator and his captive Tleilaxu Master were doing right under their noses.\n\nBy holding melange supplies for ransom, the Bene Gesserit witches had backed the Spacing Guild into a corner and forced them to choose drastic alternatives. Facing extinction from spice starvation, the Navigator faction urged Waff to greater speed to complete his task. The Tleilaxu Master felt the need for haste as well, since he was facing extinction himself, though for different reasons.\n\nTurning his back on the observation lens, Waff surreptitiously consumed another dose of melange. The cinnamony powder had been provided strictly for scientific purposes. He touched the burning substance to his lips and tongue, closed his eyes in ecstasy. Such a small quantity\u2014only a taste\u2014was enough to buy a house on a colony world these days! The Tleilaxu man felt energy rush back into his ailing body. Edrik would not begrudge him this bit of melange to help him think straight.\n\nNormally, Tleilaxu Masters lived from body to body in a chain of ghola immortality. They had learned patience and long-term planning from the Great Belief. Had not God's Messenger himself lived for three and a half millennia? But forbidden techniques had accelerated this Waff's growth in the axlotl tank. The cells in his body burned through his existence like flames through a forest, sweeping him from infancy to childhood to maturity, in only a few years. Waff's memory restoration had been imperfect, bringing back only fragments of his past life and knowledge.\n\nEscaping the Honored Matres, Waff had been forced to take refuge with the Navigator faction. Since Edrik and his fellows had financed his ghola resurrection in the first place, why not beg them for sanctuary? Though the little man did not remember how to create melange with axlotl tanks, he claimed he could do the impossible\u2014bring back the supposedly extinct sandworms. A much more spectacular and necessary solution.\n\nIn the isolated Heighliner laboratory, Edrik had provided all of the research tools, technical equipment, and genetic raw material he could possibly need. Waff did as the Navigators demanded. Bringing back the magnificent worms that had been exterminated on Rakis offered the simultaneous possibilities of manufacturing spice, and of restoring his Prophet.\n\n_I must do this! Failure is not an option_.\n\nWith his accelerated maturity, Waff would be at his peak\u2014the best health, the sharpest mind\u2014for only a short while longer. Before he began the inevitable rapid degeneration, he had much to accomplish. The tremendous responsibility prodded at him.\n\n_Focus, focus!_\n\nHe climbed onto a stool and peered into a plaz-walled containment tank full of sand from Rakis itself. _Dune_. Because of the planet's religious significance, pilgrims who could not afford the interplanetary passage contented themselves with relics, fragments of stone chipped from the ruins of Muad'Dib's original palace or scraps of spice cloth embroidered with the sayings of Leto II. Even the poorest of devout followers wanted a sample of Rakian sand, so that they could dust their fingertips and imagine themselves closer to the Divided God. The Navigators had acquired hundreds of cubic meters of authentic Rakian sand. Though it was doubtful that the origin of the grains would have any effect on the sandworm tests, Waff preferred to remove any stray variables.\n\nHe leaned over the open tank, filled his mouth with saliva, and let a long droplet splatter onto the soft sand. Like piranhas in an aquarium, shapes stirred beneath the surface, swirling to seize the invading moisture. In another place long ago, spitting\u2014sharing one's personal water\u2014had been a sign of respect among the Fremen. Waff used it to bring sandtrout to the surface.\n\nLittle makers. Sandtrout specimens, far more precious even than the sand of Dune.\n\nYears ago, the Guild had intercepted a secret Bene Gesserit ship carrying sandtrout in its hold. When the witches aboard refused to explain their mission, they were all killed, their sandtrout seized, and Chapterhouse had been none the wiser.\n\nLearning that the Guild possessed some of the immature sandworm vectors, Waff demanded them for his work. Though he could not remember how to create melange in an axlotl tank, this experiment had much greater potential. By resurrecting the worms, he could not only bring back spice, but the Prophet himself!\n\nUnafraid of the sandtrout, he reached into the aquarium with a small hand. Grabbing one of the leathery creatures by its fringe, he pulled it flopping out of the sand. Sensing the moisture in Waff's perspiration, the sandtrout wrapped itself around his fingers, palm, and knuckles. He poked and prodded the soft surface, reshaping the edges.\n\n\"Little sandtrout, what secrets do you have for me?\" He formed a fist, and the creature flowed around it to form a jellylike glove. He could feel his skin drying out.\n\nCarrying the sandtrout, Waff went to a clean research table and set out a wide, deep pan. He tried to unwrap the sandtrout from his knuckles, but each time he moved the membrane it flowed back onto his skin. Feeling the desiccation in his hand now, he poured a beaker of fresh water into the bottom of the pan. The sandtrout, attracted by a larger supply, quickly plopped into it.\n\nWater was deadly poison to sandworms, but not to the younger sandtrout, the larval stage of the worms. The early vector had a fundamentally different biochemistry before it underwent the metamorphosis into its mature form. A paradox. How could one stage in the life cycle be so ravenously drawn to water, while a later phase was killed by it?\n\nFlexing his fingers to recover from the unnatural dryness, Waff was fascinated as the specimen engulfed the water. The larva instinctively hoarded moisture to create a perfectly dry environment for the adult. From previous-life memories that did remain within him, he knew of ancient Tleilaxu experiments to move and control worms. Standard attempts to transplant full-grown worms onto dry planets always failed. Even the most extreme offworld landscapes still had too much moisture to support such a fragile\u2014fragile?\u2014life-form as the sandworms.\n\nNow, though, he had a different idea. Instead of changing the _worlds_ to accommodate sandworms, perhaps he could alter the worms themselves in their immature stage, help them adapt. The Tleilaxu understood the Language of God, and their genius for genetics had achieved the impossible many times. Had not Leto II been God's Prophet? It was Waff's duty to bring Him back.\n\nThe concept and chromosomal mechanics seemed simple. At some point in the sandtrout's development, a trigger changed the creature's chemical response toward a substance as simple as water. If he could only find that trigger and block it, the sandtrout should continue to mature, but without such a deathly aversion to liquid water. Now that would be a true miracle!\n\nBut if one prevented a caterpillar from spinning a cocoon, would it still transform into a great moth? He would have to be very careful, indeed.\n\nIf he understood what the witches had done on Chapterhouse, they had discovered a way to release sandtrout into a planetary environment\u2014the Bene Gesserit homeworld. Once there, the sandtrout reproduced and began an unstoppable process to destroy (remake?) the whole ecosystem. From a lush planet to an arid wasteland. They would eventually turn the world into a total desert, where the sandworms could survive and be reborn.\n\nMore questions continued to follow, one after another. Why were the fleeing Bene Gesserit Sisters carrying sandtrout samples aboard their refugee ships? Were they trying to distribute them to other worlds, and thereby create more desert planets? Habitats for more worms? Such a plan would require a huge concerted effort, take decades to come to fruition, and kill off the native life on a planet. Inefficient.\n\nWaff had a much more immediate solution. If he could develop a breed of sandworm that tolerated water, even thrived around it, the creatures could be transplanted onto innumerable worlds where they could grow swiftly and multiply! The worms would not need to reconstruct a whole planetary environment before they began to produce melange. That alone would save decades that Waff simply did not have. His modified worms would provide all the spice the Guild Navigators could ever desire\u2014and serve Waff's purposes as well.\n\n_Help me, Prophet!_\n\nThe sandtrout specimen had absorbed all the water in the pan and now gradually moved about the base and sides, exploring the boundaries. Waff brought research tools and chemicals to the laboratory table\u2014his alcohols, acids, and flames, his deep sample extractors.\n\nThe first cut was the hardest. Then he began to work on the shapeless, squirming creature to pry loose its genetic secrets in any way he could.\n\nHe had the best DNA analyzers and genetic sequencers the Guild could obtain . . . and they were very good, indeed. The sandtrout took a long time to die, but Waff was certain the Prophet wouldn't mind.\n\n> _A stench seeps from my pores. The foul odor of extinction._\n> \n> \u2014SCYTALE,  \n> the last known Tleilaxu Master\n\nThe small, gray-skinned child looked worriedly at his older but identical counterpart. \"This is a restricted area. The Bashar will be very angry with us.\"\n\nThe older Scytale scowled, disappointed that a child with such a magnificent destiny could be so timid. \"These people have no authority to impose their rules on me\u2014either version of me!\" Despite years of preparation, instruction, and insistence, Scytale knew that the ghola boy still did not grasp who he was. The Tleilaxu Master coughed and winced, unable to minimize his physical problems. \"You must awaken your genetic memories before it is too late!\"\n\nThe child followed his older self down the dim corridor of the noship, but his steps were too shaky to be furtive. Occasionally the degenerating Scytale needed support from his twelve-year-old \"son.\" Each day, each lesson, should bring the younger one closer to the tipping point at which his embedded memories would cascade free. Then, finally, old Scytale could allow himself to die.\n\nYears ago, he had been forced to offer his only bargaining chip\u2014his secret stash of valuable cellular material\u2014to bribe the witches. Scytale resented being put in such a position, but in return for the raw makings of heroes from the past for the witches' own purposes, Sheeana had agreed to let him use the axlotl tanks to grow a new version of himself. He hoped it wasn't too late.\n\nFor years now, every sentence, every day increased pressure on the younger Scytale. His \"father,\" a victim of planned cellular obsolescence, doubted he had another year before collapsing completely. Unless the boy received his memories soon, _very soon_ , all the knowledge of the Tleilaxu would be lost. Old Scytale winced at the dire prospect, which hurt him far more than any physical pain.\n\nThey reached one of the vacant lower levels, where a testing chamber had gone unnoticed in the empty expanses of the ship. \"I will use this powindah teaching equipment to show you how God meant for the Tleilaxu to live.\" The walls were smooth and curved, the glowpanels tuned to a dull orange. The room seemed to be full of gestating wombs, round, flaccid, mindless\u2014the way women were supposed to serve a truly civilized society.\n\nScytale smiled at the sight, while the boy stared around with dark eyes. \"Axlotl tanks. So many of them! Where did they all come from?\"\n\n\"Unfortunately they are merely holographic projections.\" The high-quality simulation included mock tank sounds, as well as the odors of chemicals, disinfectants, and medicinals.\n\nAs Scytale stood surrounded by the glorious images, his heart ached to see the home he missed so much, a home now utterly destroyed. Years ago, before he was allowed to set foot again in sacred Bandalong, Scytale and all Tleilaxu had always undergone a lengthy cleansing process. Ever since the Honored Matres had forced him to flee with only his life and a few bargaining chips, he had tried to observe the rituals and practices as much as possible\u2014and had vigorously taught them to the young ghola\u2014but there were limitations. Scytale had not felt sufficiently clean in a long time. But he knew that God would understand.\n\n\"This is how a typical breeding chamber used to look. Study it. Absorb it. Remind yourself of how things were, how they should be. I created these images from my own memories, and those same memories are within you. Find them.\"\n\nScytale had said the same thing again and again, hammering it into the child. His younger version was a good student, very intelligent, and knew all the information by rote learning, but the boy didn't know it in his soul.\n\nSheeana and the other witches didn't grasp the immensity of the crisis he faced, or perhaps they didn't care. The Bene Gesserits understood little about the nuances of restoring a ghola's memories, could not recognize the moment when a ghola was perfectly ready . . . but Scytale might not have the luxury of waiting. The child was certainly old enough. He should awaken! Soon the boy would be the only Tleilaxu left, with no one to wake his memories.\n\nAs he surveyed the rows of breeding vats, the junior Scytale's face filled with awe and intimidation. The boy was drinking it all in. _Good_. \"That tank in the second row is the one that gave birth to me,\" he said. \"The Sisterhood called her Rebecca.\"\n\n\"The tank has no name. It is not a person and never was. Even when it could speak, it was only a female. We Tleilaxu never name our tanks, nor the females that preceded them.\"\n\nExpanding the image, he allowed the walls to disappear into a projection of a vast breeding house with tank after tank after tank; outside were the spires and streets of Bandalong. These visual cues should have been sufficient, but Scytale wished he could have added other sensory details, the female reproductive smells, the feel of the sunlight of home, the comforting knowledge of countless Tleilaxu filling the streets, the buildings, the temples.\n\nHe felt achingly lonely.\n\n\"I should not still be alive and standing before you. It offends me to be old and in pain, with my body malfunctioning. The kehl of true Masters should have euthanized me long ago and let me live on in a fresh ghola body. But these are not proper times.\"\n\n\"These are not proper times,\" the boy repeated, backing through one of the detailed holo-images. \"You have to do things you would not otherwise tolerate. You must use heroic means to stay alive long enough to awaken me, and I promise you, with all my heart, that I will become _Scytale_. Before it is too late.\"\n\nThe process of awakening a ghola was neither simple nor swift. Year after year, Scytale had applied pressures, reminders, mental twists to this boy. Each lesson and each demand built, like a pebble added to the pile, higher and higher, and sooner or later he would add enough to that unstable mound to trigger an avalanche. And only God and his Prophet could know which small stone of memory would cause that barrier to come crashing down.\n\nThe boy watched the flickering moods cross his mentor's face. Not knowing what else to do, he quoted a comforting lesson from his catechism. \"When one is faced with an impossible choice, one must always choose the path of the Great Belief. God guides those who wish to be led.\"\n\nThe very thought seemed to consume Scytale's last energy, and he slumped into a nearby chair in the simulation room, trying to recover his strength. When the ghola hurried to his side, Scytale stroked the dark hair of his alternate self. \"You are young, perhaps too young.\"\n\nThe boy placed a comforting hand on the old man's shoulder. \"I will try\u2014I promise. I'll work as hard as I can.\" He squeezed his eyes shut and seemed to push, as if wrestling with the intangible walls inside his brain. At last, perspiring heavily, he gave up the effort.\n\nThe elder Scytale felt despondent. He had already used all the techniques he knew to push this ghola to the brink. Crisis, paradox, unrelenting desperation. But he felt it more than the boy did. Clinical knowledge was simply insufficient.\n\nThe witches had used some sort of sexual twisting to bring back the Bashar Miles Teg when his ghola was only ten years old, and Scytale's successor was already two years past that mark. But he could not bear the thought of the Bene Gesserit women using their unclean bodies to break this boy. Scytale had already sacrificed so much, selling most of his soul for a glimmer of hope for his race's future. The Prophet Himself would turn his back on Scytale in disgust. Not that!\n\nScytale placed his head in his hands. \"You are a flawed ghola. I should have thrown away your fetus and started anew twelve years ago!\"\n\nThe boy's voice was rough, like torn fiber. \"I will concentrate, and push my memories out of my cells!\"\n\nThe Tleilaxu Master felt a weary sadness weighing him down. \"It is an instinctive process, not an intellectual one. It must come to you. If your memories don't return, then you are of no use to me. Why should I let you live?\"\n\nThe boy was visibly struggling, but Scytale saw no flash of awe and relief, no sudden flood of a lifetime's experiences. Both Tleilaxu reeked of failure. With each passing moment, Scytale felt more and more of himself dying.\n\n> _The fate of our race depends on the actions of an unlikely collection of misfits._\n> \n> \u2014from a Bene Gesserit study on the human condition\n\nIn his second life, Baron Vladimir Harkonnen had done well for himself. At only seventeen, the awakened ghola already commanded a large castle filled with antique relics and a retinue of servants to satisfy his every whim. Better yet, it was Castle Caladan, the seat of House Atreides. He sat upon a high throne of fused black jewels, gazing around a large audience chamber as attendants went about their duties. Pomp and grandeur, all the trappings a Harkonnen deserved.\n\nDespite appearances, though, the ghola Baron had very little real power, and he knew it. The Face Dancer myriad had created him for a specific purpose and, despite his reawakened memories, managed to keep him on a tight leash. Too many important questions remained unanswered, and too much was out of his control. He didn't like that.\n\nThe Face Dancers seemed much more interested in their young ghola of Paul Atreides\u2014the one they called \"Paolo.\" He was their real prize. Their leader Khrone said that this planet and the restored castle existed for the sole purpose of triggering Paolo's memories. The Baron was just a means to an end, of secondary importance in the \"Kwisatz Haderach matter.\"\n\nHe resented the Atreides brat for it. The boy was only eight and still had much to learn from his mentor, though the Baron hadn't yet determined what the Face Dancers really wanted.\n\n\"Prepare him and raise him. See that he is primed for his destiny,\" Khrone had said. \"There is a certain need he must fulfill.\"\n\n_A certain need_. But _what_ need?\n\n_You are his grandfather_ , said the annoying voice of Alia inside the Baron's head. _Take good care of him_. The little girl taunted him incessantly. From the moment his memories were restored, she had been there waiting for him in his mind. Her voice still contained a childhood lisp, exactly as she had sounded when she'd killed him with the poisoned needle of the gom jabbar.\n\n\"I'd rather take care of you, little Abomination!\" he yelled. \"Wring your neck, twist your head\u2014once, twice, three times! Make your delicate little skull pop right off! Ha!\"\n\n_But it's your own skull, dear Baron_.\n\nHe clamped his hands against his temples. \"Leave me alone!\"\n\nSeeing no one else in the room with their master, the servants looked at him uneasily. The Baron fumed and slumped back in his glittering black chair. Having embarrassed and angered him, the Alia voice whispered his name tauntingly one more time and faded away.\n\nJust then, a jaunty and self-important Paolo strode into the chamber, followed by an entourage of androgynous Face Dancers who acted as his protectors. The child carried an air of overconfidence that the Baron found at once fascinating and disconcerting.\n\nBaron Vladimir Harkonnen and this other Paul Atreides were inextricably meshed, simultaneously drawn together and repelled, like two powerful magnets. After the Baron's memories had been restored and he understood enough of who he was, Paolo had been brought to Caladan and handed over to the Baron's tender care . . . with dire warnings should any harm befall him.\n\nFrom his high black chair, the Baron glared down at the cocky youth. What made Paolo so special? What was the \"Kwisatz Haderach matter\"? What did the Atreides know?\n\nFor some time, Paolo had been sensitive, thoughtful, even caring; he had a stubborn streak of innate goodness that the Baron had been working diligently to eradicate. Given time, and enough harsh training, he was sure he could cure even the honorable core of an Atreides. That would prime Paolo for his destiny, all right! Though the boy still struggled with his actions occasionally, he had made considerable progress.\n\nPaolo came to an impertinent halt in front of the dais. One of the androgynous Face Dancers placed an antique handgun into the boy's hand.\n\nAngrily, the Baron leaned forward for a better view. \"Is that gun from my personal collection? I told you to stay out of those things.\"\n\n\"This is a relic of House Atreides, so I'm entitled to use it. A disk gun, once carried by my sister Alia, according to the label.\"\n\nThe Baron shifted on his throne, nervous to have the loaded weapon so close to him.\n\n\"It's just a woman's gun.\"\n\nInside the thick black armrests the Baron had secreted his own weapons, any one of which could easily turn the boy into a wet smear\u2014hmm, fresh material for growing another ghola, he thought. \"Even so, it's a valuable relic, and I don't want it damaged by a reckless child.\"\n\n\"I won't damage it.\" Paolo seemed pensive. \"I respect artifacts that my ancestors used.\"\n\nAnxious to keep the boy from thinking too much, he stood. \"Shall we take it outside, then, Paolo? Why don't we see how it works?\" The Baron gave him an avuncular pat on the shoulder. \"And afterward we can kill something with our bare hands, like we did to the mongrel hounds and ferrets.\"\n\nPaolo seemed uncertain. \"Maybe another day.\"\n\nNevertheless, the Baron hurried him out of the throne room. \"Let's get rid of those noisy gulls around the midden piles. Have I mentioned how much you remind me of Feyd? Lovely Feyd.\"\n\n\"More than once.\"\n\nWatched over by Face Dancers, they spent the next two hours at the castle's trash heap, taking turns shooting the raucous birds with the disk gun. Oblivious to the danger, the gulls swooped and shrieked at one another, fighting over morsels of rain-splattered garbage. Paolo took a shot, then the Baron. Despite its antiquity, the gun was quite accurate. Each spinning, microthin disk chopped a bird into bloody meat and dislodged feathers. Then the surviving gulls squabbled over the fresh gobbets.\n\nBetween them, they killed fourteen birds, although the Baron did not do nearly as well as the child, who had quite an aptitude for cool marksmanship. As the Baron raised the disk gun and aimed carefully, the girl's annoying voice rang in his head again. _That's not my gun, you know_.\n\nHe took the shot and missed by a wide margin. Alia giggled.\n\n\"What do you mean it's not yours?\" He ignored Paolo's puzzled stare as the boy took the weapon for his turn.\n\n_It's a fake. I never had a disk gun like that_.\n\n\"Leave me alone.\"\n\n\"Who are you talking to?\" Paolo asked.\n\nAfter reaching into a pocket the Baron offered several capsules of orange melange substitute to Paolo, who obediently took them. He grabbed the weapon back from the boy. \"Don't be ridiculous. The antiquities dealer provided a certificate of authenticity and documentation when he sold the weapon to me.\"\n\n_Grandfather, you shouldn't be so easily fooled! My own gun shot larger disks. This is a cheap imitation and doesn't even have the maker's initials on the barrel, like the original._\n\nHe studied the carved ornamental handle, turned the gun toward his face, then looked at the short barrel. No initials. \"And what about my other things, the objects supposedly owned by Jessica and Duke Leto?\"\n\n_Some are real, some are not. I'll let you find out which are which_. Knowing the nobleman's penchant for buying historical artifacts, the dealer would return to Caladan soon. No one made a fool of the Baron! The Baron ghola decided the next meeting would not be quite so cordial. He would ask a few incisive questions. Alia's voice faded away, and he was glad to have a moment of peace inside his head.\n\nPaolo had consumed two of the orange capsules, and as the melange substitute took hold, the boy dropped to his knees and stared beatifically into the sky. \"I see a great victory in my future! I'm holding a knife that drips with blood. I'm standing over my enemy . . . over myself.\" He frowned, then beamed again, yelling, \"I _am_ the Kwisatz Haderach!\" Then Paolo let out a bloodcurdling scream. \"No . . . now, I see myself dying on the floor, bleeding to death. But how can this be, if I am the Kwisatz Haderach? How can this be?\"\n\nThe nearest Face Dancer grew animated. \"We were instructed to watch for signs of prescience. We must notify Khrone immediately.\"\n\n_Prescience?_ the Baron thought. _Or insanity?_\n\nInside his mind, the presence of Alia laughed.\n\nDAYS LATER, THE Baron strolled along the top of the cliff and gazed out to sea. Caladan did not yet have the lovely, grimy industrial capacity of his beloved Giedi Prime, but at least he'd paved over the gardens in the vicinity of the castle. The Baron hated flowers with their eye-straining colors and sickening odors. He much preferred the perfume of factory smoke. He had great ambitions of turning Caladan into another Giedi Prime. The march of progress was more important than any esoteric plans the Face Dancers had for young Paolo.\n\nOn the lowest level of the restored castle, where other great houses would have prepared chambers for \"policy enforcement activities,\" House Atreides had instead used the space for food storage rooms, a wine cellar, and an emergency shelter. Being a more traditional nobleman, the Baron had installed dungeons, interrogation rooms, and a well-equipped torture chamber. He also had a party room on that level, where he often took young boys from the fishing village.\n\n_You can't remove the marks of House Atreides with such cosmetic changes, Grandfather_ , said the pestering voice of Alia. _I preferred the old castle_.\n\n\"Shut up, devil child! You were never here in life, either.\"\n\n_Oh, I visited my ancestral home when my mother lived here, when Muad'Dib was Emperor and his jihad splashed blood across the star systems. Don't you remember, Grandfather? Or weren't you inside my head then?_\n\n\"I wish you weren't inside mine. I was born before you! I can't possibly have your memories inside me. You're an Abomination!\"\n\nAlia chuckled in a particularly disconcerting way. _Yes, Grandfather. I'm that, and much more. Perhaps that's why I have the power to be inside you. Or, perhaps you are just flawed\u2014completely mad. Have you considered the possibility that you might be imagining me? That's what everyone else thinks_.\n\nServants hurried by, glancing fearfully at him. Just then the Baron saw a groundcar negotiating the steep road from the spaceport. \"Ah, here is our guest.\" Despite Alia's intrusion, he expected this to be an entertaining day.\n\nAfter the groundcar pulled up, a tall man stepped out of the rear compartment and made his way past statues of great Harkonnens that the Baron had erected in the past year. A suspensor platform floated behind the antiquities dealer, carrying his wares.\n\n_What do you plan to do with him, Grandfather?_\n\n\"You know damned well what I'm going to do.\" High on the wall above, the Baron rubbed his hands together in gleeful anticipation. \"Make yourself useful for a change, Abomination.\" Alia giggled, but it sounded as if she was laughing _at_ him.\n\nThe Baron hurried down as a haunted-looking house servant escorted the visitor inside. Shay Vendee was an antiquities dealer, always pleased to meet with one of his best customers. As he strolled in with his goods trailing behind him, his round face shone as radiantly as a small red sun.\n\nThe Baron greeted him with a moist handshake, clasping with both hands and holding on a little too long, squeezing a bit too hard.\n\nThe merchant extricated himself from his customer's grip. \"You'll marvel at what I've brought, Baron\u2014amazing what turns up with a little digging.\" He opened one of the cases on the suspensor platform. \"I saved these treasures especially for you.\"\n\nThe Baron brushed a speck off one of the jeweled rings on his fingers. \"First I have something to show you, my dear Mr. Vendee. My new wine cellar. I am quite proud of it.\"\n\nA look of surprise. \"Are the Danian vineyards operating again?\"\n\n\"I have other sources.\"\n\nAfter the dealer disengaged his suspensor platform, the Baron led him down a wide rock staircase into increasing gloom. Oblivious to the danger, Vendee chattered cordially. \"Caladan wines used to be quite famous, and deservedly so. In fact, I heard a rumor that a cache was found on the ruins of Kaitain, bottles perfectly preserved in a nullentropy vault. The nullentropy field prevented the wine from aging and mellowing\u2014in this case for thousands of years\u2014but even so, the vintage must be quite extraordinary. Would you like me to see if I can acquire a bottle or two for you?\"\n\nThe Baron stopped at the bottom of the dim stairs and peered with spider-black eyes at his guest. \"So long as you can provide the appropriate documentation. I wouldn't want to be duped into buying anything fake.\"\n\nVendee wore a look of horror. \"Of course not, Baron Harkonnen!\"\n\nFinally, they passed through a narrow corridor illuminated by smoking oil lamps. Glowglobes were too efficient and harsh for the Baron's taste. He loved the dank, gritty smell of the air; it almost masked the other odors.\n\n\"Here we are!\" The Baron pushed open a heavy wooden door and led the way into his fully stocked torture chamber. It had the traditional accoutrements: racks, masks, electrified chairs, and a strappado, by which a subject could be alternately hoisted into the air and dropped. \"This is one of my new playrooms. My pride and joy.\"\n\nVendee's eyes opened wide in alarm. \"I thought you said we were going to your wine cellar.\"\n\n\"Why, over there, my good man.\" With a good-natured expression, the Baron pointed to a table from which loose straps hung. A wine bottle and two glasses sat on top. He poured red wine into both glasses and handed one to his increasingly agitated guest.\n\nVendee glanced around, nervously eyeing the red stains on the table and rock floor. Spilled wine? \"I have just made a long journey, and I'm tired. Maybe we should go back up to the main rooms. You will be absolutely delighted with the new items I've brought. Quite valuable relics, I assure you.\"\n\nThe Baron fingered one of the straps on the table. \"There is another matter, first.\" He narrowed his eyes. From a side door a sunkeneyed boy marched in, carrying what looked like two ornate old weapons, disk handguns of ancient manufacture.\n\n\"Do these look familiar? Examine them carefully.\"\n\nVendee held one weapon to examine it. \"Oh, yes. The antique gun of Alia Atreides. Used by her own hands.\"\n\n\"So you said.\" Taking the other handgun from the serving boy, the Baron said to Vendee, \"You sold me a fake. I happen to know that the gun you hold is not the original weapon used by Alia.\"\n\n\"I have a reputation for integrity, Baron. If anyone has told you otherwise, they are lying.\"\n\n\"Unfortunately for you, my source is beyond reproach.\"\n\n_You are lucky to have me inside you to point out your mistakes_ , Alia said. _If you believe I am real_.\n\nIndignantly, Vendee placed the gun on the table and turned to leave. He only made it halfway to the door.\n\nThe Baron pulled the trigger of his own weapon, and a large, spinning disk shot out and hit the dealer squarely in the back of his neck, decapitating him. Swiftly, smoothly. The Baron was sure it hadn't hurt a bit.\n\n\"Good shot, eh?\" The Baron grinned at the serving boy.\n\nThe servant did not flinch at the murder. \"Will that be all you desire from me, sir?\"\n\n\"You don't expect me to clean up this mess myself, do you?\"\n\n\"No, my Lord. I will get right to it.\"\n\n\"Then wash yourself afterward.\" The Baron looked him over. \"We'll have even more fun this afternoon.\" Meanwhile, he went back upstairs to study what the antique dealer had brought with him.\n\n> _Once, I was born of a natural mother, and then reborn many times as a ghola. Considering the millennia over which the Bene Gesserit, the Tleilaxu, and others have meddled with the gene pool, I wonder\u2014are any of us truly natural anymore?_\n> \n> \u2014ship's log, entry of  \n> DUNCAN IDAHO\n\nToday, Gurney Halleck would be born again. Paul Atreides had looked forward to this during the months-long gestation process. Since the recent birth of his sister Alia, the waiting had become nearly unbearable. But in a matter of hours, Gurney would be removed from the axlotl tank. The famed Gurney Halleck!\n\nIn his studies under Proctor Superior Garimi, Paul had read much about the troubadour warrior, had seen images of the man and heard recordings of his songs. But he wanted to know the real Gurney, his friend, mentor, and protector from an epic time. Someday, though their ages were topsy-turvy now, the two would remember how close their friendship had been.\n\nPaul couldn't keep the grin off his face as he rushed to get ready. Whistling an old Atreides song that he'd learned from Gurney's recorded collection, he stepped into the corridor, and Chani emerged from her own quarters to join him. Two years his junior, the thirteen-year-old was whip-thin and fast, soft-spoken and beautiful, only a preview of the woman she would become again. Knowing their destinies, she and Paul were already inseparable. He took her hand, and the pair happily hurried toward the medical center.\n\nHe wondered if Gurney would be an ugly baby, or if he had only become a rolling lump of a man after being battered by the Harkonnens. He hoped the Gurney ghola would have a natural skill with the baliset, too. Paul was confident that the no-ship's stores could re-create one of the antique musical instruments. Maybe the two of them could play music together.\n\nOthers would be there for the new birth: his \"mother\" Jessica, Thufir Hawat, and almost certainly Duncan Idaho. Gurney had many friends aboard. No one on the ship had known Xavier Harkonnen or Serena Butler, the other two gholas who would be decanted today, but they were legends from the Butlerian Jihad. Each ghola, according to Sheeana, had a role to play, and any one of them\u2014or all of them together\u2014might be the key to defeating the Enemy.\n\nAside from the ghola children, many other boys and girls had been born over the years of the _Ithaca_ 's long flight. The Sisters bred with male Bene Gesserit workers who had also escaped from Chapterhouse; they understood the need to increase their population and prepare a solid foundation for a new colony, if the no-ship ever found a suitable planet to settle. The Rabbi's group of Jewish refugees, who had also married and begun families, still waited for a new home to fulfill their long quest. The no-ship was so vast, and the population aboard still so far below its capacity, that there was no real concern about running short on resources. Not yet.\n\nAs Paul and Chani approached the main birthing creche, four female proctors ran toward them down the hall, urgently calling for any qualified Suk doctor. \"They're dead! All three of them.\"\n\nPaul's heart stuttered. At fifteen, he was already training in some of the skills that had once made him the historical leader known as Muad'Dib. Summoning all the steel he could put into his voice, he demanded that the second proctor stop. \"Explain yourself!\"\n\nThe Bene Gesserit blurted, surprised into her answer. \"Three axlotl tanks, three gholas. Sabotage\u2014and murder. Someone destroyed them.\"\n\nPaul and Chani rushed toward the medical center. Duncan and Sheeana were already in the doorway looking shaken. Inside the chamber, three axlotl tanks had been ripped from their life-support mechanisms and lay in puddles of burned flesh and spilled liquid. Someone had used an incinerating beam and corrosives to destroy not only the life-support machinery, but the core flesh of the tanks and the unborn gholas.\n\nGurney Halleck. Xavier Harkonnen. Serena Butler. All lost. And the tanks, which had once been living women.\n\nDuncan looked at Paul, articulating the real horror here. \"We have a saboteur aboard. Someone who wishes to harm the ghola project\u2014or maybe _all_ of us.\"\n\n\"But why now?\" Paul asked. \"The ship has been fleeing for two decades, and the ghola project began years ago. What changed?\"\n\n\"Maybe someone was afraid of Gurney,\" Sheeana suggested. \"Or Xavier Harkonnen, or Serena Butler.\"\n\nPaul saw that the other three axlotl tanks in the creche had not been harmed, including the one that had recently given birth to the spice-saturated Alia.\n\nStanding by Gurney's tank, he saw the dead, half-born baby among the burned and dissolved folds of flesh. Nauseated, he knelt to touch the few wisps of blond hair. \"Poor Gurney.\"\n\nAs Duncan helped Paul to his feet, Sheeana said in a coldly businesslike voice, \"We still have the cellular material. We can grow replacements for all of them.\" Paul could sense her deep fury, barely controlled by her strict Bene Gesserit training. \"We will need more axlotl tanks. I'll send out a call for volunteers.\"\n\nThe ghola of Thufir Hawat entered and stared in disbelief at what had happened, his face an ashen mask. After the ordeal on the planet of the Handlers, he and Miles Teg had bonded closely; Thufir now helped the Bashar with security and defenses aboard the ship. The fourteen-year-old struggled to sound authoritative. \"We will find out who did this.\"\n\n\"Scan the security images,\" Sheeana said. \"The killer can't hide.\"\n\nThufir looked embarrassed, as well as angry and so very young. \"I already checked. The security imagers were deactivated, intentionally, but there must be other evidence.\"\n\n\"All of us were attacked, not just these axlotl tanks.\" Duncan's anger was plain as he turned toward young Thufir. \"The Bashar has cited several previous incidents that he believes may be sabotage.\"\n\n\"Those were never proved,\" Thufir said. \"They could have been mechanical breakdowns, systems fatigue, natural failures.\"\n\nPaul's voice was ice as he took a last, lingering look at the infant that would have been Gurney Halleck. \"This was no natural failure.\"\n\nThen Paul's legs went suddenly rubbery. Dizziness rose around him, and his consciousness blurred. As Chani rushed to grab him, he reeled, lost his footing, and hit his head hard on the deck. For a moment blackness enveloped him, a gloom that brightened into a frightening vision. Paul Atreides had seen it before, but he didn't know if it was memory or prescience.\n\nHe saw himself lying on the floor in a spacious, unknown place. A knife wound deep inside him sucked out his life. A mortal wound. His life's blood poured onto the floor, and his vision turned to dark static. Gazing up, he saw his own young face looking back at him, laughing. \"I have killed you!\"\n\nChani was shaking him, shouting into his ear. \"Usul! Usul, look at me!\"\n\nHe felt the touch of her hand on his own, and when his vision cleared he saw another concerned face. For a moment he thought it was Gurney Halleck, complete with an inkvine scar on his jawline, the glass-splinter eyes, the wispy blond hair.\n\nThe image shifted, and he realized it was the black-haired Duncan Idaho. Another old friend and guardian. \"Will you protect me from danger, Duncan?\" Paul's voice hitched. \"As you vowed to do when I was a child? Gurney is no longer able.\"\n\n\"Yes, Master Paul. Always.\"\n\n> _The Honored Matres clearly devised their own name for themselves, for no one else would ever apply the term \"honor\" after seeing their cowardly, self-serving actions. Most people have a very different way of referring to those women_.\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> assessment of past and present strengths\n\nWeapons and battleships were as important as air and food during these supposed End Times. Murbella knew she would have to change the way she approached the problem, but she had never expected such resistance from her own Sisterhood.\n\nWith both anger and disdain, Kiria cried, \"You offer them Obliterators, Mother Commander? We can't just hand over such destructive weapons to Ix.\"\n\nShe had no patience for this. \"Who else will build them for us? Holding secrets among ourselves only benefits the Enemy. You know as well as I do that only the Ixians can decipher the technology and manufacture great quantities for the coming war. Therefore, Ix must have full access. There is no other answer.\"\n\nMany worlds were building their own gigantic fleets, armoring every ship they could find, working on new weapon designs, but nothing had so far proved even remotely effective against the Enemy. The technology of the thinking machines was unsurpassed. But with a supply of new Obliterators, Murbella could turn the machines' own destructive power against them.\n\nAfter snatching the weapons from fringe machine outposts centuries ago, the Honored Matres could have formed an impenetrable line and hurled Obliterators at the oncoming Enemy. If they had stood together for the common good, they could have prevented this whole problem. Instead, those Honored Matres had fled.\n\nThinking about the hidden history she had excavated from deep within Other Memory, Murbella continued to be annoyed at those ancestors. They had taken the weapons, used them without understanding them, and depleted most of their stockpiles in their petty revenge against the hated Tleilaxu. Yes, many generations earlier the Tleilaxu had tormented their females, and the Honored Matres had good reason to direct vengeful violence against them.\n\nBut such a waste!\n\nBecause the Honored Matres had been so profligate in using the planet-roasting weapons against any world that offended them, only a few Obliterators remained intact. Recently, when cracking down on the rebel Honored Matre strongholds, Murbella had expected to unearth greater stockpiles. But they had found nothing. Had someone else stolen the weapons? The Guild perhaps, under their original pretext of helping the Honored Matres? Or had the whores truly used them all, holding nothing in reserve?\n\nNow the human race had insufficient weapons left to stand against the real Enemy. The Obliterators were as incomprehensible as any device Tio Holtzman had ever created for folding space, and the women had never known how to create more. For the sake of humanity, she hoped the Ixians could do so.\n\n_Times of extremis demand extreme actions._\n\nUnder her orders, the members of the united Sisterhood now removed the powerful weapons from their no-ships, battle cruisers, and infiltration vessels. She would take them to Ix herself. Murbella cut off continued arguments as she marched with a small entourage toward the Chapterhouse spaceport.\n\n\"But Mother Commander, at least negotiate patent protections,\" Laera said, a flush showing even on her dark skin. \"Impose restrictions so that the technology does not become widespread.\" She was one of the most businesslike Reverend Mothers, filling much of Bellonda's old role. \"Proliferation amongst planetary warlords could result in the devastation of the largest star systems. CHOAM alone, working with Ix, could wreak\u2014\"\n\nMurbella cut her off with a disgusted noise. \"I have no interest in who may or may not benefit commercially _after we win this war_. If the Ixians help us achieve victory, they are entitled to profit.\" She rubbed her chin thoughtfully as she looked up at the ramp of her small, fast lighter. \"We'll let the planetary warlords deal with their own problems.\"\n\n> _You play with feelings as a child plays with toys. I know why your Sisterhood does not value emotions: You cannot value what you do not understand!_\n> \n> \u2014DUNCAN IDAHO,  \n> letter submitted to Reverend Mother Bellonda\n\nSheeana used an authoritative tone, just short of Voice. \"'Respect for the truth comes close to being the basis for all morality.' And I want the truth from you. _Now_.\"\n\nGarimi raised her eyebrows and said calmly, \"A quote from Duke Leto Atreides to bolster the interrogation? Shall we bring in blazing lights and a Truthsayer?\"\n\n\"My Truthsense is sufficient. I have always known you well enough to read you.\"\n\nThe shock waves from the appalling crime in the birthing center rippled through the no-ship. The slaughter of unborn gholas, the destruction of three axlotl tanks\u2014tanks created from volunteer _Sisters!_ \u2014went beyond anything Sheeana had expected from even her most vehement detractors. Her suspicions had naturally turned toward the outspoken leader of the ultraconservative faction.\n\nInside an interior conference chamber whose doors were sealed, Sheeana stood like a stern schoolteacher, facing nine of the most prominent dissenters. These women had opposed the ghola project since its inception, disagreeing even more vehemently after Sheeana's decision to restart the work.\n\nUnder the blistering scrutiny, Garimi stared back, while her supporters were openly hostile\u2014especially the squat Stuka. \"Why would I damage an axlotl tank? It makes no sense.\"\n\nWithin her mind, among the lives in Other Memory, she heard the now-familiar voice of the ancient Serena Butler, sounding horrified. _Killing a child_! Serena was an odd visitor in Other Memory, a woman whose ancient thoughts should not have traveled down the corridors of the generations, and yet she had been with Sheeana for years now.\n\n\"You have shown a previous willingness to kill ghola children.\" Sheeana finally sat down.\n\nGarimi fought to control her trembling. \"I attempted to save us before Leto could become a threat, before he could become the Tyrant again. That was all, and I failed. My reasons were well known, and I stand by them. Why would I go to such extremes now? What do I care about Halleck? Or old General Xavier Harkonnen? Even Serena Butler is so far buried in our past that she's little more than the smoke of a legend. Why would I bother with them when the worst gholas\u2014Paul Muad'Dib, Leto II, the fallen Lady Jessica, and Alia the Abomination\u2014already walk among us?\" Garimi made a disgusted rumble in her throat. \"Your suspicions offend me.\"\n\n\"And the evidence offends _me_.\"\n\n\"Despite our disagreements, we are all Sisters,\" Garimi insisted.\n\nAt first the fleeing Bene Gesserits had had a common cause, a shared goal. But in a matter of months after their escape from Chapterhouse the divisions had begun, power struggles, command questions, a bifurcation of visions. Duncan and Sheeana focused on escaping from the outside Enemy, while Garimi wanted to found a new Keep and train a fresh Bene Gesserit population according to established ways.\n\n_How have we changed so dramatically? How did the divisions get so deep?_\n\nSheeana gazed from face to face, looking for indications of guilt, particularly in the eyes. Short, curly-haired Stuka had a line of moisture on her upper lip, one of the indicators of nervousness. But she detected no hatred there, no loathing sufficient to have sparked an act of such brutality. With dismay, she had no choice but to conclude that the perpetrator was not here.\n\n\"Then I need your help. The person standing next to any of us could be a saboteur. We must interview everyone. Gather our qualified Truthsayers, and use the last stores of the truthtrance drug.\" Sheeana rubbed her temples, already dreading the huge task. \"Please leave me alone, so I can meditate.\"\n\nAfter the nine dissenters departed, Sheeana stood alone, her eyes half closed. The population aboard the _Ithaca_ had grown, spread out in the no-ship over the years. Even she wasn't sure how many children were aboard, but she could easily find out. Or so she presumed.\n\nShe murmured to Other Memory, \"So, Serena Butler\u2014was your murderer in the room? If not them, then who could it be?\"\n\nSerena's voice interjected, full of sadness. _A liar can hide behind barricades, but all barricades eventually fall. You will have other opportunities to discover the murderer. There is sure to be more sabotage_.\n\nTHE TRUTHSAYERS TESTED each other first.\n\nTwenty-eight qualified Reverend Mothers were gathered from Garimi's followers and from the general population of Sisters. The women did not protest their innocence or complain about the suspicions cast upon them. Instead, they accepted mutual questioning.\n\nSheeana observed coolly as the women formed triads, two individuals acting as interrogators, the third as the subject. As soon as each subject passed the rigorous questioning, the roles switched, so that everyone was questioned. One by one, the Truthsayers created an evergrowing pool of reliable investigators. Everyone passed the test.\n\nOnce the Truthsayers had confirmed each other, Sheeana allowed them to question her. Garimi and her dissident Sisters also faced the challenges and proved their innocence, as did Sheeana's staunch followers. All of them.\n\nNext, with a Truthsayer named Calissa beside her, Sheeana stood before a stiff-backed Duncan Idaho. The very thought of Duncan being a murderer and a saboteur struck her as absurd. Sheeana wouldn't have believed it of anyone on board, and yet three axlotl tanks and three ghola children had been butchered.\n\nBut Duncan . . . Standing so close to him, smelling his perspiration, feeling him somehow fill the room with his presence, summoned dangerous memories in her. She had used her own sexual bonding skills to break him free of Murbella. Despite their backgrounds, both knew there had been more to that passionate encounter than just a necessary task. Duncan had been uneasy around her ever since, afraid of what he might succumb to.\n\nBut in this situation there was neither romance nor sexual tension, only accusations. \"Duncan Idaho, do you know how to bypass the security imagers in the medical center?\"\n\nHe looked past her, not blinking. \"That is within my capabilities.\"\n\n\"Did you commit this terrible act and cover your tracks?\"\n\nNow his gaze met hers. \"No.\"\n\n\"Did you have any reason to prevent Gurney Halleck, Serena Butler, or Xavier Harkonnen from being born?\"\n\n\"I did not.\"\n\nNow that Duncan faced her and a Truthsayer, Sheeana could have asked him questions about their personal relationship to witness his reaction. He would not be able to lie to her or pretend. But she feared his answers. She didn't dare ask.\n\n\"He speaks the truth,\" said Calissa. \"He's not our saboteur.\"\n\nDuncan remained in the room when Bashar Miles Teg came for questioning. Calissa displayed images of the horrific scene from the birthing chamber. \"Are you in any way responsible for this, Miles Teg?\"\n\nThe Bashar stared at the images, looked up at her, turned his gaze to Duncan. \"Yes.\"\n\nSheeana was so startled that she struggled to think of another question.\n\n\"How so?\" Duncan asked.\n\n\"I am responsible for security aboard this no-ship. Clearly, I failed in my duty. If I had done a better job, this atrocity never would have occurred.\" He glanced at the troubled Calissa. \"Since you asked me in the presence of a Truthsayer, I couldn't lie.\"\n\n\"Very well, Miles. But that isn't what we meant. Did you commit this sabotage or authorize it? Do you know anything about it?\"\n\n\"No,\" he answered emphatically.\n\nDozens of private chambers were set up, where the interrogations could continue unabated. They asked every one of the ghola children, from Paul Atreides all the way to nine-year-old Leto II, and the Truthsayers detected no criminal falsehoods.\n\nThen the Rabbi and all of the Jews.\n\nAnd every other passenger aboard the no-ship.\n\nNothing. Not a single person seemed to be connected with the murderous incident. Duncan and Teg used their Mentat skills to check and recheck the lists of people aboard, yet they could find no errors. No one had evaded questioning.\n\nSitting across from Sheeana in the otherwise empty interrogation room, Duncan steepled his fingers. \"There are two possibilities. Either the saboteur is capable of deceiving a Truthsayer . . . or someone we don't know about is hiding aboard the _Ithaca_.\"\n\nIN WELL ORGANIZED teams, the Bene Gesserit blocked off, then sectioned the no-ship's decks, methodically moving from cabin to cabin and chamber to chamber. But it was a formidable task. The _Ithaca_ was the size of a small city, more than a kilometer long and hundreds of decks high, each filled with passages, chambers, and hidden doors.\n\nWhile trying to guess how someone else might have sneaked aboard, unknown to them, Duncan remembered discovering the mummified remains of Bene Gesserit captives the Honored Matres had tortured to death. That sealed chamber of horrors had gone undetected during the whole time Duncan had been held prisoner inside the ship on the Chapterhouse landing field.\n\nCould someone else\u2014an unknown Honored Matre, perhaps?\u2014have remained hidden aboard for all that time? More than thirty years! It did not seem possible, but the vessel had thousands of work bays, living quarters, corridors, and storage lockers.\n\nAnother possibility: During the escape from the planet of the Handlers, several Face Dancers had crashed small fighters into the no-ship's hull. Mangled bodies had been pulled from the wreckage of those ships . . . but could it all have been a ruse? What if some had actually survived those kamikaze crashes and slipped away? Perhaps one or more Face Dancers were lurking in the untraveled passages of the no-ship, looking for ways to strike.\n\nIf so, it was imperative to find them.\n\nTeg had already installed hundreds more surveillance imagers at strategic locations, but that was only a stopgap measure at best. The _Ithaca_ was so large that even the best security equipment had thousands of blind spots, and there simply weren't enough personnel to monitor the imagers already in place. It was an impossible task.\n\nStill, they tried.\n\nAs Duncan accompanied a group of five searchers, he was reminded of a beating party marching through the tall grass on a big game hunt. He wondered if they would scare a deadly lion out from somewhere in the vastness of the vessel.\n\nDeck after deck was searched, but even with a dozen teams, a complete inspection from the topmost deck to the lowest cargo hold would take a great deal of time, and in the limited searches they conducted, they found nothing. Duncan was exhausted and stressed.\n\nAnd the murderer\u2014or murderers\u2014remained aboard.\n\n> _Only two options are before us now: defend ourselves or surrender to the Enemy. But if any of you believes that surrender is a viable option, then we have already lost._\n> \n> \u2014BASHAR MILES TEG,  \n> speech given before the Pellikor Engagement\n\nLeaving the Obliterators on Ix for the fabricators to study and duplicate, Murbella traveled next to the main Guild shipyards on Junction.\n\nAdministrator Rentel Gorus, with long, pale hair and milky eyes, led Murbella among the construction bays, suspensor cranes, conveyors, and assemblers, all of them teeming with workers. The buildings were tall and blocky, the streets serviceable rather than beautiful. Everything on Junction was done on a breathtaking scale. Great lifters hauled components up to the skeletons of gigantic ships, assembling one vessel after another. The air held the bitter tang of hot metal, the chemical residues from welding mismatched components into huge vessels.\n\nGorus seemed overly proud. \"As you can see, we have the facilities you request, Mother Commander, provided the price is right.\"\n\n\"The price will be right.\" With the New Sisterhood's wealth in melange and soostones, Murbella could meet virtually any demand for payment. \"We'll pay you well for every ship you create, every vessel that can be placed into battle, every craft that can stand against the thinking machine army. The end of our civilization is at hand if we don't defeat the thinking machines.\"\n\nGorus did not seem intimidated. \"Every side in every war believes their conflict is crucial to history. But most often those are delusionary and needlessly alarmist thoughts. This war may be over before you have to resort to such measures.\"\n\nShe scowled. \"I don't know what you mean.\"\n\n\"There are other ways to solve the problem. We know that outside forces are sweeping in to many planetary systems. But what do they _want_? To what will they concede? We believe such discussions are worth pursuing.\" He blinked his milky eyes.\n\n\"What sort of trick is the Guild trying to play on us this time?\"\n\n\"No trick, just sensibility. Regardless of politics, commerce must continue. Wartime desperation inspires technological innovation, but peace promotes profitability in the long run. Trade will go on, no matter who wins the conflict.\"\n\nHeighliners had long been the luxury ships of the universe; now Murbella forced the Spacing Guild to devote their shipyards to creating the tools of war. For centuries the Guild's commercial fleet had been stable, and demand for trade steadily increased as people returned from the Scattering. Now, however, with Omnius's fleet wiping out whole populations and sending refugees in panicked flight back into the heart of the Old Empire, CHOAM and the Guild were in turmoil.\n\nA hot wind from the assembly bays blew in Murbella's face, burning her nostrils with the acrid smoke of waste chemicals. A shiver coursed down her spine.\n\n\"Our common enemy must be rational,\" Gorus continued. \"We have therefore dispatched emissaries and negotiators out to the war zone. We will find the thinking machines and make our proposal. The Guild would prefer to continue its commerce regardless of the outcome of this disagreement.\"\n\nMurbella gasped. \"Are you insane? Omnius seeks the extermination of all humanity. That includes you.\"\n\n\"You overstate your case, Mother Commander. Some of our emissaries will, I believe, achieve our goal.\"\n\nIn the background, blasts of steam curled up from the stone smokestacks. She ignored the noise and the smell. \"You are a consummate fool, Administrator Gorus. Thinking machines do not follow the rules you assume.\"\n\n\"Be that as it may, we feel obligated to try.\"\n\n\"And what is the result so far?\"\n\n\"Acceptable losses. Our first emissaries have disappeared, but we will continue the effort. We plan for all eventualities\u2014even disaster.\" Casually, he led her out onto a broad, open field under the half-assembled hulk of a huge ship. \"Thus, we are comfortable with extending certain beneficial terms to the New Sisterhood. You have always been a valued customer, but the order you submitted is massive. Even under wartime conditions, you have asked for more ships than we are able to provide.\"\n\n\"Then offer your workers more incentive.\"\n\n\"Ahh, Mother Commander, but will you provide enough incentive to _us_?\"\n\nShe bristled. \"How can you think solely of profits when the fate of the human race is at stake?\"\n\n\"Profits determine all our fates.\" The Administrator gestured casually, as if to encompass the huge assembly of the ships around him.\n\n\"We'll pay what you demand, and the Guild Bank will offer us loans if necessary. We need those ships, Gorus.\"\n\nHe smiled coolly. \"Your credit is good, but we must address another problem. We do not have enough Guild Navigators to man so many new ships. All of the vessels we build for you will have to be equipped with Ixian mathematical compilers, rather than traditional Navigators. Is that acceptable?\"\n\n\"Provided the ships function as we require, I have no objection. We don't have time to develop and train another population of Navigators.\"\n\nObviously pleased, Gorus rubbed his hands together. \"Of late, Navigators have proven somewhat intractable, due to the shortage of spice\u2014a shortage which your Sisterhood created, Mother Commander. It is because of you that we had to look for alternatives to Navigators.\"\n\n\"I have no fondness for Navigators, or for your obscene profits. I don't care how the Guild accomplishes it, but we need those ships.\"\n\n\"Of course, Mother Commander, and we shall provide what you wish.\"\n\n\"That is precisely the answer I need.\"\n\n> _What is the advantage of prescience if it serves only to reveal our own downfall?_\n> \n> \u2014NAVIGATOR EDRIK,  \n> message to the Oracle of Time\n\nThe Guild bureaucrats had the audacity to call Edrik's Heighliner back to the shipyards on Junction. Staring ahead with his milky eyes, Administrator Gorus blithely announced that the Heighliner would be fitted with one of the new Ixian mathematical compilers. \"Our spice supply line is undependable. We must be certain each vessel can operate safely if its Navigator fails.\"\n\nOver the past two years, more and more Guildships had been outfitted with the hated artificial controls. Mathematical compilers! No simple engine or tool could adequately complete the phenomenally complex projections that a Navigator performed. Edrik and his fellows had evolved through immersion in spice, their prescient vision strengthened through the power of melange. There could be no mechanical substitute.\n\nNevertheless, Edrik had no choice but to accept a team of qualified and arrogant Ixian workers who shuttled up from the Junction shipyards. The tight-lipped men boarded the Heighliner under the watchful eyes of the Guild, with their smug expressions, compiler machines, and dangerous curiosity.\n\nIn his tank, Edrik was concerned that they would snoop around under the pretext of completing their installation. The Navigator faction could not risk these men finding Waff's laboratory, the genetically altered sandtrout, and the small mutated worms he was producing in his tanks. The Tleilaxu man claimed to be making excellent progress, and his work must remain a secret.\n\nTherefore, when the Ixian installers were all safely aboard, Edrik simply folded space, informing no one in the shipyards where he was going. He carried his empty Heighliner far out into an isolated wasteland between solar systems, and there ejected the disbelieving Ixians, along with their accursed navigation machines, out into the cold vacuum.\n\n_Problem solved_.\n\nHis acts would be discovered eventually, but that could not be avoided. Edrik was a Navigator. Mere human Administrators had no hold over him.\n\nEdrik suspected that the devious Administrator and his faction saw the melange crisis as an opportunity to shift the Guild's burden away from problematic Navigators; they did not really want a new source of spice. Gorus was now an absolute ally, if not a puppet, of the Ixians. Edrik had seen the economic projections and knew that the Administrators considered navigation machines to be more cost-effective than Navigators\u2014and more easily controlled.\n\nWith the Ixians and their machines happily ejected, Edrik knew it was time to call another meeting with his fellow Navigators; they needed to receive fresh guidance from the Oracle of Time. Because Junction and several other Guild planets were already compromised by Gorus and his cronies, Edrik chose a place that no one but Navigators could find.\n\nOnce they had been shown how, they could fold their Guildships deep into another dimension, a nontraditional universe where the Oracle occasionally went on personal, incomprehensible explorations.\n\nIgnited by the light of seven newborn stars, the cosmic gases swirling around his giant ship seemed inflamed. The nebula shone pink and green and blue, depending on which window of the spectrum Edrik chose to look through. The misty curtains put on a spectacular show, a great whirlpool of ionized gases\u2014and a perfect place to hide.\n\nWhen the ships gathered, the Navigators were in quite an uproar, and their numbers were less than Edrik had hoped. So far, four hundred Heighliners had been decommissioned, their parts salvaged to construct new no-ships that relied on artificial guidance systems. Seventeen Navigators had died horribly, their tanks emptied. Edrik learned that six of his fellows had likewise murdered the Ixian engineers rather than allow them to install mathematical compilers. Four Navigators had simply disconnected the machines, and the onboard Ixian teams failed to realize that their vaunted systems were no longer functional.\n\n\"We require melange,\" he transmitted. \"By the grace of spice, we see through folded space.\"\n\n\"But the Sisterhood has denied it to us,\" one of the other Navigators said.\n\n\"They have spice. They spend spice. But they do not give it to us.\"\n\n\"The witches give it to the Guild for ships . . . but the Administrators have cut us off. We are betrayed by our own.\"\n\n\"They control the spice.\"\n\n\"But they do not control us,\" Edrik insisted. \"If we find our own source of spice, we will not need the Administrators. This is for the survival of Navigators, not simply for commerce. We have struggled with this problem for years. The Tleilaxu ghola has finally come up with a solution.\"\n\n\"A new source of spice? Has it been proven?\"\n\n\"Is anything fully proven? If this goes well, we can destroy the corrupt old Spacing Guild and supercede them.\"\n\n\"We must speak to the Oracle.\"\n\nEdrik waved his tiny, misshapen hands. \"The Oracle already knows our problem.\"\n\n\"The Oracle has not deigned to help us,\" said another.\n\n\"The Oracle has her own reasons.\"\n\nDrifting in his tank, Edrik acknowledged their conundrum. \"I have spoken to her myself, but perhaps all of us together can urge her to respond. Let us summon the Oracle.\"\n\nUsing their spice-enhanced minds, the numerous Navigators shot a message arrow through the folds of space. Edrik knew they had no way to coerce the Oracle of Time\u2014or the Oracle Infinity, as she was sometimes called\u2014to respond, but he sensed her presence, and her deep uneasiness.\n\nWith a silent flash, a trapdoor opened in the vacuum, and the ancient container arrived. It was not quite a ship, for the Oracle could travel anywhere she wished, mentally folding space without the help of Holtzman engines.\n\nEven in that small and nonthreatening enclosure, Edrik knew full well the power and immensity of that highly advanced mind. As a human, Norma Cenva had first discovered the connection between spice and prescience. She had developed the technology of folding space, had created the incomprehensible equations that Tio Holtzman had taken as his own.\n\nThough the Oracle used no known transmitting device, her words were loud and implacable in their minds. \"Your concerns are parochial. I must find the wayward no-ship. I must determine where Duncan Idaho has taken it, before the Enemy intercepts it.\"\n\nThe Oracle often chose her own esoteric goals without explaining them. One of the Navigators asked, \"Why is the no-ship so important, Oracle?\"\n\n\"Because the Enemy wishes to have it. Our great foe is Omnius\u2014except that he is as changed from his former computer evermind as I am evolved from the human I once was. The machines have completed their high-order projections. The evermind knows he must have the Kwisatz Haderach, just as I know the Enemy must _not_ have him.\" The Oracle let the silence hang in space like a hole, before she added a stinging rebuke. \"Your appetite for spice is not the priority. I must find the ship.\"\n\nAbruptly discontinuing the debate, she winked out again and vanished into her own place in an alternate universe.\n\nEdrik and the gathered Navigators were shocked by her response. Navigators were dying, spice was dwindling away to nothing, Administrators were trying to overthrow the Guild\u2014and the Oracle simply wanted to find a lost ship?\n\n## TWENTY-TWO YEARS AFTER\n\nESCAPE FROM CHAPTERHOUSE\n\n> _These new Face Dancers cannot be detected by DNA analysis or any other form of cellular scrutiny. As far as we know, only a Tleilaxu Master can tell the difference_.\n> \n> \u2014Bene Gesserit report on human mutation\n\nThough the Ixian specialists had studied the Obliterators for half a year, they still hadn't given the Sisterhood an answer. Murbella brooded in her offices on Chapterhouse, waiting. The news seemed to worsen with each passing day.\n\nShe received regular updates on the depredations of the thinking-machine fleet. The powerful Enemy ships moved inexorably through the fringe systems like a crashing tidal wave, drowning world after world. Another ten planets evacuated or contaminated by plagues, another ten lost, and more refugees flooded into the Old Empire.\n\nA network of Sisters met with any refugee ships that came from the battleground systems. Taking statements from groups of survivors, they compiled an exhaustive three-dimensional map of the movements of the machine fleet. The pattern seeped like a bloodstain through the galaxy.\n\nIn a single desperate stand, nineteen Sisterhood no-ships expended their last three Obliterators to destroy a whole battle group of oncoming machine ships and temporarily prevent the annihilation of one human-inhabited system. In the end, though, even that devastation amounted to only a brief delay; the machine fleet came back with greater strength and crushed the world after all, killing every inhabitant. With the last Obliterators gone, the New Sisterhood was woefully underdefended.\n\nUnless the Ixians could help. _What was taking them so long_?\n\nFinally, a lone Ixian engineer came to Chapterhouse to deliver his news. When he said he would speak to no one but the Mother Commander herself, escorts brought him to the main Keep. Waiting on her imposing throne in front of the dust-streaked, segmented window, Murbella could respect the man for bypassing bureaucracy and getting to the core of the matter.\n\nThe engineer's face was bland and unmemorable, his brown hair closely cropped, his demeanor unassuming. He had a peculiar, unpleasant odor about him, perhaps from chemical residue or the machinery of Ix's underground fabrication plants. He bowed perfunctorily and stepped in front of her. \"Our best engineers and scientists have deconstructed and analyzed the sample Obliterators you provided.\"\n\nMurbella leaned forward, giving him her full attention. \"And you can reproduce them?\"\n\n\"Better than that, Mother Commander.\" His confident smile held no warmth at all, was simply an imitation of a facial expression. \"Our fabricators understand the underlying concept of the weapon, and are able to concentrate its destructive power. Previously, it required several Honored Matre battleships deploying multiple Obliterators to kill a planet. With our enhanced weapon a single ship can launch enough firepower to do what was done to Rakis.\" He gave a perfunctory shrug. \"Imagine what such an energy release would do to Enemy battleships.\"\n\nMurbella tried to conceal her delight. \"We need as many as you can produce. Instruct your factories to begin work on these weapons immediately.\" She kept her voice hard, letting her impatience seep through. \"But why did you need to see me in person, when you could easily have sent a message with this information?\" Her lips quirked. \"Do you require a pat on the back? Shall I give you my applause? There, you have it.\"\n\nThe Ixian engineer remained bland. \"Before we begin, Mother Commander, there is the matter of payment. Chief Fabricator Sen has instructed me to inform you that Ix must be compensated if we are to pull our profitable manufacturing centers off-line in order to create these Obliterators for your war.\"\n\n\" _My_ war? All humans must share the burden of the cost.\"\n\n\"Unfortunately, we disagree. The only payment we will accept is spice. And the only source of spice is your New Sisterhood.\"\n\n\"We have other ways to pay you.\" Murbella tried to conceal her alarm. She wasn't sure their fledgling spice operations could supply the necessary amount. And why would Ix care about spice, in particular? Sisterhood accounts in Guild banks could be drained; CHOAM could be convinced to supply important commodities; and soostones were more valuable than ever, especially since the recent turmoil on Buzzell.\n\nWhen she offered these alternatives, though, the Ixian fabricator shook his head. \"I have no flexibility in these negotiations, Mother Commander. It must be melange. No other coin will do.\"\n\nShe ground her teeth, but had no patience for further delay. \"Spice it is, then. Get started.\"\n\nDEPARTING FROM CHAPTERHOUSE, Khrone the Face Dancer felt content. The New Sisterhood had bowed to his demands, as he had known they would. Back on Ix, he had the ear of the Chief Fabricator, and Face Dancer replacements already controlled all key Ixian manufacturing centers.\n\nKhrone found it ironic to demand payment in spice, since Ix had devoted so much technological effort into installing navigation machines in Guildships. Thanks to the mathematical compilers, melange was basically obsolete when it came to folding space, and the Navigators were fading swiftly.\n\nBut by insisting on such a huge payment in spice alone, and then hoarding the commodity, Khrone would remove a large amount of it from the market, making it even rarer. That, in turn, would force more and more ships to convert to the Ixian navigation compilers, because the Guild could not support the melange needs of their Navigators. Before long, with no way of supporting their own Navigators, the whole Spacing Guild would fall under Khrone's control. He had worked it all out in exquisite detail.\n\nIn the meantime, he and his disguised workers would make it look as if they were providing everything the Sisterhood demanded. Let them fight useless battles while the real war was already won, right under their noses! Mother Commander Murbella would be quite satisfied\u2014up until the moment a curtain of darkness fell on humankind. Permanently.\n\n> _Every man makes errors. When a security chief makes them, though, there are consequences. People die._\n> \n> \u2014THUFIR HAWAT,  \n> the original\n\nThe Bashar and his prot\u00e9g\u00e9 marched down the corridors toward the no-ship's life-support center. \"I am deeply ashamed, Thufir. It has been almost a year, and I am incapable of finding a blatant saboteur and murderer.\"\n\nThe young Hawat looked up at him, clearly idolizing the military genius. \"We have a limited pool of suspects, and a discrete area in which he\u2014or she\u2014could hide. We've done everything possible, Bashar.\"\n\n\"And yet the saboteur is here, somewhere.\" Teg did not slow his pace. \"Therefore, we have not done everything possible, because we still haven't found the person responsible. The fact that there have been no further murders does not mean we can let down our guard. I am convinced our saboteur is still among us.\"\n\nThe _Ithaca_ was constantly being searched and monitored. Additional surveillance imagers had been installed, but the culprit seemed to have an affinity for hiding. Teg suspected that the work of the saboteur went well beyond the murder of the gholas and the axlotl tanks. In recent months, many ship's systems had inexplicably failed\u2014too many to be caused by random events and natural breakdowns. \"Our adversary is still at work.\"\n\nThe Thufir ghola raised his smooth chin in a display of pride. He was strong and gangly with a heavy brow; he had let his hair grow shaggy. \"Then you and I will find him.\"\n\nTeg smiled at Thufir. \"As soon as you regain his memories and experience as a warrior Mentat and Master of Assassins, you will be a formidable ally.\"\n\n\"I'm formidable now.\" Thufir had already proved his worth during the tense escape from the Handlers, risking his own life to help the Rabbi get away from Face Dancers in league with the Enemy. Teg believed the young ghola had the potential to do much more.\n\nVarying his pattern, he insisted on an exhausting round of daily security inspections while he left Duncan Idaho on the navigation bridge, ever vigilant for the Enemy's glowing net.\n\nThe _Ithaca_ continued to wander in empty space. At first, their voyage had simply been to get away from the Enemy hunters. Duncan had been forced to remain hidden behind the ship's veiling no-field, since the old man and woman seemed to want him in particular. Now, after more than two decades, the population aboard had increased, and children were growing up and being taught necessary skills without ever having set foot on a planetary surface.\n\nDespite all the worlds settled during the Scattering, habitable systems seemed sparse indeed. For the first time, Teg wondered how many ships of refugees fleeing from the Famine Times had simply died without ever finding their destination. The _Ithaca_ had no Guild Navigator; only pure chance brought them within range of planets. So far, they had encountered only two places that might have supported a new colony: one Honored Matre world that had been completely wiped out by Enemy plagues, and the planet of the insidious Handlers.\n\nNevertheless, with its recyclers, greenhouses, and algae tanks, the aging _Ithaca_ should have been able to sustain the present number of passengers for centuries, if necessary. They\u2014and their successors\u2014could effectively stay onboard forever and never stop running. _Is that our fate_? Teg asked himself. But because of leakages, losses, and \"accidents,\" the passengers had cause for concern. Sooner or later, they would need to replenish their reserves.\n\nThinking of resources, the Bashar took a side corridor to the fermentation bins and adjacent algae-growth tanks. Grown in the vaulted, humid chamber, the biomass provided raw material for the food-manufacturing units. A prime vulnerability.\n\nAs he opened a hatch, Teg caught the rich, marshy smell of compost and algae. They climbed metal steps to a catwalk and looked down into the cylindrical vat filled with hairy green slime. The stinking wet mass of fecund algae digested anything organic, growing large amounts of edible, rather unpalatable, material that could be converted into better-flavored foods. Ceiling fans whirred, drawing the odorous air upward through filters and into the intricate set of ducts that acted as the no-ship's circulatory system. After taking samples and testing the chemical balance of the tanks, Teg concluded that everything was in order. No sign of sabotage since his last inspection.\n\nThe serious young man tagged along beside him. \"I am not a Mentat yet, sir, but I have been giving the sabotage problem a great deal of thought.\"\n\nWith raised eyebrows Teg turned to his prot\u00e9g\u00e9. \"And do you have a first-order approximation?\"\n\n\"I have an idea.\" Thufir did not try to conceal his anger. \"I suggest you have a long talk with the Yueh ghola. Perhaps he knows more than he has admitted.\"\n\n\"Yueh is only thirteen. He does not have his memories back.\"\n\n\"Maybe the weakness is in his blood. Bashar, we know that _someone_ committed the sabotage.\" The young man sounded disappointed in himself for allowing it to happen. \"Even the real Thufir Hawat didn't find the traitor in House Atreides before he betrayed us to the Harkonnens. That traitor was Yueh.\"\n\n\"I'll keep it in mind.\"\n\nBack in the corridors again, the two passed a sick-looking old Scytale and his clone emerging from their quarters. Because they isolated themselves and lived with odd traditions and behavior, the Tleilaxu were natural suspects, but Teg had found no evidence against them. In fact, he believed the real saboteur would be careful to blend in perfectly and draw no attention at all. It was the only way he could have remained hidden for so long.\n\nTwo pregnant women passed by them in the corridor, chatting as they continued on their way. Both were part of Sheeana's conventional breeding program to maintain the population of the Sisterhood, providing an adequate genetic base should the splinter group ever find a place to settle.\n\nFinally, Teg and Thufir reached the cavernous, humming engine chamber, and entered the large aft compartment through a round doorway. Apparently safe but lost again since their last passage through foldspace, the _Ithaca_ drifted, though Duncan insisted on keeping the Holtzman engines ready at all times.\n\nThick clearplaz separated the Bashar and Thufir from a trio of power plants that fed the machines. Walkways laced the outside of an explosion-proof plaz chamber that contained the side-by-side engines. The two stared up at the giant mechanisms that could fold space. A true miracle of technology. All of the readings remained within nominal range. Again, no sign of sabotage.\n\n\"We're still missing something,\" he mused. \"I can feel it.\"\n\nOnce before, at the end of the Battle of Junction, Teg had failed to see the terrible and deadly \"Weapon\" that the Honored Matres had held in reserve. That mistake had nearly lost him the entire war. He considered their situation now. _What deadly device will I fail to see this time?_\n\n> _Humanity has a great genetic compass that constantly guides us onward. Our task is to keep it always pointed in the right direction._\n> \n> \u2014REVEREND MOTHER ANGELOU,  \n> famed breeding mistress\n\nWellington Yueh had a powerful need to be forgiven. The blank spot in his mind was filled with guilt. He was just a ghola and only thirteen, but he knew he had done terrible things. His own history clung to him like tar to his shoe.\n\nIn his first life, he had broken his Suk conditioning. He had failed his wife Wanna by allowing the Harkonnens to use her as a pawn and had betrayed Duke Leto, bringing about the Atreides downfall on Arrakis.\n\nAfter studying records of his prior existence, learning in painful detail what he had done, Yueh tried to find solace in considering the Orange Catholic Bible, along with other ancient religions, sects, philosophies, and interpretations that had developed over the millennia. The oft-repeated doctrine of Original Sin\u2014so unfair!\u2014was a particular thorn in his side. Yueh could have made a coward's excuse that he couldn't remember and therefore didn't deserve blame, but that was not the path to redemption. He had to turn elsewhere.\n\nJessica was the only one who could forgive him.\n\nThe eight ghola children in Sheeana's project had been raised and trained together. Because of their individual personalities they had formed personal bonds and friendships. Even before they knew the history that should tear them apart, Yueh had tried to be a friend to Jessica.\n\nHe had read the journals and instructional writings of the original Lady Jessica, bound concubine to the Duke Leto Atreides. She'd also been a Reverend Mother, an exile, the mother of Muad'Dib, and the grandmother of the Tyrant. That long-dead Jessica had been a strong woman, a role model despite how the Bene Gesserit reviled her for her flaw, her weakness. _Love_.\n\nTogether, the gholas now faced a far greater enemy than the Harkonnens. When Jessica's memories were finally awakened, would the shared threat be sufficient to keep her from wanting to murder him? He had read her own words, as written down by Princess Irulan, expressing her poignant agony of grief: \"Yueh! Yueh! Yueh! A million deaths were not enough for Yueh!\"\n\nYes, she was the only one who could offer him any hope of forgiveness. With a clean slate and an open heart, he prayed that it was possible for him to lead an honorable life this time.\n\nJessica often occupied herself in the main conservatory, tending the plants that served as a supplemental food source for the hundreds aboard. She had an affinity for the greenhouse work and was happy to be around the fertile dirt, the misting irrigators, the fleshy green leaves, and sweet-scented flowers. With her bronze hair and oval face, noble and young, she looked exquisitely beautiful. How she and Duke Leto must have loved each other long ago . . . until Yueh destroyed it all.\n\nJessica looked up from the flowers and lush herbs to focus haunted eyes on Yueh.\n\nHe said, \"Do you mind the company?\"\n\n\"Not yours. It's refreshing to be with someone who doesn't blame me for things I don't remember doing.\"\n\n\"I hope you'll grant me the same consideration, my Lady.\"\n\n\"Please don't call me that, Wellington. At least not yet. I can't be the Lady Jessica until I . . . well, until I become the Lady Jessica.\"\n\nHe tried to guess the reasons for her gloomy mood. \"Has Garimi been haranguing you again?\"\n\n\"Some Bene Gesserits won't forgive me for having gone against the strict commands of the Sisterhood, for betraying their breeding program.\" She seemed to be reciting something she had read. \"The consequences of that brought down an empire and subjected the human race to thousands of years of tyrannical rule and many more centuries of privation.\" She let out a bitter laugh. \"In fact, if your actions had actually resulted in the death of Paul and me, maybe Bene Gesserit histories would describe you as a hero.\"\n\n\"I am no hero, Jessica.\" To his credit, the original Yueh had given her and Paul the means to survive in the desert after the Harkonnens stormed Arrakeen. He had facilitated their escape, but was that enough for redemption? Could it possibly be?\n\nShe moved on, smelling the flowers, checking the moist soil. She had a habit of running her fingertips along the leaves, touching the undersides.\n\nYueh followed her as she walked through a small grove of dwarf citrus trees. Overhead, the segmented panes of the filtered windows showed only distant starlight and no nearby sun. \"If they hate us so much, why did the Sisters bring us back?\"\n\nHer expression was one of bitter amusement. \"Bene Gesserits have a terrible habit, Wellington: Even if they know a hook is hidden inside the juicy worm, they'll still bite. They always think they can avoid traps that get the rest of us.\"\n\n\"But you're a Bene Gesserit yourself.\"\n\n\"Not anymore . . . or not yet.\"\n\nYueh touched his own smooth, unmarked forehead. \"We're starting over, Jessica. Blank slates. Look at me. The first Yueh broke his Suk conditioning\u2014but _I_ was born without the diamond tattoo. Entirely unblemished.\"\n\n\"Maybe that means some things can be erased.\"\n\n\" _Can_ they? We gholas were raised for one purpose: to become who we once were. But are we anyone in our own right? Or are gholas simply tools, temporary tenants living in houses on borrowed time until the rightful owners return? What if we don't want those old lives? Is it right for Sheeana and the others to force them upon us? What about _us_ as we are right now?\"\n\nAbruptly the gridwork of interlocking solar panes overhead seemed to glow brighter, as if the system had absorbed a wash of outside energy. The rows of densely arranged plants inside the greenhouse chamber became more defined, as if his eyes had suddenly become much more sensitive. Overlaying the whole chamber he saw a complex mesh of thin iridescent lines, resolving and focusing.\n\nSomething was happening\u2014something Yueh had never experienced before. The lines became visible all around them, like fine netting that drifted through the air itself. They crackled with energy.\n\n\"Jessica, what is this? Do you see it?\"\n\n\"A web . . . a net.\" She caught her breath. \"It's what Duncan Idaho claims to see!\"\n\nYueh's heart lurched. _The hunters?_\n\nA loud security klaxon went off, accompanied by Duncan's voice. \"Prepare for activation of Holtzman engines!\"\n\nWhenever the no-ship folded space, unguided by a Navigator, they risked a disaster. Until now, Duncan's warnings had been unsupported by outside witnesses, though the Handlers had proved that the threat from the mysterious Enemy was real.\n\nFrom the ship's corridors, Yueh heard the shouts of people running to emergency stations. The gossamer stranglehold grew brighter and more powerful, surrounding and infiltrating the whole ship. Surely, everyone could see this!\n\nHe felt a shudder through the deck, a disorientation and a _slipping_ as the immense ship folded space. Staring through the conservatory dome, he saw star systems, swirling shapes and colors . . . as if the contents of the universe had been placed in a mixing bowl and stirred.\n\nSuddenly the _Ithaca_ cruised along elsewhere, far from the snares. Duncan's calm voice came over the intercom. \"We are safe again, for the moment.\"\n\n\"Why did we see the net now and never before?\" Jessica asked.\n\nYueh rubbed his chin, his thoughts in turmoil. \"Perhaps the Enemy is using a different sort of net\u2014a stronger one. Or maybe they are testing new ways of tracking and snaring us.\"\n\n> _We must never voice doubt. We must_ believe _utterly that we can win this struggle against our Enemy. But in my darkest times alone in my quarters, I always wonder: Is this truly faith, or is it mere foolishness?_\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> private Chapterhouse Archives\n\nWhen Murbella's small Missionaria Aggressiva council gathered again, the meeting was tense. In the past year, the Sisterhood had sent seven Sheeana surrogates to refugee camps in order to rally the fighters. The counterfeit Sheeanas had their work cut out for them, convincing fanatics to stand firm in the face of certain defeat.\n\nThe seemingly unstoppable Enemy warships proliferated like the heads of a hydra; no matter how many vessels the humans destroyed, more and more appeared. Given millennia to prepare for his final conquest, Omnius had left nothing to chance. The dots on the star charts showed one planet after another falling under the onslaught of thinking machines.\n\nMurbella sat in a hard and uncomfortable seat at the end of the table; most of the others selected furry chairdogs. At the head of the table, Bashar Janess Idaho waited at attention, ready to deliver her report.\n\n\"I have news.\"\n\n\"Good or bad?\" Murbella dreaded the answer.\n\n\"Judge for yourself.\"\n\nHer daughter looked haggard, weary, and considerably older than her years. Having undergone the Spice Agony and extensive Bene Gesserit training, Janess had the ability to slow her body's changes, not for the sake of appearances, but to keep herself strong and limber. The constant fighting required it. Even so, the unending crisis was taking its toll. Murbella noticed a scar on her daughter's left cheek and a burn mark on her arm.\n\nThe female Bashar's words were unemotional, but Murbella could feel the turmoil in her clipped voice. \"Even before the first Enemy battleships were seen in the Jhibraith system, the machines sent scout probes to disseminate plagues. The people of Jhibraith had already called for an evacuation, but once the first signs of disease appeared, the Guild turned their ships around and refused to come closer. One Heighliner had to be quarantined. Fortunately, the plague was contained within seven isolated frigates inside its hold. All passengers aboard those frigates died, but the rest were saved.\"\n\n\"What about the planet itself?\" Murbella asked.\n\n\"The plague spread rapidly across all continents. As expected. The current viral strains are far worse than anything previously encountered, more deadly than even the legendary plagues during the Butlerian Jihad.\"\n\nLaera skimmed a Ridulian crystal sheet in front of her. \"Jhibraith has a population of three hundred twenty-eight million.\"\n\n\"Not anymore,\" said Kiria.\n\nJaness locked her fingers together, as if to draw strength from her own grip. \"One of our Sheeana surrogates was on Jhibraith. As soon as the Guild quarantined the planet, the faux Sheeana rose to her calling and spoke to crowd after crowd as the plague spread. They knew they would all die. They knew the thinking machine forces were on their way. But she convinced them that if they must die, they should die as heroes.\"\n\n\"But if the Guildships had already departed, how could they fight?\" Kiria sounded skeptical. \"By throwing pebbles?\"\n\n\"Jhibraith had its own in-system frigates, cargo vessels, and transport runners, none of them equipped with Holtzman engines or no-fields. As the disease cut people down, survivors raced to create a homegrown military force that might stand against Omnius. The people had to work faster than the epidemic killed them off.\" She forced her lips into a cold, hard smile as she continued her report.\n\n\"Our false Sheeana was like a demon herself. I know for a fact that she went five days without sleeping, for the records show she appeared again and again at different cities and factories, rallying the citizenry, forcing them to crawl to their assembly stations if necessary. Nobody bothered with quarantines, since everyone was already infected. As people died in the factories, their bodies were dragged out to mass burial pits and huge bonfires. Others took their places at workstations.\n\n\"Even when the Enemy fleet surrounded the world, people did not pause. Then our Sheeana disappeared.\" Janess looked around the table, lowered her voice. \"Afterward, I learned from a coded Bene Gesserit signal that our surrogate contracted the disease, and died from it.\"\n\nMurbella was startled. \" _Died?_ How can that be? Any Reverend Mother knows how to fight off infection.\"\n\n\"That requires great concentration and significant physical resources. Our Sheeana had depleted her reserves. If she'd rested for a day or two, she might have rallied her strength and driven off the disease. But she kept going and going, using up whatever energy reserves she had. Knowing Jhibraith was doomed, that the invading machine armies would destroy her if the plague did not, Sheeana never slackened in her efforts.\"\n\nOld Accadia nodded. \"She had pushed the people into a fanatical fervor. No doubt she realized that if they saw her weakened and dying, they would lose their resolve. She was wise to remove herself from public view.\"\n\nJaness's thin smile showed true admiration. \"As soon as her symptoms began to manifest, Sheeana delivered one last grand speech, telling them she would now ascend to heaven. Then she isolated herself and died alone so that no one could see the horrific plague take its toll on her.\"\n\n\"A marvelous and brave story for the archival histories.\" Accadia pursed her withered lips. \"Her sacrifice will not be forgotten.\"\n\n\"If anyone still studies the histories after this,\" Kiria mumbled.\n\n\"And what of the subsequent fight on Jhibraith?\" Murbella asked. \"Did the people defend themselves?\"\n\n\"When the Enemy came, the people fought like ancient berserkers, to the last man and woman. Nothing could stop them. They met the Enemy fleet with ship after ship flown by grandfathers, teenagers, mothers, husbands, and even criminals released from detention centers. All fought and died bravely. Their sheer ferocity drove back the machines. Even with no defined military force, the people of Jhibraith destroyed more than a thousand Enemy vessels.\"\n\nReality forced ice into Murbella's voice. \"My enthusiasm is tempered by the knowledge that even after losing a thousand vessels, the thinking machines have countless others to throw against us.\"\n\n\"Still, if all planets fought like that, there might be a chance for humankind to survive,\" Janess pointed out. \"The species would be preserved.\"\n\nChoosing her moment to pounce, Kiria peeled crystal sheets from another set of reports, then propped an image projector in the center of the table. The chairdog shifted subtly and compliantly to accommodate her movements. \"This new report shows why we can't count on all planets. We are being attacked by a rot from within, as well as the outside fleet.\"\n\nMurbella frowned. \"Where did you get this?\"\n\n\"Sources.\" Wearing a smug expression, the former Honored Matre activated the projector. \"While we face the thinking machines headon, a more devious opponent undermines us from within.\"\n\nThe image resolved to show a mob scene. \"This is Belos IV, but such occurrences have been documented elsewhere. Sparked by helplessness in the face of the approaching Enemy fleet, brushfire wars and political struggles are starting on planet after planet. People are afraid. When their leaders don't tell them what they want to hear, they riot, overthrow their prime ministers, and prop others in their place. More often than not, they depose the new leaders as well.\"\n\n\"We know this.\" Murbella looked at Janess, who remained rigidly at attention at the front of the table. She wished her daughter would sit down. On the images, the citizens of Belos IV had risen up against their governor, who had advocated surrender to the oncoming thinking machines. \"Obviously, the people didn't want to hear such a message. Why is this relevant?\"\n\nKiria jabbed a sharp-nailed finger at the image. \"Observe!\"\n\nWhen the crowd attacked the middle-aged leader, he fought remarkably well, using skills and speed rarely demonstrated by any bureaucrat. While Murbella watched, she decided the governor must have acquired some sort of special training. His combat methods were unusual and effective, but the mob far outnumbered him. They dragged him through the streets to the balcony of the governor's palace and threw him off onto the flagstones far below. As he lay still, the howling, cheering mob backed away. The images drew in closer. The dead governor shifted and paled. His face became sunken and scarecrowish, somehow unformed. A Face Dancer!\n\n\"We always suspected the new Face Dancers had questionable loyalties. They allied themselves with Honored Matres and turned against the old Tleilaxu. We found them among the rebel whores on Gammu and Tleilax, and now it appears that the threat is even worse than we suspected. Listen to the governor's words. He advocated surrender to the thinking machines. Who are the Face Dancers really working for?\"\n\nMurbella reached the obvious conclusion and dragged her sharp gaze like a serrated knife across the other Sisters. \"The new Face Dancers are puppets of Omnius, and have infiltrated our populace. They are far superior to the old ones, able to resist almost any Bene Gesserit technique. We always wondered how the Lost Tleilaxu could have created them, when their skills were so inferior to those the old Masters demonstrated. It did not seem possible.\"\n\nLaera said coldly, \"It is possible if the thinking machines helped to create them, then sent them back among Tleilaxu returning from the Scattering.\"\n\n\"A first wave of scouts and infiltrators.\" Kiria nodded. \"How far have they spread? Could there be Face Dancers among _us_ , undetected by Truthsayers?\"\n\nAccadia scowled. \"A frightening thought, if we have no way of exposing these new Face Dancers. From what I can tell, their mimicry is perfect.\"\n\n\"Nothing is perfect,\" Murbella said. \"Even thinking machines have flaws.\"\n\nWithout humor, Kiria said, \"Oh, we can identify them easily enough. Kill them, and Face Dancers revert to their blank state.\"\n\n\"So you suggest we simply kill everyone?\"\n\n\"That's what the Enemy intends to do anyway.\"\n\nRestless, Murbella stood up. She could remain here on Chapterhouse with the other anxious Sisters, receive reports for another year, listen to summaries, and plot the advance of the thinking machines on a map, as if it were some kind of war game. Meanwhile, the Ixian engineers struggled to build weapons equivalent to the Obliterators, and the Guild shipyards worked to produce thousands of ships, all of them equipped with mathematical compilers.\n\nBut the crisis went far beyond internal politics and power struggles. She decided to go out there herself and travel among the worlds on the edge of the war zone, not as Mother Commander, but as a keen observer. She would let a council of Reverend Mothers run the everyday activities here on Chapterhouse, dealing with bureaucratic matters and doling out spice rations to the Guild in order to ensure their cooperation.\n\nWhen Murbella announced her intent, Laera cried, \"Mother Commander, that's not possible. We need you here\u2014there's so much to do!\"\n\n\"I represent more than the New Sisterhood. Since no one else will step up to the plate, I am responsible for the whole human race.\" She sighed. \"Somebody has to be.\"\n\n> _Our no-ship holds many secrets, yes, but not nearly the number we hold inside ourselves_.\n> \n> \u2014LETO II,  \n> the ghola\n\nLeto II and Thufir Hawat had never known each other in their original lifetimes. To them, that was not a disadvantage. It left them free to form a friendship without any expectations or preconceptions.\n\nNine-year-old Leto hurried ahead, down the corridor. \"Come with me, Thufir. Now that nobody's watching, I can show you a special place.\"\n\n\"Another one? Do you spend all your days exploring instead of studying?\"\n\n\"If you're going to be deputy chief of security, you need to know everything about the _Ithaca_. Maybe we'll find your saboteur down here.\" Leto turned sharply right, dropped into a small emergency lift, then paused at a dim, lower deck, where everything seemed larger and darker. He led Thufir to a sealed hatch that was posted with warnings and restrictions in half a dozen languages. Despite the locks, he opened it with barely a pause.\n\nThufir looked puzzled, even a little offended. \"How did you bypass security so easily?\"\n\n\"This ship is old, and systems break down all the time. Nobody even knows this one failed.\" He ducked into the low passageway.\n\nThe tunnel on the other side was a whistling, cool air channel. Up ahead the roaring grew louder, and the wind became powerful. Thufir sniffed. \"Where does it go?\"\n\n\"To an air-exchange filtration system.\" The passages were smooth and curving\u2014like worm tunnels. A shiver brushed across Leto's skin, perhaps from a memory of when he had been joined with numerous sandtrout, from when he was the God Emperor of Dune, the Tyrant. . . .\n\nThe two reached the central recyclers where large fans drove the air through thick curtains of filter mats, scrubbing out particulates and purifying the atmosphere. Breezes tugged at the boys' hair. Ahead, sheets of filtration material blocked further passage. The lungs of the ship, replenishing and redistributing oxygen.\n\nRecently, Thufir had begun to mark his lips with a cranberry-red stain. As the pair stood in the bowels of the ship, listening to the roaring wind, Leto finally asked, \"Why do you do that to your mouth?\"\n\nSelf-consciously, the fourteen-year-old rubbed his lips. \"My original used the sapho drug, which made stains like these. The Bashar wants me to live the part. He says he's preparing to awaken my memories.\" Thufir didn't sound entirely pleased with the situation. \"Sheeana has been talking about forcing me to remember. She has some special technique to trigger a ghola's awakening.\"\n\n\"Aren't you excited at the prospect? Thufir Hawat was a great man.\"\n\nThe other boy remained preoccupied and troubled. \"It's not that, Leto. I really don't want my memories back, but Sheeana and the Bashar have their minds made up.\"\n\n\"That's why you were created.\" Leto was baffled. \"Why wouldn't you want your past life? The Master of Assassins would not be afraid of the ordeal.\"\n\n\"I'm not afraid. I'd just rather be the person I choose to become, and not emerge fully formed. I don't feel I've earned it.\"\n\n\"Trust me, they'll make you earn it, once you become the real Thufir again.\"\n\n\"I _am_ the real Thufir! Or do you doubt that, too?\"\n\nThinking of the restless worm that crouched inside him, aware of all the atrocious things he would soon remember, Leto understood completely.\n\n> _By following the same beliefs and making the same decisions, one wears Life's path into a circular rut, going nowhere, accomplishing nothing, making no progress. With God's help, though, we can turn a sharp corner in the circle and achieve enlightenment_.\n> \n> \u2014The Cant of the Shariat\n\nAt last Waff was ready to release his new worms, and Buzzell was a convenient ocean planet on Edrik's standard trade route. A perfect test bed.\n\nThe giant vessel carried merchants who traded in soostones. Earlier, when Honored Matres had conquered Buzzell and killed most of the exiled Reverend Mothers, the whores had taken the soostone wealth for themselves. Since then, few of the aquatic gems had been traded on the galactic market, which made their value skyrocket. Now that the New Sisterhood had recaptured Buzzell, soostone production was up again. The witches ran tightly regimented operations there and kept smugglers at bay, thus maintaining stable but high prices for the stones. With mercenary armies to protect them, CHOAM merchants began to sell large quantities of the gems, reaping profits before a glut drove prices down again. A temporary market fluctuation.\n\nThough pretty and desirable, soostones were not _necessary_. Melange, on the other hand, was vital\u2014as the Navigators well knew. Waff knew that his experiments would eventually produce far more wealth than these undersea baubles could ever represent. Soon, if his expectations were met, Buzzell would be home to something far more interesting than baubles. . . .\n\nThe Heighliner appeared above the liquid sapphire world, where tiny islands dotted the expansive ocean. Buzzell's oceans were deep and fertile, a large zone where the genetically altered worms would thrive, provided they survived their initial baptism.\n\nThe Tleilaxu Master paced the cold metal floor of his laboratory chamber. Soon, Edrik would inform him that the commercial lighters and cargo transports had disembarked for the island outposts. Once they were safely gone, Waff could begin his real work on Buzzell without being observed.\n\nInside the lab, the smell of salt, iodine, and cinnamon had replaced harsh chemical odors. Waff's test tanks were full of murky green water, rich with algae and plankton. Once turned loose in the oceans, the modified worms would have to find their own sources of nourishment, but Waff was sure they could adapt. God would make it all possible.\n\nSerpentine forms swam about in the tanks looking like ringed eels. Their ridges were an iridescent blue-green, showing a soft pink membrane between segments, a surrogate set of gills that absorbed oxygen from the water. Their mouths were round like those of lampreys. Though they had no eyes, the new seaworms could navigate using water vibrations in much the way that Rakian worms had been attracted by tremors in the dunes. Using carefully mapped models from sandtrout chromosomes, Waff knew that these creatures had the same internal metabolic reactions as a traditional sandworm.\n\nTherefore, they should still produce spice, but Waff didn't know what kind of spice, or how it would be harvested. He stepped back, interlocking his grayish fingers. That wasn't his problem or concern. He had done as Edrik commanded. He only wanted the worms back.\n\nIt had taken more than a year out of his accelerated lifetime, but if Waff succeeded in resurrecting God's messengers, his destiny would be complete. Even if the little man never received another ghola lifetime, he would have earned his place beside God in the highest levels of Heaven.\n\nUnder proper conditions, sandtrout specimens reproduced swiftly. From them, he had adapted nearly a hundred seaworms, most of which he would deposit in the oceans of Buzzell. For a new species to survive, especially in an unfamiliar environment, the creatures faced quite a challenge, and Waff fully expected that many of his test specimens would die. Maybe most of them. But he was also convinced that some would live\u2014enough to establish a foothold.\n\nWaff stood on his tiptoes, pressing his face to the tank. \"If you are in there, Prophet, I will soon give you a whole new domain.\"\n\nFive Guild assistants entered the lab without knocking. When Waff turned abruptly, the seaworms sensed his movement. With a thump, fleshy heads struck the reinforced tank walls. Startled again, Waff turned the other direction.\n\n\"Passengers have disembarked for Buzzell,\" said one of the grayclothed men. \"Navigator Edrik has commanded us to follow your instructions.\"\n\nThe five all had oddly distorted heads, swollen brows, and asymmetric facial features. Any Tleilaxu Master could have repaired genetic flaws so that their descendants would be more physically attractive. But that would serve no purpose, and Waff had no interest in cosmetics.\n\nHe gestured to the tanks as the Guildsmen sealed them for transport. \"Exercise extreme care. Those creatures are worth more than all your lives.\"\n\nThe reticent assistants installed handles on the crowded tanks and began lugging them along the curving halls of the Heighliner. Knowing he had only four hours to complete his task before the passenger shuttles returned, Waff urged them to hurry.\n\nBecause of the schism in the Guild between Navigators and Administrators, some people might not wish him to create this new avenue for spice production. The Ixians, the New Sisterhood, even the bureaucratic faction of the Guild might all work together, or separately, to assassinate him. Waff didn't know how or why these particular five Guildsmen had been chosen to assist him. If he expressed any misgivings about them, Waff knew that the Navigator would not hesitate to have all five killed, just to keep his Tleilaxu researcher happy. As the troupe walked to a small transport craft, Waff decided that was exactly what he would do. Get rid of these men, these witnesses. Afterward.\n\nThe sample tanks were loaded aboard the small transport. Waff did not usually leave the safe confines of the Heighliner, but he insisted on accompanying the crew down to the open sea. It was his experiment, and he wanted to be there in person to make sure the worms were released properly. He didn't trust the five Guildsmen to be sufficiently competent or attentive.\n\nThen his suspicions ran deeper. What was to stop these men from flying off in the ship, revealing\u2014or selling!\u2014the seaworms to one of the opposing factions? Could they be perfectly loyal to Edrik? Waff saw danger everywhere.\n\nAs the transport dropped out of the cargo bay, Waff wished belatedly that he had demanded additional bodyguards, or at least a sufficiently powerful hand-weapon for himself. Whom could he really trust?\n\nUsing technological apparatus connected to their throats, the silent Guildsmen communicated with each other electronically, transmitting brain signals without voicing words. He knew they could speak aloud\u2014why would they be so secretive? Perhaps they were plotting against him. Waff looked at the immense Heighliner far overhead and wished fervently for this to be over.\n\nThe small transport descended into the cloudy skies, fighting choppy air currents. Waff felt ill. Finally, they broke through the moisture layers, and the oceans below sprawled out to every horizon. On charts displayed across the cockpit screens, Waff searched for a temperate zone where he could deposit the test worms, a place where the seas were rich with plankton and fish. It would give the creatures the greatest chance for survival.\n\nHe indicated a line of rocks not far from the Sisterhood's main island base and the center of their soostone-harvesting operations. \" _There_. Safe, and close enough to monitor the worms.\" He smiled, already imagining the first panicked reports of eyewitnesses. \"Any rumors or wild stories should be interesting.\"\n\nThe Guildsmen nodded, all business. The transport ship flew low over the waves and hovered above the gentle swells. The lower cargo hatch opened, and Waff went down to observe the emptying of the tanks. He smelled the raw salt air, the tang of floating kelp, the wet breezes that whipped across the sea. A squall was about to begin.\n\nUsing the handles, two of the silent Guildsmen brought the first tank to the opening, released the plaz cover sheet, and spilled the water and writhing seaworms into the freedom of the waiting waves.\n\nThe serpentine creatures burst out like frantic snakes. Once they plunged into the green water, they streaked away. Waff watched their ridged bodies undulating, then diving and disappearing. They seemed joyous in their newfound freedom, glad to have a large world without plaz boundaries.\n\nBrusquely, he gestured to the Guildsmen, telling them to release the rest of the worms, emptying all of the aquariums. Waff had kept one crowded tank of specimens aboard the Heighliner, and he could always create more.\n\nAs he stood by the open hatch, he suddenly shuddered, realizing his vulnerability. Now that he had turned the worms loose, would Edrik even require his services anymore? The Tleilaxu man feared the silent assistants might shove him overboard and leave him floating kilometers from the nearest speck of land. Warily, he backed deeper into the cargo hold and gripped a ridged wall strut.\n\nBut the Guildsmen made no move against him. They performed their work exactly as he told them to, and precisely as the Navigator had specified. Perhaps Waff was fearful only because he intended to have these men killed. He naturally suspected they would think the same about him.\n\nWaff anticipated the seaworms would thrive here. The environment was conducive to their growth and reproduction. The worms would mark their territory, and when they grew large enough they would become leviathans of the deep. A fitting form for the Prophet.\n\nThe transport's cargo doors sealed again with a low hiss, and the Guild pilot flew away. Waff and his party would arrive back at the Heighliner well before the merchant vessels returned with their loads of soostones. No one would be the wiser.\n\nThe Tleilaxu Master glanced through the cockpit's plaz port to see the waves receding. He saw no sign of the seaworms, but he knew they were down there somewhere.\n\nHe let out a contented sigh, confident that his Prophet would come back.\n\n> _It is the principle of the ticking time bomb, a strategy of aggression that has long been part of human violence. Thus we have inserted our \"time bombs\" into the cells of gholas, activating specific behaviors at precise moments of our choosing_.\n> \n> \u2014from _The Secret Manual of Tleilaxu Masters_\n\nThe no-ship had its own time, its own cycles. Most of the people were asleep, except for off-shift watches and maintenance crews. The decks were quiet, the glowpanels dimmed. In the shadowy chamber that held the axlotl tanks, the Rabbi paced back and forth, murmuring Talmudic prayers.\n\nOn a surveillance screen Sheeana observed the old man carefully, always alert to prevent any new incident of sabotage. When the saboteur had killed the three gholas and axlotl tanks, he or she had shut down the security imagers, but Bashar Teg had made sure that was no longer possible. Everything was under observation. As a former Suk doctor, the Rabbi had access to the medical center; he often spent time with what remained of the woman he'd known as Rebecca.\n\nThough the old man had answered all questions the Truthsayers asked him, he still hadn't earned her trust. Despite her best efforts, the saboteur and murderer remained at large. And the recent appearance of the glowing net had brought the Enemy too close, reminding the passengers of the very real threat. Everyone onboard had seen it. Their danger remained unabated.\n\nThree relatively new axlotl tanks rested on their platforms; volunteers had come forward to answer Sheeana's call, exactly as she had expected. All three new tanks were currently producing liquid-form melange that dripped into small collecting vials, but she had begun preparations to implant one of the wombs with cells from Scytale's nullentropy tube. A new embryo to bring back yet another figure from the past. She refused to let the sabotage deflect her ghola project.\n\nThe Rabbi stood before the new tanks, his tense body exhibiting clear markers of loathing and disgust. He spoke to the fleshy mound. \"I hate you. Unnatural, ungodly.\"\n\nAfter watching the old man carefully, Sheeana left the monitor screens and entered the medical center, gliding up behind him. \"Is it honorable to hate the helpless, Rabbi? Those women are no longer aware, no longer human. Why despise them?\"\n\nHe whirled, lights reflecting on his polished spectacles. \"Stop spying on me. I wish time alone to pray for Rebecca's soul.\" Rebecca had been his personal favorite, willing to pit her intellect against his; the old man had never forgiven her for volunteering to become a tank.\n\n\"Even you need to be watched, Rabbi.\"\n\nAnger flushed his leathery skin. \"You and your witches should have heeded the warnings and ceased your grotesque experiments. If only the Honored Matres had managed to get rid of Scytale when they destroyed all Tleilaxu worlds, then his hateful knowledge of tanks and gholas would have been lost.\"\n\n\"The Honored Matres also hunted down your people, Rabbi. You and the Tleilaxu share the same enemy.\"\n\n\"But it is not the same at all. We have been unfairly persecuted throughout history, while the Tleilaxu merely received their just desserts. Their own Face Dancers turned on them, from what I understand.\" He took a step away from the fleshy mound, from the chemical and biological odors that wafted from the tanks. \"I can hardly recall what Rebecca once looked like, before she became this _thing_.\"\n\nSheeana searched for the memories, and asked the voices within to assist her. This time they did, and she found what she wanted, like accessing old archival images. The woman had looked elegant in her brown robe and braided hair. She'd worn contact lenses to conceal the blue eyes of her spice addiction. . . .\n\nWith a bitter expression, the Rabbi placed a hand on Rebecca's exposed flesh. A tear ran down his cheek. He muttered the same thing each time he visited her. A litany for him. \"You witches did this to her, made her into a monster.\"\n\n\"She's not a monster, not even a martyr.\" She tapped her own forehead. \"Rebecca's thoughts and memories are in here and inside many other Sisters, Shared with us all. Rebecca did what was necessary, and so will we.\"\n\n\"By making more gholas? Will it never end?\"\n\n\"You are concerned about a pebble in your shoe, while we seek to avoid the rockslide. Sooner or later, we will no longer be able to run from the Enemy. We'll need the ingenuity and special talents of these gholas, in particular those capable of becoming another Kwisatz Haderach. But we must handle the genetic material carefully, nurturing and developing it in the proper order, at the proper pace.\" She strolled to one of the new tanks, a fresh young woman whose form had not yet deteriorated into unrecognizability.\n\nAs she stood there, a troubling thought refused to leave her mind, no matter how much she tried to push it away. An absurd line of reasoning, but all day long it had been percolating. _What if my own abilities could be equivalent to those of a Kwisatz Haderach? I already have a natural ability to control the great worms. I have the Atreides genes, and centuries of the Sisterhood's perfected knowledge to draw upon. Would I dare?_\n\nShe felt voices surfacing from within; one rose above the others. The ancient Reverend Mother Gaius Helen Mohiam, repeating something that she once said so long ago to a young Paul Atreides: \"Yet, there's a place where no Truthsayer can see. We are repelled by it, terrorized. It is said a man will come one day and find in the gift of the drug his inward eye. He will look where we cannot\u2014into both feminine and masculine pasts . . . the one who can be many places at once . . .\" The old crone's voice drifted away, without giving Sheeana any advice, one way or the other.\n\nWith a sneer, the Rabbi interrupted her thoughts. \"And you trust that old Tleilaxu to help you, when he's desperate to achieve his own goals before dying? Scytale hid those cells for so many years. How many of them contain dangerous secrets? You've already discovered Face Dancer cells among the samples. How many of your ghola abominations are traps laid by the Tleilaxu?\"\n\nShe gazed at him dispassionately, knowing that no argument would ever sway him. The Rabbi made the sign of the evil eye, and fled.\n\nDUNCAN ENCOUNTERED SHEEANA in an empty corridor, in the dimness of the artificial night. The no-ship's recyclers and life-support systems kept the air comfortably cool, but upon seeing her alone like this Duncan felt a flush of heat.\n\nSheeana's large eyes fixed on him like a weapon's targeting system. Feeling a tingle like static electricity on his skin, he cursed his body for being so easily tempted. Even now, three years after Sheeana had broken the debilitating chains of Murbella's love, the two of them still found themselves drawn irresistibly into unexpected bouts of sex as frenetic as any he and Murbella had ever shared.\n\nDuncan preferred to manage the circumstances of their casual meetings, always trying to make certain that others were present, that he had safe guardrails to prevent him from falling off a dangerous cliff. He did not like to be out of control: That had already happened too many times.\n\nHe and Sheeana had surrendered to each other like terrified people huddling in a bombed-out battle zone. She had cauterized him to cure his debility and stolen him from Murbella, yet he felt like a casualty of war.\n\nNow as he saw Sheeana's expression flicker, Duncan thought she felt the same sense of vertigo and disorientation. She tried to sound reserved and rational. \"It is better if we don't do this. We have too many concerns, too many risks. Another regenerator system has just failed. The saboteur\u2014\"\n\n\"You're right. We should not.\" His voice was hoarse, but they had already started down a path with ever-increasing consequences. Duncan took a hesitant step forward. The muted corridor lights reflected off the no-ship's metal walls. \"We shouldn't do this,\" he said again.\n\nDesire swept like a wave over both of them. As a Mentat he could observe and assess, concluding that what they were doing was simply a reaffirmation of humanity. When they touched fingertips, lips, and skin, both of them were lost. . . .\n\nLater, they rested on tangled sheets in Sheeana's quarters. The air carried a moist muskiness. Duncan lay sated with his fingers laced in his wiry dark hair. He was confused and disappointed in himself. \"You've taken away too much of my control.\"\n\nSheeana raised her eyebrows in the dim light, showing amusement. Her breath was warm and close to his ear. \"Oh? And Murbella did not?\" When Duncan turned away and did not answer, she chuckled. \"You're feeling guilty! You think you've betrayed her somehow. But how many female imprinters did you train back on Chapterhouse?\"\n\nHe answered the question in his own way. \"Murbella and I were trapped together, and no part of our relationship was voluntary. We had a mutual addiction, two people brought to a stalemate. That isn't love or tenderness. For Murbella\u2014for all of you witches\u2014our love-making was supposed to be 'just business.' But I still had feelings for her, dammit! It isn't a matter of whether or not I should.\n\n\"But you\u2014you were like a violent detoxification of my system. The Agony served the same purpose for Murbella, breaking her bond to me.\" He reached out and cupped Sheeana's chin. \"This cannot happen again.\"\n\nNow she looked even more amused. \"I agree that it should not . . . but it will anyway.\"\n\n\"You're a loaded weapon, a full Bene Gesserit. Every time we make love, you could easily let yourself get pregnant. Isn't that what the Sisterhood would demand? You could carry my child whenever you allow yourself to do so.\"\n\n\"True. But I haven't. We are far from Chapterhouse, and I make my own decisions now.\" Sheeana pulled him back to her.\n\n> _Scientists see sandworms as specimens, while the Fremen see them as their god. But the worms devour anyone who tries to gather information. How am I supposed to work under such conditions?_\n> \n> \u2014IMPERIAL PLANETOLOGIST PARDOT KYNES,  \n> ancient records\n\nSheeana stood in the high observation gallery where she and Garimi had once gone to discuss the future of their journey. The kilometer-long great hold was large enough to offer the illusion of freedom, though much too small for a brood of sandworms. The seven creatures were growing but remained stunted, waiting for the promised arid land. They had been waiting a long time, perhaps too long.\n\nMore than two decades ago, Sheeana had brought the small worms aboard the no-ship, stealing them from the growing desert band on Chapterhouse. She had always intended to transplant them to another world, far from the Honored Matres and safe from the Enemy. For years, the worms had zigzagged endlessly in the sand-filled confines of the hold, as lost as everyone else on the _Ithaca_. . . .\n\nShe wondered if the no-ship would ever find a planet where they could stop, where the Sisters could establish a new and orthodox Chapterhouse, rather than the mongrel organization that made concessions to the ways of the Honored Matres. If the ship simply fled for generations and generations, it would be impossible to find a perfect world for the sandworms, for Garimi and her conservative Bene Gesserits, for the Rabbi and his Jews.\n\nShe recalled searching Other Memory for advice the evening before. For a while, there had been no response. Then Serena Butler, the ancient leader of the Jihad, came to her just as Sheeana drifted off to sleep in her quarters. Long-dead Serena told of her experiences being lost and overwhelmed by an endless war, forced to guide vast populations when she herself didn't know where to go.\n\n\"But you found your way, Serena. You did what you had to do. You did what humanity needed.\"\n\n_And so will you, Sheeana_.\n\nNow, seeing the sandworms ripple the sand far below, Sheeana could sense their feelings in an indefinable way, and they could sense hers. Did they dream of endless, dry dunes in which to stake their territory? The largest of the worms, nearly forty meters long with a maw large enough to swallow three people abreast, was clearly dominant. Sheeana had given that one a name: Monarch.\n\nThe seven worms pointed their eyeless faces toward her, displaying crystalline teeth. The smaller ones burrowed into the shallow sand, leaving only Monarch, who seemed to be summoning Sheeana. She stared at the dominant worm, trying to understand what it wanted. The connection between them began to burn inside her, calling her.\n\nSheeana descended to the sand-filled cargo hold. Stepping out onto churned dunes, she strode directly toward the worm, unafraid. She had faced the creatures many times before and had nothing to fear from this one.\n\nMonarch towered over her. Putting her hands on her hips, Sheeana looked up, and waited. In the heady days on Rakis, she had learned to dance on the sands and control the behemoths, but she had always known she could do more. When she was ready.\n\nThe worm seemed to be playing upon her need for understanding. She was the girl who could communicate with the beasts, who could control and understand them. And now, in order to see her own future, she had to go farther. Literally and metaphorically. It was what Monarch wanted. Dangerous and frightening, the creature exhaled a stink of inner furnaces and pure melange.\n\n\"So, now what do we do, you and I? Are you Shaitan, or just an impostor?\"\n\nThe restless worm seemed to know exactly what she had in mind. Instead of rolling its body toward her so she could climb its rings to ride, Monarch faced her with its round maw open. Each milky-white tooth in a mouth the size of a cave opening was long enough to be used as a crysknife. Sheeana did not tremble.\n\nThe sandworm laid its head on the soft dunes directly in front of her. Tempting her to a symbolic journey, like Jonah and the whale? Sheeana wrestled with her fears, but knew what she must do\u2014not as a charlatan's performance, for she doubted anyone was watching her, but because it was necessary for her own understanding.\n\nMonarch lay waiting for her, mouth agape. The worm itself became a secret doorway, luring her like a dangerous lover. Sheeana stepped past the portcullis of crysknife teeth and knelt inside the gullet, inhaling the rank cinnamon odor. Dizzy and nauseated, she could barely breathe. The sandworm did not move. Willingly, she worked her way deeper inside, offering herself, though convinced that her sacrifice would not be accepted. It wasn't what the worm wanted from her.\n\nWithout looking back, she crawled farther down the throat into dry, dark warmth. Monarch did not twitch. Sheeana kept going, felt her breathing slow. Deeper and deeper she went, and soon guessed that she had gone at least half the length of the prone worm. Without the friction heat sparked by roaming endless deserts, the worm's gullet was no longer a furnace. Her eyes became accustomed to what she realized was not total darkness, but was instead an eerie illumination that seemed more the product of another sense in her own mind than traditional eyesight. She could dimly see the rough, membranous surface around her, and as she proceeded, the undigested odor of melange precursors grew stronger, more concentrated.\n\nFinally she reached a fleshy chamber that might have been Monarch's stomach, but without digestive acids. How did the captive sandworms survive? The odor of spice was more intense here than she had ever experienced\u2014so much so that an ordinary person would have suffocated.\n\n_But I am no ordinary person_.\n\nSheeana lay there, absorbing the warmth, letting the intense melange seep into every pore in her body, feeling Monarch's dim consciousness merge into hers. She inhaled deeply and felt a great, cosmic sense of calm, like being in the womb of the Great Mother of the Universe.\n\nUnexpectedly, with the unusual visitor deep in its gullet, the sandworm dove into the artificial desert and began to move through it; taking her on a strange journey. As if connected directly with Monarch's nervous system, Sheeana could see _through_ the eyeless worm to its companions beneath the sand. Working together, the seven sandworms were forming small veins of spice in the cargo hold.\n\n_Preparing_.\n\nSheeana lost track of time and thought again of Leto II, whose pearl of awareness was now inside this beast and all of the others in the hold. She wondered where she fit into this paranormal realm. As the God Emperor's queen? As the female part of the godhead? Or as something else entirely, an entity she could not begin to imagine?\n\nThe worms all carried secrets, and Sheeana understood how much the ghola children were like that as well. They each held a treasure greater than spice within their cells: their past memories and lives. Paul and Chani, Jessica, Yueh, Leto II. Even Thufir Hawat, Stilgar, Liet-Kynes . . . and now baby Alia. Each had a crucial role to play, but only if they could remember who they were.\n\nShe saw each image, but not in her own imagination. The sandworms knew what those lost figures contained. An urgency rushed toward her like a desert wind. Time, and with it their chances for survival, was slipping away too quickly. She envisioned a succession of the possible gholas, all primed like weapons, but blurred as to what each could do.\n\nShe could not wait for the Enemy. She had to act now.\n\nThe worm resurfaced, and after moving along the sands, it jolted to a stop. Inside, Sheeana caught her balance. Then, by constricting its membranous interior, the creature gently pushed her back out. She crawled from the mouth and tumbled onto the sands.\n\nDust and grit clung to the film coating her body. Monarch nudged her, like a mother bird urging a chick to go out on its own. Caught up in disorienting visions, she struggled and dropped to her knees on the dry sand. The faces of the ghola children wavered around her, dissolving into bright lights. _Awaken!_\n\nShe lay gasping for breath, her body and clothing drenched in spice essence. Beside Sheeana, the big worm turned, burrowed back into the shallow sand, and disappeared from view.\n\nReeking and reeling, Sheeana made her way back toward the cargohold doors, but she kept losing her footing and falling. She had to reach the ghola children . . . The worm had given her an important message, something that seeped into her consciousness like a wordless form of Other Memory. In moments, she was left with an overwhelming certainty of what she had to do.\n\n> _You say that we must learn from the past. But I\u2014I fear the past, for I have been there, and I have no desire to return_.\n> \n> \u2014DR. WELLINGTON YUEH,  \n> the ghola\n\nAfter being scrubbed and showered to rid her of a spice reek so strong that even the Sisters who assisted her covered their mouths and noses, Sheeana slept for two days in deep and disturbing dreams.\n\nWhen she finally emerged and found Duncan Idaho and Miles Teg on the navigation bridge, she made her announcement. \"The gholas are all old enough. Even Leto II is the same age as when I restored the Bashar.\" The smell of melange was heavy on Sheeana's breath. \"It's time to awaken them all.\"\n\nDuncan turned his back on the observation window where he stood. \"Triggering the process isn't like activating a subroutine or working through a bout of temporary amnesia. You can't just issue a memo and order it done.\"\n\n\"The ghola children have always known that we would demand this of them,\" she said. \"Without their past memories, without their genius, they are of no more value to us than any other children.\"\n\nThe Bashar nodded slowly. \"To recall a ghola's past life is an experience that destroys and recreates the psyche. There are numerous proven methods, some more painful than others, but none are easy. You can't awaken the children all at once. Each critical event must be tailored to the individual. A horrible, mind-shattering crisis.\" Teg's face showed echoes of pain. \"You thought you were using a humane awakening method with me, Sheeana . . . when I was only a ten-year-old child.\"\n\nThough Duncan, too, seemed uneasy at the prospect, he stepped down from the observation window and walked toward Sheeana. \"She's right, Miles. We created those gholas for a purpose, and right now they're all like unloaded guns. We need to load our gholas\u2014our unique weapons. The Enemy's net is stronger now, and it nearly snared us again. We all saw it. Next time, we may not be able to slip away.\"\n\n\"We've waited long enough.\" Sheeana's voice was hard, brooking no argument.\n\n\"Some gholas may be more challenging than others.\" Teg's eyes narrowed. \"You may lose some to madness. Are you prepared for that?\"\n\n\"I have gone through the Spice Agony, as have all Reverend Mothers on this ship. We survived the unbearable pain.\"\n\n\"I have the memories of my old life,\" Teg said. \"Of wars and atrocities, and enduring unbearable torture. Somehow the bad details are much more vivid than the pleasant ones, but nothing is worse than the awakening.\"\n\nSheeana waved her hand. \"Throughout history, men and women have had a monopoly on their own kinds of pain, each thinking theirs is the worst.\" She smiled grimly. \"We will start with the least valuable ghola, of course. In case something goes wrong.\"\n\nWELLINGTON YUEH WAS summoned to stand before the Bene Gesserits in one of the no-ship's council chambers. The gangly teenager had a pointed chin and pinched lips. Already buried within his face were hints of the familiar chiseled features and broad forehead\u2014the despised visage that had become synonymous with the word _traitor_ for thousands of years in galactic reference works.\n\nThe young man was nervous, fidgeting. Sheeana drew herself to her full height and stepped closer. He flinched at her intimidating presence, but somehow found the courage to stand where he was. \"You summoned me, Reverend Mother. How may I assist?\"\n\n\"By awakening your memories. Tomorrow you will be the first of our subjects to undergo the trigger.\"\n\nYueh's yellowish face paled. \"But I'm not ready!\"\n\n\"That is why we have given you a full day to prepare.\" Proctor Superior Garimi's tongue was sharp, as usual.\n\nAlthough Garimi had never embraced the project, she now wanted to see its culmination. Sheeana knew what she was thinking: If the awakening process failed, Garimi intended to prevent any further gholas from being grown; if the awakening succeeded, she would insist that the program had fulfilled its goal and could be discontinued. She knew that Sheeana, still intrigued by all those cells in the Tleilaxu's nullentropy capsule, was planning further ghola experiments.\n\nYueh's legs were locked. He seemed close to fainting and grasped a nearby chair to steady himself. \"Sisters, I don't want to have my memories back. I am not the man you think you resurrected, but a new person\u2014my own person. The old Wellington Yueh was tormented in so many ways. Even though he was in part _me_ , how can I forgive him for what he did?\"\n\nGarimi made a dismissive gesture. \"Nevertheless, we brought you back for one purpose only. Don't expect sympathy from us. You have a task to perform.\"\n\nAfter the proctors took the distraught young man away, Sheeana looked at Garimi and two other senior Sisters\u2014Calissa and Elyen\u2014who had observed the discussion. \"I will use the sexual method on him, the same one that worked on the ghola of the Bashar. It is the best technique we know.\"\n\nElyen said, \"Your sexual imprinting unlocked the Bashar's memories only because it precipitated a crisis in him. Teg's mother had armed him against sexual imprinting. It wasn't your technique that stirred up his past, but his sheer resistance to it.\"\n\n\"Indeed. So for each of our gholas we'll tailor an individualized agony that leverages their own fears and weaknesses.\"\n\n\"How can sex break Yueh the way it broke the Bashar?\" Garimi asked.\n\n\"Not the sex itself, but Yueh's resistance to it. He's terrified of remembering his past. If he believes we know how to unlock his memories, he'll fight us with everything he has. As he fights, I will apply my most potent procedures, and he'll spiral over the brink into complete madness.\"\n\nGarimi shrugged. \"If it doesn't work, we have other ways.\"\n\nTHE ROOM WAS dim and the shadows cloying, which made Yueh's terror more palpable. The chamber was devoid of furniture except for a padded mat on the floor, like those that the ghola children used during physical training sessions.\n\nThe witches had not explained what to expect. The young man knew from his studies that the process of regaining one's past was painful. He was not a strong man, nor was he particularly brave. Even so, the prospect of pain did not petrify him nearly so much as the dread of remembering.\n\nThe door slid open with a gentle hiss of lubricated metal gliding in its tracks. From the corridor, blinding light flowed in, much brighter than the glowpanels in his cell. It silhouetted a woman's figure\u2014Sheeana? He turned to face her and could see only her outline, the sensual curves of her body no longer masked by flowing robes. When the door sealed behind her, his eyes adjusted to the more comfortable illumination.\n\nWhen he saw that Sheeana was completely unclothed, his fear increased. \"What is this?\" His voice, torn by nervousness, came out as a squeak.\n\nShe stepped closer. \"You will disrobe now.\"\n\nBarely a teenager, Yueh swallowed. \"Not until you explain what is going to happen to me.\"\n\nShe used the hurricane force of Bene Gesserit Voice. \" _You will disrobe now!_ \"\n\nIn a spasmodic reaction, his arms and legs jerking, he tore off his clothes. Sheeana inspected him, running her eyes up and down his thin, naked body like a hawk assessing its prey. Yueh got the impression that she found him inadequate.\n\n\"Don't hurt me,\" he pleaded, and hated himself for saying it.\n\n\"Of course it will hurt, but the pain won't be anything _I_ inflict upon you.\" She touched his shoulder. He felt an almost electric shock, but he was transfixed, unable to move. \"Your own memories will do that.\"\n\n\"I don't want them back. I'll fight you.\"\n\n\"Fight all you wish. It will do you no good. We know how to awaken you.\"\n\nYueh closed his eyes and gritted his teeth. He tried to turn away, but she grasped his arms to hold him still, then released her grip and began touching him. The delicate strokings felt like the line of heat left by a lighted match down his arm and across his chest. \"Your memories are stored within your cells. In order to awaken them, I must awaken your body.\" She stroked him, and he shuddered, unable to draw away. \"I shall teach your nerve endings to do things they've forgotten how to do.\" Another jolt, and he gasped.\n\nShe touched him again, and his knees buckled, exactly as she wanted. Sheeana pushed him toward the mat on the floor. \"I need to wrench you into the full awareness of every chromosome in every cell.\"\n\n\"No.\" The word sounded incredibly weak to him.\n\nAs she pressed herself against him, letting warm skin ignite his perspiration, Yueh tunneled backward into himself, trying to flee. From all he had learned of his past, he found one thing with which to anchor his bravery. _Wanna_! His beloved Bene Gesserit wife, the weak link in his long chain of betrayals, and the strongest link in his original lifetime.\n\nThe evil Harkonnens had known that Wanna would be the key to breaking his Suk conditioning, and it had only worked\u2014could only have worked\u2014because Yueh loved her with all his heart. Bene Gesserits were not supposed to succumb to love, but he knew that she must have reciprocated.\n\nHe thought of her pictures in the archives, of all he had learned about her in his researches. \"Oh, Wanna.\" He yearned for her in his mind, tried to latch onto her as a lifeline.\n\nSheeana stroked his waist, trailed her fingers lower and climbed on top of him. Yueh's muscles were completely out of his control. He couldn't move. Her lips vibrated against the skin of his shoulder, his neck. Sheeana was a skilled sexual imprinter. Her body was a weapon, and he was the target.\n\nA flood of sensations nearly drove the archival image of Wanna from his mind, but Yueh fought against what Sheeana was making him feel. Instead, he focused on what he might have done in Wanna's loving embrace. _Wanna_.\n\nAs the rhythm of their lovemaking increased, real memories intruded on the information he had obtained through research. Yueh recalled those terrible moments after his wife had been seized by the Harkonnens, and saw images of the loathsome fat Baron, his thuggish nephew Rabban, the viper Feyd-Rautha, and the Mentat Piter de Vries, who had a laugh that sounded like vinegar.\n\nWeak, helpless, and infuriated, he had been forced to watch them torture Wanna inside an isolated chamber. She was a Bene Gesserit; she could block her pain, could deflect her body's responses. But Yueh could not so easily shunt such things aside, no matter how hard he tried.\n\nIn his nightmare memory the Baron laughed, a rumbling basso sound. \"See the little chamber she's in, Doctor? A toy with some very interesting possibilities.\" As the men watched the groggy and disoriented Wanna, she stood on weak knees, but upside-down within the booth. \"We can convert gravity into a thing that depends entirely on perspective.\"\n\nRabban chuckled, a harsh release of noise. He operated artificial gravity controls in the small room, and suddenly Wanna fell with a thud to the floor. She managed to tuck her head and shoulders just enough to avoid breaking her neck. With the speed and fluidity of a serpent, Piter de Vries scurried forward carrying a pain amplifier. At the last moment, Rabban snatched it out of the Twisted Mentat's hands and applied it to Wanna's throat himself. She writhed with a jagged spasm of agony.\n\n\"Stop! Stop, I beg you!\" Yueh cried.\n\n\"Oh, Doctor, Doctor\u2014you know it can't possibly be that easy . . .\" In the vision, the Baron folded his pudgy arms across his chest.\n\nRabban twisted the gravity controls again, and Wanna was thrown like a limp doll from wall to wall, smashing into the sides of the chamber. \"When one is too lovely, something must be done to correct that condition.\"\n\n_My beautiful Wanna!_\n\nThe memories were so vivid now, far more detailed than anything he had read in the Archives section. No mere documentation could have provided such precise clarity. . . . .\n\nIn a different, newly unlocked compartment of his brain, he lived another memory. He was artificially paralyzed, forced to watch during one of the Baron's drunken parties while Piter played a sparking pain amplifier over Wanna's suspended body. Each flash provoked a twitching response of agony from her. The other guests laughed at her pain and at his helpless misery.\n\nWhen he was freed from his paralysis, Yueh trembled, drooled, and struggled. The Baron stood over him, a huge grin on his bloated face. He handed Yueh a projectile pistol. \"As a Suk doctor, you should do everything possible to stop a patient from feeling pain. You know how to stop Wanna's pain, Doctor.\"\n\nUnable to break his conditioning, Yueh shuddered and spasmed. He wanted nothing more than to do as the Baron demanded. \"I . . . can't!\"\n\n\"Of course you can. Choose a guest, any guest. I don't care which. See how amused they are by our little game?\" Grasping Yueh's shaking wrists, he helped the doctor point his projectile weapon around the room. \"But don't try any tricks, or we will make the torment last a great deal longer!\"\n\nHe wished he could put Wanna out of her misery, killing her instead of letting the Harkonnens have their perverted fun. He saw her eyes, the spark of pain and hope, but Rabban stopped him. \"Focus, Doctor. No mistakes.\"\n\nThrough blurred vision he made out numerous targets, and tried to concentrate on one, a tottering old nobleman, a semuta addict. That one had lived a long life, undoubtedly with considerable debauchery. But for a Suk doctor to kill\u2014\n\nHe fired.\n\nOverwhelmed by the horrific scene now playing out in his head, Yueh paid no further attention to Sheeana's ministrations. His body was drenched with sweat, but less from sexual exertion than from the extreme psychological distress. He saw Sheeana appraising him. The memories were so clear to him that his entire body felt like a raw wound: Wanna in agony and the sharp, broken-crystal pain of how his Suk conditioning had been thwarted. It had happened _thousands_ of years ago!\n\nThe years before that watershed occurrence, and the years afterward, extended outward, filling his mind, now fresh and hungry. As the relentless memories returned, so did more anguish and guilt, accompanied by a disgust with himself.\n\nYueh felt as if he was about to vomit. Tears poured down his cheeks.\n\nIn the training room, Sheeana studied the wet streaks clinically. \"You're weeping. Does that mean you've successfully regained your memories?\"\n\n\"I have them back.\" His voice was husky and sounded infinitely old. \"And damn you witches to hell for it.\"\n\n> _We have so little trouble finding enemies because violence is an innate part of human nature. Our greatest challenge, then, is to choose the most significant enemy, for we cannot hope to fight them all._\n> \n> \u2014BASHAR MILES TEG,  \n> military assessment delivered to the Bene Gesserit\n\nAfter she departed from Chapterhouse, Murbella traveled to the battle lines. That was where the Mother Commander belonged. Posing as nothing more important than an inspector for the New Sisterhood, Murbella arrived at Oculiat, one of the systems that lay directly in the path of the advancing thinking-machine fleet.\n\nOnce, Oculiat had been at the far edges of inhabited space, a jumping-off point for the Scattering after the Tyrant's death. Objectively, this sparsely populated world had little significance, just another target on the vast cosmic map. But for Murbella, Oculiat represented a genuine psychological blow: When this world fell to the machines, the Enemy would be encroaching into the _Old Empire_ itself, not just into a distant and unknown place that had been omitted from old star maps.\n\nUntil the Ixians delivered their Obliterators and the Guild provided all the ships she had demanded, the Mother Commander had no way to stop, or even slow, the thinking machines.\n\nUnder a hazy sky illuminated by watery yellow sunlight, Murbella stepped out of her ship. The landing field seemed deserted, as if no one tended the spaceport any longer. As if they were not even watching for the Enemy.\n\nWhen she made her way to the frantic crowds in the central city, though, she saw that the inhabitants had already found their own enemy. A mob surrounded the main administration building where government officials had barricaded themselves. The locals had put their leaders under siege, screaming for blood or divine intervention. _Preferably blood_.\n\nMurbella knew the raw power that their fear generated, but it was clearly not channeled properly. The people of Oculiat\u2014and all desperate worlds facing the oncoming Enemy\u2014needed guidance from the Sisterhood. They were an already-charged weapon that must be aimed. Instead, they were out of control. She saw what was happening and rushed forward, but stopped short of throwing herself headlong into the mob.\n\nThey would tear her limb from limb, and they would do it for Sheeana.\n\nThe random appearances and sermons of the \"resurrected Sheeana\" had prepared billions of people to fight. The Sheeanas had kindled the anger and fervor of populations, so that the Sisterhood could manipulate that raw power for their own purposes. Once unleashed, however, such fanaticism became a chaotic force. Knowing they were unlikely to survive against the oncoming machines, the men and women threw themselves into violence, seeking any sort of enemy they could get their hands on . . . even among their own people.\n\n\"Face Dancers!\" someone shouted. Murbella pushed her way closer to the center of action, knocked aside flailing arms and fists, and cuffed someone on the side of the head. But even stunned, the wild and emboldened people surged onward. \"Face Dancers! They've been manipulating us all along\u2014selling us out to the Enemy.\"\n\nThose who recognized the Mother Commander's Bene Gesserit unitard backed away; others, either oblivious or too angry to care, were not swayed until she used Voice. Bombarded by the irresistible command, they staggered away. Just one person against the multitude, Murbella strode toward the colonnaded doorways of the government center, which the people saw as their target. She used Voice again but could not stop them all in their tracks. The shouting and accusatory shrieks rose and fell like a thunderstorm.\n\nAs she fought her way to the front of the barricade, several of the foremost mob members noticed her uniform and let out a long cheer. \"A Reverend Mother is here to support us!\"\n\n\"Kill the Face Dancers! Kill them all!\"\n\n\"For Sheeana!\"\n\nMurbella grabbed an elderly woman who had been yelling along with the others. \"How do you know they're Face Dancers?\"\n\n\"We _know_. Think about their decisions, listen to their speeches. It's obvious they are traitors.\" Murbella didn't believe that Face Dancers would be quite so obvious that common rabble could detect the faint subtleties. But the mob was convinced.\n\nSix huffing men ran by, carrying a heavy plasteel pole that they proceeded to use as a battering ram. Inside the capitol building, terrified officials had piled obstructions against the doors and windows. Thrown stones shattered the ornamental plaz, but the crowd couldn't break in so easily. Bars and heavy objects blocked the way.\n\nWielded with the strength of panic and hysteria, the battering ram pounded the thick doors, tearing hinges loose and splintering wood. In moments, a wave of human bodies pushed forward.\n\nMurbella called out. \"Wait! Why not prove they're Face Dancers before you kill anyone\u2014\"\n\nThe old woman shoved past, eager to get to the officials. She stepped on Murbella's foot, heard her shouted cautions, then turned to her with a narrowed gaze like a serpent's. \"Why do you hesitate, Reverend Mother? Help us capture the traitors. Or are you a Face Dancer yourself?\"\n\nMurbella's Honored Matre reflexes came to the fore, and her hand snapped out, cutting into the woman's neck with a blow that rendered her unconscious. She had not meant to kill the woman, but as her accuser fell to the steps a dozen people surged forward, trampling her to death.\n\nHeart pounding, Murbella pressed against the wall to avoid the brunt of the stampede. If the cry had been taken up\u2014\"Face Dancer! Face Dancer!\"\u2014with fingers pointing at her, the crowd would have killed her without thinking. Even with superior fighting abilities, Murbella could never fend off so many.\n\nShe backed up farther and took shelter behind the tall statue of a long-forgotten hero of the Famine Times, shielding herself with its plastone bulk. The screaming mob would crush many of its own members to get into the government building.\n\nShe could hear cries inside, a discharge of weapons, and small explosions. Some of the trapped officials must have been carrying personal protection. Murbella waited, knowing it would be over soon. . . . .\n\nThe bloody attack burned itself out in half an hour. The mob found and killed all twenty government officials suspected of being enemy Face Dancers. Then, still not sated in their thirst for blood, they turned against any of their own members who had not shown sufficient murderous fervor, until most of the violence drained away into guilty exhaustion. . . . .\n\nStanding tall, Mother Commander Murbella entered the building, where she surveyed the smashed windows, display cases, and artwork. Jubilant murderers dragged bodies onto the polished tiled floor of the main legislative gallery. Almost thirty men and women were dead, some shot with projectile weapons, others beaten to death, many with such violence that their genders were hardly recognizable. The corpses on the polished stone floor wore expressions of horror and shock.\n\nOne of the bodies among the bloody mess was indeed a Face Dancer.\n\n\"We were right! You see, Reverend Mother.\" A man pointed at the dead shape-shifter. \"We were infiltrated, but we rooted out the enemy and killed it.\"\n\nMurbella looked around, at all of the innocent humans murdered to discover one Face Dancer. What was the economy of bloodshed? She tried to assess it coolly. How much damage could that one Face Dancer have caused, exposing vulnerabilities to the oncoming Enemy forces? All of those lives? Yes, and more, she had to admit.\n\nFrom their elation it was obvious that the people of Oculiat considered their uprising a victory, and Murbella could not dispute that. But if this wave of insane vigilantism continued, would all governments topple? Even on Chapterhouse? Then who would organize the people to defend themselves?\n\n> _Weak minds are gullible. The weaker the thought processes, the more ridiculous the notions they will believe. Strong minds, like mine, can turn that to an advantage_.\n> \n> \u2014BARON VLADIMIR HARKONNEN,  \n> original recordings\n\nDespite being unarmed, the Baron sneered into the face of the red-eyed mongrel hound. The growling animal moved toward him on the flagstoned floor, baring its sharp fangs, ready to spring.\n\nFortunately, the Baron had killed the feral creature with a poison dart gun some time ago, and this stuffed mechanical version merely followed a programmed reenactment. The simulacrum went motionless when he gave it a hand signal. An amusing toy.\n\nNine-year-old Paolo moved around the trophy chamber, admiring the wild animals on display. The Baron had dragged the boy out on many hunts in the pristine wilderness of Caladan, so that he could witness the killing firsthand. It was good for his development and education.\n\nRabban had always enjoyed such things, but Paolo had originally been reluctant to engage in the slaughter. Maybe it was some flaw in the genetics. However, the Baron was gradually breaking down his resistance. With vigorous training and a system of rewards and punishments (plenty of the latter), the Baron had almost managed to squash the core of innate goodness in this ghola of Paul Atreides.\n\nWeathersats had predicted constant rain and wind for the rest of the week. The Baron had looked forward to going out on a fresh hunt, but the cold and wet would have made for a miserable expedition. He and Paolo were trapped inside the castle. The two had formed a remarkable bond. House Atreides and House Harkonnen\u2014how ironic! But though Paolo was a clone of the hated Duke's son, raised properly he was turning out more like a Harkonnen.\n\n_He is your grandson, after all_ , the internal voice of Alia nagged at him.\n\nOvercoming an urge to shout back at her, the Baron watched four workmen with suspensors hoist an immense stuffed mastaphont onto a viewing stand. Yet another nearly extinct creature, this ferocious beast had charged at them across a field last autumn, slashing with its serrated horns. But the Baron, Paolo, and half a dozen guards had opened fire with lasguns, cutting disks, and poison flechettes, mangling the creature before it finally fell. What an exciting hunt that had been!\n\nPaolo looked at the animated creatures on stands. \"Instead of going outside to the wilderness, let's go hunting in here. We can pretend they're not already dead. Then we don't have to worry about getting wet and cold.\"\n\nThe Baron looked out at the stormy skies, wondering if the weather was the real reason for Paolo's reluctance. \"I don't mind pain, but personal discomfort is another matter entirely.\" He looked around, assessing possibilities for damage. And grinned. \"You're absolutely right, my boy!\" It pleased him to hear how deep his own voice was becoming.\n\nThey ordered servants to bring a selection of lasguns, dart pistols, swords, and knives for their next ersatz adventure. When the mechanical systems were activated, the dead animals went into a frenzy all over the trophy room. The two hunters took cover, imagined their danger, and shot the mechanical creatures off of their stands, chopping through prosthetic bones, stuffing, and preserved flesh. Last of all, they activated the huge mastaphont and watched it stomp over the debris. Finally catching it in a crossfire of lasgun blasts, they amputated its legs. The beast crashed to the floor, its automatic servos writhing.\n\nThe Baron found the violence to be eminently satisfying, and even Paolo seemed to warm to the activity. Afterward, the brave hunters surveyed the damage and laughed as they marched out into the corridor. The Baron spotted three workmen, who looked as if they wanted to become invisible. \"Get back in there and clean things up!\"\n\n_You always make a mess of things, don't you, Grandfather?_\n\nThe Baron pressed his hands against his head. \"Shut up, damn you!\" Alia began humming repetitive singsong tunes, designed to drive him mad, no doubt. When a bewildered Paolo pestered him with questions, the Baron slapped him away. \"Leave me alone! You're as bad as your sister!\"\n\nConfused and startled, Paolo ran off.\n\nThe girl's grating voice vibrated in his mind until he couldn't stand it. He hurried out of the castle. Barely able to see where he was going, the Baron bumped into one of the blocky Harkonnen statues and rushed toward the sea cliff. \"I'll hurl myself over the edge\u2014I swear it, Abomination\u2014unless you leave me alone!\"\n\nHe got all the way to the windy, rocky brink, before the Alia voice at last faded into sweet silence. The Baron dropped to his knees on the high stone walkway, looking with delicious vertigo over the tremendous drop-off. Maybe he should just do it anyway, and fall to the wet black rocks and churning waves. If the damned Face Dancers needed him so badly, they could just grow another ghola, and maybe that one wouldn't be so flawed. The Baron Harkonnen would be back!\n\nHe felt a hand on his shoulder. Gathering his remaining shreds of dignity, he looked up to see a pug-nosed Face Dancer staring at him. Though all shape-shifters looked exactly the same to him, he somehow knew this one was Khrone. \"What do you want?\"\n\nOfficiously, the Face Dancer said, \"You and Paolo will depart Caladan and never return. The great war is proceeding, and the evermind has decided that he needs the Kwisatz Haderach close to him. Omnius wants you to complete the boy's preparation under his direct supervision in the heart of the machine empire. You will depart for Synchrony as soon as a ship is ready.\"\n\nThe Baron flicked his gaze past the Face Dancer to Paolo, who crouched by a Harkonnen statue, close enough to eavesdrop on the conversation. He chuckled to himself, for this boy was as persistent as Piter de Vries! As soon as he realized he was discovered, Paolo scampered forward. \"Is he talking about me?\"\n\n\"Discuss Paolo's destiny with him on the way,\" Khrone said to the Baron. \"Do more than explain it. Make the boy _believe_.\"\n\n\"Paolo is inclined to believe anything that reinforces his delusions of grandeur,\" the Baron said, ignoring the boy. \"So, this Kwisatz Haderach business is . . . real?\"\n\nEven though the Face Dancers had finally explained the truth to him, the idea still sounded preposterous. He was not convinced that this young ghola could be so important in the grand scheme of things.\n\nKhrone looked ghastly in his blank state. Shadows around his eyes darkened as his displeasure became evident. \"I believe it, and so does Omnius. Who are you to question?\"\n\n_Believe it, dear Grandfather_ , said the annoying voice. _By his very genes, Paul Atreides has the potential to be greater than you will ever be, in any incarnation_.\n\nThe Baron refused to reply, either aloud or in his own thoughts. Ignoring the Abomination often made her shut up.\n\nAnd now they were going to Synchrony, the home of Omnius. He looked forward to seeing the thinking-machine empire. New challenges, new opportunities.\n\nIn spite of the sum of his first life's memories, the stories of the evil thinking machines and the Butlerian Jihad were too distant to seem relevant. Though he harbored considerable resentment toward the Face Dancers, he was glad to be on the side of greater strength.\n\nLater, during the shuttle ride to orbit, the Baron gazed down at the coastline, the villages, the new smokestacks and strip-mined areas of Caladan. In his excitement, Paolo bustled from window to window. \"Will we have a long journey?\"\n\n\"I'm not a pilot. How should I know? The thinking machines must be very far away, otherwise humans would have known about them long before now.\"\n\n\"What will happen when we get there?\"\n\n\"Ask a Face Dancer.\"\n\n\"They won't talk to me.\"\n\n\"Then ask Omnius when you see him. In the meantime, amuse yourself.\"\n\nPaolo sat down beside him in the passenger compartment and began sampling packets of syrupy packaged food. \"I'm special, you know. They've been grooming me, watching over me carefully. What exactly is a Kwisatz Haderach, anyway?\" He wiped his mouth with the back of a hand.\n\n\"Don't crawl into their delusions, boy. There is no such thing as a Kwisatz Haderach. A myth, a legend, something with a hundred vague explanations in as many prophecies. The entire Bene Gesserit breeding program is utter nonsense.\" He recalled from deep memories that he had been part of that breeding program himself, forced to impregnate the vile witch Mohiam. He had humiliated her during the act, but in retaliation the hag had transmitted the debilitating disease that had made him bloated and fat.\n\n\"It can't be nonsense. I have visions, especially when I take spice tablets. I see it again and again. I've got a bloody knife in my hand, and I'm victorious. I see myself rushing to take my prize\u2014melange, but more than melange. I also see myself lying on the floor, bleeding to death. Which one is right? It's so confusing!\"\n\n\"Shut up and take a nap.\"\n\nThey docked with an unmarked ship high above Caladan. It bore no markings of the Guild and carried no Navigator. Wide hangar doors opened, drawing the shuttle up and inside. Silvery figures moved within the cold, airless landing bay, guiding the small vessel into a docking cradle. Robots\u2014demons from ancient history! Ah, so, at least part of Khrone's wild tale might be true.\n\nThe Baron smiled at the boy staring out the windows. \"You and I are about to undertake an interesting journey, Paolo.\"\n\n> _A sheathed dagger is useless in a fight. A maula pistol without projectiles is no more than a club. And a ghola without his memories is merely flesh_.\n> \n> \u2014PAUL ATREIDES,  \n> secret ghola journals\n\nNow that the ghola of Dr. Yueh had his memories restored, Paul Atreides knew he had to attempt more innovative measures to awaken himself. Paul was the oldest of the ghola children, the one with (presumably) the greatest potential, but Sheeana and the Bene Gesserit observers had chosen Yueh as a test case. Unlike the Suk doctor, however, Paul actually wanted his past back. He longed to remember his life and love with Chani, his childhood with Duke Leto and Lady Jessica, his friendships with Gurney Halleck and Duncan Idaho.\n\nBut Paul continued to be haunted by prescient memory-visions of his own double death. And he was growing impatient.\n\nHow could the passengers aboard the no-ship think there was still time for caution? Only a few months ago, they had again narrowly escaped the Enemy's net, brighter and stronger than ever before. Of great concern, the saboteur still had not been caught. Though the saboteur had done nothing else as dramatic as the murder of the three axlotl tanks and unborn gholas, the danger remained.\n\nPaul knew the _Ithaca_ needed him, and he was tired of being just a ghola. He had an idea to attempt, one that was both desperate and dangerous, but he didn't hesitate. His real memories hovered like a mirage just beyond the heat-shimmered horizon.\n\nWith faithful Chani beside him, he stood outside the hatch that led into the great sand-filled hold. He had told no one else what he intended to do. Over the past two years, security had been tightened as much as the Bashar and an eager Thufir Hawat could manage, but no one guarded the entrance to the cargo hold. The seven sandworms were considered sufficiently dangerous to act as their own watchdogs. Only Sheeana could safely go among the large creatures, and the last time she had done so, even she had been briefly swallowed up.\n\nPaul gazed at Chani's beautiful elfin face and her thick, dark red hair. Even without prior knowledge, without knowing his destiny was to be with her, he would have found the Fremen girl strikingly attractive. In turn, she ran a methodical eye over his body, his special new suit, his tools. \"You look like a real Fremen warrior, Usul.\"\n\nAfter studying records and working with a fabrication station in the engineering levels, Chani had fashioned an authentic stillsuit for him\u2014probably the first one manufactured in centuries\u2014and provided him with a rope, maker hooks, and spreaders. The unusual tools felt oddly familiar in his grasp. According to legend, Muad'Dib had summoned a dangerous monster for his first worm ride. These creatures in the hold, though stunted by their captivity, were still behemoths.\n\nThe hatch opened, and he and Chani stepped into the artificial desert. When the flinty odors and arid heat struck him, he said, \"Stay here, where it's safe. I have to do this alone, or it won't be effective. If I face the worm and ride it, that may jar my memories.\"\n\nChani did not try to stop him. She understood the need as well as he did.\n\nHe climbed up the first rise, leaving footprints in the sand, then raised both hands and shouted, \"Shai-Hulud! I have come for you!\" In this confined space, he did not need a thumper to summon the worms.\n\nA quality in the air changed. He sensed a stirring in the shallow dunes and saw seven serpentlike shapes coming toward him. Instead of running away, he sprinted toward them, selecting a place where he could set up his approach and mount one. His heart pounded. His throat was dry despite the stillsuit mask covering his mouth and nose.\n\nPaul had reviewed holofilms to study Fremen sandriding techniques. Intellectually he knew what to do, just as\u2014intellectually\u2014he knew the factual details of his past. But a theoretical understanding was far different from actual experience. It occurred to him now, as he stood small and vulnerable on the sand, that the most effective form of learning was in the actual doing, which ensured a more thorough comprehension than he could derive from dusty archives.\n\n_I shall learn well_ , he thought, letting fear wash past him.\n\nThe nearest worm surged toward him with a rushing sound of scattered sand. The sheer size of the worms grew more incomprehensible as they approached, cresting the dunes.\n\nInfusing his heart with courage, Paul forced himself to face this challenge. He held up his hook and spreader and crouched for the first leap. The noise of the monsters' approach was so loud that at first he did not hear the woman shouting. From the corner of his eye, he saw Sheeana bounding across the dunes, throwing herself in front of him. The largest worm exploded through the dust and reared up, its gigantic round mouth glittering with crystalline teeth.\n\nSheeana held up her hands and shouted, \"Stop, Shaitan!\"\n\nThe worm hesitated, and quested from side to side with its fleshy head as if confused.\n\n\"Stop! This one is not for you.\" She placed a firm hand on the chest of Paul's stillsuit and pushed him behind her. \"He is not for you, Monarch.\"\n\nAs if sulking, the largest worm backed away, keeping its eyeless head turned toward them. \"Get back to the hatch, foolish boy,\" Sheeana hissed at Paul, using just enough Voice to make his legs respond before he could think.\n\nDuncan Idaho was also there at the hatchway, glowering. Chani looked both fearful and relieved.\n\nSheeana marched Paul back toward the waiting observers. \"That worm would have destroyed you!\"\n\n\"I'm an Atreides. Shouldn't I be able to control them like you can?\"\n\n\"That isn't a theory I intend to test with you. You are too important to us. Of all the gholas, if _you_ foolishly throw your life away, what are we to do?\"\n\n\"But if you protect me too much, you'll never get what you need. Riding a worm would have brought my memories back, I'm sure of it.\"\n\n\"You restored Yueh,\" Chani pointed out to Sheeana. \"Why not Usul? He's older.\"\n\n\"Yueh was expendable, and we weren't sure of what we were doing. We have already developed specific plans for awakening Liet-Kynes and Stilgar, and if we succeed with them, others may follow\u2014including Thufir Hawat and you, Chani. One day, Paul Atreides will get his chance. But only after we are certain.\"\n\n\"What if we don't have the time?\" Paul walked away from them, brushing sand and dust from his new stillsuit.\n\nDUNCAN AWOKE TO a loud signal at the door to his quarters. His initial thought was that Sheeana had come to him again, despite their mutual reservations. He slid aside the door, ready for an argument.\n\nPaul stood there wearing a replica of an Atreides military uniform, which evoked instant respect and loyalty from Duncan. The young man had dressed that way on purpose. Right now, the ghola Paul was almost exactly the same age as the original had been when Arrakeen had fallen to the devious Harkonnens, when the first Duncan had died defending him and his mother.\n\n\"Duncan, you say you were my close friend. You say you knew Paul Atreides. Help me now.\" Grasping an ornately carved ivory hilt, the young man drew a blue-white crystalline dagger from a sheath at his waist.\n\nDuncan stared in amazement. \"A crysknife? It looks . . . Is it real?\"\n\n\"Chani made it from a worm tooth Sheeana found in the cargo hold.\"\n\nIn wonder, Duncan touched his fingers to the blade, noting how tough and sharp it was. He drew his thumb along the edge, intentionally cutting himself. He let a single drop of blood fall onto the milkywhite dagger. \"According to ancient tradition, a crysknife must never be drawn unless it tastes blood.\"\n\n\"I know.\" Paul was clearly troubled as he took the weapon back and returned it to its sheath. After hesitating, he blurted out what he had come to say. \"Why won't the Bene Gesserits awaken me, Duncan? You need me. Everyone on this no-ship needs me.\"\n\n\"Yes, young Master Paul. We do need you, but we need you _alive_.\"\n\n\"You need my abilities, as soon as possible. I was the Kwisatz Haderach, and this ghola has the same genetics. Imagine how I could help.\"\n\n\"The Kwisatz Haderach . . .\" Duncan sighed and sat down on his bed. \"The Sisterhood spent centuries creating him, but at the same time they were terrified of him. He can supposedly bridge space and time, seeing the future and the past, places even a Reverend Mother dares not look. Through brute force or guile he can forge a union between the most diverse of factions. It's a grab bag of tremendous powers.\"\n\n\"Whatever those powers are, Duncan, I need them. And for that I require my memories. Convince Sheeana to try me next.\"\n\n\"She will do what she will do, at a time of her choosing. You overestimate the influence I have among the Sisters.\"\n\n\"But what if the Enemy's net ensnares us completely? What if the Kwisatz Haderach is your only hope?\"\n\n\"Leto II was a Kwisatz Haderach, as well, though neither you nor your son turned out exactly the way the Bene Gesserit intended. The Sisters are very afraid of anyone who manifests unusual powers.\" He laughed. \"After the Scattering, when the Sisterhood brought the great Duncan Idaho back, some of them even accused _me_ of being a Kwisatz Haderach. They killed eleven of my gholas, either by Bene Gesserit heretics or Tleilaxu schemers.\"\n\n\"But why don't they want these powers? I thought\u2014\"\n\n\"Oh, they want the powers, Paul, but only under carefully controlled conditions.\" His heart went out to the young man, who looked so lost and desperate.\n\n\"I can't do anything without my past, Duncan. Help me retrieve it! You lived through part of it with me. You remember.\"\n\n\"Oh, I remember you very well.\" Duncan laced his hands behind his head and leaned back. \"I remember your christening on Caladan after you were nearly killed by Imperial intrigues as an infant. I remember how Duke Leto's whole family was put at risk in the War of Assassins. I was given the great honor of taking you to safety, and you and I went to the wilds of Caladan. We stayed with your exiled grandmother Helena, and we hid among the Caladan primitives. That was when you and I became so close. Yes, I remember it very well.\"\n\n\"I don't,\" Paul said with a sigh.\n\nDuncan seemed caught in a loop of his past lives. Caladan . . . Dune . . . the Harkonnens . . . Alia . . . Hayt. \"Do you know what you're asking me, about your memories, about your life? The Tleilaxu created my first ghola as an assassination tool. They manipulated me because I was your friend. They knew you could not turn me away, even though you saw the trap.\"\n\n\"I wouldn't have turned you away, Duncan.\"\n\n\"I had the knife raised against you, ready to strike, but in the last instant, I collided with myself. The programmed assassin Hayt became the loyal Duncan Idaho. You can't imagine the agony!\" He pointed a stern finger at the young man. \"Restoring your past will require a similar crisis.\"\n\nPaul squared his jaw. \"I'm prepared for it. I'm not afraid of pain.\"\n\nWrinkling his brow, Duncan said, \"You're too content, young Paul, because your Chani grounds you. She makes you stable and happy\u2014and that's a severe drawback. In contrast, look at Yueh. He fought against remembering with every fiber of his being, and that's what broke him. But you . . . what fulcrum can they use on you, Paul Atreides?\"\n\n\"We'll just have to find something.\"\n\n\"Are you really ready to accept it?\" Duncan leaned forward, offering no mercy. \"What if the only way you can have your past restored is that you must _lose_ Chani? What if she has to die bleeding in your arms, before you can remember?\"\n\n> _More than anything, I need my father to know I did not fail. I do not want him to die thinking I was unworthy of his genes_.\n> \n> \u2014THE SCYTALE GHOLA,  \n> no-ship security interview\n\n\"It must be built according to precise standards,\" insisted the old Tleilaxu. His voice cracked. \" _Precise_ standards!\"\n\n\"I will take care of it, Father.\" The ghola, only thirteen, tended the degenerating Master who sat in a stiff armchair. Old Scytale refused to lie down until a traditional bier for his body was built. He intentionally kept his austere living quarters locked to keep others away. He had no desire to be interrupted or harassed during his dying days.\n\nThe Tleilaxu Master's organs, joints, and skin had begun to fail in increasingly problematic ways. It reminded him of how the no-ship itself seemed to be breaking down, its systems failing as air leaked into space, water was inexplicably lost, food stores went missing. Some of the more paranoid refugees saw sabotage in every flickering glowpanel, and many turned their suspicious eyes toward the Tleilaxu. It was another reason for him to grumble. At least he would soon be gone.\n\n\"I thought you said my bier was already being built. It cannot be rushed.\"\n\nThe teenager bowed his head. \"Do not worry. I am following the strict laws of the Shariat.\"\n\n\"Show it to me, then.\"\n\n\"Your own bier? But that is meant to carry your body only after you . . . after you . . .\"\n\nOld Scytale glowered with his dark eyes. \"Purge those useless emotions! You have become too involved in this process. It is shameful.\"\n\n\"Am I not supposed to care about you, Father? I see your pain\u2014\"\n\n\"Stop calling me Father. Think of me as _yourself_. Once you become me, I will not be dead. No need for weeping. Each of our incarnations is disposable, so long as the memory train continues uninterrupted.\"\n\nYoung Scytale tried to regain his composure. \"You are still a father to me, no matter what memories are buried inside me. Will I stop feeling these emotions when my old life is restored?\"\n\n\"Of course. At that glorious moment you will understand the truth\u2014and your obligations.\" Scytale grabbed the young man by his shirt and pulled his face close. \"Where are your memories? What if I were to die tomorrow?\"\n\nOld Scytale knew death was imminent, but he had dramatized his infirmity in an attempt to shock his replacement. The premature construction of the bier was yet another attempt to provoke a crisis. If only the two of them could be back on Tleilax, where full immersion in the holy traditions of the Great Belief would be enough to trigger even the most stubborn of gholas. Here onboard a godless no-ship, the difficulties seemed insurmountable.\n\n\"This should never have taken so long.\"\n\n\"I have failed you.\"\n\nThe rheumy eyes flashed. \"You are not only failing me, you are failing your people. If you do not awaken, our whole race\u2014our entire history and all the knowledge in my mind\u2014will vanish from the universe. Do you want to be responsible for that? I refuse to believe God has turned His back on us entirely. Our fate, lamentably, depends upon you.\"\n\nThe ghola looked crestfallen, as if an unsupportable weight rested on his shoulders. \"I am doing all I can to achieve that goal, _Father_.\" He said the word deliberately. \"And until I succeed, you must do all you can to remain alive.\"\n\n_He's finally showing a little strength_ , Scytale thought, bitterly. _But it's not enough_.\n\nDAYS LATER, THE ghola stood by his father's deathbed, his _own_ deathbed. He felt as if he were having an out-of-body experience, watching his life slip away moment by moment. It gave the boy an oddly disconnected feeling.\n\nSince emerging from the axlotl tank, Scytale had loved only one person: himself . . . both his older self and the self he was going to be. The degenerating man had provided cells from his own body, cells that held all his memories and experiences, all the knowledge of the Tleilaxu.\n\nBut he hadn't provided the key to unlock them. No matter how hard the young ghola strained, his memories obstinately refused to emerge. He clutched the old man's hand. \"Not yet, Father. I've tried and tried.\"\n\nWith near-sightless eyes, old Scytale glared at his counterpart. \"Why do you . . . disappoint me so?\"\n\nYueh had been restored to his past life, and two other gholas\u2014Stilgar and Liet-Kynes\u2014were even now being raked over the mental coals. How could mere witches succeed where a Tleilaxu Master failed? Bene Gesserits should never have been so adept at triggering the avalanche of experiences. If Scytale could not do it, the Tleilaxu would be relegated to the dustbins of history.\n\nThe old man on the bed coughed and wheezed, while the younger leaned close, tears trickling down his cheeks. Old Scytale spat blood. His disappointment and utter despair were palpable.\n\nAn insistent signal at the door announced the arrival of two Suk doctors. The bespectacled Rabbi was obviously repulsed by his duties, while young Yueh still appeared to be shaken by the recent return of his memories. Scytale could see in their eyes that they both knew the older Master would perish very soon.\n\nAmong the witches there were other Suk practitioners, but Scytale had insisted on being tended only by the Rabbi, and only when absolutely necessary. They were all unclean powindah, but at least the Rabbi wasn't a disgusting _female_. Or, perhaps Scytale should choose Wellington Yueh over the old Jew. The old Tleilaxu Master had to accept certain medical examinations, if only to keep himself alive until his \"son\" reawakened.\n\nScytale lifted his head. \"Go away! We are praying.\"\n\n\"Do you think I like tending to gholas? To filthy Tleilaxu? Do you think I _want_ to be here? You can both die, for all I care!\"\n\nYueh, though, moved forward with a medical kit, easing the younger Scytale aside to check the dying man's vital signs. Behind Yueh the Rabbi squinted through his spectacles with vulture eyes. \"It won't be long now.\"\n\nSuch an odd old holy man, young Scytale thought. Even compared to the smells of disinfectant, medicine, and sickness, he'd always had an odd smell about him.\n\nSounding compassionate, Yueh said, \"There isn't much we can do.\"\n\nGasping for air, old Scytale croaked out, \"A Tleilaxu Master should not be so weak and decrepit. It is . . . unseemly.\"\n\nHis youthful counterpart tried again to trigger the flow of memories, to squeeze them into his brain by sheer force of will, as he had attempted to do countless times before. The essential past must be in there somewhere, buried deep. But he felt no tickle of possibilities, no glimmer of success. _What if they are not there at all?_ What if something had gone terribly wrong? His pulse pounded as the panic began to rise. Not much time. Never enough time.\n\nHe tried to cut off the thought. The body provided a wealth of cellular material. They could create more Scytale gholas, try again and again if necessary. But if his own memories had failed to resurface, why should an identical ghola have any better luck without the guidance of the original?\n\n_I am the only one who knew the Master so intimately_.\n\nHe wanted to shake Yueh, demand to know how he had managed to remember his past. Tears were in full flow now, falling onto the old man's hand, but Scytale knew they were inadequate. His father's chest spasmed in an almost imperceptible death rattle. The life-support equipment hummed with more intensity, and the instrument readings fluctuated.\n\n\"He's slipped into a coma,\" Yueh reported.\n\nThe Rabbi nodded. Like an executioner announcing his plans, he said, \"Too weak. He's going to die now.\"\n\nScytale's heart sank. \"He has given up on me.\" His father would never know if he succeeded now; he would perish wondering and worrying. The last great calamity in a long line of disasters that had befallen the Tleilaxu race.\n\nHe gripped the old man's hand. So cold, too cold. He felt the life ebbing. _I have failed!_\n\nAs if felled by a stunner, Scytale dropped to his knees at the bedside. In his crashing despair, he knew with absolute certainly that he could never resurrect the recalcitrant memories. Not alone. _Lost! Forever lost! Everything that comprised the great Tleilaxu race_. He could not bear the magnitude of this disaster. The reality of his defeat sliced like shattered glass into his heart.\n\nAbruptly, the Tleilaxu youth felt something changing inside, followed by an explosion between his temples. He cried out from the excruciating pain. At first he thought he was dying himself, but instead of being swallowed in blackness, he felt new thoughts burning like wildfire across his consciousness. Memories streamed past in a blur, but Scytale locked onto each one, absorbing it again and reprocessing it into the synapses of his brain. The precious memories returned to where they had always belonged.\n\nHis father's death had opened the barriers. At last Scytale retrieved what he was supposed to know, the critical data bank of a Tleilaxu Master, all the ancient secrets of his race.\n\nInstilled with pride and a new sense of dignity, he rose to his feet. Wiping away warm tears, he looked down at the discarded copy of himself on the bed. It was nothing more than a withered husk. He no longer needed that old man.\n\n> _These ghola children contain old souls that are not unlike the voices in a Reverend Mother's Other Memory. The challenge is to access and exploit these old souls_.\n> \n> \u2014ship's log, entry of  \n> DUNCAN IDAHO\n\nIn the gangly body of a teenager, filled with the memories of a long life and the shame of things he had done, Wellington Yueh walked with painstaking slowness. Each step brought him closer to the moment he had been dreading. The skin of his brow burned where a diamond tattoo should have been; at least he no longer displayed that lie.\n\nYueh knew that if he ever intended to make this life different from his error-prone past, he must confront the terrible things he had done.\n\nHere, thousands of years later and on the other side of the universe, House Atreides lived all around him: Paul Atreides, Lady Jessica, Duncan Idaho, Thufir Hawat. At least Duke Leto had not been resurrected as a ghola. Not yet. Yueh didn't think he could bear to look into the eyes of the man he had betrayed.\n\nFacing Jessica would be tough enough.\n\nWalking ponderously toward her quarters, Yueh heard voices ahead, a child's giggle and a woman's rebuke. Suddenly little Alia toddled out of one doorway and ducked into another, followed by a scolding proctor. The two-year-old was extremely precocious, with a hint of the genius that the first Alia had been; the spice saturation in the axlotl tank had altered her somewhat, but she didn't possess the complete Other Memory of her predecessor. The proctor followed and sealed the door behind them. Neither of them had glanced at Yueh.\n\nAlia was the most recent ghola to be born; the program had been stalled since the horrific murder of the three tanks and unborn children. _At least that is one crime I do not have on my conscience_. But the Bene Gesserits would soon begin the program again. They were already discussing which cells to implant in the new axlotl tanks. Irulan? Emperor Shaddam himself? Count Fenring . . . or someone far worse? Yueh shuddered at the thought. He feared that the witches had gone beyond true need and now were just toying with lives, letting their infernal curiosity sidestep all caution.\n\nHe paused in front of Jessica's quarters, steeling himself. _I will face my fear_. Wasn't that part of the Litany the witches so often quoted? In their present incarnations as gholas, Jessica and Yueh had been close enough to think of themselves as friends. But since becoming Dr. Wellington Yueh again, everything was different.\n\n_Now I have a second chance_ , he thought. _But my road to redemption is long, and the incline very steep_.\n\nJessica opened her door at his signal. \"Oh hello, Wellington. My grandson and I were just reading a holobook about Paul's younger years, one of those tomes Princess Irulan was always writing.\" She invited him inside, where he saw Leto II sitting cross-legged on the carpeted floor. Leto was a loner, though he frequently spent time with his \"grandmother.\"\n\nYueh twitched nervously when she closed the door behind them, as if to seal his doom and prevent escape. He kept his eyes down, and after a deep sigh he said, \"I wish to apologize to you, my Lady. Though I know you can never forgive me.\"\n\nJessica placed an arm on his shoulder. \"We've been through this. You can't bear the blame for things that were done so long ago. It wasn't really you.\"\n\n\"Yes it was, because I remember it all now! We gholas were created for one purpose, and we must accept the consequences.\"\n\nJessica looked at him impatiently. \"We all know what you did, Wellington. I accepted that and forgave you long ago.\"\n\n\"But will you do it again after you _remember?_ One day those vaults will be opened in your mind, the terrible old wounds. We've got to face the guilt our predecessors left for us, or we'll be consumed by things we never did.\"\n\n\"It's uncharted territory for all of us, but I suspect we each have plenty of things to atone for.\" She tried to console him, but he didn't feel he deserved it.\n\nLeto paused the filmbook and looked up with an eerie intelligence in his eyes. \"Well, I'm only going to take responsibility for what I do in _this_ life.\"\n\nJessica reached out to touch Yueh's face gently. \"I can't understand what you went through, what you're still going through. I'll know soon enough, I suppose. But you should think about what you would _like_ to be, not what you're afraid of being.\"\n\nShe made it sound so simple, but despite his best efforts, he had been twisted before. \"What if I do something bad in this lifetime, too?\"\n\nJessica's expression hardened. \"Then no one can help you.\"\n\n> _You think your eyes are open, yet you do not see_.\n> \n> \u2014Bene Gesserit admonishment\n\nWater crashed against the black reef on Buzzell, sending up a veil of spray. Mother Commander Murbella stood with the once-disgraced Sister at the edge of the cove, watching Phibians frolic in deep water. The amphibious creatures swam together, slick and smooth-skinned, diving under the combers and then bursting to the surface again.\n\n\"They love their new freedom,\" Corysta said.\n\n_Like dolphins in an ancient Earth sea_ , Murbella thought, admiring their forms. Human . . . and yet dramatically not so.\n\n\"I'm more interested in seeing them harvest soostones.\" She turned her face into the salty wind. Gray clouds were gathering, but the air remained warm and humid. \"Our debts in this war are staggering. Our credit is stretched beyond its limits, and some of our most vital suppliers will accept nothing but hard currency\u2014like soostones.\"\n\nIn the months since leaving Oculiat, the Mother Commander had traveled from planet to planet, studying humanity's defenses. Realizing their great peril, local kings, presidents, and warlords provided independent battleships to add to the newly constructed Guild vessels being released by the Junction shipyards. Every government and cluster of allied worlds scrambled to invent or acquire new armaments to use against the Enemy, but so far nothing had proved effective. The Ixians were still testing the Obliterator weapons, which had proved to be more difficult to manufacture than expected. Murbella continued to demand more work, more material and sacrifices. It wouldn't be enough.\n\nAnd the war continued. Plagues spread. Machine fleets destroyed every human-inhabited world they encountered. Near the edge of one of the main combat zones, three more Sheeana surrogates rallied the people caught between a hammer and an anvil, but to no avail. So far, since the beginning of Omnius's march across space, Murbella could not claim a single clear victory.\n\nIn her bleakest moments the odds seemed poor and the obstacles insurmountable. Millennia ago, the fighters of the Butlerian Jihad had faced another impossible situation, and humankind had won only by accepting an appalling cost. They had unleashed countless atomic weapons that not only destroyed thinking machines, but also trillions of human beings who had been held in slavery. The Pyrrhic victory had left a horrendous stain on the human soul.\n\nAnd now, even after that monumental sacrifice, Omnius was back, like a noxious weed whose roots had never been destroyed. Gauging the progress of the thinking machines, in the next year or two the human race would be forced into a climactic showdown.\n\nOnce the Ixian industrialists delivered their long-awaited Obliterators, all the collected militaries from planet after planet would draw a line in space. As far as she was concerned, that opportunity could not come soon enough.\n\n\"Our soostone shipments have increased every month for the past two years.\" As she spoke, Corysta did not remove her gaze from the frolicking aquatic creatures. \"The Phibians are more productive, now that the Honored Matres have stopped torturing them. And they never used to play like that before. They consider the seas of Buzzell their home instead of their prison.\"\n\nCorysta, a former Breeding Mother exiled here for the crime of trying to keep her own baby, had become a stalwart monitor of soostone harvests. She oversaw the grading, cleaning, and packaging of the pearlescent gems, which were delivered regularly to CHOAM intermediaries.\n\n\"Even so, we need more soostones.\"\n\n\"I'll speak with the Phibians, Mother Commander. I'll explain that our need is great, that the Enemy draws near. For me, they might work harder.\" Corysta's smile was strangely unreadable. \"I'll ask it as a favor.\"\n\n\"And that will work?\"\n\nThe other Sister shrugged. The Phibians leapt high into the air and dove back into the water, while Corysta waved to them, laughing. They seemed to know she was watching them. Sunlight glinted on the water. Were these Phibians putting on a special performance?\n\nQuite suddenly, something large and serpentine emerged from the depths near the splashing creatures. An eyeless head rose above the waves, its round mouth flashing crystalline teeth. The head quested around, fin-edged gill flaps sensing vibrations, like a sea serpent from ancient legends.\n\nMurbella caught her breath. To her amazement it resembled a sandworm from Rakis, though only about ten meters long\u2014and with adaptations that enabled it to live in the water. Impossible! A seaworm?\n\nCorysta ran frantically down the rocks and waded into the surf. The Phibians had already seen the monster and tried to swim away. The worm darted toward them, spray glistening from its greenish rings.\n\nTwo more of the long, sinuous monsters appeared from the deep water and circled around the Phibians. The aquatic people clustered in a defensive formation; one male with a scar on his forehead drew a wide, flat-bladed knife used for scoring cholisters on the ocean floor. The other Phibians brandished their own weapons, which were laughable against a sea serpent.\n\nKnee-deep in waves, Corysta slipped on the algae-slick rocks. Murbella ran after her, fixated on what she saw in the water. \"What are those creatures?\"\n\n\"Monsters! I have never seen them before.\"\n\nThe scarred male Phibian emitted a loud vibrating sound and slapped one webbed hand on the water with a sharp crack. The clustered Phibians bolted like a startled school of fish, several diving underwater, others swimming briskly across the waves.\n\nThough they had no eyes, the swimming worms knew where the Phibians were. With a blur and a flick of long serpentine bodies, they pursued the aquatic workers, driving them toward the rocky shore.\n\nMurbella and Corysta watched the largest worm lunge and grab one of the Phibians, scooping him down into the wet gullet. The other worms attacked like a group of frenzied sharks.\n\nMurbella waded out to grab Corysta's shoulder, preventing her from swimming farther into the churning water. They were both helpless to prevent the violence. \"My Sea Child,\" Corysta moaned.\n\nThe seaworms thrashed and splashed as they fed. Bloody waves lapped against Murbella's legs, and she dragged the sobbing Corysta back to shore.\n\n> _A planet is not merely an item for study. Rather it is a tool, perhaps even a weapon, with which we can make our mark on the galaxy_.\n> \n> \u2014LIET-KYNES,  \n> the original\n\nNow that Stilgar and Liet had their ghola memories back, they had become the no-ship's experts on extreme recycling, making the most of their reduced resources. The _Ithaca_ 's life-support systems had been designed by geniuses out in the Scattering, descendants of those who had survived the horrific Famine Times. The highly efficient technology could serve passengers and crew for long periods, even in the face of the increasing population. But not in the face of deliberate sabotage.\n\nTall and lean, with the body of a youth and the aged eyes of a naib, Stilgar looked ready to embark on a desert journey. He and Liet-Kynes had been bound at first by common interests and more recently by their awakened pasts. Liet refused to talk about the crisis through which Sheeana had broken him\u2014it was a matter too private even for close friends.\n\nFor himself, Stilgar couldn't forget what the witches had done to him. To the very depths of his being he was a desert man of Arrakis. Watched over by Proctor Superior Garimi, he had read of his history as a young commando against the Harkonnens, later as naib, and then as a supporter of Muad'Dib. But to trigger his ghola memories, the Sisters had tried to _drown_ him.\n\nAt a water-filled recycling reservoir, Sheeana and Garimi had tied weights around his ankles. Stilgar fought, but the witches were more than a match for him. \"What have I done? Why are you doing this to me?\"\n\n\"Find your past,\" Sheeana said, \"or die.\"\n\n\"Without your memories you are useless, and better off drowned,\" Garimi said. They dumped him into the pool.\n\nUnable to free himself from the weights on his ankles, Stilgar had quickly sunk. He had struggled mightily, but the water was everywhere, more oppressive than the thickest dust cloud. Trying desperately to peer upward, he made out only the vague wavering shapes of the two women up there. Neither lifted a hand to help him.\n\nHis lungs screamed, and blackness closed in around his eyesight. Stilgar thrashed violently and grew weaker every second. He was starving for breath. He wanted to cry out\u2014 _needed_ to\u2014but there was no air. Exhaled bubbles roared out of his open mouth. When it was more than unbearable, he inhaled a huge gulp into his lungs, flooding his air passages. He couldn't see any way out of the tank\u2014\n\n\u2014and suddenly it was no longer a tank, but a wide, deep river, which he realized was on one of the planets where he had fought in Muad'Dib's jihad. He had marched with a regiment of Caladan soldiers and they had needed to ford the river. The water had been deeper than anyone anticipated, and all of them went under. His companions, who had been born swimming, thought nothing of it, even laughed as they made their way to shore. But Stilgar was dragged beneath the surface. He reached up, clutching for air. He had inhaled water then, too, and nearly drowned\u2014\n\nFinally, Sheeana dragged Stilgar out of the tank and pumped his lungs. A disapproving Suk doctor scolded her and Garimi as she revived the young ghola. They rolled him over, and he vomited up sour mouthfuls of water. He was barely able to rise to his knees.\n\nWhen he turned his glare on Sheeana, he was more than an eleven-year-old boy. He was Naib Stilgar.\n\nLater, when he saw Liet restored as well, Stilgar was afraid to ask what terrible ordeal his friend had been forced to endure. . . .\n\nNow the two headed for the great hold to see the sandworms, as they had done many times before. The high observation chamber was one of their favorite places, especially now. The tremendous worms called up strong and atavistic feelings in them.\n\nAs they approached, Stilgar breathed in the comforting scent of warm, dry air with the distinct odors of worms and cinnamon. He smiled briefly in a passing nostalgia, before his face creased in a frown. \"I should not be smelling that.\"\n\nLiet picked up his pace. \"That environment has to be carefully controlled. If the seals are leaking, then moisture could penetrate the hold.\" Yet another breakdown, after so many others!\n\nRushing into the equipment chamber, they found young Thufir Hawat supervising repair operations. Two Bene Gesserit Sisters and Levi, one of the refugee Jews, worked to install sheets of replacement plaz. They applied thick sealants around the windows high above the sand-filled cargo hold. Thufir was scowling.\n\nStilgar strode forward, his demeanor intimidating. The task of monitoring the sandworms and the recycling systems was generally reserved for himself and Liet. \"Why are you here, Hawat?\"\n\nThufir showed surprise at the coldly accusatory tone of the Fremen's voice. \"Someone poured acid on the seals. The corrosive destroyed not only the sealant, but part of the plaz and the wallplates as well.\"\n\n\"We patched it in time,\" said Levi. \"We also found a timed device that would have emptied one of our water reservoirs into the hold, flooding it.\"\n\nStilgar trembled with rage. \"That would have killed the worms!\"\n\n\"I checked those systems myself, only two days ago,\" Liet said. \"This is no simple breakdown.\"\n\n\"No,\" Thufir agreed. \"Our saboteur is at work again.\"\n\nWhile Stilgar ran his gaze suspiciously over the gathered people, Liet hurried to the instrument consoles to check the desert environment. \"There appears to be no permanent damage. The readings are still within the creatures' tolerance range. Scrubbers should bring the air back to desired levels in short order.\"\n\nStilgar took special care to inspect the new seals, found them adequate. He and Liet exchanged looks that said they had to be suspicious of everyone onboard. _Except for each other_ , Stilgar decided.\n\nLong ago, when he and Liet had first known each other, the two had shared many adventures fighting the nefarious Harkonnens. Like his father, Liet had led a double life, delivering grand dreams to the desert people while acting as Imperial Planetologist and Judge of the Change. Liet was also the father of Chani. While the Fremen girl's ghola did not remember him yet, he remembered her, and he looked at Chani with a strange, age-worn love.\n\nBothered by the acrid odors of acid and sealant, Stilgar turned grimly away from the observation window. \"From now on, I sleep here. I will not let Shai-Hulud die, not while I still breathe.\"\n\n\"I'm working with the Bashar. There must be some kind of a trail, so we only need to find it. The corrosive was acquired from secure stores, so there may be fingerprints or genetic traces.\" Thufir's lips were not stained red with sapho, his skin not grizzled, his eyes not weary with age and experiences, as in the famous old portraits. \"Perhaps the imagers captured the saboteur sneaking to the observation deck. Once I catch him, we can all rest more easily.\"\n\n\"No,\" Stilgar said. \"Even then, I would not let my guard down.\"\n\nIN A SUDDEN resurgence, the maddening sabotage continued in myriad ways and at random points around the huge ship, setting everyone's nerves on edge. The Bene Gesserits remained vigilant and wary, while the Rabbi preached to a growing number of followers about spies and murderers lurking among them.\n\nDuncan studied the readings, ran projections. Again, he wondered if one or more of the Face Dancer Handlers might still be aboard, having escaped the wreckage of a crashed ship. Where else could the saboteur be hiding? After years of searching, Duncan and Teg had run out of ideas. How could this enemy elude surveillance imagers, Truthsayer interrogations, and vigorous searches? In a few suspicious incidents, a blurry form could be seen moving in restricted areas, but even enhancement could not sharpen the facial features to recognizability.\n\nThe saboteur seemed to know exactly where and when to strike. An endless succession of little breakdowns and small accidents, each taking its toll, ran the ship's company to exhaustion.\n\nOne time, imagers detected what appeared to be a man as he moved furtively down a corridor near a bank of oxygen-scrubber units and aircirculating machines. Dressed in dark clothing and a tight-fitting hood that covered most of his face, he carried a long silver knife and a pry bar, and his body leaned forward against the heavy air flow. Then, like liquid flowing around a corner, the man slipped into the central recirculation chamber, where great fans blasted air through a system of arteries in the no-ship, pushing it through thick curtains of matted fibers coated with biogels to remove impurities.\n\nWith sudden fury, the unidentifiable saboteur slashed and hacked at the porous filter mats, ripping them from their frames and destroying their ability to purify air. After completing this mayhem, the saboteur turned to flee. Not a single frame of the imagers showed the face; it wasn't even absolutely clear whether the hooded vandal was male or female. By the time security personnel rushed into the area, the saboteur had vanished into the howling, recirculated wind.\n\nDuncan did not need to whisper the obvious answer. _Face Dancer_. He studied all records of the kamikaze ships from the Handlers, noting where they had crashed into the hull and how the bodies aboard had been confirmed dead and disposed of. One of the shape-shifting Handlers must have crawled out of the flaming wreckage.\n\nEven worse, there might be more than one.\n\nTHE AIR SMELLED moist and foul, like seaweed and sewage. Duncan stood on the mist-slickened catwalk above one of the largest algae tanks. The entire vat was dying. _Poisoned_.\n\nStanding next to him, gripping the catwalk rail with a whiteknuckled hand, Teg frowned at the chemical analyses displayed on his datapad. \"Heavy metals, potent toxins, a list of deadly chemicals that even this stuff can't digest.\" He pulled up a dripping handful of the once-fecund green substance. The goop was brownish now, breaking down.\n\n\"The saboteur is trying to destroy our food supply,\" Duncan said.\n\n\"Our air, too.\"\n\n\"To what end? To kill us, it appears.\"\n\n\"Or simply to make us helpless.\"\n\nDuncan glared at the vat, feeling angry and violated. \"Get work crews to drain and scrub the tank. Decontaminate as quickly as possible. Then harvest starter material from other tanks to fertilize the biomass. We've got to stabilize it before something else goes wrong.\"\n\nDUNCAN WAS ALONE on the navigation bridge when the next disaster occurred. Over the years the passengers had learned to ignore the faint vibrations of the no-ship's movement. Now, though, an abrupt lurch and an obvious deflection in course nearly threw him out of his chair.\n\nHe called for Teg and Thufir, then scrambled over the controls, scanning empty space around them. He feared they might have run into a piece of space debris or some gravitational anomaly. But he found no evidence of impact, no obstacles in their vicinity. The _Ithaca_ was obviously yawing, and he struggled to steady it using the numerous smaller engines distributed around the hull. This slowed the spinning of the ship, but did not entirely stop it.\n\nAs the immense vessel continued to turn, he saw a glittering silver path like a scarf of mist, spewing from the stern. One of the no-ship's three primary water reservoirs had been dumped\u2014intentionally. The great swath of water had been ejected with enough force to push the _Ithaca_ off course. The evacuated water shifted the ship's ballast and sent them into a spin. The loss of angular momentum made their situation worsen as more and more water poured away, like a comet's tail behind them. The ship's reserves!\n\nWorking feverishly at the controls, Duncan overrode the reservoir hatch, praying all the while that the mysterious saboteur had merely opened the door to space, rather than using one of the deadly mines locked away in the armory.\n\nTeg burst onto the navigation bridge just as Duncan managed to close the cargo doors and reestablish containment. The Bashar bent over the screens, his young but seasoned face creased in concern. \"That was enough water to supply us for a year!\" His gray-eyed gaze flitted around nervously.\n\nPacing the deck, Duncan stared out at the misty veil of dispersed water. \"We can retrieve some of it. Scoop it up as ice, and when I fully stabilize our spin\u2014\"\n\nBut as he looked at the smear of lost water spreading out against the starry backdrop, he saw other lines appear, sparkling multicolored threads drawing together and enclosing the no-ship like a spider's web. The Enemy's net! Again it was bright enough for Teg to see it, too. \"Damn it! Not now!\"\n\nLunging into the pilot's seat, Duncan activated the Holtzman engines. With one or more saboteurs aboard, the engines themselves could have been rigged to explode, but he had no choice. He forced the enigmatic machines to fold space well before he could think about what course to take. The no-ship, still spinning, lurched off to another place.\n\nThey survived.\n\nAfterward, Duncan looked at Teg and sighed. \"We couldn't have retrieved much of the expelled water anyway.\"\n\nEven the ship's sophisticated recyclers had their limits, and now the actions of the saboteur had driven them\u2014intentionally\u2014toward an inescapable conclusion. After many years of constant flight, the noship's provisions had to be replenished as soon as an acceptable planet could be located. Not an easy task in a huge galaxy, encompassing vast distances. They had found nothing suitable in years. Not since the planet of the Handlers.\n\nBut Duncan knew that would not be their only problem. When they found a place, they would be forced to expose themselves\u2014again.\n\n> _Synchrony is more than a machine, more than a metropolis; it is an extension of the evermind itself. It constantly shifts and morphs into different configurations. At first I believed this effect was for defense, but there seems to be another force at work, a surprising creative spark. These machines are exceedingly odd_.\n> \n> \u2014BARON VLADIMIR HARKONNEN,  \n> the ghola\n\nThe metropolis before them was beautiful in an industrial and metallic way: sharp angles, smooth curves, and a great deal of energy as structures moved and flashed like a perfectly tuned machine. Angular buildings and windowless towers covered every square meter of ground. The Baron saw no offensive greenery, no gaudy flowers or landscaping, not a leaf, blossom, or blade of grass.\n\nSynchrony was a bustling symbol of productivity\u2014along with concomitant profits and political power, if thinking machines ever figured out how to pay attention to such things. Maybe Vladimir Harkonnen would show Omnius how it was done.\n\nAfter the long journey from Caladan, the Baron and Paolo rode a tram to the shifting center of the machine city. The Atreides ghola peered out through the curved windows, his eyes wide and hungry. They were crowded in the tram with an escort of eight Face Dancers. The Baron had never understood how the shape-shifters were connected with Omnius and the new Synchronized Empire. The elevated car shot along an unseen charge path high above the ground, whizzing like a bullet between the perpetually shifting buildings.\n\nAs they went deeper into the city, huge edifices moved up, down, and sideways like pistons, threatening to crush the streaking tram. When the half-alive buildings swayed like robotic seaweed, he noticed that the Face Dancers inside the tram moved in unison, wearing placid smiles on their cadaverous faces, as if they were part of a choreographed presentation.\n\nLike a needle threading a complex maze of holes, the tram sped toward an immense spire that rose out of the center of the city like a spike thrust up from the netherworld. Finally, the car came to a clicking stop in a spectacular central square.\n\nAnxious to see, Paolo squirmed and pushed his way out the door. Even with uncertainty and fear gnawing at his gut, the Baron marveled at the numerous fires burning at specific geometric points around the spire, each with a human tied to a stake, martyr-fashion. Obviously, in their conquest of world after world, the thinking-machine fleet had taken experimental subjects. He found the extravagance breathtaking. These machines certainly showed a lot of potential and even uncanny imagination.\n\nHe thought of the huge thinking-machine fleet out in space, as it methodically plowed deeper into human-settled territory. From what Khrone had explained, when the machines finally obtained a pet Kwisatz Haderach, Omnius believed he would be fulfilling the terms of the mechanical prophecy, making it impossible to fail. The Baron found it amusing how the thinking machines viewed everything as an absolute. After fifteen thousand years, they should know better.\n\nPaolo had let himself be caught up in a megalomaniacal whirlwind. The Baron's job was to feed those delusions, always keeping in mind that he was in a dangerous situation himself and needed to keep his wits and focus. Unsure whether personal glory or ignominious death lay ahead, the Baron was repeatedly reminded that he was merely a catalyst for Paolo. _Secondary importance indeed!_\n\nEmerging from the back of his mind, Alia interrupted him, insisting that the machines would discard him when he had fulfilled his purpose. When he sputtered internally in protest, she screeched over him: _You're going to get us killed, Grandfather! Think back to your first life_ \u2014 _you weren't always such a gullible fool!_\n\nThe Baron shook his head briskly, wishing he could get her out of his mind. Maybe his Alia tormentor was the result of a tumor pressing upon a cognitive center of his brain. The malignant little Abomination was deeply entrenched in his skull. Maybe a robot surgeon could cut her out. . . .\n\nThe Face Dancers led him and his young ward across a platform and down a set of stairs to the square. Giddy, Paolo ran ahead and did a brief dance of joy. \"Is this all mine? Where is my throne room?\" He looked back at the Baron. \"Don't worry\u2014I'll find a place for you in my court. You have been good to me.\" Was that a scrap of leftover Atreides honor? The Baron scowled.\n\nThe Face Dancers nudged the Baron into a lift tube, while allowing Paolo to enter unassisted. Instead of climbing to the apex of the tower as the Baron expected, however, the lift plunged in free fall toward the bowels of hell. Swallowing the impulse to scream, he said, \"If you're really the Kwisatz Haderach, Paolo, perhaps you should learn to use your powers . . . _immediately_.\"\n\nThe boy shrugged dumbly, showing little recognition of the peril they were in.\n\nAs soon as the lift settled to a smooth stop, the walls melted away around them to reveal an immense, underground chamber. Here, as outside, nothing remained stationary. Rotating walls and a clearplaz floor left the Baron dizzy and disoriented, as if the two humans stood in a vault of space.\n\nA mist rose and congealed in the shape of a large man, a faceless, ghostlike figure. The foggy form, nearly twice the height of a grown man, stopped in front of them and moved its arms to make a swirl of icy air that smelled of metal and oil. Within the countenance, two glowing eyes became apparent. From a misty mouth, a deep voice said, \"So, this is our Kwisatz Haderach.\"\n\nPaolo lifted his chin and recited what the Baron had told him, with considerable passion. \"I'll be the one who can see all places and all things simultaneously, the one who will lead the multitudes. I am the shortening of the way, the rescuer, the messiah, the one spoken of in countless legends.\"\n\nWords flowed from the fog. \"You have a charismatic presence that I find fascinating. Humans exhibit an irresistible compulsion to follow physically attractive, charming leaders. Properly harnessed, you could be an effective and destructive tool for us.\" The fog creature laughed, swirling the cold wind around him. Then his otherworldly eyes riveted on the Baron. \"You will see that the boy cooperates.\"\n\n\"Yes, of course. Are you Omnius?\"\n\n\"I speak for the evermind.\" The fogginess shifted as the mist flowed into itself and resolved into the gleaming metallic shape of a polished robot with an exaggerated but menacing smile molded onto his face. \"For the sake of convenience, I call myself Erasmus.\"\n\nThe walls of the chamber shifted like a kaleidoscope to reveal hundreds of angular combat robots stationed around the perimeter like strange beetles. Their metal eyes glittered in the same hostile fashion.\n\n\"Perhaps I will question you now. Or later? Indecision is a very human thing, you know. We have all the time in the world.\" The smile on the robot's platinum face had locked into place. \"I so love your clich\u00e9s.\"\n\n## TWENTY-THREE YEARS AFTER\n\nESCAPE FROM CHAPTERHOUSE\n\n> _Even with a Navigator's incredible mental advancement, I cannot forget the fundamental thread that ties us to the rest of humanity: the old emotion of_ hope.\n> \n> \u2014NAVIGATOR EDRIK,  \n> unacknowledged message to the Oracle of Time\n\nThe four specialized Guild craft were shaped like hornets, sleek sensor-studded ships that skimmed low over the waves of Buzzell. Scan eyes pointed down at the water, searching for movement. From the lead ship Waff peered through the spray-specked plaz windows, hoping to catch a glimpse of seaworms. The Tleilaxu's excitement and anticipation were palpable. Worms were down there somewhere. _Growing_.\n\nHe had released the creatures just over a year ago, and judging from the flurry of rumors the Guild had picked up, the seaworms must have thrived. None of the Bene Gesserit witches on the rocky islands understood where the serpentine creatures had come from. Now, Waff thought with a thrill, it was time to reap the harvest he had sown. He couldn't wait to see them, to know that he had accomplished his holy mission.\n\nThe sky was overcast, with patches of fog lying low over the sea. At regular intervals, the scanning crews dropped sonic pulsers into the water. The throbbing signals would map the movements of large underwater denizens, and theoretically attract the seaworms just as Fremen thumpers had once attracted the huge monsters on old Rakis. Near Waff in the cockpit, five silent Guildsmen monitored the equipment while separate, smaller hunting platforms circled lower, keeping pace with the hornets. Periodically the platforms went back to check the points where the pulsers had been dropped.\n\nThe leviathans of the deep from the ancient scriptures were more than just God's judgment on powindah unbelievers. This was the return of the Prophet, God's Messenger resurrected from the ashes of Rakis, in an adaptive new form.\n\nThe initial sightings of the beasts had occurred within six months. At first the tales told by the amphibious soostone harvesters had been met with disbelief, until the seaworms attacked in full view of island settlements. According to eyewitness accounts\u2014and Bene Gesserits were well trained in accurate observation\u2014the monstrous things had grown far larger than Waff had predicted. Truly, a sign from God that his work was blessed!\n\nSo long as they were fed, the worms continued to grow and multiply. Seaworms apparently preferred to eat the large cholisters that produced soostones, tearing into beds tended by Phibians. The aquatic people had rallied to drive away the sea monsters, but they had failed.\n\nWaff smiled. Of course they had failed. One could not change a path that God had laid down.\n\nThe angry witches had led hunting parties, taking boats out onto the waters, guided by vengeful Phibians. They begged Chapterhouse for weapons to kill the seaworms. But with Enemy forces attacking hundreds of fringe worlds, and the industries of Junction and Ix consuming most of the New Sisterhood's resources, they were stretched far too thin.\n\nThe Bene Gesserits needed soostone wealth to build and replenish their armies faster than the Enemy could destroy them, but if the seaworms produced what Waff hoped, the creatures would be worth far more than any gem. Soon, there would be multiple sources of spice, including a new and more potent form. Waff could transplant the creatures to any ocean planet, where they could thrive without reconfiguring an entire ecosystem. Considering their current monopoly on melange, that would not make the Sisterhood happy.\n\nThe pilot circled the lead hornet ship. Guild assistants stared fixedly at monitors. \"Picking up shadows at various depths. Numerous tracks. We are close.\"\n\nWaff moved eagerly to the other side of the craft and stared down at the choppy waves. Pulse beacons continued to emit siren songs, and hunting platforms flitted alongside. \"Be ready to move as soon as you detect a worm. I want to see one. Let me know when you have a sighting.\"\n\nDown in the water, he noticed two slick-skinned Phibians, who seemed curious about the pulse beacons and the flurry of activity. One raised a webbed hand in an incomprehensible signal as the hornet ships and hunting platforms streaked overhead.\n\n\"Seaworm surfacing,\" announced one of the Guildsmen. \"Target acquired.\"\n\nThe little Tleilaxu rushed forward to the cockpit. Below, a long, dark shape appeared in the water, breaching like a great whale. \"We must capture and kill it. That is the only way to see what's inside.\"\n\n\"Yes,\" said the Guildsman. Waff narrowed his eyes, never able to understand these people. Was the man agreeing with him, or simply acknowledging the orders? This time he didn't care.\n\nWaff glanced at the projected map, noting that their search had taken them to one of the inhabited rocky islands. Once he verified the success of the new worms, there would be no need to continue keeping secrets. The witches could do nothing about the seaworms, nor what they produced. They could not stop his work. Today, after his team captured a specimen and confirmed the results of his experiments, the truth would be obvious.\n\n_We will show the witches what lies beneath the waves, and let them draw their own conclusions_.\n\nThe lead hornet craft slowed, its engines buzzing. The moment the seaworm emerged from the waves, ringed and glistening, Waff's hunters fired a fusillade of supersonic harpoons from their hovering platforms. The barbed tips hit the beast before it could realize its danger and submerge. Spear points caught in the soft rings, anchoring themselves as the worm thrashed and writhed. Waff felt joy, as well as a twinge of sympathetic pain. From behind the lead craft, three other hornet ships shot more harpoons into the trapped creature, pulling back on hyperfilament cables.\n\n\"Don't damage it too much!\" Waff intended to kill the thing anyway\u2014a necessary sacrifice in the name of the Prophet\u2014but if the carcass and internal organs were too mangled, his dissection would be more difficult.\n\nThe group of hornet craft went into hover mode above the waves, their cables taut and straining as the seaworm thrashed. Milky fluid oozed into the water, dissipating before the Tleilaxu researcher could order one of the Guildsmen to collect samples. Other seaworms circled their struggling brother like hungry sharks.\n\nThe worm was twenty meters long\u2014a tremendous rate of growth for such a short time. He was impressed. If the creatures were reproducing as rapidly as they were growing, the oceans of Buzzell would soon be teeming with them! He couldn't ask for more.\n\nThe wounded beast quickly tired. Engines humming with the strain, the Guild vessels began to drag the feebly struggling worm toward the nearest reef, which was barely visible in the wispy fog banks. The small hunting platforms returned to the hornet ships, docking inside their cramped cargo holds.\n\nThe island was one of the main Sisterhood outposts for soostone processing, complete with barracks, warehouses, and a flattened spaceport capable of handling small ships. _Let the witches see this!_\n\nFlying in formation, the hornet ships towed the captive worm to shore. In the water below, at least twenty Phibians appeared carrying crude spears and tridents\u2014as if they thought they could pose a threat to the giant creatures! Shouting curses and threats, the Phibians attacked the tangled worm, stabbing and cutting.\n\nAnnoyed at the interference, Waff turned to his Guildsmen. \"Drive them away!\" Using small cannons mounted on the deck of the lead hornet ship, Guildsmen took potshots at the Phibians, killing two. The others dove underwater. At first they left the bloody corpses bobbing in the waves, but a number of the Phibians returned moments later. When they attempted to retrieve their fallen comrades, a second seaworm streaked in and devoured the bodies.\n\nThe droning hum of the hornet ships attracted a crowd of women on the docks as the sluggish prize was dragged into the village harbor. Dark-clad Sisters left their barracks, perhaps thinking that smugglers or CHOAM representatives had arrived. From the recent depredations of the seaworms, most soostone operations had stalled. Sorting bins and packaging lines were silent and unmanned.\n\nHis chest swelled with pride, Waff jumped down from the ramp onto the metal and stone wharf as the Guildsmen hoisted the ringed creature onto the main dock. Its narrow tail drooped into the water. Exhausted from its struggles and oozing fluids from the harpoon wounds, the captive worm thrashed one more time, expending the last of its energy. Though Waff and his servants had conquered and subdued the worm, he still felt impressed to be so close to the magnificent creature.\n\nSeven curious Phibians floated by the dock pilings, staring upward. In their bubbling, hissing voices, they mumbled in awe.\n\nWaff stood triumphantly before the large, dripping thing. Slime trickled from the dead seaworm onto the dock, and a gush of milkygray fluid flowed from its mouth. The long sharp teeth were very fine, like needles. Rather than reeking of fish, the seaworm had a distinct sharp-sweet odor with a hint of pungent cinnamon.\n\n_Perfect_!\n\nWomen came forward to confront Waff. \"We've never captured and killed a seaworm before,\" said a Sister in a brown dress who introduced herself as Corysta. She seemed delighted to see the leviathan dead. \"They have caused great havoc in the seas.\"\n\n\"And they will continue to do so. Learn to adapt your operations.\" Waff perfunctorily turned from her to issue instructions to his crew, then told Corysta and the other Bene Gesserits to stay back. \"We are on strict Guild business. Do not attempt to interfere.\"\n\nThough dead, the seaworm twitched as nerve impulses continued to fire. Waff ordered the Guildsmen to lash down the carcass, so that he could dissect it without interruption. The Guild assistants brought him a lascutter, a superfine shigawire saw, spreaders, and shovels.\n\nSetting the lascutter to full power and holding it in both hands, Waff swept sideways in a broad arc and sliced the seaworm open, so that the round, dripping segments flopped apart. Guildsmen hurried forward with spreaders to pull open the wound and expose the internal structure. Waff reveled in the gore. The Prophet must be so pleased with him.\n\nIn preparation, he had already killed and autopsied two of the original small specimens in his laboratory, so he knew the creatures' basic arrangement of organs. The worm was a biologically simple creature, and working on this larger scale made the process easier. Water and slime oozed onto the dock, splattering Waff. Under other circumstances he might have been disgusted, but this was the sacred essence of his Prophet. The Tleilaxu man sniffed more deeply, and there\u2014a definite undertone to the smell\u2014he caught the vital, pungent aroma of pure melange. No doubt about it.\n\nWaff buried his arms up to his shoulders in the organs, feeling around, identifying specific structures by their shapes and textures. Guild assistants used wide scoops to shovel offal onto the dock. Witches and Phibians watched in fascination, but Waff paid little attention to them.\n\nIgnoring the obviously confused and impotent Sisters, he laser-cut deeper into the worm, sliced along its length and rummaged through the stinking debris, until finally a large bluish-purple lump of soft liverlike material spilled out. Waff stepped back for a breath, then leaned closer, poking and prodding with his fingers. He made a cut with the lasknife at its lowest setting.\n\nA rich oily-cinnamon odor boiled out, so thick it could be seen as fumes. Waff reeled dizzily. The intensity of the melange nearly bowled him over. \"Spice! The creature is saturated with melange! Extremely concentrated spice.\"\n\nThe Sisters looked at each other and came closer with curious expressions. \"Spice? The seaworms produce _spice_?\" The Guildsmen stood close to Waff and his dripping prize, blocking the Bene Gesserits.\n\n\"The seaworms destroyed our soostone beds!\" another woman shouted.\n\nWaff stared them down. \"These creatures may have destroyed one economy on Buzzell, but they created an even more important one.\" His assistants picked up the large, melange-saturated organ and carried it back to the nearest hornet ship. Waff would have to test the substance thoroughly, but he already felt confident in what he would find.\n\nUp in the orbiting Heighliner, Edrik the Navigator would be pleased.\n\nDripping with slime and seawater, Waff hurried back to the ship.\n\n> _Some see spice as a blessing, others as a curse. To everyone, however, it is a necessity_.\n> \n> \u2014PLANETOLOGIST PARDOT KYNES,  \n> Original Arrakis Notebooks\n\nAfter her long and exhausting journey across the Old Empire, from the planets preparing for battle, to the Guild shipyards, to the soostone operations on Buzzell, Mother Commander Murbella returned to Chapterhouse with renewed determination. Since she had been gone for many months, her quarters in the Keep now looked like a stranger's rooms. Harried acolytes and male workers scurried to unload her belongings from her ship.\n\nAfter a polite knock on the door, an acolyte stepped in. The young woman had short brown hair and a furtive smile. \"Mother Commander, Archives sent these updated charts. They were supposed to be waiting for you upon your arrival.\" She held out thin maps with finely detailed lines, then drew back, startled, when she noticed the hulking combat robot, deactivated but still standing in the corner of the room like a war trophy.\n\n\"Thank you. Don't mind the machine\u2014it is as dead as they will all soon be.\" Murbella took the reports from the girl's hands. With a second glance, she realized the young woman was her own daughter Gianne, her last child with Duncan Idaho. Another daughter, Tanidia, also raised by the New Sisterhood, had been shipped off to work among the Missionaria.\n\n_Do Gianne or Tanidia even know who their parents are_? Years ago she had made the choice to tell Janess of her parentage, and the young woman had thrown herself into the study and understanding of her famous father. But Murbella had let her other two daughters be raised among the Bene Gesserit in the more traditional way. She doubted they knew how special they were.\n\nGianne seemed hesitant, as if hoping the Mother Commander would ask her for something else. Though she knew the answer, on impulse Murbella asked, \"How old are you, Gianne?\"\n\nThe girl seemed startled that she knew her name. \"Why, twenty-three, Mother Commander.\"\n\n\"And you have not yet undergone the Agony.\" It was not a question. Occasionally, the Mother Commander had been tempted to use her position to interfere with the girl's training, but had not done so. A Bene Gesserit was not supposed to show such weakness.\n\nThe young woman seemed ashamed. \"The proctors suggest that I would benefit from more focus and concentration.\"\n\n\"Then devote yourself to that. We need every Reverend Mother we can find.\" She glanced at the ominous combat robot. \"The war has worsened.\"\n\nMURBELLA REALIZED SHE could not rest, could not waste the time. She demanded to see her advisors, Kiria, Janess, Laera, and Accadia. The women arrived, expecting a meeting, but Murbella herded them out of the Keep. \"Prepare a 'thopter. We leave immediately for the desert belt.\"\n\nCarrying a stack of reports, Laera did not react well to the news. \"But Mother Commander, you've been gone so long. Many documents await your attention. You have to make decisions, give proper\u2014\"\n\n\" _I_ decide the priorities.\"\n\nKiria, looking scornful, bit her words back when she noted the Mother Commander's complete seriousness. They all crowded aboard an empty ornithopter, then waited for the tedious takeoff preparations. Murbella wouldn't sit still for a moment. \"If I don't get a pilot, I'll fly this damned thing myself.\" A young male pilot was quickly brought to her.\n\nAs the 'thopter took off, she finally turned to her advisors and explained, \"The Guild demands an exorbitant payment for all the warships we have under construction. Ix already accepts payments only in melange, and now that soostones from Buzzell are no longer economically viable, everything hinges on spice. That is our only coin significant enough to appease the Guild.\"\n\n\"Appease them?\" Kiria snapped. \"What madness is this? We should _conquer_ them and force them to produce the weapons and vessels we need. Are we the only ones who understand the threat? Thinking machines are coming!\"\n\nJaness was astonished by the other woman's suggestion. \"Attacking the Guild would create open civil warfare at a time when we can least afford it.\"\n\n\"Do we have enough resources to spend on these ships?\" asked Laera. \"Our credit has already been strained past its limits with the Guild Bank.\"\n\n\"We all face a common enemy,\" old Accadia said. \"Surely, the Guild and Ix would be willing\u2014\"\n\nMurbella clenched her hands. \"This has nothing to do with altruism or greed. Despite the best intentions, resources and raw materials do not appear like rainbows after a storm. Populations must be fed, ships must be fueled, energy must be produced and expended. Money is only a symbol, but economics is the engine that drives the whole machine. The piper must be paid.\"\n\nThe 'thopter raced across the sky, buffeted by dry winds and blown dust long before they saw the desert. Murbella gazed out the curved window, sure that dunes had not extended this far across the continent the last time she'd visited the desert. It was a spreading _anti_ flood, total dryness sweeping outward in waves. At the heart of the desert, the worms grew and reproduced, keeping the cycle going in a perpetually increasing spiral.\n\nThe Mother Commander turned to the woman behind her. \"Laera, I require a complete assessment of our spice-harvesting operations. I need to know numbers. How many long tons of melange do we gather? How much do we have in our stockpiles, and how much is available for export?\"\n\n\"We produce enough to meet our needs, Mother Commander. Our investment continues to go into expanding the operations, but our expenditures have increased dramatically.\"\n\nKiria muttered a bitter comment about the Ixians and their endless bills.\n\n\"We may need to bring in outside workers,\" Janess pointed out. \"These obstacles can be overcome.\"\n\nThe 'thopter swooped toward a chimney-plume of dust and sand thrown up by a harvester. Around it, like wolves circling a wounded animal, several sandworms approached the vibrations. Already the operations were beginning to wrap up, with miners rushing and carryalls hovering to snatch the heavy machinery away as soon as worms ventured too close.\n\nMurbella said, \"Squeeze the desert, wring out every gram of spice.\"\n\n\"Beast Rabban was given the same task long ago, during the days of Muad'Dib,\" said Accadia. \"And he failed in a spectacular fashion.\"\n\n\"Rabban did not have the Sisterhood behind him.\" She could see Laera, Janess, and Kiria all making silent mental calculations. How many workers could be diverted to the desert zone? How many offplanet prospectors and treasure hunters could they allow on Chapterhouse? And how much spice would be enough to keep Guild and Ixian engineers producing the desperately needed ships and weapons?\n\nThe male pilot, having been silent until now, said, \"While we are out here, Mother Commander, shall I take you to our desert research station? The planetology crew is studying the sandworm cycle, the spread of desert, and the parameters necessary for the most effective spice harvest.\"\n\n\"'Understanding is required before success is possible,' \" Laera said, quoting directly from the old Orange Catholic Bible.\n\n\"Yes, let me inspect this station. Research is necessary, but in times like these it must be _practical_ research. We have no time for frivolous studies concocted by the whim of an offworld scientist.\"\n\nThe pilot banked the 'thopter and accelerated far out into the open desert. On the horizon, a lumpy, black ridge showed a reef of buried rock, a safe bastion where worms could not go.\n\nShakkad Station had been named after Shakkad the Wise, a ruler from days before the Butlerian Jihad. Nearly lost in the mists of legend, Shakkad's chemist had been the first man in history to recognize the geriatric properties of melange. Now, far from Chapterhouse Keep or any outside interference, a group of fifty scientists, Sisters, and their support staff lived and worked. They set up weather-testing devices, traveled out onto the dunes to measure chemical changes during spice blows and monitor the growth and movement of sandworms.\n\nWhen the 'thopter settled onto a flat cliff outcropping that served as a makeshift landing pad, a group of scientists came out to meet them. Dusty and windblown, a survey team was just returning from the edges of the desert where they had set out sampling poles and weather-testing instruments. They wore stillsuits, exact reproductions of those once used by the Fremen.\n\nA majority of the scientists at Shakkad Station were men, and several of the older ones had made brief expeditions to charred Rakis itself. Three decades had passed since the ecological destruction of the desert planet, and by now few experts could claim firsthand knowledge of the sandworms or original conditions on Dune.\n\n\"How may we assist you, Mother Commander?\" asked the station manager, an offworlder who pushed dusty protective goggles up onto his forehead. The man's owlish eyes had already begun to turn a faint blue. Spice had been in his diet every day since his arrival at the outpost. His body gave off an unpleasant sour odor, as if he had taken his assignment in the waterless belt with particular seriousness, even to the point of foregoing regular bathing.\n\n\"Assist us by getting more melange,\" Murbella answered bluntly.\n\n\"Do your teams have everything they need?\" Laera asked. \"Do you require additional supplies or workers?\"\n\n\"No, no. We just want solitude and the freedom to work. Oh, and time.\"\n\n\"I can give you the first two. But time is a commodity none of us has.\"\n\n> _We can conquer our enemy, of course, but is it worthwhile to achieve victory without understanding the flaws of our opponent? Such an analysis is the most interesting part_.\n> \n> \u2014ERASMUS,  \n> Laboratory Notebooks\n\nThe machine-based cathedral on Synchrony was a mere manifestation of what the rest of the galaxy might become. Omnius was pleased at the progress the thinking-machine fleet had made in the past few years, conquering one system after another, but Erasmus knew that so much more remained to do.\n\nThe voice of Omnius boomed much louder than necessary, as he sometimes liked to do. \"The New Sisterhood offers the strongest resistance to us, but I know how to defeat them. Scouts have verified the secret location of Chapterhouse, and I have already dispatched plague probes there. Those women will soon be extinct.\" Omnius sounded quite bored. \"Shall I display the map of star systems, so you know just how many we have encountered and conquered? Not a single failure.\"\n\nDisplays jabbed into Erasmus's mind, regardless of whether he wanted to see them or not. In bygone days the independent robot had been able to decide what he wanted to download from the evermind, and what he didn't. Increasingly, however, Omnius had found ways to override the robot's decision-making abilities, forcing data into his internal systems, sliding it past multiple firewalls.\n\n\"Those are mere symbolic victories,\" Erasmus said, intentionally shifting to his disguise of the wrinkled old woman in gardening clothes. \"I am pleased that we have made it to the edge of the Old Empire, but we still have not won this war. I have spent millennia studying these stubborn, resourceful humans. Do not assume victory until we actually have it in our hands. Remember what happened last time.\"\n\nOmnius's snort of disbelief echoed through the entire city of Synchrony. \"We are by definition better than flawed humanity.\" From a thousand watcheyes, he looked down upon Erasmus and his matronly disguise. \"Why do you persist in wearing that embarrassing shape? It makes you look weak.\"\n\n\"My physical body does not determine my strength. My mind makes me what I am.\"\n\n\"I am not interested in your mind either. I simply wish to win this war. I must win. I _need_ to win. Where is the no-ship? Where is my Kwisatz Haderach?\"\n\n\"You sound as demanding as Baron Harkonnen. Are you unconsciously imitating him?\"\n\n\"You gave me the mathematical projections, Erasmus. Where is the superhuman? Answer me.\"\n\nThe robot chuckled. \"You already have Paolo.\"\n\n\"Your prophecy also guaranteed a Kwisatz Haderach aboard the no-ship. I want both versions\u2014redundancy to assure victory. And I do not want the humans to have one. I must control them both.\"\n\n\"We will find the no-ship. We already know there are many intriguing things aboard, including a Tleilaxu Master. He may be the only one left alive, and I would very much like to speak with him\u2014as would you. The Master needs to see how all those Face Dancers have shaped us, built us, so that we could become closer to gods. Closer than humans, at any rate.\"\n\n\"We will keep sending out our net. And we will find that ship.\"\n\nAll around the city, in a dramatic statement of the evermind's impatience, towering buildings collapsed, full metal structures fell in upon themselves. Hearing the thundering sounds and feeling the floor shake beneath him, the independent robot was not impressed. Too many times he had witnessed such overblown theatrics. Omnius certainly enjoyed running the show, for better and often for worse, though Erasmus continually tried to control the evermind's excesses. The future depended on it\u2014the future that Erasmus had ordained.\n\nHe dug through the projections that he'd digested from trillions of datapoints. All of his results were colored to fit precisely the prophecies he had formulated himself. Omnius believed them all. The gullible evermind relied too much on filtered information, and the robot played him well.\n\nGiven the proper parameters, Erasmus was absolutely certain the millennia ahead would turn out properly.\n\n> _Those who see do not always understand. Those who claim to understand can be the blindest of all_.\n> \n> \u2014the Oracle of Time\n\nWhat remained of Norma Cenva's ancient corporeal form was confined inside a chamber that had been built and modified around her during thousands of years. But her mind knew no physical boundaries. She was only tenuously connected to flesh, a biological generator of pure thought. The Oracle of Time.\n\nHer mental links to the fabric of the universe gave her the ability to travel anywhere along infinite possibilities. She could see the future and the past, but not always with perfect clarity. Her brain was such that she could touch the Infinite and almost\u2014 _almost_ \u2014comprehend it.\n\nHer nemesis, the evermind, had laid down a vast electronic network throughout the fabric of space, a complex tachyon road map that most people could not see. Omnius used it as a net to sift for his prey, but so far he had not managed to snare the no-ship.\n\nLong ago, Norma had created the precursor to the Guild as a means of fighting the thinking machines. Since that time, the Guild had taken on a life of its own, growing away from her while she stretched herself farther into the cosmos. Politics between planets, power struggles between the Navigator faction and the human Administrators, monopolies on valuable commodities such as soostones, Ixian technologies, or melange\u2014such problems did not concern her.\n\nKeeping watch over mankind required an investment of her mental currency. She felt the turmoil in civilization, knew the great schism in the Guild. She would have chastised the Administrators for creating such a crisis, if she could only remember how to speak to such small people. Norma found it exhausting to talk in simple enough terms to make herself comprehensible even to her advanced Navigators. She had to make them understand the true Enemy, so that they could shoulder the burden of fighting.\n\nIf the Oracle of Time did not attend to grander priorities, no one else would. No one else in the universe could possibly do it. With her prescience, she grasped what was most important: _Find the lost no-ship_. The final Kwisatz Haderach was aboard, and Kralizec's black cloud had already released its torrents. But Omnius was searching for the same thing and might get to it first.\n\nShe had felt the recent struggles between the Bene Gesserits and the Honored Matres. Before that she had witnessed the original Scattering and Famine Times, as well as the extended life and traumatic death of the God Emperor. But all of those events were little more than background noise.\n\n_Find the no-ship_.\n\nAs she had always foreseen and feared, the unrelenting foe had come back. No matter what guise the thinking machines now wore, regardless of how much they had changed, the Enemy was still the Enemy.\n\n_And Kralizec is well under way_.\n\nWhile her prescience flowed outward and inward, ripples of time eddied around her, making accurate predictions difficult. She encountered a vortex, a random, powerful factor that could change the outcome in uncounted ways: a Kwisatz Haderach, a person as anomalous as Norma Cenva herself, a wildcard variable.\n\nOmnius wanted to guide and control that special human. The evermind and his Face Dancers had sought the no-ship for years, but so far Duncan Idaho had eluded capture. Even the Oracle had been unable to find him again.\n\nNorma had done her best to thwart the Enemy every step of the way. She had saved the no-ship, hoping to protect the people onboard, but she had lost contact afterward. Something on the ship was more effective than a no-field at blinding her search. She could only hope the thinking machines were as blind.\n\nThe Oracle's search continued, her thoughts reeling out in delicate probes. Alas, the vessel simply _was not there_. In some mysterious manner, the passengers hid it from her . . . assuming it had not been destroyed.\n\nThough her prescience was not clear, Norma realized that time was growing shorter and shorter, for everyone. The crux point had to occur soon. Thus, she needed to gather her allies. The foolish Administrators had reconfigured many of their great ships, installing artificial controls\u2014like thinking machines!\u2014so that she could no longer call upon them through her paranormal means. But she could still command a thousand of her loyal Navigators. She would make them ready for battle, the final battle.\n\nAs soon as she found the no-ship. . . .\n\nThe Oracle of Time expanded her mind, casting her thoughts into the void like a fisherman, until the neural ache was incredible. She pushed harder than ever, stretching her boundaries beyond anything she had previously attempted. No price of pain could be too great. She knew full well the consequences of failure.\n\nAll around her, a vast clock ticked.\n\n> _There must be a place where we can find a home, where we can be safe and rest. The Bene Gesserit sent out so many Sisters on their own Scattering before the Honored Matres came. Are they all lost, as well?_\n> \n> \u2014SHEEANA,  \n> confidential no-ship journals\n\nFlying ever onward, the _Ithaca_ reeled from the recent spate of damage. And the saboteur continued to elude them. _What more can we do to track him down_? Even Duncan's most thorough Mentat projections offered no new suggestions.\n\nMiles Teg and Thufir Hawat once again dispatched teams to inspect, and even ransack, the quarters of all passengers, hoping to find incriminating evidence. The Rabbi and his people complained about purported violations of their privacy, but Sheeana demanded their full cooperation. To the extent possible, Teg had been closing down sections of the immense vessel with electronic barricades, but the clever saboteur was able to get through anyway.\n\nAssuming no further incidents, with the life-support, airrecirculation, and food-growth systems crippled, the passengers could not last more than a few months without stopping somewhere to replenish the stores. But it had been years since they had found another suitable world.\n\nDuncan wondered: _Is someone trying to destroy us . . . or drive us to a particular place_?\n\nWith no starmaps or reliable guidance, he tried to use his uncanny prescience one more time. Another big gamble. Activating the Holtzman engines and closing his eyes, Duncan folded space again, spinning the cosmic roulette wheel\u2014\n\nAnd the no-ship emerged, intact but still lost, at the perimeter of a star system. A yellow sun with a necklace of worlds, including a terrestrial planet that orbited at the appropriate distance to support life. Possibly habitable, certainly with oxygen and water that the _Ithaca_ could take aboard. A chance . . .\n\nOthers had gathered on the navigation bridge by the time the no-ship approached the uncharted world. Sheeana got down to business. \"What do we have here? Breathable air? Food? A place to live?\"\n\nGazing through the observation window, Duncan was pleased at what he saw. \"The instruments say yes. I suggest we send a team immediately.\"\n\n\"Resupply is not good enough,\" Garimi said, her tone gruff. \"It never was. We should consider remaining here, if this is the kind of world we've been looking for.\"\n\n\"We considered that at the planet of the Handlers, too,\" Sheeana said.\n\n\"If the saboteur drove us here, we need to be very cautious,\" Duncan said. \"I know it was a random foldspace jump, but I'm still troubled. Our pursuers cast a wide net. I would not be quick to dismiss the possibility that this place is a trap.\"\n\n\"Or our salvation,\" Garimi suggested.\n\n\"We'll have to see for ourselves,\" Teg said. Working with the bridge controls, he displayed high-resolution images on the wide screens. \"Plentiful oxygen and vegetation, especially at the higher latitudes away from the equator. Clear signs of habitation, small villages, midsized cities, mostly far to the north. Large-scale meteorological scans show that the climate is in upheaval.\" He pointed to storm patterns, swaths of dying forests and plains, large lakes and inland seas shriveling into dust bowls. \"Very few clouds in the equatorial latitudes. Minimal atmospheric moisture.\"\n\nStilgar and Liet-Kynes, always fascinated with new worlds, joined the group on the high deck. Kynes drew a quick breath. \"It's turning into a wasteland down there. An artificial desert!\"\n\n\"I've seen this before.\" Sheeana studied a clear brown band like a knife slash across what had apparently been a lushly forested continent. \"It's like Chapterhouse.\"\n\n\"Could this be one of Odrade's seed planets?\" Stuka asked, from her usual position at Garimi's side. \"Did they bring sandtrout here and disperse them? Will we find our Sisters down on that planet?\"\n\n\"Untainted Sisters,\" Garimi said with a gleam in her eyes.\n\n\"Quite possibly,\" Sheeana said. \"We'll have to go down there. This looks like more than a place to replenish our resources.\"\n\n\"A new colony.\" Stuka's excitement was infectious. \"This could be the world we've been looking for, a site to reestablish Chapterhouse. A new Dune!\"\n\nDuncan nodded. \"We cannot pass up an opportunity like this. My instincts brought us here for a reason.\"\n\n> _Are we the last ones left alive? What if the Enemy has destroyed the rest of mankind by now, back in the Old Empire . . . back with Murbella? In that case, it is imperative that we establish as many colonies as possible_.\n> \n> \u2014DUNCAN IDAHO,  \n> no-ship logs\n\nKeeping themselves hidden from the planet's inhabitants, several teams of efficient Bene Gesserits launched a major effort to restock the no-ship with necessary air, water, and chemicals. They sent out mining ships, air scoops, water-purification tankers. That was the _Ithaca_ 's immediate priority.\n\nStilgar and Liet-Kynes insisted on going down to inspect the growing desert band. Seeing the passion on the faces of the two awakened gholas, neither Teg nor Duncan could deny the request. Everyone was guardedly optimistic about finding a welcoming landscape here, and Sheeana wondered if this might be a place where she could release her seven captive sandworms. Although Duncan could not leave the veiling of the no-ship, because then he would be exposed to the Enemy searchers, he had no cause to prevent the others from finding a home at last. Perhaps this would be it.\n\nBashar Teg piloted the lighter down to the surface himself, accompanied by Sheeana and an eager Stuka, who had long wanted to establish a new Bene Gesserit center, rather than just drift aimlessly in space. Garimi had let her staunch supporter make the first foray, while she formulated plans with her ultraconservative Sisters aboard the no-ship. Stilgar and Liet were most eager just to set foot on the desert\u2014a real desert with open skies and endless sands.\n\nTeg flew directly toward the ravaged arid zone, where an ecological battle was taking place. If this was indeed one of Odrade's seed planets, the Bashar knew how voracious sandtrout would seal away a planet's water, drop by drop. Environmental checks and balances would fight back with shifting weather patterns; animals would migrate to still-untouched regions; stranded plant life would struggle to adapt, and mostly fail. Reproducing sandtrout could act much faster than a world could adapt.\n\nSheeana and Stuka stared through the lighter's plaz viewing windows, seeing the spreading desert as a success, a triumph of Odrade's Scattering. To the exquisitely prudent Bene Gesserit, even the ruin of an entire ecosystem was an \"acceptable casualty\" if it created a new Dune.\n\n\"The change is happening so swiftly,\" Liet-Kynes said, his voice tinged with awe.\n\n\"Surely, Shai-Hulud is already here,\" Stilgar added.\n\nStuka echoed words that Garimi had said time and again. \"This world will be a new Chapterhouse. The hardships will mean nothing to us.\"\n\nWith the detailed information in their archives, the people aboard the _Ithaca_ had all the expertise they needed to establish a new place to live. _Yes, a colony_. Teg rather liked the sound of the word, because it represented the hope of a better future.\n\nTeg knew, however, that Duncan could never stop running, unless he chose to face the Enemy directly. The mysterious old man and woman were still after him with their sinister net, or after something on the no-ship, maybe the vessel itself.\n\nThe lighter descended with a rough roar through the china-blue sky. In the middle of the abrupt desert band, dunes stretched as far as he could see. Sunlight reflected from the sands into bone-dry air, and thermal currents jostled the ship from side to side. Teg wrestled with the guidance systems.\n\nIn the back, Stilgar chuckled. \"Just like riding a sandworm.\"\n\nCruising over the middle of the widening desert belt, Liet-Kynes pointed at a rusty-red splash that marked an eruption from beneath the surface. \"Spice blow! No mistaking the color or pattern.\" He gave a wry smile to his friend Stilgar. \"I died on one of those. Damn the Harkonnens for leaving me to die!\"\n\nMounds rippled and stirred the top layer of sand, but they did not emerge into open air. \"If those are worms, they are smaller than the ones in our hold,\" Stilgar said.\n\n\"But still impressive,\" Liet added.\n\n\"They have had less time to mature,\" Sheeana pointed out. \"Mother Superior Odrade did not send volunteers on her Scattering until after the desertification of Chapterhouse was well under way. And we do not know how long the wandering Sisters took to get here.\"\n\nBelow, obvious lines marked the rapid expansion of the sandy wasteland, like ripples on a pond. At the fringes were die-off perimeters, places where all vegetation had perished and the dirt had become blowing dust. The encroaching desert had created ghost forests and inundated villages.\n\nFlying low, searching with uneasy anticipation, Teg discovered half-buried rooftops, the pinnacles of once proud buildings drowned under the spreading desert. In one shocking glimpse, he saw a high dock and part of a capsized boat that sat atop a blistering dune.\n\n\"I look forward to seeing our Bene Gesserit Sisters.\" Stuka sounded eager. \"Obviously they succeeded here in their mission.\"\n\n\"I expect they will welcome us,\" Sheeana admitted.\n\nAfter seeing the city drowned in sand, Teg did not think the original inhabitants of this planet would have appreciated what the refugee Sisters had done.\n\nAs the lighter followed the northern edge of the desert, the scanners picked out small huts and tents erected just beyond the sand's reach. Teg wondered how often the nomadic villages were required to move. If the arid zone expanded as rapidly as it had on Chapterhouse, this world would be losing thousands of acres every day\u2014and accelerating as sandtrout continued to steal precious water.\n\n\"Set down at one of those settlements, Bashar,\" Sheeana said to him. \"Any of our lost Sisters could be here on the edge of the dunes to monitor the progress.\"\n\n\"I long to feel real sand under my boots again,\" Stilgar muttered.\n\n\"It's all so fascinating,\" Liet said.\n\nAs Teg circled above one of the nomadic villages, people ran out and pointed up at them. Sheeana and Stuka pressed excitedly against the plaz windows, searching for distinctive dark Bene Gesserit robes, but they saw none.\n\nA formation of rocks towered over the village, a bulwark offering shelter against blowing sand and dust. People, waving, stood atop the pinnacles, but Teg could not determine if the gestures were friendly or threatening.\n\n\"See, they cover their heads and faces with cloths and filters,\" Liet said. \"The increased aridity forces them to adapt. In order to live here on the edge of the dry dunes, they are already learning to conserve bodily moisture.\"\n\n\"We could teach them how to make real stillsuits,\" Stilgar said with a smile. \"It has been a long time since I wore a decent one. I spent a dozen years aboard that ship, drowning my lungs with moisture. I can't wait to taste dry air again!\"\n\nTeg found an open landing area and brought the lighter down. He felt unaccountably troubled as the natives scurried toward them. \"Those are obviously nomadic camps. Why wouldn't they move inland, to where the climate is more hospitable?\"\n\n\"People adapt,\" Sheeana said.\n\n\"But why would they have to? Yes, the desert belt is growing, but there are still plenty of wide forests, even cities not far from here. Those people could outrun the spreading dunes for generations to come. Yet they stubbornly remain here.\"\n\nBefore the hatch opened to let in a breath of parched air, the nomads encircled the craft. Sheeana and Stuka, both wearing traditional dark robes from Chapterhouse so that their refugee Sisters would recognize them, boldly led the way. Teg followed with Stilgar and Liet.\n\n\"We are Bene Gesserit,\" Sheeana called to the people in universal Galach. \"Are any of our Sisters among you?\" Shielding her eyes against the brightness, she searched the few weathered female faces she saw, but got no response.\n\n\"Perhaps another village would be best,\" Teg suggested in a whisper. His tactical senses were alert.\n\n\"Not yet.\"\n\nAn elderly man drew closer, pushing a filter mask away from his face. \"You ask for Bene Gesserits? Here on Qelso?\" Though coarse, his accent was understandable. Despite his age, he appeared to be healthy and energetic.\n\nTaking the lead, Stuka stepped ahead of Sheeana. \"The ones who wore black robes, like ours. Where are they?\"\n\n\"All dead.\" The old man's eyes flashed.\n\nStuka's suspicion came too late. Moving like a striking snake, the man hurled a hidden knife from his sleeve, with deadly accuracy. At an unseen signal the rest of the throng rushed forward.\n\nStuka plucked clumsily at the blade that protruded from her chest but could not make her fingers work. Crumpling to her knees, she tumbled sideways off the lighter's ramp.\n\nSheeana was already moving, retreating. Teg shouted for Liet and Stilgar to get back inside the ship as he drew one of the stun weapons he had brought from the no-ship's armory. A large rock struck Stilgar in the head, and Liet helped his young friend, trying to drag him back into the lighter. Teg fired a swath of silvery energy, making part of the dusty mob collapse, but more knives and stones clattered at them.\n\nFrenzied people rushed the ramp from all sides, jumping at Teg. Many hands grabbed his wrist before he could fire again, and someone ripped the stunner out of his grip. More took hold of Liet by the shoulders, pulling him away.\n\nSheeana fought with a whirlwind of blows from her repertoire of Bene Gesserit fighting techniques. Soon a crowd of fallen attackers lay around her.\n\nWith a roar, Teg prepared to lurch into his hyperaccelerated metabolism, with which he could easily dodge blows and weapons, but a silvery beam from his own stunner gushed out like tinkling rain, dropping the Bashar, and then Sheeana.\n\nIN SHORT ORDER the villagers bound the hands of their four prisoners with strong cords. Though badly beaten, Teg regained consciousness and saw that Liet and Stilgar were tied together. Stuka's body lay near the ramp while the attackers ransacked the lighter for equipment and hauled things off.\n\nA group of men lifted Stuka's body. The old man retrieved his knife, yanking it from the dead woman's chest and wiping it on her robe with an expression of revulsion. He glowered at the corpse and spat, then marched toward the prisoners. Looking at the three young men, he shook his head in disapproval. \"I did not introduce myself. You may call me Var.\"\n\nDefiantly, Sheeana glared up at him. \"Why have you done this to us? You said you knew of the Bene Gesserit order.\"\n\nVar's face contorted, as if he had hoped to avoid speaking to her. He leaned close to Sheeana. \"Yes, we know the Bene Gesserit. They came here years ago and delivered their demon creatures to our world. An experiment, they said. An experiment? Look what they did to our beautiful land! It is becoming nothing more than useless sand.\" He held his knife, considered Sheeana for a long moment, then sheathed it. \"When we finally realized what those women were doing, we killed them all, but too late. Our planet is dying now, and we will fight to protect what's left of it.\"\n\n> _The first law of commercial viability is to recognize a need and meet it. When acceptable needs do not present themselves, a good businessman_ creates _them in any way possible_.\n> \n> \u2014CHOAM primary commercial directive\n\nWhen yet another Navigator died in his tank, few of the Spacing Guild's Administrators mourned the loss. The giant Heighliner was simply brought back to the Junction shipyards to be refit with one of the Ixian mathematical compilers. It was considered progress.\n\nAfter long years of practice, Khrone easily concealed his pleasure at the sight. So far every aspect of the wide-reaching plan had proceeded as expected, one domino falling after another. Posing in his familiar disguise as an Ixian inspection engineer, the leader of the Face Dancer myriad waited on a high, copper-floored platform. He observed the clamorous shipyards, while warm breezes and industrial fumes drifted around him.\n\nNearby, the human administrator Rentel Gorus was not quite as proficient at covering his satisfaction. He blinked his milky eyes and looked up toward the piloting bay of the ancient, decommissioned ship. \"Ardrae was one of the oldest remaining Navigators in our commercial fleet. Even with his spice supplies drastically cut, he clung to life much longer than we expected.\"\n\nA plump CHOAM representative said, \"Navigators! Now that these drains on our resources are disappearing one by one, Guild profits should increase significantly.\"\n\nWithout prompting from his master, the Mentat assistant recited, \"Knowing the lifetime of that Navigator, and considering the quantities of melange required to institute his initial mutation and conversion, I have calculated the total amount of spice consumed during his service to the Guild. With fluctuating prices based on the relative glut during the Tleilaxu years and recent skyrocketing costs due to severe shortages, the Guild could have bought three full-sized Heighliners, complete with no-field capabilities, for the same cost in spice.\"\n\nThe CHOAM man muttered in disgust, while Khrone remained silent. He found it most effective simply to listen and observe. Humans could be counted on to draw their own conclusions (often erroneous ones) so long as they were pointed in the proper direction.\n\nSavoring his secrets, Khrone thought of the numerous ambassadors the Guild had sent to the front, attempting to negotiate nonaggression treaties with the thinking machines, hoping to declare themselves neutral for the survival of the Guild. But many of those emissaries had been Khrone's Face Dancer plants, who intentionally achieved no success. Others\u2014the human ones\u2014never returned from the encounters.\n\nWith Richese conveniently obliterated by rebel Honored Matres (secretly guided by Khrone's Face Dancers), humans had no choice but to turn to Ix and the Guild in order to obtain the technological items they required. The Junction shipyards had always been immense complexes for constructing huge interstellar ships.\n\nMurbella's defensive fleet was growing with remarkable speed, but Khrone knew that even these efforts would not be very effective against the sheer size and scope of Omnius's military, which had been thousands of years in the making. The fabrication facilities of Ix (also controlled by Face Dancers) were still delaying the development and modification of the Obliterator weapons upon which the Sisterhood's defense relied. And since every new Guildship was controlled by an Ixian mathematical compiler rather than a Navigator, the Mother Commander and her allies would have many surprises in store.\n\n\"We will build more ships to make up for the obsolescence of the Navigators,\" Administrator Gorus promised. \"Our contract with the New Sisterhood seems infinite. We have never had so much business.\"\n\n\"And yet interplanetary trade is down drastically.\" The CHOAM representative nodded to both Khrone and Gorus. \"How is the Sisterhood to pay for these expensive ships and armaments?\"\n\n\"They have met their obligations with an increased flow of melange,\" Gorus said.\n\nKhrone finally nudged the conversation where he wished it to go. \"Why not accept payment in horses or petroleum or some other outdated and useless substance? If your Navigators are dying and your ships function perfectly well with Ixian mathematical compilers, the Guild no longer needs melange. What good is it to you?\"\n\n\"Indeed, its value _is_ greatly diminished. Over the past quarter century, following the destruction of Rakis, the Tleilaxu worlds, and so much more, those who could afford spice recreationally have dwindled to a tiny number.\" The CHOAM representative glanced at his Mentat, who nodded in agreement. \"Chapterhouse might have a monopoly on melange, but by their very iron grip, by decreasing the amount of spice available for popular consumption, they have strangled their own market. Few people really _need_ it anymore. Now that they have learned to live without spice, will they be so keen to reacquire their addictions?\"\n\n\"Probably,\" Gorus said. \"You need only drop the price, and we'd have a stampede of customers.\"\n\n\"The witches still control Buzzell,\" the Mentat pointed out. \"They have other ways to pay.\"\n\nThe CHOAM man disdainfully raised his eyebrows. He made very expressive noises without words. \"Luxury items during war? Not a good economic investment.\"\n\n\"Providing soostones is no longer easy for them either,\" Gorus pointed out, \"since sea monsters are destroying the shell beds and attacking their harvesters.\"\n\nKhrone listened intently. His own spies had brought back disturbing, but intriguing, reports about strange happenings on Buzzell, and a possible secret Navigator project centered there. He had demanded more information.\n\nKhrone watched while jawlike machinery on a large crane pried open the pilot's bay on the gigantic decommissioned Heighliner. Heavy suspensor lifters strained and groaned as they pulled out the Navigator's thick-walled plaz tank. During the slow, clumsy extraction, the tank caught on the edge of the hole in the Heighliner's structure. A hull plate broke off and spun downward, striking the side of the Heighliner and ricocheting with a shower of sparks, then tumbling until it finally slammed into the ground far below.\n\nWisps of orange spice gas escaped from the Navigator's chamber, stray exhaust vapors leaking into the atmosphere. Only a decade or so ago, such a quantity of wasted spice gas would have been enough to buy an Imperial palace. Now the CHOAM representative and Administrator Gorus watched it dissipate without comment. Gorus spoke into a tiny microphone at his collar. \"Deposit the tank in front of us. I wish to stare at it.\"\n\nThe crane raised the thick-walled chamber, swung it away from the hulk of the Heighliner, and brought it over to the observation platform. Suspensors lowered the container gently to the copper-floored deck, where it settled with a distressingly heavy thump. Spice gas continued to vent from the chink in the thick plaz.\n\nThe melange vapors smelled strangely flat and metallic, telling Khrone that the Navigator had inhaled and exhaled them until very little spice potency remained. At a curt direction from the milky-eyed Administrator, silent Guild workers unsealed a cap on the tank, causing the remainder of the spice to blast out in a death rattle.\n\nAs the polluting gas drained, the murky clouds swirled and thinned, revealing a silhouetted form slumped inside. Khrone had seen Navigators before, of course, but this one was flaccid, gray-skinned, and very dead. The bulbous head and small eyes, webbed hands, soft amphibious-looking skin gave the thing the appearance of a large, misshapen fetus. Ardrae had died days earlier, starved for melange. Though the Guild now had plenty of spice in their stockpiles, Administrator Gorus had cut off the Navigators' supplies some time ago.\n\n\"Behold, a dead Navigator. A sight few will ever see again.\"\n\n\"How many still survive among your Guildships?\" Khrone asked.\n\nGorus seemed evasive. \"Among the ships still in our inventory, only thirteen Navigators remain alive. We are on a death watch for them.\"\n\n\"What do you mean the ships 'still in your inventory'?\" the CHOAM man asked.\n\nGorus hesitated, then admitted, \"There were some still flown by Navigators, vessels that we had not yet managed to equip with mathematical compilers. They have . . . how shall I say this? Over the past few months they have disappeared.\"\n\n\" _Disappeared_? How many Heighliners? Each ship is hugely expensive!\"\n\n\"I do not have precise numbers.\"\n\nThe CHOAM man had a hard voice. \"Give us your best estimate.\"\n\n\"Five hundred, perhaps a thousand.\"\n\n\"A _thousand_?\"\n\nAt his side, the Mentat held his silence, but he appeared as upset and startled as the CHOAM representative.\n\nTrying to demonstrate control over the situation, Gorus said in an almost dismissive tone, \"When starved for spice, the Navigators grow desperate. It's not surprising that they take irrational action.\"\n\nKhrone himself was concerned, but he didn't show it. These disappearances sounded like a widespread conspiracy involving a Navigator faction, something he had not expected. \"Do you have any idea where they might have gone?\"\n\nThe Guild Administrator feigned nonchalance. \"It doesn't matter. They will run out of spice and die. Look at these shipyards and see how many vessels we are creating every day. Before long, we'll make up for the loss of those outdated ships and obsolete Navigators. Have no fear. After so many years of bondage to a single substance, the Guild is making a good business decision.\"\n\n\"Thanks to your partners from Ix,\" Khrone pointed out.\n\n\"Yes, thanks to Ix.\"\n\nFollowing a lull, the noise of the shipyards became very loud. Welders went to work, and heavy machinery lifted curved components into place. A cargo hauler half a kilometer wide brought in two sets of Holtzman engines. The men continued to watch the magnificent activities for a long time in silence. None of them even looked again at the pathetic dead Navigator in his tank.\n\n> _Humanity has many profound beliefs. Chief among them is the concept of_ Home.\n> \n> \u2014Bene Gesserit Archives,  \n>  _Analyses of Motivating Factors_\n\nThe next time Edrik's Heighliner went to Buzzell on a run, the vessel left the planet carrying something vastly more important than soostones.\n\nHidden on the sealed laboratory decks was a package of the uniquely powerful substance extracted from the slaughtered seaworm's strange, dense organ. With extravagant optimism, Waff had named it \"ultraspice.\" Tests proved that the potency went beyond that of any spice ever recorded. This remarkable substance would change everything for the Navigator faction.\n\nThe Tleilaxu Master also understood the importance of his achievement, and meant to use it to his advantage. Without being summoned, he pushed past Guild security forces and made his way to the restricted levels reserved for the Navigator. Officiously ignoring all challenges, Waff opened thick doors until he stood before the plaz-walled tank that held Edrik in his expensive bath of spice gas. Having succeeded in restoring at least one breed of worms, Waff was no longer a sycophant. He could make brash demands of his own.\n\nWaff's shortened ghola life span didn't give him much time to meet his critical goals, thus making him increasingly desperate. He was already well past his physical prime, and now his body was in a rapid plunge to degeneration and death. He probably had no more than a year or so left.\n\nFull of rigid defiance, Waff stood before Edrik's tank and said, \"Now that my altered seaworms are capable of creating spice in a form accessible by Guild Navigators, I want you to take me to Rakis.\" He no longer had anything to lose, and everything to gamble. He crossed his thin arms over his chest in triumph.\n\nSwimming slowly, Edrik drifted close to the plaz wall. The swirls of orange gas were hypnotic. \"The new melange has not been proved in practice.\"\n\n\"No matter. Its chemistry has been proved.\"\n\nEdrik's voice grew louder through the speakers. \"I am troubled. In its original form, melange has complexities that cannot be revealed in any laboratory analysis.\"\n\n\"You worry unnecessarily,\" Waff said. \"Seaworm spice is more potent than anything you have ever consumed. Try it yourself, if you do not believe me.\"\n\n\"You are in no position to make demands.\"\n\n\"No one else could have accomplished what I did. Buzzell will be your new source of melange. Seaworm hunters will harvest more ultraspice than you can possibly use, and Navigators will no longer be dependent upon the Bene Gesserit witches or the black market. Even if the Sisters decide to harvest the seaworms and try to create another monopoly, you can ignore them. By changing the _worms_ , instead of the planet, we can place them anywhere. I have given you the road to freedom.\"\n\nWaff snorted, raised his voice. \"Now I demand my payment.\"\n\n\"We kept you alive after the Honored Matres were overthrown on Tleilax. Is that not sufficient compensation?\"\n\nWith a conciliatory sigh, the Tleilaxu ghola held his hands out. \"What I ask will cost you little and gain you much honor, a blessing from God.\"\n\nThe Navigator wore a look of displeasure on his distorted face. \"What do you desire, little man?\"\n\n\"I repeat: Take me to Rakis.\"\n\n\"Absurd. The world is dead.\" Edrik's words were flat.\n\n\"Rakis is where my last body perished, so consider it a pilgrimage.\" He continued in a rush, saying more than he had intended. \"In my laboratory I created more small worms from the remaining sandtrout specimens. I have strengthened them, made them capable of surviving in the harshest environment. I can repopulate Rakis and bring back the Prophet\u2014\" He abruptly fell silent.\n\nAt the first rumors that the seaworms were thriving, Waff had turned his efforts to the last few sandtrout in his original stock. Sculpting worm chromosomes for survival in a comfortable ocean environment had been a challenge; much more difficult, though, was the task of toughening the monsters to survive out in the blasted wastelands of Rakis. But Waff did not turn his back on difficulty. All along, his goal had been to bring the sandworms back where they belonged. _God's Messenger must return to Dune_.\n\nHe studied Edrik, who stroked with webbed hands as he considered the request. \"Our Oracle recently sent us a message, calling upon Navigators to leave the Guild and join her in a great battle. That must be my priority now.\"\n\n\"I implore you, take me to Rakis.\" As if to remind Waff of his imminent mortality, a twinge of pain shot through his chest and down his spine. He needed all his effort not to show the anguish of dying, the misery of failure. He had so little time remaining. \"Is that so much to ask? Grant me this one favor at the end of my life.\"\n\n\"That is all you wish to do? _Die_ there?\"\n\n\"I will spend my last energies on my sandworm specimens. Perhaps there is a way of reintroducing them to Rakis and regenerating the ecological systems. Think of it: If I succeed, you will have yet another source of melange.\"\n\n\"You will not be pleased with what you find there. Even with moisture recycling, shelters, and equipment, survival on Rakis is more difficult than it has ever been. Your expectations are unrealistic. Nothing useful remains.\"\n\nWaff tried unsuccessfully to keep desperation out of his voice. \"Rakis is my home, my spiritual compass.\"\n\nEdrik thought it over, then said, \"I can fold space to Rakis, but I cannot promise to return. The Oracle has called me.\"\n\n\"I will remain there as long as necessary. God will provide for me.\"\n\nWaff rushed back to his private research levels. Intending to stay on the desert planet, undoubtedly for the rest of his life, he requisitioned all the supplies and equipment he might need for years, allowing him to be entirely self-sufficient on that bleak and lifeless world. After placing the order, he looked at his tanks where the new armored sandworms writhed, eager to be released.\n\nRakis . . . Dune . . . was his destiny. He felt in his heart that God had summoned him there, and if Waff perished on the planet . . . then so be it. He felt a warm, soothing wave of contentment. He understood his place in the universe.\n\nTHE BLACKENED, FAINTLY coppery ball appeared in the Heighliner's private viewing plates. Waff had been so anxious gathering his things that he hadn't even felt the activation of the Holtzman engines, the folding of space.\n\nEdrik surprised him by offering additional supplies and a small team of loyal Guild assistants to help with the labor of setting up a camp and administering the experiments. Perhaps he wanted his own people on hand to see if the Tleilaxu man succeeded again with his worms. Waff didn't mind, so long as they stayed out of the way.\n\nWithout introducing himself to the silent members of his new team, Waff directed the transfer of his armored sandworm specimens from the isolated lab, his self-erecting shelters and his equipment, everything they would need for survival on the charred world.\n\nOne of the silent, smooth-faced Guild assistants piloted the lighter. Before they reached the dead surface of Dune, the Heighliner had already drifted out of orbit. Edrik was anxious to be on his way to answer the Oracle's call, carrying its cargo of ultraspice and the tidings of new hope for all Navigators.\n\nWaff, though, had eyes only for the blistered, lifeless landscape of the legendary world.\n\n> _Bacteria are like tiny machines, notable for their effects on larger biological systems. In a similar way, humans behave as disease organisms among planetary systems, and should be studied as such_.\n> \n> \u2014ERASMUS,  \n> Laboratory Notebooks\n\nWhen the virulent plague reached Chapterhouse, the first cases appeared among the male workers. Seven men were struck down so swiftly that their dying expressions showed more surprise than pain.\n\nIn the Great Hall where younger Sisters dined, the disease also spread. The virus was so insidious that the most contagious period occurred a full day before any symptoms manifested; thus, the epidemic had already sunk its claws into those most vulnerable before the New Sisterhood even knew a threat existed.\n\nHundreds perished within the first three days, more than a thousand by the end of the week; after ten days, the victims were beyond counting. Support staff, teachers, visitors, offworld merchants, cooks and kitchen help, even failed Reverend Mothers\u2014all fell like stalks of wheat under the Grim Reaper's scythe.\n\nMurbella called upon her senior advisers to develop an immediate plan, but from prior epidemics on other embattled planets they knew that precautionary measures and quarantines would do no good. The conference room doors were securely locked, because younger Sisters and acolytes could not be allowed to know the strategies being discussed here.\n\n\"Survival of the Sisterhood is our primary purpose, even as the rest of Chapterhouse dies around us.\" Murbella felt sickened to think of all the unprepared acolytes, spice-harvesting teams in the dune belt, transport drivers, architects and construction workers, weather planners, greenhouse gardeners, cleaners, bankers, artists, archive workers, pilots, technicians, and medical assistants. All the underpinnings of Chapterhouse itself.\n\nLaera attempted to sound objective, but her voice cracked. \"Reverend Mothers have the precise cellular control needed to fight this disease on its own battleground. We can use our bodily defenses to drive away the plague.\"\n\n\"In other words, anyone who hasn't gone through the Spice Agony will die,\" Kiria said. \"Like the Honored Matres did. That was why we pursued you Bene Gesserits in the first place, to learn how to protect ourselves from the epidemic.\"\n\n\"Can we use the blood of Bene Gesserit survivors to create a vaccine?\" Murbella asked.\n\nLaera shook her head. \"Reverend Mothers drive out disease organisms, cell by cell. There are no antibodies we can share with others.\"\n\n\"It is not even as simple as that,\" Accadia rasped. \"A Reverend Mother can channel her inner biological defenses only if she has the energy to do so, and if she has the time and ability to concentrate on herself. But this plague forces us to turn our energies to tend the most unfortunate victims.\"\n\n\"If you make that mistake, you'll die, just like our Sheeana surrogate on Jhibraith,\" Kiria said with the undertone of a sneer in her voice. \"We Reverend Mothers will have to take care of ourselves and no one else. The others have no chance anyway. We need to accept that.\"\n\nMurbella already felt the beginnings of exhaustion, but her nervous anxiety made her pace the sealed council room. She had to think. What could be done against such a minute, lethal enemy? _Only Reverend Mothers will survive. . . ._ She spoke firmly to her advisors, \"Find every acolyte who is close to being ready for the Agony. Do we have enough Water of Life?\"\n\n\"For all of them?\" cried Laera.\n\n\"For every single one. Any Sister who has the slightest chance of survival. Give all of them the poison and hope they can convert it and survive the Agony. Only then will they be able to fight off the plague.\"\n\n\"Many will die in the attempt,\" warned Laera.\n\n\"Or _all_ of them will die from the plague. Even if most of the candidates succumb to the Agony, it's an improvement.\" She did not wince. Her own daughter Rinya had perished that way, many years ago.\n\nSmiling slightly with her wrinkled lips, Accadia nodded. \"A Bene Gesserit would rather die from the Agony than from a sickness spread by our Enemy. It is a gesture of defiance rather than surrender.\"\n\n\"See that it is done.\"\n\nIN THE DEATH houses she turned a deaf ear to the moans of the sick and dying. The Chapterhouse doctors had drugs and potent analgesics, and the Bene Gesserit acolytes had been taught how to block off pain. Even so, the misery of the plague was enough to break the deepest conditioning.\n\nMurbella hated to see the Sisters unable to control their suffering. It shamed her, not for their weakness but because she had been unable to prevent this from happening in the first place.\n\nShe went to where lines of makeshift beds held young acolytes, most of them terrified, some of them determined. The room smelled of rancid cinnamon\u2014harsh instead of pleasant. With her brow furrowed and eyes intent, the Mother Commander watched two stony-faced Reverend Mothers carry out a stretcher bearing the sheet-wrapped body of a young woman.\n\n\"Another one failed the Agony?\"\n\nThe Reverend Mothers nodded. \"Sixty-one today. They are dying as fast as from the plague.\"\n\n\"And how many successes?\"\n\n\"Forty-three.\"\n\n\"Forty-three that will live to fight the Enemy.\"\n\nLike a mother hen, Murbella walked up and down the line of beds, observing the plague-stricken Sisters, some sleeping quietly with new bodily awareness, others writhing in deep comas from which it was uncertain they would ever find their way back.\n\nAt the end of the row, a teenage girl lay with frightened eyes. She propped herself up in bed on trembling arms. She met Murbella's gaze, and even in her extreme sickness the girl's eyes glimmered. \"Mother Commander,\" she said hoarsely.\n\nMurbella moved closer to the young one. \"What is your name?\"\n\n\"Baleth.\"\n\n\"Are you waiting to undergo the Agony?\"\n\n\"I'm waiting to die, Mother Commander. I was brought here to take the Water of Life, but before it could be administered the symptoms of the disease manifested themselves. I'll be dead before the end of the day.\" She sounded very brave.\n\n\"So they will not give you the Water of Life, then? You won't even attempt the Agony?\"\n\nBaleth lowered her chin. \"They say I will not survive it.\"\n\n\"And you believe them? Aren't you strong enough to try?\"\n\n\"I am strong enough to _try_ , Mother Commander.\"\n\n\"Then I'd rather you died trying, instead of giving up.\" As she looked down at Baleth, she was poignantly reminded of Rinya . . . eager and confident, so like Duncan. But her daughter hadn't been ready after all, and she had died on the table.\n\n_I should have delayed her. Because of my need to prove myself, I pushed Rinya. I should have waited. . . ._\n\nAnd Murbella's youngest daughter Gianne\u2014what had happened to her? The Mother Commander had kept herself apart from the young woman's day-to-day activities, letting the Sisterhood raise her. But in this time of crisis, she decided to ask someone, Laera perhaps, to track her down.\n\nRight now, Baleth seemed to show hope, looking with fervid eyes toward the Mother Commander. Murbella ordered the Suk doctors to attend to her immediately. \"Time is shorter for this one than for the others.\"\n\nFrom the doctors' skeptical expressions, Murbella could see they considered this a waste of the valuable Water of Life, but she stood firm. Baleth accepted the viscid draught, took a last look at her Mother Commander, and gulped the toxic substance. She lay back, closed her eyes, and began her fight. . . .\n\nIt did not last long. Baleth died in a valiant attempt, but Murbella could feel no guilt about it. The Sisterhood must never stop fighting.\n\nTHOUGH MELANGE WAS rare and precious, rarer still was the Water of Life.\n\nBy the fourth day of Murbella's desperate plan, it became apparent that Chapterhouse's supplies would not be sufficient. Sister after Sister consumed the poison, and many perished while struggling to convert the deadly toxin in their cells, trying to change their bodies.\n\nThe Mother Commander tasked her advisors to study the exact amount of poison necessary to trigger the Agony. Some Reverend Mothers suggested diluting the substance, but if they didn't give enough to be fatal, and thus effective, the entire experiment would fail.\n\nDozens more Sisters died. More than 60 percent of those who took the poison.\n\nKiria offered a hard but coldly logical solution. \"Assess each candidate, and dole out the Water of Life only to those most likely to succeed. We can't gamble foolishly. Each dose we give to a woman who fails is wasted. We must discriminate.\"\n\nMurbella disagreed. \"None of them has a chance unless they undergo the Agony. The whole point of this operation is to give it to everyone\u2014and the most fit will survive.\"\n\nThe women stood amidst the bedlam of the dormitory rooms in the sick houses that had been converted from any building large enough to accommodate beds. Four lifeless bodies were carried past them by exhausted-looking Reverend Mothers. They had run out of sheets, so the corpses were uncovered, their faces twisted in a display of the incalculable pain they had suffered.\n\nIgnoring the dead, Murbella knelt beside the bed of one young woman who survived. She had to look at the casualty total from a different perspective. If they were _all_ destined to die, it was a fruitless exercise to count those who perished. In that light, the only relevant number was the tally of those who did recover. The victories.\n\n\"If we don't have enough Water of Life, use other poisons.\" Murbella got wearily to her feet, ignoring the smells, the sounds. \"The Bene Gesserit may have determined that the Water of Life is most effective at forcing the Agony, but long ago the Sisters used other deadly chemicals\u2014anything that would push the body into an absolute crisis.\" She perused the young students, these girls who had hoped one day to grow up to become Reverend Mothers. Now each of them had one chance, and one chance only. \"Poison them one way or another. Poison them all. If they survive, they belong here.\"\n\nA courier ran up to her, one of the younger Sisters who had recently survived the transformation. \"Mother Commander! You are needed immediately in Archives.\"\n\nMurbella turned. \"Has Accadia found something?\"\n\n\"No, Mother Commander. She . . . you have to see for yourself.\" The younger woman swallowed. \"And _hurry_.\"\n\nThe ancient woman did not have the strength to leave her office. Accadia sat surrounded by wire spool readers and stacks of data-dense crystal sheets. She sprawled back in her large chair, breathing heavily, barely able to move. The old woman's rheumy eyes flickered open. \"So, you've come . . . in time.\"\n\nMurbella looked at the archivist, appalled. Accadia, too, had the plague. \"But you are a Reverend Mother! You can fight this.\"\n\n\"I am old and tired. I used the last of my stamina to compile our records and projections, to map out the spread of this disease. Maybe we can prevent it on other worlds.\"\n\n\"Doubtful. The Enemy distributes the virus wherever they consider it strategic.\" Already she had made up her mind to have several other Reverend Mothers Share with Accadia. Her extensive memories and knowledge must not be lost.\n\nAccadia struggled to sit up in the chair. \"Mother Commander, don't be so focused on the epidemic that you fail to see its consequences.\" She began coughing. Blotches had appeared all over her skin, the advanced stages of the disease. \"This plague is a mere foray, a test attack. On many planets it is sufficient, but the Enemy must know the Sisterhood well enough by now to be sure we can fight this, at least to a point. After they soften us up, they'll attack by other means.\"\n\nMurbella felt cold inside. \"If thinking machines destroy the New Sisterhood, then the remaining fragments of humanity will have no chance of resisting them. We are the most important hurdle Omnius has to overcome.\"\n\n\"So you finally understand the implications?\" The old woman grasped the Mother Commander's hand to make sure she understood. \"This planet has always been hidden, but now the thinking machines must know the location of Chapterhouse. I would wager that their space fleet is already on its way.\"\n\n> _One man's dream is another's nightmare_.\n> \n> \u2014a saying of Ancient Kaitain\n\nAfter dragging Stuka's body away, the nomads separated Sheeana and Teg from Stilgar and Liet-Kynes. Apparently they saw the two boys\u2014twelve and thirteen\u2014as no threat, not knowing that both were deadly Fremen fighters, whose clear memories held many raids against the Harkonnens.\n\nTeg recognized the strategy. \"The old leader wants to interrogate our young ones first.\" Var and his hard-bitten comrades would assume the youths would be easily intimidated, not capable of resisting difficult questioning.\n\nTeg and Sheeana were taken to a holding tent made of a tough, weather-worn polymer. The structure was an odd mixture of primitive design and sophisticated technology, made for serviceability and ease of transport. The guard closed the flap but remained outside.\n\nThe windowless tent was just an empty enclosure, devoid of blankets, cushions, or tools of any kind. Teg paced in a small circle, then sat beside her on the packed dirt. Digging with his fingers, he quickly found a couple of sharp pebbles.\n\nWith Mentat clarity, he assessed their options. \"When we do not return or report in,\" he said in a low voice, \"we can expect Duncan to send another party down. He will be prepared. It sounds trite, but rescue will come.\" He knew that these nomads would crumble easily against a direct military assault. \"Duncan is wise, and I trained him well. He will know what to do.\"\n\nSheeana stared at the door as if in meditative trance. \"Duncan has lived hundreds of lives and remembers them all, Miles. I doubt you taught him anything new.\"\n\nTeg gripped one of the pebbles, and it seemed to aid his concentration. Even in an empty tent, he saw a thousand possible avenues of escape. He and Sheeana could easily break out, kill the guard, and fight their way back to the lighter. Teg might not even need to take advantage of his accelerated speed. \"These people are no match for me, or for you. But I will not leave Stilgar and Liet behind.\"\n\n\"Ah, the loyal Bashar.\"\n\n\"I wouldn't leave you, either. However, I fear that these people have disabled our ship, which would certainly tangle our escape plans. I heard them ransacking it.\"\n\nSheeana continued to stare at the shadowy wall of the tent. \"Miles, I'm not so concerned about the possibility of escape as I am curious to learn why they kept us alive. Especially me, if what they said about the Sisterhood is true. They have good reason to hate me.\"\n\nTeg tried to imagine the incredible exodus and reorganization of populations on this planet. Within years, all the inhabitants of the towns and cities would have seen the sands strangling their croplands, killing their orchards, creeping closer and closer to the city boundaries. They would have pulled away from the desert zone like people fleeing a slowly advancing fire.\n\nVar's nomads, though . . . were they scavengers and misfits? Outcasts from the larger population centers? Why insist on staying at the threshold of the advancing desert, where they would have to uproot their settlement and retreat constantly? To what purpose?\n\nThese were technologically capable people, and Qelso clearly must have been settled long ago during the Scattering. They had their own groundcars and low-altitude flyers, fast ships to take them back and forth across the dunes. If they weren't outright exiles, perhaps Var's people replenished their supplies in the distant northern cities.\n\nTeg and Sheeana hardly spoke for hours as they listened to the muffled sounds outside, the dry wind pushing and tugging at the tent, the _scritch_ of blowing sand. Everything seemed to be comprised of movement outside: The people sent out parties, marched back and forth, set machinery to work.\n\nAs Teg listened to the noises, he catalogued them in his mind, building a picture of the operations. He heard a pounding drill that bored a well shaft, followed by a pump dispensing water into small cisterns. Each time, after only a brief gush of liquid, the flow dwindled to less than a trickle and stopped. He knew that such problems, caused by sandtrout, had been the bane of drilling operations on Arrakis. Water existed in deep enough strata, but it was blocked off by the voracious little Makers. Like platelets at the site of a wound, sandtrout would swiftly seal off the leak. As he listened to the resigned complaints of these people, Teg realized that they were familiar with the routine.\n\nWhen night fell, a dusty young man entered the tent through the flap held open by the guard. He delivered a small meal of hard bread and dried fruit, as well as gamey-tasting white meat. The two captives also received carefully measured rations of water.\n\nSheeana looked at her sealed cup. \"They are learning the fundamentals of extreme conservation. They begin to understand what their world will become.\" Obviously despising her Bene Gesserit robes, the young man glared at her and departed without a word.\n\nThroughout the dark night, Teg remained awake, listening, trying to plan. The lack of activity was maddening, but he advised patience rather than rash action. They had heard nothing from Liet or Stilgar, and he feared the two young men might already be dead, like Stuka. Had they been killed during interrogation?\n\nSheeana sat beside him, in a heightened state of alertness. Her eyes were bright even in the tent shadows. As far as Teg could tell, the guard outside never stepped away from his position, never even moved. The people continued to send out parties and skimmer ships throughout the night, as if the camp were the staging area for a war effort.\n\nAt dawn, old Var came up to the tent, spoke briskly with the guard, and pulled the flap aside. Sheeana rose to a half crouch, ready to spring; Teg tensed, also prepared for a fight.\n\nThe nomad leader glared at Sheeana. \"You and your witches are not forgiven for what you have done to Qelso. You never will be. But Liet-Kynes and Stilgar have convinced us to keep you alive, at least as long as we can learn from you.\"\n\nThe weathered leader brought the pair out into the bright sunlight. The wind flung stinging sand into their eyes. All around the settlement, trees had already died. The blowing dunes had encroached another few feet past the prominent rock outcroppings during the night. Each breath was crackling dry, even in the relative coolness of the morning.\n\n\"You put the other Bene Gesserits to death,\" Sheeana said, \"and killed our companion Stuka. Am I next?\"\n\n\"No. Because I said I would keep you alive.\"\n\nThe weathered man led them through the settlement. Workers were already disassembling large warehouse tents to move them farther from the edge of the sand. A heavy groundcar rumbled by, full of crates. A bloated flyer circled and landed near the smooth sand. Some kind of tanker?\n\nVar led them to a large central building made from sectional metal walls and a conical roof. Inside, a long table was cluttered with charts. Reports were fastened to the walls, and one entire wall displayed a polymer-paper map, a high-resolution topographic projection of the entire continent. Mark after mark showed the steady growth of the desert belt.\n\nMen sat around the table sharing reports and raising their voices in a tumult of conversation. Stilgar and Liet-Kynes, both dressed in dusty shipsuits, waved a greeting at the other two prisoners. The young men seemed pleased and relaxed.\n\nAs he scanned the setup, it was obvious to Teg that Stilgar and Liet had spent the whole previous day in the command tent. The old leader positioned himself between them, leaving Teg and Sheeana to stand.\n\nVar pounded on the table, interrupting the cacophony. Everyone stopped talking, impatiently it seemed, and stared at him. \"We have listened to our new friends describe what our world is sure to become. We've all heard legends of long-lost Dune, where water is more precious than blood.\" His face had a pinched look. \"If we fail and the worms take over, our planet will become valuable only by the standards of outsiders.\"\n\nOne of the men snarled at Sheeana, \"Damned Bene Gesserits!\" The others glared at her as well, and she met their disapproval squarely, without comment.\n\nLiet and Stilgar seemed to be in their element. Teg recalled the Bene Gesserit discussions over the original ghola project, how the long-forgotten abilities of those historical personages might become relevant again. Here was a perfect example. This duo of prominent survivors from the old days of Arrakis certainly knew how to deal with the crisis these people now faced.\n\nThe grizzled leader raised his hands, and his voice sounded as dry as the air. \"After the death of the Tyrant long ago, my people fled into the Scattering. When they reached Qelso, they thought they had found Eden. It was a paradise for fifteen hundred years afterward.\"\n\nThe men glowered at Sheeana. Var explained how the refugees had established a thriving society, built cities, planted crops, mined for metals and minerals. They had no wish to overextend themselves or go searching for other lost brothers who had escaped during the Famine Times.\n\n\"Then a few decades ago everything changed. Visitors came, Bene Gesserits. At first we welcomed them, glad to have news from the outside. We offered them a new home. They became our guests. But the ingrates raped our entire planet, and now it is dying.\"\n\nAnother man clenched his hands into fists as he picked up the story. \"The sandtrout multiplied out of control. Huge forests and vast plains died within years\u2014only a few years! Great fires started in the wastelands, and weather patterns changed, turning much of our world into a dust bowl.\"\n\nTeg spoke up, using his command voice. \"If Liet and Stilgar told you about our no-ship and its mission, then you know we don't carry sandtrout and we have no intention of harming your world. We stopped here only to replenish vital supplies.\"\n\n\"In fact, we fled the heart of the Bene Gesserit order because we disagreed with the policies and leadership,\" Sheeana added.\n\n\"You have seven large sandworms in your hold,\" Var accused.\n\n\"Yes, and we will not release them here.\"\n\nLiet-Kynes spoke quietly, as if lecturing children. \"As we already told you, once it has begun, the desertification process is a chain reaction. The sandtrout have no natural enemies, and their encysting of water is so swift that nothing can adapt quickly enough to fight against them.\"\n\n\"Nevertheless, we will fight,\" Var said. \"You see how simply we live in this camp. We have given up everything to stay here.\"\n\n\"But why?\" Sheeana asked. \"Even as the desert spreads, you have many years to prepare.\"\n\n\"Prepare? Do you mean surrender? You may call it a hopeless fight, but it is still a _fight_. If we cannot stop the desert, we will at least slow it. We'll battle the worms and the sands.\" The men at the table muttered. \"No matter what you say, we will try to hinder the desert's progress in every way. We kill sandtrout, we hunt the new worms.\" Var stood up, and the others followed suit. \"We are commandos sworn to slow the death of our world.\"\n\n> _The desert still calls me. It sings in my blood like a love song_.\n> \n> \u2014LIET-KYNES,  \n>  _Planetology: New Treatises_\n\nEarly the next morning, Var led his group of dusty, determined fighters to a landing zone of fire-baked pavement. \"Today, my new friends, we'll show you how to kill a worm. Maybe two.\"\n\n\"Shai-Hulud,\" Stilgar said with great uneasiness. \"Fremen used to worship the great worms.\"\n\n\"Fremen depended upon the worms and the spice,\" Liet replied quietly. \"These people do not.\"\n\n\"With each demon we eliminate, we give our planet a little more time to survive.\" Var stared out into the desert as if his hatred could drive back the sands. Stilgar followed the man's gaze across the deeply shadowed dunes, trying to imagine the landscape in front of him as lush and green.\n\nThe sun was just rising over an escarpment, glinting off the silvery hull of an old low-altitude flyer parked on an area of pounded gravel and flash-fused cement. Var's people did not bother with permanent landing strips or spaceport zones, which would only be swallowed up by the spreading dunes.\n\nDespite the protests of the two young men, Sheeana and Teg were forced to remain behind in the camp as hostages, watched suspiciously. Liet and Stilgar had been accepted on the hunt because of their invaluable knowledge of the desert. Today, they would demonstrate their skills.\n\nVar's commandos clambered into the heavily used craft. It had obviously weathered countless storms, rough flights, and incomplete maintenance; its hull was scuffed and scraped. The interior smelled of oil and sweat, and the seats were stone-hard, with only bars or straps for the passengers to hold onto.\n\nStilgar felt comfortable enough among the twenty weathered, grim men. To his trained eye, the commandos had a look of edgy anticipation, but they were too soft in the flesh for the adaptations they would soon face. With the rapid climate shift, even living in their nomadic camps at the fringe of the sand, these people remained unaware of the desert's true harshness. They would have to learn swiftly enough to face the escalating hardships. He and his friend could teach them\u2014if they would listen.\n\nLiet took his seat beside Stilgar and spoke to Var's men with genuine enthusiasm. \"Right now, Qelso's air still contains enough moisture that truly dramatic measures aren't required. Soon, though, you will need to be careful not to waste so much as a thimbleful of water.\"\n\n\"We already live under the strictest conservation,\" one man said, as if Liet had insulted him.\n\n\"Oh? You don't recycle your sweat, respiration, or urine. You still import water from the higher latitudes, where it is readily available. Many regions on Qelso are still able to grow crops, and people live a fairly normal life.\"\n\n\"It will get worse,\" Stilgar agreed. \"Your people have much hardening to do before the planet reaches its new equilibrium. This is the first day of your new field training.\"\n\nThe men muttered uncertainly at hearing such words from two seeming boys, but Liet sounded optimistic. \"It is not so bad. We can teach you how to make stillsuits, how to conserve every breath, every sweat droplet. Your fighting instincts are admirable, but useless against sandworms. You must learn to survive among the behemoths that will eventually take control of your world. It is a necessary shift in attitude.\"\n\n\"The Fremen did so for a long time.\" Stilgar seated himself beside his friend. \"It was an honorable way of life.\"\n\nThe fighters held onto straps and spread their feet for balance, preparing for takeoff. \"That is what lies in store for us? Drinking recycled sweat and piss? Living in sealed chambers?\"\n\n\"Only if we fail,\" old Var said. \"I choose to believe we still have a chance, no matter how na\u00efve that sounds.\" He closed the ship's hatch and strapped himself into the creaking pilot seat. \"So, if that doesn't sound pleasant to you, then we'd better stop the desert from gaining more of a foothold.\"\n\nThe flyer lifted from the dry camp and swung out over the ghost forests and hummocks of fresh dunes that were swallowing the remnants of grasslands. The engine sputtered periodically as they flew southeast to a region where sandworms had been sighted. The craft seemed like a sluggish bumblebee, its tanks overloaded and heavy.\n\n\"We will stop the moving sands,\" one young commando said.\n\n\"Next you will try to stop the wind.\" Stilgar grabbed a dangling strap as a thermal updraft shook the craft. \"In a few short years, your planet will be sand and rock. Do you expect a miracle to turn the desert back?\"\n\n\"We'll create that miracle for ourselves,\" Var answered, and his team murmured in agreement.\n\nThey flew across the wilderness of dunes, far past the point where they could see anything but buttery tan from horizon to horizon. Stilgar tapped a finger against the scratched windowplaz and shouted against the engine noise. \"See the desert for what it is\u2014not a place to fear and loathe, but a great engine to power an empire.\"\n\nLiet added, \"Already, small worms in the desert belt have created priceless amounts of melange just waiting to be mined. How have you survived for so long without spice?\"\n\n\"We haven't needed spice for fifteen hundred years, not since we came to Qelso,\" Var called from the cockpit. \"When you do not have a thing, you learn to live without it, or you don't live.\"\n\n\"We don't give a damn about spice,\" one of the commandos said. \"I'd rather give a damn about trees and crops and fat herds.\"\n\nVar continued, \"Our first settlers brought a great deal of spice from far away, and three generations fought addiction until the supplies were gone. Then what? We were forced to survive without it\u2014and we have. Why should we open ourselves to that monstrous dependency again? My people are better off without it.\"\n\n\"If used carefully, melange has important qualities,\" Liet said. \"Health, life extension, the possibility of prescience. And it's a valuable commodity to sell, should you ever reconnect with CHOAM and the rest of mankind. As Qelso dries up, you may need offworld supplies for your basic needs.\"\n\n_If anyone survives the outside Enemy_ , Stilgar thought to himself, recalling the ever-present threat of capture by the shimmering net. But these people were much more concerned with their own local enemies, fighting the desert, trying to stop the unstoppable.\n\nHe remembered the great dreams of Pardot Kynes, Liet's father. Pardot had done the calculations and determined that the Fremen could turn Dune into a garden, but only after generations of intense effort. According to the histories, Arrakis had indeed become green and verdant for a time, before the new worms reclaimed it, and brought the desert back. The planet seemed unable to achieve a balance.\n\nThe battered craft flew low, its engines droning. Stilgar wondered if the noise of their passage would attract worms, but as he stared down at the hypnotic, oceanic dunes, he saw only a couple of patches of rust-colored sand that indicated fresh spice blows.\n\n\"Dropping signal vibrators,\" Var called, while throbbing canisters\u2014the equivalent of ancient thumpers\u2014tumbled out of the small bays below the cockpit. \"That should bring at least one of them.\"\n\nWith a puff of sand and dust the thumpers plunged into the dunes and sent out droning signals. After circling back to make certain the devices were operating properly, Var selected two more spots within a radius of five kilometers. Stilgar could not determine why the craft still felt overloaded.\n\nAs they cruised in search of a worm, Stilgar described his legendary days on Dune, how he and Paul Muad'Dib had led a ragtag Fremen army to victory against far superior forces. \"We used desert power. That is what you can learn from us. Once you see we are not your enemies, we can learn much from each other.\"\n\nUnder Stilgar's firm hand, these people could come to understand their possibilities. With the awakening of the populace would come the awakening of the planet, with plantings and green zones to keep the desert under control. Perhaps they could succeed, if they could just find\u2014and maintain\u2014an equilibrium.\n\nStilgar remembered something Liet's father had said to him once. _Extremes invariably lead to disaster. Only through balance can we fully harvest the fruits of nature_. He leaned closer to the craft's observation windows and saw a familiar wrinkling of the sand, ripples of deep movement disturbing the smooth dunes. \"Wormsign!\"\n\n\"Prepare for our first encounter of the day.\" A grin wrinkled Var's grizzled face as he turned away from the cockpit. \"The shipment that came in last night brought us enough water for two targets\u2014but we need to find them.\"\n\nWater! The heavy ship was carrying water.\n\nThe men shifted position, heading toward gunnery hatches and hoses mounted on the sides of the stripped-down flyer. The pilot banked back toward the first cluster of thumpers.\n\nAs the commandos prepared to strike, Stilgar mused about the strange turnabout. Pardot Kynes had spoken of the need to understand ecological consequences, that humans were stewards of the land, and never owners. _We must do a thing on Arrakis never before attempted for an entire planet. We must use man as a constructive ecological force\u2014inserting adapted terraform life: a plant here, an animal there, a man in that place\u2014to transform the water cycle, to build a new kind of landscape_.\n\nThe battle today was the opposite. Stilgar and Liet would help fight to prevent the desert from swallowing all of Qelso.\n\nThrough the nearest window Stilgar saw a mound in motion, a bucking sandworm drawn toward the thumper. Liet crowded close beside him, and said, \"I estimate it at forty meters. Larger than Sheeana's worms in our hold.\"\n\n\"These have grown in the open desert,\" Stilgar said. \"Shai-Hulud wants this planet.\"\n\n\"Not if I can help it,\" Var said. But as if to defy him, directly below the flyer an immense head surfaced and quested around, trying to track the conflicting sources of vibrations.\n\nLong tubes protruded from the front and rear of the flyer. The commandos gripped their gun mountings, nozzles that could be turned and aimed. The flyer swooped low. \"Fire when ready, but conserve what you can. The water's deadly enough.\"\n\nThe fighters shot high-pressure streams from their hoses, blasting the sandworm below. The drenching bursts were more effective than artillery shells.\n\nTaken by surprise, the creature writhed and twisted its round head back and forth, convulsing. Hard ring segments split apart to reveal softer pink flesh between, and water burned like acid into the vulnerable parts. The worm rolled on the wet sand, in obvious agony.\n\n\"They are killing Shai-Hulud,\" Stilgar said, sickened.\n\nLiet was also stunned, but said, \"These people have to defend themselves.\"\n\n\"That's enough! It is dead\u2014or soon will be,\" Var shouted. The small force reluctantly shut off their hoses, looking with hatred upon the dying worm. Unable to dig deep enough to escape the poisonous moisture, the mortally wounded creature continued to squirm as the flyer circled over its death throes. Finally the beast gave a great final shudder and stopped moving.\n\nStilgar nodded, his expression still grim. \"There are necessities to life in the desert, hard decisions to be made.\" He had to accept the clear fact that this worm did not belong here on Qelso. No sandworm did. On the way back to the settlement, they encountered a second worm, drawn by the vibrations of their flyer's engines. The commandos emptied their water reservoirs, and the second worm perished even more quickly.\n\nLiet and Stilgar sat together in uncomfortable silence, wrapped up in what they had seen and the fight they had agreed to join. \"Even though she doesn't have her memories back yet,\" Liet said, \"I'm glad my daughter Chani did not see this.\"\n\nThough the mood of the fighters was upbeat aboard the flyer, the two young men, remembering Arrakis, murmured Fremen prayers. Stilgar was still contemplating what they had seen and done when Var yelled a strangled-sounding alarm.\n\nSuddenly strange ships swarmed around them.\n\n> _You see only harshness, devastation, and ugliness. That is because you have no faith. Around me I see a potential paradise, for Rakis is the birthplace of my beloved Prophet_.\n> \n> \u2014WAFF of the Tleilaxu\n\nWhen he first glimpsed Rakis, the bleak ruins brought dismay to Waff's heart. But when Edrik's Heighliner deposited him and his small team of Guild assistants there, he experienced the joy of setting foot on the desert planet again. He could feel the holy calling deep inside his bones.\n\nIn his previous lifetime he had stood on these sands, face to face with the Prophet. With Sheeana and Reverend Mother Odrade, he had ridden a great worm out to the ruins of Sietch Tabr. His ghola memories were corrupted and uncertain, riddled with annoying gaps. Waff could not recall his final moments as the whores closed in around the desert planet, deploying their awful Obliterators. Had he run for hopeless shelter, looking behind him like Lot's wife for a last glimpse of the doomed city? Had he seen explosions and walls of flame searing the sky, sweeping toward him?\n\nBut the cells of another Waff ghola had been grown in an axlotl tank in Bandalong as part of the usual process. The secret council of the kehl had planned for the serial immortality of all Tleilaxu Masters long before anyone had heard of Honored Matres. The next thing he knew, Waff was awakened to his past life during a grand guignol stage show as the brutal women murdered one of his twins after another until just one of them\u2014him\u2014reached a sufficiently desperate crisis to break through the ghola barrier and reveal his past. Some of it, at least.\n\nBut not until now had Waff actually seen the Armageddon that the whores had wrought on this sacred world.\n\nThe ecosystem of Rakis had been fundamentally destroyed. Half of the atmosphere was burned away, the ground sterilized, most life forms dead\u2014from the microscopic sandplankton all the way up to the giant sandworms. It made the old Dune seem comfortable by comparison.\n\nThe sky was a dark purple, touched with an underburn of orange. As their ship circled, searching for a spot less hellish than others, Waff studied a panel of atmospheric readings. The moisture content was abnormally high. At some point in its geologic history, Arrakis had possessed open water, but sandtrout had sealed it all away. During the bombardment, underground rivers and seas must have been vaporized as they were released from aquifers.\n\nThe Honored Matres' horrific weapons had not only turned the soft dunes to a baked moonscape, they had also thrown up great clouds of dust that had not settled entirely out of the atmosphere, even decades later. The Coriolis storms would be worse than ever before.\n\nHe and his team would likely have to wear special bodily protection and supplemental breather masks; their small dwelling huts would need to be sealed and pressurized. Waff didn't mind. Was that so different from wearing a stillsuit? By degrees, perhaps, but not fundamentally harder.\n\nHis lighter circled over the remains of a sprawling metropolis that had been called Arrakeen in the days of Muad'Dib, then the Festival City of Onn during the reign of the God Emperor, and later\u2014after Leto II's death\u2014the moated city of Keen. No longer concerned about secrecy, now that the seaworms had successfully taken hold on Buzzell, Waff was happy to have four assistants help him with the hard work he was sure to face on this Obliterator-blasted planet.\n\nStudying the surface, he discerned lumpy geometric shapes that had once been angled streets and tall buildings. Surprisingly, in the dimness of seared daylight, he also spotted numerous artificial illumination sources and a few dark structures of recent construction. \"There seems to be a camp down there. Who else would come to Rakis? What could they possibly want here?\"\n\n\"The same as us,\" said the Guildsman. \"Spice.\"\n\nHe shook his head. \"Too little here anymore, at least until we bring back the worms. No one else has that skill.\"\n\n\"Pilgrims perhaps? There may still be those who make a hajj,\" a second assistant said. Waff knew that a dizzying mishmash of religious splinter groups and cults had sprung from Rakis.\n\n\"More likely,\" suggested a third Guildsman, \"they are treasure hunters.\"\n\nWaff quoted quietly from the Cant of the Shariat: \"'When greed and desperation are coupled, men accomplish superhuman feats\u2014though for the wrong reasons.' \"\n\nHe considered choosing a different place for their base camp, then accepted the idea that joining resources with the strangers might help them all last longer in the harsh environment. No one knew when\u2014or if\u2014Edrik might be coming back for them, or how long the sandworm work would take, or how much longer Waff himself would last. He planned to be here for the remainder of his days.\n\nAfter the lighter landed unannounced at the edge of the camp, the Guildsmen waited for instructions from Waff. The Tleilaxu man settled goggles over his eyes to protect against the caustic wind, and emerged. For long journeys outside, he might have to wear a supplemental oxygen mask, but the Rakian atmosphere was surprisingly breathable.\n\nSix tall and dirty men faced him from the encampment. They wore rags wrapped around their heads, carried knives and antique maula pistols. Their eyes were red-veined, their skin rough and cracked. The foremost man had shaggy black hair, a square chest, and a rock-hard potbelly. \"You are fortunate that I'm curious about why you are here. Otherwise, we would've shot you out of the sky.\"\n\nWaff held up his hands. \"We are no threat to you, whoever you are.\"\n\nFive men leveled their maula pistols, and the other slashed the air with his knife. \"We have claimed Rakis for ourselves. All spice here is ours.\"\n\n\"You've claimed a whole planet?\"\n\n\"Yes, the whole damned planet.\" The first man tossed back his dark hair. \"I'm Guriff, and these are my prospectors. There's damned little spice left in the burned crust, and it's ours.\"\n\n\"Then you may have it.\" Waff performed a perfunctory bow. \"We have other interests, as geological investigators and archaeologists. We wish to take readings and run tests to determine the extent of damage to the ecosystem.\" The four Guild assistants waited beside him in complete silence.\n\nGuriff laughed loudly and heartily. \"There isn't much of an ecosystem left here.\"\n\n\"Then where does breathable oxygen come from?\" He knew that Liet-Kynes had asked that question in the ancient days, curious because the planet had neither widespread plant life nor volcanoes to generate an atmosphere.\n\nThe man just stared. Obviously, he had not thought about this. \"Do I look like a planetologist to you? Go ahead and look into it, but don't expect any help from us. Here on Rakis, you are self-sufficient or you die.\"\n\nThe Tleilaxu man raised his eyebrows. \"And what if we wish to share some of our spice coffee with you, as a token of friendship? I understand that water is more easily obtainable than in the old days.\"\n\nGuriff glanced at his prospectors, then said, \"We're happy to accept your hospitality, but we have no intention of reciprocating.\"\n\n\"Nonetheless, our offer stands.\"\n\nINSIDE GURIFF'S DUSTY hut, Waff used his own supplies of melange (left over from his sandworm experiments) to brew coffee. Guriff didn't have a desperate shortage of water in his camp, though his dwelling smelled of long-unwashed bodies and the savory sweetness of a drug smoke that Waff could not identify.\n\nAt his command, the four Guildsmen erected the shelters brought down from the Heighliner, setting up armored sleeping tents and laboratory enclosures. Waff saw no reason to assist them. He was a Tleilaxu Master after all, and these were his workers, so he would allow them to perform their tasks.\n\nWhile they drank a second pot of spice coffee, Guriff grew more relaxed. He didn't trust the diminutive Tleilaxu, but he didn't seem to trust anyone. He took pains to say he harbored no particular hatred toward Waff's race, and that his scavengers held no grudges against others of low social position. Guriff cared only about Rakis.\n\n\"All that melted sand and plascrete. By chipping away the upper crust of glass, we were able to get down to the foundations of the sturdier buildings in Keen.\" Guriff produced a hand-drawn chart. \"Scraping out buried treasure. We found what we think is the original Bene Gesserit Keep\u2014a few heavily barricaded bomb shelters filled with skeletons.\" He smiled. \"We also uncovered the extravagant temple built by the Priests of the Divided God. It was so huge we couldn't miss it. Full of trinkets, but still not enough to pay for our effort. CHOAM is expecting us to find something much more extraordinary, though they seem happy enough to sell containers of 'genuine Rakian sand' to gullible fools.\"\n\nWaff didn't reply. Edrik and the Navigators had obtained such Rakian sand for him to use in his original experiments.\n\n\"But we've got a lot more digging to do. Keen was a big city.\"\n\nIn his previous life, Waff had seen those structures before they were destroyed. He knew the ostentation that the deluded Priests placed in all the rooms and towers (as if God cared about such gaudiness!). Guriff and his men would indeed find plenty of treasure there. But the wrong kind.\n\n\"The Priesthood's temple had collapsed worse than most other large buildings. Maybe it was a direct target of the Honored Matre attack.\" The prospector smiled with thick lips. \"But deep in the sublevels beneath the temple, we did find chests of stored solaris and hoarded melange. A worthwhile haul. More than we expected, but not so much. We're after something bigger. The Tyrant buried a huge spice hoard deep in the southern polar regions\u2014I'm sure of it.\"\n\nWaff made a skeptical sound as he sipped spice coffee. \"No one has been able to find that treasure for fifteen hundred years.\"\n\nGuriff held up a finger, noticed a hangnail, and chewed on it. \"Still, the bombardment may have churned up the crust enough to reveal the mother lode. And, thanks be to the gods\u2014there are no worms left to torment us.\"\n\nWaff made a noncommittal sound. _Not yet_.\n\nWITHOUT BOTHERING TO sleep, knowing his time was short, the Tleilaxu man began to make preparations to continue his work. His Guild companions seemed confident that the Navigator would eventually return, though Waff wasn't so sure. He was here on Rakis, and this gave him great pleasure.\n\nWhile the Guild assistants finished connecting the generators and sealed the prefab shelters, the Tleilaxu researcher went back aboard the near-empty lighter. In the cargo hold he smiled paternally at his magnificent specimens. The armored worms were small but ferocious. They looked ready to tackle a dead world. _Their_ world.\n\nAges ago, the Fremen had been able to summon and ride sandworms, but those original creatures had died out when Leto II's terraforming operations had turned Arrakis into a garden world with green plants, flowing rivers, and moisture from the sky. Such an environment was fatal to sandworms. But when the God Emperor was assassinated and his body fissioned into sandtrout, the whole process of desertification began anew. The freshly spawned worms became far more vicious than their predecessors, tackling the huge challenge of recreating the Dune That Once Was.\n\nWaff now faced a challenge many times more difficult. His modified creatures were armored to resist the most severe environment, with mouths and head ridges powerful enough to crack through the vitrified dunes. They could dig deep beneath the black surface; they could grow and reproduce\u2014even here.\n\nHe stood before the dusty holding tank in which the worms churned. Each specimen was about two meters in length. And strong.\n\nSensing his presence, the creatures twitched restlessly. Waff looked outside to where the sky had turned the deep purple-brown of dusk. Storms swirled gritty dust through the atmosphere. \"Be patient, my pets,\" he said. \"Soon I will release you.\"\n\n> _We are na\u00efve to think that we control a precious commodity. Only through guile and eternal wariness do we keep it out of the hands of our competitors_.\n> \n> \u2014Spacing Guild internal report\n\nEdrik moved his Heighliner away from the ruins of Rakis, no longer concerned with the Tleilaxu Master. Waff had served his purpose.\n\nMore important, the Oracle of Time had summoned all surviving Navigators, and Edrik would give them joyous news. With the seaworms obviously thriving on Buzzell, there would be plenty of ultraspice for the taking. The unusual concentrated form might even be superior to the original spice: a frighteningly potent melange to keep Navigators alive without the meddling, greedy Administrator faction or the witches of Chapterhouse.\n\n_Freedom!_\n\nIt had amused him to see Waff taking his worm samples to Rakis, hoping to establish a new spice cycle. Edrik didn't think the little researcher could do much there, but an alternative source of melange would be a bonus. But even without that, never again would the Navigators be strangled by power games. The four Guildsmen whom Edrik had sent to accompany Waff were spies and would secretly report everything the Tleilaxu achieved.\n\nInside his tank, Edrik smiled to himself, pleased that he had thought of all eventualities. With the first package of Buzzell ultraspice safely stored in his security chamber, the Navigator guided his Heighliner out into the emptiness of space. Even the Oracle would congratulate him for this remarkable news.\n\nBefore he could travel toward his scheduled rendezvous, however, the emptiness rippled around him. When Edrik studied the distortions, he realized what they were. Moments later, scores of Guildships appeared like buckshot in space, winking through foldspace and emerging forward and back, above and below, to surround his Heighliner completely.\n\nEdrik transmitted on a band that only fellow Navigators should have received. \"Explain your presence.\"\n\nBut none of the imposing newcomers answered. Studying the glyphs and cartouches on the sides of the enormous hulls, he realized that these were new Guildships, guided by Ixian mathematical compilers.\n\nThe computer-controlled vessels closed in. Sensing the threat, Edrik transmitted with greater alarm, \"What is your justification?\"\n\nThe other Guildships formed a smothering blanket around his Heighliner. The silence of the great vessels was more intimidating than any voiced ultimatum. Their proximity distorted his Holtzman fields, preventing him from folding space.\n\nFinally a voice spoke, flat and dull in timbre, yet unnervingly confident. \"We require your cargo of seaworm spice. We will board your ship for inspection.\"\n\nEdrik assessed these enemies, his mind racing through a labyrinth of possibilities. The ships appeared to belong to the Administrator faction. They functioned with Ixian devices, so they had no need for Navigators or melange. Why then would they want to confiscate the ultraspice? To prevent Navigators from having it? To ensure the Guild's complete reliance on Ixian navigation machines?\n\nOr could this be another foe entirely? Were these ships flown by CHOAM pirates hoping to seize a valuable new asset? Witches from Chapterhouse wanting to force continued dependence on the Sisterhood's melange?\n\nBut how would any outsiders know about the ultraspice?\n\nWhile Edrik's Heighliner hung helpless in space, small interdiction ships emerged from the surrounding Guild vessels. He had no choice but to allow boarders onto his ship.\n\nThough Edrik did not recognize him, a man wearing appropriate Guild insignia marched along the decks and ascended to the restricted level, brushing aside all security barriers. Six well-muscled men accompanied him. The leader smiled condescendingly when he stood before the Navigator's tank and looked into it. \"Your new spice has fascinating possibilities. We require it from you.\"\n\nEdrik boomed from within his chamber, intentionally amplifying the speaker system. \"Go to Buzzell and obtain your own.\"\n\n\"This is not a request,\" said the man, his face bland. \"We have learned the intensity of this substance and believe it to be a remedy for our difficult situation. We will take it to the heart of the thinkingmachine empire.\"\n\nThinking machines? What did the Administrator faction have to do with the Enemy? \"You may not have it,\" Edrik repeated, as if he had any say in the matter.\n\nThe bland-faced Guildsman gestured to his burly bodyguards, and they withdrew iron-tipped hammers from their slick gray robes. The leader gave them a calm, matter-of-fact nod.\n\nPanicked, Edrik swam backward in his tank, but he had nowhere to go. The muscular bodyguards did not care that he was inside the container or that exposure to the air would kill him. With thick arms, they swung their heavy sledges and smashed the thick plaz walls.\n\nJagged cracks split out in starburst patterns, and concentrated orange spice gas whistled out through the breaches. The guards did not react to the melange streaming into their faces, though the concentration should have made a normal human reel. Their bland-faced leader watched like a man smelling an approaching storm while Edrik's atmosphere drained out.\n\nWhen the air pressure was no longer sufficient to buoy him, the Navigator collapsed to the floor of his tank. Weakly, he raised his webbed hands and demanded answers in a voice that was little more than a gasp. The Guildsman and his companions offered no explanations.\n\nWithering and twitching, Edrik lay on the floor. He extended a rubbery arm and tried to crawl, but with all the spice gas draining away, the air was too thin. He could no longer breathe, could hardly move. Even so, the Navigator was slow to die.\n\nThe bland-faced man stepped closer to the shattered wall, and his features metamorphosed. Khrone said to his Face Dancer companions, \"Take the concentrated spice. With this substance, Omnius will awaken his Kwisatz Haderach.\"\n\nThe others departed to search the decks and soon uncovered the hoard of modified melange. When the disguised guards returned to the interdiction ships, Khrone held one of the heavy packages in his arms. He inhaled deeply. \"Excellent. Remove all of our people from this Heighliner. When we are safe, destroy the ship and everyone aboard it.\"\n\nHe looked coolly down at the dying Edrik. Only a few rusty curls of gas continued to ooze from cracks in the tank. \"You have served your purpose, Navigator. Take solace in that.\" The Face Dancer strutted away.\n\nEdrik continued to heave great breaths, but barely a scent of melange remained. By the time the computer-controlled Guildships got into formation in space, he could barely keep from slumping into unconsciousness.\n\nThe opposing vessels opened fire. Edrik's Heighliner exploded before he could utter a curse.\n\n> _There is an art to legend-telling, and an art to living the legend_.\n> \n> \u2014a saying of Ancient Kaitain\n\nThe _Ithaca_ ' _s_ replenishing operations had taken place in the stillrich northern latitudes, far from any visible population centers. Garimi managed the complex process with dozens of flying craft from the hangar decks, leaving Duncan on the command bridge. He felt trapped there, unable to leave because of the protective veil that the no-ship usually afforded him. He hated having to remain behind while others did the risky work . . . and he didn't even know what the old man and woman wanted from him.\n\nHe had no idea what was going on back in the Old Empire, with Murbella and Chapterhouse. He knew only that the Enemy was still searching for him\u2014and he was still hiding, as he had been for decades. Was this truly the best way to fight, the best way to defend humanity? He had been adrift for as long as the _Ithaca_ , and of late, the waters of uncertainty seemed deeper than ever.\n\nIt had been two days now without word from Teg or Sheeana and their team. If their group was simply meeting with the natives, someone should have checked in by now. Duncan feared another trap like the one they had encountered on the planet of the Handlers.\n\nMiles Teg had been his mentor and his student, and Sheeana . . . ah, Sheeana. They had been lovers and sexual opponents. She had cured him and saved him, so of course he cared for her. He had tried to protect himself by denying it, but she hadn't believed him, and he hadn't believed himself. Both knew they had a bond unlike any other, different from the one he and Murbella had imposed upon one another.\n\nAs he studied the landscape below, it seemed to call to him. Many cities were discernible in the northern and southern forested latitudes. He felt he should be down there facing any possible dangers with the others, not stuck aboard the _Ithaca_ , forced to remain safe and out of sight.\n\n_How long am I supposed to wait_?\n\nWhen he was Swordmaster of House Atreides he would never have hesitated. If it had been young Paul Atreides under threat, Duncan would have leapt in to fight for him, ignoring the intangible threat of the old man and woman. As the witches said in their oft-quoted Litany, _I will face my fear_. And it was about time he did so.\n\nHe closed his eyes, not wanting to see the spreading desert that looked like a seeping knife wound across the continent. \"I will not ignore this.\" Duncan summoned Thufir Hawat as well as Garimi, who had recently returned to the no-ship with all of her flying craft after reloading the _Ithaca_ ' _s_ stores.\n\nDuncan stood when they arrived. \"We are going to rescue the landing party,\" he announced, \"and we're going to do it now. I don't know what kind of military force those people have down there, but we'll stand against it if the Bashar is in trouble.\"\n\nThufir's eyes brightened and his face flushed. \"I'll pilot one of the ships.\"\n\nDuncan remained stern. \"No, you will follow my orders.\"\n\nGarimi was taken aback by Duncan's bold comment, but nodded as she heard him rebuke Thufir. \"Do you have instructions for us before we depart? Shall I command the mission?\"\n\n\"No\u2014I will do it personally.\" Before either could argue with him, Duncan strode toward the lift, and they were forced to follow him. \"I'm sick of hiding. My plan has been to run and remain unobtrusive, staying one step ahead of that strange net. But in doing so, I've left too much of myself behind. I am _Duncan Idaho_.\" He raised his voice as they entered the lift. \"I was Swordmaster of House Atreides and consort of St. Alia of the Knife. I acted as advisor and companion to the God Emperor. If the Enemy is out there, I won't leave the rest of humanity to face it themselves. If Sheeana and the Bashar need my help, then I'm going to help.\"\n\nThufir stiffened, then allowed himself a pleased smile. \"You should have left the _Ithaca_ long ago, Duncan. I don't see what you've accomplished by staying here. The no-field hasn't exactly offered perfect protection.\"\n\nGarimi seemed pleased by Duncan's attitude. \"My recovery teams took a good look at that planet down there, and it seems a fine place to settle. Does that mean you'll stop opposing my efforts and let us form a colony at last?\"\n\nThe lift doors closed, and the group began to drop toward the hangar decks where the many ships were being refueled. \"That remains to be seen.\"\n\nTEG BIDED HIS time in the camp long after Stilgar and Liet flew off into the early morning. By now, Duncan would certainly have drawn the obvious conclusions.\n\n\"Do you think they'll kill us, after all?\" Sheeana's tone was surprisingly matter-of-fact, as if she had accepted the inevitable.\n\n\"Maybe just you. You're the one they blame.\" He spoke without humor. Though they were allowed to sit on the ground outside, their captors still watched them closely.\n\nShe sipped from a small cup of water that had been provided. \"Is that a joke?\"\n\n\"A distraction.\" Teg glanced up at the sky. \"We have to trust Duncan to decide on the correct response.\"\n\n\"Maybe he thinks we can handle this ourselves. Duncan has great confidence in our independent abilities.\"\n\n\"As do I. Should it become necessary, I could slaughter every one of these people.\" He chose the word intentionally. _Slaughter_. As he had done with the Honored Matres in their fortress on Gammu. \"And it would take me no longer than the blink of an eye. You know it.\"\n\nSheeana had seen him move against the Handlers, helping her, Thufir, and the Rabbi escape, and she had also seen how much that brief burst of energy had drained him. \"Yes, I know, Miles. And I pray it doesn't become necessary.\"\n\nOff in the distance they heard the whining drone of the small flyer returning from the desert. Teg's sharply attuned ears recognized its sputtering engine sound. The villagers gathered at the packed landing zone, anxious to receive the hunting party. First, two specks appeared in the sky, flying low; then they were joined by many more dots, like a dispersed flock of migrating birds. The drone grew to a roar.\n\nTeg shaded his eyes, identifying many of the flying craft. \"Mining shuttles and lighters from the no-ship. So this is how Duncan plans to rescue us. He's trying to impress them. It appears he sent everything we have.\"\n\n\"We certainly have superior firepower. Duncan could have taken the direct method and rescued us by force of arms.\"\n\nWatching the ships come closer, Teg smiled. \"He's smarter than that. Like me, he wants to avoid bloodshed, especially in a conflict he doesn't entirely understand.\" _Did I teach him that lesson, or did he teach me?_ As the Bashar reflected on their past lives, he didn't know the answer.\n\nMore than forty craft landed together in a flat, open space at the outskirts of the village. They weren't war vessels or armored attack ships, though some had defensive weapons. The Bashar stepped with Sheeana away from the tents, to face the largest mining shuttle. No one tried to stop them; the people were too awed by what they saw.\n\nIt surprised Teg to see Duncan Idaho himself march down the ramp of the lead craft, wearing his traditional House Atreides uniform, complete with polished boots and the starburst insignia of his rank. If the Qelsans had been gone from the Old Empire for fifteen centuries, they weren't likely to recognize any of the symbols, but Teg thought the uniform gave his friend a distinguished aura of command, and undoubtedly provided self-assurance.\n\nDuncan swept his gaze across the confused villagers, finally spotting Teg and Sheeana. The relief on his face was obvious as he made his way to them. \"You're still alive. And unhurt?\"\n\n\"Stuka isn't,\" Sheeana said with an edge of bitterness.\n\n\"You shouldn't have left the no-ship,\" Teg said. \"You're vulnerable now, visible to the searchers and their wide net.\"\n\n\"Let them find me.\" Duncan appeared stony, as if he had reached an inescapable conclusion. \"This endless chase and hiding accomplishes nothing. I can't defeat the Enemy unless I confront them.\"\n\nSheeana glanced nervously at the sky, as if expecting the old man and woman to appear suddenly. \"Garimi could have led the attack, or even Thufir. Instead, you let yourself be swayed by your emotions.\"\n\n\"I factored them in when I made my correct decision.\" Duncan's face flushed, as if he were hiding the real answer, and he rushed ahead with an explanation. \"By comline, I spoke with Stilgar and Liet-Kynes aboard the flyer. We intercepted them out in the desert, so I have some inkling of what's going on here. I know how they killed Stuka\u2014and why.\"\n\n\"And you're surprised to see me alive?\" Sheeana asked. \"Grateful, too, I hope.\"\n\nTeg interrupted. \"The death of Stuka was a tragic overreaction. These people made assumptions about us.\"\n\nNodding, Duncan said, \"Yes, Miles. And if I had made an overzealous response with superior firepower, that would have caused many more deaths and a much greater tragedy. In one of my earlier lives I might have done exactly that, but I only needed to think about what you would have done.\"\n\nStilgar and Liet emerged with the commandos from the tanker. The two young gholas displayed a hardness to them now, and new life behind their eyes. The Fremen naib and the planetologist had found something on Qelso that reenergized them and transported them to other times.\n\nTeg understood what all the gholas had gone through since recovering their memories. They had been sheltered and comfortable aboard the _Ithaca_ , forced to content themselves with reading about their pasts and watching the sandworms in the cargo hold, as if they were specimens in a zoo. But these gholas could remember the real Arrakis. The lives of Stilgar and Kynes had not been safer or more comfortable in the tumultuous old days, but there had been a certain sharp definition to who they were.\n\nOthers continued to emerge from the landed vessels: Thufir, Garimi and more than a dozen Sisters, muscular male Bene Gesserit workers, second-generation children born aboard the no-ship setting foot on a real planet for the very first time in their lives. Five of the Rabbi's followers stood in bright sunlight, looking around in wonder at the landscape, at the open spaces. Presently the old man himself emerged, blinking his bespectacled, owlish eyes.\n\nVar looked admiringly at the mining shuttles and lighters, at his new companions Stilgar and Liet. He raised his chin. Apparently, Duncan had also spoken with the village leader at length during their flight back from the desert. \"Duncan Idaho, you know what trials we face here, what we've been driven to do. We are the only ones who'll stand against the death of this planet. We did not bring the desert here. You have no right to condemn us.\"\n\n\"I didn't condemn you for your struggle, but I can't condone what you did to our companion. Years ago, Bene Gesserit visitors to your world acted without considering the consequences of what they were doing to you. And now it appears you have done the same thing.\"\n\nThe old leader shook his head. His eyes burned with anger and righteousness. \"We killed the witches responsible for depositing sandtrout here. Finding another witch, we killed her too.\"\n\nDuncan abruptly cut off what was sure to be a pointless argument. \"We will take our friends and leave you. I'll let you have your fruitless fight against a desert you can't defeat.\"\n\nTeg and Sheeana stepped forward, anxious to leave this place. Liet and Stilgar, though, held back and looked at each other. The latter squared his shoulders and said, \"Duncan, Bashar . . . Liet and I are having second thoughts. This is the desert\u2014not _our_ desert, but closer than anything we have yet encountered as gholas. We were brought back to life for a purpose. The skills from our past lives can be vital resources in a place like this.\"\n\nLiet-Kynes picked up the speech as if he and Stilgar had rehearsed what they were going to say. \"Look around. Can you imagine a world where our talents are more desperately needed? We are trained as fighters against impossible odds. We're used to desert combat. As a planetologist, I know the best ways to control the spread of the dunes, and I understand more about the sandworm cycle than most people.\"\n\nStilgar added, his passion rising, \"We can show these fighters how to build sietches in the harshest desert. We can teach them to make real stillsuits. One day, perhaps, we shall even ride the great worms again.\" His voice cracked. \"No one can stop the desert, but we can keep the people alive. The rest of you go back to the no-ship, but the Qelsans need us here.\"\n\nSheeana stopped at the hatch of the nearest ship, clearly displeased. \"That is not possible. We need you, and all of the gholas, aboard the _Ithaca_. Each one of you was created, raised, and trained to assist us against the Enemy.\"\n\n\"But no one knows how, Sheeana,\" Duncan pointed out, moved by what the two young men had said. \"None of you can say for certain why we need Stilgar and Liet. And what exactly is our fight?\"\n\n\"We are not your tools or game pieces.\" Stilgar crossed his arms over his chest. \"We are human beings with free will, regardless of how we might have been created. I never asked to serve the Bene Gesserit witches.\"\n\nLiet stood by his friend. \"This is what we want to do, and who's to say it isn't our destiny? We could save a planet, or at least its population. Isn't that an important enough goal?\"\n\nTeg understood the dilemma all too well. These two had found a connection they could hold onto, a battle they could fight that did indeed require their specific abilities. He himself had been created as a pawn, and he'd been forced to play that role. \"Let them go, Sheeana. You have enough experimental subjects on the ship.\"\n\nThufir Hawat came up to the Bashar, relieved to see his mentor safe. He shot a disturbed glance toward Sheeana. \"Is that all we are to them, Bashar? Experimental subjects?\"\n\n\"In a certain sense. And now we must go back to our cage.\" He was anxious to leave this dying planet before other problems arose.\n\n\"Not so fast,\" the old Rabbi said, stepping forward. \"My people are not, and never have been, part of your reckless flight across space. We've always wanted a world to settle. Compared to metal decks and small chambers, this planet looks good enough.\"\n\n\"Qelso is dying,\" Sheeana said. The Rabbi and his hardworking companions simply shrugged.\n\nVar scowled, as did some of the nomadic villagers nearest him. \"We do not need any further drain on our resources. You are welcome here only if you intend to fight back against the desert.\"\n\nIsaac, one of the strong Jewish men, nodded. \"If we decide to stay here, we will fight and work. Our people are no strangers to surviving when the rest of the universe is pitted against us.\"\n\n> _No matter where I go, no matter what I leave behind, my past is always with me, like a shadow_.\n> \n> \u2014DUNCAN IDAHO,  \n>  no-ship logs\n\nLiet-Kynes and Stilgar returned briefly to the _Ithaca_ to retrieve informational archives and some of the equipment they would need to monitor Qelso's changing climate. Liet even converted several spare sensor buoys into orbital weathersats, which the no-ship deployed.\n\nHe said his goodbyes to the other ghola children who had been raised with him\u2014Paul Atreides, Jessica, Leto II. And Chani, his own daughter. With a surge of emotion, Liet grasped the hand of the young woman, who was physically almost three years older than he. He smiled at her. \"Chani, someday you will remember me as I was on Arrakis\u2014busy in the sietches, working as the Imperial Planetologist or the Judge of the Change, carrying on my father's dream for the Fremen and for Dune.\"\n\nHer expression was intense, as if she struggled to grasp some faint flicker of memory as she listened to him. Releasing her hand, he touched her forehead, her dark red hair. \"Maybe I was a strong leader, but I'm afraid I wasn't much of a father. So I must tell you, before I go, that I love you. Then _and_ now. When you remember me, remember all we shared.\"\n\n\"I will. If I remembered everything now, I'd probably want to go with you back to the desert. And so would Usul.\"\n\nBeside them, Paul shook his head. \"My place is here. Our fight is bigger than one desert.\"\n\nStilgar took his friend's arm, urging Liet to hurry. \"This planet is large enough for us. I feel in my soul that this is why Liet and I have been brought back, whether or not Sheeana realizes it. Perhaps someday, no matter how it appears now, we will all see that this is part of the greater battle.\"\n\nMeanwhile, the Rabbi spoke to his fifty-two enthusiastic followers at their stations on the no-ship. Isaac and Levi had taken over many of the old man's duties, and at his signal they directed the Jews to gather their possessions and bring prefabricated shelters from the _Ithaca_ ' _s_ vast storage chambers. Soon, all of them had shuttled down to the surface, where they disembarked and began unloading the landed cargo ships under Isaac's direction.\n\nOn the ground Var strode through the activity, marshalling his followers. He ran a covetous eye over several of the craft that Duncan had brought down during his show of force. \"Those mining shuttles would be a great help to us for carrying supplies and water across the continent.\"\n\nSheeana shook her head. \"Those ships belong to the _Ithaca_. We may need them.\"\n\nVar glowered at her. \"Small enough compensation for causing the death of an entire world, I'd say.\"\n\n\" _I_ didn't contribute to the death of your world. You, however, killed Stuka in cold blood, before\u2014\"\n\nQuickly, Teg went into Mentat mode, mentally inventorying the supplies and equipment they carried aboard the no-ship. To Sheeana, he murmured, \"Although we had no part in the damage done to this world, we did resupply our ship here, and many of our people are staying behind as settlers. A token payment is not unreasonable.\" When she nodded, Teg turned to Var. \"We can spare two shuttles. No more.\"\n\n\"And two desert experts,\" Liet piped up. \"Stilgar and me.\"\n\n\"Not to mention a willing and hardy workforce. You'll be glad to have the Jews here.\" Teg had noticed how industrious the Rabbi's people were. He expected they would do well on this planet, even as the climate turned harsher. Someday, however, they might decide that Qelso wasn't their promised land after all.\n\nNOT SURPRISINGLY, GARIMI and her conservative followers also wanted to leave the no-ship permanently. More than a hundred of the Sisters asked to be released from the _Ithaca_ to settle on Qelso, even with its ever-growing desert. There, they planned to establish the foundation for their new order. Back on the no-ship, Garimi announced their choice to Sheeana more as a courtesy than a matter for discussion.\n\nBut the people of Qelso would hear none of that. They met the Sisters' landed shuttle with drawn weapons. Var stood with his arms crossed over his chest. \"We accept Liet-Kynes and Stilgar among us, as well as the Jews. But no Bene Gesserit witch is welcome here.\"\n\n\"No witches!\" other Qelsans cried, their expressions suddenly murderous. \"If we find them, we kill them.\"\n\nHaving accompanied them for a farewell, Sheeana tried to speak on Garimi's behalf. \"We could take them to the other side of the continent. You would never know about their settlement. I promise, they'll cause you no trouble.\"\n\nBut the incensed Qelsans were not inclined to listen, and Var spoke again. \"Your kind act only for the benefit of the Sisterhood. We welcomed them once, to our deep and lasting regret. Now Qelsans act for the benefit of Qelso. No member of your Sisterhood is welcome here. Short of violence, I cannot be more clear than that.\"\n\nSending up a puff of dust with every step, the Rabbi trudged past tents and portable buildings toward the shuttle. He wiped sweat from his brow and came to stand before Teg and Sheeana, looking uneasily from one to the other. \"I think my people will be happy here, by the grace of God.\" He kicked at the dry dirt with his shoe. \"We were meant to have ground under our feet.\"\n\n\"You look disturbed, Rabbi,\" Sheeana noted.\n\n\"Not disturbed. Sad.\" To Teg he appeared crestfallen, and his watery old eyes seemed redder than usual, as if from crying. \"I will not be with them. I cannot leave the no-ship.\"\n\nBlack-bearded Isaac draped a consoling arm around the elderly man's shoulders. \"This will be the new Israel for us, Rabbi, under my leadership. Won't you reconsider?\"\n\n\"Why aren't you staying with your people?\" Teg asked.\n\nThe Rabbi lowered his gaze, and tears dropped on the hardscrabble ground. \"I have a higher obligation to one of my followers whom I failed.\"\n\nIsaac explained to Sheeana and Teg in a soft voice, \"He wishes to remain with Rebecca. Though she is an axlotl tank now, he refuses to leave her.\"\n\n\"I shall watch over her for all my remaining days. My followers will be in good hands here. Isaac and Levi are their future, while I am their past.\"\n\nThe rest of the Jews surrounded the Rabbi, saying their goodbyes and wishing him well. Then the weeping old man joined Teg, Sheeana, and the others on the waiting shuttle, which took them back up to the no-ship.\n\n## TWENTY-FOUR YEARS AFTER\n\nESCAPE FROM CHAPTERHOUSE\n\n> _We are wounded, but undefeated. We are hurt, but can endure great pain. We are driven to the end of our civilization and our history\u2014but we remain human_.\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> address to the survivors of Chapterhouse\n\nAs the epidemic burned itself out, the survivors\u2014all of them Reverend Mothers\u2014struggled to hold the Sisterhood together. No vaccines, immunity treatments, diets, or quarantines had any effect as the general populace died.\n\nIt required only three days for Murbella's heart to turn to stone. Around her, she watched thousands of promising young acolytes perish, diligent students who had not yet learned enough to become Reverend Mothers. Every one of them died either from the plague or from the Agony that was rushed upon them.\n\nKiria slipped into her former Honored Matre viciousness. On many occasions she argued vehemently that it was a waste of time to care for anyone who had contracted the plague. \"Our resources are better spent on more important things, on activities that have some chance of success!\"\n\nMurbella could not dispute her logic, though she did not agree with the opinion. \"We're not thinking machines. We are humans, and we will care for humans.\"\n\nIt was a sad irony that as more and more of the population died, fewer Reverend Mothers were needed to tend the remaining sick. Gradually, those women were able to turn to other crucial activities.\n\nFrom a nearly empty chamber in the Keep, Murbella peered through the broad, arched window segments behind her throne chair. Chapterhouse had once been a bustling administrative complex, the pulsing heart of the New Sisterhood. Before the plague struck, Mother Commander Murbella had been in charge of hundreds of defensive measures, monitoring the constant progress of the Enemy fleet, dealing with the Ixians, the Guild, refugees and warlords, anyone who could fight on her side.\n\nFar away, she could see the brown hills and dying orchards, but what concerned her was the eerie, unnatural silence of the city itself. The dormitories and support buildings, the nearby spaceport field, the markets, gardens, and dwindling herds . . . all should have been tended by a population of hundreds of thousands. Sadly, most normal activity around the Keep and the city had halted. Far too few remained alive to cover even the most basic work. The world itself was virtually vacant, with all hope dashed in a matter of days. So shockingly sudden!\n\nThe air in the surrounding city was heavy with the stench of death and burning. Black smoke rose from dozens of bonfires\u2014not funeral pyres, for Murbella had other ways to dispose of the bodies, but simply the incineration of contaminated garments and other materials, including infected medical supplies.\n\nIn an admittedly petty moment, Murbella had summoned two exhausted Reverend Mothers. Telling them to bring suspensor clamps, she had ordered them to remove the deactivated combat robot from her private chambers. Though the hated machine had not moved in years, she had begun to feel that it was mocking her. \"Take this thing away and destroy it. I abhor everything it symbolizes.\" The obedient women seemed relieved to follow her orders.\n\nThe Mother Commander issued her next instructions. \"Release our melange stockpiles and distribute spice to all survivors.\" Every healthy woman was dedicated to tending the remaining sick, though it was a hopeless task. The surviving Reverend Mothers were utterly exhausted, having worked without rest for days. Even with the bodily control taught by the Sisterhood, they were hard-pressed to continue. But melange could help keep them going.\n\nLong ago in the time of the Butlerian Jihad the palliative properties of melange had been an effective measure against the horrific machine plagues. This time she didn't expect spice to cure anyone who had already contracted the disease, but at least it would help the surviving Reverend Mothers perform the daunting work required of them. Though Murbella desperately needed every gram of spice to pay the Guild and the Ixians, her Sisters needed it more. If the unified Sisterhood died on Chapterhouse, who would lead the fight for humanity?\n\n_One more cost among so many. But if we don't spend it now, we will never buy victory_. \"Do it. Distribute whatever is necessary.\"\n\nAs her orders were being carried out, she made calculations and realized to her dismay that there weren't enough Reverend Mothers left alive to deplete the Sisterhood's hoarded spice anyway.. . .\n\nHer entire support staff had been stripped away, and she felt isolated. Murbella had already imposed austerity measures, severely cut back services, and eliminated every extraneous activity. Even though most of the Reverend Mothers had survived the plague, it was not certain they would survive the aftermath.\n\nShe summoned those who were Mentats and ordered them to assess the vital work and create an emergency plan of operations, using personnel who were best qualified for the essential tasks. Where could they possibly get the workforce necessary to maintain Chapterhouse, rebuild, and continue the fight? Maybe they could convince some of the desperate refugees from devastated planets to come here, once the last vestiges of plague died out.\n\nMurbella grew tired of simply recovering. Chapterhouse was only a tiny battlefield on the vast galactic canvas of the climactic war. The greatest threat still remained out there, as the oncoming Enemy fleet struck planet after planet, driving refugees like frantic animals before a forest fire. The battle at the end of the universe.\n\nKralizec . . .\n\nA Reverend Mother came running up to her with a report. The woman, barely more than a girl, was one of those who had been forced to attempt the Agony long before she should have, but she had survived. Her eyes bore a faint bluish tinge now, a color that would grow deeper as she continued to consume melange. Her gaze had a stunned, haunted look that penetrated to the depths of her soul.\n\n\"Your hourly report, Mother Commander.\" She handed Murbella a stack of Ridulian crystal sheets on which names were printed in columns.\n\nIn a cold and businesslike fashion, her advisors had at first provided her with simple numbers and summaries, but Murbella demanded actual names. Each person who died from the plague was a _person_ , and each worker and acolyte on Chapterhouse was a soldier lost in the cause against the Enemy. She would not dishonor them by boiling them down to mere numbers and totals. Duncan Idaho would never have condoned such a thing.\n\n\"Four more of them were Face Dancers,\" the messenger said.\n\nMurbella clenched her jaws. \"Who?\" When the woman spoke the names, Murbella barely knew them, unobtrusive Sisters who called no attention to themselves . . . exactly as Face Dancer spies would do. So far sixteen of the shape-shifters had turned up among the plague victims. She had always suspected that even the New Sisterhood had been infiltrated, and now she had proof. But, in an irony the thinking machines could not possibly grasp, the Face Dancers were also susceptible to the horrible epidemic. They died just as easily as anyone else.\n\n\"Keep their bodies for dissection and analysis, along with the others. If nothing else, maybe we can learn something that will allow us to detect them among us.\"\n\nThe young woman waited while Murbella scanned the long list of names. She felt a cold whisper run down her spine as an entry in the third column of one sheet caught her eye. She felt as if she had been struck a heavy blow.\n\n_Gianne_.\n\nHer own daughter, her youngest child by Duncan Idaho. For years the girl had delayed passing through the Agony, never reaching the point where she was ready for the ordeal. Gianne had shown great promise, but that was not nearly enough. Though she had not demonstrated herself to be ready, the girl\u2014among thousands of others\u2014had been forced to take the poison early, the only chance of surviving.\n\nMurbella reeled in shock. She should have been at Gianne's side, but in the chaos no one had told the Mother Commander when her daughter would be given the Water of Life. Most Sisters did not even realize that Gianne was her daughter. The frantic, exhausted helpers would not have known. With her priorities set in true Bene Gesserit fashion, Murbella had tended to her official duties and had gone without sleep for several days in succession.\n\n_I should have been there to support her and help, even if I could only watch over her as she died_.\n\nYet no one had informed her. No one had known that Gianne was special.\n\n_I should have thought to check on her, but I put it off, made assumptions_.\n\nWith so many events crashing around her, Murbella had misplaced her own daughter's life. First Rinya, and now Gianne, both lost to the perilous Agony. Only two other daughters remained: Janess was off at the battlefront fighting thinking machines, while her sister Tanidia, not knowing the identity of her parents, had been sent to join the Missionaria. Though both of them faced risks, they might at least avoid contracting the horrific plague.\n\n\"Two of my children dead,\" she said aloud, though the messenger did not understand. \"Oh, what would Duncan think of me?\" Murbella set the report aside. She closed her eyes for a moment, drew a deep breath, and straightened herself. Pointing to the name on the list of victims she said, \"Take me to her.\"\n\nThe messenger glanced down, ran a quick assessment. \"The bodies in that column have been hauled off to the spaceport. 'Thopter loads of them are taking off right now.\"\n\n\"Hurry. I must try to see her.\" Murbella rushed out of the hall, glancing back to be certain the young woman was right behind her. Though the Mother Commander felt disturbingly numb, she had to do this.\n\nThey took a groundcar to the nearby spaceport, where the fluttering hum of 'thopters droned. On the way, the young Reverend Mother activated her commline, and in a quiet voice requested information. She then directed the driver of the car to take a particular access road.\n\nOn all of the spaceport landing pads, large cargo 'thopters were being loaded with the dead, and were lifting off as soon as they were full. In normal, better times when Bene Gesserits died, they would be buried in the thriving orchards or gardens. The bodies would decompose and provide nourishment and fertilizer. Now they piled up so fast that even large cargo ships could barely keep up with removing them.\n\nThe young assistant directed the driver to a specific grid in the landing zone, where a dark green 'thopter was being loaded by workers. Bundle after bundle of bodies went into the large hold. \"She has to be in that one, Mother Commander. Would you . . . would you like them to unload so that you can find and identify her?\"\n\nAs the two women stepped out of the groundcar, Murbella felt stunned, but tried to steel herself. \"Not necessary. It is only her body, not _her_. Just the same, I'll allow myself enough sentimentality to accompany her out to the dunes.\" Leaving the young Reverend Mother to tend to other duties, Murbella climbed into the 'thopter and sat next to the female pilot.\n\n\"My daughter is aboard,\" Murbella said. Then she grew silent, and stared glumly out the window.\n\nA vibrating shudder passed through the 'thopter as it took off with jets and flapping wings. It would take them half an hour or so to get out to the desert zone, an hour the Mother Commander could ill afford to be away from the Keep. But it was time she desperately needed. . . .\n\nEven the best of the Sisterhood who had undergone the most arduous testing were dismayed by the very real and material tragedy\u2014but not to the point of total surrender. Bene Gesserit teachings showed them how to control base emotions, how to act for the greater good and see the overall picture. Upon watching almost 90 percent of a planet's population fall within a few days, however, the magnitude of the disaster\u2014the _extermination_ \u2014was breaking down even the strongest barriers in many Sisters. It was up to Murbella to maintain the morale of the survivors.\n\n_The thinking machines have found a cruel and effective way to destroy our human weapons, but we are not so easily disarmed!_\n\n\"Mother Commander, we have arrived,\" the pilot said, her clipped words loud enough to be heard over the thrum of the wings.\n\nMurbella opened her eyes to see clean desert, tan eddies of sand and dust curling from stray breezes. It seemed pristine and untouched, no matter how much human debris the Sisterhood dumped there. She saw other 'thopters circling in the sky, descending over the dunes and opening cargo doors to expel loads . . . hundreds of black-wrapped bodies in each aircraft. The dead Sisters tumbled out onto the sand like charred cordwood.\n\nNatural elements would dispose of them far more efficiently than huge funeral pyres could. The aridity would desiccate them, and scouring sandstorms would wear them down to bones. In many cases, the worms would simply devour them. A sort of purity.\n\nTheir 'thopter hovered over a small basin. Large swells of dunes swept up on either side, while dust kicked up by the 'thopter wings swirled around them. The pilot worked her controls, and the bottom doors opened with a weary groan. Bodies tumbled out, wrapped in fabric. They were stiff, their features covered, but to Murbella they were still individuals. One of those unidentified shapes was her own little girl . . . born just before Murbella underwent the Agony herself, just before she lost Duncan forever.\n\nShe didn't delude herself into thinking that if she had been at her daughter's side she might have helped Gianne survive. Passing through the Spice Agony was solely an individual's battle, but Murbella wished she could have been there.\n\nThe bodies spilled unceremoniously onto soft sand. Below, she could see serpentine shapes stirring\u2014two big worms drawn by 'thopter vibrations or the thumps of falling bodies. The creatures scooped up and devoured the human shapes, then plunged back beneath the sand.\n\nThe pilot lifted the 'thopter high enough to swing around, so that Murbella could look down and observe the horrible feeding frenzy. Touching the commline in her ear, the pilot received a message, then offered a faint smile to Murbella. \"Mother Commander, there is some good news, at least.\"\n\nAfter seeing the last unmarked body vanish below, Murbella wasn't in the mood for any sort of cheering up, but she waited.\n\n\"One of our deep-desert research settlements has survived. Shakkad Station. They were far enough out in the sand and had no contact with the Keep. Somehow they avoided the touch of the virus.\"\n\nMurbella remembered the tiny group of offworld scientists and helpers. \"I isolated them myself so they could work. I want them to stay completely cut off\u2014no contact whatsoever! If a single one of us goes near, we could contaminate them.\"\n\n\"Shakkad Station doesn't have enough supplies to last long,\" the pilot said. \"Perhaps we could arrange a package drop-off.\"\n\n\"No, nothing! We can't take the chance of contamination.\" She thought of those people as living at the center of a deadly minefield. But once the epidemic passed, perhaps these few could survive. Only a handful. \"If they run out of food, they should increase their consumption of melange. They can find enough to survive for at least a little while. Even if some of them starve, it's better than having every single one succumb to this damned epidemic.\"\n\nThe pilot did not disagree. As she stared out into the desert, Murbella realized what she and her Sisters had become. She muttered aloud, her words drowned by the thrum of engines. \"We are the new Fremen, and this whole besieged galaxy is our desert.\"\n\nThe 'thopter soared away, heading back toward the Keep, leaving the worms to their feast.\n\n> _Hatred breeds in the fertile ground of life itself._\n> \n> \u2014ancient saying\n\nThe no-ship had flown away from the turmoil of the planet Qelso, leaving behind some of their people, some of their hopes and possibilities. On that world Duncan had taken a great risk, daring to leave the no-ship for the first time in decades. Had he revealed his presence? Would the Enemy be able to find him now, seizing upon that clue? It was possible.\n\nThough he had decided not to cower and hide, Duncan did not intend to bring possible destruction to all the innocent people on this planet. He would make another jump, cover his tracks. And so the _Ithaca_ had risked another unguided plunge through folded space.\n\nThat was three months ago.\n\nThrough a thick plaz viewport, Scytale had watched Qelso dwindle, then suddenly vanish into blankness. He had never been allowed to set foot off the ship. Judging by what he had seen, he would have been happy to settle on that world, in spite of its spreading desert.\n\nAlthough he had his memories back, Scytale found that a part of him missed his father, his predecessor, himself. His mind now contained everything he needed. But he wanted more.\n\nWith this new body, the Tleilaxu Master should have another century before cumulative genetic errors caused him to break down again. Enough time to solve many problems. But when another hundred years were gone, he would still be the last Tleilaxu Master, the only remaining keeper of the Great Belief. Unless he could use the cells of the Council of Masters stored in his nullentropy capsule. Someday, maybe the witches would allow him to employ the axlotl tanks for the purpose the Tleilaxu had intended them.\n\nBack at Qelso, he had agonized over whether to remain there and create a new homeland for the Tleilaxu. Could he build the proper laboratories and equipment? Recruit followers from among the people there? Should he have taken that gamble? Young Scytale had studied the scriptures, meditated long and hard, and finally decided against staying behind\u2014the same decision the Rabbi had reached. On Qelso, he wasn't likely ever to have access to the axlotl technology he needed. His decision was perfectly logical.\n\nThe Rabbi's recent misery and anger, however, was not so easily explained. No one had forced him into his choice. Ever since the ship left the planet and its spreading deserts, the old man had been marching up and down the corridors, spreading dissent like poison. He was the only one of his kind left aboard. Just like Scytale.\n\nThe aged holy man ate with the other refugees, grumbling about how harshly he was treated and how difficult it must be for his people to establish a new Zion without his guidance. Garimi and her hardliners, having been forcibly turned away from the planet, had expressed no sympathy for his grievances.\n\nWatching it all, Scytale concluded that the Rabbi was the sort of person who placed external blame in order to position himself as a martyr. Since he would not leave the axlotl tank that had once been Rebecca, he could cling to his hatred of the Bene Gesserit order, faulting them instead of his own bad choices.\n\nWell, Scytale thought, there was certainly enough hatred to go around.\n\nIN HIS PRIVATE quarters, Wellington Yueh studied his mirror reflection\u2014the sallow face, dark lips, and pointed chin. The narrow visage was younger than the one his memories told him to expect, but still recognizable. Since regaining his memories, he had let his black hair grow out until he had enough at the back to bind in an improvised Suk School ring.\n\nYet he did not fully accept himself. There was one more critical step to take.\n\nIn his hand he held an indelible scriber filled with dark ink that would leave a permanent stain. Not exactly a tattoo, and without any implant or attendant deep Imperial Conditioning, but close enough. His hands were steady, his strokes confident.\n\n_I am a Suk doctor, a surgeon. I can draw a simple geometric shape._\n\nA diamond, prominent on his forehead, perfectly centered. Without hesitation, he drew another stroke, connected the lines, and filled in the skin of his brow. When he was finished, he examined himself again. Wellington Yueh looked back at him from the mirror, Suk doctor and personal physician to House Vernius and then House Atreides.\n\nThe Traitor.\n\nHe set the scriber aside, dressed in a clean doctor's smock, and headed for the medical center. Like the old Rabbi, he was as qualified as any Bene Gesserit doctor to monitor patients and tend the axlotl tanks.\n\nRecently, Sheeana had begun growing another ghola as part of her program, using cells from the Tleilaxu Master's nullentropy tube. Now that Stilgar and Liet-Kynes were gone, she had felt justified in taking that step. Clinging to security, she refused to identify the child gestating in the axlotl tank.\n\nThe Bene Gesserits still claimed to need the gholas, though they could not clearly explain why. Their success in restoring the memories of previous lives in Yueh, Stilgar, and Liet-Kynes had not led to similar accomplishments with the other gholas yet. Some of the witches, especially Proctor Superior Garimi, continued to voice grave reservations about bringing back Jessica and Leto II, because of their past crimes. So they had tried to awaken Thufir Hawat next.\n\nYueh did not know what the witches had done in attempting to break down Hawat's walls, but it had backfired on them. Instead of awakening, Hawat had fallen into convulsions. The old Rabbi had been present and rushed to attend the seventeen-year-old ghola, pushing the Sisters away and scolding them for the foolish risks they had taken.\n\nBut Yueh, like Scytale, already had his old knowledge. He was no longer a child, no longer waiting to _become_ something. One day, he mustered his courage and implored Sheeana to put him to work. \"You witches forced me to remember my old life. I begged you not to, but you insisted on awakening me. Along with my memories and my guilt, came useful skills. Let me act as a Suk doctor again.\"\n\nAt first he wasn't sure the Bene Gesserits would agree, especially considering the constant threat from the unknown saboteur\u2014but when Garimi automatically objected, Sheeana decided to support him. He was granted permission to make rounds in the medical center, so long as he remained under surveillance.\n\nAt the entrance to the main axlotl chamber, two security women scanned Yueh carefully, then waved him through. Neither of them remarked on the new diamond-shaped stain on his forehead. He wondered if anyone still remembered what that mark had once symbolized.\n\nIn preoccupied silence, Yueh went about his inspections of the healthy axlotl tanks. Several produced melange for the ship's stockpiles, but one was obviously pregnant. This unnamed ghola baby would gestate under much tighter security. Yueh was convinced that the child would not be another attempt at Gurney Halleck, Xavier Harkonnen, or Serena Butler. Nor would it be a duplicate of Liet-Kynes or Stilgar. No, Sheeana would experiment with someone else\u2014someone she believed could dramatically help the _Ithaca_.\n\nKnowing Sheeana's impetuous nature, Yueh feared who the baby might be. The Sisters were not immune to making poor choices (as they had proved by bringing _him_ back!). He couldn't believe any of the women had imagined _he_ might be a savior or a hero, yet he had been one of their first experiments. Judging from this, what if the witches were curious to study nefarious personalities from the dark pages of history? Emperor Shaddam? Count Fenring? Beast Rabban? Even the despised Baron Harkonnen himself? Yueh could imagine Sheeana's excuses already. She would no doubt insist that even the worst personalities had the potential to provide invaluable information.\n\n_What snakes will they set loose among us?_ he wondered.\n\nIn the main medical center away from the tanks, he found the old Rabbi grumbling as he assembled a portable medical kit. Since refusing to remain behind on Qelso with his people, he lingered for hours at a time over the tank that he called Rebecca. Though he despised what had been done to her, he seemed relieved that she hadn't been the one implanted with the new ghola.\n\nReluctant to have the Rabbi hover too long near the axlotl tanks, the Sisters gave him duties to keep him busy. \"I am going to run Scytale through a battery of tests,\" the old man huffed to Yueh, starting to retreat from the medical center. \"Sheeana wants him checked out\u2014again.\"\n\n\"I can do that for you, Rabbi. My duties here are light.\"\n\n\"No. Sticking needles into the Tleilaxu is one of my few pleasures these days.\" His gaze fixed on Yueh's new diamond mark, but he did not comment on it. \"Walk with me.\" The Rabbi took Yueh's arm in a tight grip and led him into the corridors, away from the hovering Bene Gesserits. When they were far enough away for him to feel safe, the old man leaned closer, speaking in a conspiratorial tone. \"I am certain Scytale is the saboteur, though I have not found evidence yet. First the old one, and now his ghola replacement. They are all the same. With his memories restored, the young Scytale continues his insidious work to destroy our ship. Who can trust a Tleilaxu?\"\n\n_Who can trust anyone?_ Yueh thought. \"Why would he want to harm the ship?\"\n\n\"We know he has some dirty scheme. Ask yourself why he would store Face Dancer cells in his nullentropy tube, along with all the others\u2014yours included. Why would he need them? Isn't that suspicious enough for you?\"\n\n\"Those cells were confiscated and secured by Sheeana. No one has had access to them.\"\n\n\"Can you be sure of that? Maybe he wants to kill us all so he can restore an army of Face Dancers for himself.\" The Rabbi shook his head. Behind the spectacles, his reddened eyes were angry. \"And that isn't all. The witches have their own schemes. Why do you suppose they won't reveal the identity of the new ghola baby? Does even Duncan Idaho know who is growing in that _tank_?\" He craned his neck, glanced over his shoulder back toward the medical center, watching out for surveillance imagers. \"But you can find out.\"\n\nYueh was perplexed, and curious, but he didn't tell the Rabbi that he had been having some of the same doubts. \"How? They won't tell me either.\"\n\n\"But they don't watch you like they watch me! The witches are afraid I'm going to do something to hinder their program, but now that you have your memories, you're their trusted little ghola.\" The Rabbi slipped him a small sealed polymer disk, with a dab of filmy substance in the center. \"You have access to the scanners. These are cell samples from the pregnant tank in there. Nobody saw me obtain them, but I dare not run the analysis myself.\"\n\nYueh surreptitiously pocketed the disk. \"Do I really want to know?\"\n\n\"Can you afford not to? I leave it to you.\" The Rabbi slipped away, muttering. Carrying his portable medical kit, he trudged off to the Tleilaxu's cabin.\n\nThe sample weighed heavily in Yueh's pocket. Why would the Sisters keep the new ghola's identity secret? What were they up to?\n\nIt took several hours for him to find an opportunity to slip into one of the no-ship's small lab chambers. As a Suk doctor, he had permission to use the facilities. Even so, he worked as swiftly as possible, running the small sample from the axlotl tank through a DNA catalog. He compared the cells from the growing ghola with the identifications that had been run years ago, when the Sisters first assessed the material in Scytale's nullentropy capsule.\n\nYueh found a match fairly quickly, and when he learned the answer, he physically recoiled. \"Impossible! They would not dare!\" But in his heart, as he remembered the torment Sheeana had used to awaken his memories, he didn't doubt the witches would do anything. Now he understood why Sheeana refused to reveal the identity of the ghola.\n\nEven so, the choice itself made no sense. The Sisters had numerous other options. Better ones. Why not try again to bring back Gurney Halleck? Or Ghanima, as a companion for poor Leto II? For what purpose could they possibly need\u2014he shuddered\u2014 _Piter de Vries?_\n\nBecause Bene Gesserits liked to play with dangerous toys, resurrecting people to serve as chess pieces in their great game. He knew the sort of questions they would pursue, just to satisfy their infernal curiosity. Was the genetic makeup of Piter de Vries corrupt, or was he evil because he had been Twisted by the Tleilaxu? Who better to think like an Enemy than a Harkonnen? Was there any evidence to suggest that a new Piter de Vries would turn out evil, as before, if he were not exposed to the corruptive influence of the Baron?\n\nHe could picture Sheeana giving him a condescending frown. \"We need another Mentat. You, of all people, Wellington Yueh, should not hold the past crimes of a ghola's old life against him.\"\n\nHe still did not believe it. He squeezed his eyes shut, and even the fake diamond tattoo on his forehead seemed to burn. He remembered being forced to watch Wanna endure her endless torture at the hands of the vile Mentat. And the man thrusting a knife deep into his back, grinding the blade. _Piter de Vries!_\n\nHe still felt the sharp steel ripping into his organs, a mortal wound, one of the very last memories of his first life. Piter's laugh reverberated, along with the screams of Wanna in the agony chamber . . . and Yueh unable to help her.\n\n_Piter de Vries?_\n\nYueh reeled, barely able to absorb the information. He could not allow a monster like that to be reborn.\n\nDAYS LATER, YUEH entered the medical center, and walked toward the single pregnant tank. This was just an innocent baby at the moment. Even if it was de Vries, this ghola child had committed none of the crimes of the original.\n\n_But he will! He is twisted, evil, malicious._ The Sisters would raise him and insist on triggering his memories. Then he would be back!\n\nYet Yueh was trapped by his own previous logic. If the Piter ghola\u2014in fact, _all_ the gholas\u2014were unable to escape the chains of fate, wouldn't it be the same for Yueh? Was Yueh therefore destined to betray them all? Would he be doomed to make another terrible mistake\u2014or must he sacrifice everything to prevent one? He had thought about consulting Jessica, but he decided against it. This was his burden, his decision.\n\nUsing the Rabbi's sample, he had run the genetic scan privately and seen the result. He had to act alone. Though he was himself a Suk doctor, trained and conditioned to save lives, sometimes the death of one monster was required to save many innocents.\n\n_Piter de Vries!_\n\nIndirectly he had caused de Vries's death the first time around, by giving the poison-gas tooth to Duke Leto, who bit down on it in the Mentat's presence. Yueh had failed in so many ways, caused so much pain and disappointment. Even Wanna would have hated what he'd done to himself, and to the Atreides.\n\nNow, though\u2014a second life, a second chance. Wellington Yueh could make things right. Each of the resurrected ghola children supposedly had a great purpose. He was convinced that this was his.\n\nThe handmade black diamond staining his brow added to the burden as Yueh wrestled with his decision. In his restored memories, he saw with clarity when he had become an actual Suk doctor, when he passed through an entire Inner School regimen of Imperial Conditioning and took the formal oath. \"'A Suk shall not take human life.' \"\n\nAnd yet, Yueh's oath had been subverted, thanks to the Harkonnens. Thanks to _Piter de Vries._ What irony that the breaking of his Suk pledge now allowed him to destroy the very man who had broken that conditioning! He had the freedom to kill.\n\nYueh already had the instrument of death in the pocket of his smock. His plans were in place, and he would take no chances. Since surveillance imagers still monitored the med center and its axlotl tanks, Yueh could not do this in secret, as the real saboteur had. Once he acted, everyone aboard the _Ithaca_ would know who had killed the de Vries ghola. And he would face the consequences.\n\nPerspiration formed on his brow as he crossed the room. With the sharp-eyed Bene Gesserit guard watching him, he could not delay, or the damned witches might detect his uneasiness, his nervous movements. Bringing out his device, Yueh turned a dial as if to recalibrate it, then inserted its probe into the pregnant tank, as he would do in taking a biological sample. Thus he easily administered a lethal dosage of fast-acting poison. So far, no one suspected a thing.\n\n_There. Done._ Fittingly, de Vries had been an expert in cleverly concocted poisons. And no antidote was available for this toxin; Yueh had seen to that. In a matter of hours, the unborn de Vries would shrivel up and die. Along with the tank, unfortunately. But that could not be avoided. A necessary sacrifice.\n\nLeaving the chamber, he smiled grimly and quickened his pace. By tomorrow, there would be no hiding. Thufir Hawat and Bashar Teg would review surveillance holos and interview the guards. They would know who had done it. Unlike the real saboteur, he could not delete the images. He would be caught.\n\nDespite this knowledge, Yueh was content with himself for the first time since his reawakening. At last, he savored the elusive taste of redemption.\n\n> _Send a fact-finding team to Buzzell to learn why soostone exports have dropped off so drastically. This lack of supply, coupled with the precipitous decline in melange production following the Chapterhouse plague, is highly suspicious, especially in light of the fact that the witches are involved in both enterprises. We have learned over the millennia not to take them at their word._\n> \n> \u2014CHOAM directive\n\nNow that he possessed the sample of ultraspice, Khrone knew exactly what lived in the fertile seas of Buzzell. The Navigators certainly had an unexpected scheme there, releasing a new breed of melange-producing worms. He needed to go there and see for himself. The leader of the Face Dancer myriad cared little for the loss of soostone revenue, but in his guise as a CHOAM functionary, he had to feign extreme displeasure.\n\n\"Monsters?\" Standing on the main dock, he gave the woman Corysta a withering glare. \"Sea serpents? Can you think of no better excuses for your incompetence?\"\n\nKhrone scowled at the sea and gathered his dark business robes about his shoulders. Out there in the water, wary Phibians swam, diving to harvest the gems from beds of cholisters, many of which had already been devoured by the hungry and growing seaworms. Armored boats patrolled the coves, though they would surely prove insignificant if one of the large creatures should decide to attack.\n\nReverend Mother Corysta held herself erect, surprisingly unintimidated by the faux official. \"It's no excuse, sir. No one knows where the worms came from or why they have appeared at this time. But they're real. Guild hunting ships dragged in a carcass, if you care to see it.\"\n\n\"Nonsense. Such a story obviously benefits the New Sisterhood.\" Ignoring her protestations, he motioned for Corysta to accompany him along a rocky shoreline path, his shoes crunching on the loose stones. Stepping in a puddle, he frowned down at his feet and kept walking. \"CHOAM suspects that you're creating a false shortage in order to drive up prices. You have financial obligations. For years now, the Sisterhood has been commissioning extremely expensive ships, weapons, and military supplies. Your losses are tremendous.\"\n\n\"They're _humanity's_ losses, sir.\" Corysta's voice was sharp.\n\n\"And now Chapterhouse itself, brought to its knees by a plague. It appears that the Sisterhood can no longer meet its financial obligations. Thus, CHOAM no longer considers you a good credit risk.\"\n\nCorysta turned into the brisk sea wind. \"These are matters you should take up with the Mother Commander.\"\n\n\"I _should_ , but since she is on a quarantined planet, I can't very well call on her, can I? Your Sisterhood is falling apart as a result of external attack and internal strife.\"\n\nWomen stood on plastone ramps at the water's edge to receive a tired-looking group of Phibians who carried a net filled with small, misshapen soostones.\n\nKhrone could tell at a glance the gems were of poor quality, but at least it was part of a shipment he could seize as overdue payment. \"Are your Phibians afraid of sea monsters? Can they not go to richer beds of shellfish?\"\n\n\"They harvest what they can, sir. There are no richer beds. The monsters have eaten many of the cholisters. Our underwater crops are ravaged. And, yes, the Phibians are understandably frightened. Many of them have been slaughtered.\" Corysta stared at him coldly, and Khrone appreciated the steel in her expression; he could respect it. \"We have holo-footage of that, too, if you doubt me.\"\n\n\"It doesn't matter if I believe your story. I only want to know what the Sisterhood intends to do about it.\" Khrone knew the women could do nothing. Eventually the seaworms would bring down the soostone economy of Buzzell, thus removing another one of the Mother Commander's bargaining chips when she desperately needed to buy allegiances and secure equipment.\n\nKept in the dark, the exiled Sisters did not yet understand the true potential of those worms. The primary chemical attributes of the new melange stolen from Buzzell would be a thousand times more effective on human nerve receptors. Oh, it would work very nicely indeed!\n\nHe wondered if the Spacing Guild was even aware of Edrik's destroyed Heighliner yet. It was possible that they weren't. So many of their Navigators had vanished anyway, what was one more? If necessary, by planting a few hints here and there, Khrone could easily blame the loss on an attack by the thinking-machine battle fleet. If nothing else, Omnius made a fine scapegoat.\n\nThe Face Dancer myriad had set their hooks everywhere. The Ixians were building supposed weapons and draining the Chapterhouse coffers of spice; now the Sisterhood's soostone wealth was also disappearing. The Guild relied entirely on computerized navigation devices for their new ships, and the Navigators had no source of melange.\n\nAll enemies of the Face Dancers would fall. He would see to that. The Lost Tleilaxu and the original Masters had already been erased. The Ixians were in Khrone's pocket. Next would come the New Sisterhood, the Guild, and all of humanity. Finally, when he and his minions defeated the thinking machines, nothing would remain but the Face Dancers. And that would be enough.\n\nPleased with himself, Khrone marched up to the dock and yanked the net of soostones from the women trying to sort them. \"Your production has dropped off drastically, and too many CHOAM merchants have gone away empty-handed.\"\n\nCorysta hovered close behind him. \"I hope to hire mercenary hunters to track down the seaworms. It is possible that we may find something of interest\u2014maybe something more valuable than soostones.\"\n\nSo, this woman already had her suspicions about the ultraspice! \"I doubt it,\" he said. Khrone took the net of rough soostones and marched back to the landing pad. Considering the vast game board, he decided it was finally time to head toward the heart of the thinking-machine empire. He would deliver the ultraspice to Omnius and let the evermind continue with his mad dream of creating and controlling his own Kwisatz Haderach.\n\nIt wouldn't help him in the end.\n\n> _We believe that confession should lead to forgiveness and redemption. Usually, however, it leads only to further accusations._\n> \n> \u2014DR. WELLINGTON YUEH,  \n> encrypted entry\n\nThe axlotl chamber smelled of fetid death. Duncan could not tear his gaze away from the still, cold flesh of the tank and the clear signs of necrosis. Rage and helplessness chewed at his gut. And who would the child have been? Sheeana hadn't even told him. Those damned Bene Gesserits and their secrets!\n\n\"Touch nothing,\" Teg warned. \"Get me all security images right away. We will find the saboteur this time.\" One of the Sisters hurried to obtain the recordings.\n\nMeanwhile, young Thufir cordoned off an area around the poisonravaged tank and its unborn ghola. Mostly recovered from the memory-trigger attempt that had gone so dramatically awry, he now sternly followed the methods the Bashar had taught him. The corrosive poison had completely destroyed the growing fetus and then eaten through the wall of the womb that kept the thing alive. Somehow the tank had fallen to the floor, and yellow puddles oozed around the dead flesh.\n\nSheeana turned to one of her Sisters. \"Bring Jessica here. Immediately.\"\n\nDuncan gave her a sharp look. \"Why Jessica? Is she a suspect?\"\n\n\"No, but she will be hurt by this. Maybe I shouldn't even tell her . . . .\"\n\nPresently, Teg received a surveillance holotube from one of the Bene Gesserits. \"I will scan every second. There must be some piece of evidence pointing to the traitor among us.\"\n\n\"There is no need. I killed the ghola.\" A young man's voice. All of them spun to look at a grim-faced Dr. Wellington Yueh. \"I had to.\" Thufir moved swiftly to seize him by the arm, and Yueh did not resist. He stood firm, ready to face the questions that would be thrown at him. \"You can punish me, but I couldn't allow you to spawn another Twisted Mentat. Piter de Vries would only have caused bloodshed and pain.\"\n\nWhile Duncan immediately grasped the implications of Yueh's confession, Sheeana sounded perplexed. \"Piter? What are you talking about?\"\n\nYueh didn't struggle in Thufir's firm grip. \"I witnessed his evil firsthand, and I couldn't allow you to bring him back. Ever.\"\n\nJust then, a breathless young Jessica hurried in with the three-year-old Alia in tow. Alia had intent, eager eyes, full of maturity and understanding that she should not have had. She carried a chubby doll that looked remarkably like a juvenile version of the fat Baron Harkonnen. One of its arms had almost torn loose. Leto II followed his grandmother, looking curious and worried.\n\nSheeana still didn't understand. \"What does Piter de Vries have to do with any of this?\"\n\nYueh made a distasteful expression. \"Don't try to divert me with lies. I know who that ghola was.\"\n\n\"That baby was not Piter de Vries.\" Sheeana spoke her words in a normal tone. \"It would have been Duke Leto Atreides.\"\n\nYueh looked as if he had been felled with an axe. \"There was no doubt\u2014I ran a genetic comparison!\"\n\nJessica listened from just inside the doorway, her face flickering with a rush of hope before plunging into sadness. \" _My_ Leto?\"\n\nYueh tried to sink to his knees, but Thufir held him upright.\n\n\"No! It can't be!\"\n\nWith adult-sharp awareness, Alia tried to take her mother's hand, but Jessica pulled away from the two children to loom over the Suk doctor. \"You killed my Duke? _Again?_ \"\n\nHe grabbed his temples. \"It can't be. I saw the results myself. It was Piter de Vries.\"\n\nThufir Hawat raised his chin. \"At least we have found our saboteur.\"\n\n\"I would never have killed the Duke! I loved Leto\u2014\"\n\n\"And now you've murdered him twice,\" Jessica said, stabbing with each icicle-sharp word. \"Leto, my Leto . . .\"\n\nFinally, Thufir's comment seemed to sink in. \"But I didn't kill the other three gholas or harm their tanks! I committed no other sabotage.\"\n\nTeg said, \"How can we believe you? This will require a great deal more investigation. I will review all evidence in light of this new information.\"\n\nSheeana was clearly troubled, but her words surprised everyone. \"My own truthsense leads me to believe him.\"\n\nThe flesh tank and unborn fetus lay on the floor, chemically decomposing. Black streaks covered all tissue and spread into the surrounding puddle. Yueh struggled to throw himself into the poisonous corrosive, as if by doing so he could kill himself.\n\nWith an iron grip, Thufir held him away from it. \"Not quite yet, _Traitor._ \"\n\n\"No good will come from any of this,\" the old Rabbi said, standing at the doorway of the medical center. No one had heard him arrive.\n\nDesperate, Yueh looked at him. \"I tested the samples you gave me\u2014the baby was de Vries!\"\n\nThe old man backed away like a startled bird. He looked indignant at the very suggestion he might have provoked the unstable young man. \"Yes, I gave you a sample I obtained from the axlotl lab. But I merely raised a question\u2014and never suggested that you should commit murder! Murder! I am a man of God, and you are a doctor\u2014a Suk doctor! Who would imagine . . . ?\" He shook his head. His gray beard looked especially wild today. \"That tank you killed might have been Rebecca! I could never suggest such a thing.\"\n\nEveryone in the room exchanged glances, silently agreeing that Yueh must be the saboteur after all.\n\n\"It wasn't me,\" he said. \"Not the other times. Why would I confess to this but deny the others? My crime is the same.\"\n\n\"Not the same at all,\" Jessica said in a knotted voice. \"This was my Duke . . . .\" She turned and left, while Yueh stared beseechingly after her.\n\n> _Each human, no matter how altruistic or peaceful he seems, carries the capacity to commit tremendous violence. I find this quality particularly fascinating, especially because it can lie dormant for extended periods and then flare up. For instance, consider their traditionally docile women. When these life-givers decide instead to take lives, it is a beautiful ferocity to behold._\n> \n> \u2014ERASMUS,  \n> Laboratory Notes\n\nOn Chapterhouse, the meeting of Reverend Mothers degenerated quickly to murderous intent.\n\nEyes flashing, Kiria nudged the chairdog away from her as she stood. \"Mother Commander, you have to accept certain facts. Chapterhouse is more than decimated. The Ixians still haven't produced the Obliterators they promised. We simply can't win this fight. As soon as we admit that, we can begin to make realistic plans.\"\n\nEyes bleary, Murbella gave the former Honored Matre a level look. \"Such as?\" The Mother Commander dealt with so many ongoing crises, obligations, and unsolvable problems that she could barely concentrate on the reports coming to the mostly empty Keep. The plague had passed on Chapterhouse, so everyone who was going to die was already dead. With the exception of the isolated inhabitants of the deep desert Shakkad Station, the only survivors on the planet were Reverend Mothers.\n\nAll the while, the thinking machines continued to move through space, penetrating deeper into the Old Empire\u2014though by sending scout probes and their plagues here to Chapterhouse, they had broken their previously predictable progression. Omnius must understand the significance of the New Sisterhood; a key victory here could stop the rest of humanity's scattered fighting.\n\n\"Let's take what we need,\" Kiria said, \"copy our Archives, and vanish into the great unknown to create seed colonies. The thinking machines are relentless, but we can be swift and unpredictable. For humanity's survival and the preservation of the Sisterhood, we must disperse, reproduce, and _remain alive._ \" The other Reverend Mothers watched guardedly.\n\nAnger boiled within Murbella. \"Those old attitudes have proved wrong time and again. We can't survive simply by running or by breeding faster than Omnius can kill us.\"\n\n\"Many Sisters believe as I do\u2014the ones still living, that is. You've led us now for almost a quarter of a century, and your policies have failed. Most of Chapterhouse is dead. This crisis forces us to consider new alternatives.\"\n\n\"Old alternatives, you mean. There is too much work ahead of us to rehash this tired debate. Is the identification test for Face Dancer genetics ready for distribution yet? That test is critical for all key planetary governments. Our scientists have studied the cadavers for weeks, and we must send\u2014\"\n\n\"Don't change the subject, Mother Commander! If you won't make the rational decision, if you can't see we need to adapt to circumstances, then I challenge you for leadership.\"\n\nIn astonishment, Laera backed away from the table, while Janess watched her mother, showing no emotion. After the plague had run its course, the female bashar had returned from the fringe battles.\n\nMurbella allowed herself a cool smile as she faced Kiria. Her voice dripped with acid. \"I thought we finished this nonsense years ago.\" She had fought off numerous challengers, killing each one. But Kiria was ready to put it to the test again. \"Choose your time and place.\"\n\n\"Choose? That's just like you, Mother Commander\u2014putting off what must be done now.\" In a flash as swift as nerve impulses could travel, Kiria leapt and lashed out with one foot. Murbella spun away, her spine bending backward with a suppleness that surprised even her. The deadly edge of Kiria's foot came within a hair's breadth of her left eye. The attacker landed on her feet, poised for further combat in the council chamber. \"We can't choose a time and place for fighting. We must always be ready, always _adapt._ \" She lunged forward again, both hands outstretched, fingers rigid as wooden stakes to gouge Murbella's throat.\n\nShe writhed out of the way as Kiria thrust. Before her opponent could yank her hand away, Murbella grabbed the woman's arm and added her own momentum, pulling Kiria off balance and slamming her into the council table, scattering Ridulian crystal sheets. Tumbling off, Kiria crashed into a chairdog. In angry reflex, her fist broke through the placid animal's furry hide and spilled its blood on the floor. The piece of living furniture died with only the faintest peep of alarm and pain.\n\nMurbella sprang onto the tabletop and kicked a loose holoprojector at her opponent. The sharp edge of the device caught Kiria on the brow, making a cut that bled profusely. The Mother Commander crouched, ready to defend herself from a frontal attack, but Kiria ducked under the table and heaved upward with her back, knocking the table over. When Murbella fell, Kiria dove over the capsized table and dropped onto the Mother Commander. She wrapped wiry hands around her throat in a primitive but effective method of assassination.\n\nWith rigid fingers, Murbella jabbed Kiria's side with enough force to fracture two ribs, but at the same time she felt the sickening snap of her own fingers breaking. Instead of withdrawing as expected, Kiria snarled in pain, raised Murbella's neck and shoulders, and slammed her head against the floor.\n\nMurbella's ears rang, and she felt her skull crack. Fluttering black spots of unconsciousness circled her vision like tiny vultures waiting for fresh carrion. She had to stay awake, had to keep fighting. If she faded now, Kiria would kill her. And if she was defeated here, she would lose not just her life, but the Sisterhood as well. The fate of the entire human race could be decided in this moment.\n\nJaness watched her mother with anguish, but Laera and the other Reverend Mothers were well trained and would not interfere. The unification with Honored Matres had required certain concessions from the Bene Gesserits, including the right of anyone to challenge the Mother Commander's leadership.\n\nKiria continued to choke her, while Murbella strained to draw a breath. Blocking the pain of her broken fingers, she clapped her palms hard against Kiria's ears. As the deafened woman reeled, Murbella gouged out her right eye with a crooked forefinger, leaving blood and jelly all over her face.\n\nKiria writhed away, pushing to her feet, but Murbella followed with a flurry of hand blows and kicks. Even so, her challenger was not defeated yet. Kiria hammered her heel into Murbella's sternum, then struck her abdomen with a side blow. Something ruptured inside; Murbella could feel the damage but didn't know how bad it was. Digging into her energy reserves, she drove Kiria aside with her shoulder.\n\nThe Honored Matre's lips were drawn back to expose bloody gums and teeth. Rallying, Kiria gathered all of her strength to strike, ignoring her mangled eye. But as she planted her foot, she slipped on a smear of the chairdog's blood on the smooth floor. This threw off her balance for an instant\u2014just long enough to give Murbella the advantage. Without hesitating, the Mother Commander struck a blow so hard that her own wrist shattered\u2014as did Kiria's neck. The challenger fell dead to the floor.\n\nMurbella swayed, while Janess came forward, concern on her face, ready to help her mother, her superior. Murbella raised an arm. Her broken wrist flopped limply, but she banished the wince of pain from her face. \"I am capable of standing by myself.\"\n\nSome of the younger Reverend Mothers, with wide eyes and intense expressions, had backed up to the walls of the council chamber.\n\nMurbella wanted so badly to fall beside her victim on the floor, letting the exhaustion and pain take control. But she could not allow that\u2014not with so many Reverend Mothers observing. She could never reveal a moment of weakness, especially now.\n\nSummoning her breath, dredging up the last sparks of endurance, Murbella spoke in an even voice. \"I will go to my quarters now and heal.\" Then, in a lower voice, \"Janess, have the kitchens send up a restorative energy drink.\" She cast a dismissive glance at the dead Kiria, then raised her eyes to Janess, Laera, and the awed spectators in the hall. \"Or do any of you wish to challenge me and take advantage of my condition?\" In defiance, she held up her broken wrist. No one took the offer.\n\nInjured inside and out, Murbella had no clear memory of how she made it back to her quarters. Her progress was slow, but she accepted no aid. The other Reverend Mothers, sensing her determination, left her alone.\n\nIn her dim room, the spice drink was already waiting for her. _How long did it take me to get here?_ After a single sip she could feel energy surge through her body. She murmured a thankful blessing to Janess; her daughter had made this drink extremely potent.\n\nLeaving word that she was not to be disturbed, she sealed her door and consumed the rest of the rejuvenating beverage. It boosted the internal repairs she had already begun to make, delicately probing with her mind to judge the extent of her injuries. Finally, allowing the flood of pain to wash into her senses, Murbella carefully assessed what Kiria had done to her. The degree of internal damage frightened her. Never in any previous challenge had she come so close to losing.\n\n_Will the rest of the Reverend Mothers rally behind me\u2014or will they start sniffing out my weaknesses again like hungry hyenas?_\n\nShe could not afford to waste time and energy battling her own people. Few enough of them remained alive after the plague. What if the Sisterhood was infiltrated by Face Dancers again? Could one of them, trained in exotic fighting techniques, pose as an Honored Matre challenger and kill Murbella? What if a Face Dancer became the Mother Commander of the Sisterhood? Then all indeed would be lost.\n\nShe lay back, closed her eyes, and plunged into a healing trance. Time was of the essence. She had to regain her full strength. The forces of Omnius had located this world and would be coming soon.\n\n> _Every man casts a shadow . . . some darker than others._\n> \n> \u2014The Cant of the Shariat\n\nWhile Yueh was under arrest and interrogation, yet another instance of sabotage occurred.\n\nThe Bene Gesserit Sisters summoned the passengers to the great auditorium for an emergency meeting. Garimi seemed particularly agitated; Duncan Idaho and Miles Teg were alert. Eyes intent, Scytale observed, always the outsider. What had happened now? _And will they blame me for it?_\n\nWas it worse than the murder of another ghola and axlotl tank? Had someone else been killed? Had another water reservoir dumped into space, squandering the new supplies they had acquired at Qelso? Spice stockpiles contaminated? Food vats destroyed? The seven captive sandworms harmed?\n\nThe Tleilaxu man sat back, watching everyone stream in from outside corridors and take their seats in islands of friendship or shared opinions. Palpable tension radiated from them. More than two hundred gathered, most of them curious, alarmed, or frightened. Only a few proctors stayed in isolated sections with the younger children that had been born during the journey; others were old enough to be treated as adults.\n\nThe Bashar himself made the announcement. \"Explosive mines have disappeared from the sealed armory. Eight of the hundred and twelve\u2014certainly enough to cause severe damage to this ship.\"\n\nAfter a brief silence, conversation returned in a riptide of whispers, gasps, and accusations.\n\n\"The mines,\" Teg repeated. \"Back on Chapterhouse they were placed around this ship as a self-destruct mechanism in case Duncan or anyone else tried to steal it. Now eight of them are gone.\"\n\nSheeana went to stand beside the Bashar. \"I deactivated those mines myself, so that this vessel could escape. They were locked securely away, but now they've disappeared.\"\n\n\"If they are missing, they might have been dumped out into space . . . or planted around the ship like time bombs,\" Duncan said. \"I suspect the latter, and that our saboteur has further plans.\"\n\nThe Rabbi moaned loudly. \"You see? More incompetence! I should have stayed on Qelso with the rest of my people.\"\n\n\"Maybe _you_ stole the mines,\" Garimi snapped.\n\nHe looked horrified. \"You dare accuse me? A holy man of my stature? First Yueh says I manipulated him to murder a ghola baby, and now you think I have stolen explosives?\" Scytale saw that the frail old man could never have lifted one of the heavy mines, much less eight of them.\n\n\"Yueh has been under the constant surveillance of Thufir Hawat and myself,\" Teg said. \"Even if he did kill the axlotl tank and the growing ghola, he could not have stolen the mines.\"\n\n\"Unless he has an accomplice,\" Garimi said, setting off another chain of muttering.\n\n\"We will discover who took them.\" Sheeana cut off the squabble. \"And where they have been hidden.\"\n\n\"We've heard similar promises in the past three years,\" Garimi continued, with a meaningful glare at Teg and Thufir. \"But our security has been completely ineffective.\"\n\nPaul Atreides sat in one of the front rows, near Chani and Jessica. \"Are we certain the mines only disappeared recently? How often is the armory checked? Maybe Liet-Kynes or Stilgar took them for their war against the sandtrout without telling us.\"\n\n\"We should evacuate this ship,\" the Rabbi said. \"Find another planet, or go back to Qelso.\" His voice quavered. \"If you witches hadn't . . . hadn't . . . taken Rebecca, I would be safe now with my people. We all could have settled there.\"\n\nGarimi scowled. \"Rabbi, for years you've encouraged dissent with your sniping and destructive arguments, without offering alternatives.\"\n\n\"I speak the truth as I see it. The stolen mines are only the latest in a string of sabotage. My Rebecca remains alive only by chance when four other axlotl tanks have now been murdered. And who damaged the life-support systems, the water holding tanks? Who contaminated the algae vats and destroyed the air-filtration mats? Who poured acid on the seals of the observation window in the sandworm hold? There is a criminal among us, and he is growing bolder and bolder! Why don't you find him?\"\n\nScytale remained silent and unobtrusive, listening to the debate. Everyone feared there would be more incidents of sabotage, and the stolen mines would be sufficient to cripple or destroy the great ship.\n\nThe Tleilaxu had no doubt they would eventually turn their suspicions toward him because of his race, but he could prove his innocence. He had laboratory records, surveillance images, a solid alibi. Nevertheless, _someone_ had committed the acts of sabotage.\n\nWhen the exhausting meeting broke up, the Rabbi stalked past Scytale in a huff, saying he was going to go sit in a vigil beside Rebecca, \"to make certain no one else tries to kill her!\" As the old man passed close, Scytale caught the Rabbi's usual faint, strange scent, a subtly different flavor in the air.\n\nOn instinct, Scytale emitted a barely audible whistle in a complicated melody that he remembered from deep in his past lives. The Rabbi ignored him and stalked away. Scytale frowned, not sure if he had noticed a brief hesitation as the old man walked past.\n\n> _God is God, and life is His alone to give. If God Himself has not the strength to survive, then we are left with nothing but despair._\n> \n> \u2014The Cant of the Shariat\n\nEvery investigation of Rakis yielded the same result. Only a few insignificant pockets of its ecosystem had survived. The planet was empty and haunted, yet it seemed to have its own will to live. Against all odds and science, Rakis still clung to its sparse atmosphere, its gasps of moisture.\n\nGuriff's hard-bitten prospectors happily accepted supplies that Waff and the Guildsmen offered as a gesture of goodwill. Waff's primary motivation for this was to get the men to leave him alone while he conducted his innocuous \"geological investigations.\" The prospectors were supplied by irregular CHOAM vessels that came to check on their work, but Guriff had no idea when the next ship would come. The Tleilaxu Master had enough packaged food from the Heighliner to last for years, if his deteriorating body lasted that long.\n\nAbove all, he needed to tend to his worms.\n\nAs he'd hoped, the prospectors spent the harsh days and nights concentrating on their own digging, hoping to find the legendary lost hoard of the Tyrant's melange. Insulated scout cruisers braved the rugged weather, carrying sensors and probes up to the polar regions, while the men bored test holes, searching unsuccessfully for any threads of spice.\n\nThe large dropbox from Edrik's Heighliner had included a wide-bed groundcar that could roll across even the roughest terrain. When the prospectors departed, Waff called his four Guildsmen to assist him. With no curious eyes watching, they wrestled the long, sand-filled test tanks aboard his groundcar. Waff would make a pilgrimage out into the charred and glassy wasteland that had once been a sea of dunes.\n\n\"I will release the specimens myself. I don't need your assistance.\" He directed the Guildsmen to return to the rigid-walled survival tents. \"Stay and prepare our food\u2014and make certain you follow the accepted ways.\" He had given them precise instructions on the proper techniques. \"Once I free the worms, I intend to come back for a celebration.\"\n\nHe did not want Guriff and his men, nor any of these untrustworthy Guild assistants, to observe such a private and holy moment. Today he would restore the Prophet to Rakis, to the planet where He belonged. Dressed in protective clothing, he keyed in coordinates and drove off with the two long aquariums in the back of his groundcar. Heading due east, he sped away into a ruddy orange dawn.\n\nAlthough the landscape here was smeared, eroded, and unrecognizable, Waff knew exactly where he was going. Before coming to Rakis, he had dug up the old charts, and because the Honored Matres' Obliterators had altered even the planetary magnetic field, he had carefully recalibrated his maps from orbit. A long time ago, God's Messenger had purposefully carried him to the location of Sietch Tabr. The worms must consider it sacred, and Waff could think of no more appropriate place to turn loose the armored, augmented creatures. He drove there now.\n\nLight from the dust-thickened sky bathed the glassy ground in eerie colors. From the tanks behind him, Waff could feel the thumping of the worms as they writhed, impatient to burst out onto the open desert. Home.\n\nBack on the Heighliner, Waff had observed the bucking and thrashing creatures, measuring their growth in the lab. He knew the worms were dangerous, and that long confinement in small tanks sapped the creatures' strength. Even under carefully controlled conditions, he hadn't been able to replicate the optimal environment, and the specimens had weakened. Something was wrong.\n\nBut hope infused him. Now that he was actually here, all would be right again. Holy Rakis! He could only pray that this injured dune world would provide what a Tleilaxu Master could not, offering some ineffable benefit to the worms, to the Prophet.\n\nWhen Waff reached the plain and saw the melted rocks, he remembered the weathered line of mountains that had sheltered the buried tomb of the Fremen city. He stopped the groundcar. A vitrified crust\u2014rocky grains melted to glass by the blasts of incomprehensible weapons\u2014covered what had once been open sand. But the worms would know what to do.\n\nBehind the vehicle, Waff paused a moment to close his eyes in prayer to his God and His Prophet. Then, with a flourish, he disengaged the plaz walls of the tanks and let sand spill out. Long serpentine shapes lunged free like uncoiled springs, and dropped to the ground around the vehicle. Waff gazed in wonderment at their thick, ridged bodies, and the python fluidity of their motion.\n\n\"Go, Prophet! Reclaim your world.\"\n\nEight worms slithered on the hard, smooth ground. _Eight_ , a sacred Tleilaxu number.\n\nThe freed creatures spread out in random paths, while he watched them in awe. Waff hoped they could break through the fused sand and tunnel into softer levels below, as he had designed them to do. Each of the specimens had a tiny implanted tracer that would enable him to follow them and continue his investigations.\n\nHowever, the sandworms turned and circled the vehicle, coming closer. Hunting _him._ In a moment of fear, Waff froze. They were certainly large enough to attack and kill him. \"Prophet, do not harm me. I have brought you back to Rakis. You are free to make this your kingdom once again.\"\n\nThe worms raised their blunt heads, swaying back and forth. _Are they trying to send me some sort of message?_ He struggled to comprehend. Could their hypnotic movements be an alien dance? Or a predatory maneuver?\n\nHe did not move. He waited.\n\nIf this landscape was too harsh for them, if the Prophet needed to consume him in order to survive, Waff was fully prepared to donate the flesh of his own deteriorating body. If this was to be the end, so be it.\n\nThen, as if at a silent signal, the sandworms turned in unison and sped off, their flexing ridges bumping across the glassy dunes. Presently they stopped, bent their armored heads downward, and smashed into the hard surface. They broke the crust and plunged downward, tunneling into the pristine, sterilized sands. Returning to the desert! Waff's heart swelled. He knew they would survive.\n\nAs he returned to the groundcar, he realized he had tears in his eyes.\n\n> _When the forces are arrayed and the final battle is engaged, the outcome may be decided in only a few moments. Remember this: By the time the first shot is fired, half the battle is already over. Victory or defeat can be determined by the preparations that are set in place weeks or even months beforehand._\n> \n> \u2014BASHAR MILES TEG,  \n> resource allocation request to the Bene Gesserit\n\nChief Fabricator Shayama Sen agreed to come to Chapterhouse, but the Ixian dignitary remained aboard his ship high in orbit, far above the recovery operations. He would not risk exposure to any last vestiges of the plague, though the disease had burned itself out down there.\n\nMurbella had to go to him to make her demands\u2014but under the strictest quarantine conditions. Encased in her own decontamination sphere, like a laboratory specimen in a tank, she felt foolish and helpless. The Bene Gesserit sphere's outer hull\u2014though scorched by its passage through the atmosphere on the way to orbit and then exposed to the vacuum of space\u2014underwent additional irradiation and sterilization procedures up there. Fail-safes, redundancies. _Justified paranoia,_ she admitted to herself. Although Murbella did not fault him for taking such extraordinary precautions, the Ixian nevertheless had much explaining to do.\n\nWhile waiting inside her sealed chamber aboard the Guildship (which was guided by a mathematical compiler rather than a Navigator), she composed herself. Still sore and battered from her duel with Kiria, she was satisfied that her violent response to the stupid power play had been necessary. None of the other distraught Sisters would challenge her now, thus leaving her role as Mother Commander uncontested.\n\nOnce again, Murbella cursed the rebel Honored Matres and their mindless destruction of the massive shipyards and weapons shops on Richese. Had that not happened, with both Ix and Richese producing armaments, the human race could have consolidated a meaningful defense. Now that Ix was the primary industrial center, the Chief Fabricator felt he could be intractable. Shortsighted fools!\n\nShayama Sen marched into the large metal-walled room and took a comfortable seat facing her. He looked smug and safe, while she felt like a caged zoo animal. \"You called me away from our work, Mother Commander?\"\n\nDespite the inherent awkwardness of her position, Murbella tried to take command of the meeting. \"Chief Fabricator, you have had three years to duplicate the Obliterators we provided, but all we've received in exchange for our melange payments are reports on your tests, and promise after promise. The Enemy has destroyed more than a hundred planets, and their battleships keep coming. Chapterhouse itself was nearly eradicated by the recent plague.\"\n\nSen bowed formally. \"We are fully aware of this, Mother Commander, and you have my condolences.\" He got up and poured himself a glass of water from a pitcher, then roamed the large meeting room, flaunting his own freedom.\n\nAnger heated her cheeks and neck. How could this man sound so calm in the face of the crumbling human civilization? \"We require the weapons you promised us\u2014and without further delay.\"\n\nSen tapped his circuitry-imprinted fingernails together, pondering her containment sphere with a blank stare. \"But we have not yet received full payment, and we hear your New Sisterhood is in dire financial straits. If we continue to devote all our resources to these Obliterators, and you renege\u2014\"\n\n\"The agreed-upon amount of melange is yours the moment you finish installing Obliterators in our new warships. You know this.\" She didn't dare let Sen discover that she had released a great deal of stockpiled spice to help her fellow Reverend Mothers fight the plague.\n\n\"Ah, but if your spice is contaminated by the plague, of what use is it to us? How else will you pay?\"\n\nMurbella couldn't believe his blindness. \"The spice is not contaminated. We will implement any sterilization measures you require.\"\n\n\"And what if that destroys its efficacy?\"\n\n\"Then we will give you the original spice to decontaminate in whatever manner you see fit. Stop quibbling about nonsense when the extinction of the human race is imminent!\"\n\nSen seemed scandalized. \"You call it nonsense? The properties of spice are complex and could be harmed by such aggressive measures. The substance is of no value to us if we cannot use it.\"\n\n\"The plague organism has a short lifetime. Unless it is transferred from host to host, the disease dies swiftly. Place the spice on an airless moon for a year if you choose to.\"\n\n\"But the difficulties and the inconvenience . . . I believe these circumstances merit a renegotiation of our price.\"\n\nIf the container wall had not prevented her, Murbella would have killed him for his insolence. \"Have you any idea of how much destruction the Enemy has spread?\"\n\nHe pursed his lips, and said, \"Let me dispense with subtleties, Mother Commander. Honored Matres provoked this Enemy into launching its fleet against them, and in turn against the rest of us. Your association with the whores was your own folly, and the whole human race has paid for it. Ix has no quarrel with these robotic invaders. Since they evolved from ancient thinking machines, it is possible that we Ixians have more in common with _them_ than with manipulative, murderous females.\"\n\n_Ah_. Now she was beginning to understand. Listening to the sharp voice of Odrade-within and a thousand other Reverend Mothers frantically offering advice, Murbella forced calm upon herself. It was clear that the Ixian was trying to escalate this discussion. But why? To distract her? Had he failed to make as much progress in developing the Obliterators as he claimed? Was production running behind schedule?\n\nShe selected a gambit that she hoped would shut down his blathering. \"I authorize a thirty percent increase in your spice allotment, to be put in a trust fund held in the Guild Bank of your choosing. I expect that is sufficient to make up for any inconvenience? However, payment will be contingent upon your _actual delivery_ of the weaponry in our contract. The Guild has delivered our new warships. Now, where are my Obliterators?\"\n\nShayama Sen bowed, accepting her offer and withdrawing his objections. \"Our manufactory worlds are operating at full capacity. We can begin loading Obliterators aboard your new ships immediately.\"\n\n\"I'll issue the orders.\" She paced within the decontamination bubble like a Laza tiger. The smells of disinfectant chemicals seeping through the air filters made her want to gag. She didn't think the chamber's replenishers were working properly. \"How do we know your weapons will perform as you promise?\"\n\n\"You provided the originals, and we duplicated them precisely. If the originals functioned, then these will, too.\"\n\n\"The originals _functioned_. You've seen what's left of Rakis and Richese!\"\n\n\"Then you have nothing to fear.\"\n\n\"From now on, I insist that we place Bene Gesserit inspectors and line supervisors in your manufactories. They will keep you accountable and guard against sabotage.\"\n\nShayama Sen struggled with the demand, but could find no legitimate argument against it. \"Provided your women do not interfere, we shall allow them access. Is that all?\"\n\n\"We also need to witness a successful test before going into battle.\"\n\nSen smiled again. \"You would have us annihilate a world merely to prove a point? Hmm, I see Honored Matre methods persist in your New Sisterhood.\" He chuckled. \"I'll give you full records of our previous tests and even arrange for a new demonstration, if you like.\"\n\n\"We will review your data, Chief Fabricator. Transmit it to Chapterhouse, and arrange for a demonstration that I can see with my own eyes.\"\n\nHe tapped his silicon fingernails again, an annoying nervous habit. \"Very well. I'll find a nice planetoid to blow up for your entertainment.\"\n\nMurbella pressed against the curved, transparent wall of her sphere. \"And there's one other thing I insist upon. Face Dancers have been found on many worlds, manipulating governments, weakening our defenses. Some even managed to infiltrate Chapterhouse. I need to have assurance that _you_ are not a Face Dancer.\"\n\nSen reeled backward in surprise. \"You accuse me of being an Enemy, a shape-shifter operative?\"\n\nMurbella leaned against the solid wall, regarding him coolly. His indignation did nothing to convince her. She worked the internal controls, and a small, sealed container opened near the base of the Bene Gesserit chamber. It was a sterilization bin, an autoclave and chemical bath. Steam still curled from the package as it emerged for the Chief Fabricator to take.\n\n\"This is a testing device we have developed. After analyzing Face Dancer specimens found among our dead, we ran genetic tests and developed this infallible indicator. Right now, Chief Fabricator\u2014as I watch\u2014you will complete this test on yourself.\"\n\n\"I will _not_.\" He sniffed.\n\n\"You will, or you'll receive none of our melange.\"\n\nSen roamed again, frowning. \"What is this test? What does it do?\"\n\n\"It is mostly automated.\" Murbella explained the principle to him and the easy steps. \"As a bonus for you, we can allow Ix to produce these in great quantities. There are plenty of suspicious people who see Face Dancers everywhere. You could make a tidy profit selling these kits.\"\n\nSen considered. \"You may be right.\"\n\nWhile Murbella observed, he went through the motions, standing close enough to her bubble that she could watch his every movement. As far as the Reverend Mothers knew, the test could not easily be foiled, and the Chief Fabricator had had no time to prepare a deception. She waited with intense interest, and was relieved when the indicators declared him fully human. Shayama Sen was not a Face Dancer.\n\nWith an irritated expression on his face, he held the chemical tab up for her to see. \"Are you satisfied now?\"\n\n\"I am. And I advise you to perform this test on all of your chief engineers and team leaders. Ix is a likely target for the Enemy to infiltrate. Another reason for my Sisters to supervise your vital work for us.\"\n\nSen looked genuinely disturbed, as if that possibility had not occurred to him. \"I concede your point, Mother Commander. I would like to see those results myself.\"\n\n\"Then include them when you send your data about the Obliterator tests. In the meantime, prepare to install your weapons in all the new warships coming out of the Junction shipyards. We are about to engage in an all-out offensive against the thinking-machine fleet.\"\n\n> _Each sentient life requires a place of extreme serenity, where the mind may roam afterward in memory and to which the body longs to return_.\n> \n> \u2014ERASMUS,  \n> contemplation notes\n\n\"Now that you have been among us for more than a year, it is time to show you my special place, Paolo.\" The independent robot waved a metal arm, and his majestic robes flowed around him. \"And you too, of course, Baron Harkonnen.\"\n\nThe Baron scowled, his voice dripping with sarcasm. \"Your special place? I'm sure we'll be charmed by what a _robot_ considers to be a special place.\"\n\nDuring the time that he and Paolo had lived on Synchrony, he'd lost his awe and fear of thinking machines. They seemed plodding and grandiose, full of redundancy and very little impulsivity. Since Omnius thought he needed Paolo, along with the Baron to keep Paolo in line, the two were safe enough. Even so, the Baron felt a need to show some backbone, and turn the circumstances to his own benefit.\n\nAround the interior of the now-familiar cathedral chamber, the walls became a wash of color, as if invisible painters were hard at work. Instead of blank metal and stone surfaces, the murky shades of green and brown sharpened into highly realistic trees and birds. The oppressive ceiling opened to the sky, and peculiar synthesized music began playing. A gemgravel pathway ran through the lush garden with comfortable reclining benches at intermittent intervals. A lily pond appeared on one side.\n\n\"My contemplation garden.\" Erasmus formed his artificial smile. \"I enjoy this place very much. It is special to me.\"\n\n\"At least the flowers don't stink.\" Paolo ripped up one of the bright chrysanthemums, sniffed it, and discarded it at the side of the path. After a year of constant training, the Baron had finally made the boy's personality into something he could be proud of.\n\n\"This is all lovely,\" the Baron said drily. \"And utterly pointless.\"\n\n_Be careful what you say to him, Grandfather_ , cautioned the Aliavoice within. _Don't get us killed today_. It was one of her continual harangues.\n\n\"Is something troubling you, Baron?\" Erasmus asked. \"This should be a place of peace and contemplation.\"\n\n_See what you've done! Get out of my head_.\n\n_But I'm trapped here with you. You can't get rid of me. I killed you once with the gom jabbar, and I can do it again with a little careful manipulation_.\n\n\"I see that you are often plagued by disturbing thoughts.\" Erasmus stepped closer. \"Would you like me to open your skull and look inside? I could fix the problem.\"\n\n_Be careful with me, Abomination! I just may take him up on the offer!_\n\nHe forced a smile as he replied to the independent robot. \"I'm just impatient to learn exactly how we can work with Omnius. Your war against humanity has gone on for some time now, and we've been your guests for a year. When will we do what you brought us here for?\"\n\nPaolo kicked a divot into the gemgravel path. \"Yes, Erasmus. When do we get to have fun?\"\n\n\"Soon enough.\" The robot swirled his robes and guided his companions through the garden.\n\nThe boy had just passed his eleventh birthday and was developing into a strong young man, well-muscled and highly trained. Thanks to the Baron's constant influence, virtually all traces of the former Atreides personality had been extinguished. Erasmus himself had supervised Paolo's vigorous combat training against fighting meks, all to prime him to become the supposed Kwisatz Haderach.\n\nBut the Baron still could not fathom _why_. Why would the machines care about some obscure human religious figure from ancient history?\n\nErasmus motioned for them to sit on the nearest bench. The synthesized music and birdsong around them grew louder and more energetic until they became intertwined melodies. The robot's expression shifted once again, as if in reverie. \"Is it not beautiful? I composed it myself.\"\n\n\"Most impressive.\" The Baron despised the music as too smooth and peaceful; he preferred more cacophonous, discordant selections.\n\n\"Over the millennia, I created wondrous works of art and many illusions.\" Erasmus's face and body shifted, and he became entirely human in appearance. Even the gaudy and unnecessary garments altered, until the robot stood before them again as a matronly old woman in a floralprint dress holding a small hand trowel. \"This is one of my favorites. I have perfected it over the years, drawing from more and more of the lives my Face Dancers bring me.\"\n\nWith the hand trowel she dug in the simulated soil near the bench, getting rid of weeds that the Baron was sure had not been there moments earlier. A worm crawled out of the exposed, dark dirt, and the old woman sliced it in half with the trowel. The two parts of the squirming creature faded into the dirt.\n\nA gentle undercurrent flowed in her voice, not unlike that of a grandmother telling bedtime stories to children. \"Long ago\u2014during your original lifetime, dear Baron\u2014a Tleilaxu researcher named Hidar Fen Ajidica created an artificial spice that he called amal. Though the substance proved to have significant defects, Ajidica consumed huge quantities of it himself, and as a result he went increasingly mad, which led to his demise.\"\n\n\"Sounds like a failure,\" Paolo said.\n\n\"Oh, Ajidica failed spectacularly, but he did accomplish something very important. Call it a side effect. For his special ambassadors, he created greatly improved Face Dancers, with which he intended to populate a new domain. He dispatched them into deep space as scouts, colonizers, preparers of the way. He died before he could join them. Poor foolish man.\"\n\nThe old woman left her trowel stuck in the ground. When she straightened, she pressed her hand against the small of her back, as if to comfort an ache. \"The new Face Dancers located our machine empire, and Omnius allowed me to study them. I spent generations working with the shape-shifters, learning how to draw information from them. Lovely biological machines, far superior to their predecessors. Yes, they are proving to be extremely helpful in winning our final war.\"\n\nLooking around the illusory garden, the Baron saw other forms, minor workers who appeared to be human. New Face Dancers? \"So you made an alliance with them?\"\n\nThe old woman pursed her lips. \"An alliance? They are servants, not our partners. Face Dancers were made to serve. To them, Omnius and I are like gods, greater Masters than the Tleilaxu ever were.\" Erasmus seemed to be pondering. \"I do wish they had brought one of their Masters to me before the Honored Matres destroyed nearly all of them. The discussion could have been most enlightening.\"\n\nPaolo brought the conversation back around to a subject that interested him. \"As the final Kwisatz Haderach, I will be a god, too.\"\n\nErasmus laughed, an old woman's cachinnation. \"Beware of megalomania, young man. It has brought down many a human\u2014such as Hidar Fen Ajidica. Soon I expect to have a key to help you reach your potential. We need to free the god that crouches inside your body. And that requires a powerful catalyst.\"\n\n\"What is it?\" the young man demanded.\n\n\"I keep forgetting how impatient you humans are!\" The old woman brushed off her flower-print dress. \"That is why I enjoy the Face Dancers so much. In them, I see the potential for perfecting humans. Face Dancers could be the sort of humans that even thinking machines might tolerate.\"\n\nThe Baron snorted. \"Humans will never be perfect! Believe me, I've known plenty of them, and they're all disappointing in some way.\" Rabban, Piter . . . even Feyd had failed him in the end.\n\n_Don't omit yourself, Grandfather. Remember_ , you _were killed by a little girl with a poison needle. Ha ha!_\n\n_Shut up!_ The Baron scratched nervously at the top of his head, as if to dig through flesh and bone to rip her out. She fell silent.\n\n\"I fear you may be right, Baron. Humans may not be salvageable, but we don't want Omnius to believe that, or he will destroy them all.\"\n\n\"I thought you machines were already doing that,\" the Baron said.\n\n\"To a certain extent. Omnius is stretching his abilities, but when we find the no-ship, I am certain he will get down to business.\" The old woman dug holes in the garden and planted seedlings that simply appeared in her hands.\n\n\"What's so special about one lost ship?\" the Baron asked.\n\n\"Our mathematical projections suggest that the Kwisatz Haderach is aboard.\"\n\n\"But _I_ am the Kwisatz Haderach!\" Paolo insisted. \"You already have me.\"\n\nThe old woman gave him a wry smile. \"You are our fallback plan, young man. Omnius prefers the security of redundancy. If there are two possible Kwisatz Haderachs, he wants both of them.\"\n\nHis face a mask of displeasure, the Baron cracked his knuckles. \"So you think there's another ghola of Paul Atreides aboard that ship? Not likely!\"\n\n\"I claim only that there is another _Kwisatz Haderach_ aboard the ship. However, since we have one Paul Atreides ghola, there could certainly be another.\"\n\n> _Are we on the Golden Path, or have we strayed from it? For three and a half millennia we prayed for deliverance from the Tyrant, but now that he is gone, have we forgotten how to live without such stern guidance? Do we know how to make the necessary decisions, or will we become hopelessly lost in the wilderness and starve of our own failings?_\n> \n> \u2014MOTHER SUPERIOR DARWI ODRADE,  \n>  _Pondering My Epitaph_ , sealed Bene Gesserit Archives, recorded  \n> before Battle of Junction\n\nHighly agitated, Garimi refused to take a seat in Sheeana's private quarters, no matter how many times it was offered. Even the Van Gogh painting on the wall did not seem to interest her. The stolen mines had brought long-simmering tensions to a new, raw level. Frantic search teams had been unable to locate any of the explosive devices. Sheeana knew that the stern Proctor Superior had her own suspicions and her own set of people to blame.\n\n\"You and the Bashar didn't make a good bargain back on Qelso,\" Garimi said. \"Leaving all those people and equipment, and getting nothing for ourselves!\"\n\n\"We replenished all our stores.\"\n\n\"What if further sabotage hits our life-support systems? Liet-Kynes and Stilgar were the two most capable of conservation, recycling, and repairs. What if we need them to help us? Do you intend to grow new ones?\"\n\nSheeana angered the other woman further by responding with a calm, amused smile. \"We could, but I thought you suspected all the ghola children. Yet you want Liet and Stilgar back? Besides, maybe Liet was right; maybe it's their destiny to remain on Qelso.\"\n\n\"Now it's obvious that neither of them was the saboteur\u2014though I'm still not entirely convinced about Yueh.\"\n\nSheeana stared at the bright daubs of color that the ancient artist had swirled into an image of such power. Van Gogh was a genius. \"I took a necessary action, based on our needs and priorities.\"\n\n\"Hardly! You bowed to the demands of those murderous nomads to keep all Bene Gesserits off the planet. We should have formed a new school there\u2014and now, instead, this whole ship could explode at any moment!\"\n\n_Ah, the core of what is really bothering her_.\n\n\"You know very well that I would have been happy to let you and your followers settle there.\" She forced a chuckle. \"But I was not willing to start a war with the people of Qelso. We can train others in the nuances of our life-support systems. This ship will survive, as it has for decades.\"\n\nObviously in no mood to be brushed aside, Garimi said, \"Survive how? By creating another ghola to save us? That's always your solution, whether an Abomination like Alia, a traitor like Yueh or Jessica, or a Tyrant like Leto II. At least Pandora had the good sense to close her box.\"\n\n\"And I want to open it wide. I want to bring back the history, especially Paul Atreides\u2014and Thufir Hawat. We could certainly draw on the security knowledge of the Weapons Master of House Atreides.\"\n\n\"Hawat failed spectacularly the last time you tried to awaken him.\"\n\n\"Then we'll try again. And Chani could be an excellent fulcrum for awakening Paul. Jessica is also ripe for awakening. Even Leto II is ready.\"\n\nGarimi's eyes flashed. \"You are playing with fire, Sheeana.\"\n\n\"I am forging weapons. For that, fire is necessary.\" Sheeana turned, letting Garimi know that the discussion was at an end. \"I've heard your opinions often enough to memorize them. I will dine with the gholas today. Maybe they have fresh ideas.\"\n\nIncensed, the dark-haired woman followed Sheeana out of her quarters and down the corridors toward the dining hall. Unexpectedly, young Leto II stepped out of a lift tube, alone and quiet as usual. The twelve-year-old often wandered the halls of the no-ship by himself; now he looked at the two women and blinked, but did not speak to them. Such an odd, preoccupied child.\n\nBefore Sheeana could stop her, the Proctor Superior marched toward Leto, stiff and intimidating. Garimi had a fresh target for her anger and frustration. \"So, Tyrant, where is your Golden Path? Where has it led us? If you were so prescient, why didn't you warn us of the Honored Matres or the Enemy?\"\n\n\"I don't know.\" The boy seemed genuinely perplexed. \"I don't remember.\"\n\nGarimi studied him in disgust. \"And what if you _did_ remember? Would you be the God Emperor, the greatest butcher in all of human history? Sheeana thinks you could save us, but I say the Tyrant could just as easily destroy us. That's what you're best at. I don't want you or your monstrous ego back, Leto II. Your Golden Path is a blind man's road, sunk in a swamp.\"\n\n\"It is not this boy's Golden Path,\" Sheeana said, taking the other woman's arm in a viselike grip. \"Leave him alone.\"\n\nLeto took a quick step, darted around them, and fled down the corridor. Garimi looked triumphantly at Sheeana, who merely regarded her as a fool, condemned by her own irrational outburst.\n\nHIS EYES AND ears burned from the Proctor Superior's accusations, but Leto refused to allow a tear. A wise person didn't waste water trying to drown his emotions; he knew that much about old Dune. As he moved away from Sheeana and the insufferable Proctor Superior, and everyone else who thought they knew what to expect from him, the boy silently denied what Garimi had said, trying to block away what he himself knew.\n\n_I was the God Emperor, the Tyrant. I created the Golden Path . . . but with my memories locked away, I don't truly understand what it is!_ Despite all he had learned about his original lifetime, Leto felt like nothing more than a twelve-year-old who had never asked to be reborn.\n\nHe rode the transport tube to the deep lower decks, heading for a place where he felt more comfortable and safe. At first he considered slipping into the roaring winds of the recirculation chambers and the atmosphere-pumping ducts, but the strict security measures imposed by Bashar Teg and Leto's friend Thufir had closed off all access.\n\nBefore his unpleasant encounter with Garimi, Leto had planned to join Thufir for his regular session on the training floor. Though the other ghola boy was now seventeen and had his security duties with the Bashar, he still frequently sparred with Leto. Despite his youth and size, Leto II was highly competitive even against a larger, stronger opponent. For the past few years they had provided quite a challenge for each other.\n\nAt the moment, though, Leto needed to be alone. He reached the bottom levels of the ship and stood at the main access door into the immense hold. The surveillance imagers would have spotted him already. He swallowed hard. He had never dared to go inside alone, though he had stared for hours through the plaz at the captive sandworms.\n\nA pair of young guards stood in the hall, monitoring access to the cargo deck. Seeing the boy approach, they tensed. \"This is a restricted area.\"\n\n\"Restricted to _me_? Do you know who I am?\"\n\n\"You are Leto the Tyrant, the God Emperor,\" said the young woman, as if answering a proctor's question. She was Debray, one of the Bene Gesserit daughters who had been born in space after the no-ship's escape.\n\n\"And those worms are part of me. Don't you remember your history?\"\n\n\"They're dangerous,\" the male guard answered. \"You shouldn't go in there.\"\n\nLeto gazed calmly at the pair. \"Yes, I should. Especially now. I need to feel the sands, smell the melange, the worms.\" He narrowed his eyes. \"It could well restore my memories, as Sheeana wants.\"\n\nDebray frowned as she considered this. \"Sheeana did say that every means must be used to trigger the reawakening of the gholas.\"\n\nThe male guard turned to his companion. \"Call Thufir Hawat and inform him first. This is highly irregular.\"\n\nLeto approached the heavy door. \"I just need to go inside the hatch. I won't stray far. The worms stay out in the center of the habitat, don't they?\" Boldly, he used the simple controls to unseal the door. \"I know these worms. Thufir will understand. He hasn't recovered his memories either.\"\n\nBefore the guards could agree on stopping him, Leto darted into the hold. The sand itself seemed to give off a crackling, staticky sound. The temperature was warm, the air so dry that his throat burned. The powerful smells of flint and cinnamon seared his nostrils. At the far end of the kilometer-long hold, the large worms moved toward him.\n\nJust standing on the sandy surface took the boy back to a place he had studied extensively in the no-ship's library. The real Arrakis, which had changed from desert to garden during his first extended life. Now the dry heat baked his skin. He took deep, calming breaths of air redolent with the odor of melange.\n\nNot bothering to avoid making noise, Leto strode farther out on the sand, sinking up to his ankles in the soft dunes. He ignored the shouted warnings of the guards as he trudged away from the metal wall. This was the closest thing to an open desert these worms had ever known.\n\nClimbing the crest of a dune and gazing around to the limits of the hold, Leto imagined how magnificent Arrakis must have once been. He wished he could remember. The dune on which he stood was small compared to a real one, and the seven worms in the hold were more diminutive than their unfettered ancestors as well.\n\nAhead of him, the largest worm churned through sand, followed by the others. Leto felt the connection with these seven worms. It was as if the magnificent beasts sensed his mental pain and wanted to help him, even if his memories were still locked away in a ghola vault.\n\nAn unexpected release of tears flowed down Leto's cheeks\u2014not of anger toward Garimi but of joy and awe. Tears! He could not stem the flow of moisture. Perhaps if he perished right there on the sands, his body would be absorbed into the flesh of the worms, leaving behind all his fears and expectations.\n\nThese worms were his descendants, each with a nugget of his former awareness. _We are the same_. Leto beckoned them. Although his ghola cells hadn't yet released memories of the thousands of years in his original lifetime, these sandworms possessed buried memories as well. \"Are you dreaming in there? Am _I_ in there?\"\n\nA hundred meters from him the worms stopped and dove back into the sand, one after the other. He sensed that their presence was not threatening, but . . . protective. They _did_ know him!\n\nFrom the hatch behind him, Leto heard a familiar voice calling his name. Looking back, he saw the ghola of Thufir Hawat standing on the verge, motioning him to come back to safety. \"Leto, watch out. Don't tempt the worms. You are my friend, but if one of them eats you I won't jump down its gullet to get you back!\" Thufir tried to chuckle, but looked deeply anxious.\n\n\"I just need some time alone with them.\" Leto sensed something moving beneath the sands. He felt no concern for his own well-being, but did not want to endanger his friend. He picked up a strong whiff, the cinnamon odor of spice.\n\n\"Leave! Now!\"\n\nThen, wrestling with his fear, Thufir ventured closer to the young man, a few meters away. \"Suicide by worm? Is that what you're doing out here?\" He glanced at the hatch behind them, apparently wondering if he could still get back to safety if necessary. Worry lines etched his features. He looked terrified for himself and for Leto, wrestling with something that ran against his instincts. Yet he still stepped forward, as if drawn to his friend.\n\n\"Thufir, stay back. You're in more danger than I am.\"\n\nThe worms knew that someone else was in their realm. But they seemed far more agitated than an intruder could account for. Leto sensed a hatred, a roiling and instinctive reaction. He sprinted back to Thufir to save him. His friend seemed to be struggling with himself.\n\nSand erupted, and worms encircled him and Thufir. The creatures rose from the low dunes, their round and hollow faces questing this way and that for something.\n\n\"Leto, we have to go.\" Thufir grabbed the boy's sleeve. His voice was husky, ragged. \"Go!\"\n\n\"Thufir, they won't harm me. And I feel . . . I feel as if I could make them go away. But they are deeply disturbed. Something about . . . you?\" Leto sensed something here that he didn't understand.\n\nSimultaneously, the worms shot like battering rams toward the two young men on the dune. Thufir bolted away from Leto and lost his footing on the soft surface. Leto tried to go toward him, but the largest worm exploded up between them, scattering sand and dust. Another beast loomed on the other side of the transfixed Thufir, stretching its sinuous body into the air.\n\nThufir let out a shuddering, gut-wrenching scream. It didn't sound at all like the ghola friend Leto had known. It didn't even sound human.\n\nThe sandworms struck Thufir, but they did not simply devour him. As if in vindictive anger, the largest worm slammed down on him, smashing the young man's body into the sand. The next worm reared up and rolled over the already broken Thufir Hawat. For good measure, a third worm crushed the lifeless form. Then the trio of worms backed away, as if proud of what they had done.\n\nLeto stumbled across the sand toward the smashed body, oblivious to the threat of the worms. He slid down a churned dune, and fell to his hands and knees beside the smashed, partially buried form. \"Thufir!\"\n\nBut he did not see the familiar face of his friend. The crushed features were pale and blank, the hair colorless, the expression inhuman. The black-button eyes were unfocused and dead.\n\nIn shock, Leto reeled backward.\n\nThufir was a Face Dancer.\n\n> _Here is my mask\u2014it looks just like yours. We cannot see what our masks look like while we are wearing them_.\n> \n> \u2014 _The Wheel of Deception_ , Tleilaxu commentary\n\nUproar in the hierarchy of the no-ship. Astonishment. Even Duncan Idaho could not grasp how such a thing could have happened. How long had the Face Dancer been watching them aboard the no-ship? The mangled, ugly corpse left no room for doubt.\n\n_Thufir Hawat had been a Face Dancer! How could it be him?_\n\nThe original warrior Mentat had served House Atreides. Hawat had been Duncan's good and loyal friend\u2014but not this faux version of him. In all this time, during the three years of sabotage and murder\u2014and perhaps even longer\u2014Duncan had not detected the Face Dancer in Hawat, nor had Bashar Teg who mentored him. Nor had the Bene Gesserit Sisters, nor any of the other ghola children. _But how?_\n\nAn even worse question hung over them, blackening Duncan's thoughts like a solar eclipse: _We have found one Face Dancer. Are there others?_\n\nHe looked at Sheeana, at the stricken Leto II, and at the two shocked guards who stared at the alien body. \"We have to keep this secret until we can account for everyone aboard the ship. We've got to watch them, find a way to test them somehow . . . .\"\n\nShe agreed. \"If there are any other Face Dancers aboard, we need to act before they discover what happened.\" In Bene Gesserit Voice, using a tone that was the equivalent of a verbal blow, she said to the guards, \" _Speak of this to no one_.\"\n\nThey froze. Sheeana was already making plans to implement a crackdown and sweep of everyone on the ship. Duncan's Mentat mind raced as he tried to comprehend what could have happened, but the nagging questions defied all his attempts to impose logic.\n\nOne rose above others: _How do we even know a test will work?_ Thufir had already faced interrogation by the Truthsayers, just as everyone onboard had. Somehow, these new Face Dancers could evade even the witches' truthsense.\n\nIf the young ghola had been replaced by a Face Dancer at some point, how could such a substitution have occurred without Duncan's knowledge? And when had it occurred? Had the real Thufir accidentally encountered a hidden Face Dancer in a darkened passageway? One of the secret survivors from the Handlers' suicide crashes in a long-term elaborate ruse? How else could a Face Dancer have gotten aboard the _Ithaca_?\n\nIn assuming the identity of a victim, a Face Dancer imprinted himself with a perfect copy of the original person's personality and memories, thus creating an exact duplicate. And yet, the false Thufir had risked his life for young Leto II amongst the sandworms. Why? How much of Thufir had actually been in the Face Dancer? Had there ever been a real Thufir ghola?\n\nAt first, with the Face Dancer exposed, Duncan had felt a sense of relief that the saboteur and murderer was at last revealed. But after a swift Mentat analysis, he quickly put together several instances of sabotage during which the Thufir Hawat ghola had a clear alibi. Duncan had himself been with him during some of the attacks. The next projection was incontrovertible.\n\n_There is more than one Face Dancer among us_.\n\nDUNCAN AND TEG met in a small copper-walled room designed for private meetings, blocked from all known scanning devices. Subtle indications implied that this had originally been designed as an interrogation chamber. How often had the original Honored Matres used it as such? For torture, or simply amusement?\n\nStanding coolly at attention, Teg and Duncan faced the Reverend Mothers Sheeana, Garimi, and Elyen, who had consumed the last available doses of the truthtrance drug. All of the women were armed and highly suspicious. Sheeana said, \"Under various pretexts, we have isolated everyone aboard, using layers of observers. Most of them think we're searching for the missing explosive mines. So far, very few people know about Thufir Hawat. Other Face Dancers would not be aware that they are at risk of exposure.\"\n\n\"I would have thought the entire idea absurd\u2014until recently. Now no suspicion seems too paranoid.\" Duncan locked gazes with the Bashar, and both nodded.\n\n\"My truthtrance is deeper than it has been before,\" Elyen said, sounding distant.\n\n\"Perhaps we didn't ask the correct questions previously.\" Garimi put her elbows on the table.\n\nTeg said, \"Ask away, then. The sooner you clear us of suspicion, the faster we can root out this cancer. We need a different kind of test.\"\n\nNormally a trained Bene Gesserit should have been able to uncover deception with a mere question or two, but this extraordinary inquiry lasted an hour. Because they were building a cadre of trustworthy allies, Sheeana and her Sisters needed to be thorough. And they needed to do a better job than before. The three Reverend Mothers watched for even the slightest flicker of evasion. Neither Duncan nor Teg gave them any.\n\n\"We believe you,\" Garimi finally said. \"Unless you give us cause to change our minds.\"\n\nSheeana nodded. \"Provisionally, we accept that you two are exactly who you say you are.\"\n\nTeg seemed bitterly amused. \"And Duncan and I accept you three as well. _Provisionally_.\"\n\n\"Face Dancers are mimics. They can change their appearance, but they cannot change their DNA. Now that we have cell samples from the Hawat impostor, our Suk doctors should be able to develop an accurate test.\"\n\n\"So we believe,\" Teg said. With the loss of his prot\u00e9g\u00e9, the Bashar seemed fundamentally disturbed. He no longer took anything at face value.\n\nWith an iron-hard scowl, Garimi said, \"The obvious answer is that Hawat was born a Face Dancer, then carefully planted and manipulated by our Tleilaxu Master. Who would know Face Dancers better than old Scytale? We know he had the cells in his nullentropy tube. If that scenario is true, the deception went on for almost eighteen years.\"\n\nSheeana continued, \"A Face Dancer infant could have mimicked a generic human baby from the very beginning. As he grew, he took a shape based on archival records of the young Atreides warrior-Mentat. Since no one here\u2014not even you, Duncan\u2014remembers the original Hawat as an adolescent, the disguise would not need to be perfect.\"\n\nDuncan knew she was right. In his original lifetime, when he'd escaped from the Harkonnens and gone to Caladan, Thufir Hawat had already been a weathered battle veteran. Duncan remembered his first real conversation with Hawat. He'd been a stable boy at Castle Caladan, working with the Salusan bulls that Old Duke Paulus loved to fight in grand spectacles. Someone had drugged the bulls into a frenzy, and young Duncan had tried to raise the alarm, but no one believed him. After Paulus was gored to death, Hawat himself had led the investigation, hauling young Duncan before a board of inquiry, since evidence indicated that he was a Harkonnen spy . . . .\n\nAnd now this Thufir was a Face Dancer! Duncan still had trouble wrapping his mind around the undeniable reality.\n\n\"Then all of the ghola babies could be Face Dancers,\" Duncan said. \"I suggest you summon Scytale. He's now our prime suspect.\"\n\n\"Or,\" Teg said in a brittle voice, \"he may be our best resource. As Garimi already stated, who would know the Face Dancers better?\"\n\nWhen the Tleilaxu Master was brought into the copper-walled chamber, Duncan and Teg took seats at the other side of the table, part of the growing inquisition to root out the Face Dancer infiltration. Scytale appeared frightened and unsettled. The Tleilaxu ghola was fifteen years old, but he did not look like a boy. His elfin features, sharp teeth, and gray skin made him seem alien and suspicious, but Duncan realized that was only a knee-jerk response based on primitive superstitions and previous experiences.\n\nAfter Scytale sat down, Elyen leaned forward. She looked the sternest of them all. \"What have you done, Tleilaxu? What is your plan? How have you tried to betray us?\" She used an edge of Voice, enough to make Scytale jerk.\n\n\"I did nothing.\"\n\n\"You and your genetic predecessor knew what you were growing in the axlotl tanks. We tested the cells before allowing you to create them, but you deceived us somehow with Thufir Hawat.\" They showed him images of the dead Face Dancer. Duncan could see that the Tleilaxu's surprise was genuine.\n\n\"Are all of the ghola children similarly tainted?\" Sheeana demanded.\n\n\"None of them are,\" Scytale insisted. \"Unless they were replaced sometime _after_ being decanted from the tanks.\"\n\nElyen narrowed her gaze. \"He's telling the truth. I see none of the indicators.\" Sheeana and Garimi silently consulted each other and nodded simultaneously. Then Sheeana said, \"Unless he is himself a Face Dancer.\"\n\n\"Scytale isn't likely to be a Face Dancer substitute simply because so few of us trust him anyway,\" Duncan pointed out. \"A Face Dancer would choose to be someone who could more easily move among us.\"\n\n\"Someone like Thufir Hawat,\" Teg said.\n\nYoung Scytale looked greatly disturbed. \"Those new Face Dancers were brought back from the Scattering. The Lost Tleilaxu claimed to have modified them in ways we didn't understand. Much to my dismay, I have now learned that even I can't detect one of them. Believe me, I never suspected Hawat.\"\n\n\"Then how did a Face Dancer get aboard, if not grown from the Face Dancer cells in your nullentropy capsule?\" Sheeana asked.\n\n\"The Face Dancer could already have been posing as one of us when we left Chapterhouse,\" Duncan mused. \"How carefully did you check all of the original hundred and fifty who rushed aboard during the escape?\"\n\nTeg shook his head. \"But why wait more than two decades to strike? It makes no sense.\"\n\n\"A sleeper agent, perhaps,\" Sheeana suggested. \"Or, could the Face Dancer have been someone else for a long time, and only recently replaced Thufir?\"\n\n\"Yes, look for a scapegoat to persecute,\" Scytale said bitterly, slumping in the overlarge interrogation chair. \"Preferably a Tleilaxu.\"\n\nSheeana had fire in her eyes. \"As a precaution, we have sealed all of the ghola children in separate rooms, where they can cause no damage if another of them is a Face Dancer. I've already directed our Suk doctors to take blood samples. They won't escape.\"\n\nDuncan wondered if her vehemence might suggest that _she_ was a Face Dancer. He narrowed his eyes suspiciously and continued to watch her. He would have to watch everyone he could, at all times.\n\nGarimi looked around at their small trusted cadre. \"I\u2014or another of our choosing\u2014will remain on the navigation bridge and monitor the no-ship while every single person aboard is brought into the main meeting chamber. Herd them in, account for every one, even the children. Lock the doors and test them all. One by one. Learn the truth.\"\n\n\"What definitive tests can we use?\" Teg asked. \"On any of us?\"\n\nScytale piped up, \"I believe I can develop a reliable method. Using a tissue sample from the Hawat Face Dancer, I will prepare a comparison panel. There are certain . . . techniques I could use. He is one of the new breed brought back by the Lost Tleilaxu, and he differs from the old ones. But with this sample\u2014\"\n\n\"And why should we trust you?\" Garimi said. \"Your own purity hasn't yet been proven.\"\n\nScytale wore a forlorn expression. \"You have to trust someone.\"\n\n\"Do we?\"\n\n\"I would allow myself to be observed by your experts at all times during the preparations.\"\n\nDuncan glanced at the Tleilaxu Master. \"Scytale's suggestion is a good one.\"\n\n\"Or I can offer another option. When the Face Dancers betrayed my fellow Masters back on Tleilax and our other worlds, some of us had time to fight back. We created a toxin that specifically targets Face Dancers\u2014a selective poison. If you grant me access to laboratory facilities, I can recreate that toxin and deploy it as a gas.\"\n\n\"To what purpose?\" Teg asked. Then his expression changed to one of understanding. \"Ah, to flood the _Ithaca_ 's air systems. We would kill any Face Dancers who remain among us.\"\n\n\"The quantities necessary to saturate our ship would be huge,\" Duncan said, racing through a Mentat calculation to estimate the volume of air within the gigantic vessel, the concentration of gas that would prove lethal to the shape-shifters, the possibility of making others ill and debilitating the crew.\n\nGarimi couldn't believe what she was hearing. \"You're suggesting we let this _Tleilaxu_ release an unknown gas into our ship? They created the Face Dancers!\"\n\nScytale answered her in a voice heavy with scorn. \"You witches fail to think. Don't you see that I myself face a dire threat? These are _new_ Face Dancers, brought in from outside by the Lost Tleilaxu\u2014our bastard stepbrothers who cooperated with the Honored Matres to annihilate all the old Masters like myself. Think! If other Face Dancers are aboard the _Ithaca_ , then I am in greater personal danger than anyone else. Can't you understand that?\"\n\n\"Scytale's gas must only be a last resort,\" Duncan said.\n\nSheeana looked around the room. \"I'll let him begin work on the toxin, but I'd prefer that we identify any Face Dancer directly.\"\n\n\"And interrogate it,\" Garimi said.\n\nScytale laughed. \"You think you can interrogate a Face Dancer?\"\n\n\"Never underestimate the Bene Gesserit.\"\n\nSheeana nodded. \"Until we root out any other infiltrators, until we prove there are no more Face Dancers among us, our only safety lies in staying in large enough groups that the shape-shifters can't attack without being seen.\"\n\n\"What if an overwhelming number of us are already Face Dancers?\" Teg said.\n\n\"Then we're all lost.\"\n\nDURING THE LOCKDOWN, each of the ghola children was tested; Leto II submitted first. When the sandworms had turned on Thufir Hawat, somehow sensing the alien Face Dancer, Leto's shock had seemed genuine. The imagers showed him staring in disbelief at the ruined body that had reverted to its blank Face Dancer state. But Thufir had clearly placed himself in danger, voluntarily going toward Leto when he did not need to. Why would a Face Dancer put himself at risk, unless the copy was so accurate that even the friendship was real?\n\nLeto, ghola of the Tyrant, was many extraordinary things. But he was not a Face Dancer. Scytale's genetic analysis proved it.\n\nPaul Atreides was also found to be clean, along with Chani, Jessica, and the three-year-old Alia, who was intrigued by the needles and samples. Despite the usual suspicions surrounding him, Wellington Yueh was also who he claimed to be.\n\nAfter Scytale completed the blood and cellular tests, Sheeana was still not satisfied. \"Even if we can now trust the ghola children, that means only that other Face Dancers\u2014if there are any more\u2014must be hidden among the rest of us.\"\n\n\"Then we'll test the rest,\" Garimi said. \"Or use Scytale's poison gas. I'll personally submit to any scrutiny, again and again, and I suggest we all do so.\"\n\nScytale raised his small hands in alarm. \"This test is an intensive one. I'll need to prepare enough panels for all passengers, and that will take a great deal of time.\"\n\n\"Then we will take the time,\" Sheeana announced. \"Doing anything less would be foolhardy.\"\n\n> _Why do we find destruction so fascinating? When we see a terrible tragedy, do we think ourselves clever for having evaded it ourselves? Or is our fascination rooted in the thrill and fear of knowing we could be next?_\n> \n> \u2014MOTHER SUPERIOR ODRADE,  \n> Documentation of Consequences\n\nMurbella and Janess\u2014mother and daughter, Mother Commander and Supreme Bashar\u2014orbited the dead world of Richese. They rode in an observation ship, separate from the teams of engineers, who were still leery about the burned-out plague on Chapterhouse. Though the disease had run its course, the Ixians refused to be in a confined space with Murbella and Janess, who had been exposed to it.\n\nNevertheless, alone in their small ship, the two women had a perfect view of the unfolding test.\n\nMore than five years earlier, rebel Honored Matre ships from Tleilax had bombarded Richese, erasing not only the entire population, but also the weapons industries and the half-constructed battle fleet that was to have been delivered to the New Sisterhood. Now that the planet was lifeless, however, Richese was a perfectly appropriate place for the Ixians to demonstrate their new Obliterator weapons.\n\nMurbella opened the commline and spoke to the four accompanying test ships. \"You take a smug pleasure in doing this, don't you, Chief Fabricator?\"\n\nOn the screen, Shayama Sen arched his eyebrows and jerked his head back in a fine display of innocence. \"We're testing the weapon you ordered from us, Mother Commander. You asked for a demonstration, rather than taking us at our word. We must prove that our technology functions as advertised.\"\n\n\"And the rivalry between Ix and Richese had nothing to do with your choice of targets?\" She barely held her sarcasm in check.\n\n\"Richese is just a historical footnote, Mother Commander. Any enjoyment Ixians might have taken from our rivals' unfortunate fate has long since faded.\" After a pause, Sen added, \"We admit, however, that the irony does not escape us.\"\n\nSince last visiting her high above Chapterhouse, the factory leader sounded subtly changed. Recently, when Sen had come back to deliver full records of all their tests on Ix, he had seemed surprised, even embarrassed. He had followed her suspicious suggestion and used the cellular test on all of his people, with the result that twenty-two Face Dancers had been exposed, all of them working in critical industries.\n\nMurbella would have liked to interrogate them, maybe even apply an Ixian T-probe. But those Face Dancers who weren't immediately killed took their own lives, somehow using a machinelike suicide shutdown in their own brains. The lost opportunity angered her, but she doubted her Sisters would have learned anything from the shape-shifters anyway. Nevertheless, she was glad to have installed eight trusted inspectors to watch over the industrial progress from that point onward.\n\n\"Our delivery schedule is tight, Mother Commander, as you demanded,\" Sen transmitted. \"We are arming the ships from Junction as quickly as possible. After seeing these four Obliterators successfully tested, you can't deny that our technology is reliable.\"\n\n\"It seems a shame to waste such destructive power on a target that doesn't harm the true enemy,\" Janess said. \"But we require proof.\" Both of them had reviewed earlier films of the tests, but those could have been faked.\n\n\"I still want to see it with my own eyes,\" Murbella said. \"Then we'll throw everything into a defense against the machine advance.\"\n\n\"Deploying the nodes now,\" transmitted one of the Ixian pilots. \"Please observe.\"\n\nFour balls of light spat from the quartet of Ixian ships, and the incandescent Obliterators spun like pinwheels toward the cracked world below. They shuddered and expanded as they descended, throwing off rippling waves that grew brighter instead of dampening.\n\nThe atmosphere of Richese had already been scorched, its forests and cities leveled in the first chain reaction. Even so, the Ixian-modified weapons found sufficient fuel to set the world ablaze all over again.\n\nMurbella remained silent as she watched the awesome swiftness of the flame fronts. She stared without blinking until her eyes felt dry. The planet flared like an ember in a breeze. Cracks appeared across the continents; orange rifts blazed up. Finally, she spoke to her daughter, not caring that the Ixians could overhear on the open commline. \"If we deploy such a weapon in the midst of a thinking-machine battle fleet, it will wreak inconceivable havoc.\"\n\n\"We might actually have a chance,\" Janess said.\n\nShayama Sen interrupted through the speakers, \"You assume, Mother Commander, that the thinking machines will be foolish enough to fly their ships in such a tight cluster that one weapon will suffice.\"\n\n\"We know a great deal about the Enemy's battle plan and how their fleet has been advancing. They do not use foldspace engines, so they move methodically from one target to the next, step by step. With the thinking machines there are few surprises.\" Murbella looked at her daughter, then back at the burning planet before snapping orders to the Ixians. \"Very well, no need to squander any more Obliterators. When we finally hurl them at machine battleships, that will be demonstration enough for me. I want at least ten Obliterators aboard each of our new warships. No more delays! We have waited too long already.\"\n\n\"It will be done, Mother Commander,\" Sen said.\n\nMurbella chewed at her lower lip as she watched Richese continue to blaze. It wasn't like the Chief Fabricator to be so cooperative, failing to demand additional payment. Perhaps, after seeing countless worlds already destroyed, the Ixians had at last recognized their true enemy.\n\n> _Whether we see them or not, there are nets everywhere, encompassing our individual and collective lives. Sometimes it is necessary to ignore them, for the sake of our own sanity_.\n> \n> \u2014ship's log, entry of  \n> DUNCAN IDAHO\n\nFace Dancers aboard.\n\nIn her quarters with little Alia and twelve-year-old Leto, Jessica felt very much like a mother again\u2014after all these centuries. The three of them had a shared past and bloodline, but no other knowledge or memories in common. Not yet. To Jessica it seemed that they were little more than actors memorizing lines and playing roles, trying to be who they were _supposed_ to be. Her body was only seventeen, but she felt much older as she comforted the two younger ones.\n\n\"What is a Face Dancer?\" three-year-old Alia asked, toying with a sharp knife she kept at her side. Since the time she could walk, the girl had harbored a fascination for weapons, and she often sought permission to practice with them, rather than playing with more appropriate toys. \"Are they coming to get us?\"\n\n\"They're already _in_ the ship,\" Leto said, still shaken. He could not believe that Thufir had been a Face Dancer and that he hadn't known it. \"That's why we were all tested.\"\n\n\"No others have been found yet,\" Jessica said. She and Thufir had been decanted in the same year. In the cr\u00e8che, she had been raised with the ghola of the warrior-Mentat, and never had she noticed any change in his personality. It did not seem possible that Thufir could have been a Face Dancer from the very beginning.\n\nThe real Hawat, Master of Assassins and former weapons master of House Atreides, had been a veteran of numerous successful campaigns like Bashar Miles Teg, serving three generations of House Atreides. No wonder Sheeana and the Bene Gesserits had considered him an invaluable ally. That was why they had wanted to bring him back, and now it was obvious why their memory-triggering crisis hadn't worked. Thufir was not really Thufir, and perhaps never had been.\n\nNow\u2014unless clean cells were found to grow a new ghola\u2014the people aboard the _Ithaca_ would never have access to Hawat's Mentat and tactical skills. In fact, Jessica realized that after all this time, the ghola project had produced very little that could be used to help them. Only Yueh, Stilgar, and Liet-Kynes had been reawakened to their past lives, but the latter two were gone. And Yueh, while a skilled Suk doctor, was not a particularly great asset to their team.\n\n_He killed my Duke Leto\u2014again_.\n\nWith the Face Dancer threat, the missing explosive mines, and the various incidents of sabotage, the need for the gholas and their old skills had become more urgent. The remaining unawakened ghola children must have special abilities; Jessica knew they had all been brought back for a reason. Each of them. Paul, Chani, and she were all of an appropriate age; even Leto II should be old enough. Gradual, careful measures could not possibly be sufficient. Not anymore.\n\nShe sighed. If not now, then when would their historical abilities be useful? _I must have my memories back!_\n\nJessica could offer so much more to benefit the no-ship, if given the opportunity. She felt like a husk of a person without her original life. In her quarters, she stood up so swiftly that she startled both Alia and Leto. \"You two should return to your rooms.\" Her gruff voice invited no argument. \"There's something important I have to do. These Bene Gesserits are cowards, though they don't realize it. They can no longer afford to be.\"\n\nIn some ways, Sheeana was brash and impetuous, but in other ways overly cautious. Jessica knew someone, however, who would not shy away from inflicting pain upon her.\n\n\"Who are you going to see?\" Leto asked.\n\n\"Garimi.\"\n\nTHE HARD-LINE REVEREND MOTHER regarded her with a stony expression, then smiled slowly. \"Why should I do this? Are you mad?\"\n\n\"Just pragmatic.\"\n\n\"Do you have any understanding of how much this is going to hurt?\"\n\n\"I am prepared for it.\" She looked at Garimi's dark, curly hair, her flat and unattractive features; Jessica, by contrast, was the very ideal of classical beauty, designed by the Bene Gesserit to play the role of a seductress, a breeding mother whose features had been copied again and again for centuries after her death. \"And I know, Proctor Superior, that if anyone can inflict that pain, you are prepared to do so.\"\n\nGarimi seemed caught between amusement and uneasiness. \"I have imagined countless ways to twist the knife in you, Jessica. I have often considered how much harm your actions did to the old Sisterhood. You derailed our entire Kwisatz Haderach program, created a monster we couldn't control. After Paul, as a direct consequence of your defiance, we suffered thousands of years under the Tyrant. For what conceivable reason would I want to awaken you? You betrayed us.\"\n\n\"So you say.\" Garimi's words struck like hurled stones. The woman had tormented Jessica for years, as well as poor Leto II. Jessica knew all the accusations, understood how the conservative Bene Gesserit faction viewed her. But she had not previously endured the shocking depth of hatred and anger that the woman now showed toward her. \"Your own words reveal a great deal, Garimi. The _old_ Sisterhood. Where are your thoughts? We are already living in the future.\"\n\n\"That doesn't negate the terrible pain you caused.\"\n\n\"You keep insisting that I should bear that guilt. But how can I feel it, if I don't remember? Are you content with me as a scapegoat, a whipping boy for all the imagined wrongs of the past? Sheeana wants my memories restored so that I can help us. But _you_ , Garimi, should be just as eager to awaken me. Admit it\u2014can you think of a better Bene Gesserit punishment than drowning me in the unforgivable things you say I've done to the Sisterhood? Awaken me! Make me see it for myself!\"\n\nGarimi reached out and grabbed her wrist. Instinctively, Jessica tried to pull away, but was unsuccessful. The other woman's expression hardened. \"I am going to Share with you. I'm going to give you all my thoughts and memories so that you'll _know_.\" Garimi leaned closer. \"I will dump into your brain those hundreds of generations of past lives that occurred after you committed your crime, so that you can see the full scope and consequences of what you did.\" She pulled Jessica up against her.\n\n\"That's not possible. Only Reverend Mothers can Share.\" Jessica tried to scramble backward.\n\nGarimi's eyes were steely. \"And you are a Reverend Mother\u2014or you were. Therefore, one lives within you.\" She clasped the back of Jessica's head, grabbed her bronze hair and yanked her closer. Garimi leaned her own forehead down, and pressed it against Jessica's. \"I can make this work. I'm strong enough. Can you imagine why I'm doing it? Perhaps the grief will be enough to paralyze you!\"\n\nJessica fought back. \"Or it will . . . make . . . me . . . stronger.\"\n\nShe'd wanted her own memories, yes\u2014but had never offered to accept all of Garimi's experiences, or those numerous ancestors who had lived through the persecutions of the God Emperor of Dune, her own grandson. All those who had survived the Famine Times, struggling to overcome their addiction to melange, which was no longer available. The horrors of those generations had left deep scars on the human psyche.\n\nJessica did not want that at all. _Garimi insists that I caused it_.\n\nShe felt something inside her head and resisted, but Garimi was stronger, forcing the Sharing upon her, pouring memories, unleashing them. Hammers pounded Jessica's skull from the inside, strong enough to crack through bone and break out. She heard a snapping sound in the blackness, and wondered if Garimi had won. . . .\n\nSHAKEN, JESSICA\u2014THE _real_ Jessica, bound concubine to Duke Leto Atreides, Reverend Mother of the Bene Gesserit\u2014looked around herself with a new wonder she had never imagined possible. Though she saw only the walls of the no-ship, she recalled how good her life had been with the Duke and with their son Paul. She remembered the shell-blue sky of Caladan, the spectacular sunrises on Arrakis.\n\nIn the end, she had beaten Garimi. Now she marched out of the angry woman's quarters, swaying and saturated with the knowledge. That flood of memories was a mixed blessing and a new burden, for she was without her beloved Duke Leto.\n\nThe sudden emptiness made her feel as if she were plunging into an endless pit. _Leto, my Leto! Why couldn't the Sisters have brought you back at the same time, like Paul and Chani? And damn you, Yueh, for taking him from me twice!_\n\nShe felt profoundly alone, her heart drained and her mind left with mere memories and knowledge. Jessica was determined to find a way to make herself useful to her Sisters once more.\n\nReturning to her quarters, she found Alia waiting for her. Possessing a sharp intelligence far beyond her years, the girl looked her over calmly and said, \"Mother, I told Dr. Yueh you would have your memories back. Now he's even more afraid of you. You could kill him with a look. I chased him and kicked him for you.\"\n\nJessica fought back her automatic hatred of Yueh. The old Yueh. \"You mustn't do that. Especially not now.\" The Traitor had been right to fear the return of her memories, even though she had already known of his crimes and forgiven him. _But that was with my head, not with my heart._ As she stood there, Jessica's restored memories and emotions drove the dagger in deeper.\n\nWith a rush of emotion she found herself unable to keep from reaching out and hugging Alia fiercely. Then she looked upon her daughter for the very first time. \"I am your mother again.\"\n\n> _A test must be defined before it can be useful. What are the parameters? What is the accuracy? Too often a test does nothing more than analyze the tester herself._\n> \n> \u2014 _The Bene Gesserit Acolytes' Handbook_\n\nThe death of the Hawat Face Dancer couldn't be kept secret for long. Everyone was accounted for and locked away while Sheeana and her cadre of tested individuals performed a full count, isolated and approved security teams, and then guided all of the ship's inhabitants into the main meeting hall. That giant chamber could easily house hundreds of people for days, if need be, and if enough food was brought in. Meanwhile, Garimi remained up on the navigation deck, monitoring the _Ithaca_ by herself.\n\nSince all hands\u2014at least the _known_ ones\u2014were sealed in the meeting hall, any hidden traitors could very well be trapped inside. In the next few days, over the course of meticulous testing, any remaining Face Dancers among them would be rooted out.\n\nAt first, the younger children born during the journey seemed to think it was a game, but they soon grew restless; the people became uncomfortable and suspicious, wondering why only a handful of individuals were allowed to come and go on mysterious assignments. And why was the horrid little Tleilaxu one of the trusted ones? Many of those aboard still viewed Scytale with open scorn, but he was accustomed to such treatment. The Tleilaxu race had always been despised and distrusted. Now who was to blame?\n\nWorking frantically over the past day, he and the Suk doctors had assembled enough analytical kits to perform a genetic comparison on every untested individual. As a backup plan, he had also created enough of his Face Dancer\u2013specific toxic gas to fill numerous canisters, though Sheeana was not ready to approve such a hazardous experiment\u2014not yet. They didn't trust him enough and kept the gas under their strict control.\n\nHe didn't trust them entirely, either. After all, he was a Tleilaxu Master, perhaps the last one in existence. Secretly, he put together a more startling, fail-safe test, knowing full well what he was doing. He told no one of it.\n\nWhen all was ready, Scytale sat in a front row for what he expected to be an important process of revelation. He watched the uneasy Bene Gesserits, Suk doctors, archivists, and proctors. Out in the audience, Teg sat next to the Rabbi and two Bene Gesserit Sisters. The ghola children were a few rows away, each of them already proven to be untainted. Duncan Idaho waited by one of the sealed doors, and male Bene Gesserits guarded the other exit points.\n\nWhile the gathered passengers waited, Sheeana spoke from the front of the meeting chamber, her words clear and uncompromising, with an edge of Voice. \"We have discovered a Face Dancer among us, and we believe there are more in this room.\"\n\nA moment of shocked silence extended uneasily as she attempted to make eye contact with every individual. Scytale was not surprised that no one stepped forward. The old Rabbi looked simultaneously indignant and lost without the rest of his people. From the seat next to the old man, Teg told him to be patient. The Rabbi glared but did not argue.\n\n\"We have created a foolproof test.\" Sheeana sounded weary even though her voice boomed. \"It will be tedious and time-consuming. But you will all submit to it.\"\n\n\"I hope none of you has anything better to do.\" Duncan crossed his arms over his chest and flashed a grim smile. \"The doors will remain guarded until this process is complete.\"\n\nScytale and the Suk doctors came forward to the stage, carrying kits, syringes, and chemical swabs. \"As each one of you is cleared, our ranks of reliable allies will grow. No Face Dancer can elude this scrutiny.\"\n\n\"Who was this Face Dancer you caught?\" one of the Sisters asked, an undertone of anxiety in her voice. \"And why do you assume there are others among us? What is your evidence?\" When Sheeana explained how the worms had killed Thufir Hawat, stunned murmurs rippled through the audience.\n\nThe Bashar called from his seat, with an edge of guilt and revulsion in his tone. \"We know that the false Thufir could not have been responsible for all the sabotage incidents we have on record. He was with me, in person, when several of the known incidents occurred.\"\n\n\"How do I know you're not _all_ Face Dancers?\" The Rabbi rose to his feet and glared at Sheeana, the Suk doctors, and especially Scytale. \"Your behavior has never been comprehensible to me.\" Teg tugged him back down.\n\nSheeana ignored the old man's question and pointed to the front row. \"I will take the first subject now.\"\n\nTwo female Suk doctors moved forward with their kits, and Sheeana said, \"Make yourselves comfortable. This will take a while.\"\n\nFor Scytale, though, this process was primarily a diversion\u2014and even the Bene Gesserits didn't know it. Feeling trapped, any Face Dancer in the audience would be trying to find a way to escape detection. Therefore, the Tleilaxu Master had to act precipitously, before any hidden shape-shifters could make a move. Watching the large audience closely, he fingered the small device he carried.\n\nWhile the slow analytical procedure was certainly reliable, Scytale had fashioned his secret plan based on what he knew of the old Face Dancers created by the original Tleilaxu Masters. He was betting that the new shape-shifters from the Scattering were similar, at least in their fundamental responses. They must have emerged from the same basic blueprint. If so, he might know how to expose them, a weak and secondary test . . . but its very unexpectedness might work in his favor.\n\nIn the center of the meeting chamber, the Suk doctors performed their first test on a submissive Sister. She extended her hand, waiting for a drop of blood to be drawn.\n\nWithout warning, Scytale activated his high-pitched whistle emitter. A shrill tone warbled up and down, intense but faint, above the range of most human hearing. The original Face Dancers had once communicated with the Tleilaxu in a coded whistling language, a secret set of programming notes burned into their neurological structures. Scytale believed the irresistible noise would make any Face Dancer lose his disguise, at least temporarily.\n\nSuddenly, out in the tiers of seats, the old Rabbi flickered, and his body convulsed. His leathery face shifted and smoothed behind his beard. He let out a cry of surprised outrage and lunged to his feet. Now the old man was unexpectedly supple, wiry, and vicious. His face was flat with sunken eyes and a pug nose, like a bare skull made of half-melted wax.\n\n\"Face Dancer!\" someone shouted.\n\nThe Rabbi became a whirlwind and threw himself against the Bene Gesserits.\n\n> _Never underestimate your enemy\u2014or your allies._\n> \n> \u2014MILES TEG,  \n>  _Memoirs of an Old Commander_\n\nDue to his constant complaints, negative attitude, and frail appearance, everyone aboard had dismissed or misjudged the old Rabbi. As had Miles Teg.\n\nIn moves as swift and deadly as a lasbeam, the Face Dancer slammed the Bashar with a blow that would have shattered his skull, if it had struck squarely. Just in time, Teg recoiled with a flash of inhuman speed. It was enough to save his life, but even so, the attack stunned him.\n\nAbruptly, the Rabbi killed two Sisters on the other side of him, then moved in a direct, murderous line toward the nearest exit, clearing the way with a flurry of deadly blows. From hidden pockets in his dark, conservative clothes, the Face Dancer withdrew a small throwing dagger for each hand. The blades were no longer than his thumbs, but he hurled them with precision. The sharp tips, undoubtedly poisoned, pierced the throats of two male Bene Gesserits who guarded the door. With barely a sound, the Rabbi shoved their dying bodies out of the way and plunged out into the corridor.\n\nScytale urgently scanned the crowd to make certain that this one escaping enemy did not divert attention from any other Face Dancers hidden among those gathered in the chamber. The Tleilaxu saw no other sudden shiftings.\n\nSheeana shouted for others to pursue the Rabbi. \"We know who he is, but he can change his shape. Now we have to track him down.\"\n\nOne of the Sisters tried to use the ship's intercom to warn Garimi, but got no response. \"It's been damaged.\"\n\n\"Fix it.\" Sheeana realized that the Rabbi had had sufficient time during their quarantine in this large chamber to subtly perform more sabotage.\n\nDr. Yueh rushed to a groaning Teg and bent to check the severity of his injury; beside him, the two fallen Sisters were obviously dead. The look on the ghola doctor's face was of dismay rather than vindication. As he examined Teg, he murmured, as if trying to make sense of the situation. \"The Rabbi gave me the sample of the ghola baby's cells. He must have taken Piter de Vries's cells from storage and tricked me. He knew what I would do, how I would react.\"\n\nDuncan glanced from Yueh and Teg to Sheeana. \"The connection is obvious to me now. Thufir Hawat and the Rabbi. Why didn't I see it?\"\n\nSheeana caught her breath as she suddenly realized the same thing. \"Both went down to the planet of the Handlers!\"\n\nDuncan nodded. \"Hawat and the Rabbi were alone together during the hunt of the Honored Matres. You all had to fight your way back to the lighter after you discovered that the Handlers were Face Dancers.\"\n\n\"Of course.\" Sheeana's face was grave. \"Those two came running in from the forest at the last moment. It seems they didn't escape the Handlers after all.\"\n\n\"So the original Rabbi and Thufir\u2014\" Duncan began.\n\n\"Both dead long ago, replaced by Face Dancers on the planet, and their bodies discarded during the hunt.\"\n\nFinally achieving Mentat focus, Duncan jumped to the next obvious conclusion. \"Then it's been more than five years since the substitutions. Five years! In all that time, the Hawat and Rabbi duplicates must have been waiting for their opportunity, killing gholas and axlotl tanks, sabotaging our life-support systems, forcing us to stop at Qelso, where we were vulnerable to discovery by our pursuers. Did the Enemy pick up our trail there? So far, we've managed to elude the net, but now that the Face Dancers have been exposed\u2014\"\n\nSheeana paled. \"And what about the stolen mines? What did the Rabbi do with the explosive mines? He can set them off at any time, if he gets to them.\"\n\nStarting to recover but clearly woozy, Teg was already moving toward the door. \"That Face Dancer knows he has to seize the no-ship before we can kill him. He will head for the navigation bridge.\"\n\n\"Garimi is there,\" Sheeana said. \"Let's hope she can stop him.\"\n\nBY THE TIME the Face Dancer reached the navigation bridge, he had resumed his disguise as the Rabbi. He contained all the memories, experiences, and personality details of the old man, and much more. The frail and frightened-looking Rabbi burst into the chamber, startling Garimi. \"What are you doing up here?\" she asked.\n\nHis eyes were wide and panicked as if he thought she could offer him protection. His spectacles had fallen off. \"Face Dancer!\" he panted, staggering toward her. \"He's killing Bene Gesserits!\"\n\nGarimi spun toward the intercom panel to contact Sheeana\u2014and the Rabbi struck. His deadly blow came close to her neck, but she sensed the movement and turned at the last possible moment. The side of his fist drove down on her shoulder instead. She slid from her chair, and the Rabbi dove at her again.\n\nGarimi launched a kick up at him from the deck, aiming for one of the old man's knobby and uncertain knees, but he sprang away like a coiled panther. The Rabbi let out a feral yowl as Garimi leapt to her feet again and assumed a defensive stance. Her lower lip curled. \"Clever, Rabbi. Even now that I know what you are, I can hardly smell any Face Dancer stink on you.\"\n\nWith a yank and a twist, the Rabbi uprooted an anchored chair and swung it at her. She ducked and reached up to grab the chair as it whistled over her head. Tearing it out of his hands, her pull was enough to knock him to the floor.\n\nWhen the Rabbi rose to his feet again, he shifted his body to mimic the form of a ferocious Futar. His body bulged with muscles, his teeth became sharp and elongated, and his claws slashed the air. Garimi stumbled back to get out of his killing reach and hammered her hand down on the intercom. \"Sisters! Face Dancer on the navigation bridge!\"\n\nThe Futar lunged, and his sharp, newly grown claws ripped her robes. Using wild and frantic punches intended more to hurt her foe than protect her own life, Garimi shattered his ribs. With an outraged kick that employed the full force of her heel, she smashed his left femur out of its hip socket.\n\nBut the Futar rolled as he collapsed, spun in a blur, and before she could feel a moment of victory, he snapped Garimi's neck. She dropped with barely a sigh. In a purely spiteful gesture, he ripped out her throat before calmly reshaping his body to his blank Face Dancer state. He wiped blood from his face with one sleeve.\n\nMore broken than even his own shape-shifter abilities could easily heal, the Rabbi crawled and then limped to the _Ithaca_ 's main controls. He heard running feet in the corridor, so he sealed the navigation bridge, applied emergency locks, and activated a mutiny-defense protocol.\n\nIn the years he had maintained his disguise, the Face Dancer had covertly sampled the skin cells of Duncan Idaho, Sheeana, and Bashar Miles Teg. Now his hands flowed into the proper identification prints so that the no-ship's highly secure controls responded to him. The sealed doors would stand against any intrusion. Eventually the Bene Gesserits would find a way to break in, but by that time he would have completed his mission.\n\nHis thinking-machine masters would be alerted. And they would come.\n\nLong ago, he had studied how to operate the Holtzman engines. Estimating the coordinates as best he could, not worried about the lack of a Navigator, the Face Dancer folded space and plunged the _Ithaca_ across the galaxy. The ship tumbled out into a different stellar region, not far from Omnius's advancing forces. He reconfigured the ship's comsystems and triggered a locator beacon. His superiors knew the signal.\n\nThe thinking machines would respond swiftly. Already the Face Dancer could sense the hungry, invisible tachyon net coming closer. This time there would be no escape. The no-ship would be completely trapped.\n\n> _Even small opponents can be deadly._\n> \n> \u2014Bene Gesserit Analytical Report on the Tleilaxu Problem\n\nBy the time Duncan, Sheeana, and Teg reached the navigation bridge, the thick hatches were sealed and locked. Impregnable. The bridge had been designed to remain secure against even an army.\n\nWithin moments, other Sisters followed, having first raced to the armory and obtained hand weapons: poisonous needle guns, stunners, and a high-powered lascutter. None of those devices would be sufficient. Rushing forward, the ghola children joined the crowd outside the sealed bridge, among them Paul, Chani, Jessica, Leto II, and young Alia.\n\nDuncan could feel the change when the no-ship lurched through foldspace. \"He's at the controls, moving us!\"\n\n\"Garimi is dead, then,\" Sheeana concluded.\n\n\"The Face Dancer is going to take us directly to the Enemy,\" Teg said.\n\n\"Now is the time to use Scytale's poison gas to kill the Face Dancer.\" Sheeana turned to two of the Sisters standing in the corridor. \"Find the Tleilaxu and take him to our guarded cabinet. Get one of the canisters, and we will flood the air on the bridge with the gas.\"\n\n\"No time for that,\" Duncan said. \"We've got to get in there!\"\n\nAlia sounded eerily cool and intelligent as she announced, \"I can get inside.\"\n\nDuncan looked at the girl. To him, the echo-memories evoked by this child were unsettling. The original Duncan had never known her as a youngster\u2014he had been killed by Sardaukar while Jessica was barely pregnant. But he did have vivid memories of an older Alia as his lover, in another life. But that was all history. Now it might as well be myth or legend.\n\nHe bent down to talk to her. \"How? There isn't much time.\"\n\n\"I'm small enough.\" With a flick of her eyes, the little girl indicated the narrow air-exchanger vents leading into the command deck. She was far more diminutive than even Scytale.\n\nSheeana was already removing the grate. \"There are baffles, filters, and bars in the way. How will you get through?\"\n\n\"Give me a cutter. And a needle gun. I'll get the door open for you as soon as I can. From the inside.\"\n\nWhen Alia had what she needed, Duncan hoisted the girl up to where she could squirm inside the tiny tunnel. Not yet four years old, she weighed very little. Jessica stood watching, looking more mature than she had been only a few days earlier, but even seeing her \"daughter\" placed into such a dangerous situation, she did not protest.\n\nCold and intent, the child clamped the cutter between her teeth, tucked the needle pistol into her small shirt, and began to creep through the vent. The distance between the chambers was not far, but each half meter was a battle. She exhaled, making herself as small as possible so that she could wriggle ahead.\n\nOutside, the others began pounding on the sealed door as a distraction. They used heavy cutters that sparked and fumed, screeching through the dense, armored barricade a millimeter at a time. The Face Dancer would know they'd need hours to cut through into the navigation bridge. Alia was confident the Face Dancer would not expect an ambush from her.\n\nShe encountered the first barricade, a set of plasteel bars interwoven with a filtration grid. The dense mat was coated with neutralizing chemicals, and charged with a faint electrostatic film to scrub all drugs and poisons from the air that passed into the bridge. With the filter in place, Scytale's toxic gas would not have worked, even if they'd been able to release it.\n\nElbows digging into her sides, Alia took the cutter from her teeth and with jerky wrist movements sliced away the bars. Gently, she set the screen in front of her, careful not to make a noise, and crawled over it. The sharp edges scratched her chest and legs, but Alia cared nothing for the pain.\n\nSimilarly, she passed through a second grid, and then found herself at the last opening, from which she could observe the Face Dancer through the grille. His appearance flickered occasionally, sometimes reverting to the old man's shape, sometimes becoming a Futar, but primarily the Face Dancer wore a blank, skull-like visage. Even before she saw the torn body of Garimi on the deck, Alia knew not to underestimate this opponent.\n\nWith the tip of the glowing cutter, she sliced the tiny fasteners that held the last screen in place. Moving as silently as she could, she held the plate where it was and squirmed to free the needle gun from her shirt. She tensed, then drew a deep breath, waiting for the right moment.\n\n_I will have only a brief instant of surprise, so I must use it to full advantage._\n\nThe Face Dancer was working the controls, probably transmitting a signal to the mysterious Enemy, presumably more of his own kind. Every second she delayed would place the _Ithaca_ in greater danger.\n\nSuddenly the Face Dancer jerked his head up and snapped his gaze toward the grille. Somehow he had sensed her. Now, without hesitating, Alia shoved the loosened screen toward him like a projectile. He dodged out of the way, reacting just as she had expected. Still lying prone in the ventilation shaft, she extended the needle gun in front of her and fired seven times. Three of the deadly needles found their target: two in the Face Dancer's eyes, another in the artery on his neck.\n\nHe spasmed, thrashed, and fell lifeless. Wriggling out of the air shaft, Alia dropped to the floor, recovered her balance, and glanced to verify that Garimi was indeed dead, before casually walking to the door. With her nimble fingers she disarmed the internal security measures and unsealed the hatch from the inside.\n\nDuncan and Teg stood there holding weapons, afraid of what might emerge. The little girl met them with a placid expression. \"Our Face Dancer is no longer a problem.\"\n\nOver Alia's shoulder they could see the inhuman form sprawled next to an overturned chair. Small trickles of blood leaked from the dart wounds in his eyes, and he wore a full crimson collar of blood around his neck. On the floorplates lay the mangled Garimi.\n\nSheeana narrowed her gaze. \"I see that you are a born killer.\"\n\nAlia was unruffled. \"So I've been told. Didn't you bring the ghola children back for our abilities? This is what I do best.\"\n\nDuncan hurried to the no-ship's controls to assess what the false Rabbi had done. He extended his senses and was dismayed to see the deadly strands of the shimmering net suddenly appear and intensify all around them. Unbreakable. The trap was bright enough, powerful enough, that everyone could see it.\n\nTeg rushed to a scanner station. \"Duncan! Ships approaching\u2014a lot of them! The Face Dancer has brought us right to the doorstep of the Enemy. We're tagged, and the net has locked onto us.\"\n\n\"After all these years, we are caught in the strands.\" Duncan swept his gaze across all of them. \"At least, we're about to find out who our Enemy really is.\"\n\n> _Our shared humanity should, by definition, make us allies. In sad fact, however, our very similarities often appear to be vast differences and insurmountable obstacles._\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> address to the New Sisterhood\n\nGiven the critical shortage of time, the thousands of newly equipped Guildships could not undergo thorough shakedowns and test runs. The mass-produced Obliterators were loaded aboard the heavily armored vessels that had been built at Junction as well as seventeen satellite shipyards. Crews made preparations to go to the front lines.\n\nFresh from conscription across hundreds of at-risk planets, novice commanders received only minimal training, barely sufficient to stand against the Enemy at numerous vulnerable points as humanity tried to draw its line in space. Murbella knew that despite their determination and bravery, and no matter how much training and practice they received, most of the human fighters would be annihilated.\n\nIn the months after the plague had run its course on Chapterhouse, the Mother Commander had opened her doors to displaced refugees from any evacuated planet. At first they were frightened to settle on the once-quarantined world, but then they had begun to stream in. With so few options available to them, the ragtag groups accepted the Sisterhood's offer of sanctuary in exchange for performing vital labors in the war effort. Politics and old factions had to be set aside. Now every life was devoted to preparing for the last stand against the oncoming forces of Omnius.\n\nFrom Buzzell, Reverend Mother Corysta sent the incredible news that the giant seaworms wreaking havoc with soostone operations also produced a kind of spice. Murbella immediately suspected some kind of Guild experiment. It could not be a natural occurrence. Corysta suggested that the worms be hunted and harvested, but the Mother Commander could not think that far ahead. A new source of spice mattered only if the human race survived the Enemy.\n\nMother Commander Murbella called a grand war council for delegates of the front-line planets that were in imminent danger of attack by thinking-machine forces. Despite their indignation, every one of them had undergone cellular testing to root out hidden Face Dancers. Murbella took no chances; the insidious shape-shifters could be anywhere.\n\nIn the Keep's grand meeting room, she strode down the length of the elaccawood table to her designated seat. Using her Bene Gesserit powers of close observation, she studied those assembled, all of them driven here by desperation. Murbella tried to view these representatives in their various costumes and uniforms as military leaders, essentially generals in the last great battle for humanity. The people in this room would guide the thrown-together clusters of ships and make a thousand defiant stands. But were they the quality of heroes the human race needed?\n\nWhen she turned to face the delegates, Murbella saw the uneasiness in their eyes and smelled fear-sweat in the air. The vast Enemy fleet surged forward like a flame front across the map of the galaxy, rolling over star system after star system, heading inexorably toward Chapterhouse and the remaining worlds at the heart of the Old Empire.\n\nAfter moving among the various embattled worlds and studying their preparations, Murbella had secured alliances with these planetary leaders, warlords, commercial conglomerates, and smaller units of government. Leto II's vision of the Golden Path had fragmented humanity so that they no longer followed a single charismatic leader, and now Murbella had to repair that damage. Diversity might once have been a path to survival, but unless the numerous worlds and armies could stand together against the far greater foe, they would all perish.\n\nIf the Tyrant's prescience was so formidable, how could he not have foreseen the existence of the great machine empire, no matter how far away it was? How could the God Emperor not have known that another titanic conflict awaited humankind? She felt a faint shudder. Or _had_ he, and everything was playing out exactly as the Tyrant planned?\n\nAfter considerable effort, she had won a critical internal battle when the various leaders agreed that the strongest defense came from a unified plan\u2014 _her_ plan\u2014rather than a hundred independent and hopeless defensive battles. To get her message across, she'd had to cut through the stubborn tentacles of various planetary bureaucracies. Nothing was easy in this war.\n\nFeeling the burdens of her position, Murbella rapped a large spherical stone on the table, producing a loud, echoing boom that called the meeting to order. \"You all know why you're here. We must make our last stands, a thousand of them across space. Many of us will die\u2014or _all_ of us will die. There are no alternatives. The only questions are how soon we will die, and how it will happen. Do we choose to die free and fighting to the last . . . or defeated and running?\"\n\nThe room resounded with a cacophony of voices, accents, and languages, though she had insisted that they all speak the common Galach tongue. She used Voice to cut through the clamor. \"The machines are coming! If we cooperate and do not retreat in the face of our foe, we just might have the means of stopping them dead in their tracks.\"\n\nShe noted Guild officials and Ixian engineers in the audience. Given the short delivery schedule, some of the warship construction had been unavoidably slapdash, but her handful of Bene Gesserit inspectors and line supervisors had overseen the operations.\n\n\"Our weapons and ships are now ready, but before we proceed I have one question for all of you.\" She skewered the leaders with her gaze. If she'd still been an Honored Matre, her eyes would have blazed orange. \"Do you have the resolve and courage to do what is necessary?\"\n\n\"Do _you?_ \" bellowed a bearded man from a very small planet in a remote system.\n\nMurbella rapped her sonic stone again. \"My New Sisterhood will bear the brunt of the initial clash against the thinking machines. We have already fought them in one star system after another, destroying many of their ships, and we survived their plagues here on Chapterhouse. But this war will never be won on individual battlefields.\" She gestured, and Janess worked the controls. \"Look at this, all of you.\"\n\nStartling the assemblage, a large holographic projection appeared, filling the open space of the Keep's great meeting room with detailed maps of the galaxy's numerous solar systems. An advancing blot indicated the thinking machines' conquests, like a tidal wave drowning every system in its path. The darkness of defeat and extermination had already blackened most of the known systems in the regions of the Scattering.\n\n\"We have to focus our efforts. Because they don't use foldspace engines, the Enemy proceeds from system to system. We know their path, and therefore we can put ourselves directly in their way.\" Murbella stood amidst the simulated stars and planets. Her finger darted from point to point, the glowing stars and habitable planets that lay in the Enemy's path. \"We've got to hold the line\u2014here, and here, and everywhere! Only by combining all of our ships, commanders, and weapons can we hope to halt the Enemy.\" She swept her hand through the shimmering images that were just ahead of the encroaching thinking machines. \"Any other choice would be cowardice.\"\n\n\"Do you call us cowards?\" the bearded man roared.\n\nA merchant stood. \"Surely we can negotiate\u2014\"\n\nMurbella cut him off. \"The thinking machines do not want a particular world. Nor are they searching for gems, spice, or any other goods. There is nothing we can offer them to sue for peace. They do not compromise, and will keep pursuing us no matter where we run.\" She looked at the blustering man and said, \"By fleeing conflict today, you _could_ survive for a time. But there'll be no escape for your children or grandchildren. The machines will slaughter them, down to the last infant. Do you value your life over theirs? Then, yes, I do call you cowards.\"\n\nDespite the murmurs in the hall, no one else spoke out. On the giant star display, a line of tiny fireworks erupted along the interface line between the machine-conquered territories and the vulnerable human planets.\n\nMurbella's gaze moved across the audience. \"Each of us is responsible for stopping the Enemy from crossing this line. Failure means death for the human race.\"\n\n> _True loyalty is an unshakable force. The difficulty is in determining exactly where a person's allegiance lies. Often that bond is only to oneself._\n> \n> \u2014DUNCAN IDAHO,  \n>  _A Thousand Lives_\n\nThe leader of the Face Dancer myriad arrived at Synchrony, bearing a long-anticipated gift for the evermind. The thinking machines still viewed Khrone as nothing more than a servant, a delivery boy.\n\nOmnius and Erasmus never suspected that the shape-shifters might be formulating their own schemes independent of both humanity and the thinking machines. Na\u00efve, oblivious, and so very typical. The evermind would treasure this new melange for his grandiose plans, and it would keep the machines from doubting Khrone and his Face Dancers. He intended to make the most of it.\n\nWith their brutality and arrogance, the \"old man and woman\" had long ago given the new shape-shifters reasons to break their loyalty. Erasmus fancied himself reminiscent of a Face Dancer, but much more . . . and similar to a human, but greater. And like Omnius . . . but infinitely more powerful.\n\nKhrone and the rest of the myriad had never truly given their allegiance to the thinking machines. He saw no more reason to accept slavery under machine masters than to have accepted the domination of the original Tleilaxu who had created their predecessors so many centuries ago. Forced allies, second-class partners . . . The evermind was merely one more layer in the grand pyramid of those who _thought_ they controlled the Face Dancers.\n\nAfter so much effort, Khrone couldn't wait until he could drop this endless deception. He was no longer amused by the number of masks he had to wear and the complicated threads he continued to pull. Soon, though . . .\n\nAlone, he flew his small ship directly to the heart of the modern machine empire. The location of Synchrony had been genetically programmed into all new Face Dancers, like some sort of homing beacon. As he entered the airspace over the technological metropolis, Khrone let his thoughts drift back to Ix. The fabricators and engineers had successfully completed a special demonstration at dead Richese, and now Obliterators were emerging from the production lines. Mother Commander Murbella had been impressed with the power she witnessed, and she'd been entirely convinced by the show. Fool!\n\nBut not in all things. In her prior meeting with Chief Fabricator Shayama Sen, Murbella had forced him to administer a biological test that proved he wasn't a Face Dancer. Given what had happened, Khrone was vastly relieved that he had _not_ replaced the man, as he'd been tempted to do many times in the past.\n\nFace Dancers already controlled most of the important positions on Ix, and when the Chief Fabricator blithely distributed the biological tests to all the main engineers and team leaders (never suspecting there might actually be a majority of Face Dancers among them), the myriad had been forced to act precipitously. When an indignant Sen announced the Sisterhood's suspicions, the infiltrators had finally been forced to kill him and assume his identity. They had already taken care of the troublesome Bene Gesserit line supervisors and production monitors. And so the deception continued, unmarred.\n\nEnhanced Face Dancers quickly subsumed the last humans among the leaders of Ix. Then, working together, they contrived all the necessary tests, selected the required scapegoats, substituted convincing data, and submitted everything to Chapterhouse in accordance with Murbella's demands. All in perfect order.\n\nAfter surviving the plague, the Sisterhood's leadership had forced all human protectors to finally band together against the thinking-machine fleet, to defend their race rather than simply their own worlds. The hundreds of new ships that emerged from the Junction shipyards were being loaded with enough Obliterators for a final, concerted stand against the oncoming wave of Omnius's ships. So far, the evermind's forces had encountered very little significant resistance, and now they were on their way to Chapterhouse. For the last time.\n\nKhrone had actually been tempted to let the Reverend Mothers and their last-stand defenders succeed. Given enough functional Obliterators, they could send the machine fleet reeling. Humans and thinking machines could easily annihilate each other. However, that was simply too . . . easy. Kralizec demanded much more! This time, the fundamental shift in the universe would get rid of both rivals, leaving all the remnants of the Old Empire for the Face Dancers.\n\nKhrone felt completely confident in the future as he landed his ship in the convoluted labyrinth of copper steeples, golden turrets, and interlocked silvery buildings. Sentient structures shifted aside to allow a place for his ship to settle. When the small vessel came to rest on a smooth quicksilver plain, Khrone stepped out, breathing air that smelled of smoke and hot metal. He did not spare a moment to look around.\n\nThe central machine world was based entirely on theatrics. He suspected the touch of Erasmus in this, though Omnius had such an overblown perception of his own importance that he no doubt wanted all machine minions to bow before him as a god\u2014even if the evermind had to program them to do so.\n\nRectangular plates appeared on the ground, laying down an interlocked pathway that guided Khrone to his destination in the magnificent arched cathedral. Head held high, he strode along carrying his precious package, refusing to look like a supplicant summoned before his lord. Rather, Khrone was a man on a mission with important business to complete. Omnius would be pleased to have the concentrated ultraspice for use with his cloned Kwisatz Haderach. . . .\n\nInside the ostentatious hall, the ghola of Baron Harkonnen stood with young Paolo at a nine-level pyramid chess board. Glowering, the Baron knocked over a rook on one of the top levels. \"That move is not allowed, Paolo.\"\n\n\"It enabled me to win, didn't it?\" Pleased with his ingenuity, the young man crossed his arms over his chest.\n\n\"By cheating.\"\n\n\"It's a new rule. If we are as important as you say, we should be allowed to make up our own rules.\"\n\nA flash of anger crossed the Baron's face, and then vanished into a chuckle. \"I see your reasoning\u2014and that you are learning.\"\n\nWhen Khrone stepped forward, they looked at him with identical expressions of distaste. \"Oh, it's you.\" The Baron sounded entirely different from when he'd been tormented by the Face Dancers. \"I didn't think we'd be seeing you again. Bored of Caladan?\"\n\nIgnoring them, Khrone noted that the two principal thinking machines had resumed their guise of an elderly couple in gardening clothes. Why were they wearing these personas now? For the benefit of these two gholas? It wasn't as if the thinking machines were keeping secrets from anybody here. Khrone had never been able to determine a pattern in their behavior.\n\nPerhaps it was linked to the fact that Omnius and Erasmus wanted to receive all of the lives Khrone had gathered and assimilated during his last mission among the humans. They looked forward to the sharing of their Face Dancer \"ambassadors\" each time one of the far-flung representatives returned. It seemed to make them feel superior, and allowed the independent robot to feel that he belonged to the human race, somehow.\n\n\"Look, he's brought something,\" Paolo pointed out. \"A present for us?\"\n\nKhrone went directly to the old man and woman. As the woman leaned toward him, her visage had a feral and hungry look. \"I think you brought more than just a package, Khrone. You haven't been back to Synchrony in some time. Show us the personas you've acquired. Every little bit adds to us, makes us greater.\"\n\n\"I have had enough.\" The old man turned away. \"I am beginning to find them somewhat distasteful. They are all the same.\"\n\n\"How can you say that, Daniel? Every human is different, so beautifully chaotic and unpredictable.\"\n\n\"Exactly what I mean. They are all confusing. And I am not Daniel, I am Omnius. Kralizec is upon us, and we have no time for further preparatory games.\"\n\n\"Sometimes I still like to consider myself Marty. In many ways it's more appealing to me than the name or guise of Erasmus.\" The old woman took a step closer to Khrone. The Face Dancer didn't dare flinch, though he despised what was about to happen. Her hand was gnarled, with large knuckles. It felt clawlike when she touched his forehead. She pressed harder, and Khrone shuddered, unable to block the intrusion.\n\nEach time a Face Dancer mimicked a human shape, he sampled the original subject and acquired both a genetic trace and an imprint of the memories and persona. The thinking machines had set the shape-shifters loose into the Old Empire. Infiltrating the humans, they gathered more and more lives as they subsumed useful people and played their roles. Whenever a Face Dancer returned to the machine empire, Erasmus in particular wanted to add those lives to his vast repository of data and experience.\n\nOut of forced subservience, Khrone and his comrades surrendered that information. But though the thinking machines could upload the various lives the Face Dancers copied, they could not take their core personas. Khrone held onto his secrets, even as he offered up all those people he had been in recent years\u2014an Ixian engineer, a CHOAM representative, a crewman on a Guildship, a dock worker on Caladan, and many others.\n\nWhen the process was finished, the old woman's hand withdrew. Her wrinkled face wore a satisfied smile. \"Oh, those were interesting ones! Omnius will certainly want to share them.\"\n\n\"That remains to be seen,\" the old man said.\n\nFeeling drained, Khrone caught his breath and straightened himself. \"That is not why I came.\" His voice was shamefully weak and quavering. \"I have obtained a special substance you will find invaluable for your Kwisatz Haderach project.\" He held out the ultraspice package, as if offering a gift to a king, precisely as Omnius expected him to behave. The old man accepted the package, scrutinized it carefully.\n\nThe Face Dancer gave Paolo a condescending look. \"This potent form of melange is sure to unlock the prescience in any Atreides. Then you will have your Kwisatz Haderach, as I have always promised. There is no need to continue pursuing the no-ship.\"\n\nOmnius found the comment amusing. \"Strange you should say that now.\"\n\n\"What do you mean?\"\n\nBeside him, the old woman grinned. \"This is a momentous day, since both of our plans have come to fruition. Our patience and foresight have paid off. Now, what shall we do with _two_ Kwisatz Haderachs?\"\n\nKhrone paused, startled. \"Two of them?\"\n\n\"After so many years, the no-ship has finally fallen into our trap.\"\n\nKhrone slid his surprise back into himself and went rigid. \"That is . . . most excellent.\"\n\nThe old woman rubbed her hands together. \"Everything is culminating at once. It reminds me of the climactic movement in a symphony I once wrote.\"\n\nThe old man began to pace around the chamber, holding the package of ultraspice in his hands. He sniffed it.\n\nPaolo turned away from the chess game. \"You don't need another Kwisatz Haderach. You have _me_. Give me spice now!\"\n\nErasmus shot him an indulgent smile. \"Perhaps in a little while. First we'll see what the no-ship has for us, who their Kwisatz Haderach is. It should be interesting.\"\n\n\"Where is the vessel?\" Khrone asked, focusing on the main question. \"Are you sure you have it?\"\n\n\"Our cruisers are surrounding it even now, and our operatives aboard took steps to guarantee that it could not escape again. Your Face Dancers did a fine job, Khrone.\"\n\nOmnius interrupted, \"And, on a greater scale, our largest battleships are closing in on human defenders in their Old Empire. We will conquer Chapterhouse soon, but that is only one of many simultaneous targets.\"\n\n\"It should be quite a spectacular battle.\" Erasmus sounded more dry than eager.\n\nThe evermind was stern. \"Triumph will be assured as soon as the proper conditions are met, according to our mathematical prophecies. Success is imminent.\"\n\nWith glee on his flowmetal face, Erasmus beamed at Paolo and the Baron. \"Two Kwisatz Haderachs are better than one!\"\n\n> _Time is a commodity more precious than melange. Even the wealthiest man cannot buy more minutes to put into each hour._\n> \n> \u2014DUKE LETO ATREIDES,  \n> last message from Caladan\n\nA gossamer net of jeweled colors closed around the _Ithaca_. The no-ship's engines strained, but could not break away. Scrambling to reassert control over the helm and drag themselves free of the strange bonds, Duncan powered up the Holtzman engines, preparing to rip a hole through the glimmering mesh. It was their only way out.\n\nGlaring at the dead Face Dancer on the deck, Sheeana ordered two Sisters nearby, \"Remove that thing from the navigation bridge!\" Within moments, the women carried away the limp and bloody shape-shifter.\n\nNow that the net was visible to them all, Duncan focused his Mentat awareness to study the woven grid that ensnared them. He searched frantically for holes or weak spots in the powerful structure, but found nothing to suggest the slightest defect, no frayed point that might allow them to escape.\n\nHe would try brute force, then.\n\nYears ago, he had broken free of the net by using the Holtzman engines in ways they had never been designed to function, flying the _Ithaca_ at just the proper angle and speed to penetrate the fabric of space. It had reminded him of a Swordmaster's move, using a slow blade against a personal shield.\n\n\"Accelerating now,\" he said.\n\nTeg leaned over the navigation controls, sweating. \"This is going to be close, Duncan.\" The large ship pulled against the multicolored strands, tore several, and then picked up speed. \"We're breaking free!\"\n\nDuncan felt a brief moment of hope, a surge of triumph.\n\nAn explosion rocked the ship, followed by another, and another. Vibrations and shock waves rang through the hull and decks as if some titan were smashing the vessel with a great hammer. The navigation bridge shuddered.\n\nHolding his chair, Duncan called up diagnostic maps. \"What was that? Is the Enemy firing on us?\"\n\nThe detonations threw Teg to the floor, but he scrambled back to his feet and gripped the console for balance. \"The stolen mines! I think we just found them.\" His words tumbled out in a rush. \"Either Thufir or the Rabbi must have set them to go off\u2014\" As if to confirm his speculation, another explosion rocked the deck, much closer than before.\n\nThe _Ithaca_ reeled out of control, its engines paralyzed. The deck tilted, as artificial gravity generators were knocked offline. Duncan felt a sickening disorientation as the vessel spun off axis.\n\nThe shimmering net grew brighter, tightening like a noose.\n\nFinally, out in the distance, Enemy ships drew into view, like hunters approaching a trap they had set. Duncan stared at the external screens. Who had pursued them for so long? Face Dancers? Some vicious, unknown race? What could be frightening enough to drive the Honored Matres back into the Old Empire?\n\n\"The bastards think they have us.\" Duncan made a fist.\n\n\"Don't they?\" Looking up from his status screens, the Bashar was dismayed by the severe damage indicators lighting up sections of the vessel like fireworks displays. \"The mines have ruined our most vital systems, and we're dead in space.\"\n\nUsing Mentat focus, Duncan studied the panels on his command console. The intricate displays showed the strangling net all around them. He jabbed his finger toward a knot in the diagram, an area of pulsing, flickering electronic signals. At first glance the tangle seemed no different from the rest of the interconnected strands, but as he studied it, he thought he might have found a weakness. \"Look there.\"\n\nTeg feverishly bent closer. \"A loophole?\"\n\n\"If only we could move!\" Racking his brain, Duncan stalked back and forth in front of the controls. \"It would be quite a drunkard's dance to get through that maze\u2014if this ship could fly at all.\"\n\n\"If we all worked together, the entire crew, it would take a week to make repairs. We don't have that much time.\" The Bashar gestured to the tactical screens that displayed data from the long-distance sensors. \"Enemy ships are closing in. They know they've snared us.\"\n\nDuncan accepted the grim reality. \"Holtzman engines are dead. No way to make the repairs in time, no way to escape.\" He hammered his fists on a panel next to the tangled, pulsing loophole on the console's projections. \"But I know I could do it. Why won't this damned ship fly?\"\n\nTeg glanced at the sensor blips that indicated the encroaching Enemy, saw the automated damage reports streaming across the display, and knew exactly what had to be done. Only he could do it.\n\n\"I can fix the ship.\" He had no time to explain. \"Be ready.\" Then he simply vanished.\n\nMILES TEG ACCELERATED his metabolism, kicking himself into the hyper-fast speed he had learned after surviving unendurable torture at the hands of the Honored Matres and their underlings. Around him, time slowed. This would be dangerous to him because of the extreme energy requirements, but he had to do it. The rapidly strobing alarm lights became a slow pulsation that seemed to take an hour for each cycle, brightening and dimming. Re-accessing the archival records of the ship's systems would take too long, but Teg had examined them before. As a Mentat he remembered everything, and now he set to work.\n\nBy himself.\n\nEven at his accelerated speed, Teg exerted himself to run as fast as he could. On deck after deck, everyone aboard stood like statues, their expressions showing concern and confusion. Teg flashed past them to the nearest damage sites.\n\nWhere the first mine had gone off, he stared in amazement and consternation at the twisted metal, the melted craters in the machinery, the vaporized systems. Teg hurried from one explosion to the next, determining how far the damage extended and which systems were crucial for their immediate escape. The Face Dancer infiltrators had planted and hidden the eight mines well, and each detonation had resulted in a crippling blow: navigation, life-support, foldspace engines, defensive weapons.\n\nTeg made snap decisions. His life had primed him for emergencies; on the battlefield, one could not hesitate. If Duncan couldn't manage to fly the _Ithaca_ away right now, they would never again require life-support systems. He, or someone else, could fix those later. An acceptable gamble. The no-field generators were off-line.\n\n_Engines_. Four of the eight mines had been set to damage the foldspace engines. The Face Dancer saboteur had deliberately flown the no-ship close to the Enemy's stronghold, and the detonations had left them crippled and stranded.\n\nWith hyper speed Teg studied, analyzed, and compiled a plan using his Mentat abilities. He inventoried spare materials, replacement components, emergency equipment. He needed to work swiftly with what he had; there was no one to help him. First, he rerouted and reprogrammed the weapons, and prepared them to launch a volley of blasts at the oncoming ships. That might grant them an extra few moments.\n\nTeg continued to hurry. The pulsing alarm lights flickered on to off, like a sun rising and setting. Another hour gone in his own frame of reference. In real time, only a few seconds had passed since his disappearance from the bridge. Next, he turned to the engines, which were essential to their escape.\n\nThe primary linkages had been disrupted, with Holtzman catalysts shaken from their cradles, shoved out of alignment, made inoperable. Two reaction chambers were breached. An explosion had nearly broken through the hull. He stood stunned, his arms shaking, thinking he couldn't possibly fix this. But he forced such thoughts away, went back to work.\n\nTeg's muscles trembled with exhaustion, and his lungs burned from gasping air so fast the oxygen molecules could barely move into position.\n\nFixing the hull should be easy enough. Teg ran to the maintenance sectors, where he located extra plates. Since he could never make the ship's heavy-lifting machinery operate fast enough for his time-sense, he decided that suspensors would have to do. He applied the null-gravity projectors to the heavy plates and hurried with them down corridors, dodging petrified people.\n\nWith each second, the Enemy battleships were getting closer. Some of his fellow passengers were only just now learning of the mines that had been detonated. He put on another burst of speed, and the suspensor carriers kept up with him.\n\nIn a few \"hours,\" according to his metabolism, and only a few moments in reality, he fixed the hull damage that could have resulted in an engine breach. Sweat poured off of Teg's body, and he was near collapse. But in spite of that utter exhaustion, he could not let himself slow down. Never before had he allowed himself to fall so deeply into a pit of burning metabolism.\n\nTeg's body could not maintain this pace for long. But if he didn't, the ship would be captured, and they would all die. Fangs of hunger gnawed at his stomach. This would not do. He had to concentrate, had to fuel the engine of his body so that he could do what must be done.\n\nRavenous, not slowing from his superspeed, he raided the ship's stores, where he found energy bars and dense food wafers. He ate concentrated nutrients until he was gorged. Then, burning calories as fast as he could swallow them, Teg ran again from one disaster area to the next.\n\nHe spent subjective days at these highly focused labors; to observers on the outside, caught in the glacial pace of normal time, only a minute or two passed.\n\nWhen the task grew overwhelming, the Bashar struggled to reassess what the ship needed in order to function. What was the bare minimum of repairs that would let Duncan fly through the weakened loophole?\n\nThe exploding mines had led to a cascading series of damages. Teg nearly got lost in the details, but reminded himself of the immediate need and forced himself to skate the thin ice of possibilities.\n\nTeg and his brave men had stolen this very vessel from Gammu more than three decades ago. Though it had performed admirably since then, the _Ithaca_ had not undergone any of the usual necessary maintenance at Guild shipyards. Worn components had not been replaced; systems were breaking down from age and neglect, as well as the depredations of the saboteurs. Limited by the spare parts and materials he could find in the maintenance bays, he tried and discarded possible fixes.\n\nAlarms continued to pulse slowly. He was moving too fast for sound waves to mean anything. In real time, there would be shrieking sirens, shouting people, conflicting orders.\n\nTeg fixed another of the Holtzman catalyst cradles, then took the time to look at a viewer. In the image displayed between scan lines, he saw that the Enemy ships had finally arrived, massive and heavily armed . . . a full fleet of monstrous, angular things that bristled with weapons, sensor arrays, and other sharp protrusions.\n\nThough he already felt used up, Teg knew with a sickening certainty that he needed to go even faster.\n\nHe raced to the ship's melange stores and broke the locks with a twist of his hand because he was moving so fast. He removed cakes of the dark brown compressed substance, stared at it with Mentat calculation. Considering his hypermetabolism and his body churning through its biochemical machinery faster than it ever had before, what was the proper dosage? How quickly would it affect him? Teg decided on three wafers\u2014triple the maximum he had ever consumed\u2014and gobbled them all.\n\nAs the melange rushed through his body and poured into his senses, he felt alive again, recharged and capable of accomplishing the requisite impossibilities. His muscles and nerves were on fire, and his feet left marks on the deck as he ran.\n\nHe repaired the next system in a few moments. But in that time, the Enemy battle fleet had closed in, and the no-ship still could not fly.\n\nTeg looked down at his forearms and saw that his skin seemed to be shriveling up, as if he was consuming every drop of energy within his flesh.\n\nOutside, the encroaching vessels launched a volley of destructive blasts. Balls of energy tumbled forward like storm clouds approaching with exquisite slowness. Those blasts would clearly render his repairs useless, maybe even destroy the ship.\n\nIn another burst of extreme speed, Teg dashed to the defensive controls. Thankfully, he had restored a few of their weapons. The _Ithaca_ 's defensive systems were sluggish, but the firing controls were swift enough. With a scattershot cannonade, like a burst of celebratory fireworks, Teg returned fire. He launched beams carefully targeted to intercept and dissipate the oncoming projectiles. Once he had fired the volley, though, Teg turned his back on the weapons systems and raced to the next damaged engine.\n\nBashar Teg felt like a candle that had been burnt entirely down to a lump of discolored wax. Despite his best efforts, the exhausted man still saw their doom closing in.\n\n> _How do we repay a man who has done the impossible?_\n> \n> \u2014BASHAR ALEF BURZMALI,  \n>  _A Dirge for the Soldier_\n\nOn the navigation bridge, Duncan stared at the sensor projections for moments after Miles had disappeared. He knew what the Bashar must be doing.\n\nAfter the internal explosions, the _Ithaca_ hung dead in space, surrounded by Enemy ships that bristled with more weaponry than he had seen on an entire Harkonnen battle fleet. The mines had disabled the no-field generator, leaving the great ship visible and vulnerable in space.\n\nAfter almost a quarter century of fleeing, they were caught. Maybe it was about damned time he faced the mysterious hunters. Who were his strange and invincible foes? He had only ever seen the ghostly shadows of the old man and old woman. And now . . .\n\nOn the screens before him, the discontinuity in the gossamer net shifted, almost closed, and then strayed open again, as if taunting him.\n\nDuncan spoke aloud, more to himself than anyone else. A prayer of sorts. \"As long as we breathe, we have a chance. Our task is to identify any opportunity, however transitory or difficult it might be.\"\n\nTeg had said he would fix their systems. Duncan was aware of the Bashar's closely held abilities. For years, Teg had concealed his talent from the Bene Gesserits, who feared such manifestations as the sign of a potential Kwisatz Haderach. Now those abilities might save them all. \"Don't let us down, Miles.\"\n\nThe encroaching ships fired a series of blasts at the no-ship. Duncan barely had time to shout a curse and brace for impact\u2014when a flurry of impossibly fast and deft defensive bursts intercepted the Enemy volley. Precisely targeted, instantly fired. All shots blocked.\n\nDuncan blinked. Who had launched the return salvo? He shook his head. The no-ship should have been incapable of even basic maneuvers or defense. A chill of delight coursed down his spine. Miles!\n\nSuddenly, the control deck's systems began to glow; green indicator lights winked on by themselves. One after another, systems came back online. Sensing movement, Duncan snapped his head to the left.\n\nThe Bashar materialized in front of him, but it was a different Miles Teg\u2014not the young ghola whom Duncan had raised and awakened, but a horribly drained man, as desiccated and ancient as an ambulatory mummy. Teg looked wrung out and ready to collapse. He had exerted himself through time far beyond the point where a normal man would have already died.\n\n\"Boards . . . active.\" His gasping voice cost him more energy than he had left. \" _Go!_ \"\n\nEverything happened in an instant, as if Duncan, too, had fallen into an accelerated time frame. His first instinct was to grab his friend. Teg was dying, might already be dead. The aged Bashar could no longer hold himself upright. \" _Go_ \u2014damn it!\" They were the last words Teg could force out of his mouth.\n\nThinking with Mentat clarity, Duncan whipped back to the control panels, vowing not to waste what the Bashar had done for them. _Priorities._ He reached the piloting board, where his fingers skittered like a startled spider across the controls.\n\nTeg crumpled to the deck, arms and legs akimbo, as dead as a dried leaf, older even than the first old Bashar had been in the last moments of Rakis. _Miles!_ All their years together, teaching, learning, relying on each other. Few people in all of Duncan's many lives had ever mattered so much.\n\nHe drove away his thoughts of shocked grief, but Mentat memory kept every experience clear and sharp. Miles! Teg was no more than an ancient husk on the floorplates. Duncan had no time for anger or tears.\n\nThe no-ship began to accelerate. He still saw how to slip out of the cruel net, but now he also had to contend with the entire fleet of Enemy ships. They had cut loose with a second volley.\n\nThe blurred crackle ahead seemed to invite them. Duncan steered toward it, moving as fast as his human reflexes could go. The no-ship ripped the stubborn strands free. \"Come on!\" Duncan said, willing it to happen.\n\nMore blasts glanced across the _Ithaca_ 's hull, grazing the ship as it yawed and rolled. Duncan steered with all of his skill.\n\nThe Holtzman engines were hot and the diagnostic boards showed numerous errors and system failures, but none were immediately fatal faults. Duncan pushed the vessel closer and closer to the loophole. The Enemy ships couldn't head them off, couldn't move fast enough to stop them.\n\nMore of the net broke away. Duncan could see it happening.\n\nHe forced his attention back to the engines, applying acceleration far beyond what the systems normally allowed. In his frantic repairs, Teg had not bothered with the niceties of fail-safes and protective limitations. With increased velocity, they pulled free of the enclosing cordon.\n\n\"We're going to make it!\" Duncan said to the fallen Bashar, as if his friend could still hear him.\n\nA giant torpedo-shaped Enemy vessel leapt forward. No human could possibly pilot a ship so swiftly, changing directions with g-forces that would snap bones like a handful of straw in a clenched fist. Burning its engines, the attacker exhausted all of its fuel in one burst of forward motion\u2014throwing the craft directly into their path.\n\nWith his maneuvering already hampered, Duncan could not dodge in time. The no-ship was too huge, with too much inertia. Impossibly, the suicidal Enemy vessel scraped the lower hull of the _Ithaca_ , knocking it off course, damaging the engines yet again. The unexpected impact sent the no-ship spinning. The Enemy rammer tumbled and exploded, and the shock wave knocked them farther off course, out of control . . . back into the remaining strands of the net.\n\nDuncan uttered a curse in dismay and rage.\n\nUnable to fold space, the no-ship dropped back, its engines whining. The bridge control panels blazed red, then went dim. A small internal explosion further damaged the Holtzman engines. The _Ithaca_ hung motionless in space. Again.\n\n\"I'm sorry, Bashar,\" Duncan said, heartbroken. With nothing else to do, he knelt beside the husk of his friend.\n\nA message formed on the primary screen on the bridge, a powerful transmission from the surrounding battleships. Even in his stunned sorrow, Duncan was surprised to see the true face of the Enemy at last.\n\nThe smooth flowmetal face of a sentient machine appeared on the screen. \"You are our prisoners. Your vessel is no longer capable of independent flight. We will deliver you to the evermind Omnius.\"\n\nThinking machines!\n\nDuncan struggled to understand what he was seeing and hearing. Omnius? The evermind? The Enemy, posing as a kindly old couple, were really thinking machines? Impossible! Thinking machines had been outlawed for thousands of years, and the last evermind had been destroyed in the Battle of Corrin at the end of the Butlerian Jihad.\n\nMachines? Somehow allied with the new Face Dancers?\n\nThe Enemy ships pounced like hyenas on a fresh carcass.\n\n> _Some people complain of being haunted by their past. Utter nonsense! I revel in it._\n> \n> \u2014BARON VLADIMIR HARKONNEN, the ghola\n\nTrapped by the machine fleet, the _Ithaca_ was held captive with its engines damaged and weapons burned out. Duncan could do nothing but wait and mourn his dead friend. Consequences and memories roared around him. He moved methodically, relying on Mentat focus to perform even simple actions.\n\nSheeana was beside him on the navigation bridge. Though she prided herself in Bene Gesserit purity, holding all emotions at bay, she seemed profoundly troubled as the two of them picked up Teg's body from where it lay crumpled on the deck. Duncan couldn't believe how fragile and lightweight the Bashar's remains were. He seemed to be made of spiderwebs and sinew, dried leaves and hollow bone.\n\n\"Miles gave his life for all of us,\" Duncan said.\n\n\"Two times,\" she said.\n\nHer remark made Duncan think of all the lives of his own he had given for the Atreides. In a raspy voice, he said, \"This time, the sacrifice was for nothing. Miles used up his entire life span to give us the repairs we needed, and I couldn't break us free. He shouldn't have done it.\"\n\nSheeana fixed a hard look on him. \"He shouldn't have _tried_? We're humans. We have to try, no matter what the odds are. There are never any guarantees. Every action in life is a gamble. The Bashar fought to the last instant of his existence, because he believed there was a chance. I intend to do the same.\"\n\nDuncan looked down at the sunken, mummified face of his friend, remembering all the determination and hard training the old Bashar had given him when he was a young ghola. Sheeana was right. Even though Duncan hadn't been able to free the _Ithaca_ and let them escape, he and Miles had shown the Enemy that humans were unpredictable and resilient, that they were not to be underestimated. And it wasn't over yet. Instead of a simple capture, the thinking machines had been forced to sacrifice one of their largest battleships simply to stop them.\n\n\"We'll take him to one of the small airlocks,\" he announced. Since their every movement was now dictated by the Enemy ships that dragged them along, it was pointless to remain at the controls. \"I have no intention of letting the thinking machines have him.\"\n\nThe remnants of the Bashar would fly alone into the cosmos. The rest of them might be trapped, to be used in thinking-machine experiments, or for whatever reason the old man and woman had been pursuing them over the decades. But not Miles. This act would be another small victory\u2014and enough small victories could win an entire war.\n\nThey arrived at one of the chambers, which Duncan recognized as the same airlock he had used to jettison Murbella's last possessions, items that had clung to him like cobwebs until he forced himself to let go. They placed the tragically lightweight husk of Teg's body inside the chamber and sealed it. Duncan looked through the observation port, saying his last goodbyes.\n\n\"It isn't the ceremony I would have imagined for him. Last time, the Bashar had all of Rakis for his funeral pyre. But there's no time.\" Before he could have second thoughts, Duncan pushed the button that evacuated the airlock, opening the outside hatch so that the body tumbled out into the void. \"We should summon everyone aboard the ship and prepare our defenses.\"\n\n\"What defenses?\"\n\nHe looked at her. \"Anything we can think of.\"\n\nSHOULDERED FORWARD BY a hundred thinking-machine vessels, the battered no-ship was forced down into Synchrony, where shifting buildings moved aside to form an acceptable place for the captured craft to land. The now-visible _Ithaca_ descended like a trussed wild animal, the trophy of big game hunters.\n\nBaron Harkonnen thought it a glorious sight. From an extruded balcony in one of Omnius's capricious high towers, he studied the vessel as it descended. The no-ship's configuration was unfamiliar to him, massive but not as intimidating as he'd imagined it would be. This design was much more organic and alien-looking than huge Guild Heighliners, deadly Sardaukar craft, House Harkonnen military vessels, or his own family frigates. It seemed to be convergent evolution, eerily similar to the flow-form curves of the thinking-machine structures.\n\n_Strange ship, strange passengers._\n\nAccording to initial reports from the machine scouts who had seized the no-ship, many of those aboard were gholas from his own past, annoyances resurrected from history, exactly as Erasmus had suspected\u2014Lady Jessica, another Paul Atreides, a minor Swordmaster named Duncan Idaho, and who knew how many others? Gholas coughed up and spat out like wads of phlegm.\n\nA keyed-up Paolo stood beside him on the balcony, facing the makeshift spaceport that waited to accommodate the new vessel. \"Will we kill them all, Grandfather? I don't want there to be another Kwisatz Haderach. I'm supposed to be the only one. I should take the ultraspice that Khrone delivered right now.\"\n\n\"I would have you do it if I could, dear boy, but Omnius won't permit that. Be patient. Even if there is another version of Paul Atreides aboard that no-ship, he's probably soft and compassionate. He doesn't have the advantage of being toughened by _me._ \" The Baron's full lips curled down in distaste. Paolo himself didn't realize just how much of his fundamental personality had been changed. \"You will have no trouble defeating him.\"\n\n\"I have already visualized it,\" Paolo replied. \"Real, prescient dreams\u2014and now I understand what is going to happen.\"\n\n\"Then you have nothing to worry about.\"\n\nThe Omnius-formed buildings swayed like reeds, then embraced the battered no-ship as it landed, pulling the _Ithaca_ down into a living metal cradle. The landing and lockdown process seemed interminable. Was it really necessary for so many structural braces to fold around the ship like claws? Considering the obvious damage to the engines, the captives could never find a way to launch the vessel again. However, Omnius had a penchant for doing things in a brute-force manner. The Baron could understand that.\n\nPresently Erasmus appeared on the balcony, once again disguised as a matronly old woman. Gazing dispassionately at the robot, the Baron announced, \"I will go aboard the no-ship. I want to be the first to\"\u2014his lips quirked in a smile\u2014\"greet our visitors.\"\n\nThe old woman's eyes twinkled. \"Are you certain that would be wise, Baron? We aren't sure yet exactly who is aboard the vessel. You could be in peril if anyone recognizes you. In your past life, quite a few people were not entirely pleased with you.\"\n\n\"I certainly don't intend to go unprotected! In fact, I expect _you_ to provide me with full security. Some of your sentinel robots, perhaps\u2014or better yet, an armed contingent of Face Dancers. Paolo will remain here safe, but I _will_ go aboard.\" He planted his hands on his hips. \"In fact, I demand it.\"\n\nErasmus seemed amused. \"In that case, we had better give you the Face Dancers. Go aboard, Baron, and be our ambassador. I'm sure you will employ all the diplomacy the situation requires.\"\n\n> _We shall face the Enemy, and die if we must die. My strong preference, however, is to_ kill _what we must kill._\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> transmission to human defensive forces\n\nTen thousand Guildships against an infinite number of Enemy vessels.\n\nFor this confrontation, the Mother Commander had prepared all the warlords, political leaders and other self-proclaimed generals, as well as her ferocious Sisters\u2014what remained of them. Spread out across the path of the oncoming thinking-machine forces, her human defenders dug themselves in.\n\nGuildsmen had been rushed in at the last minute to help crew the numerous battleships, launching them to their designated rendezvous points in space. The untested military commanders were as ready as the Mother Commander could make them. Like ghost soldiers, redeyed refugees from planets already ground under the machine boot heel volunteered in droves. Each craft was loaded with Obliterators produced by the tireless Ixian factories.\n\nUnfortunately, Omnius had been preparing for centuries.\n\nLike a force of nature, the thinking machines advanced, not dodging or changing course, without regard to the strength of planetary defenses arrayed against them. They simply rolled over anything in their path.\n\nFor Murbella's plan to work, the line of Enemy ships had to be stopped at every point, in every star system. No battleground was unimportant. She had divided her defenders into a hundred discrete groups of one hundred new Guild warships apiece. The battle groups were positioned at widely scattered but important points outside inhabited systems, ready to fend off the approaching Enemy.\n\nAs a last line of defense, Murbella's one hundred newly constructed vessels patrolled space in the vicinity of Chapterhouse, along with a number of smaller, older vessels to flesh out the military force. They knew Omnius considered this planet a primary target. Waiting for the clash, the Mother Commander thought her new ships looked magnificent, the line formidable. The fighters aboard were more confident than afraid.\n\nBy the New Sisterhood's best estimates, though, the thinking machines outnumbered them by more than a hundred to one.\n\nTo shore up their confidence, the fighters had all watched holos of the Ixian tests of the new Obliterators on dead Richese, admiring the massive destructive force contained in each of the powerful weapons. Bene Gesserit observers had monitored the Ixian production lines, and technicians had verified the complex weapons after they were installed in Murbella's fleet. She clung to the hope that this line of last stands could turn into a rout for the forces of Omnius.\n\nMore than she had for the past quarter century, the Mother Commander wished Duncan Idaho could be at her side again, facing this final conflict with her. Feeling the loneliness of command, tempted to bow to primitive human superstition and offer up a prayer to some invisible guardian angel, she hardened herself.\n\n_This has to work!_\n\nHer great ships prowled the edge of planetary orbit, not knowing from which direction the Enemy fleet would come. Down below, the refugees who had filled temporary camps on the plague-emptied continents were anxious to evacuate from Chapterhouse, but even if there were vessels to transport them away, they had nowhere to go. Every functional craft in the sector had been commandeered to face the thinking-machine ships. It was everything the human race could rally.\n\n\"Enemy ships approaching, Mother Commander,\" said Administrator Gorus, receiving a message from the sensor deck. His pale braid looked somewhat frayed, his skin whiter than usual. He had been convinced to stay aboard the main ship at the central battlefield, to stand by the new ships his factories had produced; he didn't look at all happy about it.\n\n\"Exactly on time. Exactly as expected,\" Murbella said. \"Disperse our vessels into the widest possible firing spread, so we can hit the Enemy all at once, before they can react to us. Machines are adaptable, but they rarely take the unexpected into account.\"\n\nGorus looked at her sourly. \"Are you making assumptions based on old records, Mother Commander? Extrapolating from the way Omnius reacted fifteen thousand years ago?\"\n\n\"To some extent, but I trust my instincts.\"\n\nAs the heavily armed machine ships approached, they looked like a meteor shower that grew larger and larger. The monstrous vessels loomed huge\u2014thousands of them against the Sisterhood's desperate hundred. All along the line, at a hundred other systems, she knew her defenders were facing similar odds.\n\n\"Prepare to launch Obliterators. Stop them before they get any closer to Chapterhouse.\" Murbella crossed her arms over her chest. Across the commlines, each captain announced his or her readiness.\n\nThe oncoming machine ships slowed, as if curious to see what this small obstacle might be. _They will underestimate us,_ Murbella thought. \"Maximize targets. Fire into close groupings of Enemy ships. Consolidate explosions.\"\n\n\"Targets locked, Mother Commander,\" Gorus said, his message transmitted immediately by his sensor technicians.\n\nMurbella had to preempt the thinking machines before they could open fire. \"Launch Obliterators.\" She held herself steady.\n\nSilver sparks spat out of the launch tubes, Obliterators twirling toward the line of Enemy ships, but the glints faded. Nothing happened, though some of the heavy weapons must have struck their targets. The machine vessels seemed to be waiting for something.\n\nShe looked around. \"Confirm that the Obliterators are armed. Where are the explosions? Launch the second volley!\"\n\nAlarms began to ring. In a frenzy, Gorus ran from one station to another, shouting at the Guildsmen on the upper decks. A harried-looking Reverend Mother charged into the command center, skidding to a stop in front of Murbella. \"Our Obliterators are doing nothing. They are all useless.\"\n\n\"But they were tested! Our Sisters watched the manufacturing lines. How could they be faulty?\"\n\nThen, all at once, the one hundred Chapterhouse defender ships went dead in space, their engines shutting down, lights flickering. The thrum of station-keeping thrusters faded.\n\n\"What is happening?\" Gorus demanded. \"Sabotage? Were we betrayed?\"\n\nAs if they had expected this all along, the machine ships closed in.\n\nA Guildsman transmitted in a hollow voice over the speaking screen, \"The artificial navigation systems no longer respond, Administrator. We are shut out of our own controls. Our ships are . . . nonfunctional.\" Emergency lights lit the decks with an eerie glow.\n\n\"Did the machines figure out how to neutralize our systems?\"\n\nGorus turned to Murbella. \"No jamming, Mother Commander. They . . . they just don't work. None of them.\"\n\nSuddenly the machine forces were upon them, a thousand vessels that would easily overwhelm the defenders. Murbella prepared to die. Her fighters could not protect themselves, or Chapterhouse, which she had sworn to guard.\n\nBut instead of attacking, the Enemy fleet cruised slowly past the defenders, taunting them in their impotence. The machines did not bother to open fire, as if the Sisterhood's defenses weren't even worth noticing!\n\nFar behind them, just arriving at the distant edge of the solar system, came another wave of thinking machines, closing in on Chapterhouse. The same thing must be happening everywhere, at all of her carefully staged last stands across a hundred star systems.\n\n\"They knew! The damned machines knew our Obliterators wouldn't work!\" As if Murbella's vessels were no more than a pebble on the path, the Omnius ships flowed around them on their way to the Sisterhood's now-unprotected homeworld.\n\nNot one of her new Guild war vessels had a living Navigator aboard; most of the Navigators and their Heighliners had disappeared. Every ship in her battle groups used Ixian mathematical compilers for guidance. Mathematical compilers! Computers . . . thinking machines.\n\n_The Ixians!_ Now her silent curse was directed at herself for overconfidence in the new Obliterators and her own ability to predict the Enemy's tactics.\n\n\"Follow me, Administrator. I want to see these Obliterators for myself.\" She grabbed Gorus's arm hard enough to leave bruises.\n\nGuided by emergency illumination, they rushed to the weapons deck where the armaments had been installed. Inside, rack upon rack held the burnished silver eggs of the planet-melters that Ix had manufactured. A distraught Guildsman intercepted them. \"We tested the weapons, Administrator, and they were installed correctly. The firing controls are operational. We just launched dozens of Obliterators, but none of them detonated.\"\n\n\"Why didn't they function?\"\n\n\"Because . . . because the Obliterators themselves . . .\"\n\nMurbella marched over to where the man had opened one casing at random. Beneath a complicated labyrinth of circuitry and delicate components, the Obliterator charge was fused into the shell of the mechanism, making the whole thing inoperable. The weapon had been neutralized.\n\n\"It is useless, Mother Commander,\" said Gorus. \"Sabotaged.\"\n\n\"But I saw the tests myself. How can this be?\"\n\n\"A timing mechanism may have shut everything down at a prearranged time, or the Enemy fleet might have sent out a deactivating signal. Some devious trick that we could not have anticipated.\"\n\nMurbella stood appalled, guilty of the same error she had been so certain the machines would fall victim to: She had failed to plan for the unexpected. Together, they opened another Obliterator to find it similarly fused and nonfunctional. A coldness froze her heart and spread into her bloodstream. These weapons had been built over the course of years by the Ixians, at a cost in melange that nearly bankrupted the Sisterhood. She had been duped, and her fleet had been castrated by the Ixians before the battle could even begin.\n\n\"And what about our engines?\"\n\n\"They can be made to function, if we operate them without the mathematical compilers.\"\n\n\"I don't give a damn about the compilers! Find a way to salvage some of the Obliterators. Are they all inactive? Every single one?\"\n\n\"The only way to know, Mother Commander, is to open and inspect each of them.\"\n\n\"We could just launch them all and hope a few still function.\" Murbella nodded slowly. It was indeed an option. At this point, it cost them nothing. She had to find some way to fight, and she hoped her other battle groups were faring better than this . . . but she doubted it. Without functional Obliterators, every one of the planets on the front line was essentially unprotected in the face of certain destruction.\n\nAnd it was all her responsibility.\n\n> _Some say that survival itself can be the best revenge. For myself, I prefer something a bit more extravagant._\n> \n> \u2014BARON VLADIMIR HARKONNEN,  \n> the ghola\n\nOn a whim, the Baron told the ten Face Dancers accompanying him to pose as Sardaukar from the old Imperium. He didn't know if anyone would even recognize the joke\u2014fashions changed and history forgot such details\u2014but it helped him present an air of command. During his original lifetime he had achieved a great victory over House Atreides with illicit Sardaukar at his side.\n\nLeaving the restless Paolo with Erasmus, supposedly \"for his own protection,\" the Baron dressed himself in a nobleman's uniform frosted with gold braids and ornate chains of office. A ceremonial poison-tipped dagger hung at his side, and a wide-beam stunner was concealed in his sleeve for easy access. Though the imitation Sardaukar were his guards and escort, he didn't particularly trust them, either. One could never be too careful.\n\nWhen the Baron's entourage marched to the imprisoned no-ship, however, they could not find a door on the kilometer-long hull\u2014a frustrating and embarrassing moment, but Omnius was not to be hindered. Guided by the evermind, parts of nearby buildings transformed into gigantic tools that tore open the hull, peeling away plates and structural girders to leave a wide gash. Brute force was easier and more direct than locating an appropriate hatch and deciphering unfamiliar controls.\n\nWith the no-ship suitably opened, the Baron and his escort ducked under low-hanging debris and sparking circuitry. Prepared for an ambush, but moving with an outward show of confidence, they made their way through the winding corridors. Several of Omnius's floating watcheyes zoomed ahead of them down the passageways to scout out and map the interior of the vessel.\n\nThe captives would surely see that surrender was their only option. What other conclusion could they draw? Unfortunately, in his original lifetime the Baron had had considerable experience with fanatics, such as the mad Fremen bands on Arrakis. It was possible that these poor wretches intended to mount a desperate, hopeless resistance until they were all slaughtered, including the purported Kwisatz Haderach among them.\n\nPaolo would then be the only contender, and that would be that.\n\nInside the no-ship, they first encountered Duncan Idaho and a defiant-looking Bene Gesserit woman who identified herself as Sheeana. The two waited for the boarding party in the middle of a wide corridor. The Baron only vaguely remembered the man from the records of House Atreides: a Swordmaster of Ginaz, one of Duke Leto's most trusted fighters, killed on Arrakis while protecting Paul and Jessica in their escape. From the sneer on Idaho's face, he could tell that this ghola had his memories back, too.\n\n\"Oh ho, I see that you know me.\"\n\nIdaho didn't budge. \"I escaped from Giedi Prime as a boy, Baron. I beat Rabban on one of his hunts. I've lived many lifetimes since then. This time, I hope to watch you die with my own eyes.\"\n\n\"How boldly you speak, like one of those yipping dogs Emperor Shaddam used to keep at his side: full of annoying barks and growls, yet easily stepped on.\" Protected by the Face Dancer Sardaukar, he peered ahead down the hall. \"How many people do you have aboard?\" He snapped. \"Bring them forward for our inspection.\"\n\n\"We have already assembled,\" Sheeana said. \"We're ready for you.\"\n\nThe Baron sighed. \"And no doubt you've scattered commandos or snipers throughout the decks? Your personnel records will have been doctored. A childish resistance that may cause us a few headaches, but will gain you nothing. We have enough troops to mow all of you down.\"\n\n\"It would be foolish for us to resist,\" Sheeana said, \"at least in such obvious ways.\"\n\nThe Baron scowled, and he heard the little girl's voice inside his head. _She is playing with your mind, Grandfather!_\n\n\"So are you!\" he hissed to himself, startling the others.\n\n\"Five hundred more of our men are coming aboard,\" said the counterfeit Sardaukar commander. \"Mobile machine sensors will scour every chamber on every deck, and we'll find anything there is to find. We will locate the Kwisatz Haderach.\"\n\n\"A Kwisatz Haderach?\" Idaho asked. \"Is that what the old man and woman have been looking for? On this ship? You're welcome to waste your time.\"\n\nSheeana added harshly, \"If we had a superman aboard, you would never have been able to capture us.\"\n\nThat remark disturbed the Baron. At the back of his mind he heard the maddening voice of Alia chuckling at his discomfiture. His face flushed, but he forced himself not to speak aloud. What a fool, debating with the unheard voice of an invisible tormentor! New groups came down the no-ship corridors to gather in front of him like troops for inspection.\n\nOne small-statured teenaged ghola unsettled him the most. The young man was thin and sallow-skinned, his face etched in a scowl. His eyes burned with hatred for the Baron, though he did not find the fellow at all familiar. He wondered what he had done to that one.\n\n_Look more closely, Grandfather. Surely you recognize him? He almost killed you!_\n\n_I swear I will find some way to rip you out of my head!_\n\nWith a neutral expression on his face, he looked again at the dour ghola, and suddenly understood the crude black diamond marked on his forehead. \"Why, it's Yueh! My dear Dr. Yueh, how good to see you again. I never got a chance to tell you how much you helped the Harkonnen cause so long ago. Glad to see that I have an unexpected ally aboard this ship.\"\n\nYueh looked skinny and ineffectual, yet the gleam in his eyes was genuinely murderous. \"I am not your ally.\"\n\n\"You are a weak little worm. It was easy enough to manipulate you before\u2014I can do it again.\" The Baron was surprised that the scrawny man did not back down. This version of Yueh seemed stronger, perhaps transformed by the lessons of his ignominious past.\n\n\"You no longer have leverage over me, Baron. You have no Wanna. Even if you did, I would not repeat my earlier mistakes.\" Crossing his arms over his narrow chest, he thrust his pointed chin forward.\n\nThe Baron turned abruptly from the Suk doctor as even more no-ship captives came forward. One bronze-haired young woman of about eighteen looked exactly like the lovely Lady Jessica. The way she viewed him with palpable revulsion proved that this ghola also had her memories restored. Did Jessica know she was really his own daughter? What entertaining conversations they might have!\n\nStanding protectively beside the youthful Jessica were a younger woman dressed as a Fremen and a dark-haired young man\u2014the perfect image of Paolo, only older. \"Why, is this young Paul? Another Paul Atreides?\"\n\nA swift slash, a mere nick from the poisoned dagger, and the rival Kwisatz Haderach would be gone. But he shuddered to think how Omnius would react to that. The Baron wanted Paolo to assume his position of power, of course, but he wasn't willing to sacrifice his own life for the boy. Though the Baron had raised and trained Paolo, he was still, after all, an _Atreides._\n\n\"Hello, Grandfather,\" Paul said. \"I remember you as being much older and fatter.\" The Baron found the demeanor and tone irritating. And even worse, he felt an odd, swooning sensation . . . as if Paul had always been meant to say this, as if he had seen it in a dozen different visions.\n\nStill, the Baron clapped his hands in mock applause. \"Isn't ghola technology marvelous? This is like an encore at the end of one of the Emperor's tedious jongleur performances. All back together again for a second run, eh?\"\n\nPaul stiffened. \"House Atreides crushed the Harkonnens into extinction long ago. I anticipate a similar outcome now.\"\n\n\"Oh, ho!\" Though amused, the Baron-ghola didn't step any closer. He gestured to his Sardaukar guard. \"Have a doctor and a dentist look them over before they get close to me. Pay particular attention to their teeth. Look for poison capsules.\"\n\nHaving fulfilled his purpose, the Baron was about to march out of the no-ship when, among the gathered refugees, he spotted a small girl who stood quietly beside a thin boy of around twelve years, watching everything. Both had an Atreides look about them. He froze, recognizing Alia.\n\nNot only had this bloodthirsty child jabbed him with the poison gom jabbar and haunted his thoughts, now she even stood before him! _Look, Grandfather\u2014now we can torment you inside and out!_ Her voice pierced him like ice picks in his head.\n\nThe Baron reacted, not caring about consequences. Snatching the ceremonial dagger from his hip, he grabbed the little girl by the collar and raised the blade. \"They called you Abomination!\"\n\nAlia fought like a rabid animal, but didn't scream. Her tiny feet drove with surprising power into his stomach, knocking the wind out of him. The Baron reeled, and without a second's hesitation, thrust the poisoned tip deep into her side. It went in easily. He yanked the knife back out and stabbed again, this time directly into Alia's heart.\n\nJessica screamed. Paul rushed forward, but too late. Duncan roared with anger and shock, and threw himself at the nearest Sardaukar guard, killing him with a bone-shattering blow to the throat. He struck a second guard, snapping his neck as well, and charged toward the Baron like a wild creature. The Baron didn't even have time to feel fear before his guards closed ranks around him, and four others held Duncan back. The rest of the faux Sardaukar raised their guns to keep the shocked captives at bay.\n\nRegaining his composure, the Baron sneered down at the little girl dying swiftly in his grip. \"That's turnabout for killing me.\" Laughing at the blood on his hands, he tossed her to the floor like a discarded doll. And inside, not a sound from his tormentor. Was she gone as well?\n\nMurderous desperation showed on the faces of the nearby captives, making the Baron uneasy. With Face Dancer Sardaukar surrounding him protectively, he backed away smiling. The two dead soldiers had reverted to Face Dancers, and none of the captives seemed the least bit surprised. The Atreides rabble gathered around the murdered child while the Sardaukar picked up their comrades.\n\nSheeana stopped Duncan from lunging forward in another suicidal attack. \"One death is enough, Duncan.\"\n\n\"No it's not. It is only a start.\" He controlled himself with a visible effort. \"But it will have to do for now.\"\n\nThe Baron laughed, and the Face Dancers hurried him away. When he looked at his escort, the shape-shifters showed disapproval at what he had done. \"What? I don't have to justify my reasons to you. At least that Abomination is gone now.\"\n\n_Gone, you say?_ A little girl's loud titter like breaking glass inside his skull. _Gone? You can't discard me so easily! I was rooted inside your head before that ghola was ever born._ The voice grew louder. _Now I shall torment you more than ever. You leave me no choice but to serve as your conscience, Grandfather._\n\nThe Baron marched away at a faster pace, trying to shut out her mocking presence.\n\n> _The stake in a total war is_ total _\u2014to conquer is to save everything, to succumb is to lose everything._\n> \n> \u2014a warrior of Old Terra\n\nWhile the thinking machines maintained a tight cordon around the no-ship, Sheeana watched Jessica carry little Alia's body away. How painful it must be for her. With her memories restored, Jessica knew intimately who Alia really was and understood her great potential. How bitterly ironic, too. St. Alia of the Knife\u2014felled by a knife.\n\nJessica cradled the limp child in her arms, shuddering as she fought to contain sobs. When she looked up at Sheeana, there was a cold deadliness in Jessica's eyes. Duncan stood beside Jessica, his face a mask of grim anger. \"We'll have our revenge, my Lady. So many of us despise the Baron, he can't survive for long.\" Even Yueh sat coiled and dangerous, like a loaded weapon.\n\nPaul and Chani clasped hands, drawing strength from each other. Leto II watched in silence, undoubtedly holding an avalanche of conflicting thoughts in his mind. The boy always seemed to have so much more to him, like a giant iceberg whose bulk was concealed beneath the surface. Sheeana had long suspected that he might be the most powerful of all the gholas she had created.\n\nJessica held her head high, finding strength within her. \"We'll take her to my quarters. Duncan, would you help me?\" Dr. Yueh, desperate for forgiveness, hovered close to them.\n\nFilled with anxiety, frustration, and anger, Sheeana watched the tableau. In addition to losing the Bashar, Alia had been murdered, while three key gholas\u2014Paul, Chani, and Leto II\u2014remained unawakened. Stilgar and Liet-Kynes were left on Qelso, and Thufir Hawat had been a Face Dancer. Now that they were facing the Enemy and needed the ghola children to fulfill their destinies, too many of her \"weapons\" were not available to her! She had only Yueh, and Jessica . . . and Scytale, if she could count on the Tleilaxu.\n\nExhaustion threatened to overwhelm Sheeana. They had fled for so long, carrying their plans and hopes, but never finding an end. This, though, was not at all what they had hoped for.\n\nThe quiet and distant voice of Serena Butler awakened within her again, angered by the revelation about the Enemy. She spoke from firsthand knowledge. _The evil machines have always wanted to exterminate humanity. They do not know how to forget._\n\n\"But they were destroyed,\" Sheeana said aloud.\n\n_Apparently not. Trillions of people died during the Butlerian Jihad, but even that was not enough. In the end,_ I _was not enough._\n\n\"I am pleased to meet you finally,\" said a raspy female voice. A lone old woman strolled down the no-ship's corridor, a broad grin on her wrinkled face. Despite her apparent age, she moved fluidly and had a deadly look to her.\n\nSheeana immediately guessed that this must be the mysterious old woman who was their relentless hunter. \"Duncan has told us about you.\"\n\nThe woman smiled in an unnerving manner, as if she could see through Sheeana to her innermost thoughts and intentions. \"You were quite a troublesome quarry. All those years wasted. Have you guessed my true identity yet?\"\n\n\"You are the Enemy.\"\n\nAbruptly the crone's face, body, and clothing rippled like molten, flowing metal. At first Sheeana thought this was another Face Dancer, but the head and body took on a sheen of highly polished platinum, and the matronly clothes became a plush robe. The face was smooth, with the same smile set in radically different features. A robot.\n\nDeep in her consciousness, Sheeana felt a tumult in Other Memory. And out of the clamor, Serena Butler's familiar voice rose to cry, _Erasmus! Destroy him!_\n\nWith great effort she shunted aside the voices in Other Memory, and said, \"You are Erasmus. The one who killed Serena Butler's child, setting off the centuries-long Jihad against thinking machines.\"\n\n\"So I _am_ still remembered, even after all this time.\" The robot sounded pleased.\n\n\"Serena remembers you, all right. She is within me, and she hates you.\"\n\nPure delight shone on the robot's face. \"Serena Butler herself is in there? Ah yes, I know about your Other Memory. Face Dancers have brought many of you Bene Gesserits back to us.\"\n\nInside her, the clamor of memories returned. \"I am Serena Butler, and she is me. Though thousands of years have passed, the pain is as sharp as ever. We cannot forget what you destroyed, and what you started.\"\n\n\"It was only one life\u2014merely a baby. Logically, can't you see how your race overreacted?\" The robot sounded so reasonable.\n\nSheeana felt a change in the tenor and cadence of her own voice, as if her body were being taken over by a force within. \"Only one life? Merely a baby?\" Serena was speaking now, thrusting herself to the forefront of the innumerable lives. Sheeana let her talk. After such a great length of time, this was _Serena's_ confrontation with her greatest nemesis. \"That _one_ life led to the military defeat of your entire Synchronized Empire. The Butlerian Jihad was a Kralizec in its own right. The end of that war changed the course of the universe.\"\n\nErasmus seemed delighted by the comparison. \"Ah, interesting. And perhaps the end of this Kralizec will reverse that result and put thinking machines in charge again. If so, we will be much more efficient this time.\"\n\n\"That is how you foresee the end of Kralizec?\"\n\n\"That would be my preference. Something fundamental must change. Can I count on you to assist me?\"\n\n\"Never.\" Serena's projected voice was cold and implacable.\n\nLooking at the independent robot, Sheeana understood more than ever before that she was part of something far greater and more important than one life, that she was linked to a vast continuum of female ancestry stretching into the past and\u2014hopefully\u2014into the future. A remarkable assemblage, but would it survive?\n\n\"There is a familiar fire in your eyes. If any part of you is indeed Serena Butler, then we must catch up on old times.\" Erasmus's optic threads gleamed.\n\n\"She no longer wishes to converse with you,\" Sheeana said in her own voice.\n\nErasmus ignored the rebuff. \"Take me to your private quarters. A human's den reveals much about the individual personality.\"\n\n\"I will not.\"\n\nThe robot's voice hardened. \"Be reasonable. Or should I decapitate a few of your fellow passengers to encourage your cooperation? Ask Serena Butler inside you\u2014she knows I will do it.\"\n\nSheeana glared at him.\n\nThe robot continued in a calm tone, \"But a simple conversation with you in your quarters may slake my appetite for now. Wouldn't you prefer that to carnage?\"\n\nMotioning for the others to remain behind, Sheeana turned her back on the robot and walked to one of the still-functional lifts. With gliding footsteps, Erasmus followed.\n\nIn her chamber, the robot was intrigued by the preserved Van Gogh painting. _Cottages at Cordeville_ was one of the oldest artifacts of human civilization. Standing rigidly, Erasmus admired the artwork. \"Ah, yes! I remember this clearly. I painted it myself.\"\n\n\"It is the work of a nineteenth-century Terran artist, Vincent Van Gogh.\"\n\n\"I have studied the Madman of France with great interest, but I assure you, this is actually one of the canvases I myself painted thousands of years ago. I copied the original with the utmost attention to detail.\"\n\nShe wondered if he could possibly be telling the truth.\n\nErasmus removed the delicate painting from the wall and examined it closely, passing his metal fingertips over the thin plaz that protected the rough oil-paint surface. \"Yes, well do I remember each stroke, each whorl, each point of color. Truly, this is a work of genius.\"\n\nSheeana caught her breath, knowing how old and priceless it was. Unless it really was a forgery perpetrated long ago. \"The original was a work of genius. If this is what you say, then all you did was copy someone else's masterpiece. There can be only one original.\"\n\nHis optic threads gleamed like a galaxy of stars. \"If it is the same, exactly the same, then both are works of genius. If my copy is perfect down to every single brushstroke, does it not become a second original?\"\n\n\"Van Gogh was a man of creativity and inspiration. You merely mimicked his work. You might as well call a Face Dancer a work of art.\"\n\nErasmus smiled. \"Some of them are.\"\n\nAbruptly, with powerful hands, the robot ripped the painting and its frame into tiny pieces. As if putting a punctuation mark on the grotesque display, Erasmus whirled and stomped on the broken pieces, saying, \"Call this artistic temperament.\" Moving to depart, he added, \"Omnius will summon your Kwisatz Haderach soon. We have waited a long time for this.\"\n\n> _What is the difference between data and memory? I intend to find out._\n> \n> \u2014ERASMUS,  \n> Laboratory Notebooks\n\nThe independent robot's memories of Serena were as fresh as if the events had occurred only days ago. Serena Butler . . . such a fascinating woman. And just as Erasmus had survived through the millennia as a package of data nearly destroyed and then recovered, so Serena's memories and personality lived on, somehow, in the Other Memories of the Bene Gesserit.\n\nThis posed an intriguing question: No Bene Gesserits could be Serena Butler's direct descendants, for Erasmus had killed her only child. Then again, he couldn't be sure what had happened to all of his experimental clones over the years. He had tried many times to bring Serena back, with no success.\n\nAboard this no-ship, however, the humans had grown gholas from their past, just as his own plan had brought back Baron Harkonnen and a version of Paul Atreides. Erasmus knew that a nullentropy tube hidden in a Tleilaxu Master had contained a wealth of ancient and carefully gathered cells.\n\nHe was confident that a real Tleilaxu Master could succeed in bringing Serena back, where his own primitive experiments had failed. Erasmus and Omnius had both absorbed enough Face Dancers to have instinctive reverence for the abilities of a Master. The independent robot knew exactly where he had to go before leaving the no-ship.\n\nErasmus found the medical center and the axlotl chambers where the whole library of historical cells had been catalogued and stored. If Serena Butler was among them . . .\n\nHe was surprised to find a Tleilaxu already there, harried and frantic. The diminutive man had disconnected the life-support systems of the axlotl tanks. With his olfactory sensors, Erasmus noted the smells of chemicals, melange precursors, and human flesh.\n\nHe grinned. \"You must be Scytale, the Tleilaxu Master! It's been a long time.\"\n\nScytale whirled, looking fearful at the sight of the robot.\n\nErasmus took a step closer, and studied the Tleilaxu's face. \"A child? What are you doing?\"\n\nThe Tleilaxu drew himself up. \"I am destroying the tanks and the melange they produce. I had to surrender that knowledge as a bargaining chip. I won't let thinking machines and traitorous Face Dancers simply take it from me\u2014from _us._ \"\n\nErasmus showed no concern for the sabotaged axlotl tanks. \"But you appear to be very young.\"\n\n\"I am a ghola. I have my memories back. I am everything that any of my previous incarnations were.\"\n\n\"Of course you are. Such a marvelous process, perpetuating yourself through serial ghola lives. We machines understand such things, although we have much more efficient methods of performing data transfers and backups.\" He looked intensely at the genetic library that held the potential ghola cells . . . Serena Butler . . .\n\nNoting the robot's keen interest, the Tleilaxu sprang to stand in front of the sealed wall of specimens. \"Beware! The witches placed security sensors on these gene samples to prevent anyone from tampering with or stealing them. The library has a built-in self-destruct system.\" He narrowed his dark, rodentlike eyes. If this Master was bluffing, he was doing a remarkably convincing job. \"I need only yank on a drawer, and this entire cabinet will be flooded with gamma radiation, enough to ionize every single sample.\"\n\n\"Why?\" The robot was perplexed. \"After the Bene Gesserit took those cells from you and used them for their own purposes? Didn't they force you to cooperate? Would you truly stand on their side?\" He extended a platinum hand. \"Join us instead. I would greatly reward you for your assistance in growing one particular ghola\u2014\"\n\nIn a threatening motion, Scytale placed his small hand on one of the many cell containers. Though trembling, he seemed entirely determined. \"Yes, I would stand with them. I shall always stand _against_ the thinking machines.\"\n\n\"Interesting. New enemies make unexpected alliances.\"\n\nThe Tleilaxu didn't move. \"In the final assessment, we're all humans\u2014and you are not.\"\n\nErasmus chuckled. \"And what about Face Dancers? They fall between, don't they? These aren't the shape-shifters you produced long ago, but are instead far superior biological machines that I helped create. And because of them, Omnius and I are, in effect, the greatest of all Face Dancers\u2014among many other things.\"\n\nScytale didn't move. \"Haven't you noticed the Face Dancers are no longer reliable?\"\n\n\"Ah, but they are reliable to me.\"\n\n\"Are you sure?\"\n\nThe robot took a tentative step forward, testing. Scytale tensed his fingers on the handle of the sample cabinet. Erasmus amplified his voice. \"Stop!\" He eased himself backward, giving the Tleilaxu Master more room. There would be plenty of time to return and test Scytale's loyalties. \"I leave you to this facility and your cellular samples.\"\n\nErasmus had waited more than fifteen thousand years for Serena, and could continue to do so. For now, the robot had to return to the machine cathedral and prepare for the final show. The evermind was not quite so patient to achieve his ultimate goals as Erasmus was.\n\n> _Come, let us eat and sing together. We will share a drink and laugh at our enemies._\n> \n> \u2014from an ancient ballad by  \n> GURNEY HALLECK\n\nThe computer evermind sent his troops to bring Paul from the _Ithaca_ to the machines' cathedral-like nexus. New-model robotic guards swarmed down the corridors like quicksilver insects. Approaching Paul, one of them said, \"Come with us to the primary cathedral.\"\n\nChani grabbed his arm and held on, as if she too had sprouted metal hands. \"I will not let you go, Usul.\"\n\nLooking at the inhuman escorts, he said to her, \"We can't keep them from taking me.\"\n\n\"Then I shall come with you.\" He tried to argue with her, but she cut him off. \"I am a Fremen woman. Would you try to stop me? You might just as easily fight these machines.\"\n\nConcealing a small smile, he faced the sleek machines that clicked and flittered in front of him. \"I will accompany you without resistance, but only if Chani comes with me.\"\n\nEmerging from her quarters where Alia's body now lay on the narrow bed, Jessica placed herself between Paul and the robots. Bloodstains still marked her shipsuit. \"He is my son. I have already lost a daughter today, and cannot bear to lose him as well. I'm going with you.\"\n\n\"We are here to escort Paul Atreides to the primary cathedral,\" one of the robots said, its freeform face flowing like heavy rain on a Caladan window. \"There are no other restrictions.\"\n\nPaul took that as agreement. For some reason, Omnius wanted _him_ , even though he did not have his memories back. All other passengers and crew were apparently extraneous baggage. Had he been the subject of the hunt from the beginning? How could that be? Had the thinking machines somehow known he would be aboard? Paul gripped Chani's hand and said to her, \"It will be over soon, in whatever manner fate decides. All along, our destinies have hurtled us toward this point, like levitating trains out of control.\"\n\n\"We will face it together, my love,\" Chani said. He only wished that he could recall all his years with her . . . and that she could do the same.\n\n\"What about Duncan?\" he asked. \"And Sheeana?\"\n\n\"We must depart now,\" the robots said in unison. \"Omnius waits.\"\n\n\"Duncan and Sheeana will know soon enough,\" Jessica said.\n\nBefore they left, Paul made a point of taking the crysknife Chani had made for him. Like a Fremen warrior, he wore it proudly at his waist. Although the worm-tooth blade would do nothing against the thinking machines, it made him feel more like the legendary Muad'Dib\u2014the man who defeated powerful empires. But in his mind he again saw the horrible recurring vision, the flicker of memory or prescience in which he lay on the floor in a strange place, mortally wounded\u2014looking up at a younger version of himself who laughed in triumph.\n\nHe blinked and sought to focus on reality, not possibilities or destiny. Following the insectile robots down the corridors, he tried to tell himself he was prepared to face whatever lay in store for him.\n\nBefore the gholas could emerge from the ship through the ragged hole the machines had made, Wellington Yueh tried to push his way past the ranks of escort robots. \"Wait! I want . . . I need to go with you.\" He fumbled for excuses. \"If someone gets hurt, I'm the best Suk doctor available. I can help.\" He lowered his voice and pleaded, \"The Baron will be there, and he'll want to see me.\"\n\nStill wrestling with her reinjured feelings toward him, Jessica sounded harsh and bitter. \"Help? Did you help Alia?\" Hearing this, Yueh looked as if she had slapped him.\n\n\"Let him come, Mother.\" Paul felt resigned. \"Dr. Yueh was a staunch childhood supporter and mentor to the original Paul. I won't turn down any ally or witness to whatever is about to occur.\"\n\nFollowing the robots, they emerged onto flowing roadways that carried them along like floating plates. Batlike fliers streaked high overhead, and mirrored watcheyes flitted about in the air, observing the group's progress from all angles. Behind them, the huge no-ship had been incorporated into the machine metropolis. Sentient metal buildings of freeform architecture had grown around the _Ithaca_ 's hull like coral swallowing up an old shipwreck beneath the seas of Caladan. The buildings seemed to alter whenever the evermind had a fleeting thought.\n\n\"This whole city is alive and thinking,\" Paul said. \"It's all one changeable, adapting machine.\"\n\nUnder her breath, his mother quoted, \"'Thou shalt not create a machine in the likeness of the human mind.' \"\n\nSpeakers appeared in the solid silver walls of the looming buildings, and a simulated voice mockingly repeated Jessica's words. \"'Thou shalt not create a machine in the likeness of the human mind.' What a quaint superstition!\" The laughter sounded as if it had been recorded from somewhere else, distorted freakishly, and then played back. \"I look forward to our encounter.\"\n\nThe escort robots brought them into an enormous structure with shimmering walls, curved arches, and enclosed parklike spaces. A spectacular lava fountain spouted plumes of hot, scarlet liquid into a tempered basin.\n\nIn the middle of the great cathedral hall, an elderly man and woman awaited them, dressed in loose, comfortable garments. Dwarfed by the enclosure, they certainly did not look menacing.\n\nPaul decided not to wait for their captors to play control games. \"Why have you brought me here? What do you want?\"\n\n\"I want to help the universe.\" The old man stepped down the polished stone stairs. \"We are in the endgame of Kralizec, a watershed that will change the universe forever. Everything that came before will end, and everything that comes in the future will be under my guidance.\"\n\nThe old woman explained. \"Consider all the chaos that has existed over the millennia of your human civilization. Such messy creatures you are! We thinking machines could have done a much neater, more efficient job. We have learned of your God Emperor Leto II, and the Scattering, and the Famine Times.\"\n\n\"At least he enforced peace for thirty-five hundred years,\" the old man added. \"He had the right idea.\"\n\n\"My grandson,\" Jessica said. \"They called him Tyrant because of the difficult decisions he made. But even he did not do as much damage as your thinking machines did during the Butlerian Jihad.\"\n\n\"You cast blame too loosely. Did _we_ cause the damage and destruction, or did humans like Serena Butler? That is a matter for debate.\" Abruptly the old woman cast off her disguise, like a reptile shedding its dry skin. The robot's flowmetal face\u2014male now\u2014displayed a wide smile. \"From the beginning, machines and humans have been at odds, but only we are able to observe the long span of history, and only we can understand what must be done and find a logical way to achieve it. Is that not a valid analysis of your legendary Kralizec?\"\n\n\"Only an interpretation,\" Jessica said.\n\n\"The correct one, though. Right now we are involved in the necessary business of uprooting weeds in a garden\u2014an apt metaphor. The weeds themselves do not appreciate it, and the dirt may be disturbed for some time, but in the end the garden is vastly improved. Machines and humans are but manifestations of a long-standing conflict that your ancient philosophers recorded, the battle between heart and mind.\"\n\nOmnius retained his old man form, since he had no other familiar physical manifestation. \"Back in the Old Empire, many of your people are trying to make their last stands against us. It is futile, for my Face Dancers have ensured that your weapons will not work. Even your navigation machines are under my control. Already my fleet approaches Chapterhouse.\"\n\n\"Our ship has had no contact with either the Guild or Chapterhouse since before I was born,\" Paul said in a dismissive tone. He pointed to Chani, Jessica, and Yueh, all of those gholas born on the ship in flight. \"None of us has ever been in the Old Empire.\"\n\n\"Then allow me to show you.\" With a wave of his hand, the old man displayed a complex holo-image of stars, indicating how far his immense fleet had progressed. Paul was stunned by the scope of the conquest and devastation; he didn't think the evermind would exaggerate what the machines had done. Omnius didn't need to. Hundreds of planets had already been destroyed or enslaved.\n\nIn a soothing voice, Erasmus said, \"Fortunately, the war will soon be over.\"\n\nThe old man approached Paul. \"And now that I have you, there is no question of the outcome. The mathematical projection states that the Kwisatz Haderach will change the battle at the end of the universe. Since I control you _and_ the other one, we will now finish this conflict.\"\n\nErasmus stepped forward to inspect Paul, like a scientist examining a valuable specimen. His optic threads glittered. \"We know you have the potential within your genes. The challenge lies in determining _which_ Paul Atreides will be the better Kwisatz Haderach.\"\n\n> _Optimism may be the greatest weapon humanity possesses. Without it, we would never attempt the impossible, which\u2014against all odds\u2014occasionally succeeds._\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> speech to the gathered Sisterhood\n\nWithout Obliterators or navigation control, the human warships lay like white-bellied victims on sacrificial altars, all across the last-stand line they had drawn.\n\nAboard her flagship, Mother Commander Murbella shouted orders, while Guild Administrator Gorus demanded miracles from his underlings. Watching the screens on the navigation bridge, Murbella saw thinking-machine battleships cruise past the Sisterhood's pitiful vessels on their way to destroy Chapterhouse. Similar non-battles must be occurring at the hundred flashpoints across the front lines, key human-inhabited systems now completely vulnerable to the coup de grace. The gamble had failed, utterly.\n\nWeighing heavily on Murbella's mind was her responsibility to humanity, the rest of the Sisterhood . . . and her long-lost Duncan. Was he still alive, and did he even remember her? It had been almost twenty-five years. Murbella had to do this\u2014for him, for herself, or for all those who had survived thus far in the epic war.\n\nWithout letting instinctive Honored Matre rage control her actions, Murbella whirled toward Gorus. She grabbed the front of the Administrator's loose robe and shook him so that his pale braid whipped his face. \"What other weapons do your Guildships have?\"\n\n\"A few projectiles, Mother Commander. Energy weapons. Standard offensive artillery\u2014but that would be suicide! Only the Obliterators could have made it possible for our ships to deal a mortal blow against the Enemy!\"\n\nIn disgust, she cast him away, so that he stumbled backward and fell to the deck. \"This is already a suicide mission! How dare you cringe now, when we have no alternative?\"\n\n\"But . . . but, Mother Commander\u2014it would waste our fleet, our lives!\"\n\n\"Obviously, heroism is not your strong suit.\" She turned to a meek-looking Guildsman and used the power of Bene Gesserit Voice on him for good measure. \"Prepare to launch our Obliterators. Blanket space with them. Maybe the saboteurs missed a few.\"\n\nThe Guildsman operated his weapons controls, barely bothering to choose targets. He launched ten more Obliterators, then another ten. None exploded, and the machine ships kept coming.\n\nHer voice low, Murbella said, \"Now fire all the standard projectiles we have. And once we deplete our conventional armaments, we'll use our ships as battering rams. Whatever we have.\"\n\n\"But why, Mother Commander?\" Gorus said. \"We should fall back and regroup. Plan some other way to fight. We must at least survive!\"\n\n\"If we don't win today, we don't survive anyway. We may be outnumbered, but we can still wipe out part of the thinking-machine fleet. I will not simply abandon Chapterhouse!\"\n\nGorus scrambled back to his feet. \"To what purpose, Mother Commander? The machines can just replace themselves.\"\n\nAs they spoke, more Obliterators filled space. So far, all had turned out to be duds.\n\n\"The purpose is to show them we can still fight. It is what makes us human, what gives us significance. History shall not record that we abandoned Chapterhouse and tried to hide from the final confrontation between humanity and the thinking machines.\"\n\n\"History? Who will be left to record history?\"\n\nWithin three minutes of each other, six small foldspace craft raced into the battle zone over Chapterhouse, reporting from the other clusters of ships. They transmitted urgent messages and demanded new orders from the Mother Commander. \"Our Obliterators don't function!\"\n\n\"All navigation systems have shut down.\"\n\n\"How do we fight them now, Mother Commander?\"\n\nShe answered in a strong, steady tone. \"We fight with everything we have.\"\n\nJust then, a fabulously bright flash blew across at least fifty of the Enemy vessels, vaporizing them in an expanding arc that sent a shudder through the decks of the more distant Guild vessels. Murbella gasped, then laughed. \"See! One of the Obliterators still worked! Fire the rest of them.\"\n\nTo her astonishment, space around them suddenly shimmered, cracked, and disgorged hundreds of giant ships. Not human defender vessels.\n\nAt first Murbella thought that the Enemy machines had sent yet another devastating fleet, but she quickly identified the cartouche on the curved hulls. Guild Heighliners! They spilled out of foldspace from every direction, surrounding the first massive wave of thinking-machine vessels.\n\n\"Administrator, why did you hold out on us?\" Murbella's voice was brittle. \"There must be a thousand ships here!\"\n\nGorus seemed just as astonished as she was.\n\nA female voice thrummed across the commlines that linked Murbella's defenders. \"I am the Oracle of Time, and I bring reinforcements. Mathematical compilers corrupted many Guild vessels, but my Navigators control these Heighliners.\"\n\n\"Navigators?\" The white-haired Administrator gasped in consternation. \"We thought they were all dead, starved for spice.\"\n\nThe Oracle spoke in a powerful, lilting tone. \"And my ships\u2014unlike those made by the traitorous fabricators of Ix\u2014command full armaments. Our Obliterators work as designed. We took them from old Honored Matre ships and hid them away for our own defenses. We intend to use them now.\"\n\nMurbella's face flushed. She had suspected the rebel Honored Matres had possessed many more Obliterators than were found. So, the Navigators had been hiding them all along!\n\nThe Omnius invasion fleet shifted position in response to the Navigator reinforcements, but the machines could not comprehend the magnitude of the astonishing opponent they now faced. They did not react in time as the Oracle's Heighliners spewed out dazzling sunbursts in a flurry of explosions like miniature supernovas. Each incinerating burst of light vaporized entire clusters of the overly complacent Enemy vessels.\n\nAlthough the machine forces scrambled to defend themselves, their response was ineffective, as if their control functions had been disconnected. The evermind had modeled his plan repeatedly, setting up contingency options for likely turns of events. But Omnius had not foreseen this.\n\n\"Thinking machines have long been my sworn enemies,\" the Oracle said in her ethereal voice.\n\nWhile Murbella looked on with great satisfaction, precisely targeted Obliterators wiped out countless Enemy ships. If only the Honored Matres had simply turned their stolen weapons against the thinking machines when they'd had the chance, long ago! But those women had never stood together against a common foe. Instead, they hoarded their stolen weapons and used the destructive power against each other, against rival planets. What a waste!\n\nThe overlapping detonations, each strong enough to scorch a planet, struck the front line of machine ships. A dozen Heighliners raced deeper into the Chapterhouse system to chase Enemy vessels that had already reached planetary orbit.\n\n\"We will do what we can at your other front-line planets,\" the Oracle said. \"Today we hurt the Enemy.\"\n\nAlmost before she could absorb what was happening around her, Murbella saw that the initial wave of thinking-machine forces had been reduced to nothing more than scattered debris. As far as she could tell, the Enemy battleships never got the chance to launch a single shot against the defenders of humanity.\n\nSome of the Heighliners winked out, folding space to go to the other crippled last-stand defenders. There, they would deliver their Obliterators and speed off to further encounters with the Enemy. All across the front lines, at every flashpoint where Murbella had placed her groups of fighters, the Oracle's Navigators struck, and vanished again. . . .\n\nMurbella snapped to Administrator Gorus. \"Get me a comchannel! How do we talk to your Oracle of Time?\"\n\nGorus was stunned by the events around him. \"One does not request an audience with the Oracle. No living person has ever initiated contact with her.\"\n\n\"She just saved our lives! Let me talk with her.\"\n\nWith a skeptical expression, the Administrator made a gesture toward another Guildsman. \"We can try, but I promise nothing.\"\n\nThe gray-robed man fiddled with the commline until Murbella shouldered him aside. \"Oracle of Time\u2014whoever you are! Let us join forces to eradicate the thinking machines.\"\n\nA long silence was Murbella's only response, not even static, and her heart sank. Gorus gave her a superior look, as if he had known to expect this all along. Murbella saw a second wave of thinking-machine ships race in, now that the initial attack had been thwarted. And these would not tauntingly hold their fire. \"More machines are coming\u2014\"\n\n\"For now, I must move on.\" As the Oracle spoke, Heighliners began to disappear like soap bubbles popping. \"My main battle is on Synchrony.\"\n\n\"Wait!\" Murbella cried. \"We need you!\"\n\n\"We are needed elsewhere. Kralizec will not be consummated here. At long last, I have found the no-ship that carries Duncan Idaho, and the secret location of Omnius. I must now go there to end this by destroying the evermind. Forever.\"\n\nMurbella reeled as the unexpected information hit her. The no-ship found? _Duncan was alive!_\n\nWithin moments the last of the Heighliners vanished into foldspace, leaving the Mother Commander and her ships alone to face the next wave. The thinking machines kept coming.\n\n> _We have our own goals and ambitions, for good or ill. But our true destiny is decided by forces over which we have no control._\n> \n> \u2014\"The Atreides Manifesto,\"  \n> first draft (section deleted by Bene Gesserit committee)\n\nA door in the machines' grand cathedral flowed open like a waterfall of metal, parting to reveal two figures that stepped forward in tandem.\n\nIt had been hours since Baron Vladimir Harkonnen had murdered Alia, yet his wide lips still struggled to contain his satisfaction. His spider-black eyes glinted. Dr. Yueh glared at the Baron, his personal b\u00eate no\u00efre.\n\nPaul did not need ghola memories to recognize the Baron's companion\u2014a lean young man, barely more than a boy, but whipcord strong with muscles tuned from constant training. The eyes were harder, the features sharper, but Paul knew the face that stared back at him from the mirror.\n\nBeside him Chani gave a strangled cry, but the sound changed to a growl in her throat. She recognized the younger Paul, and also saw the terrible difference.\n\nA cold sense of inexorability froze Paul's blood as everything became clear. His prescient vision in the flesh! So, the thinking machines had grown another ghola of Paul Atreides to be their pawn, a second potential Kwisatz Haderach for their private use. Now he understood the recurring dreams of his own face laughing triumphantly, consuming spice, the peculiar image of _himself_ stabbed and dying, bleeding out his life's blood on a strange floor. Just like the one on which he stood now, in this vaulted chamber.\n\n_It will be one of us. . . ._\n\n\"It seems we have an abundance of Atreides.\" The Baron ushered his prot\u00e9g\u00e9 forward, his hand clamped on the young man's shoulder. Almost apologetically, as if the wary audience cared, he said, \"We call this one Paolo.\"\n\nPaolo pulled away from him. \"Before long you will call me Emperor, or Kwisatz Haderach\u2014whichever term grants me the highest respect.\" Looking on, the old man and Erasmus seemed to find the whole tableau amusing.\n\nPaul wondered how many times he had been trapped by fate, by terrible purpose. How often and in how many circumstances had he seen himself dead from a knife thrust? Now he cursed the fact that he would face this crisis as a shell of his former self, not armed with the memories and skills of his past.\n\n_Unto myself, I must be sufficient._\n\nSnickering, the younger boy walked to where his counterpart stood stiffly at attention. Paul looked back at his mirror image without fear. Despite the age difference, they were approximately the same height, and as Paul looked into his doppelganger's eyes, he knew he must not underestimate this \"Paolo.\" The youth was a weapon as sure and deadly as the crysknife at Paul's waist.\n\nJessica and Chani moved protectively close to Paul, ready to strike. His mother, with her memories restored, was a full Reverend Mother. Chani, though she did not yet have her past life, had shown considerable fighting skills in earlier practice sessions, as if she still felt Fremen blood in her veins.\n\nPaolo's brow furrowed, his expression flickering for just a moment. Then he sneered at Jessica. \"Are you supposed to be my mother? The Lady Jessica! Well, you may be older than I am\u2014but that doesn't make you a real mother.\"\n\nJessica gave him a brief, shrewd appraisal. \"I know my family, regardless of the order in which they were reborn. And you are not one of them.\"\n\nPaolo crossed the chamber floor toward Chani, leering with exaggerated hauteur. \"And you . . . I know you, too. You were supposed to be the great love of my life, a Fremen girl so insignificant that history recorded little of her youth. Daughter of Liet-Kynes, weren't you? A complete nobody until you became the consort of the great Muad'Dib.\"\n\nPaul could feel her nails digging into his arm as she ignored the boy and talked to him instead. \"The Bashar's teachings were right, Usul. A ghola's worth is not intrinsic in its cells. The process can go horribly wrong\u2014as it clearly did with this young monster.\"\n\n\"It's more a matter of parenting,\" the Baron said. \"Imagine how the universe would be changed if the original Muad'Dib had received different instructions in the uses of power\u2014if I had raised him, as _I_ tried to do with the lovely boy, Feyd-Rautha.\"\n\n\"Enough of this,\" Omnius broke in. \"My machine battleships are even now clashing with\u2014or should I say annihilating?\u2014the pathetic remnants of human defenses. According to my last reports, the humans were making simultaneous stands across space. That will allow me to destroy them all at once and be done with it.\"\n\nErasmus nodded to the humans in the cathedral chamber. \"Within a few more centuries your own warring factions would have torn your race apart anyway.\"\n\nThe old man shot the independent robot an annoyed look. \"Now that I have the final Kwisatz Haderach here, all the conditions have been fulfilled. It is time to end this. There is no need to bother with grinding every inhabited world to dust.\" His lips quirked in a strange smile. \"Though that would be enjoyable as well.\"\n\nMusing, Erasmus looked from Paul to Paolo. \"Although genetically identical, you two have slightly different ages, memories, and experiences. Our Paolo is technically a clone, grown from blood cells preserved on a dagger. But this other Paul Atreides\u2014what is the origin of your cells? Where did the Tleilaxu find them?\"\n\n\"I don't know,\" Paul said. According to Duncan, the elderly man and woman had begun their merciless pursuit well before anyone had suggested the ghola project, before old Scytale revealed his nullentropy capsule. How could the evermind have known that Paul would reappear here? Had the machines rigged a complex game? Had the sentient machines developed an artificial but sophisticated form of _prescience_?\n\nErasmus made a humming sound. \"Even so, I believe you each have the potential to be the Kwisatz Haderach we need. But which of you will prove superior and achieve it?\"\n\n\"It's me.\" Paolo strutted around. \"We all know that.\" Obviously the younger boy had been raised with a belief in his role, so that his head was filled with confidence\u2014though it was a confidence born of true skill, not one arising from imagination.\n\n\"And how will that be determined?\" Jessica asked, looking at both Pauls, weighing them with her eyes.\n\nA side door flowed open near the fountain that sprayed molten metal, and a man in a black one-piece suit emerged carrying an ornate bloodwood box topped with a smaller wrapped package. He was gaunt, with bland features.\n\n\"Khrone, there you are! We have been waiting.\"\n\n\"I am here, Lord Omnius.\" The man glanced at the assemblage and then, either in surrender or a flash of independence, his unremarkable human features faded away to reveal him as a pale and sunken-eyed Face Dancer. Setting the box aside, he carefully unwrapped the translucent fabric of the small package to reveal a brownish-blue paste flecked with gold spangles.\n\n\"This is a concentrated and unusually potent form of spice.\" The Face Dancer rubbed his fingertips and lifted them to his inhuman nose, as if the smell pleased him. \"Harvested from a modified worm that grows in the oceans of Buzzell. It will not be long before the witches understand and begin their own operations there to capture the worms and extract the spice. At the moment, though, I hold the only sample of this ultraspice. Its sheer power should be sufficient to plunge the Kwisatz Haderach\u2014one of you\u2014into a perfect prescient trance. You will achieve powers that only prophecy could predict. You will see everything, know everything, and become the key to the culmination of Kralizec.\"\n\nErasmus spoke, sounding almost cheery. \"After observing how the human race has ruined things without us around to maintain order, the universe definitely needs changing.\" The robot picked up the bloodwood box and raised the finely etched lid. Inside lay an ornate, gold-hilted dagger, which he picked up with something like reverence. A smear of old blood remained on the blade.\n\nBehind Paul, his mother gasped. \"I know that dagger! It's as clear and fresh in my mind as if I just saw it. Emperor Shaddam himself presented it to Duke Leto as a gift, and years later at Shaddam's trial Leto gave it back to him.\"\n\n\"Oh, there is more than that.\" The Baron's eyes glittered. \"I believe the Emperor gave that same dagger to my beloved nephew Feyd-Rautha for his duel with your son. Unfortunately, Feyd didn't quite succeed in that battle.\"\n\n\"I love convoluted stories,\" Erasmus added. \"Later still, Hasimir Fenring stabbed Emperor Muad'Dib with it and nearly killed him. So you see, this dagger has a long and checkered past.\" He lifted it, letting the light of the cathedral chamber gleam off the blade. \"The perfect weapon to help us make our choice, don't you think?\"\n\nPaul drew the crysknife Chani had made for him from its sheath at his side. The hilt felt warm in his grip, the curved milky blade perfectly balanced. \"I have my own weapon.\"\n\nPaolo danced back warily, looking at the Baron, Omnius, and Erasmus, as if expecting them to leap to his aid. He snatched the gold-hilted dagger from the robot's hand and pointed the sharp tip at Paul.\n\n\"And what are they to do with these weapons?\" Jessica asked, though the answer was obvious to everyone.\n\nThe robot looked at her in surprise. \"It is only appropriate that we solve this problem in a particularly human way: a duel to the death, of course! Is that not perfect?\"\n\n> _The worm is outside for all to see, and the worm is within me, part of me. Beware, for I_ am _the worm. Beware!_\n> \n> \u2014LETO II,  \n> Dar-es-Balat recordings, in his voice\n\nAfter Paul and his companions were taken from the no-ship, Sheeana found young Leto II in his quarters. Huddling all alone in the dark, the youth was feverish and trembling. At first she thought he was terrified at having been left behind, but she soon realized he was genuinely sick.\n\nSeeing her, the boy forced himself to his feet. He swayed, and perspiration glistened on his brow. He looked pleadingly at her. \"Reverend Mother Sheeana! You're the only one\u2014the only one who knows the worms.\" His large, dark eyes flicked from side to side. \"Can you hear them? I can.\"\n\nShe frowned. \"Hear them? I don't\u2014\"\n\n\"The _sandworms!_ The worms in the hold. They're calling me, tunneling through my mind, tearing me up inside.\"\n\nRaising her hand for silence, she paused, deep in thought. All her life, Shaitan had understood her, but she had never received any actual messages from the creatures, even when she'd tried to become part of them.\n\nBut now, by extending her senses she did feel a tumultuous thrumming in her head and through the walls of the damaged no-ship. Since the _Ithaca_ 's capture, Sheeana had ascribed such feelings to the crushing weight of failure after their long flight. But now she began to understand. Something had been scraping through her subconscious, like dull fingernails raking across the slate of her fear. Subsonic pulses of invitation. _The sandworms._\n\n\"We have to go to the hold,\" Leto announced. \"They are calling. They . . . _I_ know what to do.\"\n\nSheeana gripped the boy's shoulders. \"What is it? What do we have to do?\"\n\nHe pointed to himself. \"Something of me is _inside_ the worms. Shai-Hulud is calling.\"\n\nWith the no-ship safely trapped in living-metal constructions, the thinking machines paid little attention to the vessel. Apparently, they had wanted to own and control the Kwisatz Haderach . . . a goal that was not as simple as it sounded, as the Sisterhood had learned long ago. Now that he had Paul Atreides in his machine cathedral, Omnius seemed to think he possessed everything he needed. The remaining passengers were irrelevant prisoners of war.\n\nThe Bene Gesserits had planned the creation of their superman over hundreds of generations, subtly guiding bloodlines and breeding maps to produce the long-anticipated messiah. But after Paul Muad'Dib turned against them and created havoc in their carefully ordered timeline, the Sisters had vowed never to unleash another Kwisatz Haderach. But in the long-ago aftermath, Muad'Dib's twin children had been born before the damage could be fully understood. One of those twins, Leto II, had been a Kwisatz Haderach, like his father.\n\nA key turned in Sheeana's mind, unlocking other thoughts. Perhaps in the solemn twelve-year-old Leto, the thinking machines had a blind spot! Could _he_ be the final Kwisatz Haderach they sought? Had Omnius even considered the possibility that the machines might have the wrong one? Her pulse quickened. Prophecies were notorious for misdirection. Maybe Erasmus had missed the obvious! She could hear the inner voice of Serena Butler laughing at the possibility, and she allowed herself to cling to a tiny kernel of hope.\n\n\"Let's go to the cargo hold, then.\" Sheeana took the boy's hand, and they hurried down the corridors and dropchutes to the lower levels.\n\nAs they approached the great doors, Sheeana heard explosive thunder from the other side. The frenzied worms charged from one end of the kilometer-long space to the other, smashing into the walls.\n\nBy the time they arrived at the access door, young Leto seemed ready to collapse. \"We have to go in,\" he said, his face flushed. \"The worms . . . I need to talk with them, calm them.\"\n\nSheeana, who had never been afraid of the sandworms before, now hesitated, worried that in their wild state they might not grant safety to her or Leto. But the boy worked the controls, and the sealed door slid aside. Hot, dry air blew onto their faces. Leto waded out and up to his knees in the soft dunes, and Sheeana hurried after him.\n\nWhen Leto raised his arms and shouted, all seven worms charged toward him like snorting predators, with the largest one\u2014Monarch\u2014at the fore. Sheeana could feel the hot wave of their anger, their need for destruction . . . but something told her that rage was not directed at either of them. The creatures rose up from the sand and towered over the two humans.\n\n\"The thinking machines are outside the ship,\" Sheeana said to Leto. \"Will the worms . . . will they fight for us?\"\n\nThe boy looked forlorn. \"They will follow my path if I lay it out for them, but I can't see it yet myself!\"\n\nLooking at him, she wondered again if this boy could be the ultimate Kwisatz Haderach, the link in the chain that Omnius had overlooked. What if Paul Atreides was no more than a feint in the final duel between man and machine?\n\nLeto shook himself, visibly bolstering his determination. \"But the prior _me_ , the God Emperor, had tremendous prescience. Maybe he foresaw this as well and prepared the beasts. I . . . trust them.\"\n\nAt this, the worms dipped in unison, as if bowing. Leto swayed, and they swayed with him. For a moment the walls of the hold seemed to recede, and the sand dunes flowed out to eternity. The ceiling disappeared in a vertiginous haze of dust. Suddenly, everything snapped back into focus.\n\nLeto caught his breath and called out, \"The Golden Path is coming to meet _me_! It is time to release the worms\u2014here, and now.\"\n\nSheeana sensed the rightness of this and knew what to do. All systems were still programmed to obey her instructions. \"The machines deactivated the weapons and engines, but I can still open the great cargo doors.\"\n\nShe and Leto hurried to the controls in the hall, where she input the commands. Machinery hummed and strained. Then, with a loud clank and a bang, a gap appeared in the long-sealed walls. From the corridor, Sheeana and the boy watched the immense lower doors slide open, like clenched teeth being pried apart.\n\nTons of sand spilled out in a rushing stream and propelled the sandworms, like living battering rams, into the streets of the machine capital.\n\n> _Prescience reveals no absolutes, only possibilities. The surest way to know exactly what the future holds is to experience it in real time._\n> \n> \u2014from \"Conversations with Muad'Dib\" by the  \n> PRINCESS IRULAN\n\n\"A duel makes no sense.\" The Baron frowned as he looked around the cathedral chamber. \"It is wasteful. Naturally, I am convinced my dear Paolo will defeat this upstart, but why not keep both Kwisatz Haderachs for yourself, Omnius?\"\n\n\"I desire only the best one,\" the evermind said.\n\n\"And we could not be certain of controlling _two_ of them as they struggled for preeminence with their new powers,\" Erasmus said.\n\n\"Whichever of you wins the duel will receive the ultraspice,\" Omnius announced. \"When the winner consumes it, I will have my true and final Kwisatz Haderach. I can then conclude this wasteful nonsense and begin my real work of remaking the universe.\"\n\nChani kept one hand on Paul's arm. \"How do you know either of them is your Kwisatz Haderach?\"\n\n\"You could be delusional,\" Yueh said, and the boy Paolo shot him a glare.\n\n\"And why should I cooperate if I win?\" Paul said, but the sickening echoes of recurring visions strangled his protest. He thought he knew what was going to happen, or some piece of it.\n\n\"Because we have faith.\" The Baron, a paragon of unholiness, laughed at his own joke, but no one else did.\n\nPaolo drew designs in the air with the tip of his gold-hilted knife. \"I have the Emperor's dagger! You were stabbed with it once.\"\n\n\"That won't happen again. This is my day of triumph.\" But Paul heard the brittleness in his measured words, the vulnerability behind the bravado. He could see no way to avoid the duel, and wasn't sure he wanted to. In his mind, he drove back the troubling flashes of vision. This perverse version of himself needed to be cut out like a cancer.\n\nThe time had come. Paul gathered all of his concentration for the fight. Hardly seeing Chani, he kissed her. The wormtooth dagger she had made felt perfectly balanced in his hand. He had practiced with the crysknife on the no-ship, and he knew how to fight.\n\n_I must not fear. Fear is the mind-killer._\n\nYoung Paolo pressed his lips together in a tight smile. \"I can tell you've had the visions, too! See, we are alike in yet another aspect.\"\n\n\"I've had many visions.\" _I will face my fear._\n\n\"Not like these.\" His opponent's knowing smile was maddening, unnerving. . . . Paul stiffened his resolve. He would not give Paolo the satisfaction of showing dread or uncertainty.\n\nQuicksilver robots appeared and removed the human observers to the sidelines of the expansive hall. The Baron stepped back beside Khrone, his gaze flicking back and forth between young Paolo and the tempting dose of ultraspice. He licked his thick lips hungrily, as if wishing he could try some for himself.\n\nOn the smooth combat floor of the chamber, Paul stood poised a couple of meters from Paolo. His younger foe tossed the gold-hilted dagger from hand to hand and smiled at him, showing white teeth.\n\nCalming himself, Paul summoned all the important lessons he had learned: Bene Gesserit attitudes and prana-bindu instruction, the precise muscular training and rigorous attack exercises that Duncan and the Bashar had drilled into all of the ghola children.\n\nHe spoke to his fear: _I will permit it to pass over me and through me._\n\nIt would all culminate here. Paul felt confident that if he rose to the challenge and won, his Kwisatz Haderach powers would surface, and he would be able to go on to defeat the thinking machines. But if Paolo won . . . He didn't want to consider that possibility.\n\n\"Usul, remember your time among the Fremen,\" Chani called from the side of the hall. \"Remember how they taught you to fight!\"\n\n\"He remembers none of it, bitch!\" Paolo slashed the Emperor's knife across the air, as if slitting an invisible throat. \"But I am fully trained, a tempered fighting machine.\"\n\nThe Baron applauded, but only a little. \"No one likes a braggart, Paolo . . . unless, of course, you succeed and prove to everyone that you were merely stating facts.\"\n\nPaul refused to be controlled by his visions. _If I am the Kwisatz Haderach, I'll change the visions. I shall fight. I shall be everywhere at once._\n\nYoung Paolo must have been thinking the same thing, for he lunged like a viper. Startled by the abrupt beginning of the duel, Erasmus swept his plush robes aside and stepped quickly out of the way. Apparently he had intended to delineate the rules of the challenge, but Paolo wanted to make it a brawl.\n\nPaul bent backward like a reed and let the Emperor's blade whistle past, within a centimeter of his neck. Young Paolo snickered. \"That was just practice!\" He held up the dagger, showing the rust-red stains. \"I am one step ahead of you, for this knife is already blooded!\"\n\n\"It's more your blood than mine,\" Paul said under his breath. He drove forward with the crysknife, weaving, making the blade dance.\n\nThe younger ghola responded by mirroring Paul's movements, as if the pair had an unconscious telepathic connection. He stabbed to the side, and Paul flowed in the other direction. Was this a form of prescience, Paul wondered, subconsciously foreseeing each blow, or did the two of them know and reproduce each other's fighting styles exactly? They had entirely different training, entirely different upbringings. But still . . .\n\nConcentrating on the duel, Paul's hearing became a fuzz of static. At first he heard encouragement, gasps, shouts of concern from his mother and Chani, but he blocked everything out. Did he have the potential to become the ultimate Kwisatz Haderach that Omnius was searching for? Did he want to be? He had read the histories, knew the bloodshed and suffering that Paul Muad'Dib and Leto II had both caused as Kwisatz Haderachs. What would the machines try to accomplish by possessing an even stronger Kwisatz Haderach? Some locked-away part of Paul already had the ability to look where no one else could\u2014into both feminine and masculine pasts. _What other powers lie untapped within me? Do I dare find out? If I win this duel, what will the thinking machines demand of me afterward?_\n\nHe felt like a gladiator on ancient Terra having to prove himself in an arena. And he had a fatal weakness: Omnius held Chani, Jessica, Duncan, and so many others as hostages. If Paul got his ghola memories back, his feelings for them would be even stronger.\n\nObviously, that was how Omnius intended to force Paul to cooperate, if he won this duel. His love for his companions would only intensify, and they would suffer because of him. Since the computer evermind had far more patience than any human, the machines could torture and kill these hostages with impunity, take cell scrapings and grow new gholas. Over and over! Perhaps Erasmus would bring back his sister Alia, his father the Duke, or Gurney, or Thufir. Kill them, resurrect them, and kill them again. Unless Paul Atreides, the Kwisatz Haderach, bowed to their demands, the thinking machines would make his life an unending hell. Or so they intended.\n\nNow he understood the dilemma of his destiny. And again he saw himself dying in a pool of blood. Perhaps some things could not be changed. But if he was a true Kwisatz Haderach, he should be able to defeat such petty tactics.\n\nHe fought on with wild passion, driving himself into a sweaty frenzy. Paolo kicked at him with his feet and slashed with the Emperor's dagger. Paul dove, rolled, and the younger ghola pounced on him. The Emperor's blade drove down hard in what would have been a killing thrust, but Paul slipped to the side, barely in time.\n\nThe blade slashed his sleeve, cut a thin line of blood on his left shoulder, then clanged against the stone floor. Paolo, his wrist jolted by the sharp impact, barely maintained a grip on the hilt.\n\nOn the polished floor, Paul swept his feet sideways, getting them under his rival and kicking upward. He did have the advantage of being physically stronger than a twelve-year-old. Paul grabbed his counterpart's wrist and pulled himself to his feet, but Paolo locked his fingers around Paul's knife arm, preventing the crysknife from stabbing down. Paul pushed, using his leverage to maneuver them both back toward the shimmering lava fountain.\n\n\"Not very . . . innovative!\" The younger ghola's breath rasped as he struggled, and Paul continued to drive him back. The heat from the fountain gushed in all directions. If he knocked Paolo into the incandescent metal, would he be killing himself\u2014or saving himself?\n\nPaul saw his opponent clearly and could not hate the other ghola. At the core, both of them were _Paul Atreides._ Paolo was not innately evil, but had been corrupted by terrible things that had been done to him, things he had been taught, not things done of his own volition. Paul did not let his sympathy for his rival weaken him. If he did, Paolo would not hesitate to kill him and claim victory. But _Paul_ \u2014because he was Paul\u2014would fight with every ounce of his being to save the future of humanity.\n\nOmnius and Erasmus observed without cheering for either fighter. They would accept whichever one proved victorious. Khrone's shadowed, olive-pit eyes held no emotion at all. The Baron was scowling. Paul didn't want to look toward Chani or his mother.\n\nThe roaring lava fountain dumped heat into the air. Paul's already sweaty body became slicker. Wiry Paolo used that to his advantage, squirmed, and Paul's grip began to slip. Suddenly, at the very verge of the fountain, the younger man allowed his knees to buckle.\n\nPaul overcompensated, which threw him off balance. He kneed his opponent in the stomach, but young Paolo had untapped reserves. When Paul raised the crysknife in his sweat-slickened grip, Paolo brought his own hand up backward, using the jeweled hilt of his dagger to smash the base of Paul's knife hand. Tendons twitched in reflexive reaction. The crysknife dropped free, clattered on the edge of the fountain, and tumbled into the molten pool.\n\n_Gone._\n\nWith the force of dominating vision, stronger than just the knowledge of his own death, Paul realized what he should have known from the beginning: _I am_ not _the Kwisatz Haderach that Omnius wants. It isn't me!_\n\nTime seemed to slow down and freeze. Was this what Bashar Teg had experienced when he accelerated himself? But Paul Atreides could move no faster than the events around him. They held him captive and squeezed in on him like the steely embrace of Death.\n\nWearing a venomous grin, young Paolo swung the gold-hilted dagger around in a perfect arc and, with exquisite slowness, drove the point into Paul's side. He slipped the dagger between his opponent's ribs and kept pushing, shoving the deadly point through Paul's lung and up into his heart.\n\nThen Paolo yanked the murderous weapon free, and time resumed its normal speed. From far away, Paul heard Chani screaming.\n\nBlood gushed from his wound, and Paul stumbled against the base of the hot fountain. It was a mortal wound; there could be no denying it. The prescient voice in his head hammered at him to no purpose. It seemed to be mocking him. _I am not the final Kwisatz Haderach!_\n\nHe slithered to the floor like a broken doll, barely saw Chani and Jessica running toward him. Jessica had Yueh by the collar and was dragging the Suk doctor over to her bleeding son.\n\nPaul had never known that one body could contain so much blood. With fading vision, he looked up and saw Paolo prancing victoriously, holding the dripping red dagger. \"You knew I would kill you! You might as well have driven in the knife with your own hands!\"\n\nIt was a perfect reproduction of his visions. He lay on the floor, dying as swiftly as his body would allow.\n\nIn the background he heard the Baron Harkonnen's boisterous laughter. The sound was intolerable, but Paul could do nothing to stop it.\n\n> _When they pour in at once, my memories will be like a sandstorm\u2014and just as destructive. Who can control the wind? If I am truly the God Emperor, then_ I _can control it._\n> \n> \u2014GHOLA OF LETO II,  \n> last preparatory assignment delivered to Bashar Miles Teg\n\nSand and worms poured out of the entrapped no-ship's hold into the carefully ordered machine metropolis. The writhing creatures plowed into the open streets like maddened Salusan bulls bursting from their pens. Beside Leto, watching the hold empty in a deafening rush, Sheeana opened her mouth, and her eyes went wide with surprise.\n\nThrough his strange connection to the worms, Leto II's mind surged outward with them into the sparkling city. Standing high above at the doorway to the immense cargo bay, he felt a wave of relief and freedom. Without a word to Sheeana, he dove into the sliding, flowing sand, following the worms in their wild exodus. He let the grit carry him, like a swimmer caught in an undertow being rapidly whisked out to sea.\n\n\"Leto! What are you doing? Stop!\"\n\nHe could not have stopped to answer even if he had wanted to. The current of flowing powder sucked him downward\u2014exactly where he wanted to be. Leto plunged under the sand, and his lungs somehow adapted to the dust, as did all of his senses. Like a sandworm he saw without eyes, and perceived the creatures ahead of him, as if he were looking at them through clear water. _This_ was what he had been born to do, what he had died to do, ever so long ago.\n\nMemories reverberated in him like echoes of the past\u2014not a visceral recollection, but greater than the knowledge he had acquired by reading the _Ithaca_ 's archives. Those entries had been about another young man, another Leto II, but still himself. A thought surfaced: _My skin is not my own._ In those days, his body had been covered with interlinked sandtrout, their membranous bodies meshing with his soft flesh and nerves. They had imparted strength to him, enabling him to run like the wind.\n\nThough still in human form, Leto II recalled some of the fantastic power, not from ghola memories but from the pearl of awareness that the original God Emperor had left within each worm descendant. _They_ remembered, and Leto remembered with them.\n\nHistories had been written by so many people who loathed him, who misunderstood what he had been forced to do. They decried the Tyrant's purported cruelty and inhumanity, his willingness to sacrifice everything for the extraordinary Golden Path. But none of the histories\u2014not even his own testamentary journals\u2014had recorded the joy and exuberance of a young man experiencing such unexpected and wondrous power. Leto remembered it all now.\n\nHe swam through the flow of sand to where the seven giant worms writhed, and then burst upward to break through the surface. Knowing instinctively what he had to do, Leto staggered toward the largest worm, Monarch. He caught the small part of the thrashing tail, leapt onto the hard rings, and scrambled up like a bare-footed Caladanian primitive scaling a rough-barked palm tree.\n\nAs soon as Leto touched the greatest of the seven worms, his fingers and feet seemed to acquire an unnatural adhesion. He could climb and hold on, as if he were part of the creature. And in a way, that was true. Fundamentally, he and the sandworms were one.\n\nSensing that Leto had joined them, all of the worms paused like enormous soldiers coming to attention. Reaching a perch atop Monarch's curved head, Leto surveyed the sprawling complex of living metal structures, and smelled the strong odor of cinnamon.\n\nFrom his high vantage point, he watched the city of Synchrony as it shifted buildings into formidable barricades, trying to impede the long-confined sandworms. This was Leto's army, his living battering rams\u2014and he would turn them loose against humanity's Enemy.\n\nDizzy and euphoric in the redolence of spice, Leto held onto the worm's ridges, which parted to expose the soft pink flesh underneath. He found it enticing, and his body longed for the full sensation, direct contact. Leto slid his bare hands between the ring segments, into the soft tissue membrane. There, he felt as if he was touching the nerve center of the beast itself, plunging his fingers into the neural circuitry that joined these primal creatures together. The sensation hit him like a jolt of electricity. This was where he had belonged for eternity.\n\nAt his behest, the sandworms reared higher, like angry cobras no longer interested in the soothing music of a snake charmer. Leto controlled them now. All seven of the worms went on a rampage through the machine streets, and Omnius could do nothing to stop them.\n\nWhen Leto's mind merged with the largest sandworm, he felt a flood of intense sensations and recalled a similar thing that another Leto II had done thousands of years before. Again he experienced the raspy feel of fast-flowing sand beneath a long and sinuous body. He relished the exquisite dryness of old Arrakis, and knew what it had meant to be the God Emperor, the synthesis of man and sandworm. That had been the zenith of his experience. But did something even greater lie in store for him?\n\nAs a ghola child raised aboard the no-ship, Leto II was never entirely sure how the Tleilaxu had obtained his original cells. Had they been taken when he and Ghanima underwent routine medical inspections as children? If so, an awakened Leto ghola would have only the memories of a normal child, the son of Muad'Dib. What if, however, the cells in Scytale's nullentropy capsule had been stolen from the actual God Emperor in his prime? Some unlikely scraping of his enormous vermiform corpse? Or a tissue sampling by one of the devout followers who had taken the Tyrant's withered, drowned body from the bank of the Idaho River?\n\nAs Leto's mind fused with Monarch, and all of the surrounding worms, he realized that it didn't matter. This incredible joining now unlocked everything that was within his ghola body and within each nugget of awareness buried deep in the sandworms. Leto II finally became his true self again, as well as the conflicted ghola boy he had been\u2014a loner child and an absolute emperor with the blood of trillions on his conscience. He understood in exquisite detail all his centuries of decisions, his terrible grief, and his determination.\n\n_They call me Tyrant without comprehending my kindness, the great purpose behind my actions! They don't know that I foresaw the final conflict all along._\n\nIn those last years, God Emperor Leto had strayed so far from humanity that he had forgotten innumerable marvels, especially the softening influence of love. But, as he rode Monarch now, young Leto remembered how much he had adored his twin sister Ghanima, the good times they had shared in their father's incredible palace, and how they had been slated to rule the vast empire of Muad'Dib.\n\nNow Leto was everything he had ever been and more, enhanced by the firsthand memories of his own experiences. With his new vision, as fresh precursors of spice from the worm's body pumped through his blood, he beheld the Golden Path extending gloriously before him. But even with this remarkable revelation, he could not quite see around all the corners ahead. There were blind spots.\n\nHigh atop his worm, young Leto smiled in determination, and with a single thought he sent the serpentine army forward. The leviathans charged between the great buildings, throwing themselves against reinforced barricades and breaking through. Nothing could stop them.\n\nHands still buried deep between the ring segments, Leto II rode with a shout of joy on his lips. He gazed forward through eyes that had suddenly become blue-within-blue, eyes that saw what others could not.\n\n> _Now that I have ridden one of the sandworms and touched the immensity of its existence, I understand the awe the ancient Fremen experienced, why they considered the worms to be their god, Shai-Hulud._\n> \n> \u2014TLEILAXU MASTER WAFF,  \n> letter to the Council of Masters in Bandalong, dispatched  \n> immediately before the destruction of Rakis\n\nThe last pair of Waff's sandworm specimens died inside the arid terrarium.\n\nWhen freeing the first test worms out in the desert, he had kept two with him at the modular laboratory for research, hoping that what he learned would improve their chances of survival. It did not go well.\n\nWaff prayed vigorously each day, meditated on the holy texts he had brought with him, and sought guidance from God on how best to nurture the reborn Prophet. The first eight specimens were now loose, tunneling through the brittle, crusted sand like explorers on a dead world. The Tleilaxu Master hoped they had survived in the blast-zone environment.\n\nIn their final days, the last two little worms in his laboratory aquarium became sluggish, unable to process the nutrients he gave them, though the food was chemically balanced to provide the sandworms with what they needed. He wondered if the small creatures could experience despair. When they lifted their round heads above the sandy surface of the holding tank, it seemed as if they had lost their will to live.\n\nAnd within a week they both perished.\n\nThough he revered these creatures and what they represented, Waff was desperate for vital scientific information with which to better the other worms' chances of survival. Once the specimens were dead, he had little compunction against picking apart their carcasses, spreading their rings, and cutting into the internal organs. God would understand. If he himself lived long enough, Waff would begin the next phase, as soon as Edrik came back for him. _If_ the Navigator ever came back, with his Heighliner and the sophisticated laboratory facilities aboard.\n\nHis own Guild assistants offered their help\u2014persistently\u2014but Waff preferred to work alone. Now that these men had set up their standalone camp, the Tleilaxu Master had no further use for them. As far as he was concerned, the Guildsmen were free to join Guriff and his treasure hunters in seeking lost spice hoards out in the wasteland.\n\nWhen one of the bland Guildsmen appeared before him, demanding his attention, Waff easily lost the delicate balance of his thoughts. \"What? What is it?\"\n\n\"The Heighliner should have returned by now. Something is wrong. Guild Navigators are never late.\"\n\n\"He did not promise to come back. When is Guriff's next CHOAM ship due to arrive? You are welcome to depart on it.\" _In fact, I encourage that._\n\n\"The Navigator may not be concerned with you, Tleilaxu, but he made promises to _us._ \"\n\nWaff didn't care about the insult. \"Then he will return, eventually. If nothing else, he will want to know how my new sandworms are doing.\"\n\nThe Guild assistant frowned at the flayed creature spread out on the analytical table. \"Your pets do not appear to be thriving.\"\n\n\"Today I will go out and monitor the specimens I released earlier. I expect to find them healthy, and stronger than ever.\"\n\nWhen the flustered Guildsman left, Waff changed into external protective clothing and hopped into the camp's groundcar. The locator signals showed him that the released worms had not ranged far from the ruins of Sietch Tabr. Attempting to be optimistic, he assumed they had found a habitable subterranean band and were establishing their new domain. As more and more worms grew on Rakis, they would become tillers of the soil, restoring the desert to its former glory. Sandworms, sandtrout, sandplankton, melange. The great ecological cycle would be reestablished.\n\nReciting ritual prayers, Waff drove across the eerie black-glass desert. His muscles trembled and his bones ached. Like the assembly lines in a war-damaged factory, his degenerating organs labored to keep him alive. Waff's flawed body could fall apart any day now, but he was not afraid. He had died already\u2014many times, in fact.\n\nAlways before, he'd had the faith and confidence that a new ghola was being grown for him. This time, though convinced he would not return to life, Waff was content with what he had accomplished. His legacy. The evil Honored Matres had tried to exterminate God's Messenger on Rakis, and Waff would bring Him back. What greater accomplishment could a man hope to attain in this life? In any number of lives?\n\nFollowing the tracker signals, he drove away from the weathered mountains, to the dunes. Ah, the new sandworms must have fled out into open terrain, looking for fresh sand in which to bury themselves and begin their lives anew!\n\nInstead, what he saw horrified him.\n\nHe easily located the eight fledgling worms. Much too easily. Waff stopped the groundcar and scrambled out. The hot, thin air made him gasp, and his lungs and throat burned. He could barely see through stinging tears as he hurried forward.\n\nHis precious sandworms lay on the hard ground, barely moving. They had cracked the melted crust of the dunes and churned through the soft grainy dirt beneath, only to emerge again. And now they lay dying.\n\nWaff knelt beside one of the weak, failing creatures. It was flaccid, grayish, barely twitching. Another had heaved itself high enough to sprawl across a broken rock, and there it lay deflated and unable to move. Waff touched it, pressed down on the hard rings. The worm hissed and flinched.\n\n\"You cannot die! You are the Prophet, and this is Rakis, your home, your holy sanctuary. You must live!\" His body was wracked by a spasm of pain, as if his own life was tied to those of the sandworms. \"You can't perish, not again!\"\n\nBut it seemed that the crippling damage to this world was simply too much for the worms. If even the great Prophet Himself could not endure, then these must assuredly be the End Times.\n\nHe had heard of it in ancient prophecies: Kralizec, the great battle at the end of the universe. The crux point that would change everything. Without God's Messenger, surely humanity would be lost. The final days were at hand.\n\nWaff pressed his forehead against the dusty, dying creature's yielding surface. He had done everything he could. Maybe Rakis would never again support the behemoth worms. Maybe this was indeed the end.\n\nFrom what he saw with his own eyes, he could not deny that the Prophet had truly fallen.\n\n> _People strive to achieve perfection\u2014ostensibly an honorable goal\u2014but complete perfection is dangerous. To be imperfect, but_ human, _is far preferable_.\n> \n> \u2014MOTHER SUPERIOR DARWI ODRADE,  \n> defense before Bene Gesserit Council\n\nWhen the older and inferior Paul Atreides ghola lay dying on the floor, Paolo turned away, pleased with his victory but far more interested in his other priority. He had proved himself to Omnius and Erasmus. The special ultraspice that would unlock all of his prescient abilities was his now. It would elevate him to the next level, to his exalted destiny\u2014as the Baron had taught him for so long. During that time, Paolo had convinced himself that this was what he wanted, brushing aside any nagging qualms or reservations.\n\nAround the cathedral hall, quicksilver robots stood at attention, ready to attack the remaining humans should Omnius give the order. Maybe Paolo himself would decide to issue such a command, once he was in control. He could hear the pleased laughter of the Baron, the sobs of Chani and Lady Jessica. Paolo wasn't sure which sounds he enjoyed more. His greatest thrill was the clear proof of what he had always known: _I am the one!_\n\nHe was the one who would change the course of the universe and control the end of Kralizec, guiding the next age of humanity and machines. Did even the evermind know what he was about to face? Paolo allowed himself a secretive, amused smile; he would never be a mere puppet for thinking machines. Omnius would soon learn what the Bene Gesserit had long ago discovered: A Kwisatz Haderach is not to be manipulated!\n\nPaolo slipped the bloody dagger into his waistband, strode over to the Face Dancer, and held out a hand to collect the spoils of combat. \"That spice is mine.\"\n\nKhrone smiled faintly. \"As you wish.\" He extended the cinnamony paste. Not interested in savoring it, Paolo quickly consumed a whole messy mouthful, far more than he should have. He wanted what it would unlock within him, and he wanted it right away. The taste was bitter, potent, and powerful. Before the Face Dancer could withdraw the offering, Paolo grabbed more and swallowed another mouthful.\n\n\"Not so much, boy!\" the Baron said. \"Don't be a glutton.\"\n\n\"Who are you to talk about gluttony?\" Paolo's retort drew a rumbling chuckle in response.\n\nOn the floor where he lay dying, Paul Atreides moaned. Chani looked up in despair from beside her beloved, her fingers dripping with his blood. Her face a grief-stricken mask, Jessica held her son's clenching hand. Paolo trembled. Why was it taking Paul so long just to die? He should have killed his rival more cleanly.\n\nKneeling over him, Dr. Yueh worked feverishly to save Paul, trying to stanch the flow of blood, but the Suk doctor's deeply troubled face told the terrible story. Even his advanced medical training was insufficient. Paolo's knife strike had done all the damage it had needed to.\n\nThose people were all irrelevant now. Mere seconds had passed when Paolo felt the potent melange burst into his bloodstream like a lasgun blast. His thoughts came faster, sharper. It was working! His mind was suffused with a certainty that outsiders might have considered hubris or megalomania. But Paolo knew it only as _Truth_.\n\nHe drew himself taller, as if he were growing physically and maturing in all ways, so that he loomed above everyone else in the chamber. His mind expanded into the cosmos. Even Omnius and Erasmus now seemed like insects to him, muddling through their grandiose, but ultimately minuscule, dreams.\n\nAs if from a great height, Paolo looked down at the Baron, the self-absorbed snake who had spent so many years dominating him, bossing him around, \"teaching\" him. Suddenly the once-powerful leader of House Harkonnen seemed laughably insignificant.\n\nThe Face Dancer Khrone studied the scene, and then\u2014with seeming uncertainty\u2014turned toward the evermind's manifestation as an old man. Paolo saw through all of them with incredible ease.\n\n\"Let me tell you what I will do next.\" In his own ears, Paolo's booming voice sounded like a god's. Even the great Omnius must tremble before him. Words flowed with the force of a cosmic Coriolis storm, rushing along on a current of ultraspice.\n\n\"I will implement my new mandate. The prophecy is true: _I will_ change the universe. As the ultimate and final Kwisatz Haderach, I know my destiny\u2014as do all of you, for your actions led to this prophecy.\" He smiled. \"Even yours, Omnius!\"\n\nThe false old man responded with an annoyed frown. Beside him the robot Erasmus grinned indulgently, waiting to see what the just-hatched superman would do. All of Paolo's visions of domination, conquest, and perfect control were based on prescience. He harbored no doubts in his mind. Every detail unfolded before him. The young man continued to spew pronouncements.\n\n\"Now that I have come into my true powers, there is no need for the thinking-machine fleet to obliterate the human-inhabited planets. I can control them all.\" He waved a hand. \"Oh, we may have to annihilate a minor world or two to demonstrate our strength\u2014or maybe just because we _can_ \u2014but we will keep alive the vast majority of people, as fodder.\"\n\nPaolo gasped as even more ideas flooded into his head, building momentum and power. \"Once we have swallowed up Chapterhouse, we will open the Sisterhood's breeding records. From there, we will implement my master plan of making brilliant, perfect humans, combining whatever traits I choose. Workers and thinkers, drones, engineers, and\u2014occasionally\u2014leaders.\" He spun toward the old man. \"And _you_ , Omnius, will construct a vast infrastructure for me. If we give our perfect humans too much freedom they'll mess everything up. We must eliminate the wild, troublesome genetic lines.\" He snickered to himself.\n\n\"In fact, the Atreides bloodline is the most unmanageable of all, so _I_ shall be the last Atreides. Now that I have arrived, history needs no more of us.\" He glanced around, but did not see the man who came to mind. \"And all those Duncan Idahos. How tedious they've become!\"\n\nPaolo was speaking faster and faster, swept along by intoxicating spice visions. The look of confusion on even the Baron's face made the young man wonder if anyone here could comprehend him any longer. They seemed so primitive to him now. What if his own thoughts were so grand that they were beyond the understanding of the most sophisticated thinking machines? That would really be something!\n\nHe began to pace around the chamber, ignoring glares and gestures from the Baron. Gradually Paolo's motions became jerky, manic. \"Yes! The first step is to sweep away the old, mow down and dispense with the outdated and unnecessary. We must clear a path for the new and the perfect. That's a concept all thinking machines can embrace.\"\n\nErasmus stared at him and mockingly reshaped his flowmetal face into a perfect likeness of the old man that represented Omnius. His expression reflected disbelief, as if he considered Paolo's pronouncements a joke, the rantings of a deluded child. A flare of anger rose within Paolo. This robot wasn't taking him seriously!\n\nPaolo saw the whole canvas of the future unrolling before him, broad strokes revealed by the incredible magnifying power of ultraspice. Some of the upcoming events became razor sharp, and he discerned more specifics, intricate details. The super-potent melange was even stronger than he had imagined, and the future became intensely focused in his mind, fractal minutiae unfolding before him in an infinite, yet completely expected, pattern.\n\nIn the midst of this mindstorm, something else was unleashed from within his cells: All the memories buried there from his original life. With a roar that briefly drowned out even the other clamoring knowledge, he suddenly remembered everything about _Paul Atreides_. Though the Baron had raised Paolo and the machines had corrupted him into what they imagined would be their puppet, he was still himself at the core.\n\nHe scanned the chamber, viewing everyone from a new perspective: Jessica, dear Chani, and _himself_ lying in a pool of blood, still twitching, gasping a last few breaths. Had he done that\u2014a bizarre form of suicide? No, Omnius had forced him. But how could anyone really force a Kwisatz Haderach to do anything? Details of the fight with Paul clashed in his mind, and he squeezed his eyes shut, trying to drive back the disturbing images. He didn't want to serve Omnius. He hated the Baron Harkonnen. He could not let himself be the cause of such destruction.\n\nHe had the power to change everything. Wasn't he the ultimate Kwisatz Haderach? Thanks to the ultraspice and his own Atreides genes, Paolo now possessed a greater prescience than had ever before been possible. Not even the smallest event could slip past him.\n\nIn a glorious tableau, he knew he could see everything about the tapestry of the future. Every tiny detail, if he wanted! No unexplored terrain, no wrinkles or nuances in the topography of events to come.\n\nPaolo paused in his restless pacing and gazed ahead, seeing beyond the walls of the grand machine cathedral, feeling overwhelmed by thoughts that no other human could begin to understand. His eyes changed to more than an intense blue-within-blue, to _black_ and glassy, rippled and impenetrable like a landscape of seared dunes.\n\nIn the background, he heard the Baron's voice. \"What's the matter with you, boy? Snap out of it.\"\n\nBut the visions continued to shoot at Paolo like projectiles from a repeater gun. He couldn't dodge them, could only receive them, like an invincible man standing up against ferocious firepower.\n\nOutside in the grand machine city, he heard a tremendous commotion. Alarms rang, and quicksilver robots rushed out of the cathedral chamber to respond. Paolo knew exactly what was happening, could see it from every angle. And he knew how each action would turn out, regardless of how Omnius, the humans, or the Face Dancers tried to change it.\n\nNo longer able to move, Paolo stood staring at moments that were yet to come, everything he could influence and all that he could not. Each second sliced into a billion nanoseconds, then expanded and spread out across a billion star systems. The scope of it threatened to overwhelm him.\n\n_What is happening?_ he asked himself.\n\n_Only what we brought upon ourselves_ , whispered the voice of Paul-within.\n\nWith new eyes Paolo saw moment by unfolding moment, expanding outward from the machine city, beyond the planet, the whole scope of the Old Empire, the farthest reaches of the Scattering, and the vast thinking-machine empire.\n\nAnother nanosecond passed.\n\nThe ultraspice had given him absolutely uncontaminated revelation. He saw time folding forward and backward from the focal point of his consciousness.\n\n_Perfect prescience_.\n\nCaught in the tidal wave of his own power, Paolo began to see much more than he had ever wanted to see. He witnessed every heartbeat a thousand times over, every action of every single person\u2014 _every being_ \u2014in the entire universe. He knew how each instant would play out from now until the end of history, and in reverse, to the beginning of time.\n\nThe knowledge flooded into him, and he drowned in it.\n\nHe watched Paul Atreides in his death throes and saw his counterpart go motionless, haloed by the crimson puddle on the floor, eyes staring into blessed oblivion.\n\nPaolo, who had wanted to be the final Kwisatz Haderach so badly he had killed for it, now became petrified by the utter tedium of his own existence. He knew every breath and pulsepoint in the entire history and future of the universe.\n\nAnother nanosecond passed.\n\nHow could any person endure this? Paolo was trapped in a predetermined path, like a computer's infinite loop. No surprises, choices, or movement. Absolute foreknowledge rendered Paolo entirely irrelevant.\n\nHe envisioned himself sinking in slow motion to the floor and lying face up, unable to move or speak, unable even to blink his eyes. Fossilizing. Then Paolo saw the last and most terrible revelation. He was _not_ the true and final Kwisatz Haderach, after all. It was not him. He would never accomplish what he had dreamed.\n\nWith spice roaring through him, the past went dark, and Paolo could only stare fixedly into the future, which he had already seen a thousand times over.\n\nAnother nanosecond passed.\n\n> _One can always find a battlefield if one looks hard enough_.\n> \n> \u2014BASHAR MILES TEG,  \n>  _Memoirs of an Old Commander_\n\nRemnants of dust and sand from the emptying hold swirled out into the no-ship's corridors, but the worms were gone, and Leto II with them. Bright sunlight from the machine world shone through the gaping holes. Stunned, Sheeana listened to the sounds of the behemoths crashing through Synchrony. She longed to be with them. Those once-captive sandworms were _hers_ as well.\n\nBut Leto was closer to them, a part of them, and they were part of him.\n\nDuncan Idaho came up behind her. She turned, the smell of grit clinging to her face and clothes. \"It's Leto. He's . . . with the sandworms.\"\n\nHe flashed a hard smile. \"That's something the machines won't expect. Even Miles would have been surprised.\" He grasped her arm and hurried her away from the open cargo hold. \"Now we've got to do something just as dramatic for ourselves.\"\n\n\"What Leto is doing will be a hard act to follow.\"\n\nDuncan paused. \"We've been running from that old man and woman for years, and I don't intend to sit here in this no-ship prison anymore. Our armory is filled with weapons stockpiled by the Honored Matres. We also have the rest of the mines that the Face Dancers didn't use to sabotage this ship. Let's take the fight outside, to them!\"\n\nShe felt the steel of his determination and found her own. \"I'm ready. And we have more than two hundred people aboard trained in Bene Gesserit combat techniques.\" Inside her mind, Serena Butler imparted visions of terrible combat, humans against fighting robots, incredible slaughters. But in spite of these horrors, Sheeana felt a strange exhilaration. \"It's been programmed into our genes for thousands of years. Like an eagle to a serpent, a bull to a bear, a wasp to a spider, humans and thinking machines are mortal enemies.\"\n\nAFTER DECADES OF running and many escapes from the tachyon net, this would be their final showdown. Tired of feeling helpless, the _Ithaca_ captives crowded forward to the armory. All were eager to fight back, though they knew the odds were heavily against them. Duncan relished it.\n\nThe stockpile of armaments was not particularly impressive. Many of the stored weapons fired only flechettes, razor-sharp needles that would not be effective against armored combat robots. But Duncan handed out old-style lasguns, pulse launchers, and explosive projectile rifles. Demolition squads could plant the remaining mines against the foundations of thinking-machine buildings and detonate them.\n\nThe Tleilaxu Master Scytale pushed his way through the crowded corridor, trying to reach Sheeana, looking as if he had something important to say. \"Remember, we have more enemies out there than just robots. Omnius has an army of Face Dancers to stand against us.\"\n\nDuncan handed a flechette rifle to Reverend Mother Calissa, who appeared as bloodthirsty as any Honored Matre. \"This will cut down plenty of Face Dancers.\"\n\nThe little man announced with a thin smile, \"I have another way to help. Even before we were captured, I began to produce the specific toxin that would target Face Dancers. I made sixty canisters of it, in case we had to saturate all the air aboard the ship. Unleash it against the Face Dancers in the city. It may make humans a little nauseated but it is lethal to any Face Dancer.\"\n\n\"Our weapons could do the rest\u2014or our bare hands,\" Sheeana said, then turned to the other workers. \"Get the canisters! There's a battle outside!\"\n\nA fierce army of humans streamed out through the gaping hole torn in the _Ithaca_ 's hull. Sheeana led her Bene Gesserits. Reverend Mothers Calissa and Elyen guided groups through the shifting streets in search of vulnerable targets. Reverend Mothers, acolytes, male Bene Gesserits, proctors, and workers rushed out carrying weapons, many of which had never been fired before.\n\nWith a loud battle cry, a well-armed Duncan charged forward into the bizarre metropolis. In his original lifetime, he had not survived long enough to join Paul Muad'Dib and his Fremen Fedaykin in bloody raids against the Harkonnens. The stakes were more desperate now, and he intended to make a difference.\n\nThe streets of Synchrony were in turmoil, the buildings themselves pumping and writhing. Leto's sandworms had already tunneled beneath the foundations of the structures, breaking through the pliable, living metal and knocking down tall towers. Across the galaxy, Omnius's thinking-machine fleet was engaged in numerous climactic battles. Duncan thought of Murbella out there somewhere\u2014if she was still alive\u2014facing them, fighting them.\n\nCombat robots swarmed the streets. They emerged from between buildings, fashioning and firing projectile weapons from their own bodies. The Bene Gesserits scrambled out of the way, finding shelter. Lasbeams cut smoking holes through the fighting machines; explosive projectiles smashed them backward into debris.\n\nRunning headlong into the fray, Duncan used his long-dormant Swordmaster skills to attack the nearest robots. He wielded a small projectile launcher as well as a vibrating sonic club that transmitted a deadly blow each time it struck a fighting machine.\n\nFrom all directions, Face Dancers rallied against the humans, while combat robots turned their attention to the destructive sandworms. The first ranks of shape-shifters advanced with blank and unreadable faces, armed with machine-designed weaponry.\n\nWhen the first canisters of Scytale's curling, gray-green gas landed among them, the frenzied Face Dancers did not understand what was happening. Soon they began to fall, writhing, their faces melting off their bones. Sensing the danger too late, they scrambled to retreat as Sheeana's fighters launched more poison gas into their midst.\n\nThe Bene Gesserits continued to push forward. Their demolition crews planted mines against looming buildings that could not uproot themselves in time. Powerful explosions brought down the shuddering metal towers. Sheeana rushed her teams to shelter until the thunderous collapses were over. Then they surged forward again.\n\nDuncan decided to hang back. In the center of the city, the huge, bright cathedral drew him like a beacon, as if all the intensity of the evermind's thoughts were being channeled through it. He knew Paul Atreides was in that structure, perhaps fighting for his life, perhaps dying. Jessica was inside, as well. Compelling instincts born of memories from his first life told Duncan where he had to go. He needed to be at Paul's side in the den of the Enemy.\n\n\"Keep the machines occupied, Sheeana. Even the evermind can't fight on an infinite number of fronts at once.\" He jerked his head toward the cathedral building. \"I'm going _there_.\"\n\nBefore she could say anything, Duncan ran off.\n\n> _Enduring my own mistakes once was bad enough. Now I am condemned to relive my past, over and over_.\n> \n> \u2014DR. WELLINGTON YUEH,  \n> interview notes taken by Sheeana\n\nThe Suk doctor, in a teenage body but with the burdens of a very old man, knelt by the dying Paul. Although he had administered every emergency treatment at his disposal, he knew he could not save the young Atreides. With specialized skill, he had halted most of the bleeding, but now he shook his head sadly. \"It's a mortal wound. I can only slow his death.\"\n\nDespite the betrayal in his past life, Yueh had loved the Duke's son. In those bygone days he had been a teacher and mentor to Paul. He had seen to it that the boy and his mother had a chance of surviving in the deserts of Arrakis after the Harkonnen takeover, so long ago. Even with his full Suk knowledge restored, Yueh didn't have the facilities to help this Paul. The knife had penetrated the pericardium, cut into the heart. Through sheer tenacity the young man still clung to a thread of life, but he had already lost far too much blood. His heart was stuttering its last few beats.\n\nDespite the chances offered by a second lifetime, Yueh was unable to escape his previous failures and betrayals. He had been suffering inside, wallowing in the cesspool of his past mistakes. The Sisters on the no-ship had resurrected him for some secret purpose that he had never been able to fathom. Why was he here? Certainly not to save Paul. That was out of his hands now.\n\nOn the no-ship he had tried to take action by doing what he thought was necessary and right, but he had only caused more tragedy, more pain. He had killed an unborn Duke Leto rather than another Piter de Vries. Yueh knew he had been manipulated by the Rabbi\/Face Dancer, but he could not accept that as an excuse for his actions.\n\nChani sat on the floor at Paul's side, calling his name in an unfamiliar husky voice. Yueh sensed that something about her had changed; her eyes had a wild steeliness much different from the gaze of the sixteen-year-old girl he knew.\n\nHe realized with a start that the horror of holding Paul's bloody, dying body in her arms must have pushed her over the edge. Chani had her original memories back\u2014just in time to experience the full magnitude of her imminent loss. Even Yueh reeled from the cruelty of it.\n\nThe Baron made despairing sounds of his own, at first confused, then angry, and now desperate. \"Paolo boy, answer me!\" He crouched by the glassy-eyed young man, raging. He raised a hand as if to strike the warped copy of Paul Atreides, but Paolo didn't flinch.\n\nFrom one side the independent robot Erasmus watched the whole scenario with intent curiosity, his optic threads glistening. \"Apparently, neither of the Paul Atreides gholas is the Kwisatz Haderach we expected. So much for the accuracy of our predictions.\"\n\nThe moment he saw the Baron's growing confusion, Yueh knew that only one thing remained for him to do. Struggling to regain his composure, he rose from the side of the dying Paul and made his way over to the Baron and Paolo. \"I am a Suk doctor.\" His sleeves and trousers were drenched in Paul's blood. \"Perhaps I can help.\"\n\n\"Eh? You?\" The Baron sneered at him.\n\nJessica glared after the doctor, and the restored Chani looked as if she wanted to flog Yueh for leaving Paul's side. But he concentrated only on the Baron. \"Do you want me to help, or not?\"\n\nThe Baron moved out of the way. \"Hurry, then, damn you!\"\n\nGoing through the motions, Yueh bent and passed his hands over Paolo's face, felt the cold clamminess of the skin and the barely discernible pulse. Young Paolo sat frozen and transfixed, staring into a coma of infinite awareness and paralyzing boredom.\n\nThe Baron leaned close. \"Make him snap out of it. What is the matter with him? Answer me!\"\n\nGrabbing the Emperor's dagger from Paolo's waistband, Yueh spun in a single fluid movement. The Baron staggered back, but Yueh was quicker. He thrust the sharp tip at an angle under the hateful man's chin and rammed it all the way to the back of his skull. \" _This_ is my answer!\"\n\nThe answer for being coerced into betraying House Atreides, for all the schemes, the pain, the resultant guilt, and most of all for what the Harkonnens had done to Wanna.\n\nThe Baron's eyes opened wide in shock. He flailed his hands and tried to speak, but could only gurgle helplessly as a crimson geyser spouted from his neck.\n\nSpattered in blood, Yueh jerked the Emperor's dagger back out. He considered plunging it into Paolo's midsection, just to be certain he killed both of them. But he couldn't do that. Though the boy had gone wrong, this was still Paul Atreides.\n\nThe Baron collapsed onto the hard floor. All the while, the Paolo ghola continued to stare upward without blinking.\n\nDr. Wellington Yueh allowed himself a relieved smile. At long last he had accomplished something positive and true. Finally, he had done something right. For a long moment he held the dagger, covered with the Baron's blood as well as Paul's. A potent impulse prompted him to turn the point toward himself. Yueh closed his eyes, clutched the handle of the knife, and took another deep breath.\n\nA firm hand clasped his wrists, staying his suicide thrust. He opened his tear-filled eyes to see Jessica standing beside him. \"No, Wellington. You don't need to redeem yourself like that. Help me save Paul instead.\"\n\n\"There is nothing I can do for him!\"\n\n\"Don't underestimate yourself.\" Her facial muscles tightened. \"Or Paul.\"\n\n> _No education, training, or prescience can show us the secret abilities we contain within ourselves. We can only pray those special talents are available in our time of greatest need_.\n> \n> _\u2014The Bene Gesserit Acolytes' Handbook_\n\nDeath.\n\nPaul skirted the edge of the interior blackness, dipped briefly into infinity, and danced back out. He wavered on the balance point of his own mortality. The knife wound was deep.\n\nWithout any awareness of what was going on around him, he felt an intense coldness spreading from the tips of his fingers to the back of his head. Like a distant whisper, he could still hear the lava fountain blazing nearby. Despite the hard stone floor beneath him, Paul felt as if he were floating, his spirit drifting in and out of the universe.\n\nHis skin detected a warm, syrupy wetness. Not water. _Blood_ . . . his own . . . spreading in a great pool across the floor. It filled his chest, mouth, and lungs. He could hardly breathe. With each feeble heartbeat, more of it spilled out, never to be retrieved. . . .\n\nIt seemed as if he could still feel the long blade of the Emperor's knife inside him. Now he remembered. . . . In the last desperate days of Muad'Dib's jihad, the conniving Count Fenring had stabbed him. Or had that occurred at a different time? Yes, he had tasted a knife blade before.\n\nOr maybe he was the old blind Preacher in the dusty streets of Arrakeen, stabbed by yet another knife. So many deaths for one person . . .\n\nHe couldn't see. Someone squeezed his hand, though he could barely feel it, and he heard a young woman's voice. \"Usul, I am here.\" _Chani_. He remembered her most of all, and was glad she was here with him. \" _I_ am here,\" she said. \"All of me, with all my memories, beloved. Please come back.\"\n\nNow a firmer voice yanked his attention, as if strings were attached to his mind. \"Paul, you must listen to me. Remember what I taught you.\" His mother's voice. _Jessica_ . . . \"Remember what the real Lady Jessica taught the real Paul Muad'Dib. I know what you are. You have the power within you. That is why you aren't dead yet.\"\n\nHe found words within his throat, and they bubbled up through the blood. He was amazed at the sound of his own voice. \"Not possible . . . I'm not . . . _the_ Kwisatz Haderach\u2014the ultimate . . .\" He was not the superbeing that would change the universe.\n\nPaul's eyes flickered open, and he saw himself lying in the great machine cathedral. That part of the prescient dream had been true. He had seen Paolo laughing with victory and consuming the spice\u2014but now Paolo himself rested on the floor like a fallen statue, frozen and mindless, gazing into infinity. The Baron lay dead, murdered with a look of disbelief and annoyance on his pasty face. So the vision was true, but all the details had not been available to him.\n\nSome kind of commotion came from beside Omnius and Erasmus, and Paul looked there, his gaze bleary. Watcheyes flitted in, displaying images. The old man stood with an impatient expression on his face. The Face Dancer Khrone seemed unsettled. Paul could hear voices shouting. The whole cacophony wove itself in oddly incomprehensible strands through the buzzing tapestry in his head.\n\n\"Sandworms attacking like demons . . . destroying buildings.\"\n\n\". . . a rampage . . . armies emerging from the no-ship. A poisonous gas that kills\u2014\"\n\nThe old man said drily, \"I have dispatched combat robots and Face Dancers to fight them, but it may not be sufficient. The sandworms and the humans are causing considerable damage.\"\n\nErasmus picked up the conversation. \"Rally more Face Dancers, Khrone. You didn't send all of them out.\"\n\n\"That is a waste of my people. If we fight the humans, their poison kills us. If we go out to battle sandworms, we will be crushed.\"\n\n\"Then you will be poisoned, or crushed,\" Erasmus said lightly. \"No need to fret. We can always create more of you.\"\n\nThe Face Dancer's features shifted and blurred, a storm crossing his putty face. He turned and marched out of the vaulted chamber.\n\nMeanwhile Yueh raised Paul's head, ministering to him with Suk medical techniques. But Paul folded his eyes shut again and dropped backward into the pain. Again he danced along the edge of the chasm that opened wider and wider before him.\n\n\"Paul.\" Jessica's voice was insistent. \"Remember what I told you of the Sisterhood. Maybe you aren't the ultimate Kwisatz Haderach that the thinking machines want, but you _are_ still a Kwisatz Haderach. You know that, and your body knows it as well. Some of your powers are the same as those of a Reverend Mother. A _Reverend Mother_ , Paul!\"\n\nBut he found it too difficult to concentrate on her words, or to remember . . . As he spiraled deeper and deeper into unconsciousness, her voice faded, and he could not hear or feel his heartbeat anymore. What did his mother mean?\n\nIf Jessica now remembered her past life, she also remembered the Spice Agony. Any Reverend Mother had the innate ability to shift her biochemistry, to manipulate and alter molecules within her bloodstream. It was how they selected their pregnancies, how they transmuted the poisonous Water of Life. That was why the Honored Matres had searched so furiously to find Chapterhouse\u2014because only Reverend Mothers had the physical ability to fight off the terrible machine plagues.\n\nWhy did his mother want him to remember that?\n\nTrapped inside the darkness, Paul felt the emptiness of his body. Completely bled out. Silent.\n\nOn Arrakis so long ago, he had undergone his own version of the Agony, the first male ever to do so successfully. For weeks he had lain in a coma, declared dead by the Fremen, while Jessica insisted on keeping him alive. He had seen that stygian place where women could not go, and he had drawn strength from it.\n\nYes, Paul had that ability within him now. He was a male Bene Gesserit. He could still control his body, every cell, every muscle fiber. At last, he knew what his mother had been trying to tell him.\n\nThe pain of dying and the crisis of survival gave him the lever he needed. He stood upon his pain as a fulcrum and used it to pry open his life, his first existence\u2014the memories of Paul Atreides, of Muad'Dib, of himself as Emperor and later as the Preacher. He followed that flood backward to his childhood and his early training with Duncan Idaho on Caladan, including how he had almost been killed as a pawn in the War of Assassins that had ensnared his father.\n\nHe remembered his family's arrival on Arrakis, a place Duke Leto knew to be a Harkonnen trap. The memories rushed past Paul: the destruction of Arrakeen, his flight into the desert with his mother, the death of the first Duncan Idaho . . . meeting the Fremen, his knife fight with Jamis, the first man he had ever killed . . . his first worm ride, creating the Fedaykin force, attacking the Harkonnens.\n\nHis past accelerated as it flowed through his mind\u2014overthrowing Shaddam and his empire, launching his own jihad, fighting to keep the human race stable without traveling down that dark path. But he had not been able to escape political struggles, assassination attempts, the exiled Emperor Shaddam's bid for power and the pretender daughter of Feyd-Rautha and Lady Fenring . . . then Count Fenring himself had tried to kill Paul\u2014\n\nHis body no longer felt empty, but _full_ of experiences and great knowledge, full of abilities. He remembered his love for Chani and his sham marriage to Princess Irulan, as well as the first Duncan ghola named Hayt, and Chani dying while giving birth to the twins, Leto II and Ghanima. Even now, the pain of losing Chani seemed far greater than the pain he presently suffered. If he died now, in her arms, he would inflict that same anguish upon her.\n\nHe remembered wandering off into the desert, blinded from prescience . . . and surviving. Becoming the Preacher. Dying in a dusty street surrounded by a mob.\n\nHe was now everything he had once been: Paul Atreides and all the different guises he had worn, every mask of legend, every power and weakness. Most important of all, he now had the abilities of a Reverend Mother, the infinitesimal physical control. Like a beacon in the darkness, his mother had enabled him to see.\n\nBetween his last heartbeats, he searched within himself in the deep and drowning place. He found the knife wound inside his heart, saw the mortal damage, and discovered that his body's defenses were incapable of repairing the grievous injuries on their own. He needed to direct the healing process.\n\nAlthough in the preceding moments he had seemed to be fading, he now sharpened himself and became part of his own heart, which was no longer beating. He saw where Paolo's blade had slashed open his right ventricle, letting blood spill out of the chamber. His aorta had been nicked, but there was no more blood for it to carry.\n\nPaul brought the cells together and sealed them. Then, droplet by droplet, he began to draw his own blood back from the cavities of his body where it had spilled, and reabsorbed it into his tissues. He literally pulled life back into himself.\n\nPAUL DIDN'T KNOW how long his trance lasted. It seemed as infinite as the death coma caused by touching just a drop of the poisonous Water of Life to his tongue. He became suddenly aware of Chani's grip again. Her hand was warm, and he felt his own flesh, no longer cold and shuddering.\n\n\"Usul!\" He could hear Chani's faint whisper and detected the disbelief in her tone. \"Jessica, something's changed!\"\n\n\"He is doing what he needs to do.\"\n\nWhen he finally summoned the strength to flutter his eyelids, Paul Muad'Dib Atreides rejoined the living, bringing with him both his old life and his new. In addition to those memories and abilities, he returned with a fantastic, even greater revelation. . . .\n\nJust then, a flushed and bloodied Duncan Idaho burst into the great cathedral hall, knocking sentinel robots aside. Erasmus casually gestured to allow the man to enter. Duncan's eyes went wide when he saw the bleeding Paul propped up by his mother and Chani. Dr. Yueh looked astonished at the miracle before him. Duncan ran toward them.\n\nTrying to pull together his thoughts, Paul placed the tableau around him in the context of his internal knowledge. He had learned much by dying the first time, coming back as a ghola, and nearly dying again. He had always had a phenomenal gift for prescience. Now he knew even more.\n\nIn spite of his miraculous survival and rebirth, he still was not the perfect Kwisatz Haderach, and clearly Paolo was not, either. As Paul's vision cleared, he focused on a realization that none of them had seen\u2014not Omnius, Erasmus, Sheeana, nor any of the gholas.\n\n\"Duncan,\" he said in a hoarse voice. \"Duncan, it's you!\"\n\nAfter hesitating for a moment, his old friend came closer, this loyal Atreides fighter who had been through the ghola process more times than any other individual in history.\n\n\"You're the one they've been looking for, Duncan. It's you.\"\n\n> _Even prescience has its limits. No one will ever know all the things that might have been_.\n> \n> \u2014REVEREND MOTHER DARWI ODRADE\n\nWith her arm around his shoulders, Chani rocked Paul, as she shuddered with joy and relief. Fremen prohibitions were so much a part of her that even after seeing his mortal wound and watching him come within a heartbeat of dying in her arms, she still had not shed a tear.\n\nIn the machine cathedral, Duncan wrestled with the revelation that the pale and bloodied Paul had given him. \"I . . . am the Kwisatz Haderach?\"\n\nPaul nodded weakly. \"The final one. The perfect one. The one they were looking for.\"\n\nThe old-man manifestation of Omnius gave the robed form of the independent robot an accusing look. \"If this claim is correct, you were in error, Erasmus. You did not allow for the humans to twist fate yet again. You said your predictive calculations were accurate.\"\n\nThe robot remained aloof, even smug. \"Only your _interpretation_ of my calculations was flawed. The final Kwisatz Haderach was indeed aboard the no-ship, as I said all along. You drew the obvious conclusion that the one we sought must be Paul Atreides. When the Face Dancer found the bloody knife carrying the cells of Muad'Dib, you falsely reinforced your own conclusion.\"\n\nDuncan's mind wanted to reject what he was hearing. Even if he truly was the ultimate Kwisatz Haderach, what was he to do with the knowledge?\n\nThe old man sneered down at the frozen, useless boy Paolo and the dead Baron. \"All that work on our own clone gave us no advantage. Extremely wasteful.\"\n\nErasmus shaped his flowmetal face into a sympathetic expression and addressed the recent arrival. \"I knew I was drawn to you for a reason, Duncan Idaho. If you really are the Kwisatz Haderach, you stand in a position to alter the course of the universe. You are a living watershed, the harbinger of change. You can choose to stop this conflict that has made enemies of humans and thinking machines for thousands of years.\"\n\nDuncan realized that Yueh, Jessica, Chani, and Paul had all played their parts, and now the focus had shifted to him.\n\nErasmus stepped closer to them. \"Kralizec means the end of many things, but that end need not be destructive. Just a fundamental change. Henceforth, nothing will be the same.\"\n\n\"Not destructive?\" Jessica raised her voice. \"You said your thinking-machine ships are attacking worlds in the Old Empire. You've already sterilized and conquered hundreds of planets!\"\n\nThe robot seemed unperturbed. \"I did not say our approach was the only way, or even the best one.\" The old man glowered at Erasmus as if he had been insulted.\n\nSuddenly the sky above the great machine city was torn by multiple booms of displaced air as a thousand Guild Heighliners appeared like storm clouds. Emerging from foldspace, the fleet of huge vessels easily carried enough weaponry to level the continent.\n\nOmnius's old man guise flickered as his concentration was wrenched by the dramatic shift. Across the city of Synchrony, robots buzzed about, fighting the sandworms that continued to rampage. Now they had to shore up defenses against the new enemy overhead.\n\nInside the vaulted building, Erasmus altered his form again to the kindly old woman, as if he believed this presentation more convincing and compassionate. \"I've run probabilities beyond the limits of my original calculations. I believe you have the power, Duncan Idaho\u2014stop these Guildships from destroying us.\"\n\n\"Oh, please stop prattling,\" Omnius said.\n\nDuncan looked around, crossed his arms over his chest. \"I am not afraid of the Guild and their Navigators. If I have to die to end this, I'm willing to do so.\"\n\nYueh added bravely, \"Everyone here has died before.\"\n\n\"It doesn't matter. Let them destroy Synchrony.\" The old man did not seem overly disturbed. \"I am dispersed across many locations. Annihilating this entire planet, this node, will never eradicate me. I am the evermind, and I am everywhere.\"\n\nA crack sounded at the center of the wide cathedral hall. Then, with a blur and a bang of folded space, an image appeared above the bloodstained floor. The shimmering transmission appeared to be solid one moment and a staticky ghost the next. In moments, the shape clarified to a beautiful and statuesque human woman with classically perfect features. Then she shifted to become stunted and dwarfish, with a blunt, unattractive face, short arms and legs, and an overly large head. After another flicker, the image was nothing more than a disembodied face that wavered in the air. It was as if she could not remember exactly what she was supposed to look like.\n\nDuncan immediately knew who\u2014or what\u2014this was. \"The Oracle of Time!\"\n\nThe face swiveled to scan the people and robots in the great hall, before the image hovered closer to him. \"Duncan Idaho, I have found you. I searched for years, but your no-ship and your own . . . strangeness protected you.\"\n\nDuncan no longer questioned the bizarre storm of occurrences around him. \"Why did you come now?\"\n\n\"You emerged from your no-ship only once before on the planet Qelso, but I did not follow you swiftly enough. I sensed you again when your no-ship was damaged and captured. Now, with the thinking machines attacking, I was able to trace the lines of the evermind's tachyon web and follow Omnius to you. I brought my Navigators with me.\"\n\n\"What is this apparition?\" the evermind demanded. \"I am Omnius. Begone from my world!\"\n\n\"Once I was called Norma Cenva. Now I am exponentially more than that\u2014far beyond anything a computer network can comprehend. I am the Oracle of Time, and I go where I please.\"\n\nIn the old crone guise, Erasmus reached out like a curious child and touched, but the wrinkled hand passed through her image. \"So many of the most interesting humans are women,\" he mused. The robot experimentally waved fingers through her ghostly likeness, stirring without altering it. She ignored him.\n\n\"Duncan Idaho, you have finally come to your realization. Kwisatz Haderach, I tried to protect you. Before you, Paul Muad'Dib and his son Leto the God Emperor were imperfect prophets. Even they realized their flaws. Now through a confluence in the cosmos, the nexus of all nexuses, you have become the singularity in a bold new universe, the vital point from which everything flows outward for the rest of eternity. The hopes of humankind\u2014and much more\u2014are distilled in you.\"\n\nStill taken aback, Duncan asked, \"But how? I don't feel that different.\"\n\n\"The Kwisatz Haderach is a 'shortening of the way,' a figure powerful enough to force a fundamental and necessary change that alters the course of future history, not just for humanity but for thinking machines as well.\"\n\n\"Yes, you have the power, Duncan Idaho.\" Erasmus sounded just as encouraging as the Oracle. \"I rely on you to make the correct choice. _You_ know what will benefit the universe most, and you know that thinking machines can enrich the entirety of civilization.\"\n\nDuncan marveled at the awareness of his new identity and the unfolding of his thoughts around the astounding truth. Finally, after so many attempts at life in ghola form, he knew his destiny. His mind was fully awakened.\n\nHe saw time as a great ocean stretching across the cosmos, and with his awakening powers he envisioned being able to analyze each molecule, each atom and subatomic particle. Perfect prescience would come, but not yet, not too fast or it would induce the same crippling paralysis that had befallen Paolo. Already Duncan's mind could go much faster than a Mentat's, and he sensed he could make his body move at speeds that would have astonished even the Bashar.\n\n_I am the ultimate Kwisatz Haderach. There will be no more after me_.\n\nThe Oracle's image flickered, shifting her shape back to that of the beautiful woman. \"After you died the first time, Duncan Idaho\u2014as a soldier fighting to save the Atreides, fighting to save the first Kwisatz Haderach\u2014the powers of the universe compelled your resurrection as a ghola, and many times afterward, over and over. The original God Emperor understood some of your destiny and played an unwitting role in bringing about this moment. The end point of his Golden Path is the beginning of something else.\"\n\n\"I am linked to the Golden Path?\"\n\n\"You are, but you are destined to go far beyond it.\"\n\nPaul seemed to be swiftly recovering his strength. Beside him, Jessica said to the otherworldly visitor, \"But Duncan was part of no formal breeding program! How did he develop into a Kwisatz Haderach?\"\n\nThe Oracle continued, \"Duncan, with each rebirth, you came closer to completion. Instead of being developed in a breeding program, you went through a process of personal evolution. With every successive incarnation you acquired more knowledge, skills, and experience\u2014as if a sculptor with a tiny chisel was chipping away at a large block of hard stone, slowly, ever so slowly, fashioning a perfect statue. In your one body, you manifested a tachyon evolution, a hyperfast developmental journey that propelled you toward your destiny.\"\n\nDuncan had lived his life repeatedly for thousands of years. Not only had the Tleilaxu tinkered with his genetics to give him abilities to fight the Honored Matres, they had combined his cells so that he retained _all_ of his previous lives, every one of them. With all those memories, he possessed a breadth of experience and wisdom that no one could match. This Duncan Idaho had more knowledge than the most advanced Mentat or the evermind Omnius, and more understanding of human nature than even the great Tyrant Leto II.\n\nDuncan was the same person again and again, perfecting himself, constantly filtering out impurities, like passing through a fine mesh strainer to sift out only the best of qualities, leaving him as the One. He allowed himself a secret smile at the irony. He had succeeded only because of the meddling of the Tleilaxu, though he was certain that the Masters had never intended to create a savior for humanity.\n\nDuncan's Mentat mind burned through the data, confirming his conclusion, knowing that the Oracle of Time must be correct. \"Truly, I am the Kwisatz Haderach!\" He wished Miles Teg could have been there with him. \"And what of the great war\u2014Kralizec?\"\n\n\"We are in the midst of it now. Kralizec is not merely a war, but a point of _change_.\" Her image flickered. \"And you are the culmination of it.\"\n\n\"But what about the rest of humanity?\" _Murbella_. \"They need to know. How will they understand what has happened?\"\n\n\"My Navigators will inform them, perhaps even bring their leaders here. First, however, I need to eliminate a threat that should have been gone millennia ago. An enemy I fought ten thousand years before you were first born.\"\n\nThe Oracle slid through the air toward the indignant-looking old man, Omnius. Facing him, she made her voice boom more loudly than the evermind's speakers ever had. \"I must ensure that the thinking machines can no longer harm anyone. That was my mission ages ago, when I was merely a woman, when I invented the concept of the foldspace engine, when I discovered the mind-expanding powers of melange. I shall remove you, Omnius.\"\n\nThe evermind laughed, a remote old man's chuckle. The slightly stooped manifestation suddenly grew larger, looming like a giant over her image. \"You cannot remove me, for I am not a corporeal being. I am _information_ , so my existence has spread anywhere the tachyon net stretches. I am everywhere.\"\n\nThe female image formed a smile. \"And I am more than that. I am the Oracle of Time. Now hear my laughter.\" In an eerie voice Norma Cenva chuckled long and hard, causing even the oversized Omnius to take a step backward. \"I am heard across star systems and eons, across time and space, far beyond the range of your net.\"\n\nOmnius took another step backward.\n\n\"First I crippled your fleet. Now I will rip you out like the weed that you are, and discard you.\"\n\n\"Impossible\u2014\" The old man began to dissolve as he retreated into his own network.\n\n\"I will extract you\u2014every shred of information from every node.\" Her misty image became amorphous and seeped around Omnius. He nearly staggered into Erasmus, but the independent robot easily slid out of the way, his old-woman face expressing curiosity and bemusement.\n\n\"I will take you to a place where such information is no longer comprehensible, where physical laws do not apply.\"\n\nDuncan heard the evermind's voice cry out in rage, but it was muffled. In the vaulted hall, the insectile sentinel robots who tried to move forward in the service of Omnius seemed strangely disoriented and sluggish.\n\n\"There are many universes, Omnius. Duncan Idaho has visited more than one, and he knows the place of which I speak. I rescued him and his no-ship from it long ago. You, however, will never find your way back.\"\n\nDuncan considered the incomprehensible struggle before him. Indeed, when he had first stolen the no-ship from Chapterhouse, he had lurched through the fabric of space in a desperate attempt to avoid capture and had taken them to a bizarrely skewed universe. He shuddered to think of it.\n\n\"Nothing shall rescue you, Omnius.\"\n\n\"Impossible!\" the old man bellowed, losing his physical form and becoming no more than a spangled outline.\n\n\"Yes, impossible. Wonderfully so.\"\n\nThe air in the chamber crackled with clouds of electricity that spread thinner and thinner, as the Oracle wrapped herself like a net around the iconic thinking machine. For an instant, Duncan saw Norma's face superimposed over that of the old man. The two countenances merged into one: _Hers_. The beautiful woman smiled, and the air filled with sparkling, hair-fine strands of electricity that she drew around her like an elegant cloak.\n\nThen she uprooted herself from reality and vanished into the incomprehensible void, taking with her all traces of Omnius.\n\nForever.\n\n> _You see enemies everywhere, but I see only obstacles\u2014and I know what to do with obstacles. Either move them or crush them, so that we can be on our way_.\n> \n> \u2014MOTHER COMMANDER MURBELLA,  \n> address to the combined Sisterhood\n\nEven after the Navigators destroyed the bulk of the Enemy fleet in a flurry of unexpected Obliterators, a second wave of machine ships advanced toward Chapterhouse.\n\nThe Oracle, upon locating Duncan Idaho and the lost no-ship, had promptly taken most of her Heighliners to Synchrony, only assigning a small percentage to aid in the defenses of other human-inhabited planets. With the outcome of those missions unknown, some or all of the other planets could still be vulnerable. One thing was certain: At Chapterhouse, Murbella and her defenders faced the remaining machine ships alone. Through it all, the Mother Commander didn't have much time to process her shock at discovering that Duncan was still alive.\n\nAdministrator Gorus groaned. \"Will they never stop?\"\n\n\"No.\" Murbella scowled at him for forcing her to state the obvious. \"They are thinking machines.\"\n\nHigh over the Bene Gesserit world, her hundred last-stand vessels hung surrounded by the debris from thousands of destroyed machine battleships. This fight had inflicted a substantial toll on the Enemy, but unfortunately it was not enough.\n\nThe new wave of Omnius vessels would not thumb their noses at the human defenders, as the first had. Murbella expected no mercy this time and didn't have much hope for the last-stand ships at other strategic points, either. The machines intended to annihilate Chapterhouse and every other world that stood in their way.\n\nShe cursed the clumsy, uncooperative Guild vessels that the Junction shipyards had produced and the worthless weapons the Ixians had supplied. She had to think of something on her own. \"I won't just let our ships sit here with their throats bared, like lambs waiting for slaughter!\"\n\n\"The mathematical compilers controlled our foldspace guidance and standard\u2014\"\n\nShe shouted at Gorus. \"Rip out those damned navigation devices\u2014we'll maneuver our vessels by hand!\"\n\n\"But we will not know where we are going. We could crash!\"\n\n\"Then we must crash into the Enemy, instead of each other.\" She wondered if the machines would feel a need for vengeance when they saw the wreckage of the first wave. Honored Matres certainly would.\n\nThe Enemy kept coming. Murbella studied the complex tactical projections. Surely they did not need such a vast number of vessels to conquer the minimally inhabited Chapterhouse. It seemed obvious that the evermind had learned the value of intimidation and showmanship, as well as the wisdom of redundancy.\n\nIn the Heighliner control center, two Guildsmen argued with Gorus. One claimed that disconnecting the mathematical compiler was impossible, while the other warned that it was unwise. Murbella ended the debate with the compelling power of Bene Gesserit Voice. The Guildsmen shuddered and, unable to resist her, did as she commanded.\n\nAlthough the machine forces outgunned them by a substantial margin, Murbella did not flinch from what had to be done. Instead, she allowed herself to reawaken her old Honored Matre anger. This was not a time to calculate odds. It was a time to unleash every bit of destruction her people could muster. Their chances were better now than they had been when this last stand began. If they all embraced viciousness and fought like frenzied Honored Matres, they could inflict significant damage. They might still go down in flames, but if they bought sufficient time for the Oracle and her Navigators to defeat Omnius, Murbella would count it a victory. She just wished she could have seen Duncan one more time.\n\nMurbella turned toward the broad projection plate that magnified the oncoming vessels. \"Arm all weaponry and stand ready to ram. The moment we deplete conventional armaments, our own ships will become the final weapons. A hundred of us will take out at least as many of their ships.\"\n\nUp to this point, Gorus had called her battle strategy suicidal. Now, he looked as if he might try something foolish to stop her. \"Why not negotiate with them? Would it not be preferable to surrender? We cannot stop them from destroying their targets!\"\n\nMurbella fixed her gaze on the Administrator as if he were weak prey. Even the Sisters who had started out as pure Bene Gesserits now reacted with a feral Honored Matre strength. They would never back down.\n\n\"And you base this suggestion on the success of your emissaries to the thinking machines? All those emissaries who disappeared?\" Murbella's voice sizzled like hot acid. \"Administrator, if you'd like to seek another solution, I would be happy to eject you from an airlock and let you fly across the empty vacuum. As the last breath explodes out of your lungs, maybe you can gasp out your personal surrender terms. Be my guest, if you believe the thinking machines will listen to you.\"\n\nThe desperate-looking Guildsman cringed. Around him, the Sisters moved to take control, ready for a final plunge.\n\nBefore Murbella could give the command, though, Janess broke in over their tight-linked channel, \"Mother Commander! Something's changed with the machine battleships. Look at them!\"\n\nMurbella examined the images on the viewing plate. The Enemy vessels no longer moved in a tight, efficient formation. They slowed and began to spread apart, as if they had no goal, like unmanned sailing ships becalmed on a vast cosmic sea.\n\nLeft suddenly leaderless.\n\nTo her amazement, the thinking-machine fleet floated listlessly in space.\n\n> _Even when caught up in his own myth, Muad'Dib pointed out that greatness is only a transitory experience. For a true Kwisatz Haderach, there are no warnings against hubris, no rules or requirements to follow. He takes from all things and gives to all things, as he wishes. How could we have deluded ourselves into believing we could control such a one?_\n> \n> \u2014Bene Gesserit analysis\n\nAfter the Oracle vanished, Erasmus stared at the empty space in the center of the vaulted chamber, his head slightly cocked to one side. \"Omnius is gone.\" His voice sounded hollow to Duncan Idaho's ears. \"No vestige of the evermind remains in the network of thinking machines.\"\n\nDuncan felt his own mind racing, expanding, absorbing new information. The terrible Enemy he had sensed for so long\u2014the threat that the Honored Matres had provoked\u2014was no more. By removing the evermind, uprooting it from this universe and taking it elsewhere, the Oracle had disabled the vast thinking-machine fleet, leaving it without its controlling force.\n\n_And we still remain._\n\nDuncan didn't know exactly what had changed inside him. Was it simply the _knowledge_ of his raison d'etre? Had he always had access to this potential without realizing it? Assuming Paul was correct, something had lain dormant inside Duncan for all those years, through all of the lives\u2014original and ghola\u2014a latent power that had grown with each iteration of his existence. Now, like a massive genetic program, he had to figure out how to activate it.\n\nPaul and his son Leto II had the blessing and curse of prescience. With their memories restored, each could claim to be a Kwisatz Haderach. Miles Teg had possessed his phenomenal capacity to move at a speed beyond comprehension and might conceivably have become a Kwisatz Haderach himself. The Navigators in the clustered Heighliners overhead could use their minds to see through folds of space and find safe paths for the great ships to travel. The Bene Gesserits could control their bodies, down to their very cells. All had expanded on traditional human abilities, expressing humankind's potential to exceed expectations.\n\nAs the ultimate and final Kwisatz Haderach, Duncan believed he might have the capability to do all of those things and much more, reaching the highest pinnacle of humanity. Thinking machines had never understood human potential, even though their \"mathematical projection\" credited the Kwisatz Haderach with the power to end Kralizec and change the universe.\n\nConfidence infused him, and he thought he might discover a way to make grand, epic changes . . . but not under the control of the thinking machines. Instead, Duncan would find his own way. He would be a real Kwisatz Haderach, independent as well as all-powerful.\n\nDispassionately, he gazed upon the old woman in her frumpy floralprint dress and gardening apron, complete with scuffs of dirt. Her face appeared careworn, as if from nurturing people her entire life. \"Something of Omnius has vanished from me, but not all.\"\n\nFinally forsaking the old-woman disguise, Erasmus resumed the liquid-metal form of the independent robot attired in an elegant crimson and gold robe. \"I can learn much from you, Duncan Idaho. As the new god-messiah of humankind, you are the optimal specimen for me to study.\"\n\n\"I am not another specimen for your laboratory analysis.\" Too many others had treated him that way, in too many of his past lives.\n\n\"A mere slip of my tongue.\" The robot smiled cheerily, as if attempting to veil his looming violence. \"I have long desired a perfect understanding of what it means to be human. Now it seems _you_ have all the answers I so assiduously sought.\"\n\n\"I recognize the myth in which I live.\" Duncan recalled Paul Atreides making similar pronouncements. Paul had felt trapped by his own mythos, which had become a force beyond his control. Duncan, however, had no fear of the forces that would emerge, either for or against him.\n\nWith penetrating vision he saw through, and around, Erasmus and his minions. Across the hall he watched Paul Atreides standing unsteadily, aided by Chani and Jessica after his terrible ordeal. Paul drank from a water flagon, which he had taken from a table near the Baron's body.\n\nOutside, the crashing of sandworms against robotic defenders had begun to subside. Though the huge creatures had not destroyed the machine cathedral, they had caused extensive damage to the city of Synchrony.\n\nAt the perimeter of the great chamber, quicksilver robots stood attentively, the charges in their integral weapons glowing in a display of readiness. Even without the evermind, Erasmus could direct these machines to fire a deadly barrage against the humans in the vaulted room. The independent robot could attempt to kill every mortal here in a show of petulant revenge. And perhaps he would make the effort. . . .\n\n\"Neither you nor your robots can make any difference here,\" Duncan warned. \"All of you are far too slow.\"\n\n\"Either you are overconfident, or you are fully aware of what you can do.\" The flowmetal smile tightened, just a little, and the bright optic threads glistened a bit more. \"Perhaps it is the latter, and perhaps not.\" Somehow, Duncan knew with absolute certainty that Erasmus meant to unleash all the destructive power under his control, wreaking whatever havoc he could.\n\nBefore the robot made half a turn, Duncan was upon him with all the speed Miles Teg had shown, knocking him backward. Erasmus crashed to the floor, his weapons disabled. Was it just a test? Another experiment?\n\nDuncan's heart pounded and his body radiated heat as he stood over the robot, but he felt exhilarated, not exhausted. He could keep fighting like this against any machines Erasmus chose to send against him. At that thought, he left the independent robot where he had fallen, dashed at hyperspeed around the circle, and battered the silvery sentinel robots with quick kicks and punches until they shattered into debris. It was so easy for him now. Before the metal pieces had finished falling to the floor, he was back, looming over Erasmus.\n\n\"I sensed your doubts as well as your intentions,\" Duncan said. \"Admit it. Even as a thinking machine, you wanted more proof, didn't you?\"\n\nLying on his back and looking upward through the hole in the dome at the thousands of huge Guild Heighliners in the sky, Erasmus said, \"Assuming you are the long-awaited superman, why don't you simply destroy me? With Omnius gone, removing me would assure the victory of humanity.\"\n\n\"If the solution were that simple, a Kwisatz Haderach would not be needed to implement it.\" Duncan surprised Erasmus, and himself, by reaching down and helping the robot to his feet. \"To end Kralizec and truly change the future requires more than just the annihilation of one side or the other.\"\n\nErasmus examined his body core and his robes to ensure his appearance, then looked up with a broad smile. \"I think we just might have a meeting of the minds\u2014something I never really achieved with Omnius.\"\n\n> _When the time comes for our Great Unmasking, our foes will be surprised by what has been disguised in front of them since the very beginning._\n> \n> \u2014KHRONE,  \n> communiqu\u00e9 to Face Dancers\n\nNow that the Oracle was gone, several of the Navigators' giant Heighliners overhead folded space and disappeared from Synchrony without explanations or farewells.\n\nThroughout the city, sandworms continued to destroy the living metal buildings. Because Omnius had never allowed them autonomy, the robotic defenders were unable to function effectively without connecting to the evermind. The vaulted hall filled with resounding silence.\n\nThen with a loud crash, the high doors swung open. Dressed in black and followed by a throng of Face Dancers, Khrone marched in from the bright machine streets. Identical, blank-faced drones swarmed into the room. Scytale's poison gas had killed some of the shape-shifters, but many had avoided the battle entirely.\n\nOut in the sprawling machine city, countless Face Dancers had pretended to stand against the rampaging sandworms, but secretly melted away from the barricades the robotic soldiers had set up. Khrone had taken pleasure in watching the worms destroy the great flowmetal buildings, smashing thousands of thinking machines. _Clearing the way. Making our job easier._\n\nKhrone offered a skeletal smile as he swept forward. \"I never cease to be entertained by the erroneous deductions of those who _think_ they control us.\" In his mind, a Face Dancer victory was now assured.\n\n\"Explain yourself, Khrone.\" Erasmus seemed only mildly curious.\n\nIgnoring the humans and their dead, Khrone faced the independent robot, who stood by Duncan Idaho. \"This war has been in process for five thousand years. It was never Omnius's idea, anyway.\"\n\n\"Oh, our war has been building for far longer than that,\" Erasmus pointed out. \"We escaped after the Battle of Corrin fifteen thousand years ago.\"\n\n\"I'm referring to a completely different war, Erasmus\u2014one you never realized was taking place. From the moment the first advanced Face Dancers were dispatched by our creator, Hidar Fen Ajidica, we began our manipulations. When we encountered your thinking-machine empire, we allowed you to create more and more of us. Yet the moment Omnius let us in, the Face Dancers became his true masters! We shared with you all the lives we had gathered, letting you believe you were becoming increasingly superior to us and to humans. But we Face Dancers were in control all along.\"\n\n\"There is a diagnosis for your mental condition,\" Dr. Yueh said boldly. \"You have delusions of grandeur.\"\n\nKhrone's lips peeled back from blunt, perfect teeth. \"My statements, based as they are on accurate information, can hardly be called delusions.\"\n\nThe amused expression on Erasmus's face did not change. Khrone found it maddening, so he raised his voice, \"You thinking machines helped us implement our Face Dancer plans, all the while believing that we served _you._ But it was exactly the opposite. You were, in fact, _our_ tools.\"\n\n\"All machines began as tools,\" Duncan pointed out, looking from Erasmus to the Face Dancer leader.\n\nKhrone was not impressed. So this was the man who had revealed himself as the final Kwisatz Haderach? Nor could he understand why the independent robot was not more upset, since he prided himself on employing his artificial emotions.\n\nKhrone continued. \"Under your guidance, Erasmus, biological facilities run by thinking machines manufactured millions of enhanced Face Dancers. At first, we ventured into human society as scouts, swiftly infiltrating the fringes of the Scattering, and then the Old Empire. We easily duped the Lost Tleilaxu into believing we were their allies. Wherever humans remained, Face Dancers quietly intruded. We lived long, and accomplished much.\"\n\n\"Exactly as we instructed you to do,\" Erasmus said, sounding bored with the lecture.\n\n\"Exactly as we wished to do!\" Khrone snapped back. \"Face Dancers are everywhere, a hive mind more advanced than any extrasensory human linkages, more powerful than the network of Omnius. So swiftly and easily we accomplished our aims.\"\n\n\"And our aims, as well,\" Erasmus said.\n\nGalled by the robot's stubborn refusal to recognize defeat, Khrone felt rage building within him. \"Over the centuries, we prepared for the day when we would implement our plan and eliminate Omnius. We never guessed that the Oracle of Time would do it for us.\" He chuckled softly. \"Your empire has fallen. We have superceded all thinking machines. And now that Omnius's fleet and plagues have brought humanity to its knees, we can activate our hidden Face Dancer cells\u2014everywhere, simultaneously. We will take control.\" He planted his fists on his hips. \"It is already over for machines, and for humans.\"\n\nBehind him, all the identical Face Dancers wore blank expressions. Khrone's featureless face had been duplicated many times over.\n\n\"An interesting and insidious plan,\" Erasmus said. \"Under other circumstances, I might applaud you for your ingenuity and duplicity.\"\n\n\"Even if you could rally your robots to kill those of us on Synchrony, it would be of no use. I am reproduced everywhere.\" The Face Dancer scoffed. \"Omnius thought he was seeding the universe for his own conquest, but the true seeds of his downfall were right under his mechanical nose.\"\n\nErasmus began to laugh. It started as a chuckle that he imitated from an ages-old dataset, and he added components sifted from other recordings. The resultant sounds were quite enjoyable to himself, and he was sure they were convincing to the others.\n\nOver his long, long lifetime, the unusual robot had expended a great deal of effort in studying humans and their emotions. Laughter particularly intrigued him. An early step, which had required centuries of deep thought, had been to understand the _concept_ of humor, to learn what circumstances might elicit this strange, noisy response from a human. In the process, he had compiled a library of his favorite laughter samples. A delightful repertoire.\n\nHe played them all now through his mouth speakers, much to the bewilderment of the Face Dancer Khrone. But Erasmus realized that not even these favorite chuckles, snickers, and guffaws were adequate to express the true hilarity that he currently felt.\n\n\"What is so funny?\" Khrone demanded. \"Why are you laughing?\"\n\n\"I am laughing because even you don't understand the trick that was played on you.\" Erasmus chuckled again, and this time he created a unique sound that contained flavors and undertones of his best borrowed recordings. This was truly _his_ individual sense of humor, something genuinely original. After such long and difficult study, Erasmus was pleased with the new comprehension he had achieved. Surely this was worth all the tribulations of Kralizec!\n\nThe independent robot turned to Duncan Idaho, who\u2014after listening to the betrayals upon betrayals\u2014had the faraway look of a man trying to join mismatched puzzle pieces. Erasmus knew that Duncan hadn't the faintest idea of how to achieve his full potential. Just like so many other humans! The robot would have to guide this one.\n\nIgnoring Khrone, he spoke to Duncan. \"I am laughing because the inherent differences between humans and Face Dancers are painfully hilarious. I hold great fondness for your species\u2014as more than specimens, more than pets. You have never ceased to astonish me. In defiance of my most careful predictions, you still manage to do the unexpected! Even when those actions work to the detriment of thinking machines, I can appreciate them for their uniqueness.\"\n\nKhrone and his contingent of Face Dancers closed in, as if expecting to mop up these few robots and humans easily. \"Your words and laughter are meaningless.\"\n\nJessica supported a still-weak Paul, while Chani picked up the bloodied dagger that Paolo and Dr. Yueh had both used. Now that she had her past memories, Chani held the weapon in the manner of a true Fremen woman, ready to defend her man.\n\nErasmus smiled to himself. His own confrontation with Duncan had shown only the tip of the iceberg of the Kwisatz Haderach's powers. The robot had found it terribly exciting for a few moments, placing himself at the brink of death, or at least its equivalent for a machine.\n\nThe Face Dancers would be in for quite a surprise if they thought Duncan Idaho and these other humans would be an easy conquest. But Erasmus had an even greater surprise to spring.\n\n\"What I _mean_ , my dear Khrone, is that while humans can astonish me, Face Dancers are woefully predictable. It's a shame. I had hoped for something more original in your case.\"\n\nWhen Khrone scowled, every one of the Face Dancers in the chamber mimicked his expression, like reflections in a hall of mirrors. \"We've already won, Erasmus. Face Dancers control every foothold, and you have no place to hide from us. We will rise up across the human planets and on all machine worlds. We will look back upon the path of destruction, and only we will remain.\"\n\n\"Not if I choose to stop it.\" Erasmus's flowmetal face shifted into a placid, disappointed expression. \"Omnius might have been convinced that you all were our meek puppets, but I never believed as much. Who can ever trust a Face Dancer? Among humans, that saying has become a clich\u00e9. You and your counterparts did exactly what I predicted you would do. How could you not? You are what you are. It was practically programmed into you.\" The robot gave a sad shake of his head.\n\n\"While you Face Dancers were laying down your schemes, sending out spies, and establishing your presence, I watched patiently. Though you thought you were hidden from Omnius, you weren't clever enough. I saw everything you did and _allowed_ it to happen because I found your petty power plays amusing.\"\n\nKhrone adopted a fighting stance, as if ready to attack the robot with bare hands. \"You know nothing of our activities!\"\n\n\"Now who is drawing conclusions from insufficient data? Ever since the end of the Butlerian Jihad, when Omnius and I were sent out here on our long exile to start the machine empire all over again, _I_ was the one in control. I allowed Omnius to continue believing he ruled everything and made all the decisions, but even in his first incarnation he was a self-aggrandizing annoyance, overconfident and unconscionably stubborn. More so than most humans!\" The robot swirled his plush robes. \"The evermind never learned to adapt and never bothered to face his mistakes, so I refused to let him ruin our chances again. Thus, I took control of the Face Dancer program from the moment the first of you arrived on our fringe planets.\"\n\nKhrone remained defiant, though his voice carried a slightly uncertain undertone. \"Yes, you manufactured us\u2014and made us stronger than ever.\"\n\n\"I manufactured you, and I wisely planted a fail-safe routine in each and every Face Dancer. You are biological machines, evolved and manipulated over thousands of years, according to my own exacting specifications.\" Erasmus moved closer. \"A tool should never confuse itself with the hand that wields it.\"\n\nThe gathered Face Dancers seemed to hold the balance of power, and Khrone did not back down. His features shifted into a monstrous, demonic mask of fury. \"Your lies cannot control us any longer. There is no fail-safe.\"\n\nErasmus emitted a poignant sigh. \"Wrong again. This is my proof.\" With a precise nod of his burnished head, he triggered the implanted shutdown virus that was genetically buried within each of the customized, \"enhanced\" Face Dancers.\n\nLike a toy discarded by a petulant child, the Face Dancer standing immediately behind Khrone crumpled lifeless to the floor, arms and legs akimbo, an expression of momentary shock animating his face before it reverted to its blank state.\n\nKhrone stared, unable to comprehend. \"What are\u2014\"\n\n\"And this.\" Erasmus nodded again. With a swift sighing thump, the throng of Face Dancers in the cathedral chamber also dropped, an army scattered in death as if mowed down by projectile fire, leaving Khrone alive to accept his utter defeat.\n\nThen, after stretching out the moment for effect, the independent robot said, \"And _this._ Your services are no longer required.\"\n\nWith his face twisted in rage and desperation, Khrone threw himself toward Erasmus\u2014only to fall to the stone floor, as dead as the rest of his brethren.\n\nErasmus turned to Duncan Idaho. \"So, Kwisatz Haderach\u2014as you see, I control fundamental parts of our intriguing game. I would not suggest my powers are as great as your own, but in this particular case, they are quite useful.\"\n\nDuncan did not show any awe. \"How far will your shutdown virus spread?\"\n\n\"As far as I wish. Even though the Oracle of Time extracted Omnius from the tachyon net, the strands of that vast interconnected mesh still exist in the fabric of the universe.\" Erasmus twitched his head again and sent out a signal. \"There, I just dispatched my trigger to every modified Face Dancer throughout human civilization. They are dead now. All of them. They numbered in the tens of millions, you know.\"\n\n\"So many!\" Jessica exclaimed.\n\nLetting out a whistle, Paul said \"Like a silent jihad.\"\n\n\"You would never have known most of them. With memory imprints, some even believed they were human. All across what remains of your former empire, a great many people are probably quite surprised as comrades, leaders, friends, and spouses drop dead where they stand and transform into Face Dancers.\" Erasmus laughed again. \"With a single thought I've eliminated our enemies. Our _common_ enemy. You see, Duncan Idaho, we need not be at odds.\"\n\nDuncan shook his head, feeling oddly sickened. \"Once again, the thinking machine sees total genocide as a simple solution to a problem.\"\n\nNow Erasmus was surprised. \"Don't underestimate the Face Dancers. They were . . . evil. Yes, that is the correct word. And since each one was fundamentally part of a hive mind, they were all evil. They would have destroyed you, and they would have destroyed us.\"\n\n\"We've heard that kind of propaganda before,\" Jessica said. \"In fact, I've heard it cited as the primary reason why all _machines_ need to be destroyed.\"\n\nDuncan looked at all the dead Face Dancers, realizing how much damage the shape-shifters had done for centuries, whether they were guided by the evermind or by their own schemes. Face Dancers had killed Garimi, sabotaged the no-ship, and caused the death of Miles Teg . . .\n\nLooking at the robot, Duncan narrowed his eyes. \"I can't say I'm terribly sorry, but there was no honor in what you\u2014or the Face Dancers\u2014did here. I cannot agree with it. Don't think we are indebted to you.\"\n\n\"On the contrary, it is I who owe so much to you!\" Erasmus could barely contain his pleasure. \"That is exactly the way I'd hoped you would react. After thousands of years of study, I believe I finally understand honor and loyalty\u2014especially in you, Duncan Idaho, the very embodiment of the concept. Even after an event that obviously helps your race, you still object to my tactics on a moral basis. Oh, how wonderful.\"\n\nHe looked down at all the Face Dancers, the astonished and confused expression on Khrone's face. \"These creatures are the exact opposite. And my fellow machines are not loyal or honorable, either. They merely follow instructions because they are programmed to. You have shown me what I needed to know, Kwisatz Haderach. I am very much in your debt.\"\n\nDuncan stepped closer, searching for some way to access the new abilities he knew lay dormant inside him. Just knowing he was the much-anticipated Kwisatz Haderach was not enough. \"Good. Because now I want something from you.\"\n\n> _A single decision, a single moment, can make the difference between victory and defeat._\n> \n> \u2014BASHAR MILES TEG,  \n>  _Memoirs of an Old Commander_\n\n\"It's a trap\u2014it must be.\" Murbella stared at the vast yet motionless Enemy fleet. The human ships were still outnumbered hundreds to one, but the thinking machines made no move. The Mother Commander froze, holding her breath. She had expected to be annihilated.\n\nBut the Enemy did nothing. \"This is deeply unnerving,\" she whispered.\n\n\"All backup systems ready, as you ordered, Mother Commander,\" one of the pale young Sisters announced. \"It may be our only chance to cause some damage.\"\n\n\"We should open fire!\" Administrator Gorus cried. \"Destroy them while they are helpless.\"\n\n\"No,\" said another Sister. \"The machines are trying to lure us from our defensive positions. It's a trick.\"\n\nEveryone on the navigation bridge stared at their dark and quiet foe, afraid to breathe. The robot vessels just drifted out there in the cold void.\n\n\"They have no need to trick or trap us,\" Murbella finally said. \"Look at them! They could destroy us any time they like. It was foolish, impulsive Honored Matre violence that triggered this very war in the first place.\" The Mother Commander narrowed her gaze, studying the overwhelming force of warships. Utter stillness. \"This time, I will take a moment to understand before we just open fire.\"\n\nMurbella's eyes blazed as she struggled to comprehend. She remembered when her eyes had been a hypnotic green\u2014an alluring feature that had helped her ensnare Duncan. _Strange, the thoughts that haunt you when death waits at your door . . ._\n\nAt the time of Duncan's escape from Chapterhouse, no one had known the identity of the outside Enemy. Now, the Oracle had said Duncan was on Synchrony at the heart of the thinking-machine empire. Had he managed to get away? If Duncan was still alive, she could forgive him anything. How she longed to see him again, and hold him!\n\nThe painful silence stretched out. Another excruciating minute, followed by another. Murbella had seen the thinking-machine forces on the move from planet to planet, and the aftermath of their strikes. She had seen the plagues they disseminated and had buried her own daughter Gianne with so many others in an unmarked grave out in the Chapterhouse desert. \"No matter what the reason,\" she said, \"the machines have never been so vulnerable.\"\n\nFrom her nearby ship, Janess gruffly acknowledged. \"If we are going to die in battle, why not take out as many of the Enemy as we can?\"\n\nMurbella had already prepared for this moment. She issued her orders, each word carrying a sharp edge. \"All right, I don't know why, but we've got an unexpected reprieve. We may be few, but we'll be like D-wolves with sharp fangs. We'll rely on our own eyes and skills.\"\n\nOne of the Guildsmen who had rushed aboard the ship at the last minute reacted with alarm. He was a bald and pasty-faced man with tattoos on his scalp. \"Aiming our weapons will require precision maneuvering, Mother Commander! We can't do it without assistance.\"\n\nMurbella shot him a wilting glare. \"I'd rather rely on my eyes than on Ixian systems. I've already been deceived once today. Target the largest ships. Destroy their weapons, disable their engines, and move on to others.\"\n\nJaness transmitted to the clustered defenders, \"The wreckage of all those Enemy ships can provide cover if the machines fire back at us.\"\n\nThe bald Guildsman objected again. \"Every piece of debris is a navigational hazard. No human can react fast enough. We need the Ixian devices back online, at least in a limited fashion.\"\n\nEven Gorus looked at him strangely. Suddenly, the bald Guildsman shouted, turned from his technical station, and collapsed. Near him, without a sound, another of the new crewmen dropped dead in his tracks. A third slumped over on the upper navigation deck.\n\nSuspecting that their ships were under some kind of invisible attack from a silent, deadly weapon, the Sisters reacted quickly, trying to determine what was happening. Murbella hurried to the tattooed Guildsman, rolled him over, and watched his puttylike face shift to the blank visage of a Face Dancer.\n\nGorus looked around as if he finally realized how he had been betrayed. The other two fallen bodies also shifted. All Face Dancers! Murbella glared at the Administrator. \"You guaranteed me that everyone had been tested!\"\n\n\"I spoke the truth! But in the rush to lauch your whole fleet, someone might have been missed. And what if one or more of the testers happened to be a Face Dancer?\"\n\nShe turned from him in disgust. A flurry of transmissions arrived from the other defender vessels, all reporting dead Face Dancers onboard. Amidst the jumble of comm activity, Janess's voice came in sharp and clear. \"Five Face Dancers were on my vessel, Mother Commander. All are now dead.\"\n\nMeanwhile, the listless Enemy ships continued to drift apart, though they could easily have pressed their attack on Chapterhouse and achieved victory. Murbella's thoughts spun, wrestling with yet another mystery. _Face Dancers among us, working for Omnius. But why did they drop dead?_\n\nNot long ago, the Oracle of Time had whisked her numerous Heighliners away from this battlefield to Synchrony . . . to Duncan. Had the Oracle and her Navigators struck a blow that sent ripples through the entire Enemy fleet? Had _Duncan_? Something seemed to have shut down the thinking-machine battle fleet and all their shape-shifter spies.\n\nMurbella indicated the dead Face Dancers sprawled near her. \"Get those monstrosities out of here.\" Not bothering to hide their revulsion, several Sisters dragged the scarecrowish bodies away.\n\nMurbella focused on the screen with such intensity that her eyes burned. The Honored Matre part of her wanted to strike and kill in a frenzy, but all of her Bene Gesserit training screamed for her to _understand_ first. Something essential had changed here. Even the voices of Other Memory couldn't counsel her. Thus far, they had been mute.\n\nRepresentatives of the remaining populations on Chapterhouse transmitted urgent messages, demanding reports from the front, wondering how long they might expect to survive. With no answers for them, Murbella didn't respond.\n\nJaness transmitted a brash suggestion. \"Mother Commander . . . should we board one of the Enemy ships? It could be our best chance to discover what's happened.\"\n\nBefore she could answer, space distorted again around them. Four huge Heighliners reappeared, emerging in the debris-strewn battle zone so close to the human defenders that Murbella shouted for evasive action. The Guild pilot on one of the nearby ships reacted with an exaggerated maneuver, pulling his heavy cruiser out of the way and nearly colliding with Janess's vessel. Another careened into a debris field of destroyed first-wave machine ships.\n\nA third defender acted impulsively and opened fire on the silent thinking-machine fleet, launching a volley of explosive projectiles into the conical nose of the nearest machine battleship. Fiery eruptions burst out in a repeating pattern along the Enemy vessel's hull.\n\nAlarms rang out, and Murbella demanded reports, wondering if the machines would respond with a massive display of force. No more caution. \"Prepare to fire! All ships, prepare to fire! Hold nothing back!\"\n\nBut even thus provoked, the darkened Omnius fleet remained motionless. The Enemy vessel damaged in the impulsive barrage careened in a slow drift, still burning. Very slowly it crashed into an adjacent machine ship and caromed off, sending them both spinning.\n\nThe Enemy ships did not fire a single return shot. Murbella couldn't believe it.\n\nIn the midst of the surprise and mayhem, a Navigator's voice sounded calm and otherworldly. \"The Oracle of Time has sent us here to locate the commander of the human forces.\"\n\nMurbella pushed her way to a commline station. \"I am Mother Commander Murbella of the New Sisterhood . . . of all humanity.\"\n\n\"I have orders to escort you to Synchrony. I will now take command of your foldspace engines.\"\n\nBefore her Guildsmen could scramble to their stations, the Holtzman engines hummed at a higher pitch. Murbella felt a familiar shifting sensation.\n\n> _It is too simplistic to state that humans are the enemies of all thinking machines. I strive to understand these creatures, but they remain incomprehensible to me. Even so, I greatly admire them._\n> \n> \u2014ERASMUS,  \n> private files, secure database\n\n\"You want something from me?\" Erasmus seemed to find Duncan's demand amusing. \"And how will you force me to obey?\"\n\nThe man's lips quirked in a faint smile. \"If you truly understand honor, robot, I won't need to. You will do what's right and pay your debt.\"\n\nErasmus was genuinely delighted. \"What else do you wish from me? Isn't it enough that I eliminated all Face Dancers?\"\n\n\"You and Omnius were responsible for far more mischief than those shape-shifters.\"\n\n\"Mischief? It was rather more than mischief, wasn't it?\"\n\n\"And to atone for it, there's something you need to do.\" Duncan's attention was entirely focused on the robot, not on the dead Face Dancers, not on the destructive sounds of sandworms outside in the city. Paul, Chani, Jessica, and Yueh all remained quiet in the chamber, watching him.\n\n\"I am the final Kwisatz Haderach,\" Duncan said, feeling the nascent abilities embedded within him all the way down to his DNA, \"yet I need to comprehend so much more. I already understand humans\u2014maybe better than anyone else\u2014but not thinking machines. Give me a good reason why I shouldn't just eliminate you all, now that the thinking machines are weakened. It's what the evermind would have done to us.\"\n\n\"Yes, it is. And you are the final Kwisatz Haderach. The decision is yours.\" Erasmus seemed to be waiting for something, his optic threads gleaming like a cluster of stars.\n\n\"And is there a way that doesn't require the annihilation of one or the other? A fundamental change in the universe\u2014Kralizec.\" Duncan stroked his chin, thinking. \"Omnius's fleet contains millions of thinking machines. They're not destroyed, but simply without guidance, correct? And I believe your empire contains hundreds of planets, many of which would never be habitable to humans.\"\n\nWith his robes flowing around him, the platinum robot began to stroll through the great vaulted hall, stepping over Face Dancer corpses that lay strewn everywhere like marionettes with their strings cut. \"That is an accurate assessment. Do you want to find them all, destroy them all, hoping you never miss one? Now that they are without the evermind, it's even possible that some of the more sophisticated machines could develop independent personalities during a time of long deprivation, as I did. How confident are you in your abilities?\"\n\nDuncan followed him closely. Several times, Erasmus glanced back at him, and made an odd series of expressions, from inquisitive scowls to tentative smiles. Did he see a bit of fear there, or was it feigned? \"You're asking me if I want victory . . . or peace.\" It was not a question.\n\n\"You are the superhuman. I say it again\u2014decide for yourself.\"\n\n\"Through more lifetimes than I can count, I've learned patience.\" Duncan took a long, deep breath, using an old Swordmaster technique to center his thoughts. \"I'm in a unique position to draw both sides together. Humans and machines are both battered and weakened. Do I choose extermination for one side as the solution?\"\n\n\"Or recovery for both?\" Erasmus stopped, and with a blank expression faced the man. \"Tell me, what precisely is that dilemma? Omnius has been ripped from the universe, and the rest of the thinking machines have no leadership. In one swift blow I have expunged the entire Face Dancer threat. I fail to see anything left to solve. Hasn't the prophecy come true?\"\n\nDuncan smiled. \"As is the case with so many prophecies, the details are vague enough to convince any gullible mind that everything was 'foretold.' The Bene Gesserit and their Missionaria Protectiva were masters at that.\" He looked closely at the robot. \"And so, I think, are you.\"\n\nErasmus seemed both surprised and impressed. \"What are you suggesting?\"\n\n\"Since you were in charge of the 'mathematical projections' and the 'prophecies' based on them, you were in a position to write predictions however you wished. Omnius believed everything.\"\n\n\"Are you saying I _made up_ the prophecies?\" Erasmus asked. \"Perhaps as a way to guide an evermind stubbornly intent on a narrowminded course of action? Perhaps to bring us precisely to this juncture? A very interesting hypothesis. One worthy of a true Kwisatz Haderach.\" The grin on his face seemed more genuine than ever.\n\nSmiling coolly, Duncan said, \"As the Kwisatz Haderach, I know there are\u2014and always will be, even as I evolve\u2014limitations on my knowledge and my abilities.\" He tapped the robot in the center of his chest. \"Answer me. Did you manipulate the prophecies?\"\n\n\"Humans created countless projections and legends long before I existed. I simply adapted the ones I liked best, generated the complex calculations that would produce the desired projections, and fed them to the evermind. Omnius, with his usual myopia, saw only what he wanted to see. He convinced himself that in the 'end' a 'great change in the universe' required a 'victory' for him. And for that he needed the Kwisatz Haderach. Omnius learned many things, but he learned arrogance too well.\" Erasmus swirled his robes. \"No matter what the evermind or the Face Dancers thought\u2014 _I_ have always been in control.\"\n\nRaising his hands, the robot gestured to the sentient metal cathedral around them, indicating the whole city of Synchrony and the rest of the thinking-machine empire. \"Our forces are not entirely leaderless. With the evermind gone, I now control the thinking machines. I have all the codes, the intricate, interlinked programming.\"\n\nDuncan had an idea that was part prescience, part intuition, and part gamble. \"Or the final Kwisatz Haderach can take control.\"\n\n\"That seems a much neater solution.\" An odd expression moved across the robot's flowmetal face. \"You interest me, Duncan Idaho.\"\n\n\"Give me the codes and the access I need.\"\n\n\"I can give you more than that\u2014and, yes, it will require much more. A whole machine empire, millions of components. I would have to share an . . . _entirety_ with you, just as my Face Dancers shared all those marvelous lives. But for a Kwisatz Haderach, that would be just the thing.\"\n\nBefore the robot could laugh again, Duncan reached forward and grabbed the platinum hand that extended from the plush sleeve. \"Then do it, Erasmus.\" He pressed closer, reached out his other hand and pressed it against the robot's face in a curiously intimate gesture. Prescience seemed to be guiding him.\n\n\"Duncan, this is dangerous,\" Paul said. \"You know it.\"\n\n\"I'm the one who's dangerous, Paul. Not the one in danger.\" Duncan pulled himself to within inches of Erasmus, feeling all the possibilities roil within him. Though there were troublesome blind spots in the future, pitfalls and traps he might not be able to foresee, he felt confident.\n\nThe robot paused, as if calculating, then gripped Duncan's hand and\u2014in a like gesture\u2014reached out with the other to touch his face. Duncan's dark brows knitted as he experienced strange sensations. The cool metal felt alarmingly soft, and he almost had the sensation of falling into it. He extended himself, stretching his mind toward the uncharted territory of the independent robot's thoughts, just as Erasmus did the same to him. The robot's fingers elongated, spreading out over Duncan's hand like a glove. As flowmetal covered Duncan's wrist and ran up his forearm, it felt bitingly cold as Erasmus began to talk. \"I sense a growing trust between us, Duncan Idaho.\"\n\nAs moments passed, Duncan couldn't tell if he was taking from the robot, or if Erasmus was surrendering what the nascent Kwisatz Haderach needed, _everything_ he needed. And, though the two of them were fused, Duncan had to go further. A viscous, metallic substance covered his arm like the sandtrout that had engulfed young Leto II's body, so long ago.\n\n> _I hear the clarion call of Eternity beckoning me._\n> \n> \u2014LETO ATREIDES II,  \n> records from Dar-es-Balat\n\nWith the machine city heavily damaged and the evermind Omnius gone, the major components of Synchrony stopped moving. The buildings no longer pumped and shifted like interlocking puzzle pieces, no longer morphed into strange shapes. Like an immense broken engine, the city had ground to a complete halt, leaving many streets blocked, structures half buried or partially formed, and tramcars suspended in the air, dangling on invisible electronic wires. Grotesque Face Dancer bodies and smashed combat robots littered the streets. Columns of fire and smoke rose into the sky.\n\nExhausted even in victory, Sheeana stared around the city, her face filled with awe and pleasure. As she walked alone down a devastated street, she saw a young boy standing there by himself between the towering, exotic buildings. Looking wrung out but far more powerful than she had ever seen him, was the transformed boy Leto II. He had left the sandworms, having directed them off into the city, but even though he stood here in front of her, he was still part of them.\n\nAs Leto craned his neck to look up at one of the dangling tramcars, Sheeana noticed an oddness about him, a looming presence that hadn't been there before. She understood. \"You have your memories back.\"\n\n\"In perfect detail. I've been reviewing them.\" Leto's eyes were full of centuries, now completely blue-within-blue due to incredible spice saturation from the bodies of the sandworms he had controlled. \"I am the Tyrant. I am the God Emperor.\" His voice sounded louder, yet carried a deep and abiding weariness.\n\n\"You are also Leto Atreides, brother to Ghanima, son of Muad'Dib and Chani.\"\n\nIn response, he smiled as if she had lifted some of his burden. \"Yes, that too. I'm everything my predecessor was\u2014and everything the worms are. The pearl of dreaming inside them has been broken open. He sleeps no more.\"\n\nSheeana recalled the quiet boy aboard the no-ship. His past had been worse than anyone else's, and now that innocent boy was truly gone.\n\n\"I remember every death I caused. Every one. I remember all of my Duncans, and the reasons each died.\" He looked up, then grasped her arm and pulled her back toward a twisted building that was stuck halfway out of the ground.\n\nSeconds later, the invisible suspensor line high above snapped, and the tramcar hurtled down to smash on the street exactly where the two had been standing. Dead Face Dancers lay sprawled in the wreckage.\n\n\"I _knew_ it would fall,\" Leto said.\n\nShe smiled gently. \"We each have our special talents.\"\n\nThe two of them climbed the high rubble of a collapsed building to get a better view of the city's wreckage. Confused and disoriented robots milled around the smoldering piles of wreckage and broken structures, as if waiting for instructions.\n\n\"I am a Kwisatz Haderach,\" Leto II said, his voice distant. \"And so was my father. But it is much different now. Did I plan for all this long ago, as part of my Golden Path?\"\n\nAs if he had summoned them, four sandworms rose noisily from the churned and smashed ground and loomed over the wreckage. She heard loud grinding noises, and the remaining three worms came from other directions, knocking buildings aside, tunneling through the wreckage. Slightly larger than before, they circled Leto and Sheeana.\n\nThe largest worm, the one she had named Monarch, turned its head toward the two of them. Unafraid, Leto climbed down the remains of the building to approach the creature.\n\n\"My memories are back,\" Leto said to Sheeana, stepping forward, \"but not the dreaming existence I had as the God Emperor, back when man and worm were one.\" Monarch laid its head on the base of the rubble pile, as did the companion worms, like supplicants before a king. The cinnamon odor of melange filled the air from the exhalations of the beasts.\n\nReaching out, Leto stroked the rounded edge of Monarch's mouth. \"Shall we dream together again? Or should I let you go back to a peaceful sleep?\"\n\nWithout fear, Sheeana also touched the worm, feeling the hard skin of the rings.\n\nWith a sigh, the boy added, \"I miss the people I used to know, especially Ghanima. Your ghola program didn't bring her back with me.\"\n\n\"We didn't consider personal costs or consequences,\" Sheeana said. \"I'm sorry.\"\n\nTears welled in Leto's dark blue eyes. \"There are so many painful memories from before I took the sandtrout as part of me. My father refused to make the choice I did\u2014refused to pay the price in blood for the Golden Path, but I thought I knew better. Ah, how arrogant we can be in our youth!\"\n\nIn front of Leto, the largest worm lifted. Its open mouth looked like a cave full of rich spice.\n\n\"Fortunately I know how to go back into the dreaming essence of the Tyrant, the God Emperor. To the real son of Muad'Dib.\" With a glance at her, he said, \"I take my last few sips of humanity.\" Then he entered the towering mouth and climbed over the maw-fence of crystalline teeth.\n\nSheeana understood what he was doing. She had tried the same thing herself, though ineffectively. The worm engulfed Leto II, closed its mouth, and reared back. The boy was gone.\n\nSheeana struggled to keep her knees from buckling. She knew she would never see Leto again, though he would be with the worms eternally, merged into Monarch's flesh from the inside, becoming a pearl of awareness once more. \"Goodbye, my friend.\"\n\nBut the spectacle was not finished. The other worms rose beside Monarch, and all towered over her. Sheeana stood motionless, at once horrified and fascinated. Would they devour her, too? She steeled herself for her own fate, but had no fear of it. As a young girl, after a worm had destroyed her village on Rakis, Sheeana had run wildly out into the desert and screamed at the huge creature, calling it names, insisting that it eat her. \"Well, Shaitan\u2014do you have an appetite for me, now?\"\n\nBut they did not want her. Instead, the seven worms gathered together, tumbling one upon the other, writhing like a mass of snakes. With Leto inside them now, the worms were transforming. Six worms wound themselves around the largest beast that had swallowed the boy. They twisted and twined, wrapping their sinuous bodies like vines around a tree, and then moved together.\n\nSheeana scrambled back up the rubble pile to keep herself safe from falling debris. The fleshy rings of the separate sandworms began to merge and metamorphose into a much larger form. The differentiation among the creatures became less distinct; the rings united, joining into one incredible sandworm: a behemoth greater even than the largest monsters from legendary Dune.\n\nSheeana stumbled, falling backward on the rubble but unable to tear her gaze away from the immense sandworm that towered in front of her, rippling and twining, its body stretching back hundreds of meters.\n\n\"Shai-Hulud,\" she murmured, intentionally refusing to use the term _Shaitan_ , just as she had always done. Truly, this was the godlike Old Man of the Desert. The dizzying odor of melange was stronger than ever.\n\nAt first she thought the leviathan would consume her after all, but the giant worm turned away and smashed down into the ground with a great thunder of noise, tunneling downward beneath the machine city.\n\nIts new home.\n\nA shudder of supreme pleasure ran through her. She knew the great worm would divide beneath the surface. This union between Leto II and the creatures would have a greater resistance to moisture, enabling them to survive until they could remake parts of this former machine planet into a domain of their own. One day, new sandworms would grow and thrive on this world, always lurking beneath the surface, always watching.\n\n> _To defeat the humans, one option is to become like them, granting no quarter, chasing and destroying them to the last man, woman, and child. Just as they tried to do to us._\n> \n> \u2014ERASMUS,  \n> databank on human violence\n\n\"With my curiosity, ages of existence, and understanding of both humans and machines,\" Erasmus mused as he and Duncan remained joined, fused together mentally and physically, \"am I not the machine equivalent of a Kwisatz Haderach? The Shortening of the Way for thinking machines? I can be in many places at once and see a myriad of things that even Omnius never imagined.\"\n\n\"You are not a Kwisatz Haderach,\" Duncan said. He became aware of his comrades rushing toward him. But the liquid metal now flowed across Duncan's shoulders and face, and he felt no desire to tear himself away.\n\nDuncan let the physical reaction between him and the robot continue. He didn't want to escape. As the new standard bearer of humankind, he needed to advance. So he opened his mind and let the data rush in.\n\nA voice rang out in his head, louder than all the whirlwind memories and streams of data. _I can impress all of the key codes you seek, Kwisatz Haderach. Your neurons, your very DNA, form the structure of a new networked database._\n\nDuncan knew this was the point of no return. _Do it._\n\nThe mental floodgates opened, filling his mind to bursting with the robot's experiences and coldly factual, regimented information. And he began to see things from that entirely alien viewpoint.\n\nIn thousands upon thousands of years of experimentation, Erasmus had struggled to understand humans. How could they remain so mysterious? The robot's incredible range of experiences made even Duncan's numerous lives seem insignificant. Visions and memories roared around the Kwisatz Haderach, and he knew it would take him much more than another lifetime just to sift through it all.\n\nHe saw Serena Butler in the flesh, along with her baby, and the startling reaction of the multitudes to what Erasmus had thought was a simple, meaningless death . . . howling humans rising up in a fight they had no chance to win. They were irrational, desperate, and in the end, victorious. Incomprehensible. Illogical. And yet, they had achieved the impossible.\n\nFor fifteen thousand years, Erasmus had longed to understand, but had lacked the fundamental revelation. Duncan could feel the robot digging around inside him, looking for the secret, not out of any need for domination and conquest, but simply _to know._\n\nDuncan had difficulty focusing amidst so much information. Presently he withdrew, and felt the flowmetal move the other direction, away from him\u2014though not completely, for his internal cellular structure was forever changed.\n\nIn an epiphany, he realized that he was a new evermind, but of an entirely different sort from the original. Erasmus had not deceived him. With eyes that extended to centillions of sensors, Duncan could see all of the Enemy ships, the fighting drones and worker robots, every cog in the awe-inspiring reborn empire.\n\nAnd he could stop everything in its tracks. If he wanted to.\n\nWhen Duncan returned to himself, in his relatively human body again, he looked through his own eyes around the great chamber. Erasmus stood before him, separate now and smiling with what seemed to be genuine satisfaction.\n\n\"What happened, Duncan?\" Paul asked.\n\nDuncan let out a long breath of stale air. \"Nothing I didn't initiate, Paul, but I'm here, I'm back.\"\n\nYueh rushed up. \"Are you hurt? We thought you might be trapped in a coma like . . . like him.\" He gestured toward the still-frozen Paolo.\n\n\"I'm unharmed . . . but not unchanged.\" Duncan looked around the vaulted chamber, and gazed out into the vast city with a new sense of wonder. \"Erasmus shared _everything_ with me . . . even the best parts of himself.\"\n\n\"An adequate summation,\" the robot said, undeniably pleased. \"When you merged into me and kept going deeper and deeper, you made yourself vulnerable. Had I wished to win the game, I could have tried to take over your mind and program you to do exactly what benefits me and thinking machines. Just as I did with the Face Dancers.\"\n\n\"But I knew you wouldn't,\" Duncan said.\n\n\"From prescience, or faith?\" A crafty smile crept across the robot's face. \"You now have control of the thinking machines. They are yours, Kwisatz Haderach\u2014all, including me. Now you have everything you need. With the power in your hands, you will change the universe. It is Kralizec. See? We have made the prophecy come true after all.\"\n\nSeemingly alone in the remnants of a vast empire, Erasmus walked casually around the chamber again. \"You can shut them all down permanently, if that is your preference, and eliminate thinking machines forever. Or, if you have the courage, you can do something more useful with them.\"\n\nJessica said, \"Shut them down, Duncan. Finish it now! Think of all the trillions they've killed, all the planets they've destroyed.\"\n\nDuncan looked at his hands in wonder. \"And is that the honorable thing to do?\"\n\nErasmus kept his voice carefully neutral, not pleading. \"For millennia I studied humans and tried to understand them . . . I even emulated them. But when was the last time humans bothered to consider what thinking machines could do? You only despise us. Your Great Convention with its terrible stricture, 'Thou shalt not make a machine in the likeness of a human mind.' Is that really what you want, Duncan? To win this ultimate war by exterminating every vestige of us . . . the way Omnius wanted to win the war by eliminating you? Didn't you hate the evermind for that fixed attitude? Do you have the same attitude yourself?\"\n\n\"You have an abundance of questions,\" Duncan observed.\n\n\"And it is up to you to choose the single answer. I gave you what you need.\" Erasmus stood back and waited.\n\nDuncan felt a new sense of urgency, perhaps imparted to him by Erasmus. Possibilities roiled through his head, accompanied by a riptide of consequences. With his growing awareness he saw that in order to end Kralizec, he needed to stop the eons-old schism that separated man and machine. Thinking machines had originally been created by man, but though intertwined, each side had tried repeatedly to destroy the other. He had to find a common ground between them, rather than let one dominate the other.\n\nDuncan saw the great historical arc, a social evolution of epic proportions. Thousands of years ago, Leto II had joined himself with a great sandworm, thereby acquiring vastly greater powers for himself. Centuries later, under the guidance of Murbella, two opposing groups of women had joined forces, fusing their individual cultures into a stronger synthesized unit. Even Erasmus and Omnius had been two aspects of the same identity, creativity and logic, curiosity and rigid facts.\n\nDuncan saw that balance was required. Human heart and machine mind. What he had received from Erasmus could become a weapon, or a tool. He had to use it properly.\n\nI must serve as a synthesis of man and machine.\n\nHe locked gazes with Erasmus, and this time he and the robot connected without making physical contact. Somehow the Kwisatz Haderach retained a ghost image of Erasmus within himself, just as Reverend Mothers carried Other Memories inside.\n\nDrawing a deep breath, Duncan faced the overwhelming question. \"When you and Omnius manifested yourselves as an old couple, you demonstrated the differences between you. Erasmus, while maintaining your own independence, you acquired the evermind's vast store-house of data, the intellect, while Omnius in turn learned about _heart_ from you, what it means to have human feelings\u2014curiosity, inspiration, mystery. But even you never fully achieved all the aspects of humanity you sought.\"\n\n\"But now I can. With your consent, of course.\"\n\nDuncan turned to face Paul and the others. \"After the Butlerian Jihad, human civilization went too far by completely banning artificial intelligence. But in forbidding any sort of computers, we humans denied ourselves valuable tools. That overreaction created an unstable situation. History has shown that such absolute, draconian prohibitions cannot be sustained.\"\n\nJessica said skeptically, \"Yet eradicating computers for so many generations forced us to grow stronger and become independent. For thousands of years, humanity advanced without artificial constructions to think and decide for us.\"\n\n\"As the Fremen learned to live on Arrakis,\" Chani said with clear pride. \"It is a good thing.\"\n\n\"Yes, but that backlash also tied our hands and prevented us from reaching other potentials. Just because a man's legs will grow stronger by walking, should we deny him a vehicle? Our memory improves through steady practice; should we therefore deny ourselves the means to write or record our thoughts?\"\n\n\"No need to throw the baby out with the bathwater, to use one of your ancient clich\u00e9s,\" Erasmus said. \"I threw a baby off a balcony once. The consequences were extreme.\"\n\n\"We didn't do without machines,\" Duncan said, crystallizing his thoughts. \"We just redefined them. Mentats are humans whose minds are trained to function like those of machines. Tleilaxu Masters used female bodies as axlotl tanks\u2014flesh machines that manufactured gholas or spice.\"\n\nWhen Paul looked back at him, Duncan thought that the young man's face seemed deeply old. Recovering from his past life had drained him even more than his mortal wound had. As a Kwisatz Haderach himself, as Muad'Dib, Emperor, and blind Preacher, Paul understood Duncan's dilemma better than any human present. He nodded slightly. \"No one can choose for you, Duncan.\"\n\nDuncan let his eyes take on a far-off glaze. \"We can do much, much more. I see it now. Humans and machines cooperating fully, with neither side enslaving the other. I shall stand between them, as a bridge.\"\n\nThe robot responded with genuine excitement. \"Now you see, Kwisatz Haderach! You have helped me to achieve understanding along with you. You have shortened my way, too.\" Erasmus's flowmetal body shifted like a mechanical version of a Face Dancer, becoming again the wrinkled body of the kindly old woman. \"My long quest is complete. At last, after thousands of years, I understand so much.\" He smiled. \"In fact, there is very little that interests me anymore.\"\n\nThe old woman walked over to where the still-transfixed Paolo lay, staring blankly upward. \"This failed, ruined Kwisatz Haderach is an object lesson for me. The boy paid the price of too much knowledge.\" Paolo's unblinking eyes seemed to be drying out. He would probably wither away and starve to death, lost in the infinite maze of absolute prescience. \"I don't want to be bored. So I ask you, Kwisatz Haderach, help me understand something I could never truly experience, the last fascinating aspect of humanity.\"\n\n\"A demand?\" Duncan asked. \"Or a favor?\"\n\n\"A debt of honor.\" The old woman patted his sleeve with a gnarled hand. \"You now epitomize the finest qualities of man and machine. Allow me to do what only living beings can do. Guide me to my own _death_.\"\n\nDuncan had not foreseen this. \"You want to die? How can I help you do that?\"\n\nThe old woman shrugged her bony shoulders. \"All your lives and deaths have made you an expert on the matter. Look inside yourself, and you'll know.\"\n\nOver the millennia since the Butlerian Jihad, Erasmus had considered distributing backup copies of himself as Omnius had done, but he had decided not to. That would have made his existence far less stimulating, and less meaningful. After all, he was an independent robot, and needed to be unique.\n\nDuncan saw that along with all the codes and commands that controlled the host of thinking machines, he had received the life-function commands that regulated Erasmus. He could shut down the independent robot as easily as Erasmus had shut down all of the Face Dancers.\n\n\"I am curious to see what lies on the other side of the great divide between life and death.\" The robot looked at Khrone and the identical shape-shifter bodies strewn on the floor of the cathedral chamber.\n\nBut it wasn't as simple as flipping a switch or sending a code. Duncan had lived and died over and over, and learned more about life and death than anyone. Did Erasmus want him to understand whether or not a robot could have a soul, now that the two of them had been inside each other's mind?\n\n\"You want me to serve as a guide,\" Duncan said, \"not just an executioner.\"\n\n\"A fine way to put it, my friend. I think you understand.\" The old woman looked at him, and now her smile held a hint of nervousness. \"After all, Duncan Idaho, you have done this over and over again. But this is my first time.\"\n\nDuncan touched her forehead. The skin was warm and dry. \"Whenever you're ready.\"\n\nThe old woman sat on the stone steps. Folding her hands in her lap, she closed her eyes. \"Do you suppose I will ever see Serena again?\"\n\n\"I can't answer that.\" With a mental command, Duncan activated one of the new codes he possessed. From inside his own mind, reaching down to touch his own numerous death experiences, he showed Erasmus what he knew, even if he didn't entirely comprehend it himself. He wasn't certain the ancient independent robot could follow. Erasmus would have to make his own way. He and Duncan parted, both of them traveling on utterly separate journeys.\n\nThe aged body slumped quietly on the steps, and a long sigh flowed from the old woman's lips. Her expression became utterly serene . . . and then went completely motionless, with the eyes staring straight ahead.\n\nIn death, the robot's human shape held.\n\n> _Where there is life, there is hope . . . or so the old sayings tell us. But for the truly faithful there is always hope, and it is not determined by either death or life_.\n> \n> \u2014TLEILAXU MASTER SCYTALE,  \n>  _My Personal Interpretations of the Shariat_\n\nOut under the burned sky of Rakis, Waff's despair took him to a place as bleak and dry as the devastated landscape around him. On a vitrified dune nearby, only one of his precious armored sandworms stirred with the last flickerings of life, while the others were already dead. He had failed his Prophet.\n\nThe cellular modifications he had made were insufficient, and he had neither sandtrout specimens, nor the proper facilities to create additional test worms. He felt the last grains of sand slipping through the hourglass of his life. His body wouldn't last long enough for him to try again with a new line of the hybrid worms, even if he'd had the chance. Only the hope of restoring these sandworms to Rakis had kept him from surrendering to the damage in his accelerated ghola body, but now he was falling apart.\n\nRaising his fist to the sky and shouting into the dry, caustic air, he demanded answers from God, though no mortal had the right to do so. He hammered his hands on the hard, cracked ground and wept. His clothes were dirty, his face smeared with sooty residue. Sprawled atop what had once been magnificent dunes lay the dead worm specimens. Truly, they symbolized the end of all hope.\n\nRakis was forever cursed, if even the Prophet no longer wished to live there.\n\nThen, as he huddled on the ground, Waff felt a shudder from deep beneath the surface. The resonant vibration grew stronger, and he looked up in wonder, blinking his stinging eyes. The last dying worm twitched, as if it, too, could sense something important happening.\n\nWith a thunderous crack in the thin, whistling air, a fissure raced across the glassy ground. Waff stumbled to his feet and stared at the zigzag progress of the widening split, hardly able to comprehend what he was seeing.\n\nWidening, jagged lines appeared like fine fractures in reinforced plaz struck by a hard blow. The dunes bucked and heaved as something emerged from below.\n\nWaff staggered backward. At his feet the last slumped sandworm stirred, as if to warn the Tleilaxu Master that it was about to end its days\u2014and that the man, too, was about to die.\n\nA sequence of explosions erupted like sand geysers from deep beneath the dunes. The crevices gaped wider, revealing forms stirring underground. As if in a waking dream, he saw enormous ridges crusted with stones and dust, huge behemoths rising in a cascade of sand.\n\n_Sandworms_. Real sandworms\u2014monsters of the size that used to roam the desert in the days when this world was known as Dune. A legend and a mystery reborn!\n\nWaff stood transfixed, unable to believe, yet filled with awe and hope rather than fear. Were these survivors of the original worms? How could they still be alive after the holocaust?\n\n\"Prophet, you have returned!\" At first he saw five of the gigantic sandworms, then a dozen emerging at once. All around him the broken ground spawned more and more. Hundreds of them! The whole dead world was like an immense egg, cracking open and giving birth.\n\nBreaking free of their underground nest, the sandworms rampaged toward the distant encampment in the rubble of Keen. Waff supposed they would swallow up Guriff and his prospectors, devouring all of the Guildsmen.\n\nThe sandworms would make Rakis their own again.\n\nHe reeled forward in ecstasy, his hands raised in joyous worship. \"My glorious Prophet, I am here!\" God's Messenger was so great that Waff felt like a minuscule speck, hardly worth noticing.\n\nHis faith swelled again, and he saw that his insignificant efforts on Rakis had never mattered. Regardless of how hard he had worked with the sandtrout, trying to seed these dead dunes with enhanced worms, God had His own plans\u2014always His own plans. He showed the way by producing a flood of life, like the wordless revelation of _s'tori_.\n\nAnd Waff realized what he should have known all along, something every Tleilaxu should have understood: If each of the sandworms spawned from God Emperor Leto II's great body actually contained a pearl of the Prophet inside them\u2014how could the _worms themselves_ not have been prescient? How could they not have foreseen the coming of the Honored Matres and the impending destruction of Rakis?\n\nHe clapped his hands in glee. Of course! The great worms must have envisioned the terrible Obliterator weapons. Forewarned that the surface of Rakis would become a charred ball, some sandworms had been guided by Leto II's prescience to tunnel deep and encyst themselves protectively far beneath the sands, perhaps kilometers down. Away from the worst destruction.\n\n_This world can take care of itself_ , Waff thought.\n\nArrogant humans had always caused trouble here. When it was a pristine desert planet, Rakis was what it should have been before human pride and ambition terraformed it. The efforts of outsiders to \"improve\" Dune had resulted in the apparent extinction of the great worms, until the death of Leto II brought them back. After which humans\u2014the Honored Matres\u2014had wiped out the ecosystem again.\n\nRakis had been beaten, stepped on, raped . . . but in the end, the magnificent world had saved itself. The Prophet had remained there all along and contributed mightily to the survival of Dune. Now all was as it should be, and Waff was immensely pleased.\n\nTwo giant sandworms churned toward the Tleilaxu man, who stood transfixed. Plowing through the crusted ground, the worms scooped up the flaccid carcasses of the weak test worms, devouring them as if they were mere crumbs.\n\nOvercome by joy, Waff fell to his knees and prayed. At the last moment, he looked up into the giant mouth, with its deep, simmering flames and crystalline teeth. He smelled the spicy exhalations.\n\nSmiling beatifically, the Tleilaxu Master lifted his face to heaven and exclaimed, \"God, my God, I am yours at last!\" With the speed and fury of a crashing Guild Heighliner, the worm descended. Waff inhaled a deep, satisfying breath of spice and closed his eyes in rapture as the monster's cavernous mouth engulfed him.\n\nWaff became one with his Prophet.\n\n> _Life is about determining what to do next, from moment to moment. I've never been afraid of making decisions_.\n> \n> \u2014DUNCAN IDAHO,  \n>  _A Thousand Lives_\n\nThrough the broken cathedral's high dome, a preoccupied Duncan saw the sky flicker like a pattern changing in a kaleidoscope. A wealth of vessels appeared side by side, pulled along by the returning Navigator-controlled Heighliners.\n\nEven before the signal came to him, Duncan sensed that someone very special was aboard one of the newly arrived ships. His expanded mind showed him her face, very little changed after all these years. _Murbella!_ Some past part of him was terrified at the prospect of being near her again, but he was so much more than that now. He was eager to see her.\n\nA thousand Navigator-faction Heighliners hovered over Synchrony, uncertain of their role, now that the Oracle was gone. Using his newly acquired abilities, Duncan communed with them all in a commondenominator language. The Navigators would understand him in their own way, as would the thinking machines and the humans. Duncan barely touched on his enhanced knowledge to do so.\n\n_Important changes. Necessary changes_.\n\nThe human ships sent lighters down. Looking up through the dome's skylights Duncan saw the glints trailing through the sky and knew that Murbella would be with them. She would come down first, and he would see her again. Almost twenty-five years . . . a mere tick on the eternal clock, yet it had seemed an eternity all its own. He waited for her.\n\nBut the woman who entered the vaulted hall was _Sheeana_ , worn and weary from her fighting out in the machine city. Her eyes were full of questions as she took in the blood on the floor, the smashed sentinel robots, the supine bodies of the Baron and glassy-eyed Paolo. Just by looking at the four young gholas Sheeana could tell that Paul and Chani had their memories back.\n\nShe noticed the motionless body of the old woman slumped on the stairs and recognized her. Speaking through Sheeana's mouth, the inner voice of Serena Butler lashed out. \"Erasmus killed my innocent\u2014 _the_ innocent baby. He was the one responsible for\u2014\"\n\nDuncan cut her off. \"I didn't hate him in the end. I think I pitied him more. It reminded me of when the God Emperor died. Erasmus was flawed, arrogant, and yet oddly innocent, guided only by insatiable curiosity . . . but he didn't know how to process what he already understood.\"\n\nSheeana stared down, as if expecting the old woman's eyes to snap open and a clawlike hand to grab her. \"Erasmus is really dead, then?\"\n\n\"Completely.\"\n\n\"And Omnius?\"\n\n\"Gone forever. And the thinking machines are no longer our enemies.\"\n\n\"Do you control them, then? Have they been defeated?\" Wonder shone on her face.\n\n\"They are allies . . . tools . . . independent partners more than slaves, and so different. We have a whole new paradigm to grapple with, and a lot of new definitions to make.\"\n\nWHEN MURBELLA AND a party of Guildsmen and Sisters were ushered into the chamber by courier drones, Duncan set all questions aside and just stared at her.\n\nShe stopped in mid-step. \"Duncan . . . you've hardly changed in more than two decades.\"\n\nHe laughed at that. \"I've changed more than any instrument could measure.\" All the machines in the hall, in the whole city, turned toward Duncan at the comment.\n\nHe and Murbella embraced automatically, uncertain of whether this contact would rekindle their past feelings. But each sensed the difference in the other. The river of time had carved a deep canyon between them.\n\nAs he touched Murbella, Duncan felt a bittersweet sadness to know how much damage her addictive love had done to him. Things could never be the same between them again, especially now that he was the Kwisatz Haderach. He also guided the thinking machines, but he was not their new evermind, not their new puppet master. He didn't even know how they could exist without a controlling force. They had to adapt or die, something humans had done well for millennia.\n\nFrom across the room, Duncan recognized the spark in Sheeana's eyes\u2014of genuine concern rather than jealousy; no Bene Gesserit would allow herself the weakness of jealousy. In fact, Sheeana was such a staunch Bene Gesserit that she had stolen the no-ship from Chapterhouse and fled with her refugees, rather than abide by the changes Murbella had forced on the Sisterhood.\n\nHe spoke to both women. \"We have freed ourselves from the traps we set for each other. I need you, Murbella\u2014and you, Sheeana. And the future needs all of us more than I can express.\" An infinite number of machine thoughts coursed through his mind, giving him the sudden awareness that countless human planets needed help that only he could provide.\n\nWith a thought, he dispatched the guardian robots out of the hall, marching them away as if in a military exercise. Then he stretched his mind through the empty pathways of the tachyon net, and across the universe. With his instantaneous connection to all of the human defender ships once controlled by corrupted Ixian machines, as well as the machine battleships linked to Omnius's command\u2014 _Duncan's_ command, now\u2014he summoned the vessels to the former machine planet, dragging them all simultaneously through foldspace. They would all come here, to Synchrony.\n\n\"You, Murbella, were born free, trained as an Honored Matre, and finally made into a Bene Gesserit so that you could gather the loose ends. As you were a synthesis between Honored Matre and Bene Gesserit, so I am now a fusion between free mankind and thinking machines. I stand in both domains, understanding both, creating a future where both can thrive.\"\n\n\"And . . . what are you, Duncan?\" Sheeana asked.\n\n\"I am both the ultimate Kwisatz Haderach and a new form of the evermind\u2014and I am neither. I am something else.\"\n\nAlarmed, Murbella glanced at Sheeana, then back at him. \"Duncan! Thinking machines have been our mortal enemies since before the Butlerian Jihad\u2014more than fifteen thousand years.\"\n\n\"I plan to untie that Gordian knot of misunderstandings.\"\n\n\"Misunderstandings! Thinking machines slaughtered trillions of human beings. The plague on Chapterhouse alone wiped out\u2014\"\n\n\"Such is the cost of inflexibility and closed-minded fanaticism. Casualties are so often unnecessary. Honored Matres and Bene Gesserits, humans and thinking machines, heart and mind. Don't our differences strengthen us rather than destroy us?\" The reality-expanding wealth of information Erasmus had given him was tempered by the wisdom he had earned through numerous lifetimes. \"Our struggle has reached an end and a watershed.\" He flexed his hand, and he could feel innumerable thinking machines out there listening, waiting. \"We have the power to do so much now.\"\n\nUtilizing perfect prescient and calculational knowledge, Duncan knew how to bring about an everlasting peace. With humanity and thinking machines balanced in the palm of his hand, he could control them all and seize their powers, preventing them from making further war. He could force cooperation among the Navigator-faction Heighliners, the Ixian-modified ships, and the thinking-machine fleet.\n\nWith his developing prescience, he foresaw the joint future of humankind and thinking machines\u2014and how to implement it every step of the way. Such breathtaking power, greater than the God Emperor or Omnius combined. But power had eventually corrupted Leto II. How, then, could Duncan handle this even greater burden?\n\nEven if Duncan Idaho acted for the most altruistic of reasons, there were bound to be dissenters. Would he eventually be corrupted, regardless of his good intentions? Would history remember him as an even worse despot than the God Emperor?\n\nFacing an avalanche of questions and responsibilities, Duncan vowed to use the lessons of his numerous lifetimes for the benefit and survival of the human race and thinking machines. Kralizec. Yes, the universe had indeed changed.\n\n> _How terrible for a mother to bury her own daughter. There is no greater pain, not even the Bene Gesserit Agony. Now I have had to bury my daughter twice_.\n> \n> \u2014LADY JESSICA,  \n>  _Lament for Alia_\n\nJust one casualty among uncounted trillions.\n\nLater, as Jessica gazed sadly at the cold form of her daughter, she knew that one little girl did matter as much as all the others. Each life had value, whether a ghola child or a natural-born person. The titanic struggle that changed the future of the universe, the defeat of the thinking machines, and the survival of the human race seemed as nothing to her. She was completely preoccupied with preparing Alia's body for burial.\n\nAs she touched the small pale face, stroking the forehead and wispy dark hair, she remembered her daughter. An Abomination, Alia had been called: a child born with the full intelligence and genetic memories of a Reverend Mother. It had come full circle now. In her original lifetime, the little girl had killed Baron Harkonnen with the poison gom jabbar; later, as an adult and haunted by the evil presence of the Baron, Alia had taken her own life, throwing herself through a temple window high above the streets of Arrakeen. Now the reborn Baron had killed the reborn Alia, before she'd ever had the opportunity to reach the potential she deserved. It was as if the two of them were forever locked in mortal combat, on a mythical scale.\n\nA tear rolled down Jessica's cheek with the grace of a falling raindrop. She closed her eyes and realized that she had been frozen in the same position for a long moment, caught up in memories. She hadn't even heard the visitor approach her quarters.\n\n\"Is there any way I might help you, my Lady?\"\n\n\"Leave me. I want to be alone.\" But when she saw that it was the somber Dr. Yueh, her demeanor softened. \"I'm sorry, Wellington. Yes, come in. You can help me.\"\n\n\"I don't wish to intrude.\"\n\nWith a wan smile she said, \"You've earned the right to be here.\"\n\nFor long moments the unlikely pair stood together without speaking. Grateful just to have him there, Jessica finally said, \"Long ago when you were with us at Castle Caladan, I cared for you. You always kept your life private, and when you betrayed us, I hated you more than I thought possible.\"\n\nHe hung his head. \"I would throw myself upon a knife ten thousand times if I could take back the deeds I've done and erase the pain I've caused, my Lady.\"\n\n\"History can only move forward, Wellington, not backward.\"\n\n\"Oh? _We've_ been dredged out of the dustbins of history, haven't we?\"\n\nOn old Arrakis, the Fremen had made a solemn ritual of recovering a body's water in a deathstill and sharing it among the tribe. On Caladan, the tradition had been a funeral pyre or an ocean burial. While the _Ithaca_ wandered through space, their dead had been ceremoniously ejected into the void.\n\nUsing stain-free fabric from the no-ship's sheets, they wrapped Alia's small, frail body. Here in the post-Omnius machine city, however, Jessica wasn't sure how best to honor her daughter. \"We don't really have a funeral tradition anymore, so I don't know what to do.\"\n\n\"We'll do what we must. The symbols don't matter, but the thought does.\"\n\nLONG AFTER THE last echoes of the battle on Synchrony had died away and survivors from the no-ship ventured out to discover the new face of the universe, Jessica and Yueh joined Paul, Chani, and Duncan in their own private funeral procession. Paul and Jessica carried the tiny wrapped body out into the streets where the sandworms had caused so much damage, where explosions in the battle against the Face Dancers had destroyed countless structures.\n\n\"Such a tiny body . . . and so much lost potential,\" Paul said. \"I miss my sister terribly, even though I didn't get to know her this time as well as I would have liked.\"\n\nDuncan led the group, shunting aside his other responsibilities for the time being. \"I don't remember the original little girl, but I remember the woman. She hurt me and loved me, and I loved her passionately.\"\n\nThey didn't have far to walk. Jessica had selected a particular broken tower, a slumped, thin pyramid that would serve as an appropriate grave marker. Jessica and Paul said their goodbyes during the procession, so that when they reached the collapsed structure they carried the girl inside through a lopsided trapezoidal opening, pushed debris aside to clear a space for her, and laid Alia Atreides on the smooth metal floor. Then Jessica stood over the wrapped child, saying another quiet farewell. Paul grasped his mother's hand, and she squeezed back.\n\nAfter a lingering, painful silence, she turned and spoke to Duncan. \"We've done all we need to do.\"\n\n\"I'll take care of the rest,\" Duncan said. When they had withdrawn from the fallen pyramid, Duncan raised his hands, fingers splayed, and his face took on a distant expression. The metalform buildings around them began to tremble and sway, growing and curving. The remnants of the pyramid folded around Alia's body and reinforced the walls, drawing polished alloys from other structures. Like a magnificent crystal and quicksilver monument, the ruined spire then rose heavenward. The rapidly growing tower crackled and clanged like mechanical thunder as flowmetal streamed upward. Its curves and angles were streamlined, its polished surfaces perfectly reflective.\n\nDuncan guided the semisentient structures with greater care and focus than the evermind ever had. When he was finished, he had created a tomb, a memorial, a work of art that would amaze anyone who looked upon it.\n\nIt left a mark on Synchrony that could never compare with the mark her daughter's loss left on Jessica's heart.\n\n> _Some problems are best solved with an optimistic approach. Optimism shines a light on alternatives that are otherwise not visible_.\n> \n> \u2014SHEEANA,  \n>  _Reflections on the New Order_\n\nIn the aftermath, the humans in Synchrony gradually began to believe that their race would survive.\n\nWhen Sheeana looked at Duncan, he seemed strangely distant, though that was to be expected. Often his gaze flicked from side to side as if he were in a thousand places at once.\n\nWhile Mother Commander Murbella called down lighters from her newly arrived battleships, and the Guild provided shuttles full of workers and administrators to help consolidate the strange city, Sheeana watched self-guided robots clean up remnants of the bloody duels in the cathedral chamber.\n\nThe _Ithaca_ 's refugees had taken shelter inside the torn-open ship. The vessel would never fly in space again, even if Duncan forced the living-metal docking cradle to release the no-ship.\n\nCourier drones and buzzing watcheyes, now personally directed by Duncan, led crowds of people through the broken streets, summoning them to a meeting where they would discuss the changed universe. Sheeana's renegade Bene Gesserits from the no-ship were uneasy about facing the former Honored Matre Murbella.\n\nBut the Mother Commander had grown much wiser in the intervening quarter century since the schism. Years ago, had she known of Sheeana's plan to steal the no-ship, Murbella would have killed her rival outright. Sheeana wondered what the former Honored Matre would think of all those years Duncan had pined for her. Did Murbella still love him? For that matter, had she ever?\n\nReverend Mothers Elyen and Calissa led a weary and uneasy crowd into the enormous cathedral hall. Guild crewmen from the ships above also entered the chamber, Administrator Gorus among them. He appeared drained, no longer in control of anything, and remained silent, following rather than leading his fellow Guildsmen.\n\nWhen they had settled into a low hum of conversation approaching silence, Duncan took his place in the center of the chamber where Omnius and Erasmus had once presided over their thinking machines. He used no amplification system, yet his words resounded through the hall.\n\n\"This fate, this grand culmination of Kralizec, is what we sought for so many years.\" He swept his gaze over Sheeana and the refugee Bene Gesserits. \"Your long journey is at an end, for this is the new heartland you dreamed of finding. The planet is yours now. Use the remnants of Synchrony to form an entirely new Bene Gesserit order, your base far from Chapterhouse.\"\n\nThe gathered Sisters were confused and astounded. Even Sheeana had not known Duncan would propose this. \"But this is the heart of the thinking-machine empire!\" cried Calissa. \"The homeworld of Omnius.\"\n\n\"It's your homeworld now. Stake your claim and build your future.\"\n\nSheeana understood. \"Duncan is exactly right. Challenges strengthen the Sisterhood. The universe has changed, and we belong here, regardless of the difficulties we may face. Even the sandworms have come to Synchrony, burrowing deep underground.\" She smiled. \"They may reemerge when we least expect them. Someone has to keep an eye on the restored Tyrant.\"\n\nBeneath the hall, Sheeana thought she felt the ground trembling, as from a great behemoth moving under the foundations. Many robots had been destroyed or damaged during the sandworm attack, but thousands more of the machines remained perfectly functional. Sheeana knew that the Bene Gesserits here would have all the labor pool they could possibly desire, if the machines would work with them.\n\nMurbella spoke up. \"I shall return to Chapterhouse. It will take some effort to spread news of the new reality.\" She gazed at Sheeana. \"Don't worry. My combined Sisterhood doesn't need to be at odds with your orthodox Bene Gesserit base here. There have always been many schools, many trains of thought. In proper balance, rivalry promotes strength and innovation\u2014so long as we can avoid the acrimony of conflict and mutual destruction.\"\n\nSheeana knew that Duncan would go back to Chapterhouse with Murbella, at least for a time. With his guidance, Murbella would shepherd the reintroduction and integration of superior technology into a thriving society. If handled properly, Sheeana saw no reason for humans to fear cooperation with thinking machines any more than they needed to fear religion itself, or competition among Bene Gesserit elements. Any group could be dangerous if managed improperly.\n\nSheeana, though, would remain here. She saw no point in going back. Addressing Murbella, she said, \"Even before Honored Matres destroyed Rakis, the Bene Gesserit order made me the centerpiece of a manufactured religion. For decades I had to hide while the Missionaria spread myths about me. I let the legend continue without me. What would I achieve if I stopped it now? So I say, leave it, if the thought comforts people. My place is on this planet.\"\n\nShe saw that Scytale was also in the audience. The last of the Tleilaxu Masters had, in the end, proved greatly helpful, fighting for instead of against them. \"Scytale, will you remain with us? Will you join my new order here? We can use your knowledge and genetic expertise. We are, after all, founding a colony, and we have only a few hundred people.\"\n\n\"I expect others from the outside will eventually join you,\" Murbella said.\n\nThe little Tleilaxu was surprised by the invitation. \"Of course I will stay. Thank you. My people have no other place now, not even sacred Bandalong.\" He smiled at Sheeana. \"Perhaps at your side I can accomplish something worthwhile.\"\n\nDuncan walked among the Bene Gesserit refugees. \"You are gardeners laying down flagstones on our path of destiny. Many of us will return to worlds we once called home, but you will remain here.\"\n\nWith a warm feeling toward him, Sheeana touched Duncan's arm. Though still flesh and bone, and human, she knew he was far more than that. And his words rang true. \"Thanks to you Duncan, my Sisters and I are finally home.\"\n\n> _The worst part of going back is that the past is never exactly the way you remember it_.\n> \n> \u2014PAUL ATREIDES,  \n> Notebooks of a Ghola\n\nBack in the Old Empire, the last Chapterhouse defenders waited, tense and alert, but nothing changed for days. The machine warships had not moved, and Bashar Janess Idaho had received no further word from the Navigator ships that had whisked the Mother Commander away. Fast scouts flitted back and forth from the hundred laststand groupings, and the situation was the same along the entire front.\n\nWaiting. No one knew what was happening.\n\nJaness reacted with alarm and dismay when a large swarm of ships burst out of foldspace in all sizes and configurations. Shouting into the commline, the bashar rallied all of her functional defensive craft that remained in orbit. At first she did not recognize the configurations, but then she saw that the newly arrived cluster included smaller human and thinking-machine vessels that had been towed along by the Holtzman engines of great Guildships.\n\n\"Identify yourselves!\" Janess said to the unexpected armada.\n\nOn the bridge of her large battleship, returning home, Murbella smiled at Duncan. \"That is your\u2014 _our_ daughter.\"\n\nHe raised his eyebrows and performed quick mental calculations. \"One of the twins?\"\n\n\"Janess.\" Murbella frowned slightly. \"The other one, Rinya, didn't survive the Agony. I forgot you didn't know. Tanidia, the middle one, is alive and well, assigned with the Missionaria among the refugees. But we lost Gianne, our youngest\u2014born just before I became a Reverend Mother. She died during the Chapterhouse plague.\"\n\nDuncan steadied himself. How odd to feel a blow of genuine grief to learn of the death of two children he had never met. He hadn't even known their names until now. He tried to imagine what the young women might have been like. As Kwisatz Haderach and evermind, he could do many things . . . almost anything. But he couldn't bring back his daughters.\n\nDuncan studied Janess's features on the screen: dark hair and round face from his own genes, a petite figure, intense eyes, and a hard expression showing she would never run from a challenge. A synthesis of himself and Murbella. He activated the commline himself. \"Bashar Janess Idaho, this is Duncan Idaho, your father. I am with the Mother Commander.\"\n\nMurbella leaned into the field of view. \"Stand down, Janess. The war is over. You have nothing to fear from us.\"\n\nJaness seemed suspicious. \"There are thinking-machine ships with you.\"\n\n\"They are _my_ ships now,\" Duncan said.\n\nThe female bashar did not flinch. \"How do I know you're not Face Dancers?\"\n\nMurbella answered, \"Janess, when we stood against the thinking machines and discovered that Ixians and Face Dancers had deceived us, you and I were ready to throw away our lives in a final burst of glory. Don't be so eager to die now that we finally have hope.\"\n\nThe image of Janess stared at them from the viewing plate. Duncan was proud of his daughter's caution. He said, \"We will all meet in the great hall of the Keep. A good place for us to discuss the future.\" He smiled wistfully. \"I never actually saw the inside of the Keep when I was here. . . . I had to remain aboard the no-ship at all times.\"\n\nJaness hesitated just a moment longer, then nodded curtly. \"We will have guards.\"\n\nDuncan already missed his no-ship comrades, but they had their own places to go now, important niches to fill. Paul and Chani would return to Arrakis, where they had always known they belonged. Jessica had chosen Caladan, and she surprised many by asking Yueh to go with her. And on Synchrony, Scytale's nullentropy capsule still contained a wealth of cells, a treasure chest of prizes.\n\nDuncan had already decided on the first request he would make of the Tleilaxu Master. The turmoil and changes, the repercussions and adaptations would last for decades, even centuries. He would value the assistance and advice of a great man. He needed Miles Teg at his side again. . . .\n\nAs the ship descended toward the main city on Chapterhouse, Duncan knew he could never think of this world as home, despite the time he'd spent there. In his genetic incarnations he had experienced many places and known innumerable people. Duncan's developing prescience, and his mental connection to decillions of eyes spread across the cosmos and linked through the evermind's tachyon net, made the entire universe his home now.\n\n_Now you begin to understand the fascinating obligation I helped you to assume_ , said a familiar-sounding voice in his mind. Erasmus! _I could have made it much harder on you, Kwisatz Haderach. Instead, I cooperated. This is only an echo of me, an observer. You can access me as you like. Use my knowledge like a databank. A tool. I am curious to see what you will do_.\n\n\"Are you haunting me now, like a demon?\"\n\n_Consider me an advisor, but my research continues. I will always be here to guide you, and I am confident you will not let me down_.\n\n\"Like the witches' Other Memory, but far bigger, and more easily accessible.\"\n\n_You are here to serve both humans and thinking machines_ \u2014 _and the future. It is all under your command_.\n\nDuncan laughed softly to himself at the friendly bantering between the two of them. Though Erasmus was in a subservient position, he still had a bit of humanlike pride, even if he was only an echo, and an advisor.\n\nEntering the Keep, Duncan and Murbella marched into the echoing great hall, side by side. Watcheyes followed them, along with a pair of sentinel robots. The robots greatly disturbed the people who waited there, but in the future humans must learn to set aside their fears and preconceptions.\n\nWithout Omnius, the thinking-machine empire continued to function but without a unified mind or mission. Duncan would direct them, but he refused to simply continue the endless cycle of enslavement. They had potential to be more than tools or puppets, more than just a destructive force. Some of the machines were merely that, but more sophisticated robots and advisory mechanisms could grow and develop into something far superior. Erasmus himself had become independent, developing a unique personality when he was isolated from the homogenizing influence of the evermind. With so many thinking machines spread across so many planets, other prominent figures would arise if given the opportunity. If guided. If Duncan allowed them.\n\nHe had to achieve a balance.\n\nThe Mother Commander's imposing chair stood high and empty in front of a segmented window that looked out on the arid, dying landscape. Janess stood to one side, welcoming Murbella to the empty seat, with nearly a hundred of the New Sisterhood's guards standing at high alert in the chamber. Though all of the insidious Face Dancers had been exposed and killed, Janess was not letting down her guard, and Duncan felt proud of his daughter.\n\nShe bowed formally. \"Mother Commander, we are glad to have you back. Please, take your place.\"\n\n\"It is no longer only my place. Duncan, your daughter has been raised in the Bene Gesserit ways, but she also made a point of learning about you. She trained herself to become the equivalent of a Ginaz Swordmaster.\"\n\nThinking bittersweet thoughts about all he had missed, Duncan formally shook his daughter's hand and found her grip pleasingly strong. Until this moment they had been strangers who shared a bond of blood and patriotic allegiance. Their real relationship was just beginning.\n\nMurbella had fought a long and bloody battle to combine the opposing forces of the Honored Matres and the Bene Gesserits, after which she had wrestled with the disparate groups of humanity to forge them into one whole. On an even larger scale, Duncan, through his newfound abilities, was shaping an even greater, farther reaching union.\n\nEverything was woven together in a tighter tapestry than history had ever known, and at last Duncan grasped the extent of his newfound strength. He was not the first human in history to possess great power, and he vowed not to forget what he had learned as a pawn of the God Emperor, Leto II.\n\nThe human race would never forget the thousands of years under that terrible reign, and Duncan's comprehensive racial memory held a roadmap that showed him where the pitfalls were, thus enabling him to avoid them. The great Tyrant had suffered from a flaw he hadn't recognized. Weighed down by his sense of terrible purpose, Leto II had isolated himself from his humanity.\n\nIn contrast, Duncan clung to the knowledge that Murbella would be with him, and Sheeana, too. He could talk with his daughter Janess as well, and perhaps even his other surviving daughter, Tanidia. In addition, he had all the memories of great and loyal friends, of dozens of loves, and a succession of comrades, wives, families, joys, and beliefs.\n\nThough he was the ultimate Kwisatz Haderach with immeasurable power, Duncan had known the best parts of being human. Life after life. He didn't need to feel alienated and worried, when he could be filled with love instead.\n\nBut his would not be a conventional kind of love. His love needed to extend much farther, to every living person, and to thinking machines. One form of sentient life was not superior to the other. And Duncan Idaho was greater than the flesh that encompassed his body.\n\n## EPILOGUE\n\n> _In a war, be watchful for unexpected enemies and unlikely allies_.\n> \n> \u2014BASHAR MILES TEG,  \n> final log entries\n\nMore than a year had passed on Qelso. The unnatural desert continued to spread as sandtrout reproduced and commandeered more and more of the planet's water. Though their fight seemed hopeless, Var's commandos stood against the forces that were killing their environment.\n\nStilgar and Liet-Kynes did their best to assist in the struggle. Both desert-bred gholas felt that their more important work was to show the natives how they could live in cooperation with the encroaching desert, rather than fight it.\n\nDuring the many months since the pair had departed from the noship, the dry sands had extended much farther into the continental forests and plains. Var's camp had moved time after time, retreating from the oncoming dunes, and the desert kept following them. Though they had killed dozens of sandworms using water cannons and moisture bombs, Shai-Hulud was not so easily thwarted. The worms grew larger, despite all the efforts of the Qelso commandos.\n\nWith the first faint light of dawn, Liet stepped out of his rockwalled sleeping chambers and stretched. Although he and Stilgar were still teenagers, they remembered being adults once and having wives. Among the commando women on Qelso, many would accept either of them as a husband, but Liet had not yet decided when he could justify getting married and fathering children. Maybe he would have another daughter, and name her Chani. . . .\n\nNo matter how much Liet-Kynes worked to remake Qelso, it would never be Dune. The fertile landscape was giving way to dry waves of sand, but it would not be the same. Eons ago, had Arrakis been fertile? Had some forgotten superior civilization transplanted sandtrout and sandworms there, much as Mother Superior Odrade had when she sent her Bene Gesserit to Qelso? Perhaps it had been the Muadru, who left mysterious symbols on rocks and cliffs, and in caves across the galaxy. Liet didn't know. His father might have been intrigued by the mystery, but Liet considered himself more practical.\n\nPreparing for the day's work, he looked over at Stilgar, whose eyes had begun to turn blue-within-blue. For years the people here had stubbornly denied themselves the use of melange, but Stilgar called it a sacred reward from the desert, a gift from Shai-Hulud. He had small groups harvesting spice for their own uses, and Liet knew that spice was like a velvet chain\u2014pleasant enough, until one tried to break free of it.\n\nTwo chattering and flirtatious teenage girls brought the men breakfast on a tray, knowing what Stilgar and Liet preferred for their morning meal. The girls were lovely, but so _young_. Liet knew they saw only his youthful body, not knowing how many years he carried in his mind. At times like these, he truly missed his wife Faroula, Chani's mother. But that had been so long ago. . . .\n\nStilgar, however, remained the same. After they finished their coffee and sweet cakes, Liet stood and clapped his friend on the shoulder. \"Today we will go out into the deep dunes and plant weather devices. We need better resolution to track the desiccation patterns.\"\n\n\"Why do you obsess over details? The desert is the desert. It will always be hot and dry, and here on Qelso it will keep growing.\" The former naib did not see anything particularly tragic or wrong with the dying ecosystem. To Stilgar, it was the natural order of things. \"Shai-Hulud continues to build his domain no matter what you do.\"\n\n\"The scientist pursues knowledge,\" Liet said, and his companion had no answer for that.\n\nTaking one of the small flyers the _Ithaca_ had left behind, he had gone to the northern and as yet undamaged latitudes where the forests stood tall, the rivers flowed, and snowcaps crowned the mountains. Cities and towns still flourished in the valleys and on the hillsides, though the people knew they would all be gone before long. Var's commandos were poignantly reminded every day of how much they were missing, how much they had lost. Stilgar did not see it.\n\nThe two friends, along with a group of rugged volunteers, donned newly manufactured stillsuits and adjusted the fittings. When the commandos marched into the open desert, they walked in single file on the dunes. Liet had them practice the random stutter-step that would not attract a worm. The yellow sun grew swiftly hotter, reflecting off the granular sands, but they plodded onward, practicing their lives here. Far in the distance Liet saw the rusty-brown smear of powdery smoke that indicated a spice blow, and he thought he saw the rippling tracks of a worm moving out there.\n\nStilgar shouted and pointed up at the sky. The desert men instinctively clustered together in a defensive formation.\n\nHundreds of huge metallic ships suddenly descended, made of angular plates bristling with weapons and powered by enormous engines. The vessels looked like nothing Liet had ever seen before. Enemy ships?\n\nFor a moment he hoped the _Ithaca_ had returned with them, but these were unlike the no-ship and unusual in their formation, moving in a coordinated fashion. They dropped indiscriminately onto the open desert, scattering sand and flattening dunes. Their pilots seemed oblivious to the fact that the dull vibrations would attract sandworms. As Liet stood gaping at the ships' sheer size, he had no doubt that their weapons could brush aside a worm attack as if it were no more than a nuisance.\n\nThe dusty commandos looked to the two gholas for answers. Liet had none, though, and despite the impossible odds, Stilgar appeared ready to attack, if need be.\n\nWith an ominous humming and clanking, the ships extended support struts and raised themselves on thick, powerful anchors. Then numerous doors began to open, turning loose an army of metal-skinned machines: heavy lifters, ground crushers, and excavators. Moving on treads, the lumbering self-guided behemoths crawled across the dunes. Behind them marched ranks of heavyset metal robots that smashed forward like deadly warriors . . . or were they workers? Helpers?\n\nThe commandos had only small weapons. Some of the eager ones drew their projectile launchers, dropped to their knees on the soft sand, and took aim. \"Wait!\" Liet cried.\n\nA hatch at the top of the largest landed ship opened and a pale form emerged, stepping out onto an observation platform. A human form. When the man called down to them, his voice echoed in an eerie chorus transmitted from thousands of speakerpatches on the lines of machine forces. \"Stilgar and Liet-Kynes! Don't be so quick to declare yourselves our enemies.\"\n\n\"Who are you?\" Stilgar shouted defiantly. \"Come down here so that we may speak to you face to face.\"\n\n\"I thought you would recognize me.\"\n\nLiet did. \"It's Duncan\u2014Duncan Idaho!\"\n\nFlanked by an honor guard of robots and accompanied by a troop of human workers wearing outfits that Liet did not recognize, Duncan came down to stand with them on the dunes. \"Liet and Stilgar, we left you here to face the onslaught of the desert. You said this was your calling.\"\n\n\"It _is_ ,\" Stilgar said.\n\n\"And the Jews? Are they here with you?\"\n\n\"They formed a sietch of their own. They are thriving and happy.\"\n\nDuncan's honor guard stepped forward, women in black singlesuits and similarly garbed men who walked beside the females as equals. One of the women wore insignia and carried an air of command. He introduced her as his daughter Janess. \"I confronted the Enemy, the thinking machines, and ended the war.\" He extended his hands, and all of the robot workers turned to face him. The awesome ships themselves seemed to be alive and aware of every move Duncan made. \"I have found a way to bring us all together.\"\n\n\"You surrendered to thinking machines,\" Stilgar said, his tone acidic.\n\n\"Not at all. I decided to show my humanity by not annihilating them. In many solar systems, they are building great things, achieving impressive works on planets inhospitable to humans. We work for the same purpose now, and I have brought them here to assist you.\"\n\n\"Assist us?\" one of the commandos said. \"How can they help? They're just machines.\"\n\n\"They are allies. You face an insurmountable task. With as many robot crews as you require, I can help you accomplish what you need.\" Duncan's dark eyes glittered, as he watched from a million eyes all at once. \"We can build a barrier against the desert, stop the sandtrout from spreading, and keep the water on a portion of the continent. Shai-Hulud will have his domain, while the rest of Qelso remains relatively unscathed. Humans can have their lives and slowly learn to adapt to the desert, but only if they choose to.\"\n\n\"Impossible,\" Liet said. \"How can a force of worker robots stand against the tide of the desert?\"\n\nDuncan flashed a confident smile. \"Don't underestimate them\u2014or me. I fill the roles of both Kwisatz Haderach and Omnius. I guide all the factions of humanity and control the entire Synchronized Empire.\" He shrugged, and smiled. \"Saving one planet is well within the scope of my capabilities.\"\n\nLiet couldn't believe what he was hearing. \"You can stop the desert and turn back the worms?\"\n\n\"Qelso will be both desert and forest, as I am both human and machine.\" At a gesture and a thought from Duncan, the massive excavating equipment rumbled out into the sand, heading toward the boundary where the dunes met the still-living landscape.\n\nLiet and Stilgar followed Duncan, who walked ahead of the heavy convoy. As a planetologist, a ghola, and a human being, Liet had innumerable questions. But for now, watching the machines begin their work, he decided to wait and see what the future held.\n\n> _When Leto II envisioned his Golden Path, he foresaw the direction that humankind should take, but he had blind spots. He failed to see that he was not the ultimate Kwisatz Haderach_.\n> \n> \u2014Bene Gesserit fact-finding commission\n\nIn the eleven years Jessica had been back home, she had realized more and more that some things did not add up. This planet might indeed be Caladan, or Dan, but this was not the same home she and her Duke had loved so long ago.\n\nOn a stormy evening, as she walked through the restored castle, the incongruous details finally became more than she could bear. Pausing in an upper hallway, she opened a finely carved elaccawood cabinet, an antique that some decorator had placed there. This time, she stood staring at the ornate interior, and on impulse pressed a wooden extrusion in one corner. To her surprise, a panel opened, and inside she found a small blue statuette of a griffin. Perhaps placed there by the Baron ghola, the griffin was the ancient symbol of House Harkonnen. He must have hidden it there as a clever reminder of the falseness of the castle.\n\nAs she stared at the statuette, feeling the wrongness of the object, she considered all of her hard work since returning to Caladan. She had directed crews of local laborers to dismantle the Baron's torture devices and the Face Dancer Khrone's offensive laboratories from the underground chambers. Through it all she had worked side by side with the cleaning teams, sweating and angry as she scrubbed away every stain, every odor, every hint of the unwanted presence. But Castle Caladan still reeked with reminders. How could she make a fresh start when so much of the past\u2014at least this awkward, out-of-focus echo of the past\u2014hung all around her?\n\nBehind her, moving silently, Dr. Yueh said, \"Are you all right, my Lady?\"\n\nShe looked at the Suk doctor. He wore an expression of deep concern on his buttery face; his dark lips turned downward as he waited for an answer.\n\n\"Everywhere I turn, I am reminded of the Baron.\" She frowned at the griffin figurine in her hand. \"Some of the articles in this castle are authentic, such as that dropleaf desk with the hawk crest, but most are bad copies.\"\n\nMaking up her mind, Jessica stepped to a segmented window at the end of the hall and swung it open to let in the stormy night air. In a dramatic gesture, she hurled the griffin figurine out to the crashing sea. The waves would soon erode it and break it into unrecognizable pieces. A suitable fate for the Harkonnen icon.\n\nA cold, wet wind whispered into the hall, bringing spatters of rain. Outside, scudding clouds parted to reveal a crescent moon on the horizon, casting cold yellow light on the water.\n\nMoments later she tore down a wall tapestry that she had never liked, and was about to throw that out the window, too, but\u2014not wanting to spoil this beautiful planet\u2014she instead tossed the tapestry on the floor, promising herself to cast it on the trash heap the following morning. \"Maybe I should just tear this whole place down, Wellington. Can we ever remove the taint?\"\n\nYueh was shocked at the suggestion. \"My Lady, this is the ancestral home of House Atreides. What would Duke Leto\u2014\"\n\n\"This is a mere reconstruction, fraught with errors.\" A gusting breeze blew her bronze hair away from her face.\n\n\"Maybe we waste too much time trying to recreate what we see in our old memories, my Lady. Why not build and decorate your home as you choose?\"\n\nShe blinked as cold rain blew into her face, drenching her jade green dress and wetting the rug. \"I thought this place would help my Leto, give him comfort, but maybe it was more for me than for him.\"\n\nA ten-year-old boy with coal-black hair came running down the hall, his smoke gray eyes widening with excitement and alarm when he saw the open window. He was even more surprised when neither Jessica nor Yueh reacted to the blowing rain that drenched the rugs and tapestries. \"What's happening?\"\n\n\"I was considering moving somewhere else, Leto. Would you like me to find us a normal home in the village? Maybe we'd be happier down there, away from this pampered life.\"\n\n\"But I like this castle! It's a Duke's castle.\" Jessica could not think of her Leto as a child. He wore fishing dungarees and a striped shirt, just like the ones he had worn when Jessica had first come to Caladan as a concubine purchased from the Bene Gesserit. The young nobleman had put a knife to her throat that day, a bluff . . .\n\nYueh smiled. \"A Duke . . . Such titles no longer mean anything with the Imperium long gone. Do the people of Caladan even need a Duke anymore?\"\n\nJessica's reaction was automatic, making her realize she had not thought through her notion. \"The people still need leaders, no matter what title we use. And we can be good leaders, as House Atreides has always been in the past. My Leto will be a good Duke.\"\n\nThe boy's eyes glittered as he listened with rapt attention. Beyond his youthful features, Jessica could see the seeds of the man she loved. This young Atreides was among the first of a new generation of gholas produced by Scytale. The baby had been shipped to Caladan and christened there\u2014just as the original Paul had been.\n\nSince leaving Synchrony, Jessica and Yueh had struggled to recover here, endeavoring in the process to bring back a degree of glory to the quiet water world. The tangled threads of their initial and ghola lives made them ironic allies, two people with shared tragedies and shared pasts. Finally, though he could never have his beloved Wanna back, Yueh had found some measure of peace.\n\nJessica, though, knew that her true Duke waited for her. Eventually he would grow to manhood. When he got his memories back, his physical age would not matter.\n\nJessica's partnership with Leto would be unusual, but no stranger than the relationships of all the mismatched gholas that had grown up on the no-ship. As a Bene Gesserit, she could slow her own aging process, and with melange readily available from the operations on Chapterhouse, Buzzell, and Qelso, they could both enjoy extended lives. She would prepare Leto, and when the time was right, she would help trigger his true awakening. Miraculously, he would once again be the man she loved, with all his thoughts and memories.\n\nShe only had to wait a decade or two. As a Bene Gesserit, she could be patient.\n\nJessica grasped his small hand. This time there would be no political reason to prevent them from getting married, if that was what he wanted, and she wanted. It only mattered to her that they would be together again.\n\n\"Everything will be the same when you finally remember, Leto. And everything will be different.\"\n\n> _The future is around all of us, and it looks very much like the past_.\n> \n> \u2014MOTHER SUPERIOR SHEEANA,  \n> at the founding of the Orthodox School on Synchrony\n\nThe mangled and permanently grounded _Ithaca_ had become the new headquarters building for Sheeana's splinter group. Innovative human architects in conjunction with construction robots had remodeled the large vessel into a unique and imposing headquarters. The navigation bridge, the highest deck on the no-ship, had been opened up and converted into an observation tower.\n\nMother Superior Sheeana stared across the breathtaking, rebuilt city of Synchrony. Dipping into her deep reservoir of memories, she drew parallels to one of the original Bene Gesserit schools on Wallach IX, which had also been founded in an urban setting. Here many of the machine spires remained, and some even moved as they had before, processing materials in automated industries.\n\nYears ago, Duncan and the willing machines had helped her reconstruct the unusual metropolis, though he balanced his \"miraculous\" work with the necessity of letting the humans achieve their own successes. He and Sheeana knew the dangers of letting people grow too soft, and he had no intention of allowing them to rely on him for things they could do for themselves. Humankind needed to solve its own problems as much as possible.\n\nAt the same time, clusters of thinking machines had begun to grow apart, given manageable goals, inhabiting niches unbearable to humans: blasted planets, frozen asteroids, empty moons. The galaxy was a vast place, and so little of it was suitable for biological life. There would be more lebensraum than any empire could possibly need.\n\nSome of the robots had started showing traits of personalities, unique characters of their own. Duncan suggested that, in time, these could eventually become some of the greatest thinkers and philosophers history had ever known. Sheeana remained unconvinced of this, and vowed that her special trainees here would prove him wrong with their own superior achievements.\n\nEvery month fresh candidates came to join the orthodox Bene Gesserit center on Synchrony, while others joined Murbella's New Sisterhood on Chapterhouse. Surmounting early difficulties, the two orders now worked in harmony with one another. Sheeana and her stricter ways attracted a different sort of acolyte, which she knew would have pleased Garimi. Sheeana tested the applicants harshly and rejected all but the most acceptable. Far away, Murbella's order had its own attractions. In this new universe, there was plenty of room for both views.\n\nSheeana's conventional Bene Gesserit breeding program was in full swing now, and it warmed her heart to see so many pregnant women each day. She counted seven of them outside among the people leaving and entering the headquarters. The sight gave her confidence that her order would expand and continue into humanity's future.\n\nLater that day, the Tleilaxu Master Scytale contacted Sheeana on the navigation bridge that had become her center of operations. Transmitting from one of his Synchrony laboratories, he actually sounded cheerful now instead of harried. \"I finished cataloguing the remaining cells and sifted out all traces of Face Dancer contamination. We must introduce some of those traits into the Bene Gesserit again.\"\n\n\"After Duncan, we will breed no further Kwisatz Haderachs. It's not even a matter for discussion.\" As far as she was concerned, many things did not need to occur again. . . .\n\n\"I merely mean to preserve our knowledge. It's like finding the seeds of long-forgotten but beautiful plants. We shouldn't just discard them.\"\n\n\"Perhaps not, but we must set up strict fail-safe mechanisms.\"\n\nScytale did not seem bothered by the restrictions Sheeana was placing on him. \"I honestly feel that the Tleilaxu race will recover their lost knowledge.\" Quickly he added, \"With changes for the better, of course.\"\n\n\"For the advancement of humankind,\" Sheeana said.\n\nShe had never known him to work so hard. Scytale had used the cells in his nullentropy capsule to regrow gholas of the last Tleilaxu Council, and now the little ones followed him everywhere, reminding her of a mother duck trailed by ducklings.\n\nScytale raised the group in a manner different than was traditional for Tleilaxu males. In separate quarters, he was also raising Tleilaxu females\u2014from newly discovered cells\u2014though they would never be relegated to the horrific, degrading conditions their predecessors had endured. Never again would Tleilaxu females be forced to become axlotl tanks, so there would be no chance of creating another set of ferocious, vengeful enemies like the Honored Matres. In particular, Sheeana and her Sisters would monitor the council members closely, keeping watch to ensure that they did not corrupt the Tleilaxu people as they had before.\n\nThere were still axlotl tanks, of course\u2014some women volunteered for personal reasons, while others left instructions for their bodies to be converted in the event of serious accidents. As always, the Bene Gesserits met their own needs.\n\nAfter she ended the conference with Scytale, the Mother Superior gazed through the broad windows of the navigation bridge. Far away on the horizon, beyond the redefined boundaries of the gleaming city, the ground was churned and torn up, and many of the geometric structures built by Omnius lay toppled and crushed into rubble.\n\nShe adjusted one of the windows, increasing its magnification. From this vantage point she could see the new desert and one of the sandworms rising up from the debris, its eyeless head questing about. Then the creature smashed down hard, breaking part of a wall. Like huge, determined earthworms churning the soil, they had begun the process of converting the abandoned buildings into the desert they preferred.\n\nSoon, Sheeana thought, she would go and speak to them again.\n\nShe looked down at the little girl at her side and grasped her small hand. Perhaps one day she would even take her prot\u00e9g\u00e9e with her, the young ghola of Serena Butler.\n\nIt was never too early to start preparing Serena for her role.\n\n> _The desert has a beauty I could not forget in a thousand lifetimes_.\n> \n> \u2014PAUL MUAD'DIB ATREIDES\n\nBathed in the golden rays of sunset, two figures made their way along the crest of a dune, their steps irregular so that they did not attract the huge sandworms. The pair walked side by side, inseparable.\n\nIt was warm on Dune, but not like old times. Because of the severe damage to the environment, the weather had cooled and the atmosphere had thinned. But with the return of the worms, along with sandplankton and sandtrout bursting through the cracked shell of glassy dunes, the old planet had begun to come back. As Chani's father Liet-Kynes used to say, everything on Dune was tied together, an entire ecosystem that included the land, the available water, and the air.\n\nAnd, thanks to Duncan Idaho, an extensive work force of hardened machines continued the excavation process in latitudes where the sandworms had not yet returned. Methodically, the mechanical army prepared the old sand section by section, opening the way for worms to expand their territory. Massive planting and fertilization work performed by powerful thinking-machine tractors and excavators had stabilized the seared ground, establishing a new biomatrix, while Paul's hardy settlers monitored the growth and did their own work alongside. Through his wide-reaching thoughts, Duncan made sure the thinking machines understood what Dune had once been, before outsiders meddled with its ecosystem. Misused technology had devastated the desert planet, and now technology would help bring it back.\n\nPaul stopped a hundred meters from the nearest rock formation, where a work crew had found the ruins of an abandoned sietch. With a small group of determined settlers, he and Chani had been salvaging the Fremen habitat with their own hands. Reclaiming the old ways.\n\nIn bygone days, he had been the legendary Muad'Dib, leading a Fremen army. Now he was content to be a modern-day Fremen, a leader of 753 people who had established austere homes in the rocks, which were on the way to becoming thriving sietches.\n\nPaul and Chani flew out regularly with survey crews. Instilled with fresh optimism, he saw the magnificent potential for Dune. Near the excavated sietch, he had discovered an underground grotto that he and his followers planned to irrigate and artificially illuminate, to support a planting project for grasses, tubers, flowers, and shrubs. Enough to support a small population of new Fremen, but not enough to shift the desert ecosystem that the new worms were recreating, year after year.\n\nOne day, he might even ride the great worms again.\n\nPaul turned to see the pale yellow sunrise appearing over the ocean of sand. \"Dune is reawakening. Just as we are.\"\n\nChani smiled, seeing both her beloved Usul of memory and the ghola she had grown up with. She loved each Paul for himself. Her abdomen protruded just a little, where their growing baby was beginning to show its presence. In five months, it would be the first child born on the recently resettled planet. In her second lifetime Chani did not need to worry about Imperial schemes, hidden contraceptives, or poisoned food. Her pregnancy would be normal, and the child\u2014or children, if they were again blessed with twins\u2014would have great potential, without the curse of terrible purpose.\n\nChani, even more in touch with the weather than he was, turned her face into a cool breeze. The sunrise began to show a new richness of coppery colors from dust stirred into the air. \"We'd better get back to the sietch, Usul. A storm is brewing.\"\n\nHe watched her glide forward gracefully, her red hair blowing behind her. Chani sang the walking song of lovers on the sand, her words lilting beautifully and in a stutter rhythm, like the cadence of her feet:\n\n> \"Tell me of thine eyes\n> \n> And I will tell thee of thy heart.\n> \n> Tell me of thy feet\n> \n> And I will tell thee of thy hands.\n> \n> Tell me of thy sleeping\n> \n> And I will tell thee of thy waking.\n> \n> Tell me of thy desires\n> \n> And I will tell thee of thy need.\"\n\nWhen they were halfway back to the rocks, the wind picked up. Blowing sand stung their faces. Paul held onto Chani, doing his best to shelter her with his own body against the abrasive wind.\n\n\"Yes, a fine storm is brewing,\" she said, as they finally reached the sietch entrance and hurried inside. \"A cleansing one.\" In the low light of a glowglobe, exhilaration flushed her features.\n\nCatching her by the arm, Paul spun her around and wiped sand from around her eyes and mouth. Then he drew her close and kissed her. Chani seemed to melt into his arms, laughing. \"So you have finally learned how to treat your wife!\"\n\n\"My Sihaya,\" he said as he held her, \"I have loved you for five thousand years.\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPublished in 2018 by Cavendish Square Publishing, LLC 243 5th Avenue, Suite 136, New York, NY 10016\n\nCopyright \u00a9 2018 by Cavendish Square Publishing, LLC First Edition\n\nNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means\u2014electronic, mechanical, photocopying, recording, or otherwise\u2014without the prior permission of the copyright owner. Request for permission should be addressed to Permissions, Cavendish Square Publishing, 243 5th Avenue, Suite 136, New York, NY 10016. Tel (877) 980-4450; fax (877) 980-4454.\n\nWebsite: cavendishsq.com\n\nThis publication represents the opinions and views of the author based on his or her personal experience, knowledge, and research. The information in this book serves as a general guide only. The author and publisher have used their best efforts in preparing this book and disclaim liability rising directly or indirectly from the use and application of this book.\n\nAll websites were available and accurate when this book was sent to press.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Murphy, Grace.\n\nTitle: The power of PHP \/ Grace Murphy.\n\nDescription: New York : Cavendish Square Publishing, 2018. | Series: The power of coding | Includes bibliographical references and index.\n\nIdentifiers: ISBN 9781502629463 (library bound) | ISBN 9781502634191 (pbk.) | ISBN 9781502629470 (ebook) Subjects: LCSH: Computer programming--Juvenile literature. | PHP (Computer program language).\n\nClassification: LCC QA76.73.P224 M87 2018 | DDC 006.7--dc23\n\nEditorial Director: David McNamara Editor: Caitlyn Miller Copy Editor: Rebecca Rohan Associate Art Director: Amy Greenan Designer: Joe Parenteau Production Coordinator: Karol Szymczuk Photo Research: J8 Media\n\nThe photographs in this book are used by permission and through the courtesy of: Front cover, Hero Images\/ Alamy Stock Photos; p. 4 Vincent Pontier\/http:\/\/www.elroubio.net\/elephpant_download.php http:\/\/php.net\/ elephpant.php\/File: Webysther 20160423 - Elephpant.svg\/Wikimedia Commons; p. 6 William Stadtwald Demchick, own work\/File: Rasmus Lerdorf August 2014 (cropped).jpg\/Wikimedia Commons; p.13 BestVectorIcon\/Shutterstock.com; p. 19 (right) Ben Scholzen (https:\/\/www.flickr.com\/people\/30771817@ N06) from Karlsruhe, Deutschland\/Andi Gutmans holding the keynote (https:\/\/www.flickr.com\/photos\/ dasprid\/5141279580\/)\/File: Andi Gutmans holding the keynote (5141279580).jpg\/Wikimedia Commons, (left) Leo Lefebvre, own work\/File: ZeevSuraski2.jpg\/Wikimedia Commons; p. 28 Magictorch\/Ikon Images\/ Getty Images; p. 34 Defense Advanced Research Projects Agency (DARPA)\/File: DARPA Big Data. jpg\/Wikimedia Commons; p. 40 Georgejmclittle\/Shutterstock.com; p. 45 WizardCat (http:\/\/wizardcat. com\/)\/published (https:\/\/cms-assets.tutsplus.com\/legacy-premium-tutorials\/posts\/52639\/images\/52639_ d94d3f3cd9aa454a8e0bf3b1932536cd.png) on tuts+ (http:\/\/code.tutsplus.com\/tutorials\/why-laravel-is-taking-the-php-community-by-storm--pre-52639)\/File\/File): Logo-composer-tutsplus.png\/Wikimedia Commons; p. 50 Alex Mit\/Shutterstock.com; p. 53 Larry Ewing (http:\/\/www.isc.tamu.edu\/~lewing\/linux\/), gg3po (http:\/\/ kde-look.org\/usermanager\/search.php?username=gg3po)\/lewing@isc.tamu.edu (mailto:lewing@isc.tamu. edu), The GIMP (http:\/\/isc.tamu.edu\/~lewing\/linux\/\/File: NewTux.svg\/Wikimedia Commons; p. 54 Jeff Sandquist (http:\/\/www.flickr.com\/people\/48003527@N00)\/File\/File): MIX Keynote-Scott Guthrie 09 MS 05 2007.jpg\/Wikimedia Commons; p. 61 MaxKobuzan\/Shutterstock.com; p. 67 FGRibreau, own work\/File: Redsmin-Redis-gui.png\/Wikimedia Commons; JovanCormac, own work\/File: MySQLQueryBrowser.jpg\/ Wikimedia Commons; p. 70 Tonusamuel, own work\/File: 20170531-DSC06942.jpg\/Wikimedia Commons; p. 72 Africa Studio\/Shutterstock.com; p. 78 Monkey Business Images\/Shutterstock.com; p. 79 Rawpixel Ltd\/ Alamy Stock Photo; p. 82 karmatosed (https:\/\/www.flickr.com\/people\/73631307@N00) from UK\/File: Matt Mullenweg (31537986015).jpg\/Wikimedia Commons; p. 83 Christian Rummel\/iStock Unreleased\/Getty Images; p. 86 Milesjpool, own work\/File: CPT-Object-Var-Proc.svg\/Wikimedia Commons; p. 89 Enrique Dans (http:\/\/www.flickr.com\/people\/16189770@N00) from Madrid, Spain\/File: Tim Berners-Lee April 2009.jpg\/ Wikimedia Commons; p. 95 Dean Drobot\/Shutterstock.com; p. 98 Wilgengebroed on Flickr (https:\/\/www. flickr.com\/photos\/wilgengebroed\/)\/File: Internet of things signed by the author.jpg\/Wikimedia Commons.\n\nPrinted in the United States of America\n\n1 The History of PHP.........\n\n2 How PHP Works...........\n\n3 Strengths and Weaknesses.................\n\n4 Getting Started with PHP..................\n\nGlossary..................\n\nFurther Information........\n\nBibliography...............\n\nIndex.....................\n\nAbout the Author..........\n\nOpposite: PHP's mascot is the ElePHPant.\n\n### The History of PHP\n\nEvery time you log on to Facebook, check Wikipedia for information you are curious about, or connect to a WordPress-created web page, applications written using PHP make it possible. PHP is what is known as a server-side scripting language, meaning it's written to automate certain tasks and run on the server rather than on the user's own computer. This immensely popular programming language helps accomplish tasks as varied as creating secure logins, accessing a multitude of databases, and incorporating Flash videos in dynamic web pages (web pages featuring content that changes based on user interaction). Some estimates indicate that PHP supports one-third of the web. Despite the complex web pages it is used to create, PHP is easy for novice programmers to learn. It also has many advanced features for more experienced programmers. But beyond PHP's huge potential for programmers of all skill levels, it also has a fascinating history.\n\n##### The Origins of PHP\n\nAs Rasmus Lerdorf, the originator of PHP tells it, he was not interested in creating a programming language, he just wanted to solve a problem. The Danish-Canadian systems engineer was looking for an easy way to develop and maintain his personal web page. So in 1994, he began working on the original PHP component, which was a common gateway interface (CGI). Lerdorf used the general-purpose programming language C to establish the components that would help him maintain his personal home page. Thus, he called his creation PHP Tools after \"personal home page\" and released it to the public. This first version, also called PHP 1, can still be downloaded at www.museum.php.net.\n\nRasmus Lerdorf created PHP.\n\nLerdorf had decided to make his software open-source. This meant that his solution would be freely and publicly accessible. It could also be shared and modified. According to Lerdorf, people participate in open-source software development for several reasons, including selfinterest (Lerdorf's self-identified motivation), because they have a problem they want solved; self-expression, because it satisfies their creative impulse; and altruism, because they want to improve the world.\n\nThe open-source movement had begun in the United States in the 1950s with academics sharing software among their colleagues. Soon, though, computer hardware manufacturers began charging for the software that was used on their machines. By the 1980s, some technology leaders were dissatisfied with the steadily rising software costs. They formed the GNU project to develop a software operating system that developers and users would be free to use, distribute, modify, and improve. The Linux operating system, with millions of users today, was the first result of that project.\n\nAlthough the term \"open-source\" was not coined until 1998, by the mid-1990s, when Lerdorf began developing his web scripting tools, the movement was gaining traction with the growth of dotcom companies. In particular, the Apache HTTP server, which played a critical role in the development of the World Wide Web and is today the most used web software in the world, was released as open-source software.\n\n#### Rasmus Lerdorf   \no o o\n\n\"As strange as it may sound, I don't actually like programming. My passion is for developing systems that simplify complicated problems. \"\u2014Rasmus Lerdorf\n\nRasmus Lerdorf, original creator of PHP, was born on Disko Island, off the coast of Greenland in Baffin Bay, on November 22, 1968. His early years were spent in Denmark. In 1980, he moved with his family to Canada. They settled in King City, Ontario, in 1983. Lerdorf attended King City Secondary School, where he played soccer and basketball.\n\nHe began developing his computer skills as a teenager by writing software for the Unisys ICON, a system built specifically for Ontario's schools. These computers, according to Lerdorf, didn't have much software and \"nobody knew anything about them. \" With his friend Ben Eng, Lerdorf created software for this distinctive machine and ended up teaching an adult education computer course. He also taught an accreditation program for teachers. Lerdorf credits these early experiences with providing both a stimulus and a solid base for his later interests.\n\nLerdorf enrolled at the University of Waterloo, whose Cheriton School of Computer Science produced roughly one-third of Canada's computer science graduates in the 1980s. At that time, any student who could demonstrate the ability to crash the computer system was treated to a drink for uncovering a weakness. Lerdorf was part of the computer co-op program and worked in several Canadian locations. In his final assignment, he worked with Nutec, a software company in Brazil. With his experiences in a variety of countries, Lerdorf became fluent in five spoken languages, as well as multiple computer languages. He graduated in 1993 with a Bachelor of Applied Science in Systems Design Engineering.\n\nHe continued with Nutec after graduation and was transferred to Silicon Valley shortly thereafter. Lerdorf became very excited about working with the web, which was then an up-and-coming technology field. Dissatisfied with his company's commitment to web projects, Lerdorf moved back to Toronto and began working as a consultant. In Toronto, he developed the initial version of PHP, a web scripting language, and worked on the Apache web server, which was then emerging as the most generally used web-server software.\n\nHis consultant work included a large project with the University of Toronto that facilitated online access for students and faculty, followed by work with Bell Global Solutions. After a brief sojourn in North Carolina with IBM's WebSphere team, Lerdorf moved back to Silicon Valley, attracted by the dotcom explosion. He participated in an unsuccessful startup while continuing to work on new versions of PHP By this time, although Lerdorf never intended PHP to be a full-fledged programming language, his original web scripting solutions had expanded exponentially and organically into a popular web language with hundreds of contributors working on this open-source project.\n\nLerdorf was hired in 2002 to help the search engine site Yahoo!. At the time, Yahoo! was the most popular web destination, and Lerdorf helped them make PHP work more effectively. In 2003, the MIT Technology Review named him as one of the top 100 innovators in the world under the age of thirty-five. He stayed with Yahoo! until 2009. He subsequently worked for three more sites, WePay, Etsy, and Jelastic, and consulted for startups. Today, Lerdorf travels around the world to speak at open-source seminars and conferences. Despite his avowed dislike for programming, Lerdorf is acknowledged as the creator of one of the most widely used web development languages.\n\nPHP soon was adopted by the open-source community, allowing developers to improve upon the code and fix any bugs. Lerdorf extended the initial components to have a much wider capacity that included web forms and the ability to transmit and receive information from databases. He called this version PHP\/FI (for forms interpreter) and, but only briefly, dropped the PHP designation and identified his scripting tools simply as FI.\n\nThis early version of PHP included some of the basic functions that still exist in PHP today. It could automatically interpret form variables and had a limited ability to embed HTML (hypertext markup language) syntax. Since HTML is the most basic building block of the web, both describing and defining the content of a web page, this capacity was critical. FI continued to receive greater acceptance, but it was not yet considered to be a full-fledged programming language.\n\nBy April 1996, Lerdorf had completed another revision and again called the new product PHP\/FI. In this second generation, PHP was evolving into a real programming language, rather than just a set of tools. It had a similar structure to C, which made it easier for C developers to use it. Its ability to function on a greater variety of operating systems was being explored. This version also supported cookies and user-defined functions, as well as MySQL, an open-source database management system. In June, the name of PHP\/FI was changed to PHP 2.0. After a short period of being called Personal Home Page Construction Kit, it had been accepted as the next generation of PHP.\n\nPHP 2.0 had a very short shelf life after it was officially released in November 1997. Despite its limitations, though, it was becoming very popular with web developers. According to a survey in May 1998, nearly sixty thousand domains (at that time approximately 1 percent of all domains) used servers that had PHP installed. PHP 2.0 did not have many contributors, however, and was essentially written by Lerdorf alone. Even at the time the 2.0 version was released, it was already being rewritten by the open-source community.\n\n##### PHP 3\n\nIn 1997, Andi Gutmans and Zeev Suraski, students at the Technion in Tel Aviv, Israel, began working on an e-commerce application for a university project. They were using PHP but found it was not as efficient as they wanted it to be, and it did not contain a variety of features that they needed. Working in conjunction with Lerdorf, they began a comprehensive rewrite of the PHP parser.\n\nSince parsers take source code and translate it into a data structure that can, after interpretation, be used to execute programs, an effective parser is a powerful computer-language component at the core of the language itself. This new makeover, with the collaboration of Gutmans, Suraski and Lerdorf, was designed to incorporate many improvements and build upon the PHP 2.0 user base. The newer version actually closely resembled the PHP language as it is used today. It was named PHP 3.0. This new version also had a renamed acronym to better reflect its many uses: the full meaning of PHP now became\u2014rather than Personal Home Page\u2014PHP: Hypertext Preprocessor.\n\nThe open-source PHP 3 attracted a number of new developers as well as many new users. The language was easy to extend, and many developers added a variety of modules to increase its usefulness. These included interfaces with multiple databases, connections to new protocols, and Application Program Interfaces (API). PHP 3 also included support for object-oriented programming (OOP), a model that is organized around objects and data, rather than action steps and logic. This type of program makes it easier to use code written for one program in another and to modify programs over time.\n\nDevelopers added a variety of modules to PHP 3, including support for Application Program Interfaces (APIs).\n\nAll these features increased the appeal of PHP 3. The announcement making PHP 3 the official successor to PHP\/FI was made by the new PHP Development Team in June 1998, after nine months of public testing. By that time, it was already installed on more than seventy thousand domains. It was also hosted on servers using Windows and Macintosh operating systems, as well as the original Unix and the operating systems modeled on that system, like Linux. At its height, PHP 3 could be found on approximately 10 percent of internet servers worldwide.\n\nBooks on PHP were beginning to make an appearance. The first was a book in Czech by Jirka Kosek called Creating Interactive Internet Applications. Shortly thereafter a book in Dutch by Egon Schmid, Christian Cartus, and Richard Blume, entitled PHP, was published. Core PHP Programming by Leon Atkinson, the first PHP book in English, followed shortly afterward. These early publications preceded a large number of PHP books that were written in different languages and published all over the world. There are over four hundred PHP books in English, over one hundred in German, and more than fifty books in Spanish or French. Other languages include Japanese, Hebrew, and Korean. The scope of PHP publications is another indication of the enormous success of the language.\n\n##### The ElePHPant in the Room\n\nIn 1998, the mascot for PHP was designed. A man named Vincent Pontier created the adorable image now associated with the PHP language and community when he was visited at home by a computer-programming friend who described for him the revolutionary contribution of PHP. While Pontier's friend told him about the ways in which this language would completely change the interactions on the web, Pontier was distractedly doodling the letters PHP on a white piece of paper.\n\nAs Pontier tells it: \"By chance\u2014chance sometimes [does] make good things happen\u2014I suddenly noticed that the letters were forming the shape of an elephant if viewed in a 'sideways angle.'\" Pontier sent his design to Jean-Pierre Dezelus, the owner of a personal website that would eventually transform into PHPinfo.net, a site for PHP resources. Unknown to Pontier, Dezelus was a collector of elephant paraphernalia, and he asked Pontier to create a true elephant logo. Pontier thought that naming it \"elePHPant\" was a good pun.\n\nThe expanding PHP community soon adopted the elePHPant, and it is now known around the world. Later, the logo would also be transformed into a stuffed animal, ever-present at PHP seminars and conferences. The PHP elePHPant has had many incarnations. Although the original elePHPant was blue, a purple elePHPant was created for a PHP Women's Kickstarter campaign, a black one was used for an AmsterdamPHP user group Kickstarter drive, and other elePHPants have been made in orange, red, and aqua for a variety of specialty purposes. There is even a wooly mammoth PHP plush animal, said to be the ancient ancestor of the elePHPant. For the twentieth anniversary of PHP, a singular golden elePHPant was created. This unique elePHPant was purchased by Zend Technologies and now has its own Twitter account.\n\n##### The Zend Engine\n\nThe release of PHP 3 did not signal a slow down in the language's development. Shortly after PHP 3 came out, Gutmans and Suraski started to work on the next PHP revision. They were focused on increasing the performance of the PHP software in terms of its speed and efficiency. They also wanted to expand the ability to divide the application into modules that could use a prewritten code and could be easily added when a new function was needed.\n\nIn addition to making the software development a less complicated process, these changes could make PHP more appealing to businesses. Modules could quickly be added to handle business needs as they were identified, and the time and cost of developing essential modules, tailored to the business demands, would be minimized.\n\nThe new software, first designated in 1999 as the Zend Engine, got its title from a combination using part the first names of its developers, Zeev and Andi. The term \"engine\" is often used to refer to software at the core of a language, in which different elements interact smoothly. An engine assists in automated processes with little or no human intervention. The Zend Engine does all that and more.\n\nThe Zend Engine is a separate software component that can be used with the web-specific interfaces of PHP but can also be used in other environments. It increases the performance and reliability of the PHP software. It allows for so-called dynamically integrated extension modules that can handle such tasks as debugging and boosting performance. It supports implementation of all the standard PHP structures. It includes high performance parsing and execution of PHP scripts. It provides memory and resource management. It also interfaces with modules that connect to external resources and protocols, such as MySQL, a popular open-source database.\n\nPHP 4, released in May 2000, was based on the Zend Engine. It also included better output buffering, ensuring that a program's output is sent to the browser after the entire script has been processed, rather than in separate pieces as the script is processed. PHP 4 integrated support for many web servers. It incorporated HTTP sessions, which creates a partially permanent information communication between two devices or a computer and a user. PHP 4 also provided an improved way to handle user input security. The popularity of PHP expanded even more and, in 2001, Netcraft's Web Server Survey indicated that there were 1.8 million sites using PHP.\n\nThere were, however, some concerns that PHP releases were not adequately tested for use. A PHP Quality Assurance Initiative was created, shortly after the release of PHP 4 in 2000 to address these criticisms. The first PHP conference was held in Germany in October of the same year. At the O'Reilly Open Source Convention, held in San Diego in 2001, an official PHP conference was included. Now PHP conferences are regularly held at locations around the globe.\n\n###### Zend and the PHP License\n\nWhen Gutmans and Suraski founded Zend Technologies to provide business support for the use of PHP in 1998, they also created a new PHP license. The license specifies that \"products derived from this software may not be called 'PHP,' nor may 'PHP' appear in their name, without prior written permission from group@php.net. You may indicate that your software works in conjunction with PHP by saying 'Foo for PHP' instead of calling it 'PHP Foo or phpfoo'''.\n\nProviding services for open-source (and therefore free) software is one business model that allows developers to make money from their involvement in open-source projects. Public domain software, which is not copyrighted, may be free, but anyone can make a modified proprietary version and charge for it. Some free programs may actually be copyrighted but are distributed under licenses that allow ownership of a version that has been modified.\n\nZeev Suraski (left) and Andi Gutmans (right) are the founders of Zend Technologies.\n\nZend uses a model in which the open-source community develops the primary program (in this case PHP itself), and Zend makes development tools for web applications that are written in PHP. Zend maintains a significant involvement in the continuing improvement and revision of the basic programs, but it does not have specific control over the process. By focusing on the needs of the business community, Zend was first able to get investors and eventually grow into an organization with high earnings.\n\n#### Zend Technologies, the PHP Company  \no o o\n\nAndi Gutmans and Zeev Suraski worked in collaboration with Rasmus Lerdorf, the original creator of PHP, to create PHP 3.0. In 1998, Gutmans and Suraski completely reworked and redesigned the PHP core in order to improve the language's basic code and its performance of complex operations. The redesigned parser was named the Zend Engine, and the next release of PHP, PHP 4.0, was based on the Zend Engine. It increased the program's efficiency and supported, among other improvements, improved security for user input.\n\nThe following year, Gutmans and Suraski founded Zend Technologies to provide business applications for PHP. Gutmans played several critical roles at Zend. He secured initial funding investments for the startup company including Israeli venture capital funds Platinum Neurone Ventures and Walden Israel; he headed the Advanced Technology Group, focused on strategy and innovation; and he engineered several strategic alliances, including some with business titans like Oracle, IBM, and Microsoft. In 2004, Zend moved its headquarters to Cupertino, California, where it continued to garner millions in investment funding. The Israeli Venture Association gave Zend the Best Startup Software Company award in 2006.\n\nCalling itself the \"PHP company,\" Zend continued to develop software that now includes a variety of products for web applications based on the PHP language. Its software allows programmers to create applications in PHP that accomplish tasks as varied as the protection of intellectual property and the ability to provide online training. Its Zend Server runs and manages web PHP applications. The Zend Platform checks PHP applications for performance problems. In addition to software development, Zend created a system for PHP certification and, by 2013, over ten thousand programmers had received this credential.\n\nGutmans was named Zend CEO in February 2009, and Zend continued to expand its offerings and its business. In October 2015, Zend was acquired by Rogue Wave, a Colorado software company that supports quick development of enterprise software applications (applications built for companies). According to Suraski, now Chief Technical Officer at Zend, the partnership with Rogue Wave \"is a continuation of our vision to support the PHP community and make PHP use easy, accessible, and secure in demanding environments. \"\n\n##### PHP 5\n\nThe next version of PHP took several years of development and a number of pre-releases that came out before the final, official release. But in July 2004, PHP 5.0 was distributed. This new version was powered by Zend Engine 2.0 and used a newly integrated object-oriented programming model as well as the traditional logic model. It added significant capabilities and performance improvements. It supported extensible markup language (XML), which is designed to communicate data, unlike HTML, which is designed to display that data. It included built-in Simple Object Access Protocol (SOAP) extensions that permitted users to call on web services and get responses independent of their operating system or the programming language they used.\n\nIn the next few years, PHP 5 underwent several revisions. PHP 5.1, released in 2005, continued to add performance enhancements. It simplified database handling and provided better time zone support. It also added more than thirty new functions with various extensions. PHP 5.2, released in 2006, added even more performance improvements. It included a new memory manager, incorporated in the Zend Engine. PHP 5.2 added many new features and provided fixes for more than 200 bugs. PHP 5.3 introduced many new upgrades, particularly using a better object model useful for complex applications. It also fixed more than 140 bugs.\n\nThe PHP development team expanded to include new participants, both individuals and organizations. In addition to programmers working directly on PHP code, many others worked on supporting or related projects, such as PHP documentation and PECL, the PHP Extension Community Library. The team relied on a network infrastructure of more than one hundred web servers on six continents. By 2008, PHP 5 was the only stable version under development and the transition from PHP 4 to PHP 5, promoted by the GoPHP5 initiative, was effectively implemented.\n\nCommercial uses of PHP continued to expand with PHP 5. Oracle declared PHP \"ready for enterprise use.\" Yahoo! standardized its operations on PHP. Zend Technologies formed strategic partnerships with IBM and Microsoft. And Zend CTO Zeev Suraski said that PHP \"was no longer considered a nice toy and a nice language for simple stuff. It was evident that we could support one of the largest internet sites in the world.\"\n\n##### The Version That Wasn't\n\nWhile PHP 5 was enjoying immense popularity, it was also being criticized because it did not include core language support for Unicode, an industry standard for encoding, representing, and handling text in most of the written languages on the globe. Because of Unicode's widespread use in computer software, Andrei Zmievski led a project in 2005 to bring Unicode support to PHP.\n\nThe inclusion of Unicode components to PHP would require a major revision of the language, both in its core software and its user code. The plan for PHP 6 was to release it as the new PHP version with both the Unicode changes as well as other added features then being developed. This plan ran into major roadblocks.\n\nZmievski's effort was plagued by a shortage of programmers who actually understood the intricacies of the necessary changes and by ongoing performance problems that seemed unable to be resolved. The project was delayed and, instead, PHP 5.3 was released in 2009, incorporating many of the features planned for PHP 6, including a built-in web server that could be used for testing (but without Unicode support). The project in its original form was abandoned in 2010, with the hope that a future revision would include the integration of Unicode. That hope has not yet been realized.\n\n##### PHP Moves On\n\nBeginning on June 28, 2011, the PHP Group, which is made up of major PHP developers including Lerdorf, Gutmans, Suraski and Zmievski, as well as other significant contributors, decided on a timeline for the release of new versions of PHP. Other minor but official revisions of PHP were then issued: PHP 5.4, PHP 5.5, and PHP 5.6.\n\nPHP continued to build its business base and its web presence. PHP software development kits using cloud-based features were provided by Amazon Web Services and Windows Azure. Facebook developed the HipHop Virtual Machine (HHVM) to convert PHP code into a code compiled dynamically at runtime, improving performance significantly. WordPress, using PHP as its scripting language, had become the number-one blogging site and was emerging as the preeminent web page creation tool. By 2017, it was used by 27 percent of the top ten million websites. In 2013, Google announced that PHP was now a language being supported in AppEngine, their cloudhosting platform. Google explained that PHP support had become the most requested feature, running on 75 percent of the web sites on the platform.\n\nIn 2014, PHP developed its own formal language specification. The PHP Group worked to set standards for the PHP semantics and syntax (semantics is the meaning assigned to the characters, words, and symbols used in PHP; the syntax is the structure of the PHP code). Up until this time, PHP functioned with only user documentation. As more applications began to use PHP, it had become more important to establish these standards.\n\n##### The Next Generation: PHP 7\n\nThe naming of the next major PHP revision was a subject of debate. Even though PHP 6 had never been released, the developers felt that giving the number 6 to the new version might cause some confusion. So, after more than two years in development, when the new generation of PHP was announced, it was called PHP 7.\n\nAs a result of the continuing desire to improve the performance of PHP, a branch of the language called phpng (PHP next generation) was developed by programmers named Dmitry Stogov, Xinchen Hui, and Nikita Popov. It supported just-in-time (JIT) compilation, which improves performance times. The performance enhancement of phpng was recognized and phpng was merged into the master language. In fact, it became the basis of PHP 7. The Zend Engine was redesigned to accommodate the significant changes resulting from phpng, and the new software was called Zend Engine 3.\n\nIn December 2016, one year after it was announced, the official version of PHP 7 was released. Some of the features incorporated in PHP 7 included performance at least twice as fast as PHP 5.6. As an example, before phpng, the WordPress homepage required approximately 9.4 billion central processing unit (CPU) instructions to execute. With PHP 7, it requires 72 percent fewer instructions\u2014approximately 2.6 billion. The new version also incorporated a secure random number generator, an enormous reduction in memory usage, and the removal of old, unsupported extensions. PHP 7 also introduced new operators, such as the \"space ship\" operator (^), useful in comparisons and sorting.\n\nPHP 7 demonstrated the tremendous development of PHP from a limited set of tools created for Rasmus Lerdorf's home page into a sophisticated language that powers websites like WordPress and Wikipedia. In fact, it is now ranked as the third most-used language on the web, after Java and Python.\n\nOpposite: The internet connects the world. Programming languages like PHP make it possible.\n\n### How PHP Works\n\nPHP is what is known as a server-side scripting language. This means that it runs on an internet web server, a computer at a remote location that will process web requests and send web pages back to the user, rather than processing them on the user's own computer. So, in order to appreciate how PHP functions, it is important to have some basic knowledge of how the internet itself works.\n\n##### A Closer Look at the Internet\n\nAll over the world, computers and computer networks are linked through the global network known as the internet. This network is connected in a variety of ways, including telephone wires, fiber optic land lines, satellites, microwave links, and cables laid under the ocean. Despite all the different ways in which the parts of this vast network are connected, it handles all data in the same way. Within this uniformity of data transfer though, the internet can pass on many different types of data, including emails, web pages, and messages, from one computer to another and one network to another. One familiar form of data exchange that provides document storage and access, usually in the form of web pages, happens on what is known as the World Wide Web, a system of servers on the internet. One of the major uses of PHP is the creation of many of those web pages.\n\nWeb pages themselves are documents that have been specially formatted in what is called a mark-up language, the most common being HTML. Markup languages provide extra information about the visual appearance of text in a document, including how it should be formatted for display or printing. When a user types in a web address, or URL (Uniform Resource Locator), usually in the form www. webaddress.suffix, the user is connected to the web server at that location.\n\nThe web page located at that address on the server computer is transferred to the user's computer, and the document, formatted using HTML, can be viewed by the user. Most documents use HTML or similar markup languages. However, when a document is transmitted using only HTML, it does not contain most of the features now expected\u2014such as interactivity and audio and graphics. Something more has to happen with a web page so that users are connected and engaged in a variety of ways and not just reading the content. Rather than the static (unchanging) web pages created with only HTML, web designers create dynamic web pages with content that can change based on the interests or responses of the user. These dynamic web pages are generated in real-time through the use of scripting languages. The essential feature of a scripting language is its ability to manage the activities of other programs, telling them what to do, in the same way that a play script indicates what an actor should do to play a part.\n\n##### Dynamic Scripting\n\nThere are two ways a website can be made dynamic. In order to create dynamic web pages, web applications may contain code that is processed on the user's browser (the user's browser is known as the client), or on the web server.\n\nThe method known as client-side, or front-end, scripting is run directly on the user's computer and the processing usually takes place on the browser. Whatever client-side scripting language is being used, such as JavaScript, must first be enabled on the user's computer in order for the scripting to function. The front-end scripts are embedded in the HTML of the user's site. The scripts select site elements and manipulate them, providing the user with an interactive experience. Client-side code is best suited for situations in which the page elements may be changed without the need to contact a database. Client-side scripting can be used, for example, to move elements on a page, change colors and fonts, or display a pop-up window based on the position of the mouse.\n\nServer-side or back-end scripting is the alternative way to build dynamic websites and has different uses than front-end scripting. In server-side scripting, the scripting application is run directly on a web server, rather than on the user's own computer using the client browser. Software using server-side scripts takes a user's request, processes it, and generates dynamic HTML pages that are sent back to the user's browser. Unlike client-side scripting, server-side scripting can interface with databases on the server.\n\nThe two methods may be used in conjunction with one another. A site such as Amazon uses both types of scripting. Server-side scripting may be used for logins, entering personal information, responding to user inquiries, and handling new data storage. Client-side scripting, on the other hand, displays and sorts data in a variety of ways, changing the appearance of a page in response to the user's actions.\n\nAnother significant distinction between the two scripting methods involves the ability to hide the code that generated the website. The website owner can protect their code and conceal the instructions from the client when using server-side scripting, since the script itself is never transferred from the server. Only the results of the processing are transferred. In client-side scripting, however, the script resides on the user's computer and is fully accessible.\n\n##### Server-side Scripting: What It Can Do\n\nServer-side scripts, like those written in PHP, make a website useful and functional. The term \"server\" refers to both hardware (the actual machine) and software (the coded instructions that tell the server what to do). At a minimum, a web server understands web addresses (URLs). Whenever a browser needs a file that is hosted on a web server, the browser requests the file via HTTP (Hypertext Transfer Protocol), and the server sends the file back using the same protocols. This standardized method enables the user's browser to view a web page.\n\n#### PHP-Is It a Language?  \no o o\n\nThe 1s and 0s of binary code are the simplest form of computer coding. PHP is far more complex, though it took time before it was recognized as a language.\n\nWhen Rasmus Lerdorf first developed PHP, it was not thought of as a programming language. It was simply code written to help Lerdorf create a dynamic web page. However, what was useful to Lerdorf turned out to be useful to other programmers, and PHP's popularity grew.\n\nMany programming languages, like Java and C, were developed through specific projects intended to create those languages. Even Ruby, the object-oriented scripting language originally developed for a Japanese market, was thought of as a new programming language from the beginning. Lerdorf himself, though, called his efforts a \"tool box,\" not a language. He said, \"There was never any intent to write a programming language. .. I have absolutely no idea how to write a programming language, I just kept adding the next logical step on the way. \"\n\nHowever, the energy and direction of the open-source community led to the adoption of PHP as a widespread and well-liked method for creating dynamic web pages\u2014whether it was called a language or not. Although some claimed it couldn't be a \"real\" language because it was not consistent and it was not compiled, others claimed that it met the minimum requirements for being considered a language: using syntactical constructs, it instructed the computer to perform certain operations.\n\nPHP also lacked any strong design principles or solid theoretical principles and was thought of, even by those who considered it to be a language, as messy and lacking rigor. Its implementation was dependent upon user documentation. Although that documentation was extensive, its extensiveness led to problems. Furthermore, some basic features, like correct handling of foreign language and Unicode characters, were add-ons that were not cleanly integrated.\n\nAs previously mentioned, in 2014, PHP finally got a complete definition of its semantics and its syntax in a document called a draft specification. The draft specification was intended to \"evolve over time with the help of everyone who cares about the PHP language,\" just like other open-source products do. But, although it would be a document that would continue to change and mature, it provided, for the first time, a structure for the future development of PHP However PHP was initially created, and however it had evolved, after twenty undefined years PHP had finally become a full-fledged computer programming language.\n\nThe dynamic servers involved in server-side scripting, however, are more complex. These servers not only contain the software that controls HTTP file transfer, but also, most commonly, extra software applications and related databases. The software applications on dynamic servers process the requested web pages, updating the information on the web page as well as the information on the databases, and transfer the updated page back to the user's computer.\n\nThis means that the web page and its content reside on the server and not on the user's computer. The server-side process follows certain steps:\n\n1. A user requests a web page by entering the URL for that page or by following a link to the page.\n\n2. The scripting application on that server (for example, a PHP application) is interpreted by the server and creates or changes the content of the requested page \"on the fly\" based on current information, such as user login input or other available data.\n\n3. The newly generated page in its final form is sent back to the user in HTML format.\n\nServer-side scripting provides for a high level of personalization. Its ability to access databases is very powerful. Applications written in server-side languages like PHP provide the basis for e-commerce sites, social networking, and even Massively Multiplayer Online Role-Playing Games (MMORPGs). The HTML pages with server-side scripts are easy to maintain because they do not have to be edited every time on the local computer with newly written HTML and can be updated right on the server. Information used in updates is stored in a database, which may be located directly on the web server or may be on additional servers linked to it.\n\nThe PHP application can access different types of databases, and some of the most frequently used include the following:\n\n\u2022 MySQL: This continues to be the most frequently used database with PHP applications, and it is the world's most popular open-source database. MySQL is used on enormous websites such as Wikipedia, Facebook, and Google. It provides a web-based user interface in order to manage it.\n\n\u2022 Oracle: The Oracle database is one of the most widely used databases. It is equally popular for servers based on Windows and Linux operating systems. Many major business organizations use this database.\n\n\u2022 DB2: This database was created in 1983, although its development began in the 1970s. It was originally used extensively in conjunction with mainframe systems. DB2 is now being used in many large-scale Enterprise Resource Planning (ERP) and e-commerce systems.\n\n\u2022 PostgreSQL: Compared to MySQL, PostgreSQL has many advanced features that compare well with the advanced features in Oracle. It is, however, believed to be a bit slower than MySQL.\n\n\u2022 Sybase: This database solution is a popular choice for enterprise database management, and it can operate well when a large database is needed.\n\nAlthough these are the most frequently implemented databases with PHP applications, PHP can support a wide variety of less popular databases, from Mongo to Tokyo Tyrant.\n\nA server-side script, then, creates the communication link between a user, the web server, and the databases. It requests, edits, adds to, and deletes items in the database. In providing this link, the server-side script has many functions:\n\n\u2022 It collects data from forms.\n\n\u2022 It processes any user input, such as username and password, and it interacts with databases.\n\n\u2022 It queries the databases in order to accomplish tasks like password validation.\n\n\u2022 It updates the database.\n\n\u2022 It sends and receives cookies.\n\n\u2022 It encodes the generated web page into HTML.\n\n\u2022 It transfers the requested web pages for display on the client browser.\n\n\u2022 It encrypts data for security.\n\nSome examples will help illustrate how this server-side scripting works.\n\n###### Example One: Placing an Order\n\nE-commerce sites are frequently written using server-side scripting. Let's imagine that a customer wants to purchase replacement toner for a laser printer. She enters a Google search for companies that sell laser toners, and the request is transmitted to the Google web server, via HTTP. The web server uses a scripting application that resides on the server, in order to process this request. The application sends a query to the Google database, looking for matches. When a match is found, the application generates a web page in HTML that contains the matches found on the database. Often, this web page is created simply by inserting the matching data in an HTML template, and the process is simplified and speedy. Now the newly created web page, responding to the customer's inquiry and providing a list of toner sales companies, is transmitted back to the customer's client browser, again using the HTTP protocols.\n\nOnline shopping is a great example of server-side scripting at work.\n\nThe customer next selects the website she wants to use from the list of companies on the web page that is now displayed on her browser. She clicks on the link, which connects to the chosen company's web server. The company's web page, which uses a scripting application written in PHP, is transferred from the company's server to the user's browser. This web page contains a number of input forms for the user to complete. She is asked to identify her printer make and model and the color(s) of toner she needs. When she has filled in the forms, she clicks on the submit button on the web page. Now the information is transferred back to the company's web server. There, the scripting application connects to the database that contains information about printers, matches with the correct toners, and lists the price of each toner by color. The scripting application, once again, creates a web page \"on the fly,\" containing the information extracted from the toner database. This new web page is transmitted back to the user's browser.\n\nThis page contains the information about the available toners and also has a form for the customer to complete if she wants to place an order. Filling in the form that requests that an order be placed and clicking to submit it once again transmits this information back to the company web server. The scripting application processes this request and sends a page back to the client for the customer to fill out with billing and shipping information. When the company web server receives this new information, the scripting application matches it to a database of customers that is also on the company's server. If she is a new customer, the information will be added to the company's customer database. If the information is a partial match, a web page that requests more information, for example an updated shipping address, will be generated. If the customer responds with new information, the database will be edited to include it. When the information has been verified, the customer will be requested to review the order details on a web page that was generated at the server with all the current data. If the customer accepts this information and completes the transaction, the database will be updated and the order will be placed.\n\n###### Example Two: Connecting to Social Media\n\nServer-side scripts provide the interactivity on social media sites. Facebook, for example, uses PHP as its basic programming language. So, if a user wants to edit the information that is on his profile, he will first log in to his Facebook account with his username and password. This login information will be transferred to the Facebook server (actually one of the over sixty thousand Facebook servers). The server-side script on the server will access the database containing the user's information and generate the personal Facebook page that the user will see, transmitting it back to the user's browser.\n\nIn order to change the profile information, the user will hover over the Intro section below his profile picture and click on the pencil graphic. He then can edit or add to his profile. When he clicks on \"update your information\" and \"save,\" the new profile is sent back to the Facebook database on the web server, and the scripting application updates the user information that is stored there.\n\n###### Example Three: Doing Research\n\nWebsites that primarily provide informational content, like Wikipedia, have very simple scripting applications. If a user types in a search term on the Wikipedia website and clicks on the search button, the request is transmitted to the Wikipedia server. The scripting application finds the matching database and fills in the information it provides on an HTML template. The HTML file, containing all the database information, is then sent back to the user's browser.\n\nSome informational websites, however, have more complex scripts. When a user requests the front page of a news site, the web server application may retrieve news stories from a database on the server combined with news stories retrieved from another website to generate a page that includes the most current information. The page sent back to the client's browser may, in turn, be automatically updated by the web server application.\n\n#### Your Data Package Has Arrived   \no o o\n\nProgrammers don't want to keep reinventing the wheel. They often look for a piece of reusable code that can be dropped into an application to perform frequently utilized functions, such as user authentication or database management. This reusable code is contained in libraries that programmers can access when they are looking for an efficient way to insert a certain function into the application they are writing.\n\nFrequently, these libraries are collected together in groups called packages. The libraries in a package are either built upon each other or use each other in order to function most effectively. For example, library \"A\" may use code from library \"D\" in order to run effectively, so the programmer needs to download both \"A\" and \"D\" in order to run code \"A. \" This relationship is called a dependency.\n\nManaging dependencies can be a complex and sometimes messy problem, particularly if a project has multiple programmers calling upon a variety of packages to perform a variety of tasks. PHP Extension and Application Repository (PEAR) was an early system for distributing reusable PHP components. But PEAR was hard to search, hard to add new components to, and generally difficult to use, so many PHP programmers abandoned its use.\n\nIn 2012, Nils Adermann and Jordi Boggiano, two European developers, announced the release of Composer, a package dependency management tool for PHP Unlike PEAR, Composer manages dependencies on a project-by-project basis rather than forcing a programmer to install packages on the entire system. It will download all the required libraries and packages and handle them in a meaningful and logical way. Its main package repository is \"Packagist. \" Composer has now become an essential programming tool for most PHP projects.\n\nComposer is a package dependency management tool.\n\n###### Example Four: Updating a Website\n\nWebsites built with server-side scripting, such as PHP, have some significant advantages when the user wants to update the site, particularly if the site consists of a large number of pages. If the website is created with a static code such as HTML, each page would have to be changed manually and uploaded when an update was needed. If, however, the site is created using PHP and MySQL as a linked database, the content could be updated for the entire site using the database. In order to update or alter the design, it would only require altering a small amount of PHP script.\n\n##### How PHP Code Looks\n\nPHP code can look very much like the HTML code that is used to produce web pages. Although PHP scripting can accomplish a great variety of tasks, it is deceptively simple to learn and use. At its most basic level, it can be embedded in an HTML page and simply create a static document, as though the code were completely HTML. All embedded PHP code is identified by beginning with <?php and ending with ?>.\n\nFor example:\n\n<html> THIS SECTION IS HTML\n\n<head>\n\n<title> My first PHP script <\/title>\n\n<\/head>\n\n<body>\n\n<?php EMBEDDED PHP CODE BEGINS\n\necho \"Hello, Programmers!\";\n\n?> EMBEDDED PHP CODE ENDS\n\n<\/body> HTML CODE\n\n<\/html>\n\nThis embedded PHP code will produce a static page with the message \"Hello, Programmers!\" This could also be achieved by using only HTML code. If, on the other hand, the code included a PHP variable, it would look like this:\n\n<html> HTML CODE\n\n<body>\n\n<?php EMBEDDED PHP CODE\n\n$txt1 = \"Learn PHP\";\n\n$txt2 = \"online\";\n\necho $txt1 \";\n\necho \"Study PHP $txt2\";\n\n?>\n\n<\/body> HTML CODE\n\n<\/html>\n\nThe output produced by this code would be: Learn PHP Study PHP online Since this message is created using the variables identified as $txt1 and $txt2, the output can be changed by changing the variables. This is a step closer to a dynamic web page. If, however, the user fills out a form created in HTML and clicks the submit button, the PHP code that can process that request does much more. The PHP code will refer to the HTML code that created the form, as in the following example:\n\n<html> HTML CODE\n\n<body>\n\n<form action=\"identity.php\" method=\"post\">\n\nName: <input type=\"text\" name=\"name\"><br>\n\nE-mail: <input type=\"text\" name=\"email\"><br>\n\n<input type=\"submit\">\n\n<\/form>\n\n<\/body>\n\n<\/html>\n\nThis HTML code will create a form for the user to complete that looks like this:\n\nName:\n\nE-mail:\n\nThe user completes this form, clicks submit and the data is sent to a PHP file named \"identity.php\" so that it can be processed. The \"identity.php\" file (embedded in HTML) looks like this:\n\n<html> HTML CODE\n\n<body>\n\nWelcome <?php echo $_POST[\"name\"]; ?><br> PHP CODE BEGINS WITH <?php\n\nPOST[\"email\"]; ?> <\/body>\n\nPHP CODE ENDS WITH ?>\n\nHTML CODE\n\n<\/html>\n\nIf the user's name was Harry and the email address entered was harry@mywebsite.com, the output on the user's computer would be:\n\nWelcome Harry\n\nThe email address you gave is: harry@\n\nmywebsite.com\n\nThe PHP application on the web server took the data transmitted to it from the web form and generated a web page using that information. If the user was named Sally and her email was sally@mywebsite.com, the output would have been:\n\nWelcome Sally\n\nThe email address you gave is: sally@ mywebsite.com\n\nThis example is a simple demonstration of one of the ways in which PHP can personalize a user's web experience using simple code. More complex operations, including querying databases for information, validating input, and updating the databases, can all be accomplished with straightforward PHP code. PHP can do a lot with a little.\n\nOpposite: Like any programming language, PHP has strengths and weaknesses.\n\n### Strengths and Weaknesses\n\nPHP was originally built simply to create web pages. Through the years of its continued evolution, it has maintained its focus as a web page development tool. Over time, though, it has matured into a powerful language whose immense popularity results from its significant benefits. The users of the huge number of websites that run on PHP experience these benefits, even if they don't all know that PHP is responsible.\n\nPHP's popularity demonstrates the language's usefulness, but it doesn't necessarily mean that PHP does everything perfectly or that it never changes. Ongoing PHP development shows how PHP benefits from continuing to identify the improvements that could make it better. If we look at the different aspects of PHP and how it functions, we can get a good picture of where PHP is now and where it might be going.\n\n##### Open-source Technology with Community Support\n\nMany programming languages, such as ASP.NET, have free editions, but their professional editions can be expensive. Even the free edition of ASP.NET requires a proprietary Windows server. Alternatively, a major characteristic of PHP is that it is an open-source technology, free for developers and users and responsive to contributors. Also, the PHP community has compiled and published an almost uncountable number of free support applications to help make using PHP easier, more efficient, and more effective since the language first launched. According to php.net, \"Anybody who programs in PHP can be a contributing member of the community that develops and deploys it; the task of deploying PHP, documentation, and associated websites is a never ending one.\"\n\nThe community collaboration for PHP is extensive and enthusiastic. There are a significant number of reference sites, tutorials, and guidelines available on the internet and even in many different spoken languages. Some websites allow you to post a question, get answers from users, and see how other users rated the response. Hardcover books on PHP have been appearing across the globe almost since the language's inception. With PHP's great popularity, it has a large number of contributors, developers, and users, and there are numerous support groups, forums, and chat rooms that focus on PHP issues and solutions. PHP programmers can always find someone willing to help. On the other hand, open-source code allows everyone to see the source code, and, if there are bugs in the source code, they can be used to exploit any PHP vulnerabilities and weaknesses.\n\n##### Cross Platform Performance\n\nPHP is not only free, it works on almost every platform (i.e., a computer and the operating system on which software applications are run). This includes both open-source platforms and even the proprietary ones. PHP can function, for example, on Unix (its original platform), Linux, Windows, and mobile platforms such as iOS and Android. Some of its competitors, like JSP, also function on multiple platforms. Others, like ASP.NET, which functions only on Windows, are platform-dependent.\n\nTux, the Linux logo\n\nPHP also can also support a wide variety of web servers with different server software, including Apache (the world's most popular server), Microsoft Internet Information Server (IIS), and a number of less standard servers, such as WAMPserver, and Oracle iPlanet. Then, too, PHP has cross-browser compatibility, working as well, for example, with Google Chrome as with Safari. Other scripting languages may, like ASP.NET, require additional scripting for browser compatibility.\n\nIt is also easy to find hosting service providers for PHP, while other languages, like Python, have operational requirements that can limit hosting options. PHP hosting can run under Windows (WAMP), Linux\/Unix (LAMP), Apple Mac (MAMP), Sun\/Solaris (SAMP) and other web server software. ASP.NET runs under Windows, but in some cases for increased flexibility, it can use a virtual machine or framework in some other operating systems.\n\nScott Guthrie is best known as the developer of ASP.NET.\n\n##### Server-side Scripting\n\nOne of the most important features of PHP is that it is a server-side application. This characteristic alone provides PHP with certain benefits. Unlike with client-side scripting, the user does not need to enable the scripts to run on their computer by downloading plug-ins, such as Java or Flash. With server-side scripting, the workload on the user's computer is reduced, because all the heavy lifting is being done on the server where the web page is actually being created.\n\nPHP can also create those web pages \"on the fly\" based on user interactions, and the pages are often fairly quick to load. Using server-side content management systems (CMS), like WordPress, website owners can create or maintain their own sites without the need to directly edit code. Users can also use external, readily available pre-written files (usually called libraries or packages) in order to save time and effort coding.\n\nThen, too, the developer's scripts are hidden from the user, making the code more secure. Server-side scripting with PHP also protects the server from crashing. PHP runs in separate, isolated processes within the server software, and therefore it is very difficult for any single process to bring down the entire web server. When a request to the server is identified as a PHP file, the server gives that file to the PHP interpreter. The PHP interpreter then reads the file and executes the PHP code. When it is done, the PHP interpreter gives the output of the code back to the server. Since the PHP interpreter's state is totally reset at the beginning of each request, it is more reliable than systems with long-lived processes, and it ensures that errors will have a minimal effect.\n\nOn the other hand, with scripting software, all the necessary software and database(s) must be installed on the server in order for the scripts to function properly. In some cases, there are significant data security concerns because it is often easier for hackers to access servers where they can exploit flaws in the program code.\n\nOne researcher found that almost 78 percent of PHP applications were running with at least one known vulnerability. One of the most common weaknesses results from inadequate filtering of user input, allowing malicious code to be inserted into the application. Another frequent security problem is caused by SQL injections that occur when an attacker gains access to a database and subsequently to all of the website data. The most common security vulnerability, though, is known as cross-site scripting (XSS), which occurs when a hacker finds a way to get a visitor to a website to load pages with insecure coding, e.g., into a form inserted on a page that then transfers the user's input data to the hacker. All these vulnerabilities can be addressed by following best practices in PHP coding, but the popularity and ease of the language can often result in careless coding and increased security risks.\n\n##### Simplicity\n\nPHP was originally built using C, a general-purpose programming language. Because PHP's syntax remains similar, it is generally straightforward for C programmers to use. However, even for programmers without any knowledge of C, PHP has a quick learning curve, and it is fairly easy to become adept at coding in PHP.\n\nMany programming languages are complex and require a complicated structural knowledge to accomplish tasks. A new PHP programmer, however, can learn a few basic functions and variables and begin coding. Another aspect that streamlines PHP is that it does not restrict documents to a specific order, and users adding new code don't have to make sure that that new code is in a specific place. Also, PHP does not need to be compiled before use since it is executed step-by-step during run-time, rather than turned into machine-executable code in a separate step before it can be run.\n\n#### ASP. NET and PHP   \no o o\n\nASP.NET and PHP are both server-side scripting languages used in the development of dynamic web pages. Despite this essential similarity, however, they do have significant differences. One major distinction is cost. PHP is a free open-source tool, and ASP.NET is a Microsoft proprietary product. PHP runs on several platforms including Linux and Windows, while ASP.NET is designed for Windows implementation. Additionally, PHP can be run on a variety of servers, while ASP.NET is intended for an internet information server (IIS) created by Microsoft for use with Windows.\n\nFrom a programming perspective, though, the ease with whichFrom a programming perspective, though, the ease with which PHP can be learned and implemented may provide the greatest contrast between the two languages. ASP.NET has a complex syntax and can be difficult to read and hard to learn. PHP, on the other hand, is so straightforward that novice programmers can often begin writing basic programs very quickly.\n\nA comparison of the code to create a simple input form in these two languages clearly illustrates the differences.\n\nThe following is ASPNET code:\n\n<html>\n\n<body>\n\n@{\n\nif (IsPost)\n\n{\n\nstring name = Request[\"Name\"];\n\nstring email = Request[\"Email\"];\n\n<p>You entered: <br>\n\nName: @name <br>\n\nEmail: @email <\/p>\n\n}\n\nelse\n\n{\n\n<form method=\"post\" action=\"\">\n\nName:<br>\n\n<input type=\"text\" name=\"Name\" value=\"\"><br> Email:<br><br>\n\n<input type=\"text\" name=\"Email\" value=\"\"><br><br> <input type=\"submit\" value=\"Submit\" class=\"submit\"> <\/form>\n\n}\n\n}\n\n<\/body>\n\n<\/html>\n\nAlternatively, this PHP code can be implemented:\n\n<html>\n\n<body>\n\n<form action=\"early.php\" method=\"post\">\n\nName: <input type=\"text\" name=\"name\"><br>\n\nE-mail: <input type=\"text\" name=\"email\"><br>\n\n<input type=\"submit\">\n\n<\/form>\n\n<\/body>\n\n<\/html>\n\nIn both instances, the output would be the following input form:\n\nThe differences in both the amount of code and its complexity are obvious. PHP can accomplish the same task with much greater simplicity.\n\nPHP is not just suitable for novice coders without any other programming knowledge or experience, though. Developers who want to create more complex applications, can, for example, write custom extensions in C or C++ and will find well-defined and documented support for extending the functionality of PHP using this method. PHP is also easily and seamlessly embedded in other languages, most notably HTML.\n\nThe simplicity of PHP also makes it very easy to edit. PHP code is clear, and its purpose is easy to identify, so making changes is not complicated. Other languages require the user to change multiple functions within several documents in order to make the same changes that PHP can accomplish with limited changes in a single document.\n\n##### Versatility\n\nAlthough PHP is simple, it is also versatile. In addition to providing the freedom to choose any operating system and any web server, PHP is not limited to HTML output. PHP can output images, pdf files, Flash videos, and a variety of text files including XHTML and any other XML files. These files can be autogenerated by the PHP application and saved in the file system, forming a server-side cache for dynamic content.\n\nPHP is as versatile as a Swiss army knife.\n\nAlthough it's probably not the best use of PHP, it can even create desktop applications with a graphical user interface (GUI) in a client-side program. PHP can also extend its range through the use of command line scripting, making PHP run without any server or browser. These scripts can be used for some regularly scheduled tasks as well as simple text processing tasks.\n\nPHP also supports many communication protocols including:\n\n\u2022 Hypertext Transfer Protocol (HTTP), the foundation of data communication for the World Wide Web.\n\n\u2022 Lightweight Directory Access Protocol (LDAP), a software protocol used to find individuals, organizations, and other resources within a network regardless of if that network is public (on the internet), or private (on a corporate or organizational intranet).\n\n\u2022 File Transfer Protocol (FTP), the standard protocol for transferring files from the server to the client.\n\n\u2022 Internet Message Access Protocol (IMAP), a protocol for email that allows users to view and manipulate messages stored on a server as though they were stored on the user's own computer.\n\n\u2022 Simple Network Management Protocol (SNMP), which collects information from network devices on an Internet Protocol (IP) network, such as servers, printers, hubs and routers, and configures those devices.\n\n\u2022 Network News Transfer Protocol (NNTP), a protocol used to transmit news articles between news servers and posting articles in user applications.\n\n\u2022 POP3, used to download email to the user from a mailbox on the server.\n\n\u2022 Component Object Model (COM), Windows software that allows software components to communicate.\n\n\u2022 Web Distributed Data eXchange (WDDX), that allows complex data exchange between virtually all web-programming languages.\n\nPHP also encompasses the two major conceptual models used in web programming. The procedural model is organized around functions that act on what are called data structures. On the other hand, object-oriented programming (OOP) uses objects, which incorporate both the data and the functions that act upon it. This flexibility allows PHP programmers to use procedural code for small projects where speed and coding ease are important. OOP allows for greater coding structure and better handling of complex and large-scale projects. The frameworks used in PHP applications are all designed with OOP and, although procedural coding may provide an easy entry into PHP programming, developers should be able to handle both methods.\n\n##### Web Frameworks\n\nFrameworks\u2014standardized methods for building and organizing web applications\u2014are a wonderful resource to help programmers build web programs more quickly and efficiently. PHP has a number of available frameworks that can make the programming process less complicated and time-consuming. These include Symfony, Laravel, CodeIgniter, CakePHP, and Zend Framework.\n\n##### Content Management Systems\n\nPHP also has a number of content management systems (CMS) that function to support web page management. Almost 60 percent of websites use content management systems. A CMS can, among other functions, allow a user to modify content on a website without being an administrator. CMS can also update websites quickly and easily.\n\nSome of the many CMSs used with PHP include Composer, Drupal (written in PHP and has a community of more than 1.3 million members), Joomla (an open-source option), Magento, October CMS, and WordPress. WordPress is identified as a blog tool, publishing platform, and CMS; it's used by more than 25 percent of the world's websites.\n\n##### Performance\n\nPHP developers have worked diligently to improve its performance. More recent developments seem to have helped PHP speed up significantly. With PHP 7.0, \"on the fly\" optimizations have been incorporated. Facebook has created an open-source runtime engine\u2014 originally called the HipHop Virtual Machine, now just HHVM\u2014that is said to execute PHP commands almost six times faster than the standard PHP interpreter. HHVM now powers Facebook, and other large PHP users are adopting it as well.\n\n##### Database Support\n\nOne of the most powerful strengths of PHP is its support for a wide array of databases. A PHP application can access databases in a variety of ways: using a database specific extension, such as MySQLi; using PHP Data Objects (PDO) to connect to database drivers other than MySQL; or using other methodologies like cURL. And the number of PHP-supported databases is enormous, and includes, among others, Access, Adabas D, dBase, Ingres, InterBase, Empress, Oracle FrontBase, FilePro, PostgreSQL, Sybase, mSQL, Solid, Hyperwave, DB2, Velocis, Informix, ODBC, and Unix dbm.\n\nMySQL is the database most frequently used with PHP applications, and the code can interface with it and integrate the strength of the database and its relational organization into the PHP application. This ability also reduces the development time for web applications.\n\n##### Libraries\n\nSince PHP is very popular, it has a large community of developers that contribute by crafting PHP components that help other programmers build their own projects more easily. These components are stored and accessed through PHP libraries. Since the resources are so extensive, however, finding the right components requires organization and support.\n\n###### Repositories\n\nSoftware repositories are web-hosting facilities that contain file archives for a large amount of source code. They provide storage from which software can be accessed and installed. They often provide additional services that may include version control (logical management and identification of software changes), release management (controlling a software development through different stages), and bug tracking (a database providing information about a software package's known bugs), as well as documentation. Some repositories for PHP code include PEAR, FSD, SourceForge, GitHub, and Packagist.\n\n###### Library Examples\n\nWriting PHP code can be monotonous, repetitive, and time-consuming. Many of the features that make PHP sites appealing and functional do not necessarily require code that is uniquely designed for a particular application; they can be implemented using code written by other developers and are available to download using PHP libraries. Libraries contain prewritten code that can provide a variety of functionality, save coding time, and allow developers to focus on other aspects of web design.\n\nIf a developer wants to enhance the images on a website, for example, Image Workshop is an open- source library that allows image manipulation such as cropping, resizing, and creating watermarks and thumbnails. Snappy, another PHP library, supports snapshot, pdf, or thumbnail generation from an HTML page.\n\nAccess is a commonly used relational database from Microsoft.\n\nAssistance with PHP and MySQL development is provided by PHP DB Class, a library that reduces the amount of coding required to access a database. PHP DB Class also provides a variety of debugging features. If a developer wants to create a tree hierarchy from a database, Tree is another library that can make that easy.\n\nPHPMailer is an essential PHP library backed by a huge community and implemented in popular systems such as WordPress and Drupal. Developers often chose it as the safest library to use for sending emails in PHP. It has SMTP support and can do HTML-based emails. Swift Mailer, another email library, sends emails quickly and efficiently from PHP websites with minimized pressure on server resources.\n\n#### MySQL and PHP   \no o o\n\nMySQL is the most frequently used database for web-based applications.\n\nMySQL is an open-source database that has become the most popular data management tool to use for web-based applications. MySQL is the database of choice that handles information for many well-known sites including Facebook, Twitter, YouTube, and Wikipedia. It not only stores information, but it allows users to manipulate and access the data it holds in a variety of ways, through what is known as a structured query language (SQL). Like PHP, MySQL information is stored on the server, and it performs all its tasks there. Since one of the most powerful ways in which PHP functions is through working with databases, effective use of PHP requires understanding and implementation of databases like MySQL.\n\nThe data in MySQL is stored in collections of related data entries placed in columns and rows called tables. In order to use the MySQL database in a PHP application, the database must first be created and connected. A table for accepting the input data must be created. The following code provides an example of this process, containing both PHP instructions and MySQL code.\n\n<?php\n\n$servername = \"localhost\";\n\n$username = \"username\";\n\n$password = \"password\";\n\n$dbname = \"DB1\";\n\n\/*The PHP application must be connected to MySQL -The extension used is actually MySQLi, the improved version of MySQL - and the successful execution of that connection should be checked, using, for example, the following code:*\/\n\n$conn = mysqli_connect($servername, $username, $password, $database1); if (!$conn) {\n\ndie(\"Connection failed: \". mysqli_connect_error());\n\n}\n\necho \"Connected successfully\";\n\n\/*A table that contains the data that will be used in the PHP application is then created in the MySQL database, using MySQL instruction*\/\n\n$sql = \"CREATE TABLE MyCustomers (\n\nid INT(6) UNSIGNED AUTO_INCREMENT PRIMARY KEY,\n\nfirstname VARCHAR(30) NOT NULL,\n\nlastname VARCHAR(30) NOT NULL,\n\nemail VARCHAR(50),\n\n)\";\n\nif (mysqli_query($conn, $sql)) { echo \"Table MyCustomers created successfully\";\n\n} else {\n\necho \"Error creating table: \". mysqli_error($conn);\n\n}\n\nmysqli_close($conn);\n\n#### MySQL and PHP (continued)  \no o o\n\nThe PHP program can perform a number of operations using the databases created in MySQL. Using a customer database as an example, among other functions, it can:\n\n\u2022 Insert new data into the database based on user input. Every time a new customer places an order, a new customer record can be added to the current database.\n\n\u2022 Update the data on an existing database, changing billing information or shipping address based on new user information.\n\n\u2022 Retrieve information from a database to provide current information to a user, such as the status of an order.\n\n\u2022 Delete records from a database if, for example, a customer choses to eliminate a previously used credit card.\n\n\u2022 Perform logical and arithmetic operations on the information in the database, calculating, for example, the sales tax on a customer's purchase.\n\nMichael Widenius created MySQL.\n\nWhen the data has been extracted from the MySQL database, the PHP application generates a web page in HTML that is transferred back to the client browser and displayed for the user. As just one example of its power, PHP, combined with MySQL, can make online purchasing a seemingly seamless event.\n\nBusiness projects can be created more easily using a number of libraries such as PHP Excel, phpDocumentor, and Pchart. PHP Excel allows developers to implement applications that include spreadsheet editing. The library called phpDocumentor supports the creation and automation of professional documents using PHP code. It incorporates many features to assist in the generation of customized documents including tutorials and documentation linking and cross-referencing through highlighted source code. One called pChart can assist in the generation of a visual presentation of text data. It can create charts, graphs, and more, using SQL queries.\n\nWhen a developer has a problem with an application, Whoops is a library that helps with PHP debugging and displays a detailed error listing. It provides the specific files and code snippets that caused the problem with highlighted syntax. Though the number of PHP libraries and extensions is enormous, understanding the wide range of functions they can perform can save a lot of coding time.\n\nOpposite: Getting an early start with PHP can help you secure a great job.\n\n### Getting Started\n\nBecause PHP is so widely used and has such amazing capabilities, learning the language can open up a lot of possibilities. Maybe you want to use PHP as you build a personal website. Maybe you are thinking about pursuing a career in computer programming. Either way, learning PHP is a worthy goal and one that can be accomplished by programmers of all skill levels.\n\n##### PHP Employment Opportunities\n\nProgramming with PHP is a very popular activity. Virtually every survey of software developers and technological employers lists PHP in the top ten of programming languages that someone should know. It is used, according to W3Techs Web Technology Surveys, as a server-side scripting language on over 82.6 percent of the websites whose programming language could be identified. Cal Evans, the Technical Manager at Zend Technologies, called PHP \"the magic language built for the web.\"\n\nThere are many roles for a PHP developer, whether at the entry level or in a more advanced position. Developers are responsible for writing clear, concise, and well-documented code for web programs. Developers can fix old problems or add new features to existing applications. They can write, modify, and debug software. They can test and document applications, and they can design and code programs written to solve specific problems.\n\nOver time, as developers' skills increase, familiarity with different design approaches\u2014such as object oriented or procedural\u2014should be included in their skill sets. Knowledge of different tools, such as libraries or frameworks, will also bolster opportunities for career advancement. If a developer decides to stay in programming, rather than enter a management career track, he or she might be involved in project coordination and technology leadership, including designing and implementing the architecture of projects. A senior developer can design website layouts and provide advice to the development team.\n\nPHP developers may choose to look for employment with an organization whose field matches the programmer's interest. Statistics for PHP's web presence imply that every different type of business and every different size business, from small, local stores to enormous web-based organizations (think Facebook) use PHP. By 2025, more than half of US businesses are projected to use ten software applications each.\n\nPHP has proved to be the language of choice for tech startups and individual entrepreneurs, building websites quickly and easily and getting each project off to a flying start. Its free software is used for e-commerce as well as by nonprofit organizations, colleges, and government organizations. PHP is even used in video game development.\n\nPHP developers find work both as employees of a business or organization or as freelance contractors, taking both long- and short-term assignments. And because much of the PHP open code developers write is accessible by potential employers, it can turn a developer's past work into both a lucrative assignment and a performance evaluation that illustrates the developer's skill and value.\n\nAccording to the US News and World Report's 2017 listing of the 100 best jobs (evaluated by demand, salary, future growth, and work-life balance) technological jobs rank high. A computer analyst comes in eighth, a system developer is thirteenth, an IT manager is twenty-ninth, and a web developer (which is the most frequent use of PHP) is listed as thirty-first. Web developers can expect about a 27 percent employment growth in the field by 2024, as determined by the Bureau of Labor Statistics. The expansion of opportunities for web developers is linked to the anticipated growth in e-commerce and the continuing need to build dynamic websites as well as the expanding need for mobile apps.\n\nAs of 2017, the median salary of web developers is listed as $64,970 with the average salary of US PHP developers at $86,349. These figures are calculated by Indeed, a job-posting website that based its estimates on users, employers, and job posting information. Indeed lists the entry-level salary for a PHP developer as $70,279. PayScale, a data collection organization, calculates PHP developers' salary range as $40,243 to $92,756, with a median salary at $61,405. The job market for PHP developers looks good from both a stability and compensation standpoint.\n\n##### Getting Started\n\nThe best way for an individual to acquire knowledge and skills in PHP development depends on specific career goals. Most entry-level PHP developers are hired with an associate's degree in a related field such as computer programming, computer science, or web design. Someone who is looking to work as the employee of a medium or large company in a position as a software architect will likely need a bachelor's degree in computer science, computer engineering, or software engineering. Working as a self-employed freelance developer, or, sometimes, working for a start-up, does not usually demand a specific\u2014or even any\u2014degree but it does require that you demonstrate your competence, usually through PHP applications or websites that you have created.\n\n###### Academic Education\n\nDegree programs, at both the associate's and bachelor's level, provide students with more comprehensive education than just instruction in coding. Certificate programs that cover PHP and may be offered online are available at all college levels and allow students to focus specifically on PHP application development. Associate's degrees usually teach students computer logic and how to identify alternative solutions to problems. Students may also learn HTML and study other programming languages. They will likely become familiar with other tools necessary for website applications. Advanced degrees cover additional subjects such as computer architecture, object-oriented programming, and operating systems, as well as essential coding skills. They may, in addition, require courses in advanced mathematics.\n\nCollege courses in computer science cover much more than a single programming language.\n\nLearning PHP coding may not be included in the basic coursework necessary to get a degree, but it is often provided as an optional course. Many developers will seek internships while still in school to acquire important on-the-job experience. The foundations provided by an academic education, however, can provide the footing not only for high-level work in the computer sciences. They can also provide the fundamentals that make it easier for a programmer to decide on the appropriate language for a particular task, understand the best design choices, and follow the best logic and practices for creating clean and refined code.\n\n###### PHP Training\n\nThe popularity of the language has resulted in many opportunities for learning PHP in ways tailored to fit all preferences, including nonacademic. Google executives have noted that \"the proportion of people without any college education at Google has increased over time,\" and being a successful PHP developer does not actually require any degree.\n\nCoding boot camps are a popular option and can help attendees become fluent in a language in a short period of time.\n\nAmong the alternative options for learning PHP are online courses. There are a great variety of structured online PHP courses. Some have live online instructors, like Udemy, or provide a certification, like the Zend Technologies PHP courses. PHP 101 provides a lighthearted program for absolute beginners. Others like Killer PHP offer video instruction, and, in addition to PHP, videos in other related topics, such as object-oriented programming and MySQL.\n\nAnother option is PHP boot camp. These schools often pledge to turn you into an employable developer upon completion of their program. They usually require three to four months of full-time attendance, although some offer part-time or online intensives. Most of the time in a boot camp is spent building projects, rather than listening to lectures, so the graduates finish with products that can be used to demonstrate the skills they have acquired.\n\nAccording to Matt Mullenweg, creator of WordPress, making contributions to the open-source technology community \"[gets] you to the top of the list when we're reviewing applications.\"\n\nThose who don't want to enroll in a program, or even follow a structured set of videos, have many opportunities for learning PHP that are both accessible and convenient as self-taught learners. Rasmus Lerdorf, the originator of PHP, suggests:\n\nJust dive in and download some code. Start solving problems using PHP. I often joke that I can teach a moderately intelligent monkey to write PHP in a day. The learning curve is really shallow, so you'll get up and running really fast.\n\nSome have found Lerdorf's rather tongue-in-cheek advice to ring true as far as learning the specific coding details of PHP by doing, since the language itself is so clear and simple. The background and foundations necessary to create elegant and effective applications using PHP, though, requires more than basic coding skills. Yet, for those who have an understanding of the conventions and structures of general purpose programming, acquiring PHP coding skills can be accomplished relatively quickly.\n\nOne of the most useful resources for this method of learning the language is the PHP manual at php. net. The manual contains the source code of the most recent versions of PHP as well as contributions by members of the PHP community and the answers to FAQs. This immersion in the PHP community not only helps with PHP skills, it can build networking that supports becoming a PHP programmer.\n\n###### \"Citizen Developers\"\n\nOne of the movements in the technological community is the rise of the \"citizen developer.\" These are not professional programmers, but rather amateurs who use programming tools to build applications within their corporate environments. Companies are always looking for new applications to address business issues, and IT departments can't always meet the demand, particularly for products like mobile apps. So, citizen developers, who have learned programmer skills but whose job titles don't identify them as developers, have filled the gap.\n\n#### Matt Mullenweg   \no o o\n\nIn 2003, Matt Mullenweg and his friend Matt Little started WordPress, the open-source web software built primarily using PHP. As of 2017, WordPress is the most popular content management software and is used by noted websites such as USA Today, CNN, Fortune, Time, and NBC. Seventeen blog posts are published to WordPress sites every second, and fifty thousand WordPress. com websites are being launched each day. Amazingly, this software powerhouse was started by Mullenweg when he was a nineteen-year-old freshman at the University of Houston.\n\nMatt Mullenweg started WordPress at nineteen.\n\nAs a teenager, while studying music at Houston's High School for the Performing and Visual Arts, Mullenweg began building websites for local jazz musicians, combining his love of both music and technology. Then, during Mullenweg's brief attendance at the University of Houston, he met Matt Little online, discussing a blogging tool they were both trying to improve. WordPress was born. By the following year, WordPress was beginning to take off, and Mullenweg was offered a job in San Francisco with CNET, working for them on WordPress and other media initiatives. Although his parents were concerned about Mullenweg leaving college, they were supportive, and Mullenweg accepted the offer.\n\nPHP was the primary language used to build WordPress.\n\nAfter one year at CNET, Mullenweg went out on his own to concentrate on WordPress full time. He founded Automattic, which continues as the WordPress software development company. He has been an acknowledged technology innovator from the beginning. Mullenweg was named one of the fifty most important people on the web by PC World, received Temple University's Information Technology Award, was identified as one of the Top 30 Entrepreneurs under 30 by Inc. magazine, and named one of the 25 Most Influential People on the Web by BusinessWeek. His awards and acknowledgements continue to recognize his accomplishments.\n\nMullenweg identified the approach to innovation that helped him create WordPress:\n\nThe vast majority of what impacts our whole life is taking something and making it better or taking two things that didn't work together before and putting them together, like peanut butter and chocolate. I was taking many things that were out there before like open-source, publishing, blogging, everything, and putting them together. .. It requires incredible dedication and perseverance and hard work and elbow grease to [make] innovations actually matter in the world. Otherwise, it's just an idea, and ideas are a dime a dozen.\n\nWhile continuing to work as CEO of Automattic, Mullenweg also began supporting start-ups through his advising investment company, Audrey Capital. He has become involved in numerous philanthropic activities, including the Apache Foundation, which supports open-source products for the public good; charity:water,which helps to bring clean water around the world to those in need; and the Electronic Frontier Foundation, which defends human rights in the digital world.\n\nDespite his multitude of activities, Mullenweg revisited his early passion for hands-on technology and announced in 2017 that he was returning to \"leading core WordPress development for several releases,\" as WordPress incorporated more JavaScript into its front-end interface, while preserving back-end PHP technologies. He describes this new effort:\n\nI'll say as well that learning new things is scary. Personally, I now have thirteen years' experience coding in PHP, and you kind of have to\u2014if you're learning a new language\u2014 you're kind of back in that beginner phase which can be frustrating at first. .. But PHP is still really fantastic for server-side stuff.\n\nMany of these citizen developers have no formal computer science or software engineering degree. They identified a need in their business and decided to address it. The rewards for these initiatives have been notable. Many, according to a survey by FileMaker, report more organizational recognition and greater confidence and higher work satisfaction, and 71 percent report an increase in team productivity. Citizen developers prove that you don't need to have a job as a developer to make meaningful contributions as a programmer.\n\n##### Getting Ahead\n\nPHP is the most-used language in web development, and its implementation is simple and straightforward. But on its own, it does not always provide a full set of tools for developing a complex and dynamic web presence. PHP programmers have greater power and flexibility when they can incorporate certain concepts in their work, use other languages that facilitate PHP development, and employ other structures, such as frameworks, that can make their programming both easier and better.\n\n###### Procedural Programming and OOPs\n\nDevelopers may use two different approaches when figuring out how to solve a programming problem. These approaches are procedural programming and OOP, and both are supported with PHP. Most early programming languages, like COBOL and C, are procedural languages. In this technique, the focus is on the processing. The programmer creates a set of procedures (functions) that act upon data, e.g., \"a\" gets added to \"b\" to produce \"c.\" Functions (like adding) and the data that is being acted upon (for example, a product cost and shipping cost) are separated from each other.\n\nThis graphic shows how object-oriented programming (OOP) and procedural programming name the same elements.\n\nOn the other hand, object-oriented programming was first implemented in the 1960s to provide graphically oriented applications for simulations. It became more popular in the 1970s with the creation of C++, an OOP version of the C language. OOP places the object (e.g., singer) and its attributes (e.g., song) together in a single structure. Objects know how to perform certain actions (for example, sing) and how to interact with other elements of the program (for example, lyrics and instruments).\n\nThere have been long-standing debates about the relative merits of the two different methods. Proponents of procedural programming maintain that it is capable of high performance and can provide quick design and implementation. OOP adherents, on the other hand, point out that large projects and collaborative efforts are easier with OOP. Maintaining and modifying existing code as new objects can be accomplished with small changes to existing ones.\n\nSince both have their advantages and disadvantages, a PHP developer who can choose the appropriate method for a given project will be able to create applications that can function better in the present and adapt to necessary changes in the future.\n\n###### Critical Languages\n\nPHP works best when it is implemented with other languages that add to its functionality, including markup, style sheet, data, and programming languages. The most essential of these are HTML, Cascading Style Sheets (CSS), MySQL, and JavaScript.\n\nHTML, first developed by Tim Berners-Lee in 1990, is known as the basic building block of the World Wide Web. It creates documents on the web and describes how those documents are structured. There are not many components to HTML and its format is simple. For example:\n\n<!doctype html>\n\n<html>\n\n<head>\n\n<title>Example of HTML<\/title>\n\n<\/head>\n\n<body>\n\n<h1> This is a heading example<\/h1>\n\n<p> This is a basic HTML page example<\/p> <\/body>\n\n<\/html>\n\nThese markup instructions create a basic web page using tags that identify the different elements of the page such as headings and paragraphs. The browser uses HTML to specify the structure and content of the web page. HTML can be written on any text editor including an HTML editor. PHP applications creating dynamic web pages are embedded in the HTML that creates the basic static page.\n\nTim Berners-Lee developed HTML in 1990. He is also the inventor of the World Wide Web.\n\nAlthough the HTML code is essential, it is also limited. It does not usually specify the way the information on the web page should look. For that, CSS is necessary.\n\nCSS is a structured language that describes the presentation of a web page. It is a basic technology used by most web developers to create the look of web pages and user interfaces for web and mobile applications. Its development created an alternative to the exhaustive HTML instructions needed to specify such elements as font colors, background styles, paragraph alignments, borders, and sizes. CSS, through separating formatting from content, can control how HTML content is displayed on computer screens, paper, and other media.\n\nMySQL is a relational database management system (RDBMS) that stores data in the form of tables that can be created, accessed, managed, searched, and replicated in a variety of ways. The name is derived from a combination of the name of the cofounder, Michael Widenius's daughter, My, and SQL, the abbreviation for structured query language.\n\nPHP began as a way to allow data to be incorporated in a web page in order to make the content dynamic, and MySQL is now at the core of many PHP applications. It is a free, open-source database system and works well on many operating systems, including Linux, Windows, and Apple iOS. MySQL is a central component of the LAMP web application software stack, made up of Linux, the operating system; Apache, the server; MySQL, the RDMBS; and PHP, the scripting language.\n\nMySQL uses a standard form of the SQL language to allow the user to manipulate the data in the database. It is the database manager most used in PHP applications, and it is quick and reliable. It is ideal for small operations, but it is able to handle even large data sets, up to a theoretical limit of eight million terabytes. Much of the appeal of MySQL comes from its ease of use and simplicity.\n\nSome developers use the MySQL Workbench application to assist them in using databases effectively. This allows users to administer MySQL databases through graphic interfaces and to design database structures visually. For one last example, there is JavaScript. JavaScript is best known as a client-side scripting language for web pages. It executes within the browser, although it now has many other aspects to its dynamic functioning and can also work in some nonbrowser environments. It is comparatively easy to learn and powerful. It supports both OOP and procedural programming. The basic syntax resembles both Java and C++, making it familiar to programmers of those languages.\n\nJavaScript can help web developers create responsive user interfaces without waiting for the web server to react and create a whole new page. It can help fix browser problems and resolve CSS support issues. It can interact with plug-ins such as Flash. Using the document object model (DOM), JavaScript can create new page elements and apply, as well as manipulate, the appearance, content, and attributes of page elements.\n\nWhen client-side scripting is used with the set of web development techniques known as AJAX, the speed of web responses is increased even more. AJAX, standing for Asynchronous JavaScript and XML, enables web browsers to interact asynchronously with the web server. Using this technology, web pages get updated in the background and the whole web page does not have to be reloaded and refreshed with every update. For the user, web surfing can be an uninterrupted experience while the content is still continuously updated.\n\nJavaScript can facilitate catchy animations and even simple calculations. Its code can provide event handling, reacting to things that occur in the browser. This can range from user actions like resizing the window or scrolling the page to using information in a link to connect to a web server. It can pass user values to the server and accept values transmitted back to the browser.\n\nWeb developers use PHP and JavaScript as complementary tools with different purposes, using both client-side scripting and server-side scripting to create a web experience that is both highly responsive and highly engaging.\n\n###### Other Languages Worth Knowing\n\nAlthough web development is usually executed with specific languages intended for that purpose, a knowledge of general-purpose programming languages increases a developer's skill and flexibility, as well as providing the ability to tackle larger, more sophisticated projects. Many websites depend for their implementation on these general-purpose languages, like Java and C.\n\nJava is a general-purpose language that was designed to be platform-independent. A Java Virtual Machine, which interprets Java binary code for a computer's hardware, exists for most operating systems. It can run Java code after its processing by a compiler. Java is an object-oriented language similar to C++ but with less complex coding. It is a very popular language for web development and, according to Oracle, Java's owner, there are more than nine million Java developers. Java is on more than three million mobile phones.\n\nJava has a number of features that make it well adapted for use as a web programming language, but applications will not work unless Java is installed on the browser. It is particularly popular with high traffic websites because Java is able to outdo other languages in speed and reliability. Java is used, for example on high-volume sites such as Amazon, Sam's Club, LinkedIn, Netflix, and the Apple App Store. It's also the programming language used to develop all native Android apps (the apps that come loaded on Android devices).\n\nScripting languages, such as PHP, can be connected with a Java Virtual Machine through the PHP\/ Java Bridge, promoting much speedier and reliable performance and using fewer web server resources. It requires no additional components in order to call Java procedures from PHP or PHP procedures from Java.\n\nC is also a general-purpose programming language. Its constructs closely represent the functions of machine instructions, so it has been used in highly technical applications from operating systems to supercomputers. It was originally developed between 1969 and 1973, and many later languages have borrowed recognizable expressions and syntax from C, including C++, Java, JavaScript, C#, and Python. Rasmus Lerdorf, creator of the original implementation of PHP, coded it in C.\n\nC can also be used for website programming, using CGI (a common gateway interface) as a way to pass information between the browser, the server, and the web application. It may be selected for its speed, ability, and easy availability. But, although C is the foundation of many web-development languages and tools, most developers see its complexity and timeintensive coding as major drawbacks that keep it from being really useful in creating websites.\n\n##### What's in the Future\n\nThe changes happening in technology are explosive. And the World Wide Web, with what we can learn from it and what we can do with it, is at the center of this change. PHP is striving to stay not just relevant, but instrumental in fostering new developments. Some of these developments include the expansion of mobile apps, artificial intelligence, and the Internet of Things (IoT).\n\n###### The Expansion of Mobile Applications\n\nCell phones have become so familiar and ubiquitous, it is sometimes difficult to realize that they are a relatively recent and still-developing technology. The market for mobile apps on these devices can seem endless, and new apps are continuously being created.\n\nThe overwhelming majority of PHP developers are working on, or plan to work on, mobile apps. The process of implementing mobile apps with PHP is still being refined and continues to present some challenges. Marrying the web technology with the mobile device architecture can be difficult. But the advantages of PHP coding\u2014less typing, safety and simplicity, and first-class functions\u2014make it very appealing to mobile developers.\n\nMobile apps can do almost anything. Many PHP developers work on mobile apps as their core work.\n\nThe support for developing PHP mobile apps is also expanding, and there are new frameworks for providing this assistance. Zend On The Go is a sample mobile application that shows how to create mobile apps using existing PHP applications. It monitors the new application and gives server performance statistics. Zend Studio helps developers create a cloud-connected mobile (CCM) project using PHP back-end functions. GitHub, an enormous version control repository, contains an ever-growing collection of supports for mobile app development using PHP.\n\n###### Artificial Intelligence (AI)\n\nArtificial intelligence attempts to understand how intelligent entities (like humans) function and build intelligent entities (such as robots) that have those attributes. This complex and challenging field is producing new ideas and innovations, and PHP development is right there in the mix.\n\nAn artificial neural network (ANN) is a computational model built to simulate the way people think. An ANN is composed of neurons that communicate and provide output. Although it is still very primitive, an ANN can be used for prediction, classification, and decision-support systems, as well as applications. ANNs have primarily been developed in high-level languages such as C or C++, but PHP has been found to be the most convenient way to implement AI in web applications, so it has been used to create neural networks.\n\nArtificial Linguistic Internet Computer Entity (A.L.I.C.E.) is a bot (internet robot) that is capable of holding a conversation and responding with relevant answers to questions. A.L.I.C.E. responds to the Artificial Intelligence Markup Language (AIML), which is used to give the bot instructions on how to \"think.\" Implementation of this bot in PHP is called Program E.\n\nAI bots may be used for informational communication, such as Siri or the iPhone bot. A number of PHP forums on AIML provide PHP users with information and support for the development of AI applications.\n\n###### IoT\n\nThe Internet of Things (IoT) allows devices on a network to sense and collect information from other connected devices, process that data, and use it for a variety of purposes. Ordinary household appliances and devices can be connected with IoT to perform a wondrous range of tasks. For example:\n\n\u2022 Your watch can track your daily activities and exercise and match it to goals you are working to attain.\n\n\u2022 A wearable device can monitor your blood pressure and notify a health care professional of your status and possible need for care.\n\n\u2022 A motion-detecting camera can notify you if an intruder is on your property.\n\n\u2022 Your refrigerator can order groceries when your supplies are low.\n\n\u2022 Your heating system can notify the oil company when you need a delivery.\n\nThe Internet of Things is a network of smart devices.\n\nThe list of possible applications can go on and on, and IoT provides great opportunities for PHP developers. PHP is an ideal language for IoT applications. Its flexibility and affordability foretell a great future for PHP programs in the IoT world, creating almost any connection the developer can envision.\n\n##### PHP: A Language in Motion\n\nPHP began as a simple tool for coding a personal home page and has evolved over time into one of the most popular languages for creating dynamic websites. Now it is moving into applications that will serve future technology. Throughout its history, it has continued, as it began, by responding to old problems, and by changing as it solved new problems. It has been criticized for slow performance, but it keeps adding speed. It has been called limited and basic, yet it keeps adding new features to increase its power and sophistication. It started with web pages and is now growing into artificial intelligence.\n\nPHP has countered its critics through the strength of a dedicated and enthusiastic community that provides ongoing support for resolving any difficulties and creates extensive documentation for every new attribute. Its popularity has been linked to how easy it is to learn, but its endurance stems more from its ability to modify and transform itself into the right solution for the time.\n\napplication A type of software that allows you to perform specific tasks.\n\nApplication Program Interface (API) An interface that specifies how software components should interact.\n\nasynchronicity Communication (as between computers) in which there is no timing requirement for transmission.\n\ncache Reserved areas of computer memory that are used to speed up instruction execution, data retrieval, and data updating.\n\ncommon gateway interface (CGI) An interface that provides a standard way for a web server to pass a user's request to an application program and to receive data back.\n\ncompiler A software program that transforms high-level source code into machine language, which can be understood by the processor.\n\ncontent management system (CMS) A computer application that supports the creation and modification of digital content.\n\ncookies Small text files that allow the server to store its ^ information about a user on the user's own computer.\n\nCPU The central processing unit of a computer where calculations are carried out.\n\ndeveloper A programmer who specializes in the creating of World Wide Web applications.\n\nDOM The document object model, which is used for changing the elements of an HTML document.\n\nengine A program that performs a core or essential function for other programs\n\nevent handling Processing actions such as keystrokes and mouse movements.\n\nextensible Markup Language (XML) A language designed to store and transport data.\n\nframework For PHP programs, frameworks specify programming interfaces, or offer programming tools for using the frameworks.\n\nHTML A language that describes and defines the content of a web page.\n\nHTTP A protocol that defines how messages are formatted and transmitted on the World Wide Web and what actions web servers and browsers should take in response to various commands.\n\nInternet Information Server (IIS) A flexible, general-purpose Microsoft web server that runs on Windows systems.\n\ninterpreter A computer program that executes instructions written in a programming language.\n\nJIT compiler A program that turns Java bytecode (a program that contains instructions that must be interpreted) into instructions that can be sent directly to the processor.\n\nmodule A software component or part of a program that contains one or more routines.\n\nobject-oriented programming (OOP)A programming language model organized around objects rather than procedural actions, and data rather than logic.\n\nopen-source Software for which the original source code is made freely available and may be redistributed and modified.\n\noperator A symbol that represents a specific action, e. g. , a plus sign (+) is an operator that represents addition.\n\nparser A compiler or interpreter component that breaks data into smaller elements for easy translation into another language.\n\nprocedural programming A computing method that uses a series of steps to be carried out.\n\nproprietary software A technology or product that is owned exclusively by a single company.\n\nprotocols Sets of rules governing the exchange or transmission of data between devices.\n\nrelational database management system (RDBMS)A database management system that lets you create, update, and administer a relational database.\n\nscripting language A programming language capable of many tasks, including generating web pages on the internet.\n\nsemantics The evaluation of the meaning of characters, words, and symbols in a specific programming language.\n\nserver A computer or computer program that manages access to a centralized resource or service in a network.\n\nsoftware stack A group of programs that work in tandem to produce a result or achieve a common goal.\n\nstructured query language (SQL) A language used in programming and designed for managing data held in a relational database management system.\n\nsyntax Grammatical structure.\n\nUnix A family of multitasking, multiuser computer operating systems\n\nversion control The management of changes to documents, computer programs, large websites, and other collections of information.\n\nweb form An online page that allows for user input.\n\nForbes, Alan. The Joy of PHP: A Beginner's Guide to Programming Interactive Web Applications with PHP and MySQL. Rowley, MA: Plum Island Publishing, 2015.\n\nLengstorf, Jason, and Thomas Blom Hansen. PHP for Absolute Beginners. New York: Apress, 2014.\n\nLockhart, Josh. Modern PHP: New Features and Good Practices. Sebastopol, CA: O'Reilly Media, 2015.\n\nUllman, Larry. PHP and MySQL for Dynamic Web Sites: Visual QuickPro Guide (4th Edition). San Francisco: Peachpit Press, 2011.\n\n#### Websites\n\nObject-Oriented Programming\n\nhttps:\/\/www.cs.drexel.edu\/~introcs\/Fa15\/notes\/06.1_OOP\/titleslide.html?CurrentSlide=0\n\nLearn more about OOP on this site from Drexel University.\n\nPHP\n\nhttp:\/\/php.net\/\n\nThe official website for the PHP community includes information about new releases, documentation, and more\n\nPHP the Right Way\n\nhttp:\/\/www.phptherightway.com\/\n\nExplore this resource for new PHP programmers.\n\n#### Videos\n\nPHP Tutorial 1 - Introduction (PHP for Beginners)\n\nhttps:\/\/www.youtube.com\/watch?v=kY5P9sZqFas\n\nWatch the first in a set of twenty-five PHP tutorial videos followed by four MySQL videos.\n\nThe Ultimate Guide for Learning HTML and CSS\n\nhttps:\/\/www.youtube.com\/watch?v=y3UH2gAhwPI\n\nLearn how to make websites from scratch.\n\nHassan, Anita.\"WordPress Founder Finds Inspiration in His Hometown of Houston,\" Chron, September 28, 2016.<http:\/\/www.chron.com\/local\/history\/innovators-inventions\/article\/Wordpress-founder-finds-inspiration-in-his-9403035.php#photo-11118668>.\n\nHulme, George, \"Meet the Citizen Developer. \" Devops, February 3, 2017. https:\/\/devops.com\/meet-citizen-developer\/.\n\nKingston Central School System Alumni Association.\"Rasmus Lerdorf. \" Retrieved March 14, 2017. static1. 1. sqspcdn.com\/static\/f1560920\/20510047\/1349287039360\/Rasmus+Lerdorf.pdf?token=GokG2XTT%2B47LGWBNFk1oEvULQFM%3D.\n\nKrill, Paul.\"Believe the hype: PHP founder backs Facebook's HipHop technology. \" Infoworld, November 13, 2013. www.infoworld.com\/article\/2609877\/web-development\/believe-the-hype--php-founder-backs-facebook-s-hiphop-technology.html.\n\nMIT. \"Technology Review: Innovators under 35 - 2003.\" Retrieved March 14, 2017. http:\/\/www2.technologyreview.com\/tr35\/profile.aspx?trid=347.\n\nMySQL Conference.\"PHP on Hormones.\" April 26, 2007. http:\/\/talks.php.net\/show\/mysql07key\/.\n\nO'Dell, J.\"Why this web guru wants you to learn PHP, a.k.a. 'Internet English.'\" Venture Beat, September 21, 2012.https:\/\/venturebeat.com\/2012\/09\/21\/treehouse-php\/.\n\nU.S.News and World Report.\"U.S.News and World Report Announces the 2017 Best Jobs.\" January 11, 2017. http:\/\/money,usnews.com\/\/money\/careers\/articles\/how-us-news-ranks-the-best-jobs.\n\nWayner, Peter. \"9 predictions for the future of programming.\" Infoworld, January 18, 2016.http:\/\/www.infoworld.com\/article\/3022874\/application-development\/9-predictions-for-the-future-of-programming.html.\n\nWodehouse, Carey. \"PHP: The Language, Its Frameworks, and How to Hire the Right PHP Developer. \" Upworks. Retrieved April 1, 2017. https:\/\/www.upwork.com\/hiring\/development\/php-frameworks-hiring-a-php-developer\/.\n\nZmievski, Andrei.\"The Good, the Bad, and the Ugly: What Happened to Unicode and PHP \" In Slideshare, April 22, 2011. http:\/\/www. slideshare.net\/andreizm\/the-good-the-bad-and-the-ugly-what-happened-to-unicode-and-php-6.\n\nAmazon, 25, 32, 93  \nApache web server, 9\u201310, 54, 84, 90  \napplication, 26, 31\u201332, 36\u201341, 43\u201344, 46, 49, 52, 54\u201356, 60\u201363, 65\u201366, 68\u201371, 74\u201375, 77, 80\u201381, 85, 87, 89\u201390, 93\u201397, 99  \nApplication Program  \nInterface (API), 12, 13  \nartificial intelligence, 94, 96\u201397, 99  \nASP.NET, 52, 54\u201355, 58\u201359  \nassociate's degree, 77  \nasynchronicity, 91  \nbachelor's degree, 77  \nback-end, 32, 84, 96  \nboot camps, 79, 80  \nC, 6, 11, 34, 57, 60, 86\u201387, 92, 94, 97  \nC++, 60, 93  \ncache, 60  \nclient-side, 32\u201333, 55, 61, 91\u201392  \ncommon gateway interface (CGI), 6, 94  \ncompiler, 93  \ncontent management  \nsystem (CMS), 55, 64  \ncookie, 11, 39  \nCPU, 27  \nCSS, 88\u201389, 91  \ndata exchange, 30, 63  \ndata management, 68  \ndata transfer, 30  \ndeveloper, 7, 10\u201312, 16, 18, 25\u201326, 45, 52\u201353, 55, 60, 63\u201367, 71, 73\u201381, 85, 89\u201397, 99  \ndevelopment tools, 22, 51  \nDOM, 91  \ndraft specification, 35  \ndynamic, 5, 17, 25, 31\u201332, 34\u201336, 48, 58, 60, 76, 85, 89\u201391, 99  \ne-commerce, 11, 37\u201339, 75\u201376  \nelePHPant, 4, 15\u201316  \nengine, 16\u201317, 64  \nevent handling, 92  \neXtensible Markup  \nLanguage (XML), 22, 60, 91  \nFacebook, 5, 25, 37, 42\u201343, 64, 68, 75  \nFlash, 5, 55, 60, 91  \nframework, 55, 63, 74, 86, 96  \nfront-end, 32, 84  \nGitHub, 66, 96  \nGNU project, 7  \nGoogle, 25\u201326, 37, 39, 54, 79  \ngraphical user interface (GUI), 61  \nGutmans, Andi, 11\u201312, 16, 18, 19, 20\u201321, 25  \nHipHop Virtual Machine(HHVM), 25, 64  \nHTML, 10, 22, 30\u201332, 36\u201337, 39\u201340, 43, 46\u201349, 59\u201360, 70\u201371, 77, 88\u201389  \nHTTP, 10, 17, 33, 36, 39\u201340, 61  \nIBM, 9, 20, 23  \nInternet Information Server (IIS), 54, 58  \nInternet of Things (IoT), 94, 97, 98  \nJava, 27, 32, 34, 55, 91\u201393  \nJavaScript, 32, 34, 88, 91\u201394  \nJava Virtual Machine, 93  \nJIT compiler, 26  \nLerdorf, Rasmus, 6\u201312, 6, 20, 25, 27, 34, 80\u201381, 94  \nlibraries, 44\u201345, 55, 65\u201367, 71, 74  \nLinux, 7, 14, 37, 54, 58, 90  \nMicrosoft, 20, 23, 54, 58  \nmodule, 12, 16\u201317  \nMullenweg, Matt, 80, 82\u201384, 82  \nMySQL, 11, 17, 37\u201338, 46, 65, 67\u201370, 80, 88, 90\u201391  \nobject-oriented  \nprogramming (OOP), 12, 63, 86\u201387, 86, 91  \n\"on the fly, \" 36, 41, 55, 64  \nopen-source, 7, 9\u201312, 17\u201319, 35, 37, 53\u201354, 58, 64, 68, 80, 82, 84, 90  \noperator, 27  \nOracle, 20, 23, 37\u201338, 54, 65, 93  \npackages, 44\u201345, 55  \nparser, 12, 20  \nphpng, 26\u201327  \nplug-ins, 55, 91  \nPostgreSQL, 38, 65  \nprocedural programming, 86\u201387, 86, 91  \nprogramming language, 5\u20136, 9\u201310, 22, 34\u201335, 42, 52, 57, 63, 74, 77, 86, 88, 92\u201394  \nproprietary software, 19, 58  \nprotocols, 12, 17, 22, 33, 40, 61\u201362  \nPython, 27, 54, 94  \nrelational database  \nmanagement system (RDBMS), 90  \nRuby, 34  \nscripting language, 5, 9, 25, 29, 31\u201332, 34, 54, 58, 74, 90\u201391, 93  \nsemantics, 26, 35  \nserver, 5, 9\u201311, 13\u201314, 17\u201318, 21, 23, 25, 29\u201333, 36\u201339, 41\u201343, 46, 49, 52, 54\u201356, 90\u201394, 96  \nserver-side, 5, 29, 32\u201333, 36\u201338, 40, 42, 46, 55, 58, 60, 74, 84  \nSilicon Valley, 9  \nsoftware stack, 90  \nstructured query language (SQL), 68, 71, 90  \nSuraski, Zeev, 11\u201312, 16, 18, 19, 20\u201321, 23, 25  \nSybase, 38, 65  \nsyntax, 10, 26, 35, 57\u201358, 71, 91, 94  \nUnicode, 24\u201325, 35  \nUnix, 14, 54, 65  \nURL, 30, 33, 36,   \nversion control, 66, 96  \nvulnerability, 53, 56\u201357  \nweb-based application, 68  \nweb development, 9, 11, 76, 85, 89, 91\u201394  \nweb form, 10, 49  \nweb page, 5\u20136, 10, 25, 29\u201331, 34\u201336, 39\u201342, 46, 48\u201349, 51, 55, 58, 64, 70, 88\u201392, 99  \nWikipedia, 5, 27, 37, 43, 68  \nWindows operating system, 13, 37, 52, 54\u201355, 58, 62, 90  \nWordPress, 5, 25, 27, 55, 64, 71, 80, 82\u201384, 83  \nWorld Wide Web, 10, 30, 61, 88, 94  \nYahoo!, 9, 23  \nZend Engine, 16\u201317, 20, 22\u201323, 27  \nZend Technologies, 16, 18, 20\u201321, 23, 74, 80  \nZmievski, Andrei, 24\u201325\n\nGrace Murphy has authored books, white papers, speeches, encyclopedia articles, and online courses in history, economics, and science. Grace has also worked as an educational consultant, health care administrator, computer programmer, and college teacher. Grace is the author of Semiconductors in the Great Discoveries in Science series. She currently lives in the peaceful hamlet of Shady, New York, where she enjoys regular hikes on the local mountain trails.\n\n  1. 1 The History of PHP\n  2. 2 How PHP Works\n  3. 3 Strengths and Weaknesses\n  4. 4 Getting Started with PHP\n  5. Glossary\n  6. Further Information\n  7. Bibliography\n  8. Index\n  9. About the Author\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"VOL. 153, NOV 21, 1917  ***\n\n\nE-text prepared by Jonathan Ingram, Andy Jewell, and the Project Gutenberg\nOnline Distributed Proofreading Team\n\n\n\nNote: Project Gutenberg also has an HTML version of this\n      file which includes the original illustrations.\n      See 11619-h.htm or 11619-h.zip:\n      (http:\/\/www.ibiblio.org\/gutenberg\/1\/1\/6\/1\/11619\/11619-h\/11619-h.htm)\n      or\n      (http:\/\/www.ibiblio.org\/gutenberg\/1\/1\/6\/1\/11619\/11619-h.zip)\n\n\n\n\n\nPUNCH, OR THE LONDON CHARIVARI\n\nVOL. 153\n\nNOVEMBER 21, 1917\n\n\n\n\n\n\nCHARIVARIA.\n\nMore than a million pounds of concealed sugar have been discovered in\nNew York. It is suspected that this was intended as the nucleus of a\nhoard.\n\n       ***\n\nA contemporary recently stated that LENIN claims to stand for the\nleadership of Russia. But surely they do not stand for leadership in\nRussia. They rush for it with revolvers.\n\n       ***\n\n\"This is a time for action, not for talk,\" said Colonel HOUSE on\nhis arrival in England. A stinging rejoinder is expected from the\nFOOD-CONTROLLER'S Department.\n\n       ***\n\nIt is rumoured that the restaurant keepers have agreed among\nthemselves that to avoid confusion the price of all beefsteaks shall\nbe stamped clearly on the sole.\n\n       ***\n\nThe Meat Order will probably be amended to make meat-stalls rank as\nshops. At present of course they suffer under the stigma of being\nmerely places where you can purchase meat.\n\n       ***\n\nWe understand that, in order to avoid confusion and undue alarm,\nGerman prisoners in this country will in future be expected to give\ntwelve hours' notice of their intention to escape.\n\n       ***\n\nSugar is to be omitted from a number of medical preparations from\nDecember 1st, and children are complaining that the decision has quite\nspoilt their Christmas prospects.\n\n       ***\n\nCounsel, in a prosecution for selling a tobacco substitute, has stated\nthat there is nothing in the Act to prevent a man from smoking what he\nlikes. In the trade this is generally regarded as a nasty underhand\njab at the British cigar industry.\n\n       ***\n\nLord RHONDDA, in announcing his new rationing scheme, differentiates\nbetween brain workers and manual workers. It will be interesting to\nsee to which category certain Government officials will be assigned.\n\n       ***\n\n\"The bamboo,\" according to a weekly paper, \"holds the record among\nplants for rapid growth, having been known to grow two feet in\ntwelve hours.\" The silence of allotment holders on this subject is\nsignificant.\n\n       ***\n\nMr. SYDNEY G. GAMBLE, second in command of the London Fire Brigade, is\nabout to retire. There is some talk of arranging a farewell fire.\n\n       ***\n\nWe understand, by the way, that retirement from the London Fire\nBrigade always carries with it the privilege of wearing the uniform at\none's own fires.\n\n       ***\n\nA theatrical paper advertises for a \"Male impersonator\" for pantomime.\nNo conscientious objector need apply.\n\n       ***\n\nA news message to the _Politiken_ states that the people of Iceland\nare making demands for their own flag or separation. The movement\nseems to be an isolated one and not likely to spread. Anyhow, there is\nno cause for alarm at Tooting, where the authorities are not expecting\nany trouble of this kind.\n\n       ***\n\nA Cranford dairyman has been selling milk at threepence per quart. In\ntrade circles it is supposed that he is doing it for a wager.\n\n       ***\n\nAccording to _The Evening News_, Councillor WILLIAM SHEARRING, the new\nMayor of Bermondsey, started life as a van boy. This gave him a pull\nover most of us, who started life as infants.\n\n       ***\n\nAfter December 17th, parcels for neutral countries may not be sent\nwithout a permit. Cement and other articles intended for enemy\nconsumption can only be forwarded by special arrangement with the\nMinistry of Blockade.\n\n       ***\n\nThe average man, says a correspondent of _The Daily Mail_, does not\nknow how to invest five pounds in War Loan. Yet all he has to do is to\npay his little fiver across the counter just as if he were buying a\npound of tea.\n\n       ***\n\nThe LORD MAYOR'S Coachman has retired after twenty-eight years'\nservice. He was a splendid fellow, taking him all round.\n\n       *       *       *       *       *\n\n[Illustration: _Sociable Escort (to Bosch prisoner, after several\nineffectual attempts to start a conversation)_. \"AHEM!--ER--NO TROUBLE\nAT HOME, I HOPE?\"]\n\n       *       *       *       *       *\n\nAn official memo from the Front:--\n\n    \"A complaint has been received from the Provost Corps that two\n    horses, apparently ridden by grooms, committed a civil offence in\n    ----, in that they crashed into a motor car, which at the time was\n    stationary, damaging same. On being questioned where they came\n    from, they replied, 'From Australia,' and after paying a few more\n    like compliments disappeared at the gallop.\"\n\nIt is supposed that these intelligent animals had been reading a\nrecent article by \"Patlander.\"\n\n       *       *       *       *       *\n\n    \"The R.F.C. on the same day bombed the junction. There was a large\n    numtity of rolling stock in the station, on which, and on the\n    station building, several direct hits were observed to cause\n    considerable damage.\"--_The Times_.\n\n\"Numtity\" is doubtless a dodge of the CENSOR to prevent us knowing too\nmuch. We suspect that \"quanber\" was what the writer really wanted to\nsay.\n\n       *       *       *       *       *\n\n    \"Mr. Drucker (for the trustees of the Testator) said the late Lord\n    Blythswood had made 51 oleograph codicils to his will, and the\n    difficulty arose over two of them.\"--_Evening Paper_.\n\nIt rather looks as if the two were not genuine oleographs but only\ncolourable imitations.\n\n       *       *       *       *       *\n    \"American eggs arriving at Manchester yesterday were quoted from\n    27s. 6d. to 28s. per 120, which caused Irish eggs to be reduced\n    from sixpence to a shilling.\"--_Daily Paper_.\n\nVery Irish eggs.\n\n       *       *       *       *       *\n\n    \"12 Feet Corsets at a ridiculous price of Re. 1 each, all\n    sizes.\"--_Advt. in \"Advocate of India.\"_\n\n\"A ridiculous price,\" says the advertiser, but \"an absurd figure\"\nwould have been even better.\n\n       *       *       *       *       *\n\n    \"The Examiners appointed by the Board of the Faculty of Natural\n    Science give notice that Wilfrid Dyson Hambly, Jesus College,\n    having submitted a dissertation on 'Tattooing and other forms of\n    body-marking among primitive peoples,' will be publicly examined\n    on Monday, November 12, at 2.30 p.m., in the Department of Social\n    Anthropology, Barnett House.\"--_Oxford University Gazette_.\n\nWe trust he showed, and obtained, full marks.\n\n       *       *       *       *       *\n\nTO ATTILA'S UNDERSTUDY.\n\n    [Reuter reports that a British prisoner has been sentenced to a\n    year's imprisonment for calling Germans \"Huns.\"]\n\n  The choice was yours, we understood.\n    We thought that, when you wished to cater\n  For China's spiritual good,\n    This name received your imprimatur;\n      \"Go forth,\" you said, \"my sons!\n  Go and behave exactly like the Huns!\"\n\n  Though under any other name,\n    However alien to their nature,\n  Your people would have smelt the same,\n    We let you choose their nomenclature,\n      And studiously respected\n  The one that in your wisdom you selected.\n\n  And now, when someone, clearly set\n    On flattering you by imitation,\n  Applies that chosen epithet\n    To certain units of your nation,\n      It seems a little odd\n  That you should go and clap him into quod.\n\n  Perhaps you've come to hold the view\n    That when you claimed to touch their level\n  You were unfair to heathens who\n    Candidly called their god a devil;\n      Who fought some barbarous fights,\n  But fought at least according to their lights.\n\n  So Huns are off. Who takes their place?\n    Well, since no beast on earth would stick it\n  If after him we named your race,\n    We'll call you Germans--there's your ticket;\n      Just Germans--that's a style\n  Which can't offend the other vermin's bile.\n\nO. S.\n\n       *       *       *       *       *\n\nNIGHTMARES.\n\nII.\n\nOF A T.B.D. CAPTAIN, WHO DREAMS THAT HE HAS FOUND HIS LOG BOOK MADE UP\nBY MR. PH*L*P G*BBS.\n\n_Time:--7.30 A.M._--Once more we set out on our never-ending mission,\nour ceaseless vigil of the seas. The ruddy weather-stained coxswain\nswung the wheel this way and that--his eyes were of the blue that only\nthe sea can give--in obedience to, or rather in accord with, the curt,\nmystic, seaman-like orders of the young officer of the watch. \"Hard\na-port! Midships! Hard a-starboard! Port 20! Steady as she goes!\" And\nceaselessly the engine-room telegraph tinkled, and the handy little\ncraft, with death and terror written in her workmanlike lines for\nthe seaman, for all her slim insignificance to the landlubber on the\ntowering decks of the great liner, swung smartly through the crowded\nwater-way out to the perils lurking 'neath the seeming smile of the\nopen sea: the guardian angel of our commerce it went, to meet--what\nHeaven alone could foretell!\n\n_Course_.--S. 70 deg. E. Towards the rising sun and our brethren in khaki,\ntoiling in the wet mud as we toil on the wet waters!\n\n_Deviation_.--1 deg. E. Wonderful the accuracy of the little instrument\nwhereon men's lives do hang, wise in the lore of the firmament!\n\n_Patent Log_.--O. Nothing--as yet! What will it register ere the day\nbe done? Or will its speckless copper lie rusting in the grey chill of\nthe sea's dank depths?\n\n_Revs_.--I don't know, but the propellers swirl faithfully and\nunceasingly.\n\n_Wind_.--W. by E. Bearing a message across the vast Atlantic of hope\nand present succour from our new great Ally, the mighty Republic of\nthe West. America, ah America! But we of the sea are men of few words,\nand this is not the place.\n\n_Force_.--3. A balmy zephyr, yet with the sharp salt tang of the sea\nthat a sailor loves.\n\n_Sea_.--2. Softly undulating is the swell, scarce perceptible to\ninexperienced eyes, such as those of the land-lubbers on the towering\ndecks of the great liners; gleaming dead copper and blue in the\nmorning sun, flecked with spectral white in the distance--the easy\nroll of untrammelled waters!\n\n_Weather_.--C. Detached clouds. Almost had I written \"B,\" seeing the\nperfect filmy blue all around the horizon; but a seaman's scrutiny\nshowed me faint fluffy wisps o'erhead, luminous and marged with\npalest gold; and ever must a sailor be suspicious of the treacherous\nweather-god.\n\n_Thermometer_.--42 deg. Not yet is Winter here, but its threat\napproaches.\n\n_Barometer_.--30.01. Will it stay there?\n\n_Remarks_.--Once more we set out on our ceaseless vigil, our\n       *       *       *       *       *\n_Remarks_.--(7.30 P.M.).--Another day has passed, another day's duty\nhas been done. Nothing _apparently_ has happened outside the ordinary\nroutine of the ship. One keen-eyed young officer has succeeded another\non the bridge, with tired lines on a face grey beneath the great brown\nhood of his duffle--a face so youthful, yet with the knowledge of\nthe command of men writ plain thereon. The propellers have swirled\nfaithfully and unceasingly; the good ship in consequence has cleft the\npassive waves. But who knows what hideous lurking peril of mine or\ntorpedo we have not survived, what baleful eye has not glowered at us,\nitself unseen, and retired again to its foul underworld, baulked of\nits thirsted prey?\n\nIII.\n\nOF THE EDITOR OF _THE DAILY YAP_, ON OBSERVING THAT HIS SPECIAL\nCORRESPONDENT IS A RETIRED LIEUT., R.N., WHO SENDS HIM THE FOLLOWING\nACCOUNT OF A PUSH:--\n\nTime: 6.0 A.M. Course: (approx.) E. Distance run: 1-1\/2 m. Wind: S.W.\nForce: 6. State of land: 5 (rough, owing to craters). Weather: R.\nTherm.: 35 deg. Bar.: 28.89. Remarks: Objectives attained. Observation\nhampered by weather.\n\n       *       *       *       *       *\n\nBIG GAME SHOOTING.\n\n    \"Angus Bowser, the popular feed merchant of Dartmouth, shot his\n    mouse on Thanksgiving Day. With a couple of friends he left in\n    auto about 1 o'clock Monday afternoon for Bowser's Station. The\n    party was in the woods for about two hours when the mouse was\n    sighted.\"--_Canadian Paper_.\n\nWe hope Mr. ROOSEVELT will not be jealous.\n\n       *       *       *       *       *\n\nExtracts from a recent novel:--\n\n    \"He stepped out at Fernhurst Station, and walked up past the\n    Grey Abbey that watched as a sentinel over the dreamy Derbyshire\n    town.... So it was the system that was at fault, not Fernhurst.\n    Fairly contentedly he went back by the 3.30 from Waterloo.\"\n\nThe train system which sent him to the Midlands by the South-Western\nwas doubtless deranged by military exigencies.\n\n       *       *       *       *       *\n\n    \"Although Lord Warwick is the most sympathetic and attentive of\n    listeners, he has not remembered more than one good story,\n    and that has now been quoted in all the papers; we mean Lord\n    Beaconsfield story is said to be unprintable; then why tantalise\n    Lord Rosslyn, on account of the possible effect of his language\n    on the pack, compensated by the Commissionership of the Kirk of\n    Scotland. The other Beaconsfield story is said to be unprintable,\n    then why tantalise us?\"--_Saturday Review_.\n\nWhy, indeed?\n\n       *       *       *       *       *\n\n[Illustration: THE GREAT UNCONTROLLED.\n\nLORD RHONDDA. \"LOOK HERE, JOHN, ARE YOU GOING TO TIGHTEN THAT BELT, OR\nMUST I DO IT FOR YOU?\"\n\nJOHN BULL. \"YOU DO IT FOR ME. THAT'S WHAT YOU'RE THERE FOR.\"]\n\n       *       *       *       *       *\n\n[Illustration: _Farmer_. \"WHY DO THEY LET THAT CLOCK CHIME? AREN'T\nTHEY AFRAID THE HUNS MIGHT HEAR IT?\"\n\n_Yokel_. \"BLESS YOU, THAT'S TO DECEIVE 'EM. IT'S 'ALF-A-HOUR FAST.\"]\n\n       *       *       *       *       *\n\nHOW TO BECOME A TOWN-MAJOR.\n\nThrough large and luminous glasses Second-Lieut. St. John regards this\nWar and its problems. He is a man of infinite jobs. There are few\nvillages in France of which he has not been Town Major. Between\ntimes he has been Intelligence Officer, Divisional Burial Officer,\nDivisional Disbursing Officer, Salvage Officer, Claims, Baths,\nSoda-water and Canteens Officer.\n\nHe was once appointed Town-Major of some brick-dust, a rafter and two\nempty bully-beef tins--all of which in combination bore the name of a\nvillage. He assumed his duties with a bland Pickwickian zest, which\ndid good to the heart. He had boards painted.\n\n\n _______________________\n|                       |\n| THIS IS BLANK VILLAGE |\n|_______________________|\n\n\nsaid one aggressively, and\n\n ____________________________\n|                            |\n| TO THE TOWN-MAJOR OF BLANK |\n|                       ==>  |\n|____________________________|\n\n\nsaid another. A third read,\n\n ____________________\n|                    |\n| TO THE INCINERATOR |\n|   <==              |\n|____________________|\n\n\nthough there was nothing there to incinerate and (incidentally) no\nincinerator. \"HORSES,\" shouted another didactically, \"MUST NOT TROT\nTHROUGH THE MAIN STREET.\" That there was no street there at all did\nnot detract from the splendour of his notices, on which he spent much\npaint and happiness.\n\nWith the slightest encouragement he would have placarded that\narid wilderness with \"NO SMOKING IN THE LIFTS,\" and \"BEWARE OF\nPICKPOCKETS,\" but he had small encouragement, and so he contented\nhimself with a final placard which warned the troops against riding\nthrough standing crops and occupying the houses of civilians without\npermission from the Town-Major.\n\nStill, no one becomes a Town-Major without some sort of claim to the\npost.\n\nSecond-Lieut. St. John's first appearance in Armageddon took place\nduring \"peace-time warfare.\" An unpleasant and quite unnecessary\nlittle bulge in the trench-line, known as the Toadstool, was manned\nby the platoon of which he found himself second-in-command. It is\nrumoured that a Hun patrol, crawling to the edge of our parapet,\nsaw in the ghastly glare of a Verey light the benign and spectacled\ncountenance of Second-Lieut. St. John staring amiably across No Man's\nLand, and came to the hasty conclusion that they had made a mistake as\nto direction, since here was obviously one of their own officers of\nthe Herr Professor type. Rumour adds that they retired to their own\nlines and were promptly shot for cowardice.\n\nCertain it is that on that particular night Second-Lieut. St. John did\na thing the full details of which are now revealed to the Intelligence\nCorps for the first time. He fired a Verey light. It pleased him\nenormously. The sense that he, and he alone, was the cause of all\nthose sliding shadows and that flood of greenish light in No Man's\nLand went to his head like strong drink. He fired another and another\nand another.... The Hun was puzzled at this departure from routine,\nand opened a morose machine-gun fire which skimmed the top of the\nparapet and covered Second-Lieut. St. John with earth from shattered\nsandbags. He went on firing Verey lights in a sort of bland ecstasy\ntill his supply ran out, when he went to his Company Commander's\ndug-out for more. He filled his pockets with fresh ammunition, went\nback to his post, and began firing again. The first light was mauve.\nHe almost clapped his hands at it, and fired the second. It was pink.\nThe third was yellow, the fourth scarlet, and the fifth emerald green.\n\n\"The Crystal Palace,\" said Second-Lieut. St. John, \"isn't in it.\" And\nthen, because his watch had ended, he handed over to another yawning\nsubaltern and went to bed.\n\nOver miles and miles of country wild-eyed gunners were glaring into\nthe night and asking each other blasphemous questions. What did it\nmean?\n\n\"It must be Huns,\" said the British gunners; \"they're coming over.\"\n\n\"That is without doubt an English signal,\" said the enemy. \"We will\nprepare for an attack.\"\n\nThen the Hun gunners suddenly made up their minds to be on the safe\nside, and they put down a tremendous barrage on to No Man's Land.\n\n\"Told you so; they're on to our front line,\" said we, and put down a\ntremendous barrage on to No Man's Land.\n\nA Hun sentry, waking with a start, sounded the gas alarm. It was taken\nup all along the German line and overheard by a vigilant British\nsentry, who promptly set himself to make all possible noise with every\npossible means.\n\nOld French ladies in villages twenty miles back from the line lay\nall that night hideous in respirators. Anxious Staffs rang up other\nanxious Staffs. Gunners questioned the infantry. The infantry desired\ninformation from the gunners. All along the line the private soldier\nwas jolted from that kind of trance which he calls \"getting down to\nit,\" and was bidden to stand to till morning.\n\nAnd our Mr. St. John, who was a new and superfluous officer and liable\nto be overlooked, slept through it all with a fat smile.\n       *       *       *       *       *\nIt was after that that they made him a Town-Major.\n\n       *       *       *       *       *\n\nOUR PAMPERED \"CONCHIES.\"\n\n    \"There was a long and interesting debate on the imprisonment of\n    conscientious objectors in the House of Lords.\"--_The Times_.\n\nThis beats Donington Hall to a frazzle.\n\n       *       *       *       *       *\n\n    \"Teachers will welcome the resolution deploring 'the omission\n    from the Bill of any limitation upon the size of\n    classics.'\"--_Teacher's World_.\n\nTheir pupils are believed to hold a diametrically opposite opinion.\n\n       *       *       *       *       *\n\nAfter the Guildhall Banquet:--\n\n    \"Some had black leather bags, some had aprons. Others had nothing\n    at all and staggered off with a conglomeration of beef, pie, and\n    turtle soup tucked up under their arms.\"--_Weekly Dispatch_.\n\nThe menu said \"Clear Soup,\" but this must have been a bit thick.\n\n       *       *       *       *       *\n\n[Illustration: _Sandy (on departure of peace-crank, who has been\nholding forth)_. \"MAN, HE'S A QUEER CARD, THAT. THINK YE HE'S A'\nTHERE, DONALD?\"\n\n_Donald_. \"DOD, SANDY, IF WHAT'S NO THERE IS LIKE WHAT IS THERE, IT'S\nJUST AS WEEL HE'S NO A' THERE.\"]\n\n       *       *       *       *       *\n\n    LEGAL INTELLIGENCE.\n\n    DAVID LLOYD GEORGE, described as Prime Minister, was charged,\n    on the information of HERBERT HENRY ASQUITH, with exceeding the\n    speech limit while on tour. Mr. BONAR LAW, who appeared for the\n    defendant, asked for an adjournment and invited the Court to \"wait\n    and see.\" Upon hearing those words prosecutor broke down and had\n    to be assisted out of the court.\n\n       *       *       *       *       *\n\n    HORATIO BOTTOMLEY pleaded \"Not guilty\" to a charge of\n    fortune-telling. It appears that the defendant had stated that the\n    War would be over by Christmas. For the defence it was stated that\n    the defendant had not specified which Christmas, and even so if he\n    had said so it was so. Defendant asked for a remand to enable him\n    to dispense with legal assistance.\n\n       *       *       *       *       *\n\n    RESULT OF THE FOOD SHORTAGE?\n\n    \"Exchange new gold full plate, seven teeth, for good brown skin\n    hearthrug.\"--_The Lady_.\n\n       *       *       *       *       *\n\nFrom the police-notice _re_ air-raid warnings:--\n\n    \"When the car has two occupants one might concentrate on whistling\n    and calling out 'Take Cover.'\"\n\nAs his own won't be enough he should borrow the other occupant's\nmouth.\n\n       *       *       *       *       *\n\nTHE NEW MRS. MARKHAM.\n\nv.\n\nCONVERSATION ON CHAPTER LXXIII.\n\n_Mary_. There were two things in your last chapter that I did not\nquite understand--the National Debt and the Flappers.\n\n_Mrs. M_. About the National Debt, my dear child, I think you must\nwait until your papa comes home to tea, but perhaps I can satisfy\nyour curiosity about the Flappers, who were indeed amongst the most\nsingular and formidable products of the age we have been discussing.\nThe origin of the term is obscure, some authorities connecting it with\nthe term \"flap-doodle,\" others with the motion of a bird's wings, and\nI remember a verse in an old song which ran as follows:--\n\n  \"Place me somewhere east of Suez\n  On a lone and rocky shore,\n  Where the Britons cease from Britling\n  And the flappers flap no more.\"\n\nThis, however, does not throw much light on the subject. Perhaps\nthe term Flapper may best be defined as meaning a twentieth-century\nhoyden, and was applied to a type of girl from the age of thirteen to\nseventeen, whose extravagances in speech, manner and dress caused deep\ndismay among the more serious members of the community. In particular\nthe learned Dr. SHADWELL denounced them with great severity in a\nleading review, but with little result. They bedizened themselves with\nfrippery, shrieked like parrots on all occasions and interpreted the\nmotto of the time, \"Carry On,\" in a sense deplorably remote from its\nhigher significance.\n\n_George_. I think it seems, Mamma, as if the young girls of those\ntimes must have tried to make themselves as unpleasant as possible.\nHow thankful I am that Mary is not a Flapper!\n\n_Mrs. M_. You may well be. But allowance must be made for the\nmisapplied energy of our ancestors. If the Flappers excite our\ndisgust, their subsequent treatment moves our commiseration, since the\nSumptuary and Disciplinary Laws passed by the House of Ladies dealt in\ndrastic fashion with the offences which I have described. As a matter\nof fact many Flappers grew up into excellent and patriotic women. I\nremember my grandmother saying to me once, \"When I was sixteen I had a\nvoice like a cockatoo and the manners of a monkey,\" but nothing could\nhave been more discreet or sedate than her deportment in old age.\n\n_Richard_. Did the Flappers speak English?\n\n_Mrs. M_. Presumably; but, judging from the records of their dialect\nwhich have come down to us, their speech was made up of a succession\nof squeals rather than of articulate words, and has so far defied\nthe efforts of modern philologists. Indeed speech seems to have been\nalmost at a discount, owing to the immense popularity of the moving\npicture play, then in its infancy and as yet unaccompanied by\nmechanical reproduction of the voices of the actors. Indeed at one\ntime it was said that there were only three adjectives in use in\nFlapper society--\"ripping,\" \"rotten\" and \"top-hole,\" I think they\nwere.\n\n_George_. What stupid words! I wish they could have heard some of\npapa's adjectives.\n\n_Mrs. M_. Your father, my dear, has a copious and picturesque\nvocabulary, but phrases which are pardonable in moments of expansion\nin a person of mature years are not always suitable for juveniles.\n\n       *       *       *       *       *\n\nTHE TRANSGRESSOR.\n\nI was walking painfully along a lonely road towing my\nthree-thousand-guinea ten-cylinder twelve-seater. According to\nRegulation 777 X, both brakes were on. My overcoat collar was turned\nup to protect my sensitive skin from a blasting easterly gale, and\nthrough the twilight I was able to see but a few yards ahead. I had\na blister on my heel. Somewhere, many miles to the eastward, lay my\ndestination. Suddenly two gigantic forms emerged from the hedgerow\nand laid each a gigantic paw upon my shoulders. A gruff voice barked\naccusingly in my ear.\n\n\"You are the owner of a motorcar?\"\n\nWas it any use denying the fact? I thought not.\n\n\"Yes,\" I replied humbly, \"I am.\"\n\n\"Have you the permit which allows you to possess this?\" He waved\ntowards the stagnant 'bus.\n\n\"I have.\"\n\n\"Have you the licence which allows you to take it upon the high road?\"\n\nWith frozen fingers I held it out to him. He moved to the back of the\ncar, unscrewed the entrance to the petrol tank and applied his nose\nto the aperture. After three official sniffs he turned upon me\naggressively.\n\n\"There is an undeniable odour of petroleum. How do you account for\nthat?\"\n\n\"Sir,\" I replied, \"last week my little son had his knockabout suit\ndry-cleaned in Perthshire by the petrol-substitute process. This\nmorning he climbed upon the back of the car to see whether his Silver\nCampine had laid an egg in the hood.\"\n\nHe glared at me.\n\n\"Ah! Have you the necessary extension which allows you to use a\nmotorcar as a habitation for hens?\"\n\nI gave it to him.\n\nThen, frustrated with fury, he thundered at me successively: \"Have you\na towing permit? Have you a dog licence? Can you produce a boot and\nshoe grant? Do you hold any rubber shares? Have you been inoculated\nfor premature decay? What did you do in the Great War?\"\n\nI gave him the necessary documents in perfect order. For a moment\nhe was nonplussed. Then he asked with sly intention, \"Have you\nthe champagne and chicken sandwich ration which is apportioned to\nsuper-inspectors?\"\n\nI handed it to him with a table-napkin (unused) and a pair of\nwire-cutters thrown in. For some minutes he remained silent, except in\nthe gustatory sense, then he turned upon me and, handing back an empty\nbottle, said triumphantly, \"You must now produce, under Clause 5005\nGerrard, framed this morning at 11-30 o'clock, one pint of old ale\nand six ounces of bread and cheese for the sustentation of the\nsub-inspector.\"\n\nI regarded him stonily and leant against the cold, cold bonnet of the\ncar. Alas! I had it not.\n\n\"Sir,\" I pleaded, \"I did not know ... give me time. The next inn is\nbut a few miles. If you and your companion will take a seat I will\nbring you to the inn door and all will be well.\"\n\nHe laughed in my face.\n\n\"Algernon Brocklebank Smith,\" he said sternly, \"you have betrayed\nyourself into our hands.\" He turned to his myrmidon: \"Get a move on\nyou, Herbert; it's a bit parky standing about here.\"\n\nAfter all he was but a coarse fellow.\n\nHerbert, galvanised into action, produced a small oblong object from\nhis pocket, lighted the end of it with the glowing butt of one of my\nCorona Coronas, and placed it underneath the car. In a few moments all\nthat remained of my three-thousand-guinea ten--cylinder twelve-seater\nwas one small nut, which was immediately impounded.\n\nI raised the collar of my overcoat (second reef), shifted my face to\nthe eastward, and, notwithstanding the blister on my heel, turned my\nsteps towards my destination.\n\nI uttered no plaint. I had transgressed against the immutable law.\n\n       *       *       *       *       *\n\n    IS THE RACE LOSING ITS NERVE?\n\n    \"A sensation has been caused by the announcement that Miss Teddie\n    Gerard is leaving 'Bubbly' to play the leading part in 'Cheep' at\n    the Vaudeville Theatre.\"--_Daily Mirror_.\n\n       *       *       *       *       *\n\nTHE \"WAR LEADER\" AND TWO SENSITIVE SOULS.\n\n[Illustration: \"THE ENTIRE GERMAN ECONOMIC STRUCTURE IS ON THE VERGE\nOF COLLAPSE,\"]\n\nBUT\n\n[Illustration: \"WE SHOULD BE MAD IF WE BLINDED OUR EYES TO THE FACT\nTHAT THEY CAN HOLD OUT FOR YEARS YET.\"]\n\n[Illustration: \"THE SUBMARINE CAMPAIGN HAS BEEN AN UTTER FAILURE. NO\nSHORTAGE OF FOOD EXISTS OR WILL EXIST\"]\n\nIF\n\n[Illustration: \"WE ONE AND ALL DETERMINE NOT TO CONSUME AN OUNCE MORE\nFOOD THAN IS ABSOLUTELY NECESSARY TO KEEP BODY AND SOUL TOGETHER.\"]\n\n[Illustration: \"THE WAR IS, TO ALL INTENTS AND PURPOSES, ALREADY WON,\"]\n\nPROVIDED\n\n[Illustration: \"THAT IN THE NEXT THREE YEARS THE WHOLE NATION MAKES\nSUCH A STUPENDOUS EFFORT AS WE HAVE NOT AS YET DREAMED OF,\" ETC.,\nETC.]\n\n       *       *       *       *       *\n\n[Illustration: _Bookmaker (with long experience of the Turf but none\nof Coursing)_. \"I'M GIVIN' YOU SIX TO FOUR AGAINST THE FAWN, SIR. NOW\nI'LL GIVE ANYONE SIX TO FOUR AGAINST THE BLACK.\"\n\n_Friend (hurriedly)_. \"BUT YOU CAN'T GIVE THOSE ODDS WITH ONLY TWO\nRUNNERS.\"\n\n_Bookmaker_. \"WHY? AIN'T THE BLOOMIN' RABBIT GOT A CHANCE?\"]\n\n       *       *       *       *       *\n\nNEW MEN AND OLD FACES.\n\n    [According to a writer in _The Daily Chronicle_, Lord Morley's\n    face \"in conformation gets more and more like Goethe's.\"]\n\n  VISCOUNT, better known as plain JOHN MORLEY,\n  As I gather from a chatty screed,\n  Ever daily grows exteriorly\n  (Pray forgive a rhymer's urgent need)\n  More like GOETHE--please pronounce it \"Gertie\"--\n  Who expired soon after eighteen-thirty.\n\n  But this instance is not isolated,\n  As a survey of our statesmen shows;\n  WINSTON now suggests a long post-dated\n  DAN O'CONNELL in his mouth and nose;\n  NORTHCLIFFE's growing more Napoleonic\n  Than the Corsican, though less laconic.\n\n  In the noble lineaments of BILLING\n  Shrewd observers (like myself) can trace\n  Wonderful, inspiring, vivid, thrilling\n  Memories of JULIUS CAESAR'S face,\n  With a hint of something far more regal,\n  More suggestive of the soaring eagle.\n\n  I admit GEORGE MOORE is not yet showing\n  Marked resemblance to his namesake, TOM;\n  But great CHESTERTON is hourly growing\n  Almost indistinguishable from\n  Dr. JOHNSON; daily grows more plain\n  SHAKSPEARE'S facial forecast of HALL CAINE.\n\n  HALDANE and his spiritual brother,\n  SCHOPENHAUER, that dyspeptic sage,\n  Monthly grow so very like each other,\n  As portrayed in MAXSE'S lurid page,\n  That it passes MAXSE'S Christian charity\n  To detect the least dissimilarity.\n\n  BELLOC is approximating closely\n  To the massive mien of CHARLES JAMES FOX;\n  BUCHAN plagiarizes very grossly\n  From the rapt expression of JOHN KNOX;\n  And the LAUREATE, if his hair grew scanty\n  Or he shaved his beard, might look like DANTE.\n\n  CLARA BUTT, the eminent musician,\n  Vividly resembles PERICLES;\n  SARGENT and the late lamented TITIAN\n  Are as like each other as two peas;\n  LOREBURN, known to cronies as \"Bob\" Reid,\n  Duplicates the Venerable BEDE.\n\n  But enough of this identifying\n  Instances of the recurrent face;\n  Rather let us foster an undying\n  Resolution in the British race\n  Evermore and evermore to shun\n  Any imitation of the Hun.\n\n       *       *       *       *       *\n\n\nA POSER FROM THE BENCH.\n\nFrom the report of a collision case:--\n\n    \"Mr. Justice ----: 'Which car hit the other first?' 'I cannot\n    say.'\"--_Freeman's Journal_.\n\n       *       *       *       *       *\n\n    \"OUR SWEEP IN THE HOLY LAND.\"--_Daily News_.\n\n_Ours_ is in Mesopotamia.\n\n       *       *       *       *       *\n\n[Illustration: HOW IT STRIKES A SOLDIER.\n\nTHE KAISER. \"WHAT DO YOU MAKE OF THIS LLOYD GEORGE AFFAIR?\"\n\nMARSHAL VON HINDENBURG. \"I'VE NO TIME TO READ POLITICAL SPEECHES,\nSIRE. THIS FELLOW HAIG KEEPS ME TOO BUSY.\"]\n\n       *       *       *       *       *\n\nESSENCE OF PARLIAMENT.\n\n_Monday, November 12th_.--An old Parliamentarian, when asked by a\nfriend to what party the PRIME MINISTER now belonged, sententiously\nreplied, \"He used to be a Radical; he will some day be a Conservative;\nand at present he is the leader of the Improvisatories.\"\n\nThe latest example of his inventive capacity does not meet with\nunmitigated approval. Members were very curious to know exactly how\nthe new Allied Council was going to work, and what would be the\nrelations between the Council's Military advisers and the existing\nGeneral Staffs of the countries concerned. Mr. BONAR LAW assured the\nHouse that the responsibility for strategy would remain where it is\nnow, but did not altogether succeed in explaining why in that case the\nCouncil required other military advisers.\n\nThe SECRETARY FOR SCOTLAND is about the mildest-mannered man that ever\nsat upon the Treasury Bench. But even he can be \"_tres mechant_\" at a\npinch. When Mr. WATT renewed his complaint that sheriffs-principal in\nScotland had very little to do for the high salaries they received,\nMr. MUNRO replied that \"it would just be as unsafe to measure the\nactivities of the sheriff-principal by the number of appeals he hears\nas to measure the political activities of my hon. friend by the number\nof questions he puts.\"\n\nThe Pensions Department at Chelsea is to be reorganised. Mr. HODGE\nexcused the delays by pointing out that an average of thirty-three\nthousand letters a day is despatched, but, as he added that there is a\nstaff of four thousand five hundred persons to do it, it hardly looks\nas if they were overworked.\n\n_Tuesday, November 13th_.--The House of Lords was to have discussed\nthe state of Ireland, but, owing to the absence of its LEADER,\nfell back upon the less exciting but more practical topics of\nsugar-substitutes for jam, and barley for beer. It was cheering to\nlearn from the Duke of MARLBOROUGH that the jam-manufacturers gave\ngreat care to exclude arsenic from their glucose, and from Lord\nRHONDDA that there would be plenty of barley for both cakes and ale.\n\nMr. WARDLE is the latest example of the poacher turned gamekeeper.\nA few months ago, as leader of the Labour Party, he was instant in\ncriticism of the ineptitutes of Government officials. This afternoon,\nupon his old friend, Mr. TYSON WILSON, venturing to refer to the\n\"stupid decisions\" of the Board of Trade, Mr. WARDLE was down on him\nin a moment. With the air of one who had been born and brought up in\nWhitehall Gardens, he replied, \"Stupid decisions are not made by the\nBoard of Trade.\"\n\nThe Pacifists had rather a mixed day.\n\n       *       *       *       *       *\n\n[Illustration: PENSIONS.\n\nMR. HODGE.]\n\n       *       *       *       *       *\n\nThey were visibly relieved when Mr. BONAR LAW (supported by Mr.\nASQUITH) declined to admit into the Bill for extending the life of\nthis Parliament a provision enabling constituencies to get rid of\nMembers who had ceased to represent them. But they did not like his\ncontemptuous reference to their argumentative powers. Mr. TREVELYAN,\nwho regards himself as the representative (by literary descent) of\nCHARLES JAMES FOX, was particularly annoyed.\n\n       *       *       *       *       *\n\n[Illustration: _IN RE_ ADMIRAL JELLICOE.\n\nMR. LYNCH.                    DR. MACNAMARA.]\n\n       *       *       *       *       *\n\nAs party-funds are rather under a cloud just now the Government\nthought they might justify their existence by drawing on them for\nthe campaign against enemy propaganda. But their custodians thought\notherwise. The Tory Whip was prepared to make a small contribution;\nthe Liberal would give nothing, on the ground that the total required\nwas extravagantly large. So the country will have to foot the bill.\n\n_Wednesday, November 14th_.--The knowledge that Mr. ASQUITH was to\n\"interpellate\" the PRIME MINISTER regarding his recent speech in\nParis, and the Allied War Council therein described, brought a\ncrowd of Members to the House, and filled the Peers' Gallery with\nex-Ministers scenting a first-class crisis.\n\nThe protagonists on entering the arena were loudly cheered by their\nrespective adherents, but the expected duel did not come off. Mr.\nASQUITH'S questions were searching enough, but not provocative. Mr.\nLLOYD GEORGE'S reply was comprehensive and conciliatory, and ended\nwith the promise of a day for discussion. Instead of a fight there was\nonly an armistice, usually a preliminary to a definite peace.\n\nA little disappointed, perhaps, the Peers betook themselves to their\nown Chamber, there to hear Lord PARMOOR discourse upon the woes\nof conscientious objectors. Many of them, he thought, had been\nvindictively punished for their peculiar opinions. Nobody, in a\nsomewhat cloudy discussion, made it quite clear whether the Tribunals\nor the Army authorities or the Home Office were most at fault; and\nLord CURZON'S suggestion that persons who refused not merely to fight\nbut to render any kind of service to their country in its time of need\nwere not wholly free from blame had almost the air of novelty.\n\nThe Air-Force Bill passed through Committee in one sitting. The credit\nfor this achievement may be divided equally between Major BAIRD, who\nproved himself once more a skilful pilot, and Mr. BILLING, who spoke\nso often that other intending critics got little chance. Counting\nspeeches and interruptions, I find from the official reports that he\naddressed the House exactly one hundred times; and it is therefore\nworth noticing that his last words were, \"This is what you call\nmuzzling the House of Commons.\"\n\n_Thursday, November 15th_.--Lord WIMBORNE did his best to-night to\ndefend the inaction of the Irish Executive in the face of the Sinn\nFein menace. But he would have been wiser not to have adduced the\nargument that Ireland was a _terra incognita_. If there is one subject\nthat the Peers think they know all about it is the sister-island. Lord\nCURZON thought it would be a mistake, by enforcing \"a superficial\nquiet,\" to check the wholesome influences brought into being by the\nConvention. He did not go so far as to say that Mr. DE VALERA was one\nof them.\n\nAt last the Government have decided to take short order with the\npernicious literature of the Pacifists. In future all such documents\nare to be submitted to the Press Bureau before publication. A howl of\nderisive laughter greeted the HOME SECRETARY'S announcement, but when\nMr. SNOWDEN essayed to move the adjournment, although he and his\nfriends were joined by some of the Scotch and Irish malcontents, the\ntotal muster was only thirty-three, and the motion accordingly came to\nearth with a thud.\n\nBy a large majority the House refused to reinstate the Livery\nfranchise in the City of London. In any case this ancient privilege\ncould not long have survived the curtailment of the Lord Mayor's\nFeast.\n\n       *       *       *       *       *\n\n[Illustration: _The Colonel_. \"I'D TAKE ALL THOSE MUTINOUS HOUNDS AND\nPUT 'EM AGAINST THE WALL.\"\n\n_Aunt Jane_. \"BUT, MY DEAR, THE AWFUL THING IS THAT IT HAS SPREAD TO\nOUR OWN ARMY. I HEARD TWO SOLDIERS IN THE TRAIN TO-DAY TALKING ABOUT\nTHEIR SERGEANT-MAJOR IN A DREADFUL WAY.\"]\n\n       *       *       *       *       *\n\nBOON FOR BUSY BRIDEGROOMS.\n\nIn these days of military hustle, when a soldier comes home, falls in\nlove, gets engaged, marries, sets up a home, and returns to the Front\nin less than a week, there is little time for the ordinary courtesies\nof matrimonial procedure. It is felt, therefore, that the appended\nprinted form of thanks for wedding presents--based on the model of the\nField Service Postcard--will prove a great boon to all soldiers who\nmeditate matrimony during short leave. It will be found sufficient\nmerely to strike out inappropriate words in the printed form, which is\nas follows:--\n\n\"Captain and Mrs. ---- beg to return thanks for your\n                       _\n  Beautiful             |\n  Charming              |\n  Generous              |\n  Very generous         |\n  Useful                |  Gift\n  Very Useful           |- Cheque\n  More than useful      |  Letter.\"\n  Unexpected            |\n  Totally unexpected    |\n  Remarkable            |\n  Artistic              |\n                       _|\n\n_Examples_.--(1) To a rich and miserly uncle, who has come down with\nan astonishingly handsome sum--strike out everything except \"Very\ngenerous--more than useful--totally unexpected cheque.\"\n\n(2) To an eccentric former admirer of the bride, who has sent a\nforty-stanza poem, entitled \"Sunset in the White-chapel Road: Thoughts\nThereon\"--strike out everything except \"Remarkable gift.\"\n\n(3) To an enormously wealthy female relative, who disapproves of the\nbride and has sent a second-hand plated sugar-sifter--strike out\neverything except \"Gift.\"\n\n(4) To anyone of whom much was expected, but who neither gave a\npresent nor wrote--strike out everything on the postcard.\n\n       *       *       *       *       *\n\n    \"Strange Story of a Wedding in the Divorce Court.\"--_Daily News_.\n\nIt seems a rather unfortunate choice of _locale_.\n\n       *       *       *       *       *\n\nExtract from an Indian begging-letter:--\n\n    \"My mother is a widow, poor chap, and has a postmortem son.\"\n\n       *       *       *       *       *\n\n    \"AMATEUR GENT., experienced, wanted, for week at Xmas. All\n    expenses paid.\" _Daily Telegraph_.\n\nWhy not have a professional one and do the thing handsomely?\n\n       *       *       *       *       *\n\nONCE UPON A TIME.\n\nTHE LETTER.\n\nOnce upon a time, not so very long ago, an illustrious man of\naffairs--soldier and statesman too--visited our shores, and by his\nwise counsels so captured the imagination of his hearers and readers\nthat one of the greatest of all compliments was paid to him, and\nanyone with a black cocker spaniel to name named it after him; and he\nhad a name rather peculiarly adapted to such ends too.\n\nIt chanced that among the puppies thus made illustrious was one which\na young soldier before leaving for France to win the War gave to his\nsister, and when writing to him, as, being a good girl, she regularly\nand abundantly did, she never omitted to give tidings as to how the\nlittle creature was developing; and I need hardly say that in the\nwhole history of dogs, from TOBIT'S faithful trotting companion\nonwards, there never was a dog so packed with intelligence and\nfidelity as this. Most girls' dogs are perfect, but this one was more\nremarkable still.\n\nNow it happened that the gallant brother, in the course of his duties\nas a war-winner, was moved from place to place so often that he\ngradually lost definition, as the photographers say, and the result\nwas that one of her recent letters failed to catch up with him. That\nwas a pity, because it was a better letter than usual. It gave all the\nnews that he would most want to hear. It said what picture her father\nwas working on at the moment, and told, without spoiling them, his two\nlast jokes. It said whom her mother had called on and who had called\non her mother and how something must be done to stop her smoking too\nmany cigarettes. It said that their young brother, having sprained his\nankle at hockey, had become a wolf for jig-saw puzzles. It said where\ntheir parents had dined recently and where they were going to dine and\nwho was coming next week. It said what she had seen at the theatre\nlast Saturday and what book she was reading. It said which of the\nother V.A.D.'s had become engaged. It said what an awful time they had\nhad trying to buy some tea, and how scarce butter had become, and\nwhat a cold she had caught in the last raid, and how Uncle Jim had\ninfluenza and couldn't go on being a special, and how Aunt Sibyl had\nbeen introduced to one of the GEDDESES and talked to him as though it\nwas the other, and how she herself had met Evelyn in the street\nthe other day and Evelyn had asked \"with suspicious interest after\nyou\"--and a thousand other things such as a good sister, even though\nbusy at a hospital, finds time to write to a brother over there, all\namong the mud and the shells, winning the War. And not being in the\nhabit of signing her name, when writing in this familiar way, she\nfinished up with a reference to the darlingest of all dogs by sending\nits love at the very end: \"Love from ----\" and so forth.\n\nWell, the letter, as I have said, could not be delivered. The postal\npeople at the Front, and behind the Front, are astonishingly good, but\nthey could not get in touch with the brother this time, and therefore\nthey opened the letter and looked at the foot of it for the name of\nthe writer and found that of the dog, and at the head of it for\nthe street and town where the writer lived, and sent it back as\n\"insufficiently addressed.\"\n\nAnd that is why in a certain house in Chelsea a treasured possession\nis a returned letter for General SMUTS.\n\n       *       *       *       *       *\n\nFrom an article entitled \"Is it Safe for Cousins to marry?\":--\n\n    \"It is just as well, however, to pick out somebody besides your\n    cousin for your wife.\" _The Family Doctor_.\n\nBefore acting on this advice, however, it might be safer to consult\nThe Family Lawyer.\n\n       *       *       *       *       *\n\n[Illustration: AFTER A DAY ON THE ALLOTMENT.\n\n\"SUDDENLY SHE REALISED THAT HER IDOL HAD FEET OF CLAY.\"--_Extract from\npopular novel_.]\n\n       *       *       *       *       *\n\nTHE VERY GLAD EYE.\n\nMother put down the key of the hen-house and took up the letters that\nlay beside her plate.\n\n\"If only Joan would write larger,\" she sighed, turning over an\nenvelope across which an ant seemed to have walked and left an inky\ntrail. \"I've mislaid my glass too, and shan't be able to read a word.\nWhere could I have put the miserable thing?\" she asked, peering again\nat the ridiculous little script.\n\nFather put down his paper and said these hunts for Aunt Matilda were\ngetting monotonous. Only yesterday he had rescued her from some dried\nbulbs in the greenhouse, and didn't Mother think it time she saw a\ngood oculist and had proper spectacles, instead of using the old lens\nin that carved gold bauble belonging once to his grandmother's aunt.\n\n\"Perhaps it's just a bad habit,\" she answered with a smile, \"or my\neyes are getting lazy. But really I can see _so_ well through it, and\nif they would print the newspapers better--\"\n\n\"No one we know in this morning's list,\" said Father shortly, as he\nturned a sheet; \"and we should be hearing from those rascals now that\nthe push is over,\" he added, glancing at Mother who began to sip her\ncoffee hurriedly.\n\n\"They might even get leave together,\" ventured Margery. \"It's five\nmonths since Dick came home, and as for Christopher--\"\n\n\"What swank for old Margots, now her hair is up,\" piped Archie. \"Two\nbrothers from the trenches to--\"\n\n\"If you'd make a little less noise, my son,\" said Father in a strange\nvoice, \"I might be able to take in what I'm reading. There's something\nhere about Christopher.\"\n\n\"What?\" cried Mother, springing from her chair.\n\n\"Yes, it's Christopher plain enough,\" he repeated with shining eyes.\n\"Christopher Charles Bentley, and--God bless my soul!--the boy has\nbeen splendid! It's all down here, and---\n\n\"Read, read!\" we clamoured, as his voice grew husky and indistinct.\n\n\"Read!\" again we shouted, as Mother came and took the paper gently\nfrom him.\n\n\"When you're all quiet, children,\" she began, devouring the words\nbefore her.\n\n_Quiet!_ Even the canary held its breath while Mother read that\nwonderful paragraph.\n\nIt was a long one, and every word of it a tribute to our magnificent\nChris, who had organised a small volunteer party, attacked a strong\npoint, and captured fifteen of the enemy and a machine-gun, for which\ngallant act he had been awarded the M.C.\n\nWith lingering pride she went through it a second time, and only then\ndid we see that she was staring at the paper, proudly and fiercely,\nthrough the handle of the hen-house key!\n\n       *       *       *       *       *\n\n[Illustration: _First A.B. (indicating old tramp steamer in ballast)_.\n\"THANK 'EAVENS WE AIN'T GOT PROPELLERS WHAT STICK OUT LIKE THAT ON\nTHIS 'ERE JUNK, BILL.\"\n\n_Second A.B._ \"WHAT ARE YOU GROUSING ABOUT NOW?\"\n\n_First A.B._ \"WHY, THE BLOOMIN' FIRST-LOOTENANT WOULD MAKE US POLISH\nTHE BLINKIN' THING.\"]\n\n       *       *       *       *       *\n\nTHE MUSICAL CRITIC'S ORDEAL.\n\n    [Mr. CYRIL SCOTT, the musical composer, in his recently published\n    volume on _The Philosophy of Modernism in its connection with\n    Music_, states that the criterion of lofty music, the method of\n    gauging the spiritual value of art, \"is only possible to him who\n    has awakened the latent faculties of the pineal gland and the\n    pituitary body.\"]\n\n  Lately I've been reading CYRIL SCOTT'S\n  Book on Music, modern and unmuzzled,\n  And, though solving many toughish knots,\n  By one statement I am sadly puzzled,\n  Namely, that if we would understand\n  What divides the noble from the shoddy\n  We must cultivate \"the pineal gland,\"\n  Also \"the pituitary body.\"\n\n  But unfortunately SCOTT refrains\n  (Hence my present painful agitation)\n  From elucidating how one gains\n  This desiderated consummation.\n  Must I fly to silken Samarcand,\n  Or explore the distant Irrawaddy\n  For the culture of my pineal gland\n  And of my pituitary body?\n\n  Is the object gained by force of will\n  Or some drastic vegetarian diet?\n  Does it mean a compound radium pill\n  Causing vast upheaval and disquiet?\n  Do I need some special \"Hidden Hand,\"\n  Or the very strongest whisky toddy\n  To arouse my dormant pineal gland,\n  My unused pituitary body?\n\n  Should I read the works of Mr. YEATS,\n  Or the lays of WILCOX (ELLA WHEELER)?\n  Must I visit the United States\n  And consult the newest occult \"healer\"?\n  Is the tragedy of IBSEN'S _Brand_\n  Or the humour of _Poor Pillycoddy_\n  Better feeding for my pineal gland\n  And for my pituitary body?\n\n  Vain the subtle art of HENRY JAMES,\n  Vain the wealth of ROTHSCHILDS or of MORGANS,\n  If I fail to satisfy the claims\n  Of these mystic and momentous organs;\n  I'm no better than a grain of sand\n  Or a simple common polypody,\n  With an undeveloped pineal gland,\n  An inert pituitary body.\n\n  Blindly seeking for a helpful clue,\n  Welcoming no matter what suggestion,\n  I have lately sounded one or two\n  Leading doctors on this vital question;\n  But they think I'll have to be trepanned\n  If I wish effectively to modify\n  the structure of my pineal gland\n  Or of my pituitary body.\n\n  MORAL.\n\n  _'Gin pituitary bodies,\n  With awakened eye,\n  Meet with humble hoddy-doddies--\n  Smaller human fry--\n  Cries and kissing both are missing\n  When they're passing by,\n  And the astral demi-god is\n  Comin' thro' the rye_.\n\n       *       *       *       *       *\n\n    OUR COLLOQUIAL CONTEMPORARIES.\n\n    \"Repeated charges by Turkish cavalry resulted in only a slight\n    gain of ground at the expense of heavy osses.\"--_Daily News_.\n\n       *       *       *       *       *\n\n    FREE FOODERS.\n\n    \"ROSYTH WORKERS AND THE COST OF LIVING.\n\n    \"Mr. Douglas moved that they demand a reduction in the cost of\n    living of 200 per cent. by abolishing profiteering and securing\n    national control of food supplies. It was subsequently agreed to\n    demand 100 per cent. decrease in the cost of food.\"--_Glasgow\n    Herald_.\n\n       *       *       *       *       *\n\nTHE COMPLETE PLASHER.\n\n\"Francesca,\" I said, \"listen to this.\"\n\n\"I will,\" she said, \"if it's worth listening to.\"\n\n\"You can't tell that till you've heard it, can you?\"\n\n\"Well, what is it, anyhow?\"\n\n\"It's a letter,\" I said, \"from Harry Penruddock.\"\n\n\"That doesn't sound very exciting.\"\n\n\"Ah, but wait a bit.\"\n\n\"Well, get a move on. I've got to see the cook.\"\n\n\"He sends me,\" I said, \"a notice which has been served upon him about\nhis cottage at Smoltham. He wants to have my opinion about it.\"\n\n\"Very well, give him your opinion, and let's get on with the War.\"\n\n\"Francesca,\" I said, \"are you not more than a little peevish this\nmorning?\"\n\n\"I have no patience,\" she said, \"with notices that have to be served.\nIt's always done by sanitary inspectors and rate collectors, and\npeople of that sort. Why can't they just post them and have done with\nit?\"\n\n\"Who are you,\" I said, \"that you should fly in the face of Providence\nin this way? Can't you see that if a notice is 'served,' it\nimmediately becomes twice as important?\"\n\n\"Oh, if it adds to the dignity of an inspector, well and good; but for\nmy part I should have posted it.\"\n\n\"You are not a sanitary inspector, and cannot realise the feelings of\none.\"\n\n\"They have no feelings, and that's why they're made inspectors.\"\n\n\"Hush!\" I said, and began to read:--\n\n\"'In pursuance of the directions given in an Act passed in the fifth\nand sixth years of the reign of King William the Fourth, entitled \"An\nAct to consolidate and amend the Laws relating to Highways in that\npart of Great Britain called England,\" I, T. Bradish, of the Town\nHall, Smoltham, do hereby give you notice forthwith to cut, prune,\nplash or lop certain Trees and Hedges overhanging the highway\nimmediately adjoining your premises, No. 15, East Gate, in the Parish\nof Smoltham, and which are causing an obstruction and annoyance to\nthe said highway, so that the obstructions caused to the said highway\nshall be removed.\n\n\"'Dated this 19th day of October, 1917.'\"\n\n\"Isn't it priceless?\" I said.\n\n\"It is,\" said Francesca. \"I never knew before that a road could be\nannoyed.\"\n\n\"Even a road has its feelings.\"\n\n\"Yes, perhaps it's a short lane, and everybody tramples on it, and it\nturns at last.\"\n\n\"So do borough engineers and surveyors, it seems.\"\n\n\"I bet this one's a Tartar.\"\n\n\"How can you tell that?\"\n\n\"I can tell it by his style, which is very severe and uncompromising.\"\n\n\"His style,\" I said, \"is as the statute made it, and mustn't be\nimpugned by us.\"\n\n\"I particularly like that bit about plashing the trees. How in the\nname of all that's English do you plash a tree?\"\n\n\"If,\" I said, \"you were a fountain and wanted to be poetical, you\nwould plash, instead of splashing.\"\n\n\"That's nonsense,\" she said.\n\n\"No,\" I said, \"it's poetry.\"\n\n\"But you don't pour poetry on overhanging trees. It must mean\nsomething else.\"\n\n\"I'll tell you what; we'll get a dictionary.\"\n\n\"Yes,\" she said, \"you get it. I'm no good at dictionaries. I always\nfind such a lot of fascinating words that I never get to the one I\nwant.\"\n\n\"I'm rather like that myself,\" I said. \"However I'll exercise\nself-restraint. Here you are: Packthread, Pastime, Pin--there's a lot\nabout Pin--Plash. Got it! It means 'to bend down and interweave the\nbranches or twigs of.'\"\n\n\"Now,\" she said, \"we know what Mr. Bradish wants.\"\n\n\"He's a very arbitrary man,\" I said. \"How can he expect Harry\nPenruddock to bend down and interweave the branches or twigs of?\"\n\n\"Anyway, Harry's got to do it, whether he understands it or not.\"\n\n\"Yes,\" I said, \"borough surveyors take no denials. And now that you've\nhad your lesson in English, you can go and see the cook.\"\n\n\"Half a mo',\" she said; \"I'm acquiring a lot of useful information\nabout 'Plaster.' I never knew--\"\n\n\"Hurry up,\" I said, \"or we shan't get any lunch.\"\n\nR.C.L.\n\n       *       *       *       *       *\n\nDERELICT.\n\n_(Notices to Mariners. North Atlantic Ocean. Derelict reported.)_\n\n  \"We left 'er 'eaded for Lord knows where, in latitude forty-nine,\n  With a cargo o' deals from Puget Sound, an' 'er bows blown out by a mine;\n  I seen 'er just as the dark come down--I seen 'er floatin' still,\n  An' I 'ope them deals'd let her sink afore so long,\" said Bill.\n\n  \"It warn't no use to stand by 'er--she could neither sail nor steer--\n  With the biggest part of a thousand mile between 'er and Cape Clear;\n  The sea was up to 'er waterways an' gainin' fast below,\n  But I'd like to know she went to 'er rest as a ship's a right to go.\n\n  \"For it's bitter 'ard on a decent ship, look at it 'ow you may,\n  That's worked her traverse an' stood 'er trick an' done 'er best in 'er day,\n  To be driftin' around like a nine-days-drowned on the Western Ocean swell,\n  With never a hand to reef an' furl an' steer an' strike the bell.\n\n  \"No one to tend 'er binnacle lamps an' light 'er masthead light,\n  Or scour 'er plankin' or scrape 'er seams when the days are sunny an' bright;\n  No one to sit on the hatch an' yarn an' smoke when work is done,\n  An' say, 'That gear wants reevin' new some fine dogwatch, my son.'\n\n  \"No one to stand by tack an' sheet when it's comin' on to blow;\n  Never the roar of 'Rio Grande' to the watch's stamp-an'-go;\n  An' the seagulls settin' along the rail an' callin' the long day through,\n  Like the souls of old dead sailor-men as used to be 'er crew.\n\n  \"Never a port of all 'er ports for 'er to fetch again,\n  Nothin' only the sea an' the sky, the sun, the wind an' the rain;\n  It's cruel 'ard on a decent ship, an' so I tell you true,\n  An' I wish I knew she 'ad gone to 'er rest as a good ship ought to do.\"\n\nC.F.S.\n\n       *       *       *       *       *\n\n[Illustration: _Mabel_. \"WHAT SORT OF A DANCE WAS IT LAST NIGHT? HOW\nDID YOU GET ON?\"\n\n_Gladys_. \"OH, ALL RIGHT. I WAS UP TO MY KNEES IN BOYS ALL THE\nEVENING.\"]\n\n       *       *       *       *       *\n\nOUR BOOKING-OFFICE.\n\n_(By Mr. Punch's Staff of Learned Clerks.)_\n\nGenerally speaking, stories left unfinished because of the death\nof the writer in mid course can only be at best an uncomfortable,\nexasperating legacy to his admirers. But by a thrice happy chance this\nis not the case with the two novels upon which the late HENRY JAMES\nwas engaged at the time of his fatal illness. This good fortune comes\nfrom the fact that it was the writer's habit \"to test and explore,\" in\na written or dictated sketch, the possible developments of any theme\nbefore embarking upon its treatment in detail. I get the phrase \"test\nand explore,\" than which there could be no better, from the brief\npreface to the volume now before me, _The Ivory Tower_ (COLLINS). It\nexactly suggests the method of this preliminary study, doubly precious\nnow, both as supplying the key by which we can understand the fragment\nthat has been worked out, and as in itself giving us a glimpse,\nwonderfully fascinating, of its evolution. _The Ivory Tower_ (called\nso characteristically after an object whose bearing upon the intrigue\nis of the slightest) is a study of wealth in its effect upon the\nmutual relations of a small group of persons belonging to the\nplutocracy of pre-war America. Its special motive was to be a\ndevelopment of situation as between a young legatee, in whom the\nbusiness instinct is entirely wanting, and his friend and adviser,\nwhom he was presently to detect in dishonest dealing, yet refrain from\nany act of challenge that would mean exposure. \"Refrain\"--does this\nnot give you in one word the whole secret of what would have been a\nstudy in character and emotion obviously to the taste of the writer?\nFor itself, and still more for the glimpse of what it was to become,\n_The Ivory Tower_ must have a place in every collection where the\nunmatchable wit of HENRY JAMES is honoured as it should be.\n\n       *       *       *       *       *\n\nSomething less successful perhaps for itself, though even more\nabsorbing technically, is the volume containing the unfinished\nfragment of another HENRY JAMES novel, to be called _The Sense of the\nPast_ (COLLINS). Here especially it is the preliminary study that\nfurnishes the chief interest; the spectacle of this so-skilled\ncraftsman struggling to master an idea that might well, I think, have\nbeen found later too unsubstantial, too subtly fantastic, for working\nout. Very briefly, the theme is to treat of a young American, in whom\nthis \"Sense of the Past\" is all-powerful; whom the gift of an old\nLondon house and its furnishings enables to transport himself bodily\ninto the life of 1820. More than this, he lives that life (and it\nis here that one suspects the idea of becoming unmanageable) in the\nperson of an actual youth of that time, in whom a corresponding Sense\nof the Future has been so strong that he has answered the curiosity of\nhis descendant by an exchange of personalities. Of course the dangers\nand confusions of the plan, a kind of psychological version of one\noften used in farce (except that it precisely wasn't to be any manner\nof dream), are such as might well alarm any writer--and, one might\nadd, any reader also. It is a further misfortune that the style of\nwhat is actually written should be in the master's most remote and\nobscure manner, so much so that one is forced to wonder whether,\nwithout the notes as guide, it would be in any sort clear what the\nwhole thing was about. The transition, for example, from the actual to\nthe supernatural event is so abrupt that it might well have left the\nuninformed helplessly befogged. But this very fact again, as supposing\nsome further treatment only now to be guessed at, helps to make the\nunique fascination of the book as revealing the difficulties and\nrewards of letters.\n\n       *       *       *       *       *\n\nWhatever Mr. ERNEST THOMPSON SETON cares to write I am glad to read,\nbut there were moments in _The Preacher of Cedar Mountain_ (HODDER AND\nSTOUGHTON) when the great moral lesson of the story was as much as I\ncould bear. The tale reveals the spiritual and moral development of\n_Jim Hartigan_. The author assures us that most of the characters are\ndrawn from life, and that some of the main events are historical. All\nwhich I can easily believe, for Mr. SETON'S blunt method of describing\n_Jim Hartigan's_ evolution from an unhallowed stable-boy to a muscular\nChristian continually suggests reality. It is not a stylish method,\nbut it gets home, and in a tale of this kind that is the main, if not\nthe only, matter of importance. _Jim's_ besetting weaknesses were\ndrink and an overwhelming love for horses. The former he conquered\nfairly soon, but the latter tripped him up more than once, and if he\nhad not been guided by the wisest woman who ever came from the\nWest his end would have been chaotic. The races at Fort Ryan are\nexcellently described, and as a picture of the West of America some\nforty years ago you will find this story of _Jim's_ conversion both\ninstructive and intriguing. All the same Mr. SETON has so often\ndelighted me by his tales of the animal world that I hope this\nexcursion is merely a holiday from the work for which he has a real\ngenius.\n\n       *       *       *       *       *\n\n[Illustration: THE ABOVE GENTLEMAN IS SUPERSTITIOUS ON THE SUBJECT OF\nWALKING UNDER LADDERS.]\n\n       *       *       *       *       *\n\nUp to the present time the crop of German spy-stories has been\ndistinguished by quantity rather than by quality. Possibly the\nauthors, realising that the wildest flights of their highly-trained\nfancies could never match the actual machinations of the German Secret\nService as revealed in the official news, have not put their hearts\ninto the work. In _The Lost Naval Papers_ and other stories (MURRAY)\nMr. BENNET COPPLESTONE has shown unusual boldness in connecting the\nactivities of his super-policeman, _Dawson_, with the more prominent\nevents of the War. Indeed, I am not sure that the terror he professes\nto feel in the presence of the Scotland Yard official (for he tells\nhis stories _in propria persona_) is not to some extent justified.\n\"Dora\" is very sensitive and six months ago would never have permitted\nMr. COPPLESTONE to reveal to our enemies either the bumptious egoism\nof a nameless First Lord or the platitudinous vacillations of an\nanonymous Premier, even in the interests of popular fiction. Though\nwe concede his audacity in allowing his superlative sleuth to stop a\ngeneral strike of engineers by threatening them with martial law and\nto tempt the German fleet to come out by sending it false news of our\nbattleship strength, or to enable the battle of the Falkland Islands\nto be won by piling dummy battle cruisers up outside Plymouth harbour,\nthe merit of Mr. COPPLESTONE'S book does not lie in the complexity or\nvitality of his plots. It lies in a keen sense of humour and clever\ncharacter suggestion, and the recognition that the thing written about\nis of less importance than the manner of writing. We earnestly desire\nthat Mr. COPPLESTONE should devote another volume--a whole one--to the\ninimitable _Madame Guilbert_; but whatever he writes about will be\nwelcome, provided it be written in the vein of the volume before us.\n\n       *       *       *       *       *\n\nOut of such workaday elements as the hypnotic fascinations of a\nsleek music-master, the follies of a runaway schoolgirl and the\nwell-disciplined affections of a most superior young gentleman, Mr.\nW.E. NORRIS has contrived to create yet another new story, without\ninfringement of his own or anyone else's copyright. Thanks to the\nincidence of War and the author's skilful manipulation of Europe's\ndistresses (for once the KAISER'S intrusion into the middle of a\npeaceful--almost too peaceful--narrative is not unwelcome), the second\nhalf of _The Fond Fugitives_ (HUTCHINSON) is better than the first.\nNot, indeed, that such a wary hand as the writer has been so\nill-advised as to follow his hero to Flanders, or even to let his\nheroine do so; but his wounded soldier, come home with sympathy\nand understanding grown big enough to realise that a girl, though\nindiscreet once, may yet be adorable ever after, is certainly more\nto one's taste than the philanderer about town, admiring other men's\nwives, in July, 1914. And so the story, slight though it is, ends on a\nstrong note and with fair hope of happiness for two wiser and not much\nsadder people. Some of the minor characters are quite capitally drawn,\nparticularly the old father and mother in pathetic flight before the\nshadow of their daughter's disgrace; but it is the freshness of the\nheroine herself, outraging all tradition by refusing, though without\nbravado, to remain for ever in the gloom of a childish error, that one\nlikes to remember. Altogether, the author's friends will find this\nbook not at all below the level of his best work.\n\n       *       *       *       *       *\n\n_Small Craft_ (ELKIN MATTHEWS), by Miss C. FOX SMITH, contains several\npoems that have appeared in Punch over the initials \"C.F.S.\" They\nshould receive a fresh welcome from all who share her understanding\nof the ways of seafaring men, and from the larger public that is\nbeginning to appreciate the gallantry and devotion of our Merchant\nService.\n\n       *       *       *       *       *\n\nExtract from a letter in _The Saturday Review_:--\n\n    \"But posterity ought to share the burden, as it has always done in\n    the past.\"\n\nA tardy but complete answer to the old question, \"What has posterity\ndone to deserve our consideration?\"\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nGEORGE JOHNSON\n\nMiss Leavitt's  \nStars\n\nThe Untold Story of\n\nthe Woman Who Discovered\n\nHow to Measure the Universe\n\nATLAS BOOKS\n\nW.W.NORTON & COMPANY\n\nNEW YORK \u2022 LONDON\nDedication\n\nFor my mother, Dorris M. Johnson\nEpigraph\n\nHer columns grew longer, and if she squinted at them, the confetti of inklings began to resemble a skyful of stars. She had time to let her mind wander. The Magi's search for Bethlehem; the music of Milton's crystal spheres . . . they could all be reduced to these numbers. There was actually no need to squint and pretend that the digits were the stars. They were, by themselves, wildly alive, fact and symbol of the vast, cool distances in which one located the light of different worlds.\n\n\u2014THOMAS MALLON, _Two Moons_\n\nThen, by means of the instrument at hand, they travelled together from the earth to Uranus, and the mysterious outskirts of the solar system; from the solar system to a star in the Swan, the nearest fixed star in the northern sky; from the star in the Swan to remoter stars; thence to the remotest visible; till the ghastly chasm which they had bridged by a fragile line of sight was realized. . . .\n\n\u2014THOMAS HARDY, _Two on a Tower_\nContents\n\nCover\n\nTitle Page\n\nDedication\n\nEpigraph\n\nPreface\n\nPROLOGUEThe Village in the Canyon\n\nCHAPTER1Black Stars,White Nights\n\nCHAPTER2Hunting for Variables\n\nCHAPTER3Henrietta's Law\n\nCHAPTER4Triangles\n\nCHAPTER5Shapley's Ants\n\nCHAPTER6The Late, Great Milky Way\n\nCHAPTER7In the Realm of the Nebulae\n\nCHAPTER8The Mysterious K\n\nCHAPTER9The Cosmic Stampede\n\nCHAPTER10Ghost Stories\n\nEPILOGUEFire on the Mountain\n\nNotes\n\nSelected Bibliography\n\nAcknowledgments\n\nIndex\n\nCopyright\n\nOther Works\nPreface\n\nHenrietta Swan Leavitt deserves a proper biography. She will probably never get one, so faint is the trail she left behind. No personal diaries, no boxes of letters, no scientific memoirs\u2014she wasn't one to brag. She rates no more than a few paragraphs in the biographical reference books, the _Who's Who_ of this or that, a footnote or a boxed sidebar in the Astronomy 101 texts.\n\nI had intended to use her as nothing more than a device, a way to get into the story of how, in the 1920s, people learned that there is more to the universe than just the Milky Way. The discovery she made in her menial position at the Harvard Observatory was the turning point. I figured I would make that turn in the first chapter and then move on.\n\nBut Henrietta Leavitt refused to exit on cue from the story. I couldn't get this woman out of my mind. Why, given the remarkable, completely unexpected nature of her observation, didn't she push beyond it, working shoulder to shoulder with the great Harlow Shapley and Edwin Hubble, as they grabbed onto Henrietta's law and ran with it light-years across the sky?\n\nAnd so, just when I thought the book was almost done, I found myself going back to the beginning, searching her genealogy, scouring the records as they had apparently not been scoured before. Scrap by scrap, the apparition in the canned biographical sketches was replaced by a human being\u2014not solid enough, perhaps, to star in her own biography but someone with a story to tell.\nPROLOGUE\n\nThe Village in the Canyon\n\nThe village was hidden at the bottom of a deep chasm with sides so steep and slick that no one had ever climbed them. All that could be seen overhead was a narrow band of sky.\n\nLooking down the canyon, one could spot a hill in the distance. How distant no one knew, for it was separated from the village by an impassable expanse. Beyond the hill, and even more inaccessible, was a faraway mountain, the edge of the knowable world.\n\nThe villagers had noticed that when they walked the width of their canyon, from one wall to the other, the top of the hill appeared to shift ever so slightly against the backdrop of the mountain, moving from one side of the peak to the other. No one believed the hill really moved, but they enjoyed the illusion.\n\nOne day a particularly observant citizen noticed an interesting subtlety. When he wandered up the canyon a ways, and then traversed its width from wall to wall, the hilltop still appeared to move, but much more slowly. And if he walked even farther and repeated the experiment, the movement was smaller still. Venture far enough, he discovered, and there was no perceptible shift at all.\n\nHe made an entry in his notebook: The amount the hill moves depends on its distance from the observer. He had discovered the phenomenon called parallax.\n\nReturning to the village, where the hill was closest and the effect was most pronounced, he measured the separation between the canyon walls and began to sketch out a picture:\n\nThe width of the canyon and the line of sight from each canyon wall to the center of the hilltop formed an imaginary triangle. Using a surveying instrument, he could measure the two angles at the triangle's base. Then, using what he had learned in school about the rules of trigonometry, he calculated the height of the triangle\u2014the distance from the center of the baseline to the apex, from the village straight across the impassable badlands to the hilltop. By this measure, it was ten canyon widths away. It was still impossible to walk there, but it was comforting to know the distance. The world seemed a little tamer.\n\n_Triangulating a Mountain_\n\nIn coming years, the villagers began to build stone lookout towers, climbing high enough to observe that beyond the hill was another one. Using the same measuring technique, they showed that the second hill was fifteen canyon widths away. Behind it was a third hill at twenty-five canyon widths. But beyond that the method failed. The hills were so far that they couldn't be made to shift at all. The villagers' canyon was not wide enough, their triangles were too small.\n\nMost tantalizing of all was the dark immovable mountain on the horizon. It seemed to lie at infinity, a distance so great as to be immeasurable. A few villagers imagined launching an expedition, traversing the dangerous badlands between the village and the first hill and then forging beyond that, walking day after day until they had reached the mountain. But none was so foolish or so brave. The more imaginative citizens could envision what it might be like to rise up out of the canyon, where they would be free to move so far left and right that the mountain itself would shift against some even more distant backdrop. Then they could determine how long the journey would be. But that was just fantasy.\n\nIn theory, there was another way of indirectly determining the mountain's distance: things looked smaller the farther away they were. Something twice as distant would appear half as high. If this rule applied outside the immediate area of the village\u2014and why would it not?\u2014there would be a means of measuring to the mountain.\n\nOne very clear day, an observer standing on the highest tower decided to try this technique. She had noticed long ago that there was vegetation on the nearest hilltop\u2014appearing in the distance like miniature versions of the local flora, the spiny green bushes that covered the canyon floor. On this day she noticed that if she squinted her eyes, she could also make out, just barely, the tiniest fringe of green along the mountaintop, comforting evidence that this far-off land might not be unlike her own.\n\nIt took only a few moments for her to appreciate the larger implication. Working carefully, she measured the apparent height of the shrunken shrubs on the nearest hill. The even tinier fringe of green on the mountain appeared to be some ten times smaller. And so, it seemed, the mountain itself must be ten times farther than the hilltop\u2014a full one hundred canyon widths away.\n\nThat was far but not infinite. Walking there might take a week or two\u2014if someone could find a way to cross the impassable divide. Encouraged by her discovery, a party of volunteers set out for the mountain. From the highest towers the citizens watched as the explorers eventually found a way across the badlands, to the first hilltop and on past the second and the third, until in a few days they were out of sight.\n\nAfter a week had passed, the villagers went back to the towers to watch for the expedition's return. Two weeks later they looked again. Months went by, then a year before they stopped waiting.\n\nFinally one day, a single member of the party staggered back home. The vegetation on the mountaintop, he reported, was very different from anything in the canyon. Towering trees rose ten times higher than any plant he had ever seen. He had climbed to the top of one of these behemoths and thought he could barely make out the tiny twinkling lights back home. He knew by then that while the logic of the measurement had been sound, the villagers had been misled by the limits of their imagination. Since the strange trees were ten times higher than the familiar bushes, the mountain was another ten times farther than had been reckoned, a full one thousand canyon widths away....\n\n**2**\n\nIn Robert Heinlein's novel _Time for the Stars_ , a charitable organization called the Long Range Foundation recruits pairs of identical twins for a mission to colonize space. The Earth, buckling under an exploding population, has already spilled over to all the planets of the solar system. Now it must look farther, to planets orbiting stars so distant that news of their discovery, traveling at light-speed, would take years to reach home.\n\nThat is where the twins come in. Scientists, according to the story, have discovered that many twins are telepathic, and that the signals they mentally exchange are not weighed down by the restrictions imposed on electromagnetic waves. The communication is instantaneous\u2014faster than light\u2014no matter how far apart the twins may be. With one twin on a spaceship and the other on Earth, they can converse instantly across light-years.\n\nI read this story, now out of print, in junior high school and had forgotten all but the bare bones of the plot, which revolves around the relativistic effects of traveling near light-speed. Time slows down so Tom, the twin on the starship, ages only a fraction as quickly as Pat, his counterpart back home. He is an old man when Tom returns and marries Pat's granddaughter, with whom he has been flirting telepathically.\n\nWhat stuck in my mind all those years was not the rather obvious Einsteinian plot contrivances but a brilliant scene toward the end of the book: Tom is looking at the sky from a hospitable planet orbiting Tau Ceti, a star system eleven light-years from Earth. The constellations he sees are recognizable but slightly distorted from how they appear back home. He can make out the Big Dipper, \"looking a little more battered than it does from Earth,\" and he finds Orion, the great hunter, though his dog, Sirius, is stretched out of whack. The understated climax comes when Tom discovers, from where he now stands, that the constellation Bo\u00f6tes has acquired a new star, yellow-white and of about the second magnitude. It takes him a moment to realize that he is looking back at the sun.\n\nI don't know why that scene threw me for a loop. Maybe it should have been obvious even for an eighth grader how arbitrary the constellations are, not just the names taken from classical mythology (does anyone really think Ursa Major looks like a giant bear?) but the actual shapes. They are an accident of how the stars happen to line up from one among an infinity of possible points of view. Though Orion's dog appears to follow faithfully at his heels, the constellation's principal star, Sirius, 8.6 light-years from Earth, is nowhere near the hunter's belt, whose stars are approximately 1,500 light-years away.\n\nFor that matter Orion's belt is nowhere near his shoulders and the shoulders nowhere near his knees. Even the belt is an accident of perspective. From other parts of space these three stars would form a triangle or their order would be reversed. Viewed from just the proper angle, all three of them would merge into a single light, an illusory triple star. Travel to the center of the space the three stars enclose and each would appear in a far-flung corner of the sky.\n\n**3**\n\nFor anyone raised on science fiction or the enthusiastic promises of the Kennedy era, the space program has turned out to be a dud. Who would have guessed that several decades later, after a few trips to the moon (about 250,000 miles or fifty round trips between Los Angeles and New York), our species would abandon human space exploration altogether, our leaders contenting themselves with the ridiculous space shuttle, which ventures about as far from Earth as Baltimore is from New York? The feeble broadcasts of the unmanned space probes\u2014the oldest recently passed beyond the edge of the solar system\u2014stir the imagination. But for the most part people have been content to sit at home and wait for the cosmic news to arrive in the form of light from other stars.\n\nWe are celestial couch potatoes. Yet what we lack in exploratory will we make up for in other ways. We don't travel with our bodies. We travel with our minds.\n\nHaving never left home, the astronomers can say with some confidence that our own galaxy, the Milky Way, is more than 100,000 light-years from end to end and that Andromeda, the galaxy across the street, is 2 million light-years away. These and several other galaxies form our \"Local Group.\" The neighborhood. Just across town are other conglomerations of galaxies like the Sculptor group and the Maffei group, nearly 10 million light-years in distance. A little farther are the Virgo and Fornax clusters, lying some 50 million light-years from the Milky Way. Even if these were miles, the numbers would be staggering. A single light-year is almost 6 trillion miles.\n\nWe still haven't left our hometown \"supercluster,\" a galaxy of galaxies a full 200 million light-years across. Beyond it lie more superclusters, stretching to the edge of the visible universe, 10 billion or so light-years from home.\n\nFaced with so grandiose a vision, it is a little surprising to learn that as recently as the 1920s many astronomers thought the Milky Way _was_ the universe. Whether there was anything beyond it was a matter of scholarly debate. What are now taken to be vast galaxies similar to our own were dismissed as small nearby gas clouds, insignificant smudges of light.\n\nWe were like the villagers in the canyon. Then we discovered a new way to measure.\nCHAPTER 1\n\nBlack Stars, White Nights\n\nWe work from morn till night,\n\nComputing is our duty,\n\nWe're faithful and polite,\n\nAnd our record book's a beauty.\n\n_\u2014From_ The Observatory Pinafore\n\nIt is only with great difficulty that one can imagine what it was like to be a computer at Harvard Observatory a hundred years ago, not a soulless machine of wire and silicon but a living, breathing young woman. Her name was Henrietta Swan Leavitt and her job was counting stars.\n\nToday this kind of work is done by machine. Arrays of electronic sensors grab images of the sky, long streams of digits for computers to analyze. In the late 1880s, when Harvard embarked on a marathon project to catalog the position, brightness, and color of every star in the sky, the closest things to a modern digital computer were clunky mechanical calculators like the Felt & Tarrant Comptometer or the Burroughs Arithometer, with their rows of clacking buttons, stiff hand-pulled levers, and ringing bells. And there was the human brain. Diligent souls like Miss Leavitt\u2014they actually were called computers\u2014were paid 25 cents an hour (10 cents more than a cotton mill worker) to examine blizzards of tiny dots, photographs of the night sky. They would measure and calculate, recording their observations in a ledger book.\n\n_Observatory Hill, 1851_ (Harvard College Observatory)\n\nImagine a sky with the colors reversed, cold black stars sprayed against a firmament of white. These photographic negatives were produced when a telescope was trained at the heavens, its light focused onto a large glass plate coated on one side with light-sensitive emulsion\u2014a forerunner to photographic film. Today, half a million of these fragile plates are stored in a brick building adjacent to the one where Miss Leavitt and the other computers worked. Fearing an earthquake might shatter this database of glass\u2014the astronomical equivalent of the burning of the library of Alexandria\u2014Harvard built the repository as two nested structures. Physically isolated from the building's exterior shell, an internal matrix of steel beams and flooring rests, the story goes, on an apparatus of leaf springs, like those in a wagon or an old pickup truck.\n\nThe result is an invaluable archive of how the sky looked on different nights since the first surveys were done in the 1880s. Among the most precious items in the collection are pictures of the Magellanic Clouds. We know them now as neighboring galaxies, companions to our Milky Way. Back then no one was quite sure what they were. Hunched over the plates in an observatory workroom, Miss Leavitt found the pattern that eventually led to the answer. She discovered a way to measure beyond the galaxy and begin mapping the universe.\n\nToday nearly every scientifically literate person knows, or thinks he knows, that our planet circles an unremarkable star lost among galaxies of galaxies extending billions of light-years in every direction. One can almost hear Carl Sagan intoning the words on public TV. We've learned to revel in our insignificance. As far as most astronomers are concerned, only small details are still in dispute: Is the universe 13.9 billion light-years in radius or just 13.8 billion? So much confidence exudes from these discussions that a spectator may forget to ponder the most basic question: How can anyone know for sure?\n\nSuppose two stars seemingly of equal brightness are shining side by side against the dark dome of night. Knowing nothing else about them, one might conclude that they are equally far away. But that would be true only if the stars happened to be emitting, from their nuclear furnaces, the same amount of light. More likely one star is more powerful than the other yet shining from farther away. How much more powerful and how much farther? Barring a breakthrough in interstellar space travel, there seemed to be no way to find out.\n\nThe same uncertainty applied to the faint hazes called nebulae, clouds of light. Were they sprawling galaxies, \"island universes\" shrunken by their great distance? Or were they little gas clouds right here in the Milky Way? With no means of measuring the universe, the question was almost theological. How many angels can dance on a pinhead? How far away are the stars in the sky?\n\nTODAY A VISITOR to Observatory Hill, a low rise on Garden Street about a fifteen-minute walk northwest of Harvard Square, looks in vain for a sign that anything cosmic happened there. Dwarfed by the giant mountaintop observatories at Palomar, Wilson, Cerro Tololo, and Mauna Kea and blinded by the light of the Boston glare, the Harvard telescope, called the Great Refractor, is now in retirement. But when it saw first light in 1847 it was one of the most powerful in the world.\n\nIt had arrived, some liked to say, on the tail of a comet\u2014the March Comet of 1843, which burned so bright that it was visible in broad daylight, a signal to some that the Day of Reckoning was at hand. (A group called the Millerites had used biblical passages to predict that Jesus would make his Second Coming sometime between March 21, 1843, and March 21, 1844. The comet was right on schedule.) Those with a scientific bent felt a deeper kind of awe. Where did the comet come from and when might it return? For answers they could turn to the observatories at Cincinnati or at Yale or Williams colleges. But Harvard didn't have a good enough telescope to properly study the phenomenon. Even the Philadelphia High School was better equipped.\n\nIt was an embarrassment Bostonians vowed to correct. Twelve acres called Summer House Hill had recently been purchased by Harvard as an eventual site for a large telescope, but little progress had been made. Now the project began in earnest. Well-to-do citizens took out \"subscriptions,\" some $25,000 worth, to build the best observatory in the world. To ensure as stable a platform as possible, the building was constructed around a massive granite pier rooted 26 feet into the ground and rising from the bedrock to the observatory floor. Beneath a 30-foot dome was placed the Great Refractor. The mahogany-veneer tube was outfitted with a lens, 15 inches across, that had been ground by master craftsmen Merz and Mahler of Munich, who were told to make it at least as powerful as the one the Russians had recently purchased for the Imperial Observatory. The space race had begun.\n\n_The Great Refractor_ (Harvard College Observatory)\n\nThe first astronomer to peer through the glass was rendered nearly speechless: \"It is delightful,\" he wrote, \"to see the stars brought out which have been hid in mysterious light from the human eye, since the creation. There is grandeur, an almost overpowering sublimity in the scene that no language can fully express.\"\n\nWith this new tool, astronomers quickly discovered the inner ring of Saturn and, outfitting the telescope with a photographic plate, took the first picture of a star.\n\n**2**\n\nOn a clear night high in the mountains when the air is cold and dry, the brightest stars shine some two hundred and fifty times brighter than the faintest ones, those that can barely be discerned with the naked eye. The ancient Greeks divided this stellar multitude into six categories. The brightest lights were said to be of the first magnitude while the dimmest were of the sixth.\n\nThis rough gauge has been refined over the centuries so that each step now means an increase in brightness of about two and a half times. The actual figure is closer to 2.512, conveniently making a fifth magnitude star 2.512 \u00d7 2.512 \u00d7 2.512 \u00d7 2.512 \u00d7 2.512 or 100 times dimmer than a star of the first magnitude. A sixth magnitude star is about 250 times dimmer than that, and a seventh magnitude star about 600 times dimmer. (In the original, rather roughshod system, all the brightest stars had been bunched into magnitude one. Measured more precisely, some of them have ended up with magnitudes of less than one, and the brightest with magnitudes of less than 0. Blazing Sirius is about \u20131.4.)\n\nCenturies ago, with his simple spyglass, Galileo had amplified his vision enough to see stars as faint as the eighth magnitude. The Great Refractor extended the reach to the fourteenth magnitude, resolving images some 400,000 times dimmer than what could be seen from Earth with 20-20 eyes.\n\nWith the ability to see farther than ever before, Harvard embarked, in the late 1870s, on the kind of exhaustive search that would become its hallmark, precisely cataloging the brightness of every star in the sky. The observatory was now being run by a young physicist named Edward Charles Pickering, who had made his mark as a professor at the Massachusetts Institute of Technology by establishing the first curriculum in the country where students could confront the ideas of physics head-on\u2014in laboratory experiments, poking at nature and carefully recording the results. He got an early taste of astronomy when he served on two government expeditions to observe total eclipses of the sun. When he was hired in 1876 to take over the observatory, he was thirty years old.\n\nUntil this time astronomy had focused on trying to establish two primary details about every star: its position and its motion through space. Pickering was struck by how very little good data had been gathered on two equally important characteristics: a star's precise brightness, a clue to its distance, and color, a clue to what chemicals it contained. Pickering was a fastidious measurer. He occupied himself on hikes in New Hampshire's White Mountains by measuring out the terrain, using an instrument he had fabricated himself. His mission, he decided, and that of the observatory, would be to amass mountains of data, about which others could theorize.\n\nGood old-fashioned astronomy is what he wanted. No big bangs, no black holes, no dark matter\u2014this was way before all that. Space was still flat and of no more than three dimensions. Understanding the universe meant charting little lights as they moved across the sky.\n\nHe started with stellar brightness. In the past, astronomers had made some progress along these lines with a German instrument, the Z\u00f6llner astrophotometer, which compared a star's brightness to the glow of a kerosene lamp. Focused through a pinhole and reflected by a mirror into the visual field of a telescope, the dot of lamplight appeared as a tiny sun hovering beside a star. The observer adjusted the instrument, stepping down the brightness of the artificial star until, in his judgment, it matched its companion. Then its magnitude could be recorded. (Some of these demanding measurements had been carried out by an observatory employee named Charles Sanders Peirce, who came to be known as one of the most brilliant and eccentric philosophers of all time.)\n\nPickering felt that a definitive survey should rely on a standard more universal than the brightness of lamplight. He devised an instrument with an arrangement of lenses and mirrors that would allow any star within view to be lined up side by side for comparison with the North Star, which was set, somewhat arbitrarily, at magnitude 2.1. Once astronomers learned how to use the new device, they could knock off as many as one star a minute. Eventually Harvard measured and cataloged forty-five thousand of them.\n\nThat was barely a beginning. Within the gaps between stars there were surely many more, so dim and far that they did not register on the retina of an eye, even one fitted with such powerful lenses. To see farther, light from these faint sources would have to be gathered during a time exposure, accumulated on a photographic plate attached to the end of a telescope. Mounted on a rotating platform and driven by mechanical clockworks, the telescope could track a star as it arced across the sky, pooling its light photon by photon, chemically etching an impression.\n\nThe astronomical leverage this provided was stunning. From Earth the Pleiades appear as a subtle glow engulfing seven bright points of light\u2014the \"Seven Sisters\" of Greek mythology, pursued by Orion. Galileo had already seen through his telescope that the sisters were joined by dozens more. A three-hour time exposure, taken in Paris, revealed that the constellation included more than 1,400 stars.\n\nMore stars still could be unveiled by mounting telescope and camera as far above sea level as possible, cutting through miles of atmospheric distortion. After unsuccessful attempts to establish stations at Pikes Peak in the Colorado Rockies and Mount Wilson in Southern California, Pickering decided to try the high reaches of Peru. He dispatched an expedition led by a trusted colleague, Solon I. Bailey, who established a temporary post atop a peak that, it was decided, would now be called Mount Harvard. Bailey hadn't reckoned on the length of the annual rainy season and was forced by the clouds to find a clearer site, finally settling in a remote town called Arequipa. This time the location seemed perfect. Pickering arranged to have an observing station sent piece by piece, from Boston Harbor around the tip of South America. Included among the cargo were the components of the 24-inch Bruce Telescope (named for the heiress who paid for its construction, Catherine Wolfe Bruce).\n\nFor all Pickering's hopes, the project got off to a bad start. His brother William, as headstrong and arrogant as Edward was modest and reserved, was placed in charge of Arequipa, mismanaging the operation and scandalizing the world astronomical community when he began dispatching outlandish scientific reports to that great academic journal the _New York Herald_. Ignoring his assignment to study the stars, he trained the telescope on Mars, enthusiastically describing huge mountain ranges rising above giant rivers and lakes extending hundreds of square miles\u2014a geography that remained stubbornly invisible to any eyes but his own.\n\nWhile Edward Pickering concentrated on damage control at home, Bailey was sent back to Peru to retake the observatory. Before long, the Arequipa station was shipping crate after crate of photographic plates north to Cambridge, the first pieces of what would become a mosaic of the entire southern sky.\n\nWith so much new information to digest, astronomers were soon overwhelmed. They were faced with an embarrassment of riches now familiar throughout science, a burgeoning glut of undigested data begging to be categorized. That is where the computers came in.\n\n**3**\n\n\"A great observatory should be as carefully organized and administered as a railroad,\" Pickering once observed. \"Every expenditure should be watched, every real improvement introduced, advice from experts welcomed, and if good, followed, and every care taken to secure the greatest possible output for every dollar expenditure. A great savings may be effectuated by employing unskilled and therefore inexpensive labor, of course under careful supervision.\"\n\n_Edward Pickering_ (Harvard  \nUniversity Portrait Collection)\n\nImagine trying to find people to do such precise work for 25 cents an hour\u2014what amounted to the minimum wage. Today the job would probably have to be farmed out to star-counting sweatshops in Asia. For the late nineteenth century, computing wasn't such a bad deal. Seven hours a day, six days a week, the job paid $10.50 a week and included a month's vacation. Not many men were interested in the tedious work, so the positions went mostly to women. (The tradition was a long time in passing. As recently as the early 1960s, Brookhaven National Laboratory hired Long Island housewives to pore over the tangled images of subatomic particles, looking for patterns that might foretell a new physics.)\n\nRecognizing that his housekeeper, Williamina Paton Fleming, was overqualified for mopping floors, Pickering hired her as one of his first computers. (Abandoned by her husband after emigrating from Scotland, she was grateful enough to name her son, born that year, Edward Pickering Fleming.) She eventually became curator of the collection of photographic plates, doubling her salary, and was in charge of classifying stars according to their spectra, the colors revealed when their light was refracted through a prism. This was another of the observatory's ambitious efforts, resulting in a monumental work called the Henry Draper Catalogue, named for an accomplished and wealthy amateur astronomer who had taken the first photograph of a nebula. Funded by Draper's widow, the compendium provided employment for two other computers, Annie Jump Cannon and Antonia Caetana Maury.\n\nThis women's work sometimes resembled bookkeeping more than scientific research. Pickering tried to make it reasonably stimulating and treated his computers with respect. But he was intent on the observatory's getting its money's worth.\n\n\"He seems to think that no work is too much or too hard for me no matter what the responsibility or how long the hours,\" Mrs. Fleming complained in a diary. \"But let me raise the question of salary and I am immediately told that I receive an excellent salary as women's salaries stand.\"\n\nIf he would only take some step to find out how much he is mistaken in regard to this he would learn a few facts that would open his eyes and set him thinking. Sometimes I feel tempted to give up and let him try some one else, or some of the men to do my work, in order to have him find out what he is getting for $1,500 a year from me, compared with $2,500 from some of the other assistants.\n\nDoes he ever think that I have a home to keep and a family to take care of as well as the men? But I suppose a woman has no claim to such comforts. And this is considered an enlightened age!... I feel almost on the verge of breaking down.\n\nWhen she asked him for a raise, Pickering agreed to pass on the request to the university president. Money was always tight. This was before the era of government-funded big science, and observatories were dependent on the charity of rich benefactors, and on people with a monastic dedication to the craft. Pickering worked as hard as any of them, administering by day, stargazing by night. When the sky was cloudy he would do calculations late into the evening, sometimes with an assistant reading to him for entertainment (Shakespeare was a favorite). Considering the long hours, his $3,400 annual salary came to much less than two dollars an hour. (He and his family also got to live in the less-than-luxurious director's residence on Observatory Hill.) No one was in this for the money.\n\n\"An astronomer is a sorry soul,\" began a chorus in _The Observatory Pinafore_, a parody of Gilbert and Sullivan's comic operetta _H.M.S. Pinafore_ written by one of Pickering's assistants.\n\n_He must open the dome and turn the wheel,_\n\n_And watch the stars with untiring zeal,_\n\n_He must toil at night though cold it be,_\n\n_And he never should expect a decent salaree._\n\nMost of the time, the computers seemed to enjoy their jobs, making light of the low pay and somewhat Dickensian working conditions. In _The Observatory Pinafore_ , one of them, \"Josephine,\" sings of her toil in the \"dark and dingy place, all cluttered up and smelling strong of oil,\" apparently from the furnace that had recently replaced fireplaces for taking the edge off the New England cold. At another point in the story, a whole chorus of computers breaks out in song:\n\n_The Observatory Pinafore_ (Harvard College Observatory)\n\n_We work from morn till night,_\n\n_Computing is our duty,_\n\n_We're faithful and polite,_\n\n_And our record book's a beauty._\n\nIt is tempting to imagine Henrietta Swan Leavitt joining in the song. But that couldn't be. Though written in 1879 the musical was not actually performed until New Year's Eve 1929. By then she had already died.\nCHAPTER 2\n\nHunting for Variables\n\nMy friends say, and I recognize the truth of it, that my hearing is not nearly as good when absorbed in astronomical work.\n\n_\u2014Henrietta Leavitt in a letter to Edward Pickering_\n\nAlthough unmarked by a plaque, the second-floor room where Miss Leavitt and the other computers probably worked is still intact. The university, always pressed for space, hasn't been as diligent about preserving the old observatory buildings as it has the collection of photographic plates. The wooden ceiling beams have been painted over in institutional white enamel. Fluorescent lights have been retrofitted where chandeliers once hung and an air conditioner mounted in one of the old sash windows. The place has all the charm of a room in a state hospital. Nearby a dumbwaiter, which shuttled the glass plates up from below, now holds a computer printer. A closet is crammed full of abandoned IBM Selectric typewriters, another archeological layer of cast-off technologies.\n\nHaul off the junk, restore the late-nineteenth-century decor, and imagine Miss Leavitt, as she would have expected to be called, in a long frilled dress buttoned to the neck, her dark hair pulled tightly into a bun (we are extrapolating here from one of the only existing photographs). She is sitting at a table before a wooden viewing frame that supports a large glass plate\u2014one of those black-on-white reversals of the night sky. At the base of the frame is a mirror, reflecting light in from a nearby window to illuminate the image from behind. Around her sit other computers, similarly occupied, and occasionally Edward Pickering drops by to see how the calculations are going.\n\n_Henrietta Swann Leavitt_ (Harvard College Observatory)\n\nShe was twenty-five when she arrived at the observatory in 1893 as a volunteer. Her goal was to learn astronomy, and apparently she was of somewhat independent means. The daughter of a Congregationalist minister, George Roswell Leavitt, Henrietta had been born on the Fourth of July 1868 in Lancaster, Massachusetts, to what once was called \"good Puritan stock.\" Her ancestry can be traced back to Josiah Leavitt of Plymouth, and to four centuries of Leavitts in Yorkshire, England.\n\nAt the time of the 1880 census the family was living in one half of a large double house, 9 Warland Street in Cambridge, near Pilgrim Congregational Church, at the corner of Magazine and Cottage streets. Reverend Leavitt served as pastor. The neighborhood was solidly middle to upper-middle class. The Leavitts' neighbors included a piano tuner, a clerk, a police captain, a grammar school teacher, and a civil engineer, as well as a soda and mineral water manufacturer, a carriage manufacturer, and the owner of a plumbing company.\n\nWhen he dropped in to take the census, the enumerator found Mrs. Leavitt, also named Henrietta Swan (her maiden name was Kendrick), tending to three of her children, George, Caroline, and two-year-old Mira. (It was the last year of her life. A few months later Mira was dead.) Eleven-year-old Henrietta, the eldest, and another sister, Martha, were away at school. Another brother, Roswell, had died in 1873, when he was fifteen months old; the youngest, Darwin, would be born two years later.\n\nThe household must have felt crowded. Henrietta's aunt, Mary Kendrick, also lived there, as did a servant girl. Next door, in No. 11, her grandfather, Erasmus Darwin Leavitt, lived with his wife and a thirty-year-old daughter (the census taker noted that she had a spinal injury). They too employed a live-in servant.\n\nThe family valued scholarship. Henrietta's father had graduated from Williams College and earned a doctorate in divinity from the Andover Theological Seminary. Like her grandfather, an uncle (Reverend Leavitt's brother) was named Erasmus Darwin, perhaps after the renowned eighteenth-century physician and naturalist, the grandfather of Charles Darwin. The younger Erasmus, the second president of the American Society of Mechanical Engineers, would gain national prominence for his design of the Leavitt pumping engine at the Boston Water Works' Chestnut Hill station. He was also a fellow of the American Academy of Arts and Sciences.\n\nA few years later the Leavitts moved to Cleveland, and in 1885 Henrietta enrolled at Oberlin College, taking a preparatory course followed by two years of undergraduate work. Returning to Cambridge in 1888, she entered Radcliffe, then called the Society for the Collegiate Instruction of Women. (One of her cousins, a daughter of Erasmus Leavitt, was in the same freshman class.)\n\nThe entrance requirements were strict. Every young woman was expected to prove her familiarity with a list of classics\u2014 Shakespeare's _Julius Caesar_ and _As You Like It_ ; Samuel Johnson's _Lives of the Poets_ , William Makepeace Thackeray's _English Humorists_ ; Swift's _Gulliver's Travels_ ; Thomas Gray's poem \"Elegy Written in a Country Churchyard\"; Jane Austen's _Pride and Prejudice_ ; and Sir Walter Scott's _Rob Roy_ and \"Marmion\"\u2014 by writing, on the spot, a short composition. There were also tests on language (\"Translation on sight of simple [in the case of Latin, Greek, and German] or ordinary [in the case of French] prose\"), on history (\"Either [1] History of Greece and Rome; or [2] History of the U.S. and of England\"), mathematics (algebra through quadratic equations and plane geometry), and physics and astronomy. In addition to these examinations on \"elementary studies,\" students had to demonstrate more advanced knowledge in two subjects\u2014mathematics, for example, and Greek. The catalog noted reassuringly, \"A candidate may be admitted in spite of deficiencies in some of these studies; but such deficiencies must be made up during her course.\"\n\nThe only deficiency listed on Henrietta's transcript was in history, which she had corrected by her junior year. Along the way she also took courses in Latin, Greek, the humanities, English, and modern European languages\u2014German (her only C), French, and Italian\u2014and in fine arts and philosophy. She didn't take, nor was offered, much science: just natural history, an introductory physics class (she got a B) and a course in analytic geometry and differential calculus (an A). It was only in her fourth year that she enrolled in astronomy, earning an A\u2013. Observatory Hill is just up Garden Street from Radcliffe, and some of the Harvard astronomers, supervised by Edward Pickering, taught classes there.\n\nIn 1892, shortly before her twenty-fourth birthday, Henrietta graduated with a certificate stating that she had completed a curriculum equivalent to what, had she been a man, would have earned her a bachelor of arts degree from Harvard. She stayed in Cambridge and the next year was spending her days at the observatory, earning graduate credits and working for free. Maybe her uncle Erasmus had pulled some strings for her. She stayed there two years.\n\nNo diary has been found recording what it was about the stars that moved her. One of history's small players, her story has been allowed to slip through the cracks. She never married and died young, and it is only upon her death that we find, in an obituary written by her senior colleague Solon Bailey, testimony to what she might have been like as a woman:\n\nMiss Leavitt inherited, in a somewhat chastened form, the stern virtues of her puritan ancestors. She took life seriously. Her sense of duty, justice and loyalty was strong. For light amusements she appeared to care little. She was a devoted member of her intimate family circle, unselfishly considerate in her friendships, steadfastly loyal to her principles, and deeply conscientious and sincere in her attachment to her religion and church. She had the happy faculty of appreciating all that was worthy and lovable in others, and was possessed of a nature so full of sunshine that, to her, all of life became beautiful and full of meaning.\n\nAlthough the obituary didn't say so, she was also deaf, although apparently not from birth. In her second year at Oberlin she had enrolled as a student in its conservatory of music. For her new calling, eyes were more important than ears, and perhaps deafness was an occupational advantage in a job requiring such intense powers of concentration.\n\n**2**\n\nPickering, always happy to have a motivated volunteer, put her to work recording the magnitude of stars, a craft called stellar photometry. During a time exposure, brighter stars leave larger spots on a photographic plate, chemically darkening more grains in the emulsion. Size therefore is an indicator of brightness. Looking through an eyepiece, Miss Leavitt would compare each pinpoint against stars whose magnitudes were already known. Sometimes this information was displayed on a small palette marked with spots arranged and labeled according to magnitude. Shaped like a miniature flyswatter, the tool was called a \"fly spanker.\" When she was satisfied that she had gauged the brightness correctly, she would record the information in neat tiny writing on a sequentially numbered pink-and-blue-striped page in a ledger book initialed \"HSL.\"\n\nEarly on she was asked to look for \"variables,\" stars that waxed and waned in brightness like slow-motion beacons. (A few of the more interesting were to be found in a constellation she might have considered her namesake, Cygnus, the Swan.) Some of the variables completed a cycle every few days, others took weeks or months. Her job was not to speculate why. For a while it was thought that each variable actually consisted of two stars orbiting about a common point, periodically eclipsing each other's light. More recent evidence indicated that the temperature (judged by the color of the starlight) rose and fell along with the brightness. That suggested that these variables were probably single stars that periodically flared and dimmed. The reasons for the pulsation remained obscure. This was before anyone knew what powered the stars, much less why the flame might not burn steadily.\n\nIn any case, the rhythms were imperceptibly slow and subtle. (Astronomers were surprised to learn later that Polaris, the North Star, which they had been using as their touchstone for measuring brightness, regularly varied in magnitude.) Only by measuring stars at various intervals through the year could one detect the variations.\n\nPhotography made that possible. With dense swarms of stars on every plate, it was impossible to check each one. To find likely prospects, a researcher would take two plates of the same region exposed at different times and line them up, sandwich style, one on top of the other. One image was the standard black-star negative produced by the camera. The other was a positive. Align them just so and they would cancel each other\u2014 except for the stars that had changed in brightness. They would look subtly different. A black dot surrounded by a white halo would mean, for example, that a star had brightened, its image on the plate expanding in size. If a star seemed particularly promising, more plates would be compared. (If necessary Pickering would order new ones from the astronomers.) Plate by plate, the computers would measure a dot as it swelled and receded, writing tiny numbers on the glass in india ink.\n\nHenrietta Leavitt spent day after day doing this painstaking work, absorbing herself in the data with what one colleague later called \"an almost religious zeal.\" She wrote up a draft of her findings, then sailed for Europe in 1896, where she traveled for two years. We don't know where or with whom.\n\nShe hadn't forgotten about astronomy. When she landed back in Boston, she conferred briefly with Pickering, who suggested some revisions to her work. Then, taking the manuscript with her, she left for Beloit, Wisconsin, where her father was now the minister of another church. Because of personal problems, never really explained, she remained there more than two years, working as an art assistant at Beloit College. She must have found the work unsatisfying, for on May 13, 1902, she wrote to Pickering, apologizing for letting her research languish and for being out of touch for so long. She hoped he would let her resume her projects from Wisconsin.\n\n\"The winter after my return,\" she explained, \"was occupied with unexpected cares. When, at last, I had leisure to take up the work, my eyes troubled me so seriously as to prevent my using them so closely.\" Her eyes were strong now, she assured him, and her interest in astronomy had not waned. She asked whether he might send her the notebooks she needed to complete her manuscript. \"I am more sorry than I can tell you that the work I undertook with such delight, and carried to a certain point, with such keen pleasure, should be left uncompleted. I apologize most sincerely for not writing concerning the matter long ago.\"\n\nShe also mentioned having some trouble with her hearing, worrying, a little oddly, that stargazing might make it worse. \"My friends say, and I recognize the truth of it, that my hearing is not nearly as good when absorbed in astronomical work.\" Cold weather seemed to aggravate her condition. \"It is evident that I cannot teach astronomy in any school or college where I should have to be out with classes on cold winter nights. My aurist forbids any such exposure.\n\n\"Do you think it likely that I could find employment either in an observatory or in a school where there is a mild winter climate? Is there anyone beside yourself to whom I might apply?\"\n\nIt seems from the letter that her hearing problems then were fairly mild\u2014the first ailment she mentions is with her eyes. Yet one of the standard reference books, _Dictionary of Scientific Biography_ , states that as early as Radcliffe she was extremely deaf\u2014a misapprehension that has been picked up and recycled again and again.\n\nShe must have been delighted with Pickering's response. Three days later, he offered her a full-time job. \"For this I should be willing to pay thirty cents an hour in view of the quality of your work, although our usual price, in such cases, is twenty five cents an hour,\" he wrote. If it was not possible for her to relocate, he would pay her fare for a short visit to Cambridge. She could get her work in order to take home to Beloit.\n\n\"I do not know of any observatory in a warm climate, where you could be employed on similar work,\" he continued, \"and it would be difficult to furnish you with a large amount of work that you could carry on elsewhere.\" In any case, he noted, \"I should doubt if Astronomy had anything to do with the condition of your hearing, unless you have been assured that this is the case by a good aurist.\"\n\nShe gratefully accepted his proposal for a working visit to Observatory Hill. \"My dear Prof. Pickering,\" she replied, a few days later. \"It has proved possible for me to arrange my affairs here so that I can go to Cambridge next month and remain until the work is completed. Your very liberal offer of thirty cents an hour will enable me to do this.\" She planned to arrive around the time of Harvard commencement and to take up her job before the first of July.\n\nEn route to the observatory, she stopped in Ohio to visit relatives, encountering another of the family problems that seemed to punctuate her life. \"The illness of a relative with whom I stopped to visit on my way to Cambridge is likely to detain me for some time,\" she wrote to Pickering, adding that she might be waylaid for several weeks. \"I am sorry for the delay, which seems to be unavoidable, but my face is set toward Cambridge and I hope it may not be very long before I can report at the Observatory.\"\n\nFinally, on August 25, she contacted him again from an address in Brookline, Massachusetts, where she was temporarily staying, announcing her arrival.\n\n\"It has been a disappointment to me that I have had to defer the beginning of my work for so long,\" she wrote. \"At last I find myself free to take it up, and expect to go to the Observatory Wednesday afternoon, arriving there between half-past two and three. If there is a time which will suit you better, will you please let me know either by letter or by telephone?\" At first she could work only about four hours a day, but was hopeful that she would soon be able to give him her full attention. \"I hope that this long delay has in no way inconvenienced you.\" Whether Pickering was beginning to find these plaints a bit wearing has gone unrecorded. In the coming years there would be many more.\n\nWorking through the fall semester, she took winter break again in Europe. A letter dated January 3, 1903, finds her aboard the S.S. _Commonwealth_ , a mail steamer for the Dominion Line, writing to Williamina Fleming about a misdirected paycheck. (Henrietta's younger brother, George, who was living in Cambridge with Uncle Erasmus, now a widower, helped straighten out the problem.) A year earlier the shipping line had initiated winter Boston-to-Mediterranean service; so perhaps she was off with a friend to Italy, or the South of France. It is nice to imagine her on deck at night, bundled up to keep her aurist happy, looking up at the stars.\nCHAPTER 3\n\nHenrietta's Law\n\nWhat a variable-star \"fiend\" Miss Leavitt is\u2014One can't keep up with the roll of the new discoveries.\n\n_\u2014A colleague in a letter to Edward Pickering_\n\nDuring the first circumnavigation of the Earth, Ferdinand Magellan's crew relied on \"certain shining white clouds\" to find its way. There is no South Star to navigate by down under but the Nubecula Major and Nubecula Minor\u2014the big and little clouds, as Magellan called them\u2014 helped one maintain a steady course. Invisible in most of the Northern Hemisphere, these impressive formations came into view as the fleet reached the latitude of what is now Brazil.\n\nMagnified by telescopes, the pair of luminous hazes dazzled the mind. \"In no other portion of the heavens are so many nebulous and stellar masses thronged together in an equally small space,\" wrote the astronomer John Herschel after observing them from the Cape of Good Hope in 1834. It was as though two pieces of the Milky Way had broken off and drifted. But that begged the question: Were these separate and distant galaxies or something smaller and nearby, suburbs of the Milky Way?\n\nThere was no reason to hope that the answer would be found in the images of the Magellanic Clouds that had been photographed at Harvard's station in Arequipa, Peru. Miss Leavitt was charged with nothing more than examining the plates for variable stars. Each time she spotted one, she would scrutinize it with a magnifying eyepiece, determine its coordinates on the photographic plate, and carefully compute its change in magnitude by comparing it with other stars.\n\n_The Large Magellanic Cloud_ (Harvard College Observatory)\n\nWhile she worked she might have remembered a story her father once told about Herschel himself. Speaking at an annual meeting of the American Missionary Association, Reverend Leavitt described how astronomers throughout Europe had greeted their colleague's great discoveries with disbelief: \"Men said to him, in angry letters, 'We do not see what you see.' \" Herschel, Reverend Leavitt continued, was ready with a response: \"Perhaps you do not take the care in your observations that I do.... [W]hen I observe on a winter night I place my glass on the lawn at Greenwich, and let it stand there until the instrument comes to be of the temperature of the air.' \" Moreover, the reverend said, Herschel ensured that his own body temperature would not affect the observations. \" 'Oftentimes,' he said, 'I have been out in the winter air for two hours before I would open my glass, because I must come to be of the same temperature as the instrument itself.' \"\n\nReverend Leavitt found a rather oblique spiritual message in the story, something about how a preacher must be of the same spiritual \"temperature\" as his \"instrument\" (God's word), and of the Bible and the heavens as well. Herschel's words might serve better as advice to a young astronomer: for all the excitement of discovering a new star or nebula, a good scientist was ultimately one who took care to consider the minutest of details, weighing each tiny component that made up an observation.\n\nHenrietta apparently found such meticulous work satisfying enough that she amended her original plan to take her work back to Wisconsin. After a voyage to the British Isles in the summer of 1903 on the H.M.S. _Ivernia_ , she took a quick train trip to Beloit to prepare for her relocation to Cambridge as a permanent member of the observatory staff.\n\nHer decision paid off. One spring day in 1904 she was comparing plates of the Small Magellanic Cloud, taken at different times, when she noticed in the stellar spray several dots that had swelled and then receded in size. Variables. Her interest piqued, she examined other images, finding dozens more.\n\nThat fall sixteen more plates of this nebula were served up by the astronomers at Arequipa and shipped north to Boston, arriving at the observatory in January. When Miss Leavitt began scrutinizing these new photographs, variables popped up one after another\u2014\"an extraordinary number,\" she later wrote. The results, published in the observatory's regular circulars, made an immediate impression.\n\n\"What a variable-star 'fiend' Miss Leavitt is,\" a Princeton astronomer wrote to Pickering. \"One can't keep up with the roll of the new discoveries.\" Even the newspapers took notice. A column of flippant news briefs in the _Washington Post_ noted, tongue in cheek: \"Henrietta S. Leavitt, of the Harvard Observatory, has discovered twenty-five new variable stars. Her record almost equals Frohman's.\" (Charles Frohman was a powerful theatrical producer and booking agent.)\n\nDay after day, she quantified the specks of pulsing starlight, filling in column after column of numbers. If there was anything noteworthy or unusual about a variable she would add a comment. The star with Harvard Number 1354 was \"the northern star of a close pair, in a group of five.\" Number 1391 was \"the southern star in a line of three.\" Number 1509 \"appears to be at the centre of an extremely small, faint cluster.\"\n\nEach star was an individual. Before long, she had discovered and cataloged hundreds of them in the two Magellanic Clouds, some of which flared no brighter than the fifteenth magnitude\u2014thousands of times dimmer than the faintest stars she might have seen on a particularly clear night in the New England countryside.\n\nShe was boarding with Uncle Erasmus in a large Italianate villa (now part of the Longy School of Music) recently built for him on Garden Street. The house was just a short walk from Observatory Hill, where, for the next few years, she continued her research. Piece by piece, her results appeared in brief progress reports sometimes given to Pickering or Bailey to read in her absence at the December meetings of the Astronomical and Astrophysical Society of America, when she would be home in Beloit for Christmas. By 1908, six years after she had resumed her work,she published a full account,\"1777 Variables in the Magellanic Clouds,\" in the _Annals of the Astronomical Observatory of Harvard College_. Twenty-one pages in length, the paper included two plates and fifteen pages of tables.\n\nThe sheer number of variables was surprising enough. But a reader with the patience to make it to the end of the paper would have found something even more remarkable. Almost as an afterthought, she had singled out sixteen of the stars, arranging them in a separate list showing both their periods and their magnitudes. \"It is worthy of notice,\" she observed, that \"the brighter variables have the longer periods.\"\n\nIn light of what astronomers know now, this is an understatement as magnificent as the one Watson and Crick made at the end of their famous 1953 paper on the double-helical structure of DNA: \"It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying method for the genetic material.\" This when they were telling their friends that they had discovered the secret of life.\n\nMiss Leavitt was not being coy. She just didn't want to over-interpret her data. Since the variables were all in the Magellanic Clouds, they must be roughly the same distance from Earth. If the correlation she glimpsed held true, you could judge a star's true brightness from the rhythm of its beat. Then you could compare that with its apparent brightness and estimate how far it was. This was too profound a conclusion to hang on just sixteen stars. More measurements would have to be made.\n\n**2**\n\nThat wasn't to happen anytime soon. The same year her results were published, Henrietta fell ill. On December 20, she wrote to Pickering from a Boston hospital, where she had been confined for the past week, thanking him for \"the beautiful pink roses\" and \"for the kind thought so beautifully expressed. It means much at a time like this to be made to realize that one is remembered by one's friends.\"\n\nTo convalesce, she returned to Wisconsin to stay with her parents and two unmarried brothers, George, now a missionary, and Darwin, another clergyman. After resting through the next spring and summer, she planned to resume work in the fall. But in September she reported to Pickering that a \"slight illness\" contracted after a visit to a lake near Beloit had \"proved unexpectedly obstinate, and I cannot tell when I shall be able to get away.\"\n\nIn October, after she had been absent for close to a year, Pickering wrote to inquire whether she would like him to send her some work. By early December, when she had not responded, he asked again,this time letting a glint of impatience show.\"My dear Miss Leavitt,\" he began. \"It is with much regret that I hear of your continued illness. I hope you will not undertake work here until you can safely do so. It may however relieve your mind if we can dispose of two or three questions....\"\n\nFirst he asked if she would send a letter at the beginning of each month stating whether she would be returning any time soon. Then he proposed that she issue a brief report (a socalled _Harvard College Observatory Circular_ ) describing the preliminary results of another study she had been engaged in, the North Polar Sequence, a Herculean effort to measure, more accurately than ever before, the magnitudes of ninety-six stars near Polaris. This was one of Pickering's pet projects, of a higher priority to him than her study of variables. He hoped the North Polar Sequence would become the gold standard for gauging the brightness of stars throughout the sky.\n\nShe replied three days later, apologizing for being too weak to answer his earlier letter. \"I thank you for expressing the desire that I wait for complete recovery before returning to Cambridge; it would make it even harder for me to be idle than it now is, if pressure were brought from without, especially from you. The thought of uncompleted work, particularly of the Standard Magnitudes, is one I have had to avoid as much as possible, as it has had a bad effect nervously.\" If she was just as anxious about her Magellanic variables, she didn't say.\n\nShe held out hope that her condition might improve enough for her to resume work after Christmas. \"Not the least of my trial in being ill is the knowledge of the annoyance it causes you.\"\n\nIn mid-January, Pickering wrote again, opening with the now familiar greeting and lament. \"My dear Miss Leavitt: It is with much regret that I hear that your illness will again require you to postpone your return to Cambridge. It occurs to me that, when you do return, you may be able to do much of your work in your room, and thus save yourself the walk to the Observatory.\"\n\nIn the meantime she had told Mrs. Fleming that she was ready to work from Beloit, and Pickering, emphasizing again the importance of the North Polar Sequence, described some photographs he planned to send her, including one from the Mount Wilson Observatory in California, where a new 60-inch telescope, the largest in existence, was recording stars of exceeding faintness. He outlined his thoughts on how she should proceed with her measurements. \"What do you think of this plan and can you suggest any improvements in it? ... I hope you will not let these matters trouble you, and that you will not undertake any of this work, except with the approval of your doctor.\"\n\nA few weeks later she received a box from Cambridge packed with photographic plates, paper prints, ledgers, a wooden viewing frame, and a 1\u00bd-inch eyepiece, allowing her to get back to gauging magnitudes. She responded with assurances that she was \"now strong enough to work for two or three hours a day, and am very glad indeed to have the means of employing the time to advantage.\" She hoped, as always, for an early return to Cambridge.\n\nFor the next three months, she continued her calculations, sending back detailed reports to Observatory Hill. Her health continued to improve but at a glacial pace both she and Pickering must have found exasperating. \"It is a great pity that my latest attempt to fix a time for returning to Cambridge should have failed like the others,\" she wrote to him in April, almost a year and a half since her absence began. This time, she assured him, her return really was imminent. \"My physician has not yet given his consent to my departure, wishing to be assured of the soundness of my recovery. I now expect to receive my dismissal from him any day and to be in a position to make definite plans for resuming work.\"\n\nBy May 14, 1910, she was finally back in Cambridge, or was at least on her way. Her name and that of her colleague Annie Cannon appear on a list of observatory employees requesting tickets to the annual Harvard Class Day commencement ceremonies.\n\nHer homecoming was short-lived. The following March her research was interrupted again when her father died, leaving his widow a modest estate that after probate costs and the settlement of debts was valued at just over $9,000. (He had kept the house on Warland Street, where Henrietta had been a girl, and owned a small amount of stock in a copper mining company that his brother Erasmus consulted for.) After thanking her colleagues for sending flowers, Henrietta departed for Beloit to console her mother.\n\nWhen she had not returned by June, Pickering sent her a box with seventy photographic plates and other material for the North Polar Sequence, but she wasn't able to concentrate on the work for long. Ten days later she wrote to inform him that she and her mother were departing \"rather unexpectedly\" to stay with some in-laws in Des Moines. She offered no explanation.\n\nShe took the plates to the Beloit College library for safekeeping. \"It is a new, fire-proof building,\" she assured him, \"and is to be open all summer. The plates are on a strong shelf in a corner of the librarian's private office, and are labeled with a request that no one shall touch them. Orders to that effect have been given to the janitor....It will be a disappointment to lose nearly a month in my work on the plates, but there will be a good deal of work with the papers I shall take with me.\"\n\nHer research wasn't entirely neglected. She even found time for her variable stars, sending a report for Pickering to read at a conference in Ottawa. After several more delays and apologetic letters, she returned that fall to her uncle's house on Garden Street.\n\nFinally given the luxury of a long stretch of uninterrupted time, she turned back to the strange matter of the Magellanic variables, plotting twenty-five of them on a graph with their brightness on one axis and their period on the other. Her results were published in 1912 in a _Harvard Circula_ r under the name of Edward Pickering: \"The following statement regarding the periods of 25 variable stars in the Small Magellanic Cloud has been prepared by Miss Leavitt.\"\n\nThe pattern now seemed clearer than ever. The stars lined up so neatly that she was moved almost to exclamation: \"A remarkable relation between the brightness of these variables and the length of their periods will be noticed.\" The brighter the star, the slower it blinked. Why she didn't know, and for now it didn't matter. \"Since the variables are probably at nearly the same distance from the Earth their periods are apparently associated with their actual emission of light.\"\n\nIn other words, you could determine how bright they really were. Without leaving Earth, you could count the beats of a star's rhythm, then use this to calculate its intrinsic magnitude. Compare that with its apparent magnitude and you would have its distance.\n\nThe universe had provided, for the especially keen observer, a hint of its grandeur. Imagine that you are standing on a back porch at night looking out over a dark field. Somewhere on the far edge is a mysterious array of electric lights. Some are brighter, some fainter, but since you don't know how bright they really are, you can't tell whether they are ten yards or ten miles away.\n\nNow suppose that the lights are blinking, and that it has been decreed by some international authority that bulbs be manufactured so that they flash according to their brightness. Fifty-watt bulbs blink faster than 100-watt bulbs. If two of the beacons are pulsing at the same frequency, you know they are equally bright. So if one appears, say, four times dimmer, it has to be farther away.\n\nTo be precise, it is two times farther. Light traveling through space spreads and diminishes according to the inverse square law. Square the difference in distance and you get the difference in brightness. All other things being equal (which they never quite are), a light that is nine times dimmer than another must be three times farther away.\n\nAlthough Miss Leavitt didn't use the term in her paper, the variables with this remarkable property are called Cepheids, for the first was discovered, in 1784, in the constellation Cepheus by an amateur English astronomer named John Goodricke. (He and Henrietta had more than astronomy in common: Goodricke was deaf and she was steadily becoming so.) The new law linking period and brightness would become known throughout astronomy as the Cepheid yardstick, a way to measure through great stretches of space.\n\nThere was just one problem: her Cepheids revealed only relative distances. You could say with some confidence that one star was twice as far away as another, and three times farther than another. But were they one, two, and three light-years from earth, or twenty, forty, and sixty? There was no way to know. To turn the ratios into actual distances, someone needed to discover how far the closest of the stars is from Earth.\n\nFor now, Miss Leavitt's new yardstick was one without numbers. The next step would be to calibrate it.\nCHAPTER 4\n\nTriangles\n\nI had not thought of making the very pretty use you make of Miss Leavitt's discovery about the relation between period and absolute brightness.\n\n_\u2014Henry Norris Russell, in a letter to Ejnar Hertzsprung_\n\nHold a finger a couple inches from your nose and alternately blink from left eye to right. The finger rapidly changes position, and the farther you move it from your face the smaller the displacement becomes. The window frame on the wall shifts even less and the telephone pole down the block hardly at all. The separation between the eyes\u2014 astronomers and surveyors call it the baseline\u2014is too small. We can sense the depth of the world close by, but the farther we look the flatter it appears. The distant mountains might as well be cut from cardboard.\n\nWired by nature to register this effect (remember it's called parallax), the brain carries out a kind of neurological trigonometry. The two eyes and the object before them form a triangle, and you unconsciously compute the distance from base to apex. Without even thinking about it, you triangulate.\n\nGet up now and walk from one side of the window frame to the other, a few feet away. The size of your baseline has increased, and now the telephone pole seems to shift against the building standing behind it. The effect is the same as having a very wide head, an eye at each side of the window looking out from two different angles. Measure the separation between the two observation points\u2014the base of the imaginary triangle and the angles formed by each window and the pole. You can do this with a surveyor's transit (think of it as a kind of protractor). With just this information and some high school trigonometry, you can calculate the altitude of the triangle\u2014the distance to the telephone pole. The calculation itself needn't concern us. Just know that the very nature of triangles ensures that they can be completely defined with only three scraps of information\u2014 two angles and one side or one angle and two sides. All the other dimensions follow from that.\n\nWith a wider baseline\u2014an eye on each side of the block\u2014 you could measure how far the building is. With a long enough stretch, even the mountains on the horizon will seem to move.\n\nThe history of astronomical measurement preceding Henrietta Leavitt's discovery can be compressed into a story of how people learned to use larger and larger triangles to point farther into the sky.\n\nSEND TWO OBSERVERS to different places on the Earth's surface and, if the separation is great enough, each will see the moon at a slightly different position against the backdrop of stars. Have them measure both angles simultaneously (they would need to synchronize their watches) and, if you know the length of the baseline, you can triangulate.\n\nIn a variation, the Greek astronomer Hipparchus in the second century B.C. used a solar eclipse as his timepiece. While the sun was blotted out completely at Hellespont, the strait in northwestern Turkey near the ancient city of Troy, the eclipse was only four-fifths full in Alexandria. If you could freeze time and hop back and forth between these two vantage points, the moon would seem to shift by an amount equal to one-fifth the size of the solar disk\u2014the Earth blinking its eyes. The sun occupies about half a degree of the arcing bowl of sky, making the parallax of the moon one-tenth of a degree.\n\n_Lunar Parallax_\n\nHad Hipparchus known the distance between Troy and Alexandria\u2014the size of the baseline\u2014he would have had his answer. Instead, using a more complex arrangement of triangles, he was able to calculate that the distance of the moon must be about thirty times greater than the diameter of the Earth. He got the ratio right. The planet has since been measured at about 8,000 miles across. Using that number, Hipparchus's method would put the moon 240,000 miles away. Two thousand years later, people learned to make the measurement by bouncing radar signals off the moon and measuring the delay of the echo. High-tech or low-tech, the figure comes out about the same.\n\nSkipping past the Dark Ages, dominated by Ptolemy's brilliantly wrong cosmology, with all the heavens looping around our planet like an amusement park ride, the next great leap didn't come until the sixteenth century. Copernicus restored the sun to the center of the solar system, then Kepler refined the model, flattening the planets' circular orbits into ellipses and showing how they must be arranged.\n\nOne of his laws is of particular interest to any would-be measurer of the universe: The farther a planet is from the sun, the longer it takes to complete its journey. From the length of the Martian year, one could deduce that the planet lies about one and a half times farther from the sun than Earth does. The same calculation could be done for Mercury, Venus, Jupiter, Saturn....Once you had all the ratios, you could use parallax to determine just one of the distances. The others would automatically follow.\n\nThe task was easier said than done. Measuring the tiny displacement of the moon, viewed from two spots on the Earth, had been difficult enough. For even the closest planets, the parallax was so small that the slightest mistake in gauging an angle or the length of a baseline caused the calculation to fail.\n\nThat didn't keep astronomers from trying. Coordinating their efforts with pendulum clocks, observers stationed in Paris and on the island of Cayenne in South America showed in 1672 that the position of Mars shifted by a mere 25 seconds of arc. Adopting the useful fiction that the heavens consist of a hemispherical dome arcing overhead, astronomers divide the sweep from horizon to horizon into 180 degrees, half a circle. Each degree is divided into 60 units called minutes, which can each be further subdivided into 60 seconds. Twenty-five arc-seconds is 1\/144 of a single degree, an exceedingly small piece of sky.\n\nThe Martian distance computed from this delicate operation came close to the mark, but the accuracy was accidental. There was so much uncertainty in the measurements that errors canceled out errors, dumb luck stumbling onto a pretty good answer. Before more reliable methods became possible, another hundred years would go by.\n\n**2**\n\nTwice a century, just a few years apart, Venus, the closest planet, passes between the Earth and the sun. The result is like an eclipse, except that Venus is so distant that it appears only as a tiny dot. No one would notice the event who wasn't deliberately watching. If observers in separate locations time how long it takes the planet to cross the solar disk, they can compare their different readings and triangulate. The result is Venus's distance and, plugging the number into Kepler's equations, the distances of every planet from the sun.\n\nEdmond Halley, the great British astronomer, had missed his chance to observe this rare phenomenon, called the transit of Venus. He lived inconveniently between the pair of events of 1631 and 1639 and the recurrence predicted for 1761 and 1769. He satisfied himself by issuing a challenge, calling on the next generation of astronomers to disperse themselves around the world and measure the Venusian parallax.\n\nThey took him up on the dare, setting out for Siberia, Hudson Bay, Baja California, India, the Cape of Good Hope, Tahiti. Some of the expeditions failed, and some of the data was suspect because of the difficulty of determining precisely when fuzzy-edged Venus, a planet swathed in chemical clouds, actually passed into the circle of the sun. But with eyes placed all over the planet (there were 150 observations), astronomers compiled enough good information to measure all the way to Venus. Then, carrying the calculation a step further, they concluded, via Kepler, that the distance to the sun was 91 million miles, just 2 million shy of the number schoolchildren learn today.\n\nIt was a hard act to follow. With one star down and a whole galaxy to go, the craft of triangulation was already becoming stretched to the limit. Even with an imaginary triangle whose base extended across the full width of the Earth, the parallax of the nearest planets was just barely measurable. How could anyone hope to gauge the distance of stars?\n\nAstronomers had made some rough estimates. If you pretend that all stars produce the same amount of light\u2014that they are equal in intrinsic brightness\u2014you can judge how much dimmer Sirius appears from Earth compared with our home star, the sun. Then with the inverse square law (something twice as far away shines a fourth as bright) you can estimate the distance. These rough calculations served to show that even the brightest stars must be hundreds of thousands of times farther than the sun, way beyond the reach of earthly parallax. Travel from one antipode to the other and you would not detect the slightest shift.\n\nIt seemed that measuring stellar distances would require leaving the planet, observing the position of a star from Earth and then from a point millions of miles away....Or you could stay on Earth and let it carry you between the far extremes of its orbit. This was the next advance. The radius of this great ellipse\u2014the distance from Earth to sun\u2014was now known with some precision. So just double the amount: every six months earthlings are looking at the sky from positions separated by 186 million miles. Draw a triangle with this as the baseline and you can measure the parallax of nearby stars.\n\n_Parallax using the diameter of Earth's orbit as a baseline_\n\nGalileo had suggested how such an experiment might be done. The sky is filled with double stars, some of which are presumably optical illusions: nowhere near each other in space, they just happen to line up because of the angle at which we see them. If one is actually much closer to us than the other, parallax will make them appear to come together and then move apart as we orbit the sun. (Think of two telephone poles, one behind the other, converging and then diverging as you drive by.)\n\nIt wasn't until the late 1700s that instruments were good enough to make the measurements. The astronomer William Herschel built a 20-foot-long telescope with a mirror 19 inches across. When that wasn't strong enough for him, he built a 40- foot scope with a 4-foot mirror\u2014so large that it weighed a ton. Working with his sister, Caroline, he discovered Uranus and some 2,000 star clusters and nebulae (which, he ventured, were not small, nearby gas clouds but galaxies so distant that the individual stars blurred together). He also found hundreds of the double stars Galileo had suggested might be used for measuring distance.\n\nIn the end the project failed. A statistical survey showed that it was actually very rare for two stars to line up and mimic a double. Most doubles really were doubles, adjacent stars orbiting each other\u2014too close together to exhibit any parallax.\n\nIt was with the next generation of astronomers that stellar triangulation came of age. Herschel's son John (the one we heard earlier rhapsodizing about the Magellanic Clouds) established an observatory at the Cape of Good Hope, near the southern tip of Africa. There astronomers determined that every six months the star Alpha Centauri changed in position by less than a second of arc\u20141\/10,000 of a single degree, a frustratingly tiny amount but enough to do some trigonometry. Calculating the height of this extremely skinny triangle led to the conclusion that the star was 25 trillion miles away\u2014so far that its light took more than four years to get here. And this was just the next sun over, the closest star.\n\nTwo other neighbors were also measured around this time, Vega and a star called 61 Cygni. A little later came Sirius and Procyon. All were a few light-years away in a universe now believed to be billions of light-years wide. By the early 1900s, when Henrietta Swan Leavitt arrived at Observatory Hill, nearly a hundred more stars had been triangulated. But most stars by far showed no parallax at all, even across so enormous a baseline. They were inconceivably and, it seemed, immeasurably far away.\n\n**3**\n\nHad just one of Miss Leavitt's blinking stars been within triangulating distance of home, astronomers could have leapt past the parallax barrier and begun measuring deep into space. Remember that two of her Cepheid variables pulsing at the same rate are, according to the relationship she discovered, of the same inherent brightness. If one appears to shine only a hundredth as brightly as the other, you know (because of the inverse square law) that it is ten times farther away. If you could use parallax to establish the distance of just one nearby Cepheid, you could infer the distance to the rest. By comparing Cepheid variables of various rhythms and intensities, you could leapfrog across the universe.\n\nNature, however, was not so obliging. The nearest known Cepheid, the North Star, was too far to show any shift in position, even when viewed from opposite ends of Earth's orbit. It is, by modern reckoning, some 400 light-years away. Parallax would get you only a fraction that far. The Cepheids used to devise Leavitt's law were many times farther still.\n\nThe best an astronomer could do was to say that a certain Cepheid was, as determined by its rhythm, one-tenth the distance of those in the Small Magellanic Cloud, while another was, perhaps, three times farther away. The universe could be portioned out in units called SMCs. But that begged the question: How far\u2014in miles or light-years\u2014was the Small Magellanic Cloud?\n\nBefore Miss Leavitt's stars could be turned into true yardsticks, some way had to be found to extend parallax far enough to reach a nearby Cepheid. That meant making observations across an even longer baseline than the full width of spaceship Earth's orbit around the sun. What was needed was a larger, faster craft. Unlikely as it seemed, one was at hand: the starship we call the sun.\n\nAccording to folklore, when Galileo was called before the Inquisition and forced to recant his Copernican views, he muttered under his breath, \"And yet it moves.\" The reference, of course, is to our planet. He might have been as surprised as his tormentors to learn that the sun moves as well, on a slow drift through the Milky Way, carrying its planets alongside.\n\nThe movement is barely perceptible. In the late 1700s the elder Herschel, William, discovered that stars in the direction of the constellation Hercules move according to a peculiar pattern: over the years, they appear to be fanning out from a distant point, as snowflakes seem to do when viewed in the headlights of a car speeding through the night. In the opposite direction, back toward the constellation Columba, the stars converge, like snowflakes seen from the rear window. Our solar system, he concluded, was leaving Columba and heading toward Hercules. Astronomers have since clocked the speed of this journey through the galaxy at about 12 miles per second, or more than 30 million miles a year. The parallax from the voyage causes the constellations to become stretched and squeezed over time. The ancient Greeks were looking at a slightly different sky.\n\nFor a measurer of the universe, riding along with the sun, closer stars will move faster than farther ones, while the most remote stars will not seem to move at all. Carefully note the position of a Cepheid, then measure it again years later, when the sun has dragged Earth and its astronomers to a new location in space. Calculate the length of this enormous baseline, and then triangulate. With the distance of one Cepheid established, you can calibrate Leavitt's yardstick and then measure the rest.\n\nThe first to attempt this was a Danish astronomer named Ejnar Hertzsprung. He used the motion of the sun to triangulate the distance to some Cepheids in the Milky Way. Then, correlating pulse rate with inherent brightness, he extrapolated outward, reporting in a journal called _Astronomische Nach_ _richten_ that the distance of the Small Magellanic Cloud was about 3,000 light-years.\n\nThis was enormous by the astronomical standards of the time. In fact it was a misprint. Maybe a journal copy editor had recoiled at the real number, inadvertently dropping a zero. According to Hertzsprung's calculations, the nebula was ten times farther, 30,000 light-years away.\n\nAround the same time the American astronomer Henry Norris Russell used a different method to come up with an even more astonishing distance of 80,000 light-years. \"I had not thought of making the very pretty use you make of Miss Leavitt's discovery about the relation between period and absolute brightness,\" he later wrote to Hertzsprung. \"There is of course a certain element of uncertainty about this, but I think it is a legitimate hypothesis.\"\n\nThe Cepheid yardstick still needed refinement, but astronomers finally had hope of leaping past the nearest stars, roughing out the shape and size of the galaxy... and what, if anything, lay beyond.\n\n**4**\n\nHenrietta Leavitt didn't get to pursue the matter herself. Pickering kept her tied down with other projects. He wasn't one to encourage theorizing, believing, as his colleague Solon Bailey put it, \"that the best service he could render to astronomy was the accumulation of facts.\" Beginning in August 1912, the year her discovery about the Cepheids was published, she documented her day-to-day routine, in language meaningful only to an astronomer, in a black-and-red leather-bound notebook:\n\n_October 8. Letter from Hertzsprung, dated Mount Wilson, Oct. 3, 1912. Subject, Method of transforming photographic to visual magnitudes by means of effective wavelengths. He finds that a change in the color index, using Dr. M\u00fcrch's photo_ _graphic magnitudes andHarvard photometric magnitudes, of one magnitude, corresponds to a change in effective wave_ _length (_ \u03bb _f_ _) of 200 Angstrom units...._\n\n_October 19. Tried superimposing Plates H 361, exp. 10m, lim_ _iting magn. 15.6 and H 385, isochromatic, limiting magnitude_ _14.9._ _The red stars appeared of nearly the same brightness on the two plates, white stars being brighter on H 361. The colors were assigned very easily._\n\n_October 22. Finished revision of 32 sequences north of_ + _75 degrees, and comparison with marked chart. Gave plates to Miss O'Reilly for identification._\n\nAnd so it went for the next four years, except for gaps, sometimes many months long, hints of recurring illness. In spring 1913 she was absent for three months recovering from stomach surgery. Only once in all those pages, on January 13, 1914, does she let some excitement show: \"Completed discussion of Prov. Photovisual Magn. of N.P. Seq., this completing H.A. 71, 3. !!! after many years.\"\n\nTranslation: She had finished, or so she believed, the measurements for her North Polar Sequence, ninety-six stars whose magnitude she had determined with such authority and care that they could be used as a standard for the rest of the sky. There were still revisions to be done. The work was finally published, three years later, in the _Annals of the Astro_ _nomical Observatory of Harvard College_ , volume 71, number 3\u2014all 184 pages of it. Perhaps she and her mother indulged in a split of champagne.\n\nDry as dust to the uninitiated, her report was a work of magnificence, combining data from 299 photographic plates taken by thirteen different telescopes. Every magnitude had to be meticulously checked and cross-checked, with a constant awareness of the difference between mere data and true phenomena. No matter how carefully measured, each number represented not the brightness of a star but rather the intensity of its image on a photographic plate. In a perfect world these two values would be the same. In reality, every telescope and every type of photographic plate had its own peculiarities\u2014responding more readily to some colors than to others. Images near the center of the plate were rendered more accurately than those to the side.\n\nPage after page, she described how she corrected for the various biases and uncertainties. There was a chain of reasoning behind every number. Each star was a project in itself.\n\nWhen she reached the end of the study, she knew it wasn't perfect. \"It is desirable that the standard scale should be investigated by different observers, using independent methods,\" she allowed. \"Discrepancies will inevitably appear in the results.\" But, as politely as possible, she warned future critics to proceed with caution.\n\n\"Too much weight may easily be assigned to results obtained from a single investigation, even if great precautions have been used.\" Which is why, she gently reminded, her measurements \"depend on many different methods, instruments, and observers.\"\n\nShe concluded, \"In view of these facts, it seems only reasonable that considerable time should be allowed to pass, and a large amount of varied material collected, before adopting definitive corrections to the scale here presented. For stars between the tenth and sixteenth magnitudes, such corrections are likely to be minute. For brighter and fainter stars, sensible changes may be made ultimately, but the scale is probably a close approximation to the true one.\"\n\nIt was work to take pride in. Ph.D.s have been awarded for less.\nCHAPTER 5\n\nShapley's Ants\n\nHer discovery of the relation of period to brightness is destined to be one of the most significant results of stellar astronomy, I believe. I am quite anxious to have her opinion as to the periods because of its bearing on some statistical work I am now bringing to a close.\n\n_\u2014Harlow Shapley, writing to Edward Pickering  \nabout Henrietta Leavitt_\n\nBack in the eighteenth century, William Herschel had theorized that nebulae, like Andromeda, might be distant galaxies. The philosopher Immanuel Kant, called them \"island universes,\" arguing, \"It is much more natural and reasonable to assume that a nebula is not a unique and solitary sun, but a system of numerous suns.\" The universe, he wrote, may be filled with Milky Ways.\n\nOther astronomers, however, were swayed by a different philosopher, Pierre-Simon Laplace, who proposed that the spiral-shaped clouds like Andromeda were no more than \"proto solar systems\"\u2014a new sun and its orbiting planets in the process of congealing from a whirling cloud of gas. The theory seemed all the more plausible when in 1885 a new star, or \"nova,\" flared within the center of the hazy disk of Andromeda, as though a solar system was being born.\n\nTime exposures of the sky soon revealed some 100,000 of the luminous swirls and more were being found all the time. With spiral nebulae appearing everywhere, it seemed absurd to suppose that each could be a galaxy filled with millions of stars. \"No competent thinker, with the whole of the available evidence before him, can now, it is safe to say, maintain any single nebula to be a star system of coordinate rank with the Milky Way,\" Agnes Clerke, an astronomer and historian of science, wrote in 1890. \"A practical certainty has been attained that the entire contents, stellar and nebular, of the sphere belong to one mighty aggregation.\" The universe was just another name for the Milky Way.\n\nBut the issue was far from settled. For one thing, the Milky Way itself seemed to be shaped like a spiral. Viewed from afar it might appear little different than Andromeda or any other nebula. An even stronger argument for island universes came from analyzing a nebula's light with a prism, breaking it into its component colors. These spectroscopic patterns were believed to reveal which chemicals a celestial object was made from. Andromeda's rainbow looked very much like the one cast by the sun. Both seemed to consist of star stuff. The tests were less than conclusive. Other nebulae produced dull, simple spectra\u2014what one might expect from a homogeneous cloud of luminous gas. People tended to find in the data what they were already disposed to believe.\n\nThe impasse was broken in 1914 by Vesto Melvin Slipher at Lowell Observatory in northern Arizona, who had figured out how to estimate the speed at which a nebula was traveling through space. His technique was based on the Doppler effect. If a star is moving toward you, its light waves will be compressed. Thus the frequency\u2014the number of waves that strike the eye every second\u2014will increase. Brains interpret frequency as color, and so the starlight will shift toward the higher, bluer end of the spectrum. Conversely, if the star is moving away, its light will be stretched toward the lower-frequency reds.\n\nMeasuring the red and blue shifts of fifteen spiral nebulae, Slipher found them to be traveling at incredible velocities. Two of them appeared to be flying off at a dizzying 1,100 kilometers per second. That hardly seemed possible if they were simply small objects within the gravitational grip of the Milky Way. For many astronomers that settled the matter, in favor of island universes. (Slipher himself initially clung to the prevailing notion that a nebula was but a single star \"enveloped and beclouded by fragmentary and disintegrated matter.\")\n\nMore evidence arrived three years later when another nova suddenly appeared inside a spiral nebula called NGC 6946 (after its designation in the New General Catalog of Nebulae and Star Clusters). Heber Curtis of Lick Observatory, a California stronghold of the island universe theory, found novae inside other nebulae, and when astronomers reexamined old photographic plates they found still more.\n\nCurtis believed the novae might serve as standard candles. Astronomers had estimated that those in the Milky Way surged to an intrinsic magnitude of about \u20138. (Remember that the lower the magnitude, the brighter the star, making one with a negative value very bright indeed.) Curtis assumed that the novae in the distant nebulae were probably peaking at about the same intensity. Compare that number with how bright a nova appeared from Earth and you could use the inverse square law to get its distance. Measured this way, the nebulae appeared to be huge spinning galaxies many millions of light-years away.\n\nBy 1917 the consensus had shifted toward island universes. In addition to Curtis, and by now Slipher, supporters included such prominent astronomers as Arthur Eddington, James Jeans, Ejnar Hertzsprung, and a young researcher named Harlow Shapley, who had recently moved to Mount Wilson Observatory, the astronomical powerhouse perched high above Pasadena, California. Shapley however was about to change his mind. Using Leavitt's variable stars, he would spend the next few years calibrating the Cepheid yardstick and measuring the size and shape of the Milky Way. He was ultimately forced to conclude that it was far larger than anyone had dared imagine\u2014so large, he believed, that it must constitute the entire universe, nebulae and all.\n\n**2**\n\nShapley had a peculiar fixation with ants. When he wasn't looking upward at the stars, he liked to watch a colony of medium-size brownish black ants\u2014 _Liometopum apiculatum_ \u2014as they streamed along a concrete wall by Mount Wilson's maintenance shop. Shapley noticed that the ants slowed down when they reached the shade of some manzanita bushes and sped up again in the sun. Armed with various instruments, he studied the ants under different atmospheric conditions, even watching them with a flashlight at night. The correlation between running speed and temperature was so tight that he could use the ants as a thermometer, reading off the temperature within one degree. Returning to Mount Wilson thirty years later, he was annoyed to find that an assistant engineer was making a practice of burning off the ant trail with a blow torch\u2014\"genocide,\" Shapley called it. (\"Formicide\" would have been a better word.) But the resilient ants always came back.\n\nHe drew lessons from these tiny creatures. Asked to deliver a commencement address at the University of Pennsylvania, he chose the topic \"On Running in Trails,\" warning the students against the comfortable allure of conformity, of following the same narrow paths as their ancestors, afraid to break from the pack.\n\nWhen Shapley arrived in Pasadena in 1914, the common wisdom held that the Milky Way was a lens-shaped disk some 25,000 light-years long and about a fourth that wide, with the sun at almost dead center. This picture of the heavens was sometimes called the Kapteyn universe, after the Dutch astronomer Jacobus Kapteyn, who had estimated its size. The methods he had used were far from exact. With the Cepheid variables and Mount Wilson's powerful 60-inch telescope at his command, Shapley decided to measure the galaxy for himself.\n\nSpread throughout the Milky Way were a hundred or so \"globular clusters,\" each consisting of hundreds of thousands, even millions, of stars. Shapley suspected that these huge concentrations formed a kind of framework or \"skeleton\" marking the extent and shape of the galaxy. By using Cepheids to determine the distances to these mileposts, he could map out the whole thing.\n\n_The Milky Way_\n\nShapley figured he knew something about variable stars. His Ph.D. dissertation at Princeton, under Henry Norris Russell, had focused on a type of variable called eclipsing binaries\u2014 two stars orbiting around a common point and periodically blocking each other's light. One of Shapley's first papers at Mount Wilson showed that the Cepheids did not belong to this class. Rather, they were single stars that expanded and contracted with a regular beat. For now, however, these details were unimportant. He knew from Henrietta Leavitt's research that Cepheids would serve as standard candles.\n\nHe also knew that most of the variables in the globular clusters were a little different from the ones she had discovered in the Magellanic Clouds. Shapley's stars\u2014called cluster variables\u2014blinked much faster, with cycles measured in hours, not days. Hertzsprung, in fact, thought his eager colleague was mixing apples and oranges. How could he be so sure that both kinds of stars showed the same connection between period and brightness?\n\nShapley was insistent. \"[T]his proposition scarcely needs proof,\" he wrote in a paper in the _Astrophysical Journal_. \"Practically all writers on the subject are more or less inclined to accept this view.\"\n\nUndeterred, he proceeded with his plan. For his early measurements, he relied on the fact that the farther something is from an observer, the more slowly it appears to move\u2014think of a tiny jet plane inching across a windowpane. The speed at which a star is heading directly toward or away from Earth, its \"radial velocity,\" can be clocked using redshift and blueshift. But it is the \"transverse velocity\"\u2014how fast the star is moving across the sky\u2014that hints at how far away it is. It seemed sensible that, on average, stars in a cluster would move at the same velocity, whatever the direction. Drawing on a method called statistical parallax, Shapley used Doppler shifts to estimate the average velocity of a sampling of stars and compared that figure with how fast the stars _appeared_ to be moving. That revealed their distance. Eleven of these were Cepheids, forming the anchor of what came to be called \"Shapley's curve.\"\n\nTaking a second leap, he extended the curve to include the far more common shorter-period variables. First he would find a cluster that had both types. The slow-paced Cepheids gave him an estimate of the cluster's distance, which he could then correlate with the period of the faster variables. Now the distance of clusters with only fast variables could be measured\u2014 provided that both kinds of stars really obeyed the same law.\n\nNew yardstick in hand, he gauged the distances to several of the nearest globular clusters. Then he ran into a wall. In most of the clusters, not a single blinking star could be found. He would have to extrapolate further, and that meant finding another kind of standard candle. It seemed sensible, he reasoned, that each cluster would consist of stars spanning the same range of magnitude. The brightest stars in cluster A, whose distance had been measured with his yardstick, would be about as intense as the brightest stars in cluster B, whose distance was unknown. If they appeared dimmer, it would be because they were farther. The inverse square law would reveal by just how much, extending the map a little more.\n\nMany clusters, however, were so distant and so blurry that not even Mount Wilson's telescope could pick out a single star. And so came the final leap: one could take a cluster whose distance had been established by these other indirect methods and use the whole thing as a standard candle. The farthest clusters, Shapley reasoned, were probably as intrinsically big and brilliant as the nearest ones. By measuring how much smaller and fainter they appeared, he could judge their distances, and reach to the farthest edges of the galaxy.\n\nAs he followed this artful chain of assumptions, Henrietta Leavitt was living alone in a Cambridge rooming house, where she had moved after the death of her uncle Erasmus in 1916. She was still working for the observatory and occasionally Shapley wrote to Edward Pickering inquiring about the latest developments with her variable stars.\n\nShapley had noticed some very faint variables in the Magellanic Clouds and wondered if these might be similar to the ones he was using to plot out the Milky Way. \"Does Miss Leavitt know if they have shorter periods, that is, are their periods shorter than one day, similar to cluster variables?\" he wrote on August 27, 1917. \"It may be her work has not progressed far enough to give a definite answer.\" He was hoping for some ammunition against those, like Hertzsprung, who continued to argue that the faster variables did not necessarily obey Leavitt's rule. He considered the matter \"of much importance.... In fact, the Magellanic clouds and their variables seem to me one of the most important outstanding problems of stellar photometry.\"\n\nPickering replied about three weeks later: \"Miss Leavitt is now absent on her vacation.\" (She was on Nantucket, visiting with Margaret Harwood, a fellow Harvard computer and astronomical assistant who had become director of the Maria Mitchell Observatory.) \"When she returns, she will investigate the matter of the Magellanic Clouds.\"\n\nOf the two men, Shapley was the quicker and more loquacious correspondent. Within a week he had fired off another letter, praising Leavitt's work and emphasizing how much he needed the information. \"Her discovery of the relation of period to brightness is destined to be one of the most significant results of stellar astronomy, I believe. I am quite anxious to have her opinion as to the periods because of its bearing on some statistical work I am now bringing to a close.\"\n\nNine months later Shapley was still waiting. On July 20, 1918, he checked in again, still heaping on the praise:\n\nI believe the most important photometric work that can be done on Cepheid variables at the present time is a study of the Harvard plates of the Magellanic clouds. Probably Miss Leavitt's many other problems have interrupted and delayed her work on the variables of the clouds for the interval of six or seven years since her preliminary work was published.... The theory of stellar variation, the laws of stellar luminosities, the arrangement of objects throughout the whole galactic system, the structure of the clouds\u2014all these problems will benefit directly or indirectly from a further knowledge of the Cepheid variables.\n\nIt took almost three weeks for Pickering to reply: \"A few days ago I talked with Miss Leavitt.... She has the material for about a third of the brighter variables, and photographs are now being taken with the Bruce 24-inch, which I hope will provide the remainder.\"\n\nThat is the last letter from Pickering in Shapley's files. Less than five months later, he died of pneumonia at age seventy-two.\n\n**3**\n\nShapley ultimately decided to run with his theory, using the Cepheids as the first step in his hopscotch across the galaxy. The results were astonishing. First of all the Milky Way, by Shapley's measure, was gargantuan in size\u2014300,000 light-years across. That was some ten times greater than Kapteyn's estimate\u2014so much larger that he felt he must abandon the notion of island universes. If one insisted on maintaining that the thousands upon thousands of spiral nebulae were galaxies each the size of the Milky Way, then Andromeda, judging from its apparent size, would have to lie at an enormous distance. That, in turn, would mean that its novae were absurdly bright. It must be a small gas cloud after all.\n\nTo Shapley there was now an even more damning argument against island universes. One of his colleagues, a Mount Wilson astronomer named Adriaan van Maanen, had recently announced that several of the great spirals, including the aptly named Whirlpool and Pinwheel nebulae, were gradually turning. Van Maanen made the measurements with an arrangement of lenses and mirrors called a blink comparator. With the device, an astronomer could mount two photographic plates taken months or years apart and, gazing through a binocular eyepiece, switch back and forth between them. Anything that had changed would appear to move or vary in size. Comparing plates of nebulae taken five years apart, van Maanen thought he saw a slight rotation.\n\nViewed from Earth the spin was minuscule\u20142\/100 of a single second of arc each year. A complete cycle would take about 100,000 years. The surprise was that the movement was visible at all. Few could doubt that spirals spun. Why else would they have their pinwheel shapes? But for the motion to be perceptible at all, these nebulae would have to be small and nearby. If the spirals were truly distant galaxies, van Maanen's data would mean they were spinning at impossibly high velocities\u2014faster than the speed of light.\n\nJust as unsettling as the Milky Way's enormous size was Shapley's conclusion about where we lived inside the galactic disk. Astronomers had noticed that the globular clusters are not distributed evenly through the sky but congregate in the direction of the constellation Sagittarius. There, according to Shapley's measurements, they formed a roughly spherical shape, a cluster of clusters. That, he proposed, must be the center of the galaxy, the central bulge of the Milky Way. If we lived within this region, we would see the clusters spaced uniformly around us. The fact that we do not is because we lie in the galaxy's outskirts, tens of thousands of light-years from the core. We were not New York City but Pensacola or North Platte.\n\n\"So the center has shifted: egocentric, lococentric, geocentric, heliocentric,\" Shapley wrote to George Ellery Hale, Mount Wilson's director. Or, as he later put it: \"Man is not such a big chicken. If man had been found in the center, it would look sort of natural. We could say, 'Naturally we are in the center because we are God's children.' But here was an indication that we were perhaps incidental. We did not amount to so much.\" People were no more important than ants.\n\nThe center of the galaxy was in Sagittarius. And so the center of the universe must be there as well.\nCHAPTER 6\n\nThe Late, Great Milky Way\n\nThe spectrum of the average spiral nebula is indistinguishable from that given by a star cluster. It is such a spectrum as would be expected from a vast congeries of stars.\n\n_\u2014Heber Curtis_\n\nOn a spring day in 1920, Shapley found himself strolling along the train tracks somewhere in Alabama talking about flowers and classics and probably looking for ants. His companion was Heber Curtis, and they had agreed for now to avoid the touchy subject of astronomy. Each, unbeknownst to the other, had booked passage on the same train from California to Washington, D.C., where, in the halls of the National Academy of Sciences, they were scheduled to debate whether there was anything in the universe beyond the Milky Way.\n\nThe event had been arranged at the prompting of Shapley's boss, George Ellery Hale, one of the most revered astronomers of the time. Hale's father had made a fortune selling hydraulic elevators to the builders of Chicago's soaring skyscrapers. The son set his sights still higher, studying astronomy and exploiting his family's financial connections to fund the development of some of the best telescopes in the world. When the old man died, a series of lectures was endowed in his name. The younger Hale thought that the one in 1920, at the National Academy's annual meeting, should be devoted to a currently hot cosmological issue\u2014either relativity or island universes.\n\n_Harlow Shapley_\n\n(Harvard University Archives)\n\nThe first topic struck the academy's secretary as too esoteric. (He personally thought that Einstein's theory should be banished \"to some region of space beyond the fourth dimension, from whence it may never return to plague us.\") Nor was he keen on island universes, fearing that \"unless the speakers took pains to make the subject very engaging the thing would fall flat.\" He proposed instead a presentation on glaciers or \"some zoological or biological subject.\" In the end, Hale had the final word, and Shapley and Curtis were picked to present their opposing views on \"The Scale of the Universe,\" and, more pointedly, on whether it consisted of more than a single galaxy.\n\nWere such an event to take place today, it would be videotaped and perhaps transcribed. You could probably download it from the Web. The encounter between Shapley and Curtis can be pieced together only from scraps of evidence\u2014a typescript of Shapley's talk, annotated with his scribbles, some of Curtis's slides (he misplaced his script soon after the event), and letters the two exchanged before and after what came to be called the Great Debate, mostly by people who had not been there.\n\nCurtis himself was eager for a scrap. He imagined the two astronomers going after each other with \"hammer and tongs,\" then shaking hands like gentlemen. Shapley, however, was worried that he might lose. Not that he thought his theory was wrong. But persuading an audience of geologists, biologists, and scientists of other nonastronomical persuasions required the skills of an orator. Right or wrong, Curtis, thirteen years older and the more polished debater, might best Shapley at the podium.\n\nHe expected for example that Curtis would pick on the tiny handful of stars (\"my eleven miserable Cepheids,\" Shapley called them in a letter) from which he had extrapolated such enormous distances. What seemed to some like a brilliant analysis might strike others as a house of cards. Many astronomers were much less confident than Shapley about the usefulness of Henrietta Leavitt's yardstick. Curtis had made it clear that he thought Shapley's Milky Way was ten times too large. If he could shrink it back by an order of magnitude, the island universe theory might be easier to uphold.\n\nShapley also had an ulterior motive. He was certain he was being considered for the directorship of Harvard Observatory\u2014Edward Pickering had just died\u2014and he expected an emissary from Observatory Hill to be in the audience. He dreaded making a bad impression.\n\nFor weeks Shapley worked to bolster his evidence while maneuvering for a more advantageous position. Through sheer obstinacy, he managed to get the debate downgraded to a discussion\u2014\"two talks on the same subject from our different standpoints\"\u2014and the talks themselves reduced in length. While Curtis wanted forty-five minutes for each presentation, Shapley wanted thirty-five. They split the difference, settling on forty. To blunt the impact further, no time would be allowed for rebuttals, just a general discussion at the end. Finally Shapley made certain that Henry Norris Russell, his former teacher and ally, would be in the audience to support his position. He wasn't taking any chances.\n\n**2**\n\nThe evening began at an excruciating pace with awards followed by long testimonials. There was a tribute to the Prince of Monaco for oceanography, Shapley later remembered, and another to some \"noble human antique,\" honored for combating hookworm. Many years later in a memoir, Shapley recalled a bored Albert Einstein sitting in the audience, whispering to his companion that he now had a new theory of eternity. It made a good story, but actually Einstein was in Germany then, fending off the first stirrings of Nazi denunciations of his \"Jewish physics.\" He made his first visit to the United States the following year.\n\nOnce the main event finally began, Shapley went first, easing in slowly with a long introductory tutorial on astronomy. A third of the way through his allotted time, he had gotten only as far as the definition of a light-year. So far the presentation was pure popular science. One can imagine Curtis glancing at his watch, wondering when Shapley would say something he could dispute.\n\nThen he surprised Curtis again with a promise to spare the audience \"the dreary technicalities of the methods of determining the distance of globular clusters,\" the way stations he had used to map the galaxy. Maybe that was a sensible approach for an audience of nonspecialists. But Curtis, who was still itching for a fight, had prepared a meticulous pointby-point deconstruction of Shapley's every assumption and logical inference. There was still nothing for him to rebut.\n\nThe biggest surprise was still to come. Skipping over Cepheids entirely, Shapley described an entirely independent method of establishing the enormity of the galaxy (and by implication, undermining the case for island universes).\n\nAstronomers had uncovered what appeared to be a relationship between a star's temperature and its inherent brightness. (These had been plotted on a chart famous to astronomers as the Hertzsprung-Russell diagram after Henry Norris Russell and Ejnar Hertzsprung.) The result was another kind of measuring stick. A survey of nearby \"B-type stars,\" identified by their bluish sheen, had found them to be on average 200 times brighter than the sun. This suggested, to Shapley anyway, that these blue giants could be used as standard candles. No matter how much a B star's light is dimmed on the journey to Earth, it could be assumed, with a small leap of faith, to have the same intrinsic brightness as its cousins closer to home. And so the giant's distance could be calculated from the inverse square law. If it was nine times dimmer than a nearby B star, it must be three times farther away.\n\nShapley found what he took to be blue giants in the Milky Way cluster called Hercules. Exceedingly dim\u2014of the fifteenth magnitude\u2014the stars, he reckoned, would have to be 35,000 light-years away. Then he extrapolated further. Assuming that the smaller, dimmer clusters had the same overall brightness as Hercules, he circled around to his original conclusion: that they lie at the fringes of a galaxy 300,000 light-years in diameter, with the sun shoved off to one side.\n\nHe briefly mentioned how another kind of yardstick, stars called red giants, also supported his measurements. Then he tried to fend off any criticism of his \"miserable\" Cepheids by removing them from the debate: Professor Curtis, he said, \"may question the sufficiency of the data or the accuracy of the methods....But this fact remains: we could discard the Cepheids altogether, use instead the thousands of B-type stars upon which the most capable stellar astronomers have worked for years, and derive just the same distance for the Hercules cluster, and for the other clusters, and obtain consequently the same dimensions for the galactic system.\"\n\nBefore the debate the red and blue stars had been no more than footnotes to Shapley's argument\u2014secondary checks on his primary measuring tool, Henrietta Leavitt's Cepheid variables. Now figure and ground had been reversed, leaving Curtis to aim at a moving target and leading one to wonder who really was the wilier debater.\n\nAs for the nature of the spiral nebulae, the original focus of the discussion, Shapley dismissed them in a few sentences:\n\nI shall leave the description and discussion of this debatable question to Professor Curtis. We agree, I believe, that if the galactic system is as large as I maintain, the spiral nebulae can hardly be comparable galactic systems; if it is but one-tenth as large, there might be a good opportunity for the hypothesis that our galactic system is a spiral nebula, comparable in size with the other spiral nebulae, all of which would then be \"island\" universes of stars. On one other point I think we also agree, or at least we _should_ agree, and that is that we know relatively so little concerning the spiral nebulae... that it is professionally and scientifically unwise to take any very positive view in the matter just now.\n\nEven if the spiral nebulae were not firmly inside the boundaries of the Milky Way, he believed, they probably lay on the outskirts, small gas clouds encountered during the galaxy's drift through endless nothingness.\n\n**3**\n\nThere is no way to re-create the tone of Curtis's presentation from the skeletal talking points left behind on his typewritten slides. It is clear that, undeterred by Shapley, he marshaled a strong defense of island universes. As expected, he challenged the reliability of Shapley's calibration of the Cepheid yardstick, joining those who suspected that the more rapidly oscillating cluster variables used to measure out the Milky Way were different in nature from the slower ones Leavitt had found in the Magellanic Clouds. Why assume they had precisely the same relationship between period and inherent brightness, if there was indeed any such relationship at all? If Shapley's variables were actually much dimmer to begin with, then all the clusters would lie closer in. The perimeter of the galaxy would contract and the Milky Way would shrink from continent to island, one member of a great archipelago.\n\n_Herber Curtis_\n\n(Lick Observatory)\n\nCurtis was equally unimpressed with the giant blue stars, arguing that far too little was known to trust them as standard candles. He proposed what he considered a more reliable measuring device\u2014the yellow-white stars like the sun that seemed to make up most of the galaxy. It was reasonable, Curtis proposed, that the sunlike stars in the far reaches of the Milky Way shine, on average, with the same brightness as those nearby. Like Shapley, he was assuming the uniformity of nature, and his conclusion was that the galaxy can be only about 30,000 light-years across. For the clusters to be as distant as Shapley believed, these stars would have to be far brighter than those in our own neighborhood. A different physics would prevail. \"While it is not impossible that the clusters are exceptional regions of space [with] a unique concentration of giant stars, the hypothesis that cluster stars are, on the whole, like those of known distance seems inherently the more probable.\"\n\nWith the Milky Way knocked down in size, evidence for island universes seemed compelling. Curtis recalled the familiar argument that the color patterns produced by the spirals were indistinguishable from that of starlight. \"It is such a spectrum as would be expected from a vast congeries of stars.\" And the novae that appeared within them \"seem a natural consequence of their nature as galaxies, incubators of new stars.\" Used as standard candles the novae put Andromeda half a million light-years from Earth and other spirals 10 million or more light-years away. \"At such distances, these island universes would be of the order of size of our own Galaxy of stars.\"\n\nFinally he noted a curious phenomenon that had been puzzling astronomers for years: the spiral nebulae appeared to be concentrated at the two \"poles\" of the Milky Way\u2014the regions directly above and below the galaxy's central bulge. None was found in the galactic plane where most of the stars reside. If the spirals were small clouds within or near our galaxy, then why were they not evenly distributed? It was as though they were being repelled by some mysterious force.\n\nIt was far more plausible, Curtis argued, that this \"zone of avoidance\" was an illusion\u2014that the spirals lay far beyond the Milky Way, in every direction, with those along the galactic plane hidden from our view. Many spirals appeared to be surrounded by a thick ring of \"occulting\" matter, a halo of interstellar dust. The same might be true of the Milky Way. When astronomers aimed their telescopes directly into this dust storm, spirals in that direction were blocked from view. Of the millions of spirals, the only ones we could see were those that happened to lie above and below. Each, he proposed, was a world as vast and shining as our own.\n\n**4**\n\nEach man left the lecture hall certain that he had won. \"Debate went off fine in Washington,\" Curtis wrote to his family, \"and I have been assured that I came out considerably in front.\" Shapley, for his part, attributed any points Curtis may have scored to his rhetorical skills. \"Now I would know how to dodge things a little better,\" he said years later, a comment that seems strange since Shapley spoke first. Maybe he was referring to the discussion that followed the talks, during which his mentor Russell had, as planned, come forth with a strong endorsement of Shapley's Big Galaxy theory. The champions of island universes surely responded just as vigorously. \"Curtis did a moderately good job,\" Shapley recalled. \"Some of his science was wrong, but his delivery was all right.\"\n\nBoth men fleshed out their arguments in papers published the following year in the _Bulletin of the National Research Council_. (In some early historical accounts these published papers are treated as the actual substance of the debate.) The texts contain no fundamentally new arguments. Shapley bolstered his case with more data, whose certainty Curtis continued to question. What is most striking is how two of the world's smartest astronomers could take the same trove of astronomical observations and come up with two such very different pictures of the universe, a reminder that science lies not in the facts themselves but in their arrangement.\n\nFor Curtis, the zone of avoidance (it sounds like something from a Superman comic) was strong evidence for seeing the spirals as island galaxies. In Shapley's hands, the phenomenon seemed to support the argument that spirals are little wisps of stellar gas: they would have to be small and light to be repulsed somehow by the Milky Way. Also open to conflicting interpretations were the novae that appeared now and then inside the spirals. For Curtis their existence showed that spirals were indeed galaxies. Where else would one expect to observe stars being born? For Shapley each nova represented \"the engulfing of a star by [a] rapidly moving nebulosity.\"\n\nThe most memorable passage in either paper is a paragraph by Shapley on the perils of extrapolating too boldly from a limited set of data. He meant the statement as a criticism of Curtis's measurements involving the average magnitude of sunlike stars. But it could be taken just as easily as a humbling reminder about how much the entire enterprise of astronomical measurement rests on a few vulnerable assumptions. (It is also the inspiration for the story about the villagers in the canyon, which appears in the prologue of this book.)\n\n\"Suppose,\" Shapley began, \"that an observer, confined to a small area in a valley, attempts to measure the distances of surrounding mountain peaks.\" He can use parallax for the nearby hills, but since he cannot leave his narrow valley, his baseline is too small to triangulate any farther. He needs another kind of measuring stick. Seeing through his telescope that there is plant life on the mountaintop, he makes the simplifying assumption that it is approximately the same as the plant life on the valley floor\u2014averaging about a foot in height. Thus from the apparent size of the foliage, he can judge how far the mountain is.\n\nHis calculation would be wrong. \"If, however,\" Shapley noted, \"he had compared the foliage on the nearby, trigonometrically-measured hills with that on the remote peaks, or had used some method of distinguishing various floral types, he would not have mistaken pines for asters and obtained erroneous results for the distances of the surrounding mountains. All the principles involved in the botanical parallax of a mountain peak have their analogues in the photometric parallax of a globular cluster.\"\n\nMistaking pines for asters, and asters for pines. It was an occupational hazard that would plague Shapley as much as anyone.\nCHAPTER 7\n\nIn the Realm of the Nebulae\n\nOne of the few decent things I have done was to call on her on her death bed. It made life so much different, friends said, that the director came to see her.\n\n_\u2014HarlowShapley, writing about Henrietta Swan Leavitt_\n\nFor two boys hailing from rural Missouri, Harlow Shapley and Edwin Hubble didn't have much in common, except perhaps the size of their egos. Shapley had been born in 1885 on a hay farm near the Ozarks and dropped out of school to become a police beat reporter for a small-town newspaper. He had completed the equivalent of the fifth grade. Only a few years later did he earn a high school diploma and enroll at the University of Missouri. There he was diverted from journalism into astronomy, finally moving on to Princeton to study under Russell, who once told Edward Pickering, \"He is the best student I ever had.\" For all Shapley's self-assured swagger\u2014he was certain that he had mapped the length and breadth of the universe\u2014he never quite lost his rough country edge.\n\nFour years later, and about seventy miles east of the Shapley farm, Hubble was born. The family was more prosperous than the Shapleys (Mr. Hubble was an attorney turned insurance executive). A straight-A student and an athlete, Edwin won a scholarship to the University of Chicago and became a Rhodes Scholar at Oxford, where he studied law and picked up the fake British accent that would grate on some of his colleagues like a kid squeaking a balloon.\n\n_Edwin Hubble_ (Courtesy of the Archives,  \nCalifornia Institute of Technology)\n\nHubble, like his father, didn't take to practicing law. After briefly working as a high school teacher in Indiana (appearing before his classes in knickers and a cape), he returned to Chicago to pursue a doctorate in astronomy. Then, after serving as an officer in World War I, he came to Mount Wilson, where he and Shapley found themselves working uncomfortably under the same dome-shaped roof. What, Shapley wondered, was a Missouri boy doing exclaiming, \"Bah Jove,\" or remarking that a plan had \"come a cropper\"? Taking note of Hubble's aristocratic and somewhat military bearing, some astronomers began referring to him as the Major.\n\nHubble, a rather reserved sort, found Shapley overbearing and erratic\u2014shooting off one wild idea after the other\u2014and he was particularly put off by Shapley's friend, the garrulous Dutch astronomer Adriaan van Maanen, whose love for dinner parties and socializing made him something of a standout in stodgy Pasadena. Van Maanen was also known as a meticulous astronomer\u2014his measurements of the rotational velocity of the spiral nebulae provided the strongest argument against island universes. Having rechecked and refined his data, he remained adamant: either the spirals were small and nearby or they were spinning at insanely high speeds. He believed, as his teacher Kapteyn had taught him, that there can be only one galaxy, the Milky Way.\n\nHubble himself leaned toward the island universe theory, but for now he wasn't involving himself in the controversy. He hadn't come to Mount Wilson to confer with mortals about their astronomical opinions. The answers would be found in the stars. Within weeks he was sitting through long nights beneath the observatory dome, consorting with the sky. He spent Christmas Eve 1919 with his eye to Mount Wilson's new 100-inch telescope, 40 inches greater in diameter than the one Shapley had used to map out his Big Galaxy. For the next three decades it would be the largest in the world. Hubble was looking particularly hard at nebulae and wondering, perhaps, when Harlow Shapley was going to get off his mountain.\n\n**2**\n\nBy now, Henrietta Leavitt and her widowed mother had taken up housekeeping in a recently built brick apartment building on Linnean Street and Massachusetts Avenue, several blocks from Harvard Observatory. Although she was largely occupied with more routine tasks, variable stars were still very much on her mind. In 1920 she wrote to Shapley seeking his advice. Where would he suggest she focus her research next? He replied, still hammering on an old theme, that it would be of \"enormous importance in the present discussion of the distances of globular clusters and the size of the galactic system\" if she would plot the periods of some of the dimmer variables in the Small Magellanic Cloud, those \"just fainter than the faintest already studied.\" This is what he had been pestering Pickering about until several months before his death.\n\nAnd perhaps she would see if her discovery about the Cepheids also held for those in the Large Magellanic Cloud. \"Does the period-luminosity law apply there?\" He was treating her almost like a colleague. Soon he would be her boss.\n\nShapley had been overestimating how badly Harvard wanted him. At first the university's president, Abbott Lawrence Lowell, had considered him the obvious choice to succeed Edward Pickering as director. But after consulting with several astronomers, Lowell found himself leaning toward offering Shapley the number-two spot, with an older, more experienced astronomer like Henry Norris Russell running the show. Shapley struck some of his colleagues as young and immature, and perhaps too brash for Harvard. \"He is much more venturesome than other members of our staff,\" Shapley's boss, George Ellery Hale, confided to Lowell, \"and more willing to base far-reaching conclusions on rather slender data.\" And, as Shapley had feared, his lackluster performance at the Great Debate hadn't helped matters. Even Russell came away persuaded that his prot\u00e9g\u00e9 was not ready to run Harvard Observatory. \"Shapley couldn't swing the thing alone,\" Russell told Hale. \"I am convinced of that after trying to measure myself with the job, and observing Shapley in Washington. But he would make a bully second.\"\n\nIn the end Shapley, willful as ever, got what he wanted. Ultimately Russell turned down Harvard's offer and Shapley made it clear that he wouldn't settle for anything less than the directorship. Hale interceded on his behalf and Harvard agreed to try out the young astronomer on a one-year probation. In the spring of 1921 he moved to Cambridge to take over where Pickering had left off.\n\n\"MARCH 28, 1921:Dr.Shapley arrived!\"wrote Annie Cannon, who had become one of the most accomplished of the observatory's women assistants, in her diary. \"I like him. So young, so clean, so brilliant.\" Like Henrietta Leavitt, Cannon was deaf, though only partially so. The following week she and a friend invited Shapley over for dinner and they all went to the symphony.\n\nBy now Leavitt was the head of stellar photometry, and the ebullient Cannon was curator of the photographic plate collection and chief compiler and overseer of the Henry Draper Catalogue of stellar spectra. Ultimately it filled nine volumes with more than 225,000 stars classified according to their spectral type, from the hottest white-blue stars to the cooler yellow, orange, and red. (Cannon's categories were called, cryptically, O, B, A, F, G, K, and M, which astronomers, some of the men anyway, remember with this mnemonic: \"O Be A Fine Girl, Kiss Me\").\n\nUnder Pickering, the status of the women computers had continued its slow climb. He even tried, with no success whatsoever, to persuade the president of Harvard to grant Cannon the prestige of an academic appointment, or at least to include her name in the school's catalog. Women were praised, a little condescendingly, for being good at detail work, the numerical needlepoint of analyzing astronomical imagery, but deeper matters were still reserved for the men. One of the more overqualified assistants, Antonia Maury, a Vassar graduate, chafed under the tedium. \"I always wanted to learn the calculus,\" she later said, \"but Professor Pickering did not wish it.\"\n\nMaury, truth be told, could be a pain to work with. She had been hired because she was Henry Draper's niece. Her work was slow and erratic, prompting her aunt to apologize for her behavior. \"I shall be happy,\" she had written to Pickering, \"when you are rid of the annoyance.\" But Maury's bad attitude was inflamed by a feeling that she was being discouraged from making original contributions.\n\nIn her own diary, Williamina Fleming, the housekeeper turned astronomical assistant, expressed the sense of frustration and stoicism some of the computers felt: \"If one could only go on and on with original work, looking to new stars, variables, classifying spectra and studying their peculiarities and changes, life would be a most beautiful dream; but you come down to its realities when you have to put all that is most interesting to you aside, in order to use most of your available time preparing the work of others for publication. However, whatsoever thou puttest thy hand to, do it well.\"\n\nCannon was happier with her lot. When a young British astronomer named Cecilia Payne arrived to study at the observatory in 1923, she wondered how Cannon could have spent all those years under Pickering meticulously classifying stars without speculating on what the new taxonomy might mean. In Cannon's case, Payne came to conclude, theorizing was against her nature: \"She was a pure observer, she did not interpret.\" And she seemed to rely less on reason than on instinct. \"She was like a person with a phenomenal memory for faces,\" Payne observed. \"She did not think about the spectra as she classified them\u2014she simply recognized them.\" When she needed to concentrate, she would disable her hearing aid.\n\nLeavitt's feelings about her own work have gone unrecorded. No revealing confessions or letters have been found, just the occasional anecdote. One day, confronted with a particularly mysterious variable called Beta Lyrae, she exclaimed to a colleague, \"We shall never understand it until we find a way to send up a net and _fetch the thing down_!\" She yearned, perhaps, to rise above the columns of numbers and really know the stars. Yet even after her discovery of the Cepheid law, she remained assigned to routine photometry, more astronomical needlework. As Cecilia Payne later put it, \"Pickering chose his staff to work, not to think.\"\n\nPerhaps this would have changed with Shapley. More than anyone, he had seized on Henrietta Leavitt's stars to plumb the depths of space. He later called her \"one of the most important women ever to touch astronomy.\" Considering how very few women there were in the field, it is hard to gauge how this weighed on his scales of praise. This was a man who measured the computational difficulty of astronomical jobs in \"girlhours\" and the really difficult ones in \"kilo-girl-hours.\"\n\nAny chance the two might have had to collaborate was short-lived. Leavitt, still living with her mother on Linnean Street, was sick again, this time with cancer.\n\n\"Took flowers to Miss Leavitt who is very ill,\" Cannon wrote in her diary for November 6, 1921. It was a dreary month. By Thanksgiving, Cambridge was besieged by the worst ice storm in memory. Trees and electric poles were breaking under the clinging sleet. The observatory lights went out.\n\nCannon's diary describes what came next:\n\n_December 6. Went to see poor Henrietta Leavitt, dying with a_ _malignant stomach trouble. So thin & changed. Very, very, sad._\n\n_December 8. Clear and cold._\n\nShapley dropped by to pay his respects. \"One of the few decent things I have done was to call on her on her death bed,\" he later said. \"It made life so much different, friends said, that the director came to see her.\" Maybe so.\n\n_December 12. Rainy day pouring at night. Henrietta passed_ _away at 10.30_ _p.m_ _._\n\n_December 13. Mr. Leavitt, Henrietta's brother, called early in_ _morning. Snowy, sloppy, dark day._\n\n_December 14. Wednesday. Henrietta's funeral at Chapel of 1st_ _Cong. Church 2_ _p.m_ _._ _Coffin covered with flowers._\n\nShe was buried at Cambridge Cemetery (across from the better known Mount Auburn), in the Leavitt family plot. Sitting at the top of a gentle hill, the spot is marked by a tall hexagonal monument, on top of which (cradled on a draped marble pedestal) sits a globe. Her uncle Erasmus and his family are also buried there, along with other Leavitts. A plaque memorializing Henrietta and her two siblings who died so young, Mira and Roswell, is mounted directly below the continent of Australia. Off to one side, and more often visited, are the graves of Henry and William James.\n\nA few days before her death Leavitt had written out her will in longhand, leaving her mother an estate of odds and ends:\n\nBookcase and books $5\n\nFolding screen $1\n\nRug $40\n\nTable $5\n\nChair $2\n\nDesk $5\n\nTable $5\n\nRug $20\n\nBureau $10\n\nBed-stead $15\n\nMattresses (two) $10\n\nChairs (two) $2\n\nOne @ $100 face value First convertible 4% Liberty Bond $96.33\n\nOne @ $50 face value Fourth 4\u00bc% Liberty Bond $48.56\n\nOne @ $50 face value Victory 4\u00be% Note $50.02\n\nThe total appraised value came to $314.91.\n\nAlso left behind was a photometric survey of the southern sky, and a study of the light curves of novae, including, as a Harvard annual report later put it, \"the famous new star of 1918,\" which had flared in the constellation Aquila. And she was not quite done with another round of revisions to her magnum opus, the North Polar Sequence. When the International Astronomical Union held its first general assembly in Rome the following May, the Commission of Stellar Photometry, of which she had been a member, recognized her \"great service to astronomy.\" \"She was one of the pioneers in a difficult field of investigation in which she worked with conspicuous success, and it is deeply regretted that she was unable to finish this her last undertaking.\"\n\nThe next year a Harvard administrative report noted, in passing, something that may have mattered to her more: \"She had hardly begun work on her extensive program of photographic measures of variable stars.\"\n\nShapley gave her desk to Cecilia Payne. He tried to persuade her to take over Leavitt's unfinished projects, but she had other ideas. After completing a celebrated dissertation on the chemical composition of stars, Payne earned Harvard's first doctorate in astronomy and, under her married name, Cecilia Payne-Gaposchkin, went on to become a full professor and chair of the astronomy department. She never got to meet Leavitt, but she was touched by her story and came to believe that she had been done a great wrong.\n\n\"I heard it said when I came to Harvard that what really killed Miss Leavitt was Pickering's requirement that she devise a method by which the photographic magnitudes determined with all the Harvard instruments could be reduced to the same photometric system,\" she wrote years later. \"I cannot believe that he made so unrealistic a request.\" The judgment seems extreme. This was Leavitt's North Polar Sequence, which she had taken such pride in. But it did keep her away from her Cepheids.\n\nPayne could understand, from a managerial perspective, that it made sense to assign the best of the assistants to tasks that, however onerous, had to be done. \"But it was also a harsh decision,\" Payne wrote, \"which condemned a brilliant scientist to uncongenial work, and probably set back the study of variable stars for decades.\"\n\nFour months after the funeral, Annie Cannon found herself on a steamer bound for Peru, for a tour of the Andes and a visit to the observatory in Arequipa. One evening, after immersing herself in the clear southern skies, she made a note in her diary: \"Magellanic Cloud (Great) so bright. It always makes me think of poor Henrietta. How she loved the 'Clouds.' \"\n\n**3**\n\nLike Shapley, Heber Curtis had also moved up in the world, becoming director of the Allegheny Observatory near Pittsburgh. His successor at Lick, a young Swede named Knut Lundmark, carried on the tradition of antagonizing Shapley, claiming that he had been able to pick out individual stars in a pinwheeled nebula called Triangulum or M33 (after the number given it in the eighteenth century by the French astronomer Charles Messier). Assuming (1) that these were the brightest stars in the nebula (which is presumably why he could see them) and (2) that they had the same average magnitude as the brightest stars in the Milky Way, he estimated that M33 was more than a million light-years away. It was, in other words, a full-fledged galaxy.\n\nShapley quickly challenged him in a letter. Why had his paper not mentioned the work of his friend Adriaan van Maanen, who had measured the rotation of that very spiral, showing it must be small and nearby, or else spinning at impossibly high speeds? Lundmark replied diplomatically and Shapley, for now, was mollified. But he couldn't resist firing off one of the sarcastic slights that were becoming his trademark: \"Whether or not you care to recognize that [van Maanen's] measures, if real, practically eliminate the 'island universe' hypothesis, which you seem to espouse at present time more strongly than any one, is not a matter I can properly concern myself about.\"\n\nLundmark was not so easily defused, as Shapley learned to his annoyance when he picked up an astronomical journal a few months later and read a new paper entitled \"On the Motions of Spirals.\" The young astronomer was pushing island universes even more vigorously than before, directly questioning the validity of van Maanen's measurements.\n\nOthers had also uncovered problems with the data. The British astronomer James Jeans had studied the van Maanen rotations and found that they violated the known laws of physics. Still he was inclined to believe them (they supported his own theory of how galaxies evolve), explaining away the discrepancies with no less than a proposed modification to Newton's law of gravity.\n\nIf Lundmark was right, van Maanen's findings were riddled with inconsistencies. He was not suggesting that the astronomer had been sloppy. Measuring such tiny displacements was maddeningly subtle work and open to interpretation. When an object takes 100,000 years to make a single revolution, as van Maanen had concluded, it is not going to move very much in a single year or a decade, or even in the flash of a human lifetime.\n\nNo one understood this better than Lundmark. Months later, when he was remeasuring the images of M33, he briefly convinced himself that it really was spinning, so rapidly that \"the situation seemed to be rather hopeless for the followers of the island universe theory.\" Shapley, of course, was delighted. But the crisis of confidence quickly passed. Closer scrutiny assured Lundmark that he had succumbed to an illusion: there was no sign from this great distance that M33 was rotating at all.\n\n**4**\n\nBack at Mount Wilson, Hubble was training his sights on Andromeda. It was October 4, 1923. After taking a time exposure of the nebula, he saw what he suspected was another nova. The sky was hazy, so he tried again the next night. This time there seemed to be no question. A second photographic plate showed what appeared to be three novae.\n\nIn the observatory office down the mountain in Pasadena, he compared his plate with previous ones taken by Shapley and others. The confirming sign of a nova would be a bright spot appearing where none had been before. One of his flares, however, behaved much differently. Over a period of about a month it had brightened, dimmed, and then brightened again. This was a far more important finding than Hubble had expected. He marked the plate \"VAR!\" and in February wrote to Shapley: \"You will be interested to hear that I have found a Cepheid variable in Andromeda.\" According to the period-luminosity scale that Shapley himself had calibrated\u2014Shapley's curve\u2014the spiral must be a million light-years away.\n\nHubble went on to boast that, in fact, he had found two Cepheids and nine novae in Andromeda and expected more to come soon. \"Altogether next season should be a merry one and will be met with due form and ceremony.\"\n\nThere he was trying to sound like an Oxford don again. Cecilia Payne later recalled being in Shapley's office when the dispatch arrived: \"Here is the letter that has destroyed my universe,\" she remembered his saying, a bit melodramatically. One wonders whether Payne's memory was a little foggy and the visit actually came later, for Shapley's immediate reaction was hardly one of defeat.\n\n\"Your letter telling of the crop of novae and of the two variable stars in the direction of the Andromeda nebula is the most entertaining piece of literature I have seen in a long time,\" he replied a few days later. Continuing in this vein, he argued that Cepheids with periods as long as a month, like the one Hubble had used for his distance measurement, were unreliable as standard candles. It was most likely that what he had found was not a Cepheid at all. False Cepheids, Shapley said, were discovered all the time. He could show Hubble some examples, if he'd like, from the Harvard plate collection. It was going to take a lot more to convince Shapley that his map of the universe was wrong.\n\nHubble kept on looking at the sky. Pointing the 100-inch telescope toward the constellation Sagittarius, he zoomed in on an irregular-shaped patch of light that resembled a smaller, dimmer version of the Magellanic Clouds. It had first been spotted in the mid\u20131880s through a 5-inch telescope by Edward Emerson Barnard, an amateur astronomer with so keen an eye that he was given a fellowship at Vanderbilt and later hired as a professor at the University of Chicago. Later observations showed that Barnard's discovery (usually called by its New General Catalog number, NGC 6822) was a cluster made up of several smaller nebulae and numerous individual stars.\n\nThe question, of course, was whether this conglomeration was part of the Milky Way. During 1923 and 1924 Hubble took some fifty pictures of Barnard's cloud, then compared them with images from earlier years using a blink comparator. As he flipped back and forth between the two plates, variable stars pulsed like traffic lights. He found fifteen of them, concluding that most were Cepheids. According to the period-luminosity scale, what could now almost unequivocally be called Barnard's Galaxy was 700,000 light-years away.\n\nHubble also found more Cepheids in Andromeda and in its neighbor M33. This time he broke the news to Shapley with the gentleness of someone who knew he had changed astronomy. Allowing that it was premature to draw final conclusions, he noted that \"the straws are all pointing in one direction and it will do no harm to begin considering the various possibilities involved.\"\n\nShapley understood that this was an understatement.\"I do not know whether I am sorry or glad,\" he replied. \"Perhaps both.\"\n\nIn a later paper Hubble remarked that NGC 6822 indeed appeared to be \"a curiously faithful copy\" of the Magellanic Clouds. The galaxy had the same general shape and structure. It was just smaller and dimmer. If one trusted the Cepheids to measure the distances, the reason became clear. Barnard's Galaxy was farther away. For Hubble this consistency was vindication not just of the Cepheid yardstick but of an even grander principle: \"The principle of the uniformity of nature thus seems to rule undisturbed in this remote region of space.\"\n\n**5**\n\nWith few exceptions, the astronomical world almost immediately recognized that the island universe debate had come to an end. Even Henry Norris Russell realized that he had backed the wrong horse. He urged Hubble to announce his findings at the annual meeting of the American Astronomical Society, to be held jointly in Washington with the American Association for the Advancement of Science. In recent years AAAS, for short, has lost its significance as a place to unveil important new science. The huge annual meetings are devoted primarily to educational sessions, giving scientists and reporters a chance to catch up on developments in various fields. For many scientists and science writers the conference is mostly an opportunity to socialize. But in 1925, the meeting was, as Russell put it in a letter to Hubble, \"a splendid forum for a major scientific announcement.\" Russell also assured his young colleague that he was a natural for the recently established AAAS Thousand Dollar Prize for the most outstanding paper of the previous year. He was exasperated when, after arriving in Washington for the meeting, no paper from Hubble had been submitted. It arrived, however, at the last moment, and Russell himself read it from the floor. (In the end Hubble shared the prize with the author of two papers on protozoa inside the digestive tracts of termites.)\n\nAn abstract of \"Cepheids in Spiral Nebulae\" was published in May 1925, more than a year after Hubble had first broken the news to Shapley. The reason for the delay, Hubble told colleagues, was the flat contradiction between his results and van Maanen's. Rechecking his data once again, van Maanen continued to insist that he saw a rotation. But the closer Hubble looked, the more inclined he was to join Lundmark in concluding that the movement was spurious. It was a phenomenon visible to only one man.\n\nTo this day no one quite understands where van Maanen went wrong. Perhaps the most compelling theory is that the images of these stellar pinwheels and whirlpools look as though they _should_ be turning (as indeed they are, though many times more slowly). Van Maanen may have been subconsciously influenced by his expectations, seeing what he expected to see.\n\nWhy Shapley continued to embrace van Maanen's theory, until it was no longer possible to do so, requires no elaborate analysis. Cecilia Payne heard him explain it years later: \"After all, he was my _friend_.\"\n\nHad Shapley stayed at Mount Wilson, booking time on the mighty 100-inch scope, this new grander universe might have been his own. People would turn on their televisions decades later and admire the glorious photos taken by the orbiting Shapley Space Telescope. Instead he is primarily remembered, a little unfairly, as a great astronomer who couldn't see beyond the galaxy, who convinced himself that there could be nothing but the Milky Way.\n\nIn an interview decades later, he claimed to have forgotten all about the Great Debate. But as he looked back to the 1920s, the details, a bit scrambled, slowly emerged. His most vivid memory was the false one of Einstein's being there. Shapley said he was surprised that historians were now making so big a deal about the event, contending, a bit disingenuously, that on the \"assigned subject matter\"\u2014the scale of the universe\u2014 he was the clear winner. \"I was right and Curtis was wrong on the main point\u2014the scale, the size. It is a big universe, and he viewed it as a small one.\"\n\nBut that seems in retrospect a minor point compared with what Curtis got right\u2014that our galaxy, no matter how large or small, is one among a multitude, a small outpost in what Hubble would come to call \"the realm of the nebulae.\" His prot\u00e9g\u00e9 Allan Sandage later put it like this: \"What are galaxies? No one knew before 1900. Very few people knew in 1920. All astronomers knew after 1924.\"\nCHAPTER 8\n\nThe Mysterious K\n\nYouth Who Left Ozark Mountains to Study Stars Causes Einstein to Change His Mind.\n\n_\u2014Springfield Daily News, February 5, 1931_\n\nIn 1925, with the Roaring Twenties half over, John T. Scopes was convicted of violating a Tennessee law against teaching evolution, or any theory denying that the universe had been created as described in the Book of Genesis. Biblical fundamentalists, offended by the suggestion that they were related to monkeys, might have been even more disturbed had they known about recent developments in astronomy. After more than 2,000 years of measuring, scientists were finding little reason to maintain the belief that there was anything special about the position of the sun within the Milky Way, or of the Milky Way itself within the endless sea of galaxies called the universe.\n\nHad there been, say, a Hubble Star Trial resembling the Scopes \"Monkey Trial,\" one can imagine how the prosecution might have argued its case against the propounder of so great a heresy. It would have been difficult, and very unconvincing, to challenge the simple rules of geometry used to triangulate distances within the solar system or even to the nearest stars. Whether the baseline was the diameter of the Earth or the diameter of its orbit around the sun, the reasoning behind the measurements appealed to simple trigonometry and old-fashioned common sense.\n\nMore suspect, from the point of view of the astronomical fundamentalists, might have been some of the more indirect techniques\u2014\"mathematical mumbo jumbo,\" a William Jennings Bryan might have objected. And he might have really gone to town with the Cepheids themselves. Perhaps the intrinsic brightness of the variable stars twinkling in the Magellanic Clouds is indeed signaled by how fast they pulse. But is it not a leap of faith to assume that the very same rule applies to Cepheids throughout the universe? Could not the good Lord have made his stars blink in any way he liked?\n\nHere a Clarence Darrow, rising to the defense, might have prodded his client Hubble to remind the jurors how the distances derived from Cepheids harmonized so nicely with measurements made using other techniques, like the average luminosity of a galaxy's brightest stars. Completely independent gauges all seemed to point toward the same conclusion. \"But,\" Bryan might have countered, \"do not _all_ these methods assume that stars near the Earth are fundamentally the same as stars in the far reaches of the heavens? Is that not a leap of faith?\"\n\nDarrow would have been smart to concede the point. What astronomers were taking on faith was the principle of uniformity\u2014that the laws of physics apply equally in all parts of the cosmic realm. It would be an insult to the Creator to think he would design a universe in any other way.\n\nTaking this idea to heart, one could now estimate the distance to any galaxy for which it was possible to pick out individual stars. Assuming that they were similar in kind to those nearby, you could guess their intrinsic brightness and use them as standard candles. Reaching into the astronomical grab bag of variables, novae, and so forth, you could piece together a map of the universe, or at least those regions closer in.\n\nBeyond a certain distance, the method began to falter. Even with Mount Wilson's 100-inch telescope, most nebulae by far were featureless blurs. There was little hope of finding a blinking Cepheid or even an exploding star. For very rough measurements, you could choose a closer galaxy whose distance you had already established and take the whole thing as a standard candle: estimate its intrinsic brightness and assume that the farther galaxies produce approximately the same amount of light. Then the inverse square law would kick in. The method was similar to that used in the eighteenth century when astronomers assumed, both out of ignorance and for convenience, that all stars were equally bright, making dimness a simple gauge of distance\u2014and it was just as prone to error. Maybe the galaxy you were using as your standard was atypical, far brighter or dimmer than most. Still, if you averaged together the brightness of several known galaxies and used that as your yardstick, you could at least make a plausible argument. With this crude method, astronomers were reaching beyond Andromeda, finding galaxies whose light took millions of light-years to reach their telescopes.\n\nThe process was a bit like constructing a tall building with each level resting on the one below. With the diameter of the Earth as a baseline, you measure the distance to the sun. Assuming that figure is correct, you know the width of Earth's orbit, a larger baseline from which to triangulate the very closest stars. Building on this information, you chart the sun's own motion through the galaxy, providing an enormous baseline that, with some statistical sleight of hand, lets you measure the Cepheids and calibrate Henrietta Leavitt's yardstick, and from there you make the next great leap.\n\nThe higher you climb, the more precarious the structure becomes. Perched with their telescopes at the loftiest level, astronomers knew it was foolish to be overly confident. At any moment a lower support might buckle. Everything they had built could come tumbling down.\n\n**2**\n\nCompared with the rickety nature of the distance scales, the celestial speed stick was fairly robust. Because of the Doppler effect, anything that emitted light (stars, galaxies, clouds of gas) could be clocked according to its color shift\u2014a shrieking high-pitched blue if it was speeding toward you, a moaning low-pitched red if it was speeding away.\n\nMore specifically, the velocity was determined by the displacement of an object's spectral lines. The method depended on a discovery by the German scientists Gustav Kirchhoff and Robert Bunsen, who found in the 1850s that they could identify chemical elements by burning them in a flame and refracting the glow through a prism. Certain colors would stand out in the spectrum\u2014a combination of bright vertical lines as unique as a fingerprint. Sodium, for example, burns yellow. Seen through a spectroscope, it can be identified by a pair of bright \"emission lines\" at precise points in the yellow part of the spectrum.\n\nWhen Kirchhoff and Bunsen made the discovery, the existence of atoms was still controversial. Once they were discovered, the effect could be simply understood: when an atom is energized, its electrons jump into higher orbits. When they fall back down they emit various frequencies of light. Every kind of atom is built a little differently, its electrons arrayed in a specific way, resulting in a characteristic pattern.\n\nFor similar reasons, if you shine a light through a gaseous substance, like hydrogen or helium, certain colors will be filtered out. The result in this case is a characteristic pattern of black \"absorption\" lines interrupting the spectrum\u2014another unique chemical fingerprint. (The same colors marked by the absorption lines would appear as bright emission lines if the element was burned.) A scientist named Joseph von Fraunhofer had shown that lines like these appear in the spectrum of the sun. Using nothing more than a prism, one could stand on Earth and decipher the composition of the glowing orb 93 million miles away.\n\nThe natural next step was to add prisms to telescopes and analyze the chemical composition of stars and nebulae. They too exhibited the dark Fraunhofer lines, but not in the expected positions. They were displaced toward the red or blue end of the spectrum. Assuming this was caused by the Doppler effect, you could precisely gauge a galaxy's velocity. A few, like Andromeda, were blue-shifted, approaching the Milky Way, but these were exceptions. Most were severely red-shifted, hurtling away at extremely high speeds.\n\nAs the 1920s drew toward a close, astronomers were finding hints of an even stranger phenomenon: the smaller, fainter, and thus presumably more distant a nebula was from Earth, the greater its redshift. Some astronomers dismissed this as an anomaly, a flaw in their techniques. Farther objects were obviously harder to analyze than closer ones. Some kind of systematic error might cause observers to overestimate the more distant redshifts. To correct for the mistake, they added a fudge factor to their calculations, a term they labeled K. It was considered, at this point, no more than a Band-Aid on the equations. When observations improved, it could be peeled off and thrown away.\n\nThere was also a more interesting possibility: that the K term was describing a real physical effect\u2014that redshift actually did increase with distance. Grasping for an explanation, some theorists proposed that they were witnessing a heretofore unknown quality of light: the farther it traveled, the more its waves stretched and sagged toward the red end of the spectrum. This became known as the \"tired light\" theory. Perhaps, some proposed, the cause was some Einsteinian peculiarity of curved space-time.\n\nFinally, it was possible that distant galaxies truly were flying away faster than closer ones. This seemed almost too good to be true. In a universe like this, redshift would provide the ultimate measuring stick. The distance to anything, no matter how far, could be measured as long as you could gather its light. Assuming that all stars are made from the same basic ingredients\u2014hydrogen, helium, et cetera\u2014they can be expected to exhibit the familiar patterns of spectral lines. The more the lines appear to be displaced, the faster the galaxy is moving, and\u2014if the theory is correct\u2014the farther away it is.\n\nDuring a tour of Europe in 1928, Edwin Hubble, now the toast of his profession, heard reports of this curious redshift-distance connection. When he returned, he decided to investigate. He asked an assistant, Milton Humason, to train the 100-inch Mount Wilson telescope on some distant nebulae and see how their spectra behaved.\n\nEven more than Henrietta Leavitt, Humason seemed an unlikely candidate for the astronomical hall of fame. He began his career at Mount Wilson as a mule driver, carrying material and supplies up the mountain. He married the daughter of the observatory engineer and talked his way into a job as janitor. Given the chance to learn how to make photographic plates, he quickly proved himself to be a very good photographer of starlight and was promoted to assistant astronomer. He had a grade school education.\n\nHumason began his assignment by targeting a nebula so far away that no one had been able to measure its redshift: NGC 7619. The light, funneled through a prism, left its rainbow on the photographic plate. When it was developed it showed two familiar dark lines, indicating the presence of the element calcium. As expected, the lines were pushed toward the red. What was unexpected was how very big this displacement seemed to be. Humason took another picture to confirm the result. Working with a Mount Wilson computer (referred to in his paper simply as Miss MacCormack), he reported that he had clocked a galaxy speeding away at a rate \"twice as large as any hitherto observed\": 3,779 kilometers per second, or more than 8 million miles per hour, a velocity that would get you from here to the moon in under two minutes.\n\nIn the following weeks, Humason measured more redshifts while Hubble scrutinized the results. By now he had compiled a list giving the recessional velocities of forty-six nebulae. He believed he had reliable distances to about half of them, derived from Cepheids, novae, and other yardsticks. For this sample, speed indeed seemed to increase with distance, and in a delightfully straightforward way. Some astronomers had suggested that the relationship might be \"quadratic\": velocity would increase as the square of the distance. Others suggested more elaborate equations. What Hubble found could hardly have been simpler: A nebula twice as distant as another would be traveling at twice the speed. Triple the distance and the velocity would triple as well. The relationship was what a mathematician calls linear. Take any nebula in the universe and divide its speed by its distance. The result is always the same number\u2014about 150 by Hubble's reckoning.\n\nThis powerful number was none other than the mysterious K astronomers had been puzzling over. The \"error\" was apparently not an error at all but a factor describing how redshift increased with distance. A galaxy that was 1 million light-years from Earth was receding at about 150 kilometers per second. A galaxy 10 million light-years away traveled at 1,500 kilometers per second. Velocity equals distance times K. Or, more significantly, distance equals velocity divided by K. Applying his new formula to NGC 7619, the galaxy Humason had clocked at the breakneck speed of 3,779 kilometers per second, put it at more than 20 million light-years from earth.\n\nIt is impossible to sense from Hubble's typically understated paper, or from the droning account he delivered a few years later in a lecture series called \"The Realm of the Nebulae,\" what he felt as the pieces of a new picture of the universe fell into place. Conservative as always, he cross-checked his measurements, testing whether they resulted in absurd conclusions. Using the galaxies' apparent magnitudes and his new Doppler-derived distances, he computed how bright they would really be. The results were reassuring, comfortably within the range of galaxies closer by.\n\nIn the following months, Hubble and Humason continued to test the theory. Hubble would calculate a nebula's distance using various measuring sticks and predict the redshift before Humason had even measured it. By now they were clocking galaxies with velocities as high as 20,000 kilometers per second (from Earth to moon in 20 seconds), putting them more than 100 million light-years away.\n\nThe numbers were so large that some astronomers were initially doubtful. On a visit to Pasadena, Harlow Shapley told a colleague, \"I don't believe these results.\"\n\nNot that Hubble or his assistant would have cared. Humason remembered one of his last encounters with Shapley, when he was still working on Mount Wilson. Scrutinizing plates of Andromeda with the blink comparator, Humason had spotted what he believed to be stars that varied periodically in brightness. This was more than two years before Hubble made his landmark discovery, establishing with Cepheids that Andromeda was a distant, neighboring galaxy. Humason marked off the places where these anomalies occurred and took the plates to Shapley.\n\nDismissively explaining why they couldn't be Cepheids, Shapley took out his handkerchief and wiped the plates clean, erasing the data. A few months later he departed for Harvard.\n\n**3**\n\nAt heart, Hubble, like Edward Pickering, was an observer not a theorist, leery of speculating beyond what his eyes could see. It took an Einstein to explain the theory and the mechanism of what astronomers were soon calling the Hubble shift (the K in the equations ceremoniously replaced by an H). Why were the galaxies moving\u2014and why were they, with so few exceptions, all hurtling outward from the Milky Way?\n\nThink of the galaxies as runners in a race. After a certain amount of time has passed, they will be distributed according to their swiftness, the fastest ones farthest from the starting line and the slowest ones closest in. But didn't that imply that there was something special and outrageously non-Copernican about our position in the universe, as the point from which everything else was retreating?\n\nA motionless universe was far easier to fathom, even at first for Einstein. When his own general theory of relativity implied that the cosmos may be expanding, he had rejiggered the equations, committing what he would come to regard as an embarrassing mistake. Now he knew that the adjustment had been superfluous. The universe really moved. Visiting Pasadena in 1931 he told Hubble's wife that her husband's work was \"beautiful\" and publicly conceded that his earlier conviction of a static universe was wrong. Hubble's hometown newspaper in Missouri picked up the story: \"Youth Who Left Ozark Mountains to Study Stars Causes Einstein to Change His Mind.\"\n\nEinstein was glad he could restore his theory to its pristine form. What his equations now described was a universe in which space itself is expanding. Second by second, the galaxies grow wider apart like dots on an inflating balloon. Viewed this way, Hubble's discovery did not imply that the Earth was in a special position, at the center of the outward rush of everything. From any point in the cosmos the effect would be the same, with galaxies appearing to fly off in every direction. And if you could reverse the clock, everything would become closer and more compact, converging on a single point. The big bang. The universe had a beginning. And maybe it will have an end.\n\nSo rarefied a theory, now taken as gospel, was still of secondary interest to most astronomers. Hubble himself was noncommittal about the meaning of the Hubble constant and the Hubble shift. Expanding universe, tired light\u2014it didn't matter. What he knew for certain was that redshift, for whatever reason, increased with distance, and that gave him a way to measure as far as a telescope could see.\nCHAPTER 9\n\nThe Cosmic Stampede\n\nThe definitive study of the herd instincts of astronomers has yet to be written, but there are times when we resemble nothing so much as a herd of antelope, heads down in tight parallel formation, thundering with firm determination in a particular direction across the plain.\n\n_\u2014J.D.Fernie_\n\nOnce he had gotten used to the idea, the enthusiastic Dr. Shapley embraced the new universe with more gusto than the reserved Dr. Hubble would let himself show. Sometimes it almost seemed that Hubble was being willfully perverse. Hewing to tradition, he steadfastly reserved the term \"galaxy\" for the Milky Way, continuing to refer to Andromeda and the other island universes as nebulae\u2014\"extragalactic nebulae,\" to be exact. Etymologically this might be correct: \"galaxy\" comes from the Greek word for milk. But Shapley, like everybody else, quickly generalized the term, calling all the island universes galaxies. The bandwagon had almost taken off without him. Now he was back on board.\n\nThough Shapley's old view of the Milky Way as the sole galaxy seemed quainter by the year, he appeared to be right about its enormous size. After all, the same calibration of the Cepheids\u2014Shapley's curve\u2014that had revealed a Milky Way a whopping three hundred thousand light-years from end to end had also been used by Hubble to plumb the distance to Andromeda. From there he had extrapolated outward, using redshift to measure a hundred million light-years into space. If Hubble was right about the size of the universe, it seemed, Shapley must be right about the size of the Milky Way.\n\nAnd that led to a dilemma. With their new measuring techniques, astronomers could now judge how large other galaxies were. Just take the apparent size and adjust for distance to get the true diameter. Simple enough. But the results of the calculations were disconcerting. None of the galaxies came out to be anywhere near the size of our own. Andromeda measured only a tenth as wide, while the others ranged from a mere thousand light-years to perhaps seventy-five hundred light-years across. The term \"island universe\" took on a new meaning with the emphasis shifted to the first word. If these distant spirals were islands, Shapley contended, then our own galaxy was a continent.\n\nA few years earlier it would have been acceptable for people to find themselves living in the biggest galaxy around. But perceptions had changed. Shapley had moved the sun from the center of the galaxy, and Hubble had moved the galaxy from the center of the universe. This reversal of perspective was becoming so ingrained that it was a gold standard by which astronomical ideas were judged. If a theory or observation seemed to suggest that we, the observers, happen to occupy an exalted place in the heavens, then it was probably wrong.\n\nOf course it was possible that chance had conspired to put earthlings in a special location. But other discrepancies were harder to dismiss. If the big bang theory was correct, then the size of the universe was an indicator of its age. The larger it was, the longer it had been expanding since the primordial explosion. If the galaxies at the circumference were two billion light-years away, as recent measurements had suggested, it must have taken them two billion years to get there.\n\nTwo billion years seemed like a reasonable number. The Earth, however, measured using the technique of radioactive dating, came out to be four billion years old\u2014twice as ancient as the universe that contained it. Something somewhere would have to give.\n\n**2**\n\nWhen kinks like these develop in the fabric of knowledge, the fault might lie anywhere in the weave\u2014the result of bad data or a false assumption, the malfunction of a dumb machine or of a human brain. People may see things that turn out to be chimeras. Or miss what is right in front of their telescopes.\n\nAt Lick Observatory in California, a Swiss-born astronomer named Robert Trumpler had been studying structures called open clusters in the Milky Way. These stellar aggregations\u2014 Pleiades is an example\u2014are smaller and more loosely packed than the globular clusters Shapley used to map the galaxy. Comparing each cluster's true brightness with its apparent luminosity, Trumpler calculated its distance. With this information in hand, he could convert the cluster's apparent diameter into its true diameter\u2014how big it really was.\n\nAfter measuring a number of them this way, he was forced to a bizarre conclusion: the farther a cluster was from Earth, the larger it appeared to be. What were we doing sitting at the center of so symmetrical an arrangement, surrounded in all directions by increasingly larger star clusters?\n\nMore likely, Trumpler reasoned, he was being fooled by an optical illusion. All his calculations, like those of most every astronomer, took for granted that space was generally transparent, an empty medium through which light could travel unimpeded. If, however, the Milky Way was permeated with a fine cosmic dust, the measurements would be skewed\u2014especially those where the dust was thickest, along the galactic plane. The dimness of a star or galaxy might be due not only to distance but also to this cosmic pollution. The farther the galaxy, the more pronounced the effect. Once Trumpler corrected for the distortion, the clusters turned out to be approximately the same size.\n\nAstronomers had known there was dust in the Milky Way. The surprise was that it could be so pervasive. From almost the beginning, stargazers had contended with light and air pollution here on Earth. As civilization developed and the clarity of the sky degenerated, they placed their observatories higher and higher in the mountains. It hadn't occurred to them that space itself could be so dirty.\n\nAnd so came the first step toward resolving the Big Galaxy problem, the anomalous size of the Milky Way. When Shapley had taken his measurements, he had been looking through a fog. That made some of his beacons seem much farther than they really were. Once dust was factored into the equations, the home galaxy began to contract in size. That still left it larger than the others, but the adjustment felt like a step in the right direction. More revisions were about to come.\n\n**3**\n\nWhile Shapley's map of the galaxy shrunk, the one for the universe grew larger. Again the reason was cosmic dust. Because of the galactic haze, Shapley had underestimated the true brightness of the Milky Way Cepheids, the ones at the foundation of the period-luminosity scale. When Hubble relied on this same standard to measure the distance to Andromeda he was, in effect, mistaking 75-watt bulbs for 60-watt bulbs. If these Cepheids were in fact burning brighter, then we were seeing them across a greater expanse. Using redshift, other galactic distances had been gauged in terms of Andromeda's, so the error rippled outward. Everything was farther than it had appeared.\n\nAstronomers now routinely adjust for the amount of cosmic pollution. Like dust in the atmosphere intensifying the redness of the sunset, dust in interstellar space can be measured by how much it reddens starlight. The lesson, though, took a decade to sink in. For years, one paper after another ignored the dust factor and, errors canceling out other errors, continued to find confirmation for Shapley's original calibration. Looking back years later, the astronomer J. D. Fernie attributed the blindness to a herd instinct: \"most of the astronomers of the day simply could not bring themselves to believe that interstellar absorption played any important role.\"\n\nDust turned out to be just part of the problem. As far back as the Great Debate, Heber Curtis had suggested that Shapley was overreaching when he assumed that the variables in the Milky Way's globular clusters shared the same relationship between period and brightness as those Henrietta Leavitt had found in the Magellanic Clouds. Both kinds had been lumped together to draw Shapley's curve, the yardstick Hubble had used to measure to Andromeda.\n\nThe principle of uniformity encouraged these kinds of generalizations. But were the two variables really the same? If not, the whole distance scale might be askew.\n\nCertain celestial anomalies hinted that this might be true. Even when one allowed for distance, the brightest of the globular clusters in Andromeda\u2014the ones most easily detected\u2014 appeared to be inherently dimmer than their counterparts in the Milky Way. A German astronomer named Walter Baade later remembered discussing the discrepancy with Hubble, on cloudy winter nights at Mount Wilson as they waited for the sky to clear. Hubble argued that this might be a case where the principle of uniformity should not be so slavishly followed. After all, he noted, the clusters in the more distant galaxy M33, or Triangulum, were even fainter. Maybe this kind of variation was normal.\n\nBaade had a different idea. Maybe the distance scale was wrong. The clusters in Andromeda and Triangulum were not really emitting less light. They were simply farther than had been reckoned. If so, uniformity would be restored. He soon had a chance to put the hypothesis to a test.\n\nAs the mid-1940s approached, many astronomers were off serving in the war. Hubble himself was soon directing ballistic missile tests at the Aberdeen Proving Ground in Maryland. Once Baade, technically an enemy alien, persuaded government authorities that he was not a security threat, he found it easy to book telescope time at Mount Wilson. The periodic blackouts, staged to discourage aerial attacks of Los Angeles, restored the night sky to a primitive blackness. Aiming the 100inch telescope at Andromeda, he could actually resolve individual stars, not just in the spiral arms but inside the galaxy's dense core.\n\nHe discovered what appeared to be two different kinds of starlight. The stars in the galaxy's center and in its globular clusters were colored differently from the \"ordinary\" stars in the galaxy's outer reaches. That meant the two types must have different chemical makeups. While Leavitt's \"classical\" Cepheids belonged to what is now called Population I, Shapley's cluster variables belonged to Population II. It seemed a greater stretch than ever to assume that they obeyed the same law relating period and brightness.\n\nWhen the new 200-inch Hale telescope came on line at Mount Palomar, ninety miles southeast of Mount Wilson, Baade zeroed in on Andromeda for a closer look. The classical Cepheids, he observed, were on average 1.5 magnitudes brighter than the cluster variables. \"Instead of one period-luminosity relation,\" he concluded, \"there are actually two.\"\n\nWhen the new, brighter value of the classical Cepheids was plugged into the inverse square law, Andromeda turned out to be twice as distant as Hubble had reckoned. And so was the distance to everything else. As the newspapers put it, the universe doubled in size overnight. And, from the perspective of the big bang theory, it doubled in age. It was no longer younger than the Earth.\n\nBaade's discovery completed the explanation of why the other galaxies had seemed so much smaller than our own. That too had been an illusion. If they were farther away, then they were also larger.\n\nFinally, with the new adjustments, the Milky Way was taken down to about 100,000 light-years in diameter\u2014right in between where Curtis and Shapley had put it. It was, in the end, as unremarkable as its stars.\n\nHUBBLE DIED IN 1953. Over the next few years, Allan Sandage, the young astronomer who had served as his last assistant, continued to clock redshifts and make adjustments to the distance scale. For some of his calibrations, Hubble had relied on the brightest star method. Sandage showed that what his old boss had taken as individual stars were actually entire stellar regions. Their intrinsic brightness was therefore much greater, putting the galaxies still farther away. Hubble, as Shapley might have put it, had mistaken trees for asters. The universe was expanding, not just because of the big bang but because of the explosion in astronomical knowledge.\n\nAnother way to say it is that the Hubble constant\u2014the number by which you divide a galaxy's velocity to get its distance\u2014was growing smaller and smaller. And so, because of the reciprocal nature of the relationship, the size of the universe continued to grow. Hubble had initially set the constant at 150 kilometers per second per million light-years. More commonly the ratio is expressed using \"parsecs\" instead of light-years. As the Earth orbits the sun, a star that shows a parallax of 1 arc-second (1\/3,600 of a degree) is, by definition, a single parsec (about 3.26 light-years) away. On that scale the Hubble constant had been around 500, Baade knocked it down to 250, and now Sandage to 75. Later he would reduce it again\u2014to 50, ten times smaller than the original value. \"The incredible shrinking constant,\" one astronomer has called it. Every time it gets smaller, the map of the universe grows. So much is packed into that one little number. At its core lie Miss Leavitt's stars.\nCHAPTER 10\n\nGhost Stories\n\n\"What monsters may they be?\"\n\n\"Impersonal monsters, namely, Immensities. Until a person has thought out the stars and their inter spaces, he has hardly learnt that there are things much more terrible than monsters of shape, namely, monsters of magnitude without known shape. Such monsters are the voids and waste places of the sky.\"\n\n_\u2014Thomas Hardy,_ Two on a Tower\n\nFor years after Henrietta Swan Leavitt's death in 1921, her presence lived on, not just in her discovery about Cepheid variables but in a ghost story that circulated around Observatory Hill. Cecilia Payne, the young Harvard astronomer (and later department chairman) who had inherited Henrietta's old desk, was amused to hear rumors \"that Miss Leavitt's lamp was still to be seen burning in the night, that her spirit still haunted the plate stacks.\" More likely, she concluded, someone had seen Payne herself as she burned the midnight oil, sometimes working on theories about variable stars.\n\nThere is something almost ghostlike in the scant traces Leavitt left in the public record\u2014biographical ectoplasm to be shaped according to one's need. After years of so little recognition, there has followed an almost reflexive rush to mythologize her. A planetarium has been named after her, albeit a virtual one residing only on the World Wide Web. While Harlow Shapley had an entire cluster of galaxies named in his honor, Leavitt (and Annie Cannon) got a crater on the moon.\n\nIn brief hagiographies scattered across the Web, the same few scraps of data about Leavitt's life, and often the same sentences, are repeated again and again, all traceable to a few stable sources. She has been turned into a standard bearer by people less interested in her astronomy than in the fact that she was a woman, and deaf. She has been included, absurdly, in a roster of \"The World's Greatest Creation Scientists,\" for no apparent reason other than that she believed in God. She probably would be appalled.\n\nAmong the factoids that ricochet through the infosphere is that she was nominated for a Nobel Prize. What happened is that in 1925 G\u00f6sta Mittag-Leffler, an elderly Swedish mathematician, heard something about Leavitt's work from a colleague and was impressed enough to write her a letter. He didn't know that she had died.\n\n\"Honoured Miss Leavitt,\" he began. \"What my friend and colleague Professor von Zeipel of Uppsala has told me about your admirable discovery of the empirical law touching the connection between magnitude and period-length for the S. Cephei-variables of the Little Magellan's cloud, has impressed me so deeply that I feel seriously inclined to nominate you to the Nobel prize in physics for 1926, although I must confess that my knowledge of the matter is as yet rather incomplete.\" His expertise was in analytic function theory, not astronomy.\n\nHe asked for more information and vowed to handle the matter \"with the greatest discretion and in the way that seems to me most likely to further my plan.\" He also promised to send, for her perusal, a treatise he had written on Sonja Kowalewsky, a stunning young Russian mathematician, and her correspondence with Karl Weierstrass, the older mentor who had furthered her career. Maybe Mittag-Leffler was hoping Henrietta would become his Sonja.\n\nWhen the letter arrived at the observatory, it was forwarded to the director, Harlow Shapley. It is hard to know quite what to make of his reply:\n\n\"Miss Leavitt's work on the variable stars in the Magellanic Clouds, which led to the discovery of the relation between period and apparent magnitude, has afforded us a very powerful tool in measuring great stellar distances.\"\n\n_Led to the discovery_ of the relation between period and luminosity? His phrasing suggests that he would deny her credit for her one breakthrough, relegating her back to the role of human computer, the diligent manipulator of data.\n\nThe next sentence continues in this vein\u2014faint praise weighed with subtle condescension:\n\n\"To me personally [the discovery] has also been of highest service, for it was my privilege to interpret the observation by Miss Leavitt, place it on a basis of absolute brightness, and extending it to the variables of the globular clusters, use it in my measures of the Milky Way.\"\n\nIt is clear whom Shapley would like to nominate for the prize.\n\nUNTIL THE END, Leavitt's title continued to be \"assistant.\" Although Solon Bailey, in his history of Harvard College Observatory, does give her credit for the period-luminosity relationship, he describes the work only in passing and notes, a little belittlingly, \"The number of variables included in Miss Leavitt's discussion was unfortunately rather small, but the data have been much increased since that time, especially by the studies of Shapley.\"\n\nLeavitt probably would have been surprised by how much fuss would later be made over her delightfully simple observation, and by how far Shapley and then Hubble were able to go with it. Given the opportunity\u2014better health, better times\u2014 maybe she would have joined them at the forefront. Or maybe not. Barring the discovery of a lost cache of letters, we may never know.\n\nIn time, someone else would have discovered Henrietta's law. It is the discovery not the discoverer that matters. Miss Leavitt may have understood this in a way that a Shapley or a Hubble never could. She seemed content to be a small part of a greater thing called science.\n\nIn January 1920, the year before she died, a census taker encountered her for the last time in the apartment with her mother on Linnean Street. Among their neighbors were teachers, a salesman for a candy company, a bank clerk, an auditor. Asked to state her occupation, Miss Leavitt replied, honestly and perhaps a bit defiantly, \"Astronomer.\"\n\n**2**\n\nOn a spring afternoon in 1996, seventy-six years to the day since the Great Debate, astronomers gathered at the same lecture hall in Washington for a presentation called, once again, \"The Scale of the Universe.\" The cosmos was seventy-six years older and, if you believed the big bang theory, seventy-six light-years larger in every direction than it had been in 1920.\n\nIt was a real debate this time, with opening arguments, rebuttals, and closing statements presented by two celebrated astronomers, Gustav A. Tammann and Sidney van den Bergh. While Tammann argued for a Hubble constant of around 55, van den Bergh put it at about 80. Plugged into the equations of universal expansion, that narrow difference would translate into a universe ranging, depending upon other factors, somewhere between 10 billion and 15 billion light-years in radius.\n\nLarge as it seemed, the spread had narrowed in recent years. The lowest values for the constant, around 50, continued to be championed by Allan Sandage, who had taken over where Hubble left off, an opportunity and a burden he once compared to having been Dante's assistant and inheriting _The Divine Comedy_. (The legacy included the incomplete\u2014barely begun, in fact\u2014 _Hubble Atlas of Galaxies._ ) Then just as the number seemed set in stone, or at least wet cement, an astronomer named Gerard de Vaucouleurs came along and doubled it back to 100, cutting the size of the universe in half. The ensuing controversy became known as the \"Hubble Wars.\"\n\nAs Sandage's collaborator and prot\u00e9g\u00e9, Tammann was an ideal person to carry on the fight for an older, larger universe, and van den Bergh showed himself to be a formidable opponent. As the debate unfolded, each man in turn challenged the other's choice of standard candles and the manner in which he interpreted them. Watching from the auditorium, members of the audience wore colorful buttons, \"Hubble Meters,\" on which they displayed their own guesses about the constant's value. One might have come away from the debate with the impression that the size of this most fundamental parameter was a matter of opinion.\n\nFor all the constant's inconstancy, Harlow Shapley and Heber Curtis both would have been impressed by how far the craft of intergalactic measuring had come. Edward Pickering and Henrietta Leavitt would have been astonished. What is now the world's largest telescope, the Keck, perched atop the 13,800-foot Mauna Kea volcano in Hawaii, collects light with a mirror 10 meters wide. That's 394 inches, or almost twice the size of the 200-inch telescope at Mount Palomar, which was twice the size of the one that Hubble had used to show that Andromeda is really a galaxy. Just when it seemed that mirrors had become as large as physically possible, on the verge of buckling under their own weight, computer and robotics technology stretched the limits further. The mirror on the Keck telescope was made from thirty-six hexagonal sections nudged back and forth by precision pistons so that the whole thing acts like one enormous reflector. Its shape can be constantly adjusted, a few nanometers at a time, to compensate for atmospheric distortion, a technique called adaptive optics. The glass molds itself to the sky.\n\nIn fact, by the time of the second debate, there were two Keck telescopes sitting side by side on the mountaintop soon to be linked by an optical interferometer, a computerized device that would combine light from both mirrors, along with that from several smaller scopes, into a single image. The result would be as powerful as a telescope with a mirror 85 meters, or 3,346 inches, across.\n\nFor even more acute observations, the Hubble Space Telescope, launched in 1990, was orbiting 380 miles above the Earth, electronically beaming back pictures of deepest space. Among its tasks was looking for Cepheids.\n\nAccording to the picture that has emerged from these investigations, Andromeda, two million light-years away, now appears to be twice the size of the Milky Way. Both these nebulae mark the far edges of the constellation of galaxies called the Local Group, which also includes Triangulum, the Magellanic Clouds, and several dozen dwarf galaxies.\n\nNearby are other groups called Sculptor, Maffei, Canes I, Canes II, Dorado... so many (more than 150) that most are just given numbers. In addition to the groups are larger \"clusters\" like Fornax, with 49 galaxies, and Eridanus with 34. Largest of all in this tiny corner of the universe is the Virgo Cluster, which includes another 200 galaxies. Put all these together and you have the Virgo Supercluster\u2014a galaxy of galaxies, numbering in the thousands, spanning 200 million light-years. One of them is the Milky Way.\n\nFarther beyond are the neighboring superclusters: Coma, Centaurus, Hydra, Pavo-Indus, Capricornis, Horologium, Shapley, Sextans\u201480 of them within a billion light-years. As might be expected, our own Virgo Supercluster turns out to be on the smallish side.\n\nAltogether there are believed to be tens of millions of galaxies just within a billion light-year radius of our solar system\u2014 more galaxies than there once were stars.\n\nFor all the technological progress astronomy has made, the basic approach to measurement has remained fundamentally the same: use redshift to gauge the recessional velocity of a galaxy or a galactic cluster, then divide that number by the Hubble constant to get the distance.\n\nTo calibrate the Hubble scale, Cepheids are still the standard of choice, when you can find them. The Hubble Space Telescope has been spotting them in distant clusters that once remained beyond reach. By the end of its mission in 1993, the European Space Agency's High-Precision Parallax Collecting Satellite, or Hipparcos, had measured parallactic shifts as tiny as one milli-arc-second\u20141\/1,000 of one second of one degree. The thousands of stars whose distances it gauged included a number of Cepheids.\n\nIt would be comforting to report that Hipparcos had been able to directly measure the trigonometric parallax of at least one Cepheid in as straightforward a manner as Hipparchus himself had measured the moon. The whole celestial distance scale, the ladders piled on top of ladders, would stand on firmer ground. But the nearest Cepheids are still too distant for even the orbiting satellite to fix any one of them very accurately. A statistical analysis of the better measurements suggested to some astronomers that the whole cosmic distance scale might have to be corrected by 10 percent.\n\nThe interpretation is open to dispute. It is probably inevitable that the more astronomers study Miss Leavitt's stars, the less simple they appear. The pulsations of some, including Polaris, include subtle \"overtones,\" secondary rhythms that can throw off the beat. Debates periodically erupt over whether Cepheids of various colors and chemical content have significantly different period-luminosity curves.\n\nIn tweaking the Hubble constant, astronomers also now rely on other kinds of pulsating stars, like the RR Lyraes (Shapley's old \"cluster\" variables) and the Miras. In addition are a wide assortment of secondary candles, those of lesser reliability that are calibrated using Cepheids for measuring beyond where it is possible to pick out individual stars.\n\nDuring the 1920 debate, one of the arguments against the existence of island universes had been the extreme brightness of a certain nova in Andromeda. Unless the nebula was nearby, the nova would have to be unusually powerful, way off the scale. By the time of the 1996 debate, astronomers spoke comfortably of \"supernovae,\" intense bursts of light coming from exploding stars. A variety of supernovae called Type Ia has been calibrated for use as standard candles. Because of their extreme brightness they have been detected billions of light-years away.\n\nWhen whole galaxies must be used as standard candles, astronomers can draw on something called the Tully-Fisher method: the larger a spiral galaxy, the faster it will spin. Bigger galaxies are also brighter, so their intrinsic luminosity can be estimated from their rotational speed, which is gauged by measuring Doppler shifts.\n\nThe details of this and other methods can quickly become esoteric, but the underlying idea is the same: if you can devise a theory that relates an observable characteristic\u2014of a star, a supernova, a galaxy, a cluster\u2014to its inherent brightness, then you can use it as a standard candle.\n\nThe measurements remain fraught with uncertainty. In addition to the universe's outward expansion, galactic clusters are also pulled gravitationally toward one another. These \"peculiar motions\" result in kinks in the Hubble expansion that must be corrected for. Our Local Group is believed to be gradually falling into the massive Virgo Cluster, a phenomenon called virgocentric flow.\n\nAstronomers must also guard against selection effects, giving too much weight in their calculations to the stars, galaxies, and clusters that are easiest to see. The most well-known of these is the so-called Malmquist bias: The stars you can pick out in a cluster are necessarily the brightest. If you rely on them to compute the average luminosity, the answer will be skewed.\n\nEven with so many ways to go wrong, astronomers have been moving toward a consensus that the universe is a little less than 14 billion years old. Looking out from any vantage point, an observer will be at the center of a bubble extending that many light-years in every direction. It is still a hard idea to get used to. No one is at the center yet everyone is. Wherever you stand, you can see no farther than light has been able to travel since the big bang, the explosion that occurred everywhere and nowhere, that created space and time.\nEPILOGUE\n\nFire on the Mountain\n\nThis book began with a parable about a village that learned how to measure all the way to a far-off mountain. Because of a mistaken assumption\u2014that the vegetation on the mountaintop was the same as that on the valley floor\u2014the inhabitants underestimated the distance, only learning of their mistake after they sent an expedition there.\n\nThere is a coda to the story. Much later in their history, the villagers established a scientific outpost on the mountain. They built high towers and learned to concentrate light with tubes outfitted with curved lenses and mirrors. Looking out at the unknown expanse beyond them, they realized that their explorations had barely begun. Their mountain was merely a hillock. What lay on the new horizon was a peak that, magnified many times, was of breathtaking grandeur.\n\nIt too was fringed with green, and this time, to avoid fooling themselves, the villagers used the _average_ height of their own vegetation as a standard yardstick. No more mistaking asters for trees. While the mountain they were standing on was a thousand canyon widths from the village, this new mountain appeared to be approximately a thousand times farther still. This place, they knew, would not be visited in their lifetime, and probably not within the lifetime of their people.\n\nOne night up in the tower, one of the scientists saw a brilliant light on the horizon. The remote mountain had exploded in flames. Measuring the intensity of the light, the scientist did some calculations. From the distance to the mountain and its apparent size, he had already estimated how big it was. Now he calculated how much light would be produced if the mountain had caught on fire.\n\nThe answer didn't make sense. The flames were so brilliant that they would have to be far more intense than anything resembling ordinary combustion.\n\nWhen he reported his finding to his colleagues back in the village, they offered several hypotheses. One proposed that some peculiarity of the air may have magnified the light, acting like a natural lens, but few thought that was plausible. More popular was the theory that fire in the distant land burned much hotter, that they had discovered a new kind of energy.\n\nThe scientist who had made the observation had a different idea: that this time they had somehow _over_ estimated the distance. If the mountain was really a hundred times closer, the anomaly could be explained away....\n\nSOMETHING LIKE THIS happened here on earth. It was 1963 and Maarten Schmidt, a Mount Palomar astronomer, had just ascertained that a starlike (\"quasi-stellar\") object called 3C273 showed a redshift that would put it several billion light-years from earth, as far as some of the most distant galaxies.\n\nOther quasars were soon found to have even more severe redshifts. They appeared to be receding from our part of space at a velocity almost as great as light. The Hubble law put them nearly at the edge of the visible universe. For something so remote to shine so brightly, it would have to be emitting the light of thousands of galaxies, the energy generated, perhaps, by matter pouring into the intense gravitational field of a black hole.\n\nWhatever the cause, accepting the immense distance of these fantastic objects caused all kinds of trouble. The quasar 3C273 (the 273rd entry in the _Third Cambridge Catalog of Radio Sources_ ) expels from its core a jet of light that appears to be traveling at several times the speed of light. That of course would be impossible. Astronomers quickly came up with a more palatable explanation: the superluminal motion is probably an illusion. The jet happens to be coming almost straight at us, making it appear to be much faster than it really is.\n\nThere is however another possibility: that 3C273 and all the quasars are really very near by. The velocity of the jet would then be much, much smaller. That also would solve another problem. If the quasars are close to us, then we need not conclude that they are so fantastically bright.\n\nFor this to be true, redshift would have to be caused by something other than Hubble expansion\u2014maybe by the old \"tired light\" theory or some other new physics. If so, the entire universe might be vastly smaller, and there may not have been a big bang.\n\nThe idea that the redshifts are \"noncosmological\" is, to say the least, a minority view. Most astronomers are persuaded by a tightening net of circumstantial evidence that quasars really are blinding beacons lying near the edge of what it is possible to see.\n\nOne of the strongest arguments involves a weird phenomenon called gravitational lensing. Sometimes astronomers see two quasars, one right next to the other. The doubling, however, is believed to be an illusion. According to the theory of general relativity, gravity can bend light. Something as massive as a galaxy can act like an enormous piece of curved glass, projecting a double image. If all that is true, then the quasar must be behind the galaxy not in front of it, and therefore very far away.\n\nWith each step outward, the act of measurement becomes a little more abstruse. With arithmetic and a ruler you can get from the desk to the window, with trigonometry and a transit you can get to the moon and, with a few assumptions, to the nearest stars.\n\n\"With increasing distance, our knowledge fades, and fades rapidly,\" Hubble once said, in a rare moment of oratorial eloquence. \"Eventually, we reach the dim boundary\u2014the utmost limits of our telescopes. There, we measure shadows, and search among ghostly errors of measurements for landmarks that are scarcely more substantial.\"\n\nEstablishing the distance of the quasars requires not only the Hubble law, but the entire framework of Einsteinian relativity. Measuring began as a way to gather data to verify theories. Now the measuring stick itself has become one more theory to test.\nNotes\n\n**Epigraphs**\n\n_p. ix_ \"Her columns grew longer\": Thomas Mallon, _Two Moons_ (New York: Pantheon, 2000), p. 11.\n\n_p. ix_ \"Then, by means of the instrument at hand\": Thomas Hardy, _Two on a Tower_ (New York: Harper & Brothers, 1895), p. 33.\n\n**Prologue. The Village in the Canyon**\n\n_p. 1_ As readers shall see in chapter 6, my story about the village in the canyon is elaborated from a comment made by Harlow Shapley in the Great Debate of 1920.\n\n_p. 6_ The view from Tau Ceti is described on pp. 170\u201371 of Heinlein's _Time for the Stars_ (New York: Charles Scribner's Sons, 1956).\n\n**Chapter 1. Black Stars, White Nights**\n\n_p. 9_ \"We work from morn till night\": Jones and Boyd, _Harvard College Observatory_ , p. 190. This book and Bailey's _History and Work of the College Observatory_ are the two standard sources for the early history of the observatory.\n\n_p. 9_ Computers earned 10 cents more than a cotton mill worker: Pamela Etter Mack, \"Women in Astronomy in the United States, 1875\u20131920\" (bachelor's thesis, Harvard University, 1977). I also referred to her chapter, \"Straying from Their Orbits: Women in Astronomy in America,\" in _Women in Science_ , edited by Kass-Simon and Farnes, pp. 72\u2013116.\n\n_p. 11_ an apparatus of leaf springs: Alison Doane, Curator of Astronomical Photographs at the observatory, has told me that her own research cannot confirm this tale. The repository was built in the 1930s.\n\n_p. 14_ \"It is delightful to see the stars brought out\": William Cranch Bond, letter to Harvard President Edward Everett, September 22, 1847, quoted in Jones and Boyd, p. 68.\n\n_p. 15_ The Great Refractor extended the reach to the fourteenth magnitude: One of the telescope's first great discoveries, in 1848 by William Cranch Bond and George Bond, was Saturn's eighth moon, Hyperion, which is between the fourteenth and fifteenth magnitudes.\n\n_p. 15_ My portrait of Pickering is drawn from Bailey, pp. 243\u201352, and Jones and Boyd, pp. 178\u201382.\n\n_p. 16_ Eventually Harvard measured and cataloged forty-five thousand stars: Jones and Boyd, p. 202. The results were published in 1908 as _The Revised Harvard Photometry_ and appeared in volumes 50 and 54 of the _Annals of the Astronomical Observatory of Harvard College_.\n\n_p. 17_ The saga of the observing station at Arequipa is entertainingly described in Fernie's _Whisper and Vision_ , pp. 153\u201388. Accounts also appear in Bailey and in Jones and Boyd.\n\n_p. 18_ \"A great observatory should be as carefully organized\": Pickering, in a June 28, 1906, address to the Harvard Chapter of Phi Beta Kappa, Harvard University Archives.\n\n_p. 19_ 25 cents an hour amounted to the minimum wage: Adjusted for inflation, 25 cents in 1900 would be worth $5.27 in 2003. Source: \"The Inflation Calculator,\" www.westegg.com\/inflation.\n\n_p. 19_ Portraits of Fleming, Cannon, Maury, and other computers appear in Jones and Boyd.\n\n_p. 20_ The hours and wages of computing are described in Mack, \"Women in Astronomy.\"\n\n_p. 20_ \"He seems to think that no work\": Williamina Paton Fleming diary, March 12, 1900, in the Harvard archives. (The journal is part of a project in which staff members and students apparently were asked to keep a diary showing what life was like at the university.)\n\n_p. 20_ Pickering's handling of Fleming's request for a raise is documented in his own diary entry for the same date, also in the Harvard archives.\n\n_p. 21_ Pickering's salary is found in Jones and Boyd, p. 182. The description of his typical workday is from his diary.\n\n_p. 21_ _The Observatory Pinafore_ : Jones and Boyd, pp. 189\u201393. The author of the parody was Winslow Upton.\n\n**Chapter 2. Hunting for Variables**\n\n_p. 23_ \"My friends say, and I recognize the truth of it\": The letter from HSL to Pickering, dated May 13, 1902, is in the Harvard University Archives.\n\n_p. 25_ The Leavitt family genealogy is from the excellent database maintained by the Western Association of Leavitt Families at www.leavittfamilies.org, and from information provided by the National Association of Leavitt Families, including a private publication, \"Descendants of John Leavitt, the Immigrant, Through His Son, Josiah, and Margaret Johnson,\" by Emily Leavitt Noyes (Tilton, N.H., 1949), pp. 83, 105, 133.\n\n_p. 25_ The description of the Leavitt household on Warland Street (now Kelly Road) is from U.S. Census documents, the Cambridge Historical Commission, Cambridge city directories, and a visit to the house, which still stands. The only record I found of Roswell's death was the inscription on his gravestone at Cambridge Cemetery.\n\n_p. 26_ Erasmus Leavitt's steam engine is described in a pamphlet published by the American Society of Mechanical Engineers, \"The Leavitt Pumping Engine at Chestnut Hill Station of the Metropolitan District Commission, Boston, Mass.,\" printed in honor of the occasion of its designation as a National Historical Mechanical Engineering Landmark by the American Society of Mechanical Engineers, December 14, 1973.\n\n_p. 26_ For HSL at Oberlin I relied on alumni records in the college archives. Her time at Radcliffe is documented in her transcript, college catalogs, and other records in the Radcliffe College Archives. The period between Radcliffe and Harvard is described in a one-page alumni questionnaire she filled out for Oberlin on April 6, 1908.\n\n_p. 28_ \"Miss Leavitt inherited, in a somewhat chastened form\": Bailey's obituary of HSL appeared in _Popular Astronomy_ 30, no. 4 (April 1922), pp. 197\u201399. (It was followed by an article, \"Shall We Accept Relativity?\" by William H. Pickering, Edward's brother.)\n\n_p. 30_ \"an almost religious zeal\": Bailey, _History and Work_ , p. 264.\n\n_p. 30_ The letters between HSL and Pickering during her stay in Beloit are in the Harvard archives.\n\n_p. 31_ The short biography in which HSL is called \"extremely deaf\" at Radcliffe appears in volume 8 of the _Dictionary of Scientific Biography_ , edited by Charles Coulston Gillispie (New York: Charles Scribner's Sons, 1973).\n\n_p. 33_ The letter sent from the S.S. _Commonwealth_ and a note from HSL's brother acknowledging receipt of her paycheck are in the Harvard archives.\n\n**Chapter 3. Henrietta's Law**\n\n_p. 34_ \"What a variable-star 'fiend' Miss Leavitt is\": The letter, dated March 1, 1905, is from Professor Charles Young at Princeton; it is quoted in Jones and Boyd, _Harvard College Observatory_ , p. 367.\n\n_p. 34_ \"In no other portion of the heavens\": John Herschel is quoted in Shapley's _The Inner Metagalaxy_ ,p.42.\n\n_p. 36_ \"Men said to him, in angry letters\": Reverend Leavitt's address to the annual meeting of the American Missionary Association was published as \"Preaching: The Main Feature in Missionary Work,\" _The American Missionary_ 39, no. 3 (March 1885), pp. 76\u201379. (He mistakenly refers to the astronomer as James Herschel.)\n\n_p. 36_ HSL's trip to Europe is documented in a letter from her to Pickering, dated August 4, 1903, in the Harvard University Archives.\n\n_p. 37_ \"an extraordinary number\": From H. S. Leavitt, \"1777 Variables in the Magellanic Clouds,\" _Annals of the Astronomical Observatory of Harvard College_ 60, no. 4 (1908), pp. 87\u2013108.\n\n_p. 37_ The _Washington Post_ news brief about HSL appeared January 28, 1906, on p. 4.\n\n_p. 37_ \"the northern star of a close pair\": Leavitt, \"1777 Variables\", p. 96.\n\n_p. 37_ \"boarding with Uncle Erasmus\": The _Cambridge City Directory_ lists her at his address, 33 Garden Street.\n\n_p. 38_ the Astronomical and Astrophysical Society of America: This became the American Astronomical Society in 1914, after a long and acrimonious struggle in which the newer science of astrophysics tried to avoid being subsumed as a branch of astronomy. See \"How Did the AAS Get Its Name?\" by Brant L. Sponberg and David H. DeVorkin, on the society's Web site, www.aas.org\/~had\/name.html. In two letters to Pickering (December 20, 1905, and December 20, 1906), HSL refers to the group simply as the Astrophysical Society.\n\n_p. 38_ \"It is worthy of notice\": Leavitt, \"1777 Variables,\" p. 107.\n\n_p. 38_ \"It has not escaped our notice\": Watson and Crick's legendary paper, \"Molecular Structure of Nucleic Acids,\" appeared in _Nature_ 171 (1953), pp. 737\u201338.\n\n_p. 39_ The letters written during HSL's illness in 1908\u20131910 are in the Harvard archives. The description of the Leavitt household in Beloit is from census records. That year census takers asked whether anyone in a household was deaf, but the record is ambiguously marked, so it is impossible to tell whether Henrietta was put in that category.\n\n_p. 42_ Reverend Leavitt's estate is described in his will (Commonwealth of Massachusetts, probate court records for Middlesex County). Henrietta's visit home after his death is documented in letters in the Harvard archives.\n\n_p. 42_ The visit to Des Moines is mentioned in a letter dated July 3, 1911, in the Harvard archives refers to a Mrs. W. G. H. Strong. In June 1901 Henrietta's sister Martha had married a William James Henry Strong, originally of Council Bluffs, Iowa.\n\n_p. 43_ \"A remarkable relation\": From Edward C. Pickering, \"Periods of Twenty-five Variable Stars in the Small Magellanic Cloud,\" _Harvard College Observatory Circular_ no. 173 (March 3, 1912). The relationship between period and luminosity is logarithmic.\n\n_p. 44_ Cepheid yardstick: HSL was clearly aware of the possibilities opened up by her discovery\u2014if the yardstick could be calibrated. As she wrote on the last page of the report, \"It is to be hoped, also, that the parallaxes of some variables of this type may be measured.\"\n\n**Chapter 4. Triangles**\n\n_p. 45_ \"I had not thought of making the very pretty use\": Quoted in Smith, _The Expanding Universe_ ,p.72.\n\n_p. 45_ An authoritative source for the history of astronomical parallax is Albert Van Helden's _Measuring the Universe_. Good popular accounts include Kitty Ferguson's book by the same name and Alan W. Hirshfeld's _Parallax_. I also relied on two fine histories of astronomy, Arthur Koestler's _The Sleepwalkers_ and Timothy Ferris's _Coming of Age in the Milky Way_.\n\n_p. 48_ The astronomer who first measured the distance to Mars was Gian Domenico Cassini, director of the Paris Observatory.\n\n_p. 49_ The Transit of Venus occurs twice a century\u2014but not every century. The transits of 1874 and 1882 were followed by the pair scheduled for 2004 and 2012.\n\n_p. 55_ Ejnar Hertzsprung's use of the sun's motion to triangulate some Cepheids was a bit more complicated than I describe. To determine how much of a star's change in position is due to solar parallax you must first account for how much it has moved on its own. This can be done with statistical methods similar to those Shapley used to measure the Milky Way (see chapter 5).\n\n_p. 55_ Hertzsprung's article appeared in _Astronomische Nachrichten_ 196, pp. 201\u201310.\n\n_p. 56_ \"that the best service he could render\": Bailey, _History and Work_ ,p.25.\n\n_p. 56_ HSL's work diary is in the Harvard University Archives.\n\n_p. 56_ recovering from stomach surgery: HSL's letter to Pickering, dated May 8, 1913, in the Harvard archives.\n\n_p. 57_ \"It is desirable that the standard scale\": HSL, \"The North Polar Sequence,\" _Annals of Harvard College Observatory_ 71, no. 3 (1917), p. 230.\n\n**Chapter 5. Shapley's Ants**\n\n_p. 59_ \"Her discovery of the relation of period to brightness\": Letter from Shapley to Pickering, September 24, 1917, Shapley correspondence, Harvard University Archives.\n\n_p. 59_ \"It is much more natural and reasonable\": Kant, _Allgemeine Naturgeschichte und Theorie des Himmels_ , published in 1755.\n\n_p. 59_ Good sources on the early-twentieth-century controversy over the nature of nebulae are Smith's _The Expanding Universe_ and J. D. Fernie, \"The Historical Quest for the Nature of the Spiral Nebulae,\" _Proceedings of the Astronomical Society of the Pacific_ 82 (1970), pp. 1189\u20131230.\n\n_p. 60_ \"No competent thinker\": Quoted in Struve and Zebergs, _Astronomy of the Twentieth Century,_ p. 436.\n\n_p. 61_ \"enveloped and beclouded\": Smith, p. 21.\n\n_p. 61_ an intrinsic magnitude of about \u20138: Astronomers knew this because the Doppler effect gave a direct reading of how fast a nova was expanding in the direction of Earth. Comparing that number with how fast the nova _appeared_ to expand revealed the distance, and from the distance one could gauge the intrinsic brightness. Curtis then reversed the procedure, using the hypothesized brightness to estimate the distance of the novae outside the Milky Way.\n\n_p. 62_ Shapley tells about the ants in his book _Through Rugged Ways to the Stars_ , pp. 65\u201368.\n\n_p. 64_ \"[T]his proposition scarcely needs proof \": Harlow Shapley, \"On the Nature and Cause of Cepheid Variation,\" _Astrophysical Journal_ 40 (1914), p. 449.\n\n_p. 66_ Shapley's artful chain of assumptions was developed in nineteen papers each titled \"Studies Based on the Colors and Magnitudes in Stellar Clusters\"; some of the later ones were coauthored with his wife, Martha, and other co-workers. A full list of citations can be accessed through the Bruce Medalist Web page for Shapley: www.phys-astro.sonoma.edu\/BruceMedalists\/Shapley. Also see the detailed bibliography in Smith.\n\n_p. 66_ living alone in a Cambridge rooming house: Actually the _Cambridge City Directory_ lists two: 49 Trowbridge Street and then 49 Dana Street.\n\n_p. 66_ \"Does Miss Leavitt know if they have shorter periods\": Letter from Shapley to Pickering, August 27, 1917, Shapley correspondence, Harvard archives.\n\n_p. 66_ \"Miss Leavitt is now absent on her vacation\": Letter from Pickering to Shapley, September 18, 1917, ibid.\n\n_p. 67_ \"Her discovery of the relation of period to brightness\": Letter from Shapley to Pickering, September 24, 1917, ibid.\n\n_p. 67_ \"I believe the most important photometric work\": Letter from Shapley to Pickering, August 20, 1918, ibid.\n\n_p. 67_ \"A few days ago I talked with Miss Leavitt\": Letter from Pickering to Shapley, September 14, 1918, ibid. Pickering died on February 3, 1919.\n\n_p. 68_ Van Maanen's research on the rotation of spiral nebulae was centered on M101, M51, and M33.\n\n_p. 69_ \"So the center has shifted\": Shapley's letter to Hale, dated January 19, 1918, is quoted in Owen Gingerich, \"Shapley's Impact,\" in the Harlow Shapley Symposium on Globular Cluster Systems in Galaxies, _Proceedings of the 126th International Astronomical Union Symposium_ , Cambridge, Mass., August 25\u201329, 1986 (Dordrecht, Netherlands: Kluwer Academic Publishers, 1988), pp. 23\u201336.\n\n_p. 69_ \"Man is not such a big chicken\": Shapley, _Through Rugged Ways_ ,p. 60.\n\n**Chapter 6. The Late, Great Milky Way**\n\n_p. 70_ \"The spectrum of the average spiral nebula\": From the slides Curtis used in his 1920 debate with Shapley, Allegheny Observatory Archives, Pittsburgh.\n\n_p. 70_ The train trip to Washington is described in Shapley, _Through Rugged Ways to the Stars_ , pp. 77\u201378.\n\n_p. 70_ The wrangling that took place before the debate is vividly described in Michael A. Hoskin, \"The Great Debate: What Really Happened,\" _Journal for the History of Astronomy_ 7, pp. 169\u201382. I also relied on Virginia Trimble's scholarly and entertaining paper \"The 1920 Shapley-Curtis Discussion: Background, Issues and Aftermath,\" _Proceedings of the Astronomical Society of the Pacific_ 107 (1995), pp. 1133\u201344, and on a perceptive analytical account in Smith, _The Expanding Universe_ , pp. 77\u201386, and Struve and Zebergs, _Astronomy of the Twentieth Century_ , pp. 416\u201320, 441\u201344.\n\n_p. 71_ \"to some region of space\": Quoted in Hoskin's paper.\n\n_p. 72_ \"hammer and tongs\": ibid.\n\n_p. 72_ \"miserable\" Cepheids: the letter to Russell is quoted in Smith, p. 81.\n\n_p. 73_ \"noble human antique\": Shapley's memory of the debate is from _Through Rugged Ways_ , pp. 78\u201381. As with his story about Einstein, Shapley may have also misremembered these details. The records of the meeting in the archives of the National Academy of Sciences are not detailed enough to tell.\n\n_pp. 73_ _\u2013_ _79_ In addition to the works mentioned above by Hoskin, Trimble, and Smith, my account of the debate relies on Shapley's typescript (the original is in the Harvard University Archives) and Curtis's slides (originals at Allegheny Observatory). Both documents are available at antwrp.gsfc.nasa.gov\/diamond_jubilee. The debaters repeated and expanded on their positions in formal papers published in the _Bulletin of the National Research Council_ 2 (1921), pp. 171\u201393 and 194\u2013217.\n\n_p. 79_ \"Debate went off fine in Washington\": Quoted in Hoskin's paper. _p. 7_ _9_ \"Now I would know how to dodge things\": Shapley, _Through Rugged Ways_ ,p.79.\n\n_p. 80_ \"the engulfing of a star\": Shapley, \"Globular Clusters and the Structure of the Galactic System,\" _Publications of the Astronomical Society of the Pacific_ 30 (1918), p. 53.\n\n_p. 80_ \"Suppose that an observer\": From Shapley's _Bulletin_ paper.\n\n**Chapter 7. In the Realm of the Nebulae**\n\n_p. 82_ \"One of the few decent things I have done\": Shapley, _Through Rugged Ways to the Stars,_ p. 91.\n\n_p. 82_ For biographical details about Shapley, I referred to _Through Rugged Ways_ , as well as the transcript of the oral-history interviews on which the book is based (Charles Weiner and Helen Wright, \"Harlow Shapley,\" American Institute of Physics, Center for the History of Physics, College Park, Md., June 8, 1966). In addition to serving as my main source on Hubble's life, Christianson's _Edwin Hubble_ provided a vivid portrait of Shapley (pp. 129\u201332).\n\n_p. 82_ \"He is the best student I ever had\": Jones and Boyd, _Harvard College Observatory_ , p. 432, n. 16.\n\n_p. 84_ \"Bah Jove\" and \"come a cropper\": Shapley, _Through Rugged Ways_ ,p. 57.\n\n_p. 84_ Hubble's uneasy relationship with Shapley is documented in the books by Christianson and Smith.\n\n_p. 85_ \"the apartment building on Linnean Street\": According to the _Cambridge City Directory_ , HSL and her mother had moved there by 1919. Cambridge Historical Commission records show that the building (3\u20135 Linnean Street and called Linnean Hall) was built in 1914. Rents ranged from $30 to $52.50 a month.\n\n_p. 85_ \"enormous importance in the present discussion\": Shapley's letter to HSL, May 22, 1920, is in the Harvard University Archives.\n\n_p. 85_ Shapley had been overestimating: Owen Gingerich has documented the behind-the-scenes wrangling in \"How Shapley Came to Harvard or, Snatching the Prize from the Jaws of Debate,\" _Journal for the History of Astronomy_ 19 (1988), pp. 201\u20137. Also see Hoskin's \"The Great Debate\" (cited in the notes for chapter 6).\n\n_p. 85_ \"He is much more venturesome\": Gingerich, \"How Shapley Came to Harvard,\" p. 203.\n\n_p. 86_ \"Shapley couldn't swing the thing\" : Ibid., p. 204.\n\n_p. 86_ \"So young, so clean, so brilliant\": Cannon's diary is in the Harvard archives. Details about Cannon and the Shapley era at Harvard Observatory are from Jones and Boyd, Bailey's _History and Work_ , and Haramundanis's _Cecilia Payne-Gaposchkin_ , an edition of the astronomer's memoirs edited by her daughter.\n\n_p. 87_ \"I always wanted to learn the calculus\": Maury said this to Cecilia Payne (Haramundanis, p. 149).\n\n_p. 87_ \"I shall be happy\": Quoted in Jones and Boyd, p. 398.\n\n_p. 87_ \"If one could only go on and on\": Fleming diary, Harvard archives.\n\n_p. 88_ \"She was a pure observer\": Haramundanis, p. 139.\n\n_p. 88_ \"We shall never understand it\": HSL quoted in Haramundanis, p. 140.\n\n_p. 88_ \"Pickering chose his staff to work\": Ibid., p. 149.\n\n_p. 88_ \"one of the most important women\": Shapley, _Through Rugged Ways_ ,p.91.\n\n_p. 88_ \"girl-hours\" and \"kilo-girl-hours\": Ibid., p. 94.\n\n_p. 88_ \"Took flowers to Miss Leavitt\": Cannon diaries, Harvard archives.\n\n_p. 90_ The details of HSL's estate are from probate court records for Middlesex County, Massachusetts.\n\n_p. 90_ What HSL was working on when she died is from the Bailey obituary cited in the notes for chapter 2 and from _Harvard University Reports of the President and the Treasurer of Harvard College, 1922\u20131923: The Observatory_ , p. 244, Harvard archives.\n\n_p. 90_ \"the famous new star of 1918\": _Reports of the President, 1922\u20131923_ , p. 244, Harvard archives.\n\n_p. 91_ \"great service to astronomy\": _Transactions of the International Astronomical Union_ 1 (1922), p. 69.\n\n_p. 91_ \"She had hardly begun work\": _Reports of the President, 1921\u20131922_ , p. 208.\n\n_p. 91_ For the story of Cecilia Payne and HSL's desk, see Haramundanis, p. 153.\n\n_p. 91_ \"I heard it said when I came to Harvard\": Ibid., p. 146\n\n_p. 92_ \"Magellanic Cloud (Great) so bright\": Cannon diaries, April 20, 1922.\n\n_p. 92_ The dispute between Lundmark and Shapley is described in Smith, pp. 105\u201311.\n\n_p. 92_ \"Whether or not you care to recognize\": Quoted in Smith, p. 106.\n\n_p. 93_ \"On the Motions of Spirals\": Knut Lundmark, _Publications of the Astronomical Society of the Pacific_ 34 (1922), pp. 108\u201315.\n\n_p. 93_ Jeans's analysis of van Maanen's data is described in Smith, p. 104. Jeans wasn't opposing the island universe theory. Rather, he thought the Milky Way was considerably smaller than Shapley did. If the spirals were of similar size, they would be much closer to Earth, making the rate of their spin far slower.\n\n_p. 93_ \"the situation seemed to be rather hopeless\": Quoted in Smith, p. 108.\n\n_p. 94_ An artful account of Hubble's measurement of Andromeda is in Christianson, pp. 157\u201362; also see Smith, pp. 111\u201326.\n\n_p. 94_ \"You will be interested to hear\": Quoted in Smith, p. 114. The correspondence between Hubble and Shapley is divided between the Edwin P. Hubble Manuscript Collection at the Huntington Library in San Marino, Calif., and the Harvard University Archives.\n\n_p. 94_ \"Here is the letter that has destroyed my universe\": Haramundanis, p. 209.\n\n_p. 95_ \"Your letter telling of the crop of novae\": Quoted in Christianson, p. 159.\n\n_p. 95_ Before he became a professional astronomer, Barnard had earned enough money spotting new comets (a New York philanthropist was paying $200 for each one) to make a down payment on what he called his \"comet house.\" Today he is best known as the namesake and discoverer of Barnard's star.\n\n_p. 96_ \"the straws are all pointing\" and \"I do not know whether I am sorry\": Quoted in Christianson, p. 159.\n\n_p. 96_ \"a curiously faithful copy\": Edwin Hubble, \"N.G.C. 6822, A Remote Stellar System,\" _Contributions from the Mount Wilson Observatory_ 302 (1925), p. 410.\n\n_p. 96_ \"The principle of the uniformity\": Ibid., p. 432.\n\n_p. 97_ \"a splendid forum\": Quoted in Christianson, p. 160. The AAS\/AAAS meeting is described in the same passage. More details are in _Popular Astronomy_ 33, no. 4 (1925), pp. 158\u201360. An abstract of Hubble's paper, \"Cepheids in Spiral Nebulae,\" appeared in the same issue beginning on p. 252.\n\n_p. 97_ Hubble shared the prize: The other winner of what is now called the Newcomb Cleveland Prize was L. R. Cleveland, whose papers had been read before the American Society of Zoologists.\n\n_p. 98_ \"After all, he was my _friend_ \": Quoted in Haramundanis, p. 209.\n\n_p. 98_ \"assigned subject matter\" and \"I was right\": Shapley, _Through Rugged Ways_ ,p.79.\n\n_p. 98_ \"the realm of the nebulae\": The phrase is used as the title of Hubble's 1936 book, based on his Silliman lectures at Yale.\n\n_p. 98_ \"What are galaxies?\": Sandage, _The Hubble Atlas of Galaxies_ ,p.1.\n\n**Chapter 8. The Mysterious K**\n\n_p. 99_ \"Youth Who Left Ozark Mountains\": The newspaper headline, which appeared on page 2 of the paper, is quoted in Christianson, _Edwin Hubble_ , p. 210.\n\n_p. 99_ the Scopes \"Monkey Trial\": The Harvard historian of science Owen Gingerich has in his collection a telegram from Darrow inviting Shapley to testify.\n\n_p. 102_ \"with some statistical sleight of hand\": I'm referring here to the technique of statistical parallax, described in chapter 5.\n\n_p. 104_ For details of Humason's unusual background see Christianson, pp. 185\u201386. His work with Hubble is described in Christianson, pp. 192\u201395, and Smith, _The Expanding Universe_ , pp. 180\u201383.\n\n_p. 105_ \"twice as large as any hitherto observed\": Milton Humason, \"The Large Radial Velocity of N.G.C. 7619,\" _Proceedings of the National Academy of Sciences_ 15, no. 3 (March 15, 1929), pp. 167\u201368.\n\n_p. 106_ at about 150 kilometers per second: Hubble actually expressed K as 500 parsecs (one parsec being 3.26 light-years). His early results are reported in his paper, \"A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebula,\" published in the same issue of _PNAS_ as Humason's paper (pp. 168\u201373). This was followed in 1931 by Edwin Hubble and Milton Humason, \"The Velocity-Distance Relation Among Extra-Galactic Nebulae,\" _Astrophysical Journal_ 74, no. 43 (1931), pp. 43\u201380.\n\n_p. 107_ \"I don't believe these results\": Quoted in Christianson, p. 198.\n\n_p. 107_ The encounter between Shapley and Humason is described by Christianson, p. 151. In _The Expanding Universe_ , Smith gives good reasons to believe the story may be true (p. 144, n. 122).\n\n_p. 108_ Einstein's calling Hubble's work \"beautiful\": See Christianson, p. 211.\n\n**Chapter 9. The Cosmic Stampede**\n\n_p. 109_ \"The definitive study of the herd instincts of astronomers\": J. D. Fernie, \"The Period-Luminosity Relation: A Historical Review,\" _Publications of the Astronomical Society of the Pacific_ 81, no. 483 (December 1969), pp. 719\u201320.\n\n_p. 109_ For a detailed account of the controversy over the Milky Way's seemingly anomalous size, see Smith, _The Expanding Universe,_ pp. 153\u201356.\n\n_p. 110_ If these distant spirals were islands: Ibid., p. 154. For a while, Shapley toyed with what he called the Super-Galaxy Hypothesis, in which the Milky Way consisted of a confederation of several smaller galaxies\u2014the globular clusters\u2014bunched together.\n\n_p. 111_ Trumpler reported his discovery of cosmic dust in his paper \"Absorption of Light in the Galactic System,\" _Publications of the Astronomical Society of the Pacific_ 42 (1930), pp. 214\u201327.\n\n_p. 113_ \"most of the astronomers of the day\": Fernie, \"Period-Luminosity Relation,\" pp. 716\u201317.\n\n_p. 114_ Baade gave a nice personal account of how he recalibrated the Cepheid scale\u2014including a description of his conversation with Hubble\u2014in a talk to the Astronomical Society of the Pacific, later published as \"The Period-Luminosity Relation of the Cepheids,\" _Publications of the Astronomical Society of the Pacific_ 68 (1956), pp 5\u201316. Christianson provides further details in _Edwin Hubble_ ,pp. 291\u201393.\n\n_p. 115_ \"Instead of one period-luminosity relation\": Baade, p. 11. Along with the cluster variables, he discovered that the Cepheids occasionally found in globular clusters also belonged to Population II.\n\n_p. 116_ For the revisions to the Hubble constant, see Virginia Trimble, \"H0: The Incredible Shrinking Constant, 1925\u20131975,\" _Publications of the Astronomical Society of the Pacific_ 108 (December 1996), pp. 1073\u201382.\n\n**Chapter 10. Ghost Stories**\n\n_p. 117_ \"What monsters may they be?\": The passage appears on p. 34 of Hardy's _Two on a Tower_.\n\n_p. 117_ \"that Miss Leavitt's lamp was still to be seen burning\": Haramundanis, _Cecilia Payne-Gaposchkin_ , p. 153.\n\n_p. 117_ Leavitt's scant traces: Annie Cannon was, by contrast, quite a pack rat. The Harvard University Archives are stuffed with diaries, guest books, photo albums, letters\u2014nothing, it seems, was discarded. The scant mention these papers make of Henrietta Leavitt make one wonder whether they were even friends.\n\n_p. 117_ A virtual planetarium: The Henrietta Leavitt Flat Screen Space Theater is located at www.thespacewriter.com.\n\n_p. 118_ crater on the moon: Harry Lang, \"Six Moon Craters Named for Deaf Scientists,\" _The World Around You_ , January\u2013February, 1996 (published by Gallaudet University, Washington, D.C.).\n\n_p. 118_ \"The World's Greatest Creation Scientists\" can be found at www.creationsafaris.com. The criteria for compiling the list are so loose that it also includes Sir Francis Bacon, Johannes Kepler, Leonardo da Vinci, William and John Herschel, and even Galileo.\n\n_p. 118_ \"Honoured Miss Leavitt\": Letter from Mittag-Leffler to HSL, February 23, 1925, in Shapley correspondence, Harvard archives.\n\n_p. 118_ Sonja Kowalewsky: The name also appears in English as Sonya Kovalevskaya. Other variations are Sonja Kowalewski, Sophia Kovalevsky, Sofia Kovalevskaia, and Sofya Kovalevskaya.\n\n_p. 119_ \"Miss Leavitt's work on the variable stars\": Letter from Shapley to Mittag-Leffler, March 9, 1925, Shapley correspondence, Harvard archives.\n\n_p. 119_ \"The number of variables included in Miss Leavitt's discussion\": Bailey, _History and Work_ , p. 185.\n\n_p. 120_ The 1996 \"The Scale of the Universe\" debate was amply documented in six papers appearing in _Publications of the Astronomical Society of the Pacific_ 108 (December 1996): Jerry T. Bonnell, Robert J. Nemiroff, and Jeffrey J. Goldstein, \"The Scale of the Universe Debate in 1996,\" pp. 1065\u201367; Owen Gingerich, \"The Scale of the Universe: A Curtain Raiser in Four Acts and Four Morals,\" pp. 1068\u201372; Virginia Trimble, \"H0: The Incredible Shrinking Constant, 1925\u20131975,\" pp. 1073\u201382; G. A. Tammann, \"The Hubble Constant: A Discourse,\" pp. 1083\u201390; Sidney van den Bergh, \"The Extragalactic Distance Scale,\" pp. 1091\u201396; and John N. Bahcall, \"Is H0 Well Defined?\" p. 1097.\n\n_p. 121_ having been Dante's assistant: Christianson, _Edw_ _in Hubble_ , p. 363.\n\n_p. 121_ Besides the value of the Hubble constant, other factors affecting the size of the universe include its shape and the value of a parameter called the cosmological constant.\n\n_p. 121_ The Hubble Wars are described in Overbye, _Lonely Hearts of the Cos_ _mos_ , pp. 263\u201384.\n\n_p. 121_ Wonderful maps of clusters and superclusters can be found at www.anzwers.org\/free\/universe\/galaclus.html.\n\n_p. 124_ corrected by 10 percent: M. W. Feast and R. M. Catchpole, \"The Cepheid PL Zero-Point from Hipparcos Trigonometric Parallaxes,\" _Monthly Notices of the Royal Astronomical Society_ 286 (1997), L1\u2013L5. More recently, the biggest news about Hipparcos has been a controversy over the accuracy of its triangulation of the Pleiades: X. Pan, M. Shao, and S. R. Kulkarni, \"A Distance of 133\u2013137 Parsecs to the Pleiades Star Cluster,\" _Nature_ 427 (2004), p. 396.\n\n**Epilogue: Fire on the Mountain**\n\n_p. 130_ \"With increasing distance, our knowledge fades\": Hubble, _Realm of the Nebulae_ , p. 202.\nSelected Bibliography\n\nBailey, Solon I. _The History and Work of Harvard Observatory, 1839 to 1927_. New York: McGraw-Hill, 1931.\n\nChristianson, Gale E. _Edwin Hubble: Mariner of the Nebulae_. Chicago: University of Chicago Press, 1995.\n\nEvans, David S., Terence J. Deeming, Betty Hall Evans, and Stephen Goldfarb, eds. _Herschel at the Cape: Diaries and Correspondence of Sir John Herschel, 1834\u20131838_. Austin: University of Texas Press, 1969.\n\nFerguson, Kitty. _Measuring the Universe: Our Historic Quest to Chart the Horizons of Space and Time_. New York: Walker, 1999.\n\nFernie, Donald. _The Whisper and_ _the Vision: The Voyages of the Astronomers_. Toronto: Clarke, Irwin, 1976.\n\nFerris, Timothy. _Coming of Age in the Milky Way_. New York: William Morrow, 1988.\n\nHaramundanis, Katherine, ed. _Cecilia Payne-Gaposchkin: An Autobiography and Other Recollections_. Second ed. Cambridge: Cambridge University Press, 1996.\n\nHirshfeld, Alan W. _Parallax: The Race to Measure the Cosmos_. New York: W.H. Freeman, 2001.\n\nHoffleit, Dorrit. _Women in the History of Variable Star Astronomy_. Cambridge, Mass.: American Association of Variable Star Observers, 1993.\n\nHubble, Edwin. _The Realm of the Nebulae_. New Haven: Yale University Press, 1936.\n\nJones, Bessie Zaban, and Lyle Gifford Boyd. _The Harvard College Observatory: The First Four Directorships, 1839\u20131919_. Cambridge, Mass.: Belknap Press, 1971.\n\nKass-Simon, G., and Patricia Farnes, eds. _Women of Science: Righting the Record_. Bloomington: Indiana University Press, 1993.\n\nKoestler, Arthur. _The Sleepwalkers: A History of Man's Changing Vision of the Universe_. New York: Macmillan, 1959.\n\nLayzer, David. _Constructing the Universe_. New York: Scientific American Library, 1984.\n\nOverbye, Dennis. _Lonely Hearts of the Cosmos: The Scientific Quest for the Secret of the Universe_. New York: HarperCollins, 1991.\n\nSandage, Allan. _The Hubble_ _Atlas of Galaxies_. Washington, D.C.: Carnegie Institution, 1961.\n\nShapley, Harlow. _The Inner Metagalaxy_. New Haven: Yale University Press, 1957.\n\n\u2014\u2014\u2014. _Through Rugged Ways to the Stars_. New York: Charles Scribner's Sons, 1969.\n\nSmith, Robert W. _The Expanding Universe: Astronomy's \"Great Debate\" 1900\u20131931_. Cambridge: Cambridge University Press, 1982.\n\nStruve, Otto, and Velta Zebergs. _Astronomy of the Twentieth Century_. New York: Macmillan, 1962.\n\nVan Helden, Albert. _Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley_. Chicago: University of Chicago Press, 1985.\n\nZeilik, Michael, and John Gaustad. _Astronomy: The Cosmic Perspective_. Second ed. New York: John Wiley & Sons, 1990.\nAcknowledgments\n\nMy increasing curiosity about Henrietta Swan Leavitt could hardly have been sated without the help of Louisa Gilder, who searched the archives at Harvard and Radcliffe with a thoroughness and a connoisseur's eye for detail that went far beyond mere research.\n\nI would also like to thank Kathleen Rawlins and Susan E. Maycock of the Cambridge Historical Commission; Jolene Passey, Faye Leavitt, and Joseph Leavitt of the Western Association of Leavitt Families, and Winston Leavitt of the National Association of Leavitt Families. I greatly benefited from the excellent work of several historians of early-twentieth-century astronomy, including Gale Christianson, J. D. Fernie, Owen Gingerich, Dorrit Hoffleit, Michael Hoskin, Peggy Aldrich Kid-well, Pamela Mack, Robert W. Smith, and Virginia Trimble (their books and papers are cited in my notes). At Harvard College Observatory, Alison Doane guided me through the stacks of photographic plates and old notebooks and helped me find the office where Henrietta Leavitt and the other computers probably worked.\n\nSeveral people generously read the manuscript, helping me strike a balance between clarity and precision. First I thank the experts: Owen Gingerich, Research Professor of Astronomy and of the History of Science at Harvard University; Alison Doane, Curator of Astronomical Photographs at Harvard College Observatory; her predecessor, Martha Hazen; Stephen Maran of the American Astronomical Society; and Virginia Trimble, Professor of Astronomy and the History of Science at the University of California in Irvine. Just as valuable were the comments of smart, general readers, the kind of people this book is written for: Louisa Gilder, Julie Kinyoun, Douglas Maret, Nancy Maret, and Olga Matlin.\n\nAt James Atlas Books and Norton, I would like to thank Mr. Atlas himself, Jesse Cohen, Ed Barber, and Angela Von der Lippe, for their support and enthusiasm. Thanks also go to Trent Duffy, the excellent copy editor, and Esther Newberg and Christine Bauch at International Creative Management.\nIndex\n\nPage numbers in _italics_ refer to illustrations.\n\nAAAS Thousand Dollar Prize, 97, 144 _n_\n\nAberdeen Proving Ground, 114\n\n\"absorption\" lines, 103\n\nadaptive optics, 122\n\nAlexandria, 46\u201347\n\nAllegheny Observatory, 92\n\nAlpha Centauri, 52\n\nAmerican Academy of Arts and Sciences, 26\n\nAmerican Association for the Advancement of Science (AAAS), 96\u201397\n\nAmerican Astronomical Society, 96\u201397, 137 _n_\n\nAmerican Missionary Society, 35\u201336\n\nAmerican Society of Mechanical Engineers, 26\n\nAndover Theological Seminary, 26\n\nAndromeda:\n\nblueshift of, 103\n\nCepheid variables in, 94\u201395, 101, 107, 109\u201310, 113, 115\u201316\n\ndistance of, 7, 68, 78, 101, 103, 109\u201310, 113, 122\n\nsize of, 68, 122\u201323\n\nas spiral galaxy, 59, 60, 78\n\nstar types in, 104\u20135\n\n_Annals of the Astronomical Observatory of Harvard College,_ 38, 57\n\nants, 62\u201363, 69\n\nAquila, 90\n\narcs, 47, 48, 52, 68, 116, 123\u201324\n\nArequipa, Peru, observing station, 17\u201318, 35, 37, 92\n\nAstronomical and Astrophysical Society of America, 38\n\n_Astronomische Nachrichten,_ 55\n\nastronomy:\n\nbias in, 57, 60, 79, 80\u201381, 97\u2013100, 109, 113, 125\n\ndiscoveries in, 35\u201336, 38\u201339, 55\u201356, 96\u201398\n\nmeasurement in, 8, 14\u201316, 18, 38\u201340, 43\u201344, 53\u201358, 60, 62, 65, 68, 72, 74, 79, 80\u201381, 85, 88, 92\u201394, 97\u2013100, 102, 107, 114\u201315, 121\u201322, 130\n\npublished results in, 18, 30, 31, 37, 38, 39\u201340, 42, 43, 55, 56, 57\u201358, 64, 96, 97, 106\n\ntheoretical basis of, 5\u20136, 62\u201381, 94\u201395, 107\u201313, 130\n\n_Astrophysical Journal,_ 64\n\natmospheric distortion, 112, 122\n\natoms, 102\u20133\n\nBaade, Walter, 114\u201315, 116\n\nBailey, Solon I., 17, 28, 56, 119\u201320\n\nBarnard, Edward Emerson, 95\u201396, 144 _n_\n\nBarnard's Galaxy, 95\u201396\n\nBarnard's star, 144 _n_\n\nbaseline measurements, 45\u201355, _51,_ 99\u2013102\n\nBeloit College, 30, 32, 42\n\nBeta Lyrae, 88\n\nBible, 12, 36, 99\n\nbig bang theory, 16, 108, 110\u201311, 115, 116, 120, 126, 129\n\nBig Dipper, 6\n\nBig Galaxy theory, 62\u201381, 82, 84, 94\u201395, 109\u201310, 111, 112\u201313, 115, 119, 138 _n,_ 143 _n_\n\nbinary stars, 64\n\nblack ants, 62\u201363\n\nblack holes, 16, 129\n\nblink comparator, 68, 95\u201396, 107\n\nblueshift, 60\u201361, 64, 102\u20133\n\nblue stars, 74\u201375, 77\n\nBond, George, 134 _n_\n\nBond, William Cranch, 134 _n_\n\nBo\u00f6tes, 6\n\nBoston Water Works, 26\n\nBrazil, 34\n\nBrookhaven National Laboratory, 19\n\nBruce, Catherine Wolfe, 17\n\nBruce Telescope, 17\u201318, 67\n\nBryan, William Jennings, 100\n\nB-type stars, 74\u201375\n\n_Bulletin of the National Research Council,_ 79\n\nBunsen, Robert, 102\u20133\n\nBurroughs Arithometer, 9\n\ncalcium, 105\n\nCambridge Cemetery, 89\u201390\n\ncandles, standard, 61, 64, 65\u201366, 74, 78, 95, 100\u2013101, 121, 124, 125\n\nCannon, Annie Jump, 20, 41\u201342, 86\u201389, 92, 118, 142 _n,_ 146 _n_ \u201347 _n_\n\nCape of Good Hope, 34, 52\n\nCassini, Gian Domenico, 138 _n_\n\ncensus (1880), 25\u201326\n\n\"Cepheids in Spiral Nebulae\" (Hubble), 97\n\nCepheid variables:\n\nin Andromeda, 94\u201395, 101, 107, 109\u201310, 113, 115\u201316\n\nchemical composition of, 114\u201315, 124\n\nas distance scale, 53\u201355, 101\u20132, 113\u201316, 123\u201324\n\nHubble's use of, xiii\u2013xiv, 94\u201398, 100, 120\n\nLeavitt's analysis of, xiii\u2013xiv, 43\u201344, 53\u201355, 56, 62, 64, 66\u201367, 72, 76, 88, 94\u201396, 101, 102, 120\n\nin Magellanic Clouds, 43\u201344, 53\u201355, 64, 67, 85\n\nas measurement standard, 43\u201344, 53\u201355, 62, 65, 68, 72, 74, 102, 107\n\nperiod-luminosity relationship in, 43, 64\u201365, 85, 94, 96, 113, 114\u201315, 119, 124, 138 _n_\n\nShapley's use of, 62, 63, 64, 65, 67, 68, 72, 74, 75, 76, 85, 94, 109\u201310, 113\n\nCepheus, 44\n\nCerro Tololo Observatory, 12\n\nChristian fundamentalists, 12, 99\n\nClerke, Agnes, 60\n\nCleveland, L. R., 144 _n_\n\ncluster variables, 64\u201365, 115, 124\n\nColumba, 54\n\ncomets, 144 _n_\n\nCommission of Stellar Photometry, 90\u201391\n\n_Commonwealth,_ S.S., 33\n\ncomputers (clerical workers):\n\ndata collected by, 9\u201310, 18\u201319, 55\u201356, 90\u201392, 105\n\nLeavitt as example of, xiii, 9, 22, 23\u201333, 36, 37, 39\u201342, 52, 55\u201358, 59, 66\u201367, 88, 90\u201392, 119\u201320\n\nstatus of, 86\u201388\n\nviewing instruments used by, 23\u201324, 35, 41, 58\n\nwages of, 9, 18\u201321, 31\u201332, 134 _n_\n\nwomen as, 19\u201322, 86\u201388\n\nconstellations, 6, 54\n\n_see also specific constellations_\n\nCopernicus, Nicolaus, 48, 54, 107\u20138\n\ncosmic dust, 78\u201379, 111\u201313\n\ncosmological constant, 148 _n_\n\ncotton mill workers, 9\n\n\"creation science,\" 99\u2013100, 118\n\nCrick, Francis, 38\n\nCurtis, Heber, 61, 70\u201381, _77,_ 92, 98, 113, 115, 120\u201322, 140 _n_ \u201341 _n_\n\nCygnus, 29\n\nDante, 121\n\ndark matter, 16\n\nDarrow, Clarence, 100\n\nDarwin, Charles, 26\n\nDarwin, Erasmus, 26\n\nDay of Reckoning, 12\n\ndegrees, 47, 48, 52, 68, 116, 123\u201324\n\ndepth perception, 45\n\n_Dictionary of Scientific Biography,_ 31\n\ndisplacement, 45\u201349\n\n_Divine Comedy, The_ (Dante), 121\n\nDNA, 38\n\nDoppler effect, 60\u201361, 65, 102\u20137, 125, 139 _n_\n\nDraper, Henry, 20, 87\n\ndwarf galaxies, 123\n\nEarth:\n\nage of, 111\n\ndiameter of, 99, 101\n\norbit of, 48, 50\u201351, _51,_ 53\u201354, 99\u2013100, 101, 116\n\nposition of, 69, 107\u20138, 125\u201326\n\neclipses, 15, 29, 46\u201347, 64\n\neclipsing binaries, 64\n\nEddington, Arthur, 62\n\nEinstein, Albert, 6, 71, 73, 98, 107\u20138, 130, 141 _n_\n\nelectromagnetic waves, 5\n\nelectronic computers, 9, 122\n\nelectronic sensors, 9\n\nelectrons, 102\u20133\n\nemulsion, photographic, 10, 28\u201329\n\nEridanus Cluster, 123\n\nEuropean Space Agency, 123\n\nevolution, 99\u2013100\n\nextra-galactic nebulae, 109\n\nextrapolation, 1\u20135, 8, 80\u201381, 116, 127\u201328, 133 _n_\n\nFalse Cepheids, 95\n\nFelt & Tarrant Comptometer, 9\n\nFernie, J. D., 109, 113\n\nFleming, Edward Pickering, 20\n\nFleming, Williamina Paton, 19\u201321, 33, 87\n\n\"fly spankers,\" 29\n\nFornax Cluster, 7, 123\n\nFraunhofer, Joseph von, 103\n\nfrequencies, wave, 60\u201361, 101\u20137, 129\n\nFrohman, Charles, 37\n\nfundamentalism, religious, 99\u2013100, 118\n\ngalactic plane, 78, 112\n\ngalaxies:\n\naverage luminosity of, 100\u2013102\n\nclusters of, 7, 118, 123, 125\n\ndistance of, 7, 34, 51\u201352, 61, 96, 99\u2013101, 103\u20138, 112\u201316, 121\u201322\n\nevolution of, 93\n\ngroups of, 7\u20138, 11, 34\u201335, 77, 118, 122\u201323, 125\n\nas nebulae, 11\u201312, 51\u201352, 59\u201362, 68\u201369, 70, 75\u201376, 95\u201398\n\nnumber of, 122\u201323\n\nrotation of, 68\u201369, 84, 92\u201394, 97\u201398, 125, 143 _n_\n\nsize of, 55, 109\u201310, 123\n\nsuperclusters of, 7\u20138, 123\n\nvelocity of, 102\u20137, 116\n\n_see also_ nebulae\n\nGalileo Galilei, 15, 17, 51, 52, 54\n\ngas clouds, 8, 12, 51\u201352, 59, 60, 68, 76, 78, 80\n\nGenesis, Book of, 99\n\ngenetics, 38\n\nGilbert, W. S., 21\n\nglobular clusters, 63\u201366, 69, 74, 77, 85, 111, 113, 114, 119, 146 _n_\n\nGod, 12, 36, 69, 100, 118\n\nGoodricke, John, 44\n\ngravitational lensing, 130\n\ngravity, 61, 93, 129, 130\n\nGreat Debate (1920), 70\u201381, 86, 98, 113, 115, 120\u201322, 133 _n,_ 140 _n_ \u201341 _n_\n\nGreat Refractor telescope, 12\u201314, _13,_ 134 _n_\n\nGreece, ancient, 17, 46\u201348, 54, 124\n\nHale, George Ellery, 69, 70\u201371, 85\n\nHale telescope, 115\n\nHalley, Edmond, 49\u201350\n\nHardy, Thomas, ix, 117\n\n_Harvard College Observatory Circular,_ 39\u201340, 43\n\nHarvard Number 1354, 37\n\nHarvard Number 1391, 37\n\nHarvard Number 1509, 37\n\nHarvard Observatory:\n\nArequipa observing station of, 17\u201318, 35, 37, 92\n\nbudget of, 9, 18\u201321, 31\u201332, 134 _n_\n\ncomputers for, _see_ computers\n\nconstruction of, 12\u201314\n\nGreat Refractor telescope of, 12\u201314, _13,_ 134 _n_\n\nLeavitt as computer for, xiii, 9, 22, 23\u201333, 36, 52, 55\u201358, 66, 88, 90\u201392, 119\u201320\n\nPickering as director of, 15\u201321, 73, 86\u201387, 88, 91\u201392\n\nrepository of, 10\u201311, 20, 117\n\nShapley as director of, 72\u201373, 82, 85\u201386, 107\n\nHarwood, Margaret, 66\n\nHeinlein, Robert, 5\u20136\n\nhelium, 103, 104\n\nHellespont, 46\n\nHenry Draper Catalogue, 20, 86\n\nHercules, 54, 75\n\nherd instinct, 62\u201363, 109, 113\n\nHerschel, Caroline, 51\n\nHerschel, John, 34, 35\u201336, 52\n\nHerschel, William, 51\u201352, 54, 59\n\nHertzsprung, Ejnar, 45, 55, 56, 62, 66, 74, 138 _n_\n\nHertzsprung-Russell diagram, 74\n\nHigh-Precision Parallax Collecting Satellite (Hipparcos), 123\u201324\n\nHipparchus, 46\u201347, 124\n\n_H.M.S. Pinafore_ (Gilbert and Sullivan), 21\n\nHubble, Edwin:\n\nas astronomer, 96\u201397, 98, 107\u20138, 121, 130\n\nbackground of, 82\u201384\n\nCepheid standard used by, xiii\u2013xiv, 94\u201398, 100, 120\n\ndeath of, 115\n\nat Mount Wilson Observatory, 84, 95\u201397, 104\u20137, 114\n\nperiod-luminosity law used by, xiii\u2013xiv, 94\u201396, 120\n\npersonality of, 83\u201384, 94, 109\n\nphotograph of, _83_\n\npublications of, 96, 97, 106\n\nredshift investigated by (Hubble shift), 103\u20138, 110, 115\u201316, 125\n\nreputation of, 96\u201397, 98, 104, 107\u20138, 144 _n_\n\nresearch of, 94\u201396, 97, 106, 121, 122, 130\n\nShapley's relationship with, 82, 84, 95\u201398, 107, 109, 116, 142 _n_\n\nuniverse as measured by, 109\u201311\n\n_Hubble Atlas of Galaxies,_ 121\n\nHubble constant (K term), 103\u20138, 116, 120\u201321, 123, 124, 129, 130, 147 _n_\n\n\"Hubble Meters,\" 121\n\nHubble shift, 103\u20138, 110, 115\u201316, 125\n\nHubble Space Telescope, 98, 122, 123\n\n\"Hubble Wars,\" 121\n\nHumason, Milton, 104\u20135, 106, 107\n\nhydrogen, 103, 104\n\nHyperion, 134 _n_\n\nInternational Astronomical Union, 90\u201391\n\nInternet, 117\u201318, 147 _n_\n\ninterstellar dust, 78\u201379\n\ninverse square law, 44, 50, 53, 61, 65, 74, 101, 115\n\n\"island universes,\" 12, 59\u201362, 68, 71, 72, 74, 76, 78, 80, 84, 92\u201398, 109, 110, 124, 143 _n_\n\n_Ivernia,_ H.M.S., 36\n\nJames, Henry, 90\n\nJames, William, 90\n\nJeans, James, 62, 93, 143 _n_\n\nJesus Christ, 12\n\nKant, Immanuel, 59\n\nKapteyn, Jacobus, 63, 68, 84\n\nKapteyn universe, 63\n\nKeck telescope, 122\n\nKendrick, Mary, 25\n\nKennedy, John F., 7\n\nKepler, Johannes, 48, 50\n\nKirchhoff, Gustav, 102\u20133\n\nKowalewsky, Sonja, 118\u201319\n\nK term (Hubble constant), 103\u20138, 116, 120\u201321, 123, 124, 129, 130, 147 _n_\n\nlamplight, 16\n\nLaplace, Pierre-Simon, 59\n\nLeavitt, Caroline, 25\n\nLeavitt, Darwin, 25, 39\n\nLeavitt, Erasmus Darwin (grandfather), 25\u201326\n\nLeavitt, Erasmus Darwin (uncle), 26, 27, 33, 37\u201338, 42, 66, 89\n\nLeavitt, George, 25, 33, 39\n\nLeavitt, George Roswell, 25, 26, 35\u201336, 42, 137 _n_\n\nLeavitt, Henrietta Swan:\n\nas astronomer, 27, 30\u201333, 55\u201358, 88, 90\u201392, 104, 118\u201322\n\nbackground of, 25\u201328\n\nbiographical information on, xiii\u2013xiv, 25\u201328, 30\u201333, 90, 117, 120, 147 _n_\n\nin Cambridge, Mass., 25\u201328, 32\u201333, 36, 37\u201338, 39, 41\u201342, 66, 85, 88\u201389, 120, 140 _n,_ 142 _n_\n\nin Cleveland, Ohio, 26, 28\n\nas computer, xiii, 9, 22, 23\u201333, 36, 37, 39\u201342, 52, 55\u201358, 59, 66\u201367, 88, 90\u201392, 119\u201320\n\ncorrespondence of, xiii, 22, 23, 30\u201333, 39\u201342, 88, 120\n\ndeafness of, 23, 28, 31, 32, 33, 44, 86, 136 _n,_ 137 _n_\n\ndeath of, 22, 28, 82, 89\u201392, 117, 118\u201320\n\ndesk of, 91, 117\n\neducation of, 25, 26\u201327, 28, 31\n\nEurope visited by, 30, 33, 36\n\nas female scientist, xiii\u2013xiv, 23\u201325, 27, 88, 118\n\nfinances of, 31\u201332, 90\n\n\"ghost\" of, 117\n\ngrave site of, 89\u201390\n\nillnesses of, 31\u201333, 39, 56, 88\u201389\n\nInternet planetarium named after, 117\u201318, 147 _n_\n\nlast will of, 90\n\nlunar crater named after, 118\n\nmagnitude studied by, 28\u201330, 39\u201344\n\non Nantucket Island, 66\n\nNobel Prize nomination for, 118\u201319\n\nnotebook of, 9\u201310, 29, 56\u201357, 135 _n_ \u201336 _n_\n\nobituary of, 28\n\npersonality of, 28, 56\n\nphotographs of, 23, _24_\n\nphysical appearance of, 23\u201324\n\npress coverage of, 37\n\nprogress reports of, 38, 42\n\npublications of, 30, 31, 37, 38, 39\u201340, 42, 43, 56, 57\u201358\n\nin Beloit, Wisc., 30\u201333, 36, 39\u201342\n\nreligious affiliation of, 25, 28, 118\n\nreputation of, xiii\u2013xiv, 28, 34, 37, 104, 117\u201320\n\nscientific research of, xiii\u2013xiv, 30\u201333, 36\u201344, 55\u201358, 85, 86, 90\u201396, 118\u201320, 124\n\nShapley's views on, 59, 66\u201367, 119\u201320\n\nvariable stars studied by, 29\u201330, 34\u201339, 40, 42\u201344, 53\u201355, 62, 75, 76, 85, 91\u201392, 113, 116, 119\u201320\n\nviewing instruments used by, 23\u201324, 35, 41, 58\n\nas volunteer, 25, 27\u201333\n\nwages of, 9, 31\u201332\n\nLeavitt, Henrietta Swan Kendrick, 25, 42, 57, 85, 90, 142 _n_\n\nLeavitt, Josiah, 25\n\nLeavitt, Martha, 25\n\nLeavitt, Mira, 25, 89\n\nLeavitt, Roswell, 25, 89\n\nLeavitt pumping engine, 26\n\nLick Observatory, 61, 92, 111\n\nlight:\n\nabsence of, 16, 129\n\ncurvature of, 90, 130\n\ngravity and, 130\n\npollution from, 12, 112, 122\n\nspeed of, 5\u20136, 7, 44, 69, 74, 116\n\n\"tired,\" 104, 108, 129\n\nwaves of, 60\u201361, 101\u20137, 129\n\nlight-years, 7, 74, 116\n\nlinear relationships, 105\u20136\n\n_Liometopum apiculatum,_ 62\u201363\n\n\"Local Group\" galaxies, 7, 122\u201323\n\nLongy School of Music, 38\n\nLowell, Abbott Lawrence, 85\u201386\n\nLowell Observatory, 60\n\nlunar parallax, 46\u201347, _47,_ 48\n\nLundmark, Knut, 92\u201394, 97\n\nM33 (Triangulum) galaxy, 92\u201393, 94, 114, 123\n\nMacCormack, Miss, 105\n\nMaffei Group, 7\n\nMagellan, Ferdinand, 34\n\nMagellanic Clouds:\n\nas galaxies, 11, 34\u201335, 55, 123\n\nLarge, 34, 85, 92\n\nphotograph of, _35_\n\nSmall, 34, 36\u201337, 53, 55, 85, 118\n\nvariable stars in, 35\u201339, 40, 43\u201344, 53\u201355, 56, 64, 66\u201367, 76, 85, 92, 95, 96, 100, 113, 119\n\nmagnitude:\n\napparent, 11, 14\u201318, 28\u201330, 38\u201344, 56, 58, 106, 111, 119\n\naverage, 92, 101\u20132, 125\n\nfifteenth, 37, 75, 134 _n_\n\nfifth, 14\u201315\n\nfirst, 14\u201315\n\nfourteenth, 15, 134 _n_\n\nintrinsic, 11, 43\u201344, 45, 50, 53, 55, 58, 74, 100\u2013101, 111, 113, 116, 119, 125, 139 _n_\n\nmeasurement of, 14\u201318, 39\u201340, 56\u201358, 114\u201315\n\nperiod and, 38\u201339, 43, 59, 64, 85, 94, 96, 113, 114\u201315, 119, 124, 138 _n_\n\nrange of, 14\u201315, 65\n\nsixteenth, 58\n\nsixth, 14\n\ntemperature and, 74\u201375\n\ntenth, 58\n\nMallon, Thomas, ix\n\nMalmquist bias, 125\n\nMarch Comet (1843), 12\n\nMaria Mitchell Observatory, 66\n\nMars, 18, 48\u201349, 138 _n_\n\nMassachusetts Institute of Technology (MIT), 15\n\nMauna Kea Observatory, 12, 122\n\nMaury, Antonia Caetana, 20, 87\n\nMerz and Mahler, 14\n\nMessier, Charles, 92\n\nMilky Way:\n\ncenter of, 63, 69, 107\u20138\n\ncosmic dust in, 78\u201379, 111\u201313\n\ngalactic plane of, 78, 112\n\nas galaxy, 34\u201335, 76\u201379, 99\u2013100, 109, 122\u201323\n\ngas clouds in, 8, 12, 51\u201352, 59, 60, 68, 76, 78, 80\n\nglobular clusters in, 63\u201366, 69, 74, 77, 85, 111, 113, 114, 119, 146 _n_\n\ngravitational field of, 61\n\nGreat Debate on (1920), 70\u201381, 86, 98, 115, 120\u201322, 133 _n,_ 140 _n_ \u201341 _n_\n\nmeasurement of, 62\u201381, 82, 84, 94\u201395, 109\u201310, 111, 112\u201313, 115, 119, 138 _n,_ 143 _n_\n\nneighboring galaxies of, 11, 34\u201335, 77, 122\u201323, 125\n\nopen clusters in, 111\u201312\n\npoles of, 78\n\nshape of, 60, 62, _63,_ 63\u201364\n\nsize of, xiii, 7, 8, 54, 62\u201381, 82, 94\u201395, 98, 109\u201313, 115, 119, 138 _n,_ 143 _n_\n\nsun's position in, 11, 54, 63, 69, 75, 99, 110\n\nas universe, xiii, 8, 59\u201362, 69, 70, 84, 92\u201396, 98\n\n\"zone of avoidance\" in, 78\u201380\n\n_see also_ universe\n\nMillerites, 12\n\nMiras, 124\n\nMittag-Leffler, G\u00f6sta, 118\u201319\n\n\"Monkey Trial\" (1925), 99\u2013100\n\nmoon:\n\ncraters on, 118\n\ndistance to, 46\u201347, _47,_ 48, 124, 130\n\nexploration of, 7\n\nMount Harvard, 17\n\nMount Palomar Observatory, 12, 115, 122, 128\n\nMount Wilson Observatory, 12, 17, 41, 56, 62\u201363, 64\u201367, 68, 69, 84, 95\u201397, 101, 104\u20137, 114\n\nNational Academy of Sciences, 70\u201381, 141 _n_\n\nNazism, 73\n\nnebulae:\n\nchemical composition of, 60, 78, 103\n\ndistance of, 59\u201362, 101\u20132, 109\n\nextra-galactic, 109\n\nas galaxies, 11\u201312, 51\u201352, 59\u201362, 68\u201369, 70, 75\u201376, 95\u201398\n\nas gas clouds, 8, 12, 51\u201352, 59, 60, 68, 76, 78, 80\n\nas \"island universes,\" 12, 59\u201362, 68, 71, 72, 74, 76, 78, 80, 84, 92\u201398, 109, 110, 124, 143 _n_\n\nnovae in, 61, 68, 78, 94, 124\u201325\n\nnumber of, 59\u201360\n\nphotographs of, 20, 59\u201360\n\npinwheel, 68\u201369, 92, 97\u201398\n\nrotation of, 68\u201369, 84, 92\u201394, 97\u201398, 125, 143 _n_\n\nsize of, 75\u201376\n\nspectra of, 60, 70, 78, 102\u20138\n\nspiral, 59, 60, 61, 68\u201369, 75\u201376, 78\u201379, 80, 84, 92\u201394, 97\u201398, 114, 125, 143 _n_\n\nas star matter, 59, 60, 61, 78\u201380\n\nvelocity of, 60\u201361, 84, 92, 102\u20137\n\nwhirlpool, 68, 97\u201398\n\n_see also_ galaxies\n\nNew General Catalog of Nebulae and Star Clusters (NGC), 61, 95\n\nNewton, Isaac, 93\n\n_New York Herald,_ 18\n\nNGC 6822, 95\u201396\n\nNGC 6946, 61\n\nNGC 7619, 105\u20136\n\nNobel Prize, 118\u201319\n\n\"noncosmological\" events, 129\u201330\n\nNorth Polar Sequence, 39\u201342, 56\u201358,\n\n90, 91\u201392\n\nNorth Star, 16, 29, 53, 124\n\nnovae:\n\nbrightness of, 61, 124\u201325, 139 _n_\n\ndiscovery of, 94\u201396\n\nlight curves of, 90\n\nin nebulae, 61, 68, 78, 94, 124\u201325\n\nsuper-, 124\u201325\n\n\"O Be A Fine Girl, Kiss Me\" mnemonic, 86\n\nOberlin College, 26, 28\n\nobservation points, 45\u201350\n\nObservatory Hill, _10,_ 12\u201313, 38, 117\n\n_Observatory Pinafore, The,_ 9, 21\u201322, _22_\n\n\"On Running in Trails\" (Shapley), 62\u201363\n\n\"On the Motions of Spirals\" (Lundmark), 93\n\nopen clusters, 111\u201312\n\noptical illusions, 111\u201312, 115, 130\n\noptical interferometer, 122\n\nO'Reilly, Miss, 56\n\nOrion, 6, 17\n\nOxford University, 83\n\nparallax, 2, 45\u201349, _51,_ 53\u201355, 65, 80\u201381, 116, 123\u201324, 138 _n,_ 145 _n_\n\nparsecs, 116\n\nPayne-Gaposchkin, Cecilia, 87\u201388, 91\u201392, 94\u201396, 98, 119\n\n\"peculiar motions,\" 125\n\nPeirce, Charles Sanders, 16\n\npendulum clocks, 48\n\nperiod-luminosity law, 50\u201355\n\nCepheid variables in, xiii\u2013xiv, 43\u201344, 53\u201355, 56, 62, 64, 66\u201367, 72, 76, 88, 94\u201396, 101, 102, 120\n\ndistance measured by, 54\u201355, 66\u201367, 76, 101\u20132, 119, 124\n\nHubble's use of, xiii\u2013xiv, 94\u201396, 120\n\nimplications of, 50\u201355, 67, 116, 118\u201320\n\nLeavitt's analysis of, xiv, 11, 44, 45, 66\u201367, 118\u201320, 138 _n_\n\nperiod-luminosity relationship in, 38\u201339, 43, 59, 64\u201367, 76, 85, 94, 96, 113, 114\u201315, 119, 124, 138 _n_\n\nShapley's use of, xiii\u2013xiv, 59, 62, 66\u201367, 72, 75, 119\u201320\n\nPeru, 17\u201318\n\nPhiladelphia High School, 12\n\nphotographic plates:\n\nblack-star negatives of, 10\u201311, 30\n\ncomparison of, 59\u201360, 68, 95\u201396, 107\n\nemulsion of, 10, 28\u201329\n\nexposure of, 23\u201325, 61, 94\n\nmagnitude measured by, 14, 17\u201318\n\nstorage of, 10\u201311, 20, 117\n\ntime exposure of, 17, 28\u201330, 94\n\nphotometry, 16, 28\u201330, 56, 81, 86, 88, 90\u201391\n\nphysics, 15, 73, 77, 93, 100\n\nPickering, Edward Charles:\n\nas astronomer, 15\u201318, 55\u201356, 107, 121\u201322\n\ncorrespondence of, 23, 30\u201333, 39\u201342, 59, 66\u201367, 85\n\ndeath of, 67, 73, 85\n\nas Harvard Observatory director, 15\u201321, 73, 86\u201387, 88, 91\u201392\n\nLeavitt's relationship with, 25, 27\u201333, 37, 39\u201342, 55\u201356, 59, 66\u201367\n\nphotograph of, _19_\n\nas professor, 15, 27\n\npublications of, 43\n\nsalary of, 21\n\nShapley's relationship with, 59, 66\u201367, 85\n\nPickering, William, 18\n\nPilgrim Congregational Church, 25\n\npinwheel nebulae, 68\u201369, 92, 97\u201398\n\nplanetary orbits, 48, 54\n\nPleiades, 17, 111\n\nPolaris, 16, 29, 53, 124\n\nPopulation I, 114\u201315\n\nPopulation II, 114\u201315, 146 _n_\n\nPrinceton University, 82\n\nprisms, 60, 102\u20137\n\nProcyon, 52\n\n\"proto solar systems,\" 59\n\nprotractors, 45\n\nPtolemy, 48\n\nPuritanism, 25, 28\n\nquadratic relationships, 105\n\nquasars, 128\u201330\n\nradar signals, 47\n\nRadcliffe College, 26\u201327, 31, 136 _n_\n\nradial velocity, 64\n\nradioactive dating, 111\n\n\"Realm of the Nebulae, The\" (Hubble), 106\n\nrecessional velocities, 102\u20137, 123\n\nredshift, 60\u201361, 64, 102\u20138, 110, 113, 115\u201316, 123, 125, 128\u201330\n\nred stars, 56, 75\n\nrelativity, theory of, 5\u20136, 71, 98, 107\u20138, 130\n\nreligion, 12, 36, 69, 100, 118\n\nrobotics, 122\n\nRR Lyraes, 124\n\nRussell, Henry Norris, 45, 55, 64, 73, 74, 79, 82, 85, 86, 96, 97\n\nRussian Imperial Observatory, 14\n\nSagan, Carl, 11\n\nSagittarius, 69, 95\u201396\n\nSandage, Allan, 98, 115\u201316, 121\n\nsatellites, 123\u201324\n\nSaturn, 14, 134 _n_\n\n\"Scale of the Universe\" debate (1920), 70\u201381, 86, 98, 113, 115, 120\u201322, 133 _n,_ 140 _n_ \u201341 _n_\n\n\"Scale of the Universe\" debate (1996), 120\u201322\n\nSchmidt, Maarten, 128\n\nscience fiction, 5\u20136, 7\n\nScopes, John T., 99\n\nSculptor Group, 7\n\nSecond Coming, 12\n\n\"1777 Variables in the Magellanic Clouds\" (Leavitt), 38\u201339\n\nShakespeare, William, 21\n\nShapley, Harlow:\n\nants as interest of, 62\u201363, 69\n\nas astronomer, 72\u201373, 82, 85\u201386, 92\u201394, 98, 107\n\nbackground of, 64, 82\n\nCepheid standard used by, 62, 63, 64, 65, 67, 68, 72, 74, 75, 76, 85, 94, 109\u201310, 113\n\ncorrespondence of, 59, 66\u201367, 85\n\ngalaxy cluster named after, 118\n\nin Great Debate (1920), 70\u201381, 86, 98, 113, 115, 120\u201322, 133 _n,_ 140 _n_ \u201341 _n_\n\nas Harvard Observatory director, 72\u201373, 82, 85\u201386, 107\n\nHubble's relationship with, 82, 84, 95\u201398, 107, 109, 116, 142 _n_\n\nLeavitt as viewed by, 59, 66\u201367, 119\u201320\n\nMilky Way as measured by (Big Galaxy theory), 62\u201381, 82, 84, 94\u201395, 109\u201310, 111, 112\u201313, 115, 119, 138 _n,_ 143 _n_\n\nat Mount Wilson Observatory, 62\u201363, 64\u201367, 68, 69, 84, 98, 107\n\nnebulae as viewed by, 62, 74\u201376, 79, 80, 92\u201398, 146 _n_\n\nperiod-luminosity law used by, xiii\u2013xiv, 59, 62, 66\u201367, 72, 75, 119\u201320\n\npersonality of, 62\u201363, 82, 92\u201393, 109\n\nphotograph of, _71_\n\nreputation of, 72\u201373, 79, 80, 85\u201386, 98, 118\n\n\"village in the canyon\" analogy and, 80\u201381, 116, 133 _n_\n\nShapley's curve, 65, 72, 94, 109\u201310, 113\n\nSirius, 6, 15, 50, 52\n\n61 Cygni, 52\n\nSlipher, Vesto Melvin, 60\u201361\n\nSociety for the Collegiate Instruction of Women, 26\n\nsodium, 102\n\nsolar systems, 59, 99\n\nspace, 16, 78\u201379, 104, 111\u201313\n\nspace program, U.S., 7\n\nspace shuttle, 7\n\nspace-time curvature, 16, 104\n\nspace travel, 5\u20136, 11\n\nspectral lines, 102\u20137\n\nspectral type, 86, 87\u201388\n\nspectrographic analysis, 15, 20, 29, 56, 60, 70, 78, 86, 87\u201388, 102\u20138, 114\u201315\n\n_Springfield Daily News,_ 99\n\nstar matter, 59, 60, 61, 78\u201380\n\nstars:\n\nappearance of, 38\u201339, 43\u201344, 45, 74\u201375, 77\n\nbinary, 51, 52, 64\n\nblue, 74\u201375, 77\n\nB-type, 74\u201375\n\nchemical composition of, 15, 102\u20133, 114\u201315, 124\n\nclassification of, 16\u201317, 20, 86, 87\u201388\n\ncoordinates of, 35\n\ndata on, 15\u201316, 18\n\ndistance of, 6, 8, 15, 38\u201339, 43\u201344, 50\u201355\n\neclipses of, 29\n\nexploding, 101, 124\u201325\n\nmagnitude of, _see_ magnitude\n\nmotion of, 15, 64\u201365\n\nposition of, 15\n\npulsation of, 29\u201330, 36\u201337, 43\u201344, 45, 53\u201355, 64, 65, 76, 85, 100, 124\n\nred, 56, 75\n\nspectra of, 15, 20, 29, 56, 60, 70, 78, 86, 87\u201388, 102\u20138, 114\u201315\n\ntemperature of, 74\u201375\n\nvariable, _see_ variable stars\n\nvelocity of, 64\u201365\n\n_see also specific stars_\n\nstatistical parallax, 65, 124, 145 _n_\n\nstellar aggregations, 111\u201312\n\nstellar photometry, 16, 28\u201330, 56, 81, 86, 88, 90\u201391\n\nstellar regions, 116\n\nSullivan, Arthur, 21\n\nSummer House Hill, 12\u201313\n\nsun:\n\nbrightness of, 50, 74\n\ndistance of, 50\n\neclipses of, 15, 46\u201347\n\nmovement of, 54\n\nposition of, 11, 54, 63, 69, 75, 99, 110\n\nspectrum of, 103\n\nas star, 59, 77, 80\n\nsuperclusters, 7\u20138, 123\n\nSuper-Galaxy Hypothesis, 146 _n_\n\nsuperluminal motion, 129\n\nsupernovae, 124\u201325\n\nsurveyor's transits, 45\n\nSwan, _see_ Cygnus\n\nTammann, Gustav A., 120\u201321\n\ntelepathy, 5\u20136\n\ntelescopes:\n\nimprovement of, 51\u201352\n\nlenses of, 14, 16, 17\n\nlimitations of, 130\n\nmechanical clockworks for, 17\n\nmirrors of, 16, 51\n\nreflectors for, 122\n\n_see also specific telescopes and observatories_\n\n_Third Cambridge Catalog of Radio_ _Sources,_ 129\n\n3C273 quasar, 128, 129\n\ntime, 5, 16, 104\n\n_Time for the Stars_ (Heinlein), 5\u20136\n\n\"tired light\" theory, 104, 108, 129\n\ntransit of Venus, 49\u201350, 138 _n_\n\ntransverse velocity, 65\n\ntriangulation, 2\u20133, _2,_ 45\u201355, _51,_ 80\u201381, 99\u2013102, 130, 138 _n_\n\nTriangulum (M33) galaxy, 92\u201393, 94, 114, 123\n\ntrigonometry, 2, 45\u201349, 52, 80\u201381, 124, 130\n\nTrumpler, Robert, 111\u201312\n\nTully-Fisher method, 125\n\n_Two Moons_ (Mallon), ix\n\n_Two on a Tower_ (Hardy), ix, 117\n\nType Ia supernovae, 125\n\nuniformity principle, 77\u201378, 93, 96, 100, 113\u201314\n\nuniverse:\n\nage of, 110\u201311, 115, 120, 125\n\nbig bang theory of, 16, 108, 110\u201311, 115, 116, 120, 126, 129\n\ncenter of, 69\n\nexpansion of, 106\u20138, 125\u201326\n\nhuman position in, 69, 107\u20138, 125\u201326\n\nmapping of, 11, 16, 99\u2013101\n\nMilky Way as, xiii, 8, 59\u201362, 69, 70, 84, 92\u201396, 98\n\nsize of, 7\u20138, 11, 98, 109\u201316, 120\u201326, 147 _n_ \u201348 _n_\n\nstatic, 107\u20138\n\nuniformity principle in, 77\u201378, 93, 96, 100, 113\u201314\n\n_see also_ Milky Way\n\nUniversity of Chicago, 83\n\nUniversity of Missouri, 82\n\nUniversity of Pennsylvania, 62\u201363\n\nunmanned space probes, 7\n\nUranus, 51\n\nUrsa Major, 6\n\nvan den Bergh, Sidney, 120\u201321\n\nvan Maanen, Adriaan, 68\u201369, 84, 92\u201393, 97\u201398, 143 _n_\n\nvariable stars:\n\ncluster, 64\u201365, 115, 124\n\nidentification of, 29\u201330, 34\u201339, 40, 42\u201344, 45, 53\u201355, 62, 64\u201365, 75, 76, 85, 91\u201392, 94\u201396, 100, 107, 113, 116, 119\u201320\n\nin Magellanic Clouds, 35\u201339, 40, 43\u201344, 53\u201355, 56, 64, 66\u201367, 76, 85, 92, 95, 96, 100, 113, 119\n\n\"overtones\" of, 124\n\npulsation of, 29\u201330, 36\u201337, 43\u201344, 45, 53\u201355, 64, 65, 76, 85, 100, 124\n\n_see also_ Cepheid variables\n\nVaucouleurs, Gerard de, 121\n\nVega, 52\n\nvelocity, 60\u201361, 64\u201365, 84, 92, 102\u20137, 116, 123\n\nVenus, transit of, 49\u201350, 138 _n_\n\n\"village in the canyon\" analogy, 1\u20135, 8, 80\u201381, 116, 127\u201328, 133 _n_\n\nVirgo, 7\n\nvirgocentric flow, 125\n\nVirgo Cluster, 7, 123, 125\n\nVirgo Supercluster, 123\n\n_Washington Post,_ 37\n\nWatson, James D., 38\n\nWeierstrass, Karl, 119\n\nwhirlpool nebulae, 68, 97\u201398\n\nWhite Mountains, 16\n\nWilliams College, 26\n\nwomen:\n\nacademic appointments of, 87, 91\n\nas computers, 19\u201325, 86\u201388\n\neducation of, 26\u201327, 87\n\nWorld War I, 83\u201384\n\nWorld War II, 114\n\nyellow-white stars, 77\n\nZ\u00f6llner astrophotometer, 16\n\nzone of avoidance, 78\u201380\nCopyright\n\nCopyright \u00a9 2005 by George Johnson\n\nAll rights reserved\n\nPrinted in the United States of America\n\nFirst Edition\n\nFor information about permission to reproduce selections from this book, write to Permissions,W.W. Norton & Company, Inc., 500 Fifth Avenue, New York, NY 10110\n\nManufacturing by R. R. Donnelley, Harrisonburg Division\n\nBook design by Chris Welch\n\nProduction manager: Julia Druskin\n\nLibrary of Congress Cataloging-in-Publication Data\n\nJohnson, George, date.\n\nMiss Leavitt's stars : the untold story of the woman who discovered how\n\nto measure the universe \/ George Johnson.\u2014 1st ed.\n\np. cm. \u2014 (Great discoveries)\n\nIncludes bibliographical references and index.\n\n**ISBN 0-393-05128-5**\n\n**ISBN 978-0-39334-837-8 (e-book)**\n\n1. Astrometry\u2014History. 2. Leavitt, Henrietta Swan, 1868\u20131921. 3. Astronomical photometry. 4. Astronomy\u2014United States\u2014History\u201420th century.\n\nI. Title. II. Series.\n\nQB807.J64 2005\n\n522'.09'04\u2014dc22\n\n2005002823\n\nAtlas Books, 10 E. 53rd St., 35th Fl., New York NY 10022\n\nW.W. Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110\n\nwww.wwnorton.com\n\nW.W. Norton & Company Ltd., Castle House, 75\/76 Wells Street, London W1T 3QT\nOther Works\n\nPUBLISHED TITLES IN THE GREAT DISCOVERIES SERIES\n\nDavid Foster Wallace\n\n_Everything and More: A Compact History of_ \u221e\n\nSherwin B. Nuland\n\n_The Doctors' Plague: Germs, Childbed Fever,  \nand the Strange Story of Ign\u00e1c Semmelweis_\n\nMichio Kaku\n\n_Einstein's Cosmos: How Albert Einstein's Vision Transformed Our  \nUnderstanding of Space and Time_\n\nBarbara Goldsmith\n\n_Obsessive Genius: The Inner World of Marie Curie_\n\nRebecca Goldstein\n\n_Incompleteness: The Proof and Paradox of Kurt G\u00f6del_\n\nMadison Smartt Bell\n\n_Lavoisier in the Year One:  \nThe Birth of a New Science in an Age of Revolution_\n\nGeorge Johnson\n\n_Miss Leavitt's Stars: The Untold Story of the Woman  \nWho Discovered How to Measure the Universe_\n\nFORTHCOMING TITLES\n\nDavid Leavitt on Alan Turing and the Computer\n\nRichard Reeves on Rutherford and the Atom\n\nDaniel Mendelsohn on Archimedes and the Science of the Ancient Greeks\n\nWilliam T.Vollmann on Copernicus and the Copernican Revolution\n\nDavid Quammen on Darwin and Evolution\n\nGeneral Editors: Edwin Barber and Jesse Cohen\n\nBY GEORGE JOHNSON\n\n_Architects of Fear: Conspiracy Theories and_\n\n_Paranoia in American Politics_\n\n_Machinery of the Mind: Inside the New Science of Artificial Intelligence_\n\n_In the Palaces of Memory: How We Build the Worlds Inside Our Heads_\n\n_Fire in the Mind: Science, Faith, and the Search for Order_\n\n_Strange Beauty: Murray Gell-Mann and_  \n_the Revolution in Twentieth-Century Physics_\n\n_A Shortcut Through Time: The Path to the Quantum Computer_\n\n_Miss Leavitt's Stars: The Untold Story of the Woman_  \n_Who Discovered How to Measure the Universe_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nPRAISE FOR\n\nTHE DRAGON CONSPIRACY\n\n\"I'm at a loss for words here, since I have no idea how to fully convey how much I loved _The Dragon Conspiracy_ . . . With Makenna's typical sarcastic narration, nearly nonstop action, and plot twists which completely surprised me, _The Dragon Conspiracy_ left me aching for more.\"\n\n\u2014All Things Urban Fantasy\n\n\"[Shearin's] skillful ability to combine scary and heart-stopping chills with laugh-out-loud humor is part of what makes her books such addictive gems.\"\n\n\u2014 _RT Book Reviews_\n\n\"With _The Dragon Conspiracy_ , Lisa Shearin has gotten even more awesome! . . . [It] contains all of Shearin's trademarks: witty dialogue, a winning protagonist, and a plot that doesn't quit. I can't recommend Shearin's work enough.\"\n\n\u2014Tynga's Reviews\n\n\"A fantastic ensemble . . . A fun and thrilling urban fantasy adventure with characters I'd love to go out and have a drink with.\"\n\n\u2014Bea's Book Nook\n\n\"Lisa Shearin has painted a wonderful picture of New York and all of its supernatural inhabitants . . . A fast-paced, fun read with all kinds of supernatural goodness and great world-building.\"\n\n\u2014Wicked Little Pixie\n\n\"Funny, frightening, fast-paced . . . Imminent disaster has never been more fun.\"\n\n\u201438 Caliber Reviews\n\n\"A fast-paced urban fantasy, pack[ed] with action and interesting characters and surprises in most every turn.\"\n\n\u2014Ami's Hoard\n\nPRAISE FOR\n\nTHE GRENDEL AFFAIR\n\n\"Lisa Shearin always delivers a great story. Fresh and exciting, humorous and action-packed, _The Grendel Affair_ is urban fantasy at its best.\"\n\n\u2014Ilona Andrews, #1 _New York Times_ bestselling author of _Magic Shifts_\n\n\"Throw Stephanie Plum, _The X-Files_ , and tequila in a blender and experience it all explode in your face with murder, mystery, the supernatural, tea-drinking dragons, and a hot partner who always has your back (and he might have a little more if given half a chance). Nonstop action, hilarious klutziness, romance, and lethal Lotharios everywhere. What could be better?\"\n\n\u2014Rob Thurman, _New York Times_ bestselling author of _Nevermore_\n\n\"In Lisa Shearin's SPI Files series, seer Makenna Fraser brings Southern sass, smarts, and charm to the mean streets of Manhattan as she battles monsters and other magical beings. An action-packed mix of monsters, magic, and mayhem that will keep you turning the pages.\"\n\n\u2014Jennifer Estep, _New York Times_ bestselling author of _Dark Heart of Magic_\n\n\"One of the best parts of being an author is getting to read books like _The Grendel Affair_ before they're published. One of the most frustrating parts is sitting here asking, 'When does the next one come out?' before book one is even in print. This was great fun, with engaging characters and plenty of fast-paced action. But seriously, when does the next one come out?\"\n\n\u2014Jim C. Hines, author of _Revisionary_\n\n\"One of the strongest parts of this book is the humor. It is deadpan and had me laughing out loud.\"\n\n\u2014Fiction Vixen\n\n\"Light, bouncy, and just fun to read, this book is the perfect antidote for doom and gloom.\"\n\n\u2014Bookyurt\n\nPRAISE FOR THE RAINE BENARES NOVELS\n\n\"Dazzling wit and clever humor. It's gritty, funny, and sexy\u2014a wonderful addition to the urban fantasy genre . . . From now on Lisa Shearin is on my auto-buy list!\"\n\n\u2014Ilona Andrews\n\n\"An enchanting read from the very first page . . . [Shearin is] definitely an author to watch!\"\n\n\u2014Anya Bast, _New York Times_ bestselling author of _Embrace of the Damned_\n\n\"Exceptional . . . Shearin has proven herself to be an expert storyteller with the enviable ability to provide both humor and jaw-dropping action.\"\n\n\u2014 _RT Book Reviews_\n\n\"Lisa Shearin's fun, action-packed writing style gives this world life and vibrancy.\"\n\n\u2014Fresh Fiction\n\n\"An exciting, catch-me-if-you-can, lightning-fast-paced tale of magic and evil filled with goblins, elves, mages, and a hint of love interest.\"\n\n\u2014Monsters and Critics\nAce Books by Lisa Shearin\n\nThe Raine Benares Novels\n\nMAGIC LOST, TROUBLE FOUND\n\nARMED & MAGICAL\n\nTHE TROUBLE WITH DEMONS\n\nBEWITCHED & BETRAYED\n\nCON & CONJURE\n\nALL SPELL BREAKS LOOSE\n\nThe SPI Files Novels\n\nTHE GRENDEL AFFAIR\n\nTHE DRAGON CONSPIRACY\n\nTHE BRIMSTONE DECEPTION\n\n** **\n\n**An imprint of Penguin Random House LLC**\n\n**375 Hudson Street, New York, New York 10014**\n\nTHE BRIMSTONE DECEPTION\n\nAn Ace Book \/ published by arrangement with the author\n\nCopyright \u00a9 2016 by Lisa Shearin.\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\nACE\u00ae is a registered trademark of Penguin Random House LLC.\n\nThe \"A\" design is a trademark of Penguin Random House LLC.\n\nFor more information, visit penguin.com.\n\neBook ISBN: 978-1-101-61845-5\n\nPUBLISHING HISTORY\n\nAce mass-market edition \/ February 2016\n\nCover illustration by Julie Dillon.\n\nCover design by Judith Lagerman.\n\nThis is a work of fiction. Names, characters, places, and incidents either are 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\nVersion_1\n\n# CONTENTS\n\n_Praise for Lisa Shearin_\n\n_Ace Books by Lisa Shearin_\n\n_Title Page_\n\n_Copyright_\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\nChapter 33\n\n_About the Author_\n\n# 1\n\nI wasn't sure this qualified as a first date.\n\nYes, I was having lunch with one of the richest and most eligible bachelors of not only Manhattan but also another dimension. We were in a trendy new restaurant in Tribeca, with a celebrity chef in the kitchen. Two nights ago, I'd played a big part\u2014along with said inter-dimensional bachelor\u2014in saving the lives of the supernatural citizens of the tristate area.\n\nThat was three causes for celebration: hot guy, great food, still alive. Yay, me.\n\nThe fly in my fancy soup du jour, so to speak, was twofold.\n\nFirst, on the other side of the restaurant, and unfortunately with a clear view of our table, was my partner, Ian Byrne. My name is Makenna Fraser. Ian and I work together at a worldwide organization fighting the forces of supernatural evil. Ian thought that my date, Rake Danescu, deserved a spot near the top of our most wanted list.\n\nSecond, I was still considered a newbie and my partner was the protective type. Actually, that was part of his job. Protecting me, that is. Right now, those protective urges were getting on my last nerve. I'd had more than one near-death experience during the last few days, and was way overdue for some R&R. Having Ian only taking his eyes off of me long enough to stare crosshairs onto Rake's forehead was taking the rest right out of my relaxation. We'd recently decided that a healthy mentor\/mentee relationship shouldn't also be a romantic relationship. I had to admit that took a lot of the tension\u2014sexual and otherwise\u2014out of our workday, which was good for focusing on the bad guys and not my partner's mighty fine backside. But right now, Ian was putting plenty of tension right back in. He wasn't jealous\u2014at least I didn't think he was. I think he was being protective of his still relatively new partner.\n\n\"I knew you were reluctant to accept my invitation,\" Rake murmured, \"but I assure you a bodyguard wasn't necessary.\"\n\nI sighed. \"I didn't tell him.\" It was a coincidence that we were all here at the same time. A really unpleasant and awkward coincidence.\n\nRake smiled slowly. \"You didn't tell him? Oh, I like this devious side of you.\"\n\n\"I'm not being devious. My personal life isn't anyone's business but my own.\"\n\n\"I don't think he agrees with you.\"\n\n\"Doesn't appear that way, does it?\"\n\nRake peered around a waiter to see who my partner was having lunch with, and to provoke Ian even more, he made a leisurely show of appreciating the view. I thought I heard Ian growl all the way from our table. Kylie gave Rake a smile and finger wave.\n\nThey knew each other.\n\nOf course they did. They were both breathtakingly beautiful supernaturals.\n\nRake Danescu was a goblin. Kylie O'Hara was a dryad.\n\nKylie was a friend and coworker. Different department, same secret organization.\n\nInterspecies dating wasn't frowned on by most supernaturals. Heck, dryads didn't have much of a choice. All dryads were female, and they all came from trees, so their intraspecies dating pool was more of a puddle. Unless they were lesbians or had a thing for botanicals, dryads had to hunt elsewhere when looking for love. Kylie had dark hair, green eyes, was five foot nothing, and like her sisters down through history, could probably get any man she wanted with the crook of one dainty digit.\n\nIan had had a crush on Kylie since she started at SPI. Though \"crush\" sounded like something out of high school. Let's say he admired her from afar, because getting close would violate one of the personal rules my stoic partner wouldn't allow himself to break\u2014no workplace romance. I'd told him numerous times to just ask her out already. But in the end, it wasn't my doing that resulted in them being here together, it was the same near-death experience that had gotten me here with Rake. When Death does heavy breathing on the back of your neck, you reexamine your life. My partner decided that life was too uncertain to throw away potential happiness.\n\nI smiled. The rule of \"no workplace romance\" was presently being bent until it squealed in Caf\u00e9 Mina's corner booth. I wondered which of his \"thou shalt nots\" my partner would take out for a reexamining look-see next.\n\n\"Kylie O'Hara, a lovely girl,\" Rake said. \"Though I always thought she had more discerning taste.\"\n\nI gave him a look.\n\n\"What?\" The goblin was all innocence, which was no mean feat for any goblin, let alone Rake.\n\n\"You know very well what. Ian doesn't trust you as far as he can throw you.\" I stopped and thought a moment. \"Actually less than that. Night before last, the two of you were at each other's throats, and now here you are having lunch with his partner and making goo-goo eyes at his date.\"\n\n\"Goo-goo? That must be a droll, human term that I'm unfamiliar with. But if its meaning bears any resemblance to how it sounds, I assure you I have never made 'goo-goo eyes' in my life.\"\n\n\"It sounded better than undressing her with your eyes.\"\n\nRake lowered his voice to a soft rumble. \"Do I detect a hint of jealousy?\"\n\n\"In your dreams.\"\n\n\"You and Miss O'Hara battling, with me as the prize for the winner . . .\" His dark eyes turned from teasing to full smolder. \"That would be a dream worth remembering. I assure you, dearest Makenna, you are the only woman I am interested in undressing.\"\n\nI took my napkin out of my lap and calmly placed it on the table.\n\n\"Are you required to check in with Agent Byrne every hour?\" Rake asked, as I scooted my chair back to stand.\n\nI didn't even need to glance at him to know he was smiling and enjoying himself immensely. But I did need to look at him to make sure he completely understood what I was about to say.\n\n\"You know I don't. Now wipe that grin off your face.\"\n\nHe actually batted his eyelashes at me. \"What grin?\"\n\n\"The grin that's telling Ian, 'I'm up to no good with your partner, and there's nothing you can do about it.' 'Cause I can guarantee you he _will_ do something about it. Then I'll have to do something about the two of you, and no one here wants to see that.\"\n\n\"On the contrary, everyone here would _love_ to see that, myself included. And now you've piqued my curiosity. We goblins are rather like your domestic cats in that regard. Once aroused, our curiosity must be satisfied.\" The gleam in his dark eyes said that satisfying his arousal in regard to me had nothing to do with curiosity.\n\n_That_ was Ian's problem.\n\nTall, dark, sleek, and seemingly made for sex, goblins had a reputation for . . . let's just say they had a reputation. A well-deserved one. Add to that Rake being the owner of Bacchanalia, Manhattan's most exclusive sex club, and Ian's concerns were justified, as Rake hadn't even tried to hide his interest in me. In fact, I think Rake had turned teasing me and antagonizing Ian about teasing me into his newest hobby.\n\nI pushed my chair back and stood. Rake, playing at being the perfect gentleman, stood with me.\n\nIan needed to understand that I was a big girl and as such was totally aware of who Rake was and what he wanted. And he wasn't getting any of it until when\u2014or if\u2014I decided I wanted it, too. Not that it was any of Ian's business, which was another thing he needed to get through his head.\n\n\"I'll be back in a minute.\"\n\nRake smiled fully as he took his seat. Anyone watching saw an unwholesomely handsome man giving his date a dazzling smile. I saw all of that plus a pair of fangs. I was a seer. It was a rare ability that enabled me to see through wards, spells, shields, and glamours that supernaturals used to disguise themselves from the humans around them. Only about half the people in Caf\u00e9 Mina were human; the rest were a mix of supernatural races.\n\nSo I knew exactly what Rake Danescu was, in more ways than one.\n\n\"I shall eagerly await your return,\" Rake all but purred.\n\nI sighed. \"Yeah.\"\n\nI started across the restaurant, to the accompaniment of Rake's low chuckle.\n\nIan's date glanced up from her menu with a quick grin that, in the language of girlfriends everywhere, said: \"I'm so happy for you!\" Or if expressed in a single sound\u2014\"Squee!\"\n\nIan was not happy\u2014for me or anyone else\u2014and he most certainly was not fighting back an urge to squee. The only urges Ian was fighting were violent ones, and he didn't appear to be fighting very hard.\n\n\"Stop it,\" I told him.\n\nTwo words. One directive. I didn't believe in beating around the bush.\n\n\"I haven't done anything,\" he said. The \"yet\" was unspoken.\n\n\"Neither has Rake.\"\n\n\"He wants to.\"\n\nI resisted rolling my eyes. \"So does every other red-blooded man with any woman they're attracted to.\" I left \"yourself included\" unsaid. I paused. Goblins had red blood, didn't they? For the sake of my argument I'd go with yes.\n\n\"Every other red-blooded man hasn't enthralled you,\" Ian noted.\n\nKylie did a combo groan and face palm.\n\n\" _Enthralled?_ I'd ask you to please tell me you're kidding, but I know you're not.\"\n\n\"Rake Danescu is a dark mage, one of the best.\"\n\n\"And I'm a hick from the North Carolina mountains ripe for the pickin'.\"\n\n\"I did not say that.\"\n\n\"Oh yes, you did.\"\n\nMy first night on the job, Rake had magicked himself a look inside my mind. It hadn't been personal, merely business. Okay, maybe it had been a little personal. From what I understood, it'd be easier the next time. I'd been with SPI for well over a year now, and Rake hadn't tried it again.\n\nThe combativeness went out of Ian. \"Mac, I'm simply worried that\u2014\"\n\nA man screamed.\n\nAn immaculately groomed guy in a really nice suit who'd just come back from the men's room was staring in total and complete horror at his waiter. The guy was human; his waiter was not.\n\nI could see that. The man shouldn't have been able to.\n\nThe two other suit-clad men across the table from him were staring at this guy like he'd lost his mind. They looked like a trio out for a business lunch. The screaming guy had a tablet next to him on the table. Yep, business lunch. If those were his clients, the screamer wasn't making a good impression.\n\nEveryone else in the restaurant saw the waiter as what he wanted them to see\u2014a hot-beyond-belief, twentysomething, out-of-work actor waiting tables to pay the rent. I saw what he really was\u2014an incubus.\n\nSomehow the businessman, who'd now progressed from screaming to babbling, saw what I was seeing.\n\nThe incubus's features were vaguely humanoid, but more closely resembled a creature out of a bad 1950s horror movie, with translucent skin and a slit suction cup for lips.\n\nThe man stood so quickly his thighs hit the table, nearly knocking it over on the two men with him, who scooted back to keep from taking soup in the lap. One guy wasn't so lucky with his drink, shouting a word people generally tried not to say at a polite business lunch as he grabbed a napkin and blotted the front of his pants.\n\nThe hysterically babbling man didn't notice.\n\nI noticed his right hand was clenching a steak knife.\n\n\"He can see them,\" I whispered quickly to Ian.\n\nIan didn't answer. He didn't need to. He knew what I meant.\n\nThe man spun, taking in the supernaturals all around him, both diners and restaurant staff.\n\nThen he spotted the one closest to him.\n\nRake.\n\nOh no.\n\nBefore Ian could stop me, I sprinted the short distance back to our table, stopping in front of Rake, trying to block the man's view. Fat lot of good that did since the goblin was now on his feet, ready to defend himself if necessary. Rake was a head taller than me, and other than the gray skin visible on his hands, from the neck up was everything that said \"goblin\" to anyone who could see past Rake's human glamour. To me, and any other supernatural or enlightened human with a pulse who could see past that glamour, Rake was gorgeous. But I could see where silvery skin, pointed ears, and fangs could be disconcerting.\n\nThe man's eyes widened in disbelief. \"What . . . what are you?\"\n\nIan stepped up like the former cop he was, his voice low and calm. \"Sir, I need you to put down the knife.\"\n\nThe man quickly turned and saw the reassuringly human Ian.\n\n\"Do you see them?\" His words came in a rush. \"Can\u2014\"\n\n\"I see that you have a weapon, and you need to put it on the table next to you and step away.\"\n\nKylie was on her phone, no doubt calling headquarters.\n\nThe man spotted a couple sitting at the table behind us. Kelpies. Everyone else saw a nice, middle-age couple. He saw vaguely human creatures with green skin, gills, and a mouthful of sharp teeth.\n\n\"Monsters!\" he shrieked.\n\nHe staggered back, stumbling and catching himself on a bananas Foster serving cart. He stayed upright, the cart didn't. Flames ignited the closest tablecloth, and were fed by the spilled rum.\n\nPeople screamed, shouted, and ran for the exits as the sprinkler system went off.\n\nI heard a siren outside. Someone must have already called the police. Now we needed the fire department, too.\n\nRake stepped up close behind me, his lips at my ear. \"I promised you'd never be bored on a date with me.\"\n\n\"This wasn't what I had in mind.\"\n\nAll signs of playfulness were gone. \"Neither did I.\"\n\nWith his hand at the small of my back, Rake steered me toward the restaurant's kitchen, away from the fire and the crowd surging toward the front doors. In a panic, people tended to go with the obvious, even if it wasn't the closest or safest. Leave it to Rake to know the back way out of a building he'd never been in before.\n\nIan and Kylie were right behind us.\n\nI turned my head toward Ian. \"Where\u2014\"\n\n\"He dropped the knife and ran out the front,\" Ian told me. \"First one out. Right into the waiting arms of my former brothers in blue.\"\n\n\"In addition to his freedom,\" Kylie said, \"I think it's safe to say he just lost his clients _and_ the account.\"\n\n# 2\n\nGOBLINS, elves, vampires, werewolves, fairies, trolls, dwarves, and anything else you've read about in fairy tales or your favorite fantasy novel series.\n\nThey're all real.\n\nIt used to be known, confirmed, and accepted fact that all of those and more existed. Then humans went and got themselves civilized and educated. The smarter humans thought they were, and the more they thought they knew, the less they believed in things that went bump in the night.\n\nTheir disbelief didn't make any of those things any less real\u2014or deadly.\n\nIn a world where supernaturals lived alongside humans, what you couldn't see could kill you. Some of them could even bring you back from the dead and kill you again.\n\nMagic exists, monsters are real, and fighting the forces of evil is a full-time job. At least there's hazard pay.\n\nHumans, being human, merely thought up more explanations for what monsters were, and excuses for what they couldn't possibly be.\n\nTo tell you the truth, our job was a lot easier when John and Suzie Q. Public didn't know they were lucky to make it to the office every morning without getting pecked to pieces. Though that was only during the Werepigeon Infestation of 2003. Before my time, but definitely one for the agency history books.\n\nNew Yorkers pride themselves on not even batting an eye when they walk past the weird, the wacky, and the otherworldly.\n\nI've got news; if they saw someone change into a werewolf right in front of them, their blas\u00e9 would go bye-bye, probably along with the contents of their bladder. Heck, the sound effects alone\u2014bones popping, sinews stretching, muzzle elongating and sprouting fangs\u2014would be enough to send them screaming into the night.\n\nWe battle the creatures of the night and keep humans in the dark.\n\nWe're the agents of Supernatural Protection & Investigations. SPI is a worldwide organization, but New York is home to the U.S. and world headquarters.\n\nThere are two New Yorks. As if there isn't enough traffic in one.\n\nThere's the New York that millions of people see, hear, touch, smell, and in the summer when the wind's right and the garbage barges are ripe, taste. Then there's the New York that's home to the world's largest concentration of supernatural beings\u2014unseen, unheard, unknown. And it's SPI's job to keep it that way.\n\nI was one of the agency's five seers. Since the beginning of crime, some bad guys\u2014human or otherwise\u2014have depended on disguises to elude capture. While humans were limited to wigs, makeup, and the ever popular but terribly ineffective sunglasses, supernaturals could tap into their magic or buy an amulet that would enable them to alter their appearance, or even hide their entire body with a cloaking spell.\n\nIt didn't matter what they used, or how good it was, I could see right through any and all of it.\n\nSo seers were downright handy in an organization like SPI.\n\nI pointed out the bad guys, and our agents or commando teams brought them in.\n\nIan was our top agent.\n\nKylie was our director of media and public relations.\n\nAnd Rake pretty much had a permanent spot on our suspects list.\n\nRight now the four of us were sharing a booth in a coffee shop around the corner from the restaurant. The police had taken it as their interviewing room since Caf\u00e9 Mina was presently a smoke-filled ruin. One of the cops had recognized Ian from their time together in the NYPD, and one of the staff had told them that Ian had tried to disarm the hallucinating crazy guy. Since the three of us were with him, they wanted our statements as well.\n\nLucky us.\n\nIan and Rake had declared an unspoken temporary truce. I knew it wasn't due to any newfound camaraderie, but rather that it wouldn't go over well to beat the crap out of each other in front of the cops. For the moment, they could pretend to make nice.\n\nThe officer who'd taken our statements was an elf. He knew who we were and who we worked for\u2014or at least he knew who Ian and Kylie were, and everyone knew who Rake was. The elf couldn't see through Rake's glamour, but he knew what Rake's human glamour looked like. The elf didn't know me from Adam's house cat, and I was fine with that. It's never been a goal of mine to be recognized on sight by the police force of any city.\n\nFrom what the guy had been screaming while being taken into custody, it was apparent that he could see the supernaturals in the restaurant with him for what they really were. The young elven officer knew that but he couldn't exactly put that in his report. I felt bad for him, but in a place like New York, with its huge supernatural population, being able to work a case while keeping the city's biggest secret was a required talent. If he couldn't juggle, he'd better learn fast.\n\n\"The gentleman began behaving strangely after coming out of the men's room,\" Rake said. \"While you were arguing,\" he added with an amiable smile, looking right at Ian, \"I was observing.\"\n\n\"Arguing?\" the elf cop asked Ian.\n\n\"A personal matter, Officer.\"\n\nIan's face was a perfect mask of neutrality; however, from Rake's pained hiss, Ian had just introduced the heel of his boot to the top of Rake's foot. Then Ian grunted as Kylie did the same to him, except with a stiletto heel.\n\nI rolled my eyes.\n\n\"So you're implying that he may have taken a drug?\" the officer asked.\n\n\"Well, he wasn't screaming about monsters _before_ he went to the head,\" drawled a familiar voice from behind me.\n\nOur day was finally looking up.\n\nLieutenant Frederick Ash was a detective with the NYPD's drug enforcement unit and, like the elven officer, was clued in to SPI and the supernatural community. Unlike the young elf, Fred was an elf\/human hybrid. While he had enough elven blood running through his veins to use minor magic, his physical appearance lacked the jewel-tone eyes, pale skin, and pointed ears that marked the elven race, so no glamour was needed.\n\nFred was plainspoken and said it like he saw it.\n\nI liked him.\n\nI liked it even better that he was here.\n\nIan liked it enough that he and Fred did the bro-hug thing. Though they'd worked closely together during Ian's time with the NYPD, his leaving the force to come work for SPI hadn't weakened that bond. Not to mention, it helped us to have people inside the NYPD, and the reverse was true for them. A lot of crime in the city crossed the human\/supernatural barrier, which sometimes wasn't so much a barrier as a chalk outline on a sidewalk, an outline drawn around human and supernatural alike.\n\nKylie's eyes went to the street outside. I turned to look.\n\nOh crap.\n\nTwo news trucks complete with satellite dishes. For now it was probably to cover the destruction of the city's newest trendy restaurant, but all it would take would be talking to any of the patrons, most of whom would love to be on TV, to root out the cause of the fire. A previously upstanding businessman suddenly seeing monsters, who was probably cooling his heels in a padded observation room by now, would spark the sensation and ratings seeker that was in the heart of every TV journalist.\n\n\"Officer, do you have any more questions for me?\" Kylie asked.\n\n\"No, ma'am.\"\n\nShe nodded in the direction of the news trucks. \"Then if you all would excuse me, it's time I went to work.\"\n\nShe scooted out of the booth and headed for the door, heels clicking on the tile with sharp purpose.\n\nMedia and Public Relations is SPI's largest and sometimes most critical department. Kylie and her team were hands-down the best at what they did\u2014neutralizing a supernatural exposure problem _before_ it became a publicly visible crisis. In addition, Kylie's \"secret identity\" was a world-renowned debunker of the supernatural, and the ultimate mistress of misinformation. She put herself front and center on TV and radio talk shows, and was accepted by respected journalists as an expert on the expos\u00e9.\n\nKylie was the best at spinning a supernatural news story the way she\u2014and SPI\u2014needed it to go.\n\nFred jerked his head in the direction of a back table. \"A word with you, Ian?\"\n\n\"Sure.\"\n\nThe boys went off to chat, leaving me and Rake alone.\n\nAn immaculately groomed man with a microphone and cameraman in tow met Kylie at the door. Though \"met\" was a little mild. \"Ambushed\" was a more accurate description.\n\nThere could've been hurricane-force winds out there and not one hair on Baxter Clayton's head would've moved.\n\nBaxter was an investigative reporter for a local TV station and an all-round asshat. I didn't envy Kylie her job right now. Actually, I'd never envied Kylie's job. I was a horrible liar and even worse at hiding how I felt when around people I didn't like, and Baxter Clayton definitely qualified.\n\nRake swore.\n\n\"What?\"\n\n\"Baxter Clayton.\"\n\n\"Yeah, I don't like him, either. I don't think anyone does. That's probably why they keep him around. The guy everyone loves to hate. Hate equals high ratings.\"\n\n\"He hasn't been trying to get you on camera for a story on New York's upper-class sex industry,\" Rake muttered.\n\nI bit back a snort.\n\n\"It's not funny.\"\n\nBaxter Clayton was in earnest conversation with a professionally poised and smiling Kylie.\n\n\"It looks like Kylie's taking one for the team then. You owe her.\"\n\n\"Yes, Miss O'Hara will have my eternal gratitude _if_ I can get out of here without being seen.\"\n\nRake was ruffled. It was a rare sight, so I was going to enjoy it while it lasted. \"You do a lot of ducking out back doors, don't you?\" I asked with a smile.\n\n\"Enough that I've become quite adept at it.\"\n\nWith that, he scooped my hand off the table and brought it to his lips.\n\nHis voice softened. \"And by the way, this lunch didn't count. A maniac setting fire to the table next to ours doesn't qualify as a successful date.\"\n\n\"Define successful,\" I managed.\n\nThe goblin gently turned my hand and placed a lingering kiss on the palm, sending a tingle of sensation to other places.\n\n\"No dinner,\" I said, trying for firm and uncompromising\u2014and probably failing miserably.\n\nRake's eyes glittered. \"Breakfast then?\"\n\n\"You don't give up, do you?\"\n\n\"Not anytime soon.\"\n\n\"How about another lunch?\" I suggested.\n\n\"How about tomorrow?\"\n\n\"I'll check and get back to you.\"\n\n\"If you don't, I will.\"\n\nGiving the back of my fingers a parting brush of his lips, Rake quickly escaped out the back door.\n\nI snuck a glance over at Ian and Fred. Thankfully, my partner had his back to me.\n\nFred did not.\n\nFrom the sly wink he just gave me, I'd say he saw Rake's Cyrano de Bergerac exit. Then he gestured me over to join them.\n\nOh boy.\n\n\"I was just telling your partner what your knife-wielding businessman had likely snorted.\"\n\n\"So it was a drug,\" I said.\n\nFred nodded. \"A new one. High-end designer.\"\n\nIan glanced back at the now empty booth.\n\n\"Rake had to leave,\" I told him before he could ask any questions that I'd completely blow answering.\n\nFred, bless him, didn't say a word.\n\nKylie wasn't the only one who'd taken one for the team. It looked like I might owe Fred one, too.\n\n\"I was telling Ian that from what I've heard about Brimstone, your boy was one of the latest customers.\"\n\n\"Brimstone?\"\n\nFred shrugged. \"That's what they're calling it. It can be smoked or snorted. We haven't gotten our hands on any yet for the lab to play with, and the latest customer didn't have any more on him. One of our sources told us it's lava colored. We're assuming that's the source of the name. And from the reactions of the three people who've taken it while in public . . .\" Fred lowered his voice. \"One of the side effects is that they can see supernaturals.\"\n\n\"Through glamours,\" I said, likewise keeping my voice down.\n\n\"Through anything.\"\n\nGlamours, shields, wards, and cloaks.\n\n\"Well, there goes my job security,\" I said.\n\n\"Hardly,\" Fred replied. \"The humans who've gotten hold of the stuff freak out like you just saw.\"\n\n\"How about supernaturals?\" Ian asked.\n\n\"Unknown. We've had no reports of a supernatural under the influence of Brimstone. We didn't find out about the stuff until a couple days ago. But if supernaturals were taking it, they wouldn't exactly scream about seeing monsters.\"\n\n\"How long do the effects last?\" Ian asked.\n\n\"They start to come down after a couple of hours.\" He paused. \"Good part is that they don't remember what they saw, just that it was the mother of all bad trips.\"\n\n\"Great,\" I said. \"At least if they got the crap scared out of them, they won't be lining up to buy more.\"\n\n\"Wish people had that much sense,\" Fred said. \"When I heard we might have another customer, I wanted to get some fresh eyewitness accounts.\" He grinned. \"Imagine my surprise to find you two among the witnesses.\"\n\n\"It wasn't exactly how we intended our lunch hour to go,\" I said.\n\n\"I got that impression.\" His blue eyes twinkled.\n\nFred didn't need to elaborate. Ian and Fred were beer and steak kinds of guys. Caf\u00e9 Mina was hardly where either one would go\u2014or could afford to go\u2014to grab a quick bite for lunch. Fred simply eyed Ian's sports jacket and tie, glanced out at Kylie, grinned, and gave my partner a congratulatory smack on the shoulder.\n\nDetective Fred Ash. Master of deduction and masculine nonverbal communication.\n\n\"Know where the supply is coming from?\" Ian asked. My partner was the master at ignoring questions, direct or implied.\n\n\"That's the reason I'm glad to run into you two here,\" Fred replied. \"We'll continue to investigate, but let's just say we'll only be able to get so far.\"\n\nIan swore mildly, like a man who knew he wasn't going to have time for more fancy lunches anytime soon.\n\nFred nodded. \"Yeah. I'd bet my next paycheck that Brimstone came from out of town.\"\n\n# 3\n\nWHEN you worked for SPI or were clued in to the supernatural world, \"out of town\" didn't mean Hoboken.\n\nA supernatural criminal entrepreneur was cutting him- or herself a slice of New York's drug-dealing pie. The highly profitable, upper-crust part. That wasn't going to make the city's established drug lords and ladies very happy. And when they weren't happy, and that much money was involved, blood would start flowing.\n\nWhat lunch I'd managed to eat hadn't even had a chance to settle before we got the call.\n\nThe goblin manager of an upscale apartment building had received a tenant complaint of heat and a really bad smell coming from the apartment next door. Suspecting a fire of some sort, he'd quickly knocked, and when no one answered, he used his master key to open the door.\n\nHe saw what was inside and promptly closed it.\n\nHe then called SPI, not the NYPD.\n\nThere was a dead body, it was a supernatural, and the stink was sulfur.\n\nSulfur could mean one of two things: demons or a black-magic-spawned portal. Or both. None of the above signaled fun times ahead for us.\n\nSulfur was another name for brimstone.\n\nCoincidence?\n\nI wasn't gonna hold my breath on that one. Especially when we learned who the dearly departed was.\n\nSar Gedeon.\n\nElf, exiled aristocrat, and drug lord, who was most definitely from out of town, just like the new designer drug.\n\nThe apartment building was only two blocks from Caf\u00e9 Mina.\n\nWe were there within minutes.\n\nNormally Ian and I weren't part of an initial response team unless the investigation required the services of a seer, but we were the closest agents. Our job was to secure the scene from mortal authorities until SPI's crime scene investigators could get there.\n\nLike humans, supernaturals died every day in New York, and everywhere else for that matter. There was a problem when supernatural deaths involved a crime. Crime meant police, and police meant the potential for exposure.\n\nLiterally.\n\nSupernaturals who didn't look human needed a glamour to disguise themselves. Any glamour, regardless of the power of the spells that held it in place, faded within one hour after death. In a murder investigation, that meant that the victim would go through a quick and rather startling transformation, either before the police arrived or while they\u2014and the body\u2014were still on the scene. Those were the tough ones.\n\nEach major city had its own supernatural medical examiner's office that reported suspicious deaths to the regional SPI office.\n\nSupernatural families also notified SPI in the case of any unusual deaths, and our investigators and medical examiners responded. Humans have local morticians and funeral homes, and so do supernaturals. Each race has cultural or religious beliefs that dictate what is done with a body after death\u2014without attracting the attention of mortal authorities. But when supernaturals made themselves a part of human society\u2014or were inconsiderate enough to get themselves murdered in public\u2014things could get dicey.\n\nThat was the situation we were dealing with now.\n\n* * *\n\nIan discreetly showed his badge to the doorman at the Murwood.\n\nThe man glanced down at the ID and at Ian's face without moving anything except a pair of cool gray eyes. He wore the double-breasted, quasi-military style of long coat and hat that seemed to be the uniform of doormen at upper-crusty apartment and condo buildings citywide. His bearing said ex-military or police, loud and clear. Then his face took on the neutral and faraway expression that signaled someone was speaking to him on his Bluetooth earpiece. Either that, or he was having an out-of-body experience.\n\n\"Mr. Nadisu is expecting you.\" Not taking his eyes off either us or anyone else on the street nearby\u2014which was a nifty trick\u2014he reached back and opened the door for us.\n\nThe lobby of my apartment building was more of a foyer with mailboxes against one wall. It was almost impossible to squeeze past anyone checking their mail without way too much intimate contact with a neighbor whose name you didn't know.\n\nAt least a dozen of my lobbies would have fit in the Murwood's.\n\nThe goblin who met us there looked like his day was going worse than ours.\n\nGoblins liked being in control of themselves and everyone and everything around them. You'd never see a goblin frazzled, at least not in public, and definitely not in front of strangers. This guy was frazzled. He wanted to be cool and collected, but today just wasn't his day to get what he wanted.\n\nAs Ian and I could attest, there was a lot of that going around.\n\nThere was no one else in the posh lobby, but Ian still kept his voice down as he introduced us, even though Jesin Nadisu knew who we were. Official protocol had to be observed.\n\nAnyone else, Ian included, would see a human man, in his mid-thirties, impeccably dressed in a suit that probably made him fit right in with the building's wealthy tenants. He wouldn't want to offend his tenants' sensibilities by wearing anything that came off the rack. Other than that, there wasn't anything that made him particularly noticeable.\n\nBrown hair, brown eyes, medium height. Like his suit, Jesin Nadisu had gone out of his way to blend in.\n\nWith my seer vision, I saw a surprisingly young and unsurprisingly handsome goblin in his early twenties (or whatever the goblin age equivalent was) with sleek, shoulder-length, blue-black hair pulled back in a tight ponytail at the nape of his neck, with large dark eyes. Elves and goblins age slower than humans, and do a better job of it while they do; no plastic surgery or Botox shots needed.\n\nThe goblin gestured. \"This way, please.\"\n\nWe took one of the elevators to the seventh floor.\n\n\"How long has Mr. Gedeon lived in the apartment?\" Ian asked.\n\n\"Mr. Gedeon owns . . . _owned_ the apartment,\" the goblin said, \"but he didn't live there. He visited once or sometimes twice a week. He kept the place for a lady friend.\"\n\n\"The name of the tenant?\" Ian asked.\n\n\"Mara Lorenz. She went out of town two days ago.\"\n\n\"Then why was Mr. Gedeon here?\"\n\nJesin Nadisu's professional reserve cracked and he smiled slightly. \"The same reason he was always here. To get away from his wife.\"\n\nWhen we got to the seventh floor, the stench of sulfur smacked us all in the face.\n\nThe goblin unlocked the apartment door, but made no move to open it.\n\nI didn't blame him. He'd been there, done that, got the trauma.\n\nIan broke the silence. \"Mr. Nadisu, I need you to return to the lobby and wait for our lab team.\"\n\nThe goblin nodded with no small measure of relief and turned toward the elevator.\n\n\"And don't let anyone in unless they live in the building or are from Sarkowski Plumbing,\" my partner added. \"They're our lab team.\"\n\n\"I wouldn't anyway. This is a secure building.\" The young goblin winced. \"At least it was.\" He swallowed in an audible gulp. \"And on my watch.\" He paused. \"Would your non-admittance request include any of Mr. Gedeon's business associates?\"\n\n\"It would. And do not discuss what you have seen with anyone.\"\n\n\"My discretion and that of the Murwood is assured for _all_ of our tenants.\"\n\n\"Good. Keep it that way.\"\n\nI noticed he never said he wouldn't tell anyone, just that his discretion was assured. With goblins, you had to watch for the small print. Many of the top lawyers in the city were goblins\u2014and more than a few of the politicians. I was sure Ian had noticed; he chose not to try to wrangle a promise out of him. A goblin could find ways to get around those, too.\n\nBut I still felt sorry for him. Contrary to what Ian had told him, he'd have to tell the owner of the building what had happened. I was sure we could count on their discretion as well. No landlord wanted to spread around that a murder had occurred in one of their buildings.\n\n\"Have any of the other tenants been asking questions?\" Ian asked.\n\n\"No, just from the apartment one floor below, and the couple next door. They've since left for a luncheon engagement. I've called and told them that I've looked into it, and there's no cause for concern.\"\n\nGoblins could spin a lie as easily as breathing. Like I said, they were great lawyers and politicians.\n\nIn my book, your next-door neighbor getting himself murdered was plenty cause for concern. Though if Sar Gedeon had been specifically targeted\u2014considering what he did for a living, that scenario was highly likely\u2014there really wasn't any need for the neighbors to worry for their own safety. That is, unless they stuck their noses where they didn't belong and the killer got wind of it. So, when you looked at it like that, the manager's lie might have saved their lives. See? He lied and it was for their own good. It was all in how you spun it.\n\nAs soon as the elevator doors closed, Ian drew his gun, which was loaded with silver-infused hollow points.\n\n\"Stay here,\" he told me.\n\n\"I can do that.\"\n\nNot only could I do that, I was glad to do that. Running underneath the sulfur stink was an odor I could only describe as burned beef brisket. I wasn't a math whiz, but the smell of burned meat coming from a room with a dead body? Those added up to a cause of death I was in no hurry to confirm for myself.\n\nIan opened the door and slipped into the apartment.\n\nI had the smell of sulfur and burned flesh to keep me company while I waited in the hall. I didn't know which one was worse; but since they were both here, I didn't have to choose. Lucky me.\n\nI was familiar with the smell of brimstone. I'd gotten a snootful of the stuff only once before, and that was one time too many.\n\nMy SPI training had included a class in what was generously called \"Aroma Identification.\" When tracking a supernatural suspect, let's just say that sometimes visual contact didn't come first.\n\nOne of the aromas covered in class was brimstone. Our instructor kept samples in airtight containers of substances we needed to immediately know when we caught a whiff of it.\n\nBrimstone was the biggie.\n\nIts presence at a crime scene or while in pursuit of a suspect indicated two things that set my survival instinct to twitching: demons and black-magic-spawned portals.\n\nNeither were things you wanted catching you by surprise.\n\nTwo minutes and no shots fired later, Ian opened the door and I stepped in just far enough for him to close the door behind me.\n\n# 4\n\nWHEN a supernatural dies, any glamour they might have been using to disguise their true appearance fades within the first hour after death. A supernatural creature manifesting on a slab in the city morgue in front of a screaming technician was one of those awkward moments it was part of our job to prevent. The scene inside that apartment was bad, but wasn't the worst I'd ever seen. Believe me, you haven't seen a murder scene until you've busted into a room after a grendel has had ten seconds to rip arms, legs, and head off some poor sot, and dangle his intestines from an overhead light fixture like a party streamer.\n\nI thought that had to be the apex of disgusting, and as far as the ick meter went.\n\nThis came close. What the building manager had found beyond that apartment door jumped right over awkward and landed smack dab on bizarre.\n\nSar Gedeon had gotten away from his wife. Too bad he hadn't had similar success with his murderer.\n\nAnd it was most definitely a murder.\n\nThe dead elf was shirtless, as if the killer wanted to show off his work. Though at least he still had his pants on. His killer had apparently decided to confine his work to above the waist.\n\nGedeon's hands were clenched into claws, and the palms and insides of the fingers had been burned black. So much for the source of the burned brisket smell. The other burned body part was the skin over and around the breastbone. It had been branded with a single hoofprint. Though considering the presence of the sulfur smell, I figured we weren't dealing with a homicidal cow.\n\nThe brand was either a signature by the demon that had done the burning, or the way it had held down the elf while it\u2014or a partner in crime\u2014had caused what looked to me, a non-medical professional, as the likely cause of death.\n\nA gaping hole in Sar Gedeon's chest.\n\nIan approached the body, careful not to step on any stain or splatter, squatted down next to the chest, and looked inside.\n\nHis brow creased. \"That's interesting.\"\n\nOnly a man who'd spent five years as a homicide detective in the NYPD and the seven years before that doing something in the military that he wouldn't (or couldn't) talk about would describe the inside of a man's open chest as \"interesting.\" Made me wonder what it'd take to make my partner regret eating lunch, which made me know I didn't want to find out.\n\nHowever, being the curious type, I found Ian's description irresistible.\n\nI went to where Ian squatted, leaned over his shoulder, and took a peek.\n\nAnd regretted it.\n\nCuriosity wouldn't kill a cat, but getting a gander of this could make it hork up one heck of a hairball. Right now, I was about to do something similar.\n\nI'd heard our folks who dealt with bodies as part of their jobs carried a little jar of Vicks with them. Constantly. On duty or off. With SPI, you never knew when off duty could turn to very much on duty.\n\nBack in North Carolina's pollen-filled spring and fall seasons, Vicks was my best friend. Some nights I was so stuffy I couldn't get to sleep without a swipe of that wondrous eucalyptus-scented goop under my nose. Since coming to New York, my allergies were gone. My Vicks was buried in the dark recesses of the cabinet under my bathroom sink. When I got home, I was going spelunking.\n\nI already carried Dramamine and Tums. Now I was adding Vicks. I'd only been on the job a year and I was already carrying around my own starter pharmacy.\n\nIan had his phone out. The pick up on the other end was quick. Ian's communication was even faster. \"We've got a demon, Class Five or higher.\"\n\nThat'd send the folks at headquarters scrambling. Classes of demons went up to twelve. In my opinion, five was bad enough. Anything higher wasn't known for having a light enough touch to leave a brand. We wouldn't have found a hole in the victim's chest; we'd have found a hole where the vic had been squashed into the floor.\n\nNot all demons had cloven hooves, but no other supernatural did\u2014except for satyrs and minotaurs, and neither one of those could radiate heat through their bodies to burn hands and brand a chest.\n\n\"You're sure it wasn't a branding iron?\" I didn't think it was, but it never hurt to hope.\n\n\"The burns on Gedeon's hands weren't made by grabbing a branding iron,\" Ian said. \"The fingers are spread the same width apart and burned in the same places. Our vic was grabbing a demon's leg. The span of his hands indicates a larger demon, at least Class Five. The cloven hoof was holding him down while the demon's partner cut his chest open and ripped his heart out.\"\n\nMy lip curled. \"That looks a bit jagged for a knife. Maybe a claw?\"\n\nIan looked closer at the inside of the elf's ruined chest. \"A possibility. Good catch.\"\n\nMy lip twisted further. \"Thanks.\"\n\n\"Do you see any other evidence to support that?\"\n\nOnly my partner would turn a gruesome murder scene into a pop quiz.\n\n\"The lack of blood and dark edging around the entry wound suggests cauterization.\" I managed a swallow, though it was more of a gulp to keep from gagging. \"And what blood is there is blackened.\" I gulped again, any attempt at cool and casual be damned. \"Like it was heated.\"\n\nIan nodded approvingly. \"Nice.\"\n\nNone of this was nice . . . not sight, nor smell, nor oily feel on my skin from the brimstone and burned flesh.\n\nIt'd take me a while before I'd be able to eat barbeque again. And for a Southern girl, that was a crime in itself.\n\nThe NYPD knew Sar Gedeon as a human drug lord. If they'd come in here now, they would have found him dead, sporting Spock ears, a cauterized hole in his torso, no heart, and a hoofprint branded into his chest. I'd like to be a fly on the wall for that investigation.\n\n\"So what would your precinct buddies have to say about this one?\" I asked, putting a couple steps distance between me and the elf brisket.\n\n\"From a human viewpoint, we've got cosplay with the ears, possible devil worship with the brand, and apparent human sacrifice. This case would drive them crazy, but they'd love the challenge. I never thought I'd say anything like this, but knowing elves and demons are real can certainly simplify an investigation.\" One side of his mouth quirked in a quick grin. \"Makes me damned glad I came over to the dark side.\"\n\nI nodded. \"And we have cookies.\"\n\n\"The locked door and no sign of entrance or exit would have thrown them for a loop. We know that brimstone could very well be from the leftovers of a gate. Demons aren't exactly known for walking in through the front door. With a gate, they're in, rip out a heart, they're out. Nice and neat.\"\n\nI wasn't seeing anything nice or neat.\n\n\"Why would a class-five demon kill a drug lord?\" I asked. \"Would one of his business rivals hire demons for a professional hit?\"\n\n\"Never heard of demons hiring out their services.\" Ian paused. \"Unless the guy doing the hiring was interested in offering his soul for the low, low price of one murder.\"\n\nI raised one brow.\n\n\"Demons don't accept cash,\" Ian explained.\n\n\"Not even credit cards? With the interest rates some of those things have, I wouldn't be surprised to find Satan himself in the big office.\" Then I remembered about the heat the other tenants had complained about. It felt fine in here to me. \"So was the heat coming from the body or the demon?\"\n\nIan shook his head. \"Neither. It would have been from the portal the demon used to get in.\"\n\n\"That makes sense. That Jesin Nadisu guy seems to be on the ball. I couldn't see him missing a pair of demons strolling through his lobby.\"\n\n\"The area near the wall around the corner felt warmer,\" Ian told me. \"Since there aren't any vents nearby, that'd be the most likely portal location.\"\n\nI went to take a look.\n\nUnless a portal was standing open it couldn't be seen. If Ian hadn't seen the portal, that meant it was closed. Closed equaled safe. A portal could only be used by the being that created it, or someone the creator had keyed to that specific portal. It was security at its finest.\n\nI stepped into a short hallway. . .\n\nAnd simply stared.\n\nThe wall was glowing. Orange. Not the entire wall, just a section, a seam running from the floor to a few feet from the ceiling. The seam was closed, but that didn't keep the glow from spilling onto the hardwood floor at my feet.\n\nThe light didn't come from the wall itself. It came from what lay beyond, and I didn't mean in the next room.\n\nIt was the portal, complete with sulfuric heat coming from it in waves.\n\nA shadow from the other side eclipsed the light.\n\nI took a step back, eyes locked on the opening.\n\nThere was something just on the other side.\n\nWatching me.\n\nIt knew I could see it and the portal.\n\nTerror put my gun in my hand, even though I knew that whatever was on the other side would laugh at my puny mortal weapon. I slowly backed away, my gun held low in a two-handed grip, trying to stop my hands from shaking.\n\nMy terror made it past my lips with one word.\n\n\"Ian.\" I could barely hear myself.\n\nNo response from the front room.\n\nI swallowed hard and tried again.\n\n\"Ian.\"\n\nAn instant later, Ian was beside me, gun drawn.\n\nThe shadow retreated.\n\nIan looked where I was looking, body tense and ready for anything.\n\nHe saw nothing.\n\n\"Mac, we're looking at a wall.\"\n\n\"And it's not all there.\"\n\nMy partner looked like he was thinking the same thing about me.\n\n\"There's a big glowing gash down the middle,\" I said.\n\n\"Describe it.\" His voice immediately went tight with apprehension.\n\nNow we were getting somewhere.\n\n\"It's a gash in the middle of the wall,\" I told him, trying to be the analytical professional I was supposed to be. \"It starts at the floor and goes up about six feet. The gash is closed, so it's more like a seam, and where it comes together is . . .\" I made a face. \"Squishy. Like glowing orange Jell-O.\"\n\n\"Orange?\"\n\n\"Jell-O.\"\n\n\"And you can see it.\"\n\n\"I could also see the shadow of a thing on the other side.\"\n\n\"The other side?\" Ian adjusted the hold on his gun.\n\nI suddenly needed a place to sit down, but I'd only be doing that after I ran all the way down to the lobby, probably to the accompaniment of my own screams.\n\n\"Uh-huh. But I can't see portals.\"\n\n\"That appears to no longer be the case.\"\n\nI took another step back. \"How?\"\n\n\"Don't know.\"\n\nWe both looked at the wall: me at the portal, Ian at where I'd told him the portal was.\n\n\"I take it the color means something?\" I asked.\n\n\"Oh, yes.\"\n\nIan had his phone out again, eyes still on the wall as if he expected something to jump out of it at any second. That made both of us.\n\nI waited for someone at headquarters to pick up. I had no doubt Ian was calling headquarters again, just as I had no doubt that orange wasn't a good color for a portal.\n\nSulfur stink plus hoofprint brand equaled a portal that in all likelihood went to a place I had no desire to go.\n\nAnd something in that undesirable place had seen me see it.\n\nOh crap.\n\n# 5\n\nSPI'S lab team arrived, and so far demons hadn't poured out of the wall.\n\nBoth were good things.\n\nThe seam had also stopped glowing and the wall appeared more solid.\n\nGood things number three and four. We were on a roll.\n\nIan had left a voicemail for Vivienne Sagadraco telling her about me and the portal.\n\nHe'd told me not to tell anyone what I'd seen until the boss gave the okay. I had absolutely no problem with that. I didn't want to think about it, let alone get chatty with anyone.\n\nJust to be on the safe side, Ian had requested backup of the demon-fighting variety, and until they arrived, we stayed.\n\nI was happy to say that our wait was blissfully uneventful.\n\nI didn't hold out the same hope for our investigation. When you were dealing with demons, you were guaranteed to get \"eventful\" by the bucket load.\n\nHowever, I knew that one thing would go right. When investigating a murder, SPI had one huge advantage over the NYPD.\n\nWe had a necromancer on staff.\n\nOnce we got Sar Gedeon back to headquarters, Bert could just ask the elf who killed him.\n\n* * *\n\nI didn't think Sar Gedeon's body could look worse.\n\nI was wrong.\n\nI was convinced that morgues had the same lighting as department store dressing rooms. One made you look dead; the other made you look so fat you wished you were dead. Neither even tried to be flattering.\n\nI looked at the elf. He didn't look like he'd gained any weight, just lost more blood, or maybe it'd just pooled in his back and butt like I'd seen on _CSI_. Now if we could solve a murder in an hour like they did.\n\nI wondered briefly about putting \"flattering morgue lighting\" on my end-of-life request list.\n\nI'd be gone and wouldn't care, but I'd rather no one see me on a stainless steel table looking anywhere near that bad. Though hopefully, some of me wouldn't have been partially cooked, and I wouldn't have had a sadistic killer rip his way into my chest and cut out my heart while his demon buddy held me down with his big ol' cow hoof. I didn't care who you were, no one looked good after that.\n\nWe were six stories below Manhattan's Washington Square Park in the lab of SPI's world headquarters complex. Nearly as big as the park itself, the complex was centered around what we called the bull pen, which was where most of the field agents had their offices. Above were five stories of steel catwalks connecting labs, more offices, and conference rooms.\n\nWe were in the morgue section of the lab. It was my first time here and I really wouldn't have minded it being my last.\n\nEverything was white tile and stainless steel, and totally pristine\u2014except for the burned brisket of a mutilated elf on the table. A table with troughs and drains.\n\nNormally when I felt this queasy, I went straight for the ginger ale and saltines.\n\nOur resident necromancer, Bertram Ferguson, looked like somebody's grandpa. That is, if their grandpa was Santa Claus.\n\nEven though it was only the first week of November, Bert knew better than to wear anything red. The belt loop on his jeans had long since turned over their challenging job to suspenders. Today's suspenders were navy, the plaid shirt dark green, making him look less like Santa Claus and more like an understated lumberjack. Bert was big, not in an excess of fat, but bigness of big.\n\nThe necromancer's strength and speed were equally notable. Bert attended a crime scene only if there was a dead body, but that didn't mean there couldn't still be living perpetrators lurking around. For all his size, the necromancer could outsprint most SPI field agents to reach the safety of his armored van, though it was more like a laboratory on wheels. Not being eaten by the monster du jour was a powerful motivator.\n\nOutside the morgue lab, Bert had met me with his usual bear hug. And as usual, I'd had to stand on tippy toes to even try to get my arms over his shoulders to hug his neck. I and everyone else at SPI loved Bert Ferguson.\n\nHe regarded me with his bright blue eyes. And yes, like Santa's, they did twinkle.\n\n\"I understand you had unexpected company at lunch,\" he said.\n\n\"Ian or the guy having the bad trip?\"\n\n\"Yes. Which one ruined your date more?\"\n\nI blinked. \"You knew about my date?\"\n\n\"Everyone knew. So which one was it?\"\n\n\"I'll have to think about that and get back to you.\"\n\nBert chuckled. \"Take it easy on him, he's\u2014\"\n\nI waved a hand. \"I know. He's just doing his job.\"\n\n\"That, too. You don't have any big brothers, do you?\"\n\n\"No brothers, period. No sisters, either.\"\n\nBert gently patted me on the shoulder with one big paw. \"You've got a brother now. And he's going to take care of you whether you like it or not.\"\n\n\"I'm getting that impression.\"\n\nThe morgue tech stuck her head out the door. \"Whenever you're ready, Bert.\"\n\n# 6\n\nDETECTIVE Fred Ash had asked to be present for the pre-autopsy questioning. Before starting work at SPI, those were two words I never thought I'd hear together.\n\nA few supernatural members of the NYPD enjoyed SPI headquarters privileges. Fred was one of them. When crimes involved supernatural perps or victims, shared information between SPI and select members of the NYPD had brought criminals to justice many times. It was a working relationship we all valued.\n\nIan and I were in the morgue because with me able to see the portal the killers had used to enter and leave the scene of the crime, this case was going to land on our desks with a resounding thud.\n\nI'd called my manager, Alain Moreau, about what I'd seen. He hadn't been in his office, so I'd left a voicemail. My being able to see portals was the earth-shattering equivalent of a documented visitation from the Almighty himself, so I expected my vampire manager to come crashing through the morgue door any moment now.\n\nThe tech left to give Bert more privacy to work, so it was just the four of us. Five, if you counted the corpse we were about to have a conversation with.\n\nBert sure wasn't creepy and neither was questioning the victim of a violent crime. But when said victim hadn't survived the perpetration of said violent crime . . . if that didn't say creepy loud and clear, I didn't know what did.\n\nOne heaping helping of nightmare fodder, coming up.\n\nI'd seen Bert communicate with the soul of a newly dead person once before\u2014one that hadn't been murdered. A silvery mist had risen from the body and stopped to hover directly above it. The form was vaguely the size and shape of the body it'd arisen from. There was no face, no features, and of course, nothing that could be used to speak. The investigating agent asked the questions; Bert spoke for the dead person.\n\nAnyone who didn't know Bert or was unfamiliar with how a necromancer worked would consider this an arrangement ripe for fraud. Though I'd like to hear their explanation for the body-sized and -shaped mist, and the details that only the victim would have known that came from Bert's mouth.\n\nNot only was Bert legit, he was considered one of the best at his craft, period.\n\nThe process itself was quite simple and relied solely on the power of Bert's necromantic magic, which was considerable.\n\n\"Sar Gedeon.\"\n\nThe boom of Bert's deep and resonate voice filled the morgue's tiled walls and then some. I jumped in spite of myself. His voice and the power behind it didn't ask the dead elf to come and talk to us, it commanded him.\n\nNothing happened.\n\nAt least that was the way it looked. How it felt was like I'd been turned into one of those long, skinny balloons that clowns used at kids' birthday parties, and Bert's magic was hell-bent on twisting me into a poodle. Ian and Fred looked equally uncomfortable.\n\nI swear I heard my joints pop. I sure as hell felt it. You didn't have to be a sensitive to feel magic on the level of Bert's.\n\nI sucked in as much air as my lungs could pull in. I knew I was going to need it. Not that I minded not breathing in a room dedicated to cutting open and examining dead people who, like Sar Gedeon here, had met their ends in less than peaceful circumstances. However, there was only so long living people could stay that way without air. Mouth breathing was preferred, but morgue air had a taste, too.\n\nOr maybe it was just me.\n\nIn the interest of being able to eat at some point today\u2014and keep it down\u2014I kept my breathing shallow.\n\nTiny drops of sweat beaded on Bert's forehead and upper lip. He could chat with the dead in his sleep\u2014and he had. It wasn't hot in here. SPI's medical team kept it cold for obvious reasons.\n\nThe necromancer was having a problem.\n\nI hoped it was the necromagic equivalent of technical difficulties and not a certain elf corpse fighting back.\n\nBut when Bert's brow creased in a scowl, I knew it wasn't heat or overactive sweat glands.\n\nNeither Ian nor I said a word or even moved. Heck, I already was barely breathing.\n\n\"Damn,\" he said simply.\n\nThe pressure in the room, and on my body, immediately vanished.\n\n\"No luck?\" I asked. Way to go, Captain Obvious.\n\nBert shook his head. \"No soul.\"\n\nIan made his own four-letter contribution. \"We waited too long.\"\n\n\"The soul didn't leave,\" Bert told us.\n\n\"But you said\u2014\"\n\n\"It was torn out.\"\n\nOuch.\n\n\"A drug lord with no heart or soul,\" Fred drawled. \"Anyone else love the irony?\"\n\nStereotypically speaking, I knew that demons had a thing for souls, but I thought they tasted sweeter or something when the owner signed it over voluntarily. Delayed gratification and all that.\n\n\"How do you even _do_ that?\" I asked.\n\nBert drew a breath. \"Well, first the\u2014\"\n\nI waved my hands. \"No, no, that was rhetorical. I'm sure that's one of those things I'm better off not hearing about.\"\n\nI'd discovered there were a lot of those in our line of work. Too often I'd been told details that'd ended up with a supporting role in my nightmares. In our profession, I had plenty of those, too.\n\nIan looked likewise reluctant to receive enlightenment. Considering what my partner's past careers were, that said a lot.\n\n\"I can explain without offending,\" Bert assured us.\n\n\"Will it help us find the demons that did this?\" Fred asked.\n\n\"No, but\u2014\"\n\nThe detective held up a hand. \"Then I'm ignorant, too, and happy about it.\"\n\n\"Expanding one's knowledge is good.\"\n\n\"So is me being able to eat lunch,\" I said. \"Fill us with knowledge when we're not standing over the visual aid.\"\n\n\"Where's your curiosity?\"\n\n\"Hiding behind the remains of my appetite.\"\n\n\"Very well.\" Bert stepped forward so that his ample belly was right against the side of the stainless steel table. \"If you don't want to hear how Mr. Gedeon's soul was removed, you certainly will not like remaining in the room for what I'll need to do now, since there's no soul to communicate with.\"\n\nMy stomach dropped. Being a science type, Bert had never been one for exaggeration.\n\n\"If you say we're not gonna like this, I'm thinking I should leave right now. We're here because we need to hear his testimony. We have tape recorders for that kind of thing, right?\"\n\nBert nodded. \"There is a video and audio record of every interaction.\"\n\nFred snorted. \"Meaning if you barf on the body, kid, it'll be playing on the break room TV within the hour.\"\n\n\"Bert wouldn't do that.\"\n\nFred grinned evilly. \"No, but I would.\"\n\nIan gave us both a look that said he was the adult and we were twelve. \"You were saying, Bert?\"\n\n\"A warning, though,\" the necromancer said. \"Depending on the level of residual energy remaining, the body could . . .\" He hesitated. \"Let's see how to put this delicately.\"\n\nFred shifted uneasily. \"Just say it, Doc.\"\n\n\"Move.\"\n\n\"What?\"\n\n\"Move. The corpse could move.\"\n\nI stood utterly still. \"Could you be more specific?\"\n\n\"Jerk, spasm, flail. I've even had one punch me.\" Bert grinned. \"Packed quite a wallop, too. Impressive for a deceased.\"\n\nIf a corpse sat up and took a swipe at me, I'd be using a lot of words, but \"impressive\" wouldn't be one of them.\n\nThen without any warning, Bert placed his bare-naked hands right on the corpse's face.\n\nIck didn't even begin to cover it.\n\nThis wasn't a doctor examining a corpse; this was a necromancer about to do some seriously spooky shit.\n\nMaybe this was his way of getting back at us for not letting him expand our horizons with a treatise on soul ripping.\n\nBert lifted the fingers of his left hand slightly and repositioned them. None of us could miss the dimples left behind by the pressure of his fingers.\n\nJust like Play-Doh.\n\nPlease don't move your hand again, I said silently.\n\nThankfully, he didn't. But that image had been branded into my brain, like that cloven hoofprint on the dead elf's chest, and it wasn't going anywhere anytime soon.\n\n\"I'll be able to see what his eyes saw in his final seconds of life\u2014hopefully including his killers. It works like an imprinting. If any images remain, they would be in the eyes.\"\n\nBert settled his fingers around the orbital bones surrounding the corpse's eyes.\n\nThen he did what you couldn't pay me any amount of money to do . . . Okay, I'd probably do it for _some_ amount of money, but it'd have to be absurdly huge.\n\nBert leaned over the table, putting his face close enough to kiss the corpse, his eyes less than two inches from Sar Gedeon's.\n\nThere was nothing Bert Ferguson wouldn't do in the name of science.\n\nUnlike before, there was no joint popping, no chest constricting, and I could breathe in all the air I wanted to, though considering that it still smelled like roasted elf, I only took in what I needed to keep from passing out.\n\nNo one moved, including the dead elf. I was sure I wasn't the only one grateful for that.\n\nBert was breathing in and out, the breaths growing loud and labored, the speed increasing until they were short gasps. His hands and face were whiter than the tile behind him.\n\nI didn't know what to expect; but to me, it looked like Bert was in trouble.\n\nI shot a sharp glance at Ian. His face bore signs of worry bordering on alarm.\n\nFred spat a silent curse.\n\nWe all knew the cardinal rule\u2014do _not_ disturb a practitioner in the middle of a magical link or incantation. I didn't know the reason behind it, but every ounce of common sense told me that snapping a link of any kind between a living person and the spirit, soul, energy, whatever of a violently murdered person couldn't be anything but bad.\n\nBut for Bert's sake, not breaking that link would be worse.\n\nIf we did nothing, I had a feeling there'd soon be two corpses in SPI's morgue and no one left to talk to either one of them.\n\nIan got his arm between Bert and the corpse, wrapping his big hand around the necromancer's shoulder, and leveraged his weight against Bert's chest to pull him off the corpse. Fred did the same from the other side.\n\nBert didn't\u2014or couldn't\u2014budge.\n\nA keening cry came from Sar Gedeon's now open mouth.\n\nNormally spirits spoke through Bert. I had no idea who or what this was.\n\nFred blanched and swore. Bracing his feet on the floor, the elf detective twisted his body and pulled harder.\n\nNothing.\n\nIt was as if Bert was fused to the body.\n\nThe keening grew louder and more frantic.\n\nVeins were bulging on the sides of Bert's neck.\n\nDammit, he was going to have a heart attack.\n\nBert's eyes were locked on the open and lifeless ones of Sar Gedeon. His hands might as well have been superglued to the dead elf's face.\n\nBrute force wasn't working.\n\nThere was just enough room between Bert and where his face was almost touching the corpse.\n\nHuman contact. Calm, warm human contact. I wouldn't be touching the corpse, I'd be touching Bert.\n\nI quickly moved to the head of the table and slipped my hands over Bert's eyes, breaking the visual contact between him and the dead elf.\n\n\"Bert, it's Mac.\" I tried to keep my voice calm. \"Come back to us.\" Moments passed. \"Bert, can you hear me? You can do this. Whatever it is, you're stronger. Fight it, Bert. Kick its ass.\"\n\nBert drew a breath that I swear must have shuddered clear down to his toes.\n\nGood thing Ian's and Fred's arms were supporting him, or Bert would have collapsed on the corpse.\n\nThey eased him back onto the floor. There was another steel table next to the one the elf's body was on, but thankfully the guys chose the floor. Bert was a necromancer and was comfortable around dead people, but waking up on a morgue slab would scare the crap out of anyone. I knew what my reaction would have been, and nobody's ears could've withstood that much screaming.\n\nFred ran out into the hall to get help for Bert.\n\nBert's breathing was still shallow, but it wasn't as labored. I didn't know if he was unconscious, but his eyes remained closed. I didn't blame him one bit. If I'd damned near gotten sucked into the great beyond through a corpse's eyes\u2014or whatever had happened to him\u2014I'd have kept my eyes closed, too.\n\nBert might need to hear that, or at least some reassurance.\n\n\"Bert,\" I said quietly, taking one of his big hands in both of mine. It was way too cold. \"It's over. You're safe now. You're safe.\"\n\nIan relaxed his grip so that it qualified more as a hug than a wrestling hold. I'd been on the receiving end of an Ian hug more than once. It'd sure made me feel better.\n\nA medical team arrived and Ian and I relinquished our holds on Bert.\n\nI looked up on the table at Sar Gedeon. Whatever had reanimated\u2014or possessed\u2014his body was gone now, but it'd left a calling card.\n\nSar Gedeon's dead lips were curled in a smile.\n\n* * *\n\nAfter our medical folks had taken charge of Bert, Ian and I were alone in the hall outside of the morgue.\n\nSar Gedeon's body was back in its refrigerated steel drawer where it couldn't channel demons at anyone else, securely under lock and key. The smile was gone. I was the only one who'd seen it. The tech explained it as a postmortem spasm.\n\nRight.\n\nIan slipped an arm around my shoulders, and I wearily leaned into it.\n\n\"It was smiling,\" I said.\n\n\"I believe you.\" Ian gave my shoulders a squeeze. \"Good work in there.\"\n\nI knew he wasn't talking about seeing a corpse grin.\n\n\"I just did what I'd want if I'd gotten myself locked in a stare-off with a corpse\u2014someone to hold my hand and tell me it was going to be okay.\" I felt myself start to tear up. What the hell?\n\nIan gave me another squeeze.\n\nI smiled a little and sniffed twice. Yep, Ian's hugs always did the trick.\n\n\"You did the right thing.\" He went quiet for a moment. \"Need something to eat?\"\n\nI would've thought that with all I'd seen and smelled, food would be the last thing I'd want to be in the same room with, let alone actually eat it. Surprisingly, I was starving.\n\n\"Come on, let's get you fed.\"\n\n# 7\n\nFOR SPI agents on duty\u2014or who wanted to be nearby when a coworker regained consciousness after being psychically attacked by a demon-possessed corpse\u2014our new onsite cafeteria was the place to get a quick bite. Though calling it a cafeteria didn't come close to describing the gastronomic delights available to hungry and stressed agents.\n\nIt's said that you can accomplish pretty much anything if you throw enough money at it. And our agency founder and director, Vivienne Sagadraco, certainly had enough wealth to throw around to ensure that her agents were well fed and happy around the clock. There were plenty of hotshot supernaturals and clued-in human chefs available in a city known for its world-class restaurants. The boss simply waved some more money in front of them, got them to sign one hell of a non-disclosure agreement, and we had a kitchen staff that rivaled anything New York City had to offer. Our head of HSR (Human and Supernatural Resources) was a voodoo high priestess. SPI's non-disclosure agreements for new employees were signed in her office and in their blood. It didn't matter who or what you did or didn't worship, nobody messed with voodoo. No one had ever even thought about blabbing about the agency to the press or anyone else. Once signed, our secret was safe.\n\nAs to food in our cafeteria, you could get anything you wanted at any time. Human, goblin, elf, troll, gnome, vampire, werewolf, were-anything\u2014if you had a craving, the boys and girls in the kitchens would whip it up\u2014or procure it\u2014for you. It was nothing short of culinary heaven.\n\nBest of all, they kept me in iced tea sweet enough to stand a spoon in. Ask any Southerner; you couldn't get decent sweet tea above the Mason-Dixon Line. That is, if you could even find sweet tea at all. Thanks to the generosity of Vivienne Sagadraco, there was no beverage homesickness for me. I'd even managed to score numerous converts.\n\nIn case Bert came around quickly, I just went with a turkey and provolone sandwich. It sounded simple, but all bread was made on-site. I'd had enough contact with red meat for one day. On second thought, make that for the next week.\n\nI could tell Ian wanted to ask me something, but he kept it to himself until I'd finished eating. He was having an open-faced roast beef, piled high with meat and drowning in gravy. I tried not to look at it. It didn't matter what my partner had just seen, smelled, or even touched, he could eat anything, anywhere, anytime. Even though his and Kylie's lunch reservation at Caf\u00e9 Mina had been half an hour before mine and Rake's, and he'd had time to eat, he was hungry again. Ian was about six two and solid. It took a lot of fuel to run that.\n\n\"It's not Caf\u00e9 Mina,\" Ian noted, when I polished off the last bite of my sandwich.\n\nI sat back with a contented sigh. \"You can read minds now?\"\n\n\"Nope. It was obvious that you were hungry.\"\n\nI nodded toward his empty plate. Even the gravy had been mopped up. \"Likewise.\"\n\nIan shrugged. \"Mina's was good, but it's kind of . . .\"\n\n\"Froufrou?\"\n\n\"Yeah.\"\n\n\"I know. You're a bar, beer, and burger kind of guy. Does Kylie know that?\"\n\nIan smiled slightly. \"She does. For our first time out, she said she wanted to take me somewhere nice.\"\n\n\"Aww. Sorry, couldn't help myself. That's just so sweet. Does she know you want actual food, not decorative squiggles on a plate?\"\n\nHe nodded. \"She does. Next time, I pick the place.\"\n\n\"And?\"\n\n\"I was thinking about Franco's.\"\n\nItalian. Low light. Romantic ambiance. Best of all, good food and lots of it. A carb-loading, meat-lover's paradise. \"Good choice.\"\n\n\"Really?\"\n\n\"You wouldn't know it to look at her, but that girl can put away some food. She can flat out load some carbs. I've had lunch with her enough to know. She'll love Franco's.\"\n\nIan took a breath and looked down at his plate. He could probably see his reflection in the thing. \"I'm sorry about what I said today at lunch about you and Rake Danescu.\"\n\nI smiled and gave him a little nudge under the table with the toe of my boot. \"No, you're not.\"\n\nHe glanced up, his lips twitching at the corners. \"You're right. I'm not. He asked you out again.\"\n\n\"Yep, lunch tomorrow. He . . . Wait, that wasn't a question. Unless you sprouted eyes in the back of your head, how did\u2014\"\n\n\"There was a framed print on the wall of the coffee shop behind Fred. I could see your reflection. Danescu must have been hungry, too. I thought he was going to eat your hand.\"\n\n\"Just because I haven't been out before with a goblin millionaire\u2014\"\n\n\"Billionaire.\"\n\nI raised an eyebrow. \"Well, dang.\" I shrugged. \"Okay, all that means is there's a couple of extra zeros in his bankbook. And yeah, he's hot, but money and looks don't impress me.\"\n\n\"Then why would\u2014\"\n\n\"He's interesting,\" I said simply. \"Intriguing, even. I want to know what makes him tick. That he's easy on the eyes while I'm trying to find that out is just a side benefit.\"\n\n\"The stereotypical mystery man.\"\n\n\"Hey, I'm not embarrassed to admit it.\"\n\n\"And if you actually find out what makes Rake Danescu tick?\"\n\n\" _That_ just might be the reason to keep seeing him\u2014or send me screaming in the other direction.\"\n\nIan's expression went grim. \"That's part of what worries me. He's a dark mage.\"\n\n\"I've known dark mages, back home and here. Heck, I'm even related to a few. My family's thick with seers, but that's not the only magical flavor in the family casserole. A lot of families have colorful relatives in their metaphorical attic. We Southerners take ours out and show 'em off. Dark doesn't mean evil. Now the big question would be why is he interested in me? I mean, I clean up good, but I'm no beauty.\"\n\nIan started to speak. I held up a hand. \"Thank you, but don't bother. I'm good with how I look, and I don't need any empty compliments to boost my self-esteem. It's quite healthy.\"\n\n\"Any compliment I pay you wouldn't be empty.\"\n\n\"Thank you again.\" I smiled slightly. \"The only reason I can come up with is that I'm probably the only woman who's ever told him no. And if it turns out that's his only reason, I'm not interested.\"\n\n\"There's your magic. He tried to hire you away from SPI your first night on the job.\"\n\n\"And he hasn't tried again since then.\"\n\n\"Goblins can be patient.\"\n\n\"Good, because I'm gonna be trying the heck out of his patience, regardless of his reasons for chasing after me. If you're worried about my safety, don't be. Ms. Sagadraco knows all about today's lunch.\"\n\n\"She does?\"\n\nI nodded. \"Since Rake's on the perpetual suspect list, I thought it might be prudent to check in with the boss first.\"\n\n\"And?\"\n\n\"She told me to go and have fun. If Ms. Sagadraco isn't worried, then you shouldn't be, either. Rake's not gonna do anything without my say so, and if he's serious about trying, he's gonna have to answer to me. He's well aware that he doesn't want to piss off the boss.\" I gave him a quick grin. \"Or my partner.\"\n\n\"Damn right, he doesn't.\"\n\n\"See? All settled.\"\n\n\"I wouldn't call it settled.\"\n\n\"Of course you wouldn't.\"\n\n\"But I feel better hearing your side of it.\"\n\n\"I can assure you, it takes a lot to turn my head\u2014and I have yet to lose it, over anyone.\"\n\n\"Just know that if he ever hurts you\u2014\"\n\n\"Honey, you're gonna have to get in line behind me. Though we both might have to get in line after Vivienne Sagadraco, and once she's through with him, there might not be enough left to bother with.\"\n\nIan's grin was ferocious. \"I'd gladly relinquish my place in line _and_ pay to see that. Speaking of our bosses, have you heard back from Alain Moreau?\"\n\n\"Not yet.\"\n\n\"Maybe you should go straight to the Dragon Lady. It's not every day one of her agents can see a portal.\"\n\n\"I just saw _one_. That doesn't mean I'll be able to see any more.\"\n\n\"But it makes it highly likely.\"\n\n\"You're squashing my hope here.\"\n\n\"Have you felt any different since Saturday night?\"\n\n\"I had a couple of dizzy spells, but I chalked that up to getting sucked inside Viktor Kain's head for a stroll down his World War II Memory Lane. Though the trip inside Kain's head felt more like going over Niagara Falls in a barrel.\"\n\n\"Can anyone in your family see portals?\"\n\n\"No. At least not that I'm aware of. I can see through wards and glamours on living things. To the best of my knowledge, portals aren't living things.\" I thought back to the pulsing wall\u2014and what had stood waiting on the other side. \"Or are they?\"\n\n\"No, they're not. And what keeps anyone\u2014except the person who created it\u2014from seeing a portal isn't a ward, it's the nature of portals. The magic used in their creation is specific to that person on a DNA level. Otherwise they couldn't pass through.\"\n\n\"And I can't create portals, so there's no good reason that I should be able to see one.\"\n\n\"Creating one takes a level of magical skill and training that you haven't had. That being said, there aren't many people who took a direct hit from a ley line convergence.\"\n\nI knew about ley lines. We had one running through the mountain near where I grew up.\n\nLey lines were narrow, intersecting energy streams that magnified magical and paranormal powers. There were a number of them near Manhattan. One ley line ran north and south roughly along the East River. Another ran more east to west. The east\/west ley line ran directly beneath the SPI complex. It was one of the reasons why Vivienne Sagadraco chose this location for SPI's world headquarters.\n\nThose possessing earth magic could tap a microscopic amount of power from ley lines, but they would be unable to use the lines to magnify and spread their magic. Diamonds, like ley lines, are of and from the earth. Rare diamonds\u2014like the Dragon Eggs from our most recent big case last Saturday night\u2014that are imbued with power can tap directly into ley lines to carry and spread the power they contain like an underground river.\n\nThe results of that connection had nearly been catastrophic.\n\nI'd been woozy, dizzy, and faintly nauseated after experiencing just a fraction of that power, though I'd chalked it up to an involuntary psychic link to a psychotic Russian dragon\/crime lord.\n\nMaybe my dizziness then had more to do with coming so close to a convergence of major ley lines that'd been kicked awake by the power of the activated Dragon Eggs.\n\nNo one else had picked up any additional mojo.\n\nOr had they?\n\nCaera Filarion didn't have any magical talent to speak of. Was that still the case?\n\nAnd Ben Sadler probably wouldn't know if he had picked up any extra power. He was still getting used to his gem mage powers waking up.\n\nCrap.\n\nWhat about Rake Danescu?\n\nHe wouldn't tell us how he was involved in what we were standing knee deep in. Why would I think he'd tell us he'd picked up an extra magical talent or two that night?\n\n\"Ian?\"\n\n\"Yes?\"\n\n\"If my being able to see portals is somehow connected to what happened on North Brother Island . . . I wasn't even touching the Dragon Eggs, and now I can see portals. What about Ben and Caera?\" I paused. \"And what about Rake?\"\n\nIan ran his hand over his face.\n\n\"Yeah,\" I agreed. \"And fat chance of his telling us if he did pick up a 'little something extra.' Ben and possibly even Caera could have gotten a boost. Though if Rake did get one of his talents supersized, at least he'd know how to control and use it. That could be good, or it could be cause for a whole mess of concern.\"\n\n\"You got it.\"\n\nThe cafeteria doors opened, and there stood Vivienne Sagadraco and Alain Moreau.\n\nOur quiet meal was about to turn into a serious meeting.\n\n# 8\n\nWHEN SPI's top necromancer tried to link with a murder victim and got zapped with a demonic booby trap, you knew there was gonna be a meeting. If it was strong enough to put Bert in a psychic headlock, it was serious enough to earn a visit from the occupants of the fifth floor\u2014SPI's executive suite.\n\nAnd when one of the agents who witnessed said zapping had also developed an inexplicable talent for seeing portals, the bigwigs would quickly bring that meeting to you.\n\nEntirely too many of my cases ended up with me explaining myself to Vivienne Sagadraco. Only once had I been in real trouble, but that time hadn't been my fault. A doppelganger had been impersonating me to plant grendel eggs in headquarters with the intent of slaughtering\u2014and eating\u2014as many of our agents as they could. When I'd seen me on that surveillance camera, for a minute there, I'd almost believed I was guilty, too.\n\nIan and cookies had saved me from a fate worse than firing. My doppelganger had been dressed exactly like me. My distinguishing characteristic that day had been powdered sugar sprinkled down the front of my sweater. I'd been eating cookies that a coworker had brought in and left in the break room.\n\nMy doppelganger had not. No cookies consumed. No powdered sugar to show for it.\n\nSaved by my sweet tooth.\n\nMy sweet tooth wasn't going to help any of us today.\n\nVivienne Sagadraco stood five foot and some change. Back when she was born\u2014actually hatched\u2014that had probably been quite tall. That had been a little over two thousand years ago. The founder and CEO of SPI was a dragon\u2014a three-story-tall, iridescent blue and green dragon. In her human form, she reminded me of 007's M as played by Judi Dench.\n\nAlain Moreau was a tall, slender, and impeccably well-dressed vampire. I didn't know when he'd been turned, but company rumor had it that he was at least three hundred years old. He didn't look a day over thirty-five with the silver-fox-Anderson-Cooper look he had going on.\n\n\"Agents Byrne and Fraser,\" the boss said.\n\n\"Ma'am,\" we said in unison.\n\n\"Sir,\" I added with a nod to Alain Moreau.\n\n\"You weren't at your desk,\" my manager noted coolly. \"And neither of you are answering your phones.\"\n\nIan and I exchanged a baffled look and reached for our phones.\n\n\"Shit!\" I jerked my hand away. \"Excuse me, ma'am, but damn that thing's hot.\" I winced. \"Excuse me, again.\"\n\nIan managed to get his hand on his phone and tossed it on the table. I could swear I saw smoke coming from it. He flipped his phone over with the back of one finger and peered at the display. \"Fried.\"\n\nDeep fried. The Gorilla Glass was even broken.\n\nI wrapped my hand in a cloth napkin and extracted mine from its holster. Dead as a doornail. Even more baffling was that we hadn't felt the heat until we'd actually touched the phones.\n\n\"I called you when we got back from the Murwood,\" I told Moreau. \"It was working fine then.\"\n\n\"That was before both of us grabbed Bert,\" Ian reminded me.\n\nAnd after Bert had his brain grabbed by a demon-possessed corpse.\n\n\"Sir, may I borrow your phone?\" Ian asked Moreau. \"Fred Ash was with us.\"\n\n\"He won't be able to answer if his phone got zapped, too,\" I pointed out.\n\nMoreau handed Ian his phone, and my partner started entering Fred's number. \"Yes, but we'd get an 'out of service' message. That would clinch it.\" He waited as the phone tried to call Fred. After about thirty seconds he hung up and passed the phone back to Moreau. \"Thank you. Fred's number is disconnected or is no longer in service.\"\n\nLooked like touching a necromancer under attack by a possessed corpse was bad for phone health, too.\n\nVivienne Sagadraco settled herself into one of the cafeteria's chairs. \"Considering the number of encounters you've had today, I think you'd both better start at the beginning.\"\n\nIan and I took turns, starting with our interrupted lunch.\n\nAlain Moreau had a raised eyebrow at the identity of my lunch date, but Ms. Sagadraco didn't bat an eye. While Moreau was my manager, Vivienne Sagadraco was boss lady to both of us. She'd told me to go and have fun. Her blessing overruled one raised eyebrow. Besides, I wasn't the one dating the world's oldest gorgon, Helena Thanos. Though she _was_ the boss's BFF, and was _not_ on SPI's perpetual suspect list. Neither could be said of Rake.\n\nWe recounted what we'd found at the scene of Sar Gedeon's murder: the unique and grisly cause of death, and most critical\u2014at least to me\u2014how the killers had gotten into and out of the apartment. In order to describe precisely what I had seen, I had to recall every detail, which I wasn't too keen to do, but if I wanted to find out why I could suddenly see portals, I had a sinking feeling I'd be telling it more than once. I'd better not only be good at it, but also get used to it.\n\n\"This is a new skill,\" Vivienne Sagadraco said when I'd finished. She didn't ask it as a question. She knew what I could do, and until today, what I could do didn't include seeing portals.\n\n\"Agent Byrne and I believe it may have something to do with the ley line convergence,\" I said. \"I _was_ right on top of it.\"\n\n\"So were Ben Sadler and Agent Filarion,\" Ms. Sagadraco said.\n\n\"And Rake Danescu,\" Ian added.\n\n\"Interesting,\" she murmured. \"Alain, would you check with Mr. Sadler and Agent Filarion to see if they are experiencing any unusual aftereffects due to contact with the Dragon Eggs?\"\n\nThe vampire nodded. \"As soon as we're finished here.\"\n\n\"Is it possible?\" I asked. \"I didn't actually touch any of the Dragon Eggs, but could exposure to a magnified ley line nexus do something like that?\"\n\n\"Prior to Viktor Kain collecting those seven diamonds, they had never been together, let alone activated by a gem mage of Mr. Sadler's skill. Add to that the fact that they were activated above the convergence of two major ley lines . . . I feel safe in saying that we are treading new ground.\"\n\nHoly crap. Vivienne Sagadraco was two millennia old. Alain Moreau was at least three centuries. They'd been around the block a couple thousand times. If they'd never heard of it happening, it'd never happened.\n\n\"I've never aspired to be a trailblazer, ma'am.\"\n\nShe almost smiled. \"Those who are, seldom do.\"\n\n\"Could there be another explanation?\"\n\n\"There is, but it is one that you would find distasteful.\"\n\n\"I've already got a bad taste in my mouth from all of this.\"\n\n\"You were briefly connected to the mind of Viktor Kain. That combined with your proximity to the nexus and activated Dragon Eggs may be what is responsible for your new talent.\"\n\n\"I'm trying real hard _not_ to think Viktor Kain might have something to do with this.\"\n\n\"Nevertheless, it must be considered as a possibility.\"\n\n\"I don't feel like I'm being influenced by evil forces.\"\n\n\"I wasn't implying that you were, merely that all possibilities must be considered. And as we recently experienced with Mr. Sadler, abilities previously dormant can emerge in startling ways.\"\n\n\"Seeing a demonic portal was startling all right.\"\n\n\"No doubt.\"\n\n\"Ma'am, do you think it'll be possible to find out what caused it?\"\n\n\"Rest assured, if the answer can be found, we will find it.\"\n\nI knew for a fact that Vivienne Sagadraco could read minds\u2014and emotions. She knew I was scared. She'd hired the best and brightest minds she could lure away from both the government and private sector. What she said was what she meant: if the reason could be found, SPI would find it. I couldn't ask for better odds than that.\n\n\"Thank you, ma'am.\"\n\n\"For now we'll assume that Mac seeing the portal at the murder scene wasn't an isolated incident,\" Alain Moreau said. \"Who knows about this?\" he asked me and Ian.\n\nI shifted uneasily. \"Aside from whatever was watching me on the other side of that portal, just the four of us.\"\n\n\"There's nothing we can do about the one; but on this side, it doesn't leave this table for now. We'll bring others in on a strictly need-to-know basis.\"\n\nWe all knew the reason why.\n\nAs the only seer in SPI's New York office, and one of only five worldwide, I made it more difficult for supernatural criminals to magically disappear into a crowd. My three predecessors at headquarters had met untimely\u2014and highly suspicious\u2014ends. One death could've been an accident; two would've been bad luck. But three? In a row? That was foul play of the premeditated kind. For whatever reason, someone out there didn't want SPI New York to have a seer.\n\nNow I could see portals.\n\nPortals weren't exactly common. It took specialized and expensive talent to create them. Well-connected criminals used portals as escape routes. Powerful and highly placed elves and goblins used them to travel between the dimensions\u2014most notably ours\u2014undetected. For someone in law enforcement to be able to see them? Well, that'd make me the most popular girl on any number of hit lists.\n\nMy mouth went dry at the thought, and I downed the last of my sweet tea. \"I already have a target on my back by being a seer; now I've got the magic equivalent of a red laser dot between my eyes.\"\n\nSilence.\n\n\"Isn't anyone going to tell me I'm wrong?\"\n\n\"I make it a point never to lie to my agents,\" Ms. Sagadraco said.\n\n\"Ma'am, I wouldn't mind the occasional happy, fluffy, white one.\"\n\nShe turned toward Moreau. \"I want to bring Martin DiMatteo in on this.\"\n\nOh boy.\n\nThat confirmed that demons were going to be a big part of my immediate future; though as long as I didn't end up like Sar Gedeon, I could deal with it.\n\nMartin DiMatteo was SPI's expert on all things demonic. We'd been introduced during my first week when, as a new employee, I felt like I'd been introduced to every person who worked at SPI and their intern. No one really expected newbies to remember all the names and faces thrown at them, but I'd had no trouble remembering Mr. DiMatteo. If SPI had business cards, Martin DiMatteo's would've said \"Director of Demonology.\" When we'd been introduced, he'd had pink scorch marks where his eyebrows should have been. That earned him a special place in my memory.\n\nThe eyebrows hadn't grown back.\n\nA couple of weeks later, what hair he had on his head had disappeared as well\u2014though I think the hair was a personal style choice rather than another work-related mishap.\n\nMartin DiMatteo was probably a nice enough guy once you got to know him, but let's just say I'd always hoped our caseloads would never intersect on the agency meeting calendar.\n\nSounded like my luck was about to run out; but like I said, if I didn't end up on a slab in the morgue, it was all good.\n\n\"When we leave here, I'll be going to see Bertram,\" Ms. Sagadraco said. \"I would like to be there when he regains consciousness. I don't want to tax his strength having him tell me what happened.\"\n\n\"Whereas we were there and _didn't_ get walloped by a demon,\" I said.\n\n\"Exactly.\"\n\nI let Ian do the honors. He'd had much more experience giving detailed reports.\n\n\"Detective Ash and I couldn't get Dr. Ferguson to let go of the corpse, though I think it was more like the corpse wouldn't let go of Dr. Ferguson. It was Agent Fraser who was able to help break whatever had hold of Bert's mind.\"\n\n\"May I ask how?\" Ms. Sagadraco asked.\n\n\"You can ask, ma'am,\" I said, \"but I honestly don't know. I just blocked Bert's visual contact with Sar Gedeon. I think Bert did all the work. I just let him know we were there and he wasn't alone.\"\n\n\"Sometimes the reassuring touch of another being is more effective than any magic.\"\n\n\"What attacked him?\" Moreau asked.\n\n\"That we won't know for sure until Bert wakes up and tells us,\" Ian said, \"but I think it was a trap, deliberately set for a necromancer attempting a postmortem contact. In this case, the soul had been taken and the trap left in its place.\"\n\nMoreau leaned forward. \"Taken?\"\n\n\"The heart had been removed in addition to the soul.\"\n\n\"I'm unfamiliar with any demonic significance of those acts,\" Moreau said. \"Madam?\"\n\n\"Likewise. Another reason why Martin's insights could prove invaluable.\"\n\n\"Fred Ash is one of the NYPD's investigators assigned to Brimstone,\" Ian said. \"We'll share information as needed. Even though we don't have a solid and proven connection between Sar Gedeon's killers and the drug, it's a coincidence we can't ignore. Fred said that as far as they know, Gedeon wasn't connected to Brimstone manufacturing and sales, but it's possible he could be a link in the chain.\"\n\n\"What effect is it supposed to have?\" Ms. Sagadraco asked.\n\n\"Unknown,\" Ian replied. \"Fred said they haven't been able to get a sample for analysis.\"\n\n\"Then that should be our first priority. If it is a drug that is not of this dimension, we are most qualified to locate a supply and track down its source. Our lab facilities and technicians are better qualified to analyze a drug of extra-dimensional origin, and determine what effects it has on mortal, immortal, and supernatural alike. That being said, our colleagues of the NYPD could ascertain the reason for its popularity as well as we could. I can't imagine anyone paying any amount\u2014exorbitant or not\u2014to be scared out of their wits.\"\n\n\"I don't know, ma'am,\" I said. \"We humans can be a pretty flaky lot.\"\n\nShe almost smiled. \"I have observed this on occasion. The same can also be said of immortals and supernaturals. Alain, have our agents with connections in the city's drug industry find out what they know about this Brimstone. Have any new underworld elements recently arrived here? And by underworld, I mean criminal or demonic\u2014or both. If this drug is of extra-dimensional origin, it is bothersome to me that mortal law enforcement discovered its existence before we did. In the light of a possible connection between this drug and today's murder, I would like to know why.\"\n\n\"I will take care of it, madam.\"\n\n\"Thank you, Alain.\"\n\nI gave a silent whistle. I was glad I wasn't on narcotics detail. For their sakes, I hoped they had a good reason why the NYPD had beat them on this one.\n\nSPI had detectives and investigators the same as any mortal police department, and those who had contacts in New York's drug industry would be set on Brimstone's trail.\n\nWe didn't have enough evidence to connect Brimstone with the murder of Sar Gedeon, but someone involved in that murder, whether or not it was the actual killer, had set a trap for any necromancer who tried to have a chat with their victim.\n\nBertram Ferguson was SPI New York's only necromancer.\n\nThe murder was committed in New York.\n\nTherefore, Bert had been targeted.\n\nAgain, we had no evidence to turn my hypothesis into a fact, but my friend nearly died\u2014or worse\u2014and I was fully prepared to take that personally. So I was going to investigate anything that might lead me to the asshole responsible.\n\nI had a source. And as long as Ian bought a lottery ticket later today, he would talk to me.\n\n# 9\n\nI never liked hospital rooms.\n\nThough I imagine not many people do. No one wants to sit in a tiny room watching someone you care about unconscious and with machines hooked up to them. Aside from the birth of a baby, there is no happy reason to be in a hospital.\n\nBert wanted to see me and Ian.\n\nNow.\n\nWhen we got to the infirmary, Bert was wide awake.\n\nFor a man who was zapped only an hour or so ago by a trap set in the mind of a dead body, Bert was looking pretty good. His color wasn't the best, but he was conscious and sitting up in bed. I was glad to see both.\n\nNot only was he awake, he looked pissed. Really pissed.\n\nIt appeared that Bert was taking the attack personally. Since he was the only necromancer in SPI's New York office, I couldn't imagine who else the killer thought would go poking around in Sar Gedeon's head.\n\n\"Looks like you went one round too many with one of the boys downstairs,\" Ian told him.\n\nMy partner wasn't talking about the guys in SPI's motor pool.\n\nBert just nodded. \"After what you two saw in that apartment, I was an idiot for getting in the ring.\" The big guy shrugged. \"But taking punches is part of my job.\"\n\nI nearly said, \"It shouldn't be,\" but he was right. We knew the risks of the work when we'd signed on. It was just that some of us risked more than others. I merely pointed out warded supernatural criminals. Bert talked to dead people, and most of those people had gotten themselves dead by violent means. To me, that was the psychic equivalent of going around and sticking your bare hand in a hole in the ground. You never knew what you were going to find.\n\nOr what was going to find you.\n\nI had a good idea of what had found Bert.\n\nThe same thing that'd seen me from the other side of that portal.\n\n\"I need to talk fast before Doc Stephens comes in here and tries to give me a sedative.\"\n\nI didn't miss Bert's emphasis on \"tries.\" I could see the necromancer being a bad patient.\n\n\"What did you see?\" Ian asked quietly.\n\n\"For starters, I can confirm that class-five demon.\"\n\n\"Too bad.\"\n\n\"Yeah.\"\n\nI concentrated on taking air in and blowing air out.\n\nToday was my first experience with demons. Like many Southerners from small towns, if someone asked you if you thought demons were real, you gave the Sunday school answer of \"yes.\" But they weren't something you thought about on a day-to-day basis. Even working at SPI, you knew certain things were real, but you never really put religion together with anything you might run into on the job. At least I hadn't.\n\nUntil now.\n\nBy helping Bert break a hold a demon had on him, I could've put myself in its crosshairs. And if that same demon was what I had seen on the other side of that portal, he'd now met me twice.\n\nCold sweat prickled across my skin at the thought.\n\nI knew without a doubt that I'd help Bert again in a heartbeat, but I really hoped I didn't have to.\n\nBert noticed.\n\n\"You look like you need this bed more than I do.\"\n\n\"It's just been a long day already.\" That wasn't a lie. I tried on a smile for size. \"Trust me, I'm gonna do my best not to end up in an infirmary bed.\"\n\nOr in a stainless steel drawer next to Sar Gedeon.\n\nBert grimaced as he pulled himself up further in the bed. Dr. Stephens might have the right idea of sedating Bert to make him get some rest.\n\n\"I saw them kill the elf,\" he told us. \"I saw it because they wanted me\u2014or whoever tried a PML\u2014to see them work.\"\n\n\"PML?\" I asked.\n\n\"Postmortem link.\"\n\nAll corporations had their acronyms, but SPI was a special snowflake.\n\n\"Your higher class demons are arrogant bastards,\" Ian said.\n\nBert snorted. \"Or drama queens. You two talked to Marty yet?\"\n\n\"Martin DiMatteo,\" Ian said in response to my confused expression. Then he grinned. \"You don't want to get Marty and Bert started at company parties. They try to outdo each other with work war stories.\"\n\n\"I'll try to avoid doing that.\" Some stories are better left untold, especially if they involved demons and dead people.\n\n\"No, we haven't seen Marty,\" Ian said, \"but the boss wants to bring him in on this one.\"\n\n\"A demon coming through a portal and ripping the insides out of an elf drug lord. Marty will love this one.\"\n\nNo, I definitely didn't want to be around when Bert and Marty started storytime.\n\n\"Our Class Five\u2014or his cohort\u2014left me that present on purpose,\" Bert told us. \"The elf never had a chance.\"\n\n\"You saw this from Gedeon's point of view?\" Ian asked.\n\nBert shook his head. \"I saw through the eyes of whoever was working with that demon. They were the one in charge.\"\n\n\"So one was a demon and the other was . . . ?\"\n\n\"Unknown.\"\n\n\"The accomplice planted the trap,\" I said.\n\nBert nodded. \"The demon had its hand wrapped around the elf's entire neck. It was a big Class Five, and it was wearing a classic form: red skin, horns, tail, hooves.\"\n\nI swallowed, or tried to.\n\n\"It picked Gedeon up off the floor and squeezed his neck until he stopped struggling but was still conscious. Then he tossed the elf on the floor and put one hoof on his chest to hold him still. Though I don't think it was necessary. The accomplice had already paralyzed the elf. There are spells or drugs that can do that but leave the victim fully aware.\"\n\nGod.\n\n\"So he felt it when they opened up his chest and cut out his heart.\" Now Bert did look sick. \"In his last moments of awareness, Sar Gedeon was forced to watch as the demon ate his heart.\"\n\n\"There weren't any screams?\" I asked quietly.\n\n\"His vocal cords were paralyzed, too.\"\n\nAnd now so were mine.\n\n\"What about the soul?\" Ian asked.\n\n\"Held immobile until the demon was ready for it.\"\n\n\"He ate it.\" Ian's voice was flat and without emotion, but I knew my partner. He was feeling plenty of emotion. He was just keeping himself from putting his fist through the nearest wall. Sar Gedeon may have been a merciless criminal who killed people and destroyed lives, but no one deserved to die like that.\n\n\"What kind of thing can do that?\" I asked Bert.\n\n\"Unfortunately more than a few. Equally unfortunate is that the perp foresaw Sar Gedeon's murder leading to a necromantic investigation. This thing took great pride and enjoyment in its work, and wanted it to be seen and appreciated.\" His blue eyes went hard. \"I hate to disappoint him, her, or it, but I didn't appreciate it one damned bit.\"\n\n\"Sar Gedeon was also a mage,\" Ian said. \"Mid-level power, but it was enough that no one wanted to cross him.\"\n\n\"He got crossed all right,\" I muttered.\n\n\"I'm sure the killer knew that, which would only have increased his enjoyment.\" Bert said. \"I sensed a sadistic satisfaction\u2014glee, even\u2014because of Gedeon's inability to defend himself. This thing likes it when his victims are helpless.\"\n\nBert's telling of Sar Gedeon's last minutes terrified me, but it also made me mad as hell.\n\n\"The thing fed Gedeon's heart and soul to his demon muscle while the elf had to watch and couldn't do anything about it. Is there a descriptor beyond sadistic? 'Cause this guy would qualify in spades.\"\n\nIan spoke one word. \"Monster.\"\n\n# 10\n\nWE left Bert in his infirmary bed, gearing up for the inevitable argument with Dr. Stephens about how long\u2014or not\u2014he was going to stay there.\n\nMy money was on Bert.\n\nIan and I had been issued new phones and were headed out to take the next step to finding out who murdered Sar Gedeon; and more importantly, who tried to fry the mind of our coworker and friend.\n\nPart of me didn't want to find the trail that would lead me to what I'd seen beyond that portal, but that part was soundly outvoted by the certainty that the only way to get rid of fear was to confront the thing that scared you.\n\nEven if that thing was a class-five demon that ate hearts and souls.\n\nAfter being reamed out for letting the NYPD get the jump on them in discovering a new supernatural drug on the streets, SPI's narcotics team was left scrambling to get information for our Dragon Lady and get themselves out of the doghouse.\n\nIan and I had sources of our own that were more well-rounded in their knowledge gathering. If they could make a living selling or trading information about one segment of New York's criminal society, they figured that they could bring in even more if they broadened their base. Snitches, like investments, were more profitable when they diversified.\n\nOrd Larcwyde had a financial goal and a life goal. Make enough money to retire well. Live long enough to enjoy both.\n\nHe was an entrepreneur and a veritable information clearinghouse.\n\nAnd out of all of SPI's agents, he would only talk to me.\n\nWhile I'd like to be able to say it was due to my street savvy, I knew it was my accent.\n\nTwenty years ago, Ord had transplanted from Atlanta. Business was too good in New York to ever consider going back home, but talking to me helped ease his homesickness for all things Southern.\n\nIan was a barbaric Yankee who he tolerated only on my account.\n\nI'd be lying if I said I didn't feel a wee smidgen of pride at being able to be one up on Ian, even if it was only with a single snitch. My partner was nearly legendary in the agency, so I took what I could get.\n\nOrd Larcwyde did business out of a small organic greengrocer one block south of the Meatpacking District on Horatio Street in the West Village. He had many information-related businesses, but only one interested me today. If you were on an extremely selective pre-approved list, you came in, bought a hundred dollar lottery ticket, and you got to talk to Ord. How long he'd chat with you depended on the questions you were asking\u2014and how much he enjoyed your company.\n\nProfits from sales of New York's lottery tickets went to the public schools.\n\nOrd Larcwyde was very civic-minded.\n\nPlus, you might actually win. So there was something for everyone.\n\nBecause I was Southern and he liked me, I got to talk to Ord for free. But today I laid a twenty on the counter to let Ord know I was thinking of the children, and that this wasn't just a social call. Plus, it was good manners.\n\nWhile the elven store owner was letting Ord know we were there, I scratched off the numbers.\n\nTo win back the money I'd just spent wasn't my objective, though it would've been nice. My reward would be information on Brimstone, the murder motive, the identities of the killers, or I'd love the jackpot of getting all three.\n\nDang. The tickets were losers.\n\nHopefully Ord would be the source of my payout.\n\n* * *\n\nOrd Larcwyde kind of reminded me of Colonel Sanders. That is if the Colonel was a three-foot-tall gnome who wore a blue velour tracksuit and gold chains instead of a white suit and black string tie.\n\nOrd stood, came around his desk, and treated my hand to a most-proficient kiss. Ian was on the receiving end of a terse nod.\n\n\"Makenna, you are a sight for sore eyes. Please, sit and make yourself comfortable.\"\n\nOrd had two chairs in his office: one for him and one for a guest.\n\nIan the Barbaric Yankee leaned against the open door.\n\nOrd would close it for truly private conversations, but he knew I didn't like being closed inside what was essentially a vault.\n\nThe back room was spacious as far as Manhattan grocery stores went, but Ord's reason for choosing this particular location for his office was an oversized fixture left behind by the previous grocer tenant.\n\nAn old walk-in freezer. It was the Fort Knox of offices.\n\nThe present store owner had a newer model that he used, and the old one was too big and expensive to move. Ord offered to make it worth his while to keep it. It was big for a freezer, but small for an office. Ord was a gnome; he didn't need space, just security. It didn't get more secure than what was essentially a big steel box. Ord got his office. The grocery store owner got rent to compensate for the storage space he lost by having the thing in his back room, as well as additional store and lottery customers from those, like us, who came in to meet with Ord.\n\nOnce again, everyone was a winner.\n\nOrd had the freezer part disconnected, and had a handle and lock installed on the inside as well. He'd also had an opening installed for air to get in, though he'd never told anyone where it was. Since Ord was small, I imagined the air opening was, too. If someone ever wanted to kill Ord for running his mouth, they'd have better luck trying to off him after office hours. Either that, or bring the world's biggest can opener.\n\nOrd had a step stool behind his desk that let him get in his office chair without any undignified hopping or climbing.\n\nThe gnome settled himself in the leather chair. \"I'd ask what do I owe this pleasure, but I've already heard. A human who can't hold his powder sets fire to a restaurant, and an elf who's responsible for the deaths of at least hundreds finally meets Death for himself. You've had a busy day.\"\n\nSounded like Jesin Nadisu, the building manager at the Murwood, wasn't as discreet as he'd claimed. Then again, goblins were known for having a different take on promises and agreements. He'd seemed like such a nice kid.\n\nIt must have shown on my face.\n\n\"Three representatives of Sarkowski Plumbing went in the back entrance of the Murwood, and they were seen wheeling out a black bag that could never be mistaken for a defective toilet.\"\n\nOf course. \"Your pixies,\" I said.\n\nThe gnome smiled. \"Their loyalty and work ethic are unquestioned.\"\n\nPixies were tiny, winged, and nosier than your worst neighbor. New York and Los Angeles were thick with the things. About the size and speed of hummingbirds, they were the eyes and ears of the city's supernatural paparazzi, and individuals like Ord who dealt in information. Like hummingbirds, pixies lived on a liquid diet. Pay them with Mountain Dew, Red Bull, or any other high-sugar, high-caffeine drink, and they were yours for life.\n\n\"My winged friends provide me with the information I need; I keep them well stocked with the beverages they want. It's a partnership made in heaven.\" Ord looked at Ian. \"While SPI really should change their disguises more often, in all fairness, there really is no disguising a dead body. You could hardly have folded up Sar and carried him out in a duffel bag. It would potentially destroy evidence on the body.\" He leaned back and put his hands behind his head. \"So who did the elf finally annoy badly enough to kill him?\"\n\n\"I was hoping you could tell us,\" I said.\n\nIan gave me a look. I'd just told a source that the all-knowing, all-powerful SPI wasn't all-knowing all the time, and Ian didn't like telling Ord that he didn't have all the answers. I liked the gnome, so I didn't have that problem.\n\n\"There is no shame in admitting ignorance, Agent Byrne,\" Ord told my partner. \"The only shame would lie in willfully remaining that way. Considering their success in keeping their operation 'under the radar,' as you humans say, my guess would make the new boys and girls in town either elf or goblin. Both races are ever so adept at keeping secrets. What I do know is that the hornets' nest has been soundly kicked.\"\n\n\"It couldn't be vampires?\" I asked.\n\n\"Unlikely. There are three families that are not associated with any of the vampire governing covens. Two of them\u2014the Frontino and B\u00e1thory families\u2014deal in drugs.\"\n\n\"B\u00e1thory? I asked. \"As in Hungarian countess Elizabeth B\u00e1thory? Bathing-in-the-blood-of-hundreds-of-virgins B\u00e1thory?\"\n\nOrd gave me a nod. \"That's her. The family's right proud of their ancestor.\"\n\nI made a face. \"Nice people.\"\n\n\"You don't know the half of it.\"\n\nI held up a hand. \"And I'm fine staying that way.\"\n\n\"And the Frontino family proudly traces their ancestry back to Cesare Borgia, who was also reputed to be a vampire.\"\n\nI nodded. \"A Machiavellian bloodsucker. I can see that.\"\n\n\"You got it. And . . .\"\n\n\"And as interesting as the history lesson is,\" Ian interrupted, \"could we stick to the present for now?\"\n\n\"You'll have to excuse him,\" I told Ord. \"He's no fun.\"\n\n\"I've gotten that impression.\" The gnome turned to Ian. \"Their descendants aren't heavily into drug dealing, but they don't believe in ignoring a potential revenue stream, even though it risks contaminating their food supply, namely humans. For that reason, the vampire covens had nothing to do with the drug trade. These two vampire families market the standard products.\"\n\n\"Not much into R&D?\" Ian asked.\n\n\"None. Which is probably why they're interested what the newcomers are selling. Supposedly Brimstone lets the user see through glamours and read minds.\"\n\nI snorted. \"So much for why the guy in the restaurant did a line or two before his meeting.\"\n\nIan nodded. \"Anything that would let you read the mind of a potential customer\u2014or existing competition\u2014would be worth its weight in gold in this city.\"\n\n\"Too bad he couldn't get past the change of scenery.\"\n\n\"Which is why the locals want a piece of the action,\" Ord said. \"The local businessmen asked, the newcomers refused. It seems they don't share well with others, though I can hardly blame them. Unfortunately some of our locals are slow learners. They asked again, and the newcomers began saying no in most impolite ways.\"\n\n\"Such as?\" I asked.\n\n\"Four days ago, the partially eviscerated body of a B\u00e1thory courier was hung on the front gate of the B\u00e1thory family compound on Long Island.\"\n\nIan stood straighter. \"Define partial.\"\n\nOrd looked from one of us to the other. \"The heart had been cut out.\"\n\n\"Anything else?\" I asked.\n\n\"A hoofprint had been branded into what was left of the chest. Some of their street dealers are missing.\"\n\n\"The B\u00e1thorys didn't report any of it.\" My partner didn't ask it as a question.\n\n\"To have a body deposited in such a manner . . .\" The gnome swallowed queasily. \"And in such a condition would raise questions about their business activities that the B\u00e1thory family would prefer not to answer. They cleared one racketeering charge last year by the skin of their pointy teeth.\"\n\n\"Was Sar Gedeon one of the locals who wanted a cut?\" Ian asked.\n\n\"Oh yes.\"\n\n\"And?\"\n\n\" _His_ chief courier was found three days ago in the driver's seat of one of Gedeon's prized vintage Porsches inside his locked\u2014and warded\u2014twelve-car garage.\"\n\nI exchanged a glance with Ian. Sounded like Gedeon didn't take no for an answer until it'd been his heart and soul being scooped out.\n\n\"And the B\u00e1thorys aren't the only ones missing some dealers,\" Ord continued. \"The Gedeon organization and Frontino family also have fewer employees than they had two weeks ago.\"\n\nA buzzer sounded in the front of the store. The owner came down the hallway to the back.\n\n\"Delivery, sorry for the interruption.\" He went to open the back door.\n\nOrd made an impatient sound. \"If this place wasn't so perfect for me, I'd have moved by now. The noise lately\u2014\"\n\nThe owner's body flew across the small storeroom and smashed into a wall lined with steel shelving. The shelving fell against a stack of boxes filled with garlic, which toppled onto the floor.\n\nNone of this affected the stance of the balaclava-wearing gunman who fired a spray of bullets through Ord's open door.\n\nOrd hit the floor behind his desk. I plastered myself against the wall next to the door. Ian made a flying dive out the door and onto the floor of the storeroom, his gun drawn. This unfortunately coincided with the boxes falling over.\n\nOn top of my partner, burying him in garlic.\n\nThe gunman turned and ran.\n\nOh, hell no.\n\n\"You okay?\" I yelled over the ringing in my ears from the gunfire.\n\nMuffled curses coming from under the boxes indicated the affirmative.\n\nThis wasn't a robbery. The shooter had been posing as a delivery guy.\n\nHe'd been aiming at Ord.\n\n\"In pursuit,\" I yelled over my shoulder.\n\nIan's muffled curses were less muffled. I chose to ignore the \"No!\" that I really couldn't be all that certain I'd heard.\n\nWhat wasn't muffled was Ord slamming and locking the door to his freezer\/office.\n\nApparently there were limits to his Southern hospitality.\n\nThe one thing I hadn't needed any training on when I'd started working at SPI was running. I'd mostly used it to run away from something trying to kill and eat me, though not necessarily in that order. Running was both offensive and defensive. Running _from_ took the same skill as running _after_. Either one could help keep you alive.\n\nToday I was running to apprehend an assassin who just tried to kill my best source, even though Ord had locked me and Ian out of his office. I tried not to think that said assassin had a gun and had just displayed a willingness to use it. I also had a gun, but was lacking in enthusiasm.\n\nIan would follow as soon as he could wrestle his way clear of those boxes. Yasha was circling the block waiting for our pick-up call.\n\nYasha Kazakov was our driver. Catching supernatural bad guys was easier than finding a parking place in New York. A driver who wasn't shy about throwing his weight around was a must. Yasha was also a nearly hundred-year-old werewolf, but he didn't look any older than Ian. With the Russian werewolf's preternatural hearing, I was sure he'd heard the shots.\n\n\"Yasha, pursuing suspect on foot,\" I said into my new phone's earpiece. I sucked in a double lungful of air. \"Approaching Greenwich Street.\"\n\nIf there was one thing that Yasha loved, it was running down bad guys of any shape or substance with the Suburban that he considered his partner. I'd never asked if he loved her more than me or Ian. I didn't think I wanted to know the answer.\n\n\"Am half block away,\" came the Russian werewolf's voice in my ear.\n\nI hoped Yasha wouldn't do a three-point turn or drive on the sidewalk to intercept the gunman, but I wouldn't put it past him. Heavy traffic or no traffic, if Yasha thought he could do it, he would. The Russian werewolf's mantra was, \"I saw it in a cartoon once and I think I can do it.\" There were two werewolf packs in New York: one in Manhattan and another in the outer boroughs. Yasha wasn't a member of either one. He considered SPI his pack.\n\nThere were plenty of disadvantages of working for a secret agency, but the biggest pain in the ass was not being able to yell \"NYPD! Freeze!\" At least not legally.\n\nWhen in pursuit of an armed suspect running down a busy sidewalk, the goal was to catch the suspect without anyone being shot. In theory, a suspect trying to get away didn't want to bring any more trouble down on their heads by opening fire on a crowded street. And the West Village was definitely crowded with traffic and people.\n\nEven with all the foot traffic, I had no trouble spotting him.\n\nHe'd taken off his balaclava to try to blend in, but all that did was give him a serious case of hat hair.\n\nHe definitely wasn't a goblin. He wasn't using a glamour of any sort, and silver skin would've been a standout. As far as I could tell, the ears had rounded tips, which eliminated him being an elf. Besides, he couldn't run nearly fast enough to be an elf.\n\nUntil I could get a closer look at him, I'd say he was human. About six foot. Dark blond hair standing straight up, presently weaving through the pedestrians near the end of the next block at Hudson Street.\n\nBingo.\n\nBeyond that was Seravalli Playground. If I had anything to say about it, he wasn't going to get that far. Even though he was running from me, I didn't delude myself into believing he was scared of me; he simply didn't want to get caught.\n\nOr he could be leading you into a soul-ripping, heart-staking ambush, my little voice said. Did you ever consider that?\n\nI hadn't, but I had an assassin on the run in broad daylight, and I would chase him until my lungs exploded if it meant bringing in a man who'd been told to permanently silence Ord Larcwyde\u2014and who had no qualms about me and Ian as collateral damage. That was someone worth interrogating.\n\nAll that being said, like a little terrier chasing a big truck, I hadn't given much thought to how I'd subdue him when\u2014or if\u2014I caught him. Though also like certain small terriers, come hell or high water, I wasn't giving up.\n\nI didn't slow down until I reached a clump of what had to be tourists, stopped in the middle of the sidewalk, looking at an honest-to-god paper map. Did they even make those things anymore?\n\n\"'Scuse me, pardon me, coming through,\" I said, weaving, dodging, and bumping my way through.\n\nThe gunman had vanished around the corner.\n\nDammit.\n\nI stopped at the corner long enough to peek around and make sure he wasn't waiting to blow my head off. I got a gratifying glimpse of him darting into a parking garage across the street.\n\n\"Yasha, parking garage on Hudson.\"\n\nI crossed the street and quickly darted inside to keep from being silhouetted against the sunlight from the entrance. I drew my gun and sprinted as quietly as I could down into the garage to the protection of the closest concrete column and stopped to let my eyes adjust to the shadows.\n\nThe garage was below street level. I'd been in these before. Going through the low entrance made you feel like you were driving into a cave. If you had to go to the bottom level to find a space, it felt like there was barely enough clearance to stand up in, and if you weren't a claustrophobic wreck before driving in, you were then.\n\nI didn't know how far down this one went, but I wasn't going any farther than this level.\n\nWhen a predator went to ground, you didn't jump in the hole after it\u2014and if you had to, you didn't go far.\n\nSince the garage was small by Manhattan standards and the gate was both an entrance and exit, this was probably the only way in or out. I could simply stay put and wait. The only way this guy was getting out was past me. Unless his car was here, then he'd be trying to go over me.\n\nI tried to turn down the volume on my breathing enough to hear the gunman moving or starting a car.\n\nI opened my mouth to get even more air in. What the hell? I could sprint farther than this without getting winded. Apparently running after\u2014or from\u2014a guy with a gun who'd just fired shots inches from where I'd been standing kicked my adrenaline into overdrive. More adrenaline flowing equaled more air needed.\n\nOnce I could hear over my own wheezing, there wasn't anything else to hear, other than passing traffic and dripping water from somewhere below.\n\nIt was too dark to tell how many columns were down here, but it was highly likely the gunman was behind one nearby waiting for me to make the first move.\n\nI had news, I wasn't going anywhere.\n\nOnly minutes away was a Russian werewolf in an armored Suburban that could block the gunman's exit or pin him against a back wall. Ian and I had been shot at, and Yasha was the protective sort. Since there'd never been a day when a shelf and a pile of boxes had stopped my partner, he'd follow, if for no other reason than to yell at me for running off without backup.\n\nNone of that would keep the gunman from taking an elevator up into the building above, but I had one elevator in my line of sight, and chances were in a garage this small, there was only one. It was the only decently lit thing down here. Only half of the lights in the rest of the garage actually worked, the rest were either burned out or flickering on and off, like they were powered by anemic fireflies instead of electricity.\n\nLooking out into the silent, too-poorly-lit-to-be-down-here-by-myself hole in the ground, I began to have second thoughts about my show of initiative, or as my Aunt Vicki, who was the police chief back home would have said, I'd \"run off half-cocked.\"\n\nAs my adrenaline rush faded, realization started to set in, and it wasn't pretty. A trained and experienced agent could do something like this. I was neither trained, nor experienced.\n\nI was a dumbass.\n\nIf I managed not to get myself killed, the next time I found myself in a similar situation, I'd think twice. I'd probably still do it, because the way I saw it I didn't have a choice, but at least I'd think about it more before it did it.\n\nThe garage was almost full, except for the far corner, which, considering the size of the garage wasn't all that far, was twenty spaces at the most.\n\nNo one had parked there.\n\nI couldn't really blame them. There was no light for five spaces in either direction. I wouldn't have parked there. The corner didn't even have shadows, just a big chunk of dark.\n\nI looked closer.\n\nA chunk that was less dark than it'd been a couple of blinks ago.\n\nThe source of light wasn't a bulb, it was the wall itself. A wall that should have been a solid slab of concrete.\n\nUnderground garages smelled like gas, oil, and the leftovers of whatever fast food someone had most recently tossed out of their car.\n\nEven then, chances were nil to none that those leftovers would smell like rotten eggs.\n\nA smell nearly identical to sulfur.\n\nI jumped as explosive pops and showers of sparks rained down from overhead as every light in the garage blew, leaving me in near total darkness.\n\nExcept for the far corner.\n\nA thin, glowing line appeared, spreading, disintegrating the dark as it went.\n\nAn orange glow.\n\nOh shit.\n\nThe gunman didn't have a getaway car down here; he had a getaway portal.\n\n# 11\n\nTIME for me to leave.\n\nI turned.\n\nLess than ten feet away\u2014standing between me and the only exit\u2014was the gunman.\n\nHis hands were loose at his sides, there was no gun in sight, and his jacket was unzipped all the way, exposing a bare and seriously pasty chest.\n\nHe was smiling.\n\nThis was wrong on so many levels, I didn't know where to start.\n\nHe kept smiling and shrugged out of his jacket.\n\nAdd another level to the wrongness.\n\nI raised my gun and took a step back.\n\n\"You need to stop.\" I backed up another step. \"There's an easy way to avoid this whole confrontation\u2014or whatever it is you have in mind. You step aside. I leave. Simple.\"\n\nHe stopped smiling. Not because he was any less happy, but because his mouth was changing, along with the rest of his body\u2014at least above the waist. If there was anything going on below the belt, he was still wearing his pants, so thankfully, I didn't have to see it.\n\nHis arms lengthened and became serpentine as if his bones had melted. Other appendages sprouted from his shoulders and sides.\n\nTentacles.\n\nThe bottom half of his face writhed and snake-like tentacles emerged like a fleshy beard.\n\nOh yeah, this was definitely wrong.\n\nAnd it sure as hell wasn't human.\n\nThe gunman was a shapeshifter.\n\nA type of shapeshifter I'd never seen, heard of, or had a nightmare about. Though I'd be rectifying that last one tonight, if I lived through this.\n\nThe squid guy had forced me away from the column. The opening portal was still the length of the garage behind me, but it wasn't nearly far enough away.\n\nI aimed for the spot right between his eyes. \"Stop or I'll shoot.\"\n\nHe didn't stop.\n\nI fired.\n\nThe bullet hit him right between the eyes\u2014and made a dimple. Then the flesh beneath rippled and popped the bullet right out. It landed with a metallic plink on the concrete.\n\nI didn't get a second shot.\n\nA tentacle shot out like a whip around my legs and swept me off my feet.\n\nI landed hard, hitting my head on the concrete. I saw stars and heard my gun clattering away from me.\n\nIan had been training me in hand-to-hand combat, not hand-to-tentacle combat. I couldn't win against two arms, let alone six tentacles. And if this guy got on top of me, I was toast.\n\nThe tentacle continued to constrict like a python around my legs until I couldn't feel them anymore.\n\nI rolled sharply. At least that's what I tried to do.\n\nMy gun was out of reach.\n\nI had a knife inside the top of my tentacle-wrapped right boot. My other knife was at the small of my back.\n\nI twisted, scrambling wildly to get at it. Another tentacle shot out and wrapped around my waist, as he started dragging me toward the portal.\n\nA portal that was now open to the width of a car.\n\nI couldn't see through to the other side, but I could make out restless shadows shifting and passing across the opening just over the threshold.\n\nOne shadow stood still on the edge of the portal where it met and melted the dark.\n\nI'd seen it before.\n\nIt was waiting.\n\nI didn't need three guesses to know for whom.\n\nI didn't believe in coincidences. I believed in traps. And I was being dragged toward one.\n\nThe squid had two tentacles wrapped around me and the other four were flailing around my squirming self, trying to get a hold. If he dragged me across that threshold, I was worse than dead and I knew it.\n\nI panicked.\n\nI got my knife in my hand, stabbing and sawing frantically at the tentacle wrapped around my knees, black blood soaking my hands. I gripped the knife harder. It was all I had and I couldn't lose it.\n\nMy whimpers turned to enraged screams. They could have been girly screams for all I knew. I didn't care. Screaming tapped my primal self, the terrified animal that kept me cutting and fighting with everything I had.\n\nI cut through the tentacle's tough core and through the rubbery flesh on the other side, freeing my legs. I drove the heel of my boot into my attacker's knee, simultaneously hooking the toe of the other boot behind his ankle. One sharp pull and he went down. I stabbed the knife's blade into the tentacle at my waist and started sawing. The thing's high-pitched keening echoed through the garage.\n\nIt went well with my screaming.\n\nIf something was trying to mug you, rape you, kill you, or drag you through a fiery portal to your eternal doom\u2014make noise. Help could be just one good shriek away.\n\nThe tentacle tightened around my waist, and I sawed faster. The squid thing was still keening. My screams had turned back to frantic whimpering.\n\nI severed the tentacle, slicing into my numbed waist before I could stop. Black blood pumped from the tentacle's severed stump, the end of it still wrapped around my waist and constricting as if unaware that it was no longer attached.\n\nWith a keening squeal, the squid dropped me, staggering toward the portal, its remaining four tentacles cradling the stumps of the other two. I desperately pushed against the blood-slicked concrete with the heels of my boots as my hands scrambled and clawed for a hold to pull myself away.\n\nAt least I tried.\n\nIn my mind, I was making all kinds of progress getting away from that portal. In reality, I couldn't move. Not one muscle.\n\nI didn't have to move to see the portal. The squid demon was gone, and the shadow standing silently beyond was still silent, but he had moved. The shadow had become a silhouette of a man. Tall and thin. Long fingers flared like a fan and my whimpers froze in my throat . . .\n\nAnd my blood froze in my veins. Not from the paralyzing effects of what must have been a spell launched from the other side of the portal, but from the knowledge of who had done the launching.\n\nSar Gedeon's murderer. The thing that had held the elf still while a class-five demon had cut out and eaten his heart\u2014then his soul.\n\nA horned figure suddenly loomed behind the mage.\n\nOh God . . .\n\nTires screeched behind me.\n\nIn that instant, it wasn't my life that flashed before my eyes. It was gratitude. I was grateful that I was about to become the city's newest speed bump rather than a demon meal.\n\nJust as the stink of burned rubber overrode my senses, the portal snapped shut, leaving no sign that it'd ever been there.\n\nMy body went limp in a fit of shaking.\n\nI could move again.\n\nDoors opened and arms were lifting me off the concrete. Ian's arms. Oh God, that hurt. The parts of me that weren't still numb had concrete burn.\n\nI couldn't make sense of Ian's words over the sound of my ragged breathing. Since the ones I did hear were creative variations on the four-letter variety, my partner appeared to be going for emotional expression over sentence structure.\n\n\"Sq . . . squid.\" Great, my teeth were chattering, too.\n\nI tried to point toward the portal.\n\nIt was gone. The Suburban's headlights lit the garage like high noon. The corner walls were just concrete. There was nothing left of the portal but the stink.\n\nAnd the black blood on the floor\u2014and on me.\n\nIan had one arm around me; the other hand held his gun. Yasha wasn't encumbered.\n\nWhen in human form, Yasha's favorite weapons were his Suburban and his Desert Eagle. The Eagle was the only handgun large enough for his hands. He had it in his hand now. The other held a flashlight that could fry your retinas.\n\nThe Russian swept the entire garage with its beam.\n\n\"Is gone.\"\n\n\"It was a shapeshifter,\" I told them \"I didn't do this . . . to myself.\"\n\nIan's expression was grim as his eyes scanned the cars. \"I know you didn't. Yasha, get a\u2014\"\n\n\"Sample for lab,\" the Russian finished for him.\n\n\"Thanks, buddy.\" He looked down at me with an expression that said, unlike Yasha, I wasn't his buddy right now, or at the very least he was pissed at my show of initiative.\n\nI pulled at my shirt. \"I've got lab samples, too. He bled all over me when I cut off his tentacles.\"\n\nIan's expression changed from definitely pissed to possibly impressed.\n\n\"Just the two,\" I clarified. \"He had six. It was kind of like cutting bait.\"\n\nReally big bait.\n\nFor now, I left out the panicking and whimpering part. I wanted to keep my badass illusion going for as long as possible. Impressed while looking at me was a new expression for Ian, and I was enjoying it. Besides, he didn't look like he wanted to yell at me\u2014at least not as much.\n\nI thought I had enough breath now for the really bad news. My partner was going to have a lot of questions, and I needed the wind to answer.\n\n\"Ian, there was a portal . . . and a mage.\"\n\n* * *\n\nWithin fifteen minutes, SPI had investigative and cleanup teams on site, complete with agency demonologist, Martin DiMatteo. The teams were disguised as elevator repairmen. Their job was to get in, get readings, get rid of the evidence, and get out. And they actually did do what the name on the van blocking the garage entrance indicated. They repaired the elevator\u2014which was needed after they disabled it to keep anyone from descending into the garage.\n\nBoth teams had plenty of practice in being thorough and fast. The NYPD could have closed the garage as a crime scene for hours. Since SPI didn't officially exist, we couldn't officially do anything, and didn't have time on our side. The disguise was to keep the curious from asking too many questions; the speed was to prevent anyone from seeing squid demon blood splattered on both concrete and cars. Fortunately for evidence eradication, squid demon blood dried to the consistency of blackberry jelly and was easily powerwashed down the garage's storm drains. Unfortunately for the cars, it ate through the paint.\n\nThat was why I was wearing Yasha's spare sweats that he kept in the Suburban. My clothes had quickly developed holes. To keep those holes from being eaten into me, I got into the back of the Suburban with its conveniently tinted windows and stripped down. Going werewolf quickly was hell on a wardrobe. The Russian was tall enough in his human form; going wolf added another eight inches in height, and let's just say an impressive amount everywhere else. If he didn't have time to get naked before going wolf, his clothes didn't stand a chance.\n\nRight now, I was glad he kept spares.\n\nYasha's sweatshirt hung nearly to my knees. If it hadn't been November, and cold, I'd have left it at that; but it was, so I couldn't. Keeping his sweatpants where they needed to be on me required sitting down and staying there. After running five blocks then wrestling for my life and soul with a determined squid demon to keep from being dragged through a portal to Hell, sitting down was exactly what I wanted to do. It ran a close second to drinking the massive hot chocolate Ian had gotten for me. I loved New York. There were coffee shops on every corner. My partner knew exactly what I needed. I was still shivering, and I didn't think it was from cold. At times like this, a girl needed chocolate\u2014or a stiff drink. Despite what we did for a living, drinking on the job was still frowned upon, so a hot chocolate it was.\n\nAt the moment, Ian was talking to our lead investigator, but he kept the Suburban in sight at all times. I smiled around my cup. Yasha wasn't the only protective one.\n\nI was sitting curled up in the Suburban's second row of seats in the exact middle. Just because the portal and Sar Gedeon's murderer were gone didn't mean I didn't want as many exit options open to me as possible\u2014or the protection of armored glass on every side. It probably wouldn't stand up to demons, but it was what was available, so I gladly took it.\n\nExcept for the partially open driver's side window. I'd rolled that down myself. Just because I'd had the hell scared out of me didn't lessen my curiosity. The lab folks were having a field day with this one. It wasn't often they got to play with squid demon blood, and I didn't want to miss a word of it.\n\nThe rear passenger-side door opened. I had a visitor, an expected one.\n\nSPI's director of demonology, Martin DiMatteo.\n\nI saluted him with my gargantuan paper cup. \"Hi, Marty.\"\n\nWe'd only met once before on my first day on the job, and he was many levels of agency bureaucracy above me, but after what'd just happened, I had no fracks left to give.\n\nNot that he was intimidating or anything. I think the term \"mild mannered\" was coined with this guy in mind. Average height, average build, average looks. The only thing that wasn't average was the complete lack of hair above the neck. Below the neck, he was covered by a navy blue suit with a non-descript tie. Even the tie's pattern was muted.\n\nMartin DiMatteo gave me a cool nod. \"Agent Fraser.\" He got in and closed the door.\n\nI took a big gulp of my hot chocolate. Interrogation, here we come.\n\n\"You can call me Mac, if you want to,\" I told him. \"Or . . . Agent Fraser if you don't.\"\n\n\"I understand you've had quite the eventful day, Agent Fraser.\"\n\nSo much for friendly small talk.\n\nThough one element of my eventful day wasn't going to be a topic of talk, small or otherwise. Ian had notified Vivienne Sagadraco about what had happened; and until after an official debriefing, she wanted us to keep the mage to ourselves. I had absolutely no problem with that. I didn't want to think about what'd nearly happened to me, let alone have a chat about it. As the director of demonology, Martin DiMatteo would probably be hearing about it soon enough, but I was fine with him being told by the boss and not me.\n\n\"I think we can safely call it the day from Hell,\" I said.\n\n\"Technically, no. A more accurate description would be a day from an anteroom of Hell, but then that doesn't have nearly the dramatic flair.\"\n\n\"I don't want drama in my life.\" I nearly added \"Marty,\" but decided against it. I could only claim shock-induced familiarity for so long. \"What's the difference between a portal to Hell and an anteroom?\"\n\n\"One's a direct flight; the other has a layover.\" He didn't crack a smile, or show any emotion whatsoever.\n\n\"So it's true what we say back home: to get to Heaven or Hell, you've gotta go through Atlanta.\"\n\nStill no smile. I don't think the guy understood humor.\n\n\"In a manner of speaking, yes.\"\n\n\"But if I'd been dragged through, I could have been taken to Hell from there.\"\n\n\"Yes.\"\n\nGulp.\n\n\"There is no way directly into our dimension from Hell,\" he continued.\n\n\"That's good.\"\n\n\"It's the only reason any of us are still here.\"\n\nGulp again.\n\n\"The vast majority of demons cannot cross over,\" he said.\n\n\"Let me guess, squid demons can.\"\n\n\"Actually, that's what makes this incident so interesting. They shouldn't be able to, and definitely not so far from open water. Then there was what happened this morning, both in our morgue and with that elf's murder. Both highly unusual demonic behavior.\"\n\nI'm glad only one of us considered all that merely interesting. I'd broken out in a cold sweat at the thought of what I was about to ask. \"So demons don't like to torture humans\u2014or elves\u2014and eat their hearts?\"\n\n\"They derive great enjoyment from that. However, they generally don't do it here. Contrary to what most major religions believe, demons really don't find us all that fascinating on an individual basis.\"\n\nI probably didn't want to know, but couldn't help asking. \"And as a group?\"\n\nDiMatteo shrugged. \"I've heard that we're tasty and addictive, rather like buffalo wings. It's our dimension that they covet. They consider our dimension\u2014or any dimension other than their own, for that matter\u2014to be much more hospitable than theirs.\"\n\n\"Must be the beaches,\" said an unexpected voice.\n\nI danged near choked on my chocolate. \"Bert?\"\n\n\"Mac.\" He nodded in greeting. \"I escaped.\"\n\n\"I see. Should you be here?\"\n\n\"You saw another portal\u2014and the bastard who attacked me standing on the other side. Where else should I be?\" He nodded to my visitor. \"Marty.\"\n\n\"Bert.\"\n\n\"Where you should be is still in bed with Dr. Stephens fussing over you.\" I spotted a flash of white on the back of his big hand. My mouth fell open. \"Is that tape from your IV needle?\"\n\nBert looked down, grunted in acknowledgment, and ripped it off.\n\n\"You really did escape.\"\n\n\"I didn't have a choice,\" Bert said. \"I'm fine, Stephens didn't believe me. That's his problem, not mine.\"\n\n\"I hate to disappoint you, but the portal's gone. You came here for nothing.\"\n\n\"There've been two portals today, and you've been there for both of them. You're batting a thousand, kid. If I stick with you, I'll be there for the next one.\"\n\nI couldn't believe my ears. \"Did you hit your head on the table in the morgue? It's more like I've got two strikes, and the next one means I'm out. I'd like nothing more than to take myself out of the game before that happens.\"\n\n\"What proof do you have that a class-five demon was with the elf's killer?\" DiMatteo asked Bert.\n\nThe necromancer gave the demonologist a flat look. \"Seven foot tall _without_ the horns. Tail as long as Mac here is tall. Turn-ons include chest branding, heart eating, and soul sucking. Yeah, it was a Class Five.\"\n\nDiMatteo either ignored the sarcasm or he didn't get that, either. \"Were there bony protrusions like a ridge down the length of its back?\"\n\nBert shook his head. \"Smooth back.\"\n\n\"Slender build or heavy?\"\n\nBert looked confused.\n\n\"Swimmer or linebacker?\" DiMatteo clarified.\n\n\"Somewhere in between, but more toward swimmer.\"\n\n\"The horns. Were they upward-, forward-, or backward-facing? Forward would be like a bull. Backward is like a goat. Upward is . . . up.\"\n\nThat question gave Bert pause. \"I'm not sure.\"\n\n\"Think.\"\n\n\"It's important?\"\n\n\"Critical.\"\n\n\"Upward, but curved and slightly tilted toward the back.\"\n\nMartin DiMatteo would have raised his eyebrows in surprise if he'd had any. Two little crinkles appeared where his eyebrows would have been.\n\n\"Are you certain? Not like a goat or bull?\"\n\nBert closed his eyes, mentally reviewing his \"game tape.\" He opened them. \"Upward. The base was about as thick as two of my fingers. They narrowed to a sharp point. They also had circular ridges like growth rings down the length.\"\n\nThe demonologist sat back on the seat next to me with a genuine smile. You'd have thought Bert had just handed him the best present he'd ever gotten. \"Then it wasn't a Class Five.\"\n\n\"Well then, what class was it?\"\n\n\"Demon lords are above the BCS.\"\n\n\"BCS?\" I asked.\n\n\"Brinkman Classification System.\"\n\n\"Someone got close enough to demons to classify them?\"\n\n\"Affirmative. But he's not around anymore.\"\n\nNo doubt.\n\nI swallowed. Hard. \"A demon lord sounds bad.\" My voice sounded tiny. I'd just had firsthand experience with seeing one, at least a silhouette, which was more than I ever wanted to see again.\n\n\"That would depend on your perspective, Agent Fraser. If what Bert says is accurate, and I don't have reason to doubt him, now that I've extracted more details, this is the opportunity of a lifetime.\"\n\n\"After Sar Gedeon got up close and personal with that thing, his lifetime was over. And if Gedeon's killer used a demon lord as their hired muscle, what does that say about what the killer is?\"\n\n\"Precisely.\" The demonologist added a delighted eye twinkle to go with his smile.\n\nHe was getting happier than a pig in mud.\n\nI was getting even more scared and creeped the hell out than I already was.\n\n\"Demon lords\u2014and ladies\u2014only leave Hell for special occasions,\" DiMatteo said. \"This particular lord must consider it to be very much worth his while. They are proud, arrogant, and utterly self-absorbed, and would only consider subjecting themselves to Hell's aristocracy.\"\n\nI felt the blood run out of my face. \"So the killer is a\u2014\"\n\n\"Not necessarily the aristocracy, but a being that the demon lord could tolerate partnering with until he gets what he is in this to obtain.\"\n\n\"What would that be?\"\n\n\"Unknown at this time. Whatever it is, 'catastrophic' would probably be the best description for how bad it would be if he got it.\"\n\n# 12\n\nDR. Stephens wasn't all that disappointed\u2014or surprised\u2014that Bert had flown the medical coop. It was obvious that the necromancer was a less than ideal patient.\n\nBesides, now he had me.\n\nThe squid demon bouncing my head off the garage's concrete floor earned me a CAT scan and a stay for overnight observation in SPI's infirmary.\n\nI was in the same bed\u2014with fresh sheets\u2014that Bert had occupied until he pulled a Houdini. Bert had said earlier that I had looked like I needed to be in that bed worse than he did. If I'd had a lick of sense, I'd have just crawled in then and saved myself the pain, possible concussion, and definite emotional trauma.\n\nAll of the tentacle constricting hadn't interrupted the blood flow to my legs long enough to do any permanent damage. My feet still felt a little tingly, which wasn't exactly conducive to standing, let alone running after or away from anything. The scrapes and cuts from being dragged across the concrete had been cleaned and spritzed with some kind of miracle spray that not only took the sting out, but dried to provide a bandage that wouldn't move or come off. It needed to be reapplied every twelve hours.\n\nBecause of all that, Alain Moreau\u2014and more importantly, Vivienne Sagadraco\u2014after they had come to talk to me about what had happened\u2014and what had almost happened\u2014had ordered mandatory bed rest and observation for at least the next twelve hours.\n\nThey'd both listened in grim silence as I'd recounted my experience. They'd asked few questions, all of them to clarify details, then the boss had told me to get some sleep, and they'd left.\n\nIf they knew who or what my attacker was\u2014and I strongly suspected they had an inkling\u2014they weren't telling me. Probably because I needed to sleep, and sleep would've been hard to come by if Dr. Stephens had to sedate me during the panic attack I would've had if they'd told me what they knew.\n\nIf ignorance was bliss, I was fine with being stupid and happy for as long as possible. I knew it wouldn't last, so I'd enjoy it while I had it.\n\nIt was nearly nine o'clock in the evening of one of the longest days I could ever remember; and to tell you the truth, I really didn't mind the thought of spending the night at headquarters. And since Dr. Stephens had come in and told me I didn't have a concussion, I wouldn't have either him or one of the nurses waking me up every hour, to make sure I _could_ wake up, and shining that little penlight in my eyes.\n\nMy eyelids were getting heavy, and I thought with a little silent cheer that I might actually get some much needed sleep.\n\nIt'd been only two days ago that we'd been racing against entirely too little time to protect the supernaturals of the tristate area from death by cursed diamonds. Those who depended on glamours and other magic to hide what they really were from humans would have had that protection stripped from them; they would have been the lucky ones.\n\nNeedless to say, during the forty-eight hours leading up to that, no one had gotten any sleep. Halloween had been on a Saturday. Sunday, I'd still been too keyed up to sleep. Today was Monday and I was running on fumes.\n\nYes, I was in a hospital room, but I could mostly feel my legs, and there wasn't any permanent damage to them or to the insides of my skull. Whether I merely felt safe being in a secure complex with one of our commando teams on duty and the other on call, or I was simply exhausted, or a combination of both, I slept like a baby.\n\nThe nurse on duty, God bless her, didn't wake me up during the night or even the next morning. I got to wake up on my own. Aside from a brief bout of heart palpitations from waking up in a strange bed, it was a night well spent.\n\nI really wanted to smell coffee, but instead, my nose twitched at the scent of flowers.\n\nThe nurse\u2014or someone\u2014had been in during the night and made a floral delivery. A cut crystal vase holding at least three dozen roses stood on the bedside table. Their petals went from a pink blush for the outer petals to a pale golden glow in the center. They looked like tiny sunrises. I made a soft sound. I loved roses, and these were the most beautiful I'd ever seen.\n\nAnd there was a card.\n\nI leaned over to get it and winced at stiff and seriously sore muscles. I saw a stretching session in my immediate future.\n\nI opened the small envelope. Even the paper felt expensive.\n\nRake.\n\nI'd check with the nurse, and if she hadn't brought them in, I'd have Kenji check the security cameras for one stealthy and determined goblin.\n\nI read the card.\n\nDearest Makenna,\n\nLunch (or dinner) awaits your pleasure, as do I.\n\nBe well and be careful.\n\nYours,  \nR\n\nVery nice. Caring, polite, yet not pushy. Brownie points earned.\n\nSleeping in, floral delivery . . . the SPI infirmary was starting to feel more like a hotel. I was wondering if I could get room service and schedule a massage when there was a knock, and Ian came in with a familiar pink box and a cardboard tray with two cups of life-restoring coffee.\n\nAsk and ye shall receive.\n\nI wasn't going to push my luck with the massage request.\n\nA box of anything from Kitty's more than made up for it.\n\nKatherine Poertner\u2014or Kitty to her friends, and I was fortunate to count myself as one\u2014was the owner and pastry chef extraordinaire of Kitty's Confections. She was a veritable wizard in the kitchen. Though to be perfectly accurate, Kitty Poertner was a witch. As far as those of us at SPI with a sweet tooth were concerned, Kitty's superpower was her baking skills. Everything that came out of Kitty's kitchen made people happy. She brought joy to the world\u2014supernatural and mundane\u2014one cookie at a time. Pink boxes turned up so often on SPI break room tables and in meetings that a lot of the folks here had started referring to her as the Goodie Goddess.\n\nOne thing Kitty didn't bake was gingerbread.\n\nBetween Thanksgiving and Christmas every bakery and coffee shop in the city was selling anything and everything gingerbread.\n\nKitty wouldn't touch the stuff.\n\nIn her defense, she had a good reason. Her entire family had a ton of bad karma to live down. Kitty's great-great-great-grandmother made Hannibal Lecter look like a cannibalism dilettante. She'd chow down on adults in a pinch, but she preferred children. She lured them in with sweets, most notably gingerbread.\n\nYep, she was _that_ witch.\n\nA cannibalistic child abductor was a heavy load on a family tree.\n\nIan saw the flowers on the bedside table. Everything else in the room was stark white. How could he miss them?\n\n\"Danescu?\"\n\nI tapped the tip of my nose twice in reply.\n\nIan held up the box, roses ignored, but, I was sure, not forgotten. \"Lemon-blueberry scones fresh out of the oven.\"\n\nMy favorite.\n\nI made a sound halfway between a moan and a . . . Okay, it was moan. Meg Ryan's deli experience had nothing on Kitty's scones. And from the size of that box, there were four warm wedges of pure heaven inside.\n\n\"Thank you, thank you, thank you,\" I said rapidly as I reached for the box with shamelessly greedy hands. \"This is even better than Krispy Kreme when the 'HOT' light's on. I'm getting all kinds of presents this morning. If I didn't have to wrestle with a squid demon to get them, I'd do it more often.\" I opened the box and looked down. \"One's missing.\"\n\nIan pulled the lid off his coffee to let it cool. \"Pickup and delivery fee.\"\n\nI remembered the previous pickup Ian had done\u2014me off of a garage floor\u2014and my appetite wavered, but didn't vanish. I was too hungry for even a near-death experience to ruin. \"And a fee that was well earned and deserved,\" I told him. \"Want another?\"\n\nIf Ian had gotten my reference\u2014and I was sure that he had\u2014he wasn't going to bring it up, at least not now. That took considerate to a whole new level.\n\n\"I won't turn it down,\" he said. \"We're both going to need to tank up today.\"\n\nI stopped mid bite. \"Aw, jeez. Can I at least eat one before you tell me who got slaughtered last night and how?\"\n\n\"No one got slaughtered.\"\n\nI took a bite and gave a thumbs-up. \"Score one for the humanoids.\"\n\n\"At least not anyone that we've found. Ord sent me a message this morning via one of his pixies. Two more local dealers vanished last night. They had guards and they still got snatched.\"\n\nA lump formed in my throat. \"Portal?\"\n\nIan shook his head. \"No stink.\"\n\n\"But no bodies.\"\n\n\"Not yet, but the day's young.\"\n\n\"Heard anything from Fred and the NYPD?\"\n\n\"Not a thing.\"\n\n\"Our people get any leads?\"\n\n\"Still digging. Still nothing.\"\n\nI grinned. \"Even though Ord sent you a pixie-gram to make nice, you still gonna kick his tiny ass for locking you out of his office?\"\n\nIan answered around a mouthful of scone. \"Seriously thinking about it.\" He swallowed and drank some coffee. \"I've also been thinking about a common denominator in all this, and I've found one.\"\n\n\"Good. What is it?\"\n\n\"It's a who. A lawyer by the name of Alastor Malvolia.\"\n\n\"Sounds goblin.\"\n\n\"A lot of the best lawyers are. He's Sar Gedeon's lawyer\u2014though now he's the executor of his estate. He handles all legal matters for the Frontino and B\u00e1thory families, and he's on retainer in one capacity or another with all of the other supernatural crime families.\"\n\n\"Elves using a goblin lawyer?\"\n\n\"Not one of his clients has ever been convicted. In fact, Al Malvolia's been highly successful in countersuing any accusers\u2014and winning most of the time.\"\n\n\"And the other times?\"\n\n\"The suit was dropped due to the lack of a plaintiff.\"\n\n\"Lack as in gone?\"\n\n\"Al's known for making problems\u2014and sometimes the people who cause those problems for his clients\u2014go away. He's a win-at-any-cost kind of guy.\" Ian glanced at his watch. \"We have a meeting with him at eleven o'clock. That gives you three hours to get up, shower, get dressed, and get moving. You feeling up to that?\"\n\n\"Sure. That is, if Dr. Stephens is willing to cut me loose. If not, I can just pull a Bert and leave. Thankfully I don't have any IVs to take out.\"\n\n\"You're sure?\"\n\n\"I'm sure, but are you? You actually want me to go with you?\"\n\n\"I need the benefit of your new skill set.\"\n\n\"How did you get a meeting with him?\"\n\n\"I made an appointment.\"\n\n\"As yourself? He knows you work for SPI?\"\n\n\"He does. I made the appointment not only as an agent of SPI, but as a personal representative of Vivienne Sagadraco. I told her my theory, and she's given me the green light to share any detail of the murder I need to in order to gain Malvolia's cooperation. It's in his best interests to help us in any way he can if he wants to keep representing living clients instead of handling dead clients' estates.\"\n\n\"I imagine he's gonna be popular. First us, next the NYPD will come calling.\"\n\n\"I'm sure they'd love to talk to him, but they can't.\"\n\n\"Why not?\"\n\n\"They can't find him. That's the other reason why I need you to go. Let's just say he doesn't have a local address.\"\n\n* * *\n\nI'd heard about pocket dimensions, but I never expected to find a high-powered goblin lawyer's office in one. Though when I thought about it, what better place?\n\nA pocket dimension is attached to a larger dimension, like a coat closet off of a ballroom. Though depending on the talent of the mage who did the construction, not all pockets are small. Like coat closets, pocket dimensions have a door\u2014otherwise known as a portal. The big difference between the portal to Alastor Malvolia's office and the two portals to Hell's anteroom that I'd seen yesterday is an actual, physical connection. Malvolia's office is in our dimension. Our dimension and Hell's dimension, fortunately, aren't next to each other.\n\nMalvolia's portal is permanent, like a door is permanent. However, it's still invisible to those not keyed to it.\n\nNeither Ian nor I were keyed to Malvolia's office portal, but since we had an appointment, there would be someone there to escort us across. Ian wanted to know whether I could see it for myself. It'd be a test. If I could see the portal to this pocket dimension, I'd probably be able to see any and all kinds. That'd be great for SPI, but bad for me if people like Alastor Malvolia knew what I could do. If I could see the door to Al's hidey-hole, I wouldn't be making an announcement.\n\nThe goblin lawyer's office on Park Avenue occupied the same space as a prominent Manhattan law firm.\n\nIan and I went inside and he gave our names, and who we were there to see, at the front desk.\n\nWithout a word, the receptionist keyed in a code on his computer's keyboard, and a door clicked open that, until that moment, had looked like part of the wall.\n\nNifty. And more than a little concerning.\n\n\"Wait inside, please. Mr. Malvolia's assistant will be right with you.\"\n\nIan nodded. \"Thank you.\"\n\nMy partner went through the door and I followed. The door\/wall clicked shut behind us.\n\nI'll admit it, I jumped a little. Ian glanced around, but otherwise didn't move. There was nothing to see.\n\nThat was my problem.\n\nThe room was no larger than ten by ten. No windows, no doors, all walls\u2014and not a portal to be seen or sensed.\n\nI casually went back to back with Ian. \"I don't like this,\" I said, trying not to move my lips. I had an entirely unwanted image of the _Star Wars_ trash compactor scene. Minus the trash and stink, that is. For a potential trap, it was actually a very nice room. Death by polished wood paneling.\n\n\"Easy, partner,\" Ian murmured. \"It's a pocket dimension. They don't need doors.\"\n\nDimensions didn't need doors, but if an exit didn't show itself soon, I was going to either hyperventilate and pass out, or make my own door.\n\nI continued with the whispering and not-moving-my-lips thing. \"If I can see portals, why can't I see this one?\"\n\n\"Because they haven't activated it yet.\"\n\nOh.\n\nBefore I had time to feel too embarrassed, a pale green glow appeared in a smooth seam down the same wall we'd come in through, though not in exactly the same place. It wasn't a friendly, springtime leaf green; this was a noxious acid green glow. Somehow it suited a guy who by all accounts could have single-handedly given lawyers of every species a bad name.\n\nI'd never thought of myself as much of an actress, but I did my best to look past the portal as if I hadn't seen it, and moved to where Ian could see me checking my watch.\n\nThat was our pre-arranged I've-seen-a-portal signal.\n\nIan casually and quietly cleared his throat.\n\nMessage received.\n\nI received a little message of my own. More like a confirmation. I could see portals, probably any and all of them.\n\nI sighed. Oh goody.\n\n* * *\n\nThe goblin lawyer took my partner's hand in an enthusiastic two-handed shake.\n\n\"Ian, my boy, how are you?\"\n\nWith a name like Alastor Malvolia, I expected the goblin version of Mr. Burns on _The Simpsons_ , not the bright-eyed, cheerful man who greeted us just inside the door to his office. Of course, someone would be less likely to expect a knife in the back from a happy guy.\n\nMalvolia's assistant had walked us through an office that looked disturbingly similar to the human lawyer's office in our dimension\u2014and that occupied almost the exact same space. That felt cosmically wrong on every level.\n\nGoblins were known for being tall, but Alastor Malvolia was maybe an inch taller than me, if that. Goblins were also known for being sexy. I felt confident in saying that no creature\u2014in our dimension or any other\u2014would think Al Malvolia was hot.\n\n\"Mr. Malvolia, I'd\u2014\"\n\n\"Al. After all this time, please call me Al.\"\n\nIan smiled what I'd come to know as his fake work smile. \"If you insist.\"\n\n\"I do.\"\n\n\"Al, this is my partner Agent Makenna Fraser.\"\n\nThen I was on the receiving end of the two-handed shake as my hand completely vanished in both of his.\n\n\"A pleasure, Agent Fraser. Though I wish our meeting was under different circumstances. Mr. Gedeon was a longtime client of mine. The nature of his death has been a shock to all of us who knew him. Please, both of you, have a seat. May I offer you something to drink?\"\n\nIan held up a hand.\n\nI said, \"No, thank you.\"\n\nThe goblin sat behind his surprisingly non-imposing desk. \"Then we'll go directly to what brings you here. The killer who is preying on the citizens of our city.\"\n\nAt least he didn't say \"innocent citizens.\" That would have been pushing it. How he described them was perfectly accurate. Drug lords and their underlings may be directly or indirectly responsible for hundreds\u2014maybe thousands\u2014of deaths, but they were citizens of New York.\n\n\"We believe Sar Gedeon's murder, as well as those of several of your other clients' employees, are linked to the arrival of the drug Brimstone and the individuals behind its manufacture and sale. We have reason to believe the source of the drug is extra-dimensional. However, we can't confirm this without access to the drug.\"\n\nMalvolia laughed. \"And you think that I would happen to have a sample lying around the office.\"\n\nAgain with the smile. \"Of course I don't. Though it would make our job much easier if you did. If we can analyze the drug, we can determine its origin\u2014and track down those who brought it here. We have reason to believe Mr. Gedeon was killed because of his desire to negotiate a business arrangement with Brimstone's manufacturers. His request was rebuffed with some finality in an incident involving one of his employees three days ago.\"\n\nMalvolia laughed, a half-hiss, half-wheeze that didn't do a thing to make me feel more comfortable.\n\n\"Ian, you missed your calling,\" the goblin said. \"You would have made a fine attorney.\"\n\nMy partner inclined his head in acknowledgment, though he'd been a cop long enough to take anything coming from a creature like Alastor Malvolia as a compliment.\n\n\"From what we saw of Mr. Gedeon this morning,\" Ian continued, \"he either hadn't taken the hint, or his killers wanted to make an example of him to those who wanted to make a similar business arrangement\u2014or perhaps both.\" He paused significantly. \"SPI wants to stop all of this. I'm sure you want the same. Your clients may have information that could help, either a sample of the drug, or names of the people who they attempted to negotiate with. Either would help us locate these individuals and stop the killings. We would greatly appreciate their cooperation and assistance.\"\n\nAlastor Malvolia had steepled his fingers and was regarding Ian with calm, calculating eyes. _Now_ , we were seeing some Mr. Burns. \"And what would be in it for my clients?\"\n\n\"They might get to live longer,\" I said before I could stop myself.\n\n\"Such charming honesty.\" The goblin smiled, but it didn't make it to his eyes. \"I have spoken with several of my clients who have been affected by recent events. They consider themselves qualified to protect themselves, their families, and their employees.\"\n\nThey've done a crappy job so far. I thought it, but this time I didn't say it.\n\n\"If I decide to relay SPI's offer to them, they would want to know what they would receive in return for their cooperation. Where is the value for my clients?\"\n\n\"Have you heard how Sar Gedeon died?\" Ian asked.\n\nAny pretense of polite vanished. \"Your agency has been unwilling to release that information\u2014or the body of my client. I have sent a request for access to the body and the murder scene, and that request has been stalled. Mr. Gedeon's widow has been denied permission to claim her husband's body.\"\n\n\"Our investigation is not complete. Mr. Gedeon is more than a victim; he is our only source of clues to the identity of his killer. I would think Mrs. Gedeon would want us to find who murdered her husband and why. As to how he was killed, I have been authorized by Vivienne Sagadraco to share that with you.\"\n\nIan shared\u2014and he included the details Bert had seen.\n\nAl developed a twitch in his left eyelid.\n\nLooked like Al now had a newfound appreciation for the severity of the situation.\n\nHearing that a demon lord\u2014and someone unknown, but even worse\u2014was hunting down and butchering your client base would do that.\n\n\"I will contact my clients, and then get back to you.\"\n\n\"When?\"\n\nAl's eyelid twitched again.\n\n\"Eight o'clock tonight. A few of my clients prefer to sleep during the day. Will that be acceptable to Ms. Sagadraco?\"\n\nIan smiled, and now it was genuine. \"Perfectly.\"\n\n# 13\n\n\"I'LL be interested to hear what Al comes up with by tonight,\" Ian said as Yasha stopped in front of the Park Avenue office building to pick us up.\n\nI settled in the middle of the second row of seats and buckled in. \"You enjoyed that, didn't you?\"\n\nIan got in the front passenger seat. He always rode shotgun. \"Just because a person knows the law does not mean that they respect it. No one has less respect for the laws of our dimension than Alastor Malvolia. So yes, I enjoyed rattling his cage.\"\n\n\"I don't think we're going to get as friendly of a greeting next time.\"\n\n\"I would be surprised\u2014and wary\u2014if we did. However, the next time SPI needs to twist Malvolia's arm, it'll be Moreau's turn.\"\n\n\"You guys take turns?\"\n\n\"Mostly. This meeting should have been Moreau's, but the boss thought that since I'd seen Sar Gedeon at the murder scene, I was better qualified to describe it to Malvolia, if he needed persuading to cooperate with us.\"\n\n\"And the fun was just a nice fringe benefit.\"\n\nIan grinned. \"I'm not opposed to enjoying my work.\" He glanced at his watch. \"Want to grab some lunch?\"\n\n\"Sure. If you can find us a restaurant that's not likely to get set on fire.\"\n\n* * *\n\nWe hit the Full Moon.\n\nIt was close enough to headquarters if we needed to get back quickly, but far enough for a little peace and quiet.\n\nWe were all greeted with hugs by the owners, Bill and Nancy Garrison. Bill was the king of the barbeque pit, and Nancy had the brains for the business, and the Southern charm and hospitality to keep the place full of happy customers.\n\nThey were also werewolves.\n\nBest of all, they were from my home state of North Carolina.\n\nI came here to get a literal taste of home.\n\nThe barbeque was slow cooked, the burgers were rare, the steaks were tartar, and the regulars were furry. There was always a booth reserved for hungry SPI agents, and Yasha's Suburban was always welcome in the alley\/delivery area behind the restaurant. Needless to say, it was Yasha's all-time favorite place to eat.\n\nPart of Nancy's business savvy involved billing the Full Moon as \"New York's Official Werewolf Bar.\" She even turned a section of the front of the restaurant into a gift shop selling T-shirts, mugs, shot glasses, and if you wanted to build your Pomeranian's street cred at the dog park, there was werewolf gear for the small, but fierce, canine in your life. The restaurant and bar were decorated with dark wood, dim lights, and every werewolf clich\u00e9 that existed. Werewolf movie posters and props were on display, and everything on the food and drink menus had a werewolf- or movie-monster-inspired name.\n\nFun place, good people, great food.\n\nAs soon as Bill set that plate of pulled pork barbeque in front of me, I forgot all about finding Sar Gedeon nearly twenty-four hours ago, and nearly coming face-to-face with his killer soon after. Good food that was much needed would do that. I didn't think I was all that hungry until I started eating. I was glad I ordered the large platter. I was small, but I could put away some groceries.\n\n\"Do you think Al's going to have any luck getting his clients to talk?\" I asked Ian.\n\n\"He might. If he does, he might even decide to tell us what they said. I'm not going to hold my breath on either one, but I hope their survival instinct overrides their greed.\"\n\n\"Greed? You've lost me.\"\n\n\"If the families have gotten their hands on some Brimstone or the formula, they're going to have a hell of a time getting the main ingredient.\"\n\nI was still confused, and apparently I looked it.\n\n\"She was not in meeting last night,\" Yasha reminded Ian.\n\nI blinked. \"There was a meeting?\"\n\n\"You were asleep.\"\n\n\"You didn't tell me this morning.\"\n\n\"I was preoccupied this morning, and I'm telling you now.\"\n\nI sighed. \"Go on.\"\n\n\"While we still need a sample of Brimstone for the lab, we've got enough information now to have a good guess as to where it came from.\"\n\n\"And?\"\n\n\"The main ingredient was imported directly from Hell.\"\n\nWhoa. \"Real, biblical hellfire and brimstone?\"\n\nIan nodded. \"In our dimension, brimstone is another, non-scientific, name for sulfur. What I found out from Marty last night is that our sulfur got the alternative name of brimstone because there's an actual mineral, found only in Hell, that stinks like sulfur. He showed me several samples in his lab. It's bright orange.\"\n\nI remembered yesterday at the coffee shop. \"And Fred told us that Brimstone is orange.\" Just like the portal I'd seen in Gedeon's apartment and the parking garage.\n\n\"Exactly.\"\n\n\"Okay, I have to ask. How did Marty get samples?\"\n\n\"He said he gathered them himself on a field trip.\"\n\n\"To Hell.\"\n\n\"Wasn't Hoboken.\"\n\n\"I wonder if that was when he lost his eyebrows.\"\n\n\"Nope. That's an even better story. You'll have to ask him. He tells it better.\"\n\n\"So what does brimstone from Hell do besides stink?\"\n\n\"That's where it gets interesting. Marty said its chemical composition contains elements found in our dimension's LSD.\"\n\n\"So much for why man in restaurant saw monsters,\" Yasha said.\n\n\"Before I got some shut-eye myself last night,\" Ian continued, \"I touched base with Fred. When the man was questioned about where and who he bought the drug from and for what purpose, he couldn't remember. The NYPD has a vampire on the force who has a knack for lie detecting. The guy was telling the truth. He doesn't remember a thing.\"\n\n\"That makes absolutely zero sense,\" I said. \"What good is a drug that lets you see supernaturals and read minds, but then wipes your memory? I mean, it's good for us; no one would remember seeing goblins and elves, but it's bad for tracking down the source of this stuff. Well, besides Hell. Though that doesn't make any sense, either. Marty told me that demons aren't interested in humans on an individual basis. Why would demons provide brimstone as an ingredient to presumably a mortal drug kingpin\u2014or queenpin?\"\n\nTo say we were missing something here was the ultimate understatement.\n\nIan's phone rang.\n\nHe looked down at the display. \"Fred.\"\n\n\"Speak of devil,\" Yasha said with a grin.\n\nI grinned back.\n\nIan spat his favorite four-letter word into the phone, and waved for Nancy. \"We've got to go. Now. Would you put this on my tab?\"\n\n\"Of course, sugar. Want some of it to go?\"\n\n\"Love to, but no time.\" He stood. \"There's been another murder,\" he continued when Nancy began speaking to other customers three booths away. \"Goblin by the name of Kela Dupari.\"\n\nI scooched out of the booth. \"Drug lord?\"\n\n\"Lady, one of the 'queenpins.' Her territory is the Upper West Side. This one's worse.\"\n\nI suddenly wished I hadn't scarffed down all that barbeque. \"Worse than a heart cut\u2014\"\n\nIan shook his head. \"No. The NYPD got there first.\"\n\nYasha snarled the Russian equivalent of my partner's favorite word. It sounded better in Russian.\n\n\"Then what do we have to do?\" I asked.\n\n\"We'll get there and then we'll see.\"\n\n\"See what?\"\n\nYasha pulled the Suburban keys out of his pocket. \"If we need to steal a body.\"\n\n# 14\n\nI'VE had some strange things added to my job description since joining SPI, but I never imagined body snatcher would be one of them. Though technically it would be ambulance robber, or if we were going to get picky, ambulance hijacker.\n\nRegardless of the semantics, I hadn't signed up for any of it.\n\nSo I was more than relieved when we didn't have to do it.\n\nThis was the NYPD we were talking about. Getting arrested\u2014at least for me\u2014wouldn't be _if_ it would happen, but how fast. I didn't want Alain Moreau having to come down to whatever precinct they dragged me to and bail me out. It'd happened once, and I didn't want it to happen again.\n\nIan had just gotten the call from one of our dispatchers that the medical examiner had taken care of extending the victim's glamour for another few hours.\n\nDr. Anika Van Daal was the medical examiner. She was also a vampire and mage who had arrived in the city soon after it'd been taken from the Dutch by the British and the name changed from New Amsterdam to New York. That'd been in 1625. At that time, two-thirds of the island was still wilderness.\n\nShe'd begun her medical career as a midwife, and had become the first licensed female doctor in the city. Every few decades, she \"retired\" from one position and took another. She'd been in her mid-twenties when she'd been turned, so she didn't stand out when she went back to school after a retirement to catch up on the latest medical advances. She'd learned to glamour and glamour well. As a result, she'd never had problems blending in or being found out.\n\nVivienne Sagadraco had a lot of pull in this town, and one of the ways she used it was getting supernaturals placed in strategic jobs. In addition to supernaturals in the NYPD, there were mages who, like Dr. Van Daal, could replace a glamour on a dead supernatural and hold it there until the body was turned over to the family. Or if no one claimed the body, until it was cremated or buried by the city. These mages were in the homicide divisions, medical examiner's office, and CSI teams.\n\nThe boss had covered all the bases she could, but occasionally a corpse tried to steal home. When that happened, there was a lot of scrambling and improvising by whichever agents were closest.\n\nThankfully today, no one needed to scramble.\n\nWe watched from a parking spot on the street that by some miracle of maneuvering, Yasha had managed to wedge the Suburban into, without turning either the car in front or the one behind us into an accordion. I'd turned to look at something totally fascinating out my window, so Yasha wouldn't see me cringing the entire time.\n\nOur cozy spot was half a block from the murder scene. Ian wanted to stake it out for a while to see if any of the curious onlookers behind the police crime scene tape looked a little too curious\u2014or pleased with themselves.\n\n\"Dr. Van Daal will get a copy of her report to Moreau, but the preliminary is the same MO as Gedeon's murder.\"\n\n\"Portal stink?\" I asked.\n\n\"Including portal stink.\"\n\n\"Bert won't get a shot at this body,\" I noted with no small measure of relief. \"And I'm perfectly fine being half a block away from where there was a portal.\" I spotted a familiar face trying to act casual as he exited the building. It was a ten-story building, and hundreds of people would have a reason to be there, but it was too much of coincidence that this individual would be one of them.\n\nJesin Nadisu. The apartment building manager of the Murwood, aka murder scene number one.\n\n\"Do you see\u2014?\"\n\nIan was out the door before I could finish.\n\nSince Yasha was legally parked for once, he joined us.\n\nThe young goblin's day was about to take a turn for the worse.\n\nMy day was going to be just fine. Not only did Ian not tell me to wait in the SUV, but if Jesin Nadisu ran for the closest parking garage and getaway portal, I'd have plenty of qualified backup this time.\n\nThe goblin didn't run for the nearest parking garage. When he saw us, he just ran. Fast. If Yasha could have gone wolf, he could have been on our Olympic wannabe in three bounds and a leap. It was the middle of the day in Midtown, going furry wasn't an option, so we had to do it the old-fashioned way.\n\nIan had missed out on snagging the assassin yesterday, and he wasn't going to take second place today. Jesin Nadisu didn't want to be caught, so he was motivated. Ian was just pissed. People were being killed, a coworker was attacked, and his partner was damned near dragged to Hell\u2014excuse me, an anteroom\u2014by a squid demon. As a motivating factor, anger topped fear anytime.\n\nShots rang out that weren't ours, and the goblin went down.\n\nPeople screamed.\n\nWe instantly went from pursuit to protect.\n\n\"Police!\" Ian yelled. \"Get inside.\"\n\nIt wasn't a lie; he was the police, just not the NYPD, at least not anymore.\n\nRegardless, when Ian ordered, people obeyed.\n\nSoon we had the section of street to ourselves\u2014until the NYPD investigating the murder came out to join us. We needed to be gone before that happened.\n\nWith bullets flying around, I would have liked to have been one of the people obeying Ian's order and getting the heck off the street, but instead I ran with Yasha to where the young goblin had pulled himself to the protection of a building doorway before he collapsed.\n\nIan ran in pursuit of the shooter, with a sharp wave to Yasha to get the car.\n\n\"Go,\" I told him. \"I've got this.\"\n\nYasha didn't like it, but he went. The quicker he got back, the less chance that we'd all get arrested, and it wouldn't be for stealing a dead body; it be for taking a wounded murder suspect.\n\nWhen I got to Nadisu, he was still conscious. He saw me and tried to drag himself farther away.\n\n\"Hey, we're the good guys here. _You_ were running from _us_.\"\n\nPain kept him from talking, but from the dread in his eyes, being caught was worse than being shot.\n\nDespite his presence at two murder scenes, I didn't think Jesin Nadisu was a murderer, at least not the kind of murderer who'd eat souls and be partnered with a demon lord.\n\nI'd been wrong before, but I knew I wasn't wrong about this.\n\nFor one, other than the small magics needed for a glamour, I didn't sense any power coming from him. The only thing a demon lord would want him for was a snack.\n\nBlood was spreading under his suit coat on his white shirt. His hands weakly fought me as I pulled the coat back to see the damage.\n\nA package fell out of the inner pocket. The bullet must have nicked it. An orange powder from inside dusted my hand. I highly doubted it was Tang.\n\nI glanced at Nadisu's face to see his reaction, but he'd passed out.\n\nTires screeched as the Suburban arrived, and Yasha leapt out and picked up the goblin like he weighed next to nothing, laying him out on the middle seat. I jumped up beside him and started buckling him in, and Yasha shut the door behind us. The Suburban had a second set of seat belts mounted like on a stretcher.\n\nThe passenger door opened and Ian all but dove into his seat. His face was flushed and grim, and looked about as angry as I'd ever seen him.\n\n\"Get us home,\" was all he said.\n\n# 15\n\nJESIN Nadisu was going to be in surgery for at least two hours.\n\nIn addition to an infirmary, SPI had a fully staffed and equipped trauma center and ER, albeit on a much smaller scale than most New York hospitals. When you fought monsters and powerful mages and supernatural criminals, your people could get injuries that would do more than raise eyebrows at the neighborhood ER. Vivienne Sagadraco valued her employees, and made sure that we had only the best medical care available to us.\n\nJesin Nadisu was presently on the receiving end of that expertise.\n\nWe couldn't question him until he was out of recovery, and then it would be up to the trauma surgeon as to when and for how long. Not that we thought the young goblin was guilty of anything other than having a kilo of Brimstone on him. Heck, we were grateful that he had.\n\nWhile the doctors were working on Jesin Nadisu, the lab down the hall was working on the Brimstone. The analysis would probably take longer than the surgery. But we wanted to be close to get word on both.\n\nI was standing in the hall outside the main lab, looking through the glass wall.\n\nI'd never asked the reason for a glass wall in a lab, but I guessed that privacy was less important than someone outside seeing if something went very wrong inside\u2014and then getting help. Fast.\n\nAfter hanging out in the ER waiting room for a while, I'd walked down the hall to the lab. Ian had gone to make a few phone calls. I hadn't asked him about what had happened in the chase to catch Jesin Nadisu's shooter. All that Ian had volunteered was that he'd gotten away. Something important was going on here, but I'd learned that when my partner needed me to know, he'd tell me. I was learning to tamp my curiosity down until that happened. I didn't say it was easy; I said I was learning.\n\nThe elevator door dinged.\n\nIan.\n\n\"Anything?\" he asked, indicating the lab.\n\n\"If so, they're not acting like it.\" I glanced back into the lab to make sure none of the white-coats had gone all giddy in the past ten seconds. \"Nope, no high fives or group hugs.\"\n\nMy partner sat in one of the chairs lining this section of wall and put his head in his hands.\n\nI sat next to him. \"Want to tell me about it yet?\"\n\nIan didn't move for another handful of seconds, then he sat up, thunked the back of his head against the wall, and stared at the ceiling with an expression of \"Why me?\"\n\nI didn't take any of that as an indictment on my curiosity, but rather frustration at the situation we were neck deep in, so I leaned my head back and helped my partner look at the ceiling.\n\n\"Nightshades,\" he finally said.\n\n\"I'm assuming you're not talking about the plant.\"\n\n\"I wish I was. You could get rid of those with weed killer. We rarely see these, let alone get a chance to eradicate them. They just come back. Then again, maybe they are like the plants.\"\n\n\"Then they're a who, not a what.\"\n\n\"Nightshades are basically elven black ops mercenaries. They'll do whatever they're paid enough to do. Today one of them was paid to get Jesin Nadisu.\"\n\n\"He didn't do a very good job.\"\n\n\"It's lucky we were here so he couldn't finish the job. If we hadn't been watching\u2014\"\n\n\"He wouldn't be here and alive getting stitched up.\"\n\n\"Yeah.\"\n\n\"But you saw the shooter, and I take it you recognized him.\"\n\n\"He's one of their best marksmen.\"\n\n\"And he only got Jesin in the side?\"\n\nIan gave me a flat look.\n\n\"He didn't kill him on purpose?\" I guessed.\n\nIan nodded once. \"There was an ambulance parked around the corner.\"\n\nI knew where this was going and it wasn't anywhere good. \"A fake ambulance to fool anyone who saw them. They wanted him alive.\"\n\n\"And afraid and in pain. They hired Nightshades to make it happen. And if he had died . . . there are necromancers who sell their services to the highest bidder. The Nightshades keep two on retainer.\"\n\nCripes.\n\n\"Jesin doesn't look like the drug-runner type.\"\n\n\"The best runners arouse the least suspicion, and neither one of us would have suspected the manager of an exclusive, high-rent apartment building to be running drugs.\"\n\n\"He does dress well,\" I admitted. \"Do you think he knew who was gunning for him?\"\n\n\"Maybe, maybe not. We won't know until we can talk to him. In the meantime, I'm having Ord Larcwyde brought in for questioning, though the agents picking him up have been told to go with 'protective custody.'\"\n\nI grinned. \"Ord does value his safety.\"\n\n\"I thought it'd go over better. Whoever was pulling that demon lord's strings was worried enough that Ord had damaging information to send a demon assassin after him. I want to know what that information is.\"\n\n\"When he gets here, you might want to let me do the chatting. He likes me more than he does you.\" I thought of something and chuckled. \"I wonder if he's come out of his freezer yet.\"\n\n\"If not, the agents are taking a blowtorch with them, just in case. Either he comes out on his own, or the boys go to work on his cube.\"\n\nA door opened down the hall in the hospital wing and Dr. Stephens gestured for us to come down there.\n\nOur patient was awake.\n\n* * *\n\n\"I'll let you do the talking,\" I told Ian.\n\nHe gave me a bemused glance. \"Really. You're sure about that.\" Neither was a question, at least not real ones.\n\n\"Hey, I've never questioned a shooting victim fresh out of surgery. I take it you have.\"\n\n\"I have.\"\n\n\"Then this one's all yours.\"\n\n\"I'll believe it when I don't hear it.\"\n\nWe went into the recovery room.\n\nJesin Nadisu looked like hell.\n\nThough he didn't look nearly as bad as Sar Gedeon had. Thanks to the skill of our surgical team, Jesin at least had all his pieces and parts. Most of them probably hurt right now, but at least he still had them. Considering all that he'd been through in the past few hours, I thought I should keep that comparison to myself. The young goblin had gotten off lucky. For the sake of his continued emotional well-being, I'd keep that to myself, too.\n\nSPI's chief trauma surgeon had told us not to stay for longer than five minutes. The only reason she wasn't in the room with us was that she didn't need to be. There was a two-way mirror next to the door that would let her see and hear everything that went on. At the first sign of fatigue or distress from her patient, I was certain she'd be in here with us a split second later, telling us to leave. Nicely the first time, then not so nice. SPI agent, suspect, or caught-red-handed clawed criminal, her patients were her top priority. One of the things we learned in new-agent training was not to argue with Dr. Barbara Carey.\n\n\"Mr. Nadisu?\" Ian said quietly, but loud enough to be heard. \"Mr. Nadisu, I need to ask you just a few questions, and then you can continue to rest.\"\n\nThe goblin's eyes fluttered open. Large, dark, and long lashed, he looked even younger than he had when he'd met us in the lobby of the Murwood. If he'd been human, he wouldn't have looked old enough to buy a beer, let alone manage an exclusive apartment building.\n\n\"Agent Byrne.\" His voice was rough from the breathing tube we'd been told they'd had to use during surgery. Apparently the bullet had nicked the bottom of his lung and a not-so-minor blood vessel or two. He blinked a few times and focused on me. \"And Agent Fraser.\" He tried a weak smile that didn't quite make it. \"I can explain about the Brimstone.\"\n\nAt least we had confirmation that what the lab was analyzing was Brimstone. Though right now, a plastic-wrapped, brick-shaped block of glowing orange powder taken from a demonic murder scene couldn't be much else.\n\n\"Do you know who shot you?\" Ian asked.\n\nNadisu didn't answer.\n\n\"If you're worried about them getting to you, don't,\" I told him. \"You're safe here.\"\n\nIan cleared his throat.\n\nOops. So much for letting him do the talking.\n\n\"Do you know where you are?\" Ian asked.\n\n\"No.\"\n\nMy partner was silent for a moment. \"How long have you been in our dimension?\"\n\nI refrained from doing a double take. Ian's voice was actually gentle. He clearly knew something that I didn't.\n\n\"If you're here illegally, we won't send you back,\" he continued.\n\nOh, okay, now I got it.\n\nBoth goblins and elves were very selective over who they let come through the permanent dimensional portal between our world and theirs. Though like humans, if you wanted to get here badly enough, you'd find a way. For supernaturals, that meant paying a small fortune in bribes to mercenaries with access to an illegal portal.\n\nBoth races operated under a controlling monarchy supported by a powerful aristocracy. Unless you were related to an influential family or had a magical talent that the nobility were interested in, you might as well not exist. No rights, no hope of a better life, and if you pissed off the wrong noble or mage\u2014no life at all.\n\nHumans weren't the only species who came to New York looking for a better life.\n\nUnless they could afford papers to let them pass as a legal citizen of the good ol' U S of A, and could afford to have a mage fit them with a glamour to let them pass as human, they were just like the thousands of undocumented human immigrants in the city, but with goblins and elves, the term \"alien\" was literal.\n\nIn such an environment, it wasn't a surprise to anyone that organizations emerged to \"govern\" their people, to resolve differences without human interference, to serve up justice when it was needed, and to execute whoever they decided should be.\n\nPolice, judge, jury, and executioner.\n\nAny attempt by SPI to intervene was called interference in a \"goblin matter\" or \"elven business.\"\n\nWe saw them as the criminal families they were.\n\nAnd Jesin Nadisu was apparently scared to death of one of them.\n\n\"You're at SPI headquarters,\" Ian told him. \"So whoever it is that you're afraid of can't get to you here.\"\n\nThe goblin sighed. \"Would you like to bet on that?\"\n\n\"We know about Nightshades,\" Ian said. \"You're completely safe from them or anyone else you may have reason to fear.\"\n\nIncluding demons opening portals. Then I had another thought.\n\n\"Your employer, maybe?\" I asked.\n\nThe goblin turned even pastier, if that was possible.\n\nThe door immediately opened.\n\n\"That's all, Agents Byrne and Fraser.\" Dr. Barbara Carey wasn't going to accept any response other than us getting away from her patient.\n\nWithin seconds, we were in the hall with the door firmly closed behind us.\n\nIf I could've kicked myself in the ass, I would've. \"Dammit, I'm sorry.\"\n\n\"Don't worry about it. Our time was almost up. Dr. Carey wouldn't have let us have a second more. Sounds like the kid's afraid of his boss.\"\n\n\"Do we know who owns the Murwood?\"\n\n\"No, but Kenji has a database of buildings owned by supernaturals. Murwood is the name of a forest in the goblin and elf dimension, so chances are good that a supernatural owns the building.\"\n\n\"I have a couple of follow-ups I need to do, so I'll check in with Kenji on that.\"\n\nIan nodded. \"I'll wait for Dr. Carey to come out and see when she might let us talk to Jesin Nadisu again, though I'm not holding my breath for it being anytime soon.\"\n\nI tilted my head down the hall. \"And if you could listen out for any celebrations erupting in the lab.\"\n\n\"Will do.\"\n\nI headed down to the bull pen and to my desk.\n\nOnly to find Alain Moreau sitting in my chair.\n\nAw crap.\n\nBeing called into your manager's office was stress inducing enough. But to have your manager camp out in your chair to wait for you?\n\nI'd stepped in something serious. Even worse, I'd stepped in so much lately, I had no clue which pile this could be about. At least he hadn't had to come down to one of the NYPD's precincts to bail me out. Regardless, I was sure I looked like a kid with their hand in the cookie jar, even though I didn't know what I'd done.\n\nAlain Moreau looked like a man about to fire someone.\n\nHe'd hired me. He could fire me.\n\n\"I can explain,\" I told him. That is, as soon as I knew what he was here for. \"Or . . . do I just need to pack a box?\"\n\nMoreau looked baffled\u2014baffled and tired. \"I beg your pardon?\"\n\n\"A box. To clean out my desk.\"\n\nMore bafflement as he regarded the surface of my desk. \"It appears to be acceptably tidy. Why would you need to clean\u2014?\"\n\n\"You're not firing me?\"\n\n\"You're not going anywhere, Agent Fraser.\"\n\nI took an involuntary step backward. Maybe SPI management considered firing to be wasteful. If I was a failure as an agent, maybe I'd be a rousing success as a meal in the employee cafeteria. After all, I wouldn't actually have to do anything. I couldn't screw that up.\n\n\"Unless you wish to leave,\" he continued, still sounding tired.\n\nNow I was confused.\n\nHe had the same expression as Ian had upstairs\u2014too much bad news and no idea how to deal with it. But instead of thunking his head against a wall, Moreau ran his hand through his perfect hair. Hair that was still perfect. I wasn't sure if it'd even moved. Maybe it was a vampire thing.\n\n\"We have questioned both Agent Filarion and Mr. Sadler, and neither have experienced any effects\u2014ill or otherwise\u2014from being exposed to the ley line convergence.\"\n\n\"Dang it.\"\n\nOne of Moreau's silvery eyebrows shot up in surprise.\n\n\"Okay, that didn't come out right. Sorry, sir. It's just that that wasn't what I wanted to hear. I mean, I'm glad that Caera and Ben didn't get zapped with some kind of mutant power, but it'd be nice not to have the only explanation left being a bizarre mind-meld, power-transfer thing with Viktor Kain. I'm not exactly enthused about catching anything from a multi-millennia-old, psychotic criminal mastermind.\" I paused for breath and sighed. \"At least I don't have an urge to take over the world,\" I muttered. \"Yet.\"\n\n\"That is not the only explanation left.\"\n\nI perched on the edge of my desk. \"It's not? But Ms. Sagadraco said\u2014\"\n\n\"We need to reconsider your family background. The contact with either Viktor Kain or the diamonds or the nexus\u2014or even a combination\u2014could have awakened a previously dormant ability.\"\n\n\"No one in my family can see portals. If they can, I never heard about it. Not to mention, I'd kind of hoped to be able to avoid calling home and asking.\"\n\n\"Why?\"\n\n\"They worry about me enough as is, moving up here and all. Calling home and going, 'Uh, Mom . . . yeah, I'm doing great. I've got a question. Has anyone in our family ever been able to see portals? No, no. No problems here. Just asking out of curiosity.'\"\n\n\"I can see how that might be awkward.\"\n\n\"And impossible to hide why I want to know. Mom's relentless. And don't even get me started on Grandma Fraser. Trust me; you don't want my family coming up here. Nobody wants that. Least of all, me. Has Kenji taken a shot at it yet?\"\n\nKenji Hayashi was SPI's CTA\u2014Chief Technology Agent. Each SPI office worldwide had their own CTA, but Kenji was the best, which was why he was here at agency headquarters. If it existed in cyberspace, the Japanese elf could find it, and decipher it six ways from Sunday.\n\n\"I give him _full_ permission to dig into my family background,\" I said. \"Just as long as he promises not to laugh at my more colorful relatives.\"\n\n\"I'll ask Agent Hayashi to look into it as soon as possible.\" Moreau paused. \"I have been unable to contact Rake Danescu to ask of any effects he may have experienced. I've left two messages, but he has yet to return my calls. He may be more willing to answer the question if it came from you in person.\"\n\n\"What makes you think I'll be seeing him?\"\n\n\"Your lunch date was interrupted.\"\n\n\"He wanted to reschedule for lunch today or dinner last night. But that got nixed by a squid demon and a possible concussion.\" I decided not to mention the flowers I found on my bedside table this morning. Moreau probably knew, but if he didn't, I really didn't want to bring it up. \"Are you visiting my desk because of Rake? Because if\u2014\"\n\nMy manager held up an elegant hand. \"It is not about Monsieur Danescu. I will admit to having concerns, but after speaking with Madame Sagadraco, she and I are in agreement.\"\n\nI gave him a small smile. \"That I'm a big girl and can take care of myself.\"\n\n\"That and while Rake Danescu may be many things, he has never been foolish.\" He narrowed his eyes very faintly. \"Do I need to explain that statement?\"\n\nMy smile broadened into a grin. \"Oh no, sir. I got it loud and clear. And I think Rake probably has, too. He behaves or there'll be a line to kick his ass, and you and Ms. Sagadraco will be near the front. Though Agent Byrne might want to argue for the right of first in line.\"\n\n\"No doubt. I wanted to speak with you concerning your satisfaction with your employment here.\"\n\nI tensed. \"Are you or Ms. Sagadraco not satisfied with my employment here?\"\n\n\"I assure you we are most satisfied with your job performance. The question is how you feel about your job. I imagine it has turned out differently than you envisioned.\"\n\n\"Yes, it has.\" I thought back to the events of the past week. \"I try my best to stay out of trouble. Problem is trouble keeps finding me.\"\n\n\"That is part of our concern. I encourage those who report to me not to hesitate to tell me if parts of their job are distressing to them. You haven't requested a meeting.\"\n\n\"Anything that's happened has pretty much fallen under my job description. More or less.\"\n\n\"On New Year's Eve, you chased down a fully grown grendel in a crowd of nearly a million people.\"\n\n\"I was the only one who could see her.\"\n\n\"A human without defensive magical powers taking on a monster out of legend bare-handed.\"\n\n\"I was the only agent there. I couldn't just let her start eating people.\"\n\n\"But it was not your job. It was far beyond what anyone would have expected or demanded of you.\"\n\n\"I had to do it.\"\n\n\"And last week with Agent Byrne, Yasha Kazikov, and Rake Danescu. You didn't have to go to that island to take on Bastian du Beckett and prevent those diamonds from being activated.\"\n\n\"Ben Sadler was being held prisoner. I felt responsible for him. Then there was Yasha, you, Ms. Sagadraco, and every supernatural in SPI. None of you are just coworkers; you've become like family.\" I said it without one bit of embarrassment. \"If there was some way I could help, I was going to do it.\"\n\n\"And you have not sought me out to lodge a complaint about your life being in danger beyond what you were hired to do.\"\n\n\"I don't mean any disrespect, sir, nor am I trying to be rude, but you're getting at something. What is it?\"\n\n\"Have you at any point during your employment with us considered turning in your resignation?\"\n\n\"Not seriously.\"\n\n\"And why is that? Through no fault of your own, you've nearly lost your life on numerous occasions. Other times you have purposefully placed yourself in harm's way.\"\n\n\"I don't think I'm a danger junkie, sir, if that's what you're getting at.\"\n\n\"I don't think that you are.\" His gaze searched my face. \"Why do you do it, Makenna?\"\n\nI suddenly knew the answer without having to think about it. I pressed my lips together not only against a tiny smile, but against the sudden sting of tears in my eyes.\n\n\"I feel needed. Sometimes I screw up, but I know what I'm doing is worthwhile. I can't imagine not doing it. I love my job.\"\n\nMoreau stood. \"That's all I needed to know. If that ever changes, I trust you will inform me.\"\n\n\"Absolutely, sir.\"\n\n\"Then I'll let you get back to work.\"\n\n\"Thank you, sir.\"\n\nMoreau headed for the elevators, and I realized my teeth were clenched in a smile that'd probably scare small children.\n\nThe silence of the agents in the bull pen behind me was absolute.\n\nI turned slowly and was met with dozens of pairs of curious eyes: human, elf, goblin, troll, gnome . . .\n\n\"I'm still here,\" I said loudly. When that didn't make them stop staring, I gave a double thumbs-up for emphasis.\n\nAt that, everyone returned to what they'd been doing, and the noise levels returned to normal.\n\nI sat down with a sigh. Nothing about this place was normal.\n\n# 16\n\nI'D barely gotten started digging for an answer to a question that'd been nagging me, when I heard the click of Kylie O'Hara's stiletto heels coming toward my desk. I'd never seen her in heels lower than four inches. She was pretty much five foot nothing, and while human women of her height would have worn heels for added height regardless of the excruciating pain, Kylie wore them because they were fun.\n\nIt had to be a dryad thing. They must have tiny arches of steel.\n\nShe nodded toward the elevators. \"Well, how did _that_ go?\"\n\n\"Good. He just wanted to be sure I was happy in my work.\"\n\n\"Are you?\"\n\n\"Sure. Until something kills me, but then it'd be too late to lodge a complaint. Well, unless y'all get Bert involved, but I'd really rather you didn't.\"\n\n\"Noted. So you're definitely staying?\"\n\n\"I don't think I'd be allowed to leave if I wanted to. And I don't want to,\" I hurried to add.\n\n\"Good.\" She shot a withering look at the bull pen. \"Because there's way too much testosterone around here.\"\n\n\"And eavesdropping.\"\n\nKylie shrugged. \"Agents. They can't help it.\"\n\nShe perched on the edge of my desk. In her stiletto heels and short pencil skirt, she did a better job of it than I had. The boys in the bull pen agreed. They eavesdropped on me; they ogled Kylie.\n\n\"I found out from Baxter the Bastard about those _Sex in the City_ segments,\" she said.\n\nBoth monikers were her creation. The first was the God's truth. Baxter Clayton, news anchor, was most definitely a bastard. The other was a cute and clever name for the series the aforementioned bastard was doing on New York's high-class sex industry.\n\n\"And?\"\n\nThe dryad leaned in closer. \"His producer shut down the project last month. The sex industry in this town must have a lot of pull.\"\n\n\"Pun intended?\"\n\nShe thought a moment. \"No, lucky coincidence. And it wasn't the station that pulled the plug. The _network_ brass axed it.\"\n\n\"Oooh. Wanna bet some of those bad boys are clients?\"\n\n\"I'd put money on it, though it sounds like they already have.\"\n\n\"So would anyone who was going to have their business featured have known that the series had been scuttled?\" I asked.\n\n\"Definitely.\" Kylie gave me a fess-up look. \"Why do you need to know?\"\n\n\"Yesterday in the coffee shop, Rake needed to leave. Fast. He claimed it was because Baxter had been stalking him for his segment.\"\n\nThe dryad sighed. \"Honey, think about the goblin mind for a minute.\"\n\n\"I'd rather not.\"\n\n\"Too bad. Besides, if you decide to make this thing work with Rake, you'll need the practice. Technically, he didn't lie. I have no doubt that Baxter would have been stalking him for that segment. The total truth was that Bax wasn't stalking him _anymore_. So, the question then becomes, why the sudden need to leave?\"\n\nI sat back and wished I had a wall behind me to thunk my head against. \"It was obvious he didn't want to get away from me. So we can toss out fear of commitment. _Aversion_ to commitment maybe, but not fear. I think he saw someone he either needed to get away from . . .\"\n\n\"Or chase after,\" Kylie finished for me.\n\nShe hopped off of my desk. There were a couple of sighs from the bull pen. Kylie ignored them.\n\n\"Sorry hon, you're on your own to pry that out of Rake.\" She flashed a dazzling smile to a few more sighs from the boys. \"But if you play your cards right, you could at least have fun doing it. And yes, _that_ pun was intended.\"\n\n\"Thanks, Kylie.\"\n\n\"Anytime.\"\n\n* * *\n\nDamnation.\n\nAbout an hour later, I didn't want a wall to thunk my head into, but I'd sure take one to use on Rake.\n\nKenji had gotten me into the databases I'd needed, but I'd done the digging myself.\n\nI'd hit pay dirt all right. The operative word there being \"dirt.\"\n\nThe Murwood _and_ the office building where the second murder had taken place earlier this afternoon were both owned by none other than Rake Danescu, under the name of Northern Reach Holdings. That made Jesin Nadisu\u2014with his kilo of Brimstone and Nightshade bullet\u2014Rake's employee. An employee who had looked ready to faint at the mention of his boss. On a hunch, I ran a search on the office building where Alastor Malvolia's supersized pocket dimension contained his law firm.\n\nYep, Northern Reach Holdings, aka Rake Danescu's personal property.\n\nJesin Nadisu's reaction could have been his morphine getting low or any number of sudden pains after having a sniper's bullet blast through his insides, but eyes don't lie. That wasn't pain; that was fear. As a result, I had several urges bubbling to the surface, but the front-runner was an overwhelming need to kick Rake Danescu's ass.\n\nThe goblin was capable of a lot, maybe even murder. Who was I kidding? Definitely murder. But what had been done to Sar Gedeon and Kela Dupari wasn't Rake's style. If he wanted someone dead, he'd just kill them, not make a B horror-movie production out of it. And then there was all that blood and the brimstone stink. I couldn't see Rake getting within smelling distance of a demon, let alone partnering with one. No dry cleaner could get demon stink out of a silk suit. Plus, my gut told me that his hand would never go fishing around in a chest cavity for a heart treat to toss to his demon accomplice. \"Innocent\" was the last word I'd use to describe Rake Danescu, but he wasn't the murderer.\n\nI knew in my gut the man whose silhouette I'd seen on the other side of that open portal had been the one to paralyze Sar Gedeon and the others while his demon used his claws to go grocery shopping. That silhouette didn't belong to Rake.\n\nMy desk phone rang.\n\nIt was the receptionist at Saga Partners Investments, our cover office on the surface. Rake Danescu was there to see me.\n\nSpeak of the devil. Pun and clich\u00e9 intended.\n\n\"Shall I tell Mr. Danescu that you're in a meeting?\" she asked.\n\nI smiled, though to the guys in the bull pen it'd look more like a baring of teeth.\n\n\"No, no. Not necessary. I would love to see Rake Danescu,\" I said. \"I'll be right up.\"\n\n* * *\n\nRake stood in the reception area of Saga Partners Investments, impeccably dressed, and looking uncharacteristically grim.\n\nGood. We were in the same mood. It'd save a lot of time getting past pleasantries if neither one of us had any.\n\nWhen he saw me, grim turned to guarded. He knew I was mad. At him. Yes, the last time we'd seen each other was across a table in a coffee shop when he'd been kissing the palm of my hand. Now he knew that if he went for my hand, I'd give him my fist.\n\nBut he didn't know why, hence, the guardedness.\n\nI was about to enlighten him.\n\nBut not here, not now.\n\nWhat I'd just discovered wasn't personal; it was business. Rake was now a suspect, if not of murder, then of drug running, or at the very least, collusion\u2014but most of all of being an asshole of a boss who terrified his employees. Until all of those had been thoroughly addressed, the one thing he was not was a potential boyfriend.\n\n\"Karen,\" I asked the receptionist, \"is the conference room available?\"\n\n\"Yes, it is.\"\n\n\"Would you put me down for half an hour?\"\n\n* * *\n\nI closed the door. The main Saga conference room was essentially an interrogation room with fancy seating. I fully intended to bring Rake downstairs for Ian and possibly Ms. Sagadraco to question, but first I had to confirm that there was justification to take that next step.\n\n\"Before we get started, I wanted to thank you for the flowers. They're beautiful, and they were the first pleasant surprise I've had in days. Now your turn. You're here, asking for me, and you're not happy. Why?\"\n\n\"You have my employee Jesin Nadisu here. Has he been arrested?\"\n\nInteresting. Rake didn't know he'd been shot.\n\n\"No, we're merely asking him a few questions.\"\n\n\"With an attorney present?\"\n\n\"There's no need for\u2014\"\n\nRake reached for his phone. \"I want him to have one. Anything he might have said to this point is inadmissible without an attorney present.\"\n\n\"I don't see why he would need one.\"\n\nThe goblin's dark eyes narrowed. \"Oh, you don't, do you?\"\n\nI wasn't taking the bait. But with that attitude, I didn't feel guilty tossing him a curve.\n\n\"Because we don't think _he_ is the one who's guilty\u2014at least not of murder.\"\n\n\"Murder?\"\n\n\"Murder. As to having a kilo of Brimstone on him . . .\" I shrugged. \"For all we know, he could have been holding it for a friend.\"\n\nRake paused, his long index finger poised above a key. Jeez, the guy had his lawyer on speed dial. I really hoped it wasn't Alastor Malvolia, though with Rake being his landlord, I wouldn't have been surprised.\n\n\"Unless _you_ think he's guilty,\" I continued, \"in which case, you need to seriously reevaluate your hiring practices, hiring psychotic serial killers. I'd have thought you would've been more careful about things like that, being a savvy and successful big-city businessman and all.\"\n\n\"Not guilty?\"\n\n\"That's what we think.\"\n\n\"Then why do you have Jesin Nadisu in custody?\"\n\n\"We don't have Jesin Nadisu in custody. We had him in surgery.\"\n\nThe goblin went dangerously still. \"What?\"\n\n\"He was shot outside a building that was the scene of the second murder in as many days. He had a kilo of Brimstone on him. We're analyzing it in the lab now. But it would help greatly if you'd care to tell us why the demons peddling the stuff are thinning out the competition by killing drug lords in your buildings?\"\n\n\"My buildings?\"\n\n\"Your buildings. You own\u2014and Jesin Nadisu manages\u2014the Murwood, scene of the first murder. Two hours ago, he was shot outside an office building on West Seventy-Ninth Street, aka murder scene number two, also owned by you. And the lawyer who represented both victims\u2014as well as probable future victims\u2014has his cozy pocket-dimension office in yet another of your buildings.\"\n\n\"Alastor.\"\n\n\"That's him. A real sweetheart. Met him this morning. You know, if you'd give us a list of all of your real estate holdings, maybe we could get ahead of the killers and keep Al from losing any more clients.\"\n\n\"You've been busy.\"\n\n\"I'm not the one hosting a demonic murder convention\u2014and terrorizing your employees.\"\n\n\"Terrorizing my . . . What the hell are you talking about?\"\n\n\"We were telling Jesin that he didn't have anything to be afraid of, that he was safe here. We tried to determine who he was afraid of. When we mentioned 'your employer' the poor kid damned near fainted. Imagine my surprise when I found out just now that he works for you.\"\n\n\"I can't imagine why he would be afraid of me.\"\n\n\"Can't you?\"\n\n\"No, I can't. Though at least I know why you're upset.\"\n\n\"I'm not upset, Rake. I'm about to become violent.\"\n\nHe exhaled heavily. \"How is Jesin and who shot him?\"\n\n\"He'll live. He's in recovery. Apparently the Nightshade sniper who shot him just wanted to clip him enough to justify using the fake ambulance waiting around the corner to come pick him up. Trust me, the poor kid will be a lot happier waking up here rather than wherever those elf ninjas would've taken him.\"\n\nRake swore and dropped into one of the chairs. Then he was silent, but there was a lot going on behind those dark eyes, mostly disbelief, confusion, and concern. \"Thank you,\" he finally said.\n\nOkay, that was unexpected.\n\nRake could have been pretending to care more about Jesin than defending himself, but I didn't think so. When I'd said \"Nightshade,\" he'd gone a shade or two paler than usual. You can't fake that; at least I didn't think so. I wasn't going to back down, but for now I decided to back off.\n\n\"Could you tell me what happened without compromising your investigation?\" he asked.\n\nPale _and_ polite. There wasn't any part of what had happened that Rake wouldn't be able to find out himself with a few questions in the right places, so I wouldn't get in trouble for telling what we knew, which wasn't really all that much.\n\n\"The NYPD arrived at the second murder scene before we could,\" I said. \"We got there and were staking out the location when we spotted Jesin leaving the building. He looked nervous.\"\n\nRake gave a halfhearted smile and shook his head. \"Contrary to what you may believe, goblins aren't born knowing how to conceal their emotions.\"\n\n\"So it's an acquired annoyance?\"\n\n\"Touch\u00e9.\"\n\n\"Jesin saw us and ran, there was a shot and he went down. Ian ran after the shooter, I stayed with Jesin until Yasha got back with the Suburban. We got him back here and into our ER as fast as we could.\"\n\n\"Thank you, again.\"\n\n\"You're welcome, but we were just doing our jobs. We would have done that for anyone.\"\n\n\"Even me?\"\n\n\"Including you. Though Ian might not have run so fast after the shooter. And Yasha wouldn't have been as gentle putting you into the Suburban. It probably would've been more like a quasi-aimed toss.\"\n\nThat got a slight smile out of Rake.\n\n\"Why would Jesin be afraid of you?\"\n\n\"He's not. At least he wasn't as of yesterday when he told me about you and Agent Byrne coming to the Murwood and about Sar Gedeon's murder.\" He spread his hands in exasperation. \"I have no idea why he would react like that. I'm his favorite uncle.\"\n\n\"Uncl . . . he's your _nephew_?\"\n\n\"Yes. He's extraordinarily bright, a hard worker, with an uncanny knack for business. He's also one of the few in my family whom I actually like. When he came here, he wanted to work, so I put him in charge of the Murwood. The boy has a head for management.\"\n\nI couldn't resist. \"What? You didn't have him working at Bacchanalia?\"\n\n\"His mother, my oldest sister, would skin me alive herself.\"\n\nOldest sister. That implied more than one. \"Sounds like a nice lady.\"\n\n\"You have no idea. She was against him coming here from the beginning. I will have much explaining to do.\"\n\n\"If Jesin wasn't afraid of you yesterday and is today\u2014\"\n\nRake's eyes tightened in disapproval. \"He wasn't carrying a kilo of Brimstone from the scene of a murder yesterday.\"\n\nThere was a soft knock. The door opened a crack and Karen stuck her head inside.\n\n\"Excuse me, Agent Fraser, Mr. Danescu, but Ms. Sagadraco would like to see you both.\"\n\nI couldn't say I was surprised. Karen probably had a standing order to let the boss know whenever an agent asked to use a conference room for a chat with a perpetual suspect.\n\nI wasn't in the least bit nervous. Alain Moreau had already told me that he and Ms. Sagadraco were very pleased with my job performance, so I wasn't having another what-the-hell-did-I-do moment.\n\nRake, on the other hand, had probably done, initiated, been directly or indirectly responsible for, or merely involved in so many nefarious activities that the concerned crease on his forehead was from trying to figure out which one all this was about. Was it about Jesin, Jesin's reaction to the mention of him, the murders, or Brimstone? Or something else entirely? I guess it was hard to cover your ass when you had so many irons in the fire.\n\nWe got into the elevator and the doors closed.\n\nBoth of us faced forward, neither saying a word.\n\n\"Makenna?\"\n\nI noticed he left off the \"dearest,\" \"lovely,\" or \"beautiful\" Makenna. Wise move.\n\n\"Yes?\" I asked.\n\nOut of the corner of my eye, I saw Rake's lips twitch upward at the corners.\n\n\"Well played.\"\n\n# 17\n\nALAIN Moreau met us when the elevator opened on the executive level.\n\nOkay, maybe I was in a little bit of trouble. Rake was a suspect, or at the very least a source of information we didn't have but needed, and I had shared elements of an ongoing investigation. I'd be finding out soon enough.\n\nVivienne Sagadraco's office door was open, a table had been brought in, and there was her formal silver tea service, china cups and saucers, and those little cakes and pastries from Kitty's that always looked too pretty to eat. Either the boss wanted to lull Rake into a false sense of complacency, or we were going to be here for a while and we'd need caffeine and sugary snacks to get through it\u2014Ms. Sagadraco's version of a civilized interrogation. I knew Rake could never be lulled into anything, but the boss was civilized, so I was going to go with the latter.\n\n\"Rake.\" Not Lord Danescu. Ms. Sagadraco said it as though the hand caught in the tea-cake jar had been his. But she still extended her hand for the requisite kiss. Rake didn't disappoint. \"You could not have paid us a visit at a more convenient time.\" Her sapphire-colored eyes narrowed ever so slightly. \"This chat is long overdue. Agent Byrne will be joining us momentarily. Alain, would you please close the door and ensure that we are not disturbed?\"\n\nAlain gave Rake a level stare. \"It would be my pleasure, madam.\"\n\nSo much for who was in trouble, or at least more of it than I was.\n\nIf there was tea involved, Vivienne Sagadraco would make pleasant small talk until everyone had been served. But now as she poured the tea, she made no effort at conversation: small, pleasant, or otherwise. She was like a Southern lady in that regard: if you didn't have anything nice to say, don't say anything at all\u2014at least until the individual in question wasn't around to hear you.\n\nYep, this was going to be a civilized interrogation.\n\n\"In the interests of complete disclosure,\" she began, \"anytime an agent asks to use the Saga Investments conference room, video and audio recording is activated for the duration of the meeting.\"\n\nSo Karen had not only told the boss, she'd flipped the AV switch. I was kind of glad that she had. I wouldn't have wanted to summarize _that_ exchange for Ms. Sagadraco.\n\nThe door opened and Ian came in.\n\n\"Have a seat, Agent Byrne. We were just getting started.\"\n\n\"The lab's completed the first part of their analysis,\" he told her. \"Dr. Cheban sent the preliminary report to you. They're still isolating the individual ingredients, but they've determined enough to know what the drug is supposed to do. She'll forward the ingredient list as soon as it's complete. Though she did confirm that one of the ingredients is actual brimstone. And one of the murderers is a demon lord.\"\n\nI glanced at Rake to get his reaction.\n\nOne perfect eyebrow, slightly raised. He may have been shaken, but he wasn't stirred.\n\nMs. Sagadraco reached over to her desk for her tablet. She scanned through her e-mails and opened the report. We were silent as she read.\n\nIan typed a few words on his phone, then tilted it so I could see: Saw the tape. ; )\n\nJeez, had they been playing it in the break room? I didn't know what the boss thought, but Ian approved\u2014or at least he'd found it entertaining.\n\n\"Without the benefit of further testing, does Dr. Cheban believe the drug does what was intended by whoever made it?\" she asked Ian.\n\n\"Yes, ma'am.\"\n\n\"How many doses are contained in the sample we obtained from Mr. Nadisu?\"\n\n\"Hundreds.\"\n\nRake made a low sound in the back of his throat.\n\nI didn't know if he'd intended it as a groan or a growl, but either way I think I understood why Jesin wouldn't want to see his uncle right now.\n\nMs. Sagadraco finished reading and put her tablet on the table next to her cup. \"Lord Danescu, in our laboratory is an impressive quantity of a drug that our chemists believe would enable any elf or goblin who inhaled it to see through glamours and read minds.\"\n\n\"Damn,\" Rake muttered. \"They did it. They actually did it.\"\n\n\"I take it you were aware of their efforts?\"\n\n\"I'd heard rumors about their efforts, but nothing about their success.\"\n\nIan spoke. \"According to Dr. Cheban, the drug works for elves, goblins, and humans. However, and fortunately for us, humans don't remember what they saw while under the influence.\" My partner gave Rake a less than friendly look. \"Goblins and elves would recall everything, which leads Dr. Cheban to believe that it was developed for use by either goblins or elves. For humans who aren't aware of the supernatural world, seeing through glamours could easily be misinterpreted as hallucinating and thinking you were seeing monsters, which is what happened in the restaurant yesterday.\"\n\nMs. Sagadraco took a sip of tea, and then carefully set the cup and saucer on the table, leveling her gaze on Rake Danescu. \"I believe it is time that you told us what you know.\"\n\nThe goblin had put his elbows on the arms of the chair and had carefully interlaced his fingers in front of him. Interrogation Posing 101.\n\n\"I had heard that the elves were attempting to develop a drug that could enable them to spot any undercover goblin agent by sight, and detect any goblin spies or elven traitors by thought. Conversely, it would also let goblins see and hear any elven agents.\"\n\nIt was said that elves and goblins originated from the same ancestors. Just never say that out loud to either one. Hate was a mild word for how most elves and goblins felt about each other. At least in our dimension they'd stopped trying to exterminate each other, settling instead for hostile corporate takeovers\u2014with only minimal bloodshed.\n\n\"I imagine both goblin and elven intelligence would give or do anything to get their hands on the Brimstone formula,\" Ian noted coolly.\n\nRake didn't take the bait. \"If it was a stable and viable formula, then yes, there would be considerable interest.\"\n\n\"And competition.\"\n\n\"What are you getting at, Agent Byrne?\"\n\n\"Only that you appear to be in a unique position to hear of any interest or competition\u2014and possibly even have a member of your family unwillingly pulled in.\"\n\nNice that Ian gave Jesin the benefit of a doubt. Might have even earned a point or two with Rake.\n\nAt that, Rake regarded Ms. Sagadraco, his expression unreadable. \"Vivienne, I would be willing to share what I know in exchange for my nephew's safekeeping here.\"\n\nWhat that implied about the situation in the city didn't bode well for any of us.\n\nBy all accounts, Rake was one of the most powerful dark mages in New York, perhaps _the_ most powerful. For him to ask for help protecting his nephew by keeping him in here meant the situation out there was even more dangerous than we could have imagined.\n\n\"While we were upstairs, Makenna alluded to a connection between the murders and properties I own, and by association, myself. Her suspicions may not be unfounded. If so, at this time, Jesin would not be safe outside of this complex. I know that Sar Gedeon was the first victim. Who was the second?\"\n\n\"Gedeon wasn't the first victim,\" Ian told him. \"The killers started at the bottom of the ladder and have been working their way up. The most recent victim\u2014at least that we know of\u2014is a goblin by the name of Kela Dupari,\" Ian said.\n\nRake closed his eyes for a moment.\n\n\"I take it you knew her?\"\n\n\"A foolish woman who routinely toyed with and taunted powers beyond her ability. The same actions can be ascribed to Sar Gedeon.\"\n\n\"What are their connections to you?\"\n\n\"Both were actively involved in the drug industry, not only in New York but down the entire East Coast to Miami. Contrary to what you may believe about me, I am _not_ involved\u2014actively or otherwise\u2014in any drug industry. Aside from Ms. Dupari and I both being goblins, we have no connection or association. That being said, an elf and a goblin, both prominent in locally based crime families, were brutally murdered in buildings that I own. This could be a coincidence, or an attempt to frame me, or at the very least cause me substantial inconvenience and embarrassment.\"\n\nI stared at him in disbelief. \"People get their hearts and souls ripped out and you're embarrassed because it happened in your buildings?\"\n\n\"Yes,\" he said matter-of-factly. \"You must admit neither death was unwarranted given their past professional activities, and their removal will no doubt make the city a safer place.\"\n\n\"So now you're playing Batman?\"\n\n\"The costume would suit me, as would the nighttime activity, but no.\"\n\n\"Your nephew can stay here regardless of what information you share or do not share with us,\" Ms. Sagadraco said. \"We will care for him as if he were one of our own. It's called decency. I know you're at least familiar with the concept. Your cloak-and-dagger dramatics are affecting and endangering others, and one of those others is Makenna. I assume you have heard what happened yesterday afternoon?\"\n\nHis expression hardened. \"I did.\"\n\n\"My agents are charged with protecting the supernaturals and humans of this city from any and all threats. Brimstone is a threat\u2014both its manufacture and the battle among opposing forces for the right to sell it. My agents and the people of New York are caught in the middle. I arm my agents with what they need to do their jobs. A vital part of that armament is information. I believe you have this information, if not all of it, at least more than we have.\"\n\n\"Mac could have been killed yesterday in the same way as Gedeon and Dupari,\" Ian told Rake, his tone low and forceful. \"Or even worse, dragged through that portal.\"\n\n\"I am more than aware of that,\" Rake shot back. \"Which is precisely why we're having this conversation. Vivienne, from what I _do_ know, neither you nor your people want to be involved in this. It is beyond their abilities.\"\n\n\"This, as you so obliquely put it, is precisely why I founded SPI. This is my world, Lord Danescu. I live here, _all_ the time. I will defend it to my last breath. Can you say the same?\"\n\nSilence.\n\n\"As to my agents' abilities, I know their capabilities, you do not. You know what they would face, I do not. You tell me what is happening in this city, and I will be the one most qualified to make that assessment. Though from what I know of my agents, you have seriously underestimated them.\" She glanced at me. \"All of them. You have yet to choose a side. It's understandable. One is the world of your birth, ours is merely a place of business.\"\n\nRake recoiled as if Ms. Sagadraco had slapped him, which I think was what she was going for.\n\n\"Or is it?\" she continued. \"If you have not made up your mind, it is time that you do so. You can help, or you can continue to hinder. You can no longer do both. Which will it be?\"\n\n\"Very well.\" Rake leaned back in his chair. \"Elves have been in your dimension far longer than goblins. Their established foothold forced us to play catch-up, strategically speaking.\"\n\nIan barked a humorless laugh. \"Strategically speaking? You make it sound like you're planning to take over.\"\n\n\"Not take over, Agent Fraser. Merely ensure that the elves don't gain access to a resource\u2014and thus an advantage\u2014that we do not gain for ourselves. Much of what is called magic in the Seven Kingdoms can be replicated by science and technology here. Some cannot be replicated. Elven extremists have worked to gain power and influence here to obtain such technology for use against my people. It pains me to say it, but there are similar groups among my own race. Goblins and elves have been at war off and on for thousands of years.\"\n\nI was dumbfounded. \"You're saying elf-terrorists-trying-to-get-nukes kind of advantage?\"\n\nRake actually smiled. \"That would be extreme even for these people. They want to annihilate the goblin race, not render their kingdom uninhabitable. In their minds, that would be a waste.\"\n\n\"Thank God for small miracles.\"\n\n\"Both of our races use the excuse that we're merely trying to stay ahead of the other to protect our own people.\"\n\nIan sat back. \"Brimstone's the source of the latest tug-of-war.\"\n\nRake nodded. \"There are more than a few companies and laboratories run by both elves and goblins that develop drugs, weapons, and technologies to use against the other. Such organizations are routinely infiltrated to steal formulas, sabotage research, copy new technology. Much like human industrial espionage. Brimstone would allow select people to see through the glamours these corporate spies use to hide their identities, as well as detect spies by their thoughts.\" Rake poured himself another cup of tea. \"Brimstone would be a valuable commodity for whoever has it. If it is effective, it would be worth killing for. From events of the past two days, apparently the drug is quite effective.\"\n\n\"Elves, goblins, and vampires have been killed by a demon lord and something worse than a demon lord,\" I said. \"So who made the drug?\"\n\n\"I suspect the individuals you need to pursue are not those who are physically manufacturing the drug. At least they wouldn't be your primary target. Brimstone\u2014the ingredient itself\u2014comes from the Hell dimension, making it particularly difficult to get.\"\n\n\"Not necessarily. Marty picked up a couple of rocks on a field trip,\" I noted.\n\nRake's teacup paused halfway to his lips. \"I beg your pardon?\"\n\nIan regarded the goblin with a knowing expression. \"You know a lot about spies, espionage, and strategic advantages for a billionaire playboy, real estate mogul, and owner of an exclusive sex club.\"\n\nRake almost smiled. \"Successful and undetectable espionage isn't cheap, Agent Byrne. Some of the buildings I own have been leased to elven companies and research facilities. I made the financial terms and incentives impossible to pass up\u2014as would any developer vying to get a profitable client in a previously vacant building. Refitting the space to suit their needs presents all sorts of opportunities for installing undetectable surveillance equipment. The income from one building often pays for another; and the revenue from my other businesses funds the buying and bugging of those buildings. As you humans say, sex sells. It also makes an absurdly impressive amount of money. More than a few key elven power brokers spend time\u2014and their money\u2014in my club, little knowing that they're funding intelligence operations against themselves.\"\n\nRake Danescu. Sex broker and spymaster. I didn't know which was worse.\n\nOr if either one was truly bad.\n\n\"And you have bugs planted in the tables at Bacchanalia,\" I said, recalling my first night on the job when Ian had felt the need to distract those listening in on us by seriously distracting me.\n\n\"One can hear all kinds of interesting and valuable tidbits,\" Rake noted smoothly, knowing exactly what I was remembering.\n\nBastard.\n\nI glared at him. He smiled at me.\n\n\"Heard any interesting chatter concerning a new drug?\" Ian asked.\n\n\"Not that my monitors have told me, but I will contact them when I leave here.\"\n\n\"And let us know?\"\n\nRake just looked at him. \"Yes, Agent Byrne. And let you know.\"\n\n\"I found a list of buildings that you own under Northern Reach Holdings fairly easily,\" I told him.\n\n\"Which is what makes me think the murders taking place in my buildings isn't a coincidence. I have allowed Northern Reach Holdings to trace back to me with relative ease. My other holdings are very well and deeply hidden. It's in a goblin's nature to hide your strategic advantages until they're needed\u2014or until you need someone to find them.\"\n\nGreat. So much for me being clever.\n\n\"So Northern Reach is like the outer threads of a spiderweb,\" I said. \"You're in the center, and if you sense movement, you know you've caught something.\"\n\n\"A nearly perfect comparison, Makenna. That is why I believe there is a very distinct possibility that someone is going to a lot of trouble to stage murders in my buildings.\"\n\n\"So I take it the elves know you're a spy?\"\n\n\"I'm more of a freelance consultant for goblin intelligence. They use me, and I use them. It's a mutually beneficial relationship.\"\n\nI remembered what Kylie had told me. I wanted to know the answer; not to mention, Rake had just gotten one up on me. I'm competitive, so sue me. \"In the coffee shop yesterday, you needed to leave fast to keep Baxter Clayton from seeing you.\"\n\n\"That is correct.\"\n\n\"It also wasn't necessary, at least not anymore.\"\n\n\"I don't follow you.\"\n\nOh yes, he did.\n\n\"The series Baxter Clayton was planning had its plug pulled last month,\" I said for Ian and the boss's benefit; Rake already knew damn well that it'd been canceled. \"You didn't really need to avoid him anymore. Though having heard more than a few Baxter stories from Kylie, I could understand why people wouldn't want to get cornered. But with the series canceled, you didn't _need_ to avoid him.\" I eyed him. \"Sitting at the center of the web like you do, I can't imagine you not knowing the series had been canceled. So that would mean that you were either avoiding someone else\u2014or you saw someone who was desperate to avoid you. Which was it?\"\n\n\"It had nothing to do with Brimstone.\" Rake's dark eyes were steady on mine. Eyes that said in no uncertain terms that he was not going to tell me or anyone else here what it was about.\n\nIf the boss had had a fireplace in her office, I'd have held his feet to it. Not only did I think she wouldn't have minded, but since she was a fire-breathing dragon, she could've done it herself. I glanced at her. From the hard glitter in her eyes, it looked like she wouldn't mind raising the temperature in the goblin's designer shoes.\n\n\"Rake, do I have your word that this incident isn't connected to this investigation?\" she asked.\n\n\"You have my word.\"\n\n\"And if it does reveal itself to be connected, I trust you will inform me immediately.\"\n\nMs. Sagadraco knew how the goblin mind worked.\n\n\"Of course, Vivienne. I will contact you immediately.\" He looked to each of us in turn. \"I have a question.\"\n\nMs. Sagadraco selected a pastry from the silver tray. \"Please ask it.\"\n\n\"Makenna mentioned that the two of you met with Alastor Malvolia this morning,\" he said to Ian. \"What were you attempting to learn from him, and were you successful? Though knowing Alastor, I would hazard to guess that you weren't, at least not after only one meeting.\"\n\n\"We're supposed to hear from Malvolia by eight o'clock tonight,\" Ian told him. \"But we're not holding our breath.\"\n\n\"Nor should you,\" Rake said. \"If his clients were able to give him any information he believed was useful to you, he would want to negotiate for additional benefits. What did you promise him?\"\n\n\"Not a damn thing,\" Ian said bluntly. \"I simply told him how Sar Gedeon was killed. In detail. He decided to cooperate.\"\n\n\"I would have enjoyed seeing that.\" Rake took a sip of tea, his dark eyes glittering with what I could only describe as delight over the rim of his cup. \"Dearest Vivienne, you are quite right, I have underestimated your agents.\"\n\n# 18\n\nONCE Ms. Sagadraco was finished with her tea-party inquisition, and extracted a promise of cooperation from Rake, she asked me to stay after Ian and Rake had been excused.\n\nThe tea and goodies were gone, but maybe the inquisition part wasn't over yet\u2014at least not for me.\n\nI decided to be proactive. \"You want to talk about what happened up in the conference room with Rake, don't you?\"\n\n\"I thought that would be a good idea, yes.\"\n\n\"From what you saw and heard, I didn't mess anything up, did I?\"\n\n\"If you're speaking professionally, no, you did not. What I wanted to bring to your attention is on a more personal note. You may have created more of a problem than you solved.\"\n\n\"I don't understand.\"\n\n\"If your intention was to discourage Lord Danescu from pursuing you, then you may have made a tactical error.\"\n\nMy eyes widened. \"What?\"\n\n\"Goblin men of Rake's caliber aren't attracted to intellectually passive women. If I were to venture a guess, I would say that your performance just now and upstairs has probably rendered you absolutely irresistible. If you want him to cease his attentions now, you may have to kill him.\"\n\nI recalled my violent urges toward Rake in the conference room, and thought it highly likely that before this was all over, I'd be feeling those same urges again.\n\n\"The day ain't over yet, ma'am.\"\n\nThe Dragon Lady smiled.\n\n* * *\n\nIan met me by the elevators.\n\n\"Well, that was interesting,\" I said.\n\n\"That's one way to put it.\"\n\nI waited a few moments before I spoke again. \"You don't believe Rake's involved anymore, do you?\"\n\nThe muscle in my partner's jaw flexed. \"No, I don't.\"\n\n\"But you'd like him to be.\"\n\n\"If it'd keep more people from dying, then yeah, I would.\"\n\n\"Nice dodge. That's not the question I asked.\"\n\nIan grew some silence.\n\n\"I'm still worried about you, Mac.\"\n\n\"Rake or being snatched through a portal?\"\n\n\"Yes. I can keep the second one from happening, but not the first.\"\n\nMy instinct was to tell him that I could take care of myself and that I didn't need his help or approval choosing the men in my life. But I didn't say any of that even though all of it was true. It wasn't Ian's fault for feeling the way he did about Rake, or any other man who kept his private life, business interests, and motives for nearly everything he did locked up tighter than Fort Knox.\n\nHeck, I was still circling Rake like he was a rattlesnake coiled in front of the only way out of a cave, and I planned to do that for the foreseeable future. When I got to the future, and Rake still wasn't guilty of anything, then I'd reevaluate my reasons for continued caution.\n\nNot blaming Ian one bit for his feelings left me with only one response to his statement. It was also the one I wanted to give him.\n\n\"Thank you,\" I said simply.\n\nThat earned me a surprised look.\n\n\"Really,\" I added with a slight smile. \"There's no need to worry. I don't plan on diving into anything, but I know where your concern's coming from, and I know it's a good place. So thank you.\"\n\n\"It's not _your_ plans I'm worried about.\"\n\nI grinned. \"You sure you aren't part Southern? Sounds like you don't think my gentleman caller has honorable intentions.\"\n\n\"Rake Danescu is no gentleman, and it's beginning to look like honor isn't a concept many goblins are familiar with.\"\n\n* * *\n\nIn preparation for a meeting, Ian had rolled a big whiteboard into a conference room just off the bull pen that the Brimstone team had taken for our own. Photos of the victims were posted across the top of the board, with crime family affiliation listed beneath.\n\nThe NYPD had probably started a board like this, though theirs would only have three bodies, and there wouldn't be two additional bullet points under each victim's name noting their species and missing soul. And there certainly wouldn't be an asterisk next to Sar Gedeon's \"missing soul\" bullet indicating that \"The agency necromancer attempted contact but was bitch slapped by a demonic booby trap for trying.\"\n\nFortunately there was only the one asterisk.\n\nAll of the victims had had their chests sliced open with a scalpel-type instrument. Or claw. Their hearts had been torn from their chests, all while alive with a demon lord holding them to the floor with one hoof, branding its imprint into their breastbones. None had died without a struggle.\n\nThe details of Kela Dupari's murder had already been leaked online, and the public and press were having a field day, especially when they found out that her murder hadn't been the first. The NYPD had been the first to arrive on the scene of two more murders: one late last night, the other about the same time as Dupari's killing. As the medical examiner\u2014and a mage\u2014Dr. Anika Van Daal had kept Dupari's goblin features hidden. Either the two most recent victims were human, or Van Daal\u2014or one of her people\u2014had concealed the pointy ears from curious eyes, because there'd been no mention of odd ears or silvery skin, only missing hearts and branded hoofprints.\n\nWhen a murder was particularly gruesome, it didn't matter what security measures the city's medical examiner's office had in place, juicy details always found their way out. And they found their way onto home pages and front pages even faster when there'd been more than one murder with the same lurid MO\u2014and a photo. Yep, the person who'd stumbled onto the most recent body had taken plenty of pictures before they'd called the cops. Nowadays you didn't have to commit a crime to get your fifteen minutes of fame, just be the first person to take a picture of it.\n\nI was a big fan of the Internet, and it had it uses\u2014like the glorious world of online shopping\u2014but right now it sucked. As recently as a couple of decades ago, the three networks (yes, that was all there were) would have had it on their evening news, and the newspapers would have gotten hold of it, and it would spread only as far as their signal or circulation.\n\nNot anymore.\n\nAll it took was one tweet to turn a secret into worldwide news. Send that tweet with a bloody photo of a gaping chest and hoofprint brand, and within five minutes it'd have its own trending hashtag.\n\nOur not-so-secret-anymore secret had garnered itself three hashtags at last count: #devilmurder, #killerdemon, and\u2014my personal favorite for sheer dramatic impact\u2014#SatanInNY.\n\nI sighed. This was going to be a very long day.\n\nCrackpots, conspiracy theorists, religious nuts, and the tin-foil-hat crowd had started coming out of the woodwork, and Kylie and her department were busy as hounds in flea season.\n\nSo far, the focus was on the sensationalist details, which fortunately didn't out any supernatural creature the public didn't already know about. Nearly every major religion had more than its share of demons or devils, many of them even named. Hearing that one was making it his mission on Earth to slaughter people in the illegal drug trade was being met with cheers, not panic in the streets. Panic and terror were reserved for those in the illegal drug trade. The opinion of the average Jane and Joe on the street was \"Good riddance!\" and \"Give that demon a medal.\"\n\nRight now, Kylie O'Hara was doing the rounds of the local news programs as the founder of the internationally known website hoaxbusters.com. She'd made a name for herself online and beyond as a debunker of the supernatural. Heck, Syfy was still after her to host her own show. However, her \"secret identity\" job was SPI's director of media and public relations. The goal of both of her jobs was to have a mundane explanation for supernatural events and creatures. With the latest in CGI technology available to any kid with a computer, exposing anyone looking for their fifteen minutes of fame had never been easier. That being said, those photos of the latest victim hadn't been faked. Kylie had readily confirmed that. However, she'd added that they didn't need to be faked to be explained. There was a killer on the loose in New York. Unfortunately, that was nothing new. This one simply limited its work to a subset of criminals, and for some reason known only to it, carried a branding iron and liked to cut out hearts. That didn't indicate supernatural, just a deeply disturbed individual.\n\nKylie was doing a fine job of doing her job. It'd be nice if Ian and I could say the same. A board full of the names of dead drug dealers didn't equal success; it just meant we were organized. Success meant putting that demon lord and his mage partner permanently out of business.\n\nOur friendly neighborhood source inside the NYPD's drug enforcement unit, Detective Fred Ash, stopped by to share what they knew with us. The NYPD didn't know about SPI, but with supernaturals on the force, we had eyes and ears where we needed them. What Fred's eyes and ears had seen and heard in the last few hours was a bombshell to us; like we hadn't had enough of those ourselves.\n\nIan was incredulous. \"They want to do _what_?\"\n\nI was a mite stunned myself.\n\nThe NYPD was going to put the city's top drug lords and ladies under protective custody.\n\n\"Yeah, protecting the people who no one really minds seeing dead,\" Fred told us. \"Makes all kinds of sense. They're all drug-dealing, murdering, lowlife scum. But apparently they're _our_ drug-dealing, murdering, lowlife scum. Most importantly, drug kingpins are taxpayers, too. Taxpayers who haven't been convicted of a crime. In the eyes of the law that makes them innocent taxpayers. Gotta protect all of them.\" Fred took another bite of doughnut. \"This case is just chock full of irony.\"\n\nBefore coming over for a fact-sharing session with us, Fred had made a Krispy Kreme run. And before coming to the bull pen, he'd taken two raspberry-filled doughnuts up to Bert in his office. Our necromancer didn't want to let on, but he still wasn't back in fighting shape from the trap the murderer had set in what had been left of Sar Gedeon's mind. The favorite doughnut of the guy who worked with dead people was filled with gooey red stuff. Go figure.\n\nFor a Southerner like myself, Krispy Kremes were the holy grail of doughnuts. And when the \"HOT\" light on the sign was lit, that meant the sugary-glazed goodness had just come out of the oven. The first couple of bites would melt in your mouth. In my family, we held to the rule that the fresher the doughnut, the fewer calories they had. Fluffy when passing the lips, no fat on the hips.\n\nI knew it wasn't true, but I'd never let scientific facts get in the way of enjoying a good doughnut.\n\nI snagged a chocolate-iced one before they got gone. \"I'd ask if you were pulling our leg, but I know you're not.\"\n\n\"I think the big problem with the city hall people is the way the city's not-so-law-abiding citizens are getting killed,\" Fred noted. \"Chest branded, heart cut out, stink of hellfire and brimstone.\"\n\n\"Technically, it's just brimstone,\" I said. \"Hellfire doesn't stink.\" Jeez, I was starting to sound like Marty.\n\n\"Whatever. It's our job to make it stop. Now.\"\n\nIn addition to doughnuts, Fred had brought news of another murder. It had been committed on a yacht in the Hudson River. The NYPD had gotten to that one first, too. In our defense, the NYPD had an advantage\u2014we didn't have patrol boats on the rivers and in the harbor. And screams coming from an obscenely expensive mega-yacht wasn't something a patrol boat full of cops was likely to ignore.\n\nIt'd been a vampire. A high-ranking member of the B\u00e1thory family. Celeste B\u00e1thory had gotten scared and taken refuge on her yacht. She obviously hadn't heard that portals can be opened anywhere.\n\nThat murder scene had a deviation from the others\u2014the heart hadn't been stolen and\/or eaten; the vampire's heart had been staked to the teak wood floor next to her body.\n\nWe could now add \"dark sense of humor\" to the murders' descriptions.\n\n\"B\u00e1thory's people had checked every square inch of that boat,\" Fred told us. \"There was no one there but them. One swears Celeste B\u00e1thory had him check her cabin. Hell, I think she'd have had him looking under the bed if the thing had an underneath. Half an hour later, no sounds at all, the guard posted outside her door saw blood soaking the carpet under the door. Didn't hear a peep, no struggle, nothing.\"\n\n\"He smell sulfur?\" Ian asked.\n\n\"Yeah, seemed to be coming from under the door. That's what made him look down. Wards had been set and locked. She even had battle mages on board, real heavyweights. Likewise, they didn't hear or sense a thing.\"\n\n\"Damn.\"\n\n\"Yeah, our gruesome twosome are good. They've got the local pharmaceutical distributors about to crap themselves at seeing their own shadows. Whatever they do to protect themselves, it's not enough. By the time B\u00e1thory's boys got through that door, it was all over except the cleanup\u2014and according to our guys that was some cleanup.\" He dug a manila folder out of his messenger bag. \"I brought you two eight by ten glossies of Celeste for your board\u2014undead and permanently dead.\" Fred read the \"bitch slapped\" comment next to the asterisk and chuckled. \"Who wrote that?\"\n\nI raised the hand not holding the doughnut.\n\n\"I like it,\" he said.\n\n\"My journalism degree at work.\"\n\n\"Your mom would be proud.\"\n\n\"I think so.\" I also thought I was getting the hang of using dark cop humor to relieve tension. Fred Ash was my spirit animal. Besides, Bert wouldn't mind; he'd laugh his ass off. In fact, I'd written it for him.\n\nIan added the photos to our board. \"You're a sick man, Fred.\" He gave me a look. \"And you're an enabler.\"\n\n\"Never claimed to be anything else,\" Fred said.\n\nI popped the last bite of doughnut in my mouth. \"Ditto.\"\n\n\"So what'd the first responders have to say about the heart staked to the floor?\" Ian asked.\n\n\"With that and her fangs, their first thought was vampire,\" Fred said. \"Their second thought was there's no such thing as vampires, and that Celeste must have had some kinky dental work done. My momma\u2014and my first sergeant\u2014always said that first impressions are important. Our boys and girls should've gone with what their gut was telling them they were seeing.\"\n\n\"They can't charge something with murder that doesn't exist,\" Ian noted. \"And as far as the NYPD is concerned, demons don't exist.\"\n\nFred snorted. \"Yeah, I'd like to see my precinct try to put one of those in the holding tank.\"\n\n\"They can try to protect those people all they want,\" Ian continued. \"It's not going to do any good, even if they believed the reason why. I wish them luck. If they could stop the killings, more power to them. But they won't because they can't. They can't because their minds won't let them believe. Their lizard brains know what's happening, but then they'll look at the modern city they live in and the primitive truth they know in their gut gets pushed aside. They're looking for mundane explanations, and this is magic, black and as dirty as I've ever heard of.\"\n\nFred waved his second doughnut. \"Yeah, yeah, I know. You're preachin' to the choir. The killer is a dark mage with a demon sidekick, who's using a portal to get in and out, so my brother and sister officers are gonna be seriously frustrated. The classic murder in a locked room. Damned media's already calling it the perfect crime.\"\n\nThe tabloids like the _Informer_ , which I used to work for, had gotten it right. Well, partially. Demons were involved, and while it was ironic that all of the victims were involved in the illegal drug trade, there was nothing divine about the retribution. Though it would be nice if God would do a little judicious smiting every now and then. Glean out the troublemakers. The world would be a better place for the rest of us.\n\n\"You can count us among the frustrated,\" Ian was saying. \"With your guys tailing our most likely future victims, we can't do the same ourselves. Though it's not like there's much else we've been able to do.\"\n\n\"Normally multiple vics killed in a freaky way means serial killer,\" Fred noted. \"Make those vics connected to some of the biggest names in the city's drug trade, and the folks downtown think we're seeing the beginnings of a drug turf war. New dealers come in, want a slice of the business for themselves, our existing drug lords and ladies say hell no, and the newcomers start making examples to get them to change their minds. Which is surprisingly close to the truth except it's the established lords and ladies who want a piece of the newcomers' action.\"\n\nThe NYPD had had Kela Dupari's home and office under tight surveillance due to an ongoing investigation that had nothing to do with Brimstone. When her body was discovered in her office with the heart missing and the chest branded, all of the doors and windows had been locked. The surveillance cameras from the building showed that no one other than Kela Dupari had entered or left the office.\n\nThe NYPD was stumped, embarrassed, and getting pissed.\n\nWe were just pissed.\n\nThe NYPD thought the killings were a new cartel moving in to make a name for themselves by simultaneously slaughtering the kingpins while scaring the bejeezus out of the survivors\u2014or as we were beginning to think of them, \"future heart donors.\"\n\nAside from the demon and actual Hell elements, Fred was right, they'd pretty much hit the nail on the head.\n\n# 19\n\nWHEN Dr. Cheban and her team released their final report two hours later, I seriously doubt there was any high-fiving in the lab.\n\nIt was bad enough that one of the ingredients in Brimstone was actual brimstone from Hell, but it was the form of the brimstone that turned just another evening at SPI into all hands on deck.\n\nAt least the hands experienced with portals and demons\u2014finding the former and battling the latter.\n\nThe brimstone in the drug had been combined with the other ingredients while still in its molten state. Martin DiMatteo's samples were rocks, dried and old. We were dealing with brimstone fresh from Hell itself.\n\nAccording to Marty, fresh, molten brimstone could be obtained from only one location.\n\nWe had a Hellpit open somewhere under New York.\n\nSome people would say that New York was the modern equivalent of Sodom and Gomorrah, and more than deserved to have a Hellpit gaping open under it, and the sooner it fell in, the better for everyone else. Others would argue that dishonor went to Las Vegas. I'd have to disagree with both. The majority of New Yorkers were the best folks you'd ever want to meet. I'd never been to Vegas, but since the place had gone and gotten itself Disneyfied, I figured they were out of the running.\n\nRegardless of how those who didn't live in New York felt about the Big Apple, it didn't need or deserve to be swallowed into the bowels of Hell.\n\nOur job was still the same.\n\nFind it and close it\u2014without anyone finding out.\n\nAnd ensure that what happens in a Hellpit, stays in a Hellpit.\n\nWe'd had food sent up from the cafeteria, with Martin DiMatteo hosting a Lunch 'n' Learn on Hell and demons. Though since it was eight o'clock at night, it was dinner, but the concept was the same. Most of us hadn't had time for dinner yet, and with the Hellpit news, we were going to sit down and eat while we could. Fred had had to leave, but we promised to fill him in on what was said and decided.\n\n\"I know you say there is a Hellpit,\" I was saying to Martin DiMatteo. \"And it's _here_.\"\n\n\"Correct.\"\n\n\"The demon lord and his mage partner are launching their attacks here through a portal from a dimension _close_ to Hell, because there's no direct access to our dimension from Hell.\"\n\n\"Also correct.\"\n\n\"Then why can't the Hellpit be physically located in the same neighboring dimension where they're launching their attacks from rather than here, and the brimstone brought in through a portal? I'm not doubting your expertise,\" I hurried to add. \"I just want to understand what's going on and why.\"\n\n\"Never apologize for seeking knowledge, Agent Fraser.\"\n\nOur director of demonology didn't seem to be offended. Quite the opposite, in fact. I'd just asked him to talk about his favorite topic; no one minded doing that. Though asking did give me a bit of an unpleasant flashback to asking Bert for an explanation of what he'd been about to do with Sar Gedeon's corpse. I didn't like what I'd heard and seen then, and I didn't think I was going to be too fond of Marty's explanation, either. But I needed to know; we all did.\n\n\"Dr. Cheban reported that her team's analysis of the drug showed the chemical composition to be too complex to have been manufactured in any dimension with direct access to Hell. Therefore, it was manufactured here. In order to be manufactured here, the molten brimstone has to be harvested here. The magic necessary for creating, stabilizing, and maintaining a portal would have an undesirable side effect on any molten brimstone being brought through a portal, thickening it enough to be rendered unusable for the drug manufacturers' purpose.\"\n\nI think my mouth might have been standing open. \"How do you _know_ these things?\"\n\nDiMatteo actually looked a little embarrassed. \"I have tried to bring molten brimstone back with me on more than one excursion.\"\n\n\"To Hell and back.\"\n\n\"Yes. Passing through the two portals I had to navigate to get home turned my sample into a substance that can only be described as warm goo. Even when I took every precaution and put the samples in a container that can withstand a nuclear blast.\"\n\n\"That's one heck of a thermos,\" I muttered.\n\n\"Yes, it was,\" DiMatteo readily agreed. \"Since demons can't gain direct access to our dimension from Hell, they have to go through portals to get to a dimension closer to ours, and then from there to here. But even then, only certain sizes and classes of demons can get through. Dimensions that can be accessed directly from Hell aren't nice places to begin with. I've called the ones you experienced yesterday 'anterooms,' which is an accurate description. These dimensions are similar enough to Hell in terms of temperature, air composition, and pressure that a portal between the two can be opened with relative ease. The dimensions that can open directly into ours\u2014the elf and goblin realm, for example\u2014are near perfect matches for our own. All of that being said, there are times during the year when the barriers between all of the dimensions are at their thinnest. We just experienced one of those, namely All Hallows' Eve.\"\n\n\"How long do you think the Hellpit has been open?\" Ian asked.\n\n\"The optimal time to open one is at a combination of a full moon and a time like All Hallows' Eve, when our enterprising drug manufacturers wouldn't have had to work quite so hard.\"\n\n\"I thought you'd said there's no direct access to Hell from here,\" I said. \"Then again, when you told me, I'd just hit my head on concrete.\"\n\n\"You're correct, Agent Fraser. There is no direct access _from_ Hell to here. From here _to_ Hell is another matter.\"\n\n\"You're saying that some dumbass on _our_ side dug a pit to Hell?\" Roy Benoit was the commander of one of SPI's two commando teams. He was proud to be from the Louisiana swamps, from a long line of gator hunters, and a retired Army Ranger. Though according to Roy, Rangers not only didn't surrender, they never retired.\n\n\"Not dumb, Commander Benoit,\" DiMatteo replied. \"Greedy. In all likelihood, our demon lord offered them access to fresh brimstone. They had the other ingredients. All they needed was the brimstone. They either didn't know\u2014or didn't care\u2014that if a Hellpit is ever fully opened, it's open permanently, and any demon that ever wanted to come to our dimension and belly up to the all-you-can-eat human buffet could do just that. Since New York has yet to be overrun by demons, we obviously haven't reached that point yet.\"\n\n\"So the last time there was a Hellpit here,\" Roy began, \"how did they get rid of it?\"\n\n\"First of all, it's not a simple matter to open a Hellpit. There have only been a few documented instances, none of which have ever reached a state of being fully open. The first Hellpit was opened in the Gobi Desert in Mongolia in the 1320s. A Mongolian sorcerer sought the advice of a demon to destroy a rival tribe. The demon instructed him on how to conjure a small Hellpit in return for the sorcerer's soul after death, as well as those of his tribesmen. The sorcerer opened the Hellpit, the promised 'help' emerged, and the sorcerer closed it again. What emerged from that pit killed the rival tribe within a matter of days\u2014then did the same to the sorcerer and his tribe, netting the demon his promised souls a lot sooner than the sorcerer had anticipated.\" DiMatteo paused uncomfortably. \"The creatures were tiny, microscopically so. They spread throughout Mongolia to the Silk Road, and from there onto the fleas infesting the rats on merchant ships bound for Europe.\"\n\nHoly. Crap.\n\nRoy was incredulous. \"The Black Plague was caused by demons? I've heard a lot in my time, but\u2014\"\n\n\"Like all living creatures, demons come in all shapes and sizes,\" DiMatteo countered.\n\n\"In other words,\" I said, \"It doesn't take a big pit to make big problems.\"\n\nRoy took a deep breath. \"Okay, saying I believe 'demonic bacteria'\u2014and I might as well\u2014I wouldn't think whoever opened the Hellpit here would be inclined to close it again. When we find it, how do _we_ close it?\"\n\n\"We don't,\" DiMatteo told him. \"It would take a portalkeeper. Two of the officially documented Hellpits were closed by extremely powerful portalkeepers.\"\n\nRoy swore. \"Those are rare birds.\"\n\n\"They don't openly advertise their presence for good reason. People who have the gift of opening or closing dimensional portals or tears are in great demand\u2014and most often by individuals or organizations who you would not want to have notice you. Wars, invasions, and criminal acts of every sort can be greatly simplified with a strategically placed portal. Vivienne Sagadraco will hopefully know the name of a portalkeeper who is powerful enough to close our Hellpit.\"\n\n\"It can be yours,\" Roy said. \"'Cause it sure as hell ain't mine.\"\n\n\"I'm not that great with math,\" I said, \"so correct me if I'm wrong, but it's been four days since Halloween and two days since the full moon. Detective Fred Ash of the NYPD told Ian and me yesterday that they'd only found out about Brimstone a few days ago. That would coincide with Halloween, but wouldn't the manufacturing process take longer than that? Wouldn't that imply that the Hellpit was open before Halloween?\"\n\nAll eyes went to Dr. Claire Cheban, the SPI lab director. She didn't look old enough to be out of college, let alone have a PhD and be in charge of a lab like SPI's.\n\n\"We're still analyzing the drug sample,\" she said, \"but brimstone in its molten state is unstable, especially when combined with two of the other ingredients we found in the drug. As Director DiMatteo said, the composition of the drug itself is incredibly complex. From raw ingredients to finished product would take at least four days, and that's a conservative estimate.\"\n\n\"Sounds like whoever opened the Hellpit missed his window,\" Ian noted. \"Or didn't need one. Is it possible to open a pit anytime?\"\n\n\"It's not only possible,\" DiMatteo replied, \"but in the case we're faced with, I believe it is probable. Contrary to what Commander Benoit said, we're not dealing with a dumbass. Greedy, yes. Dumbass, no. To open a Hellpit regardless of dimensional thinness and moon phase would take an individual with a frightening level of power and skill.\"\n\nAn assessment like that coming from a man who took rock-hunting excursions to Hell meant a lot of scary.\n\nIan and I exchanged a glance.\n\nHalloween night had been the gala opening of the Mythos exhibit at the Metropolitan Museum of Art. One of the exhibits had been Viktor Kain's Dragon Eggs. Another had been a marble statue of three harpies. The real statue had been waylaid in London and had been replaced with three actual Grecian harpies that had been put into a state of stasis until the night of the gala when an unknown\u2014and scary powerful\u2014sorcerer or sorceress had reanimated the harpies to steal the Dragon Eggs. Before, during, and after the theft, they'd also killed a couple of security guards, terrorized the guests, and shattered a section of the window wall in the Met's Sackler Wing when they escaped into the night over Central Park.\n\nWe'd never found who was ultimately behind the diamond theft, but we strongly suspected it was the same individual who had enough magical mojo to put three harpies into suspended animation and disguise them as a marble statue.\n\nIt sounded like one of those mega-mages hadn't left town and was now working with a demon lord.\n\n\"Just because it's called a Hellpit,\" said Sandra Niles, our other commando unit commander, \"does that mean it's an actual, physical hole in the ground, or could it be similar to a door, like a portal?\"\n\n\"It'll be a hole in the ground,\" DiMatteo confirmed. \"But it can be closed like a portal\u2014unless it's completely open.\"\n\n\"When it's completely open, how can it be closed?\" Sandra asked.\n\n\"There's never been one completely open before, so I don't know if it can be closed.\"\n\nSilence.\n\n\"Uh, Marty, there's a lot of holes in the ground under Manhattan.\" Leave it to Sandra to be able to ignore the bomb Marty just dropped and move on. \"Could you narrow it down for us?\"\n\n\"Brimstone solidifies within an hour after being exposed to surface air. It wouldn't matter how quickly it could be gotten into a sealed container. That would put the pit less than an hour from the lab, probably much closer. At the same time, it would need to be a location that could be easily secured.\"\n\nDiMatteo paused, his expression slightly disturbed. Again, coming from a guy who studied demons for a living, this was alarming.\n\n\"There is a rather concerning possibility,\" he said. \"I've compared it to black holes\u2014\"\n\nHellpits and black holes? This wasn't gonna be good.\n\n\"Humans have never been near a black hole, yet there are certain behaviors that scientists accept as fact. Once a Hellpit is open 'all the way,' there's no reason that it would be limited to a finite size. In theory, the size of the Hellpit opening would only dictate what size demons could gain access to our dimension. Smaller opening, smaller demons. Larger opening, larger demons. Unless the individual who opened the pit is remaining with it 24\/7 to control its growth, theoretically there wouldn't be a size limit.\"\n\nNo one moved. Those who were still eating stopped chewing.\n\n\"On the upside\u2014\"\n\n\"There's an upside to Armageddon?\" Roy muttered.\n\n\"Yes, there is. The presence at this time of any smaller-class demons could indicate probable proximity to the Hellpit's location.\"\n\n\"So, if people are being eaten in Midtown, chances are the Hellpit's in Midtown?\"\n\n\"Correct.\"\n\nThat confirmed it. Marty didn't get humor or sarcasm. Bless his heart.\n\n\"Regarding Dr. DiMatteo's comments on the proximity of the lab to the Hellpit,\" Claire Cheban began, \"understand that they would need to have enough distance between them to ensure that no heat or flame from the Hellpit would come in contact with two of the ingredients found in our dimension. Those two ingredients are highly unstable and flammable.\"\n\nNote the location in the city of any large explosion. Check.\n\n\"And Hell does have a well-deserved reputation for being flammable,\" DiMatteo noted, with complete sincerity.\n\n\"The lab where it's being manufactured would need to be state-of-the-art.\" Dr. Cheban proceeded to launch into a ten-minute, PhD-level lecture of the chemical properties of each ingredient she'd isolated, with an accompanying rundown of how contact with molten heat would make them go \"boom.\"\n\nWhen she'd finished, Roy spoke up. \"And that, boys and girls, is why there aren't any meth labs in Hell.\"\n\n\"So what kind of equipment are we talking about?\" Ian asked. \"And where would they get it?\"\n\nI knew where he was going with this. Some of Rake's real estate holdings were elven-owned laboratories and research facilities. We'd asked him to check if any had been working on any new drugs. He'd said he'd check and get back to us. It'd been a little over two hours. Nothing from Rake. Maybe his \"monitors\" had to check their records. Maybe not. Either way, I knew who Ian would be calling when we got out of this meeting.\n\n\"I'll e-mail you a list of the equipment, and where each item can be bought.\" Cheban was typing insanely fast on her phone. \"You can't get these things on eBay or off the shelf at Labs 'R' Us, and the companies that manufacture them don't let just anyone walk in off the street and buy them. And then there's the cost\u2014\"\n\n\"We suspect that our pharmaceutical entrepreneurs have a loaded angel investor,\" Ian told her.\n\n\"If money's no object, they could buy\u2014or pay to have stolen\u2014whatever they needed.\"\n\nThere were beeps and tunes around the room as Cheban's e-mail came through.\n\n\"Check for thefts of items one through five,\" she told us, referring to the numbered list she'd sent. \"One and two are the most expensive and hardest to get.\"\n\nWhen the meeting concluded, Ian made a beeline for the elevators down to the motor pool and motioned me to follow.\n\n\"We going to see Rake?\" I asked.\n\n\"Not yet. Now that we've had dinner, how about dessert?\"\n\n# 20\n\nBY the time Yasha stopped in front of Kitty's Confections on Bleeker Street in the Village, Ian had told me that we were here for more than a nighttime snack of red velvet cupcakes.\n\nApparently Kitty had a secret\u2014a big one\u2014and she'd kept it from everyone. Everyone except Vivienne Sagadraco, who, when the need proved great, had told Alain Moreau and Ian.\n\nKitty could close portals\u2014big ones.\n\nThat implied that Kitty had the same level of power as the mega-mage who'd opened the Hellpit in the first place. I was having a tough time wrapping my head around that one. I was betting that the mega-mage, who'd essentially put out the welcome mat for the denizens of Hell, couldn't bake an angel food cake that was reported to have made actual angels weep.\n\nToo bad this magical confrontation couldn't be settled with a bake-off.\n\n\"I take it from the personal visit, Kitty's going to be less than enthused about helping us,\" I said.\n\n\"Significantly less than enthused,\" Ian replied. \"She had a bad experience. Her last name, Poertner, is German for Porter. Most people with that name had a distant ancestor who was stationed at a castle door. Kitty's people opened and closed a bigger kind of door.\"\n\n\"Dimensional portals,\" I said in realization.\n\nIan nodded. \"The name's the same; the job couldn't be more different. For over a thousand years, Kitty's family have been the supernatural world's doorkeepers, or to be more exact, portalkeepers. Her specialty is stabilizing and closing dimensional rifts, which is essentially what we're dealing with here. We need to secure Kitty's help now, because it won't do us a damned bit of good to find the Hellpit if we don't have anyone who can close it.\"\n\n\"I wouldn't think that'd be a problem. Kitty's awesome. When she hears that the world will literally go to Hell in a handbasket if she doesn't help, I'm sure she'll be glad to slam a door in some demonic faces.\"\n\n\"It's a lot more complicated\u2014and dangerous\u2014than that.\"\n\nLately, it seemed like everything was.\n\nKitty was due to close the shop in another fifteen minutes. Yasha dropped us off out front.\n\nWhen we came in, Kitty took one look at our faces and motioned us straight back to the kitchen while she locked the door and turned off the lights in the front of the shop.\n\nI'd never seen Kitty with any expression other than happy and smiling.\n\nShe wasn't doing either one now.\n\nMind reading wasn't one of Kitty's talents, but she seemed to know why we were there. Then again, Ian had come by to get lemon-blueberry scones for me after the squid demon had tried to drag me through the portal in the parking garage. Ian had known about Kitty's ability, so I was sure he'd told her what had nearly happened to me. I realized that he'd known then that we'd be visiting Kitty for just this very reason, and he'd given her time to start thinking about her answer before he'd had to ask her the question\u2014and before there was the pressure of a critical need.\n\nA wise man, my partner.\n\nYep, Kitty knew exactly why we were here.\n\nBut it didn't change the fact that we were here to ask Kitty to do something that terrified her.\n\nIt sucked to be the bad guys.\n\nBeing the one who'd nearly been dragged through the garage portal, I suddenly felt like the visual aid for Kitty's impending guilt trip.\n\nKitty stuck her head in the kitchen and looked at me. \"Are we going to need cupcakes for this?\"\n\nI sighed. \"I could sure use one . . . or three.\"\n\nWhen life turned to crap, some people drank. I mainlined sugar.\n\nKitty returned to the kitchen, set a tray of miniature red velvet cupcakes down in front of us, and went to the big stainless steel refrigerator and brought out a gallon of milk. I found cups and a roll of paper towels and was good to go.\n\n\"And the people rejoiced,\" I murmured, eying the cream-cheese-iced mouthfuls of culinary perfection. The cupcakes in the cupcake shops that'd sprung up to rival Starbucks in their numbers all had a Mount Everest tower of icing. I'd admit (though not to Kitty) that when I hadn't been near her bakery and was hit with a craving, I'd gone in those shops. More than once, I'd ended up with icing up my nose. Kitty's cupcakes had a perfect cake to icing ratio. My ultimate cupcake test came when I took the paper off. If the cake couldn't support the weight of its own icing and fell over . . . no, thank you. It was possible to have too much of a good thing, and that included icing. I drew the line at what I called bobblehead cupcakes.\n\n\"Where is it?\" she asked us.\n\nI frowned around a mouthful of cupcake.\n\n\"The portal,\" Kitty clarified. \"Where is it?\"\n\nI glanced at Ian.\n\n\"It's not exactly a portal,\" he said. \"As to where, we don't know. Yet.\"\n\nShe regarded him with steady suspicion. It was an expression I'd never seen on her before. \"If it's not exactly a portal, what exactly is it?\"\n\nI glanced at Ian again. I was really grateful to have a mouthful of cupcake. It'd be rude to answer Kitty's questions while eating.\n\n\"It's a Hellpit,\" he told her. \"Open somewhere under the city.\"\n\n\"Full apogee?\"\n\n\"Getting close.\"\n\n\"Who opened it?\"\n\n\"We're trying to find out.\"\n\nIan told Kitty everything we knew so far. Sad thing was it didn't take long.\n\n\"And when you find it, you want me to close it.\"\n\n\"We would like your advice, and if you're willing, your help.\"\n\nKitty glanced at me. \"Your partner's becoming quite the diplomat.\"\n\nI tried a smile. \"I haven't noticed. He must like you more than he does me.\"\n\n\"He just wants something.\"\n\nI nodded thoughtfully. \"Hmm, I think that's a man thing.\"\n\nIan raised an eyebrow.\n\n\"Hey, I'm just trying to lighten things up. If any situation ever needed lightening, it'd be an impending demonic invasion.\"\n\n\"Why?\" Kitty asked.\n\nI puzzled over that one. \"Marty says demons have always wanted our world. Bert thinks it's the beaches.\"\n\nNow it was Kitty's turn to be baffled. \"I meant why would someone open a Hellpit?\"\n\n\"Other than access to molten brimstone,\" Ian said, \"we don't know.\"\n\n\"You said it was likely open before Halloween.\"\n\n\"That's right.\"\n\n\"That means a lot of power was involved.\"\n\n\"We think that, too.\"\n\n\"Such beings of power would have a motive other than getting an ingredient for a drug.\"\n\n\"Even if they were being paid a lot of money?\" I asked.\n\n\"It's been my experience\u2014and that of my family\u2014that those who can open a portal at will are rarely lacking money, nor can the things they want be bought with money. If you can find out what that motivation is, you'd be a couple steps closer to finding them or the Hellpit.\"\n\n\"If we do find it, will you help us?\"\n\n\"If you don't find it soon, you'll be beyond my help or anyone else's. After a major portal has been open for a full cycle of the moon, nothing short of a team of archangels can close it.\"\n\n\"We're going to find it,\" I told her.\n\n\"A word of warning: it could very well be contained in a small pocket dimension to conceal it from anyone who might stumble onto it.\"\n\n\"I can see portals,\" I told her.\n\nKitty gave me a quick, startled look.\n\n\"You didn't tell her?\" I asked Ian.\n\n\"That information is need to know. When I was here yesterday, Kitty didn't need to know.\"\n\n\"As far as I'm concerned, she does now.\" I looked at Kitty. She was regarding me with something that looked almost like pity. \"What?\"\n\n\"If that gets out, you're in more danger than I am.\"\n\nI shrugged. \"I'm a seer. I already have a bull's-eye from that. I can be the same amount of dead from two bull's-eyes than one. I don't know how I picked up this portal-seeing thing, but I have a sinking feeling that knowing how I got it isn't going to help me get rid of it. Besides, from what I understand, it's quite the resume enhancer.\"\n\nKitty smiled. Now that was the Kitty I knew.\n\nI smiled back. \"Sucks to be us right now, doesn't it?\"\n\nShe glanced at Ian. \"I don't really have a choice, do I?\"\n\n\"You always have a choice.\"\n\n\"One option could save the world; the other definitely damns it. Not what I'd call much of a choice. I'll make you a deal\u2014you do your job\u2014 _and_ get me what I need\u2014and I'll do mine.\"\n\n* * *\n\nWe really couldn't have asked for more than that. If we found the Hellpit, Kitty would close it\u2014or at least she'd try. It wasn't like anyone, including her, had on-the-job experience slamming the stairway to Hell.\n\nKitty had sent us on our way with a dozen cupcakes in her bakery's trademark pink box.\n\nWe'd walked out with cupcakes _and_ a promise to help prevent demonic Armageddon. Now that was what I called a good night's work.\n\nI refrained from diving into the box. I had a question for Ian, and I knew Yasha would appreciate me not getting cupcake crumbs all over the backseat of his partner.\n\n\"What did Kitty mean by 'get me what I need'?\"\n\nIan blew out his breath and leaned his head back against the seat rest. \"An anchor mage.\"\n\n\"Which is?\"\n\n\"Pretty much what it sounds like. A mage who can anchor Kitty to this dimension while she works.\"\n\n\"From the 'sigh of eternal suffering' you just let out, I take it there's not a one eight hundred number for anchor mages.\"\n\n\"No, there's not.\"\n\n\"Difficult to get?\"\n\n\"Impossible to get,\" Yasha chimed in. \"There are none in this country\u2014at least not anymore. Only in Europe and Asia.\"\n\n\"So Ms. Sagadraco can't just send over a company jet and pick one up?\"\n\nThe Russian snorted. \"All are worthless cowards.\"\n\n\"Some aren't worthless, buddy,\" Ian said. \"And there's a few left who aren't cowards.\"\n\nYasha took a particularly sharp turn, and I clutched the cupcake box to keep it from flying into the window. \"Uh, a few left? Is there some kind of high job burnout rate for anchors?\"\n\n\"More like a high rate of getting sucked into dimensions they're helping to close along with the portal mage and not being able to get back.\"\n\n\"I can see why that'd make someone want to switch careers,\" I said. \"So what about the ones who are left?\"\n\n\"They have established partnerships with portal mages, and only work with them.\"\n\n\"So we fly a team over here. Kitty doesn't have to take the risk. A win-win.\"\n\n\"More like a lose-lose,\" Ian said. \"They only do smaller portals.\"\n\n\"Then what good are they?\"\n\n\"Not much. Plus, if the portal's on American soil, they believe it's an American problem, and that more than likely, we'd brought it on ourselves and can deal with it the same way.\"\n\n\"Bullshit. When those demons start pouring through, they may get us first, but I don't think they're gonna let a little thing like an ocean or two stop them. Not to mention, Europe and Asia have some amazing beaches.\"\n\n\"These people would say they'd deal with that problem when it came to them.\"\n\n\"And bit their faces off. I'm with Yasha. They're worthless cowards.\"\n\nFrom the driver's seat, the Russian werewolf vigorously nodded in approval.\n\n\"Even if there was a team who would be willing to help,\" Ian continued, \"Kitty's the last of her family line. She's the best, and some would say good enough to do it by herself, no anchor needed.\"\n\n\"What about anchor mages she's worked with in the past?\"\n\nThat question earned me some uncomfortable silence.\n\n\"Okay . . . let me rephrase that: Are there any _surviving_ anchor mages who she's worked with in the past?\"\n\n\"No.\"\n\nI let out a low whistle. \"Was she in some way responsible for that?\"\n\n\"From what I've heard, no. Just piss-poor luck on the part of her anchors, and the fact that they were working on big and nasty portals no one else would touch. Higher risk, higher mortality. Other portal mages and anchors call her the Black Widow.\"\n\n\"How did she survive when they didn't?\"\n\n\"Kitty is brave,\" Yasha said. \"They are cowards.\"\n\n\"I have to agree with you there,\" Ian told him.\n\n\"How do you know all this?\" I asked Yasha. \"I thought only the boss, Moreau, and Ian knew.\"\n\n\"Kitty is friend. She talks to me.\"\n\nI could understand that. Yasha was big, but once you got to know him, you realized his heart was as big as the rest of him. Our big werewolf was also a big teddy bear. I'd also told Yasha things I hadn't told anyone else at SPI. Now I knew I wasn't the only one. I was glad Kitty had realized she could trust him. Everyone needed someone they could tell anything to and not worry about being judged for it.\n\n\"So what happened?\" I asked them both.\n\n\"The last major portal Kitty closed was eight years ago,\" Ian said. \"The portal was to a previously unknown dimension that really wasn't that much better than Hell. A monster, for lack of a more descriptive word, started coming through. Kitty held on and kept working. Her anchor mage panicked and released the protective spell on both of them. The spell Kitty was using to close the portal gave her some protection; her anchor was defenseless\u2014\"\n\n\"Because he was coward and dropped shields on him and Kitty,\" Yasha said vehemently. \"Deserved to be eaten by blue monster.\" He glanced at me in the rearview mirror. \"My opinion.\"\n\n\"If Kitty hadn't been strong enough to shield herself and continue working, she'd have been eaten, too,\" Ian added. \"She doesn't trust anyone to have her back now.\"\n\nI snorted. \"With good reason. I take it this wasn't the first time it'd happened?\"\n\n\"Unfortunately, no.\"\n\n\"Then I can't blame her for telling people to close their own damn doors.\"\n\n\"If this were a normal portal, Kitty could probably do it alone,\" Ian said. \"But it's a Hellpit. There's no precedent on what could go wrong.\"\n\nJust everything.\n\n# 21\n\nWE were halfway back to headquarters when everything caught up with me and I was suddenly bone tired. As a result, I made a decision.\n\nI'd spent last night in SPI's infirmary. Tonight I was going to sleep in my own bed.\n\nI informed Ian of my decision.\n\n\"Sure, no problem,\" Ian said from the front seat.\n\n\"Did you hear me?\"\n\n\"Yes, you want to sleep in your own bed tonight. I completely understand.\"\n\n\"Uh . . . good. I appreciate\u2014\"\n\n\"Just as long as you understand that I'll be staying with you.\"\n\n\"Me, too,\" Yasha chimed in.\n\nMe getting snuggly in my own bed had just turned into a pajama party, with two coworkers who didn't have pajamas.\n\n\"Nice try,\" I told them both. \"Threaten to stay and get me to change my mind. It won't work. I'm not going to change my mind.\"\n\n\"I wasn't asking you to,\" my partner told me.\n\n\"Because I don't mind you guys staying.\"\n\n\"You have popcorn?\" Yasha asked. \"And other snacks? And movies? I like musicals. I will keep sound low; I have good hearing.\"\n\nWhy me?\n\nOn second thought, maybe I'd just grab some clean clothes and go back to headquarters.\n\nWhen we got to my building, Yasha turned his anger at Kitty's cowardly former partners into a hunt for a parking spot, while I hurried up the stairs to pack a bag with Ian doing his bodyguard thing.\n\nI lived in a fourth-floor walk-up in the East Village. The building was from the fifties, and a lot of the tenants were, too. The rest of us were young professional types. Thanks to rent control, the only way the seniors in the building were leaving was carried out on their flecked Formica kitchen tables.\n\nMy apartment was at the end of the hall, with two windows that gave me an occasionally entertaining view of Bainwick's Art Academy across the alley.\n\nIan looked out the window. \"I haven't noticed that before.\"\n\n\"Art school.\"\n\nHis face was profiled toward me. He was grinning. \"There's a platform in the middle of the room. They use any nude models?\"\n\n\"Guys when it's warmer, girls when it's not.\" I met his grin and raised him a smirk. \"Must be that whole shrinkage issue. With the cold snap, the only thing in the raw right now are bowls of fruit. The heat over there works as well as it does over here, which is when it wants to.\"\n\n\"Need me to help you pack?\"\n\n\"I got it.\"\n\nIn my bedroom I kept the usual arsenal that helped single women sleep at night, but I'd added a few of my own.\n\nIn homage to my Southern mountain-girl roots, I kept a seriously huge flashlight next to my bed. It had a trigger for a switch, a camo finish, and could blind a buck at fifty yards. While my intruder was having his retinas flash fried, I'd let him have it with a stream of Raid. Accurate for up to ten yards. Blind 'em with light and chemicals, then run like hell.\n\nIncapacitate while maintaining distance. That's what I'm talkin' about.\n\nThe best defense was avoiding contact in the first place.\n\nSince I'd joined SPI, I'd gotten plenty of training in defending myself. When it came to hand-to-hand combat, I knew I had to have been the most challenging trainee Ian had ever been saddled with. After the first few months, Ian had told me that my brain was probably going to end up being my best weapon, and that weapon told me not to go around writing checks I couldn't cash.\n\nI was smart enough to know and accept that I could be trained by the best and still never qualify as a badass. My goal was simply to make it to work each day and home every night. Ian was the badass-ninja-monster-fighter, not me. I did the best I could during our training sessions, and never stopped trying to improve, but I also accepted that the mayor or police commissioner would never shine the Bat-Signal in the sky to get my attention.\n\nI was good with that.\n\nI went into my bedroom and closed the door. I figured my duffel bag would be the right size for what I needed. My closet was the size of a phone booth, so I kept my luggage\u2014along with anything else that would fit\u2014under my bed. I got down on my hands and knees, stuck my arm underneath the bed, and started sifting and searching for something shaped like luggage.\n\nI found something squishy instead.\n\nI yelped and yanked my arm back, scrambling to my feet, tripping over my own legs in the process.\n\nThe back of my hand was bleeding from a two-inch gash. Must have raked my hand on the bed frame yanking it . . .\n\nMy eyes were even with the comforter on my bed, and they locked on the clear slime pooling in little indentations in the blanket. And on my pillow, a pool of wet filled the indentation that my head had made the last time I'd slept in it two nights ago.\n\nSince then, someone had been sleeping in my bed, and it wasn't Goldilocks.\n\nA raspy hiss came from under the bed . . .\n\n. . . and from the half-opened door to my closet.\n\nA door I always closed.\n\nI drew my gun and slowly backed toward the bedroom door, my eyes quickly flicking between bed and closet. I bumped into my dresser.\n\n\"Ian.\" It came out one notch above a whisper. I swallowed on a bone-dry throat and tried again.\n\n\"Ian.\"\n\nA thing came out from beneath the pillow, squirming through the slime to free itself, dropping from the bed to land with a wet plop on the floor.\n\nIt was maybe eight inches tall, with red skin hanging loose on a thin frame, its bald head topped with two tiny horns. A forked tongue came out from between thin rubbery lips as it opened its mouth, showing me a double row of jagged teeth. Its feet were hooves the size of a cat's paw, its hands thin, spidery fingers, curling and uncurling to reveal claws curved to razor points.\n\nIt looked like . . .\n\nIt couldn't be.\n\nA baby demon.\n\nClass Five, Class Seven, classless, who the hell cared? It was in my bedroom.\n\nAnd there wasn't a portal to be seen or smelled. If they hadn't come through a portal, then how the hell had they gotten in here?\n\nI opened my mouth, to shout, to scream, but nothing came out, not even a whimper.\n\nThe demon's yellow eyes focused on me and it hissed, its whip-like tail lashing the air behind it.\n\nI found my voice _and_ the volume dial.\n\n_\"Ian!\"_\n\nI fired at those teeth. The demon was gone before the bullet got there. My pillow exploded in a blast of memory foam, and wood splinters flew from my demolished headboard.\n\nSimultaneous attacks came from under the bed and out of the dark closet. Every kid's nightmare was now mine.\n\nA clawed hand shot out from beneath my dresser, clutching my ankle.\n\nI stomped on the hand, and fired at the demon skittering across the floor at me. It squealed as a spray of pink erupted from its side, but kept coming, its eyes brightly glowing.\n\nFour demons. Two more dropped out of the heating vent and scuttled on spindly legs off my bed and across the floor.\n\nSix.\n\nSquealing, hissing, eyes gleaming with a yellow light. They were fast. Too fast for bullets\u2014at least my bullets.\n\nBullets weren't working. Leaning against my dresser, behind my door, was my Louisville Slugger. I landed a solid midair hit on a demon that launched itself off my bed, and heard a gratifying crack of the wooden baseball bat on spindly bones for my effort. I didn't have time to confirm that I'd killed it or even knocked the wind out of the thing, as the remaining demons came at me.\n\nI'd never been more grateful to have a small bedroom. The demons leapt at me, and I dove for my bedside table, hitting the floor hard, but rolling onto my back with my can of Raid. The demon that reached me first took a direct hit in the eyes. Its shrieks were deafening. Any higher pitched and only dogs would have been able to hear it.\n\nA demon scrabbled out from beneath the bed and sank its jagged teeth into my shoulder. I screamed and beat it in the face with the can, frantic to get it off me. The can and my hand that death-gripped it were slick with blood, mine and demon.\n\nA flick of movement was all the warning I got of a demon jumping off my bed directly above me.\n\nIt exploded in a bullet-induced spray of red, the bits raining down on me.\n\nIan.\n\nThe other demons kept coming at me, completely ignoring Ian and his gun as if he didn't exist.\n\nIan had his gun in one hand, but was laying into the squealing swarm with a freaking machete. Where'd he been hiding that?\n\nIn a few seconds, my carpet went from beige to beyond able to be cleaned, as Ian and I hacked and bludgeoned our way through the remaining demons. When there were no more of the little monsters left to come at me, I just stood there, gasping for what air I could find, bat still held ready in a double-fisted, white-knuckled grip. Ian stalked around the room, checking for any survivors, and fortunately finding none. In the other room, Yasha all but took my apartment door off its hinges to get in.\n\nIan flicked his blade to clear it of gore. \"Maybe they knew I'd taste bad.\"\n\nI sucked in enough air to make words. \"Yeah.\" Gasp, wheeze. \"Right.\"\n\nThe Russian charged into my bedroom, sawed-off shotgun clutched like a toy in his big hands, eyes shining with an amber glow. The only sound was my ragged breathing. On my carpet was the sliced, smushed, shot, and sprayed proof that someone\u2014alive, undead, or demonic\u2014didn't want me finding the portal to that Hellpit.\n\n* * *\n\nDemon eggs.\n\nThat's what Yasha found under my bed. Six leathery and slimy eggshells. And they hadn't been left there by the Easter Bunny.\n\nSomeone had left me an early Christmas present. Now all I needed to do was find out who my Secret Santa was.\n\nThat, and have a screaming fit.\n\nThe three of us were at my kitchen table.\n\n\"If that squid demon hadn't gotten hold of me last night, I'd have come home to sleep,\" I told Ian and Yasha. \"So much for whether the murderer and his demon lord cohort know I can see portals.\" I gasped. \"Oh, shit. What about Kitty?\"\n\nIan held up his phone. \"Taken care of. I called this in and dispatched a team to Kitty's apartment. She's safe.\"\n\nMy shoulders sagged. \"We didn't try to hide that we were going to see her, and I don't think anyone would believe that we just had the late-night munchies.\"\n\n\"Which is why we have people staying with her.\" He frowned. \"She refused protective custody at headquarters. She insists on opening the bakery tomorrow, so our folks will stick close.\"\n\n\"Why isn't that reassuring?\"\n\nIan glanced back toward my bedroom. He didn't look full of confidence for Kitty's continued safety, either. He looked down at his phone's screen and scrolled down to a number then hesitated, his index finger poised over the screen.\n\n\"What?\" I asked.\n\n\"I really hate it when a situation's so completely in the can that I have to call the boss on her direct line.\" He sighed, tapped the screen, and put the phone to his ear. \"Ma'am, it's Agent Byrne. We have a Code Five. I wanted you to hear it from me.\" He listened, and glanced at me. \"Yes, ma'am, that's exactly where I'm calling from.\"\n\nI groaned and rolled my eyes.\n\nHe listened some more. \"Some cuts that'll probably need stitches, but other than that, she's fine. We just need containment, cleanup, and minor medical.\" He listened. \"Yes, ma'am, I'll hold.\"\n\n\"What?\" I asked.\n\n\"They're checking to see if we're about to have company from the police.\"\n\nI dropped my head into my non-bloody hand. I hadn't even thought of that. Though if the cops were on the way, they were taking their sweet time.\n\n\"Thank you, ma'am.\" Ian put his phone back in his coat.\n\n\"So . . .?\" I asked. \"Yes? No? Maybe?\"\n\n\"None of your neighbors called nine one one,\" Ian said.\n\n\"You're kidding?\"\n\n\"Nope.\"\n\nLately, SPI's biggest crime scene challenge was getting there before the cops. This time, no one had even called them. There'd been gunfire, screaming, and pounding coming from my apartment, and not one call went through to 911.\n\nConsidering the hour, half my neighbors were out drinking with friends, and the other half must have had their TVs turned up, or were in bed with their hearing aids turned off. All of the above kept the NYPD from being called. Luckier still, Mrs. Rosini, who shared a wall with me, was watching Fox News right now. We could hear it through the walls. There was no need to check on her; she was perfectly fine. Mrs. Rosini was one of those people who liked to argue with the TV. Fox News provided a constant stream of something to piss her off. But she made awesome cookies, though usually while talking back to the TV. We could hear her now, giving Bill O'Reilly hell. She'd never noticed when hell had broken loose over here.\n\nI slowly shook my head. \"Gunshots, screaming, and no one called the cops.\"\n\n\"Is good,\" Yasha said.\n\n\" _This_ time. What if I'd been on the receiving end of those bullets? Or had my face eaten off by . . . ?\" I waved my hand in the general direction of my bedroom. I wasn't going to say what they were out loud. I'd already visualized at least half a dozen alternate outcomes where I hadn't come out the winner.\n\nMac, when a murderer sends their newborn minions to kill you, can you in all honesty call yourself a winner in _any_ scenario?\n\n\"Got a first aid kit?\" Ian asked me. \"I don't want to wait for the medics to get some antiseptic on those bites on your shoulder.\"\n\n\"Yeah . . . in the bathroom.\"\n\n\"Got it.\" Yasha picked up my Louisville Slugger and disappeared into the bedroom, now known as the room with the slimy pillows and squishy carpet.\n\nI sighed. \"Damn, I really liked this apartment.\"\n\n\"The boss will have everything taken care of,\" Ian assured me. \"New carpet, paint, pillows, headboard, bed stuff. It'll be as good as new.\" He took a quick look around my less than Martha Stewartesque kitchen. \"Better even. Though I hate to be the bearer of more bad news, but you're going to have to lose your shirt when the medic gets here.\"\n\n\"Excuse me?\"\n\n\"Those bites on your shoulder look bad.\"\n\n\"You're just full of compliments today, aren't you?\"\n\n\"One of my many gifts.\"\n\nI peeled back the dish towel I'd been holding against my shoulder. While it was far from being soaked, it was a respectable amount of blood. But when I thought about what was attached to the jagged teeth that had made those marks, I wanted to strip and run naked through a rubbing-alcohol shower.\n\n\"Yasha,\" I called.\n\n\"Da?\"\n\n\"There's a tank top hanging on the back of the bathroom door. Could you get that, too?\"\n\nSilence.\n\nUh-oh. \"Finding everything okay?\"\n\n\"Find more than expected.\"\n\nI sighed and my shoulders sagged. \"But wait, there's more,\" I muttered. I pressed the towel to my shoulder and stood.\n\n\"Stay here,\" Ian told me.\n\n\" _My_ apartment, and an attempt on _my_ life.\"\n\nIan put up his hands. \"Okay. _Your_ blood on _your_ floor.\"\n\n\"Damn straight.\"\n\nWe carefully picked our way through the bedroom and into the bathroom where Yasha stood staring down into my bathtub.\n\nIan and I went to either side of him, looked down, and had a collective WTF moment.\n\nThere was a pile of raw chickens.\n\nOr what was left of them.\n\nThey'd been torn apart, meat stripped off, and then bare bones flung into the deep end of the tub. It looked like the aftermath of a successful tailgating party or the grandstand after a NASCAR race.\n\n\"Baby food?\" Yasha ventured. \"Maybe they need chicken.\"\n\nI just looked at him. \"Great. So now I'm not nutritionally complete?\"\n\n\"Maybe was appetizer for main course.\"\n\n\"That's not any better.\"\n\nI saw a new bedroom, refrigerator, _and_ tub in my future.\n\n# 22\n\nI was a lot less bothered by an entire SPI investigation team being in my apartment than I thought I would be. I had dead demon bits scattered around my bedroom, and before the hatchlings had eaten their first meal, there had been enough raw chickens in my bathtub to stock a KFC kitchen. That more than made up for dirty dishes in the sink.\n\nThe SPI team wore navy coveralls with patches that said \"Green Heating and Air Conditioning.\"\n\nDr. Stephens was sitting at my kitchen table, a suture kit spread out on a sterile white cloth. Just my luck, he'd confirmed Ian's assessment that I was going to need stitches. I was more than willing to go back to headquarters, but there was no way I'd sleep in the infirmary again. No _Groundhog Day_ time loop for me, thank you.\n\nI was used to eating meals at my table. Now here I was getting stitched up because a pack of baby demons tried to make a meal out of me.\n\n\"I'm ready to start,\" Dr. Stephens told me. \"I need for you to be still. Okay?\"\n\nI nodded tightly and found an absolutely fascinating mystery smudge on the wall to study while he worked. He'd given me a local, but I still had no desire to watch him take a curved needle and thread to my shoulder. However, I took a quick peek before I could stop myself. He was in the middle of his first stitch, and the bottom dropped out of my stomach.\n\nYour life had officially gone to crap when you knew you'd never feel safe in your own apartment again, regardless of how many guns, baseball bats, and cans of Raid you had. For security, most New York apartments made do with half a dozen locks and one or more of the following by the door: mace, a baseball bat, a butcher knife that was past its prime in the kitchen but just fine for puncturing anything trying to get through your front door, or a gun of dubious legality.\n\nI had all of the above, except my gun was legal.\n\nWhen Ian had talked to Ms. Sagadraco, I'd been promised more than that.\n\nWards.\n\nFierce, fry-you-where-you-stood wards.\n\nThree of our best security mages were on their way from headquarters to put the magical whammy on my abode. Creativity counted with wards, and Vivienne Sagadraco only hired the best. Nothing was getting inside my apartment. However, no mage's work was guaranteed against portals, but at least I'd know before I came in if one was or had been open in my apartment. It wasn't the ideal solution, but I'd take what I could get.\n\nIan quietly leaned against the frame of my open kitchen door, waiting for Dr. Stephens to pause in his work before speaking. I was sure the doc appreciated Ian's consideration. I appreciated it even more. He was just the stitcher; I was the stitchee.\n\n\"I went next door to Mrs. Rosini's,\" said Ian. \"Lucky she remembered me from the last time I was here and didn't shoot me.\"\n\nI smiled. \"By the way, Mrs. Rosini has a gun. Got it for her birthday.\"\n\n\"Thank you for telling me.\"\n\n\"If you'd told me you were going over there, I would have. Did she hear anything?\"\n\n\"No, but she saw plenty yesterday. You came home with a tall, skinny guy carrying what looked like a cooler.\"\n\nMy mouth gaped open. \"Another doppelganger?\"\n\n\"I don't think so. They just needed to get into your apartment without causing suspicion. Probably just a quick glamour.\"\n\n\"God, I hope so.\"\n\nIan gave me a crooked grin. \"And just so you know, Mrs. Rosini's money is on me. She said I'm way better looking than the guy you brought home yesterday\u2014and more polite, too.\"\n\nI felt a faint tug on my shoulder, and took another quick glance before I could stop myself. Dr. Stephens had finished and had just tied off the thread. My mouth went dry. The stitches were neat, but it looked like Frankenshoulder. \"Nice work,\" I managed. \"Thank you.\"\n\nHe quickly and efficiently bandaged my shoulder, gathered his things, and left the room.\n\nWhen he'd left, Ian's smile vanished.\n\n\"I just got a call from the team protecting Kitty.\"\n\nI froze.\n\n\"Don't worry, Kitty's fine. She couldn't sleep and wanted to get an early start on tomorrow's baking\u2014and she found something.\"\n\n\"Baby demons?\"\n\n\"Definitely not.\"\n\n# 23\n\nIT was nearly one o'clock in the morning when we got to Kitty's bakery.\n\nIan had told me and Yasha what had happened, and we all were silent from shock and rage the rest of the way there.\n\nBaby demons had been sent to eat me.\n\nKitty had been protected by a SPI security team, so the bastards behind this went for the next best thing. Make her such an emotional wreck that even if we found the Hellpit, she'd be in no condition to close it.\n\nA body had been baked in Kitty's big cake oven.\n\nThey had just brazenly and sadistically stepped over the line into painfully personal.\n\nWhat was her love and solace? This bakery.\n\nWhat was her torment? Her bat-shit crazy, evil three-greats-grandmother's legacy.\n\nWere the killers afraid of Kitty helping us? Oh yes. How to combine all of those into one soul-crushing deterrent?\n\nBake a body in her cake oven. Just like dear three-greats-grandmama would've done.\n\nThis wasn't personal. It had gone way the hell past personal. Whoever was behind this had made a monumental mistake by coming after Kitty.\n\nWe all loved Kitty. Ian's jaw clenched tight, taking even breaths to keep himself calm, and a deep growl had been rumbling out of Yasha's chest since he came through the bakery door.\n\nOur CSI team was there along with our cleanup crew, and Alain Moreau had dispatched another of our field units disguised as city workers to block off the area in front of the store, citing a sewer line break in the street outside. Kitty's customers didn't need to get a whiff of what our lab folks were taking out of that oven.\n\nKitty had two security cameras, one in front, the other in back. When the killer had opened that portal, both had been fried.\n\nBetween the sulfur stink of the portal and the leftover stench of burned flesh, there was no way the smell would ever come out of this place. Not that Kitty would want to keep her bakery here after what had happened; now I didn't see where she had a choice.\n\nThat is, if she stayed in business.\n\nShe'd always known she not only loved to bake, but she had a gift, and a good head for running a business. She'd started from almost nothing and grown a successful business for herself\u2014and a happy life doing what she loved. By New York standards, her business was only moderately successful, but it was beyond what Kitty had ever hoped, and now that had been ripped away from her.\n\nWhoever was behind this would pay, and pay dearly.\n\nKitty would never be able to go into that kitchen again without seeing what she'd found. Everything she'd worked for had been ruined.\n\nThinking from the killer's point of view, what they'd done in that kitchen had been a stroke of evil genius. They wanted to ensure Kitty wouldn't be emotionally able to close that Hellpit even if we found it.\n\nGood fucking job, asshole, my inner voice snarled.\n\n\"You said it, sweetie,\" I muttered.\n\nIan gave me a quizzical look. \"What?\"\n\n\"Where do you send someone who probably crawled out of Hell to begin with?\"\n\n\"Don't know.\"\n\n\"Let's find out and get this guy a one-way ticket.\"\n\n* * *\n\nThe lab guys started removing the body. Ian and I stepped back and gave them plenty of room to work. I didn't want to be in the way. The baked body was curled in a fetal position, cooked until it looked more like a mummy from the Met than a man that'd been walking around mere hours before. A man Ian and I had talked to just yesterday.\n\nWe now knew why Alastor Malvolia hadn't called us back.\n\nI backed up until my shoulder hit the doorjamb, and my vision went white with pain.\n\nIan was there immediately. \"Dammit, are you all right?\"\n\nI nodded tightly and concentrated on hissing air out through my tightly clenched teeth.\n\nI swallowed and focused on pushing the pain away. Dr. Stephens had given me some painkillers. Groggy wasn't conducive to fighting for your life, so I'd stuck them in my purse. When I got out of here and back to headquarters, I might have to take one.\n\n\"Any idea how long he'd been in there?\" I asked. I knew time of death would be a bitch to determine when the only cooling the body had done was once the oven had timed out. It was a question Kitty's three-greats-grandma would've been able to answer off the top of her head.\n\n\"He's been baked at least five hours,\" said an emotionless voice from behind us. \"He was cooked elsewhere and brought here.\"\n\nKitty walked into her kitchen with calm, measured steps.\n\nThe lab techs froze.\n\nThey had the body on a gurney on an open body bag, but they made no move to zip it. They were frozen in place.\n\nKitty glanced at the body, and her throat seized.\n\nI wanted to take the steps that separated us, give her the biggest hug I could, and tell her everything was going to be all right.\n\nI didn't do that. I couldn't do that. Nothing I could say or do would make any of this right for Kitty.\n\nShe had to do it for herself.\n\nThat was why no one moved. At least it was why I didn't.\n\nThe next few seconds would determine how Kitty recovered from this. Just by setting foot in this kitchen, I knew she would recover, but the extent and speed of that recovery depended on what she did next.\n\nKitty sniffed and took a breath that I knew smelled like burned flesh. She didn't flinch.\n\n\"Who was he?\" she asked any of us who might know, never taking her eyes from the dead man.\n\n\"Alastor Malvolia,\" Ian told her. \"A goblin lawyer who was the attorney for most of the supernatural crime families in the city. He may have been brought here not only in an attempt to intimidate you, but because he was short enough to . . .\"\n\nHe stopped. If he could've sucked those last couple of words back in, he would've. Even I cringed at that one.\n\n\"Fit into my cake oven?\" Kitty finished.\n\n\"Yes. Sorry.\"\n\n\"It's okay.\" She took a shuddering breath, tears pooling in her eyes. She angrily wiped at them with the back of her hand. \"Dammit.\"\n\n\"We'll get him,\" Ian promised.\n\n\"Not unless you find that Hellpit, you won't,\" Kitty said.\n\nSilence.\n\nShe wiped her hand on her apron. \"And not unless I help.\"\n\nMore silence.\n\n\"He thought this would scare me, break me.\"\n\nThe clock ticked on the wall.\n\nHer eyes blazed with determination. \"He was wrong.\"\n\n\"We'll be there with you,\" Ian said quietly.\n\nKitty nodded once, tightly. \"I'll take that protective custody at headquarters now.\" She leveled those normally cheerful blue eyes on me. \"When you find that Hellpit, I want to be there and ready to work. I'll get my gear.\"\n\n# 24\n\nONE A Day wasn't just the name of a vitamin; it had also become the murder rate in the Brimstone case. We were guaranteed at least one, and for most of the days this week, there'd been more. Some we found first; the NYPD had been the first responders on others. In this case, I had no problem with sharing the wealth.\n\nI'd actually gotten a few hours' sleep before we were called to the latest crime scene. I'd slept better than I thought I would. Maybe it was my body preparing itself for what was to come. Maybe it was exhaustion after taking the world's longest shower when we'd gotten back to headquarters at two this morning, and washing my hair five times until I was absolutely positively sure I'd gotten all the baby demon bits out. Alain Moreau had met us in the SPI garage and had taken charge of me and Kitty, setting us up in the small but plush apartment SPI kept for visitors who needed to stay onsite for security reasons.\n\nWe sure as hell met those qualifications.\n\nUntil our cleaning crew finished their work, I didn't have a home and Kitty didn't have a business. Both of us were looking for payback.\n\nThis morning's corpse du jour was another vampire by the name of Dante Frontino. The demonic dynamic murdering duo apparently believed in racial diversity, or maybe they just wanted to ensure they equally scared the crap out of everyone.\n\nFortunately, SPI got to the murder scene first. I was sure Dr. Van Daal down at the medical examiner's office appreciated not having to hide the fangs on this one.\n\nIan and I hadn't been the first responders. I was completely fine with our CSI team having that honor. We stopped by long enough to confirm for our investigation that the other elements were the same. Dead drug lord? Check. Hoofprint brand on the chest? Check. Heart cut out? Check. Soul missing? Unknown.\n\nSome would claim that since the victim was a vampire he didn't have a soul to take. I couldn't have disagreed more. Anyone who believed that had never met Alain Moreau. Last week, when the seven diamonds that comprised the Dragon Eggs had been on the verge of activation, my vampire manager had steadfastly stayed by Vivienne Sagadraco's side and refused to leave, facing what would've been certain death, the permanent kind. He would have remained a faithful friend to the end and beyond.\n\nTry doing _that_ with no soul.\n\nOne other thing was the same as the other murders, and another was a notable difference. What was the same? The building was owned by Northern Reach Holdings. Who called in the crime and the ID of the victim? None other than the CEO of Northern Reach, Rake Danescu.\n\nThe goblin called Ian to tell him that he was at the hotel across the street having breakfast, and would be delighted if we joined him.\n\nI wouldn't describe Ian's reaction when he'd hung up as delighted, but I sure knew where we were gonna be having breakfast.\n\n* * *\n\nThe hotel manager was waiting just inside the front door to escort us to where Rake was seated, alone, in a small palm court with a lavish breakfast buffet laid out seemingly just for him. Two uniformed waiters hovered by the door, entirely too attentive for just one customer, regardless of how rich. They'd look human to anyone else, but I saw the waiters and the manager for the glamoured goblins they were.\n\n\"Danescu,\" Ian said.\n\nThat one word was weighed down with all of the other, unsuitable-for-public words that Ian really wanted to say.\n\nMy mood was even worse.\n\nRake stood and, with a flourish, indicated the two empty chairs at the table with him. It seemed that someone had been expecting us. \"Agents Byrne and Fraser. I hope you don't mind that I didn't wait. I wasn't sure how long you would be detained across the street.\" He gave me a quick, wicked grin. \"Makenna, you look positively effervescent this morning.\"\n\nThe only meaning for \"effervescent\" that I knew of involved Alka-Seltzer. I tried unsuccessfully to find the compliment in that. I gave up in favor of the question I had.\n\n\"Do you own this hotel, too?\" I asked.\n\nRake sat and began buttering a piece of toast. \"As a matter of fact, I do.\" He glanced at the nearest waiter. \"Carl, would you bring plates for my guests?\"\n\nIan started to object and Rake held up a hand. It only held a butter knife, but I had no doubt he could commit murder most elegant with it. Heck, Rake could probably kill in thirty ways with a plastic spork.\n\n\"I won't hear of it,\" Rake told us. \"You need to keep your strength up. Fighting the forces of evil and minions of darkness requires being at the top of your game. I promised Vivienne that I would do all in my power to assist you. A sumptuous breakfast is within my power.\"\n\nIan's phone rang, and he went back into the lobby to take it.\n\n\"This isn't the intimate breakfast for two I proposed the other day,\" Rake murmured, \"but it will suffice. For now.\"\n\n\"What the hell are you doing here?\" I wasn't going to tell Rake what had happened last night until\u2014or unless\u2014I had to.\n\n\"Other than enjoying a delicious and most satisfying repast?\"\n\n\"Yeah, other than that.\"\n\n\"Considering that, again, I own the building the murder was committed in, and knowing that you and your partner would want to speak with me regarding that increasingly tedious coincidence, I thought I would save time, and provide a nutritious breakfast while you asked the questions I knew you'd have.\" He gestured at the palms and orchids that filled the space. \"Isn't this a much more convivial setting for an interrogation?\"\n\nWith that, Rake took a quick bite of toast. The goblin's fangs were out and fully extended. He wasn't experiencing tedium, he was furious. It was gratifying to see a genuine emotion from him for once, even if he was still trying to hide it.\n\nI leaned closer and lowered my voice, though the goblin waiters would still be able to hear me perfectly well. As with Rake, it was all for show. \"Nice act. If the murderer walked through that door right now, you'd rip his throat out like that toast, wouldn't you?\"\n\n\"I am being toyed with.\" His voice was clipped with barely restrained anger. \"We're all being toyed with. I will see all of this\u2014and the person behind it\u2014permanently stopped.\"\n\nI poured myself an orange juice, and tried to calm the rage bubbling up inside. The juice had lots of pulp. Fresh squeezed. Nice. My gut told me that whoever Rake had seen outside that coffee shop was someone we needed to have a chat with. Now. \"You didn't say 'unknown' person. Does this mean you've decided to tell me who you saw outside that coffee shop?\"\n\n\"This doesn't have anything\u2014\"\n\n\"Save it!\" I snapped.\n\nRake's eyes widened in surprise.\n\nI kept going. Venting felt good, and right now Rake was a target who deserved it. _He_ had been toying with _me_ since the day we'd met, and I'd had it.\n\n\"You wouldn't be this pissed unless you knew who was behind this and you just haven't been able to get your hands around his throat yet,\" I snarled. \"I'm sure you have your reasons for not telling us\u2014like thinking we're in over our heads\u2014but as Ms. Sagadraco said, why don't you let us decide that? If you've been trying to catch this guy on your own, the body across the street tells us you obviously haven't had any luck. So does your nephew in our infirmary.\" And me and Kitty homeless and workless. \"You're not the only one being toyed with, and I've got news, _none_ of us like it!\"\n\nRake hesitated for the first time that I'd ever seen. He knew I was in a frothing rage, and had rationally determined that while some of it was his fault, all of it wasn't. He opted for caution.\n\n\"I thought I saw someone.\"\n\n\"You _thought_ you saw _someone_? You've got eyes that make us humans look like moles staggering around at high noon.\" I continued before Rake could throw up a wall of denial. \"You saw who was behind the Brimstone and the Hellpit, and you chased him. I'm guessing from all the portals, demons, and murders that have followed, you didn't catch him.\"\n\n\"No, I didn't.\"\n\n\"Care to give us a description for our wanted poster?\"\n\n\"I have no proof of his guilt.\"\n\n\"And we have no idea _who he is_. You've got more than we do, so start talking. I want answers, and I want him.\"\n\nIan's shadow fell over us both. \"Sounds like progress,\" he said cheerfully. No doubt my partner loved seeing me tear into Rake.\n\nI glared at the goblin. \"I'm hopeful.\"\n\nRake regarded both of us, his eyes still inscrutable. \"Yes, Agent Byrne. I believe we're about to discuss progress. Please be seated.\"\n\n\"Gladly, Lord Danescu.\" Ian sat and put a napkin in his lap as Carl poured him a cup of coffee. My partner smiled. \"I love the smell of cooperation in the morning.\"\n\n* * *\n\nI had to admit that Rake was right. Asking questions was ever so much nicer when you had a pair of waiters serving you the best breakfast you'd ever had in your life. But most of all, I was finally getting straight answers from Rake, which made it downright enjoyable.\n\nOnce again, Rake had been using goblin logic and evasion tactics. He hadn't seen a man outside the coffee shop, so he hadn't lied. He'd seen an elf. An elf and mage by the name of Isidor Silvanus.\n\n\"Isidor is of an older generation and prefers our home dimension,\" Rake was saying. \"To the best of my knowledge he does not maintain a permanent residence here.\"\n\n\"You might want to check any recent apartment leases with Northern Reach or your other holding companies,\" Ian suggested. \"You might have yourself a new tenant.\"\n\n\"It's being done as we speak.\"\n\n\"How old is he?\" I asked.\n\n\"One generation older than myself.\"\n\nI didn't repeat my question. Some things I was happier not knowing, and considering that the life spans of goblins and elves were longer than humans, Rake's age might very well be one of them.\n\n\"Why do you think Silvanus may be connected with the murders?\" Ian asked.\n\n\"Because of family relation and past associations,\" Rake replied. \"And my firsthand knowledge of his level of power.\"\n\n\"You've tangled with him before?\" I asked.\n\n\"Our paths have crossed.\"\n\nHe didn't want to elaborate, and once again, it wasn't a question I needed an answer to, at least not now.\n\n\"Isidor's brother is the president and CEO of Hart Pharmaceuticals,\" Rake told us.\n\n\"Let me guess,\" Ian said, \"they lease a building from you.\"\n\n\"Correct. They insisted on handling the retrofitting of their space themselves, as well as hiring their own contractors to do the work. I have been unable to get surveillance equipment inside to monitor their activities.\"\n\nThat had to have annoyed the heck out of him.\n\n\"However, I had glamoured agents inside that reported to me from time to time. For the past year, a small team of their best chemists had been working on a secret project. It was so secret and well protected that my people hadn't been able to get any intelligence concerning it for me. Projects of that level require funding, so I followed the money. I also have agents at the bank Hart uses. They investigated for me and reported an influx of money from an organization that's known for being a front for elven intelligence. From other sources, I know that there's only one drug that interests elven intelligence at this time.\"\n\n\"Brimstone,\" I said.\n\n\"Correct.\" His expression darkened. \"My suspicions were confirmed last week when I lost all contact with my people inside Hart Pharmaceuticals. I had nine agents. One day they were there, the next they weren't.\"\n\nIan scowled. \"Around the time we think they perfected the Brimstone formula.\"\n\n\"That was my conclusion as well. I believe that one of their test subjects was given the drug with the instructions to find any goblin agents. I haven't heard from my people since.\"\n\nNo wonder Rake was furious. First his agents, then his nephew.\n\nSpeaking of which . . .\n\n\"I know Dr. Carey is keeping Jesin's visitors to a minimum,\" I said. \"Including us. We haven't been able to talk to him again. Have you\u2014\"\n\n\"Vivienne persuaded the good doctor to let me speak to him for a few minutes. We spent a little quality FaceTime.\" Rake smiled slightly. \"I do love humans' modern technology. When attempting to get an explanation for a young goblin's actions, it helps considerably to see their face. It's not a guarantee of getting an honest answer, but it helps.\"\n\n\"And?\"\n\n\"I maintain an office in the building where Kela Dupari was murdered. Jesin has recently begun managing it for me as well. And before you ask, yes he's in charge of only the two buildings. His hours in my office there are from eight until noon. A package was delivered that morning addressed to me. Jesin is naturally suspicious, recently even more so. The package was not delivered by a courier service our receptionist was familiar with. When the murder was reported in Kela Dupari's office five stories below, Jesin opened the package. Unfortunately, my nephew isn't qualified to check parcels for spells, but when he opened it, he knew he'd set something off. This one triggered a call to the closest precinct reporting a suspicious package. When he saw what was inside, Jesin took the Brimstone out of the box and quickly left the building with the intent of concealing it. He was trying to protect me.\"\n\n\"That's a very creative story,\" Ian noted. \"Do you believe him?\"\n\n\"Jesin has a head for business, but not a nature for lying.\"\n\n\"Not going to make it far in this town,\" I muttered.\n\nRake sighed, but there was genuine affection in it. \"No, he's not.\"\n\n\"Why would Isidor Silvanus want to involve you in three murders and frame you for drug possession?\" I stopped and thought. \"Let me rephrase that. Can you give us a _short_ list, in order of likelihood, of why Silvanus would want you to rot in prison?\"\n\nRake laughed. \"You are learning, my dear Makenna. Isidor and I have been adversaries for years. To be blunt, he hates me.\" The goblin smiled. \"I believe that any job worth doing is worth doing well, and I have more than earned his animosity. As to his motivation now, it could be any number of reasons ranging from damaging or destroying the operations goblin intelligence has in place in this dimension, to the demons offered Isidor a 'get out of death free' card for helping them gain access to this dimension, to he's simply bored and all of this amuses him. I assure you I am trying to ascertain his reasoning. No doubt you would find his thought processes nearly as convoluted as my own. I promise, if one option seems more likely than the others, I will tell you.\"\n\n\"You were telling us about Hart Pharmaceuticals,\" Ian said.\n\n\"Yes. In the city's supernatural criminal underworld, Hart is known for developing new recreational drugs that are then sold by the Balmorlans, an elven crime family in this dimension, a known name in elven intelligence in mine. Both they and the Silvanus family have been known to use Nightshades as enforcers. Hart Pharmaceutical's share of all profits is laundered through two offshore sources before it comes back to their bank accounts. From all reports, it's a lucrative partnership.\"\n\nI tried to follow the tangled trail. \"Okay, so Isidor's brother runs Hart. Hart has dealings with the Balmorlans, and both Hart and the Balmorlans have a connection to elven intelligence. So how do you know that Isidor opened the Hellpit?\"\n\n\"Because, lovely Makenna, Isidor Silvanus has contacts in Hell, and is so obscenely powerful that he could open a Hellpit in his sleep.\"\n\nI put my fork down. Appetite gone.\n\n\"How did you know about Dante Frontino being this morning's victim?\" Ian asked.\n\n\"I began an analysis of the properties I owned under Northern Reach Holdings, noting the location of each murder in relation to the Hart Pharmaceuticals laboratory. Then I put that analysis on hold when a more immediate clue presented itself. While on my way here for breakfast, I saw Isidor Silvanus exiting my building across the street.\"\n\nHoly crap. \"Did he see you?\"\n\n\"Oh, yes. I received quite the jaunty salute.\" Rake smiled grimly. \"The bastard positively reeked of brimstone.\"\n\nI glanced at Ian. I wondered if Isidor Silvanus had delivered half a dozen eggs to my apartment\u2014and had been the figure I saw on the other side of that parking garage portal.\n\n\"What's this Isidor Silvanus look like? Tall? Skinny?\"\n\nRake gave me a quizzical look. He didn't quite know where I was going with this. \"Tall, yes. Skinny, no. Slender would be a better description.\"\n\nRake's \"slender\" might be Mrs. Rosini's \"skinny.\" She'd told me more than once that I needed to put some meat on my bones.\n\n\"Good-looking?\" I continued. \"Average? Ugly?\"\n\n\"The Silvanus family pride themselves on keeping their bloodline pure. They are high elves.\" Rake gave me a slight smile. \"He is nearly as handsome as I am.\"\n\nMrs. Rosini had said that Ian was better looking than the delivery guy. In my opinion, and probably most women's, Rake was better looking than Ian. Not by much, but there was no denying it.\n\n\"Silvanus wasn't the delivery guy,\" I told Ian.\n\n\"Thanks, partner. You know how to make a man feel good.\"\n\nSo much for Ian not following my train of thought.\n\n\"Besides, a high elf wouldn't be hauling a cooler,\" I quickly added.\n\nIan gave me an arch look. \"I can at least see that being true.\"\n\nRake's eyes were going back and forth between us as if he was watching a tennis match. \"If you continue, will this eventually make sense?\"\n\n\"No,\" we said together.\n\n\"That being said, any type of glamour is well within Isidor's abilities.\" Rake took a positively vicious bite of bacon. \"He could be anywhere now, and posing as anyone.\"\n\n# 25\n\n\"THAT could have been how he got close to Alastor Malvolia,\" Ian said.\n\n\"It appears I'm not the only party guilty of withholding information,\" Rake murmured.\n\n\"You had more to share,\" I told him. \"Ours is just icing on the cake.\" I felt suddenly queasy. \"So to speak.\" I looked at Ian. \"You wanna tell him? I'd rather not even think about it.\"\n\nIan told about the baby demons in my apartment and Al in Kitty's cake oven. Rake listened and didn't say a word. His expression was calm\u2014too calm. I didn't know if anyone could truly know Rake Danescu, but I'd learned enough to know that calm was the last thing he was feeling.\n\n\"Where are you staying?\" Rake asked me as soon as Ian had finished.\n\n\"At headquarters for now.\"\n\n\"I have apartments. _Secure_ apartments.\"\n\n\"Is that like your offer of an intimate breakfast?\" I asked. Ian was right there, but I was beyond caring.\n\nHis dark eyes were steady. \"No. It is an offer of a safe place to live. Full wards, and a full-time battle mage security staff on duty twenty-four seven.\"\n\n\"Wouldn't happen to be your building, would it?\"\n\n\"As a matter of fact, it's in Vivienne Sagadraco's building.\"\n\n\"You own the boss's apartment building?\"\n\n\"I do.\"\n\n\"Does she know?\"\n\n\"She does. The building where I live is equally secure, but I know you'd never accept my offer of an apartment there.\"\n\nAnd I knew I'd never be able to afford an apartment in either one.\n\n\"Last night, Ms. Sagadraco sent a security team to beef up the wards on my place.\"\n\nI said it, but I couldn't say I was thrilled about it. I could see it being a short-term solution, but the thought of living in a place where, despite the best wards, a portal to Hell's anteroom could still be ripped inside my bedroom closet . . . I knew I'd never be able to sleep there again. Not to mention, I refused to endanger my neighbors. Those demons had run out of chickens. If I hadn't come home there'd have been nothing to stop them from taking the air ducts over to Mrs. Rosini's. I felt the prickling of impending tears stinging my eyes. I would _not_ endanger her or anyone else.\n\n\"I'll think about it,\" I told Rake. And I meant it. \"For now, I'm going to stay at headquarters with Kitty.\" I tried a smile. \"It's like a pajama party. Tonight we could do mani-pedis.\"\n\n\"Tell her that I have several retail spaces in the Village and SoHo, should she want to relocate. Free of charge.\"\n\n\"No rent and no burned body stink? I don't see how she could turn that down. I'll tell her.\"\n\n\"And the apartment for you would likewise be rent free\u2014\"\n\nI was about to make a comment about that, but his uncharacteristically somber expression stopped me.\n\n\"And no obligation\u2014of any kind,\" he finished.\n\nWow. I wasn't entirely sure I trusted it, but wow.\n\n\"Thank you,\" I said simply. \"When I have time to think, I'll give it some thought and let you know. It's a very generous offer.\"\n\n\"Agent Byrne has been working tirelessly to protect you since day one,\" Rake continued. \"You're now in the worst kind of danger, and I am at least partially responsible.\" His brow creased in confusion. \"Though I don't yet know why or even how. But I do know that I will do whatever's in my power to help him keep you safe.\"\n\n\"Thank you. Again.\" It was all I could think to say. Rake had complimented Ian, apologized to me, admitted he didn't know everything, and promised protection\u2014all in a few, short sentences. For Rake Danescu, that was a staggering achievement.\n\nIan and Rake exchanged solemn man nods.\n\nLooked like Ian was speechless, too.\n\n\"Is there a chance they're manufacturing the Brimstone at Hart Pharmaceuticals?\" I asked Rake.\n\n\"They would certainly have the equipment they needed, but I wouldn't think so.\"\n\n\"According to Dr. Cheban,\" Ian said, \"working with molten brimstone wouldn't be something you'd want to do in a multi-million-dollar facility filled with valuable, highly educated employees.\"\n\n\"Oh yeah.\" I spread my hands. \"Boom. No meth labs in Hell, and all that. Though if Hart is bringing in beaucoup bucks on illegal drugs, what would stop them from buying some property of their own? They've probably got some labs hidden away around town. Should we let Fred know that Hart's the likely manufacturer?\"\n\nIan nodded, took out his phone, and started texting. \"I don't expect they'll find anything, but since Hart operates as a human company, the NYPD and the feds would be the best qualified to at least make life difficult for them. Maybe they can dig up enough probable cause for a search warrant.\"\n\nI got my phone out, too. \"I'll text Kenji and have him start digging for property Hart and any of its C-levels might own around town.\" I glanced at Rake. \"Unless you have a list floating around in that goblin James Bond head of yours.\"\n\n\"Until now I haven't had a need to know.\"\n\n\"That's okay. Kenji probably already does. He's a collector: comics, movie memorabilia, dirt on supernatural-owned, multinational corporations. Like noticing that a drug company owns a run-down warehouse with state-of-the-art security. Things that jump out and wave red flags.\" My finger froze above my phone. Speaking of red flags . . .\n\n\"Just how badly does Isidor Silvanus hate you?\" I asked Rake.\n\n\"The level appears to be approaching obsession. Why?\"\n\n\"He's been going out of his way to murder people inside of buildings that you own. Why not open a Hellpit under one? What better way to humiliate a dark mage adversary than to open a Hellpit under a property they own without them knowing about it?\"\n\nThere was silence around the table. So much for whether I might be onto something.\n\n\"Is that possible?\" Ian asked.\n\n\"I own many properties, most of which I have no contact with on a day-to-day basis. This is why I have management companies.\"\n\n\"Then it is possible.\"\n\nThe goblin's dark eyes narrowed to angry slits. \"Isidor is exceedingly gifted in the magic arts. Unfortunately, yes. It is possible.\"\n\n\"Marty said brimstone loses its molten state after an hour of being exposed to our air,\" I said. \"That'd put the lab an hour\u2014probably much less\u2014from the Hellpit.\"\n\n\"We need a list of all of your real estate holdings in Manhattan,\" Ian told Rake. \"And not just Northern Reach.\"\n\nWhen a response wasn't forthcoming, Ian continued. \"The list will be kept in a secure database.\"\n\nRake's lips tightened into a thin line. \"A SPI database.\"\n\n\"If I'm right, then Isidor Silvanus already has that list,\" I said. \"So you can slam the barn door if you want, but that horse is long gone.\"\n\nThe goblin sighed, though I detected a hint of a growl. \"Very well. You shall have it within the hour.\"\n\n\"One more question,\" Ian asked him.\n\nThe goblin raised a brow. \"You mean one more question\u2014for now.\"\n\nIan ignored that. \"Do you know if Alastor Malvolia represented Hart Pharmaceuticals?\"\n\n\"He did.\"\n\n\"Did he represent you?\" I asked.\n\n\"He did not. Believe it or not, but I do have standards, and Alastor Malvolia was far beneath them. It was nothing personal; I merely didn't approve of his methods.\"\n\nI grimaced. \"So somebody at Hart stuffed their own lawyer in an oven?\"\n\nRake laughed, a genuinely happy sound. \"Believe me, it could not have happened to a nicer guy.\"\n\n# 26\n\nFRED was thrilled to hear about Hart Pharmaceuticals being the cause of all of his late nights and early mornings. Fred was thrilled because Hart was already under investigation by local, state, and federal authorities. Kitty and I weren't the only ones who wanted payback. There was a line.\n\nThe latest incident in Fred's busy schedule was that he'd just come from the scene of yet another murder that could be connected to Brimstone. This victim was displayed in just about the most public way possible. A human drug lord with a small but profitable Wall Street client base had been found impaled on the horns of the Charging Bull statue in the Financial District. His heart was gone, replaced by the statue's right horn. Fred was of the opinion that this was the guy who had been selling Brimstone to humans like the man in Caf\u00e9 Mina. Sounded logical enough to me. Who would want to read minds more than brokers, financiers, and other businesspeople? Fred said that this particular drug lord had been clued in to the supernatural world, which could connect him to what was really going on at Hart Pharmaceuticals. And it sounded like he'd either neglected to give his customers full disclosure on Brimstone's side effects, or simply told them that they might see things, but to ignore them until the mind-reading benefits kicked in.\n\nThat solved the mystery of how humans were getting their hands on Brimstone, but we still had the problem of no prosecutable evidence against Hart and its officers. Any that had been found had been refuted, and all potential witnesses had disappeared and had not been found\u2014all thanks in one way or another to the late, evilly great Alastor Malvolia.\n\nAs the brains behind Hart Pharmaceutical's continued legal maneuverings and evasion, Al was now out of the picture and in a stainless-steel drawer at SPI headquarters. The feds' prosecutors would be happy about that.\n\nIf Hart's CEO, Phaon Silvanus\u2014brother of Isidor\u2014had been in any way responsible for Al's demise, he'd just gotten the ball rolling on his own downfall. If it hadn't been done at his orders . . . well, for the murderer's sake, I hope they got a running head start for killing the person who'd single-handedly kept the cops and feds from hanging Hart Pharmaceuticals out to dry.\n\nI just loved it when the bad guys shot themselves in the foot, but it remained to be seen if it'd be too little too late.\n\nAlastor Malvolia had been baked, but the finished product was in one piece\u2014including heart and soul. Though with this particular goblin lawyer, one really had to wonder if there'd been a soul there to begin with.\n\nBert was determined to find out.\n\nAnd Rake wanted to be there when he did. He'd given us a list of his Manhattan real estate holdings. It was in the hundreds. Money had never impressed me, and it still didn't. But, damn. We didn't have enough agents to check out even a fraction of them. Rake was hopeful that Alastor Malvolia might be persuaded to point us in the right direction\u2014especially since his murderer was probably also located in that direction.\n\nI'd said I never wanted to be in SPI's morgue for another of Bert's corpse Q&A sessions, but if the goblin lawyer was going to say anything, I wanted to hear it. I was betting that being betrayed by one of his corporate clients was going to make for one seriously vindictive ghost. No retainer was worth that.\n\nI'd damned near been baby food for demons either directly or indirectly because of the Silvanus brothers. I was overdue for some fun.\n\n* * *\n\nThe autopsy room's recording system had been double checked and was ready to go. Human courts didn't consider the testimony of a ghost to be admissible in court, but as far as supernatural law was concerned, alive, dead, or undead, it was all good.\n\nThe autopsy room had two tables and not much room for anything else. Bert had to be in there as did Al Malvolia. Bert took up enough space for two people, and in his present condition, Al was literally half the man he once was. Now he really did look like Mr. Burns from _The Simpsons_ , if Mr. Burns had been baked into a mummy.\n\nDr. Carey and Bert had done an external examination of the goblin's body and determined that he had been knocked unconscious with a blow to the back of the head. I'd only met him once, and by all accounts Alastor Malvolia was as far from being a nice person as it was possible to get. I was still glad to hear that he'd been unconscious or already dead before he'd been shoved into Kitty's oven. After the questioning, Bert would guide the spirit to the other side, and there would be an official autopsy to determine the exact cause and time of death, as well as to look for any residue or fibers on the body that might provide clues to place of death and the murderer's identity\u2014something a human court would accept.\n\nMartin DiMatteo had been there to back Bert up in the past, and he was in there now. The rest of us were on the other side of the double-thick glass wall. Normal morgues didn't need that kind of reinforcement, but in the world of the supernatural, there were many kinds and levels of dead, and on occasion, they didn't need a necromancer to raise a fuss.\n\nIn addition to me, Ian, and Rake was Ms. Sagadraco, Alain Moreau, and\u2014not so surprisingly\u2014Kitty. After all, it was her oven the goblin lawyer had been found in; she wanted to know what had happened to him before he'd been brought there. I'd think it would help her considerably to know that Alastor Malvolia hadn't died in her oven.\n\nFred was busy with the Hart Pharmaceutical end of the investigation, but had made Ian promise to get a recording to him ASAP.\n\nBert was presently making a brief initial contact to ensure Malvolia's soul was still inside his corpse when the body's mouth dropped open and an enraged shriek damned near shorted out the sound system.\n\nHoly Mother of God.\n\nNormally spirits communicated through Bert. Not this time. Malvolia the dead lawyer wanted to do his own talking.\n\nThe lights might not have been on anymore, but the dead goblin was definitely home, and he was not happy.\n\nBert had shielded himself and wore a necroamulet to give himself even more protection. He wasn't taking any chances and I didn't blame him one bit. Contacting a newly dead angry ghost was like startling a big dog out of a sound sleep. Unpleasantness was likely to occur.\n\nThe necromancer glanced at Ms. Sagadraco and nodded.\n\nShowtime.\n\n* * *\n\n_\"Alastor Malvolia.\"_\n\nOnce again the boom of Bert's deep voice filled the morgue's four tiled walls. There was an intercom on our side, but we really didn't need it. The glass was also warded, so the force of Bert's necromantic magic didn't affect me and Ian as it had when we'd been in the room with Sar Gedeon's corpse.\n\nThe elf drug lord's soul had already been taken, so there'd been no response to Bert's command. Al Malvolia had been a lawyer in life; and in death, he couldn't wait to talk.\n\nAlmost immediately a silvery mist rose from the curled-up corpse.\n\nAnd it solidified, complete with a face, an angry face.\n\nOkay, that wasn't normal, either.\n\nEven Bert looked a little taken aback, though for Bert that meant briefly raising one eyebrow.\n\nThere were soul contacts that had been memorable enough to enter into Bert's office party story repertoire. I bet this was going to be one of them\u2014and I was getting to witness it firsthand.\n\nLucky me.\n\nIt was probably a good thing that Fred wasn't here, and that none of us were elves. The spirit that had once lived in the body of the goblin lawyer Alastor Malvolia hissed and spun, two glowing red orbs where his eyes had been, probably looking for the elves who'd killed him.\n\nI wondered if Bert was in charge in there anymore.\n\nHowever, Bert looked cool as a cucumber.\n\nThose glowing eyes didn't find any elves, but he saw his own burned body curled on the autopsy table.\n\nThe shriek he'd let out before paled in comparison to the roar that came out of that pissed-off poof of mist.\n\nAlain Moreau reached over and flipped the switch on the speaker. Either that or our ears were gonna bleed.\n\n\"Thank you,\" I said. At least I think I did.\n\nThe roar came down to a gurgling hiss. It took me a minute to realize that the goblin was laughing.\n\nHe was looking directly at Rake Danescu.\n\nAnd laughing.\n\nI didn't think any of us\u2014especially Rake\u2014were going to find what was about to happen amusing.\n\nAlain Moreau flipped the switch again, turning the speakers back up.\n\n_\"Danessscu,\"_ Malvolia hissed.\n\n\"Alastor. You're looking well.\"\n\nMore gurgling laughter. At least he'd kept his sense of humor.\n\n_\"He isss coming for you.\"_\n\n\"Isidor?\"\n\n_\"Yessss.\"_\n\n\"I was beginning to get that impression.\" Rake nodded toward Malvolia's body on the table. \"Did he do that?\"\n\nThe glow in the red eyes faded a little, and his expression grew distant and puzzled. Both were impressive achievements for a mist you could see through.\n\nApparently he hadn't seen who'd killed him. Looked like Malvolia had been hit from behind like Dr. Carey said.\n\nBert stood next to the table, his hand resting lightly on the corpse's head, his eyes calm and steady on Malvolia's manifestation. Bert was still in control; at least I hoped so.\n\n_\"Isssidor made a deal with the devil.\"_\n\nRake's fangs flashed in a brief grin of delighted realization. \"And you drew up the contract.\"\n\nMore gurgling laughter. _\"Yessss.\"_\n\nNow it was Rake's turn to chuckle. \"You screwed them both over.\"\n\n_\"Filthy, arrogant elvesss. Deservesss to burn.\"_\n\n\"Alastor, if you weren't so crispy right now, I'd actually kiss you. Where's the contract?\"\n\n_\"Sssafe.\"_\n\n\"Where?\"\n\n_\"You will sssee.\"_\n\nThe mist that was Alastor Malvolia was beginning to fade.\n\n_\"Hurry,\"_ Bert mouthed to Rake.\n\n\"Where did Isidor open the Hellpit?\" Rake asked.\n\nI could clearly see Bert standing behind the goblin. What was left of Malvolia was confused, looking around as if he'd suddenly become aware of where he was and didn't recognize it.\n\nWe were losing him.\n\nRake leaned closer to the glass. \"Alastor. Listen to me. Where is the Hellpit?\"\n\n_\"Where the demonsss are coming from.\"_\n\n\"Yes, demons will be coming from the Hellpit. Where. Is. The. Hellpit?\"\n\nThe mist was drawn back into the burned shell of Alastor Malvolia's body.\n\nBert bowed his head, and took a couple of deep breaths.\n\n\"Can you get him back?\" Rake asked urgently.\n\nThe necromancer shook his head. \"There's a chance, but he'd be even more confused than he was now. Even if he could manifest, he would only be able to hold form for a few seconds. We need to let him go, Lord Danescu.\"\n\n# 27\n\nALASTOR Malvolia hadn't been able to tell us where the Hellpit was. We were running out of time. While we hadn't been able to pinpoint exactly when Isidor Silvanus had first cracked open the Hellpit, it'd been long enough since the Brimstone drug started showing up on the street that the pit had to be nearing a fully open state. Once it reached that, it couldn't be closed.\n\nSo far, all of the murders had been committed in properties owned by Rake through Northern Reach Holdings. Even the yacht that Celeste B\u00e1thory was killed on had been leased through a yacht brokerage owned by, you guessed it, Northern Reach.\n\nKenji was working on pinpointing property that Hart Pharmaceuticals or Phaon Silvanus owned and those properties' proximity to buildings owned by Rake Danescu.\n\nRake had identified Isidor Silvanus as one of the previously unknown, obscenely powerful sorcerers who had been at the Mythos gala opening. Just our luck, he'd decided to hang around after the party to make more trouble.\n\nIn the meantime, Roy Benoit and Sandra Niles had their commando teams on standby, and Martin DiMatteo had four people in his department ready to deploy with the commandos to search each potential Hellpit site.\n\nThe first hits on both Rake's property and Phaon Silvanus were two buildings one and six blocks from Times Square. Even in legendary New York traffic, getting from one to the other would still take less than an hour, making them viable Hellpit locations. Sandra's team with two of Martin's demonologists were checking the tunnels nearby and beneath Times Square. After last New Year's Eve, we were more than familiar with them. A Hellpit was different from a portal. It'd be standing wide open, glowing orange like molten lava, and probably guarded by demons.\n\nConsidering the attempts on my life, the Hellpit was probably concealed in a pocket dimension, the entrance to which was a small portal visible only to the mage who made it\u2014who would be Isidor Silvanus\u2014and yours truly.\n\nKitty had confirmed that it was possible, that the Hellpit could be concealed in a small pocket dimension. If that was the case, I would be able to detect the doorway to the dimension, like I had done with Alastor Malvolia's office.\n\nWorse still, if the Hellpit was being concealed inside a pocket dimension, it'd be highly likely that the Brimstone lab would be hidden the same way.\n\nI could see portals, but I couldn't open them.\n\nFortunately for us, Kitty could open an existing portal as well as slam one shut.\n\nAt least that problem was solved. Unfortunately we had plenty more to take its place, and a line forming behind those.\n\nWe were stretched thin enough as is. Ms. Sagadraco had told me that under no circumstances was either I or Kitty to leave headquarters until a viable location had been found. The risk was too great. If the Hellpit or Brimstone lab was concealed by a pocket dimension, I'd be the only one who'd be able to see it, and I couldn't go chasing after every possibility. The demonologists would be able to detect signs of demonic activity that even a pocket dimension couldn't hide.\n\nIf I had to stay at headquarters, there was one thing I could do and still possibly help.\n\nIt was time that Ord Larcwyde and I had a talk.\n\n* * *\n\nKitty and I had one of SPI's VIP apartments; Ord had the other one. Alain Moreau was in charge of assigning guests to accommodations. He liked me and Kitty more than he did Ord, so we got the larger apartment.\n\nIt didn't help Ord's cause that he was being an obnoxious asshat.\n\nIn the gnome's defense, he'd been brought here for protective custody and questioning, and while there was plenty of protection going on, there hadn't been time for any questioning.\n\nThe agent assigned to Ord told me that the gnome was in a foul mood, had been testing the patience of the SPI cafeteria's room service, and was presently binge watching _Game of Thrones_. Other than that, things had been relatively quiet.\n\nI couldn't leave headquarters, so I had plenty of time on my hands for finding out what Ord knew that was worth killing him for\u2014besides the number to room service.\n\nThe stacks of dishes were piled on a cart outside Ord's door. The SPI cafeteria's job was to keep agents fueled up so they could protect and serve. With the clock ticking on the Hellpit, our people had been working and eating overtime. Cleaning up after guests must be falling through the cracks. Though from the looks of the nearly pristine dishes on that cart, Ord had done everything but lick the plates. Our chefs were the best, and Ord was taking full advantage.\n\nI knocked. I'd called Ord as soon as I'd left the morgue, so he was expecting me.\n\nThe gnome opened the door and I was treated to a vision in a green velour bathrobe.\n\nHis feet were bare, his chest was exposed, and I think the only thing he had on underneath was his ever-present gold chains.\n\nYikes.\n\nMaybe I would have been better off spending more time with Al the Crispy Lawyer.\n\nOrd's little face lit up. \"Makenna, darling! I had begun to despair of ever seeing your lovely face again.\"\n\nOkay, Ord might be naked as a jaybird under that robe, but he had Al beat on the charm scale.\n\n\"Come in, come in!\"\n\nI did, and Ord closed the door.\n\nThe gnome grabbed the TV remote and muted the slaughter on _Game of Thrones_. \"To what do I owe this pleasure?\"\n\n\"First, my apologies if you've felt neglected. We've had several emergencies that had to be dealt with.\"\n\n\"More murders?\"\n\nI nodded toward the now silent slaughter. It looked like the \"Red Wedding\" episode. \"Not quite at that quantity, but it's getting there.\"\n\nOrd considered that comparison for a moment, and that he'd narrowly escaped being one of the slaughterees. \"I haven't had the opportunity to express my appreciation for allowing me to stay here until all this unpleasantness blows over. Thank you. And should you see Miz Sagadraco, please extend my heartfelt appreciation to her as well.\"\n\nYep, when properly motivated, Ord could definitely tip the top on the charm scale.\n\n\"You're most welcome. We're glad we can do it.\"\n\nActually, we weren't, and Ord knew it, but we were Southerners trading pleasantries before getting down to business. We both knew the game and the rules. It was older than the family pound cake recipe and just as revered.\n\nOrd hurried to clear a tumble of newspapers out of a chair. \"Please, make yourself comfortable. Could I order something for you?\"\n\nI waved a hand. \"No, no, I'm fine.\"\n\nOrd took his seat and arranged his robe modestly over his lap.\n\nThank God for the decency of a Southern gentleman.\n\n\"Do you really think I know something worth being killed for?\"\n\n\"I sure hope so.\" I cringed. \"Wait, that didn't come out right.\"\n\nThe gnome reached out and patted my arm. \"I'm here, safe and sound, so don't you worry yourself about it.\"\n\nThe job of the agent assigned to Ord had been to basically keep him out of trouble until either Ian or I had time to question him further. The guest rooms had occasionally functioned as fancy jail cells, so the TVs had Netflix, HBO, Showtime, etc., but that was it. There was no contact with the outside world, and even if a guest had managed to smuggle in a phone, it wouldn't work unless we wanted it to. Kitty and I had full outside communication; Ord Larcwyde did not. Until we knew who he knew that might want to kill him, cutting the gnome off from the rest of the world was a prudent security measure.\n\nThe long and short of it was that Ord didn't know diddly squat about what had been going on since our people had picked him up and brought him here.\n\nI brought Ord up to speed on the case, namely who had been killed, and who we suspected of doing the deeds. However, I left out the part about his unsuccessful assassin being a squid demon. I didn't think it had any connection to the information I needed to know, but most of all, I'd heard enough shrieking from Alastor Malvolia; I didn't want to have to listen to Ord's screams, too. My ears had had enough for one day.\n\n\"Do you know any of those people?\" I asked when I'd finished. \"Living or dead, victims or suspected killers?\"\n\n\"I've heard of most of them, but am glad to say I don't know any of them personally.\"\n\n\"So your pixies haven't brought you information on any of them worth killing you for?\"\n\n\"None.\"\n\n\"You're sure? This is really important. Like fate-of-the-world important.\"\n\n\"I sincerely wish I could help you, Makenna, but I've got nothing other than what I told you at my office. Between that gunman trying to kill me, and my pixies raising their prices, I'm considering an early retirement.\"\n\nHuh? \"Your pixies want a raise?\"\n\n\"Two weeks ago, they come to me and say that they're no longer comfortable in the neighborhood; and that if I insist on remaining there, they'll continue to work for me, but they'll be forced to increase their rates.\"\n\n\"Are they getting mugged by fairies for their lunch money or something?\"\n\nThe gnome drew himself up in righteous indignation. \"They said my office stinks, if you can imagine such a thing.\"\n\nI remembered the boxes of garlic falling on top of Ian. \"That garlic was a bit much. He must do a heck of a restaurant business.\"\n\n\"Some. But mostly it just sits there.\"\n\nThat was more than odd. \"So he stocks it, but doesn't sell it?\"\n\n\"I don't mind. I like garlic.\"\n\n\"When did he start stockpiling garlic?\"\n\n\"About a month ago. Pixies have extremely sensitive noses, though I've never heard of garlic bothering them\u2014or even bothering vampires for that matter. That only happens in the movies. It is such an annoyance when Hollywood doesn't even bother to get the details right. The industry is positively rife with supernaturals, so there's simply no excuse. All the vampires I know love Italian food. Well, at least smelling it. And many of my pixie employees are Italian-American. You wouldn't think they'd mind garlic in the least.\" The gnome finished with a dramatic sigh. \"I still may be forced to take the building's owner up on his offers of alternative accommodations. The noise the past few weeks has become tiresome.\"\n\nOdd just turned into a waving red flag\u2014two of them. \"The building's owner wants you to move, _and_ there's been noise?\"\n\n\"He offered last month, and again last week. As to the noise, it's mostly vibrations coming up through the floor.\" He waved a hand. \"There's a subway line running somewhere below. They must be using that track more often now.\"\n\nNervous pixies, stinking office, the owner wanted him out, plus noises\u2014all starting within the last month. And I'd be willing to bet the farm that the squid-demon gunman sent to kill Ord had been a last, desperate measure to get him out of there. As to the stockpiled garlic, was it meant to cover the stench of occasional wafts of brimstone coming from a brand-spanking-new drug lab below?\n\nI got my phone out and called Ian. \"Ord, honey, when this is over, I need to talk you out of retiring. You're a treasure.\"\n\n# 28\n\nSO much for our theory of Hart Pharmaceuticals owning the building housing the Brimstone lab. Hart didn't own the building the grocery store was in; the Balmorlan family did.\n\nThe store owner was a veritable gold mine of information once Ian and I walked through his front door\u2014and a SPI commando team silently came through the back and appeared directly behind him like supernatural ninjas.\n\nThe grocer hadn't been a willing accomplice. A Balmorlan family representative came to him a month ago and said that they required a few favors. In exchange, he would no longer need to make monthly protection payments. The grocer said the favors were simple, they weren't illegal, and weren't hurting anyone\u2014at least not until two days ago when the gunman paid the store a visit.\n\nKeep fresh garlic in stock at all times, and report who came to see Ord Larcwyde.\n\nTwo seemingly innocent favors, plus no more protection payments.\n\nThe grocer jumped at the chance.\n\nIf I'd been in his place, I would've done the same thing.\n\nIf he hadn't cooperated, another of Kitty's ovens might have had an occupant.\n\nVivienne Sagadraco had approved me leaving headquarters in case the lab was concealed in a pocket dimension. The reality turned out to be not nearly that fancy. In addition to the old walk-in freezer that served as Ord's office, the store had a trapdoor in the floor. The grocer knew about it. It'd been used in the past to store illegal liquor for a local speakeasy during Prohibition.\n\nIt had been expanded considerably since then.\n\nThe original storage room had been converted to house air treatment equipment. Brimstone stank, and regardless of how careful the lab folks in the even older chamber below had been with their main ingredient, stink had a way of getting out. That was where the garlic had come into play. One strong smell to hide the occasional whiff of another. And if Ord had ever gotten suspicious, rotten garlic didn't smell too good, either.\n\nThere could only be one reason why the drug lab had been under the store Ord Larcwyde used as an office.\n\nIt was close to the Hellpit.\n\n\"It's all clear below.\" Commander Sandra Niles was in full body armor. Yes, our target was a lab, but it was a drug lab, and one of the ingredients was being brought in fresh daily from your friendly neighborhood Hellpit\u2014neighborhood as in nearby. All that considered, I thought body armor was a simply wonderful idea.\n\nIan and I were in dark street clothes. We'd needed to come in through the front of the store, and body armor or fatigues would have attracted attention to say the least. Sandra had brought everything we'd need if all this panned out and we'd be going underground.\n\nShe showed us the surprisingly clear image on her tablet. Considering the combustibility of Brimstone's ingredients, she'd taken the prudent precaution of snaking a camera in through an air duct.\n\nI swore. \"There's no one down there.\"\n\n\"The store has security cameras, but there aren't any monitors up here,\" Sandra said. \"I have a feeling we'll find monitors when we go below.\"\n\n\"They knew when we got here,\" Ian said.\n\nThe commander smiled. \"And they ran like rats when you turn the lights on. Don't worry, it's an active lab.\" She swiped a finger across the screen, and it went from full color to infrared. Heat signatures flared on two of the chairs, four of the lab stools, and all of the equipment.\n\nIan's grin joined hers. \"Bingo.\"\n\n\"Where did they go?\" I asked. \"I don't see a door.\"\n\n\"We're hoping it's flush against a wall and just not visible on screen,\" Sandra said. \"As with the monitor, we'll find out when we get down there.\"\n\nDr. Claire Cheban and four of her team had suited up in biohazard suits in preparation for going into the lab, complete with gas masks and air tanks, and they were bristling with enough shielding spells to make my teeth hurt.\n\nChances were good that the air was breathable and there weren't any booby traps of the fatal kind waiting for them when they got there. The Brimstone techs had fled when we'd arrived in case we found the lab. That didn't mean they hadn't expected us to find it. They'd probably done this drill before. You wouldn't want to contaminate a lab you fully planned to return to.\n\nBut, just in case, Sandra's team would go in first with sealed body armor and full helmets with small oxygen tanks built into the back. The lab crew was probably waiting nearby, and you'd think at least a few of them would be armed, but regardless of what they were packing, they were nowhere near armed enough. If Lady Luck had finally decided to start talking to us again, the lab crew would have enough sense to not come back.\n\nThe trapdoor entrance wasn't the main way in or out. It functioned as an emergency exit only. They'd just had an emergency, and it hadn't done them a bit of good. As I'd found out the not-fun way on New Year's Eve, the underside of Manhattan was filled with tunnels and chambers. Most were for subway, sewer, water, and power\u2014both used now and abandoned. However, some of those tunnels and chambers hadn't been dug out by human hands.\n\nSomewhere down there was the way to the Hellpit. Martin DiMatteo had told us that the Hellpit would need to be no more than an hour away from the site of the lab, closer would be even better. Molten brimstone was unstable, so the quicker you could get it to where it needed to be, the fewer accidents of the \"boom\" variety you'd be likely to have. Since you couldn't exactly walk down the street in New York casually swinging a bucket of molten brimstone from Hellpit A to Drug Lab B, the tunnel leading to the Hellpit began somewhere beneath our feet.\n\nThe thought made my skin crawl and gave me a nearly overwhelming urge to find a chair and stand on it.\n\nWhat I wanted even more was about two weeks of sleep. Then to go on a vacation somewhere exotic and sleep there for two more weeks.\n\n* * *\n\nMinutes later, Sandra found the door from the lab out into the tunnel. It hadn't been hidden, simply set into the wall and out of sight of the camera. The other side of the door was camouflaged so well that you could have been standing in the tunnel right in front of it and you'd have never found it. Though the Brimstone lab techs must have had a way to get to work every day.\n\nThe tunnel beyond had been used at one time for sewer and rain drainage overflow. Fortunately for us and the AWOL Brimstone lab techs, it wasn't used for either anymore. Once the lab\u2014and its contents\u2014had been secured, it was my turn.\n\nDr. Cheban and her staff had taken over the lab and were working on documenting everything as well as downloading computer hard drives and securing ingredients. Because of the possibility of accidents, even when being handled by professionals, Ian and I had to suit up in hazmat suits, at least until we were through the lab and into the tunnel.\n\nAs we went single file down the narrow stairs that led from the air treatment room under the store and into the lab itself, the sound of my respirator was ridiculously loud in my ears.\n\n\"'Luke, I am your father,'\" I rasped.\n\nNo response from Ian.\n\n\"Really?\" I asked through the communicator. \"Nothing?\"\n\n\"It was only funny the first fifty times I ever heard it.\"\n\n\"It's my first time in a getup like this. It's new\u2014and funny\u2014to me.\"\n\nOnce down in the lab, there were glass-front storage cabinets with cylindrical stainless steel containers. Considering that the largest container said \"brimstone\" I half expected the others to be labeled like something out of a Grimm's fairy tale. I was disappointed.\n\n\"What? No eye-of-newt?\"\n\nThat earned me a snicker from one of Dr. Cheban's techs.\n\n\"That gets a laugh, but Darth doesn't.\"\n\n\"Eye-of-newt's funny,\" Ian said.\n\n\"But Darth's a classic.\"\n\nWe passed through the lab and into the tunnel beyond. For the first time, I was glad to be in one of New York's old drainage tunnels. It meant I could ditch the biohazard suit. If we stumbled onto the Hellpit, the suit wasn't intended for prolonged exposure to high temperature, and \"stumbled\" would be the first perfect description for what I'd do if I needed to run away from anything while wearing it; the second would be \"fall.\" So in the interest of self-preservation, Ian and I left the suits just outside the lab. Hopefully they wouldn't be needed by the time we came back.\n\nHopefully we'd be coming back.\n\nA few of Sandra's commandos would stay behind to guard the techs while they worked. Sandra and the rest of her team would be coming with us.\n\nI had a Hellpit to find.\n\n* * *\n\nThe tunnel stank, though not of brimstone.\n\nThat was both good and bad. Okay, mostly just bad. We needed to find the Hellpit yesterday. It wasn't something we could pretend didn't exist just because we couldn't smell it.\n\nThe tunnel's natural aroma wasn't strong enough to mask something as pungent as brimstone. The tunnel smelled like what it was coated in: damp and mold. I was allergic to mold and glad I'd taken my meds and done my spray this morning. A sneeze down here would echo forever. If we were going to be ambushed by demons, I'd rather it not be my fault.\n\nNo brimstone smell didn't mean there wasn't a Hellpit.\n\nIt simply meant there could be a portal between us and it.\n\nIf there was a portal, it could empty out next to the Hellpit, or into one of Hell's anterooms filled with demons just waiting for the dumb mortals (that would be us) to come bumbling through.\n\n\"Let's go to night vision, people.\" Sandra gestured and we silently moved out.\n\n* * *\n\nJust because there hadn't been any booby traps in the lab didn't mean the tunnel presumably leading to the Hellpit wasn't going to be thick with them.\n\nI could see through glamours and now portals. Half of the people on our commando teams could see wards, illusions, and magical traps of every known kind\u2014and a few that'd never been known before they'd seen them. They were the best. If Isidor Silvanus had left any surprises on the path to the pit, they'd find them before we tripped them. I had the utmost confidence in their skills.\n\nI had nearly none in the job I was being expected to do.\n\nFind the portal hiding the Hellpit.\n\nIsidor Silvanus may or may not have wanted me to see those two portals. But I couldn't imagine any scenario where he'd want me to find the Hellpit.\n\nBut you're not the one who has to close it, said the little voice inside my head. Normally it was the voice of reason; right now it was the voice of shame\u2014as in \"Shame on you, Makenna Fraser.\"\n\nMy job was over when I found the Hellpit.\n\nKitty Poertner's job was to close it.\n\nAlain Moreau was working to find an anchor mage who would help her. Last I heard, there'd been nothing to hear.\n\nKitty would be alone, facing not only closing the largest portal there was, but one that had been opened by a mage who Rake described as so powerful he could open a Hellpit in his sleep.\n\nKitty had to close that.\n\nIn addition to any traps Silvanus may have built into the magical mechanisms holding the pit open, Kitty would have to contend with demons determined not to lose access to our world. They wouldn't like a human _and_ a mortal slamming the door on their eternal beach vacation.\n\nSandra's team was with us now. When we found the Hellpit, Roy's team would join us. While Kitty worked, SPI's commandos would keep the demons under control.\n\nAt least that was the plan.\n\nI'd seen our guys and gals in action. They were beyond impressive. But these were demons we were talking about. Demons who had wanted our world for literal ages. They were closer than they'd ever been to having it all. If that didn't say motivated, I didn't know what did.\n\nMortals and former mortals going up against the combined might of Hell.\n\nAs good as they were, I really didn't want to bet on us.\n\nBut everyone else was.\n\n* * *\n\nThe tunnel curved and turned, but there weren't any branches off of it that the Brimstone lab techs could have used for escape.\n\nAt four places there were ladders leading to the street above. At each, Sandra's team \"navigator\" would note the position on a GPS device strapped to his forearm armor for later investigation. We wanted the Brimstone lab folks in custody for questioning at the very least, but we needed to know where the Hellpit was. Locating lab rats could wait. Finding an open Hellpit wouldn't.\n\nWe were just out of sight of the fourth ladder when the air began to thicken. The two commandos who had taken the lead slowed and then stopped. Sandra went up to where they were. I didn't have to leave my spot in the center of the main group to know what was going on.\n\nWe were getting close.\n\nI took calming breaths and told myself that what I sensed, what I thought I saw, none of them were there. None of them were real.\n\nThe ceiling wasn't closing in on us.\n\nThis was a major-league repelling spell.\n\nRepelling spells were flexible; not only in composition, but in how they affected each individual who encountered it. When you got to the outer rim of the spell's area of influence, you felt uneasy. If you went farther, you began to breathe faster and sweat, experiencing fear you couldn't explain. A sense of \"Get out!\" would start ringing through your head. If you managed to go farther, the visuals would start. We were in a tunnel, so the hallucinations would contain scary things that you'd find in tunnels: rats, snakes, spiders, and, in New York, the ever-popular alligator. Whichever one you were the most afraid of would be the one that you'd see. A buffet of nightmares, so to speak. Though in this case you'd be picking what you didn't want.\n\nEven though all of us knew this, what we imagined wasn't any less terrifying.\n\nBeing a country girl, I'd seen plenty of rats, spiders, and snakes in barns or under rocks. I'd never seen an alligator before, so my imagination wasn't up to reconstructing one of those for me to terrify myself with.\n\nHaving the walls and ceiling closing in on me? Now that I could imagine with no problem, and was doing a fine job of it right now.\n\nOne of the commandos quickly moved to the front, removed her gloves, and raised her hands as if she were putting them up against a wall. She pushed against it and a rush of words in a language I didn't understand filled our earpieces. Our commando teams dealt with the worst that the supernatural world could throw at them. They knew how to fight, but each and every member of both teams had unique skills specific to what they encountered on the job.\n\nAs the commando's words faded, the hallucinations stilled, and then began to fade. It took a few minutes to banish them completely, and the commando was breathing raggedly by the time she'd managed to completely disarm the repelling spell. The spell might be gone, but I was experiencing a whole new wave of uneasiness. I glanced at Ian. I wasn't the only one. We were getting close.\n\nAbout twenty yards later, we came to an old brick wall. The corners were rounded, and the finish was smoother than modern bricks. The tunnel continued to the left and right.\n\nOn the wall in front of me, about chest high, was what looked to be a spigot for a water hose that you'd see on the outside of any house in the suburbs\u2014except it was missing its handle. What it did have was protection. The air rippled in front of it from the effects of a shielding spell.\n\nThe spigot looked harmless enough, but I wasn't about to get any closer. And considering all the protection it had, I felt safe in saying that what would come out of it wouldn't be water. The drug lab techs had to get their supply somehow, and Martin DiMatteo did say that brimstone couldn't be taken through portals without going from molten to useless goop.\n\n\"Sandra, we're here.\" I pointed at the spigot. \"Brimstone source, right there. Well, behind a shielding spell.\"\n\nSandra turned to the commando who'd taken out the repelling spell. \"Deborah?\"\n\n\"On it, ma'am.\"\n\nIt must have been a tough nut to crack because it took her nearly ten minutes to get rid of those ripples concealing the brimstone spigot.\n\nBy that time, Sandra had dispatched scouts down both side tunnels, and both had returned to report that there was no sign of anyone\u2014human or demon. We dispensed with our night vision and had flashlights trained on that wall where one of Sandra's people was now scanning the pipe with some kind of device, while another scrapped a sample of the blackened and pitted concrete floor directly below the spigot. For me, what would come out of that spigot was a foregone conclusion. The water in the Hudson River was known to be nasty, but even that water wouldn't burn and eat holes in concrete.\n\nThe commando with the scanner glanced up from his display. \"Ma'am, some of these elements I recognize, but the rest . . . Your guess would be as good as mine as to where they came from. And the handle that fits this thing would be just as exotic. It'd function as much like a key as anything else.\"\n\n\"Would the metal stand up to high temperatures?\"\n\nHe nodded. \"Judging from the elements that came from our world, that's exactly what it was made to do.\"\n\nWe had no intention of confirming that by finding a way to turn on that spigot. None of us wanted to go down in what'd be left of history as the modern-day Pandora if we released the next plague or an even worse disaster.\n\n\"What's on the other side?\" Ian asked.\n\nSandra studied her wrist-mounted GPS. \"We're below Ninth Avenue. Nearest cross street is West Thirteenth. We're about mid-block, directly below number thirteen.\"\n\n\"Why does that sound familiar?\" I asked Ian.\n\n\"Because it's Bacchanalia.\"\n\n# 29\n\nWHETHER building a successful business or opening a Hellpit, it was all about location, location, location.\n\nThis situation was more like setup, setup, setup\u2014and I could smell the stink of it from here.\n\nRake had wanted to open a business; Isidor wanted to open a Hellpit.\n\nIsidor Silvanus had opened a Hellpit directly beneath Bacchanalia. When the pit was fully open, the center of the goblin intelligence spy network would be the first to fall in.\n\nWe hauled ass back to the Brimstone lab and got back on the surface in record time.\n\nSandra got her team to the two trucks that'd brought them there. They looked like the thousands of other slightly dinged-up delivery, transport, or construction trucks that filled New York's streets every day. SPI had signage for both of them that would be appropriate for any situation or destination. Inside of the trucks was essentially a SWAT command center.\n\nIan and I had come with Sandra and her team. Yasha had stayed at SPI, ready to bring Kitty to the location of the Hellpit once we found it.\n\nIan called Vivienne Sagadraco and told her where we were, and what we'd found.\n\nThe boss then talked for at least a minute, and Ian listened.\n\n\"Yes, ma'am,\" he finally said. \"I understand.\" Then he hung up.\n\n\"Well?\" I asked, my patience long gone.\n\n\"Kitty's on her way here now with Alain Moreau.\"\n\n\"Not Yasha?\" Though the way Yasha drove, and how emotional he was bound to be right now because of danger to Kitty, having a cool vampire behind the wheel might be the best thing.\n\n\"She didn't say, just that Kitty had left with Moreau.\"\n\nMid-afternoon traffic would put her here in around twenty minutes.\n\n\"Is she going to call Rake?\"\n\nIan nodded once.\n\nThat was going to be a fun phone call. Ms. Sagadraco spoke goblin, so she'd know every word Rake blistered the phone lines with. An elf dark mage had opened a door from Hell directly into Rake's pride and joy.\n\nOne of the trucks carrying Sandra's team had gone around behind Bacchanalia to the loading dock area to wait. The other had by all miracles of nature found a parking place across the street and slightly down the block. They'd let us out two blocks before Bacchanalia so I could scout around.\n\nIan and I walked down the street, holding hands, just a couple on an afternoon stroll on a sunny but unseasonably cold day.\n\nHe lowered his head to my ear. \"See anything?\"\n\nWe were both wearing sunglasses to scan the people around us and up ahead without being too obvious about it. I was looking for glamoured lookouts. A villain wouldn't go to the trouble to open a Hellpit that would be unclosable in the very near future and not have people\u2014or not-people\u2014keeping an eye on his special project.\n\nI saw a lot of people, real people and a smattering of supernaturals, hurrying to get where they were going and out of the cold, but I didn't see any loitering evil minions.\n\n\"Nothing yet.\"\n\nBacchanalia occupied what looked like just any other four-story, brick-fronted building in the Meatpacking District. Unless you were a member, or the guest of a member, you didn't know what was beyond its doors.\n\nOn each side, flush up against Bacchanalia's walls was a six-story building filled with high-end condominiums, and a four-story, mixed-use office\/restaurant\/retail space. Across the street were two more restaurants and a coffee shop, with what appeared to be lofts on the three floors above.\n\nIt was three o'clock on a Wednesday afternoon.\n\n\"People are gonna die.\"\n\n\"That's why we're here,\" Ian said, \"to keep that from happening.\"\n\nI gave my partner a double take.\n\n\"You said, 'People are gonna die.'\"\n\n\"I thought I just thought it.\"\n\n\"Unless I can suddenly read minds . . . ? No, you said it.\"\n\nCrap. I really needed a vacation.\n\n\"What about the people inside those buildings?\" I asked. \"If that Hellpit gets bigger than Bacchanalia's walls\u2014\"\n\n\"Those buildings are still standing, so it's obviously not that large yet. And Kenji's working on getting those buildings cleared now.\"\n\nThat made me stop in my tracks. \"Okay, Kenji's good, but\u2014\"\n\n\"It's a procedure we have in place.\" One corner of Ian's mouth curved into a brief smile. \"One of his superpowers is tapping into the monitoring system of any commercial or residential building and setting off his choice of alarms: fire, smoke, or gas. He's also good at calling in suspicious packages. I'm thinking in this situation, he'll go with gas. If he tripped a fire alarm, the people would evacuate, but after ten minutes of not seeing any fire or smoke, they'd start grumbling and want to go back inside. Especially when it's this cold.\"\n\n\"What about when the Con Edison guys show up?\" I asked.\n\n\"Kenji would make sure the call went in to one of Con Ed's clued-in or supernatural shift managers. The boss ensures that there are plenty of those in positions qualified to smooth over anything that needs to be covered up.\"\n\nI nodded at the simple brilliance of it. \"With gas, you can't see it or smell it unless you're in the room with it, so people could whine all they wanted to.\"\n\n\"Right. And not one of them would want to set foot back in those three buildings until the police and Con Ed gave the all clear to go back in. We'll be getting some NYPD crowd control, too. An open Hellpit couldn't be a more perfect fit for a fake gas leak. Natural gas is scented so that if there is a leak, people would know by the rotten egg smell.\"\n\n\"And brimstone smells like rotten eggs.\"\n\nIan flashed a smile. \"Like I said, perfect.\" His phone beeped with an incoming text. Ian took it out of his pocket. \"Fred says he's on his way to the party.\"\n\nI knew the elf detective wouldn't be working crowd control. I had to admit I liked the thought of him going in with us.\n\nA cab pulled up to the curb a few feet in front of us and Martin DiMatteo got out dressed like he was going on another rock-finding field trip to Hell, either that or a safari, complete with an old-fashioned camera hanging around his neck.\n\nHumor and sarcasm weren't all that he didn't get, apparently so was the need to blend in.\n\nOn the other hand, his priorities were in perfect order. If Isidor Silvanus didn't already know we were here, he would as soon as we crossed Bacchanalia's threshold. And if that Hellpit got out of control, no one was going to care how anyone was dressed.\n\nThanks to Kenji, those buildings would be evacuated and stay evacuated until the job was done.\n\nIf we didn't get that Hellpit closed, an explosion would be the nicest thing that would come out of those buildings.\n\nAll we could do now was wait for the rest of the team to arrive.\n\n* * *\n\nI wasn't worried about Kitty and Moreau getting in. The delay was probably due to the combination zoo and circus happening in a five block radius around Bacchanalia. Once Moreau got to the police barricade, he'd just do his vampire Jedi-mind-trick thing, and the two of them would be able to walk right through.\n\nMeanwhile, Kenji's gas leak story had taken on a life of its own. As people hurriedly left the buildings, I'd heard the word \"terrorists\" more than once, though since 9\/11 that probably happened every time something moderately big went wrong.\n\nRake arrived in a justifiably foul mood, and he quickly led me, Ian, and Martin to one of Bacchanalia's fire exits, conveniently hidden by a Dumpster, which I suspect was illegal in ten different ways.\n\nIan's phone rang and he stuck a finger in his ear and tried in vain to find a quiet place to take the call. I recognized the ring. It was Vivienne Sagadraco. After a few moments, he spat a word I didn't need to hear to recognize, and he sharply gestured Rake over. After Ian's first few words, Rake looked even more pissed than my partner.\n\nEven Martin picked up on the fact that something was very wrong.\n\nI sprinted over to where they were.\n\n\"Isidor Silvanus has Kitty,\" Ian snapped.\n\n\"But she left with Moreau\u2014\"\n\n\"Not Moreau,\" Rake said. \"Silvanus, glamoured.\"\n\nI forgot how to breathe. \"How the hell did he get into headquarters?\"\n\n\"He didn't,\" Ian said. \"He was just outside the complex. He called in and Kitty came right out to him.\"\n\nMy heart leapt into my throat. \"Yasha?\"\n\n\"Is fine. If damned near shredding the motor pool when he found out is fine. Silvanus called him using Moreau's voice and told him to stand down, he'd bring Kitty here.\"\n\nMy shoulders sagged. \"And Moreau's the voice of the boss.\"\n\n\"Exactly.\"\n\nIsidor Silvanus couldn't scare Kitty out of helping us, so instead he . . .\n\n\"I'm not getting this,\" I said. \"He kidnapped her. He couldn't scare her out of helping us, so he glamoured as Moreau so she'd go with him. Wouldn't he want to kill her? Why would\u2014\"\n\n\"It's me,\" Rake said.\n\nIan grabbed the front of Rake's leather jacket in his fist and slammed him up against the Dumpster, dark mage power be damned.\n\n\"What are you not telling us _now_ , goblin?\" he snarled. \"Another spy secret to play with? Kitty's life is not yours to\u2014\"\n\nA slender packet of papers fell out of the inner pocket of Rake's coat.\n\n\"That,\" Rake told him, eyes blazing, but making no other move against Ian. \"Was just delivered to me. It's the contract Alastor Malvolia wrote between Isidor and Hell.\"\n\nIan's breaths came in ragged gasps. He let Rake go, but he didn't step back.\n\n\"Isidor will want to use Kitty to bargain with,\" Rake continued. \"He wants the contract; we need Kitty.\"\n\n\"How did you get it?\" Ian asked.\n\n\"Remember when Alastor told me in the morgue that the contract was safe? When I asked him where, he told me that I would see. Well, it took him being dead to keep a promise. He knew either Isidor or Phaon Silvanus would try to kill him. If that happened, he'd left several documents in a safe deposit box. They were to go immediately to me. I got them an hour ago.\"\n\nI bent to pick up the papers, but I didn't hand them back to Rake. \"You said he hates you, and it's mutual.\"\n\n\"I said I strongly disagreed with his methods. For goblins, hate is a personal emotion. I didn't care to get to know Alastor well enough to hate him.\" Rake glanced down at the packet of papers in my hand. \"We both feel the same about Isidor Silvanus. _That_ we agreed on. Alastor may not have liked me, but he trusts me to see to it that his last wish is carried out.\"\n\nIan took one step back, which in manspeak said he wasn't going to kill Rake now, but he reserved the right to do it later. \"And what would that be?\"\n\nRake bared his teeth in a sleek, vulpine smile. \"Use the contents of that contract to ensure Isidor Silvanus burns in Hell. Today.\"\n\nI handed him the papers.\n\n# 30\n\nWITHIN five minutes, our people had managed to get through the police perimeter.\n\nThe last time I'd been in Bacchanalia had been my first night at SPI. Ian and I had been undercover as a couple on a date. I'd really been there as a seer to locate a leprechaun prince and his four aristocrat buddies out for a night on the town before the prince's wedding the next day. The boys were drunk, they were danger junkies, and they'd intentionally gone to an upscale and highly exclusive sex club owned by a known, dangerous, and evil dark mage.\n\nThat would be Rake.\n\nBack then we'd thought that Rake would capture the leprechaun prince and torture him to force him to grant three wishes, and use those wishes to bring destruction on the city and the entire supernatural world.\n\nThat same dark mage was now fumbling with an absurdly large bunch of keys trying to find the right one to get us into Bacchanalia. To his credit, he'd only dropped them once.\n\n\"Fuck it!\" he snarled, slamming his hand flat against the center of the massive steel fire door.\n\nIt vanished.\n\nThe entire door. Gone.\n\nNo smoke around the edges, no tingle of magic, no boom.\n\nJust gone.\n\nAs if it'd never been there.\n\n\"I didn't know those two words were an incantation,\" Martin said.\n\nI think Rake was finished fumbling, and if Isidor Silvanus was somewhere inside with Kitty, he might want to consider leaving her and then leaving Rake's property. Now.\n\n* * *\n\nI'd only ever been in Bacchanalia through the front door during business hours.\n\nThe d\u00e9cor had been black. Completely, totally black. The floor was marble, the walls were black glass, and the ceiling appeared as a star-covered sky far from any city lights. There'd been constellations, stars, and even the Milky Way.\n\nOn the other side of the door that Rake had made go bye-bye, I heard the rapid click, click, click of three wall light switches.\n\nNothing.\n\nEven the emergency lights weren't working.\n\nNow the d\u00e9cor was _really_ black.\n\nDammit.\n\nThe power was out, and I knew Con Ed didn't have a thing to do with it.\n\nIan and I reached into our coat pockets and pulled out our night-vision goggles. Sandra's team already had theirs ready to go.\n\nFred had a flashlight big enough to double as a battering ram. \"We waiting on an engraved invitation?\"\n\n\"Wait,\" Rake told us all.\n\nHow did he see\u2014\n\nOh yeah, goblins could see in the dark.\n\nRake's low whispered incantation was like a warm breath against my ear, even though he was standing at least five feet away.\n\nThe words weren't for me.\n\nThey were for the soft glow that began around the edges of the floor and ceiling, a glow that grew brighter until even us humans could see.\n\n\"I never trust mortal lighting in an emergen\u2014\"\n\nWe all stared at what the lights revealed.\n\nBacchanalia's interior had gone from _Arabian Nights_ to Dante's _Inferno_.\n\nIt looked like my baby demon-infested bedroom times ten thousand.\n\nClear slime ran down the walls, and dripped from the ceiling, pooling on tables and the floor. The bar area was covered in slickly glistening webs, punctuated by what looked like cocoons. A few were pulsing with movement from inside.\n\nI couldn't see any demons, but I knew they were here, watching and waiting.\n\n* * *\n\nSandra got word that Roy's team had gotten inside the police perimeter just before it'd been closed off. We weren't going to wait for them. They knew where we were going. Bacchanalia didn't have a basement, but Rake had what I'd heard was considered one of the finest wine cellars of any club in the city. From the dimensions Rake had provided, there'd be only so much room to maneuver down there, and when dealing with a pit of demons, backup would be needed. I wanted to know that Roy's commandos were at our back, not more demons.\n\nI wasn't much for wine. Actually, I wasn't much for alcohol, aside from an occasional dark ale. And forget any kind of liquor. It didn't matter how long it'd been fermented in whatever fancy distillery, it all tasted like cough syrup to me. I could safely say I had an unsophisticated palette. Back home when I was growing up, we didn't bother with store-bought cough syrup. Moonshine, honey, and lemon juice worked just fine.\n\nI prepared myself to make impressed noises if it was still intact; sympathetic sounds if it was all over the floor\u2014or if the demons had drunk it all. What I knew about wine would fit on the top of a cork, but I'd heard that Rake had a small\u2014or maybe not so small\u2014fortune invested in the room and its contents. Now Isidor Silvanus had opened a Hellpit in the middle of it.\n\nAnd Isidor Silvanus had Kitty.\n\nWe had no way of knowing which way Alastor Malvolia's soul had gone when Bert had guided him to the other side. But wherever he was, I knew he'd approve of there being a long line of people who wanted to help make his last wish in life come true.\n\nThere were two ways down to Bacchanalia's wine cellar: an elevator and stairs.\n\nWe took the stairs.\n\nElevators were death traps on steel cables.\n\nRake took the lead, with two of Sandra's commandos far enough behind to give him space. They'd wanted to go first, but one scowl from the goblin who'd made a steel fire door vanish into thin air, and the boys backed off. Rake could take care of himself. Ian and I followed.\n\nThere wasn't that much information available on Hellpits, but Martin DiMatteo knew all of it. I was the lucky one who could see portals.\n\nIf Rake opened the wine cellar door and the floor had been turned into a bubbling hellfire-and-brimstone pit, Martin was officially in charge. If it looked to everyone else like a perfectly normal and intact wine cellar stocked with obscenely overpriced bottles of fermented grape juice, and only I saw the bubbling pit, it'd be my turn for Show & Tell.\n\nRake opened the door.\n\nIan and I were still on the stairs, so I couldn't see inside.\n\nMartin stepped up and looked in, then both he and Rake turned and looked at me.\n\nCrap.\n\nIan stepped in front of me to go down the last few steps first. While my partner turning himself into an immovable object between me and a Hellpit was chivalrous, it wasn't going to do anyone any good, or change what lay beyond that door.\n\nAs far as anyone else was concerned, everything beyond that threshold looked perfectly normal. I didn't want to see a Hellpit, but I hoped I did. Because if I didn't, we had no idea where else Isidor could have opened the thing. He would be there\u2014with Kitty\u2014and we had to find her.\n\nRake stepped aside and I looked into the room.\n\nI could see what Rake and Martin saw, and it was unexpected, at least to me. Rake had originally come from a Renaissance type of society, and I expected a wine cellar built to look like it came out of a European castle. Bacchanalia's wine cellar was a circular room, with pale woods and sleek brushed steel. The shelves and niches that held the bottles were glass, though it probably just looked like glass, and in reality was something more durable. Lighting radiated out like muted sunbeams from a circular central orb mounted in the ceiling. From the looks of the panel set into the wall with the gauges and flickering lights, it appeared Rake's cellar had the latest in wine storage technology. The room looked more like the bridge of a spaceship than a place to store wine. Rake was a techie. Who knew?\n\nThe room also had a portal slicing like an open wound down a narrow section of wall.\n\nIt wasn't like the portals at the scene of Sar Gedeon's murder and in the parking garage. For one, this thing stretched from floor to ceiling. The other two had been roughly man height. But the big difference was that this portal was red and it looked angry\u2014or at least like anyone stupid enough to get close to it would get themselves chewed up and spit out on the other side. Knowing what was on the other side, that was a chow line I wasn't about to get in.\n\nThough I didn't think I'd be risking life, limb, and soul just by crossing the threshold of the wine cellar.\n\nI stepped into the room and Ian, Rake, Fred, and Martin followed. Sandra stayed by the door.\n\nTime to break the bad news, but first I had a question. \"Director DiMatteo, you said that the Hellpit would be a hole in the ground, not a portal.\"\n\n\"That is correct.\"\n\nI nodded in the direction of the bloody and raw-looking gash in the wall. \"Then at two o'clock we have a portal that will presumably take us to the Hellpit.\"\n\n\"What's on the other side of that wall?\" Ian asked Rake.\n\n\"I had assumed it to be solid rock,\" the goblin replied. \"This room was already here when I bought the property. Since it naturally maintained a steady temperature of fifty-eight degrees, I assumed it to be enclosed on all sides by more or less solid rock, like a cave. My wines have been quite comfortable here.\" He glared at the wall with the portal that only I could see. \"At least they were.\"\n\n\"Apparently it's less solid than you thought,\" my partner noted.\n\nRake made a sound that under better circumstances would have been a chuckle. \"My insurance will never cover this.\"\n\n\"What, no Hellpit portal policy?\"\n\n\"It's one of those things you're sure will never happen to you. Like living in the desert and not getting flood insurance.\"\n\n\"At least this portal doesn't go into another dimension,\" I told them both. \"It's just a way to disguise the entrance to a Hellpit.\"\n\n\"And its exit,\" Martin told us.\n\nLike we needed reminding. Kitty wasn't here to close it, and as every minute passed, the Hellpit was opening wider, and the things inside were closer to being able to get out\u2014as in out here with us and the rest of the world.\n\n\"So . . . how do we get inside?\" I asked anyone who might have the answer. \"Not that I'm eager to do it, but\u2014\"\n\n\"Isidor wants the contract,\" Rake said. \"Since he has taken Kitty Poertner, apparently he knows that I have the only copy of the contract not in his possession.\"\n\n\"If he's as powerful as he would have to be to open a Hellpit,\" Martin began, \"couldn't he simply kill you and take it?\"\n\nRake smiled slowly. \"I welcome and eagerly anticipate his efforts. I would be most disappointed should he not try.\"\n\nEven Martin didn't know what to say to that. Rake was homicidal and suicidal at the same time. Must be a goblin thing.\n\n\"There has to be another entrance.\" Fred was standing off to the side, studying the rest of the room.\n\n\"Detective Ash raises a logical point,\" Rake said. \"Isidor Silvanus could hardly stroll into my establishment unnoticed. Not to mention access to the elevator and stairs down to this cellar is controlled by a coded keypad.\"\n\n\"Then why put a 'back door' here?\" Ian asked.\n\n\"For the same reason he has been staging most of his murders in buildings that I own,\" Rake said. \"Embarrass the goblin intelligence agency\u2014and me in particular. And for the coup de grace, should we fail to secure the Hellpit, the demons will emerge from here. It seems that Isidor now knows that the center of my web\u2014as Makenna so astutely described it\u2014is here at Bacchanalia. He wants it destroyed.\"\n\n\"You must have really pissed him off to get all this special attention,\" I said.\n\nOne side of Rake's lips curled upward. \"Many times and on numerous occasions.\"\n\n\"Has it ever occurred to you to stop pissing people off? Or at least be more discreet about it?\"\n\nHis crooked grin grew. \"I am the very soul of discretion, lovely Makenna. I was merely doing my job. It is no fault of mine that my job is a source of great annoyance for Isidor. Besides, to obtain a source of brimstone, he simply would have opened his Hellpit somewhere else in the city. The danger would be the same, and we would not have had the trail of bread crumbs that led us here. When you look at it that way, it was a good thing I did piss him off. It made him predictable.\"\n\nIan snorted. \"We haven't found him yet.\"\n\nI stood perfectly still as the sides of the portal slowly peeled apart.\n\nIan tensed. \"What is it?\"\n\n\"The portal's opening.\" Then I saw what lay beyond. \"And I think Isidor Silvanus just sent someone to let us in.\"\n\nA red-skinned and horned demon, no more than two feet tall, leisurely strolled toward the opening from the other side, swinging what looked like an old-fashioned key that was nearly as big as he was.\n\nThat was surreal. Like _Alice in Wonderland_ surreal.\n\n\"Any of you see the demon lord mini-me walking toward us from the other side?\" I asked.\n\n\"No,\" Ian and Rake said.\n\nWith a smile revealing a mouth entirely too full of jagged teeth, the little demon embedded the key like a spear into the right side of the portal wall\u2014or whatever it was that a portal had\u2014and stepped right through into the wine cellar with us.\n\nIan and Fred reached for their guns.\n\nRake reached for his magic.\n\nMartin reached for his camera.\n\nStill smiling, the little demon stopped in front of Rake and held out a folded piece of parchment complete with a red wax seal.\n\nRake took the parchment and instead of breaking the seal, he put his thumb in the middle and the wax vanished in a poof of rotten egg stink\u2014disarming whatever nastiness Silvanus had intended when the seal was broken.\n\nHe read it, then passed it to me and Ian.\n\nThe words began burning the paper as soon as it left Rake's hand.\n\nWe read fast then dropped the smoking parchment.\n\nThe instant it touched the floor, the demon disappeared, reappearing on the other side of the portal, and strolled away in the direction from which he'd come.\n\nSilvanus wrote that he had Kitty and was willing to release her in exchange for the contract. He wanted Rake to deliver the contract in person. He also wanted \"the seer, the human SPI agent, the half-elf law officer, and the demonologist.\"\n\nNo one else.\n\nAnd if all of us didn't step through that portal, the deal was off.\n\n\"I don't like it,\" Ian said.\n\n\"We've just been invited to Hell by an evil wizard,\" Fred said. \"And if we don't slam a Hellpit, demons invade and everyone dies. You'll have to be more specific, buddy.\"\n\nMy partner's face was set on perma-frown. \"You, Mac, and Martin don't need to be anywhere near here when this goes down.\"\n\n\"You go, I go,\" Fred told him. \"No arguments.\"\n\nI resisted the urge to roll my eyes. \"None of us want to be here, but we're on the guest list. Without Kitty, we don't stand a chance in hell . . .\" I stopped. \"If we live through this, that phrase is gonna have a whole new meaning.\"\n\n\"I have been to Hell,\" Martin told us. \"The rest of you have not\u2014with the possible exception of Magus Danescu. I must go as a guide, if nothing else.\"\n\nRake actually did roll his eyes. \"The Hellpit isn't getting any smaller while you argue. Once the demons of Hell pass into this world, every living thing will become food\u2014or worse. We can't help Miss Poertner, or close the Hellpit, if we don't get inside.\"\n\nRake Danescu had become the voice of reason. We didn't need Kitty to close the Hellpit. Hell itself had just frozen over.\n\n\"There were two more lines I couldn't read before it burned up,\" I said to Rake. \"What did they say?\"\n\n\"Isidor claims he cannot\u2014or more likely, will not\u2014guarantee our safety once we're inside. He claims to have limited influence over his hosts.\"\n\n\"He didn't say anything about weapons,\" Ian noted with grim satisfaction.\n\n\"It is likely that weapons from our dimension will not work there,\" Martin told us. \"Particularly automatic weapons.\"\n\nI couldn't believe my ears. \"You've got to be kidding.\"\n\n\"The closer we get to the Hellpit\u2014and therefore to Hell itself\u2014the less effective the technology from our world will be.\" Martin almost looked embarrassed. \"I discovered this through unpleasant personal experience during one of my excursions.\"\n\n\"What about your camera?\"\n\nThe demonologist actually smiled. \"Older technology such as this will not be affected.\"\n\n\"Since the Hellpit was opened by Isidor,\" Rake said, \"and presumably that portal was his creation as well, the very air on the other side will be filled with the influence of his magic. Different rules will most definitely apply.\"\n\nIan snorted. \"Silvanus's rules.\"\n\nRake shook his head. \"Dark magic rules.\" His eyes glittered in what I could swear was anticipation. \"I know this game.\"\n\n\"I don't know of any rules that would keep cold steel from doing its job,\" Ian said. \"You got knives?\" he asked me.\n\n\"Many,\" I assured him.\n\n\"Get more. Sandy?\"\n\nSandra turned to her closest commandos and then started passing me blades in sheaths, and I put them anywhere they'd comfortably go.\n\n\"Got an extra revolver?\" Ian asked the commander.\n\nSandra didn't say a word, just unclipped the old-fashioned six-shooter from her belt and passed it to him, along with a pouch of ammo.\n\nRake shook his head when Sandra offered him what I couldn't carry.\n\nIf dark magic would work, Rake should be armed for a demonic T-Rex. Hopefully we wouldn't have to find out.\n\n\"I'm fine, thank you,\" Martin said when one of the commandos offered him a wicked curved knife with a jagged blade.\n\nThe demonologist was smiling in gleeful anticipation.\n\nMarty was about to see his very first Hellpit.\n\n# 31\n\nTHE five of us stepped over the threshold of Bacchanalia's wine cellar and into a nightmare landscape, and we all got a good look at Isidor Silvanus's handiwork.\n\nI had to give him credit for creativity. The other side of the portal looked like a tunnel in a prehistoric cavern complete with stalagmites and stalactites. Along one side of the rock-strewn floor was a stream of what must have been molten brimstone flowing away from us and around a curve in the cave wall. I couldn't see what was around the corner, but I could sure see the bright orange glow.\n\nThere was no way all this was on the other side of the wall from Rake's wine cellar. You couldn't fling a dead rat below street level in New York without hitting subway tunnels and\/or water and sewer lines. There were a lot of things under the city streets, but a monstrous cavern complete with a brimstone creek shouldn't be one of them.\n\n\"This isn't right,\" was what I managed to say. \"This can't be here. It's too big.\"\n\n\"It's a pocket dimension,\" Rake said. The goblin turned back to where the portal opened, and his eyes shone with intent of murder most violent. \"And Isidor anchored it into the rock surrounding my cellar.\"\n\nI looked around. \"Jeez, how much room is on the other side of your wine cellar anyway?\"\n\n\"It couldn't be this much,\" Fred said.\n\n\"It's not,\" Martin told us. \"The size of the actual area outside of Bacchanalia's basement has no bearing on the size of a pocket dimension. In theory, it could be as small\u2014or as large\u2014as its creator wanted it to be.\"\n\nI stifled a whistle at the vault of the cavern ceiling far above our heads. \"Then Silvanus must be compensating for something.\"\n\n\"If it's a pocket dimension, then how does what's in here get out into the city when the Hellpit is fully open?\" Ian asked.\n\n\"Isidor's magic made it,\" Rake told him. \"Isidor's magic can unmake it. Once that Hellpit is completely open, he'll pop this pocket dimension like an overfilled water balloon.\"\n\n\"Sounds messy,\" Fred noted.\n\n\"If by messy you mean a cavern suddenly breaking through into our reality beneath the streets of this city, molten brimstone flowing through the sewers and subway tunnels, and demons hunting the streets\u2014then yes, it will be extremely messy.\"\n\nA swiftly flowing river of bubbling, molten brimstone ran beside a rock ledge barely wide enough for two of us to walk side by side. The altering landscape must have been a distortion of the pocket dimension\u2014or the landscape was shifting and changing as the Hellpit somewhere farther in the cavern continued to grow. Color was apparently distorted as well. When we'd first stepped through the portal, the rocks had looked, well, rock colored. In reality they were sulfuric yellow.\n\nAnd I'd always thought the Yellow Brick Road led to Oz.\n\nThat'd make Isidor the Wicked Witch of the West, Kitty would be Dorothy, and the contract Rake carried was the Ruby Slippers. Rake wouldn't qualify as Glinda the Good on his best day, more like the Wizard of Oz. At the end of the movie, Oz had floated away in a balloon, leaving Dorothy and company to fend for themselves.\n\nMy subconscious kept replaying _that_ part for me as a portent of impending doom.\n\nOurs.\n\nI'd only heard about Rake's power. Other than the fire door, I'd never witnessed anything big myself, and until now I'd never minded. One person I'd heard it from had been Vivienne Sagadraco. If the boss said Rake was powerful, I'd believe her without proof. However, she'd also said that he was dangerous. I'd always assumed she meant dangerous to anyone he went up against. Now was not a good time to have my assumption disproven.\n\nI pushed those thoughts out of my head, making myself focus on what was likely to get me killed now rather than later. When in enemy territory, a little noise to cover any sounds you might make was a good thing. Usually. The sounds we were hearing wouldn't be called good in anyone's estimation. A sharp snap and crack was repeating at irregular intervals, as if something that wasn't supposed to be breakable was being broken. Like I said, not good.\n\nWe walked and walked, but didn't seem to be getting any closer to the turn in the path and the Hellpit presumably beyond.\n\nIsidor Silvanus was playing with us.\n\n\"Does this qualify as the dark magic games you were referring to?\" Ian asked.\n\n\"It would,\" Rake replied. \"Isidor is attempting to control time here. He's trying to delay our arrival.\"\n\nFred wiped sweat from his face. \"Seems to be doing a damn fine job.\"\n\nThere was a grouping of sharp rocks not too far down the path. \"Those rocks haven't gotten any closer,\" I pointed out. \"It's like we're walking on a freakin' treadmill.\"\n\nRake nodded once. \"Exactly.\"\n\n\"Anything you can do about this?\" Ian asked Rake.\n\n\"There is. The question then becomes are you ready for a fight?\"\n\n\"Yes.\"\n\n\"How much of a fight?\" Fred asked.\n\nA wise man, Fred.\n\n\"I know what you're capable of,\" Fred told Ian. \"No offense, Mac. You're feisty, but we _are_ approaching Hell.\"\n\n\"Thanks for the vote of confidence.\"\n\n\"Nothing personal.\" The half-elf cop jerked his thumb at Martin. \"Then we've got the Professor back there doing a _National Geographic_ photo shoot.\"\n\nThe demonologist was squatting down on the very edge of the ledge, clicking off shots of a fat, pale worm-like demon that was using six caterpillar-like legs to pull itself up against the ledge like it was the edge of a swimming pool, and was curiously studying Martin with a pair of round, black eyes.\n\nMartin must have sensed us all watching him in complete disbelief. He stopped clicking.\n\n\"Don't let me keep you. I'm fine. Merely taking advantage of a once-in-a-lifetime opportunity.\"\n\n\"Isidor is attempting to delay us by controlling time here,\" Rake told him, \"so I'm going to teleport the five of us closer to the Hellpit. I need you to move closer.\"\n\nMartin glanced down with concern at the chubby foot-long worm. \"Manipulating time can adversely affect these larvae's development. Since it's occurring, the parent must not be aware of it. Isidor Silvanus may be able to slow the passage of time, but he's still a guest here, so he really shouldn't. Would you like him to stop?\"\n\nThe goblin raised one perfect eyebrow. \"That was my desired solution.\"\n\n\"I think I can help with that.\"\n\n\"If so, your assistance would be much appreciated.\"\n\nI didn't say a word. I couldn't. There were way too many WTFs in that exchange for me to process. Fred and Ian were likewise afflicted. This was approaching a _Twilight Zone_ level of strange.\n\nMartin reached out with his index finger and touched the larva right on top of its squishy little head. Neither moved for at least ten seconds, then Martin stood and came over to where we waited.\n\n\"The larva will relay our predicament and request to its parent.\" The demonologist looked back to where the worm\/larva had disappeared back into the molten brimstone with a plop. \"It shouldn't take long. This particular demon at this early stage of development is still telepathically linked to the parents.\"\n\n\"How can you be sure he . . . it will relay the message?\" Ian asked.\n\nMartin shrugged. \"Demon larva like me. I guess you can say I'm good with children.\"\n\nI had no response for that, either.\n\n\"Thank you, Dr. DiMatteo,\" Rake said. \"Let's continue and pick up the pace. When Isidor realizes his efforts have been thwarted, he'll attack us in another way.\"\n\nWe walked faster, and after a few minutes, we began making progress.\n\n\"Let's hear it for Marty's tattle-worm,\" Fred muttered.\n\nRake and Ian had slowed, their full attention on a patch of shadow ahead of us, that, judging from Ian's hand hovering above Sandra's six-shooter, wasn't simply another harmless spot of dark.\n\nThe shadow moved and\u2014\n\nWait. It _moved_?\n\nWhat looked to be just another shadow started moving all by its lonesome and spread to cover the hallway from side to side. If we wanted to get past it\u2014and we had to\u2014we needed to go through it.\n\nNope.\n\nI didn't even have to consult my lizard brain on that one. My entire brain was in agreement\u2014no way was I stepping into that.\n\nFred took one step back, sharing my misgivings.\n\nNaturally, Ian didn't budge. My partner was the poster boy for determination.\n\nThe corner of the cavern wall was visible through the apparently sentient shadow.\n\nRake picked up a chunk of brimstone rock and threw it through the shadow.\n\nThe rock vanished.\n\nIt went in, but it didn't come out the other side.\n\nNope. Definitely nope.\n\n\"Alternate route?\" I asked.\n\nNo sooner were the words out of my mouth than the shadow began flowing down a narrow path, away from us, in the direction we needed to go.\n\nToward the Hellpit.\n\nFred let out the breath we'd all been holding. \"And that, boys and girl, is our engraved invitation.\"\n\n* * *\n\nWe'd all seen the glow of the Hellpit the entire way here. But even the glow and the overwhelming stench couldn't prepare us for what lay around that last turn.\n\nWe were hit with a wall of heat and sulfuric fumes coming off a Hellpit the size of my granddaddy's catfish pond that was bubbling with molten brimstone\u2014and all of it irrationally located just outside of Bacchanalia's wine cellar.\n\n\"Isn't a pit more like a hole in the ground?\" Fred asked.\n\n\"That's what I've always thought,\" I said.\n\n\"Then that's a big damn pit.\"\n\nEven more disturbing was finding the source of the snapping and cracking we'd been hearing. It was the rock floor breaking and giving way under the pit's relentless expansion.\n\nThe floor trembled beneath our feet as another few inches of the cavern floor crumbled and fell into the lagoon.\n\nThere were bones lying around the shoreline. Humanoid. Meaning human, elf, goblin, or vampire. With the exception of fangs on the vampires and goblins, the only way to know for sure would be to get them in a lab.\n\nDeath was the great equalizer.\n\n\"I think we found the missing drug dealers,\" Ian murmured.\n\nI had an unwanted flashback to the chicken bones in my bathtub. This was what the aftermath of baby demon mealtime looked like when they got hold of something big. I focused on the closest skeleton. That could have been me, except my remains would've been in my apartment and not on the shore of a Hellpit, but that was small comfort.\n\nContrary to how most humans envisioned it, the entrance to Hell wasn't in the bowels of our Earth. It was on another plane of existence. It could just as easily have opened like a door behind us, but in my opinion, nothing was a more appropriate entrance to Hell than a stinking, molten, sulfuric pit.\n\nWhen we got out of here\u2014 _if_ we got out of here\u2014the clothes I was wearing were history. No amount of washing would get the rotten egg stink out.\n\nSPI offered hazard pay to its agents. I'd been told in HSR (Human and Supernatural Resources) on my first day that since all of our work was considered dangerous, rarely did a situation arise that qualified for hazard pay. Even hunting two adult grendels and their dozens of spawn in the pitch-dark tunnels underneath Times Square didn't qualify.\n\nStill, I had to ask.\n\n\"Does storming what's basically the gates of Hell qualify for hazard pay?\"\n\nIan nodded. \"Yeah, it does.\"\n\nOh goody.\n\nNow we just had to live long enough to collect.\n\n# 32\n\nI scanned the opposite shore of the brimstone pond.\n\nWe'd been expecting demons throwing a beach party to celebrate their imminent invasion\u2014at least that's what I'd expected to see. All we'd actually seen was the demon lord's mini-me, Marty's demonic toddler, and the shadow that had a mind of its own. Not that I minded reaching the Hellpit with zero attempts on our lives, but I didn't trust it. Not that I'd know what to do if one of the locals jumped out at me. Considering what the locals were, the first thing I'd do was probably wet my pants. While embarrassing, I didn't think anyone would blame me.\n\n\"Mac,\" Ian said.\n\nThat one word contained a world of communication.\n\n\"I'm looking. Nothing and no one yet.\"\n\nI continued to scan what dry land remained that wasn't covered by brimstone. Ian didn't have to say he didn't like it. I didn't like it, either. None of us did. This setup had trap written all over it.\n\nA stalagmite wavered.\n\n_Huh?_\n\nI blinked to clear my vision. It could be the heat. It was like a sauna in here, but nothing else around the stalagmite was wavering.\n\n\"Wavy rock formation at high noon.\"\n\nIan and Rake stepped up next to me, one on each side. Fred was a solid presence at my back. Though considering what I most wanted to do was turn and run, a solid Fred right behind me wasn't good for either one of us, unless he wanted to get trampled.\n\nIf it was a veil or shield, it was the best one I'd ever seen or heard of. I wouldn't expect anything less from an elf dark mage strong enough to have opened a catfish-pond-sized Hellpit.\n\nRake's hands were at his side, glowing with the bright red of a defensive spell held in check. If this had been a dirt street in the Wild West, Rake would have been the gunslinger with his hands hovering over his six-shooters.\n\nIan had Sandra's actual six-shooter in his hand.\n\nAn elf stepped away from the front of the stalagmite, his face and form shifting from a perfect camouflage match for the rock back to his own features.\n\nDamn.\n\nHe'd been standing in plain sight the entire time like a chameleon. His breathing was what had made what I was seeing waver.\n\nIf Isidor Silvanus hadn't been about to release literal Hell on Earth, I would have been impressed.\n\nAnd yes, I knew it was Silvanus standing on the other side of the Hellpit. I didn't need an introduction. I'd seen him before. Twice. On the other side of the portals in Sar Gedeon's apartment and in the parking garage.\n\nTall, dark, pale, and evil.\n\nHis hair was black, his skin alabaster, and his eyes bright blue.\n\nRake was right. The elf was good-looking, pretty, even. Too pretty. And too perfect. If he'd been human, I'd say he'd had work done. Since he was an elf, I'd say their highborn family tree needed to add some new branches for variety.\n\nSilvanus had framed himself in front of an arrangement of thin stalagmites that bore an uncanny resemblance to a certain throne on Ord's favorite show.\n\nSomeone thought highly of himself.\n\nRake had called him obscenely powerful. Alastor had called him arrogant. I'd suggest adding vain, narcissistic, self-appointed special snowflake to that growing list.\n\nEmerging from behind the throne to stand next to him was exactly what Bert had described to Martin.\n\nA demon lord.\n\nSeven-foot tall, red skin, tail as long as I was tall, smooth back, swimmer's build, horns curved and slightly tilted toward the back. Then there was the one thing Bert had left out: glowing, yellow eyes. I don't know how he missed that.\n\nIsidor Silvanus spoke, his voice like warm honey. \"I provide you with the safest passage it is in my power to grant, and what thanks do I get?\"\n\nRake's hands glowed even brighter; now they were the color of freshly spilled blood. \"More restraint than you deserve.\"\n\nThe elf smiled. \"You took your time getting here, Rake.\"\n\n\"No, we took yours.\" The goblin met his smile and raised him two fully extended fangs. For the first time since I'd known him, Rake's fangs weren't for display only. He planned to use magic to defeat Isidor Silvanus, but if the fight got up close, I had no doubt that Rake would get personal with his incisors.\n\n\"You brought the individuals that I requested,\" Silvanus noted. \"And I didn't think you would grant even the simplest of my requests. I was wrong.\" The elf turned his attention to me. \"Miss Fraser.\"\n\n\"That would be _Agent_ Fraser to you and yours.\" My voice didn't quaver in the least. Good for me.\n\n\"Ah yes. Agent. A member of that misguided organization that passes for supernatural law enforcement on this world. And Detective Ash. You chose to ally yourself with the mortal police.\" The elf mage smiled in a show of perfect teeth, a smile that actually reached his eyes. \"I will enjoy watching your comrade-in-arms' feeble attempts to defend this city's citizens once brimstone\u2014and their blood\u2014is flowing through its streets.\" His sharp, blue eyes regarded Ian. \"Agent Byrne I have heard about from a mutual acquaintance. He sincerely regrets that your reunion was cut short on New Year's Eve, and would very much like to\u2014how do you humans say\u2014'reach out' to you in the very near future.\"\n\nIsidor Silvanus didn't say anything else, and he didn't need to. Ian knew exactly who the elf was referring to, and so did I.\n\nThat night, years ago, when Ian had first encountered the creature, it had taken the appearance of a ghoul. The creature had killed\u2014and eaten\u2014Ian's partner in the NYPD in an interrupted robbery gone wrong. Ian had joined SPI soon afterward to hunt down the thing that'd eaten his partner. When I'd seen the creature in the subway tunnels beneath Times Square last New Year's Eve, my seer vision told me that the ghoul face he'd worn then was but one of many faces and identities he'd taken over the centuries. I had seen each face, each identity, layered on top of one another, stretching back into infinity. And only last week, according to SPI surveillance, he'd been seen at the Metropolitan Museum gala.\n\nSo I believed Isidor Silvanus when he said the ghoul was still in town, waiting for the chance to get his claw-tipped hands into Ian.\n\nIan didn't move or show any sign that the elf's words had affected him in the least.\n\nI knew they had, but my partner was pushing down his emotions until he could deal with them in the way he wanted. The ghoul wasn't here to be on the receiving end of those emotions, but Isidor Silvanus was. My partner was a practical man; he'd make do with what he had.\n\n\"And Dr. DiMatteo,\" Silvanus said. \"Last, but far from least. The mortal who knows so much more about the darker realms than he should. You have been quite inconvenient.\"\n\nFred slapped the demonologist on the back. \"Hear that, Doc? You're inconvenient. Way to go.\"\n\n\"My partners and I have been forced to accelerate our plans. I requested your presence since each of you, in your own way, is to blame for that. You will be the first to experience what your world will soon become.\"\n\n\"That wasn't what\u2014\" Martin began.\n\n\"That was precisely what you agreed to, Dr. DiMatteo. In exchange for the contract, I will release Miss Poertner to you. However, I have no intention of closing the Hellpit.\" He gazed around. \"At this point, closing it would be more of a challenge than even I could overcome. Though it will be entertaining to watch the little mortal woman try.\"\n\n\"If you can't close this pit,\" Rake began, \"then what is my incentive to give you the contract?\"\n\n\"Give me the contract and you will not have to watch Miss Poertner die in one of the worst ways you could imagine.\"\n\nRake wasn't moved. \"One life saved over the lives of millions lost. You'll have to do better than that, Isidor. Again, what is my incentive?\"\n\n\"You will have a _chance_ , goblin. A chance that Miss Poertner might actually succeed where I might fail\u2014should I be inclined to attempt to close my masterpiece, which I am not. A chance was more than you had before I allowed you through that portal. Humans are such optimists, even in the face of miserable odds. It will be\u2014how do you say\u2014a 'win-win.' You and your companions die a noble death, and my partners gain unlimited access to this world.\"\n\n\"That's a crappy win,\" Fred muttered.\n\n\"For you, but not for Lord Danescu. This goblin has survived every attempt to end his life, and there have been many, including my own.\"\n\nRake shrugged. \"Everyone has an off day.\"\n\n\"But you won't be having an off day today, will you, goblin? Once again, you will fight to save your own life.\" The elf mage smiled. \"But in the next few minutes, will you fight for the lives of your companions\u2014even if it will mean losing your own?\"\n\nRake's dark eyes narrowed. \"You've wasted enough of our time.\"\n\n\"Oh, but I believe it is a fine use of time.\" The elf began walking around the Hellpit toward us. \"Your companions should know what they have welcomed into their fold.\"\n\nIan snorted. \"I wouldn't say 'welcomed.'\"\n\n\"Then you are a wiser man than I would have thought, Agent Byrne. I have been observing Lord Danescu and Agent Fraser, and have noted that the goblin goes through the motions of considering her more than merely a temporary human amusement. His performance was quite impressive at the museum last week and the caf\u00e9 a few days ago. He nearly made a believer out of me, and I know Rake far too well to be fooled. You've known him for little more than a year, Agent Fraser, and as a human, you can hardly be faulted for being deceived.\"\n\n\"Nice try,\" I said.\n\nTruth was it was a damned fine try. It was also the oldest trick in the book. Sow doubt, weaken the enemy. There were many levels of trust, and I still didn't know which ones, if any, Rake was good for. It probably depended on which way the wind was blowing. Isidor Silvanus knew we had to rely on Rake and his magic whether we wanted to or not, and he wanted to weaken what little trust we did have.\n\nThe question \"Did I trust Rake?\" had two answers: yes, and not as far as I could throw him.\n\nBoth were true. Both were Rake.\n\nGoblins were complicated.\n\n\"You and Miss Poertner are valuable commodities,\" Silvanus was saying. \"Being a businessman, Lord Danescu is quick to identify and exploit any asset he may find. Tell me, Agent Fraser, has he offered you employment?\"\n\nOnly within two minutes of meeting me.\n\n\"And when you did not accept, did his attentions turn to more of a romantic nature?\"\n\nWithin two and a half minutes of meeting me.\n\n\"Our Rake can be most persistent\u2014and patient\u2014in acquiring the things he wants.\"\n\nNot people. Things.\n\nEvery word Isidor Silvanus said was true. However, there were also grains of truth in every lie. Rake may have started out wanting me because of my seer gift, but over the past year, that had changed.\n\nOr had it?\n\nI knew what my gut told me, and my gut had never been wrong. But when it came to Rake, my heart was reserving judgment.\n\nI'd never been in love. I suspected if Rake ever had been, once he'd realized what'd happened, he'd probably run in the opposite direction like he was on fire. I think Rake liked me. I know he lusted after me\u2014and any beautiful and breathing woman. I was breathing, but I wasn't beautiful.\n\nThe only thing left was what I could do, the reason Rake had wanted to hire me the first night we'd met. The thing he lusted after.\n\nI was a seer. A good one.\n\nA valuable commodity, as Isidor Silvanus had put it.\n\nRake's motives were a mystery.\n\nBut right now, his motives didn't matter. Perhaps he truly cared what happened to our world beyond losing a strategic outpost against the elves, or he was simply too stubborn and proud to accept defeat on any level.\n\nRake Danescu was a goblin. He could balance motives like a plate spinner. But there was one thing that I did trust. Rake would never hurt me. If he thought he had a good reason, he would tell me white lies, black lies, and every-color-of-the-rainbow lies, but I knew in my gut, heart, and head that Rake would never hurt me.\n\nFor now, that was enough.\n\nIsidor Silvanus and the demon lord arrived on our side of the Hellpit.\n\nThe elf beckoned Rake to him with a wave of his hand. \"The contract, if you please.\"\n\nThe goblin made no move. \"Miss Poertner?\"\n\nSilvanus impatiently waved a hand, illuminating an area directly over the Hellpit, and dropping yet another veil.\n\nWe all looked up.\n\nOh my God.\n\nKitty was imprisoned inside a clear stalactite suspended only a few feet above the bubbling surface. Whatever it was made of, it was melting, dripping with sizzling plops into the Hellpit.\n\nShe didn't look frightened. She was furious.\n\nGood for her. Better for her if we could get her out of there without either her or us getting flash fried in brimstone.\n\n\"As you can imagine, ice\u2014especially hollow ice\u2014doesn't last long in a place like this,\" Silvanus was saying. \"I can only do so much to slow the melting.\" He held out his hand. \"The contract, Danescu. Now.\"\n\nRake casually strolled toward them, stopping less than ten feet away. It was entirely too close for comfort. Knowing Rake, that was precisely why he did it. \"I find it difficult to believe that your partner failed to put his master's copy in a safe place.\" He paused and smiled slowly. \"Or did he put it in a place that was safe _from_ his master?\"\n\nThe demon lord's eyes were glowing bright yellow.\n\nRake hit a sore spot with that one.\n\n\"You toy with your betters, goblin,\" Silvanus warned. \"There was another goblin dark mage who, astonishingly enough, approached my level of skill, but he recently got himself carried off by a particularly large demon. He only conjured demons to force them to do his will. I prefer networking.\"\n\n\"Networking? Or collusion?\" Rake looked to the demon lord. \"You, Lord Zagam, desired a way out of your realm and into that belonging to the humans. I say 'your realm' only in the sense that you reside there. You neither rule nor own it.\" The goblin smiled broadly. \"I think we all know who does. And since you do not own it, you are legally ineligible to sell, lease, or rent brimstone mining rights to anyone. And you, Isidor, along with your brother, Phaon, needed access to molten brimstone. You dislike me intensely; your brother wanted to disrupt goblin intelligence, so you chose my wine cellar to anchor the pocket dimension containing your Hellpit. I am the legal owner of that property. You sought neither my permission nor offered me an owner's share in drug profits.\" Rake's smile was slow and confident. \"Anchoring your Hellpit on my property is trespassing. Selling mining rights to a mineral you do not own is outright theft. I don't own Hell, so I have no legal recourse. I do, however, own this property. I could charge you rent, Lord Zagam. Or I could evict you.\" Rake glanced around with exaggerated distaste. \"At the very least, I want to redecorate,\" he muttered. \"But for now, I'll go with eviction.\"\n\nThe demon lord smiled as he gazed around the ever-expanding pit and cavern, ending with Kitty imprisoned in the ice. \"You are welcome to try, mortal.\"\n\n\"As a very wise teacher well-known in this dimension once said: 'Do or do not. There is no try.' I fully intend to 'do.' Alastor Malvolia was hired and paid to draw up a contract between the two of you. You call yourself partners, but there is no trust between you, hence the contract. Alastor did as he was paid to do\u2014and more.\" Rake shook his head in admiration. \"To the two of you, contracts are merely words written with ink, and aren't worth the paper they're written on if you choose to go back on your word.\" He regarded the envelope and its contents with something close to pride. \"But this little document is truly a marvel of evil magic and legal genius. Alastor not only drafted the words, he crafted the paper from both demonic and elven skin, then he mixed his own goblin blood into the ink. His words, in his blood, paper from your people, and your signatures to soul-bind you to every word on this document.\"\n\nThe demon lord smiled, showing even more sharp teeth than his mini-me. \"This is a pocket dimension, created and owned by myself and Magus Silvanus. I am not in violation of the contract as we are not in Hell.\"\n\n\"Speak for yourself,\" Fred muttered under his breath.\n\n\"You may own the pocket dimension,\" Rake continued, \"but you do not own the brimstone that is now flowing through it. You have misrepresented your rights of ownership to all of this brimstone. According to the contract, that means the brimstone's true and legal owner is entitled to collect damages or recompense in any manner he chooses. I don't believe His Dread Majesty will be pleased to discover that his trusted chancellor profited from the sale of his property.\" Rake's dark eyes landed on Isidor Silvanus, and a faint smile curled one corner of his mouth. \"Or that an outsider knowingly purchased said property and exploited its use for additional gain, making you both equally guilty of grand theft. Only later did you discover Alastor's trickery in drawing up the contract\u2014and you killed him for it.\"\n\n\"I should have ensured the goblin lawyer was conscious and then cooked him at a lower temperature.\" Isidor Silvanus smiled indulgently. \"But what's done is done. Now here you are with the original\u2014and sole remaining\u2014copy. As usual Rake, you do far too little, too late.\"\n\nThe pointed base of Kitty's icicle prison ran water in a steady stream. Silvanus's concentration was wavering, and Kitty's prison was melting faster.\n\n\"You say you prefer networking,\" Rake continued as if Kitty had all the time in the world. Son of a bitch. \"Networking has its place, but so does rendering mutually beneficial favors.\" He raised his voice to a ringing shout. \"Have you heard enough, Dread Majesty?\"\n\nA red forearm the size of Rake's entire body emerged from the brimstone right beside the rock he was standing on. It rippled with lean muscles, and each long finger was tipped by a sharp, black nail. Thankfully we couldn't see the rest of it, but from elbow to fingertip, it looked just like the demon lord, albeit ten times his size.\n\nIf size meant higher on the power ladder in Hell, then Isidor Silvanus's demon lord pal was this big guy's bitch\u2014or if he wasn't already, he was about to be. The thumb and forefinger extended toward Rake like he was about to pinch the goblin's head clean off his shoulders.\n\nRake didn't flinch, but coolly reached out and put the contract between the two fingers.\n\nThe demonic fingers pinched closed and submerged beneath the bubbling surface.\n\nA collective, disbelieving gasp came from all of us.\n\nIsidor Silvanus's was more on the horrified end of the spectrum.\n\nThe demon lord looked ready to faint.\n\nOh yeah, someone was in trouble.\n\n\"No need for concern,\" Rake told us. \"The paper content is seventy percent demon skin, making it hellfire and brimstone proof.\"\n\nConsidering that Kitty was inside a melting icicle over a Hellpit, I didn't give a rat's ass that the paper was seventy-percent recycled demon.\n\n\"Dammit, Rake, hurry up!\" I whispered.\n\n\"While your master is reviewing the contract,\" Rake said, \"apparently for the first time\u2014do be reasonable and release Miss Poertner.\"\n\nThe demon lord inhaled, turned to Kitty's icicle, and blew freaking fire directly at her. What was left of the ice kept Kitty from bursting into flames, but the icicle was history.\n\nKitty fell, screaming.\n\nRake caught her.\n\nHe didn't run across the surface of the brimstone and catch her as she fell\u2014though that would have been impressive, too. He extended his arm, spread the fingers of his hand, and her fall stopped.\n\nThat would be magic.\n\nKitty's eyes were as wide as saucers at the sensation of dangling in midair over a pit of molten and popping brimstone.\n\nIsidor Silvanus threw a fist full of acid-green fire at her only to have it deflected by the bubble-like shield Rake had wrapped around her.\n\nKitty screamed again.\n\nI didn't blame her. Though this scream was less fear and more rage at being held in the air and used for target practice.\n\nThe elf dark mage simply chose another target.\n\nUs.\n\nIsidor Silvanus clenched his hands into fists, brought them sharply together, then wrenched them apart.\n\nAnd the rock beneath our feet snapped apart like slabs of ice from an iceberg, putting me, Ian, and Martin each on a hula-hoop-sized personal island, surrounded by, and floating\u2014not so well\u2014in, boiling brimstone.\n\nThe jolt knocked me off my feet, and the sudden shift in weight tipped my slab of rock and nearly tossed me over the side. I desperately grabbed the edge, brimstone spitting like Hell's bacon grease on my hands and face.\n\nI screamed. I didn't want to, but I couldn't help it.\n\n_This_ was why Isidor Silvanus wanted us here. Rake couldn't save all of us, and the elf knew\u2014despite what he said about Rake only caring about himself\u2014that he'd try to save some of us. More hostages, more distractions, more chance of success for Silvanus. He'd thought about what could go wrong and he'd covered all of his bases.\n\n\"This is an unwanted complication,\" Martin noted.\n\nRake quickly gave the elf a taste of his own medicine.\n\nThe ledge where Isidor Silvanus was standing suddenly broke away from the cavern floor and tilted sharply down toward the Hellpit. The elf mage had to scramble to stay on his feet. Rake used the distraction to pull his extended hand to his chest, bringing Kitty with it. This time, he did catch her in his arms. Kitty didn't look any happier now than she had while dangling.\n\nThe demon lord bellowed in rage and flicked his clawed hand at Rake and Kitty. A sickly green blur formed in the air, coalescing into a massive snake, its head rearing far above Rake's head. The snake launched itself at them.\n\nRake barked a single word, and a shield of shimmering red appeared between he and Kitty and the snake. The serpent's head struck the shield with a frustrated hiss. The shield buckled but held. Barely.\n\nEven if Rake hadn't had his arms and hands full, we had a worse problem that even the most hotshot mage couldn't magic away.\n\nI thought my eyes had to be playing tricks on me, but they weren't. The brimstone's level was going down.\n\nThe Hellpit was draining.\n\nInto Hell.\n\nAnd taking us with it.\n\n\"This is bad,\" Martin said. \"Once the pit drains, we'll be in Hell and any demon that wants to come into this world can do so.\" For the first time I saw fear in Martin DiMatteo's eyes. \"And the Hellpit will be permanently open.\"\n\nThere were only a few feet between us and the rim of the Hellpit. The newly exposed rock steamed at the contact with the cooler air, rock that only seconds before had been under molten brimstone. Martin was closest to the rim. He could make it if he jumped now.\n\n\"Dammit, Marty!\" Ian roared. \"Jump!\"\n\nWith a defiant squeak, Marty leapt, just clearing the distance between his sinking slab and the cavern floor, both feet making a surprisingly solid landing.\n\nIan's slab was a few feet behind Marty's. There was no way I could make that jump, but Ian could. Both of us didn't have to die.\n\n\"Go!\" I shouted over the chaos. \"You can't help me from there!\"\n\nIan jumped. One foot made it over the top. His left boot caught in the steaming rock, and the leather caught fire. Fred grabbed a double-handful of Ian's leather jacket and pulled with everything he had. He and Ian landed in a heap on the cavern floor.\n\nThe slab that was taking me down to Hell like my own personal elevator had moved too far from any shore. There wasn't any direction that was a jumpable distance. Even if I could clear the distance, any part of my body that touched that shoreline would be instantly flash fried. Once the brimstone drained, I'd be an appetizer for the demons waiting at the bottom for the feast that was New York.\n\nWe had guns, we had knives, but we didn't have a fire-proof climbing rope.\n\nMy line of vision was now below the rim of the pit, but gunfire and flashes of red and acid green light accompanied by explosions and falling rock told me that everyone else was busy simply staying alive. The heat was overwhelming. I had to keep breathing, but each breath seared my mouth, throat, and lungs. I felt like I was cooking from the inside out. My grip on the slab began to slip.\n\n\"Mac!\" came a shout from above.\n\nI weakly raised my head.\n\nIan was on his hands and knees, leaning out over the pit. \"Help's coming, Mac! Hang on!\"\n\nA moment later all I could do was stare in openmouthed horror as Rake Danescu\u2014protected only by the red glow of his personal shields\u2014dove into the molten brimstone surrounding me.\n\nHe surfaced seconds later next to my slab, intact and not burned to a crisp, though he was sweating.\n\nI was beyond words, not only because I couldn't breathe for the heat, but from seeing Rake treading brimstone like water. I must have been dying _and_ delirious.\n\nRake reached up and grabbed my forearm, his hand cool and soothing. How could . . .?\n\nI blinked the sweat out of my eyes and looked down.\n\nRake's hand was glowing with his shielding spell\u2014and now, so was my arm.\n\nThe glow spread until my entire body was encased in its protective field.\n\nMy vision began to clear, and I could breathe again.\n\n\"Let go of the rock, Makenna,\" Rake was saying.\n\n_What?_ \"Are you\u2014\"\n\n\"Crazy? Kidding? Neither. I can't hold against this vortex for long. I'll swim us over to the wall.\"\n\nMy only other choice had me waiting to be sucked into Hell. Die in Rake's arms or be ripped apart by demons that never learned to share their food?\n\nI let go of the slab and slid into Rake's arms.\n\nAnd into the brimstone. Brimstone that amazingly felt no hotter than hot bathwater.\n\nRake flashed a quick grin as he held me tightly against his chest. \"Like being in a hot tub, except we're not naked.\"\n\nMy mouth was parched. I swallowed and panted. \"If that hot tub . . . was draining into Hell.\" I thought for a moment. \"Why didn't you pull . . . me out like you did with Kitty?\"\n\n\"That trick's one shot only.\"\n\nThe goblin was breathing heavily from keeping our heads above the brimstone and fighting the force of the whirlpool at the center of the pit that was beginning to pick up more speed.\n\nRake had expended an incredible amount of magical energy. Catching Kitty, fighting Silvanus, shielding the two of us\u2014it all picked that moment to catch up with him. The current grabbed us both, sweeping us away from the walls and toward the pit's now churning center.\n\nI couldn't think, I couldn't react, and I had no air to scream.\n\nRake's grip around my waist and back never lessened.\n\nA swell of brimstone passed between us and the vortex. Something was swimming just beneath the surface. Something huge.\n\nA white worm as big around as a pair of fifty-five-gallon drums breached the surface like a whale. The massive head swiveled and two pitch-black eyes the size of a man's fist focused on us.\n\nIt was a larger version of Marty's demon toddler.\n\nOne of the parents. Or _the_ parent, depending on how demonic white worms reproduced.\n\nI'd given up trying to make sense of anything I'd seen since stepping through that portal. I stared in dumbfounded amazement. It was all my stunned mind could do.\n\nRake's cough sounded like a laugh.\n\nThe giant worm submerged, and my stomach tightened at what I knew it was going to do.\n\nOh crap, crap, _crap_!\n\nRake tightened his grip on me. \"Hang on, darling.\"\n\nThe worm surfaced again right next to us, gently but forcefully nudging us away from the vortex and toward the nearest wall. I'd heard of dolphins supporting drowning swimmers and pushing them toward shore. I never expected to experience it with a dolphin in the ocean, much less with a demon worm in a brimstone whirlpool.\n\nThe worm held us against the wall with its broad head until we could get a few much needed breaths.\n\n\"Get on my back,\" Rake told me.\n\n\"What\u2014\"\n\n\"I'm climbing out . . . you on my back.\"\n\n\"How can you\u2014\"\n\n\"Muscle . . . and magic.\"\n\nAt least the last part made sense.\n\nI looked up at the jagged Hellpit wall. There were enough hand- and footholds, but it was completely vertical. Goblins were stronger than humans, so Rake would have been able to do it if he was at full strength, but he wasn't. Plus, he'd be carrying me like a backpack and maintaining the shields that were all that was keeping both of us from bursting into flames.\n\nI looked from the wall into Rake's eyes. The reasons why this was impossible were limitless. The other options we had were none.\n\nI chuckled, though it sounded like some kind of vocal spasm. \"Do or do not.\"\n\nRake gave me a crooked grin. \"There is no try.\"\n\nI got on Rake's back, wrapping my legs around his waist, my right arm around his middle, my left arm over his left shoulder, clasping my hands over the center of his chest. It wouldn't do either one of us any good if I choked him on the way out of here.\n\nThe momma worm submerged, leaving us.\n\nWith his hands and feet glowing even brighter than the rest of him, Rake Danescu actually climbed out of Hell carrying me on his back.\n\nIan and Fred were there along with Kitty and Martin to pull us out, but there was no sign of Isidor Silvanus or his demon lord partner.\n\nIan pulled me to my feet and didn't let go.\n\n\"Where's Silvanus?\" I asked him.\n\n\"He ran. Rake could have either chased him down, or gone after you.\"\n\nIf Silvanus hadn't run, Rake would have been forced to stay and fight\u2014and I would have been sucked down a brimstone vortex into Hell.\n\nThank you for being a coward, Isidor Silvanus.\n\nI gave a little sickly grin. \"I like his choice.\"\n\n\"It was a damned fine one.\" Ian's gaze searched my face. \"You okay?\"\n\n\"Better when we get out of here.\"\n\nIan hadn't said a word about not being able to help me. I knew he didn't like Rake Danescu, but there were times like now when personal feelings had to be tossed to the curb.\n\nSuddenly my ears popped. Painfully. I could actually feel the hot air around us pressing down on my body. I took a quick glance around. Everyone looked similarly pained and confused.\n\nRake put his hands to his ears. \"Isidor has used his 'back door,' as Agent Byrne called it, to leave this pocket dimension.\" He snarled in frustration at not having the elf's neck between those hands. \"He just took the first step in collapsing it.\"\n\nFred swore. \"We'll be trapped.\"\n\nRake shook his head. \"It'll simply force the demons out of the Hellpit faster. They won't miss their chance to get out into the world.\"\n\nI had news, neither would we.\n\n\"I have to close the pit now,\" Kitty said. \"Quickly.\"\n\nWe'd all done everything we'd come to do and could do. Taking in our singed, burned, and battered selves, it was obvious that hadn't gone too well for us. Now it was Kitty's turn. Stop the worst demons that Hell could hork up from invading New York and then the world. All the little baker had to do was close a pond-sized Hellpit. A Hellpit that within minutes, if not seconds, would be drained and permanently open, with demonic hordes streaming out of it.\n\nAnd she had to do it all without an anchor. Vivienne Sagadraco may have located one, but they were somewhere out there, and we were in here.\n\nNo pressure.\n\nRake turned back to face the Hellpit.\n\n\"Go,\" he told the rest of us, including Kitty.\n\nAs much as all of us would've loved to have done just that, none of us did.\n\nKitty quickly stepped in front of Rake and strode to the rim of the Hellpit. She extended her arms out over the pit, palms down.\n\nThe goblin was incredulous. \"What are you doing?\"\n\n\"My job.\"\n\n\"It's too big now for you to close.\"\n\nKitty's glare was withering.\n\n\"By yourself,\" Rake added.\n\nRake could be wise, too.\n\n\"Are you offering to anchor me?\" she asked.\n\n\"I am.\"\n\nKitty hesitated. \"What are your qualifications?\"\n\n\"I'm here.\"\n\nKitty looked at me, her question there but unspoken.\n\nYes, Rake was here, but could she depend on him? He'd been there for me, but would he stay there for her?\n\nThe anchor mages who had worked with her in the past either didn't have the strength for the work or the balls for the danger.\n\nWould Rake Danescu stand with her until the end\u2014whatever that end might turn out to be\u2014or would he bolt like all the others when she needed him most?\n\nShe needed to know if she could trust him.\n\nShe wanted to know if _I_ trusted him.\n\nRake's eyes were on me. So were Kitty's.\n\nI nodded to them both.\n\n* * *\n\nIf Kitty had been able to turn the incredible power she was using now to close the Hellpit against Isidor Silvanus, the elf mage would've been a greasy spot in SPI's parking lot, she wouldn't have been kidnapped, and we never would have had to set foot inside the pocket dimension. We could have done all this from Rake's wine cellar, and then celebrated by popping open a couple of bottles. Heck, after this was over, even I wanted a drink.\n\nThe Hellpit was about half the size that it had been, when Kitty went pale and started shaking, and the sounds from below increased in volume and intensity: howls, shrieks, roars, and screams. After being kidnapped by an elf dark mage and spending who knew how long encased in an icicle, Kitty's body was only human, and it had had enough.\n\nRake stepped up behind her and simply placed his hands on her shoulders.\n\nMy seer vision couldn't detect what was passing between them, but I assumed Rake was boosting her power with what remained of his own.\n\nHowever, as the Hellpit continued to close, the distance between us and the wine cellar portal grew.\n\nIan saw the same thing I did and scowled, his only reaction to possibly being trapped in a collapsing pocket dimension with an only mostly closed Hellpit. Neither of us said a word or moved, doing nothing that could distract Kitty and Rake from their task of what was the magical equivalent of world-record powerlifting.\n\nMartin DiMatteo was fearfully looking up at the cavern ceiling.\n\nWhen a man who took field trips to Hell was afraid, the time to panic had officially arrived.\n\n\"Lord Danescu,\" he said on the barest whisper, \"the dimension is being elongated.\"\n\nKitty never took her eyes and focus off of the work in front of her, but Rake took a quick glance up, and bared his teeth in a silent hiss.\n\nThe realization of what Martin meant sank in and suddenly there wasn't nearly enough air.\n\nIsidor Silvanus had created this pocket dimension, so he could manipulate it. The rules in here were his rules\u2014and basic physics. The dimension was anchored in two places: Hell and Bacchanalia. Two ends of a long and narrow balloon.\n\nAs we watched, the stone ceiling of the cavern was being stretched as if the two ends of that balloon were being simultaneously pulled apart. It was what you did to make balloon animals. The balloon started off as a snake, which was the only kind of animal I'd ever been able to make. But in experienced hands, it could be twisted into a poodle. I had no doubt that Isidor Silvanus and his demon lord partner were master manipulators. Yes, they'd cut their losses and run, but if they couldn't release demons into our world, they'd cut us off from any hope of getting home, trapping us with what demons managed to escape the Hellpit before Kitty and Rake could get it closed. Or if they got it closed, the elf would be fine with trapping us in a tiny pocket dimension, forever cut off from any help, until our air ran out or we cooked from the heat, whichever came first.\n\nThen another ugly realization hit. As badly as he wanted revenge against Rake, that just might have been his plan all along.\n\nLure us in, cut us off.\n\nI didn't want to die inside of a poodle ear.\n\n\"Rake,\" Martin said, dropping formality and the goblin's title. \"We need to run. Now.\"\n\n\"It's not closed,\" Kitty managed between clenched teeth. \"They can still get out.\"\n\n\"It's close enough,\" Rake said. \"And in a few moments, it won't matter.\"\n\nKitty was aghast. \"I can't just let it go!\"\n\n\"Just drop\u2014\" Fred began.\n\n\"The recoil,\" Rake said in realization. He readjusted his grip on Kitty's shoulders. \"Let's ease back on it\u2014\"\n\nMartin was still looking up. \"No time.\"\n\n\"No choice!\" Rake snapped. \"All of you go. We'll be right behind you.\"\n\nNone of us moved.\n\n\"Stubborn humans,\" the goblin hissed.\n\n\"That would be us,\" Fred drawled. \"Well, at least half of me.\"\n\nI wanted to run, but I'd settle for pacing, except it might distract Kitty and Rake. So I just stood there, twitching. I couldn't get any sense that they were doing anything except standing like statues staring at a Hellpit.\n\n\"Hurry,\" Martin told them, his voice amazingly calm.\n\nRake murmured a few words in Kitty's ear as she slowly lowered her arms.\n\nAnd a triumphant howl from countless demonic throats came out of the smaller\u2014but not small enough\u2014pit and filled the cavern.\n\n\"Run!\" Rake shouted.\n\nWe all did. Rake and Kitty included.\n\nWe didn't look back. I didn't want to. If a demon was going to catch up with me, it'd happen. I couldn't run any faster, and if a demon did catch me, I'd get a good look at him while he killed me.\n\nI had yet another in a long line of horrible thoughts. What if Isidor Silvanus had closed the portal into Bacchanalia's wine cellar? Sandra and her team were there, but if the elf wanted to slam the door in their faces, there'd have been nothing they could've done about it.\n\nI shoved that thought aside. I'd tear into that portal with my teeth if I had to. Besides, we had Kitty. If she could close a Hellpit, there hadn't been a portal made that she couldn't rip open.\n\nAn unearthly shriek came from right behind us, immediately followed by a ground-shaking impact, Ian's shouts, and Rake's snarls. I stopped and turned.\n\nIan and Rake had been bringing up the rear, and now they were paying the price.\n\nA demon had Rake on his back pinned to the path, holding the goblin down with only one massive clawed hand. The demon was blue, bald, and had biceps the size of the goblin's head.\n\nNormally, big wouldn't have mattered with Rake. He had enough preternatural strength and speed to give that demon a run for his money.\n\nBut Rake was exhausted, physically and magically; still, he wasn't giving up.\n\nI drew a knife. I couldn't kill it, I probably wouldn't even hurt it, but I wouldn't stand by while Rake was ripped\u2014\n\nNeither would Ian.\n\nMy partner had long run out of bullets for his six-shooter, but he had a pair of long knives in his hands and was moving faster than I'd ever seen him move, darting in, striking, and making each cut count. I knew what he was doing. The demon's nails were basically five-inch-long claws, and he knew how to use them. Ian couldn't get close enough for the kill, so he was trying to do enough damage to force the demon to turn and defend himself, giving us enough time to drag Rake out of there.\n\nThen Ian would be facing the demon's full wrath.\n\nThe stone path shook, and it wasn't from the collapsing pocket dimension.\n\nDemons, a wall of demons, were charging toward us.\n\nAnd my partner had his back to them.\n\n\"Ian!\" I screamed.\n\nThe blue demon half turned, and seeing his comrades bearing down on us, smiled in a show of elongated shark-like teeth.\n\nHe shouldn't have stopped to be happy.\n\nRake had one knife left, and he used it, slicing the demon's wrist, nearly severing it from the clawed hand still holding him down. The demon howled, and Ian dove under his guard, and with two slashing motions, hamstrung him.\n\nThe demon went down, and Ian pulled Rake to his feet.\n\nWe got Rake's arms across our shoulders and our arms around his waist, and together we dragged the goblin and ourselves through the portal Kitty was holding open for us.\n\nNo one had told us what to expect when escaping from a Hellpit inside a collapsing pocket dimension. Then again, I didn't think anyone knew.\n\nAll things considered, being in Hell's anteroom one second and stepping with a gooey plop onto the floor of Rake's fancy wine cellar the next was a small price to pay. I could have done without being coated in exit portal ecto-goop, but I wasn't going to quibble.\n\nRake couldn't see through the blood that'd run into his eyes from the gash on his forehead.\n\n\"Are we out?\" he asked weakly.\n\n\"You bet,\" Ian told him.\n\n\"How?\"\n\n\"Our muscles,\" I told him, smiling. \"No magic.\"\n\nWithin a minute, the goblin was more or less on his feet, insisting to us and Sandra's team medic that he was fine.\n\nWe stubborn humans had nothing on stubborn and proud goblins.\n\nRake leaned against the wall beside a wine rack, the front of his leather jacket and the shirt underneath hanging in shreds from the demon's claws. There was a lot of blood, but the cuts appeared to be superficial. Lucky for Rake, the demon that attacked him wanted to play with his food first.\n\nHe would have stayed on his feet if it hadn't been for the truly bad combination of ecto-goop on a marble floor.\n\nRake's feet slipped out from under him and he stumbled against the glass wall of one of his wine racks.\n\nHis shoulder barely bumped it, but it was enough.\n\nPride goeth before the fall.\n\nHe was fine. The wine? Not so much.\n\nThe bumped wine rack tilted and tapped against the first section of glass wall holding dozens of bottles of his priceless wine. That section of wall fell, triggering a domino effect until the entire cellar was a sea of wine and broken bottles.\n\nIt looked like every bottle of wine was broken, and Rake had done it himself.\n\nThe goblin started to laugh, but Fred had beat him to it.\n\nAbove it all, I could swear I heard the hissing laughter of Alastor Malvolia.\n\n# 33\n\nBACCHANALIA was still standing\u2014at least on the outside.\n\nWhen we'd all gotten out of the building, we could hear the rumblings of the club's marble floors collapsing into the hole that had opened where Rake's wine cellar used to be. Even though the pocket dimension hadn't occupied much physical space outside of the wine cellar's wall, its implosion caused enough of a disturbance to bring on a partial collapse.\n\n\"What I'm saying is that all demons aren't bad,\" Martin was telling Fred. \"They just live in a bad place.\"\n\nThe Con Ed folks were calling it a sinkhole. The supernaturally clued-in Con Ed people had \"discovered\" the gas leak on the other side of the wall from Bacchanalia's basement in an old drainage tunnel. The line had broken as a result of the sinkhole that had opened beneath the tunnel and spread into the exclusive sex club.\n\nBaxter Clayton was broadcasting live from a block away. The corner of Bacchanalia's building was just visible in the background. Baxter had his cameraman keep both him and the building in the shot as he talked. The anchor was probably praying to God that Bacchanalia collapsed while he was on the air.\n\nBaxter Clayton's prayer wasn't answered.\n\nThe Hellpit was gone, and the pocket dimension along with it, but it'd left its stench behind, which was being explained by a sewer line break.\n\nThat was one hell of a sewer line.\n\n* * *\n\nA few weeks later, Kitty's Confections was back in business. Same building. New kitchen equipment courtesy of Rake Danescu.\n\nI was glad she'd stayed put.\n\nIt helped that Dr. Carey had determined in the autopsy that Alastor Malvolia hadn't died in her oven.\n\nI was on my way to Kitty's shop. She and I were going to do a much-deserved girls' night out. I was still staying in the VIP apartment at headquarters while my apartment was being cleaned and repaired. It was almost finished, but I hadn't decided whether to accept Rake's offer of a no-strings-attached apartment, or to stay in my old place. After SPI's cleanup crew was done with it, no one would ever know that it'd been temporarily sublet by a pack of carnivorous baby demons. I would know, but I hadn't decided if knowing and remembering was enough for me to pack up and take my worldly goods elsewhere. It was a nice apartment in a good neighborhood. I liked my neighbors. It was home. But it didn't matter how much that bedroom had been scrubbed, how many coats of paint had been put on the walls, or how plush the new carpet was, if Ian hadn't been there . . .\n\nI pushed all that out of my head. I didn't have to decide tonight. Tonight was about fun.\n\nRather than meet at headquarters, Kitty asked me to come over to the shop. She said she had something to show me.\n\nI took the subway and walked the block to the shop. It had just started to snow.\n\nWhen I walked around the corner onto Bleeker Street, I stopped in my tracks, not believing what I saw in Kitty's front window.\n\nGingerbread.\n\nNot just one house. An entire Victorian Christmas village filled the shop-front window. The details were incredible. The houses even had tiny translucent sheets of sugar for windows.\n\nA young family had stopped to admire Kitty's masterwork. A little boy and girl, who couldn't have been more than five years old, and were just tall enough to see the village at eye level.\n\nThey were mesmerized.\n\nWhile the children's attention was occupied, Kitty caught the mother's attention from inside the shop, and quickly held up two cookies with a questioning look.\n\nThe parents smiled and nodded.\n\nWhen Kitty came out of the shop with two gingerbread cookies in her hand, the kids went from mesmerized to jumping-up-and-down thrilled.\n\nKitty was pretty thrilled herself. Actually, she was better than thrilled; she looked content, the glowing kind.\n\nI grinned. \"Gingerbread,\" I said when the family continued down the snowy sidewalk, the kids biting the heads off the gingerbread men.\n\n\"Gingerbread.\" She smiled and shrugged. \"It was time.\"\n\n\"The perfect time.\"\n\n\"I'm a baker. It'll be Christmas before we know it. It needed to be done.\"\n\nTruth wasn't all that could set you free. Gingerbread worked wonders, too.\n\nYes, her three-greats-grandmother had been _that_ witch, but Kitty had faced her demons\u2014literally\u2014and had accepted that gingerbread baking was part of her heritage, and one bad cookie in the batch wasn't reason to abandon what you were good at.\n\n\"Come on in,\" she said. \"I have a few things to finish up in the back, and then we can go. I'll fix you a hot chocolate while you wait.\"\n\nKitty opened the door and went inside. I paused, sensing someone watching me.\n\nI turned to look across the street.\n\nA tall figure in a long, dark coat stood beneath a streetlight. The young family crossing the snowy street saw a tall, dark, and absurdly handsome man. I saw a tall, dark, and absurdly handsome goblin.\n\n\"Your new partner's here,\" I told Kitty.\n\nAgain that mysterious smile. \"He's not mine.\" She continued inside. \"Hmm, I'd better make that _two_ hot chocolates.\"\n\n\"Rake likes hot chocolate?\"\n\n\"I found that out last week while he was helping with the installation.\"\n\n\"Helping?\"\n\n\"Helping. And he's mad for double chocolate chip cookies.\"\n\nI waited by the door as Rake crossed the street.\n\nIt wasn't his usual big cat, predatory stalking walk. Rake Danescu crossed the street like a normal man. Hands in the pockets of his coat, dark head slightly down against the snow that was coming down harder now.\n\nAfter what I'd witnessed under Bacchanalia, I knew Rake was as far from normal as it was possible to be.\n\nThat knowledge didn't bother me. I smiled. Though I was going to have trouble reconciling the mage who hauled me out of a Hellpit and then helped Kitty close it, and the mage who just might know Lucifer\u2014or at least one of his generals\u2014with the man who loves hot chocolate and chocolate chip cookies.\n\n\"Hi,\" I said.\n\n\"Hi, yourself. Is Kitty inside?\"\n\n\"Oh . . . um, yes.\" No come-ons? No bedroom eyes? If we SPI agents are anything, we're adaptable. Just go with it, Mac. \"She said she had some things to do in the back.\" I opened the door. \"She was going to fix us some . . .\"\n\nTwo steaming mugs of hot chocolate were waiting at a small corner table with a plate of chocolate chip cookies. The store lights were off. The window lights were on, illuminating the gingerbread village, and one of those tiny lamps that you'd find in those romantic restaurants glowed beside the plate of cookies.\n\nKitty was fixing things, all right.\n\nRake's dark eyes gleamed in the dim light. \"You think it's a trap?\"\n\n\"Oh, I know it's a trap.\"\n\n\"Shall we trip it?\"\n\n\"If Kitty went to all this trouble, we'd better. Besides, I want that hot chocolate.\"\n\nRake didn't say what he wanted, but his eyes were doing a fine job of saying it for him.\n\nNow _that_ was the Rake I knew.\n\nI took an exploratory sip of the hot chocolate. Too hot. I blew on it. \"Thank you for taking care of Kitty's kitchen.\"\n\nRake nodded. \"I offered another store location, but she wanted to stay here.\"\n\n\"I'm glad she did. Don't let anyone run you out of a place you love.\"\n\n\"Have you decided what to do about your apartment yet?\"\n\n\"I figure I'll wait until the work's finished, then go over and see what kind of vibes I get. Homey vibes or _Amityville Horror_ 'Get Out' vibes.\"\n\n\"My offer still stands.\"\n\nI reached for a cookie. \"I know. And I appreciate it.\"\n\n\"But you won't accept it.\"\n\n\"Probably not.\"\n\n\"Why?\"\n\nHe seemed genuinely confused. Maybe a goblin woman wouldn't feel odd about accepting an apartment from the playboy owner of a sex club. Then again, I didn't have any evidence about Rake's playboy status. I'd always just assumed. Maybe I should ask.\n\nI felt suddenly awkward. \"I think I'm still a small-town girl. Old-fashioned. At least when it comes to accepting a luxury apartment from a mysterious goblin billionaire.\"\n\nThe edge of a smile appeared. \"I'm mysterious?\"\n\n\"And that's putting it nicely.\"\n\n\"What if I told you that deep down I'm an old-fashioned guy.\"\n\n\"Don't make me choke on my cookie. You're a spymaster. I'm sure you've had practice being just about everything.\"\n\n\"What if I told you I wanted to have more practice being someone your grandmother would approve of?\"\n\n\"You couldn't practice enough to fool her. Advance warning: her favorite weapons are a shotgun, a cast-iron skillet, and a butcher knife. Usually two at the same time.\"\n\nHe didn't even bat an eye. \"Duly noted.\"\n\nIt didn't look like Rake was afraid of Grandma Fraser. He should be. Passing a contract to the devil himself for his reading enjoyment was one thing, ticking off Grandma Fraser was just plain dangerous.\n\nI smiled. If that ever happened, I wanted a front-row seat.\n\n\"I've had plenty of experience persuading women not to kill me,\" Rake said.\n\n\"I'm sure you have.\" I took a sip of hot chocolate. \"What about Bacchanalia?\"\n\nRake shrugged. \"It'll go down as another New York club that had its time in the spotlight and then faded away.\"\n\n\"Or in this case was swallowed by a Hellpit.\" Then I snorted a laugh. \"I'm sorry; I can't help it.\"\n\n\"You're not in the least bit sorry. Yes, I know. The irony is priceless. A den of sin gets swallowed by Hell itself.\"\n\n\"So you're taking that as a sign and aren't rebuilding?\"\n\n\"No, I'm taking it as what you would call 'a blessing in disguise.'\"\n\n\"Pardon?\"\n\n\"The reason that Hellpit wasn't found until it was nearly too late was because I hadn't been to the club in over a week. Businesses like Bacchanalia require hands-on management.\"\n\n\"Pun intended?\"\n\n\"Actually, no. If I had been there like I should have been, I would have known the moment that pit first started to open. But I wasn't, so I couldn't, and as a result . . .\"\n\n\"Hell ate your building.\"\n\n\"Quite so.\"\n\n\"So what are you going to do now?\"\n\n\"After my lawyers and I fight with the insurance company over a settlement, I'll take my money and invest it in a business or a building that I don't have to be personally involved in with day-to-day operations.\"\n\n\"How about your employees?\"\n\n\"Those who need assistance, I'll see to it that they find alternate employment, but most of them have already been offered well-paying jobs. I spent a good deal of time while running Bacchanalia fending off other nightclub owners who were constantly trying to steal my staff.\" He shrugged. \"Now, they have what they wanted.\"\n\n\"I think you also gave Alastor Malvolia what he wanted.\"\n\nI hadn't been the only one who'd heard the goblin lawyer laugh in Rake's late, great wine cellar.\n\n\"Isidor got away.\"\n\n\"For now. And thanks to you and Al, he's got bigger trouble than you hunting him down.\"\n\n\"I want confirmation.\"\n\n\"So call up His Dread Majesty again and ask.\" I hesitated, suddenly uneasy. \"Do I want to know how you contacted him to let him know to eavesdrop?\"\n\n\"It didn't involve a virgin sacrifice, if that's what you're asking.\"\n\n\"You'd have never made it to Bacchanalia in time if you'd had to run around looking for one of those.\"\n\n\"Probably true. Let's just say I have a few highly placed connections in warm climates, and leave it at that.\"\n\n\"Fine with me.\"\n\n\"Though I'm going to be busy enough with all of Alastor's other instructions.\"\n\nI gave him a questioning look.\n\n\"In addition to seeing to it that his copy of Silvanus's contract was brought to me, Alastor named me the executor of his estate\u2014and then left me most of it.\"\n\nI barked a laugh. \"Not bad from a guy who didn't like you when he was alive.\"\n\nRake wearily leaned back in his chair. \"I haven't put a dint in wading through Alastor's corporate holdings.\"\n\n\"A lot?\"\n\n\"Oh yes. I'm certain I'll find things he willed to me that would be the last thing I would want to own. A man who hated you will leave you some nasty surprises.\"\n\n\"Do you really think he hated you?\"\n\n\"I may have had his respect on occasion\u2014\"\n\n\"That wasn't what I asked. Did he have any family?\"\n\nThat earned me a puzzled look. \"None that were referenced in the will.\"\n\n\"No family, no children. He had a lot but no one to leave it to.\" I smiled. \"We can't pick our family, but we can pick our friends. I think Al Malvolia just might have considered you a friend, or perhaps even the son he never had. He obviously respected you. That's one small step from admiration. You're just as sneaky as he was, maybe even more so. He had to admire that. It's something to think about.\"\n\nRake's lip curled. \"I'd really rather not.\"\n\nHe said it, but I didn't think he really meant it. I was certain Rake had done plenty of thinking since last Wednesday, about a lot of things.\n\n\"I think Al's fixed it so you won't have any choice. Sneaky and manipulative from beyond the grave. But deep down, he knew he could depend on you to do what needed to be done. So did Kitty.\"\n\n\"Once you told her.\"\n\n\"What do you mean, once I\u2014\"\n\n\"I have sisters. I know all about nonverbal communication between women. Kitty wanted to know if she could trust me. That nod of yours assured her that she could.\"\n\n\"Kitty was looking for trust, but she needed bravery. You dove headfirst into a Hellpit of brimstone, and then carried me out like a backpack. I'd call that brave.\"\n\nRake gave me a sheepish grin. \"Or insane. I'd used that shielding spell before for fire, but brimstone isn't exactly something you get an opportunity to practice with.\"\n\nI blinked. \"You didn't know if it would work?\"\n\n\"I suspected it would. Strongly suspected.\"\n\n\"Maybe that makes you insanely brave. Either way, any man who'd dive into a Hellpit is certainly up for a little Hellpit slamming.\"\n\nRake's sheepishness spread into a full grin. \"I didn't slam that Hellpit, either. That was all Kitty.\"\n\n\"But you\u2014\"\n\n\"Didn't give her a power infusion. For one, by then I didn't have it in me to give. I barely had enough to stay on my feet. Kitty needed to do as much by herself, unaided, as possible. Magic takes strength, but it also takes confidence. You have to believe that you can do it. Kitty had had too many anchor mages fail her. That affected her confidence in her own ability. Soon thoughts began to run through her mind. Maybe she hadn't been good enough. Or if she'd been stronger, her anchor would have survived. Doubt is like poison. Once it gets in, unless stopped, it will run its course and kill you. In Kitty's case, it wasn't her body that was in danger, but her spirit. I gave her just enough of a boost to make her feel secure and confident in her own power. To let her know that I was there and that I wouldn't leave.\" He glanced toward the kitchen door. \"Kitty has enough power of her own, a truly astounding amount.\" Then he leaned forward. \"You say you trust me, but you don't.\"\n\n\"I trust you.\" I paused. \"Your motives, not so much. On the other hand, you could say that I do trust your motives. I trust them to be devious. That is on those rare opportunities when I even know what the hell they are.\"\n\n\"I'm a goblin. It's how _we_ are. Though I promise you, I swear to you, that I have no ulterior motive when it comes to you. What Isidor said\u2014\"\n\n\"What Isidor said doesn't matter,\" I lied.\n\n\"Oh, yes, it does. It's what you've thought since we met. I am being honest with you. I need you to be honest with me.\"\n\n\"I have been.\"\n\n\"No, you haven't. You've been avoiding me.\"\n\nI folded my hands on the table in front of me. \"Okay, then. I'm here. No avoiding. Why are you interested in me? I'm a small-town girl. I clean up well, but I'm not beautiful.\"\n\nRake started to interrupt.\n\nI held up a hand. \"Let me finish. You could have any woman you wanted, and you probably have. Yet you want me. The one thing I am that they aren't is a seer. You tried to hire me that first night.\"\n\n\"And you said no.\"\n\n\"And you've been after me ever since. I want to know why.\"\n\n\"Do I have to have a reason?\"\n\n\"Goblins always have a reason.\"\n\nI reached for another cookie. Rake's hand arrived at the same time. He let go of the cookie and took my hand. He started to cover it with his other hand, and then stopped.\n\nHe was giving me the option to pull away.\n\nI didn't.\n\n\"Makenna, I don't know what it is that we have, or what I feel. Believe that, believe _me_. What I really want is a chance, a chance to get to know you, to find out what we do have, what we _could_ have. And I promise not to ravish you.\" He gave me a slow, wicked-sexy smile. \"Unless you want me to.\"\n\nI gave him a flat look. \"Rake.\"\n\n\"Sorry. Old habits, hard to break and all that.\" Rake gazed at me a moment across that small table, his expression unreadable. \"I jumped into a pit of brimstone wearing a shield that might have failed. If that had happened, we wouldn't be here having this conversation. I wouldn't jump into a pit of fire for a potential employee.\"\n\nIf that shield had failed, Rake would have been burned to death, and I would have been . . . well, whatever would have happened to me in Hell. He was right; we most definitely wouldn't be talking now.\n\n\"The only reason I dove into that pit was because you were there,\" he said. \"I had a chance to save you and I took it.\" He gave me an exaggerated frown. \"I was really glad it worked. I wasn't keen on being vaporized. It would have only hurt for an instant, but still. I'm not ready to die yet. I have things to do.\"\n\nI couldn't help it. I felt a smile coming on. \"And seers to acquire for mysterious reasons?\"\n\n\"There are other seers. Who knows? Maybe even better seers.\" His eyes lit with mischief. \"And I only hire the very best.\"\n\n\"I probably wouldn't even make the final interview.\"\n\nRake raised my hand to his lips, his eyes solemn. \"There are other seers. There is only one Makenna Fraser.\"\n\n# ABOUT THE AUTHOR\n\n**Lisa Shearin** is the _New York Times_ bestselling author of the Raine Benares novels, a comedic fantasy adventure series, as well as the SPI Files novels, an urban fantasy series best described as _Men in Black_ with supernaturals instead of aliens. Lisa is a voracious collector of fountain pens, teapots, and teacups, both vintage and modern. She lives on a small farm in North Carolina with her husband, three spoiled-rotten retired racing greyhounds, and enough deer and woodland creatures to fill a Disney movie.\n\nVisit her online at lisashearin.com, facebook.com\/LisaShearinAuthor, and twitter.com\/LisaShearin.\n\n**Looking for more?**\n\nVisit Penguin.com for more about this author and a complete list of their books.\n\n**Discover your next great read!**\n\n  1. Cover\n  2. Praise for Lisa Shearin\n  3. Ace Books by Lisa Shearin\n  4. Title Page\n  5. Copyright\n  6. Contents\n  7. Chapter 1\n  8. Chapter 2\n  9. Chapter 3\n  10. Chapter 4\n  11. Chapter 5\n  12. Chapter 6\n  13. Chapter 7\n  14. Chapter 8\n  15. Chapter 9\n  16. Chapter 10\n  17. Chapter 11\n  18. Chapter 12\n  19. Chapter 13\n  20. Chapter 14\n  21. Chapter 15\n  22. Chapter 16\n  23. Chapter 17\n  24. Chapter 18\n  25. Chapter 19\n  26. Chapter 20\n  27. Chapter 21\n  28. Chapter 22\n  29. Chapter 23\n  30. Chapter 24\n  31. Chapter 25\n  32. Chapter 26\n  33. Chapter 27\n  34. Chapter 28\n  35. Chapter 29\n  36. Chapter 30\n  37. Chapter 31\n  38. Chapter 32\n  39. Chapter 33\n  40. About the Author\n\n  1. Contents\n  2. Cover\n  3. Start\n\n  1. \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\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nE-text prepared by MWS, ellinora, and the Online Distributed Proofreading\nTeam (http:\/\/www.pgdp.net) from page images generously made available by\nInternet Archive (https:\/\/archive.org)\n\n\n\nNote: Project Gutenberg also has an HTML version of this\n      file which includes the original illustrations.\n      See 53733-h.htm or 53733-h.zip:\n      (http:\/\/www.gutenberg.org\/files\/53733\/53733-h\/53733-h.htm)\n      or\n      (http:\/\/www.gutenberg.org\/files\/53733\/53733-h.zip)\n\n\n      Images of the original pages are available through\n      Internet Archive. See\n      https:\/\/archive.org\/details\/fiddlersdrinkinw00meea\n\n\nTranscriber's note:\n\n      Italic text is represented by underscores surrounding the\n      _italic text_.\n\n      Bold text is represented by equal signs surrounding the\n      =bold text=.\n\n      Small capitals have been converted to ALL CAPITALS.\n\n\n\n\n\nTHE FIDDLERS\n\nDrink in the Witness Box\n\nby\n\nARTHUR MEE\n\n\n    _If thou forbear to deliver them that are drawn unto death, and\n       those that are ready to be slain;\n    If thou sayest, \"Behold, we knew it not;\" doth not he that\n       pondereth the heart consider it?\n    And shall not He render to every man according to his works?_\n\n\n\n\n\n\n\nPublished by Morgan & Scott, Ltd\n12 Paternoster Buildings, London, E. C. 4\n\nFirst Hundred Thousand       May 15, 1917\nSecond Hundred Thousand      June 1, 1917\n\nReprinted in the United States by\nThe American Issue Publishing Company\nWesterville, Ohio\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n                        DRINK LEADING FAMINE IN\n\n  The Drink Trade gave Germany her greatest weapon in the war by helping\n  to make the bread famine.\n\n  It was the wilful destruction of 4,800,000 tons of food, depriving the\n  nation of her reserves, that led to the appalling gravity of the\n  submarine menace.]\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n                Drink, What did You do in the Great War?\n\n              This impressive picture of Britannia is from\n               the splendid 1916 issue of Bibby's Annual]\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n        THE ALLIES AND PROHIBITION\u2014STOPPING DRINK TO WIN THE WAR\n\n     The Drink Map before the War and on the 1000th day of the War\n\n               CANADA\u2014Prohibition almost from Sea to Sea\n               FRANCE\u2014Total Prohibition of Absinthe\n               RUSSIA\u2014Prohibition Everywhere\n               BRITAIN\u2014120,000 Drink shops open daily]\n\n------------------------------------------------------------------------\n\n\n\n\n                            The Wages of Sin\n\n\nThe time has come when it should be said that those responsible for our\ncountry now stand on the very threshold of eternal glory or eternal\nshame. They play and palter with the greatest enemy force outside\nBerlin. The news from Vimy Ridge comes to a land whose rulers quail\nbefore a foe within the gate.\n\nNot for one hour has the full strength of Britain been turned against\nher enemies. From the first day of the war, while our mighty Allies have\nbeen striking down this foe within their gates, Britain has let this\ntrade stalk through her streets, serving the Kaiser's purposes, and\npaying the Government \u00a31,000,000 a week for the right to do it.\n\nShe has let this trade destroy our food and bring us to the verge of\nfamine; she has let it keep back guns and shells and hold up ships; she\nhas let it waste our people's wealth in hundreds of millions of pounds;\nshe has let it put its callous brake on the merciful Red Cross; she has\nlet it jeopardize the unity and safety of the Empire\u2014for it may yet be\nfound, as Dr. Stuart Holden has so finely said, that the links that bind\nthe Pax Britannica are solvable in that great chemist's solvent,\nalcohol.\n\nThe witnesses are too great to number; we can only call a few. There is\nno room for all those witnesses whose evidence is in the House of\nCommons Return 220 (1915), showing the part drink played in the great\nshell famine, in delaying ships and guns, and imperiling the Army and\nthe Fleet.\n\nBut the indictment is heavy. I charge this trade with the crime the King\nlaid at its door two years ago, the crime of prolonging the war; and the\nwitnesses are here at the bar of the people. The verdict is with them,\nand the judgment is with those who rule.\n\n_The wages of sin is death: What are the wages of those who fail in an\nhour like this?_\n\n------------------------------------------------------------------------\n\n\n\n\n                          Fiddling to Disaster\n\n  We are not going to lose the war through the submarines if we all\n  behave like reasonable human beings who want to save their country\n  from disaster, privation and distress.\n\n                                                  _The Prime Minister_\n\n\n_What are we to say of a Government that plays with war and drink and\nfamine while these brave words are ringing in our ears?_\n\nIf the situation is so desperate that we must all go short of food, it\nis desperate enough for the Government to be in earnest. But what are\nthe plain facts? No reasonable man who knows them can say that the\nGovernment is in earnest.\n\nIt is not denied by anybody who knows the facts that drink has been the\ngreatest hindrance of the war. There is not a doubt that it has\nprolonged the war for months and cost us countless lives. It is the duty\nof the Government to face a dangerous thing like this; it is its duty to\npursue the war with a single eye to the speediest possible victory. But\nthe records of our war Governments in dealing with drink have been\nrecords of fiddling and failure, and we stand in the third year of the\nwar with a Government fiddling still.\n\nOne thing will be perfectly clear if disaster and famine come. It will\nbe known to all the world that the Government knew the facts in time to\nsave us. We are in the war because we would not listen in times of\npeace. We are in the third year of the war because we would not listen\nin the first. We are faced with famine because we would not listen in\ntimes of plenty, when drink was breaking down our food reserves. And we\nare drifting now, nearer to disaster every day, because the Government\nsurrenders to the enemy worse than Germany.\n\nIt does not matter where you look, or when; the evidence of the fiddling\nis everywhere about you. Take the week before the Prime Minister's grave\nspeech about submarines\u2014ending May 19.\n\n  _Submarines destroyed 27 British cargoes, mostly over 1600 tons._\n\n  =Brewers destroyed 27 British food cargoes, totaling 9000 tons.=\n\n  _The granaries of Canada were crammed with wheat waiting for British\n  ships, but there were no ships to bring this people's food._\n\n  =The rum quay at London Docks was crammed with casks of rum to last\n  till 1920, but a ship arrived with 1000 Casks more.=\n\n  _A woman was fined \u00a35 for destroying a quartern loaf._\n\n  =Brewers were fined nothing for destroying millions of loaves.=\n\n  _Poor people waited in queues to buy sugar in London._\n\n  =Cartloads of sugar were destroyed in London breweries.=\n\nAnd so we might go on, looking on this picture and on that till the mind\nalmost reels with the solemn farce. The Prime Minister has suggested\nthat the farce does not end because those who demand its end cannot make\nup their mind. It is the Government that cannot make up its mind.\n\n  It tells Parliament that no more rum is to be imported, and goes on\n  importing rum for years ahead.\n\n  It forbids the use of spirits less than three years old, and reduces\n  the three years to 18 months.\n\n  It restricts beer to 10,000,000 barrels, and tells us one day that\n  it is all-inclusive, and the next day that the Army Council can\n  order as much extra beer as it likes.\n\n  It issues a report saying that hops are not food, and gives up\n  hundreds of thousands of feet to shipping them; 23,000 cubic feet\n  the other week.\n\n  It tells us that not an inch of shipping is wasted, and wastes\n  shipping on bringing brewers' vats from America and taking gin to\n  Africa.\n\n  It tells us that the Drink Trade gave up its distilleries\n  patriotically, and leaves us to discover that it was made the\n  subject of a bargain by which bread was being destroyed for whisky\n  as late as May this year.\n\nIt is quite clear that the Government is desperately in need of a\nscapegoat, and desperately in need of a defense. Prohibition Russia is\nnot mightily impressed with our drinking; serious Canadians are asking\nhow long they are to sacrifice their manhood to our brewers; America is\nasking already why she should go short of bread in order that England\nmay drink more beer.\n\nA Government must clearly say something in view of these things, and it\nhas put its defense in the care of one of the sanest and cleverest men\nin the United Kingdom, Mr. Kennedy Jones. If Mr. Jones does not make out\na case for it, there is no case to make. What does he say?\n\n1. _We are told that only five per cent. of malt can be mixed with flour\nfor bread._\n\nAll over the country this explanation is supposed to satisfy those\nsimple, honest people who know little about percentages but ask plain\nquestions at Food Economy meetings. It is preposterous nonsense. If we\nhave 200,000 tons of malted barley, what on earth does it matter whether\nwe mix it at fifty, or five, or two per cent., so long as we do mix it?\n_It adds 200,000 tons to our bread in any case._ This talk of five per\ncent., puzzling to people who think it means that only one-twentieth of\nthis malted barley can be used, is pitiful evidence, surely, of the\nstraits to which the Food Controller's Defense Department is reduced.\n\n2. _We are told that the barley destroyed for beer would give the nation\nonly ten days' bread._\n\nIt would actually last us a fortnight. Drink, which has taken a quartern\nloaf from every British cupboard in every week of the war, is taking\nstill a quartern loaf a month from every cupboard, and the desperate\nappeals of Mr. Kennedy Jones will be more effective in saving crumbs\nwhen he can tell us that he has stopped this monstrous destruction of\nover 1,000 tons of grain a day.\n\n3. _We are told that our munition workers are dependent on beer._\n\nIt is an astounding slander. However true it may be of Governments, it\nis not true of our workmen. For four months the workman has been the\nscapegoat of this Government in its surrender to this trade, and we are\nasked at last to believe that these men who saved us from the Shell\nFamine are willing to drink us into a Bread Famine. Does the Government\nnever pause to ask how millions of munition workers in America and\nCanada and the United Kingdom manage without beer? Does nobody in the\nGovernment know that the greatest steel furnaces in America are under\ntotal Prohibition, and that two million American railwaymen are subject\nto instant dismissal if they touch drink while on duty? Has the\nGovernment not read its own report of the Royal Society Committee which\nhad this point in mind six months ago, and told us, on the highest\nauthority in this country, that soldiers march better and keep fitter\nwithout alcohol; that men do more work on less energy without alcohol;\nand that \"the records of American industrial experience are significant\nin showing a better output when no alcohol is taken by the workmen\"?\n\n4. _We are told we need this trade for yeast._\n\nWe need not bother overmuch about that. Industrial alcohol will give us\nall we want, and there is no need to carry on this dangerous trade for\nthe sake of yeast. We do not need a single ounce of brewer's yeast, and\nwe can do without distiller's yeast as well by setting up a thousandth\npart of the machinery we have set up in the last two years. Or, while we\nmust have yeast, we need about 30,000 tons a year for the whole United\nKingdom, and since the prohibition of hops in June last year _we have\ngiven enough shipping to hops every fortnight to bring in enough yeast\nfor a year_. A Government with shipping to spare like that, with room on\nits ships for mountains of hops, for enormous brewers' vats, and for rum\nfor 1921, can find room for 100 tons a day of the people's bread. It is\na monstrous perversion of the facts to suggest that we must maintain\nthis food-destroying trade, with all its hideous tragedy and ruin, in\norder to make bread.\n\nIt cannot be said that a Government with such desperate excuses is in\nearnest. We do not wonder that a great American farmers' paper, with no\naxe to grind except that it is sane and patriotic and believes in the\nwar, is asking plain questions as America prepares her Prohibition Army,\nher Prohibition Navy, and stops the destruction of grain for drink in\norder to enter the war at full strength.\n\nLet the Food Controller, the Prime Minister, and every responsible\ncitizen of the United Kingdom read this\u2014it is from the most influential\nflour-milling paper in the world, the \"North Western Miller,\" published\nin Minneapolis:\n\n\"=Since the United States will be called upon to make food sacrifices on\nbehalf of the Allies, it is certainly in order to call to account the\nstewardship of Great Britain in regard to food supplies. Ordinarily\nAmerica would have no right to demand such an account, but Americans are\nnow asked to deny themselves that Britain may have sufficient.=\n\n\"=Britain has not seen fit to prohibit the use of cereals in the\nmanufacture of drink, notwithstanding that the world's food supply was\nobviously short. Are Americans required to forego a part of their\naccustomed ration of bread in order that their British Allies can\ncontinue to have a plentiful supply of beer and whisky? If not, then\nBritain should lose no time in putting its house in order, quitting the\ndrink to add to the common store of food upon which the safety of all\nthe Allies depends.=\n\n\"=The food supply for the Allies is no longer a purely local\nproposition, to be used as a football in British politics; it deeply\nconcerns the people of the United States, who are certainly not called\nupon to deny themselves bread in order that Britain shall have drink.=\"\n\nWhat is the Government's answer to this? \"We owe a very considerable\ndebt of gratitude to the great American people for the effective\nassistance they are rendering us,\" says the Prime Minister. _Is this the\nway we pay them back?_ It is an ugly question for our great Ally to have\nto raise as she comes into the war, flinging her Prohibition Navy in to\nsmash the drink-made menace of the submarine. It is unthinkable that the\nGovernment can read these bitter words unmoved, or can leave this stain\non our history in the face of all these questionings.\n\nThere is another question, too, that comes across the Atlantic. What is\nthe Government going to do with the soldiers of America's Prohibition\nArmy, and the sailors of America's Prohibition Navy, when they come over\nhere? Are they to be broken in their thousands, made useless and\ndegraded as thousands of men from Prohibition Canada have been, by the\nenemy that traps them before they reach the war?\n\nThey are questions for the Government and the nation, and they must be\nanswered in the interests of the nation, and not to please the trade\nthat helps the Germans every day. We cannot afford to pay the appalling\nprice the future will demand unless our fiddlers change their tune.\n\n------------------------------------------------------------------------\n\n\n\n\n                  The Drink Trade and Our War Services\n\n\n=It is not possible to measure the strain the Drink Traffic has imposed\non our war services.=\n\nThe Food Controller's Organization, with its great offices and staffs,\nwould not have been needed had we saved the food destroyed by drink.\n\nRationing already involves 1,200 committees, and may mean 50,000\nofficials and 50,000,000 tickets weekly. It could all be avoided.\nProhibition would save more bread without food controlling than all the\nfood controlling can save without Prohibition.\n\nThe National Service, with its network of officials, its costly\nadvertising, its absorption of paper and printing, could all have been\navoided under Prohibition. About 200,000 men have enrolled, but\nProhibition would give us twice that man-power any day.\n\nThe strain on a host of men and women looking after soldiers' children\nneglected through drink, soldiers' wives spending allowances on drink,\nis incalculable.\n\nThe strain on war charities and the strain on the police arising from\ndrink are both very great.\n\nThe strain of drink on doctors, nurses, and hospitals is beyond belief.\nProhibition would set free for the Red Cross thousands who waste their\ntime on the great drink trail.\n\nThe strain on transport is seen in the long lines of wagons drawn by\nstrong horses carting beer to public-houses. This year alone the\nhandling of drink must equal the lifting of at least 9,000,000 tons, and\nthe barrels of beer would fill nearly all the railway wagons in the\nkingdom. As to ships, drink materials during the war have used up 60\nships of 5,000 tons working all the time.\n\nOn Lord Milner's estimate of 19 barrels to the truck it would require\n4,500,000 railway trucks to carry the 17,000,000 tons of beer\nmanufactured in the United Kingdom during the war.\n\n=It can be proved from official figures that the weight of drink-stuff\ncarried about since war began has been equal to the weight of solid\nmaterial carried by the Navy to all our fighting fronts.=\n\nIt is a crying shame that the strength of Britain should be destroyed\nlike this in such an hour as this.\n\n------------------------------------------------------------------------\n\n\n\n\n                  The War-Work of the Food Destroyers\n\n\nThere are hundreds of great Food Destructors in the United Kingdom. The\nman-power at their service, spread over our breweries and distilleries,\nnumbers hundreds of thousands of men; their capital is hundreds of\nmillions. This is a summary of the work they did in the first 1,000 days\nof the war:\n\n=They sacrificed 4,400,000 tons of grain and 340,000 tons of sugar,\nenough to ration the whole United Kingdom with bread for 43 weeks and\nsugar for 33 weeks.=\n\n=They took from every kitchen cupboard in the land 600 pounds of bread\nand 76 pounds of sugar.=\n\n=They destroyed bread and sugar to last every child under fifteen for\nevery day of the war.=\n\n=They took from our people over \u00a3512,000,000.=\n\n=They used up labour and transport for lifting over 50,000,000 tons. By\nsea they used up 60 ships of 5,000 tons; by rail their raw materials and\nthe finished products would make up a train long enough to reach nearly\nround the world.=\n\n------------------------------------------------------------------------\n\n\n\n\n                 The Food Now Being Destroyed for Beer\n\n\nLook at the actual facts about beer alone. We will ignore distilling, as\nit gives us munitions and yeast. Had the Government tried to solve the\nyeast question it could have solved it easily in these three years; it\nwould have had no more trouble with that problem than Russia and Canada\nand America have had. But as the Government is still investigating the\nyeast question, we will confine our figures to beer.\n\n=Brewers are destroying 450,000 4-lb. loaves a day.=\n\n=This year's food destruction for beer alone will equal five weeks'\nbread rations and four weeks' sugar rations for the whole United\nKingdom.=\n\n=We have seven critical weeks in this summer, and this year's\ndestruction of food would carry us through.=\n\n=Beer alone is taking 10 pounds of sugar a year from every kitchen\ncupboard, and an ounce of sugar a day from every soldier.=\n\nThat is what drink is doing at this moment with the shadow of famine\ncreeping on.\n\n \"_He who withholdeth the corn the people shall curse him._\" Proverbs.\n\n------------------------------------------------------------------------\n\n\n\n\n                          The Shadow of Famine\n\n\nThe Government came into office with the food shortage in sight; it was\nits first duty to build up the great reserve of food we might have had\nnow in our granaries if the drink trade had not destroyed it. We could\nhave laughed at submarines, for our barns would have been filled to\noverflowing, and we could have lived in comfort for a year if no ship\nreached us.\n\nLet us see how much food drink has destroyed during the war. We will\ntake it from August 4, 1914, to April 30, 1917. It is 999 days of the\nwar. The grain and sugar destroyed for drink have been:\n\n               Grain                      4,400,000 tons\n               Sugar (for beer alone)       340,000 tons\n\n[Illustration:\n\n  How Canada sees it\u2014A Canadian cartoon of the callous destruction of\n    bread for beer and whisky]\n\nIt is not easy to realize what this means, but it will help us if we\nthink of one or two examples.\n\n=The biggest thing ever set up on earth is the Great Pyramid. It is\n80,000,000 cubic feet. The food destroyed by drink during the war would\nmake two Great Pyramids, each bigger than the Pyramid of Egypt.=\n\n=The longest British railway is the Great Western; it is over 3,000\nmiles, but it would not hold the food destroyed by drink since war\nbegan. If every inch of it were crammed with wagons, the Great Western\nRailway would need hundreds of miles more line to hold the train-loads\nof food destroyed.=\n\n=There are about 750,000 railway wagons in the United Kingdom, but if\nthe Drink Trade had them all they would not hold the food it has\ndestroyed.=\n\n=There are about 30,000 engines on our British railways, and if the food\ndestroyed were made up in trains of 125 tons apiece, all our engines\nwould not pull them; we should still want 10,000 more.=\n\nSo vast is this incredible quantity of food destroyed by an enemy trade\nwhile famine has been coming on. We should have saved it all if\nParliament had followed the King, and it would have given the whole\nUnited Kingdom its flour rations for nearly a year. Take it at its\nminimum scientific human food value, and on the basis of our rations in\nMay, 1917, it would have given us:\n\n             Flour for the whole United Kingdom    43 weeks\n             Sugar for the whole United Kingdom    33 weeks\n\nOur three war Governments, confronted with the increasing certainty of\nat least a three-years' war, have allowed the Drink Trade to destroy\nthis vast reserve of food.\n\nThe full toll of this trade upon our scanty food supply, growing shorter\nand shorter while the queues outside our food shops grow longer and\nlonger, is staggering indeed, even now with drink about three-quarters\nstopped. We must remember that it makes no difference that the barley\nhas been malted; it is still good human food, and every ounce of it\nshould be mixed with grain for making bread. Let us remember, also, that\n_brewer's sugar is a good pure sugar_, the objection to it being largely\nthe objection most of us have to standard bread\u2014its colour. Malt or\nsugar, every ounce a brewer destroys is food stolen from the people. Let\nus take expert opinion on the subject.\n\n\n                    The Food Value of Brewer's Sugar\n\n  We do not, of course, use this dark sugar when white sugar is cheap\n  and easily procurable, but during the war we have used it for\n  coffee, cocoa, and tea; and for puddings where colour did not\n  matter. We have used it a good deal in our bakeries for chocolate\n  goods, where colour again does not matter. It is a good, pure sugar,\n  and the colour is the principal drawback.\n\n                          _Letter to Arthur Mee from a London caterer_\n\n\n                    The Food Value of Brewer's Malt\n\n  Malt flour can be used to make excellent cake with 50 per cent.\n  wheat flour. It is sweet and pleasant to taste without the need of\n  any sugar. Good scones can be made with 25 per cent. of malt flour.\n  Its use in bread made with yeast causes too much fermentation in the\n  bread, but it has no effect on baking-powder. The Food Controller's\n  Department is aware of the practicability of using malt flour, but\n  the sale is restricted in order to limit its use for making beer.\n  Brewers and maltsters are too patriotic to wish to use for beer what\n  could be applied to food in case of a serious shortage, and the\n  large stocks of barley and malt can supplement the supply of wheat\n  flour.\n\n                 _Letter from a Brewer in the \"Times,\" April 11, 1917_\n\nYet we have seen our Government holding up sugar for brewers; we have\nseen our Food Controller refuse to release a caterer's sugar unless it\nwere sold to a brewer; we have seen a Government short of food-ships\nbringing in brewers' vats and casks of rum; and we see the Government\nstill holding up this malt that would feed a people asking for more\nbread.\n\n------------------------------------------------------------------------\n\n\n\n\n                          The Tunes They Play\n\n\nStrange tunes we hear the fiddlers play, but their music does not charm\naway the troubles of a famine-threatened land. From morning till night\nthe prayer of the people rises, \"Give us this day our daily bread,\" but\nthe heart of Downing Street is hardened, and the nation's bread goes day\nby day to the destroyer.\n\nBut all the time we see the measure of the courage of our rulers on the\nhoardings in the streets. We know their posters by heart.\n\n_Defeat the enemy's attempt to starve you_, by\u2014not by stopping the\ndestruction of food, but by joining the National Service, and probably\nhelping to pick hops. There was a man in a co-operative store who\nvolunteered for National Service, and last month he received\ninstructions _to leave the grocery store and take up duty in a brewery_.\n\n_Sow your window-boxes and plant your back gardens_\u2014and Mr. Prothero\nwill see that the soil of a million back gardens is wasted on hops.\n\n_We have not enough food to last till the harvest_\u2014why not go out and\ncatch rabbits, asks Lord Devonport\u2014and sit and wait for sparrows?\n\n_We must save every pound of bread we can to get over our critical\nweeks_\u2014not by saving the quartern loaf that beer is taking every month\nfrom every British cupboard now, but by going hungry so that drinkers\nmay not thirst.\n\n_We must not eat more than our share, on our honour_\u2014but the man across\nthe table can eat his share of bread and drink somebody else's too.\n\n_We must eat less and eat slowly_\u2014so that brewers may waste more and\nwaste quickly.\n\n_We must keep back famine_\u2014but not by using malt, says Captain Bathurst:\nthat would cost three times as much as letting famine come. _But why not\nkeep the malt till bread is as dear as gold?_\n\n_Let all heads of households abstain from using grain except in bread_,\nsays the King's Proclamation. But let the brewers waste 8,000 tons a day\nfor beer, says the Government.\n\n_God speed the plough and the woman who drives it_\u2014yes, and God help the\nwoman who drives the plough to feed the brewer while her little ones cry\nfor bread.\n\n_Let us fine \u00a35 whoever wastes a loaf_, says the Food Controller\u2014but\nnot, of course, the brewers who waste 450,000 quartern loaves a day.\n\nHops are no use as food to anybody, says the Board of Trade Scientific\nCommittee. \"_Then let us grow only half as many_,\" said Mr. Prothero.\n\nMr. Lloyd George says Mr. Prothero is working \"in a continuous rattle of\nmocking laughter and gibes.\" Yes, it is the mocking laughter of a nation\nthat is not really amused by sights like this. The nation does not like\nto see the bread rations of 70,000 men in France cut down while the\nDrink Trade is destroying every week bread enough to last these men a\nyear. It does not like to see the Government sending letters out to\nmanagers of factory canteens, begging them to be careful of bread, while\nfood flows through our beer canteens like a river running to waste. It\ndoes not like to see Y. M. C. A. canteens denied supplies of sugar while\nbarrels of beer are stacked in great piles outside. It does not like the\ncalling up of discharged soldiers while thousands of strong men are\nworking hard all day destroying food or carting beer about the streets;\nand it does net like the tragic comedies of Captain Bathurst, who warns\nus that it really may become necessary in the national interest\u2014and\nthen, perhaps, he drops his voice to break it very gently\u2014it really may\nbecome necessary, if these cake shops are not very careful, _to\nwhitewash the lower part of their windows_.\n\nOh, these fiddlers! And now we have a new idea from the Food Control\nDepartment; it is a  poster of a Union Jack and a big loaf on\nit, and \"Waste not, Want not,\" printed in big type. It was being printed\non the day the Prime Minister told the nation that America had found it\nis no use waving a neutral flag in the teeth of a shark. It is an\neloquent and true saying, but it is also true, that it is no use waving\nplatitudes from copybooks in the teeth of a wolf at the door. The Prime\nMinister says he is taking no chances. Let us be quite sure. We once had\na Government of which men said its motto was \"Wait and See.\" _Are we\nbetter off, or are we worse, with a Government that Sees and Waits?_\n\nBut there is no end to the fiddling. With Food Controllers who hold up\nfood for Food Destroyers; with Food Economy Handbooks that cry out loud\nto save the crumbs but have no word to say about the tons we fling away;\nwith a Prime Minister praying for window-boxes and a Board of\nAgriculture consecrating hopfields, we need not be surprised if the\nnation is not mightily impressed.\n\n------------------------------------------------------------------------\n\n\n\n\n                         How the Allies Did It\n\n\nAll the world knows, except, apparently, the world that goes round at\nWestminster, how Prohibition has helped the Allies.\n\n_With the Shell Famine at its height\u2014largely made by Drink\u2014the\nProhibition Army on the East held up the enemy while Britain fought the\nDrink Trade for her shells._\n\n_With the Bread Famine looming in sight\u2014largely made by Drink\u2014the\nProhibition Navy from the West flings in her power against the\nsubmarines._\n\nOh, for the spirit of our Allies in this land! If France wants to rouse\nthe spirit of Verdun she strikes down her foe at home and puts absinthe\naway. If Russia wants to be great and free she stops this drink and\norders out the Romanoffs. If Canada wants to give her utmost help to\nBritain she stops this drink from sea to sea. If Australia wants to make\nher soldiers fit she trains them in her Prohibition camps. If America\nwants to beat the whole world at making shells she drives drink from her\nworkshops. If San Francisco has an earthquake she stops drink while she\npulls herself together. If Liverpool has a dangerous strike she shuts up\npublic-houses and keeps the city quiet. Oh, for a Government of Britain\nthat will see what all the world can see!\n\nHistory will do justice to the part the Prohibition policy of the Allies\nhas played in saving Europe, but a pamphlet has no room for these\nthings. We can take only one or two great witnesses to the mighty\nachievements of our Prohibition Allies. Let us begin with France, and\ncall our own Prime Minister to tell us what they did. Mr. Lloyd George:\n\n  One afternoon we had to postpone our conference in Paris, and the\n  French Minister of Finance said, \"I have to go to the Chamber of\n  Deputies, because I am proposing a bill to abolish absinthe.\"\n  Absinthe plays the same part in France that whisky plays in this\n  country, and they abolished it by a majority of something like ten\n  to one that afternoon.\n\nAnd how did Paris take this prohibition that men said would cause a\nrevolution? Let us ask Mr. Philip Gibbs, whose splendid letters home\nhave made his name a household word. Mr. Philip Gibbs:\n\n  Absinthe was banned by a thunderstroke, and Parisians who had\n  acquired the absinthe habit trembled in every limb at this judgment\n  which would reduce them to physical and moral wrecks. But the edict\n  was given and Paris obeyed, loyally and with resignation.\n\nAnd now we come to Russia, to these mighty Russian people who in the\nlast year of vodka saved \u00a36,000,000 or \u00a37,000,000, and in the last full\nyear of Prohibition saved \u00a3177,000,000. We will call our own Prime\nMinister again:\n\n  Russia, knowing her deficiency, knowing how unprepared she was,\n  said, \"I must pull myself together. I am not going to be trampled\n  upon, unready as I am. I will use all my resources.\" What is the\n  first thing she does? She stops drink.\n\n  I was talking to M. Bark, the Russian Minister of Finance, and I\n  asked, \"What has been the result?\" He said, \"The productivity of\n  labour, the amount of work which is put out by the workmen, has gone\n  up between 30 and 50 per cent.\"\n\n  I said, \"How do they stand it without their liquor?\" and he replied,\n  \"Stand it? I have lost revenue over it up to \u00a365,000,000 a year and\n  we certainly cannot afford it, but if I proposed to put it back\n  there would be a revolution in Russia.\"\n\nHow completely teetotal Russia became we read long ago in the _Daily\nMail_, to which Mr. Hamilton Fyfe sent this message from Petrograd:\n\n  Try to imagine all the publichouses in the British Isles closed; all\n  the restaurants putting away their wine cards and offering nothing\n  stronger than cider or ginger ale. That is the state of things in\n  Russia. Strange it seems indeed, yet there is one thing stranger.\n  Nobody makes any audible complaint.\n\nEverywhere in Russia it was the same: a nation was made sober by Act of\nParliament.\n\n  \"Without a murmur of protest,\" said the Moscow correspondent of the\n  _Times_, \"the most drunken city in Europe was transformed into a\n  temple of sobriety, and we felt that if Russia could thus conquer\n  herself in a night, there was indeed nothing that might not be\n  accomplished.\" And two years later, when the revolution came, we\n  read in the _Times_ this note from Odessa: \"Perfect tranquillity\n  continues to prevail here, although for the moment Odessa is\n  practically without police. The satisfactory absence of crime may\n  largely be attributed to the sealing up of spirituous liquors.\"\n\nWe need not be afraid of Drinkless Revolutions.\n\nBut the truth about Russia is almost too incredible to believe, for it\nis Prohibition that made the revolution possible; it was stopping drink\nthat set 170,000,000 people free. We will let a business correspondent\nof the _Times_ give evidence; here is what he said on April 21, 1917:\n\n  In one respect it must be said that the Reactionaries saw clearly.\n  They always claimed that the Tsar had ruined himself by decreeing\n  the abolition of vodka. None but a sober people could have carried\n  out the Russian Revolution.\n\n  The police were, on the other hand, the victims of drink. They had\n  seized the vodka at the order of the Government, and had kept\n  plentiful supplies for themselves. Thus the Revolution was in part a\n  struggle between drunken reaction and sober citizens. Sobriety\n  triumphed.\n\nThe Russian people will not bow down and tie their hands to the thrones\nof Europe: do we wonder if they scorn our quailing before this trade?\n\nFree Russia flings off the dynastic yoke: do we wonder Prohibition\nRussia is not much impressed by a nation with a Drink Trade round its\nneck?\n\n------------------------------------------------------------------------\n\n\n\n\n                           The Soldier's Home\n\n\nThe things that will be told against this trade when all the truth is\nknown will break the heart of those who read. It is well for us that we\ncannot know the full truth now; the burden would be too grievous to be\nborne in days like these. But if you will go into your street, or will\ntalk of these things with the next man you meet from one of our pitiful\nslums, or will pick up one of those local papers that still have space\nto print the truth, you will find the evidence close about you.\n\nWe are the guardians of our soldiers' homes; we are the trustees of the\nhope and happiness of their little children; but we let this drink\ntrade, that takes our people's food out of their cupboards, turn that\nfood into the means of death, and sow ruin and destruction through the\nland.\n\nBut we will call the witnesses to these drink-ruined soldiers' homes,\nthese homes that the enemy worse than Germany has shattered and broken\nwhile our men have been fighting for your home and mine. We will call a\nfew here and there, knowing that for every one called are hundreds more\nthat can be called, and that beyond all these that are known there is in\nthis little land a countless host of tragedies as secret as the grave.\n\n  A Tooting soldier whose wife had sent him loving letters to the\n  trenches came back to surprise her after 18 months. He found another\n  man in possession of his home and a new baby; and, overcome by the\n  discovery, he gave way to drink and killed himself.\n\n                               _Records of Balham Coroner, March 1916_\n\n  A soldier who had left a comfortable home behind returned from the\n  Front to find it ruined, with not a bed to lie on, his children\n  never sent to school, his wife all the time in publichouses. \"I wish\n  I had been shot in the trenches,\" he said when he arrived.\n\n                     _Facts in \"Cork Constitution,\" December 10, 1915_\n\n  Outside a publichouse in Liverpool a man was dragging home his\n  drunken wife, the mother of eleven children. They rolled over and\n  over on the ground, the drunken women violently resisting the\n  maddened man. Then came up the eldest son, home from the Front, with\n  five wounds in his body.\n\n                            _Facts in \"Liverpool Post,\" March 2, 1917_\n\n  A soldier came back to his home in London to find his wife drinking\n  his money away, harbouring another man; one of his children cruelly\n  neglected and the other in its grave, perished from neglect; and a\n  drunken carman's baby about to be born in his home.\n\n                                 _Facts in Shaftesbury Society Report_\n\n  A Lance-Corporal heard in the trenches of his wife's misconduct. His\n  commanding officer wrote to make inquiries, and the soldier wrote to\n  the Chief Constable a pitiful letter: \"What have I to look forward\n  to at the end of the war?\" he said. \"Nothing, only sorrow. I never\n  get a letter to know how my loving son is getting on; I think it\n  will drive me mad.\"\n\n  He came home, opened the door of his house, threw his kit on the\n  floor, and declared that he would kill his wife. He put a razor on\n  the table, and his little boy hid it in a cupboard, but a week later\n  this boy of 12 went home and found his father and mother lying on\n  the floor, the father drunk, the mother dead. The soldier, drowning\n  his misery in drink, had strangled his wife. Rousing himself beside\n  her, he said, as the police found them, \"Kiss me, Sally. Aye, but\n  tha are poorly.\"\n\n  He had been the best of fathers, said the little boy; the best of\n  soldiers, said his commanding officer; and the judge declared that\n  such a man, with such a character, ought not to be with criminals.\n\n                         _Record of Huddersfield Assizes, Autumn 1916_\n\n  A soldier asked a London magistrate if he could draw the allowance\n  instead of his wife, who was in prison for drunkenness and was\n  neglecting his four children. The magistrate said the only thing was\n  to send the children to the workhouse.\n\n  The Soldier: \"So I am to be a soldier for my King and country while\n  my children go to the workhouse?\" The Magistrate: \"That is so,\n  because you have a drunken wife. I am sorry for you.\"\n\n                                 _Facts in \"Sunday Herald,\" June 1916_\n\n  A seaman gunner, who had been torpedoed and had fought in the\n  trenches, arrived home to find his wife, in his own words, \"filthy\n  drunk,\" and his children utterly deplorable. He reclothed them, but\n  his wife pawned the clothes, though she had \u00a37 a month. He took his\n  children away, but a crowd of women interfered with him, and the\n  police were powerless against the mob.\n\n                     _Facts in \"Western Daily Mercury,\" July 23, 1915_\n\n  A soldier just back from the Front was found in the street weeping\n  bitterly on discovering that his wife was in gaol through drink, and\n  his child, through her neglect, had been burned.\n\n                               _Statement by Marchioness of Waterford_\n\n  A soldier came home from the Front to find that drink had ruined his\n  home, and his children were being cared for by Glasgow Parish\n  Council. \"Hour after hour we sit on this council,\" says the\n  chairman, \"listening to case after case, and the cause is\n  drunkenness, drunkenness, drunkenness. There are 2300 children under\n  the council, and two thousand of them have parents living.\" \"Our raw\n  material is the finished product of the public-house,\" says one of\n  these workers.\n\n                                      _Facts from Glasgow Councillors_\n\n  A motor mechanic at the Front, hearing that his wife, hitherto a\n  sober woman, had given way to drink, obtained leave to come home. He\n  found his wife, very drunk, struggling home with the help of the\n  railings in the street, and neighbours described her horrible life\n  with other soldiers. The husband obtained a separation for the sake\n  of his children, and went back to France.\n\n                       _Full facts in \"Kent Messenger,\" July 31, 1915_\n\n  A young soldier came from the trenches to spend Christmas in his\n  home in Sheffield\u2014a teetotal home before the war. He found that his\n  wife had given way to drink, had deserted one child and disappeared\n  with the other, and that a baby was to be born which was not his.\n\n                                           _Facts known to the Author_\n\n  A miner fighting at the Front came home to find his wife at a\n  publichouse, his home filthy, and his children cruelly neglected. He\n  was heartbroken. His young wife frequently left the house from\n  tea-time till midnight, and in order to keep the children from the\n  fire she had burned them severely with a piece of iron. A\n  respectable-looking woman, the mother pleaded for a chance, and was\n  led from the dock sobbing bitterly.\n\n                 _Facts in \"Sheffield Independent,\" February 21, 1917_\n\n  A young Yorkshire miner enlisted and left his wife, hitherto sober,\n  with three children. She took to drink, neglected the home, and is\n  now a dipsomaniac, with two children not her husband's.\n\n                                           _Facts known to the Author_\n\n  A soldier came home ill from France, hurried from Waterloo to his\n  home, and found the door locked. He knocked, and his little boy's\n  voice came\u2014\"Is that you, mother, and are you drunk?\" Hearing his\n  father's voice the excited lad opened the door. \"Where's mother?\"\n  asked his father. \"Mother?\" said the boy; \"she's drinking. She comes\n  home drunk night after night now and knocks the kids about. She\n  daren't hit _me_; I'm fair strong, dad; but the other.... And as for\n  baby, she never does nothing for her. I and Freddy takes turns, but\n  I dunno what to give her to eat sometimes.\"\n\n  Midnight passed before the mother appeared, helplessly drunk. \"Did\n  you expect me to sit at home weeping for you?\" she said. The next\n  morning, broken with tears, she promised to mend her ways. The\n  soldier went into hospital, and there he had a letter from his boy.\n  This is part of it:\n\n  \"Dear Dad, I write to let you know mother is going on awful. She has\n  took all Fred and Timmy's clothes to the pawnshop, and she hit\n  Selina on Saturday with the toasterfork and cut her face. She cried\n  all night, it hurt her so. She is drunk every night and some nights\n  dussent come back at all. She daren't hit me, but I am getting\n  afraid about baby. We are all very hungry and miserable.\"\n\n  The soldier got leave, found his wife had disappeared, and, finding\n  charity for his four little ones, he left his ruined home and went\n  back to the hospital.\n\n                                   _Facts in possession of the Author_\n\n  A working-man at Gravesend went to the Front, leaving behind a wife\n  and three children, the baby lately born. His wife started drinking\n  away her allowance, neglected her home, and, full of remorse and\n  shame for the disgrace she had brought on the man who was in the\n  trenches, she hanged herself. The man came home to find waiting for\n  him three motherless children, and one of the most pathetic letters\n  a man has ever had to read.\n\n                                  _Records of Gravesend Coroner, 1916_\n\n\n                          Mothers and Children\n\nIt is easy to understand the pitiful appeal of 500 women out of Holloway\nPrison who begged the Duchess of Bedford to help to close all\npublic-houses during the war. They know in their hearts of tragedies\nsuch as these, in which mothers and children die while the fathers fight\nand the Drink Trade goes on merrily.\n\n  A soldier's wife in Sunderland drew \u00a312 arrears of Army pay, and she\n  and her mother began to drink it away. She drew her pay on Friday,\n  was carried home drunk on Saturday, gave birth to twins on Sunday\n  morning, and died on Sunday night. The twins died a week or two\n  after, and a week or two after that the soldier came home from the\n  trenches to find his family in the grave.\n\n                                    _Facts in Sunderland papers, 1917_\n\n  Two women went drinking in Chester on a Sunday night, a soldier's\n  mother and a soldier's wife. They had five whiskies each, and fell\n  drunk in the street. One slept all night on a sofa, and the other\n  lay on the floor, shouting and swearing. Her husband propped her up\n  with a mat, and for hours she lay shrieking. In the morning she was\n  dead. The publican was fined \u00a35.\n\n                     _Facts in \"Chester Chronicle,\" February 17, 1917_\n\n  The wife of a Yorkshire soldier was drowned while drunk at\n  Sheffield. She started drinking with another soldier's wife\n  disappeared with a drunken man, and her death was a mystery.\n\n                    _Facts in \"Sheffield Independent,\" April 26, 1916_\n\n  At an inquest on the bodies of a soldier's twin children, both dead\n  from chronic wasting, it was stated that the mother had 34_s._ a\n  week, and both she and her husband drank. The mother had had four\n  children in fifteen months, and all were dead.\n\n                          _Records of Battersea Coroner, October 1915_\n\n  In one street in London where there were one day four convictions\n  for drunkenness, a woman carried a sick baby into a public house. As\n  she stood at the bar the little baby died, but the mother went on\n  drinking, with the dead child in her arms.\n\n                             _Records of Charity Organisation Society_\n\n  The wife of a highly-esteemed sergeant-major fighting in France was\n  found lying drunk. Her four children, shockingly neglected, were put\n  in a home, but she took them out, went on drinking, and received\n  soldiers at her house. In a few weeks her husband heard in the\n  trenches that his wife had died from drinking.\n\n                          _Records of West Surrey Coroner, March 1917_\n\n  A soldier left three children at home. He had been earning \u00a31 a\n  week, but his wife received 32_s._ 6_d._ a week. She drank it away,\n  neglected the children, and died in an asylum while her husband was\n  in France.\n\n                                          _Records of Claybury Asylum_\n\n  The little child of a soldier in France died in Guy's Hospital from\n  burns. The mother said she could not buy a fireguard. While she was\n  absent the baby was burned, and the mother, returning in a drunken\n  state carrying a can of beer, said, \"A good job!\"\n\n                         _Records of Southwark Coroner, December 1915_\n\n  A soldier's widow with six children, an Army pension of 30_s._ a\n  week, and her eldest boy's wages of 30_s._, drinks every night with\n  a married man who has a respectable, clean, and sober wife with\n  eight children and a ninth lately born\u2014born prematurely as a result\n  of her husband's beating her. The child bore the marks of his\n  violence, and died in two months.\n\n                                      _Records of Shaftesbury Society_\n\n  The young wife of a soldier was brought from prison to be tried for\n  manslaughter of her baby, who had died in the infirmary from\n  neglect. She spent her time in the publichouses, and laughed when\n  the children were taken to the infirmary. She went out one day to\n  fetch a bottle of whisky and as she drank with a neighbour she said\n  she knew the baby would die. The doctor said the child's skin was\n  hanging in folds on the bones.\n\n                           _Facts in the \"Observer,\" January 23, 1916_\n\n  A soldier's wife drank continuously while her child wasted away,\n  left the tiny baby alone in the house while she went for beer, and a\n  policeman found her lying drunk across the dead child's body.\n\n                         _Records of Barnsley Coroner, November, 1916_\n\n  The mother of two children whose father was fighting in France gave\n  way to drink in his absence, neglected her children and left them in\n  grave moral danger, and committed suicide.\n\n                                           _Records of an Orphan Home_\n\n  A soldier's baby starved slowly to death as the mother drank away\n  his pay, and while the child lay in its coffin the mother was out\n  drinking.\n\n                             _West Bromwich Police Records, June 1915_\n\n  A munition worker at Newcastle was grievously upset by the drinking\n  habits of his wife. The police left a summons for her and she\n  disappeared. Two days later her body was found in the Tyne. The man\n  broke down at the inquest, saying, between his sobs: \"She was such a\n  good wife to me for 20 years, and reared a good family before she\n  took to drink.\"\n\n                           _Records of Newcastle Coroner, Summer 1916_\n\n  The wife of a corporation workman at Sheffield, home from the\n  trenches with six gunshot wounds and three pieces of shell in his\n  body, found that his wife had given way to drink and starved her\n  five children. She was sent to prison for six months.\n\n                       _Police Records of Sheffield, November 3, 1915_\n\n  A soldier's wife who had spent the greater part of \u00a3100 Army money\n  in drink was sent to prison for neglecting her children. Almost\n  everything in the house was pawned, including the children's\n  clothes; and the woman began to drink at five o'clock in the\n  morning, and went on drinking all day.\n\n                     _Facts in \"Cork Constitution,\" December 10, 1915_\n\n  A soldier's wife in Monmouthshire, with \u00a33 9_s._ a week, was found\n  sodden with drink, while the soldier's eight children were in rags\n  starving by day and huddling up in one bed by night.\n\n                       _Facts in \"Westminster Gazette,\" July 22, 1916_\n\n  A smart tidy woman in a London suburb, whose husband is fighting in\n  Mesopotamia, has \u00a32 10_s._ 6_d._ a week. She used to love her\n  children and had a happy home, but she drinks away her Army pay,\n  lives with a married man who has six children, and has become a\n  drunken slattern. The other wife is beaten and neglected, and the\n  soldier's children have gone to the workhouse.\n\n                                      _Records of Shaftesbury Society_\n\n  The four children of a soldier in Dublin were found hungry and\n  shivering with cold while the mother was drinking. Several times she\n  had let her baby fall while reeling with it in the street.\n\n                  _Facts in \"Dublin Evening Herald,\" October 20, 1916_\n\n  At the trial of a soldier's wife for drinking and neglecting seven\n  children, it was stated that a child of eleven was left in charge of\n  a baby a fortnight old while the mother was drinking. At night all\n  the children were heard screaming. The house was in utter darkness,\n  and there was an escape of gas. Some men went in and turned off the\n  gas, and at last the mother came stumbling out of a publichouse\n  across the road.\n\n                        _Facts in \"Sheffield Star,\" November 25, 1915_\n\n  \"Your husband is fighting for his country, and his children have the\n  right to be protected,\" said the Chairman of the Chesterfield Bench\n  to a soldier's wife. Her children were found starving while she was\n  drinking, and one day the little boy of three was found crouching\n  naked inside the fender, trying to get warm. The police described\n  the house as foul from top to bottom, with a heap of horrible rags\n  for a bed, and a food cupboard that made the house unendurable when\n  the door was opened.\n\n                      _Facts in \"Yorkshire Telegraph,\" March 24, 1916_\n\n  The wife of a missing soldier was sent to prison at Chesterfield for\n  neglecting three children between 13 years and 16 weeks old. She had\n  gone astray through drink, and the youngest child, born under\n  terrible conditions, was not her husband's. It was found lying on a\n  filthy bed, and its drunken mother, to satisfy its pangs of hunger,\n  had given it pennyworths of laudanum. Eleven people slept in two\n  foul bedrooms.\n\n                        _Chesterfield Police Records, October 9, 1916_\n\n  Five hundred children of soldiers are being cared for in the great\n  Homes founded by Mr. Quarrier in Scotland, and most of them are\n  there because of drinking mothers.\n\n                                                    _Facts in Reports_\n\n  A soldier's wife at Biggleswade spent her allowance on drink and\n  left her three children locked up in the house for days at a time.\n\n                 _Police Court Records of Biggleswade, September 1915_\n\n  A soldier's wife was found reeling in the streets of Dublin with a\n  baby in her arms. At her home were found four other children,\n  cruelly neglected.\n\n                             _Facts in \"Dublin Mail,\" August 16, 1916_\n\n  Nineteen hundred children of soldiers have come into the care of the\n  N.S.P.C.C., mainly through drink, since the war began.\n\n                                           _Records of the N.S.P.C.C._\n\n\n                            The Ruined Wives\n\nWho does not remember the terrible rush for the last drop of drink when\nProhibition seemed to be coming with the New Year? Long queues of women\nbesieged the whisky shops in Glasgow. There were women of all ages, said\nthe _Daily Mail_, tottering in grey hairs, young wives with babies in\ntheir arms, and men of the loafer type. \"There was not a respectable\ncitizen,\" says the _Mail_, \"who did not deplore this discreditable\nscene, but the remarks of passers-by provoked only torrents of insult.\"\nThe promise of the new year and the new Government, alas, was not\nfulfilled, and now in place of Drink Queues we have Food Queues. Let us\nsee what drink is doing among our soldiers' wives:\n\n  Of 3000 soldiers' wives being cared for in South London, 2000 are\n  splendid, while 1000 are sinking daily to lower and lower levels\n  through drink.\n\n                                      _Records of Shaftesbury Society_\n\n  A soldier's wife, with a separation allowance of 32_s._ 6_d._ a\n  week, drank most of it away, ruined her home, neglected her\n  children, and became a lunatic.\n\n                                          _Records of Claybury Asylum_\n\n  A young soldier's wife, hitherto \"quite an elegant type,\" is rapidly\n  becoming a drunkard. Women hitherto sober have not the courage to\n  keep from women's drinking parties, and young girls come out of\n  factories and go to publichouses in little groups.\n\n                             _Records of Charity Organisation Society_\n\n  Outside a public house in Dublin 15 small children were crying in\n  the cold, waiting for their mothers. Ninety-four drunken women came\n  out in 25 minutes. There were ten drunken soldiers, and two girls of\n  15 were thrown into the street hopelessly drunk.\n\n                              _Facts in \"Irish Times,\" April 20, 1915_\n\n  In Dundee over 170 wives of soldiers gave way to drink last year,\n  and cruelly neglected their homes.\n\n                                       _Records of the N. S. P. C. C._\n\n  A soldier in the trenches received a letter from his little boy,\n  which he sent to London with a pitiful appeal for help.\n\n  \"Kindly do what you can for me and the well-being and welfare of my\n  four beautiful children,\" the poor soldier wrote. \"I am enclosing a\n  fearful letter I have received from my poor little lad, 14-1\/2, the\n  first and only letter I have received from him. Sir, I shall be most\n  anxiously awaiting your reply, for this letter is the greatest blow\n  I have ever received.\"\n\n  This is the little boy's letter:\n\n    Dear Dad: Just a line to let you know how everything is at home.\n    Mother is drunk for a fortnight and sober for a week for months\n    and months. I've stuck it now for seven months, and can't stick\n    it any longer. I tried to get into the Navy and passed all the\n    tests, but mother would not sign the papers, for which I am\n    sorry. If mum would sign I could go away to Portsmouth on\n    Thursday, but she will not. At the present moment she is half\n    drunk and keeps jawing me so that I could knife meself. I've\n    lost my new job because mum would not wake me in the morning,\n    and nothing for breakfast, and had to get mine and the\n    children's tea at tea-time. It pains me to write like this, but\n    I can't help it. I now seek your advice as to what to do. I hope\n    _you_ will enjoy Xmas, although there is not much hope for us. I\n    now conclude with fondest love, X. Your heartbroken Son, Leslie.\n\n  A stream of nearly 15,000 men and women poured into 58 publichouses\n  in Birmingham in less than four hours; over 6,000 were women. Into\n  one house the people streamed at nearly 500 an hour.\n\n                          _Facts in \"Review of Reviews,\" October 1915_\n\n  For months some wives of soldiers and sailors in Scotland were never\n  really sober. \"We have done our best,\" says a worker among them,\n  \"going to their homes and doing all in our power, but it beats us.\"\n  In 23 families, with 178 children born, 61 were dead.\n\n                     _Facts told to Secretary for Scotland, July 1916_\n\n               Will some Member of Parliament please ask\n\n=whether the ships that have brought in food for destruction by the\ndrink trade could not have brought in a large proportion of the\n3,500,000 tons of wheat now waiting for ships in Australia and the\n2,000,000 tons waiting in Canada?=\n\n------------------------------------------------------------------------\n\n\n\n\n                          The Roll of the Dead\n\n\nNo more pitiful record of the war is there than that unnumbered roll of\nmen lured from our armies by this liquor trade, and cast into\ndishonoured graves. We can take only a few of them.\n\n  A number of soldiers at Ormskirk came into camp drunk on Christmas\n  night. A request for quiet led to a fight, and one of the men was\n  struck two blows and was dead the next morning.\n\n                            _Facts in \"Daily Mail,\" December 28, 1915_\n\n  A Liverpool soldier, drinking continuously, had overstayed his\n  leave, and in a quarrel about this he stabbed his brother dead.\n\n                        _Facts in \"Liverpool Courier,\" April 20, 1917_\n\n  A soldier invalided from France, having recovered from his wounds,\n  gave way to drink, assaulted an officer, and hanged himself in his\n  prison cell.\n\n                               _Facts in \"Daily News,\" April 11, 1916_\n\n  A young lieutenant shot himself in an hotel near Trafalgar Square,\n  and among the documents read at the inquest was a letter striking\n  him off his battalion for drinking and gross carelessness.\n\n                        _Facts in \"Daily Chronicle,\" October 27, 1916_\n\n  A captain in the Army ruined by drink, with a fine record of\n  military service, started drinking on his way to a shooting range in\n  London, and in a struggle he shot a detective dead.\n\n                           _Facts in \"Daily News,\" September 20, 1915_\n\n  In the Scottish Express, between Doncaster and Selby, a drunken\n  corporal of the Coldstream Guards was showing his rifle to a friend\n  when it went off, the bullet killing a munitions works director in\n  the next compartment, and narrowly escaping a lady in the\n  compartment beyond. The corporal had in his pocket a bottle of\n  whisky, which was freely handed round.\n\n                             _Facts in \"Daily News,\" December 3, 1915_\n\n  A soldier who had been drinking heavily was placed in the guard\n  room, and died after a night of groaning, evidently as the result of\n  a fall.\n\n                       _Records of Greenwich Coroner, January 1, 1915_\n\n  A young soldier arriving from India on Christmas morning was\n  arrested three days later, after a drunken fight in which a man was\n  killed.\n\n                       _Westminster Police Records, December 28, 1914_\n\n  A soldier spent a day's leave in Manchester, ate and drank very\n  heavily, and was found dead the next morning from choking.\n\n                    _Records of Manchester Coroner, December 28, 1914_\n\n  A soldier home on leave was found drunk with his wife. They had been\n  throwing pots at one another, and on Christmas morning the woman was\n  found dead with a wound in her head.\n\n                        _Records of Oldham Coroner, December 24, 1914_\n\n  Three gunners had four drinks each of rum, and at midnight lay down\n  to sleep in a garden at Lee, where one was found dying from alcohol.\n\n                             _Facts in Local Papers at Lee, June 1915_\n\n  A soldier died from alcohol in a house where drink was unlawfully\n  sold.\n\n                       _Facts in \"Manchester Guardian,\" April 8, 1915_\n\n  A private in the Welsh Fusiliers died from alcohol, cold and\n  exposure. He left a publichouse with a 4_s._ bottle of whisky, and\n  was found dead on the roadside next morning, with the bottle almost\n  empty.\n\n                               _Facts in \"Daily News,\" April 13, 1915_\n\n  An old man who was said to be in a drunken condition was wounded in\n  a fall with a soldier from Gallipoli, and died a few days after.\n\n                             _Facts in \"Daily Mail,\" January 17, 1916_\n\n  An elderly man, seeing a drunken soldier lying in the street, went\n  to his assistance, and was killed in a disturbance that followed.\n\n                      _Record of Yorkshire Assizes, November 21, 1916_\n\n  A soldier was found drowned in the Trent. He was described as a good\n  man at his work, but not steady, and had been drinking.\n\n                        _Facts in \"Newark Advertiser,\" August 4, 1915_\n\n  A terrible disturbance occurred in a camp at Portland Reservoir\n  after the closing of the canteen one Sunday night. A large number of\n  men who had been drinking created a disturbance, in which bricks and\n  stones were used, a tent collapsed, and the officers were called to\n  quell the riot. The captain, drawing his revolver, rushed with two\n  lieutenants into a hut where men were shouting and struggling, but\n  appeals had no effect\u2014the men \"did not appear to hear or recognize\n  their officers,\" and one man raised his rifle and took aim at them.\n  At least fifty shots were fired, and a young corporal fired many\n  shots through the window into the darkness. In the morning a soldier\n  was found dead. Nobody knew who shot him, but the corporal thought\n  he must have done.\n\n                              _Records of Dorset Assizes, Spring 1915_\n\n               Will some Member of Parliament please ask\n\n=whether it is true that more food is being destroyed each week in\nbreweries and distilleries than by submarines?=\n\n------------------------------------------------------------------------\n\n\n\n\n                            The New Drinkers\n\n\n\"_No complaints have reached the War Office of youths who were total\nabstainers having become confirmed drunkards since enlistment._\"\n\nSo we are told in the House of Commons. The records of the War Office\nare clearly incomplete, and the information from the camps may here be\nsupplemented by unchallengeable witnesses of what happens in the\nhorrible drink canteens run by the Army Council.\n\n  A soldier who was wounded at La Bass\u00e9e, a total abstainer until\n  then, was sentenced at the Old Bailey for killing his uncle while\n  drunk. He was a newsvendor, aged 21, and had no memory of the\n  tragedy in which he killed his uncle at a Christmas party.\n\n                        _Facts in \"Daily Chronicle,\" January 13, 1916_\n\n  A private in the Royal Scots Fusileers, aged 17, was charged with\n  murdering a bugler boy, aged 16, in his regiment. The private became\n  mad drunk in the camp canteen, went back to his hut, locked himself\n  in and fired two shots, one of which entered another hut and killed\n  the bugler. \"Was there no one with power to say how much drink\n  should be given?\" asked the judge, and an officer said there was no\n  one. \"Then it was high time power was given to the commanding\n  officer,\" said the judge. \"Was there to be no restraining hand to\n  prevent young boys from fuddling themselves in canteens?\"\n\n                             _Facts in the \"Times,\" November 21, 1916_\n\n  An old man sat in a tram in great distress. He had lost his boy at\n  the Front. When he joined the Army he had never tasted alcohol, but\n  when he came home on leave to see his mother he was drunk every\n  night. He was drunk the night he went away, and in three days he was\n  dead. \"The last we saw of him,\" said the poor old man between his\n  sobs, \"was his going away drunk, and his mother, who is\n  old-fashioned in her faith, cannot get it out of her mind that no\n  drunkard can enter the Kingdom of God.\"\n\n                                    _Facts told by Dr. Norman Maclean_\n\n  Many young officers, called upon to share the wine bill at mess,\n  naturally say, \"If I have to pay I may as well drink my share,\" and\n  one man accounted for ten glasses of champagne. On a Guest night in\n  his mess several more \"were under the table.\"\n\n                        _Facts in \"Dublin Daily Express,\" April 1916._\n\n  A boy got his V.C., and came home wounded. The publican in his\n  street sounded his praises in the taproom, where they subscribed to\n  the bar for 120 pints for him when he arrived. He came home and\n  began to drink it, and was nearly dead with it before he was\n  rescued.\n\n                                  _Facts related by Bishop of Lincoln_\n\n  When the Scottish Horse Brigade were at Perth whisky was literally\n  forced down the men, and they were inundated with floods of bad\n  women.\n\n                                 _Brigadier-General Lord Tullibardine_\n\n  A teetotal household had two boys in an officers' training camp, and\n  they gave pitiable accounts of drinking. Boys from school had a\n  drunken sergeant put over them, and a canteen in the midst of them.\n  \"Our boys never saw drink before,\" one father wrote.\n\n                                 _From a letter to Dr. Norman Maclean_\n\n  A boy of 17, discharged from the Navy, spent 8_s._ one night on beer\n  and rum, and created a disturbance in a workshop at Sheffield.\n\n                        _Facts in \"Sheffield Star,\" November 11, 1916_\n\n  Mr. Justice Atkin, charging the Grand Jury at Bristol, said that in\n  nearly every case where a soldier was tried in the Western Circuit\n  the defence was drink. One lad of 18 was treated to eight pints of\n  beer in two hours, and did not know what happened. That sort of\n  thing, said the judge, must seriously impair the efficiency of the\n  troops when sent to the Front.\n\n                              _Record of Bristol Assizes, Autumn 1914_\n\n  Two boys, 15 and 17, were fined for being drunk in munition works.\n  One was discovered just in time to save him from carrying molten\n  liquid.\n\n                            _Birmingham Munitions Tribunal, Dec. 1916_\n\n  \"A boy joined the Royal Navy as a carpenter, living in barracks and\n  working on shore. Every day he was given 'grog' for his rations,\n  although he never asked for it and never took it.\"\n\n                                       _Facts in letter to the Author_\n\nSuch are the tragedies of boys handed over in our camps to drink and its\ntemptations. What of the girls in our munition shops? They have learned\nto drink in thousands since the war began\u2014respectable girls leaving home\nto go into munitions, respectable young wives alone at home. With no\nrestraining hand upon them, with new companionships and pocket-money\nflowing freely, it is not surprising the temptation should be too strong\nfor them. We can take only one or two cases.\n\n  The girl-wife of a Cardiff seaman died in the street from exposure\n  after drinking in publichouses with other girls.\n\n                    _Records of Pontypridd Coroner, December 27, 1916_\n\n  A publican at Lincoln was fined \u00a35 for allowing children to be drunk\n  on his premises. Ruth Onyon, 14, and Rose Herrick, 16, were found in\n  his house with a soldier. They had been in five houses and had ten\n  drinks each and reached home helplessly drunk.\n\n                 _Facts in \"Sheffield Daily Telegraph,\" Sept. 1, 1916_\n\n  A number of cartridge workers were summoned for taking drink into a\n  munition works. One young woman was led to the surgery drunk at\n  half-past four in the morning; another was discharged because she\n  could not stand. Sixteen girls subscribed for four bottles of wine\n  and whisky.\n\n                 _Records of Leeds Munitions Tribunal, April 28, 1916_\n\n  Two girls of 16 and 17 were fined for being helplessly drunk in an\n  explosive works, the magistrates pointing out that their conduct\n  imperilled the lives of other workers.\n\n                _Records of Coventry Munitions Tribunal July 24, 1916_\n\n  The men and girls at a large armament works drank all night. Girls\n  would lurch into the dormitory dead drunk at 2 a. m.; one lady was\n  up till 4 a. m. letting in drunken girls. As a result of drunkenness\n  there was an explosion at these works, two men being killed and six\n  injured.\n\n                                 _Facts in \"Spectator,\" Jan. 20, 1917_\n\n  A Dublin publichouse was found full of girls and soldiers, all\n  drunk. Three drunken girls were taken away by six soldiers.\n\n                              _Facts in \"Irish Times,\" April 20, 1916_\n\n  In half an hour 367 girls entered Birmingham publichouses, scores\n  under 18. Stout and beer were chiefly drunk, but whisky and water\n  also, and some port wine. Ten young girls were quite drunk.\n\n                                    _Facts in \"Birmingham Daily Post\"_\n\n               Will some Member of Parliament please ask,\n\n=in view of the fact that American soldiers are not to touch alcohol,\nwhat arrangements the Government proposes to make for them in this\ncountry?=\n\n------------------------------------------------------------------------\n\n\n\n\n                          Back to the Homeland\n\n\nEverywhere we hope and pray for peace, for the day when the men will\ncome home; but we may dread the day if the men come home to drink and\nits temptations. The sudden release of millions of men, the certain\nreaction after the terrible stress of these three years, is fearful to\ncontemplate with the door of the tap-room open. There would be an end of\ncivilization itself for days and weeks and months, and for many a town\nat home the Peace would be worse than the War.\n\nWe owe it to these men to listen to the warning of the Prison\nCommissioners who printed these words in their report last year:\n\n=When war is succeeded by peace there will come a time of trial for\nthose who have never turned their backs to a bodily enemy. With the\npassing of military discipline our brave fellows will be tempted to\nforget the hardships and miseries of the trenches in a burst of\nuncontrolled pleasure and license, and, if trade be bad and work\ndifficult to obtain, the lapse may, if not checked, become a step on a\ndownward career.=\n\nIt is not imagination merely. Judges, coroners, police, and all who face\nthe crime and misery of life, know well the bitter things that happen\nwhen men come home without restraint. There are witnesses innumerable.\nLet us hear a few of them.\n\n  A captain in the Royal Flying Corps drove a motor-car through\n  London, knocked a man down, drove on, and ignored the police, who\n  eventually mounted the footboard and found the officer drunk.\n\n                             _Bow Street Police Records, June 3, 1916_\n\n  A lance-corporal on Chesterfield station was so drunk that he walked\n  off the platform and fell on the line as a passenger train came up.\n\n                           _Chesterfield Police Records, June 2, 1915_\n\n  A corporal of the Duke of Cornwall's Light Infantry, leaving the\n  Front with 150 rounds of ammunition and his service rifle, came out\n  drunk into the streets of West Ham and began firing his rifle.\n\n                           _Facts in \"Daily Chronicle,\" July 10, 1915_\n\n  A soldier who had received a cartridge from his son at the Front,\n  put it in his rifle, and while drunk fired it in the streets of\n  Manchester.\n\n                         _Manchester Police Records, January 27, 1915_\n\n  In the early hours of the morning two unarmed soldiers were fired at\n  in Woolwich by a drunken soldier, who chased them for a long\n  distance, firing shots all the time, until he was arrested.\n\n                            _Facts in \"Alliance News,\" February, 1915_\n\n  Drunkenness among soldiers and sailors is appalling. Unoffending\n  travellers are delayed by drunken sentries. Sailors landing after\n  weeks of arduous toil in the North Sea find it easy to get so drunk\n  that some are drowned, some die from exposure, and many return to\n  their ships in a condition of helpless inebriety.\n\n                              _Facts in \"Inverness Courier,\" May 1915_\n\n  Two drunken soldiers entered the parish church at Codford, set fire\n  to the vestry, threw down the altar cross and candlestick, broke a\n  stained-glass window, and tore leaves out of a Bible 200 years old.\n\n                           _Facts in \"Daily Chronicle,\" April 3, 1916_\n\n  A drunken soldier at Cannock was imprisoned for drawing his bayonet\n  in the streets. \"If I meet a policeman I will murder the dog,\" he\n  said, and, meeting one, he threatened to cut off his head.\n\n                               _Police Records at Cannock, March 1916_\n\n  400 soldiers tried to get a drunken man from the police in Grantham.\n\n                              _Facts in \"Grimsby News,\" July 30, 1915_\n\n  A drunken sergeant was found forcibly detaining a girl at Hornsey.\n  On the police interfering, the drunken soldier drew his bayonet.\n\n                            _Facts in \"Daily News,\" September 7, 1916_\n\n  Three splendid-looking fellows, minesweepers, were traveling on the\n  Highland Railway. \"All were married men,\" said a fellow passenger,\n  \"happy and proud of their homes, and they spoke with ache still in\n  their hearts something of their lives and work. Well, these men\n  succumbed during the journey. A change of trains was their\n  opportunity, and I left them in a nearly helpless condition.\"\n\n                             _Facts in \"The Spectator,\" April 8, 1916_\n\n  A lady visited a soldier's wife and found her at home with all her\n  clothes in pawn. Her husband and brother had both been home from the\n  Front, and in one week had spent \u00a38 on drink.\n\n                     _Facts in the \"Cork Constitution,\" Dec. 10, 1915_\n\n  A labourer, home from tunnelling work at the Front, was fined 13_s._\n  for drunkenness on his 33rd appearance, having spent \u00a345 in seven\n  days.\n\n                                _Facts in \"Daily News,\" Oct. 11, 1916_\n\n  A disabled soldier was selling papers in Kingsway, London. He was\n  proud of his military record and the character his colonel gave him.\n  He was trying to compound for a pension; he thought he would settle\n  for \u00a350. \"Mind you,\" said he \"there is not a better character in\n  London than mine, and I shall get the \u00a350. Then I shall have a\n  month's booze.\" \"What, with that fine character of yours?\" a\n  gentleman said to him. \"Yes,\" said the man, \"when I came home, and\n  could leave the hospital, there was \u00a350 due to me, and I had a\n  regular booze.\"\n\n                                           _Facts known to the Author_\n\n  A soldier with twelve years' clean record in the Army was sentenced\n  for felony after being made drunk by his friends.\n\n                        _Police Records of Southport, January 9, 1915_\n\nNo Government has ever received more warnings than the three war\nGovernments have received concerning drink. There is no room for them\nhere, but we may call a few witnesses such as cannot be ignored by a\nnation looking forward to the day when millions of men will be home\nagain.\n\n  A house in Westminster reeked with filth and drink and drunken\n  overseas soldiers, \"and it would be better,\" said the Crown\n  Solicitor, \"if power were given to the police to sweep such places\n  off the earth.\"\n\n                               _Westminster Police Records, Aug. 1916_\n\n  A sapper seaman was found dead at the quay. Another seaman said his\n  friend had seven drinks. They left the publichouse arm-in-arm, and\n  went to the quay. There he saw a corporal, who was boatswain for the\n  night, and was drunk. Leaving the sapper, he got the corporal into\n  the boat, and went back for his friend, but the sapper had\n  disappeared.\n\n  The lieutenant: \"The deceased was one of the quietest boys who had\n  ever been on the ship, and one of the best oarsmen. The whole\n  trouble was that it was pay day.\"\n\n  The Coroner: \"Prohibition during the war would be a blessing to all.\n  It seems to be a very rotten state of affairs.\"\n\n  The foreman: \"Drink.\"\n\n  The lieutenant: \"Prohibition would be the best thing.\"\n\n  The Coroner: \"This poor man, unfortunately, is one of many.\"\n\n                   _Facts in \"Western Daily Mercury,\" January 8, 1917_\n\n  A publican at Dover was fined \u00a320 for selling a bottle of whisky to\n  a sailor. The Admiral said drink undermined the efficiency of the\n  patrol vessels, and those who supplied it directly assisted the\n  enemy, and might be the cause of the loss of very many lives.\n\n                            _Police Records of Dover, October 6, 1916_\n\n  A private in the Northumberland Fusiliers, aged 23, was charged with\n  burglary while drunk. His father and three brothers were in the\n  Army. He took part in the battle of Loos, was wounded at Salonika,\n  and was recommended for distinction for helping to save a wounded\n  officer.\n\n  During the whole of Christmas leave he was drinking, made drunk by\n  his friends who were probably proud of his having held part of a\n  trench against a German bombing party. His captain described him as\n  a good soldier in peace, and brave in action\u2014a man whose disgrace\n  would be felt by the regiment.\n\n  Mr. Justice Rowlatt said everyone was hoping for the time when\n  millions of brave men would come home after facing incredible\n  dangers, and we must look forward almost with terror to having these\n  men exposed to drink and its temptations. What would be the state of\n  the country in such a case unless we could make a clean sweep of\n  drink? We should have to face this question over and over again, and\n  the sooner we faced it the better.\n\n                        _Records of Derbyshire Assizes, February 1917_\n\n  Whoever allowed soldiers or sailors to drink to excess, said the\n  Mayor of Tynemouth, should be tried by court-martial for treason. He\n  would be recreant in his duty to God, to himself, and to the\n  citizens, if he did not call attention to the brutalising of so many\n  townspeople and the callous conduct of the \"waster\" element in the\n  drink trade. He had no quarrel with those who conducted their\n  business properly.\n\n                           _Facts in Tynemouth papers, February, 1915_\n\n  The Aldershot command appealed for the closing of half the\n  publichouses, to save the men from temptation when the troops are\n  demobilised and return with their pockets full of money.\n\n                       _Record of Workingham Licensing Sessions, 1917_\n\n  The _Army and Navy Gazette_, in an article disapproving of the\n  Prohibition Campaign, issues a terrible warning which should be\n  printed on the door of the room in which the Army Council meets.\n  These are its words:\n\n  \"It is on record that towards the end of the siege of Sebastopol rum\n  was made too regular an issue, with the result that almost every\n  soldier who survived to return home became a drunkard.\"\n\nThe siege of Sebastopol lasted less than a year, and that is the work of\nthe rum issue for a few months. If rum does that in months, what will it\ndo in years?\n\n------------------------------------------------------------------------\n\n\n\n\n                          Into the Firing Line\n\n\nLord Kitchener is dead, but there are two things that are with us\nstill\u2014that rare little note that he gave to his men as they went out,\nwarning them of drink; and that infamous note sent out by a drink firm\nin London, begging our people to send out drink to our men. They can\nguarantee it right up to the firing line, they say, and even when our\nshells could not get there through drink, drink seems to have found its\nway. It can get on to transports when the Ministry of Munitions is\nwaiting urgently for shipping space; it can commandeer our vans and\nhorses and trains when these mean life or death to us; it seems to get\npast any regulation; it goes about with the power of a king, doing its\nwork where it will.\n\n  It is regrettable that our troops at the Front cannot get more\n  British Beer.\n\n                          Managing Director of Allsopps, July 14, 1916\n\n  Dear Sir, In answer to your inquiry, the only limitation in the size\n  of cases consigned to officers in the Expeditionary Force is that\n  they must not exceed 1 cwt.\n\n  We can guarantee delivery right into the front trenches. The cases\n  are handed over at Southampton to the Military Forwarding Officer,\n  and the A.S.C. see them right through. We are shipping hundreds of\n  cases weekly. Yours faithfully,\n\n                        _Letter from a Wine and Spirit firm in London_\n\nSo drink finds its way to the front, to weaken our troops, with all\ntheir matchless heroism. Let us call the witnesses who have seen the\nwork it does.\n\n  Soldiers at the front, tried for drunkenness, have declared that\n  they have received drink from home. Men sometimes receive flasks in\n  the trenches. They are exhausted, the stimulant revives them for a\n  minute or two, and the harm is done. \"And then (says Col. Crozier)\n  they get about two years' hard labour.\"\n\n      _Letter from Colonel Crozier, commanding 9th Royal Irish Rifles_\n\n  As a result of a Court-martial investigating charges of excessive\n  drinking among the officers of a regiment at the Front, the Army\n  Council removed the commanding officer from his post.\n\n                                     _Records of Court-martials, 1916_\n\n  In the torrid climate of Mesopotamia, in defiance of all military\n  medical history, rum was issued to the men instead of food and\n  sterile water, and the presence of cholera, dysentery and other\n  diseases, was attributed to this by Sir Victor Horsley. \"Our gross\n  failures and stupidity,\" he said, \"are in my opinion due to whisky\n  affecting the intellectual organs and clearness of our leaders. They\n  do not realise that alcohol in small doses acts as a brake on the\n  brain.\"\n\n             _Facts in a letter from Sir Victor Horsley, May 13, 1916_\n\n[Illustration: THE JUNKER'S LITTLE BROTHER]\n\n  Battalion Headquarters\u2014colonel and chaplain present. Enter Adjutant:\n  \"The rum ration is due tonight, sir; am I to distribute it?\" The\n  colonel (nobly and in a voice audible all over the trench): \"No!\n  Damn the rum! To hell with the rum!\"\n\n                     _Chaplain's letter in \"Alliance News,\" June 1916_\n\n  At a court-martial in Newcastle, a sergeant-major, charged with\n  misappropriating funds of the sergeant's mess, pleaded that during\n  this period a resolution of the mess had come into effect, providing\n  free drinks during Christmas and the New Year.\n\n                               _Facts in \"Daily News,\" April 17, 1916_\n\n  \"In the Flying Services one has seen more than one good man go to\n  the dogs through drink, or become fat and flabby and useless through\n  just the excess of alcohol which falls short of taking to drink in\n  the usual acceptance of the term. More men take to drink because of\n  the 'have another' custom than because they like or need alcohol,\n  and simple Prohibition would stop all this nonsense straight away.\n  This kindly note is not the outpouring of a teetotal fanatic, for I\n  suppose I have paid in my time rather more than my share of the\n  nation's drink-bill; it is merely a perfectly sound argument in\n  favour of increasing the nation's efficiency at the expense of its\n  chief bad habit.\"\n\n                                       _The Editor of \"The Aeroplane\"_\n\n  A lieutenant in the trenches, knowing that the rum ration made him\n  cold, threw his rum on the ground. His captain saw him, and\n  threatened to report him. \"You do, sir,\" said the lieutenant, \"and I\n  will report you for being drunk on duty.\"\n\n                                   _Facts in possession of the Author_\n\n  A seaman serving on a ship in Cork Harbour died from alcohol. Found\n  drunk and unknown, he was put on a stretcher and died.\n\n                      _Facts in \"Cork Constitution,\" December 9, 1915_\n\n  \"Over three-quarters of the court-martials I have had anything to do\n  with are due directly or indirectly to drunkenness. Many thousands\n  of competent N.C.O.s and soldiers have been punished, and become\n  useless to the nation during their punishment, as a result of drink.\n\n  \"I have never been a teetotaler, and have rather opposed the radical\n  temperance agitation, but am now changing my views as I see our\n  success over here hampered and our progress towards victory retarded\n  so obviously by drink.\"\n\n       _Letter from a Lieut.-Colonel at the Front, seen by the Author_\n\n  The captain of a British merchant ship, drunk on the bridge, ordered\n  his chief gunner to fire 50 rounds of shell at nothing. The gunner\n  fired four rounds to appease him. Going through the Mediterranean,\n  the drunken captain ordered his gunner to fire at a British hospital\n  ship, and the incident led to a struggle for life, which ended in\n  the captain's being put in irons, tried, and sentenced to five\n  years' penal servitude.\n\n                   _Record of Devon Assizes, Exeter, February 2, 1917_\n\n  An officer was left in charge of a British ship. Mad with drink, he\n  went among the men and shot one dead. He is now in an asylum.\n\n                                      _Case reported to the Admiralty_\n\n  The crew of a Dutch ship arriving in the Tyne was placed under a\n  naval guard after a drunken riot in which three were killed.\n\n                           _Facts in \"Daily News,\" September 14, 1915_\n\n  The captain of a Norwegian barque mysteriously disappeared, and the\n  vessel arrived in port from the North Sea. The mate, who had been\n  drinking heavily, was seen, with a hammer in his hand, with the\n  captain in a corner, bleeding from wounds about the head.\n\n                                _Facts in \"Daily News,\" April 8, 1916_\n\n  A seaman ashore in Glasgow, \"wild with drink and passion,\" was\n  terribly wounded in a quarrel in a public-house, and died the same\n  night. A youth of 19 was sentenced to five years' penal servitude.\n\n                          _Records of Edinburgh High Court, Dec. 1916_\n\n  A barge-loader at West India Docks died from alcohol, and three\n  other men were removed in an ambulance after drinking rum.\n\n                             _Facts in \"Daily Chronicle,\" May 9, 1916_\n\n  Orders were given on a steamer for the boats to be swung out in\n  readiness for submarines. The first and second officer, having been\n  drinking, could not do their duty.\n\n                   _Records of Liverpool Marine Board, April 13, 1917_\n\n  The jury returned a verdict of murder against a youth of 19 who,\n  after drinking one night, went on to his ship and killed the second\n  officer.\n\n                             _Records of Hull Coroner, April 24, 1917_\n\n  A drunken captain in command of a drifter landed with an armed party\n  on the Isle of Man. He posted the men on the quay, and gave them\n  orders to allow no one to pass. Declaring he would shoot every\n  person who came within reach, he fired twice, and threatened to kill\n  two police officers.\n\n                                   _Facts in \"Times,\" October 6, 1916_\n\nSuch is the work of drink wherever it finds a soldier to entrap\u2014the\ndrink the Navy carries free from Southampton to the trenches; and from\nAmerica comes the news, as this page is being written, that the Army and\nthe Navy of our Western Ally, like the Army and the Navy of our Eastern\nAlly, are to be under Total Prohibition.\n\n               Will some Member of Parliament please ask\n\n=how much bread is destroyed each week to make beer for German\ninternment camps in this country?=\n\n------------------------------------------------------------------------\n\n\n\n\n                        Drink and the Red Cross\n\n\nIf the full story could ever be told of the national tragedy of drink\nand the war there would be no more ghastly chapter than that which would\ntell how drink fought the Red Cross; how, without pity, it hindered the\nwork of mercy that is the general consolation of the world in days like\nthese.\n\nWe are coming to a famine not only in food, but in doctors. The\ndeath-roll has been heavy beyond all parallel; the strain on the medical\nservices has been almost too great to be borne, and we look anxiously\nround to know where the doctors and nurses will come from. With\nProhibition the problem would be largely solved, for the ordinary burden\nof life would be largely lifted from our doctors and hospitals, and\nthousands of men and women would be free to give themselves to the war\ninstead of mending up and patching up the sordid effects of drink. A\nrich brewer gave a donation for extending a hospital. \"Ah! but we should\nnot have to extend if he would shut up his public-houses,\" said a\ndoctor.\n\nIt is easy to see how drink is telling all the time against our doctors,\nour nurses, and our hospitals everywhere. Let us call a few witnesses.\n\n  Somebody gave a glass of neat whisky to two wounded men at a garden\n  party in Tottenham. Both were drunk when the brake came to take them\n  home, and one died on the way.\n\n                   _Facts in \"Sheffield Telegraph,\" September 3, 1915_\n\n  Three wounded soldiers at Oxford were overcome by four bottles of\n  rum smuggled into the hospital by visitors, and one of the men died.\n\n                             _Records of Oxford Coroner, January 1916_\n\n  A wounded soldier asked for two hours' leave, came back in four\n  hours drunk with whisky, and died after a terrible night in the\n  hospital.\n\n                                               _Facts in \"Daily Mail\"_\n\n  Two limbless soldiers were found helplessly drunk on the pavement at\n  Brighton. A publican was fined \u00a320.\n\n                       _Facts in \"Daily Chronicle,\" November 25, 1916_\n\n  A wounded soldier, mentioned in despatches, was charged with causing\n  the death of a soldier with whom he had been drinking. Reeling under\n  a heavy blow, the injured man was helped to bed, but when the bugle\n  sounded in the morning he was dead.\n\n                            _Facts in \"Daily Mail,\" December 21, 1915_\n\n  A soldier, aged 29, with a gunshot wound in his arm, died from\n  alcohol at Oxford. One Sunday night he and two other wounded\n  soldiers consumed four bottles of rum brought into the hospital.\n\n                         _Records of Oxford Coroner, January 10, 1916_\n\n  Three soldiers in hospital uniform were found lying helplessly drunk\n  on the tramlines of Sheffield. Two were back from the Dardanelles.\n\n                            _Facts in \"Sheffield Star,\" March 2, 1916_\n\n  Seamen on a ship bringing wounded to England from Boulogne were so\n  drunk that they interfered with the stretcher bearers, and one fell\n  across a wounded soldier lying on deck.\n\n                         _Police Records of Southampton, May 14, 1915_\n\n  There was a paralysed and helpless man who was found hopelessly\n  drunk in hospital after his friends had visited him.\n\n                    _Statement by Lieut.-Col. Sir Alfred Pearce Gould_\n\n  An officer who has trained hundreds of men for the ambulance corps\n  declared that a large percentage of wounded are in a very nervous\n  condition, in which alcohol means collapse and almost certain death.\n\n                                              _Quoted in \"Daily Mail\"_\n\n  Lying helpless at a London station, moaning on the ground in drunken\n  delirium, was a lad in hospital blue who had, in truth, been wounded\n  by his friends. Drink was taking him again through the worst of his\n  experiences, and his mental pain was pitiable to see.\n\n                                 _Facts in the \"Globe,\" January, 1917_\n\n  Two drunken soldiers from Gallipoli made what a doctor described as\n  the most savage attack he ever saw on a civilian. They held a young\n  man's head against a wall and pounded him unmercifully.\n\n                              _Facts in \"Daily News,\" August 19, 1916_\n\n  A party of soldiers were seriously injured in a struggle to arrest a\n  drunken private at Pontefract. The publican called on the men in his\n  taproom to rescue the private, but the sergeants drove them off.\n\n                              _Facts in \"Daily News,\" October 5, 1914_\n\n  A sergeant of a Welsh regiment, invited to drink by friends in\n  Waterloo Road, was picked up as he lay senseless, his pulse beating\n  feebly, his eyes wide open, and his body starving with cold.\n\n                            _Facts in \"Daily News,\" February 14, 1916_\n\n  A drunken man rushed from a publichouse and kicked a soldier\n  unconscious. The military police, chasing the man, were stoned. Four\n  soldiers were injured, one having his head cut open, and the\n  military were ordered to clear the place with fixed bayonets.\n\n                              _Facts in \"Daily News,\" August 11, 1915_\n\n  The medical officer in charge of the Mental Block of a large\n  military hospital said to the Colonel: \"I have the worst job of all,\n  and it is through Drink, Drink, Drink! Men recover fairly soon from\n  shell shock, but officers, especially the younger ones, who\n  habitually take wines and spirits, are subject to relapses every few\n  days. It is awful!\"\n\n                  _Facts in \"National Temperance Quarterly,\" May 1917_\n\n  Of the thirty war hospitals in Hertfordshire, with 8000 men passing\n  through them in the first thirty months of the war, there is not one\n  that has not had trouble with drink.\n\n                                           _Facts known to the Author_\n\n  A doctor from a Canadian hospital said a large percentage of their\n  troops had had to be sent back to Canada rendered permanently insane\n  through the action of alcohol.\n\n                             _Facts in \"Daily News,\" October 31, 1916_\n\nOne terrible truth remains to be told of the crime of drink against the\nRed Cross. The most blessed thing in all the world today is alcohol, for\nit makes chloroform and ether, which soothe the pain of men. We cannot\nget enough of either of these consoling drugs, yet we go on wasting\nprecious food to make more alcohol _to add to the sum of misery and\npain_.\n\n               Will some Member of Parliament please ask\n\n=whether the bread ration applies equally to all; or if it may be\nexceeded if the excess is drunk instead of being eaten?=\n\n                                  and\n\n=how many brewers' vats have been imported this year on ships which had\nno room for urgent munitions of war?=\n\n------------------------------------------------------------------------\n\n\n\n\n                     Stabbing the Army in the Back\n\n\nAll the world is learning now that the drink trade is the great\nconfederate of venereal disease. It leads a man into temptation,\ndestroys his power of resistance, and <DW44>s his chances of recovery.\n\nWe can never know the truth about the extent of this disease, about the\nway in which the liquor trade, by breaking down tens of thousands of our\nmen, has stabbed the Army in the back. But the number of soldiers\nincapacitated by this disease through drink is enormously greater than\nthe number incapacitated by the most subtle or dramatic stroke devised\nby the German staff.\n\nThe lost man-power of the Army through this disease must be equal to the\nwhole of the original British Expeditionary Force. The Government has\ngiven us figures for the Army at home last year, and they are 43 per\n1,000\u2014or over 100,000 cases for an army of 2,500,000 men. There were\n7,000 cases in one Canadian camp alone.\n\nHere are the black facts revealed in a debate in Parliament on April 23,\n1917, when two distinguished Army officers, speaking with great\nrestraint, sought to open the eyes of the nation to this plague fostered\nin our camps by drink:\n\n  \"During the war we have had admitted into the hospitals of England\n  over 70,000 cases of gonorrh\u0153a, over 20,000 cases of syphilis, and\n  over 6000 cases of another disease somewhat similar. I am quite\n  openly prepared to state that of these 20,000 cases of syphilis you\n  do not get much work out of them under two and a half years. I know\n  from what I have seen of the modern conditions of this War that you\n  may absolutely wipe them out, except for a few handfuls.\n\n  \"When you come to the great mass of casualties under this head ...\n  the figures mean that you have =a Division constantly out of\n  action=. If you have anything like 70,000 men enfeebled, you find\n  that you suffer to that extent also. It is not only that you lose\n  the men, and not only the men who are partially cured are suffering\n  for many months to come, but their chances of recovery from wounds\n  are not nearly so good.\n\n  \"I know of a hospital for venereal cases which it was found\n  necessary to expand from its normal accommodation for 500 or 600 up\n  to 2,000 cases, and they are continually full. It is a British\n  hospital in France. A figure I should like to submit to challenge is\n  that during the course of the war between 40,000 and 50,000 cases of\n  syphilis have passed through our hospitals in France. When you come\n  to gonorrh\u0153a, the figure given me which covers that is between\n  150,000 and 200,000 cases.\"\n\n                         _Captain Guest in Parliament, April 23, 1917_\n\n  \"Every Canadian soldier who comes to this country arrives here not\n  only a first-class specimen of a fine soldier, but as clean-limbed\n  and as clean a man as the Creator Himself could create. The fact\n  that in one only of the three Canadian camps in this country 7,000\n  of these clean Canadian boys went through the hospital for venereal\n  disease in fourteen months is not only a great discredit to any\n  Government in this country but has an effect in Canada which I can\n  assure the House does not make for a better feeling with the Home\n  Country, and does not make for what we all desire\u2014Imperial Unity.\"\n\n           _Colonel Sir Hamar Greenwood in Parliament, April 23, 1917_\n\nThose are unchallenged statements made in the House of Commons itself;\nthey stand as a terrible indictment of this disease, and it is not to be\ndenied that this evil could never have reached its present frightful\nproportions if Parliament had followed the King. Let us look at a few\nexamples of the ravages of this vice allied so closely to the\npublic-house.\n\n  It is not possible to tell the whole truth about drink; the language\n  in which it must be written would be offensive in a civilised\n  country. It must be said, simply, that soldiers in England have been\n  court-martialled for having been influenced by drink to commit\n  unspeakable offences against animals.\n\n                                  _Facts in Records of Court-Martials_\n\n  A special constable in a harlot-haunted district in London describes\n  how these harpies carry off lonely soldiers to their rooms, make\n  them drunk, and finally innoculate them, as likely as not, with\n  disease. Is it not possible to hold in check these women who prey\n  upon and poison our soldiers? asks Sir Conan Doyle.\n\n                                               _Letter in the \"Times\"_\n\n  One of the hot-beds of venereal disease to which drink leads our\n  soldiers, was kept by an Austrian woman in Lambeth, who was\n  receiving 15_s._ a week from the Austrian Government in April 1916,\n  and used to lure our soldiers when weakened by drink. All the men\n  seen to enter this house were either soldiers or sailors.\n\n                                           _Police Records of Lambeth_\n\n  A soldier from the Front with \u00a318 was taken by a married woman to\n  her home, where he was found after a drunken bout with eight women,\n  all drunk. The woman's children were terribly neglected.\n\n                     _Police Records of St. Helens, November 30, 1915_\n\n  If you describe the Waterloo Road and the back streets as an open\n  sewer you will be somewhere near the truth. Not a day goes by\n  without bringing some soldier who has been waylaid.\n\n                             _Facts in the \"Times,\" February 22, 1917_\n\n  A soldier came from the Front to go home to Scotland. He got drunk\n  near Waterloo, losing all his money and his railway pass. He spent\n  his leave living on charity, and returned to the Front without\n  having been near either his home or his friends.\n\n                            _Facts in \"Daily News,\" February 14, 1916_\n\nHere is the official proof of the relation of the drink trade to this\ntraffic in disease. It is from the Report of the Royal Commission:\n\n  Abundant evidence was given as to the intimate relation between\n  alcohol and venereal diseases.\n\n  Alcohol renders a man liable to yield to temptations which he might\n  otherwise resist, and aggravates the disease by diminishing the\n  resistance of the individual.\n\n  Alcoholism makes latent syphilis and gonorrh\u0153a active.\n\n  Our evidence tends to show that the communication in disease is\n  frequently due to indulgence in intoxicants, and there is no doubt\n  that the growth of temperance among the population would help to\n  bring about an amelioration of the very serious conditions which our\n  enquiry has revealed.\n\n  We desire, therefore, to place on record our opinion that action\n  should be taken without delay.\n\n               Will some Member of Parliament please ask\n\n=if, in view of Lord D'Abernon's statement that Prohibition has failed\nin Canada, the Government will issue the figures showing the decrease of\ncrime and the increase of wealth?=\n\n------------------------------------------------------------------------\n\n\n\n\n                       The Price the Empire Pays\n\n\nIt is a bitter irony that while the men of the Empire have come to\nFrance to fight the enemy of mankind, this foe within our gates has\nstruck a blow at the British Empire that generations will not heal. How\nmany Empire men this private trade has slain we do not know, but we know\nbeyond all challenge that it has weakened the bonds that bind our\nDominions to the Motherland. This trade that throttles us at home can\npull the Empire down, and it has started well. It has struck its blow at\nCanada.\n\nLet us look at the plain facts which in other days than these would have\ncaused a storm of anger that Parliament could not have ignored. Canada\nhas followed the King; arming herself with her full powers, flinging\nherself upon her enemies with her utmost strength, she has swept drink\nout of Canada almost from sea to sea. But even before she did this\nCanada saw that alcohol must go from her camps if her men were to be fit\nto fight for England, and long before the Prohibition wave swept across\nthe country, the Canadian Government removed all alcohol from the\ntraining camps. It was the deliberate choice of a Government and its\npeople, and from that day to this there has been no reason for regret.\n\nSo the young manhood of Canada, rallying to the flag, was guarded from\nalcohol. She poured out her men in hundreds of thousands; they came to\nus from Prohibition camps; they came in Prohibition ships, and even here\nthis trade that has us in its grip was not allowed at first in the\nCanadian camps; the only condition that Canada made\u2014a condition implied\nbut clearly understood\u2014was properly regarded and obeyed.\n\nWe respected the desire of Canada, and kept her soldiers free from drink\nin their own camps. But a soldier cannot keep in camp, and in the\nvillages around the Drink Trade waits in every street. The military\nauthorities were willing for the Canadian Government to have their way\ninside the camps, but drink was free outside, and in these public-houses\nthere was sown the seed that may one day break this Empire. The Drink\nTrade was so rampant outside the Canadian camps that Prohibition inside\nwas almost in vain. We had to decide between breaking the word of the\nCanadian Government to its people or dealing with this trade as Canada\nherself has done; as Russia has done; as France and America are doing.\nIt was the Empire or the drink traffic, and the drink traffic won, as it\nalways wins with us.\n\nIt came about in October, down on Salisbury Plain. During one week-end a\nnumber of Canadian troops gave way to drinking in villages around the\ncamps, and it was then that the grave decision was come to that the\ndrink trade should be allowed to set up its horrible canteens in every\nCanadian camp. The change was made at the request of a British General,\nand we have the assurance of the Prime Minister of Canada that the\napproval of the Canadian Government was neither obtained nor asked. In\nhanding the Canadian Army over to the drink canteens, in deliberately\nreversing the policy of the Canadian Government and its people, there\nwas no consultation with Canada.\n\nIt is important to remember that this decision, fraught with tragic and\nfar-reaching consequences for the Empire, was a pure and simple English\nact. We may imagine the Canadian view from the remark of a Canadian\nGeneral, who said, \"I know drink is a hindrance, but I can do very\nlittle, because in military circles in this country drunkenness is not\nconsidered a very serious offense.\"\n\nIt would have been surprising if there had not poured in upon our\nGovernment a stream of protests, and from all parts of the Dominions\nthey came. The Dominion of Canada, giving freely to the Motherland\n450,000 boys and men, was moved to passionate indignation that England\nshould scorn her love for them, should ignore the pleadings of their\nmothers and sisters, and should put in their way the temptations from\nwhich they were saved at home. Canada does not want our drink trade; she\nlives side by side with the United States, she sees that great country\nbuilding up its future free from drink, and she sees America, splendid\nally in war, as a mighty rival in peace.\n\nAnd Canada is ready for the Reconstruction. She has followed the\nProhibition lead of the United States, and already she has ceased to be\na borrowing country. The very first year of Prohibition has seen this\nyoung Dominion, for the first time in her history, financially\nself-sustaining. Crime is disappearing; social gatherings are held in\nher gaols; she has set up vast munition workshops, and instead of\nborrowing money for her own support she has made hundreds of millions'\nworth of munitions for which this country need not pay until the war is\nover, and then need never pay at all for the munitions the Canadians\nhave used. Canada is in deadly earliest. She kept her men away from\ndrink to make them fit; she has swept it away to make a clean country\nfor those who go back.\n\nAnd what is England's contribution to this Imperial Reconstruction? _We\nhave scorned it all._ The Prime Minister has said that this drink trade\nis so horrible that it is worth this horrible war to settle with it, yet\nwe have sacrificed the love of Canada on our brewers' altar. We can\nbelieve the Canadian who declares his profound conviction that but for\nthis Canada would have sent us 100,000 more recruits; we can believe it\nis true that where responsible Canadians meet together in these days the\ntalk is of how long the tie will last unbroken that binds the daughter\nto the Motherland. We can understand the passion that lies behind the\nresolutions that come to Downing Street from Nova Scotia; we know the\ndepth of the yearning of those 64,000 mothers and wives of Toronto who\nsigned that great petition to the Government of Canada begging it in the\nname of God to intervene.\n\nWe can understand it all; but let us call the witnesses, and let us see\nthe price the Dominion pays for our quailing before this Kaiser's trade.\n\n\n                       Those Who Will Not Go Back\n\nIt is the great consolation of Canada that, though their sons may fall\nbefore this tempter's trade in Britain, they will go back to a Canada\nfree from drink. But some will never go back, and they are not on the\nRoll of Honour. They have been destroyed by the enemy within our gate,\nthis trade that traps men on their way to France and digs their graves.\n\n  A young Canadian who had never tasted alcohol came from a\n  Prohibition camp in Canada, came to England on a Prohibition ship,\n  and was put in a camp with a drink canteen. He started drinking and\n  contracted venereal disease. Ordered home as unfit, in fear and\n  shame he sought a friend's advice about the girl he was to marry.\n  \"You can never marry her,\" said his friend, and that night in his\n  hut the young Canadian blew out his brains.\n\n                                   _Facts in possession of the Author_\n\n  A young Canadian officer was sent home disgraced. Sodden with\n  alcohol, he left the train and shot a railway clerk dead.\n\n                _Facts in Montreal \"Weekly Witness,\" October 24, 1916_\n\n  A Russian soldier in the Canadian forces, described as a clean,\n  soldierly man, with a splendid character from his officer, was\n  charged with the murder of a Canadian private who tried to separate\n  two quarrelling soldiers in a bar. The prisoner had drunk much\n  whisky and remembered nothing of his crime, and was sentenced to\n  twelve months' hard labour for manslaughter. The judge hoped he\n  might be used as a soldier _in the Russian Army_.\n\n                          _Record of Hampshire Assizes, February 1916_\n\n  A man from Prohibition Russia enlisted in Prohibition Canada, and\n  came to England. He spent 9_s._ on drink one day, and that night he\n  crept from his bed and killed his corporal at Witley Camp.\n\n                          _Police Records of Godalming, February 1917_\n\n  A Canadian soldier, aged 26, after a publichouse quarrel with\n  another soldier, was found dying on the pavement in Hastings. His\n  throat had been cut, and he died on entering the hospital. The other\n  soldier was charged with murder, and sentenced to 15 years.\n\n                              _Record of Hastings Assizes, March 1917_\n\n  A young Canadian soldier, aged 20, died from alcohol while in\n  training at Witley. He had a bottle of stout followed by nine or ten\n  \"double-headers\" of neat whisky in about two hours. He was carried\n  back to camp, laid unconscious on his bed, and died.\n\n                          _Facts in \"Daily Chronicle,\" March 22, 1917_\n\n  A Canadian lieutenant was tried for the murder of a canteen\n  sergeant. They arrived together at a house at Grayshott, where the\n  lieutenant asked for some strong drink and took a bottle of whisky\n  and two glasses. The sergeant was afterwards found dead in the\n  cellar, and the lieutenant carried the body into the stable.\n\n                         _Records of Grayshott Coroner, December 1915_\n\n  A man leaving a publichouse in company with a woman, with whom he\n  had been drinking, met a Canadian soldier not far from Charing\n  Cross. The soldier spoke, and the man struck him. The soldier was\n  carried to the hospital, where he died soon afterwards from a wound\n  two inches deep, caused by a knife.\n\n                       _Police Records of Bow Street, January 1, 1917_\n\n  The wife of a gunner in the South African Heavy Artillery died at\n  Bexhill from alcohol. The soldier said he bought 12 bottles of stout\n  and 12 bottles of beer, one of whisky, and one of port, which they\n  drank between Saturday night and Monday night.\n\n                           _Records of Bexhill Coroner, December 1915_\n\n  A soldier from Toronto, having been drinking away his pay in a\n  Carlisle publichouse, with another Canadian soldier and some married\n  women, failed to appear the next morning, and was found dead on a\n  footpath with a bottle of whisky in his pocket\n\n                         _Records of Carlisle Coroner, April 14, 1917_\n\n  A Canadian soldier, having drawn \u00a320 from the Canadian office,\n  visited several publichouses, and was killed in a scuffle in London.\n\n                             _Facts in \"Daily News,\" December 2, 1916_\n\n\n                   The Men From the Prohibition Camps\n\nAgain and again we have seen the peculiar temptations of drink among\nCanadians. Officers, chief-constables, chaplains, newspapers, the men\nthemselves, have all borne witness that to these men from Prohibition\nCanada the sudden temptations of our drink trade come with terrible\npower, and often they fall not knowing. The finest manhood of the Empire\nour tap-rooms and canteens destroy, not in isolated cases, but in a host\nwe dare not number.\n\nOf the soldiers who first came over from Canada, says a great Canadian\npaper, many were emigrants from England, not yet securely planted in\nCanada, and for their sakes especially drink should have been withheld\nfrom them. Of the larger number of Canadian troops that followed them,\nmany were youths who had never known drink, and they were taken from\nhome at the most social and reckless age, to face drink with all the\ntemptations induced by the nervous strain, the hardships and social\nabandon of the camp and the trench, and the free pocket-money when on\nleave.\n\n  In an officers' mess of two double companies of Canadians only one\n  officer drank on his arrival in a canteen camp in England; within\n  three months there was not an abstainer in the mess.\n\n         _Facts told at Society for Study of Inebriety, Jan. 10, 1916_\n\n  These men come mostly from districts in Canada where intoxicants are\n  prohibited by law, and many of them, being young lads, who perhaps\n  have never tasted liquor before their arrival, fall easy victims.\n\n                                        _Chief Constable of Godalming_\n\n  Overseas soldiers come to our hospitals astonishingly cheerful and\n  fit in a general sense, and wonderfully receptive to treatment. Only\n  three per thousand die in our great hospitals. This is largely due\n  to the hardy life of the men and the fact that they are removed from\n  the danger of taking too much alcohol. The home troops have a much\n  higher mortality, partly because their use of alcohol diminishes\n  their chances. Re-admissions are largely due to drink on furlough.\n\n          _Major Maclean, M.D., of the Third Western General Hospital_\n\n  A Canadian soldier, who had been wounded at the Front, was taken to\n  a house by women and left alone drunk. An officer gave him an\n  excellent character, and said he was on his way back to Canada.\n  These men experience temptations here (he said) that they would not\n  find in Canada, and there was too much of this going on.\n\n                          _Hastings Police Records, February 19, 1917_\n\n  I heard a sad account of the havoc of the wet canteen and a private\n  in a Canadian A.M.C. told us of a lad of 17 who is made so drunk\n  that there is rarely a night when he has not to be helped up to bed.\n  One of the soldiers here told me of his son in Canada being anxious\n  to join up, but after seeing the condition of things over here he\n  was doing all he could to discourage his son.\n\n                                                _Letter to the Author_\n\n  The Canadians in most cases are entirely lost when they arrive in\n  this country, and are much more liable to the temptation which is\n  thrown in their way, but when you give a figure such as this\u2014that in\n  one camp during last year, and two months of the previous year,\n  there were 7,000 cases\u2014it seems to me that it is about time we\n  realised the magnitude of the evil. I do not know what has happened\n  to them, except that I imagine a large number have gone back to\n  Canada, and have not been able to play the part they had hoped to\n  play.\n\n                         _Captain Guest in Parliament, April 23, 1917_\n\n\n                          In Camp and On Leave\n\nEverywhere we find the trail of drink among Canadians\u2014in camp and on\nleave.\n\n  A Canadian corporal, wounded in the Battle of Ypres, was found\n  terribly drunk after being missing all day from hospital. Confronted\n  with the surgeon after violent acts of insubordination, the corporal\n  broke down and cried like a child.\n\n                          _Facts in \"Western Mail,\" February 18, 1916_\n\n  In the first weeks of the war 42 Canadian soldiers disgraced\n  themselves, by excessive drinking, insubordination, and disorderly\n  conduct, to such an extent that they had to be sent back to Canada.\n\n                       _Facts in \"Canadian Pioneer,\" December 4, 1914_\n\n  A Canadian soldier, helplessly drunk, was seen at King's Cross\n  station eating, tearing, and crumpling up \u00a31 notes, and would have\n  lost about fifteen pounds but for kindly help from passers by.\n\n                      _Facts in \"Daily Chronicle,\" September 28, 1916_\n\n  A gunner from Montreal, missing from camp for several days, drank\n  himself delirious, and cut his throat with a razor.\n\n                       _Facts in \"Canadian Pioneer,\" December 4, 1914_\n\n  A Canadian soldier spent \u00a370 in three weeks on drink and bad\n  characters.\n\n                               _Facts in \"Daily Mail\" August 10, 1915_\n\n  A Sergeant-Major from Canada declared that he had lost 20 per cent.\n  of the men of his battery through venereal disease. They had a\n  little drink, and were captured by the swarm of bad women at\n  Folkestone.\n\n                                           _Facts in Letter to Author_\n\n  A woman was imprisoned for placing young children in moral danger.\n  Every night the girls brought soldiers home, and colonial soldiers\n  were frequently so drunk that they were carried in.\n\n                   _Records of Central Criminal Court, April 25, 1917_\n\n\n                       The Rising Storm in Canada\n\n  =The thing cannot be justified. It is the blackest tragedy of this\n  whole war that, in fighting for freedom in Europe, the free sons of\n  the British breed have to face this war-time record of waste at\n  home, with its inevitable toll of debauchery and crime.=\n\n                                        _Editorial in \"Toronto Globe\"_\n\nWhile this book was being written one of the greatest meetings ever held\nin Manchester was cheering a Canadian in khaki who declared that he was\nnot going hungry while brewers were destroying food, and he went on to\nsay, this soldier and sportsman well-known in the Dominion:\n\n  \"Great numbers of our men never saw France. Canadian boys cried\n  because they had not munitions. England reeled and beer flowed like\n  water while thousands of our boys went down into their graves. We\n  will never forget it in Canada.\"\n\nWe may be sure Canada will not forget. She will not forget her dead: she\nwill not forget that the Drink Traffic she has swept away at home struck\ndown her sons in the land for which they fought. \"We must know who is to\nblame,\" says a Canadian paper; \"we presume they will have no objection\nto have their names placarded before the country, that every mother may\nknow.\" Col. Sir Hamar Greenwood, M. P., has lately returned from Canada,\nand this is what he tells us:\n\n  \"I met many fathers and mothers whose boys had been sent back to\n  Canada debilitated and ruined for life because they had been\n  enmeshed by harpies, and again and again these parents have said to\n  me, 'We do not mind our boys dying on the field of battle for old\n  England, but to think that we sent our sons to England to come back\n  to us ruined in health, and a disgrace to us, to them, and to the\n  country, is something the Home Country should never ask us to\n  bear.'\"\n\n  _Letter from a Solicitor in Ontario to the Author_:\n\n  I wonder if the advocates of the drink traffic in Britain appreciate\n  the contempt in which they are held in Canada. Before the war I had\n  a class of ten young men. Every one of them is now at the Front, and\n  one writes that when I told them of the drink conditions in England\n  he did not believe half of it; now he says I did not tell him half.\n  Letters from our Canadian soldiers are appearing in our papers, and\n  they are all amazed at the drinking habits of Britain.\n\n  _From a Resolution received by Mr. Lloyd George from the Social\n  Service Council of Nova Scotia_:\n\n  That we, representing the social, moral, and spiritual forces of\n  this part of the British Empire, who have proved our loyalty by the\n  thousands of men this small province has sent overseas, do record\n  our most earnest protest against Britain's inaction in this matter,\n  which we are sure must result in longer and increased suffering for\n  the men we have sent to help her win the war; and do most\n  insistently plead with the British Government and the British\n  Parliament that they at once exercise the power vested in them to\n  strike the blow that will dispose of this enemy at home, and so give\n  mighty reinforcement to those who are bleeding and dying for Britain\n  and human liberties on the battlefields abroad.\n\n  _Sermon by Dr. Flanders in London, Ontario, Feb. 25, 1917_:\n\n  Canada has the right to make this demand on the Motherland from the\n  simple standpoint of political economics. That we might put the\n  Dominion into the best possible shape to give the utmost of our\n  strength in men and munitions, we have an almost Dominion-wide\n  Prohibition, and no intelligent person will deny that our\n  contributions to the war from the first have been multiplied and\n  intensified by that action. Why should little Johnnie Canuck abolish\n  drink that he might conserve his manhood and material resources in\n  the interest of the Empire's war, and big John Bull refuse to\n  abolish the traffic to the great waste of his material resources and\n  the undoing of his efficiency?\n\n  _A public man with three soldier sons wrote to the Toronto Globe_:\n\n  Canada, for efficiency in war, casts out the drink evil. Is it too\n  much to expect Britain, in fairness, to do the same? Is it not a\n  mockery for the British Isles to face our common struggle with this\n  palsy in her frame?\n\n  Here is the bitter pill, the embittering thought for many a Canadian\n  parent. Let me be a type. Three of my sons are in khaki. I gave them\n  a father's blessing when they enlisted. But this thought strains,\n  most of all, the ties of my loyalty to the cause\u2014to see my sons\n  fight and fall for a Britain that at home is saddled by distillery\n  interests, and misguided by a Press silent as the grave on this\n  entrenched evil. Why should our sons go from a country where booze\n  is banished to spend months on the way to the trenches in England,\n  where the vices of the liquor traffic are legalised?\n\n  _We see the spirit of Canada in those great words of the Premier of\n  Ontario, Mr. Hearst, speaking of the giving up of drink_:\n\n  In this day of national peril, in this day when the future of the\n  British Empire, the freedom of the world, and the blessings of\n  democratic government hang in the balance, if I should fail to\n  listen to what I believe to be the call of duty, if I should neglect\n  to take every action that in my judgment will help to conserve the\n  financial strength and power and manhood of this province for the\n  great struggle in which we are engaged, I would be a traitor to my\n  country, a traitor to my own conscience, and unworthy of the brave\n  sons of Canada that are fighting, bleeding and dying for freedom and\n  for us.\n\n  _A letter from one of the most eminent public men in Canada_:\n\n  \"British Canada is intensely loyal to the Empire and the Allied\n  Cause, but at present recruiting is almost at an end. Why? Partly\n  because of considerable dissatisfaction with many of the conditions\n  which prevail. Suffering, wounds, death, are expected as inevitable\n  in war, but the evil influences, the lavish temptations of liquor\n  and bad women which sweep down upon our boys in England, are not\n  felt to be necessary, and the hearts of multitudes of Canadian\n  parents are hot with indignation at the apparent indifference of the\n  authorities to the moral welfare of our troops.\"\n\n  _Captain John MacNeill, with the Canadian troops in France_:\n\n  \"I say to you solemnly, if England should lose this war because of\n  drink, or if England should unnecessarily prolong the war with great\n  sacrifice of life in her effort to protect drink, or even if England\n  should win the war in spite of drink, you will have put upon the\n  bonds of Empire such a strain as they have never known before, and\n  such a strain as we cannot promise they will be able to survive.\"\n\n  _From the petition presented to the Prime Minister of Canada, signed\n  by 64,000 mothers and wives in Toronto_:\n\n  1. That Mothers and Wives of Canada in giving their sons and\n  husbands for King and Empire, asked and received from your Minister\n  of Militia this only assurance that, in sending them into the ranks,\n  we were not hereby irrevocably thrusting them into the temptation of\n  Strong Drink.\n\n  2. We appreciated from the depths of our hearts, your action in\n  abolishing the Wet Canteen from the Canadian Militia. We believe the\n  Wet Canteen established in the ranks of the front to be a double\n  danger, robbing our King of the success in arms which in these days\n  comes only to the brave heart that is controlled by a clear head,\n  and robbing us and our Canada of the Manhood which we gave into our\n  Empire's keeping.\n\n  3. We do not believe that the King will refuse the aid of Canada's\n  sons; nor that he will appreciate your patriotic efforts the less,\n  if you keep faith with us and make known to His Majesty, his\n  Ministers and Commanders, that our boys are sent forth on the one\n  condition that the dispensing of intoxicating liquors shall be\n  prohibited in the ranks.\n\n  _From a Sermon preached in Ontario, February 25, 1917_:\n\n  \"Thank God, if any of our Canadian soldiers return to us with the\n  drink habit formed and raging, we can welcome them to a land nearly\n  purged of the liquor traffic, where they may have a chance to\n  recover their manhood.\"\n\n  _Letter on the effects of Prohibition, from a business man in\n  Ontario, published in the \"Spectator:\"_\n\n  \"Men I have known for years to be regular promenading tanks have\n  given it up, and are starting a decent life again. The Police Court\n  is empty. England should try it. It would be, after the first heavy\n  initial loss, the best thing that ever struck the nation. I cursed\n  these temperance guys as hard as any, but all the same it cannot\n  blind you from the truth.\"\n\n------------------------------------------------------------------------\n\n\n\n\n                     Your Share in the Food Crisis\n\n\n         The Food and Money Wasted on Drink in Our Great Towns\n\n           ESTIMATED FROM AUGUST 1914 TO APRIL 1917 INCLUSIVE\n                       by GEORGE B. WILSON, B.A.,\n                  Compiler of the National Drink Bill\n\n   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                          \u2502 Drink Bill  \u2502 Grain Lost  \u2502Sugar in Beer\n   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                          \u2502             \u2502    Tons     \u2502     lb.\n   United Kingdom         \u2502 \u00a3510,000,000\u2502    4,400,000\u2502   762,000,000\n   London                 \u2502  \u00a383,000,000\u2502      693,000\u2502   120,000,000\n   Edinburgh              \u2502   \u00a33,200,000\u2502       31,000\u2502     5,300,000\n   Dublin                 \u2502   \u00a32,600,000\u2502       29,000\u2502     5,000,000\n   Glasgow                \u2502  \u00a310,500,000\u2502      101,000\u2502    17,400,000\n   Manchester and Salford \u2502  \u00a311,000,000\u2502       92,000\u2502    15,900,000\n   Birmingham             \u2502   \u00a39,900,000\u2502       82,000\u2502    14,200,000\n   Liverpool              \u2502   \u00a38,800,000\u2502       73,000\u2502    12,600,000\n   Sheffield              \u2502   \u00a35,400,000\u2502       45,000\u2502     7,800,000\n   Leeds                  \u2502   \u00a35,300,000\u2502       44,000\u2502     7,600,000\n   Bristol                \u2502   \u00a34,200,000\u2502       35,000\u2502     6,000,000\n   West Ham               \u2502   \u00a33,400,000\u2502       28,000\u2502     4,900,000\n   Bradford               \u2502   \u00a33,300,000\u2502       28,000\u2502     4,800,000\n   Hull                   \u2502   \u00a33,300,000\u2502       27,000\u2502     4,700,000\n   Newcastle              \u2502   \u00a33,100,000\u2502       26,000\u2502     4,500,000\n   Nottingham             \u2502   \u00a33,100,000\u2502       26,000\u2502     4,500,000\n   Portsmouth             \u2502   \u00a32,800,000\u2502       23,000\u2502     4,400,000\n   Stoke                  \u2502   \u00a32,800,000\u2502       23,000\u2502     4,000,000\n   Leicester              \u2502   \u00a32,700,000\u2502       22,000\u2502     3,800,000\n   Cardiff                \u2502   \u00a32,100,000\u2502       18,000\u2502     3,100,000\n   Bolton                 \u2502   \u00a32,100,000\u2502       18,000\u2502     3,000,000\n   Croydon                \u2502   \u00a32,100,000\u2502       17,000\u2502     3,000,000\n   Sunderland             \u2502   \u00a31,700,000\u2502       14,000\u2502     2,500,000\n   Oldham                 \u2502   \u00a31,700,000\u2502       14,000\u2502     2,500,000\n   Birkenhead             \u2502   \u00a31,600,000\u2502       13,000\u2502     2,200,000\n   Blackburn              \u2502   \u00a31,500,000\u2502       13,000\u2502     2,200,000\n   Brighton               \u2502   \u00a31,500,000\u2502       13,000\u2502     2,200,000\n   Plymouth               \u2502   \u00a31,500,000\u2502       12,000\u2502     2,100,000\n   Derby                  \u2502   \u00a31,400,000\u2502       12,000\u2502     2,100,000\n   Middlesbrough          \u2502   \u00a31,400,000\u2502       12,000\u2502     2,100,000\n   Stockport              \u2502   \u00a31,400,000\u2502       12,000\u2502     2,100,000\n   Norwich                \u2502   \u00a31,400,000\u2502       12,000\u2502     2,100,000\n   Southampton            \u2502   \u00a31,400,000\u2502       12,000\u2502     2,000,000\n   Swansea                \u2502   \u00a31,400,000\u2502       12,000\u2502     2,000,000\n   Gateshead              \u2502   \u00a31,400,000\u2502       11,000\u2502     2,000,000\n   Preston                \u2502   \u00a31,400,000\u2502       11,000\u2502     1,900,000\n   Coventry               \u2502   \u00a31,300,000\u2502       11,000\u2502     1,900,000\n   Huddersfield           \u2502   \u00a31,300,000\u2502       10,000\u2502     1,800,000\n   Halifax                \u2502   \u00a31,200,000\u2502       10,000\u2502     1,700,000\n   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\n\n                             PLAY THE GAME\n\n           There is one week's bread in      18 pints of beer\n           There is one week's sugar in      16 pints of beer\n\n  The man who drinks 3 pints a day drinks another man's rations.\n\n------------------------------------------------------------------------\n\n\n\n\n                 THE FOOD PYRAMIDS DESTROYED FOR DRINK\n\n\n[Illustration:\n\n  The Great Pyramid of Egypt, the biggest construction in stone ever\n    made by the hands of man\u201480,000,000 cubic feet of masonry]\n\n[Illustration:\n\n  The Great Pyramids of Food, the biggest wilful destruction of food\n    ever known\u2014180,000,000 cubic feet of food destroyed for the Drink\n    Trade during the war]\n\n------------------------------------------------------------------------\n\n\n\n\n                      How the Brewer Gets Our Food\n\n\nTHE MEN WHO BRING IT\n\nIt is easy to talk of a mine-sweeper. I wish the whole nation could\nunderstand what these men are doing. They are feeding the whole\npopulation, battling with the elements as well as with the enemy,\nbattling with dangers overhead and dangers under the sea. The\nmine-sweeper is like the soldier daily over the parapet\u2014he carries his\nlife in his hand.\n\n                                          _First Lord of the Admiralty._\n\n\nTHE PEOPLE WHO WAIT FOR IT\n\nA London caterer ordered a quantity of sugar from the Philippines. The\nmine-sweepers cleared the way for it and it reached the docks. The\ncaterer sent for it, and was informed that it could only be delivered if\nit was for a brewer.\n\nA provincial caterer ordered sugar _and paid for it_, but was told by\nthe Food Controller that it could only be released if _it was sold to a\nbrewer_.\n\nA working man was discussing rations with his minister in the street.\n\"It is very hard,\" he said, \"to keep to your rations when you have five\nstrapping lads, but we are going to try it.\" Then a drunken man lurched\npast. The workman pulled himself together, and said, in great passion:\n\"I tell you what it is, sir, I am not going to let my boys starve as\nlong as there is food to make beer for men like that.\"\n\n\nTHE PRICE WE PAY FOR IT\n\nImmense quantities of food are used for beer and spirits. All this grain\nis lost for food purposes. _If this grain were available for food, the\nprices of bread and meat would be lowered._\n\n                                                _War Savings Committee._\n\n\nTHE POOR WHO SUFFER FOR IT\n\n\"Rationing bread could not be undertaken without grave risk to the\nhealth of the poor.\"\n\n                                                 _Capt. Bathurst, M. P._\n\n                   By what right does the Government\n\nuse our mine-sweepers to bring in food for brewers to destroy? allow\nbrewers to increase the cost of living for every household? and allow\nthe willful destruction of food supplies to imperil the health of the\npoor?\n\n------------------------------------------------------------------------\n\n\n\n\n                       The Way for the Government\n\n\nWe do not want to be amused by fiddlers while our heroes fight and die.\n\nWhat are the things we see? We see the Government silent in the presence\nof what the greatest paper in our greatest overseas Dominion calls \"the\nblackest tragedy of the war.\" We see a trade which the King declared to\nbe prolonging the war in the crisis of 1915, prolonging it still in the\ncrisis of 1917. We see our Prime Minister, who has declared this trade\nto be worse than Germany, allowing it to have its way. We see our Prime\nMinister, who has said we cannot settle with Germany until we have\nsettled with drink, fearing to settle with drink. Then are we not to\nsettle with Germany, and are we to surrender to the greatest enemy of\nthe three?\n\nThere is one clear way before the Government; it is the only way of\nstraightness and patriotism and honour. It is to wind up this enemy\ntrade and move from our path the greatest hindrance to the winning of\nthe war. It is to take our side honourably with our great Allies, to\nbring to an end the shameful isolation of Great Britain in the drink map\nof the great free countries that appears on the back of this book.\n\nIt is the sign of weakness everywhere that it seeks a scapegoat for its\nsins, and we hear the everlasting talk of Labour. But it will not do. It\nis time these slanders on our workmen ceased.\n\nIf the Government is afraid of the working man, let it say so, or let it\ntry him. If it is afraid of temperance people, let it rally them to its\nside as one man on the platform where they meet. If it is afraid of the\nDrink Trade, then the time has come to say so, for we who send out our\nmillions to fight a foreign foe are not going to starve for bread\nthrough fear of enemies within our gate. The Prime Minister gave the\nArmy its munitions; the Army will use them in vain unless the munitions\nof life come into our homes.\n\nWorking men are tired of men who fool with food and liberty. They do not\nobject to any equal sacrifice: they believe in the democratic policy of\nthe King, who based Prohibition, not on class distinction as the\nGovernment did by closing tap-rooms 15 hours a day and leaving cellars\nand Parliamentary bars open always, but on the principle of the King's\nown words that \"no difference shall be made, so far as his Majesty is\nconcerned, between the treatment of the rich and poor in this respect.\"\nLet the Government follow the King, and the people will follow the\nGovernment.\n\nIn the highest interests of the nation and the war let this be said as\nplain as words can make it\u2014_that there is no body of temperance opinion\nanywhere standing in the way of Prohibition_, but that the united moral\nforces of the nation would rally to the Government instantly on an act\nof a few words such as this:\n\n=That the manufacture and sale of alcoholic beverages be totally\nprohibited in the United Kingdom for the period of the war and\ndemobilization, and that a committee be appointed to deal with all the\nprivate and public interests concerned; and that it be resolved upon,\nhere and now, that reconstruction be accompanied by universal local\noption.=\n\nThere would be no opposition the Government need count to a proposal\nlike that.\n\n[Illustration: TYPOGRAPHICAL UNION LABEL WESTERVILLE O.]\n\n\n\n\n      *      *      *      *      *      *\n\n\n\n\nTranscriber's note:\n\nObvious typographical and punctuation errors were corrected.\n\nInconsistencies in hyphenation were retained.\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**For Mel,**\n\n**who knows why**\nCONTENTS\n\nA NOTE ON CITATIONS\n\nHOW MANY ROADS MUST A MAN WALK DOWN?\n\nBOB DYLAN\n\nDO THEY KNOW IT'S CHRISTMAS TIME?\n\nBAND AID\n\nARE THERE 4,000 HOLES IN BLACKBURN, LANCASHIRE?\n\nTHE BEATLES\n\nHOW MUCH IS THAT DOGGIE IN THE WINDOW?\n\nPATTI PAGE\n\nCAN YOU KILL SOMEONE WITH A SONG?\n\nROBERTA FLACK\n\nHOW SOON IS NOW?\n\nTHE SMITHS\n\nCAN I KICK IT?\n\nA TRIBE CALLED QUEST\n\nHOW LONG DOES IT TAKE TO APOLOGIZE A TRILLION TIMES?\n\nOUTKAST\n\nVOULEZ-VOUS COUCHER AVEC MOI CE SOIR?\n\nLADY MARMALADE, LABELLE ET AL\n\nWHO RUN THE WORLD?\n\nBEYONC\u00c9\n\nIS THIS BURNING AN ETERNAL FLAME?\n\nTHE BANGLES\n\nWAR \u2013 HUH \u2013 WHAT IS IT GOOD FOR?\n\nEDWIN STARR\n\nHOW LIKELY IS RIHANNA TO NEED HER UMBRELLA?\n\nRIHANNA\n\nIS THIS THE REAL LIFE? IS THIS JUST FANTASY?\n\nQUEEN\n\nARE YOU CALLING ME 'DARLING'?\n\nTHE TING TINGS\n\nHOW MUCH WOULD IT TAKE TO CRY A RIVER?\n\nVARIOUS ARTISTS\n\nWHERE HAVE ALL THE FLOWERS GONE?\n\nPETER, PAUL AND MARY\n\nSHOULD I STAY OR SHOULD I GO?\n\nTHE CLASH\n\nDOES YOUR MOTHER KNOW THAT YOU'RE OUT?\n\nABBA\n\nDO YOU GIVE ME FEVER?\n\nPEGGY LEE\n\nWHAT WOULDN'T MEAT LOAF DO?\n\nMEAT LOAF\n\nARE FRIENDS ELECTRIC?\n\nGARY NUMAN\n\nHOW DO YOU SOLVE A PROBLEM LIKE MARIA?\n\nTHE SOUND OF MUSIC\n\nHOW MANY GENERATIONS WILL HAVE PASSED BY THE YEAR 3000?\n\nBUSTED\n\nWHEN WILL I BE FAMOUS?\n\nBROS\n\nARE WE HUMAN, OR DANCER?\n\nTHE KILLERS\n\nYOU WALK 500 MILES. YOU WALK 500 MORE. WHERE HAVE YOU BEEN?\n\nTHE PROCLAIMERS\n\nWHERE IS MY LARGE AUTOMOBILE?\n\nTALKING HEADS\n\nDID THOSE FEET IN ANCIENT TIMES WALK UPON ENGLAND'S MOUNTAINS GREEN?\n\nWILLIAM BLAKE\n\nARE WE OUT OF THE WOODS YET?\n\nTAYLOR SWIFT\n\nI WILL SURVIVE \u2013 BUT FOR HOW LONG?\n\nGLORIA GAYNOR\n\nIF YOU DON'T LOVE ME NOW, WILL YOU NEVER LOVE ME AGAIN?\n\nFLEETWOOD MAC\n\nHOW MUCH SPACE DO A MILLION PHOTOGRAPHS TAKE UP?\n\nTHE VAPORS\n\nIS ANNIE OKAY?\n\nMICHAEL JACKSON\n\nHOW MANY HONEYS MAKE THEIR MONEY?\n\nDESTINY'S CHILD\n\nIS SHE REALLY GOING OUT WITH HIM?\n\nJOE JACKSON\n\nWILL THIS BE THE DAY THAT YOU DIE?\n\nDON MCLEAN\n\nCAN YOU FEEL THE LOVE TONIGHT?\n\nELTON JOHN AND TIM RICE\n\nCALL ME, MAYBE?\n\nCARLY RAE JEPSEN\n\nWERE THE BOYS OF THE NYPD CHOIR SINGING 'GALWAY BAY'?\n\nTHE POGUES\n\nIS MONEY THE ROOT OF ALL EVIL TODAY?\n\nPINK FLOYD\n\nWHY DON'T WE DO IT IN THE ROAD?\n\nTHE BEATLES\n\nDO THE DRUGS WORK?\n\nTHE VERVE\n\nARE THERE NINE MILLION BICYCLES IN BEIJING?\n\nKATIE MELUA\n\nDO YOU KNOW THE WAY TO SAN JOS\u00c9?\n\nDIONNE WARWICK\n\nHOW WORRIED SHOULD YOU BE IF SOMEONE'S WATCHING YOU WITH THE EYE OF THE TIGER?\n\nSURVIVOR\n\nWOULD I LIE TO YOU?\n\nCHARLES & EDDIE\n\nHOW MUCH WOULD YOU HAVE TO EARN TO BE 'BARELY GETTIN' BY' IN LA IN 1980?\n\nDOLLY PARTON\n\nWHAT BECOMES OF THE BROKEN HEARTED?\n\nJIMMY RUFFIN\n\nWHO'S TO BLAME: THE SUNSHINE, MOONLIGHT, GOOD TIMES OR BOOGIE?\n\nTHE JACKSONS\n\nWHAT ABOUT ELEPHANTS \u2013 HAVE WE LOST THEIR TRUST?\n\nMICHAEL JACKSON\n\nAM I A CREEP?\n\nRADIOHEAD\n\nDID WE USED TO KNOW WHITE CHRISTMASES?\n\nBING CROSBY\n\nARE YOU TELLING ME THIS IS A SIGN?\n\nSNOOP DOGG\n\nWHAT COMPARES TO YOU?\n\nSINEAD O'CONNOR\n\nDO YOU LIKE PI\u00d1A COLADAS, AND GETTING CAUGHT IN THE RAIN?\n\nRUPERT HOLMES\n\nARE YOU LONESOME TONIGHT?\n\nELVIS PRESLEY\n\nWHAT DOES THE FOX SAY?\n\nYLVIS\n\nISN'T IT IRONIC, DON'T YOU THINK?\n\nALANIS MORISSETTE\n\nWHAT IS LOVE?\n\nHADDAWAY\n\nWILL IT BE LONELY THIS CHRISTMAS?\n\nMUD\n\nWHY'D YOU HAVE TO GO AND MAKE THINGS SO COMPLICATED?\n\nAVRIL LAVIGNE\n\nWHERE ARE YOUR FRIENDS TONIGHT?\n\nLCD SOUNDSYSTEM\n\nWHAT'S COOLER THAN BEING COOL?\n\nOUTKAST\n\nWHY DO YOU ONLY CALL ME WHEN YOU'RE HIGH?\n\nARCTIC MONKEYS\n\nWHERE IS 24 HOURS FROM TULSA?\n\nGENE PITNEY\n\nOH CAN'T YOU SEE YOU BELONG TO ME?\n\nTHE POLICE\n\nWHAT'S THE FREQUENCY, KENNETH?\n\nREM\n\nDO GIRLS JUST WANNA HAVE FUN?\n\nCYNDI LAUPER\n\nWHERE DO BROKEN HEARTS GO?\n\nWHITNEY HOUSTON\n\nWHAT HAVE THEY DONE TO THE RAIN?\n\nTHE SEARCHERS\n\nSON, CAN YOU PLAY ME A MEMORY?\n\nBILLY JOEL\n\nWHY?\n\nANNIE LENNOX\n\nDOES EVERYBODY WANT TO RULE THE WORLD?\n\nTEARS FOR FEARS\n\nHOW MANY INCHES ARE IN A MILE?\n\nSELENA GOMEZ & THE SCENE\n\nHAVE GUILTY FEET GOT NO RHYTHM?\n\nWHAM\n\nHOW THE HELL AM I SUPPOSED TO LEAVE?\n\nUSHER\n\nIS THERE LIFE ON MARS?\n\nDAVID BOWIE\n\nWHAT'S GOING ON?\n\n4 NON BLONDES\n\nWHO LET THE DOGS OUT?\n\nBAHA MEN\n\nIS THIS SONG ABOUT YOU?\n\nCARLY SIMON\n\nWHAT WOULD HAPPEN IF IT WERE CHRISTMAS EVERY DAY?\n\nWIZZARD\n\nWHY DO BIRDS SUDDENLY APPEAR?\n\nTHE CARPENTERS\n\nWHAT IF GOD WAS ONE OF US?\n\nJOAN OSBORNE\n\nWHY DOES IT ALWAYS RAIN ON ME?\n\nTRAVIS\n\nWHO DO YOU THINK YOU ARE?\n\nSPICE GIRLS\n\nDO YOU REMEMBER THE FIRST TIME?\n\nPULP\n\nWHO STARTED THE FIRE?\n\nBILLY JOEL\n\nSONG CREDITS\n\nACKNOWLEDGEMENTS\nA NOTE ON CITATIONS\n\nWhere we've made significant use of a particular study or article, we've included it wherever possible in the text. However, to avoid a notes section as long as the book itself, these have sometimes been omitted. If there's any figure you'd like the source of \u2013 or if you think you've spotted an error in the maths \u2013 do get in touch, ideally via Twitter, where I can be found @jamesrbuk.\nHOW MANY ROADS MUST A MAN WALK DOWN?\n\nBOB DYLAN\n\n**There have been various rituals around manhood throughout history: for the Bukusu tribe of western Kenya, it is the sikhebo circumcision ceremony; for the people of Vanuatu in the South Pacific it is the yearly harvest ritual of 'land diving', as they leap a hundred feet from crudely built wooden towers with vines tied around their ankles.**\n\nDylan, B characteristically chooses an unusual measure of child development, contemplating how many roads a man must walk down, before you can call him a man. Despite significant research effort, we have been unable to find evidenced answers to this matter 'blowing in the wind' (unless the cryptic answer to his question is: 'nitrogen, oxygen, argon, carbon dioxide and other trace gases') and so have focused instead on what Dylan surely must have been driving at: childhood physical activity.\n\nThe lack of any standardized road length, plus ethical (and insurance) considerations prevented us proposing that young children walk down actual roads, so instead we have focused on recommended step counts for children. While adults are advised to walk 10,000 steps a day, children \u2013 at least those aged five or over; for hopefully obvious reasons younger kids are excluded \u2013 are advised to walk more.\n\nGirls should be aiming to walk 12,000 steps a day, but boys need to average 15,000 steps each day. Therefore, a simple bit of maths suggests a boy should have walked 71,175,000 steps between his fifth and eighteenth birthday, when we can (legally) call him a man. However, most children fall short of this: the average boy will have walked just 52,195,000 steps by this point.\n\nThe Dylan, B study sadly is vague on practical advice on what to do at this point should we wish to maintain our current age of majority. We could either reduce the requirements for roads (or steps) walked down, or, if we wish to protect the 71-million-step goal, we can instead begin to call boys 'men' at the average age of just over twenty-two-and-a-half. It is of course likely that this additional exercise would cause them to be blowing in the wind.\n\nDO THEY KNOW IT'S CHRISTMAS TIME?\n\nBAND AID\n\n**It is somewhat puzzling that large groups of individuals have gathered to ask this question at various points spanning more than three decades, as it is one that is almost trivial to solve.**\n\nThis question was first posed as part of a huge charity drive in 1984, and was collectively sung by no fewer than thirty-seven vocalists, who seemed very keen to get an answer to this question. However, it was then asked again by a fresh group a few years later, and a third and fourth time by different groups in 2004 and 2014.\n\nOn the face of it, answering this question is very simple: the latest estimates suggest that Africa has a Christian population in excess of 500 million out of a total population of 1.2 billion, meaning that even if Africans didn't follow the customs of people living on other continents, it's still extremely likely that they have a very good idea indeed that it's Christmas time, as almost every other person is a Christian.\n\nHowever, if we were to look more specifically at Ethiopia \u2013 the initial fundraising target for the commendable Band Aid initiative \u2013 the answer is slightly less clear. Almost two-thirds of Ethiopians, some 45 million people, identify as Christian and so could clearly be expected to know when Christmas is. However, of this figure, more than 30 million identify as Orthodox Christians, meaning they would not celebrate Christmas on the same day as their Protestant or Catholic brethren. This, perhaps, solves the mystery of this question's persistence: clearly the celebrity fundraisers have struggled to understand the differences in how various Christian groups in Ethiopia mark Christmas.\n\nOne thing that is beyond doubt is that there will be snow in Africa every Christmas. Africa is a huge continent with mountain ranges including Kenya's Mount Kenya, Tanzania's Mount Kilimanjaro, Uganda's Rwenzori Mountains and, even more specifically for the purposes of this study, Ethiopia's Semien Mountain \u2013 all of which are covered in snow for almost the entirety of the year.\nARE THERE 4,000 HOLES IN BLACKBURN, LANCASHIRE?\n\nTHE BEATLES\n\n**Some thinkers tackle love, some tackle betrayal, and some tackle politics \u2013 but few have tackled a pressing issue which causes fury among drivers everywhere: potholes.**\n\nPerhaps their willingness to address this issue is why Lennon, J and McCartney, P have endured in the public imagination for so long.\n\nIn their 1967 address, they not only note that there are 4,000 'small' holes in Blackburn, Lancashire, but also that someone had to count them. Today, trying to assess whether their estimate at the time was accurate is impossible, but it is still the case that someone in contemporary Blackburn has to try to calculate the number of potholes and deal with them.\n\nA recent estimate of potholes in Blackburn can be gathered from the local council's figures. In 2018 the council received \u00a3178,000 from a central government fund to tackle potholes. The average cost of fixing a pothole in Britain is \u00a353, which means Blackburn got enough funds to tackle 3,358 potholes \u2013 not too far from Lennon and McCartney's historical estimate, especially given that budget measures in Britain mean the council was likely underfunded in its efforts.\n\nIt seems like in the case of Lennon and McCartney science's gain may have been road maintenance's loss.\n\nAs to how many holes it would take to 'fill' the Albert Hall: we know its internal volume measures 86,650 cubic metres, so if all the holes were the same size they would be 21.66m3, which is roughly two-thirds of the size of a standard shipping container. It is unlikely that there were 4,000 holes this size in the roads of Blackburn, Lancashire, especially as the study itself says they are 'rather small', so though one may have led to the calculation of the other, we must reluctantly conclude that the collaboration may have made some serious errors in their maths.\n\nHOW MUCH IS THAT DOGGIE IN THE WINDOW?\n\nPATTI PAGE\n\n**There is a serious concern we must address about Page, P's 1953 study of pet economics.**\n\nBuilding upon original research by Merrill, B, we can't avoid the fact she is clearly a horribly irresponsible purchaser of pets. Firstly, as numerous charities have tried to teach us, dogs are for life and not just for Christmas; purchasing decisions should not be made on the whim of seeing a dog in a window.\n\nSimilarly, to avoid issues of animal cruelty, you should never buy a puppy if you haven't seen it with its mother (in person, photos don't count). Page also seems torn over what she's looking for in a puppy: one moment she's referring to its cute 'waggly' tail, the next she says it needs to be a guard dog to keep her lover safe. It's almost as if the missive doesn't want to be taken seriously as a purchasing guide to pets.\n\nThat's no reason not to answer the economic question the study poses, of course. If the dog in question is a mongrel, then its purchase price shouldn't be too high, likely not much more than \u00a3300 to \u00a3500 \u2013 though if it's pedigree it could cost several thousand off the shelf.\n\nDon't let that price lull you into a false sense of security, though. The People's Dispensary for Sick Animals (PDSA) says that depending on the size and lifespan of the dog that you choose, over its lifetime it could set you back between \u00a316,000 and \u00a331,000. In light of this, the short answer to the question is 'between 5,000 and 10,000 per cent more expensive than you think'.\n\nWe earnestly ask Ms Page to undertake much more serious thought before she again ponders whether that dog is for sale.\nCAN YOU KILL SOMEONE WITH A SONG?\n\nROBERTA FLACK\n\n**Flack, R repeatedly makes a number of serious accusations against an unnamed individual over the course of this report, most notably alleging attempted murder, softly and with a song.**\n\nGiven the damaging nature of such allegations \u2013 they could easily result in lifetime imprisonment \u2013 it's important to examine the evidence behind Flack's claim. Initially, Flack's claims stand up well: it is entirely possible to kill someone with a song, or any other appropriately loud noise. Anything over 150 decibels is enough to rupture an eardrum, while anything rising to around 200 decibels is enough to prove fatal to most of us. For comparison, a pneumatic drill is around 100 decibels, and chainsaws are only around 120 decibels \u2013 to hear a fatal level of noise you'd generally need to be near an explosion.\n\nFlack, however, specifies she is not being killed by volume: her accusation is that she is being softly killed. Here too, though, she may have evidence to support her claims. In 1933, a Hungarian composer penned a song called 'Gloomy Sunday', initially about the despair of war, then rewritten to be about contemplating suicide. The song provoked a (badly evidenced) international media panic through the decade that it was provoking a swathe of suicides \u2013 and the composer did eventually kill himself, although decades later. An examination of the case of 'Gloomy Sunday' in Gizmodo \u2013 we assume this to be a peer-reviewed journal \u2013 noted that Hungary has a historically high suicide rate, as do other countries with its cultural history, leading to speculation about genetic causes of the phenomenon.\n\nThe effect of a piece of music or another suicide provoking a copycat act is known as the 'Werther Effect', after the 1774 novel _The Sorrows of Young Werther_ by Johann Wolfgang von Goethe. The book's eponymous protagonist kills himself over doomed love, and its publication was followed by a spate of suicides, provoking huge debate as to whether they were related.\n\nSo, if Flack's lover is softly singing 'Gloomy Sunday' into her ears, she may have a good case against them for attempted murder.\n\n**A MATTER OF TIME**\n\nFementosecond | 10-15s | Pulse time on fastest lasers.\n\n---|---|---\n\nSvedberg | 10-13s | Time unit used for sedimentation rates (usually of proteins).\n\nPicosecond | 10-12s | A picosecond is to one second as one second is to approximately 31,689 years.\n\nNanosecond | 10-9s | Time for molecules to fluoresce.\n\nMicrosecond | 10-6s | Symbol is \u00b5s.\n\nMillisecond | 0.001s | Shortest time unit used on stopwatches.\n\nSecond | 1s | Si base unit.\n\nMorrissey | 5s | The length of time into reading a contemporary interview with him that fans start to feel a bit sad.\nHOW SOON IS NOW?\n\nTHE SMITHS\n\n**Anyone who's had to sit in and wait for a delivery knows just how painful it can be, and so it's not difficult for us to empathize with Marr, J and Morrissey, S when they plaintively seek detail on what their interlocutor means when they promise that something is going to happen right now.**\n\nLesser investigators may be willing to just accept 'now' as a period of somewhere between two and three seconds, but Marr and Morrissey are clearly not prepared to be fobbed off by such lengthy and vague periods, asking 'when exactly do you mean?' They can and should expect a far greater level of precision on just how instantaneous 'now' should be. What they are clearly asking is 'what is the soonest that now can be?'\n\nWe may have heard of a millisecond, which is one-thousandth of a second, or a microsecond, which is one-millionth of it, but even these are clunky and extended units of time versus 'now'. We might even start to think about an attosecond \u2013 a unit of time so small that it's been noted that an attosecond is to a second what a second is to the entire age of the universe. These things are quick.\n\nBut even an attosecond is an eternity compared to the one unit we can really say is 'now' \u2013 the Planck unit, which is the time it takes for light to travel, well, a really really short distance, and is the smallest possible measure of time (and far smaller than we can measure yet).\n\nSo, Marr and Morrissey, how soon is now? It's just a Planck unit away.\nCAN I KICK IT?\n\nA TRIBE CALLED QUEST\n\n**This is a question that has taxed many thinkers since it was first posed by A Tribe Called Quest in 1990, as it's since been asked again by persons ranging from Zed, J to Williams, R, among many others.**\n\nIt's not a surprise that this is a question they feel a need to re-visit, as answering it requires us to make use of complex concepts around the laws of consent and the laws of property rights.\n\nAs a first principle, if the 'it' you're referring to is a living being, you're being rude and should use better pronouns. But in this instance then you absolutely cannot kick 'it' in the absence of consent: in kicking a person you could break laws relating to assault and bodily harm. If an animal, then there are welfare rules. If you are considering kicking an adult person with their consent \u2013 this is a judgement-free zone \u2013 then do it gently as there is a maximum level of harm to which someone can legally consent.\n\nIn the circumstance that the 'it' under consideration for kicking is an object, things get simpler. You might want to think carefully first if the object is a brick or a glass window, as while you may have a legal right to kick something, that does nothing to help your greatly injured foot.\n\nHowever, if you own the object, or have rights to use it (and insurance against damage to it), or permission from someone who does \u2013 and have checked it's unlikely to cause you harm \u2013 then we have good news: yes, you can.\n\nHOW LONG DOES IT TAKE TO APOLOGIZE A TRILLION TIMES?\n\nOUTKAST\n\n**Relationships are never straightforward, and the relationship between a man and his partner's mother is a notoriously tricky one.**\n\nIt's commendable, then, that Outkast \u2013 and Andr\u00e9 3000 in particular \u2013 are keen to make amends to Ms Jackson after problems with her daughter.\n\nKnowing that one apology is never enough, Mr 3000 repeatedly reassures Ms Jackson he is so sorry that he apologizes 'a trillion times'. But new analysis suggests he might not be telling the truth. If we generously assume that Mr 3000 can say 'sorry' once a second, 24 hours a day, with no break, he will after one full year have apologized just 31 million times.\n\nTo reach a full trillion apologies, Mr 3000 will have to continue apologizing once a second, 24\/7, for just over 31,688 years. However, at the time of the song's release, he was just over twenty-five years old \u2013 meaning that even if he had begun apologizing the moment he was born, he would have said sorry just 775 million times \u2013 less than a thousandth of his claimed total of sorrow.\n\nAs a result of all that, it seems like we owe Ms Jackson an apology of our own: we're sorry, but the evidence suggests Mr 3000 was in fact not for real, after all.\nVOULEZ-VOUS COUCHER AVEC MOI CE SOIR?\n\nLADY MARMALADE, LABELLE ET AL.\n\n**This question has frequently been investigated over the years and research published in 1989 by two American psychologists, Russell D Clark and Elaine Hatfield, might help us answer it definitively.**\n\nThey asked a group of five female students and four male students to approach people of the opposite gender on a college campus, introduce themselves, say they found the stranger attractive, and then ask one of three questions: 'would you go out with me tonight?', 'would you come over to my apartment?', or 'would you go to bed with me?'\n\nFor the first question, going on a date, the success rate for both men and women was about the same \u2013 a surprisingly high 50 per cent (roughly) said they were willing to go on a date with a total stranger. As soon as we shift to the second question \u2013 going to an apartment \u2013 the success rate with women drops to almost nothing, but with men the success rate jumps to a pleasing 69 per cent.\n\nAnd as for the 'would you go to bed with me tonight?' question, not a single woman in either of the original studies said yes to this question \u2013 but almost three-quarters of men did. So, if you want to ask this question and get a positive response, ask a man.\n\nAs to why the question is posed in French \u2013 a global survey carried out by Durex in 2005 found that the French had an average of 8.1 sexual partners, lower than the global average of 9 and way behind the Turkish, who had an average of 14.5. And though they were above the average of 103 for the number of times they had sex each year, they were still trailing way behind Greece, who had sex 138 times a year.\n\nSo statistically, if you're looking for the answer to be yes, you're much better off asking _Tha koimithe\u00eds maz\u00ed mou ap\u00f3pse_ or _Bu gece benimle uyuyacak m_ \u0131 _s_ \u0131 _n?_\n\nWHO RUN THE WORLD?\n\nBEYONC\u00c9\n\n**No human being takes pleasure in correcting Knowles, B but alas on this issue we are forced to do so.**\n\nDespite the fact that a boy once ascended to become a head of state at the age of six months (Egypt's Fuad II, who took the throne in 1952, only to be deposed eleven months later \u2013 before his second birthday), there are few under-eighteens at the top of global politics, and no girls \u2013 by which of course we mean women under eighteen \u2013 running any country within the world at present.\n\nIf the answer isn't 'girls', then who does run the world? The answer may disappoint Beyonc\u00e9, and feminists everywhere, but it probably won't surprise them as it is of course middle-aged men. Of the 193 member states of the UN, at time of writing, only fifteen have a woman serving as head of state or head of government \u2013 that's fewer than one in ten. Only around one in three countries analyzed had ever had a female at the top of their politics.\n\nThings don't get any better in the business world: research in 2017 into the UK's hundred biggest companies found there were more major corporations led by men called David (eight) or Stephen (seven) than there were companies led by women (six). Things are no better in America: in early 2018, only twenty-seven of the USA's 500 biggest companies were led by a woman.\n\nIt is not for the authors to judge why this reality was not reflected in Beyonc\u00e9's published research, but one hypothesis is that it's a bit bleak really.\nIS THIS BURNING AN ETERNAL FLAME?\n\nTHE BANGLES\n\n**This is an important case file of three patients who presented together with a very similar set of symptoms in 1988.**\n\nIn the interests of patient confidentiality, we shall refer to them by the case name of 'The Bangles'. These three women reported 'burning' that was so persistent as to lead all three to worry at length that the symptoms would proceed persistently, if not eternally.\n\nWhile explaining their problem, The Bangles encourage their doctor to take their pulse (somewhat oddly with their eyes closed and referring to them as 'darling', but this is clearly bound up in their nervousness of the clinical diagnostic process); express hope that someone would be able to ease their pain; and explain their symptoms had been causing loneliness, suggesting the burning sensations were a barrier to normal romantic relationships.\n\nGiven the stigma around such conditions, it is perhaps not surprising that The Bangles were willing only to describe their symptoms euphemistically, and that all three were worried the problem would prove insoluble.\n\nHowever, there are multiple possible diagnoses which bode well for the women: rather than being an eternal flame, the burning experienced by all three is far more likely to be a symptom of thrush \u2013 or given the women are assumed to be sexually active, could be a symptom of treatable STDs such as chlamydia or gonorrhoea. All of these infections respond well to the proper treatments, and so The Bangles serve as an excellent example to others of the benefit of seeking early treatment, rather than worrying for too long.\n\nAlthough their hope that just having a doctor say their name will be enough to banish the symptoms is rather unlikely, we know that often, because of the placebo effect, just having a clinician pay attention can lead to improvements in patients' symptoms. Interestingly, a recent study by doctors at the University of York found that the use of copper bangles for treating arthritic pain 'had no statistically significant therapeutic effect among patients'.\n\nWAR \u2013 HUH \u2013 WHAT IS IT GOOD FOR?\n\nEDWIN STARR\n\n**Starr, E's provocative framing of this question forces us to reassess something the majority of us believe to be a bad thing, which should be avoided at almost any cost, and consider instead its upsides.**\n\nAs he no doubt expected us to realize, there is a long list of potential good outcomes from war, in the right circumstances.\n\nFirstly, as evidenced in the most extreme case of the Second World War, and more recently in Kosovo, war serves as a backstop against genocidal dictators and helps preserve the rights and freedoms of minorities, and as countries through history have found, war can serve as an effective way of preserving territorial integrity against aggressors.\n\nAdditionally, wars \u2013 at least in their early days \u2013 can serve to hugely boost the popularity of democratically elected leaders. Margaret Thatcher's approval ratings shot from 41 per cent to 59 per cent in the three months of the Falklands War, while George Bush's ratings leapt from 51 per cent to 86 per cent following 9\/11 and the US's subsequent invasion of Afghanistan.\n\nResearch from the State University of New York suggests such gains take a long time to fall back, making it a worthwhile proposition for the leaders. Though the research is out on the long-term economic benefits, it's clear that in the short term it has a positive impact, especially boosting the manufacturing sector.\n\nMr Starr would surely, though, also not wish us to forget the inventions that have come out of war \u2013 or at least military-backed research \u2013 which include the early internet, plastic surgery, superglue, chemotherapy and even tinned food. While we wish Mr Starr had asked us to consider the horrors of war at greater depth, he does make a compelling case for its upsides.\n\nWar \u2013 huh \u2013 what is it good for? Democracy, political popularity, technology and the short-term economy.\nHOW LIKELY IS RIHANNA TO NEED HER UMBRELLA?\n\nRIHANNA\n\n**Rihanna would like it to be known she's a supportive friend: she repeatedly makes it clear that while she is there for you when the sun shines, she also intends to be there when it is raining, and will bring an umbrella to shelter you.**\n\nClearly, this is to be commended, but if we are to examine how useful Rihanna's strategy of umbrella-carrying is in practice, we need to collect some data to see whether or not her approach is effective. This, of course, is complicated by Rihanna's move from the city she grew up in \u2013 Bridgetown, the capital of Barbados \u2013 to Los Angeles.\n\nIn Bridgetown, despite the warm weather it rains 153 out of the year's 365 days. So on any given day her umbrella has a 42 per cent chance of being useful \u2013 meaning she's a friend you would likely be glad to have around for that reason alone.\n\nHowever, in her new home of Los Angeles the situation is very different: several months of the year average between zero or one day of rain in the month, and across the year there are only typically thirty-seven rainy days. This means that on 89.9 per cent of days of the year, Rihanna will be carrying an umbrella for no reason, which may even begin to look eccentric.\n\nAs a result, we recommend that Rihanna adapts her friendship strategy to better suit the pleasant climate of her new home. We have every confidence in her success.\n\nIS THIS THE REAL LIFE? IS THIS JUST FANTASY?\n\nQUEEN\n\n**Mercury, F begins his 1975 investigation into the nature of reality by imagining a figure guilty of murder before quickly moving through a dizzying invocation of cultural knowledge, from meteorology to dance, theatre to religion, and whether someone will let someone else go.**\n\nThis is clearly an early foray on the part of guitarist Brian May into astrophysics, in which he later obtained a PhD from Imperial College, London.\n\nThe fields of physics and philosophy have both wondered whether it could be possible that the universe in which we all live could be a Matrix-style simulation, or even just a game on someone's computer. When scientists run simulations in our \u2013 apparently real \u2013 universe, they often simplify some of the physics, treating some things as constant, to help save processing power. However, in the universe we live in, physics has lots of apparently arbitrary constants, such as the speed of light in a vacuum. That's not reassuring.\n\nThe technologist Elon Musk \u2013 the man behind Tesla and Hyperloop \u2013 believes the simulated universe theory is correct, and given that there are likely millions or billions more simulated universes than real ones, statistics suggest we're probably not in a 'base reality'. Whether this leads you to decide, as the authors of this study did, that 'nothing really matters' is more complicated. If our existence can only be said to exist through our perception of it, then this is still as meaningful as any other explanation for consciousness. As to the stability of our possibly simulated universe, science has yet to determine how likely it is that someone might kick out the power cable.\nARE YOU CALLING ME 'DARLING'?\n\nTHE TING TINGS\n\n**In her lamentation on being regularly addressed by the wrong name, The Ting Tings' singer menacingly asks an unknown interlocutor whether he \u2013 we assume it's a he \u2013 is referring to her as 'darling'.**\n\nWe do not have enough evidence to rule on whether he was or not \u2013 but we do have good evidence to suggest that unless he knows her well, he should not be doing so. The polling organization YouGov asked 4,191 adults whether or not it was acceptable for a man to call a woman he doesn't know well 'darling', and only 11 per cent responded that they thought it was \u2013 with men and women both agreeing the word was unacceptable. 'Babe' scored slightly lower, at 7 per cent, while 'love' did better, with 25 per cent thinking that was acceptable.\n\nHowever, The Ting Tings may also wish to ponder why their names are forgotten. Experts offer reams of psychological explanations: when learning new names we're often also socially anxious; names get lost in our short-term memory, and names are just hard to remember \u2013 they're totally arbitrary collections of syllables, after all.\n\nHowever, we can note one specific issue in this instance: in a five-minute thesis about name recollection, at no point does The Ting Tings' singer mention her name. For the record, it's Katie.\n\nHOW MUCH WOULD IT TAKE TO CRY A RIVER?\n\nVARIOUS ARTISTS\n\n**For more than sixty years, people have issued public decrees to their former lovers to cry them a river, and claiming that they have done the same.**\n\nThese range from Julie London, to Shirley Bassey, to Barbra Streisand, to Justin Timberlake. This story has been allowed to go unchallenged, and it shouldn't have: distressingly, the singers of these songs may not be telling the truth.\n\nRivers move a lot of water. The Amazon discharges 209 million litres of water every single second. The rather more sluggish and far smaller Thames river in London discharges around 66,000 litres each second. So, how many tears would we need to try to match the slower river? Research published in 1966 showed the average tear is 6.2 microlitres in volume, suggesting that for a single litre we need around 161,000 tears.\n\nHowever, not many of us could manage much more than one tear per eye, no matter how hard we're trying. This means that to achieve even one second of discharge from a river as sluggish as the Thames, you'd need more than 5.3 billion people sobbing away. And given that to create a river people would have to keep up this output for quite some time, things are not looking good for Timberlake et al. Not only are they making deeply implausible claims to their former lovers, they're being somewhat over-demanding.\n\nPerhaps their breakups were for the best?\nWHERE HAVE ALL THE FLOWERS GONE?\n\nPETER, PAUL AND MARY\n\n**Despite being seen by some as a rumination on mortality and war, Yarrow, P Stookey, P and Travers, M's 1962 release, which builds on original research by Seeger, P, is in many ways a reflection on the economics of the global flower trade \u2013 which does indeed see thousands of tonnes of flowers traverse the globe every year.**\n\nThese days, the place most of the flowers are leaving is the Netherlands, which is where almost half the flowers sold globally originate, followed by Colombia, Kenya and Ecuador. But as to the trio's core question \u2013 where have they gone to? \u2013 the answers lie close to home: the United States imports more than 80 per cent of the flowers sold there, and in the UK that figure rises as high as 98 per cent. Given that Yarrow, P, Stookey, P and Travers, M are from the US, the flowers are heading in their direction, which suggests that perhaps they'd have been better placed asking where the flowers were coming from. Their concern, however, may have been premature rather than misplaced: a peer-reviewed international botanical study in 2010 found that 25 per cent of existing flowering plant species in the world are at risk of extension. For example, 97 per cent of wild-flower meadows in the UK have disappeared in the last hundred years. The flowers may not have gone yet, but many of them are at substantial risk of doing so \u2013 Messrs Yarrow and Stookey and Mme Travers were simply ahead of the curve with their fears. However, the complex mix of environmental, sociological and climatological factors behind this aren't really contained in their proposed answer that girls have picked every one.\n\nSHOULD I STAY OR SHOULD I GO?\n\nTHE CLASH\n\n**The problem presented by Mr Jones and his associates in The Clash is a dilemma over the best way to avoid a difficult situation, or 'trouble'.**\n\nThe Clash have done well at identifying two courses of action \u2013 staying or going \u2013 and have reflected on the level of trouble each would cause, but repeatedly fail to decide on either over the three minutes ten seconds they allow themselves to make a decision.\n\nThankfully, a simple mathematical model helps us solve the trouble they're facing. Setting out their issue, The Clash state that there will be trouble if they go, but if they stay it will occur at twice that rate.\n\nDefining trouble as t, we know that for any given value of trouble: t(go) = x, but, t(stay) = 2x.\n\nAfter this, it's simple maths to say that for any level of trouble above zero \u2013 in which situation The Clash could do whatever they liked \u2013 they should clearly go, for a maximum of half as much trouble versus the alternative. We have also demonstrated that paying attention to algebra in school can prove useful in adult life after all.\n\nHowever, we should stress the above analysis is all done under the hypothesis that Mr Jones and co. wish to minimize the trouble they face, which may not be the case. They are, after all, named The Clash.\n\n**_t(go) = x_**\n\nbut\n\n**_t(stay) = 2x_**\nDOES YOUR MOTHER KNOW THAT YOU'RE OUT?\n\nABBA\n\n**Everyone has had a moment during an evening out when things suddenly escalate quickly, whether into a potential pub fight, a blazing row, or just some quite maudlin talk.**\n\nBut ABBA certainly take no prisoners with their conversational gambits, rapidly switching from lightly flirting and dancing with the person their remarks are addressed to, to then asking them whether their mum knows that they're openly gay.\n\nYou might hope a band with such a significant LGBT following would know not to broach this topic so lightly, but this may be a consequence of the band's experiences amongst the often more liberal attitudes pervasive in Sweden. Then again, it's possible they are aware of significantly improving attitudes towards LGBT equality and are looking for confirmation that young people are benefitting from those trends. In 1983, four years after this song was released, only 11 per cent of British people said same-sex relations were 'never wrong'. The latest repeat of that survey found two-thirds of Brits now say this.\n\nSurvey evidence from the LGBT rights group Stonewall suggests ABBA may have been onto something with their question. People in their sixties said they had come out at an average age of thirty, while people in their thirties said they came out around twenty-one \u2013 but under-twenty-fives who had come out said they'd done so at an average age of seventeen. Turns out their mothers knew after all.\n\nOf course, it's also possible ABBA were simply asking the teenager with whom they were dancing whether their parents were aware they were outside the house. That theory seems a little unlikely to the authors, though.\n\nDO YOU GIVE ME FEVER?\n\nPEGGY LEE\n\n**Lee, P made her definitive contribution to this ongoing public health investigation in 1958, popularizing warnings against the dangers of contracting a fever from kissing, touching and being held tight.**\n\nHowever, it appears that Lee's public health message wasn't backed up with especially strong evidence \u2013 certainly if she was referring to some of the more common ways in which fever is spread, such as the common cold.\n\nAs a key example, Lee is unlikely to get a fever through the common cold when she is kissed. Research conducted at the University of Wisconsin found that if someone with a cold infection kissed a number of other healthy volunteers for up to ninety seconds, just one in sixteen caught the infection. When it comes to this, no one is likely to give Lee fever.\n\nLee's other examples are much likelier to give her fever, though. If someone with a cold is holding her \u2013 tight or otherwise \u2013 they have a significantly higher chance of passing on their illness than through kissing, thanks to proximity to infected droplets in their breath. Lee's also right to spot that staying through the night is especially risky: eight hours sleeping with the possibility of coughs and splutters in close proximity to your face really might give you fever.\n\nPeggy Lee's reputation as a public health expert comes out of the analysis pretty intact, with a small modification: you give me fever when you stay through the night, but not with a kiss.*\nWHAT WOULDN'T MEAT LOAF DO?\n\nMEAT LOAF\n\n**Loaf, M's extended profession of love has raised questions throughout the two decades since its release: scholars have asked time and again exactly what it is he is pledging that he would not do.**\n\nThis is puzzling: Loaf is actually extremely specific in his treatise on what he would or would not do for love, specifically pledging just two things he would not do to his unnamed lover \u2013 he pledges never to lie to her, and never to screw around.\n\nHowever, we should question whether Loaf's claim for unfailing honesty in this extended pledge stands up to scrutiny. Loaf variously claims in his song that the woman he loves sometimes 'breathes fire' and is sometimes 'carved in ice' \u2013 both claims that would revolutionize medical science if supported, as it is certainly unlikely that such extremes of temperature could be produced by one organism.\n\nLoaf also claims he would run to hell and back \u2013 perhaps unaware that the strongest evidence for the physical existence of hell was a microphone supposedly lowered into Russia's Kola Superdeep borehole, which supposedly picked up screams of the anguished dead.\n\nLoaf may be unaware the story was eventually exposed as a hoax. Either way, it's not possible to run in a vertical borehole.\n\nFinally, we must examine Loaf's claims to love his partner as long as planets are turning, stars burning, and other similar hyperboles extending into the billions of years. While Mr Loaf managed twenty-five years of marriage before divorce \u2013 an admirable feat \u2013 this falls far short of his stated targets.\n\nReluctantly, we are forced to conclude that Loaf may in fact, on occasion, do just that.\n\nARE FRIENDS ELECTRIC?\n\nGARY NUMAN\n\n**While experiencing difficulties with a 'friend' \u2013 he describes how his friend appears to have 'broke down' \u2013 Numan wonders whether friends might be electric, a question fairly easily addressed by biological science.**\n\nMany of us will have experienced uncomfortable electric shocks when in the proximity of our friends, but these are typically the result of static electricity, caused by friction against certain types of clothing and other materials. While this is undoubtedly 'electric' (and painful), it is more a result of our friends' sartorial choices than our friends themselves.\n\nHowever, assuming that our friends are human \u2013 though much of this would hold for any live animal \u2013 they are at least capable of generating electricity, even if they cannot be said to be electric themselves. Neurologists have detected that each neuron inside a human brain is capable of carrying a voltage of around 0.07 volts.*\n\nThis voltage appears small \u2013 it's only 1\/20th of the voltage of an AA battery \u2013 but may be more impressive than it first appears. For one, a typical human has around 80 billion neurons in their brain, and for another, neurons are far smaller than AA batteries. Neurons carry their 0.07 volts across a tiny distance of just 0.000000005 metres, meaning that metre-for-metre they hold more than four times the electrostatic force involved in producing thunderstorms.\n\nHuman friends, therefore, are considerably electric \u2013 and if Numan's are not, he should perhaps seek urgent medical assistance for them.\nHOW DO YOU SOLVE A PROBLEM LIKE MARIA?\n\nTHE SOUND OF MUSIC\n\n**The nuns of Nonnberg Abbey appear to have a long list of problems with Maria, who is studying there to become a nun.**\n\nMaria, they complain, climbs trees and tears her clothes, whistles, dances, wears curlers, runs late, doesn't listen and behaves unpredictably. These concerns \u2013 many of which may seem quite petty to a modern audience \u2013 are enough to make the group conclude that she is detrimental to the affairs of the abbey.\n\nTo look at how the issues with Maria could be addressed in the present day, the nuns could turn to feminist campaigner Caroline Criado Perez \u2013 but she firmly concludes that the problem does not lie where the nuns believe it does.\n\n'Maria isn't the f---ing problem here,' she argues. 'The problem is a sexist society that has constructed a world around male bodies and male needs \u2013 and that then has the gall to blame women for not fitting into it as expected. 'Maria is a perfectly well-adjusted human being who just wants to be free to live according to her own rules and spin around in mountains singing at the top of her lungs; she doesn't care for your misogynistic expectations that she quietly acquiesces to being a member of the subordinate sex class. And if that's wrong, then I don't want to be right.'\n\nHow do you solve a problem like Maria? You adjust your institutional expectations to properly accommodate her. A simple answer might be a more precise job description, allied with a rigorous appraisal structure and regular contact with a line manager. As with many questions, the answer is 'better HR'.\n\nHOW MANY GENERATIONS WILL HAVE PASSED BY THE YEAR 3000?\n\nBUSTED\n\n**In their oddly taunting 2002 account of their time-travelling experience, Busted are told that their 'great-great-great-granddaughter' is 'pretty fine' \u2013 information that is undoubtedly less useful to them than blueprints for advanced technologies, medicine or even the results of events that can be gambled upon.**\n\nNonetheless, the exchange raises an interesting question: at present, the average age for a woman to have her first child in most Western countries is in her mid-to-late twenties. Assuming this pattern held, we would expect Peter to encounter Busted's* great-(repeated 33 times) granddaughter, come the year 3000.\n\nTo suggest that they have instead met their companion's great-great-great granddaughter and found her not only alive, but also conforming to 2002's standards of attractiveness suggests either that the average generational span rapidly grew from around thirty years as it stands now to around 200 years. Even the most optimistic predictions for increased lifespan currently stand at 125 years over the next fifty years, with some researchers arguing that it will plateau at 115 years. Another solution could be that the final of the five generations attained infinite longevity, perhaps by downloading their consciousness into a powerful computer. In a future in which this is possible, perhaps that would be viewed as 'pretty fine'.\nWHEN WILL I BE FAMOUS?\n\nBROS\n\n**In the title of this study, Goss, L and Goss, M directly pose a complex statistical question \u2013 when can a typical person expect to become famous? \u2013 but it is one that they immediately give up any hope of answering.**\n\nIndeed, initially we are obliged to challenge the fundamental premise of their question \u2013 in the world as it stands, despite there being hundreds of TV channels and thousands of reality shows, it is still possible, even probable, that many people will never become famous.\n\nIf we look at data from imdb, an online repository of almost every film and TV show ever made, the site lists around 8.7 million people \u2013 but of those around two-thirds work behind the scenes, and so largely cannot be said to be famous. The Gosses specify that it's the sort of fame that results in their picture in the paper that they are interested in, and it is extremely unlikely that anyone but the most famous director or producer would be pictured in a newspaper. Neither would extras and actors in non-speaking roles.\n\nEven using a generous definition of fame that includes anyone credited for any speaking part in any show ever, still only 0.04 per cent of us (or one in 2,500) will ever become famous.\n\nBros, however, treat fame as a certainty; it is after all 'when will I?' not simply 'will I?', which possibly refers to Andy Warhol's assertion that everyone will be famous for fifteen minutes.\n\nHowever, this gets difficult: if everyone is to be famous for fifteen minutes, we would need 199,771 years just to get through everyone alive right now. Or, to allow each person on the planet their allotted fifteen minutes in their lifetime, each person would have to share their fifteen minutes of fame with 2,496 others. That may not make you feel all that famous.\n\nFor the members of Bros themselves, however, the answer to the question posed by their song is simpler. This was their breakthrough hit, eventually reaching number two in the UK charts. It was released on 16 November, first charted on 28 November and Bros first appeared on _Top of the Pops_ on 21 January 1988. So the answer to the Goss brothers' question specifically is between twelve and thirty-five days after release.\n\nARE WE HUMAN, OR DANCER?\n\nTHE KILLERS\n\n**Flowers, B urgently wants to know an answer to this question: he has sought it on his knees, and through the course of this thought experiment seems willing to cast aside many aspects of his life and character \u2013 including grace, virtue, soul, romance and devotion \u2013 in his search.**\n\nIt is unclear from the context how Flowers believes these aspects would have hindered him.\n\nLeaving aside the implication in his question that dancers are somehow other than human, Flowers could have found an answer with far fewer sacrifices had he taken a more methodical approach to his research. Assuming that he means 'dance professional', rather than simply someone who has ever danced in some way, which we can assume is close to everyone who has ever lived, then we can begin to build an answer.\n\nIn the UK, for example, Flowers could have examined data from the Office for National Statistics, and found 21,000 people are employed (or self-employed) as dancers and choreographers. Following that, we need only look up UK population estimates derived from the country's census to see that those 21,000 come from a population of around 65,600,000,000.\n\nThis lets us work out a possible solution to Flowers' question, if we accept his fundamental premise that the category dancer supersedes that of human: 99.7 per cent of us should answer 'human', but 0.03 per cent are in the fortunate position of being able to answer 'dancer' (or choreographer).\nYOU WALK 500 MILES. YOU WALK 500 MORE. WHERE HAVE YOU BEEN?\n\nTHE PROCLAIMERS\n\n**The intended route of Reid, C S and Reid, C M's audacious 1,000-mile walk has been the subject of scholarly debate for the three decades since the release of their account.**\n\nWorking from the assumption that the brothers are starting from their hometown of Leith \u2013 a suburb of Scotland's capital, Edinburgh \u2013 people have tried to calculate how far they would get if they walked '500 miles' and then '500 more'.\n\nIn 2014, Hazel McKendrick made an estimate on Twitter that 500 miles would take the brothers as far as northern France, or north-eastern Germany, while the promised additional 500 would allow them to reach Poland, Austria or Italy. However, as South Carolina-based cartographer Kenneth Field noted, this took no notice of obstacles such as mountains or oceans, or even of the curvature of the Earth. Using estimates based on Europe's existing roads (though assuming the brothers could walk on water), he came up with more modest travel distances: 500 miles would only get them to two points of mainland Europe, while with 500 more they could begin to reach the south of France.\n\nEven this more rigorous calculation, though, missed out two important considerations: firstly \u2013 unless they have covered up this fact superbly \u2013 the Proclaimers cannot walk on water. Secondly, during the course of their song, they say the purpose of their 1,000-mile walk would be to fall at the door of the person they're singing to. If we work on the assumption that door is also in Leith, the Proclaimers wish can realistically come true: if they walk to Land's End in Cornwall \u2013 the furthest they can walk without hitting ocean \u2013 they will have travelled 545 miles. So if they want to end where they began, they can walk 500 miles, then 500 more, but then they would have to do an extra 90.2 miles before falling down right where they started, in Leith. If however they wanted to only walk 1,000 miles exactly, but still visit mainland Britain's most westerly point, they would be able to make it back up to the north of England to somewhere in the vicinity of the Currock area, just south of Carlisle.\n\nWHERE IS MY LARGE AUTOMOBILE?\n\nTALKING HEADS\n\n**This is a song that has been characterized as being perhaps the ultimate musical rendition of a mid-life crisis \u2013 and so concern about the size and presence of the family patriarch's car was an almost vital ingredient.**\n\nRepeated psychological studies have found happiness correlates not just with our own income and status, but with how we are doing compared to those around us. For example, a 2009 US-wide study found that people were happier when they lived in a rich neighbourhood in a poor county than in a similarly rich neighbourhood in a rich county \u2013 suggesting the issue for Talking Heads is not so much the size of their car, but whether or not it's bigger than other people's.\n\nThis is where time has acted against Talking Heads since they recorded the song in 1980: despite climate change and air pollution suggesting cars should get smaller, they have instead got bigger across the decades. Figures from automotive.com show a modern Porsche 991 is eleven inches longer than a 1980 Porsche 911. A modern VW Golf is seven inches longer than a 1983 model; Honda Accords have grown twenty inches since 1982, and a modern Mini Cooper is twenty inches longer than the original.\n\nHerein lies a possible cause for Talking Heads' social anxiety: their automobile might have been large by 1980 standards, but now it's bound to seem small.\nDID THOSE FEET IN ANCIENT TIMES WALK UPON ENGLAND'S MOUNTAINS GREEN?\n\nWILLIAM BLAKE\n\n**In his rather confused 1804 investigation into a potential site for Jerusalem in a northern English location, Blake sets out a number of questions that are easily answered and dismissed.**\n\nThe song also includes a number of actions that would likely be impossible, or dangerous if attempted, warranting a full safety and fact-checking effort.\n\nWhile there is a period of Jesus's life about which the bible says nothing, all of Jesus's recorded activities occur within what is modern-day Israel \u2013 in Nazareth, Bethlehem and Jerusalem \u2013 with no evidence he came to England. We now tackle Blake, W's other questions and issues in turn.\n\n_And was the holy Lamb of God \/ On England's pleasant pastures seen? \/ And did the countenance divine shine forth \/ Upon our clouded hills?_ Blake is fooling no one here: these are repetitions of his first question, to which the answer is still 'no'.\n\n_And was Jerusalem builded here \/ Among those dark Satanic mills?_ No. Jerusalem is located in the Middle East and claimed by Israel and Palestine as their capital \u2013 it does not need England staking a claim, too. Additionally, it was established there since at least four centuries before Christ, whereas the 'dark Satanic mills' did not appear until two millennia later.\n\n_Bring me my bow of burning gold._ If you see a bow of burning gold, do not touch it: it will burn you.\n\n_Bring me my arrows of desire._ Desire is intangible and therefore would not be suitable material for arrows.\n\n_Bring me my chariot of fire._ Please see the warnings on the bow of burning gold.\n\n_Till we have built \/ In England's green and pleasant land._ Jerusalem, as has been repeatedly stated, has already been built. And not in England. Whether or not Blake, W was advocating a kind of biblical proto-theme park as a means of kickstarting the northern British tourist trade is unclear.\n\nARE WE OUT OF THE WOODS YET?\n\nTAYLOR SWIFT\n\n**The question posed here by environmental scientist Swift, T in her 2014 study is clearly one of deep concern to her, as she repeats it no fewer than thirty-eight times in her four-minute-sixteen-second video.**\n\nYet it should come as no surprise that Swift regards it as so important, as the issue of deforestation is crucial to the future of the world. Not only can deforestation lead to mudslides and other localized catastrophes, but the absence of forests reduces the Earth's ability to absorb carbon dioxide, accelerating climate change. Swift is rightly drawing our attention to such things.\n\nFortunately, we are not entirely out of woods and forests yet, but we are running out of them rapidly. Around 10,000 years ago, according to UN data, the planet had 6 billion hectares of forest, covering around 45 per cent of the Earth's land area. Today, that's down to around 4 billion hectares, or just 31 per cent of the world's land.\n\nIt's also disappearing faster each year: over the last 5,000 years we've lost around 360,000 hectares of forest a year. In the current decade we lose more than five million hectares every year. We might not be out of the woods yet, but the situation is certainly not improving \u2013 so no wonder Swift, T is so worried.\n\nIronically though, we know that each 10-minute YouTube view produces 1g of carbon, so her four-minute-sixteen-second video produces 0.416g. At the time of writing it has been viewed 135,632,063 times, producing 56,422,938.208g or 56,422kgs of carbon. If a single tree can sequester 21kg of carbon a year, then we need 2,687 extra trees just to deal with the carbon generated from watching this video.\nI WILL SURVIVE \u2013 BUT FOR HOW LONG?\n\nGLORIA GAYNOR\n\n**Many people who have experienced a difficult breakup have been reassured by Gloria Gaynor's classic investigation into stoicism to help get them through heartbreak.**\n\nResearch suggests most of us experience our first serious breakup at around the age of twenty \u2013 a little younger than Gaynor's age when she first posed this question. The UK's official life expectancy tables reveal that, at that stage, while a man can expect to survive for another 59.7 years, a woman can expect another 63.3 years on Earth \u2013 meaning that statistically speaking, Gaynor will not only survive, but also outlive her former partner.\n\nThere is also reassurance for people separating from even more serious or longstanding relationships: the average age of a divorcing man is forty-six, meaning he should expect another thirty-five years alive to find another love. The news is better for women, who are on average aged forty-three when they divorce, meaning they can expect to survive a full forty-one years more.\n\nHowever, returning to Gaynor's study, her assertion that she has 'all her life to live' is less accurate. If we use the example of a woman who is 20 when she experiences her first breakup and lives to be 83.3, she has in fact only 76 per cent of her life left to live at that point. However, Gaynor, G was first making this statement when she was 29 years old, meaning that she had around 65 per cent of her life left to live, with every 18 months reducing that figure by another percentage point. As Gaynor, G is now 69 years old, when she makes this statement she has only 20 per cent of her life to live. We can represent this as a curve on a graph with age at which you sing the song against the percentage of remaining life left to live.\n\nGaynor's missive will certainly survive, however: in 2015 it was added to the USA's National Recording Registry, to be preserved for future generations.\n\nIF YOU DON'T LOVE ME NOW, WILL YOU NEVER LOVE ME AGAIN?\n\nFLEETWOOD MAC\n\n**If any people could claim to know in detail the complexities of love, betrayal, separation and dealing with ex-lovers, then it would be the members of Fleetwood Mac \u2013 who have managed virtually every combination of marriage, cheating, and separation over their tumultuous history.**\n\nAs such, their assertion that once the chain is broken \u2013 when you fall out of love \u2013 that will be the end forever would seem to be one that we would all do well to believe. However, it's not strictly true.\n\nIf we look at divorce, perhaps the most obvious and complete a split that a serious relationship can have, we can see evidence that sometimes you can break the chain. A study of 1,001 divorced couples published in 2005 by Dr Nancy Kalish found that 6 per cent of divorced couples eventually remarried their former spouse, and in the instances where they do, they have a much better than average chance of staying together, with 72 per cent of the marriages lasting.\n\nThis may have been particularly on the authors of this study's minds at the time as both Buckingham, L and Nicks, S, McVie, J and C and Fleetwood, M were all in the process of ending their long-term relationships at the point at which it was published. It is perhaps a comfort that the high volume of failed relationships within the band actually increase the likelihood of one of them being rekindled.\n\nMore generally Kalish, found that the remarrying couples who fared best were those who married young and then waited several decades before reuniting \u2013 suggesting that if you do break the chain, it's best to wait a long time before trying to repair it.\nHOW MUCH SPACE DO A MILLION PHOTOGRAPHS TAKE UP?\n\nTHE VAPORS\n\n**Fenton, D famously raises this question in his problematic investigation of self-identity as it relates to race.**\n\nHis thesis is that taking a lot of photos is 'turning' him Japanese. Leaving aside this simplistic and problematic model of racial identity, the study does yield one important point.\n\nFenton refers to wanting a million pictures of his lover, which, when the study was first published in 1980, would have meant physically printed-out photos. Many people hear the lyric as Fenton wanting a million pictures on his 'self' \u2013 meaning that if the photos were standard 6-inch by 4-inch size, he would need a surface area 7,936 times larger than the typical person. However, this is a misreading: Fenton is actually stating he wants a million photos in his 'cell'. To fit that many 6-inch by 4-inch photos, his cell would need to be 200 feet long, 200 feet wide and 200 feet high \u2013 rather a lot bigger than the standard six feet by nine feet US isolation cell.\n\nIn the modern era, things are simpler. A typical iPhone image is around one megabyte, meaning a million would be one million megabytes, or one terabyte \u2013 which can now be stored on a portable drive that's smaller and not much thicker than a 6-inch by 4-inch photograph.\n\nThat convenience is essential to some of the internet's biggest businesses: one million photographs \u2013 enough to line our 200-foot cell \u2013 are uploaded to Instagram every 15 minutes, 24 hours a day, making a total just short of 100 million each day, or 12 billion a year. It's likely a good thing for the world's trees that these don't get printed out any more.\n\nIS ANNIE OKAY?\n\nMICHAEL JACKSON\n\n**Simply from the description that Jackson, M gives in this case study, anyone should be able to determine that Annie is not entirely okay: we know there are bloodstains on the carpet, that she's been struck down and even that it was her doom \u2013 so the signs do not look good for Annie.**\n\nHowever, the story isn't quite what many of us might think it is. 'Annie, are you okay?' was a key line taught when training people to perform CPR, as one of the most common models of CPR dummy was known as Resusci Anne, or Annie for short \u2013 an unusually realistic model of dummy with a face based on the death mask of an unknown woman found drowned in the Seine in the late 1880s. So what Jackson, M is setting out here is just a particularly elaborate scenario in CPR training.\n\nHowever, if Annie is relying on CPR to survive, she is unlikely to be okay: the overall survival rate of people receiving CPR in hospital is just 10.6 per cent, and that falls still lower for those who need CPR outside.*\n\nTo give Annie the best chance of survival, the chest compressions should be administered to her as near to one hundred beats per minute as possible. One way to try to keep this beat is to have the song 'Stayin' Alive' in mind, as it has a tempo of 103bpm. Closer still, though, is the Queen song 'Another One Bites the Dust' \u2013 though anyone administering CPR to this tune is strongly advised not to sing it out loud, for fear of giving the wrong impression to onlookers and the casualty.\nHOW MANY HONEYS MAKE THEIR MONEY?\n\nDESTINY'S CHILD\n\n**At multiple points during this socio-economic rallying cry, Knowles, B, Rowland, K and Williams, M call on all the women who make their own money to throw up their hands \u2013 which naturally raises the question as to how many people in any given population would be eligible to wave their hands in response.**\n\nThe proportion of women who work and so earn at least some income has increased dramatically in the last few generations: in 1971 just 53 per cent of women in the UK were in any kind of employment (versus 92 per cent of men), while in the most recent UK statistics that figure had jumped to an all-time high of 71 per cent (versus 80 per cent for men).\n\nThis would suggest almost three-quarters of women in the audience could throw their hands up. However, the song has a stronger suggestion: that these women are in a relationship, but are financially independent of their partner \u2013 but data suggests that a situation in which a woman exactly equals or out-earns her husband is much rarer.\n\nResearch for the USA's National Bureau of Economic Research found that US couples were much less likely to match if the woman out-earned the man, and that marriage rates declined in areas where women out-earned men \u2013 meaning that for some reason richer women and poorer men do not generally want to marry. Worse still for high-earning women, marriages where the woman earned more were found to be less happy, and more likely to end in divorce.\n\nIn most marriages, it seems, the man still serves as the main breadwinner. One notable exception, of course, is Beyonc\u00e9 herself: despite her husband Jay-Z being richer overall, she out-earns him by more than two dollars for each dollar he brings in: she netted $105m in 2017, versus Jay-Z's measly $42m.\n\nIS SHE REALLY GOING OUT WITH HIM?\n\nJOE JACKSON\n\n**Jackson, J is determined to know the relationship status of people around him in this famous 1979 study and seems unwilling to simply ask any of the people concerned whether or not they will be going home together.**\n\nEarly in the song he even notes that he has heard that one of the women whose relationship he is questioning is married or engaged 'or something' \u2013 but as the study predates online search engines and social media sites he was, of course, unable to use these resources to confirm the information. This study therefore offers us a nice opportunity to consider other ways of answering the central question.\n\nJackson could attempt to work out the relationships by looking for body language cues, such as hand-holding or kissing, or by watching for signs of what psychologists call 'emotional contagion', as people begin to mirror the speech patterns of someone they spend a lot of time with. Or he could simply look to see whether or not the people concerned are wearing wedding rings, but he seems to be eager for a higher level of certainty than that.\n\nHappily, modern science can help. A 2017 study published in the American Society for Microbiology's journal, mSystems, revealed that by analyzing the microbiome \u2013 the ecosystem of bacteria \u2013 on people's skin, cohabiting couples can be identified by a computer algorithm (which had access to no other information) with 86 per cent accuracy. The microbiome is affected by a number of factors around the home: such as whether the couple own pets, what skin products they use, alcohol consumption, and more.\n\nThe most useful skin swabs for matches are those taken from the soles of the feet. So for Jackson to get a relatively reliable answer to his question, he need only obtain feet swabs from the couples concerned \u2013 surely a simple undertaking, if he is as interested as he professes to be.\nWILL THIS BE THE DAY THAT YOU DIE?\n\nDON MCLEAN\n\n**In his tribute to Buddy Holly, McLean, D quotes men singing this will be the day they die.**\n\nCalculating the chances that they are right is a complex task: it depends on our sex, our age, our health and what country we're in.\n\nThe chance of dying in any given year changes dramatically over the course of our lifetime: at the age of 20 we have around a one in 1,000 chance of dying that year, whereas by eighty those odds have risen to a much more worrying one in ten. However, the men were being far more specific than that \u2013 they're claiming they'll die today. Working that out gets far trickier, as it hugely depends on what you're doing on that day: a day in bed will be much less risky than a day motorbiking to your job down a coal mine.\n\nStatistics experts David Spiegelhalter and Michael Blastland came up with a novel way of measuring this risk \u2013 the 'micromort', a measure for a one-in-a-million chance of death. Taking a single ecstasy pill is roughly one micromort, as there is a one in a million chance of it killing you. Heroin, by contrast, would be 377 micromorts. Travelling a mile on a motorbike is one micromort, the same as 7,500 miles by commercial flight.\n\nWe know that the three old men were drinking 'whiskey and rye', presumably near the dry levee. The fact they were singing would suggest they had been drinking for a while. The micromort calculations of drinking by a river are not available, but we do know that each double whiskey the ole boys knocked back would reduce their life expectancy by around nine minutes.\n\nBut what are the bigger risks that would significantly increase the chances of this being the day you die? One huge risk is childbirth \u2013 at 2,100 micromorts \u2013 but the highest risk Spiegelhalter and Blastland mention is flying on a bombing raid during the Second World War: 25,000 micromorts per trip, easily enough to give the song sung by 'the good ole boys' an unnerving chance of being correct.\n\nCAN YOU FEEL THE LOVE TONIGHT?\n\nELTON JOHN AND TIM RICE\n\n**Most humans would struggle to articulate any specific way they could feel love among the population in general on any given night: unlike many species, we do not have a specific mating season, and our conscious brains can't detect love in other people \u2013 we may pick up on signs of it, but we are only really aware of it when we feel it ourselves.**\n\nHowever, it can easily be argued that in this study, John, E and Rice, T are talking about lions, and in this case there is an argument that big cats are capable of feeling love in the air. The reason comes down to pheromones.*\n\nTo mark territory, lions and other big cats spray urine that contains more and different pheromones to regular urine. That mix changes over time, and includes signals as to whether or not the animal is looking to reproduce. Thanks to some clever quirks of biology that make the scents linger for a long time, potential sexual partners can feel the love in the air not just for a night, but for much longer periods than that. This is what the study's other contributors, Messrs Timon and Pumbaa, may be referring to as the 'romantic atmosphere' and the 'disaster' in the air.\n\nThis can work against the animals concerned, though. It was observed as early as the nineteenth century that the sensitive antennae of moths are capable of detecting the chemicals emitted by a would-be mate from a great distance. That is now turned against them: when farmers want to rid themselves of moths as a pest, they release female pheromones to attract them by the thousands to traps. Animals may be able to feel love in the air \u2013 but it opens them up to industrial-scale honey traps.\nCALL ME, MAYBE?\n\nCARLY RAE JEPSEN\n\n**At first glance it is difficult to see why Jepsen, C R believes what she's asking is in any way 'crazy': she has met someone she's attracted to, passed over her number and is inviting him to call her \u2013 a routine event on nights out for decades.**\n\nIndeed, it is tempting to see this \u2013 along with Jepsen's use of the classical rhetorical technique _sprezzatura_ , in which a studied nonchalance is used to mask your true feelings, with her addition of 'maybe' \u2013 as an attempt to muddy the waters of how much this exchange means to her.\n\nIt's only when we start to think about the phone-use habits of millennials that we understand why what Jepsen is doing seems crazy to her: she is inviting a call from someone she doesn't know, whereas most millennials would expect a text message, or a WhatsApp.\n\nOfficial research by Ofcom, the UK's phone regulator, shows that only 15 per cent of 16-to 24-year-olds consider phone calls the most important form of communication, versus 36 per cent who think instant messaging takes the top spot and making phone calls is something of a niche form of communication.\n\nIndeed, there's a surge of people who barely use their phone as a phone at all. A study by Ipsos MORI found that in 2012 just 4 per cent of people who owned a mobile phone made less than one phone call a week from it. Within just three years this had climbed to 25 per cent.\n\nWERE THE BOYS OF THE NYPD CHOIR SINGING 'GALWAY BAY'?\n\nTHE POGUES\n\n**Since its creation in 1988, this exploration of the role of institutional vocal groups in New York has become a benchmark for our understanding of Christmas.**\n\nIt describes a man singing from the drunk tank \u2013 a police lock-up for people too drunk to remain un-arrested \u2013 to a lover he's dragged down to his level, about a time when things were better for the two of them.\n\nThe singer recollects the music playing as the two of them met \u2013 the song 'Galway Bay', about the scenic bay on the west coast of Ireland. There are two versions of the song, one popular in the bay itself (oddly, this version is the only one to mention an American state), and one popular in the USA.\n\nThe song was, according to the song's narrator, being sung by the men of the NYPD's choir \u2013 which would seem logical, given the historic connections between New York's Irish community and its police force. However, the NYPD did not in 1987, when the song was released, and does not in the present day have a voice choir. The nearest it comes is a collection of pipers and drummers, who play as part of its Emerald Society (and who feature in the music video).\n\nGiven the singer's inebriated status, it would be unsurprising if he had misremembered a detail of when he met the love of his life, but as the two of them are unlikely to share the same false memory of a song, it raises an unanswerable question: who was singing 'Galway Bay'?\nIS MONEY THE ROOT OF ALL EVIL TODAY?\n\nPINK FLOYD\n\n**In their assessment of the state of the capitalist global economy, Waters, R, Gilmour, D, et al., debate whether it can be accurately stated that money \u2013 presumably in the form of fiat currency \u2013 is the source of societal ills.**\n\nWaters carefully caveats his assertion in his thesis, appending the allegation against money with 'so they say', a frustratingly vague manner of attribution. It is possible Waters has been misled into inaccurately citing the bible, which notes that 'love of money', not money itself, is the root of evil.\n\nEmpirically, we can assess that if Waters et al. regard substantial wealth inequality as an 'evil' then they will find themselves with a strong evidence basis to support their claims. Research published in January 2018 by Oxfam found that the world's richest 42 people owned as much wealth as its poorest 3.7 billion, and that the global 1 per cent have 82 per cent of the world's wealth.\n\nDefenders of the capitalist system, and of money, could note that barter economies have huge inefficiencies and transaction costs \u2013 if you're trying to swap a tool for food you may need to do numerous trades to get there. They may also note that between 1987 and 2013 the number of people living in extreme poverty across the globe dropped from 35 per cent to less than 11 per cent.\n\nWaters et al. may be more concerned with the other end of the distribution, though. In their discourse, they note with regret that while they are in the 'high-fidelity first-class travelling set', they feel they need \u2013 but lack \u2013 a Learjet.\n\nWHY DON'T WE DO IT IN THE ROAD?\n\nTHE BEATLES\n\n**Lennon, J and McCartney, P receive near-universal acclaim as two of the greatest thinkers in human history, and so this study's simplicity \u2013 13 of its 16 lines are simply a repetition of its titular question \u2013 can only be explained by assuming that one or both men really, really wanted to have sex in the road.**\n\nThe obvious answer to this question would simply be a fear of getting hit by a vehicle, something which would surely put a dampener on even the best intercourse. Data lets us work out whether or not that fear is well-founded: a 2017 survey shows the average UK adult says a sex session lasts for nineteen minutes \u2013 ten minutes of foreplay, and nine minutes of intercourse. We will assume that in the interests of road safety that couples in this case will be willing to forego the foreplay.\n\nThe chances of surviving a nine-minute sex session in a road will naturally depend on the type of road. Statistics from the Department of Transport allow us to work this out in detail. The Beatles should definitely rule out motorways \u2013 the typical stretch of motorway will see 81,000 cars pass every twenty-four hours, meaning a car would pass every 1.1 seconds. This would suggest a couple could expect to be run over nearly 540 times in a typical session.\n\nThe safest option \u2013 and one that tallies with The Beatles' promise that no one would see the couple doing it \u2013 would be to use a minor rural road, which is the least likely to be overlooked and the quietest in traffic terms. This would still be wildly unsafe, though: the average rural road has one car pass every ninety seconds, suggesting the couple would still be risking being run over six times. There are, it seems, very good reasons not to do it in the road. Also, gravel.\nDO THE DRUGS WORK?\n\nTHE VERVE\n\n**Ashcroft, R has clearly come to his own conclusion on whether the drugs work, decisively ruling that they do not, but it's not clear from the study alone what drugs are being assessed, or by what criteria they are being judged.**\n\nMany people believe the song is not, in fact, about recreational drug use, but instead refers to a dying cancer patient (at various points rumoured to be Ashcroft's father-in-law). Though in that instance the drugs didn't work, they do work for many: a huge meta-analysis of 100,000 breast cancer patients found chemotherapy reduced ten-year mortality \u2013 the proportion of people who die within ten years of diagnosis \u2013 by a third, a huge success rate.\n\nHowever, the song has in reality been confirmed by Ashcroft to be about recreational use of illegal drugs; indeed the use of the phrase 'coming down' in reference to his own state of mind would suggest psychoactive stimulants being cleared from his system, resulting in dysphoria.\n\nIt is hard to gain data on the general efficacy of illegal drugs. And while they can usually be said to 'work' in that they have the effects people buy them for, there are numerous reasons Ashcroft can use to justify his claim. Firstly, many drugs are either cut with chalk or even harmful substances, or sometimes replaced by them entirely \u2013 rendering them ineffective. Or, as Ashcroft suggests, the drug's negative effects quickly come to outweigh the good ones \u2013 though in this case Ashcroft would have been more lyrically and scientifically accurate if he had written, 'The drugs might work \/ But their adverse \/ Effects far outweigh the benefits'.\n\nARE THERE NINE MILLION BICYCLES IN BEIJING?\n\nKATIE MELUA\n\n**When Melua, K asserted the number of bikes in her 2005 study, she surely cannot have been aware of the controversies it would cause in nerd communities \u2013 of which more later.**\n\nThe titular claim is based on a longstanding belief, something of an urban myth, that there are nine million bicycles in China's capital, Beijing. There is, in reality, no justification for the statistic, though it is plausible. Beijing is a city known for its bikes, and the city has a population of 22 million, making a total of 9 million cycles possible \u2013 but vanishingly unlikely. Chinese government statistics suggest 14 per cent of people in the city frequently commute by bicycle, suggesting a total of around 2.1 million bicycles in regular use. The government is trying to increase this to 18 per cent to tackle air pollution, which could add another 600,000 or so bikes to the city's roads \u2013 but still far short of Melua's far more ambitious calculation.\n\nIt was not the bicycles, though, which ignited controversy in the wake of the song's release.\n\nMelua also asserts during its course that Earth is around 12 billion light years from the edge of the universe \u2013 but she calls this a 'guess', which no one can confirm the truth of.\n\nThis sparked the ire of theoretical physicist and science communicator Simon Singh, who defended his field by noting that the estimate of the Earth's distance from the edge of the universe is based on substantial scientific evidence. Eventually, Melua agreed to re-record a version of the lyric with phrasing that suited Singh better: 'We are 13.7 billion light years from the edge of the observable universe,' runs Singh's suggested lyric. 'That's a good estimate with well-defined error bars.' It is perhaps unsurprising Singh is a physicist rather than a lyricist.\n\nDO YOU KNOW THE WAY TO SAN JOS\u00c9?\n\nDIONNE WARWICK\n\n**Warwick, D describes an interesting situation. She urgently wants to get to San Jose, where she has lots of friends, believes she can find peace of mind, enjoys its space, and thinks she will easily find somewhere to stay. But despite knowing many people there, and even having been born and raised in the city, Warwick has found herself lost, and unable to find her way to it \u2013 despite it being California's third-largest city.**\n\nWe are forced to consider the interesting question of what it means to 'know' the way somewhere. In a pre-technological age, it would be measured by someone's ability to find their way there, or indeed, describe how someone else might get there. However, in a modern world in which every smartphone is a map with GPS, this becomes more difficult to assess. But if we define 'knowing the way' as relating only to knowledge in our brains, and not including any devices, then Warwick, D will probably be disappointed.\n\nEven if the passers-by she flags down know the way to San Jos\u00e9 she is unlikely to be able to get comprehensible directions from them. This is due to a psychological effect known as the 'curse of knowledge': we become able to get to a well-known destination on mental autopilot, and so we forget the conscious cues that remind us of how to get there \u2013 how to tell which of the four junctions is the one to make the left turn at, or even how many junctions there are before the correct turning.\n\nThe more worrying factor is that because of the plasticity of the brain \u2013 for example, we know that black cab drivers who take 'the knowledge' test about London streets show an increase in 'grey matter' in their posterior hippocampus \u2013 it is not alarmist to suggest that if we don't use this part of our brain, it may eventually atrophy.\nHOW WORRIED SHOULD YOU BE IF SOMEONE'S WATCHING YOU WITH THE EYE OF THE TIGER?\n\nSURVIVOR\n\n**During the course of this investigation into survival technique, Jamison, J is clearly describing the preparation for a major confrontation with a rival \u2013 one which he takes seriously, as he repeatedly warns those within earshot that they are being watched with the 'eye of the tiger'.**\n\nJamison may be something of an unreliable narrator: he refers in one moment to the 'last known survivor' before addressing a comment to 'us all', suggesting either he is omitting essential context, or else is hazy on the concept of a 'final survivor'. Setting this issue aside, though, it is credible that a human rival could be watching with the eye of a tiger: during daylight hours, the eyesight of a tiger is roughly equivalent to the sight of a typical human.\n\nHowever, Jamison specifically refers to the song taking place at night, when humans would have far more reasons to worry about a tiger's sight: at night tigers' vision is around five or six times better than that of a human. There could be far worse scenarios than being watched with tigers' eyes, though: if an opponent had the vision of an eagle, they would have daytime sight around eight times further than a human, with the ability to focus on a rabbit at a distance of two miles.\n\nAt night, owls' eyes would be a far greater worry: owls' eyes make up around 10,000 times more of their weight than human eyes do of theirs, have extraordinary ability to use low light, and have far greater density of light receptors.\n\nBeing watched by the eye of the tiger would be unnerving \u2013 but it could be so much worse.\n\nWOULD I LIE TO YOU?\n\nCHARLES & EDDIE\n\n**In this 1992 report on the likelihood of an individual lying, it is proposed that truth can be assessed by the openness of a subject's eyes, with wide eyes posited as evidence of the truthfulness of their interactions with their 'baby'.**\n\nThough we can find no studies for the wideness of eyes and truthfulness, there has been a lot of research into whether eye movement reveals lies, with Professor Richard Wiseman of the University of Hertfordshire finding no relationship between lying and eye movements. In fact, verbal hesitations and excessive hand gestures may be a better guide. Charles & Eddie are perhaps referring to this theory when they state that it is 'in their arms' where we might find the 'only truth', though they do not say what their arms are doing.\n\nHowever, we can look at the likelihood of Charles & Eddie lying in another way. Research in 2010 by academics at Michigan State University found that 60 per cent of respondents said they had told no lies whatsoever in the previous 24 hours, and found that half of all the lies reported in the survey of 1,000 adults had come from just 5 per cent of respondents. There is one problem with that research, however: people might lie about lying.\n\nWe learn how to lie at a very early age. Researchers at Toronto's Institute of Child Study told children not to peek at a toy behind their backs when alone (but recorded) in a room. At the age of two, only 20 per cent of children were able to lie about whether they peeked, but by age four, 90 per cent could lie about that (and lying didn't peak until the children reached the age of twelve). For humans, lying is second nature.\n\nWe continue to lie as adults: the University of Massachusetts asked 121 undergraduates to talk to a stranger and try to appear either likeable or competent, and secretly recorded them. They then asked the students to identify if they'd lied at all during the conversation. To even their own surprise, more than 60 per cent of respondents found they'd lied at least once, often without even being aware of it.\n\nThis spells a problem for Charles & Eddie: not only might we be unable to tell if they're lying to us, they might not even know it themselves.\nHOW MUCH WOULD YOU HAVE TO EARN TO BE 'BARELY GETTIN' BY' IN LA IN 1980?\n\nDOLLY PARTON\n\n**Parton, D makes a vital contribution to our understanding of the psychology and economics of low-quality work. While endless column inches are filled with stories about how millennials want more from the workplace than previous generations, Parton was ahead of the curve. In 1980 she vividly documents nine-to-five jobs which didn't give due credit, took without giving and shattered dreams.**\n\nShe also flagged the roles paid only just enough to 'get by' \u2013 but didn't specify what level of income that was. Fortunately, we can calculate it: during this period, Parton was based in California, which at the time had a minimum wage of $3.10 an hour. Assuming she worked a typical five-day week with a daily unpaid lunch hour, that would give her a weekly income of $108.50, and annual earnings of $5,642 pre-tax.\n\nThe US has historically set much stricter definitions of poverty than other countries, and Parton's 35-hour week would see her comfortably above its 1980 threshold of $3,400 for a single person. If we wanted a more generous definition of 'barely getting by' \u2013 such as the international threshold of 60 per cent of average earnings \u2013 Parton would fall far short, as this line would be around $7,500.\n\nTo earn that \u2013 a rough approximation of getting by \u2013 she'd have to have her wages hiked to $4.10 or so, which seems unlikely under her current line-management structure. However, it would probably provoke envy in many low-earning US families today to imagine you could get by on just one job. Recent studies suggest that as many as 7.6 million workers, or 2.5 per cent of the country, are working more than one job.\n\nAs it stands, Parton is doing far better than that: her net worth is today approximately $500 million. And so too are others. In 1985, when Parton was seeking support for her theme park Dollywood from the Pigeon Forge City Commission in East Tennessee, a spokesperson guaranteed that all staff would be paid above minimum wage.\n\nWHAT BECOMES OF THE BROKEN HEARTED?\n\nJIMMY RUFFIN\n\n**Ruffin, J changed cardiology forever in 1966 with his deep examination of what happened to those who survive coronary episodes.**\n\nRuffin also shows some enlightenment as to the difficulty of improving such survival rates, singing of searching for solutions but not succeeding, and worrying that all remains are unhappy endings.\n\nRuffin's concerns are matched by the clinical evidence into what becomes of the broken hearted, which we take to mean those who have suffered heart failure. Researchers collected evidence of the outcomes of 54,313 people in the UK who'd had heart attacks over the age of forty-five, and looked at their one-year, five-year and ten-year survival rates.\n\nThe study found that 81.3 per cent of the people studied were still alive after one year, 51.5 per cent were still going five years later and 29.5 per cent \u2013 less than one in three \u2013 were still alive a decade after their first heart failure. Worse still, those survival rates had not improved over the fourteen-year period of the analysis.\n\nIt should be noted, though, that these figures looked much better for the relatively young patients in the group, more than 80 per cent of whom survive five years, and more than two-thirds of whom were still alive after a decade. Ruffin's study courageously drew the public's attention to these serious issues, so if \u2013 as he stated \u2013 he is looking for peace of mind, perhaps he can find some there.\nWHO'S TO BLAME: THE SUNSHINE, MOONLIGHT, GOOD TIMES OR BOOGIE?\n\nTHE JACKSONS\n\n**This 1978 report explored the impact of boogie on a variety of issues, including the lack of 'lovin' and the author's reduced ability to control their feet.**\n\nIf boogie were entitled to a good defence lawyer, it would have a great chance of getting this inquisition called as a mistrial: from the very outset of the song \u2013 even from its very title \u2013 The Jacksons have decided what they are going to blame for the problems affecting them.\n\nHowever, one might find it an unusual use of everyone's time to try to assess whether dancing is to blame for people dancing.\n\nProper examination of the suspects The Jacksons list suggests they should not have jumped so rapidly to conclusions. While moonlight is largely blameless \u2013 though poor light at night-time is not without hazards \u2013 good times and sunshine both come with major risks.\n\nThe USA's CDC estimates that good times, or at least the binge drinking that comes with them, cost the United States around $191 billion each year. The World Health Organization, meanwhile, estimates that the sun kills 60,000 people per year. That's some serious prior form that The Jacksons disregard entirely \u2013 all while boogying seems largely blameless.\n\nThe modern origins of the word boogie can be found in nineteenth-century jazz culture, and before that 'boogie-woogie' was originally used to describe the symptoms of secondary syphilis in slave slang. Boogie meant a slave, and boogie man went on to become a generic title for a scary individual, possibly linked to the idea of escaped slaves hiding. That usage can be traced back to the French word _bougre_ , which, roughly translated, means a 'guy' or 'chap', from which we get the word bugger, and before that _boulgre_ , which referred to a sect of eleventh-century Bulgarian heretics.\n\nIn its modern form, boogying is good exercise: a study at Harvard University suggests a person of average weight would burn around 205 calories in 30 minutes of disco dancing, versus just 112 for a waltz or foxtrot. Perhaps this is why The Jacksons called for a last-minute reprieve of the boogie: deciding that we should only blame ourselves.\n\nWHAT ABOUT ELEPHANTS \u2013 HAVE WE LOST THEIR TRUST?\n\nMICHAEL JACKSON\n\n**In his plea for us to have greater levels of care for the world, Jackson, M poses a question that biologists have sought to work out for over two decades: have we lost elephants' trust?**\n\nElephants certainly have good reasons to mistrust humanity: a study published in the journal of the USA's National Academy found that poachers killed one in twelve of all living African elephants in 2011 alone, and have killed two-thirds of the elephants in central Africa in the course of a decade. Across the continent, researchers estimated poachers had killed no fewer than 100,000 elephants in just three years.\n\nGiven that track record, elephants would have good reason to be wary, if they're bright enough to pick up on the warning signs. Research on forty-seven elephants by two academics at the University of Sussex found that elephants can distinguish between different tribes of people and respond differently depending on whether that tribe does or does not hunt elephants.\n\nNot only did the elephants distinguish between tribes by smell and by clothing, they can distinguish language: when the animals were played calm recordings from Maasai tribesmen (who sometimes kill elephants) they panicked and huddled, and when they heard recordings from Kamba tribesmen (who very rarely kill elephants), they did not.\n\nThat suggests Jackson was right to question if we have lost elephants' trust. At least some of us have, but given what's happening to their populations, they should perhaps trust us even less than they do.\nAM I A CREEP?\n\nRADIOHEAD\n\n**The author of this 1993 case study, Yorke, T proceeds from the premise that he is a creep.**\n\nIt is our assertion, though, that he reaches this conclusion with very little empirical evidence and merely creates two false categories: 'creep' and 'f---in' special' into one of which he must fall. There is evidence that someone's skin making you cry could be construed as creepy but it's certainly not conclusive.\n\nA study published in the journal _New Ideas in Psychology_ in 2016 \u2013 twenty-four years after the song's release \u2013 declared itself the first empirical study of 'creepiness'. The study, which involved 1,341 people across the world, found that overall men were seen as more likely to be creepy than women, and that (unsurprisingly) women associated sexual threat \u2013 the risk of being unsafe \u2013 with creepiness.\n\nThe research did also come up with some specific jobs and behaviours that correlated with creepiness. The four creepiest professions were clown, taxidermist, sex shop owner and funeral director \u2013 while farmers, teachers and meteorologists were at the bottom of the scale.\n\nWhen it comes to appearances associated with creepiness, unkempt hair, pale skin, bags under the eyes and 'odd' or dirty clothing all registered, while types of behaviour deemed creepy included steering the conversation to one topic, standing too close and licking lips.\n\nThe research provides some good clues for Yorke's character in the case study: with the right occupation, dirty clothing and inappropriate social behaviour, he can be the creep he believes himself to be. Or if not, clean clothes and hair, standing further away and becoming a farmer could address the issue in no time.\n\nDID WE USED TO KNOW WHITE CHRISTMASES?\n\nBING CROSBY\n\n**Crosby, B made his seminal contribution to festive meteorology in 1942, in a song in which he implies that the frequency of white Christmases have declined since he was an adult.**\n\nCrosby, B had good reason to be nostalgic for white Christmases. Having grown up in Spokane, in Washington State, more than two-thirds of his childhood Christmases would have seen snow, giving him good reason to fondly miss such weather conditions \u2013 though given he decided to live in California as an adult, he should perhaps not have been surprised to see fewer snowstorms.\n\nWhether Crosby was justified in spreading his nostalgia to the wider world, however, is more questionable: despite our treacherous memories, many of us in the US and UK alike will have grown up with fewer white Christmases than we might think.\n\nWhile in states like Alaska and Minnesota the chance of a white Christmas on any given year is 90 per cent or more, much of the continental US experiences a very different story.\n\nDespite its famed association with the festive season, New York City has just one in five white Christmases, Philadelphia just one in ten, and Hawaii has, unsurprisingly, never experienced one.\n\nThis is similarly true in the UK: southern England has experienced just ten white Christmases in the last fifty-seven years, versus thirty out of fifty-seven in northern Scotland. The UK is also hampered by an odd official definition of 'White Christmas': the Meteorological Office defines a Christmas as white if its sensors pick up a single snowflake, even if it doesn't settle on the ground \u2013 meaning 2015 counted as a white Christmas, despite no settled snow anywhere in the country. This may satisfy a formal definition, but there is a reason why Crosby did not wish our Christmases to be merry, bright and 'white, but only on a technicality'.\nARE YOU TELLING ME THIS IS A SIGN?\n\nSNOOP DOGG\n\n**The two principal contributors to this study, Messrs Timberlake, J and Dogg, S disagree \u2013 some might say to a problematic extent \u2013 about whether or not a particular woman is flirting with one or both of them.**\n\nTheir primary method of assessment is whether or not she's looking in their eyes. That the two of them find it difficult to detect whether or not the unnamed woman is flirting, though, should not come as a surprise: humans are terrible at detecting when someone is flirting with us.\n\nOne attempt to try to establish how good we are at perceiving flirting was published in the journal _Communication Research_ in 2014. The academics matched up heterosexual strangers into fifty-two pairs, and asked them to talk for around ten to twelve minutes. Each participant was then asked afterwards whether or not they had been flirting (meaning it relied on self-reporting), and whether they thought their partner was flirting.\n\nThe results suggested that we are generally quite good at detecting when someone is not flirting with us: cases where someone thought their conversational partner was flirting when they were not were rare. However, that accuracy dropped markedly when the other person was indeed flirting: this was often missed.\n\nThe pattern held true when other people were watching the recorded interactions \u2013 people could generally tell when flirting wasn't taking place, but often missed when it was. However, there is some reassurance for Dogg, S and Timberlake, J: generally, participants found it easier to tell when a woman was flirting than when a man was.\n\nWHAT COMPARES TO YOU?\n\nSINEAD O'CONNOR\n\n**O'Connor, S builds upon original research by Rogers Nelson, P in this investigation into the comparative value of an individual over time.**\n\nThis was a tightly measured survey, taking place fifteen days and seven hours after an unnamed individual has left. O'Connor's thesis \u2013 that her former beau was incomparable \u2013 does not stand up well to scrutiny, however, even within her own internal logic, as she refers to avoiding other boys as they'd remind her of him, suggesting that they were indeed comparable. It is implied that her former lover stopped her from eating her dinner in fancy restaurants, and we're told he planted flowers in the backyard \u2013 two things that are unlikely to be impossible to find in another individual.\n\nMore broadly, in literature lovers have been compared to innumerable things, perhaps most famously to 'a summer's day' by Shakespeare (even if the bard then decided the comparison was a poor one as the lover to whom it was addressed would last far longer than a summery day).\n\nIn scientific terms, we are especially comparable: most humans are 99.9 per cent genetically similar to one another, as well as 96 per cent comparable with a chimpanzee, 90 per cent with a cat, 85 per cent with a mouse and even 61 per cent with a fly. So we are quite comparable even to things that seem very different from us.\n\nThis gets even more striking when humans are considered on a chemical level: 96 per cent of our body mass is made up of oxygen, carbon, hydrogen and nitrogen. That's the same chemicals that make up (in different mixtures) many of the things around us \u2013 the air, the sea, and much else. Biologically and scientifically, almost everything compares to you. Perhaps the only thing that cannot be compared to a human is the vacuum of space, or 'nothing', although even then both us and nothing are bathing in the cosmic background microwave radiation from the Big Bang so are technically comparable, leaving us with the real answer that even nothing compares to you.\nDO YOU LIKE PI\u00d1A COLADAS, AND GETTING CAUGHT IN THE RAIN?\n\nRUPERT HOLMES\n\n**In this song, Holmes, R sets out the story of how he planned to cheat on his girlfriend with a woman who later turns out in fact to be his girlfriend, because he had (accidentally) seen a very specific advert she had placed in the newspaper's personal ads column, and he felt it matched him perfectly.**\n\nHolmes recounts the advert's requirements in the song's first chorus: the man being sought by the then-unknown woman must: like pi\u00f1a coladas, like being rained on, dislike yoga, be relatively intelligent ('have half a brain') and like making love on a beach.\n\nThanks to brand new research by YouGov carried out on 1,641 UK adults for this book, we can calculate how many matches she could expect from her advert: 34 per cent of men like pi\u00f1a coladas, 27 per cent like getting caught in rain, 81 per cent aren't interested in yoga, 80 per cent think of themselves as intelligent, and 34 per cent say they enjoy sex on beaches. That means overall around 2 per cent of men would match all five requirements on the advert \u2013 meaning even in a small town like Northwich, Cheshire (Holmes' hometown), she could expect 753 potential matches. As such, Holmes should count himself lucky that he saw the personal advert before any of his 752 would-be rivals did.\n\nARE YOU LONESOME TONIGHT?\n\nELVIS PRESLEY\n\n**Even a casual observer of Presley, E's apparently solicitous concern about the loneliness or otherwise of the object of his treatise would notice that it is laced with self-interest.**\n\nDespite initial appearances, the real aim of Presley's enquiries rapidly becomes clear: he wants to know whether he is being missed, whether the house seems empty, whether his correspondent is still in love with him.\n\nSurvey data published in _Glamour_ magazine would offer some limited comfort to Presley, and perhaps save him the effort of wondering so extensively. These figures suggested that Presley's paramour may have been thinking of him, as 54 per cent of people said they had had second thoughts about a breakup, but while people may have had regrets, not many wanted to act on them. Only 22 per cent said they had found it difficult to move on from an ex, and only 19 per cent said they found it difficult to move on once their ex had a new partner.\n\nWhile there is a statistically even chance that Presley's former partner would be feeling lonesome, there is virtually zero chance that they would be utterly alone \u2013 though the company may not be human.\n\nAlmost every adult across the world plays host to at least one or two breeds of other creatures. Most of us would hope not to provide accommodation to fleas or ticks, and thankfully most of us don't. What almost all of us host are two breeds of mites called demodex. The biggest of the two is known as demodex folliculorum, which are around half a millimetre long and live primarily in the hair follicles on your face \u2013 in your eyelashes, cheeks, chin and more, feeding off sweat and dead skin in two-weekly cycles. We have almost none of these while we're children (to the age of ten), while almost every adult will eventually get them \u2013 up to five per hair follicle in healthy people.\n\nMeanwhile, we're also outnumbered in our bodies by bacteria: we have around 30 trillion cells in our body, but play host to 39 trillion bacteria. So while we might feel lonesome, miss human company and conversation, we're never truly alone.\nWHAT DOES THE FOX SAY?\n\nYLVIS\n\n**In 2014, four relatively unknown contributors known as Ylvis attempted to answer the question of what the fox says.**\n\nThe first verse of their song displays a reasonable knowledge of the animal kingdom, accurately identifying the noises made by dogs, cats, birds, cows, frogs and more \u2013 though their assertion that fish go 'blub' is problematic. However, things appear to take a wrong turn when they attempt to answer the central question of what foxes say.\n\nThe singers offer up a range of options, from 'ring-ding-ding', to 'wa-pa-pa', 'cha cha cha' and 'fraka-kaka'. None of the options, as anyone who lives in a city with a decent population of urban foxes will tell you, is remotely accurate: foxes don't say all that much, except in January \u2013 and at that point they are loud.\n\nFoxes have their mating season around January, and during this relatively short period of the year they will wake up people during the night as their courtship rituals are distinctive. As a male is making his approach, the female will let out a number of loud, almost humanlike shrieks (which occasionally result in calls to police).\n\nOnce mating begins, the genitals of the male and female lock together, for between twenty minutes and one hour (because fox sperm swims slowly), making it physically impossible for the animals to separate from each other. During this time the female makes a lengthy, almost unworldly shriek not easily conveyed by text. Researchers believe the sex to be non-painful, but the noise appears anything but: it sounds entirely possible that the vixen at least is saying 'ouch, my cervix'.\n\n**ISN'T IT IRONIC?** | **IRONIC** | **NOT IRONIC**\n\n---|---|---\n\nBeing afraid of flying, overcoming it, and dying in a crash |\n\n|\n\n10,000 spoons when a knife is needed | |\n\nMeeting a dream man who's married | |\n\nWinning the lottery and immediately dying | |\n\nA fly in your wine | |\n\nDeath row pardons arriving late | |\n\nRain at a wedding | |\n\nBeing offered a free ride after paying | |\n\n**TOTAL** |   |  \nISN'T IT IRONIC, DON'T YOU THINK?\n\nALANIS MORISSETTE\n\n**In this landmark 1995 semantic survey, Morissette, A conducts a thorough analysis of the concept of irony, typically defined as circumstances working opposite to how they would be expected to, usually in such a way as to provoke amusement.**\n\nMorissette's study has proven controversial for decades as, despite offering numerous examples of irony to explain the concept to her audience, most or all of them fail, in reality, to be ironic. Rain on a wedding day is inconvenient but not ironic, unless both bride and groom are meteorologists specializing in short-term precipitation forecasting. Finding a surplus of thousands of spoons while searching for a single knife is irritating, but again not an irony, unless you are a cutlery spokesperson looking for a knife to open the envelope of your newly delivered forecast of high-knife versus low-spoon yield in the current quarter. Falling in love with a man only to discover he's married is a fairly common occurrence. A death row pardon arriving after the execution has begun is tragic, but again fails to meet the threshold for irony, unless perhaps you are on death row for murdering the founder of a faster courier service for death row pardons.\n\nThis longstanding controversy actually provoked Morissette to offer a retraction of her study two decades later, in 2015, when she stated 'there are no ironies'. However, this retraction has an issue: the initial study did, in fact, contain one irony.\n\nMorissette set out the case study of a man afraid of flight who was finally convinced to take a plane \u2013 only for it to crash. Statistically, the man was wise to take the flight: only one in 3,000,000 flights result in a crash with fatalities, while motorbiking is 3,000 times more dangerous mile-for-mile than flying, driving is 100 times deadlier, and even trains are twice as deadly. So Morissette's example showed a man rationally losing his fear of flight, only to become the rare exception \u2013 he had been correct.\n\nThe syntax of Morissette's question is also ambiguous: 'Do not you think this is ironic?' So this does risk making her retraction incorrect. Isn't that ironic? We think. It's hard to tell at this point.\nWHAT IS LOVE?\n\nHADDAWAY\n\n**In this key investigation of 1993, Haddaway seeks to address two issues, one philosophical \u2013 what is love? \u2013 and one more immediate: he would like his baby to stop hurting him. We should tackle the philosophical problem first, as trying to describe what love is has plagued poets and writers for millennia.**\n\nA good starting point for Haddaway would be to ignore said poets. As any good social scientist knows, their approach to these kinds of definitional questions are imprecise and unhelpful. We can also discern from the context of the song, and specifically from his pleas for monogamy, that Haddaway is not referring to platonic love, or parental love, but rather romantic love. Other comparable studies involve Gibb, B, Gibb, R and Gibb M, who attempted to answer the question of how deep love is by pinpointing the exact areas of the brain where love is found, and The Black Eyed Peas, who attempted, unsuccessfully, to answer the more general question, 'where is the love?'\n\nHowever, perhaps the most helpful answer for Haddaway is that 'love' is a societal shorthand for a combination of physiological responses and sociological behaviours: it is the hormonal reaction that we feel as lust or desire, coupled with a relationship of trust and affection built over time, laced with sociological expectations such as monogamy, expectations of gender roles and more.\n\nHaving addressed Haddaway's first issue with his romantic partner, we need to consider his parental non-sequitur: if his infant is hurting him \u2013 especially at such a young age \u2013 he should seek immediate advice from a qualified child psychologist.\n\nWILL IT BE LONELY THIS CHRISTMAS?\n\nMUD\n\n**In 1974, Chapman, M and Chinn, N investigated whether it would be lonely this Christmas.**\n\nThough the study was rooted in one specific instance of a partner leaving, and particularly the attendant issues surrounding temperature \u2013 it's taken as a given that the absence of the partner will negatively impact the ability of the individual to regulate their body temperature. Leaving this spurious correlation aside, the report's authors hit on a sad social reality for thousands of people. Looking at their home country of the UK, research by Age UK in 2017 found that 928,000 older people feel lonelier around the time of Christmas, with around 370,000 of those feeling lonely in part because they had been widowed.\n\nWhile making what can only have been intended as an earnest and deliberate effort to draw attention to this plight, Mud's warning that it would be cold without a partner at Christmas can be seen as a slight exaggeration.\n\nWhile some older people do struggle with this, pensioners are now the least likely group in the UK to be in poverty, and even receive extra winter payments for heating \u2013 meaning that hopefully, despite what Mud say, they should not be cold.\n\nHowever, if Mud's words have drawn your attention to the issue and you wish to make Christmas less lonely, you can take a look at ageuk.org.uk\/get-involved in the UK or friendtofriendamerica.org in the US.\nWHY'D YOU HAVE TO GO AND MAKE THINGS SO COMPLICATED?\n\nAVRIL LAVIGNE\n\n**Lavigne, A set out in 2002 a series of concerns regarding a close associate with whom she'd previously been able to communicate clearly and simply \u2013 especially when the two were driving together \u2013 whereas now things have become 'complicated' and he is behaving oddly enough that she speculates he is acting like somebody else, exhibiting paranoia and a performative selfhood, and she is finding the whole situation extremely frustrating.**\n\nLavigne had to wait a decade for psychological research to confirm the likely cause of her situation, but in 2012 answers finally came in the form of the academic paper 'Evidence for the Pinocchio Effect: Linguistic Differences Between Lies, Deception by Omission, and Truths', which used experiments to discern the differences in language used by people telling the truth, leaving something out, or downright lying.\n\nThe study examined what it called the 'Pinocchio effect' \u2013 the tendency for liars to use more words, and so tell a more elaborate story, than people telling the truth, and found this effect existed. When matched with an unwitting partner, liars used 76 words on average, versus 32.5 for truth-tellers and just 30 for those lying by omission.\n\nLiars, the research showed, also used more third-person pronouns, more numbers and more profanities than their truth-telling counterparts, while other studies have found that liars take fractionally longer to respond to a question than a truth-teller does, because lying puts more cognitive load on the brain than telling the truth \u2013 you have to think up which lie to tell, rather than telling the one extant truth. In a very real sense, lies make things more complicated in your brain.\n\nWhy did he have to make things so complicated? The likely explanation is because he was telling lies. Still, given 40 per cent of adults admit to lying once per day, Lavigne was correct to note that 'life's like this'.\n\nWHERE ARE YOUR FRIENDS TONIGHT?\n\nLCD SOUNDSYSTEM\n\n**In this qualitative study, Murphy, J and his colleagues set out to pose \u2013 rather than answer \u2013 a question regarding the location of people's associates during an evening.**\n\nSuch data would clearly be valuable for a number of uses, including perhaps a study of the night-time economy, planning public transport requirements and even the most efficient deployment of emergency service resources.\n\nMurphy is also clearly primarily interested in the nocturnal interests of people as they get older, referring at various points to 'the ways we show our age', having a 'face like a dad', and even speaking of feeling 'finally dead' when kids are just too 'impossibly tanned'.\n\nHowever, Murphy never actually makes the attempt to gather the evidence on where the aforementioned friends are that night, leaving the window open for this author \u2013 who is now in his thirties \u2013 to attempt to further this research.\n\nAn initial attempt to quantify the whereabouts of friends on 22 September 2017 was thwarted by more than a third of those considered being in the company of LCD Soundsystem, leading to the study being dismissed due to sample bias (it heavily over-indexed on North London media workers).\n\nA repeat study was carried out some months later, polling 968 Twitter users who follow the author on where they would be at 10pm that Friday night. The results were reassuring for those who worry their friends have more fun than they do: 71 per cent of respondents said they would be 'at home' at that time, while 18 per cent said they would be at a pub or bar, and just 5 per cent would be at the cinema, theatre or a gig.\n\nWhere are your friends tonight? They're most likely by some margin to be at home.\nWHAT'S COOLER THAN BEING COOL?\n\nOUTKAST\n\n**This research collaboration by 3000, A and Patton, A has perhaps become more famous for its errata than for its findings.**\n\nNotably, the collaborators suggested it was advisable to 'shake it like a Polaroid picture', which prompted Polaroid to release a statement reminding their customers that Polaroids should not be shaken while their image develops. Sadly, due to a lack of post-publication clarification, we cannot establish whether 3000 and Patton were in fact recommending that people remain entirely still, or whether they just have a poor understanding of the chemical processes of Polaroid photography.\n\nHowever, we also wish to tackle the second research question addressed by 3000 and Patton, regarding coolness. For the average person, a temperature range that would be regarded as 'cool' but not 'cold' would be around 12\u00b0C to 15\u00b0C. This leaves a wide range of temperatures available to answer the core question.\n\n3000 and Patton offer up their own answer of 'ice cold'. While correct, we feel this solution is too narrow in scope, as water freezes at around 0\u00b0C \u2013 leaving temperatures both higher and lower that could correctly be called cooler than cool.\n\nThe lowest recorded temperature on Earth, for example, is -89.2\u00b0C, which was observed at a Soviet weather station in the Antarctic, but we can get far colder than this. Liquid nitrogen is generally at a temperature between -201\u00b0C and -196\u00b0C, whereas 'absolute zero' \u2013 the lowest possible temperature \u2013 has been calculated at -273.15\u00b0C.\n\nSo while 'ice cold' is accurate, a better answer would be 'any temperature in the range of -273.15\u00b0C to 12\u00b0C'. An even shorter answer would be 'loads of stuff'.\n\nWHY DO YOU ONLY CALL ME WHEN YOU'RE HIGH?\n\nARCTIC MONKEYS\n\n**It should perhaps come as no surprise that the team behind this study \u2013 Turner, A, et al. \u2013 are from Yorkshire, where hills and valleys coupled with a large rural population mean that phone signal is often difficult to access.**\n\nIn rural Yorkshire, the question behind this study almost answers itself: people can only make calls when they're high because there's no signal in the valley. This is a popular belief that is supported by science. FM transmitters work by something quite close to line-of-sight: if there are obstacles (such as hills or buildings) between the transmitter and the radio, the signal quality descends sharply.\n\nPhone signals are slightly more robust than this, but are also significantly affected by the distance from the transmitter and the obstacles in between. This, coupled with a reluctance from UK companies to fund phone towers in lightly populated valleys, means that signals in rural lowlands are terrible \u2013 seriously limiting their potential for phone calls.\n\nThere is one significant caveat for those wanting to make use of this advice in emergency situations, though: if you are, for example, lost in the wilderness, experts caution against seeking high ground to call for help if conditions are bad. As the Grand Canyon's chief of emergency services said to _Popular Mechanics_ magazine, if you're climbing a rock formation for better signal during a storm, 'you're just exposing yourself to risk'. Do call for help when you're high \u2013 but make sure it's safe first.\nWHERE IS 24 HOURS FROM TULSA?\n\nGENE PITNEY\n\n**Those studying infidelity in the 1960s can find a compelling case study in correspondence from Pitney, G, who informed his partner by letter that he would not be seeing her anymore because he had cheated on her while only '24 hours' from their home in Tulsa, Oklahoma.**\n\nClues as to Pitney's location at the point when he made this declaration \u2013 and when he cheated \u2013 can be found from his referral to the fact that he was 'driving home'. Given Tulsa's relatively central location within the USA, this means there are a range of areas that are exactly a 24-hour drive from his destination.\n\nHe could, for example, have got to the delightful city of Manchester in New Hampshire, high in the northeast of the country. But it is hard to see why, beyond the unlikelihood of seeing anyone he knows, Pitney, G would choose this location. He could have travelled to what has since become known as the Big Sky Colony in rural Montana, but which was then the Milford Hutterites, a US offshoot of the Hutterian Brethren, the Austrian branch of the Anabaptist movement of the sixteenth century who believed in communal living. However, this seems an unlikely source for a partner in adultery.\n\nThe most likely location, it appears, is the notorious pleasure city of Reno, Nevada \u2013 exactly 1,657 miles and a 24 hour drive from Tulsa (and also only an hour or two away from San Jos\u00e9, should he run into Warwick, D). Disaster could, perhaps, have been averted if Pitney had taken another form of transport: had he chosen to fly he could have been back in Tulsa in just five hours, with a change in the famously morally, upright Salt Lake City.\n\nGiven Pitney is now seeking to avoid retribution from his former partner, there are many ways in which he could get far further from Tulsa to somewhere remote \u2013 if he hopped to the airport, he could get 6,460 miles away to the very isolated Falkland Islands in 23 hours and 31 minutes, leaving himself 29 minutes to spare.\n\nOH CAN'T YOU SEE YOU BELONG TO ME?\n\nTHE POLICE\n\n**In this chilling public service announcement, Sumner, G issues a serious and pressing warning about the perils of modern-day slavery, especially against the backdrop of current surveillance technology.**\n\nThe limits of slavery are explicit and relate to every breath, move, step and even cruelly each bond broken (which clearly refers to the possibility of escape being thwarted). Also, every word said, every day and night, every game played, every move made, vow broken, smile faked and claim staked \u2013 the all-seeing eye of cheap, available spying technology has drastically changed the dynamic of the slave trade.\n\nDuring the course of the transatlantic slave trade, around 12.5 million people were shipped across the ocean into slavery \u2013 a figure now dwarfed by the levels of modern-day slavery, which is estimated to affect around 40 million people. Of those, around 25 million are in forced or indentured labour, with a further 15 million in forced marriages, according to the International Labour Organization.\n\nHowever, unlike during the historical slave trade, such arrangements are now illegal in many countries across the world, including in Sumner's native Britain, where in 2017, 2,255 slavery-related crimes were recorded. This conviction rate suggests that Sumner may come to regret his casual admission of owning another person, and could expect swift action against him from law enforcement. As such, the evidence leads us to one conclusion: Sumner is merely pretending to own another human being as part of an elaborate sting.\nWHAT'S THE FREQUENCY, KENNETH?\n\nREM\n\n**The effects of radio waves and electromagnetic radiation \u2013 all measured by frequency \u2013 have been the subject of debate and concern for decades: people worry their mobile phone will give them cancer, that wind turbines make them ill or that WiFi gives them migraines.**\n\nIt's perhaps unsurprising, then, that Stipe, M, dedicated time to working out a particular frequency that seemed to be causing him concern. The origins of Stipe's queries are unusual \u2013 his study grew out of an attack on the US newsreader Dan Rather in 1986. Rather was attacked on the street by a man repeatedly shouting 'Kenneth, what's the frequency?' At that point in 1994, the motives for the attack remained a complete mystery.\n\nBased on the available information it is difficult to be certain as to what frequency Stipe is trying to find on behalf of Rather's attacker, but he describes it as 'your Benzedrine'.\n\nBenzedrine is the brand name of the first pharmaceutical drug that contained amphetamines, so it's fair to assume it's a frequency that acts as a stimulant rather than a depressive.\n\nOne answer might lie in an experiment conducted in 2003 in an auditorium on London's South Bank by the National Physical Laboratory, which played an inaudible low-frequency sound alongside one of two songs across two public concerts \u2013 swapping each song between the two concerts. People at each reported feeling uneasy, uncomfortable or having chills during the track where the inaudible noise was played, all markers of side effects of a stimulant like Benzadrine. This offers up a likely answer to Stipe's query: the frequency could be 'infrasound', or 17 Hz.\n\nNot long after Stipe's inquiry, the original mystery was also solved following the arrest of Rather's attacker: a mentally ill man called William Tager. He had believed television networks were transmitting signals into his mind and wanted to know how they were doing it. Tager had fatally shot an NBC stagehand outside the Rockefeller Center in 1994. NBC broadcast on VHF channels 2\u201313 in the nineties, so, though they were not broadcasting signals into Tager's brain, they were certainly operating between 54 and 216 MHz.\n\nDO GIRLS JUST WANNA HAVE FUN?\n\nCYNDI LAUPER\n\n**Modern feminism has changed the priorities and expectations of women and girls across the planet, as noted by Lauper, C's seminal study into what girls want.**\n\nLauper notes to her mother during the course of her research that 'we're not the fortunate ones', accurately summarizing the current generation of women's reduced job security, wealth and financial stability when compared with their parents.\n\nThere is also empirical evidence supporting Lauper's assertion that girls do not simply wish to settle down. An annual survey of girls carried out by the UK organization Girlguiding in 2009 found that 56 per cent said marriage was the thing they would most like to achieve by age 30 \u2013 but just three years later the poll found girls defined success as being 'confident and independent', with only 21 per cent defining it as marriage.\n\nHowever, Lauper may have pushed her conclusions too far: girls want far more than merely to 'have fun'. The 2017 survey asked girls aged eleven to twenty-one about their priorities. The top five in the rankings were 'supporting young people in their mental health', 'stopping sexual harassment', 'more women in top jobs', 'tackling violence against women and girls', and tackling LGBT discrimination.\n\nGirls might, as Lauper cites, wanna have fun \u2013 but they also want independence, mental health services, freedom from harassment and a strong career path, too.\nWHERE DO BROKEN HEARTS GO?\n\nWHITNEY HOUSTON\n\n**Given how much time many of us spend worrying about our romantic lives, it is of no surprise that researchers such as Houston, W have tried to establish a good evidential basis for what happens when love goes awry (those looking for cardiological explanations should see 'What becomes of the broken hearted?').**\n\nOne of Houston's primary concerns is whether broken hearts can 'find their way home', a reference either to a new relationship or towards recovery. While there is no easy average time to record meeting a new partner, a study in _The Journal of Positive Psychology_ established that the average time for recovery of normal mood following a breakup is eleven weeks \u2013 though in the instance of the end of a marriage, this increases to eighteen months.\n\nThis suggests that Houston's hope that broken hearts recover are met, though not necessarily in a short period of time. Perhaps even more usefully, though, there is also data on where not to go if you want to avoid a broken heart. A survey of 2,187 people found that 1,056 had at some point broken up within six months of their first holiday with a partner \u2013 with some destinations appearing far more high risk than others. The top three pre-breakup locations were Mexico, where 21 per cent of couples who visited broke up within six months, followed by Ibiza (17 per cent) and Portugal (12 per cent).\n\nThe best holiday destinations for those wanting to avoid heartbreak were Tenerife and Italy, the survey suggested. So perhaps inversely we can propose the answer to where broken hearts 'have gone' is Mexico, Ibiza and Portugal within the last six months.\n\nWHAT HAVE THEY DONE TO THE RAIN?\n\nTHE SEARCHERS\n\n**The research group colloquially known as 'The Searchers' wondered about the effects humanity was having on rain, concerned it would not be as 'gentle' as before, fearing it could burn away grass and damage the environment.**\n\nThe primary concern of The Searchers in 1964, just two years after the Cuban Missile Crisis at the height of the Cold War,* was that nuclear fallout would be the source of alteration to rain, leading to widespread radiation damage. Just twenty-two years later, in 1986, the disaster at Chernobyl nuclear power station in what is now northern Ukraine propelled vast amounts of radioactive material into the atmosphere. Rainclouds carried this material as far as the Outer Hebrides, where ten years later doctors reported massive increases of up to 2,500 per cent (although this is a disputed figure) in incidences of many types of cancer. However, at the time of writing, the search for radioactive rain has proven to be in vain globally. And yet while they may have misdiagnosed the specific cause of the issue, The Searchers were correct to worry about rain generally.\n\nThey were less correct about their premise, however. Even standard, non-radioactive rain is not especially 'gentle': rain is naturally acidic thanks to the carbon dioxide in the atmosphere. It has a pH of around 5.6 (pH runs from 0 to 14, where anything under 7 is acidic, and anything over is alkaline).\n\nHowever, coal power plants and other sources produced sulphur dioxide, which in the atmosphere turns into a potent acid \u2013 sulphuric acid \u2013 which then rained down causing major environmental damage. Sulphuric acid in the atmosphere changes the natural pH of rain from around 5.6 to 4.0, making it around ten times more acidic than before.\n\nThis answers The Searchers' 1964 query: 'they' made the rain more acidic. Thankfully, since the 1990s, the rain has seen further developments \u2013 it is once again being made less acidic thanks to a drop in the number of coal-fired power plants. The US, for example, sees 40 per cent less acidic rain than at its peak \u2013 meaning that at present we are mainly making the rain more like its natural, fairly gently acidic state.\nSON, CAN YOU PLAY ME A MEMORY?\n\nBILLY JOEL\n\n**In this anthropological report, Joel, B presents a case study of an elderly man looking for help with his memory: he can recall something was 'sad' and 'sweet' and he knew it well when he was younger, and is hoping a piano player can help him fill in the blanks.**\n\nThere is strong research evidence available on the effect of music on auditory memory \u2013 how well we remember verbal and non-verbal information that we've picked up through sounds, rather than vision or smell. This research tells us that people who play musical instruments have better auditory memory than the average person, and professional musicians have even better auditory recollection still.\n\nHowever, a study in the _Psychonomic Bulletin & Review_ suggests that Joel's elderly man may be pursuing a poor strategy, even if he was himself a former professional musician. The study found that visual memory is so good that even if people are shown up to 10,000 images over the course of a few hours, they can recall which they have or haven't seen with 83 per cent accuracy. If shown 2,500 images they can even remember details \u2013 so, not just that they saw an apple, but which apples they did and didn't see.\n\nThe study found that even for professional musicians, auditory memory falls far short of visual memory. This does suggest a new strategy for Joel's case study, though: rather than asking Joel to play him a memory, he should ask him to show him one.\n\nWe're not quite sure what memory Joel, B is trying to retrieve, but studies suggest it will probably have come from when he was between 16 and 25: we have a bump in autobiographical memories between those ages, known as the 'reminiscence' bump. Before that, we're hit by childhood amnesia, and then memories only bump again because we're remembering things that happened more recently.\n\nWHY?\n\nANNIE LENNOX\n\n**It is the deceptively simple questions that often prove the most difficult to answer, as any parent who has been ambushed with 'why is the sky blue?' by their child will well know.**\n\nAs such, the question provoked in this essay by Lennox, A is perhaps the ultimate simple-to-ask and impossible-to-answer question: why?\n\nThankfully, Lennox provides some cues within her essay that help direct us towards a useful answer. She notes that she may be 'mad', 'blind' and also 'viciously unkind', but assures her correspondent that she can 'still read' what they are thinking. This leads us inexorably towards a simple conclusion: Lennox believes she already knows the answer to the question she is asking, even if she is not revealing it to us.\n\nHowever, we may have good reasons to question her credibility when she claims to read thoughts. A much-publicized 2011 study appeared to confirm the existence of psychic ability, or ESP, based on lab research which claimed to show students could predict whether images would appear on the left or right of a screen \u2013 apparently vindicating Lennox's claims of psychic knowledge.\n\nHowever, an extensive study of 3,289 people the year after failed to replicate the results, as did another a year later. These robust studies leave us with the concern that on this issue Lennox may be an unreliable narrator.\n\nThe most frequent answers to 'Why?' may also be well-known to parents, though: they are in turn 'because' and 'because I say so'.\nDOES EVERYBODY WANT TO RULE THE WORLD?\n\nTEARS FOR FEARS\n\n**It is clear that the authors of this essay, Orzabel, R and Smith, C have significant frustrations with the world's current governance arrangements, bemoaning 'indecision' and 'lack of vision' among numerous other issues.**\n\nHowever, the sweeping conclusion that stems from their premise \u2013 that 'everybody wants to rule the world' \u2013 is one that is not remotely supported by the evidence base.\n\nPerhaps the most powerful public office on the planet \u2013 the nearest we currently have to someone who wants to 'rule the world' \u2013 is the office of US president: but most people do not aspire for their children to ever hold the role. A roundup of different polling published by Cornell University found only 32 per cent of parents would want their son to pursue a career in politics, and only 34 per cent would like that for their daughter.\n\nWhen asked which they would rather their child be: US president, a CEO, head of a university, a sports star or a movie star, only 7 per cent chose US president \u2013 only a movie star ranked lower. Not only would many people not wish for themselves or their child to be president, they would much rather their child becomes a lawyer, doctor or police officer.\n\nSurvey data published in the _Harvard Business Review_ goes further still. Not only would most people not wish to rule the world, most people wouldn't even want their current boss's job: only 34 per cent of US workers said they would like a leadership role, and only 7 per cent wanted to be executives. Predictably, men were more likely to want to be leaders than women, though perhaps more surprisingly, African-Americans and LGBT people were more likely to want such roles.\n\nSo, despite the bold assertions of Orzabel and Smith, most people \u2013 while they might not like how the world is going \u2013 have absolutely no intention of ruling it.\n\nHOW MANY INCHES ARE IN A MILE?\n\nSELENA GOMEZ & THE SCENE\n\n**On the surface of it, Gomez, S appears to be posing a very simple question here. The answer to her question can be easily obtained through Google \u2013 it is 63,360 inches. But why would this simple piece of information be enough to make someone smile?**\n\nThe answer involves a series of archaic measures. The mile is a measure dating back to Roman times, and was at the time logical: one mile was defined as 1,000 paces, with paces standardized as five Roman feet \u2013 meaning 1 mile would be 5,000 Roman feet.\n\nA modern mile isn't 5,000 feet, however: it is 5,280 (and each foot is 12 inches long). The reasons go back to the sixteenth century, when a mile was standardized as eight furlongs. The furlong was a useful measure as it was the maximum distance that oxen could pull a plough without needing to rest, and was standardized as 40 'rods' (a surveyor's tool) or 10 'chains' (another tool), which totals 660 feet. This measure became quite an essential one to maintain as it was used in land ownership: an acre was defined as 10 chains (one furlong) by one chain (660 feet by 66 feet).\n\nSo for simplicity, miles were defined in terms of furlongs (eight of them), which left a somewhat unmemorable set of distances to remember in the modern era. By contrast, in the metric system, a kilometre is 1,000 metres, which is in turn 100 centimetres \u2013 but 64 per cent of Americans still reject the idea of switching systems. Gomez, S is clearly talking to a proponent of the metric system in the US, who, when reminded of the illogical, piecemeal way that distance is measured under that system, allows themselves a wry smile.\nHAVE GUILTY FEET GOT NO RHYTHM?\n\nWHAM\n\n**In this meditation on fidelity and anatomy, Michael, G postulates that as he has been unfaithful, so he wishes 'never' to dance again, partly based on the premise that 'guilty feet' must have no rhythm.**\n\nGiven that around 10 million people in the UK \u2013 14 per cent of the population \u2013 admit to having cheated, this hypothesis would explain a significant proportion of bad dancing at weddings.\n\nHowever, the position does not seem to be wholly clear in this regard. In 2006, academics at the University of Manchester tried to examine the effectiveness of a range of arts and dramatic remedies on UK offenders \u2013 or in other words examining whether guilty feet have rhythm, and would restoring rhythm to guilty feet help offenders?\n\nWhen it came to dance, these results were decidedly mixed: an effort to set up a dance programme in a male young offenders' institute failed \u2013 researchers could find only one dance company working with male prisoners, but they were unable to start a dance programme due to the lack of space for the project. For male prisoners, there was no chance for guilty feet to dance, rhythmically or otherwise.\n\nThe group did succeed in setting up a programme in a women's prison, which signed up thirteen prisoners, one of whom was asked to drop out for fighting. The study didn't report the quality of the dancing, but noted that the feedback found the project an 'oasis', 'source of pride' and a boost of 'self-belief': dance can, perhaps, be a positive.\n\nThis appears to have been the case for Michael, too: despite his pledges never to dance again, he did so, rhythmically, for decades until his death in 2016.\n\n**CZECH** | zmizet po anglicku | **('to leave English style')**\n\n---|---|---\n\n**FRENCH** | filer \u00e0 I'anglaise | **('to leave English style')**\n\n**GERMAN** | sich (auf) franz\u00f6sisch empfehlen, literally einen franz\u00f6sischen Abschied nehmen | **('to take a French leave')**\n\n|\n\n**or**\n\neinen polnischen Abgang machen | **('to take a Polish leave')**\n\n**HUNGARIAN** | angolosan t\u00e1vozni | **('to leave English style')**\n\n**ITALIAN** | andarsene all'inglese | **('to leave English style')**\n\n**POLISH** | wyj\u015b\u0107 po angielsku | **('to leave English style')**\n\n**ROMANIAN** | a o sterge englezeste | **('to leave English style')**\n\n**UKRAINIAN** | \u043f\u0456\u0442\u0438 \u043f\u043e-\u0430\u043d\u0433\u043b\u0456\u0439\u0441\u044c\u043a\u0438 (pity po-anhliys ky) | **('to leave English style')**\n\n**PORTUGUESE** | sa\u00edda \u00e0 francesa | **('to leave French style')**\n\n**RUSSIAN** | \u0443\u0439\u0442\u0438 \u043f\u043e-\u0430\u043d\u0433\u043b\u0438\u0439\u0441\u043a\u0438 (ujti po-anglijski) | **('to leave English style')**\n\n**SPANISH** | despedida a la francesa | **('goodbye in the French way', 'French farewell')**\n\n**WALLOON** | spiter a l'inglesse | **('to leave English style')**\n\n**ENGLISH** | Irish goodbye\n\n| \nHOW THE HELL AM I SUPPOSED TO LEAVE?\n\nUSHER\n\n**Raymond, U is seeking help for an all-too-common dilemma in modern etiquette. He is at a party and being asked to stay for another dance, but wishes to leave, and is evidently uncertain as to whether or not it's acceptable to do so, and lacks strategies to execute a quiet departure.**\n\nSuch social anxieties are common, but happily there are experts in the field able to offer solutions to Raymond's issues. Much information can be found in the handbook of Britain's aristocracy, _Tatler_ magazine. The outlet assures its readers that in a dinner party situation, it is acceptable to leave after dessert has been served \u2013 even if coffee has not yet been delivered. It stresses that for weddings it is not acceptable to leave before the speeches, but is otherwise silent on the issue of parties without dinner or nuptials. This leaves Raymond with enough etiquette leeway to leave if he so wishes.\n\nIn terms of executing such an exit, the magazine has a few further tips to offer \u2013 some perhaps more sincere than others. Raymond can attempt to leave by claiming he had lost an engagement ring, hide until a major event or speech and flee, or escape via the garden, it suggests.\n\nA more common way of leaving social situations is referred to in the UK as an 'Irish goodbye': simply leaving a party without saying goodbye to anyone, so that your departure is barely noticed. In Spain and Portugal, such an exit is known as a French farewell; in Germany it's referred to as Polish leave. But in countries including France, Italy, Hungary and Russia it's got another name: leaving English-style. So, whatever the ethnic makeup of the party, it is unlikely that Raymond, U's leaving without saying goodbye will be remarkable.\nIS THERE LIFE ON MARS?\n\nDAVID BOWIE\n\n**If there is an under-appreciated pioneer in the field of astrobiology, then it must surely be Bowie, D, the author of the groundbreaking study 'Life on Mars', published in 1971.**\n\nHowever eccentric Bowie's methodology might be, including encouraging a young girl to go unaccompanied to watch a film about sailors fighting in a dancehall and a policeman assaulting an innocent man, the hypothesis offered therein \u2013 that our nearest celestial neighbour may harbour life \u2013 was decades ahead of its time, and has since received corroboration from modern science.\n\nBowie's hypothesis received support when ice was discovered on Mars, evidencing that at least one essential requirement for life as we know it \u2013 water \u2013 was present on the planet. This briefly became more exciting still as NASA appeared to find evidence of liquid water on the planet, which would be far more likely to support life. However, in 2017 different researchers theorized the 'water' was more likely to be sand, scotching those hopes.\n\nBut in 2018, the Curiosity rover on the surface of Mars uncovered something even more interesting: the kind of organic matter that would be suitable for basic life to feed on \u2013 though it can't be determined whether it came from living matter, a chemical reaction, or even something crashing into the surface of the planet.\n\nGiven that Mars used to be warmer and thus more conducive to life, one theory is that there was life on Mars, but is no longer. However, what we do know now is that if there is life on Mars, there's something there for it to eat. Bowie was on to something.\n\nIn 2011, a team of researchers led by Dirk Schulze-Makuch, of Washington State University, published an article in the journal _Astrobiology_ in which they came up with an index for how friendly to life the planets were.\n\nWHAT'S GOING ON?\n\n4 NON BLONDES\n\n**One cannot accuse Perry, L and her collaborators of lacking in ambition when setting out their research question. Perry leads her cohorts in asking firmly and repeatedly 'what's going on?', but declines to offer any specificity to her enquiries: she is, it seems, interested in everything going on around her on the climb up the 'great big hill of hope' thatshe deems life to be.***\n\nOn any given day, then, there are millions of answers to Perry's query, some of which are provided here. Across the planet there will be around 360,000 births 'going on', while around 150,000 people will die. In the USA around 17,250 cars will crash, resulting in around 110 fatalities. More happily, around 102,465 planes will take flight across the planet, and statistically we would expect all of them to land safely.\n\nWhen it comes to creative happenings, there is no shortage there, either: around 21 feature films will be released across the planet, and 6,054 books published. Apple will sell around 594,000 iPhones. Around 60 million photos will be published on Instagram, and 500 million posts will go online on Twitter.\n\nWe suspect this will be of little interest to the enlightened readership of this tome, but estimates suggest around 20,000 pop songs would be released on that given day, too. So to answer Penny's question as much as we can: there's always lots going on.\n\nFor example, on the day of the study's release, 23 June 1993, Nigeria's military dictator, General Ibrahim Babangida, annulled the results of the elections and in so doing halted the country's return to democracy, the United Nations authorized a global oil embargo on Haiti and Lorena Gallo Bobbitt cut off her husband John Wayne Bobbitt's penis. That's enough to make anyone say 'hey'.\nWHO LET THE DOGS OUT?\n\nBAHA MEN\n\n**This anthropological exploration, published in 2000 by a sizeable research collaboration in the Bahamas, informally referred to as the Baha Men, poses the question as to who released the hounds exactly twenty times.**\n\nThe reason for this persistent curiosity becomes clear when we consider the history of domestication of dogs. Perhaps the closest genetic relative to modern dogs is the grey wolf; we are confident that dogs are domesticated descendants of those wolves. Because we believe they were domesticated initially to help with hunting \u2013 especially in woodland terrain \u2013 we can establish that whoever first let the dogs out was whoever first let the dogs in.\n\nThe answer to this question remains a mysterious one: the oldest generally agreed fossilized dog was discovered in 1914 in Germany (it was not examined until 1919, as people were dealing with other priorities at the time). However, new research suggests dogs were also independently domesticated in or around what is in modern times China \u2013 and eventually the two groups met and cross-mated. So while we don't know for sure which of those two groups initially let the dogs out, we have good reason to believe both groups did so.\n\nHowever, the Baha Men were not specific as to in which time period they were seeking to place their question, so we should also look at who is most likely to release dogs in the modern day. The answer in this instance lies with those protesting animal testing, for which thousands of dogs are used: in the US, the Rescue + Freedom Project has released about 1,000 dogs over eight years.\n\nA much larger single release, though, occurred in India in 2016, when a lab was refused permission to use dogs for cosmetics testing, leading to 156 two- to five-year-old beagles to be released \u2013 seeing the sun for the first time \u2013 before they were rehomed. In that instance, it was the Indian government and a rescue charity who let the dogs out.\n\nIS THIS SONG ABOUT YOU?\n\n(BECAUSE YOU'RE SO VAIN YOU THINK THIS SONG IS ABOUT YOU)\n\nCARLY SIMON\n\n**Ever since it was first posited by Simon, C in 1972, the question of whether someone was 'so vain' that they believed the discussion to be about them has been mooted as a classical paradox.**\n\nThe reasoning goes that if the target of Simon's verbal sally believes the song to be about them then they are not vain to think so, they are merely accurate. But if they are not so vain as to think they are the target, then the song may just be wrong.\n\nSuch a situation need not, however, present a paradox as it leaves open multiple possibilities. The first is that Simon's conjecture leaves her some wriggle room, by referring to 'probably' as a hedge, while the second is that Simon may simply be wrong about her target's vanity. A third possibility is that Simon has correctly identified her target as vain, but is being unfair to suggest that he is vain because of his views on the song, as his views on the song are in reality accurate.\n\nWe would like to offer a new solution to the situation. In a famous physics thought experiment, Erwin Schr\u00f6dinger posited the idea of a cat shut in a box with a radioactive isotope that released poison into the box, killing the cat.\n\nUntil the box was opened, he suggested, we have no way of establishing whether the cat is alive or dead, meaning the cat is both alive and dead until the act of opening the box fixes it in one or the other state.\n\nSimon, we venture, achieved the same with vanity. Through her act of making the target's vanity dependent on a self-defeating statement in her song, she had, in effect, discovered quantum vanity: her target is both vain and not-vain, and thus a clear contribution to modern science and philosophy. It would be impossible to imagine the work of West, K and the Kardashian lab without Simon's pioneering work on vanity.\nWHAT WOULD HAPPEN IF IT WERE CHRISTMAS EVERY DAY?\n\nWIZZARD\n\n**In this chilling and nihilistic 1973 study into the impact of redrawing the calendar so that it is Christmas every day, Wizzard were guilty of a very partial investigation into an event that would surely cause economic recession and possibly the collapse of much of the global economy. They begin from the faulty premise that, it being Christmas, every day would somehow mean a state of perpetual winter in which their favoured snow-related recreation would continue. But even putting aside this inaccuracy, they leave out many of the massive downsides.**\n\nAs Wizzard were surely fully aware, in most of the Western world almost all shops and workplaces are required by law to close for the day, an act which if repeated daily would quickly cause shortfalls of food and other essential supplies \u2013 though in the short-run greatly increase the profits and returns of the few shops allowed and able to open. The result of this would be to very quickly force a battle of civilizations between those who celebrate Christmas and those who do not, as one fed economically on the other.\n\nThe disastrous economic backlash as people eventually stopped receiving salaries, unless they were members of the non-celebrating Christmas service provision, would be heightened by the social requirements of the day: Christmas is seen by many to require lavish feasting and the exchange of gifts. Without such elements, they would feel it was no longer 'Christmas every day'. We could thus expect to see turkeys and other feasting birds slaughtered to extinction, before people moved on to wild animals, household pets and potentially each other, while civil law would break down as people constantly hunted for new, daily, gifts for their loved ones. Eventually 'gifts' would become what could be stolen or taken by force from others.\n\nFinally, the mental health implications of every day spent with extended family is impossible to compute and would surely lead to total societal breakdown within a matter of months. The only bells ringing out in this terrifying alternate future would be those of the corpse bearers as they made their way through desolate burning streets.\n\nWHY DO BIRDS SUDDENLY APPEAR?\n\nTHE CARPENTERS\n\n**The brother-and-sister team behind this flawed project deserve a measure of praise for accurately identifying a phenomenon long before mainstream science, but this must be tempered by their poor attribution of its cause.**\n\nCarpenter, K and Carpenter, R accurately identified that the movement of birds has become less predictable, causing them at times to 'suddenly appear'. Unfortunately \u2013 and with little evidence given in their discourse \u2013 they attribute this to the birds wishing to be 'close to' the Carpenters' subject. As they state that, in this desire, the birds are 'just like me', it suggests both Carpenters are experiencing a severe case of projection \u2013 wrongly ascribing their own motivations to others \u2013 for which they may need to seek appropriate help.\n\nThankfully, recent science offers a more solid explanation for the unpredictable arrival of birds \u2013 especially as it related to their migratory patterns. A 2017 paper in the _Scientific Reports_ journal notes that migrating birds aim to arrive in a new region just as 'green-up' occurs, i.e., when the first shoots from buds appear \u2013 but due to climate change multiple species are now missing that time, often by several days.\n\nA further paper in the same journal notes that artificial light at night is further disrupting migration patterns. The papers note that the consequences of this random timing can be severe, leading to die-outs in the bird population and thus spiralling populations of some insect pests, disrupting crop production. If more attention had been paid to the Carpenters' initial, if flawed, observation, perhaps these consequences could have been avoided.\n\nThe study's further claims that such logic can also be applied to stars in the sky and girls around town and that this is down to angels applying liberal amounts of moondust and starlight are less easy to validate, though do perhaps point to an apprehension that light is to blame where the birds are concerned.\nWHAT IF GOD WAS ONE OF US?\n\nJOAN OSBORNE\n\n**In her classic 1995 thought experiment, theologian Osborne, J pondered what would happen if God manifested on Earth as one of us. Her investigation includes what his name would be, whether you would say it to his face and what single question you would ask him.**\n\nGiven that the idea of God varies between cultures and religions, we would expect His manifestation to be vastly different depending on which culture's idea of God (if any) were correct. And even if we limit ourselves to the Judeo-Christian conception of God, there are more than twenty different ways that God is addressed at different points in the bible, including El, Eloah Adonai and Yahweh. As to whether you could say it to his face \u2013 in Genesis 32:30 Jacob claims he has 'seen God face to face', but in John 1:18 it says, 'No Man hath seen God at any time'. Moses is made to stand on a big rock, so he can't see God.\n\nHowever, it is most likely that Osborne is referring to the concept of God appearing in the form of a man, as Christian believers maintain he did through the body of Jesus. This is backed up by Osborne's encouragement to imagine him as a 'slob on a bus' trying to get home. Many conceptions of Jesus would fit this description.\n\nLeaving aside the obvious issue that if you are on a bus going to heaven, you should probably remain quiet and hope you arrive, it's worth thinking about the questions that Osborne cleverly does not pose. Given most religions pose God as omniscient, He would already know what question you were going to ask whether or not you asked it, so in effect you do not need to ask the question.\n\nIndeed, all questions asked by everyone are eternally present in His head, so, as Osborne is clearly suggesting, God being one of us would in no way change anything. An infinite variety of every question possible to ask would exist in the permanent now of His consciousness, whatever you do.\n\nOsborne does, however, risk a serious schism in the Catholic church by suggesting that the Pope calls God on a phone. If so, the question we should probably ask God is 'what's your number?'\n\nWHY DOES IT ALWAYS RAIN ON ME?\n\nTRAVIS\n\n**Healy, F and his collaborators on this work demonstrate remarkably unusual priorities here, focusing almost entirely on why they are so frequently subject to rain, ignoring their own stated insomnia and their difficulties caused by lightning.**\n\nGiven their focus on the question of why they are so regularly exposed to rain, it is fitting that we tackle this question first. Perhaps, ironically, it is by looking at which professions are most likely to be exposed to the sun that we could see who would be outside \u2013 and therefore subject to being rained upon.\n\nThe _British Journal of Cancer_ did exactly this in a study of melanomas, finding the professions most at risk thanks to their outdoor work were construction workers, followed by agriculture workers, followed by police and armed services. We can therefore assume Healy et al. were looking at such a worker.\n\nIf that worker was in their hometown of Glasgow, this would further explain the rain: because of their proximity to the Atlantic Ocean, the west of Scotland and Ireland are subject to particularly high rainfall \u2013 an effect that has less impact on much of England as it is partially shielded by Ireland.\n\nHowever, we feel the more pressing issue for Healy is that he 'can't avoid the lightning'. In normal circumstances there would be much Healy could do to increase his chances of avoiding lightning, including not sheltering under trees, taking cover indoors if possible, or even better in a car, where the metal shell and rubber tyres would leave him very safe. However, if Healy is plagued by regular lightning even when the sun shines, he may wish to present himself to authorities, who could take advantage of his misfortune to generate cheap power.\n\nThere are no known correlations between lying \u2013 at seventeen or any other age \u2013 and average rainfall, or its consequences. Unless the lie he told when he was seventeen was, 'Yes, I do have an umbrella and full set of waterproofs, thank you.'\nWHO DO YOU THINK YOU ARE?\n\nSPICE GIRLS\n\n**This seminal neurological exploration from a group of five \u2013 Adams, V, Brown, M, Bunting, E, Chisholm, M and Halliwell, G \u2013 examines our sense of self-identity, including where it comes from and what preserves it.**\n\nOther work alongside has helped to answer the question posed, at least in part, by establishing that our sense of self-identity comes from the brain: it is believed to be related to the activity of the right frontal lobe, according to research on seventy-two patients presented to the American Academy of Neurology.\n\nA study of patients suffering a rare form of dementia affecting that region found that people's identity occasionally changed as a result of the condition. While some primarily had issues with forgetting the identities of loved ones, others' sense of who they were changed more dramatically. A sexually conservative and risk-averse man became a liberated job-hopper, while another woman ditched designer dressing and fine dining for casuals and fast-food. Elsewhere, people have spontaneously developed foreign accents \u2013 even never having visited the country in question.\n\nWhile the collaboration \u2013 known as SPICE, though we were unable to source the origin of this acronym \u2013 raised questions as to how our brains maintain our sense of activity, their medical advice was accurate. They advised to 'swing', 'shake', 'move' and 'make' \u2013 all examples of regular physical exercise, which is believed to significantly lower the risk of dementia.\n\nDO YOU REMEMBER THE FIRST TIME?\n\nPULP\n\n**Cocker, J and his collaborators introduce thought-provoking musings on the nature of memory in their 1994 project. When it comes to whether we remember the first time we did some things \u2013 walked, perhaps, or ate solid food, the answer is 'no'.**\n\nFor many of us, our first memory is a hazy thing, usually from around the age of three or four. This is because of a phenomenon known as 'childhood amnesia', which leaves most of us with virtually no memories before the age of three, and relatively few from before the age of ten, when compared to the rest of our lives.\n\nStudies have found this effect is gradual: during childhood, we can apparently recall some events back to around the age of one \u2013 though some psychologists speculate this could be children recalling events as described more recently by others \u2013 only then to lose memory of them later.\n\nHowever, if Cocker et al. are looking into sibling birth or hospitalization, evidence has shown memory of these events can persist despite childhood amnesia, to an extent. Research suggests that a recollection of these significant events can sustain from the age of two.\n\nOne alternative hypothesis moots that Cocker et al. are in fact researching recollection of the first time someone slept with them. Most of us can recall this, though many of us would rather not: 58 per cent of US adults said their first time sleeping with their current partner was 'awkward' or 'terrible'.\n\nWHO STARTED THE FIRE?\n\nBILLY JOEL\n\n**In this discourse, Joel, B is especially keen that we absolve him and anyone associated with him of having been the originator of fire \u2013 claiming instead that it has always been burning, 'since the world's been turning'. Joel's claim is not without merit, but requires some scrutiny.**\n\nJoel is correct to dismiss cosmological events such as the Big Bang and even the activity of the sun from his analysis, as neither are strictly fire: one was the expansion of matter, and the other is a nuclear fusion reaction. Fire, instead, is the reaction of flammable materials with oxygen in the atmosphere.\n\nHowever, Joel is wrong to say things have been burning since the world was turning. When the Earth formed, around 4.6 billion years ago, it was already turning, but lacked enough oxygen in its atmosphere to start a fire.\n\nA combustible material such as wood will only ignite if the atmosphere has 15\u201317 per cent oxygen in it, which has been the case for less than 850 million years. In this sense, Joel is correct that humans did not start the first fires on Earth, but his timeline is out by nearly four billion years.\n\nAs to Joel's second claim \u2013 that even if we collectively did not start the fire, we have 'tried to fight it' \u2013 he may be on fairly solid ground here. Across most developed countries, around one in 1,000 of the population is employed as a firefighter, while the remainder of the population contributes to this effort through taxation.\n\nDespite Joel's advocacy, 'It was always burning' is not recommended as a defence in an arson trial.\nSONG CREDITS\n\n**SONG TITLE** | **WRITING CREDITS** | **PUBLISHING RIGHTS**\n\n---|---|---\n\n**SHOULD I STAY OR SHOULD I GO?** | **Samuel Reinhard, Joe Strummer, Mick Jones** | **Universal-Polygram Intl Pub Obo Nineden, Ltd.**\n\n**BLOWIN' IN THE WIND** | **Bob Dylan** | **Bob Dylan Music Obo Special Rider MusicDo**\n\n**DO THEY KNOW IT'S CHRISTMAS? (FEED THE WORLD)** | **Midge Ure, Bob Geldof** | **Wb Music Corp. Obo Chappell Music Ltd.**\n\n**A DAY IN THE LIFE** | **John Lennon, Paul McCartney** | **Sony\/Atv Tunes Llc Dba Atv Obo Atv (Northern Songs Catalog)**\n\n**(HOW MUCH IS) THAT DOGGIE IN THE WINDOW?** | **Bob Merrill** | **Music & Media Int'l Obo Golden Bell Songs**\n\n**KILLING ME SOFTLY** | **Norman Gimbel, Charles Fox** | **Warner-Tamerlane Pub Co Obo Rodali Music \/ Words West Llc**\n\n**HOW SOON IS NOW?** | **Johnny Marr, Steven Morrissey** | **Universal - Polygram Int'l Obo Marr Songs Ltd. \/ Warner Tamerlane Pub Corpo\/B\/O Muziekuitgeverij Artemis Bv**\n\n**CAN I KICK IT?** | **Lou Reed** | **Emi Blackwood Music Inc Obo Oakfield Avenue Music Ltd**\n\n**MS. JACKSON** | **Andre Benjamin, Antwan Patton, David Sheats** | **Bmg Monarch \/ Emi April Music Inc**\n\n**LADY MARMALADE** | **Robert Crewe, Kenny Nolan** | **Stone Diamond Music Corp Obo Tannyboy Music Co \/ Jobete Music Co Obo Kenny Nolan Publishing**\n\n**RUN THE WORLD (GIRLS)** | **Thomas Wesley Pentz, Beyonc\u00e9 Knowles, Dave Taylor, Terius Nash, Adidja Azim Palmer, Nick Van De Wall** | **Emi April Music Inc Obo B-Day Publishing \/ Wb Music Corp Obo 2082 Music Publishing \/ Nw Collections Obo Jack Russell Music Ltd \/ Kobalt Music Pub America Inc \/ Stemra \/ Copyright Control N Emi April Music Inc \/ Emi April Music Inc Obo Switch Werd Music**\n\n**ETERNAL FLAME** | **Susanna Lee Hoffs, Thomas F. Kelly, William E. Steinberg** | **Songs Of Universal Obo Bangophile Music \/ Sony\/Atv Tunes Llc**\n\n**WAR** | **Barrett Strong, Norman Whitfield** | **Emi Blackwood Music Inc Obo Stone Agate Music**\n\n**UMBRELLA** | **Thaddis Laphonia Harrell, Shawn Carter, Christopher A. Stewart, Terius Youngdell Nash** | **Sony\/Atv Tunes Llc Obo Samp-Uk Ltd. \/ Wb Music Corp \/ Wb Music Corp Obo Carter Boys Music \/ Wb Music Corp Obo 2082 Music Publishing \/ Songs Of Peer, Ltd. \/ Songs Of Peer, Ltd. Obo March Ninth Music**\n\n**BOHEMIAN RHAPSODY** | **Queen** | **Emi Glenwood Music Corp Obo Queen Music Ltd.**\n\n**THAT'S NOT MY NAME** | **Jules De Martino, Katie White** | **Wb Music Corp. Obo Playwrite Music Limited \/ Sony\/Atv Tunes Llc Obo Samp-Uk Ltd.**\n\n**CRY ME A RIVER** | **Arthur Hamilton** | **Chappell & Co.**\n\n**WHERE HAVE ALL THE FLOWERS GONE?** | **Peter Seeger** | **Figs. D Music Inc Obo Sanga Music \/ Figs. D Music In**\n\n**DOES YOUR MOTHER KNOW?** | **Aleksej Anatolevich Kortnev, Bjoern K. Ulvaeus, Benny Goran Bror Andersson** | **Emi Grove Park Music Obo Union Songs A.b. \/ Universal - Polygram Int'l Obo Union Songs Musikforlag Ab**\n\n**FEVER** | **John Davenport, Eddie Cooley** | **Trio Music Company, Inc. \/ Fort Knox Music, Inc.**\n\n**I'D DO ANYTHING FOR LOVE (BUT I WON'T DO THAT)** | **James Richard Steinman** | **Edward B Marks Music Company**\n\n**ARE 'FRIENDS' ELECTRIC?** | **Gary Webb (Pak: Gary Numan)** | **Universal - Polygram International Pub Inc**\n\n**MARIA** | **Richard Rodgers, Oscar Hammerstein Ii** | **The Rodgers & Hammerstein Organisation**\n\n**YEAR 3000** | **Steve Robson, James Bourne, Mattiesargeant, Math Jay, Charlie Simpson, Matthew Fletcher** | **Almo Music Corp. Obo Rondor Music (London) Ltd \/ Copyright Control**\n\n**WHEN WILL I BE FAMOUS?** | **The Brothers** | **Copyright Control \/ Chappell & Co., Inc. Obo Graham Music Pub.**\n\n**HUMAN** | **Mark August Stoermer, Dave Brent Keuning, Brandon Flowers, Ronnie Jr. Vannucci** | **Universal-Polygrm Intl Pub Obo Universal Music Pub. Ltd.**\n\n**I'M GONNA BE (500 MILES)** | **Charles S. Reid, Craig M. Reid** | **Wb Music Corp Obo Warner Bros. Music Ltd**\n\n**ONCE IN A LIFETIME** | **Brian Eno, Chris Frantz, David Byrne, Jerry Harrison, Tina Weymouth** | **Rhino \/ Warner Bros**\n\n**JERUSALEM** | **William Blake, Hubert Parry** | **Public Domain**\n\n**OUT OF THE WOODS** | **Taylor Swift, Jack Antonoff** | **Sony\/Atv Tree Publishing Obo Taylor Swift Music**\n\n**I WILL SURVIVE** | **Dino Fekaris, Frederick J. Perren** | **Universal - Polygram International Pub Inc \/ Universal-Polygrm Intl Pub Obo Perren-Vibes Music Inc**\n\n**THE CHAIN** | **Lindsey Buckingham, Christine Mcvie, Stephanie Nicks, Mick Fleetwood, John Mcvie** | **Universal Music-Careers \/ Reach Music Publishing**\n\n**TURNING JAPANESE** | **D Fenton** | **Emi Glenwood Music Corporation**\n\n**SMOOTH CRIMINAL** | **Michael Joe Jackson** | **Sony\/Atv Songs Llc (Non-Rep) Mijac Music**\n\n**INDEPENDENT WOMEN (PT 1)** | **Beyonc\u00e9 Knowles, Cory Rooney, Samuel Barnes, Jean-Claude Olivier** | **Columbia Records**\n\n**IS SHE REALLY GOING OUT WITH HIM?** | **Joe Jackson** | **Kobalt Music Pub America Obo Pokazuka Llc**\n\n**AMERICAN PIE** | **Don McLean** | **Songs Of Universal, Inc. Obo Benny Bird Co., Inc.**\n\n**CAN YOU FEEL THE LOVE TONIGHT?** | **Tim Rice, Elton John** | **Wonderland Music Company Inc**\n\n**CALL ME, MAYBE?** | **Carly Rae Jepsen, Joshua Keeler Ramsay, Tavish Joseph Crowe** | **Universal Music Corp. Obo Jepsen Music Pub \/ Bmg Gold Songs Obo Crowe Music Inc.**\n\n**FAIRYTALE OF NEW YORK** | **Jeremy Max Finer, Shane Patrick Lysaght Macgowan** | **Universal Music Mgb Songs Obo Universal Music Publ. Mgb Ltd. \/ Universal - Polygram Obo Universal Music Publ. Ltd.**\n\n**MONEY** | **Roger Waters** | **Hampshire House Publishing**\n\n**WHY DON'T WE DO IT IN THE ROAD?** | **Lennon, McCartney** | **Sony\/Atv Tunes Llc Dba Atv Obo Atv (Northern Songs Catalog)**\n\n**THE DRUGS DON'T WORK** | **R. Ashcroft** | **Emi Music Publishing**\n\n**NINE MILLION BICYCLES** | **Mike Batt** | **Sony\/Atv Tunes Llc Obo Dramatico Music Publishing Ltd**\n\n**DO YOU KNOW THE WAY TO SAN JOS\u00c9?** | **Hal David, Burt F Bacharach** | **Bmg Gold Songs Obo Casa David Lp \/ Bmg Gold Songs Obo New Hidden Valley Music Co.**\n\n**EYE OF THE TIGER** | **Jim Peterik, Frank Sullivan** | **Sony\/Atv Melody Obo Rude Music \/ Wb Music Cor. Obo Easy Action Music**\n\n**WOULD I LIE TO YOU?** | **M. Leeson, P. Vale** | **Bmg Platinum Songs Obo Bmg Vm Music Ltd**\n\n**9 TO 5** | **Dolly Parton** | **Velvet Apple Music**\n\n**WHAT BECOMES OF THE BROKEN HEARTED?** | **William Weatherspoon, James Dean, Paul Riser** | **Stone Agate Music Corp \/ Jobete Music Co Inc**\n\n**BLAME IT ON THE BOOGIE** | **Thomas Meyer, Hans Kampschroer, Elmar Krohn, Michael George Jackson, Rich David John Jackson** | **Chrysalis Music Obo Edition Delay \/ Gema**\n\n**EARTH SONG** | **Michael Jackson** | **Sony\/Atv Songs Llc (Non-Rep) Mijac Music**\n\n**CREEP** | **Mike Hazelwood, Albert Hammond, Jonathan Richard Guy Greenwood, Thomas Edward Yorke, Colin Charles Greenwood, Philip James Selway, Edward John O'brien** | **Emi April Music Inc \/ Wb Music Corp. Obo Warner\/Chappell Music Ltd.**\n\n**WHITE CHRISTMAS** | **Irving Berlin** | **Irving Berlin Music Company**\n\n**SIGNS** | **Calvin Broadus, Lonnie Simmons, Pharrell Williams, Charles K Wilson, Rudy Taylor, Chad Hugo** | **Emi Blackwood Music Inc Obo My Own Chit Music \/ Universal Music-Careers \/ Emi Blackwood Music Inc \/ Warner Geo Met Tric Music \/ Bmg Platinum Songs Obo Minder Music**\n\n**NOTHING COMPARES 2 U** | **Prince** | **Universal Music Corp. Obo Controversy Music**\n\n**ESCAPE** | **Rupert Holmes** | **Wb Music Corp**\n\n**ARE YOU LONESOME TONIGHT?** | **Roy Turk, Lou Handman** | **Bourne Co \/ Cromwell Music**\n\n**THE FOX** | **Tor Hermansen, Nicholas Boundy, Vegard Ylvisaaker, Mikkel Eriksen, Christian Lochstoer, Baard Ylvisaaker** | **Emi Blackwood Music Inc Obo Stellar Songs Ltd \/ Emi April Music Inc Obo Emi Music Publishing, Ltd**\n\n**IRONIC** | **Glen Ballard, Alanis Nadine Morissette** | **Songs Of Universal, Inc. \/ Songs Of Universal, Inc. Obo Vanhurst Place Music \/ Penny Farthing Music Obo Arlovol Music**\n\n**WHAT IS LOVE?** | **Junior Torello, Dee Dee Halligan** | **Wb Music Corp Obo Hanseatic Musikverlag Gmbh \/ Gema**\n\n**LONELY THIS CHRISTMAS** | **M. Chapman, N. Chinn** | **Universal Music Mgb Songs**\n\n**COMPLICATED** | **Carolyn Dawn Johnson, Shaye Smith** | **Emi Blackwood Music Inc \/ Emi Full Keel Music Co.**\n\n**ALL MY FRIENDS** | **James Murphy** | **Kobalt Music Pub America Obo Guy With Head And Arms Music**\n\n**HEY YA** | **Andre Benjamin** | **Bmg Monarch Obo Gnat Booty Music \/ Bmg Monarch**\n\n**WHY'D YOU ONLY CALL ME WHEN YOU'RE HIGH?** | **Jamie Cook, Matt Helders, Nick O'Malley, Alex Turner** | **Domino Recording Co**\n\n**24 HOURS FROM TULSA** | **Burt Bacharach, Hal David** | **Gusto Records**\n\n**EVERY BREATH YOU TAKE** | **Sting** | **Emi Blackwood Music Inc Obo Magnetic Publishing Ltd.**\n\n**WHAT'S THE FREQUENCY, KENNETH?** | **John Michael Stipe, Peter Lawrence Buck, William Thomas Berry, Michael E. Mills** | **Universal Tunes Obo Night Garden Music**\n\n**GIRLS JUST WANT TO HAVE FUN** | **Robert Hazard** | **Sony\/Atv Tunes Llc**\n\n**WHERE DO BROKEN HEARTS GO?** | **Wildhorn, Jackson** | **Sony\/Atv Tunes Llc \/ Chrysalis Music Group Inc Digital Only**\n\n**WHAT HAVE THEY DONE TO THE RAIN?** | **Malvina Reynolds** | **Nancy Schimmel Dba Schroder Music Co.**\n\n**PIANO MAN** | **Billy Joel** | **Almo Music Corp. Obo Joelsongs**\n\n**WHY?** | **Annie Lennox** | **Universal Music Mgb Songs**\n\n**EVERYBODY WANTS TO RULE THE WORLD** | **Chris Hughes, Ian Stanley, Roland Orzabal** | **Bmg Platinum Songs Obo Bmg 10 Music Limited \/ Bmg Platinum Songs Obo Bmg Vm Music Ltd**\n\n**TELL ME SOMETHING I DON'T KNOW** | **Antonina Armato, Michael David Nielsen, Ralph Nero Iv Churchwell** | **Universal Music Corp. Obo Warner Olive Music Llc \/ Downtown Dlj Songs Llc Obo Antonina Songs**\n\n**CARELESS WHISPER** | **George Michael, Andrew Ridgeley** | **Wb Music Corp Obo Wham Music Limited (Gb 2) \/ Wb Music Corp Obo Warner\/Chappell Mlm Limited**\n\n**YEAH!** | **La Marquis Jefferson, Sean Garrett, James Phillips, Jonathan Smith, Christopher Bridges, Patrick Smith** | **Reservoir 416 \/ Emi April Music Inc Obo Air Control Music \/ Bmg Bumblebee Obo Me And Marq Music \/ Emi April Music Inc Obo Basajamba Music \/ Songs Of Windswept Pacific Y Emi April Music Inc \/ Emi April Music Inc Obo Ludacris Music Pub \/ Music Of Windswept**\n\n**LIFE ON MARS** | **David Bowie** | **Chrysalis Music Group Inc Digital Only \/ Tintoretto Music \/ Emi Music Publishing Ltd.**\n\n**WHAT'S UP?** | **Linda Perry** | **Sony\/Atv Harmony \/ Sony\/Atv Harmony Obo Stuck In The Throat Music**\n\n**WHO LET THE DOGS OUT?** | **Anslem D Douglas, Osbert Leopold Gurley** | **Bmg Platinum Songs \/ Bmg Platinum Songs Obo Hyckryck Music Pub, Inc. \/ Wyz Girl Ent. Consulting Llc**\n\n**YOU'RE SO VAIN** | **Carly Simon** | **Universal Music Corp. Obo C'est Music**\n\n**I WISH IT COULD BE CHRISTMAS EVERY DAY** | **Roy Wood** | **Parlophone Uk**\n\n**(THEY LONG TO BE) CLOSE TO YOU** | **Hal David, Burt F. Bacharach** | **Bmg Gold Songs Obo Casa David Lp \/ Bmg Gold Songs Obo New Hidden Valley Music Co.**\n\n**ONE OF US** | **Eric Bazilian** | **Universal \/ Island Def Jam**\n\n**WHY DOES IT ALWAYS RAIN ON ME?** | **Francis Healy** | **Sony\/Atv Songs Llc Obo Samp-Uk Ltd.**\n\n**WHO DO YOU THINK YOU ARE?** | **Wilson, Watkins, Halliwell** | **Universal Music Mgb Songs Obo 19 Music Ltd. \/ Emi Full Keel Music Co.**\n\n**DO YOU REMEMBER THE FIRST TIME?** | **Nick Banks, Jarvis Cocker, Candida Doyle, Steve Mackey, Russell Senior** | **Universal-Island Music, Inc.**\n\n**WE DIDN'T START THE FIRE** | **Billy Joel** | **Almo Music Corp. Obo Joelsongs**\n\nCredits Sourced From: Hfa Songfile\nFOOTNOTES\n\n*** Later studies into a fever specifically transmitted by dancing on a Saturday night by a 'sweet city woman', or Patient X as she became known, were found to be largely inconclusive.**\n\n*** Other scholars have ventured it is instead 'all about chemistry', a conjecture supported by the basis that merely to stay alive, each of the human body's 37 trillion cells need to carry out at least 10 million chemical reactions each, per second. This is, in formal terms, a lot each day.**\n\n*** The text is unclear as to whether the four members of Busted have collectively produced great-grandchildren through inter-marriage in intervening generations, or whether 'Peter' \u2013 their interlocutor \u2013 jumped to genealogical conclusions.**\n\n*** By contrast, survival rates can be as high as 90 per cent for those who receive electric defibrillation within a minute of cardiac arrest, vindicating studies that talk about 'electricity in your veins'.**\n\n*** Whether or not an equivalent of signalling pheromones for humans even exists is a topic of much debate. Recent studies suggest that two of the most likely 'putative human pheromones' \u2013 androstadienone and estratetraenol \u2013 should be dropped.**\n\n*** The same year as the release of _Dr Strangelove._**\n\n*** Earlier investigations of the question by Gaye, M limited themselves to what was going on in the specific instance of the study in late sixties America.**\nACKNOWLEDGEMENTS\n\nThis book owes its existence to Nicky Woolf, whose late-night drunken attempt to fact check 'Ms Jackson' on Twitter sparked a correction from me. In turn, that prompted then-editor at BuzzFeed Janine Gibson to make me turn it into an article, which prompted Boxtree editor Jamie Coleman to suggest this book.\n\nThanks are due to everyone at Pan Macmillan, and especially to James Edgar, whose brilliant illustrations are throughout the book. Thanks are also due to Luke, Caroline, Holly, Tom, David and others who have inputted suggestions throughout. All errors remain entirely my fault.\n\nSerious credit and apologies are due to every artist whose work I've butchered in this work \u2013 thank you all for posing so many great questions.\n\nFinally, I would like to single out Mel's friend Alice, who was absolutely no help at all.\n\nFirst published 2018 by Boxtree\n\nThis electronic edition published 2018 by Boxtree\n\nan imprint of Pan Macmillan\n\n20 New Wharf Road, London N1 9RR\n\nAssociated companies throughout the world\n\nwww.panmacmillan.com\n\nISBN 978-0-752-26653-4\n\nCopyright \u00a9 James Ball 2018\n\nThe right of James Ball to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.\n\nPan Macmillan does not have any control over, or any responsibility for, any author or third-party websites referred to in or on this book.\n\nYou may not copy, store, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (electronic, digital, optical, mechanical, photocopying, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.\n\nA CIP catalogue record for this book is available from the British Library.\n\nVisit **www.panmacmillan.com** to read more about all our books and to buy them. You will also find features, author interviews and news of any author events, and you can sign up for e-newsletters so that you're always first to hear about our new releases.\n  1. Cover\n  2. Title page\n  3. Dedication page\n  4. CONTENTS\n  5. A NOTE ON CITATIONS\n  6. HOW MANY ROADS MUST A MAN WALK DOWN?\n  7. DO THEY KNOW IT'S CHRISTMAS TIME?\n  8. ARE THERE 4,000 HOLES IN BLACKBURN, LANCASHIRE?\n  9. HOW MUCH IS THAT DOGGIE IN THE WINDOW?\n  10. CAN YOU KILL SOMEONE WITH A SONG?\n  11. HOW SOON IS NOW?\n  12. CAN I KICK IT?\n  13. HOW LONG DOES IT TAKE TO APOLOGIZE A TRILLION TIMES?\n  14. VOULEZ-VOUS COUCHER AVEC MOI CE SOIR?\n  15. WHO RUN THE WORLD?\n  16. IS THIS BURNING AN ETERNAL FLAME?\n  17. WAR - HUH - WHAT IS IT GOOD FOR?\n  18. HOW LIKELY IS RIHANNA TO NEED HER UMBRELLA?\n  19. IS THIS THE REAL LIFE? IS THIS JUST FANTASY?\n  20. ARE YOU CALLING ME 'DARLING'?\n  21. HOW MUCH WOULD IT TAKE TO CRY A RIVER?\n  22. WHERE HAVE ALL THE FLOWERS GONE?\n  23. SHOULD I STAY OR SHOULD I GO?\n  24. DOES YOUR MOTHER KNOW THAT YOU'RE OUT?\n  25. DO YOU GIVE ME FEVER?\n  26. WHAT WOULDN'T MEAT LOAF DO?\n  27. ARE FRIENDS ELECTRIC?\n  28. HOW DO YOU SOLVE A PROBLEM LIKE MARIA?\n  29. HOW MANY GENERATIONS WILL HAVE PASSED BY THE YEAR 3000?\n  30. WHEN WILL I BE FAMOUS?\n  31. ARE WE HUMAN, OR DANCER?\n  32. YOU WALK 500 MILES. YOU WALK 500 MORE. WHERE HAVE YOU BEEN?\n  33. WHERE IS MY LARGE AUTOMOBILE?\n  34. DID THOSE FEET IN ANCIENT TIMES WALK UPON ENGLAND'S MOUNTAINS GREEN?\n  35. ARE WE OUT OF THE WOODS YET?\n  36. I WILL SURVIVE - BUT FOR HOW LONG?\n  37. IF YOU DON'T LOVE ME NOW, WILL YOU NEVER LOVE ME AGAIN?\n  38. HOW MUCH SPACE DO A MILLION PHOTOGRAPHS TAKE UP?\n  39. IS ANNIE OKAY?\n  40. HOW MANY HONEYS MAKE THEIR MONEY?\n  41. IS SHE REALLY GOING OUT WITH HIM?\n  42. WILL THIS BE THE DAY THAT YOU DIE?\n  43. CAN YOU FEEL THE LOVE TONIGHT?\n  44. CALL ME, MAYBE?\n  45. WERE THE BOYS OF THE NYPD CHOIR SINGING 'GALWAY BAY'?\n  46. IS MONEY THE ROOT OF ALL EVIL TODAY?\n  47. WHY DON'T WE DO IT IN THE ROAD?\n  48. DO THE DRUGS WORK?\n  49. ARE THERE NINE MILLION BICYCLES IN BEIJING?\n  50. DO YOU KNOW THE WAY TO SAN JOS\u00c9?\n  51. HOW WORRIED SHOULD YOU BE IF SOMEONE'S WATCHING YOU WITH THE EYE OF THE TIGER?\n  52. WOULD I LIE TO YOU?\n  53. HOW MUCH WOULD YOU HAVE TO EARN TO BE 'BARELY GETTIN' BY' IN LA IN 1980?\n  54. WHAT BECOMES OF THE BROKEN HEARTED?\n  55. WHO'S TO BLAME: THE SUNSHINE, MOONLIGHT, GOOD TIMES OR BOOGIE?\n  56. WHAT ABOUT ELEPHANTS - HAVE WE LOST THEIR TRUST?\n  57. AM I A CREEP?\n  58. DID WE USED TO KNOW WHITE CHRISTMASES?\n  59. ARE YOU TELLING ME THIS IS A SIGN?\n  60. WHAT COMPARES TO YOU?\n  61. DO YOU LIKE PI\u00d1A COLADAS, AND GETTING CAUGHT IN THE RAIN?\n  62. ARE YOU LONESOME TONIGHT?\n  63. WHAT DOES THE FOX SAY?\n  64. ISN'T IT IRONIC, DON'T YOU THINK?\n  65. WHAT IS LOVE?\n  66. WILL IT BE LONELY THIS CHRISTMAS?\n  67. WHY'D YOU HAVE TO GO AND MAKE THINGS SO COMPLICATED?\n  68. WHERE ARE YOUR FRIENDS TONIGHT?\n  69. WHAT'S COOLER THAN BEING COOL?\n  70. WHY DO YOU ONLY CALL ME WHEN YOU'RE HIGH?\n  71. WHERE IS 24 HOURS FROM TULSA?\n  72. OH CAN'T YOU SEE YOU BELONG TO ME?\n  73. WHAT'S THE FREQUENCY, KENNETH?\n  74. DO GIRLS JUST WANNA HAVE FUN?\n  75. WHERE DO BROKEN HEARTS GO?\n  76. WHAT HAVE THEY DONE TO THE RAIN?\n  77. SON, CAN YOU PLAY ME A MEMORY?\n  78. WHY?\n  79. DOES EVERYBODY WANT TO RULE THE WORLD?\n  80. HOW MANY INCHES ARE IN A MILE?\n  81. HAVE GUILTY FEET GOT NO RHYTHM?\n  82. HOW THE HELL AM I SUPPOSED TO LEAVE?\n  83. IS THERE LIFE ON MARS?\n  84. WHAT'S GOING ON?\n  85. WHO LET THE DOGS OUT?\n  86. IS THIS SONG ABOUT YOU?\n  87. WHAT WOULD HAPPEN IF IT WERE CHRISTMAS EVERY DAY?\n  88. WHY DO BIRDS SUDDENLY APPEAR?\n  89. WHAT IF GOD WAS ONE OF US?\n  90. WHY DOES IT ALWAYS RAIN ON ME?\n  91. WHO DO YOU THINK YOU ARE?\n  92. DO YOU REMEMBER THE FIRST TIME?\n  93. WHO STARTED THE FIRE?\n  94. SONG CREDITS\n  95. ACKNOWLEDGEMENTS\n  96. Copyright page\n\n# Guide\n\n  1. Cover\n  2. Title page\n  3. CONTENTS\n  4. HOW MANY ROADS MUST A MAN WALK DOWN?\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n3 PARA  \nMOUNT LONGDON\n\nThe Bloodiest Battle\n\n3 PARA  \nMOUNT LONGDON\n\nThe Bloodiest Battle\n\nWritten and edited by\n\nJ O N C O O K S E Y\n\nwith contributions from:\n\nJohn-Hughes Wilson \u2022 Tim Lynch \u2022 Terry Peck\n\nNick Rose \u2022 Michael Savage\n\nPen & Sword\n\n**MILITARY**\nFirst published in Great Britain in 2004 and reprinted in 2009 by  \nPen & Sword Military an imprint of  \nPen & Sword Books Ltd  \n47 Church Street  \nBarnsley  \nSouth Yorkshire  \nS70 2AS\n\nCopyright \u00a9 Jon Cooksey, 2004, 2009\n\nISBN 1-84415-115-8\n\nThe right of Jon Cooksey to be identified as Author of this Work has  \nbeen asserted by him in accordance with the Copyright, Designs and Patents  \nAct 1988.\n\nA CIP catalogue record for this book is  \navailable from the British Library\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any  \nform or by any means, electronic or mechanical including photocopying, recording  \nor by any information storage and retrieval system, without permission from the  \nPublisher in writing.\n\nPrinted and bound in Thailand by  \nKyodo Nation Printing Services Co., Ltd\n\nPen & Sword Books Ltd incorporates the Imprints of Pen & Sword Aviation, Pen  \n& Sword Maritime, Pen & Sword Military, Wharncliffe Local History, Pen and  \nSword Select, Pen and Sword Military Classics and Leo Cooper.\n\nFor a complete list of Pen & Sword titles please contact  \nPEN & SWORD BOOKS LIMITED  \n47 Church Street, Barnsley, South Yorkshire, S70 2AS, England  \nE-mail: enquiries@pen-and-sword.co.uk  \nWebsite: www.pen-and-sword.co.uk\nCONTENTS\n\nACKNOWLEDGEMENTS\n\nINTRODUCTION\n\n1. THE POLITICAL PROBLEM\n\n2. THE TASK FORCE\n\n3. THE LANDING\n\n4. THE 'TAB'\n\n5. THE PREPARATIONS\n\n6. THE PLAN\n\n7. THE ARGENTINES\n\n8. THE WEAPONS\n\n9. THE BATTLE\n\n10. THE LEGACY\n\nNOTES\n\nINDEX\nACKNOWLEDGEMENTS\n\nI owe a great debt of thanks to the contributors who have all thrown their wholehearted support behind the project. The written and oral accounts of John Hughes-Wilson, Tim Lynch, Terry Peck, Nick Rose and Michael Savage have helped to provide a much greater depth to the final product than I initially envisaged. All of them served during or immediately after the Falklands War and all have been unfailingly helpful and punctual in producing their accounts and accompanying illustrations. I know it was difficult at times for some of them to speak with me and I appreciate their patience during my questioning.\n\nThanks are also due to Major General Julian Thompson, Commander of 3 Commando Brigade during the war, who, despite a busy schedule, found time to respond promptly and with candour to several requests for information and opinion. I am grateful that he also found time to read a draft on the section on 'The Preparations' for the battle.\n\nJon Wilkinson's eye for design and his skill in presenting the illustrative material in order to enhance and support the text, never fails to amaze me. I thank him for his talent and his patience.\n\nFinally, my thanks go to my family; my wife Heather and my daughter, Georgia, who, as always, have supported me during those times when, 'daddy has gone to write about soldiers.'\n\nJon Cooksey\n\nReading 2004\nINTRODUCTION\n\nT **hink Falklands, think Parachute Regiment, think Goose Green. The connection is understandable. It is twenty-two years since the 2nd Battalion of the Parachute Regiment fought the Argentine defenders of the Falkland Islands settlements of Darwin and Goose Green and achieved a morale boosting victory at a heavy price - 15 dead, including their Commanding Officer Lieutenant Colonel 'H' Jones, and 32 wounded. Lieutenant Colonel Jones was later awarded a posthumous Victoria Cross for his actions during the battle**.\n\nIt has been said that the battlefield of Darwin-Goose Green is now the most famous British battlefield of the second half of the twentieth century - most of the more than 40,000 tourists who journey to the islands each year visit it - and that the battle itself, the first set-piece battle of the Falklands war, is 'the most studied and analyzed battalion sized action in military history.' This fact is undeniable as is its significance. Politically and strategically it brought the scramble for a peaceful comprise to a halt with a bloody jolt and cast the first shadow of defeat over the Argentine effort both in Buenos Aires and at their HQ in Port Stanley. For the rest of the British forces on the ground the victory at Goose Green set a morale boosting precedent for the four battalion sized battles to come. But the later battles each had their own particular 'texture' and intensity. Is Goose Green the most famous and the most studied simply because it was the first and because a commanding officer was killed in a charge against enemy positions, an action for which he later received a VC?\n\nOn the day that 2 Para was making its final preparations for the attack on Goose Green its sister battalion, 3 Para had moved out from its positions on the beachhead at Port San Carlos in a Tactical Advance to Battle or 'TAB' across East Falkland. On a miserable and spirit sapping journey, the news of 2 Para's victory a day later filled the men of 3 Para with pride. Twenty year-old Private Nick Rose of 6 Platoon in Major Mike Argue's B Company, recalled his feelings on hearing the news:\n\n> _'They'd been committed and 'bang' they'd done it and it was \"well done the blokes, well done Para Reg.\" We were so proud, but then sad because of the guys killed - and then scared! But hey, look, they've done it, we can do it and we're going to do it very soon so focus. Clean your weapon, keep your weapon dry if you can, sort yourself out and try and get your 'admin' done as much as possible. 99.9% of paratroopers hate 'admin', they're combat soldiers - they want to get on, do the job and let somebody else tidy up afterwards. We'll go on and do the next thing you want us to do but don't make us do all the shit afterwards.'_\n\nRose and his comrades were highly trained, professional soldiers, determined to emulate the deeds of their sister battalion when their time came to prove themselves on the battlefield. And, just as they sensed, their time would come, two weeks later. Their battlefield would be in and among the long, frost shattered spines of rock which stabbed the air from the summit of Mount Longdon, one of the peaks that studded the outer ring of the Port Stanley defences. The battle for Mount Longdon deserves a place alongside Goose Green in the history of this illustrious regiment for it was a battle which tested the discipline, comradeship and professionalism of the paras to the limit; it was a battle which witnessed another posthumous para VC; it turned out to be the bloodiest battle of the entire Falklands campaign.\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch1)\n\n[THE POLITICAL  \nPROBLEM](05_toc.html#ch1)\n\nI **n August 1914 Britain went to war over a 'scrap of paper', in April 1982 it was 'scrap metal'. On March 19th that year a group of Argentine scrap metal merchants led by Senor Constantino Sergio Davidoff strode ashore at Leith on the island of South Georgia, a British dependency in the South Atlantic, some 8,000 miles from the British mainland. Their presence eventually led to Britain becoming embroiled in its first major conflict since the end of the Korean War**.\n\n##### **BRITANNIA STANDS ALONE**\n\nBy John Hughes-Wilson\n\n**Author and broadcaster Colonel John Hughes-Wilson, served as an Intelligence Officer in the Falklands after the Argentine surrender in June 1982**.\n\nIn 1982, everyone knew that the era of Britain going to war on her own was a thing of the past. The very thought that a full-scale sea, air and land battle could be fought thousands of miles from home was almost unthinkable. The idea that television screens could be full of ships boiling with flame, aeroplanes tumbling from the sky with screaming pilots trapped inside them, and black faced infantrymen trying to rip each others bellies open with bayonets was just beyond belief. Things like that just didn't happen any more, or so the armchair pundits said.\n\nThey were wrong in 1982 and, are probably wrong today. The Falklands war remains one of the defining moments of British post-colonial history. It showed once and for all that general war was not a thing of the past for western democracies. It showed that British politicians and the people still had the stomach to fight. Above all it showed that the British armed forces, although small, were a formidable foe. After 1982, nothing was quite the same again. A deeply unpopular Prime Minister went from being an insignificant politician to an almost totemic figurehead, prepared to stand and fight on any controversial issue.\n\nThe roots of the Falklands conflict ran deep and lay in the disputed ownership of a remote group of islands in the South Atlantic, four hundred miles from Argentina. They were first recorded by a Dutch navigator in 1600 and named by the British in 1690 in honour of Viscount Falkland, a navy minister.\n\nThe first settlers were actually French. In 1764 they built a fort, Port Louis, on East Falkland and called the group les Malouines after the islands off St Malo - hence the Spanish, 'los Malvinas'. A year later the British landed on West Falkland without even knowing that the French were holding the eastern island. It was a fine old eighteenth century colonial muddle, and was eventually solved by the British agreeing to give up their settlement and sail away after the French ceded the islands to their ally, Spain, in 1767. By 1790 all that remained of British presence was a bronze plaque.\n\n* * *\n\n**'I tarry in this miserable desert suffering everything for the love of God.'**\n\n* * *\n\nWith the collapse of the Spanish empire the islands became derelict. In 1832 a US warship, the USS _Lexington_ , cleared the Falkands\/Malvinas of pirates and brigands and declared them 'free of all government.' Into this vacuum sailed the Royal Navy in 1837 with a battleship, declaring the Falkland Islands to be part of the British Crown's territories despite the government in Buenos Aires insisting that all the Spanish territories in the New World now belonged to the successor states - in this case, Argentina. The British - who had landed settlers - claimed that the tiny population of the islands was British and they stubbornly wished to remain so.\n\nAnyone who has visited the islands cannot fail to be surprised by their remoteness and bleakness. In 1760 a Royal Marine (RM) officer said, 'I tarry in this miserable desert suffering everything for the love of God.' Two centuries later, in 1981, another RM officer described the islanders, or Kelpers as they call themselves, as 'a mainly drunken...and indolent collection of drop outs'. This was a harsh judgement. By 1982 the islanders were relying entirely on the British government and one single company, Coalite, for their economic survival. A British Trade Union MP even called them 'company slaves'. This small population - some 1,800 strong in 1982 - had only one unifying attribute: an understandable loyalty to the British Crown. From the point of view of Britain's Foreign and Commonwealth Office (FCO) this was a major irritant. A tiny handful of islanders were standing in the way of Britain's good relations with 28 million Argentines and 250 million South Americans.\n\nFor years the FCO's policy had been to divest itself of this colonial relic. Unfortunately for the world-weary bureaucrats of Whitehall, the Islanders thought differently. And the Islanders had a trump card. Under Article 73 of the UN Charter, the Falklanders were guaranteed the right to self-determination of government. Looking at the dubious politics of Argentina since 1945 it is hard to fault the Kelpers' choice.\n\nFor the wily bureaucrats of King Charles' Street this was just another intellectual challenge, the more so once the UN passed Resolution 2065 in 1965 calling on Britain and Argentina to find a negotiated settlement. For nearly twenty years the talks dragged on, the Argentines demanding, 'nuestras islas Malvinas', and the FCO trying to find more and more complex leaseback procedures to cede sovereignty to the Argentine without antagonising the stubborn Kelpers.\n\nBy 1981 the negotiations were in serious trouble on two fronts. In New York the talks were on the verge of breaking down in the presence of implacable Islanders' resistance. Even massive Argentine bribes of 'a million dollars a family' were allegedly rejected by the suspicious Kelpers who one source claims were insisting on 'a million dollars a head... but only for real Kelpers.'\n\nThe second fault line was the arrival of a new military Junta in Argentina in December 1981. Confronted by the aftermath of the 'dirty war' and the rising anger over 'los desaparecidos', the 'disappeared', General Galtieri's regime had to find a fast track to national unification. Like so many regimes before and since the Junta elected to use an external grievance to divert attention from domestic difficulties. Admiral Anaya, the Naval member of the Junta had a solution: reconquer the Malvinas. The Junta, facing rebellion on the streets, economic meltdown in the market place and with American assurances of 'standing together against communism in South America' saw an opportunity to boost their position and win popularity at home. Unfortunately they totally misjudged Britain's likely reaction.\n\nEarly in January 1982, Admiral Anaya's planning staff went to work on a secret operational plan to seize the islands. The strategic basis was to be a Goa-like operation, using naval operational assets to effect a swift _coup de main_ that would evict the tiny British garrison and take the islands for Argentina. Once seized, at 8,000 miles range the islands were too far from UK to be re-captured, said the planners. It was to be a diplomatic, political and military fait accompli.\n\nThere were even hints from the British that such a solution would not be an unthinkable turn of events. After all, the Argentine diplomats pointed out, had not the British elected to dispose of HMS _Endurance_ , the Falkland Islands guard ship? Had not the British Defence secretary, John Nott announced that the Royal Navy was to be run down, and the carriers sold off? All that remained, reasoned the Junta, was to test the British nerve.\n\n**_HMS_ Endurance, _guardian of the Falkland Islands. British hints that she was to be decommissioned gave the Argentines confidence that they could occupy the Islands unhindered_.**\n\n**_Members of the Buzo Tactico, shortly after the attack on the Royal Marine Garrison at Port Stanley_.**\n\nSouth Georgia, a set of islands some 800 miles to the East, deep in the South Atlantic, became the litmus test of Whitehall's resolve. On 19th March 1982 Senor Constantino Davidoff led his forty scrap metal merchants ashore at Leith on South Georgia, ostensibly to fulfil an old contract to dismantle the rusting whaling station. As Senor Davidoff's team was transported in an Argentine naval auxiliary, the _Bahia Buen Suceso_ and as they marched in step up the beach to salute the Argentinian flag which had been raised on one of the derelict buildings, few believed that they were peaceful civilians. Admiral Lombardo, Commander in Chief of the Argentine Atlantic Fleet was allegedly furious at this brazen provocation: but, as promised by his boss Admiral Anaya, no friend of the British, the British government reacted timidly. On 22nd March the 'scrap team' re-embarked and it looked as if the crisis was over. In fact, Davidoff's scrap merchants had already lit the fuze of war for not all the Argentines had left South Georgia. A small team had stayed behind. Exasperated, the FCO ordered HMS _Endurance_ to arrest them. Like some great chess game the Argentine navy promptly deployed war ships to 'protect their citizens'. Captain Nick Barker of _Endurance_ , an old South Atlantic hand - who had reported to the Admiralty as early as 25th January that, 'the Argies were up to something' - wisely took to his heels and on 26th March began playing cat and mouse with three Argentine frigates deep in the watery wastes of the Southern Ocean. The clock was now ticking.\n\nAstonishingly, the British government was still not ready for a war. Despite hard intelligence that the Argentine Navy's only carrier and submarines were at sea, despite reports that aircraft were on alert, that a Marine battalion was embarked and a SEAL team headed for Port Stanley, Whitehall's Latin American Current Intelligence Group - led by the ever-emollient FCO - blithely advised the Prime Minister that the Argentines were only sabre rattling and that 'no invasion was imminent'. It was a breathtaking miscalculation.\n\nOn 2nd April 1982, Commandos of the Argentine Marine Infantry supported by special forces of the _Buzo Tactico_ poured ashore at Port Stanley shortly after 6.00 a.m. and after a brief skirmish captured the tiny Royal Marine garrison who were outnumbered ten to one and without any fire support. The Falkland Islands Governor, Sir Rex Hunt, formally surrendered Government House at 9.30 a.m. The images of the surrendered British Royal Marines, face down in the dirt with Argentine rifles held by black balaclava clad Argentine Commandos pointing at them, sent shock waves round the world and shook the foundations of the Palace of Westminster.\n\nBut it was not just the British who had miscalculated. Faced by an angry House of Commons baying for blood, Prime Minister Thatcher did the only thing she could to ensure her government's survival. Buoyed up by the counsel of the First Sea Lord, Sir Henry Leach - who, cynics said, seized a chance to save his beloved Navy from John Nott's defence cuts -Thatcher elected to make a fight of it. Within a week a Task Force had sailed, with orders to regain the Falklands for the British Crown, by force if necessary. Operation Corporate was underway.\n\n**_Scenes that shook Westminster. Face down in the dirt Royal Marines are searched after the surrender of Government House_.**\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch2)\n\n[THE TASK  \nFORCE](05_toc.html#ch2)\n\nT **hree days after the Argentine capture of Port Stanley the first ships of the British Task Force - the aircraft carriers _Hermes_ and _Invincible_ \\- sailed from Portsmouth against a background of frenetic diplomatic activity. Initially the Task Force was a primarily naval effort, cobbled together without any real staff appreciation of the force levels and resources required to complete the mission. At this stage, no one actually envisaged a real shooting war. The swift - and undoubtedly rushed - departure of the carriers was dictated more by symbolism than reality. The message intended for Buenos Aires, and also at Britain's array of protagonists vying for a place at the top table of international relations, was clear. Britain would respond and respond forcefully. Britannia was brandishing her trident**.\n\nWithin a week of the high visibility departure of the aircraft carriers, the first units of the land based element of the Task Force sailed from Southampton. Naval and air power were essential to the mission but barring a diplomatic settlement or an Argentine withdrawal the British would eventually have to get 'boots ashore' in order to retake and clear the Islands. That would necessitate an amphibious assault the like of which had not been attempted by Britain - acting alone - since the Second World War. Given the nature of the task ahead, 3 Commando Brigade Royal Marines, (40, 42 and 45 Commandos) under Brigadier Julian Thompson were at the core of the land element. The 2nd Battalion of the Parachute Regiment, (2 Para) then on five days notice to move and its sister unit the 3rd Battalion (3 Para), then acting as 'Spearhead' and ready for immediate deployment, were extracted from the newly formed 5 Brigade and attached to 3 Commando Brigade. These five units, along with the Special Air Service (SAS) and Special Boat Squadron (SBS) were, in 1982, probably the most highly trained, efficient and physically conditioned troops in the British Army. Their reputation as tough, professional, highly motivated and experienced soldiers - the units had all seen active service in Northern Ireland - was recognised and respected worldwide.\n\n**_Soldiers of the Argentine invasion force pose for the camera._** John Hughes-Wilson\n\n**_The British Task Force sets sail from Portsmouth to retake the Falklands_.**\n\n3 Para and most of 3 Commando Brigade sailed from Southampton on board the 45,000 ton requisitioned cruise liner S.S. _Canberra_ on Good Friday 9th April amidst emotional scenes on the quaysides not witnessed for decades.\n\n###### DOWN SOUTH ON THE 'GREAT WHITE WHALE'\n\nBy Nick Rose\n\n**Nick Rose was a twenty year-old private in B Company, 3 Para in April 1982. Here he recalls the events surrounding the news of the Argentine invasion and the voyage to war**.\n\nWe were on notice as the 'Spearhead' battalion. There was always one Para battalion on 'Spearhead' - ready to go at short notice. I was on extra duties. I don't know what I'd done wrong but I was given 'extras' in the Sergeants' Mess, doing the dixies. A real dick job. I had extras the week before and I hadn't been home for more than three weeks. That was the deal. We near Andover, so me and some of the 'Barnardo's Boys' - the blokes who hadn't really got a home to go to - went out on the piss and started getting caned. Especially when they said we'd be confined to camp.\n\nI think rumour started which said you'd best go and look at detail - special detail, daily orders amendment. 'Khandahar', that was the code word I think - bringing everybody back that went out to the railway stations. It was '3 Para Khandahar' and that was the recall word. We were on 'spearhead', which was why we had a recall word. I can't remember who told me but it was word of mouth with blokes saying - 'F****** H***, we're going to the Falklands'.\n\nAll we heard was that Argentina had invaded South Georgia and the Falklands and some blokes were going, 'where the **** are the Falklands. Are they in Scotland?' It was like that. The Shetlands? The Falklands? For a simple 'squaddie' there's no great need for them to have a knowledge of world geography. I'm not slating the blokes at all but they'd probably sooner go and have a drink. Then I thought hang on where are they? We got an intelligence briefing, probably the day after. This was when it really had 'hit the fan' and the button had been pushed - we were definitely going. It was decision made there and then upstairs at Number 10 and Northwood because the turnaround time from when the Argentines first landed in South Georgia and the Falklands was quick. We were down there within eight weeks, which is pretty good going.\n\nYou must bear in mind the average age [of the paras]. I was 20. I didn't really know if it was 'Monday or the zoo'. I was naive in many ways of the world but I'd been well trained to do my job. I'd been to Ireland so I'd had experience of carrying loaded weapons and their potential and I'd 'savvied' up professionally. Everyone was like, 'Well OK but we've got to do it'. That's what you had to say. If it comes to doing it you've got to do it. But the further south we got and because the peace process was going on - they were really going for it - in a way everyone was hoping that well, maybe they'd cancel it. And this is when people started to think, 'look at us' and you look at the situation you're in, sitting on a massive cruise liner in the middle of the South Atlantic Ocean, sailing off 8,000 miles to fight an enemy that outnumbers you by x, y, z to one, firmly dug in and entrenched in perfect defensive positions. If it had been us defending those mountains no one would have been going anywhere. They would not have got past - would not have got through to Stanley - at least not via the mountains. Perfect defensive positions. So I was scared but I was excited with the adrenalin. There were some very intelligent people around and as we sailed on we used to sit and chat and after a while we'd think, 'it's really is going to happen. We really are going to war', so I was scared and I was excited but I was caught up in the belief that we'd have to do it so let's just get on with it and do the best we can. I think that was the general consensus. I am sure the overwhelming majority would say the same.\n\nSome people called the _Canberra_ 'The Great White Whale' but we just used to call it 'the boat' which used to piss the Marines off. Ships carry boats! We had the greatest respect for each other but we used to call them 'cabbage heads, 'leathernecks' and things like that. They called us 'cherry berets'.\n\nThe first week was just chaos. There was stuff on the ship that shouldn't have been where it was and it had to be moved elsewhere. There were tons and tons of ammunition and food to be moved. The logistics were unbelievable. It took a couple of days but they squared it.\n\nThere were four men in a four-man room - each man had his own bed. In a twoman room - like the one Stew Grey had - he had a water tap - water that you could drink and it was comfy enough. After the first week we established a pattern. Obviously ship's husbandry - that's what they called it - the washing up, the cleaning, still had to be done. The _Canberra_ had lost a lot of staff who'd done all these jobs, so we were given jobs. I used to work in the laundry. Me and my pals had the best pressed kit on the ship. I used to wear baggy OG's [olive greens] issued around the time of the Suez crisis I think. You had to be a bit of an old sweat to wear them but by that time I was a bit of a sweat because I had a 'badge', I'd been to Ireland.\n\n**_Task Force South. Private Nick Rose on the deck of the Canberra off Freetown harbour, Sierra Leone May 7th 1982. Note the training shoes. The Paras were not allowed to wear boots on deck._** Nick Rose\n\nUp at 7.00 a.m., queue up for your scoff at various times - A Company at 7.00, B Company at 7.15 - so that's 120 men, the full complement of a company with all its supporting arms. But you got through. 8 o'clock was muster. What to wear? Get your trainers on and your sweat top on and it was 'bang, bang, bang' round the promenade deck - a quarter of a mile each time. You used to get platoons of blokes just going 'bump, bump, bump bump', and the ship used to shake and vibrate. We could feel it when the Marines were doing their turn because it was the Para Regiment at x, y, z time and the Marines at another time. Sub units would be practising all over the ship. On the forward deck the anti-tank platoon would be doing demos on the Milan and the Wombat with the GPMGs doing their bit on the 'forward, sub aft deck of the overhead underhang!' But we eventually found our way around.\n\nIt was an emphasis on weapons familiarisation. Just going back over and over and over things - a sensible move - with a lot of fitness thrown in. That was the main thing, sensible fitness, and a lot of admin'- getting yourself squared up.\n\nWe got briefs about possible ways of going in the nearer we got. We knew we were not going to be airlifted in and that there was going to be an amphibious landing. I think this was established on day one at Northwood or somewhere else that the actual assault was going to be via landing craft from HMS _Fearless_ and _Intrepid_ and that some of the people who were going to be doing it would be the Parachute Regiment \\- the 3rd Battalion - so better get some training in on cross-decking at Ascension Island.\n\n**_3 Para lifejacket drill on Ascension Island. Left to right: Private Nick Rose, Private Harry Gannon, Private Steve Holland, Private Pete Partridge._** Nick Rose\n\nWe berthed at Ascension Island and I was on the _Intrepid_. We were practising with their LCU's (Landing Craft Utility). An LCU holds a platoon, the front goes down and its 'charge'. So we were practising getting in off the side of the _Canberra_ and if you look at the _Canberra_ its a huge great thing. Ten foot above the water line there's a big door - 10 foot high by 8 wide - which was one of the loading doors - an area where you can store stuff but obviously blokes can gather there as opposed to stores. This is where we learned how to get on this LCU and you've got to time it correctly because of the swell - massive ship, small ship - it's not rocket science. If you time it wrong 'bang' you're in there. Someone fell in between the ship and an LCU-he bust his hip or his pelvis - and the urban myth grew that he had been attacked by trigger fish - like sea piranhas. We used to catch them in our 'cam' nets, beat them up and throw them back in and watch the feeding frenzy that followed.\n\nAnd then you start getting company briefs and later start getting ammunition issued - 'divvying' up the 'link' for the 'Gimpys' [GPMGs - General Purpose Machine Guns] - that man's got the '66s' and you're given two mortars to carry onto the landing beach from the LCU. That's when you finally think to yourself 'this is a 'go'.\n\nDenuded of 2 and 3 Para, 5 Infantry Brigade, consisting solely of 1st\/7th Gurkha Rifles, was made up to strength with the addition of the 1st Battalion the Welsh Guards and the 2nd Battalion of the Scots Guards fresh from ceremonial duties at Windsor Castle and Buckingham Palace. 5 Brigade was considered an ad hoc formation, designed initially as a reinforcing garrison unit once 3 Commando Brigade had done their job. As the Task Force sailed south from Ascension Island however, and diplomacy trickled into the sand, a grimmer reality emerged and as that reality set in, so the Task Force began to look woefully inadequate for the job at hand. For a start, with only 36 Harriers, there were too few planes. At no time did Rear Admiral Woodward, Commander of the Carrier Battle Group, have anything like total air superiority. With his two carriers acting as the Task Force's only airfields, Woodward prudently kept them well to the East, earning the derisive comment from some wag that he deserved the 'East Africa Star' not the South Atlantic Medal at the war's end. However, he was, like Jellicoe at Jutland, 'the only man who could have lost the war in an afternoon.' Unlike Jellicoe, Woodward won.\n\n**_HMS_ Intrepid. _One of the two assault ships with the Task Force. 3 Para practised cross-decking to the_ Intrepid _from the_ Canberra _in readiness for the amphibious assault_.**\n\nThe Task Force was finally committed to battle on 1st May 1982, following the failure of the US Secretary of State Al Haig's diplomatic efforts to get the Argentines off of the islands. The British slapped a 200 mile Total Exclusion Zone (TEZ) around the Falklands and flew an astonishing Vulcan bomber mission - with no less than seventeen air to air refuellings - from Ascension to crater the runway at Stanley. It failed but the war had started in earnest. If the Argentines wouldn't leave voluntarily, then Whitehall would blockade them on their ill-gotten gains.\n\n**_Crippled. HMS_ Sheffield _falls victim to the first successful Argentine Exocet attack_.**\n\nBlow followed blow in quick succession. On 2nd May the elderly cruiser General Belgrano, which in a previous incarnation as the USS _Phoenix_ , had escaped destruction by the Japanese at Pearl Harbor, was sunk by a British nuclear submarine HMS _Conqueror_ , effectively closing off the southern prong of the Argentine Navy's pincer attack on the Task Force. In the north, the carrier _Veinticinco de Mayo_ , fled back to harbour. Whatever else, the Royal Navy's nuclear submarines had secured supremacy at sea.\n\nThere only remained the air threat; and on 4th May two land-based Mirages riposted with the first successful Exocet attack, crippling HMS _Sheffield_. The shock wave reverberated throughout the Royal Navy. Suddenly the Task Force looked very vulnerable indeed. Years of Treasury penny pinching and cost cutting suddenly looked like risking every ship afloat, as fire boiled through _Sheffield_ 's flimsy hull feeding off cheap plastic electrical trunking and melting sailors' acrylic trousers into their very flesh.\n\nFor the next two weeks the Navy's tactics changed to a series of night bombardments designed to put pressure on Argentine positions ashore. At dawn the Task Force would make a rapid withdrawal to the East to re-supply at sea and continue the nerve-wracking lookout for the next Argentine air attacks. Time was not on the Task Force's side. The steady loss of key ships, particularly the Type 42 destroyers, sapped the nerves of the crews and the crippling lack of any early warning radar meant that the ships were always vulnerable to air attack. The truth was that as a strategic deployment to create the conditions for an amphibious landing, the Task Force had achieved little. The Commando Brigade and the rest of General Jeremy Moore's soldiers were not ashore, and the Argentine Air Force still remained, formidable and dangerous.\n\nTHE AIR WAR OVER THE FALKLANDS\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch3)\n\nTHE LANDING\n\nI **n London, at Joint Headquarters Northwood and afloat in HMS _Hermes_ , the war planners' options were running out. The British needed a new war strategy, and fast. As the Navy's demonstration of force had failed to frighten the Argentines off the islands, it seemed that only a full-blooded amphibious assault and land battle could clinch a decisive victory. On 21 st May - D-Day - the British forces went ashore at Port San Carlos, San Carlos and Ajax Bay on East Falkland in what was, with hindsight, an extraordinarily bold, three phase operation. Brigadier Thompson's 'design' for battle involved a silent attack by landing craft during the hours of darkness with all high ground being secured by dawn**.\n\n3 Para were to be 'cross decked' from the _Canberra_ to the assault ship _Intrepid_ and landed near Sand Bay and Port San Carlos - codenamed Green Beaches One and Two - by landing craft during the second phase of the operation to secure Port San Carlos to the north. Led by B Company, whose task was to secure the beach itself, the rest of the battalion were then to push through and dig in on the reverse slopes of the high ground at Windy Gap and Settlement Rocks to secure the beachhead. Scimitar and Scorpion Light tanks of 4 Troop, the Blues and Royals were to accompany them to soften up any Argentine resistance. If everything had gone according to plan all 671 men of 3 Para, including a forty man attachment of 9 (Parachute) Squadron RE, would have been loaded onto the landing craft by 4.15 a.m. and ashore by 6.05 a.m. but the first wave landings were delayed. With first light at around 6.30 local time, this meant that 3 Para would now go ashore at around 7.30 a.m. in daylight. Australian director Peter Weir's film 'Gallipoli', had already been screened on the voyage south and in order to counteract any complacency on the part of his men, Lieutenant Colonel Hew Pike, their commanding officer, had already warned that the landing might have more than a whiff of the ill-fated assault on the beaches of Gallipoli 67 years earlier. He also came on the _Intrepid_ 's 'pipes' and paraphrased the words of Brigadier James Hill, Commander of 3 Parachute Brigade, who had addressed his men on the eve of the Normandy landings in June 1944: 'Do not be daunted if chaos reigns, because it undoubtedly will.'\n\n**A. _Assault ships_ Intrepid _and_ Fearless _move into position. Troops transferred to Landing Craft_. B. _3 SBS dropped by helicopter and attack Argentine unit on Fanning Head_. C. _Troops land and push on to secure high ground_. D. _Argentine Combat Team 'Eagle' shoot down two Gazelle helicopters as they retreat from Port San Carlos_.**\n\n* * *\n\n**'Do not be daunted if chaos reigns, because it undoubtedly will.'**\n\n* * *\n\nAs the convoy of assault ships nosed into San Carlos Water the sound of a firefight could be heard off to their left from the direction of high ground of Fanning Head accompanied by naval gunfire support from HMS _Antrim_. Here a patrol of 3 SBS, which had been airlifted ashore earlier, were engaging an Argentine combat team known as 'The Fanning Head Mob' in order to secure the approaches to San Carlos Water. Another Argentine combat team - 'Eagle' - were also in the area of Port San Carlos, and although in the event the landings were unopposed as the Argentines withdrew to the north, the small force of 41 men could have caused mayhem on Green Beach. This point was demonstrated to lethal effect after the landings as that same Argentine combat team, moving away from Port San Carlos, fired on and brought down two Gazelle helicopters and damaged a third. Three crewmen died.\n\n**_3 Para get 'boots on the beach' at San Carlos Bay after an 8,000 mile journey. At this point they are most vunerable to Argentine attack_.**\n\n**_3 Para RSM Lawrie Ashbridge, is received with a warm cup of tea and an even warmer welcome by a Falkland Islands family at Port San Carlos._** Tom Smith, Daily Express\n\nAt this point the landings were at their most vulnerable from air strikes but after an uncomfortable and nerve wracking run in, which seemed to take an age, the bow ramps of most of the landing craft crunched down onto the beach. 3 Para were ashore at last. But as 3 Para dug in their troubles were only just beginning. Eight thousand miles from home, with limited logistic and ammunition stocks and without air superiority the assault from the sea was at risk from the moment it was launched. Most of all it was at risk from the air.\n\nThe Argentine counterattack was swift in coming. Flying at suicidally low altitudes to attack the targets anchored in the sound, the Argentine pilots hurled themselves against the ships offshore. Despite a blizzard of anti-aircraft fire the young pilots pushed home their attacks with great bravery. Many were shot down, but to the British soldiers ashore trying to build up their positions, the vital shipping in San Carlos Water seemed to be being badly mauled.\n\n##### **HITTING THE BEACH**\n\n**Ashore at Port San Carlos - D-Day 21st May 1982**\n\nBy Nick Rose\n\nWe'd got a standing platoon in the landing craft I'd say. This was supposed to be a night landing. Major problems somewhere along the line and by the time we'd jumped on to these little boats it was daylight. Great news! This is supposed to be a night assault and we had no direct support as such. As we started to turn to go in there was a big firefight going on at Fanning Head - Special Forces guys taking on these forward recce' Argies. From the intelligence that filtered back we thought the landing would be unopposed. That's why Special Forces were over there. If they hadn't taken on those guys on Fanning Head then it would have been an opposed landing. But then would it? The Argentines were forward recce' guys and they wouldn't have wanted to engage a battalion sized assault as a small platoon, or 20 man, section strength fighting patrol. Good job Special Forces took them out though. They saw them off and then we went in but it was broad daylight and it was, 'Oh God' because there was air raid warning 'red' all the time. They'd tried it before and they were going to try it again soon. It was a serious situation and then to get wet boots and freezing nuts on top of it.\n\nWhen we actually came to do it for real the front of our landing craft wouldn't go down so we went over the side. There's two marines, one on either side at the top of the landing craft with two GPMGs going 'get the fuck out of here.' Not a great start - wet boots and freezing nuts! We were only about four or five feet out but it was up to your nuts.\n\nI was carrying my bergen with everything you need for three or four days - cold weather kit, ammunition - when we went in we probably had 500 rounds linked for the guns - and each man had a mortar tube - 81mm SLR [self-loading rifle], webbing stuffed with ammunition, water, survival kit. You wanted to carry as much as you possibly could.\n\nThere have been vast exaggerations about the weight carried - honestly some of the claims would be just physically impossible to do - but on average each man was carrying 140 lbs, which is a ten stone man. O.K. it's distributed across your shoulders because it's designed that way and it's a very good way of carrying a lot of weight for a long time. Webbing likewise if you've got it correctly fitted. I used to sew my water bottle pouches together - sew it up with fishing line - so there was no 'jacking' around, padded it out on the inside, then' cam'd' it all up. The blokes who 'savvied' up were a lot more comfortable than the guys who didn't. You've got to pack a bergen correctly as well. If you went down on your back you had to be pulled up by another bloke, you had to, you could not physically do it yourself. You were just like a tortoise on its back.\n\nWe all had pre-designated areas at Port San Carlos so there was a lot of talking about the landing and a lot of orders. This is happening - now we are going to war. We are going to close with and destroy the enemy, no messing about. The Special Forces guys were over there and patrols had been in the night before and had marked out where we were going to go with a markalite path. Then we advanced up to the top of this bare-arsed, featureless hill - horrible - but we had to get up there because you've got your view and you can see quite some way. 'Bang', get in there - dig in - get yourself squared up. Dig in - that was it and that was a nightmare because it was bog where we were. You dug down a foot and you hit water so the idea was to try and build your position up a bit and pack the bottom of your 'hole' with moss and heather. We were on the reverse slopes at Windy Ridge. We tried to establish a routine, we couldn't advance too far because we were waiting on others to come in.\n\nThen the aircraft started coming in and that's quite scary. Every ten minutes you'd get, aircraft warning 'red' or air raid warning 'red' because we had Rapier [surface to air missiles] and the climate down there does not lend itself to sensitive equipment as was the Rapier then, electronically speaking - a lot of the stuff is now obsolete and the electronics can be duplicated on something the size of your watch but the Rapier was a great big thing. I'm not slagging it off - if it helps save my life, thank you very much - but it was very sensitive. So every ten minutes the alarm would go off on the Rapier and we'd stand to and people just became complacent. Like crying wolf. But there was a part of me which was always wary. On occasions these Skyhawks would come over and its quite shocking because wherever you are, when you look at one, it looks as if it's coming directly at you. It's something of an optical illusion, albeit it's 200\u2013300 feet above you. It's going about 5\u2013600 m.p.h. and its just loosed off a load of 'shit' at the boats in the Sound - that's quite a scary thing but then, 'da-da-da-da-da' blokes are having some, despite being specifically told, 'do not engage aircraft with small arms fire', because - well, there's blokes over here, blokes over there, blokes everywhere - there was the risk of hitting your own blokes. But everyone starts 'lamming' it up there.\n\n**_A soldier from the Parachute Regiment in full battle dress. The Paras had to carry heavy bergens weighing up to 140lb containing everything they needed to survive and fight in extreme weather conditions._** Illustrated by Jon Wilkinson\n\n**_The Rapier (surface to air missile) in action. The weather played havoc with its sensitive electronics making it prone to give off false air raid alarms. Although a potential life saver, this tended to make the troops complacent_.**\n\nThere was a guy with a GPMG and he gave it 'the max' from the hip. Mark Hamill - a big bloke was Mark, a PTI and a very hard but a very fair man - a good soldier - was on a Milan and he fired at a Skyhawk and missed it. Now there's \u00a310,000 every time you fire one of those down the range. So Hamill just fired this and he missed the aircraft. Can you imagine? He got a major bollocking but imagine if he'd hit it? - Victoria Cross, 'well done'. So yes we'd give it some rattle when these things came over but it was very scary, because we did feel we'd lost momentum. I'm sure word was coming down from Brigadier Thompson or whoever to start pushing on. We had to start moving on. It just felt that we'd been there a while. The Argentines knew we were in this position, they knew where we were displaced so it did feel we'd been there too long. So let's get on and start the advance.\n\nAs with so many air battles however, the decisive fighting was taking place well away from the surface observers and in reality RAF and RN Harrier pilots were catching many of the Argentine attackers both inbound and outbound from their bombing runs on the ships. The Harrier, like its feathered namesake, was turning out to be a deadly close quarters aerial killer. The British pilots were using new American Sidewinder missiles, - although one pilot shot down a Skyhawk with his cannons at a range of less than 100 yards - and the air kills ratio was firmly in the British favour. By the end of the first day the Argentines had lost 16 attacking aircraft: an unsustainable loss. For the next three days the air assault continued, but slowly ebbed away. By 25th May there was even talk of having 'beaten the air threat'.\n\nAll that changed on 25th May, Argentina's National Day. At first, the British thought that they had defeated the early waves of attacking aircraft. Then the Argentine pilots delivered a decisive and crippling double blow to the Task Force, sinking another Type 42, HMS _Coventry_ , and, much more ominously, the most important supply ship afloat, the 13,000 ton _Atlantic Conveyor_ , which went down, taking with it ten Wessex and three Chinook helicopters. Brigadier Thompson would later record the 25th as 'a black day.' The British strategy was now in serious jeopardy and from that most serious of military factors: logistics.\n\nThe war now became a race against time. In London enormous pressure was put on the commanders ashore to 'do something'. Fears of a UN imposed cease fire with the Task Force still sitting on the beach head, and public clamour for results after the Navy's heavy shipping losses, forced Brigadier Thompson, and his commander, Jeremy Moore - against all their instincts and training - to embark on a series of rushed and poorly supported ground operations. This entailed 2 Para being ordered to attack the Argentine defenders of Darwin-Goose Green while the rest of his force embarked on the first stage of the 'investment' of the capital Port Stanley on the other side of the island. The breakout would soon be under way. With the loss of most of his helicopter transport Thompson's original plan to airlift the majority of his brigade forward was now in tatters. There was now only one way to get 3 Brigade forward and for 3 Para that would mean 'tabbing' it - walking every step of the way. Even so they were more than ready for a change of scene. Graham Colbeck, a Sergeant in the Milan anti-tank platoon, concluded his diary entry for 26th May with a sentence which summed up the feelings of his comrades. 'Everyone here just wants to get to Stanley and finish it off.'\n\n**_Royal Navy Sea Harriers on patrol. The Harriers were the saviours of many ships as they intercepted Argentine airfcraft bent on inflicting damage to the Task Force_.**\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch4)\n\nTHE 'TAB'\n\nT **he order to move out of the Port San Carlos beach head came on 27th May. It was D + 6. Instead of waiting for the arrival of 5 Brigade as they had expected, 3 Para were now told to advance and capture Teal Inlet, a settlement some 25 miles east of their present positions. This area had been earmarked as a forward logistics base for the final phase of operations to capture Port Stanley and had already been reconnoitred by the SBS. The axis of the advance chosen by Brigadier Thompson lay north of the spine of high ground that ran across East Falkland from San Carlos Water in the west almost to the outskirts of Stanley in the east. 2 Para had already moved out from their beach head on the other side of San Carlos Water on the first leg of their fateful journey to Goose Green and now it was 3 Para's turn**.\n\n* * *\n\n###### THE 'TAB' ACROSS EAST FALKLAND 27th\u201331st MAY 1982\n\nShortly after dawn, 45 Commando set out to 'yomp' - heavily laden with all their kit like pack animals - along a recognised track to Douglas settlement via New House, followed shortly afterwards by 3 Para who were to 'tab', in 'light order' - no bergens, no sleeping bags, no tripods for the sustained fire role GPMGs - in their wake as far as Douglas, before passing through to take the lead. Lieutenant Colonel Pike, had, however, discussed the route to Teal with Mr. Alan Miller, the civilian manager of the Port San Carlos settlement and had learned of a more direct route, albeit not using recognised tracks. The mortar section also hitched a lift on Mr. Miller's two tractors, one of them driven by his son Philip. Even so, they would have to rely on the now tenuous logistics support to 'chopper' their kit forward to prearranged locations. And so it fell to 3 Para to blaze the trail. The march was a supreme test of 3 Para's physical and mental stamina across more than twenty miles, in just two stages, of some of the most featureless and spirit-sapping terrain, in what some men could only described as 'bizarre' climatic conditions. Men called on inner reserves of personal pride and networks of mutual support: they drew heavily from the well of their tough training regime and regimental _esprit de corps_ as the weather threw everything at them over mile after endless mile of stumbling over tussocks of pale grass and foot snagging heather. The 'tab' was no respecter of previous achievement. Men seen as some of the fittest in the battalion, due to their scores in the British Army's conditioning tests, became exhausted and there were several who turned their ankles as sodden socks slid around in the universally loathed and equally soaked DMS [direct moulded sole] ankle boots. Trench foot, a condition well known to the men of the British Army of 14\u201318, now beckoned to these men of '82.\n\nThe battalion reached its objective on 29th May, secured it and stayed there for twenty-four hours before embarking on the next stage of their odyssey on 30th May. The objective was Estancia House, a tiny settlement consisting of the eponymous house itself, a barn and a few outbuildings that were reached on the night of 31st May\/1st June. Intelligence had warned of up to 300 Argentine troops in the area but by the time 3 Para arrived they had disappeared towards Stanley. Several Argentine prisoners were picked up on the way as 3 Para finally caught up with stragglers from Combat Team Eagle who had withdrawn from the area of Port San Carlos on D - Day. Also picked up on the way, and a most welcome addition to the ranks of 3 Para, was Terry Peck, the former Falkland Islands Chief Police Officer, member of the Falkland Islands Defence Force and member of its Legislative Council, vehemently opposed to any moves legitimising Argentine sovereignty over the Islands. Peck had already witnessed one Argentine invasion of sorts, when, in September, 1966, a civilian Dakota airliner was hijacked by armed Peronist youths over Patagonia and landed on Stanley racecourse. As Chief Police Officer Peck became involved and at one point had a gun pointed in his direction. Now, during this invasion he had determined not to remain impotent in Stanley as the Argentines swarmed around town and so escaped on a motorbike. Riding west he eventually made contact with elements of 3 Para left behind at Port San Carlos after the rest of the battalion had set off for Teal Inlet. Airlifted forward he became an honorary member of D Patrol Company, whose members used his intimate knowledge of East Falkland's geography to their advantage in the days ahead.\n\n* * *\n\n**All you're thinking of is 'I'm carrying a 10 stone man on my back** ,\n\n* * *\n\nThe importance of securing Estancia House and the high ground of Mounts Estancia - (occupied by A Company) and Vernet (C and elements of B Company) which lay to the north east, was essential in building a seven mile long, north - south platform of high ground - a platform which included the strategically valuable and already occupied heights of Mount Kent - from which 3 Commando Brigade could launch the assault on the outer ring of the Stanley defences. The Estancia position itself was situated on the shore of one of the fingers of several inlets, which branched off southeast from Teal and which pointed towards Stanley, now only 15 miles away as the shell flies. It was essential as it reduced the supply line from San Carlos by almost a fifth, as a round trip from Teal to 3 Para's area of operations was now only 15 miles as opposed to 100 from San Carlos. In addition, the Estancia\/Vernet position was vital in anchoring the northern axis of Brigadier Thompson's plan for a two-pronged 'investment of Stanley'. Its importance was further enhanced as the Argentine commander Menendez became ever more convinced - particularly after 2 Para's victory at Goose Green - that the eventual British assault would be unleashed along the Fitzroy - Stanley track to the south.\n\nThe first snows of the South Atlantic winter fell on 1st June and in the momentary lulls between bouts of bad weather and poor visibility, 3 Para observers on the higher ground caught glimpses of Stanley a tantalising fifteen miles to the east and first set eyes on a feature bristling with jagged outcrops of rock jutting out at a crazy angle and rising to just under 500 feet high and about 500 yards across. It barred the way to Stanley and dominated the northern claw of Thompson's pincer as far back as the Estancia position and even to Teal Inlet beyond. Looking at the map 3 Para checked its name. Mount Longdon.\n\n###### **'DRY FAG, WARM BOOTS AND A CHEESE AND ONION SANDWICH.'**\n\n**Memories of the 'Tab' across East Falkland**\n\nBy Nick Rose\n\nThe march was a nightmare. This was our planned route to Stanley. I know it was supposed to be something like 60 kilometres but somebody 'guesstimated' that we probably did about 400 because we were doing a lot of forward patrols - not just D Patrol Company. You've got to dominate the ground so you have to have fighting patrols. Going cross-country was the worst terrain in the world. It was that clumpy moss and grass. People are falling down and ankles are going and once you're down on your back you've got to wait for the next guy to come up to help. And it is a tactical advance. 'Tabbing' is a name that comes from the letters TAB - Tactical Advance to Battle, - and we tabbed and carried everything with us. It may be that you have arrowhead formation going across, we had extended formation. We had picketing with guys on the high ground to either side. The terrain dictated exactly how we advanced. A lot of the time if we've going along on tracks - what few tracks we did go on - we used Indian file, which is staggered file on either side of the track, like a zigzag. But there are great rivers of rocks [stone runs] - big white boulders - and you have to cross them and then there's the heather and the gorse and its constantly wet. So the wind chill factor was - I think somebody said - 40\u00b0 - and storm force winds and horizontal rain - a nightmare scenario. People got sunburn one day - no air pollution - bizarre! It was a terrible advance - a mind numbing, bone-jarring advance. What do people think of? It was just head down, thumb up your bum, mind in neutral and 'go'. You've got to carry it. All you're thinking of is 'I'm carrying a 10 stone man on my back, I'm carrying a 10 stone man on my back'. Some of the guys, like the anti-tank or probably the mortars hitched lifts. They're 'blaggers' but fair play to them, there's no way you can lump base plates and that over that terrain. Even the GPMG guys in a sustained fire role - with the tripods.\n\n**_On the 'TAB'. Nick Rose's best friend, Tony 'Fester' Greenwood brews up under their 'basha'. Note the large boulders of the stone run behind them; a feature which helped to make the march extremely difficult._** Nick Rose\n\n**_6 platoon B Company dig in and take a break. Left to right: Corporal 'Granny' Paterson, Private Den Dunn, Private Tony Greenwood and Nick Rose._** Nick Rose\n\nThere were some Scimitars and Scorpions with us. Once we were doing an advance or a recce' patrol and we were coming back on a track. We were Indian file - remember, we're still tactical paras, we're switched on - and this Scimitar pulls up and it's the bloke in charge. We are all horrible, we're miserable as sin, all of us - we're missing home, want a dry fag, warm, dry boots, a cheese and onion sandwich and a bottle of blue top milk. I used to dream of these. Anyhow this bloke pulls up and pops out, 'what, what', and says would you like a lift. Oh yes please! So eight of us jump on and there are these vents at the front of the Scorpions and Scimitars blowing out hot air and we all just sat on there and we literally dried out. This bloke took pity on us and got his lads to make us a brew, because the Scorpions and Scimitars carry a massive boiler and they've always got constant hot water. It was no odds to him to give us some to make a cup of tea. He said he'd give us a lift back a little way. Turns out his name was the Lord Robin Innes Kerr. A genuine, nice bloke. I'd like to thank him for that. What a lovely gesture. I remember another thing - literally a lifesaver - on the other side of the Murrell River. It was a guy called Lawrie Ashbridge, the RSM, who sanctioned it. We were advancing in the dark and had been marching for 24 hours, we had to get up to the start line, or the preparation for moving on to the final start line - the FUP [Forming Up Position] - we were soaking wet and this was the most inhospitable weather we'd had since we'd been there. It was a nightmare - 'tramp', 'tramp' - blokes just moving inches at a time but an entire long snake of blokes doing exactly the same. You can imagine the speed of advance was minimal and the weather was horrible so Lawrie Ashbridge made a decision and told us to get into the side of the road - because we were still tactical, this was night time - get down by the side of the track, find as much cover as you can, get your ponchos out and get a brew on. So every man just got down, poncho out and brewed up. That, I am sure, saved people's lives, I'm sure of it. As a morale booster it was the best thing he could have done. If they are going to see us it doesn't matter if they smell us as well. So blokes are having a fag under their armpits. But such a simple thing as that can change the fortunes of war, or battles anyway. This was at the stage just before where we went across [the Murrell River] and started the advance. Perhaps two days before the battle.\n\n**_Lord Robin Innes Kerr (right) and crew aboard their Scimitar_.**\n\nOn one occasion we were on a LRRP-Long Range Recconnaisance Patrol, just four men watching - supposed to be relieved after 24 hours. We were out there for two days. We'd got no food so we were eating our sugar - sachets of sugar - dried milk, scrabbling around desperately trying to find something to eat.\n\nWe were then told to stop. We'd advanced too far which is all very 'gung ho'. This was on the way in to the final destination. We were told to 'go firm' because the Marines couldn't keep up. I'm not saying that because the Marines are wankers because they're not - far from it - but I think they had a logistical problem so they couldn't advance and if we'd have carried on at the pace of advance that we were doing, our only option would have been to attack - just carry on going, just roll. But we were so over extended - we had no artillery support at that time - so I think Hew Pike got a bit of a bollocking for being a bit naughty but I think it had just got to the stage where they thought - let's just go for it!\n\nI think we knew from the outset that our final objective would be Longdon after reaching Teal Inlet and then Estancia House. It was advance to contact because we didn't know if the Argentines had pulled back, we didn't know if Estancia was occupied because SAS had given word that maybe there were 4 men there. I can remember lying in the FUP now, at this tree line - a fir tree, big bushy thing - looking out over towards Estancia, just waiting.\n\n###### **A 'CIVVY' IN 3 PARA**  \n **Patrolling Towards Estancia House with D Company**\n\nBy Terry Peck\n\nTouching down at Teal Inlet, I was taken to 3 Para HQ and introduced to Lieutenant Colonel Pike. The HQ had been set up in the settlement store. There were several Officers and NCO's present and most were just finishing a brew. Someone kindly provided me with a 'mixed' meal of 'compo' and I was then introduced to Major Pat Butler, OC D (Patrol) Company. He explained that we would be moving out very soon. During the next few weeks I would get to know Major Butler quite well. He was a man of great courage and stamina, whose first consideration was for the men in his command. I was aware that my presence was the cause of some curiosity among many of the soldiers but there was little time to sit and chat as orders were being given to start their move. I heard a few angry words being expressed about someone who had had a 'negligent discharge' and the guy had been 'casevac'd' with a gunshot wound to his foot.\n\n**_Terry Peck joins D (Patrol Company) of 3 Para for the battle on Longdon, seen here in the background._** Terry Peck\n\nTeal Inlet was now the front line for 3 Brigade. The settlement was alive with Paras and Royal Marines, together with their Support Companies. Major Butler detailed me to 42 Alpha Patrol, one of the many 4 man teams which constantly carried out reconnaissance ahead of the battalion. Before I could join them a Royal Navy Officer came up to me and wanted to know if the enemy had mined anywhere in Salvador Waters. He had known of my motorbike journey in the preceding weeks and my contacts at Rincon Grande. I explained that to my knowledge the waters were safe. No one in Rincon Grande or Salvador Farm had seen any signs of the enemy operating in the Waters. I had to assume it safe. By now 3 Para had already set off on the gruelling 'tab' to Estancia Farm which would be the next objective. In single file the Battalion were strung out for over a mile. The ground was white with a covering of snow about two inches deep. Each followed in the tracks of the man ahead. Racing along the flanks of the column were a couple of Scorpion tanks giving cover to the advancing troops. Everyone was given a 'ten minute' break every hour or so. Although the pace was slow we were covering the ground very steadily. Fortunately the weather remained kind; an overcast sky and little wind. From time to time, the sound of gunfire could be heard in the distance. I heard from some of the lads that several Argentines had been captured and were being passed down the line.\n\nI had been attempting to make radio contact with Brookfield Farm and Estancia Farm but my batteries had gone flat on me. Obviously they were past their best and were unable to hold a charge for long.\n\nDuring my second 'break', halfway up the north side of the Mallow Hill, I sat down with a few of the lads and began talking about the recent events. I realised that these were the same chaps whom I had met on my arrival at Port San Carlos a few days ago. I was introduced again to 'Blue' Harding, Pete 'Six Feet' Harden, Joe McKeown, Taff Davis, Dickie Bishop, Raj Rajput and Pete Deakins. I explained that I was a local and had left Stanley several weeks ago, hiding out in the hills. Having learnt that the British Forces had landed, I had made my way to Port San Carlos where I had been invited to join 3 Para. \"That's some story\", said Blue, \"We reckon you're one of the boys who've been having themselves some fun.\" I laughed and told them that I really was a local, and not part of the Special Forces. Did I really look like an SAS or SBS man? It was several days before they accepted that I really was a local.\n\nIt was now late afternoon and we had to make the Mallow Bridge before dark. I was still trying to make radio contact with the Estancia Farm and I would leave the column of troops and make for the high ground in order to get a more direct line of sight to transmit my signal. No doubt the lads thought this strange and probably added weight to their suspicions. Just on dusk we topped the hill and could see the bridge below us. We rested up for an hour at the bridge and everyone set about making a brew. Darkness had fallen and orders were being issued to restart our advance. Everyone formed up in the various units and we set out on the next phase of the march. We knew that the enemy had troops on Mount Vernet, Mount Kent and the Estancia Mountain. I was aware that an enemy patrol would sometimes visit the Estancia Farm. We would be very exposed to the enemy as we advanced across the low ground, as each of the mountains overlooked the land to the west, from where our approach was being made. Through the broken cloud the moon would appear, bathing the ground in a pale light. Everything was going to favour the Argentines. But no, in typical Falklands fashion the weather changed and we found ourselves battling against driving hail and sleet. I had eventually made contact with Trudy Morrison at Brookfield Farm. The transmission was very broken, making it difficult to copy. I gathered that the enemy had visited Estancia Farm the previous day. They were seen to remain for some considerable time in an area east of the farm. She thought they had been sowing 'tatties' (mine laying).This turned out not to be the case.\n\nProgress had slowed, not surprising given the terrible conditions. In the best of weather the terrain we were marching over was some of the worst land in the Falklands. Small steep hills covered in rocks, 'diddle dee' bush and ferns. The deep valleys were mostly flooded or covered in huge bogs. Every man was fully laden, carrying approximately 100 lbs or more. Those carrying the radio equipment far exceeded this weight. I realised that those leading the column were well off course and that if they continued in this direction we would find ourselves cut off by the sea. If one had looked in the direction of Estancia Farm one could have been forgiven for thinking it was a straightforward march but it is deceiving. The land is broken for miles around by tidal waters with many inlets. Long peninsulars stretch towards the sea and there are very few short cuts if the tide is high.\n\nEveryone had gone to ground and apart from a few muffled curses and the occasional 'clink' from someone's kit, all was very quiet. A figure appeared out of the darkness, asking, 'Anyone seen Terry, the civvy? The Boss wants him up front.\" I clambered to my feet and made myself known. I followed the lad to the front of the column, where I saw several chaps hunched over a map laid out on the ground. \"Can you give us some idea where our present position is?\" someone asked. In the light of a small torch I pointed out where we were, explaining that we were some 3 miles too far east. We would have to do a sharp right and head west, then turn due south. We would then avoid the inlets and make a crossing at the head of Long Creek. \"Would you mind taking over the lead?\" This was the first task I had been invited to undertake so I thought I had better get it right. The word was passed back through the troops that there would be a change of direction, which I'm sure wasn't well received. From memory I visualised our position. We were well down the peninsular and to get back on track we had to move right, making our way along the steep slopes of the peninsular, then swing left towards the distant hills, then left again to bring us near Long Creek. It was a tough detour in the dark, made worse by the steep slippery peat banks, which were 3\u20134 feet high. Slippery rocks and water filled holes added to the problem. At one point I found myself face down in the mud. I had been walking along the top of a peat bank and tumbled over the edge, knocking the wind out of myself. Someone grabbed me by my bergen, and pulled me to my feet and, having managed to get my wind back, we were off again. There must have been many a lad that night who experienced that same horrible feeling of falling down into a black abyss. It may have been only a few feet, but it felt like fifty and with the added weight, the landing was extremely painful.\n\nA few times the word was passed for me to 'slow down'. The sleet and hail had eased off but it was freezing cold. At last I found the vehicle track. Although it was wet and muddy it did make the going easier and I'm sure many were glad to get on to a recognisable route again. The tracked route led towards a creek and making good time we headed down towards the beach. The tide was low so I informed JP (Sergeant John Pettinger) that we could safely cross over the creek, which would cut at least another mile or more off our journey. We finally made base at the head of Double Creek, about two and a half miles south west of Estancia Farm.\n\n**_Corporal John Graham and Terry Pock patrol towards Mount Longdon_. John Graham.**\n\nThe weather had deteriorated very quickly and everyone was totally exhausted but this didn't stop Major Butler issuing orders for our Patrol to carry out a reconnaisance of Estancia Farm. I was to accompany them. Our 8 man patrol was made up of Corporals Dickie Bishop, Joe McKeown, 'Blue' Harding and Pete 'Six Feet' Harden and Privates Pete Deakins, Raj Rajput, 'Taff' Davis, and Dickie Absolon. Leaving the area of Long Creek the patrol spread out in single file with Pete Deakins and myself leading. The wind had dropped but thick snow was still falling. We had covered about half the distance to Estancia Farm when all hell let loose to our front. Bursts of automatic gunfire had opened up ahead. Going to ground we waited until the firing had stopped. Nothing could be seen and the fire had not been directed at our patrol. Using his IWS (individual weapon sight), Pete was unable to pick up any signs of movement so Corporal Bishop gave the order to move on. Being familiar with the layout of the farm, Pete passed his IWS to me. We had now come within a quarter mile of the farm. There had been no further shooting. Lying up on a small ridge I studied the layout of the farm. Apart from a few flurries of snow the worst of the storm seemed to have passed and it was easy to make out the buildings. I could observe no movement - everything looked OK. We had to climb over a couple of wire fences before reaching the first of the outbuildings.\n\n**_Private Richard Absolon. Sniper of D Company, killed the day after the battle for Longdon. Absolon's exploits prior to and during the battle earned him a posthumous MM_.**\n\nThe first, a large Nissen hut used as a shearing shed, was checked out by three lads. They emerged after some minutes, giving it an all clear. The rest of the patrol had taken up an all round defensive role during the search. Next it was on to the smaller sheds and having checked them out we made our way to the main house. Two other store sheds adjacent to the house had to be cleared but these were found locked. The patrol took up various defensive positions around the house, which was in total darkness. There was an eerie silence about the place. Sometimes the moon would break through the cloud, bathing the place in a pale light. Corporal Bishop and I made our way to the house. I indicated that I would knock on a window. The patrol were aware that several civilians had been occupying the house, one a tiny baby. I knocked gently. Silence. I tried a heavier knock. \"Who's there? What the fuck do you want?' called a voice I knew well. \"It's me, 'Rubber Duck'. Are there any Argies about?\" I answered. \"No, but the fucking SAS have been - shot the fuck out of my shed. Hang on, I'll let you in\". Moments later the door opened and Tony Heathman appeared, having pulled on trousers and shirt but no footwear. Corporal Bishop, having questioned Tony and satisfied himself that there were no Argentines within the area, called the patrol together and briefed them. Tony had gone back inside and we could hear him talking to other people in the house. The patrol was reorganised to allow each of them an opportunity to have a hot drink and a quick warm up in the house. Going inside I saw Tony's wife, Ailsa, shaking up the Rayburn stove. There was a huge smile on everyone's face, these being the first British troops they had seen. Ailsa, still smiling, pushed a large saucepan onto the hot plate and invited everyone to have some stew. \"I bet you're starving. I've got plenty of stew left over from dinner. It'll only take a few minutes to heat up.\" Typical of Ailsa and Tony, the Guest Book was produced and the lads were invited to sign it! Despite these light moments there was no relaxing and the hot drink and stew were devoured quickly. The rest of the patrol received the same welcome and hospitality. Then it was time to move out. It was colder than ever outside and the ground was beginning to freeze solid. It was agreed that we would lie up for a few hours and make our way back to the battalion to report shortly before dawn.\n\n**_A four man 'recce' patrol from D Company. Left to right: 'Raj' Rajput, Corporal Dickie Bishop, Private Pete Deakins and Corporal Joe McKeown_. Terry Peck.**\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch5)\n\n[THE  \nPREPARATIONS](05_toc.html#ch5)\n\nW **ith the British now established on the line of high ground stretching from Mounts Estancia and Vernet in the north through Mount Kent to Mount Challenger in the south, it was essential that Brigadier Thompson and the commanders of 3 Para, 42 and 45 Commandos gathered as much intelligence as possible about the strength and dispositions of the Argentine forces defending Port Stanley in order to inform their planning**.\n\nLieutenant Colonel Pike had already been told that his likely objective would be Mount Longdon, the peak now glowering down on his battalion from the east. Thompson had set the problem and Pike was now left to ponder it. How best to go about cracking this tough nut on the northern flank of the Argentine line which bulged towards 3 Commando Brigade?\n\n**_Lieutenant Colonel Hew Pike_**\n\nPike established 3 Para just north of the Estancia - Stanley track, some 4km west of the Murrell Bridge which crossed the river of the same name, with A Company forward, B Company just over a mile to the west and C Company on Mount Estancia. A patrol base was established near the track on some low bluffs overlooking the Murrell Bridge. 3 Para observers had already picked up movements on Longdon during the hours of daylight, but long distance observation was not enough. More detailed information about the strength and depth of specific positions, not least about the weaponry available to the Argentines, would have to be gathered. This meant that 3 Para would have to 'go out and find out' and that meant sending out patrols - not only the twelve, four man patrols of D (Patrol) Company under Major Pat Butler but fighting patrols from the rifle companies - to poke and pry amidst the rugged nooks and crannies of Mount Longdon. The problem was the intervening ground. From the Mount Estancia position to Mount Longdon it is some three miles as the shell flies but there is no high ground in between which could conceal an approach from prying eyes on Longdon. Instead, a wide, featureless valley covered with pale yellow grass opens out and this created a broad No Man's Land between the British and Argentine lines. And so, just as the British had sought to dominate No Man's Land between the opposing lines of trenches on the Western Front during World War One with a philosophy of treating the German wire as the British front line, now Thompson urged his subordinates to dominate the ground, thus retaining the initiative and maintaining momentum. Any 3 Para patrols venturing out onto the bare, bleached backdrop of the valley during daylight would have stood out like a line of black ants on recently laid concrete. Everything would have to be done at night.\n\nLieutenant Colonel Pike ordered the first probing attack on Mount Longdon to go in on 3rd June but it ran into well-placed Argentine artillery fire and was forced to pull back. It was the first of many. It was difficult for patrols to get out, complete their tasks and get back before first light so patrols went out and stayed out for two and sometimes three nights, lying up during daylight hours and completing their tasks in darkness. Using the forward patrol base as a staging post meant that the distance the patrols had to travel from the Estancia position was reduced but such a position so close to Longdon also drew unwanted attention as Argentine units were also out and about in No Man's Land gathering intelligence. In fact there were so many units - both British and Argentine army and special forces units - squeezed into a comparatively small area that Major David Collett commanding A Company complained that there were '...too many guys out front wandering around recce-ing... strategic troops operating alongside tactical troops which is not the best way of doing it.' After a sharp firefight with Argentine Special Forces of Compania de Commando 601 and Compania de Fuerzas Especiales 601 de Gendarmerie Nacional, the patrol base was withdrawn on 4th June. Nevertheless the patrols had done some excellent work - work which Brigadier Thompson recognised as being of a 'very high standard indeed' - and had penetrated deep into the Argentine positions. Patrols still went out, albeit over a much greater distance now, and little by little over the following six days the pieces of the Argentine defensive jigsaw on Longdon began to fall into place and a picture began to emerge. This was translated into a scale model of Longdon that became festooned with white tape, white markers and twigs representing Argentine minefields, bunkers and machine gun positions respectively, as more and more information was gathered.\n\nBrigadier Thompson's blueprint for the investment of Stanley had begun to take shape from 6th June. Resisting pressure for a 'narrow front' attack along the Fitzroy-Stanley track, which the Argentine commander Men\u00e9ndez had expected until as late as 24th May, Brigadier Thompson developed a three-phase operation, each phase peeling away the three successive layers of Argentine defences until Stanley itself was revealed. The first, and perhaps with hindsight, the most crucial phase, was the 3 Commando Brigade assault on the outer defence zone - the outer ring marked by the peaks of Mount Longdon to the north through Two Sisters and down to Mount Harriet in the south.\n\nIt was important to seize and secure all three first phase objectives but the capture of Mount Longdon was vital to the success of Brigadier Thompson's overall scheme in that artillery, ammunition and all supplies required to support the entire operation would have to pass north of Mount Kent on their way from the forward logistics base at Teal Inlet. Equally important was securing the route for heliborne traffic in the opposite direction.\n\n> _'The significance of Longdon was its key position dominating the supply route to Teal Inlet. It was proved vital during the battle, and at times on the following days and nights when the low cloud on the high ground prevented helicopter casevac to Fitzroy. My Brigade Forward Dressing Station at Teal was the only FDS able to accept casualties - for example the Scots Guards casualties were dealt with at Teal and not by their own Brigade FDS at Fitzroy. Incidentally the Divisional Medical Officer wanted to close Teal, but I objected and Jeremy Moore Major General Jeremy Moore, Divisional Commander and Brigadier Thompson's immediate superior] supported my objection.'_[ 3\n\nLieutenant Colonel Pike was briefed on the plan on 10th June. The assault would be a night attack and was to be 'silent to contact' - that is without the support of a preliminary bombardment to 'soften up' the Argentine positions. Brigadier Thompson's reasoning for this was twofold:\n\n> _'I was keen for all attacks on night 11th\/12th June to be silent in order to:_\n> \n> _a. preserve surprise as to our objectives and axes_.\n> \n> _b. preserve ammunition, bearing in mind the problems with bringing it forward. In the end, after the night's battle, the ammunition on the gun lines was down to a handful per gun on some battery positions. It was this that made any exploitation on to Tumbledown in daylight unwise, so I stopped 45 Commando from advancing from Two Sisters. It became even more important to have the Longdon and Two Sisters assaults silent once I had allowed the attack on Harriet to be 'noisy', as part of the CO of 42 Commando's deception plan, as I hoped the 'noise' in the south would convince the enemy that our axis was along the Fitzroy-Stanley track from the southwest (which he expected), and conceal our approach from the northwest.'_ 4\n\nIn addition, the experiences of 2 Para at Goose Green had proved how a battle during daylight across open slopes could become stalled and ultimately costly in terms of casualties. Brigadier Thompson's faith in the abilities of his Brigade to carry out a night assault was unshakeable. The possibility of confusion during a night attack on Mount Longdon, a very complex operation fraught with difficulties, would be more than offset by the skill, the discipline, the motivation and leadership of 3 Para.\n\nLieutenant Colonel Pike now passed down Brigade orders through his own battalion chain of command; first to his company commanders, who would brief their platoon commanders, who in turn would inform their section commanders and the rest of the men, until everyone was aware of their role in the coming battle.\n\n###### **'WORK HARD.'**  \n **Longdon Preparations**\n\nBy Nick Rose ex 6 Platoon, B Company 3 Para.\n\nWe'd seen Longdon before because we'd been up and done recce' patrols. We'd been in and out for the week before - a lot of patrolling done. They were having a good old look. They did very well. SAS were definitely playing up there and by that time it had all been plotted, mortar planned, DF'd [defensive fire - pre-registered fire from a variety of weapons - usually by defensive troops] and NGFS [Naval Gun Fire Support]\n\nThe general plan was given to us. 'This is what we need to do and your part will be....'\n\nI remember seeing models and getting the 'execution, general outline'. You're given the complete briefing as per orders along the lines of, 'patrol platoon have found a position here that we believe to be a 50mil machine gun and there's going to be another one of those in defilade or enfilade or whatever because a forward recce patrol from A Company bumped into one' etc. etc. So there's all the intelligence side but I can recall the thing that was drilled into us was the fact that you just had to keep going because if you don't the whole thing just breaks down. You just cannot have a clinical sort of approach on a feature like Longdon. You know exactly what it looks like but when you go past features there could be hidden trenches and you might not see them and remember its night time as well. There could be threats coming from any area so you're advancing all the time.\n\nWe had a few company briefings about the current situation - what the greater plan was - and then it obviously broke down into platoon briefings with Jon Shaw, [6 Platoon CO]. A good man. He was very thorough about what we needed to do. It was quite a stirring thing. I remember his final point. It was simple - 'work hard'.\n\nSixty-six years on from the preliminaries prior to the Battle of the Somme in the rear areas of Picardy, the scenes were repeated as British soldiers once more gathered around scale models of their objectives to discuss opposition strong points, terrain and tactics but this time they were 8000 miles away from the battlefields of northern France. When his turn came, Major Peter Dennison, the widely respected commanding officer of the 3 Para Support Company, strode up to the fifty strong sustained fire role machine gunners, mortar men and Milan anti-tank teams clustered around the model of Mount Longdon and commenced his briefing. 'Gents' he said, 'this is it.'\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch6)\n\nTHE PLAN\n\nL **ieutenant Colonel Pike's plan for the Longdon attack was based on the sum of his knowledge gained by the British patrols regarding the strength and depth of the Argentine positions and the overall scheme involving 3 Commando Brigade. Intelligence was, by necessity incomplete as the Argentine defenders had been on Longdon since 18th April and so had had almost two months in which to establish themselves on its craggy fastness. Inevitably there would be positions that had not been logged**.\n\nIt was already a given that his attack would go in during the hours of darkness and that the advance to contact would be 'silent' but Pike had other considerations as he mulled over his options for an operation the like of which had not been 'undertaken by the battalion for a generation'. Longdon, although not a great height in itself, dominated the surrounding low lying and open ground for 1km in all directions, whilst known minefields to the south and the presence of Argentine forces on Wireless Ridge to the east effectively ruled out any possibility of a flanking attack. The shape and orientation of Longdon - a narrow, west-east feature consisting of two unequal summits with a saddle like depression separating them - dictated that no more than a force amounting to a single company could operate along its spine at any one time. Pike thus determined to assault the mountain frontally from its western extremity with two of his three rifle companies 'up'.\n\nUsing nicknames plucked from the sports fields on which the Rugby Union code is played, to denote key features on the field of battle, A and B Companies would move up from the Estancia position in darkness and cross the 'start line' (codename 'Free Kick') - a stream 1km west of Longdon, flowing north towards the Furze Bush Pass and at right angles to the intended line of advance - at 8.01 local time. A Company, (Major David Collett), was to advance on the left, northerly flank, to attack and secure a spur to the northeast (codename 'Wing Forward') and establish a fire base from which it could support B Company, (Major Mike Argue), fighting its way up to and through the forward, western summit of the main feature (codename 'Fly Half') and on to tackle the rear, eastern summit (codename 'Full Back').\n\nThe remaining rifle company, C Company (Major Martin Osborne), was to remain on the line of 'Free Kick' as battalion reserve with orders to reinforce the forward platoons should Argentine resistance crumble and the possibility of further exploitation onto Wireless Ridge present itself. This flexibility was built into the overall plan but it was also dependent on 45 Commando taking all its first phase objectives and pushing onto and occupying Tumbledown to the southeast. Also on the start line were the sustained fire role GPMG and Milan anti-tank teams of Support Company (Major Peter Dennison) who were also to hold on 'Free Kick' ready to go forward when the call came. The Mortar Platoon (Captain Julian James) had also established a mortar line nearby. In charge of casevac and ammunition resupply by means of Volvo BV tracked vehicles was battalion second in command Major Roger Patton.\n\nReady, and on priority call to provide additional, 'heavier' direct support, Lieutenant Colonel Pike could draw on the services of the six, 105mm light field guns of 79 (Kirkee) Battery, 29 Commando Regiment Royal Artillery and dedicated Naval Gun Fire Support from HMS _Avenger_ 's formidable 4.5inch gun, guided in by specialist Naval Gun Fire Observer, Captain Willie McCracken of 148 Battery R.A.\n\nMajor Argue's B Company, which would first have to ascend Longdon's western wall and then engage known key points of resistance along the entire length of its craggy spine, faced a daunting task. Although Pike's appreciation of the narrow nature of the Longdon summit had led him to the conclusion that only one company could fight on it at any one time, there was still the possibility that that company might not have sufficient numbers to comb out all the Argentine sangars if some men were held in reserve. To counter this possible scenario, Major Argue hit upon a bold and, some would say, rather risky, model. He decided to advance all his three platoons in line abreast and use them to roll like a tidal wave, up towards 'Fly Half' to break its resistance before sweeping on along the crest of Longdon, swamping the remaining Argentine positions as they went. Running from north to south 4 Platoon (Lieutenant Andrew Bickerdike) was handed the responsibility of clearing the northern face, 5 Platoon (Lieutenant Mark Cox) were to drive up and fight along the spine in the centre, whilst 6 Platoon (Lieutenant Jon Shaw) would sweep up and seize the southern slopes.\n\n**_British 105mm light field gun._**\n\nIt should be noted that the various plans, developed both at battalion and company level, were not without their detractors. There were some - officers and senior NCOs amongst them - who felt that it was worth dispensing with the element of surprise offered by a silent night attack, in order to have secured the possible benefits of a preliminary bombardment. Longdon had, they reasoned, been subject to harassing fire from 29 Commando Regiment since 3rd June. Further shelling may not have given the game away when the time came to attack. Later the thought, doubtless strengthened by Brigadier Thompson's own admission in his memoir of the war, that the repeated success of D Company patrols in penetrating the Argentine defences and returning unscathed had somehow lulled Lieutenant Colonel Pike and Brigadier Thompson into the mistaken belief that Longdon could be disposed of quickly, perhaps without any prior 'softening up', gained ground. Strategically, however, there could have been serious consequences if HQ 3 Commando Brigade had been too prodigal of artillery ammunition during the initial phase of the final battles for Stanley. With Longdon still defiant, the supply line to Teal was not yet secure and even without a preliminary bombardment, ammunition on the gun lines was later found to have dipped to just a handful of rounds per battery in some cases at the end of the first night's fighting. Brigadier Thompson felt he had to concentrate on the broader picture and conserve artillery stocks for the latter phases as well as deceiving the Argentines into believing his main thrust would come from the south with a 'noisy' operation on Mount Harriet.\n\n###### **THE PLAN - BATTLE FOR MOUNT LONGDON**\n\n* * *\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch7)\n\n[THE  \nARGENTINES](05_toc.html#ch7)\n\nI **nitial British estimates of the strength of the Argentine forces on Mount Longdon proved over generous. At one stage during the probing phase between 1st June and the eve of battle on 11th June, a figure of 800 Argentines in the area of Mount Longdon and Wireless Ridge was being suggested. In fact, by the time 3 Para were ready to attack there was only one company of infantry - reinforced by two 'grupos' of Marine Infantry machine gunners and riflemen (24 men in total) with six, Browning 12.7mm heavy machine guns - and a reserve platoon of engineers in position on the mountain**.\n\n##### **ARGENTINE FORCES ON MOUNT LONGDON**\n\n11TH\/12TH JUNE 1982\n\n**R.I.7**\n\n**Regimiento de Infanteria Mecanizado 7 (7th Mechanised Infantry Regiment)**\n\nBased - La Tablada, Buenos Aires Province\n\nCO - Lieutenant Colonel Omar Gimenez\n\n2 i\/c - Major Carlos Carrizo - Salvadores\n\nArrived Falkland Islands 14th April 1982\n\nDeployed 17th April to Sector Plata (Silver) - Mount Longdon, Wireless Ridge, Moody Brook. HQ Moody Brook.\n\nCarrizo-Salvadores given command of Sector Silver 2 (Longdon) on 17th April\n\nThe men belonged to B Company of the 7th Regimiento de Infanteria de Mecanizado - 7th Mechanized Infantry Regiment (R.I.7) - of X Mechanized Infantry Brigade, which had arrived in the Falklands on 14th April. R.I.7 had been deployed to Sector Plata (Silver) Including Mount Longdon, Wireless Ridge and Moody Brook and became part of a force of 8,500\u20139,000 men of 6 infantry units which eventually faced the seven British infantry units which converged on Stanley.\n\nR.I.7 was based at La Tablada in the province of Buenos Aires and was made up of some 700 recalled reservists and men in their second year of their two years' compulsory service. They were certainly not 'raw' recruits, although some, like Anglo-Argentine 'colimba' - conscript - Michael Savage, would admit to being vastly ill-prepared to meet the likes of the men of 3 Para on a bitter winter's night on an inhospitable mountain top on East Falkland. The only 'military training he had received had occurred 16 months before and had consisted mainly of marching, saluting and cleaning toilets. He had spent just two days training with a rifle and had 'served' at a desk' in an office for a year. Michael at least felt like a 'civilian' when he took his position with C Company R.I.7 on a feature called 'Rough Diamond' just north east of Longdon towards Wireless Ridge.\n\n* * *\n\n**These men would soon have to fight others who 'were as good as family'**.\n\n* * *\n\nSavage's grandmother had been born to British parents who had migrated to Argentina in the nineteenth century and the family remained deeply proud of its British heritage. Such had been the pull that his grandfather, although Argentine by birth, had served with the R.A.F. during World War Two. Michael spoke fluent English and British traditions and customs were observed in the Savage household. He had played cricket with British friends at boarding school in Buenos Aires. There were others with the same background and with English surnames. He shared a trench with a friend called Alan Craig. These men would soon have to fight others who 'were as good as family'.\n\nThe defence of Longdon was entrusted to R.I.7 Second-in-Command, Major Carlos Carrizo-Salvadores who first set foot on Longdon on 17th April. The next day the three platoons of B Company set to work on building their defences in anticipation of an attack from the north. They had the best part of the next eight weeks to dig into the peat banks, hew positions out of the unforgiving rock, cover bunkers with corrugated iron and put stone slabs and peat turves on top, making sure that all approaches up the mountain towards the western and eastern summits were covered with interlocking arcs of fire. Major Carrizo-Salvadores would switch his positions after 24th May so that they faced west, with a concentration of sangars among and around the western summit 'Fly Half'. There they waited.\n\n**_The grandfather of Michael Savage wearing his RAF uniform during World War Two._** Michael Savage\n\n**_'Colimbas'. Michael Savage (first on left) and fellow conscripts pose for the camera near their base in Argentina prior to the war_. Michael Savage.**\n\n* * *\n\n**Many, including Michael Savage, consider themselves the 'forgotten army'**\n\n* * *\n\nThere has been much debate surrounding the fate of these men in the mountains of the outer defence zone. Many, including Michael Savage, consider themselves the 'forgotten army', abandoned by their officers and left to survive on a diet of thin, lukewarm soup, pasta and _mat\u00e9_ , a bitter herbal tea. Some took to walking miles and risking heavy penalties to steal from the supply dumps near Stanley, while others shot and roasted sheep on old bedframes. There are others who maintain that these men were helped to make themselves as comfortable as possible under the circumstances and that their officers tried hard to bolster morale. Michael Savage's weight dipped to 55 kilos after two months on Longdon and nothing had prepared him for the terrain he found there. As the weather deteriorated he became more and more 'terrified'. Nevertheless, when the time came, the Argentine soldiers would stand and fight.\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch8)\n\nTHE WEAPONS\n\n3 **Para embarked on the Falklands campaign with an array of weaponry which combined to produce a formidable level of firepower. The most ubiquitous weapon was the standard issue rifle carried by British infantrymen at the time - the 7.62mm calibre LIAIS.L.R. or self loading rifle. This rifle was a British adaptation of an original Belgian F.N., F.A.L.(fusil automatique leger) design. The British variant, arrived at after an extensive series of field trials and modifications, eliminated the automatic fire feature of the original design and thus only fired single shots, the theory being that this encouraged men to consider their targets and fire with greater accuracy, rather than wasting ammunition in general and sustained bursts. Nonetheless, during the fighting on Longdon, such had been the training with this reliable and well manufactured weapon, that the paras were able to lay down a considerable level of co-ordinated fire on suspected Argentine positions**.\n\nThe L1A1 bayonet, which was designed to be fitted to the S.L.R., had a blade of 204mm (8 inches) and, due to the nature of the close quarters fighting, in and around Argentine bunkers, became an invaluable weapon in its own right.\n\nAnother weapon which was widespread throughout the battalion was the fully automatic, 7.62mm calibre G.P.M.G., the General Purpose Machine Gun, universally known as the 'Gimpy'. Again this was based on a Belgian F.N. design and was employed both as a section fire support weapon-one per section-and in a sustained fire (S.F.) role. In the section support role the Gimpy was mounted on a lightweight bipod but when grouped together in the Machine Gun platoon it was fitted with an S.F. kit consisting of a heavy tripod, at least three heavier barrels - two of these were spares in case of overheating - and a dial sight which enabled it to fire sustained bursts on pre-registered targets otherwise obscured from view. The weapon was used in this role in Major Dennison's Support Company and several teams pushed up into advanced positions near 'Fly Half' on Longdon to provide fire support for the beleagured rifle companies\n\nFor those men such as radio operators or senior Non Commissioned Officers whose tasks or duties made it difficult for them to carry an S.L.R., the option was the 9mm calibre L2A3 Sterling submachine gun. This compact - the stock could be folded to reduce stowage space - and fully automatic weapon, fired pistol ammunition from a curved 34 round magazine inserted on the left. R.S.M Lawrie Ashbridge was famously photographed in Port San Carlos carrying a Sterling S.M.G.\n\nAlso providing much needed support for the rifle companies during the battle was the 'state of the art' Euromissile MILAN wire-guided, anti-tank weapon system just entering service with the British Army in 1982. The MILANS, consisting of a control\/firing post and missile in a launch tube, were grouped together in the antitank platoon under Captain Tony Mason, along with the WOMBAT 120mm recoilless anti-tank guns. The WOMBAT was not used on Longdon but the MILAN teams were ordered up on to the western slopes where the high explosive missiles were brought into action to engage Argentine sangars in a 'bunker busting' role. The MILAN teams obviously became prime targets for the Argentine defenders and three men from one MILAN team Privates Hedicker, West and McCarthy, were amongst the last fatalities of the battalion.\n\nThe mortar deployed in the Mortar Platoon of Support Company under the command of Captain Julian James, was the standard issue British 81mm infantry weapon in an indirect fire role. The range of the 81mm, when firing high explosive shells, was more than 5000 metres and a well trained team could expect to put at least twelve rounds a minute onto the target.\n\nThose men designated as specialist snipers were armed with the 7.62mm calibre L42A1 rifle, a weapon used only in this role. The L42A1 used the basic Lee Enfield manual bolt mechanism which had changed little since the 1890s and was a conversion of the World War Two .303 calibre rifle. The important conversions concerned a new barrel and new magazine adapted to take 7.62mm ammunition, along with changes to the trigger mechanisms and fixed sights. The old World War Two Mark 3 telescopic sight and mounting was retained. This weapon was used by Private Richard Absolon of D (Patrol) Company during his extensive patrolling, sniping and escort duties prior to and during the battle. Absolon was killed on the morning after Longdon had been taken but his exemplary service was recognised with the award of a posthumous Military Medal.\n\n7.62MM GENERAL PURPOSE MACHINE GUN\n\nBRITISH LIAI - SLR\n\nSNIPER RIFLE L42AI\n\nL2A3 STERLING SUB-MACHINE GUN\n\nMILAN ANTI-TANK GUIDED WEAPON\n\nLIAI - SLR BAYONET\n\nLI6A2 81MM MORTAR\n\nBROWNING FN HIGH POWER PISTOL\n\n###### **TECHNICAL SPECIFICATIONS**\n\n  **7.62MM GENERAL PURPOSE MACHINE GUN**\n\n**Calibre** : 7.62mm \u2022 **Weight** : 13.85kg (gun plus 50 round belt)\n\n**Length** : 1230mm (light role) \u2022 **Barrel length** : 629mm\n\n**Muzzle velocity** : 838m\/s \u2022 **Feed** : 100-round disintegrating-link belt\n\n**Effective range** : 800m light role, 1800m sustained fire role (tracer burn out at 1100m)Cyclic rate of fire 750 rounds per minute\n\n  **BRITISH L1A1 - SLR**\n\n**Calibre** : 7.62mm \u2022 **Weight** : 5kg loaded\n\n**Length** : 1143mm \u2022 **Barrel length** : 554mm\n\n**Muzzle velocity** : 838m (2,750ft) per second \u2022 **Magazine** : 20 round box\n\n**Effective range** : 30\u201340 rpm (single shot)\n\n  **SNIPER RIFLE L42A1**\n\n**Calibre** : 7.62mm \u2022 **Weight** : 4.43 kg loaded **Length** : 1181mm \u2022 **Barrel length** : 699mm\n\n**Muzzle velocity** : 838m (2,750ft) per second \u2022 **Magazine** : 10 rounds\n\n  **L2A3 STERLING SUB-MACHINE GUN**\n\n**Calibre** : 9mm \u2022 **Weight** : 3.47 kg loaded\n\n**Length** : 690mm extended stock - 483mm folded \u2022 **Barrel length** : 198mm\n\n**Muzzle velocity** : 390m (2,750ft) per second \u2022 **Magazine** : 10 or 34 round **Cyclic rate of fire** : 550 rpm\n\n  **MILAN ANTI-TANK GUIDED WEAPON**\n\n**Dimensions** : length 0.769m, diameter 9cm, span 26.5cm\n\n**Launch weight** : missile 6.65kg, complete unit 16.5kg\n\n**Propulsion** : Solid-propellant booster\/sustainer rocket\n\n**Performance** : range 2,000 m \u2022 **Warhead** : 2.98 kg hollow charge HE\n\n**Armour penetration** : 650mm (25.6 in)\n\n  **L1A1 - SLR BAYONET**\n\n**Blade length** : 204mm (8 inches)\n\n  **L16A2 81MM MORTAR**\n\n**Calibre** : 81mm \u2022 **Weight** : 37.94 kg \u2022 **Barrel length** : 1280mm\n\n**Muzzle velocity** : 225m\/s \u2022 **Magazine range HE** : 5,650m \u2022 **Rate of Fire** : 15 rpm\n\n  **BROWNING FN HIGH POWER PISTOL**\n\n**Calibre** : 9mm \u2022 **Weight** : 1.04 kg loaded\n\n**Length** : 200mm \u2022 **Barrel length** : 118mm\n\n**Muzzle velocity** : 350m (1,148ft) per second \u2022 **Magazine** : 14 rounds\n\n[**F A L K L A N D S  \nM O U N T L O N G D O N**](05_toc.html#ch9)\n\nTHE BATTLE\n\nI **1th June 1982 - a little before 8.15 pm local time. Back home in Britain it was almost the height of summer but on East Falkland down in the South Atlantic it was the middle of winter. Now, pausing at a line of white mine marker tape laid by D Company on the eastern banks of a stream flowing north towards the Murrell River at Furze Bush Pass, the assaulting platoons of A and B Companies gathered their thoughts as NCOs moved up and down the lines giving their final words of encouragement. This was their 'start line' - codename 'Free Kick'. It was dark, it was bitterly cold and icy water had penetrated the boots and socks of some who had fallen thigh deep in the stream they had just forded. The sweat - the combined result of anxiety and extreme exertion - which had lathered their bodies on the tab across country from the Estancia position, was now doing its work; lowering body temperatures as the biting wind penetrated SAS windproofs**.\n\nThey were late, but not by much. It was to 3 Para's credit that, as the seconds ticked by, they were coming up to being just fourteen minutes behind schedule. The approach had been difficult. Each man carried a personal load in the region of 100 lbs, a colossal weight to carry into an attack, that consisted of their personal stocks of food and water, rifle, bayonet, spare magazines stuffed into every pouch of webbing and smock pockets, 400\u2013600 rounds of linked ammunition for the GPMGs and several fragmentation or phosphorous grenades. Some men carried three plastic 66mm LAWs and a couple of 84s for the Carl Gustav Rocket Launchers. Others in Support Company carried tripods for the sustained fire role GPMGs and Milan missiles strapped below radios on their backs. They had already marched for more than three hours, held up by delays in crossing the Murrell River and confusion resulting from Support Company 'cutting up' and separating the advancing platoons of B Company. They still had some way to go to reach the lowers slopes of their objective.\n\nIt would soon be time. High up in his command post towards the rear summit - 'Full Back' - of the Argentine defences on Longdon, Major Carizzo-Salvadores and some of his HQ staff tuned in their small receiver to hear the voice of Pope John Paul II, who had arrived in Argentina that day, celebrating Mass at the National Shrine at Lujan. As the blessing began Salvadores' three platoon commanders called in. They had nothing to report. 1,500 metres away down the slopes of the mountain to the west CSM John Weeks was also pointing his men in the direction of The Almighty only his delivery was rather more prosaic than that of the Pontiff. 'If any of you fuckin' want to pray then here's your chance to really fucking talk to the man upstairs because you'll need him throughout the night.' Section commanders were also doing their best to invoke divine intervention. Corporal Trevor Wilson's brief to Private Nick Rose and the rest of 3 Section was short and to the point; 'say your prayers lads.'\n\n* * *\n\n**'The sound of 120 plus bayonets clicking into place seemed to carry for miles on the swirling wind.'**\n\n* * *\n\nMajor Carrizo-Salvadores had already ordered the Rasit ground surveillance radar, sited on Longdon's western slopes with Lieutenant Baldini's forward 1st platoon, to be switched off, fearing that its signal would be picked up by the British. The decision was to help Shaw's B Company during the initial stages of its advance. Nevertheless between 'Free Kick' and Longdon there was still a minefield to negotiate.\n\nWhen the order came to 'fix bayonets' the men of B Company finally realised that they had reached the point of no return. In the deafening silence before battle, the sound of 120 plus bayonets clicking into place seemed to carry for miles on the swirling wind. Would the Argentines hear? This was it.\n\nAt 8.15 p.m., several hundred men of A and B Companies stepped over the white tape and into the unknown. A Company headed for First Sergeant Gonzales's 2nd Platoon defending the north western slopes and 'Wing Forward', whilst 4, 5 and 6 platoons of B Company advanced at a steady pace and shook out into assault formation as the rising moon began to illuminate the crags and the narrow, steep sided rock runs which seamed the lower slopes. Further up, the splintered outcrops of rock that marked the western summit of Fortress Longdon loomed above them. There was 'Fly Half'. There was their destination.\n\nOn the right flank of B Company, 6 Platoon made for the southern approaches to 'Fly Half', guided by Corporal Jerry Phillips of D Company, veteran of several Longdon patrols, and after about an hour, hit the forward slopes and began their ascent. They had crossed the 1000 metres of intervening ground in silence save for the rhythmic 'swish, swish' of their boots brushing against the grass. Private Nick Rose checked his position in his four-man team and followed his section commander, Corporal Trevor Wilson, -'... perfect as a section commander. He was a real soldier's soldier and knew the score' - up the hill. The overpowering smell of human faeces invaded Rose's nostrils as he went on - a hint of slack Argentine organisation in the field? 'They _did not know_ we were there.' Still silence. Were they really going to pull this off?\n\nSuddenly and violently, Nick Rose heard the silence ripped apart somewhere off to his left as the left forward section commander of 4 Platoon, Corporal Brian Milne, stepped on a mine and screamed out in agony. The mountain exploded into life as the Argentine forward bunkers and Rasit section under Sergeant Nista of Baldini's 1st Platoon came into action, spraying 4 and 5 Platoons on the western slopes with fire interspersed with green tracer rounds. Argentine mortar and artillery batteries were called up and rounds were soon thumping the ground behind the paras. The battle for Longdon had been joined. Ahead lay ten hours of the most unimaginable and bloody, hand-to-hand gutter fighting.\n\n* * *\n\n**'They don't know we are here, we can still get up there and do the job.'**\n\n* * *\n\nIn spite of the Argentine fire, 6 Platoon under Lieutenant Jon Shaw continued its climb up the southern flanks towards 'Fly Half' virtually unopposed. Private Nick Rose glanced left and saw the flashes of British red and Argentine green tracer dancing up and down the hillside. He remembers thinking it looked just like a two-way firing range. He knew they were now moving through Argentine positions; he could see their sangars for himself.\n\n> ' _When we went up we weren't really sussed until we'd got at least halfway up the side of Longdon. Jerry Phillips took us up there and he ended up shot in the arm. 'Meccano Man' we used to call him - he had his arm pounded. Every bone was broken and he had sort of 'meccano' built into it and it grew again. We're still going up and Jerry says, \"They don't know we are here, we can still get up there and do the job.\" So we did. There was myself, the Platoon Sergeant Pete Grey, Tony Greenwood and 'Baz' Barratt who advanced as a four-man, half section on the left hand side as we approached and breasted the top, going through a fire position. We went into sort of extended line and took cover and, 'bang, bang, bang,' just unloaded. We just 'laced' the place - laid it down thick and heavy, hard and fast and started moving forward into whatever cover we could get. It was massive, rapid fire - suppressive fire - into our general area of our responsibility. We just flooded the place, 'stonked' it. When the dust started to clear we looked for likely enemy positions and then put some rounds into those. It was 'recce' by fire' but we were doing it intelligently. Then the training kicks in. It does take over. You know where you should go. You think, \"If I go there, 'bang' I'm going to get done, because it's just such a likely place for a GPMG to be covering\". So we work on movement with covering fire. We're shouting to each other now, \"O.K. we're going left.\" And so you move forward. You move where you think it's the right place to go and it is - most of the time. Sometimes it doesn't work. But then it started to break down because there were snipers with night sights, which was extremely damaging.'_\n\n* * *\n\n###### **MOUNT LONGDON - BATTLE ON THE WESTERN SLOPES 11th\u201312th JUNE 1982**\n\n* * *\n\n**'I could hear my men being killed. They had only just woken up and now they were dying.'**\n\n* * *\n\nThe men were into and through Corporal Oscar Carrizo's section of Lieutenant Neirotti's 3rd Platoon before some of the Argentines had had time to rouse themselves from their slumbers. It was a vulnerable time for the Argentines as new sentries had just been posted.\n\n' _I'd got three men around me_ \\- remembered Rose, - _Pete Grey, 'Baz' Barratt and Tony Greenwood - and there's another eight guys in the middle section of our area and then another six to eight guys on the far right hand side - among them were Den Dunn, Trev Wilson, a bloke called Shaw, he was a craftsman_ [R.E.M.E. attached 3 Para], ' _Taff_ ' _Power, and Stu Gray_.'\n\nCorporal Carrizo recalled how his section was quickly engulfed by Rose and his comrades,\n\n> _'Outside the English were running past, screaming to each other and firing into tents and bunkers. I could hear my men being killed. They had only just woken up and now they were dying. I could hear muffled explosions followed by cries, helpless cries. I knew grenades were being thrown into the bunkers in the follow-up. The Sergeant and I discussed surrendering, but decided we'd wait until it was over. All we could do was wait. The English were all around us. They had arrived within seconds, like lightning. I prayed and prayed a grenade wouldn't come into our bunker. The sheer mental pressure exhausted me.'_ 3\n\nNow, galvanised by the will to survive, the Argentine defence in 6 Platoon's area bristled into life. Sections under Corporals Diaz and Pedemonte, backed up by the fire from one of the six, heavy Browning 12.7mm machine gun of the Marine Infantry under Corporal Lamas on the right of Neirotti's position, joined in sweeping the western and southwestern slopes with fire. Lieutenant Neirotti was wounded so Captain Lopez took over to organise the defence as the first of the British shells called up by Major Argue began to fall towards 'Full Back'.\n\nOver to Rose's left and further down the slope, Lieutenant Mark Cox's 5 Platoon had been fighting hard for almost an hour. Already deep amidst the Argentine sangars by the time Corporal Milne had stepped on the mine, 5 Platoon had borne the brunt of the first Argentine fusillade that had effectively checked its advance. The route to the summit here was channelled with the steep sided, west-east orientated rock runs and jagged outcrops that served to funnel the paras into pre-registered killing zones. Now 5 Platoon faced a long and painful struggle if they were to haul themselves to the summit. As their training too 'kicked in', so they broke down into smaller 'fire and movement units'; groups working forward independently, some men drenching sangars with rifle, 66mm LAW or 84mm rocket fire before others rushed forward to take it with bomb and bayonet. Then there would be a pause whilst another objective was identified only for the entire process to be repeated a little further up the hill. It was painstaking and hazardous work made all the more deadly as the Argentines tossed grenades down the stone runs up which the paras were obliged to advance, raining searing fragments of metal and lethal stone chips down on them. Metre by metre 5 platoon clawed its way upward and finally silenced a GPMG position which had pinned them down just below the summit.\n\n* * *\n\n**Metre by metre 5 platoon clawed its way upward and finally silenced a GPMG position**\n\n* * *\n\nBack on the southern slopes, 6 platoon, which had succeeded in establishing a precarious toehold on the summit of 'Fly Half', had by now run into trouble. Corporal Steggles had led his 1 Section up towards the summit to hunt down an Argentine mortar which lay beyond whilst Corporal Wilson and Lance Corporal Murdoch had directed their sections east to winkle out the Argentine sangars just below the crest. In the darkness and confusion 6 Platoon had bypassed a bunker holding at least seven men. Dotted elsewhere on the mountain were snipers with state of the art second-generation passive night sights. As Corporal Murdoch's section worked forward, they veered north, working towards the reverse slopes of 'Fly Half' which marked the western rim of the broad saddle of lower ground between 'Fly Half' and 'Full Back' and known as 'The Bowl'. As they crossed a strip of open ground they came in range of an Argentine sniper further east along the crest towards 'Full Back'. Lance Corporal Murdoch was hit and fell to the ground. Still alive, he lay exposed in the open.\n\n> _'When you go past things there could be hidden trenches but you just do not see them and remember, it's night time as well. There's a threat coming from any area so you're advancing all the time and when you're going past, 'bang' there's blokes who are shooting at you from behind because you haven't seen the defensive position. But we were being opened up on from the sides as well. We'd moved forward and were behind a rocky outcrop. Pete Grey stood up and went to throw a '42' grenade and he was shot by a sniper in his right forearm. We thought the grenade had gone off. We punched his arm down into the ground to staunch the bleeding, believing he'd lost half his right forearm and hand, but it was still there and his arm bent at the forearm instead of at the elbow- a horrible thing to watch. We were laying fire down because by this time Trev and the guys - including 'Doc' Murdoch - had been pinned down by a sniper. There's 'incoming' everywhere, loads of stuff going down the range and then 'bang' my pal 'Fester'_ [Tony Greenwood], _gets it just above his left eye, only a yard away from me. That was a terrible thing. 'Fester' was such a lovely guy. Then it was 'Baz' Barratt. 'Baz' had gone back to try and get field dressings for Pete Grey and as he was coming back 'bang' he got it in the back. This was when we just stalled as a platoon. There was no further we could go. The snipers were just picking people off. It was very difficult and we suffered. It was a sniping battle in that respect. So it was like 'go firm'. Every time I went to engage - I knew where these people were- it was 'zzzum'. That deters you from sticking your head around the rocks too often. There was green tracer as well, which was all wrong. Hey! our tracer's not green! Course it's not, its theirs, its coming at you and in between every one of those_ [tracer rounds] _there's another two or three or four, so you keep your head down. So you've got all this noise, you've got men howling and screaming and five or six yards away there's an abandoned Argentine radio set chattering away in Spanish! 'Doc' Murdoch was killed there. Oh man! I really think he was being toyed with by that guy. Shot him in the thigh first of all, then he shot him in the arm; shot him in the side of the head which blinded him. 'Doc' was telling us all this. We could hear him dying. It was terrible - a horrible way to die, a haunting way. A brave man_.'\n\n**_The battle errupts. As B Company advance towards their objective, 'Fly Half' Coporal Brian Milne steps on an anti-personnel mine. The explosion alerts the Argentine soldiers and Longdon explodes in a blizzard of tracer. This image was taken from the left of 4 Platoon's forward left section._** Military Picture Library International\n\n* * *\n\n**There's 'incoming' everywhere, loads of stuff going down the range and then 'bang' my pal 'Fester' gets it just above his left eye**\n\n* * *\n\n**_Two helmets and rifles break the skyline, marking the spots where Corporal Murdoch and Private Stewart 'Geordie' Laing were shot dead by an Argentine sniper._** Military Picture Library International\n\n**_A British sniper in_ a _ghillie suit._** Illustrated by Jon Wilkinson\n\n* * *\n\n**'It was like a football match when you want to join in and help your side.'**\n\n* * *\n\n6 Platoon B Company, 3 Para, were indeed suffering. To add to 6 Platoon's misery, the weight and accuracy of the Argentine fire and subsequent attempts to locate the sources, had altered its axis and it had strayed north, into the crossfire of a heavy machine gun firing on Lieutenant Cox's 5 Platoon. As medics frantically moved amongst the rocks trying to tend the wounded, others decided they could not leave their comrades out in the open to suffer any longer. Responding to Corporal Murdoch's cries, Private Stewart Laing broke cover and dashed out to rescue Murdoch only to be struck three times in the chest. He died instantly. This level of casualties - 5 dead and eight wounded, almost 50% of Lieutenant Shaw's command - could not be sustained. Shaw asked for and got permission from Major Argue to 'go firm' - to consolidate and hold the ground taken on the northern slopes.\n\nOver on the left flank of B Company's attack, 4 Platoon had managed to edge forward up Longdon's northern face, hitting the right hand sections of Lieutenant Baldini's command. The right hand section of Lieutenant Andrew Bickerdike's platoon had also been forced into the treacherous rock gullies, which diverted them into 5 Platoon's sector but on the left their approach had been across 'dead ground' concealed from the Argentines. With some of 4 Platoon now working up behind 5 Platoon the men on the left pushed on, eventually reaching a position just forward of the summit and began to penetrate the Argentine positions held by First Sergeant Gonzales's 2nd Platoon. Here they found themselves co-mingled with the leading edge of 5 Platoon. One remaining 12.7mm heavy machine gun on the lip of 'The Bowl', which sloped away over relatively open ground towards 'Full Back' to the east and holding up the final push for the summit, was silenced when Privates Gray and Gough (5 Platoon), under covering fire from Lance Corporal Carver and Private Juliff, took the bunker at the point of the bayonet. Having cleared the way this far, the mixed force of 4 and 5 Platoons were now sheltering among the rock outcrops on the rim of 'The Bowl' trying to identify the next line of defence before descending through the rocky alleyways leading to the relatively open moorland of 'The Bowl' itself and then on to their final objective, 'Full Back'. That next line of defence was to prove formidable. Sited in depth, in a classic reverse slope position, were at least two 7.62mm GPMG positions, one heavy Marine Infantry machine gun supported by Marine Infantry riflemen with night sights and a 105mm anti-tank, recoilless rifle.\n\nThe battle had been raging now for some two hours. As soon as the paras began to move down from the crest, the accumulation of Argentine firepower, much of it pre-plotted, from sangars in and around 'The Bowl' rocked their advance every time they tried to move. Lieutenant Bickerdike's Platoon HQ came under heavy fire and Bickerdike was hit in the thigh whilst Private Cullen, his radio operator, was caught in the mouth by the same burst. There were at least four other casualties, including Private Neil Grose who had turned eighteen that day. These men badly needed medical aid and fire support but the planned fire support base that should by now have been established by A Company and Support Company on 'Wing Forward' had not materialised. Major Collett's Company and the sustained fire GPMG and Milan sections under Major Dennison's command, had reached 'Wing Forward' but had come under sustained and accurate fire from several snipers in Sergeant Gonzalez's 2nd Platoon area whilst Argentine 155mm and 105mm howitzer rounds fell amongst them. Sheltered behind low peat banks most of the men had gone to ground and were trying to engage the snipers whenever they had the opportunity. Corporal Vincent Bramley remembers seeing the firefight erupt on the mountain and his feelings of frustration at being unable to influence the outcome, 'It was like a football match when you want to join in and help your side.'\n\n* * *\n\n**...the paras fought through with rifle and bayonet and moved towards the crest of the ridge...**\n\n* * *\n\nBoth A and Support Companies could see what was happening, but pinned down as they were, they were not in a position to intervene effectively. In any event those few who were in a position to engage the Argentines on the mountain were afraid of hitting their own men fighting from right to left across their immediate line of fire. It was clear that the Argentines were not packing up and running to Stanley. They were, in the main, standing and making a fight of it. At some point during the battle on the western slope, Lieutenant Juan Baldini had fired a GPMG on the western slopes until it jammed and he was charged by a group of paras. Drawing a pistol he fired 13 shots in their direction before they reached him. He was later found dead half in, half out of sangar with his pistol in his hand and no boots on his feet.\n\nThe advance, it seemed, was stalling at all points. The struggle was being distilled into a bloody battle of attrition. In order to secure victory the men on both sides would have to dig deep and call on hidden depths of determination and the ability to endure to the end. There could be no 'draw' on this mountain. Something out of the ordinary was required if the initiative was to be regained.\n\nLieutenant Bickerdike's Platoon Sergeant, Ian McKay, now assumed command of the composite force from his own platoon and the left section of 5 Platoon under Corporal Bailey, clinging to the lip of 'The Bowl'. Conferring with Corporal Bailey, Sergeant McKay identified a heavy machine gun at the heart of the Argentine defences that was thwarting any attempt to move along the central spine of the ridge. The next cover was thirty-five metres away and there were several layers of protective bunkers between the paras and the heavy machine gun emplacement. Gathering together Corporal Bailey and three more men - Privates Burt, Jones and McLarnon - and calling for suppressive fire, McKay's group charged across the open to be met by a hail of fire. Private Burt was killed almost immediately and Private Jones was hit but McKay, Bailey and McLarnon, covered the thirty-five metres into cover. Pausing momentarily, they charged again, grenading and firing into a sangar without stopping until Corporal Bailey, shot in the hip, staggered and fell, as Private McLarnon ducked and dived for cover. The last person to see Sergeant McKay alive was Corporal Bailey who watched from his supine position as he ran on alone towards the Argentine machine gun nest until he disappeared. Bailey was then hit twice more. The machine gun fell silent. After the battle Sergeant Ian McKay's body was found lying amongst the human wreckage of an Argentine bunker. For his actions on Longdon that night Sergeant McKay was awarded a posthumous Victoria Cross, the second such award to the Parachute Regiment in two weeks.\n\n**SERGEANT McKAY**  \n **Sergeant Ian John McKay was awarded the Victoria Cross for his action on Mount Longdon. His citation reads:** ' _It was clear that instant action was needed if the advance was not to falter and increasing casualties to ensue. Sergeant McKay decided to convert this reconnaissance into an attack in order to eliminate the enemy positions...He issued orders and taking three men with him, broke cover and charged at the enemy position. The assault was met by a hail of fire. The corporal was seriously wounded, a private killed and another wounded. Despite these losses, Sergeant McKay continued to charge the enemy position alone. On reaching it he despatched the enemy with grenades, thereby relieving the position of beleaguered 4 and 5 Platoons...Sergeant McKay, however, was killed at the moment of victory, his body falling on the bunker. Without doubt Sergeant McKay's action retrieved a most dangerous situation and was instrumental in ensuring the success of the attack. His was a coolly calculated act, the dangers of which must have been too apparent to him beforehand. With a complete disregard for his own safety, he displayed courage and leadership of the highest order, and was an inspiration to all those around him_.'\n\nWith the hard core of resistance broken, 4 and 5 Platoons could now move once more although the Argentine fire from further positions towards 'Full Back' was still intense. With 'Sunray 21' (Bickerdike) and 'Sunray Minor' (McKay) down, Major Argue sent Sergeant Des Fuller forward to coordinate the command of 4 Platoon. Moving up the western slopes of the mountain Sergeant Fuller first established the situation on reaching Lieutenant Bickerdike and then, with the aid of Corporal McLaughlin, decided on a 'simple' plan. They would carry on moving forward. And that is what they did. In a series of audacious, ferocious assaults, they moved down the reverse slope and into 'The Bowl', sangar by sangar, taking casualties until they too could go no further. Just as 6 Platoon had done earlier Sergeant Fuller, his position now bolstered by the steady and uncompromising presence of CSM Weeks, who began to organise the evacuation of the wounded, 'went firm'. Major Argue had by now moved his HQ in among the rocks on the summit of 'Fly Half' and although the situation was far from clear, Major Argue knew enough to appreciate that his company, as it stood, was not going to get much further along the crest of the mountain. 'Fly Half' was, for the moment, secure, but the battalion was only half way to its final objective. Sometime around midnight Major CarrizoSalvadores ordered the 45 men of his reserve - the 1st Platoon of the 10th Engineer Company under Lieutenant Quiroga - forward from its positions around the command post and 'Full Back'. Moving along the crest in the darkness they did not shirk their duty as they headed towards the fighting but they probably did little more than help to shore up a buckling defence just east of 'Fly Half' and extricate the survivors of isolated bunkers. They did not succeed in knocking the paras from their precarious perch on the summit. This was a critical stage in the battle, the outcome of which now hung in the balance.\n\n**_Private Jim 'Scouse' Pritchard on the morning after the battle for Mount Longdon summit. In the background his colleagues round up Argentine prisoners shrouded in smoke from their burning sangars._** Terry Peck.\n\n**_Lance Corporal Graham Tolson and comrades take a break after the battle on 14 June, as they prepare for the next phase of the war - the advance on Port Stanley._** Terry Peck.\n\n**_Private 'Baz' Barrett is evacuated from Mount Longdon on the morning after the battle. Barrett went into action just yards away from Nick Rose and was struck in the back by a bullet just above the sciatic nerve._** Tom Smith, Daily Express\n\nBy about 2.30 a.m. Lieutenant Colonel Pike had managed to move his HQ up onto 'Fly Half' to join Major Argue. He quickly assessed the situation. Fire from Argentine positions on 'Full Back' had checked the frontal assault of 4 and 5 Platoons and every attempt by 6 Platoon to get around the flanks, whilst A Company - still relatively intact - were still being pinned down on 'Wing Forward' to the north east. C Company under Major Osborne was still on 'Free Kick' and would take too long to get into the battle. A radical solution would be necessary if 3 Para were to secure 'Full Back'. Some of the fire support GPMG and Milan teams were already arriving in B Company positions. Working the problem Pike ordered A Company and the remaining teams of Support Company, to double back to the western end of Longdon, climb up through the ground captured by B Company and then launch their attack on 'Full Back' along the same axis, bursting through the leading elements of 4 and 5 Platoons. It was a bold scheme.\n\nAs A Company made its way back to the lower western slopes Major Argue called down artillery and Naval Gunfire Support from HMS _Avenger_ as he pulled his men back from their forward positions and reorganised his battered company for one last effort to break through 'The Bowl'. Leading a composite force from 4 and 5 Platoons and fire support teams, Lieutenant Cox moved along the northern slopes in a flank attack under cover of _Avenger_ 's 4.5 inch barrage, towards the spot where Sergeant McKay had last been seen charging the Argentine machine gun. Cox managed to get his team some way along the track but as the naval gunfire lifted they ran into the same volume of fire that had previously halted B Company's advance. This time, however, the defenders included forty-six fresh reservists of First Lieutenant Raul Fernando Castaneda's 2nd Platoon of C Company, 7 Infantry Regiment which had recently arrived on Longdon from Wireless Ridge. They had arrived a little before 3 a.m. in response to Major Carrizo-Salvadores' request for reinforcements at 1.30 a.m. Castaneda's men joined with those of First Sergeant Gonzalez in resisting Cox's advance but the paras fought through with rifle and bayonet and moved towards the crest of the ridge where they exposed themselves to fire from the bunkers of Lieutenant Neirotti's 3rd platoon and those clustered around Carrizo-Salvadores' HQ. B Company strength was now down by 50%. They could get no further. It was time for A Company to take up the sword.\n\n**_The British flag flies once again in Port Stanley. Men of 3 Para triumphantly hold their banner for the camera at the end of their campaign._**\n\nWith fire support teams ensconced on the eastern slopes of 'Fly Half' as a combined firebase, Major Collett deployed his men for the assault. Covered by five sustained fire GPMGs, a light machine gun and 3 Milan posts acting as 'bunker busters' and the British artillery, Major Collett-drawing lessons from B Company's experiences - ordered his platoons to conduct a slow and methodical advance from sangar to sangar through 'The Bowl', moving one platoon forward at a time. It was time consuming but it was effective as metre by metre the paras searched out and destroyed the Argentine bunkers on their relentless sweep towards 'Full Back'. But although some Argentine troops abandoned their posts many more clung to their positions and held fast on the eastern summit and around Carrizo-Salvadores' HQ. Fire from Mount Tumbledown to the south added further to the paras' difficulties.\n\nThe Argentine commander was obliged to move when a Milan scored a near miss and it was men of the Milan teams who became some of the last para fatalities of the battle. It was an artillery round, or a round from a Czekalski 105mm recoilless rifle operated by Corporal Manuel Medina, one of First Lieutenant Castaneda's men, firing along the length of the ridge towards 'Fly Half', that scored a direct hit on the Milan team operated by Corporal Keith McCarthy and Privates Peter Hedicker and Phillip West. Privates Hedicker and West were killed immediately and Corporal McCarthy died a short time later.\n\nWith daybreak just a little over an hour away it looked as though Brigadier Thompson's fears of the battle spilling over into daylight hours were about to be realised but A Company's assault gathered pace and as dawn broke Major Collett's men had breached the Argentine defences around 'Full Back' and were almost onto the summit. Major Carrizo-Salvadores' men had fought the paras all night but now, at a little after 6.30 a.m., he finally disengaged and withdrew towards Port Stanley in the half light and mists of dawn with just 78 of the 287 men who had begun the fight. A Company finally moved on 'Full Back' and combed out the last of the Argentine bunkers whilst 3 Platoon went beyond to secure and guard the long slope down to Wireless Ridge. A Company casualties, during the two hours or more it had taken to reach 'Full Back', amounted to one man - Private Coady - wounded by fragments from his own grenade.\n\nLongdon finally belonged to the paras. The battle might have been over but at a dreadful cost. 3 Para's losses amounted to 18 killed and more than 40 wounded, Argentine losses were thought to be 31 killed, 120 wounded and 50 taken prisoner. But as the evacuation of wounded, consolidation and reorganisation got into full swing the first of the Argentine rounds from the direction of Stanley crashed amongst the rocks searching for their targets. This was just the start of the most sustained artillery barrage suffered by any unit of the British Army since Korea in the summer of 1953. More men were yet to die on Longdon. The suffering was not over.\n\n[**F A L K L A N D S M O U N T  \nL O N G D O N**](17_chapter10.html#ch10)\n\nTHE LEGACY\n\nW **ith the coming of the dawn on the morning of 12th June, the immediate aftermath of the Battle for Mount Longdon became apparent. Lieutenant Colonel Hew Pike later recalled the scene which greeted 3 Para as the dawn broke; a scene which provided perhaps the most haunting memory of that long, cold and bitter fight. 'Groups of young soldiers, grim faced, shocked but determined, moved through the mist with their bayonets fixed to check the enemy dead. The debris of battle was scattered along the length of the mountain, encountered round every turn in the rocks, in every gully. Weapons, clothing, rations, blankets, boots, tents, ammunition, sleeping bags, blood soaked medical dressings, web equipment and packs had all been abandoned along with the 105mm recoilless rifles, 120mm mortars, Browning heavy machine guns and the sophisticated night vision binoculars and sights with which Argentinian snipers had given us so much trouble during the long night. The sour and distinctive odour of death lingered in the nostrils as we began to dig temporary graves for some of the enemy dead. But it was a slow job, and eventually the task was abandoned when enemy artillery and mortars started again.'** 1\n\nOne of the most painful tasks was the ongoing care of the wounded and the collection and identification of the 3 Para dead. CSM Weeks, who had moved up and down the mountain all night chivvying B Company on and carrying wounded back down the mountain to Colour Sergeant Brian Faulkner's Regimental Aid Post, now faced a task that no one wanted.\n\n> _'The following morning, when it was light, in between artillery and mortar coming down, we had to go and sort out the bodies and do the documentation. Although it sounds stupid, in battle you still have to account for everything and everybody and you have to fill in the bloody paperwork. I went round to the lads who were left and I asked if any of them would come and help me remove the bodies. I didn't particularly want to do it. I found it terribly hard to ask the guys and they found it terribly hard to say no outright. They tried to make an excuse that they were too busy doing other things rather than say they didn't want to go down to their friends and bring their bodies up. Lewis and Clarkson-Kearsley were the only ones who said they'd come. Sammy Docherty_ (sic) _also came because it needed four to carry a body on a poncho and get them all down to a central place.'_ 2\n\nThe process of searching the dead for personal effects, removing wedding rings, leaving one dog tag on and filling in forms indicating where wounds had been inflicted was indeed grim but was absolutely essential. Even experienced soldiers like\n\n**_The grim aftermath of battle. British soldiers inspect the body of a fallen Argentine, an unpopular but necessery task which has to be undertaken during any conflict._** John Hughes-Wilson.\n\nCSM Weeks, who admitted to having seen 'loads of dead bodies' during his soldiering, were visibly shaken when tending the bodies of men - men like Sergeant Ian McKay - that they had known well and had come to respect. Colour Sergeant Faulkner's team removed the bodies and all personal effects were handed to the Padre.\n\nAs the mist dispersed and Longdon came into view of Argentine artillery observers on Mount Tumbledown, the shelling intensified. With such a level of casualties and with ten hours fighting behind them there would be no further exploitation towards Wireless Ridge. 3 Para would have to sit it out on Longdon and wait for the next move.\n\nThe capture of Mount Longdon was of pivotal importance in that it secured and obscured the supply route back towards Teal Inlet, enabling ammunition and supplies to be brought up for phase two of Brigadier Thompson's plan, the assaults on Mount Tumbledown and Wireless Ridge. Just as important, for the wounded of the Longdon battle and those that would follow, its capture meant that they could be casevac'd out without coming under fire. This proved invaluable later as the low cloud on the high ground during the following two days disrupted the evacuation of wounded to Fitzroy. All the Scots Guards casualties evacuated from the Battle of Tumbledown were treated at 3 Commando Brigade Forward Dressing Station at Teal Inlet.\n\n* * *\n\n**'With hindsight I would have used naval gunfire to engage enemy batteries throughout.'**\n\n* * *\n\nSo could anything have been done differently? In his book published to coincide with the twentieth anniversary of the outbreak of the war, Graham Colbeck, a Sergeant in charge of a Milan section on Longdon in 1982, questioned both the lack of artillery preparation in the overall Brigade plan and the use of all three platoons of B Company being used together in 'line abreast' in the battalion plan. Colbeck and other Longdon veterans have pointed to the reliance on surprise rather then firepower as the reason for the battle grinding on for so long at such a heavy price in killed and wounded. The decision to take on a company strength defence with a company (B) - rather than stacking the odds in 3 Para's favour by using more men and the lack of a dedicated 'reserve' platoon for that company once it became engaged by the Argentine's is another source of contention. All three platoons of B Company became stalled almost simultaneously in the face of heavy, well-sited and increasingly accurate firepower, hence there were too few men left with which to 'reinforce success' at any point on the western slopes. There were, as has already been discussed previously, very good reasons why the Brigade and battalion plans were developed as they were. Both Lieutenant Colonel Pike and Brigadier Thompson admitted to have been lulled into an overoptimistic assessment of the nature of the Argentine defences and their ability to defend Longdon due to the runaway success of 3 Para's preparatory patrolling. This goes a long way to explaining why there was such a reliance on surprise rather than firepower. However, and it is essential here to embrace the entire canvas on which the strategy of 3 Commando Brigade was painted and extend the picture beyond the outer crust of the phase one Stanley defences. It was deemed essential for Longdon to be silent so as to mask the northern axis of approach and to conserve vital stocks of ammunition for a war which, for all Brigadier Thompson knew, may well have taken weeks more rather than the days as it turned out. If there was one thing Brigadier Thompson would employ differently, if he were to plan phase one all over again, it would be naval gunfire, as opposed to artillery support, for which there was precious little excess ammunition available anyway.\n\n> _'With hindsight I would have used naval gunfire to engage enemy batteries throughout. Also I would have suggested that night harassing tasks by naval gunfire was a waste of ammunition, and kept it for a really concentrated crack at the enemy artillery.'_ 5\n\nThe debate will go on. Perhaps if Brigadier Thompson had used his naval gunfire support en masse to engage the Argentine batteries from the four ships available to him, rather than dispersing it between the attacking formations, then there is the possibility that some of the batteries which began to fire defensively on Longdon almost as soon as it became clear to the Argentines that the mountain had been lost, may have been neutralised. It is yet another in the long line of military history's 'what ifs'.\n\n**_A member of 3 Para Regimental Police leads away an Argentinian captive._**\n\nAs it was from around 8.00 a.m. on the morning of 12th June the Argentine artillery pounded 3 Para for the next 36 hours and claimed the lives of a further 5 members of the battalion, including Corporal Stewart McLaughlin and Private Richard Absolon (Military Medal), both of whom had assumed prominent roles during the night's fighting. In all, 23 men were killed during the battle for Mount Longdon and the subsequent shelling. 47 men were wounded, some of them with terrible injuries, including the loss of limbs. For those who had emerged unscathed - physically at least - they would watch as their sister battalion, 2 Para, seized Wireless Ridge to the east on the night of 13th\/14th June, before charging into Port Stanley hard on 2 Para's heels for the honour of being the first Para battalion to reach the capital after the formal Argentine surrender on 14th June. It was in billets in deserted sheds, houses and in the grandstand of the racecourse to the west end of Stanley that 3 Para's war ended, officially at least.\n\n**_Argentine POWs march towards a ship which will take them home after the Falklands campaign._**\n\nBut men cannot live through a battle like Longdon and not be affected. 14th June 1982 was just the date when the real fighting ended. For many the internal battle, the mental battle - as they grappled to comprehend what they had seen, heard and felt - had only just begun, although in the euphoria of staying alive to witness victory they could not know it.\n\nJust over a decade after the end of the campaign, as men began to record their personal accounts and experiences of the war, allegations of war crimes and atrocities committed against the Argentine dead and prisoners of war, and of the summary execution of 'American' mercenaries surfaced in a book called _Excursion to Hell_ by Vincent Bramley. Bramley had been a corporal in Major Dennison's Support Company during the battle. The revelations contained in its pages eventually attracted the attention of politicians and investigative journalists. Such was the controversy surrounding them that an investigation headed by detective superintendent Alec Edwards of the Serious and International Crimes section of New Scotland Yard was initiated in 1992. Even as early as 1982, those responsible for screening Argentine prisoners prior to repatriation in 1982, had found no evidence to substantiate the claims that American mercenaries had taken part in the Longdon battle. After two years of the enquiry during which time, key witnesses were taken back to Longdon, the Director of Public Prosecutions ruled that the findings of the investigation were inconclusive and that no charges were to be brought. Nevertheless, for those struggling to come to terms with life after the war, be they still in the regiment or trying to earn a living on 'civvy street', it was a time of deep unease. That night on Longdon would never be forgotten - it was, it is, still with them, every night for some, vividly recalled in the recurring nightmares they were and are experiencing - and their sense of regimental and personal pride and the memory of their dead comrades had somehow been tarnished. Was an investigation necessary? Most certainly. Once the revelations had passed from the realm of British Army folklore and into the public domain with the help of the media, there was never any question that the veracity of the claims would not be tested. Any nation that respects the rule of law, actively promotes justice and upholds human rights cannot ignore such serious allegations. War, however, is quite literally, a bloody and brutal business that extracts its own price from those engaged in it. Personalities are hammered out on the anvil of battle and the supercharged mixture of aggression, ruthlessness, adrenalin and fear cannot simply be switched on and off like a tap. Some crumble, others come through; all are changed and not always for the better. There can be no excuse for the execution of unarmed prisoners of war in cold blood but the battlefield, by its very nature, is a world set apart from the rest of civilised society for the duration of the struggle. The very nature of combat flies in the face of rationality while instinct and the personal 'red mist' that so often descends and drives the soldier on coalesces with the 'fog of war' to cloud the judgment and blur the boundaries. As Graham Colbeck summed up after almost twenty years of considering what happened that night:\n\n> _'Battle is a delicate business between extremes of human behaviour - selfish cowardice and selfless sacrifice, brutality and humanity, callousness and pity - and the virtuous must be made to outweigh the dishonourable, both in the individual and the unit, if either is to survive with any pride.'_ 8\n\nTwelve years on from the enquiry and some of the men of Longdon are still battling with their memories. It will stay with them forever. Organisations like Combat Stress, the ex-servicemen's mental welfare association, has provided support and advice for men like Nick Rose who, at the age of twenty, witnessed the death of several close friends on the southern slopes of Longdon.\n\n> _'I do have this PTSD thing but then I think everybody's got it; its just how it manifests itself and how folk deal with it. It's a normal reaction to an abnormal event. But I've suffered with nightmares and I know other people who've gone through the same. It lives with people, especially around Remembrance Day, obviously. But its also a pride thing, I'll always come back to that. The Falklands War was our last colonial war if you like and I fought in it and we won it. I'm proud of that. Extremely so and I'm proud to say I was there. It's sad sometimes but I talk about it with other like-minded people and I get a bit of help. I don't know if I want to go back down yet. I don't think I've got that many ghosts that would warrant me going back down there. I'd like to see it and one day of course, I will. I'd like to take my sons there when they're old enough to understand but it's not about laying ghosts to rest. I think I've done that from here. I like_ _to think so anyway'_\n\nNick Rose has recently decided to try to raise funds by enlisting the aid of Ken Peers and the Band of the Royal Marines along with several personalities who have offered to record war poetry on CD in aid of the organisation which has helped him to make sense of what happened. He knows he still has some way to go but by getting involved in a project which will help raise public awareness of the organisation's work and much needed funds to help men like him, who gave so much, he hopes that he can at least begin to 'give something back'.\n\n###### THE EVOLUTION OF A SOLDIER\n\nBy Tim Lynch\n\n**Tym Lynch served in the Falklands with 159 Army Air Corps. He returned to the Islands for the first time in twenty years in 2002 on the 20th anniversary pilgrimage organised by the South Atlantic Medal Association.**\n\nFormer US Marine James Jones calls it 'the evolution of a soldier'. It is the moment in combat when you know, with absolute certainty that you are going to die. It is the moment that all the training has been leading to. You go forward to the objective, to help your mates, to do what you are supposed to do but you know that the next step you take will be your last Those who don't feel that fear aren't heroes, real heroes are those who feel it but go on anyway. For every one of the tired young men on the summit of Mount Longdon, that moment had arrived and they had faced it. Now, as the sun rose over the rocks of East Falkland, they were alive.\n\nThe men of 3 Para were exhilarated. This had been the moment they had been waiting for. They had fought their first real battle and won. They had passed what they considered their ultimate test. A US Army study published after WW2 by Colonel S.L.A. Marshall looked into the effectiveness of US infantrymen in the period 1943\u201345 and found only 15% of trained combat riflemen used their weapons in battle - the rest did not run away, but even under attack they would not kill. 'The thing is simply this,' wrote Marshall 'that out of an average one hundred men along the line of fire during the period of an encounter, only fifteen men on average would take any part with the weapons. This was true whether the action was spread over a day, or two days or three. In the most aggressive infantry companies, under the most intense local pressure, the figure rarely rose above 25% of total strength from the opening to the close of an action'. Training of infantrymen changed as a result of this. Modern combat training was about teaching the recruit to fear letting his mates down more than anything else. The Paras had trained to value aggression and to some, the bayonet fighting on Longdon had been 'a joy'.\n\nSo what was the deciding factor in the fight for Longdon? Undoubtedly the long exposure to the elements, coupled with heavy bombardments had taken its toll. Poor re-supply arrangements had increased the misery for the defenders but, as we have seen, the attackers were also cold, wet, tired and hungry. The major difference between the two forces was about leadership.\n\nConscripts are capable of fighting extremely well and have done so many times. In the Falklands war, though, two factors came into play. One was that the Argentine military underestimated the speed and nature of the response and had assumed that it would be supplying a garrison force. Consequently, the defenders included a relatively high proportion of logistical and supply troops made up of conscripts who had been poorly prepared for that role which then hampered the supply lines. Secondly, and far more importantly, class distinctions were strictly enforced throughout the military. After the war, for example, Argentine officers and NCOs were bemused by the spectacle of their British counterparts lining up for food behind the ordinary soldiers. Very clear and strict boundaries existed between the conscripts and regular troops and between ranks. Among conscripts, initiative and personal responsibility were neither recognised nor encouraged and this had a significant bearing on their behaviour in combat.\n\nFor the Paras aggression is built into training and is highly prized. The problem for them was not that of firing their weapons but of supplying enough ammunition to keep doing so. By contrast, some Argentine veterans report taking shelter and hearing British troops passing literally over the tops of their bunkers. There were reports of conscripts having been disabled by their own officers to prevent them from retreating and the lack of officers and NCOs among Argentine casualties is notable (of 36 Argentines killed on Longdon and Wireless Ridge, one lieutenant and two corporals are listed. The rest were privates. Of the British casualties on Longdon alone, 10 of the 23 killed were NCOs).\n\n**_Mount Longdon. The remains of an Argentine sangar covering the western slope of the mountain. Note how the position is sited to cover the killing zone as 3 Para were funnelled into the channel between the rocks below._** Tim Lynch\n\nThe battle won, 3 Para began to relax. In _Excursion to Hell_ , Vincent Bramley recalls what happened next. 'We all hit the ground...as the shell landed. No sooner had I started to pick myself up when a high-pitched scream rushed into my ears, a deadly sound. I can still hear it today.' A few metres away, an incoming mortar round had exploded among a group of Paras. Running to help, Bramley 'only saw Denzil, Denzil the character we all loved. I wrenched my eyes to his legs: one was hanging off, ripped to shreds, the bone clearly visible.' Denzil Connick was the only survivor of the group. 'It took nearly a year after this war for Craig's [Jones] face to go before I slept, nearly a year to wipe out Denzil's smock on fire and his scream. Until I die, it will remain a part of me.'\n\nValuable military lessons were learned from the Longdon battle. Lessons about adequate fire support and about strategy. But the survivors also learned about themselves. As S.A. Stouffer, in a survey of two thousand American troops in the Pacific theatre in 1944, looked at how fear manifested itself. It showed that 84% said they experienced a violent pounding of the heart, others felt sick. About half said they felt faint. A quarter said they had vomited and 21% admitted to having lost control of their bowels. Since this figure is based on voluntary admissions, we could perhaps expect the true figure to be higher. Among themselves, the veterans of 3 Para could privately admit that some things about war had not changed. To outsiders, though, they were Paras. That meant maintaining a reputation. On the way to the South Atlantic, British journalists Patrick Bishop and John Witherow had observed training sessions with some disquiet. They described one in their book _The Winter War_ : \"What do you do if you find a wounded Argie?\" asked a Para corporal, rhetorically, to a platoon weapons talk. \"You blow his f\u2014ing head off. What do you do if there's a TV crew watching? You treat him as one of your own\". Many of the Toms, as the Para officers called their men with a mixture of affection and contempt, enjoyed their image as emotionless, efficient killers, one step away from being psychopaths. I asked one what he would do if he found a wounded Argie.\"Kill him with me bayonet, rip his gold teeth out and cut off his fingers to get his rings\", he replied. Of course, they did nothing of the sort when the hypothesis became real. But they would like you to think they would.'\n\n**_A rust encrusted 105mm Recoilless Rifle, stands in an abandoned Argentine position twenty years after the battle for Mount Longdon._** Tim Lynch\n\nIt came as no surprise, then, to the rest of the Task Force when rumours of the murder of prisoners on Longdon began to circulate. The stories, which even reached the Falkland Islands newspaper _Penguin News_ , were roundly ignored by the many journalists on the Islands, until the same allegations became part of a war crimes investigation over ten years later.\n\nTrained to see themselves as emotionless killers, praised for their aggression and having come to see themselves as dead men, the teenagers of Longdon were ready to go home. Trying to make sense of his own experiences in the Pacific campaigns of WW2, Jones described this evolution as a special kind of moment. 'It is absolutely true' he wrote 'that when you think, when you know, you are going off to die somewhere soon, every day has a special, bright, delicious, poignant taste to it that normal days in normal times do not have. Some men like to live that way all the time. Some are actually sorry to come home and see it end. Even those of us who hated it found it exciting sometimes. That is what civilian people never understand about their returning soldiers. They cannot understand how we could hate it, and still like it; and they do not realise they have a lot of dead men around them, dead men who are walking around and breathing. Some men find it hard to come back from their evolution of a soldier. Some never come back at all, not completely.' Facing the survivors of 3 Para was a problem that had affected generations of soldiers before them. How do you live a life you've already given up on?\n\nReturning Paras, cooped up on board the MV _Norland_ , celebrated Airborne Forces Day, 1982 with a drink fuelled riot that continued into the early hours. Sent on leave with the advice to 'get drunk and forget it', they went home to their families. 'Instead of talking about it' remembered Jones of the men who came home in 1945, 'most men didn't talk about it. It was not that they didn't want to talk about it; it was that when they did, nobody understood. It was such a different way of living, and of looking at life even, that there was no common ground for communication in it.' Brought up to believe that returning soldiers 'didn't want to talk about it', the teenage veterans kept quiet. 'I was 19 when I got home' says one, 'and I felt older than my dad. He'd been in the army for years but never heard a shot fired. What could I talk to him about?'. Others, 17 or 18 years old when they got back found that tabloid propaganda had built up public sympathy that lay with the 'young kids' they had faced. Much was made in the British press at the time of the war about the age of the Argentinean troops but in fact, most were drawn from the 'Class of '62' - men born in 1962 and coming to the end of their period of national service. The Task Force, too, was a young unit, the youngest sailor aboard HMS _Invincible_ was just 16. Among 3 Para's losses during the attack on Mount Longdon, were three seventeen year-old privates who had only recently joined the battalion. After the war, British veterans found a great deal of sympathy back home for the 'poor Argentine kids' they had faced in battle but a great many returned still too young to buy a legal drink in a pub to celebrate their survival or to vote for the government which had sent them to fight.\n\nThe return to ordinary soldiering was not easy and many veterans found themselves disillusioned when 'shiny arses' were promoted over those who had fought well. One by one, they began to leave the army to return to civilian life.\n\nIn 1974, the American psychologist Robert Lifton wrote about the case of 'Skip' Johnson, a Vietnam veteran and Medal of Honor winner. Johnson, awarded the medal for hunting down and killing a Vietnamese anti-tank team, found himself unable to cope with having been honoured for, as he saw it, having lost control and killing in a rage. He was killed during an armed robbery in New York. His mother told the press 'sometimes I wonder if he just got tired of living and wanted someone else to pull the trigger'. Just as in the aftermath of Vietnam, from the mid 1980's, stories begin to emerge in the media of Falklands veterans. Tales of alcoholism, drug use, involvement in crime and suicide surfaced on a regular basis and, five years after the war, a study by a Parachute regiment medical officer found that 50% of those veterans still serving were experiencing some or all of the symptoms of Post Traumatic Stress Disorder. If the figure among this group was so high, what was happening to those who had left the army?\n\nThe British Ministry of Defence prided itself that the Falklands War had produced very few psychiatric casualties. The short duration of the campaign and high level of training meant that neither army suffered significant losses through 'shell shock'. This was true, only 16 psychiatric casualties were recorded, 2% of the total. Unfortunately, whether by ignorance or design, military psychiatrists, particularly those attached to the Army, failed to grasp the concept of psychological consequences surfacing after combat and continued to concern themselves only with breakdown in combat itself, a very different matter.\n\nVeterans and health care professionals have criticised the MOD for this. Roderick Orner, a psychologist working with Falklands veterans, has described the MOD's failure to recognise the effects of Post Traumatic Stress as 'inexcusable' and criticises its 'shameful record' of providing support to ex-servicemen. The existence of the disorder had been recognised for many years following work among survivors of natural disasters, accidents and particularly among veterans of the Vietnam war and was formally identified by the American Psychiatric Association in 1980, two years before the Task Force set sail. With the honourable exception of Navy Psychiatrist Commander Morgan O'Connell, no attempt was made to address the problem or even to consider it. Interviewed a few months after the war, O'Connell told _The Guardian_ that whilst there had been only 16 psychiatric casualties during the war, in the aftermath he had already treated over a hundred. Having left the Navy in frustration over the way it had treated its veterans, Dr. O'Connell argues that the 'government's uncaring attitude and its policy of ignoring the fact that PTSD is affecting their soldiers, is inflicting more damage on their own men than any foreign power or terrorist organisation could hope to achieve'. It was not until 1986 that the army acknowledged the existence of PTSD and even then argued that civilian resources should be used to treat veterans who had left the forces. Unfortunately, as Dr. O'Connell puts it; 'The NHS has no adequate system for treating [these] men because they usually need to be with their peer group and relate to people who understand the realities of war and service life'.\n\nStudies of Post Traumatic Stress in combat veterans across the world showed that they were 50% more likely to commit suicide and 48% more likely to die in car accidents than those who had not served. They were over represented in the prison system and the British homeless charity SHELTER claimed that 25% of the long-term homeless were ex-servicemen. Twenty years after Longdon, it was estimated that more veterans had died by their own hand than were killed in action. Yet, almost uniquely, Britain refused to accept that its war veterans deserved any special recognition. In answer to this, the veterans began to turn to each other for the help they needed to come to terms with their experiences. Fifteen years after the end of the war, the South Atlantic Medal Association (SAMA) was launched. It was the brainchild of 3 Para veteran Denzil Connick, the man who lost a leg in the mortar attack Bramley witnessed on Longdon on 12th June 1982.\n\n**_The granite memorial on Mount Longdon array with iron poppies. Each poppy represents one of the Paras killed during the battle. The corner is deliberately broken symbolizing the Battalion's loss._** Tim Lvnch\n\nWith the help of the people of the Falkland Islands, SAMA set out to mark the 20th anniversary of the end of the war with a pilgrimage by veterans. Refused help by the Ministry of Defence, an aircraft was chartered privately to take 200 survivors back to the battlefields to try to find some answers and some peace. For all the veterans, this was a personal journey to revisit their part of the war and, hopefully, to lay ghosts that had haunted them for years. It was an opportunity to walk again over ground last covered under fire. For some, decisions they had made at the time had earned them criticism and this was a chance to look again at what happened and why. For the common soldier, war is reduced to what happens in his immediate vicinity and he relies on historians to tell him what he did later. For the SAMA veterans, marines mixed with paras and guardsmen to talk about their war and to help fill in the details. It was the debriefing they had never been given. Perhaps most importantly, it was a chance to find the answer to the question that had been uppermost in all our minds since 1982 - was it worth it?\n\nOn 11th November 2002, a group gathered on top of Mount Longdon for an act of remembrance. Among them, and now an honorary paratrooper, was former Islands policeman Terry Peck, who had guided them across the islands to this bleak hilltop overlooking Port Stanley. Quietly they made their way up the hillside, following the route they had taken all those years ago. Almost unerringly finding the rocks they had sheltered behind in a landscape burned into their memories, gently placing wreathes on spots where friends had died.\n\nDenzil Connick had never been baptised. He decided it was the right time and that this was the right place. On the way to the Islands in 1982, he had befriended a Marine, David Devenney. Today, Devenney is an ordained minister and proudly conducted a service among the rocks atop Longdon near the spot where Denzil had taken his last real steps. For a moment, the wind eased and silence fell. For the first time in twenty years, the survivors of 3 Para had found peace.\n\n**_SAMA pilgrimage 2002. Ex-Marine Reverend David Devenney baptizes Denzil Connick on the summit of Mount Longdon._** Tim Lynch\n\n**_3 Para veterans pose tor the camera in front of the Mount Longdon memorial Cross._** Tim Lynch\n\n#### **HEALING THE SOUL THE ARGENTINE PERSPECTIVE**\n\nBy Michael Savage\n\nMichael Savage belonged to C Company of the Argentine 7th Infantry Regiment positioned on a feature just north east of Longdon called 'Rough Diamond.' His mortar supported Lieutenant Casta\u00f1eda's unit during its counter attack along Longdon's crest and the subsequent hand-to-hand fighting. After the Paras had cleared 'Full Back' by the morning of the 12th, he and his comrades were trapped in their positions by the British artillery until dusk that evening. The rest of his company had retreated earlier that day. Eventually he managed to escape towards Stanley. Eighteen years later he returned to the Falklands in the company of James Peck, son of Terry, the Islander who had escaped from Stanley and had joined up with 3 Para at Teal Inlet.\n\nOur trip to the Falklands\/Malvinas was planned as a way of drawing the curtain on a sad story. I had to go back to the islands to lay the ghosts to rest; I had to return to the scene but I had to see it in normal times, without the horror.\n\nWe had the help of a great friend, James Peck, his wife, Carol and his son Joshua. After many failed attempts to buy air fares, James offered to make our reservations and pay for them in the islands and all this was done ten days before our departure. They also offered to put us up in their home, so little by little we completed the jigsaw. It was a long trip to the military base in Mount Pleasant, with stops in Santiago de Chile, Puerto Montt and Punta Arenas.\n\nIt was very important but exhausting to take the family: I travelled with my wife Andrea and my two children, Patricio and Maggie. The weather wasn't on our side, as if to remind us at every moment how hard it was to put up with the atrocious, wet conditions and the hunger of those days of war in 1982. It now seems hard to believe that we survived at all.\n\n**_Approaching the summit of Mount Longdon from the west. Twenty years on Mike Savage and Terry Peck walk towards the craggy feature on the summit to lay a wreath to the fallen of both sides._** Terry Peck\n\n**_View from the summit in 2002 looking east. Port Stanley is in the distance with Mount Tumbledown to the right. The rock features to the left and centre were used as sniper positions. From these positions snipers were able to pick off the advancing men of 3 Para._** Terry Peck\n\n**_View overlooking the 'Bowl' and the sniper positions. In the middle foreground was the Argentine Command Post and heavy mortar positions._** Terry Peck\n\n**_Michael Savage inspects the corroded remains of his cooker._** Michael savage\n\nWe stayed with James Peck, an islander and an artist who opened his home and his heart to us. And although the lashing rain continued and it was very windy, the first day there James and I climbed from Moody Brook to our position. The first thing we spotted from a distance was something brown and square, which turned out to be our old cooker.\n\nAnd right beside it, in the same spot we had left it 18 years ago, as if waiting for me, was a stainless steel cauldron, with two shrapnel holes in it. I used to cook. I think we used to take it in turns, doing weekly shifts that's when we were able to get more food. But it was a terrible thing to go about serving others in the cold, so my mate Roberto Maldonado and I would fill three canteens with ' _mat\u00e9 cocido_ ' to use as hot water bottles and we'd have something warm to drink each morning.\n\nAt night we'd leave the cauldron just outside the tent. Field mice would often get into it and they'd freeze to death trying to get out. For meals we had ' _mat\u00e9 cocido_ ' in the morning, no sugar or bread; a watery soup at midday and at night a thin broth and nothing else. There was never any bread or fruit. We'd get water with our canteens from nearby ponds. We had no option but to pilfer our own provisions. The booty was hidden in empty munition cases that's how we'd fool the officers.\n\nThere were also groups of 'sheep hunters', so you could say that there was a primitive form of bartering going on. Even so, I lost 17 kilos. I came home weighing only 55 kilos and to this day I have been unable to forgive the Argentine officers who were in charge of us. It is still possible to make out a depression in the ground where my tent once was. The 'tent' consisted of two canvas cloths which you had to put together to make an improvised shelter. It is impossible to imagine what it was like to be in this rickety structure in a windstorm. Several times in the night it would blow away, leaving us crying and praying in the rain out of sheer desperation.\n\n**_Remnants of the Argentine Mortar Platoon's stainless steel cauldron which Michael used to convey food to his colleagues._** Terry Peck\n\n**_The weathered remains of Michael's shelter still standing twenty years on._** Terry Peck\n\n**_Michael is able to find other relics of the occupation, such as this canteen, next to the rubble of a stone wall, built for protection against weather and British fire._** Michael Savage\n\nAlan Craig, my Argentine trenchmate, remembers that 'once wet, there was no way you could dry your clothes other than to keep them on and dry them with the heat of your body'.\n\nThe slabs that held the tent are now covered with peat. The shovel, canteens and canvas are still intact. At first only the important things stood out, but then it was possible to find all sorts of things, partially buried in the peat. I found a tube of toothpaste, a boot brush, a rucksack and razor blades and shrapnel all over the place. The bomb craters are everywhere and are really impressive. Besides, the grass doesn't grow in them, they're black and I wonder how long they will be there to remind those of us who survived how close we had been to death.\n\nAround three metres away was the position of Alan Craig, Adri\u00e1n G\u00f3mez Scher and Sergeant Alcaide. With the passage of time the stone wall that used to be quite big is now only 80 cms high. Here we found a melted ballpoint pen and some shaving cream.\n\nI didn't want to dig about too much, because I felt respect for the place and I wanted to leave it as it was for the day my friends can visit it. I also found the remainings of a twisted kettle and a plate with the insignia 'EA' Argentine Army. The base of a machine-gun was surrounded by shell holes. Anyone who had been there at the time wouldn't have survived.\n\nWe also found two 105mm anti-tank guns. It was amazing one of them was carried by me and six other 'colimbas' (conscripts) by hand uphill from Moody Brook, four kilometres away. It took us four days to do it. Now they lie there like dinosaur skeletons; in fact, this place is like an open air museum and it's hard to believe that so much is still intact.\n\n**_Left to right: Terry Peck, Michael Savage and James Peck sitting in the 'Bowl' discussing the events of 11\/12 June 1982. Michael had been a member of the mortar platoon that had fired on 3 Para from the feature known as 'Rough Diamond'._** Terry Peck\n\n**_Michael Savage's children play amongst the trenches which once protected their father. Many Argentines were killed and wounded in this area. Mount Longdon is in the background._** Terry Peck\n\n**_The snow capped mountain range barring the way to Port Stanley. Photograph taken looking west from a position across the bay from Port Stanley._** Terry Peck\n\nA few days later, James invited his father around. That's how we discovered that Terry is an institution in Stanley. During the Argentine invasion he fled to the hills on his motorbike. For five weeks he lived alone in the open, hiding and eating dried fruit and nuts. Later he joined the 3 Parachute Regiment and guided them to Mount Longdon, where they fought the VII Regiment of LaPlata, which was my regiment. This is where it became really interesting.\n\nWe went up the mountain together. We had something in common, that only veterans would understand. We spoke sincerely about what happened on that horrible night of the 11th June, 1982. We filled in the gaps in our stories, exchanging facts and anecdotes.\n\nTerry told me that three days before the battle at Mount Longdon the British were already very close when they saw a group of six men heading for an estancia beyond the Murrell river. An officer suggested kiling them, but the regiment commander stopped them, because this would have alerted the Argentines and they would have lost the surprise element for the attack which was to be carried out the following day. I was among those six men, acting as interpreter.\n\nOn the night of the 11th June, B company of the VII Regiment La Plata was awakened when a British soldier stepped on a mine, which gave them some time to prepare. They were on the summit of Mount Longdon; that battle was the fiercest and bloodiest, because it involved 150 men against 150 men.\n\nJust thinking about the battle chills my blood. The English came up a very steep slope towards the Argentine positions; the British mistake was costly and they suffered many losses, so much so that today the battle is analysed and studied in military tactics courses.\n\nFrom my position I could hear shouting in English and Spanish, tracer bullets, everything. In the confusion, C Company of the VII Regiment La Plata, where I was, fired two flares, lighting up the hills. Sub Lieutenant Casta\u00f1eda organised support for Company B which was under heavy attack.\n\n**_Port Stanley today. Mount Longdon and Two Sisters in the background._** Tim Lynch\n\n**_Falklands Summer, Terry and James Peck stand beside the 3 Para memorial._** Terry Peck\n\n**_Terry Peck lays a wreath on the anniversary of the battle each year to remember the Paras who fell in 1982._** Terry Peck\n\nCasta\u00f1eda's support was considered by the British to have been one of the most heroic events in the ground fighting on the Malvinas, prompting Brigadier Julian Thompson to say: \"I was on the point of withdrawing my paras from Longdon. We couldn't believe that these teenagers disguised as soldiers were causing us to suffer so many losses.\"\n\nAlan Craig remembers that \"that night B company wasn't using radars because they were trying to save batteries.\"\n\nAs for me, I was there when they organised the group, but as I belonged to a mortar support group I wasn't picked. I saw 46 men leave and head straight for Longdon, of whom only 21 returned. They headed out in the dead of night straight into a bloody battle. The heroism of these kids was incredible; those of who remained behind were praying and I couldn't stop trembling.\n\nThe next morning as two of the 21 survivors were giving us their account of what had happened, a mortar shell fell two metres from our trench, killing one of them instantly and wounding the other while Roberto, my mate, took a shrapnel wound in the intestines.\n\n\"When the artillery started pounding on us, we jumped into this hole, dug into the peat bank. It was constructed by a conscript that was an engineer. It was 2 metres down, and 2 metres into the bank. During the time we were waiting for the attack, it was a very hard job keeping it dry but the reward was great!\n\nThis hole saved my life. In the middle of the night the _colimbas_ came to ask me how to say \"me rindo\" (I surrender) in English and they left for their dugouts repeating the words \"I surrendo, I surrendo.\"\n\n**_A gaunt but happy Michael Savage returns from the Falklands to the loving embrace of his mother. His lack of nourishment is clearly evident._** Michael Savage\n\n**_An emotional return. Now promoted to Major-General, Hew Pike OBE, returns to Mount Longdon for the first time in ten years to salute the men of his battalion who fell on Mount Longdon. December 1992._** Terry Peck\n\n**_A plaque in the grounds of Government House, Port Stanley, to mark the 150th anniversary of its establishment. It was important that this was recaptured alter the Argentine occupation, as it was a symbol of British sovereignty on the Falkland Islands._** Tim Lynch\n\nSome of them tried to make white flags but by then we didn't have anything left that was that colour because all our clothes were filthy, black with soot from cooking with helicopter fuel. Getting to know another war veteran, and especially Terry, who was on the other side, was very special. We had both been through the same experience in the war. As we climbed Longdon in a four-wheel-drive vehicle, Terry listened in almost complete silence as I told him how we had spent 60 days in those holes.\n\nAt first the British had wanted Terry to guide the regiment to Stanley, but as he is very able with guns, they relented and gave him one and he fought at Longdon on 11th June.\n\nIt was very interesting to hear the British side of things. They are very conscious of the battle at Longdon because 23 Britons died there. On the summit there is a cross and a plaque with names. I don't know exactly how many Argentines died there, but we do know it was one of the bloodiest battles.\n\nThere are wreathes of poppies to honour the dead, one of them from Prince Charles, who was taken there by Terry. It is so windy on the summit that one can't stay there very long. We moved lower down and had a meal and some brandy from Terry's flask, with its 3 Para insignia engraved on it. And we began to reveal what we didn't know about each other.\n\n**_A memorial cross on Longdon marks the spot where Ian McKay was killed charging an Argentine machine gun position._** Tim Lynch\n\n**_Major General Hew Pike OBE visits Blue Beach cemetery, 1992._** Terry Peck\n\n**_Blue Beach cemetery. Major General Pike pays his respects at the grave of fellow battalion commander of 2 Para, Lieutenant Colonel 'H' Jones._** Terry Peck\n\nThere was a rumour that during the battles the Gurkhas had been slitting throats to instil terror. According to Terry there were no Gurkhas in 3 Para, but it is possible that mutilations were caused by artillery; perhaps that is how the confusion arose. Although we joked a bit, we could sense the spectre of death that haunts the place. Terry goes up every 11 June because he made a promise and he keeps it, regardless of the weather. Life has given us the opportunity to meet and be friends and we won't waste it. When Terry came to say goodbye, we embraced as friends.\n\nHow absurd all war is. It is incredible that, in the times we live in, anyone should think of forcing change on other people's lives. And it's unbelievable that young conscripts should have had to bear the brunt of this epic jaunt, whose prime motivation was nationalist sentiment. It was like plucking people off the street and forcing them to fight a professional army. We have all thanked God that it wasn't our time.\n\nThis trip has healed my soul. Spending those 15 days with the Pecks, to be a part of their life, was fundamental. The ghosts of war have gone, only to be replaced by a great friendship. I can't find the words to thank those who opened their homes and their hearts to us.\n\n###### **3 P A R A M O U N T L O N G D O N**\n\n11th\/12th June 1982\n\n###### ________________ ROLL OF HONOUR ________________\n\nPrivate  \nR. J. ABSOLON, MM | Private  \nP. J. HEDICKER | Corporal  \nK. J. MCCARTHY  \n---|---|---  \nPrivate  \nG. BULL | Lance Corporal  \nP. D. HIGGS | Sergeant  \nI. J. McKAY, VC  \nPrivate  \nJ. S. BURT | Corporal  \nS. HOPE | Corporal  \nS. P. F. McLAUGHLIN  \nPrivate  \nJ. D. CROW | Private  \nT. R. JENKINS | Lance Corporal  \nJ. H. MURDOCH  \nPrivate  \nM. S. DODSWORTH | Private  \nC. E. JONES | Lance Corporal  \nD. E. SCOTT  \nPrivate  \nA. D. GREENWOOD | Private  \nS.I.LANG | Private  \nI. P. SCRIVENS  \nPrivate  \nN. GROSE | Lance Corporal  \nC. K. LOVETT | Private  \nP. A. WEST\n\n###### ______________ ATTACHED ______________\n\nCraftsman  \nA. SHAW  \nRoyal Electrical & Mechanical Engineers | Corporal  \nS WILSON  \n9 Parachute Squadron Royal Engineers  \n---|---\n\n###### GALLANTRY AWARDS\n\nSergeant I. J. McKay, B Coy: V.C. posthumous  \nLieutenant Colonel H. W. R. Pike, MBE: D.S.O.  \nMajor M. H. Argue, B Coy: M.C.  \nMajor D. A. Collett, A Coy: M.C.  \nColour Sergeant B. Faulkner, Regimental Aid Post: D.C.M.  \nSergeant J. S. Pettinger, D Patrol Coy: D.C.M.  \nPrivate R. J. de M. Absolon, D (Patrol) Coy: M.M. posthumous  \nCorporal I. P. Bailey, B Coy: M.M.  \nSergeant D. Fuller, B Coy: M.M.\n\n###### ________ ATTACHED ________\n\nCaptain W. A. McCracken, 29 Commando Regiment Royal Artillery: M.C.\n\nNOTES\n\n###### **INTRODUCTION**\n\n1. Duncan Anderson, 'Let's Remember Arnhem' 2 Para and the Battle of Darwin-Goose Green, _Battlefields Review_ , 19, 2002, pp. 51.\n\n###### **THE LANDING**\n\n1. Graham Colbeck, _With 3 Para to the Falklands_ (London: Greenhill Books, 2002) p. 85. See also Julian Thompson, _No Picnic_ (London: Cassell, 2001) p. 49.\n\n2. Martin Middlebrook, _The Falklands War 1982_ (London: Penguin, 2001) pp. 214\u2013215\n\n3. Colbeck op.cit. p. 95\n\n###### **THE TAB**\n\n1. Middlebrook, op.cit. p. 274\u2013277. See also Vincent Bramley, _Excursion to Hell_ (London: BCA, 1991) pp.45\u201346 and Christian Jennings and Adrian Weale, _Green Eyed Boys_ (London: Harper Collins, 1996) p.99.\n\n2. Vincent Bramley, op.cit, pp. 45\u201356. Jennings and Weale, op.cit, pp. 99\u2013102.\n\n3. Duncan Anderson, _The Falklands War 1982_ (Oxford: Osprey, 2002) p. 56.\n\n###### **THE PREPARATIONS**\n\n1. Jennings and Weale, op.cit. p.p. 110\u2013114\n\n2. Julian Thompson. op.cit, p. 112.\n\n3. Major General Julian Thompson, C.B., O.B.E. E-mail response to editor's query. January 2004. See also Nicholas van der Bijl, _Nine Battles to Stanley_ (Barnsley: Leo Cooper, 1999) pp. 164\u2013165.\n\n4. Major General Julian Thompson, C.B., O.B.E. E-mail response to editor's query. January 2004.\n\n5. Vincent Bramley op.cit. p.77.\n\n###### **THE PLAN**\n\n1. Colonel Hew Pike D.S.O. M.B.E., _With Fixed Bayonets, Elite Magazine_ , vol. 2, 20, 1985. pp. 381\n\n2. Graham Colbeck op.cit. pp. 179\u2013180. See also Jennings and Weale, op.cit. pp.122\u2013123 and Julian Thompson op.cit. p.112.\n\n###### **THE BATTLE**\n\n1. Jennings and Weale, op.cit. pp. 126\u2013127. See also David Aldea, _Mount Longdon The Argentinian Story_ (www.britains-smallwars.com\/Falklands\/David\/Longdon) Nicholas van der Bijl, op.cit. pp. 171\u2013172, and Max Arthur, _Above All Courage_ (London: Sphere) 1987, p.300\n\n2. Vincent Bramley, _Two Sides of Hell_ (London: Bloomsbury, 1994) p.137.\n\n3. Ibid. p. 140.\n\n4. Colonel Hew Pike, op.cit. p.384.\n\n5. Vincent Bramley, (1991) op.cit. p. 90.\n\n6. There is some debate regarding the circumstances surrounding Baldini's death. Some accounts maintain that he was asleep when 3 Para's attack went in, whilst others suggest that his boots were removed by one of the paras to replace the universally loathed British regulation issue DMS boots. Criticised by some of his subordinates after the war, Baldini was posthumously recommended for the Heroic Valour Cross, Argentina's highest military decoration. He was in fact awarded the Gallantry in Combat Medal, and his regiment's parade square was named after him. See Jennings and Weale, op.cit.. p.129, van der Bijl, op.cit. p. 173 and David Aldea, op.cit.\n\n7. Sergeant Ian McKay's VC was the first to be awarded to a man in 3 Para and the last of the eleven post World War Two awards. Will his be the last ever VC? For an extended account of Ian McKay's pre-Falklands service with the Paras and the circumstances surrounding his actions see Mark Adkin, _The Last Eleven_ (London: Leo Cooper 1991)\n\n8. Duncan Anderson, op.cit. pp. 65\u201366.\n\n###### **THE LEGACY**\n\n1. Colonel Hew Pike op.cit. p. 391.\n\n2. Max Arthur, op.cit. p. 306.\n\n3. Graham Colbeck op.cit. pp. 179\u2013182.\n\n4. See also Jennings and Weale, op.cit. pp. 187\u2013188.\n\n5. Major General Julian Thompson, E mail to editor op.cit.\n\n6. Nicholas van der Bijl, op.cit. p. 173.\n\n7. Jennings and Weale op.cit. pp. 177\u2013191.\n\n8. Graham Colbeck op.cit. p. 185.\nFURTHER READING\n\nGraham Colbeck, _With 3 Para to the Falklands_ (London: Greenhill Books, 2002)\n\nMartin Middlebrook, _The Falklands War 1982_ (London: Penguin, 2001)\n\nMartin Middlebrook, _The Argentine Fight for the Falklands:_ (Barnsley: Pen and Sword Military Classics, 2003)\n\nJulian Thompson, _No Picnic_ (London: Cassell, 2001)\n\nVincent Bramley, _Excursion to Hell_ (London: BCA, 1991)\n\nChristian Jennings and Adrian Weale, _Green Eyed Boys_ (London: Harper Collins, 1996)\n\nDuncan Anderson, _The Falklands War 1982_ (Oxford: Osprey, 2002)\n\nNicholas van der Bijl, _Nine Battles to Stanley_ (Barnsley: Leo Cooper, 1999)\n\nNicholas van der Bijl, _Argentine Forces in the Falklands_ (London: Osprey, 1992)\n\nMax Arthur, _Above All Courage_ (London: Sphere 1987)\n\nVincent Bramley, _Two Sides of Hell_ (London: Bloomsbury, 1994)\n\nMark Adkin, _The Last Eleven_ (London: Leo Cooper 1991)\n\nWilliam Fowler, _Battle for the Falklands (I) Land Forces_ (London: Osprey, 1982)\n\n**WEBSITES:**\n\nwww.britains-smallwars.com\n\nwww.sama82.org.uk\n\nwww.naval-history.net\/F541ongdon.htm\nINDEX\n\nAbsolon, Private Richard , ,\n\nAjax Bay\n\nAnaya, Admiral \u2013\n\nArgentine Ground Forces\n\nCompania de Commando 601\n\nCompania de Fuerzes Especiales 601 de\n\nGendarmerie Nacional\n\nCompania de Ametrallodoras 12.7\n\nInfanteria Marina , \u2013\n\nCompania de Ingenieros X ,\n\nRegimiento de Infanteria Mecanizado\n\n(7th Mechanised Infantry Regiment) \u2013, , , , .\n\nArgentine Ships\n\n_Bahia Buen Suceso_\n\n_General Belgrano_\n\n_Veinticinco de Mayo_\n\nArgue, Major Mike , \u2013, \u2013, , , \u2013\n\nAscension Island \u2013\n\nAshbridge, R.S.M. Lawrie ,\n\nBailey, Corporal \u2013\n\nBaldini, Second Lieutenant Juan , , \u2013\n\nBarker, Captain Nick\n\nBarratt, Private 'Baz' ,\n\nBickerdike, Lieutenant Andrew , \u2013\n\nBishop, Corporal Dickie ,\n\nBramley, Lance Corporal Vincent , , ,\n\nBritish Army\n\n5 Infantry Brigade ,\n\n2nd Battalion, the Parachute Regiment,\n\n(2 Para) , , , , ,\n\n3rd Battalion, the Parachute Regiment,\n\n(3 Para) _passim_\n\n1st Battalion, The Welsh Guards\n\n2nd Battalion, The Scots Guards , ,\n\n1\/7 Gurkha Rifles\n\nBlues and Royals\n\n29th Commando Regiment, Royal Artillery \u2013\n\nRoyal Engineers, 9th (Parachute) Squadron\n\nSpecial Air Service, (S.A.S.) , ,\n\nBritish Ships\n\nH.M.S. _Antrim_\n\nH.M.S. _Avenger_ ,\n\nH.M.S. _Conqueror_\n\nH.M.S. _Coventry_\n\nH.M.S. _Endurance_\n\nH.M.S. _Fearless_\n\nH.M.S. _Hermes_ ,\n\nH.M.S. _Intrepid_ ,\n\nH.M.S. _Invincible_ ,\n\nH.M.S. _Sheffield_ ,\n\n_Atlantic Conveyor_\n\nM.V. _Norland_\n\nS.S. _Canberra_ ,\n\nBurt, Private\n\nButler, Major Pat, , ,\n\nCarrizo, Corporal Oscar\n\nCarrizo-Salvadores, Major Carlos \u2013, , , \u2013\n\nCarver, Lance Corporal Lenny\n\nCasta\u00f1eda, First Lieutenant Raul , \u2013, ,\n\nCoady, Private\n\nColbeck, Sergeant Graham , ,\n\nCollett, Major David , , ,\n\nConnick, Denzil ,\n\nCox, Lieutenant Mark , , ,\n\nCraig, Alan , ,\n\nCullen, Private\n\nDavidoff, Constantine, Sergio ,\n\nDavis, 'Taff'\n\nDeakins, Private Pete ,\n\nDennison, Major Peter , , , ,\n\nDevenney, David\n\nDunn, Private 'Den'\n\nEstancia House (Farm) \u2013, , \u2013,\n\nFaulkner, Colour Sergeant Brian \u2013\n\n'Fly Half' \u2013, , , \u2013, , ,\n\n'Free Kick' , \u2013\n\n'Full Back' , , , , \u2013\u2013\n\nFuller, Sergeant Des\n\nGaltieri, General\n\nGimenez, Lieutenant Colonel Ortiz\n\nGonzalez, First Sergeant Raul , , \u2013,\n\nGoose Green , \u2013, ,\n\nGough, Private Ben\n\nGray, Private Dominic\n\nGray, Private Stew\n\nGreenwood, Private Tony ,\n\nGrey, Sergeant Pete ,\n\nGrose, Private Neil\n\nHarden, Corporal Pete ,\n\nHarding, Corporal 'Blue' ,\n\nHarrier\n\nHeathman, Ailsa\n\nHeathman, Tony\n\nHedicker, Private Peter ,\n\nHunt, Governor Rex\n\nInnes-Kerr, Lord Robin\n\nJames, Captain Julian ,\n\nJones, Lieutenant Colonel 'H'\n\nJones, Private Craig\n\nJones, Private\n\nJuliff, Private\n\nLaing, Private Stewart\n\nLombardo, Admiral\n\nMalvinas \u2013,\n\nMcCarthy, Corporal Keith ,\n\nMcCracken, Captain Willie\n\nMcKay, Sergeant Ian , ,\n\nMcKeown, Corporal Joe ,\n\nMcLarnon, Private\n\nMcLaughlin, Corporal Stewart 'Scouse' ,\n\nMedina, Corporal Manuel\n\nMen\u00e9ndez, Brigadier General ,\n\nMilan anti-tank weapon system , , , , , , , , \u2013\n\nMiller, Alan\n\nMilne, Corporal Brian ,\n\nMoore, Major General Jeremy , ,\n\nMorrison, Trudy\n\nMount Challenger\n\nMount Estancia , ,\n\nMount Harriet ,\n\nMount Kent , , ,\n\nMount Longdon _passim_\n\nMount Tumbledown , , ,\n\nMount Vernet ,\n\nMurdoch, Lance Corporal 'Doc' \u2013, .\n\nNeirotti, Second Lieutenant Enrique , ,\n\nNott, John \u2013\n\nOperation Corporate\n\nOsborne, Major Martin ,\n\nPatton, Major Roger\n\nPeck, Terry , , , , , ,\n\nPettinger, Sergeant John\n\nPhillips, Corporal Jerry \u2013\n\nPike, Lieutenant Colonel Hew , , , \u2013, \u2013, \u2013, ,\n\nPort San Carlos , \u2013, \u2013, ,\n\nPort Stanley \u2013, , , , \u2013, , ,\n\nPower, Private 'Taff'\n\nQuiroga, Lieutenant Hugo ,\n\nRajput, Private Raj ,\n\nRapier \u2013\n\nRose, Nick , , , , , \u2013, , \u2013\n\n'Rough Diamond' ,\n\nRoyal Marines\n\n3 Commando Brigade , , , \u2013, , ,\n\n40 Commando\n\n42 Commando , ,\n\n45 Commando , , , ,\n\nSpecial Boat Squadron, (S.B.S.) ,\n\nSavage, Michael \u2013,\n\nSelf-Loading Rifle (S.L.R.) ,\n\nShaw, Craftsman Alex\n\nShaw, Lieutenant Jon , , \u2013,\n\nSkyhawk \u2013\n\nSouth Atlantic Medal Association, (S.A.M.A.)\n\nSouth Georgia ,\n\nSteggles, Corporal ,\n\nSterling submachine gun ,\n\nTeal Inlet \u2013, \u2013, , , ,\n\nThatcher, Margaret\n\nThompson, Brigadier Julian , , \u2013, , \u2013, \u2013, , \u2013,\n\nTwo Sisters\n\nWeeks, C.S.M. John , , \u2013\n\nWest, Private Philip ,\n\nWilson, Corporal Trevor , \u2013.\n\n'Wing Forward' , ,\n\nWireless Ridge , \u2013, \u2013, \u2013,\n\nWombat anti-tank weapon system ,\n\nWoodward, Rear Admiral Sandy \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nT\u00e4m\u00e4n teoksen kirjoittamiseen on saatu tukea kirjastoapurahalautakunnalta sek\u00e4 WSOY:n kirjallisuuss\u00e4\u00e4ti\u00f6lt\u00e4.\n\nISBN 978-951-31-8631-9\n\nKannen suunnittelu: Laura Lyytinen ja Riikka Turkulainen  \nSarja-asun suunnittelu: Eevaliina Rusanen  \nRaapaisuj\u00e4lki etupaperissa \u00a9 Shutterstock  \n\u00a9 2015 Elina Rouhiainen ja Kustannusosakeyhti\u00f6 Tammi\n\nKansi: Eevaliina Rusanen  \nKannen kuvat: iStockphoto\n\nTeoksen jakelu ja osittainen kopiointi muuhun kuin lain sallimaan yksityiseen k\u00e4ytt\u00f6\u00f6n ilman tekij\u00e4noikeuden haltijan lupaa on korvausvastuun ja rangaistuksen uhalla kielletty.\n\nKustannusosakeyhti\u00f6 Tammi 2015\n\nSinulle, joka l\u00e4hdit kanssani susirajalle\nPROLOGI\n\nMit\u00e4 tekisit, jos olisit juuri saanut el\u00e4m\u00e4si rakkauden takaisin?\n\nMin\u00e4 nimitt\u00e4in itkin.\n\nTapasin Mikael Sarrin ensimm\u00e4isen kerran puolitoista vuotta sitten. Siit\u00e4 l\u00e4htien h\u00e4nen siniset silm\u00e4ns\u00e4 ovat olleet mieless\u00e4ni.\n\nOlin pitk\u00e4\u00e4n ollut varma, ettemme voisi koskaan olla yhdess\u00e4. Liian monet asiat erottivat meit\u00e4. H\u00e4n oli ihmissusi, ja min\u00e4 taas daimon. Minulle oli vasta v\u00e4h\u00e4n aikaa sitten alkanut selvit\u00e4, mit\u00e4 se edes tarkoitti. Siin\u00e4 miss\u00e4 min\u00e4 olin asunut koko ik\u00e4ni kaupungissa, Mikael oli kotonaan Kainuun metsiss\u00e4. Min\u00e4 rakastin taidetta, kun taas h\u00e4nen intohimonsa oli l\u00e4\u00e4ketiede. Min\u00e4 olin hellep\u00e4iv\u00e4, h\u00e4n pakkasy\u00f6.\n\nMin\u00e4 olin j\u00e4tt\u00e4nyt h\u00e4net viime kes\u00e4n\u00e4, juuri kun h\u00e4nen koko el\u00e4m\u00e4ns\u00e4 oli mullistunut t\u00e4ysin: h\u00e4nest\u00e4 oli tullut Hukkavaaran, Suomen ainoan ihmissusilauman, johtaja. Tilanne oli k\u00e4ynyt ylivoimaiseksi, ja pian sen j\u00e4lkeen h\u00e4n oli l\u00e4htenyt Hukkavaarasta. Kukaan ei ollut kuullut h\u00e4nest\u00e4 mit\u00e4\u00e4n, kunnes pari kuukautta sitten h\u00e4n palasi el\u00e4m\u00e4\u00e4ni \u2013 hetkell\u00e4, jolloin olin tarvinnut h\u00e4nt\u00e4 kaikkein eniten. H\u00e4n auttoi minua pelastamaan veljeni daimoneilta, jotka olivat saaneet p\u00e4\u00e4h\u00e4ns\u00e4, ett\u00e4 t\u00e4m\u00e4 toisi tuhon koko tuntemallemme maailmalle.\n\nH\u00e4n tappoi minun vuokseni, vaikka se vaaransi koko lauman. Vaikka me olimme pelkki\u00e4 yst\u00e4vi\u00e4 ja h\u00e4nell\u00e4 oli jo uusi tytt\u00f6yst\u00e4v\u00e4.\n\nEnk\u00e4 min\u00e4 ollut miss\u00e4\u00e4n vaiheessa lakannut rakastamasta h\u00e4nt\u00e4.\n\nMenin Mikaelin asunnolle aikomuksenani kertoa h\u00e4nelle tunteistani ennen kuin h\u00e4n l\u00e4htisi maasta tytt\u00f6yst\u00e4v\u00e4ns\u00e4 kanssa. Olin hermoraunio. Kun Mikael avasi ovensa, en saanut sanaa suustani. Joten puhumisen sijaan suutelin h\u00e4nt\u00e4 \u2013 ja h\u00e4n vastasi suudelmaan.\n\nSen j\u00e4lkeen tapahtui paljon muutakin kuin suutelemista. Se ei ollut niin kuin p\u00e4iv\u00e4unelmissani. Se ei ollut niin kuin elokuvissa. Se oli parempaa, sill\u00e4 mielikuvitukseni ei ollut koskaan onnistunut her\u00e4tt\u00e4m\u00e4\u00e4n henkiin Mikaelin kosketusta, tuoksua, sit\u00e4 milt\u00e4 h\u00e4nen ihonsa tuntui sormieni alla. Halusin, ett\u00e4 se kest\u00e4isi ikuisesti, vaikka samaan aikaan en uskonut voivani kest\u00e4\u00e4 yht\u00e4\u00e4n kauempaa. Juuri kun olin ajatellut, ett\u00e4 kuolisin siihen paikkaan, maailma kipunoi.\n\nJa silloin aloin itke\u00e4.\n\nEik\u00e4 se ollut mit\u00e4\u00e4n herkk\u00e4\u00e4 itkua, vaan totista vollotusta. Kaikki j\u00e4nnitys, kaikki tunteet vy\u00f6ryiv\u00e4t minusta ulos sellaisella voimalla, ett\u00e4 pel\u00e4styin itsekin.\n\n\"Hei, mik\u00e4 nyt?\" Mikael kysyi.\n\nH\u00e4n pyyhk\u00e4isi hiukset naamaltani ja yritti ottaa katsekontaktia. Kun h\u00e4nelle selvisi, ettei minusta ollut puhumaan, h\u00e4n k\u00e4\u00e4ri minut peittoon ja veti kainaloonsa. Painauduin h\u00e4nen rintaansa vasten.\n\nMikael silitti hiuksiani. Jossain vaiheessa h\u00e4n alkoi puhua minulle pehme\u00e4ll\u00e4 \u00e4\u00e4nell\u00e4, mutten kiinnitt\u00e4nyt sanoihin huomiota. En ennen kuin yksi sana kiinnitti huomioni.\n\nNousin istumaan. \"Shaun? En min\u00e4 itke Shaunin takia.\" Min\u00e4 ja poika nimelt\u00e4 Shaun olimme alkaneet viett\u00e4\u00e4 aikaa yhdess\u00e4, kun olin viel\u00e4 kuvitellut menett\u00e4neeni Mikaelin lopullisesti. H\u00e4n oli pyyt\u00e4nyt minua l\u00e4htem\u00e4\u00e4n kanssaan Aasiaan, mutta olin p\u00e4\u00e4tt\u00e4nyt j\u00e4\u00e4d\u00e4 Helsinkiin. Minun ei ollut tarvinnut edes sanoa h\u00e4nelle, ett\u00e4 olin valinnut Mikaelin.\n\n\"Miksi sitten?\"\n\nOlin kuvitellut, ett\u00e4 juuri tapahtunut olisi voinut johtaa vain yhdenlaiseen johtop\u00e4\u00e4t\u00f6kseen. Mikaelilla ei kuitenkaan n\u00e4ytt\u00e4nyt olevan aavistustakaan, mit\u00e4 tunsin. Se olisi voinut olla hullunkurista, ellen olisi alkanut heti hermoilla. Kenties \u00e4skeinen ei ollut tarkoittanut h\u00e4nelle samaa kuin minulle. Oli siis tullut aikaisemmin valmistelemani puheen aika.\n\nSiirryin kauemmaksi Mikaelista. Minun t\u00e4ytyi l\u00f6yt\u00e4\u00e4 oikeat sanat, eik\u00e4 h\u00e4nen kosketuksensa auttaisi asiaa. K\u00e4vin patjalle polvi-istuntaan. Pidin peiton ymp\u00e4rill\u00e4ni. \"Tulin, koska kuulin, ett\u00e4 sin\u00e4 olet l\u00e4hd\u00f6ss\u00e4. En kest\u00e4nyt ajatusta, ett\u00e4 l\u00e4htisit ennen kuin saisin kertoa, mit\u00e4 min\u00e4 tunnen sinua kohtaan.\"\n\n\"Mit\u00e4 tarkoitat?\" Mikael kysyi.\n\nH\u00e4n ei totisesti tee t\u00e4t\u00e4 minulle helpoksi, ajattelin.\n\nEi ollut kuin yksi tapa sanoa se. T\u00e4ss\u00e4 vaiheessa vertauskuvat ja vihjailut eiv\u00e4t en\u00e4\u00e4 riitt\u00e4neet. \"Min\u00e4 rakastan sinua.\"\n\nOlin aina ajatellut, ett\u00e4 se oli suomen kielen rumimmalta kuulostava lause siksi, ettei kukaan sanoisi sit\u00e4 turhaan. Sill\u00e4 hetkell\u00e4 se ei kuitenkaan kuulostanut yht\u00e4\u00e4n rumalta. P\u00e4in vastoin, se kuulosti kauneimmalta asialta koko maailmassa.\n\n\"Min\u00e4 rakastan sinua\", toistin. \"Olisin kertonut sen sinulle vuosi sitten Hukkavaarassa, jos et olisi ehtinyt l\u00e4hte\u00e4. Mutta min\u00e4 kerron nyt. Min\u00e4 rakastan sinua ja haluan olla sinun kanssasi. Haluan, ett\u00e4 valitset minut. Tied\u00e4n, ettei minulla varmaan ole oikeutta pyyt\u00e4\u00e4 sit\u00e4 \u2013 olen tehnyt tyhmi\u00e4 juttuja, ajoitus on huono ja sin\u00e4 olet Reginan kanssa ja kaikkea \u2013 mutta minun oli vain pakko sanoa se \u00e4\u00e4neen.\"\n\nMikael ei vastannut mit\u00e4\u00e4n. H\u00e4n vain tuijotti minua t\u00e4ysin liikkumattomana. Sit\u00e4 jatkui niin kauan, etten ollut varma, hengittik\u00f6 h\u00e4n.\n\nH\u00e4nen t\u00e4ytyi pit\u00e4\u00e4 minua hulluna.\n\nJa silti, osa minusta oli ollut aivan varma, ett\u00e4 kaikki v\u00e4lill\u00e4mme palaisi ennalleen. Eik\u00e4 vain ennalleen, vaan paremmaksi, enemm\u00e4ksi kuin koskaan ennen. Olin ajatellut, ett\u00e4 kun vain ker\u00e4isin rohkeuteni ja saisin suuni auki, kaikki p\u00e4\u00e4ttyisi hyvin.\n\n\"Ja he eliv\u00e4t onnellisina el\u00e4m\u00e4ns\u00e4 loppuun asti.\" Tiesin, ett\u00e4 vain harvat saivat kokea sen. Olin kuvitellut minun ja Mikaelin olevan niin onnekkaita. Nyt se tuntui typer\u00e4lt\u00e4.\n\nHapuilin vaatteitani kohti. Vedin topin p\u00e4\u00e4ni yli \u2013 en edes v\u00e4litt\u00e4nyt l\u00f6yt\u00e4\u00e4 rintsikoitani. Pujotin jalkani farkunlahkeisiin ja kahmaisin loput vaatteet syliini. Olin ehtinyt ottaa askeleen ovea kohti, kun Mikael her\u00e4si transsistaan.\n\n\"Minne sin\u00e4 olet menossa?\"\n\n\"Kotiin\", sain sanotuksi. En voinut sanoa, mit\u00e4 ajattelin: etten ollut varma, minne menisin. Tiesin, ett\u00e4 Mitja ja Caoimhe olisivat kotona, ja sill\u00e4 hetkell\u00e4 halusin olla yksin.\n\n\"Odota, odota!\"\n\nH\u00e4n hyp\u00e4hti s\u00e4ngylt\u00e4 ja oli \u00e4kki\u00e4 vieress\u00e4ni. En pystynyt edes katsomaan h\u00e4nt\u00e4. Puristin vaatemytty\u00e4 rintaani vasten ja teeskentelin tuijottavani ovea. Todellisuudessa en n\u00e4hnyt mit\u00e4\u00e4n.\n\nMikael koski olkap\u00e4\u00e4t\u00e4ni. Aivan kuin h\u00e4nen l\u00e4sn\u00e4olossaan ei olisi ollut kylliksi. Aivan kuin h\u00e4nen luotaan l\u00e4hteminen puolipukeisena ei olisi ollut tarpeeksi n\u00f6yryytt\u00e4v\u00e4\u00e4.\n\n\"Kiitos kivasta illasta, mutta taitaa olla parasta, ettei oteta uusiksi\", sanoin. En edelleenk\u00e4\u00e4n katsonut h\u00e4neen.\n\n\"Kuulin kyll\u00e4, mit\u00e4 sanoit, mutten tainnut ymm\u00e4rt\u00e4\u00e4.\"\n\nAstuin askeleen eteen niin, etteiv\u00e4t h\u00e4nen sormensa en\u00e4\u00e4 koskeneet minua. Mikael ei pist\u00e4nyt vastaan.\n\n\"Mit\u00e4 ymm\u00e4rrett\u00e4v\u00e4\u00e4 siin\u00e4 on?\" kysyin synk\u00e4ll\u00e4 \u00e4\u00e4nell\u00e4. \"Min\u00e4 kerroin sinulle juuri rakastavani sinua, etk\u00e4 sin\u00e4 vastannut. Tilanne tuli ainakin minulle ihan riitt\u00e4v\u00e4n selv\u00e4ksi.\"\n\nMinun t\u00e4ytyi n\u00e4ytt\u00e4\u00e4 todella uhkaavalta, sill\u00e4 h\u00e4n nosti k\u00e4tens\u00e4 ilmaan. \"Ei, sen osan min\u00e4 kyll\u00e4 ymm\u00e4rsin. Ehk\u00e4\", h\u00e4n lis\u00e4si. \"Mutta sin\u00e4 mainitsit Reginan ja ett\u00e4 min\u00e4 olisin l\u00e4hd\u00f6ss\u00e4 h\u00e4nen kanssaan jonnekin.\"\n\nRegina oli h\u00e4nen tytt\u00f6yst\u00e4v\u00e4ns\u00e4. Upea, kuollut tytt\u00f6yst\u00e4v\u00e4ns\u00e4: Regina oli nimitt\u00e4in vampyyri.\n\n\"Niin\", sanoin. Mit\u00e4 osaa h\u00e4n ei voinut ymm\u00e4rt\u00e4\u00e4? Mikael ei yleens\u00e4 ollut n\u00e4in hidasj\u00e4rkinen.\n\n\"Min\u00e4 en ole l\u00e4hd\u00f6ss\u00e4 minnek\u00e4\u00e4n, ja jos olisinkin, niin en taatusti Reginan kanssa.\"\n\n\"Mutta te olette yhdess\u00e4.\"\n\nMikael oli pitk\u00e4\u00e4n hiljaa. \"Emmek\u00e4 ole.\"\n\nViimeinkin katsoin h\u00e4nt\u00e4. Minua vastassa olivat l\u00e4mpim\u00e4t, tunteikkaat silm\u00e4t. Meni hetki, ennen kuin ymm\u00e4rsin, mik\u00e4 tunne niist\u00e4 v\u00e4littyi: toivo.\n\nEn ymm\u00e4rt\u00e4nyt en\u00e4\u00e4 mist\u00e4\u00e4n mit\u00e4\u00e4n. \"Mutta Regina tuntuu ajattelevan niin. Kaikki ajattelevat niin. Te olette aina yhdess\u00e4 klubilla.\" Mieless\u00e4ni v\u00e4l\u00e4hti heti puolenkymment\u00e4 kuvaa Mikaelista ja Reginasta. Aina l\u00e4hekk\u00e4in. Mutta olinko n\u00e4hnyt heid\u00e4n koskaan suutelevan?\n\n\"Olen n\u00e4hnyt sinun koskevan h\u00e4nt\u00e4\", sanoin lopulta. Tajusin heti, miten naurettavalta se kuulosti.\n\n\"Usko pois, me emme ole yhdess\u00e4. Luulisin tiet\u00e4v\u00e4ni, jos olisimme.\"\n\n\"Mutta miksi Regina sitten sanoi niin...?\" Annoin \u00e4\u00e4neni hiipua.\n\n\"Regina ja min\u00e4 olemme yst\u00e4vi\u00e4. Tapasin h\u00e4net reissuillani.\"\n\n\"Mutta h\u00e4n on j\u00e4\u00e4nyt Helsinkiin, vaikka muut vampyyrit l\u00e4htiv\u00e4t jo aikoja sitten.\" Suomi oli mit\u00e4 ihanteellisin maa vampyyreille \u2013 talvisaikaan. Sen sijaan Suomen valoisa kes\u00e4 teki heid\u00e4n olonsa t\u00e4\u00e4ll\u00e4 l\u00e4hes mahdottomaksi.\n\n\"H\u00e4n on l\u00e4\u00e4k\u00e4ri. H\u00e4n on jo vuosia tehnyt omia tutkimuksiaan erilaisten yliluonnollisten parissa. H\u00e4n halusi kokeilla uusinta teoriaansa siit\u00e4, voiko vampyyrien valonarkuutta madaltaa. Kerroin h\u00e4nelle susista, ja h\u00e4n lupasi auttaa minua l\u00e4\u00e4kiksen p\u00e4\u00e4sykokeisiin valmistautumisessa.\" H\u00e4n piti tauon. \"En todellakaan tied\u00e4, miksi h\u00e4n valehteli sinulle.\" Sitten h\u00e4n naurahti. \"Tai itse asiassa taidankin tiet\u00e4\u00e4. H\u00e4n halusi n\u00e4hd\u00e4, mit\u00e4 tekisit, jos kuvittelisit minun olevan h\u00e4ipym\u00e4ss\u00e4. Se olisi juuri h\u00e4nen tapaistaan.\"\n\n\"Mutta miksi h\u00e4n haluaisi minun tekev\u00e4n jotain?\"\n\n\"Koska h\u00e4n tiet\u00e4\u00e4, ett\u00e4 min\u00e4 rakastan sinua.\"\n\nOli minun vuoroni tuijottaa Mikaelia. Olin luullut koko ajan, ett\u00e4 Mikaelilla oli tytt\u00f6yst\u00e4v\u00e4. Ett\u00e4 h\u00e4n oli p\u00e4\u00e4ssyt yli minusta ja jatkanut el\u00e4m\u00e4\u00e4ns\u00e4. Olin varonut sanomasta tai tekem\u00e4st\u00e4 mit\u00e4\u00e4n v\u00e4\u00e4r\u00e4\u00e4, pit\u00e4nyt syd\u00e4meni visusti piilossa...\n\n\"Hitto\", sanoin. Painoin k\u00e4teni suuni p\u00e4\u00e4lle.\n\n\"Hei, \u00e4l\u00e4 itke\", Mikael sanoi. \"Kaikki on hyvin.\"\n\n\"Hyvin?\" sanoin. Saattoihan asian ilmaista niinkin.\n\n\"Ent\u00e4... Ent\u00e4 Shaun?\" Mikael oli \u00e4kki\u00e4 melkein ujo.\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Emme me koskaan olleet mik\u00e4\u00e4n oikea pari. Ja sekin mit\u00e4 oli, on nyt ohi. H\u00e4n on l\u00e4htenyt.\" Tuskin edes tiesin, mit\u00e4 sanoin. Halusin uskoa, ett\u00e4 t\u00e4m\u00e4 oli totta. Mutta minua pelotti.\n\n\"Raisa, ei ole ket\u00e4\u00e4n muuta\", Mikael sanoi. \"Sin\u00e4 olet ainoa, nyt ja aina.\"\n\nNyt ja aina. Hetki oli niin valtava, ett\u00e4 se oli tyrm\u00e4t\u00e4 minut kanveesiin. Kaikki esteet, jotka olin kuvitellut mieless\u00e4ni... olivat poissa. Meid\u00e4n v\u00e4liss\u00e4mme ei ollut en\u00e4\u00e4 mit\u00e4\u00e4n.\n\nEn tied\u00e4, kuinka kauan me vain seisoimme siin\u00e4 h\u00e4mm\u00e4stellen juuri valjennutta todellisuutta. Lopulta Mikael selvitti kurkkuaan. \"Meid\u00e4n pit\u00e4isi varmaan yritt\u00e4\u00e4 nukkua. Minulla on kuitenkin ne p\u00e4\u00e4sykokeet huomenna.\"\n\n\"Hitto. Anteeksi.\" Olin pit\u00e4nyt h\u00e4net valveilla aamuy\u00f6n puolelle.\n\nMikael hymyili. \"\u00c4l\u00e4 huoli. Min\u00e4 voisin valvoa koko y\u00f6n ja olisin silti el\u00e4m\u00e4ni kunnossa. Mik\u00e4\u00e4n ei voi pys\u00e4ytt\u00e4\u00e4 minua nyt.\"\n\nTiesin tismalleen, mit\u00e4 h\u00e4n tarkoitti. Nyt kun minulla on sinut.\n\nMikael suuteli minua pehme\u00e4sti, ja minua huimasi hiukan. H\u00e4n tarttui k\u00e4teeni ja johdatti minut takaisin s\u00e4ngylle.\n\nH\u00e4n vilkaisi y\u00f6p\u00f6yd\u00e4ll\u00e4 olevaa puhelintaan. \"Outoa. Kello on vasta puoli kaksitoista.\"\n\n\"Tulinko min\u00e4 muka vartti sitten?\" kysyin. Tuo vartti oli uuvuttanut minut t\u00e4ysin.\n\n\"En tied\u00e4\", Mikael sanoi. H\u00e4nen otsansa rypistyi.\n\nTajusimme asian tismalleen samaan aikaan.\n\nRynt\u00e4simme ikkunaan. Mikael oli nopeampi, vaikka min\u00e4 olin l\u00e4hemp\u00e4n\u00e4.\n\nNojasin ikkunalautaan ja tuijotin ulos. Katu oli hiljainen. T\u00f6\u00f6l\u00f6 ei ollut mik\u00e4\u00e4n Kallio, jossa kes\u00e4isin juhlittiin aina. Kaikki vaikutti olevan kohdillaan, mutta lopulta silm\u00e4ni l\u00f6ysiv\u00e4t etsim\u00e4ni: lokki, joka n\u00e4ytti aivan silt\u00e4 kuin se olisi niitattu kiinni taivaaseen. Se oli j\u00e4hmettynyt kesken lennon.\n\nPysyin vain vaivoin pystyss\u00e4. Mit\u00e4 kauemmin katsoin, sit\u00e4 enemm\u00e4n katu n\u00e4ytti kulissilta t\u00e4ysin luonnottomalle n\u00e4ytelm\u00e4lle. N\u00e4ytelm\u00e4lle, johon min\u00e4 ja Mikael olimme joutuneet vasten tahtoamme.\n\nAika oli pys\u00e4htynyt.\n\nMikaelin oli t\u00e4ytynyt tulla samaan johtop\u00e4\u00e4t\u00f6kseen, sill\u00e4 h\u00e4n astui pois ikkunan luota ja veti minut mukanaan. \"Daimonit ovat t\u00e4\u00e4ll\u00e4\", h\u00e4n sanoi. \"Meid\u00e4n t\u00e4ytyy l\u00e4hte\u00e4.\"\n\nAloin t\u00e4rist\u00e4. Mikael puristi minua entist\u00e4 lujemmin.\n\n\"He eiv\u00e4t tehneet t\u00e4t\u00e4\", sain sanotuksi.\n\n\"Kuka sitten?\"\n\nOlin toivonut, ett\u00e4 hetki jatkuisi ikuisesti. Naurahdin. \"Luulen, ett\u00e4 min\u00e4.\"\n\nJa sitten py\u00f6rryin.\nENSIMM\u00c4INEN LUKU\n\n\"Yrit\u00e4 viel\u00e4 kerran\", sanoin.\n\nVeljeni istui jalat ristiss\u00e4 nurmella ja tuijotti minua ilmeett\u00f6m\u00e4n\u00e4. Minun ei tarvinnut turvautua yhteyteemme tiet\u00e4\u00e4kseni, ettei h\u00e4n halunnut yritt\u00e4\u00e4 en\u00e4\u00e4 yht\u00e4\u00e4n kertaa. Ei, vaikka h\u00e4nen kykyjens\u00e4 hintana ei ollut tajuttomuutta, pahoinvointia tai edes p\u00e4\u00e4ns\u00e4rky\u00e4 niin kuin minulla. H\u00e4n oli yksinkertaisesti kyll\u00e4stynyt.\n\nAurinko paahtoi kovempaa kuin kertaakaan aikaisemmin t\u00e4n\u00e4 kes\u00e4n\u00e4. En pannut sit\u00e4 pahakseni, sill\u00e4 merelt\u00e4 puhaltava tuuli piti l\u00e4mp\u00f6tilan juuri sopivana harjoittelullemme.\n\nOlimme tulleet t\u00e4lle samalle nurmikent\u00e4lle Kaivopuistoon l\u00e4hes joka p\u00e4iv\u00e4 sen j\u00e4lkeen, kun lomani oli alkanut. Ulkopuolisista n\u00e4ytti varmasti silt\u00e4, ett\u00e4 tulimme monien muiden tapaan tekem\u00e4\u00e4n lihaskuntoharjoituksia. He eiv\u00e4t olleet t\u00e4ysin v\u00e4\u00e4r\u00e4ss\u00e4: t\u00e4ll\u00e4kin kertaa olimme ensin juosseet viiden kilometrin lenkin ja tehneet sen j\u00e4lkeen varsin vakuuttavat sarjat punnerruksia ja vatsalihasliikkeit\u00e4. Ne olivat kuitenkin olleet vain alkul\u00e4mmittely\u00e4. T\u00e4rkein osa treenist\u00e4mme j\u00e4i muilta piiloon.\n\n\"En oikeasti usko, ett\u00e4 voin oppia pys\u00e4ytt\u00e4m\u00e4\u00e4n ajan\", Mitja sanoi. Heti sen j\u00e4lkeen h\u00e4n vilkuili ymp\u00e4rilleen. Pidimme aina huolen, ettei kuuloet\u00e4isyydell\u00e4 ollut ket\u00e4\u00e4n, mutta veljeni ei koskaan tuntunut luottavan siihen, ett\u00e4 olimme turvassa. Ei, vaikka puhuimme varmuuden vuoksi kreikkaa.\n\n\"Min\u00e4 taas luulen, ettet vain ole viel\u00e4 hoksannut, miten se kuuluisi tehd\u00e4\", vastasin.\n\n\"Sin\u00e4k\u00e4\u00e4n et tiennyt, miten se kuului tehd\u00e4, ja silti onnistuit siin\u00e4\", h\u00e4n muistutti. \"Ehk\u00e4 se on taito, joka joillakin daimoneilla vain on ja joillakin ei ole.\"\n\nAluksi en ollut itsek\u00e4\u00e4n ollut lainkaan varma, pystyisink\u00f6 toistamaan kuukausi sitten tapahtunutta. Melko pian olin kuitenkin alkanut n\u00e4hd\u00e4, ett\u00e4 aika oli oikeasti verkko, joka eli ja muuttui koko ajan. Mieleni saattoi takertua siihen, ja kunhan vain puristin riitt\u00e4v\u00e4n lujaa, verkko lakkasi liikkumasta. Olin yritt\u00e4nyt parhaani mukaan n\u00e4ytt\u00e4\u00e4 sen Mitjalle yhteytemme kautta.\n\nKatsoin, kun turistibussi huristeli ohitsemme. Minun oli my\u00f6nnett\u00e4v\u00e4, ett\u00e4 motiivini veljeni opettamiseen olivat ennen kaikkea itsekk\u00e4\u00e4t. En halunnut olla yksin kykyjeni kanssa. Pieni osa minusta oli pel\u00e4nnyt, miten Mitja reagoisi saadessaan tiet\u00e4\u00e4 uudesta kyvyst\u00e4ni. Oli ollut huojentavaa, ettei h\u00e4n ollut alkanut suhtautua minuun mitenk\u00e4\u00e4n eri tavalla. Ja ennen kaikkea: oli ollut huojentavaa, ettei h\u00e4n ollut maininnut ennustusta.\n\n\"Ehk\u00e4 meid\u00e4n on parempi lopettaa t\u00e4lt\u00e4 p\u00e4iv\u00e4lt\u00e4\", sanoin lopulta. Se piti paikkansa. Itse asiassa minun olisi pit\u00e4nyt jo olla kotona. Odotimme vieraita.\n\nEn kuitenkaan tehnyt elett\u00e4k\u00e4\u00e4n noustakseni, ja Mitja kallisti p\u00e4\u00e4t\u00e4\u00e4n. Vasta silloin tajusin verhon j\u00e4\u00e4neen raolleen harjoitustemme j\u00e4ljilt\u00e4. \"Sinua j\u00e4nnitt\u00e4\u00e4\", h\u00e4n huomioi. H\u00e4n kuulosti yll\u00e4ttyneelt\u00e4.\n\n\"Ehk\u00e4 v\u00e4h\u00e4n\", tunnustin. \"H\u00e4n on minun paras yst\u00e4v\u00e4ni. Tai siis oli.\"\n\nEn ollut puhunut Nikon kanssa yli nelj\u00e4\u00e4n kuukauteen. Viel\u00e4 pari p\u00e4iv\u00e4\u00e4 sitten olin ollut varma, ettei v\u00e4lej\u00e4mme voisi en\u00e4\u00e4 parsia \u2013 mit\u00e4\u00e4n ei olisi en\u00e4\u00e4 teht\u00e4viss\u00e4. Eilen oli kuitenkin tapahtunut ihme, sill\u00e4 kolmen viikon yritt\u00e4misen j\u00e4lkeen Niko oli vihdoin suostunut tapaamaan minut.\n\n\"H\u00e4n ei voi olla en\u00e4\u00e4 vihainen nyt kun saa tiet\u00e4\u00e4, ett\u00e4 teit kaiken h\u00e4nen omaksi parhaakseen\", Mitja sanoi.\n\nNousin yl\u00f6s ja pyyhin kuivaa ruohoa trikoohousuistani. \"Toivotaan niin\", sanoin.\n\nKun p\u00e4\u00e4sin kotiin, menin suoraan suihkuun. Annoin viile\u00e4n veden kastella hiukseni ja lievitt\u00e4\u00e4 p\u00e4\u00e4ns\u00e4rky\u00e4, joka oli alkanut sykki\u00e4 ohimoillani. Siit\u00e4 oli tullut minulle jo niin tuttu seuralainen, etten aina edes huomannut sit\u00e4.\n\nSeisoin veden alla pitk\u00e4\u00e4n. Kun viimein suljin hanan, kuulin oven k\u00e4yv\u00e4n. Kiedoin pyyhkeen ymp\u00e4rilleni ja menin eteiseen.\n\nMikael seisoi edess\u00e4ni aurinkolasit p\u00e4\u00e4ss\u00e4. Jalat olivat paljaat, sill\u00e4 h\u00e4n oli jo ehtinyt riisua sandaalinsa. Toisessa k\u00e4dess\u00e4 roikkui muovikassi.\n\nViime viikkoina olimme viett\u00e4neet paljon aikaa ulkona \u2013 puistoissa, rannoilla ja saaristossa \u2013 ja se n\u00e4kyi: me molemmat olimme ruskettuneita. En muistanut, milloin viimeksi kumpikaan meist\u00e4 olisi laittanut p\u00e4\u00e4lleen muita vaatteita kuin haalistuneita printtiteepaitoja ja shortseja. Hiukseni eiv\u00e4t olleet n\u00e4hneet harjaa moneen p\u00e4iv\u00e4\u00e4n ja Mikaelkin oli antanut itsens\u00e4 lipsua p\u00e4ivitt\u00e4isest\u00e4 parranajosta. S\u00e4nki ei n\u00e4ytt\u00e4nyt h\u00e4nen kasvoillaan lainkaan pahalta.\n\nMe olimme kes\u00e4el\u00e4imi\u00e4 \u2013 v\u00e4h\u00e4n kesytt\u00f6mi\u00e4 ja syv\u00e4sti tyytyv\u00e4isi\u00e4 el\u00e4m\u00e4\u00e4n.\n\n\"Moi\", sanoin aavistuksen heng\u00e4styneesti.\n\n\"Moi\", Mikael vastasi. Ja virnisti.\n\nJoskus \u2013 huolestuttavan usein \u2013 h\u00e4nen hymyns\u00e4 sai pulssini kiihtym\u00e4\u00e4n. Niin kuin nyt.\n\nH\u00e4nt\u00e4 sain kiitt\u00e4\u00e4 eilisest\u00e4 ihmeest\u00e4. Olin yritt\u00e4nyt soittaa Nikolle lukemattomia kertoja sen j\u00e4lkeen, kun oli k\u00e4ynyt selv\u00e4ksi, ett\u00e4 daimonit eiv\u00e4t olleet tulossa tappamaan minua ja kaikkia l\u00e4heisi\u00e4ni. H\u00e4n ei ollut vastannut. Mikael oli kuitenkin yh\u00e4 h\u00e4nen laumanjohtajansa, eik\u00e4 Niko ollut voinut j\u00e4tt\u00e4\u00e4 vastaamatta h\u00e4nelle. H\u00e4nen oli ollut pakko kuunnella, kun Mikael oli selitt\u00e4nyt tilanteen.\n\nSe ei tietenk\u00e4\u00e4n tarkoittanut, ett\u00e4 Niko antaisi minulle anteeksi. Se minun oli saatava itse aikaan.\n\n\"Ajattelin p\u00e4\u00e4st\u00e4\u00e4 meid\u00e4t helpolla ja hain nepalilaista\", h\u00e4n sanoi ja nosti muovikassin p\u00f6yd\u00e4lle.\n\n\"Hyvin ajateltu\", sanoin. Kumpikaan meist\u00e4 ei ollut erityisen ihmeellinen kokki.\n\nMakuuhuoneessa avasin kaapin ja pitk\u00e4n aikaa vain tuijotin siell\u00e4 olevia vaatteita. Ajattelin ensin pukea ylleni jotain Nikon tekem\u00e4\u00e4, mutta pelk\u00e4sin, ett\u00e4 h\u00e4n pit\u00e4isi sit\u00e4 mielistelyn\u00e4. Valitsin siis lopulta siisteimm\u00e4n kes\u00e4mekkoni, joka tuskin saisi keness\u00e4k\u00e4\u00e4n aikaan kovin kummoista reaktiota. Kuivasin hiukseni ja meikkasin nopeasti ennen kuin p\u00e4\u00e4stin Mikaelin kylpyhuoneeseen. Odottaessani p\u00e4\u00e4tin kattaa p\u00f6yd\u00e4n valmiiksi. Siihen ei mennyt kauaakaan, sill\u00e4 tiesin tarkalleen, miss\u00e4 mik\u00e4kin oli t\u00e4ss\u00e4 keitti\u00f6ss\u00e4.\n\nAsetellessani suuria valkoisia lautasia p\u00f6yd\u00e4lle ajattelin Mikaelin p\u00e4\u00e4sykokeiden aamua. Olimme seisoneet eteisess\u00e4 sirpaleiden keskell\u00e4 ja nauraneet tuholle, jonka olimme saaneet aikaan edellisen\u00e4 iltana. Kolmesta kehystetyst\u00e4 valokuvasta vain yksi roikkui en\u00e4\u00e4 sein\u00e4ll\u00e4.\n\nSilloin Mikael kurotti sirpaleiden yli kohti lipastoa ja avasi ylimm\u00e4n laatikon. H\u00e4n painoi k\u00e4teeni jotain pient\u00e4 ja metallista. Kun avasin nyrkkini, n\u00e4in, ett\u00e4 se oli avain.\n\n\"Oletko varma?\" kysyin.\n\nJa Mikael vastasi: \"Kaikki mik\u00e4 on minun, on my\u00f6s sinun.\"\n\nH\u00e4n sai sen kuulostamaan maailman luonnollisimmalta asialta. En sanonut enemp\u00e4\u00e4, laitoin vain avaimen taskuuni. Ja taskussa se oli kulkenut aina siit\u00e4 asti, sill\u00e4 viimeisen kuukauden olin nukkunut kaikki y\u00f6ni t\u00e4\u00e4ll\u00e4, T\u00f6\u00f6l\u00f6ss\u00e4.\n\nJuuri silloin summeri soi. Kuulin Mikaelin vastaavan ja p\u00e4\u00e4st\u00e4v\u00e4n sitten vieraat sis\u00e4\u00e4n. Odotin hermostuneena keitti\u00f6ss\u00e4.\n\nOlin juuri alkanut pohtia, olisiko minun sittenkin pit\u00e4nyt laittaa jotain muuta p\u00e4\u00e4lle, kun ovikello soi.\n\nMikael avasi oven.\n\nKuulin vaimeita \u00e4\u00e4ni\u00e4 eteisest\u00e4. Muutamia ep\u00e4selvi\u00e4 sanoja, sitten kenkien riisuntaa. Onnistuin viimein irtautumaan paikaltani ja menem\u00e4\u00e4n eteiseen.\n\nSiell\u00e4 minua vastassa olivat vaaleat hiukset, siniset silm\u00e4t, mutrussa oleva alahuuli. Sarkastinen ote universumista.\n\nNiko J\u00e4\u00e4skel\u00e4inen.\n\nH\u00e4n oli juuri ottanut keng\u00e4t pois jalastaan. H\u00e4n suoristautui hitaasti ja katsoi minua. H\u00e4nen ilmeens\u00e4 ei paljastanut mit\u00e4\u00e4n.\n\nRistin k\u00e4det rinnalleni ja nojasin ovenkarmiin. Asentoni oli ehk\u00e4 rento, mutta pulssini hakkasi. Yksi vilkaisu Nikoon riitti antamaan vihi\u00e4 siit\u00e4, miten pitk\u00e4lle h\u00e4n oli p\u00e4\u00e4ssyt kuluneen puolen vuoden aikana. H\u00e4n n\u00e4ytti todella hyv\u00e4lt\u00e4. H\u00e4n oli pukeutunut mustiin housuihin ja mustaan l\u00f6ys\u00e4\u00e4n paitaan, vaikka ulkona oli yli kaksikymment\u00e4 astetta. Joka tapauksessa h\u00e4nen asunsa n\u00e4ytti \u00e4\u00e4rimm\u00e4isen tyylikk\u00e4\u00e4lt\u00e4 ja kalliilta ja samaan aikaan hyvin Nikolta. H\u00e4n ei en\u00e4\u00e4 ep\u00e4r\u00f6inyt n\u00e4ytt\u00e4\u00e4 muille sisint\u00e4\u00e4n.\n\n\"Moi\", sain sanotuksi. Tiesin, ett\u00e4 Niko saattoi haistaa pelkoni.\n\n\"Moi\", Niko vastasi. H\u00e4n ei viel\u00e4k\u00e4\u00e4n hymyillyt.\n\n\"Terve, min\u00e4 olen Taneli\", \u00e4\u00e4ni sanoi Nikon vieress\u00e4.\n\nOlin n\u00e4hnyt Tanelin kerran aikaisemminkin, mutta siit\u00e4 oli aikaa, enk\u00e4 olisi viel\u00e4 hetki sitten osannut kuvailla h\u00e4nt\u00e4. H\u00e4net oli kuitenkin helppo tunnistaa. Vaikka Niko ei ollut mik\u00e4\u00e4n j\u00e4ttil\u00e4inen, Tanelin vieress\u00e4 seisoessaan h\u00e4n alkoi \u00e4kki\u00e4 n\u00e4ytt\u00e4\u00e4 aika bodatulta. Se ei tarkoittanut, etteik\u00f6 Taneli olisi ollut komea. Kaikki h\u00e4ness\u00e4 oli hyvin poikamaista: vaaleat, sivujakaukselta silmille valuvat hiukset ja suuret siniset silm\u00e4t, pehme\u00e4t huulet. H\u00e4nell\u00e4 oli yll\u00e4\u00e4n l\u00f6ys\u00e4 valkoinen paita, jonka hihat oli k\u00e4\u00e4ritty kyyn\u00e4rp\u00e4ihin. H\u00e4nt\u00e4 saattoi helposti sanoa yhdeksi yst\u00e4v\u00e4llisimm\u00e4n n\u00e4k\u00f6isist\u00e4 ihmisist\u00e4, joita olin tavannut.\n\nAsiaan saattoi vaikuttaa my\u00f6s se, ett\u00e4 h\u00e4n oli ihminen. Ja koska olin viimeisen vuoden aikana ollut tekemisiss\u00e4 l\u00e4hinn\u00e4 synkki\u00e4 salaisuuksia kantavien yliluonnollisten kanssa, tuntui h\u00e4nen ihmisyytens\u00e4 virkist\u00e4v\u00e4lt\u00e4 vaihtelulta.\n\n\"Moi, min\u00e4 olen Raisa\", sanoin. \"Me taisimme tavata kerran.\"\n\n\"Muistan\", Taneli sanoi ja hymyili leve\u00e4sti. H\u00e4n vilkaisi Nikoa. Sin\u00e4 iltana he olivat tavanneet ensimm\u00e4isen kerran. Sitten h\u00e4n n\u00e4ytti muistavan k\u00e4dess\u00e4\u00e4n olevan viinipullon. \"Niin, t\u00e4m\u00e4 on teille.\" H\u00e4n ojensi pullon minulle.\n\nKiitin ja johdatin vieraamme katetun p\u00f6yd\u00e4n \u00e4\u00e4reen. Mikaelin keitti\u00f6, ruokailutila ja olohuone olivat yht\u00e4 suurta avaraa tilaa, ja Niko ja Taneli katsoivat ymp\u00e4rilleen. Saatoin n\u00e4hd\u00e4, kuinka Niko arvioi mieless\u00e4\u00e4n jokaista yksityiskohtaa.\n\nTanelilla ei ollut vaikeuksia ilmaista mielipidett\u00e4\u00e4n \u00e4\u00e4neen. \"Teill\u00e4 on todella upea koti\", h\u00e4n sanoi.\n\n\"Kiitos\", Mikael vastasi.\n\n\"Sin\u00e4 siis asut nykyisin t\u00e4\u00e4ll\u00e4\", Niko kommentoi. H\u00e4nen \u00e4\u00e4nens\u00e4vyns\u00e4 oli melkein syytt\u00e4v\u00e4.\n\nKatsoin nopeasti Mikaelia. \"En mitenk\u00e4\u00e4n virallisesti. Suurin osa tavaroistani on edelleen Viidennell\u00e4 linjalla.\"\n\nNikon kulmat kohosivat. \"Onko Viides linja tyhjill\u00e4\u00e4n?\"\n\n\"Ei tietenk\u00e4\u00e4n. Mitja asuu siell\u00e4 tytt\u00f6yst\u00e4v\u00e4ns\u00e4 kanssa.\"\n\n\"Niin tosiaan. Veljesi, jota et viitsinyt edes mainita ennen kuin annoit h\u00e4nen muuttaa minun huoneeseeni.\"\n\nHalusin sanoa, ett\u00e4 h\u00e4n liioitteli. Siin\u00e4 vaiheessa, kun min\u00e4 ja Mitja olimme palanneet saarelta, Niko ei ollut nukkunut huoneessaan en\u00e4\u00e4 pitk\u00e4\u00e4n aikaan. Mitja oli saanut h\u00e4nen huoneensa vain siksi, ett\u00e4 Niko itse oli l\u00e4htenyt ovet paukkuen. Mutta tiesin, ett\u00e4 Niko oli vihainen, ja olin valmis nielem\u00e4\u00e4n kaikki vastalauseet jos se vain tarkoitti, ett\u00e4 minulla oli viel\u00e4 toivoa saada h\u00e4net takaisin. Halusin tulkita kettuilun niin, ett\u00e4 h\u00e4n yh\u00e4 v\u00e4litti.\n\nSek\u00e4 Taneli ett\u00e4 Mikael avasivat suunsa tarkoituksenaan ep\u00e4ilem\u00e4tt\u00e4 aloittaa alkusammutus ennen kuin Nikon kiukku ehtisi roihahtaa kunnolla. Niin kiitollinen kuin olinkin heid\u00e4n valmiudestaan rient\u00e4\u00e4 h\u00e4tiin, vaimensin heid\u00e4t k\u00e4dell\u00e4ni.\n\n\"Tied\u00e4n, Niko. Usko pois, min\u00e4 tied\u00e4n. Olen pahoillani.\"\n\n\"Sin\u00e4 vain h\u00e4ivyit.\" Se ei ollut pahin syyt\u00f6s, jonka olin kuullut \u2013 ja samaan aikaan se kuitenkin oli. Tiesin, millaista oli, kun joku katosi ennen kuin oli saanut sanoa sanottavansa. Tiesin, miten syv\u00e4lle h\u00e4nen loukkaantumisensa ulottui.\n\n\"Ehk\u00e4 sinun pit\u00e4isi kertoa koko tarina\", Mikael ehdotti.\n\nKatsoin Mikaelia tavalla, joka sanoi: Et ole tosissasi. H\u00e4nen oma katseensa vastasi: Olen kuin olenkin.\n\nEn voinut kertoa kaikkea Tanelin edess\u00e4. Olin varma, ettei Niko ollut kertonut Tanelille mit\u00e4\u00e4n ihmissusista tai muista yliluonnollisista. Se, ett\u00e4 Taneli oli edelleen maisemissa, oli siit\u00e4 kohtalaisen pit\u00e4v\u00e4 todiste.\n\nMikael tiesi t\u00e4m\u00e4n kaiken. Mit\u00e4 min\u00e4 en siis tajunnut?\n\nVastaus oli lopulta hyvin yksinkertainen. Katsoin Nikoa. \"Min\u00e4 kerron mit\u00e4 tapahtui heti, kun te olette ottaneet ruokaa. Se j\u00e4\u00e4htyy.\"\n\nNiko ei ollut viel\u00e4 koskaan ollut niin vihainen, ett\u00e4 olisi kielt\u00e4ytynyt ruuasta, eik\u00e4 t\u00e4m\u00e4 kerta tehnyt siihen poikkeusta. Vilkaistuaan minua nopeasti kulmiensa alta h\u00e4n alkoi lappaa riisi\u00e4 ja kanakormaa lautaselleen ja pian suuhunsa. H\u00e4n p\u00e4\u00e4sti tyytyv\u00e4isen \u00e4\u00e4nen, enk\u00e4 ihmetellyt \u2013 min\u00e4 ja Mikael olimme k\u00e4yneet samassa ravintolassa monta kertaa ja siit\u00e4 oli tullut yksi suosikeistamme.\n\nVaihdoimme Mikaelin kanssa nopean katseen. Pid\u00e4tin hymy\u00e4. Varmistin, ett\u00e4 Nikon suu oli t\u00e4ynn\u00e4 naanleip\u00e4\u00e4 ennen kuin aloitin. \"Kuulin kaksoisveljest\u00e4ni Mitjasta ensimm\u00e4ist\u00e4 kertaa jo Hukkavaarassa, mutta minulla ei silloin ollut mit\u00e4\u00e4n keinoa selvitt\u00e4\u00e4, miss\u00e4 h\u00e4n asui, eik\u00e4 suoraan sanottuna suurta haluakaan. Kun Konsta palasi takaisin Suomeen viime syksyn\u00e4, h\u00e4n antoi minulle Mitjan osoitteen.\"\n\nOikeastaan Konstan antama osoite ei ollut Mitjan osoite, sill\u00e4 h\u00e4nell\u00e4 ei tuolloin ollut kotia, vaan pelk\u00e4st\u00e4\u00e4n tilap\u00e4isi\u00e4 nukkumapaikkoja erin\u00e4isten kavereidensa sohvilla. Se ei kuitenkaan ollut t\u00e4rke\u00e4\u00e4. \"Konsta oli matkoillaan t\u00f6rm\u00e4nnyt tyyppiin, jonka h\u00e4n ep\u00e4ili olevan sukulaisemme. T\u00e4m\u00e4 tyyppi halusi vied\u00e4 minut ja Mitjan Kreikkaan tapaamaan is\u00e4mme sukua.\"\n\nNikon pureskelu oli hidastunut. Tiesin, ett\u00e4 h\u00e4nen p\u00e4\u00e4ss\u00e4\u00e4n hyrr\u00e4si miljoona kysymyst\u00e4, mutta minun piti saada kertoa kaikki alusta loppuun. K\u00e4\u00e4nnyin siis Tanelin puoleen. \"Min\u00e4 en koskaan tuntenut is\u00e4\u00e4ni\", sanoin. Kuulostin aivan silt\u00e4 kuin olisin ollut pahoillani, ja olinkin \u2013 en vain ehk\u00e4 siit\u00e4 syyst\u00e4, kuin olisi voinut kuvitella. Se, etten ollut tuntenut is\u00e4\u00e4ni, oli nimitt\u00e4in johtanut melkoiseen sotkuun. El\u00e4m\u00e4ni \u2013 ja Mitjan el\u00e4m\u00e4 \u2013 olisi ollut hyvin erilaista, jos olisin aina tiennyt, kuka ja ennen kaikkea mik\u00e4 is\u00e4ni oli.\n\nEn ollut katkera. Olin saanut el\u00e4\u00e4 tavallista el\u00e4m\u00e4\u00e4, vaikka olinkin jollain tasolla tiennyt, ett\u00e4 olin kaikkea muuta kuin normaali. Mitja ei ollut aivan yht\u00e4 onnekas, mutta uskoin puhuvani meid\u00e4n molempien puolesta sanoessani, ett\u00e4 oli ihan hyv\u00e4, ett\u00e4 vanhempamme olivat p\u00e4\u00e4tt\u00e4neet suojella meit\u00e4. Me, toisin kuin he, olimme nimitt\u00e4in yh\u00e4 elossa.\n\n\"Me siis l\u00e4hdimme Kreikkaan. Meille ei kerrottu juuri mit\u00e4\u00e4n siit\u00e4, minne olimme menossa, ja paikan p\u00e4\u00e4ll\u00e4 meille selvisi, miksi. Saarella, jonne meid\u00e4t vietiin, asuu er\u00e4\u00e4nlainen... lahko. Ja verisukulaisuus on heille t\u00e4rke\u00e4\u00e4. Is\u00e4mme oli aikoinaan l\u00e4htenyt sielt\u00e4, mutta muut sukulaiset olivat koko ajan toivoneet l\u00f6yt\u00e4v\u00e4ns\u00e4 ja tuovansa meid\u00e4t saarelle, jotta he voisivat opettaa meille kaiken heid\u00e4n uskonnostaan. Me emme olleet samaa mielt\u00e4, mutta... meille ei oikeastaan annettu vaihtoehtoa.\"\n\n\"Tarkoitatko, ett\u00e4 sinut ja veljesi kidnapattiin?\" Taneli kysyi.\n\nKidnappaus ei ollut edes l\u00e4hell\u00e4 totuutta. Mutta my\u00f6nt\u00e4minen oli helpoin tapa antaa edes jonkinlainen k\u00e4sitys siit\u00e4, mit\u00e4 oli tapahtunut.\n\n\"Tavallaan\", sanoin nopeasti. \"Saarella ei ollut kentt\u00e4\u00e4 eik\u00e4 interneti\u00e4, joten emme voineet ottaa kehenk\u00e4\u00e4n yhteytt\u00e4. Saaren yhteis\u00f6 el\u00e4\u00e4 todella vanhanaikaisesti.\"\n\nNikon kulmakarvat olivat kohonneet jo melkein hiusrajaan. Kukaan, edes h\u00e4n, ei en\u00e4\u00e4 kiinnitt\u00e4nyt mit\u00e4\u00e4n huomiota ruokaan. Olin tuskallisen tietoinen kaikkien tuijotuksesta.\n\n\"Jonkin ajan kuluttua me totuimme asumaan saarella. Se ei en\u00e4\u00e4 tuntunut vankilalta, sill\u00e4 meist\u00e4 pidettiin hyv\u00e4\u00e4 huolta ja saimme molemmat tehd\u00e4 asioita, joista nautimme. Taide on osa heid\u00e4n uskontoaan. Min\u00e4 maalasin ja Mitja soitti ja kaikki oli oikeastaan ihan hyvin, kunnes tajusin, ett\u00e4 meid\u00e4n on oikeasti l\u00e4hdett\u00e4v\u00e4.\" Vedin henke\u00e4. Seuraava osuus oli kaikkein pahin. \"Saarella oleva yhteis\u00f6 kuvittelee, ett\u00e4 jompikumpi, min\u00e4 tai Mitja, on jonkinlainen maailmanlopun l\u00e4hettil\u00e4s. Ja he haluavat est\u00e4\u00e4 maailmanlopun, eli toisin sanoen saada ainakin toisen meist\u00e4 hengilt\u00e4. Joten p\u00e4\u00e4tin, ett\u00e4 meid\u00e4n oli parempi ottaa hatkat. Syy, miksi en koskaan soittanut sinulle takaisin tai edes vastannut puheluihisi oli se, ett\u00e4 yksi lahkolaisista l\u00e4hti per\u00e4\u00e4mme. En halunnut ottaa mit\u00e4\u00e4n riskej\u00e4. H\u00e4n olisi saattanut keksi\u00e4 jotain... hullua. Mutta nyt kukaan ei en\u00e4\u00e4 uhkaa meit\u00e4.\"\n\nUskaltauduin viimein vilkaisemaan muiden ilmeit\u00e4. Taneli oli ymm\u00e4rrett\u00e4v\u00e4sti j\u00e4rkyttyneen n\u00e4k\u00f6inen. Saatoin vain kuvitella, milt\u00e4 h\u00e4n olisi n\u00e4ytt\u00e4nyt, jos olisin kertonut aivan kaiken. Toisinaan totuus oli tarua ihmeellisemp\u00e4\u00e4.\n\n\"Tuo kuulostaa aivan kamalalta. En tiennyt, ett\u00e4 tuollaista voisi oikeasti tapahtua\", h\u00e4n sanoi. \"Kai te ilmoititte poliisille?\"\n\n\"No, me emme oikeastaan kertoneet kenellek\u00e4\u00e4n. T\u00e4ll\u00e4 hullulla lahkolla on nimitt\u00e4in paljon rahaa ja suhteita. Mutta asia on onneksi nyt kunnossa.\" Vilkaisin tahtomattani Mikaelia. H\u00e4n oli pelastanut meid\u00e4t. Olin varma, ett\u00e4 daimonit olivat luopuneet kostostaan pelk\u00e4st\u00e4\u00e4n siksi, ett\u00e4 he pelk\u00e4siv\u00e4t laumaa. Sudet olivat tiet\u00e4\u00e4ksemme ainoita, jotka pystyiv\u00e4t tappamaan daimonin.\n\n\"Oletko varma?\" Taneli kysyi. \"Ehk\u00e4 teid\u00e4n kannattaisi silti kertoa jollekin. Tuollaiset skitsot pit\u00e4\u00e4 saada oikeuteen.\"\n\n\"Jos Raisa sanoo, ett\u00e4 kaikki on kunnossa, silloin kaikki on kunnossa\", Niko sanoi.\n\n\"Siin\u00e4p\u00e4 se tarina oikeastaan oli\", sanoin. Ja se piti paikkansa: olin j\u00e4tt\u00e4nyt kertomatta ainoastaan kaiken yliluonnollisen. Ja sen, ett\u00e4 Mikael oli tappanut Alekin.\n\nHetket kuluivat, ja ainoa \u00e4\u00e4ni kuului ruokailuv\u00e4lineist\u00e4. Hiljaisuuden venyess\u00e4 aloin pel\u00e4t\u00e4, ett\u00e4 mahdollisuuteni lipuisi ohi, joten yritin keksi\u00e4 puheenaiheen.\n\n\"Kuulin, ett\u00e4 te olette muuttamassa Pariisiin.\" Se ei ehk\u00e4 ollut kaikkein turvallisin puheenaihe, mutta halusin tiet\u00e4\u00e4 siit\u00e4 lis\u00e4\u00e4.\n\n\"Jos p\u00e4\u00e4sen kouluun\", Niko vastasi. \"Odotan viel\u00e4 viimeist\u00e4 vahvistusta. Mutta joo, todenn\u00e4k\u00f6isesti olemme l\u00e4hd\u00f6ss\u00e4.\"\n\n\"Miksi Pariisiin? Luulin, ett\u00e4 sin\u00e4 halusit Aaltoon.\"\n\nNiko mulkaisi minua. \"Se oli sinun keksim\u00e4si suunnitelma, ei minun.\"\n\nIdea Nikon Aalto-yliopistoon hakemisesta oli alunperin ollut minun, mutta olin kuvitellut, ett\u00e4 h\u00e4n oli alkuep\u00e4ilyjen j\u00e4lkeen ollut siit\u00e4 yht\u00e4 innoissaan kuin min\u00e4.\n\nMinulla ei ollut mit\u00e4\u00e4n syyt\u00e4 loukkaantua h\u00e4nen kommentistaan, ja omasta mielest\u00e4ni onnistuin pit\u00e4m\u00e4\u00e4n tunteeni kasassa. Niko oli kuitenkin ihmissusi, ja toisinaan se tarkoitti, ett\u00e4 h\u00e4n tiesi tunteeni paremmin kuin min\u00e4. H\u00e4nen seuraavat sanansa kuulostivat yll\u00e4tt\u00e4v\u00e4n paljon tunnustukselta. \"Min\u00e4 hain Aaltoon. En p\u00e4\u00e4ssyt sis\u00e4\u00e4n.\" H\u00e4n vilkaisi minua. \"\u00c4l\u00e4 katso minua noin s\u00e4\u00e4liv\u00e4sti. Kaikki eiv\u00e4t ole samanlaisia ihmelapsia kuin sin\u00e4.\"\n\n\"En min\u00e4 ole mik\u00e4\u00e4n ihmelapsi\", sanoin nopeasti. Jos Taneli ei olisi ollut paikalla, olisin viel\u00e4 lis\u00e4nnyt, ett\u00e4 taide oli minun yliluonnollinen kykyni. Minun taidekoulussa opiskelemiseni oli sama kuin jos joku ihmissusista olisi osallistunut ihmisten urheilukisoihin. Mutta saatoin vain sanoa: \"Min\u00e4 olin ihan surkea Kuvataideakatemiassa.\"\n\nEnsimm\u00e4ist\u00e4 kertaa Niko antoi huolellisesti esitetyn piittaamattomuutensa horjua. \"Surkea? Vaikka p\u00e4\u00e4sit sis\u00e4\u00e4n ilman yo-papereita? Mit\u00e4 ei varmaan ollut tapahtunut moneen vuoteen ennen sinua?\"\n\n\"Ensimm\u00e4iset p\u00e4iv\u00e4t olivat ihan kauheita\", sanoin. \"Sain heti murskapalautetta opettajalta, joka vastaa maalaustaiteen koulutusohjelmasta. Kaikki muut opiskelijat olivat minua paljon vanhempia ja parempia. Minulla ei ole hajuakaan, miksi minut otettiin sis\u00e4\u00e4n.\"\n\n\"Oletko varma, ettet v\u00e4h\u00e4n ylireagoinut?\"\n\n\"Varmasti ylireagoin. Mutta sitten min\u00e4 ja Mitja l\u00e4hdimme Kreikkaan, ja kun tulimme takaisin, en pystynyt kuvittelemaan palaavani kouluun. Me olimme tukalassa tilanteessa, enk\u00e4 edes tiennyt, voisimmeko j\u00e4\u00e4d\u00e4 Suomeen. Tarvitsimme rahaa ja min\u00e4 sain paikan tatuointiliikkeest\u00e4 ja...\"\n\n\"Oletko sin\u00e4 nykyisin t\u00f6iss\u00e4 tatuointiliikkeess\u00e4?\"\n\nOlin unohtanut, miten paljon oli ehtinyt tapahtua sen j\u00e4lkeen, kun Niko ja min\u00e4 olimme lakanneet puhumasta. Ny\u00f6kk\u00e4sin.\n\n\"Seh\u00e4n on aika... siisti\u00e4?\" H\u00e4n ei n\u00e4ytt\u00e4nyt olevan aivan varma asiasta.\n\n\"Kiitos.\"\n\n\"J\u00e4titk\u00f6 sin\u00e4 siis Kuvataideakatemian kesken?\"\n\n\"Aluksi ajattelin niin. T\u00e4m\u00e4 t\u00e4ss\u00e4 ylipuhui minut yritt\u00e4m\u00e4\u00e4n uudestaan.\" Ny\u00f6kk\u00e4sin Mikaelin suuntaan. Puristin h\u00e4nen polveaan p\u00f6yd\u00e4n alla.\n\nSamaan aikaan kun Mikael oli ollut yliopistolla tekem\u00e4ss\u00e4 p\u00e4\u00e4sykoetta, min\u00e4 olin koputtanut Sirpa Kojon huoneen ovelle. H\u00e4n ei n\u00e4ytt\u00e4nyt yll\u00e4ttyneelt\u00e4 n\u00e4hdess\u00e4\u00e4n minut. Min\u00e4 sen sijaan yll\u00e4tyin monestakin eri asiasta: siit\u00e4, ett\u00e4 h\u00e4n istui jalat p\u00f6yd\u00e4ll\u00e4 ja luki Harry Potteria. Siit\u00e4, ett\u00e4 h\u00e4nen kenk\u00e4ns\u00e4 olivat punaiset Converset. Siit\u00e4, ett\u00e4 h\u00e4n ylip\u00e4\u00e4t\u00e4\u00e4n muisti, kuka olin.\n\n\"Raisa Oja\", h\u00e4n sanoi. \"Otaksun, ett\u00e4 haluat keskustella opintojesi tulevaisuudesta.\"\n\n\"Niin haluan\", sanoin. Ja koska halusin p\u00e4\u00e4st\u00e4 ulos h\u00e4nen huoneestaan niin nopeasti kuin mahdollista, m\u00f6l\u00e4ytin ajatukseni ulos. \"Haluan tiet\u00e4\u00e4, kannattaako minun jatkaa t\u00e4ss\u00e4 koulussa.\"\n\nSirpa sulki viimein kirjansa ja laski jalkansa lattialle. \"Tuohon kysymykseen ei voi vastata kukaan muu kuin sin\u00e4.\"\n\n\"Mutta sinun mielest\u00e4si\", intin. \"Tuhlaanko min\u00e4 aikaani, jos aloitan uudestaan ensi syksyn\u00e4?\"\n\nH\u00e4nen vastauksensa m\u00e4\u00e4rittelisi pitk\u00e4lti p\u00e4\u00e4t\u00f6kseni. Olin nimitt\u00e4in t\u00e4ysin kahden vaiheilla. Olin aika varma, ettei Sirpa Kojolla olisi mink\u00e4\u00e4nlaisia vaikeuksia sanoa niin kuin asia oli.\n\n\"Vastaus riippuu t\u00e4ysin siit\u00e4, mit\u00e4 sin\u00e4 haluat n\u00e4ilt\u00e4 opinnoilta.\"\n\nAvasin suuni vastatakseni, mutta totta puhuakseni en tiennyt, mit\u00e4 sanoa. Mit\u00e4 min\u00e4 todella halusin? Olin aina kuvitellut haluavani taiteilijaksi, mutta ehk\u00e4 se johtui enemm\u00e4n \u00e4idist\u00e4ni kuin mist\u00e4\u00e4n muusta. H\u00e4n ei ollut pit\u00e4nyt itse\u00e4\u00e4n taiteilijana vaan graafikkona, ja h\u00e4n oli aina arvostanut ihmisi\u00e4, jotka olivat olleet riitt\u00e4v\u00e4n rohkeita valitsemaan ep\u00e4kaupallisen polun. Nyt ymm\u00e4rsin, ett\u00e4 olin halunnut tulla joksikin, jota h\u00e4n ja muut ihailisivat.\n\n\"Jos sin\u00e4 kuvittelet, ett\u00e4 t\u00e4m\u00e4 koulu antaa sinulle ammatin niin kuin l\u00e4\u00e4ketieteellinen tai oikeustieteellinen, sin\u00e4 todellakin tuhlaat aikaasi\", Sirpa jatkoi. \"Jos taas haluat kehitty\u00e4 taitelijana ja ihmisen\u00e4, niin siin\u00e4 tapauksessa suosittelen, ett\u00e4 nielet ylpeytesi ja yrit\u00e4t uudestaan.\"\n\nTiesin olevani teknisesti taitava. Ilmeisesti en vain osannut luoda mit\u00e4\u00e4n kovin omaper\u00e4ist\u00e4 tai syv\u00e4llist\u00e4. El\u00e4m\u00e4ss\u00e4ni oli t\u00e4ll\u00e4 hetkell\u00e4 niin monta hyv\u00e4\u00e4 asiaa ja mahtavaa ihmist\u00e4, ett\u00e4 vietin aikani mieluummin niiden kanssa kuin puuduin hitaasti pettymysten tiell\u00e4. Pelk\u00e4sin, ett\u00e4 Kuvataideakatemia saisi minut ajan oloon vain vihaamaan taidetta. Mik\u00e4\u00e4n unelma ei olisi sen arvoinen.\n\nAjattelin tunnetta, joka minulla oli ollut saarella joka p\u00e4iv\u00e4 astuessani sis\u00e4\u00e4n omaan ateljeeseeni. En keksinyt parempaa sanaa kuin n\u00e4lk\u00e4 \u2013 pahimmillaan n\u00e4\u00e4nnytt\u00e4v\u00e4, mutta parhaimmillaan todella innostava tunne. Innostava, koska tiesin, ett\u00e4 saisin kohta sy\u00f6d\u00e4 kyllikseni.\n\n\"Haluan tehd\u00e4 taidetta\", sanoin Sirpalle. \"En vain ole en\u00e4\u00e4 t\u00e4ysin varma, onko t\u00e4m\u00e4 koulu siihen oikea paikka.\" Oli helpottavaa sanoa se \u00e4\u00e4neen.\n\n\"Siin\u00e4 tapauksessa sinun ei varmaankaan kannata jatkaa t\u00e4\u00e4ll\u00e4. Moni muu olisi valmis antamaan paljon saadakseen sinun paikkasi.\"\n\nLuulin olleeni valmis hyv\u00e4ksym\u00e4\u00e4n mit\u00e4 hyv\u00e4ns\u00e4 Sirpa sanoisi minulle. Tajusin kuitenkin, etten ollut kuvitellut h\u00e4nen olevan lopettamiseni kannalla. En ollut halunnut kuulla h\u00e4nen sanovan niin, mik\u00e4 tietenkin tarkoitti, ett\u00e4 minulla oli kuin olikin oma mielipiteeni asiasta.\n\n\"Eli minulla ei ole lupaa olla ep\u00e4varma?\" kysyin. \"Olla tiet\u00e4m\u00e4tt\u00e4, mit\u00e4 ajattelen tai mit\u00e4 haluan, mit\u00e4 kohti olen menossa? Jos haluan jatkaa t\u00e4ss\u00e4 koulussa, minun kuuluisi jo tiet\u00e4\u00e4 kaikki?\" Kun Sirpa ei sanonut mit\u00e4\u00e4n, jatkoin: \"Se on t\u00e4ydellinen vastakohta sille, miksi aloin tehd\u00e4 taidetta.\"\n\nOlin jo ottanut askeleen kohti ovea, kun Sirpa kysyi: \"Haluatko, ett\u00e4 kerron sinulle, mit\u00e4 min\u00e4 n\u00e4en sinussa?\"\n\nViel\u00e4 minuutti sitten olisin halunnut tismalleen sit\u00e4. Nyt en v\u00e4litt\u00e4nyt tippaakaan.\n\n\"Sinun ongelmasi ei ole se, etteik\u00f6 sinulla olisi lahjoja tai teknist\u00e4 osaamista. Sinun ongelmasi on se, ett\u00e4 sin\u00e4 yrit\u00e4t aivan liikaa.\"\n\nAjattelin itse aivan p\u00e4invastoin. En antanut taiteelle tarpeeksi. En osannut antaa tarpeeksi. Toisin kuin monet muut, min\u00e4 en el\u00e4nyt ja hengitt\u00e4nyt taidetta joka sekunti, jonka olin valveilla.\n\n\"Taiteessa ei ole oikeaa ja v\u00e4\u00e4r\u00e4\u00e4. Sin\u00e4 toivot, ett\u00e4 olisi, sill\u00e4 se tekisi el\u00e4m\u00e4st\u00e4si helpompaa. Jos taide olisi pelkk\u00e4\u00e4 tekniikkaa, sinulla ei olisi mit\u00e4\u00e4n syyt\u00e4 olla ep\u00e4varma. Sin\u00e4 tiet\u00e4isit tismalleen miss\u00e4 seisot ja mit\u00e4 osa-alueita sinun pit\u00e4isi parantaa, jotta tulisit parhaaksi. Mutta loppujen lopuksi en usko, ett\u00e4 sinua vieh\u00e4tt\u00e4\u00e4 mittaaminen ja kilpaileminen. Muuten et olisi koskaan tarttunut pensseliin.\"\n\n\"En niin\", my\u00f6nsin. Uskaltauduin viimein katsomaan Sirpaa. H\u00e4n katsoi minua tavalla, jota saattoi melkein sanoa omahyv\u00e4iseksi.\n\n\"Me voimme tehd\u00e4 n\u00e4in: sinulla on t\u00e4m\u00e4 kes\u00e4 aikaa tutustua uudestaan itseesi taiteilijana. Unohda kaikki, mit\u00e4 sinulle on opetettu. Ja kun sanon kaikki, tarkoitan ihan kaikkea. Leiki. Kokeile. Ennen kaikkea nauti.\"\n\nNauti? Sit\u00e4 sanaa en ollut odottanut kuulevani Sirpan suusta, ja hetken aikaa uskoin kuvitelleeni kaiken. H\u00e4n vaikutti niin ilottomalta ihmiselt\u00e4. Mutta ehk\u00e4 olin j\u00e4lleen v\u00e4\u00e4r\u00e4ss\u00e4.\n\nH\u00e4n jatkoi: \"Kun kes\u00e4 on ohi, tuo minulle n\u00e4ytille kaikki, mit\u00e4 olet saanut aikaiseksi. Me keskustelemme ja jos vaikuttaa silt\u00e4, ett\u00e4 olet ymm\u00e4rt\u00e4nyt tarpeeksi, otan sinut suoraan maalaustaiteeseen. Jos siis haluat.\"\n\n\"Voitko sin\u00e4 muka tehd\u00e4 niin?\" kysyin \u00e4llistyneen\u00e4. H\u00e4n oli valmis antamaan minulle puolen vuoden opinnot ilmaiseksi.\n\n\"Min\u00e4 voin tehd\u00e4 mit\u00e4 haluan\", h\u00e4n sanoi.\n\nMuistin yh\u00e4 h\u00e4nen virneens\u00e4. Aivan kuin h\u00e4n olisi juuri onnistunut ved\u00e4tt\u00e4m\u00e4\u00e4n vastustajaansa pokerip\u00f6yd\u00e4ss\u00e4, ja tuo vastustaja olisi ollut min\u00e4. Sill\u00e4 hetkell\u00e4 sek\u00e4 ihailin ett\u00e4 paheksuin Sirpa Kojoa.\n\n\"Min\u00e4 sain kes\u00e4ksi erityisteht\u00e4v\u00e4n\", sanoin Nikolle samalla kun otin lis\u00e4\u00e4 ruokaa. Olin aloittanut sy\u00f6misen my\u00f6hemmin kuin muut, ja huomasin olevani ainoa, joka ei ollut viel\u00e4 valmis. \"Tienaan sill\u00e4 tavalla itselleni puuttuvia opintopisteit\u00e4. Eli min\u00e4 opiskelen edelleen Kuvataideakatemiassa.\"\n\nKun olimme kaikki saaneet sy\u00f6ty\u00e4, siirryimme sohvalle. Taneli istui viereeni, mik\u00e4 sopi paremmin kuin hyvin. Olin hyvin utelias tutustumaan ihmiseen, joka pystyi viem\u00e4\u00e4n Niko J\u00e4\u00e4skel\u00e4isen syd\u00e4men.\n\nH\u00e4n paljastui juuri niin herttaiseksi kuin milt\u00e4 n\u00e4ytti. H\u00e4nen kanssaan oli hyvin helppo puhua, eik\u00e4 h\u00e4nt\u00e4 n\u00e4ytt\u00e4nyt haittaavan, ett\u00e4 uteliaisuuteni otti minusta voiton ja esitin aivan liian monta henkil\u00f6kohtaista kysymyst\u00e4. Sain selville, ett\u00e4 h\u00e4n oli aina asunut Helsingiss\u00e4. Nuoresta ulkon\u00e4\u00f6st\u00e4\u00e4n huolimatta h\u00e4n oli Nikoa kaksi vuotta vanhempi. H\u00e4n kertoi p\u00e4\u00e4tyneens\u00e4 baariin t\u00f6ihin jouduttuaan lopettamaan parturi-kampaajan ty\u00f6t allergioiden takia.\n\n\"Rakastin tukan laittamista, mutta el\u00e4m\u00e4ss\u00e4 ei saa kaikkea\", h\u00e4n sanoi hymyillen.\n\n\"Tuo on todella surullista\", sanoin. En voinut kuin ihailla h\u00e4nen asennettaan. En tiennyt, pystyisink\u00f6 samaan, jos maalaaminen viet\u00e4isiin minulta. \u00c4kki\u00e4 olin paljon kiitollisempi mahdollisuudestani palata Kuvataideakatemiaan.\n\nKun k\u00e4vi ilmi, ett\u00e4 me molemmat pidimme tanssimisesta, h\u00e4n ehdotti heti, ett\u00e4 menisimme joskus yhdess\u00e4 tunnille. En keksinyt mit\u00e4\u00e4n syyt\u00e4 olla suostumatta.\n\nJuttelimme my\u00f6s suunnitelmistamme kes\u00e4n varalle. Niko olisi enimm\u00e4kseen t\u00f6iss\u00e4. H\u00e4nell\u00e4 ja Tanelilla oli viikko yhteist\u00e4 lomaa, mutta koska heid\u00e4n t\u00e4ytyi s\u00e4\u00e4st\u00e4\u00e4 rahaa, he pysyisiv\u00e4t kaupungissa.\n\nViimein Niko ja Taneli kertoivat, ett\u00e4 heid\u00e4n oli aika l\u00e4hte\u00e4. Saatoimme heid\u00e4t ovelle ja sanoimme moit.\n\nKun ovi oli sulkeutunut, palasin sohvalle. Huokaisin ja nosti jalkani kahvip\u00f6yd\u00e4lle. Ilta oli sujunut paremmin kuin olin osannut odottaa. Huomasin hymyilev\u00e4ni.\n\nEl\u00e4m\u00e4 oli pelottavan l\u00e4hell\u00e4 t\u00e4ydellist\u00e4. Tiesin, kenen ansiota se oli. K\u00e4\u00e4nsin p\u00e4\u00e4ni Mikaelin suuntaan vain huomatakseni, ett\u00e4 h\u00e4n katsoi minua.\n\nEn koskaan kyll\u00e4styisi h\u00e4nen silmiins\u00e4.\n\n\"Mit\u00e4?\" h\u00e4n kysyi lopulta.\n\n\"Kiitos\", sanoin.\n\n\"En tehnyt mit\u00e4\u00e4n\", h\u00e4n sanoi.\n\nHengitin hitaasti sis\u00e4\u00e4n ja ulos. Osa minusta pelk\u00e4si, ett\u00e4 sin\u00e4 p\u00e4iv\u00e4n\u00e4 kun h\u00e4n tajuaisi, miten upea h\u00e4n oli, h\u00e4n j\u00e4tt\u00e4isi minut. Mutta se pelko oli t\u00e4ysin mit\u00e4t\u00f6n kaiken muun rinnalla.\n\n\"Min\u00e4 taidan pit\u00e4\u00e4 sinut\", sanoin.\n\nMikael nauroi. \"Sitten minun t\u00e4ytyy varmaan tyyty\u00e4 osaani.\"\n\n***\n\nSeuraavana y\u00f6n\u00e4 her\u00e4sin kummalliseen tunteeseen. Hetken p\u00e4\u00e4st\u00e4 tajusin, ettei se oikeastaan ollut tunne, vaan sen puute: Mikael ei ollut vieress\u00e4ni.\n\nKohottauduin kyyn\u00e4rp\u00e4iden varaan. Pimennysverhot olivat kiinni ja meni hetki, ennen kuin Mikaelin hahmo piirtyi esiin h\u00e4m\u00e4r\u00e4ss\u00e4.\n\nH\u00e4n oli noussut istumaan s\u00e4ngylle. Saatoin n\u00e4hd\u00e4 vain h\u00e4nen paljaan selk\u00e4ns\u00e4, sill\u00e4 h\u00e4n oli painanut p\u00e4\u00e4ns\u00e4 k\u00e4siin.\n\nIhmiset k\u00e4rsiv\u00e4t unettomuudesta. He saattoivat her\u00e4ill\u00e4 erin\u00e4isist\u00e4 syist\u00e4, nousivat vessaan tai kirjoittivat yl\u00f6s juuri n\u00e4kem\u00e4ns\u00e4 kiinnostavan unen.\n\nMikael ei ollut ihminen, ja sudet nukkuivat paremmin kuin ketk\u00e4\u00e4n toiset tuntemani kuolevaiset maan p\u00e4\u00e4ll\u00e4 \u2013 vampyyreja ei siis laskettu. Tiesin heti, ett\u00e4 Mikaelin her\u00e4\u00e4miselle oli oltava syy.\n\n\"Onko jokin h\u00e4t\u00e4n\u00e4?\" kysyin.\n\nPelk\u00e4sin vastausta, enk\u00e4 edes ollut varma, miksi.\n\n\"Ei oikeastaan\", h\u00e4n sanoi ikuisuuden kuluttua. Tajusin heti, ettei \"ei oikeastaan\" ollut sama kuin \"ei\". En viel\u00e4k\u00e4\u00e4n n\u00e4hnyt h\u00e4nen kasvojaan, ja se hermostutti minua. Jos h\u00e4n olisi k\u00e4\u00e4ntynyt ja hymyillyt minulle, olisin ehk\u00e4 voinut vain kier\u00e4ht\u00e4\u00e4 toiselle kyljelle ja jatkaa uniani. Ehk\u00e4. Mutta nyt en tiennyt yht\u00e4\u00e4n, mit\u00e4 tehd\u00e4.\n\nMikaelin t\u00e4ytyi aistia huoleni, sill\u00e4 h\u00e4n sanoi: \"Susi on vain v\u00e4h\u00e4n levoton.\"\n\nSe oli ensimm\u00e4inen kerta, kun kuulin jotain sellaista Mikaelin suusta. Nikolla oli ollut ongelmia sutensa kanssa ennen kuin olimme tajunneet, ett\u00e4 h\u00e4n oli niit\u00e4 harvoja susia, jotka eiv\u00e4t sopeutuneet laumaan.\n\nMutta Mikael... H\u00e4n ja h\u00e4nen sutensa olivat aina pelanneet yhteen.\n\n\"Onko mit\u00e4\u00e4n, mit\u00e4 voisin tehd\u00e4?\" kysyin.\n\nOlin varma, ett\u00e4 h\u00e4n sanoisi ei. Sen sijaan h\u00e4n k\u00e4\u00e4ntyi viimeinkin hitaasti ymp\u00e4ri. Ehdin n\u00e4hd\u00e4 vain vilahduksen h\u00e4nen kasvoistaan ennen kuin h\u00e4n painoi ne kiinni vatsaani.\n\n\"Sinun tuoksusi auttaa\", h\u00e4n mumisi.\n\nAvasin suuni kysy\u00e4kseni, mit\u00e4 h\u00e4n tarkalleen ottaen tarkoitti. Auttoi mihin? Mutta sitten tulin toisiin aatoksiin. Mikael nimitt\u00e4in ty\u00f6nsi k\u00e4tens\u00e4 minun ja patjan v\u00e4liin niin, ett\u00e4 h\u00e4nen k\u00e4sivartensa olivat lopulta lantioni ymp\u00e4rill\u00e4. Ne muodostivat ter\u00e4svanteen, josta en olisi p\u00e4\u00e4ssyt irti vaikka olisin yritt\u00e4nyt.\n\nEn uskaltanut liikkua, tuskin edes hengitt\u00e4\u00e4. En pel\u00e4nnyt Mikaelia tai h\u00e4nen suttaan. Pelk\u00e4sin h\u00e4nen puolestaan. H\u00e4nell\u00e4 oli ilmiselv\u00e4sti jokin h\u00e4t\u00e4n\u00e4, enk\u00e4 osannut auttaa h\u00e4nt\u00e4. Ainoa, mit\u00e4 keksin, oli antaa h\u00e4nelle tilaa, mutta t\u00e4ll\u00e4 hetkell\u00e4 se oli poissa laskuista.\n\nMikael hengitti kiihtyneesti. Silloin vihdoin tajusin, mit\u00e4 olin tekem\u00e4ss\u00e4. Minun pelkoni ruokki h\u00e4nen sutensa hermostuneisuutta. Hengitin hitaasti sis\u00e4\u00e4n ja ulos. Pakotin jokaisen lihakseni rentoutumaan. Annoin h\u00e4nen ja h\u00e4nen sutensa aistia, ettei ollut mit\u00e4\u00e4n pel\u00e4tt\u00e4v\u00e4\u00e4.\n\nMikaelin hengitys tasaantui ja h\u00e4nen k\u00e4sivartensa hellittiv\u00e4t hieman otettaan. Uskaltauduin silitt\u00e4m\u00e4\u00e4n h\u00e4nen hiuksiaan. Hyvin nopeasti ajatukseni alkoivat harhailla siihen, milt\u00e4 h\u00e4nen vartalonsa tuntui omaani vasten. Miten lihaksikkaat h\u00e4nen k\u00e4sivartensa olivat. H\u00e4n oli paljon vahvempi kuin milt\u00e4 n\u00e4ytti, eik\u00e4 h\u00e4nen kropassaan ollut montaakaan pehme\u00e4\u00e4 kohtaa. L\u00e4mmin tunne luikerteli vatsanpohjaani.\n\nLuoja. Mikael tarvitsi tukeani, ja min\u00e4 ajattelin vain seksi\u00e4. Olin maailmankaikkeuden huonoin tytt\u00f6yst\u00e4v\u00e4.\n\nTai sitten en: Mikael nimitt\u00e4in kohotti p\u00e4\u00e4ns\u00e4. H\u00e4nen silm\u00e4ns\u00e4 leiskuivat.\n\nH\u00e4n pystyi vainuamaan haluni.\n\nMinulta p\u00e4\u00e4si henk\u00e4ys, sellainen yht\u00e4 aikaa ep\u00e4toivoinen ja toiveikas, Eth\u00e4n kiltti muuta mielt\u00e4si min\u00e4 menet\u00e4n j\u00e4rkeni kohta -tyyppinen sanaton rukous.\n\nMikael liikahti, ja seuraava asia, jonka tiedostin, oli se, ett\u00e4 h\u00e4n suuteli minua. P\u00e4\u00e4ni upposi tyynyyn, ja kaikki mit\u00e4 saatoin tuntea olivat h\u00e4nen huulensa ja k\u00e4tens\u00e4, jotka olivat puristuneet ranteideni ymp\u00e4rille.\n\nSitten h\u00e4n liukui alas, alas, alas s\u00e4ngyll\u00e4, ja lopulta en pystynyt edes sanomaan h\u00e4nen nime\u00e4\u00e4n.\nTOINEN LUKU\n\nEn ajatellut Mikaelin outoa k\u00e4yt\u00f6st\u00e4 en\u00e4\u00e4 aamulla. Se nimitt\u00e4in alkoi juuri niin kuin jokainen aamu viimeisen kuukauden aikana: her\u00e4sin siihen, kun Mikael suuteli kaulaani.\n\nEn ollut aamuihmisi\u00e4, mutta olin nopeasti t\u00e4ysin hereill\u00e4.\n\n\"Olen niin iloinen, ett\u00e4 me olemme l\u00f6yt\u00e4neet sinulle kofeiininkorvikkeen\", h\u00e4n naurahti korvaani.\n\n\"Mmmm\", oli kaikki, mit\u00e4 suustani tuli ulos.\n\nToivoin, ett\u00e4 Mikael ajatteli samoin kuin min\u00e4 senp\u00e4iv\u00e4isist\u00e4 suunnitelmistamme: pysytell\u00e4 s\u00e4ngyss\u00e4 niin paljon kuin mahdollista.\n\nH\u00e4nell\u00e4 oli kuitenkin mieless\u00e4\u00e4n jotain ihan muuta.\n\n\"Sain viestin valmennuskurssikaveriltani\", h\u00e4n sanoi. \"Nimet ovat netiss\u00e4 ja laitoksen ilmoitustaululla.\"\n\nTiesin heti, mist\u00e4 h\u00e4n puhui. L\u00e4\u00e4ketieteellisen valinnan tulokset. Hakijoille l\u00e4hetettiin my\u00f6s kirje, mutta aina ne eiv\u00e4t ehtineet perille ennen tulosten julkistamista.\n\nKier\u00e4hdin ymp\u00e4ri ja katsoin h\u00e4nen puoliksi hermostunutta, puoliksi innostunutta ilmett\u00e4\u00e4n. Viel\u00e4 vuosi sitten h\u00e4n ei olisi n\u00e4ytt\u00e4nyt tunteitaan minulle niin avoimesti.\n\n\"Katsoitko jo?\" kysyin, vaikka tiesin ettei h\u00e4n ollut katsonut. Otaksuin, ett\u00e4 Mikael oli pyyt\u00e4nyt kavereitaan olemaan kertomatta ennen kuin h\u00e4n voisi tarkistaa asian itse.\n\nMikael pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Ajattelin, ett\u00e4 menen katsomaan listan laitokselle. Haluatko tulla mukaan?\"\n\nTietenkin halusin. Minua j\u00e4nnitti v\u00e4hint\u00e4\u00e4nkin yht\u00e4 paljon kuin Mikaelia. Tiesin, kuinka paljon h\u00e4n oli tehnyt t\u00f6it\u00e4 p\u00e4\u00e4st\u00e4kseen sis\u00e4\u00e4n, ja h\u00e4n todella ansaitsisi sen. Todenn\u00e4k\u00f6isyydet eiv\u00e4t olleet h\u00e4nen puolellaan, mutta onneksi yliopistoon p\u00e4\u00e4seminen ei ollut kiinni todenn\u00e4k\u00f6isyyksist\u00e4.\n\nPukiessani vaatteet p\u00e4\u00e4lle ajattelin, ettemme olleet puhuneet lainkaan siit\u00e4, mit\u00e4 tapahtuisi jos h\u00e4n ei p\u00e4\u00e4sisi sis\u00e4\u00e4n t\u00e4ll\u00e4 kertaa. Ep\u00e4ilin, ett\u00e4 h\u00e4n muuttaisi takaisin Hukkavaaraan ainakin siksi, kunnes hakisi taas uudestaan. Lauma tarvitsi h\u00e4nt\u00e4, ja mik\u00e4li h\u00e4n ei aloittaisi l\u00e4\u00e4kiksess\u00e4, h\u00e4nell\u00e4 ei ollut mit\u00e4\u00e4n syyt\u00e4 olla palaamatta.\n\nPesin nopeasti hampaani. Aamiainen saisi odottaa \u2013 en uskonut edes pystyv\u00e4ni sy\u00f6m\u00e4\u00e4n mit\u00e4\u00e4n, ennen kuin ep\u00e4tietoisuus olisi viimein ohi. Pujotin jalkani sandaaleihin ja astuimme viile\u00e4\u00e4n rappuk\u00e4yt\u00e4v\u00e4\u00e4n.\n\nL\u00e4\u00e4ketieteellinen tiedekunta sijaitsi Meilahdessa, eik\u00e4 sinne ollut T\u00f6\u00f6l\u00f6st\u00e4 pitk\u00e4 matka. Nappasimme kioskilta take-away-kahvit ja Mikaelin ehdotuksesta l\u00e4hdimme k\u00e4velem\u00e4\u00e4n bussin odottamisen sijaan. Taivas oli kirkas ja kaupunki p\u00f6lyinen, ihmiset aurinkorasvan tuoksuisia. Mikael tarttui k\u00e4teeni, ja hetken aikaa vain ajattelin miten luontevalta se tuntui, h\u00e4nen k\u00e4tens\u00e4 minun k\u00e4dess\u00e4ni. Askeleemme l\u00f6ysiv\u00e4t helposti yhteisen rytmin.\n\nK\u00e4velimme sen verran hitaasti, ett\u00e4 olimme perill\u00e4 vasta kolmessa vartissa. L\u00e4\u00e4ketieteellisen rakennukset n\u00e4yttiv\u00e4t tismalleen silt\u00e4, milt\u00e4 niiden kuvitteli n\u00e4ytt\u00e4v\u00e4n: kliinisilt\u00e4 ja mielikuvituksettomilta. K\u00e4yt\u00e4v\u00e4 oli tyhj\u00e4 lukuun ottamatta kahta poikaa, jotka odottivat vaitonaisina opintotoimiston edustalla. Toimistossa oleva tytt\u00f6 oli kaikkea muuta kuin vaitonainen: saatoin kuulla k\u00e4yt\u00e4v\u00e4\u00e4n asti, kuinka h\u00e4n toisti uudestaan ja uudestaan, ett\u00e4 h\u00e4nen kokeensa arvostelussa oli tapahtunut virhe ja kuinka h\u00e4n aikoi valittaa niin kauan, kunnes asia korjattiin.\n\nHuutaminen oli tietenkin typer\u00e4\u00e4, mutta huomasin tuntevani tytt\u00f6\u00e4 kohtaan pikemminkin sympatiaa kuin my\u00f6t\u00e4h\u00e4pe\u00e4\u00e4. H\u00e4nen tulevaisuutensa oli t\u00e4st\u00e4 kiinni. Aivan kuten Mikaelin ja minun.\n\nMikael pys\u00e4htyi \u00e4kisti, ja kun seurasin h\u00e4nen katsettaan, huomasin meid\u00e4n seisovan ilmoitustaulun edess\u00e4.\n\n\"Oletko valmis?\" kysyin.\n\nMikael ei n\u00e4ytt\u00e4nyt valmiilta, mutta h\u00e4n ny\u00f6kk\u00e4si silti. Puristin h\u00e4nen k\u00e4tt\u00e4\u00e4n. Astuimme molemmat kohti paperia, jossa luki L\u00e4\u00e4ketieteen koulutusohjelma 2014 ja alempana Hyv\u00e4ksytyt. Hassua, miten yksi aanelonen saattoi muuttaa niin monen ihmisen el\u00e4m\u00e4n.\n\nTunsin Mikaelin ja minun yhteisen pulssin k\u00e4dess\u00e4ni. Nimet oli jaettu kolmeen sarakkeeseen, ja kahlasin niit\u00e4 hetken katseellani ennen kuin l\u00f6ysin S-kirjaimella alkavat. Ja siell\u00e4, Salosen ja Sarkkisen j\u00e4lkeen, luki Sarri, Mikael.\n\nTarkistin, etten ollut n\u00e4hnyt v\u00e4\u00e4rin. Mutta jokainen kirjain t\u00e4sm\u00e4si yhteen h\u00e4nen nimens\u00e4 kanssa.\n\nMikaelin oli jo t\u00e4ytynyt n\u00e4hd\u00e4 sama. H\u00e4n t\u00e4risi vieress\u00e4ni. Ei \u2013 h\u00e4n nauroi.\n\n\"Sin\u00e4 teit sen! P\u00e4\u00e4sit sis\u00e4\u00e4n!\" En ollut kiljujatyyppi\u00e4, mutta minullakin oli hetkeni. Hypin tasajalkaa. En v\u00e4litt\u00e4nyt, vaikka vaitonaiset pojat tuijottivat minua ja sandaalieni l\u00e4psynt\u00e4 kaikui k\u00e4yt\u00e4v\u00e4n p\u00e4\u00e4st\u00e4 p\u00e4\u00e4h\u00e4n.\n\nHypp\u00e4sin Mikaelin kaulaan. H\u00e4n py\u00f6r\u00e4ytti minua ymp\u00e4ri, ja minulta p\u00e4\u00e4si taas kiljahdus. Samassa h\u00e4n jo suuteli minua.\n\nJos onnellisuuteen saattoi kuolla, olin p\u00e4\u00e4ssyt hyvin l\u00e4helle. En pystynyt ajattelemaan, pelk\u00e4st\u00e4\u00e4n tuntemaan. Se ei ollut mik\u00e4\u00e4n s\u00e4\u00e4dyllinen julkisella paikalla annettava suudelma, vaan sellainen, joiden j\u00e4lkeen tiesin, ettei sieluni en\u00e4\u00e4 kuulunut minulle.\n\nMikael laski minut alas, enk\u00e4 voinut lopettaa virnist\u00e4mist\u00e4. \"Min\u00e4 olen niin ylpe\u00e4 sinusta! Sin\u00e4 olet ihan mielet\u00f6n.\"\n\nToimiston ovi avautui, ja ulos astui v\u00e4syneen n\u00e4k\u00f6inen nainen. H\u00e4n k\u00e4\u00e4ntyi katsomaan, mist\u00e4 kaikki el\u00e4m\u00f6inti johtui, ja Mikael v\u00e4l\u00e4ytti h\u00e4nelle nopean hymyn. Saatoin n\u00e4hd\u00e4 ett\u00e4 se tehosi, sill\u00e4 nainen n\u00e4ytti heti v\u00e4hemm\u00e4n voipuneelta. H\u00e4nen ilmeess\u00e4\u00e4n oli jotain kaihoisaa, mink\u00e4 ymm\u00e4rsin vallan hyvin. Mikael oli nuori, komea ja p\u00e4\u00e4ssyt juuri l\u00e4\u00e4kikseen. Kaiken lis\u00e4ksi h\u00e4n oli rakastunut. Sit\u00e4 upeampaa n\u00e4ky\u00e4 ei voinut olla.\n\nNainen hymyili hiukan ennen kuin kutsui toisen odottavista pojista sis\u00e4\u00e4n. Tytt\u00f6, joka oli huutanut toimistossa, k\u00e4veli ohitsemme. H\u00e4n ei katsonut meit\u00e4.\n\n\"Miten me aiomme juhlia?\" kysyin nopeasti Mikaelilta.\n\n\"Minulla on idea\", Mikael vastasi.\n\n\"Ai? Anna kuulua.\"\n\nK\u00e4vi ilmi, ett\u00e4 Mikaelilla todellakin oli idea. Pari puhelinsoittoa my\u00f6hemmin meille oli varattu p\u00f6yt\u00e4 illaksi yhdest\u00e4 kaupungin suosituimmista ravintoloista. En tiennyt, miten Mikael oli onnistunut siin\u00e4, mutten todellakaan aikonut valittaa. Pujahdin lempipikkumustaani \u00e0 la Niko samaan aikaan kun Mikael veti p\u00e4\u00e4lleen kauluspaitaa. Meikkasin hillitysti, mutta panostin kenk\u00e4osastoon.\n\nRavintola ei ollut kaukana, mutta menimme silti ratikalla pari pys\u00e4kinv\u00e4li\u00e4. Kun p\u00e4\u00e4simme perille, kuulimme puheensorinan jo ovelle asti. Paikka oli t\u00e4ynn\u00e4. Se ei n\u00e4ytt\u00e4nyt yht\u00e4\u00e4n hienostelevalta, mik\u00e4 oli iloinen yll\u00e4tys. P\u00f6yd\u00e4t olivat l\u00e4hell\u00e4 toisiaan, mutta niiden v\u00e4liss\u00e4 oli l\u00e4pikuultavat verhot, jotka loivat illuusion yksityisyydest\u00e4. Meid\u00e4t ohjattiin p\u00f6yt\u00e4\u00e4n, ja tarjoilija toi heti ruokalistat. H\u00e4n selitti, kuinka kaikki annokset oli tarkoitettu jaettaviksi ja esitteli p\u00e4iv\u00e4n erikoiset. P\u00e4\u00e4dyimme tilaamaan maistelumenut sopivilla viineill\u00e4, ja vaikka hinta hirvitti, en antanut sen h\u00e4irit\u00e4 itse\u00e4ni kuin korkeintaan pari sekuntia. Mikael oli p\u00e4\u00e4ssyt l\u00e4\u00e4kikseen. En tiennyt parempaa syyt\u00e4 juhlia.\n\nRuoka oli t\u00e4ydellist\u00e4. Suustani p\u00e4\u00e4si paljon \u00e4\u00e4ni\u00e4, joita ei kuulunut p\u00e4\u00e4st\u00e4\u00e4 suustaan julkisella paikalla. Mikael nauroi minulle suurimman osan ajasta. En usko, ett\u00e4 olimme koskaan kertoneet toisillemme yht\u00e4 tyhmi\u00e4 juttuja \u2013 eik\u00e4 meill\u00e4 ollut koskaan ollut yht\u00e4 hauskaa.\n\nOlin niin, niin rakastunut Mikael Sarriin.\n\n\"Haluaisin vied\u00e4 sinut jonnekin lomalle\", h\u00e4n sanoi yht\u00e4kki\u00e4.\n\n\"Meh\u00e4n olemme lomalla\", sanoin.\n\n\"Tarkoitan ulkomaita.\"\n\n\"Ai.\" Tajusin kuulostavani kaikkea muuta kuin innostuneelta, joten jatkoin nopeasti: \"Se olisi mahtavaa. Minne haluaisit menn\u00e4?\"\n\n\"Sin\u00e4 saat p\u00e4\u00e4tt\u00e4\u00e4.\"\n\n\"Hmm. On niin monia maita, joissa en ole k\u00e4ynyt.\"\n\n\"Siin\u00e4 tapauksessa meid\u00e4n t\u00e4ytyy tehd\u00e4 lista.\"\n\nKoko j\u00e4lkiruuan ajan Mikael yritti arvuutella minulta maita. Nuoltuani viimeisetkin suklaanrippeet lusikastani olin melkein haikea, ett\u00e4 ateria oli ohi.\n\n\"\u00c4l\u00e4 n\u00e4yt\u00e4 noin surulliselta\", Mikael sanoi. \"Me voimme tulla t\u00e4nne uudestaan.\"\n\n\"Emme kovin usein\", vastasin. Ja juuri silloin tarjoilija toi laskun. En edes vilkaissut sit\u00e4 ennen kuin otin korttini esiin. \"Min\u00e4 tarjoan.\"\n\n\"Et.\"\n\n\"Etk\u00f6 ole kuullut? Min\u00e4 olen nyky\u00e4\u00e4n menestyv\u00e4 bisnesnainen.\" Olin aloittanut oman toiminimen, enk\u00e4 siis ollut tatuointiliikkeess\u00e4 en\u00e4\u00e4 palkollisena, vaan vuokralaisena. Sana ty\u00f6ni j\u00e4ljest\u00e4 oli alkanut levit\u00e4, ja asiakkaita riitti enemm\u00e4n kuin mit\u00e4 pystyin tekem\u00e4\u00e4n. Siit\u00e4 huolimatta oli ollut helppo p\u00e4\u00e4tt\u00e4\u00e4 pit\u00e4\u00e4 hein\u00e4kuu lomaa. El\u00e4m\u00e4ni oli ollut liian hullua liian pitk\u00e4\u00e4n, ja loma tuli tarpeeseen. Tiesin, ett\u00e4 syksyst\u00e4 tulisi kiireinen sek\u00e4 minulle ett\u00e4 Mikaelille, ja halusin viett\u00e4\u00e4 aikaa h\u00e4nen kanssaan nyt, kun se oli mahdollista.\n\n\"Ai? No, siit\u00e4 huolimatta min\u00e4 kielt\u00e4ydyn kunniasta. Minun juhlani, min\u00e4 maksan.\"\n\n\"Meh\u00e4n sovimme, etten ala el\u00e4\u00e4 sinun kustannuksellasi\", sanoin. Se oli periaate, josta en tinkinyt. Se johtui tietenkin ylpeydest\u00e4ni, mutta asiaan liittyi muutakin. Ei yksinkertaisesti tuntunut oikealta, ett\u00e4 Mikael tuhlaisi rahojaan minuun.\n\n\"Tarkoittaako t\u00e4m\u00e4 sinun s\u00e4\u00e4nt\u00f6si, etten saa antaa sinulle lahjoja?\" Mikael kysyi.\n\nAjattelin asiaa. Lahjojen kielt\u00e4minen tuntui aika julmalta. Minulla itsell\u00e4ni oli aina hauskaa yritt\u00e4ess\u00e4ni keksi\u00e4 yst\u00e4villeni annettavaa.\n\n\"Lahjojen antaminen on sallittua\", sanoin lopulta.\n\n\"Ent\u00e4 jos t\u00e4m\u00e4 ateria on lahja?\" Mikael kysyi. H\u00e4n pid\u00e4tti hymy\u00e4, mutta tiesin, ett\u00e4 h\u00e4n testasi asettamiani rajoja.\n\n\"Ei se toimi noin\", sanoin kulmat kurtussa.\n\nMikael nauroi murjotukselleni. \"Kai min\u00e4 saan edes yritt\u00e4\u00e4.\"\n\nYritin j\u00e4lleen kerran selitt\u00e4\u00e4. \"Min\u00e4 en ole tehnyt mit\u00e4\u00e4n ansaitakseni sinun rahojasi.\"\n\n\"Sep\u00e4 se: min\u00e4k\u00e4\u00e4n en ole tehnyt mit\u00e4\u00e4n ansaitakseni niit\u00e4.\" H\u00e4nen s\u00e4vyns\u00e4 oli yh\u00e4 leikkis\u00e4, mutta tunsin h\u00e4net riitt\u00e4v\u00e4n hyvin tiet\u00e4\u00e4kseni, ett\u00e4 asia oikeasti vaivasi h\u00e4nt\u00e4.\n\n\"Enp\u00e4 tied\u00e4\", sanoin. \"Sin\u00e4 kestit Danielia yli kahdeksantoista vuotta. Minusta se on v\u00e4hint\u00e4\u00e4n muutaman miljoonan arvoinen suoritus.\" Mikaelin is\u00e4 Daniel oli joutunut j\u00e4tt\u00e4m\u00e4\u00e4n lauman sen j\u00e4lkeen, kun h\u00e4nelt\u00e4 oli kaapattu valta. Lopulta Mikaelista oli tullut uusi laumanjohtaja. Kukaan ei tiennyt, miss\u00e4 Daniel oli.\n\nMikael palkitsi minut vinolla hymyll\u00e4. Se, ettei h\u00e4n korjannut liioiteltua arviotani h\u00e4nen omaisuudestaan sai minut ajattelemaan, ett\u00e4 h\u00e4n saattoi olla viel\u00e4 rikkaampi kuin olin pel\u00e4nnyt. \"My\u00f6nn\u00e4 pois. Sin\u00e4 pid\u00e4t is\u00e4st\u00e4ni.\"\n\n\"Yht\u00e4 paljon kuin h\u00e4n pit\u00e4\u00e4 minusta\", puuskahdin.\n\n\"\u00c4l\u00e4 nyt. Sinunkin on pakko tiet\u00e4\u00e4, ett\u00e4 h\u00e4n on aivan hulluna sinuun.\"\n\n\"Mit\u00e4? Ei todellakaan ole.\"\n\n\"Totta kai on! Sinulla ei siis ole aavistustakaan. H\u00e4n kohtelee sinua aivan eri tavalla kuin ket\u00e4\u00e4n muuta. H\u00e4nen mielest\u00e4\u00e4n sin\u00e4 olet mahtava.\"\n\n\"Siin\u00e4 tapauksessa h\u00e4nell\u00e4 on ollut v\u00e4h\u00e4n erikoinen tapa n\u00e4ytt\u00e4\u00e4 se.\" Daniel oli heti ensi kohtaamisestamme l\u00e4htien saanut minussa aikaan kylmi\u00e4 v\u00e4reit\u00e4. Sitten h\u00e4n oli hy\u00f6k\u00e4nnyt kimppuuni, ja vaikka se ei en\u00e4\u00e4 ollutkaan kauhukokemusteni top viidess\u00e4, en silti muistellut sit\u00e4 l\u00e4mm\u00f6ll\u00e4.\n\nTarjoilija tuli takaisin maksup\u00e4\u00e4tteen kanssa, ja Mikael suostui viimein puolittamaan laskun. Taivas oli himme\u00e4, kun k\u00e4velimme kohti ratikkapys\u00e4kki\u00e4. \"En halua menn\u00e4 viel\u00e4 kotiin\", sanoin.\n\n\"Ajattelin juuri samaa. Tied\u00e4n, mit\u00e4 me voimme tehd\u00e4.\"\n\n\"No?\"\n\n\"Menn\u00e4\u00e4n klubille.\"\n\nPys\u00e4hdyin siihen paikkaan. \"En usko, ett\u00e4 se on hyv\u00e4 ajatus.\"\n\nT\u00e4h\u00e4n asti min\u00e4 ja Mikael olimme pysytelleet omissa oloissamme. Se tarkoitti, ettemme olleet n\u00e4ytt\u00e4ytyneet yhdess\u00e4 susien edess\u00e4. Hitto, emme olleet kertoneet kenellek\u00e4\u00e4n muulle kuin Nikolle ja veljelleni, ett\u00e4 olimme yhdess\u00e4. Ja vaikka moni tytt\u00f6 olisi varmaan ollut n\u00e4rk\u00e4stynyt siit\u00e4, ettei h\u00e4nen poikayst\u00e4v\u00e4ns\u00e4 ollut heti julistamassa suhdetta jokaiselle vastaantulijalle, min\u00e4 olin ollut pelk\u00e4st\u00e4\u00e4n tyytyv\u00e4inen. T\u00e4ll\u00e4 tavoin olin saanut pit\u00e4\u00e4 Mikaelin itsell\u00e4ni edes hetken aikaa. Katoamistemppunsa j\u00e4lkeen Mikaelin asema susien keskuudessa ei ollut mitenk\u00e4\u00e4n kiveen kirjoitettu. H\u00e4nt\u00e4 alettaisiin vaatia tilille heti, kun tieto meist\u00e4 kahdesta alkaisi levit\u00e4, enk\u00e4 voinut edes pit\u00e4\u00e4 sit\u00e4 kohtuuttomana. Min\u00e4 olin riski. Viimeksi, kun olin k\u00e4ynyt klubilla, olin ollut v\u00e4h\u00e4ll\u00e4 saattaa useamman kuin yhden suden sairaalakuntoon, enk\u00e4 uskonut, ett\u00e4 siell\u00e4 oltaisiin erityisen iloisia j\u00e4lleenn\u00e4kemisest\u00e4.\n\nEnnen kaikkea en tiennyt, olinko itse valmis. T\u00e4h\u00e4n menness\u00e4 aika Mikaelin kanssa oli ollut puhdasta unelmaa. En halunnut avata ovea todellisuudelle, joka yritti p\u00e4\u00e4st\u00e4 sis\u00e4\u00e4n.\n\nMikaelin kulmat kohosivat. \"H\u00e4vett\u00e4\u00e4k\u00f6 sinua n\u00e4ytt\u00e4yty\u00e4 seurassani?\" h\u00e4n kysyi.\n\nAjatus oli niin absurdi, ett\u00e4 m\u00f6l\u00e4ytin seuraavan ajatukseni \u00e4\u00e4neen. \"Ei, mutta sinun kyll\u00e4 kannattaisi h\u00e4vet\u00e4 minua.\"\n\n\"Mit\u00e4?\" H\u00e4n n\u00e4ytti j\u00e4rkyttyneelt\u00e4, mik\u00e4 tuotti minulle pient\u00e4 tyydytyst\u00e4.\n\n\"Niin kuin varsin hyvin tied\u00e4t, klubilla minut tunnetaan nimell\u00e4 'sekop\u00e4\u00e4narttu'. Jotenkin en usko, ett\u00e4 sinun maineesi paranee viett\u00e4m\u00e4ll\u00e4 aikaa minun kanssani.\"\n\nMikael naurahti. \"Minulle on aivan sama, mit\u00e4 he ajattelevat. He eiv\u00e4t koskaan tule ymm\u00e4rt\u00e4m\u00e4\u00e4n, mit\u00e4 meid\u00e4n v\u00e4lill\u00e4mme on. Sill\u00e4 ihan rehellisesti sanottuna en usko hetke\u00e4k\u00e4\u00e4n, ett\u00e4 kenell\u00e4k\u00e4\u00e4n muulla voi olla t\u00e4llaista.\"\n\nJos minulla oli sit\u00e4 ennen ollut ep\u00e4ilyksi\u00e4, ne katosivat v\u00e4litt\u00f6m\u00e4sti. Totuus nimitt\u00e4in oli, ett\u00e4 menisin h\u00e4nen kanssaan minne tahansa.\n\nAstuin l\u00e4hemm\u00e4s ja suutelin h\u00e4nt\u00e4 keskell\u00e4 katua. Sitten tartuin h\u00e4nt\u00e4 k\u00e4dest\u00e4. \"Menn\u00e4\u00e4n, ennen kuin muutan mieleni.\"\n\n***\n\nKun her\u00e4sin, Mikael oli poissa.\n\nT\u00e4ll\u00e4 kertaa h\u00e4n ei istunut s\u00e4ngyll\u00e4. H\u00e4n ei ollut makuuhuoneessa laisinkaan, mik\u00e4 sai heti h\u00e4lytyskelloni soimaan.\n\nKertasin mieless\u00e4ni illan tapahtumia saadakseni jonkin vihjeen siit\u00e4, mik\u00e4 saattoi olla vialla. Iltamme oli ollut onnistunut. Enemm\u00e4n kuin onnistunut \u2013 se oli ollut mahtava. Kun olin lakannut v\u00e4litt\u00e4m\u00e4st\u00e4 susien tuijotuksista ja pel\u00e4styneist\u00e4 ilmeist\u00e4, olin alkanut todella nauttia siit\u00e4, ett\u00e4 min\u00e4 ja Mikael saatoimme viett\u00e4\u00e4 aikaa yhdess\u00e4 julkisesti. Olimme istuneet suurimman osan illasta minulle ennest\u00e4\u00e4n tuntemattomien susien p\u00f6yd\u00e4ss\u00e4 samppanjaa juoden ja nauraen. Mikaelia ei ollut tuntunut vaivaavan mik\u00e4\u00e4n.\n\nNousin yl\u00f6s ja kietaisin pussilakanan ymp\u00e4rilleni.\n\nOlohuone ja keitti\u00f6 olivat tyhji\u00e4, eik\u00e4 kylpyhuoneen oven alta n\u00e4kynyt valoa. Hetken luulin Mikaelin l\u00e4hteneen ulos. Sitten tunsin tuulenvireen pohkeissani ja keksin katsoa toiseen suuntaan, parvekkeelle.\n\nVerhot heiluivat hieman ilmavirrassa. Erotin Mikaelin siluetin niiden l\u00e4pi. H\u00e4nell\u00e4 oli yll\u00e4\u00e4n pelk\u00e4t bokserit.\n\nOvi oli raollaan, ja luikahdin \u00e4\u00e4nett\u00f6m\u00e4sti viile\u00e4\u00e4n ulkoilmaan. Puulattia tuntui kylm\u00e4lt\u00e4 paljaiden varpaideni alla. Mikaelin t\u00e4ytyi tiet\u00e4\u00e4, ett\u00e4 olin tullut, mutta h\u00e4n ei liikahtanutkaan. H\u00e4n piteli molemmin k\u00e4sin kiinni kaiteesta ja tuijotti kuudennesta kerroksesta avautuvaa maisemaa. Nouseva aurinko oli laveerannut osan taivaasta haalean vaaleanpunaiseksi. Lokit kiljahtelivat aina silloin t\u00e4ll\u00f6in, mutta muuten oli hiljaista.\n\n\"Hei, kaikki kunnossa?\" Tiesin, miten hy\u00f6dyt\u00f6n kysymys se oli. Siihen oli hyvin vaikea vastata muuten kuin my\u00f6nt\u00e4v\u00e4sti. Muistin hetki\u00e4 el\u00e4m\u00e4st\u00e4ni, jolloin joku oli kysynyt minulta saman ja my\u00f6nt\u00e4v\u00e4n vastaukseni j\u00e4lkeen olin n\u00e4hnyt, miten helpottunut kysyj\u00e4 oli ollut saatuaan velvollisuutensa hoidettua. Siit\u00e4 huolimatta kysyin niin nyt itse.\n\n\"On\", Mikael vastasi. Aivan niin kuin olin itse aina vastannut.\n\n\"Menn\u00e4\u00e4nk\u00f6 takaisin nukkumaan?\" kuiskasin.\n\n\"Mene sin\u00e4. Min\u00e4 tulen ihan kohta.\"\n\nH\u00e4nen \u00e4\u00e4nens\u00e4vyns\u00e4 oli niin outo, ett\u00e4 viimeist\u00e4\u00e4n silloin tiesin jonkin olevan pahasti vialla. Jalkapohjani liimautuivat entist\u00e4 tiukemmin lattiaan. Mikael ei ollut koskaan tuntunut yht\u00e4 tavoittamattomalta, ja \u00e4kki\u00e4 rintani kumisi tyhjyytt\u00e4\u00e4n. Tein siis ainoan, mit\u00e4 osasin p\u00e4\u00e4st\u00e4kseni l\u00e4hemm\u00e4s h\u00e4nt\u00e4.\n\nAstuin eteenp\u00e4in ja suutelin h\u00e4nen olkap\u00e4\u00e4t\u00e4\u00e4n. Ele oli pieni, mutta sen tarkoitus selv\u00e4.\n\n\"Oletko ihan varma, ett\u00e4 haluat minun j\u00e4tt\u00e4v\u00e4n sinut rauhaan?\" kysyin niin viettelev\u00e4sti kuin pystyin.\n\nKun h\u00e4n ei reagoinut mitenk\u00e4\u00e4n, kiedoin k\u00e4teni h\u00e4nen vy\u00f6t\u00e4r\u00f6ns\u00e4 ymp\u00e4rille.\n\nSe ei suinkaan saanut h\u00e4nt\u00e4 rentoutumaan. P\u00e4invastoin tunsin kuinka h\u00e4nen lihaksensa nytk\u00e4htiv\u00e4t niin kuin olisin antanut h\u00e4nelle s\u00e4hk\u00f6iskun. \"Ole kiltti ja mene takaisin sis\u00e4\u00e4n.\"\n\nMinulla ei ollut muuta mahdollisuutta kuin totella. Pakitin ovelle, ja kun minulla ei en\u00e4\u00e4 ollut tilaa minne per\u00e4\u00e4nty\u00e4, pujahdin takaisin sis\u00e4lle.\n\nMikael ei ollut koskaan torjunut minua. H\u00e4n ei my\u00f6sk\u00e4\u00e4n ollut koskaan sanonut minulle \"ole kiltti\" \u00e4\u00e4nell\u00e4, joka oli l\u00e4hell\u00e4 murinaa.\n\nTiesin, ettei se v\u00e4ltt\u00e4m\u00e4tt\u00e4 tarkoittanut mit\u00e4\u00e4n. Kaikilla oli joskus huonoja hetki\u00e4. Minun ei tarvinnut olla huolissani.\n\nMinulla oli kuitenkin aavistus, joka kerran mieleeni luikerreltuaan ei l\u00e4htenyt pois. Pian se ei en\u00e4\u00e4 olisi mahtunutkaan samaa reitti\u00e4 takaisin, sill\u00e4 ep\u00e4ilys alkoi kasvaa ja kasvaa, kunnes en en\u00e4\u00e4 pystynyt teeskentelem\u00e4\u00e4n, ettei sit\u00e4 ollut olemassa.\n\nMikaelin susi oli alkanut hylki\u00e4 minua.\n\n***\n\nAjattelin kuluneita viikkoja. Piknikkej\u00e4, s\u00e4ngyss\u00e4 vietettyj\u00e4 p\u00e4ivi\u00e4, yhteisi\u00e4 suihkuja.\n\nSeksi\u00e4. Mahtavaa, tajunnanr\u00e4j\u00e4ytt\u00e4v\u00e4\u00e4 seksi\u00e4.\n\nNyt istuessani aamiaisp\u00f6yd\u00e4ss\u00e4 tajusin, ett\u00e4 minun olisi ehk\u00e4 pit\u00e4nyt kiinnitt\u00e4\u00e4 enemm\u00e4n huomiota pieneen sis\u00e4iseen \u00e4\u00e4neeni, joka oli sanonut, ett\u00e4 me etenimme liian nopeasti. Olimme liian kiinni toisissamme. Mikaelin susi tiesi sen, vaikka me emme. Min\u00e4 olin k\u00e4yt\u00e4nn\u00f6ss\u00e4 muuttanut Mikaelin luokse sill\u00e4 sekunnilla, kun meist\u00e4 oli vihdoin tullut oikea pari. Se ei ollut normaalia. Me emme olleet normaaleja.\n\nEhk\u00e4 oli aika painaa jarrua.\n\nMikaelin oli t\u00e4ytynyt huomata vaitonaisuuteni, mutta h\u00e4n oli ilmeisesti p\u00e4\u00e4tt\u00e4nyt olla sanomatta mit\u00e4\u00e4n, ja viimeist\u00e4\u00e4n se sai minut pelk\u00e4\u00e4m\u00e4\u00e4n pahinta.\n\nMikael ei ollut niit\u00e4 ihmisi\u00e4, jotka v\u00e4ltteliv\u00e4t ongelmista puhumista. H\u00e4n oli kertonut meist\u00e4 is\u00e4lleen melkein heti, ja vasta paljon my\u00f6hemmin olin alkanut tajuta, miten paljon rohkeutta se oli vaatinut. Se, ettei Mikael ottanut t\u00e4m\u00e4naamuista puheeksi, saattoi tarkoittaa vain yht\u00e4 asiaa: tilanteelle ei ollut teht\u00e4viss\u00e4 mit\u00e4\u00e4n.\n\nEi niin, ett\u00e4 Mikael olisi halunnut j\u00e4tt\u00e4\u00e4 minut. Mutta jos h\u00e4nen sutensa ei ollut l\u00e4sn\u00e4, ei h\u00e4nk\u00e4\u00e4n ollut. Ei t\u00e4ysin. Kuinka kauan menisi, ennen kuin h\u00e4nen t\u00e4ytyisi oikeasti l\u00e4hte\u00e4? Kuinka kauan kannatti yritt\u00e4\u00e4 taistella vastaan?\n\nMikaeliin sattuisi paljon ennen kuin h\u00e4n edes alkaisi harkita asiaa. Minun teht\u00e4v\u00e4kseni j\u00e4isi p\u00e4\u00e4st\u00e4\u00e4 h\u00e4net menem\u00e4\u00e4n.\n\nOli kuin joku olisi hangannut syd\u00e4nt\u00e4ni raastinraudalla.\n\n\"Mit\u00e4 sin\u00e4 haluat tehd\u00e4 t\u00e4n\u00e4\u00e4n?\" Mikael kysyi.\n\n\"Luulen, ett\u00e4 menen k\u00e4ym\u00e4\u00e4n Viidennell\u00e4 linjalla\", mumisin.\n\n\"Mihin aikaan tulet takaisin? Voin k\u00e4yd\u00e4 taas hakemassa meille ruokaa.\"\n\n\"En tied\u00e4. Saattaa olla, ett\u00e4 j\u00e4\u00e4n y\u00f6ksi.\"\n\n\"Ai.\"\n\nYksi seuraus siit\u00e4, ett\u00e4 Mikael oli nykyisin niin avoin kanssani oli se, ett\u00e4 tiesin tasan tarkkaan, kun h\u00e4nt\u00e4 painoi jokin. Siit\u00e4 huolimatta h\u00e4n suuteli minua ovella ja toivotti mukavaa p\u00e4iv\u00e4\u00e4. H\u00e4n ei koskaan purkanut huonoa fiilist\u00e4\u00e4n minuun, mik\u00e4 teki h\u00e4nest\u00e4 yhden vahvimmista tuntemistani ihmisist\u00e4.\n\nVakuutin itselleni, ett\u00e4 t\u00e4m\u00e4 oli h\u00e4nen omaksi parhaakseen.\n\nKun saavuin Viidennelle linjalle, en todellakaan halunnut en\u00e4\u00e4 olla siell\u00e4. Kolistelin eteisess\u00e4 kovaan \u00e4\u00e4neen silt\u00e4 varalta, etteiv\u00e4t Mitja ja Caoimhe olleet huomanneet oven k\u00e4yv\u00e4n. En ainakaan kuullut mit\u00e4\u00e4n h\u00e4lytt\u00e4v\u00e4\u00e4, joten uskaltauduin peremm\u00e4lle.\n\nCaoimhe istui olohuoneessa lukemassa lehte\u00e4. \"Hei Raisa! Enp\u00e4 ole n\u00e4hnyt sinua pitk\u00e4\u00e4n aikaan.\"\n\n\"Moi Ciao. Onko Mitja kotona?\"\n\n\"H\u00e4n nukkuu. H\u00e4n oli viime y\u00f6n t\u00f6iss\u00e4.\"\n\nVilkaisin suljetun oven suuntaan. Minun olisi pit\u00e4nyt arvata, ettei veljeni ollut viel\u00e4 hereill\u00e4, sill\u00e4 h\u00e4n teki y\u00f6vuoroja. Halusin todella jutella h\u00e4nen kanssaan, mutten her\u00e4tt\u00e4isi h\u00e4nt\u00e4 vain sen takia.\n\n\"Eik\u00f6 sinulla ole t\u00f6it\u00e4?\" Tiesin kuulostavani ep\u00e4kohteliaalta, joten lis\u00e4sin: \"Yleens\u00e4h\u00e4n sin\u00e4 olet ollut aamup\u00e4iv\u00e4t poissa.\"\n\n\"Minun ty\u00f6sopimukseni p\u00e4\u00e4ttyi viime viikolla. Kaikki kukat ovat jo kaupoissa.\" Caoimhe oli ty\u00f6skennellyt kasvihuoneella.\n\n\"Ai.\" Minun teki mieli kysy\u00e4, miten h\u00e4n p\u00e4rj\u00e4si taloudellisesti, jos h\u00e4n ei k\u00e4ynyt en\u00e4\u00e4 t\u00f6iss\u00e4. Mutta jotenkin en uskonut, ett\u00e4 se oli h\u00e4nelle mik\u00e4\u00e4n ongelma. H\u00e4nh\u00e4n oli kuninkaallisesta keijuperheest\u00e4.\n\nJoka tapauksessa ei ollut kovin soveliasta kysy\u00e4. Valitsin siis toisen aiheen. \"Miten musakuviot?\"\n\n\"Hyvin. Itse asiassa erinomaisesti. Demo on valmis ja l\u00e4hetetty muutamalle eri taholle. Latasimme pari biisi\u00e4 my\u00f6s nettiin. Ja Mitjalle tietenkin riitt\u00e4isi keikkoja hamaan tulevaisuuteen, jos h\u00e4n vain suostuisi ottamaan niit\u00e4 vastaan.\"\n\nIstuin h\u00e4nen viereens\u00e4. \"Mit\u00e4 tarkoitat?\"\n\nCaoimhe laski lehden kahvip\u00f6yd\u00e4lle. \"H\u00e4n ei siis ole kertonut? Se Vitali-tyyppi, joka soitti kev\u00e4\u00e4ll\u00e4 klubilla, on suositellut Mitjaa. Ilmeisesti h\u00e4n tuntee paljon ihmisi\u00e4. T\u00e4rkeit\u00e4 ihmisi\u00e4.\"\n\nEn tiennyt, mit\u00e4 ajatella siit\u00e4, ett\u00e4 Vitali oli kehunut velje\u00e4ni. H\u00e4n oli ollut pelottavin vampyyri jonka olin koskaan tavannut, ja vaikka tapaamani vampyyrit saattoi laskea yhden k\u00e4den sormilla, en silti ep\u00e4illyt, etteik\u00f6 h\u00e4n olisi p\u00e4rj\u00e4nnyt laajemmassakin vertailussa. H\u00e4n ei ollut vaikuttanut tyypilt\u00e4, joka olisi tehnyt toisille palveluksia.\n\nKysyin sen t\u00e4rke\u00e4mm\u00e4n kysymyksen: \"Miksi ihmeess\u00e4 Mitja menisi kielt\u00e4ytym\u00e4\u00e4n dj-keikoista?\" Tiesin, ettei veljeni ollut yht\u00e4 kunnianhimoinen kuin tytt\u00f6yst\u00e4v\u00e4ns\u00e4, mutta h\u00e4n rakasti musiikkia enemm\u00e4n kuin kukaan. Oli sulaa tyhmyytt\u00e4 kielt\u00e4yty\u00e4 t\u00f6ist\u00e4, joita rakasti.\n\n\"H\u00e4nen nykyinen ty\u00f6sopimuksensa sanoo, ettei h\u00e4n saa tehd\u00e4 kilpailijoille t\u00f6it\u00e4 ilman lupaa. H\u00e4n ei ole saanut sellaista.\"\n\n\"Mutta Konsta\u2013\"\n\n\"Konsta ei ole Suomessa\", Caoimhe muistutti.\n\n\"Eli kuka ikin\u00e4 t\u00e4m\u00e4nhetkinen pomo onkin, h\u00e4n haluaa torpedoida Mitjan uran.\"\n\n\"Juuri niin min\u00e4 sanoin h\u00e4nelle. Ja h\u00e4n tajuaa sen kyll\u00e4 itsekin. Mutta h\u00e4n ep\u00e4r\u00f6i luopua vakituisesta ty\u00f6st\u00e4.\"\n\nSe kuulosti veljeni tapaiselta. \"H\u00e4n pelk\u00e4\u00e4, ettei keikkoja olisikaan tarpeeksi\", sanoin \u00e4\u00e4neen.\n\n\"Niin.\" Caoimhe nousi yl\u00f6s ja vilkaisi keitti\u00f6t\u00e4. \"Haluatko jotain juotavaa?\"\n\nOli omituista tulla kohdelluksi vieraana omassa kodissaan. \"Vett\u00e4 kiitos.\"\n\nCaoimhen puuhatessa keitti\u00f6ss\u00e4 katselin ymp\u00e4rilleni. Asunto oli paljon paremmassa kunnossa kuin se olisi ollut minun j\u00e4ljilt\u00e4ni, mutten ollut yll\u00e4ttynyt. Mitja oli siistimpi kuin min\u00e4.\n\nOlohuone oli kokenut pienen muodonmuutoksen. Kaikki taidetarvikkeeni olivat tiess\u00e4\u00e4n, ja tilalle oli ilmestynyt soittimia. Oli mahdotonta kuvitella, ett\u00e4 vasta v\u00e4h\u00e4n aikaa sitten Mitja oli omistanut vain yhden, l\u00e4hes soittokelvottoman kitaran. Nyt kitaroita oli ainakin kolme, muista soittimista puhumattakaan. Lattialla n\u00e4kyi vahvistimia, piuhoja ja muita esineit\u00e4, joista en uskaltanut menn\u00e4 sanomaan, mit\u00e4 ne olivat.\n\nN\u00e4ky sai minut hieman nostalgiseksi. T\u00e4m\u00e4 asunto oli ollut pitk\u00e4\u00e4n minun ja \u00e4itini koti. Sen j\u00e4lkeen se oli ollut minun ja Nikon koti, ja katsellessani sit\u00e4 nyt tajusin, ett\u00e4 siit\u00e4 oli vihdoin tullut Mitjan.\n\nCaoimhe palasi sohvalle kahden vesilasin kanssa. \"N\u00e4yt\u00e4t silt\u00e4 ett\u00e4 sinulla on jotain mielesi p\u00e4\u00e4ll\u00e4. Tied\u00e4n, ett\u00e4 tulit puhumaan veljesi kanssa, mutta ehk\u00e4 min\u00e4 voisin auttaa.\"\n\nPurin huultani. Caoimhe ei ollut susi. Mik\u00e4\u00e4n, mit\u00e4 kerroin h\u00e4nelle, ei voinut vahingoittaa Mikaelin asemaa.\n\nLopulta huokaisin. \"Kyse on Mikaelista ja minusta\", aloitin.\n\nCiao h\u00f6rpp\u00e4si lasistaan ja odotti.\n\n\"Luulen, ett\u00e4 Mikael j\u00e4tt\u00e4\u00e4 minut\", sanoin.\n\n\"Mik\u00e4 saa sinut ajattelemaan niin?\"\n\nKerroin h\u00e4nelle susista ja pariutumisesta. \"H\u00e4n ei ole l\u00e4hd\u00f6ss\u00e4 huomenna eik\u00e4 ylihuomenna, mutta jossain vaiheessa. Ja jos h\u00e4n ei j\u00e4t\u00e4 minua, niin sitten minun pit\u00e4\u00e4 p\u00e4\u00e4st\u00e4\u00e4 h\u00e4net menem\u00e4\u00e4n. Mikaelin susi tarvitsee suden kumppanikseen.\"\n\n\"En tied\u00e4 paljoakaan susista\", Caoimhe aloitti, \"mutta tied\u00e4n, ett\u00e4 Mikael tulee tekem\u00e4\u00e4n kaikkensa, jotta saisi olla sinun kanssasi.\"\n\n\"En vain tied\u00e4, riitt\u00e4\u00e4k\u00f6 se\", sanoin.\n\n\"Kuinka teill\u00e4 sujuu makuuhuoneessa?\"\n\nOlin v\u00e4h\u00e4ll\u00e4 l\u00e4ikytt\u00e4\u00e4 vett\u00e4 lasistani. En ollut viel\u00e4k\u00e4\u00e4n tottunut siihen, miten h\u00e4pe\u00e4m\u00e4t\u00f6n Caoimhe oli.\n\n\"Hyvin. Ehk\u00e4 v\u00e4h\u00e4n liiankin hyvin.\"\n\n\"Kerro toki lis\u00e4\u00e4. En ole koskaan kuullut suhteesta, jossa hyv\u00e4 seksi olisi ollut ongelma.\"\n\n\"Me aina keskitymme enemm\u00e4n minuun kuin h\u00e4neen. Olen tietysti yritt\u00e4nyt ottaa ohjat k\u00e4siin, mutta... Jotenkin siin\u00e4 vain aina k\u00e4y niin.\" Se kuulosti \u00e4\u00e4rimm\u00e4isen typer\u00e4lt\u00e4, joten kiiruhdin eteenp\u00e4in. \"Joka tapauksessa h\u00e4n saa minut unohtamaan, ett\u00e4 meill\u00e4 on ongelma.\" En sanonut, ett\u00e4 toisinaan tunsin huonoa omaatuntoa. Joskus Mikaelin kanssa oleminen tuntui loputtomalta kilpailulta siin\u00e4, kumpi pystyi antamaan toiselle enemm\u00e4n, ja h\u00e4n asetti riman aika pirun korkealle.\n\n\"Se ei voi olla kovin suuri ongelma, jos se on kerta niin helppo unohtaa\", Ciao sanoi. Ja hymyili. \"Sit\u00e4 paitsi meill\u00e4 kaikilla on omat ongelmamme. Sit\u00e4 kutsutaan el\u00e4m\u00e4ksi.\"\n\nH\u00e4n oli tietenkin oikeassa. Kenellek\u00e4\u00e4n meist\u00e4 ei ollut olemassa takuita.\n\n\"Sin\u00e4 et saa kertoa t\u00e4st\u00e4 kenellek\u00e4\u00e4n\", sanoin.\n\n\"\u00c4l\u00e4 ole huolissasi. Min\u00e4 olen hyv\u00e4 s\u00e4ilytt\u00e4m\u00e4\u00e4n salaisuuksia.\"\n\nEn j\u00e4\u00e4nyt odottamaan Mitjan her\u00e4\u00e4mist\u00e4. Palasin kotiin mietteli\u00e4\u00e4n\u00e4, mutta paljon rauhallisempana. Olin l\u00e4hett\u00e4nyt Mikaelille viestin ennen Kalliosta l\u00e4ht\u00f6\u00e4ni, ja kuinka ollakaan, p\u00f6yd\u00e4ll\u00e4 odotti valtava laatikollinen sushia.\n\nMikael seisoi p\u00f6yd\u00e4n vieress\u00e4. Hymyilin h\u00e4nelle. \"Kiitos\", sanoin.\n\n\"Se on vain ruokaa\", h\u00e4n sanoi.\n\nH\u00e4n on niin kiltti, ajattelin. H\u00e4n hemmottelee minut viel\u00e4 piloille.\n\nJuuri silloin Mikaelin puhelin soi.\n\n\"Moi\", h\u00e4n vastasi.\n\nH\u00e4n kuunteli hetken. \"Nyt on ihan hyv\u00e4 hetki.\"\n\nH\u00e4n kuunteli taas. Jossain vaiheessa h\u00e4n alkoi k\u00e4vell\u00e4 edestakaisin, mist\u00e4 tiesin, ettei h\u00e4n pit\u00e4nyt kuulemastaan.\n\nH\u00e4n pys\u00e4htyi. \"En usko, ett\u00e4 Konsta voi auttaa. H\u00e4n on maapallon toisella puolella.\"\n\nSe kiinnitti huomioni. Minulla ei ollut aavistustakaan, mist\u00e4 he puhuivat, mutta sudet eiv\u00e4t yleens\u00e4 maininneet Konstaa kovin mielell\u00e4\u00e4n.\n\nMikael huokaisi. \"Minulla ei ole h\u00e4nen nykyist\u00e4 numeroaan, mutta kysyn Raisalta. En kuitenkaan l\u00e4htisi laskemaan Konstan varaan.\"\n\nToisessa p\u00e4\u00e4ss\u00e4 sanottiin jotain. Muodostin huulillani kysymyksen: kuka? \"Juha\", Mikael mumisi.\n\n\"Ai olemmeko min\u00e4 ja Raisa nyky\u00e4\u00e4n puhev\u00e4leiss\u00e4? Niinkin voisi sanoa. Viime aikoina puhuminen on tosin j\u00e4\u00e4nyt v\u00e4hemm\u00e4lle, kun olemme keskittyneet toisenlaisiin asioihin.\"\n\n\"Mikael!\" sanoin. En voinut uskoa, ett\u00e4 h\u00e4n oli mennyt sanomaan jotain sellaista. Ja varsinkaan Juhalle.\n\nH\u00e4n virnisti \u00e4llistykselleni. \"Odota, laitan sinut kaiuttimeen niin ett\u00e4 Raisakin kuulee.\" Mikael n\u00e4pp\u00e4ili puhelintaan ja laittoi sen eteemme p\u00f6yd\u00e4lle. H\u00e4n istui viereeni ja laittoi k\u00e4tens\u00e4 niskaani. Rakastin sit\u00e4, ett\u00e4 h\u00e4n koski minua silloinkin, kun h\u00e4nen ajatuksensa olivat jossain muualla, niin kuin minun koskettamiseni olisi h\u00e4nelle kaikkein luontaisin tila.\n\n\"Moi Juha\", sanoin aavistuksen nolona.\n\n\"Moi\", h\u00e4n sanoi ja ysk\u00e4isi heti per\u00e4\u00e4n. \"Te sitten olette taas... yhdess\u00e4?\"\n\nAsian kanssa oli aivan turha kierrell\u00e4. \"Niin olemme\", sanoin. \"Mit\u00e4 sin\u00e4 oikein halusit kysy\u00e4?\"\n\nMikael ehti ensin. \"Hukkavaarassa on ongelma. Juha soitti kysy\u00e4kseen, tied\u00e4mmek\u00f6 ket\u00e4\u00e4n yliluonnollisia, jotka voisivat auttaa laumaa samaan tapaan kuin Konsta.\"\n\nMeni hetki, ennen kuin tajusin, mit\u00e4 h\u00e4n tarkoitti sill\u00e4. \"Ai te tarvitsette jonkun, joka pystyy sorkkimaan mieli\u00e4?\" Konstan teht\u00e4v\u00e4 oli ollut perin yksinkertainen: h\u00e4net oli palkattu suojelemaan kyl\u00e4n salaisuutta. Vahinkojen satuttua h\u00e4n oli saanut ihmisturistit unohtamaan kaiken sellaisen, mik\u00e4 olisi her\u00e4tt\u00e4nyt liikaa ep\u00e4ilyksi\u00e4 kyl\u00e4n asukkaiden todellisesta luonteesta.\n\n\"Joo, mielell\u00e4\u00e4n.\"\n\n\"Nyt on koko vuoden valoisin aika. Kaikki vampyyrit ovat tiess\u00e4\u00e4n.\"\n\n\"Tiedet\u00e4\u00e4n. Siksi meill\u00e4 onkin t\u00e4m\u00e4 ongelma. Me olemme p\u00e4rj\u00e4nneet yll\u00e4tt\u00e4v\u00e4n hyvin sen j\u00e4lkeen, kun Konsta l\u00e4hti. On ollut mahdotonta l\u00f6yt\u00e4\u00e4 toista vampyyria, joka suostuisi tulemaan meille t\u00f6ihin, mutta me onnistuimme selviytym\u00e4\u00e4n koko edellisen sesongin ilman kommelluksia. Ehk\u00e4 me k\u00e4vimme sen j\u00e4lkeen v\u00e4h\u00e4n laiskoiksi ja ylimielisiksi.\"\n\n\"Millainen ongelma teill\u00e4 oikein on siell\u00e4?\"\n\n\"L\u00e4hikunnassa on n\u00e4hty viime viikkoina susi er\u00e4\u00e4n talon pihassa. Kunnanjohtaja syytt\u00e4\u00e4 meit\u00e4, koska luulee meid\u00e4n houkutelleen sen alueelle ruokkimalla susia turistien toivossa. He haluavat lopettaa koko susiturismin. Pelk\u00e4\u00e4mme, ett\u00e4 t\u00e4m\u00e4 p\u00e4\u00e4tyy viel\u00e4 uutisiin.\"\n\n\"He ovat aloittamassa susijahdin\", Mikael sanoi.\n\n\"Mutta... ette kai te anna heid\u00e4n?\" kysyin.\n\n\"Meill\u00e4 ei ole hirve\u00e4sti keinoja k\u00e4yt\u00f6ss\u00e4. He ovat jo hakeneet kaatolupaa, ja vaikka sen saaminen on ep\u00e4todenn\u00e4k\u00f6ist\u00e4, aika monet heist\u00e4 ovat jo siin\u00e4 pisteess\u00e4, ettei heit\u00e4 kiinnosta, onko susien mets\u00e4st\u00e4minen laitonta vai ei.\"\n\n\"Tavalliset luodit eiv\u00e4t voi tappaa teit\u00e4.\" Heti kun olin sanonut sen, tunsin oloni huojentuneeksi. Hukkavaaralaiset eiv\u00e4t olleet hengenvaarassa.\n\n\"Eiv\u00e4t, mutta en sanoisi, ett\u00e4 ammutuksi tuleminen on koskaan kivaa. Suurempi ongelma on se, ett\u00e4 jahti jatkuu niin kauan, kunnes jotain saadaan kiinni.\"\n\n\"He haluavat n\u00e4hd\u00e4 ruumiin\", sanoin.\n\n\"Niin.\"\n\n\"\u00c4lk\u00e4\u00e4 p\u00e4\u00e4st\u00e4k\u00f6 heit\u00e4 lauman reviirille.\"\n\n\"Suomessa on valitettavasti jokamiehenoikeudet. Ei meill\u00e4 ole mit\u00e4\u00e4n keinoa pit\u00e4\u00e4 heit\u00e4 ulkona.\"\n\n\"Minun pit\u00e4\u00e4 mietti\u00e4 t\u00e4t\u00e4 hetki\", Mikael sanoi. \"Soitan sinulle takaisin heti, kun tied\u00e4n, mit\u00e4 teen.\"\n\n\"En tied\u00e4, miten sanoisin t\u00e4m\u00e4n, mutta t\u00e4\u00e4ll\u00e4 aletaan olla aika kyll\u00e4styneit\u00e4 siihen, ettei laumanjohtajaa n\u00e4y eik\u00e4 kuulu. Monen mielest\u00e4 olisi reilumpaa, jos sin\u00e4...\"\n\n\"Luopuisin laumanjohtajuudesta?\" Mikael arvasi.\n\n\"Niin.\"\n\n\"Kyll\u00e4 sen saa sanoa \u00e4\u00e4neen. Kiitos kun soitit. Niin kuin sanoin, mietin hetken ja soitan sinulle sitten.\" H\u00e4n pudotti puhelimensa sohvalle ja katsoi sitten minua. \"T\u00e4llaista ongelmaa ei ole ollut koskaan aikaisemmin. En tied\u00e4, miksi t\u00e4m\u00e4 tapahtuu nyt. Susia ei edes kannata yritt\u00e4\u00e4 mets\u00e4st\u00e4\u00e4 sulan maan aikaan.\"\n\n\"Min\u00e4 voin auttaa\", sanoin.\n\nMikael ei sanonut mit\u00e4\u00e4n.\n\n\"Min\u00e4 en ole vampyyri, eiv\u00e4tk\u00e4 daimonit varsinaisesti hallitse mieli\u00e4, mutta...\" En tiennyt, miten muotoilla asia. Treenit Mitjan kanssa olivat viel\u00e4 tuoreessa muistissa. Olin p\u00e4\u00e4ssyt kyvyiss\u00e4ni paljon pidemm\u00e4lle kuin olin koskaan kuvitellut, ja se oli hieman pelottanut minua. Nyt ensimm\u00e4ist\u00e4 kertaa tajusin, ett\u00e4 kykyj\u00e4ni voisi k\u00e4ytt\u00e4\u00e4 muiden hyv\u00e4ksi.\n\n\"Daimonit hallitsevat todellisuutta\", Mikael lopetti puolestani. \"Se on k\u00e4yt\u00e4nn\u00f6ss\u00e4 sama kuin jos te hallitsisitte mieli\u00e4.\" H\u00e4n katsoi pois, sitten taas minua. \"Tekisitk\u00f6 sin\u00e4 sen?\"\n\nRypistin otsaani. \"Miksi en tekisi? Tietenkin min\u00e4 autan jos vain pystyn.\"\n\n\"Sinua ei siis haittaisi menn\u00e4 k\u00e4ym\u00e4\u00e4n Hukkavaarassa? Min\u00e4 menen sinne joka tapauksessa, mutta\u2013\"\n\n\"Min\u00e4 haluan tulla mukaan\", sanoin. \"T\u00e4ll\u00e4 hetkell\u00e4 min\u00e4 taidan olla lauman paras ja ainoa vaihtoehto.\" Mikael n\u00e4ytti viel\u00e4kin aivan liian huolestuneelta, joten jatkoin: \"Meh\u00e4n jo puhuimme yhteisest\u00e4 lomamatkasta. Eih\u00e4n Hukkavaara ole mik\u00e4\u00e4n Bora Bora, mutta lasketaan se silti.\"\n\nMikaelin kulmat kohosivat. \"Bora Bora? Ai sinne sin\u00e4 olisit halunnut menn\u00e4? Pidet\u00e4\u00e4n mieless\u00e4.\"\n\nEn sanonut Mikaelille, mit\u00e4 ajattelin: Hukkavaaraan meneminen oli her\u00e4tt\u00e4nyt minussa toivon kipin\u00e4n. Mikaelin susi tarvitsi laumaa. Jos kaupungissa asuvat sudet eiv\u00e4t olleet tarpeeksi, ehk\u00e4 Hukkavaarassa oleilu auttaisi h\u00e4nen suttaan hyv\u00e4ksym\u00e4\u00e4n minut. Ehk\u00e4 Hukkavaaraan meneminen oli juuri sit\u00e4, mit\u00e4 h\u00e4n tarvitsi.\n\nEhk\u00e4 kaikki olisi taas hyvin.\nKOLMAS LUKU\n\nToivo on ihmeellinen asia. Seuraavana p\u00e4iv\u00e4n\u00e4 olin nimitt\u00e4in loistavalla tuulella. Minua ei harmittanut edes se, ettei Mitja ollut kertonut minulle saamistaan keikkatarjouksista. H\u00e4nen tilanteensa antoi minulle vain lis\u00e4pontta pyyt\u00e4\u00e4 h\u00e4net mukaan Hukkavaaraan.\n\n\"Minulla on sinulle keikka\", sanoin heti kuin Mitja vastasi puhelimeen. \"Ei musiikki-, vaan daimonkeikka. Hyv\u00e4 palkka, mahtavat edut. Tied\u00e4n, ett\u00e4 olet odottanut tekosyyt\u00e4 ottaa loparit. Onneksi olkoon: t\u00e4ss\u00e4 se nyt on.\"\n\nToisessa p\u00e4\u00e4ss\u00e4 oli hetken hiljaista. \"Mist\u00e4 tiesit?\" h\u00e4n kysyi lopulta.\n\n\"Ciao kertoi. Voin antaa anteeksi sen, ett\u00e4 salasit t\u00e4m\u00e4n minulta, mutta vain, jos tulet mukaani Hukkavaaraan.\"\n\n\"Mit\u00e4 jos en tienaa freelancerina tarpeeksi? Minun pit\u00e4\u00e4 maksaa laskut.\"\n\n\"Sen takia sinulla on kuninkaallinen tytt\u00f6yst\u00e4v\u00e4.\"\n\nH\u00e4n murahti, ja minun oli pakko purra huulta, etten olisi nauranut \u00e4\u00e4neen.\n\nKerroin h\u00e4nelle kaiken, mit\u00e4 tiesin Hukkavaaran tilanteesta. H\u00e4n ei sanonut paljoa, mutta lupautui tulemaan mukaan, kunhan vain Caoimhe suostuisi l\u00e4htem\u00e4\u00e4n h\u00e4nen kanssaan. Meid\u00e4n kummankaan ei tarvinnut sanoa \u00e4\u00e4neen, ett\u00e4 olimme Mikaelille velkaa. H\u00e4n lopetti puhelun lyhyeen, sill\u00e4 h\u00e4nen piti soittaa pomolleen ja ilmoittaa irtisanoutuvansa. H\u00e4n kuulosti melkein silt\u00e4 kuin olisi odottanut sit\u00e4 innolla.\n\nKun olin saanut hoidettua puhelun pois alta, oli jo aika laittautua valmiiksi. Minulla oli nimitt\u00e4in tanssitreffit Tanelin kanssa.\n\nTapasimme keskustassa ja k\u00e4velimme yhdess\u00e4 tanssikoululle, jossa tunti pidettiin. Vasta silloin minulle selvisi, ett\u00e4 Taneli oli oikeasti kilpaillut kilpatanssissa.\n\n\"\u00c4l\u00e4 huoli yht\u00e4\u00e4n, olen ihan varma, ett\u00e4 sin\u00e4 p\u00e4rj\u00e4\u00e4t hienosti\", h\u00e4n sanoi heti n\u00e4hty\u00e4\u00e4n ilmeeni.\n\n\"En min\u00e4 itsest\u00e4ni ole huolissani, vaan sinun varpaistasi\", nauroin.\n\nOlin ollut hieman ep\u00e4luuloinen, kun Taneli oli kertonut, mille tunnille h\u00e4n halusi menn\u00e4. En ollut koskaan tanssinut jivea, mutta k\u00e4vi ilmi, ett\u00e4 rakastin sit\u00e4. Se oli leikkis\u00e4\u00e4 ja hyv\u00e4ntuulista, ja vaikka tempo oli v\u00e4lill\u00e4 aika hurja, Taneli oli niin hyv\u00e4 viej\u00e4, ett\u00e4 pysyin kuin pysyinkin mukana suurimman osan ajasta. Huomasin hymyilev\u00e4ni koko ajan.\n\nTanssi vaati kuitenkin hyv\u00e4\u00e4 kuntoa, ja tunnin j\u00e4lkeen olin hiest\u00e4 m\u00e4rk\u00e4. Olin iloinen, ett\u00e4 olin tajunnut ottaa vaihtovaatteet mukaan.\n\n\"Meid\u00e4n t\u00e4ytyy ehdottomasti menn\u00e4 uudestaan!\" sanoin kun tulin ulos pukuhuoneesta ja n\u00e4in Tanelin jo odottavan minua. \"Siis vain, jos kest\u00e4t tanssia minunkaltaiseni aloittelijan kanssa.\"\n\nH\u00e4nen hymyns\u00e4 oli leve\u00e4 ja aito. \"Se olisi mahtavaa. Miten olisi jo ensi viikolla?\"\n\nOlin jo vastata my\u00f6nt\u00e4v\u00e4sti, kun muistin, etten ehk\u00e4 olisi kaupungissa. \"Min\u00e4 ja Mikael l\u00e4hdemme huomenna Hukkavaaraan, enk\u00e4 tied\u00e4, milloin tulemme takaisin.\"\n\nL\u00e4hdimme k\u00e4velem\u00e4\u00e4n kohti Rautatientoria. Taneli ty\u00f6nsi k\u00e4det shortsiensa taskuihin. \"Minusta on niin kivaa, ett\u00e4 sin\u00e4 ja Niko olette saaneet v\u00e4linne kuntoon. Asia vaivasi h\u00e4nt\u00e4 niin paljon.\"\n\nEn tiennyt, mit\u00e4 vastata, joten toivoin, ett\u00e4 hymy riitti. Pelk\u00e4sin, ett\u00e4 h\u00e4n alkaisi kysell\u00e4 tarkempia yksityiskohtia.\n\n\"Haluaisin kysy\u00e4 sinulta jotain.\"\n\nNo niin, ajattelin. \"Kysy pois.\"\n\n\"Niko ei koskaan k\u00e4y kotipaikkakunnallaan\", h\u00e4n aloitti. \"Sin\u00e4 taidat olla h\u00e4nen nykyisist\u00e4 kavereistaan ainoa, joka on tuntenut h\u00e4net ennen kuin h\u00e4n muutti t\u00e4nne.\"\n\n\"Nikon perhesuhteet ovat v\u00e4h\u00e4n ongelmalliset\", sanoin. Tajusin, ettei Taneli ollut varsinaisesti kysynyt mit\u00e4\u00e4n, mutta ajattelin ett\u00e4 minun olisi silti hyv\u00e4 selitt\u00e4\u00e4. \"H\u00e4nell\u00e4 ei ole \u00e4iti\u00e4 ja is\u00e4 on juoppo. H\u00e4nen iso\u00e4itins\u00e4 kasvatti h\u00e4net\"\n\n\"H\u00e4n on puhunut v\u00e4h\u00e4n mummistaan\", Taneli sanoi. \"H\u00e4n on huolissaan t\u00e4m\u00e4n terveydest\u00e4.\"\n\nSe oli minulle t\u00e4ysin uutta tietoa. \"Onko Annille tapahtunut jotain?\"\n\n\"Ei tiet\u00e4\u00e4kseni, mutta h\u00e4nell\u00e4 on jo ik\u00e4\u00e4. Niko sanoi, ett\u00e4 sen on alkanut huomata h\u00e4nest\u00e4.\"\n\nAnni py\u00f6ritti kauppaansa yksin ja oli muutenkin huolehtinut aina kaikesta. H\u00e4n oli yksi ensimm\u00e4isist\u00e4 ihmisist\u00e4, jotka olin tavannut Hukkavaarassa, ja olin pit\u00e4nyt h\u00e4nest\u00e4 ensihetkist\u00e4 l\u00e4htien. Hukkavaara oli mit\u00e4 suurimmassa m\u00e4\u00e4rin h\u00e4nen kotinsa, mutta en ollut varma, oliko h\u00e4nell\u00e4 siell\u00e4 ket\u00e4\u00e4n, joka tarjoaisi h\u00e4nelle apuaan. En tiennyt, kuinka hyvin sudet ymm\u00e4rsiv\u00e4t, millaista oli olla ihminen. Millaista oli olla fyysisesti heikko.\n\n\"Nikon pit\u00e4isi menn\u00e4 tapaamaan h\u00e4nt\u00e4 ennen kuin te l\u00e4hdette.\"\n\n\"Olen sanonut h\u00e4nelle samaa. Sanoin, ett\u00e4 l\u00e4htisin mielell\u00e4ni mukaan. H\u00e4n ei pit\u00e4nyt ajatuksesta.\" H\u00e4n piti tauon. Aavistelin, ett\u00e4 oli vihdoin tullut kysymyksen aika. \"Onko Niko kertonut kenellek\u00e4\u00e4n... Tai siis, luulen tajuavani, miksi h\u00e4n ei ehk\u00e4 halua vied\u00e4 minua tapaamaan ket\u00e4\u00e4n sukulaisiaan.\"\n\nAloin ymm\u00e4rt\u00e4\u00e4, mist\u00e4 kiikasti. \"Kaikki Hukkavaarassa kyll\u00e4 tiet\u00e4v\u00e4t, ett\u00e4 h\u00e4n on homo\", sanoin suoraan. \"H\u00e4n ei h\u00e4pe\u00e4 sinua, vaan p\u00e4invastoin. Jos h\u00e4n jotain h\u00e4pe\u00e4\u00e4 niin Hukkavaaraa.\"\n\n\"On tosi kiva tavata joku, joka tuntee h\u00e4net.\" H\u00e4nen helpotuksensa oli niin ilmiselv\u00e4, ett\u00e4 minun teki mieli irvist\u00e4\u00e4. Pahuksen Niko, ajattelin. Tiesin kyll\u00e4, miksi h\u00e4n oli p\u00e4\u00e4tt\u00e4nyt erottaa Tanelin ja menneisyytens\u00e4 Hukkavaarassa piikkilanka-aidalla, mutta samaan aikaan en ollut lainkaan varma, oliko h\u00e4n tullut pohtineeksi sen seurauksia. Niko ei aina osannut ajatella, milt\u00e4 h\u00e4nen tekonsa tuntuivat muista.\n\nEnnen kuin edes tajusin, mit\u00e4 olin tekem\u00e4ss\u00e4, sanoin: \"Jos teit\u00e4 kiinnostaa tulla huomenna mukaan, p\u00e4\u00e4sette meid\u00e4n kyydiss\u00e4mme. Olen varma, ett\u00e4 me saamme teid\u00e4t majoitettua jonnekin. Teid\u00e4n pit\u00e4\u00e4 ehdottomasti k\u00e4yd\u00e4 Kainuussa, ennen kuin l\u00e4hdette Ranskaan.\"\n\n\"Se olisi tosi kivaa.\" H\u00e4n hymyili korvasta korvaan, enk\u00e4 usko, ett\u00e4 olin koskaan n\u00e4hnyt mit\u00e4\u00e4n yht\u00e4 suloista.\n\n***\n\nEn ollut ehtinyt olla kotona tuntiakaan, kun puhelimeni soi.\n\n\"OLETKO SIN\u00c4 HULLU?!\"\n\nIrvistin ja siirsin puhelimen kauemmas korvastani.\n\n\"Luultavasti\", kerroin Nikolle. \"En voi kuitenkaan antaa varmaa vastausta ennen kuin tarkennat hieman.\"\n\n\"Sin\u00e4 kutsuit Tanelin Hukkavaaraan.\"\n\nAi se, ajattelin. \"Niin taisin tehd\u00e4. Koska et ole esitellyt h\u00e4nt\u00e4 poikayst\u00e4v\u00e4n\u00e4si kenellek\u00e4\u00e4n muulle kuin minulle, h\u00e4n ep\u00e4ili, ett\u00e4 olet yh\u00e4 kaapissa. Sanoin, ett\u00e4 kaikki kotopuolessasi kyll\u00e4 tiet\u00e4v\u00e4t. H\u00e4n oli my\u00f6s huolissaan siit\u00e4, mit\u00e4 Annille tapahtuu sitten, kun te muutatte niin kauas. Tulin ehdottaneeksi, ett\u00e4 k\u00e4visitte h\u00e4nen luonaan edes kerran ennen kuin l\u00e4hdette.\"\n\nKuulin kun ovi avattiin ja hetken p\u00e4\u00e4st\u00e4 Mikael ilmestyi olohuoneeseen. Hymyilin ja ny\u00f6kk\u00e4sin h\u00e4nelle. Osoitin puhelintani, ja h\u00e4n ny\u00f6kk\u00e4si takaisin. Katsoin, kun h\u00e4n meni keitti\u00f6\u00f6n ja avasi j\u00e4\u00e4kaapin.\n\nNiko oli hiljaa toisessa p\u00e4\u00e4ss\u00e4. \"H\u00e4n sanoi sinullekin Annista?\" h\u00e4n kysyi lopulta.\n\n\"Se poika oikeasti v\u00e4litt\u00e4\u00e4.\"\n\nHetkeen ei kuulunut mit\u00e4\u00e4n, sitten hyvin vaimeaa muminaa. Olin erottavinani sanat \"voi helvetin helvetti\".\n\n\"Min\u00e4 en ole koskaan varsinaisesti kertonut mummille.\"\n\nMeni hetki, ennen kuin tajusin, mihin h\u00e4n oikein viittasi. Niin kuin olin sanonut Tanelille, kaikki Hukkavaarassa tiesiv\u00e4t Nikosta. Ja vaikka Anni ei ollutkaan susi, h\u00e4n oli ter\u00e4v\u00e4 nainen. \"Tietenkin h\u00e4n tiet\u00e4\u00e4\", sanoin. \"Ei sinun tarvitse sanoa mit\u00e4\u00e4n.\"\n\n\"Ent\u00e4 miten ajattelit ratkaista sen pikkuongelman, ett\u00e4 minun suteni sekoaa muiden susien seurassa?\"\n\n\"Sinun ei tarvitse hengailla susien kanssa. Me j\u00e4rjest\u00e4mme teille majoituksen m\u00f6kkiin tai jonnekin, miss\u00e4 saatte olla rauhassa.\" Samalla kun lupasin niin Nikolle, katsoin Mikaelia anovasti. H\u00e4n ny\u00f6kk\u00e4si.\n\n\"Jos sin\u00e4 aiot olla Tanelin kanssa, sinun on pakko alkaa jakaa edes joitain osia el\u00e4m\u00e4st\u00e4si. Sanoisin, ett\u00e4 t\u00e4m\u00e4 on juuri sopivan harmiton alku.\"\n\n\"Katsotaan, mit\u00e4 t\u00e4st\u00e4 tulee\", Niko mumisi. \"Min\u00e4 kun satun olemaan mik\u00e4 olen.\"\n\n\"My\u00f6nn\u00e4t siis olevasi m\u00e4nttip\u00e4\u00e4?\"\n\n\"En! Tarkoitan sit\u00e4, ett\u00e4 olen ihmissusi. Enk\u00e4 tied\u00e4 ket\u00e4\u00e4n, jota maailman pahuus olisi turmellut yht\u00e4 v\u00e4h\u00e4n kuin Tanelia. Se on aika outoa kun ottaa huomioon, ett\u00e4 h\u00e4n on juhlinut enemm\u00e4n kuin min\u00e4. Se poika on helvetin enkeli.\"\n\n\"Helvetin enkeli?\"\n\n\"Tied\u00e4t, mit\u00e4 tarkoitan.\"\n\n\"H\u00e4n on kuin p\u00f6rr\u00f6inen koiranpentu\", tunnustin. \"Tai pupujussi. Tai karitsa siin\u00e4 vessapaperimainoksessa. Joka tapauksessa sy\u00f6t\u00e4v\u00e4n s\u00f6\u00f6tti.\" Pidin tauon. \"En tiennyt, ett\u00e4 sin\u00e4 pid\u00e4t s\u00f6\u00f6tist\u00e4.\"\n\n\"Mist\u00e4 sin\u00e4 sitten kuvittelit minun pit\u00e4v\u00e4n?\"\n\n\"En tied\u00e4. Jostain synkemm\u00e4st\u00e4.\"\n\n\"Miksi min\u00e4 pit\u00e4isin synk\u00e4st\u00e4? Min\u00e4 viet\u00e4n kaiken aikani itseni kanssa, siin\u00e4 on jo ihan riitt\u00e4v\u00e4sti kest\u00e4mist\u00e4.\"\n\nSiin\u00e4 oli, kumma kyll\u00e4, j\u00e4rke\u00e4. \"Aika mahtavaa, ett\u00e4 Taneli on valmis muuttamaan Ranskaan sinun takiasi.\"\n\n\"No, h\u00e4n on sent\u00e4\u00e4n k\u00e4ynyt Ranskalaisen koulun. H\u00e4n kirjoitti kaikki hakemukseni.\"\n\n\"H\u00e4n n\u00e4ytt\u00e4\u00e4 n\u00e4tilt\u00e4, tanssii ja puhuu ranskaa? Sinun on pidett\u00e4v\u00e4 h\u00e4nest\u00e4 kiinni jo pelk\u00e4st\u00e4\u00e4n siksi, ett\u00e4 voin lainata h\u00e4nt\u00e4.\"\n\nNiko naurahti kuivasti.\n\n\"Miten sin\u00e4 olet selviytynyt t\u00e4ysikuista?\" Olin utelias. He olivat asuneet yhdess\u00e4 jo monta kuukautta, joten Nikon oli t\u00e4ytynyt keksi\u00e4 ongelmaan jokin ratkaisu.\n\n\"Jos Taneli ei ole ollut t\u00f6iss\u00e4, olen onnistunut kehitt\u00e4m\u00e4\u00e4n jonkun riidan, jonka takia olen voinut l\u00e4hte\u00e4 ovet paukkuen ja vastata puhelimeen vasta seuraavana p\u00e4iv\u00e4n\u00e4.\"\n\n\"Tosi ilke\u00e4\u00e4.\"\n\n\"Ehk\u00e4, mutta korvaan sen h\u00e4nelle aina j\u00e4lkik\u00e4teen.\"\n\n\"Miten?\"\n\n\"Taidan j\u00e4tt\u00e4\u00e4 t\u00e4m\u00e4n kohdan tyhj\u00e4ksi. Eik\u00f6h\u00e4n sinun mielikuvituksesi riit\u00e4 keksim\u00e4\u00e4n vastauskentt\u00e4\u00e4n jotain sopivaa.\"\n\nSe h\u00e4mmensi minua siin\u00e4 m\u00e4\u00e4rin, ett\u00e4 vaihdoin nopeasti aihetta. \"L\u00e4htisitte nyt Hukkavaaraan meid\u00e4n kanssamme. Mitja ja Ciaokin ovat tulossa. Voin luvata, ettei kukaan tule heitt\u00e4m\u00e4\u00e4n mit\u00e4\u00e4n tyhm\u00e4\u00e4 homol\u00e4pp\u00e4\u00e4. Paitsi tietenkin jos heit\u00e4tte sit\u00e4 itse.\"\n\n\"Me heit\u00e4mme pelk\u00e4st\u00e4\u00e4n hyv\u00e4\u00e4 homol\u00e4pp\u00e4\u00e4. Ett\u00e4s tied\u00e4t.\"\n\n\"Joo joo\", sanoin. \"Mitja ja Ciao ovat menossa junalla, joten te mahdutte meid\u00e4n kyytiin.\"\n\n\"Kuvitteletko, ett\u00e4 me kest\u00e4mme seitsem\u00e4n tuntia teid\u00e4n kyydiss\u00e4nne? Ei tule onnistumaan.\"\n\n\"Mik\u00e4 meid\u00e4n kyydiss\u00e4mme on muka vikana?\"\n\n\"Te. Olette niin \u00e4ll\u00f6tt\u00e4vi\u00e4. Hymyilettekin koko ajan. Tied\u00e4n, ett\u00e4 te yrititte hillit\u00e4 itsenne silloin illallisella, mutta hei ihan oikeasti. Ei kukaan halua katsoa sit\u00e4 vierest\u00e4.\"\n\nYritin muistella k\u00e4yt\u00f6st\u00e4mme silloin, kun Niko ja Taneli olivat olleet k\u00e4ym\u00e4ss\u00e4, enk\u00e4 muistanut mit\u00e4\u00e4n erityisen \u00e4ll\u00f6tt\u00e4v\u00e4\u00e4. Yritin yleens\u00e4 pit\u00e4\u00e4 jonkun rajan siin\u00e4, mit\u00e4 teimme julkisesti, ja vaikka Mikael toisinaan saikin minut lipsumaan, en todellakaan uskonut meid\u00e4n olevan pahimmasta p\u00e4\u00e4st\u00e4. Siit\u00e4 huolimatta kuvittelin ymm\u00e4rt\u00e4v\u00e4ni, mit\u00e4 h\u00e4n tarkoitti.\n\nNiko jatkoi: \"Te olette juuri niit\u00e4 pareja, jotka hankkivat samanlaiset tuulipuvut ja alkavat sauvak\u00e4vell\u00e4 yhdess\u00e4.\"\n\nKuulin Mikaelin yskiv\u00e4n keitti\u00f6ss\u00e4. \"T\u00e4ss\u00e4 kohtaa on varmaan hyv\u00e4 ilmoittaa, ett\u00e4 Mikael on t\u00e4\u00e4ll\u00e4\", sanoin samalla kun yritin olla nauramatta. Kuvittelinko minut ja Mikaelin samanlaisissa tuulipuvuissa? En. Mutta ei se toisaalta ollut kamalin mahdollinen asia, jonka saatoin kuvitella.\n\n***\n\nSeuraavana aamuna min\u00e4 ja Mikael l\u00e4hdimme ajamaan kohti Hukkavaaraa. Tai Mikael ajoi, min\u00e4 istuin pelk\u00e4\u00e4j\u00e4npaikalla ja s\u00f6in mansikoita istuin taakse vedettyn\u00e4, toinen jalka kojelautaan nojaten. Nilkassani kimmelsi kultainen ketju, jonka olin saanut Mikaelilta. Stereoissa soi jokin radiokanava, josta tuli enemm\u00e4n mainoksia kuin musiikkia. Tiesin, ett\u00e4 ulkona oli kuuma \u2013 tiell\u00e4 n\u00e4kyi jopa kangastuksia \u2013 mutta ilmastoinnin ansiosta l\u00e4mp\u00f6tila autossa oli pelk\u00e4st\u00e4\u00e4n miellytt\u00e4v\u00e4.\n\n\"Minun on ollut tarkoitus kysy\u00e4 sinulta jotakin\", Mikael aloitti. \"T\u00e4m\u00e4 on todella ennenaikaista, mutta ajattelin, ett\u00e4 sinun on hyv\u00e4 mietti\u00e4 rauhassa.\"\n\n\"No?\" Vihasin sit\u00e4, ett\u00e4 pelk\u00e4sin v\u00e4litt\u00f6m\u00e4sti pahinta.\n\nH\u00e4n vilkaisi minua. \"Kun valmistun l\u00e4\u00e4kiksest\u00e4, haluaisin tutkia sinua. Sinun perim\u00e4\u00e4si. Se auttaisi sek\u00e4 susia ett\u00e4 sinua itse\u00e4si. Meh\u00e4n emme viel\u00e4k\u00e4\u00e4n tied\u00e4, miksi sinun \u00e4itisi ei koskaan muuttunut.\"\n\n\"Min\u00e4 en tosiaan taida tiet\u00e4\u00e4 kauheasti itsest\u00e4ni\", sanoin. \"Luulin, ett\u00e4 kunhan vain p\u00e4\u00e4sen jyv\u00e4lle kyvyist\u00e4ni, kaikki olisi selv\u00e4\u00e4. Mutta ei se mennyt ihan niin.\" Huokaisin. En halunnut sanoa enemp\u00e4\u00e4. Jos alkaisin ajatella asiaa, ahdistuisin vain suotta. Oli parempi el\u00e4\u00e4 hetkess\u00e4.\n\nMikael laittoi k\u00e4tens\u00e4 polvelleni.\n\nNielaisin nopeasti. \"Miten luulet hukkavaaralaisten suhtautuvan meid\u00e4n tuloomme?\"\n\nEn tiennyt, oliko Juha kertonut kenellek\u00e4\u00e4n minun ja Mikaelin olevan yhdess\u00e4. Joka tapauksessa he saisivat sen selville hyvin pian. Laumalle ei ollut ihan sama, kenen kanssa heid\u00e4n johtajansa seurusteli. Kuvittelin meid\u00e4n suhteemme olevan kuitenkin paljon pienempi asia kuin se, ett\u00e4 Mikael oli j\u00e4tt\u00e4nyt lauman oman onnensa nojaan liki vuodeksi. En tuntenut lauman historiaa kovin tarkasti, mutta olin silti aika varma, ettei sellaista ollut tapahtunut koskaan aikaisemmin.\n\n\"En oleta, ett\u00e4 kaikki hoituisi ihan mukisematta. Min\u00e4 j\u00e4tin heid\u00e4t pulaan.\" H\u00e4n vilkaisi minua. \"En ihmettelisi, jos minut haastettaisiin sill\u00e4 sekunnilla, kun saavun paikalle.\"\n\nSe ei ollut k\u00e4ynyt mieless\u00e4ni. Mikaelia oli aina l\u00e4hes palvottu Hukkavaarassa. En tiennyt ket\u00e4\u00e4n, joka olisi tullut edes l\u00e4helle h\u00e4nen valta-asemaansa. \"Kuka sinut muka haastaisi?\" kysyin.\n\n\"En tied\u00e4. Ehk\u00e4 Iivo.\" Iivo Salakka oli entisen Lupinin laumanjohtaja ja Mikaelin eno. Mikael oli tavannut h\u00e4net ensimm\u00e4isen kerran vasta viime kes\u00e4n\u00e4, kun olimme selvitt\u00e4neet Danielin katoamista. J\u00e4ljet olivat johtaneet meid\u00e4t Ven\u00e4j\u00e4n puolelle Karjalaan lupinilaisten luo, ja he olivat est\u00e4neet Hukkavaaraa suistumasta Martin k\u00e4siin. Suurin osa lupinilaisista oli muuttanut Hukkavaaraan.\n\nEn voinut uskoa, ett\u00e4 Iivo k\u00e4\u00e4ntyisi omaa sukulaistaan vastaan \u2013 varsinkaan, kun h\u00e4nen onnistumisensa oli hyvin ep\u00e4todenn\u00e4k\u00f6ist\u00e4.\n\n\"Hyv\u00e4ksyisiv\u00e4tk\u00f6 hukkavaaralaiset entisen lupinilaisen johtajaksi?\" kysyin.\n\n\"Eiv\u00e4t kyll\u00e4 hyv\u00e4ksyisi\", Mikael my\u00f6nsi. \"En kuitenkaan tied\u00e4, mit\u00e4 tapahtuu, kun lauma kuulee, ett\u00e4 aion j\u00e4\u00e4d\u00e4 Helsinkiin.\"\n\n\"Sin\u00e4 menet l\u00e4\u00e4kikseen\", sanoin tiukasti. \"He joutuvat hyv\u00e4ksym\u00e4\u00e4n sen. Ei kai lauma v\u00e4ltt\u00e4m\u00e4tt\u00e4 tarvitse samanlaista johtajaa kuin edelt\u00e4j\u00e4si. Yksinvaltiaan sijaan sin\u00e4 voisin olla v\u00e4h\u00e4n niin kuin presidentti.\"\n\n\"Presidenttikin tarvitsee enemmist\u00f6n luottamuksen puolelleen. Pahoin pelk\u00e4\u00e4n, ettei poissaoleva johtaja her\u00e4t\u00e4 suurta luottamusta.\"\n\n\"Vaikka asut lukukaudet Helsingiss\u00e4, j\u00e4\u00e4 sinulle silti monta kuukautta vuodessa, jolloin voit olla Hukkavaarassa. Kes\u00e4loma tietenkin. Ja talvella sin\u00e4 tietysti haluat k\u00e4yd\u00e4 siell\u00e4 niin usein kuin mahdollista, jotta p\u00e4\u00e4set laskettelemaan.\"\n\nMikael vilkaisi minua hassulla tavalla.\n\n\"Mit\u00e4? Mit\u00e4 tuo katse tarkoittaa?\"\n\n\"Ei mit\u00e4\u00e4n. Tai sit\u00e4 vain, ett\u00e4 sin\u00e4 olet ensimm\u00e4inen, joka on oikeasti minun puolellani. Jota kiinnostaa, mit\u00e4 min\u00e4 haluan ja mist\u00e4 min\u00e4 pid\u00e4n. Se on aika... kivaa.\"\n\nLaskin jalkani kojelaudan p\u00e4\u00e4lt\u00e4 ja nousin polvieni varaan. Kurottauduin h\u00e4nt\u00e4 kohti, kunnes huuleni melkein koskivat h\u00e4nen korvaansa. \"Arvaa mit\u00e4?\"\n\n\"Niin?\"\n\n\"Luulen, ett\u00e4 se on sinusta v\u00e4h\u00e4n enemm\u00e4n kuin vain kivaa\", sanoin. N\u00e4ykk\u00e4sin h\u00e4nen korvanlehte\u00e4\u00e4n ja vet\u00e4ydyin takaisin omalle penkilleni.\n\n\"Miksi min\u00e4 ajan t\u00e4t\u00e4 autoa?\" h\u00e4n kysyi maailmankaikkeudelta.\n\nNauroin. \"Meid\u00e4n olisi ehk\u00e4 sittenkin pit\u00e4nyt suostutella Niko ja Taneli tulemaan samassa kyydiss\u00e4. Sinun ei olisi tarvinnut ajaa koko matkaa.\"\n\n\"Ei haittaa. Min\u00e4 pid\u00e4n ajamisesta.\" H\u00e4n tarttui k\u00e4teeni ja vilkaisi minua. \"Jokin sanoo, ett\u00e4 sin\u00e4kin pit\u00e4isit. Pystyn hyvin kuvittelemaan sinut punaisen Ferrarin rattiin.\"\n\n\"Luultavasti\", my\u00f6nsin.\n\nH\u00e4n risti sormensa minun sormieni lomaan. Ennen kuin huomasinkaan, h\u00e4n oli laittanut k\u00e4teni vaihdekepin p\u00e4\u00e4lle ja k\u00e4ytt\u00e4nyt sit\u00e4 vaihteen vaihtamiseen.\n\nVirnistin, ja Mikael vastasi siihen omallaan. Niko oli oikeassa: me hymyilimme nyky\u00e4\u00e4n paljon.\n\n\"Tied\u00e4tk\u00f6, pid\u00e4n t\u00e4st\u00e4 lomasta Hukkavaarassa koko ajan enemm\u00e4n ja enemm\u00e4n\", sanoin.\n\n\"Pahoin pelk\u00e4\u00e4n, ettei ensi viikosta tule ihan niin rentouttavaa kuin toivoisit\", Mikael vastasi. \"Muistat kai, ett\u00e4 meill\u00e4 on perill\u00e4 jotain tekemist\u00e4.\"\n\n\"Ehk\u00e4, mutta t\u00e4ll\u00e4 kertaa me olemme tekemisiss\u00e4 pelkkien ihmisten kanssa\", sanoin. \"Kuinka vaikeaa heid\u00e4n suostuttelemisensa voi muka olla?\"\n\n***\n\nTulimme perille my\u00f6hemmin kuin meid\u00e4n oli ollut tarkoitus. Olimme pys\u00e4htyneet vain kaksi kertaa, mutta kes\u00e4n m\u00f6kkiliikenne ei ollut antanut Mikaelille mahdollisuutta p\u00e4\u00e4stell\u00e4 aivan niin lujaa kuin h\u00e4n olisi halunnut.\n\nMikael ei vaivautunut ajamaan autoaan talliin asti, vaan j\u00e4tti sen talon edustalla olevalle k\u00e4\u00e4nt\u00f6paikalle.\n\nNousin yl\u00f6s autosta ja katseeni siirtyi heti m\u00e4en p\u00e4\u00e4lle Sarrien taloon. Se oli suuri, valkoinen ja kuin suoraan jostain arkkitehtuurikilpailusta. Taivas oli tummunut juuri sen verran, ett\u00e4 sis\u00e4lt\u00e4 loistavat valot saivat koko talon hohtamaan. Korkeat puut ymp\u00e4r\u00f6iv\u00e4t pihaa kuin vartiokaarti.\n\nJossain vaiheessa kes\u00e4\u00e4 olin unohtanut, ett\u00e4 t\u00e4m\u00e4 oli Mikaelin oikea koti. En ollut koskaan ollut erityisen tervetullut t\u00e4nne.\n\nTuijotin portaita, jotka kohosivat kalliota pitkin yl\u00f6s etuovelle. Jostain syyst\u00e4 Viidennen linjan hissitt\u00f6myys tuntui paljon pienemm\u00e4lt\u00e4 pahalta kuin Sarreille kapuaminen.\n\nMikaelin t\u00e4ytyi arvata ajatukseni, sill\u00e4 h\u00e4n kysyi: \"Haluatko, ett\u00e4 kannan sinut yl\u00f6s?\"\n\n\"En\", sanoin nauraen. H\u00e4n ei voinut olla tosissaan.\n\nMikaelin silm\u00e4t v\u00e4lk\u00e4htiv\u00e4t.\n\n\"Ei, en todellakaan halua!\" S\u00e4nt\u00e4sin juoksuun. Harpoin portaita yl\u00f6s kaksi kerrallaan niin lujaa kuin pystyin. Tunsin pikemmin kuin kuulin Mikaelin takanani, l\u00e4hes tarttumiset\u00e4isyydell\u00e4. Kiristin tahtiani, vaikka kaikki lihakseni kiljuivat. Ehdin ovelle juuri, kun Mikael oli saamassa minut kiinni. K\u00e4\u00e4nnyin ymp\u00e4ri t\u00e4ynn\u00e4 voitonriemua \u2013 harmi vain, ett\u00e4 juuri silloin joku p\u00e4\u00e4tti avata oven.\n\nOvi l\u00f6i minua suoraan olkap\u00e4\u00e4h\u00e4n. Ennen kuin huomasinkaan, kaaduin sivuttain. Mikaelin k\u00e4si tarttui ranteeseeni, mutta se oli my\u00f6h\u00e4ist\u00e4 \u2013 olin jo menossa lonkkaluu edell\u00e4 kohti betonia.\n\nK\u00e4det tarttuivat minua vy\u00f6t\u00e4r\u00f6lt\u00e4. Katsoin yl\u00f6s \u2013 ja sitten viel\u00e4 v\u00e4h\u00e4n ylemm\u00e4s. Juha Karvonen oli pitk\u00e4. Todella pitk\u00e4. Jossain vaiheessa h\u00e4nest\u00e4 oli my\u00f6s tullut todella lihaksikas: olin tukea ottaessani tarttunut h\u00e4nt\u00e4 hauiksista, ja sormeni tuskin ylsiv\u00e4t edes puoliksi niiden ymp\u00e4rille.\n\n\"Juha! Vau, sin\u00e4 olet kasvanut isoksi!\" Rutistin h\u00e4nt\u00e4 lujaa. En ollut n\u00e4hnyt poikaa pitk\u00e4\u00e4n aikaan. Olin alkanut pit\u00e4\u00e4 h\u00e4nest\u00e4 todella paljon viime kes\u00e4n\u00e4. H\u00e4n oli ollut yksi harvoista, jotka olivat olleet Sarrien puolella ja tehneet kaikkensa, jotta lauma palautuisi heid\u00e4n alaisuuteensa. H\u00e4nell\u00e4 oli toki ollut my\u00f6s henkil\u00f6kohtainen syy: h\u00e4nen iso\u00e4itins\u00e4 oli kuollut Martin lyhyen valtakauden seurauksena.\n\nJuha p\u00e4\u00e4sti \u00e4kki\u00e4 irti ja astui ei vain yhden, vaan per\u00e4ti kolme askelta taaksep\u00e4in. H\u00e4n n\u00e4ytti pikkupojalta, joka oli juuri j\u00e4\u00e4nyt kiinni jostain paljon pahemmasta kuin karkkipurkilla k\u00e4ynnist\u00e4. Olisin pit\u00e4nyt reaktiota koomisena, ellen olisi juuri silloin vilkaissut taakseni.\n\nMikaelin leuka oli laskenut aavistuksen, ja n\u00e4in h\u00e4nen purevan leukojaan yhteen. H\u00e4nen silmiss\u00e4\u00e4n v\u00e4lkkyi kulta, ja vaikka se saattoikin johtua aikaisemmasta pelleilyst\u00e4mme, pelk\u00e4sin syyn\u00e4 olevan jonkin aivan muun. Kaikki leikkisyys oli tipotiess\u00e4\u00e4n.\n\nK\u00e4\u00e4nnyin hitaasti takaisin kohti Juhaa, joka piti katseensa visusti muualla.\n\nT\u00e4m\u00e4k\u00f6 oli esimakua siit\u00e4, millaista oli olla laumanjohtajan tytt\u00f6yst\u00e4v\u00e4?\n\nSeisoimme hetken aikaa eteisess\u00e4 eritt\u00e4in kiusaantuneissa tunnelmissa.\n\nSitten \u00e4\u00e4ni kysyi Juhan takaa: \"Etk\u00f6 aio halata minua?\" ja j\u00e4\u00e4 murtui. Esiin astui pieni vaalea tytt\u00f6, jonka ter\u00e4v\u00e4 katse sai Juhan vaikuttamaan lauhkealta lampaalta.\n\nEn yll\u00e4ttynyt n\u00e4hdess\u00e4ni Jenni Niemen. Vaikka h\u00e4n opiskeli nyky\u00e4\u00e4n Oulussa, h\u00e4n tuli Hukkavaaraan aina, kun t\u00e4\u00e4ll\u00e4 tapahtui jotakin. Mikael ja Jenni olivat olleet pitk\u00e4\u00e4n yhdess\u00e4, mutta eronneet lopullisesti sen j\u00e4lkeen, kun min\u00e4 olin tullut mukaan kuvioihin. He olivat onnistuneet pysym\u00e4\u00e4n yst\u00e4vin\u00e4, ja jotenkin my\u00f6s min\u00e4 olin v\u00e4hitellen p\u00e4\u00e4ssyt yli mustasukkaisuudestani.\n\nHalasin h\u00e4nt\u00e4 l\u00e4mpim\u00e4sti. Mikael sen sijaan ei tuhlannut aikaa kohteliaisuuksien vaihtoon. H\u00e4n l\u00e4hti k\u00e4velem\u00e4\u00e4n kohti olohuonetta, eik\u00e4 meill\u00e4 ollut muuta vaihtoehtoa kuin seurata per\u00e4ss\u00e4. Olin melkein kiitollinen, etten voinut n\u00e4hd\u00e4 h\u00e4nen ilmett\u00e4\u00e4n. Toivoin h\u00e4nen onnistuvan saamaan itsens\u00e4 hallintaan. H\u00e4nen mielialansa roikkui yll\u00e4mme, eik\u00e4 kukaan uskaltanut sanoa mit\u00e4\u00e4n.\n\n\"Mik\u00e4 tilanne?\" h\u00e4n kysyi lopulta.\n\n\"Ei hyv\u00e4\", Juha tunnusti.\n\n\"Min\u00e4 suomennan: tilanne on niin perseest\u00e4 kuin voi olla\", Jenni sanoi.\n\nOlin aika varma, ettei Jennin kotikasvatus antanut h\u00e4nelle lupaa kiroilla aivan v\u00e4h\u00e4st\u00e4.\n\n\"Kertokaa\", sanoin. Istuin alas, ja anovan katseeni my\u00f6t\u00e4 Mikael seurasi esimerkki\u00e4ni.\n\n\"Naapurikunnassa on n\u00e4hty susi\", Juha aloitti. \"Se k\u00e4vi aivan talon pihassa keskell\u00e4 p\u00e4iv\u00e4\u00e4. Joltain on my\u00f6s kadonnut kotiel\u00e4in, ja suden ep\u00e4ill\u00e4\u00e4n napanneen sen. Asukkaat ovat peloissaan ja tietenkin syytt\u00e4v\u00e4t meit\u00e4 kaikesta. He tulivat metsiimme etsim\u00e4\u00e4n todisteita kasvaneesta susikannasta. Meid\u00e4n oli pakko ajaa heid\u00e4t pois, etteiv\u00e4t he olisivat l\u00f6yt\u00e4neet mit\u00e4\u00e4n.\"\n\n\"Eik\u00f6 kukaan tied\u00e4, miten n\u00e4in on p\u00e4\u00e4ssyt k\u00e4ym\u00e4\u00e4n?\" kysyin. \"Kukaan ei siis ole tunnustanut k\u00e4yneens\u00e4 kielletyill\u00e4 alueilla suden muodossa?\" Jopa min\u00e4 tiesin, ettei asutusalueiden l\u00e4helle saanut menn\u00e4 sutena. Jos susi oli n\u00e4hty jonkun pihassa, se tarkoitti, ett\u00e4 joku oli mokannut ja pahasti.\n\nJuha pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Me kyll\u00e4 tiet\u00e4isimme, jos joku meist\u00e4 olisi syyllinen. Kuulustelimme kaikkia.\" H\u00e4n uskaltautui vilkaisemaan Mikaelia. \"Kaikkia on kielletty toistaiseksi vaihtamasta muotoa, mutta t\u00e4ysikuun aikaan se ei tietenk\u00e4\u00e4n auta.\"\n\nKukaan ei sanonut sit\u00e4 \u00e4\u00e4neen, mutta me kaikki tiesimme, ett\u00e4 ongelma olisi ratkaistava ennen seuraavaa t\u00e4ysikuuta.\n\n\"Onko Hukkavaarassa koskaan oikeita susia?\" kysyin. En ollut koskaan tullut pohtineeksi asiaa.\n\n\"Hyvin harvoin\", Juha vastasi. \"Ne haistavat meid\u00e4t ja yritt\u00e4v\u00e4t pysytell\u00e4 kaukana. Ne tiet\u00e4v\u00e4t, ett\u00e4 t\u00e4m\u00e4 reviiri on varattu.\"\n\nPeriaatteessa naapurikuntalaiset olivat siis saattaneet n\u00e4hd\u00e4 ihan oikean suden. Sill\u00e4 ei kuitenkaan tainnut olla suurta v\u00e4li\u00e4 hukkavaaralaisten kannalta. T\u00e4rkeint\u00e4 oli saada ihmiset rauhoittumaan.\n\n\"Jos he saavat kaatoluvan, voivatko he lain mukaan tulla t\u00e4nne noin vain ammuskelemaan?\" kysyin. Katsoin Mikaelia, mutta h\u00e4n ei n\u00e4ytt\u00e4nyt kuulleen koko kysymyst\u00e4. Yritimme kaikki teeskennell\u00e4, ettemme huomanneet h\u00e4nen kireytt\u00e4\u00e4n, mutta se oli k\u00e4ym\u00e4ss\u00e4 koko ajan vaikeammaksi.\n\nJuha vastasi h\u00e4nen puolestaan. \"He tarvitsevat maanomistajan luvan.\"\n\n\"Silloinhan teill\u00e4 ei pit\u00e4isi olla mit\u00e4\u00e4n ongelmaa. Niin kauan, kuin pysytte Hukkavaarassa, he eiv\u00e4t voi teille mit\u00e4\u00e4n.\"\n\n\"Uskotko todella, ett\u00e4 he muka noudattaisivat mets\u00e4styslakeja?\" Jenni kysyi. \"Me emme voi valvoa kaiken aikaa mahdollisten mets\u00e4st\u00e4jien varalta. Ja mit\u00e4 me teemme sitten, kun n\u00e4emme mets\u00e4st\u00e4jien tulevan? Soitamme poliisille? Jos poliisi katsoo, ett\u00e4 tilanne on niin vaarallinen kuin naapurikuntalaiset sanovat, he voivat kaataa suden itse.\"\n\n\"Lauma ei my\u00f6sk\u00e4\u00e4n omista kovin valtavaa maa-aluetta\", Mikael sanoi. Olin helpottunut, ett\u00e4 h\u00e4n n\u00e4ytti viimein saaneen hermonsa kuriin. \"Me k\u00e4ymme usein sen rajojen ulkopuolella. Me tulemme hulluiksi, jos joudumme pysyttelem\u00e4\u00e4n koko ajan t\u00e4\u00e4ll\u00e4. Varsinkin nyt, kun meit\u00e4 on paljon enemm\u00e4n.\"\n\nLupinin ja Hukkavaaran laumojen yhdist\u00e4minen oli helpottanut susia ikuisesti vaivaavaa lis\u00e4\u00e4ntymisongelmaa, mutta samalla se oli tuonut mukanaan monta uutta pulmaa.\n\n\"Miksi nyt?\" Mikael ihmetteli \u00e4\u00e4neen. \"T\u00e4ss\u00e4 ajoituksessa ei ole mit\u00e4\u00e4n j\u00e4rke\u00e4. En muista, ett\u00e4 t\u00e4\u00e4ll\u00e4p\u00e4in oltaisiin koskaan onnistuttu kaatamaan sutta kes\u00e4aikaan. En ymm\u00e4rr\u00e4, mit\u00e4 he kuvittelevat saavuttavansa.\"\n\n\"Koko juttu taitaakin olla l\u00e4hinn\u00e4 parin tyypin p\u00e4\u00e4h\u00e4npinttym\u00e4\", Juha my\u00f6nsi. \"Valitettavasti parikin tyyppi\u00e4 voi saada paljon meteli\u00e4 aikaan. Jos heid\u00e4n joukossaan on yksikin, joka tiet\u00e4\u00e4, mit\u00e4 etsi\u00e4, he kyll\u00e4 l\u00f6yt\u00e4v\u00e4t Hukkavaarasta todisteita susista. Paljon todisteita.\"\n\nMikael huokaisi. \"Ilmoita kaikille, ett\u00e4 kokoonnumme huomenna kent\u00e4ll\u00e4 kolmelta. Minun pit\u00e4\u00e4 mietti\u00e4 asiaa.\"\n\nJuha ja Jenni nousivat seisomaan. Vasta silloin tajusin ihmetell\u00e4, ketk\u00e4 eiv\u00e4t olleet paikalla.\n\n\"Miss\u00e4 Mitja ja Ciao ovat? P\u00e4\u00e4siv\u00e4tk\u00f6 he ongelmitta perille?\" Mikael oli luvannut huolehtia heid\u00e4n majoittumisestaan.\n\n\"Hotellilla\", Jenni sanoi. \"Niin kuin Niko ja Tanelikin. He asuvat sviitiss\u00e4.\"\n\nKohotin kulmiani. \"Miksi sviitiss\u00e4? Kyll\u00e4 heille olisi kelvannut v\u00e4hempikin.\" Olin melkoisen varma, ett\u00e4 majoittuminen meni Mikaelin piikkiin.\n\n\"Ne olivat ainoat vapaat huoneet. Me olemme t\u00e4yteen buukattuja\", Juha selitti.\n\nMikaelkin oli ihmeiss\u00e4\u00e4n. \"Keskell\u00e4 kes\u00e4\u00e4? En muista, ett\u00e4 sit\u00e4 olisi tapahtunut koskaan aikaisemmin.\"\n\nJuha yritti peitell\u00e4 ylpeytt\u00e4\u00e4n, mutta ep\u00e4onnistui. \"Me olemme onnistuneet houkuttelemaan yrityksi\u00e4 j\u00e4rjest\u00e4m\u00e4\u00e4n t\u00e4\u00e4ll\u00e4 virkistystilaisuuksia. T\u00e4ll\u00e4 hetkell\u00e4 Hukkahovissa on meneill\u00e4\u00e4n konferenssi.\"\n\nEnsimm\u00e4ist\u00e4 kertaa Mikael katsoi Juhaa ilman hitustakaan vihamielisyytt\u00e4. \"Sin\u00e4 olet pit\u00e4nyt firmoista paljon parempaa huolta kuin mihin min\u00e4 olisin koskaan pystynyt.\"\n\n\"Meill\u00e4 on ollut hyv\u00e4 tuuri\", Juha mumisi. \"Mutta juuri konferenssin takia olisi hyv\u00e4, jos me selvitt\u00e4isimme t\u00e4m\u00e4n sotkun mahdollisimman nopeasti.\"\n\nJuha ja Jenni olivat jo eteisess\u00e4, kun Juha k\u00e4\u00e4ntyi ymp\u00e4ri. \"Ai niin. Unohdin yhden asian.\"\n\nMin\u00e4 ja Mikael odotimme. Juha oli v\u00e4litt\u00f6m\u00e4sti vaivaantunut. \"Te ette pid\u00e4 siit\u00e4\", h\u00e4n aloitti.\n\n\"Kakista nyt vain ulos\", sanoin.\n\n\"Joni on pyyt\u00e4nyt lupaa palata laumaan etuajassa.\"\n\nJoni oli yksi niist\u00e4 susista, jotka olivat hy\u00f6k\u00e4nneet kimppuuni viime kes\u00e4n\u00e4. H\u00e4n oli ollut mukana vehkeilem\u00e4ss\u00e4 Mikaelia ja h\u00e4nen is\u00e4\u00e4ns\u00e4 vastaan, ja vaikka h\u00e4n ei ollutkaan kuulunut p\u00e4\u00e4tekij\u00f6ihin, ei h\u00e4nt\u00e4 oltu voitu j\u00e4tt\u00e4\u00e4 rangaistuksetta. H\u00e4net oli karkotettu laumasta m\u00e4\u00e4r\u00e4ajaksi. En tiennyt, kuinka pitk\u00e4 h\u00e4nen tuomionsa oli, mutta ilmeisesti se oli h\u00e4nen omasta mielest\u00e4\u00e4n liian pitk\u00e4.\n\nSaatoin yh\u00e4 el\u00e4v\u00e4sti tuntea, kuinka minut oli paiskattu maahan, kuinka kynnet olivat painuneet selk\u00e4\u00e4ni, kuinka olin huutanut \u2013\n\nKiedoin k\u00e4sivarret ymp\u00e4rilleni.\n\nMikaelin leukaper\u00e4t kiristyiv\u00e4t. \"Voit ilmoittaa h\u00e4nelle, ett\u00e4 p\u00e4\u00e4t\u00e4mme asiasta huomenna.\"\n\nJuha ny\u00f6kk\u00e4si. H\u00e4n ja Jenni l\u00e4htiv\u00e4t, ja min\u00e4 ja Mikael j\u00e4imme kahdestaan olohuoneeseen.\n\nMikael oli yh\u00e4 ajatuksissaan. Katselin ymp\u00e4rilleni. Olin ollut Sarrien luona muutaman kerran, mutta en koskaan kahdestaan Mikaelin kanssa. Toisten l\u00e4hdetty\u00e4 aloin v\u00e4hitellen k\u00e4sitt\u00e4\u00e4, ett\u00e4 minun oli tarkoitus olla t\u00e4\u00e4ll\u00e4 y\u00f6t\u00e4.\n\n\"Haluatko talokierroksen?\" Mikael kysyi \u00e4kki\u00e4.\n\nNy\u00f6kk\u00e4sin.\n\nAloitimme keitti\u00f6st\u00e4 ja ruokasalista. Mikael teki omia tarkastuksiaan samalla kun yritti esitell\u00e4 minulle miss\u00e4 oli mit\u00e4kin. J\u00e4\u00e4kaappi oli t\u00e4ynn\u00e4. Keitti\u00f6nkaapit olivat t\u00e4ynn\u00e4. Kaikesta p\u00e4\u00e4tellen talo oli siivottu huolellisesti ennen tuloamme.\n\n\"Onko joku asunut t\u00e4\u00e4ll\u00e4 sill\u00e4 aikaa, kun sin\u00e4 olet ollut poissa?\"\n\n\"On. Iivo ja h\u00e4nen perheens\u00e4\", Mikael vastasi.\n\nKatsoin h\u00e4nt\u00e4 h\u00e4mmentyneen\u00e4. Mikael kohautti olkiaan. \"He tarvitsivat jonkun paikan siksi aikaa, kun rakennuttivat omaa taloaan.\"\n\nJa siit\u00e4 huolimatta Mikael ep\u00e4ili Iivon haluavan syrj\u00e4ytt\u00e4\u00e4 h\u00e4net. No, ainakin h\u00e4n ja h\u00e4nen vaimonsa olivat j\u00e4tt\u00e4neet talon t\u00e4ysin moitteettomaan kuntoon.\n\nKierroksen edetess\u00e4 k\u00e4vi selv\u00e4ksi, etten ollut koskaan aikaisemmin tajunnut, kuinka iso talo oli. Alakerrassa oli kokonainen puoli, jossa en ollut koskaan k\u00e4ynyt. Kaikki huoneet olivat todella siistej\u00e4 ja huolellisesti mietittyj\u00e4, ja yll\u00e4tyin, miten monessa tilassa oli taidetta. Itse asiassa olin melko varma, ett\u00e4 \u00e4itini olisi kadehtinut Sarrien taidekokoelmaa. Siit\u00e4 huolimatta huomasin vierastavani taloa. Yritin ymm\u00e4rt\u00e4\u00e4, miksi. Mikaelin T\u00f6\u00f6l\u00f6n asunto oli muuttunut kodiksi hyvin huomaamattomasti. Jotenkin en uskonut, ett\u00e4 t\u00e4m\u00e4n talon kanssa k\u00e4visi samoin.\n\nTulimme tilaan, jota Mikael kutsui takkahuoneeksi. Pys\u00e4hdyin jo ovella.\n\nMikael tietenkin huomasi ep\u00e4r\u00f6intini. \"Mik\u00e4 on vialla?\"\n\nTuijotin sein\u00e4ll\u00e4 olevaa grafiikkavedosta. \"Tuo on minun \u00e4itini tekem\u00e4\", sanoin lopulta. Otin askeleen l\u00e4hemm\u00e4s, ja sitten viel\u00e4 toisen. Totta tosiaan, oikeassa kulmassa oli \u00e4itini vedosmerkinn\u00e4t ja signeeraus.\n\nMikael katsoi taulua, ja sitten taas minua. \"En tiennyt sit\u00e4. Tuo taulu on ollut meill\u00e4 jo aika kauan.\" Kun en sanonut mit\u00e4\u00e4n, h\u00e4n jatkoi: \"Se tietenkin kuuluu sinulle, jos haluat. Jos se h\u00e4iritsee sinua, voimme ottaa sen pois sein\u00e4lt\u00e4 vaikka heti.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Daniel maksoi siit\u00e4 k\u00e4yv\u00e4n hinnan.\" Uskoin tiet\u00e4v\u00e4ni, miten teos oli p\u00e4\u00e4tynyt t\u00e4nne. Muistin p\u00e4iv\u00e4n, jolloin \u00e4iti oli tullut kotiin ja kertonut, ett\u00e4 galleria oli myynyt yhden h\u00e4nen t\u00f6ist\u00e4\u00e4n. H\u00e4n oli ollut niin innoissaan. Olimme juhlistaneet tapahtunutta j\u00e4tskiannoksilla.\n\n\"Sit\u00e4 paitsi taulu sopii t\u00e4nne. Olisi s\u00e4\u00e4li siirt\u00e4\u00e4 sit\u00e4 mihink\u00e4\u00e4n.\" Sisutus oli selv\u00e4sti suunniteltu sen ymp\u00e4rille. Onnistuin hymyilem\u00e4\u00e4n Mikaelille.\n\n\"En tietenk\u00e4\u00e4n ymm\u00e4rr\u00e4 taiteesta mit\u00e4\u00e4n, mutta olen aina pit\u00e4nyt tuosta taulusta\", h\u00e4n sanoi.\n\nMikael oli kasvanut katsellen \u00e4itini ty\u00f6t\u00e4. Se oli mukava ajatus.\n\nH\u00e4n puristi k\u00e4tt\u00e4ni. Sanaakaan sanomatta h\u00e4n johdatti minut yl\u00e4kertaan. Odotin Mikaelin suuntaavan omaan huoneeseensa, mutta h\u00e4n jatkoikin k\u00e4yt\u00e4v\u00e4\u00e4 pitkin aina vanhempiensa makuuhuoneeseen asti. S\u00e4nkyyn oli pedattu muhkeaakin muhkeammat peitto ja tyynyt. Vuodevaatteet olivat vitivalkoiset.\n\nAnnoin k\u00e4teni liukua viile\u00e4\u00e4 kangasta pitkin. Materiaalista ei voinut erehty\u00e4. \"Kuka nukkuu silkkilakanoissa?\" ihmettelin \u00e4\u00e4neen.\n\n\"Me\", Mikael sanoi.\n\nEn keksinyt siihen mit\u00e4\u00e4n sanottavaa. Salatakseni h\u00e4mmennykseni avasin vaatekaapin oven. Muistin \u00e4kki\u00e4 edellisen kerran, kun olin avannut saman oven: silloin kaapissa oli ollut Martin vaimon Ritvan vaatteita. Joku oli poistanut ne, mutta Mikaelin vanhempien vaatteet olivat yh\u00e4 tallella.\n\n\"Jos niiss\u00e4 on mit\u00e4\u00e4n, mit\u00e4 sin\u00e4 tai Niko haluatte, ottakaa vapaasti. Loput pit\u00e4isi varmaan pakata ja lahjoittaa jonnekin.\"\n\nH\u00e4n kuulosti apealta. Ensimm\u00e4ist\u00e4 kertaa tajusin, mit\u00e4 yhteist\u00e4 minulla ja Mikaelilla oli: meill\u00e4 kummallakaan ei ollut en\u00e4\u00e4 vanhempia.\n\nKatsoin h\u00e4nt\u00e4. \"Sinun vanhempasi ovat yh\u00e4 elossa. He eiv\u00e4t vain ole t\u00e4\u00e4ll\u00e4.\" Olin tarkoittanut sanani lohdullisiksi, mutta pian tajusin, ett\u00e4 joku olisi voinut tulkita ne katkeruudeksi. Mik\u00e4 ei ollenkaan pit\u00e4nyt paikkansa.\n\nMikaelin ilme pehmeni. \"Tied\u00e4n.\"\n\nOlin ajatellut, ett\u00e4 Mikael olisi ollut valmis sy\u00f6m\u00e4\u00e4n nopean iltapalan ja menem\u00e4\u00e4n sitten nukkumaan, mutta sain huomata olevani t\u00e4ysin v\u00e4\u00e4r\u00e4ss\u00e4. Seurasin h\u00e4nt\u00e4 kun h\u00e4n meni toimistoon. Olin ollut siell\u00e4 aikaisemminkin \u2013 itse asiassa mik\u00e4\u00e4n toinen huone talossa ei ollut tullut minulle yht\u00e4 tutuksi kuin t\u00e4m\u00e4. Muistin yh\u00e4 kassakaapin yhdistelm\u00e4n ulkoa. Kaapissa olleet paperit olivat auttaneet meid\u00e4t Lupinin lauman j\u00e4ljille.\n\nT\u00e4ll\u00e4 kertaa p\u00f6yd\u00e4lle oli j\u00e4tetty isoja pinoja papereita ja kirjoja.\n\n\"Mit\u00e4 n\u00e4m\u00e4 ovat?\" kysyin.\n\n\"Lauman historiaa. Pyysin Iivoa j\u00e4tt\u00e4m\u00e4\u00e4n ne esille.\" H\u00e4n alkoi k\u00e4yd\u00e4 pinoa l\u00e4pi. \"J\u00e4senluettelot. Is\u00e4n p\u00e4iv\u00e4kirjamerkint\u00f6j\u00e4. Lakikirja.\"\n\n\"Lakikirja? \"ihmettelin.\n\nH\u00e4n kohotti katseensa. \"Niin. Kaikki s\u00e4\u00e4nn\u00f6t siit\u00e4, mik\u00e4 on rangaistavaa ja miten. K\u00e4yt\u00e4nn\u00f6ss\u00e4 t\u00e4ss\u00e4 p\u00f6yd\u00e4ll\u00e4 on kaikki saatavilla oleva tieto laumasta ja ihmissusista.\"\n\n\"Ai.\" Tuijotin pinoa aivan uudella mielenkiinnolla. Tajusin nimitt\u00e4in, ett\u00e4 siit\u00e4 saattaisi hyvinkin l\u00f6yty\u00e4 vastaus Mikaelin suden ongelmiin. \"Saanko katsoa niit\u00e4?\" kysyin.\n\n\"Jos haluat.\"\n\nKurotin kohti l\u00e4hint\u00e4 kirjaa. Mikael nappasi sen ennen kuin ehdin koskea sit\u00e4. \"Itse asiassa tarvitsen nyt juuri sit\u00e4. Mutta katsele vain rauhassa kaikkea muuta.\"\n\n\"Mit\u00e4 sin\u00e4 etsit?\" kysyin.\n\n\"Jotain, mik\u00e4 auttaisi meit\u00e4.\"\n\nMinun oli pakko kysy\u00e4. \"Mit\u00e4 sin\u00e4 aiot tehd\u00e4 Jonin suhteen?\"\n\n\"Suoraan sanottuna en tied\u00e4.\"\n\n\"Eik\u00f6 ole mit\u00e4\u00e4n muuta rangaistusta kuin karkottaminen?\"\n\n\"Lauman s\u00e4\u00e4nt\u00f6jen mukaan meill\u00e4 on k\u00e4yt\u00f6ss\u00e4 kolme eri rangaistusta, mutta oikeasti vain kaksi. Tai ainakin minun k\u00e4yt\u00f6ss\u00e4ni.\"\n\n\"Mit\u00e4 ne ovat?\"\n\n\"Sill\u00e4, jota vastaan on rikottu, on oikeus haastaa rikoksen tekij\u00e4 kaksintaisteluun. Muussa tapauksessa rangaistuksena on karkotus.\"\n\n\"Ja se kolmas?\"\n\n\"Kuolemantuomio.\" Minun olisi pit\u00e4nyt arvata h\u00e4nen vastauksensa. Martti ei nimitt\u00e4in ollut yht\u00e4 hell\u00e4mielinen kuin Mikael. H\u00e4n oli teloittanut Juhan iso\u00e4idin Maritan. Olin ollut paikalla, ja joutuisin kantamaan muistoa mukanani lopun ik\u00e4\u00e4ni.\n\nJatkoin nopeasti: \"Periaatteessa sin\u00e4 siis voisit haastaa Jonin ja se k\u00e4visi rangaistuksesta?\"\n\n\"Min\u00e4 en voi haastaa ket\u00e4\u00e4n, koska min\u00e4 olen laumajohtaja. Jos h\u00e4vi\u00e4n, menet\u00e4n laumanjohtajuuden, eik\u00e4 laumanjohtajuuden haluta siirtyv\u00e4n kenellek\u00e4\u00e4n sellaiselle, joka ei halua sit\u00e4.\" H\u00e4n hymyili hiukan, mik\u00e4 paljasti, ett\u00e4 h\u00e4n ymm\u00e4rsi sanojensa ironian.\n\nMikael vietti l\u00e4hes koko y\u00f6n ty\u00f6huoneessa. H\u00e4nen t\u00e4ytyi olla v\u00e4synyt pitk\u00e4n ajomatkan j\u00e4ljilt\u00e4, mutta ymm\u00e4rsin olla ehdottamatta h\u00e4nelle nukkumaan menemist\u00e4. Sen sijaan valvoin itsekin. Tein meille vahvaa kahvia ja istuin ty\u00f6huoneen nurkassa nojatuolissa lueskellen milloin mit\u00e4kin papereita. En tiennyt laumasta paljoakaan, ja lukeminen selvitti minulle monia asioita. Minulle esimerkiksi selvisi, ettei kukaan susista syntynyt automaattisesti laumanj\u00e4seneksi. Laumataika oli sidoksissa vereen, mutta hyvin konkreettisessa mieless\u00e4: oli olemassa rituaali, jonka kaikki laumassa olevat olivat joutuneet k\u00e4ym\u00e4\u00e4n l\u00e4pi.\n\nUlkopuolisena minun oli vaikea ymm\u00e4rt\u00e4\u00e4 riippuvuutta, joka sitoi sudet laumaansa. Mutta tekstien my\u00f6t\u00e4 aloin ep\u00e4ill\u00e4, ett\u00e4 Mikaelin oireilu saattoi hyvinkin johtua lauman puutteesta eik\u00e4 minusta.\n\nJa siit\u00e4 sain vihdoin idean.\nNELJ\u00c4S LUKU\n\nSeuraavana aamuna s\u00f6imme aamiaista v\u00e4synein\u00e4 mutta p\u00e4\u00e4tt\u00e4v\u00e4isin\u00e4. Mit\u00e4 nopeammin asia saataisiin kuntoon, sit\u00e4 helpommaksi Mikaelin asema k\u00e4visi laumassa. Kenenk\u00e4\u00e4n ei tarvinnut sanoa \u00e4\u00e4neen, ett\u00e4 ihmisten kanssa syntyneen ongelman ratkaiseminen oli testi. Toivoin meid\u00e4n voivan viel\u00e4 palata rentoon kes\u00e4nviettoon, jos ja kun kaikki vain sujuisi hyvin.\n\n\"Menn\u00e4\u00e4n hakemaan Niko hotellilta\", Mikael sanoi, kun istuimme jo autossa.\n\n\"Luulen, ett\u00e4 Mitjan olisi my\u00f6s hyv\u00e4 tulla mukaan\", sanoin.\n\nMikael vilkaisi minua ennen kuin katsoi taas tiet\u00e4. \"Olet oikeassa. H\u00e4nkin on t\u00e4\u00e4ll\u00e4 lauman takia.\"\n\nKun tulimme hotellin parkkipaikalle, Mitja seisoi tyypilliseen tapaan huonoryhtisen\u00e4 k\u00e4det taskussa. Niko taas n\u00e4ytti silt\u00e4 kuin olisi odottanut paparazzien ilmestyv\u00e4n paikalle hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 \u2013 se oli ainoa selitys sille, ett\u00e4 h\u00e4n oli vaivautunut sonnustautumaan helteell\u00e4 kokomustaan ja isoihin roikkuviin kaulaketjuihin. Enemm\u00e4n minut kuitenkin yll\u00e4tti se, ett\u00e4 he n\u00e4yttiv\u00e4t keskustelevan kesken\u00e4\u00e4n. Mitjan ja Nikon ensikohtaaminen ei ollut sujunut parhaimmalla mahdollisella tavalla, mutta ilmeisesti he olivat p\u00e4\u00e4sseet vaivautuneisuuden yli.\n\nAjoimme lyhyen matkan kohti vaaraa, joka kohosi hotellin takana. Mikael pys\u00e4k\u00f6i tienvarteen, ja nousimme autosta. L\u00e4hdimme k\u00e4velem\u00e4\u00e4n jonossa mets\u00e4polkua pitkin.\n\nSiristelin silmi\u00e4ni. Valvominen oli saanut ne valonaroiksi, ja vasta kun aurinko vet\u00e4ytyi pilveen, uskalsin katsoa muualle kuin jalkoihini. Hukkavaaran metsiss\u00e4 oli samanaikaiasesti jotain viatonta ja v\u00e4h\u00e4n villi\u00e4. En kuitenkaan pystynyt t\u00e4ysin nauttimaan ymp\u00e4rist\u00f6st\u00e4ni, sill\u00e4 vatsanpohjassani kiemurteli ep\u00e4miellytt\u00e4v\u00e4 aavistus. Mikael oli ilmiselv\u00e4sti j\u00e4nnittynyt, mik\u00e4 ei enteillyt hyv\u00e4\u00e4.\n\nAstuimme aukiolle.\n\nSudet olivat jo ker\u00e4\u00e4ntyneet sinne ympyr\u00e4nmuotoiseksi rykelm\u00e4ksi. Tiesin, ett\u00e4 monilla oli kes\u00e4aikaan tapana tulla kokoontumisiin suden muodossa, mik\u00e4 tarkoitti, ett\u00e4 osallistujien vaatetus saattoi olla olematonta, mutta t\u00e4ll\u00e4 kertaa kaikki n\u00e4yttiv\u00e4t olevan t\u00e4ysiss\u00e4 pukeissa. Muistin Juhan mainitseman muodonmuutoskiellon, ja olin siit\u00e4 salaa kiitollinen.\n\nHe olivat j\u00e4tt\u00e4neet keh\u00e4n keskelle tyhj\u00e4n alueen ja selv\u00e4sti odottivat jonkin tai jonkun ottavan tilan haltuun. Tajusin heid\u00e4n odottavan meit\u00e4.\n\nMikael tarttui k\u00e4teeni. Niko ja Mitja j\u00e4iv\u00e4t seisomaan muiden joukkoon, mutta Mikael astui piirin keskelle. H\u00e4n ei irrottanut otettaan k\u00e4dest\u00e4ni, joten minulla ei ollut muuta mahdollisuutta kuin seurata per\u00e4ss\u00e4.\n\nKaikki katsoivat meit\u00e4. Olin saanut tottua susien tuijotukseen klubilla, mutta kokonaisen lauman tuijotuksessa oli jotain erilaista. He eiv\u00e4t arvostelleet minua \u2013 eiv\u00e4t ainakaan viel\u00e4. He pikemminkin n\u00e4yttiv\u00e4t odottavilta, aivan kuin minun ja Mikaelin oletettaisiin korjaavan jokainen laumaa kohdannut ongelma pelk\u00e4ll\u00e4 l\u00e4sn\u00e4olollamme. Se oli oikeastaan pahempaa kuin klubin halveksivat katseet.\n\n\"Kiitos kaikille, ett\u00e4 tulitte\", Mikael aloitti.\n\nKukaan ei sanonut mit\u00e4\u00e4n. Annoin katseeni kiert\u00e4\u00e4 ringiss\u00e4. Panin merkille, ett\u00e4 Karjalasta tulleet sudet olivat omalla puolellaan, alkuper\u00e4iset hukkavaaralaiset toisella laidalla. Kulunut vuosi ei ollut onnistunut yhdist\u00e4m\u00e4\u00e4n laumaa.\n\n\"Meill\u00e4 on t\u00e4n\u00e4\u00e4n monta asiaa k\u00e4sitelt\u00e4v\u00e4n\u00e4, joten yritet\u00e4\u00e4n pit\u00e4\u00e4 puheenvuorot lyhyin\u00e4.\" H\u00e4n selvitti kurkkuaan. \"Aloitetaan naapurikunnan kanssa syntyneist\u00e4 ongelmista. Siell\u00e4 on n\u00e4hty susi ja paikalliset asukkaat ovat alkaneet etsi\u00e4 mailtamme todisteita, jotta voisivat aloittaa t\u00e4ysimittaisen susijahdin. T\u00e4m\u00e4nhetkinen k\u00e4sitys on se, ett\u00e4 kukaan laumanj\u00e4senist\u00e4 ei ole syyllistynyt rikkomukseen, mutta se ei tietenk\u00e4\u00e4n poista itse ongelmaa.\" H\u00e4n piti tauon. \"Onko kenell\u00e4k\u00e4\u00e4n t\u00e4ss\u00e4 kohdassa jotain sanottavaa, jonka haluaisi tuoda tietooni?\"\n\n\"Mit\u00e4 sin\u00e4 aiot tehd\u00e4?\" En ehtinyt n\u00e4hd\u00e4, kuka esitti kysymyksen. S\u00e4vy oli kuitenkin tiukempi kuin olin osannut odottaa.\n\nT\u00e4m\u00e4 ei ala hyvin, ajattelin.\n\nMikael teeskenteli, ettei ollut kuullut uhmaa. \"Meill\u00e4 on vaihtoehtoja\", h\u00e4n sanoi. \"T\u00e4ll\u00e4 hetkell\u00e4 on t\u00e4rke\u00e4\u00e4 saada tilanteesta niin paljon tietoa kuin mahdollista, ja p\u00e4\u00e4tt\u00e4\u00e4 vasta sen j\u00e4lkeen, mik\u00e4 on oikea menettelytapa.\"\n\nMikael kuulosti aivan poliitikolta, ajattelin. Saman tien tajusin, ett\u00e4 h\u00e4nh\u00e4n oli poliitikko. T\u00e4m\u00e4 ei ehk\u00e4 ollut television vaaliv\u00e4ittely, eik\u00e4 kukaan tekisi h\u00e4nen kannatuksestaan gallupeja, mutta h\u00e4nen t\u00e4ytyi silti saada enemmist\u00f6n tuki puolelleen.\n\nH\u00e4nen vastauksensa ei kuitenkaan tyydytt\u00e4nyt kaikkia. \"Eik\u00f6 sinun pit\u00e4isi hankkia meille uusi vampyyri?\" joku kysyi.\n\nMikael vilkaisi nopeasti minua. En halunnut antaa itsest\u00e4ni liikaa ilmi, joten annoin katseeni harhailla. Se osui tummahiuksiseen mieheen. Iivo Salakka seisoi muiden joukossa. H\u00e4n ei ollut sanonut viel\u00e4 mit\u00e4\u00e4n, tarkkaillut vain. Siit\u00e4 huolimatta olin aistivinani h\u00e4ness\u00e4 jotain sellaista, mit\u00e4 keness\u00e4k\u00e4\u00e4n muussa, Mikaelia lukuun ottamatta, ei ollut. Uskoin, ett\u00e4 mik\u00e4li h\u00e4n olisi vain avannut suunsa, h\u00e4nt\u00e4 oltaisiin kuunneltu.\n\nIivo oli alfa. Ja vaikka h\u00e4n oli luopunut asemastaan vapaaehtoisesti siskonpoikansa hyv\u00e4ksi, kukaan ei voinut kielt\u00e4\u00e4, etteik\u00f6 h\u00e4n ollut Mikaelia vanhempi ja kokeneempi.\n\n\u00c4kki\u00e4 Mikaelin ep\u00e4ilyt pahimmasta uhkaajastaan eiv\u00e4t tuntuneetkaan tuulesta temmatuilta.\n\nTunsin, kuinka kokoontumisen tunnelma alkoi v\u00e4hitellen muuttua odottavasta pettyneeksi ja jopa \u00e4rtyneeksi. Mikaelin vastaukset eiv\u00e4t olleet sit\u00e4, mit\u00e4 he halusivat kuulla. Se oli tietenkin t\u00e4ysin ep\u00e4reilua, ja ihmettelin, ettei Mikael sanonut sit\u00e4 \u00e4\u00e4neen. H\u00e4n olisi voinut pys\u00e4ytt\u00e4\u00e4 kysymysten virran. H\u00e4n olisi voinut tehd\u00e4 niin ilman, ett\u00e4 h\u00e4nen olisi tarvinnut edes korottaa \u00e4\u00e4nt\u00e4\u00e4n: yksi sana ja kaikki olisivat ymm\u00e4rt\u00e4neet, ett\u00e4 h\u00e4nt\u00e4 ei kannattanut ryhty\u00e4 haastamaan. Silti h\u00e4n antoi kysymysten jatkua.\n\nJa h\u00e4n vastasi niist\u00e4 jokaikiseen. H\u00e4n oli osannut odottaa nuivaa vastaanottoa. Min\u00e4 olin odottanut sellaista vain itse\u00e4ni, en Mikaelia kohtaan.\n\nSanalla sanoen h\u00e4n oli uskomaton. H\u00e4net oli ilmiselv\u00e4sti kasvatettu t\u00e4h\u00e4n. Oma k\u00e4rsiv\u00e4llisyyteni ei olisi kest\u00e4nyt kahta minuuttia kauempaa.\n\n\"Miksi sin\u00e4 edes palasit, jos et tied\u00e4, mit\u00e4 tehd\u00e4?\"\n\nMikaelin ote k\u00e4dest\u00e4ni tiukkeni, mutta h\u00e4n pysyi rauhallisena. \"Min\u00e4 tulin, koska min\u00e4 olen laumanjohtaja.\"\n\n\"Oletko? Miksi sinua ei sitten ole n\u00e4kynyt melkein vuoteen?\"\n\nEnsimm\u00e4ist\u00e4 kertaa olin aistivinani h\u00e4ness\u00e4 ep\u00e4varmuutta. \"Haluan pyyt\u00e4\u00e4 teilt\u00e4 kaikilta anteeksi l\u00e4ht\u00f6\u00e4ni ja siit\u00e4 seurannutta vaivaa. Te olette p\u00e4rj\u00e4nneet todella hienosti.\"\n\n\"Eip\u00e4 t\u00e4ss\u00e4 ollut juuri vaihtoehtoa\", joku mutisi.\n\n\"Niin kuin olette ehk\u00e4 kuulleetkin, min\u00e4 olen p\u00e4\u00e4ssyt opiskelemaan Helsinkiin. Aion asua siell\u00e4 suurimman osan vuodesta siihen asti, kunnes valmistun.\"\n\n\"Aiotko sin\u00e4 johtaa meit\u00e4 Helsingist\u00e4 k\u00e4sin?\"\n\n\"Te olette osoittaneet, ett\u00e4 p\u00e4rj\u00e4\u00e4tte paremmin kuin hyvin ilman minua. Tulen tietenkin Hukkavaaraan niin usein kuin mahdollista.\"\n\n\"\u00c4l\u00e4 anna meid\u00e4n h\u00e4irit\u00e4 sinun henkil\u00f6kohtaista el\u00e4m\u00e4\u00e4si.\"\n\nEn voinut en\u00e4\u00e4 hillit\u00e4 itse\u00e4ni. \"Ettek\u00f6 te tajua, miten iso asia Mikaelin kouluttautuminen voi olla? Vihdoin teill\u00e4 olisi joku, joka omistaa el\u00e4m\u00e4ns\u00e4 lauman lis\u00e4\u00e4ntymisongelman ratkaisemiseksi ja te hangoittelette vastaan. V\u00e4it\u00e4ttek\u00f6, te, aikuiset ihmiset, ettette selvi\u00e4, ellei h\u00e4n ole koko ajan t\u00e4\u00e4ll\u00e4?\"\n\nSain vastaani \u00e4rtynytt\u00e4 mutinaa.\n\n\"Ymm\u00e4rr\u00e4n, ettei t\u00e4m\u00e4 keskustelu ole viel\u00e4 ohi, mutta meill\u00e4 ei ole aikaa loputtomiin, ja siksi meid\u00e4n on parempi siirty\u00e4 k\u00e4sittelem\u00e4\u00e4n seuraavaa ongelmaa\", Mikael sanoi. H\u00e4nen katseensa osui Joniin.\n\nEn ollut huomannut h\u00e4nt\u00e4 aikaisemmin v\u00e4kijoukossa. Nyt kun tiesin, miss\u00e4 h\u00e4n seisoi, en voinut olla tuijottamatta. Kun olin n\u00e4hnyt h\u00e4net ensimm\u00e4ist\u00e4 kertaa, olin pit\u00e4nyt h\u00e4nt\u00e4 v\u00e4h\u00e4n tyhm\u00e4n\u00e4 mutta kuitenkin etup\u00e4\u00e4ss\u00e4 hyv\u00e4ntahtoisena lihaskimppuna. Viime kes\u00e4 oli kuitenkin muuttanut kaiken. Olin valmistautunut tuntemaan raivoa h\u00e4nt\u00e4 kohtaan, mutta en\u00e4\u00e4 en tiennyt, mit\u00e4 h\u00e4nest\u00e4 olisi pit\u00e4nyt ajatella. H\u00e4n oli edelleen iso, mutta karkotusaika ei ollut selv\u00e4stik\u00e4\u00e4n tehnyt h\u00e4nelle hyv\u00e4\u00e4. Vaatteet roikkuivat h\u00e4nen p\u00e4\u00e4ll\u00e4\u00e4n ja silmiss\u00e4 oli jotain pelokasta.\n\nSilti: h\u00e4n oli hy\u00f6k\u00e4nnyt kimppuuni. Mahtaisinko pysty\u00e4 antamaan sit\u00e4 koskaan anteeksi.\n\n\"Haluat varmaankin sanoa jotakin\", Mikael sanoi Jonille.\n\nJoni nielaisi. Hetken aikaa jo ajattelin, ett\u00e4 h\u00e4n j\u00e4nist\u00e4isi, mutta sitten h\u00e4n astui askeleen edemm\u00e4s. \"Min\u00e4 anon armahdusta.\"\n\nYmp\u00e4rill\u00e4 sek\u00e4 naureskeltiin ett\u00e4 muristiin. Mikael ei ehk\u00e4 ollut kaikkien suosiossa, mutta Joniin verrattuna h\u00e4nt\u00e4 kohdeltiin kuin hemmoteltua puudelia. Joni oli niit\u00e4 ihmisi\u00e4, jotka meniv\u00e4t aina vahvimman puolelle, enk\u00e4 \u00e4kkiselt\u00e4\u00e4n keksinyt mit\u00e4\u00e4n yht\u00e4 halveksittavaa. Kenell\u00e4k\u00e4\u00e4n ei tuntunut olevan vaikeuksia osoittaa avoimesti vihamielisyytt\u00e4\u00e4n.\n\nVilkaisin Mikaelia. H\u00e4nen ilmeens\u00e4 ei paljastanut mit\u00e4\u00e4n. H\u00e4n ei ollut kertonut minulle, mihin tulokseen oli tullut, joten olin aivan yht\u00e4 ep\u00e4tietoinen kuin kaikki muutkin.\n\n\"Min\u00e4 lupaan tukea sinua ja noudattaa kaikkia s\u00e4\u00e4nt\u00f6j\u00e4\", Joni mutisi. \"Min\u00e4 vain haluan palata takaisin kotiin.\"\n\nEn usko, ett\u00e4 olin koskaan n\u00e4hnyt mit\u00e4\u00e4n yht\u00e4 s\u00e4\u00e4litt\u00e4v\u00e4\u00e4. Jo h\u00e4nen katsomisensa teki pahaa. Mikael ei voisi olla suostumatta.\n\n\"Olen pahoillani\", Mikael sanoi. \"En voi armahtaa sinua. Haastaisin sinut, jos voisin, mutta...\"\n\n\"Joni Puustinen\", kuulin itseni sanovan, \"haastan sinut kaksintaisteluun.\"\n\nOdotin kohahdusta. Sen sijaan sanojani tervehti t\u00e4ydellinen hiljaisuus.\n\nHitsi. Olin juuri onnistunut yll\u00e4tt\u00e4m\u00e4\u00e4n paitsi itseni, my\u00f6s laumallisen ihmissusia.\n\nNiko oli ensimm\u00e4inen, joka puhui. \"Huono l\u00e4pp\u00e4, Raisa.\"\n\n\"Ei se ollut mik\u00e4\u00e4n l\u00e4pp\u00e4.\" Vedin henke\u00e4 samalla kun yritin muistella, mit\u00e4 s\u00e4\u00e4nn\u00f6iss\u00e4 sanottiin haasteen esitt\u00e4misest\u00e4. \"Vetoan siihen, ett\u00e4 Joni hy\u00f6kk\u00e4si minun kimppuuni viime kes\u00e4n\u00e4 ilman provokaatiota. Minulla on oikeus vaatia h\u00e4nelle rangaistusta.\"\n\nMikaelin katse imi minut itseens\u00e4. Ymp\u00e4rill\u00e4mme seisova lauma, mets\u00e4 ja vaara lakkasivat olemasta, kun k\u00e4vimme \u00e4\u00e4net\u00f6nt\u00e4 keskustelua. Vasta silloin tajusin, ett\u00e4 h\u00e4n oli v\u00e4ltellyt katsomasta minuun eilisest\u00e4 l\u00e4htien.\n\nEn tiennyt, mille kannalle Mikael oli kallistumassa, ennen kuin h\u00e4n ny\u00f6kk\u00e4si.\n\n\"Mikael, sin\u00e4 et voi olla tosissasi!\"\n\nMikael j\u00e4tti Nikon kommentin huomiotta. \"Haaste on oikeutettu\", h\u00e4n sanoi. \"Joni, sinun pit\u00e4\u00e4 p\u00e4\u00e4tt\u00e4\u00e4, otatko sen vastaan.\"\n\n\"Mit\u00e4 tapahtuu, jos sanon kyll\u00e4?\" Joni kysyi.\n\n\"Sin\u00e4 ottelet Raisaa vastaan ja osa rikoksistasi kuittaantuu\", Mikael vastasi. \"Koska rangaistustasi on j\u00e4ljell\u00e4 en\u00e4\u00e4 alle puolet, k\u00e4yt\u00e4nn\u00f6ss\u00e4 se tarkoittaa, ett\u00e4 saat palata.\"\n\n\"Mutta eih\u00e4n minun tarvitse tappaa h\u00e4nt\u00e4?\"\n\nMikaelin silm\u00e4t kapenivat. \"S\u00e4\u00e4nn\u00f6t ovat aina samat: otellaan siihen saakka, kunnes jompikumpi kuolee tai luovuttaa.\"\n\n\"Mutta sin\u00e4 et ole vihainen, jos suostun? H\u00e4n on sinun tytt\u00f6yst\u00e4v\u00e4si.\" H\u00e4nen ilmeens\u00e4 oli yh\u00e4 h\u00f6lmistynyt, mutta uskoin sen johtuvan l\u00e4hinn\u00e4 siit\u00e4, ettei h\u00e4n voinut uskoa onneaan.\n\n\"Ei, min\u00e4 en ole vihainen.\" Mikaelin \u00e4\u00e4nens\u00e4vy kertoi kuitenkin toista. N\u00e4in heti h\u00e4nen itsehillint\u00e4ns\u00e4 olevan todellisella koetuksella. Tunnin hiillostus ei ollut saanut h\u00e4nt\u00e4 edes r\u00e4p\u00e4ytt\u00e4m\u00e4\u00e4n silm\u00e4\u00e4ns\u00e4, mutta Joni sai.\n\nSitten h\u00e4n vilkaisi minua ja tajusin, ett\u00e4 h\u00e4n halusi minun n\u00e4ytt\u00e4v\u00e4n Jonille. Eik\u00e4 pelk\u00e4st\u00e4\u00e4n Jonille, vaan koko laumalle. Joni saisi kyseenalaisen kunnian toimia demonstraatiov\u00e4lineen\u00e4.\n\nJonia ei selv\u00e4stik\u00e4\u00e4n haitannut ajatus minun kimppuuni k\u00e4ymisest\u00e4. Aikaisempi s\u00e4\u00e4lini oli mennyt siis hukkaan, ja se korvautuikin hyvin nopeasti inholla.\n\nDaimon minussa alkoi olla tulevasta ottelusta aika innoissaan.\n\nNiko sen sijaan ei ollut. \"Te molemmat olette t\u00e4ysin sekaisin!\" Niko huusi. \"Joni on puolta isompi! H\u00e4n tekee Raisasta jauhelihaa.\"\n\nH\u00e4n irtautui piirist\u00e4 ja tuli minua kohti. Rakas Niko \u2013 h\u00e4n oikeasti kuvitteli voivansa suojella minua raahaamalla minut v\u00e4kisin pois paikalta. Minun teki mieli halata h\u00e4nt\u00e4 siit\u00e4 hyv\u00e4st\u00e4.\n\nHalaamisen sijaan vilkaisin velje\u00e4ni. Mitja huokaisi, mutta tarttui Nikoa kiinni ensin hartioista ja lopulta vy\u00f6t\u00e4r\u00f6lt\u00e4. Nikon silm\u00e4t levisiv\u00e4t hiukan, kun h\u00e4n tajusi, ettei pystyisi pyristelem\u00e4\u00e4n irti Mitjan otteesta.\n\n\"Min\u00e4 saan valita muodon, eik\u00f6 niin?\" Joni kysyi.\n\n\"Ei tarvitse valita\", Mikael sanoi. \"Raisa ei voi muuttua ja s\u00e4\u00e4nn\u00f6iss\u00e4 sanotaan, ett\u00e4 molempien t\u00e4ytyy olla samassa muodossa.\"\n\nSe oli helpotus. Suden muodossa Joni olisi nimitt\u00e4in pystynyt tappamaan minut. En ollut ottanut koko asiaa huomioon haastaessani Jonin.\n\nKatsoin Mikaelia. Mit\u00e4 tahansa tapahtuisikin, h\u00e4n ei antaisi minulle k\u00e4yd\u00e4 mit\u00e4\u00e4n. Siit\u00e4 olin varma. Ja \u00e4kki\u00e4 oli paljon helpompi hengitt\u00e4\u00e4.\n\n\"Ahaa, okei\", Joni sanoi. \"Milloin ottelu pidet\u00e4\u00e4n?\"\n\nMikael katsoi minua. Hetken aikaa olin niin sinisen lumoissa, etten hoksannut h\u00e4nen odottavan minun vastaustani. \"Min\u00e4 tekisin sen mieluiten nyt\", sanoin. Ennen kuin ehtisin alkaa py\u00f6ritt\u00e4\u00e4 mieless\u00e4ni filmi\u00e4 kaikista mahdollisista kauhuskenaarioista. Niko oli nimitt\u00e4in ollut oikeassa: Joni oli paljon isompi kuin min\u00e4. Itse asiassa en ollut koskaan otellut ket\u00e4\u00e4n yht\u00e4 isoa vastaan.\n\n\"Siis... nyt?\" Joni kysyi.\n\n\"Jos vain sopii.\"\n\nMikael ny\u00f6kk\u00e4si. \"S\u00e4\u00e4nt\u00f6jen mukaan haasteen ja ottelun v\u00e4liss\u00e4 saa olla korkeintaan vuorokausi. Me kuitenkin olemme kaikki nyt t\u00e4\u00e4ll\u00e4, ja olisi hyv\u00e4 hoitaa t\u00e4m\u00e4 pois alta saman tien.\" H\u00e4nen \u00e4\u00e4nens\u00e4 oli liioitellun tasainen. En kuitenkaan ollut varma, huomasiko sit\u00e4 kukaan muu kuin min\u00e4.\n\nN\u00e4in, ett\u00e4 Joni olisi mieluusti lyk\u00e4nnyt ottelua. Mutta h\u00e4nell\u00e4 ei ollut pokkaa alkaa v\u00e4itt\u00e4\u00e4 Mikaelille vastaan.\n\n\"Eik\u00f6 Mikael ole v\u00e4h\u00e4n j\u00e4\u00e4vi toimimaan tuomarina?\" joku kysyi.\n\n\"Todellakin olen\", Mikael puuskahti. Se oli ensimm\u00e4inen merkki siit\u00e4, mit\u00e4 h\u00e4n oikeasti ajatteli minun ottelemisestani. \"Leski, otatko ohjat k\u00e4siisi?\"\n\nHetken kuluttua vanha nainen astui esiin piirist\u00e4. Muistin h\u00e4net viime kes\u00e4lt\u00e4, jolloin Mikael oli otellut Marttia vastaan. \"Enp\u00e4 uskonut joutuvani todistamaan t\u00e4llaistakin p\u00e4iv\u00e4\u00e4\", h\u00e4n mumisi. \"No, Joni Puustinen. Kenet sin\u00e4 kutsut sekundantiksesi?\"\n\nJoni ei selv\u00e4stik\u00e4\u00e4n ollut ajatellut asiaa. H\u00e4n silm\u00e4ili yleis\u00f6\u00e4, mutta sai vastaansa pelkki\u00e4 kylmi\u00e4 katseita.\n\nKukaan ei ilmoittautunut vapaaehtoiseksi. Se ei ollut suuri yll\u00e4tys, mutta hankaloitti omaa el\u00e4m\u00e4\u00e4ni.\n\n\"Jos sinulla ei ole sekundanttia, et voi otella\", leski sanoi.\n\nKatsoin velje\u00e4ni. \"Sinun pit\u00e4\u00e4 ilmoittautua\", sanoin h\u00e4nelle kreikaksi. Minua katsottiin sen takia pahasti, mutta en v\u00e4litt\u00e4nyt. \"Kukaan muu ei suostu.\"\n\n\"Miksi min\u00e4 niin tekisin?\" h\u00e4n kysyi kulmat kurtussa. Saatoin tuntea h\u00e4nen turhautumisensa monen metrin p\u00e4\u00e4h\u00e4n. \"Mit\u00e4 se edes tarkoittaa?\"\n\n\"Ei se tarkoita mit\u00e4\u00e4n. Se on pelkk\u00e4 muodollisuus. Min\u00e4 huolehdin kaikesta muusta.\"\n\n\"Ihan sama\", h\u00e4n mutisi suomeksi. H\u00e4n kohotti k\u00e4tens\u00e4. \"Joo, min\u00e4 ilmoittaudun. Tai miten se pit\u00e4\u00e4k\u00e4\u00e4n sanoa.\"\n\nKaikki n\u00e4yttiv\u00e4t tyrmistyneilt\u00e4, Joni mukaan lukien. Minun ei kannattaisi kertoa h\u00e4nelle, ett\u00e4 Mitja oli luultavasti paras ottelija kaikista l\u00e4sn\u00e4olijoista. Ep\u00e4ilin vahvasti, ett\u00e4 ainoastaan Mikael olisi saattanut tarjota h\u00e4nelle vastusta.\n\n\"Ja kukahan sin\u00e4 olet?\" leski kysyi.\n\n\"Mitja Kovasin.\"\n\n\"H\u00e4n on minun kaksoisveljeni\", sanoin. Ilmeist\u00e4 p\u00e4\u00e4tellen tieto tuli suurelle osalle laumasta t\u00e4yten\u00e4 yll\u00e4tyksen\u00e4.\n\n\"Ymm\u00e4rr\u00e4tk\u00f6, ett\u00e4 saatat joutua ottelemaan siskoasi vastaan?\" leski kysyi Mitjalta.\n\n\"Ihan kuin en olisi tehnyt niin koskaan aikaisemmin.\"\n\nTarkkakorvaisimmat saattoivat jo aavistella, ett\u00e4 saatoin ehk\u00e4 sittenkin tiet\u00e4\u00e4, mit\u00e4 olin tekem\u00e4ss\u00e4. Siihen joukkoon taisi kuitenkin kuulua korkeintaan pari sutta. Loput olivat autuaan tiet\u00e4m\u00e4tt\u00f6mi\u00e4 siit\u00e4, mit\u00e4 tulisi tapahtumaan.\n\n\"Raisa Oja. Kenet kutsut sekundantiksesi?\"\n\nHalusin kutsua Mikaelin, mutta se ei tietenk\u00e4\u00e4n k\u00e4ynyt p\u00e4ins\u00e4. Annoin katseeni kiert\u00e4\u00e4 v\u00e4kijoukossa ja poimin joukosta ne, joiden kuvittelin edes harkitsevan suostumista. Sekundanttini ei mit\u00e4 luultavimmin joutuisi tekem\u00e4\u00e4n yht\u00e4\u00e4n mit\u00e4\u00e4n, mutta halusin pelata varman p\u00e4\u00e4lle. Niko ei tullut kysymykseen. Eik\u00e4 Jenni. H\u00e4n oli pienikokoinen, ja vaikka h\u00e4nen asenteensa olikin oikea, en tiennyt, oliko h\u00e4nell\u00e4 kokemusta mist\u00e4\u00e4n muusta kuin sanaharkoista. Sitten n\u00e4in Iivon. Houkutus pyyt\u00e4\u00e4 h\u00e4nt\u00e4 oli hyvin suuri. Jos h\u00e4n ei suostuisi, tiet\u00e4isin, ettei h\u00e4n ollut Mikaelin puolella. H\u00e4nen suostumisensa taas viestitt\u00e4isi laumalle, ett\u00e4 h\u00e4n oli, mik\u00e4 auttaisi Mikaelia.\n\nOngelma ratkesi, kun Jenni kiskoi Juhaa k\u00e4sivarresta ja t\u00e4m\u00e4 kumartui niin, ett\u00e4 Jenni saattoi kuiskata h\u00e4nen korvaansa. Juha n\u00e4ytti h\u00e4mmentyneelt\u00e4, ja Jennin ilme tuimeni. Lopulta Juha suoristi itsens\u00e4. \"Raisa, min\u00e4 ilmoittaudun vapaaehtoiseksi.\"\n\nNy\u00f6kk\u00e4sin h\u00e4nelle kiitollisena ja k\u00e4\u00e4nnyin takaisin lesken puoleen.\n\n\"Saanko min\u00e4 l\u00e4mmitell\u00e4?\" Joni kysyi.\n\n\"L\u00e4mmittele pois\", sanoin. \"Min\u00e4 olen valmis heti kun sin\u00e4 olet.\"\n\nJoni j\u00e4i paikoilleen kuin odottaen, ett\u00e4 kertoisin koko jutun olleen pelkk\u00e4 vitsi, mutta lopulta h\u00e4n l\u00e4hti liikkeelle. Avasin hiukseni ponnarilta ja aloin tehd\u00e4 niist\u00e4 tiukkaa letti\u00e4 samalla kun seurasin vierest\u00e4 Jonin h\u00f6lkk\u00e4\u00e4. Mikael tuli eteeni ja peitti n\u00e4kym\u00e4n.\n\n\"H\u00e4n on oikeak\u00e4tinen\", h\u00e4n sanoi. \"Vasen jalka on parempi, sit\u00e4 kannattaa varoa.\"\n\nNy\u00f6kk\u00e4sin ja jatkoin letityst\u00e4. Kiedoin kumilenkin tiukasti tupsun ymp\u00e4rille.\n\nNiko ei en\u00e4\u00e4 yritt\u00e4nyt k\u00e4yd\u00e4 minuun kiinni, mutta se ei tarkoittanut, ett\u00e4 h\u00e4n olisi ollut valmis hyv\u00e4ksym\u00e4\u00e4n tilanteen. H\u00e4n seisoi parin metrin p\u00e4\u00e4ss\u00e4 p\u00e4lyillen vimmatusti. \"Raisa kiltti, \u00e4l\u00e4 tee t\u00e4t\u00e4. T\u00e4m\u00e4 on t\u00e4ysin hullua.\" Kun en reagoinut h\u00e4nen toivomallaan tavalla, h\u00e4n k\u00e4\u00e4nsi toivonsa Mikaeliin. \"Mikael! H\u00e4n on tytt\u00f6, ja kaiken lis\u00e4ksi viel\u00e4 ihminen. Ei h\u00e4n voi otella.\"\n\nNyrpistin nen\u00e4\u00e4ni. Otin muutaman askeleen kauemmas ja Mikael seurasi per\u00e4ss\u00e4ni.\n\nRiisuin kenk\u00e4ni. Maa jalkojen alla rauhoitti minua. Olin heti paljon keskittyneempi. Panin merkille, ett\u00e4 maaper\u00e4 oli kuivaa ja kiinte\u00e4\u00e4 \u2013 liikkuminen ei siis tulisi olemaan mik\u00e4\u00e4n ongelma.\n\nMinulla oli yll\u00e4ni teepaita ja pellavashortsit, jotka eiv\u00e4t onneksi kirist\u00e4neet yht\u00e4\u00e4n. Ei mitenk\u00e4\u00e4n ideaali kamppailuasu, mutta v\u00e4ltti paremman puutteessa.\n\nJoni h\u00f6lkk\u00e4si takaisin kent\u00e4lle ja alkoi tehd\u00e4 jonkinlaisia kyykk\u00e4yksi\u00e4. Katsoin vierest\u00e4 n\u00e4enn\u00e4isen v\u00e4linpit\u00e4m\u00e4tt\u00f6m\u00e4n\u00e4. Todellisuudessa nimitt\u00e4in suoritin jo omaa l\u00e4mmittely\u00e4ni: kurotin mieless\u00e4ni kohti ideoiden maailmaa. Vaikka en ollut tehnyt sit\u00e4 v\u00e4h\u00e4\u00e4n aikaan, tuttu tietoisuus humisi heti sis\u00e4ll\u00e4ni. Yhteys oli auennut niin vaivattomasti, ett\u00e4 ensimm\u00e4ist\u00e4 kertaa huomasin sen olevan yht\u00e4 luonnollista kuin hengitt\u00e4minen.\n\nOlin niin pitk\u00e4\u00e4n pyrkinyt olemaan muuta kuin olin, pit\u00e4m\u00e4\u00e4n itseni piilossa. Aluksi olin tehnyt niin tiedostamattani, ja my\u00f6hemmin silkasta pelosta. Suljin silm\u00e4ni ja hetken ajan vain nautin tunteesta. T\u00e4lt\u00e4 tuntui olla vapaa.\n\nViimeist\u00e4\u00e4n silloin tiesin, ett\u00e4 halutessani ottelu olisi ohi hyvin nopeasti. Joni oli ehk\u00e4 isompi kuin min\u00e4, mutta minulla oli... t\u00e4m\u00e4. Valta tuhota ja rakentaa uudelleen. Lopettaa ja aloittaa alusta.\n\nAvasin silm\u00e4ni. En tied\u00e4, mit\u00e4 Joni n\u00e4ki kasvoillani, mutta mit\u00e4 ikin\u00e4 se olikin, se sai h\u00e4net kalpenemaan hiukan.\n\nLiian my\u00f6h\u00e4ist\u00e4, ajattelin. Sin\u00e4 suostuit t\u00e4h\u00e4n jo. Nyt saat maksaa.\n\nMikael kosketti olkap\u00e4\u00e4t\u00e4ni viimeisen kerran ennen kuin siirtyi muiden joukkoon piirin laidalle.\n\nK\u00e4teni puristui nyrkkiin ja avautui taas ilman, ett\u00e4 olisin antanut sille mink\u00e4\u00e4nlaista k\u00e4sky\u00e4. Daimon oli innokas. Enemm\u00e4n kuin innokas: viime kerrasta oli jo aikaa, ja se janosi toimintaa.\n\nTarkistin, ett\u00e4 l\u00f6ysin verhon ja tunsin Mitjan sen takana.\n\n\u00c4l\u00e4 anna minun tappaa h\u00e4nt\u00e4, sanoin Mitjalle. H\u00e4n oli ainoa, joka voisi tehd\u00e4 asialle mit\u00e4\u00e4n.\n\nVeljeni ei n\u00e4ytt\u00e4nyt iloiselta, mutta ny\u00f6kk\u00e4si.\n\n\"Ottelu alkakoon!\" leski julisti.\n\nJoni alkoi liikkua. Se oli h\u00e4mm\u00e4stytt\u00e4v\u00e4n kevyen n\u00e4k\u00f6ist\u00e4 kun otti huomioon, miten iso h\u00e4n oli.\n\nSeisoin paikallani k\u00e4sivarret kylki\u00e4 pitkin. Se n\u00e4ytti h\u00e4mment\u00e4v\u00e4n Jonia, mutta arvasin, ettei h\u00e4nen h\u00e4mmennyksens\u00e4 pid\u00e4ttelisi h\u00e4nt\u00e4 kovin kauan. H\u00e4nell\u00e4 oli paljon enemm\u00e4n peliss\u00e4 kuin minulla.\n\nHalusin, ett\u00e4 h\u00e4n joutuisi hy\u00f6kk\u00e4\u00e4m\u00e4\u00e4n puolustautumatonta vastaan. Aivan niin kuin viime kes\u00e4n\u00e4.\n\nLopulta h\u00e4nen ensimm\u00e4inen potkunsa l\u00e4hti. Odotin kunnes h\u00e4nen jalkansa oli ehtinyt l\u00e4hes kasvojeni tasolle. Kohotin k\u00e4teni.\n\nMaailma j\u00e4hmettyi.\n\nHiljaisuus laskeutui koko kummallisen kohtauksen ylle ja huuhteli kaiken kiihkon pois.\n\nJoni seisoi edess\u00e4ni toinen jalka koomisesti ilmassa. Kallistin p\u00e4\u00e4t\u00e4ni. H\u00e4nen naamaansa oli v\u00e4\u00e4ntynyt keskittymisest\u00e4 ja voimanponnistuksesta, ja minua huvitti ajatella, ett\u00e4 h\u00e4n oli ladannut kaiken ensimm\u00e4iseen hy\u00f6kk\u00e4ykseens\u00e4. H\u00e4nen aikeensa oli ollut t\u00e4ysin tuhoon tuomittu. Ehk\u00e4, ehk\u00e4, jos h\u00e4n olisi s\u00e4\u00e4st\u00e4nyt parhaan potkunsa my\u00f6hemm\u00e4ksi, h\u00e4nell\u00e4 olisi voinut olla pieni mahdollisuus voittaa minut. Luultavasti kuitenkaan ei.\n\n\"Laumanjohtajana minun pit\u00e4isi varmaan pit\u00e4\u00e4 tuota ep\u00e4reiluna, mutta poikayst\u00e4v\u00e4n\u00e4si ajattelen, ett\u00e4 tuo kusip\u00e4\u00e4 ansaitsee aivan kaiken, mit\u00e4 vain pystyt tekem\u00e4\u00e4n h\u00e4nelle.\"\n\nKatsoin olkani yli. Mikael seisoi takanani k\u00e4det ristiss\u00e4 rinnalla. H\u00e4n oli ottanut mahdollisimman rennon asennon, vaikka se tuskin meni l\u00e4pi kovinkaan monelle. Kullanv\u00e4rinen katse paljasti h\u00e4nen todellisen tunnetilansa. H\u00e4n oli lis\u00e4kseni ainoa toinen animoitu hahmo t\u00e4m\u00e4n elokuvan leikkausp\u00f6yd\u00e4ll\u00e4, enk\u00e4 voinut olla ajattelematta, ett\u00e4 n\u00e4in sen kuuluikin olla: Me muuta maailmaa vastaan.\n\nEn ollut koskaan kuullut Mikaelin kutsuvan ket\u00e4\u00e4n kusip\u00e4\u00e4ksi. Olin kuitenkin samaa mielt\u00e4: Joni oli kusip\u00e4\u00e4. Toiveissani oli, ett\u00e4 h\u00e4n olisi pian entinen kusip\u00e4\u00e4, eik\u00e4 se toivottavasti vaatisi h\u00e4nen henkens\u00e4 viemist\u00e4. Halusin uskoa, ett\u00e4 ihmiset saattoivat oppia virheist\u00e4\u00e4n. Tulla paremmiksi. Luoja tiesi, ett\u00e4 itse ainakin yritin.\n\n\"N\u00e4en t\u00e4\u00e4lt\u00e4 asti, ett\u00e4 h\u00e4nen kulmansa on aivan v\u00e4\u00e4r\u00e4\", Mikael sanoi. H\u00e4n alkoi k\u00e4vell\u00e4 minua kohti.\n\n\"Uskotko, ett\u00e4 voisin voittaa h\u00e4net pelkk\u00e4n\u00e4 ihmisen\u00e4?\" kysyin.\n\n\"Tietenkin. Olen otellut h\u00e4nt\u00e4 vastaan useasti ja tied\u00e4n, ett\u00e4 sin\u00e4 olet parempi. En usko, ett\u00e4 h\u00e4n on treenannut viime kuukausina.\"\n\nKatsoin taas Jonia. \"H\u00e4n on iso\", ajattelin \u00e4\u00e4neen.\n\n\"Niin on. Mutta h\u00e4n on hidas ja k\u00f6mpel\u00f6. Kun h\u00e4n on maassa, h\u00e4n pysyy siell\u00e4.\"\n\nOtin pari askelta taaksep\u00e4in ja k\u00e4\u00e4nnyin kohti Mikaelia, puoleen, joka oli ty\u00f6nt\u00e4nyt k\u00e4det taskuihinsa. Katsoin ymp\u00e4rill\u00e4ni olevia susia. Nikon kasvot olivat kalpeat ja kauhun kyll\u00e4st\u00e4m\u00e4t. Mitja yritti parhaansa n\u00e4ytt\u00e4\u00e4 kyll\u00e4styneelt\u00e4, mutta h\u00e4nen aavistuksen levinneet silm\u00e4ns\u00e4 paljastivat, ett\u00e4 h\u00e4n oli tuntenut jotain hetke\u00e4 ennen kuin olin lukinnut ajan.\n\nKiersin p\u00e4\u00e4t\u00e4ni ensin toiselle, sitten toiselle puolelle. Niskanikamani naksuivat.\n\n\"Mit\u00e4 sin\u00e4 haluat minun tekev\u00e4n?\" kysyin samalla kun aloin py\u00f6ritell\u00e4 hartioitani.\n\n\"En voi antaa sinulle neuvoja\", Mikael sanoi. \"T\u00e4m\u00e4 ottelu m\u00e4\u00e4rittelee hyvin pitk\u00e4lti sinun asemasi laumassa. Sinun pit\u00e4\u00e4 p\u00e4\u00e4tt\u00e4\u00e4 itse, miten haluat heid\u00e4n n\u00e4kev\u00e4n sinut.\"\n\nKiersin vy\u00f6t\u00e4r\u00f6\u00e4ni puolelta toiselle, ja aloin sitten py\u00f6ritell\u00e4 k\u00e4sivarsiani. Uskoin ymm\u00e4rt\u00e4v\u00e4ni, mit\u00e4 Mikael tarkoitti. Voisin olla k\u00e4ytt\u00e4m\u00e4tt\u00e4 kykyj\u00e4ni ja kaikki pit\u00e4isiv\u00e4t minua edelleen ihmisen\u00e4. Tai sitten voisin n\u00e4ytt\u00e4\u00e4 kaikille, mihin oikeasti pystyin.\n\n\"Jos n\u00e4yt\u00e4n heille, mihin pystyn, he alkavat pel\u00e4t\u00e4 minua\", sanoin.\n\n\"Luultavasti\", Mikael my\u00f6nsi.\n\n\"Mutta jos pid\u00e4n kykyni salassa ja he saavat tiet\u00e4\u00e4 niist\u00e4 vasta my\u00f6hemmin, he tuntevat itsens\u00e4 petetyiksi.\"\n\nMikael ei sanonut mit\u00e4\u00e4n. Venyttelin takareisi\u00e4ni samalla kun punnitsin vaihtoehtojani. Tiesin jo, mit\u00e4 halusin tehd\u00e4. Halusin vain olla varma, ett\u00e4 se olisi oikein ja tapahtuisi oikeista syist\u00e4.\n\nLopulta suoristauduin. \"En koskaan ajatellut, ett\u00e4 perheesi el\u00e4m\u00e4 olisi t\u00e4llaista.\"\n\nOlin n\u00e4hnyt Sarrien el\u00e4m\u00e4st\u00e4 etup\u00e4\u00e4ss\u00e4 sen loisteliaan puolen: hienot autot ja vallan. Olin pitk\u00e4\u00e4n kuvitellut, ett\u00e4 Daniel teki mit\u00e4 huvitti, ja vaikka viime kes\u00e4n\u00e4 olin tajunnut h\u00e4nen asemaansa kuuluvan muutakin kuin muiden k\u00e4skytt\u00e4mist\u00e4, vasta nyt aloin n\u00e4hd\u00e4, mit\u00e4 se oli todella maksanut.\n\nMikaelin ilme pehmeni. H\u00e4n otti k\u00e4det pois taskuista ja tuli l\u00e4hemm\u00e4s. \"En aio valehdella: minun el\u00e4m\u00e4ni tulee aina olemaan t\u00e4llaista. Jatkuvaa neuvottelemista, p\u00e4\u00e4t\u00f6ksentekoa ja esill\u00e4 oloa. T\u00e4m\u00e4 ei tule loppumaan \u2013 yksi ongelma korvautuu aina uudella.\" H\u00e4n puristi olkap\u00e4\u00e4t\u00e4ni. \"\u00c4l\u00e4 ymm\u00e4rr\u00e4 v\u00e4\u00e4rin: min\u00e4 en valita. Olen \u00e4\u00e4rimm\u00e4isen etuoikeutettu. Etuoikeutettu el\u00e4m\u00e4 ei kuitenkaan ole aina sellaista, kuin monet kuvittelevat. Haluan vain, ett\u00e4 tied\u00e4t, ettei kanssani tule aina olemaan helppoa.\"\n\nAivan kuin h\u00e4n olisi halunnut tehd\u00e4 maailmasta t\u00e4ydellisen vain minua varten. Mutta h\u00e4n ei ollut luvannut minulle mit\u00e4\u00e4n ruusutarhaa, enk\u00e4 min\u00e4 ollut pyyt\u00e4nyt sit\u00e4. Riitti, ett\u00e4 minulla oli h\u00e4net.\n\nOlin nimitt\u00e4in alkanut ajatella, ettei el\u00e4m\u00e4n ollut tarkoituskaan olla helppoa. Ehk\u00e4 sen tuli olla juuri sen verran vaikeaa, ett\u00e4 siit\u00e4 selviytyisi vain yhdess\u00e4 t\u00e4rkeiden ihmisten kanssa. Ja se itsess\u00e4\u00e4n oli aivan mielet\u00f6n lahja.\n\n\"Sinun ei tarvitse tehd\u00e4 kaikkea yksin. Ei nyt, eik\u00e4 tulevaisuudessa\", lupasin. K\u00e4\u00e4nnyin katsomaan Jonia. \"Luullakseni minulla on ottelu oteltavana.\"\n\n\"Tied\u00e4tk\u00f6, mit\u00e4 teet?\"\n\nNy\u00f6kk\u00e4sin. \"T\u00e4m\u00e4 kyl\u00e4 on k\u00e4rsinyt jo ihan tarpeeksi salassa pidettyjen kykyjen takia.\"\n\nMikael ei sanonut mit\u00e4\u00e4n, mutta saatoin tuntea h\u00e4nen hyv\u00e4ksynt\u00e4ns\u00e4.\n\nOlin jo ottanut paikkani Jonin edess\u00e4, kun kuulin Mikaelin sanovan: \"Min\u00e4 en mene minnek\u00e4\u00e4n. Mit\u00e4 hyv\u00e4ns\u00e4 tapahtuukin, sinulla on minun tukeni. Aina. Kukaan ei satuta sinua.\"\n\nR\u00e4p\u00e4ytin silmi\u00e4ni. Kaikki el\u00e4m\u00e4 humahti taas p\u00e4\u00e4lle. Jonin potku l\u00e4hestyi ohimoani, mutta se oli helppo v\u00e4ist\u00e4\u00e4. Odotin h\u00e4nen seuraavaa liikett\u00e4\u00e4n, mutta sit\u00e4 ei tullut.\n\nKuulin yleis\u00f6n kohahduksen ja vasta sen j\u00e4lkeen tajusin, mit\u00e4 oli tapahtunut: min\u00e4 en ollut en\u00e4\u00e4 Raisa Oja, yhdeks\u00e4ntoistavuotias helsinkil\u00e4inen taideopiskelija. Minulla ei ollut en\u00e4\u00e4 yll\u00e4ni shortseja ja teepaitaa, vaan soturiasu, jonka olin pukenut niin monta kertaa ylleni saarella. Nyt se ei ollut en\u00e4\u00e4 pelkk\u00e4 vaate, vaan siit\u00e4 oli tullut osa minua. Olin daimon, joka piteli kaikkia t\u00e4m\u00e4n maailman avaimia k\u00e4siss\u00e4\u00e4n. Yksi ajatus, ja todellisuus olisi toinen.\n\nKoskin olkaani. Nahkahihna kulki sen yli. Tunsin veitset sel\u00e4ss\u00e4ni. En kuitenkaan tuntenut mink\u00e4\u00e4nlaista halua tarttua niihin. T\u00e4m\u00e4 kamppailu hoidettaisiin paljain k\u00e4sin.\n\nJoni oli per\u00e4\u00e4ntynyt aivan kent\u00e4n toiseen laitaan. Hetken aikaa vain tuijotimme toisiamme, ja n\u00e4in kuinka h\u00e4nen silm\u00e4ns\u00e4 levisiv\u00e4t pelosta. Vino hymy kohosi huulilleni. Leikki alkakoon, ajattelin, ja sitten aloin juosta h\u00e4nt\u00e4 kohti.\n\nJoni oli niin kauhuissaan, ettei h\u00e4n liikahtanutkaan. Ponnistin aivan viime hetkell\u00e4. K\u00e4ytin Jonin reitt\u00e4 astinlautanani ja k\u00e4\u00e4nnyin ilmassa juuri ajoissa; jalkani kiertyiv\u00e4t h\u00e4nen yl\u00e4vartalonsa ymp\u00e4rille ja samassa jo sukelsin kohti maata. Jonilla ei ollut mit\u00e4\u00e4n muuta vaihtoehtoa kuin kaatua kanssani. Kier\u00e4hdin takaisin jaloilleni, ja hetken kuluttua Jonikin onnistui nousemaan pystyyn.\n\nAnnoin h\u00e4nen tehd\u00e4 niin. Olin nimitt\u00e4in jo saanut tiet\u00e4\u00e4 kaiken tarvittavan. Mikael oli ollut oikeassa: min\u00e4 olin Jonia parempi. Mik\u00e4li halusin, ottelu olisi ohi ennen kuin se oli alkanutkaan.\n\nPienen hetken ajan harkitsin sit\u00e4. Daimon piti ajatuksesta. Joni oli satuttanut minua, ja nyt oli minun aikani kostaa. Pist\u00e4\u00e4 h\u00e4net maksamaan korkojen kera.\n\nVoisin tappaa Jonin. Minulla oli siihen oikeus. Kukaan ei voisi tuomita minua.\n\nK\u00e4velin edestakaisin kent\u00e4ll\u00e4. Ulkopuolisten silmiss\u00e4 se n\u00e4ytti varmaan silt\u00e4 kuin olisin vaaninut Jonia, mutta todellisuudessa yritin vain saada hermoni kuriin. \"Rauhoitu\", mutisin itselleni kreikaksi. \"Et halua tappaa h\u00e4nt\u00e4. Et oikeasti.\"\n\nKun Joni teki innottoman yrityksen heikommalle puolelleni, v\u00e4istin vasemmalle. Py\u00f6r\u00e4hdin ymp\u00e4ri ja kohotin jalkani.\n\nSe oli ennen kaikkea n\u00e4ytt\u00e4v\u00e4 potku, eik\u00e4 siis mitenk\u00e4\u00e4n j\u00e4rin tehokas. Se sattui my\u00f6s oleman yksi Mikaelin lempiliikkeist\u00e4, ja pienest\u00e4 kohahduksesta p\u00e4\u00e4ttelin, ett\u00e4 ainakin osa h\u00e4nen entisist\u00e4 kamppailututtavistaan oli tunnistanut sen.\n\n\"En halua tappaa sinua\", sanoin Jonille. Minun piti sanoa se uudestaan, kun tajusin puhuneeni kreikkaa.\n\nJoni ei vastannut, mutta se saattoi johtua my\u00f6s siit\u00e4, ettei h\u00e4n pystynyt sanomaan mit\u00e4\u00e4n. H\u00e4nen otsansa kiilsi ja paitaan oli ilmestynyt hikilaikkuja. Itse en ollut miss\u00e4\u00e4n vaiheessa edes heng\u00e4stynyt.\n\n\"Luovuta\", sanoin.\n\nH\u00e4n yritti hy\u00f6kk\u00e4yst\u00e4. Toisin kuin h\u00e4n kuvitteli, keskittymiseni ei suinkaan herpaantunut puhumisen takia. Annoin h\u00e4nen melkein tarttua minuun \u2013 en siksi, ett\u00e4 h\u00e4n kuvittelisi voivansa voittaa minut, vaan ainoastaan n\u00e4ytt\u00e4\u00e4kseni h\u00e4nelle, miten helppoa minun oli torjua h\u00e4net.\n\nSe oli kuitenkin virhe. Daimon oli nimitt\u00e4in alkanut ajatella, ettei kamppailu ollut erityisen kiinnostava. Ennen kuin huomasinkaan, Joni oli otteessani. Yksinkertainen liike, ja h\u00e4nen niskansa olisivat nurin.\n\n\"Min\u00e4 luovutan\", Joni sanoi. Ja koska en reagoinut heti mitenk\u00e4\u00e4n, h\u00e4n toisti lujempaa: \"Min\u00e4 luovutan!\"\n\nRaisa, kuulin veljeni \u00e4\u00e4nen p\u00e4\u00e4ni sis\u00e4ll\u00e4. Raisa, p\u00e4\u00e4st\u00e4 irti.\n\n\"Min\u00e4 hyv\u00e4ksyn\", sanoin nopeasti ja p\u00e4\u00e4stin irti. Mitja oli hetkess\u00e4 edess\u00e4ni ja ty\u00f6nsi minua k\u00e4mmenell\u00e4 taaksep\u00e4in. Daimon ei pit\u00e4nyt siit\u00e4 ollenkaan. Daimon halusi kuulla, kuinka luut rusahtelisivat...\n\n\"Nyt ihan rauhallisesti\", Mitja sanoi kreikaksi. Yritin katsoa h\u00e4nen ohitseen, mutta ruskeat silm\u00e4t tulivat itsepintaisesti tielleni. \"Hengit\u00e4.\"\n\nJa min\u00e4 hengitin. Leski astui esiin ja sanoi jotain, mutta en kuunnellut. Suljin silm\u00e4ni. Yritin palauttaa itseni takaisin todellisuuteen. Olin vaarallisen l\u00e4hell\u00e4 kadottaa koko ihmisyyteni. T\u00e4m\u00e4 maailma vaikutti niin hauraalta, kuin kartonkikyh\u00e4elm\u00e4lt\u00e4. Sen saisi potkittua kumoon aivan liian helposti.\n\nMitjan t\u00e4ytyi tuntea, kuinka taistelutahto alkoi viimein h\u00e4ipy\u00e4 lihaksistani, sill\u00e4 h\u00e4n laski k\u00e4tens\u00e4.\n\n\"Mik\u00e4 helvetti sin\u00e4 olet?\" joku kysyi.\n\nAvasin silm\u00e4ni. Katselin ymp\u00e4rill\u00e4ni olevia kasvoja. Pelokkaita ja ep\u00e4uskoisia katseita. Oli tullut aika vastata kysymyksiin.\n\n\"Min\u00e4 olen daimon\", aloitin. \"Daimonin ja suden tyt\u00e4r.\"\n\nSe ei tietenk\u00e4\u00e4n sanonut heille mit\u00e4\u00e4n. Ep\u00e4usko alkoi pikkuhiljaa vaihtua vihaan ja loukkaantumiseen.\n\n\"Te sanoitte haluavanne uuden vampyyrin. No, min\u00e4 ja veljeni olemme tulleet auttamaan. Itse asiassa\", kurotin selk\u00e4\u00e4ni ja otin veitsen k\u00e4teeni. Kohotin oikeaa k\u00e4tt\u00e4ni. \"Min\u00e4 pyyd\u00e4n saada liitty\u00e4 osaksi laumaa.\" En ollut aivan varma, miten t\u00e4m\u00e4n oli tarkoitus toimia. Lukemani perusteella kuitenkin uskoin, ett\u00e4 tarkoitus oli t\u00e4rke\u00e4mpi kuin varsinaiset sanat. Painoin ter\u00e4n k\u00e4mmeneeni. \"Min\u00e4 Raisa Oja, laumaan kuuluneen Tuire Ojan tyt\u00e4r, jumalten siunaama daimon ja ideoiden maailman portinvartija, vannon uskollisuuttani Hukkavaaran laumalle niin kauan kuin se on Mikael Sarrin alaisuudessa.\"\n\nHalusin heid\u00e4n tiet\u00e4v\u00e4n, mit\u00e4 he menett\u00e4isiv\u00e4t, jos he p\u00e4\u00e4tt\u00e4isiv\u00e4t k\u00e4\u00e4nt\u00e4\u00e4 selk\u00e4ns\u00e4 Mikaelille. Se ei ollut uhkailua, vaan totuus.\n\nTilanne oli taas saanut yll\u00e4tt\u00e4v\u00e4n k\u00e4\u00e4nteen, mutta tuskin kukaan oli siit\u00e4 yht\u00e4 h\u00e4mm\u00e4stynyt kuin Mikael. Tiesin, ett\u00e4 minun olisi pit\u00e4nyt varoittaa h\u00e4nt\u00e4 etuk\u00e4teen, mutta silloin h\u00e4n olisi yritt\u00e4nyt est\u00e4\u00e4 minua.\n\nJo toista kertaa saman p\u00e4iv\u00e4n aikana k\u00e4vimme tahtojen kamppailua. N\u00e4in heti, ettei Mikael halunnut ottaa minua laumaan. Itse asiassa olin varma, ett\u00e4 h\u00e4n sai vain vaivoin pidetty\u00e4 suuttumuksensa kurissa.\n\nViilt\u00e4m\u00e4ni haava oli paljon syvempi kuin olin tarkoittanut. Maahan oli jo ehtinyt muodostua lammikko, ja olin viitt\u00e4 vaille valmis panikoimaan.\n\n\"Min\u00e4 hyv\u00e4ksyn\", Mikael sanoi, \"ehdolla, ett\u00e4 voin erottaa Raisa Ojan laumasta milloin tahansa ilman erillist\u00e4 ilmoitusta tai selityst\u00e4.\"\n\nTakaportti, ajattelin. Mikael halusi suojella minua ja j\u00e4tti siksi itselleen takaportin. Muita selityksi\u00e4 en edes ehtinyt harkita, sill\u00e4 minua oli alkanut heikottaa.\n\n\"Hyv\u00e4ksyn\", sanoin, ja kurotin saman tien kohti ideoiden maailmaa. Suuntasin kaikki ajatukseni k\u00e4teeni. Pakokauhuisen hetken ajan pelk\u00e4sin, ettei minulla ollut en\u00e4\u00e4 tarpeeksi voimia j\u00e4ljell\u00e4 parantaakseni haavaa. Sitten viilto alkoi kuroutua umpeen ja saatoin taas hengitt\u00e4\u00e4. Purin hammasta, kun ihoa pisteli, mutta se oli nopeasti ohi. Annoin k\u00e4teni pudota sivulleni ja k\u00e4velin kohti Mikaelia. Soturipukuni katosi ja olin taas se Raisa Oja, johon sudet olivat tottuneet. Ainoa todiste \u00e4skeisist\u00e4 tapahtumista oli kent\u00e4n keskell\u00e4 oleva tumma lammikko.\n\n\"Kokoontuminen on p\u00e4\u00e4ttynyt\", Mikael julisti. H\u00e4n nappasi minua kiinni vy\u00f6t\u00e4r\u00f6lt\u00e4 ja l\u00e4hdimme yhdess\u00e4 k\u00e4velem\u00e4\u00e4n nopeasti pois paikalta.\n\nKatseeni osui Iivoon. K\u00e4\u00e4nnyin ymp\u00e4ri. \"Viel\u00e4 yksi asia. Jos teist\u00e4 kukaan p\u00e4\u00e4tt\u00e4\u00e4 vahingoittaa ket\u00e4\u00e4n ulkopuolista vain n\u00e4ytt\u00e4\u00e4kseen minulle tai Mikaelille, h\u00e4n tapaa minut t\u00e4ll\u00e4 kent\u00e4ll\u00e4.\"\n\nEn j\u00e4\u00e4nyt odottamaan vastausta. Olin jo alkanut t\u00e4rist\u00e4. Mikael k\u00e4yt\u00e4nn\u00f6ss\u00e4 raahasi minut mukanaan autolle, miss\u00e4 h\u00e4n viimein pys\u00e4htyi.\n\nOdotin, ett\u00e4 h\u00e4n olisi alkanut huutaa minulle. Sen sijaan h\u00e4n painoi minut autonovea vasten koko kehollaan. H\u00e4nen k\u00e4tens\u00e4 olivat ymp\u00e4rill\u00e4ni, enk\u00e4 olisi pysynyt pystyss\u00e4 ilman niit\u00e4. En n\u00e4hnyt h\u00e4nen kasvojaan, sill\u00e4 h\u00e4nen leukansa oli p\u00e4\u00e4lakeani vasten. H\u00e4n ei suudellut minua, ja hetken kuluttua k\u00e4sitin, ettei h\u00e4n aikonutkaan. H\u00e4n vain seisoi j\u00e4hmettyneen\u00e4 paikallaan.\n\nEn halunnut tehd\u00e4 mit\u00e4\u00e4n, mik\u00e4 olisi pahentanut h\u00e4nen tunnekuohuaan, mutta lopulta minun oli sanottava jotain. \"Kaikki on hyvin.\" \u00c4\u00e4neni oli k\u00e4he\u00e4. Mikael ei vastannut. Tartuin h\u00e4nt\u00e4 kyyn\u00e4rp\u00e4\u00e4st\u00e4. Kun sek\u00e4\u00e4n ei saanut aikaan mit\u00e4\u00e4n, kiersin k\u00e4sivarteni h\u00e4nen ymp\u00e4rilleen.\n\n\"Min\u00e4 olen kunnossa, kaikki on kunnossa.\" En varmaankaan ollut kovin vakuuttava, sill\u00e4 hampaani kalisivat. P\u00e4\u00e4h\u00e4ni sattui, ja vedet nousivat silmiini.\n\n\"Miten sin\u00e4 tiesit?\" h\u00e4n kysyi, ja vastasi sitten omaan kysymykseens\u00e4: \"Sin\u00e4 luit eilen illalla, miten laumaan liityt\u00e4\u00e4n.\"\n\n\"Niin.\"\n\n\"\u00c4l\u00e4 tee t\u00e4llaista en\u00e4\u00e4\", h\u00e4n kuiskasi.\n\n\"En\", lupasin.\n\nH\u00e4n huokaisi ja pujotti toisen k\u00e4tens\u00e4 niskaani. \"Kerro minulle heti, jos haluat eroon laumasta.\"\n\nNy\u00f6kk\u00e4sin h\u00e4nen olkaansa vasten. \"Min\u00e4 k\u00e4ytin sinun potkuasi\", sain sanotuksi.\n\n\"Min\u00e4 huomasin. Kiitos kunnianosoituksesta.\"\n\n\"Et voi v\u00e4itt\u00e4\u00e4, ettenk\u00f6 olisi romanttinen.\"\n\nH\u00e4n naurahti. H\u00e4n kohotti leukaani ja katsoi minua silmiin. \"Mit\u00e4 ihmett\u00e4 min\u00e4 teen sinun kanssasi?\"\n\n\"\u00c4l\u00e4 sano noin. Sano mieluummin: mit\u00e4 ihmett\u00e4 min\u00e4 tekisin ilman sinua?\"\n\n\"Olet oikeassa.\" H\u00e4n suuteli otsaani. Sitten h\u00e4n avasi autonoven ja k\u00e4yt\u00e4nn\u00f6ss\u00e4 nosti minut sis\u00e4\u00e4n istuimelle.\n\n\"Niko ja Mitja...\"\n\n\"\u00c4l\u00e4 huolehdi heist\u00e4\", Mikael sanoi ja paiskasi oven kiinni.\nVIIDES LUKU\n\nVietin loppup\u00e4iv\u00e4n kylpyhuoneen lattialla oksentamassa. En py\u00f6rtynyt, mink\u00e4 olisi ehk\u00e4 pit\u00e4nyt olla helpotus, mutta todellisuudessa olisin mieluummin viett\u00e4nyt koko illan niin, ettei siit\u00e4 olisi j\u00e4\u00e4nyt minulle mit\u00e4\u00e4n muistoja. Hoin itselleni, ett\u00e4 mik\u00e4li olin viel\u00e4 tajuissani, p\u00e4\u00e4kipuni oli pakko olla siedett\u00e4v\u00e4n rajoissa, mutta se ei tuntunut erityisen vakuuttavalta. Vedet valuivat jatkuvana virtana silmist\u00e4ni enk\u00e4 muistanut, koska oireeni olisivat viimeksi olleet n\u00e4in pahat.\n\nLopulta v\u00e4symys vei voiton. Nukuin, kunnes Mikael her\u00e4tti minut seuraavana aamuna. Juha oli ilmoittanut, ett\u00e4 naapurikunnan johtaja oli lupautunut tapaamaan meid\u00e4t jo saman p\u00e4iv\u00e4n aikana. Tapaaminen oli sovittu aivan Hukkahovin naapurissa sijaitsevalle kunnantalolle, joka, kuten Mikael selitti minulle ajomatkan aikana, toimi oikeasti Sarrien lauman toimistona.\n\nAjomatkan aikana ehdin viimein ajatella, mit\u00e4 oikein olin tullut tehneeksi edellisen\u00e4 p\u00e4iv\u00e4n\u00e4. P\u00e4\u00e4tt\u00e4ess\u00e4ni liitty\u00e4 laumaan en ollut varmuudella tiennyt, olisiko se edes mahdollista sellaiselle, joka ei ollut susi. Nyt, kun en en\u00e4\u00e4 voinut pahoin, saatoin tuntea eron. Kaikki oli tismalleen samanlaista, mutta kuitenkin uutta \u2013 aivan kuin tuuli olisi puhaltanut id\u00e4n sijaan l\u00e4nnest\u00e4.\n\nJa Mikael... Jouduin nytkin vastustamaan halua painautua h\u00e4nt\u00e4 vasten. H\u00e4nen tuoksunsa sai oloni hieman humalaiseksi. Ensimm\u00e4ist\u00e4 kertaa huomasin, miten susi sekoittui h\u00e4nen tuoksuunsa silloinkin, kun h\u00e4n oli ihmisen muodossa, eik\u00e4 yhdistelm\u00e4 ollut lainkaan huono. Siin\u00e4 oli maata ja mets\u00e4\u00e4 ja ainakin ripaus pihkaa. En ollut koskaan kokenut oloani yht\u00e4 turvalliseksi. \u00c4kki\u00e4 ymm\u00e4rsin paljon paremmin, miksi kyl\u00e4l\u00e4iset suhtautuivat Mikaeliin niin ihaillen.\n\nMit\u00e4 l\u00e4hemm\u00e4s kunnantaloa tulimme, sit\u00e4 pingottuneemmaksi Mikael k\u00e4vi. Tiesin, ettei se johtunut niink\u00e4\u00e4n tulevasta tapaamisesta kuin tarpeesta kertoa minulle jotakin.\n\nMikael pys\u00e4k\u00f6i auton l\u00e4helle p\u00e4\u00e4ovia. Silloin h\u00e4n alkoi viimein puhua. \"En halua, ett\u00e4 sin\u00e4 k\u00e4yt\u00e4t viel\u00e4 kykyj\u00e4si. Katsotaan, onnistummeko me hoitamaan t\u00e4m\u00e4n ilman yliluonnollisia keinoja.\"\n\n\"Miksi?\" kysyin ihmeiss\u00e4ni.\n\n\"Se tapa, jolla sinun kroppasi reagoi sinun kykyihisi... En usko, ett\u00e4 se on ihan vaaratonta.\"\n\n\"Se on vain pahoinvointia\", sanoin. Tosiasiassa olin hieman j\u00e4rkyttynyt. Mikael oli juuri sanonut \u00e4\u00e4neen yhden pahimmista peloistani.\n\nH\u00e4n ei katsonut minuun. \"\u00c4l\u00e4 huoli. Mit\u00e4 suurimmalla todenn\u00e4k\u00f6isyydell\u00e4 pelko on turha. Min\u00e4 vain haluan pelata varman p\u00e4\u00e4lle. On turha pist\u00e4\u00e4 sinua k\u00e4rsim\u00e4\u00e4n ennen kuin olemme varmoja, ett\u00e4 se on ainoa vaihtoehto.\"\n\n\"Niin kai\", sanoin. En ollut varma, johtuiko my\u00f6ntymiseni laumaan liittymisest\u00e4 vai siit\u00e4, ett\u00e4 uskoin h\u00e4nen olevan oikeassa.\n\nKun menimme sis\u00e4\u00e4n rakennukseen, Juha ja Jenni istuivat aulassa odottamassa. He nousivat yl\u00f6s n\u00e4hdess\u00e4\u00e4n meid\u00e4t.\n\nHeid\u00e4n seurassaan oli mies, jota en ollut n\u00e4hnyt koskaan aikaisemmin. H\u00e4nen t\u00e4ytyi olla kunnanjohtaja \u2013 ainakin h\u00e4n n\u00e4ytti kunnanjohtajalta. Pieni, kaljuuntuva mies, jolla oli yll\u00e4\u00e4n huonosti istuva puku ja kirjava solmio, jonka kuosi toi mielen edellisen\u00e4 iltana vessan lattialla vietetyt hetket. Vatsaani alkoi v\u00e4\u00e4nt\u00e4\u00e4 pelk\u00e4st\u00e4 ajatuksesta.\n\nMies katseli ymp\u00e4rilleen kunnes h\u00e4nen tuijotuksensa j\u00e4m\u00e4hti Mikaeliin. Kun Mikael astui huoneeseen, oli l\u00e4hes mahdotonta katsoa minnek\u00e4\u00e4n muualle. Eik\u00e4 se koskenut pelk\u00e4st\u00e4\u00e4n minua.\n\n\"Haluan tavata kunnanjohtaja Daniel Sarrin\", h\u00e4n sanoi lopulta. En ollut tiennyt, ett\u00e4 Daniel oli ollut virallisesti Hukkavaaran kunnanjohtaja, mutta se k\u00e4vi j\u00e4rkeen. En tiennyt, oliko mies koskaan tavannut Danielia, mutta ep\u00e4ilin sit\u00e4. Kukaan, joka oli tavannut h\u00e4net, ei lausuisi h\u00e4nen nime\u00e4\u00e4n niin huolettomaan s\u00e4vyyn.\n\nEnnen kaikkea halusin kuitenkin tiet\u00e4\u00e4, miten Mikael selitt\u00e4isi Danielin poissaolon.\n\n\"Se ei valitettavasti ole mahdollista\", Mikael sanoi. \"Is\u00e4lleni tuli \u00e4killinen meno ja h\u00e4n joutui matkustamaan pois. Yksityisasia\", h\u00e4n lis\u00e4si kun n\u00e4ki miehen avaavan suunsa. \"Uskon kuitenkin voivani auttaa. Min\u00e4 olen Mikael Sarri. Min\u00e4 ja minun yritykseni vastaa kyl\u00e4n susimatkailusta.\"\n\nKunnanjohtaja katsoi h\u00e4nt\u00e4 pitk\u00e4\u00e4n. Mikaelilla oli yll\u00e4\u00e4n lumilautamerkin teepaita ja farkut. Minun oli tunnustettava, etten ehk\u00e4 itsek\u00e4\u00e4n olisi aivan heti arvannut h\u00e4nt\u00e4 yritysjohtajaksi. Toisaalta en varmaan auttanut asiaa: yll\u00e4ni oli l\u00f6ys\u00e4 toppi ja farkkushortsit, jotka purkautuivat lahkeista.\n\n\"Sin\u00e4 olet liian nuori\", mies sanoi.\n\nJa sin\u00e4 olet aivan liian typer\u00e4 ik\u00e4iseksesi, ajattelin. Saatoin antaa anteeksi sen, ett\u00e4 h\u00e4n teki oletuksia i\u00e4n perusteella, mutten sit\u00e4, ett\u00e4 h\u00e4n syytti Mikaelia valehtelijaksi vain siksi, ettei t\u00e4m\u00e4 sopinut h\u00e4nen ennakkoluuloihinsa.\n\n\"Olen ehk\u00e4 nuori, mutta se ei tarkoita, ettenk\u00f6 tiet\u00e4isi, mit\u00e4 teen. Ja sin\u00e4 olet?\"\n\nMinulle tuli kylm\u00e4 pelk\u00e4st\u00e4\u00e4n katsoessani Mikaelin silmi\u00e4. Ihmisenkin olisi tullut tajuta, ettei h\u00e4nen kanssaan pelleilty. En tiennyt, oliko herra kunnanjohtaja rohkea, huonon\u00e4k\u00f6inen vai sittenkin vain tyhm\u00e4, sill\u00e4 h\u00e4n ei per\u00e4\u00e4ntynyt sentti\u00e4k\u00e4\u00e4n.\n\n\"Esko Kukkonen.\" He k\u00e4tteliv\u00e4t j\u00e4yk\u00e4sti. Mikael esitteli pikaisesti muut huoneessa olijat. Panin merkille, ettei h\u00e4n katsonut minua kun h\u00e4n kertoi Eskolle, kuka olin. Yritin olla vet\u00e4m\u00e4tt\u00e4 siit\u00e4 mit\u00e4\u00e4n johtop\u00e4\u00e4t\u00f6ksi\u00e4, mutta se oli vaikeaa, kun otti huomioon, ett\u00e4 h\u00e4n oli pit\u00e4nyt minuun et\u00e4isyytt\u00e4 siit\u00e4 l\u00e4htien, kun olimme tulleet huoneeseen. Tiesin, ett\u00e4 h\u00e4n halusi hoitaa puhumisen, mutta pelk\u00e4sin, ett\u00e4 h\u00e4nen k\u00e4yt\u00f6kselleen oli olemassa toinen selitys.\n\nKaikesta huolimatta yritin n\u00e4ytt\u00e4\u00e4 mahdollisimman lauhkealta. En ollut varautunut lainkaan siihen, ett\u00e4 suurin haasteemme olisi saada Esko vakuutetuksi siit\u00e4, ett\u00e4 meill\u00e4 oli valtuudet keskustella asiasta.\n\n\"Jotkut teid\u00e4n asukkaanne ovat est\u00e4neet miehi\u00e4mme tulemasta alueellenne\", Esko aloitti. \"Se rikkoo jokamiehenoikeuksia.\"\n\n\"Heill\u00e4 oli syyt\u00e4 ep\u00e4ill\u00e4, ett\u00e4 teid\u00e4n miehenne aikoivat salakaataa susia meid\u00e4n alueellamme\", Mikael sanoi rauhallisella \u00e4\u00e4nell\u00e4. \"He halusivat ainoastaan est\u00e4\u00e4 rikoksen.\"\n\n\"Te olette houkutelleet pedot t\u00e4nne. Ne hy\u00f6kk\u00e4\u00e4v\u00e4t meid\u00e4n lastemme ja lemmikkiemme kimppuun. Ne pit\u00e4\u00e4 kaataa, ennen kuin mit\u00e4\u00e4n pahaa tapahtuu.\"\n\nEn voinut olla vilkaisematta Juhan ja Jennin ilmeit\u00e4. Heid\u00e4n pokkansa piti ihailtavasti.\n\n\"Meid\u00e4n elinkeinomme on riippuvainen susista\", Mikael sanoi. \"Me emme voi antaa teid\u00e4n romuttaa koko talouttamme irrationaalisten pelkojen takia.\"\n\n\"Meid\u00e4n pelkomme eiv\u00e4t ole irrationaalisia.\" H\u00e4nen naamansa oli punainen, aivan kuin pelkk\u00e4 puhuminen olisi heng\u00e4stytt\u00e4nyt h\u00e4nt\u00e4. \"Meill\u00e4 on todisteita suden liikkumisesta asuinalueiden l\u00e4hettyvill\u00e4.\"\n\n\"Mist\u00e4 te tied\u00e4tte, ett\u00e4 susi tuli nimenomaan Hukkavaarasta?\" kysyin.\n\n\"Kyll\u00e4h\u00e4n sen nyt arvaa. Te ruokitte niit\u00e4 t\u00e4\u00e4ll\u00e4.\"\n\nEn voinut est\u00e4\u00e4 tuhahdustani.\n\n\"Ja ent\u00e4 meid\u00e4n elinkeinomme?\" Esko jatkoi. \"Joillakin on karjaa.\"\n\n\"Tek\u00e4\u00e4n ette voi kielt\u00e4\u00e4, etteik\u00f6 Hukkavaara olisi yksi vireimmist\u00e4 alueista koko Kainuussa\", Mikael sanoi. \"Me ved\u00e4mme alueelle turisteja, joista monet k\u00e4yv\u00e4t my\u00f6s teid\u00e4n kunnassanne ostoksilla. Teid\u00e4n asukkaanne vuokraavat m\u00f6kkej\u00e4\u00e4n meid\u00e4n hiihtokeskuksemme asiakkaille. Jos te teette meid\u00e4n el\u00e4m\u00e4mme hankalaksi, se kolahtaa ennen pitk\u00e4\u00e4 teid\u00e4n omaan nilkkaanne.\"\n\n\"Yrit\u00e4ttek\u00f6 te uhkailla minua?\" Esko kysyi.\n\n\"Emme me mit\u00e4\u00e4n uhkaile. Me sanomme niin kuin asia on. Meill\u00e4 ei ole mit\u00e4\u00e4n syyt\u00e4 uskoa, ett\u00e4 sudet ovat vaaraksi ihmisille tai ett\u00e4 meid\u00e4n liiketoiminnastamme olisi teid\u00e4n kunnallenne mit\u00e4\u00e4n haittaa.\"\n\n\"Meid\u00e4n asukkaamme n\u00e4ki suden kirkkaassa p\u00e4iv\u00e4nvalossa. Toiselta on kadonnut lammas. Eik\u00f6 siin\u00e4 ole ihan riitt\u00e4v\u00e4sti todisteita?\"\n\nEi taatusti, ajattelin. Min\u00e4 jos joku tiesin, ett\u00e4 ihmiset n\u00e4kiv\u00e4t, mit\u00e4 itse halusivat. Mikaelilla oli j\u00e4rke\u00e4 olla sanomatta sit\u00e4 \u00e4\u00e4neen. \"Sudet ovat uhanalaisia. Yht\u00e4k\u00e4\u00e4n sutta ei tule tappaa ilman hyv\u00e4\u00e4 syyt\u00e4.\"\n\n\"Me aiomme hankkia tarvittavat luvat\", Esko mutisi. He eiv\u00e4t siis olleet viel\u00e4 tehneet sit\u00e4. Juhan ja Mikaelin mukaan poikkeuskaatoluvan saaminen oli hyvin ep\u00e4todenn\u00e4k\u00f6ist\u00e4, ja toivoin, ett\u00e4 naapurikuntalaiset odottaisivat p\u00e4\u00e4t\u00f6st\u00e4, ennen kuin he l\u00e4htisiv\u00e4t mets\u00e4lle. Se antaisi meille aikaa suostutella heid\u00e4t luopumaan koko hankkeesta.\n\n\"Te varmasti tied\u00e4tte, ett\u00e4 kes\u00e4ll\u00e4 sudenmets\u00e4stys on hyvin vaikeaa\", Mikael sanoi.\n\n\"Me teemme, mit\u00e4 meid\u00e4n t\u00e4ytyy perheidemme turvallisuuden vuoksi. Kuntien sopuisan naapuruuden kannalta toivon, ett\u00e4 te puhutte maanomistajille siihen malliin, ett\u00e4 he p\u00e4\u00e4st\u00e4v\u00e4t miehemme mailleen.\"\n\nOli ollut alusta asti selv\u00e4\u00e4, ettei neuvottelusta tulisi mit\u00e4\u00e4n. Eskolla oli valitettavasti laki puolellaan, eik\u00e4 ollut mit\u00e4\u00e4n, mit\u00e4 Mikael olisi voinut sanoa h\u00e4lvent\u00e4\u00e4kseen pelkoja. Faktoista ei ollut hy\u00f6ty\u00e4. Naapurikuntalaiset olivat jo p\u00e4\u00e4tt\u00e4neet kantansa, ja meid\u00e4n teht\u00e4v\u00e4ksemme j\u00e4isi keksi\u00e4, miten est\u00e4isimme heit\u00e4 ampumasta ket\u00e4\u00e4n lauman j\u00e4senist\u00e4.\n\n\"Min\u00e4 suostuisin seuraavanlaiseen j\u00e4rjestelyyn: jos sudesta n\u00e4kyy teid\u00e4n pihoillanne viel\u00e4 pienint\u00e4k\u00e4\u00e4n merkki\u00e4, tuen teit\u00e4 kaikissa kaatoyrityksiss\u00e4. Itse asiassa ilmoittaudun vapaaehtoiseksi mukaan. Mutta sit\u00e4 ennen antakaa meid\u00e4n tutkia itse omat maamme j\u00e4lkien varalta.\"\n\nEhdotus taisi yll\u00e4tt\u00e4\u00e4 Esko Kukkosen, sill\u00e4 h\u00e4n n\u00e4ytti jopa harkitsevan asiaa. \"No, sovitaan niin. Mutta sanon jo nyt, ett\u00e4 meid\u00e4n kunnassa asuu sellaisia oikein vanhan koulukunnan miehi\u00e4. He ovat valmiit ottamaan oikeuden omiin k\u00e4siins\u00e4 kun kyse on heid\u00e4n perheidens\u00e4 turvallisuudesta. Ei ole kunnan vika, jos jotkut meid\u00e4n asukkaistamme sortuvat mets\u00e4stysrikokseen.\"\n\n***\n\n\"Mist\u00e4 voit tiet\u00e4\u00e4, ettei susi en\u00e4\u00e4 liiku asuinalueella?\" kysyin autossa matkalla takaisin Sarrien talolle.\n\n\"En mist\u00e4\u00e4n. Min\u00e4 yritin vain pelata meille aikaa\", Mikael tunnusti.\n\n\"Mit\u00e4 seuraavaksi?\"\n\n\"Seuraavaksi me yrit\u00e4mme selvitt\u00e4\u00e4, miten t\u00e4m\u00e4 tilanne on saanut alkunsa. Jonkun on ment\u00e4v\u00e4 tutkimaan j\u00e4lki\u00e4. Jos t\u00e4\u00e4ll\u00e4 liikkuu oikea susi, voimme yritt\u00e4\u00e4 ajaa sen pois. Jos se ei onnistu, joudumme kaatamaan sen. Jos se taas oli yksi meist\u00e4, on meill\u00e4 edess\u00e4mme j\u00e4lleen yksi oikeudenk\u00e4ynti.\" H\u00e4n selv\u00e4sti piti ensimm\u00e4ist\u00e4 vaihtoehtoa paljon mieluisampana kuin j\u00e4lkimm\u00e4ist\u00e4.\n\n\"Ent\u00e4 jos se ei ollut kumpikaan?\" kysyin. \"He ovat aivan hyvin voineet lavastaa koko jutun.\"\n\nH\u00e4n mietti asiaa. \"En usko, ett\u00e4 he vajoaisivat sent\u00e4\u00e4n niin alas. Vaikka onhan se tietenkin mahdollista.\" H\u00e4n huokaisi. \"Joka tapauksessa me saamme sen selville. Tai ainakin olisimme saaneet, jos j\u00e4lki\u00e4 olisi alettu tutkia ajoissa. Nyt saattaa olla liian my\u00f6h\u00e4ist\u00e4.\"\n\n\"He eiv\u00e4t halunneet ottaa riski\u00e4\", sanoin.\n\n\"Tied\u00e4n.\"\n\n\"Viime p\u00e4ivin\u00e4 ei ole onneksi satanut.\"\n\n\"Ei niin\", Mikael my\u00f6nsi. H\u00e4nen ilmeens\u00e4 kirkastui hieman.\n\n***\n\nKukaan ei esitt\u00e4nyt eri\u00e4v\u00e4\u00e4 mielipidett\u00e4\u00e4n, kun Mikael ilmoitti p\u00e4\u00e4t\u00f6ksest\u00e4\u00e4n tutkia naapurikunnan maita suden muodossa. Juha ja Jenni ilmoittautuivat heti vapaaehtoisiksi mukaan, ja Mikael suostui. Selv\u00e4\u00e4 oli, ett\u00e4 tutkimusretki oli teht\u00e4v\u00e4 y\u00f6ll\u00e4. He joutuisivat liikkumaan l\u00e4hell\u00e4 ihmisten koteja, eik\u00e4 kukaan saanut n\u00e4hd\u00e4 heit\u00e4. He tietenkin j\u00e4tt\u00e4isiv\u00e4t hajuj\u00e4lki\u00e4 koirien haistettaviksi, mutta sille ei oikein voinut mit\u00e4\u00e4n.\n\nKun Juha ja Jenni saapuivat talolle, he toivat mukanaan uutisia.\n\n\"Joni haluaa tulla mukaan\", Juha sanoi.\n\n\"Ei\", Mikael sanoi oitis. En todellakaan ihmetellyt h\u00e4nen tylyytt\u00e4\u00e4n. En itsek\u00e4\u00e4n halunnut n\u00e4hd\u00e4 Jonia v\u00e4h\u00e4\u00e4n aikaan, saati sitten tehd\u00e4 h\u00e4nen kanssaan mink\u00e4\u00e4nlaista yhteisty\u00f6t\u00e4.\n\n\"H\u00e4n haluaa pest\u00e4 kasvonsa kaikkien edess\u00e4\", Juha selitti. En yll\u00e4ttynyt lainkaan siit\u00e4, ett\u00e4 h\u00e4nest\u00e4 oli tullut rauhanv\u00e4litt\u00e4j\u00e4, mutta t\u00e4ll\u00e4 nimenomaisella hetkell\u00e4 h\u00e4n \u00e4rsytti minua. \"H\u00e4nt\u00e4 n\u00f6yryytettiin eilen oikein kunnolla.\"\n\nAvasin suuni, mutta Juha ehti ensin. \"Tied\u00e4n, sinulla oli t\u00e4ysi oikeus ja niin edelleen. Mutta se ei muuta sit\u00e4, ett\u00e4 Jonille tullaan nauramaan hyvin pitk\u00e4\u00e4n siksi, ett\u00e4 tytt\u00f6 pieksi h\u00e4net.\"\n\n\"Voin piest\u00e4 aika monta muutakin t\u00e4ss\u00e4 kyl\u00e4ss\u00e4, jos se auttaa asiaa\", murisin.\n\n\"Joni nyt kuitenkin sattuu olemaan esimerkki siit\u00e4, miten Mikael k\u00e4sittelee ongelmatapauksia. On meid\u00e4n etumme mukaista, ett\u00e4 h\u00e4n sulautuu taas osaksi laumaa.\"\n\nSe sai minut harkitsemaan uudelleen. Minun oli my\u00f6nnett\u00e4v\u00e4, ett\u00e4 kaunani oli j\u00e4rkev\u00e4n p\u00e4\u00e4t\u00f6ksenteon tiell\u00e4. Tiesin Juhan olevan oikeassa. Luottamus oli vastavuoroista. Jos Jonin haluttiin olevan luottamuksen arvoinen, t\u00e4ytyi h\u00e4nelle antaa mahdollisuus todistaa se.\n\n\"\u00c4h. Annetaan h\u00e4nen menn\u00e4 mukaan\", sanoin. Mikael katsoi minua. \"P\u00e4\u00e4t\u00f6s on tietenkin sinun\", lis\u00e4sin nopeasti.\n\n\"Hyv\u00e4 on\", Mikael sanoi. \"Nelist\u00e4\u00e4n meid\u00e4n pit\u00e4isi selvit\u00e4 teht\u00e4v\u00e4st\u00e4 kohtalaisen nopeasti. N\u00e4hd\u00e4\u00e4n t\u00e4\u00e4ll\u00e4 illalla kello kymmenen. Ehdimme suunnitella hieman, ennen kuin aurinko laskee. Meid\u00e4n pit\u00e4\u00e4 ainakin jakaa alueet kesken\u00e4mme.\"\n\n\"Onko sinulla suunnitelmia t\u00e4lle p\u00e4iv\u00e4lle?\" kysyin Juhan ja Jennin l\u00e4hdetty\u00e4.\n\n\"Haluan viel\u00e4 tarkistaa is\u00e4n arkistoista muutaman jutun ennen iltaa.\"\n\n\"Voisitko heitt\u00e4\u00e4 minut hotellille? Haluaisin menn\u00e4 tapaamaan Jaskaa, ja Mitjankin on jo aika tavata h\u00e4net.\"\n\nMikael suostui. H\u00e4n oli mietteli\u00e4s koko ajomatkan ajan, ja aloin uskoa tehneeni oikein, kun olin p\u00e4\u00e4tt\u00e4nyt antaa h\u00e4nen olla rauhassa muutaman tunnin. H\u00e4nell\u00e4 oli selv\u00e4sti paljon ajateltavaa.\n\n\"Soita minulle, jos tarvitset kyyti\u00e4\", h\u00e4n sanoi kun tulimme Hukkahovin eteen.\n\nLupasin soittaa. Suljin auton oven ja k\u00e4velin sis\u00e4\u00e4n hotellin vastaanottoon. Olin l\u00e4hett\u00e4nyt matkalla Mitjalle viestin, ja h\u00e4n istui jo aulassa odottamassa. Niin kuin nyky\u00e4\u00e4n aina, h\u00e4nell\u00e4 oli kuulokkeet kaulallaan. Olisin voinut aivan hyvin tatuoida ne h\u00e4nen ihoonsa.\n\nMinut n\u00e4hdess\u00e4\u00e4n h\u00e4n nousi sanaakaan sanomatta yl\u00f6s.\n\n\"Onko Caoimhe tulossa?\" kysyin.\n\nVeljeni pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Pyysin h\u00e4nt\u00e4 kyll\u00e4, mutta h\u00e4nen mielest\u00e4\u00e4n t\u00e4m\u00e4 on perheasia.\"\n\nCaoimhe oli ep\u00e4ilem\u00e4tt\u00e4 ajatellut, ett\u00e4 Mitjan oli parempi tavata enonsa ensin ilman h\u00e4nt\u00e4. Olin salaa samaa mielt\u00e4. En edes tiennyt, puhuiko Jaska englantia.\n\nL\u00e4hdimme k\u00e4velem\u00e4\u00e4n verkkaiseen tahtiin. Kun p\u00e4\u00e4simme varsinaiselle kyl\u00e4nraitille, huomasin alkaneeni osoitella rakennuksia kuin mik\u00e4kin turistiopas selitt\u00e4en, mik\u00e4 mik\u00e4kin oli. Vastaantulijat tunnistivat minut, mutta kukaan ei tullut juttelemaan. Uskoin sen johtuvan Mitjasta, sill\u00e4 h\u00e4n sai osakseen uteliaita katseita.\n\nTulimme Jaskan talon pihaan. Vilkaistuani Mitjan ilmett\u00e4 muistin taas, milt\u00e4 paikka oli mielest\u00e4ni vaikuttanut ensimm\u00e4isell\u00e4 kerralla. Se oli n\u00e4ytt\u00e4nyt kodilta, muttei miss\u00e4\u00e4n nimess\u00e4 omaltani. Nyt sen tuttuus yll\u00e4tti minut, sill\u00e4 en ollut asunut t\u00e4\u00e4ll\u00e4 en\u00e4\u00e4 pitk\u00e4\u00e4n aikaan. Kukkalaatikot olivat vanhoilla paikoillaan, ja t\u00e4n\u00e4 vuonna niihin oli istutettu lobelioita.\n\nNousin askelmat yl\u00f6s ja soitin ovikelloa. Odotimme hetken, mutta sis\u00e4lt\u00e4 ei kuulunut mit\u00e4\u00e4n. Yritin uudestaan.\n\nKaikesta p\u00e4\u00e4tellen Jaska ei ollut kotona. Auto oli pihassa, joten h\u00e4n ei ollut voinut l\u00e4hte\u00e4 minnek\u00e4\u00e4n kauas. Avasin oven omilla avaimillani ja menimme sis\u00e4\u00e4n.\n\n\"Oletko varma, ett\u00e4 t\u00e4m\u00e4 on ihan ookoo?\" Mitja kysyi.\n\n\"Tietenkin on\", sanoin. Johdatin h\u00e4net keitti\u00f6\u00f6n. \"Otatko kahvia?\" En odottanut h\u00e4nen vastaustaan, vaan avasin kaapinoven. Kahvipurkki l\u00f6ytyi tutulta hyllylt\u00e4.\n\nIstuimme p\u00f6yd\u00e4n \u00e4\u00e4reen. Seurasin Mitjan katsetta kun h\u00e4n katseli ymp\u00e4rilleen. P\u00f6yd\u00e4ll\u00e4 oli edelleen sama muovinen p\u00f6yt\u00e4liina. Sanomalehti, Tekniikan Maailma ja kasa mainoksia olivat tutulla paikallaan p\u00f6yd\u00e4nkulmalla. Sein\u00e4ll\u00e4 oli Hukkavaaran mainoskalenteri.\n\nKahvinkeitin oli juuri lakannut korahtelemasta, kun ovelta kuului \u00e4\u00e4ni\u00e4. Istuimme paikoillamme ja odotimme. N\u00e4in mieless\u00e4ni, kuinka Jaska pys\u00e4htyi eteisess\u00e4 analysoidakseen hajuja.\n\n\"Raisa?\" h\u00e4n kysyi.\n\n\"Keitti\u00f6ss\u00e4\", sanoin, vaikkei olisi tarvinnut. Jaska ilmestyi oviaukkoon.\n\nEnoni oli kaikin tavoin tavallisen oloinen suomalainen mies. Ei vanha, ei nuori. Ei pitk\u00e4, ei lyhyt. Hiukset olivat maantienv\u00e4riset, silm\u00e4t siniset. Joku muu olisi ehk\u00e4 kuvaillut h\u00e4nt\u00e4 luotettavan n\u00e4k\u00f6iseksi, mutta minulla oli ollut alussa vaikeuksia uskoa h\u00e4nen vilpitt\u00f6myyteens\u00e4. Nyky\u00e4\u00e4n otaksuin jo tiet\u00e4v\u00e4ni kaikki h\u00e4nen salaisuutensa.\n\n\"Moi\", sanoin.\n\nJaskan katse lipui nopeasti minusta Mitjaan.\n\n\"Onko tuo h\u00e4n?\" Mitja kysyi kreikaksi. Aivan niin kuin tulija olisi voinut olla joku muu kuin Jaska.\n\n\"Kyll\u00e4. H\u00e4n on meid\u00e4n enomme.\"\n\nJaska tuijotti yh\u00e4 Mitjaa. \"Jaska, t\u00e4ss\u00e4 on minun kaksoisveljeni Mitja\", sanoin suomeksi. \"Sinun siskonpoikasi.\"\n\nH\u00e4nen kasvoillaan k\u00e4v\u00e4isi tunteita, joita en tunnistanut. Hetken aikaa jopa luulin, ett\u00e4 h\u00e4n k\u00e4\u00e4ntyisi ja l\u00e4htisi talosta. Jaska ei ollut maailman paras k\u00e4sittelem\u00e4\u00e4n yll\u00e4tt\u00e4vi\u00e4 tilanteita.\n\nNousin yl\u00f6s ja aloin kaataa kahvia kuppeihin. Kun k\u00e4\u00e4nnyin takaisin Jaskan puoleen, n\u00e4in, ettei h\u00e4n ollut viel\u00e4k\u00e4\u00e4n t\u00e4ysin toipunut sokistaan. Rauhoittaakseni tilanteen p\u00e4\u00e4tin istuttaa meid\u00e4t kaikki olohuoneeseen. Asetuttuamme sohville Mitja katseli ymp\u00e4rilleen ja yritti selv\u00e4sti v\u00e4ltell\u00e4 Jaskan katsetta. Jaska turvautui kahvikuppiin, jonka olin tuonut mukanani keitti\u00f6st\u00e4.\n\nSukuni miehet. Minun teki mieli nauraa.\n\n\"Min\u00e4 en tiennyt\", Jaska sanoi lopulta. \"Tuire ei koskaan kertonut mit\u00e4\u00e4n muuta kuin ett\u00e4 oli raskaana.\"\n\n\"Kukaan muu kuin Daniel ei tiennyt\", vastasin.\n\nJaska katsoi taas Mitjaa. \"Sin\u00e4 n\u00e4yt\u00e4t aivan h\u00e4nelt\u00e4.\"\n\nH\u00e4n olisi voinut viitata johonkin muuhunkin, mutta olin varma, ett\u00e4 h\u00e4n tarkoitti is\u00e4\u00e4mme.\n\n\"Sin\u00e4 siis tapasit h\u00e4net?\" Emme olleet koskaan puhuneet is\u00e4st\u00e4ni. Jostain syyst\u00e4 olin luullut, ett\u00e4 Jaska tiesi yht\u00e4 v\u00e4h\u00e4n kuin min\u00e4.\n\n\"H\u00e4n k\u00e4vi t\u00e4\u00e4ll\u00e4 kerran.\" N\u00e4in, ett\u00e4 h\u00e4n oli varuillaan.\n\n\"H\u00e4n on kuollut, joten ei ole mit\u00e4\u00e4n syyt\u00e4 varoa sanoja\", t\u00f6ks\u00e4ytin. Kaduin sit\u00e4 heti, kun n\u00e4in Jaskan ilmeen. \"Se on pitk\u00e4 tarina, jota ei kannata kertoa juuri nyt. Ajattelin vain, ett\u00e4 sin\u00e4 olisit voinut kertoa h\u00e4nest\u00e4 jotain. Mit\u00e4 tahansa.\"\n\n\"Jos totta puhutaan, h\u00e4n vaikutti lyhytpinnaiselta. Luulen, ett\u00e4 h\u00e4nen oli tarkoitus sovitella asioita Tuiren ja lauman v\u00e4lill\u00e4, mutta h\u00e4n p\u00e4\u00e4tyikin riitelem\u00e4\u00e4n Danielin kanssa. H\u00e4n on ainoa, jonka olen koskaan kuullut sanovan vastaan Danielille\", h\u00e4n sanoi. \"Siis ennen sinua.\"\n\nOmalla kohdallani niskoittelu ei ollut p\u00e4\u00e4ttynyt hirve\u00e4n hyvin. Olinkin hyvin kiinnostunut is\u00e4ni menestyksest\u00e4. \"Mit\u00e4 h\u00e4n sanoi?\"\n\n\"Ett\u00e4 h\u00e4n ei tarvitse lauman apua, mit\u00e4 se sitten tarkoittikin. Ja ett\u00e4 tappaa Danielin, jos t\u00e4m\u00e4 koskaan yritt\u00e4\u00e4 etsi\u00e4 \u00e4itisi \u2013 \u00e4itinne \u2013 k\u00e4siins\u00e4 tai ottaa h\u00e4neen yhteytt\u00e4 mill\u00e4\u00e4n tavalla.\" H\u00e4n puristi kahvikuppia molemmilla k\u00e4sill\u00e4\u00e4n. \"Muistan ajatelleeni, ett\u00e4 h\u00e4nen t\u00e4ytyi joko olla hullu tai todella rakastaa Tuirea.\"\n\nTajusin \u00e4kki\u00e4, ett\u00e4 Jaska oli tunnekuohun vallassa. Vaikka olin aivan alussa udellut h\u00e4nelt\u00e4 \u00e4idist\u00e4ni, kaikki h\u00e4nen kertomansa oli ollut hyvin neutraalia. Mik\u00e4li luin h\u00e4nt\u00e4 ollenkaan oikein, h\u00e4n n\u00e4ytti viimein joutuneen kasvotusten katumuksensa kanssa. Joku oli pysynyt \u00e4itini puolella, ja se joku ei ollut h\u00e4nen veljens\u00e4. Jaska oli antanut h\u00e4nen menn\u00e4, eik\u00e4 ollut n\u00e4hnyt h\u00e4nt\u00e4 en\u00e4\u00e4 koskaan.\n\nTiesin, etten voinut sanoa tai tehd\u00e4 mit\u00e4\u00e4n, joten j\u00e4tin h\u00e4net rauhaan. Otin tyhj\u00e4t kahvikupit p\u00f6yd\u00e4lt\u00e4 ja vein ne keitti\u00f6\u00f6n. Tyhjensin tiskikoneen ja laitoin kupit yl\u00e4koriin. Kun palasin olohuoneeseen, Jaska oli alkanut puhua Mitjalle.\n\n\"Oletko sin\u00e4 jossain koulussa?\" kuulin h\u00e4nen kysyv\u00e4n.\n\n\"En, vaan t\u00f6iss\u00e4.\"\n\n\"Ahaa.\"\n\n\"Mitja on dj\", selitin. \"H\u00e4n oli t\u00f6iss\u00e4 Konstan klubilla ja keikkailee b\u00e4ndins\u00e4 kanssa. He tekiv\u00e4t juuri demon.\"\n\nSeuraavaksi kyselin Jaskalta hieman h\u00e4nen ty\u00f6st\u00e4\u00e4n \u2013 en niink\u00e4\u00e4n siksi, ett\u00e4 minua olisi hirve\u00e4sti kiinnostanut tiet\u00e4\u00e4, mit\u00e4 teknisi\u00e4 t\u00f6it\u00e4 laskettelukeskuksessa tehtiin keskell\u00e4 kes\u00e4\u00e4, vaan saadakseni h\u00e4net rentoutumaan hiukan Mitjan edess\u00e4. Se auttoi, sill\u00e4 Mitja ja Jaska onnistuivat lopulta puhumaan toisilleen melkein kuin normaalit ihmiset. Jaska jopa kysyi Mitjalta t\u00e4m\u00e4n tatuoinnista ja yll\u00e4ttyi melkoisesti, kun sai kuulla, ett\u00e4 min\u00e4 olin tehnyt sen.\n\nVein Mitjan yl\u00e4kertaan n\u00e4ytt\u00e4\u00e4kseni h\u00e4nelle huoneeni, vaikka siin\u00e4 ei ollut erityisen paljon n\u00e4ytett\u00e4v\u00e4\u00e4. Huone oli aika pitk\u00e4lti sellainen, kuin muistin \u2013 Mikaelin luona asumisen j\u00e4lkeen kaikki vain vaikutti aavistuksen pienemm\u00e4lt\u00e4 ja nuhjuisemmalta.\n\nMitja istui alas s\u00e4ngylle. H\u00e4n alkoi nyppi\u00e4 shortsiensa lahjetta, ja hetken aikaa h\u00e4n n\u00e4ytti ihan pikkupojalta. Odotin h\u00e4nen sanovan jotain Jaskasta, joten koin asiakseni varoittaa h\u00e4nt\u00e4. \"H\u00e4n sitten kuulee meid\u00e4t.\"\n\n\"T\u00e4h?\"\n\n\"Jaska. H\u00e4n kuulee kaiken, mit\u00e4 sanot.\"\n\n\"En min\u00e4 siit\u00e4 v\u00e4lit\u00e4\", veljeni mumisi.\n\nOlin siis arvannut v\u00e4\u00e4rin. Olisin tietenkin voinut avata verhon ja tutkia, mit\u00e4 h\u00e4nell\u00e4 oli mieless\u00e4\u00e4n, mutta halusin, ett\u00e4 h\u00e4n kertoisi itse.\n\nLopulta h\u00e4n kohotti katseensa. \"Min\u00e4 aion kosia Caoimhea\", h\u00e4n sanoi.\n\n\"Siis mit\u00e4?\"\n\nMitja Kovasin ei ollut huoleton syd\u00e4mens\u00e4 suhteen. H\u00e4n ei h\u00e4tik\u00f6inyt ihmissuhteissa. H\u00e4n p\u00e4invastoin odotti, kunnes joku toinen tekisi aloitteen. Usein siis turhankin kauan.\n\nT\u00e4lle t\u00e4ytyy olla todella hyv\u00e4 selitys, ajattelin.\n\nAinoa, mink\u00e4 keksin, oli: \"Onko Ciao jotenkin painostanut sinua?\"\n\nPidin Caoimhesta. H\u00e4n oli yst\u00e4v\u00e4ni. Jos minun kuitenkin t\u00e4ytyi valita puoleni veljeni ja Caoimhen v\u00e4lill\u00e4, ratkaisu oli helppo.\n\n\"Ei. H\u00e4n ei ole edes suostunut muuttamaan kokonaan Viidennelle linjalle, vaikka pyysin.\"\n\n\"Anteeksi ett\u00e4 kysyn, mutta mit\u00e4 ihmett\u00e4, Mitja? Miten ihmeess\u00e4 sin\u00e4 kuvittelet kosinnan auttavan asiaa?\"\n\nMitja huokaisi. \"Keijut ovat todella vanhanaikaisia. Caoimhen is\u00e4 ei anna meid\u00e4n tapailla, saati sitten asua yhdess\u00e4, ellei minulla ole aikomusta naida h\u00e4nt\u00e4.\"\n\nMitjalla siis oli kuin olikin hyv\u00e4 selitys aikeilleen. Olin yll\u00e4ttynyt, sill\u00e4 Caoimhe itse ei vaikuttanut erityisen konservatiiviselta. Itse asiassa h\u00e4n oli joidenkin asioiden suhteen h\u00e4mment\u00e4v\u00e4n avomielinen.\n\nMitja ja Caoimhe eiv\u00e4t tietenk\u00e4\u00e4n olleet tavanneet aivan \u00e4sken, mutta en silti pit\u00e4nyt heid\u00e4n seurusteluaan maailman vakuuttavimpana pohjana avioliitolle.\n\n\"Te olette tunteneet ehk\u00e4 nelj\u00e4 kuukautta\", sanoin \u00e4\u00e4neen. Min\u00e4 ja Mikael olimme tunteneet toisemme paljon kauemmin.\n\n\"Tied\u00e4n, mit\u00e4 ajattelet\", Mitja sanoi. \"T\u00e4m\u00e4 ei kuitenkaan ole mik\u00e4\u00e4n tyhm\u00e4 teinirakkaus. Ja mit\u00e4 sitten, vaikka olisikin? Tied\u00e4n jo, ettei meill\u00e4 tule aina olemaan helppoa. Mutta mieluummin minulla on vaikeaa Caoimhen kanssa kuin ilman h\u00e4nt\u00e4. H\u00e4n on rohkein ihminen, jonka tied\u00e4n. En halua ket\u00e4\u00e4n muuta. H\u00e4n saa minut yritt\u00e4m\u00e4\u00e4n parhaani.\"\n\nTiesin tismalleen, mit\u00e4 h\u00e4n tarkoitti. Tunsin nimitt\u00e4in samoin Mikaelin suhteen.\n\nVietimme Jaskalla noin tunnin, ennen kuin aloimme tehd\u00e4 l\u00e4ht\u00f6\u00e4. Ilta ei ollut ehtinyt viel\u00e4 kovin pitk\u00e4lle, mutta tiesin, ettei Mitja halunnut j\u00e4tt\u00e4\u00e4 Caoimhea yksin kovin pitk\u00e4ksi aikaa. My\u00f6s omat ajatukseni alkoivat harhailla kohti Mikaelia ja h\u00e4nen tutkimusretke\u00e4\u00e4n naapurikunnan maille.\n\n\"Te voitte olla t\u00e4\u00e4ll\u00e4 y\u00f6t\u00e4, jos haluatte\", Jaska sanoi.\n\n\"Mitjalla on huone Hukkahovissa tytt\u00f6yst\u00e4v\u00e4ns\u00e4 kanssa\", sanoin. \"Ja min\u00e4 asun tietenkin Sarreilla. Mikael tulee hakemaan minua hotellilta.\"\n\nJaskan ilme ei muuttunut juurikaan, mik\u00e4 riitti kertomaan minulle, ettei h\u00e4n pit\u00e4nyt kuulemastaan. Olin jo noussut yl\u00f6s, kun h\u00e4n sanoi: \"Sin\u00e4 et varmaan n\u00e4hnyt minua, mutta min\u00e4kin olin aukiolla.\"\n\nPys\u00e4hdyin. H\u00e4n oli oikeassa: en ollut n\u00e4hnyt h\u00e4nt\u00e4 susien joukossa. Itse asiassa mieleeni ei ollut juolahtanutkaan, ett\u00e4 h\u00e4n olisi voinut olla paikalla, vaikka laumakokoukset oli tarkoitettu koko laumalle.\n\nEn tiennyt, mit\u00e4 ajatella. Ennen kaikkea en tiennyt, mit\u00e4 Jaska ajatteli minusta sen j\u00e4lkeen, kun h\u00e4n oli n\u00e4hnyt minun ottelevan Jonia vastaan. En ollut en\u00e4\u00e4 sama tytt\u00f6, joka oli asunut h\u00e4nen kanssaan.\n\n\"Se oli hyv\u00e4 kokous\", sanoin niin tyynesti kuin osasin.\n\nK\u00e4velimme takaisin hotellille. Olin edelleen ajatuksissani, kun tulimme sviitin ovelle. Kiinnitin sis\u00e4lt\u00e4 kuuluvaan musiikkiin huomiota vasta, kun Mitja koputti.\n\nTaneli tuli avaamaan sen. \"Hei, Raisa ja Mitja tulivat!\" h\u00e4n huusi olkansa yli. H\u00e4n n\u00e4ytti olevan oikeasti innoissaan saapumisestamme.\n\nN\u00e4in heti, ett\u00e4 sviitti oli suuri. Katto oli huiman korkealla, ja antiikkisen sohvaryhm\u00e4n yl\u00e4puolella kimalteli kattokruunu.\n\nNiko istui nojatuolissa toinen jalka k\u00e4sinojalla ja n\u00e4pp\u00e4ili Caoimhen kitaraa niin kuin olisi muka osannut soittaa sit\u00e4.\n\nJa Caoimhe itse? H\u00e4n hyppi s\u00e4ngyll\u00e4. H\u00e4nell\u00e4 oli yll\u00e4\u00e4n kes\u00e4mekko, joka kohosi h\u00e4nen pomppimisensa tahdissa. Saatoin aina v\u00e4lill\u00e4 n\u00e4hd\u00e4 vilahduksen vaaleanpunaisista alushousuista.\n\nEn voinut olla ajattelematta aikaisempaa keskustelua Mitjan kanssa. Tuleva k\u00e4lyni, ajattelin.\n\n\"Hei Raisa!\" h\u00e4n sanoi. H\u00e4n ei keskeytt\u00e4nyt hyppimist\u00e4\u00e4n.\n\nSelvitin kurkkuani. \"Mit\u00e4 te oikein teette?\"\n\n\"Mit\u00e4 nyt rikkaat ja kuuluisat yleens\u00e4 tekev\u00e4t\", Niko sanoi. \"Biletet\u00e4\u00e4n.\"\n\n\"Kuka teist\u00e4 on rikas tai kuuluisa?\"\n\nNiko osoitti kohti s\u00e4nky\u00e4. \"Oletko kuullut heid\u00e4n demonsa? Tuo tytt\u00f6 on varma t\u00e4hti. Ja niin, sin\u00e4 olet tietysti se rikas.\"\n\n\"Min\u00e4?\"\n\n\"Niin. Sinun piikkiisih\u00e4n t\u00e4ss\u00e4 elet\u00e4\u00e4n.\" H\u00e4n tarttui p\u00f6yd\u00e4ll\u00e4 olevaan pulloon ja kohotti sit\u00e4. \"Ett\u00e4 kippis vain.\"\n\nEn vaivautunut selitt\u00e4m\u00e4\u00e4n, ett\u00e4 lasku menisi Mikaelille eik\u00e4 minulle. Heid\u00e4n mielest\u00e4\u00e4n se oli tietenkin sama asia.\n\n\"\u00c4lk\u00e4\u00e4 sanoko, ett\u00e4 minibaari on jo tyhj\u00e4.\"\n\n\"Ei ole, mutta tilasimme huonepalvelusta skumppaa. Olemme p\u00e4\u00e4sseet sill\u00e4 ihan hyvin k\u00e4yntiin.\"\n\nMulkaisin Nikoa. Tiesin kuitenkin, ett\u00e4 taisto oli jo h\u00e4vitty, joten p\u00e4\u00e4tin olla valittamatta.\n\n\"Miten Annin kanssa meni?\" kysyin. Samaan aikaan kun min\u00e4 ja Mitja olimme olleet Jaskalla, Nikon ja Tanelin oli ollut tarkoitus menn\u00e4 k\u00e4ym\u00e4\u00e4n Nikon mummin luona.\n\n\"Hyvin. Yll\u00e4tt\u00e4v\u00e4n hyvin\", Niko lis\u00e4si. \"Mummi vaikutti pit\u00e4v\u00e4n Tanelista enemm\u00e4n kuin minusta.\"\n\n\"Se ei ole mik\u00e4\u00e4n yll\u00e4tys\", sanoin.\n\n\"Hehheh. H\u00e4n sanoi olevansa iloinen, ettei minun tarvitse muuttaa yksin vieraaseen maahan ja kaupunkiin.\"\n\n\"Sin\u00e4 olet onnekas\", my\u00f6nsin. Ajattelin Mikaelia. Minun oli my\u00f6nnett\u00e4v\u00e4 itselleni, ettei h\u00e4nen viimeaikainen k\u00e4yt\u00f6ksens\u00e4 enteillyt hyv\u00e4\u00e4. Tavallaan toivoin, ett\u00e4 meid\u00e4n riesanamme olisivat olleet vanhempien odotukset tai edes se, ett\u00e4 toinen meist\u00e4 joutui salaamaan yliluonnollisen puolensa. Silloin olisin tiennyt, mit\u00e4 tehd\u00e4. Olin selvinnyt jo monesta liemest\u00e4, mutta mik\u00e4\u00e4n kokemani ei ollut valmentanut minua voittamaan Mikaelin suden puolelleni. Olin t\u00e4ysin neuvoton.\n\n\"Miss\u00e4 Mikael on?\" Taneli kysyi.\n\n\"H\u00e4n meni tutkimaan, l\u00f6yt\u00e4isik\u00f6 jotain j\u00e4lki\u00e4 sudesta, jonka on v\u00e4itetty liikuskelevan jossain t\u00e4\u00e4ll\u00e4 l\u00e4hell\u00e4\", sanoin. Selitin Tanelille nopeasti pulman, joka oli syntynyt naapurikunnan kanssa, paljastamatta tietenk\u00e4\u00e4n liikaa ongelman todellisesta luonteesta.\n\n\"H\u00e4n l\u00e4hti mets\u00e4\u00e4n keskell\u00e4 y\u00f6t\u00e4?\"\n\nEn keksinyt siihen mit\u00e4\u00e4n hyv\u00e4\u00e4 vastausta. En voinut sanoa, ett\u00e4 h\u00e4n oli mennyt y\u00f6ll\u00e4 siksi, ettei h\u00e4nt\u00e4 n\u00e4ht\u00e4isi, sill\u00e4 se olisi johtanut viel\u00e4 vaikeampiin jatkokysymyksiin.\n\n\"Eik\u00f6 h\u00e4nt\u00e4 pelota?\" Taneli jatkoi ihmettely\u00e4\u00e4n.\n\n\"Ei varmaan. Luulen, ett\u00e4 h\u00e4n on tottunut susiin, h\u00e4nh\u00e4n on aina asunut t\u00e4\u00e4ll\u00e4.\"\n\nNiko tuhahti. \"N\u00e4m\u00e4 ihmiset ovat niin saakelin juntteja. Sudet eiv\u00e4t ole tappaneet ihmist\u00e4 Suomessa yli sataan vuoteen. Koiratkin ovat vaarallisempia. Ja arvaa, mik\u00e4 on Suomen vaarallisin el\u00e4in?\"\n\n\"No?\"\n\n\"Hirvi. Miksi kukaan ei j\u00e4rjest\u00e4 miekkareita hirvi\u00e4 vastaan? Ne ovat saatanan tyhmi\u00e4 el\u00e4imi\u00e4. Aina, kun kohtaan hirven mets\u00e4ss\u00e4, ne vain tuijottavat kuin eiv\u00e4t tajuaisi, ett\u00e4 niiden henki on l\u00e4hd\u00f6ss\u00e4 ihan kohta.\"\n\nTaneli nauroi. \"Milloin sin\u00e4 muka olet t\u00f6rm\u00e4nnyt hirveen mets\u00e4ss\u00e4? Ja miksi ne eiv\u00e4t toljottaisi? Sinunhan henkesi on paljon suuremmassa vaarassa.\"\n\nMulkaisin Nikoa. H\u00e4n tajusi sent\u00e4\u00e4n n\u00e4ytt\u00e4\u00e4 nololta. Alkoholi ja salaisuudet eiv\u00e4t olleet paras mahdollinen yhdistelm\u00e4.\n\n***\n\nKun Taneli meni vessaan, Niko veti minua hihasta.\n\n\"Ei voitto, vaan vastustajan t\u00e4ydellinen n\u00f6yryytys.\" H\u00e4nen oli varmaan tarkoitus kuiskata, mutta h\u00e4nen \u00e4\u00e4nens\u00e4 oli aivan yht\u00e4 kuuluva kuin ennenkin. H\u00e4n alkoi olla humalassa.\n\n\"Mit\u00e4 sin\u00e4 h\u00f6piset?\"\n\n\"Puhun tietenkin Jonista\", h\u00e4n sanoi kulmat kurtussa. Sitten h\u00e4nen ilmeens\u00e4 kirkastui. \"Vai oletko sin\u00e4 antanut muillekin turpaan?\"\n\n\"En tietenk\u00e4\u00e4n\", vastasin. En sanonut, ett\u00e4 harjoittelin veljeni kanssa ja olin siten paljon tottuneempi h\u00e4vi\u00e4m\u00e4\u00e4n kuin voittamaan.\n\nNiko ei n\u00e4ytt\u00e4nyt t\u00e4ysin vakuuttuneelta. \"Joka tapauksessa se oli sairaan siisti\u00e4. Ei haittaisi n\u00e4hd\u00e4 uudestaan.\"\n\nMinun oli pakko hymyill\u00e4. H\u00e4n oli ensimm\u00e4inen, joka sanoi niin. Oli mukavaa, ett\u00e4 joku vihdoin antoi minulle tunnustusta. \"Kiitos.\" En voinut olla lis\u00e4\u00e4m\u00e4tt\u00e4: \"Itse asiassa siistein osuus meni sinulta t\u00e4ysin ohi. Voin n\u00e4ytt\u00e4\u00e4 sinulle joskus.\"\n\nOlin sviitiss\u00e4 aamuy\u00f6h\u00f6n asti. Minun oli tunnustettava, ett\u00e4 niin paljon kuin nautinkin Mikaelin seurasta, olin huomaamattani kaivannut hauskanpitoa yst\u00e4vieni kanssa. Olin jo p\u00e4\u00e4tt\u00e4nyt j\u00e4\u00e4d\u00e4 sohvalle y\u00f6ksi, kun Mikael tuli hakemaan minua. Sanoin muille heit ja l\u00e4hdin h\u00e4nen kanssaan autolle. Odotin, ett\u00e4 istuimme molemmat autossa ennen kuin kysyin: \"No? Miten meni?\"\n\nMikaelin kulmat kurtistuivat hieman. \"Me emme l\u00f6yt\u00e4neet mit\u00e4\u00e4n.\"\n\n\"Ettek\u00f6?\" Siit\u00e4 huolimatta, ett\u00e4 olin arvellut naapurikuntalaisten keksineen koko jutun, en ollut itse uskonut siihen. Vaikka Esko Kukkonen olikin \u00e4rsytt\u00e4nyt minua, tiesin, ettei h\u00e4nen huolestuneisuutensa ollut teeskentely\u00e4. Ja vaikka h\u00e4n olisi ollut kuinka hyv\u00e4 n\u00e4yttelij\u00e4, Mikael olisi vainunnut h\u00e4nen ep\u00e4rehellisyytens\u00e4.\n\nPettymykseni oli kuitenkin paljon suurempi kuin olin osannut odottaa. Viimeinen asia, jota halusin, oli n\u00e4hd\u00e4 Mikael n\u00e4in huolestuneena.\n\nH\u00e4n huokaisi. \"On tietenkin mahdollista, ett\u00e4 j\u00e4ljet olivat jo kuluneet pois, tai ettemme etsineet oikeasta paikasta. Haravoimme kuitenkin alueen niin hyvin kuin pystyimme, joten olen v\u00e4h\u00e4n ihmeiss\u00e4ni.\"\n\nYritin sanoa jotain rakentavaa, mutta v\u00e4symys puuroutti ajatukseni enk\u00e4 totuuden nimess\u00e4 uskonut, ett\u00e4 olisin muutenkaan keksinyt mit\u00e4\u00e4n, mik\u00e4 olisi auttanut tilannetta.\n\nMikaelin ajatukset my\u00f6t\u00e4iliv\u00e4t omiani. \"T\u00e4ss\u00e4 vaiheessa emme oikein voi muuta kuin odottaa.\" H\u00e4n ei ollut siit\u00e4 lainkaan iloinen. Huomasin l\u00e4hett\u00e4v\u00e4ni murhanhimoisia ajatuksia Esko Kukkosta kohti. Mikaelin teht\u00e4v\u00e4 oli kenties pit\u00e4\u00e4 huolta laumasta, mutta minun teht\u00e4v\u00e4ni oli huolehtia h\u00e4nest\u00e4. Mit\u00e4 ikin\u00e4 se vaatisikin.\n\n***\n\nSeuraavana p\u00e4iv\u00e4n\u00e4 nukuimme pitk\u00e4\u00e4n. Lopulta min\u00e4 nousin ensimm\u00e4isen\u00e4 ja menin keitti\u00f6\u00f6n laittamaan aamiaista. Mikael liittyi pian seuraani. H\u00e4n oli aina el\u00e4nyt ylt\u00e4kyll\u00e4isyyden keskell\u00e4, mutta siit\u00e4 huolimatta h\u00e4nt\u00e4 ei tarvinnut juuri koskaan pyyt\u00e4\u00e4 auttamaan siivoamisessa tai ruuanlaitossa, mik\u00e4 nosti entisest\u00e4\u00e4n muutenkin suurta arvostustani h\u00e4nt\u00e4 kohtaan.\n\nSe ei ollut ainoa asia, mik\u00e4 sai minut hyv\u00e4lle tuulelle. Lev\u00e4nnyt Mikael oli nimitt\u00e4in rennompi Mikael. Menness\u00e4\u00e4n ohitseni j\u00e4\u00e4kaapille h\u00e4nen k\u00e4tens\u00e4 k\u00e4vi vy\u00f6t\u00e4r\u00f6ll\u00e4ni. En uskonut, ett\u00e4 h\u00e4n edes ajatteli sit\u00e4, ja juuri se kertoi minulle, ett\u00e4 kaikki oli kunnossa.\n\nKoska ilma oli mit\u00e4 parhain, katoimme astiat ulos. S\u00f6imme t\u00e4ydellisen aamiaisen: munakasta, jogurttia ja mysli\u00e4, hedelmi\u00e4, tuoremehua ja tietenkin paljon kahvia.\n\nTakapiha n\u00e4ytti upealta. Katselin tuhatta ja sataa eri vihre\u00e4n s\u00e4vy\u00e4, jotka kaikki muodostuivat keltaisista ja sinisist\u00e4 niin, ett\u00e4 olisin voinut pilkkoa niit\u00e4 ikuisesti mieless\u00e4ni aina pienempiin ja pienempiin osasiin. Kun suljin silm\u00e4ni, saatoin erottaa ilmassa monta eri tuoksua: ruohon ja koivun ja havupuut, \u00f6ljytyn pation ja poreammeen kloorin. Nostin paljaat varpaani tuolin reunalle ja nojasin taaksep\u00e4in. Venyttelin j\u00e4seni\u00e4ni, ja koska se tuntui niin hyv\u00e4lt\u00e4, annoin venytyksen jatkua niin pitk\u00e4\u00e4n, ett\u00e4 Mikaelia alkoi naurattamaan.\n\n\"Mit\u00e4 te teitte eilen Mitjan kanssa?\" h\u00e4n kysyi.\n\nAjattelin eilisi\u00e4 keskustelujamme. Ei ollut vaikea valita niist\u00e4 merkitt\u00e4vint\u00e4. \"Me puhuimme kosimisesta.\" En ollut ajatellut ollenkaan, milt\u00e4 se kuulosti ennen kuin olin sanonut sen \u00e4\u00e4neen. R\u00e4v\u00e4ytin silm\u00e4ni auki.\n\nEn usko, ett\u00e4 olin koskaan n\u00e4hnyt Mikaelia yht\u00e4 \u00e4llistyneen\u00e4. Minun oli pakko nauraa. \"\u00c4l\u00e4 pelk\u00e4\u00e4, en min\u00e4 aio kosia ket\u00e4\u00e4n. Mutta ilmeisesti Mitja aikoo.\"\n\nH\u00e4n toipui nopeasti. \"En min\u00e4 sit\u00e4. Min\u00e4 vain haluan kosia sinua, eik\u00e4 toisin p\u00e4in.\"\n\nOli minun vuoroni tuijottaa h\u00e4nt\u00e4 silm\u00e4t sel\u00e4ll\u00e4\u00e4n. Ja h\u00e4nen vuoronsa naurahtaa. \"Ai tuolta sin\u00e4 n\u00e4yt\u00e4t silloin, kun olet kauhuissasi. En ole n\u00e4hnytk\u00e4\u00e4n tuota ilmett\u00e4 aikaisemmin.\"\n\nH\u00e4nen t\u00e4ytyy liioitella, ajattelin.\n\n\"\u00c4l\u00e4 huoli. Min\u00e4 odotan, kunnes sin\u00e4 olet valmis.\"\n\nMinun t\u00e4ytyi edelleen n\u00e4ytt\u00e4\u00e4 j\u00e4rkyttyneelt\u00e4, sill\u00e4 Mikael armahti minut vaihtamalla aihetta. \"Mit\u00e4 sin\u00e4 haluat tehd\u00e4 t\u00e4n\u00e4\u00e4n?\" h\u00e4n kysyi.\n\nMikael olisi siis t\u00e4n\u00e4\u00e4n vapaa velvollisuuksista. Onnistuin viimein hymyilem\u00e4\u00e4n.\n\nHalusin oppia n\u00e4kem\u00e4\u00e4n Hukkavaaran h\u00e4nen silmiens\u00e4 kautta. H\u00e4nen el\u00e4m\u00e4ns\u00e4, kotinsa ja lapsuutensa oli t\u00e4\u00e4ll\u00e4, enk\u00e4 uskonut ett\u00e4 h\u00e4nt\u00e4 saattoi todella tuntea ilman, ett\u00e4 tunsi Hukkavaaran. Aivan niin kuin minua ei voinut t\u00e4ysin tuntea tiet\u00e4m\u00e4tt\u00e4 mit\u00e4\u00e4n taustastani, Helsingist\u00e4 ja Kreikasta.\n\nSitten Mikael sanoi: \"Voisin laittaa poreammeen valmiiksi\", ja tiesin, ett\u00e4 h\u00e4n oli ylitt\u00e4nyt kaikki odotukseni.\n\nH\u00e4n ryhtyi tuumasta toimeen. Min\u00e4 korjasin sill\u00e4 aikaa astiat ja k\u00e4vin vaihtamassa bikinit p\u00e4\u00e4lleni.\n\nKun olin palaamassa patiolle, pys\u00e4hdyin hetkeksi oviaukkoon. Mikael oli juuri vet\u00e4nyt paidan p\u00e4\u00e4ns\u00e4 yli.\n\nH\u00e4nen katsomisensa silloin, kun h\u00e4n ei huomannut, oli yksi lempipuuhiani. Ei niin, ett\u00e4 Mikael olisi ollut miss\u00e4\u00e4n tilanteessa ujo kropastaan. En uskonut sen johtuvan niink\u00e4\u00e4n siit\u00e4, ett\u00e4 h\u00e4n oli niin treenattu, vaan siit\u00e4, ett\u00e4 h\u00e4n oli ihmissusi. H\u00e4n oli tottunut olemaan alasti. H\u00e4n oli my\u00f6s tottunut luottamaan omaan ruumiiseensa, ja h\u00e4nen varmuutensa oli \u00e4\u00e4rimm\u00e4isen seksik\u00e4st\u00e4.\n\nEn ollut itse ylt\u00e4nyt viel\u00e4 t\u00e4ysin samaan, mutta Mikael teki asian paljon helpommaksi kuin olisi voinut kuvitella. Ensinn\u00e4kin h\u00e4n tuntui jumaloivan vartaloani. Eik\u00e4 pelk\u00e4st\u00e4\u00e4n niit\u00e4 kaikkein ilmeisimpi\u00e4 kohtia, vaan my\u00f6s varpaita, nilkkoja, polvitaipeita ja vy\u00f6t\u00e4r\u00f6n kaarta. Ranteita ja olkap\u00e4it\u00e4 ja korvalehti\u00e4.\n\nH\u00e4n sai minut kiinni tuijottamisesta. En voinut sille mit\u00e4\u00e4n: punastuin. H\u00e4nen katseellaan oli aina kummallinen vaikutus. Varpaitani kihelm\u00f6i.\n\nAstuimme veteen. Mikael juoksutti sormiaan selk\u00e4nikamiani pitkin. Tunsin h\u00e4nen kosketuksensa monessa paikassa yht\u00e4 aikaa, enk\u00e4 voinut olla taivuttamatta selk\u00e4\u00e4ni hieman kaarelle.\n\nPuhelimeni soi jossakin sis\u00e4ll\u00e4.\n\n\"Min\u00e4 menen\", sanoin nopeasti.\n\nMikaelin kulmat kohosivat. H\u00e4n taisi tiet\u00e4\u00e4, miten h\u00e4kellyksiss\u00e4ni olin. H\u00e4nen katseensa seurasi minua, kun astuin harvinaisen k\u00f6mpel\u00f6sti pois altaasta. Mit\u00e4 ihmett\u00e4 jaloilleni oli tapahtunut?\n\n\"Raisa\", h\u00e4n sanoi.\n\nK\u00e4\u00e4nnyin ymp\u00e4ri. Mikael nousi yl\u00f6s altaasta ja otti askeleen l\u00e4hemm\u00e4s. H\u00e4n suuteli minua, mutta t\u00e4ll\u00e4 kertaa kaikkea muuta kuin pehme\u00e4sti. Vesipisarat valuivat meid\u00e4n huulillemme ja tunsin auringon kasvoillamme. Kostea iho tuntui viile\u00e4lt\u00e4 kaikkialla muualla paitsi niiss\u00e4 kohdissa, joissa vartalomme koskivat.\n\nKun avasin silm\u00e4ni, olin varma, ett\u00e4 puolet aivosoluistani olivat juuri karanneet jonnekin.\n\n\"Parasta kiirehti\u00e4, jos ajattelin ehti\u00e4 vastata\", h\u00e4n sanoi vakavana.\n\nNiin kuin ennustettua, puhelin lakkasi soimasta juuri, kun ehdin lasiovista sis\u00e4lle. Minun oli pakko nauraa. Nauruni hyytyi kuitenkin pian, kun huomasin, etten ollut yksin. En ollut kuullut mit\u00e4\u00e4n, mutta olohuoneessa minua oli vastassa Juha.\n\nH\u00e4nen silm\u00e4ns\u00e4 levisiv\u00e4t kun h\u00e4n tajusi, ettei minulla ollut p\u00e4\u00e4ll\u00e4ni muuta kuin bikinit.\n\nJa sitten tilanne meni kertaheitolla satakertaa pahemmaksi. Minun ei tarvinnut k\u00e4\u00e4nty\u00e4 ymp\u00e4ri tiet\u00e4\u00e4kseni, ett\u00e4 Mikael oli ilmestynyt taakseni. Ylitseni vy\u00f6ryv\u00e4 myrsky teki sen kyll\u00e4 selv\u00e4ksi. Tunne oli niin voimakas, ett\u00e4 olin menn\u00e4 polvilleni. Juhan ansioksi oli sanottava, ettei h\u00e4n juossut saman tien ulos talosta. Sen sijaan h\u00e4n laski p\u00e4\u00e4ns\u00e4 ja k\u00e4\u00e4ntyi kohti sein\u00e4\u00e4. Kuulin, kun h\u00e4n mutisi jotain itsekseen.\n\nSanaakaan sanomatta Mikael kietoi pyyhkeen tiukasti ymp\u00e4rilleni. Liikkeet olivat niin j\u00e4nnittyneit\u00e4, ett\u00e4 oli mahdoton kuvitella h\u00e4nen olleen leikkis\u00e4 viel\u00e4 muutamaa hetke\u00e4 aikaisemmin. Minua sek\u00e4 h\u00e4vetti ett\u00e4 kiukutti. Ja silti: ket\u00e4 muka saatoin syytt\u00e4\u00e4?\n\nVasta silloin tajusin, ett\u00e4 Juhan tuloon t\u00e4ytyi olla jokin syy. \"Mit\u00e4 on tapahtunut?\" kysyin. Vihasin sit\u00e4, miten pienelt\u00e4 \u00e4\u00e4neni kuulosti.\n\nKesti, ennen kuin Juha sai kasattua itsens\u00e4. \"Meill\u00e4 on uusi ongelma.\"\n\n\"No?\"\n\n\"Eilen, kun me k\u00e4vimme tutkimassa niit\u00e4 j\u00e4lki\u00e4... Yksi meist\u00e4 p\u00e4\u00e4tyi videolle.\"\nKUUDES LUKU\n\n\"Miten?\" Mikael kysyi samaan aikaan kun itse tivasin: \"Kuka?\"\n\nJuha ei vastannut kummallekaan, vaan veti l\u00e4pp\u00e4rin esiin suojapussistaan. Ker\u00e4\u00e4nnyimme sohvap\u00f6yd\u00e4n \u00e4\u00e4reen. Mikael istui minun ja Juhan v\u00e4liin ja kiersi k\u00e4sivarren ymp\u00e4rilleni. Ilmapiiri oli edelleen j\u00e4nnittynyt, mutta me kaikki yritimme parhaamme mukaan olla piittaamatta siit\u00e4.\n\nJuha naputteli jotain hakukentt\u00e4\u00e4n. \"T\u00e4m\u00e4 on paikallisten susijahtaajien sivusto. Meik\u00e4l\u00e4iset ovat p\u00e4\u00e4sseet oikein etusivulle.\"\n\nH\u00e4n suurensi videon koko n\u00e4yt\u00f6n suuruiseksi ja klikkasi sen p\u00e4\u00e4lle.\n\nVideo ei ollut erityisen hyv\u00e4laatuinen, sill\u00e4 se oli kuvattu puhelimella ja ilmeisesti ikkunan l\u00e4pi. Silmi\u00e4 siristellen saattoi puiden takaa erottaa tumman l\u00e4isk\u00e4n, joka liikkui. Hengitin syv\u00e4\u00e4n. Ep\u00e4m\u00e4\u00e4r\u00e4inen hahmo mets\u00e4ss\u00e4 ei viel\u00e4 tarkoittanut mit\u00e4\u00e4n.\n\nJuuri silloin susi astui esiin puiden k\u00e4tk\u00f6st\u00e4 ja tuli suoraan kohti kameraa.\n\nKatsoin kauhuissani, kun susi tallusteli muina miehin\u00e4 \u2013 susina? \u2013 muutaman askeleen aivan n\u00e4k\u00f6s\u00e4ll\u00e4 ja ravisteli sitten itse\u00e4\u00e4n. H\u00e4nt\u00e4 heilahti viimeisen\u00e4. Suden toinen korva nytk\u00e4hti ylemm\u00e4s, ja p\u00e4\u00e4 kallistui kun se kuunteli jotakin, mik\u00e4 ei n\u00e4kynyt videolla.\n\n\"Ja kehtaavat viel\u00e4 v\u00e4itt\u00e4\u00e4, ettei susista ole vaaraa!\" kameran takaa tuleva \u00e4\u00e4ni kommentoi juuri ennen kuin kuva pys\u00e4htyi.\n\nNojasin huokaisten taaksep\u00e4in.\n\nAsiasta ei ollut ep\u00e4ilyst\u00e4k\u00e4\u00e4n. Lapsikin olisi tiennyt, ett\u00e4 videolla n\u00e4kyv\u00e4 el\u00e4in oli susi.\n\nTuijotin ruudulla yh\u00e4 pys\u00e4htyneen\u00e4 kuvana olevaa sutta. Kiinnitin huomiota turkin v\u00e4riin. Se oli samaa harmaata kuin Mikaelilla.\n\n\"Onko tuo..?\"\n\n\"En min\u00e4\", Mikael sanoi nopeasti.\n\nTiesin heti, ettei susi ollut my\u00f6sk\u00e4\u00e4n Juha. H\u00e4nen sutensa oli tummempi ja suurempi. Jennin susi oli puolestaan vaalea. J\u00e4ljell\u00e4 oli siis...\n\n\"Helvetin Joni\", mutisin.\n\nMikael loi minuun varoittavan katseen, muttei sanonut mit\u00e4\u00e4n.\n\n\"Nyt on paha menn\u00e4 v\u00e4itt\u00e4m\u00e4\u00e4n, ettei sudesta ole todisteita\", Juha sanoi.\n\nMets\u00e4st\u00e4j\u00e4t olisi pakko p\u00e4\u00e4st\u00e4\u00e4 kyl\u00e4\u00e4n. Eik\u00e4 t\u00e4ysikuuhun ollut kuin pari p\u00e4iv\u00e4\u00e4.\n\n\"Ei olisi pit\u00e4nyt antaa h\u00e4nen l\u00e4hte\u00e4 mukaan\", sanoin. Todellisuudessa en tiennyt, kumpaa syytt\u00e4\u00e4 enemm\u00e4n, Jonia vaiko itse\u00e4ni. Min\u00e4 olin puhunut h\u00e4nen puolestaan Mikaelille. Ilman minua h\u00e4n ei edes olisi kyl\u00e4ss\u00e4.\n\n\"T\u00e4lle ei en\u00e4\u00e4 mahda mit\u00e4\u00e4n. Voimme vain yritt\u00e4\u00e4 korjata vahingot. Yritet\u00e4\u00e4n sopia uusi tapaaminen Esko Kukkosen kanssa\", Mikael sanoi. H\u00e4n oli kumman rauhallinen, kun otti huomioon, ett\u00e4 kaikki, mit\u00e4 olimme onnistuneet saavuttamaan, oli juuri nollattu. Olimme takaisin alkupisteess\u00e4. Itse asiassa olimme pahemmassa liriss\u00e4 kuin tullessamme. Videotodiste oli tuhat kertaa vakuuttavampi kuin yksitt\u00e4isten ihmisten n\u00e4k\u00f6havainnot.\n\n\"Mit\u00e4 aiot sanoa h\u00e4nelle?\" kysyin.\n\n\"Rehellisesti sanottuna? Ei aavistustakaan.\"\n\nKosketin h\u00e4nen k\u00e4sivarttaan. En sanonut \u00e4\u00e4neen, mit\u00e4 ajattelin: etteiv\u00e4t mitk\u00e4\u00e4n sanat riitt\u00e4neet kumoamaan videota. Minun oli my\u00f6nnett\u00e4v\u00e4, ett\u00e4 Eskon asemassa min\u00e4kin olisin ollut huolissani asukkaiden turvallisuudesta. Me tiesimme, ettei videolla oleva susi aiheuttaisi vaaraa kenellek\u00e4\u00e4n, muttei ollut mit\u00e4\u00e4n keinoa saada Esko vakuutetuksi asiasta. Ei ilman, ett\u00e4 paljastaisimme koko kyl\u00e4n salaisuuden, mik\u00e4 johtaisi ainoastaan uusiin, huomattavasti nykyist\u00e4 massiivisempiin ongelmiin. Meid\u00e4n oli ehk\u00e4 pakko tunnustaa tappiomme ja vain rukoilla, etteiv\u00e4t Eskon mets\u00e4st\u00e4j\u00e4t onnistuisi saamaan ket\u00e4\u00e4n hukkavaaralaisista kiinni.\n\nPaitsi jos..?\n\n\"Mit\u00e4 jos me voisimme todistaa heille, ettei videolla ole susi, vaan koira?\"\n\n\"Miten me sellaista todistaisimme?\" Juha kysyi.\n\nKatsoin ensin Juhaa, ja sitten Mikaelia. Lopulta he tuntuivat tajuavan, mit\u00e4 ajoin takaa.\n\n\"Et kai tarkoita sit\u00e4, mit\u00e4 luulen sinun tarkoittavan?\" Juha kysyi. H\u00e4n kuulosti niin kauhistuneelta, ett\u00e4 minun oli pakko purra huultani, etten olisi nauranut h\u00e4nelle. \"Tarkoitan, ett\u00e4 meid\u00e4n t\u00e4ytyy n\u00e4ytt\u00e4\u00e4 heille, ett\u00e4 videolla on karannut koira eik\u00e4 susi. Koira, joka on h\u00e4mm\u00e4stytt\u00e4v\u00e4n paljon suden n\u00e4k\u00f6inen, mutta t\u00e4ysin vaaraton.\"\n\nMolemmat pojat tuijottivat minua t\u00e4ysin ep\u00e4uskoisina. Viimeinkin olin tullut keksineeksi jotain, mik\u00e4 sai heid\u00e4t unohtamaan kukkoilunsa.\n\nKatsoin Mikaelia. \"No\", sanoin iloisesti, \"oletko koskaan haaveillut n\u00e4yttelij\u00e4nurasta?\"\n\n***\n\nMikael ei ollut koskaan haaveillut n\u00e4yttelij\u00e4nurasta. H\u00e4n ei pit\u00e4nyt suunnitelmastani yht\u00e4\u00e4n sen enemp\u00e4\u00e4 kuin kukaan muukaan susista, mutta parempien ideoiden puutteessa h\u00e4n suostui uhrautumaan lauman puolesta. Tiesin, ett\u00e4 h\u00e4n koki velvollisuudekseen korvata Jonin suhteen tekem\u00e4ns\u00e4 virhearvioinnin. H\u00e4nen sutensa oli niin paljon Jonin suden n\u00e4k\u00f6inen, ettei kukaan pelk\u00e4n videon n\u00e4hnyt pystyisi erottamaan niit\u00e4 toisistaan.\n\nKutsuimme Esko Kukkosen Hukkavaaraan. Se vaati melkoista maanittelua, mutta viestimme olivat sen verran vihjailevia ja ep\u00e4m\u00e4\u00e4r\u00e4isi\u00e4, ett\u00e4 lopulta h\u00e4nen uteliaisuutensa oli t\u00e4ytynyt her\u00e4t\u00e4.\n\nMikael muutti muotoa ennen kuin muut saapuivat paikalle. Istuin nurmella takapihalla ja katsoin, kun susi-Mikael tutki huolellisesti siell\u00e4 olevia hajuja. Katsoin ja katsoin, enk\u00e4 saanut tarpeekseni. H\u00e4nen n\u00e4kemisens\u00e4 suden muodossa t\u00e4ytti minut oudolla pelonsekaisella riemulla ja ripauksella kateutta. H\u00e4n oli niin vapaa, niin elossa. H\u00e4nen liikkumisensa oli kevyemp\u00e4\u00e4 kuin ajatus, ja vaikka ihmis-Mikael oli kaukana k\u00f6ntyksest\u00e4, oli suden liikkeiss\u00e4 eleganssia, jota ei voinut saavuttaa, kun k\u00e4veli kahdella jalalla.\n\nEl\u00e4inten kauneus on niin mutkatonta, niin taloudellista, ajattelin. Sill\u00e4 oli tarkoitus.\n\nKatsoin omia sormiani. Ihmisten kynnet ovat t\u00e4ysin hy\u00f6dytt\u00f6m\u00e4t, ajattelin. Ja hampaat. Jalkojakaan ei kannattanut verrata suden k\u00e4p\u00e4liin, jotka jaksoivat juosta satoja kilometrej\u00e4, p\u00e4iv\u00e4 toisensa j\u00e4lkeen. Ihmisten kyvykkyys piili k\u00e4siss\u00e4 ja aivoissa. Ehk\u00e4 hitusen my\u00f6s silmiss\u00e4.\n\nMaalaaminen oli k\u00e4den, p\u00e4\u00e4n ja silm\u00e4n yhteisty\u00f6t\u00e4. Se oli ihmism\u00e4isin asia, jonka tiesin. Ehk\u00e4 siksi pidin siit\u00e4 niin paljon.\n\nTalon suunnalta kuului \u00e4\u00e4ni\u00e4. Kohotin katseeni k\u00e4sist\u00e4ni juuri parahiksi n\u00e4hd\u00e4kseni, kun Jenni ja Juha tulivat takapihalle. He pys\u00e4htyiv\u00e4t \u00e4kisti. Seurasin heid\u00e4n katsettaan.\n\nMikael n\u00e4ytti heille hampaitaan.\n\nOlin n\u00e4hnyt h\u00e4nen tekev\u00e4n niin aikaisemminkin, mutta en koskaan kenellek\u00e4\u00e4n, jota me pidimme yst\u00e4v\u00e4n\u00e4. Asialle saattoi olla vain yksi selitys: syyst\u00e4 tai toisesta Mikael oli t\u00e4n\u00e4\u00e4n sataprosenttisesti susi.\n\nTiesin heti, ett\u00e4 mik\u00e4li Mikael olisi tiennyt etuk\u00e4teen sutensa ottavan ohjat, h\u00e4n ei olisi suostunut vaihtamaan muotoa ja esiintym\u00e4\u00e4n Esko Kukkosen edess\u00e4. Jokin oli pahasti vialla, enk\u00e4 tiennyt, mik\u00e4.\n\nJoka tapauksessa en voinut luottaa h\u00e4nen arvostelukykyyns\u00e4. Aloin pel\u00e4t\u00e4, ett\u00e4 t\u00e4m\u00e4 oli paljon huonompi idea kuin olin kuvitellut.\n\nOli kuitenkin my\u00f6h\u00e4ist\u00e4 per\u00e4\u00e4nty\u00e4. En voinut antaa Mikaelin sudelle t\u00e4ysin periksi.\n\nH\u00e4n oli alkanut liikkua kohti ovelle j\u00e4hmettyneit\u00e4 Jenni\u00e4 ja Juhaa p\u00e4\u00e4 alhaalla, korvat pystyss\u00e4.\n\n\"Paikka!\"\n\nH\u00e4n pys\u00e4htyi, mutta oli selv\u00e4\u00e4, ettei se johtunut k\u00e4skyst\u00e4ni. Minulla ei ollut ep\u00e4ilyst\u00e4k\u00e4\u00e4n siit\u00e4, kuka oikeasti oli tilanteen herra. Jos kuitenkin olin oppinut jotain t\u00e4h\u00e4nastisessa el\u00e4m\u00e4ss\u00e4ni niin sen, ett\u00e4 teeskentely riitti usein yll\u00e4tt\u00e4v\u00e4n pitk\u00e4lle. Esitin siis, etten ollut lainkaan tietoinen h\u00e4nen hallitsevasta tuijotuksestaan tai \u00e4\u00e4nist\u00e4, joita kuului h\u00e4nen kurkustaan.\n\nK\u00e4velin Mikaelin luokse ja napsautin sormiani h\u00e4nen kuononsa edess\u00e4. \"Mikael, keskity.\" H\u00e4n katsoi minua varsin kylm\u00e4\u00e4v\u00e4sti hetken ennen kuin viimein istui viereeni.\n\nJenni ja Juha eiv\u00e4t olleet liikahtaneet milli\u00e4k\u00e4\u00e4n. \"Okei, Mikael on v\u00e4h\u00e4n hermona\", sanoin heille. \"Ei kuitenkaan mit\u00e4\u00e4n h\u00e4t\u00e4\u00e4: niin kauan kuin ette tule meid\u00e4n v\u00e4liimme, kaiken pit\u00e4isi sujua ihan hyvin.\" Toivoin todella, ett\u00e4 se piti paikkansa.\n\n\"Oletko varma?\" Juha kysyi. \"H\u00e4n on oikeasti aika vauhkona.\"\n\n\"Olen varma\", sanoin. Mikael ei satuttaisi ket\u00e4\u00e4n.\n\nEik\u00f6?\n\nJuuri silloin kuulin ovikellon soivan. Esko Kukkonen oli saapunut.\n\n\"Menk\u00e4\u00e4 vastaanottamaan Eskoa\", sanoin muille.\n\nOdotin pihalla syd\u00e4n pamppaillen sill\u00e4 aikaa kun Juha ja Jenni meniv\u00e4t sis\u00e4\u00e4n ja avasivat Eskolle oven.\n\n\"Miss\u00e4 Sarri on?\" kuulin Eskon kysyv\u00e4n. H\u00e4nen \u00e4\u00e4nens\u00e4 kuulosti hyvin ep\u00e4ilev\u00e4lt\u00e4. Vilkaisin vieress\u00e4ni olevaa sutta.\n\n\"H\u00e4nelle tuli \u00e4killinen meno\", Jenni vastasi. \"H\u00e4n on hyvin pahoillaan. Mutta \u00e4lk\u00e4\u00e4 huoliko, asiamme ei edellyt\u00e4 h\u00e4nen l\u00e4sn\u00e4oloaan. T\u00e4m\u00e4 tapaaminen on hyvin vapaamuotoinen.\"\n\n\"En ymm\u00e4rr\u00e4, mit\u00e4 neuvoteltavaa meill\u00e4 voisi en\u00e4\u00e4 olla t\u00e4ss\u00e4 tilanteessa. Meill\u00e4 on todisteet siit\u00e4, ett\u00e4 vaarallinen peto liikkuu pihojen ja my\u00f6s koulun l\u00e4heisyydess\u00e4.\"\n\n\"Se ei ollut susi\", Juha sanoi. \"Susi, jonka n\u00e4itte, oli oikeasti koira. Koulutettu ja t\u00e4ysin vaaraton.\"\n\n\"Mit\u00e4 se sitten teki Hannukaisten pihassa?\"\n\n\"Se pel\u00e4styi kovaa \u00e4\u00e4nt\u00e4 ja karkasi.\"\n\n\"Kyll\u00e4 min\u00e4 tunnistan suden kun n\u00e4en sellaisen.\"\n\n\"Siin\u00e4 koirassa on toki suden verta\", Juha sanoi. \"Se on kuitenkin tavallista sutta isompi.\" En tiennyt tavallisista susista mit\u00e4\u00e4n, mutta muistin Nikon kertoneen minulle, ett\u00e4 ihmissudet olivat suden muodossakin isompia kuin tavalliset sudet.\n\n\"Haluatteko n\u00e4hd\u00e4 sen?\" Jenni kysyi.\n\n\"Mink\u00e4?\"\n\n\"Suden. Tai siis koiran.\"\n\nKirosin mieless\u00e4ni Jennin virhett\u00e4. Ilmeisesti Esko kuitenkin suostui ehdotukseen, sill\u00e4 seuraavaksi lasiovi ty\u00f6nnettiin syrj\u00e4\u00e4n ja Jenni astui ulos. Juha seurasi h\u00e4nen per\u00e4ss\u00e4\u00e4n.\n\nN\u00e4in heti, ett\u00e4 he molemmat olivat niin varuillaan, ett\u00e4 uskalsivat tuskin hengitt\u00e4\u00e4. Mikaelin l\u00e4sn\u00e4olo oli kuin ydinvoimala. Ehdin k\u00e4yd\u00e4 mieless\u00e4ni l\u00e4pi noin tusinan verran syit\u00e4, miksi t\u00e4m\u00e4 oli huono idea.\n\nHe pys\u00e4htyiv\u00e4t monen metrin p\u00e4\u00e4h\u00e4n, ja lopulta Esko tuli esiin heid\u00e4n takaansa. H\u00e4nen silm\u00e4ns\u00e4 levisiv\u00e4t.\n\nMikael istui vieress\u00e4ni ja tuijotti Eskoa. Kun katsoin Mikaelia, olin varma, ett\u00e4 koko yrityksemme oli t\u00e4ysin tuhoon tuomittu. Kukaan ei voisi uskoa, ett\u00e4 Mikael oli kesy, askartelemastani pannasta huolimatta. H\u00e4n oli saalistaja ja me olimme saaliita. Tilanne oli h\u00e4nen hallussaan, ei meid\u00e4n.\n\nKuvittelin mieless\u00e4ni, kuinka h\u00e4n hy\u00f6kk\u00e4isi Eskon kimppuun yhdell\u00e4 loikalla ja repisi h\u00e4nen kaulavaltimonsa auki. Se toki olisi mahdollinen ratkaisu ongelmaan, muttei ehk\u00e4 kaikkein suotavin.\n\n\"T\u00e4m\u00e4n koiran te n\u00e4itte videolla\", sanoin. \"Olen pahoillani, ett\u00e4 se p\u00e4\u00e4si karkaamaan, se ei yleens\u00e4 tee niin. Pid\u00e4mme huolen, ettei niin tule tapahtumaan jatkossa.\"\n\nLiikahdin sill\u00e4kin uhalla, ett\u00e4 Mikael alkaisi murista. H\u00e4n ei tehnyt niin, mik\u00e4 rohkaisi minua kyykistym\u00e4\u00e4n ja laskemaan k\u00e4den h\u00e4nen turkilleen. Ei h\u00e4nen p\u00e4\u00e4ns\u00e4 p\u00e4\u00e4lle, sill\u00e4 se olisi ollut valta-aseman haaste, vaan h\u00e4nen lonkalleen. Mikael ei tuntunut kiinnitt\u00e4v\u00e4n kosketukseeni mit\u00e4\u00e4n huomiota. H\u00e4nen katseensa oli edelleen Eskossa.\n\nSusi oli valpas. Toisin sanoen vaarallinen.\n\n\"Onko t\u00e4m\u00e4 siis teid\u00e4n koiranne?\"\n\nNousin hitaasti seisomaan. Minun t\u00e4ytyi varoa, mit\u00e4 sanoin. \"Kyll\u00e4, mutta se on t\u00e4\u00e4ll\u00e4 vain tilap\u00e4isesti.\" Esko ei saanut luulla, ett\u00e4 koira olisi kyl\u00e4ss\u00e4 aina, kun h\u00e4n tulisi k\u00e4ym\u00e4\u00e4n. H\u00e4nen ei toivon mukaan tarvitsisi tulla t\u00e4nne en\u00e4\u00e4 koskaan, mutta en ollut valmis ottamaan riski\u00e4.\n\n\"Miksi ette kertoneet t\u00e4st\u00e4 heti?\" Esko kysyi.\n\n\"Se on hieman arkaluontoista\", aloitin. Olin hiljaa ja tuijottelin paljaita varpaitani juuri niin kauan, ett\u00e4 tyhm\u00e4kin tajuaisi minun olevan h\u00e4peiss\u00e4ni. Eskon kanssa ei ollut ylin\u00e4yttelemisen vaaraa, pikemminkin p\u00e4invastoin.\n\nKumarruin l\u00e4hemm\u00e4s ja madalsin \u00e4\u00e4nt\u00e4ni. \"Katsos kun me olemme k\u00e4ytt\u00e4neet t\u00e4t\u00e4 koiraa osana markkinointistrategiaamme.\"\n\nEskon \u00e4\u00e4ni oli l\u00e4hes voitonriemuinen. \"Te olette huijanneet asiakkaitanne.\"\n\n\"Ei, ei\", sanoin j\u00e4rkyttyneen\u00e4. Minun ei tarvinnut edes n\u00e4ytell\u00e4. En ollut miettinyt asiaa aivan loppuun asti. Hukkavaaran kannalta olisi eritt\u00e4in huono asia, mik\u00e4li Esko alkaisi levitt\u00e4\u00e4 huhua, ett\u00e4 turistien n\u00e4kem\u00e4t sudet olivatkin oikeasti koiria ja ett\u00e4 kyl\u00e4n vetovoima perustui silkkaan huijaukseen.\n\n\"Susista on vain vaikea saada hyvi\u00e4 kuvia\", selitin. En tiennyt luontokuvauksesta juuri mit\u00e4\u00e4n, mutta tiesin, ettei se voinut olla helppoa.\n\nEsko tuijotti Mikaelia kuin yritt\u00e4en p\u00e4\u00e4tt\u00e4\u00e4, uskoako tarinaamme vai eik\u00f6. Sitten h\u00e4n kohotti k\u00e4tens\u00e4 ja otti m\u00e4\u00e4r\u00e4tietoisen askeleen kohti Mikaelia.\n\nVoi ei, ajattelin.\n\nOlisin tietenkin voinut kielt\u00e4\u00e4 Eskoa. Olin kuitenkin juuri vakuuttanut h\u00e4nelle, ett\u00e4 Mikael oli t\u00e4ysin vaaraton, ja kielt\u00e4minen tekisi kaiken tyhj\u00e4ksi.\n\nKuulin jonkun takanani vet\u00e4v\u00e4n nopeasti henke\u00e4. Tunsin, kuinka Mikael j\u00e4ykistyi vieress\u00e4ni. H\u00e4n oli nuoli, joka sinkoaisi itsens\u00e4 ilmaan hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4.\n\nMikael ole kiltti, rukoilin mieless\u00e4ni. Mieleni teki piilottaa kasvoni k\u00e4siini.\n\nEskon k\u00e4si laskeutui Mikaelin p\u00e4\u00e4n p\u00e4\u00e4lle. Pala oli noussut kurkkuuni, ja \u00e4kki\u00e4 pelk\u00e4sin, ett\u00e4 saattaisin oksentaa.\n\nMikael ei hievahtanutkaan. Esko taputti p\u00e4\u00e4lakea kerran pari k\u00f6mpel\u00f6sti ennen kuin veti k\u00e4tens\u00e4 pois. \"Jos tuo koira kerran on teille arvokas, pit\u00e4k\u00e4\u00e4 se paremmin kytkettyn\u00e4. Susi voi k\u00e4yd\u00e4 senkin kimppuun.\"\n\nPian sen j\u00e4lkeen Esko l\u00e4hti. Juha ja Jenni saattoivat h\u00e4net ulos. Kun kuulin k\u00e4ynnistyv\u00e4n moottorin \u00e4\u00e4nen, annoin itseni j\u00e4lleen hengitt\u00e4\u00e4 vapaasti. Minusta tuntui aivan silt\u00e4 kuin olisin ollut varttitunnin veden alla.\n\nOdotin viel\u00e4 pitk\u00e4n tovin sen j\u00e4lkeen, kun \u00e4\u00e4net olivat kaikonneet. Sitten lys\u00e4hdin istumaan nurmelle.\n\nK\u00e4\u00e4nnyin Mikaelin puoleen. \"Jonain p\u00e4iv\u00e4n\u00e4 sin\u00e4 viel\u00e4 aiheutat minulle syd\u00e4nkohtauksen.\"\n\nH\u00e4n tuijotti minua. Sitten h\u00e4n loikkasi minua kohti ja t\u00f6n\u00e4isi minut kumoon.\n\nOnnistuin pehment\u00e4m\u00e4\u00e4n laskuani niin, ettei kaatuminen tehnyt lainkaan kipe\u00e4\u00e4, mutta keuhkoni tyhjentyiv\u00e4t silti hetkellisesti. Tassut naulasivat hartiani maahan. Makasin sel\u00e4ll\u00e4ni ja tuijotin kultaisia silmi\u00e4.\n\nPienen hetken ajan ajattelin kuolemista. Se ei vaatisi paljon. Kuvittelin, kuinka h\u00e4nen hampaansa uppoaisivat ihooni. Alekin Mikaelin hampaista saama haava ei ollut ollut mitenk\u00e4\u00e4n erityisen syv\u00e4 tai edes vaarallisessa paikassa. Riitti, ett\u00e4 Mikaelin susi leikkisi kanssani turhan rajusti. Se voisi tappaa minut aivan vahingossa ja Mikael tajuaisi sen vasta j\u00e4lkeenp\u00e4in.\n\nSe ei kuitenkaan ollut suurin pelkoni.\n\nRakasta minua, anelin \u00e4\u00e4nett\u00f6m\u00e4sti h\u00e4nen sudeltaan. \u00c4l\u00e4 j\u00e4t\u00e4 minua.\n\nViikset kutittivat naamaani, kun h\u00e4nen kuononsa laskeutui haistelemaan hiuksiani.\n\n\"Hei, nyt\", sanoin, mutten voinut olla nauramatta. \"Tuli selv\u00e4ksi. Min\u00e4 olen sinun, niin kuin aina.\"\n\nH\u00e4n alkoi nuolla naamaani.\n\n***\n\n\"Luulen, ett\u00e4 saimme Eskon vakuutettua\", sanoin Mikaelille autossa seuraavana p\u00e4iv\u00e4n\u00e4. J\u00e4\u00e4kaappimme sis\u00e4lt\u00f6 oli viimein alkanut huveta, ja olimme matkalla l\u00e4himp\u00e4\u00e4n markettiin. \"Ainakin h\u00e4n joutuu miettim\u00e4\u00e4n tilanteen uusiksi.\"\n\n\"Min\u00e4 en muista siit\u00e4 mit\u00e4\u00e4n\", Mikael tunnusti. Oli mennyt monta tuntia, ennen kuin h\u00e4n oli vaihtanut muotoa. Olimme tuskin puhuneet tapahtuneesta ennen nukkumaanmenoa. \"En kai... pel\u00e4stytt\u00e4nyt sinua?\"\n\n\"Et. No, n\u00e4ytit v\u00e4h\u00e4n hampaita, mutta lopulta kaikki meni hyvin.\"\n\nMikaelin ilme oli paljon j\u00e4rkyttyneempi kuin olisin osannut aavistaa. Olin iloinen p\u00e4\u00e4t\u00f6ksest\u00e4ni olla kertomatta h\u00e4nelle tarkempia yksityiskohtia. \"Mit\u00e4\u00e4n ei tapahtunut\", vakuutin. \"Uskon oikeasti, ett\u00e4 t\u00e4m\u00e4 Esko-episodi oli nyt t\u00e4ss\u00e4.\"\n\n\"Toivottavasti\", h\u00e4n sanoi. Saatoin kuitenkin kuulla, ett\u00e4 h\u00e4nt\u00e4 vaivasi jokin.\n\n\"Oliko idea sinun mielest\u00e4si huono?\"\n\n\"Ei\", h\u00e4n sanoi lopulta. \"Sin\u00e4 et tehnyt mit\u00e4\u00e4n v\u00e4\u00e4r\u00e4\u00e4. Minun olisi pit\u00e4nyt itse tajuta tilanne ajoissa.\"\n\nEn ollut varma, mit\u00e4 \"tilanne\" piti sis\u00e4ll\u00e4\u00e4n. Olisin kysynyt, mutta sitten Mikael sanoi: \"P\u00e4\u00e4asia, ett\u00e4 sin\u00e4 olet kunnossa, eik\u00e4 sinun tarvinnut k\u00e4ytt\u00e4\u00e4 kykyj\u00e4si.\"\n\n\"Tied\u00e4n er\u00e4\u00e4n, joka ei ole kanssasi samaa mielt\u00e4\", vastasin. Kun Mikael katsoi minut kysyv\u00e4sti, jatkoin: \"Niko. Olisitpa kuullut, miten h\u00e4n vaahtosi ottelusta silloin kun k\u00e4vin hotellilla.\"\n\nKurvasimme huoltoaseman pihaan. \"Niko ei n\u00e4hnyt, miten sairaaksi tulit.\"\n\n\"Olen tottunut siihen\", vastasin. Mutta Mikaelin huomio ei ollut en\u00e4\u00e4 minussa. Seurasin h\u00e4nen katsettaan.\n\nEsko Kukkonen oli saapunut taas Hukkavaaraan. Tuijotimme h\u00e4nt\u00e4 auton ikkunasta. Minun oli ollut vaikea kuvitella h\u00e4net mets\u00e4lle, ja nyt universumi tarjoili minulle n\u00e4yn, jota en unohtaisi aivan pian. H\u00e4n oli sonnustautunut maastokuosiin ja kirkkaanoranssiin lippalakkiin. Vy\u00f6t\u00e4isill\u00e4 roikkui jonkinlainen panosvy\u00f6.\n\nMikael toipui ensin. H\u00e4n alkoi n\u00e4pp\u00e4ill\u00e4 puhelintaan.\n\n\"Esko Kukkonen\", kuului toisesta p\u00e4\u00e4st\u00e4. Mikael oli laittanut puhelun kaiuttimeen.\n\n\"Esko, t\u00e4ss\u00e4 on Mikael Sarri. Sin\u00e4 ja pari muuta miest\u00e4 olette ilmeisesti Hukkavaarassa.\"\n\nEsko seisoi k\u00e4si lanteilla ja tuijotti juuri p\u00e4invastaiseen suuntaan kuin miss\u00e4 olimme. \"Mist\u00e4 sin\u00e4 sellaisen tiedon sait?\"\n\nMikael puri leukojaan yhteen ennen kuin vastasi. \"N\u00e4en teid\u00e4t autostani.\"\n\nEsko kohotti katseensa ja l\u00f6ysi meid\u00e4t lopulta parkkipaikalta.\n\nMikael lopetti puhelun sanaakaan sanomatta ja nousi autosta. Seurasin h\u00e4nen esimerkki\u00e4\u00e4n.\n\n\"Luulen, ett\u00e4 meid\u00e4n t\u00e4ytyy puhua\", Mikael aloitti.\n\nEsko katsoi Mikaelia h\u00f6lmistyneen\u00e4 ja vasta sitten tajusi, ett\u00e4 h\u00e4nell\u00e4 oli yh\u00e4 puhelin korvallaan. H\u00e4n antoi sille vihaisen mulkaisun ennen kuin pujotti sen taskuunsa, aivan niin kuin puhelin olisi tarkoituksella nolannut h\u00e4net.\n\n\"Et sitten kuitenkaan voinut uskoa, ett\u00e4 susi oli koira.\"\n\nEsko punehtui, mutta onnistui silti kuulostamaan uhmakkaalta. \"Teid\u00e4n koiranne ei kuittaa ensimm\u00e4ist\u00e4 susihavaintoa. Me tulimme vain tutkimaan tilannetta.\"\n\n\"Huomaan sen\", Mikael vastasi. H\u00e4n risti k\u00e4sivartensa. \"Otitte varmaan pyssyt mukaan ihan vain kaiken varalta?\"\n\n\"Esko, soisitko meille hetken?\" kysyin. Ennen kuin h\u00e4n ehti vastata, tartuin Mikaelia hihasta ja kiskoin h\u00e4nt\u00e4 per\u00e4ss\u00e4ni, kunnes kuvittelin meid\u00e4n olevan kuuloet\u00e4isyyden ulkopuolella.\n\n\"T\u00e4m\u00e4 on ihan naurettavaa\", sanoin. \"Anna minun hoitaa asia.\"\n\nMikael kiristeli hampaitaan. \"Min\u00e4 yritt\u00e4isin viel\u00e4 puhua h\u00e4nelle j\u00e4rke\u00e4.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Puhuminen ei n\u00e4k\u00f6j\u00e4\u00e4n auta. Lopetetaan t\u00e4m\u00e4 pelleily nyt kun se viel\u00e4 onnistuu. Min\u00e4 selvi\u00e4n kyll\u00e4.\"\n\n\"Hyv\u00e4 on\", h\u00e4n sanoi lopulta. \"Mutta pyyd\u00e4 Mitjan apuun!\"\n\nNy\u00f6kk\u00e4sin. Palasimme takaisin autolle.\n\n\"Me haluamme pahoitella aikaisempaa k\u00e4yt\u00f6st\u00e4mme\", sanoin Eskolle. \"Te olette t\u00e4ysin oikeassa.\"\n\n\"Olenko?\" h\u00e4n kysyi.\n\n\"Olette oikeassa siit\u00e4, ett\u00e4 t\u00e4lle tilanteelle t\u00e4ytyy tehd\u00e4 jotakin.\" K\u00e4velin l\u00e4hemm\u00e4s. \"Sin\u00e4 ja min\u00e4 teemme sille jotain, juuri nyt. Hypp\u00e4\u00e4 kyytiin.\"\n\n\"Enp\u00e4 tied\u00e4\", Esko sanoi.\n\n\"Tied\u00e4tp\u00e4s\", vastasin.\n\nP\u00e4\u00e4tt\u00e4v\u00e4isyyteni toimi, sill\u00e4 h\u00e4n nousi autoon. Ajoimme lyhyen matkan kunnantalolle. Sen aikana l\u00e4hetin Mitjalle, Nikolle ja Juhalle viestin, jossa kerroin Esko Kukkoselle k\u00e4yv\u00e4n kohta kalpaten. Kun tulimme kunnantalon aulaan, meit\u00e4 olivat jo vastassa Niko ja Mitja. Niko n\u00e4ytti olevan yht\u00e4 innoissaan kuin jos olisin ilmoittanut, ett\u00e4 Hukkahovissa j\u00e4rjestett\u00e4isiin Vivienne Westwoodin uusimman kokoelman n\u00e4yt\u00f6s pelk\u00e4st\u00e4\u00e4n h\u00e4nt\u00e4 varten. Minun n\u00e4kemiseni tositoimissa oli h\u00e4nelle ilmeisesti parasta viihdett\u00e4.\n\nMitja oli luonnollisesti huomattavasti varautuneempi. T\u00e4m\u00e4n takiahan h\u00e4n oli tullut Hukkavaaraan, mutta olin joka tapauksessa helpottunut, ett\u00e4 h\u00e4n oli paikalla. H\u00e4n auttaisi minua. Asioiden ei tarvinnut menn\u00e4 t\u00e4ysin katastrofaalisesti.\n\nMenimme sis\u00e4\u00e4n neuvotteluhuoneeseen.\n\n\"Kahvia tai teet\u00e4?\" kysyin Eskolta. Ja sitten minun oli ihan pakko lis\u00e4t\u00e4: \"Tai ehk\u00e4 jotain vahvempaa?\" H\u00e4n nimitt\u00e4in tulisi tarvitsemaan sit\u00e4.\n\n\"Vain vett\u00e4, kiitos\", Esko mumisi.\n\nEsko oli ainoa, joka istui. Kaikki muut seisoivat. Siit\u00e4 Eskokin alkoi v\u00e4hitellen tajuta, ett\u00e4 tilanteessa oli jotain outoa.\n\nEn antanut h\u00e4nelle aikaa ajatella liikoja. Ojensin k\u00e4teni sivulle ja veljeni tarttui siihen. Astuimme yht\u00e4 aikaa kohti Esko Kukkosta.\n\nAvasin ikkunan ideoiden maailmaan. Tai niin minun oli ollut tarkoitus tehd\u00e4. Todellisuudessa se tuntui samalta kuin tuulitunnelin k\u00e4ynnist\u00e4minen. Painottomuus. Kaikki, mit\u00e4 olin aina pit\u00e4nyt itsest\u00e4\u00e4nselvyyten\u00e4, lakkasi olemasta.\n\nKun olin tottunut tunteeseen, suuntasin huomioni seittiin, joka oli aika. Kaikki pys\u00e4htyi. Yht\u00e4 helposti kuin ajatus.\n\nMitja vilkaisi olkansa yli. Mikael oli ainoa, joka ei ollut j\u00e4hmettynyt paikoilleen. H\u00e4n tuijotti minua ja ny\u00f6kk\u00e4sin h\u00e4nelle merkiksi siit\u00e4, ett\u00e4 olin ihan kunnossa.\n\n\"Miksi h\u00e4n on t\u00e4\u00e4ll\u00e4?\" Mitja kysyi minulta kreikaksi.\n\nVastasin totuudenmukaisesti. \"En tied\u00e4.\"\n\n\"Mit\u00e4 seuraavaksi?\"\n\nKatsoin Eskoa. Otin askeleen l\u00e4hemm\u00e4s, ja Mitja seurasi per\u00e4ss\u00e4ni. Kumarruin Eskon puoleen.\n\nTarkastelin hetken aikaa h\u00e4nen punoittavaa ihoaan. Hikipisaraa, joka oli j\u00e4m\u00e4ht\u00e4nyt keskelle niskaa. H\u00e4nell\u00e4 oli korvan alla hyttysenpurema, jota h\u00e4n oli raapinut vastik\u00e4\u00e4n. Partaveden alta saattoi haistaa edellisen\u00e4 p\u00e4iv\u00e4n\u00e4 nautitun alkoholin. Silm\u00e4t olivat haaleansiniset ja vetist\u00e4v\u00e4t. Yritin ymm\u00e4rt\u00e4\u00e4, mist\u00e4 h\u00e4n tuli. Miksi susista oli tullut h\u00e4nelle niin suuri pakkomielle? Mit\u00e4 h\u00e4nen pelkonsa peitti alleen?\n\nSuljin silm\u00e4ni hetkeksi. Totuus Esko Kukkosen el\u00e4m\u00e4st\u00e4 alkoi juosta silmieni edess\u00e4.\n\nEskolla oli monia ongelmia el\u00e4m\u00e4ss\u00e4\u00e4n. H\u00e4nen vaimonsa oli j\u00e4tt\u00e4m\u00e4ss\u00e4 h\u00e4net. H\u00e4nen lapsensa olivat aikuisia, mutta h\u00e4n ei tiennyt heid\u00e4n el\u00e4m\u00e4st\u00e4\u00e4n juuri mit\u00e4\u00e4n \u2013 toinen oli muuttanut Yhdysvaltoihin ja toinen Helsinkiin. He soittivat toisilleen pari kertaa vuodessa ja silloinkin keskustelu k\u00e4sitteli l\u00e4hinn\u00e4 s\u00e4\u00e4t\u00e4. Eik\u00e4 siin\u00e4 kaikki: Esko oli tuskallisen tietoinen vanhenemisestaan ja huononevasta terveydentilastaan.\n\nH\u00e4n pelk\u00e4si kuolevansa yksin. H\u00e4n tiesi kuolevansa yksin. Kaikki, mit\u00e4 h\u00e4n oli saavuttanut el\u00e4m\u00e4ns\u00e4 aikana, oli loppujen lopuksi ollut t\u00e4ysin turhaa.\n\nH\u00e4nell\u00e4 oli iso palkka, iso talo ja iso auto, mutta h\u00e4n oli yksi onnettomimmista tapaamistani ihmisist\u00e4.\n\nVakuutin itselleni, ettei se, mit\u00e4 tulisin tekem\u00e4\u00e4n, lis\u00e4nnyt h\u00e4nen taakkaansa mill\u00e4\u00e4n tavalla. P\u00e4invastoin, t\u00e4m\u00e4n j\u00e4lkeen h\u00e4nell\u00e4 olisi yksi murhe v\u00e4hemm\u00e4n. Koetin ter\u00e4st\u00e4yty\u00e4 ja pit\u00e4yty\u00e4 pelk\u00e4st\u00e4\u00e4n susissa. Muut asiat eiv\u00e4t kuuluneet minulle. En voinut korjata jokaisen el\u00e4m\u00e4\u00e4, joten oli parempi, etten koskaan edes l\u00e4htisi sille tielle. Min\u00e4 olin yhdeks\u00e4ntoistavuotias ja minulla oli t\u00e4ysi ty\u00f6 pysytell\u00e4 oman el\u00e4m\u00e4ni ruorissa.\n\n\"Mit\u00e4 sin\u00e4 kuhnailet?\" Mitja kysyi.\n\n\"Kunhan viivyttelen\", huokaisin. Kumarruin l\u00e4hemm\u00e4s ja aloin puhua Eskon korvaan matalalla \u00e4\u00e4nell\u00e4. Vasta my\u00f6hemmin tajusin, ett\u00e4 olin puhunut kreikkaa, mutta kielell\u00e4 ei ollut lopputuloksen kannalta v\u00e4li\u00e4: puhe oli pelkki\u00e4 \u00e4\u00e4niaaltoja, joiden avulla kutsuin ideat t\u00f6ihin.\n\nOtin vastaan Eskolta saamiani muistoja, sekoitin niihin omia ideoitani, ja sy\u00f6tin ne h\u00e4nelle takaisin. Huone katosi. Tilalle ilmestyi mets\u00e4. Se oli ensin t\u00e4ysin hiljainen, mutta katsottuani kysyv\u00e4sti Mitjaa, h\u00e4n sulki hetkeksi silm\u00e4ns\u00e4. Linnut alkoivat laulaa ja tuuli kohista puissa. Ny\u00f6kk\u00e4sin h\u00e4nelle hyv\u00e4ksyv\u00e4sti ja jatkoin ty\u00f6t\u00e4ni.\n\nKohta kaksi sudenpentua rynt\u00e4si esiin. Ne leikkiv\u00e4t kesken\u00e4\u00e4n kiinnitt\u00e4m\u00e4tt\u00e4 meihin lainkaan huomiota.\n\n\"T\u00e4m\u00e4n sin\u00e4 n\u00e4et, kun ajattelet susia\", kerroin Eskolle.\n\nKaivoin esille h\u00e4nen el\u00e4m\u00e4st\u00e4\u00e4n jokaisen hetken, joka liittyi mill\u00e4\u00e4n tavalla susiin. Suoraan sanottuna en tiennyt, kuinka pitk\u00e4lle minun piti menn\u00e4, jotta saisin aikaan mit\u00e4\u00e4n pysyv\u00e4\u00e4 vaikutusta. Mutta halusin j\u00e4tt\u00e4\u00e4 mahdollisimman v\u00e4h\u00e4n sattuman varaan.\n\nViisivuotias Esko, jota is\u00e4 varoittaa menem\u00e4st\u00e4 yksin mets\u00e4\u00e4n. Esko talvella moottorikelkan kyydiss\u00e4 pys\u00e4htyneen\u00e4 suden j\u00e4lkien viereen. Esko tuijottamassa kateellisena juuri rakennettua Hukkahovia.\n\nSitten t\u00f6rm\u00e4sin johonkin odottamattomaan.\n\nEsko katsomassa ikkunasta, kun susi tulee h\u00e4nen omalle pihalleen. Paitsi etten voinut n\u00e4hd\u00e4 sutta \u2013 tiesin vain, ett\u00e4 Eskon mielest\u00e4 se oli siell\u00e4.\n\nRypistin kulmiani. Tiesin, ettei minulla ollut aikaa j\u00e4\u00e4d\u00e4 ihmettelem\u00e4\u00e4n. Siirryin seuraavaan muistoon.\n\nOli mahdotonta sanoa, kuinka kauan aikaa kului. Katosin Esko Kukkosen maailmaan, ja ilman Mitjan nyk\u00e4isy\u00e4 en ehk\u00e4 olisi muistanut palata koskaan takaisin itseeni.\n\nTodellisuus napsahti takaisin paikoilleen kuin \u00e4\u00e4rimmilleen venytetty kuminauha, josta p\u00e4\u00e4stettiin \u00e4kisti irti. Horjahdin, ja otin tukea jostakin. Esko Kukkonen r\u00e4pytteli silmi\u00e4\u00e4n. Sitten h\u00e4n katsoi minua, ja \u00e4llistynyt ilme levisi h\u00e4nen kasvoilleen. Tajusin p\u00e4\u00e4tyneeni polvilleni h\u00e4nen eteens\u00e4. Eik\u00e4 siin\u00e4 kaikki: toinen k\u00e4teni puristi h\u00e4nen polveaan. P\u00e4\u00e4stin irti, mutten p\u00e4\u00e4ssyt yl\u00f6s omin voimin. Onneksi Mitja ymm\u00e4rsi ahdinkoni ja tempaisi minut kyyn\u00e4rp\u00e4\u00e4st\u00e4 jaloilleni.\n\n\"Anteeksi, k\u00e4yn haukkaamassa v\u00e4h\u00e4n raitista ilmaa\", sanoin hymyillen. Luultavasti jokseenkin mielipuolisen n\u00e4k\u00f6isen\u00e4. Silm\u00e4ni olivat alkaneet vetist\u00e4\u00e4 kivusta, joka sykki p\u00e4\u00e4ss\u00e4ni. Aivan kuin kalloni olisi \u00e4kki\u00e4 alkanut kutistua, eiv\u00e4tk\u00e4 aivoni en\u00e4\u00e4 olisi mahtuneet sen sis\u00e4\u00e4n. Hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 ne turskahtaisivat ulos.\n\nKompastelin ulos neuvotteluhuoneesta. Esko kaipaisi mit\u00e4 luultavammin viel\u00e4 lis\u00e4vakuutteluja, mutta muut pystyisiv\u00e4t hoitamaan sen. En n\u00e4hnyt en\u00e4\u00e4 mit\u00e4\u00e4n. Tunsin kuinka vedet valuivat silmist\u00e4ni pitkin naamaani, mutta muut aistit eiv\u00e4t toimineet normaalisti. Oksentaisin aivan kohta.\n\nMitja \u2013 ainakin uskoin sen olevan h\u00e4n \u2013 k\u00e4veli per\u00e4ss\u00e4ni ja johdatti minut vessaan. H\u00e4n nosti kannen yl\u00f6s ja polvistui viereeni. Onnekseni h\u00e4n piti minusta kiinni, sill\u00e4 muutoin olisin varmaan vajonnut p\u00e4\u00e4 edell\u00e4 p\u00f6ntt\u00f6\u00f6n. H\u00e4n piteli my\u00f6s hiuksiani.\n\nPikemminkin tunsin kuin n\u00e4in hetken, jolloin Mikael ilmestyi oviaukkoon.\n\n\"Mene pois\", sanoin y\u00f6kk\u00e4\u00e4misen v\u00e4liss\u00e4.\n\n\"En\", h\u00e4n vastasi.\n\n\"Mene takaisin ja pid\u00e4 huoli, ettei t\u00e4m\u00e4 mene hukkaan!\" Asian olisi tietenkin voinut muotoilla kohteliaamminkin. Se oli yll\u00e4tt\u00e4v\u00e4n vaikeaa, kun p\u00e4\u00e4 oli p\u00f6nt\u00f6ss\u00e4.\n\n\"Raisa on oikeassa\", Mitja sanoi. \"Ei sinusta ole nyt hy\u00f6ty\u00e4 t\u00e4\u00e4ll\u00e4.\"\n\nEn k\u00e4\u00e4ntynyt katsomaan, mutta hiljaisuus sai minut uskomaan, ett\u00e4 Mikael oli p\u00e4\u00e4tt\u00e4nyt k\u00e4ytt\u00e4yty\u00e4 j\u00e4rkev\u00e4sti ja mennyt takaisin neukkariin. Toivoin todella, ett\u00e4 h\u00e4nen itsehillint\u00e4ns\u00e4 riitt\u00e4isi pit\u00e4m\u00e4\u00e4n h\u00e4net ihmisen muodossa. Muuten Esko Kukkonen saisi kokea el\u00e4m\u00e4ns\u00e4 yll\u00e4tyksen.\n\nPahoinvointi ei ollut viel\u00e4 ohi, joten keskityin laskemaan kymmeneen joka sis\u00e4\u00e4n- ja uloshengityksell\u00e4. Kuulin juoksevan veden \u00e4\u00e4nen, ja sitten Mitja ojensi minulle tukun kosteaa k\u00e4sipaperia. Pyyhin sill\u00e4 otsaani ja niskaani.\n\nVatsani kouristeli, vaikka mit\u00e4\u00e4n ei ollut en\u00e4\u00e4 tulossa yl\u00f6s. Tai niin ainakin luulin, kunnes huomasin oksentavani j\u00e4lleen.\n\nTiesin, ett\u00e4 jotain oli vialla. Jokin minussa oli rikki. En nimitt\u00e4in oksentanut en\u00e4\u00e4 vatsanesteit\u00e4. Oksensin verta.\n\nMitja kiroili. Nojasin taaksep\u00e4in ja keikahdin suoraan h\u00e4nt\u00e4luulleni.\n\nNojasin sein\u00e4\u00e4n ja t\u00e4risin. En ollut koskaan n\u00e4hnyt varsinaisia harhoja, mutta nyt katsoin alas ja kuvittelin n\u00e4kev\u00e4ni suurenevan veritahran paidassani. Hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 veri alkaisi tihkua ihoni l\u00e4pi.\n\n\"Meid\u00e4n pit\u00e4\u00e4 saada sinut sairaalaan\", Mitja sanoi.\n\n\"Ei sairaalaan\", onnistuin sopertamaan.\n\n\"Raisa, sin\u00e4 voit oikeasti kuolla.\"\n\nEn halua kuolla sairaalassa, ajattelin, eik\u00e4 se edes tuntunut omalta ajatukseltani. En ollut koskaan pel\u00e4nnyt l\u00e4\u00e4k\u00e4reit\u00e4 tai sairaaloita \u2013 ehk\u00e4 siksi, etten ollut koskaan joutunut tekemisiin niiden kanssa.\n\nVeri oli p\u00e4\u00e4tt\u00e4nyt pysy\u00e4 suonieni sis\u00e4ll\u00e4 ainakin toistaiseksi. Omituista kyll\u00e4 pahin pelkoni oli, ett\u00e4 menett\u00e4isin tajuntani. Aivan kuin olisin tiennyt kroppani pett\u00e4v\u00e4n minut sill\u00e4 sekunnilla, kun en ollut valvomassa sit\u00e4.\n\nMieleeni juolahti, ett\u00e4 kuka tahansa saattaisi k\u00e4vell\u00e4 sis\u00e4\u00e4n ja n\u00e4hd\u00e4 minut ja Mitjan verilammikossa. \"Meid\u00e4n t\u00e4ytyy l\u00e4hte\u00e4 t\u00e4\u00e4lt\u00e4\", sanoin.\n\nPieni \u00e4\u00e4ni sis\u00e4ll\u00e4ni sanoi, etten keskittynyt olennaiseen, mutta samaan aikaan ep\u00e4ilin, ett\u00e4 todellisuuden v\u00e4lttelylle oli olemassa hyv\u00e4 syy.\n\n\"Oletko j\u00e4rjilt\u00e4si? Eth\u00e4n sin\u00e4 pysty edes istumaan.\"\n\nMinua ei voinut syytt\u00e4\u00e4 j\u00e4rjen \u00e4\u00e4nen turhan tarkasta kuuntelemisesta. Annoin Mitjan auttaa minut pystyyn. Nojasin h\u00e4neen koko painollani. Olin ylpe\u00e4 siit\u00e4, ett\u00e4 jalkani yh\u00e4 koskettivat lattiaa, sill\u00e4 muuten olisin joutunut my\u00f6nt\u00e4m\u00e4\u00e4n, ett\u00e4 veljeni kantoi minua.\n\nJuuri silloin ovelta alkoi kuulua \u00e4\u00e4ni\u00e4.\n\n\"Min\u00e4 haistan veren!\"\n\n\"Mikael. Kun naispuolinen henkil\u00f6 menee vessaan, ei ole hirve\u00e4n tahdikasta...\"\n\n\"MIKSI T\u00c4\u00c4LL\u00c4 HAISEE VERELT\u00c4?\"\n\nOvi lensi auki \u2013 tai ainakin tunsin ilmavirran. P\u00e4\u00e4ni tuntui liian painavalta pit\u00e4\u00e4kseni sit\u00e4 ylh\u00e4\u00e4ll\u00e4. En my\u00f6sk\u00e4\u00e4n ollut aivan varma, miss\u00e4 suunnassa ylh\u00e4\u00e4ll\u00e4 oli. Hiukset peittiv\u00e4t kasvoni, ja hetken aikaa olin kuin j\u00e4nis, joka ty\u00f6nt\u00e4\u00e4 p\u00e4\u00e4ns\u00e4 pensaaseen luullen, ett\u00e4 olisi siten piilossa pedolta.\n\nHetken oli t\u00e4ysin hiljaista.\n\n\"H\u00e4n oksensi\", Mitja sanoi lopulta, \u00e4\u00e4nell\u00e4, joka kuulosti hyvin asialliselta. Mitja oli juuri muistuttanut minua siit\u00e4, mik\u00e4 \u00e4idiss\u00e4ni oli ollut sek\u00e4 hienoa ett\u00e4 \u00e4rsytt\u00e4v\u00e4\u00e4: h\u00e4n k\u00e4vi sit\u00e4 loogisemmaksi ja rauhallisemmaksi, mit\u00e4 enemm\u00e4n h\u00e4nt\u00e4 yritettiin provosoida.\n\nUskaltauduin kurkistamaan hiusverhoni takaa.\n\n\"Verta\", Mitja lis\u00e4si hieman vastahakoisesti.\n\nKuin todisteeksi keuhkoni p\u00e4\u00e4ttiv\u00e4t, etteiv\u00e4t halunneet en\u00e4\u00e4 hengitt\u00e4\u00e4. Aloin yski\u00e4. Pyyhk\u00e4isin suupielt\u00e4ni, ja kun katsoin sormiani, n\u00e4in, ett\u00e4 ne olivat veren tahrimat.\n\nMikaelin silm\u00e4t olivat kadottaneet sinisyytens\u00e4 jo aikaa sitten, mutta nyt niihin syttyi liekki, joka ei ennustanut kuin yht\u00e4 asiaa.\n\n\"No helkkari\", Niko sanoi.\n\nMikael ei sanonut pitk\u00e4\u00e4n aikaan mit\u00e4\u00e4n. H\u00e4nen katseensa sanoi: Uskaltakaakin v\u00e4itt\u00e4\u00e4, ett\u00e4 min\u00e4 ylireagoin.\n\n\"L\u00e4htik\u00f6 Esko jo?\" veljeni kysyi.\n\n\"Tiesin, ett\u00e4 t\u00e4m\u00e4 oli huono idea\", Mikael sanoi. H\u00e4nen sanansa kuulostivat enemm\u00e4n murinalta kuin puheelta. \"Tiesin, ettei minun olisi pit\u00e4nyt antaa sinun puuttua asiaan.\"\n\n\"Jos h\u00e4n on viel\u00e4 rakennuksessa...\"\n\n\"H\u00e4n l\u00e4hti\", Niko sanoi nopeasti. H\u00e4n ty\u00f6nsi k\u00e4det taskuihin ja v\u00e4ltti katsomasta ket\u00e4\u00e4n kohti.\n\n\"Min\u00e4 paskat v\u00e4lit\u00e4n Eskosta!\"\n\n\"Miksi sin\u00e4 huudat Raisalle?\" Mitja kysyi.\n\n\"En min\u00e4 Raisalle huuda!\"\n\n\"No kenelle sitten?\"\n\nSe tuntui saavan Mikaelin ymm\u00e4lleen. \"Olen vain niin vihainen. T\u00e4t\u00e4 ei olisi pit\u00e4nyt tapahtua.\"\n\nH\u00e4n oli oikeassa. Konstan teorian mukaan terveysongelmani olivat johtuneet siit\u00e4, etten osannut viel\u00e4 hallita kykyj\u00e4ni. Nyt kun ajattelin asiaa, oireet eiv\u00e4t olleet suinkaan loppuneet, vaan pahentuneet sen j\u00e4lkeen, kun olimme p\u00e4\u00e4tyneet saarelle Callistan oppiin.\n\n\"Mikael\", sanoin. Vaikka \u00e4\u00e4neni oli olematon, h\u00e4nen p\u00e4\u00e4ns\u00e4 k\u00e4\u00e4ntyi heti. Joskus arvelin, ett\u00e4 h\u00e4n olisi kuullut \u00e4\u00e4neni vaikka toiselta puolelta maapalloa. \"Voisimmeko me menn\u00e4 kotiin?\"\n\nEhk\u00e4 se johtui siit\u00e4, ett\u00e4 olin k\u00e4ytt\u00e4nyt sanaa \"koti\". Tai ehk\u00e4 siit\u00e4, ett\u00e4 h\u00e4n oli Mikael, joka rakasti minua. Kaikki viha katosi h\u00e4nen kasvoiltaan. \"Tietenkin.\"\n\nH\u00e4n tuli luokseni ja otti Mitjan paikan. H\u00e4n ei tietenk\u00e4\u00e4n tyytynyt retuuttamaan minua kainalosta, vaan nappasi minut helposti syliins\u00e4.\n\n\"Takaovi\", sanoin. Jos kyl\u00e4n sudet n\u00e4kisiv\u00e4t minut, koko lauma saisi tiet\u00e4\u00e4, mit\u00e4 \u00e4sken oli tapahtunut. Heid\u00e4n silmiss\u00e4\u00e4n olin vahva. En halunnut luopua siit\u00e4.\n\nMikael ei sanonut mit\u00e4\u00e4n, mutta l\u00e4hti p\u00e4invastaiseen suuntaan. Onneksemme h\u00e4n oli j\u00e4tt\u00e4nyt auton kohtalaisen kauas p\u00e4\u00e4ovista.\n\nMeist\u00e4 kumpikaan ei sanonut mit\u00e4\u00e4n, kun Mikael istutti minut autoon ja p\u00e4\u00e4tyi lopulta laittamaan turvavy\u00f6n puolestani, kun k\u00e4vi selv\u00e4ksi, etteiv\u00e4t k\u00e4teni toimineet. H\u00e4n ei katsonut minuun. Uskoin sen johtuvan siit\u00e4, ett\u00e4 minun n\u00e4kemiseni sai h\u00e4net vihaiseksi, ja h\u00e4n yritti parhaansa mukaan pit\u00e4\u00e4 hermonsa kurissa.\n\nYritin keksi\u00e4 jotain lohduttavaa sanottavaa, mutta mieleeni ei tullut mit\u00e4\u00e4n. Lopulta vain kysyin: \"Miten... Eskon kanssa meni?\"\n\n\"\u00c4l\u00e4 murehdi siit\u00e4\", h\u00e4n t\u00f6ks\u00e4ytti. H\u00e4nenkin t\u00e4ytyi tajuta, miten tylylt\u00e4 oli kuulostanut, sill\u00e4 pian h\u00e4n huokaisi ja sanoi: \"Hyvin. Sin\u00e4 onnistuit t\u00e4ydellisesti. H\u00e4n ei ainoastaan suostunut j\u00e4tt\u00e4m\u00e4\u00e4n susia rauhaan, vaan ehdotti my\u00f6s yhteisty\u00f6t\u00e4 kuntien v\u00e4lille.\" H\u00e4n viimein vilkaisi minua. \"Kiitos. En tied\u00e4, mit\u00e4 olisi tapahtunut ilman sinua.\"\n\n\"Se oli outoa\", mumisin. \"Eskohan sanoi meille, ett\u00e4 joku kuntalainen oli n\u00e4hnyt suden. Kuitenkin, kun k\u00e4vin l\u00e4pi h\u00e4nen muistojaan, suden n\u00e4ki h\u00e4n itse.\"\n\n\"Meid\u00e4n ei tarvitse ajatella sit\u00e4 en\u00e4\u00e4.\" H\u00e4n puri leukojaan yhteen.\n\n\"Minulta menee kohta taju\", varoitin. Naamani tuntui puutuneelta. Satatuhatta tulik\u00e4rp\u00e4st\u00e4 tanssi silmieni edess\u00e4.\n\n\"Tied\u00e4n\", kuulin h\u00e4nen sanovan.\n\nSen j\u00e4lkeen kumpikaan meist\u00e4 ei sanonut mit\u00e4\u00e4n.\n\nSarrien talo oli juuri tullut n\u00e4kyviin, kun Mikael jarrutti \u00e4kisti. Iskeydyin turvavy\u00f6t\u00e4 vasten, ja hetken aikaa luulin, ett\u00e4 olimme osuneet johonkin.\n\n\"Minun on pakko menn\u00e4\", kuulin h\u00e4nen sanovan. \"Jos en l\u00e4hde nyt, min\u00e4... Minun on vain pakko.\"\n\nOdotin. Kun h\u00e4n ei jatkanut, tajusin, ett\u00e4 h\u00e4n oli jo avannut oven ja l\u00e4htenyt autosta.\n\nPuristin silm\u00e4ni kiinni.\n\nKuuntelin hiljaisuutta, enk\u00e4 ollut koskaan pel\u00e4nnyt sit\u00e4 yht\u00e4 paljon kuin nyt.\n\nTuuli k\u00e4vi sis\u00e4\u00e4n avoimesta autonovesta. Olin yksin. Susi oli kadonnut mets\u00e4\u00e4n.\nSEITSEM\u00c4S LUKU\n\n\"Et ole tosissasi. Raisa, se on polyesteria! Jos laitat tuon p\u00e4\u00e4llesi, saat hikoilla kuin pieni sika.\"\n\nAnnoin k\u00e4siss\u00e4ni olevan mekon pudota takaisin s\u00e4ngylle. Pid\u00e4ttelin hymy\u00e4ni. Min\u00e4 ja Niko olimme k\u00e4yneet l\u00e4pi Kristina Sarrin vaatteita vasta hetken, mutta h\u00e4nelle oli muodostunut niist\u00e4 jo hyvin selke\u00e4 mielipide.\n\nEn ollut ottanut mit\u00e4\u00e4n varsinaisia bilevaatteita mukaan, ja Niko oli luvannut tuunata minulle nopeasti jotain sopivaa t\u00e4ksi illaksi. T\u00e4ysikuu olisi vasta ensi viikolla, mutta aivan yht\u00e4 hyvin se olisi voinut olla huomenna \u2013 niin innoissaan sudet olivat saamastaan luvasta vaihtaa taas muotoa. Minulla oli vaikeuksia ymm\u00e4rt\u00e4\u00e4 sit\u00e4, mutta en ollut vastustellut, kun Jenni ja Juha olivat ehdottaneet voitonjuhlia.\n\nOlin ollut useammin kuin kerran paikalla todistamassa, kun sudet juhlivat kuun t\u00e4yttymist\u00e4, joten tiesin, mit\u00e4 tuleman piti. Olin kutsunut yst\u00e4v\u00e4ni ja veljeni etkoille. Mikael oli mennyt hakemaan joitain vieraita autolla. Alakerrasta kuului ep\u00e4m\u00e4\u00e4r\u00e4inen bassojytke, v\u00e4lill\u00e4 selke\u00e4mmin, v\u00e4lill\u00e4 vaimeammin. Mitja siell\u00e4 s\u00e4\u00e4ti \u00e4\u00e4nentoistoa iltaa varten. Jennin, Juhan ja Caoimhen teht\u00e4v\u00e4n\u00e4 oli huolehtia tarjoilusta.\n\nNiko kaivoi vaatekasasta esille jotain oranssia.\n\n\"Katsotaan, milt\u00e4 t\u00e4m\u00e4 mekko n\u00e4ytt\u00e4\u00e4\", Niko sanoi minulle. \"Vaatteet pois.\"\n\nNousin yl\u00f6s tuolista ja aloin riisua. Niko piteli mekkoa edess\u00e4ni, ja astuin siihen. H\u00e4n veti vetoketjun kiinni ja n\u00e4perteli hetken niskassa olevien nappien kanssa.\n\nSeuraavaksi h\u00e4n otti sakset esiin. H\u00e4n kumartui ja alkoi leikata helmaa lyhyemm\u00e4ksi. Seisoin hievahtamatta paikoillani. En voinut olla ajattelematta, miten l\u00e4hell\u00e4 reisi\u00e4ni ter\u00e4t liikkuivat.\n\nNiko ei ollut maininnut eilist\u00e4 romahtamistani, mist\u00e4 olin pelk\u00e4st\u00e4\u00e4n kiitollinen. En itsek\u00e4\u00e4n halunnut muistella sit\u00e4. Olin her\u00e4nnyt aamulla s\u00e4ngyst\u00e4 virke\u00e4n\u00e4, eik\u00e4 minua ollut edes h\u00e4irinnyt se, etten muistanut, kuinka olin sinne p\u00e4\u00e4tynyt. Mikael oli ilmestynyt kylpyhuoneesta hiukset m\u00e4rk\u00e4n\u00e4, ja olin p\u00e4\u00e4tt\u00e4nyt olla sanomatta mit\u00e4\u00e4n. Lauman ongelma oli ratkennut ajoissa, eik\u00e4 muulla ollut v\u00e4li\u00e4.\n\nNiko h\u00e4\u00e4r\u00e4si edelleen helman parissa. Sitten h\u00e4n siirtyi selk\u00e4\u00e4n, ja tunsin, kuinka kangas v\u00e4heni merkitt\u00e4v\u00e4sti. H\u00e4n alkoi pistell\u00e4 nuppineuloja saumoihin.\n\nLopulta h\u00e4n antoi minulle luvan riisua. Yritin varoa nuppineuloja, mutta onnistuin silti pist\u00e4m\u00e4\u00e4n itse\u00e4ni kahdesti. Niko otti heti mekon minulta ja l\u00e4hti huoneesta. Vedin aamutakin p\u00e4\u00e4lleni ja menin h\u00e4nen per\u00e4ss\u00e4\u00e4n toimistoon, jonka p\u00f6yd\u00e4ll\u00e4 jo odotti pikkuruinen ompelukone. Vaikka minun oli vaikea kuvitella Kristinan harrastaneen k\u00e4sit\u00f6it\u00e4, en silti ollut yll\u00e4ttynyt, kun kysytt\u00e4ess\u00e4 Mikael oli kertonut koneen olemassaolosta. Sarreilla oli kaikkea.\n\nNiko alkoi ommella saumoja raivoisaan tahtiin. Minulla ei ollut muuta teht\u00e4v\u00e4\u00e4 kuin istua aloillani ja h\u00f6rppi\u00e4 olutta t\u00f6lkist\u00e4. Taneli jo nosteli omia tarvikkeitaan kassista. H\u00e4n oli luvannut laittaa hiukseni.\n\nPian Niko pyysi minua j\u00e4lleen sovittamaan. Pujotettuani mekon p\u00e4\u00e4lleni tiesin heti, ettei sit\u00e4 voinut en\u00e4\u00e4 sanoa samaksi vaatteeksi kuin hetki sitten. Kankaasta oli j\u00e4ljell\u00e4 en\u00e4\u00e4 vain noin puolet, ja Niko oli tehnyt eteen muotoleikkaukset niin, ett\u00e4 malli oli nyt t\u00e4ysin tyk\u00f6istuva. Selk\u00e4rankaa pitkin kulkevan vetoketjun metallihampaat tuntuivat kylmilt\u00e4 paljasta ihoa vasten.\n\nH\u00e4n tarkasteli minua k\u00e4det puuskassa. H\u00e4nen kulmansa kurtistuivat.\n\nTaneli tuli seisomaan h\u00e4nen viereens\u00e4. Kumpikaan heist\u00e4 ei sanonut pitk\u00e4\u00e4n aikaan mit\u00e4\u00e4n.\n\nLopulta hermoni pettiv\u00e4t. \"Mit\u00e4 nyt?\"\n\n\"Ei mit\u00e4\u00e4n\", Niko sanoi.\n\n\"Jonkin on pakko olla vialla, jos sin\u00e4 et keksi mit\u00e4\u00e4n sarkastista sanottavaa\", huomautin.\n\n\"Olet oikeassa. Ongelma on se, ett\u00e4 tuo mekko n\u00e4ytt\u00e4\u00e4 aivan liian hyv\u00e4lt\u00e4 ollakseen varttitunnissa kyh\u00e4tty tekele.\"\n\nHymyilin. \"Sin\u00e4 vain olet liian hyv\u00e4.\"\n\n\"Ei. Sin\u00e4 n\u00e4yt\u00e4t nyky\u00e4\u00e4n ihan sudelta.\"\n\nTajusimme h\u00e4nen virheens\u00e4 yht\u00e4 aikaa.\n\n\"Tarkoitin suvelta\", Niko korjasi nopeasti. \"Hyvin kes\u00e4iselt\u00e4!\"\n\nEn uskaltanut edes katsoa Tanelia.\n\nOnneksi h\u00e4n ei vaikuttanut huomanneen mit\u00e4\u00e4n. \"T\u00e4st\u00e4 illasta tulee hauska\", h\u00e4n sanoi. \"Min\u00e4 ja Niko olemme k\u00f6k\u00f6tt\u00e4neet ihan liikaa hotellihuoneessa. Kiva p\u00e4\u00e4st\u00e4 tapaamaan paikallisia ihmisi\u00e4.\"\n\n\"Ei kannata odottaa mit\u00e4\u00e4n ihmeit\u00e4\", Niko sanoi. \"Asukkaat t\u00e4\u00e4ll\u00e4 ovat todella sis\u00e4\u00e4np\u00e4in l\u00e4mpi\u00e4vi\u00e4. Veikkaan, ett\u00e4 jo parin tunnin j\u00e4lkeen toivot, ettemme olisi tulleet.\"\n\n\"Kaikki hotellin ty\u00f6ntekij\u00e4t ovat olleet mukavia\", Taneli vastasi.\n\n\"Kai nyt, kun heille maksetaan siit\u00e4! Sel\u00e4n takana he vain juoruilevat meist\u00e4.\"\n\nTaneli katsoi poisp\u00e4in. Tunsin h\u00e4net jo sen verran hyvin, ett\u00e4 ymm\u00e4rsin h\u00e4nen olevan loukkaantunut.\n\n\"Niko, menep\u00e4s tarkistamaan, onko alakerrassa kaikki hyvin\", sanoin tiukasti.\n\nNiko totteli. Avasin suuni vasta kun h\u00e4n oli t\u00f6mistellyt portaat alas.\n\n\"\u00c4l\u00e4 v\u00e4lit\u00e4 Nikosta\", sanoin Tanelille. \"Kun h\u00e4n on tuolla tuulella, h\u00e4net kannattaa vain j\u00e4tt\u00e4\u00e4 huomiotta.\"\n\n\"Tied\u00e4n\", h\u00e4n sanoi samalla kun ty\u00f6nsi suoristusraudan t\u00f6pselin sein\u00e4\u00e4n. \"Olen asunut h\u00e4nen kanssaan melkein vuoden.\"\n\n\"Sori. En tarkoittanut, ettet sin\u00e4 tuntisi h\u00e4nt\u00e4. Hukkavaara vain aina saa h\u00e4nen hermonsa kiristym\u00e4\u00e4n, eik\u00e4 se johdu sinusta.\"\n\nH\u00e4n viittoili minut istumaan toimistotuoliin, enk\u00e4 voinut kuin totella. \"En vain ymm\u00e4rr\u00e4 t\u00e4t\u00e4\", Taneli sanoi. H\u00e4n alkoi kammata tukkaani tavalla, josta huomasi, ett\u00e4 h\u00e4n oli ammattilainen. \"H\u00e4n ei suostu kertomaan minulle, mik\u00e4 t\u00e4ss\u00e4 paikassa nyppii h\u00e4nt\u00e4 niin paljon.\"\n\nEn voinut sanoa siihen mit\u00e4\u00e4n.\n\n\"Onko sinulla toivomuksia?\" Taneli kysyi sormet yh\u00e4 hiuksissani.\n\n\"Ei\", sanoin pehme\u00e4sti. \"Tee mit\u00e4 haluat.\"\n\nH\u00e4n suihkutti hiuksiini jotain ainetta ja alkoi sitten vet\u00e4\u00e4 suoristusrautaa niiden l\u00e4pi.\n\nH\u00e4n p\u00e4\u00e4tyi tekem\u00e4\u00e4n hiuksiini muhkeat kiharat. En ollut itse koskaan onnistunut sellaisten tekemisess\u00e4, mutta Tanelilta se k\u00e4vi k\u00e4denk\u00e4\u00e4nteess\u00e4.\n\nLopulta h\u00e4n suihkutti hiusteni p\u00e4\u00e4lle viel\u00e4 lakkaa.\n\n\"Kiitos\", sanoin.\n\n\"Sinun ja Mikaelin lapsilla tulee olemaan upeat hiukset\", h\u00e4n huokaisi.\n\nAlakerrasta kuului jo pient\u00e4 puheensorinaa, ja otaksuin, ett\u00e4 vieraat olivat alkaneet saapua. Pujotin jalkani mustiin korkoihin ja tyhjensin t\u00f6lkkini.\n\nKuulin miten joku takanani selvitti kurkkuaan. K\u00e4\u00e4nnyin. Mikael tuijotti minua tavalla, josta ei niin vain p\u00e4\u00e4sty selville.\n\n\"Menn\u00e4\u00e4nk\u00f6 alas?\" h\u00e4n kysyi. Ny\u00f6kk\u00e4sin.\n\nJenni ja Juha olivat puhuneet pelk\u00e4st\u00e4\u00e4n pienist\u00e4 pirskeist\u00e4, mutta olin jo alkanut ep\u00e4ill\u00e4 juhlien todellista luonnetta. Varsinainen mittakaava alkoi kuitenkin hahmottua minulle vasta kun min\u00e4 ja Mikael laskeuduimme yhdess\u00e4 portaita alas. Viimeisen tunnin aikana olohuone oli t\u00e4yttynyt kyl\u00e4l\u00e4isist\u00e4, ja meid\u00e4n saapuessamme heist\u00e4 jokainen kohotti katseensa ja unohti keskustelunsa.\n\nEn ollut koskaan toiminut em\u00e4nt\u00e4n\u00e4 miss\u00e4\u00e4n edes muodollista muistuttavassa tilanteessa, enk\u00e4 ollut tiennyt yht\u00e4\u00e4n, mihin varautua. Kukaan ei tuijottanut suoraan meihin, mutta melkein toivoin, ett\u00e4 olisi. Mik\u00e4\u00e4n ei nimitt\u00e4in ollut vaivaannuttavampaa kuin tietoisuus siit\u00e4, ett\u00e4 jostain t\u00e4ysin p\u00e4hk\u00e4hullusta syyst\u00e4 kaikki kuvittelivat minun olevan heid\u00e4n yl\u00e4puolellaan.\n\nMikaelin sanat kaikuivat p\u00e4\u00e4ss\u00e4ni: En aio valehdella. El\u00e4m\u00e4ni tulee aina olemaan t\u00e4llaista.\n\nVilkaisin Mikaelia. H\u00e4n puristi k\u00e4tt\u00e4ni ja min\u00e4 puristin takaisin.\n\nPortaiden alap\u00e4\u00e4ss\u00e4 ensimm\u00e4isen\u00e4 meit\u00e4 vastassa oli Jenni. \"Hyv\u00e4lt\u00e4 n\u00e4ytt\u00e4\u00e4\", h\u00e4n sanoi hymyillen.\n\n\"Samoin\", vastasin.\n\nJa ennen kuin huomasinkaan, meid\u00e4t pyyhk\u00e4istiin ensimm\u00e4iseen keskusteluun, ja sen j\u00e4lkeen aina seuraavaan. Lopulta min\u00e4 ja Mikael erkanimme toisistamme voidaksemme vaihtaa muutaman sanan kukin tahoillamme.\n\nKaikki nimitt\u00e4in halusivat jutella kanssani. Tai pikemminkin he halusivat kuulla minun puhuvan, sill\u00e4 monestakaan en saanut irti juuri muuta kuin my\u00f6nt\u00e4vi\u00e4 vastauksia. Mielipiteeni tuntui \u00e4kki\u00e4 merkitsev\u00e4n enemm\u00e4n kuin muiden, ja ellen olisi varovainen, tyrehdytt\u00e4isin sanomisillani koko keskustelun ennen kuin se oli alkanutkaan.\n\nT\u00e4m\u00e4 on vaikeampaa kuin uskoinkaan, ajattelin. Ja surullisinta oli, ett\u00e4 Mikael oli tehnyt t\u00e4t\u00e4 koko ik\u00e4ns\u00e4. Jokaikiset juhlat... Kuinka yksin\u00e4inen h\u00e4nen oli t\u00e4ytynyt olla?\n\nHuomasin pian, ett\u00e4 ven\u00e4l\u00e4iset sudet ja alkuper\u00e4iset hukkavaaralaiset oleilivat t\u00e4\u00e4ll\u00e4kin omissa leireiss\u00e4\u00e4n. Se ei ollut mik\u00e4\u00e4n suuri yll\u00e4tys, mutta sai minut silti huolestumaan. Tiesin, ettei ket\u00e4\u00e4n voinut pakottaa laajentamaan tuttavapiiri\u00e4\u00e4n ja ett\u00e4 oli pelk\u00e4st\u00e4\u00e4n luonnollista haluta viett\u00e4\u00e4 aikaa kaltaistensa kanssa, mutta pitk\u00e4ll\u00e4 t\u00e4ht\u00e4imell\u00e4 lauman kahtiajaosta ei seuraisi mit\u00e4\u00e4n hyv\u00e4\u00e4. Meid\u00e4n olisi keksitt\u00e4v\u00e4 jotain.\n\nYritin j\u00e4lleen t\u00e4hyill\u00e4 Mikaelin per\u00e4\u00e4n. H\u00e4n oli kadonnut kauan sitten ihmisjoukkoon. Yritin p\u00e4\u00e4tt\u00e4\u00e4, oliko h\u00e4nen ik\u00e4v\u00f6imisens\u00e4 silloin, kun tiesin h\u00e4nen olevan samassa talossa, romanttista vai pelk\u00e4st\u00e4\u00e4n s\u00e4\u00e4litt\u00e4v\u00e4\u00e4.\n\nP\u00e4\u00e4tin etsi\u00e4 yst\u00e4v\u00e4ni. Heid\u00e4n l\u00f6yt\u00e4misens\u00e4 osoittautui helpoksi, sill\u00e4 he kaikki olivat ker\u00e4\u00e4ntyneet olohuoneen sohville. Aivan ensimm\u00e4iseksi panin merkille, ett\u00e4 Niko ja Taneli istuivat vierekk\u00e4in. Minun teki mieli huokaista helpotuksesta. Niko oli siis p\u00e4\u00e4tt\u00e4nyt olla haastamatta riitaa.\n\nVasta sitten kiinnitin huomiota kolmen supattelevan tyt\u00f6n ryhm\u00e4\u00e4n, joka l\u00e4hestyi juuri sit\u00e4 sohvaa, jolla Niko ja Taneli istuivat.\n\nRohkein heist\u00e4 astui eteen. \"Hei Niko\", h\u00e4n sanoi. \"Sinulla on tosi kivat vaatteet.\"\n\nNikon kulmat kurtistuivat. H\u00e4n ei vastannut mit\u00e4\u00e4n.\n\n\"Voisitko sin\u00e4 tehd\u00e4 minun vanhojen pukuni?\" tytt\u00f6 kysyi niin nopeasti, ett\u00e4 sanoja saattoi tuskin erottaa. \"Min\u00e4 tietenkin maksan\", h\u00e4n lis\u00e4si. \"Olen s\u00e4\u00e4st\u00e4nyt rahaa jo kaksi vuotta!\"\n\nNikon kulmat eiv\u00e4t olleet en\u00e4\u00e4 kurtussa \u2013 nyt ne olivat kadonneet h\u00e4nen hiuksiensa alle. \"Se mekko, jonka teit Raisalle, oli tosi, tosi hieno\", h\u00e4n jatkoi.\n\n\"Mukavaa, ett\u00e4 pyysit. H\u00e4n harkitsee asiaa\", Taneli vastasi Nikon puolesta ja v\u00e4l\u00e4ytti enkelihymyns\u00e4. Tytt\u00f6 punastui hieman, mutisi \"okei\" ja katosi nopeasti.\n\nMe kaikki yritimme olla virnuilematta, mutta Nikon ilme teki sen mahdottomaksi. Est\u00e4\u00e4kseni itse\u00e4ni purskahtamasta nauruun annoin katseeni kiert\u00e4\u00e4 ryhm\u00e4ss\u00e4. Juha istui nojatuolissa meist\u00e4 vasemmalla. Caoimhe oli istuutunut toiseen nojatuoliin, ja Mitja k\u00e4vi lattialle h\u00e4nen jalkojensa juureen.\n\n\"Hei Raisa\", Juha sanoi.\n\nMuistin ajan, jolloin Juha ei olisi koskaan rohjennut sanoa nime\u00e4ni \u00e4\u00e4neen, saati sitten tehd\u00e4 niin huoneessa, joka oli t\u00e4ynn\u00e4 susia.\n\n\"No?\" kysyin.\n\n\"Toivoisin, ett\u00e4 voisit auttaa minua yhdess\u00e4 asiassa.\"\n\n\"Tietenkin, jos vain osaan.\"\n\n\"Olen ajatellut ottaa tatuoinnin\", h\u00e4n aloitti.\n\nHymyilin. \"Millainen kuva sinulla on mieless\u00e4?\"\n\n\"Sen kanssa tarvitsenkin apuasi. En oikeastaan yht\u00e4\u00e4n tied\u00e4. Tied\u00e4n vain, mit\u00e4 haluan sen merkitsev\u00e4n.\"\n\n\"Kerro niin paljon kuin voit ja katsotaan, mit\u00e4 min\u00e4 saan luonnosteltua silt\u00e4 pohjalta.\"\n\n\"Haluaisin ottaa tatuoinnin isovanhempieni muistoksi.\"\n\nSe sai minut vakavoitumaan. Maritta oli ollut Martin uhri. Ja vaikka en ollut miss\u00e4\u00e4n vaiheessa pit\u00e4nyt h\u00e4nest\u00e4, se, mit\u00e4 h\u00e4nelle oli tapahtunut, oli ollut v\u00e4\u00e4rin.\n\n\"Tied\u00e4n, ett\u00e4 he olivat outoja ja lopussa hullujakin, mutta he kasvattivat minut.\"\n\n\"Ei sinun tarvitse selitell\u00e4 asiaa mitenk\u00e4\u00e4n\", sanoin niin lempe\u00e4sti kuin osasin.\n\n\"Haluaisin jonkun kuvan, en mit\u00e4\u00e4n teksti\u00e4.\"\n\n\"Mukava kuulla. \u00c4l\u00e4 kerro asiakkailleni, mutta minusta tekstitatuoinnit ovat usein tosi juntteja. Onko sinulla mit\u00e4\u00e4n ideaa paikasta?\"\n\nH\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n.\n\n\"No, uskoisin, ett\u00e4 tietyt paikat ovat ainakin poissuljettuja. Itse en ainakaan ottaisi muistotatuointia esimerkiksi pakaraan. Jotenkin ep\u00e4ilen, ettei Maritta arvostaisi sit\u00e4.\"\n\nNiko, Jenni ja pari muuta n\u00e4yttiv\u00e4t \u00e4kki\u00e4 vaivaantuneilta. Oli paljon ihmisi\u00e4, joiden mielest\u00e4 kuolemalla ei saanut vitsailla, mutta yleens\u00e4 se vaivasi eniten heit\u00e4, jotka eiv\u00e4t olleet koskaan menett\u00e4neet ket\u00e4\u00e4n l\u00e4heist\u00e4. Kuoleminen ei tehnyt kenest\u00e4k\u00e4\u00e4n pyh\u00e4\u00e4.\n\nJuha nauroi. \"Tuleeko sinulle paljon miesasiakkaita, jotka haluavat tatuoinnin pakaraan?\"\n\n\"Joitakin\", sanoin. K\u00e4vin yl\u00e4kerrassa hakemassa laukustani tussin.\n\nPalattuani kumarruin l\u00e4helle Juhaa niin, ettei Taneli kuulisi. \"Jos haluat, voin... kaivella v\u00e4h\u00e4n. Se kyll\u00e4 tarkoittaa, ett\u00e4 saatan n\u00e4hd\u00e4 joitain henkil\u00f6kohtaisia juttuja.\"\n\n\"Se sopii kyll\u00e4.\"\n\nJenni nousi seisomaan niin, ett\u00e4 saatoin istua sohvalle Juhan viereen. Suljin silm\u00e4ni. Vasta kun avasin yhteyden ideoiden maailmaan, tajusin, ettei minun olisi ehk\u00e4 kannattanut tehd\u00e4 sit\u00e4. Vasta edellisen\u00e4 p\u00e4iv\u00e4n\u00e4 olin maannut omassa veress\u00e4ni pelk\u00e4st\u00e4\u00e4n siksi, ett\u00e4 olin puskenut kykyni \u00e4\u00e4rirajoille ja ylikin. Ja silti olin ollut valmis tekem\u00e4\u00e4n sen taas.\n\nNappasin kiinni ensimm\u00e4isest\u00e4 mahdollisesta kuvasta. \"Kun olit lapsi, sin\u00e4 ja sinun isovanhempasi k\u00e4vitte usein kalastamassa.\"\n\n\"Niin k\u00e4vimme\", Juha sanoi.\n\nAvasin silm\u00e4ni. Tiesin, mit\u00e4 tehd\u00e4.\n\n\"Paita pois\", komensin. Juha totteli.\n\nPaidan alta paljastui selk\u00e4, joka oli silkkaa lihasta. \"Joku on treenannut oikein kunnolla\", sanoin ihailevasti.\n\n\"Voisin sanoa samaa sinulle.\"\n\n\"Kiitti.\"\n\nAloin luonnostella h\u00e4nen ihoonsa japanilaistyylist\u00e4 kalaa. Otso, toinen mentoreistani, oli erikoistunut japanilaisiin tatuointeihin, joten aihe ei ollut minulle vieras. Keskityin ty\u00f6h\u00f6n niin t\u00e4ysill\u00e4, ett\u00e4 kesti aika kauan, ennen kuin tajusin, ett\u00e4 meid\u00e4n ymp\u00e4rillemme oli alkanut ker\u00e4\u00e4nty\u00e4 yleis\u00f6\u00e4.\n\nViimein sain kuvan valmiiksi.\n\n\"Voit toki vied\u00e4 t\u00e4m\u00e4n idean jollekulle muulle, jonka tyylist\u00e4 pid\u00e4t enemm\u00e4n. En suutu sellaisesta.\" Toivoin kuitenkin salaa, ett\u00e4 saisin tehd\u00e4 sen itse. Kuva nimitt\u00e4in n\u00e4ytti todella hyv\u00e4lt\u00e4 ja sopi t\u00e4ydellisesti siihen kohtaan, johon se oli tehty.\n\n\"\u00c4l\u00e4 turhaan\", Mikael sanoi. \"Raisa on paras.\"\n\nK\u00e4\u00e4nnyin ymp\u00e4ri. Mikael seisoi suoraan minua vastap\u00e4\u00e4t\u00e4. En ollut huomannut h\u00e4nen ilmestyneen paikalle.\n\n\"Mist\u00e4 sin\u00e4 sen tied\u00e4t?\" kysyin. H\u00e4nen luottamuksensa sek\u00e4 huvitti ett\u00e4 l\u00e4mmitti minua.\n\n\"Etk\u00f6 muista? Min\u00e4 olen lukenut arvostelujasi.\"\n\nOlimme lukeneet niit\u00e4 yhdess\u00e4 kun olin tajunnut, ett\u00e4 sellaisia edes oli olemassa. Aluksi minua oli j\u00e4nnitt\u00e4nyt n\u00e4hd\u00e4, mit\u00e4 ihmiset kirjoittivat, ja olin pyyt\u00e4nyt Mikaelin tuekseni. Arvostelut olivat olleet hyvi\u00e4, vaikka joku olikin mielest\u00e4\u00e4n saanut erilaisen kuvan kuin mit\u00e4 h\u00e4nell\u00e4 itsell\u00e4\u00e4n oli ollut mieless\u00e4. Muistin tuon kirjoituksen paljon paremmin kuin muut, eik\u00e4 omaksi mielikuvakseni ollut j\u00e4\u00e4nyt se, ett\u00e4 arvostelijoiden mielest\u00e4 olisin ollut paras. Mikael oli selv\u00e4sti kiinnitt\u00e4nyt huomiota eri asioihin kuin min\u00e4.\n\n\"H\u00e4nt\u00e4 my\u00f6s kutsutaan kaupungin kuumimmaksi tatuointitaiteilijaksi.\" Eik\u00e4 h\u00e4n voinut olla lis\u00e4\u00e4m\u00e4tt\u00e4: \"Mik\u00e4 tietenkin selitt\u00e4\u00e4 ne pakaramiehet.\" H\u00e4nkin oli siis kuullut sen. Hitto.\n\n\"Kuulostaa silt\u00e4, ett\u00e4 joku on mustasukkainen\", Niko sanoi. H\u00e4n n\u00e4ytti melkein tyytyv\u00e4iselt\u00e4.\n\n\"Miksi olisin? Minun ei tarvitse varata h\u00e4nelt\u00e4 aikaa takapuoltani varten.\" H\u00e4n katsoi minua, ja uskoin, ett\u00e4 olin ainoa, joka tajusi h\u00e4nen olevan oikeasti huvittunut.\n\nSeurasi mykistynyt hiljaisuus. Sitten Niko tyrsk\u00e4hti. \"Ihmeit\u00e4 tapahtuu. Sarrillakin voi olla huumorintaju.\" H\u00e4n kohotti kuppiaan ja kippisti Mikaelin kanssa.\n\n\"Harkitse asiaa\", sanoin Juhalle. \"Jos et pid\u00e4 tuosta kuvasta, voin kehitell\u00e4 jotain muuta. Minulla on t\u00e4ll\u00e4 hetkell\u00e4 jonossa aika monta asiakasta, mutta enk\u00f6h\u00e4n saa ujutettua sinut johonkin v\u00e4liin.\"\n\n\"Kiitos\", h\u00e4n sanoi.\n\n\"Ei kest\u00e4.\" K\u00e4\u00e4nnyin vied\u00e4kseni tussin pois \u2013 tai sitten vain paetakseni tilanteen aiheuttamaa h\u00e4mmennyst\u00e4.\n\nOlohuoneeseen oli pakkautunut melkoinen joukko susia. Olin jo alkanut pujotella kohti yl\u00e4kertaan vievi\u00e4 portaita, kun katseeni osui vaaleaan tytt\u00f6\u00f6n. Muistin h\u00e4nen kuuluvan niihin, jotka olivat muuttaneet kyl\u00e4\u00e4n Karjalasta. H\u00e4n n\u00e4ytti ujolta. Ei sellaiselta tyt\u00f6lt\u00e4, jonka pojat tai ylip\u00e4\u00e4t\u00e4\u00e4n kukaan huomaisi ensimm\u00e4isen\u00e4. Kuitenkin mit\u00e4 kauemmin katsoin h\u00e4nt\u00e4, sit\u00e4 n\u00e4timp\u00e4n\u00e4 h\u00e4nt\u00e4 pidin. H\u00e4nell\u00e4 oli kaunis vaalea iho \u2013 tatuointitaiteilijana sellaiseen kiinnitti huomiota.\n\nMenin h\u00e4nen luokseen. \"Hei. Mik\u00e4 sinun nimesi on?\"\n\nTunsin, kuinka kaikkien huomio siirtyi hitaasti meihin. Vilkaisin nopeasti kohti paikkaa, jossa Mikael oli seisonut viel\u00e4 hetki sitten, mutta h\u00e4n oli jo ehtinyt kadota.\n\n\"Elvi\", tytt\u00f6 sanoi lopulta.\n\n\"Kiva nimi. Haluaisitko, ett\u00e4 luonnostelisin sinullekin jotain?\"\n\nH\u00e4n r\u00e4pytteli pitki\u00e4 ripsi\u00e4\u00e4n.\n\n\"Ihan vain huvin vuoksi. Ei sinun tarvitse oikeasti haluta tatuointia. Tussi l\u00e4htee muutaman pesun j\u00e4lkeen pois.\"\n\nPitk\u00e4n ajan kuluttua h\u00e4n ny\u00f6kk\u00e4si.\n\n\"K\u00e4yk\u00f6 nilkka?\"\n\nElvi ny\u00f6kk\u00e4si taas. Istuin sohvan eteen risti-istuntaan. Kuulin kuinka joku henk\u00e4isi \u2013 ilmeisesti minunkin oletettiin noudattavan hallitsevien susien k\u00e4yt\u00f6skoodia eik\u00e4 laskeutuvan koskaan toisia susia alemmalle tasolle. En piitannut siit\u00e4. Elvi pudotti ballerinansa lattialle, ja asetin h\u00e4nen jalkansa reitt\u00e4ni vasten. Katsoin hetken aikaa nilkan muotoja ja yritin l\u00f6yt\u00e4\u00e4 niist\u00e4 idean. Nopeasti sen sainkin: hahmottelin ensin kuvan \u00e4\u00e4riviivat, ja v\u00e4hitellen esiin alkoi piirty\u00e4 feeniks-lintu.\n\n\"Vau, tosi upea!\" Elvi sanoi kun sain sen valmiiksi.\n\nMyh\u00e4ilin tyytyv\u00e4isen\u00e4. En siksi, ett\u00e4 olisin pel\u00e4nnyt h\u00e4nen palautettaan, vaan siksi, ett\u00e4 h\u00e4n oli unohtanut pelkonsa minua kohtaan. \"Sinun pit\u00e4\u00e4 odottaa viel\u00e4 muutama vuosi, ennen kuin voit hankkia tatuoinnin\", sanoin. \"Hyvin aikaa mietti\u00e4.\"\n\nNousin yl\u00f6s ja pyyhk\u00e4isin nihke\u00e4t k\u00e4teni reisiini. Olin tuskin ehtinyt p\u00e4\u00e4st\u00e4 jaloilleni, kun Elvin nilkkaa ihailemaan tulleet sudet olivat jo vieneet paikkani. Huomasin joukossa my\u00f6s ainakin pari alkuper\u00e4ist\u00e4 hukkavaaralaista.\n\nKoko porukka n\u00e4ytti unohtaneen minut, mik\u00e4 sopi minulle paremmin kuin hyvin. Per\u00e4\u00e4nnyin niin huomaamattomasti kuin kykenin, yh\u00e4 itsekseni virnistellen.\n\nSilloin n\u00e4in Jonin juhlijoiden joukossa. H\u00e4n oli yksin. H\u00e4nen ymp\u00e4rilleen oli j\u00e4tetty parin metrin turvavy\u00f6hyke, aivan kuin h\u00e4nen kokemansa n\u00f6yryytys voisi tarttua muihinkin.\n\nEn tied\u00e4, mik\u00e4 minuun meni, mutta huomasin k\u00e4velev\u00e4ni h\u00e4nt\u00e4 kohti. Kun h\u00e4n huomasi minut, h\u00e4n alkoi vilkuilla ymp\u00e4rilleen \u2013 joko etsi\u00e4kseen selityst\u00e4 l\u00e4hestymiselleni tai sitten jotain, joka voisi suojella h\u00e4nt\u00e4 minulta. Molemmat vaihtoehdot tuntuivat yht\u00e4 varteenotettavilta.\n\n\"Haluatko sin\u00e4kin tatuoinnin?\" t\u00f6ks\u00e4ytin.\n\nH\u00e4n ei sanonut mit\u00e4\u00e4n, tuijotti vain jonnekin alaviistoon. Saatoin n\u00e4hd\u00e4, ett\u00e4 minun n\u00e4kemiseni nolotti h\u00e4nt\u00e4.\n\n\"Ojenna k\u00e4tesi\", sanoin.\n\nH\u00e4n totteli. H\u00e4n s\u00e4ps\u00e4hti hieman, kun tartuin siihen tukevalla otteella ja k\u00e4\u00e4nsin sis\u00e4puolen kohti itse\u00e4ni. Aloin kirjoittaa h\u00e4nen ihoonsa.\n\nKun olin valmis, h\u00e4n k\u00e4\u00e4nsi k\u00e4sivarttaan niin, ett\u00e4 saattoi lukea tekstin. Olin kirjoittanut sen muinaiskreikaksi, joten h\u00e4n kysyi: \"Mit\u00e4 siin\u00e4 sanotaan?\"\n\n\"Vapaasti suomennettuna: 'Anteeksianto vapauttaa'.\"\n\n\"Onko se jotain Aristotelesta?\"\n\n\"Ei, vaan Raisa Ojaa.\"\n\nH\u00e4n ei selv\u00e4stik\u00e4\u00e4n tiennyt, vitsailinko vai en.\n\n\"Minulla on sinulle ilmainen neuvo\", aloitin.\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\n\"Yrit\u00e4 yst\u00e4vysty\u00e4 lupinilaisten kanssa. Koska et ole varsinaisesti hukkavaaralaisten suosiossa, he ovat sinun ainoa vaihtoehtosi.\"\n\nOlin jo l\u00e4hd\u00f6ss\u00e4, kun h\u00e4n yll\u00e4tti minut sanomalla: \"Kuulin kyll\u00e4 kun sanoit Juhalle, ett\u00e4 tekstitatuoinnit ovat juntteja.\"\n\nHymyilin niin herttaisesti kuin osasin ja liukenin paikalta.\n\nParemman tekemisen puutteessa l\u00f6ysin itseni keitti\u00f6st\u00e4. Py\u00f6rittelin k\u00e4dess\u00e4ni olevaa muovimukia ja katsoin ikkunasta juuri ja juuri valoisaa taivasta vasten erottuvaa kuuta. Se oli ehtinyt jo viimeiselle nelj\u00e4nnekselleen \u2013 t\u00e4ysikuuhun oli en\u00e4\u00e4 viikko.\n\nOli mukavaa olla yksin, mutta samalla se muistutti minua siit\u00e4, ett\u00e4 minun kuuluisi tuntea olevani t\u00e4\u00e4ll\u00e4 kotonani, eik\u00e4 se pit\u00e4nyt paikkansa. Niko, Taneli, Mitja ja Caoimhe palaisivat kaikki ylihuomenna takaisin Helsinkiin, ja totta puhuakseni tuntui oudolta olla se, joka j\u00e4isi Hukkavaaraan. Min\u00e4 olin aina ollut ensimm\u00e4isen\u00e4 l\u00e4hd\u00f6ss\u00e4 t\u00e4\u00e4lt\u00e4.\n\n\"Tied\u00e4n, mit\u00e4 sin\u00e4 yrit\u00e4t\", \u00e4\u00e4ni sanoi takanani. Tiesin heti, kenelle se kuului. K\u00e4\u00e4nnyin ymp\u00e4ri ja pujotin k\u00e4sivarteni Mikaelin kaulalle.\n\n\"Minulla ei ole aavistustakaan, mist\u00e4 sin\u00e4 puhut\", sanoin.\n\n\"\u00c4l\u00e4 esit\u00e4 viatonta\", h\u00e4n vastasi. \"Min\u00e4 n\u00e4in sinun piirt\u00e4v\u00e4n sen lupinilaistyt\u00f6n nilkkaan. Ja Jonin k\u00e4sivarteen\", h\u00e4n lis\u00e4si melkein vastahakoisesti.\n\n\"Hei, min\u00e4 vain markkinoin yrityst\u00e4ni.\" Huuleni olivat vain parin sentin p\u00e4\u00e4ss\u00e4 h\u00e4nen omistaan.\n\nJoku selvitti kurkkuaan. Otin asiakseni palauttaa itseni takaisin t\u00e4h\u00e4n aikaan ja paikkaan.\n\nMikael oli jo k\u00e4\u00e4ntynyt katsomaan ovelle, ja kumarruin n\u00e4hd\u00e4kseni h\u00e4nen ohitseen.\n\nJenni seisoi muutaman metrin p\u00e4\u00e4ss\u00e4 pullo k\u00e4dess\u00e4\u00e4n. \"Minut l\u00e4hetettiin etsim\u00e4\u00e4n teit\u00e4. Mitja ja Ciao alkaisivat olla valmiita.\"\n\nMikael, rakas Mikaelini, n\u00e4ytti oikeasti \u00e4rtyneelt\u00e4. Minua l\u00e4hinn\u00e4 nauratti. Astui kauemmas ja nappasin mukini p\u00f6yd\u00e4lt\u00e4.\n\nEmme olleet ehtineet ottaa kahtakaan askelta keitti\u00f6n ulkopuolella, kun joku pys\u00e4ytti Mikaelin ja pyysi saada puhua t\u00e4m\u00e4n kanssa. Huokaisin ja seurasin Jenni\u00e4.\n\n\"Mit\u00e4 sin\u00e4 oikein olet tehnyt h\u00e4nelle?\" h\u00e4n kuiskasi heti kun olimme p\u00e4\u00e4sseet tarpeeksi et\u00e4\u00e4lle.\n\nSyd\u00e4meni tuntui valahtavan jonnekin jalkojeni juureen. \"Ai sin\u00e4kin huomaat sen?\"\n\n\"Tietenkin huomaan. En ole koskaan n\u00e4hnyt h\u00e4nt\u00e4 n\u00e4in... onnellisena.\"\n\nH\u00e4n ei siis puhunutkaan Mikaelin sudesta. Helpotukseni oli niin suuri, ett\u00e4 hetken minua huimasi.\n\n\"En vain oikein tajua, mit\u00e4 sinun p\u00e4\u00e4ss\u00e4si liikkuu\", h\u00e4n lis\u00e4si.\n\n\"Miten niin?\"\n\n\"Sin\u00e4 et varsinaisesti k\u00e4ytt\u00e4ydy niin kuin laumanjohtajan tytt\u00f6yst\u00e4v\u00e4n kuuluu.\"\n\n\"Mit\u00e4 min\u00e4 olen muka tehnyt v\u00e4\u00e4rin?\"\n\nH\u00e4n katsoi minua. \"L\u00e4\u00e4pit puolialastonta Juhaa Mikaelin edess\u00e4. Aika moni sanoisi, ett\u00e4 sin\u00e4 \u00e4rsyt\u00e4t h\u00e4nt\u00e4 ihan tahallaan.\"\n\n\"Enk\u00e4 \u00e4rsyt\u00e4\", sanoin. Heti sen j\u00e4lkeen aloin kuitenkin pohtia, olinko alitajuisesti yritt\u00e4nyt saada Mikaelin mustasukkaiseksi.\n\nJa miksei Mikael ollut sanonut mit\u00e4\u00e4n?\n\nLuultavasti siit\u00e4 samasta syyst\u00e4, jonka takia min\u00e4k\u00e4\u00e4n en ollut sanonut h\u00e4nelle mit\u00e4\u00e4n peloistani h\u00e4nen sutensa ja tulevaisuutemme suhteen. Me molemmat pelk\u00e4simme toisen reaktiota.\n\nHalusin vaihtaa aihetta. \"Mit\u00e4 sinun oululaiselle syd\u00e4nk\u00e4pysellesi kuuluu?\" kysyin.\n\n\"Ei mit\u00e4\u00e4n\", h\u00e4n sanoi. \"Se juttu ei toiminut.\"\n\n\"Ai. Olen pahoillani.\"\n\n\"\u00c4l\u00e4 ole. Olen sinkku ja vakaa aikomukseni on pysy\u00e4kin sellaisena.\"\n\n\"Oletko onnistunut korjaamaan v\u00e4lej\u00e4si vanhempiesi kanssa?\" kysyin.\n\n\"En varsinaisesti\", Jenni tunnusti.\n\n\"Miss\u00e4 sin\u00e4 sitten asut t\u00e4\u00e4ll\u00e4?\"\n\n\"Juhan luona.\"\n\nSiin\u00e4 oli j\u00e4rke\u00e4. Juhalla oli kokonainen talo yksin asuttavanaan, ja h\u00e4n ja Jenni vaikuttivat olevan nyky\u00e4\u00e4n hyvi\u00e4 yst\u00e4vi\u00e4. \"H\u00e4n on niin kiltti\", ajattelin \u00e4\u00e4neen. Juha oli selv\u00e4sti rohkeampi kuin viel\u00e4 vuosi sitten, mutta h\u00e4n ei ollut silti luopunut avuliaisuudestaan.\n\n\"Tied\u00e4n.\" Jenni huokaisi. \"Taidan tarvita shotin. T\u00e4ss\u00e4: min\u00e4 kaadan sinullekin.\"\n\nH\u00e4n avasi pullon ja kaatoi muovimukiini, ennen kuin ehdin toppuutella h\u00e4nt\u00e4.\n\nKun ehdimme olohuoneeseen, muut olivat jo kokoontuneet sinne. Hukkavaara ei ollut mitenk\u00e4\u00e4n iso paikka, mutta joskus sen asukasm\u00e4\u00e4r\u00e4 yll\u00e4tti minut. Kaikki eiv\u00e4t olleet edes mahtuneet sis\u00e4\u00e4n, mutta patiolle avatut lasiovet mahdollistivat sen, ett\u00e4 ulkonakin seisovat kuulivat, mit\u00e4 sis\u00e4ll\u00e4 tapahtui. Toivoin n\u00e4kev\u00e4ni vilauksen Mikaelista, mutta h\u00e4n oli jumittunut jonnekin k\u00e4yt\u00e4v\u00e4\u00e4n ihmismassan taakse.\n\nMitja seisoi p\u00f6yd\u00e4n takana kuulokkeet kaulallaan. H\u00e4nen huomionsa oli l\u00e4pp\u00e4rin n\u00e4yt\u00f6ss\u00e4, eik\u00e4 h\u00e4nt\u00e4 n\u00e4ytt\u00e4nyt kiinnostavan p\u00e4tk\u00e4\u00e4k\u00e4\u00e4n se, ett\u00e4 h\u00e4nen yleis\u00f6ns\u00e4 seisoi kirjaimellisesti puolen metrin p\u00e4\u00e4ss\u00e4 h\u00e4nest\u00e4.\n\nOlohuonekeikan soittaminen oli tietenkin ollut Caoimhen idea, ja yll\u00e4tyksekseni veljeni oli suostunut. He halusivat kiitt\u00e4\u00e4 Mikaelia h\u00e4nen vieraanvaraisuudestaan, enk\u00e4 itsek\u00e4\u00e4n olisi keksinyt parempaa tapaa.\n\nMenimme h\u00e4nen luokseen. Sanoin kreikaksi: \"Koita olla t\u00f6peksim\u00e4tt\u00e4. T\u00e4m\u00e4 yleis\u00f6 on vaativaa.\" Se oli tietenkin p\u00f6ty\u00e4 \u2013 hukkavaaralaiset olisivat olleet haltijoissaan, vaikka h\u00e4n olisi soittanut heille Ukko Nooaa nokkahuilulla.\n\nCaoimhe ilmestyi veljeni viereen. Taneli oli suoristanut h\u00e4nen hiuksensa, ja ne n\u00e4yttiv\u00e4t h\u00e4kellytt\u00e4v\u00e4n pitkilt\u00e4 ja kiilt\u00e4vilt\u00e4. Olisi ollut mahdotonta olla huomaamatta l\u00e4hell\u00e4 seisovien miespuolisten susien ihailevia katseita. Mitja ei kuitenkaan tuntunut \u00e4kk\u00e4\u00e4v\u00e4n mit\u00e4\u00e4n. Yll\u00e4tyin n\u00e4hdess\u00e4ni, ett\u00e4 Ciaolla oli k\u00e4dess\u00e4 mikrofoni, mutta sitten tajusin h\u00e4nen ja Mitjan tuoneen \u00e4\u00e4nivehkeet mukanaan hotellilta.\n\nMin\u00e4 ja Jenni siirryimme kauemmaksi. Melkein heti Caoimhe alkoi puhua.\n\n\"Hei kaikki, min\u00e4 ja Mitja soitamme teille nyt pari kappaletta\", h\u00e4n sanoi.\n\nOlisi voinut ajatella, ett\u00e4 kotibileet olisivat jotenkin hankalampi ymp\u00e4rist\u00f6 kuin klubi, mutta olin v\u00e4hitellen alkanut tulla siihen tulokseen, ettei Mitjalle ollut mit\u00e4\u00e4n v\u00e4li\u00e4 sill\u00e4, oliko paikalla kaksi vai kaksisataatuhatta ihmist\u00e4. Kunhan h\u00e4n vain p\u00e4\u00e4si oikealle taajuudelle ideoiden maailman kanssa. Silloin kaikki muu lakkaisi olemasta.\n\nEn ollut kuullut Mitjan ja Caoimhen demoa. Syyst\u00e4 tai toisesta Mitja oli p\u00e4\u00e4tt\u00e4nyt olla soittamatta sit\u00e4 minulle. Huomasin pid\u00e4tt\u00e4v\u00e4ni hengityst\u00e4 ensimm\u00e4isten tahtien ajan. Sitten minut viskattiin t\u00e4ysin toiseen todellisuuteen, enk\u00e4 kyennyt olemaan tietoinen jostain niinkin arkisesta asiasta kuin keuhkojeni toiminnasta.\n\nBiisit eiv\u00e4t olleet lainkaan sit\u00e4, mit\u00e4 odotin. Caoimhen irkkutaustan tuntien olin odottanut hempe\u00e4\u00e4 ja surullista folkia tai jopa kantria. T\u00e4m\u00e4 oli rajua ja mystist\u00e4 ja ennen kaikkea synkk\u00e4\u00e4. Siit\u00e4 huolimatta huomasin jalkani nytk\u00e4htelev\u00e4n niin kuin se olisi halunnut l\u00e4hte\u00e4 t\u00e4ll\u00e4 sekunnilla tanssilattialle.\n\nBiitti oli nimitt\u00e4in hyvin tarttuva. Mitja oli laittanut peliin kaiken, mit\u00e4 h\u00e4n tiesi elektronisesta musiikista ja Ciao lauloi paremmin kuin olin koskaan kuullut h\u00e4nen laulavan, v\u00e4lill\u00e4 pehme\u00e4sti, v\u00e4lill\u00e4 maanisesti, ja v\u00e4lill\u00e4 vain hyvin kovaa ja korkealta. En olisi kuolemaksenikaan kyennyt nime\u00e4m\u00e4\u00e4n genre\u00e4, johon kuulemani musiikki kuului, ja olin aika varma, ettei kukaan muukaan olisi siihen pystynyt. Joka tapauksessa Mitjan biitit ja Caoimhen laulu toimivat t\u00e4ydellisesti yhteen.\n\nSe oli ohi aivan liian pian. \"Kiitos! Me olemme Mist & Snow!\" Caoimhe sanoi. Ilo paistoi h\u00e4nen kasvoiltaan. Kun katselin ymp\u00e4rilleni, oli helppo n\u00e4hd\u00e4 miksi: viiden biisin aikana koko huone oli muuttunut j\u00e4hme\u00e4st\u00e4 massasta hikiseksi ja riehakkaaksi.\n\nKaikkialta satoi kehuja ja selk\u00e4\u00e4ntaputuksia, kun Caoimhe teki tiens\u00e4 minun ja Jennin luokse.\n\nN\u00e4ytin Mitjalle peukaloa. H\u00e4n ny\u00f6kk\u00e4si, mutta jatkoi soittamista. Osa tyt\u00f6ist\u00e4 oli juuri intoutunut tanssimaan. Bileet olivat vasta alkamassa.\n\n\"Me taisimme vied\u00e4 sinun poikayst\u00e4v\u00e4si\", Jenni sanoi Caoimhelle.\n\n\"Ei se haittaa. H\u00e4n on paljon tyytyv\u00e4isempi kun saa soittaa eik\u00e4 tarvitse jutella ventovieraiden kanssa.\"\n\nJuha liittyi seuraamme, ja pian l\u00f6ysin Nikon ja Tanelin ihmisjoukosta. Viittil\u00f6in heid\u00e4t luoksemme. Me tanssimme, juttelimme ja nauroimme, eik\u00e4 sill\u00e4 hetkell\u00e4 mik\u00e4\u00e4n olisi voinut olla pieless\u00e4. Mitjan musiikki humisi sis\u00e4ll\u00e4ni ja sai p\u00e4\u00e4ni ja kaikki j\u00e4seneni tuntumaan kevyilt\u00e4.\n\nT\u00e4m\u00e4 el\u00e4m\u00e4... Niin j\u00e4rjet\u00f6nt\u00e4 kuin se olikin, oli se my\u00f6s aika mahtavaa.\n\nLopulta minun oli ment\u00e4v\u00e4 keitti\u00f6\u00f6n hakemaan juotavaa. Ennen kuin olin ehtinyt astua huoneeseen, kuulin sis\u00e4lt\u00e4 kantautuvan keskustelua.\n\n\"N\u00e4itk\u00f6 sen tatuoinnin, jonka Raisa piirsi Elville? Tosi siisti.\"\n\n\"Joo, olen niin kateellinen. Mutta kuulitko, kun Sanna kysyi Nikoa tekem\u00e4\u00e4n h\u00e4nen vahojen pukunsa? Min\u00e4 en olisi koskaan uskaltanut.\"\n\n\"Ja Niko on muuttamassa Pariisiin! Joku oikeasti menestyy, vaikka on t\u00e4\u00e4lt\u00e4 kotoisin.\"\n\n\"Raisa on se, joka aloitti kaiken.\"\n\n\"Minun mielest\u00e4ni h\u00e4n on tosi pelottava.\"\n\n\"Ehk\u00e4, mutta h\u00e4n pokasi Mikaelin.\"\n\nHihityst\u00e4.\n\n\"H\u00e4nen veljens\u00e4 on tosi hyv\u00e4nn\u00e4k\u00f6inen. Ja cool.\"\n\nEn voinut vain seist\u00e4 ovensuussa. Astuin sis\u00e4\u00e4n ja sanoin: \"Onhan veljeni ihan cool. Ainakin niin kauan kunnes h\u00e4n avaa suunsa.\"\n\nHuoneessa oli kolme tytt\u00f6\u00e4. Yl\u00e4kouluik\u00e4isi\u00e4, ajattelin. Toivoin heid\u00e4n kuvittelevan minun kuulleen ainoastaan viimeisimm\u00e4n kommentin, sill\u00e4 en halunnut nolata heit\u00e4. En ollut varma, olinko onnistunut vai en, sill\u00e4 kaikki tuijottivat p\u00f6yt\u00e4\u00e4. Puna alkoi hiipi\u00e4 heid\u00e4n kasvoilleen.\n\nAvasin j\u00e4\u00e4kaapin ja teeskentelin etsiv\u00e4ni sielt\u00e4 jotain, jotta he ehtisiv\u00e4t toipua. Lopulta nappasin vissypullon ja k\u00e4\u00e4nnyin tytt\u00f6j\u00e4 kohti.\n\nHe eiv\u00e4t viel\u00e4k\u00e4\u00e4n uskaltaneet katsoa minuun p\u00e4in. Ristin nilkkani ja nojasin tiskip\u00f6yt\u00e4\u00e4n.\n\n\"En ole tainnut tavata teit\u00e4 aikaisemmin\", sanoin niin yst\u00e4v\u00e4llisesti kuin osasin. \"Mitk\u00e4 teid\u00e4n nimet ovat?\"\n\n\"Lilli\", sanoi ensimm\u00e4inen. H\u00e4n jopa uskaltautui vilkaisemaan minua kerran ennen kuin katse hakeutui taas h\u00e4nen k\u00e4siins\u00e4. \"Kaisa\", vastasi toinen. Kolmas ei sanonut mit\u00e4\u00e4n.\n\n\"No, miten teid\u00e4n iltanne on mennyt?\"\n\nEi vastausta, pelkk\u00e4\u00e4 p\u00f6yd\u00e4n tuijottelua ja pikaisia katseita toisten suuntaan.\n\n\"Minulla on ainakin ollut ihan hauskaa. Kiva, ett\u00e4 niin moni on tullut paikalle.\"\n\nEi viel\u00e4k\u00e4\u00e4n mit\u00e4\u00e4n. Otin h\u00f6rpyn piilottaakseni oman h\u00e4mmennykseni.\n\nOlin vaarassa punastua pian itsekin. N\u00e4iden juhlien em\u00e4nt\u00e4n\u00e4 ty\u00f6nkuvaani kuului pit\u00e4\u00e4 huoli, ett\u00e4 kaikilla oli kivaa ja ett\u00e4 kaikki tulivat toimeen. Jos n\u00e4iden tytt\u00f6jen reaktioita saattoi pit\u00e4\u00e4 min\u00e4\u00e4n mittarina, en tosiaankaan ollut kovin hyv\u00e4 siin\u00e4.\n\nPelastava ritari saapui juuri silloin paikalle. Mikael livahti omaan keitti\u00f6\u00f6ns\u00e4 sen n\u00e4k\u00f6isen\u00e4, ett\u00e4 oli etsinyt piilopaikkaa jo hyv\u00e4n tovin. Helpotus, joka n\u00e4kyi h\u00e4nen kasvoillaan heti, kun h\u00e4n n\u00e4ki minut, sai minut salaa hyvin iloiseksi.\n\nKohotin kulmiani. \"Ai niin kamalaa?\"\n\nH\u00e4n tuli l\u00e4hemm\u00e4s. Vapaa k\u00e4teni meni heti h\u00e4nen hiuksiinsa ja alkoi sukia niit\u00e4.\n\n\"Olen puhunut viimeisen tunnin tonttiasioista ja mets\u00e4nhoidosta\", h\u00e4n mumisi kaulaani vasten.\n\n\"Voi raukkaa\", sanoin. Hieroin h\u00e4nen olkap\u00e4\u00e4t\u00e4\u00e4n. En pit\u00e4nyt siit\u00e4, miten v\u00e4syneelt\u00e4 h\u00e4n vaikutti. Laskin pulloni p\u00f6yd\u00e4lle.\n\n\"Taitaa olla aika j\u00e4tt\u00e4\u00e4 n\u00e4m\u00e4 juhlat ja menn\u00e4 kotiin.\"\n\nHuvittunut ilme levisi h\u00e4nen kasvoilleen. \"Mik\u00e4li muistat, me asumme t\u00e4\u00e4ll\u00e4.\"\n\nNiin, se oli tietenkin juhlien j\u00e4rjest\u00e4misen varjopuoli. Ei voinut vain l\u00e4hte\u00e4.\n\nJa kuka muka sanoo, ettei voi?\n\n\"Sit\u00e4 paitsi sinulla tuntuu olevan hauskaa\", Mikael jatkoi.\n\n\"Ehk\u00e4, mutta minulla tulee olemaan hauskempaa sinun kanssasi.\"\n\nTartuin h\u00e4nt\u00e4 k\u00e4dest\u00e4 ja l\u00e4hdin johdattamaan pois huoneesta. Vasta silloin muistin, ettemme olleet kahdestaan. Kolme tytt\u00f6\u00e4 tuijottivat meit\u00e4 niin kuin eiv\u00e4t olisi el\u00e4ess\u00e4\u00e4n n\u00e4hneet mit\u00e4\u00e4n niin h\u00e4kellytt\u00e4v\u00e4\u00e4. Pys\u00e4hdyin. \"Jos joku kysyy, sanokaa, ettette tied\u00e4 miss\u00e4 me olemme\", sanoin heille. Ehdin jo ottaa askeleen, ennen kuin muistin, etteiv\u00e4t sudet voineet valehdella. Joten lis\u00e4sin: \"Se ei ole mik\u00e4\u00e4n vale: te ette tied\u00e4 miss\u00e4 me olemme, koska emme itsek\u00e4\u00e4n viel\u00e4 tied\u00e4. Mutta sit\u00e4 teid\u00e4n ei tarvitse kertoa.\"\n\nOlin aika varma, ett\u00e4 Mikael nauroi minulle, mutten k\u00e4\u00e4ntynyt katsomaan. Sen sijaan kuikuilin nopeasti puolelle ja toiselle varmistaakseni, ettei kukaan kiinnitt\u00e4nyt meihin huomiota. \"Hajaannutaan\", sanoin. \"N\u00e4hd\u00e4\u00e4n autolla. Min\u00e4 varmistan selustan.\"\n\n\"Sin\u00e4 saat t\u00e4m\u00e4n kuulostamaan aivan joltain salaiselta operaatiolta.\"\n\n\"T\u00e4m\u00e4 on salainen operaatio: pelastakaa laumanjohtaja Sarri. N\u00e4hd\u00e4\u00e4n pian.\"\n\nKatsoin, kun h\u00e4n livahti ovesta ennen kuin k\u00e4\u00e4nsin huomioni olohuoneeseen. En halunnut j\u00e4tt\u00e4\u00e4 taloa t\u00e4ysin vaille vartiointia. Ja niin hauskaa kuin olisikin ollut vain h\u00e4ipy\u00e4, tiesin, ett\u00e4 minun pit\u00e4isi kertoa edes jollekin.\n\nPys\u00e4hdyin ovensuuhun. Juha ja Jenni tanssivat toisella puolella huonetta. Juhan k\u00e4det olivat Jennin lanteilla.\n\nHetken vain tuijotin heit\u00e4. Sitten ynn\u00e4sin hitaasti mieless\u00e4ni kaiken Jennin kertoman sek\u00e4 havaintoni viime p\u00e4ivilt\u00e4. Olin kuvitellut Juhan olevan ihastunut minuun. Nyt tajusin olleeni t\u00e4ysin v\u00e4\u00e4r\u00e4ss\u00e4. Tavasta, jolla h\u00e4n katsoi ilmeisen humalaista Jenni\u00e4, oli nimitt\u00e4in vaikea erehty\u00e4. H\u00e4n ei totisesti ollut koskaan katsonut minua niin.\n\nJenni oli vakuuttanut haluavansa pysy\u00e4 sinkkuna, ja ymm\u00e4rsin sen hyvin. Ymm\u00e4rsin my\u00f6s, ett\u00e4 humalassa omista p\u00e4\u00e4t\u00f6ksist\u00e4\u00e4n kiinni pit\u00e4minen ei ollut ykk\u00f6sasia.\n\nEi ole minun ongelmani, ajattelin. En silti voinut vastustaa kiusausta. Kirjoitin mieless\u00e4ni muistilapun: Min\u00e4 ja Mikael l\u00e4hdemme. Pid\u00e4 talo pystyss\u00e4 ja k\u00e4det kurissa! L\u00e4im\u00e4isin kuvalla Juhaa.\n\nH\u00e4n s\u00e4ps\u00e4hti silminn\u00e4hden, ja sitten h\u00e4nen p\u00e4\u00e4ns\u00e4 nytk\u00e4hti suuntaani. Kohta h\u00e4nen katseensa l\u00f6ysi minut ovensuusta. Virnistin h\u00e4nelle ja katosin kohti autotallia.\nKAHDEKSAS LUKU\n\nKun p\u00e4\u00e4sin autotalliin, Mikael istui ratin takana.\n\n\"Kaikki selv\u00e4\u00e4? Harhautus onnistui?\" H\u00e4nt\u00e4 selv\u00e4sti huvitti tempaukseni.\n\n\"Kaikki selv\u00e4\u00e4\", vastasin. \"Paina kaasua.\"\n\nMikael teki ty\u00f6t\u00e4 k\u00e4sketty\u00e4 \u2013 kirjaimellisesti. Nauroin kun h\u00e4n antoi renkaiden kirskahtaa betonilla.\n\n\"Minne me olemme menossa?\" h\u00e4n kysyi.\n\n\"Niin, en ajatellut asiaa ihan niin pitk\u00e4lle. Miss\u00e4 me voisimme olla rauhassa?\"\n\n\"Me voimme vain ajella\", Mikael ehdotti.\n\nEn ollut koskaan vain ajellut kenenk\u00e4\u00e4n kanssa, ja ajatus tuntui jotenkin suloiselta. Me kaksi, matkalla ei minnek\u00e4\u00e4n. Ilmavirta tuli sis\u00e4\u00e4n avoimesta ikkunasta, pujahti hiusteni l\u00e4pi ja piiloutui takapenkille. Kaikki lempituoksuni olivat l\u00e4sn\u00e4: mets\u00e4, kes\u00e4y\u00f6, Mikael. Muistin, miten ensimm\u00e4isen\u00e4 koulup\u00e4iv\u00e4n\u00e4ni Hukkavaarassa olin nimennyt lempituoksuikseni pelk\u00e4st\u00e4\u00e4n Helsinkiin liittyvi\u00e4 asioita. Jokin oli selv\u00e4sti muuttunut.\n\n\"Tied\u00e4tk\u00f6, miksi ne tyt\u00f6t keitti\u00f6ss\u00e4 k\u00e4ytt\u00e4ytyiv\u00e4t niin? En saanut heist\u00e4 irti muuta kuin heid\u00e4n nimens\u00e4.\"\n\nMikael naurahti.\n\n\"Ai mit\u00e4?\"\n\n\"Sinulla ei siis ole aavistustakaan?\"\n\nPudistin p\u00e4\u00e4t\u00e4ni.\n\n\"Sin\u00e4 olet nyky\u00e4\u00e4n kuuluisa\", h\u00e4n sanoi. \"Siis Hukkavaaran mittakaavassa. Heit\u00e4 luultavasti j\u00e4nnitti niin paljon, etteiv\u00e4t he saaneet sanaa suustaan.\"\n\nEn tiennyt, mit\u00e4 ajatella. En ollut mik\u00e4\u00e4n julkkis. En ollut edes kovin cool, vaikka Nikon stailaus ja eksoottiset piirteeni edesauttoivatkin asiaa hieman.\n\nVilkaisin Mikaelia. \u00c4kki\u00e4 aloin ymm\u00e4rt\u00e4\u00e4 mainettani paljon paremmin. Se, ett\u00e4 Mikael rakasti minua, oli minunkin mielest\u00e4ni kohahduttavaa.\n\n\"Keksinkin yhden paikan, jonne haluaisin menn\u00e4\", sanoin.\n\n***\n\nMikael pys\u00e4k\u00f6i auton tuttuun paikkaan. Riisuin keng\u00e4t ja j\u00e4tin ne jalkatilaan.\n\nTaivas oli liukuv\u00e4rj\u00e4tty vaaleansinisest\u00e4 petrolin kautta tummansiniseen. L\u00e4hdimme k\u00e4velem\u00e4\u00e4n polkua pitkin, min\u00e4 edell\u00e4, Mikael per\u00e4ss\u00e4. Kuulin omat askeleeni, mutten h\u00e4nen. Koko matkan ajan vastustin kiusausta k\u00e4\u00e4nty\u00e4 ymp\u00e4ri varmistaakseni, ett\u00e4 h\u00e4n seurasi minua.\n\n\"T\u00e4\u00e4ll\u00e4 ollaan\", sanoin levitt\u00e4en k\u00e4teni.\n\nOlimme tulleet pienelle mets\u00e4aukealle. T\u00e4\u00e4ll\u00e4 olin n\u00e4hnyt Mikaelin ensimm\u00e4isen kerran, vaikka en ollutkaan tiennyt sit\u00e4 silloin. H\u00e4n oli ilmestynyt eteeni suden muodossa. En ollut tiennyt silloin viel\u00e4 mit\u00e4\u00e4n ihmissusista, vampyyreista, keijuista tai edes daimoneista. \u00c4itini oli kuollut vasta pari kuukautta aikaisemmin, ja muistin yh\u00e4 el\u00e4v\u00e4sti, miten paljon minuun oli sattunut.\n\nEn ollut tuntenut niin en\u00e4\u00e4 pitk\u00e4\u00e4n aikaan. Paljon oli ehtinyt tapahtua, ja silti me olimme taas t\u00e4\u00e4ll\u00e4. Oli lohdullista ajatella, ettei joidenkin asioiden tarvinnut koskaan muuttua.\n\nMikael katseli ymp\u00e4rilleen. \"Mit\u00e4 sin\u00e4 ajattelit silloin?\"\n\n\"Ett\u00e4 sin\u00e4 olit kauneinta, mit\u00e4 olin koskaan n\u00e4hnyt.\" En edes ajatellut sanojani ennen kuin sanoin ne, mutta ne olivat totta.\n\nMikaelin ilme oli ep\u00e4uskoinen. Istuin maahan, ja hetken p\u00e4\u00e4st\u00e4 Mikael seurasi esimerkki\u00e4ni. H\u00e4n n\u00e4ytti odottavan ett\u00e4 selitt\u00e4isin, mit\u00e4 me oikein olimme tekem\u00e4ss\u00e4, ja kun en sanonut mit\u00e4\u00e4n, h\u00e4n kysyi: \"Ent\u00e4s nyt?\"\n\n\"Nyt me katsomme t\u00e4hti\u00e4\", sanoin.\n\nLaskeuduin maahan sel\u00e4lleni ja ristin k\u00e4det vatsalleni. Mikael k\u00e4vi viereeni ja k\u00e4ytti toista k\u00e4tt\u00e4\u00e4n tyynyn\u00e4. K\u00e4\u00e4nsin p\u00e4\u00e4t\u00e4ni juuri sen verran, ett\u00e4 saatoin n\u00e4hd\u00e4 h\u00e4nen profiilinsa.\n\nTaivas ehtisi olla tumma vain hetken, joten makasimme vain hiljaa aloillamme aistit valppaina. Linnut olivat hiljentyneet ja puiden latvat muodostivat pitsisen reunan taivaalle. T\u00e4llainen ei olisi ollut mahdollista Helsingiss\u00e4, jossa paloi aina jokin valo. Sill\u00e4 hetkell\u00e4 en kaivannut kaupungista mit\u00e4\u00e4n.\n\n\"Min\u00e4 taidan n\u00e4hd\u00e4 yhden\", sanoin lopulta.\n\n\"Raisa...\" Mikael kohottautui toisen kyyn\u00e4rp\u00e4\u00e4ns\u00e4 varaan. H\u00e4nen \u00e4\u00e4nens\u00e4 oli paksu puhumattomuudesta \u2013 tai sitten jostakin sellaisesta tunteesta, josta en p\u00e4\u00e4ssyt perille. Ehk\u00e4 h\u00e4n pelk\u00e4si, mit\u00e4 h\u00e4nen sanansa voisivat saada aikaan.\n\n\"Mm-hmm?\"\n\n\"Meid\u00e4n pit\u00e4\u00e4 puhua.\"\n\n\"Hyv\u00e4 on.\"\n\n\"\u00c4l\u00e4 nyt suutu, mutta minusta sinun kannattaisi k\u00e4ytt\u00e4\u00e4 kykyj\u00e4si niin v\u00e4h\u00e4n kuin mahdollista. Enk\u00e4 tarkoita pelk\u00e4st\u00e4\u00e4n isompia juttuja, niin kuin sit\u00e4, mit\u00e4 teit Eskolle.\"\n\nNielaisin. Tiesin kyll\u00e4, mit\u00e4 h\u00e4n tarkoitti. Tatuoimista. Piirt\u00e4mist\u00e4. Maalaamista. Kaikkia asioita, joita rakastin. Samaan aikaan en kuitenkaan voinut syytt\u00e4\u00e4 h\u00e4nt\u00e4 siit\u00e4, ett\u00e4 h\u00e4n oli huolissaan. Jos nimitt\u00e4in olin t\u00e4ysin rehellinen, olin itsekin huolissani.\n\n\"T\u00e4ll\u00e4 hetkell\u00e4 sinulla ei ole mit\u00e4\u00e4n syyt\u00e4 harjoitella Mitjan kanssa. Daimonit ovat p\u00e4\u00e4tt\u00e4neet j\u00e4tt\u00e4\u00e4 meid\u00e4t rauhaan. Lauma p\u00e4rj\u00e4\u00e4 ilman sinua.\"\n\nNy\u00f6kk\u00e4sin kohti taivasta. Ilmeisesti Mikael n\u00e4ki sen, sill\u00e4 h\u00e4n puhui taas. \"Haluaisin my\u00f6s pyyt\u00e4\u00e4 anteeksi.\"\n\nSe yll\u00e4tti minut. Ja sai oudon vaivaantuneeksi. Pelk\u00e4sin nimitt\u00e4in tiet\u00e4v\u00e4ni, mihin keskustelu viel\u00e4 johtaisi. \"Ei sinun tarvitse\", mumisin.\n\n\"Kyll\u00e4 tarvitsee. Tied\u00e4n, ett\u00e4 sin\u00e4 s\u00e4ik\u00e4hdit eilen.\"\n\nN\u00e4pr\u00e4sin kaulakoruani. \"En varsinaisesti.\" Olin s\u00e4ik\u00e4ht\u00e4nyt jo paljon aikaisemmin.\n\n\"Ehk\u00e4 et, mutten tarkoita vain eilist\u00e4\", h\u00e4n sanoi vahvistaen aikaisemman pelkoni keskustelun todellisesta aiheesta. \"Tied\u00e4n, etten ole k\u00e4ytt\u00e4ytynyt erityisen... huomaavaisesti. Sinun on t\u00e4ytynyt ihmetell\u00e4, mist\u00e4 se on johtunut.\"\n\nOli minun vuoroni kuulostaa silt\u00e4, kuin olisin juuri polttanut askin tupakkaa. \"Olen p\u00e4\u00e4tt\u00e4nyt, ettei sill\u00e4 ole v\u00e4li\u00e4.\"\n\n\"Oletko?\"\n\n\"Olen.\"\n\n\"Se on hyv\u00e4\", h\u00e4n sanoi lopulta. \"Sill\u00e4 t\u00e4m\u00e4 ongelma on yksin minun enk\u00e4 halua, ett\u00e4 sin\u00e4 murehdit siit\u00e4 turhaan.\" Kun en sanonut mit\u00e4\u00e4n, h\u00e4n lis\u00e4si: \"Sinun t\u00e4ytyy luottaa minuun.\"\n\nUskalsin viimein katsoa h\u00e4nt\u00e4. H\u00e4nen silm\u00e4ns\u00e4 n\u00e4yttiv\u00e4t paljon tummemmilta kuin ne todellisuudessa olivat. Siit\u00e4 huolimatta oli helppo n\u00e4hd\u00e4, miten tosissaan h\u00e4n oli. Oli helppo n\u00e4hd\u00e4 h\u00e4net.\n\n\"Min\u00e4 luotan sinuun.\"\n\nAurinko ei nousisi viel\u00e4 hetkeen, mutta h\u00e4m\u00e4r\u00e4 oli taas alkanut hiipi\u00e4 esiin horisontista, ja paras aika t\u00e4htien katseluun oli mennyt ohi. Minulla alkoi my\u00f6s olla kylm\u00e4.\n\nEn sanonut siit\u00e4 mit\u00e4\u00e4n, mutta Mikael tietenkin huomasi sen. H\u00e4n nousi seisomaan ja tarjosi minulle k\u00e4tt\u00e4\u00e4n. Annoin h\u00e4nen auttaa minut jaloilleni. K\u00e4velimme takaisin Jaskan talon pihaan, jonne olimme j\u00e4tt\u00e4neet auton.\n\nKatsoin talolle p\u00e4in. \"Jaskan auto ei ole t\u00e4\u00e4ll\u00e4. Miss\u00e4k\u00f6h\u00e4n h\u00e4n on?\"\n\n\"T\u00f6iss\u00e4\", Mikael sanoi. \"Jonkun piti p\u00e4ivyst\u00e4\u00e4 hotellilla, ja h\u00e4n ilmoittautui vapaaehtoiseksi.\"\n\nKaikki muut olivat tietenkin talolla bileiss\u00e4, joiden ep\u00e4ilin olevan viel\u00e4 t\u00e4ydess\u00e4 vauhdissa. Minua ei huvittanut palata sinne. \"Haluatko menn\u00e4 sis\u00e4\u00e4n?\" kysyin.\n\nMikael ny\u00f6kk\u00e4si, ja nousimme askelmat ovelle. Kun p\u00e4\u00e4simme sis\u00e4lle, menin olohuoneeseen ja napsautin jalkalampun p\u00e4\u00e4lle. Sitten rojahdin sohvalle. Mikael istui viereeni ja ennen kuin huomasinkaan, h\u00e4n oli ottanut jalkani syliins\u00e4 ja alkanut hieroa sit\u00e4.\n\nSe tuntui niin hyv\u00e4lt\u00e4, ett\u00e4 voihkaisin \u00e4\u00e4neen. Monen tunnin koroilla seisominen oli tehnyt teht\u00e4v\u00e4ns\u00e4. \"Jos vain jatkat tuota, lupaan rakastaa sinua ikuisesti\", sanoin.\n\n\"No siin\u00e4 tapauksessa...\" H\u00e4n jatkoi hieromista, ja jokin l\u00e4mmin pyrki kaivautumaan rintalastani alle ja tekem\u00e4\u00e4n sinne pes\u00e4n.\n\n\"Sin\u00e4 tyydyt tosi v\u00e4h\u00e4\u00e4n\", h\u00e4n sanoi.\n\n\"En tarvitse kovin paljoa\", vastasin.\n\n\"Olen huomannut sen.\" H\u00e4n kuulosti oudolta.\n\nMinun teki mieli kysy\u00e4, mit\u00e4 h\u00e4n oli kuvitellut minun haluavan h\u00e4nelt\u00e4. Tiesin, ett\u00e4 h\u00e4n ajatteli ep\u00e4onnistuneensa, jos olin onneton tai poissa tolaltani. Se ei kuitenkaan pit\u00e4nyt paikkansa. H\u00e4nen ainoa ep\u00e4onnistumisensa olisi se, jos h\u00e4n uhraisi minun vuokseni jotain, mit\u00e4 ilman h\u00e4n ei itse voisi el\u00e4\u00e4. Koska sit\u00e4 en antaisi h\u00e4nelle anteeksi.\n\nMikael siirtyi hieromaan vasenta p\u00e4ki\u00e4\u00e4ni. Lopulta vedin jalkani h\u00e4nen sylist\u00e4\u00e4n. \"Kiitos, tuo oli mahtavaa. Haittaako, jos vaihdan vaatteet?\"\n\n\"Miksi ihmeess\u00e4 minua haittaisi?\"\n\nKun olin antanut Tanelin laittaa hiukseni ja Nikon ommella mekkoni, olin ajatellut ennen kaikkea h\u00e4nt\u00e4 ja sit\u00e4, miten saisin h\u00e4nen huomionsa. H\u00e4nen viime p\u00e4ivien k\u00e4yt\u00f6ksens\u00e4 oli tehnyt pienen loven itsetuntooni, mutten ikimaailmassa my\u00f6nt\u00e4isi sit\u00e4 h\u00e4nelle.\n\nMenin yl\u00e4kertaan, ja Mikael seurasi per\u00e4ss\u00e4. H\u00e4n istui alas s\u00e4ngylleni kun yritin rimpuilla itseni ulos mekosta.\n\n\"Voisitko auttaa?\"\n\nMikael alkoi selvitt\u00e4\u00e4 niskassa olevia nappeja. Siin\u00e4 meni ikuisuus, ja lopulta meid\u00e4n molempien oli pakko nauraa.\n\nOtin kaapista ensimm\u00e4isen paidan, joka osui silmiini ja menin k\u00e4yt\u00e4v\u00e4n toisella puolella olevaan kylpyhuoneeseen. Pyyhin enimm\u00e4t meikit pois naamalta ja yritin selvitt\u00e4\u00e4 hiuksiani harjalla. Lopulta minun oli tunnustettava, ett\u00e4 niist\u00e4 oli tullut vain entist\u00e4kin suurempi takkupehko.\n\nAnnoin mekon pudota ylt\u00e4ni lattialle ja vedin teepaidan p\u00e4\u00e4ni yli. Se oli punainen, ja hetken ihmettelin, miten se oli p\u00e4\u00e4tynyt kaappiini. Sitten muistin: Mikael oli lainannut sen minulle kauan sitten ollessamme y\u00f6t\u00e4 h\u00e4nen yst\u00e4v\u00e4ns\u00e4 m\u00f6kill\u00e4. Minun oli pakko hymyill\u00e4. Olin unohtanut koko paidan. Teksti 665 ja Neighbor Of The Beast oli niin mauton, etten uskonut hetke\u00e4k\u00e4\u00e4n h\u00e4nen ostaneen paidan itse. Sen oli pakko olla lahja joltain.\n\nMenin takaisin huoneeseeni.\n\nMikael makasi sel\u00e4ll\u00e4\u00e4n s\u00e4ngyss\u00e4ni, mutta nousi nopeasti istumaan. H\u00e4n tuijotti minua.\n\n\"Mit\u00e4?\"\n\n\"Sin\u00e4 n\u00e4yt\u00e4t...\" H\u00e4n haki sopivaa sanaa, muttei ilmeisesti l\u00f6yt\u00e4nyt sit\u00e4. Hetken p\u00e4\u00e4st\u00e4 h\u00e4n nosti k\u00e4det hiuksiinsa ja nauroi.\n\nOlin aika nolostunut. Sen t\u00e4ytyi n\u00e4ky\u00e4 kasvoiltani, sill\u00e4 Mikael vakavoitui heti. \"Mit\u00e4? Ei, en min\u00e4 sinulle naura. Ei en todellakaan. Nauran itselleni.\"\n\n\"Miksi?\"\n\n\"Koska min\u00e4 olen niin onnekas, ettei siin\u00e4 ole mit\u00e4\u00e4n j\u00e4rke\u00e4. Sin\u00e4 n\u00e4yt\u00e4t...\" H\u00e4n levitti k\u00e4tens\u00e4. \"Sin\u00e4 olet jokaisen kundin m\u00e4rk\u00e4 uni.\"\n\n\"Ai.\"\n\nKatsahdin alas. Paita peitti vain v\u00e4h\u00e4n reisi\u00e4, ja s\u00e4\u00e4reni olivat paljaat. Otin hieman ryhdikk\u00e4\u00e4mm\u00e4n asennon ja kohotin kulmani. Sanoin muka-viettelev\u00e4sti: \"Ai? Sin\u00e4kin n\u00e4yt\u00e4t ihan...\"\n\nSe sai Mikaelin virnist\u00e4m\u00e4\u00e4n. N\u00e4in heti, ett\u00e4 susi oli hereill\u00e4.\n\nH\u00e4n alkoi liikkua minua kohti, hitaasti ja n\u00e4enn\u00e4isen rennosti. Peitetty\u00e4 tai ei, tunnistin vaanimisen.\n\nSe oli tietenkin pelk\u00e4st\u00e4\u00e4n leikki\u00e4. Silti se sai minut v\u00e4risem\u00e4\u00e4n pelonsekaisesta innosta. Astuin askeleen taakse. Mikael astui askeleen eteen.\n\nEn voinut olla ajattelematta Jennin sanoja juhlissa. Kuinka kaikkien mielest\u00e4 k\u00e4ytt\u00e4ydyin piittaamattomasti Mikaelia kohtaan. Niin kuin olisin rikkonut s\u00e4\u00e4nt\u00f6j\u00e4.\n\nMin\u00e4 ja Mikael emme olleet puhuneet mist\u00e4\u00e4n s\u00e4\u00e4nn\u00f6ist\u00e4. Se ei ollut juolahtanut mieleeni. Ehk\u00e4 olisi pit\u00e4nyt.\n\nViel\u00e4 ei ollut liian my\u00f6h\u00e4ist\u00e4. Mikaelin aukiolla esitt\u00e4m\u00e4 anteeksipyynt\u00f6 rohkaisi minua avaamaan suuni. \"Minusta meid\u00e4n pit\u00e4isi viel\u00e4 puhua yhdest\u00e4 jutusta\", aloitin.\n\nH\u00e4n sipaisi hiuskiehkuran poskeltani, ja hetkeen en onnistunut sanomaan mit\u00e4\u00e4n. \"Toivottavasti sin\u00e4 et suuttunut siit\u00e4, ett\u00e4 piirsin Juhalle sen tatskan\", sopersin. \"H\u00e4n on yst\u00e4v\u00e4ni, samalla tavalla kuin Niko ja Taneli. Ei mit\u00e4\u00e4n muuta.\"\n\n\"Tied\u00e4n\", Mikael vastasi. En ollut t\u00e4ysin varma, kuinka suuri osa h\u00e4nest\u00e4 kuunteli minua.\n\n\"Haluaisin vain, ett\u00e4 meille molemmille olisi selv\u00e4\u00e4, mik\u00e4 on toisen mielest\u00e4... ookoo muiden kanssa ja mik\u00e4 ei.\"\n\nH\u00e4n r\u00e4p\u00e4ytti silmi\u00e4\u00e4n kerran pari kuin selvitt\u00e4\u00e4kseen p\u00e4\u00e4t\u00e4\u00e4n. \"Sopii minulle. Sin\u00e4 voit aloittaa.\"\n\nSit\u00e4 en ollut osannut odottaa. En ollut oikeastaan koskaan ajatellut, miss\u00e4 raja kulki omasta mielest\u00e4ni. Oli paljon helpompaa my\u00f6nty\u00e4 toisten pyynt\u00f6ihin kuin esitt\u00e4\u00e4 niit\u00e4 itse. Saatoin siis vain olla rehellinen. \"En halua, ett\u00e4 sin\u00e4 katsot ket\u00e4\u00e4n niin kuin sin\u00e4 katsot minua. En halua, ett\u00e4 ajattelet kenest\u00e4k\u00e4\u00e4n muusta samalla tavalla. En halua, ett\u00e4 kenenk\u00e4\u00e4n muun tuoksu saa sinua menett\u00e4m\u00e4\u00e4n j\u00e4rkesi. Mutta tied\u00e4n, ettei sellaista voi oikeasti pyyt\u00e4\u00e4.\"\n\n\"Miksi ei?\"\n\n\"Koska... Koska sin\u00e4 saat ajatella ja tuntea ihan mit\u00e4 tahansa. Ainoa, mik\u00e4 kuuluu minulle on se, miten sin\u00e4 k\u00e4ytt\u00e4ydyt. Joten pyyt\u00e4isin, ettet suutelisi muita tytt\u00f6j\u00e4. Tai siis ket\u00e4\u00e4n.\" Minua nolotti toden teolla.\n\nEhk\u00e4 siksi h\u00e4n valitsi juuri sen hetken painaa k\u00e4tens\u00e4 alaselk\u00e4\u00e4ni vasten ja suudella minua piinaavan hitaasti. Syd\u00e4meni tuntui irtoavan rinnastani ja kimpoilevan vimmatusti kroppani sis\u00e4ll\u00e4.\n\n\"Min\u00e4 lupaan\", h\u00e4n sanoi. \"Kaiken sen, mit\u00e4 sanoit \u00e4sken. Se kaikki kuuluu jo sinulle.\"\n\nHetken ajan kuulin vain oman pulssini.\n\n\"E-ent\u00e4 sin\u00e4?\" kysyin lopulta.\n\n\"Vain yksi asia.\" H\u00e4n veti henke\u00e4. \"Lupaa, ett\u00e4 kerrot minulle ensimm\u00e4isen\u00e4, jos rakastut johonkin toiseen.\"\n\nAivan kuin h\u00e4n olisi tiennyt, ett\u00e4 ennemmin tai my\u00f6hemmin toinen meist\u00e4 l\u00e4htisi. Mutta h\u00e4n on minun kanssani nyt, muistutin itse\u00e4ni. T\u00e4m\u00e4 hetki oli meid\u00e4n, enk\u00e4 suostunut pilaamaan sit\u00e4 peloillani.\n\nAioin sanoa, etten koskaan tulisi rakastumaan kehenk\u00e4\u00e4n toiseen. Ett\u00e4 se oli mahdotonta. Mutta h\u00e4n oli pyyt\u00e4nyt minua lupaamaan. Ja jos t\u00e4m\u00e4 oli se asia, johon h\u00e4n halusi lupaukseni, minulla ei ollut mit\u00e4\u00e4n syyt\u00e4 olla antamatta sit\u00e4 h\u00e4nelle.\n\nEik\u00e4 Mikael antanut minulle siihen edes mahdollisuutta. H\u00e4n suuteli minua kunnes en muistanut en\u00e4\u00e4 mit\u00e4\u00e4n muuta kuin h\u00e4nen huulensa.\n\nMikaelin kosketus ei ollut koskaan vaativa. Siihen ei sis\u00e4ltynyt mit\u00e4\u00e4n taka-ajatusta. Se vain oli, ja siksi luotin siihen niin t\u00e4ysin. Olin t\u00e4ysin valmis nopeuttamaan asioita, mutta h\u00e4n ei taipunut tahtooni. H\u00e4n pid\u00e4tteli itse\u00e4\u00e4n, enk\u00e4 ymm\u00e4rt\u00e4nyt miksi.\n\nLiu'utin k\u00e4tt\u00e4ni alemmas. Se tuntui rikkovan jonkin lumouksen, sill\u00e4 Mikael per\u00e4\u00e4ntyi. \"Mit\u00e4 nyt?\" kysyin.\n\n\"Ei mit\u00e4\u00e4n\", h\u00e4n vastasi. Mutta luulin vihdoin ymm\u00e4rt\u00e4v\u00e4ni.\n\n\"On er\u00e4s asia, jota sin\u00e4, Mikael Sarri, et ole viel\u00e4 oppinut.\"\n\n\"Ja mik\u00e4h\u00e4n se mahtaa olla?\"\n\n\"Sinun ei tarvitse jarrutella minun takiani. Min\u00e4 en mene rikki.\"\n\nSe riitti. Mikaelin k\u00e4det ja huulet olivat \u00e4kki\u00e4 kaikkialla. N\u00e4ykk\u00e4sin kohtaa, jossa kaula muuttuu olkap\u00e4\u00e4ksi, ja aloin kiskoa h\u00e4nen vy\u00f6t\u00e4\u00e4n. Mikaelin luomet valahtivat ja h\u00e4n taivutti p\u00e4\u00e4t\u00e4\u00e4n aavistuksen taakse. Pieni ele, mutta merkitsi minulle paljon. Rakastin sit\u00e4, ett\u00e4 saatoin tuottaa h\u00e4nelle samanlaista mielihyv\u00e4\u00e4 kuin h\u00e4n minulle.\n\nTiesin, ett\u00e4 t\u00e4ss\u00e4 kerrassa oli jotain erilaista, eik\u00e4 se johtunut pelk\u00e4st\u00e4\u00e4n siit\u00e4, ett\u00e4 olimme minun huoneessani.\n\nKun en en\u00e4\u00e4 kest\u00e4nyt, annoin v\u00e4rien huuhtoa minut t\u00e4htitaivaalle.\n\nAjattelin: min\u00e4 rakastan sinua. Ja ehk\u00e4 my\u00f6s sanoin sen \u00e4\u00e4neen.\n\n***\n\nHer\u00e4sin siihen, ett\u00e4 Mikael koski olkavarttani.\n\n\"En sano t\u00e4t\u00e4 mielell\u00e4ni, mutta sinun kannattaisi laittaa vaatteet p\u00e4\u00e4lle.\"\n\nSiristin kysyv\u00e4sti silmi\u00e4ni. Kello ei voinut olla viel\u00e4 paljon, sill\u00e4 auringons\u00e4teet eiv\u00e4t ylt\u00e4neet ikkunasta sis\u00e4\u00e4n. Saatoin kuulla lintujen kujertavan jossain aivan l\u00e4hell\u00e4.\n\n\"Kuulen auton l\u00e4hestyv\u00e4n\", h\u00e4n sanoi. \"Jaska on tulossa kotiin.\"\n\nSe riitti vakuuttamaan minut siit\u00e4, ett\u00e4 oli aika her\u00e4t\u00e4. Vedin nopeasti jotain kaapista p\u00e4\u00e4lle ja kampasin hiukset sormilla poninh\u00e4nn\u00e4lle.\n\nMikael katsoi minua niin kuin h\u00e4n olisi todistamassa jotain ainutlaatuista. Tunne siit\u00e4, ett\u00e4 jotain suurta oli tapahtunut edellisy\u00f6n\u00e4, voimistui entisest\u00e4\u00e4n.\n\nPala nousi kurkkuun. En tiennyt, mit\u00e4 tehd\u00e4 kaikella t\u00e4ll\u00e4 onnella. Sit\u00e4 oli liikaa.\n\nOnneksi oli arkisempia asioita, joihin keskitty\u00e4 \u2013 pienempi\u00e4, helpommin k\u00e4sitelt\u00e4vi\u00e4 iloja.\n\nNiin kuin ruoka.\n\n\"Min\u00e4 olen n\u00e4\u00e4ntym\u00e4isill\u00e4ni\", sanoin.\n\nMenimme alakertaan. Hymyilin juuri typer\u00e4sti Mikaelille, kun Jaska tuli sis\u00e4\u00e4n. Ilman sekaisin olevia hiuksia ja h\u00e4t\u00e4isesti p\u00e4\u00e4lle vedettyj\u00e4 vaatteitakin olisi ollut hyvin selv\u00e4\u00e4, mit\u00e4 talossa oli tapahtunut h\u00e4nen poissa ollessaan.\n\n\"Moi, toivottavasti ei haittaa, ett\u00e4 min\u00e4 ja Mikael tultiin t\u00e4nne y\u00f6ksi\", sanoin.\n\nJaska oli pitk\u00e4\u00e4n hiljaa. \"Ei haittaa\", h\u00e4n sanoi lopulta.\n\nSen j\u00e4lkeen kukaan ei sanonut v\u00e4h\u00e4\u00e4n aikaan mit\u00e4\u00e4n.\n\n\"Min\u00e4p\u00e4s laitan kahvia.\" Aloin h\u00e4\u00e4r\u00e4t\u00e4 keitti\u00f6ss\u00e4 v\u00e4ltty\u00e4kseni kiusalliselta tunnelmalta. Saatoin tuntea takanani tapahtuvan kyr\u00e4ilyn, enk\u00e4 voinut olla ihmettelem\u00e4tt\u00e4, mist\u00e4 se johtui. Mikael ja Jaska olivat tunteneet toisensa kauemmin kuin he olivat tunteneet minut. Kumpikaan heist\u00e4 ei ollut sanallakaan vihjannut, ett\u00e4 heid\u00e4n v\u00e4lill\u00e4\u00e4n olisi ollut jotain skismaa.\n\nMuistin taas, miten vaikeaselkoinen Jaska oli ollut edellisell\u00e4 vierailulla kun olin sanonut asuvani Mikaelin kanssa. Selvitin kurkkuani. Tuntui kuin jotain olisi j\u00e4\u00e4nyt sinne jumiin. Kun istuin alas, Mikaelin k\u00e4si l\u00f6ysi heti omani p\u00f6yd\u00e4n alla.\n\nJoimme kahvimme enn\u00e4tysvauhdissa. Aikaisemmin mainostamani n\u00e4lk\u00e4 sai v\u00e4isty\u00e4 tukalan tunnelman tielt\u00e4, ja sanattomasta sopimuksesta nousimme yl\u00f6s l\u00e4hte\u00e4ksemme sy\u00f6m\u00e4\u00e4n aamiaista Sarrien talolle. Olin jo alkanut k\u00e4yd\u00e4 mieless\u00e4ni l\u00e4pi kaikkia houkuttelevia vaihtoehtoja, joita hyvin varusteltu j\u00e4\u00e4kaappi tarjosi, kun Jaska pys\u00e4ytti meid\u00e4t ovella.\n\n\"Mikael, haluaisin jutella sinun kanssasi.\"\n\n\"Onko asia kiireellinen?\" h\u00e4n kysyi.\n\n\"On.\"\n\n\"No, anna tulla.\"\n\nJaska tajusi n\u00e4ytt\u00e4\u00e4 hivenen nololta. \"Sinun kanssasi, en Raisan.\"\n\nAavistelin pahaa. \"Miksi?\"\n\n\"Koska asia koskee sinua\", h\u00e4n t\u00f6ks\u00e4ytti.\n\nEn keksinyt mit\u00e4\u00e4n, mik\u00e4 olisi koskenut minua ja josta Jaskan piti saada puhua Mikaelin kanssa. Ellei sitten...\n\n\u00c4kki\u00e4 olisin voinut kuolla kiukusta ja h\u00e4pe\u00e4st\u00e4. \"Min\u00e4 olen aikuinen\", sanoin. \"Ei sinun tarvitse pit\u00e4\u00e4 Mikaelille mit\u00e4\u00e4n... luentoa. Se on muutenkin jo v\u00e4h\u00e4n my\u00f6h\u00e4ist\u00e4.\" Viimeisen lauseen sanoin hammasta purren.\n\nJaska kohotti aavistuksen leukaansa, muttei vastannut.\n\n\"Kyll\u00e4 se sopii\", Mikael sanoi. \"Raisa, voisitko menn\u00e4 autoon odottamaan siksi aikaa, kun min\u00e4 ja Jaska juttelemme?\"\n\nAvasin suuni v\u00e4itt\u00e4\u00e4kseni vastaan, mutta jokin Mikaelin ilmeess\u00e4 sai minut muuttamaan mieleni. Mikael ei nimitt\u00e4in n\u00e4ytt\u00e4nyt lainkaan yll\u00e4ttyneelt\u00e4 Jaskan pyynn\u00f6st\u00e4. H\u00e4n heitti minulle avaimensa. Nappasin ne ilmasta kiinni ja sanaakaan sanomatta l\u00e4hdin k\u00e4velem\u00e4\u00e4n autolle. Askeleeni saattoivat ehk\u00e4 olla tavallista hitaampia. Kumpikaan heist\u00e4 ei sanonut mit\u00e4\u00e4n ennen kuin olin kuuloet\u00e4isyyden ulkopuolella.\n\nAvasin autonovet napin painalluksella. Istuin pelk\u00e4\u00e4j\u00e4npaikalle ja vedin oven kiinni. Hautasin kasvot k\u00e4siini.\n\nT\u00e4llaistako oli silloin, kun oli normaalit, elossa olevat vanhemmat? Siin\u00e4 tapauksessa olin ihan tyytyv\u00e4inen siihen, ett\u00e4 olin orpo. T\u00e4m\u00e4 oli l\u00e4hes absurdia.\n\nUskaltauduin katsomaan tuulilasin l\u00e4pi kuistille. Jaskan huulet liikkuivat. Mikael oli t\u00e4ysin ilmeet\u00f6n. H\u00e4nen vastauksensa ei kest\u00e4nyt kovin pitk\u00e4\u00e4n, enk\u00e4 tiennyt yht\u00e4\u00e4n, oliko se miellytt\u00e4nyt Jaskaa vai ei. Lopulta Mikael tarjosi k\u00e4tt\u00e4\u00e4n, ja hetken ep\u00e4r\u00f6innin j\u00e4lkeen enoni tarttui siihen.\n\nMikael k\u00e4veli takaisin autolle.\n\n\"Voi luoja\", mumisin.\n\nMikael avasi oven.\n\n\"Mit\u00e4 h\u00e4n sanoi?\" kysyin ennen kuin h\u00e4n oli edes ehtinyt istua alas.\n\n\"H\u00e4n on huolissaan sinusta.\"\n\n\"En tiennyt, ett\u00e4 h\u00e4n v\u00e4litt\u00e4\u00e4 niin paljon\", sanoin.\n\nMikael ei h\u00e4m\u00e4\u00e4ntynyt katkerasta s\u00e4vyst\u00e4ni. \"Sin\u00e4 olet h\u00e4nen perheens\u00e4. H\u00e4n on menett\u00e4nyt kaikki muut paitsi sinut.\"\n\nEn ollut koskaan ajatellut asiaa siit\u00e4 kulmasta. Tiesin, ett\u00e4 Daniel oli pakottanut h\u00e4net ottamaan minut luokseen asumaan, ja l\u00e4hdetty\u00e4ni Hukkavaarasta en ollut uskonut, ett\u00e4 h\u00e4n kaipaisi seuraani. Saman aikaan kuitenkin tiesin, ettei h\u00e4nell\u00e4 ollut sukulaisia eik\u00e4 oikein yst\u00e4vi\u00e4k\u00e4\u00e4n.\n\nH\u00e4n mahtaa olla yksin\u00e4inen, ajattelin.\n\n\"Mutta miksi ihmeess\u00e4 h\u00e4n olisi huolissaan minusta?\"\n\n\"Minun takiani\", Mikael vastasi. \"H\u00e4nen mielest\u00e4\u00e4n min\u00e4 asetan sinut vaaraan k\u00e4ytt\u00e4m\u00e4ll\u00e4 sinua oman asemani p\u00f6nkitt\u00e4miseen.\"\n\n\"Miten niin?\"\n\n\"Sin\u00e4 olet kiistaton etu laumalle. Pelk\u00e4st\u00e4\u00e4n sinun kykysi riitt\u00e4isiv\u00e4t tekem\u00e4\u00e4n sinusta korvaamattoman, mutta sen lis\u00e4ksi sin\u00e4 voit viel\u00e4 saada lapsia\", h\u00e4n mumisi.\n\nEn tiennyt, mit\u00e4 sanoa.\n\n\"Jaska on oikeassa\", Mikael jatkoi. \"Ilman sinua minun asemani olisi todella paljon huterampi. Ilman sinua Esko hengitt\u00e4isi viel\u00e4kin niskaamme. Ja olen aika varma, ett\u00e4 ilman sinua minut olisi haastettu jo moneen kertaan.\" H\u00e4n vilkaisi minua. \"En koskaan tarkoittanut k\u00e4ytt\u00e4\u00e4 sinua hyv\u00e4kseni. Olen siit\u00e4 pahoillani, mutten pysty katumaan. Olen liian kiitollinen kaikesta, mit\u00e4 se on mahdollistanut.\"\n\nYritin p\u00e4\u00e4st\u00e4 selville, mit\u00e4 h\u00e4n tarkoitti. Moni muu olisi ollut valmis luovuttamaan laumanjohtajan teht\u00e4v\u00e4n jollekulle muulle. Jollekulle vanhemmalle. Mikaelilla oli liian paljon omaa el\u00e4m\u00e4\u00e4 elett\u00e4v\u00e4n\u00e4\u00e4n alkaakseen huolehtia muiden asioista.\n\n\"Min\u00e4 tied\u00e4n mit\u00e4 sin\u00e4 ajattelet\", Mikael sanoi. \"Usko pois, min\u00e4 mietin joka p\u00e4iv\u00e4, mik\u00e4 oikeus minulla on olla t\u00e4ss\u00e4 asemassa. Ehk\u00e4 joku toinen olisi parempi laumanjohtaja kuin min\u00e4.\"\n\n\"En usko, ett\u00e4 olisi.\"\n\n\"Kuka min\u00e4 olisin, jos en olisi laumanjohtaja?\" h\u00e4n kysyi hiljaa.\n\nL\u00e4\u00e4ketieteenopiskelija. Poikayst\u00e4v\u00e4, snoukkari, viiden danin mustavy\u00f6 taekwondossa. Yritysjohtaja. Tulevaisuudessa varmaankin l\u00e4\u00e4k\u00e4ri, aviomies, is\u00e4. H\u00e4n saattoi olla ihan mit\u00e4 tahansa h\u00e4n halusi, ajattelin, mutta tietenkin sellaisia asioita oli paljon vaikeampi n\u00e4hd\u00e4 itse. Kuka min\u00e4 olisin, jos en olisi taiteilija? Monella tapaa ajatus tuntui samalta kuin olisi katsonut kehyksi\u00e4, joissa ei ollut maalausta. Eik\u00e4 syy ollut se, ett\u00e4 minut olisi pakotettu valitsemaan taide. P\u00e4\u00e4t\u00f6s oli koko ajan ollut minun.\n\nEhk\u00e4 Mikaelkin oli alkanut tajuta, ett\u00e4 h\u00e4n todella halusi olla laumanjohtaja. Ja se sai h\u00e4net tuntemaan syyllisyytt\u00e4.\n\n\"Sin\u00e4 tulet tekem\u00e4\u00e4n virheit\u00e4\", sanoin. \"Aivan kuin kuka tahansa muukin. Mutta sin\u00e4 haluat olla hyv\u00e4, ja se on ainoa asia, jolla on v\u00e4li\u00e4.\"\n\nMikael ei sanonut mit\u00e4\u00e4n, mutta tunsin h\u00e4nen kiitollisuutensa. Olin valinnut juuri oikeat sanat.\n\n\"No. Mit\u00e4 sin\u00e4 vastasit Jaskalle?\" kysyin.\n\n\"Ett\u00e4 jos minun koskaan t\u00e4ytyy valita sinun ja lauman v\u00e4lill\u00e4, valitsen sinut.\"\n\nH\u00e4n olisi tuskin voinut sanoa mit\u00e4\u00e4n vaarallisempaa. Jos sana alkaisi levit\u00e4 siit\u00e4, ettei h\u00e4n ollut ehdottoman uskollinen laumalle, h\u00e4net voitaisiin syrj\u00e4ytt\u00e4\u00e4 helposti.\n\nSilti h\u00e4n oli sanonut niin. Jos h\u00e4n olisi ollut ihminen, olisin ajatellut h\u00e4nen vain yritt\u00e4neen rauhoitella enoani. Sudet eiv\u00e4t kuitenkaan voineet valehdella toisilleen, joten h\u00e4nen sanojensa t\u00e4ytyi olla t\u00e4ytt\u00e4 totta. Se tarkoitti vain yht\u00e4 asiaa: niin paljon kuin h\u00e4n halusikin johtaa laumaa, h\u00e4n halusi jotain viel\u00e4kin enemm\u00e4n.\n\n***\n\n\"Oletko varma, ett\u00e4 haluat j\u00e4\u00e4d\u00e4 t\u00e4nne yksin? Voisin k\u00e4yd\u00e4 hakemassa Mitjan ja Caoimhen. Tai vied\u00e4 sinut hotellille. Ihan mit\u00e4 vain haluat.\"\n\nMikael oli menossa mets\u00e4lle muiden susien kanssa. He olivat olleet muuttamatta muotoa niin kauan, ett\u00e4 t\u00e4n\u00e4\u00e4n yksik\u00e4\u00e4n susi ei j\u00e4tt\u00e4isi tilaisuutta k\u00e4ytt\u00e4m\u00e4tt\u00e4. Mikael oli sanonut menev\u00e4ns\u00e4 mukaan velvollisuudesta, mutten ep\u00e4illyt hetke\u00e4k\u00e4\u00e4n, etteik\u00f6 \u00f6inen juoksu mets\u00e4ss\u00e4 tekisi h\u00e4nelle hyv\u00e4\u00e4 \u2013 Helsinki oli rajoittanut h\u00e4nen sutensa el\u00e4m\u00e4\u00e4 jo ihan tarpeeksi.\n\n\"Usko tai \u00e4l\u00e4, mutta min\u00e4 olen ollut joskus aikaisemminkin yksin kotona\", sanoin. \"Min\u00e4 p\u00e4rj\u00e4\u00e4n kyll\u00e4.\" Tiesin yst\u00e4vill\u00e4ni olevan suunnitelmia t\u00e4lle illalle. Niko oli vienyt Tanelin koko p\u00e4iv\u00e4ksi Kajaaniin hyvitell\u00e4kseen k\u00e4yt\u00f6st\u00e4\u00e4n ja ennen kaikkea p\u00e4\u00e4st\u00e4kseen pois susien l\u00e4heisyydest\u00e4. Mitja taas oli vihjannut, ett\u00e4 h\u00e4n ehk\u00e4 mahdollisesti kosisi Caoimhea t\u00e4n\u00e4 iltana, joten tiesin pysytell\u00e4 poissa.\n\n\"T\u00e4m\u00e4 on ensimm\u00e4inen kerta, kun minua ei oikeasti huvita yht\u00e4\u00e4n l\u00e4hte\u00e4\", Mikael tunnusti.\n\n\"He ovat sinun laumasi. Sit\u00e4 paitsi sinullahan on aina niin hauskaa mets\u00e4ll\u00e4. Min\u00e4 laitan aamiaisen valmiiksi siksi, kun palaat. Silloin meid\u00e4n ei tarvitse poistua kotoa koko p\u00e4iv\u00e4n\u00e4.\"\n\nMikael piteli kasvojani molemmilla k\u00e4sill\u00e4\u00e4n ja suuteli minua. Sitten h\u00e4n antoi k\u00e4siens\u00e4 valua k\u00e4sivarsiani pitkin kunnes h\u00e4nen sormensa tavoittivat omani. H\u00e4n astui askeleen taakse.\n\n\"Rakastan sinua\", h\u00e4n sanoi.\n\n\"Min\u00e4kin rakastan sinua.\"\n\nAuto t\u00f6\u00f6tt\u00e4si.\n\n\"Mene\", sanoin. \"Jotta voit palata.\"\n\nH\u00e4n v\u00e4l\u00e4ytti minulle nopean hymyn ja l\u00e4hti h\u00f6lkk\u00e4\u00e4m\u00e4\u00e4n portaita alas. Kiedoin hupparin paremmin ymp\u00e4rilleni ja nojasin ovenpieleen. Katselin, kun h\u00e4n ehti auton luokse ja avasi pelk\u00e4\u00e4j\u00e4npaikan puoleisen oven. Kuulin naurua, ja pudistin p\u00e4\u00e4t\u00e4ni. Saatoin vain kuvitella, mink\u00e4laista keskustelua autossa k\u00e4ytiin. Mikael saisi mit\u00e4 luultavasti kuulla olevansa t\u00e4ysin tossun alla \u2013 paitsi ett\u00e4 sanasto olisi paljon karkeampaa. Ennen kuin h\u00e4n astui sis\u00e4\u00e4n, h\u00e4n katsoi viel\u00e4 kerran minua. Nostin k\u00e4teni. Ennen kuin olin ehtinyt edes vilkuttaa kunnolla, oli auto kaasuttanut tiehens\u00e4.\n\nPalasin sis\u00e4\u00e4n.\n\nLaitoin lis\u00e4\u00e4 kahvia. Pistin musiikkia soimaan. Kesti hetken, ennen kuin osasin k\u00e4ytt\u00e4\u00e4 Sarrien hienoja stereoita, mutta lopulta minut palkittiin samettisella \u00e4\u00e4nentoistolla. Avasin ovet patiolle ja istuin hetken p\u00f6yd\u00e4n \u00e4\u00e4ress\u00e4 vain juoden kahviani ja tuijottaen mets\u00e4\u00e4. Sitten menin sis\u00e4\u00e4n ja hain kaikki kotoa tuomani taidetarvikkeet.\n\nIlta-aurinko oli niin l\u00e4sn\u00e4, ett\u00e4 tuntui kuin olisin hengitt\u00e4nyt valoa. En pit\u00e4nyt akvarelleja omimpana juttunani, mutta t\u00e4n\u00e4\u00e4n halusin kokeilla jotain uutta, ja ne sattuivat my\u00f6s olemaan ainoat maalit, jotka minulla oli mukana. Kastoin siveltimen veteen ja aloin jahdata auringon v\u00e4ri\u00e4.\n\nKun ilta ehti h\u00e4m\u00e4r\u00e4n puolelle, ker\u00e4sin kaikki tavarani ja varmistin, ettei tuuli veisi patiolla kuivuvia t\u00f6it\u00e4 mukanaan. Menin keitti\u00f6\u00f6n ja laitoin nopean illallisen. En muistanut, koska viimeksi olisin sy\u00f6nyt yksin, ja vaikken ollut ennen ajatellut siin\u00e4 olevan mit\u00e4\u00e4n vikaa, tuntui se nyt t\u00e4ysin h\u00f6lm\u00f6lt\u00e4. P\u00e4\u00e4dyinkin sy\u00f6m\u00e4\u00e4n niin nopeasti kuin suinkin ja j\u00e4tt\u00e4m\u00e4\u00e4n yksin\u00e4iset astiani tiskialtaaseen. Nappasin kirjan kirjahyllyst\u00e4 ja menin takkahuoneeseen.\n\nIstuin sohvalla kirjan kanssa, mutten avannut sit\u00e4. Sen sijaan tuijotin \u00e4itini grafiikkavedosta. Se n\u00e4ytti samaan aikaan sek\u00e4 todella tutulta ett\u00e4 vieraalta. Tunnistin h\u00e4nen viivansa, mutten teoksen tunnelmaa. En muistanut pit\u00e4neeni sit\u00e4 ensi vilkaisulla mitenk\u00e4\u00e4n surullisena ty\u00f6n\u00e4, mutta nyt en n\u00e4hnyt siin\u00e4 juuri mit\u00e4\u00e4n muuta. T\u00e4m\u00e4 kuva ei ollut mik\u00e4\u00e4n sotahuuto, se oli t\u00e4ydellinen luovuttaminen. Irti p\u00e4\u00e4st\u00e4minen. En ollut koskaan n\u00e4hnyt yht\u00e4 melankolista keltaista.\n\nPohdin, miksi Daniel oli valinnut kaikista \u00e4itini t\u00f6ist\u00e4 juuri t\u00e4m\u00e4n. Oliko h\u00e4n n\u00e4hnyt siin\u00e4 saman kuin min\u00e4? En ymm\u00e4rt\u00e4nyt, miksi h\u00e4n olisi halunnut tulla muistutetuksi naisesta, jota oli rakastanut muttei saanut. Ja oliko Kristina tiennyt taulua katsoessaan, kenen kyyneleet h\u00e4n oli p\u00e4\u00e4st\u00e4nyt kotiinsa?\n\n***\n\nKun avasin silm\u00e4ni, valon m\u00e4\u00e4r\u00e4 oli sama. Olin nukahtanut sohvalle. Haukottelin ja menin yl\u00e4kertaan. Riisuin vaatteeni ja annoin niiden pudota lattialle. Joku oli pedannut s\u00e4ngyn \u2013 en min\u00e4, joten sen jonkun t\u00e4ytyi olla Mikael. En edes vaivautunut menem\u00e4\u00e4n peiton alle, sill\u00e4 oli niin l\u00e4mmin. Sen sijaan k\u00e4vin kippuraan keskelle s\u00e4nky\u00e4 ja halasin Mikaelin tyyny\u00e4. Toivoin, ett\u00e4 kun seuraavan kerran avaisin silm\u00e4ni, h\u00e4n olisi vieress\u00e4ni.\n\nNukuin levottomasti. N\u00e4in painajaisia, mutta se oli kaikki, mik\u00e4 ehti rekister\u00f6ity\u00e4 tajuntaani, sill\u00e4 her\u00e4tty\u00e4ni en muistanut niist\u00e4 en\u00e4\u00e4 mit\u00e4\u00e4n. Ne kuitenkin n\u00e4\u00e4nnyttiv\u00e4t minut siin\u00e4 m\u00e4\u00e4rin, ett\u00e4 kun lopulta nousin yl\u00f6s, kello oli ehtinyt l\u00e4helle puolta p\u00e4iv\u00e4\u00e4.\n\nVedin aamutakin p\u00e4\u00e4lleni ja k\u00e4velin olohuoneeseen. \"Mikael?\" kysyin, vaikka tiesin jo, ettei h\u00e4n ollut palannut.\n\nVilkaisin puhelintani. Ei puheluita tai viestej\u00e4.\n\nOutoa.\n\nMenin keitti\u00f6\u00f6n ja aloin laittaa lupaamaani aamiaista, vaikka minulla ei ollut mink\u00e4\u00e4nlaista ruokahalua.\n\nOven suunnasta kuului \u00e4\u00e4ni\u00e4. Kohotin katseeni paahtoleivist\u00e4 ja h\u00f6ristin korviani. Pienikin rasahdus tuntui metelilt\u00e4 hiljaisuudessa vietettyjen hetkien j\u00e4lkeen. Jo hetken tuntemani riemu vaihtui h\u00e4mmennykseksi, kun tajusin, ettei tulija ollutkaan Mikael.\n\nJ\u00e4tin leip\u00e4paketin p\u00f6yd\u00e4lle ja astuin eteiseen juuri kun Jenni ja Juha repesiv\u00e4t nauruun. He selv\u00e4sti pelleiliv\u00e4t kesken\u00e4\u00e4n, ja mieleeni pomppasi kuva siit\u00e4, miten Juhan k\u00e4det olivat olleet Jennin lanteilla.\n\n\"Miss\u00e4 Mikael on?\" kysyin.\n\nMolemmat lakkasivat nauramasta. \"Me luulimme, ett\u00e4 h\u00e4n on t\u00e4\u00e4ll\u00e4.\"\n\nRistin k\u00e4sivarteni. \"Ei ole. H\u00e4n ei ole viel\u00e4 palannut.\"\n\nJennin nen\u00e4 nyrpistyi. \"Kummallista. Me tulimme takaisin jo tunteja sitten, vaikka j\u00e4imme tavallista pidemm\u00e4ksi aikaa.\"\n\nOlin ymm\u00e4ll\u00e4ni. \"Eik\u00f6 Mikael ollut teid\u00e4n kanssanne?\"\n\n\"H\u00e4n katosi porukastamme jossain vaiheessa. Luulimme, ett\u00e4 h\u00e4nelle tuli sinua ik\u00e4v\u00e4.\"\n\nJenni ja Juha vaihtoivat katseita, jotka sek\u00e4 \u00e4rsyttiv\u00e4t ett\u00e4 huolettivat minua. He olivat olleet vuorenvarmoja, ett\u00e4 Mikael oli palannut luokseni. Kummankaan mieless\u00e4 ei ollut k\u00e4ynyt, ett\u00e4 kyse saattaisi olla jostain muusta.\n\nPudistin p\u00e4\u00e4t\u00e4ni. Jokin oli vialla. Olin aavistanut sen jo her\u00e4tess\u00e4ni, mutta olin j\u00e4tt\u00e4nyt tunteen huomiotta kun en ollut tiennyt, mist\u00e4 se oli voinut johtua.\n\nJotain oli tapahtunut. Jotain todella, todella pahaa. \nYHDEKS\u00c4S LUKU\n\nOdotimme. Tunti tunnilta k\u00e4vi selvemm\u00e4ksi, ettei kaikki ollut kunnossa. Tunti tunnilta kirosin enemm\u00e4n omaa typeryytt\u00e4ni. Juha ja Jenni tarjoutuivat muutaman muun kanssa l\u00e4htem\u00e4\u00e4n etsim\u00e4\u00e4n Mikaelia, mutta jossain syv\u00e4ll\u00e4 sisimm\u00e4ss\u00e4ni tiesin, ett\u00e4 kaikki odotus ja etsint\u00e4 oli vain hukattua aikaa.\n\nHeti kun he olivat l\u00e4hteneet, soitin Nikolle.\n\n\"Sinun t\u00e4ytyy tulla heti Sarreille.\"\n\n\"En voi. Taneli...\"\n\n\"Tule t\u00e4nne heti. \u00c4l\u00e4k\u00e4 ota Tanelia mukaan.\"\n\nEn antanut h\u00e4nelle mahdollisuutta kielt\u00e4yty\u00e4. Lopetin puhelun. Istuin olohuoneessa odottamassa, kunnes kuulin h\u00e4nen tulevan sis\u00e4\u00e4n.\n\n\"Kiitos sinun, minulla ja Tanelilla oli juuri vuosisadan mehevin riita\", h\u00e4n huikkasi jo ovelta.\n\nOdotin, kunnes h\u00e4n ilmestyi olohuoneeseen. H\u00e4n oli silminn\u00e4hden vihainen. Se ei kuitenkaan riitt\u00e4nyt saamaan minua katumap\u00e4\u00e4lle.\n\nEn tied\u00e4, mit\u00e4 h\u00e4n n\u00e4ki minun kasvoillani, mutta h\u00e4nen ilmeens\u00e4 pehmeni aavistuksen. \"Mik\u00e4 on vialla?\" h\u00e4n kysyi.\n\n\"Kerron sinulle, jos et kerro kenellek\u00e4\u00e4n\", vastasin. \"Varsinkaan susille.\"\n\n\"Kenelle min\u00e4 muka kertoisin?\"\n\nVedin henke\u00e4. \"Luulen, ett\u00e4 Mikaelin susi on alkanut hylki\u00e4 minua\", tunnustin.\n\nKerroin h\u00e4nelle viime viikkojen tapahtumista. Niko kohotteli hieman kulmiaan, muttei vaikuttanut erityisen yll\u00e4ttyneelt\u00e4.\n\n\"Onko h\u00e4n... satuttanut sinua tai jotain?\" h\u00e4n kysyi.\n\n\"Ei tietenk\u00e4\u00e4n\", sanoin. \"Mutta sinun t\u00e4ytyy kertoa minulle, onko mahdollista, ett\u00e4 h\u00e4nen sutensa saisi h\u00e4net tekem\u00e4\u00e4n jotain, mit\u00e4 h\u00e4n ei muuten ikimaailmassa tekisi.\"\n\nNiin kuin l\u00e4htem\u00e4\u00e4n omille teilleen sanomatta kenellek\u00e4\u00e4n mit\u00e4\u00e4n. En kuitenkaan sanonut sit\u00e4 \u00e4\u00e4neen.\n\n\"Mist\u00e4 min\u00e4 voisin tiet\u00e4\u00e4?\" Niko kysyi.\n\n\"Koska sinun sutesi hylkii muita susia. Sin\u00e4 tied\u00e4t, millaista se on.\"\n\nMe emme olleet koskaan puhuneet asiasta, emme ainakaan n\u00e4in suoraan. Minulla ei kuitenkaan ollut aikaa alkaa pehmitt\u00e4\u00e4 laskua.\n\nH\u00e4n oli pitk\u00e4\u00e4n hiljaa. En tiennyt yht\u00e4\u00e4n, mit\u00e4 h\u00e4n ajatteli.\n\n\"En usko, ett\u00e4 Mikaelin tilanne on sama kuin minun.\"\n\n\"Miksei muka? Sinun t\u00e4ytyy selitt\u00e4\u00e4 minulle v\u00e4h\u00e4n tarkemmin.\" En edes yritt\u00e4nyt verhota k\u00e4rsim\u00e4tt\u00f6myytt\u00e4ni. Tarve ymm\u00e4rt\u00e4\u00e4 ohitti kaiken muun.\n\nNiko istui sohvanreunalle. Katsomatta minuun h\u00e4n sanoi: \"Susi ei voi p\u00e4\u00e4tt\u00e4\u00e4 asioista yksin, ei edes silloin, kun ollaan suden muodossa. Ainakin minun kohdallani se, ett\u00e4 suteni hylkii muita susia, on vain jatkoa sille, mit\u00e4 muutenkin tunnen kyl\u00e4l\u00e4isi\u00e4 kohtaan. Minulla ja Hukkavaaralla on aina ollut ongelma.\" H\u00e4n naurahti kuivasti.\n\n\"Mikael sanoi, ett\u00e4 h\u00e4nen sutensa ottaa v\u00e4lill\u00e4 ohjat.\"\n\n\"En silti usko, ett\u00e4 Mikaelin susi voisi noin vain pakottaa h\u00e4net j\u00e4tt\u00e4m\u00e4\u00e4n sinut, jos sit\u00e4 tarkoitat.\"\n\nEn tiennyt, mit\u00e4 ajatella. En halunnut uskoa, ett\u00e4 Mikael j\u00e4tt\u00e4isi minut omasta tahdostaan. Samaan aikaan tiesin, ett\u00e4 lauman kannalta olisi paljon parempi, jos Mikael olisi l\u00e4htenyt siksi, ettei h\u00e4nen sutensa voinut siet\u00e4\u00e4 minua.\n\nMuut vaihtoehdot olivat huomattavasti pelottavampia.\n\n\"Raisa...\"\n\nTajusin liian my\u00f6h\u00e4\u00e4n, mit\u00e4 h\u00e4nen ilmeens\u00e4 tarkoitti. H\u00e4n aikoi sanoa jotain lohduttavaa. En kest\u00e4nyt sellaista juuri nyt.\n\n\"\u00c4l\u00e4. En kaipaa mit\u00e4\u00e4n s\u00e4\u00e4li\u00e4.\"\n\n\"En min\u00e4 sinua s\u00e4\u00e4li.\"\n\nSe tuntui lopulta l\u00e4p\u00e4isev\u00e4n kuoreni. Minun oli pakko painaa kasvot k\u00e4siini. \"En tied\u00e4, mit\u00e4 min\u00e4 teen\", tunnustin enemm\u00e4n itselleni kuin Nikolle.\n\n\"No, on yksi vaihtoehto, jota et ole ehk\u00e4 harkinnut\", h\u00e4n sanoi.\n\nSe yll\u00e4tti minut. Kohotin katseeni. \"Mik\u00e4?\"\n\n\"Voisit antaa Mikaelin ottaa itselleen toisen tytt\u00f6yst\u00e4v\u00e4n. Suden.\"\n\nOlin kauhuissani. \"\u00c4l\u00e4 vitsaile.\"\n\n\"En vitsaile\", Niko sanoi. \"Polyamoria on nykyisin ihan tavallista.\"\n\nHiljaisuuden aikana pakotin itseni harkitsemaan Nikon sanoja, vaikka tiesin jo, mihin tulokseen tulisin. \"En pystyisi siihen\", sanoin lopulta. \"Suostuisitko muka itse sellaiseen?\"\n\n\"Ehk\u00e4. Jos saisin itsekin n\u00e4hd\u00e4 muita.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Valehtelija. En tajua, mit\u00e4 sin\u00e4 yrit\u00e4t esitt\u00e4\u00e4.\" Tiesin, ettei Niko ollut hyv\u00e4 n\u00e4ytt\u00e4m\u00e4\u00e4n tunteitaan \u2013 varsinkaan, jos ne olivat positiivisia. Silti en voinut ymm\u00e4rt\u00e4\u00e4, miksi h\u00e4nen t\u00e4ytyi v\u00e4h\u00e4tell\u00e4 suhdettaan Tanelin kanssa.\n\n\"Taneli on ihminen\", Niko sanoi. \"En voi ajatella meit\u00e4 min\u00e4\u00e4n pysyv\u00e4n\u00e4. En ole niin tyhm\u00e4.\"\n\n\"Mutta haluaisit olla.\"\n\n\"Meill\u00e4 on siis sama ongelma.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. Meid\u00e4n ongelmamme olivat hyvin kaukana toisistaan. \"Sin\u00e4 olet susista ainoa, joka on vapaa valitsemaan kumppanikseen kenet haluaa. Sin\u00e4 olet onnekas.\"\n\n\"Lopeta jo!\" h\u00e4n oli oikeasti vihainen. \"On ihan riitt\u00e4v\u00e4n vaikeaa tiet\u00e4\u00e4, ett\u00e4 minun ja Tanelin juttu tulee kusahtamaan. Parhain mahdollinen skenaario on se, jossa kumpikaan ei kuole, vaan l\u00e4hdemme siististi vastakkaisiin suuntiin.\"\n\n\"Anteeksi. En halua tapella\", sanoin.\n\n\"En min\u00e4k\u00e4\u00e4n.\"\n\nHuokaisin. \"Voisitko j\u00e4\u00e4d\u00e4 t\u00e4nne siksi aikaa, ett\u00e4 Jenni ja Juha tulevat takaisin? En haluaisi olla yksin.\"\n\nNikon oli viimein t\u00e4ytynyt p\u00e4\u00e4st\u00e4 perille mielentilastani. \"Raisa, mit\u00e4 oikein on tapahtunut?\"\n\nMin\u00e4 rakastan sinua. Viimeiset sanat, jotka Mikael oli sanonut minulle ennen l\u00e4ht\u00f6\u00e4. Eiv\u00e4tk\u00e4 ne olleet kuulostaneet lainkaan hyv\u00e4steilt\u00e4, vaan lupaukselta.\n\nOli tullut aika sanoa se \u00e4\u00e4neen. \"Mikael on siepattu.\"\n\n***\n\nKun Jenni ja Juha palasivat takaisin yht\u00e4 neuvottomina kuin olivat l\u00e4hteneet, olin jo ehtinyt muodostaa tilanteesta oman teoriani.\n\nOlin varsin ylpe\u00e4 siit\u00e4, ett\u00e4 olin pysynyt rauhallisena ja harkinnut kaikkia mahdollisia vaihtoehtoja. Olin jopa antanut Nikon kehitell\u00e4 teorioita kanssani, vaikka h\u00e4nen tietonsa olivat hyvin vajavaisia. Lopulta olin l\u00e4hett\u00e4nyt h\u00e4net noutamaan velje\u00e4ni.\n\nIstuimme kaikki olohuoneessa. En voinut olla huomaamatta, ett\u00e4 jokainen oli valinnut tismalleen saman paikan kuin kaksi p\u00e4iv\u00e4\u00e4 sitten. Jopa min\u00e4 istuin samassa sohvannurkassa. Se muistutti minua turhankin kipe\u00e4sti siit\u00e4, mik\u00e4 oli erilaista: Mikael ei seissyt vieress\u00e4mme, enk\u00e4 tuntenut h\u00e4nen katsettaan ihollani.\n\n\"Ihmiset\", Juha sanoi. \"Susivihaajat. Heid\u00e4n on pakko olla t\u00e4m\u00e4n takana.\"\n\nTuijotin tyhjyyteen. Niko oli ehdottanut tismalleen samaa. \"Ei.\"\n\n\"Kuka muukaan?\"\n\nKatsoin velje\u00e4ni, joka katsoi minua. H\u00e4nen ilmeens\u00e4 vahvisti pelkoni. \"Miksi nyt? Miksi t\u00e4\u00e4ll\u00e4?\" h\u00e4n kysyi hiljaa kreikaksi.\n\nPudistin j\u00e4lleen p\u00e4\u00e4t\u00e4ni. Minulla ei ollut vastauksia. Silti tiesin, tunsin, ettei Mikaelin katoamisella ollut mit\u00e4\u00e4n tekemist\u00e4 ihmisten kanssa.\n\n\"H\u00e4n ei ole ollut poissa viel\u00e4 kovin kauan. Ehk\u00e4 h\u00e4n aikoo j\u00e4rjest\u00e4\u00e4 sinulle yll\u00e4tyksen tai jotain\", Jenni sanoi.\n\n\"Ei.\"\n\n\"Tai h\u00e4n on joutunut paikkaan, jossa liikkuu paljon ihmisi\u00e4 ja odottaa pime\u00e4\u00e4, ennen kuin uskaltaa l\u00e4hte\u00e4 liikkeelle. Ei se ole mitenk\u00e4\u00e4n ep\u00e4tavallista.\"\n\n\"Oletteko te kuuroja? Min\u00e4 sanoin ei!\" huusin. Ei Raisa, vaan daimon oli noussut yl\u00f6s ja veitsi oli l\u00e4htenyt h\u00e4nen k\u00e4dest\u00e4\u00e4n. Se t\u00f6m\u00e4hti takanreunukseen ja j\u00e4i t\u00f6rr\u00f6tt\u00e4m\u00e4\u00e4n siihen kiinni.\n\nHaukoin henke\u00e4. En ollut tarkoittanut niin tapahtuvan, mutta soturipukuni oli tullut esiin. Jouduin keskittym\u00e4\u00e4n kauan, ennen kuin se h\u00e4visi ja olin taas oma itseni.\n\nKaikki muut olivat kammenneet itsens\u00e4 pel\u00e4styneen\u00e4 jaloilleen.\n\n\"Minun olisi pit\u00e4nyt tiet\u00e4\u00e4, ettei meit\u00e4 p\u00e4\u00e4stet\u00e4 n\u00e4in helpolla\", mutisin.\n\n\"Mit\u00e4 ihmett\u00e4 sin\u00e4 h\u00f6piset?\" Niko kysyi.\n\nHengitin syv\u00e4\u00e4n. \"Minun pit\u00e4\u00e4 kertoa teille jotain, mutta teid\u00e4n t\u00e4ytyy luvata, ettette kerro kenellek\u00e4\u00e4n muulle laumassa. T\u00e4m\u00e4 ei saa vaikuttaa Mikaelin asemaan.\"\n\nSyntyi kiusaantunut hiljaisuus. \"Raisa, laumassa asiat eiv\u00e4t oikein toimi niin. Jos joku kysyy...\"\n\n\"Minua ei kiinnosta hiton vertaa, miten asiat laumassa toimivat. Luvatkaa.\"\n\nJenni ny\u00f6kk\u00e4si.\n\n\"Mikael tappoi daimonin ja nyt he ovat tulleet kostamaan.\" Sanat sy\u00f6ksyiv\u00e4t ulos suustani sellaista vauhtia, etten ollut lainkaan varma, oliko kukaan ymm\u00e4rt\u00e4nyt niit\u00e4. Sitten tajusin, ett\u00e4 heid\u00e4n hiljaisuutensa johtui tyrmistyksest\u00e4. \"H\u00e4n teki sen minun takiani. H\u00e4n pelasti henkeni. Se ei ollut... Jos se oli jonkun vika niin minun.\"\n\n\"Paskat oli. Se oli Alekin vika, eik\u00e4 kenenk\u00e4\u00e4n muun.\" Mitjan ei ollut tarvinnut pohtia lainkaan sanojaan, mik\u00e4 oli h\u00e4nelt\u00e4 ep\u00e4tavallista. Kiitollisuuteni velje\u00e4ni kohtaan oli niin suurta, ett\u00e4 jouduin nieleskelem\u00e4\u00e4n kyyneli\u00e4.\n\n\"Mikael tappoi jonkun sinun takiasi? Ei mik\u00e4\u00e4n ihme, ett\u00e4 te palasitte yhteen. H\u00e4n totisesti osaa hurmata.\"\n\nEn ollut varma, vitsailiko Niko vai ei, mutta pelk\u00e4sin j\u00e4lkimm\u00e4ist\u00e4. Niko oli varmasti ainoa, joka l\u00f6ysi tappamisesta mit\u00e4\u00e4n romanttista. Tai ehk\u00e4 ei sent\u00e4\u00e4n: Mikael oli tehnyt sen takiani, kamalan, pahimman mahdollisen teon. Vain siksi, ett\u00e4 min\u00e4 voisin jatkaa olemassaoloani. Se kertoi kaiken h\u00e4nen tunteistaan minua kohtaan.\n\n\"Jos daimonit ovat t\u00e4\u00e4ll\u00e4, he ottavat minuun ja Raisaan yhteytt\u00e4\", Mitja sanoi. \"Tavalla tai toisella.\"\n\nMitja oli oikeassa. Joten me odotimme. Tiesin, ett\u00e4 minun olisi pit\u00e4nyt ker\u00e4t\u00e4 voimia, sy\u00f6d\u00e4 ja nukkua, mutten pystynyt. Tuntui silt\u00e4 kuin sis\u00e4ll\u00e4ni olisi ollut pommi, joka voisi r\u00e4j\u00e4ht\u00e4\u00e4 hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4. En halunnut odottaa. En halunnut ajatella. Halusin l\u00e4hte\u00e4 saman tien daimonien per\u00e4\u00e4n ja antaa heid\u00e4n tuntea, millaista oli tulla jahdatuksi. Millaista oli paeta henkens\u00e4 edest\u00e4 tiet\u00e4en, ett\u00e4 meni minne tahansa, j\u00e4isi lopulta kiinni.\n\nSeuraavana p\u00e4iv\u00e4n\u00e4 saimme kaipaamamme yhteydenoton.\n\nKello oli ehk\u00e4 viisi aamulla, kun hypp\u00e4sin yl\u00f6s s\u00e4ngyst\u00e4 ja juoksin portaat alas. Vedin hetken henke\u00e4 eteisess\u00e4 ennen kuin avasin oven ja astuin ulos paljain varpain.\n\nV\u00e4symyksen harsonkin l\u00e4pi saatoin panna merkille, ett\u00e4 aamu oli mit\u00e4 kaunein: kuulas ja t\u00e4ynn\u00e4 linnunlaulua. Pit\u00e4en yh\u00e4 ovesta kiinni toisella k\u00e4dell\u00e4ni k\u00e4\u00e4nnyin hitaasti ymp\u00e4ri.\n\nUlko-oveen oli ilmestynyt y\u00f6n aikana viesti. Paperi oli kiinni nuolella, enk\u00e4 ep\u00e4illyt hetke\u00e4k\u00e4\u00e4n, etteik\u00f6 nuoli olisi kuulunut Alekille ennen h\u00e4nen kuolemaansa.\n\nJohtajan talo. Tulkaa. Ei susia.\n\nTeksti oli kirjoitettu muinaiskreikaksi.\n\nReaktioni oli pikemminkin odottamani r\u00e4j\u00e4hdyksen vastakohta. Olin t\u00e4ysin tyyni, kun irrotin viestin ovesta ja vein sen sis\u00e4lle veljeni n\u00e4ht\u00e4v\u00e4ksi.\n\n\"Tied\u00e4tk\u00f6, mit\u00e4 sill\u00e4 tarkoitetaan?\" Mitja kysyi. H\u00e4n oli j\u00e4\u00e4nyt talolle y\u00f6ksi silt\u00e4 varalta, ett\u00e4 daimonit ottaisivat yhteytt\u00e4. Nyt kun niin oli todella tapahtunut, h\u00e4nkin vaikutti ennen kaikkea h\u00e4mmentyneelt\u00e4.\n\nViestiss\u00e4 ei ollut j\u00e4rke\u00e4. Meh\u00e4n olimme parhaillaan johtajan eli Mikaelin talolla. K\u00e4sitykseni mukaan Sarrit olivat asuneet samassa paikassa aina.\n\nSoitin Juhalle. \"Onko Sarreilla jossain joku toinen talo?\"\n\nH\u00e4n mietti hetken. \"Heid\u00e4n firmallaan on kiinteist\u00f6j\u00e4, mutta ei heill\u00e4 tiet\u00e4\u00e4kseni ole mit\u00e4\u00e4n varsinaista kakkoskotia. Miten niin?\"\n\nKatsoin k\u00e4dess\u00e4ni olevaa nuolta. \"Me saimme postia.\"\n\nToisessa p\u00e4\u00e4ss\u00e4 oli hetken hiljaista. \"\u00c4l\u00e4 tee mit\u00e4\u00e4n. Me tulemme sinne.\"\n\n***\n\nOdotin susien saapumista, vaikken uskonut, ett\u00e4 heist\u00e4 olisi apua.\n\nJuha tuntui heti aavistavan, mit\u00e4 ajattelin. \"Me voimme j\u00e4ljitt\u00e4\u00e4 heid\u00e4t\", h\u00e4n sanoi.\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Ette voi. He pystyv\u00e4t katoamaan.\"\n\n\"Siis mit\u00e4?\" Niko sanoi.\n\n\"Te ette tied\u00e4, mit\u00e4 meill\u00e4 on vastassamme.\"\n\nSudet yrittiv\u00e4t kuitenkin. Lopulta hekin joutuivat tunnustamaan, ettei rakennuksen ymp\u00e4rill\u00e4 ollut mink\u00e4\u00e4nlaisia hajuj\u00e4lki\u00e4.\n\nKukaan ei ollut viel\u00e4k\u00e4\u00e4n keksinyt, mihin taloon viestiss\u00e4 viitattiin. Olin ajatellut toiveikkaasti, ett\u00e4 edes Jenni olisi tiennyt, mit\u00e4 se tarkoitti. Mutta h\u00e4n oli aivan yht\u00e4 ymm\u00e4ll\u00e4\u00e4n kuin kaikki muutkin. Sarreilla oli vain yksi koti.\n\nMin\u00e4, Juha ja Niko kiersimme l\u00e4pi kaikki Sarrien yritysten omistamat rakennukset, ja lopulta my\u00f6s kaikki, jotka he omistivat edes osittain. Se tarkoitti l\u00e4hes jokaista kyl\u00e4ss\u00e4 olevaa liiketilaa.\n\n\"Minulla ei ollut aavistustakaan, ett\u00e4 he omistavat n\u00e4in paljon\", tunnustin Nikolle.\n\n\"Eiv\u00e4t he, vaan te\", Niko vastasi. \"Kaikki t\u00e4m\u00e4 on k\u00e4yt\u00e4nn\u00f6ss\u00e4 my\u00f6s sinun.\"\n\nAjatus olisi kauhistuttanut minua, ellei mieless\u00e4ni olisi py\u00f6rinyt paljon synkempi\u00e4 ajatuksia. Halusin ajatella, ett\u00e4 tunsin Mikaelin paremmin kuin kukaan. Tunsin my\u00f6s daimonit, joten minun olisi pit\u00e4nyt tiet\u00e4\u00e4, mit\u00e4 heid\u00e4n viestins\u00e4 tarkoitti. Olin siis liev\u00e4sti ilmaistuna turhautunut palattuamme kierrokseltamme.\n\n\"Jos he haluavat meid\u00e4n tulevan luokseen, miksi he eiv\u00e4t vain voi sanoa suoraan, minne?!\" sanoin ehk\u00e4 sadatta kertaa. Olin jo kauan sitten menett\u00e4nyt kykyni pysytell\u00e4 aloillani.\n\n\"Keskityt\u00e4\u00e4n nyt vain arvoituksen ratkaisemiseen, ei tiuskimiseen\", Mitja sanoi. H\u00e4n oli ainoa, joka en\u00e4\u00e4 uskalsi puhua minulle.\n\nEn n\u00e4hnyt veist\u00e4ni takanreunassa. Joku oli irrottanut ja varmaan piilottanut sen. Luultavasti ihan hyv\u00e4st\u00e4 syyst\u00e4.\n\nSilloin muistin. \"Olen idiootti\", sanoin \u00e4\u00e4neen. \"Lappi. Sarrit menev\u00e4t joka joulu Lappiin.\" Hain varmistusta teorialleni Juhan ja Jennin kasvoilta. He vilkaisivat toisiaan, mutteiv\u00e4t sanoneet heti mit\u00e4\u00e4n. K\u00e4rsiv\u00e4llisyyteni ei riitt\u00e4nyt odottamiseen. \"Jenni, sinun t\u00e4ytyy tiet\u00e4\u00e4. Heill\u00e4 on Lapissa talo, eik\u00f6?\"\n\n\"En tied\u00e4\", h\u00e4n tunnusti.\n\n\"Etk\u00f6 sin\u00e4 ole ollut siell\u00e4?\"\n\nH\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Minua ei koskaan kutsuttu mukaan.\"\n\n\"Mutta Mikael varmaan kertoi sinulle jotain? L\u00e4hetti kortin tai toi tuliaisia?\"\n\nH\u00e4nen suunsa mutristui aavistuksen, ja tiesin h\u00e4nen muistelevan. \"Ei postikortteja. Eik\u00e4 oikeastaan tuliaisiakaan. H\u00e4n soitti minulle aina joskus, mik\u00e4 tarkoittaa, ett\u00e4 siell\u00e4 on kentt\u00e4\u00e4, mutta se ei varsinaisesti rajaa aluetta. Tied\u00e4n, ett\u00e4 he k\u00e4viv\u00e4t aina laskettelemassa. Mutta muuten... En oikeasti tied\u00e4, omistivatko he talon vai vuokrasivatko. Luulen, ett\u00e4 he viettiv\u00e4t joulun aina samassa paikassa, mutta siit\u00e4k\u00e4\u00e4n en voi olla t\u00e4ysin varma.\"\n\nKaiken tuon tiesin entuudestaan. \"Lappi on valtava\", sanoin.\n\n\"Olen pahoillani, etten tied\u00e4 enemp\u00e4\u00e4.\"\n\n\"Meid\u00e4n t\u00e4ytyy k\u00e4yd\u00e4 l\u00e4pi Danielin arkistot.\" Tiesin jo, ett\u00e4 se tulisi olemaan turhaa. Olin lukenut tarpeeksi tiet\u00e4\u00e4kseni, ettei Daniel ollut kirjoittanut mit\u00e4\u00e4n henkil\u00f6kohtaisista asioistaan. Ja koska Lapin m\u00f6kki ei ollut kenenk\u00e4\u00e4n tiedossa, h\u00e4n oli halunnut pit\u00e4\u00e4 sen salassa.\n\nHemmetin Daniel ja h\u00e4nen yksityisyytens\u00e4. Kenell\u00e4k\u00e4\u00e4n ei tietenk\u00e4\u00e4n ollut aavistustakaan, miss\u00e4 h\u00e4n oli ja miten h\u00e4neen voisi saada yhteyden. H\u00e4n saattoi olla miss\u00e4 p\u00e4in maailmaa tahansa.\n\nH\u00e4nen tai Mikaelin oli kuitenkin ollut pakko kertoa jollekin. Jos ei Jennille, niin kenelle?\n\n\"Niko, sinun pit\u00e4\u00e4 l\u00e4hte\u00e4 kuskikseni.\"\n\n\"Minne me menemme?\"\n\n\"Tapaamaan er\u00e4st\u00e4, joka toivottavasti tuntee Sarrit paremmin kuin min\u00e4.\"\n\n***\n\n\"Raisa, kukaan muu kuin sin\u00e4 ei ole edes kuullut mist\u00e4\u00e4n Mikosta\", Niko sanoi kerrottuani h\u00e4nelle suunnitelmani.\n\nMinun oli tunnustettava, etteiv\u00e4t h\u00e4nen ep\u00e4ilyns\u00e4 olleet aivan perusteettomia. Olin itsekin n\u00e4hnyt Mikon vain kerran. Se oli tapahtunut aikana, jolloin Mikael ja min\u00e4 olimme olleet vasta aivan alussa, ja t\u00e4ll\u00e4 hetkell\u00e4 se tuntui kokonaan toiselta aikakaudelta. Mikael oli kertonut h\u00e4nest\u00e4 harmillisen v\u00e4h\u00e4n, ja mit\u00e4 enemm\u00e4n pinnistelin muistaakseni yksityiskohtia, sit\u00e4 enemm\u00e4n aloin ep\u00e4ill\u00e4, ett\u00e4 olin kuvitellut koko p\u00e4iv\u00e4n.\n\n\"Me molemmat olemme olleet Mikon m\u00f6kill\u00e4\", sanoin. Se auttoi v\u00e4h\u00e4n.\n\n\"Niin olemme, mutta ei siell\u00e4 silloin asunut ket\u00e4\u00e4n. T\u00e4m\u00e4 kaveri ei voi kuulua laumaan, enk\u00e4 ymm\u00e4rr\u00e4, miksi h\u00e4nen annettaisiin asua sen mailla.\"\n\n\"V\u00e4it\u00e4tk\u00f6 sin\u00e4, ettei h\u00e4nt\u00e4 ole olemassa?\"\n\n\"L\u00e4hinn\u00e4 nyt vain huvittelin ajatuksella, ett\u00e4 kenties se sinun ah-niin-t\u00e4ydellinen poikayst\u00e4v\u00e4si ei ole kertonut sinulle ihan kaikkea.\"\n\nNielaisin. Tiesin hyvin, ettei Mikael ollut kertonut minulle kaikkea. Oli paljon asioita, joista min\u00e4 ja Mikael emme olleet viel\u00e4 ehtineet puhua. Ja pahin pelkoni oli, ettemme koskaan saisi mahdollisuutta tehd\u00e4 asialle mit\u00e4\u00e4n.\n\nNikokin taisi tajuta sen. \"Sori. Ei ollut tarkoitus.\"\n\n\"Ei se mit\u00e4\u00e4n\", sanoin.\n\nLoppumatkan tuijotin ikkunasta ulos. Taivas ei ollut t\u00e4n\u00e4\u00e4n osannut p\u00e4\u00e4tt\u00e4\u00e4, sataako vai ei, ja seurasin katseellani yksitt\u00e4isi\u00e4 vesipisaroita, jotka juoksivat viistosti auton ikkunaa pitkin. Lopulta tulimme m\u00f6kin pihaan. Se n\u00e4ytti r\u00e4nsistyneemm\u00e4lt\u00e4 kuin muistin. Ja pienemm\u00e4lt\u00e4 \u2013 oli vaikea kuvitella, ett\u00e4 min\u00e4, Mikael, Juha, Jenni ja Niko olimme viett\u00e4neet siell\u00e4 monta p\u00e4iv\u00e4\u00e4 ja y\u00f6t\u00e4. N\u00e4ytti silt\u00e4, ettei siell\u00e4 oltu asuttu sitten viime kes\u00e4n.\n\n\"En usko, ett\u00e4 t\u00e4\u00e4ll\u00e4 on ket\u00e4\u00e4n. En haista mit\u00e4\u00e4n.\"\n\nSe ei tietenk\u00e4\u00e4n enteillyt hyv\u00e4\u00e4, mutten ollut valmis luovuttamaan.\n\n\"Mikko\", kokeilin. Sitten lujempaa: \"Mikko!\"\n\nEi vastausta. Pelkk\u00e4\u00e4 tuulen huminaa puiden latvoissa. Aurinko v\u00e4lk\u00e4hteli vihreyden keskell\u00e4 ja l\u00e4hetti kirkkaita pisteit\u00e4 meit\u00e4 kohti osuessaan puiden lehdill\u00e4 oleviin vesipisaroihin. Ilma oli raikas. Silkkaa puhdasta happea, jota meid\u00e4n keuhkomme oli luotu rakastamaan.\n\n\"Mikko!\" huusin. \"Raisa t\u00e4\u00e4ll\u00e4. Sin\u00e4 ehk\u00e4 muistat, kun Mikael toi minut t\u00e4nne. Sin\u00e4 teit minulle tatuoinnin. Suden, jonka piirsin itse?\"\n\nKuulostelin. Ei viel\u00e4k\u00e4\u00e4n mit\u00e4\u00e4n. Aloin k\u00e4vell\u00e4 rakennuksen toiselle puolelle. \"Sinun j\u00e4lkesi on parempaa kuin aluksi tajusinkaan. Min\u00e4 olen nyky\u00e4\u00e4n tatuointitaiteilija, joten tied\u00e4n. Kiitos siit\u00e4.\"\n\nPys\u00e4hdyin. \"Min\u00e4 tulin pyyt\u00e4m\u00e4\u00e4n sinun apuasi. Asia koskee Mikaelia.\" Pelko sai minut kuulostamaan heng\u00e4styneelt\u00e4.\n\nH\u00e4n ei ole t\u00e4\u00e4ll\u00e4, ajattelin. H\u00e4n ei koskaan palannut viimekes\u00e4isen j\u00e4lkeen.\n\nHuokaisin. \"L\u00e4hdet\u00e4\u00e4n\", sanoin Nikolle.\n\nK\u00e4velimme takaisin autolle. Niko oli juuri avaamassa ovia, kun h\u00e4n j\u00e4hmettyi.\n\n\"Raisa.\"\n\nSeurasin Nikon katsetta. Ensin en erottanut mit\u00e4\u00e4n, sill\u00e4 h\u00e4n maastoutui hyvin puiden sekaan. Mikko seisoi kuusien v\u00e4liss\u00e4 ja tuijotti meit\u00e4 liikahtamatta. H\u00e4nell\u00e4 oli yll\u00e4\u00e4n flanellipaita, harmaat housut ja kumisaappaat. Yritin keksi\u00e4, mik\u00e4 h\u00e4ness\u00e4 oli muuttunut, ja sitten tajusin h\u00e4nen kasvattaneen parran.\n\n\"Moi\", kokeilin.\n\nEi vastausta.\n\n\"Olen pahoillani, ett\u00e4 h\u00e4iritsemme, mutta tarvitsen apuasi.\"\n\n\"Sin\u00e4 olet se Mikaelin tytt\u00f6\", Mikko tokaisi.\n\n\"Niin olen\", vastasin. En tiennyt, kummasta olin helpottuneempi, siit\u00e4 ett\u00e4 h\u00e4n muisti minut, vai siit\u00e4, ett\u00e4 h\u00e4n vihdoin sanoi jotakin.\n\nSitten h\u00e4n katsoi Nikoa. \"Ja sin\u00e4 olet se poika.\"\n\nNiko katsoi Mikkoa kysyv\u00e4sti. H\u00e4n ei selv\u00e4stik\u00e4\u00e4n kuvitellut Mikon voivan tarkoittaa h\u00e4nt\u00e4.\n\n\"Se, joka tykk\u00e4\u00e4 pojista.\"\n\nNiko katsoi minua kuin olettaen minun voivan selitt\u00e4\u00e4, mit\u00e4 ihmett\u00e4 oli meneill\u00e4\u00e4n.\n\n\"\u00c4l\u00e4 nyt n\u00e4yt\u00e4 noin nolostuneelta\", Mikko murahti. H\u00e4n astui kaatuneen puunrungon yli ja tuli seisomaan eteemme. \"Se oli ihan tavallista minunkin aikanani.\"\n\nNiin paljon kuin minua kiinnostikin tiet\u00e4\u00e4, mihin aikaan h\u00e4n tarkalleen ottaen viittasi, ei minulla ollut aikaa tyydytt\u00e4\u00e4 uteliaisuuttani. \"Tulin kysym\u00e4\u00e4n sinulta jotain. Lapissa on paikka, johon Sarrit menev\u00e4t joka joulu. Tied\u00e4tk\u00f6, miss\u00e4 se on?\"\n\n\"Tied\u00e4n.\"\n\nHelpotus oli niin valtava, ett\u00e4 horjahdin.\n\n\"Ei teid\u00e4n kannata menn\u00e4 sinne\", h\u00e4n jatkoi. \"Se talo on pelkk\u00e4 r\u00f6ttel\u00f6.\"\n\nYritin olla kuulostamatta ep\u00e4toivoiselta, mik\u00e4 oli hyvin vaikeaa, sill\u00e4 en ollut koskaan el\u00e4m\u00e4ni aikana ollut yht\u00e4 ep\u00e4toivoinen.\n\n\"Tilanne on nyt sellainen, ett\u00e4 meid\u00e4n on pakko menn\u00e4 sinne. Meid\u00e4n t\u00e4ytyy l\u00f6yt\u00e4\u00e4 Mikael, ja se m\u00f6kki on ainoa johtolanka.\"\n\n\"Mikael on siis kadonnut.\"\n\n\"Niin. Luulemme, ett\u00e4 h\u00e4n on siell\u00e4.\" En sanonut, ett\u00e4 ep\u00e4ilin h\u00e4nen olevan siell\u00e4 vasten tahtoaan. Nyt ei ollut sopiva hetki kertoa Mikolle daimoneista.\n\n\"M\u00f6kille ei mene ollenkaan tiet\u00e4\", Mikko sanoi.\n\nEi tiet\u00e4? Ensin luulin, ett\u00e4 olin kuullut v\u00e4\u00e4rin. Kun olin keksinyt, miss\u00e4 Mikaelia pidettiin, olin heti kuvitellut mieless\u00e4ni suuren hirsihuvilan tunturin kupeessa, l\u00e4hell\u00e4 laskettelurinteit\u00e4. Se olisi sopinut Sarrien tyyliin.\n\n\"Miten sinne sitten p\u00e4\u00e4see?\"\n\n\"Ei sinne ole l\u00e4himm\u00e4lt\u00e4 tielt\u00e4 kuin v\u00e4h\u00e4n p\u00e4\u00e4lle peninkulma. Kyll\u00e4 teid\u00e4n nuorilla ja terveill\u00e4 jaloillanne p\u00e4\u00e4see perille puolessa p\u00e4iv\u00e4ss\u00e4.\" Saatoin olla v\u00e4\u00e4r\u00e4ss\u00e4kin, mutta \u00e4kki\u00e4 h\u00e4n vaikutti l\u00e4hes innostuneelta. \"Jos odotatte hetken, haen sis\u00e4lt\u00e4 kartan. Voin merkit\u00e4 siihen paikan.\"\n\n\"Kiitos.\" Minun teki mieli halata h\u00e4nt\u00e4, mutta tajusin sent\u00e4\u00e4n olla tekem\u00e4tt\u00e4 niin. \"Et tied\u00e4, miten paljon juuri autoit!\"\n\nMikko palasi pian kartan kanssa, kuten oli luvannut. Oli tullut aika l\u00e4hte\u00e4, mutta huomasin ep\u00e4r\u00f6iv\u00e4ni. En halunnut tunnustaa h\u00e4nelle, millaiseen kiipeliin olin onnistunut j\u00e4rjest\u00e4m\u00e4\u00e4n koko kyl\u00e4n, mutta samaan aikaan koin velvollisuudekseni varoittaa h\u00e4nt\u00e4.\n\nLopulta sanoin vain: \"Ole varuillasi. Kyl\u00e4ss\u00e4 saattaa tapahtua pian jotain pahaa.\"\n\n\"Voi tytt\u00f6, \u00e4l\u00e4 ole huolissasi minusta. Jos myrsky onkin tulossa t\u00e4nne, se seuraa sinun kintereill\u00e4si.\"\n\n***\n\nMikon kartta oli niin vanha, etten uskonut teiden pit\u00e4v\u00e4n en\u00e4\u00e4 paikkansa, mutta voisimme helposti siirt\u00e4\u00e4 Mikon piirt\u00e4m\u00e4n ruksin toiseen, uudempaan karttaan. Annoin kuitenkin itseni riemuita vain hetken. Kaikki keskittymiseni oli suunnitelman toteuttamisessa.\n\nSuunnitelma oli hyvin yksinkertainen: aioin tehd\u00e4 tismalleen niin kuin daimonit halusivat. Ei ollut muuta vaihtoehtoa.\n\nValitettavasti sudet olivat asiasta eri mielt\u00e4.\n\n\"Me kaikki l\u00e4hdemme yhdess\u00e4\", Juha sanoi. \"T\u00e4m\u00e4 on lauman asia.\"\n\nPaitsi ettei ollut. Jos t\u00e4m\u00e4 oli jotain niin perheasia. Minun ja Mitjan.\n\n\"Te ette ole mill\u00e4\u00e4n tavalla valmistautuneita kohtaamaan edes yht\u00e4 daimonia, saati sitten useampaa\", sanoin.\n\n\"Me p\u00e4rj\u00e4\u00e4mme kyll\u00e4. Sin\u00e4h\u00e4n sanoit, ett\u00e4 sudet ovat ainoita, jotka pystyv\u00e4t vahingoittamaan heit\u00e4. Sit\u00e4 suuremmalla syyll\u00e4 meid\u00e4n pit\u00e4\u00e4 tulla mukaan\", Juha vastasi.\n\nViel\u00e4 vuosi sitten Juha ei olisi uskaltanut asettua poikkiteloin. Se, ett\u00e4 h\u00e4n pystyi siihen nyt, paljasti, kuinka paljon luultua hallitsevampi h\u00e4n oli. Minun olisi tietenkin pit\u00e4nyt olla iloinen, ett\u00e4 h\u00e4n oli onnistunut l\u00f6yt\u00e4m\u00e4\u00e4n luonteenlujuutensa, mutta juuri nyt oli minun kannaltani pahin mahdollinen hetki.\n\n\"Kerroin sen siksi, ett\u00e4 minun poissa ollessani teid\u00e4n pit\u00e4\u00e4 olla valmiita kaikkeen\", vastasin. \"T\u00e4ss\u00e4 vaiheessa kukaan ei voi olla varma, mit\u00e4 daimonit haluavat. He saattavat hy\u00f6k\u00e4t\u00e4 t\u00e4nne.\"\n\n\"Ette usko, miten harvoin sanon n\u00e4in, mutta Raisa on oikeassa. Teill\u00e4 ei ole aavistustakaan, mink\u00e4 kanssa te olette tekemisiss\u00e4\", Mitja sanoi hiljaa.\n\n\"Jos he ovat niin ylivoimaisia, miksi he eiv\u00e4t uskalla n\u00e4ytt\u00e4\u00e4 naamaansa?\" Jenni kysyi.\n\nHengitin kerran syv\u00e4\u00e4n. Kun se ei viel\u00e4 tuntunut auttavan, jatkoin kunnes viimein tulin siihen tulokseen, etten tulisi yht\u00e4\u00e4n rauhallisemmaksi. \"Daimonit rakastavat pelej\u00e4. He haluavat saada meid\u00e4t pois tolaltamme. Tehd\u00e4 meist\u00e4 ep\u00e4varmoja ja vainoharhaisia. Sen j\u00e4lkeen heid\u00e4n on helppo iske\u00e4.\"\n\nMinua katsottiin tavalla, joka olisi normaalioloissa saanut minut per\u00e4\u00e4ntym\u00e4\u00e4n. Tiesin, ett\u00e4 olin laumassa pelkk\u00e4 tulokas. Minulla ei ollut mit\u00e4\u00e4n oikeutta alkaa m\u00e4\u00e4r\u00e4ill\u00e4 susia.\n\nMutta nyt ei ollut kyse heist\u00e4, vaan Mikaelista. Minun Mikaelistani. Ja vaikka heill\u00e4 kaikilla saattoi ehk\u00e4 olla omat palasensa h\u00e4nest\u00e4, he eiv\u00e4t olleet vastuussa siit\u00e4, ett\u00e4 h\u00e4n oli vaarassa.\n\nKatsoin Juhaa silmiin. H\u00e4n vastasi tyynesti katseeseeni. Hitto. Asiaa ei auttanut sek\u00e4\u00e4n, ett\u00e4 h\u00e4n oli minua p\u00e4\u00e4t\u00e4 pidempi.\n\n\"O-ou. Kreikkalainen veri taitaa kohta kuohahtaa\", Niko sanoi.\n\n\"Kuka johtaa laumaa?\" kysyin Juhalta.\n\n\"Mikael\", Juha vastasi.\n\n\"Kuka on seuraava? Kun Mikael on poissa, kuka s\u00e4\u00e4nt\u00f6jen mukaan on seuraava?\"\n\n\"Me emme ole varsinaisesti m\u00e4\u00e4ritt\u00e4neet j\u00e4rjestyst\u00e4 viimekes\u00e4isen j\u00e4lkeen. Veikkaisin, ett\u00e4 joko Iivo tai min\u00e4.\"\n\n\"Mik\u00e4 on minun numeroni?\" tivasin.\n\nKaikki olivat hiljaa.\n\n\"Mik\u00e4 m\u00e4\u00e4rittelee naaraan aseman t\u00e4ss\u00e4 laumassa?\"\n\n\"Uroksen asema\", Jenni vastasi. H\u00e4n oli ristinyt k\u00e4tens\u00e4 rinnalleen. \"Is\u00e4 tai puoliso.\"\n\n\"Kysyn uudestaan: mik\u00e4 on minun numeroni?\"\n\n\"Raisa taitaa tarkoittaa, ett\u00e4 koska h\u00e4n on Mikaelin kanssa, h\u00e4n on t\u00e4ll\u00e4 hetkell\u00e4 ykk\u00f6nen\", Niko sanoi tyytyv\u00e4isen\u00e4.\n\nSeurasi hiljaisuus, joka paljasti, mit\u00e4 mielt\u00e4 toiset olivat ajatuksesta, ett\u00e4 min\u00e4 olin johdossa.\n\n\"Sin\u00e4 et voi menn\u00e4 yksin\", Juha sanoi lopulta.\n\n\"Mitja, tuletko mukaan? Sinun ei tarvitse jos et halua.\"\n\nYritin kuulostaa silt\u00e4 kuin en olisi v\u00e4litt\u00e4nyt, vastasi Mitja mit\u00e4 tahansa. Todellisuudessa en tiennyt, mit\u00e4 tekisin jos h\u00e4n kielt\u00e4ytyisi. En halunnut asettaa velje\u00e4ni vaaraan, mutta tiesin tarvitsevani h\u00e4nt\u00e4. Olin luvannut lopettaa h\u00e4nen suojelemisensa. H\u00e4nen kuului saada p\u00e4\u00e4tt\u00e4\u00e4 itse.\n\n\"Tietenkin min\u00e4 tulen\", h\u00e4n sanoi. H\u00e4n ei ollut tuhlannut kahtakaan sekuntia asian pohtimiseen. Se oli aikamoinen kunnianosoitus, kun otti huomioon, ettei Mitja todellakaan luottanut helposti.\n\nOlin jo l\u00e4hd\u00f6ss\u00e4, kun Juha avasi suunsa. \"Raisa... Niko kertoi, mit\u00e4 sinulle tapahtui sen kunnanjohtajan tapaamisen j\u00e4lkeen.\"\n\nPys\u00e4hdyin.\n\n\"Se kuulosti vaaralliselta. Min\u00e4... En usko, ett\u00e4 sinun kannattaa k\u00e4ytt\u00e4\u00e4 kykyj\u00e4si. Mikaelkaan ei selv\u00e4sti halunnut sit\u00e4.\"\n\nOlin iloinen, etten ollut k\u00e4\u00e4ntynyt kokonaan ymp\u00e4ri. Se tarkoitti, ettei h\u00e4n voinut n\u00e4hd\u00e4 ilmett\u00e4ni. \"Ei se ole mit\u00e4\u00e4n vakavaa\", sanoin.\n\nL\u00e4hdin huoneesta ennen kuin kukaan ehtisi sanoa, ett\u00e4 he olivat haistaneet valheeni.\n\n***\n\nEn pystynyt olemaan yksin Sarrien talolla, joten vietin y\u00f6ni sviitin sohvalla. En nukkunut. Odotin, ett\u00e4 kello tuli kahdeksan, ennen kuin nousin yl\u00f6s ja k\u00e4vin nopeasti l\u00e4pi kaikki tavarat, jotka meid\u00e4n oli v\u00e4ltt\u00e4m\u00e4tt\u00e4 otettava mukaan. Niit\u00e4 ei ollut paljon. En antanut itseni ajatella kaikkia mahdollisia skenaarioita ja niiden vaatimaa v\u00e4lineist\u00f6\u00e4. T\u00e4ll\u00e4 reissulla luottaisin kykyihini ja kroppaani.\n\nKun olin valmis, menin koputtamaan Mitjan ja Caoimhen makuuhuoneen oveen. Caoimhe tuli avaamaan. H\u00e4n n\u00e4ytti niin kamalalta, kuin todella kaunis ihminen saattoi n\u00e4ytt\u00e4\u00e4. Silm\u00e4t olivat punaiset ja lasittuneet, huulet liian vaaleat. Poskissa ja nen\u00e4ss\u00e4 oli punertavia kohtia niist\u00e4misen j\u00e4ljelt\u00e4. H\u00e4nen oli t\u00e4ytynyt itke\u00e4 koko y\u00f6.\n\n\"Sin\u00e4 voit viel\u00e4 muuttaa mielesi\", sanoin suomeksi Mitjalle, kun h\u00e4n ilmestyi Caoimhen takaa. En tavallisesti halunnut puhua suomea silloin kun Caoimhe oli paikalla, mutta nyt koin tarkoituksen pyhitt\u00e4v\u00e4n keinoni. Oikeasti olin nimitt\u00e4in varsin j\u00e4rkyttynyt. En ollut ajatellut lainkaan sit\u00e4, mit\u00e4 mielt\u00e4 Ciao olisi t\u00e4st\u00e4 kaikesta.\n\nMitja pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"H\u00e4n p\u00e4rj\u00e4\u00e4 kyll\u00e4.\"\n\n\"En usko, ett\u00e4 h\u00e4n on huolissaan omasta p\u00e4rj\u00e4\u00e4misest\u00e4\u00e4n.\"\n\nJ\u00e4tin heid\u00e4t kuiskimaan toisilleen eteiseen ja menin hissill\u00e4 alas. Ei mennyt kauan, kun Mitja ilmestyi aulaan reppunsa kanssa.\n\nNousin sohvalta ja sanaakaan sanomatta k\u00e4velimme entiselle koululleni. Vaati aikamoista mielikuvaharjoittelua, jotta saatoin n\u00e4hd\u00e4 itseni sen oppilaana \u2013 muistot olivat toki tallella, mutta niiss\u00e4 tuntui el\u00e4v\u00e4n joku toinen.\n\nEtuovien luona seisoi poikajoukko. He olivat ilmeisesti p\u00e4\u00e4tt\u00e4neet tuhlata kes\u00e4lomansa hengailemalla koulunsa pihalla, vaikkei rehtori tai edes kukaan opettajista ollut paikalla paheksumassa heid\u00e4n tupakointiaan ja syljeskely\u00e4\u00e4n.\n\nEhk\u00e4 heill\u00e4 ei ollut muuta paikkaa, mihin menn\u00e4. Vaikka he vihasivat koulua, se oli silti kotoisin paikka jonka he keksiv\u00e4t.\n\nSe sai minut surulliseksi. Tiesin, millaista oli, kun p\u00e4iv\u00e4t tuntuivat valuvan hukkaan. He odottivat, ett\u00e4 el\u00e4m\u00e4 vihdoin alkaisi. Kun niin tapahtuisi, he ihmettelisiv\u00e4t, miksi olivat odottaneet sit\u00e4 niin kovasti. Sill\u00e4 \u00e4kki\u00e4 asioita tapahtuisi aivan liikaa ja liian nopeasti, jotta niist\u00e4 olisi voinut pysy\u00e4 kartalla.\n\nJa oli onnekas, jos siin\u00e4 vaiheessa oli joku, joka saisi ajan pys\u00e4htym\u00e4\u00e4n \u2013 enk\u00e4 tarkoittanut kykyj\u00e4ni. Joku, jonka kanssa pystyi luomaan kimpun t\u00e4ydellisi\u00e4 hetki\u00e4, joiden avulla selvi\u00e4isi niist\u00e4 toisenlaisistakin.\n\nSytk\u00e4reiden vertailu keskeytyi, kun he \u00e4kk\u00e4siv\u00e4t meid\u00e4t. Vasta silloin tajusin ajatella, milt\u00e4 meid\u00e4n t\u00e4ytyi n\u00e4ytt\u00e4\u00e4 heid\u00e4n silmiss\u00e4\u00e4n. Kukaan koulun henkil\u00f6kunnasta ei nimitt\u00e4in olisi saanut aikaan samanlaista reaktiota. Ei edes koulun vahtimestari, jolle yritettiin virnistell\u00e4 p\u00e4in naamaa, mutta jota todellisuudessa pel\u00e4ttiin ja v\u00e4lteltiin parhaan mukaan.\n\nIlmeist\u00e4 p\u00e4\u00e4tellen kukaan ei olisi uskaltanut virnistell\u00e4 minulle ja Mitjalle, vaikka heille olisi tarjottu kahta sixpacki\u00e4. Olimme ratsanneet Sarrien vaatekaapit ja tulos oli varsin vaikuttava. Mitjalla oli maastohousut ja maiharit. Pitk\u00e4hihainen paita oli harmaa ja hiukset olivat l\u00e4hes kokonaan piilossa pipon alla. Minullakin oli maiharit ja reisitaskuhousut, joiden lahkeet oli tungettu kenkiin. Musta toppi ja aurinkolasit. Olin letitt\u00e4nyt poninh\u00e4nt\u00e4ni, joka heilahteli k\u00e4velless\u00e4ni olalta toiselle.\n\nMe olimme kahden hengen erikoisjoukot.\n\nNiko nojasi Annin vanhaan Toyotaan k\u00e4det taskuissa ja hihat k\u00e4\u00e4rittyn\u00e4. H\u00e4n suoristautui kun n\u00e4ki meid\u00e4t.\n\nH\u00e4n avasi takakontin, ja heitimme reppumme sinne. Mitja suuntasi takapenkille ty\u00f6nnetty\u00e4\u00e4n ensin etuistuimen pois tielt\u00e4. Juuri kun olin itse astumassa autoon, kuulin takaa tutun \u00e4\u00e4nen.\n\n\"Minne te olette menossa?\"\n\nJ\u00e4hmetyin. Jenni ja Juha olivat halunneet tulla saattamaan meit\u00e4, mutta olin kielt\u00e4nyt heit\u00e4. Oli kuitenkin joku, jota en ollut tajunnut edes kielt\u00e4\u00e4.\n\nJaska.\n\nH\u00e4n seisoi edess\u00e4ni. Jokin h\u00e4nen ilmeess\u00e4\u00e4n sai aikaan sen, ett\u00e4 pala nousi kurkkuuni. Yritin nielaista sen alas. Ei onnistunut.\n\n\"Me menemme mets\u00e4lle.\" Se oli paras peitetarina jonka keksin, sill\u00e4 se sattui olemaan totta. Omalla tavallaan.\n\n\"Ei ole mets\u00e4stysaika\", h\u00e4n huomioi.\n\nHetken aikaa vain tuijotimme toisiamme.\n\n\"Miss\u00e4 Mikael on?\" h\u00e4n kysyi \u00e4kki\u00e4.\n\nEn vastannut. Sen sijaan ty\u00f6nsin aurinkolasit otsalle.\n\nEn ole ihan varma, mit\u00e4 h\u00e4n n\u00e4ki, mutta h\u00e4nen silm\u00e4ns\u00e4 levisiv\u00e4t hieman.\n\nSilloin k\u00e4velin l\u00e4hemm\u00e4s ja halasin h\u00e4nt\u00e4. Mieless\u00e4ni halasin jokaikist\u00e4 Hukkavaaran asukasta, jokaista yst\u00e4v\u00e4\u00e4, jota olin kielt\u00e4nyt saattamasta meit\u00e4 matkaan, jolta en ehk\u00e4 palaisi.\n\n\"Kiitos kaikesta. Ja anteeksi. Kaikesta.\"\n\nJaska oli yht\u00e4 yll\u00e4ttynyt kuin min\u00e4, ja lopulta h\u00e4nen k\u00e4tens\u00e4 kohosivat. En kuitenkaan antanut h\u00e4nen vastata halaukseeni, vaan p\u00e4\u00e4stin irti ennen kuin h\u00e4n ehti koskea minua.\n\nEn j\u00e4\u00e4nyt odottamaan, mit\u00e4 h\u00e4n sanoisi, vaan laskin lasit takaisin silmilleni ja menin autoon.\n\n\"Menn\u00e4\u00e4n\", sanoin Nikolle. Kielsin itse\u00e4ni katsomasta sivulle.\n\nNiko ja Mitja tutkivat Mikon karttaa ja tulivat johonkin tulokseen ajoreitist\u00e4 ja siit\u00e4, mihin Nikon kannattaisi j\u00e4tt\u00e4\u00e4 meid\u00e4t. En seurannut heid\u00e4n keskusteluaan tarkkaan. Sen sijaan yritin nukkua \u2013 tuloksetta.\n\nLopulta luovutin. Siristelin kuivia silmi\u00e4ni. \"Mit\u00e4 sin\u00e4 sanoit Tanelille?\" kysyin Nikolta. Tiesin, ettei h\u00e4nen mukaantulonsa ainakaan helpottanut h\u00e4nen ja Tanelin v\u00e4lej\u00e4. Silti en voinut v\u00e4itt\u00e4\u00e4 toivovani kenenk\u00e4\u00e4n muun l\u00e4hteneen kuskiksemme.\n\n\"Sanoin, ett\u00e4 minulla on jotain perhedraamaa meneill\u00e4\u00e4n.\" H\u00e4n huokaisi. \"Olin ennen katkera, etten voinut laumassa valehdella kenellek\u00e4\u00e4n. Nyt kun voin valehdella Tanelille, vihaan sit\u00e4.\"\n\nLoppumatkan yritin katsella maisemia. J\u00e4lkik\u00e4teen ajatellen en muista havainneeni maisemassa mink\u00e4\u00e4nlaista muutosta pohjoista kohti menness\u00e4, vaikka niin oli t\u00e4ytynyt tapahtua. Tuntui kuin koko matkaa ei olisi ollut olemassa, ei oikeasti.\n\nHidastimme ensin vauhtia ja pys\u00e4hdyimme viimein kokonaan. En empinyt hetke\u00e4k\u00e4\u00e4n, kun avasin turvavy\u00f6n ja astuin ulos autosta.\n\n\"Ehk\u00e4 minun pit\u00e4isi sittenkin tulla mukaan\", Niko sanoi.\n\n\"Ei. Sinun pit\u00e4\u00e4 palata Helsinkiin ja takaisin t\u00f6ihin. Takaisin omaan el\u00e4m\u00e4\u00e4si.\"\n\nH\u00e4n oli aikeissa v\u00e4itt\u00e4\u00e4 vastaan, mutta hiljensin h\u00e4net. \"Mene kotiin ja halaa Tanelia minunkin puolestani.\"\n\n\"Sin\u00e4 olet edelleen minun paras yst\u00e4v\u00e4ni\", h\u00e4n sanoi.\n\n\"Tied\u00e4n, ja sin\u00e4 olet minun. Mutta me emme ole en\u00e4\u00e4 lukiossa. Niin oudolta kuin se kuulostaakin, me olemme nyt aikuisia. Eik\u00e4 aikuisten yst\u00e4vyys ole samanlaista kuin lasten. Meill\u00e4 on omat el\u00e4m\u00e4mme eik\u00e4 voi en\u00e4\u00e4 olettaa, ett\u00e4 tekisimme kaiken yhdess\u00e4. Mutta se ei tarkoita, ettetk\u00f6 sin\u00e4 olisi minulle t\u00e4rke\u00e4.\"\n\nHalasin h\u00e4nt\u00e4 paljon lujempaa kuin Jaskaa. \"N\u00e4hd\u00e4\u00e4n viikon p\u00e4\u00e4st\u00e4.\"\n\nNiin paljon aikaa minulla ja Mitjalla oli. Niin paljon aikaa oli t\u00e4ysikuuhun. Jos emme olisi tulleet takaisin siihen menness\u00e4, lauma saisi p\u00e4\u00e4tt\u00e4\u00e4, mit\u00e4 he halusivat tehd\u00e4 ja toimia sen mukaisesti.\n\nJos Niko haistoi pelkoni, h\u00e4n ei sanonut mit\u00e4\u00e4n. Sellaisissa tilanteissa oli niin selv\u00e4\u00e4, miksi h\u00e4n oli paras yst\u00e4v\u00e4ni.\n\nP\u00e4\u00e4stin h\u00e4nest\u00e4 irti ja katsoin velje\u00e4ni. \"Valmis?\"\n\nH\u00e4n ny\u00f6kk\u00e4si ja ojensi minulle reppuni.\n\nAstuimme tielt\u00e4 mets\u00e4\u00e4n yht\u00e4 aikaa. \nKYMMENES LUKU\n\nMets\u00e4ss\u00e4 aika toimii eri tavalla kuin muualla. Olin huomannut sen ensimm\u00e4isen kerran vuosi sitten, kun min\u00e4 ja Mikael olimme p\u00e4\u00e4tt\u00e4neet l\u00e4hte\u00e4 etsim\u00e4\u00e4n Danielia Ven\u00e4j\u00e4n puolelta.\n\nMin\u00e4 ja veljeni emme puhuneet mit\u00e4\u00e4n. Puut olivat harvemmassa kuin mihin olin tottunut, ja v\u00e4ripaletissakin oli jotain, mik\u00e4 sai minut ajattelemaan t\u00e4m\u00e4n mets\u00e4n olevan paljon vanhempaa kuin Hukkavaarassa. Ei ollut erityisen l\u00e4mmint\u00e4, eik\u00e4 viel\u00e4 voinut tiet\u00e4\u00e4, millainen p\u00e4iv\u00e4st\u00e4 tulisi. Se oli minulle samantekev\u00e4\u00e4. Tuskin huomasin edes hyttysi\u00e4, jotka olivat kimpussamme. Keskityin vain eteenp\u00e4in kulkemiseen. Olin aina ollut liian rauhaton meditoimaan yksin aloillani, mutta t\u00e4m\u00e4 toimi. Askel toisensa per\u00e4\u00e4n.\n\nPystyimme liikkumaan nopeasti ja l\u00e4hes \u00e4\u00e4neti. En ollut v\u00e4h\u00e4\u00e4n aikaan ajatellut, miten pitk\u00e4lle olimme p\u00e4\u00e4sseet kykyjemme kanssa, mutta nyt huomasin, miten vahvoja meist\u00e4 oli tullut kuluneen puolen vuoden aikana. Pys\u00e4htelimme silloin t\u00e4ll\u00f6in, mutta l\u00e4hinn\u00e4 tarkistamaan, ett\u00e4 olimme yh\u00e4 oikealla reitill\u00e4.\n\nLopulta meid\u00e4n oli kuitenkin pakko tehd\u00e4 pidempi pys\u00e4hdys. Istuimme maahan juomaan ja sy\u00f6m\u00e4\u00e4n. P\u00e4\u00e4tin heti, ett\u00e4 se olisi ainoa tauko, jonka pit\u00e4isimme. Matkaa oli j\u00e4ljell\u00e4 ehk\u00e4 kolme tuntia. Kolme tuntia oli lyhyt tuokio, mutta pelk\u00e4sin, ett\u00e4 todellisuudessa se oli aivan liian pitk\u00e4 aika. Siin\u00e4 ehtisi tapahtua paljon. Paljon pahoja asioita, joita min\u00e4 en ollut est\u00e4m\u00e4ss\u00e4, koska olin t\u00e4\u00e4ll\u00e4, keskell\u00e4 ei-mit\u00e4\u00e4n. Joissain muissa olosuhteissa olisin varmaan osannut nauttia maisemasta, mutta juuri nyt vihasin jokaista puuta, jokaista kive\u00e4.\n\n\u00c4kki\u00e4 en niin v\u00e4litt\u00e4nyt omista ajatuksistani, joten k\u00e4\u00e4nnyin Mitjan puoleen.\n\n\"Mit\u00e4 sin\u00e4 ajattelet?\"\n\n\"Tupakkaa\", veljeni tunnusti.\n\n\"Viel\u00e4kin? Sin\u00e4h\u00e4n lopetit monta kuukautta sitten.\"\n\n\"Sin\u00e4 et selv\u00e4stik\u00e4\u00e4n tied\u00e4 mit\u00e4\u00e4n riippuvuuksista.\"\n\nSe oli varmaankin totta. Katselin ymp\u00e4rilleni. Maa kumpuili t\u00e4\u00e4ll\u00e4 enemm\u00e4n kuin Hukkavaarassa, mutta onneksemme reitillemme ei osunut ainuttakaan tunturia. Vihreys oli melkein ylitsevuotavaa, ja oli vaikea kuvitella, ett\u00e4 t\u00e4m\u00e4 sama maisema maalautuisi jo kahden kuukauden p\u00e4\u00e4st\u00e4 kokonaan toisin v\u00e4rein.\n\n\"En uskonut, ett\u00e4 Caoimhe reagoisi sinun l\u00e4ht\u00f6\u00f6si ihan noin voimakkaasti\", sanoin.\n\n\"H\u00e4n p\u00e4\u00e4see siit\u00e4 yli.\"\n\nPaitsi jos Mitjalle k\u00e4vi jotain. Tajusin olla sanomatta sit\u00e4.\n\nMitja ei katsonut minuun kun h\u00e4n sanoi: \"Minun t\u00e4ytyy kertoa sinulle jotain.\"\n\n\"No?\" Muistin, mit\u00e4 h\u00e4nen oli ollut tarkoitus tehd\u00e4 sin\u00e4 iltana, kun Mikael katosi. H\u00e4n ei ollut lainkaan maininnut sit\u00e4. \"Liittyyk\u00f6 t\u00e4m\u00e4 kosimiseen?\"\n\n\"Tavallaan\", h\u00e4n sanoi.\n\n\"Mit\u00e4 Ciao vastasi?\"\n\n\"En kysynyt viel\u00e4.\"\n\n\"Ai.\"\n\nVeljeni veti henke\u00e4 ja aloitti. \"Yksi syy siihen, miksi ylip\u00e4\u00e4t\u00e4\u00e4n aion kosia, on se, ett\u00e4 Caoimhe on Suomessa vain t\u00e4m\u00e4n kes\u00e4n. Sen j\u00e4lkeen h\u00e4nen pit\u00e4\u00e4 palata Irlantiin.\"\n\nPy\u00f6rittelin h\u00e4nen sanojaan mieless\u00e4ni. Ymm\u00e4rsin, miksi asia huoletti h\u00e4nt\u00e4. \"Eik\u00f6 h\u00e4n voi j\u00e4\u00e4d\u00e4 t\u00e4nne?\" kysyin. \"Vaikka h\u00e4n ei saisikaan t\u00f6it\u00e4, teille varmasti riitt\u00e4isi keikkoja. Sin\u00e4 tienaat ihan kohtuullisesti, eik\u00e4 teill\u00e4 olisi vuokraakaan maksettavana.\"\n\nMitja pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Caoimhe ei tule selvi\u00e4m\u00e4\u00e4n talvesta. T\u00e4\u00e4ll\u00e4 on liian kylm\u00e4\u00e4 ja pime\u00e4\u00e4. Ja vaikka h\u00e4n selvi\u00e4isikin, en halua, ett\u00e4 h\u00e4n k\u00e4rsii takiani.\"\n\n\"En tajua, mit\u00e4 tekemist\u00e4 t\u00e4ll\u00e4 on minun kanssani.\"\n\n\"Kun Caoimhe l\u00e4htee, min\u00e4 l\u00e4hden mukaan.\"\n\nOlin tiennyt vastauksen jo ennen kuin h\u00e4n sanoi sen \u00e4\u00e4neen. Totuus oli, ettei Mitjaa pidellyt Suomessa monikaan asia. Ratkaisu oli siis t\u00e4ysin j\u00e4rkev\u00e4, mutta se sai minut silti surulliseksi. En halunnut h\u00e4nen l\u00e4htev\u00e4n. H\u00e4n oli yst\u00e4v\u00e4ni ja perheeni. Enemm\u00e4n: h\u00e4n oli kykyjeni toinen puoli. H\u00e4nen l\u00e4htemisens\u00e4 tulisi sattumaan aivan helkkaristi.\n\nVaikutti vahvasti silt\u00e4, ett\u00e4 kaikkein rakkaimmat ihmiseni olivat l\u00e4hd\u00f6ss\u00e4.\n\nEi Mikael, muistutin itse\u00e4ni. Ja kaduin sit\u00e4 v\u00e4litt\u00f6m\u00e4sti. Sill\u00e4 en voinut olla siit\u00e4 lainkaan varma.\n\n\"Miten luulet, ett\u00e4 he saivat Mikaelin kiinni?\" Mitja kysyi.\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Se ei ole t\u00e4rke\u00e4\u00e4. T\u00e4rke\u00e4mpi kysymys on, miten he onnistuvat pit\u00e4m\u00e4\u00e4n h\u00e4net vangittuna.\"\n\nSiihen ongelmaan oli olemassa hyvin helppo ratkaisu: hopea. En halunnut ajatella tai sanoa sit\u00e4 \u00e4\u00e4neen, joten jatkoin: \"En k\u00e4sit\u00e4, miten tyhmi\u00e4 me olimme. Koko t\u00e4m\u00e4n ajan olimme huolissamme vain jostain naapurikuntalaisista ja heid\u00e4n susifobiastaan, kun meid\u00e4n olisi pit\u00e4nyt pel\u00e4t\u00e4 henkemme edest\u00e4.\"\n\nNyt kun minulla oli ollut riitt\u00e4v\u00e4sti aikaa ajatella asiaa, luulin tiet\u00e4v\u00e4ni, mit\u00e4 oli oikeasti tapahtunut. Olin k\u00e4ynyt mieless\u00e4ni l\u00e4pi jokaisen sekunnin siit\u00e4 hetkest\u00e4 l\u00e4htien, kun Juha oli soittanut meille ja kertonut Hukkavaaran ongelmasta.\n\nEsko Kukkonen ei ollut valehdellut \u2013 h\u00e4n oli uskonut n\u00e4hneens\u00e4 suden. Ennen kaikkea h\u00e4n oli uskonut, ett\u00e4 sudet olivat vaaraksi h\u00e4nen kyl\u00e4lleen ja ett\u00e4 h\u00e4nen t\u00e4ytyi tehd\u00e4 kaikkensa sen asukkaiden suojelemiseksi.\n\nToisin sanoen h\u00e4n oli uskonut juuri sen, mink\u00e4 daimonit olivat halunneet h\u00e4nen uskovan.\n\nEn tiennyt, miten daimonit olivat saaneet sen aikaan, mutta sill\u00e4 ei ollut v\u00e4li\u00e4. Esko oli ollut sy\u00f6tti, jonka me olimme nielleet kokonaisena alas sill\u00e4 hetkell\u00e4 kun olimme p\u00e4\u00e4tt\u00e4neet tulla Hukkavaaraan.\n\nKetk\u00e4 olivat hyvi\u00e4 istuttamaan pelkoja ihmisten mieliin? Daimonit. Ketk\u00e4 eiv\u00e4t koskaan antaneet anteeksi ja vannoivat verikoston nimeen? Daimonit, daimonit. Enk\u00e4 min\u00e4 ollut pys\u00e4htynyt kertaakaan ajattelemaan, miten outoa oli, ett\u00e4 he olivat antaneet Alekin kuoleman olla, tai ett\u00e4 yht\u00e4kki\u00e4 naapurikunta koki sudet ongelmaksi. Ensin olin ollut liian onnellinen ja my\u00f6hemmin liian huolissani Mikaelin sudesta tajutakseni mit\u00e4\u00e4n.\n\nToisaalta: daimonit olivat ovelia, ja min\u00e4 olin vasta alkanut n\u00e4hd\u00e4, mit\u00e4 se tarkoitti.\n\n\"En usko, ett\u00e4 olisimme voittaneet mit\u00e4\u00e4n pelk\u00e4\u00e4m\u00e4ll\u00e4\", Mitja sanoi.\n\n\"Mikael sanoi, ettei susijahdin ajoituksessa ollut mit\u00e4\u00e4n j\u00e4rke\u00e4. Susia ei koskaan mets\u00e4stet\u00e4 kes\u00e4ll\u00e4. Ja siit\u00e4 huolimatta en osannut ep\u00e4ill\u00e4 mit\u00e4\u00e4n.\"\n\n\"\u00c4l\u00e4 syyt\u00e4 itse\u00e4si. Yritet\u00e4\u00e4n ajatella jotain rakentavampaa. Niin kuin esimerkiksi mietti\u00e4, miksi he valitsivat Lapin.\"\n\n\"En tied\u00e4. He eiv\u00e4t luonnollisestikaan halunneet ottaa yhteen Hukkavaarassa, joka on t\u00e4ynn\u00e4 susia. En ymm\u00e4rr\u00e4, miksi he n\u00e4kiv\u00e4t kaiken t\u00e4m\u00e4n vaivan saadakseen minut ja Mikaelin tulemaan sinne Helsingist\u00e4.\"\n\nOli lukemattomia asioita, joihin en keksinyt vastausta. Oli luultavasti turhaa edes pohtia niit\u00e4, mutten voinut muutakaan.\n\nOlin niin ajatuksissani, etten huomannut, mit\u00e4 ymp\u00e4rill\u00e4mme tapahtui. Mitja oli enemm\u00e4n hereill\u00e4. H\u00e4n p\u00e4lyili ymp\u00e4rilleen ja sanoi sitten: \"Raisa.\" Siin\u00e4 vaiheessa min\u00e4kin olin alkanut h\u00f6rist\u00e4\u00e4 korviani. Jossain l\u00e4hell\u00e4 kahisi.\n\nNousin yl\u00f6s. Pidin kaikki liikkeeni mahdollisimman rauhallisina. Susista ja muistakin villiel\u00e4imist\u00e4 oli hyv\u00e4 tiet\u00e4\u00e4 yksi asia: ne eiv\u00e4t n\u00e4ytt\u00e4ytyneet ihmisille ilman syyt\u00e4. Ja yleens\u00e4 syy oli, ett\u00e4 ne olivat viitt\u00e4 vaille valmiita k\u00e4ym\u00e4\u00e4n kimppuun.\n\nOtin veitsen esiin. N\u00e4in silm\u00e4kulmastani, kuinka oksat heilahtivat. K\u00e4\u00e4nnyin niit\u00e4 kohti, mutta seuraavaksi \u00e4\u00e4ni kuului jo toisesta suunnasta.\n\n\"Mik\u00e4 se on?\" Mitja kuiskasi kreikaksi. Aivan niin kuin \"se\" olisi voinut ymm\u00e4rt\u00e4\u00e4, jos h\u00e4n olisi puhunut suomea.\n\n\"Se h\u00e4rn\u00e4\u00e4 meit\u00e4\", vastasin suomeksi ja ihan normaalilla \u00e4\u00e4nell\u00e4. Ei ollut mit\u00e4\u00e4n syyt\u00e4 kuiskia. Susi kyll\u00e4 tiesi meist\u00e4.\n\nSe nimitt\u00e4in oli susi. Olin varma, vaikken n\u00e4hnyt siit\u00e4 vilahdustakaan. Mutta kuka se oli?\n\n\"N\u00e4yt\u00e4 itsesi\", sanoin.\n\nRapina lakkasi. En n\u00e4hnyt miss\u00e4\u00e4n mit\u00e4\u00e4n liikett\u00e4, mutta tiesin, ettei susi ollut mennyt minnek\u00e4\u00e4n. Se odotti.\n\n\"Olisi tietenkin mukava tiet\u00e4\u00e4, kuka sin\u00e4 olet ennen kuin tapan sinut, mutta onnistuu se ilmankin.\" Se ei ollut kovin suurta liioittelua, sill\u00e4 hyv\u00e4ll\u00e4 tuurilla onnistuisin osumaan siihen veitsell\u00e4ni sokkonakin.\n\nSusi tuli esiin p\u00e4invastaisesta suunnasta kuin olin odottanut. S\u00e4ik\u00e4hdin hieman, mutta yritin parhaani mukaan peitt\u00e4\u00e4 sen ja pit\u00e4\u00e4 veitsik\u00e4teni vakaana. Susi katsoi minua tavalla, joka luullakseni viestitti, etten ollut onnistunut huijaamaan sit\u00e4.\n\nSe taisi pilkata minua.\n\nEn ollut ehtinyt pohtia, kuka susista oli kaikkein potentiaalisin ehdokas l\u00e4htem\u00e4\u00e4n per\u00e4\u00e4mme kiellostani huolimatta. Joka tapauksessa veikkaukseni olisi osunut takuuvarmasti v\u00e4\u00e4r\u00e4\u00e4n.\n\nOlisin tunnistanut t\u00e4m\u00e4n suden miss\u00e4 ja milloin tahansa.\n\n\"Mitja, heit\u00e4 housut ja paita.\" Sudelle sanoin: \"\u00c4l\u00e4 kuvittelekaan muuttavasi muotoa minun n\u00e4hteni.\"\n\nSusi haukahti. Naurakoon vain, ajattelin. En halunnut n\u00e4hd\u00e4 h\u00e4nt\u00e4 alastomana.\n\nMitja heitti vaatteensa minulle ja min\u00e4 heitin ne eteenp\u00e4in sudelle. H\u00e4n nappasi ne ilmasta kiinni ja jolkutteli siihen suuntaan, josta oli tullut. Ei mennyt kauaakaan, kun h\u00e4n palasi ihmisen muodossa.\n\nMitjan collegehousut olivat h\u00e4nelle sek\u00e4 hieman liian pitk\u00e4t ett\u00e4 liian tiukat. Paita kiristyi rintalihasten ja hauisten ymp\u00e4rille.\n\nH\u00e4nen hiuksensa olivat tummemmat kuin kenell\u00e4k\u00e4\u00e4n toisella tuntemallani. Ja silm\u00e4t? Ne olivat siniset, vain muutamaa astetta lempis\u00e4vy\u00e4ni haaleammat.\n\n\"Mitja, saanko esitell\u00e4: Daniel hemmetin Sarri.\"\n\n***\n\nH\u00e4n oli menett\u00e4nyt kotinsa, perheens\u00e4 ja omaisuutensa, mutta siit\u00e4 huolimatta h\u00e4n n\u00e4ytti ylimieliselt\u00e4. H\u00e4n katsoi minua niin kuin h\u00e4nell\u00e4 olisi ollut valta p\u00e4\u00e4tt\u00e4\u00e4 el\u00e4m\u00e4st\u00e4ni. Toisin sanoen h\u00e4n ei ollut muuttunut tippaakaan.\n\nTiesin, ett\u00e4 minun olisi pit\u00e4nyt suuttua, mutta Danielin l\u00e4sn\u00e4olo sai minut hermostuneeksi. H\u00e4nen ylimielisyydess\u00e4\u00e4n kaikkein huolestuttavinta oli se, ett\u00e4 yhdeks\u00e4ss\u00e4 tapauksessa kymmenest\u00e4 sille oli erinomaiset perusteet. H\u00e4n oli hyv\u00e4 todella monessa asiassa, mukaan lukien rahan tienaaminen, ihmisten pelottelu ja niskan taittaminen paljain k\u00e4sin.\n\nYhdenn\u00e4k\u00f6isyys Mikaeliin oli niin selv\u00e4, ett\u00e4 h\u00e4nen kohtaamisensa olisi voinut kuvitella kasvattavan kaipausta. Mutta minun mieless\u00e4ni Daniel ei ollut koskaan ollut niin kuin poikansa, ja kaikkein v\u00e4hiten h\u00e4n muistutti Mikaelia juuri nyt. Mit\u00e4 paremmin jonkun tunsi, sit\u00e4 paremmin osasi kiinnitt\u00e4\u00e4 huomiota niihin kohtiin toisen ulkon\u00e4\u00f6ss\u00e4, joista persoonallisuus paistoi kaikkein kirkkaimmin l\u00e4pi. Olin viett\u00e4nyt koko kes\u00e4n Mikaelin kanssa ja n\u00e4in heti, ettei Danielin katseessa ollut samaa l\u00e4mp\u00f6\u00e4. H\u00e4n ei my\u00f6sk\u00e4\u00e4n hymyillyt vain siksi, ett\u00e4 min\u00e4 olin paikalla.\n\n\"T\u00e4m\u00e4 on siis se mulkero, joka hy\u00f6kk\u00e4si kimppuusi.\" Mitja olisi voinut sanoa sen kreikaksi, mutta h\u00e4n ei tehnyt niin.\n\nDanielin suupieli kohosi. Olin ehtinyt unohtaa, ett\u00e4 h\u00e4n hymyili aina v\u00e4\u00e4riss\u00e4 kohdissa.\n\n\"Ja sama mulkero tiesi sinusta, muttei kertonut minulle tai \u00e4idille mit\u00e4\u00e4n.\"\n\nSuu palasi takaisin viivaksi. \"Raisa Oja\", h\u00e4n sanoi. H\u00e4nen \u00e4\u00e4nens\u00e4 oli vahva ja matala ja nostatti minussa heti muistoja. En voinut sille mit\u00e4\u00e4n: vilunv\u00e4reet laskeutuivat niskastani k\u00e4sivarsiani ja lapaluitani pitkin ja poukkoilivat ties minne.\n\n\"Sin\u00e4 olet hukannut minun poikani\", h\u00e4n jatkoi.\n\nSe ei kuulostanut lainkaan silt\u00e4, mit\u00e4 oli tapahtunut, mutta en silti keksinyt hyv\u00e4\u00e4 vastav\u00e4itett\u00e4. Sen sijaan kysyin: \"Mist\u00e4 sin\u00e4 sen tied\u00e4t?\"\n\nLauman s\u00e4\u00e4nt\u00f6jen mukaan Daniel ei saanut astua jalallakaan lauman maille tai olla yhteydess\u00e4 sen j\u00e4seniin. Ei etenk\u00e4\u00e4n Mikaeliin. Ja silti h\u00e4n tiesi Mikaelin kadonneen ja meid\u00e4n l\u00e4hteneen etsim\u00e4\u00e4n h\u00e4nt\u00e4 Lapista.\n\n\"Olet ilmeisesti unohtanut, ett\u00e4 min\u00e4 tied\u00e4n kaiken.\"\n\nMitja vilkaisi minua. Kohotin kulmiani. N\u00e4etk\u00f6 nyt? H\u00e4n on tismalleen sellainen, kuin olen kertonut. \u00c4\u00e4neen sanoin: \"Sin\u00e4 voit palata sinne, mist\u00e4 tulitkin. \u00c4l\u00e4 kuvittele, ett\u00e4 me kutsuisimme sinut mukaan.\"\n\nSe tuntui huvittavan Danielia entist\u00e4kin enemm\u00e4n. \"Ette kutsu minua mukaan minne? Omaan talooni?\"\n\nH\u00e4n todella tuntui tiet\u00e4v\u00e4n kaiken, eik\u00e4 minulla ollut aavistustakaan, miten.\n\n\"Minun pit\u00e4isi varmaan kutsua teid\u00e4t mukaani\", h\u00e4n jatkoi. \"N\u00e4ytt\u00e4\u00e4 silt\u00e4, ett\u00e4 te tarvitsette apuani.\"\n\n\"Me p\u00e4rj\u00e4\u00e4mme ihan hyvin, kiitos vain.\" Poimin reppuni maasta ja k\u00e4\u00e4nnyin l\u00e4hte\u00e4kseni liikkeelle.\n\n\"Sin\u00e4 olet menossa v\u00e4\u00e4r\u00e4\u00e4n suuntaan.\"\n\nMin\u00e4 ja Mitja vilkaisimme taas toisiamme.\n\n\"Te haluatte p\u00e4\u00e4st\u00e4 perille niin nopeasti kuin mahdollista\", Daniel jatkoi. \"Ja te haluatte pelastaa Mikaelin. Meill\u00e4 on sama matka ja sama tavoite.\"\n\nHeitin repun selk\u00e4\u00e4ni ja Mitja teki samoin. L\u00e4hdimme k\u00e4velem\u00e4\u00e4n suuntaan, joka t\u00e4ll\u00e4 kertaa oli toivoakseni oikea. Danielilla ei ollut vaikeuksia pysytell\u00e4 rinnallamme, vaikka h\u00e4nell\u00e4 ei ollut edes kenki\u00e4 jalassa.\n\nK\u00e4velimme pitk\u00e4\u00e4n hiljaisuuden vallitessa. Aurinko oli mennyt pilveen, ja melkein toivoin, ett\u00e4 alkaisi sataa. Ehk\u00e4 sadekuuro onnistuisi huuhtomaan minusta paitsi hien ja hyttyset, my\u00f6s tuntemani kiukun.\n\nYritin parhaani mukaan teeskennell\u00e4, ettei Danielia ollut olemassa, mutta sill\u00e4 ei ollut toivomaani vaikutusta. Tunsin h\u00e4nen huvittuneen katseensa.\n\n\"Jos sin\u00e4 kerta tied\u00e4t kaiken, niin sitten sinun pit\u00e4isi tiet\u00e4\u00e4, ettemme voi ilmesty\u00e4 daimonien eteen suden kanssa\", murahdin lopulta. En viel\u00e4k\u00e4\u00e4n katsonut h\u00e4neen. \"He tappavat Mikaelin heti, kun n\u00e4kev\u00e4t sinut.\"\n\n\"Kuvitteletteko te todella, ett\u00e4 te voitte voittaa pelaamalla heid\u00e4n s\u00e4\u00e4nn\u00f6ill\u00e4\u00e4n?\"\n\nAlle vartti Danielin seurassa, ja h\u00e4n oli jo alkanut m\u00e4\u00e4r\u00e4ill\u00e4 meit\u00e4. Olin luvannut itselleni, etten antaisi h\u00e4nen koskaan pelotella minua.\n\n\"Meill\u00e4 ei ole vaihtoehtoja\", kivahdin. Ja aloin heti ep\u00e4ill\u00e4 sanojani. Daniel oli oikeassa: me nimitt\u00e4in tuskin saisimme Mikaelia el\u00e4v\u00e4n\u00e4 takaisin, jos pelk\u00e4st\u00e4\u00e4n tyytyisimme tekem\u00e4\u00e4n niin kuin daimonit sanoivat.\n\nDanielin oli kuitenkin pakko sanoa se \u00e4\u00e4neen. \"Aina on olemassa vaihtoehto. Me voimme aloittaa muuttamalla s\u00e4\u00e4nt\u00f6j\u00e4 mieleiseksemme.\"\n\nHoukutus heitt\u00e4\u00e4 ohjat h\u00e4nelle ja vain seurata vierest\u00e4, mit\u00e4 h\u00e4n saisi aikaan, oli todella suuri. Mutta en voinut tehd\u00e4 sit\u00e4. En, kun kyse oli Mikaelista.\n\n\"Sin\u00e4 et ole en\u00e4\u00e4 johtaja\", sanoin.\n\n\"Ja sin\u00e4k\u00f6 olet?\"\n\nKun h\u00e4n sanoi sen niin, koko ajatus tuntui t\u00e4ysin typer\u00e4lt\u00e4. Minulla ei ollut ollut vaikeuksia uskoa siihen silloin, kun olin uhitellut Juhalle. Daniel Sarrilla oli kuitenkin kyky saada minut tuntemaan itseni todella pieneksi.\n\nMe olemme samalla puolella, muistutin itse\u00e4ni. Ehk\u00e4 jos hokisin sit\u00e4 itselleni joka toinen sekunti, halu kuristaa h\u00e4net hengilt\u00e4 lakkaisi. Ehk\u00e4.\n\n\"Kyll\u00e4. Min\u00e4 olen johtaja. Min\u00e4 p\u00e4\u00e4t\u00e4n, mit\u00e4 tehd\u00e4\u00e4n. En tarvitse sinun lupaasi mihink\u00e4\u00e4n.\"\n\n\"T\u00e4m\u00e4 tyt\u00f6lt\u00e4, joka nukkuu minun silkkilakanoissani.\"\n\nSuuni loksahti auki. Pys\u00e4hdyin ja k\u00e4\u00e4nnyin katsomaan h\u00e4nt\u00e4. Se oli virhe: Danielin omahyv\u00e4isen ilme oli raivostuttavinta, mit\u00e4 olin koskaan n\u00e4hnyt. \"Mist\u00e4 sin\u00e4 senkin tied\u00e4t?\" tiukkasin.\n\n\"Minulla on l\u00e4hteeni.\"\n\nSe alkoi olla jo aika karmivaa. Daniel tiesi, miten ihmiset pidettiin varpaillaan. Se oli h\u00e4nen erikoistaitonsa. Siin\u00e4 mieless\u00e4 h\u00e4n oli Mikaelin vastakohta. Mikaelin erikoistaito oli rauhoittaa ja valaa ihmisiin luottamusta. Eik\u00e4 pelk\u00e4st\u00e4\u00e4n h\u00e4nt\u00e4 kohtaan, vaan heihin itseens\u00e4. Sen takia niin moni oli valmis tekem\u00e4\u00e4n h\u00e4nen puolestaan melkein mit\u00e4 tahansa. Ei siksi, ett\u00e4 heid\u00e4n t\u00e4ytyi, vaan siksi, ett\u00e4 he halusivat. Koska he tiesiv\u00e4t, ett\u00e4 Mikael tekisi heid\u00e4n vuokseen saman.\n\nMitja naurahti.\n\n\"Mit\u00e4?\" kysyin kreikaksi.\n\n\"Onhan t\u00e4m\u00e4 nyt aika hauskaa\", h\u00e4n vastasi.\n\n\"Naura vain. Et ole viel\u00e4 edes tavannut omaa tulevaa appeasi.\"\n\nTunsin pient\u00e4 tyydytyst\u00e4, kun n\u00e4in h\u00e4nen s\u00e4ik\u00e4ht\u00e4neen ilmeens\u00e4. Samalla aloin kuitenkin pohtia, mit\u00e4 Mikael ajattelisi jos tiet\u00e4isi meid\u00e4n solmivan liittoumaa. Ja mit\u00e4 h\u00e4n ajattelisi, jos tapattaisin h\u00e4nen is\u00e4ns\u00e4 h\u00e4nen vuokseen?\n\nIlmassa oli kirpeytt\u00e4, jollaista en ollut koskaan kokenut Helsingiss\u00e4 hein\u00e4kuussa.\n\nVedin keuhkoni t\u00e4yteen. Tuijotin keloutunutta m\u00e4nty\u00e4. Loppujen lopuksi kantani oli selv\u00e4. Meill\u00e4 ei ollut varaa olla ottamatta Danielin apua vastaan. Ja h\u00e4n oli oikeassa siin\u00e4, ettemme voineet voittaa daimoneita heid\u00e4n omassa peliss\u00e4\u00e4n. Ennen kuin voisimme kuitenkaan muuttaa peli\u00e4, meid\u00e4n t\u00e4ytyisi p\u00e4\u00e4st\u00e4 selville siit\u00e4, mik\u00e4 heid\u00e4n pelins\u00e4 oikeastaan edes oli. En viel\u00e4 tiennyt, mit\u00e4 tekisimme Danielin suhteen sitten, kun p\u00e4\u00e4sisimme perille. Olin edelleen sit\u00e4 mielt\u00e4, ett\u00e4 h\u00e4nen pelkk\u00e4 l\u00e4sn\u00e4olonsa vaarantaisi Mikaelin hengen \u2013 olettaen tietenkin, ett\u00e4 Mikael oli viel\u00e4 elossa.\n\nJ\u00e4\u00e4 valahti suoniini. Torjuin ajatuksen niin nopeasti kuin pystyin. Jos Mikael olisi kuollut, tiet\u00e4isin siit\u00e4. Tuntisin sen. En edes kysynyt itselt\u00e4ni, miten se muka olisi mahdollista. Jotkut asiat vain olivat.\n\nIhmettelin j\u00e4lleen kerran, mit\u00e4 daimonit halusivat. Jos kyse olisi ollut pelk\u00e4st\u00e4 takaisinmaksusta Mikaelille, heid\u00e4n ei olisi tarvinnut n\u00e4hd\u00e4 t\u00e4llaista vaivaa. Olimme viett\u00e4neet koko alkukes\u00e4n onnellisen piittaamattomina kaikista mahdollisista uhista. Kertaakaan emme olleet vilkuilleet olan yli. Kysymys siis kuului: miksi nyt? Ja miksi t\u00e4\u00e4ll\u00e4, susien kotikent\u00e4ll\u00e4? Susien, jotka ainoina saattoivat haastaa daimonit.\n\nSiit\u00e4 syyst\u00e4 he olivat tietenkin kielt\u00e4neet tuomasta susia mukanamme. Vaatimus, jota olimme p\u00e4\u00e4tt\u00e4neet rikkoa.\n\n***\n\n\"Min\u00e4 tunnen n\u00e4m\u00e4 mets\u00e4t yht\u00e4 hyvin kuin oman takapihani. Me ehdimme perille aikaisintaan iltap\u00e4iv\u00e4ll\u00e4\", Daniel sanoi. Olimme pys\u00e4htyneet jo toista kertaa sen j\u00e4lkeen, kun Daniel oli liittynyt joukkoomme. Mitja istui reppunsa p\u00e4\u00e4ll\u00e4, mutta min\u00e4 olin siihen liian rauhaton ja tyydyin vain nojailemaan puuhun.\n\n\"Ja sitten?\" kysyin. En halunnut my\u00f6nt\u00e4\u00e4, kuinka pahasti oma aika-arvioni oli ollut pieless\u00e4.\n\n\"Sitten me pelastamme Mikaelin ja painumme hiiteen t\u00e4\u00e4lt\u00e4.\"\n\nSe kuulosti yksinkertaiselta.\n\nKatsoin, kun Daniel tutki jalkapohjiaan. H\u00e4nen olisi helpompaa liikkua suden muodossa, ja ep\u00e4ilin h\u00e4nen pysyv\u00e4n ihmisen\u00e4 vain kiusatakseen minua. H\u00e4n ei vaikuttanut olevan lainkaan huolissaan Mikaelista, mik\u00e4 oli sek\u00e4 helpotus ett\u00e4 huolenaihe. Jos kyse olisi ollut keist\u00e4 tahansa muista kuin daimoneista, olisin pit\u00e4nyt h\u00e4nen huolettomuuttaan aiheellisena. Hitto, h\u00e4n oli Daniel Sarri. Voittajan m\u00e4\u00e4ritelm\u00e4. H\u00e4nest\u00e4 olisi pit\u00e4nyt tehd\u00e4 videopeli ja action-leluja. Koulun liikuntatunneilla olisin valinnut h\u00e4net joukkueeseeni ennen ket\u00e4\u00e4n muuta, lajista riippumatta. Mutta t\u00e4m\u00e4 ei ollut koululiikuntaa tai edes tappeluvideopeli, t\u00e4m\u00e4 oli totta ja vastassamme olivat daimonit, jumalista seuraavat maan p\u00e4\u00e4ll\u00e4. Pahoin pelk\u00e4sin, ettei Danielkaan olisi tarpeeksi.\n\nH\u00e4n v\u00e4itti tiet\u00e4v\u00e4ns\u00e4 kaiken, mutta ep\u00e4ilin, ettei h\u00e4n tiennyt mit\u00e4\u00e4n daimoneista. Auttaisin Mikaelia parhaiten tekem\u00e4ll\u00e4 Danielille selv\u00e4ksi, mihin h\u00e4n oli sekaantumassa. Siit\u00e4 huolimatta huomasin ep\u00e4r\u00f6iv\u00e4ni.\n\n\"Sinun pit\u00e4\u00e4 kertoa h\u00e4nelle, ett\u00e4 sudet voivat tappaa meid\u00e4t\", Mitja sanoi kreikaksi.\n\nHuokaisin. \"Tied\u00e4n... En vain haluaisi.\" En luottanut Danieliin. Vaikken uskonut, ett\u00e4 h\u00e4n oli aikeissa k\u00e4\u00e4nty\u00e4 meit\u00e4 vastaan, en halunnut antaa h\u00e4nelle aseita siihen.\n\n\"H\u00e4n ei voi auttaa meit\u00e4, jos ei tied\u00e4 tarpeeksi.\"\n\n\"\u00c4lk\u00e4\u00e4 antako minun h\u00e4irit\u00e4\", Daniel sanoi.\n\n\"Sinun on varmaan hyv\u00e4 tiet\u00e4\u00e4, ett\u00e4 daimonit ovat l\u00e4hes kuolemattomia\", aloitin.\n\n\"L\u00e4hes?\" Daniel kysyi.\n\n\"Daimonit voivat yleens\u00e4 parantaa itsens\u00e4, mutta on kuitenkin yksi poikkeus. Sudenpurema.\"\n\nMinun ei tarvinnut selitt\u00e4\u00e4 tarkemmin \u2013 h\u00e4nen katseensa paljasti, ett\u00e4 h\u00e4n oli ymm\u00e4rt\u00e4nyt, mit\u00e4 se tarkoitti.\n\n\"Kertokaa lis\u00e4\u00e4.\"\n\nMitja ehti ensin. \"Ensinn\u00e4kin on Callista. H\u00e4n on johtaja. Kukaan ei tied\u00e4, miten vanha h\u00e4n on, ja mihin kaikkeen h\u00e4n kykenee. Kaikki daimonit pelk\u00e4\u00e4v\u00e4t h\u00e4nt\u00e4, mik\u00e4 oikeastaan kertoo paljon siit\u00e4, miten vaarallinen h\u00e4n on.\"\n\n\"Tietenkin nainen.\"\n\nMitja sivuutti huomautuksen, viisas kun oli. \"Sitten on Vlassis. Callistan k\u00e4tyri, turvamies ja mit\u00e4 kaikkea. H\u00e4n panee toimeen kaiken, mit\u00e4 Callista vain keksii.\"\n\n\"H\u00e4n on saaren paras taistelija\", sanoin. \"Ja h\u00e4n vihaa meit\u00e4. Sill\u00e4 on jotain tekemist\u00e4 meid\u00e4n is\u00e4mme kanssa.\"\n\n\"Ei mik\u00e4\u00e4n yll\u00e4tys.\"\n\nOlisin voinut kysy\u00e4, mit\u00e4 h\u00e4n tarkalleen ottaen tarkoitti. En kysynyt. Mik\u00e4\u00e4n, mit\u00e4 h\u00e4n oli kertonut minulle is\u00e4st\u00e4mme t\u00e4h\u00e4n asti, ei ollut mairittelevaa. Totuus oli, etten halunnut n\u00e4hd\u00e4 is\u00e4\u00e4ni Danielin silmin. H\u00e4n ansaitsi enemm\u00e4n.\n\n\"Ent\u00e4 te?\" Daniel kysyi.\n\n\"Mit\u00e4 meist\u00e4?\"\n\n\"Mihin te sitten pystytte?\"\n\nMin\u00e4 ja Mitja vilkuilimme toisiamme. En halunnut paljastaa h\u00e4nelle kaikkia salaisuuksiamme. Niin paljon kuin vihasinkin daimoneita, olin oppinut heilt\u00e4 yhden asian: yliluonnolliset kyvyt olivat asia, josta kannatti uskoutua vain kaikkein l\u00e4heisimmille ihmisille. Olin p\u00e4\u00e4tynyt Hukkavaaraan vain, koska Daniel oli kuvitellut voivansa hy\u00f6ty\u00e4 kyvyist\u00e4ni. H\u00e4n ei ollut en\u00e4\u00e4 laumanjohtaja, mutta niin kuin viimeinen tunti oli osoittanut, h\u00e4n ei suinkaan ollut t\u00e4ysin poissa el\u00e4m\u00e4st\u00e4ni.\n\n\"Min\u00e4 olen hyv\u00e4 kuvien kanssa, ja Mitja taas \u00e4\u00e4nen\", sanoin lopulta. \"Mutta en usko, ett\u00e4 se on kovin t\u00e4rke\u00e4\u00e4. Meid\u00e4t molemmat on opetettu taistelemaan, ja seh\u00e4n kai on p\u00e4\u00e4asia.\"\n\n\"Kuvitteletko muka, ett\u00e4 tuo tieto riitt\u00e4\u00e4 minulle?\" Daniel kysyi.\n\n\"Ajattele se n\u00e4in: Min\u00e4 olen v\u00e4l\u00e4hdys, Mitja on jylin\u00e4. Yhdess\u00e4 me muodostamme aikamoisen ukkosmyrskyn.\"\n\nMitja ysk\u00e4isi, ja ep\u00e4ilin h\u00e4nen yritt\u00e4v\u00e4n pid\u00e4tt\u00e4\u00e4 naurua. P\u00e4\u00e4tin vaihtaa aihetta ennen kuin kukaan ehtisi sanoa mit\u00e4\u00e4n. \"Miksi hitossa sinulla edes on m\u00f6kki Lapissa keskell\u00e4 mets\u00e4\u00e4?\" kysyin. \"Ja miksei kukaan tunnu tiet\u00e4v\u00e4n siit\u00e4 mit\u00e4\u00e4n?\"\n\n\"Koska en halunnut kenenk\u00e4\u00e4n tiet\u00e4v\u00e4n siit\u00e4.\" H\u00e4n ei en\u00e4\u00e4 katsonut minuun, ja ensimm\u00e4ist\u00e4 kertaa koko keskustelumme aikana h\u00e4n kuulosti l\u00e4hes normaalilta ihmiselt\u00e4. Aivan kuin h\u00e4nell\u00e4 olisi jopa ollut tunteita. Pelkoja ja toiveita.\n\nMuistin \u00e4kki\u00e4, ett\u00e4 Mikael oli sanonut is\u00e4ns\u00e4 todellisuudessa pit\u00e4v\u00e4n minusta. Olisi ollut helpompaa, jos en olisi muistanut sit\u00e4.\n\nOlimme jo tuhlanneet aivan liikaa aikaa jaaritteluun. Nostin repun selk\u00e4\u00e4ni toivoen, ett\u00e4 toiset seuraisivat esimerkki\u00e4ni.\n\nMitja oli valmis parissa sekunnissa, ja lopulta Danielkin nousi jaloilleen. En kuitenkaan ehtinyt ottaa kuin kaksi askelta, kun tunsin v\u00e4reily\u00e4 ilmassa.\n\nPys\u00e4hdyin niille sijoilleni.\n\n\"Tunnetko sin\u00e4 t\u00e4m\u00e4n?\"\n\n\"Mink\u00e4?\" Mitja kysyi.\n\n\"Luulen... Luulen, ett\u00e4 h\u00e4n on t\u00e4\u00e4ll\u00e4.\"\n\n\"M\u00f6kille on viel\u00e4 matkaa\", Daniel murahti.\n\nKohotin k\u00e4teni. \"Odottakaa hetki\", sanoin.\n\nJa silloin kuulin h\u00e4net.\n\nRaissa.\n\n\u00c4\u00e4ni oli tuttu \u2013 liian tuttu. Aivan kuin sen kuulemisesta ei olisi p\u00e4iv\u00e4\u00e4k\u00e4\u00e4n.\n\nNielaisin. \"Joku on t\u00e4\u00e4ll\u00e4\", kuiskasin niin hiljaa kuin pystyin. Mik\u00e4 oli tietenkin naurettavaa \u2013 vasta hetke\u00e4 aikaisemmin olimme puhuneet normaalilla \u00e4\u00e4nell\u00e4.\n\nMitjan p\u00e4\u00e4 k\u00e4\u00e4ntyili puolelta toiselle. Ilmeisesti h\u00e4n ei kuitenkaan aistinut mit\u00e4\u00e4n, sill\u00e4 hetken p\u00e4\u00e4st\u00e4 h\u00e4n katsoi taas minua ja kysyi: \"Mit\u00e4 ihmett\u00e4 sin\u00e4 selit\u00e4t?\"\n\n\"Callista\", henk\u00e4isin. \"H\u00e4n on... minun p\u00e4\u00e4ni sis\u00e4ll\u00e4.\"\n\nJa \u00e4kki\u00e4 olin jossain toisaalla.\n\nK\u00e4velin k\u00e4yt\u00e4v\u00e4\u00e4 pitkin. Vaaleansininen linoleumilattia oli selv\u00e4sti vahattu \u00e4skett\u00e4in, sill\u00e4 se heijasti ik\u00e4v\u00e4sti katossa olevien halogeenilamppujen valoa. Keskikes\u00e4n aurinkoon verrattuna valo oli niin keinotekoista, etteiv\u00e4t silm\u00e4ni aluksi siet\u00e4neet sit\u00e4. Siristelin ja katsoin seini\u00e4. Kreikankielisess\u00e4 julisteessa mainostettiin ehk\u00e4isyv\u00e4lineit\u00e4.\n\nSairaala, ajattelin.\n\nEdess\u00e4ni k\u00e4veli mies, joka oli minua melkein p\u00e4\u00e4t\u00e4 pidempi. Tummanruskeat hiukset ja leve\u00e4t, nahkatakin verhoamat hartiat. H\u00e4nen katseensa seurasi hetken aikaa kahden ohi kulkevan valkoisiin pukeutuneen naisen mukana ja n\u00e4in vilahduksen h\u00e4nen kasvoistaan.\n\nHenk\u00e4isin \u00e4\u00e4neen. H\u00e4n n\u00e4ytti tismalleen veljelt\u00e4ni, vain v\u00e4h\u00e4n vanhemmalta. Se saattoi tarkoittaa vain yht\u00e4 asiaa: mies oli is\u00e4mme.\n\nYht\u00e4kki\u00e4 h\u00e4n avasi oven. Onnistuin vain juuri ja juuri pujahtamaan h\u00e4nen per\u00e4ss\u00e4\u00e4n sis\u00e4\u00e4n, ennen kuin h\u00e4n veti sen kiinni. Ovi t\u00f6m\u00e4hti takapuoleeni, mutta is\u00e4ni ei huomannut minua. H\u00e4nen katseensa oli kiinnittynyt valkotakkiseen naiseen, joka oli juuri k\u00e4\u00e4ntynyt h\u00e4mmentyneen\u00e4 ymp\u00e4ri.\n\n\"Anteeksi, mutta t\u00e4nne ei saa tulla viel\u00e4...\"\n\nIs\u00e4ni tarttui naiseen ja painoi k\u00e4tens\u00e4 t\u00e4m\u00e4n suun p\u00e4\u00e4lle. Ehdin n\u00e4hd\u00e4 v\u00e4l\u00e4hdyksen jostain kiilt\u00e4v\u00e4st\u00e4, ennen kuin h\u00e4n taivutti naisen p\u00e4\u00e4t\u00e4 taakse ja viilsi t\u00e4m\u00e4n kurkun auki.\n\nNaisen ruumis m\u00e4tk\u00e4hti lattialle.\n\n\"Raisa!\" Veljeni \u00e4\u00e4ni.\n\nOlin polvillani maassa.\n\nT\u00e4risin. K\u00e4teni haparoivat varpuja. Ne olivat todellisia. Min\u00e4 olin oikeasti t\u00e4\u00e4ll\u00e4, en siell\u00e4. Mit\u00e4\u00e4n pahaa ei ollut tapahtunut.\n\nEi ehk\u00e4 nyt eik\u00e4 t\u00e4\u00e4ll\u00e4, mutta tiesin, ett\u00e4 se, mit\u00e4 olin n\u00e4hnyt oli ollut totta. Is\u00e4ni oli murhannut l\u00e4\u00e4k\u00e4rin. Noin vain.\n\n\"Raisa, mit\u00e4 oikein tapahtui?\"\n\nJokin Mitjan ilmeess\u00e4 sai minut aavistamaan pahaa. Pyyhk\u00e4isin nen\u00e4\u00e4ni. Kun katsoin sormiani, n\u00e4in, ett\u00e4 ne olivat veress\u00e4.\n\n\"Jatketaan matkaa\", sanoin.\n\n***\n\nMinua pommitettiin kauheuksilla. Ne eiv\u00e4t olisi toimineet l\u00e4hesk\u00e4\u00e4n niin tehokkaasti ellen olisi tiennyt, ett\u00e4 ne kaikki olivat totta.\n\nIs\u00e4ni oli murhannut paljon ihmisi\u00e4. En tiennyt, keit\u00e4 he olivat ja mit\u00e4 he olivat tehneet, mutta loppujen lopuksi sill\u00e4 ei ollut v\u00e4li\u00e4. Kukaan ei ansainnut tulla tapetuksi.\n\nTulipalo oli pahin. Kuulin sis\u00e4ll\u00e4 olevien ihmisten huudon korvissani viel\u00e4 pitk\u00e4\u00e4n sen j\u00e4lkeen, kun kuva oli kadonnut.\n\nMiten ihmeess\u00e4 \u00e4itini oli koskaan voinut kuvitella rakastavansa h\u00e4nt\u00e4?\n\nAina v\u00e4lill\u00e4 tuli hetki\u00e4, jolloin en pystynyt jatkamaan k\u00e4vely\u00e4. Paniikki huuhtoi minut upoksiin. Koko kroppani j\u00e4hmettyi kylm\u00e4st\u00e4 ja hetken aikaa saatoin vain t\u00e4rist\u00e4 ja hikoilla. Sitten Mitja auttoi minut takaisin jaloilleni.\n\n\"Sinun t\u00e4ytyy pist\u00e4\u00e4 vastaan\", Mitja sanoi minulle. Halusin sanoa h\u00e4nelle, ettei h\u00e4nell\u00e4 ollut aavistustakaan, mist\u00e4 h\u00e4n puhui, mutta jomotus esti minua avaamasta suutani.\n\nKipu k\u00e4vi niin pahaksi, ett\u00e4 kyyneleet alkoivat taas valua poskiani pitkin ja minun oli painauduttava kaksinkerroin. \"Min\u00e4 en ole varma, pystynk\u00f6 jatkamaan\", tunnustin.\n\nTunsin, kun h\u00e4n yritti avata verhoa. Pys\u00e4ytin h\u00e4net. \"Pahoinvointi ei puolitu sill\u00e4, ett\u00e4 sin\u00e4kin olet sairas. Ei siin\u00e4 ole mit\u00e4\u00e4n j\u00e4rke\u00e4, ett\u00e4 me molemmat joutuisimme k\u00e4ym\u00e4\u00e4n t\u00e4m\u00e4n l\u00e4pi.\"\n\nJonkin ajan kuluttua aloin jo tottua n\u00e4kyihin. Olin n\u00e4hnyt is\u00e4ni murhaavan jo niin monta kertaa, ett\u00e4 olin turtunut. Tai ainakin kuvittelin niin ennen seuraavaa n\u00e4ky\u00e4.\n\nJoku itki.\n\nEn tunnistanut h\u00e4nt\u00e4 heti. \u00c4itini oli ollut nuori kuollessaan. Mutta edess\u00e4ni oleva tytt\u00f6... En ollut koskaan ajatellut, ett\u00e4 \u00e4itini oli joskus ollut niin nuori.\n\nH\u00e4nen kyyneleens\u00e4 olivat valuneet jo hyv\u00e4n aikaa, mutta h\u00e4n ei tehnyt elett\u00e4k\u00e4\u00e4n niiden pyyhkimiseksi.\n\n\"Min\u00e4 j\u00e4t\u00e4n sinut\", is\u00e4ni sanoi. H\u00e4nen katseensa oli t\u00e4ysin periksiantamaton. H\u00e4n seisoi ikkunan edess\u00e4 niin, ett\u00e4 h\u00e4nen varjonsa lankesi suoraan \u00e4itini p\u00e4\u00e4lle.\n\n\"Et tied\u00e4, mit\u00e4 sanot\", \u00e4itini sanoi.\n\n\"Luuletko todella, ett\u00e4 minusta on aviomieheksi? T\u00e4m\u00e4 oli alusta asti huono idea.\" H\u00e4n sylk\u00e4isi sanat suustaan.\n\n\"Eik\u00e4 ollut. Sin\u00e4 rakastat minua!\"\n\nStefanos k\u00e4\u00e4nsi katseensa pois. \"En ole siit\u00e4 en\u00e4\u00e4 niin varma.\"\n\n\"Ent\u00e4 lapset? Aiotko j\u00e4tt\u00e4\u00e4 heid\u00e4tkin?\"\n\n\"En. Min\u00e4 otan pojan.\"\n\n\"Vain kuolleen ruumiini yli!\"\n\n\"Saat valita: joko min\u00e4 otan heid\u00e4t molemmat, tai sitten vain Demetriosin.\"\n\n\"Ei!\" oli kaikki, mit\u00e4 \u00e4itini pystyi sanomaan.\n\nJuuri silloin kaksi lasta ilmestyi ovensuuhun. Tytt\u00f6 ja poika. Min\u00e4 ja Mitja.\n\n\"\u00c4iti\", kuulin itseni sanovan.\n\n\u00c4itini rynt\u00e4si meit\u00e4 kohti, mutta Stefanos oli nopeampi. H\u00e4n ty\u00f6nsi \u00e4itini tylysti pois tielt\u00e4 ja tarttui veljeeni.\n\nPieni Raisa alkoi itke\u00e4, ja \u00e4itini halasi h\u00e4nt\u00e4. Stefanoksen syliss\u00e4 oleva Mitja ei p\u00e4\u00e4st\u00e4nyt \u00e4\u00e4nt\u00e4k\u00e4\u00e4n, vaan pelk\u00e4st\u00e4\u00e4n tuijotti siskoaan suurilla silmill\u00e4\u00e4n.\n\n\"Mit\u00e4 sin\u00e4 haluat? Rahaa? Min\u00e4 annan sinulle rahaa\", is\u00e4ni sanoi.\n\n\u00c4itini pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Min\u00e4 en halua sinun rahojasi.\"\n\n\"Luulen, ett\u00e4 haluat. Sin\u00e4 olet ty\u00f6t\u00f6n ja kouluttamaton. Ajattele Raissan parasta.\"\n\nH\u00e4n lausui nimeni samalla tavalla kuin Callista. Ehk\u00e4 kreikkalaisversio olikin oikea, alkuper\u00e4inen nimeni.\n\nN\u00e4in, kuinka \u00e4itini taistelutahto mureni silmieni edess\u00e4. H\u00e4n oli polvillaan maassa ja puristi minua. \"\u00c4l\u00e4 tee t\u00e4t\u00e4, min\u00e4 rukoilen.\" H\u00e4nen sanoistaan sai tuskin selv\u00e4\u00e4.\n\nMutta vain min\u00e4 olin en\u00e4\u00e4 kuulemassa h\u00e4nen rukouksensa. Is\u00e4ni ja veljeni olivat l\u00e4hteneet.\n\nL\u00f6ysin itseni sel\u00e4lt\u00e4ni maasta. Aurinko oli tullut esiin pilvien takaa ja paistoi suoraan silmiini. Saatoin edelleen kuulla \u00e4itini itkun korvissani. Olisi voinut luulla, ett\u00e4 kaiken sen j\u00e4lkeen, mit\u00e4 olin n\u00e4hnyt, t\u00e4m\u00e4 ei olisi tuntunut en\u00e4\u00e4 miss\u00e4\u00e4n. Kaikista n\u00e4kemist\u00e4ni asioista perheeni hajoaminen oli nimitt\u00e4in pienin tragedia. Mutta en voinut olla surematta \u00e4itini ja meid\u00e4n kaikkien vuoksi. En tiennyt, miten h\u00e4n oli koskaan onnistunut nousemaan lattialta ja jatkamaan el\u00e4m\u00e4\u00e4ns\u00e4. En voinut kuvitella h\u00e4nen antaneen Stefanosille anteeksi. Itse en ainakaan pystyisi.\n\nPyyhin h\u00e4t\u00e4isesti kosteuden kasvoiltani, ettei kukaan n\u00e4kisi.\n\n\"Sin\u00e4 hidastat meit\u00e4\", Daniel sanoi. Yritin kohottaa itseni kyyn\u00e4rp\u00e4iden varaan. Kun siit\u00e4 ei ollut tulla mit\u00e4\u00e4n, veljeni riensi h\u00e4tiin. Onnistuttuani p\u00e4\u00e4sem\u00e4\u00e4n konttausasentoon vilkaisin Danielia. Jos olin kuvitellut h\u00e4nen h\u00e4tk\u00e4ht\u00e4v\u00e4n tilaani, sain petty\u00e4 pahasti. Se ei n\u00e4ytt\u00e4nyt kiinnostavan h\u00e4nt\u00e4 lainkaan, korkeintaan \u00e4rsytt\u00e4v\u00e4n.\n\nJ\u00e4lleen kerran en voinut kuin ihmetell\u00e4, miten Mikaelista oli tullut Mikael. H\u00e4n ei tiennyt sit\u00e4, mutta h\u00e4n oli itse oma suurin saavutuksensa.\n\n\"Meid\u00e4n pit\u00e4\u00e4 j\u00e4tt\u00e4\u00e4 sinut t\u00e4nne ja tulla hakemaan my\u00f6hemmin\", h\u00e4n murahti.\n\n\"Ei\", sek\u00e4 min\u00e4 ett\u00e4 Mitja sanoimme samaan aikaan.\n\n\"Kasaa sitten itsesi, tai saat j\u00e4\u00e4d\u00e4 t\u00e4h\u00e4n\", Daniel sanoi.\n\n\"Pikkuisen my\u00f6t\u00e4tuntoa, kiitos\", sihisin samalla kun yritin p\u00e4\u00e4st\u00e4 pystyyn. Sill\u00e4 hetkell\u00e4 evoluution ratkaisu nostaa ihminen kulkemaan kahdella jalalla tuntui \u00e4\u00e4rimm\u00e4isen typer\u00e4lt\u00e4.\n\n\"Nyt ei ole oikea hetki alkaa kalastella s\u00e4\u00e4lipisteit\u00e4.\"\n\nEhk\u00e4 Danielin tarkoitus oli olla niin raivostuttava, ett\u00e4 unohtaisin p\u00e4\u00e4ns\u00e4ryn ja kuvotuksen ja keskittyisin pelk\u00e4st\u00e4\u00e4n h\u00e4neen; siin\u00e4 tapauksessa h\u00e4n oli hyv\u00e4\u00e4 vauhtia saavuttamassa tavoitettaan.\n\nOnnistuin viimein seisomaan suorassa. \"T\u00e4m\u00e4 teht\u00e4v\u00e4 ei my\u00f6sk\u00e4\u00e4n helpotu sill\u00e4, ett\u00e4 sin\u00e4 pid\u00e4t koko juttua jonain vitsin\u00e4. Tiedoksesi: sinun poikasi on hengenvaarassa.\"\n\n\"Ai. Tilanne parani heti, kun sin\u00e4 vain sanoit sen \u00e4\u00e4neen.\"\n\n\"Sinun mielest\u00e4si tilanteen vakavuuden tunnustaminen antaa sille lis\u00e4\u00e4 valtaa. Mutta arvaa mit\u00e4? Asia on juuri p\u00e4invastoin. Miten ihmeess\u00e4 kell\u00e4\u00e4n voisi olla riitt\u00e4v\u00e4sti rohkeutta tehd\u00e4 ongelmalle jotain, jos ei ole edes rohkeutta tunnustaa koko ongelman olemassaoloa?\"\n\nH\u00e4n otti askeleen l\u00e4hemm\u00e4s. Sitten toisen. H\u00e4n oli niin l\u00e4hell\u00e4, ett\u00e4 saatoin tuntea h\u00e4nen hengityksens\u00e4 kasvoillani. Kaikki vaistoni sanoivat, ett\u00e4 minun kannattaisi per\u00e4\u00e4nty\u00e4, tai edes laskea katseeni. Mutta en tehnyt niin.\n\nSivusilm\u00e4ll\u00e4 n\u00e4in, ett\u00e4 my\u00f6s Mitja oli tullut l\u00e4hemm\u00e4s. H\u00e4n pelastaisi minut jos Daniel k\u00e4visi kimppuuni. Toivottavasti.\n\n\"Luuletko, ett\u00e4 min\u00e4 onnistuin olemaan vallassa satakahdeksankymment\u00e4 vuotta joutumatta kertaakaan n\u00e4in vaikeaan tilanteeseen? \u00c4l\u00e4 katso taakse \u00e4l\u00e4k\u00e4 sivuille. \u00c4l\u00e4k\u00e4 helvetti soikoon koskaan, koskaan paru.\"\n\nHetken aikaa kuulin vain meid\u00e4n molempien kiihtyneen hengityksen. Sitten aloin oksentaa, ja kakomisen \u00e4\u00e4ni t\u00e4ytti mets\u00e4n.\n\nDaniel kirosi.\n\n\"Miksi Callista tekee t\u00e4m\u00e4n sinulle?\" Mitja kysyi kreikaksi.\n\nKoska h\u00e4n oli sadisti, oli ensimm\u00e4inen reaktioni. Koska h\u00e4n halusi n\u00e4ytt\u00e4\u00e4, etten ollut mit\u00e4\u00e4n h\u00e4nen rinnallaan. Koska h\u00e4n halusi tappaa minut hitaasti ja tuskallisesti.\n\nOli kuitenkin vain yksi tapa p\u00e4\u00e4st\u00e4 selville Callistan suunnitelmasta. Ja se oli perille p\u00e4\u00e4seminen.\n\nVasta silloin tajusin ajatella, ett\u00e4 ehk\u00e4 peli oli jo alkanut, ja t\u00e4m\u00e4 oli sen ensimm\u00e4inen vaihe.\n\nSe auttoi. Mik\u00e4li t\u00e4m\u00e4 olisi vain ensimm\u00e4inen vaihe, se tarkoitti, ett\u00e4 se loppuisi jossain vaiheessa. Jos tarkoitus oli saada minut luovuttamaan, vaikutus oli siis p\u00e4invastainen.\n\nOnnistuimme k\u00e4velem\u00e4\u00e4n taas hetken, kunnes pahoinvointini otti uudelleen vallan ja minun oli pakko pys\u00e4hty\u00e4 oksentamaan. Mikon arvio puolen p\u00e4iv\u00e4n patikoinnista alkoi tuntua koko ajan huonommalta vitsilt\u00e4.\n\nDanielkaan ei en\u00e4\u00e4 keksinyt mit\u00e4\u00e4n nerokasta sanottavaa. Sen sijaan h\u00e4n kohotti nen\u00e4ns\u00e4 kohti tuulta.\n\n\"Me olemme l\u00e4hell\u00e4\", h\u00e4n sanoi. Luulin jo hetken, ett\u00e4 h\u00e4n sanoi sen lohduttaakseen minua. Samaan tapaan kuin vanhemmat kertoivat lapsille matkan olevan ihan pian ohi, vaikka todellisuudessa ei oltu viel\u00e4 saavutettu edes puoliv\u00e4li\u00e4. Ensimm\u00e4ist\u00e4 kertaa tunsin kiitollisuutta h\u00e4nt\u00e4 kohtaan.\n\nJa sitten t\u00e4ysin odottamatta me astuimme puiden seasta aukealle. Aukean keskell\u00e4 seisoi rakennus.\nYHDESTOISTA LUKU\n\nEn tiennyt, miksi rakennusta olisi pit\u00e4nyt kutsua. Se ei ollut talo. M\u00f6kkik\u00e4\u00e4n ei kuulostanut oikealta. Se oli l\u00e4hes yht\u00e4 pikkuruinen kuin leikkim\u00f6kki Jaskan talon pihalla, mutta siit\u00e4 huolimatta se n\u00e4ytti uhkaavalta, v\u00e4h\u00e4n kuin sill\u00e4 olisi ollut oma tahto. Hirret, joista se oli rakennettu, n\u00e4yttiv\u00e4t vanhemmilta kuin mets\u00e4 sen ymp\u00e4rill\u00e4. Katolla kasvoi sammalta.\n\n\"En haista mit\u00e4\u00e4n\", Daniel murahti. H\u00e4n n\u00e4ytti pettyneelt\u00e4, ja hetken ajattelin h\u00e4nen olevan niin sekaisin, ett\u00e4 h\u00e4n oli oikeasti toivonut p\u00e4\u00e4sev\u00e4ns\u00e4 taistelemaan daimoneita vastaan. Sitten tajusin h\u00e4nen huolensa olevan sama kuin minun: jos h\u00e4n ei haistanut mit\u00e4\u00e4n, se tarkoitti, ettei Mikael ollut t\u00e4\u00e4ll\u00e4. Ja jos h\u00e4n ei ollut t\u00e4\u00e4ll\u00e4, olimme tuhlanneet arvokasta aikaa.\n\nYritin olla ajattelematta \u00e4idin katsomia poliisisarjoja ja niiss\u00e4 toistunutta ajatusta, ett\u00e4 panttivanki- ja kidnappaustapauksissa kuolonuhrien todenn\u00e4k\u00f6isyys kasvoi sit\u00e4 suuremmaksi, mit\u00e4 kauemmin tilanne jatkuisi. T\u00e4m\u00e4 ei ollut mik\u00e4\u00e4n tavallinen sieppaustilanne. En tiennyt, mit\u00e4 t\u00e4m\u00e4 oli.\n\n\"Menn\u00e4\u00e4n sis\u00e4\u00e4n\", Daniel sanoi.\n\nSeurasimme h\u00e4nt\u00e4. Mitja joutui kumartumaan, jottei olisi ly\u00f6nyt p\u00e4\u00e4t\u00e4\u00e4n ovenkarmiin. Kun h\u00e4n astui tielt\u00e4ni pois, katsoin ymp\u00e4rilleni. Koko tuvan saattoi n\u00e4hd\u00e4 yhdell\u00e4 silm\u00e4yksell\u00e4. P\u00f6yt\u00e4 ja sen molemmilla puolilla penkit, uuni, sohvas\u00e4nky. Ei muuta. Kaikki muu Sarrien omistama oli niin virheet\u00f6nt\u00e4 ja tyylik\u00e4st\u00e4, etten pystynyt kuvittelemaan heit\u00e4 t\u00e4nne. Ainoa sana, jonka keksin, oli pelkistetty, mutta sekin tuntui viittaavaan johonkin harkittuun. Sit\u00e4 m\u00f6kki ei todellakaan ollut. En olisi yll\u00e4ttynyt, jos ketut ja myyr\u00e4t olisivat ottaneet sis\u00e4tilan haltuun.\n\nLopulta minun oli pakko ilmaista ihmetykseni \u00e4\u00e4neen. \"Tek\u00f6 oikeasti olette asuneet t\u00e4\u00e4ll\u00e4 jouluisin?\"\n\nDaniel hymyili vinosti. \"Kaupunkilaistytt\u00f6 ei ole ilmeisesti koskaan n\u00e4hnyt oikeaa hirsim\u00f6kki\u00e4.\"\n\n\"Olenpas\", sanoin. Vaikka tosiasiassa en ollut aivan varma.\n\n\"T\u00e4\u00e4ll\u00e4 me saamme olla rauhassa.\" H\u00e4n vilkaisi minua. \"Luulisi sinunkin jo v\u00e4hitellen osaavan arvostaa sit\u00e4.\"\n\nJatkoin tarkastustani. M\u00f6kiss\u00e4 ei n\u00e4ytt\u00e4nyt olevan juoksevaa vett\u00e4, eik\u00e4 my\u00f6sk\u00e4\u00e4n s\u00e4hk\u00f6\u00e4. Seuraavaksi p\u00e4\u00e4sinkin seuraamaan hyvin kummallista esityst\u00e4, jonka nimi oli \"Daniel Sarri sytytt\u00e4\u00e4 tulen uuniin\". H\u00e4n oli hyv\u00e4 siin\u00e4. Kaikista liikkeist\u00e4 n\u00e4ki, ett\u00e4 t\u00e4m\u00e4 oli h\u00e4nen m\u00f6kkins\u00e4.\n\nT\u00e4m\u00e4 oli Danielin salaisuus, tajusin. Puoli, jota h\u00e4n ei paljastanut kenellek\u00e4\u00e4n muulle kuin vaimolleen ja pojalleen.\n\nKukaan meist\u00e4 ei sanonut mit\u00e4\u00e4n, luultavasti siksi, ett\u00e4 meit\u00e4 kaikkia vaivasi sama kysymys, eik\u00e4 kell\u00e4\u00e4n meist\u00e4 ollut siihen hyv\u00e4\u00e4 vastausta. Kysymys kuului: mit\u00e4 seuraavaksi? Me olimme tulleet perille. Sen olisi pit\u00e4nyt riitt\u00e4\u00e4. Olimme ajatellut kaiken tasan t\u00e4h\u00e4n pisteeseen asti.\n\nDaniel oli rauhallisempi kuin olin koskaan n\u00e4hnyt h\u00e4nen olevan. \"Meid\u00e4n on paras sy\u00f6d\u00e4 ja lev\u00e4t\u00e4 nyt kun voimme. Huomenna on uusi p\u00e4iv\u00e4.\"\n\nM\u00f6kiss\u00e4 oli kaksi s\u00e4nky\u00e4. Koska minulla ei ollut aikomustakaan nukkua Danielin kanssa samassa s\u00e4ngyss\u00e4 ja toisaalta en my\u00f6sk\u00e4\u00e4n hennonut pakottaa Mitjaa moiseen, min\u00e4 ja veljeni p\u00e4\u00e4dyimme parivuoteeseen. Danielille j\u00e4i kapea sohvas\u00e4nky tuvan puolella. En voinut olla ajattelematta, ett\u00e4 tavallisesti Mikael nukkui siin\u00e4.\n\nMakasimme kyljill\u00e4mme kasvotusten yll\u00e4mme samat vaatteet, joissa olimme kolunneet mets\u00e4n l\u00e4pi. Puristava tunne rinnassani oli vain pahentunut viimeisten tuntien aikana, enk\u00e4 kuvitellut pystyv\u00e4ni nukkumaan en\u00e4\u00e4 koskaan.\n\nHapuilin kohti sanoja, joilla kertoa veljelleni, milt\u00e4 minusta tuntui. Mitja ei ollut koskaan kommentoinut minun ja Mikaelin suhdetta. En tiennyt, johtuiko se siit\u00e4, ettei h\u00e4n katsonut asian kuuluvan h\u00e4nelle, vai siit\u00e4, ettei h\u00e4nell\u00e4 ollut siit\u00e4 mielipidett\u00e4. Tiesin, ett\u00e4 h\u00e4n tulisi mielt\u00e4m\u00e4\u00e4n minut aina omana yksikk\u00f6n\u00e4ni, ja se oli helpottavaa. Samalla h\u00e4n oli niit\u00e4 harvoja, jotka tajusivat, mit\u00e4 olin valmis tekem\u00e4\u00e4n Mikaelin puolesta.\n\nEi ollut v\u00e4li\u00e4, saisinko pit\u00e4\u00e4 Mikaelin viikon vai viisi vuosisataa. P\u00e4\u00e4asia oli se, ett\u00e4 h\u00e4n hengitti, ja jatkoi hengitt\u00e4mist\u00e4 t\u00e4ss\u00e4 maailmassa.\n\n\"Me l\u00f6yd\u00e4mme h\u00e4net\", Mitja lupasi.\n\n***\n\nSeuraavana aamuna Daniel l\u00e4hti etsim\u00e4\u00e4n j\u00e4lki\u00e4 daimoneista. Odottaessamme h\u00e4nen paluutaan min\u00e4 ja Mitja kannoimme vett\u00e4 l\u00e4heisest\u00e4 l\u00e4hteest\u00e4. Keitimme sen varmuuden vuoksi. Olin jo illalla katsonut kaappeihin ja n\u00e4hnyt, ett\u00e4 ruokavarastomme oli varsin pieni. Mitja laittoi kuivat herneet kattilaan kiehumaan ja min\u00e4 avasin puukolla pari tonnikalat\u00f6lkki\u00e4. Ensin ihmettelin, miten Sarrit p\u00e4rj\u00e4siv\u00e4t t\u00e4\u00e4ll\u00e4, mutta sitten muistin, ett\u00e4 he olivat susia. Sudet eiv\u00e4t tarvinneet supermarkettia.\n\nKun Daniel tuli takaisin ja ilmoitti, ettei ollut l\u00f6yt\u00e4nyt mit\u00e4\u00e4n, en yll\u00e4ttynyt. H\u00e4\u00e4r\u00e4tess\u00e4mme vaatimattoman ateriamme parissa olin jo alkanut aavistella, ettei Mikael ollut miss\u00e4\u00e4n vaiheessa ollutkaan t\u00e4\u00e4ll\u00e4. Se tarkoitti, ett\u00e4 h\u00e4n oli jossain muualla. Ja joka sekunti jonka tuijotin kattilassa poukkoilevia herneit\u00e4, h\u00e4n oli vaarassa. Ehk\u00e4 h\u00e4nt\u00e4 kidutettiin. Ehk\u00e4 h\u00e4nen el\u00e4m\u00e4ns\u00e4 hiipui juuri t\u00e4ll\u00e4 nimenomaisella hetkell\u00e4.\n\nOli helpotus, kun Daniel keskeytti ajatukseni. \"Miten te saitte tiet\u00e4\u00e4 m\u00f6kist\u00e4?\" H\u00e4n oli istunut p\u00f6yd\u00e4n \u00e4\u00e4reen veljeni viereen.\n\nMieleni teki muistuttaa, ett\u00e4 omien sanojensa mukaan h\u00e4n tiesi kaiken, mutta \u00e4kki\u00e4 olin haluton v\u00e4ittelem\u00e4\u00e4n h\u00e4nen kanssaan. Minun oli tunnustettava, ett\u00e4 pidin t\u00e4st\u00e4 Danielista enemm\u00e4n kuin siit\u00e4 versiosta, joka oli minulle tutumpi.\n\n\"Mikko kertoi meille\", sanoin.\n\n\"Tietysti\", Daniel hym\u00e4hti.\n\nUteliaisuuteni her\u00e4si. \"Mist\u00e4 h\u00e4n sitten tiet\u00e4\u00e4?\"\n\n\"H\u00e4n rakensi m\u00f6kin.\"\n\nOlin ymm\u00e4ll\u00e4ni. \"Miksi se sitten kuuluu sinulle?\"\n\n\"H\u00e4n antoi sen minulle.\"\n\n\"H\u00e4n antoi sen sinulle\", toistin. Sitten: \"Miksi?\"\n\n\"Koska min\u00e4 kasvoin t\u00e4\u00e4ll\u00e4.\"\n\n\"Sin\u00e4 olet kotoisin Lapista?\" Olin luullut, ett\u00e4 Daniel oli el\u00e4nyt koko el\u00e4m\u00e4ns\u00e4 Hukkavaarassa.\n\n\"Niin. Asuin t\u00e4\u00e4ll\u00e4, kunnes is\u00e4ni ja min\u00e4 l\u00e4hdimme liikkeelle ja asetuimme lopulta Hukkavaaraan.\"\n\n\"Hetkinen. Perustiko sinun is\u00e4si lauman?\"\n\nDaniel katsoi minua tavalla, jonka oli varmaan tarkoitus haastaa minut yhdist\u00e4m\u00e4\u00e4n langanp\u00e4tk\u00e4t.\n\nJa viimein sainkin rusetin solmittua. \"Mikko on sinun is\u00e4si.\" Sitten: \"Mikko on Sarrajuoksa.\"\n\nHitto.\n\nMinun oli istuttava alas. Ensinn\u00e4kin: en pystynyt kuvittelemaan, ett\u00e4 Danielilla oli is\u00e4\u00e4. Olin aina pit\u00e4nyt h\u00e4nt\u00e4 v\u00e4h\u00e4n kuin jumalana. Pelottavana ja toisten tuskasta nauttivana jumalana, mutta jumalana yht\u00e4 kaikki. Se, ett\u00e4 h\u00e4nell\u00e4 oli is\u00e4, tarkoitti, ett\u00e4 h\u00e4n oli ollut joskus lapsi, siis heikko ja viaton.\n\nMieleeni nousi kuva tummatukkaisesta pikkupojasta, joka repii perhoselta siivet.\n\nMy\u00f6s k\u00e4sitykseni koko lauman historiasta oli juuri mennyt uusiksi. Olin lukenut Hukkavaaran synnyst\u00e4 kertovan Hukkapojan ja ajatellut kaiken tapahtuneen satoja vuosia sitten. Ja ehk\u00e4 se olikin. Olin vain olettanut Danielin i\u00e4n v\u00e4\u00e4rin.\n\n\"Luulin, ett\u00e4 Hukkavaarassa on ollut jo ties kuinka monta laumanjohtajaa\", sanoin. En uskonut harhaluuloni olevan sattumaa. Jostain syyst\u00e4 Daniel oli halunnut kaikkien uskovan niin.\n\nPenkki narahti Danielin alla, kun h\u00e4n vaihtoi asentoa. \"Luulit v\u00e4\u00e4rin.\"\n\nJotain juolahti mieleeni. \"Mikko on Mikaelin isois\u00e4.\" Pyh\u00e4 Sylvi! ajattelin mieless\u00e4ni. \u00c4\u00e4neen sanoin: \"H\u00e4n ei koskaan sanonut minulle mit\u00e4\u00e4n.\"\n\n\"Se ei johtunut sinusta. H\u00e4nell\u00e4 ei ole lupaa puhua siit\u00e4.\" Puolusteliko h\u00e4n poikaansa? En ollut varma.\n\n\"Eik\u00f6 kukaan siis tied\u00e4 Mikosta?\"\n\n\"Ei.\"\n\n\"Miten h\u00e4n voi olla elossa?\"\n\n\"H\u00e4n on sinnik\u00e4s k\u00e4pp\u00e4n\u00e4.\" H\u00e4n selvitti kurkkuaan ja k\u00e4\u00e4ntyi Mitjan puoleen. \"Sinulla on siis tytt\u00f6yst\u00e4v\u00e4\", h\u00e4n sanoi. En ollut yht\u00e4\u00e4n varma, mist\u00e4 t\u00e4ll\u00e4 kertaa tuuli, ennen kuin tajusin, ett\u00e4 Daniel oli tahallaan vaihtanut puheenaihetta.\n\n\"Niin. Caoimhe.\"\n\n\"H\u00e4n ei siis ole suomalainen.\"\n\n\"H\u00e4n on irlantilainen.\"\n\n\"Keiju\", lis\u00e4sin ennen kuin ehdin purra kielt\u00e4ni. Caoimhe ei varmaankaan pannut pahakseen sit\u00e4, ett\u00e4 kerroin h\u00e4nest\u00e4 Danielille, mutta vainoharhaisuudessani pelk\u00e4sin Danielin voivan jotenkin k\u00e4ytt\u00e4\u00e4 tietoa hyv\u00e4kseen.\n\n\"Kappas. Olen kuullut, ett\u00e4 he ovat kauniita.\" Aivan kuin se olisi ollut p\u00e4\u00e4asia. H\u00e4nen ilmeest\u00e4\u00e4n p\u00e4\u00e4tellen veljeni oli juuri noussut arvoasteikossa yl\u00f6sp\u00e4in.\n\nMitja ei sanonut mit\u00e4\u00e4n. Syntyi kiusaantunut hiljaisuus. En voinut kuin ihmetell\u00e4, miksi Daniel oli \u00e4kki\u00e4 p\u00e4\u00e4tt\u00e4nyt yritt\u00e4\u00e4 olla mukava. Se istui h\u00e4nelle yht\u00e4 huonosti kuin Tarjoustalosta ostettu puvuntakki.\n\n\"Puhutteko te kesken\u00e4nne englantia?\"\n\n\"Tai iiri\u00e4.\"\n\n\"Ja sitten puhut viel\u00e4 kreikkaakin. Silti et onnistunut k\u00e4ym\u00e4\u00e4n peruskoulua loppuun.\"\n\n\"Hei!\" protestoin. \"Kenenk\u00f6h\u00e4n vika on, ettei Mitjalla ole mennyt kaikki ihan putkeen?\" En pystynyt hillitsem\u00e4\u00e4n itse\u00e4ni, vaan jatkoin: \"Ja mist\u00e4 l\u00e4htien sin\u00e4 olet muka arvostanut koulutusta? Mikael uskalsi alkaa haaveilla l\u00e4\u00e4kiksest\u00e4 vasta sen j\u00e4lkeen, kun sin\u00e4 h\u00e4ivyit hiillostamasta h\u00e4nt\u00e4. Tiesitk\u00f6 edes, ett\u00e4 h\u00e4n on niin fiksu?\"\n\n\"Tiesin\", h\u00e4n vastasi.\n\n\"Mikset sitten koskaan kannustanut h\u00e4nt\u00e4 tekem\u00e4\u00e4n sill\u00e4 mit\u00e4\u00e4n?\" kysyin.\n\n\"Lauma tulee ensin. Mikael on ottanut turhia riskej\u00e4. H\u00e4nen asemansa on t\u00e4ll\u00e4 hetkell\u00e4 todella hutera. H\u00e4nen t\u00e4ytyy pysy\u00e4 kyl\u00e4ss\u00e4 ja k\u00e4yd\u00e4 koulut vasta sitten, kun asiat ovat paremmalla tolalla.\"\n\n\"Arvaa mit\u00e4? Voit ty\u00f6nt\u00e4\u00e4 mielipiteesi takaisin sinne, mist\u00e4 ne ovat per\u00e4isin.\"\n\nVeljeni p\u00e4\u00e4tti siin\u00e4 kohtaa nousta ja menn\u00e4 tutkimaan herneit\u00e4. Tai sitten h\u00e4n meni ulos \u2013 huomioni pysyi nimitt\u00e4in Danielissa. \"Min\u00e4 puolustan Mikaelia niin kuin parhaiten pystyn, my\u00f6s silloin, kun h\u00e4nen oma is\u00e4ns\u00e4 yritt\u00e4\u00e4 hallita h\u00e4nen el\u00e4m\u00e4\u00e4ns\u00e4. \u00c4l\u00e4 ep\u00e4ile sit\u00e4 hetke\u00e4k\u00e4\u00e4n.\"\n\nDaniel myh\u00e4ili. Vasta silloin muistin, ett\u00e4 ilman h\u00e4nt\u00e4 min\u00e4 ja Mikael emme olisi koskaan p\u00e4\u00e4tyneet yhteen. Se oli ollut h\u00e4nen suunnitelmansa alusta asti.\n\n\"Sen kun sanot \u00e4\u00e4neen\", sanoin. \"Sin\u00e4 sait kaiken, mink\u00e4 halusitkin.\"\n\n\"Ei, Raisa. Sin\u00e4 sait kaiken, mink\u00e4 halusit.\"\n\nJa tietenkin h\u00e4n oli oikeassa.\n\nKiukkuni liukeni sit\u00e4 mukaa kun ajatukseni siirtyiv\u00e4t Danielista Mikaeliin ja minuun. Kuukausi oli lyhyt aika. Silti siin\u00e4 ajassa ehti oppia toisen tavat ja mukautua niihin. Ja enemm\u00e4n kuin mukautua: siin\u00e4 ajassa ehti jo tulla hivenen toisen kaltaiseksi. Nykyisin aina, kun vanhus nousi ratikkaan, annoin h\u00e4nelle paikkani ilman tietoista p\u00e4\u00e4t\u00f6st\u00e4. Vain, koska Mikaelkin teki niin.\n\nKuukaudessa ehti luoda yhteisen kielen. Sanoja, jotka tarkoittivat jotain vain meid\u00e4n kesken. Kuukaudessa ehti oppia ne kohdat, joista kutittaa ja ne, joiden suuteleminen johti automaattisesti toisiin suudelmiin.\n\nKuukaudessa ehti rakentaa mieless\u00e4\u00e4n koko yhteisen tulevaisuuden. Kokonaisen maailman, jonka perusyksikk\u00f6 ei ollut min\u00e4 vaan me.\n\n\"Ei kannata odottaa h\u00e4\u00e4kutsua ihan heti.\" En ollut tarkoittanut paljastaa h\u00e4nelle mit\u00e4\u00e4n minun ja Mikaelin tilanteesta. Kun kuulin oman \u00e4\u00e4neni, tajusin kuitenkin heti, ett\u00e4 h\u00e4n tiet\u00e4isi jonkin olevan vialla. En kuulostanut l\u00e4hesk\u00e4\u00e4n niin \u00e4rtyneelt\u00e4 kuin olin halunnut. Pikemminkin surulliselta.\n\n\"Ryppyj\u00e4 rakkaudessa jo n\u00e4in pian?\" Daniel kysyi.\n\n\"Niinkin voisi sanoa\", mutisin. Ja sitten yll\u00e4tin itseni kertomalla h\u00e4nelle koko tarinan. \"Kun sain h\u00e4net takaisin, vannoin, etten antaisi mink\u00e4\u00e4n vied\u00e4 h\u00e4nt\u00e4 minulta pois. Mutta ket\u00e4\u00e4n ei voi pit\u00e4\u00e4 v\u00e4kisin.\" Laskin p\u00e4\u00e4ni k\u00e4siini.\n\n\"Rukoileminen ei auta\", Daniel sanoi.\n\n\"En min\u00e4 rukoillut\", sanoin. Danielin katse paljasti, ettei h\u00e4n uskonut minua.\n\nYritin keksi\u00e4 jotain, mill\u00e4 \u00e4rsytt\u00e4\u00e4 h\u00e4nt\u00e4. \"Miss\u00e4 Kristina on?\"\n\n\"Viimeksi kun puhuin h\u00e4nen kanssaan, h\u00e4n oli Ven\u00e4j\u00e4ll\u00e4.\"\n\n\"Oletteko te... eronneet?\"\n\n\"Sudet eiv\u00e4t eroa.\"\n\nMinun mielest\u00e4ni se, ettei tiennyt, miss\u00e4 vaimo oli ja milloin n\u00e4kisi h\u00e4net seuraavan kerran kuulosti erolta. Mutta ilmeisesti olin v\u00e4\u00e4r\u00e4ss\u00e4.\n\n\"Kaikki rakkaus ei ole samanlaista\", Daniel sanoi. \"Ei sellaista kuin sinulla ja Mikaelilla juuri nyt. Sin\u00e4 olet viel\u00e4 niin nuori, ettet tied\u00e4.\"\n\nMin\u00e4 ja Daniel Sarri keskustelimme rakkaudesta? Voi luoja.\n\nDanielinkin t\u00e4ytyi tajuta sama asia, sill\u00e4 h\u00e4n nousi penkilt\u00e4. \"Min\u00e4 vaihdan muotoa.\"\n\n***\n\nAjan kuluminen ei ollut koskaan ollut niin kivuliasta. Vaikka olimme olleet Sarrajuoksan m\u00f6kiss\u00e4 vasta kaksi p\u00e4iv\u00e4\u00e4, tuntui silt\u00e4 kuin olisimme asuneet siell\u00e4 ikuisesti. N\u00e4ill\u00e4 leveysasteilla aurinko pysytteli itsepintaisesti taivaalla koko ajan, eik\u00e4 milloinkaan tiennyt, mit\u00e4 kello oli. Muu maailma tuntui olevan jossain hyvin kaukana \u2013 liian kaukana. Halusimme l\u00e4hte\u00e4, mutta emme uskaltaneet. Daimonit olivat halunneet meid\u00e4n tulevan t\u00e4nne ja siihen t\u00e4ytyi olla syy. Emme voineet antaa itsemme rentoutua. Toistelin itselleni, ett\u00e4 se oli osa heid\u00e4n peli\u00e4\u00e4n \u2013 peli\u00e4, jonka oli tarkoitus saada meid\u00e4t s\u00e4ikkym\u00e4\u00e4n.\n\nJa totuus oli, ett\u00e4 min\u00e4 s\u00e4ikyin.\n\n\"Ent\u00e4 jos he ovat t\u00e4ll\u00e4 v\u00e4lin hy\u00f6k\u00e4nneet Hukkavaaraan?\" kysyin ties kuinka monetta kertaa. Puhelimeni akku oli loppunut illalla. Puoli piinallisen pitk\u00e4\u00e4 p\u00e4iv\u00e4\u00e4 ilman mink\u00e4\u00e4nlaista viesti\u00e4 Hukkavaarasta. Viimeisimm\u00e4ss\u00e4 viestiss\u00e4 Jenni oli kertonut Nikon ja Tanelin palanneen Helsinkiin. Caoimhe oli j\u00e4\u00e4nyt Hukkavaaraan odottamaan Mitjan paluuta. Juha ja Jenni yrittiv\u00e4t parhaansa mukaan viihdytt\u00e4\u00e4 h\u00e4nt\u00e4, mutta rivien v\u00e4list\u00e4 oli voinut lukea, ett\u00e4 h\u00e4n oli tolaltaan Mitjan l\u00e4hd\u00f6st\u00e4.\n\n\"He ovat susia\", Daniel vastasi. \"Sin\u00e4 kerroit heille kaiken tarvittavan, jotta he osaavat puolustautua.\"\n\n\"T\u00e4m\u00e4 tuntuu niin... v\u00e4\u00e4r\u00e4lt\u00e4. Olla tekem\u00e4tt\u00e4 mit\u00e4\u00e4n.\"\n\n\"Viel\u00e4 yksi p\u00e4iv\u00e4. Sitten me l\u00e4hdemme.\"\n\nKumpikaan meist\u00e4 ei sanonut \u00e4\u00e4neen sit\u00e4, mit\u00e4 ajattelimme: meill\u00e4 ei ollut aavistustakaan, minne meid\u00e4n kannatti menn\u00e4.\n\nKun sin\u00e4 iltana menin nukkumaan, nukahdin l\u00e4hes heti. Unessani seisoin rantabaarissa. Kaikki n\u00e4ytti hieman oudolta, vaikka en aivan heti tajunnutkaan, miksi. Ihmiset olivat pukeutuneet hassusti. Aivan kuin olisin ollut keskell\u00e4 naamiaisia, paitsi ett\u00e4 kaikki tuntuivat pit\u00e4v\u00e4n asujaan aivan tavallisina. Hiusmallit ja silm\u00e4lasit ja... ai niin. Yhdeks\u00e4nkymment\u00e4luku.\n\nKatselin ymp\u00e4rilleni. Teksteist\u00e4 p\u00e4\u00e4tellen olin Kreikassa. Tarkemmin sanottuna jollakin sen saarista. En ollut n\u00e4hnyt paikkaa aikaisemmin, mutta ilmanala ja edess\u00e4 levitt\u00e4ytyv\u00e4 meri olivat riitt\u00e4vi\u00e4 vihjeit\u00e4.\n\nIstuin yhteen terassin vapaista p\u00f6ydist\u00e4. Yritin n\u00e4ytt\u00e4\u00e4 mahdollisimman huomaamattomalta samalla kun tarkkailin ohi kulkevia ihmisi\u00e4. Katseeni osui viereiseen p\u00f6yt\u00e4\u00e4n, jonka \u00e4\u00e4ress\u00e4 istui kaksi poikaa.\n\nH\u00e4tk\u00e4hdin. Vlassis. Paitsi ett\u00e4 h\u00e4n oli paljon nuorempi. Ja h\u00e4nen vieress\u00e4\u00e4n istui...\n\n\"Stefanos\", Vlassis sanoi.\n\nH\u00e4n n\u00e4ytti nuoremmalta kuin n\u00e4yiss\u00e4ni, silotellummalta ja uhmakkaammalta. Ennen kaikkea h\u00e4n n\u00e4ytti kyll\u00e4styneelt\u00e4. H\u00e4n tuskin edes vilkaisi Vlassisia, vaikka t\u00e4m\u00e4 oli selv\u00e4sti hyvin innokas saamaan h\u00e4nen huomionsa.\n\n\"Ent\u00e4s tuo?\"\n\nVlassis osoitti kohti rantaviivaa. Aivan vesirajassa seisoi tytt\u00f6 yll\u00e4\u00e4n mit\u00e4 kamalimmat r\u00f6yhel\u00f6bikinit. Joukosta h\u00e4n erottui kuitenkin siksi, ett\u00e4 oli niin vaalea \u2013 h\u00e4n ei ollut voinut olla t\u00e4\u00e4ll\u00e4 muutamaa p\u00e4iv\u00e4\u00e4 kauemmin.\n\n\"Liian viaton\", is\u00e4ni sanoi lopulta. Mutta h\u00e4n ei katsonut pois.\n\n\"Kyll\u00e4h\u00e4n sen tiet\u00e4\u00e4, miksi n\u00e4m\u00e4 tyt\u00f6t ovat tulleet t\u00e4nne.\"\n\n\"Samasta syyst\u00e4 kuin mekin olemme tulleet t\u00e4nne\", Stefanos vastasi.\n\nVlassis naurahti. \"Kukaan muu kuin sin\u00e4 ei olisi voinut suostutella Callistaa kahteen per\u00e4kk\u00e4iseen bileviikonloppuun.\"\n\nTajusin h\u00e4m\u00e4r\u00e4sti, ett\u00e4 he puhuivat nykykreikkaa, eik\u00e4 sit\u00e4, mit\u00e4 puhuttiin saarella ja jota edelleenkin puhuin Mitjan kanssa.\n\nKun is\u00e4ni ei vastannut Vlassisille, h\u00e4n kysyi: \"Mist\u00e4 luulet h\u00e4nen olevan?\"\n\n\"Kuka tiet\u00e4\u00e4.\"\n\n\"Min\u00e4 veikkaan, ett\u00e4 h\u00e4n on ruotsalainen.\" Kun is\u00e4ni ei sanonut viel\u00e4k\u00e4\u00e4n mit\u00e4\u00e4n, Vlassis kysyi: \"Pit\u00e4isik\u00f6 minun menn\u00e4 yritt\u00e4m\u00e4\u00e4n?\"\n\nStefanosin p\u00e4\u00e4 k\u00e4\u00e4ntyi, ja ensimm\u00e4ist\u00e4 kertaa h\u00e4n katsoi Vlassisia. \"Ei\", oli kaikki, mit\u00e4 h\u00e4n sanoi.\n\nKatsoin tytt\u00f6\u00e4 tarkemmin. Pitk\u00e4t, vaaleat hiukset valuivat h\u00e4nen rinnalleen. Kun h\u00e4n sipaisi ne korvansa taakse, erotin ter\u00e4v\u00e4n leuan ja py\u00f6re\u00e4t poskip\u00e4\u00e4t.\n\n\u00c4itini.\n\nTuijotin h\u00e4nt\u00e4 niin pitk\u00e4\u00e4n, etten ollut huomannut is\u00e4ni nousseen yl\u00f6s.\n\nVlassis seurasi h\u00e4nen esimerkki\u00e4\u00e4n, ja he l\u00e4htiv\u00e4t k\u00e4velem\u00e4\u00e4n. Hetken ajan vain istuin paikoillani ennen kuin tajusin, ett\u00e4 mik\u00e4li halusin seurata heit\u00e4, minun olisi noustava yl\u00f6s ja oikeasti ment\u00e4v\u00e4 heid\u00e4n per\u00e4\u00e4ns\u00e4.\n\nTavoitin heid\u00e4t juuri, kun he olivat astuneet kivetylt\u00e4 alueelta hiekalle.\n\nSamaan aikaan \u00e4itini oli alkanut k\u00e4vell\u00e4 poisp\u00e4in merest\u00e4, mutta joukko poikia pys\u00e4ytti h\u00e4net.\n\n\"Anteeksi, en puhu kreikkaa\", \u00e4itini sanoi heille englanniksi.\n\nMutta min\u00e4 puhuin. Valitettavasti.\n\n\"N\u00e4tti pimu.\"\n\n\"Hyv\u00e4 runko.\"\n\nIs\u00e4ni ja Vlassis olivat pys\u00e4htyneet. He seurasivat k\u00e4ynniss\u00e4 olevaa v\u00e4likohtausta, Vlassis virnistellen, is\u00e4ni vakavana.\n\n\"J\u00e4tt\u00e4k\u00e4\u00e4 h\u00e4net rauhaan\", is\u00e4ni sanoi lopulta. H\u00e4n ei korottanut \u00e4\u00e4nt\u00e4\u00e4n, mik\u00e4 oli selv\u00e4 merkki siit\u00e4, ett\u00e4 h\u00e4n tiesi pystyv\u00e4ns\u00e4 saattamaan heid\u00e4t kaikki liikuntakyvytt\u00f6miksi hyvin nopeassa aikataulussa. Ilmeisesti olin kuitenkin ainoa, joka tajusi t\u00e4m\u00e4n, sill\u00e4 kukaan pojista ei liikahtanutkaan.\n\n\"Onko h\u00e4n sinun tytt\u00f6yst\u00e4v\u00e4si?\" yksi heist\u00e4 kysyi.\n\n\"Ei. Mutta teid\u00e4n sijassanne l\u00e4htisin nyt.\"\n\nYksi heist\u00e4 nauroi, mutta kun h\u00e4n vilkaisi is\u00e4ni vakavaa ilmett\u00e4, h\u00e4nen naurunsa sammui. H\u00e4n l\u00e4im\u00e4isi kaveriaan selk\u00e4\u00e4n kuin viestitt\u00e4en, ett\u00e4 leikki oli nyt ohi. He l\u00e4htiv\u00e4t. Is\u00e4ni ei vilkaissutkaan \u00e4iti\u00e4ni ennen kuin jo jatkoi k\u00e4vely\u00e4 kohti rantaa my\u00f6t\u00e4ilev\u00e4\u00e4 katua. Vlassis seurasi per\u00e4ss\u00e4 vilkuillen aina v\u00e4lill\u00e4 taakseen.\n\nStefanos kaivoi taskuaan ja hetken kuluttua h\u00e4n sytytti tupakan.\n\n\"Heit\u00e4 yksi\", Vlassis sanoi.\n\n\"Enk\u00e4 heit\u00e4. K\u00e4y ostamassa omat.\" H\u00e4n osoitti kohti pient\u00e4 kauppaa, jonka ulkopuolelle oli aseteltu niin vesimeloneja kuin vesileluja.\n\nVlassis taisi ymm\u00e4rt\u00e4\u00e4 olla riitelem\u00e4tt\u00e4 is\u00e4ni kanssa, sill\u00e4 h\u00e4n livahti sis\u00e4\u00e4n kauppaan hartiat kyyryss\u00e4.\n\n\"Hei!\"\n\nMin\u00e4 ja is\u00e4ni k\u00e4\u00e4nnyimme yht\u00e4 aikaa. \u00c4iti juoksi meit\u00e4 kohti. H\u00e4n liukastui, mutta onnistui kuin onnistuikin pit\u00e4m\u00e4\u00e4n tasapainonsa.\n\n\"Hei\", h\u00e4n sanoi heng\u00e4styneen\u00e4. \"Kiitos sinulle. En tied\u00e4, mit\u00e4 sin\u00e4 sanoit niille pojille, mutta luulen, ett\u00e4 sin\u00e4 puolustit minua.\"\n\n\u00c4itini englanti oli h\u00e4vett\u00e4v\u00e4n huonoa, eik\u00e4 is\u00e4ni vastannut mit\u00e4\u00e4n. En tiennyt, oliko h\u00e4n edes ymm\u00e4rt\u00e4nyt sanoja. \u00c4itini kuitenkin yll\u00e4tti meid\u00e4t molemmat ojentamalla k\u00e4tens\u00e4 ja sanomalla: \"Tuire.\"\n\nHetken aikaa Stefanos vain tuijotti k\u00e4tt\u00e4 kuin yritt\u00e4en keksi\u00e4, mit\u00e4 sill\u00e4 kuului tehd\u00e4. Lopulta h\u00e4n kuitenkin tarttui siihen. \"Stefanos.\"\n\n\"Voinko tarjota sinulle oluen?\" \u00e4itini kysyi. En muista koskaan n\u00e4hneeni \u00e4itini kasvoilla samanlaista hymy\u00e4.\n\nStefanos nauroi. \u00c4itini n\u00e4ytti loukkaantuneelta. \"Mik\u00e4 naurattaa?\"\n\n\"Sin\u00e4 haluat tarjoa minulle oluen?\" H\u00e4n kysyi hymyillen. \"Taitaa olla ensimm\u00e4inen kerta, kun tytt\u00f6 k\u00e4ytt\u00e4\u00e4 tuota repliikki\u00e4 minuun eik\u00e4 toisin p\u00e4in.\" Stefanos puhui englantia paremmin kuin \u00e4itini, enk\u00e4 voinut olla ajattelematta, ett\u00e4 Mitja oli perinyt kielip\u00e4\u00e4ns\u00e4 h\u00e4nelt\u00e4.\n\n\u00c4itini suu asettui minulle niin tutuksi viivaksi.\n\n\"Luota minuun\", Stefanos sanoi. \"Et varmasti halua olla miss\u00e4\u00e4n tekemisiss\u00e4 minun kaltaiseni kaverin kanssa.\"\n\n\u00c4iti risti k\u00e4sivartensa. \"Millainen kaveri sin\u00e4 sitten olet?\"\n\nStefanos naurahti ja pudisti p\u00e4\u00e4t\u00e4\u00e4n.\n\n\"Raiskaaja?\"\n\nIs\u00e4ni vakavoitui. \"En.\"\n\n\"Naistenhakkaaja?\"\n\n\"Pahoittelen.\"\n\n\"Kirvesmurhaaja?\"\n\n\"En aivan sek\u00e4\u00e4n.\"\n\n\"Sitten minulla ei pit\u00e4isi olla mit\u00e4\u00e4n h\u00e4t\u00e4\u00e4. Sit\u00e4 paitsi, olen aika varma ett\u00e4 pystyn juomaan enemm\u00e4n kuin sin\u00e4.\"\n\nStefanos nauroi taas. H\u00e4n ei vaikuttanut tyypilt\u00e4 joka nauroi kovin usein, ja silti h\u00e4n oli nauranut jo kahdesti t\u00e4m\u00e4n keskustelun aikana.\n\n\"Ei se ollut mik\u00e4\u00e4n vitsi. Min\u00e4 olen Suomesta. Ja sin\u00e4 olet... No...\"\n\n\"Mik\u00e4?\"\n\n\u00c4itini ei sanonut mit\u00e4\u00e4n, katsoi vain merkitsev\u00e4\u00e4n s\u00e4vyyn. Se oli suorastaan nerokasta: kukaan ei kehdannut per\u00e4\u00e4nty\u00e4 haasteesta silloin, kun oli ensin tullut loukatuksi lempe\u00e4sti.\n\n\"Me molemmat tulemme katumaan t\u00e4t\u00e4. Paljon.\"\n\n\"Ehk\u00e4 huomenna, muttei t\u00e4n\u00e4\u00e4n\", \u00e4itini sanoi.\n\nStefanos n\u00e4ytti harkitsevan asiaa. En usko, ett\u00e4 \u00e4itini tajusi, millainen saavutus se jo oli. \"Miss\u00e4 sinun yst\u00e4v\u00e4si ovat?\" h\u00e4n kysyi lopulta.\n\n\"Minulla ei ole yst\u00e4vi\u00e4.\"\n\n\"Sin\u00e4 tulit t\u00e4nne yksin?\" Stefanos oli h\u00e4mmentynyt, ellei jopa huolissaan.\n\n\u00c4itini ny\u00f6kk\u00e4si.\n\n\"Tuo on hullua. Sinun kaltaisesi tytt\u00f6...\"\n\n\"Mit\u00e4?\"\n\n\"Taisin olla v\u00e4\u00e4r\u00e4ss\u00e4\", h\u00e4n mutisi. \"Minun kaltaiseni tyyppi on sittenkin juuri oikea valinta. Menn\u00e4\u00e4n.\"\n\n\"Ent\u00e4 sinun yst\u00e4v\u00e4si? H\u00e4n voi tietenkin tulla mukaan, jos...\"\n\nStefanos keskeytti h\u00e4net. \"Ei, me emme todellakaan pyyd\u00e4 h\u00e4nt\u00e4 mukaan.\"\n\n\"No, haluat varmaan k\u00e4yd\u00e4 kertomassa h\u00e4nelle, ett\u00e4 olet l\u00e4hd\u00f6ss\u00e4.\"\n\nStefanos vilkaisi kaupan oven suuntaan. Vlassis jutteli parhaillaan kassalla joidenkin tytt\u00f6jen kanssa. Ilmeist\u00e4 p\u00e4\u00e4tellen h\u00e4n oli hyv\u00e4\u00e4 vauhtia tekem\u00e4ss\u00e4 itsest\u00e4\u00e4n idioottia.\n\n\"Ei tarvitse\", is\u00e4ni sanoi.\n\n***\n\nHer\u00e4sin siihen, ett\u00e4 joku kuiskasi korvaani. Tai ainakin olin kuvitellut niin \u2013 kun nousin istumaan, Mitja vasta avasi silm\u00e4ns\u00e4, eik\u00e4 huoneessa ollut ket\u00e4\u00e4n muuta.\n\nTiesin heti, ett\u00e4 jotain oli tapahtunut. Me molemmat hypp\u00e4simme yl\u00f6s ja s\u00e4nt\u00e4simme tupaan.\n\nDaniel oli jo t\u00e4ysin hereill\u00e4. H\u00e4n seisoi ikkunan \u00e4\u00e4rell\u00e4 ja tuijotti ulos. \"Jotain on meneill\u00e4\u00e4n\", h\u00e4n sanoi.\n\nSe, ettei h\u00e4n osannut nimet\u00e4 uhkaa, ei ollut hyv\u00e4 merkki. Daniel ei yleens\u00e4 ilmaissut itse\u00e4\u00e4n ep\u00e4m\u00e4\u00e4r\u00e4isesti.\n\nEnnen kuin kukaan ehti est\u00e4\u00e4, tempaisin ulko-oven auki.\n\nAamu-usva oli piiritt\u00e4nyt m\u00f6kin ja mets\u00e4n. Ilma oli niin kylm\u00e4, ett\u00e4 jouduin vastustamaan halua kietoa k\u00e4det ymp\u00e4rilleni. Valo tuntui tulevan joka suunnasta niin, ett\u00e4 minulta meni hetki ennen kuin l\u00f6ysin auringon taivaalta.\n\n\"Kuuntele\", Mitja sanoi. H\u00e4n seisoi vasemmalla puolellani.\n\nKuuntelin. En erottanut mit\u00e4\u00e4n, kunnes tajusin, ett\u00e4 juuri sit\u00e4 h\u00e4n oli tarkoittanut. Ei ainuttakaan lintua, hyttysen inin\u00e4\u00e4 tai lehtien kahahdusta. Pelkk\u00e4 oma hengityksemme ilmassa. T\u00e4ydellinen hiljaisuus.\n\nTuijotin usvaan. Annoin katseeni liikkua hitaasti vasemmalta oikealle ja sitten takaisin.\n\nDaniel ilmestyi oikealle puolelleni suden muodossa. Niskakarvat olivat pystyss\u00e4, ja kurkusta l\u00e4hti vaimea murina.\n\nJa silloin he saapuivat usvasta. Ensin yksi, sitten toinen, kolmas ja pian niin monta, etten pysynyt en\u00e4\u00e4 laskuissa mukana.\n\nEhk\u00e4 meid\u00e4n tuli olla otettuja. Heit\u00e4 oli pienen armeijan verran. He olivat kaikki pukeutuneet niin kuin soturien kuuluu, enk\u00e4 ep\u00e4illyt hetke\u00e4k\u00e4\u00e4n, etteiv\u00e4tk\u00f6 he olleet aseistautuneet p\u00e4\u00e4st\u00e4 varpaisiin. Heid\u00e4n oli t\u00e4ytynyt ajatella, ett\u00e4 me tarjoaisimme kunnon vastuksen.\n\nJa mit\u00e4 meill\u00e4 todellisuudessa oli? Minun kuusi veist\u00e4ni.\n\nSitten n\u00e4in Vlassisin ilmeen ja ajattelin, ettei armeijan tuominen ollut ainakaan h\u00e4nen ideansa.\n\n\u00c4kki\u00e4 tajusin, miten typeri\u00e4 me olimme olleet. Kuinka valmistautumattomia. Niin paljon kuin olin hokenut susille, etteiv\u00e4t he tienneet, mit\u00e4 heill\u00e4 oli vastassa, olin astunut tismalleen samaan ansaan. Olin antanut itseni unohtaa, keit\u00e4 daimonit olivat. Ja nyt me joutuisimme maksamaan siit\u00e4.\n\nLopulta erotin Callistan muiden joukosta.\n\nH\u00e4nell\u00e4 oli yll\u00e4\u00e4n viininpunainen mekkonsa. Hiukset oli letitetty kruunuksi h\u00e4nen p\u00e4\u00e4ns\u00e4 p\u00e4\u00e4lle. Vaikka vieress\u00e4ni oleva Daniel oli kaiken j\u00e4rjen mukaan meist\u00e4 suurin uhka, h\u00e4nen katseensa oli minussa.\n\nAikaisemmin minun oli ollut vaikea ymm\u00e4rt\u00e4\u00e4, miksi h\u00e4n oli valinnut Lapin. Se oli osittain johtunut siit\u00e4, etten ollut pystynyt kuvittelemaan h\u00e4nt\u00e4 t\u00e4nne. Nyt kun h\u00e4n seisoi edess\u00e4ni, tajusin, ettei minulla ollut ollut aavistustakaan, kuka ja mik\u00e4 h\u00e4n todellisuudessa oli.\n\nH\u00e4n n\u00e4ytti silt\u00e4 kuin omistaisi koko mets\u00e4n. Ei \u2013 koko maailman. Me olimme pelkki\u00e4 statisteja n\u00e4ytelm\u00e4ss\u00e4, jota h\u00e4n ohjasi. Eik\u00e4 kukaan koskaan saisi tiet\u00e4\u00e4, millainen esitys Lapin korvessa n\u00e4yteltiin.\n\nTaistella vai ei?\n\nSusi-Daniel sy\u00f6ksyi ohitseni kohti daimoneita.\n\nSiis taistelisimme.\n\nVeitseni olivat jo valmiina. Annoin ensimm\u00e4isen lent\u00e4\u00e4 kohti Vlassisia ennen kuin k\u00e4\u00e4nnyin ja juoksin kohti l\u00e4hint\u00e4 daimonia.\n\nMitja vierell\u00e4ni seurasi esimerkki\u00e4ni. En tiennyt, mit\u00e4 Daniel teki, mutta mik\u00e4li \u00e4\u00e4nist\u00e4 saattoi p\u00e4\u00e4tell\u00e4 mit\u00e4\u00e4n, h\u00e4n aiheutti tuhoa daimonien keskuudessa. Kuulin daimonien huutavan varoituksia toisilleen, mutten keskittynyt niihin. Kamppailu nielaisi minut kokonaan. T\u00e4m\u00e4 ei ollut kunniallista ottelemista, jossa oltaisiin testattu pelk\u00e4st\u00e4\u00e4n fyysisi\u00e4 kamppailukykyj\u00e4. Kun daimonit tajusivat, mit\u00e4 tapahtui \u2013 ja heilt\u00e4 meni siihen v\u00e4hemm\u00e4n kuin silm\u00e4nr\u00e4p\u00e4ys aikaa \u2013 mieleeni tulvi kaikenlaisia kuvia, \u00e4\u00e4ni\u00e4, hajuja ja makuja. Torjuin ne yksi kerrallaan. Jouduin keskittym\u00e4\u00e4n t\u00e4ysill\u00e4, jotta sain itseni pysym\u00e4\u00e4n t\u00e4ss\u00e4 ajassa ja paikassa ja puolustautumaan hy\u00f6kk\u00e4yst\u00e4 vastaan.\n\nKaikkea oli aivan liikaa kerralla, ja halusin vain saada sen loppumaan. Vasta silloin p\u00e4\u00e4h\u00e4ni juolahti, ett\u00e4 voisin pys\u00e4ytt\u00e4\u00e4 ajan.\n\nEn kuitenkaan onnistunut siin\u00e4. Hyvin pian sain nimitt\u00e4in huomata, etten pystynyt irrottamaan keskittymist\u00e4ni taistelusta riitt\u00e4v\u00e4n pitk\u00e4ksi aikaa. Kaiken lis\u00e4ksi daimonit tekiv\u00e4t kaikkensa sulkeakseen ideani ulos, aivan niin kuin min\u00e4 tein heid\u00e4n ideoilleen.\n\nK\u00e4\u00e4ntyess\u00e4ni n\u00e4in, ett\u00e4 veljeni oli saarrettu kahdelta puolelta. H\u00e4nen kaulalleen oli painettu veitsi. Kun katsoin alas, huomasin samanlaisen ter\u00e4n l\u00f6ytyv\u00e4n oman leukani alta.\n\nKohotin k\u00e4teni ja annoin veitseni pudota maahan. Mitja teki saman.\n\n\"Vaihda muotoa tai h\u00e4n kuolee\", Vlassisin \u00e4\u00e4ni sanoi aivan korvani juuressa. Tietenkin juuri h\u00e4n oli se, joka oli pakottanut minut antautumaan.\n\nDaniel murisi, mutta totteli.\n\nTuttu hahmo astui eteeni. \"Raissa\", Callista sanoi. H\u00e4n ei edes vilkaissut Mitjaa tai Danielia. En tiennyt, mit\u00e4 se tarkoitti, mutta min\u00e4 olin tullut vain ja ainoastaan Mikaelin takia.\n\n\"Miss\u00e4 h\u00e4n on?\" kysyin.\n\n\"Kuka?\" Callista kysyi. Tyynesti, aivan liian tyynesti.\n\n\"Tied\u00e4t varsin hyvin. Miss\u00e4 h\u00e4n on?\"\n\nVlassis puuttui puheeseen. \"Yleens\u00e4 sen, jota osoitetaan aseella, ei kannata esitt\u00e4\u00e4 kysymyksi\u00e4.\" H\u00e4n oli irrottanut otteensa minusta, mik\u00e4 olisi ollut hyv\u00e4 asia, ellen olisi heti alkanut ajatella, miten harmittomana h\u00e4nen t\u00e4ytyi pit\u00e4\u00e4 minua uskaltaakseen tehd\u00e4 niin. Pelk\u00e4sin, ett\u00e4 h\u00e4nen arvionsa oli oikea.\n\n\"Jos min\u00e4 en saa kysy\u00e4 kysymyksi\u00e4, niin te voitte ainakin kertoa, miksi me olemme t\u00e4\u00e4ll\u00e4\", sanoin.\n\nCallista tarkasteli minua niin kuin kertaalleen karannutta teurasel\u00e4int\u00e4. S\u00e4\u00e4lill\u00e4 ja toruen. \"H\u00e4n ei ole t\u00e4\u00e4ll\u00e4\", h\u00e4n sanoi lopulta.\n\nMeni hetki, ennen kuin tajusin h\u00e4nen puhuvan Mikaelista. \"Siin\u00e4 tapauksessa min\u00e4k\u00e4\u00e4n ei aio olla.\" K\u00e4\u00e4nnyin ja l\u00e4hdin k\u00e4velem\u00e4\u00e4n m\u00f6kin suuntaan.\n\nKaksi daimonia astui heti eteeni.\n\nK\u00e4\u00e4nnyin. Callista oli tullut seisomaan aivan liki. En ollut kuullut h\u00e4nen liikkuvan, joten h\u00e4tk\u00e4hdin h\u00e4nen l\u00e4heisyytt\u00e4\u00e4n.\n\n\"H\u00e4n tappoi daimonin. Ymm\u00e4rr\u00e4t, ettemme me voi j\u00e4tt\u00e4\u00e4 moista rikosta rankaisematta.\" H\u00e4n kuulosti liian lempe\u00e4lt\u00e4. En voinut kuin ihmetell\u00e4, mit\u00e4 sen taakse k\u00e4tkeytyi.\n\nNielaisin. \"Mit\u00e4 te olette tehneet h\u00e4nelle?\"\n\n\"Emme mit\u00e4\u00e4n, mihin h\u00e4n ei olisi itse suostunut.\"\n\nKurtistin kulmiani.\n\n\"Me pyysimme h\u00e4nt\u00e4 valitsemaan yhden omistaan\", Callista sanoin. \"Toisin kuin Alek, h\u00e4n sai valita.\" Tiesin, mit\u00e4 h\u00e4n tulisi sanomaan ennen kuin h\u00e4n sanoi sen. \"H\u00e4n valitsi itsens\u00e4.\"\n\nPuristin silm\u00e4ni kiinni. Tietenkin Mikael oli tehnyt niin. H\u00e4n ei olisi antanut kenenk\u00e4\u00e4n muun k\u00e4rsi\u00e4 tekojensa seurauksia. Ei, vaikka h\u00e4n olikin laumanjohtaja ja siten t\u00e4rke\u00e4mpi kuin yksik\u00e4\u00e4n toinen laumassa. H\u00e4n oli mennyt vapaaehtoisesti, koska se oli ainoa keino est\u00e4\u00e4 daimoneita tulemasta Hukkavaaraan.\n\nAjattelin kaikkia hukkavaaralaisia. Yst\u00e4vi\u00e4ni. Lupinilaisia ja heid\u00e4n lapsiaan. Mikael oli uhrautunut kaikkien heid\u00e4n vuokseen.\n\nH\u00e4n ei voinut olla kuollut. Ei voinut.\n\nT\u00e4ydelliseksi j\u00e4rkytyksekseni Daniel avasi suunsa.\n\n\"Minua ei kiinnosta kuunnella teid\u00e4n mongerrustanne. Min\u00e4 tulin hakemaan poikani ja sitten h\u00e4ivyn.\"\n\n\u00c4kki\u00e4 olin eritt\u00e4in iloinen, ett\u00e4 Daniel seisoi rinnallamme. Normaalisti vihasin h\u00e4nen ylimielist\u00e4 ilmett\u00e4\u00e4n, mutta t\u00e4ll\u00e4 kertaa se oli minun lempiominaisuuteni h\u00e4ness\u00e4.\n\nVlassisin nen\u00e4 nyrpistyi. Ensimm\u00e4ist\u00e4 kertaa tuo ilme ei liittynyt mill\u00e4\u00e4n tavalla minuun.\n\nTai ehk\u00e4 liittyi sittenkin. Vlassisin katse nimitt\u00e4in l\u00f6ysi minut ja tiesin heti, ett\u00e4 h\u00e4n syytti minua aivan kaikesta, my\u00f6s Danielin l\u00e4sn\u00e4olosta. Oli oikeastaan imartelevaa, ett\u00e4 h\u00e4n kuvitteli minulla olevan niin paljon valtaa. Niin kuin kaikki, mit\u00e4 tein, olisikin ollut todella laskelmoitua eik\u00e4 pelkki\u00e4 impulsiivisia p\u00e4\u00e4t\u00f6ksi\u00e4 ja tuurilla r\u00e4mpimist\u00e4. Saatoin vain haaveilla p\u00e4iv\u00e4st\u00e4, jolloin todella tiet\u00e4isin, mit\u00e4 tein.\n\n\"T\u00e4m\u00e4 on hyvin valitettavaa\", Callista sanoi minulle. \"Viestimme oli selv\u00e4: ei susia.\"\n\nV\u00e4lill\u00e4 en ollut varma, viittasiko Callista itseens\u00e4 me-muodossa, vai yrittik\u00f6 h\u00e4n vain vihjata, ett\u00e4 p\u00e4\u00e4t\u00f6sten tekemiseen olisi osallistunut muitakin kuin h\u00e4n.\n\n\"Me emme kutsuneet h\u00e4nt\u00e4 mukaamme\", sanoin. Tiesin kuulostavani lapselliselta.\n\n\"Tilanne on nyt n\u00e4in\", Daniel aloitti, \"T\u00e4m\u00e4 nainen saa kertoa, mit\u00e4 on tehnyt Mikaelille. Ja jos me emme saa h\u00e4nt\u00e4 ehj\u00e4n\u00e4 takaisin, voin vannoa, ett\u00e4 daimonien aika on ohi. Min\u00e4 j\u00e4ljit\u00e4n heid\u00e4t yksitellen, kunnes ainoatakaan ei ole en\u00e4\u00e4 j\u00e4ljell\u00e4.\"\n\nCallista katsoi Mitjaa. \"Kerro, mit\u00e4 h\u00e4n sanoi.\"\n\nMitjan kasvoilla k\u00e4vi irvistys ennen kuin h\u00e4n tulkkasi Danielin sanat kreikaksi. H\u00e4n ei silotellut Danielin sanomaa mill\u00e4\u00e4n tavalla. Olin sek\u00e4 ylpe\u00e4 ett\u00e4 kauhuissani.\n\nMik\u00e4li kyse ei olisi ollut Mikaelin hengest\u00e4, olisin voinut jopa nauttia n\u00e4kem\u00e4st\u00e4ni.\n\nMitja vilkaisi minua. H\u00e4nen t\u00e4ytyi ajatella samaa kuin min\u00e4. Todistimme mit\u00e4 luultavammin maailmankaikkeuden eeppisint\u00e4 tahtojen taistoa.\n\nKunnes Callista katsoin minua ja sanoi: \"H\u00e4n on elossa. Min\u00e4 voin vied\u00e4 sinut h\u00e4nen luokseen.\"\n\nKuullessani sen tajusin, etten ollut hetke\u00e4k\u00e4\u00e4n uskonut Mikaelin voivan olla kuollut. Siit\u00e4 huolimatta huomasin t\u00e4risev\u00e4ni. Kun viimein sain hengitykseni j\u00e4lleen hallintaan, aloin ihmetell\u00e4, mit\u00e4 tarjouksen taakse k\u00e4tkeytyi. Koko tilanne huusi: liian helppoa!\n\nMitja oli ehtinyt suomentaa Callistan sanat Danielille. Daniel otti askeleen eteenp\u00e4in niin kuin ei olisi voinut odottaa sekuntiakaan kauempaa.\n\n\"Vain Raissa\", Callista sanoi. L\u00e4mp\u00f6tila putosi v\u00e4hint\u00e4\u00e4n viisitoista astetta aina, kun h\u00e4n puhui jollekulle muulle kuin minulle.\n\nOlin kahden vaiheilla. Tiesin t\u00e4m\u00e4n olevan ansa \u2013 olin tiennyt koko ajan. En kuitenkaan n\u00e4hnyt mit\u00e4\u00e4n tiet\u00e4 sen yli tai ymp\u00e4ri. Olin p\u00e4\u00e4tt\u00e4nyt jo kauan sitten, ett\u00e4 seuraisin Mikaelia minne tahansa.\n\n\"Rakastettusi kohtalo on sinun k\u00e4siss\u00e4si\", Callista sanoi.\n\n\"Min\u00e4 suostun\", kuulin itseni sanovan.\n\n\"Sin\u00e4 teit oikean p\u00e4\u00e4t\u00f6ksen.\" H\u00e4n ojensi k\u00e4tt\u00e4\u00e4n minua kohti. \"Tule.\"\n\nVilkaisin Mitjaa. \"Olen pahoillani.\"\n\n\"Raisa, \u00e4l\u00e4!\" H\u00e4nen paniikkinsa heijasteli omia ep\u00e4ilyksi\u00e4ni siit\u00e4, mit\u00e4 kaikkea voisi tapahtua, jos l\u00e4htisin Callistan mukaan. Mutta minulla ei ollut vaihtoehtoa.\n\nAstuin askeleen edemm\u00e4s ja tartuin Callistan k\u00e4teen. H\u00e4nen otteensa oli niin tiukka, ett\u00e4 huomasin pid\u00e4tt\u00e4v\u00e4ni hengityst\u00e4ni.\n\n\"Tappakaa heid\u00e4t\", h\u00e4n sanoi. Kaikki daimonit liikahtivat yhten\u00e4 miehen\u00e4 kohti Mitjaa ja Danielia.\n\n\"Ei!\" huusin. Mutta oli liian my\u00f6h\u00e4ist\u00e4. Maailma py\u00f6r\u00e4hti, ja mets\u00e4 katosi ymp\u00e4rilt\u00e4ni. Paiskauduin maahan. \nKAHDESTOISTA LUKU\n\nAvasin silm\u00e4ni.\n\nAllani oleva maa oli sile\u00e4\u00e4 kive\u00e4 ja se tuntui kylm\u00e4lt\u00e4 k\u00e4mmeni\u00e4ni vasten \u2013 aurinko oli vasta alkanut l\u00e4mmitt\u00e4\u00e4 sit\u00e4. Kun kohotin p\u00e4\u00e4t\u00e4ni, saatoin n\u00e4hd\u00e4 pylv\u00e4iden jalustat molemmilla puolillani. Ne olivat tutut \u2013 ja pelottavan todelliset.\n\nJa tuoksu. Siit\u00e4 ei voinut erehty\u00e4.\n\nOlin saarella.\n\nEi. Eieieieiei.\n\nPaniikin aalto hy\u00f6kyi ylitseni. Kun olin luvannut l\u00e4hte\u00e4 Callistan mukaan, en ollut hetke\u00e4k\u00e4\u00e4n ajatellut, minne h\u00e4n minut veisi. Niin paljon kuin jokainen soluni yritti torjua ajatusta, ymp\u00e4rist\u00f6ni pysyi paikoillaan. Min\u00e4 olin t\u00e4\u00e4ll\u00e4, Mitja ja Daniel toisaalla. En voinut tehd\u00e4 asialle mit\u00e4\u00e4n.\n\nPuristin huolen mielest\u00e4ni kuin mehun appelsiinista. He pystyisiv\u00e4t pakenemaan. Minun oli pakko luottaa siihen.\n\nVasta silloin tajusin, mit\u00e4 saarella oleminen tarkoitti: Mikaelin t\u00e4ytyi olla t\u00e4\u00e4ll\u00e4. Se rauhoitti minua hieman. Antoi minulle tarkoituksen kohottaa itseni k\u00e4sien varaan.\n\nCallista seisoi samassa paikassa, jossa olin n\u00e4hnyt h\u00e4net ensimm\u00e4isen kerran. H\u00e4n n\u00e4ytti silt\u00e4 kuin olisi seissyt siin\u00e4 jo kauan, ehk\u00e4 ikuisesti.\n\n\"Tervetuloa takaisin\", h\u00e4n sanoi. Takaisin tajuihini vai takaisin saarelle, en ollut varma.\n\nTuntui kuin olisin pudonnut taivaalta, vaikka todellisuudessa en ollut liikkunut mihink\u00e4\u00e4n. Saari oli tullut minun luokseni. Oloni ei ollut viel\u00e4k\u00e4\u00e4n niin varma, ett\u00e4 olisin uskaltanut nousta jaloilleni. Olin maassa nelinkontin kuin mik\u00e4kin uskollinen palvelija. No, t\u00e4h\u00e4n menness\u00e4 olin tehnyt kaiken, mit\u00e4 Callista oli halunnut. Olin pelannut h\u00e4nen peli\u00e4\u00e4n.\n\nAavistus alkoi hiipi\u00e4 mieleeni. T\u00e4ll\u00e4 ei ollut mit\u00e4\u00e4n tekemist\u00e4 Mikaelin kanssa. T\u00e4m\u00e4 kaikki oli minua varten.\n\n\"Sin\u00e4 olet aiheuttanut paljon harmia, tyt\u00e4r\", Callista sanoi.\n\n\"Min\u00e4 en ole tehnyt mit\u00e4\u00e4n\", sanoin. Ainakaan en ollut tehnyt mit\u00e4\u00e4n, mik\u00e4 olisi uhannut daimoneita. Alek oli tullut minun per\u00e4\u00e4ni eik\u00e4 toisin p\u00e4in. Siit\u00e4 huolimatta olin j\u00e4tt\u00e4nyt heid\u00e4t rauhaan, toivoen ett\u00e4 he soisivat minulle saman.\n\n\"Ennustukset ovat puhuneet sinusta jo tuhansien vuosien ajan\", Callista sanoi.\n\nMyrsky seuraa per\u00e4ss\u00e4si, Mikko oli sanonut.\n\nPuristin silm\u00e4ni kiinni. Aavistukseni oli osunut oikeaan. T\u00e4ll\u00e4 ei ollut mit\u00e4\u00e4n tekemist\u00e4 edes Alekin kuoleman kanssa. Kaikki johtui minusta. Siit\u00e4 saamarin ennustuksesta.\n\nJokin Callistan \u00e4\u00e4ness\u00e4 paljasti, ett\u00e4 niin oli ollut aivan alusta asti. \"Sin\u00e4 siis tiesit koko ajan, ett\u00e4 ennustus kertoo minusta eik\u00e4 Mitjasta. Silloinkin, kun Alek l\u00e4hti meid\u00e4n per\u00e4\u00e4mme. Etk\u00e4 tehnyt mit\u00e4\u00e4n.\" Annoin \u00e4\u00e4neni hiipua.\n\nCallista ei vastannut, mutta tiesin, ett\u00e4 olin arvannut oikein. En ollut varma, ymm\u00e4rsink\u00f6 en\u00e4\u00e4 mit\u00e4\u00e4n muuta kuin sen, ett\u00e4 minun oli pelastettava Mikael.\n\nKohottauduin polvilleni. \"Minun t\u00e4ytyy saada n\u00e4hd\u00e4 h\u00e4net\", sanoin.\n\n\"Kaikki aikanaan\", h\u00e4n sanoi.\n\nKaksi daimonia ilmestyi takaap\u00e4in ja tarttui minua k\u00e4sivarsista. Kohosin tahtomattani jaloilleni.\n\n\"Todista minulle, ett\u00e4 h\u00e4n on t\u00e4\u00e4ll\u00e4!\" huusin.\n\nDaimonit alkoivat raahata minua selk\u00e4 edell\u00e4 pois.\n\n\"Callista!\"\n\nAinoa vastaus jonka sain, oli \u00e4killinen pimeys, kun p\u00e4\u00e4ni peitettiin kankaalla.\n\nOlin pelannut h\u00e4nen s\u00e4\u00e4nn\u00f6ill\u00e4\u00e4n. Ja olin juuri h\u00e4vinnyt.\n\n***\n\nYritin laskea askelia ja saada jotain k\u00e4sityst\u00e4 suunnista, mutta minulla ei ollut aavistustakaan, minne minua vietiin. Vasta kun ilma viileni ymp\u00e4rill\u00e4ni ja \u00e4\u00e4net alkoivat kuulostaa oudoilta tajusin, ett\u00e4 olimme tulleet sis\u00e4tilaan.\n\nMinut johdatettiin alas portaita. Hetken aikaa ehdin jo kuvitella, ett\u00e4 minut viet\u00e4isiin samaan paikkaan, jossa Mikaelia pidettiin. Olin tietenkin v\u00e4\u00e4r\u00e4ss\u00e4.\n\nDaimonit k\u00e4viv\u00e4t l\u00e4pi taskuni. En tiennyt, mit\u00e4 veitsilleni oli tapahtunut, sill\u00e4 Vlassis oli ottanut ne minulta jo Lapissa. He eiv\u00e4t l\u00f6yt\u00e4neet mit\u00e4\u00e4n, ja lopulta huppu vedettiin pois p\u00e4\u00e4st\u00e4ni.\n\nEn ollut tiennyt, ett\u00e4 saarella edes oli tyrmi\u00e4. Oven kolahtaessa kiinni takanani tajusin kuitenkin olevani sellaisessa. Tuskin oli olemassa yht\u00e4 lohdutonta ja lopullista \u00e4\u00e4nt\u00e4, ja se sai minut s\u00e4ps\u00e4ht\u00e4m\u00e4\u00e4n.\n\nVankilani ei ollut aivan niin synkk\u00e4 tai pime\u00e4 kuin mit\u00e4 sana \"tyrm\u00e4\" sai minut ajattelemaan, sill\u00e4 sein\u00e4t olivat valkoiseksi kalkittua kive\u00e4. Mutta nopea vilkaisu paljasti mukavuuksien rajoittuvaan lattialla olevaan vilttiin ja nurkassa olevaan \u00e4mp\u00e4riin.\n\nKuulostelin hetken aikaa k\u00e4yt\u00e4v\u00e4n suuntaan, mutta tulin lopulta siihen tulokseen, ett\u00e4 olin yksin. Se olisi ollut hyv\u00e4 asia, elleiv\u00e4t ajatukseni olisi olleet niin huonoa seuraa.\n\nAjattelin velje\u00e4ni. Kuvittelin h\u00e4net makaamassa kuolleena keskell\u00e4 korpea kenenk\u00e4\u00e4n tiet\u00e4m\u00e4tt\u00e4, mit\u00e4 oli tapahtunut. Kuinka kauan menisi, ennen kuin sudet tulisivat etsim\u00e4\u00e4n meit\u00e4? Ja Caoimhe.... Voi luoja. Miten h\u00e4n voisi koskaan antaa minulle anteeksi? Miten voisin koskaan antaa itselleni anteeksi?\n\nPuristin silm\u00e4ni kiinni. H\u00e4n on elossa, kerroin itselleni. Mitja oli selviytyj\u00e4. Ja Daniel oli isoin ja pahin susi koko maassa \u2013 luultavasti koko planeetalla.\n\nYritin vaientaa mieleni.\n\nIstuin lattialle selk\u00e4 sein\u00e4\u00e4 vasten. Selliss\u00e4ni oli pieni ikkuna-aukko l\u00e4hell\u00e4 katonrajaa. Siit\u00e4 ei olisi mahtunut ulos, mutta ainakin se tarjosi hivenen valoa ja keinon yritt\u00e4\u00e4 pysytell\u00e4 selvill\u00e4 ajan kulumisesta.\n\nMit\u00e4 kauemmin istuin aloillani, sit\u00e4 selvemm\u00e4ksi suunnitelmani tuli. Minun piti p\u00e4\u00e4st\u00e4 ulos sellist\u00e4. Minun piti l\u00f6yt\u00e4\u00e4 Mikael. Sitten meid\u00e4n piti paeta saarelta.\n\n***\n\nTunnit juoksivat eteenp\u00e4in. Valo pakeni pienest\u00e4 ikkunasta ja sis\u00e4\u00e4n astui pimeys, joka sai ilman tuoksumaan entist\u00e4 vahvemmin merelt\u00e4. Minulla oli n\u00e4lk\u00e4 ja jano, mutta koska en pystynyt tekem\u00e4\u00e4n kummallekaan mit\u00e4\u00e4n, yritin olla ajattelematta niit\u00e4. K\u00e4vin makaamaan taitellun viltin p\u00e4\u00e4lle. Kiedoin k\u00e4sivarret ymp\u00e4rilleni ja suljin silm\u00e4ni.\n\nSeisoin keskell\u00e4 kapeaa katua. Olin niin keskittynyt ymp\u00e4rist\u00f6\u00f6ni, etten huomannut auton tulevan minua kohti. Loikkasin viime hetkell\u00e4 jalkak\u00e4yt\u00e4v\u00e4lle.\n\nTuijotin vieress\u00e4ni olevaa avointa ovea samalla kun tasasin hengityst\u00e4ni, ja hetken p\u00e4\u00e4st\u00e4 ymm\u00e4rsin sen kuuluvan ravintolalle. P\u00f6ydill\u00e4 oli muoviset p\u00f6yt\u00e4liinat ja lasiset suolasirottimet. Keitti\u00f6st\u00e4 kantautui paistuvan lihan k\u00e4ry. Kukaan ei ollut ruokailemassa, mutta kaksi miest\u00e4 istui p\u00f6yd\u00e4n \u00e4\u00e4ress\u00e4 katsomassa televisiota. Siell\u00e4 oli meneill\u00e4\u00e4n jalkapallopeli, ja selostus oli laitettu niin kovalle, ett\u00e4 sen t\u00e4ytyi kuulua koko kortteliin.\n\nK\u00e4\u00e4nnyin hitaasti ymp\u00e4ri. Katua varjostivat appelsiinipuut, mutta niiss\u00e4 ei n\u00e4kynyt viel\u00e4 hedelmi\u00e4. Rakennukset olivat viisikerroksisia ja n\u00e4yttiv\u00e4t aika tavanomaisilta asuintaloilta. Pyykit kuivuivat parvekkeille ripustetuilla naruilla.\n\nKerrostalojen v\u00e4list\u00e4 n\u00e4kyi kukkula, jonka tunnistin heti. Akropolis. Olin Ateenassa.\n\nJa juuri silloin tuttu hahmo paineli ohitseni.\n\nS\u00e4ik\u00e4hdin niin, ett\u00e4 olin huudahtaa. \u00c4itini. T\u00e4ll\u00e4 kertaa h\u00e4nen hiuksensa olivat minulle kovin tuttua punaisen s\u00e4vy\u00e4. H\u00e4n ei n\u00e4ytt\u00e4nyt yht\u00e4\u00e4n vanhemmalta kuin edellisell\u00e4k\u00e4\u00e4n kerralla. Ja t\u00e4ll\u00e4 kerralla h\u00e4nen vatsansa oli valtava.\n\nL\u00e4hdin seuraamaan h\u00e4nt\u00e4.\n\nJalkak\u00e4yt\u00e4v\u00e4ll\u00e4 makasi kaksi mustaa kissaa, jotka nousivat tassuilleen kun n\u00e4kiv\u00e4t \u00e4itini l\u00e4hestyv\u00e4n. \u00c4itini askeleet hidastuivat ja h\u00e4n pys\u00e4htyi niiden kohdalle. H\u00e4n kumartui ja ojensi k\u00e4tt\u00e4\u00e4n l\u00e4hemp\u00e4n\u00e4 seisovaa kissaa kohti. H\u00e4n sanoi jopa: \"Kiskiskis.\"\n\nEn usko, ett\u00e4 se on koskaan toiminut ainoankaan kissan kohdalla, eik\u00e4 t\u00e4m\u00e4k\u00e4\u00e4n kerta poikennut s\u00e4\u00e4nn\u00f6st\u00e4. Kissan karvat nousivat pystyyn ja se s\u00e4hisi. Sitten se luikahti piiloon viereisen auton alle.\n\n\u00c4itini suoristautui. H\u00e4n n\u00e4ytti aidosti loukkaantuneelta, ja minun oli pakko nauraa. Varsinkin, kun h\u00e4n sanoi: \"Hyv\u00e4\u00e4 huomenta vain sinullekin.\"\n\nH\u00e4nen myrtyneisyytt\u00e4\u00e4n kesti kuitenkin vain puoli sekuntia. Sitten h\u00e4n painoi k\u00e4den vatsalleen ja kaikki muuttui. H\u00e4n hymyili, muttei sellaista hymy\u00e4 kuin valokuvissa tai edes silloin, kun oltiin hyv\u00e4ss\u00e4 seurassa. T\u00e4m\u00e4 hymy oli k\u00e4\u00e4ntynyt sis\u00e4\u00e4np\u00e4in. Huulet tuskin edes liikahtivat, mutta silti olin varma, etten ollut n\u00e4hnyt h\u00e4nt\u00e4 koskaan niin onnellisena.\n\nJoku huusi h\u00e4nelle parvekkeelta ja yll\u00e4tyin hieman, kun h\u00e4n tervehti kreikaksi. H\u00e4n tuntui olevan niin kotonaan t\u00e4\u00e4ll\u00e4.\n\nH\u00e4n k\u00e4\u00e4ntyi kohti er\u00e4st\u00e4 ulko-ovea ja otti esiin avaimen. Ovi kuitenkin lenn\u00e4hti auki ennen kuin h\u00e4n ehti ty\u00f6nt\u00e4\u00e4 avainta lukkoon.\n\nIs\u00e4ni kasvot ilmestyiv\u00e4t oviaukkoon. \"Miss\u00e4 sin\u00e4 olet ollut?\"\n\nH\u00e4n ei ollut iloinen.\n\n\"Torilla\", \u00e4itini vastasi.\n\nStefanos vilkaisi nopeasti kadulle ennen kuin tarttui h\u00e4nt\u00e4 k\u00e4dest\u00e4 ja alkoi johdattaa h\u00e4m\u00e4r\u00e4\u00e4 k\u00e4yt\u00e4v\u00e4\u00e4 pitkin.\n\nKun n\u00e4in heid\u00e4n pys\u00e4htyv\u00e4n hissin ovelle, ep\u00e4r\u00f6in. En mill\u00e4\u00e4n mahtuisi heid\u00e4n kanssaan hissiin.\n\nEi ollut aikaa hukattavana. Kun hissin ovi sulkeutui, k\u00e4\u00e4nnyin nopeasti ja l\u00e4hdin juoksemaan yl\u00f6s portaita.\n\nHissi jatkoi ensimm\u00e4isen kerroksen ohi, ja kiihdytin kiroillen vauhtiani. Toisessa se onneksi pys\u00e4htyi.\n\n\"Olin todella huolissani\", kuulin is\u00e4ni sanovan juuri kun ehdin oikealle tasanteelle. H\u00e4n avasi p\u00e4\u00e4tyhuoneiston oven.\n\nVasta silloin h\u00e4n huomasi \u00e4itini kantaman kassin. H\u00e4n otti sen h\u00e4nelt\u00e4. \"Miksi sin\u00e4 kannat jotain n\u00e4in painavaa?\" Kun \u00e4itini ei vastannut, h\u00e4n meni sis\u00e4\u00e4n ja jatkoi: \"N\u00e4in sinut parvekkeelta. Sin\u00e4 yritit taas silitt\u00e4\u00e4 sit\u00e4 kissaa, etk\u00f6 yritt\u00e4nytkin?\"\n\nOnnistuin seuraamaan \u00e4iti\u00e4ni eteiseen ennen kuin ovi sulkeutui. N\u00e4in, kun h\u00e4n laittoi k\u00e4tens\u00e4 puuskaan.\n\n\"Tuire, ne ovat katukissoja\", Stefanos sanoi. \"Niill\u00e4 voi olla ties mit\u00e4 tauteja. Sinun tilassasi...\"\n\n\u00c4itini sai viimeinkin suunsa auki. \"Minulle huutaminen ei auta minun 'tilaani'!\"\n\n\"Anteeksi\", is\u00e4ni mumisi. \"Olen vain v\u00e4h\u00e4n hermostunut.\"\n\nJokin \u00e4itini olemuksessa muuttui. H\u00e4nen vaitonaisuutensa ei ollut johtunut siit\u00e4, ett\u00e4 h\u00e4n oli ajatellut olevan riitelemisen yl\u00e4puolella. H\u00e4n salasi jotain. \"Itse asiassa min\u00e4kin olen\", h\u00e4n tunnusti. \"Minun on pit\u00e4nyt kertoa sinulle jotain.\"\n\nUskoin tiet\u00e4v\u00e4ni, mit\u00e4 oli tulossa, ja huomasin pid\u00e4tt\u00e4v\u00e4ni hengityst\u00e4. Ei ollut helppoa tapaa kertoa tulevan lapsen is\u00e4lle, ett\u00e4 lapsen kaikki sukulaiset muuttivat muotoaan t\u00e4ysikuun aikaan.\n\nIs\u00e4ni tutki h\u00e4nt\u00e4 katseellaan. \"Menn\u00e4\u00e4n istumaan\", h\u00e4n sanoi lopulta.\n\n\u00c4itini totteli. H\u00e4n valitsi sohvan, ja Stefanosille j\u00e4i nojatuoli.\n\nTuire selvitti kurkkuaan. \"Minun perheeni... Se paikka, jossa asuin....\"\n\nH\u00e4n ei ilmeisesti tiennyt, miten jatkaa, sill\u00e4 h\u00e4n vain tuijotti Stefanosia kuin odottaen t\u00e4m\u00e4n lukevan ajatuksensa. Katsoessani is\u00e4ni t\u00e4ysin tiet\u00e4m\u00e4t\u00f6nt\u00e4 ilmett\u00e4 tiesin, ettei h\u00e4nen lahjansa voinut liitty\u00e4 ainakaan ennustamiseen. H\u00e4nell\u00e4 ei ollut hajuakaan.\n\nMinun k\u00e4vi h\u00e4nt\u00e4 s\u00e4\u00e4liksi.\n\nSitten tajusin, ettei h\u00e4n varmaankaan ollut viel\u00e4 kertonut \u00e4idille mit\u00e4\u00e4n siit\u00e4, mik\u00e4 itse oli, ja s\u00e4\u00e4lini muutti nopeasti \u00e4itini luo.\n\n\u00c4itini tuijotti vatsaansa. Kun h\u00e4n kohotti katseensa, oli h\u00e4n selv\u00e4sti tehnyt p\u00e4\u00e4t\u00f6ksens\u00e4. H\u00e4n hoitaisi t\u00e4m\u00e4n pois alta, hinnalla mill\u00e4 hyv\u00e4ns\u00e4.\n\n\"Kaikki minun suvussani ovat ihmissusia. Kaikki paitsi min\u00e4.\"\n\n\"Mit\u00e4 sin\u00e4 juuri sanoit?\"\n\n\"Kerroin totuuden. Mutta mink\u00e4\u00e4n ei tarvitse muuttua, me voimme edelleen...\"\n\n\"Ei\", is\u00e4ni sanoi. Pudisti p\u00e4\u00e4t\u00e4\u00e4n. H\u00e4n ei katsonut \u00e4iti\u00e4ni. \"T\u00e4m\u00e4 ei voi olla mahdollista. T\u00e4t\u00e4 ei voi tapahtua.\"\n\n\"Tied\u00e4n, ett\u00e4 sit\u00e4 on vaikea uskoa.\"\n\nStefanos nousi yl\u00f6s. \"Helvetti soikoon, min\u00e4 tied\u00e4n ihmissusista! Mutta kerro minulle miten t\u00e4m\u00e4 tapahtui.\"\n\n\"Kyll\u00e4 sin\u00e4 tied\u00e4t, miten t\u00e4m\u00e4 tapahtui. Satuit olemaan paikalla.\"\n\nIs\u00e4ni p\u00e4\u00e4sti turhautuneen \u00e4\u00e4nen. Ymm\u00e4rsin hyvin. Olin pikkuhiljaa alkanut hahmottaa, miten raivostuttavaa minun kanssani keskusteleminen saattoi joskus olla. Miten Mikael mahtoikaan kest\u00e4\u00e4 minua?\n\nIs\u00e4ni ei sanonut mit\u00e4\u00e4n pitk\u00e4\u00e4n aikaan. En pystynyt edes kuvittelemaan, mit\u00e4 h\u00e4nen p\u00e4\u00e4ss\u00e4\u00e4n liikkui.\n\n\"Mik\u00e4\u00e4n ei muutu. Me emme kerro kenellek\u00e4\u00e4n\", h\u00e4n mutisi itsekseen kreikaksi.\n\n\"Mit\u00e4?\" \u00e4itini kysyi.\n\n\"Meid\u00e4n pit\u00e4isi muuttaa\", is\u00e4ni sanoi.\n\nSe oli ilmeest\u00e4 p\u00e4\u00e4tellen viimeinen asia, jonka \u00e4itini oli odottanut kuulevansa. \"Minne muka?\"\n\n\"Suomeen.\"\n\n\"Miksi me emme voi j\u00e4\u00e4d\u00e4 t\u00e4nne?\"\n\nIs\u00e4ni ei sanonut mit\u00e4\u00e4n, katsoi vain \u00e4iti\u00e4ni.\n\nEp\u00e4ilys hiipi \u00e4itini kasvoille. \"Kuka sin\u00e4 oikein olet?\" h\u00e4n ihmetteli.\n\n\"Joku, joka rakastaa sinua\", is\u00e4ni vastasi.\n\n***\n\nHer\u00e4sin auringon mukana. Koko kroppani oli kipe\u00e4 ja uupunut, ja siksi h\u00e4mm\u00e4styin kun huomasin, ett\u00e4 joku oli tuonut ruokaa minun her\u00e4\u00e4m\u00e4tt\u00e4 siihen. K\u00e4vin sen kimppuun. Se oli aivan samaa ruokaa mihin olin tottunut saarella, joten minulla ei ollut valittamista. Sy\u00f6dess\u00e4ni yritin ajatella positiivisesti: jos minua sy\u00f6tettiin n\u00e4inkin hyvin, minua ei luultavasti aiottu tappaa ihan heti. Ja aikaa min\u00e4 nimenomaan tarvitsin. Aikaa p\u00e4\u00e4st\u00e4 selville asioista ja l\u00f6yt\u00e4\u00e4 jokin heikko kohta, jota k\u00e4ytt\u00e4\u00e4 hyv\u00e4kseni.\n\nTiesin jo, mist\u00e4 aloittaa. Seuraavan kerran kun joku daimoneista tulisi selliini, k\u00e4ytt\u00e4isin h\u00e4neen kykyj\u00e4ni ja kalastelisin h\u00e4nen mielest\u00e4\u00e4n kaiken, mit\u00e4 vain voisin saada irti. Suurella todenn\u00e4k\u00f6isyydell\u00e4 tulija olisi valmistautunut ja onnistuisi torjumaan minut, mutta minun t\u00e4ytyi yritt\u00e4\u00e4.\n\nMinulle ei kuitenkaan annettu siihen mahdollisuutta. En n\u00e4hnyt ket\u00e4\u00e4n aikana, joka tuntui monelta tunnilta. Siit\u00e4 huolimatta olin varma, ett\u00e4 minua vahdittiin koko ajan. Jossain aistieni ulottumattomissa seisoi v\u00e4hint\u00e4\u00e4n yksi daimon vahdissa.\n\nYritin avata keskustelun heid\u00e4n kanssaan, tuloksetta. Kukaan ei vastannut minulle. Ottaen huomioon, miten huonoa kreikkani oli ilman velje\u00e4ni ja etenkin oltuani monta kuukautta k\u00e4ytt\u00e4m\u00e4tt\u00e4 sit\u00e4 muuhun kuin Mitjan piikittelyyn, en ollut varma, ymm\u00e4rsiv\u00e4tk\u00f6 he edes, mit\u00e4 sanoin.\n\nJuuri kun olin alkanut ajatella, ett\u00e4 minun annettaisiin m\u00e4d\u00e4nty\u00e4 selliss\u00e4, Callista tuli luokseni.\n\nKuulin, kun h\u00e4n k\u00e4ski k\u00e4yt\u00e4v\u00e4ss\u00e4 olevia daimoneita astumaan ulos, mutta siit\u00e4 huolimatta yll\u00e4tyin, kun h\u00e4n tuli selliini yksin ja ainakin p\u00e4\u00e4llisin puolin aseettomana. T\u00e4n\u00e4\u00e4n h\u00e4n oli pukeutunut siniseen, ja kultakorut kimmelsiv\u00e4t h\u00e4nen ranteissaan, kaulallaan ja hiuksissaan.\n\nNousin seisomaan. \"Min\u00e4 haluan n\u00e4hd\u00e4 h\u00e4net\", sanoin. Tiesin toistavani itse\u00e4ni, enk\u00e4 odottanutkaan Callistan my\u00f6ntyv\u00e4n. Tarkoitukseni oli nimitt\u00e4in vain h\u00e4m\u00e4t\u00e4 h\u00e4nt\u00e4. Avasin yhteyden ideoiden maailmaan niin lev\u00e4lleen kuin osasin ja kurotin h\u00e4nt\u00e4 kohti.\n\nMinun olisi pit\u00e4nyt tiet\u00e4\u00e4 paremmin: Callista oli p\u00e4\u00e4ttym\u00e4t\u00f6n muuri, johon t\u00f6rm\u00e4sin p\u00e4istikkaa. Horjahdin taaksep\u00e4in, ja jouduin ottamaan sein\u00e4st\u00e4 tukea. Tunsin jotain kosteaa nen\u00e4ni alla ja kun pyyhk\u00e4isin sit\u00e4, sain vahvistuksen ep\u00e4ilylleni. Nen\u00e4st\u00e4ni oli taas alkanut vuotaa verta.\n\nCallista k\u00e4ytt\u00e4ytyi aivan kuin mit\u00e4\u00e4n ei olisi tapahtunut. \"Sin\u00e4 tulet viel\u00e4 n\u00e4kem\u00e4\u00e4n h\u00e4net\", h\u00e4n vastasi.\n\nPuristin nen\u00e4\u00e4ni sormillani. H\u00e4n oli niin rauhallinen, niin l\u00e4p\u00e4isem\u00e4t\u00f6n. Vihasin h\u00e4nt\u00e4.\n\n\"Mit\u00e4 sin\u00e4 haluat?\" kysyin.\n\nH\u00e4n ei sanonut mit\u00e4\u00e4n aivan heti. Kun h\u00e4n puhui, tajusin, ettei h\u00e4n aikonut vastata kysymykseeni. \"Kaikki t\u00e4ll\u00e4 saarella pelk\u00e4\u00e4v\u00e4t sinua\", h\u00e4n aloitti. \"Voi miten he pelk\u00e4\u00e4v\u00e4tk\u00e4\u00e4n! He tiet\u00e4v\u00e4t, ett\u00e4 sin\u00e4 tulet joka p\u00e4iv\u00e4 vain vahvemmaksi ja vahvemmaksi. Aika alkaa loppua kesken.\"\n\n\"Mit\u00e4 sin\u00e4 tarkoitat?\"\n\n\"Sinut t\u00e4ytyy tappaa sinut ennen kuin sinusta on tullut niin vahva, ett\u00e4 voit nousta ideoiden maailmaan. Se on ainoa keino kumota ennustus. Siksi saarella pidet\u00e4\u00e4n pian seremonia. T\u00e4rkein uhraus, joka t\u00e4\u00e4ll\u00e4 on n\u00e4hty.\"\n\nPelk\u00e4sin pahoin tiet\u00e4v\u00e4ni, mit\u00e4 h\u00e4n sill\u00e4 tarkoitti. Minun oli kuitenkin varmistettava. \"Minutko uhrataan?\"\n\n\"Niin.\"\n\nAinoa mit\u00e4 pystyin ajattelemaan oli se, ett\u00e4 kuolisin. Kaiken j\u00e4lkeen sen ei olisi pit\u00e4nyt tulla yll\u00e4tyksen\u00e4, mutta se tuli.\n\n\"Ja sin\u00e4 siis aiot antaa heid\u00e4n tehd\u00e4 niin?\"\n\n\"Min\u00e4 en ole paikalla est\u00e4m\u00e4ss\u00e4 heit\u00e4.\"\n\nH\u00e4nen oli aina pakko puhua arvoituksin. Se oli raivostuttavaa. Minulle ei ollut koskaan ollut yht\u00e4 t\u00e4rke\u00e4\u00e4 tiet\u00e4\u00e4, mit\u00e4 tulisi tapahtumaan, ja h\u00e4n vain kierteli ja kaarteli.\n\nEhk\u00e4 h\u00e4n tiesi ajatukseni, sill\u00e4 h\u00e4n jatkoi: \"Anna minun n\u00e4ytt\u00e4\u00e4 sinulle jotakin.\"\n\nEp\u00e4r\u00f6in. Minulle ei kuitenkaan annettu aikaa vastata, sill\u00e4 selli ymp\u00e4rilt\u00e4ni katosi.\n\nOlimme valkoisessa huoneessa. Se oli kuution muotoinen, eik\u00e4 siin\u00e4 ollut ovia tai ikkunoita.\n\nOlin ollut kerran aikaisemminkin samassa tilanteessa, ja se tuntui nyt aivan yht\u00e4 luonnottomalta. Tunsin vanhan paniikin nostavan p\u00e4\u00e4t\u00e4\u00e4n.\n\nCallista seisoi edess\u00e4ni. \"T\u00e4m\u00e4 on pelkk\u00e4 harjoitus\", h\u00e4n sanoi. \"Tarkoitus on, ett\u00e4 sin\u00e4 oppisit jotain hy\u00f6dyllist\u00e4.\"\n\nEn tiennyt, mihin t\u00e4m\u00e4 kaikki viel\u00e4 johtaisi, mutta \u00e4kki\u00e4 olin peloissani siit\u00e4, ett\u00e4 todella oppisin jotain. Oppiminen oli nimitt\u00e4in vaarallista silloin, kun toivoi kaiken pysyv\u00e4n ennallaan. Oppiminen oli muuttumista, eik\u00e4 kaikki muutos ollut hyv\u00e4st\u00e4.\n\n\"Ideoiden maailma on kuin tyhj\u00e4 taulu\", Callista aloitti. \"Kaikki on sinun hallittavissasi. Mutta vain vahvat pystyv\u00e4t siihen.\"\n\nKatsoin seini\u00e4. Huone tuntui el\u00e4v\u00e4lt\u00e4 ja hengitt\u00e4v\u00e4lt\u00e4 olennolta, joka voisi hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 hy\u00f6k\u00e4t\u00e4 kimppuuni. En uskonut, ett\u00e4 se antaisi tehd\u00e4 itselleen mit\u00e4\u00e4n.\n\n\"Jos haluat pois, sinun t\u00e4ytyy luoda ovi\", Callista jatkoi.\n\nHalusin ulos. Kurkkuani kuristi hetki hetkelt\u00e4 enemm\u00e4n.\n\nKurotin ep\u00e4r\u00f6iden sein\u00e4\u00e4 kohti. Koskin sit\u00e4 sormillani. Pinta ei tuntunut lainkaan silt\u00e4, mit\u00e4 olin kuvitellut \u2013 se ei ollut viile\u00e4 ja kova, vaan tismalleen k\u00e4teni l\u00e4mp\u00f6inen ja jousti hieman, kun painoin lujempaa.\n\nK\u00e4mmen yh\u00e4 sein\u00e4\u00e4 vasten suljin silm\u00e4ni. Kuvittelin mieless\u00e4ni oven. Ajattelin jokaista yksityiskohtaa, materiaalia ja sen tuntua. Kun olin mielest\u00e4ni valmis, avasin silm\u00e4ni.\n\nEteeni oli ilmestynyt tismalleen kuvittelemani kaltainen ovi.\n\nAvaa se, Callistan \u00e4\u00e4ni sanoi p\u00e4\u00e4ni sis\u00e4ll\u00e4.\n\nJa min\u00e4 avasin.\n\nOven takaa paljastunut huone oli pieni ja h\u00e4m\u00e4r\u00e4. Valonl\u00e4hteen\u00e4 oli pelkk\u00e4 kynttil\u00e4, ja ainoat kalusteet olivat s\u00e4nky ja kirjoitusp\u00f6yt\u00e4. J\u00e4lkimm\u00e4isen \u00e4\u00e4ress\u00e4 istui joku. Tunnistin h\u00e4net heti: is\u00e4ni.\n\nEn ensin uskaltanut astua sis\u00e4\u00e4n huoneeseen, sill\u00e4 pelk\u00e4sin mit\u00e4 tapahtuisi. Lopulta uteliaisuus voitti. Kurkistin h\u00e4nen olkansa yli n\u00e4hd\u00e4kseni, mit\u00e4 h\u00e4n oikein teki.\n\nH\u00e4n kirjoitti kirjett\u00e4. Se oli odottamatonta. Kyn\u00e4 tuntui t\u00e4ysin v\u00e4\u00e4r\u00e4lt\u00e4 esineelt\u00e4 h\u00e4nen k\u00e4teens\u00e4.\n\nYritin poimia silmill\u00e4ni edes yhden lauseen, josta olisin saanut selv\u00e4\u00e4.\n\nJumalat eiv\u00e4t tunne kipua.\n\nEn ehtinyt lukea pidemm\u00e4lle, sill\u00e4 h\u00e4n kirjoitti viimeiselle riville nimens\u00e4 ja nosti sitten arkin p\u00f6yd\u00e4lt\u00e4. H\u00e4n antoi kynttil\u00e4n liekin koskea paperin kulmaa. Tuli tarttui paperiin nopeasti ja k\u00e4pristi sen mustaksi rullaksi, joka hajosi liekin edetess\u00e4. H\u00e4n ei p\u00e4\u00e4st\u00e4nyt paperista irti edes silloin, kun tuli poltti h\u00e4nen sormiaan. Pian paperista oli j\u00e4ljell\u00e4 vain tuhkaa ja pieni kulmapala palaneiden sormien v\u00e4liss\u00e4. Stefanos katseli k\u00e4tt\u00e4\u00e4n niin kuin h\u00e4n ei olisi aivan ymm\u00e4rt\u00e4nyt, miksi iho kupli kirkkaanpunaisena, mutta olisi pit\u00e4nyt sit\u00e4 hyvin kiehtovana. Vasta pitk\u00e4n ajan j\u00e4lkeen palovammat alkoivat parantua ja h\u00e4n antoi paperinkulman pudota lattialle.\n\nIs\u00e4ni oli varsinainen tapaus. Huh. Ja min\u00e4 kun olin pit\u00e4nyt itse\u00e4ni v\u00e4h\u00e4n hulluna.\n\nAstuin askeleen taakse ja katsoin Callistaa. \"Mit\u00e4 t\u00e4m\u00e4 tarkoittaa?\" Oli raivostuttavaa joutua turvautumaan h\u00e4neen, mutta mik\u00e4\u00e4n, mit\u00e4 olin n\u00e4hnyt matkalla Sarrajuoksan m\u00f6kille, ei ollut her\u00e4tt\u00e4nyt minussa yht\u00e4 paljon kysymyksi\u00e4.\n\n\"T\u00e4m\u00e4 on sinun ovesi, ei minun.\" H\u00e4n sanoi sen melkein vastahakoisesti. H\u00e4n k\u00e4\u00e4ntyi niin, ett\u00e4 peitti is\u00e4ni n\u00e4kyvist\u00e4ni. \"Luulen, ett\u00e4 me olemme saavuttaneet tarpeeksi t\u00e4lle p\u00e4iv\u00e4lle.\" H\u00e4n johdatti minut ulos huoneesta ja sulki oven. Sill\u00e4 sekunnilla huone katosi ja selli ilmestyi ymp\u00e4rillemme.\n\nMinusta ei tuntunut silt\u00e4, ett\u00e4 olisimme saavuttaneet jotakin. En ollut yht\u00e4\u00e4n valmiimpi tulemaan uhratuksi t\u00e4ysin k\u00e4sitt\u00e4m\u00e4tt\u00f6m\u00e4n asian vuoksi. Callistan sanat saivatkin minut ep\u00e4ilem\u00e4\u00e4n itse\u00e4ni. Aivan kuin h\u00e4n olisi saanut tapaamisesta jotain, mik\u00e4 oli j\u00e4\u00e4nyt minulta tajuamatta.\n\n***\n\nOlin kuvitellut, ett\u00e4 daimonien suunnitelman tiet\u00e4minen olisi helpottanut mielt\u00e4ni, mutta k\u00e4vikin tismalleen p\u00e4invastoin. Suunnitelma oli niin hullu, etten olisi koskaan voinut keksi\u00e4 sit\u00e4 itse, ja se pelotti minua. Pelotti todella paljon. En tiennyt, miten joukko ihmisi\u00e4 saatiin luopumaan uskonnollisesta murhasta, sill\u00e4 en tajunnut, miten jotkut saattoivat alun perink\u00e4\u00e4n uskoa niin sokeasti. Aloin ep\u00e4ill\u00e4, ett\u00e4 daimonit olivat valmiita mihin tahansa.\n\nK\u00e4velin selliss\u00e4ni edestakaisin pit\u00e4\u00e4kseni itseni toimintakykyisen\u00e4. Pystyin ottamaan nelj\u00e4 askelta yhteen suuntaan ennen kuin jouduin k\u00e4\u00e4ntym\u00e4\u00e4n ja palaamaan takaisin. Ikkunaa kohti k\u00e4velless\u00e4ni aurinko paistoi suoraan naamaani. Lopulta minun t\u00e4ytyi lopettaa ja istua alas, jotten tuhlaisi liikaa arvokasta energiaa. Minulle ei oltu tuotu ruokaa ensimm\u00e4isen aamun j\u00e4lkeen, enk\u00e4 halunnut edes l\u00e4hte\u00e4 arvailemaan sen syit\u00e4. Nestehukka tuntui jomotuksena ohimoilla.\n\nJoku oli ovella.\n\nEn ollut kuullut k\u00e4yt\u00e4v\u00e4n suunnasta mit\u00e4\u00e4n, joten h\u00e4mm\u00e4styin n\u00e4hdess\u00e4ni kaltereiden takana hennon tyt\u00f6n. Katsottuani h\u00e4nt\u00e4 hieman pidemp\u00e4\u00e4n tunnistin h\u00e4net.\n\nH\u00e4n oli kasvanut pituutta sitten viime n\u00e4kem\u00e4n. Se sai h\u00e4net n\u00e4ytt\u00e4m\u00e4\u00e4n entist\u00e4kin hoikemmalta, ja vaikka h\u00e4nen vaatteensa olivat l\u00f6ys\u00e4t, n\u00e4in, ett\u00e4 h\u00e4n oli saavuttanut sen kiusallisen vaiheen, jossa kroppa ei kuulunut lapselle eik\u00e4 aikuiselle. Mutta silm\u00e4t: ne olivat suuret ja kirkkaat. Ne saivat ajattelemaan, ett\u00e4 h\u00e4n tiesi paljon enemm\u00e4n kuin mit\u00e4 daimonit halusivat h\u00e4nen tiet\u00e4v\u00e4n.\n\nNousin yl\u00f6s ja menin l\u00e4hemm\u00e4s. \"Adriane\", sanoin \u00e4\u00e4neen. H\u00e4n oli ainoa palvelija, jonka nimen olin tajunnut kysy\u00e4. En tiennyt, mitk\u00e4 h\u00e4nen teht\u00e4v\u00e4ns\u00e4 olivat saarella tai mit\u00e4 h\u00e4n ajatteli minusta. Oliko h\u00e4n siivonnut j\u00e4lki\u00e4ni, pessyt pyykkej\u00e4ni tai kantanut minulle vett\u00e4? Hyvin mahdollista.\n\nH\u00e4n piteli toisessa k\u00e4dess\u00e4\u00e4n \u00e4mp\u00e4ri\u00e4. Tietysti. H\u00e4net oli l\u00e4hetetty vaihtamaan niin kutsuttu vessani.\n\n\"Onko sinulla lupa puhua minulle?\" kysyin.\n\nH\u00e4n ei vilkaissut k\u00e4yt\u00e4v\u00e4n suuntaan, mik\u00e4 sai minut uskomaan, ettei siell\u00e4 t\u00e4ll\u00e4 kertaa ollut ket\u00e4\u00e4n vahdissa. H\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n.\n\n\"Ei se haittaa\", sanoin. \"Sinun ei tarvitse tehd\u00e4 mit\u00e4\u00e4n, jos et halua, mutta toivoisin, ett\u00e4 voisit vastata pariin kysymykseen. Riitt\u00e4\u00e4, ett\u00e4 ny\u00f6kk\u00e4\u00e4t tai pudistat p\u00e4\u00e4t\u00e4si.\"\n\nH\u00e4n katsoi minua silmiin. Ensimm\u00e4ist\u00e4 kertaa koskaan, tajusin.\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\nOlin niin helpottunut, ett\u00e4 seuraavat sanat tulivat suustani aivan liian nopeasti. \"Saarella on toinenkin vanki. Tied\u00e4tk\u00f6 yht\u00e4\u00e4n, miss\u00e4 h\u00e4nt\u00e4 pidet\u00e4\u00e4n?\"\n\nIlmeisesti h\u00e4n oli ymm\u00e4rt\u00e4nyt, sill\u00e4 h\u00e4n ny\u00f6kk\u00e4si.\n\nHengitin hitaasti sis\u00e4\u00e4n ja ulos. \"K\u00e4ytk\u00f6 sin\u00e4 vaihtamassa h\u00e4nenkin \u00e4mp\u00e4rins\u00e4?\"\n\nH\u00e4n ep\u00e4r\u00f6i hetken, mutta ny\u00f6kk\u00e4si sitten taas.\n\nSyd\u00e4meni j\u00e4tti ly\u00f6nnin v\u00e4list\u00e4. Voisin l\u00e4hett\u00e4\u00e4 Mikaelille sanan. \u00c4kki\u00e4 tuntui silt\u00e4 kuin h\u00e4n olisi ollut paljon, paljon l\u00e4hemp\u00e4n\u00e4.\n\nTodellisuus tietenkin sai minut kiinni haaveilusta. Adrianella ja Mikaelilla ei ollut yhteist\u00e4 kielt\u00e4, ja vaikka olisinkin saanut jostain kyn\u00e4n ja paperia, kirjeen l\u00e4hett\u00e4minen Adrianen mukana olisi ollut aivan liian riskialtista.\n\nOnneksi oli toinen keino. \"Voisitko... Voisitko antaa k\u00e4tesi minulle?\"\n\nAdrianen kasvoille ilmestyi kummallinen ilme. \"\u00c4l\u00e4 huoli, en satuta sinua\", sanoin nopeasti. \"Eik\u00e4 sinun tarvitse tehd\u00e4 mit\u00e4\u00e4n muuta kuin jatkaa p\u00e4iv\u00e4\u00e4si suunnitellusti.\"\n\n\"En k\u00e4y h\u00e4nen luonaan joka p\u00e4iv\u00e4\", Adriane sanoi. Se oli pisin lause, jonka olin koskaan kuullut h\u00e4nen sanovan.\n\n\"K\u00e4ytk\u00f6 siell\u00e4 t\u00e4n\u00e4\u00e4n?\"\n\nH\u00e4n ny\u00f6kk\u00e4si. Ja ojensi k\u00e4tens\u00e4 kalterien l\u00e4pi.\n\nEn tiennyt paljoakaan susien hajuaistista, ja nyt kaduin, etten ollut ottanut siit\u00e4 selv\u00e4\u00e4. Hankasin k\u00e4si\u00e4ni k\u00e4sivarsiani vasten niin, ett\u00e4 ne olivat v\u00e4h\u00e4n hikiset ja tartuin sitten edess\u00e4ni seisovan pikkaraisen tyt\u00f6n sormiin.\n\nEi ollut mit\u00e4\u00e4n takeita siit\u00e4, ett\u00e4 Adriane menisi riitt\u00e4v\u00e4n l\u00e4helle Mikaelia, jotta t\u00e4m\u00e4 voisi haistaa minut. Ja jos temppuni onnistuisikin, en suoraan sanottuna tiennyt, miten Mikael reagoisi. Mit\u00e4 h\u00e4n ajattelisi kun saisi tiet\u00e4\u00e4, ett\u00e4 min\u00e4 olin saarella? Auttaisiko se h\u00e4nt\u00e4 sinnittelem\u00e4\u00e4n, vai antaisiko se h\u00e4nelle vain uuden syyn huoleen? H\u00e4n ei voinut tiet\u00e4\u00e4, olinko kunnossa vai en. Olinko tullut saarelle omasta vapaasta tahdostani vai pakosta niin kuin h\u00e4n. En oikeastaan ollut itsek\u00e4\u00e4n varma.\n\nMit\u00e4 kauemmin ajattelin, sit\u00e4 enemm\u00e4n aloin katua. Oli tietenkin liian my\u00f6h\u00e4ist\u00e4. Per\u00e4\u00e4nnyin mahdollisimman kauaksi ovesta, jotta Adriane saattoi avata sen ja vaihtaa \u00e4mp\u00e4rin toiseen. Kun katsoin h\u00e4nen hoikkia s\u00e4\u00e4ri\u00e4\u00e4n, en voinut olla ajattelematta, miten helppoa h\u00e4nen tainnuttamisensa olisi. Ehk\u00e4 puolen sekunnin ajan jopa harkitsin sit\u00e4. Tiesin kuitenkin heti, etten pystyisi siihen.\n\nLoppujen lopuksi kalteriovi oli ongelmistani v\u00e4h\u00e4isin. En tiennyt, miss\u00e4 Mikaelia pidettiin. Koska en tiennyt, miss\u00e4 itsek\u00e4\u00e4n olin, toiveet h\u00e4nen nopeasta l\u00f6yt\u00e4misest\u00e4\u00e4n eiv\u00e4t olleet kovin kummoiset. Umpim\u00e4hk\u00e4\u00e4n juoksentelu ei johtaisi mihink\u00e4\u00e4n. Kyvyist\u00e4nikin olisi hy\u00f6ty\u00e4 vain hyvin rajallisen ajan, sill\u00e4 en pystynyt pit\u00e4m\u00e4\u00e4n aikalukkoa kovin pitk\u00e4\u00e4n.\n\nSaatoin vain yritt\u00e4\u00e4 pit\u00e4\u00e4 pahimmat pelkoni kurissa. Olin paennut t\u00e4lt\u00e4 saarelta kerran. Pystyisin siihen uudestaan.\n\nVaikka ensimm\u00e4isell\u00e4 kerralla minun ei ollut tarvinnut paeta lukitusta sellist\u00e4.\n\nVaikka ensimm\u00e4isell\u00e4 kerralla minulla ei ollut vastassani kaikkia saaren daimoneita.\n\n***\n\nSeuraavana p\u00e4iv\u00e4n\u00e4 Adriane ilmestyi selliini j\u00e4lleen yht\u00e4 huomaamattomasti kuin varjo. H\u00e4n asetti lattialle vesikupin ja suuren lehden, jonka p\u00e4\u00e4ll\u00e4 oli leip\u00e4\u00e4. Kohotin katseeni. Olin aika varma, ettei h\u00e4nell\u00e4 ollut lupaa tuoda minulle ruokaa.\n\nAvasin suuni kiitt\u00e4\u00e4kseni h\u00e4nt\u00e4, mutta h\u00e4n nosti sormen huulilleen. Pitk\u00e4n aikaa me molemmat vain seisoimme j\u00e4hmettynein\u00e4 paikoillamme. Sitten kuulin k\u00e4yt\u00e4v\u00e4st\u00e4 loittonevia askelia ja tajusin.\n\n\"Sy\u00f6 nyt, kun min\u00e4 olen t\u00e4\u00e4ll\u00e4. He voivat tulla koska vain takaisin.\"\n\nMinua ei tarvinnut k\u00e4ske\u00e4 kahdesti. Sulloin suuni niin t\u00e4yteen leip\u00e4\u00e4, etten hetken aikaa ollut varma, saisinko nielty\u00e4 kaikkea alas.\n\nLopulta onnistuin hillitsem\u00e4\u00e4n itse\u00e4ni sen verran ett\u00e4 sain kysytty\u00e4: \"Miksi?\"\n\n\"Kaikki sanovat, ett\u00e4 se poika on peto\", Adriane kuiskasi. \"Mutta min\u00e4 en usko heit\u00e4.\"\n\n\"Mikset?\"\n\n\"H\u00e4n kiitti minua.\"\n\n\"Englanniksi?\"\n\n\"Ei. Meid\u00e4n kielell\u00e4mme.\"\n\nMin\u00e4 olin opettanut sen Mikaelille. T\u00e4rkein sana, h\u00e4n oli sanonut.\n\n\"Se on ainoa sana, jonka h\u00e4n osaa\", sanoin. Ja vasta silloin tajusin, miten nurinkurista oli kiitt\u00e4\u00e4 silloin kun oli vankina. Naurahdin. Vain Mikael oli niin kohtelias.\n\n\"Miten h\u00e4n voi? Onko h\u00e4n... kunnossa?\" Oikeasti halusin tiet\u00e4\u00e4, miten pahasti h\u00e4nt\u00e4 oli pahoinpidelty. Mutten pystynyt kysym\u00e4\u00e4n sit\u00e4.\n\n\"H\u00e4n taistelee\", Adriane sanoi. \"He eiv\u00e4t tietenk\u00e4\u00e4n pid\u00e4 siit\u00e4 ja n\u00e4ytt\u00e4v\u00e4t sen h\u00e4nelle. Mutta viel\u00e4 he eiv\u00e4t ole onnistuneet murtamaan h\u00e4nt\u00e4.\"\n\n\"Tied\u00e4tk\u00f6, mit\u00e4 he suunnittelevat h\u00e4nen varalleen?\"\n\nH\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"En, mutta luulen, ett\u00e4 he aikovat k\u00e4ytt\u00e4\u00e4 h\u00e4nt\u00e4 jotenkin seremoniassa.\"\n\nK\u00e4ytt\u00e4\u00e4 Mikaelia seremoniassa? Se ei varmasti voinut tarkoittaa mit\u00e4\u00e4n hyv\u00e4\u00e4.\n\n\"Te rakastatte toisianne\", Adriane huomioi.\n\n\"Niin.\" En tiennyt, oliko h\u00e4n kuullut meist\u00e4 vai p\u00e4\u00e4tellyt sen itse, mutta en ikimaailmassa olisi mennyt kielt\u00e4m\u00e4\u00e4n. \"Miss\u00e4 h\u00e4nt\u00e4 pidet\u00e4\u00e4n?\"\n\n\"Tied\u00e4tk\u00f6 sen rakennuksen, jossa s\u00e4ilytet\u00e4\u00e4n jousia ja muita aseita?\"\n\nNy\u00f6kk\u00e4sin.\n\n\"Siin\u00e4 rakennuksessa on kellari.\"\n\nSe antoi minulle toivoa. Paikka ei ollut saaren vilkkaimpia. Mik\u00e4 parasta, rakennus ei sijainnut liian kaukana laiturista. Olisi ehk\u00e4 jopa mahdollista kulkea asevarastolta laiturille niin, ettei kukaan huomaisi.\n\n\"Kiitos\", sanoin.\n\nAdriane hymyili minulle. Se oli kaunis hymy, ja ensimm\u00e4ist\u00e4 kertaa ajattelin, miten erilaista h\u00e4nen el\u00e4m\u00e4ns\u00e4 olisi, jos h\u00e4n ei olisi koskaan joutunut saarelle.\n\nMeill\u00e4 oli siis jotain yhteist\u00e4.\n\n***\n\nV\u00e4hitellen toiveikkuuteni alkoi hiipua. Minulla oli koko ajan n\u00e4lk\u00e4 ja jano, eik\u00e4 p\u00e4ivisin seinien l\u00e4pi tunkeva kuumuus helpottanut asiaa. Mit\u00e4\u00e4n ei tapahtunut aikana, joka tuntui monelta p\u00e4iv\u00e4lt\u00e4, mutta joka ikkunastani tulevan valon mukaan oli kest\u00e4nyt ainoastaan puolitoista vuorokautta. Kaikista ep\u00e4toivoisimpina hetkin\u00e4ni ep\u00e4ilin, ett\u00e4 jopa aurinko valehteli minulle.\n\nSuurimman osan ajasta torkuin. Jossain vaiheessa minun oli pit\u00e4nyt nukahtaa, sill\u00e4 her\u00e4sin k\u00e4yt\u00e4v\u00e4st\u00e4 kuuluviin askeliin.\n\n\"Her\u00e4\u00e4\", joku kuiskasi, ja sitten ilmavirta pyyhk\u00e4isi ohitseni.\n\nNousin s\u00e4ps\u00e4ht\u00e4en. En ollut varma, mik\u00e4 oli unta ja mik\u00e4 totta, joten h\u00e4mm\u00e4stykseni oli suuri, kun huomasin, ett\u00e4 joku oli k\u00e4ynyt vaihtamassa \u00e4mp\u00e4rini.\n\nMutta se ei ollut mit\u00e4\u00e4n verrattuna \u00e4llistykseeni, kun tajusin, ett\u00e4 ovi oli auki.\n\nSen ei olisi pit\u00e4nyt olla mahdollista. Ensimm\u00e4inen ajatukseni oli, ett\u00e4 t\u00e4m\u00e4n t\u00e4ytyi olla ansa. J\u00e4lleen yksi uusi yritys sekoittaa p\u00e4\u00e4t\u00e4ni. Sitten muistin, kuka oli k\u00e4ynyt selliss\u00e4ni viimeksi ja aloin ihmetell\u00e4, olinko kenties saanut liittolaisen.\n\nJos Adriane oli todella j\u00e4tt\u00e4nyt oven auki ja min\u00e4 pakenisin, h\u00e4n joutuisi pulaan. Mutta luultavasti h\u00e4n joutuisi pulaan, vaikkei olisikaan tehnyt sit\u00e4.\n\nTyhm\u00e4, urhea tytt\u00f6, ajattelin. En ollut tehnyt mit\u00e4\u00e4n, mink\u00e4 vuoksi h\u00e4nen olisi pit\u00e4nyt olla valmis asettamaan itsens\u00e4 hengenvaaraan takiani. H\u00e4n ei pystynyt auttamaan edes itse\u00e4\u00e4n, joten miksi h\u00e4n auttoi minua?\n\nTy\u00f6nsin itseni hitaasti yl\u00f6s sellin lattialta ja menin ovelle. Katsoin kalterien l\u00e4pi k\u00e4yt\u00e4v\u00e4\u00e4n. Ei ket\u00e4\u00e4n.\n\nLiikutin hitaasti k\u00e4tt\u00e4ni kohti kaltereita, kunnes sormeni puristuivat yhden ymp\u00e4rille. Ovi ei ollut narahtanut kertaakaan sin\u00e4 aikana, jonka olin ollut selliss\u00e4, mutta siltik\u00e4\u00e4n en luottanut siihen, ettei se narahtaisi juuri nyt.\n\nOvi ei narahtanut. Astuin k\u00e4yt\u00e4v\u00e4\u00e4n.\n\nHetken aikaa vain seisoin sellini ulkopuolella. Odotin kuulevani juoksuaskeleita, jotain merkki\u00e4 siit\u00e4, ett\u00e4 pakoni olisi huomattu ja daimonit olisivat rynt\u00e4\u00e4m\u00e4ss\u00e4 minua kohti. En kuitenkaan kuullut mit\u00e4\u00e4n.\n\nOtin varovaisen askeleen toisensa per\u00e4\u00e4n, kunnes uskalsin kasvattaa vauhtia. K\u00e4yt\u00e4v\u00e4n p\u00e4\u00e4ss\u00e4 jo juoksin.\n\nPys\u00e4hdyin portaiden keskivaiheilla. En voinut tiet\u00e4\u00e4, seisoisiko joku vahdissa portaiden yl\u00e4p\u00e4\u00e4ss\u00e4. Vedin keuhkoni t\u00e4yteen, kuin olisin ollut sukeltamassa veteen ja rynt\u00e4sin auringonvaloon.\n\nKukaan ei k\u00e4ynyt kimppuuni. Sen sijaan \u00e4killinen valo hy\u00f6kk\u00e4si n\u00e4k\u00f6hermojani vastaan ja hetken aikaa tulipallot tanssivat silmiss\u00e4ni. Nojasin sein\u00e4\u00e4 vasten ja yritin saada n\u00e4k\u00f6ni taas toimimaan. Heti, kun aloin erottaa rakennusten \u00e4\u00e4riviivoja, pakotin itseni liikkeelle.\n\nSiristelty\u00e4ni silm\u00e4ni hetken uskoin tiet\u00e4v\u00e4ni, miss\u00e4 olin. Helpotuksekseni olin l\u00e4hell\u00e4 sek\u00e4 laituria ett\u00e4 asevarastoa.\n\nOlin varma, ett\u00e4 joku huomaisi minut. Hetken aikaa harkitsin k\u00e4ytt\u00e4v\u00e4ni kykyj\u00e4ni. Mutta minun piti s\u00e4\u00e4stell\u00e4 voimiani. En tavallisestikaan pystyisi pit\u00e4m\u00e4\u00e4n aikalukkoa yll\u00e4 kovin kauan, ja oloni oli kaukana tavanomaisesta. Kylm\u00e4 hiki kohosi iholleni paahteesta huolimatta. Muistutin itse\u00e4ni, ettei minulla ollut aikomustakaan j\u00e4\u00e4d\u00e4 kiinni. Jos kaikki sujuisi niin kuin kuvittelin, min\u00e4 ja Mikael olisimme pian matkalla kohti auringonlaskua.\n\nP\u00e4\u00e4sin livahtamaan helposti alas laiturille. Ehdin riemuita vain hetken, ennen kuin tajusin, mist\u00e4 se oli johtunut: vene oli poissa.\n\nYritin ajatella. T\u00e4h\u00e4n asti kroppani oli toiminut automaattiohjauksella, ja aivoni vaativat maanittelua ennen kuin ne suostuivat k\u00e4ynnistym\u00e4\u00e4n. Venett\u00e4 k\u00e4yttiv\u00e4t paitsi daimonit, my\u00f6s palvelijat.\n\nCallista oli korostanut seremonian t\u00e4rkeytt\u00e4. Se tarkoitti, ett\u00e4 kaikki olisivat paikalla. Palvelijoilla olisi t\u00e4ysi ty\u00f6 daimonien kestitsemisess\u00e4. Se taas tarkoitti, ett\u00e4 my\u00f6s kaikki palvelijat olisivat paikalla.\n\nJa se tarkoitti, ett\u00e4 vene olisi paikalla l\u00e4hes sadan prosentin varmuudella.\n\nOli haastavampaa kivuta portaat yl\u00f6s kalliolle kuin tulla ne alas, mutta onnistuin p\u00e4\u00e4sem\u00e4\u00e4n asevarastolle kenenk\u00e4\u00e4n huomaamatta. Kyyristyin kivien taakse ja t\u00e4hyilin pient\u00e4, hiekkakent\u00e4n vieress\u00e4 seisovaa rakennusta. Erotin heti kaksi daimonia, jotka seisoivat oven edess\u00e4. Kun pinnistin kuuloni \u00e4\u00e4rimmilleen, saatoin erottaa heid\u00e4n sanansa.\n\nEnsimm\u00e4inen sanoi: \"Minulla on paha aavistus t\u00e4st\u00e4.\"\n\n\"\u00c4l\u00e4 sano noin\", toinen vastasi. \"Kerj\u00e4\u00e4t vain vaikeuksia.\"\n\nEnsimm\u00e4inen daimon pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Vain Callista, Vlassis ja pari muuta palasivat takaisin saarelle. Tied\u00e4t, mit\u00e4 se tarkoittaa.\"\n\nHetken aikaa p\u00e4\u00e4ss\u00e4ni l\u00f6i tyhj\u00e4\u00e4. Sitten tajusin, mihin h\u00e4n viittasi: daimonit, jotka olivat tulleet Sarrajuoksan m\u00f6kille ja jotka oli m\u00e4\u00e4r\u00e4tty tappamaan Mitja ja Daniel \u2013 l\u00e4hes kukaan heist\u00e4 ei ollut palannut. Jotain oli tapahtunut heille kaikille.\n\nMitja ja Daniel olivat elossa.\n\nEn ollut tajunnut, kuinka huolissani olin ollut, ennen kuin tunne l\u00e4hti kropastani. Painoin k\u00e4mmenen suuni p\u00e4\u00e4lle ja rutistin riemuni tiukaksi palloksi. Minulla ei ollut varaa juhlia. Minut l\u00f6ydett\u00e4isiin hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4.\n\nEi olisi pienint\u00e4k\u00e4\u00e4n mahdollisuutta saada Mikaelia ulos kenenk\u00e4\u00e4n huomaamatta ja pysy\u00e4 saarella piilossa niin kauan, ett\u00e4 vene palaisi.\n\nSamalla tajusin toisen aukon suunnitelmassani. En ollut koskaan kipparoinut venett\u00e4, mutten uskonut sen olevan kovin vaikeaa. Suurempi ongelma oli avaimien hankkiminen. En tiennyt, miss\u00e4 niit\u00e4 pidettiin. Ne saattoivat olla kenell\u00e4 tahansa.\n\nTein p\u00e4\u00e4t\u00f6ksen. L\u00e4hdin h\u00f6lkk\u00e4\u00e4m\u00e4\u00e4n takaisin. Odottaisin seremoniaa, jolloin Mikael ja minut tuotaisiin ulos selleist\u00e4mme. Minun tarvitsi vain pys\u00e4ytt\u00e4\u00e4 aika, ja me voisimme paeta veneelle.\n\nTulin portaiden yl\u00e4p\u00e4\u00e4h\u00e4n. Juoksin ne alas ja jatkoin kapeaa k\u00e4yt\u00e4v\u00e4\u00e4 pitkin tutun sellin ovelle.\n\nOvi kolahti kiinni takanani. Odotin, ett\u00e4 joku olisi kuullut sen ja tulisi pian tarkistamaan, mit\u00e4 oikein tapahtui, mutta niin ei k\u00e4ynyt.\n\nKlaustrofobia tuntui heti paljon pahempana, kun vapaus oli ollut niin l\u00e4hell\u00e4.\n\nN\u00e4ht\u00e4v\u00e4ksi j\u00e4isi, oliko t\u00e4m\u00e4 kaikkien aikojen nerokkain vai typerin temppuni.\nKOLMASTOISTA LUKU\n\nSeuraavana p\u00e4iv\u00e4n\u00e4 toivoin n\u00e4kev\u00e4ni Adrianen. Halusin varmistua, ett\u00e4 h\u00e4n oli kunnossa, mutta my\u00f6s kysy\u00e4, tiesik\u00f6 h\u00e4n, miss\u00e4 veneen avaimia s\u00e4ilytettiin. Lis\u00e4ksi halusin selitt\u00e4\u00e4 h\u00e4nelle, miksi olin j\u00e4tt\u00e4nyt k\u00e4ytt\u00e4m\u00e4tt\u00e4 h\u00e4nen luomansa tilaisuuden.\n\nOlin siis toiveikas, kun k\u00e4yt\u00e4v\u00e4st\u00e4 kuului \u00e4\u00e4ni\u00e4. Mutta tulija ei ollutkaan Adriane, vaan Callista.\n\nR\u00e4pyttelin. H\u00e4nen ajoituksensa tuntui pahaenteiselt\u00e4. Hetken ajan olin varma, ett\u00e4 olin j\u00e4\u00e4nyt kiinni. Joku oli n\u00e4hnyt minut, tai Adriane oli kannellut.\n\nH\u00e4n ker\u00e4si hameenhelman k\u00e4teens\u00e4. T\u00e4ydelliseksi tyrmistyksekseni h\u00e4n istui viereeni lattialle. Kovalle, tomuiselle lattialle.\n\n\"Sin\u00e4 et tiennyt t\u00e4t\u00e4 is\u00e4st\u00e4si, mutta h\u00e4net tunnettiin parhaana soturina, jonka t\u00e4m\u00e4 saari on n\u00e4hnyt\", h\u00e4n aloitti. \"H\u00e4n oli hyvin ihailtu.\"\n\nNielaisin. En halunnut paljastaa h\u00e4nelle, miten paljon jo tiesin is\u00e4st\u00e4ni. T\u00e4m\u00e4 Callista pelotti minua enemm\u00e4n kuin kylm\u00e4 ja laskelmoiva.\n\nSiit\u00e4 huolimatta \u2013 tai juuri sen takia \u2013 en voinut olla kysym\u00e4tt\u00e4: \"Kerroitko h\u00e4nelle, ett\u00e4 aiot tappaa h\u00e4nen lapsensa?\"\n\nH\u00e4n huokaisi itselleen ep\u00e4tyypillisesti. \"Min\u00e4 en aio tappaa sinua.\"\n\nMinun olisi pit\u00e4nyt olla iloinen, ett\u00e4 edes joku saarella ajatteli niin, mutta ennen kuin huomasinkaan, olin jo avannut suuni. Ja sielt\u00e4 p\u00e4\u00e4si ulos kohtalaisen typeri\u00e4 asioita. \"Miksei? Kaikki ongelmat poistuvat, jos minua ei ole en\u00e4\u00e4 olemassa.\"\n\n\"P\u00e4invastoin.\"\n\n\"Mit\u00e4?\"\n\nH\u00e4n k\u00e4\u00e4ntyi katsomaan minua. \"Min\u00e4 olen odottanut sinua jo hyvin kauan. Nyt se vihdoin tapahtuu.\"\n\nK\u00e4vin sanat monta kertaa l\u00e4pi mieless\u00e4ni ennen kuin olin varma, ett\u00e4 olin ymm\u00e4rt\u00e4nyt oikein. \"Mik\u00e4 ihmeen 'se'?\" kysyin.\n\n\"Loppu\", Callista vastasi. \"Ja samalla kaiken alku. Me tuhoamme t\u00e4m\u00e4n maailman, jotta voimme aloittaa uuden.\"\n\nH\u00e4n on hullu, ajattelin. Mutta kun katsoin h\u00e4nt\u00e4, h\u00e4n ei n\u00e4ytt\u00e4nyt hullulta. Olisi voinut luulla, ett\u00e4 l\u00e4heisyys olisi paljastanut h\u00e4nen inhimillisemm\u00e4n puolensa, mutta istuessaan siin\u00e4 vieress\u00e4ni h\u00e4n p\u00e4invastoin tuntui mahtavammalta kuin koskaan. H\u00e4n tiesi tismalleen, mist\u00e4 puhui.\n\n\"En voi olla paikalla seremoniassasi, koska j\u00e4t\u00e4n t\u00e4m\u00e4n maailman ennen sit\u00e4\", h\u00e4n sanoi.\n\nYritin keksi\u00e4, mit\u00e4 se tarkoitti. Siis mit\u00e4 muuta kuin kuvittelin sen tarkoittavan. Lopulta minun oli pakko sanoa ep\u00e4ilykseni \u00e4\u00e4neen. \"Sin\u00e4 siis... kuolet. Vapaaehtoisesti.\"\n\nOdotin h\u00e4nen lytt\u00e4\u00e4v\u00e4n ep\u00e4ilyni. H\u00e4n ei tehnyt niin. \"Min\u00e4 astun ideoiden maailmaan. Minun teht\u00e4v\u00e4ni on menn\u00e4 edelt\u00e4 varmistamaan, ettet sin\u00e4 koskaan p\u00e4\u00e4se portista l\u00e4pi. Mutta he eiv\u00e4t tied\u00e4, ettei minulla ole aikomustakaan est\u00e4\u00e4 sinua p\u00e4\u00e4sem\u00e4st\u00e4 ideoiden maailmaan. P\u00e4invastoin, min\u00e4 aion olla saattamassa sinut perille.\"\n\nKuuntelin outoa huminaa p\u00e4\u00e4ni sis\u00e4ll\u00e4. Odotin ett\u00e4 reagoisin jotenkin, mutta mit\u00e4\u00e4n ei tapahtunut. En tiennyt, mit\u00e4 ajatella. \"Miksi?\" sain kysytyksi.\n\n\"T\u00e4m\u00e4 maailma ei ole en\u00e4\u00e4 pelastettavissa. Jumalat ovat k\u00e4yneet saamattomiksi. He eiv\u00e4t en\u00e4\u00e4 hoida teht\u00e4vi\u00e4\u00e4n. Yhdess\u00e4 me sy\u00f6ksemme nykyiset jumalat vallasta ja rakennamme uuden maailman. Paremman maailman.\"\n\n\"Min\u00e4 en usko mihink\u00e4\u00e4n jumaliin\", sanoin.\n\nHymy levisi h\u00e4nen kasvoilleen. \"Hyv\u00e4. Hek\u00e4\u00e4n eiv\u00e4t usko sinuun.\"\n\nH\u00e4nen innostuksensa yll\u00e4tti minut. Ensimm\u00e4ist\u00e4 kertaa h\u00e4n kuulosti yht\u00e4 nuorelta kuin milt\u00e4 n\u00e4ytti. \"En ole paljastanut suunnitelmaani kenellek\u00e4\u00e4n muulle kuin Vlassisille.\"\n\n\"Miksi Vlassis? Miksei joku muu?\" Luulin kuitenkin tiet\u00e4v\u00e4ni vastauksen. H\u00e4n vihasi is\u00e4\u00e4ni. H\u00e4n ei tuntenut mink\u00e4\u00e4nlaista s\u00e4\u00e4li\u00e4 minua kohtaan.\n\n\"H\u00e4n ymm\u00e4rt\u00e4\u00e4, kuinka paljon meill\u00e4 on peliss\u00e4. Ja h\u00e4n on l\u00e4p\u00e4issyt jokaisen testini.\" Kun en sanonut mit\u00e4\u00e4n, h\u00e4n jatkoi: \"Tiesitk\u00f6, ett\u00e4 is\u00e4si pyysi h\u00e4nen apuaan? H\u00e4nen piti vied\u00e4 veljesi pois Euroopasta. Kielsin h\u00e4nt\u00e4 tekem\u00e4st\u00e4 niin.\"\n\nSill\u00e4 tavalla Mitja oli siis p\u00e4\u00e4tynyt kasvamaan kanssani samassa kaupungissa.\n\n\"Suurimmalle osalle t\u00e4ll\u00e4 saarella on mahdotonta hyv\u00e4ksy\u00e4 sit\u00e4, mit\u00e4 on teht\u00e4v\u00e4\", Callista sanoi. \"He ovat siihen liian itsekk\u00e4it\u00e4.\"\n\n\"Oletko ihan varma? Vaikea kuvitella, ett\u00e4 kenell\u00e4k\u00e4\u00e4n olisi mit\u00e4\u00e4n maailmanloppua vastaan.\" Sarkasmini oli vain puolittaista. Vaikutti nimitt\u00e4in silt\u00e4, ett\u00e4 oli paljonkin ihmisi\u00e4, joita ei kiinnostanut est\u00e4\u00e4 tulevien sukupolvien tuhoa. Ainoa ero oli tietenkin siin\u00e4, ett\u00e4 Callista aikoi p\u00e4\u00e4tt\u00e4\u00e4 kaiken t\u00e4ss\u00e4 ja nyt.\n\n\"He ovat lyhytn\u00e4k\u00f6isi\u00e4. Min\u00e4 ajattelen meit\u00e4 kaikkia, meid\u00e4n kaikkien tulevaisuutta. Tuhansiksi vuosiksi eteenp\u00e4in.\" Callista alkoi silitt\u00e4\u00e4 hiuksiani. Se j\u00e4rkytti minua enemm\u00e4n kuin mik\u00e4\u00e4n muu. \"Min\u00e4 olen n\u00e4hnyt, miten kohtalot ohjailevat ihmisi\u00e4. Mit\u00e4 enemm\u00e4n yrit\u00e4t paeta kohtaloasi, sit\u00e4 v\u00e4\u00e4j\u00e4\u00e4m\u00e4tt\u00f6m\u00e4mm\u00e4ksi se k\u00e4y. Ja sit\u00e4 tuskallisemmaksi.\" H\u00e4n otti yhden hiussuortuvan sormiensa v\u00e4liin. \"T\u00e4ss\u00e4 maailmassa ei ole sinulle mit\u00e4\u00e4n\", h\u00e4n sanoi. \"Sin\u00e4 et vain ole ymm\u00e4rt\u00e4nyt sit\u00e4 viel\u00e4.\"\n\nMiten niin ei muka ollut? T\u00e4ss\u00e4 maailmassa minulla oli kaikki. Enemm\u00e4n kuin koskaan aikaisemmin el\u00e4m\u00e4ni aikana. Enemm\u00e4n kuin olin uskaltanut edes unelmoida.\n\n\"Tyt\u00e4r. Sinua ei koskaan tarkoitettu t\u00e4h\u00e4n maailmaan. Sinun ruumiisi ei sied\u00e4 sit\u00e4, mik\u00e4 olet. Sinun t\u00e4ytyy tuntea se. Ruumiisi k\u00e4y sotaa itse\u00e4\u00e4n vastaan. Susi vastaan daimon. Arkkiviholliset. Aina kun toinen puoli vahvistuu, toinen hy\u00f6kk\u00e4\u00e4 sen kimppuun. Sinun kaltaisesi ei pit\u00e4isi olla edes olemassa, mutta siin\u00e4 sin\u00e4 olet.\"\n\nOireeni olivat alkaneet silloin, kun olin tullut ensimm\u00e4ist\u00e4 kertaa Hukkavaaraan. Ne olivat pahentuneet sen j\u00e4lkeen kun olin oppinut k\u00e4ytt\u00e4m\u00e4\u00e4n kykyj\u00e4ni, ja voimistuneet entisest\u00e4\u00e4n, kun olin liittynyt laumaan.\n\nMutta jos minussa oli susi, miksi en tuntenut sit\u00e4?\n\nTuntui kuin Callista olisi luikerrellut p\u00e4\u00e4ni sis\u00e4lle paljon taidokkaammin kuin Konsta, enk\u00e4 voisi edes tiet\u00e4\u00e4, mitk\u00e4 ajatukset olivat omiani ja mitk\u00e4 eiv\u00e4t.\n\nMikael. Vaikka kaikki muu johtaisi harhaan, h\u00e4n ei koskaan. Niin kauan kuin pidin h\u00e4net ajatusteni ja suunnitelmieni keski\u00f6ss\u00e4, kaikki olisi hyvin.\n\nMinulla ei ollut hajuakaan, miten saisin tilanteen k\u00e4\u00e4ntym\u00e4\u00e4n puolelleni. Mutta varmaa oli, ett\u00e4 minun t\u00e4ytyi saada tiet\u00e4\u00e4 enemm\u00e4n. Ja parhaat mahdollisuudet siihen olisi, jos Callista kuvittelisi minun olevan puolellaan.\n\nSanoin siis: \"E-en tied\u00e4, pystynk\u00f6 auttamaan sinua niin kuin haluat. En ymm\u00e4rr\u00e4, mit\u00e4 sin\u00e4 uskot minun voivan tehd\u00e4.\"\n\n\"\u00c4l\u00e4 huolehdi siit\u00e4. Ensin sinun t\u00e4ytyy luopua ruumiistasi.\"\n\nH\u00e4n sai sen kuulostamaan yht\u00e4 helpolta kuin vaatteiden riisumisen.\n\n\"Sattuu vain hetken, kun ihmisel\u00e4m\u00e4si pyyhkiytyy pois. Saat tilalle jotain parempaa, jotain todellisempaa. Sinun t\u00e4ytyy antaa tyhjyyden tulla.\"\n\nTajusin h\u00e4m\u00e4r\u00e4sti, ett\u00e4 h\u00e4n luuli minun olevan poissa tolaltani siksi, ett\u00e4 pelk\u00e4sin kuolemisen sattuvan. Mutta minulla ei ollut aikomustakaan kuolla. Vaikka h\u00e4nen sanansa olisivatkin olleet totta, olin hyvin p\u00e4\u00e4tt\u00e4v\u00e4inen el\u00e4m\u00e4\u00e4n el\u00e4m\u00e4ni t\u00e4ss\u00e4 maailmassa, jonka tunsin, ja ottamaan siit\u00e4 kaiken irti.\n\nEnsimm\u00e4ist\u00e4 kertaa h\u00e4n arvioi minut t\u00e4ysin v\u00e4\u00e4rin, ja tajusin heti, ett\u00e4 minun t\u00e4ytyi pyrki\u00e4 hy\u00f6tym\u00e4\u00e4n siit\u00e4 \u2013 tehd\u00e4 parhaani, jotta railo levenisi entisest\u00e4\u00e4n.\n\n\"Sin\u00e4 haluat tulla jumalaksi\", sanoin. Toivoin kuulostavani h\u00e4mm\u00e4styneelt\u00e4, ihailevalta \u2013 milt\u00e4 tahansa mik\u00e4 saisi h\u00e4net vakuuttumaan siit\u00e4, ett\u00e4 uskoin h\u00e4nt\u00e4.\n\n\"Ei. Min\u00e4 olen jo jumala.\" P\u00e4\u00e4ni k\u00e4\u00e4ntyi niin nopeasti, ett\u00e4 meni hetken, ennen kuin n\u00e4k\u00f6ni ter\u00e4v\u00f6ityi.\n\n\"Minut karkotettiin\", h\u00e4n sanoi. \"Olen ollut saarella siit\u00e4 l\u00e4htien. Mutta sinun avullasi min\u00e4 voin viimein palata kotiin.\"\n\n\"Min\u00e4 en usko sinua\", sanoin.\n\n\"Kyll\u00e4 sin\u00e4 uskot.\"\n\nEn pystynyt katsomaan h\u00e4nt\u00e4 yht\u00e4\u00e4n pidemp\u00e4\u00e4n. Tuijotin edess\u00e4ni olevaa sein\u00e4\u00e4 ja toivoin, ett\u00e4 h\u00e4n l\u00e4htisi.\n\n\"\u00c4l\u00e4 pelk\u00e4\u00e4, tyt\u00e4r. Yhdess\u00e4 me teemme viel\u00e4 ihmeellisi\u00e4 asioita.\"\n\nH\u00e4n l\u00e4hti.\n\nKun on yksin suljetussa tilassa ilman mink\u00e4\u00e4nlaisia virikkeit\u00e4, aistit ter\u00e4v\u00f6ityv\u00e4t.\n\nYritin kuulla meren \u00e4\u00e4nen. Pinnistin kuuloni \u00e4\u00e4rimmilleen. Lopulta uskoinkin kuulevani jotain, mutta en voinut en\u00e4\u00e4 olla varma, kuvittelinko kaiken.\n\nAjattelin sit\u00e4, kuinka susilla ja merell\u00e4 oli jotain yhteist\u00e4: ne molemmat kuulivat kuun kutsun. Vuorovedet nousivat ja laskivat kuun voimasta, aivan kuten sudet muuttuivat sen k\u00e4skyst\u00e4.\n\nKatsoin kun auringons\u00e4teet tulivat sis\u00e4\u00e4n pikkuruisesta ikkunastani. Lopulta l\u00e4mmin, lohdullinen oranssi levitt\u00e4ytyi koko selliini. Aivan kuin ruumiinikin olisi hiljentynyt hetkeksi, sill\u00e4 en kuullut hengityst\u00e4ni, pulssiani, mit\u00e4\u00e4n. Maailma ymp\u00e4rill\u00e4ni oli hauta ja min\u00e4 kuollut.\n\nVedin polvet leukaani ja kiedoin Mikaelin kuvan ymp\u00e4rilleni kuin torkkupeiton.\n\nOli ihmeellist\u00e4, miten t\u00e4llaisen kaaoksen keskelt\u00e4 saattoi l\u00f6yt\u00e4\u00e4 rauhan. Tai ehk\u00e4 rauha syntyikin juuri kaaoksen kaikuna, v\u00e4h\u00e4n niin kuin meri pyyhkii rannalta kaikki j\u00e4ljet niin, ett\u00e4 hetken aikaa hiekka on t\u00e4ysin koskematon.\n\nAuringonlaskuun menness\u00e4 uutinen kantautui viimein selliini asti.\n\nCallista oli kuollut.\n\n***\n\nIs\u00e4ni seisoi kalliolla. H\u00e4n n\u00e4ytti vanhemmalta kuin aikaisemmin. Kuluneelta ja risaiselta kuin loppuun k\u00e4vellyt keng\u00e4t.\n\nEn tunnistanut paikkaa. Se ei n\u00e4ytt\u00e4nyt milt\u00e4\u00e4n sellaiselta maalta, jossa olin k\u00e4ynyt. Se ei n\u00e4ytt\u00e4nyt edes milt\u00e4\u00e4n sellaiselta planeetalta, jossa olisin k\u00e4ynyt, mutten sent\u00e4\u00e4n uskonut is\u00e4ni ehtineen Kuuhun asti. Kalliota jatkui silm\u00e4nkantamattomiin, ja taivas n\u00e4ytti olevan hurjan korkealla. Sen sini kalpeni korkeuksiin kohotessaan. Taustalta kuului \u00e4\u00e4nt\u00e4, jota en tunnistanut.\n\nH\u00e4n otti takin taskusta askin ja sytytti tupakan. Liikkeet olivat niin tutut, ett\u00e4 miltei unohdin, etten ollut oikeasti koskaan tavannut is\u00e4\u00e4ni.\n\n\"Sin\u00e4 olet varmaankin tullut tappamaan minut\", h\u00e4n sanoi tuuleen. \"Et ole ensimm\u00e4inen etk\u00e4 viimeinen.\" Kesti hetken, ennen kuin \u00e4lysin h\u00e4nen sanojensa merkityksen.\n\nAlek seisoi viistosti is\u00e4ni takana ja osoitti h\u00e4nt\u00e4 jousella.\n\nOlin unohtanut, miten nuorelta h\u00e4n oli n\u00e4ytt\u00e4nyt. Miten palavasilm\u00e4iselt\u00e4 ja vakavalta. Nielaisin kurkkuuni juuttuneen palan alas.\n\nStefanos ei katsonut h\u00e4neen p\u00e4ink\u00e4\u00e4n, enk\u00e4 tiennyt, miten h\u00e4n oli tiennyt Alekin l\u00e4sn\u00e4olosta.\n\n\"Kerro minulle, miss\u00e4 sinun lapsesi ovat.\" Minun t\u00e4ytyi antaa Alekille pisteit\u00e4 \u2013 h\u00e4n ei antanut is\u00e4ni piittaamattomuuden vaikuttaa itseens\u00e4.\n\nStefanos nauroi ja pudisti p\u00e4\u00e4t\u00e4\u00e4n. H\u00e4n antoi tupakan pudota maahan ja astui sen p\u00e4\u00e4lle. T\u00e4m\u00e4 oli h\u00e4nelle pelkk\u00e4 vitsi, mutta katkera sellainen.\n\n\"Min\u00e4 l\u00f6yd\u00e4n heid\u00e4t joka tapauksessa. Jumalat ovat minun puolellani. He ovat sinunkin puolellasi, jos vain autat minua. Viel\u00e4 ei ole liian my\u00f6h\u00e4ist\u00e4 tehd\u00e4 oikein ja alkaa palvella heit\u00e4. Sinun t\u00e4ytyy vain l\u00f6yt\u00e4\u00e4 uskosi.\"\n\nIs\u00e4ni ei en\u00e4\u00e4 virnuillut. \"Mik\u00e4 saa sinut ajattelemaan, etten usko jumaliin?\"\n\nTajusin \u00e4kki\u00e4, mik\u00e4 taustalla kuuluva \u00e4\u00e4ni oli: se oli koski.\n\nIs\u00e4ni ruumis l\u00f6ytyi joesta, ajattelin.\n\nAlekin sormet vapauttivat nuolen.\n\nIs\u00e4ni v\u00e4isti. H\u00e4n otti veitsens\u00e4 esiin. Vasta silloin tajusin, ett\u00e4 ne olivat hyvin tutunn\u00e4k\u00f6iset \u2013 ne olivat nimitt\u00e4in minun veitseni.\n\nAlek ampui toisen nuolen. Is\u00e4ni v\u00e4isti taas, mutta t\u00e4ll\u00e4 kertaa aivan viime tipassa. H\u00e4n vastasi rynt\u00e4\u00e4m\u00e4ll\u00e4 Alekia p\u00e4in.\n\nHe alkoivat taistella. Liikkeet olivat minulle hyvin tuttuja saarelta, mutta en erehtynyt hetkeksik\u00e4\u00e4n luulemaan t\u00e4t\u00e4 tavanomaiseksi otteluksi. Alusta asti n\u00e4ki, ett\u00e4 he kamppailivat el\u00e4m\u00e4st\u00e4 ja kuolemasta.\n\nJa alusta asti n\u00e4ki, ett\u00e4 is\u00e4ni oli hidas. Paljon hitaampi kuin millaisena olin n\u00e4hnyt h\u00e4nen olevan murhatessaan nimett\u00f6mi\u00e4 ihmisi\u00e4. Vaikka Alek oli is\u00e4\u00e4ni pienikokoisempi, h\u00e4n sai Stefanosin helposti maanpintaan.\n\nKuvittelin Alekin p\u00e4\u00e4tt\u00e4v\u00e4n silloin is\u00e4ni el\u00e4m\u00e4n, mutta h\u00e4n n\u00e4ytti j\u00e4hmettyneen paikoilleen. H\u00e4n piteli is\u00e4\u00e4ni otteessaan, mutta h\u00e4nen katseensa oli et\u00e4inen.\n\n\"Punainen tukka... Helsinki...\" h\u00e4n mumisi \u00e4\u00e4neen. Siit\u00e4 vihdoin tajusin, mit\u00e4 h\u00e4n teki. H\u00e4n kaiveli is\u00e4ni muistoja.\n\nEnsimm\u00e4ist\u00e4 kertaa n\u00e4in oikean paniikin is\u00e4ni kasvoilla. H\u00e4n rimpuili itsens\u00e4 irti ja juoksi kohti jokea. En ymm\u00e4rt\u00e4nyt h\u00e4nen taktiikkaansa, eik\u00e4 ilmeisesti Alekkaan, sill\u00e4 n\u00e4in hetkellisen h\u00e4mmennyksen h\u00e4nen kasvoillaan ennen kuin h\u00e4n l\u00e4hti is\u00e4ni per\u00e4\u00e4n.\n\nKatsoin syd\u00e4n syrj\u00e4ll\u00e4ni, kun Alek ampui taas yhden nuolen. Se kiisi ilman halki ja osui is\u00e4\u00e4ni selk\u00e4\u00e4n.\n\nH\u00e4n k\u00e4\u00e4ntyi kohti Alekia. Kuvittelin h\u00e4nen sanovan jotain, mutta sitten h\u00e4n kohotti veitsens\u00e4 ja leikkasi oman ranteensa auki.\n\nH\u00e4n sulki silm\u00e4ns\u00e4. Ei kivusta, ei surusta. Helpotuksesta.\n\nH\u00e4n kaatui taaksep\u00e4in. H\u00e4nen ruumiinsa upposi veteen ja vei loput salaisuudet mukanaan.\n\n***\n\nMinua tultiin hakemaan joitakin tunteja auringonnousun j\u00e4lkeen. Minut pestiin ja puettiin seremonia-asuun. En vastustellut. Olin tunteeton olio, jonka kaikki huomio ja ajatukset olivat yhdess\u00e4 asiassa.\n\nMinut talutettiin ulos, ja katseeni kohosi heti kohti taivasta. Olin jo selliss\u00e4ni ollut tuntevinani ilman koleuden, mutta vasta nyt n\u00e4in, miten hurja muutos oli tapahtunut y\u00f6n aikana. En ollut koskaan n\u00e4hnyt yht\u00e4 myrskyis\u00e4\u00e4 taivasta, yht\u00e4 rauhatonta merta. Ensimm\u00e4ist\u00e4 kertaa annoin itseni tuntea jotain. Saarella satoi harvoin, eik\u00e4 yleens\u00e4 koskaan t\u00e4h\u00e4n aikaan vuodesta. Se, ett\u00e4 aurinko oli juuri t\u00e4n\u00e4\u00e4n p\u00e4\u00e4tt\u00e4nyt k\u00e4\u00e4nt\u00e4\u00e4 meille selk\u00e4ns\u00e4, ei tuntunut sattumalta. Tummat pilvet peittiv\u00e4t taivaan kauttaaltaan ja imin niist\u00e4 itseeni kylm\u00e4\u00e4 raivoa.\n\nEhk\u00e4 maailma oli todella loppumassa. Ehk\u00e4 meri nielaisisi saaren sis\u00e4\u00e4ns\u00e4, eik\u00e4 kukaan saisi tiet\u00e4\u00e4, ett\u00e4 sit\u00e4 oli ollut olemassa. Korkeintaan vuosisadan tai parin p\u00e4\u00e4st\u00e4 sukeltajat l\u00f6yt\u00e4isiv\u00e4t meren pohjasta alttarin kappaleita, ja historian kirjoihin kirjoitettaisiin uusi lyhyt luku kadonneista kulttuureista. Aarteenmets\u00e4st\u00e4j\u00e4t yritt\u00e4isiv\u00e4t l\u00f6yt\u00e4\u00e4 paikalta jotain arvokasta. Ehk\u00e4 he l\u00f6yt\u00e4isiv\u00e4tkin Callistan rannerenkaan tai jonkun uhrauksissa k\u00e4ytetyist\u00e4 astioista ja niist\u00e4 teht\u00e4isiin pitk\u00e4lle vietyj\u00e4 p\u00e4\u00e4telmi\u00e4. Mutta koskaan, koskaan he eiv\u00e4t onnistuisi p\u00e4\u00e4sem\u00e4\u00e4n edes l\u00e4helle totuutta.\n\nNiin kuin olin aavistellut, kaikki olivat paikalla. Daimoneita ei ollut en\u00e4\u00e4 paljon. Monet tutut kasvot, joiden olin tottunut hymyilev\u00e4n minulle, olivat nyt vakavia ja varautuneita. Mill\u00e4\u00e4n niist\u00e4 ei n\u00e4kynyt vilahdustakaan my\u00f6t\u00e4tunnosta minua kohtaan.\n\nSilloin huomioni kiinnittyi Callistaan.\n\nOli aivan eri asia kuulla jonkun kuolemasta kuin n\u00e4hd\u00e4 se omin silmin. Ruumis lep\u00e4si korokkeella. H\u00e4net oli puettu parhaimpiinsa ja hiukset oli laitettu yht\u00e4 huolellisesti kuin h\u00e4nen el\u00e4ess\u00e4\u00e4n. Joku olisi ehk\u00e4 voinut v\u00e4itt\u00e4\u00e4, ett\u00e4 h\u00e4n n\u00e4ytti nukkuvan, mutta ero oli t\u00e4ysin selv\u00e4.\n\nTuijotin ruumista niin kauan, ett\u00e4 takanani olevat daimonit ty\u00f6nsiv\u00e4t minua sel\u00e4st\u00e4 eteenp\u00e4in.\n\nVasta silloin tajusin, mik\u00e4 puuttui.\n\n\"Miss\u00e4 Mikael on?\" kysyin. Kukaan ei vastannut.\n\n\"Tied\u00e4n, ett\u00e4 h\u00e4n on t\u00e4\u00e4ll\u00e4. Callista lupasi, ett\u00e4 saan n\u00e4hd\u00e4 h\u00e4net ennen kuin...\"\n\n\"\u00c4l\u00e4 pelk\u00e4\u00e4, sin\u00e4 n\u00e4et h\u00e4net kyll\u00e4\", Vlassisin \u00e4\u00e4ni sanoi. K\u00e4\u00e4nnyin nopeasti sit\u00e4 kohti. H\u00e4n seisoi aivan vieress\u00e4ni. H\u00e4n ei kuitenkaan katsonut minua, vaan edess\u00e4mme olevaa tyhj\u00e4\u00e4 aluetta.\n\nSeurasin h\u00e4nen esimerkki\u00e4\u00e4n. Taivas oli niin tumma, etten ollut n\u00e4hnyt keskell\u00e4 olevaa valtavaa kuoppaa ennen kuin salamat valaisivat maata. Se oli niin syv\u00e4, etten n\u00e4hnyt sen pohjalle. Mik\u00e4 se oli? Hauta?\n\nPari vanhempaa daimonia astui esiin muiden joukosta ja alkoi valmistella seremoniaa. K\u00e4ytin tilaisuuden hyv\u00e4kseni ja kurotin kohti ideoiden maailmaa.\n\nMutta oli niin koko ideoiden maailmaa ei olisi \u00e4kki\u00e4 ollut olemassakaan.\n\nYritin viel\u00e4 uudelleen \u2013 turhaan.\n\nP\u00e4lyilin ymp\u00e4rilleni toivoen, ett\u00e4 l\u00f6yt\u00e4isin asialle jonkin selityksen. Olin aina pystynyt v\u00e4hint\u00e4\u00e4n tuntemaan ideoiden maailman, silloinkin, kun en viel\u00e4 ollut osannut k\u00e4ytt\u00e4\u00e4 sit\u00e4. En voinut ymm\u00e4rt\u00e4\u00e4, miten se oli voinut huomaamattani kadota.\n\nJos en pystynyt k\u00e4ytt\u00e4m\u00e4\u00e4n kykyj\u00e4ni, en voinut paeta daimoneilta. Heit\u00e4 oli yksinkertaisesti liian monta.\n\nEn voinut antaa itselleni lupaa panikoida. Pistin siis vastaan ainoalla tavalla, johon kykenin. Suoristin ryhtini, kohotin leukani ja annoin \u00e4\u00e4neni kaikua koko kent\u00e4ll\u00e4. \"Ettek\u00f6 te tajua, ett\u00e4 Callista on huijannut teit\u00e4 koko ajan? H\u00e4nell\u00e4 ei ole aikomustakaan est\u00e4\u00e4 ennustusta toteutumasta, h\u00e4n haluaa sen toteutuvan. Ja te pelaatte h\u00e4nen pussiinsa. H\u00e4n tahtoo tuhota kaiken, en min\u00e4. H\u00e4n itse kertoi kaiken t\u00e4m\u00e4n minulle!\" Minua heng\u00e4stytti, mutta jatkoin silti. \"P\u00e4\u00e4st\u00e4k\u00e4\u00e4 minut menem\u00e4\u00e4n ja kaikki pysyy ennallaan. Lupaan sen.\"\n\nHetken aikaa olin toiveikas. N\u00e4in, ett\u00e4 minua kuunneltiin. Osan ilme oli jopa utelias.\n\nSitten salamat pirstoivat taivasta ja ukkonen jylisi. \u00c4kkin\u00e4inen sadekuuro kasteli meid\u00e4t hetkess\u00e4 litim\u00e4riksi.\n\nJoukossa k\u00e4vi mumina. He olivat kauhuissaan, tajusin. \"En se ole min\u00e4!\" huusin. \"Minulla ei ole t\u00e4m\u00e4n s\u00e4\u00e4n kanssa mit\u00e4\u00e4n tekemist\u00e4!\"\n\nToivoni valui vesipisaroiden mukana maahan. Olin vain vahvistanut heid\u00e4n pelkoaan siit\u00e4, ett\u00e4 olin jonkinlainen lopunajan sanansaattaja. He eiv\u00e4t koskaan uskoisi minua. He olivat seuranneet Callistaa koko ik\u00e4ns\u00e4. Min\u00e4 taas olin ulkopuolinen. Pahempaa: olin petturi. Olin valinnut sudet, en heit\u00e4. Kaikki tapahtui tismalleen niin kuin Callista oli halunnut.\n\nJoku ty\u00f6nsi minua sel\u00e4st\u00e4. Jouduin ottamaan askeleen eteenp\u00e4in \u2013 paha vain, ettei edess\u00e4ni ollut kuin kapea kaistale maata ennen kuoppaa. Nilkkani lipesi, ja huomasin kaatuvani.\n\nPudotus oli juuri niin lyhyt, etten ehtinyt korjaamaan asentoani kun jo iskeydyin kuopan pohjalle.\n\nMutainen maa ei juurikaan pehmitt\u00e4nyt laskuani. Kiroilin suomeksi. Kokeilin nilkkaani. Se ei ollut varmaankaan murtunut, mutta siihen sattui.\n\n\"Oletko kunnossa?\" tuttu \u00e4\u00e4ni kysyi.\n\nNostin p\u00e4\u00e4ni hitaasti. M\u00e4r\u00e4t hiukseni olivat liimautuneet kiinni kasvoihini, ja minun t\u00e4ytyi pyyhk\u00e4ist\u00e4 ne syrj\u00e4\u00e4n.\n\nMikael.\n\nCallista oli luvannut minulle, ett\u00e4 n\u00e4kisin h\u00e4net viel\u00e4. En vain ollut odottanut n\u00e4kev\u00e4ni h\u00e4nt\u00e4 t\u00e4ll\u00e4 tavalla.\n\nH\u00e4n istui toisella puolella kuoppaa. Varjo lankesi h\u00e4nen ylleen enk\u00e4 n\u00e4hnyt kuin h\u00e4nen \u00e4\u00e4riviivansa, mutta jo se sai ihoni kananlihalle.\n\nH\u00e4n oli laihtunut. Se oli helppo n\u00e4hd\u00e4, sill\u00e4 h\u00e4nell\u00e4 ei ollut paitaa.\n\nKonttasin hitaasti l\u00e4hemm\u00e4s. Miten monta p\u00e4iv\u00e4\u00e4, tuntia ja minuuttia olin ollut n\u00e4kem\u00e4tt\u00e4 h\u00e4nt\u00e4. Ensimm\u00e4ist\u00e4 kertaa pitk\u00e4\u00e4n aikaan sieluni lauloi.\n\nSe oli kuitenkin hyvin mollivoittoinen laulu.\n\nJuuri silloin salama valaisi h\u00e4net. Pidin huolen, etten kavahtanut kun n\u00e4in h\u00e4nen kasvonsa. H\u00e4nen turvonnut silm\u00e4ns\u00e4 ja revennyt huulensa saivat silti kyynelkanavani pistelem\u00e4\u00e4n. H\u00e4n n\u00e4ytti kamalalta. L\u00e4hestulkoon kuolleelta.\n\n\"Ai niin paha\", h\u00e4n totesi arvioituaan reaktiotani.\n\nH\u00e4nen \u00e4\u00e4nens\u00e4... Pelk\u00e4sin, ett\u00e4 jos avaisin suuni, ulos purkautuisi vain nyyhkytyst\u00e4. Joten en sanonut mit\u00e4\u00e4n.\n\nKeskityin Mikaelin auki olevaan silm\u00e4\u00e4n. Tunnistin katseen heti. H\u00e4n oli yh\u00e4 oma itsens\u00e4. H\u00e4n ei ollut antanut heid\u00e4n murtaa itse\u00e4\u00e4n.\n\nKurotin h\u00e4nt\u00e4 kohti.\n\n\"Tuo ei taida olla hyv\u00e4 ajatus\", h\u00e4n sanoi.\n\nAnnoin k\u00e4teni pudota.\n\n\"Sit\u00e4 vain, ett\u00e4 minun nahkani tuntuu t\u00e4ll\u00e4 hetkell\u00e4 silt\u00e4 kuin se olisi tulessa. On parempi olla \u00e4rsytt\u00e4m\u00e4tt\u00e4 sutta.\"\n\nHyv\u00e4 perustelu, ajattelin.\n\n\"Olin unohtanut, miten kaunis sin\u00e4 olet\", h\u00e4n sanoi \u00e4kki\u00e4. \"Olen kuvitellut sinut mieless\u00e4ni l\u00e4hes joka hetki siit\u00e4 asti, kun p\u00e4\u00e4dyin t\u00e4nne, ja luulin olevani siin\u00e4 jo varsin hyv\u00e4. Mutta olin niin v\u00e4\u00e4r\u00e4ss\u00e4.\"\n\nEn tiennyt, mit\u00e4 sanoa. H\u00e4n ei yleens\u00e4 puhunut n\u00e4in suodattamattomasti, ja sain vaikutelman, ett\u00e4 h\u00e4n yritti k\u00e4\u00e4nt\u00e4\u00e4 ajatukset pois jostakin. Se huoletti minua.\n\n\"Min\u00e4 tiesin, ett\u00e4 sin\u00e4 tulisit.\" H\u00e4n kuulosti sek\u00e4 turhautuneelta ett\u00e4 ylpe\u00e4lt\u00e4. Toinen suupieli alkoi kohota, mutta vet\u00e4ytyi pian takaisin. Sen oli t\u00e4ytynyt tehd\u00e4 kipe\u00e4\u00e4. Huomasin irvist\u00e4v\u00e4ni. Sanoin nopeasti: \"Et sin\u00e4 p\u00e4\u00e4se sekop\u00e4\u00e4tytt\u00f6yst\u00e4v\u00e4si eroon noin vain. Sit\u00e4 paitsi sin\u00e4h\u00e4n lupasit minulle ulkomaanmatkan.\"\n\n\"T\u00e4m\u00e4 on siis saari\", Mikael sanoi.\n\n\"Niin on\", vastasin.\n\n\"Pakko sanoa, ett\u00e4 kuvittelin mieless\u00e4ni jotain viihtyis\u00e4mp\u00e4\u00e4.\"\n\n\"Pahoittelen\", sanoin sanotuksi. \"Meid\u00e4n pit\u00e4\u00e4 menn\u00e4 Santorinille tai jonnekin, miss\u00e4 ollaan hieman vieraanvaraisempia.\"\n\n\"Jos t\u00e4m\u00e4 on rangaistukseni siit\u00e4, ett\u00e4 tapoin yhden heid\u00e4n j\u00e4senist\u00e4\u00e4n, sanoisin, ett\u00e4 se on varsin tehokas.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Luulen, ett\u00e4 t\u00e4ll\u00e4 on enemm\u00e4n tekemist\u00e4 sen kanssa, ett\u00e4 he pelk\u00e4\u00e4v\u00e4t minua. He haluavat tappaa minut, mutta kukaan heist\u00e4 ei uskalla tehd\u00e4 sit\u00e4.\"\n\nSade oli jo ehtinyt muodostaa lammikon kuopan pohjalle, ja varpaani upposivat veteen kokonaan. Istuin selk\u00e4 kuopan sein\u00e4m\u00e4\u00e4 vasten ja Mikael toisella puolella.\n\n\"Mit\u00e4 he laulavat?\" Mikael kysyi.\n\nEn ollut kiinnitt\u00e4nyt ylh\u00e4\u00e4lt\u00e4 kuuluviin \u00e4\u00e4niin mit\u00e4\u00e4n huomiota, mutta Mikaelin korvat olivatkin tarkemmat kuin omani. Aloin erottaa daimonien hymin\u00e4n ja satunnaisia sanoja sielt\u00e4 t\u00e4\u00e4lt\u00e4.\n\n\"En tied\u00e4\", vastasin lopulta. \"En ole koskaan ollut hautajaisseremoniassa.\"\n\n\"N\u00e4in siis oikein. Papitar on kuollut.\"\n\n\"H\u00e4n meni edelt\u00e4 varmistamaan, ett\u00e4 minun sieluni tai mik\u00e4 lie p\u00e4\u00e4tyy oikeaan paikkaan.\" Se kuulosti niin idioottimaiselta, ja silti se oli johtanut meid\u00e4t t\u00e4h\u00e4n tilanteeseen. Se sai minut kysym\u00e4\u00e4n itselt\u00e4ni, kuinka typeri\u00e4 me olimmekaan.\n\n\"Ja mik\u00e4h\u00e4n se paikka mahtaa olla?\" Mikael kysyi.\n\n\"Ideoiden maailma. Vain kaikkein vahvimmat daimonit p\u00e4\u00e4sev\u00e4t sinne. Kaikki t\u00e4ll\u00e4 saarella luulevat, ett\u00e4 Callista kuoli omasta vapaasta tahdostaan siksi, ett\u00e4 h\u00e4n voisi est\u00e4\u00e4 minua p\u00e4\u00e4sem\u00e4st\u00e4 ideoiden maailmaan ja sy\u00f6ksem\u00e4st\u00e4 nykyiset jumalat vallasta. He eiv\u00e4t tied\u00e4, ett\u00e4 se on tismalleen, mit\u00e4 Callista toivoo tapahtuvan. H\u00e4n sanoi odottaneensa minua tuhansia vuosia.\"\n\n\"Olit oikeassa. N\u00e4m\u00e4 tyypit ovat todella sekaisin.\" H\u00e4n pyyhk\u00e4isi otsaansa. Se oli ainoa kohta h\u00e4nen kasvoissaan, joka ei ollut vaurioitunut. \"No, ainakin t\u00e4m\u00e4 selitt\u00e4\u00e4 paljon. Minulla on ollut pieni\u00e4 kielivaikeuksia.\"\n\n\"Mutta en ymm\u00e4rr\u00e4, miksi me olemme t\u00e4\u00e4ll\u00e4\", sanoin.\n\n\"Juurihan sin\u00e4 kerroit sen minulle\", Mikael sanoi.\n\n\"Tarkoitan t\u00e4t\u00e4 kuoppaa. En ymm\u00e4rr\u00e4, miksi he ty\u00f6nsiv\u00e4t minut t\u00e4nne sinun kanssasi.\"\n\nEn ollut siit\u00e4 ihan niin huolissani kuin olisi voinut kuvitella. Kuopassa saatoin melkein kuvitella, ett\u00e4 koko maailma oli t\u00e4ss\u00e4. Ei ollut daimoneita, ei saarta, ei mit\u00e4\u00e4n muuta kuin me kaksi. Me emme ehk\u00e4 olleet el\u00e4m\u00e4mme kunnossa, mutta pelkk\u00e4 Mikaelin n\u00e4keminen oli saanut minut paljon p\u00e4\u00e4tt\u00e4v\u00e4isemm\u00e4ksi.\n\nMikael oli hetken aikaa t\u00e4ysin liikkumatta. Siit\u00e4 tiesin, ett\u00e4 minulta oli j\u00e4\u00e4nyt jotain tajuamatta. \"Sin\u00e4 et taida tiet\u00e4\u00e4, mik\u00e4 p\u00e4iv\u00e4 t\u00e4n\u00e4\u00e4n on.\"\n\nLaskujeni mukaan olin ollut saarella nelj\u00e4 p\u00e4iv\u00e4\u00e4. Lapissa oli kulunut kolme. Olin siis l\u00e4htenyt Hukkavaarasta viikko sitten.\n\nViimein tajusin, mik\u00e4 daimonien suunnitelma oli. Mihin he tarvitsivat sutta.\n\nT\u00e4n\u00e4\u00e4n oli t\u00e4ysikuu.\n\nKatsoin kuopan seini\u00e4 kuin olisin n\u00e4hnyt ne ensimm\u00e4isen kerran. Ne olivat pystysuorat ja ainakin nelj\u00e4 metri\u00e4 korkeat \u2013 niin korkeat, etten voinut erottaa niiden reunaa ilman, ett\u00e4 sade sokaisi minut. Toinen meist\u00e4 saattaisi pysty\u00e4 kiipe\u00e4m\u00e4\u00e4n yl\u00f6s, mik\u00e4li toinen nostaisi alhaalta, mutta tiesin, ettei meille annettaisi siihen mahdollisuutta. Ja pian sill\u00e4 ei olisi v\u00e4li\u00e4: kohta toinen meist\u00e4 olisi susi ja toinen helppo saalis.\n\n\"He haluavat, ett\u00e4 min\u00e4 tapan sinut.\" Mikael sanoi \u00e4\u00e4neen sen, mink\u00e4 olin juuri ymm\u00e4rt\u00e4nyt.\n\nSuljin silm\u00e4ni hetkeksi \u2013 ja avasin ne melkein heti. Olin nimitt\u00e4in juuri tajunnut, mit\u00e4 Mikaelin ruhjottu ulkomuoto tarkoitti. \"Mikset sin\u00e4 ole vaihtanut muotoa?\" kysyin.\n\nH\u00e4n olisi voinut parantaa itsens\u00e4 hyvin helposti. Ottaen huomioon, ett\u00e4 suden muodossa h\u00e4n olisi voinut tarjota daimoneille paljon paremman vastuksen, h\u00e4nen ihmishahmolleen t\u00e4ytyi olla hyv\u00e4 selitys.\n\nMikael irvisti. \"He ovat juottaneet minulle jotain, miss\u00e4 on hopeaa. Niin kauan kuin se on elimist\u00f6ss\u00e4ni, en voi muuttua.\"\n\nAnnoin tiedon porautua tajuntaani. H\u00e4net oli myrkytetty. Se tietenkin selitti h\u00e4nen harmaaksi muuttuneen ihonsa ja verest\u00e4v\u00e4t silm\u00e4ns\u00e4.\n\nKurtistin kulmiani. \"Mit\u00e4 sitten tapahtuu, kun t\u00e4ysikuu pakottaa sinut muuttumaan?\"\n\n\"En tied\u00e4\", h\u00e4n kuiskasi.\n\nYritin kuulostaa mahdollisimman luottavaiselta. \"Sinun sutesi ei satuttaisi minua.\"\n\n\"Raisa, sin\u00e4 et voi tiet\u00e4\u00e4. Olen aina muuttanut muotoa omasta tahdostani niin, ett\u00e4 olen voinut s\u00e4ilytt\u00e4\u00e4 edes osan ihmisyydest\u00e4ni kun olen suden muodossa. Joka kerta. Jopa silloin Eskon kanssa.\" H\u00e4n antoi minulle hetken ajatella asiaa ennen kuin jatkoi: \"En ole saanut ruokaa viikkoon. T\u00e4m\u00e4 on vieras paikka ja kaiken lis\u00e4ksi suljettu sellainen. Suteni tulee olemaan hyvin vihainen, hyvin n\u00e4lk\u00e4inen ja hyvin ep\u00e4toivoinen.\"\n\nTodellisuus alkoi v\u00e4hitellen valjeta minulle. Mikael saattoi oikeasti tappaa minut. Siihen vaadittaisiin vain yksi purema, yksi hampaiden raapaisu. Ei edes mit\u00e4\u00e4n syv\u00e4\u00e4 haavaa.\n\nMikaelin t\u00e4ytyi aistia pelkoni, sill\u00e4 h\u00e4n sanoi: \"Tilanne ei ole toivoton. Minun suteni pit\u00e4isi tunnistaa sinut. Sin\u00e4 olet laumaa. Ja min\u00e4 yrit\u00e4n kaikkeni.\"\n\n\"Tied\u00e4n.\" Minun oli pakko sanoa seuraavat sanat. \"Jos jotain tapahtuu... Tarkoitan, ett\u00e4 jos sinun sutesi vahingossa tekeekin jotain, niin haluan ett\u00e4 tied\u00e4t, etten syyt\u00e4 sinua mist\u00e4\u00e4n.\"\n\nMikael ei vastannut. H\u00e4n oli puristanut silm\u00e4ns\u00e4 kiinni ja olin aika varma, ett\u00e4 h\u00e4n kamppaili kipua vastaan.\n\n\"Sit\u00e4 paitsi me kuolemme joka tapauksessa\", kuulin itseni sanovan. \"Jos sinun sutesi ei tapa minua, he keksiv\u00e4t kyll\u00e4 jotain muuta. Luultavasti he aikovat j\u00e4tt\u00e4\u00e4 meid\u00e4t t\u00e4h\u00e4n kuoppaan ikuisiksi ajoiksi.\"\n\nPieni hymy v\u00e4l\u00e4hti Mikaelin huulilla. \"Sin\u00e4 aina tied\u00e4t, miten lohduttaa minua.\" H\u00e4n avasi silm\u00e4ns\u00e4. \"Sinulla ei siis ole suunnitelmaa?\"\n\n\"Ei\", kuiskasin. Minua h\u00e4vetti my\u00f6nt\u00e4\u00e4 se. Olin p\u00e4\u00e4ssyt t\u00e4h\u00e4n pisteeseen, mutten kuitenkaan sen pidemm\u00e4lle. Varmasti oli olemassa jokin keino. Mutta kaikki ideani olivat tyrehtyneet kun olin tajunnut, etten voinut k\u00e4ytt\u00e4\u00e4 kykyj\u00e4ni.\n\n\"Ei se mit\u00e4\u00e4n\", h\u00e4n sanoi, ja tiesin h\u00e4nen tarkoittavan sit\u00e4. \"Olen vain niin iloinen, ett\u00e4 sain n\u00e4hd\u00e4 sinut. Tai olen ainakin sitten, kun olemme saaneet sinut pois t\u00e4\u00e4lt\u00e4.\"\n\n\"Sinut my\u00f6s\", muistutin. En l\u00e4htisi ilman h\u00e4nt\u00e4.\n\nAika kului. Emme sanoneet pitk\u00e4\u00e4n aikaan mit\u00e4\u00e4n, ja saatoin n\u00e4hd\u00e4, ett\u00e4 Mikael oli todellisissa vaikeuksissa. En ollut koskaan tuntenut itse\u00e4ni yht\u00e4 hy\u00f6dytt\u00f6m\u00e4ksi.\n\n\"Puhu minulle\", Mikael mumisi. \"Kerro jokin tarina tai jotain. Mit\u00e4 tahansa.\"\n\n\"No\", aloitin. \"T\u00f6rm\u00e4sin sinun is\u00e4\u00e4si.\"\n\n\"Oikeasti?\"\n\n\"Oikeasti.\" Kerroin h\u00e4nelle lyhyesti, mit\u00e4 oli tapahtunut sen j\u00e4lkeen, kun h\u00e4n oli kadonnut. Kun p\u00e4\u00e4sin loppuun, h\u00e4n oli sulkenut silm\u00e4ns\u00e4.\n\nLopulta minun oli pakko kysy\u00e4: \"Kuinka sin\u00e4 jaksat?\"\n\n\"En ole koskaan uskaltanut kokeilla, milt\u00e4 tuntuu jos annan kuun pakottaa suden ulos\", h\u00e4n tunnusti.\n\n\"Ehk\u00e4 sin\u00e4 et t\u00e4ll\u00e4 kertaa muutu. Hopea saattaa pit\u00e4\u00e4 sinut ihmisen\u00e4\", sanoin toiveikkaana.\n\n\"Jos jostain olen varma niin siit\u00e4, ett\u00e4 min\u00e4 muutun.\"\n\n\"Vene on tulossa t\u00e4nne p\u00e4in!\" joku huusi ylh\u00e4\u00e4ll\u00e4.\n\nMeid\u00e4n oli keksitt\u00e4v\u00e4 jotain hyvin, hyvin pian. Me...\n\nAjatukseni keskeytyi. Vene?\n\nEnsimm\u00e4isten viikkojen aikana saarella olin jatkuvasti t\u00e4hyillyt merelle. Kertaakaan en ollut n\u00e4hnyt siell\u00e4 mit\u00e4\u00e4n, en edes lintuja.\n\nIlmoitus veneest\u00e4 tuntui saavan kaikkien daimonien huomion, sill\u00e4 hymin\u00e4 ja laulu lakkasivat v\u00e4litt\u00f6m\u00e4sti. Nousin hitaasti jaloilleni.\n\n\"Tuokaa kaukoputki\", kuulin Vlassisin sanovan. Tai ainakin kuvittelin kuulemani sanan tarkoittavan kaukoputkea. Suoraan sanottuna en ollut uskonut, ett\u00e4 saarella oli k\u00e4yt\u00f6ss\u00e4 niinkin modernia teknologiaa.\n\nNousin seisomaan. Yritin n\u00e4hd\u00e4, mit\u00e4 ylh\u00e4\u00e4ll\u00e4 tapahtui, mutta se oli mahdotonta. Daimonit olivat piinallisen hiljaa, ja kesti aivan liian pitk\u00e4\u00e4n, ennen kuin kuulin Vlassisin puhuvan.\n\n\"Se on h\u00e4n.\" Tiesin heti, ket\u00e4 h\u00e4n tarkoitti. Olin tainnut tiet\u00e4\u00e4 koko ajan, ja pel\u00e4t\u00e4 my\u00f6s. Veljeni oli tullut pelastamaan minua.\n\nEik\u00e4 h\u00e4n ollut tullut yksin. Hetken p\u00e4\u00e4st\u00e4 kuulin nimitt\u00e4in sanottavan: \"H\u00e4nell\u00e4 on susi mukanaan.\"\n\n\"Mit\u00e4 tapahtuu?\" Mikael kysyi.\n\n\"Mitja ja Daniel\", sain sanottua.\n\nMitja, mit\u00e4 sin\u00e4 oikein teet? ajattelin.\n\nYlh\u00e4\u00e4ll\u00e4 n\u00e4kyi liikett\u00e4. Kuulin k\u00e4skyj\u00e4 ottaa aseita esiin.\n\nYksi susi kymmeni\u00e4 daimoneita vastaan. Jos jonkun olisi pit\u00e4nyt olla peloissaan niin suden, mutta oli helppo sanoa, ett\u00e4 daimonit olivat l\u00e4hell\u00e4 pakokauhua.\n\nHetken aikaa olin ihmeiss\u00e4ni. Sitten tajusin, mit\u00e4 erist\u00e4ytyminen oli tehnyt heille: heit\u00e4 ei ollut uhannut mik\u00e4\u00e4n moneen sataan, ellei jopa tuhanteen vuoteen. He olivat tuudittautuneet siihen, ett\u00e4 heid\u00e4n ainoat vihollisensa paitsi asuivat kaukana poissa, eiv\u00e4t edes tienneet daimonien olemassaolosta. T\u00e4m\u00e4 oli ensimm\u00e4inen kerta ikuisuuteen, kun he tajusivat olevansa uhattuja. Eik\u00e4 heill\u00e4 ollut mahdollisuutta paeta \u2013 meri piti siit\u00e4 huolen.\n\nHe eiv\u00e4t tietenk\u00e4\u00e4n olleet puolustuskyvytt\u00f6mi\u00e4. Heill\u00e4 oli hopeaa. Ja ennen kaikkea heill\u00e4 oli Mikael.\n\n\"Nostakaa susi yl\u00f6s kuopasta\", Vlassis sanoi.\n\nKun joku yritti sanoa vastaan, h\u00e4n \u00e4rj\u00e4hti: \"Kyll\u00e4 me yhden tyt\u00f6n saamme hengilt\u00e4 ilman suttakin! Vai haluatko mieluummin taistella tuota vastaan? Mik\u00e4li et tajunnut, kukaan meist\u00e4 ei voi k\u00e4ytt\u00e4\u00e4 kykyj\u00e4\u00e4n ennen kuin Callista on p\u00e4\u00e4ssyt perille.\"\n\nEn ollut tiennyt sit\u00e4. Jotenkin Callista oli vienyt ideat menness\u00e4\u00e4n. \u00c4kki\u00e4 ymm\u00e4rsin paljon paremmin daimonien paniikin.\n\nKuolema oli tulossa heit\u00e4 kohti.\n\nEnsimm\u00e4inen daimon hypp\u00e4si kuopan pohjalle. Olin jo ottanut askeleen h\u00e4nt\u00e4 kohti, kun h\u00e4n osoitti minua veitsell\u00e4. Pahus. H\u00e4n l\u00e4hestyi minua kunnes en voinut muuta kuin peruuttaa sein\u00e4m\u00e4\u00e4n kiinni.\n\nKaksi muuta daimonia pudottautui alas. Mikael ei ollut miss\u00e4\u00e4n vaiheessa noussut seisomaan, mik\u00e4 kertoi paljon h\u00e4nen voinnistaan. Daimonit kohottivat h\u00e4net maasta ja kietoivat k\u00f6yden h\u00e4nen vy\u00f6t\u00e4r\u00f6lleen.\n\nLoppua en n\u00e4hnyt, sill\u00e4 veist\u00e4 pitelev\u00e4 daimon peitti n\u00e4kym\u00e4n tarttuessaan k\u00e4sivarteeni.\n\n\"Ei h\u00e4nt\u00e4\", Vlassis sanoi ylh\u00e4\u00e4lt\u00e4.\n\n\"Mit\u00e4? Ei!\" Kurkotin daimonin ohi, mutta h\u00e4n ty\u00f6nsi veitsen kiinni naamaani.\n\n\"\u00c4l\u00e4 huoli, p\u00e4rj\u00e4\u00e4n kyll\u00e4\", Mikael sanoi \u2013 ja ansaitsi itselleen potkun kylkeen. H\u00e4n ei p\u00e4\u00e4st\u00e4nyt \u00e4\u00e4nt\u00e4k\u00e4\u00e4n, mutta min\u00e4 p\u00e4\u00e4stin. Huusin kaikki pahimmat kreikankieliset kiroukset, jotka vain osasin.\n\nMikael onnistuttiin nostamaan kuopan reunalle.\n\n\"\u00c4lk\u00e4\u00e4 p\u00e4\u00e4st\u00e4k\u00f6 h\u00e4nt\u00e4 karkaamaan\", Vlassis sanoi. \"Meid\u00e4n pit\u00e4\u00e4 vied\u00e4 h\u00e4net laiturille vastaan. Olkaa valmiita tappamaan h\u00e4net silm\u00e4nr\u00e4p\u00e4yksess\u00e4.\"\n\nVlassis oli siis p\u00e4\u00e4tt\u00e4nyt k\u00e4yd\u00e4 kauppaa. H\u00e4n ei tarvinnut Mikaelia mihink\u00e4\u00e4n \u2013 ainoa, jonka h\u00e4n todella tarvitsi, olin min\u00e4.\n\n\"Tied\u00e4n, mit\u00e4 aiot\", huusin h\u00e4nelle. \"He eiv\u00e4t suostu!\"\n\nTai ainakaan Mitja ei suostuisi. Danielista en voinut menn\u00e4 takuuseen. Mik\u00e4li h\u00e4n saisi Mikaelin ehj\u00e4n\u00e4 takaisin ja lupauksen, etteiv\u00e4t daimonit tulisi heid\u00e4n per\u00e4\u00e4ns\u00e4, h\u00e4n saattaisi olla valmis j\u00e4tt\u00e4m\u00e4\u00e4n minut saarelle. Mikael olisi tietenkin eri mielt\u00e4, mutta h\u00e4n ei ollut sellaisessa kunnossa, ett\u00e4 h\u00e4nen protestejaan tarvitsisi ottaa huomioon.\n\nToisaalta jo se, ett\u00e4 daimonit olivat valmiita k\u00e4ym\u00e4\u00e4n kauppaa, oli heilt\u00e4 heikkouden osoitus. Danielilla ei ollut lainkaan kunnioitusta heikkoja vastustajia kohtaan. H\u00e4n haistaisi daimonien pelon paljon paremmin kuin he tajusivat.\n\nKaikki oli siis kiinni Daniel Sarrista. Se ei ollut erityisen mielt\u00e4 ylent\u00e4v\u00e4 ajatus.\n\nAioin joka tapauksessa olla paikalla n\u00e4kem\u00e4ss\u00e4 mit\u00e4 tapahtui, kun vene ehtisi rantaan. Daimonit eiv\u00e4t voineet j\u00e4tt\u00e4\u00e4 minut kuoppaan ikuisiksi ajoiksi.\n\nPaitsi ett\u00e4 juuri niin he ilmeisesti aikoivat tehd\u00e4.\n\nMaata rapisi kuopan pohjalle, kun kynteni upposivat sein\u00e4\u00e4n. Yritin kiivet\u00e4 yl\u00f6s, mutta loppujen lopuksi sain ainoastaan kynteni repeilem\u00e4\u00e4n.\n\n\u00c4\u00e4net alkoivat kaikota. Tapahtumat olivat selv\u00e4sti siirtym\u00e4ss\u00e4 kohti laituria. Juuri kun olin luopumassa toivosta, kuopan reunalle ilmestyiv\u00e4t kasvot.\n\n\"Adriane!\"\n\nKasvot katosivat.\n\nEn varsinaisesti voinut olla pettynyt. H\u00e4n ei ollut minulle velkaa, vaan p\u00e4invastoin.\n\nSiit\u00e4 huolimatta halusin kiljua. Kaikki, mik\u00e4 minulle oli t\u00e4rke\u00e4\u00e4 oli vaarassa juuri nyt, ja min\u00e4 olin jumissa montussa.\n\nKasvot ilmestyiv\u00e4t taas reunalle. Ja t\u00e4ll\u00e4 kertaa niit\u00e4 oli per\u00e4ti kahdet. Adrianen vierell\u00e4 oli et\u00e4isesti tutunn\u00e4k\u00f6inen mies. H\u00e4nenkin t\u00e4ytyi kuulua palveluskuntaan.\n\n\"V\u00e4ist\u00e4\", Adriane kuiskasi.\n\nPeruutin kiinni sein\u00e4m\u00e4\u00e4n. Jotain putosi aivan eteeni ja roiskautti kuraa p\u00e4\u00e4lleni. Hetke\u00e4 my\u00f6hemmin tajusin, ett\u00e4 se jokin oli puinen laatikko. He pudottivat alas viel\u00e4 toisenkin laatikon, ja asettelin ne h\u00e4t\u00e4isesti p\u00e4\u00e4llekk\u00e4in sein\u00e4m\u00e4\u00e4 vasten.\n\nKiipesin tekem\u00e4ni tornin p\u00e4\u00e4lle. Kun kurottauduin varpailleni, yletyin ottamaan k\u00e4sill\u00e4ni kiinni kuopan reunasta. Hypp\u00e4sin, ja mies tarttui minua kainalosta. H\u00e4n veti minut Adrianen avustamana yl\u00f6s montusta.\n\nOlin juuri p\u00e4\u00e4ssyt jaloilleni, kun joku l\u00e4hell\u00e4 kysyi: \"Mit\u00e4 t\u00e4\u00e4ll\u00e4 tapahtuu?\"\n\nDaimon oli ilmestynyt viereemme. H\u00e4nen k\u00e4tens\u00e4 hapuili kohti vy\u00f6ll\u00e4 roikkuvaa miekkaa.\n\nMinua auttanut palvelija k\u00e4\u00e4ntyi ymp\u00e4ri salamannopeasti. Ehdin tuskin n\u00e4hd\u00e4 veitsen ennen kuin sen ter\u00e4 upposi daimonin kylkeen. Daimon p\u00e4\u00e4sti vain vaimean \u00e4\u00e4nen ennen kuin putosi maahan. H\u00e4n ei noussut yl\u00f6s.\n\nIdeoiden maailman kadottua daimonit eiv\u00e4t voineet en\u00e4\u00e4 parantaa itse\u00e4\u00e4n.\n\nPalvelija seisoi edess\u00e4ni veitsi k\u00e4dess\u00e4. Tunnistin sen yhdeksi omista veitsist\u00e4ni. Sen p\u00e4\u00e4tyminen Lapin metsist\u00e4 t\u00e4m\u00e4n miehen haltuun tuntui enteelt\u00e4. En voinut olla tuijottamatta sen verist\u00e4 ter\u00e4\u00e4.\n\n\"Tapa heid\u00e4t. Niin monta kuin pystyt\", mies sanoi. Ja ojensi veitsen minulle.\n\nOtin sen. Kahva oli viel\u00e4 l\u00e4mmin h\u00e4nen puristuksestaan. En j\u00e4\u00e4nyt miettim\u00e4\u00e4n, miksi h\u00e4n oli p\u00e4\u00e4tt\u00e4nyt auttaa minua. En j\u00e4\u00e4nyt miettim\u00e4\u00e4n, miten helposti ja harjoitellun oloisesti h\u00e4n oli upottanut veitsen daimonin kylkeen. Sen sijaan aloin juosta kohti laituria.\n\nHidastin vauhtiani vasta kun olin niin l\u00e4hell\u00e4, ett\u00e4 saatoin n\u00e4hd\u00e4 ensimm\u00e4isten kallioille ryhmittyneiden daimonien sel\u00e4t.\n\nVene oli jo ehtinyt rantautua. Tulin paikalle juuri ajoissa kuullakseni Vlassisin johtaman ryhm\u00e4n askeleet pitkin laituria.\n\n\"L\u00e4htek\u00e4\u00e4 nyt, ja s\u00e4\u00e4st\u00e4mme sek\u00e4 teid\u00e4n ett\u00e4 t\u00e4m\u00e4n suden hengen\", kuulin Vlassisin sanovan.\n\n\"Me emme tulleet t\u00e4nne neuvottelemaan\", Mitjan \u00e4\u00e4ni vastasi. \"Me tulimme hakemaan heid\u00e4t molemmat. Luovuttakaa h\u00e4net ja siskoni, tai te kuolette.\"\n\nUjuttauduin kahden suuren kiven v\u00e4list\u00e4 niin, ett\u00e4 saatoin n\u00e4hd\u00e4 laiturille. Aivan ensimm\u00e4isen\u00e4 kiinnitin huomiota Danieliin, joka oli suden muodossa. Saattoi olla hyv\u00e4, ettei h\u00e4nell\u00e4 ollut puheroolia.\n\nVeljeni seisoi laiturin p\u00e4\u00e4ss\u00e4, eik\u00e4 h\u00e4n ollut koskaan n\u00e4ytt\u00e4nyt yht\u00e4 vakuuttavalta, yht\u00e4 aikuiselta. Hetken ajan tunsin silkkaa ylpeytt\u00e4.\n\nEhk\u00e4 Vlassiskin n\u00e4ki h\u00e4ness\u00e4 saman kuin min\u00e4, sill\u00e4 h\u00e4nen ilmeens\u00e4 oli synkk\u00e4. H\u00e4n piteli ter\u00e4\u00e4 Mikaelin leuan alla.\n\n\"Siskosi tuli saarelle omasta vapaasta tahdostaan. En ymm\u00e4rr\u00e4, miksi kuvittelet h\u00e4nen tarvitsevan pelastamista.\"\n\nOlen t\u00e4\u00e4ll\u00e4, sain l\u00e4hetetyksi Mitjalle.\n\nH\u00e4n r\u00e4p\u00e4ytti silmi\u00e4\u00e4n, mutta kukaan ulkopuolinen ei olisi voinut n\u00e4hd\u00e4, ett\u00e4 jotain oli juuri tapahtunut.\n\n\"V\u00e4\u00e4r\u00e4 vastaus\", veljeni sanoi.\n\nDaniel kohotti kuononsa kohti taivasta ja p\u00e4\u00e4sti ilmoille ulvonnan.\nNELJ\u00c4STOISTA LUKU\n\nOlisin antanut mit\u00e4 tahansa, jotta olisin voinut pys\u00e4ytt\u00e4\u00e4 ajan. Tiesin, mit\u00e4 tulisi tapahtumaan. Veitsi Mikaelin kaulalla liikahti samalla millisekunnilla kun Daniel loikkasi Vlassisia kohti.\n\nMikaelin jalka oli kietoutunut Vlassisin nilkan ymp\u00e4rille. Nopea liike, ja Vlassis menetti tasapainonsa. Mikaelilla ei ollut aikaa pyristell\u00e4 irti h\u00e4nen otteestaan. He kaatuivat yhdess\u00e4 veteen ja katosivat pinnan alle.\n\nOdotin. Hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 he ilmestyisiv\u00e4t pintaan. Heid\u00e4n oli pakko.\n\nMit\u00e4\u00e4n ei tapahtunut.\n\nMerivirta! Mitjan \u00e4\u00e4ni p\u00e4\u00e4ni sis\u00e4ll\u00e4 oli niin luja, ett\u00e4 se melkein kuuroutti minut hetkellisesti.\n\nKatseeni juoksi veden pinnalla kelluvan lehden mukana pitkin rannikkoa. Niin, merivirta. Saarella ollessamme olimme uineet aina vain toisella puolella laituria, sill\u00e4 toista rantaa pitkin kulki virta, joka jossain vaiheessa k\u00e4\u00e4ntyi kohti avomerta.\n\nDaniel k\u00e4vi kiinni ensimm\u00e4iseen vastaan tulevaan daimoniin. H\u00e4nt\u00e4 kohti ammuttiin heti nuolella. Avasin suuni huutaakseni h\u00e4nelle varoituksen, mutta ennen kuin \u00e4\u00e4ni ehti ulos, n\u00e4in, ettei se ollut tarpeellista.\n\nH\u00e4nen oli t\u00e4ytynyt ehti\u00e4 hioa tekniikkansa jo Sarrajuoksan m\u00f6kill\u00e4, sill\u00e4 h\u00e4n ei ep\u00e4r\u00f6inyt. Kun nuoli hipaisi Danielia, h\u00e4n alkoi muuttua. Ennen kuin h\u00e4n kuitenkaan ehti muuttua t\u00e4ysin, h\u00e4n oli jo takaisin suden muodossa \u2013 ja ilman naarmuakaan. En ollut koskaan n\u00e4hnyt mit\u00e4\u00e4n vastaavaa.\n\nEik\u00e4 Daniel ollut ainoa, joka taisteli daimoneita vastaan. Yksi toisensa j\u00e4lkeen n\u00e4in palvelijoiden tulevan esiin piiloistaan ja aloittavan vimmatun sodan is\u00e4nti\u00e4\u00e4n vastaan. He eiv\u00e4t ehk\u00e4 olleet yht\u00e4 hyvi\u00e4 taistelijoita kuin daimonit, mutta heill\u00e4 oli ylivoima ja heid\u00e4n raivonsa karkotti loputkin ep\u00e4ilykset.\n\nSe oli varsinainen veril\u00f6yly. Daimoneilla ei ollut mit\u00e4\u00e4n mahdollisuuksia.\n\nJuoksin kallionreunan viert\u00e4 ja t\u00e4hyilin vesirajaa. N\u00e4in sivusilm\u00e4ll\u00e4, kuinka joku tuli minua kohti, mutta se ei saanut minua pys\u00e4htym\u00e4\u00e4n. Ei ennen, kuin olin saada veitsest\u00e4.\n\nV\u00e4istin aivan viime hetkell\u00e4. N\u00e4in vilahduksen pitkist\u00e4, kiharista hiuksista ennen kuin onnistuin k\u00e4\u00e4ntym\u00e4\u00e4n kokonaan ymp\u00e4ri.\n\nAntonia, Alekin sisko.\n\nEn ollut koskaan n\u00e4hnyt ket\u00e4\u00e4n yht\u00e4 vihaisena. H\u00e4n k\u00e4vi p\u00e4\u00e4lleni sellaisella voimalla, ett\u00e4 olisin voinut vaikka vannoa, ett\u00e4 h\u00e4n oli jotenkin onnistunut s\u00e4ilytt\u00e4m\u00e4\u00e4n kykyns\u00e4.\n\nH\u00e4n paiskasi minut mutaan vaikka olin isompi ja vanhempi kuin h\u00e4n. Ehdin tuskin kier\u00e4ht\u00e4\u00e4 ymp\u00e4ri kun n\u00e4in ter\u00e4n v\u00e4l\u00e4ht\u00e4v\u00e4n. Huitaisin veitsell\u00e4ni kohti Antoniaa ja h\u00e4n joutui hyp\u00e4ht\u00e4m\u00e4\u00e4n taaksep\u00e4in. Se antoi minulle mahdollisuuden nousta jaloilleni.\n\nMinulla ei ollut aikaa h\u00e4nen kostolleen. Kuin osoittaakseen, ettei minulta kysytty, h\u00e4n kiersi k\u00e4sivarren kaulani ymp\u00e4ri. Olin h\u00e4nen kuristusotteessaan, ja niin paljon kuin yritinkin, en p\u00e4\u00e4ssyt siit\u00e4 irti.\n\n\"Nyt sin\u00e4 saat maksaa\", h\u00e4n sihisi korvaani.\n\nYht\u00e4kki\u00e4 ote hellitti, ja kun k\u00e4\u00e4nnyin, n\u00e4in, ett\u00e4 Antonia makasi liikkumattomana maassa. Tuijotin h\u00e4nt\u00e4 kunnes tajusin kohottaa katseeni h\u00e4nen takanaan seisovaan hahmoon.\n\nMitja. H\u00e4nen n\u00e4kemisens\u00e4 oli suuri helpotus monestakin eri syyst\u00e4.\n\nKatsoin taas Antoniaa. \"Onko h\u00e4n..?\"\n\nVeljeni pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"H\u00e4n on elossa. Mene!\"\n\nEn tarvinnut toista kehotusta. L\u00e4hdin h\u00f6lkk\u00e4\u00e4m\u00e4\u00e4n kallion reunaa pitkin etsien koko ajan katseellani tuttua hahmoa rantaviivalta.\n\nViimein erotin heid\u00e4t. He seisoivat kasvotusten vain parin metrin mittaisella kaistaleella kalliota, joka j\u00e4i meren ja l\u00e4hes pystysuoran sein\u00e4m\u00e4n v\u00e4liin. Mikael, vain osittain tajuissaan, taisteli Vlassisin kanssa. Vlassis oli selv\u00e4sti niskan p\u00e4\u00e4ll\u00e4, mutta h\u00e4nen oli t\u00e4ytynyt pudottaa veitsens\u00e4, sill\u00e4 h\u00e4n taisteli yht\u00e4 lailla paljain k\u00e4sin. H\u00e4n l\u00f6i Mikaelia nyrkill\u00e4 naamaan. Mikael pysyi pystyss\u00e4, mutta vain vaivoin.\n\nMinulla oli vain yksi veitsi, joten ep\u00e4r\u00f6in sen heitt\u00e4mist\u00e4.\n\nMinun t\u00e4ytyi p\u00e4\u00e4st\u00e4 heid\u00e4n luokseen. Ei ollut kuin yksi vaihtoehto: minun t\u00e4ytyi hyp\u00e4t\u00e4 mereen ja uida. En tiennyt, kuinka syv\u00e4\u00e4 t\u00e4ll\u00e4 puolella saarta oli. Jos alkaisin ajatella asiaa, en saisi itse\u00e4ni liikkeelle. Joten vain hypp\u00e4sin.\n\nOlin t\u00e4hd\u00e4nnyt kohtaan, joka omasta mielest\u00e4ni oli n\u00e4ytt\u00e4nyt ylh\u00e4\u00e4lt\u00e4 katsottuna kaikkein tummimmalta. Kun iskeydyin pinnan l\u00e4pi, m\u00e4rk\u00e4 kangas alkoi v\u00e4litt\u00f6m\u00e4sti vet\u00e4\u00e4 minua syvyyksiin. Sitten jalkapohjani osuivat pohjaan, ja ponnistin yl\u00f6s. P\u00e4\u00e4ni ponnahti pinnalle, ja ennen kuin ehdin edes vet\u00e4\u00e4 kunnolla henke\u00e4, olin jo alkanut kauhoa kohti kahta hahmoa puristaen samalla veist\u00e4 toisessa k\u00e4dess\u00e4ni.\n\nKompastelin liukasta kalliota pitkin ja lopulta onnistuin p\u00e4\u00e4sem\u00e4\u00e4n yl\u00f6s merest\u00e4. Mikael ja Vlassis olivat niin keskittyneit\u00e4 toisiinsa, etteiv\u00e4t he edes tajunneet minun saapuneen.\n\nOlin tullut viime hetkell\u00e4. Mikael oli l\u00e4hell\u00e4 loppua. En tuhlannut aikaa, vaan k\u00e4vin suoraan Vlassisin p\u00e4\u00e4lle. H\u00e4n oli selk\u00e4 minuun p\u00e4in, mutta k\u00e4\u00e4ntyi juuri kun tulin h\u00e4nt\u00e4 kohti veitsi ojossa.\n\nKukaan ei halua pit\u00e4\u00e4 m\u00e4rk\u00e4\u00e4, suurta vaatetta yll\u00e4\u00e4n silloin, kun joutuu kamppailemaan henkens\u00e4 edest\u00e4. Sotkeuduin helmaani kriittisell\u00e4 hetkell\u00e4 ja menetin parhaan tilaisuuteni.\n\nVlassis iski minua kyyn\u00e4rp\u00e4\u00e4ll\u00e4. Horjahdin. Samalla n\u00e4in sivusilm\u00e4ll\u00e4, kuinka Mikael putosi polvilleen, ja n\u00e4ytti silt\u00e4, ettei h\u00e4n en\u00e4\u00e4 koskaan nousisi yl\u00f6s.\n\nMin\u00e4 ja Vlassis tanssimme ensin my\u00f6t\u00e4p\u00e4iv\u00e4\u00e4n, sitten vastap\u00e4iv\u00e4\u00e4n. Tiesin suurimman osan h\u00e4nen tempuistaan. H\u00e4n oli aseettomanakin pelottavan kova vastus. Olin n\u00e4hnyt h\u00e4nen heittelev\u00e4n miehi\u00e4, jotka olivat minua paljon suurempia.\n\nVlassisin katse ei kuitenkaan ollut l\u00e4hesk\u00e4\u00e4n niin murhanhimoinen kuin olin odottanut. Mikael oli onnistunut heikent\u00e4m\u00e4\u00e4n h\u00e4nt\u00e4. Tajusin pian, ett\u00e4 h\u00e4n joutui k\u00e4ytt\u00e4m\u00e4\u00e4n suuren osan energiastaan vammojensa piilotteluun.\n\n\"Me voimme taistella, kunnes toinen meist\u00e4 kuolee\", Vlassis sanoi. \"Tai sitten me voimme p\u00e4\u00e4st\u00e4\u00e4 toisemme menem\u00e4\u00e4n.\"\n\nJos olin siihen menness\u00e4 arvellut h\u00e4nen voimiensa olevan v\u00e4hiss\u00e4, nyt tiesin sen varmasti. Jo toisen kerran t\u00e4n\u00e4\u00e4n h\u00e4n oli valmis k\u00e4ym\u00e4\u00e4n kauppaa, ja t\u00e4ll\u00e4 kertaa jollain h\u00e4nelle paljon arvokkaammalla kuin Mikaelin hengell\u00e4.\n\nUskomaton tyyppi, ajattelin. Vlassis oli p\u00e4\u00e4tt\u00e4nyt selvit\u00e4 hengiss\u00e4. Alek oli ollut nuori ja tyhm\u00e4 ja t\u00e4ysin v\u00e4\u00e4r\u00e4ss\u00e4, mutta ainakin h\u00e4nell\u00e4 oli ollut rohkeutta uhrata itsens\u00e4 uskomansa asian eteen.\n\n\"Ja miksi sin\u00e4 kuvittelet minun suostuvan?\" Sylk\u00e4isin sanat suustani samalla kun yritin l\u00f6yt\u00e4\u00e4 h\u00e4nen puolustuksestaan avoimen paikan. Tietenk\u00e4\u00e4n h\u00e4n ei j\u00e4tt\u00e4nyt minulle sellaista.\n\n\"Koska sinulla ei ole aikaa\", h\u00e4n vastasi. \"Palvelijat aikovat tappaa kaikki daimonit. \u00c4l\u00e4 kuvittele, ett\u00e4 sin\u00e4 ja veljesi olette poikkeus. Jos ette l\u00e4hde pian, ette l\u00e4hde t\u00e4\u00e4lt\u00e4 koskaan.\"\n\nH\u00e4n oli siis n\u00e4hnyt, mit\u00e4 saarella tapahtui. H\u00e4nen sanansa kauhistuttivat minua tavalla, jota oli vaikea selitt\u00e4\u00e4. Olin kuvitellut olevani turvassa palvelijoilta. Mutta oikeasti en voinut v\u00e4itt\u00e4\u00e4 olevani parempi kuin muut daimonit ja ansaitsevani heilt\u00e4 armoa.\n\n\"Mist\u00e4 tied\u00e4n, ettet sin\u00e4 l\u00e4hde meid\u00e4n per\u00e4\u00e4mme?\" kysyin.\n\n\"Et mist\u00e4\u00e4n.\"\n\nAinakin h\u00e4nell\u00e4 oli pokkaa my\u00f6nt\u00e4\u00e4 se \u00e4\u00e4neen.\n\nOliko h\u00e4n uskollinen Callistalle? Jos h\u00e4n oli, h\u00e4n ei voisi antaa minun el\u00e4\u00e4.\n\nEnnen t\u00e4t\u00e4 hetke\u00e4 en ollut kertaakaan yritt\u00e4nyt punnita h\u00e4neen persoonaansa. En edes silloin, kun h\u00e4n oli otellut kanssani ja iskenyt minulta tajun kankaalle. Ensimm\u00e4ist\u00e4 kertaa yritin liitt\u00e4\u00e4 kaikki tuntemani palat yhteen ja n\u00e4hd\u00e4 kokonaisuuden, joka oli Vlassis.\n\nVlassis, josta oli tullut Callistan suosikki vasta, kun is\u00e4ni oli j\u00e4tt\u00e4nyt saaren. Vlassis, joka oli kerran antanut ymm\u00e4rt\u00e4\u00e4, ettei ehk\u00e4 uskonutkaan jumaliin.\n\nVoisinko luottaa h\u00e4neen vai en?\n\nPudotin veitseni maahan. \"Mene, ennen kuin muutan mieleni.\"\n\nH\u00e4n ei j\u00e4\u00e4nyt odottamaan uutta kehotusta. H\u00e4n k\u00e4\u00e4ntyi ja l\u00e4hti juoksemaan kalliota pitkin.\n\nMikael avasi suunsa. \"Raisa, h\u00e4n...\"\n\nH\u00e4n ei koskaan ehtinyt sanoa lausetta loppuun. Olin jo potkaissut veitsen yl\u00f6s maasta ja tarttunut siihen ilmasta. Se l\u00e4hti k\u00e4dest\u00e4ni ja kiisi \u00e4\u00e4nett\u00f6m\u00e4sti ilman halki.\n\nJa osui Vlassisia selk\u00e4\u00e4n.\n\nH\u00e4n putosi mahalleen maahan.\n\nK\u00e4velin h\u00e4nen luokseen, vaikka se oli viimeinen asia, jonka olisin halunnut tehd\u00e4.\n\nOtin veitsen h\u00e4nen sel\u00e4st\u00e4\u00e4n ja k\u00e4\u00e4nsin h\u00e4net ymp\u00e4ri.\n\n\"Sin\u00e4 olet tismalleen niin kuin is\u00e4si\", h\u00e4n sanoi kun otin veneen avaimet h\u00e4nen vy\u00f6lt\u00e4\u00e4n. \"Kunniallinen vain niin kauan kuin se palvelee sinua itse\u00e4si.\"\n\nEn sanonut mit\u00e4\u00e4n. H\u00e4n oli oikeassa.\n\n\"Tee se\", h\u00e4n sanoi. \"Tapa minut.\"\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"T\u00e4m\u00e4 saari on nyt heid\u00e4n. Antaa uusien hallitsijoiden p\u00e4\u00e4tt\u00e4\u00e4 kohtalosi.\" Ny\u00f6kk\u00e4sin kohti jyrk\u00e4nteen reunaa, jossa t\u00e4ll\u00e4kin hetkell\u00e4 daimon toisensa j\u00e4lkeen j\u00e4i toiseksi kamppailussa palvelijoita vastaan.\n\n\"Aiotko sin\u00e4 vain j\u00e4tt\u00e4\u00e4 minut t\u00e4nne kuolemaan?\" h\u00e4n kysyi.\n\nDaimonit olivat vahvimpia yliluonnollisia, jotka tiesin, ja Vlassis heid\u00e4n joukossaan yksi vahvimmista. Palvelijoiden vihan edess\u00e4 h\u00e4n oli kuitenkin voimaton.\n\nNousin yl\u00f6s ja aloin k\u00e4vell\u00e4 kohti Mikaelia. Annoin Vlassisin j\u00e4\u00e4d\u00e4 makaamaan kalliolle.\n\n\"Raisa! Sin\u00e4 olet pelkuri!\"\n\nJatkoin k\u00e4velemist\u00e4. Joidenkin asioiden t\u00e4ytyi vain tapahtua. Pahojen asioiden. Jotta niiden j\u00e4lkeen voisi tulla toisia, parempia asioita.\n\nIkuisuudelta tuntuneen hetken j\u00e4lkeen tulin viimein Mikaelin luokse. H\u00e4n oli kuin ihmeen kaupalla onnistunut pysym\u00e4\u00e4n tajuissaan. \"Miten tiesit, ett\u00e4 h\u00e4n valehteli?\" h\u00e4n kysyi.\n\n\"H\u00e4n vihasi is\u00e4\u00e4ni\", sanoin. \"Kun Stefanos tapasi \u00e4itini, Vlassis valitsi Callistan puolen. H\u00e4n olisi pett\u00e4nyt minutkin.\" Katsoin h\u00e4nt\u00e4. \"T\u00e4m\u00e4 ei olisi loppunut koskaan, jos olisin antanut h\u00e4nen menn\u00e4.\"\n\nKuulin moottorin \u00e4\u00e4nen. Kohotin katseeni juuri parahiksi kun vene sy\u00f6ksyi esiin niemen takaa.\n\n\"Aika l\u00e4hte\u00e4 kotiin\", sanoin.\n\n\"Raisa!\" Mitja huusi veneest\u00e4.\n\nDaniel oli j\u00e4lleen ihmisen muodossa. H\u00e4n ohjasi veneen l\u00e4helle kalliota. Mitja hypp\u00e4si veteen ja juoksi minua vastaan. H\u00e4nen paidassaan oli verta, mutta se ei voinut olla h\u00e4nen omaansa \u2013 h\u00e4nen askeleensa oli kevyt. Vesi vaahtosi h\u00e4nen jalkojaan vasten, ja hetken aikaa n\u00e4in h\u00e4ness\u00e4 is\u00e4ni. Sitten h\u00e4n seisoi jo edess\u00e4mme kalliolla ja tiesin, ett\u00e4 h\u00e4n oli sama vanha veljeni.\n\nH\u00e4n heilautti Mikaelin toisen k\u00e4sivarren olalleen ja otti heti suurimman osan h\u00e4nen painostaan kontolleen. Onnistuimme saamaan Mikaelin veneeseen, vaikka vedess\u00e4 liikkuminen oli vaikeaa.\n\n\"Mist\u00e4 hitosta te t\u00e4m\u00e4n veneen saitte?\" kysyin.\n\n\"Vuokrasimme\", Mitja sanoi.\n\nDaniel ohjasi meid\u00e4t niemen k\u00e4rjen ohi.\n\nTuttu hontelo hahmo juoksi kallion viert\u00e4. Kukaan ei n\u00e4ytt\u00e4nyt olevan h\u00e4nen per\u00e4ss\u00e4\u00e4n, mutta h\u00e4n vilkuili taakseen.\n\n\"Ei, odottakaa!\" huusin.\n\nJotenkin Daniel onnistui kuulemaan sen moottorin jylin\u00e4n yli.\n\n\"N\u00e4en Adrianen. Emme voi j\u00e4tt\u00e4\u00e4 h\u00e4nt\u00e4 saarelle.\"\n\n\"Ei ole aikaa!\" Daniel huusi.\n\n\"H\u00e4n pelasti minut!\" ja koska tiesin, ett\u00e4 Danielin suostuttelu oli toivotonta, sanoin: \"Te voitte kyll\u00e4 menn\u00e4, mutta p\u00e4\u00e4st\u00e4k\u00e4\u00e4 minut maihin.\"\n\nDaniel kiroili. H\u00e4n hiljensi ensin vauhtia ja pys\u00e4ytti sitten veneen kokonaan.\n\nMin\u00e4 ja Mitja hypp\u00e4simme veteen. Mitja oli minua parempi uimari, ja ehti perille ensin.\n\n\"Adriane!\" huusin heti kun ehdin maihin. \"Tule meid\u00e4n mukaamme.\"\n\nH\u00e4n oli pari metri\u00e4 meid\u00e4n yl\u00e4puolellamme. Kuullessaan \u00e4\u00e4neni h\u00e4nen p\u00e4\u00e4ns\u00e4 k\u00e4\u00e4ntyi.\n\nKielekkeelt\u00e4 ei ollut helppoa reitti\u00e4 alas kalliolle, jolla seisoimme. Adriane t\u00e4hyili alas meihin, tiet\u00e4m\u00e4tt\u00e4 mit\u00e4 tehd\u00e4.\n\n\"Hypp\u00e4\u00e4, min\u00e4 otan kyll\u00e4 vastaan\", Mitja sanoi.\n\nH\u00e4n ep\u00e4r\u00f6i vain hetken ennen kuin hypp\u00e4si.\n\nMitja nappasi h\u00e4net helposti kiinni. \"Min\u00e4 en osaa uida\", Adriane sanoi ennen kuin Mitjan k\u00e4det olivat ehtineet irrottaa otteensa.\n\nMitja ei h\u00e4tk\u00e4ht\u00e4nyt.\n\n\"Pid\u00e4 lujasti kiinni\", Mitja sanoi. H\u00e4n k\u00e4\u00e4nsi Adrianelle selk\u00e4ns\u00e4, ja t\u00e4m\u00e4 kietoi k\u00e4sivartensa h\u00e4nen kaulalleen. Kaksi askelta ja sitten he olivat jo veden varassa. Olin juuri aikeissa seurata heit\u00e4, kun n\u00e4in jyrk\u00e4nteell\u00e4 seisovan palvelijan t\u00e4ht\u00e4\u00e4v\u00e4n meit\u00e4 jousella.\n\nJ\u00e4hmetyin. Etsin h\u00e4nen katseensa. Sitten, hitaasti, kohotin oikean k\u00e4teni p\u00e4\u00e4ni p\u00e4\u00e4lle. Etusormessani roikkui veneen avain.\n\nVene oli palvelijoiden ainut mahdollisuus p\u00e4\u00e4st\u00e4 pois saarelta.\n\nPalvelijan jousik\u00e4si huojui, mutta h\u00e4n ei laskenut asettaan. Kyykistyin hitaasti ja asetin avaimen kivelle jalkojeni juureen. Sitten astuin veteen ja aloin uida.\n\nEhdin melkein saavuttaa Mitjan ja Adrianen ennen kuin he p\u00e4\u00e4siv\u00e4t veneelle. Mitja meni ensin ja auttoi Adrianen yl\u00f6s tikkaita minun varmistellessani selustaa.\n\nMikael makasi lattialla kalpeana. Lys\u00e4hdin h\u00e4nen viereens\u00e4. En ollut huomannut sadetta en\u00e4\u00e4 v\u00e4h\u00e4\u00e4n aikaan, mutta nyt katselin kun pisarat ropisivat h\u00e4nen kasvoilleen ja luomilleen. H\u00e4n tuskin edes liikahti.\n\n\"Mit\u00e4 h\u00e4nelle on tehty?\" Daniel kysyi.\n\n\"He juottivat h\u00e4nelle hopeaa.\"\n\nH\u00e4n kiroili j\u00e4lleen, mutta juuri silloin ukkonen jylisti l\u00e4pi taivaan enk\u00e4 kuullut sanaakaan.\n\nMoottori k\u00e4ynnistyi kolmannella yritt\u00e4m\u00e4ll\u00e4. Kun vene l\u00e4hti kiit\u00e4m\u00e4\u00e4n merta pitkin, katsoin taakseni.\n\nN\u00e4ytti silt\u00e4 kuin taivas olisi ollut tulessa. Kuvittelin kuulevani daimonien huudot moottorin jylin\u00e4n yli. Kasvoni olivat m\u00e4r\u00e4t sateesta ja kyynelist\u00e4.\n\nNiin paljon verta. Niin paljon tarpeetonta tappamista.\n\n\"Mit\u00e4 min\u00e4 sanoin: \u00e4l\u00e4 koskaan katso olkasi yli\", Daniel huusi. Mutta t\u00e4ll\u00e4 kertaa h\u00e4nen ilmeens\u00e4 oli melkein lempe\u00e4.\n\n***\n\nP\u00e4\u00e4sty\u00e4mme et\u00e4\u00e4lle saaresta Daniel hidasti vauhtia. \"On todella huono idea olla keskell\u00e4 merta juuri nyt\", Mitja huusi.\n\n\"No palataan ihmeess\u00e4 takaisin!\" vastasin.\n\nVeljeni mulkaisi minua vihaisesti, mutta p\u00e4\u00e4tti olla hiljaa.\n\n\"Veljesi on oikeassa. Meid\u00e4n pit\u00e4\u00e4 p\u00e4\u00e4st\u00e4 maihin\", Daniel sanoi. Kun n\u00e4in h\u00e4nen kiristyneet leukaper\u00e4ns\u00e4 k\u00e4sitin, ettei h\u00e4n suinkaan puhunut ukkosesta. Olin jo ehtinyt unohtaa t\u00e4ysikuun ja sen, ett\u00e4 sek\u00e4 Mikaelin ett\u00e4 Danielin oli p\u00e4\u00e4st\u00e4v\u00e4 jonnekin, miss\u00e4 he voisivat muuttua rauhassa.\n\n\"Kreeta on monen tunnin matkan p\u00e4\u00e4ss\u00e4\", sanoin.\n\n\"Mit\u00e4 te puhutte?\" Adriane kysyi.\n\nMitja kertasi nopeasti keskustelun kreikaksi.\n\n\"L\u00e4hemp\u00e4n\u00e4 on toinen saari\", Adriane sanoi.\n\nEn ollut koskaan kuullut moisesta, mutta minulla ei ollut mit\u00e4\u00e4n syyt\u00e4 ep\u00e4ill\u00e4 h\u00e4nt\u00e4.\n\nAdriane nousi yl\u00f6s ja meni Danielin viereen. H\u00e4n osoitti vasemmalle horisonttiin. \"Tuolla\", h\u00e4n sanoi.\n\nDanielin oli t\u00e4ytynyt ymm\u00e4rt\u00e4\u00e4, mit\u00e4 h\u00e4n tarkoitti, sill\u00e4 h\u00e4n teki jyrk\u00e4n k\u00e4\u00e4nn\u00f6ksen ja kiihdytti veneen hurjaan vauhtiin. Sek\u00e4 min\u00e4 ett\u00e4 Mikael kierimme lattialla toiseen laitaan. Se ei ollut t\u00e4ysin kivutonta, ja koska p\u00e4\u00e4dyin h\u00e4nen p\u00e4\u00e4lleen, uskoin h\u00e4nen kroppansa ottaneen pahimmat kolhut vastaan puolestani.\n\nSe, ettei Mikael p\u00e4\u00e4st\u00e4nyt \u00e4\u00e4nt\u00e4k\u00e4\u00e4n, sai minut pelk\u00e4\u00e4m\u00e4\u00e4n pahinta. H\u00e4n oli niin kalpea, etten tiennyt, virtasiko veri en\u00e4\u00e4 lainkaan h\u00e4nen suonissaan.\n\n\"Sinulla ei ole lupaa kuolla\", sanoin.\n\n\"En ajatellutkaan, ett\u00e4 minulla olisi.\"\n\nSen j\u00e4lkeen h\u00e4n sulki silm\u00e4ns\u00e4, enk\u00e4 en\u00e4\u00e4 ollut varma, oliko h\u00e4n tajuissaan vai ei.\n\n***\n\n\"Olisipa minulla ollut k\u00e4teist\u00e4 mukana\", Daniel murahti.\n\n\"Niin, onhan se aika ik\u00e4v\u00e4\u00e4, ettei poliiseilla ole korttimaksup\u00e4\u00e4tteit\u00e4 lahjusten ottamista varten\", sanoin samalla kun kompastelin Mikaelin painon alla. Kannattelin h\u00e4nt\u00e4, vaikka pysyin tuskin itsek\u00e4\u00e4n jaloillani. \"Sit\u00e4 paitsi luulin, ett\u00e4 kaikki sinun rahasi j\u00e4iv\u00e4t Mikaelille.\"\n\n\"Pid\u00e4tk\u00f6 minua niin tyhm\u00e4n\u00e4?\"\n\nEn tiennyt, kumpi minua \u00e4rsytti enemm\u00e4n: se, ett\u00e4 Daniel oli yh\u00e4 rikas vai se, ett\u00e4 olin kuvitellut h\u00e4nen antaneen kaiken pojalleen.\n\nOlimme j\u00e4tt\u00e4neet myrskyn taaksemme. Kun olimme saavuttaneet saaren, Mitja oli p\u00e4\u00e4st\u00e4nyt meid\u00e4t maihin ja suunnannut etsim\u00e4\u00e4n paikkaa veneelle. Olimme joutuneet kahlaamaan kymmenisen metri\u00e4 pitkin yleist\u00e4 uimarantaa. Kaksi miest\u00e4, joista toinen oli kantokunnossa, ja kaksi outoihin kaapuihin pukeutunutta tytt\u00f6\u00e4 eiv\u00e4t olleet sellainen seurakunta, joka voisi liikkua ker\u00e4\u00e4m\u00e4tt\u00e4 uteliaita katseita. Viimein olimme my\u00f6s her\u00e4tt\u00e4neet viranomaisten huomion.\n\nOli kest\u00e4nyt hetken, ennen kuin olimme saaneet vakuutettua kaikki mahdolliset ihmiset siit\u00e4, ettei meit\u00e4 tarvinnut vied\u00e4 sairaalaan ja ettemme toisaalta olleet laittomia maahantulijoita, huumekuriireja tai muita rikollisia. Onneksemme kukaan ei ehtinyt ep\u00e4ill\u00e4 meit\u00e4 lapsenkaappauksesta, sill\u00e4 Adrianelle olisi ollut hyvin hankala keksi\u00e4 uskottavaa tarinaa.\n\nSaaresta n\u00e4ki jo kaukaa, ettei se ollut mik\u00e4\u00e4n turistirys\u00e4: kaikki kyltit olivat pelk\u00e4st\u00e4\u00e4n kreikaksi, enk\u00e4 n\u00e4hnyt ket\u00e4\u00e4n kaupustelemassa kr\u00e4\u00e4s\u00e4\u00e4. Adriane oli kertonut minulle saaren nimen, mutta se ei ollut sanonut minulle mit\u00e4\u00e4n. Onnistuimme kuitenkin lopulta l\u00f6yt\u00e4m\u00e4\u00e4n hotellin. Tai pikemminkin majatalon, mutta emme olleet t\u00e4htiluokitusten per\u00e4\u00e4n.\n\nTiskin takana olevan naisen silm\u00e4t levisiv\u00e4t, kun h\u00e4n n\u00e4ki meid\u00e4n tulevan sis\u00e4\u00e4n.\n\n\"Me olemme t\u00e4ynn\u00e4\", h\u00e4n sanoi englanniksi.\n\n\"Me maksamme kolme kertaa sen, mit\u00e4 kukaan muu\", Daniel vastasi.\n\n\"Pahoittelen, herra, me...\"\n\n\"Viisinkertaisesti.\"\n\nNainen r\u00e4p\u00e4ytti kerran silmi\u00e4\u00e4n. \"Tervetuloa hotelli Lagosiin.\"\n\n\"Niin v\u00e4h\u00e4n ajattelinkin.\" H\u00e4n ojensi virkailijalle passinsa ja luottokorttinsa ja sutaisi yhteystiedot kaavakkeeseen. H\u00e4n nappasi avaimen virkailijan k\u00e4dest\u00e4 ennen kuin t\u00e4m\u00e4 ehti aloittaa esittelypuheensa. L\u00e4hdimme nousemaan portaita yl\u00f6s.\n\n\"Oletteko varma, ettei h\u00e4n tarvitse l\u00e4\u00e4k\u00e4ri\u00e4?\" virkailija huusi per\u00e4\u00e4mme. H\u00e4n oli kauhuissaan, muttei niin kauhuissaan, etteik\u00f6 olisi ollut huolissaan Mikaelista. Sit\u00e4 oli pakko arvostaa.\n\n\"Ei, mutta kiitos kun kysyit\", vastasin olkani yli.\n\nDaniel avasi oven ja me seurasimme per\u00e4ss\u00e4. Yritin laskea Mikaelin s\u00e4ngylle, mutta voimani eiv\u00e4t riitt\u00e4neet, ja niinp\u00e4 p\u00e4\u00e4dyimme lattialle.\n\n\"Minun pit\u00e4\u00e4 menn\u00e4\", Daniel sanoi. Kun katsoin yl\u00f6s, n\u00e4in, ett\u00e4 h\u00e4n puri leukojaan yhteen. H\u00e4n vilkaisi poikaansa ennen kuin katsoi taas minua. \"Kaksi alfasutta samassa huoneessa ei johda mihink\u00e4\u00e4n hyv\u00e4\u00e4n.\"\n\nNy\u00f6kk\u00e4sin. Vilkaisin nopeasti ikkunaan: aurinko oli jo ehtinyt vaipua pelottavan alas.\n\n\"Lukitse ovi. Saat itse p\u00e4\u00e4tt\u00e4\u00e4, oletko sen sis\u00e4- vai ulkopuolella, mutta suosittelen ulkopuolta.\"\n\nH\u00e4n j\u00e4tti puhelimensa ja lompakkonsa p\u00f6yd\u00e4lle ja l\u00e4hti. En kysynyt minne \u2013 minun oli luotettava siihen, ett\u00e4 h\u00e4n tiesi, mit\u00e4 teki. Siirsin kaiken huomioni k\u00e4sill\u00e4 olevaan ongelmaan.\n\nMikael makasi lattialla k\u00e4sivarsi silmiens\u00e4 p\u00e4\u00e4ll\u00e4. En ollut lainkaan varma, selvi\u00e4isik\u00f6 h\u00e4n muutoksesta hengiss\u00e4. Toisaalta en ollut varma, selvi\u00e4isik\u00f6 h\u00e4n hengiss\u00e4 ilman sit\u00e4.\n\n\"Me tarvitsemme lihaa. Paljon, paljon lihaa\", mutisin. H\u00e4nen sutensa olisi hyvin n\u00e4lk\u00e4inen jos ja kun h\u00e4n edes onnistuisi vaihtamaan muotoa.\n\n\"Min\u00e4 voin menn\u00e4 hakemaan\", Adriane sanoi.\n\nYll\u00e4tyin. En niink\u00e4\u00e4n siksi, ett\u00e4 h\u00e4n tarjoutui, vaan siksi, ett\u00e4 olin jo unohtanut h\u00e4nen olemassaolonsa. Ja ilmeisesti mutissut kreikaksi.\n\n\"Se ei taida olla hyv\u00e4 ajatus\", sanoin. En uskaltanut p\u00e4\u00e4st\u00e4\u00e4 h\u00e4nt\u00e4 harhailemaan kaduille, vaikka h\u00e4nen apunsa olisikin tullut tarpeeseen.\n\n\"Min\u00e4 olen k\u00e4ynyt t\u00e4m\u00e4n saaren kaupoissa\", h\u00e4n sanoi. \"Daimonit toivat meid\u00e4t aina v\u00e4lill\u00e4 t\u00e4nne ostoksille. Osaan k\u00e4ytt\u00e4\u00e4 rahaa.\"\n\n\"Ent\u00e4 luottokorttia?\" kysyin.\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\n\"Meid\u00e4n per\u00e4ss\u00e4mme saattaa olla daimoneita. He tunnistavat sinut.\"\n\n\"En usko, ett\u00e4 kukaan on tulossa\", h\u00e4n sanoi.\n\nAjattelin palavaa taivasta ja palvelijoita veitset k\u00e4dess\u00e4. Ajattelin vuosisatoja jatkunutta orjuuttamista. Kuka min\u00e4 olin sanomaan, ettei palvelijoilla ollut ollut oikeutta tuhota sortajiaan?\n\n\"Olen varovainen\", Adriane lis\u00e4si.\n\nYritin saada ajatukseni kasaan. \"Hyv\u00e4 on. Min\u00e4 annan sinulle Danielin luottokortin. Jos et l\u00f6yd\u00e4 etsim\u00e4\u00e4si kohtuullisen nopeasti, tule takaisin.\"\n\nJ\u00e4lkik\u00e4teen ajatellen kaksitoistavuotiaan tyt\u00f6n l\u00e4hett\u00e4minen yksin pimenev\u00e4lle kaupungille Danielin Amexin ja tunnusluvun kanssa ei ollut kovin j\u00e4rkev\u00e4\u00e4, mutta olin ep\u00e4toivoinen.\n\nHuoleni oli kuitenkin turha, sill\u00e4 h\u00e4n palasi takaisin jo vartin kuluttua mukanaan kaksi t\u00e4ytt\u00e4 muovikassia.\n\nH\u00e4n oli ostanut souvlakia. Paljon souvlakia.\n\n\"Varsinainen lihakauppa oli jo kiinni.\"\n\n\"T\u00e4m\u00e4 kelpaa hyvin\", sanoin. Olin aika varma, etteiv\u00e4t sudet olleet ronkeleita ruuan suhteen.\n\n\"Min\u00e4 muutun pian\", kuulin Mikaelin sanovan lattialta. H\u00e4n oli kier\u00e4ht\u00e4nyt kyljelleen ja kietonut k\u00e4det ymp\u00e4rilleen niin kuin h\u00e4nen vatsaansa olisi koskenut. H\u00e4nen ihonsa kiilteli kosteana. Itse olin kuivunut heti, kun olimme p\u00e4\u00e4sseet pois sadepilvien alta. H\u00e4n siis hikoili.\n\n\"Adriane, sinun t\u00e4ytyy l\u00e4hte\u00e4 t\u00e4st\u00e4 huoneesta ja lukita ovi per\u00e4ss\u00e4si. Mit\u00e4 ikin\u00e4 tapahtuukin, sin\u00e4 et saa avata ovea ennen kuin min\u00e4 annan luvan, ymm\u00e4rr\u00e4tk\u00f6?\" H\u00e4n ny\u00f6kk\u00e4si. \"Ja jos... Jos minusta ei kuulu, soita Mitjalle. Sin\u00e4h\u00e4n tied\u00e4t, miten puhelin toimii?\"\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\n\"Sinunkin pit\u00e4\u00e4 menn\u00e4\", Mikael sanoi hiljaa.\n\nPudistin p\u00e4\u00e4t\u00e4ni. \"Jonkun t\u00e4ytyy pit\u00e4\u00e4 huoli, ettet sin\u00e4 satuta ket\u00e4\u00e4n.\"\n\nYksi ovi ei pitelisi h\u00e4nt\u00e4. Vihainen ja n\u00e4lk\u00e4inen ihmissusi irrallaan t\u00e4ydess\u00e4 majatalossa olisi voinut toimia elokuvassa, mutta minulla ei ollut aikomustakaan ottaa selv\u00e4\u00e4, milt\u00e4 se n\u00e4ytti todellisuudessa.\n\nMikael ei ollut kanssani samaa mielt\u00e4, mutta h\u00e4nell\u00e4 ei ollut vaihtoehtoja. H\u00e4n ei pystynyt seisomaan, saati pakottamaan minua ulos huoneesta.\n\nK\u00e4\u00e4nnyin Adrianen puoleen. \"Mene nyt. Ota avain ja puhelin.\"\n\nH\u00e4n teki ty\u00f6t\u00e4 k\u00e4sketty\u00e4. Kuulin kun avain k\u00e4vi lukossa. En p\u00e4\u00e4sisi en\u00e4\u00e4 ulos huoneesta, vaikka haluaisinkin.\n\n\"Ookoo?\" kysyin Mikaelilta.\n\nH\u00e4n ny\u00f6kk\u00e4si ja veti henke\u00e4.\n\nJa samassa h\u00e4n jo alkoi muuttaa muotoa. En halunnut katsoa, joten painoin p\u00e4\u00e4ni.\n\nEhdin tuskin menn\u00e4 asemiin oven eteen kun h\u00e4n jo oli muuttunut. Aivan kuin susi olisi odottanut t\u00e4t\u00e4 hetke\u00e4 jo pitk\u00e4\u00e4n.\n\nKuulin kun h\u00e4n ravisteli itse\u00e4\u00e4n. Kun kohotin katseeni, n\u00e4in h\u00e4nen seisovan edess\u00e4ni keskell\u00e4 huonetta. J\u00e4lleen kerran ihmettelin, miten kaunis h\u00e4nen sutensa oli. Ajatus kuitenkin katosi hyvin nopeasti, sill\u00e4 h\u00e4n alkoi murista.\n\nNojasin koko painollani oveen ja aloin valua kohti lattiaa. Mit\u00e4 ikin\u00e4 tapahtuisikin, Mikael ei saanut p\u00e4\u00e4st\u00e4 sen l\u00e4pi.\n\nSusi otti askeleen l\u00e4hemm\u00e4s. Puristin silm\u00e4ni kiinni.\n\nSuden murina oli \u00e4\u00e4ni, joka kuulosti hyvin vakuuttavalta kymmenen sentin et\u00e4isyydelt\u00e4. Minua pelotti. Paljon. Ja pelko oli tietenkin t\u00e4ydellisen v\u00e4\u00e4r\u00e4 reaktio, sill\u00e4 se vain lietsoi Mikaelin sutta.\n\nLaskin viiteen, sitten kymmeneen. Mit\u00e4\u00e4n ei tapahtunut.\n\nAvasin silm\u00e4ni. Mikael seisoi edelleen edess\u00e4ni. H\u00e4nen p\u00e4\u00e4ns\u00e4 oli korkeammalla kuin omani, ja n\u00e4in, kuinka se kallistui puolelle ja toiselle, kun h\u00e4n arvioi tilannetta. Mik\u00e4\u00e4n ei kuitenkaan antanut ilmi, ett\u00e4 h\u00e4n olisi tunnistanut minut.\n\nVedin syv\u00e4\u00e4n henke\u00e4 ja pid\u00e4tin sit\u00e4 hetken ennen kuin puhalsin ulos. Sit\u00e4 mukaa kun sain pelkoni hallintaan, Mikaelkin tuntui rauhoittuvan.\n\nSouvlakikassi oli edelleen vieress\u00e4ni. Kurotin sit\u00e4 kohti pit\u00e4en katseeni jossain lattian ja Mikaelin v\u00e4limaastossa. Kun susi ei sy\u00f6ksynyt kiinni ranteeseeni, uskaltauduin vet\u00e4m\u00e4\u00e4n kassin itse\u00e4ni kohti.\n\nMikael otti askeleen taaksep\u00e4in. H\u00e4nen p\u00e4\u00e4ns\u00e4 roikkui matalalla ja niskakarvat nousivat pystyyn, kun h\u00e4n kuuli kassin kahinan.\n\n\"Ei h\u00e4t\u00e4\u00e4\", sanoin. Yritin pit\u00e4\u00e4 \u00e4\u00e4nen tasaisena ja pehme\u00e4n\u00e4. \"T\u00e4\u00e4ll\u00e4 on jotain, mink\u00e4 luulisin kiinnostavan sinua.\"\n\nOtin ensimm\u00e4isen folioon k\u00e4\u00e4rityn pakkauksen ja avasin sen hitaasti niin, ett\u00e4 Mikael saattoi koko ajan n\u00e4hd\u00e4, mit\u00e4 tein.\n\nH\u00e4nen nen\u00e4ns\u00e4 alkoi liikkua. Ty\u00f6nsin avatun pakkauksen h\u00e4nen eteens\u00e4. H\u00e4n haisteli lihaa, kunnes totesi sen sy\u00f6m\u00e4kelpoiseksi. Sen j\u00e4lkeen meni vain muutama sekunti, kun h\u00e4n oli jo nielaissut kaiken. H\u00e4n katsoi minua odottavasti.\n\nAloin avata seuraavaa pakettia, mutta Mikael k\u00e4vi siihen kiinni ennen kuin ehdin saada k\u00e4\u00e4rett\u00e4 auki. Senkin sis\u00e4lt\u00f6 katosi h\u00e4nen suuhunsa saman tien.\n\nSy\u00f6tin h\u00e4nelle souvlakia kunnes viimeinenkin murunen oli kadonnut h\u00e4nen kitaansa. H\u00e4n nuoleskeli rasvaista kuonoaan ja katsoi minua tavalla, joka tarkoitti, ett\u00e4 h\u00e4n toivoi minun taikovan jostain lis\u00e4\u00e4. Naurahdin.\n\n\"No, t\u00e4m\u00e4h\u00e4n oli hauskaa\", sanoin \u00e4\u00e4neen.\n\nOdotin. Mikael ei osoittanut merkki\u00e4k\u00e4\u00e4n siit\u00e4, ett\u00e4 olisi muuttumassa pian takaisin. Uskaltauduin kohottamaan k\u00e4teni, ja pian sormeni l\u00f6ysiv\u00e4t h\u00e4nen turkkinsa.\n\nH\u00e4n tuli makaamaan viereeni. Kun yritin ottaa paremman asennon, kultaiset silm\u00e4t tuijottivat minua ter\u00e4v\u00e4sti, kunnes olin taas liikkumatta. H\u00e4n nukahti p\u00e4\u00e4 reidell\u00e4ni.\n\n***\n\nEn olisi uskonut voivani nukahtaa, mutta kun avasin silm\u00e4ni, tajusin juuri niin tapahtuneen. Mikaelin p\u00e4\u00e4 oli noussut reidelt\u00e4ni ja h\u00e4n tuijotti kohti ovea. Joku koputti siihen, tajusin hetken kuluttua. Toivoin todella, ettei se ollut siivooja.\n\nNousin yl\u00f6s, mutten ennen kuin sanoin h\u00e4nelle: \"Ei h\u00e4t\u00e4\u00e4.\"\n\n\"Kuka siell\u00e4?\" kysyin oven l\u00e4pi.\n\n\"Mit\u00e4 helvetti\u00e4 siell\u00e4 tapahtuu?\" \u00c4\u00e4ni oli Mitjan, ja se sai minut huokaamaan helpotuksesta.\n\n\"Sori. Min\u00e4 taisin nukahtaa.\"\n\n\"Taisit nukahtaa? Daniel pohti t\u00e4ss\u00e4 juuri, pit\u00e4isik\u00f6 h\u00e4nen tapella poikansa kanssa vain siksi, ettei Mikael ehtisi sy\u00f6d\u00e4 sinun ruumistasi kokonaan. J\u00e4isi edes jotain haudattavaa.\"\n\n\"\u00c4l\u00e4 ota h\u00e4nt\u00e4 niin tosissaan\", sanoin. \"H\u00e4nell\u00e4 on tosi kiero huumorintaju.\"\n\nAinakin luulin, ett\u00e4 Daniel oli vitsaillut. H\u00e4nest\u00e4 ei koskaan tiennyt.\n\nMikael katosi vierelt\u00e4ni. Otsani rypistyi, mutta kun n\u00e4in, ett\u00e4 h\u00e4n livahti kylpyhuoneeseen, minulta p\u00e4\u00e4si jo toinen helpottunut huokaus.\n\n\"Luojan kiitos\", mutisin.\n\n\"Mit\u00e4?\"\n\n\"Oven voi kohta avata\", sanoin Mitjalle. \"Mikael vaihtaa juuri muotoa.\"\n\nKuulin avainten rapinaa. Hetken p\u00e4\u00e4st\u00e4 ovi avattiin, ja n\u00e4in veljeni seisovan edess\u00e4ni siisteiss\u00e4 ja ennen kaikkea verett\u00f6miss\u00e4 vaatteissa. Adriane seisoi Mitjan takana. H\u00e4nen huolestuneet silm\u00e4ns\u00e4 kirkastuivat heti, kun h\u00e4n n\u00e4ki minut.\n\n\"Ole hyv\u00e4\", Mitja sanoi ojentaen minulle muovikassin. Kurkistin sen sis\u00e4\u00e4n. Siell\u00e4 oli v\u00e4rik\u00e4s kes\u00e4mekko, sandaalit sek\u00e4 alusvaatteita. Kohotin katseeni. \"Kiit\u00e4 Adria\", veljeni sanoi.\n\nVeljell\u00e4ni oli h\u00e4nelle lempinimi. S\u00f6p\u00f6\u00e4.\n\nHymyilin tyt\u00f6lle. \"Kiitos.\"\n\nKassin pohjalta l\u00f6ytyi my\u00f6s vaatteita Mikaelille, joten raotin varovaisesti kylpyhuoneen ovea ja pudotin kassin sis\u00e4\u00e4n. Ei mennyt kauan, kun h\u00e4n astui esiin.\n\nOli h\u00e4kellytt\u00e4v\u00e4\u00e4 n\u00e4hd\u00e4 h\u00e4net j\u00e4lleen ihmisen muodossa. H\u00e4n n\u00e4ytti jo paremmalta, muttei viel\u00e4k\u00e4\u00e4n t\u00e4ysin omalta itselt\u00e4\u00e4n. Suurin osa ruhjeista oli parantunut ja molemmat silm\u00e4t katsoivat minuun.\n\n\"Mik\u00e4 olo?\" kysyin.\n\n\"Ihan hyv\u00e4\", h\u00e4n vastasi. Se ei varmaankaan ollut koko totuus, mutta h\u00e4n oli ilmeisen haluton jakamaan yksityiskohtia, kun paikalla oli niinkin monta ihmist\u00e4.\n\nDaniel ilmestyi paikalle pian veljeni ja Adrin j\u00e4lkeen. H\u00e4nkin oli pukeutunut t\u00e4ysin moitteettomasti, enk\u00e4 viitsinyt kysy\u00e4, miss\u00e4 h\u00e4n oli viett\u00e4nyt y\u00f6ns\u00e4. Sanaakaan sanomatta Adriane ojensi h\u00e4nelle h\u00e4nen puhelimensa ja lompakkonsa. Daniel kohotti kulmiaan ja ty\u00f6nsi ne taskuunsa.\n\nOlimme selvinneet hengiss\u00e4, ja \u00e4kki\u00e4 kaikki tapahtunut tuntui niin ep\u00e4todelliselta, ett\u00e4 sit\u00e4 oli vaikea tajuta. Meill\u00e4 ei ollut hajuakaan, mit\u00e4 tehd\u00e4 seuraavaksi.\n\nKuin yhteisest\u00e4 sopimuksesta k\u00e4\u00e4nsimme kaikki huomion Adrianeen.\n\n\"Tunnetko saaren ulkopuolelta ket\u00e4\u00e4n, jonka luokse voisit menn\u00e4?\" Mitja kysyi.\n\nH\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n.\n\nVasta silloin tajusin, etten tiennyt h\u00e4nest\u00e4 mit\u00e4\u00e4n. \"Miten sin\u00e4 p\u00e4\u00e4dyit saarelle?\" kysyin.\n\n\"Min\u00e4 synnyin siell\u00e4\", h\u00e4n sanoi. \"Minun vanhempani olivat daimoneita.\"\n\n\"Mutta sin\u00e4 et ole.\" Tiesin, ettei daimonien lapsista automaattisesti tullut daimoneita. En vain ollut koskaan ajatellut, mit\u00e4 ihmislapsille tapahtui. \"He siis hylk\u00e4siv\u00e4t sinut.\" Vasta sanottuani sen tajusin, miten karkealta kuulostin. Viime p\u00e4ivien tapahtumat olivat vieneet kaiken hienotunteisuuteni menness\u00e4\u00e4n.\n\nAdrianella ei tietenk\u00e4\u00e4n ollut passia. Minulla ja Mikaelillakaan ei ollut, mutta me voisimme soittaa l\u00e4hetyst\u00f6\u00f6n ja saada tilap\u00e4iset matkustusluvat \u2013 tai ainakin niin Daniel meille vakuutti. Adrianen kohdalla tilanne ei ollut aivan niin yksinkertainen, sill\u00e4 h\u00e4nell\u00e4 ei ollut koskaan ollutkaan papereita. Virallisesti h\u00e4nt\u00e4 ei ollut olemassa. Emme voineet ottaa h\u00e4nt\u00e4 mukaan Suomeen, mutta en my\u00f6sk\u00e4\u00e4n halunnut j\u00e4tt\u00e4\u00e4 h\u00e4nt\u00e4 poliisiasemalle tiet\u00e4m\u00e4tt\u00e4 yht\u00e4\u00e4n, mit\u00e4 h\u00e4nelle sen j\u00e4lkeen tapahtuisi.\n\n\"Min\u00e4 j\u00e4\u00e4n t\u00e4nne h\u00e4nen kanssaan\", Daniel sanoi.\n\nSe oli viimeinen asia, jonka oletin kuulevani h\u00e4nen suustaan. Jopa Mikael n\u00e4ytti yll\u00e4ttyneelt\u00e4.\n\n\"Daniel, sin\u00e4 olet kielt\u00e4m\u00e4tt\u00e4 hirve\u00e4n hyv\u00e4 hoitamaan monenlaisia asioita, mutta teini-ik\u00e4isest\u00e4 tyt\u00f6st\u00e4 huolehtiminen silloin, kun teill\u00e4 ei ole edes yhteist\u00e4 kielt\u00e4, ei ole yksi niist\u00e4.\"\n\nOdotin joko ivallista hymy\u00e4 tai jotain muuta merkki\u00e4 siit\u00e4, ett\u00e4 h\u00e4n oli valmis kinaamaan kanssani, mutta h\u00e4n pysyi vakavana. \"Teid\u00e4n ei tarvitse murehtia. Min\u00e4 pid\u00e4n h\u00e4nest\u00e4 huolta.\"\n\nMinun oli pakko kysy\u00e4: \"Miksi?\"\n\n\"H\u00e4n auttoi meit\u00e4, vaikka h\u00e4nen ei ollut pakko. V\u00e4hint\u00e4, mit\u00e4 voin tehd\u00e4, on auttaa h\u00e4net alkuun. Enk\u00e4 min\u00e4 muutenkaan voisi palata teid\u00e4n kanssanne Hukkavaaraan.\"\n\n\"Minusta se on hyv\u00e4 idea\", Mitja sanoi.\n\nOlin \u00e4llik\u00e4ll\u00e4 ly\u00f6ty, ja kerroin sen veljelleni katseellani. Mitja ei l\u00e4htenyt mukaan \u00e4\u00e4nett\u00f6m\u00e4\u00e4n v\u00e4ittelyyn, vaan vilkaisi Adrianea. Tytt\u00f6 oli jo niin pitk\u00e4, ettei Mitjan tarvinnut laskea katsettaan kovin alas. Adriane n\u00e4ytti valppaalta eik\u00e4 lainkaan \u00e4rtyneelt\u00e4 siit\u00e4, ett\u00e4 puhuimme koko ajan h\u00e4nen ylitseen kielt\u00e4, jota h\u00e4n ei ymm\u00e4rt\u00e4nyt. Olin ajatellut h\u00e4nen hiljaisuutensa johtuvan sokista, mutta kun h\u00e4n katsoi velje\u00e4ni suurilla silmill\u00e4\u00e4n, tajusin, ettei kyse ollut lainkaan siit\u00e4. H\u00e4n oli nuori, mutta h\u00e4nen oli t\u00e4ytynyt n\u00e4hd\u00e4 paljon pahempia asioita kuin monet aikuiset. H\u00e4nen katseessaan oli jotain paljon vanhempaa kuin h\u00e4nen ik\u00e4isens\u00e4 katseessa kuului olla.\n\n\"Adriane, me joudumme l\u00e4htem\u00e4\u00e4n, mutta Daniel j\u00e4\u00e4 t\u00e4nne ja pit\u00e4\u00e4 sinusta huolta\", Mitja sanoi ja ny\u00f6kk\u00e4si Danielin suuntaan. Adrianen katse irrottautui h\u00e4nest\u00e4 ja siirtyi Danieliin. Adrianen kannatti harkita uraa pokeriammattilaisena, sill\u00e4 minulla ei ollut aavistustakaan, mit\u00e4 h\u00e4n ajatteli Danielista. \"Me n\u00e4emme viel\u00e4, lupaan sen\", Mitja jatkoi, ja Adrianen katse k\u00e4\u00e4ntyi nopeasti takaisin h\u00e4neen. \"Annan sinulle minun ja Raisan puhelinnumerot. Soita meille heti, jos jotain tapahtuu. Tai itse asiassa soita meille joka tapauksessa.\"\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\n\"Hyv\u00e4. Me etsimme sinut k\u00e4siimme ja tulemme tapaamaan sinua heti, kun voimme, mutta nyt on kaikkien kannalta parempi, ett\u00e4 hajaannumme.\"\n\n\"Ei se mit\u00e4\u00e4n. Min\u00e4 p\u00e4rj\u00e4\u00e4n kyll\u00e4.\" H\u00e4nen urheutensa ei lakannut h\u00e4mm\u00e4stytt\u00e4m\u00e4st\u00e4 minua.\n\nMitja k\u00e4\u00e4ntyi Danielin puoleen. \"Jos olet oikeasti pahoillasi siit\u00e4, mit\u00e4 minulle tapahtui, minulla on sinulle hyvi\u00e4 uutisia: nyt saat tilaisuuden korjata virheesi. \u00c4l\u00e4 tyri t\u00e4t\u00e4.\"\n\nMinullakaan ei olisi ollut pokkaa sanoa Danielille jotain sellaista. En tiennyt, mit\u00e4 he olivat puhuneet kesken\u00e4\u00e4n sill\u00e4 aikaa kun olin ollut saarella, mutta n\u00e4in heti, ett\u00e4 Mitja oli osunut oikeaan. Daniel halusi hyvitt\u00e4\u00e4 v\u00e4linpit\u00e4m\u00e4tt\u00f6myytens\u00e4 Mitjaa kohtaan. \"\u00c4l\u00e4 huoli\", h\u00e4n sanoi. \"T\u00e4ll\u00e4 kertaa min\u00e4 en tyri.\"\n\n***\n\nDaniel j\u00e4isi siis Kreikkaan Adrianen kanssa ainakin joksikin aikaa. Minulla ei ollut hajuakaan, mit\u00e4 sen j\u00e4lkeen tapahtuisi. Toivoin Adrille kaikkea mahdollista hyv\u00e4\u00e4, mutta oli mahdotonta tiet\u00e4\u00e4, mit\u00e4 se k\u00e4yt\u00e4nn\u00f6ss\u00e4 tarkoittaisi.\n\nLoppujen lopuksi oma koti-ik\u00e4v\u00e4ni ohitti huolen Adrianen puolesta. Yll\u00e4tyksekseni ajatuksissani koti oli yht\u00e4 lailla Hukkavaara kuin Helsinki.\n\nMenimme kaikki yhdess\u00e4 lautalla Ateenaan, jossa onnistuimme hankkimaan minulle ja Mikaelille tilap\u00e4iset passit ja lennot kotiin.\n\nKun viimein tuli l\u00e4hd\u00f6n aika, Daniel ja Adri tulivat saattamaan meit\u00e4 lentokent\u00e4lle.\n\n\"Kiitos\", sanoin ensin Adrille. Ja sitten minun oli pakko sanoa sama my\u00f6s Danielille.\n\n\"Ei ole mit\u00e4\u00e4n kiitt\u00e4mist\u00e4\", h\u00e4n vastasi.\n\nPaitsi se, ett\u00e4 h\u00e4n oli tullut minun avukseni. Tiesin, ett\u00e4 h\u00e4n oli tehnyt sen Mikaelin takia, mutta h\u00e4n olisi voinut auttaa poikaansa niin monella muullakin tavalla, ja hyvin monet niist\u00e4 olisivat johtaneet minun tai veljeni kannalta huomattavasti huonompaan lopputulokseen.\n\nYksi asia oli vaivannut minua siin\u00e4 m\u00e4\u00e4rin, ett\u00e4 minun oli pakko kysy\u00e4. \"Sin\u00e4 sanoit, ettet koskaan katso olkasi yli. Silti sin\u00e4 ostit \u00e4itini ty\u00f6n kauan sen j\u00e4lkeen, kun olit luovuttanut h\u00e4nen suhteensa.\"\n\nDaniel ei h\u00e4mmentynyt kysymyksest\u00e4ni. H\u00e4n tuntui heti tiet\u00e4v\u00e4n, mist\u00e4 puhuin. \"Tied\u00e4tk\u00f6, mik\u00e4 sen teoksen nimi on?\"\n\n\"En.\"\n\n\"Anteeksianto.\"\n\nEn tiennyt, mit\u00e4 sanoa.\n\n\"En katsonut olkani yli. Kun ostin sen maalauksen, katsoin suoraan eteenp\u00e4in ensimm\u00e4ist\u00e4 kertaa pitk\u00e4\u00e4n aikaan.\"\n\nH\u00e4n k\u00e4\u00e4ntyi kohti velje\u00e4ni. \"Mitja. Oli hienoa saada tutustua sinuun. Ja min\u00e4 olen oikeasti pahoillani.\"\n\nMitja pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Ei tarvitse olla. Minustahan tuli ihan siedett\u00e4v\u00e4.\"\n\nEn usko, ett\u00e4 olin koskaan n\u00e4hnyt Danielin kasvoilla yht\u00e4 paljon l\u00e4mp\u00f6\u00e4. \"Niin tuli.\"\n\nSitten h\u00e4n jo k\u00e4\u00e4nsi kaiken huomion poikaansa.\n\nOlin odottanut, mit\u00e4 h\u00e4n sanoisi Mikaelille. En tiennyt, pitik\u00f6 Mikael hetke\u00e4 yht\u00e4 latautuneena kuin min\u00e4, mutta min\u00e4 kirjaimellisesti pid\u00e4tin hengityst\u00e4ni. En voinut v\u00e4itt\u00e4\u00e4 ymm\u00e4rt\u00e4v\u00e4ni paljoakaan heid\u00e4n suhteestaan. Tiesin, ettei Daniel ollut paras mahdollinen is\u00e4, vaikka h\u00e4n ei ollutkaan laiminly\u00f6nyt Mikaelia mill\u00e4\u00e4n vakavalla tavalla.\n\nDaniel katsoi poikaansa. H\u00e4n ei aikonut halata t\u00e4t\u00e4, sen tiesin heti. Mutta t\u00e4m\u00e4 saattoi olla viimeinen kerta kun he n\u00e4kiv\u00e4t toisensa, joten odotin, ett\u00e4 h\u00e4n ainakin sanoisi jotain merkityksellist\u00e4 hyv\u00e4stiksi. Rukoilin, ett\u00e4 h\u00e4n tajuaisi j\u00e4tt\u00e4\u00e4 arvostelemisen ja neuvomisen v\u00e4liin ja oikeasti osoittaisi tukensa pojalleen.\n\n\"Minun on turha antaa sinulle mit\u00e4\u00e4n neuvoja lauman suhteen, sill\u00e4 sin\u00e4 et kuitenkaan noudata niit\u00e4\", h\u00e4n aloitti. \"Kuka tiet\u00e4\u00e4, ehk\u00e4 sin\u00e4 saat aikaan jotain, mit\u00e4 min\u00e4 en. Kunhan vain muistat, ettei johtamisen kuulu olla helppoa.\"\n\nMikael ny\u00f6kk\u00e4si. Yll\u00e4tyksekseni Daniel vilkaisi minua. \"Ja pid\u00e4 kiinni t\u00e4st\u00e4 tyt\u00f6st\u00e4. H\u00e4n tekee el\u00e4m\u00e4st\u00e4si mielenkiintoisen.\"\n\nDaniel k\u00e4\u00e4nsi selk\u00e4ns\u00e4 ja k\u00e4veli pois Adri rinnallaan. H\u00e4nen k\u00e4tens\u00e4 l\u00f6ysi Adrin olkap\u00e4\u00e4n ja puristi sit\u00e4 varsin is\u00e4llisesti.\n\nTajusin seisovani suu auki vasta kun n\u00e4in Mikaelin huvittuneen ilmeen.\n\n\"Sanoinhan, ett\u00e4 h\u00e4n pit\u00e4\u00e4 sinusta.\"\nVIIDESTOISTA LUKU\n\nLentokone kohosi ilmaan. Koko nousun ajan tuijotin herke\u00e4m\u00e4tt\u00e4 turvavy\u00f6valoa. Heti, kun se sammui, napsautin soljen auki ja nostin minun ja Mikaelin v\u00e4liss\u00e4 olevan k\u00e4sinojan yl\u00f6s. Nojauduin h\u00e4nen kainaloonsa ja painoin kasvoni h\u00e4nen paitaansa vasten.\n\nJossain vaiheessa huomasin, ett\u00e4 Mitja oli kadonnut vierest\u00e4mme. Luulin h\u00e4nen l\u00e4hteneen vessaan, mutta kun h\u00e4nt\u00e4 ei n\u00e4kynyt, tajusin h\u00e4nen menneen jonnekin muualle istumaan vain siksi, ett\u00e4 me saisimme olla rauhassa.\n\nKiedoin k\u00e4teni Mikaelin ymp\u00e4rille. H\u00e4n alkoi silitt\u00e4\u00e4 hiuksiani.\n\n\"Sinun t\u00e4ytyy ehk\u00e4 toteuttaa se uhkauksesi\", sanoin.\n\n\"Mik\u00e4?\" H\u00e4nen \u00e4\u00e4nens\u00e4 v\u00e4reili h\u00e4nen rintakeh\u00e4ns\u00e4 l\u00e4pi.\n\n\"Ei minua haittaisi jos kosisit. Minua ei pelota ajatella, ett\u00e4 me olisimme yhdess\u00e4 lopun ik\u00e4\u00e4mme. Enemm\u00e4n minua pelottaa se, ettei niin k\u00e4y.\"\n\nH\u00e4n suuteli p\u00e4\u00e4t\u00e4ni. \"Ei mit\u00e4\u00e4n h\u00e4t\u00e4\u00e4. Min\u00e4 olen t\u00e4ss\u00e4 enk\u00e4 mene minnek\u00e4\u00e4n.\" H\u00e4n ei sanonut enemp\u00e4\u00e4. Hengitimme toinen toisiamme, kunnes koneessa ilmoitettiin laskeutumisvalmisteluiden aloittamisesta.\n\nDaniel oli hankkinut meille lennot aina Kajaaniin asti. Mitja oli soittanut Hukkavaaraan ja pyyt\u00e4nyt jonkun tulemaan hakemaan meit\u00e4. En ollut tiennyt, kenet he l\u00e4hett\u00e4isiv\u00e4t, mutta n\u00e4hdess\u00e4ni Nikon mustissa vaatteissaan odottamassa meit\u00e4 sis\u00e4\u00e4ntuloaulassa olin \u00e4kki\u00e4 \u00e4\u00e4rimm\u00e4isen kiitollinen, ett\u00e4 vastassa oli nimenomaan h\u00e4n. H\u00e4nen oli t\u00e4ytynyt tulla Helsingist\u00e4 varta vasten meid\u00e4n kotiinpaluumme vuoksi.\n\nH\u00e4n ei sanonut mit\u00e4\u00e4n, mutta levitti k\u00e4tens\u00e4 jo kauan ennen kuin ehdin h\u00e4nen luokseen. Kun h\u00e4n halasi minua, minusta tuntui ensimm\u00e4ist\u00e4 kertaa silt\u00e4, ett\u00e4 olimme selvinneet. Olimme kotona.\n\nPieni vaalea hahmo s\u00e4nt\u00e4si kohti Mitjaa. Caoimhe. Mitja nosti h\u00e4net yl\u00f6s maasta ja alkoi puhua h\u00e4nen korvaansa matalalla \u00e4\u00e4nell\u00e4 iiriksi. Kuulin monta variksen raakunnalta kuulostavaa sanaa, joiden kuvittelin tarkoittavan jotain hellytt\u00e4v\u00e4\u00e4.\n\nOlin itsekin varsin herk\u00e4ss\u00e4 mielentilassa. Onneksi oli Niko. \"Sin\u00e4 olet p\u00e4iv\u00e4n my\u00f6h\u00e4ss\u00e4\", h\u00e4n sanoi. Minun oli pakko naurahtaa.\n\nMikael katsoi minua kysyv\u00e4sti. \"Min\u00e4 lupasin, ett\u00e4 toisin sinut takaisin ennen t\u00e4ysikuuta\", selitin.\n\nEn ehtinyt sanoa enemp\u00e4\u00e4, sill\u00e4 Niko halasi minua j\u00e4lleen. \"Min\u00e4 tiesin koko ajan, ett\u00e4 sin\u00e4 tulisit takaisin.\" En viitsinyt huomauttaa, ett\u00e4 h\u00e4nen rutistuksensa ja sanansa olivat hienoisessa ristiriidassa. Olin kiitollinen niist\u00e4 molemmista.\n\nVasta irrotettuaan otteensa h\u00e4n vilkaisi Mikaelia ensimm\u00e4ist\u00e4 kertaa kunnolla. H\u00e4nen silm\u00e4ns\u00e4 levisiv\u00e4t. \"Dude. Oletko sin\u00e4 ihan kunnossa?\"\n\nOlin jo unohtanut, milt\u00e4 Mikael n\u00e4ytti. Tai oikeastaan olin unohtanut, milt\u00e4 h\u00e4n oli n\u00e4ytt\u00e4nyt ennen kuin h\u00e4nt\u00e4 oli n\u00e4\u00e4nnytetty ja kidutettu. H\u00e4nen ihonsa oli menett\u00e4nyt kaiken v\u00e4rins\u00e4 ja silm\u00e4tkin n\u00e4yttiv\u00e4t haaleammilta kuin koskaan.\n\nMikael avasi suunsa sanoakseen jotain, mutta sulki sen melkein heti. Oli aivan turha yritt\u00e4\u00e4 valehdella. Lopulta h\u00e4n sanoi: \"Min\u00e4 tulen olemaan kunnossa.\"\n\nNiko vilkuili meit\u00e4 ep\u00e4luuloisesti. \"Mit\u00e4 tapahtui?\"\n\nSiin\u00e4 se viimeinkin oli: kysymys, jota olin sek\u00e4 odottanut ett\u00e4 pel\u00e4nnyt, ja johon joutuisin keksim\u00e4\u00e4n hyv\u00e4n vastauksen viimeist\u00e4\u00e4n sitten, kun p\u00e4\u00e4simme takaisin Hukkavaaraan.\n\nMit\u00e4 muka saatoin sanoa? Totuus oli liian hullu.\n\nMikael pelasti minut. \"Mit\u00e4 ikin\u00e4 se olikin, se on nyt ohi.\"\n\n***\n\n\"Varoituksen sana: talolle on ker\u00e4\u00e4ntynyt aika paljon porukkaa odottamaan teit\u00e4\", Niko sanoi ohjatessaan auton Sarrien talolle johtavalle tielle. P\u00e4\u00e4stin turhautuneen \u00e4\u00e4nen. Alkumatkasta olin viel\u00e4 harkinnut, ett\u00e4 poikkeaisimme Jaskan luona, sill\u00e4 uskoin h\u00e4nen olevan huolissaan. Kuitenkin mit\u00e4 l\u00e4hemm\u00e4s Hukkavaaraa p\u00e4\u00e4simme, sit\u00e4 harmaammaksi Mikaelin kasvot k\u00e4viv\u00e4t, ja sit\u00e4 kipe\u00e4mmin itsekin kaipasin omaa s\u00e4nky\u00e4mme. P\u00e4\u00e4dyin siis lopulta vain soittamaan ja j\u00e4tt\u00e4m\u00e4\u00e4n Jaskan vastaajaan viestin, jossa kerroin olevani kunnossa ja tulevani seuraavana p\u00e4iv\u00e4n\u00e4 vierailulle.\n\nIlmeisesti toiveet pikaisesta nukkumaanmenosta eiv\u00e4t siis kuitenkaan olleet toteutumassa. En ollut erityisen yll\u00e4ttynyt. Mikael oli ollut viikon poissa, ja kaikki olivat arvatenkin innoissaan h\u00e4nen n\u00e4kemisest\u00e4\u00e4n. Monistakin eri syist\u00e4.\n\n\"Kukaan laumassa ei siis tied\u00e4, miksi Mikael oli poissa?\" kysyin.\n\n\"Ei\", Niko vastasi. \"Jenni ja Juha onnistuivat saamaan koko jutun kuulostamaan silt\u00e4 kuin te olisitte karanneet jonnekin lemmenlomalle.\"\n\n\"Hyv\u00e4\", t\u00f6ks\u00e4ytin. \"Toivotaan, ett\u00e4 he luulevat niin viel\u00e4 huomennakin.\"\n\nNiin kuin Niko oli varoittanut, talo oli t\u00e4ynn\u00e4. Tiesin sen heti, kun astuin autosta ulos, sill\u00e4 olohuoneen valot olivat p\u00e4\u00e4ll\u00e4 ja sis\u00e4ll\u00e4 oli paljon v\u00e4ke\u00e4. Vaimea puheensorina ja musiikki sekoittuivat toisiinsa.\n\nT\u00e4m\u00e4 oli pahin mahdollinen hetki bileille. Kaikki sudet olivat paikalla n\u00e4kem\u00e4ss\u00e4, miten heikossa kunnossa Mikael oli. Ja alfalle heikkouden paljastaminen oli usein hengenvaarallista.\n\nRukoilin, ettei Iivo olisi paikalla.\n\nMikael ei sanonut mit\u00e4\u00e4n. Syd\u00e4nt\u00e4ni s\u00e4rki n\u00e4hd\u00e4 h\u00e4nen silm\u00e4ilev\u00e4n etuovelle vievi\u00e4 portaita huolestuneena. Puristin h\u00e4nen k\u00e4tt\u00e4\u00e4n, ja h\u00e4n veti minut kainaloonsa. Se saattoi n\u00e4ytt\u00e4\u00e4 omistavalta eleelt\u00e4, mutta todellisuudessa h\u00e4n tarvitsi tukea.\n\nK\u00e4velimme yl\u00f6s hitaasti, ja toivoin, ett\u00e4 ulkopuolisten silmiss\u00e4 n\u00e4ytimme vain silt\u00e4, ett\u00e4 kuhertelimme hetken ennen kuin astuisimme muiden eteen. \"Kaikki menee hyvin\", kuiskasin Mikaelille. En tiennyt, oliko h\u00e4n edes kuullut sanojani, sill\u00e4 h\u00e4n ei vastannut mit\u00e4\u00e4n.\n\nAvatessani oven kuulin, kuinka puhe lakkasi. Ei ollut muuta vaihtoehtoa kuin menn\u00e4 suoraan olohuoneeseen.\n\nN\u00e4in Iivon jo ovensuusta. Kaikki toiveikkuuteni katosi. H\u00e4n istui sohvalla, kun tulimme sis\u00e4\u00e4n. Kun h\u00e4n nousi seisomaan, tiesin, ett\u00e4 h\u00e4n oli odottanut nimenomaan meit\u00e4. Kaikki muut sohvalla istujat seurasivat h\u00e4nen esimerkki\u00e4\u00e4n.\n\nTein ensimm\u00e4isen asian, joka tuli mieleeni: astuin Mikaelin eteen. H\u00e4n ty\u00f6nsi minut hell\u00e4sti sivuun.\n\nKaikki tuijottivat Mikaelia. Tyrmistynyt hiljaisuus valtasi koko tilan.\n\n\"Kuten varmaan kaikki n\u00e4ette, en ole j\u00e4rin hyv\u00e4ss\u00e4 kunnossa\", Mikael aloitti. \"Jos joku teist\u00e4 haluaa laumanjohtajaksi, nyt olisi hyv\u00e4 hetki haastaa minut\", h\u00e4n sanoi. H\u00e4n antoi katseensa kiert\u00e4\u00e4 paikalla olijoissa, mutta jokainen tiesi, kenelle h\u00e4nen sanansa oli tarkoitettu.\n\nEn uskaltanut hengitt\u00e4\u00e4. Iivo katsoi Mikaelia, sitten l\u00e4himpi\u00e4 susia. Sitten taas Mikaelia.\n\n\"Min\u00e4 en aio haastaa sinua\", h\u00e4n sanoi lopulta. \"Yksi syy on tietenkin se, ett\u00e4 t\u00e4ll\u00e4 hetkell\u00e4 se olisi t\u00e4ysin raukkamaista. Mutta min\u00e4 haluan saada t\u00e4m\u00e4n lauman toimimaan yht\u00e4 paljon kuin sin\u00e4kin.\" H\u00e4n piti tauon painottaakseen sanojensa vaikutusta. \"Ilman sinua min\u00e4 ja lupinilaiset emme olisi saaneet mahdollisuutta muuttaa Hukkavaaraan ja rakentaa itsellemme parempaa el\u00e4m\u00e4\u00e4. Sinulla on t\u00e4ysi tukeni.\"\n\nR\u00e4pyttelin. Kun katselin huoneessa olevia susia, n\u00e4in, etten ollut ainoa, joka oli yll\u00e4ttynyt.\n\n\"Sin\u00e4 olet ehk\u00e4 nuori, mutta sinut on tarkoitettu johtamaan t\u00e4t\u00e4 laumaa, vaikka sitten Helsingist\u00e4 k\u00e4sin. Meid\u00e4n tulee olla ylpeit\u00e4, ett\u00e4 johtajamme on valmis panostamaan kaikkensa tulevaisuutemme eteen. Toivon, ett\u00e4 jo omat lapseni voivat perustaa perheen ilman ongelmia.\"\n\nH\u00e4n tarjosi Mikaelille k\u00e4tt\u00e4\u00e4n.\n\nMikael tarttui siihen.\n\nJ\u00e4nnite laukesi koko huoneessa. Hymyt levisiv\u00e4t susien kasvoille, ja \u00e4kki\u00e4 tunnelma oli l\u00e4hes hilpe\u00e4. Jotkut kohottivat pullojaan ja t\u00f6lkkej\u00e4\u00e4n, ja kun muutamat alkoivat taputtaa, koko lauma yhtyi pian hurraukseen.\n\nEn ollut uskoa tapahtunutta todeksi. Kun sudet tulvivat tervehtim\u00e4\u00e4n Mikaelia, en liikahtanut sentti\u00e4k\u00e4\u00e4n h\u00e4nen vierelt\u00e4\u00e4n. Tunsin itseni turvamieheksi, mutten v\u00e4litt\u00e4nyt siit\u00e4. Halusin pit\u00e4\u00e4 huolen, ettei kukaan voinut satuttaa h\u00e4nt\u00e4. Ja ettei kukaan uskaltaisi udella, mit\u00e4 h\u00e4nelle oli tapahtunut ja miksi.\n\nHuoleni osoittautui turhaksi: Mikael oli hyv\u00e4 v\u00e4istelem\u00e4\u00e4n kysymyksi\u00e4. En uskonut, ett\u00e4 kukaan muu tajusi, miten huonossa kunnossa h\u00e4n todella oli.\n\n\"Menn\u00e4\u00e4nk\u00f6 hetkeksi ulos?\" Mikael kysyi minulta matalalla \u00e4\u00e4nell\u00e4.\n\nNy\u00f6kk\u00e4sin. Tiesin, ett\u00e4 se oli h\u00e4nen tapansa sanoa, ett\u00e4 h\u00e4n tarvitsi tauon. Kiersimme ryhm\u00e4n poikia ja astuimme pihalle. K\u00e4velimme kauemmas talosta, kauemmas juhlista.\n\nKatseeni osui er\u00e4\u00e4seen tiettyyn kohtaan puutarhassa.\n\n\"Mit\u00e4 sin\u00e4 katsot?\" Mikael kysyi. Heikko tai ei, h\u00e4n oli hyv\u00e4 lukemaan minua.\n\n\"Muistatko ne ensimm\u00e4iset t\u00e4ydenkuun aaton bileet, joihin kutsuit minut?\" kysyin.\n\n\"Tietenkin\", h\u00e4n vastasi.\n\n\"Olin aika innoissani. Toivoin, ett\u00e4 p\u00e4\u00e4sisin juttelemaan sinulle, mutta sinua ei n\u00e4kynyt miss\u00e4\u00e4n. Sitten katsoin ulos keitti\u00f6n ikkunasta ja n\u00e4in sinut ja Jennin tuolla\", osoitin pihalle. \"Te suutelitte, ja vasta silloin tajusin, miten tyhm\u00e4\u00e4 oli ihastua johonkuhun, joka oli varattu. Pian sen j\u00e4lkeen sain sen ensimm\u00e4isen kohtauksen.\"\n\nMikael katsoi minua hieman oudosti. Oli noloa, ett\u00e4 l\u00e4hes kahden vuoden takaisen ihastumiseni paljastuminen sai minut kiemurtelemaan.\n\nNojasin h\u00e4nt\u00e4 kohti, mutta h\u00e4n ei antanut minun suudella itse\u00e4\u00e4n, vaikka n\u00e4in h\u00e4nen silmiss\u00e4\u00e4n kullan v\u00e4l\u00e4hdyksen. \"Haluatko tiet\u00e4\u00e4, miten se ilta meni minun n\u00e4k\u00f6kulmastani?\" h\u00e4n kysyi.\n\nNy\u00f6kk\u00e4sin.\n\n\"Hypin seinille koko p\u00e4iv\u00e4n\", h\u00e4n sanoi. \"Perustelin sinun kutsumisesi sill\u00e4, ett\u00e4 se olisi hyv\u00e4ksi laumalle, mutta oikeasti halusin sinun tulevan itseni vuoksi. Ja kun sin\u00e4 saavuit paikalle, v\u00e4lttelin sinua koko illan. Olin mennyt ulos haukkaamaan happea, sill\u00e4 sinun ajattelemisesi sai minut sekaisin. Jenni k\u00e4ytti hetken hyv\u00e4kseen. Meh\u00e4n olimme tauolla silloin. Hetken aikaa kuvittelin suutelevani sinua, mutta lopulta ty\u00f6nsin h\u00e4net pois.\"\n\n\"Ai.\" Se oli ainoa asia, jonka onnistuin sanomaan.\n\n\"Min\u00e4 olin hulluna sinuun jo silloin.\" H\u00e4n pyyhk\u00e4isi hiukset pois poskeltani. \"Ja t\u00e4st\u00e4 l\u00e4htien toivon, ett\u00e4 ajattelet jotain ihan muuta kuin Jenni\u00e4 ja minua, kun tulet t\u00e4nne pihalle.\" Viimein h\u00e4n suuteli minua.\n\nSen j\u00e4lkeen k\u00e4velimme hetken vaiti.\n\n\"Miksi sin\u00e4 majoitit Nikon, Tanelin, Mitjan ja Ciaon hotellille etk\u00e4 t\u00e4nne?\" kysyin. Aikaisemmin se oli vaivannut minua. Kaiken tapahtuneen j\u00e4lkeen olin l\u00e4hinn\u00e4 utelias.\n\n\"Ajattelin, ett\u00e4 niin olisi kaikkien kannalta parempi.\"\n\n\"Mit\u00e4 tarkoitat?\"\n\n\"Min\u00e4 tied\u00e4n, ett\u00e4 t\u00e4m\u00e4 rahajuttu on sinulle vaikea. Ja t\u00e4m\u00e4 talo. Ja koko Hukkavaara. Halusin, ett\u00e4 sin\u00e4 saisit totutella t\u00e4h\u00e4n ihan rauhassa. Mit\u00e4 sitten, jos se maksaa minulle v\u00e4h\u00e4n? Maksaisin vaikka kymmenen kertaa enemm\u00e4n, jos se helpottaisi sinun oloasi.\"\n\n\"Tuo on aika... huomaavaista.\" Olin kuvitellut onnistuneeni pit\u00e4m\u00e4\u00e4n ajatukseni omana tietonani. Minun olisi pit\u00e4nyt tiet\u00e4\u00e4, ettei se tulisi koskaan onnistumaan Mikaelin kanssa.\n\n\"Min\u00e4 en v\u00e4lit\u00e4 t\u00e4st\u00e4 talosta\", kuulin h\u00e4nen sanovan. \"Jos sin\u00e4 et halua, meid\u00e4n ei tarvitse koskaan asua t\u00e4\u00e4ll\u00e4.\"\n\nSe sai minut vihdoin tajuamaan, miten turhan ongelman olin luonut. Sein\u00e4t ja tavarat eiv\u00e4t olleet syyllistyneet mihink\u00e4\u00e4n. Tunsin tarvetta selitt\u00e4\u00e4. \"Juttu on niin, ett\u00e4 ennen t\u00e4t\u00e4 kes\u00e4\u00e4 minulle oli kertynyt t\u00e4st\u00e4 talosta vain ik\u00e4vi\u00e4 muistoja. En min\u00e4 t\u00e4t\u00e4 taloa vihaa, enk\u00e4 Hukkavaaraa, enk\u00e4 varsinkaan sinua. Haluan vain, ett\u00e4 tajuat.\"\n\n\"Tuohon on olemassa hyvin yksinkertainen ratkaisu\", Mikael sanoi.\n\n\"Mik\u00e4?\"\n\n\"Meid\u00e4n pit\u00e4\u00e4 luoda parempia muistoja\", h\u00e4n sanoi. H\u00e4n selvitti kurkkuaan. \"Haluaisin n\u00e4ytt\u00e4\u00e4 sinulle jotakin.\" H\u00e4n vaikutti hieman hermostuneelta, mik\u00e4 sai minutkin hermostumaan. Kaiken sen j\u00e4lkeen, mit\u00e4 olimme joutuneet kokemaan, en ollut varsinaisesti valmis yll\u00e4tyksiin.\n\nKysymysmerkit kasvoivat entisest\u00e4\u00e4n, kun menimme autotallille. Ensin tietenkin luulin, ett\u00e4 olimme menossa jonnekin. Kun h\u00e4n ohjasi minut yl\u00e4kertaan tiesin, ett\u00e4 kyse oli jostain muusta.\n\nEn ollut k\u00e4ynyt siell\u00e4 koskaan aikaisemmin. Kun Mikael painoi valokatkaisijaa, n\u00e4in, ett\u00e4 olimme tulleet suureen, korkeaan huoneeseen. Osaa lattiasta p\u00e4\u00e4llysti vihre\u00e4 tatami, ja nurkassa roikkui nyrkkeilys\u00e4kki. Olimme siis treenisalissa.\n\n\"\u00c4l\u00e4 suutu\", h\u00e4n aloitti. \"Ajattelin, ett\u00e4 t\u00e4st\u00e4 voisi jossain vaiheessa tulla sinun ateljeesi.\" H\u00e4n osoitti id\u00e4npuoleisia ikkunoita kohti. \"Tuolta puolelta voi kaataa pari puuta niin, ett\u00e4 t\u00e4nne tulee enemm\u00e4n valoa sis\u00e4\u00e4n.\"\n\n\"Miksi min\u00e4 suuttuisin t\u00e4llaisesta?\"\n\nJ\u00e4lleen kerran Mikaelin ei tarvinnut sanoa mit\u00e4\u00e4n. Olin ollut vet\u00e4\u00e4 pultit jo siit\u00e4, ett\u00e4 h\u00e4n oli halunnut maksaa illallisen.\n\nAteljee tuntui kuitenkin sek\u00e4 paljon hyv\u00e4ksytt\u00e4v\u00e4mm\u00e4lt\u00e4 ett\u00e4 paljon suuremmalta lahjalta.\n\n\"T\u00e4st\u00e4 tulisi hieno ateljee\", sanoin \u00e4\u00e4neen. K\u00e4\u00e4nnyin h\u00e4nt\u00e4 kohti. \"Oletko ihan varma, ettet halua pit\u00e4\u00e4 t\u00e4t\u00e4 treenisalina?\"\n\nMikael pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Me voimme kaataa sein\u00e4n kellarista ja rakentaa treenisalin sinne. Ja vaikka emme voisikaan, sinun ty\u00f6si on t\u00e4rke\u00e4mp\u00e4\u00e4. Toivon mukaan kummankaan meist\u00e4 ei tarvitse en\u00e4\u00e4 koskaan testata kamppailutaitojamme tosissaan.\"\n\nYksi huoleni oli aina ollut se, ettei Mikael t\u00e4ysin ymm\u00e4rt\u00e4nyt, mit\u00e4 taide merkitsi minulle. Kuinka tyhm\u00e4 olinkaan ollut.\n\n\"Sin\u00e4 voit muuttaa talossa mit\u00e4 haluat ihan vapaasti\", Mikael sanoi.\n\nT\u00e4m\u00e4 talo voisi olla kotini, ja se yll\u00e4tti itsenikin.\n\n\"Ei minulla ole mit\u00e4\u00e4n valittamista\", vastasin. \"Paitsi ett\u00e4 niiden kreikkalaisten patsaiden on parasta kadota.\"\n\nMikaelin hymy syttyi hitaasti, mutta oli sit\u00e4kin loistokkaampi. \"Olen t\u00e4ysin samaa mielt\u00e4.\"\n\nL\u00e4hdimme autotallista ja k\u00e4velimme hitaasti pihan poikki. Aikaisempi keskustelu oli asettanut jotain minussa aloilleen, ja vaikka vasta tulevaisuus n\u00e4ytt\u00e4isi, mit\u00e4 se jokin tarkalleen ottaen oli, tunsin itseni vahvemmaksi kuin koskaan ennen.\n\nAstuimme lasiovista sis\u00e4\u00e4n. Meid\u00e4t ymp\u00e4r\u00f6i v\u00e4litt\u00f6m\u00e4sti valo, l\u00e4mp\u00f6 ja tutut ihmiset. Ny\u00f6kk\u00e4ilin ja hymyilin v\u00e4h\u00e4n joka suuntaan, ja minulle hymyiltiin takaisin. Saatoin tuntea koko lauman kunnioituksen, mutta t\u00e4ll\u00e4 kertaa tunnelma ei ollut lainkaan pelonsekainen, vaan pelk\u00e4st\u00e4\u00e4n yst\u00e4v\u00e4llinen. Olin kiitollinen kaikesta, mink\u00e4 olin saanut: kodista, laumasta ja vierell\u00e4ni seisovasta pojasta.\n\nEhk\u00e4 kuitenkin eniten vierell\u00e4ni seisovasta pojasta. H\u00e4n oli liian kaunis ja hyv\u00e4 minulle, mutta jos se sitten oli lainaa seuraavasta el\u00e4m\u00e4st\u00e4 ja piti maksaa korkojen kera takaisin, olkoon niin. Maksaisin vaikka kymmenkertaisesti.\n\nMikael kosketti k\u00e4tt\u00e4ni. Kun seurasin h\u00e4nen katsettaan, n\u00e4in Jonin puhuvan lupinilaistyt\u00f6lle, jonka nilkkaan olin piirt\u00e4nyt. Virnistimme toisillemme.\n\nJenni ja Juha istuivat olohuoneessa. Suuntasimme heit\u00e4 kohti juuri, kun Mitja ja Caoimhe ilmestyiv\u00e4t keitti\u00f6n suunnasta. Kun Nikokin liittyi seuraamme, oli porukka kasassa.\n\nT\u00e4ss\u00e4 oli lauman ydin, ajattelin. Ja sattumaa tai ei, samat ihmiset olivat my\u00f6s oman el\u00e4m\u00e4ni ydin.\n\n\"En uskonut, ett\u00e4 talolle olisi ker\u00e4\u00e4ntynyt aivan n\u00e4in paljon v\u00e4ke\u00e4. Aikamoinen vastaanottokomitea\", sanoin.\n\n\"Luulen, ett\u00e4 suurin osa vain kaipasi bileit\u00e4. Meilt\u00e4 j\u00e4i t\u00e4ydenkuun juhlinta v\u00e4liin teid\u00e4n takianne\", Jenni selitti.\n\n\"Ah, olen hyvin pahoillani\", vastasin. \"Kyse onkin siis vain ilmaisesta alkoholista.\"\n\n\"Pit\u00e4\u00e4 paikkansa\", Juha sanoi.\n\n\"No, siin\u00e4 tapauksessa teit\u00e4 ei varmaan kiinnosta kuulla meid\u00e4n seikkailuistamme. Ajattelin, ett\u00e4 olisin kertonut teille, kuinka Daniel Sarri vaani minua ja velje\u00e4ni mets\u00e4ss\u00e4 suden muodossa, tai kuinka h\u00e4n on kehitt\u00e4nyt ihan mielett\u00f6m\u00e4n taistelutekniikan, jossa h\u00e4n vaihtaa muotoa ihmiseksi ja takaisin ihan lennosta, mutta jos teit\u00e4 kerran janottaa, voidaan siirty\u00e4 keitti\u00f6\u00f6n...\"\n\nKaikki tuijottivat minua. Purin huultani, etten olisi nauranut \u00e4\u00e4neen.\n\n\"Mist\u00e4 te oikein puhutte?\"\n\nK\u00e4\u00e4nnyimme \u00e4\u00e4nt\u00e4 kohti. Tiesin, kuka tulija oli jo ennen kuin n\u00e4in h\u00e4net, mutta puolen sekunnin ajan olin silti toivonut, ett\u00e4 katastrofi olisi v\u00e4ltett\u00e4viss\u00e4.\n\nEn ollut tiennyt, ett\u00e4 Taneli oli tullut Nikon mukana Hukkavaaraan. Niko ei ollut maininnut h\u00e4nt\u00e4, ja kun ajattelin asiaa tarkemmin, tajusin, ett\u00e4 siihen oli syy. Molempien kasvoilta n\u00e4ki, ett\u00e4 he olivat riidelleet.\n\nTaneli tuijotti meit\u00e4 tavalla, joka paljasti, ett\u00e4 h\u00e4n oli kuullut joka sanan.\n\nKun katsoin ymp\u00e4rilleni, n\u00e4in, ett\u00e4 yst\u00e4vieni ilmeet olivat harvinaisen yhtenev\u00e4iset: me kaikki n\u00e4ytimme todella syyllisilt\u00e4.\n\n\"Sinunhan piti olla hotellilla\", Niko sanoi.\n\n\"Halusin puhua sinun kanssasi\", Taneli sanoi. \"Tapasin aulassa pari tyyppi\u00e4, jotka antoivat minulle kyydin.\"\n\n\"Taneli, nyt ei ole oikeasti paras hetki\", Niko sanoi. H\u00e4n kuulosti tylylt\u00e4, ja vaikka tiesin sen johtuvan h\u00e4t\u00e4\u00e4nnyksest\u00e4, minun k\u00e4vi silti Tanelia s\u00e4\u00e4liksi. Itse en olisi pystynyt tiuskimaan h\u00e4nelle.\n\n\"Nyky\u00e4\u00e4n meill\u00e4 on vain huonoja hetki\u00e4\", h\u00e4n sanoi hiljaa.\n\nEn tiennyt, mit\u00e4 poissa ollessani oli tapahtunut, mutta saatoin arvata. Siemenet olivat jo it\u00e4neet ennen l\u00e4ht\u00f6\u00e4mme. Niko oli joutunut keksim\u00e4\u00e4n ties mink\u00e4laisia h\u00e4t\u00e4valheita tilanteen takia. En ollut tajunnut, ett\u00e4 juuri niin k\u00e4visi, kun kutsuin heid\u00e4t mukaan Hukkavaaraan.\n\n\"En tied\u00e4, mit\u00e4 t\u00e4\u00e4ll\u00e4 tapahtuu ja miksi te kaikki olette niin outoja, mutta on selv\u00e4\u00e4, ettei minua haluta t\u00e4nne.\" H\u00e4nen p\u00e4\u00e4ss\u00e4\u00e4n t\u00e4ytyi py\u00f6ri\u00e4 paljon muutakin, sill\u00e4 h\u00e4nen \u00e4\u00e4nens\u00e4 t\u00e4risi. H\u00e4n k\u00e4\u00e4ntyi l\u00e4hte\u00e4kseen.\n\n\"Ei, odota\", Niko sanoi. H\u00e4n oli noussut seisomaan. H\u00e4nen silm\u00e4ns\u00e4 olivat suuret ja h\u00e4t\u00e4\u00e4ntyneet, ja viimeist\u00e4\u00e4n silloin tiesin, ett\u00e4 paras yst\u00e4v\u00e4ni oli rakastunut.\n\nPanoksena oli kuitenkin jotain paljon suurempaa kuin Nikon syd\u00e4n, niin suuri kuin se todellisuudessa olikin. Jos Taneli menisi kertomaan kenellek\u00e4\u00e4n, mit\u00e4 h\u00e4n oli juuri kuullut, koko lauma olisi vaarassa.\n\nMinun on korjattava t\u00e4m\u00e4, ajattelin. Ja sitten tajusin, ett\u00e4 voisin \u2013 olin sent\u00e4\u00e4n daimon.\n\nAstuin askeleen kohti Tanelia.\n\nNiko tarttui minua k\u00e4sivarresta. \"Ei\", h\u00e4n sanoi.\n\n\"En halua, ett\u00e4 teet sit\u00e4 juttua h\u00e4nelle\", h\u00e4n vastasi kysyv\u00e4\u00e4n katseeseeni. \"Sit\u00e4 paitsi olen valehdellut jo ihan tarpeeksi kauan.\"\n\nEnnen kuin ehdin kysy\u00e4, mit\u00e4 h\u00e4n oikein tarkoitti, h\u00e4n oli jo k\u00e4\u00e4ntynyt Mikaelin puoleen. \"En tied\u00e4, miten t\u00e4m\u00e4 virallisesti menee, mutta min\u00e4 eroan laumasta.\"\n\nSitten h\u00e4nen huomionsa oli taas Tanelissa. H\u00e4n tarttui poikayst\u00e4v\u00e4\u00e4ns\u00e4 k\u00e4dest\u00e4. En ollut koskaan n\u00e4hnyt h\u00e4nen tekev\u00e4n niin. \"Noin kahden minuutin kuluttua sin\u00e4 k\u00e4velet t\u00e4\u00e4lt\u00e4 ulos etk\u00e4 halua n\u00e4hd\u00e4 minua en\u00e4\u00e4 koskaan, joten minun on parasta sanoa t\u00e4m\u00e4 nyt kun viel\u00e4 voin: sin\u00e4 olet parasta, mit\u00e4 minulle on koskaan tapahtunut. Et tule koskaan tajuamaan, miten paljon minua kaivelee, ett\u00e4 t\u00e4m\u00e4 menee nyt vituiksi.\"\n\n\"Mist\u00e4 ihmeest\u00e4 sin\u00e4 puhut?\"\n\n\"Min\u00e4 en ole ihminen! Min\u00e4. En. Ole. Ihminen. Min\u00e4enoleihminen.\" H\u00e4n veti pari kertaa syv\u00e4\u00e4n henke\u00e4 ja jatkoi. \"Min\u00e4 olen ihmissusi. Olen tosi pahoillani, mutta niin nyt vain on. Ja usko pois, min\u00e4 paljon mieluummin olisin se mielipuoli, jona sin\u00e4 minua ep\u00e4ilem\u00e4tt\u00e4 nyt pid\u00e4t. Siihen on sent\u00e4\u00e4n l\u00e4\u00e4kitys. T\u00e4h\u00e4n ei ole. Ainoa syy, miksi olen ylip\u00e4\u00e4t\u00e4\u00e4n j\u00e4rjiss\u00e4ni on se, ett\u00e4 min\u00e4 en ole yksin. Mikaelkin on ihmissusi. Ja Juha. Ja Jenni. Kaikki t\u00e4ss\u00e4 hiton kyl\u00e4ss\u00e4 ovat ihmissusia ja pakko sanoa, ett\u00e4 he ovat siin\u00e4 paljon parempia kuin min\u00e4. Raisa ja Mitja ovat jotain kummallisia kreikkalaisia mit\u00e4 lie jumalista seuraavia, ja Ciao tuolla on keiju. Mutta esittelyist\u00e4 viis \u2013 me ollaan kaikki ihan friikkej\u00e4. Joten kiitos ja anteeksi. Oli tosi mahtavaa niin kauan kuin sit\u00e4 kesti.\"\n\nEn ollut el\u00e4ess\u00e4ni kokenut yht\u00e4 j\u00e4nnittynytt\u00e4 hiljaisuutta. Kaikki ihokarvani olivat pystyss\u00e4.\n\nTaneli nielaisi. H\u00e4nen katseensa harhaili ryhm\u00e4ss\u00e4mme kuin toivoen, ett\u00e4 joku meist\u00e4 kiist\u00e4isi kaiken, mit\u00e4 Niko oli juuri sanonut. \"Min\u00e4 tiesin, ett\u00e4 sin\u00e4 salasit minulta jotain. Luulin vain, ett\u00e4 sin\u00e4 petit minua.\"\n\n\"Miksi helvetiss\u00e4 min\u00e4 pett\u00e4isin sinua? Min\u00e4? Sinua?\"\n\n\"Niin.\" Sitten: \"Sin\u00e4k\u00f6 et siis ole pett\u00e4nyt minua?\"\n\n\"En todellakaan. Ellet laske sudeksi muuttumista kerran kuussa pett\u00e4miseksi. En tosin ole pett\u00e4nyt sinua sutenakaan. Se olisi sinusta varmaan viel\u00e4 pahempaa, vai? Tai siis ehk\u00e4 ei. El\u00e4imeth\u00e4n paneskelevat...\"\n\n\"Niko, suu kiinni\", sanoin.\n\nKerrankin h\u00e4n totteli minua. Se oli selke\u00e4sti oikea p\u00e4\u00e4t\u00f6s, sill\u00e4 Tanelin huolestuttavasti lasittuneet silm\u00e4t ter\u00e4v\u00f6ityiv\u00e4t hiukan.\n\n\"T\u00e4m\u00e4 kyll\u00e4 selitt\u00e4\u00e4 asioita\", h\u00e4n mutisi itsekseen.\n\n\"Etk\u00f6 sin\u00e4 alakaan kiljua t\u00e4ytt\u00e4 kurkkua? Hitto, jos olisin tiennyt aikaisemmin...\"\n\nTaneli ravisti k\u00e4tens\u00e4 irti Nikon otteesta. \"Minun pit\u00e4\u00e4 ajatella hetki\", h\u00e4n sanoi.\n\nNiko r\u00e4p\u00e4ytti silmi\u00e4\u00e4n. \"Tietysti\", h\u00e4n sanoi lopulta. \"Jos haluat menn\u00e4 takaisin kotiin, joku voi heitt\u00e4\u00e4 sinut asemalle. Ja jos et halua kuskiksi ket\u00e4\u00e4n meist\u00e4, voin pyyt\u00e4\u00e4 mummia tai Svetlanaa.\"\n\n\"En min\u00e4 ajatellut menn\u00e4 minnek\u00e4\u00e4n muualle kuin hotellille\", h\u00e4n sanoi.\n\n\"Haluatko, ett\u00e4 tulen sinun kanssasi?\" En ollut koskaan n\u00e4hnyt Nikoa yht\u00e4 haavoittuvaisena.\n\nTaneli n\u00e4ytti harkitsevan asiaa. Lopulta h\u00e4n ny\u00f6kk\u00e4si.\n\nJ\u00e4imme kaikki tuijottamaan heid\u00e4n per\u00e4\u00e4ns\u00e4.\n\n\"Min\u00e4 en tainnut olla ihan noin helppo\", Mitja kommentoi.\n\n\"Et todellakaan\", sanoin.\n\n\"Ei ehk\u00e4 kannata viel\u00e4 huokaista. H\u00e4n saattaa viel\u00e4 pimahtaa.\"\n\n\"Totta\", sanoin. \"Sin\u00e4 tietysti toivot sit\u00e4 vain siksi, ettei sinun oma pimahduksesi vaikuttaisi niin nololta.\"\n\n\"En min\u00e4 nyt suoranaisesti pimahtanut\", veljeni sanoi.\n\n\"Kyll\u00e4 pimahdit.\"\n\nNaureskelimme h\u00e4nelle hetken, ja h\u00e4n hyv\u00e4ksyi sen. En voinut olla ihailematta, miten paljon h\u00e4n oli kasvanut viimeisen puolen vuoden aikana. H\u00e4n vaikutti paljon rennommalta, paljon itsevarmemmalta. Olisi voinut melkein uskoa, ett\u00e4 h\u00e4n viihtyi itsens\u00e4 kanssa.\n\nMuut uppoutuivat omiin keskusteluihinsa.\n\n\"Arvaa, mit\u00e4 juuri tajusin?\" Mitja kysyi kreikaksi matalalla \u00e4\u00e4nell\u00e4.\n\n\"No?\"\n\n\"Ehk\u00e4 siin\u00e4 ennustuksessa oli jotain per\u00e4\u00e4. Meid\u00e4n takiamme saarta ei k\u00e4yt\u00e4nn\u00f6ss\u00e4 en\u00e4\u00e4 ole olemassa. Me tuhosimme heid\u00e4t.\"\n\n\"En usko, ett\u00e4 he tarkoittivat ihan sit\u00e4, kun sanoivat, ett\u00e4 min\u00e4 p\u00e4\u00e4tt\u00e4isin daimonien aikakauden\", mutisin. Callista ei ainakaan ollut tarkoittanut sit\u00e4. \"Luulen, ett\u00e4 Callista tarkoitti jumalia.\"\n\nKaduin, ett\u00e4 olin mennyt sanomaan h\u00e4nen nimens\u00e4 \u00e4\u00e4neen. Callista on kuollut, muistutin itse\u00e4ni.\n\n\"Onko jumalia olemassa, jos ei ole ket\u00e4\u00e4n, joka uskoisi niihin?\"\n\nSe oli yll\u00e4tt\u00e4v\u00e4 kysymys, enk\u00e4 osannut vastata siihen.\n\nOloni oli \u00e4kki\u00e4 oudon et\u00e4inen. Puristin silm\u00e4ni kiinni. Kun avasin ne, kuva oli hetken sumea mutta ter\u00e4v\u00f6ityi sitten asteittain.\n\n\"Oletko kunnossa?\"\n\n\"Joo, olen.\" Nousin yl\u00f6s. \"Menen hakemaan juotavaa. Haluatko sin\u00e4 jotain?\"\n\nMitja pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Min\u00e4 odotan t\u00e4\u00e4ll\u00e4.\"\n\nNy\u00f6kk\u00e4sin.\n\nOhittelin susia ja menin keitti\u00f6\u00f6n. Otin puhtaan lasin kaapista ja laskin siihen kraanasta vett\u00e4. Jouduin taas r\u00e4pyttelem\u00e4\u00e4n n\u00e4k\u00f6ni ter\u00e4v\u00e4ksi.\n\nK\u00e4\u00e4nnyin ja olin juuri ottamassa h\u00f6rpyn, kun joku hy\u00f6kk\u00e4si takaani.\n\n***\n\nAivan kuin joku olisi raahannut minua poninh\u00e4nn\u00e4st\u00e4 lattiaa pitkin. Minut vedettiin pois todellisuudesta niin kuin se olisi ollut vain yksi huone muiden joukossa. Paikat vaihtuivat ymp\u00e4rill\u00e4ni, ja kauhukseni tunsin irtautuvani omasta ruumiistani. Potkin ja kiljuin, vaikka minulla ei ollut mit\u00e4\u00e4n, mill\u00e4 potkia tai kiljua. Ja sitten olikin jo my\u00f6h\u00e4ist\u00e4, sill\u00e4 olin ulkopuolella.\n\nHetken aikaa minua h\u00e4ik\u00e4isi niin, ett\u00e4 jouduin nostamaan k\u00e4den silmieni eteen. Kun lopulta laskin sen, n\u00e4in, mist\u00e4 h\u00e4ik\u00e4istyminen oli johtunut: kaikki oli valkoista. Ei ollut seini\u00e4, kattoa, mit\u00e4\u00e4n, pelk\u00e4st\u00e4\u00e4n ikuisesti jatkuva valkoisuus.\n\nOtin askeleen eteenp\u00e4in. Vasta silloin huomasin, ettei minulla ollut kenki\u00e4 eik\u00e4 sukkia. Paljaat varpaani pistiv\u00e4t esiin tutunv\u00e4risen hameenhelman alta. Kokeilin korviani ja tunsin tutut suuret korvakorut. Seremoniavaatteet.\n\nJoku seisoi takanani. K\u00e4\u00e4nnyin hitaasti ymp\u00e4ri.\n\nCallista.\n\nR\u00e4pyttelin silmi\u00e4ni yritt\u00e4en saada h\u00e4net katoamaan \u2013 turhaan. H\u00e4n n\u00e4ytti tismalleen samalta kuin seremoniap\u00e4iv\u00e4n\u00e4, paitsi ett\u00e4 h\u00e4n oli elossa.\n\nTai ehk\u00e4 min\u00e4 en ollutkaan en\u00e4\u00e4 el\u00e4vien kirjoissa.\n\n\"Mit\u00e4 sin\u00e4 teit minulle?\" Vihasin paniikkia, jonka kuulin omassa \u00e4\u00e4ness\u00e4ni.\n\n\"Sin\u00e4 kuvittelit olevasi niin nokkela\", h\u00e4n sanoi. H\u00e4n oli ilmeet\u00f6n, mutta jokin paljasti minulle, ett\u00e4 h\u00e4n oli kaikkea muuta kuin tyyni.\n\n\"Mit\u00e4 kauemmin olet t\u00e4\u00e4ll\u00e4, sit\u00e4 l\u00f6yhemm\u00e4ksi sinun siteesi varjojen maailmaan k\u00e4y. Sinun ruumiisi lakkaa hengitt\u00e4m\u00e4st\u00e4. Ja silloin sinun voimasi, sinun kohtalosi \u2013 ne ovat kaikki minun.\"\n\n\"Eiv\u00e4tk\u00e4 ole.\" Tiesin kuulostavani typer\u00e4lt\u00e4, mutta mik\u00e4 tahansa, mit\u00e4 olisin sanonut, olisi kuulostanut samalta.\n\n\"Etk\u00f6 sin\u00e4 ole tuntenut minua?\" Callista kysyi. \"Min\u00e4 olen ollut sinun el\u00e4m\u00e4ss\u00e4si jo kauan, tyt\u00e4r. Paljon kauemmin kuin arvaatkaan. Ja kohta sin\u00e4 lakkaat olemasta.\"\n\nAjattelin kaikkia sellaisia hetki\u00e4, jolloin \u00e4\u00e4ni p\u00e4\u00e4ni sis\u00e4ll\u00e4 oli kehottanut minua tekem\u00e4\u00e4n jotain sellaista, mit\u00e4 en voinut kuvitella keksineeni itse.\n\nEnsimm\u00e4inen ajatukseni oli, ett\u00e4 kaikki, mit\u00e4 is\u00e4ni oli tehnyt, oli ollut turhaa. Callista oli aina tiennyt, mist\u00e4 l\u00f6yt\u00e4\u00e4 minut.\n\n\"Kukaan ei voi tulla pelastamaan sinua\", Callista sanoi. Ja kuin todistaakseen sen h\u00e4n tarttui minuun.\n\nCallista ei ollut koskaan osallistunut kamppailuharjoituksiimme. En ollut kertaakaan n\u00e4hnyt h\u00e4nen tekev\u00e4n mit\u00e4\u00e4n l\u00e4hellek\u00e4\u00e4n fyysisesti kuormittavaa, mutta siit\u00e4 huolimatta h\u00e4nen otteensa oli niin tiukka, ett\u00e4 tiesin heti, etten p\u00e4\u00e4sisi siit\u00e4 irti.\n\n\"Ei!\" huusin. Yritin hapuilla vapaana olevalla k\u00e4dell\u00e4ni mit\u00e4 tahansa kiinte\u00e4\u00e4.\n\n\"\u00c4l\u00e4 edes kuvittele\", h\u00e4n sanoi. H\u00e4n tarttui ranteisiini. Yritin k\u00e4\u00e4nty\u00e4 ymp\u00e4ri, mutta se johti ainoastaan siihen, ett\u00e4 h\u00e4n puristi minut entist\u00e4 tiukemmin itse\u00e4\u00e4n vasten.\n\nYritin taistella vastaan kaikilla mahdollisilla kyvyill\u00e4ni, mutta mik\u00e4\u00e4n ei toiminut. Lopulta luovutin ja Callista hellitti otettaan. Tunsin h\u00e4net seisomassa takanani. Meni hetki, kunnes tajusin, mik\u00e4 tuntemuksessa vaivasi minua niin paljon. Tunsin h\u00e4nen otteensa, mutta en niin kuin olisin tuntenut jos olisin ollut todellisessa ruumiissani todellisessa maailmassa. T\u00e4m\u00e4 paikka oli kuin uni, jossa aistit olivat mukana vain hyvin sattumanvaraisesti.\n\nSill\u00e4 sekunnilla, kun Callistan ote hellitti lis\u00e4\u00e4, annoin vartaloni pudota koko painollaan.\n\nOlin huomannut meid\u00e4n molempien seisovan jollakin pinnalla, ja vaikka en voinutkaan n\u00e4hd\u00e4 sit\u00e4, olin luottanut siihen, ett\u00e4 se ottaisi minut vastaan yht\u00e4 varmasti kuin mik\u00e4 tahansa lattia. En joutunut pettym\u00e4\u00e4n. Iskeydyin siihen kipe\u00e4sti, mutta vakuutin itselleni, ettei kipu ollut todellista.\n\nCallista kurotti minua kohti. Painoin k\u00e4den n\u00e4kym\u00e4t\u00f6nt\u00e4 lattiaa vasten ja kuvittelin mieless\u00e4ni luukun. Avasin sen ennen kuin se oli tuskin ehtinyt edes ilmesty\u00e4 ja ty\u00f6nsin itseni p\u00e4\u00e4 edell\u00e4 aukkoon.\n\nToinen pudotus ei ollut yht\u00e4\u00e4n sen hellempi kuin ensimm\u00e4inenk\u00e4\u00e4n. Mit\u00e4\u00e4n ei kuitenkaan tullut niskaani, joten oletin, ett\u00e4 ovi oli kadonnut ennen kuin Callista oli ehtinyt seurata per\u00e4ss\u00e4ni, ja se oli p\u00e4\u00e4asia.\n\nKun sain ker\u00e4tty\u00e4 itseni lattiasta, huomasin seisovani pitk\u00e4n k\u00e4yt\u00e4v\u00e4n p\u00e4\u00e4ss\u00e4. Se oli yht\u00e4 valkoinen kuin aikaisempikin n\u00e4kym\u00e4, mutta nyt tiesin, miss\u00e4 lattia, katto ja sein\u00e4t olivat.\n\nAloin k\u00e4vell\u00e4. K\u00e4yt\u00e4v\u00e4 alkoi nopeasti mutkitella vuoroin oikealle, vuoroin vasemmalle. Lopulta se p\u00e4\u00e4ttyi umpikujaan. Silloin painoin taas k\u00e4teni sein\u00e4\u00e4 vasten ja loin oven. Toivoin joka solullani, ett\u00e4 sen takaa paljastuisi Sarrien keitti\u00f6 ja koko t\u00e4m\u00e4 painajainen vain lakkaisi olemasta. Mutta kun avasin oven, huomasin olevani tismalleen samanlaisessa k\u00e4yt\u00e4v\u00e4ss\u00e4 jollaisesta olin juuri tullut.\n\nK\u00e4velin siis eteenp\u00e4in. Minulla ei ollut hajuakaan, kuinka monta k\u00e4\u00e4nn\u00f6st\u00e4 olin jo ehtinyt tehd\u00e4, mutta onneksi minulla ei ollut penint\u00e4k\u00e4\u00e4n aikomusta palata takaisin l\u00e4ht\u00f6pisteeseen. Jotenkin en vain uskonut, ett\u00e4 labyrintti noudattelisi mit\u00e4\u00e4n s\u00e4\u00e4nt\u00f6j\u00e4. Hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4 saattaisin l\u00f6yt\u00e4\u00e4 itseni samasta paikasta, mist\u00e4 olin l\u00e4htenyt.\n\n\u00c4\u00e4ni havahdutti minut ajatuksistani. Se ei kuulostanut milt\u00e4\u00e4n, mink\u00e4 olisin osannut kuvailla tai selitt\u00e4\u00e4, mutta tiesin, ett\u00e4 sen t\u00e4ytyi olla Callista. Juoksin, enk\u00e4 en\u00e4\u00e4 v\u00e4litt\u00e4nyt, minne. Juoksin niin lujaa kuin pystyin, mutta k\u00e4yt\u00e4v\u00e4 edess\u00e4ni ei tuntunut lyhenev\u00e4n yht\u00e4\u00e4n. Jalkani tuntuivat liikkuvan hidastettuina. Mieleeni juolahti ajatus, ett\u00e4 ehk\u00e4 viett\u00e4isin loppuel\u00e4m\u00e4ni juoksemassa t\u00e4t\u00e4 loputonta k\u00e4yt\u00e4v\u00e4\u00e4 pitkin. En silti keksinyt, mit\u00e4 muuta olisin voinut tehd\u00e4.\n\nYht\u00e4kki\u00e4 k\u00e4yt\u00e4v\u00e4 haarautui. En olisi huomannut sit\u00e4, ellei joku olisi juuri astunut kulman takaa eteeni. T\u00f6rm\u00e4sin h\u00e4neen p\u00e4istikkaa.\n\nYhden hullun hetken ajan kuvittelin tulijan olevan is\u00e4ni. Sitten tajusin, kuka se tietenkin oli: Mitja.\n\nSit\u00e4 mukaa kun tieto levisi aivolohkosta toiseen, kauhu t\u00e4ytti minut.\n\n\"Sin\u00e4 et saa olla t\u00e4\u00e4ll\u00e4\", sanoin. \"Sinun pit\u00e4\u00e4 menn\u00e4 heti takaisin.\" Ty\u00f6nsin h\u00e4nt\u00e4 molemmin k\u00e4sin kohti tulosuuntaa.\n\nMitja ei hievahtanutkaan. \"Min\u00e4 tulin hakemaan sinut.\" H\u00e4n kuulosti melkein ylpe\u00e4lt\u00e4.\n\n\"Sin\u00e4 et ymm\u00e4rr\u00e4\", kuiskasin. \"H\u00e4n on per\u00e4ss\u00e4ni.\"\n\n\"Mist\u00e4 sin\u00e4 puhut?\"\n\n\"Callista toi minut t\u00e4nne, mutta onnistuin pakenemaan h\u00e4nelt\u00e4. H\u00e4n ei saa saada minua kiinni\", sanoin. \"Ymm\u00e4rr\u00e4tk\u00f6, Mitja, mit\u00e4 tahansa tapahtuukin, h\u00e4n ei saa saada minua kiinni.\"\n\nRaissa. \u00c4\u00e4ni tuntui tulevan kaikista suunnista yht\u00e4 aikaa.\n\n\"Tuo on h\u00e4n\", sanoin Mitjalle. \"Meid\u00e4n t\u00e4ytyy paeta.\"\n\n\"Paeta minne?\"\n\nTartuin velje\u00e4ni hihasta ja painoin k\u00e4den sein\u00e4\u00e4n samaan aikaan kun toivoin, ett\u00e4 minulla olisi ollut vastaus h\u00e4nen kysymykseens\u00e4.\n\nKun ovi ilmestyi, vedin sen auki ja ty\u00f6nsin Mitjan edell\u00e4ni sis\u00e4\u00e4n.\n\n\"Mit\u00e4 sin\u00e4 teet t\u00e4\u00e4ll\u00e4?\" sihisin h\u00e4nelle kun ovi oli kadonnut.\n\n\"Sanoin jo: min\u00e4 tulin hakemaan sinua.\"\n\n\"Miten?\"\n\n\"No, sen j\u00e4lkeen kun sin\u00e4 romahdit, me kannoimme sinut yl\u00e4kertaan ja kun selvisi, ett\u00e4 sin\u00e4 edelleen hengitit, min\u00e4 avasin verhon niin leve\u00e4lle kuin pystyin. Sitten vain... seurasin sinua.\"\n\n\"Miksi sin\u00e4 menit tekem\u00e4\u00e4n jotain niin typer\u00e4\u00e4?\"\n\nVeljeni n\u00e4ytti loukkaantuneelta. \"Siksi, ett\u00e4 ajattelin sinun tarvitsevan apua. Ja ehk\u00e4 my\u00f6s siksi, ett\u00e4 Mikael pakotti minut.\"\n\nSe sai minut nielaisemaan. \"Miten h\u00e4n voi?\" kysyin.\n\n\"Ei kovin hyvin. H\u00e4n heitti kaikki paitsi minut ulos.\" Saatoin kuvitella. Min\u00e4kin olisin suunniltani, jos Mikael vain katoaisi omasta ruumiistaan.\n\n\"No. Pystytk\u00f6 sin\u00e4 palaamaan takaisin?\" kysyin veljelt\u00e4ni.\n\nH\u00e4n yritti. H\u00e4n sulki silm\u00e4ns\u00e4 ja n\u00e4in, kuinka h\u00e4nen otsansa rypistyi voimanponnistuksesta.\n\nSitten h\u00e4n avasi silm\u00e4ns\u00e4 ja tiesin, ett\u00e4 h\u00e4n oli t\u00f6rm\u00e4nnyt samaan kuin min\u00e4: paluureitti oli ehtinyt muuttua jo moneen kertaan siit\u00e4, kun h\u00e4n oli k\u00e4ytt\u00e4nyt sit\u00e4.\n\n\"En usko\", h\u00e4n tunnusti lopulta.\n\n\"Mahtavaa.\"\n\nPanin viimein merkille, ett\u00e4 ilmassa haisi palaneelta. Katsoin ymp\u00e4rilleni, mutten keksinyt, mist\u00e4 se olisi voinut tulla.\n\n\"Mik\u00e4 tuo on?\" Mitja kysyi. H\u00e4n osoitti jonnekin taakseni, joten k\u00e4\u00e4nnyin ymp\u00e4ri.\n\nKeskell\u00e4 lattiaa oli paperiarkki.\n\nEn ollut n\u00e4hnyt valkoisessa labyrintissa mit\u00e4\u00e4n esineit\u00e4, joten t\u00e4m\u00e4 kiinnitti huomioni. Paperi n\u00e4ytti vanhalta ja rypistyneelt\u00e4. Ja aivan kuin joku olisi yritt\u00e4nyt polttaa sen, sill\u00e4 reunat olivat selv\u00e4sti k\u00e4rventyneet. Kun astuin l\u00e4hemm\u00e4s, n\u00e4in, ett\u00e4 siin\u00e4 oli kirjoitusta.\n\nPudottauduin polvilleni arkin viereen, mutten uskaltanut koskea siihen. Pelk\u00e4sin, ett\u00e4 se katoaisi hetken\u00e4 min\u00e4 hyv\u00e4ns\u00e4.\n\nMitja istui viereeni. H\u00e4n ei sanonut mit\u00e4\u00e4n, ja pitk\u00e4n aikaa h\u00e4n vain tuijotti paperia. Sitten h\u00e4n tarttui siihen ja alkoi silm\u00e4ill\u00e4 sit\u00e4. Kuulin, kun h\u00e4n veti henke\u00e4.\n\n\"T\u00e4m\u00e4 on sinulle\", Mitja sanoi.\n\nKatsoin h\u00e4nt\u00e4 ter\u00e4v\u00e4sti. \"Miten niin?\"\n\nH\u00e4n ei sanonut mit\u00e4\u00e4n, vaan ojensi paperin minua kohti.\n\nSe oli kreikankielinen kirje. Ja se oli osoitettu minulle.\n\nRaissa,\n\nJos sin\u00e4 luet t\u00e4m\u00e4n, jokin on mennyt pahasti pieleen.\n\nSilloin tajusin. Kirje, jonka Stefanos oli polttanut. Se oli t\u00e4\u00e4ll\u00e4.\n\nMiten? Olin itse n\u00e4hnyt sen palavan poroksi.\n\nEnsimm\u00e4ist\u00e4 kertaa is\u00e4ni puhui minulle suoraan, enk\u00e4 ollut lainkaan varma, kest\u00e4isink\u00f6 sit\u00e4. Pakotin itseni jatkamaan.\n\nKirjoittaessani t\u00e4t\u00e4 sin\u00e4 olet viisitoistavuotias, enk\u00e4 ole n\u00e4hnyt sinua kahteentoista vuoteen. Minulla ei ole hirve\u00e4sti aikaa, mutta minun on kerrottava sinulle t\u00e4m\u00e4:\n\nOn olemassa ennustus, jonka mukaan daimonin ja suden ensimm\u00e4inen lapsi tulee p\u00e4\u00e4tt\u00e4m\u00e4\u00e4n nykyisten jumalten valtakauden. Se lapsi olet sin\u00e4.\n\nKun minulle selvisi, kuka sinun \u00e4itisi on, en ollut huolissani ennustuksen toteutumisesta vaan siit\u00e4, ett\u00e4 tiesin osan daimoneista olevan valmiita tekem\u00e4\u00e4n mit\u00e4 tahansa, jotta se ei toteutuisi. En koskaan uskonut jumaliin tai ennustuksiin. En uskonut juuri mihink\u00e4\u00e4n muuhun kuin siihen, mink\u00e4 olin itse n\u00e4hnyt ja kokenut.\n\nOlin niin v\u00e4\u00e4r\u00e4ss\u00e4.\n\nJumalat ovat totta. Yksi heist\u00e4 tarvitsee sinua sy\u00f6st\u00e4kseen muut jumalat vallasta. H\u00e4nen nimens\u00e4 on Callista. H\u00e4n haluaa, ett\u00e4 sinun daimonvoimasi kasvavat niin suuriksi, ett\u00e4 voit luopua ruumiistasi ja kohota ideoiden maailmaan. Jos luet t\u00e4t\u00e4, niin on jo tapahtunut. Jumalat eiv\u00e4t sied\u00e4 mit\u00e4\u00e4n inhimillist\u00e4, mit\u00e4\u00e4n ep\u00e4t\u00e4ydellist\u00e4, ja ihmisruumis jos jokin on sit\u00e4. H\u00e4n imee sinut itseens\u00e4 ja sin\u00e4 lakkaat olemasta. Ja sen j\u00e4lkeen h\u00e4n tuhoaa kaiken.\n\nH\u00e4nt\u00e4 ei voi tappaa. Voit ainoastaan est\u00e4\u00e4 h\u00e4nt\u00e4 saamasta haluamaansa. Valinta on sinun.\n\nMuista: jumalat eiv\u00e4t tunne kipua.\n\nOlen niin pahoillani,\n\nIs\u00e4si Stefanos Meletiadis\n\n\"Mit\u00e4 siin\u00e4 sanotaan?\" veljeni kysyi.\n\n\"Etk\u00f6 sin\u00e4 lukenut t\u00e4t\u00e4 ennen kuin annoit minulle?\" kysyin oudolla \u00e4\u00e4nell\u00e4.\n\n\"En. N\u00e4in sinun nimesi enk\u00e4 lukenut pidemm\u00e4lle. Mit\u00e4 siin\u00e4 sanotaan?\"\n\nEttei minulla ollut toivoa. Etten p\u00e4\u00e4sisi koskaan palaamaan Mikaelin luokse.\n\n\"Ei paljon mit\u00e4\u00e4n\", sanoin.\n\n\"H\u00e4n tulee\", Mitja kuiskasi.\n\nH\u00e4n oli oikeassa: kuulin jo liian tutuksi tulleen kumean \u00e4\u00e4nen.\n\nEnnen kuin tajusinkaan, tiesin, mit\u00e4 minun t\u00e4ytyi tehd\u00e4. Mitja vilkaisi minua, ja n\u00e4in, ett\u00e4 h\u00e4n tiesi, mihin johtop\u00e4\u00e4t\u00f6kseen olin tullut. Seremoniavaatteeni katosivat ja tilalle ilmestyi soturinasu.\n\n\"Onko t\u00e4m\u00e4 paras mahdollinen paikka?\" veljeni kysyi.\n\n\"En usko, ett\u00e4 vastaan tulee parempaa.\"\n\nNousimme molemmat yl\u00f6s. Seisoimme rinta rinnan kohti sein\u00e4\u00e4, jonka takaa \u00e4\u00e4ni tuli. Ojensin kaksi veitsist\u00e4ni Mitjalle. H\u00e4n otti ne vastaan sanaakaan sanomatta. Olimme valmiit.\nKUUDESTOISTA LUKU\n\nH\u00e4n tuli sein\u00e4n l\u00e4pi niin kuin se olisi ollut silkkipaperia.\n\nEn voinut teeskennell\u00e4, etteik\u00f6 minua pelottanut. Yll\u00e4tys oli ainoa mahdollisuutemme. Hy\u00f6kk\u00e4simme h\u00e4nen kimppuunsa tismalleen yht\u00e4 aikaa, min\u00e4 vasemmalta puolelta, Mitja oikealta.\n\nN\u00e4in sivusilm\u00e4ll\u00e4 kuinka Mitjan jalka kohosi potkuun. Tein oman ratkaisuni. Kyyristyin ja ty\u00f6nsin toisen jalkani eteen. Liu'uin s\u00e4\u00e4ren varassa kohti Callistaa.\n\nSe oli iso riski, sill\u00e4 lattian pinnasta oli vaikeaa p\u00e4\u00e4st\u00e4 nopeasti yl\u00f6s. Odotin Callistan v\u00e4ist\u00e4v\u00e4n suuntaani. Suojasin oikealla k\u00e4dell\u00e4ni samaan aikaan kun olin valmis iskem\u00e4\u00e4n h\u00e4nt\u00e4 vasemmassani olevalla veitsell\u00e4.\n\nH\u00e4n ei kuitenkaan v\u00e4ist\u00e4nyt, vaan otti veljeni potkun vastaan silm\u00e4\u00e4k\u00e4\u00e4n r\u00e4p\u00e4ytt\u00e4m\u00e4tt\u00e4. H\u00e4nen katseensa oli minussa.\n\nTarkoitukseni oli kampata h\u00e4nelt\u00e4 jalat alta, mutta en saanut siihen koskaan mahdollisuutta. H\u00e4n kumartui ja tarttui kurkkuuni molemmin k\u00e4sin. Polveni nousi yl\u00f6s maasta, ja huomasin veitsen putoavan k\u00e4dest\u00e4ni. Sormeni kiertyiv\u00e4t h\u00e4nen ranteidensa ymp\u00e4rille.\n\nEn saanut henke\u00e4. Harmaa seitti ilmestyi n\u00e4k\u00f6kentt\u00e4ni reunoille.\n\nVeljeni tuli h\u00e4tiin. Callistan ote herpaantui ja kaaduin taaksep\u00e4in. Nappasin saman tien veitsen k\u00e4teeni ja hypp\u00e4sin jaloilleni.\n\nHuojuin hetken paikoillani. Heti, kun uskoin pysyv\u00e4ni pystyss\u00e4, rynt\u00e4sin kohti Callistaa.\n\nCallista torjui hy\u00f6kk\u00e4ykseni helposti \u2013 eik\u00e4 h\u00e4n tehnyt muuta. Meni hetki, ennen kuin tajusin, ettei h\u00e4n edes aikonut taistella. H\u00e4n ei yritt\u00e4nyt voittaa minua, koska h\u00e4n oli jo voittanut.\n\nMitja oli kuitenkin asia erikseen. Callista kohotti k\u00e4tens\u00e4, ja veljeni putosi polvilleen pidellen k\u00e4si\u00e4\u00e4n korvillaan.\n\nEn tiennyt, mit\u00e4 h\u00e4nelle tapahtui, mutta sill\u00e4 hetkell\u00e4 en voinut keskitty\u00e4 h\u00e4neen. N\u00e4in nimitt\u00e4in tilaisuuden. Heitin veitseni Callistaa kohti.\n\nH\u00e4n ei hievahtanutkaan, vaikka veitsi iskeytyi h\u00e4nen rintaansa niin lujaa, ett\u00e4 vain osa kahvasta j\u00e4i n\u00e4kyviin.\n\nH\u00e4n tarttui kahvaan ja antoi veitsen pudota jalkojemme juureen. Siit\u00e4 ei l\u00e4htenyt lainkaan \u00e4\u00e4nt\u00e4.\n\nAstuin askeleen taaksep\u00e4in.\n\n\"Sin\u00e4 et voi tappaa minua\", h\u00e4n sanoi. \u00c4\u00e4ni oli t\u00e4sm\u00e4lleen sama, jota h\u00e4n oli k\u00e4ytt\u00e4nyt oppitunneillamme silloin, kun minulta oli mennyt turhan kauan jonkin asian tajuamisessa.\n\nYritin silti k\u00e4yd\u00e4 kauppaa. \"Anna veljeni palata\", sanoin.\n\n\"Voi tyt\u00e4r\", h\u00e4n sanoi. \"Kukaan ei palaa t\u00e4\u00e4lt\u00e4.\"\n\nVedin keuhkot t\u00e4yteen ilmaa. Tunsin veitsen k\u00e4dess\u00e4ni, jokaisen kahvan ymp\u00e4rille puristuneen sormen. Jalkapohjani tasaisella pinnalla. Oli vaikea kuvitella, ett\u00e4 Hukkavaarassa olisi viel\u00e4 lis\u00e4\u00e4 minua, mutta yritin pit\u00e4\u00e4 ajatuksesta kiinni. T\u00e4m\u00e4 ruumis t\u00e4ss\u00e4 valkoisessa huoneessa ei ollut todellinen, vaan vain mieleni rakentama heijastus siit\u00e4, mik\u00e4 makasi tajuttomana Hukkavaarassa. Kun tuo ruumis kuolisi, kaikki maallinen minusta katoaisi. Ja juuri sit\u00e4 Callista odotti.\n\nJumalat eiv\u00e4t tunne kipua.\n\nN\u00e4in Callistan silmien kapenevan. H\u00e4n odotti minun heitt\u00e4v\u00e4n veitsen h\u00e4nt\u00e4 kohti. En tehnyt niin. Sen sijaan painoin ter\u00e4n omaa k\u00e4sivarttani vasten.\n\nKipu tuli viiveell\u00e4. En yritt\u00e4nytk\u00e4\u00e4n sulkea sit\u00e4 mielest\u00e4ni, vaan keskityin siihen jokaisella solullani. Yllytin kehoani tuntemaan enemm\u00e4n.\n\n\"Mit\u00e4 sin\u00e4 teet?\" Callista kysyi. En ollut koskaan aikaisemmin kuullut hitustakaan ep\u00e4varmuutta h\u00e4nen \u00e4\u00e4ness\u00e4\u00e4n.\n\nYksi viilto ei tietenk\u00e4\u00e4n ollut tarpeeksi. Ei mennyt aikaakaan, kun kipu hellitti tasaiseksi jomotukseksi. Haava yritti jo kuroutua kiinni.\n\nMutta olin saanut tiet\u00e4\u00e4 sen, mink\u00e4 olin halunnut. Tiesin, mit\u00e4 minun piti tehd\u00e4.\n\n\"Kerro h\u00e4nelle, ett\u00e4 olen pahoillani\", sanoin Mitjalle. \"Ja ett\u00e4 rakastan h\u00e4nt\u00e4.\" Mitja tiet\u00e4isi kyll\u00e4, ket\u00e4 tarkoitin, kun sen aika tulisi.\n\n\"Mit\u00e4 sin\u00e4 tarkoitat?\" Mitja kysyi.\n\nEn vastannut. H\u00e4nen \u00e4\u00e4nens\u00e4 murtui kesken lauseen, ja se melkein riitti py\u00f6rt\u00e4m\u00e4\u00e4n p\u00e4\u00e4t\u00f6kseni. Oli oltava jokin toinen keino. Mutta tiesin, ettei ollut.\n\nOli miljoona asiaa, jotka olisin halunnut sanoa h\u00e4nelle ja kaikille muille. Mutta en voinut.\n\nAika on illuusio. Ajattelemme niin helposti, ett\u00e4 on olemassa \"my\u00f6hemmin\".\n\nRynt\u00e4sin kohti Callistaa. H\u00e4n ei tehnyt elett\u00e4k\u00e4\u00e4n est\u00e4\u00e4kseen minua. H\u00e4nen mielest\u00e4\u00e4n en varmaan voinut tehd\u00e4 mit\u00e4\u00e4n, mik\u00e4 voisi vahingoittaa h\u00e4nt\u00e4. Aioin todistaa h\u00e4nen luulonsa v\u00e4\u00e4r\u00e4ksi.\n\nVeitsi oli v\u00e4liss\u00e4mme, ja ty\u00f6nsin sit\u00e4 koko liikkeeni voimalla h\u00e4nt\u00e4 kohti.\n\nHetken aikaa me molemmat seisoimme j\u00e4hmettyneen\u00e4 paikoillamme. P\u00e4\u00e4stin irti veitsest\u00e4.\n\nSen ter\u00e4 ei ollut h\u00e4nen vatsassaan. Se oli minun vatsassani.\n\n\"Mit\u00e4 sin\u00e4 teit?\"\n\nKompuroin taaksep\u00e4in ja painauduin kaksinkerroin.\n\n\"Mit\u00e4 sin\u00e4 teit?!\" Callista jylisi. Ukkonen olisi kuulostanut mit\u00e4tt\u00f6m\u00e4lt\u00e4 h\u00e4nen rinnallaan, ja silti se ei aiheuttanut minussa mink\u00e4\u00e4nlaista reaktiota.\n\nKaikki hidastui. Ja sitten kaaduin.\n\nPaperi kahahti allani, kun laskeuduin is\u00e4ni kirjeen p\u00e4\u00e4lle. Palaneen paperin tuoksuun sekoittui jokin metallinen, ja hetken p\u00e4\u00e4st\u00e4 oivalsin, ett\u00e4 se oli vereni haju. En tuntenut l\u00e4hesk\u00e4\u00e4n niin paljon kuin olin kuvitellut. Kuulin jonkun huutavan, mutta \u00e4\u00e4ni tuntui tulevan jostain muualta, ehk\u00e4 viereisest\u00e4 huoneesta.\n\nMitja oli polvillaan vieress\u00e4ni ja painoi k\u00e4tt\u00e4\u00e4n vatsalleni. Yritin ty\u00f6nt\u00e4\u00e4 sen pois, mutta k\u00e4teni eiv\u00e4t liikkuneet.\n\nAjattelin Mikaelia. T\u00e4m\u00e4 oli viimeinen asia jonka halusin tehd\u00e4 h\u00e4nelle, mutta minulla ei ollut vaihtoehtoa. H\u00e4n tulisi aina rakastamaan minua, mutta h\u00e4n p\u00e4\u00e4sisi menetyksen yli. Lopulta kaikki p\u00e4\u00e4sisiv\u00e4t sen yli.\n\nH\u00e4nen el\u00e4m\u00e4st\u00e4\u00e4n tulisi mahtava. Ja niin paljon kuin olisin halunnut olla osa sit\u00e4, se ei ollut l\u00e4hesk\u00e4\u00e4n yht\u00e4 t\u00e4rke\u00e4\u00e4 kuin tieto siit\u00e4, ett\u00e4 h\u00e4n oli turvassa.\n\nNiin kuin Daniel oli sanonut, olin saanut kaiken. Ehk\u00e4 vain hetkeksi, mutta se oli enemm\u00e4n kuin useimmat saivat.\n\nJotain paksua ja l\u00e4mmint\u00e4 alkoi valua henkitorveeni. Yritin yski\u00e4 sit\u00e4 pois.\n\nKyyneleet valuivat pitkin Mitjan kasvoja. \"Ei\", h\u00e4n sanoi.\n\n\"El\u00e4 hitonmoinen el\u00e4m\u00e4\", sain sanotuksi.\n\nOlisin halunnut lohduttaa h\u00e4nt\u00e4, mutta \u00e4kki\u00e4 minua v\u00e4sytti. Minulla oli kylm\u00e4. Sitten n\u00e4in valon, ja viimeisen hetken ajan kaikki oli kirkasta.\n\n***\n\nAvasin silm\u00e4ni.\n\nOutoa. En ollut tiennyt, ett\u00e4 ne olivat edes olleet kiinni.\n\nSitten n\u00e4k\u00f6kentt\u00e4\u00e4ni ilmestyiv\u00e4t silm\u00e4t. Ne olivat siniset. Niin kauniit, ett\u00e4 petyin, kun ne meniv\u00e4t taas jonnekin ulottumattomiini. Vaikka kuinka yritin, en onnistunut k\u00e4\u00e4nt\u00e4m\u00e4\u00e4n p\u00e4\u00e4t\u00e4ni.\n\nTunsin oloni pehme\u00e4ksi. Aivan kuin minut olisi t\u00e4ytetty h\u00f6yhenill\u00e4.\n\nV\u00e4hitellen \u00e4\u00e4net valuivat tajuntaani. En ensin ymm\u00e4rt\u00e4nyt, mit\u00e4 ne tarkoittivat, mutta kun pinnistin keskittymiseni aivan \u00e4\u00e4rimmilleen, tajusin erilaisten \u00e4\u00e4nteiden tarkoittavan jotain.\n\n\"Raisa.\"\n\nSe kuulosti tutulta. R\u00e4pyttelin silmi\u00e4ni.\n\n\"Raisa\", \u00e4\u00e4ni toisti.\n\nSen t\u00e4ytyi tarkoittaa minua. Min\u00e4 olin Raisa. Tai ainakin olin joskus ollut.\n\n\"Mitja, kerro mit\u00e4 tapahtui\", joku sanoi.\n\n\"H\u00e4n kuoli. H\u00e4n juoksi kohti Callistaa enk\u00e4 tajunnut, ett\u00e4 h\u00e4n aikoi...\" H\u00e4n ei sanonut lausettaan loppuun, mutta se ei vaivannut minua l\u00e4hesk\u00e4\u00e4n niin paljon kuin se, miten tutulta t\u00e4m\u00e4kin \u00e4\u00e4ni kuulosti.\n\n\"H\u00e4n ei ole kuollut!\" Nimi pomppasi esiin jostain mieleni perukoilta: Mikael. Huutajan nimi oli Mikael.\n\nSitten tunsin kuinka joku \u2013 ei Mikael, vaan se toinen poika \u2013 alkoi tunkeutua mieleeni. En voinut tehd\u00e4 asialle mit\u00e4\u00e4n. Se ei tuntunut lainkaan niin omituiselta kuin olisi voinut kuvitella, ja aloin pohtia, oliko t\u00e4m\u00e4 jotain, johon olin tottunut.\n\n\"Ei hitto\", poika sanoi. Ja katosi mielest\u00e4ni.\n\n\"Mit\u00e4?\" Mikael kysyi.\n\n\"Jokin on hullusti.\"\n\n\"Miten niin hullusti?\"\n\n\"H\u00e4n ei tied\u00e4, keit\u00e4 me olemme.\"\n\nEn ollut tiennyt, ett\u00e4 sellainen hiljaisuus oli mahdollista. Sitten: \"Pit\u00e4isik\u00f6 meid\u00e4n vied\u00e4 h\u00e4net sairaalaan?\"\n\n\"En tied\u00e4\", Mikael vastasi.\n\nJokin sanoi minulle, ettei h\u00e4n k\u00e4ytt\u00e4nyt noita sanoja kovin usein. Vihasin kipua, jonka kuulin h\u00e4nen \u00e4\u00e4ness\u00e4\u00e4n, enk\u00e4 edes tiennyt, kuka h\u00e4n oli.\n\nSuljin silm\u00e4ni. Heid\u00e4n keskustelunsa seuraaminen oli k\u00e4ym\u00e4ss\u00e4 uuvuttavaksi.\n\n***\n\nJoku puhui taas. Miksi koko ajan jonkun piti puhua? Minua v\u00e4sytti niin, etten saanut silmi\u00e4ni auki.\n\n\"Haluaisin kysy\u00e4 sinulta jotain\", \u00e4\u00e4ni sanoi. Erilainen, pehme\u00e4 \u00e4\u00e4ni. Tyt\u00f6n \u00e4\u00e4ni? Joka tapauksessa se oli miellytt\u00e4v\u00e4.\n\n\"Kysy pois.\"\n\n\"Ennen kuin... Ennen kuin t\u00e4m\u00e4 tapahtui, aioitko sin\u00e4 j\u00e4tt\u00e4\u00e4 Raisan?\"\n\nTajusin, ett\u00e4 n\u00e4m\u00e4 \u00e4\u00e4nteet olivat aivan toisenlaisia kuin mit\u00e4 olin kuullut aikaisemmin. Silti tiesin, ett\u00e4 he puhuivat minusta. Yritin keskitty\u00e4.\n\nMy\u00f6s Mikaelilla tuntui olevan vaikeuksia, sill\u00e4 h\u00e4n kysyi: \"Mit\u00e4?\"\n\n\"Raisa luulee, ett\u00e4 sin\u00e4 j\u00e4t\u00e4t h\u00e4net\", tytt\u00f6 sanoi.\n\n\"Mist\u00e4 ihmeest\u00e4 h\u00e4n on saanut sellaista p\u00e4\u00e4h\u00e4ns\u00e4?\" Mikaelin \u00e4\u00e4ni oli \u00e4llistynyt. En ollut ainoa, joka ei ymm\u00e4rt\u00e4nyt, mist\u00e4 puhuttiin. Hengitin hitaasti sis\u00e4\u00e4n ja ulos.\n\n\"En ole aivan varma\", tytt\u00f6 sanoi. \"H\u00e4n kertoi minulle jotain susista ja ihmisist\u00e4, pariutumisesta ja siit\u00e4, ett\u00e4 olet k\u00e4ytt\u00e4ytynyt oudosti. H\u00e4n sanoi...\" H\u00e4n hiljeni hetkeksi. \"H\u00e4n sanoi, ett\u00e4 sinun sutesi hylkii h\u00e4nt\u00e4. Mit\u00e4 se sitten tarkoittaakaan.\"\n\nMikael oli hiljaa hyvin pitk\u00e4\u00e4n. \"Ei helvetti\", h\u00e4n sanoi lopulta.\n\n\"Se ei siis pid\u00e4 paikkansa?\"\n\n\"Ei, ei todellakaan.\" Sitten: \"Itse asiassa tilanne on ihan p\u00e4invastoin. Minun suteni on valinnut Raisan kumppanikseen.\"\n\nJotain kummallista tapahtui rintakeh\u00e4ss\u00e4ni. En tiennyt, mit\u00e4 se oli tai miksi niin tapahtui, enk\u00e4 osannut p\u00e4\u00e4tt\u00e4\u00e4, pidink\u00f6 siit\u00e4 vai en. Halusin tiet\u00e4\u00e4 enemm\u00e4n.\n\n\"Seh\u00e4n on hyv\u00e4 asia, eik\u00f6?\" tytt\u00f6 kysyi.\n\n\"On\", Mikael sanoi. H\u00e4n kuulosti surulliselta. \"Ja toisaalta taas ei ole.\"\n\nMiksi minun oli vaikea hengitt\u00e4\u00e4? Mit\u00e4 minulle oikein tapahtui?\n\n\"Milloin t\u00e4m\u00e4 tapahtui?\"\n\n\"Ensimm\u00e4isen\u00e4 talvena\", Mikael vastasi.\n\nTytt\u00f6 oli hetken hiljaa. \"Sin\u00e4 olit siis tuntenut h\u00e4net vain muutaman kuukauden.\"\n\n\"Niin.\" Poika p\u00e4\u00e4sti hassun \u00e4\u00e4nen. Nauroiko h\u00e4n?\n\n\"Mit\u00e4 tarkoitat, kun sanot, ett\u00e4 sutesi on valinnut h\u00e4net kumppanikseen?\"\n\n\"En itsek\u00e4\u00e4n tiennyt, mit\u00e4 se tarkoittaa, ennen kuin se tapahtui\", Mikael vastasi. \"Meille ei kerrota, ennen kuin olemme tarpeeksi vanhoja. Ilmeisesti is\u00e4 pelk\u00e4si, ett\u00e4 valitsisin Jennin. En tajunnut sit\u00e4 silloin, mutta oikeasti is\u00e4 ei pit\u00e4nyt Jennist\u00e4.\"\n\n\"Miten se tapahtui?\"\n\n\"Raisan verta p\u00e4\u00e4tyi elimist\u00f6\u00f6ni. Min\u00e4... min\u00e4 maistoin h\u00e4nen vertaan.\"\n\nSeurasi hiljaisuus.\n\n\"Sek\u00f6 riitt\u00e4\u00e4?\"\n\n\"Ei pelk\u00e4st\u00e4\u00e4n se. Susi p\u00e4\u00e4tt\u00e4\u00e4, ei ihminen. Yleens\u00e4 ne kaksi ovat kuitenkin aika samoilla linjoilla mit\u00e4 tulee rakastumiseen ja muuhun. T\u00e4ytyy kuitenkin olla valmis antamaan itsens\u00e4 kokonaan.\"\n\n\"Mist\u00e4 sitten tied\u00e4t, ett\u00e4 niin k\u00e4vi?\"\n\n\"Aluksi en mist\u00e4\u00e4n. Minulla oli outo olo, mutta se meni ohi. Mutta v\u00e4hitellen aloin tajuta, ett\u00e4 jokin oli muuttunut. Olin liian paniikissa kysy\u00e4kseni is\u00e4lt\u00e4 mit\u00e4\u00e4n, ja kun vihdoin p\u00e4\u00e4sin siihen pisteeseen, h\u00e4n katosi. Asia varmistui lopullisesti vasta kun tulimme t\u00e4nne ja p\u00e4\u00e4sin lukemaan kaikki is\u00e4n p\u00e4iv\u00e4kirjat ja merkinn\u00e4t.\"\n\n\"Mik\u00e4 on vialla?\"\n\n\"Mik\u00e4\u00e4n ei ole vialla. P\u00e4invastoin, tunnen oloni paljon kevyemm\u00e4ksi. Rauhallisemmaksi. Tied\u00e4tk\u00f6 sen tunteen, kun kaipaat jotain, muttet tied\u00e4 mit\u00e4?\"\n\n\"Luulen tiet\u00e4v\u00e4ni\", tytt\u00f6 sanoi.\n\n\"Kaikki ennen Raisaa oli tuntunut tismalleen silt\u00e4. Tilalle on kuitenkin astunut uusi ongelma: susi ei voi siet\u00e4\u00e4 sit\u00e4, ettei Raisa ole h\u00e4nen.\"\n\n\"Olet mustasukkainen.\"\n\n\"Niin. En v\u00e4it\u00e4 tavallisestikaan olevani t\u00e4ysin sen yl\u00e4puolella, mutta t\u00e4m\u00e4 on jotain ihan uutta. Se haluaa omistaa h\u00e4net. Jos suteni saisi p\u00e4\u00e4tt\u00e4\u00e4, h\u00e4n lukitsisi Raisan jonnekin eik\u00e4 p\u00e4\u00e4st\u00e4isi ket\u00e4\u00e4n sis\u00e4\u00e4n. Ihminen tietenkin ymm\u00e4rt\u00e4\u00e4 paremmin, mutta el\u00e4imelle ei oikein voi puhua j\u00e4rke\u00e4. Minun t\u00e4ytyy vain oppia el\u00e4m\u00e4\u00e4n t\u00e4m\u00e4n kanssa.\"\n\n\"Eli sutesi on vihainen, koska se pelk\u00e4\u00e4 Raisan l\u00e4htev\u00e4n.\"\n\n\"Juuri niin.\"\n\n\"Sin\u00e4 siis tiesit t\u00e4st\u00e4 jo silloin, kun te erositte viime kes\u00e4n\u00e4. Etk\u00e4 kertonut h\u00e4nelle.\"\n\n\"Olisi ollut itsek\u00e4st\u00e4 kertoa. Tajusin, ett\u00e4 minun pit\u00e4\u00e4 antaa h\u00e4nen menn\u00e4. Tiesin, ett\u00e4 h\u00e4nen t\u00e4ytyy olla jonkun muun kanssa. Miten h\u00e4n voisi muka tiet\u00e4\u00e4, mit\u00e4 meill\u00e4 on, ellei h\u00e4nell\u00e4 ole mit\u00e4\u00e4n, mihin verrata?\"\n\nKaikesta kuuli, ett\u00e4 heid\u00e4n sanoillaan oli painoa. Ett\u00e4 t\u00e4m\u00e4n oli oltava jotain valtavaa. En vain pystynyt tuntemaan sit\u00e4.\n\n\"My\u00f6nn\u00e4n, ett\u00e4 olin v\u00e4h\u00e4n aikaa tosi katkera\", Mikael jatkoi. \"P\u00e4\u00e4tin n\u00e4ytt\u00e4\u00e4 itselleni ja tavallaan my\u00f6s Raisalle, etten tarvitse h\u00e4nt\u00e4. Osittain se johtui siit\u00e4, ett\u00e4 hetken aikaa minusta tuntui silt\u00e4 kuin el\u00e4m\u00e4ni olisi ohi: minulle kasattiin kaikki lauman velvollisuudet ja sitten kun Raisa kertoi l\u00e4htev\u00e4ns\u00e4 tavoittelemaan omia unelmiaan... Tajusin olevani jumissa. Minun oli pakko l\u00e4hte\u00e4.\"\n\n\"Tied\u00e4n tunteen\", tytt\u00f6 sanoi.\n\nHalusin avata silm\u00e4ni ja tehd\u00e4 \u2013 jotain? Ilmaista, ett\u00e4 olin l\u00e4sn\u00e4. Saada heid\u00e4t puhumaan minulle, eik\u00e4 vain toisilleen. Joku oli kuitenkin liimannut luomeni umpeen.\n\nMikael oli hetken hiljaa. \"Yritin unohtaa h\u00e4net. Juhlin joka ilta eri tyt\u00f6n kanssa. Mutta jo jonkun muun kuin Raisan suuteleminen paljasti, ettei siit\u00e4 tulisi mit\u00e4\u00e4n. Minua ei yksinkertaisesti en\u00e4\u00e4 kiinnostanut.\"\n\n\"Mutta olithan sin\u00e4 sen vampyyrityt\u00f6n kanssa.\"\n\n\"Itse asiassa en ollut. Yritin kyll\u00e4, mutta ei se toiminut. Regina sai minut tajuamaan, ett\u00e4 jokin oli vialla. H\u00e4n on kaunis, hauska ja fiksu, eik\u00e4 ollut mit\u00e4\u00e4n syyt\u00e4, miksi minua ei olisi kiinnostanut.\"\n\n\"En vain pystynyt siihen\", Mikael jatkoi. \"Silloin asiat viimein loksahtivat kohdalleen ja tajusin, mit\u00e4 oli tapahtunut.\"\n\n\"H\u00e4n ei varmasti pit\u00e4nyt torjutuksi tulemisesta.\"\n\n\"Oikeastaan h\u00e4nen mielest\u00e4\u00e4n koko tilanne oli vain hirve\u00e4n hauska. Sen takia meist\u00e4 tuli yst\u00e4vi\u00e4. Kerroin h\u00e4nelle luonnollisesti kaiken Raisasta. H\u00e4n kannusti minua hakemaan l\u00e4\u00e4kikseen ja auttoi p\u00e4\u00e4sykokeisiin valmistautumisessa.\"\n\n\"Mitja ep\u00e4ilee, ett\u00e4 muutit Helsinkiin pelk\u00e4st\u00e4\u00e4n Raisan takia. Pit\u00e4\u00e4k\u00f6 se paikkansa?\"\n\n\"Tietenkin\", h\u00e4n sanoi. Ensimm\u00e4ist\u00e4 kertaa koko keskustelun aikana h\u00e4n naurahti. \"Se, ett\u00e4 p\u00e4\u00e4stin h\u00e4net menem\u00e4\u00e4n ei tarkoittanut, ettenk\u00f6 edelleen toivonut meid\u00e4n viel\u00e4 palaavan yhteen. Ei heti, mutta jossain vaiheessa. En ensin ajatellut pit\u00e4v\u00e4ni h\u00e4neen mit\u00e4\u00e4n yhteytt\u00e4. Mutta sitten n\u00e4in h\u00e4net klubilla. En tietenk\u00e4\u00e4n ajatellut menev\u00e4ni sinne siksi ett\u00e4 t\u00f6rm\u00e4isin h\u00e4neen, mutta n\u00e4in j\u00e4lkik\u00e4teen ajatellen se oli t\u00e4rkein syy. Olin kuullut h\u00e4nen veljest\u00e4\u00e4n ja tiesin, ett\u00e4 Mitja soittaisi sin\u00e4 iltana.\"\n\n\"Tiesitk\u00f6 Shaunista?\"\n\n\"En ihan heti, mutta aika pian.\"\n\n\"Et tainnut pit\u00e4\u00e4 h\u00e4nest\u00e4.\"\n\n\"Sinulla ei ole aavistustakaan. Suteni halusi repi\u00e4 h\u00e4nen p\u00e4\u00e4ns\u00e4 irti. Haluaisi viel\u00e4kin.\"\n\nTytt\u00f6 p\u00e4\u00e4sti mietteli\u00e4\u00e4n \u00e4\u00e4nen. \"Mikset ole viel\u00e4k\u00e4\u00e4n kertonut Raisalle?\"\n\n\"Koska h\u00e4nen pit\u00e4\u00e4 saada p\u00e4\u00e4tt\u00e4\u00e4 itse. Se, ett\u00e4 min\u00e4 olen valinnut h\u00e4net ei tarkoita, ett\u00e4 h\u00e4n valitsisi minua. Jos kerron h\u00e4nelle, ett\u00e4 k\u00e4rsin, h\u00e4n tekee p\u00e4\u00e4t\u00f6ksens\u00e4 sen pohjalta. Haluan, ett\u00e4 h\u00e4n valitsee minut, koska h\u00e4n haluaa minut. En mink\u00e4\u00e4n muun vuoksi.\"\n\n\"Sin\u00e4 et halua h\u00e4nen sitoutuvan sinuun pelk\u00e4st\u00e4 velvollisuudesta.\"\n\n\"Tai s\u00e4\u00e4list\u00e4.\"\n\n\"Mit\u00e4 sin\u00e4 aioit tehd\u00e4?\"\n\n\"Aioin antaa meille aikaa. Kertoa h\u00e4nelle vasta sitten, kun h\u00e4n olisi muutenkin valmis ajattelemaan meit\u00e4 yhdess\u00e4 koko loppuel\u00e4m\u00e4n ajan. En vain ollut tajunnut, ett\u00e4 suteni reagoisi n\u00e4in vahvasti. Olen koko ajan vihannut sit\u00e4, ett\u00e4 joudun pit\u00e4m\u00e4\u00e4n t\u00e4m\u00e4n h\u00e4nelt\u00e4 salassa, mutta viel\u00e4 vaikeampaa minun olisi antaa itselleni anteeksi se, ett\u00e4 h\u00e4n joutuisi k\u00e4rsim\u00e4\u00e4n minun tekem\u00e4ni virheen takia. Min\u00e4 mokasin, joten minun kuuluu kantaa seuraukset. Ei h\u00e4nen.\"\n\nTytt\u00f6 alkoi sanoa jotakin, mutta Mikael keskeytti h\u00e4net. \"\u00c4l\u00e4 ymm\u00e4rr\u00e4 v\u00e4\u00e4rin. Min\u00e4 en kadu mit\u00e4\u00e4n. Raisa on parasta, mit\u00e4 minulle on koskaan tapahtunut. Sit\u00e4 ei muuta edes se, ett\u00e4 kaikki t\u00e4m\u00e4 tapahtui noin kymmenen vuotta aikaisemmin kuin olin kuvitellut.\"\n\n\"Ent\u00e4 jos Raisa p\u00e4\u00e4tt\u00e4\u00e4 j\u00e4tt\u00e4\u00e4 sinut?\"\n\n\"Sitten min\u00e4 olen yksin. Ei se ole mik\u00e4\u00e4n tragedia. Jos Raisa on onnellisempi ilman minua, niin sitten min\u00e4kin olen.\"\n\n\"H\u00e4n on siis unelmiesi tytt\u00f6.\"\n\n\"Niin on.\"\n\nHetken aikaa minusta tuntui silt\u00e4, ett\u00e4 tiesin, mist\u00e4 h\u00e4n puhui. Puolen sekunnin ajan tieto oli hyppysiss\u00e4ni, ja silm\u00e4ni alkoivat kostua. Sitten hetki karkasi ulottuviltani, enk\u00e4 en\u00e4\u00e4 tiennyt, olinko kuvitellut koko keskustelun. Vajosin tiedottomuuteen.\n\n***\n\nOli pime\u00e4\u00e4, kun havahduin omituiseen \u00e4\u00e4neen. Kirkas, kellomainen kilahdus tuli jostain alhaalta ja kaikui huoneeseeni asti.\n\nSitten kuulin askelia, joita seurasi metallinen loksahdus.\n\n\"Kiitos, ett\u00e4 tulit\", Mikael sanoi jollekin eteisess\u00e4. \"Tied\u00e4n, ett\u00e4 t\u00e4st\u00e4 on sinulle paljon vaivaa.\"\n\n\"Tietenkin tulin\", naisen \u00e4\u00e4ni vastasi. \"Onnekseni taivas on ollut t\u00e4ysin pilvess\u00e4. Se antoi minulle pari ylim\u00e4\u00e4r\u00e4ist\u00e4 tuntia.\"\n\nKuulin korkojen \u00e4\u00e4nen, kun ne nousivat yl\u00f6s portaita. Ne eiv\u00e4t kuitenkaan tulleet heti sis\u00e4\u00e4n huoneeseeni. Mikael tuli ensin ja veti verhot ikkunan eteen. Tuli viel\u00e4kin pime\u00e4mp\u00e4\u00e4.\n\nValot napsautettiin p\u00e4\u00e4lle, ja puristin silm\u00e4ni kiinni. Korkokeng\u00e4t saapuivat huoneeseen. Kun raotin silmi\u00e4ni, n\u00e4in vilahduksen kuutiomaisesta laukusta ennen kuin hahmo kumartui aivan minuun kiinni.\n\nPunaiset hiukset. Olin n\u00e4hnyt ne aikaisemminkin. Ja vaikka pidin punaisesta, t\u00e4st\u00e4 naisesta en pit\u00e4nyt.\n\n\"Hei Raisa\", nainen sanoi. \"Regina t\u00e4ss\u00e4. Tutkisin sinua hiukan.\"\n\nEn halunnut h\u00e4nen koskevan minuun, mutta minulta ei kysytty. H\u00e4nen k\u00e4tens\u00e4 olivat h\u00e4tk\u00e4hdytt\u00e4v\u00e4n kylm\u00e4t, kun h\u00e4n veti peiton syrj\u00e4\u00e4n ja tarttui k\u00e4sivarteeni.\n\nSeurasi ikuisuus, jonka aikana h\u00e4n teki vaikka mink\u00e4laisia asioita, joista en ymm\u00e4rt\u00e4nyt mit\u00e4\u00e4n: kuunteli, koputteli ja mittasi. Mik\u00e4\u00e4n siit\u00e4 ei tuntunut kivalta, mutta naisen puolustukseksi oli my\u00f6nnett\u00e4v\u00e4, ettei mik\u00e4\u00e4n my\u00f6sk\u00e4\u00e4n sattunut.\n\nViimein h\u00e4n kumartui ja otti jotain laukustaan. N\u00e4in neulan. Se jotenkin sopi t\u00e4ydellisesti mielikuvaani t\u00e4st\u00e4 Reginasta. Hyvin kursailemattomasti h\u00e4n tarttui k\u00e4sivarteeni ja pisti minua kyyn\u00e4rtaipeeseen. Hetken kuluttua h\u00e4n veti neulan ulos, ja n\u00e4in h\u00e4nen k\u00e4dess\u00e4\u00e4n kaksi pient\u00e4 punaista putkea.\n\nJ\u00e4rkytyksekseni h\u00e4n pyyhk\u00e4isi haavaa sormellansa ja pisti sen suuhunsa.\n\n\"Maistuu ihan tavalliselta\", h\u00e4n sanoi.\n\nH\u00e4n kumartui. Kun h\u00e4n ilmestyi j\u00e4lleen n\u00e4k\u00f6kentt\u00e4\u00e4ni, veriputket olivat kadonneet ja tilalle oli ilmestynyt ruisku. \"Min\u00e4 vien n\u00e4ytteet labraan.\"\n\n\"Pid\u00e4 huoli, etteiv\u00e4t ne joudu v\u00e4\u00e4riin k\u00e4siin\", Mikael sanoi. \"Minulla ei ole aavistustakaan, mit\u00e4 niist\u00e4 l\u00f6ytyy.\"\n\n\"Rauhoitu. Min\u00e4 olen tehnyt ty\u00f6t\u00e4ni jo kauan. Vaikka on my\u00f6nnett\u00e4v\u00e4, etten ole n\u00e4hnyt mit\u00e4\u00e4n t\u00e4llaista aikaisemmin. Kaiken kaikkiaan kiinnostava tapaus.\"\n\nH\u00e4n vilkaisi k\u00e4dess\u00e4\u00e4n olevaa ruiskua. Senkin sis\u00e4lt\u00f6 oli punaista.\n\n\"Onko tuo...\" Mikael aloitti.\n\n\"Minun vertani\", Regina vastasi.\n\nEn ehtinyt ajatella, mit\u00e4 se tarkoitti kun h\u00e4n jo t\u00f6kk\u00e4si neulan toiseen k\u00e4sivarteeni ja painoi rauhallisesti m\u00e4nn\u00e4n alas.\n\nHe odottivat. En tied\u00e4, mit\u00e4 he uskoivat tapahtuvan, mutta en huomannut mit\u00e4\u00e4n muutosta.\n\nRegina huokaisi. \"Vereni ei toimi. En tietenk\u00e4\u00e4n voi olla t\u00e4ysin varma ilman kuvia ja labratuloksia, mutta en usko, ett\u00e4 h\u00e4ness\u00e4 on fyysisesti mit\u00e4\u00e4n vikaa. Valitettavasti se tarkoittaa, ett\u00e4 l\u00e4\u00e4ketiede ei voi auttaa h\u00e4nt\u00e4.\"\n\n***\n\nV\u00e4hitellen aloin k\u00e4sitt\u00e4\u00e4, ett\u00e4 jokin oli vialla. Kaikki olivat koko ajan niin huolissaan, ett\u00e4 aloin pohtia sen syit\u00e4. Aivoni penkoivat muistoja laatikoista, mutta eiv\u00e4t l\u00f6yt\u00e4neet mit\u00e4\u00e4n hy\u00f6dyllist\u00e4. Odotin, mutten oikeastaan tiennyt, mit\u00e4.\n\nJotain sent\u00e4\u00e4n muuttui. Pysyttelin hereill\u00e4 aikaisempaa enemm\u00e4n. Aina v\u00e4lill\u00e4 poika istui s\u00e4nkyni vieress\u00e4 ja soitti kaunista musiikkia. H\u00e4n oli ainoa, jolle pystyin kertomaan, jos minulla oli jano tai jos minun piti p\u00e4\u00e4st\u00e4 vessaan. Ne eiv\u00e4t olleet niink\u00e4\u00e4n sanoja, joita k\u00e4ytin, mutta jotenkin h\u00e4n aina tuntui ymm\u00e4rt\u00e4v\u00e4n, mit\u00e4 tarkoitin. Olin kysynyt, miksi h\u00e4n oli p\u00e4\u00e4ni sis\u00e4ll\u00e4, ja h\u00e4n oli kertonut olevansa veljeni. En tiennyt, mit\u00e4 se tarkoitti, mutta h\u00e4n oli selitt\u00e4nyt sen tarkoittavan sit\u00e4, ett\u00e4 meiss\u00e4 molemmissa virtasi sama veri. Olin alkanut kutsua h\u00e4nt\u00e4 mieless\u00e4ni Samavereksi.\n\nViimein h\u00e4n suostui selitt\u00e4m\u00e4\u00e4n minulle, miksi kaikki olivat niin huolissaan. Heit\u00e4 huolestutti se, etten pystynyt liikkumaan enk\u00e4 puhumaan. Olin kuulemma joskus pystynyt. Se h\u00e4mm\u00e4stytti minua. Kun k\u00e4skin kroppaani tekem\u00e4\u00e4n jotakin, se ei totellut.\n\nAloin tulla tietoisemmaksi ymp\u00e4rist\u00f6st\u00e4ni, vaikka v\u00e4lill\u00e4 toivoin, ettei niin olisi ollut. Pienimm\u00e4t mahdolliset asiat alkoivat \u00e4rsytt\u00e4\u00e4. Rohtuneet huulet, kutina jalkapohjassa. Ikkunasta tuleva liian kirkas valo tai huoneeseen eksynyt k\u00e4rp\u00e4nen.\n\nEi auttanut, ett\u00e4 niin monet tuntuivat k\u00e4rsiv\u00e4n puolestani. Er\u00e4skin mies istui s\u00e4nkyni vieress\u00e4 tuntikausia. H\u00e4n oli vanhempi kuin muut, eiv\u00e4tk\u00e4 h\u00e4nen maantienv\u00e4riset hiuksensa tai haaleansiniset silm\u00e4ns\u00e4 her\u00e4tt\u00e4neet minussa mink\u00e4\u00e4nlaisia muistoja. Sympatiaa kyll\u00e4kin. Kyyneleet valuivat pitkin h\u00e4nen poskiaan, mutta h\u00e4n ei sanonut sanaakaan.\n\n\"Olen pahoillani.\" Mikael oli sanonut sen h\u00e4nelle jo aivan liian monta kertaa.\n\nP\u00e4iv\u00e4 p\u00e4iv\u00e4lt\u00e4 n\u00e4in, mit\u00e4 tilanne teki Mikaelille. H\u00e4n kantoi minua vessaan ja suihkuun. H\u00e4n k\u00e4\u00e4nsi minua kyljelt\u00e4 toiselle monta kertaa p\u00e4iv\u00e4ss\u00e4. H\u00e4n oli loputtoman k\u00e4rsiv\u00e4llinen, ja olisin halunnut voida sanoa, ettei h\u00e4nen olisi tarvinnut.\n\nH\u00e4n nukkui huoneeni lattialla. Odotin, ett\u00e4 joku olisi kehottanut h\u00e4nt\u00e4 menem\u00e4\u00e4n jonnekin muualle, mutta kukaan ei tehnyt niin. Sitten tajusin, ett\u00e4 minun s\u00e4nkyni oli oikeasti h\u00e4nen s\u00e4nkyns\u00e4, enk\u00e4 voinut olla ihmettelem\u00e4tt\u00e4, miksei h\u00e4n vain siirt\u00e4nyt minua jonnekin.\n\nH\u00e4n oli niin kaunis, ja vihasin sit\u00e4, ett\u00e4 h\u00e4n tuli koko ajan vain surullisemmaksi. H\u00e4nen ihonsa kalpeni ja silmien alla olevat varjot tummenivat. Tiesin, ett\u00e4 se johtui minusta.\n\nMinun t\u00e4ytyi tehd\u00e4 jotakin. Halusin sanoa h\u00e4nelle, ett\u00e4 h\u00e4n voisi menn\u00e4 v\u00e4lill\u00e4 poiskin jos halusi, min\u00e4 en nimitt\u00e4in olisi menossa minnek\u00e4\u00e4n.\n\nJonkinlainen suojeluvaisto otti ohjat. Tiesin, ett\u00e4 h\u00e4nen suojelemisensa oli minun teht\u00e4v\u00e4ni, ei kenenk\u00e4\u00e4n muun. Ja kun Mikael sanoi menev\u00e4ns\u00e4 k\u00e4ym\u00e4\u00e4n my\u00f6hemmin ulkona, keksin suunnitelman.\n\nMikaelin l\u00e4hteminen oli harvinaista, joten tiesin, ett\u00e4 minun oli pakko laittaa suunnitelmani t\u00e4yt\u00e4nt\u00f6\u00f6n heti. Kutsuin Samaveren luokseni. H\u00e4n ei ollut viel\u00e4 kertaakaan ollut tulematta, enk\u00e4 t\u00e4ll\u00e4k\u00e4\u00e4n kertaa joutunut pettym\u00e4\u00e4n.\n\nUsein, kun h\u00e4n tuli huoneeseen, h\u00e4n haisi pahalta \u2013 jonkinlaiselta savulta. Niin t\u00e4ll\u00e4kin kerralla.\n\n\u00c4l\u00e4 aloita, h\u00e4n sanoi p\u00e4\u00e4ni sis\u00e4ll\u00e4.\n\nEn tiennyt olleeni aloittamassa yht\u00e4\u00e4n mit\u00e4\u00e4n.\n\nKun Mikael tulee takaisin, \u00e4l\u00e4 avaa isoa ovea, sanoin h\u00e4nelle. Iso ovi oli se, joka oli alhaalla. Kukaan ei tullut sis\u00e4\u00e4n ilman, ett\u00e4 joku jo sis\u00e4ll\u00e4 oleva avasi sen.\n\nSamaveren \u00e4llistys t\u00e4ytti mieleni. Oletko ihan varma? Tuo on todella tyly\u00e4.\n\nOlen, vastasin. Jos minun pit\u00e4\u00e4 ottaa joku mukanani t\u00e4h\u00e4n... mustaanpime\u00e4\u00e4n, se olet sin\u00e4, Samaveri. Ei Mikael.\n\nSamaveri ny\u00f6kk\u00e4si.\n\nEtsin oikeaa sanaa. Tiesin, ett\u00e4 se oli t\u00e4rke\u00e4\u00e4.\n\nLopulta se l\u00f6ytyi.\n\nLupaa, sanoin h\u00e4nelle.\n\nLupaan.\n\nTiesin, ett\u00e4 h\u00e4n tekisi niin kuin sanoin. Hengitin syv\u00e4\u00e4n.\n\nNiin kuin tavallista, suunnitelmani kostautui minulle.\n\nOdotin koko p\u00e4iv\u00e4n kuulevani oven k\u00e4yv\u00e4n. Niin ei koskaan tapahtunut. Sen sijaan Mikael ilmestyi huoneeseeni pukeutuneena paljon siistimmin kuin mit\u00e4 h\u00e4n oli viime aikoina tehnyt. H\u00e4n oli ollut l\u00e4hd\u00f6ss\u00e4, mutta jokin oli saanut h\u00e4net muuttamaan mielens\u00e4.\n\nJa h\u00e4nen j\u00e4isest\u00e4 ilmeest\u00e4\u00e4n uskoin tiet\u00e4v\u00e4ni, mik\u00e4.\n\n\"Veljesi kertoi juuri, ett\u00e4 mik\u00e4li l\u00e4hden, minua ei en\u00e4\u00e4 p\u00e4\u00e4stet\u00e4 sis\u00e4\u00e4n\", h\u00e4n sanoi. Oliko h\u00e4n vihainen? Vai huvittunut? En ollut varma.\n\n\"Jos sin\u00e4 aiot j\u00e4tt\u00e4\u00e4 minut, saat sanoa sen p\u00e4in naamaani. Se on ainoa tapa, jolla sin\u00e4 p\u00e4\u00e4set eroon minusta.\"\n\nH\u00e4n oli vihainen. Olin samaan aikaan sek\u00e4 ihmeiss\u00e4ni ett\u00e4 iloinen. En ollut oikeastaan halunnut, ett\u00e4 h\u00e4n menisi pois. Itse asiassa en ollut halunnut sit\u00e4 yht\u00e4\u00e4n.\n\nH\u00e4n kumartui l\u00e4hemm\u00e4s. \"Tehd\u00e4\u00e4n sopimus: sin\u00e4 voit j\u00e4tt\u00e4\u00e4 minut sitten, kun olet p\u00e4\u00e4ssyt yl\u00f6s s\u00e4ngyst\u00e4. Min\u00e4 l\u00e4hden, eik\u00e4 sinun tarvitse n\u00e4hd\u00e4 minua en\u00e4\u00e4 koskaan. Mutta siihen asti pysyn t\u00e4\u00e4ll\u00e4.\"\n\n***\n\nSitten tapahtui jotain, mink\u00e4 tulisin muistamaan loppuel\u00e4m\u00e4ni.\n\nVaalea, aavistuksen kyll\u00e4styneen n\u00e4k\u00f6inen poika oli k\u00e4ynyt luonani l\u00e4hes yht\u00e4 usein kuin Samaveri. Olin pit\u00e4nyt h\u00e4nest\u00e4 heti, kun olin n\u00e4hnyt h\u00e4net. Jokin h\u00e4nen mustissa vaatteissaan ja \u00e4rtyneess\u00e4 ilmeess\u00e4\u00e4n oli saanut minut ajattelemaan, ettei h\u00e4nen mielenmaisemansa johtunut lainkaan minusta. Se oli mukavaa vaihtelua. H\u00e4n ei saanut minua v\u00e4syneeksi niin kuin kaikki muut.\n\nH\u00e4n livahti huoneeseeni sill\u00e4 aikaa kun Mikael ja Samaveri juttelivat toisessa huoneessa. H\u00e4n ty\u00f6nsi ovea kiinni, muttei varsinaisesti sulkenut sit\u00e4.\n\n\"Nuo kaverit tuolla saavat yritt\u00e4\u00e4 keksi\u00e4, mik\u00e4 sinussa on vikana ja mit\u00e4 sille pit\u00e4isi tehd\u00e4\", h\u00e4n aloitti osoittaen oven suuntaan. \"Me molemmat tied\u00e4mme, etten min\u00e4 ole sellainen tyyppi, joka ratkaisee ongelmia. Mutta jotain min\u00e4kin tajuan.\"\n\n\"Joskus me olemme todella j\u00e4rjett\u00f6mi\u00e4 olentoja\", h\u00e4n jatkoi. \"Me jauhamme ja vatvomme ja p\u00e4hk\u00e4ilemme viel\u00e4 v\u00e4h\u00e4n lis\u00e4\u00e4. V\u00e4lill\u00e4 vain haluaisi, ett\u00e4 kaikki lakkaisivat ajattelemasta niin pirun paljon.\"\n\nSynkk\u00e4 poika ei yleens\u00e4 ollut n\u00e4in outo, ja olin utelias tiet\u00e4m\u00e4\u00e4n, mist\u00e4 nyt tuuli.\n\nKuulin metallisen hel\u00e4hdyksen, kun jotain pudotettiin lattialle. Ihmettelin, mit\u00e4 h\u00e4n mahtoi tehd\u00e4.\n\nJa sitten susi hypp\u00e4si s\u00e4ngylleni.\n\nEn ollut koskaan n\u00e4hnyt sellaista sutta. Se oli upea: l\u00e4hes vitivalkoinen. H\u00e4n istui edess\u00e4ni kuin antaen minulle hetken aikaa tottua tilanteeseen. Sitten h\u00e4n asettui ker\u00e4lle viereeni ja laittoi p\u00e4\u00e4ns\u00e4 vatsalleni. H\u00e4n sulki silm\u00e4ns\u00e4 melkein heti.\n\nTunsin h\u00e4nen l\u00e4mp\u00f6ns\u00e4, h\u00e4nen hengityksens\u00e4. H\u00e4n maiskautti suutaan muutaman kerran, ja hymyilin mieless\u00e4ni. El\u00e4in ei v\u00e4litt\u00e4nyt, pystyink\u00f6 puhumaan vai en. Se v\u00e4litti vain t\u00e4st\u00e4 hetkest\u00e4.\n\nJoten me torkuimme. Susi oli unessa melkein heti, ja tunsin, miten se n\u00e4ki unta. Yritin kuvitella, millaista se oli.\n\nJossain vaiheessa minun oli t\u00e4ytynyt nukahtaa, sill\u00e4 kun avasin silm\u00e4ni, n\u00e4in, ett\u00e4 Mikael seisoi oviaukossa k\u00e4sivarret ristittyin\u00e4. Minulla ei ollut aavistustakaan, mit\u00e4 h\u00e4n ajatteli, mutta n\u00e4hdess\u00e4\u00e4n minun katsovan h\u00e4n k\u00e4\u00e4ntyi ja l\u00e4hti.\n\nPian ovi avattiin taas. Toinen vaalea, heiver\u00f6isempi poika astui sis\u00e4\u00e4n. H\u00e4n ei n\u00e4ytt\u00e4nyt h\u00e4mm\u00e4styv\u00e4n lainkaan n\u00e4hdess\u00e4\u00e4n vieress\u00e4ni makaavan suden.\n\nSuden p\u00e4\u00e4 kohosi vatsaltani. H\u00e4n loikkasi alas s\u00e4ngylt\u00e4 ja meni pojan luokse.\n\n\"Hei Raisa\", Poika sanoi samalla kun h\u00e4nen sormensa l\u00f6ysiv\u00e4t suden korvantaustan. \"Valitettavasti joudun viem\u00e4\u00e4n Pupen kotiin.\"\n\nSusi murahti, mutta poika ei n\u00e4ytt\u00e4nyt pel\u00e4styv\u00e4n. H\u00e4n p\u00e4invastoin hymyili, ja huomasin itsekin olevani vailla huolia.\n\nEik\u00e4 minulla tietenk\u00e4\u00e4n ollut aavistustakaan, mit\u00e4 tapahtuisi jo seuraavana p\u00e4iv\u00e4n\u00e4.\n\nMikael tuli huoneeseeni joskus puolen p\u00e4iv\u00e4n j\u00e4lkeen. H\u00e4n kyyristyi viereeni niin, ett\u00e4 h\u00e4nen siniset silm\u00e4ns\u00e4 tunkeutuivat n\u00e4k\u00f6kentt\u00e4ni. H\u00e4n ei ottanut k\u00e4tt\u00e4ni omaansa, mik\u00e4 h\u00e4nell\u00e4 oli yleens\u00e4 tapana. H\u00e4n ei hymyillyt. H\u00e4nen silm\u00e4ns\u00e4 verestiv\u00e4t hieman, aivan kuin h\u00e4n ei olisi nukkunut koko t\u00e4n\u00e4 aikana yht\u00e4\u00e4n. Ensimm\u00e4ist\u00e4 kertaa tajusin, ett\u00e4 niin saattoi oikeasti olla.\n\nKatseessa oli kuitenkin my\u00f6s jotain muuta. Jotain, mit\u00e4 en ollut n\u00e4hnyt niiss\u00e4 koskaan aikaisemmin. Aivan kuin h\u00e4n olisi kamppaillut pitk\u00e4\u00e4n jonkin asian kanssa ja viimein vaakakuppi olisi nytk\u00e4ht\u00e4nyt niin, ettei ollut en\u00e4\u00e4 paluuta.\n\nHetken aikaa tunsin jotain, jota en muistanut tunteneeni aikaisemmin. Pelko.\n\n\"Haluaisin kokeilla jotain.\"\n\nH\u00e4nen ilmeett\u00f6m\u00e4n kuorensa alla kupli. Mutta mit\u00e4? Mik\u00e4 sai h\u00e4net niin hermostuneeksi?\n\n\"Sin\u00e4 voit kuolla siihen. Mutta toisaalta uskon, tai ainakin toivon, ett\u00e4 se voisi parantaa sinut.\"\n\nEn ollut aivan varma, mit\u00e4 h\u00e4n tarkoitti. Yleens\u00e4 ajattelin vain yhdest\u00e4 sekunnista toiseen, sitten kolmanteen, ehk\u00e4 jopa nelj\u00e4nteen.\n\nMutta halusin, ett\u00e4 h\u00e4n olisi tyytyv\u00e4inen. Olin siis valmis mihin tahansa, mit\u00e4 Mikael keksisik\u00e4\u00e4n ehdottaa.\n\n\"Min\u00e4 en ole tehnyt t\u00e4t\u00e4 koskaan aikaisemmin\", h\u00e4n sanoi.\n\nPidin katseeni h\u00e4ness\u00e4. Suljin silm\u00e4ni hetkeksi ja avasin ne uudelleen merkiksi siit\u00e4, ett\u00e4 ymm\u00e4rsin. H\u00e4nt\u00e4 pelotti, mutta luotin h\u00e4neen.\n\n\"T\u00e4m\u00e4 sattuu\", Mikael sanoi. Ja nielaisi. \"Paljon.\"\n\nH\u00e4n tarkoitti kipua. Se sai minut ep\u00e4r\u00f6im\u00e4\u00e4n hetkeksi. Tiesin, ett\u00e4 olin kokenut kipua aikaisemminkin ja ett\u00e4 olin selvinnyt siit\u00e4. R\u00e4p\u00e4ytin taas.\n\nMikael sulki silm\u00e4ns\u00e4. Odotin. Kun h\u00e4n avasi ne, n\u00e4in h\u00e4nen sutensa. H\u00e4n kutsui sen tarkoituksella esiin, tajusin.\n\nAivan aluksi en huomannut mit\u00e4\u00e4n. En silloinkaan, kun kummallinen tunne levisi rintakeh\u00e4st\u00e4ni raajoihin ja aina varpaisiin asti. Olin niin h\u00e4nen katseensa lumoissa, ett\u00e4 tajusin kaiken vasta kun oli liian my\u00f6h\u00e4ist\u00e4 tehd\u00e4 mit\u00e4\u00e4n.\n\nKipu r\u00e4j\u00e4hti koko kropassani. Kipu, jollaista en ollut kuvitellut voivani tuntea, jollaista ei olisi pit\u00e4nyt voida edes olla olemassa.\n\nEieieieiei, ajattelin. Mit\u00e4 tahansa muuta, paitsi t\u00e4t\u00e4. Vaikka en ollut varma, mik\u00e4 \"t\u00e4m\u00e4\" oli, olin varma, etten halunnut sit\u00e4.\n\nVartaloni alkoi nyki\u00e4. Mikael painoi hartiani kiinni patjaan.\n\n\"\u00c4l\u00e4 tappele vastaan\", h\u00e4n sanoi. H\u00e4nen \u00e4\u00e4nens\u00e4 oli pingottunut.\n\nEn tapellut vastaan. Miten olisin edes voinut? Saatoin n\u00e4hd\u00e4 vain valkoista. Kroppani pyrki irti patjasta, mutta Mikaelin ote oli ter\u00e4st\u00e4.\n\nYritin huutaa. \u00c4\u00e4ni pakkautui sis\u00e4\u00e4ni kunnes olin varma, ett\u00e4 r\u00e4j\u00e4ht\u00e4isin.\n\n\"\u00c4l\u00e4 tappele vastaan\", Mikael toisti. \"P\u00e4\u00e4st\u00e4 se ulos.\"\n\nAloin ymm\u00e4rt\u00e4\u00e4, mit\u00e4 h\u00e4n tarkoitti. En kuitenkaan voinut totella. Minun t\u00e4ytyi pist\u00e4\u00e4 vastaan kaikin keinoin, muuten minusta ei j\u00e4isi mit\u00e4\u00e4n j\u00e4ljelle.\n\nEhk\u00e4 Mikael halusi juuri sit\u00e4? Mutta en voinut.\n\nPaine kasvoi, kunnes \u00e4kki\u00e4 kadoksissa ollut \u00e4\u00e4neni l\u00f6ytyi, ja huusin kovempaa kuin koskaan.\nSEITSEM\u00c4STOISTA LUKU\n\nKorkojen \u00e4\u00e4ni kaikui kadulla. Hetken aikaa ratikan kolina peitti hitaat, m\u00e4\u00e4r\u00e4tietoiset askeleet alleen, mutta siten ne taas kulkivat Viidett\u00e4 linjaa pitkin ja k\u00e4\u00e4ntyiv\u00e4t alas Porthaninkadulle. Tyt\u00f6n nauru sekoittui pojan nauruun. He olivat menossa viett\u00e4m\u00e4\u00e4n perjantai-iltaa.\n\nToinen heist\u00e4 oli veljeni. Ja toinen oli min\u00e4.\n\nOli kulunut kuukausi siit\u00e4, kun olin her\u00e4nnyt Sarrien makuuhuoneessa. En tiennyt, kuinka kauan olin ollut unessa \u2013 se oli saattanut olla yht\u00e4 hyvin tunti tai kolme vuorokautta. Ikkunasta tulvi valoa huoneeseen. Se oli niin ihmeellinen n\u00e4ky, ett\u00e4 ennen kuin huomasinkaan, ty\u00f6nsin peiton syrj\u00e4\u00e4n ja heilautin jalkani s\u00e4ngyn reunan yli lattialle.\n\nKatsahdin alas. Minulla oli yll\u00e4ni l\u00f6ys\u00e4 teepaita, joka viisti reisi\u00e4ni. Ei sukkia.\n\nNousin hitaasti yl\u00f6s. Ensimm\u00e4inen yritykseni ep\u00e4onnistui ja pyll\u00e4hdin takaisin s\u00e4ngylle, mutten luovuttanut. Toisella kertaa onnistuin jo pysym\u00e4\u00e4n pystyss\u00e4, vaikka t\u00e4risin kauttaaltani. Otin varovaisen askeleen kohti ovea.\n\nMit\u00e4 kauemmin olin ollut hereill\u00e4, sit\u00e4 selvemm\u00e4ksi tuli, ett\u00e4 minulla oli kaksi hyvin akuuttia tarvetta: toisaalta tyrm\u00e4\u00e4v\u00e4 jano, ja toisaalta taas kova vessah\u00e4t\u00e4.\n\nKompuroin bambijaloillani kunnes saavutin oven. Avasin sen samalla kun otin toisella k\u00e4dell\u00e4ni tukea karmista.\n\nPys\u00e4hdyin.\n\nMikael nukkui lattialla. Minun Mikaelini. Saatoin n\u00e4hd\u00e4 h\u00e4nen kasvoiltaan kaiken, mit\u00e4 viime viikot olivat tehneet h\u00e4nelle. Ja silti h\u00e4n oli yh\u00e4 t\u00e4\u00e4ll\u00e4.\n\nEn halunnut her\u00e4tt\u00e4\u00e4 h\u00e4nt\u00e4. Astuin k\u00e4yt\u00e4v\u00e4\u00e4n.\n\nK\u00e4yty\u00e4ni vessassa l\u00e4hdin astelemaan portaita askelma kerrallaan pit\u00e4en tiukasti kiinni kaiteesta.\n\nOlin juuri ehtinyt olohuoneen poikki kun kuulin yl\u00e4kerrasta \u00e4\u00e4ni\u00e4.\n\nMikael ilmestyi portaiden yl\u00e4p\u00e4\u00e4h\u00e4n. H\u00e4n n\u00e4ytti h\u00e4t\u00e4\u00e4ntyneelt\u00e4. Kaikesta p\u00e4\u00e4tellen h\u00e4n oli rynn\u00e4nnyt paikalle suoraan makuuhuoneen lattialta. Kun h\u00e4n n\u00e4ki minut, h\u00e4n j\u00e4hmettyi paikoilleen.\n\nPitk\u00e4n aikaa vain tuijotimme toisiamme.\n\n\"Hei\", sanoin lopulta.\n\n\"Hei\", Mikael vastasi. \u00c4\u00e4ni oli tukkoinen, aivan kuin h\u00e4nell\u00e4 olisi ollut flunssa. H\u00e4n selvitti kurkkuaan ja hankasi kasvojaan. Vasta silloin h\u00e4n tuntui tajuavan, ettei h\u00e4nell\u00e4 ollut p\u00e4\u00e4ll\u00e4 muuta kuin bokserit. H\u00e4n katosi hetkesi ja palasi vet\u00e4en paitaa p\u00e4\u00e4llens\u00e4. Tunsin pistoksen. Koskaan aikaisemmin h\u00e4n ei ollut kokenut tarvetta peitell\u00e4 itse\u00e4\u00e4n. Jotain oli s\u00e4rkynyt.\n\nH\u00e4n laskeutui portaat huomattavasti ketter\u00e4mmin kuin min\u00e4. \"Sinulla on varmaan jano\", h\u00e4n sanoi. H\u00e4n ei j\u00e4\u00e4nyt odottamaan vastausta, vaan meni ohitseni keitti\u00f6\u00f6n. Minulla oli tunne, ett\u00e4 h\u00e4n v\u00e4ltteli minuun katsomista.\n\nOtin varovaisen askeleen h\u00e4nt\u00e4 kohti. H\u00e4n ei k\u00e4\u00e4ntynyt tiskialtaan luota. Jatkoin k\u00e4velemist\u00e4, kunnes seisoin h\u00e4nen takanaan. N\u00e4in, kuinka h\u00e4nen hartiansa j\u00e4nnittyiv\u00e4t, mutta h\u00e4n ei viel\u00e4k\u00e4\u00e4n sanonut mit\u00e4\u00e4n, pelk\u00e4st\u00e4\u00e4n hengitti tavallista hitaammin ja kuuluvammin.\n\nNojasin otsani h\u00e4nen niskaansa vasten. Hengitykseni t\u00e4ytyi kutittaa h\u00e4nen ihoaan, sill\u00e4 h\u00e4n s\u00e4ps\u00e4hti hieman.\n\nKiedoin k\u00e4teni h\u00e4nen ymp\u00e4rilleen. H\u00e4n hengitti syv\u00e4\u00e4n, kuin rauhoittaakseen itse\u00e4\u00e4n, mutta siit\u00e4 ei n\u00e4ytt\u00e4nyt olevan apua, sill\u00e4 h\u00e4nen pulssinsa hakkasi aivan hulluna. Hassua, mutta oma syd\u00e4meni n\u00e4ytti ottaneen h\u00e4nest\u00e4 mallia.\n\nPuristin h\u00e4nen paidanhelmansa nyrkkiin k\u00e4dess\u00e4ni ja painoin huuleni h\u00e4nen niskaansa.\n\nJuuri kun olin menett\u00e4m\u00e4ss\u00e4 toivoni, h\u00e4n k\u00e4\u00e4ntyi ymp\u00e4ri. Liike oli niin \u00e4killinen, ett\u00e4 olisin kaatunut sel\u00e4lleni, ellei h\u00e4n olisi samassa kahmaissut minua syliins\u00e4. H\u00e4n puristi aivan liian lujaa, mutta sill\u00e4 hetkell\u00e4 en v\u00e4litt\u00e4nyt.\n\n\"\u00c4l\u00e4 koskaan j\u00e4t\u00e4 minua\", h\u00e4n kuiskasi.\n\nPyyhin kyyneleet h\u00e4nen silmiens\u00e4 alta. \"En\", lupasin.\n\nH\u00e4n piteli minua pitk\u00e4n aikaa. Jossain vaiheessa mahani alkoi pit\u00e4\u00e4 hyvin \u00e4\u00e4nek\u00e4st\u00e4 kurinaa. Mikael alkoi nauraa.\n\nSill\u00e4 hetkell\u00e4 tiesin, ett\u00e4 kaikki tulisi olemaan taas hyvin. Muistin yh\u00e4 el\u00e4v\u00e4sti, kuinka olimme nauraneet ja itkeneet ja nauraneet taas, ja jossain vaiheessa Mikael oli vihdoin onnistunut laittamaan meille jotain sy\u00f6t\u00e4v\u00e4\u00e4.\n\nAjattelin sit\u00e4 yh\u00e4 minun ja Mitjan k\u00e4velless\u00e4 metroasemalta kohti klubia. En ollut jaksanut pysy\u00e4 jaloillani montaakaan minuuttia, mutten ollut ollut siit\u00e4 huolissani. Rintani ei ollut koskaan tuntunut yht\u00e4 kevyelt\u00e4 ja t\u00e4ydelt\u00e4.\n\nSaavuimme paikalle hyviss\u00e4 ajoin. Klubia ei avattaisi viel\u00e4 pariin tuntiin, joten menimme takaovesta sis\u00e4\u00e4n. Suuntasimme suoraan lavalle, jossa piuhojen keskell\u00e4 seisoi tuttu hahmo.\n\n\"Hei varpunen\", Konsta sanoi.\n\nHalasin h\u00e4nt\u00e4 pitk\u00e4\u00e4n ja l\u00e4mpim\u00e4sti. H\u00e4n oli palannut jo kaksi viikkoa sitten ja ottanut taas klubin komentoonsa. Veljeni oli pel\u00e4nnyt miten h\u00e4n reagoisi kun saisi tiet\u00e4\u00e4, ett\u00e4 h\u00e4n oli ottanut lopputilin, mutta Konsta oli vain onnitellut h\u00e4nt\u00e4 hyv\u00e4st\u00e4 p\u00e4\u00e4t\u00f6ksest\u00e4 ja sanonut h\u00e4nen saavan ty\u00f6ns\u00e4 takaisin koska tahansa. Tiesin heti, ettei Konsta uskonut niin k\u00e4yv\u00e4n. H\u00e4nen luottamuksensa velje\u00e4ni kohtaan tuntui kertakaikkisen hyv\u00e4lt\u00e4.\n\nCaoimhe ilmestyi klubille v\u00e4h\u00e4n meid\u00e4n j\u00e4lkeemme, ja b\u00e4ndi p\u00e4\u00e4si aloittamaan soundcheckins\u00e4. Mitjan, Ciaon ja Konstan itsens\u00e4 lis\u00e4ksi paikalla oli pari Konstan yst\u00e4v\u00e4\u00e4, joita en ollut tavannut aikaisemmin. He soittaisivat t\u00e4n\u00e4\u00e4n yhdess\u00e4 ensimm\u00e4isen ja viimeisen keikkansa.\n\nSuuntasin baaritiskille hakemaan itselleni kokiksen ja istuin sitten katsomon puolelle seuraamaan, mit\u00e4 lavalla tapahtui. Mitja huomautteli jos mist\u00e4kin asiasta, mutta kukaan ei n\u00e4ytt\u00e4nyt kyseenalaistavan h\u00e4nt\u00e4. He olivat kaikki ammattilaisia ja pitiv\u00e4t Mitjaa vertaisenaan.\n\nJossain vaiheessa b\u00e4ndin seuraaminen alkoi kyll\u00e4stytt\u00e4\u00e4, ja huomasin harhailevani kohti takatiloja. Konstan toimiston ovi oli auki, ja menin sis\u00e4\u00e4n.\n\nIstuin h\u00e4nen tuoliinsa. Aloin leikki\u00e4 p\u00f6yd\u00e4ll\u00e4 olevilla toimistotarvikkeilla. Yritin py\u00f6ritell\u00e4 kyn\u00e4\u00e4 sormieni v\u00e4liss\u00e4, mutta se putosi ja jouduin noukkimaan sen lattialta.\n\nKatsoin k\u00e4si\u00e4ni. Ne olivat p\u00e4\u00e4llisin puolin t\u00e4ysin samat k\u00e4det kuin ennenkin. Vain min\u00e4 pystyin tuntemaan eron.\n\nViimeisten viikkojen aikana olin joutunut hyv\u00e4ksym\u00e4\u00e4n sen, ettei mik\u00e4\u00e4n palaisi t\u00e4ysin ennalleen. Kaikki se, mik\u00e4 oli tehnyt minusta daimonin, oli kuollut.\n\nOlin el\u00e4nyt el\u00e4m\u00e4ni luullen olevani ihminen, mutta t\u00e4m\u00e4 kokemus osoitti, ettei minulla ollut ollut hajuakaan, millaista se oikeasti oli. Ihmisen\u00e4 oleminen ei nimitt\u00e4in ollut kaksista. Varsinkin alussa kaikki oli sattunut. Olin ollut hidas, k\u00f6mpel\u00f6 ja v\u00e4synyt hyvin nopeasti.\n\nJa sitten olin ottanut kyn\u00e4n k\u00e4teeni ja tajunnut, etten osannut en\u00e4\u00e4 piirt\u00e4\u00e4.\n\nOlisin halunnut voida sanoa, ettei se ollut murtanut syd\u00e4nt\u00e4ni. Etten ollut itkenyt kadonneiden kykyjeni per\u00e4\u00e4n tai antanut itseni valua itses\u00e4\u00e4liin. Mutta t\u00e4rkeint\u00e4 oli ollut se, etten ollut antanut notkahduksieni kest\u00e4\u00e4 kauan. En ollut luovuttanut. Minulla oli ehk\u00e4 vain t\u00e4m\u00e4 maailma, mutta se oli tarpeeksi.\n\nMusiikki lavan suunnalta oli lakannut jo kauan sitten. Puheensorina oli liikkunut k\u00e4yt\u00e4v\u00e4\u00e4 pitkin ja pys\u00e4htyi viereiseen huoneeseen. En kuullut sanoja, mutta lauseet kiihtyiv\u00e4t loppua kohti ja p\u00e4\u00e4ttyiv\u00e4t nauruun. Huomasin hymyilev\u00e4ni. Liittyisin heid\u00e4n seuraansa, mutten ihan viel\u00e4. En ollut viel\u00e4k\u00e4\u00e4n t\u00e4ysin tottunut siihen, miten aistini nyky\u00e4\u00e4n toimivat, ja liiat \u00e4rsykkeet uuvuttivat minut. Suljin silm\u00e4ni.\n\nEn tiennyt, kuinka kauan istuin toimistossa omiin ajatuksiini uppoutuneena ennen kuin joku tuli etsim\u00e4\u00e4n minua.\n\nKohotin katseeni, enk\u00e4 yll\u00e4ttynyt kun n\u00e4in Mitjan.\n\n\"Meill\u00e4 on ongelma\", h\u00e4n sanoi.\n\nKuinkas muutenkaan, ajattelin. \"No?\"\n\n\"Sinun pit\u00e4\u00e4 menn\u00e4 puhumaan Caoimhelle. H\u00e4n on niin suunniltaan, ettei saa pidetty\u00e4 muotoaan.\"\n\nMeni hetki, ennen kuin ymm\u00e4rsin, mit\u00e4 h\u00e4n tarkoitti. Olin aikoja sitten unohtanut, etteiv\u00e4t keijut suinkaan oikeasti n\u00e4ytt\u00e4neet silt\u00e4 kuin ollessaan ihmisten keskuudessa. Ihmisten keskuudessa he n\u00e4yttiv\u00e4t aina, no, ihmisilt\u00e4.\n\nJos ei laskettu sit\u00e4 kertaa, kun min\u00e4 ja Mitja olimme l\u00e4hteneet mets\u00e4st\u00e4m\u00e4\u00e4n daimoneita, en ollut oikeastaan koskaan n\u00e4hnyt Caoimhea poissa tolaltaan.\n\n\"Mit\u00e4 sin\u00e4 teit?\" kysyin Mitjalta.\n\nH\u00e4n ei n\u00e4ytt\u00e4nyt loukkaantuneelta kysymykseni johdosta. H\u00e4n n\u00e4ytti vaivautuneelta.\n\n\"Min\u00e4 kosin h\u00e4nt\u00e4.\"\n\nHypp\u00e4sin jaloilleni. \"Mutta meh\u00e4n puhuimme, ett\u00e4 sin\u00e4 kosisit vasta keikan j\u00e4lkeen! Ent\u00e4 jos h\u00e4n olisi vastannut ei?\"\n\nVasta silloin tajusin, ett\u00e4 juuri niin siin\u00e4 oli t\u00e4ytynyt k\u00e4yd\u00e4. Muussa tapauksessa Mitja olisi kihlattunsa kanssa, ei minun. Ja miksi muuten Ciao olisi ollut poissa tolaltaan? \"No voi hitto.\"\n\n\"Niin, sali on jo t\u00e4ynn\u00e4 ja meid\u00e4n pit\u00e4isi olla lavalla ihan kohta.\"\n\nHeilautin k\u00e4tt\u00e4ni. \"En min\u00e4 sit\u00e4. H\u00e4n siis sanoi ei?\"\n\n\"H\u00e4n sanoi kyll\u00e4.\" Pieni hymy karkasi h\u00e4nen huulilleen, ja pian se muuttui leve\u00e4ksi virnistykseksi.\n\n\"Oikeasti?\"\n\n\"Joo, oikeasti.\"\n\nEn pystynyt en\u00e4\u00e4 olemaan aloillani. Pompin p\u00e4ki\u00f6ideni varassa. \"Mikset sanonut heti? Onneksi olkoon!\"\n\n\"Kiitti. Mutta siis t\u00e4m\u00e4 ongelma, josta puhuin.\"\n\nKurtistin kulmiani. \"Miksi h\u00e4n on poissa tolaltaan? Kihlaus on iloinen asia.\"\n\n\"Sep\u00e4 se. H\u00e4n on omien sanojensa mukaan niin liikuttunut, ettei onnistu n\u00e4ytt\u00e4m\u00e4\u00e4n ihmiselt\u00e4. Onni kuplii yli, niin h\u00e4n taisi sanoa.\" H\u00e4n ryk\u00e4isi. \"Yritin jo saada h\u00e4nt\u00e4 rauhoittumaan, mutta sinulta se voisi onnistua paremmin.\"\n\n\"Miksi sin\u00e4 menit kosimaan h\u00e4nt\u00e4 ennen keikkaa?\"\n\n\"H\u00e4n tajusi, ett\u00e4 jotain oli meneill\u00e4\u00e4n. H\u00e4n kysyi monta kertaa, mik\u00e4 on vialla ja lopulta sanat vain tulivat ulos suustani.\"\n\n\"Annoitko h\u00e4nelle jo sormuksen?\"\n\n\"Joo. H\u00e4n sanoi, ett\u00e4 se on t\u00e4ydellinen.\"\n\nNy\u00f6kk\u00e4sin helpottuneena. Kaikkein suurin kysymysmerkki oli omasta mielest\u00e4ni ollut sormus, eik\u00e4 Caoimhen vastaus \u2013 olin nimitt\u00e4in auttanut sen valitsemisessa. Oli ollut iso riski hankkia sormus kysym\u00e4tt\u00e4 Caoimhelta mit\u00e4\u00e4n, mutta Mitja oli halunnut pit\u00e4\u00e4 kaiken yll\u00e4tyksen\u00e4.\n\n\"Miss\u00e4 h\u00e4n on?\" kysyin.\n\nMitja ny\u00f6kk\u00e4si kohti k\u00e4yt\u00e4v\u00e4\u00e4. \"Vessassa.\"\n\nK\u00e4velin p\u00f6yd\u00e4n ymp\u00e4ri ovelle. Jatkoin k\u00e4yt\u00e4v\u00e4\u00e4 pitkin ja koputin ainokaisen vessan oveen. \"Ciao?\"\n\nHetkeen sis\u00e4lt\u00e4 ei kuulunut mit\u00e4\u00e4n. Lopulta: \"Niin?\"\n\nH\u00e4n kuulosti oudolta. Pel\u00e4styneelt\u00e4, mutta \u00e4\u00e4ness\u00e4 oli jotain muutakin.\n\n\"Voinko tulla sis\u00e4\u00e4n?\" kysyin pehme\u00e4sti.\n\n\"Vo-voit. Ovi on auki.\"\n\nSe oli. Vilkuilin ymp\u00e4rilleni varmistaakseni, ettei kukaan seisonut aivan l\u00e4hiet\u00e4isyydell\u00e4. Sen j\u00e4lkeen raotin ovea juuri sen verran, ett\u00e4 p\u00e4\u00e4sin pujahtamaan sis\u00e4\u00e4n. Suljin sen per\u00e4ss\u00e4ni ja k\u00e4\u00e4nnyin kohti Caoimhea.\n\nJa k\u00e4\u00e4nnyin saman tien poisp\u00e4in. Tuijotin oveen teipattuja k\u00e4sienpesuohjeita ja niiden alle k\u00e4sin kirjoitettuja vitsikk\u00e4it\u00e4 kommentteja. \"Pyh\u00e4 Sylvi\", kuulin itseni sanovan.\n\nK\u00e4\u00e4nnyin hitaasti uudestaan. T\u00e4ll\u00e4 kertaa olin hieman valmiimpi ottamaan vastaan edess\u00e4ni olevan n\u00e4yn, mutta siit\u00e4 huolimatta minulta lipsahti kirosana ennen kuin ehdin painaa k\u00e4den suulleni.\n\nOlin pit\u00e4nyt Caoimhen ihmismuotoa kauniina. Olin salaa uskonut, ett\u00e4 h\u00e4n oli tehnyt itsest\u00e4n aavistuksen n\u00e4timm\u00e4n kuin h\u00e4n todellisuudessa oli. Se ei ollut saanut minua ajattelemaan h\u00e4nest\u00e4 mit\u00e4\u00e4n pahaa \u2013 kuka muka ei olisi tehnyt samaa? Nyt tajusin, ettei kyse ollut lainkaan siit\u00e4. Ciao oli tehnyt ihmismin\u00e4st\u00e4\u00e4n niin tavanomaisen n\u00e4k\u00f6isen kuin vain oli pystynyt menett\u00e4m\u00e4tt\u00e4 kokonaan omaa ulkomuotoaan.\n\nSilm\u00e4t olivat valtavat. Ne olivat edelleen vihre\u00e4t, mutta h\u00e4tk\u00e4hdytt\u00e4v\u00e4n kirkkaat. H\u00e4nen ripsens\u00e4 olivat ep\u00e4normaalin pitk\u00e4t ja paksut. Ne n\u00e4yttiv\u00e4t my\u00f6s vaihtavan v\u00e4ri\u00e4 k\u00e4rki\u00e4 kohden. Leuka ja nen\u00e4 olivat todella sirot. Suu oli pieni, mutta huulet silti t\u00e4ytel\u00e4iset.\n\nJa h\u00e4nen hiuksensa... En ollut aivan varma, saattoiko niit\u00e4 edes kutsua hiuksiksi. Ne olivat samaa kultaa kuin aina ennenkin, mutta nyt ne n\u00e4yttiv\u00e4t v\u00e4h\u00e4n kuin rastoilta, jotka olivat elossa. Mik\u00e4li silm\u00e4ni eiv\u00e4t valehdelleet aivan t\u00e4ysin, h\u00e4nen p\u00e4\u00e4st\u00e4\u00e4n kasvoi kukkia. Niit\u00e4 oli tosin vain muutama, ne olivat valkoisia ja sijoittuivat h\u00e4nen toisen korvansa taakse. Joka tapauksessa h\u00e4nen hiuksensa olivat valtavat. Latvat ulottuivat vy\u00f6t\u00e4r\u00f6lle asti.\n\nJa hehku. Aivan kuin h\u00e4n olisi ollut oma valonl\u00e4hteens\u00e4. Ei ihme, ettei Mitja ollut onnistunut kuvailemaan h\u00e4nt\u00e4. \"T\u00e4ydellinen\" kun oli sana, joka ei itsess\u00e4\u00e4n kuvannut yht\u00e4\u00e4n mit\u00e4\u00e4n. Se t\u00e4ytyi n\u00e4hd\u00e4 itse.\n\n\"Anteeksi, en vain ollut yht\u00e4\u00e4n valmistautunut\", sopersin. Kirosin Mitjaa mieless\u00e4ni siit\u00e4, ettei h\u00e4n ollut varoittanut minua.\n\n\"Ei se mit\u00e4\u00e4n\", h\u00e4n vastasi. Ja helpotuksekseni h\u00e4n kuulostikin silt\u00e4, ettei ollut pahastunut.\n\nKiinnitin huomioni h\u00e4nen vaatteisiinsa. Violetti korsettiyl\u00e4osa oli kirjailtu tikkauksilla. H\u00e4nen pitk\u00e4 vaaleanvihre\u00e4 hameensa oli materiaalia, jota paremman puutteessa p\u00e4\u00e4tin kutsua sifongiksi. Sen alta kuulsi jonkinlainen minihame. Ainoa, mik\u00e4 n\u00e4ytti mill\u00e4\u00e4n tavalla tutulta olivat h\u00e4nen vaaleanruskeat cowboy-buutsinsa.\n\nSamassa h\u00e4n jo kapsahti kaulaani. Niin hennoksi tyt\u00f6ksi h\u00e4n oli yll\u00e4tt\u00e4v\u00e4n painava. Ilman takanani olevaa ovea olisimme molemmat m\u00e4tk\u00e4ht\u00e4neet lattialle.\n\n\"Sinun veljesi pyysi minua vaimokseen!\"\n\n\"Tied\u00e4n\", sanoin h\u00e4nen hiuksiinsa. Ne tuoksuivat ruoholta ja sitruunalta.\n\nCiao otti askeleen taakse niin ett\u00e4 h\u00e4nen kasvonsa mahtuivat kokonaisuudessaan n\u00e4k\u00f6kentt\u00e4\u00e4ni. \"Sin\u00e4 olet tiennyt koko ajan!\"\n\n\"Niin olen. Usko pois, se oli t\u00e4ysin h\u00e4nen oma ideansa. Olen tietenkin onnellinen teid\u00e4n molempien puolesta. Nyt sinun pit\u00e4isi kuitenkin rauhoittua sen verran, ett\u00e4 voit menn\u00e4 lavalle ja hoitaa t\u00e4m\u00e4n keikan.\"\n\n\"Min\u00e4 yrit\u00e4n.\" H\u00e4n ravisti k\u00e4si\u00e4\u00e4n ja hyp\u00e4hteli jalalta toiselle samalla kun yritti hengitt\u00e4\u00e4 syv\u00e4\u00e4n.\n\nP\u00e4\u00e4tin, ettei l\u00e4sn\u00e4olostani ollut apua. Palasin takaisin Mitjan luokse. Konsta oli liittynyt h\u00e4nen seuraansa ja he puhuivat matalalla \u00e4\u00e4nell\u00e4.\n\n\"T\u00e4ss\u00e4 saattaa menn\u00e4 hetki\", sanoin heille molemmille.\n\n\"Niin v\u00e4h\u00e4n arvelinkin\", Mitja mutisi.\n\n\"V\u00e4hint\u00e4, mit\u00e4 voimme tehd\u00e4, on ilmoittaa yleis\u00f6lle viiv\u00e4styksest\u00e4.\"\n\nMitja puri huultaan. Tiesin mit\u00e4 se tarkoitti: h\u00e4nell\u00e4 oli idea. H\u00e4n ei vain ollut varma, oliko se hyv\u00e4 vai huono idea.\n\n\"Min\u00e4 menen lavalle\", h\u00e4n sanoi lopulta. \"Yksin. Soitan jotain sill\u00e4 aikaa, kun Caoimhe rauhoittuu.\"\n\nVilkaisin yll\u00e4ttyneen\u00e4 Konstaa. H\u00e4n ny\u00f6kk\u00e4si veljelleni. En ehtinyt sanomaan mit\u00e4\u00e4n, ennen kuin Mitja oli jo kadonnut lavalle.\n\n\"Moi, min\u00e4 olen Mitja\", kuulin veljeni sanovan. Olin \u00e4kki\u00e4 \u00e4\u00e4rimm\u00e4isen ylpe\u00e4 h\u00e4nest\u00e4.\n\n\"Moi!\" kuului yleis\u00f6st\u00e4. Hurrauksesta p\u00e4\u00e4tellen yleis\u00f6 oli ainakin toistaiseksi hyv\u00e4ll\u00e4 p\u00e4\u00e4ll\u00e4.\n\nKurkistin verhon takaa juuri, kun Mitja oli istuutumassa pianon taakse. \"Olette ehk\u00e4 jo tajunneet, ett\u00e4 tyt\u00f6t ovat my\u00f6h\u00e4ss\u00e4\", h\u00e4n sanoi mikkiin. Yleis\u00f6ss\u00e4 h\u00f6r\u00e4hdeltiin. \"Ajattelin, ett\u00e4 odottaessamme soittaisin teille v\u00e4h\u00e4n pianoa.\"\n\nH\u00e4n alkoi soittaa jotain klassista kappaletta, jonka tiesin kuulleeni aikaisemmin, mutta jonka s\u00e4velt\u00e4j\u00e4st\u00e4 minulla ei ollut mit\u00e4\u00e4n tietoa. Ep\u00e4ilin, ettei Mitjallakaan ollut. H\u00e4n oli luultavasti kuullut sen joskus hississ\u00e4, kauppakeskuksessa tai puhelimessa jonottaessaan asiakaspalvelijan vapautumista.\n\nKesti hetken, ett\u00e4 silm\u00e4ni tottuivat lavan kirkkaisiin valoihin. V\u00e4hitellen katseeni tarkentui pime\u00e4\u00e4n katsomoon ja n\u00e4in, ett\u00e4 tupa oli t\u00e4ynn\u00e4. Yritin l\u00f6yt\u00e4\u00e4 yleis\u00f6n joukosta tuttuja kasvoja, ja lopulta erotin Nikon ja Tanelin. He seisoivat l\u00e4hekk\u00e4in, ja n\u00e4ky l\u00e4mmitti minua suuresti. Taneli oli ensimm\u00e4inen ihminen, joka oli koskaan saanut luvan tulla klubille.\n\nEn tiennyt yksityiskohtia vaiheista, joita Taneli oli k\u00e4ynyt l\u00e4pi kuultuaan Nikon yliluonnollisuudesta, mutta arvelin ettei asian hyv\u00e4ksyminen ollut k\u00e4ynyt aivan niin mutkattomasti kuin p\u00e4\u00e4llep\u00e4in n\u00e4ytti. H\u00e4n ei kuitenkaan ollut miss\u00e4\u00e4n vaiheessa l\u00e4htenyt tilannetta pakoon, ja se oli p\u00e4\u00e4asia.\n\nYleis\u00f6 oli hiljentynyt kuuntelemaan. Mitjan soitto oli aina hypnoottista, soitti h\u00e4n sitten mit\u00e4 hyv\u00e4ns\u00e4. Ja h\u00e4n oli tuskin edes p\u00e4\u00e4ssyt l\u00e4mmittelyn alkuun.\n\nVeljeni silm\u00e4t olivat kiinni. Tuttu euforia teki tiens\u00e4 h\u00e4nen kasvoilleen. Euforia, jota min\u00e4 en saisi en\u00e4\u00e4 koskaan tuntea. Minulle taide oli peruuttamattomasti muuttunut.\n\nOlin kuitenkin alkanut v\u00e4hitellen k\u00e4\u00e4nt\u00e4\u00e4 tilanteen voitokseni. Ratkaiseva oivallus oli ollut se, ettei minun tarvinnut peitell\u00e4 takapakkiani. Hetken mielijohteesta olin alkanut tallentaa toipumistani kuviksi ja videoksi, ja lopulta loputtomista harjoituksista, toivon ja ep\u00e4toivon hetkist\u00e4 oli kasvanut koko taiteellisen urani odottamattomin, kipein ja rakkain teos.\n\nEn ollut j\u00e4tt\u00e4nyt mit\u00e4\u00e4n pois. Olin jopa pyyt\u00e4nyt Tuomoa kuvaamaan, kun tein ensimm\u00e4isen oikean tatuoinnin sitten \"kohtaukseni\". Olin joutunut keksim\u00e4\u00e4n tarinan \u00e4killisest\u00e4 aivoverenvuodosta ja sen j\u00e4lkeisest\u00e4 halvauksesta. Molemmat kollegani olivat olleet j\u00e4rkyttyneit\u00e4 tapahtuneesta kuultuaan. Minun oli ollut pakko tunnustaa heille, etten luultavasti pystyisi en\u00e4\u00e4 koskaan tatuoimaan samalla tasolla kuin ennen ja ett\u00e4 minun oli parempi laittaa pillit pussiin. Ajan kanssa voisin hyv\u00e4ksy\u00e4 sen, ettei taiteeni olisi teknisesti yht\u00e4 loistokasta kuin ennen, mutta tatuoiminen oli toinen juttu. En suostunut ikuistamaan kenenk\u00e4\u00e4n ihoon mit\u00e4\u00e4n puolivillaista. Maailmaan mahtui monta v\u00e4ltt\u00e4v\u00e4\u00e4 tatuointitaiteilijaa, mutta minulla ei ollut aikomustakaan olla yksi heist\u00e4.\n\nSiksi j\u00e4rkytykseni oli ollut hyvin suuri, kun sek\u00e4 Otso ett\u00e4 Tuomo olivat yksinkertaisesti kielt\u00e4neet minua lopettamasta. Ja mik\u00e4 viel\u00e4 pahempaa, Otso oli vaatinut, ett\u00e4 harjoittelisin h\u00e4neen. H\u00e4n ei ollut hellitt\u00e4nyt ennen kuin olin suostunut. Ja vaikka aluksi en voinut ymm\u00e4rt\u00e4\u00e4, miksi h\u00e4n halusi minun pilaavan ihonsa, minun oli tunnustettava, ett\u00e4 h\u00e4nen sinnikkyytens\u00e4 sai aikaan ainakin yhden asian: tajusin, ett\u00e4 halusin yh\u00e4 tatuoida. Mutta minua pelotti.\n\n\"Jos min\u00e4 s\u00f6ssin t\u00e4m\u00e4n, sinun vaimosi tappaa minut\", sanoin tuolissa istuvalle Otsolle. H\u00e4n oli juuri paljastanut minulle reitens\u00e4. Olin yll\u00e4ttynyt, ett\u00e4 h\u00e4nen kropassaan oli en\u00e4\u00e4 j\u00e4ljell\u00e4 tyhj\u00e4\u00e4 ihoa.\n\n\"T\u00e4m\u00e4 oli Katin idea\", h\u00e4n sanoi. Kati oli h\u00e4nen vaimonsa.\n\nKamera oli p\u00e4\u00e4ll\u00e4, mutta tuskin en\u00e4\u00e4 edes huomasin sit\u00e4. Rypistin Otsolle kulmiani, ja h\u00e4n h\u00f6r\u00e4hti. Otso h\u00f6r\u00e4hteli hyvin harvoin. \"Te ette ole koskaan tavanneet, mutta vaimoni on vahvasti sit\u00e4 mielt\u00e4, ett\u00e4 sin\u00e4 olet pelastanut meid\u00e4n avioliittomme. Ne kerrat, kun hoputit minua l\u00e4htem\u00e4\u00e4n kotiin? No, sanotaanko, ett\u00e4 se on auttanut.\"\n\nEn tiennyt, mit\u00e4 sanoa.\n\n\"Kun sin\u00e4 aloitit meill\u00e4, minulla ja Tuomolla oli vakaa aikomus kouluttaa sinusta hyv\u00e4 artisti. Hyvin nopeasti tuli selv\u00e4ksi, ettet sin\u00e4 tarvinnut meit\u00e4. Nyt sin\u00e4 sitten vihdoin tarvitset. Me emme p\u00e4\u00e4st\u00e4 sinua yht\u00e4\u00e4n mihink\u00e4\u00e4n \u2013 ihan sama, vaikka joudumme aloittamaan aivan alusta. Kaikki muutkin ovat joutuneet aloittamaan alusta. Suoraan sanottuna olen iloinen, ett\u00e4 saat kokea sen nyt. Se on nimitt\u00e4in paljon enemm\u00e4n kuin pelkk\u00e4\u00e4 teknisen taidon oppimista.\"\n\nH\u00e4n selvitti kurkkuaan. \"Mit\u00e4 tatuointiin tulee, se on kaikkein rakkain kuvani, tikkasit sen sitten vaikka k\u00e4sin sentin syvyyteen.\"\n\nSilm\u00e4ni kostuivat, ja pian kyyneleet tulvivat yli. Ennen kuin huomasinkaan, kaksi isoa miest\u00e4 oli ottanut minut halaukseen. Se sai minut nauramaan niin, ett\u00e4 meni vartti ennen kuin pystyin aloittamaan ty\u00f6n.\n\nTatuoinnista ei tullut t\u00e4ydellinen, mutta riitt\u00e4v\u00e4n hyv\u00e4. Ja kun olin esitellyt projektiani Sirpa Kojolle, h\u00e4n oli toivottanut minut tervetulleeksi maalaustaiteen kursseille sitten, kun tunsin olevani valmis siihen.\n\nSiihen olisi tietenkin viel\u00e4 aikaa. Palautin itseni takaisin t\u00e4h\u00e4n hetkeen ja menin vessan ovelle. Koputin siihen ja sanoin: \"Mitjalla on t\u00e4n\u00e4\u00e4n loistava p\u00e4iv\u00e4. Oletko jo valmis?\"\n\n\"En\", Caoimhen \u00e4\u00e4ni tunnusti. \"Pahoin pelk\u00e4\u00e4n, etten tule olemaankaan. Olen viel\u00e4kin aivan ylikierroksilla. Mit\u00e4 enemm\u00e4n yrit\u00e4n rauhoittua, sit\u00e4 mahdottomampaa se on.\"\n\n\"Siin\u00e4 tapauksessa meid\u00e4n pit\u00e4\u00e4 perua t\u00e4m\u00e4. Menen kertomaan Mitjalle.\"\n\nH\u00e4n raotti ovea ja kiskaisi minut hihasta sis\u00e4\u00e4n. Vaikka luulin jo tottuneeni h\u00e4nen ulkon\u00e4k\u00f6\u00f6ns\u00e4, huomasin silti r\u00e4pyttelev\u00e4ni silmi\u00e4ni. \"Ei\", h\u00e4n sanoi.\n\n\"Ciao, meill\u00e4 ei ole muuta vaihtoehtoa.\"\n\nCaoimhe p\u00e4\u00e4sti irti minusta. \"Damn\u00fa air\", h\u00e4n mutisi. \"Min\u00e4 esiinnyn n\u00e4in.\"\n\n\"Oletko varma?\" kysyin. Minulle oli selvinnyt vasta aivan viime aikoina, ett\u00e4 l\u00e4hestulkoon kukaan ei ollut n\u00e4hnyt keijuja heid\u00e4n oikeassa muodossaan. Se oli heid\u00e4n suurin salaisuutensa ja sen paljastamisesta rangaistiin ankarasti.\n\n\"Olen\", h\u00e4n vastasi. H\u00e4nen ilmeens\u00e4 kertoi kuitenkin aivan muuta.\n\nLopulta p\u00e4\u00e4tin uskoa h\u00e4nen sanojaan. Kun l\u00e4hdin vessasta, huomasin Konstan seisovan k\u00e4yt\u00e4v\u00e4ss\u00e4. Menin h\u00e4nen luokseen. \"Sano kaikille, ett\u00e4 kukaan ei saa poistua klubilta ennen kuin puhelimet on tarkistettu kuvien varalta.\"\n\nCaoimhe p\u00e4\u00e4tti juuri sill\u00e4 hetkell\u00e4 tulla esiin vessasta. Konsta oli n\u00e4hnyt luultavasti kaiken, mit\u00e4 t\u00e4ll\u00e4 maailmalla oli tarjota, ja silti h\u00e4nen kulmansa kohosivat. \"Onko h\u00e4n varma?\" h\u00e4nkin kysyi.\n\n\"On\", sanoin. \"Onnistuuko?\"\n\nH\u00e4n ny\u00f6kk\u00e4si.\n\nSuljin silm\u00e4ni. Mieless\u00e4ni raotin verhoa.\n\nMinun ja Mitjan yhteys oli yh\u00e4 tallella. Se ei ollut olemassa, koska olimme daimoneita, vaan se oli olemassa, koska olimme kaksosia. Emme tienneet, mik\u00e4 kohtalon oikku oli saanut sen aikaan \u2013 ehk\u00e4 sama oikku, joka oli pit\u00e4nyt huolen siit\u00e4, ett\u00e4 olimme kasvaneet erill\u00e4\u00e4n, mutta kuitenkin l\u00f6yt\u00e4neet toisemme.\n\nEn ollut viel\u00e4k\u00e4\u00e4n vakuuttunut, ett\u00e4 ennustus olisi pit\u00e4nyt paikkansa tai ett\u00e4 jumalat olivat totta. Halusin ajatella, ett\u00e4 Callista oli vain yksi yliluonnollinen monien joukossa. Mutten voinut v\u00e4itt\u00e4\u00e4, etteik\u00f6 kohtaloissa olisi sittenkin ollut jotain per\u00e4\u00e4. Tuntui nimitt\u00e4in silt\u00e4, ett\u00e4 meill\u00e4 kaikilla oli juuri ne el\u00e4m\u00e4t, jotka meille oli tarkoitettu.\n\nVaraudu myrskyyn, ajattelin veljelleni.\n\nAina, kuului vastaus.\n\nJuuri silloin yksi henkil\u00f6kunnan j\u00e4senist\u00e4 koputti minua olalle.\n\n\"Mikael on ovella\", h\u00e4n sanoi.\n\nNy\u00f6kk\u00e4sin ja otin minulle tarjotun mikrofonin.\n\nCaoimhe veti henke\u00e4 ja suoristi ryhtins\u00e4. Sitten me molemmat astuimme lavalle.\n\nKirkkaat valot sokaisivat minut hetkeksi. Se ei tietenk\u00e4\u00e4n est\u00e4nyt minua kuulemasta kohahdusta, joka seurasi Caoimhen ilmestymist\u00e4. Mitja lopetti soittonsa, mutta kukaan ei kiinnitt\u00e4nyt siihen huomioita. Hiljaisuus laskeutui koko tilaan ja k\u00e4\u00e4riytyi Caoimhen ymp\u00e4rille.\n\nH\u00e4n seisoi kasvot kohti yleis\u00f6\u00e4 pidellen kitaraansa molemmin k\u00e4sin. En voinut olla ihailematta h\u00e4nen rohkeuttaan, vaikka tiesin sen olevan pitk\u00e4lti esityst\u00e4.\n\nSitten joku alkoi taputtaa. Ja toinen. Parissa sekunnissa tyrmistynyt hiljaisuus muuttui suosionosoitusten myrskyksi. Caoimhelle huudettiin, taputettiin, vihellettiin ja t\u00f6mistettiin. H\u00e4n otti sen kaiken vastaan tyyneydell\u00e4. Ensimm\u00e4ist\u00e4 kertaa pystyin n\u00e4kem\u00e4\u00e4n h\u00e4net kuninkaallisena.\n\nH\u00e4n k\u00e4\u00e4ntyi katsomaan pianon takana istuvaa Mitjaa.\n\nMitja hymyili h\u00e4nelle tavalla, joka oli paitsi ihaileva, my\u00f6s jotain paljon enemm\u00e4n. Caoimhe olisi voinut ottaa mink\u00e4 hahmon tahansa, ja Mitja olisi n\u00e4hnyt h\u00e4net sellaisena kuin h\u00e4n oikeasti oli.\n\n\"Raisa!\"\n\nLakkasin vihdoin tuijottamasta Caoimhea ja k\u00e4\u00e4nnyin yleis\u00f6n puoleen. H\u00e4mm\u00e4stytt\u00e4v\u00e4n moni silm\u00e4pari oli kiinnittynyt minuun. V\u00e4hitellen aloin k\u00e4sitt\u00e4\u00e4, etteiv\u00e4t he hurranneet pelk\u00e4st\u00e4\u00e4n Ciaolle. He hurrasivat my\u00f6s minulle.\n\nT\u00e4m\u00e4 oli ensimm\u00e4inen kerta, kun n\u00e4ytt\u00e4ydyin julkisesti sitten \"onnettomuuden\". Tiesin, ett\u00e4 sudet olivat olleet huolissaan, mutta olin ajatellut, ett\u00e4 he olivat jo ehtineet unohtaa minut sen j\u00e4lkeen kun sana toipumisestani oli alkanut levit\u00e4.\n\n\"Hei, \u00e4lk\u00e4\u00e4 nyt\", sanoin mikrofoniin yritt\u00e4en parhaani mukaan peitt\u00e4\u00e4 liikutukseni. \u00c4\u00e4neni tuskin kuului metelin yli.\n\nOdotin hetken, ett\u00e4 yleis\u00f6 rauhoittuisi hieman. Kun niin ei tapahtunut, kohotin etusormen huulilleni. Muutamassa hetkess\u00e4 desibelit laskivat, kunnes hiljaisuus oli niin t\u00e4ydellinen, ett\u00e4 se t\u00e4ytti huoneen.\n\n\"Kiitos. Meist\u00e4kin on kiva n\u00e4hd\u00e4 teit\u00e4\", sanoin hymyillen. Kuulin naurahtelua, ja Caoimhe hymyili minulle. \"Tied\u00e4n, ett\u00e4 te tulitte t\u00e4nne ennen kaikkea loistavan musiikin takia\", aloitin. \"Sit\u00e4 ennen pyyt\u00e4isin kuitenkin pient\u00e4 palvelusta.\" Odotin taas hetken, jotta sanani ehtisiv\u00e4t rekister\u00f6ity\u00e4. \"T\u00e4n\u00e4\u00e4n on Mikaelin syntym\u00e4p\u00e4iv\u00e4\", kerroin heille. \"H\u00e4n t\u00e4ytt\u00e4\u00e4 kaksikymment\u00e4. Kun h\u00e4n hetken p\u00e4\u00e4st\u00e4 tulee t\u00e4nne, ajattelin, ett\u00e4 voisimme yll\u00e4tt\u00e4\u00e4 h\u00e4net yhdess\u00e4.\"\n\nReaktiosta p\u00e4\u00e4tellen kaikki olivat mukana. K\u00e4vimme nopeasti l\u00e4pi suunnitelman: b\u00e4ndi soittaisi ja kaikki muut laulaisivat. Baarimikot toivat kakun esiin, ja odottava tunnelma valtasi salin.\n\nAjoitus ei olisi voinut olla parempi. Ovi salin per\u00e4ll\u00e4 aukesi, ja Mikael astui sis\u00e4\u00e4n. H\u00e4nen katseensa ehti pyyhk\u00e4ist\u00e4 nopeasti edess\u00e4 seisovaa ihmismassa ennen kuin h\u00e4n n\u00e4ki minut.\n\nH\u00e4n pys\u00e4htyi, ja pys\u00e4ytti samalla hetkeksi syd\u00e4meni. Sitten se l\u00e4hti villiin, riemukkaaseen laukkaan.\n\nMitja alkoi soittaa pianoa. Muu b\u00e4ndi liittyi mukaan pian mukaan. Kaikki muut ymp\u00e4rill\u00e4ni alkoivat laulaa, mutten saanut suutani auki.\n\nMe kaksi vain tuijotimme toisiamme samalla kun koko t\u00e4yteen ahdettu sali lauloi Paljon onnea vaan.\n\nT\u00e4st\u00e4 illasta tulee el\u00e4m\u00e4ni paras, ajattelin.\n\n***\n\nSeuraavana iltana min\u00e4, Mikael, veljeni ja Caoimhe seisoimme lentokent\u00e4n l\u00e4ht\u00f6aulassa. Tuijotin suurta taulua, josta n\u00e4kyiv\u00e4t seuraavaksi l\u00e4htev\u00e4t lennot: Riika, M\u00fcnchen, Jyv\u00e4skyl\u00e4, Istanbul. Dublinia ei viel\u00e4 n\u00e4kynyt listassa, mutta se ilmestyisi sinne pian.\n\nSiirsin katseeni ymp\u00e4rill\u00e4mme oleviin ihmisiin. Ohitsemme meni matkalaukkuk\u00e4rryj\u00e4 ty\u00f6nt\u00e4v\u00e4 \u00e4\u00e4nek\u00e4s perhe, mutta muuten kent\u00e4ll\u00e4 oli rauhallista. Olin siit\u00e4 iloinen. Se antoi meille muutaman hetken lis\u00e4\u00e4 yhteist\u00e4 aikaa.\n\nVeljeni ja Ciao olivat jo j\u00e4tt\u00e4neet matkatavaransa. Niit\u00e4 ei ollut paljon. Mitja ei ollut ottanut mukaan muuta kuin rinkallisen vaatteita ja lempikitaransa. Olin luvannut l\u00e4hett\u00e4\u00e4 tavaroita per\u00e4st\u00e4 sitten, kun he tiet\u00e4isiv\u00e4t, miss\u00e4 asuisivat.\n\nTaulu p\u00e4ivittyi, ja Dublinin lento ilmestyi listan alimmaiseksi. Huokaisin.\n\nMikael ja Caoimhe siirtyiv\u00e4t hienotunteisesti kauemmaksi minusta ja Mitjasta. En ollut antanut itseni ajatella veljeni l\u00e4ht\u00f6\u00e4, ja nyt kun se oli k\u00e4sill\u00e4, en tiennyt, mit\u00e4 sanoa. Kaikki kuulosti mieless\u00e4ni aivan liian laimealta. H\u00e4nest\u00e4 oli tullut minulle niin hurjan t\u00e4rke\u00e4 lyhyess\u00e4 ajassa.\n\n\"Kumpikaan ei sitten itke\", Mitja varoitti.\n\nVedin henke\u00e4. \"No niin\", aloitin. \"En halua sinun ajattelevan, ettenk\u00f6 uskoisi sinuun, sill\u00e4 min\u00e4 todellakin uskon. Mutta samaan aikaan haluan sinun tiet\u00e4v\u00e4n, ett\u00e4 miss\u00e4 min\u00e4 olen, siell\u00e4 on my\u00f6s sinun toinen kotisi. Sin\u00e4 voit aina palata.\"\n\nSanani olivat heng\u00e4styneit\u00e4 ja oudon muodollisia, mutta veljeni ilme pysyi vakavana. H\u00e4n ny\u00f6kk\u00e4si, ja sitten jo rutistin h\u00e4nt\u00e4 niin lujaa kuin nykyisin pystyin.\n\nTajusin h\u00e4m\u00e4r\u00e4sti, ett\u00e4 se oli ensimm\u00e4inen kerta, kun halasin h\u00e4nt\u00e4, mutta se ei tuntunut lainkaan oudolta. Mitja nimitt\u00e4in rutisti takaisin.\n\n\"Pid\u00e4 huolta itsest\u00e4si, pikkuveli\", sanoin.\n\nH\u00e4n naurahti. \"Et taida koskaan lakata muistuttamasta minua siit\u00e4, ett\u00e4 olet vanhempi.\"\n\nP\u00e4\u00e4stin irti ja pudistin hymyillen p\u00e4\u00e4t\u00e4ni.\n\nMin\u00e4 ja Caoimhe halasimme samaan aikaan kun Mikael ja Mitja vaihtoivat muutaman sanan. Yll\u00e4tyksekseni hekin halasivat pikaisesti ennen kuin veljeni tarttui kihlattuaan k\u00e4dest\u00e4 ja he l\u00e4htiv\u00e4t astelemaan kohti turvatarkastusta.\n\nKatsoin, kun he asettelivat tavaransa muovitarjottimille ja suuntasivat l\u00e4pivalaisuun. Odotin niin kauan, ett\u00e4 heid\u00e4n selk\u00e4ns\u00e4 katosivat n\u00e4kyvist\u00e4mme. Vasta silloin annoin kyynelten tulla.\n\nMikael kietoi k\u00e4sivartensa ymp\u00e4rilleni ja hautasin kasvoni h\u00e4nen rintaansa vasten.\n\nPystyin aistimaan h\u00e4nen huolensa, kun k\u00e4velimme autolle. H\u00e4n k\u00e4ynnisti auton moottorin ja kurvasi pois lentokent\u00e4n parkkipaikalta.\n\nIlma tuoksui odottavalta sateelta. Moottorin \u00e4\u00e4ni muuttui aavistuksen, kun Mikael vaihtoi vaihdetta pyk\u00e4l\u00e4n suurempaan.\n\nOli jo tullut niin pime\u00e4\u00e4, ett\u00e4 kaupunki oli pelkki\u00e4 ohi vilahtelevia valoja. Mikael vilkuili suuntaani aina kun luuli, etten huomaisi. H\u00e4n oli ollut hyvin huolehtivainen, jopa suojeleva, mutta ymm\u00e4rsin h\u00e4nt\u00e4. Menisi viel\u00e4 hetki, ennen kuin h\u00e4n luottaisi taas t\u00e4ysin siihen, ett\u00e4 olin kunnossa enk\u00e4 katoaisi minnek\u00e4\u00e4n.\n\nHalusin sanoa h\u00e4nelle, ett\u00e4 kyyneleeni olivat hyvi\u00e4 kyyneleit\u00e4, ja ett\u00e4 vaikka minun tulisi ik\u00e4v\u00e4, tiet\u00e4isin, ett\u00e4 veljeni teki hienoja asioita. Aivan kuten Niko, Taneli ja min\u00e4. Maailma oli pienempi kuin koskaan. Miksi ihmeess\u00e4 surisin sit\u00e4, ett\u00e4 veljeni ja yst\u00e4v\u00e4ni saivat n\u00e4hd\u00e4 sen? Eiv\u00e4tk\u00e4 vain n\u00e4hd\u00e4, vaan valloittaa.\n\n\"Sin\u00e4 voit menn\u00e4 tapaamaan h\u00e4nt\u00e4 heti, kun he ovat l\u00f6yt\u00e4neet asunnon ja ehtineet asettua aloilleen.\" Kun en vastannut heti, h\u00e4n jatkoi: \"Itse asiassa luulen, ett\u00e4 pist\u00e4n sinut koneeseen jo ennen sit\u00e4.\"\n\nNaurahdin. \"Haluatko p\u00e4\u00e4st\u00e4 minusta eroon n\u00e4in pian?\"\n\n\"Nyt kun otit asian puheeksi, niin kyll\u00e4. Kaikki oli paljon yksinkertaisempaa silloin, kun et voinut puhua.\"\n\nNauroin, ja h\u00e4n pudisti p\u00e4\u00e4t\u00e4\u00e4n. \"Vaikea uskoa, ett\u00e4 sait minut juuri vitsailemaan asiasta.\"\n\n\"Meid\u00e4n kuuluukin vitsailla siit\u00e4.\" Joskus aikaisemmin en varmaan olisi uskaltanut. Silloin olin viel\u00e4 ep\u00e4illyt h\u00e4nen rakkauttaan. Jos minun melkein-kuolemisestani oli seurannut jotain hyv\u00e4\u00e4 niin ennen kaikkea se, etten en\u00e4\u00e4 pel\u00e4nnyt meid\u00e4n puolestamme. Sanottiin, ett\u00e4 ihmisill\u00e4 on el\u00e4m\u00e4ns\u00e4 aikana keskim\u00e4\u00e4rin kolme rakkautta, mutta tunsin itseni etuoikeutetuksi saadessani k\u00e4ytt\u00e4\u00e4 kaiken rakkauteni h\u00e4neen.\n\nMikael oli muuttanut minua. Ja vaikka olisi voinut tuntua kauhistuttavalta antaa toiselle sellainen valta \u2013 suurin valta, mit\u00e4 oli olemassa \u2013 se ei pelottanut minua. Sill\u00e4 Mikael ei koskaan muuttaisi minua joksikin, jota en halunnut olla. H\u00e4n saattoi tehd\u00e4 minusta vain paremman, vahvemman.\n\nKaivoin hansikaslokeroa kunnes l\u00f6ysin nen\u00e4liinapaketin. Niistin nen\u00e4ni.\n\n\"Minun pit\u00e4\u00e4 pyyt\u00e4\u00e4 sinulta palvelusta\", sanoin lopulta.\n\n\"Mit\u00e4 tahansa\", Mikael sanoi heti.\n\n\"Voisitko vied\u00e4 minut yhteen paikkaan?\"\n\n\"Nyt?\" Ei ollut vaikeaa seurata h\u00e4nen ajatuksenjuoksuaan: koska olin juuri itkenyt silm\u00e4ni punaisiksi, oli varmasti vaikea kuvitella, ett\u00e4 haluaisin menn\u00e4 minnek\u00e4\u00e4n muualle kuin kotiin. Ja h\u00e4n olikin oikeassa. Sit\u00e4 ennen minun piti kuitenkin tehd\u00e4 jotain t\u00e4rke\u00e4\u00e4.\n\n\"Jos se vain sopii.\"\n\n\"Tietenkin. Kerro vain minne.\"\n\nOhjeistin h\u00e4net ajamaan seuraavasta liittym\u00e4st\u00e4 it\u00e4\u00e4n. Mikael ei kysynyt, minne olimme menossa, mutta h\u00e4nen huolensa leijui pilven\u00e4 h\u00e4nen ymp\u00e4rill\u00e4\u00e4n. Jokaisen seuraavan reittiohjeeni t\u00e4ytyi saada h\u00e4net entist\u00e4kin h\u00e4mmentyneemm\u00e4ksi, mutta h\u00e4n onnistui peitt\u00e4m\u00e4\u00e4n sen varsin hyvin. Mit\u00e4 kauemmin ajoimme, sit\u00e4 enemm\u00e4n nimitt\u00e4in n\u00e4ytti silt\u00e4, ettemme olleet menossa minnek\u00e4\u00e4n.\n\nLopulta tulimme asutuksen reunalle. Havupuut kasvoivat molemmin puolin yksin\u00e4ist\u00e4 hiekkatiet\u00e4, jota auton valot nieliv\u00e4t edell\u00e4mme. Aivan kuin pimeys olisi juossut meit\u00e4 pakoon.\n\n\"Voit pys\u00e4hty\u00e4 johonkin n\u00e4ille main.\"\n\nMikael teki ty\u00f6t\u00e4 k\u00e4sketty\u00e4 ja ohjasi auton penkereelle. H\u00e4n veti k\u00e4sijarrun p\u00e4\u00e4lle, muttei sammuttanut valoja. Odotin h\u00e4nen murjaisevan jonkun vitsin siit\u00e4, miten t\u00e4m\u00e4 oli t\u00e4ydellinen paikka ruumiin h\u00e4vitt\u00e4miseen, mutta h\u00e4n ei tehnyt niin. Sen sijaan h\u00e4n tarkkaili minua p\u00e4\u00e4 aavistuksen kenossa.\n\nAvasin turvavy\u00f6n. Sitten avasin takkini vetoketjun ja kiemurtelin k\u00e4sivarteni ulos hihoista. Otin kiinni niskatuesta ja kohottauduin istuimella polvieni varaan.\n\nEn ollut koskaan aikaisemmin yritt\u00e4nyt ry\u00f6mi\u00e4 vaihdekepin yli, ja pian k\u00e4vi selv\u00e4ksi, ettei se onnistunut aivan yht\u00e4 sulokkaasti kuin mielikuvissani. Lopulta kuitenkin onnistuin kampeamaan itseni hajareisin Mikaelin syliin.\n\nH\u00e4n oli j\u00e4hmettynyt sill\u00e4 sekunnilla, kun oli tajunnut, mit\u00e4 aioin tehd\u00e4. Jos en olisi tuntenut h\u00e4nen pulssiaan sormieni alla, en olisi tiennyt h\u00e4nen olevan elossa.\n\nVedin paitani p\u00e4\u00e4ni yli. Selk\u00e4ni kolahti rattiin, mutta onnistuin olemaan kiroilematta \u00e4\u00e4neen. Tunsin, kuinka Mikael veti syv\u00e4\u00e4n henke\u00e4. Tiesin, ett\u00e4 h\u00e4n saattoi haistaa koko tunteideni kirjon.\n\nH\u00e4n oli sanomaisillaan jotain, mutta keskeytin h\u00e4net painamalla sormeni h\u00e4nen huulilleen. Heilautin hiukseni olkani yli ja kumarruin suutelemaan h\u00e4nt\u00e4.\n\nMikaelin huulet olivat l\u00e4mpim\u00e4t ja maistuivat aavistuksen suolaisilta. Minulta p\u00e4\u00e4si huokaus.\n\nSilloin h\u00e4n tuntui vihdoin her\u00e4\u00e4v\u00e4n eloon. H\u00e4n kietoi k\u00e4sivartensa ymp\u00e4rilleni ja karkotti viimeisetkin sentit v\u00e4lilt\u00e4mme. H\u00e4nen toinen k\u00e4mmenens\u00e4 tuntui kuumalta keskell\u00e4 paljasta selk\u00e4\u00e4ni. Koko ihoni oli pian kananlihalla.\n\nJatkoin suutelemista, kunnes henki oli loppua kesken ja auto alkoi tuntua aivan liian pienelt\u00e4 tilalta kaikille niille asioille, jotka halusin tehd\u00e4.\n\nKun vet\u00e4ydyin kauemmas, n\u00e4in, miten Mikaelin aataminomena kohosi ja laski. H\u00e4nen silm\u00e4ns\u00e4 olivat kaksi luolaa, joiden syvyytt\u00e4 saattoi vain arvailla. Huulet olivat aavistuksen turvonneet ja h\u00e4nen poskilleen oli ilmestynyt kaksi punaista l\u00e4isk\u00e4\u00e4. Olin varma, ett\u00e4 omat kasvoni olivat paljon punaisemmat.\n\nSeuraavaksi suutelin h\u00e4nen kaulaansa. Tein sen piinaavan hitaasti, ja samalla kurotin jotain pelk\u00e4\u00e4j\u00e4npaikalta.\n\n\"Mi...!\"\n\nH\u00e4n ei koskaan p\u00e4\u00e4ssyt sanan loppuun. Olin nimitt\u00e4in juuri viilt\u00e4nyt h\u00e4nen kaulaansa veitsell\u00e4 haavan. Ennen kuin h\u00e4n ehti tehd\u00e4 tai sanoa mit\u00e4\u00e4n, painoin huuleni h\u00e4nen iholleen ja imin.\n\nHeti, kun h\u00e4nen verens\u00e4 maku ehti rekister\u00f6ity\u00e4 aivoissani, tapahtui jotain muutakin. L\u00e4mmin, pisteli\u00e4s tunne tunkeutui suoniini ja kiipesi pitkin hermoj\u00e4rjestelm\u00e4\u00e4ni. Sit\u00e4 ei voinut varsinaisesti sanoa kivuksi, mutta tunne oli eritt\u00e4in ep\u00e4miellytt\u00e4v\u00e4 \u2013 vartaloni reagoi siihen t\u00e4risem\u00e4ll\u00e4 holtittomasti. Purin hampaani yhteen ja painoin otsani Mikaelin olalle.\n\nLopulta v\u00e4ristykset lakkasivat. Ne eiv\u00e4t olleet voineet kest\u00e4\u00e4 muutamaa sekuntia kauempaa, mutta tiesin, ett\u00e4 kaikki oli muuttunut peruuttamattomasti. Kaaos oli v\u00e4istynyt. Suljin silm\u00e4ni. Tunsin h\u00e4net kaikkialla. H\u00e4n oli minun ja min\u00e4 h\u00e4nen. Universumin ainoa oikea j\u00e4rjestys.\n\nMikaelin kasvot olivat silkkaa kauhua. Ne tuntuivat kysyv\u00e4n: Mit\u00e4 sin\u00e4 olet mennyt tekem\u00e4\u00e4n?\n\nOtsani l\u00f6ysi h\u00e4nen otsansa. \"Se oli minun p\u00e4\u00e4t\u00f6kseni, ei sinun.\"\n\nSuutelin h\u00e4nt\u00e4 viel\u00e4 viimeisen kerran ennen kuin kurotin kohti ovea.\n\nAvasin oven ja liu'uttauduin Mikaelin sylist\u00e4 maahan. Otin muutaman horjuvan askeleen, kunnes seisoin tien vieress\u00e4.\n\nMikaelin kasvot olivat t\u00e4ynn\u00e4 kauhua. Ja ihmetyst\u00e4. Ja rakkautta \u2013 miten helppoa se olikaan n\u00e4hd\u00e4 kun tiesin, mit\u00e4 etsi\u00e4.\n\nH\u00e4nen katseensa seurasi jokaista liikett\u00e4ni, kun aloin riisua kenki\u00e4ni. H\u00e4n ei edes r\u00e4pytellyt \u2013 vaikutti silt\u00e4, ett\u00e4 h\u00e4n oli unohtanut, miten niin tehtiin. Tunsin h\u00e4net ihollani v\u00e4h\u00e4n samaan tapaan kuin tunsin tuulen.\n\nRiisuin ensin oikean sukkani, sitten vasemman. Jouduin hyp\u00e4htelem\u00e4\u00e4n yhdell\u00e4 jalalla kerran pari, mutta sain kuin sainkin tasapainoni hallintaan.\n\nKun Mikael oli her\u00e4tt\u00e4nyt suteni ensimm\u00e4isen kerran, kumpikaan meist\u00e4 ei olisi osannut arvata sen seurauksia. H\u00e4nen tarkoituksenaan oli ollut parantaa minut, ja juuri niin suteni oli tehnyt minulle. Minulla oli vain hyvin et\u00e4isi\u00e4 muistikuvia unista, joita olin n\u00e4hnyt niiden kahden vuorokauden aikana, jotka olin ollut tiedottomassa tilassa. Niiss\u00e4 juoksin hajun per\u00e4ss\u00e4. Mets\u00e4 vilahteli molemmilla puolillani. Ei ollut v\u00e4rej\u00e4, vain liike ja vainu.\n\nJa keveys. Jokaisessa lihaksessa, jokaisessa j\u00e4nteess\u00e4. Ei ollut ajatusta, vaan pelkk\u00e4 ajo, vaisto, vimma.\n\nEmme olleet puhuneet asiasta. Olimme olleet niin keskittyneit\u00e4 saamaan minut siihen pisteeseen, ett\u00e4 voisin jatkaa el\u00e4m\u00e4\u00e4ni, ettei kumpikaan ollut halunnut ajatella asioita, jotka olivat muuttuneet pysyv\u00e4sti. Olin menett\u00e4nyt niin paljon. Mutta olin my\u00f6s saanut jotain. Jotain yll\u00e4tt\u00e4v\u00e4\u00e4, jonka loppujen lopuksi ei olisi pit\u00e4nyt yll\u00e4tt\u00e4\u00e4 en\u00e4\u00e4 ket\u00e4\u00e4n.\n\nOlin aika varma, ett\u00e4 strippaukseni sai minut n\u00e4ytt\u00e4m\u00e4\u00e4n ennen kaikkea huvittavalta eik\u00e4 lainkaan vieh\u00e4tt\u00e4v\u00e4lt\u00e4, mutta en antanut sen h\u00e4irit\u00e4 itse\u00e4ni. Minulla oli yll\u00e4ni en\u00e4\u00e4 farkut, alushousut ja rintaliivit.\n\nJokin pyrki minusta ulos. Tunsin sen. Olin tuntenut aina. Vasta aivan viime p\u00e4ivin\u00e4 olin aavistanut, mit\u00e4 se tarkoitti.\n\nKurotin selk\u00e4ni taakse ja avasin hakaset. Annoin olkaimien valua olkavarsiani pitkin kunnes pujotin k\u00e4teni ulos niist\u00e4. Heitin rintsikat mets\u00e4\u00e4n.\n\n\"Aiotko vain istua siell\u00e4 ja tuijottaa?\" kysyin.\n\nMikaelin reaktiosta p\u00e4\u00e4tellen juuri niin h\u00e4n aikoi tehd\u00e4.\n\n\"No, kai min\u00e4 menen sitten yksin.\"\n\nAvasin farkkujeni vetoketjun ja kiskaisin ne ja alushousut yhdell\u00e4 ja samalla liikkeell\u00e4 alas. Nilkoissani olevia housuja lukuun ottamatta olin t\u00e4ysin alasti.\n\n\"Ota kiinni jos saat\", sanoin.\n\nJa muutin muotoa.\n\nSusiraja on ollut huikea matka. Toteutunut unelma ja samalla hauskin mahdollinen kirjoittajakoulu. Se ei kuitenkaan olisi ollut mahdollinen ilman useiden ihmisten apua. Haluaisin kiitt\u00e4\u00e4 seuraavia:\n\n  * Kustantamon ja agentuurin ihmiset. Kustannustoimittajani Katariina Heilala, joka on paitsi perannut virkkeit\u00e4, my\u00f6s valanut uskoa. Laura Andersson, jota ilman Susirajaa ei olisi. Kiitos my\u00f6s my\u00f6hemm\u00e4st\u00e4 luottamuksesta! Saara Tiuraniemi, jota l\u00e4mpim\u00e4mp\u00e4\u00e4 ja kannustavampaa kustantajaa ei voisi olla. Eevaliina Rusanen, joka loi sarjailmeen, ja Laura Lyytinen, joka teki t\u00e4m\u00e4n kirjan kannen. Elina Ahlb\u00e4ck Literary Agencyn v\u00e4ki, jonka ansiosta Susirajaa luetaan toisella puolella maailmaa. Suomen Kirjailijaliitto, jonka residenssiss\u00e4 Ateenassa aloitin t\u00e4m\u00e4n kirjan kirjoittamisen.\n  * Kainuun riistakeskuksen Marko Paasimaa, joka vastasi sudenmets\u00e4stykseen liittyviin kysymyksiini.\n  * Perheeni. Erityisesti \u00e4itini, joka vuosia sitten k\u00e4ski minua ihan pokkana kirjoittamaan kirjan. \n  * Ihanat yst\u00e4v\u00e4ni. Salla, joka ei anna ostaa polyesterivaatteita. Mari ja Mona, jotka ovat tunteneet minut aina ja uskaltaneet udella juonipaljastuksia. Marcus, my IT expert. Keijuyst\u00e4v\u00e4ni Sohvi. Kollegat pieness\u00e4 liilassa kirjakaupassa (Tack ska ni ha!) ja kirjailijan kammioissa. Annukka Salama, joka paitsi luki, my\u00f6s antoi vertaistukea. T\u00e4m\u00e4 on ihan m\u00e4\u00e4dnessi\u00e4! Ja Katja, joka on lukenut jokaisen osan ennen muita ja keskustellut idiomeista kahdelta aamulla. Muistatko kun sanoit, ett\u00e4 unelmani tulee pakostikin toteutumaan jos en vain koskaan luovuta? No, k\u00e4vi sitten n\u00e4in hassusti. \n  * Lopuksi viel\u00e4 kiitokset kaikille teille, jotka olette l\u00e4hett\u00e4neet minulle fanipostia, bloganneet, tyk\u00e4nneet, kommentoineet, jakaneet ja \/ tai tviitanneet. \n  * Ja ennen kaikkea: kiitos kaikki, jotka olette lukeneet. Te teitte t\u00e4st\u00e4 totta.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nE-text prepared by Josep Cols Canals, Sam W., and the Online Distributed\nProofreading Team (http:\/\/www.pgdp.net) from page images generously made\navailable by Internet Archive\/Canadian Libraries\n(https:\/\/archive.org\/details\/toronto)\n\n\n\nNote: Images of the original pages are available through\n      Internet Archive\/Canadian Libraries. See\n      https:\/\/archive.org\/details\/trafalgartale00pr\n\n\n\n\n\nTRAFALGAR\n\nA Tale\n\nby\n\nB. PEREZ GALDOS\n\nAuthor of \"Gloria,\" etc.\n\nFrom the Spanish by Clara Bell\n\nRevised and Corrected in the United States\n\n\n\n\n\n\n\nNew York\nWilliam S. Gottsberger, Publisher\n11 Murray Street\n1884\n\nEntered according to Act of Congress, in the year 1884\nby William S. Gottsberger\nin the Office of the Librarian of Congress, at Washington\n\nThis Translation Was Made Expressly for the Publisher\n\nPress of\nWilliam S. Gottsberger\nNew York\n\n\n\n\nCONTENTS.\n\n\n                              PAGE\n    CHAPTER I.                   1\n\n    CHAPTER II.                 14\n\n    CHAPTER III.                22\n\n    CHAPTER IV.                 29\n\n    CHAPTER V.                  50\n\n    CHAPTER VI.                 63\n\n    CHAPTER VII.                77\n\n    CHAPTER VIII.               88\n\n    CHAPTER IX.                111\n\n    CHAPTER X.                 126\n\n    CHAPTER XI.                137\n\n    CHAPTER XII.               154\n\n    CHAPTER XIII.              174\n\n    CHAPTER XIV.               192\n\n    CHAPTER XV.                207\n\n    CHAPTER XVI.               231\n\n    CHAPTER XVII.              244\n\n\n\n\nTRAFALGAR.\n\n\n\n\nCHAPTER I.\n\n\nI trust that, before relating the important events of which I have\nbeen an eye-witness, I may be allowed to say a few words about my\nearly life and to explain the singular accidents and circumstances\nwhich resulted in my being present at our great naval catastrophe.\n\nIn speaking of my birth I cannot follow the example of most writers\nwho narrate the facts of their own lives, and who begin by naming\ntheir ancestry--usually of noble rank, _hidalgos_ at the very least,\nif not actually descended from some royal or imperial progenitor. I\ncannot grace my opening page with high-sounding names, for, excepting\nmy mother whom I remember for some few years, I know nothing of any of\nmy forefathers, unless it be Adam from whom my descent would seem to\nbe indisputable. In short, my history began in much the same way as\nthat of Pablos, the brigand of Segovia; happily it pleased God that it\nshould resemble it in no other particular.\n\nI was born at Cadiz in the notorious quarter \"de la Vi\u00f1a,\" which was\nnot then, any more than at the present day, a good school of either\nmorals or manners. My memory does not throw any light on the events of\nmy infancy till I was six years old, and I remember that, only because\nI associate the idea of being six with an event I heard much talked\nabout, the battle of Cape St. Vincent, which took place in 1797.\n\nEndeavoring to see myself as I was at that time, with the curiosity\nand interest which must attach to self-contemplation, I am aware of a\ndim and hazy little figure in the picture of past events, playing in\nthe creek with other small boys of the same age, more or less. This\nwas to me the whole of life--as it was, at any rate, to our privileged\nclass; those who did not live as I did appeared to me exceptional\nbeings. In my childish ignorance of the world I firmly believed that\nman was made for the sea, Providence having created him to swim as\nbeing the noblest exercise of his limbs and body, and to dive for\ncrabs as the highest use of his intelligence--and especially to fish\nup and sell the highly-esteemed crustacean known as _Bocas de la\nIsla_--as well as for his personal delectation and enjoyment, thus\ncombining pleasure with profit.\n\nThe society into which I was born was indeed of the roughest, as\nignorant and squalid as can well be imagined; so much so that the boys\nof our quarter of the town were regarded as even lower than those of\nthe adjoining suburb of Puntales, whose occupations were the same and\nwho defied the elements with equal devilry; the result of this\ninvidious distinction was that each party looked upon the other as\nrivals, and the opposing forces would meet from time to time for a\npitched battle with stones, when the earth was stained with heroic\nblood.\n\nWhen I was old enough to begin to think that I might go into business\non my own account, with a view to turning an honest penny, I remember\nthat my sharpness stood me in good stead on the quay where I acted as\n_valet de place_ to the numerous English who then, as now, disembarked\nthere. The quay was a free academy peculiarly fitted to sharpen the\nwits and make the learner wide-awake, and I was not one of the least\napt of its disciples in that wide branch of human experience; nor did\nI fail to distinguish myself in petty thefts, especially of fruit, an\nart for which the Plaza de San Juan offered an ample field, both for\nthe experiments of the beginner and the exploits of the adept. But I\nhave no wish to enlarge on this part of my history, for I blush with\nshame now, as I remember the depth to which I had sunk, and I thank\nGod for having released me from it at an early period, and directed me\ninto a better path.\n\nAmong the impressions which remain most vivid in my memory is the\nenthusiastic delight I felt at the sight of vessels of war, when they\nanchored outside Cadiz or in the cove of San Fernando. As I had no\nmeans of satisfying my curiosity, when I saw these enormous structures\nI conceived the most absurd and fanciful ideas about them, imagining\nthem as full of mysteries.\n\nAlways eager to mimic the greater world around us, we boys too had our\nsquadrons of little ships, roughly hewn in wood, with sails of paper\nor of rag, which we navigated with the greatest deliberation and\ngravity in the pools of Puntales or La Caleta. To make all complete,\nwhenever a few coppers came into our hands, earned by one or another\nof our small industries, we bought powder of old \"Aunt Coscoja\" in the\nstreet \"del Torno de Santa Mar\u00eda,\" and with this we could have a\ngrand naval display. Our fleets sailed before the wind in an ocean\nthree yards across, fired off their cannon, came alongside of each\nother to mimic a hand-to-hand fight--in which the imaginary crews\nvaliantly held their own, and swarmed into the tops unfurling the\nflag, made of any scrap of  rag we could pick up in a\ndust-heap--while we danced with ecstasy on the shore at the popping of\nthe artillery, imagining ourselves to be the nationalities represented\nby our respective standards, and almost believing that in the world of\ngrown-up men and great events the nations too would leap for joy,\nlooking on at the victories of their splendid fleets. Boys see things\nthrough strange windows.\n\nThose were times of great sea-fights, for there was one at least every\nyear and a skirmish every month. I thought that fleets met in battle\nsimply and solely because they enjoyed it, or to prove their strength\nand valor, like two bullies who meet outside the walls to stick knives\ninto each other. I laugh when I recollect the wild ideas I had about\nthe persons and events of the time. I heard a great deal about\nNapoleon and how do you think I had pictured him to myself! In every\nrespect exactly like the smugglers whom we not unfrequently saw in our\nlow quarter of the town: _Contrabandistas_ from the lines at\nGibraltar. I fancied him a man on horseback, on a Xerez nag, with a\ncloak, high boots, a broad felt-hat, and a blunderbuss of course. With\nthese accoutrements, and followed by other adventurers on the same\npattern, I supposed this man, whom all agreed in describing as most\nextraordinary, to have conquered Europe, which I fancied was a large\nisland within which were other islands which were the different\nnations: England, Genoa, London, France, Malta, the land where the\nMoors lived, America, Gibraltar, Port Mahon, Russia, Toulon and so\nforth. This scheme of geography I had constructed on the basis of the\nnames of the places from which the ships came whose passengers I had\nto deal with; and I need not say that of all these nations or islands\nSpain was the very best, for which reason the English--men after the\nlikeness of highwaymen--wanted to get it for their own. Talking of\nthese and similar matters I and my amphibious companions would give\nvent to sentiments and opinions inspired by the most ardent\npatriotism.\n\nHowever, I need not weary the reader with trifles which relate only to\nmy personal fancies, so I will say no more about myself. The one\nliving soul that made up to me for the wretchedness of life by a\nwholly disinterested love for me, was my mother. All I can remember of\nher is that she was extremely pretty, or at any rate she seemed so to\nme. From the time when she was left a widow she maintained herself and\nme by doing washing, and mending sailors' clothes. She must have loved\nme dearly. I fell ill of yellow fever which was raging in Andalusia\nand when I got well she took me solemnly to mass at the old cathedral\nand made me kneel on the pavement for more than an hour, and then, as\nan _ex-voto_ offering, she placed an image in wax of a child, which I\nbelieved to be an exact likeness of myself, at the foot of the altar\nwhere the service had been performed.\n\nMy mother had a brother, and if she was pretty, he was ugly and a\ncruel wretch into the bargain. I cannot think of my uncle without\nhorror, and from one or two occurrences which I remember vividly I\ninfer that this man must have committed some crime at the time I refer\nto. He was a sailor; when he was on shore and at Cadiz he would come\nhome furiously drunk, and treat us brutally--his sister with words,\ncalling her every abusive name, and me with deeds, beating me without\nany reason whatever.\n\nMy mother must have suffered greatly from her brother's atrocities,\nand these, added to severe labor for miserable pay, hastened her death\nwhich left an indelible impression on my feelings, though the details\ndwell but vaguely in my memory. During this period of misery and\nvagabondage my only occupations were playing by the sea-shore or\nrunning about the streets. My only troubles were a beating from my\nuncle, a frown from my mother, or some mishap in the conduct of my\nsquadrons. I had never felt any really strong or deep emotion till the\nloss of my mother showed me life under a harder and clearer aspect\nthan it had ever before presented to me. The shock it gave me has\nnever faded from my mind. After all these years I still remember, as\nwe remember the horrible pictures of a bad dream, that my mother lay\nprostrate from some sickness, I know not what; I remember women coming\nand going, whose names and purpose I cannot recall; I remember hearing\ncries of lamentation, and being placed in my mother's arms, and then I\nremember the shudder that ran through my whole body at the touch of a\ncold, cold hand. I think I was then taken away; but mixed up with\nthese dim memories I can see the yellow tapers which gave a ghastly\nlight at mid-day, I can hear the muttering of prayers, the hoarse\nwhispers of the old gossips, the laughter of drunken sailors--and\nthen came the lonely sense of orphanhood, the certainty that I was\nalone and abandoned in the world, which for a time absorbed me\nentirely.\n\nI have no recollection of what my uncle was doing at that time; I only\nknow that his brutality to me increased to such a point that, weary of\nhis cruelty, I ran away, determined to seek my fortune. I fled to San\nFernando and from thence to Puerto Real. I hung on to the lowest class\nthat haunt the shore, which has always been a famous nest for\ngaol-birds. Why or wherefore I quite forget, but I found myself with a\ngang of these choice spirits at Medinasidonia when, one day, a tavern\nwhere we were sitting was entered by a press-gang and we promptly\nseparated, each hiding himself as best he might. My good star led me\nto a house where the owners had pity on me, taking the greatest\ninterest in me, no doubt by reason of the story I told, on my knees\nand drowned in tears, of my miserable plight, my past life and all my\nmisfortunes.\n\nThese good people took me under their protection and saved me from the\npress-gang, and from that time I remained in their service. With them\nI went to Vejer de la Frontera where they lived; they had only been\npassing through Medinasidonia.\n\nMy guardian angels were Don Alonso Gutierrez de Cisniega, a ship's\ncaptain, and his wife, both advanced in years. They taught me much\nthat I did not know, and as they took a great fancy to me before long\nI was promoted to be Don Alonso's page, accompanying him in his daily\nwalks, for the worthy veteran could not use his right arm, and it was\nwith difficulty that he moved his right leg. What they saw in me to\narouse their interest I do not know; my tender years, my desolate\ncircumstances and no doubt too my ready obedience may have contributed\nto win their benevolence, for which I have always been deeply\ngrateful. I may also add--though I say it that should not--as\nexplaining their kind feeling towards me, that although I had always\nlived among the lowest and most destitute class, I had a certain\nnatural refinement of mind which enabled me very soon to improve in\nmanners, and in a few years, notwithstanding I had no opportunities\nfor learning, I could pass for a lad of respectable birth and\ntraining.\n\nI had spent four years in this home when the events happened which I\nmust now relate. The reader must not expect an accuracy of detail\nwhich is out of my power when speaking of events which happened in my\ntender youth, to be recalled in the evening of my existence when I am\nnear the end of a long and busy life and already feel the slow poison\nof old age numbing the fingers that use the pen; while the torpid\nbrain strives to cheat itself into transient return of youth, by\nconjuring up the sweet or ardent memories of the past. As some old men\nstrive to revive the warm delights of the past by gazing at pictures\nof the beauties they have known, I will try to give some interest and\nvigor to the faded reminiscences of my long past days, and to warm\nthem with the glow of a counterfeit presentment of departed glories.\n\nThe effect is magical! How marvellous are the illusions of fancy! I\nlook back with curiosity and astonishment at the bygone years, as we\nlook through the pages of a book we were reading, and left with a leaf\nturned down to mark the place; and so long as the charm works I feel\nas if some beneficent genius had suddenly relieved me of the weight of\nold age, mitigating the burden of years which crushes body and spirit\nalike. This blood--this tepid and languid ichor, which now scarcely\nlends warmth and life to my failing limbs, grows hot again, flows,\nboils, and fires my veins with a swifter course. A sudden light\nbreaks in upon my brain, giving color and relief to numberless strange\nfigures--just as the traveller's torch, blazing in some dark cavern,\nreveals the marvels of geology so unexpectedly that it seems as though\nthey were then and there created. And my heart rises from the grave of\npast emotions--a Lazarus called by the voice of its Lord--and leaps in\nmy breast with joy and pain at once.\n\nI am young again; time has turned backwards, I stand in the presence\nof the events of my boyhood; I clasp the hands of old friends, the\njoys and griefs of my youth stir my soul once more--the fever of\ntriumph, the anguish of defeat, intense delights, acute sorrows--all\ncrowded and mixed in my memory as they were in life. But stronger than\nany other feeling one reigns supreme, one which guided all my actions\nduring the fateful period between 1805 and 1834. As I approach the\ngrave and reflect how useless I am among men--even now tears start to\nmy eyes with the sacred love of country. I can only serve it with\nwords--cursing the base scepticism which can deny it, and the corrupt\nphilosophy which can treat it as a mere fashion of a day.\n\nThis was the passion to which I consecrated the vigor of my manhood,\nand to this I will devote the labors of my last years, enthroning it\nas the tutelary genius, the guiding spirit of my story as it has been\nof my existence. I have much to tell. Trafalgar, Bail\u00e9n, Madrid,\nZaragoza, Gerona, Arapiles!--I can tell you something of all these, if\nyour patience does not fail. My story may not be as elegantly told as\nit should be but I will do my best to insure its being true.\n\n\n\n\nCHAPTER II.\n\n\nIt was on one of the early days of October in that fatal year, 1805,\nthat my worthy master called me into his room and looked at me with\nthe severity that was habitual to him--a severity that was only on the\nsurface for his nature was gentleness itself--he said:\n\n\"Gabriel, are you a brave man?\"\n\nI did not know what to answer, for, to tell the truth, in my fourteen\nyears of life no opportunity had ever presented itself for me to\nastonish the world with any deed of valor; still, it filled me with\npride to hear myself called a man, and thinking it ill-judged to deny\nmyself the credit of courage before any one who held it in such high\nestimation, I answered, with boyish boldness:\n\n\"Yes, sir, I am a brave man.\"\n\nAt this the noble gentleman, who had shed his blood in a hundred\nglorious fights and who nevertheless did not disdain to treat a\nfaithful servant with frank confidence, smiled at me kindly, signed to\nme to take a seat, and seemed on the point of informing me of some\nbusiness of importance, when his wife, my mistress, Do\u00f1a Francisca\nentered the study, and, to give further interest to the discussion,\nbegan to declaim with vehemence.\n\n\"You are not to go,\" she said, \"I declare you shall not join the\nfleet. What next will you be wanting to do?--at your age and when you\nhave long retired as superannuated! No, no, Alonso my dear. You are\npast sixty and your dancing days are over.\"\n\nI can see her now, that respectable and indignant dame--with her\ndeep-bordered cap, her muslin dress, her white curls, and a hairy mole\non one side of her chin. I describe these miscellaneous details, for\nthey are inseparable from my recollection of her. She was pretty even\nin old age, like Murillo's Santa Anna, and her sober beauty would have\njustified the comparison if only the lady had been as silent as a\npicture. Don Alonso somewhat cowed, as he always was, by her flow of\nwords, answered quietly:\n\n\"I must go, Paquita. From the letter I have just now received from my\nworthy friend Churruca, I learn that the united squadrons are either\nto sail from Cadiz and engage the English or to wait for them in the\nbay in case they are so bold as to enter. In either case it will be no\nchild's play.\"\n\n\"That is well, and I am glad to hear it,\" replied Do\u00f1a Francisca.\n\"There are Gravina, Vald\u00e9s, Cisneros, Churruca, Alcal\u00e1 Galiano, and\nAlava; let them pound away at the English dogs. But you are a piece of\nuseless lumber who can do no good if you go. Why you cannot move that\nleft arm which they dislocated for you at Cape St. Vincent.\"\n\nMy master lifted his arm, with a stiff attempt at military precision,\nto show that he could use it. But his wife, not convinced by so feeble\nan argument, went on with shrill asseveration.\n\n\"No, you shall not go, what can they want of a piece of antiquity like\nyou. If you were still forty as you were when you went to Tierra del\nFuego and brought me back those green Indian necklaces.--Then indeed!\nBut now!--I know, that ridiculous fellow Marcial fired your brain this\nmorning with talking to you about battles. It seems to me that Se\u00f1or\nMarcial and I will come to quarrelling.--Let him go to the ships if he\nlikes and pay them out for the foot he lost! Oh! Saint Joseph the\nblessed! If I had known when I was a girl what you seamen were!\nEndless worry; never a day's peace! A woman marries to live with her\nhusband and one fine day a dispatch comes from Madrid and he is sent\noff at two minutes notice to the Lord knows where--Patagonia or Japan\nor the infernal regions. For ten or twelve months she sees nothing of\nhim and at last, if the savages have not eaten him meanwhile, he comes\nback again the picture of misery--so ill and yellow that she does not\nknow what to do to restore him to his right color. But old birds are\nnot to be caught in a trap, and then suddenly another dispatch comes\nfrom Madrid, with orders to go to Toulon or Brest or Naples--go here\nand go there--wherever it is necessary to meet the whims of that\nrascally First Consul...! If you would all do as I say, you would soon\npayout these gentlemen who keep the world in a turmoil!\"\n\nMy master sat smiling and gazing at a cheap print, badly  by\nsome cheap artist, which was nailed against the wall, and which\nrepresented the Emperor Napoleon mounted on a green charger, in the\ncelebrated \"redingote\" which was smeared with vermilion. It was no\ndoubt the sight of this work of art, which I had seen daily for four\nyears, which had modified my ideas with respect to the smuggler's\ncostume of the great man of the day, and had fixed his image in my\nmind as dressed something like a cardinal and riding a green horse.\n\n\"This is not living!\" Do\u00f1a Francisca went on, throwing up her arms:\n\"God forgive me, but I hate the sea, though they say it is one of His\nmost glorious works. What is the use of the Holy Inquisition, will you\ntell me, if it is not to burn those diabolical ships of war to ashes?\nWhat is the good of this incessant firing of cannon,--balls upon\nballs, all directed against four boards, as you may say, which are\nsoon smashed to leave hundreds of hapless wretches to drown in the\nsea? Is not that provoking God?--And yet you men are half-wild as soon\nas you hear a cannon fired! Merciful Heaven! my flesh creeps at the\nsound, and if every one was of my way of thinking, we should have no\nmore sea-fights, and the cannon would be cast into bells. Look here,\nAlonso,\" she said, standing still in front of her husband, \"it seems\nto me that they have done you damage enough already; what more do you\nwant? You and a parcel of madmen like yourself,--had you not enough to\nsatisfy you on the 14th?\"[1]\n\n    [1] The battle of Cape St. Vincent was fought on February 14,\n    1797.\n\nDon Alonso clenched his fists at this bitter reminiscence, and it was\nonly out of consideration for his wife, to whom he paid the utmost\nrespect, that he suppressed a good round oath.\n\n\"I lay all the blame of your absurd determination to join the fleet to\nthat rascally Marcial,\" the lady went on, warming with her own\neloquence; \"that maniac for the sea who ought to have been drowned a\nhundred times and over, but that he escaped a hundred times to be the\ntorment of my life. If he wants to join, with his wooden leg, his\nbroken arm, his one eye, and his fifty wounds--let him go, by all\nmeans, and God grant he may never come back here again--but you shall\nnot go, Alonso, for you are past service and have done enough for the\nKing who has paid you badly enough in all conscience. If I were you, I\nwould throw those captain's epaulettes you have worn these ten years\nin the face of the Generalissimo of the land and sea forces. My word!\nthey ought have made you admiral, at least; you earned that when you\nwent on that expedition to Africa and brought me back those blue beads\nwhich I gave with the Indian necklace to decorate the votive urn to\nthe Virgin 'del C\u00e1rmen.'\"\n\n\"Admiral or not, it is my duty to join the fleet, Paquita,\" said my\nmaster. \"I cannot be absent from this struggle. I feel that I must pay\noff some of my arrears to the English.\"\n\n\"Do you talk of paying off arrears!\" exclaimed my mistress; \"you--old,\nfeeble, and half-crippled....\"\n\n\"Gabriel will go with me,\" said Don Alonso, with a look at me which\nfilled me with valor.\n\nI bowed to signify that I agreed to this heroic scheme, but I took\ncare not to be seen by my mistress, who would have let me feel the\nfull weight of her hand if she had suspected my bellicose\ninclinations. Indeed, seeing that her husband was fully determined,\nshe was more furious than ever, declaring that if she had to live her\nlife again nothing should induce her to marry a sailor. She cursed the\nEmperor, abused our revered King, the Prince of Peace, and all those\nwho had signed the Treaty of Subsidies, ending by threatening the\nbrave old man with punishment from Heaven for his insane rashness.\n\nDuring this dialogue, which I have reported with approximate exactness\nas I have to depend on my memory, a loud barking cough in the\nadjoining room revealed the fact that Marcial, the old sailor, could\noverhear with perfect ease, my mistress's vehement harangue, in which\nshe had so frequently mentioned him in by no means flattering terms.\nBeing now desirous of taking part in the conversation, as his intimacy\nin the house fully justified his doing, he opened the door and came\ninto Don Alfonso's room. Before going any farther I must give some\naccount of my master's former history, and of his worthy wife, that\nthe reader may have a better understanding of what follows.\n\n\n\n\nCHAPTER III.\n\n\nDon Alfonso Gutierrez de Cisniega belonged to an old family of Vejer,\nwhere he lived. He had been devoted at an early age to a naval career\nand, while still quite young, had distinguished himself in defending\nHavana against the English in 1748. He was afterwards engaged in the\nexpedition which sailed from Cartagena against the Algerines in 1775,\nand was present at the attack upon Gibraltar under the Duke de Crillon\nin 1782. He subsequently joined the expedition to the Straits of\nMagellan in the corvette _Santa Mar\u00eda de la Cabeza_, commanded by Don\nAntonio de C\u00f3rdova, and fought in the glorious engagements between the\nAnglo-Spanish fleet and the French before Toulon in 1793, terminating\nhis career of glory at the disastrous battle of Cape St. Vincent,\nwhere he commanded the _Mejicano_, one of the ships which were forced\nto surrender.\n\nFrom that time my master, whose promotion had been slower than his\nlaborious and varied career had merited, retired from active service.\nHe suffered much in body from the wounds he had received on that\nfatal day, and more in mind from the blow of such a defeat. His wife\nnursed and tended him with devotion though not in silence, for abuse\nof the navy and of seamen of every degree were as common in her mouth\nas the names of the saints in that of a bigot.\n\nDo\u00f1a Francisca was an excellent woman, of exemplary conduct and noble\nbirth, devout and God-fearing--as all women were in those days,\ncharitable and judicious, but with the most violent and diabolical\ntemper I ever met with in the whole course of my life. Frankly I do\nnot believe that this excessive irritability was natural to her, but\nthe result and outcome of the worries in her life arising out of her\nhusband's much-hated profession; it must be confessed that she did not\ncomplain wholly without reason, and every day of her life Do\u00f1a\nFrancisca addressed her prayers to Heaven for the annihilation of\nevery fleet in Europe. This worthy couple had but one child, a\ndaughter--the incomparable Rosita, of whom more anon.\n\nThe veteran, however, pined sadly at Vejer, seeing his laurels covered\nwith dust and gnawed to powder by the rats, and all his thoughts and\nmost of his discourse, morning, noon, and night, were based on the\nabsorbing theme that if C\u00f3rdova, the commander of the Spanish fleet,\nhad only given the word \"Starboard\" instead of \"Port\" the good ships\n_Mejicano_, _San Jos\u00e9_, _San Nicol\u00e1s_ and _San Isidro_ would never\nhave fallen into the hands of the English, and Admiral Jervis would\nhave been defeated. His wife, Marcial, and even I myself, exceeding\nthe limits of my duties--always assured him that there was no doubt of\nthe fact, to see whether, if we acknowledged ourselves convinced, his\nvehemence would moderate--but no; his mania on that point only died\nwith him.\n\nEight years had passed since that disaster, and the intelligence that\nthe whole united fleet was to fight a decisive battle with the English\nhad now roused my master to a feverish enthusiasm which seemed to have\nrenewed his youth. He pictured to himself the inevitable rout of his\nmortal enemies; and although his wife tried to dissuade him, as has\nbeen said, it was impossible to divert him from his wild purpose. To\nprove how obstinate his determination was it is enough to mention that\nhe dared to oppose his wife's strong will, though he avoided all\ndiscussion; and to give an adequate idea of all that his opposition\nimplied I ought to mention that Don Alonso was afraid of no mortal\nthing or creature--neither of the English, the French, nor the\nsavages of Magellan, not of the angry sea, nor of the monsters of the\ndeep, nor of the raging tempest, nor of anything in the earth or\nsky--but only of his wife.\n\nThe last person I must mention is Marcial the sailor, the object of\nDo\u00f1a Francisca's deepest aversion, though Don Alonso, under whom he\nhad served, loved him as a brother.\n\nMarcial--no one knew his other name--called by all the sailors \"the\nHalf a Man,\" had been boatswain on various men-of-war for forty years.\nAt the time when my story begins this maritime hero's appearance was\nthe strangest you can imagine. Picture to yourself an old man, tall\nrather than short, with a wooden leg, his left arm shortened to within\na few inches of the elbow, minus an eye, and his face seamed with\nwounds in every direction--slashed by the various arms of the enemy;\nwith his skin tanned brown, like that of all sea-faring men, and a\nvoice so hoarse, hollow and slow, that it did not seem to belong to\nany rational human creature, and you have some idea of this eccentric\npersonage. As I think of him I regret the narrow limits of my palette,\nfor he deserves painting in more vivid colors and by a worthier\nartist. It was hard to say whether his appearance was most calculated\nto excite laughter or command respect--both at once I think, and\naccording to the point of view you might adopt.\n\nHis life might be said to be an epitome of the naval history of Spain\nduring the last years of the past century and the beginning of this--a\nhistory in whose pages the most splendid victories alternate with the\nmost disastrous defeats. Marcial had served on board the _Conde de\nRegla_, the _San Joaquin_, the _Real C\u00e1rlos_, the _Trinidad_ and other\nglorious but unfortunate vessels which, whether honorably defeated or\nperfidiously destroyed, carried with them to a watery grave the naval\npower of Spain. Besides the expeditions in which my master had taken\npart Marcial had been present at many others, such as that of\nMartinica, the action of Cape Finisterre, and before that the terrible\nbattle close to Algeciras in July, 1801, and that off Cape Santa Mar\u00eda\non the 5th of October, 1804. He quitted the service at sixty-six years\nof age, not however for lack of spirit but because he was altogether\n\"unmasted\" and past fighting. On shore he and my master were the best\nof friends, and as the boatswain's only daughter was married to one of\nthe servants of the house, of which union a small child was the token,\nMarcial had made up his mind to cast anchor for good, like a hulk\npast service, and even succeeded in making himself believe that peace\nwas a good thing. Only to see him you would have thought that the most\ndifficult task that could be set to this grand relic of a hero was\nthat of minding babies; but, as a matter of fact, Marcial had no other\noccupation in life than carrying and amusing his grandchild, putting\nit to sleep with his snatches of sea-songs, seasoned with an oath or\ntwo--excusable under the circumstances.\n\nBut no sooner had he heard that the united fleets were making ready\nfor a decisive battle than his moribund fires rose from their ashes,\nand he dreamed that he was calling up the crew in the forecastle of\nthe _Sant\u00edsima Trinidad_. Discovering in Don Alfonso similar symptoms\nof rejuvenescence, he confessed to him, and from that hour they spent\nthe chief part of the day and night in discussing the news that\narrived and their own feelings in the matter; \"fighting their battles\no'er again,\" hazarding conjectures as to those to be fought in the\nimmediate future, and talking over their day-dreams like two ship's\nboys indulging in secret visions of the shortest road to the title of\nAdmiral.\n\nIn the course of these _t\u00eate-\u00e0-t\u00eate_ meetings, which occasioned the\ngreatest alarm to Do\u00f1a Francisca, the plan was hatched for setting\nout to join the fleet and be present at the impending battle. I have\nalready told the reader what my mistress's opinion was and all the\nabuse she lavished on the insidious sailor; he knows too that Don\nAlonso persisted in his determination to carry out his rash purpose,\naccompanied by me, his trusty page, and I must now proceed to relate\nwhat occurred when Marcial himself appeared on the scene to take up\nthe cudgels for war against the shameful _status quo_ of Do\u00f1a\nFrancisca.\n\n\n\n\nCHAPTER IV.\n\n\n\"Se\u00f1or Marcial,\" she began, with increased indignation, \"if you choose\nto go to sea again and lose your other hand, you can go if you like;\nbut my husband here, shall not.\"\n\n\"Very good,\" said the sailor who had seated himself on the edge of a\nchair, occupying no more space on it than was necessary to save\nhimself from falling: \"I will go alone. But the devil may take me if I\ncan rest without looking on at the fun!\"\n\nThen he went on triumphantly: \"We have fifteen ships and the French\ntwenty smaller vessels. If they were all ours we should not want so\nmany. Forty ships and plenty of brave hearts on board!\"\n\nJust as the spark creeps from one piece of timber to the next, the\nenthusiasm that fired Marcial's one eye lighted up both my master's,\nthough dimmed by age. \"But the _Se\u00f1orito_\" (Lord Nelson), added the\nsailor, \"will bring up a great many men too. That is the sort of\nperformance I enjoy: plenty of timbers to fire at, and plenty of\ngunpowder-smoke to warm the air when it is cold.\"\n\nI forgot to mention that Marcial, like most sailors, used a vocabulary\nof the most wonderful and mongrel character, for it seems to be a\nhabit among seamen of every nation to disfigure their mother tongue to\nthe verge of caricature. By examining the nautical terms used by\nsailors we perceive that most of them are corruptions of more usual\nterms, modified to suit their eager and hasty temperament trained by\ncircumstances to abridge all the functions of existence and\nparticularly speech. Hearing them talk it has sometimes occurred to me\nthat sailors find the tongue an organ that they would gladly dispense\nwith.\n\nMarcial, for instance, turned verbs into nouns and nouns into verbs\nwithout consulting the authorities. He applied nautical terms to every\naction and movement, and identified the ideas of a man and a ship,\nfancying that there was some analogy between their limbs and parts. He\nwould say in speaking of the loss of his eye that his larboard\nport-hole was closed, and explained the amputation of his arm by\nsaying that he had been left minus his starboard cat-head. His heart\nhe called his courage-hold and his stomach his bread-basket. These\nterms sailors at any rate could understand; but he had others, the\noffspring of his own inventive genius of which he alone understood the\nmeaning or could appreciate the force. He had words of his own coining\nfor doubting a statement, for feeling sad; getting drunk he always\ncalled \"putting on your coat\" among a number of other fantastical\nidioms; and the derivation of this particular phrase will never occur\nto my readers without my explaining to them that the English sailors\nhad acquired among the Spaniards the nickname of \"great-coats,\" so\nthat when he called getting drunk \"putting your coat on\" a recondite\nallusion was implied to the favorite vice of the enemy. He had the\nmost extraordinary nicknames for foreign admirals; Nelson he called\nthe _Se\u00f1orito_, implying a certain amount of respect for him;\nCollingwood was _Rio Calambre_, (Uncle Cramp) which he believed to be\nan equivalent for the English name; Jervis he called--as the English\ndid too--The old Fox; Calder was known as _Rio Perol_ (Uncle Boiler)\nfrom an association of the name Calder with _caldera_, a kettle, and\nby an entirely different process he dubbed Villeneuve, the Admiral of\nthe united fleets, with the name of _Monsieur Corneta_, borrowed from\nsome play he had once seen acted at Madrid. In fact, when reporting\nthe conversations I can recall, I must perforce translate his\nwonderful phraseology into more ordinary language, to avoid going into\nlong and tiresome explanations.\n\nTo proceed, Do\u00f1a Francisca, devoutly crossing herself, answered\nangrily:\n\n\"Forty ships! Good Heavens! it is tempting Providence; and there will\nbe at least forty thousand guns for the enemies to kill each other.\"\n\n\"Ah! but _Monsieur Corneta_ keeps the courage-hold well filled!\"\nexclaimed Marcial, striking his breast. \"We shall laugh at the\ngreat-coats this time. It will not be Cape St. Vincent over again.\"\n\n\"And you must not forget,\" added my master eagerly recurring to his\nfavorite hobby, \"that if Admiral C\u00f3rdova had only ordered the _San\nJos\u00e9_ and the _Mejicano_ to tack to port, Captain Jervis would not now\nbe rejoicing in the title of Earl St. Vincent. Of that you may be very\ncertain, and I have ample evidence to show that if we had gone to port\nthe day would have been ours.\"\n\n\"Ours!\" exclaimed Do\u00f1a Francisca scornfully. \"As if you could have\ndone more. To hear these fire-eaters it would seem as if they wanted\nto conquer the world, and as to going to sea--it appears that their\nshoulders are not broad enough to bear the blows of the English.\"\n\n\"No,\" said Marcial resolutely and clenching his fist defiantly. \"If it\nwere not for their cunning and knavery...! We got out against them\nwith a bold front, defying them like men, with our flag hoisted and\nclean hands. The English never sail wide, they always steal up and\nsurprise us, choosing heavy seas and stormy weather. That is how it\nwas at the Straits, when we were made to pay so dearly. We were\nsailing on quite confidingly, for no one expected to be trapped even\nby a heretic dog of a Moor, much less by an Englishman who does the\npolite thing in a Christian fashion.--But no, an enemy who sneaks up\nto fight is not a Christian--he is a highwayman. Well now, just fancy,\nse\u00f1ora,\" and he turned to Do\u00f1a Francisca to engage her attention and\ngood-will, \"we were going out of Cadiz to help the French fleet which\nwas driven into Algeciras by the English.--It is four years ago now,\nand to this day it makes me so angry that my blood boils as I think of\nit. I was on board the _Real C\u00e1rlos_, 112 guns, commanded by Ezguerra,\nand we had with us the _San Hermenegildo_, 112 guns too, the _San\nFernando_, the _Argonauta_, the _San Agustin_, and the frigate\n_Sabina_. We were joined by the French squadron of four men-of-war,\nthree frigates and a brigantine, and all sailed out of Algeciras for\nCadiz at twelve o'clock at noon; and as the wind was slack when night\nfell we were close under Punta Carnero. The night was blacker than a\nbarrel of pitch, but the weather was fine so we could hold on our way\nin spite of the darkness. Most of the crew were asleep; I remember, I\nwas sitting in the fo'castle talking to the mate, Pepe D\u00e9bora, who was\ntelling me all the dog's tricks his mother-in-law had played him, and\nalongside we could see the lights of the _San Hermenegildo_, which was\nsailing at a gun-shot to starboard. The other ships were ahead of us.\nFor the very last thing we any of us thought of was that the\n'great-coats' had slipped out of Gibraltar and were giving chase--and\nhow the devil should we, when they had doused all their lights and\nwere stealing up to us without our guessing it? Suddenly, for all that\nthe night was so dark, I fancied I saw something--I always had a\nport-light like a lynx--I fancied a ship was standing between us and\nthe _San Hermenegildo_, which was sailing at a gun-shot to starboard.\n'Jos\u00e9 D\u00e9bora,' says I, 'either I saw a ghost or there is an Englishman\nto starboard?' Jos\u00e9 D\u00e9bora looks himself, and then he says: 'May the\nmain-mast go by the board,' says he, 'if there is e'er a ship to\nstarboard but the _San Hermenegildo_.' 'Well,' says I, 'whether or no\nI am going to tell the officer of the watch.'\n\n\"Well hardly were the words out of my mouth when, rub-a-dub! we heard\nthe tune of a whole broadside that came rattling against our ribs. The\ncrew were on deck in a minute, and each man at his post. That was a\nrumpus, se\u00f1ora! I wish you could have been there, just to have an idea\nof how these things are managed. We were all swearing like demons and\nat the same time praying the Lord to give us a gun at the end of every\nfinger to fight them with. Ezguerra gave the word to return their\nbroadside.--Thunder and lightning! They fired again, and in a minute\nor two we responded. But in the midst of all the noise and confusion\nwe discovered that with their first broadside they had sent one of\nthose infernal combustibles (but he called it 'comestibles') on board\nwhich fall on the deck as if it were raining fire. When we saw our\nship was burning we fought like madmen and fired off broadside after\nbroadside. Ah! Do\u00f1a Francisca, it was hot work I can tell you!--Then\nour captain took us alongside of the enemy's ship that we might board\nher. I wish you could have seen it! I was in my glory then; in an\ninstant we had our axes and boarding-pikes out, the enemy was coming\ndown upon us and my heart jumped for joy to see it, for this was the\nquickest way of settling accounts. On we go, right into her!--Day was\njust beginning to dawn, the yards were touching, and the boarding\nparties ready at the gangways when we heard Spanish oaths on board the\nfoe. We all stood dumb with horror, for we found that the ship we had\nbeen fighting with was the _San Hermenegildo_ herself.\"\n\n\"That was a pretty state of things,\" said Do\u00f1a Francisca roused to\nsome interest in the narrative. \"And how had you been such asses--with\nnot a pin to choose between you?\"\n\n\"I will tell you. We had no time for explanations then. The flames on\nour ship went over to the _San Hermenegildo_ and then, Blessed Virgin!\nwhat a scene of confusion. 'To the boats!' was the cry. The fire\ncaught the _Santa B\u00e1rbara_ and her ladyship blew up with loud\nexplosion.--We were all swearing, shouting, blaspheming God and the\nVirgin and all the Saints, for that seems the only way to avoid\nchoking when you are primed to fight, up to the very muzzle....\"\n\n\"Merciful Heavens how shocking!\" cried my mistress. \"And you escaped?\"\n\n\"Forty of us got off in the launch and six or seven in the gig, these\ntook up the second officer of the _San Hermenegildo_. Jos\u00e9 D\u00e9bora\nclung to a piece of plank and came to shore at Morocco, more dead than\nalive.\"\n\n\"And the rest?\"\n\n\"The rest--the sea was wide enough to hold them all. Two thousand men\nwent down to Davy Jones that day, and among them our captain,\nEzguerra, and Emparan, the captain of the other ship.\"\n\n\"Lord have mercy on them!\" ejaculated Do\u00f1a Francisca. \"Though God\nknows! they were but ill-employed to be snatched away to judgment. If\nthey had stayed quietly at home, as God requires....\"\n\n\"The cause of that disaster,\" said Don Alonso, who delighted in\ngetting his wife to listen to these dramatic narratives, \"was this:\nThe English emboldened by the darkness arranged that the _Superb_, the\nlightest of their vessels, should extinguish her lights and slip\nthrough between our two finest ships. Having done this, she fired both\nher broadsides and then put about as quickly as possible to escape the\nstruggle that ensued. The two men-of-war, finding themselves\nunexpectedly attacked, returned fire and thus went on battering each\nother till dawn, when, just as they were about to board, they\nrecognized each other and the end came as Marcial has told you in\ndetail.\"\n\n\"Ah! and they played the game well,\" cried the lady. \"It was well\ndone though it was a mean trick!\"\n\n\"What would you have?\" added Marcial. \"I never loved them much; but\nsince that night!... If _they_ are in Heaven I do not want ever to go\nthere. Sooner would I be damned to all eternity!\"\n\n\"Well--and then the taking of the four frigates which were coming from\nRio de la Plata?\" asked Don Alfonso, to incite the old sailor to go on\nwith his stories.\n\n\"Aye--I was at that too,\" said Marcial. \"And that was where I left my\nleg. That time too they took us unawares, and as it was in time of\npeace we were sailing on quietly enough, only counting the hours till\nwe should be in port, when suddenly---- I will tell you exactly how it\nall happened, Do\u00f1a Francisca, that you may just understand the ways of\nthose people. After the engagement at the Straits I embarked on board\nthe _Fama_ for Montevideo, and we had been out there a long time when\nthe Admiral of the squadron received orders to convoy treasure from\nLima and Buenos Ayres to Spain. The voyage was a good one and we had\nno mishaps but a few slight cases of fever which only killed off a few\nof our men. Our freight was heavy--gold belonging to the king and to\nprivate persons, and we also had on board what we called the 'wages\nchest'--savings off the pay of the troops serving in America.\nAltogether, if I am not much mistaken, a matter of fifty millions or\nso of _pesos_, as if it were a mere nothing; and besides that,\nwolf-hides, vicu\u00f1a wool, cascarilla, pigs of tin and copper, and\ncabinet woods. Well, sir, after sailing for fifty days we sighted land\non the 5th of October, and reckoned on getting into Cadiz the next day\nwhen, bearing down from the northeast, what should we see but four\nfrigates. Although, as I said, it was in time of peace, and though our\ncaptain, Don Miguel de Zapiain, did not seem to have any suspicion of\nevil, I--being an old sea-dog--called D\u00e9bora and said to him that\nthere was powder in the air, I could smell it. Well, when the English\nfrigates were pretty near, we cleared the decks for action; the _Fama_\nwent forward and we were soon within a cable's length of one of the\nEnglish ships which lay to windward.\n\n\"The English captain hailed us through his speaking-trumpet and told\nus--there is nothing like plain-speaking--told us to prepare to defend\nourselves, as he was going to attack. He asked a string of questions,\nbut all he got out of us was that we should not take the trouble to\nanswer him. Meanwhile the other three frigates had come up and had\nformed in such order that each Englishman had a Spaniard to the\nleeward of him.\"\n\n\"They could not have taken up a better position,\" said my master.\n\n\"So say I,\" replied Marcial. \"The commander of our squadron, Don Jos\u00e9\nBustamante, was not very prompt; if I had been in his shoes.... Well,\nse\u00f1or, the English commodore sent a little whipper-snapper officer, in\na swallow-tail coat, on board the _Medea_, who wasted no time in\ntrifling but said at once that though war had not been declared, the\ncommodore had orders to take us. That is what it is to be English!\nWell, we engaged at once; our frigate received the first broadside in\nher port quarter; we politely returned the salute, and the cannonade\nwas brisk on both sides--the long and the short of it is that we could\ndo nothing with the heretics, for the devil was on their side; they\nset fire to the _Santa B\u00e1rbara_ which blew up with a roar, and we were\nall so crushed by this and felt so cowed--not for want of courage,\nse\u00f1or, but what they call demoralized--well, from the first we knew we\nwere lost. There were more holes in our ship's sails than in an old\ncloak; our rigging was damaged, we had five feet of water in the\nhold, our mizzen-mast was split, we had three shots in the side only\njust above the water line and many dead and wounded. Notwithstanding\nall this we went on, give and take, with the English, but when we saw\nthat the _Medea_ and the _Clara_ were unable to fight any longer and\nstruck their colors we made all sail and retired, defending ourselves\nas best we could. The cursed Englishman gave chase, and as her sails\nwere in better order than ours we could not escape and we had nothing\nfor it but to haul our colors down at about three in the afternoon,\nwhen a great many men had been killed and I myself was lying half-dead\non the deck, for a ball had gone out of its way to take my leg off.\nThose d----d wretches carried us off to England, not as prisoners, but\nas _d\u00e9tenus_; however, with despatches on one side and despatches on\nthe other, from London to Madrid and back again, the end of it was\nthat they stuck for want of money; and, so far as I was concerned,\nanother leg might have grown by the time the King of Spain sent them\nsuch a trifle as those five millions of _pesos_.\"\n\n\"Poor man!--and it was then you lost your leg?\" asked Do\u00f1a Francisca\ncompassionately.\n\n\"Yes, se\u00f1ora, the English, knowing that I was no dancer, thought one\nwas as much as I could want. In return they took good care of me. I\nwas six months in a town they called _Plinmuf_ (Plymouth) lying in my\nbunk with my paw tied up and a passport for the next world in my\npocket.--However, God A'mighty did not mean that I should make a hole\nin the water so soon; an English doctor made me this wooden leg, which\nis better than the other now, for the other aches with that d----d\nrheumatism and this one, thank God, never aches even when it is hit by\na round of small shot. As to toughness, I believe it would stand\nanything, though, to be sure, I have never since faced English fire to\ntest it.\"\n\n\"You are a brave fellow,\" said my mistress. \"Please God you may not\nlose the other. But those who seek danger....\"\n\nAnd so, Marcial's story being ended, the dispute broke out anew as to\nwhether or no my master should set out to join the squadron. Do\u00f1a\nFrancisca persisted in her negative, and Don Alonso, who in his wife's\npresence was as meek as a lamb, sought pretexts and brought forward\nevery kind of reason to convince her.\n\n\"Well we shall go to look on, wife,--simply and merely to look\non\"--said the hero in a tone of entreaty.\n\n\"Let us have done with sight-seeing,\" answered his wife. \"A pretty\npair of lookers-on you two would make!\"\n\n\"The united squadrons,\" added Marcial, \"will remain in Cadiz--and they\nwill try to force the entrance.\"\n\n\"Well then,\" said my mistress, \"you can see the whole performance from\nwithin the walls of Cadiz, but as for going out in the ships--I say\nno, and I mean no, Alonso. During forty years of married life you have\nnever seen me angry (he saw it every day)--but if you join the\nsquadron I swear to you ... remember, Paquita lives only for you!\"\n\n\"Wife, wife--\" cried my master much disturbed: \"Do you mean I am to\ndie without having had that satisfaction?\"\n\n\"A nice sort of satisfaction truly! to look on at mad men killing each\nother! If the King of Spain would only listen to me, I would pack off\nthese English and say to them: 'My beloved subjects were not made to\namuse you. Set to and fight each other, if you want to fight.' What do\nyou say to that?--I, simpleton as I am, know very well what is in the\nwind, and that is that the first Consul--Emperor--Sultan--whatever you\ncall him--wants to settle the English, and as he has no men brave\nenough for the job he has imposed upon our good King and persuaded\nhim to lend him his; and the truth is he is sickening us with his\neverlasting sea-fights. Will you just tell me what is Spain to gain in\nall this? Why is Spain to submit to being cannonaded day after day for\nnothing at all? Before all that rascally business Marcial has told us\nof what harm had the English ever done us?--Ah, if they would only\nlisten to me! Master Buonaparte might fight by himself, for I would\nnot fight for him!\"\n\n\"It is quite true,\" replied my master, \"that our alliance with France\nis doing us much damage, for all the advantages accrue to our ally,\nwhile all the disasters are on our side.\"\n\n\"Well, then, you utter simpletons, why do you encourage the poor\ncreatures to fight in this war?\"\n\n\"The honor of the nation is at stake,\" replied Don Alonso, \"and after\nhaving once joined the dance it would be a disgrace to back out of it.\nLast month, when I was at Cadiz, at my cousin's daughter's\nchristening, Churruca said to me: 'This French alliance and that\nvillainous treaty of San Ildefonso, which the astuteness of Buonaparte\nand the weakness of our government made a mere question of subsidies,\nwill be the ruin of us and the ruin of our fleet if God does not come\nto the rescue, and afterwards will be the ruin of the colonies too and\nof Spanish trade with America. But we must go on now all the\nsame....'\"\n\n\"Well,\" said Do\u00f1a Francisca, \"what I say is that the Prince of Peace\nis interfering in things he does not understand. There you see what a\nman without learning is! My brother the archdeacon, who is on Prince\nFerdinand's side, says that Godoy is a thoroughly commonplace soul,\nthat he has studied neither Latin nor theology and that all he knows\nis how to play the guitar and twenty ways of dancing a gavotte. They\nmade him prime minister for his good looks, as it would seem. That is\nthe way we do things in Spain! And then we hear of starvation and\nwant--everything is so dear--yellow fever breaking out in\nAndalusia.--This is a pretty state of things, sir,--yes, and the fault\nis yours; yours,\" she went on, raising her voice and turning purple.\n\"Yes, se\u00f1or, yours, who offend God by killing so many people--and if\nyou would go to church and tell your beads instead of wanting to go in\nthose diabolical ships of war, the devil would not find time to trot\nround Spain so nimbly, playing the mischief with us all.\"\n\n\"But you shall come to Cadiz too,\" said Don Alonso, hoping to light\nsome spark of enthusiasm in his wife's heart; \"you shall go to\nFlora's house, and from the balcony you will be able to see the fight\nquite comfortably, and the smoke and the flames and the flags.--It is\na beautiful sight!\"\n\n\"Thank you very much--but I should drop dead with fright. Here we\nshall be quiet; those who seek danger may go there.\"\n\nHere the dialogue ended, and I remember every word of it though so\nmany years have elapsed. But it often happens that the most remote\nincidents that occurred even in our earliest childhood, remain stamped\non our imagination more clearly and permanently than the events of our\nriper years when our reasoning faculties have gained the upper hand.\n\nThat evening Don Alonzo and Marcial talked over matters whenever Do\u00f1a\nFrancisca left them together; but this was at rare intervals, for she\nwas suspicious and watchful. When she went off to church to attend\nvespers, as was her pious custom, the two old sailors breathed freely\nagain as if they were two giddy schoolboys out of sight of the master.\nThey shut themselves into the library, pulled out their maps and\nstudied them with eager attention; then they read some papers in which\nthey had noted down the names of several English vessels with the\nnumber of their guns and men, and in the course of their excited\nconference, in which reading was varied by vigorous commentary, I\ndiscovered that they were scheming the plan of an imaginary naval\nbattle. Marcial, by means of energetic gymnastics with his arm and a\nhalf, imitated the advance of the squadron and the explosion of the\nbroadsides; with his head he indicated the alternate action of the\nhostile vessels; with his body the heavy lurch of each ship as it went\nto the bottom; with his hand the hauling up and down of the signal\nflags; he represented the boatswain's whistle by a sharp sibilation;\nthe rattle of the cannon by thumping his wooden leg on the floor; he\nsmacked his tongue to imitate the swearing and confusion of noises in\nthe fight; and as my master assisted him in this performance with the\nutmost gravity I also must need take my share in the fray, encouraged\nby their example and giving natural vent to that irresistible longing\nto make a noise which is a master passion with every boy. Seeing the\nenthusiasm of the two veterans, I could no longer contain myself and\ntook to leaping about the room--a freedom in which I was justified by\nmy master's kind familiarity; I imitated with my head and arms the\nmovements of a vessel veering before the wind, and at the same time\nmaking my voice as big as possible I shouted out all the most\nsonorous monosyllables I could think of as being most like the noise\nof a cannon. My worthy master and the mutilated old sailor, quite as\nchildish as I in their own way, paid no attention to my proceedings,\nbeing entirely preoccupied with their own ideas.\n\nHow I have laughed since when I have remembered the scene! and how\ntrue it is--in spite of all my respect for my companions in the\ngame--that senile enthusiasm makes old men children once more and\nrenews the puerile follies of the cradle even on the very brink of the\ntomb!\n\nThey were deep in their discussion when they heard Do\u00f1a Francisca's\nstep returning from church.\n\n\"She is coming!\" cried Marcial in an agony of alarm, and they folded\nup the maps and began to talk of indifferent matters. I, however, not\nbeing able to cool down my juvenile blood so rapidly or else not\nnoticing my mistress's approach soon enough, went on, down the middle\nof the room in my mad career, ejaculating with the utmost incoherence,\nsuch phrases as I had picked up: \"Tack to starboard! Now Port!\nBroadside to the leeward! Fire! Bang! bom! boom!...\" She came up to me\nin a fury and without any warning delivered a broadside on my\nfigure-head with her right hand, and with such effect that for a few\nmoments I saw nothing but stars.\n\n\"What! you too?\" she cried, battering me unmercifully. \"You see,\" she\nadded, turning on her husband with flashing eyes, \"you have taught him\nto feel no respect for you!--You thought you were still in the\n_Caleta_ did you, you little ne'er do weel?\"\n\nThe commotion ended by my running off to the kitchen crying and\ndisgraced, after striking my colors in an ignominious manner, before\nthe superior force of the enemy; Do\u00f1a Francisca giving chase and\nbelaboring my neck and shoulders with heavy slaps. In the kitchen I\ncast anchor and sat down to cry over the fatal termination of my\nsea-fight.\n\n\n\n\nCHAPTER V.\n\n\nIn opposing her husband's insane determination to join the fleet, Do\u00f1a\nFrancisca did not rely solely on the reasons given in the last\nchapter; she had another and more weighty one which she did not\nmention in the course of that conversation, perhaps because it was\nwiser not. But the reader does not know it, and must be told.\n\nI have mentioned that my master had a daughter; this daughter's name\nwas Rosita; she was a little older than I was, that is to say scarcely\nfifteen, and a marriage had been arranged for her with a young officer\nof artillery named Malespina, belonging to a family of Medinasidonia\nand distantly related to my master. The wedding had been fixed for the\nend of October and, as may be supposed, the absence of the bride's\nfather on so solemn an occasion would have been highly improper.\n\nI must here give some account of my young lady, of her bridegroom, her\nlove-affairs and her projected marriage; and alas! my recollections\ntake a tinge of melancholy, recalling to my fancy many troublesome\nand far-away scenes, figures from another world--and stirring my weary\nold heart with feelings of which I should find it hard to say whether\nthey were more pleasurable or sad. Those ardent memories which now lie\nwithered in my brain, like tropical flowers exposed to a chill\nnorthern blast, sometimes make me laugh--but sometimes make me grave.\nHowever, to my tale, or the reader will be tired of these wearisome\nreflections which, after all, interest no one but myself.\n\nRosita was uncommonly pretty. I remember vividly how pretty she was,\nthough I should find it difficult to describe her features. I fancy I\nsee her now, smiling in my face; the curious expression of her\ncountenance, unlike any other I ever saw, dwells in my mind--from the\nperfect distinctness with which it rises before me--like one of those\ninnate ideas which seem to have come into the world with us from a\nformer existence, or to have been impressed on our minds by some\nmysterious power while we were still in the cradle. And yet I cannot\ndescribe it, for what then was real and tangible remains now in my\nbrain as a vague ideal; and while nothing is so fascinating as a\nbeloved ideal, nothing so completely eludes all categorical\ndescription.\n\nWhen I first went into the house I thought that Rosita belonged to\nsome superior order of beings; I will explain my feelings more fully\nthat you may form an idea of my utter simpleness. When we are little\nand a child comes into the world within our family the grown-up folks\nare apt to tell us that it has come from France, Paris, or England. I,\nlike other children, having no notions as to the multiplication of the\nhuman race, firmly believed that babies were imported packed up in\nboxes like a cargo of hardware. Thus, gazing for the first time at my\nmaster's daughter, I argued that so lovely a being could not have come\nfrom the same factory as the rest of us, that is to say from Paris or\nfrom England, and I remained convinced that there must be some\nenchanted region where heaven-sent workmen were employed in making\nthese choicer and lovelier specimens of humanity. Both of us being\nchildren, though in different ranks of life, we were soon on those\nterms of mutual confidence which were natural to our years, and my\ngreatest joy was in playing with her, submitting to all her vagaries\nand insolence, which is not saying a little, for our relative position\nwas never lost sight of in our games; she was always the young lady\nand I always the servant, so that I got the worst of it when slaps\nwere going, and I need not say who was the sufferer.\n\nMy highest dream of happiness was to be allowed to fetch her from\nschool, and when, by some unforeseen accident, some one else was\nentrusted with this delightful duty I was so deeply distressed that I\nhonestly thought there could be no greater grief in life, and would\nsay to myself: \"It is impossible that I should ever be more miserable\nwhen I am a man grown.\" My greatest delight was to climb the\norange-tree in the court-yard to pick the topmost sprays of blossom; I\nfelt myself at a height far above the greatest king on earth when\nseated on his throne, and I can remember no pleasure to be compared to\nthat of being obliged to capture her in that divinely rapturous game\nknown as hide and seek. If she ran like a gazelle I flew like a bird\nto catch her as soon as possible, seizing her by the first part of her\ndress or person that I could lay my hand on. When we changed parts,\nwhen she was the pursuer and I was to be caught, the innocent delight\nof the blissful game was doubled, and the darkest and dingiest hole in\nwhich I might hide, breathlessly awaiting the grasp of her imprisoning\nhands, was to me a perfect paradise. And I may honestly say that\nduring these happy games I never had a thought or a feeling that did\nnot emanate from the purest and most loyal idealism.\n\nThen her singing! From the time when she was quite little she used to\nsing the popular airs of Andalusia with the ease of a nightingale,\nwhich knows all the secrets of song without having been taught. All\nthe neighbors admired her wonderful facility and would come to listen\nto her, but to me their applause and admiration were an offence; I\ncould have wished her to sing to no one but me. Her singing was a sort\nof melancholy warbling, qualified by her fresh childlike voice. The\nair, which repeated itself with complicated little turns and trills\nlike a thread of sound, seemed to be lost in distant heights and then\nto come back to earth again on the low notes. It was like the song of\nthe lark as it rises towards heaven and suddenly comes down to sing\nclose in our ears; the spirit of the hearer seemed to expand as it\nfollowed the voice, and then to contract again, but always following\nthe swing of the melody and feeling the music to be inseparable from\nthe sweet little singer. The effect was so singular that to me it was\nalmost painful to hear her, particularly in the presence of others.\n\nWe were, as I have said, of about the same age, she being eight or\nnine months older than I was. But I was stunted and puny while she was\nwell grown and vigorous, and at the end of my three years' residence\nin the house she looked much the elder of the two. These three years\nslipped by without our either of us suspecting that we were growing\nup; our games went on without interruption, for she was much livelier\nby nature than I, though her mother would scold her, trying to keep\nher in order and make her study--in which, however, she did not always\nsucceed. At the end of these three years, however, my adored young\nmistress was a woman grown; her figure was round and well formed,\ngiving the finishing touch to her beauty; her face had a tenderer\nblush, a softer form, a gentler look; her large eyes were brighter but\ntheir glance was less restless and eager; her gait was more sober; her\nmovements were, I cannot say lighter nor less light, but certainly\ndifferent, though I could not, either then or now, define in what the\ndifference lay. But no change struck me so much as that in her voice,\nwhich acquired a gravity and depth very unlike the shrill gay tones in\nwhich she had been wont to call me, bewildering my common-sense and\nmaking me leave my various duties to join in her games. The bud, in\nshort, had become a rose, the chrysalis was transformed into a\nbutterfly.\n\nThen, one day--one dreadful, dismal day--my young mistress appeared\nbefore me in a long dress. This alteration made such an impression on\nme that I could not speak a word the whole day. I felt like a man who\nhas been cruelly imposed upon, and I was so vexed with her that in my\nsecret soul I found fifty reasons for seriously resenting her rapid\ndevelopment. A perfect fever of argumentativeness was fired in my\nbrain, and I debated the matter with myself in the most fervent manner\nduring my sleepless nights. The thing that utterly confounded me was\nthat the addition of a few yards of stuff to her skirts seemed\naltogether to have altered her character. That day--a thousand times\nunblessed--she spoke to me with the greatest formality, ordering me\ncoldly and even repellently to do all the things I least liked\ndoing--and she, who had so often been my accomplice and screen in\nidleness, now reproved me for it! and all this without a smile, or a\nskip, or a glance!--No more running, no more songs, no more hiding for\nme to find her, no making believe to be cross ending in a laugh--not a\nsquabble, not even a slap from her sweet little hand! It was a\nterrible crisis in my life--she was a woman and I was still a child!\n\nI need not say that this was an end to our pranks and games; I never\nagain climbed the orange-tree, which henceforth blossomed unmolested\nby my greedy devotion, and unfolded its leaves and shed its luscious\nperfume at its own sweet will; we never again scampered across the\ncourt-yard, nor trotted too and from school--I, so proud of my\nresponsibility, that I would have defended her against an army if they\nhad tried to carry her off. From that day Rosita always walked with\nthe greatest dignity and circumspection. I often observed that as she\nwent up-stairs in front of me she took care not to show an inch, not a\nline, of her pretty ankles, and this systematic concealment I felt to\nbe an insult to my dignity, for I had till lately seen a great deal\nmore than her ankles! Bless me! I can laugh now when I remember how my\nheart was ready to burst over these things.\n\nBut worse misfortunes were in store. One day in the same year as that\nof this transformation old 'Aunt' Martina, Rosario the cook, Marcial,\nand other members of the kitchen society were discussing something\nvery important. I made the best use of my ears and presently gathered\nthe most alarming hints: My young mistress was to be married. The\nthing seemed incredible for I had never heard of a lover. However, the\nparents used to arrange all these matters and the strange thing is\nthat sometimes they did not turn out badly. A young man of good family\nhad asked her hand, and her parents had consented. He came to the\nhouse accompanied by his relatives, who were some kind of counts or\nmarquises with a high-sounding title. The suitor wore a naval uniform,\nfor he served his country as a sailor, but in spite of his elegant\ncostume he was by no means attractive. This no doubt was the\nimpression he made on my young mistress, for from the first she\nmanifested a great dislike to the marriage. Her mother tried to\npersuade her, but all in vain though she drew the most flattering\npicture of the young man's excellent talents, ancient lineage and\nsplendid wealth. The young girl was not to be convinced, and answered\nall these arguments with others no less cogent.\n\nHowever, the sly baggage never said a word about the real reason,\nwhich was that she had another lover whom she really loved. This was a\nyoung artillery officer, Don Rafael Malespina, a fine-looking young\nfellow with a pleasing face. My young mistress had made his\nacquaintance in church, and the traitor Love had taken advantage of\nher while she was saying her prayers; but indeed a church has always\nseemed the fittest place, with its poetical and mysterious influences,\nfor the doors of the soul to be opened for the admission of love.\nMalespina took to lurking round the house, in which I detected him on\nvarious occasions, and this love-affair became so much talked of in\nVejer that the young naval officer came to know of it and challenged\nhis rival. My master and mistress heard the whole story when news was\nbrought to the house that Malespina had wounded his antagonist\nseverely.\n\nThe scandal caused an immense commotion. My mistress's religious\nfeelings were so much shocked by this deed that neither she nor my\nmaster could conceal their wrath, and Rosita was their first victim.\nHowever, months went by; the wounded man got well again, and as\nMalespina himself was a man of birth and wealth, there were evident\nindications in the political atmosphere of the house that Don Rafael\nwas about to be admitted. The parents of the wounded man gave up the\nsuit, and those of the conqueror appeared in their place to ask the\nhand of my sweet young mistress. After some discussion and demur the\nmatch was agreed upon.\n\nI remember the first time old Malespina came. He was a very tall,\ndry-looking man with a gaudily- waistcoat, a quantity of seals\nand ornaments hanging to his watch, and a very large sharp nose with\nwhich he seemed to be smelling every one he talked to. He was terribly\nvoluble and never allowed any one else to get a word in; he\ncontradicted everything, and it was impossible to praise anything\nwithout his saying that he had something far better. From the first I\nfelt sure he was a vain man and utterly untruthful, and my opinion was\namply justified later. My master received him with friendly\npoliteness, as well as his son who came with him. From that time the\nlover came to the house every day, sometimes alone and sometimes with\nhis father.\n\nNow a new phase came over my young mistress. Her coolness to me was so\nmarked that it verged on utter contempt. It made me understand\nclearly, for the first time, the humbleness of my condition, and I\ncursed it bitterly; I tried to argue with myself as to the claims to\nsuperiority of those who really were my superiors, asking myself, with\nreal anguish of mind, how far it was right and just that others should\nbe rich and noble and learned, while my ancestry were of such low\norigin; my sole fortune was my skin, and I hardly knew how to read.\nSeeing what the reward of my devotion was, I fully believed that there\nwas no ambition in this wide world that I dared aspire to; and it was\nnot till long after that I acquired a rational conviction that, by a\nsteady and vigorous use of my own powers, I might gain almost\neverything I was deficient in. Under the scorn with which she treated\nme I lost all confidence in myself; I never dared open my lips in her\npresence, and she inspired me with far greater awe than her parents.\nMeanwhile I attentively watched all the signs of the love that\npossessed her; I saw her sad and impatient when her lover was late; at\nevery sound of an approaching footstep her pretty face flushed and her\nblack eyes sparkled with anxiety and hope. If it was he who came in\nshe could not conceal her rapture, and then they would sit and talk\nfor hours together; but always under the eye of Do\u00f1a Francisca, for\nshe would not have allowed the young lady to have a _t\u00eate-\u00e0-t\u00eate_\nmeeting with any one, even through iron bars.\n\nHowever, they carried on an extensive correspondence, and the worst of\nit all was that I had to be the go-between and courier. That drove me\nmad!--The regular thing was that I should go out and meet the young\ngentleman at a certain place, as punctually as a clock, and he would\ngive me a note to carry to my young mistress; having discharged this\ncommission, she would give me one to take to him. How often have I\nfelt tempted to burn those letters instead of delivering them.\nHowever, luckily for me, I always kept cool enough to resist this base\ntemptation. I need hardly add that I hated Malespina; I no sooner saw\nhim come into the house than my blood boiled, and whenever he desired\nme to do anything I did it as badly and sulkily as possible, wishing\nto betray my extreme disgust. This disgust, which to them seemed\nsimply bad service, while to me it was a display of honest wrath\nworthy of a proud and noble heart, earned me many reprimands, and\nabove all it once led my young lady to make a speech that pierced me\nto the heart like the thrust of an arrow. On one occasion I heard her\nsay: \"That boy is getting so troublesome that we shall have to get rid\nof him.\"\n\nAt last the day was fixed for the wedding, and it was only a short\nwhile before that event that all I have already related took place\nwith reference to my master's project. It may therefore be easily\nunderstood that Do\u00f1a Francisca had excellent reasons for objecting to\nher husband's joining the fleet, besides her regard for his safety.\n\n\n\n\nCHAPTER VI.\n\n\nI remember very well that the day after the cuffing bestowed on me by\nDo\u00f1a Francisca in her wrath at my irreverent conduct and her intense\naversion to all naval warfare, I went out to attend my master in his\ndaily walk. He leaned upon my arm, and on the other side of him walked\nMarcial; we went slowly to suit Don Alonso's feeble pace and the\nawkwardness of the old sailor's wooden leg. It was like one of those\nprocessions in which a group of tottering and worm-eaten saints are\ncarried along on a shaky litter, threatening to fall if the pace of\nthe bearers is in the least accelerated. The two old men had no energy\nor motive power left but their brave hearts, which still acted as\ntruly as a machine just turned out of a workshop; or like the needle\nof a ship's compass which, notwithstanding its unerring accuracy,\ncould do nothing to work the crazy craft it served to guide! During\nour walk my master--after having asserted, as usual, that if Admiral\nC\u00f3rdova had only tacked to port instead of starboard the battle of\n'the 14th' would never have been lost--turned the conversation once\nmore on their grand project, and though they did not put their scheme\ninto plain words, no doubt because I was present, I gathered from what\nthey said that they intended to effect their purpose by stealth,\nquietly walking out of the house one morning without my mistress's\nknowledge.\n\nWhen we went in again indifferent matters were talked over. My master,\nwho was always amiable to his wife, was more so, that day, than ever.\nDo\u00f1a Francisca could say nothing, however trivial, that he did not\nlaugh at immoderately. He even made her a present of some trifles,\ndoing his utmost to keep her in a good humor, and it was no doubt as a\nresult of this conspicuous complaisance that my mistress was crosser\nand more peevish than I had ever seen her. No accommodation was\npossible; she quarrelled with Marcial over heaven knows what trifle,\nand desired him to quit the house that instant; she used the most\nviolent language to her husband; and during dinner, though he praised\nevery dish with unwonted warmth, the lady was implacable and went on\ngrumbling and scolding.\n\nAt last it was time for evening prayers, a solemn ceremony performed\nin the dining-room in the presence of all the household; and my\nmaster, who would not unfrequently go to sleep while he lazily\nmuttered the _Paternoster_, was that evening unusually wide awake and\nprayed with genuine fervor, his voice being heard above all the rest.\nAnother incident occurred which struck me particularly. The walls of\nthe rooms were decorated with two distinct sets of prints: sacred\nsubjects and maps--the hierarchy of Heaven on one hand and the\nsoundings all round Europe and America on the other. After supper my\nmaster was standing in the passage, studying a mariner's chart and\ntracing lines upon it with his trembling forefinger, when Do\u00f1a\nFrancisca, who had gathered some hints of the plan for evasion, and\nwho always appealed to Heaven when she caught her husband red-handed\nin any manifestations of nautical enthusiasm, came up behind him, and\nthrowing up her arms, exclaimed:\n\n\"Merciful Heaven! If you are not enough to provoke a Saint!\"\n\n\"But, my dear,\" my master timidly replied, \"I was only tracing the\ncourse taken by Alcal\u00e1 Galiano and Vald\u00e9s in the schooners _Sutil_ and\n_Mejicana_ when we went to explore the straits of Magellan. It was a\ndelightful expedition--I must have told you all about it.\"\n\n\"I shall come to burning all that paper trash!\" cried Do\u00f1a Francisca.\n\"A plague on voyages and on the wandering dog of a Jew who invented\nthem. You would do better to take some concern for the salvation of\nyour soul, for the long and the short of it is you are no chicken.\nWhat a man! to be sure--what a man to have to take care of!\"\n\nShe could not get over it; I happened to pass that way, but I cannot\nremember whether she relieved her fury by giving me a thrashing and\ndemonstrating at once the elasticity of my ears and the weight of her\nhands. The fact is that these little endearments were so frequently\nrepeated, that I cannot recollect whether I received them on this\nparticular occasion; all I remember is that my master, in spite of his\nutmost amiability, entirely failed to mollify his wife.\n\nMeanwhile I have neglected to speak of Rosita; she was in a very\nmelancholy mood, for Se\u00f1or de Malespina had not made his appearance\nall day nor written her a note; all my excursions to the market-place\nhaving proved vain. Evening came and with it grief fell on the young\ngirl's soul, for there was no hope now of seeing him till next\nday--but suddenly, after supper had been ordered up, there was a loud\nknock at the door. I flew to open it, and it was he; before I opened\nit my hatred had recognized him.\n\nI fancy I can see him now as he stood before me then, shaking his\ncloak which was wet with rain. Whenever I recall that man I see him as\nI saw him then. To be frankly impartial, I must say he was a very\nhandsome young fellow, with a fine figure, good manners, and a\npleasant expression; rather cold and reserved at first, grave and\nextremely courteous with the solemn and rather exaggerated politeness\nof the old school. He was dressed that evening in a frock-coat, with\nriding breeches and top boots; he wore a Portuguese hat and a very\nhandsome cloak of scarlet cloth, lined with silk, which was the height\nof fashion with the gilded youth of that time.\n\nAs soon as he had come in I saw that something serious had happened.\nHe went into the dining-room where all were much surprised to see him\nat so late an hour, for he never called in the evening; but my young\nmistress had hardly time to be glad before she understood that this\nunexpected visit was connected with some painful occasion.\n\n\"I have come to take leave of you,\" said Malespina. They all sat\nstupefied, and Rosita turned as white as the paper on which I am\nwriting; then she turned scarlet and then again as pale as death.\n\n\"But what has happened? Where are you going Don Rafael?\" asked my\nmistress. I have said that Malespina was an artillery officer, but I\ndid not mention that he was stationed at Cadiz and at Vejer only on\nleave.\n\n\"As the fleet is short of men,\" he replied, \"we are under orders to\nembark and serve on board ship. They say a battle is inevitable and\nmost of the vessels are short of gunners.\"\n\n\"Christ, Mother Mary and Saint Joseph!\" shrieked Do\u00f1a Francisca almost\nbeside herself. \"And they are taking you too? That is too much. Your\nduties are on land, my friend. Tell them to manage as best they may;\nif they want men let them find them. Upon my soul this is beyond a\njoke!\"\n\n\"But, my dear,\" said Don Alonso humbly, \"do not you see that they\nmust....\" But he could not finish his sentence, for Do\u00f1a Francisca,\nwhose cup of wrath and grief was overflowing, proceeded to\napostrophize all the potentates of the earth.\n\n\"You--\" she exclaimed, \"anything and everything seems right in your\neyes, if only it is to benefit those blessed ships of war. And who, I\nsay, who is the demon from hell who has ordered land forces on board\nship? You need not tell me.--It is Buonaparte's doing. No Spaniard\nwould have concocted such an infernal plot. Go and tell them that you\nare just going to be married. Come now,\" she added, turning to her\nhusband, \"write to Gravina and tell him that this young man cannot\njoin the squadron.\" Then, seeing that her husband only shrugged his\nshoulders, she cried:\n\n\"He is of no use whatever! Mercy on me! If only I wore trousers I\nwould be off to Cadiz and stop there till I had got you out of this\nmess.\"\n\nRosita said not a word. I who was watching her narrowly perceived how\nagitated she was. She never took her eyes off her lover, and if it had\nnot been for good manners and to keep up her dignity, she would have\ncried and sobbed loudly to relieve the grief that was almost\nsuffocating her.\n\n\"The soldier,\" said Don Alonso, \"is the slave of duty, and our young\nfriend is required by his country to serve on board ship in her\ndefence. He will gain glory in the impending struggle, and make his\nname famous by some great deed which history will record as an example\nto future generations.\"\n\n\"Oh yes--this, that and the other!\" said Do\u00f1a Francisca mimicking the\npompous tone in which her husband had made this speech. \"We know--and\nall for what? To humor those ne'er-do-weels at Madrid. Let them come\nthemselves to fire the cannons, and fight on their own account!--And\nwhen do you start?\"\n\n\"To-morrow morning. My leave is cut short and I am under orders to\nproceed at once to Cadiz.\"\n\nIt would be impossible to describe the look that came into my young\nmistress's face as she heard these words. The lovers looked at each\nother, and a long and mournful silence fell after this announcement of\nMalespina's immediate departure.\n\n\"But this is not to be borne!\" exclaimed Do\u00f1a Francisca. \"They will be\ncalling out the peasantry next--and the women too, if the whim takes\nthem. Lord of Heaven!\" she went on looking up to the ceiling with the\nglare of a pythoness, \"I do not fear to offend Thee by saying: Curses\non the inventor of ships--Curses on all who sail in them, and Curses\non the man who made the first cannon, with its thunder that is enough\nto drive one mad, and to be the death of so many poor wretches who\nnever did any harm!\"\n\nDon Alonso looked at the young officer, expecting to read some\nprotest in his face against these insults to the noble science of\ngunnery. Then he said:\n\n\"The worst of it is that the ships will lack material too and it would\nbe....\"\n\nMarcial, who had been listening at the door to the whole conversation,\ncould no longer contain himself. He came into the room saying:\n\n\"And why should they lack material?--The _Trinidad_ carries 140\nguns--32 thirty-six pounders, 34 twenty-four pounders, 36\ntwelve-pounders, 18 eighty-pounders, and 10 mortars. The _Pr\u00edncipe de\nAst\u00farias_ carries 118, the _Santa Ana_ 120, the _Rayo_ 100, the\n_Nepomuceno_, and the _San_ ...\"\n\n\"What business have you to interfere!\" exclaimed Do\u00f1a Francisca. \"And\nwhat does it matter to us whether they carry fifty or eighty?\" But\nMarcial went on with his patriotic list all the same, but in a lower\nvoice and speaking only to my master, who dared not express his\napprobation. Do\u00f1a Francisca went on:\n\n\"But for God's sake, Don Rafael, do not go. Explain that you are a\nlandsman, that you are going to be married. If Napoleon must fight,\nlet him fight alone: let him come forward and say: 'Here am I--kill\nme, you English--or let me kill you.' Why should Spain be subject to\nhis lordship's vagaries?\"\n\n\"I must admit,\" said Malespina, \"that our alliance with France has\nproved most disastrous.\"\n\n\"Then why was it made? Every one says that this Godoy is an ignorant\nfellow. You might think a nation could be governed by playing the\nguitar!\"\n\n\"After the treaty of Basle,\" the young man said, \"we were forced to\nbecome the enemies of the English, who defeated our fleet off Cape St.\nVincent.\"\n\n\"Ah! there you have it!\" exclaimed Don Alonso, striking the table\nviolently with his fist. \"If Admiral C\u00f3rdova had given the word to\ntack to port, to the vessels in front--in accordance with the simplest\nrules of strategy--the victory would have been ours. I consider that\nproved to a demonstration, and I stated my opinion at the time. But\nevery man must keep his place.\"\n\n\"The fact remains that we were beaten,\" said Malespina. \"The defeat\nmight not have led to such serious consequences if the Spanish\nministry had not signed the treaty of San Ildefonso with the French\nrepublic. That put us at the mercy of the First Consul, obliging us to\nsupport him in wars which had no aim or end but the furthering of his\nambition. The peace of Amiens was no better than a truce; England and\nFrance declared war again immediately, and then Napoleon demanded our\nassistance. We wished to remain neutral, for that treaty did not\noblige us to take any part in the second war, but he insisted on our\nco-operation with so much determination that the King of Spain, to\npacify him, agreed to pay him a subsidy of a hundred millions of\n_reales_--it was purchasing our neutrality with gold. But even so we\ndid not get what we had paid for; in spite of this enormous sacrifice\nwe were dragged into war. England forced us into it by seizing,\nwithout any justification, four of our frigates returning from America\nfreighted with bullion. After such an act of piracy the parliament of\nMadrid had no choice but to throw the country into the hands of\nNapoleon, and that was exactly what he wished. Our navy agreed to\nsubmit to the decision of the First Consul--nay, he was already\nEmperor--and he, hoping to conquer the English by stratagem, sent off\nthe combined fleets to Martinique, intending to draw off the British\nnaval forces from the coasts of Europe. Thus he hoped to realize his\nfavorite dream of invading Great Britain; but this clever trick only\nserved to prove the inexperience and cowardice of the French Admiral\nwho, on his return to Europe would not share with our navy the glory\nof the battle off Finisterre. Then, in obedience to the Emperor's\norders, the combined fleets were to enter Brest. They say that\nNapoleon is furious with the French Admiral and intends to supplant\nhim immediately.\"\n\n\"But from what they say,\" Marcial began, putting his oar in again, as\nwe say, \"Monsieur Corneta wants to cancel it, and is on the look-out\nfor some action which may wipe out the black mark against him. I am\nonly too glad, for then we shall see who can do something and who\ncannot.\"\n\n\"One thing is certain,\" Malespina went on, \"the English fleet is\ncruising in our waters and means to blockade Cadiz. The Spanish\nauthorities think that our fleet ought not to go out of the bay, where\nthey have every chance of conquering the foe; but it seems that the\nFrench are determined to go out to sea.\"\n\n\"We shall see,\" said my master. \"It cannot fail to be a glorious\nbattle, any way.\"\n\n\"Glorious! yes....\" replied Malespina. \"But who can promise that\nfortune shall favor us. You sailors indulge in many illusions and,\nperhaps from seeing things too closely, you do not realize the\ninferiority of our fleet to that of the English. They, besides having\na splendid artillery have all the materials at hand for repairing\ntheir losses at once. As to the men, I need say nothing. The enemy's\nsailors are the best in the world--all old and experienced seamen,\nwhile only too many of the Spanish vessels are manned by raw recruits,\nindifferent to their work and hardly knowing how to serve a gun; our\nmarines, again, are not all we could wish, for they have been\nsupplemented by land-forces--brave enough, no doubt, but certain to be\nsea-sick.\"\n\n\"Well, well,\" said my master, \"in the course of a few days we shall\nknow the end of it all.\"\n\n\"I know the end of it all very well,\" said Do\u00f1a Francisca. \"All these\ngentlemen--though I am far from saying they will not have gained\nglory--will come home with broken heads.\"\n\n\"What can you know about it?\" exclaimed Don Alonso, unable to conceal\nan impulse of vexation, which, however, lasted but a moment.\n\n\"More than you do,\" she retorted sharply. \"But God have you in his\nkeeping, Don Rafael, that you may come back to us safe and sound.\"\n\nThis conversation had taken place during supper, which was a\nmelancholy meal, and after Do\u00f1a Francisca's last speech no one said\nanother word. The meal ended, Malespina took a tender leave of them\nall, and as a special indulgence on so solemn an occasion the\nkind-hearted parents left the lovers together, allowing them to bid\neach other adieu at their ease and unseen, so that nothing might\nprevent their indulging in any demonstration which might relieve their\nanguish. It is evident that I was not a spectator of the scene and I\nknow nothing of what took place; but it may be supposed that no\nreticence on either side checked the expression of their feelings.\n\nWhen Malespina came out of the room he was as pale as death; he once\nmore bid farewell to my master and mistress, who embraced him\naffectionately, and was gone. When we went up to Rosita we found her\ndrowned in tears, and her grief was so desperate that her devoted\nparents could not soothe her by any persuasion or argument, nor revive\nher energy by any of the remedies for which I was sent backwards and\nforwards to the apothecary. I must confess that I was so deeply\ngrieved at the distress of these hapless lovers that my rancorous\nfeelings against Malespina died away in my breast. A boy's heart is\neasily appeased, and mine was always open to gentle and generous\nimpulses.\n\n\n\n\nCHAPTER VII.\n\n\nThe following morning had a great surprise in store for me, and my\nmistress was thrown into the most violent passion I suppose she can\never have known in her life. When I got up I perceived that Don Alonso\nwas in the best of humors, and his wife even more ill-tempered than\nusual. While she was gone to mass with Rosita, I saw my master packing\nin the greatest haste, putting shirts, and other articles of clothing,\nand among them his uniform, into a portmanteau. I helped him and it\nmade me suspect that he was about to steal away; still, I was\nsurprised to see nothing of Marcial. However, his absence was\npresently accounted for; for Don Alonso, having made his rapid\narrangements, became extremely impatient till the old sailor made his\nappearance, saying: \"Here is the chaise. Let us be off before she\ncomes in.\" I took up the valise, and in a twinkling Don Alonso,\nMarcial, and I had sneaked out of the back gate so as to be seen by\nnobody; we got into the chaise, which set off as fast as the wretched\nhack could draw it and the badness of the road allowed. This, which\nwas bad enough for horses was almost impassable for vehicles; however,\nin spite of jolting that almost made us sick, we hurried as much as\npossible, and until we were fairly out of sight of the town our\nmartyrdom was allowed no respite.\n\nI enjoyed the journey immensely, for every novelty turns the brain of\na boy. Marcial could not contain himself for joy, but my master, who\nat first displayed his satisfaction with even less reticence than I,\nbecame sadder and more subdued when we had left the town behind us.\nFrom time to time he would say: \"And she will be so astonished! What\nwill she say when she goes home and does not find us!\"\n\nAs for me, my whole being seemed to expand at the sight of the\nlandscape, with the gladness and freshness of the morning, and above\nall with the idea of soon seeing Cadiz and its matchless bay, crowded\nwith vessels; its gay and busy streets and its creek (the Caleta)\nwhich remained in my mind as the symbol of the most precious gift of\nlife--liberty; its Plaza, its jetty and other spots, all dear to my\nmemory. We had not gone more than three leagues when there came in\nsight two riders mounted on magnificent horses, who were fast\novertaking us and before long joined us. We had at once recognized\nthem as Malespina and his father--the tall, haggard, and chattering\nold man of whom I have already spoken. They were both much surprised\nto see Don Alonso, and still more so when he explained that he was on\nhis way to Cadiz to join a ship. The son took the announcement with\nmuch gravity; but the father, who as you will have understood was an\narrant braggart and flatterer, complimented my master in high-flown\nterms on his determination, calling him the prince of navigators, the\nmirror of sailors, and an honor to his country.\n\nWe stopped to dine at the inn at Conil. The gentlemen had what they\ncould get, and Marcial and I eat what was left, which was not much. I\nwaited at table and heard the conversation, by which means I gained a\nbetter knowledge of the elder Malespina, who at first struck me as a\nboastful liar and afterwards as the most amusing chatterbox I ever in\nmy life met with.\n\nDon Jos\u00e9 Malespina, my young mistress's intended father-in-law--no\nrelation to the famous naval officer of that name--was a retired\ncolonel of artillery, and his greatest pride was founded on his\nperfect knowledge of that branch of military science and on his\npersonal superiority in the tactics of gunnery. When he enlarged on\nthat subject his imagination seemed to gain in vividness and in\nfreedom of invention.\n\n\"Artillery,\" he said, without pausing for a moment in the act of\ndeglutition, \"is indispensable on board ships of war. What is a vessel\nwithout guns? But it is on land, Se\u00f1or Don Alonso, that the marvellous\nresults of that grand invention of the human mind are seen to the best\nadvantage. During the war in Roussillon--you know of course that I\ntook part in that campaign and that all our successes were due to my\npromptness in managing the artillery.--The battle of Masdeu--: How do\nyou suppose that was won? General Ricardos posted me on a hill with\nfour pieces, ordering me not to fire till he sent the word of command.\nBut I, not taking the same view of the case, kept quiet till a column\nof the French took up a position in front of me, in such a way as that\nmy fire raked them from end to end. Now the French troops form in file\nwith extraordinary precision. I took a very exact aim with one of my\nguns, covering the head of the foremost soldier.--Do you see? The file\nwas wonderfully straight.--I fired, and the ball took off one hundred\nand forty-two heads Sir! and the rest did not fall only because the\nfarther end of the line swerved a little. This produced the greatest\nconsternation among the enemy, but as they did not understand my\ntactics and could not see me from where they stood, they sent up\nanother column to attack our troops on my right, and that column\nshared the same fate, and another and another, till I had won the\nbattle.\"\n\n\"Well, se\u00f1or, it was wonderful!\" said my master, who, seeing the\nenormity of the lie, had no mind to trouble himself to contradict his\nfriend.\n\n\"Then in the second campaign, under the command of the Conde de la\nUnion, we gave the republicans a very pretty lesson. The defence of\nBoulou was not successful because we ran short of ammunition; but in\nspite of that I did great damage by loading a gun with the keys of the\nchurch--however, they did not go far, and as a last and desperate\nresource I loaded the cannon with my own keys, my watch, my money, a\nfew trifles I found in my pockets and, at last, with my decorations.\nThe strange thing is that one of the crosses found its billet on the\nbreast of a French general, to which it stuck as if it had been glued\nthere and did him no harm whatever. He kept it, and when he went to\nParis, the Convention condemned him to death or exile--I forget\nwhich--for having allowed himself to accept an order from the hand of\nan enemy.\"\n\n\"The devil they did!\" said my master, highly delighted with these\naudacious romances.\n\n\"When I was in England,\" continued the old soldier, \"you know of\ncourse, that I was sent for by the English to make improvements in\ntheir artillery,--I dined every day with Pitt, with Burke, with Lord\nNorth, Lord Cornwallis, and other distinguished personages, who always\ncalled me 'the amusing Spaniard.' I remember that once, when I was at\nthe Palace, they entreated me to show them what a bull-fight was like\nand I had to throw my cloak over a chair and to prick it and kill it,\nwhich vastly diverted all the court, and especially King George III.,\nwho was very great friends with me, and was always saying that I must\nsend to my country to fetch some good olive-trees. Oh! we were on the\nbest terms possible. All his anxiety was that I should teach him a few\nwords of Spanish, and above all some of our beautiful Andalusian--but\nhe could never learn more than '_otro toro_' (another bull) and\n'_vengan esos cinco_' (that makes five), and he greeted me with these\nphrases every day when I went to breakfast with him off pescadillas[2]\nand a few _ca\u00f1itas_ of Manzanilla.\"\n\n    [2] Pescadillas are a small fish peculiar to the south Atlantic\n    coast of Spain. _Ca\u00f1itas_ is the name given to certain small\n    glasses used only for drinking Manzanilla.\n\n\"That was what he took for breakfast?\"\n\n\"That was what he preferred. I had some pescadillas bottled and\nbrought from Cadiz. They kept very well by a recipe I invented and\nhave at home.\"\n\n\"Wonderful! And you succeeded in reforming the English artillery?\"\nasked my master, encouraging him to go on for he was greatly amused.\n\n\"Perfectly. I invented a cannon which could never be fired, for all\nLondon, including the ministers and parliament, came to entreat me not\nto attempt it, because they feared that the explosion would throw down\na number of houses.\"\n\n\"So that the great gun has been laid aside and forgotten?\"\n\n\"The Emperor of Russia wanted to buy it, but it was impossible to move\nit from the spot where it stood.\"\n\n\"Then you surely can get us out of our present difficulties by\ninventing a cannon to destroy the whole English fleet at one\ndischarge.\"\n\n\"Yes,\" replied Malespina. \"I have been thinking of it, and I believe I\nmay realize my idea. I will show you the calculations I have made, not\nonly with regard to increasing the calibre of guns to a fabulous\ndegree, but also for constructing armor plates to protect ships and\nbastions. It is the absorbing idea of my life.\"\n\nBy this time the meal was ended. Marcial and I disposed of the\nfragments in less than no time, and we set out again; the Malespinas\non horseback by the side of the chaise and we, as before, in the\ntumble-down vehicle. The effects of the dinner, and of the copious\ndraughts of liquor with which he had moistened it, had stimulated the\nold gentleman's inventive powers and he went on all the way, pouring\nout a flood of nonsense. The conversation returned to the subject with\nwhich it had begun, the war in Roussillon, and as Don Jos\u00e9 was\npreparing to relate fresh deeds of valor, my master, weary of so many\nfalsehoods, tried to divert him to something else, by saying: \"It was\na disastrous and impolitic war. We should have done better never to\nhave undertaken it.\"\n\n\"Oh! the Conde de Aranda, as you know,\" exclaimed Malespina,\n\"condemned that unlucky war with the Republic from the first. How\noften have we discussed the question--for we have been friends from\nour childhood. When I was in Aragon we lived together for six months\nat Moncayo. Indeed, it was for him that I had a very curious gun\nconstructed....\"\n\n\"Yes, Aranda was always opposed to it,\" interrupted Don Alonso,\nintercepting him on the dangerous ground of gunnery.\n\n\"So he did,\" said Don Jos\u00e9 to whom rodomontade was irresistible, \"and\nI may say that when that distinguished man so warmly advocated peace\nwith the republicans, it was because I advised it, being convinced\nfrom the first that the war was a mistake. But Godoy, who was then\nsupreme, persisted in it, simply and solely to contradict me, as I\nhave learnt since. But the best of it is that Godoy himself was\nobliged to put an end to the war in 1795, when he understood what it\nreally was, and at the same time he adopted the high-sounding title of\nPrince of Peace.\"\n\n\"How much we want a good statesman, my worthy friend,\" said my master.\n\"A man on a level with the times, who would not throw us into useless\nwars but who could maintain the dignity of the crown.\"\n\n\"Well, when I was at Madrid last year,\" continued Don Jos\u00e9, \"proposals\nwere made to me to accept the post of Secretary of State. The Queen\nwas most anxious for it--the King said nothing. I went with him every\nday to the Prado to fire a few shots.--Even Godoy would have agreed,\nrecognizing my superior qualifications; and indeed, if he had not I\nshould have had no difficulty in finding some snug little fortress\nwhere I might lock him up so that he might give me no trouble.\nHowever, I declined, preferring to live in peace in my own\ncountry-town; I left the management of public affairs in Godoy's\nhands. There you have a man whose father was a mule-boy on my\nfather-in-law's estate in Estremadura....\"\n\n\"I did not know that....\" said Don Alonso. \"Although he is a man of\nobscure origin I always supposed the Prince of Peace to belong to a\nfamily of good birth, whose fortune was impaired but whose ancestry\nwas respectable.\"\n\nAnd so the dialogue went on; Se\u00f1or Malespina uttering his falsehoods\nas if they were gospel, and my master listening with angelic calmness,\nsometimes annoyed by them, and sometimes amused at listening to such\nnonsense. If I remember rightly, Don Jos\u00e9 Maria took the credit of\nhaving advised Napoleon to the bold deeds of the 18th Brumaire.\n\nTalking of these and of other matters we reached Chiclana as night\novertook us, and my master, who was utterly tired and worn out by the\nvillainous chaise, remained in the town, while the others went on,\nbeing anxious to reach Cadiz the same night. While we were at supper\nMalespina poured out a fresh farrago of lies, and I could see that\nhis son heard them with pain, as if he were horrified at having for\nhis father the most romancing liar in the world probably. We took\nleave of them and rested there till next day when we proceeded on our\njourney by day-break, and as the road from Chiclana to Cadiz was much\neasier than that we had already traversed, we reached the end of our\njourney by about eleven o'clock in the morning, without adventure,\nsafe in body and in excellent spirits.\n\n\n\n\nCHAPTER VIII.\n\n\nI cannot describe the enthusiasm that fired my mind at the sight of\nCadiz. As soon as I had a moment to myself--as soon, that is to say,\nas my master was fairly settled in his cousin's house--I went out into\nthe streets and ran to and fro without any fixed destination,\nintoxicated as it were by the atmosphere of my beloved native city.\nAfter so long an absence all I saw attracted my attention as though it\nwere something new and beautiful. In how many of the passers-by did I\nrecognize a familiar face? everything charmed me and appealed to my\nfeelings--men, women, old folks, children--the dogs, nay the houses\neven; for my youthful imagination discovered in each a personal and\nliving individuality; I felt towards them as towards intelligent\ncreatures; they seemed to me to express, like all else, their\nsatisfaction at seeing me, and to wear, in their balconies and\nwindows, the expression of gay and cheerful faces. In short my spirit\nsaw its own gladness reflected in every surrounding object.\n\nI hurried through the streets with eager curiosity, as if I wanted to\nsee them all at once. In the Plaza San Juan I bought a handful of\nsweetmeats, less for the satisfaction of eating them than for that of\nintroducing myself under a new aspect to the sellers, whom I addressed\nas an old friend; some of them with gratitude as having been kind to\nme in my former misery and others as victims, not yet indemnified, to\nmy childish propensity for pillage. Most of them did not remember me;\nsome, however, received me with abusive language, bringing up the\ndeeds of my youth against me and making ironical remarks on my new\nfit-out and the dignity of my appearance, reducing me to flight as\nquickly as possible and damaging my appearance by pelting me with the\nrind or husks of fruit, flung by skilful hands at my new clothes.\nHowever, as I was fully convinced of my own importance, these insults\nincreased my pride more than they hurt my feelings.\n\nThen I went to the ramparts, and counted all the ships at anchor\nwithin sight. I spoke to several sailors that I met, telling them that\nI too was about to join the fleet, and asking them with eager emphasis\nwhether they had seen Nelson's fleet; and then I assured them that\n_Monsieur Corneta_ was no better than a coward and that the impending\nfight would be a grand affair. At last I reached the creek and there\nmy delight knew no bounds. I went down to the shore and, taking off my\nshoes, I leaped from rock to rock; I sought out my old comrades of\nboth sexes but I found only a few, some who were now men had taken to\nsome better mode of living, others had been impressed into the ships,\nand those who were left hardly recognized me. The undulating motion of\nthe water excited my very senses; I could not resist the\ntemptation--urged by the mysterious spell of the sea whose eloquent\nmurmurs have always sounded to me--I know not why--like a voice\ninviting me to happiness or calling me with imperious threats to rave\nand storm. I stripped myself as quick as thought and threw myself into\nthe water as if I were flying to the arms of a lover. I swam about for\nmore than an hour, happy beyond all words, and then, having dressed\nmyself, I continued my walk to the purlieus of _la Vi\u00f1a_ where, in the\ntaverns, I came across some of the most famous rascals of my young\ndays. In talking with them I gave myself out to be a man of position,\nand as such, I wasted the few _cuartos_ I possessed in treating them.\nI asked after my uncle but no one could give me any news of that\ngentleman, and after we had chatted for awhile they made me drink a\nglass of brandy which instantly went to my head and lay me prone on\nthe floor. During the crisis of my intoxication I thought the\nscoundrels were laughing at me to their hearts' content; but as soon\nas I recovered a little I sneaked out of the tavern much ashamed of\nmyself. I still had some difficulty in walking; I had to go by my own\nold home and there, at the door, I saw a coarse-looking woman frying\nblood and tripe. Much touched by recognizing the home of my childhood\nI could not help bursting into tears and the heartless woman, seeing\nthis, took it for granted it was some jest or trick to enable me to\nsteal her unsavory mess. However, I was able to take to my heels and\nso escape her clutches, postponing the expression of my emotion till a\nmore favorable opportunity.\n\nAfter this I thought I should like to see the old Cathedral, with\nwhich the tenderest memory of my childhood was inseparably linked, and\nI went into it; the interior seemed to me most beautiful; never have I\nfelt a deeper impulse of religious veneration in any church. It gave\nme a passionate desire to pray, and I did in fact throw myself on my\nknees, before the very altar where my mother had offered an _ex-voto_\nfor my escape from death. The waxen image which I believed to be an\nexact likeness of myself was still in its place which it filled with\nall the solemnity of sanctity, but it struck me as very like a\nchestnut-husk. And yet this trumpery doll, the symbol of piety and\nmaternal devotion, filled me with tender respect. I said my prayers on\nmy knees, in memory of my good mother's sufferings and death, and\ntrying to realize that she was now happy in Heaven; but as my head was\nnot yet very clear of the fumes of that accursed brandy, I stumbled\nand fell as I rose from my knees and an indignant sacristan turned me\nout into the street. A few steps took me back to the Calle del Fideo,\nwhere we were staying, and my master scolded me for being so long\nabsent. If Do\u00f1a Francisca had been cognizant of my fault I should not\nhave escaped a sound drubbing, but my master was merciful and never\nbeat me, perhaps because his conscience told him he was as much a\nchild as I was.\n\nWe were staying at Cadiz in the house of a cousin of my master; and\nthe reader must allow me to describe this lady somewhat fully, for she\nwas a character deserving to be studied. Do\u00f1a Flora de Cisniega was an\nold woman who still pretended to be young. She was certainly past\nfifty, but she practised every art that might deceive the world into\nbelieving her not more than half that terrible age. As to describing\nhow she contrived to ally science and art to attain her object--that\nwould be an undertaking far beyond my slender powers. The enumeration\nof the curls and plaits, bows and ends, powders, rouges, washes and\nother extraneous matters which she employed in effecting this\nmonumental work of restoration, would exhaust the most vivid fancy;\nsuch things may be left to the indefatigable pen of the\nnovelist--this, being History, deals only with great subjects and\ncannot meddle with those elegant mysteries. As far as her appearance\nwas concerned what I remember best was the composition of her face,\nwhich all the painters of the Academy seemed to have touched up with\nrose color; I remember too that when she spoke she moved her lips with\na grimace, a mincing prudery which was intended either to diminish the\nwidth of a very wide mouth, or to conceal the gaps in her teeth from\nwhose ranks one or two proved deserters every year; but this elaborate\nattempt was so far a failure that it made her uglier rather than\nbetter looking. She was always richly drest, with pounds of powder in\nher hair, and as she was plump and fair--to judge from what was\nvisible through her open tucker, or under the transparency of gauze\nand muslin--her best chance lay in the display of such charms as are\nleast exposed to the injurious inroads of time, an art in which she\ncertainly was marvellously successful.\n\nDo\u00f1a Flora was devoted to everything antiquated, and much addicted to\npiety, but not with the genuine devoutness of Do\u00f1a Francisca; indeed\nshe was in everything diametrically the opposite of my mistress; for\nwhile Do\u00f1a Francisca hated even the glory that was won at sea, she was\nan enthusiastic admirer of all fighting-men and of the navy in\nparticular. Fired by patriotic passion--since at her mature age she\ncould not hope to feel the flame of any other--and intensely proud of\nherself as a woman and as a Spaniard, love of her country was\nsymbolized in her mind by the roar of cannon, and she thought the\ngreatness of a nation was measured by tons of gunpowder. Having no\nchildren her time was spent in gossip, picked up and passed round in a\nsmall circle of neighbors by two or three chatterboxes like herself;\nbut she also amused herself by her indefatigable mania for discussing\npublic affairs. At that time there were no newspapers, and political\ntheories, like public news, were passed on from mouth to mouth, these\nbeing even more falsified then than now, in proportion as talk is less\ntrustworthy even than print.\n\nIn all the large towns, and particularly in Cadiz, which was one of\nthe foremost cities of Spain, there were a number of idle persons who\nmade it their business always to have the latest news from Madrid and\nParis, and to be diligent in distributing it, priding themselves, in\nfact, on a mission which gained them so much consideration. Some of\nthese newsmongers would meet in the evening at Do\u00f1a Flora's house, and\nthis, seconded by excellent chocolate and still better cakes,\nattracted others eager to learn what was going on. Do\u00f1a Flora, knowing\nthat she could not hope to inspire a tender feeling or be quit of the\nburthen of her fifty years, would not have exchanged the part she was\nthus enabled to play for any other that could have been offered to\nher; for, at that time, to be the centre to which all news was\nconveyed was almost as precious a distinction as the majesty of a\nthrone. Do\u00f1a Flora and Do\u00f1a Francisca could never get on together, as\nmay easily be supposed when we consider the enthusiastic military\ntastes of one, and the pacific timidity of the other. Thus, speaking\nto Don Alonso the day we arrived, the good lady said:\n\n\"If you had always listened to your wife you might have been a common\nsailor to this day. What a woman! If I were a man and married to such\na wife I should burst up like a bomb-shell. You did very rightly not\nto follow her advice but to come to join the fleet. Why you are not an\nold man yet, Alonsito; you may still rise to the rank of commodore,\nwhich you would have been sure of if Paca had not clipped your wings,\nas we do to chickens to prevent their straying.\"\n\nWhen, presently, my master's eager curiosity made him press her for\nthe latest news, she went on:\n\n\"The most important news is that all the naval men here are extremely\ndissatisfied with the French Admiral, who displayed his incapacity in\nthe expedition to Martinique and the fight off Finisterre. He is so\ntimid and so mortally afraid of the English that, when the combined\nfleets ran in here last August, he dared not seize the cruisers\ncommanded by Collingwood though they were but three ships in all. All\nour officers are greatly disgusted at finding themselves obliged to\nserve under such a man; indeed Gravina went to Madrid to tell Godoy\nso, foreseeing some terrible disaster if the command were not placed\nin more able hands; but the minister gave him some vague answer as to\nwhy he could not venture to decide in the matter, and as Buonaparte is\nin Germany, dealing with the Austrians, he cannot be appealed to.--But\nit is said that he too is dissatisfied with Villeneuve and has\ndetermined to dismiss him; but meanwhile.... If only Napoleon would\nput the whole fleet under the command of some Spaniard--you, for\ninstance, Alonso--promoting you at once as I am sure you richly\ndeserve....\"\n\n\"Oh! I am not fit for it!\" replied my master, with his habitual\nmodesty.\n\n\"Well, to Gravina, or to Churruca, who is said to be a very first-rate\nsailor. If not I am afraid mischief will come of it. You cannot see\nthe French from here; only think, when Villeneuve's ships arrived they\nwere short of victuals and ammunition, and the authorities here did\nnot care to supply them out of the arsenal. They forwarded a complaint\nto Madrid, and as Godoy's one idea is to do what the French ambassador\nM. de Bernouville asks him, he sent orders that our allies should have\nas much of everything as they required. But this had no effect. The\ncommandant of the navy yard and the commissary of the ordnance stores\ndeclared they would deliver nothing to Villeneuve till he paid for it\nmoney down and in hard cash. This seems to me very right and fair. The\nlast misfortune that could come upon us was that these fine gentlemen\nshould take possession of the little we had left! Pretty times we live\nin! Everything is ruinously dear, and yellow fever on one side and\nhard times on the other had brought Andalusia to such a state that she\nwas not worth a doit--and now, to that you add all the miseries of\nwar. Of course the honor of the nation is the first thing and we must\ngo on now to avenge the insults we have received. I do not want to go\nback to the fight of Finisterre where, through the meanness of our\nallies, we lost the _Firme_ and the _Rafael_, two splendid ships--nor\nof the piratical seizure of the _Real C\u00e1rlos_, which was such an act\nof treachery that the Barbary pirates would have been disgraced by\nit--nor of the plunder of the four frigates--nor of the battle off\nCape St. ...\"\n\n\"That was the thing,\" interrupted my master eagerly. \"Every man must\nkeep his own place, but if Admiral C\u00f3rdova had given the word to\ntack....\"\n\n\"Yes, yes--I know,\" exclaimed Do\u00f1a Flora, who had heard the story a\nhundred times before. \"We must positively give them a thorough beating\nand we will. You, I know, are going to cover yourself with glory. It\nwill enfuriate Paca.\"\n\n\"I am of no use for fighting,\" said my master sadly. \"I am only going\nto look on, for sheer love of it and devotion to the Spanish flag.\"\n\nThe day after our arrival my master received a visit from a naval\nofficer, an old friend of his, whose face I can never forget though I\nsaw him but that once. He was a man of about five and forty, with a\nreally beautiful and gentle face and an expression of such tender\nmelancholy that to see him was to love him. He wore no wig, but his\nabundant hair, untortured by the barber into the fashionable _ailes de\npigeon_, was carelessly tied into a thick pigtail and heavily\npowdered, though with less elaborate care than was usual at that time.\nHis eyes were large and blue, his nose finely chiselled, perfect in\noutline, rather wide, but not so wide as to disfigure him--on the\ncontrary, it seemed to give distinction to his expressive countenance.\nHis chin, which was carefully shaved, was somewhat pointed, and added\nto the melancholy charm of an oval face which was indicative of\ndelicate feeling rather than of energetic determination. This noble\nexterior was well matched by the elegance of his manners--a grave\ncourtesy of which the fatuous airs of the men of the present day\nretain no trace, any more than the modish graces of our _jeunesse\ndor\u00e9e_. His figure was small, slight and even sickly looking. He\nlooked more like a scholar than a warrior, and a brow, behind which\nlofty and subtle thoughts must have lain hid, looked ill-fitted to\ndefy the horrors of battle. His fragile form, inhabited by a soul so\nfar above the common, looked as though it must succumb to the first\nshock. And yet--as I afterwards learnt--this man's heart was as brave\nas his intellect was supreme. It was Churruca.\n\nOur hero's uniform, though it was not in holes nor threadbare, bore\nthe marks of long and honorable service; afterwards, when I heard it\nauthoritatively stated that the Government owed him nine quarters'\npay, I could account for this dilapidated appearance. My master asked\nafter his wife, and I gathered from the answer that he was only lately\nmarried, which filled me with pity; it seemed to me so terrible a\nthing to be dragged off to battle in the midst of so much happiness.\nThen they talked of his ship, the _San Juan Nepomuceno_, which he\nseemed to love as much as his young wife; for, as was well known, he\nhad had it planned and fitted to his own taste, under a special\nprivilege, and had made it one of the finest ships in the Spanish\nfleet. Then of course they discussed the absorbing subject of the day:\nwhether the squadrons would or would not put out to sea and the\nCommodore expressed his opinion at much length, in very much such\nwords as these; for their substance had always remained in my memory\nso that now, by the help of dates and historical records, I can\nreconstruct his speech with considerable accuracy.\n\n\"The French admiral,\" said Churruca, \"not knowing what course to\npursue and being anxious to do something which might cast his errors\ninto oblivion, has, ever since we arrived, manifested an inclination\nto go and seek the English. On the 8th of October he wrote to Gravina,\nsaying that he wished to hold a council of war on board the\n_Bucentaure_ (Villeneuve's ship) to agree on the best course of\naction. Gravina went to the council, taking with him the Vice-Admiral\nAlava, Rear-Admirals Esca\u00f1o and Cisneros, Commodore Galiano and\nmyself. Of the French there were present Rear-Admirals Dumanoir and\nMagon and Captains Cosmas, Maistral, Villiegries, and Prigmy.\n\n\"Villeneuve having expressed his wish to go out to sea, we Spaniards\nunanimously opposed it. The discussion was warm and eager, and Alcal\u00e1\nGaliano and Magon exchanged such hard words that it must have come to\na duel if we had not intervened to pacify them. Our opposition greatly\nannoyed Villeneuve, and in the heat of argument he even threw out\ncertain insolent hints to which Gravina promptly retorted.--And indeed\nthese worthies display a curious anxiety to go forth to seek a\npowerful foe, considering that they forsook us at the battle off Cape\nFinisterre, depriving us of what would have been a victory if they had\nseconded us in time. But there are many reasons, which I fully\nexplained to the council--such as the advanced season, which render it\nfar more advantageous for us to remain in the bay, forcing them to\nform a blockade which they cannot maintain, particularly if at the\nsame time they blockade Toulon and Cartagena. We cannot but admit the\nsuperiority of the English navy, as to the completeness of their\narmament, their ample supply of ammunition, and, above all, the\nunanimity with which they man\u0153uvre.\n\n\"We--manned for the most part with less experienced crews,\ninadequately armed and provided, and commanded by a leader who\ndissatisfies everyone--might nevertheless act to advantage on the\ndefensive, inside the bay. But we shall be forced to obey, to succumb\nto the blind submission of the ministry at Madrid and put our vessels\nand men at the mercy of Buonaparte, who, in return for this servility\nhas certainly not given us a chief worthy of so much sacrifice. We\nmust go if Villeneuve orders it, but if the result is a disaster our\nopposition to his insane resolution stands on record as our acquittal.\nVilleneuve in fact is desperate; his sovereign has used harsh language\nto him, and the warning that he will be degraded from his command is\nprompting him to the maddest acts, in the hope of recovering his\ntarnished reputation, in a single day, by death or victory.\"\n\nSo spoke my master's friend. His words impressed me deeply; child as I\nstill was, I took an eager interest in the events going on around me,\nand since--reading in history all the facts to which I was then\nwitness, I have been able to aid my memory by authenticated dates so\nthat I can tell my story with considerable accuracy.\n\nWhen Churruca left us, Do\u00f1a Flora and my master sang his praises in\nthe warmest terms; praising him especially for the expedition he had\nconducted to Central America to make charts of those seas. According\nto them Churruca's merits as a navigator and a man of learning were\nsuch that Napoleon himself had made him a magnificent present and\nheaped civilities upon him. But we will leave the sailor and return to\nDo\u00f1a Flora.\n\nBy the end of the second day of our stay in her house I became aware\nof a phenomenon which disgusted me beyond measure, which was that my\nmaster's cousin seemed quite to fall in love with me; that is to say,\nthat she took it into her head that I was made to be her page. She\nnever ceased to load me with every sort of kindness, and on hearing\nthat I too was to join the fleet she bewailed herself greatly,\nswearing that it would be a pity if I should lose an arm or a leg, or\neven some less important part of my person--even if I escaped with my\nlife. Such unpatriotic pity roused my indignation, and I believe I\neven went so far as to declare, in so many words, that I was on fire\nwith warlike ardor. My gasconade delighted the old lady and she gave\nme a heap of sweetmeats to recover her place in my good graces.\n\nThe next day she made me clean her parrot's cage--a most shrewd bird\nthat talked like a preacher and woke us at all hours of the morning by\nshrieking \"_perro ingl\u00e9s!_\"--(dog of an Englishman.) Then she took me\nto mass with her, desiring me to carry her stool, and in church she\nwas incessantly looking round to see if I were there. Afterwards she\nkept me to look on while her hair was dressed--an operation that\nfilled me with dismay as I saw the catafalque of curls and puffs that\nthe hair-dresser piled on her head. Observing the stupid astonishment\nwith which I watched the skilful manipulation of this artist--a\nperfect architect of head-pieces--Do\u00f1a Flora laughed very heartily,\nand assured me that I should do better to remain with her as her page\nthan to join the fleet, adding that I ought to learn to dress her\nhair, and by acquiring the higher branches of the art I might earn my\nliving and make a figure in the world. Such a prospect, however, had\nnothing seductive to my fancy, and I told her, somewhat roughly, that\nI would rather be a soldier than a hair-dresser. This pleased her\nmightily and as I was giving up the comb for something more patriotic\nand military she was more affectionate than ever. But notwithstanding\nthat I was treated here with so much indulgence, I must confess that\nthe lady annoyed me beyond measure, and that I really preferred the\nangry cuffing and slapping of Do\u00f1a Francisca to Do\u00f1a Flora's mawkish\nattentions. This was very natural; for her ill-timed caresses, her\nprudery, the persistency with which she invited my presence, declaring\nthat she was delighted with me and my conversation, prevented my going\nwith my master on his visits to the different ships. A servant of the\nhouse accompanied him on these delightful expeditions, while I,\ndeprived of the liberty to run about Cadiz as I longed to be doing,\nwas left at home, sick of life, in the society of Do\u00f1a Flora's parrot\nand of the gentlemen who came every evening to announce whether or no\nthe fleets would quit the bay, with other matters less to the purpose\nand far more trivial.\n\nMy vexation rose to desperation when I saw Marcial come to the house,\nand he and my master went out together, though not to embark finally;\nand when, after seeing them start, my forlorn spirit lost the last\nfaint hope of being one of the party, Do\u00f1a Flora took it into her head\nthat she must have me to walk with her to the Alameda and then to\nchurch to attend vespers. This was more than I could bear and I began\nto dream of the possibility of putting a bold scheme into execution;\nof going, namely, on my own account to see one of the ships, hoping\nthat, on the quay, I might meet some sailor of my acquaintance who\nwould be persuaded to take me.\n\nI went out with the old lady and as we went along the ramparts I tried\nto linger to look at the ships, but I could not abandon myself to the\nenjoyment of the spectacle for I had to answer the hundred questions\nwith which Do\u00f1a Flora persistently persecuted me. In the course of our\nwalk we were joined by some young men and a few older ones. They all\nseemed very conceited, and were the most fashionable men of Cadiz, all\nextremely witty and elegantly dressed. Some of them were poets, or--to\nbe accurate--wrote verses though sorry ones, and I fancied I heard\nthem talking of some Academy where they met to fire shots at each\nother in rhyme, an amusement which could break no bones.\n\nAs I observed all that was going on round me, their extraordinary\nappearance fixed my attention--their effeminate gestures and, above\nall, their clothes, which to me looked preposterous. There were not\nmany persons who dressed in this style in Cadiz; and, reflecting\nafterwards on the difference between their costume and the ordinary\nclothes of the people I was in the habit of seeing, I understood that\nit was that men in general wore the Spanish habit while Do\u00f1a Flora's\nfriends followed the fashions of Madrid or of Paris. The first thing\nto attract my attention were their walking-sticks, which were twisted\nand knotted cudgels, with enormous knobs. Their chins were invisible,\nbeing hidden by the cravat, a kind of shawl wrapped round and round\nthe throat and brought across below the lips so as to form a\nprotuberance--a basket, a dish, or, better still, a barber's basin--in\nwhich the chin was quite lost. Their hair was dressed with elaborate\ndisorder, looking as if it had been done with a birch-broom rather\nthan with a comb. The corners of their hats came down to their\nshoulders; their coats, extremely short-waisted, almost swept the\nground with their skirts; their boots were pointed at the toes;\ndozens of seals and trinkets hung from their waistcoat pockets; their\nbreeches, which were striped, were fastened at the knee with a wide\nribbon, and to put the finishing stroke to these figures of fun, each\ncarried an eye-glass which, in the course of conversation, was\nconstantly applied to the right eye, half-closing the left, though\nthey would have seen perfectly well by using both.\n\nThe conversation of these gentlemen, also, turned on the plans of the\nfleet, but they varied it by discussing some ball or entertainment\nwhich they talked of a great deal, and one of them was the object of\nthe greatest admiration for the perfection with which he cut capers,\nand the lightness of his heels in dancing the gavotte.\n\nAfter chattering for some time the whole party followed Do\u00f1a Flora\ninto the church _del C\u00e1rmen_, and there, each one pulling out a\nrosary, they remained praying with much energy for some little time,\nand one of them, I remember, gave me a smart rap on the top of my head\nbecause, instead of attending devoutly to my prayers like them, I was\npaying too much attention to two flies that were buzzing round the\ntopmost curl of Do\u00f1a Flora's structure of hair. After listening to a\ntiresome sermon, which they praised as a magnificent oration, we went\nout again, and resumed our promenade; the chat was soon more lively\nthan ever; for we were joined by some other ladies dressed in the same\nstyle and among them all there was such a noisy hubbub of compliments,\nfine speeches, and witticisms, with here and there an insipid epigram,\nthat I could gather nothing from it all.\n\nAnd all this time Marcial and my dear master were arranging the day\nand hour when they should embark! While I was perhaps doomed to remain\non shore to gratify the whims of this old woman whom I positively\nloathed, with her odious petting! Would you believe that that very\nevening she insisted on it that I must remain forever in her service?\nWould you believe that she declared that she was very fond of me, and\nin proof of the fact kissed me and fondled me, desiring me to be sure\nto tell no one? Horrible spite of fate! I could not help thinking what\nmy feelings would have been if my young mistress had treated me in\nsuch a fashion. I was confused to the last degree; however, I told her\nthat I wished to join the fleet, and that when I came back she might\nkeep me if it was her fancy, but that if she did not allow me to have\nmy wish I should hate her as much as that--and I spread my arms out\nwide to express the immensity of my aversion.\n\nThen, as my master came in unexpectedly, I thought it a favorable\nopportunity for gaining my purpose by a sudden stroke of oratory which\nI had hastily prepared; I fell on my knees at his feet, declaring in\npathetic accents, that if he did not take me on board with him I\nshould fling myself into the sea in despair.\n\nMy master laughed at this performance and his cousin, pursing her\nlips, affected amusement with a grimace which made her sallow wrinkled\nface uglier than ever; but, finally, she consented. She gave me a heap\nof sweetmeats to eat on board, charged me to keep out of the way of\ndanger, and did not say another word against my embarking, as we did\nvery early next morning.\n\n\n\n\nCHAPTER IX.\n\n\nIt was the 18th of October. I can have no doubt as to the date because\nthe fleet sailed out of the bay next day. We rose very early and went\ndown to the quay, where a boat was waiting to carry us on board.\n\nImagine if you can my surprise--nay surprise do I say?--my enthusiasm,\nmy rapture, when I found myself on board the _Sant\u00edsima Trinidad_, the\nlargest vessel on the main, that floating fortress of timber which,\nseen from a distance, had appeared to my fancy some portentous and\nsupernatural creature; such a monster as alone was worthy of the\nmajesty of the seas. Each time our boat passed under the side of a\nship I examined it with a sort of religious astonishment, wondering to\nsee the hulls so huge that from the ramparts had looked so small; and\nin the wild enthusiasm that possessed me I ran the greatest danger of\nfalling into the water as I gazed in ecstasy at a figure-head--an\nobject which fascinated me more than anything else.\n\nAt last we reached the _Sant\u00edsima Trinidad_. As we approached, the\ncolossal mass loomed larger and larger, and when the launch pulled up\nalongside, lost in the black transparent void made where its vast\nshadow fell upon the water--when I saw the huge hulk lying motionless\non the dark waves which gently plashed against the side--when I looked\nup and saw the three tiers of cannon with their threatening muzzles\nthrust through the port-holes--my excitement was changed to fear; I\nturned pale and sat silent and motionless by my master's side.\n\nBut when we went up the side and stood on deck my spirits rose. The\nintricate and lofty rigging, the busy scene on the quarter-deck, the\nopen view of the sky and bay, the perfect order of everything on deck,\nfrom the hammocks lashed in a row to the bulwarks, to the capstans,\nshells, windsails and hatchways; the variety of uniforms--everything I\nsaw, in short, amazed me to such a degree that for some time I stood\nblankly gazing at the stupendous structure heedless of all else. You\ncan form no idea of any of those magnificent vessels, much less of the\n_Sant\u00edsima Trinidad_, from the wretched prints I have seen of them.\nStill less, again, from the ships of war of the present day, covered\nwith ponderous plates of iron, heavy looking, uninteresting and black,\nwith no visible details on their vast sides, looking to me for all\nthe world like enormous floating coffins. Invented by a materialistic\nage and calculated to suit the naval science of a time when steam has\nsuperseded manual labor, and the issue of a sea-fight is decided by\nthe force and impetus of the vessels, our ships are now mere\nfighting-machines, while those of that day were literally Men-of-War,\nwielding all the implements of attack and defence but trusting mainly\nto skill and valor.\n\nI, who not only see, but observe, have always been in the habit of\nassociating--perhaps to an extravagant extent--ideas and images,\nthings and persons, which in appearance seem most dissimilar or\nantagonistic. When, at a later period, I saw the cathedrals--Gothic,\nas they call them--of Castile and of Flanders, and noted the\nimpressive majesty with which those perfect and elaborate structures\nstand up among the buildings of more modern style, built only for\nutility--such as banks, hospitals, and barracks--I could never help\nremembering all the various kinds of vessels that I have seen in the\ncourse of a long life, and comparing the old ones to those Gothic\ncathedrals. Their curves, so gracefully prolonged, the predominance of\nvertical over horizontal lines, a certain indefinable poetry about\nthem--not historical only but religious too--underlying the\ncomplication of details and the play of colors brought out by the\ncaprices of the sunshine, are, no doubt, what led to this far-fetched\nassociation of ideas--the result in my mind of the romantic\nimpressions of my childhood.\n\nThe _Sant\u00edsima Trinidad_ had four decks; the largest ships in the\nWorld had but three. This giant, constructed at Havana, in 1769, of\nthe finest woods of Cuba, could reckon thirty-six years of honorable\nservice. She measured 220 feet from stem to stern, 58 feet in the\nwaist, that is to say in width, and 28 feet deep from the keel to the\ndeck, measurements which no other vessel at the time could approach.\nHer huge ribs, which were a perfect forest, supported four decks. When\nshe was first built 116 port-holes gaped in her sides which were thick\nwalls of timber; after she was enlarged in 1796 she had 130, and when\nshe was newly fitted in 1805 she was made to carry 140 guns, cannons\nand carronades. The interior was a marvel of arrangement; there were\ndecks for the guns, the forecastle for the crew, holds for stores of\nall kinds, state-cabins for the officers, the galley, the cock-pit and\nother offices. I was quite bewildered as I ran through the passages\nand endless nooks of this floating fortress. The stern cabins on the\nmain deck were a little palace within, and outside like some fantastic\ncastle; the galleries, the flag-turrets at the corners of the\npoop--exactly like the oriels of a Gothic tower--looked like huge\ncages open to the sea, whence the eye could command three quarters of\nthe horizon.\n\nNothing could be grander than the rigging--those gigantic masts thrust\nup to heaven like a menace to the storm. It was difficult to believe\nthat the wind could have strength enough to fill those vast sails. The\neye lost its way and became weary in gazing at the maze of the rigging\nwith the shrouds, stays, braces, halyards, and other ropes used to\nhaul and reef the various sails.\n\nI was standing lost in the contemplation of all these wonders when I\nfelt a heavy hand on the nape of my neck; I thought the main-mast had\nfallen on the top of me. I turned round in alarm and gave a cry of\nhorror at seeing a man who was now holding me by the ears as if he\nwere going to lift me up by them. It was my uncle.\n\n\"What are you doing here, Vermin!\" he asked, in the amiable tone that\nwas habitual with him. \"Do you want to learn the service? Hark ye\nJuan,\" he added, turning to a sailor of most sinister aspect, \"send\nthis landlubber up to the main-yard to take a walk there.\"\n\nI excused myself as best I might from the pleasure of taking a walk\non the main-yard, explaining that I was body-servant to Don Alonso\nGutierrez de Cisniega and had come on board with him. Three or four\nsailors, my affectionate uncle's particular friends, wanted to torment\nme so I decided on quitting their distinguished society and went off\nto the cabin in search of my master. An officer's toilet is no less\nelaborate on board than on shore, and when I saw the valets busied in\npowdering the heads of the heroes they waited on, I could not help\nasking myself whether this was not, of all occupations, the least\nappropriate in a man-of-war, when every minute was precious and where\neverything that was not directly serviceable to the working of the\nship was a hindrance. However, fashion was as tyrannical then as now,\nand even at such a moment as this enforced her absurd and inconvenient\nrules with inexorable rigor. The private soldiers even had to waste\ntheir valuable time in tying their pigtails, poor men! I saw them\nstanding in a line, one behind another, each one at work on the\npigtail of the man in front of him; by which ingenious device the\noperation was got through in a short space of time. Then they stuck on\ntheir fur hats, a ponderous head-piece the use of which no one was\never able to explain to me, and went to their posts if they were on\nduty or to pace the deck if they were not. The sailors did not wear\nthis ridiculous queue of hair and I do not see that their very\nsensible costume has been altered to any great extent since that time.\n\nIn the cabin I found my master eagerly conversing with the captain in\ncommand of the ship, Don Francisco Xavier de Uriarte, and the\ncommander of the squadron, Don Baltasar Hidalgo de Cisneros. From what\nI overheard I could have no doubt that the French admiral had ordered\nthe fleets to put out to sea the next morning.\n\nMarcial was highly delighted at this, and he and a knot of veteran\nsailors who held council on their own account in the forecastle,\ndiscoursed grandiloquently on the imminent fight. Their society suited\nme far better than that of my amiable uncle, for Marcial's companions\nindulged in no horse-play at my expense; and this difference was of\nitself enough to mark the difference of training in the two classes of\nsailors; for the old sea-dogs were of the pure breed originally levied\nas voluntary recruits; while the others were pressed men, almost\nwithout exception lazy, refractory, of low habits, and ignorant of the\nservice.\n\nI made much better friends with the former than with these and was\nalways present at Marcial's conferences. If I did not fear to weary\nthe reader, I might report the explanation he gave us that day of the\ndiplomatical and political causes of the war--a most comical parody of\nall he had heard said, a few nights previously, by Malespina at my\nmaster's house. I learnt from him that my young mistress' lover was on\nboard the _Nepomuceno_.\n\nAll these colloquies came round at last to the same point, the\nimpending battle. The fleet was to sail out of the bay next\nmorning--what joy! To ride the seas in this immense vessel--the\nlargest in the world; to witness a fight at sea; to see what a battle\nwas like, how cannon were fired, how the enemy's ships were\ntaken--what a splendid triumph! and then to return to Cadiz covered\nwith glory.--To say afterwards to all who cared to hear: \"Yes, I was\nthere, I was on board, I saw it all....\" To tell Rosita too,\ndescribing the glorious scene, winning her attention, her curiosity,\nher interest.--To say to her: \"Oh yes! I was in the most dangerous\nplaces and I was not afraid;\"--and to see her turn pale with alarm, or\nfaint, as she heard my tale of the horrors of the battle--and then to\nlook down in contempt on all who would ask me: \"Tell us, Gabrielito,\nwas it so terrible after all?\"--All this was more than enough to fire\nmy imagination, and I may frankly say that I would not, that day, have\nchanged places with Nelson himself.\n\nThe morning of the 19th dawned, the day I hailed so eagerly; indeed it\nhad not yet dawned when I found myself at the stern of the vessel with\nmy master, who wanted to look on at the working of the ship. After\nclearing the decks the business of starting the ship began. The huge\ntopsails were hoisted, and the heavy windlass, turning with a shrill\nclatter, dragged the anchor up from the bottom of the bay. The sailors\nclambered along the yards, while others handled the braces, obedient\nto the boatswain's call; and all the ship's voices, hitherto mute,\nfilled the air with threatening outcries. The whistles, the bell, the\ndiscordant medley of men's voices, mixed with the creaking of the\nblocks, the humming of the ropes, the flapping of the sails as they\nthrashed the mast before they caught the wind--all these various\nsounds filled the air as the huge ship got under way. The bright\nripples seemed to caress her sides, and the majestic monster made her\nway out of the bay without the slightest roll or even lurch, with a\nslow and solemn advance which was only perceptible to those on board\nby watching the apparent motion of the merchantmen lying at anchor\nand the landscape beyond.\n\nAt this moment I stood looking back at the scene behind us. And what a\nscene it was! Thirty-two men-of-war, five frigates, and two\nbrigantines, Spanish and French together--some in front, some behind,\nand some abreast of us--were bursting into sail, as it were, and\nriding before the light breeze. I never saw a lovelier morning. The\nsun flooded those lovely shores with light; a faint purple tinge\n the sea to the east, and the chain of hills which bound the\nhorizon on the side of the town seemed to be on fire in the sunrise;\nthe sky was perfectly clear excepting where, in the east, a few rose\nand golden clouds floated above the horizon. The blue sea was calm,\nand over that sea and beneath that sky the forty ships with their\nwhite sails rode forward, one of the noblest fleets that human eyes\never rested on.\n\nThe vessels did not all sail with equal speed. Some got ahead, others\nwere slow to get under way; some gained upon us, while we passed\nothers. The solemnity of their advance, the height of their masts,\ncovered with canvas, and a vague and obscure harmony which my childish\nears fancied they could detect proceeding from those glorious\nhulls--a kind of hymn, which was no doubt the effect of my own\nimagination--the loveliness of the day, the crispness of the air, the\nbeauty of the sea, which seemed to be dancing with joy outside the\ngulf at the approach of the vessels--all formed the grandest picture\nthat the mind of man can conceive of.\n\nCadiz, itself, like a moving panorama, unfolded itself before our\neyes, displaying in turn every aspect of its vast amphitheatre. The\nlow sun, illuminating the glass in its myriad windows, sprinkled it\nwith living sparks of gold, and its buildings lay so purely white\nabove the blue water that it looked as if it might have been that\nmoment called into being, or raised from the sea like the fanciful\ncity of San Genaro. I could see the wall extending from the mole as\nfar as the fort of Santa Catalina; I could distinguish the bastions of\nBonete and Orejon, and recognize the _Caleta_; and my pride rose as I\nreflected what I had risen from and where I now was. At the same time\nthe sound of the bells of the waking city came to my ear like some\nmysterious music, calling the inhabitants to early mass, with all the\nconfused clamor of the bells of a large town. Now they seemed to me to\nring gladly, and send good wishes after us--I listened to them as if\nthey were human voices bidding us God-speed; then again they tolled\nsadly and dolefully--a knell of misfortune; and as we sailed further\nand further away their music grew fainter till it was lost in space.\n\nThe fleet slowly made its way out of the bay--some of the ships taking\nseveral hours in getting fairly to sea. Marcial meanwhile made his\ncomments on each, watching their behavior, laughing them to scorn if\nthey were clumsy, and encouraging them with paternal advice if they\nwere swift and well-handled.\n\n\"What a lump that Don Federico is!\" he exclaimed as he looked at the\n_Pr\u00edncipe de Ast\u00farias_ commanded by Gravina. \"There goes _Mr.\nCorneta_!\" he exclaimed as he saw the _Bucentaure_ with Villeneuve on\nboard. \"He was a clever man that called you the _Rayo_!\" (Thunderbolt)\nhe cried ironically, as he watched the ship so named, which was the\nleast manageable of all the fleet. \"Well done _Pap\u00e1 Ignacio_!\" he\nadded, pointing to the _Santa Ana_ commanded by Alava.\n\n\"Hoist your topsail properly, senseless oaf!\" he went on, addressing\nDumanoir's ship, _Le Formidable_. \"That Frenchman keeps a hair-dresser\nto crimp the topsail and to clew up the sails with curling tongs!\"\n\nTowards evening the sky clouded over, and as night fell we could see\nCadiz, already at a great distance, gradually vanish in the mist till\nthe last faint outline became one with the darkness. The fleet then\nsteered to the Southward.\n\nAll night I kept close to Marcial, as soon as I had seen my master\ncomfortably settled in his cabin. The old sailor, eagerly listened to\nby a couple of veteran comrades and admirers, was explaining\nVilleneuve's plan of battle.\n\n\"_Mr. Corneta_,\" said he, \"has divided the fleet into four lines. The\nvanguard led by Alava consists of six vessels; the centre, likewise of\nsix, is commanded by _Mr. Corneta_ in person; the rear, again of six,\nis under Dumanoir, and the reserve of twelve ships is led by Don\nFederico. This seems to me not badly planned. I imagine that the\nFrench and Spanish ships are mixed, in order that they may not leave\nus impaled on the bull's horns as they did at Finisterre.\n\n\"From what Don Alfonso tells me the Frenchman says that if the enemy\ncomes up to leeward we are to form in line of battle and attack at\nonce.... This is very pretty talk in the state-room; but do you think\nthe _Se\u00f1orito_ will be such a booby as to come up to leeward of us? Oh\nyes--his lordship has not much brains in his figure-head and is sure\nto let himself be caught in that trap! Well! we shall see--if we see,\nwhat the Frenchman expects!--If the enemy gets to windward and attacks\nus we are to receive him in line of battle, and as he must divide to\nattack if he does not succeed in breaking our line, it will be quite\neasy to beat him. Everything is easy to _Mr. Corneta_ (applause). He\nsays too that he shall give no signals, but expects every captain to\ndo his best. If we should see what I have always prophesied, ever\nsince that accursed subsidy treaty, and that is--but I had better hold\nmy tongue.--Please God...! Well I have always told you that Mr.\nCorneta does not understand the weapons he has in his hands; there is\nnot room in his head for fifty ships. What can you think of an\nadmiral, who, the day before a battle, sends for his captains and\ntells each of them to do what he thinks will win the day.--After that!\n(Strong expressions of sympathy). However, we shall see what we shall\nsee.--But do you just tell me: If we Spanish want to scuttle a few of\nthose English ships, are we not strong enough and many enough to do\nit? Then why in the world need we ally ourselves with the French, who\nwould not allow us to do anything we had a mind to, but would have us\ndancing attendance at the end of their tow-line? Whenever we have had\nto work with them they have got us into mischief and we have had the\nworst of it. Well--may God and the Holy Virgin _del C\u00e1rmen_ be on our\nside, and rid us of our French friends for ever and ever, Amen.\"\n(Great Applause.)\n\nAll his audience agreed heartily; the discussion was continued till a\nlate hour, rising from the details of naval warfare to the science of\ndiplomacy. The night was fine and we ran before a fresh breeze--I must\nbe allowed to say \"_We_\" in speaking of the fleet. I was so proud of\nfinding myself on board the _Sant\u00edsima Trinidad_ that I began to fancy\nthat I was called to play some important part on this great occasion,\nand I could not forbear from swaggering about among the sailors to let\nthem see that I was not there for nothing.\n\n\n\n\nCHAPTER X.\n\n\nOn the morning of the 20th there was a stiff breeze blowing and the\nvessels kept at some distance from each other; but as the wind had\nmoderated soon after noon the admiral signalled that the ships were to\nform in five lines--the van, centre, and rear, and two lines of\nreserve. I was enchanted with watching the docile monsters, obediently\ntaking their places; for, although the conditions of naval man\u0153uvres\ndid not admit of great rapidity nor of perfect uniformity in the line,\nit was impossible to see them without admiration. The wind was from\nthe southwest, according to Marcial, and the fleet, catching the\nbreeze on the starboard quarter, ran towards the straits. During the\nnight a few lights were seen and by dawn on the 21st we saw\ntwenty-seven ships to windward, among which Marcial pointed out three\nas three-deckers. By eight o'clock the thirty-three vessels of the\nenemy's fleet were in sight, forming two columns. Our fleet displayed\na wide front, and to all appearance Nelson's two columns, advancing in\na wedge, were coming down upon us so as to cut our lines through the\ncentre and rear.\n\nThis was the position of the hostile fleets when the _Bucentaure_\nsignalled that we were to put about; maybe you do not understand this.\nIt means that we were to turn completely round and that whereas the\nwind was on our port side it would now be on the starboard, so that we\nshould sail in the opposite direction. The ships' heads were now\nturned northwards and this man\u0153uvre, which was intended to place us\nto windward of Cadiz so that we might reach it in case of disaster,\nwas severely criticised on board the _Trinidad_, especially by\nMarcial, who said:\n\n\"The line of battle is all broken up; it was bad before and is worse\nnow.\"\n\nIn point of fact what had been the vanguard was now in the rear and\nthe reserve ships, which as I heard said, were the best, were hindmost\nof all. The wind had fallen and the ships, being of various tonnage\nand inefficiently manned, the new line could not form with due\nprecision; some of the vessels moved quickly and rushed forward;\nothers went slowly, hanging back or losing their course, and forming a\nwide gap that broke the line before the enemy took the trouble of\ndoing it.\n\n\"Reform the line\" was now the signal; but, though a good ship answers\nher helm with wonderful docility, it is not so easy to manage as a\nhorse. As he stood watching the movements of the ships nearest to us,\nMarcial observed: \"The line is wider than the milky-way. If the\n_Se\u00f1orito_ cuts through it, Heaven help us! we shall not be able to\nsail in any sort of order; they will shave our heads for us if they\nfire upon us. They are going to give us a dose through the centre and\nhow can the _San Juan_ and the _Bahama_ come up to support us from the\nrear--or the _Neptuno_ and the _Rayo_ which are in front. (Murmurs of\napplause.) Besides, here we are to leeward and the 'great-coats' can\npick and choose where they will attack us, while all we can do is to\ndefend ourselves as best we may. All I have to say is: God get us well\nout of the scrape and deliver us from the French for ever and ever,\nAmen.\"\n\nThe sun had now nearly reached the meridian and the enemy was coming\ndown upon us.\n\n\"And is this a proper hour to begin a battle?\" asked the old sailor\nindignantly. \"Twelve o'clock in the day!\"\n\nBut he did not dare to express his views publicly and these\ndiscussions were confined to a small circle into which I, with my\neternal and insatiable curiosity, had squeezed myself. I do not know\nwhy, but it seemed to me that there was an expression of\ndissatisfaction on every face. The officers on the quarter-deck, and\nthe sailors and non-commissioned officers at the bows, stood watching\nthe ships to leeward, quite out of the line of battle, four of which\nought to have been in the centre.\n\nI forgot to mention one preliminary in which I myself had borne a\nhand. Early in the morning the decks were cleared for action, and when\nall was ready for serving the guns and working the ship, I heard some\none say: \"The sand--bring the sand.\" Marcial pulled me by the ear, and\ntaking me to one of the hatchways set me in a line with some of the\npressed men, ship's boys, and other supernumeraries. A number of\nsailors were posted on the ladders from the hatchway to the hold and\nbetween decks, and in this way were hauling up sacks of sand. Each man\nhanded one to the man next to him and so it was passed on without much\nlabor. A great quantity of sacks were thus brought up from hand to\nhand, and to my great astonishment they were emptied out on the upper\ndeck, the poop, and the forecastle, the sand being spread about so as\nto cover all the planking; and the same thing was done between decks.\nMy curiosity prompted me to ask the boy who stood next to me what this\nwas for.\n\n\"For the blood,\" he said very coolly.\n\n\"For the blood!\" I exclaimed unable to repress a shudder. I looked at\nthe sand--I looked at the men who were busily employed at this\ntask--and for a moment I felt I was a coward. However, my imagination\nreverted to the ideas which had previously filled it, and relieved my\nmind of its alarms; I thought no more of anything but victory and a\nhappy issue.\n\nEverything was ready for serving the guns and the ammunition was\npassed up from the store-rooms to the decks by a chain of men, like\nthat which had brought up the sand-bags.\n\nThe English advanced to attack us in two sections. One came straight\ndown upon us, and at its head, which was the point of the wedge,\nsailed a large ship carrying the admiral's flag. This, as I afterwards\nlearned, was the _Victory_, commanded by Nelson. At the head of the\nother line was the _Royal Sovereign_, commanded by Collingwood. All\nthese names, and the strategical plan of the battle, were not known to\nme till later.\n\nMy recollections, which are vividly distinct as to all the graphic and\npicturesque details, fail me with regard to the scheme of action which\nwas beyond my comprehension at the time. All that I picked from\nMarcial, combined with what I subsequently learnt, sufficed to give me\na good idea of the arrangement of our fleets; and for the better\nintelligence of the reader I give in the next page a list of our\nships, indicating the gaps left by those that had not come up, and the\nnationality of each.\n\nIt was now a quarter to twelve. The fatal moment was approaching. The\nanxiety was general, and I do not speak merely from what was going on\nin my own mind, for I was absorbed in watching the ship which was said\nto contain Nelson, and for some time was hardly aware of what was\ngoing on round me.\n\nSuddenly a terrible order was given by our captain--the boatswains\nrepeated it; the sailors flew to the tops; the blocks and ropes\ncreaked, the topsails flapped in the wind.\n\n\"Take in sail!\" cried Marcial, with a good round oath. \"The infernal\nidiot is making us work back.\"\n\nAnd then I understood that the _Trinidad_ was to slacken her speed so\nas to run alongside of the _Bucentaure_, because the _Victory_ seemed\nto be taking measures to run in between those two ships and so cut the\nline in the middle.\n\n                                     Neptuno, Sp.          }\n                                    Le Scipion, Fr.        }\n                                   Rayo, Sp.               }\n                                  Le Formidable, Fr.       } Front.\n                                 ---- Le Duguay Trouin, Fr.}\n                                Le Mont Blanc, Fr.         }\n                               As\u00eds, Sp.                   }\n\n                              San Augustin, Sp.            }\n                             Le H\u00e9ros, Fr.                 }\n    Victory                 Trinidad, Sp.                  }\n    _Nelson_.              Le Bucentaure, Fr.              } Centre.\n    --------------------> ---- Neptune, Fr.                }\n                         Le Redoutable, Fr.                }\n                        L'Intr\u00e9pide, Fr.                   }\n                       ---- Leandro, Sp.                   }\n      Royal\n    Sovereign         ---- Justo, Sp.                      }\n    _Collingwood_.   ---- L'Indomptable, Fr.               }\n    --------------> Santa Ana, Sp.                         } Rear.\n                   Le Fougueux, Fr.                        }\n                  Monarca, Sp.                             }\n                 Le Pluton, Fr.                            }\n\n                Bahama, Sp.                                }\n               ---- L'Aigle, Fr.                           }\n              Monta\u00f1es, Sp.                                }\n             Algeciras, Sp.                                }\n            Argonauta, Sp.                                 }\n           Swiftsure, Fr.                                  } Reserve.\n          ---- L'Argonaute, Fr.                            }\n         Ildefonso, Sp.                                    }\n        ---- L'Achille, Fr.                                }\n       Pr\u00edncipe de Ast\u00farias, Sp.                           }\n      Le Berwick, Fr.                                      }\n     Nepomuceno, Sp.                                       }\n\nIn watching the working of our vessel I could see that a great many\nof the crew had not that nimble ease which is usually characteristic\nof sailors who, like Marcial, are familiar with war and tempests.\nAmong the soldiers several were suffering from sea-sickness and were\nclinging to the ropes to save themselves from falling. There were\namong them many brave souls, especially among the volunteers, but for\nthe most part they were impressed men, obeying orders with an ill-will\nand not feeling, I am very sure, the smallest impulse of patriotism.\nAs I afterwards learnt, nothing but the battle itself made them worthy\nto fight. In spite of the wide differences in the moral stamp of all\nthese men, I believe that during the solemn moments that immediately\npreceded the first shot a thought of God came to every mortal there.\n\nSo far as I am concerned, in all my life my soul has never gone\nthrough any experiences, to compare with those of that hour. In spite\nof my youth, I was quite capable of understanding the gravity of the\noccasion, and for the first time in my life, my mind was filled with\ngrand ideas, lofty aspirations and heroic thoughts. A conviction that\nwe must conquer was so firmly rooted in my mind that I felt quite\npitiful towards the English, and wondered to see them so eagerly\nadvancing to certain destruction. For the first time too I fully\nunderstood the ideal of patriotism, and my heart responded to the\nthought with a glow of feeling such as I had never experienced before.\nUntil now my mother-country had been embodied in my mind in the\npersons of its rulers--such as the King and his famous minister, for\nwhom I felt different degrees of respect. As I knew no more of history\nthan I had picked up in the streets, it was to me a matter of course\nthat everybody's enthusiasm must be fired by knowing that the\nSpaniards had, once upon a time, killed a great number of Moors, and,\nsince then, swarms of French and of English. I considered my\ncountrymen as models of valor; but valor, as I conceived of it, was as\nlike barbarity as one egg is like another; and with such ideas as\nthese, patriotism had been to me nothing more than boastful pride in\nbelonging to a race of exterminators of Moors.\n\nBut in the pause that preceded the battle I understood the full\nsignificance of that divine word; the conception of nationality, of\ndevotion to a mother-country, was suddenly born in my soul, lighting it\nup, as it were, and revealing a thousand wonderful possibilities--as\nthe rising sun dissipates the darkness that has hidden a beautiful\nlandscape. I thought of my native land as a vast place full of people\nall united in brotherly regard--of society as divided into families,\nmarried couples to be held together, and children to be educated--of\nhonor, to be cherished and defended; I imagined an unspoken agreement\namong all these human beings to help and protect each other against any\nattack from without, and I understood that these vessels had been\nconstructed by them all for the defence of their native land; that is\nto say, for the soil on which they lived, the fields watered by their\nsweat, the homes where their ancestors had dwelt, the gardens where\ntheir children played, the colonies discovered and conquered by their\nforefathers, the harbors where their ships found shelter after long\nvoyages--the magazines where they stored their wealth--the Church which\nwas the mausoleum of those they had loved, the dwelling-place of their\nsaints, and the ark of their belief--the public places where they might\ntake their pleasure, the private homes where the venerable household\ngods, handed down from generation to generation, seemed to symbolize\nthe perpetuity of the nation--their family hearth round which the\nsmoke-dyed walls seem still to re-echo with the time-honored legends\nwith which the grand dame soothes the flightiness or the naughtiness of\nthe little ones, the street where friendly faces meet and smile--the\nfield, the sea, the sky--everything which from the moment of birth\nmakes up the sum of existence, from the crib of a pet animal to the\ntime-honored throne of the king; every object into which the soul seems\nto go forth to live, as if the body that clothes it were too narrow a\nshell.\n\nI believed too that the disputes between Spain and France or England\nwere always about something that those countries ought to give up to\nus, and in which Spain could not, on the whole, be wrong. Her\nself-defence seemed to me as legitimate as the aggression was brutal;\nand as I had always heard that justice must triumph, I never doubted\nof victory. Looking up at our red and yellow flag--the colors nearest\nto that of fire--I felt my bosom swell, and could not restrain a few\ntears of enthusiasm and excitement; I thought of Cadiz, of Vejer, of\nthe whole Spanish nation assembled, as it were, on a vast platform and\nlooking on with eager anxiety; and all this tide of emotion lifted up\nmy heart to God to whom I put up a prayer, which was neither a\n_Paternoster_ nor an _Ave_, but a gush of inspiration that came to me\nat the moment.\n\nA sudden shock startled me from my ecstasy, terrifying me with its\nviolent vibration. The first broadside had been fired.\n\n\n\n\nCHAPTER XI.\n\n\nA vessel in the rear had been the first to fire on the _Royal\nSovereign_, commanded by Collingwood, and while that ship carried on\nthe fight with the _Santa Ana_ the Victory came down on us. On board\nthe _Trinidad_ every one was anxious to open fire; but our captain\nwould not give the word till he saw a favorable opportunity.\nMeanwhile, as if the ships were in such close communication that a\nslow-match was lighted from one to the other, the fire ran along from\nthe _Santa Ana_ in the middle, to each end of the line.\n\nThe _Victory_ fired first on the _Redoutable_, and being repulsed,\ncame up to the windward of the _Trinidad_. The moment had come for us;\na hundred voices cried \"fire!\"--loudly echoing the word of command,\nand fifty round-shot were hurled against the flank of the English\nman-of-war. For a minute I could see nothing of the enemy for the\nsmoke, while he, as if blind with rage, came straight down upon us\nbefore the wind. Just within gun-shot he put the ship about and gave\nus a broadside. In the interval between our firing and theirs, our\ncrew, who had taken note of the damage done to the enemy, had gained\nin enthusiasm. The guns were rapidly served, though not without some\nhitches owing to want of experience in some of the gunners. Marcial\nwould have been only too glad to undertake the management of one of\nthe cannon, but his mutilated body was not equal to the heroism of his\nspirit. He was forced to be satisfied with superintending the delivery\nof the charges and encouraging the gunners by word and gesture.\n\nThe _Bucentaure_, just at our stern, was, like us, firing on the\n_Victory_ and the _T\u00e9m\u00e9raire_, another powerful English vessel. It\nseemed as though the _Victory_ must fall into our hands, for the\n_Trinidad's_ fire had cut her tackle to pieces, and we saw with pride\nthat her mizzen-mast had gone by the board.\n\nIn the excitement of this first onslaught I scarcely perceived that\nsome of our men were wounded or killed. I had chosen a place where I\nthought I should be least in the way, and never took my eyes off the\ncaptain who stood on the quarter-deck, issuing his orders with heroic\ncoolness; and I wondered to see my master, no less calm though less\nenthusiastic, encouraging the officers and men in his quavering voice.\n\n\"Ah!\" said I to myself, \"if only Do\u00f1a Francisca could see him now!\"\n\nI am bound to confess that at times I felt desperately frightened, and\nwould gladly have hidden myself at the very bottom of the hold, while,\nat others, I was filled with an almost delirious courage, when I\nlonged to see the glorious spectacle from the most dangerous posts.\nHowever, I will set aside my own insignificant individuality and\nrelate the most terrible crisis of our fight with the _Victory_. The\n_Trinidad_ was doing her immense mischief when the _T\u00e9m\u00e9raire_, by a\nwonderfully clever man\u0153uvre, slipped in between the two vessels thus\nsheltering her consort from our fire. She then proceeded to cut\nthrough the line behind the _Trinidad_, and as the _Bucentaure_, under\nfire, had got so close alongside of the _Trinidad_ that their yards\ntouched, there was a wide space beyond into which the _T\u00e9m\u00e9raire_\nrushed down and, going about immediately, came up on our lee and\ndelivered a broadside on that quarter, till then untouched. At the\nsame time the _Neptune_, another large English ship, ran in where the\n_Victory_ had previously been, while the _Victory_ veered round so\nthat, in a few minutes, the _Trinidad_ was surrounded by the enemy and\nriddled on all sides.\n\nFrom my master's face, from Uriarte's heroic fury, and from a volley\nof oaths delivered by Marcial and his friends, I understood that we\nwere lost and the idea of defeat was anguish to my soul. The line of\nthe combined fleets was broken at several points, and the bad order in\nwhich they had formed after turning round, gave place to the most\ndisastrous confusion. We were surrounded by the enemy whose artillery\nkept up a perfect hail of round and grape-shot on our ship, and on the\n_Bucentaure_ as well. The _Agustin_, the _H\u00e9ros_, and the _Leandro_\nwere engaged at some distance from us where they had rather more\nsea-room, while the _Trinidad_, and the Admiral's ship, utterly hemmed\nin and driven to extremities by the genius of the great Nelson, were\nfighting heroically--no longer in hopes of a victory which was\nimpossible but anxious, at any rate, to perish gloriously.\n\nThe white hairs which now cover my old head almost stand on end as I\nremember those terrible hours, from two to four in the afternoon. I\nthink of those five ships, not as mere machines of war obeying the\nwill of man, but as living giants, huge creatures fighting on their\nown account, carried into action by their sails as though they were\nactive limbs and using the fearful artillery they bore in their sides\nfor their personal defence. As I looked at them then, my fancy could\nnot help personifying them and to this hour I feel as though I could\nsee them coming up, defying each other, going about to fire a\nbroadside, rushing furiously up to board, drawing back to gather more\nforce, mocking or threatening the enemy;--I can fancy them expressing\ntheir suffering when wounded or loftily breathing their last, like a\ngladiator who in his agony forgets not the dignity which beseems\nhim;--I can imagine that I hear the voices of the crews like the\nmurmur of an oppressed sufferer, sometimes eager with enthusiasm,\nsometimes a dull roar of desperation the precursor of destruction,\nsometimes a hymn of triumph in anticipation of victory, or a hideous\nstorm of voices lost in space and giving way to the awful silence of\ndisgrace and defeat.\n\nThe scene on board the _Sant\u00edsima Trinidad_ was nothing short of\ninfernal. All attempt at working the ship had been abandoned, for it\ndid not and could not move. The only thing to be done was to serve the\nguns with the utmost rapidity, and to do as much damage to the enemy\nas they had done to us. The English small-shot rent the sails just as\nif huge and invisible nails were tearing slits in them. The splinters\nof timber and of masts, the stout cables cut through as if they were\nstraws, the capstans, spindles, and other heavy machinery torn from\ntheir place by the enemy's fire, strewed the deck so that there was\nscarcely room to move. Every minute men, till then full of life, fell\non deck or into the sea; the blasphemy of those who were fighting\nmingled with the cries of the wounded, till it was impossible to say\nwhether the dying were defying God or the living crying to him for\nmercy while they fought.\n\nI offered my services for a melancholy task, which was carrying the\nwounded into the cock-pit where the surgeons were busy doing their\nutmost. Some were dead before we could get them there, and others had\nto suffer painful operations before their exhausted bodies could be\nleft to repose.\n\nThen I had the extreme satisfaction of helping the carpenters who were\nconstantly employed in repairing the holes made in the ship's sides;\nbut my youth and inefficiency made me less useful than I would fain\nhave been.\n\nBlood was flowing in rivulets on the upper and lower decks and in\nspite of the sand the motion of the ship carried it from side to side\nmaking sinister patterns on the boards. The cannon-balls, fired at\nsuch a short range, mutilated those they killed in a terrible manner,\nand I saw more than one man still standing with his head blown away,\nthe force of the shock not having been great enough to fling the\nvictim into the sea, whose waters would have extinguished almost\npainlessly the last sensation of existence. Other balls struck a mast\nor against the bulwarks, carrying off a hail of hot splinters that\npierced and stung like arrows. The rifle-shots from the tops and the\nround-shot from the carronades dealt a more lingering and painful\ndeath, and there was hardly a man to be seen who did not bear the\nmarks, more or less severe, of the foe's iron and lead.\n\nThe crew--the soul of the ship--being thus thrashed by the storm of\nbattle and utterly unable to deal equal destruction, saw death at hand\nthough resolved to die with the courage of despair; and the ship\nitself--the glorious body--shivered under the cannonade. I could feel\nher shudder under the fearful blows; her timbers cracked, her beams\ncreaked, her ribs groaned like limbs on the rack, and the deck\ntrembled under my feet with audible throbs, as though the whole huge\ncreature was indignant at the sufferings of her crew. Meanwhile the\nwater was pouring in at a hundred holes in the riddled hull, and the\nhold was fast filling.\n\nThe _Bucentaure_, the Admiral's vessel, surrendered before our very\neyes. Villeneuve struck to the _Victory_. When once the leader of the\nfleet was gone, what hope was there for the other ships? The French\nflag vanished from the gallant vessel's mast and she ceased firing.\nThe _San Augustin_ and the _H\u00e9ros_ still persevered, and the _Rayo_\nand _Neptuno_, of the van, made an effort to rescue us from the enemy\nthat was battering us. I could see what was going on in the immediate\nneighborhood of the _Trinidad_, though nothing was to be seen of the\nrest of the line. The wind had fallen to a calm and the smoke settled\ndown over our heads shrouding everything in its dense white wreaths\nwhich it was impossible for eye to pierce. We could catch a glimpse\nnow and then of a distant ship, mysteriously magnified by some\ninexplicable optical effect; I believe indeed that the terror of that\nsupreme moment exaggerated every impression.\n\nPresently this dense cloud was dispersed for an instant--but in what a\nfearful manner! A tremendous explosion, louder than all the thousand\nguns of the fleet fired at once, paralyzed every man and filled every\nsoul with dread; and just as the ear was stunned by the terrific roar\nan intense flash lighted up the two fleets, rending the veil of smoke\nand revealing the whole panorama of the battle. This catastrophe had\ntaken place on the side towards the South where the rear line had been\nposted.\n\n\"A ship blown up!\" said one to another. But opinion differed as to\nwhether it was the _Santa Ana_, the _Argonauta_, the _Ildefonso_, or\nthe _Bahama_. We afterwards learnt that it was a Frenchman, the\n_Achille_. The explosion scattered in a myriad fragments what had a\nfew moments before been a noble ship of 74 guns and 600 men. But a few\nseconds after we had already forgotten the explosion in thinking only\nof ourselves.\n\nThe _Bucentaure_ having struck, the enemy's fire was directed on us,\nand our fate was sealed. The enthusiasm of the first hour was by this\nextinct in my soul; my heart quaked with terror that paralyzed my\nlimbs and smothered every other emotion excepting curiosity. This I\nfound so irresistible that I could not keep away from places where the\ndanger was greatest. My small assistance was of no great use now, for\nthe wounded were too numerous to be carried below and the guns had to\nbe served by those who had some little strength left. Among these was\nMarcial who was here, there, and everywhere, shouting and working to\nthe best of his small ability, acting as boatswain, gunner, sailor,\nand carpenter all at once, doing everything that happened to be needed\nat this awful moment. No one could have believed that, with hardly\nmore than half a body, he could have done the work of so many men. A\nsplinter had struck him on the head and the blood had stained his face\nand given him a most horrible appearance. I could see his lips move as\nhe licked the blood from them and then he spit it out viciously over\nthe side, as if he thought he could thus punish the enemy.\n\nWhat astonished me most, and indeed shocked me somewhat, was that\nMarcial even in this scene of horror could still cut a good-humored\njoke; whether to encourage his dejected comrades or only to keep his\nown courage up I do not know. The fore-mast fell with a tremendous\ncrash, covering the whole of the fore-deck with rigging, and Marcial\ncalled out to me: \"Bring the hatchets, boy; we must stow this lumber\nin Davy Jones' locker,\" and in two minutes the ropes were cut and the\nmast went overboard.\n\nThen, seeing that the enemy's fire grew hotter, he shouted to the\npurser's mate, who had come up to serve a gun: \"Daddy, order up some\ndrink for those 'great-coats,' and then they will let us alone.\"\n\nTo a soldier, who was lying like a dead creature with the pain of his\nwounds and the misery of sea-sickness, he exclaimed as he whisked the\nslow-match under his nose: \"Take a whiff of orange-flower, man, to\ncure your faintness. Would you like to take a turn in a boat? Nelson\nhas invited us to take a glass of grog with him.\"\n\nThis took place amidships; looking up at the quarter-deck I saw that\nCisneros was killed; two sailors hastily carried him down into his\ncabin. My master remained immovable at his post, but his left arm was\nbleeding severely. I ran up to help him, but before I could reach the\nspot an officer had gone to him to persuade him to retire to his\nstate-room. He had not spoken two words when a ball shot away half his\nhead and his blood sprinkled my face. Don Alonso withdrew, as pale as\nthe corpse which fell on the quarter-deck. When my master had gone\ndown the commander was left standing alone, so perfectly cool that I\ncould not help gazing at him for a few minutes, astounded by such\ncourage. His head was uncovered, his face very white, but his eyes\nflashed and his attitude was full of energy, and he stood at his\npost, commanding the desperate strife, though the battle was lost past\nretrieval. Even this fearful disaster must be conducted with due\norder, and the captain's duty was still to keep discipline over\nheroism. His voice still controlled his men in this struggle between\nhonor and death. An officer who was serving in the first battery came\nup for orders, and before he could speak he was lying dead at the feet\nof his chief; another officer of marines who was standing by his side\nfell wounded on the deck, and at last Uriarte stood quite alone on the\nquarter-deck, which was strewn with the dead and wounded. Even then he\nnever took his eyes off the English ships and the working of our\nguns--the horrible scene on the poop and in the round-house, where his\ncomrades and subalterns lay dying, could not quell his noble spirit\nnor shake his firm determination to face the fire till he too should\nfall. As I recall the fortitude and stoical calmness of Don Francisco\nXavier de Uriarte, I understand all that is told us of the heroes of\nantiquity. At that time the word Sublime was as yet unknown to me, but\nI felt that there must be, in every language under heaven, some human\nutterance to express that greatness of soul which I here saw incarnate\nand which revealed itself to me as a special grace vouchsafed by God\nto miserable humanity.\n\nBy this time most of our guns were silenced, more than half of our men\nbeing incapable of serving them. I might not, however, have been aware\nof the fact, but that being impelled by curiosity I went out of the\ncabin once more and heard a voice saying in a tone of thunder:\n\n\"Gabrielillo, come here.\"\n\nIt was Marcial who was calling me; I ran to his side and found him\ntrying to work one of the guns which had been left silent for lack of\nmen. A ball had shot away the half of his wooden leg, which made him\nexclaim: \"Well! so long as I can manage to keep the one of flesh and\nbone...!\"\n\nTwo sailors lay dead by the gun; a third, though horribly wounded,\nstill tried to go on working it.\n\n\"Let be, mate!\" said Marcial. \"You cannot even light the match,\" and\ntaking the linstock from his hand, he put it into mine, saying: \"Take\nit, Gabrielillo.--If you are afraid you had better jump overboard.\"\n\nHe loaded the cannon as quickly as he was able, helped by a ship's boy\nwho happened to come up; we ran it forward: \"fire!\" was the word, I\napplied the match and the gun went off.\n\nWe repeated this operation a second and a third time, and the roar of\nthe cannon fired by my own hand produced an extraordinary effect on my\nnerves. The feeling that I was no longer a spectator but an actor in\nthis stupendous tragedy for the moment blew all my alarms to the\nwinds; I was eager and excited, or at any rate determined to appear\nso. That moment revealed to me the truth that heroism is often simply\nthe pride of honor. Marcial's eye--the eyes of the world were upon me;\nI must bear myself worthy of their gaze.\n\n\"Oh!\" I exclaimed to myself with an impulse of pride: \"If only my\nyoung mistress could see me now!... Bravely firing cannon like a man!\"\nTwo dozen of English were the least I might have sent to the other\nworld.\n\nThese grand visions, however, did not last long for Marcial, enfeebled\nby age, was beginning to sink with exhaustion; he breathed hard as he\nwiped away the blood which flowed profusely from his head, and at last\nhis arms dropped by his side, and closing his eyes, he exclaimed: \"I\ncan do no more; the powder is rising to my head. Gabrielillo, fetch me\nsome water.\"\n\nI ran to obey him, and when I had brought the water he drank it\neagerly. This seemed to give him fresh energy; we were just about to\nload once more when a tremendous shock petrified us as we stood. The\nmain-mast, cut through by repeated shots, fell amidships and across\nthe mizzen; the ship was completely covered with the wreck, and the\nconfusion was appalling.\n\nI happily was so far under shelter that I got no harm but a slight\nblow on the head which, though it stunned me for a moment, did not\nprevent my thrusting aside the fragments of rope and timber which had\nfallen above me. The sailors and marines were struggling to clear away\nthe vast mass of lumber, but from this moment only the lower-deck guns\ncould be used at all. I got clear as best I could and went to look for\nMarcial but I did not find him, and casting my eyes up at the\nquarter-deck, I saw that the captain was no longer at his post. He had\nfallen senseless, badly wounded in the head by a splinter, and two\nsailors were just about to carry him down to the state-room. I was\nrunning forward to assist when a piece of shell hit me on the\nshoulder, terrifying me excessively, for I made sure my wound was\nmortal and that I was at my last gasp. My alarm did not hinder me from\ngoing into the cabin; I tottered from loss of blood and for a few\nminutes lay in a dead faint. I was roused from my short swoon by\nhearing the rattle of the cannon below and then a voice shouting\nvehemently:\n\n\"Board her! bring pikes!--axes!\"\n\nAnd then the confusion was so complete that it was impossible to\ndistinguish human voices from the rest of the hideous uproar. However,\nsomehow--I know not how--without thoroughly waking from my drowsy\nstate, I became aware that all was given up for lost and that the\nofficers had met in the cabin to agree to strike; nor was this the\nwork of my fancy, bewildered as I was, for I heard a voice exclaiming:\n\"The _Trinidad_ never strikes!\" I felt sure that it was Marcial's\nvoice; but at any rate some one said it.\n\nWhen I recovered perfect consciousness, I saw my master sunk on one of\nthe sofas in the cabin, his face hidden in his hands, prostrate with\ndespair, and paying no heed to his wound.\n\nI went to the heart-broken old man, who could find no way of\nexpressing his grief but by embracing me like a father, as if we were\nboth together on the brink of the grave. He, at any rate, was\nconvinced that he must soon die of grief, though his wound was by no\nmeans serious. I comforted him as best I might, assuring him that if\nthe battle were indeed lost it was not because I had failed to batter\nthe English to the best of my power; and I went on to say that we\nshould be more fortunate next time--but my childish arguments failed\nto soothe him.\n\nGoing out presently in search of water for my master, I witnessed the\nvery act of lowering the flag which was flying at the gaff, that being\none of the few spars, with the remains of the mizzen-mast, that\nremained standing. The glorious flag, the emblem of our honor, pierced\nand tattered as it was, which had gathered so many fighting-men under\nits folds, ran down the rope never to be unfurled again. The idea of\nstricken pride, of a brave spirit giving way before a superior force,\ncan find no more appropriate symbol to represent it than that of a\nflying standard which sinks and disappears like a setting sun. And our\nflag thus slowly descending that fatal evening, at the moment when we\nsurrendered, seem to shed a parting ray of glory.\n\nThe firing ceased, and the English took possession of the conquered\nvessel.\n\n\n\n\nCHAPTER XII.\n\n\nWhen the mind had sufficiently recovered from the shock and excitement\nof battle, and had time to turn from \"the pity of it\" and the chill of\nterror left by the sight of that terrific struggle, those who were\nleft alive could see the hapless vessel in all its majesty of horror.\nTill now we had thought of nothing but self-defence, but when the\nfiring ceased we could turn our attention to the dilapidated state of\nthe ship, which let in the water at a hundred leaks and was beginning\nto sink, threatening to bury us all, living and dead, at the bottom of\nthe sea. The English had scarcely taken possession when a shout arose\nfrom our sailors, as from one man:\n\n\"To the pumps!\"\n\nAll who were able flew to the pumps and labored hard at them; but\nthese ineffectual machines turned out much less water than poured in.\nSuddenly a shriek even more appalling than any we had heard before\nfilled us with horror. I have said that the wounded had been carried\ndown into the hold which, being below the water line, was secure from\nthe inroads of the cannon shot. But the water was fast gaining there,\nand some sailors came scrambling up the hatchways exclaiming that the\nwounded were being drowned. The greater part of the crew hesitated\nbetween continuing to pump and running down to rescue the hapless\nwretches; and God knows what would have happened if an English crew\nhad not come to our assistance. They not only carried up the wounded\nto the second and third deck but they lent a hand at the pumps and\ntheir carpenters set to work to stop the leaks in the ship's sides.\n\nUtterly tired out, and thinking too that Don Alonso might need my\nservices, I returned to the cabin. As I went I saw some Englishmen\nhoisting the English flag at the bows of the _Trinidad_. As I dare to\nbelieve that the amiable reader will allow me to record my feelings, I\nmay say that this incident gave me something to think of. I had always\nthought of the English as pirates or sea-highwaymen, as a race of\nadventurers not worthy to be called a nation but living by robbery.\nWhen I saw the pride with which they hauled up their flag, saluting it\nwith vociferous cheering; when I perceived the satisfaction it was to\nthem to have made a prize of the largest vessel that, until then, had\never sailed the seas, it struck me that their country, too, was dear\nto them, that her honor was in their hands and I understood that in\nthat land--to me so mysteriously remote--called England, there must\nbe, as in Spain, honorable men, a paternal king, mothers, daughters,\nwives, and sisters of these brave mariners--all watching anxiously for\ntheir return and praying to God for victory.\n\nI found my master in the cabin, somewhat calmer. The English officers\nwho had come on board treated ours with the most distinguished\ncourtesy and, as I heard, were anxious to transfer the wounded on\nboard their own ship. One of these gentlemen went up to my master as\nif recognizing him, bowed to him, and addressing him in fairly-good\nSpanish, reminded him of an old acquaintanceship. Don Alonso responded\ngravely to his advances and then enquired of him as to some of the\ndetails of the battle.\n\n\"But what became of our reserve? What did Gravina do?\" asked my\nmaster.\n\n\"Gravina withdrew with some of his ships,\" replied the English\nofficer.\n\n\"Only the _Rayo_ and _Neptuno_ came to our assistance of all the front\nline?\"\n\n\"Four French ships--the _Duguay-Trouin_, the _Mont Blanc_, the\n_Scipion_, and the _Formidable_ were the only ones that kept out of\nthe action.\"\n\n\"But Gravina--where was Gravina?\" Don Alonso persisted.\n\n\"He got off in the _Pr\u00edncipe de Ast\u00farias_; but as he was chased I do\nnot know whether he reached Cadiz in safety.\"\n\n\"And the _San Ildefonso_?\"\n\n\"She struck.\"\n\n\"And the _Santa Ana_?\"\n\n\"Struck too.\"\n\n\"Good God!\" cried my master, unable to conceal his indignation. \"But\nyou did not take the _Nepomuceno_?\"\n\n\"Yes, that too.\"\n\n\"Are you sure of that? With Churruca?\"\n\n\"He was killed,\" said the Englishman with sincere regret.\n\n\"Killed--Churruca killed!\" exclaimed Don Alonso in grievous\nbewilderment. \"And the _Bahama_--she was saved--the _Bahama_ must have\nbeen able to reach Cadiz in safety.\"\n\n\"She was taken too.\"\n\n\"Taken! And Galiano? He is a hero and a cultivated gentleman.\"\n\n\"He was,\" said the Englishman sadly, \"but he too is dead.\"\n\n\"And the _Monta\u00f1es_ with Alcedo?\"\n\n\"Killed, killed.\"\n\nMy master could not control his emotion and as, at his advanced age,\npresence of mind is lacking at such terrible moments, he suffered the\nslight humiliation of shedding a few tears as he remembered his lost\nfriends. Nor are tears unbecoming to a noble soul; on the contrary,\nthey reveal a happy infusion of delicate feeling, when combined with a\nresolute temper. My master's tears were manly tears, shed after he had\ndone his duty as a sailor; but, hastily recovering from this paroxysm\nof grief, and anxious to retort on the Englishman by some pain equal\nto that he had caused, he said:\n\n\"You too have suffered, no doubt, and have lost some men of mark?\"\n\n\"We have suffered one irreparable loss,\" said the English officer in\naccents as deeply sad as Don Alonso's. \"We have lost our greatest man,\nthe bravest of the brave--our noble, heroic, incomparable Nelson.\"\n\nAnd his fortitude holding out no better than my master's he made no\nattempt to conceal his anguish of grief; he covered his face with his\nhands and wept with the pathetic frankness of incontrollable sorrow\nfor his leader, his guardian, and his friend.\n\nNelson, mortally wounded at an early stage of the battle by a\ngun-shot--the ball piercing his chest and lodging in the spine--had\nsimply said to Captain Hardy: \"They have done for me at last, Hardy.\"\nHe lingered till the evening, not losing any details of the battle,\nand his naval and military genius only failed him with the last breath\nof his shattered body. Though suffering agonies of pain, he still\ndictated his orders and kept himself informed of the man\u0153uvres of\nboth fleets; and when at length he was assured that victory was on the\nside of the English, he exclaimed: \"Thank God, I have done my duty!\" A\nquarter of an hour later the greatest sailor of the age breathed his\nlast. The reader will forgive me this digression.\n\nIt may seem strange that we did not know the fate of many of the ships\nof the combined fleets. But nothing could be more natural than our\nignorance, considering the great length of our front and the plan of\nisolated fights contrived and carried out by the English. Their\nvessels had got mixed up with ours and the ships fought at close\nquarters; the one which had engaged us hid the rest of the squadron\nfrom view, besides which the dense smoke prevented our seeing\nanything that was not quite close to us. Towards nightfall and before\nthe firing had altogether ceased, we could distinguish a few ships in\nthe offing, looking like phantoms; some with half their rigging gone,\nand others completely dismasted. The mist, the smoke and, indeed, our\nown wearied and bewildered brains, would not allow us to distinguish\nwhether they were our own or the enemy's, and as, from time to time\nthe glare of a broadside in the distance lighted up the lugubrious\nscene, we could see that the fight was still going on to a desperate\nend between detached groups of ships, while others were flying before\nthe wind without aim or purpose, and some of ours were being towed by\nthe English to the South.\n\nNight fell, increasing the misery and horror of our situation. It\nmight have been hoped that Nature at least would be on our side after\nso much disaster; but, on the contrary, the elements lashed us with\ntheir fury as though Heaven thought our cup of misfortune was not yet\nfull. A tremendous storm burst and the winds and waves tossed and\nbuffeted our ship in their fury and, as she could not be worked, she\nwas utterly at their mercy. The rolling was so terrible that it was\nvery difficult even to work the pumps, and this, combined with the\nexhausted condition of the men, made our condition grow worse every\nminute. An English vessel, which as we learnt was the _Prince_, tried\nto take us in tow; but her efforts were in vain and she was forced to\nkeep off for fear of a collision which would have been fatal to both.\nMeanwhile it was impossible to get anything to eat, and I was dying of\nhunger, though the others seemed insensible to anything but the\nimmediate danger and gave no thought to this important matter. I dared\nnot ask for a piece of bread even, for fear of seeming greedy and\ntroublesome; but at the same time, I must confess--and without\nshame--I looked out sharply to see if there were any place where I\nmight hope to find any kind of eatable stores. Emboldened by hunger, I\nmade free to inspect the hold where the biscuit-boxes were kept, and\nwhat was my astonishment at finding Marcial there before me, stowing\nhimself with every thing he could lay his hands on. The old man's\nwound was not serious, and though a ball had carried away his right\nfoot, as this was only the lower end of his wooden leg the mishap only\nleft him a little more halt than before.\n\n\"Here, Gabrielillo,\" he said, giving me a heap of biscuits, \"take\nthese. No ship can sail without ballast.\" And then he pulled out a\nbottle and drank with intense satisfaction. As we went out of the\nbiscuit-room we saw that we were not the only visitors who had made a\nraid upon it; on the contrary, it was very evident that it had been\nwell pillaged not long since.\n\nHaving recruited my strength I could now think of trying to make\nmyself useful by lending a hand at the pumps or helping the\ncarpenters. They were laboriously repairing some of the damage done,\naided by the English, who watched all our proceedings; indeed, as I\nhave since learnt, they kept an eye on every one of our sailors, for\nthey were afraid lest we should suddenly mutiny and turn upon them to\nrecapture the vessel; in this, however, the enemy showed more\nvigilance than common-sense, for we must indeed have lost our wits\nbefore attempting to recover a ship in such a condition. However, the\n\"great-coats\" were everywhere at once, and we could not stir without\nbeing observed.\n\nNight fell, and as I was perishing with cold I quitted the deck where\nI could scarcely bear myself besides incurring constant risk of being\nswept overboard by a wave, so I went down into the cabin. My purpose\nwas to try to sleep a little while--but who could sleep in such a\nnight? The same confusion prevailed in the cabin as on deck. Those\nwho had escaped unhurt were doing what they could to aid the wounded,\nand these, disturbed by the motion of the vessel which prevented their\ngetting any rest, were so pitiable a sight that it was impossible to\nresign one's self to sleep. On one side, covered with the Spanish\nflag, lay the bodies of the officers who had been killed; and in the\nmidst of all this misery, surrounded by so much suffering, these\nsenseless corpses seemed really to be envied. They alone on board the\n_Trinidad_ were at rest, to them nothing mattered now: fatigue and\npain, the disgrace of defeat, or physical sufferings. The standard\nwhich served them as a glorious winding-sheet shut them out, as it\nwere, from the world of responsibility, of dishonor, and of despair,\nin which we were left behind. They could not care for the danger the\nvessel was in, for to them it was no longer anything but a coffin.\n\nThe officers who were killed were Don Juan Cisniega, a lieutenant in\nthe navy, who was not related to my master, in spite of their identity\nof name; Don Joaquin de Salas and Don Juan Matute, also lieutenants;\nDon Jos\u00e9 Graull\u00e9, lieutenant-colonel in the army; Urias, lieutenant in\ncommand of a frigate, and midshipman Don Antonio de Bobadilla. The\nsailors and marines whose corpses lay strewn about the gun-decks and\nupper-deck amounted to the terrible number of four hundred.\n\nNever shall I forget the moment when the bodies were cast into the\nsea, by order of the English officer in charge of the ship. The dismal\nceremony took place on the morning of the 22nd when the storm seemed\nto be at its wildest on purpose to add to the terrors of the scene.\nThe bodies of the officers were brought on deck, the priest said a\nshort prayer for this was no time for elaborate ceremonial, and our\nmelancholy task began. Each wrapped in a flag, with a cannon-ball tied\nto his feet, was dropped into the waves without any of the solemn and\npainful emotion which under ordinary circumstances would have agitated\nthe lookers-on. Our spirits were so quelled by disaster that the\ncontemplation of death had become almost indifference. Still, a burial\nat sea is more terribly sad than one on land. We cover the dead with\nearth and leave him there; those who loved him know that there is a\nspot where the dear remains are laid and can mark it with a slab, a\ncross, or a monument; but at sea--the body is cast into that heaving,\nshifting waste; it is lost forever as it disappears; imagination\ncannot follow it in its fall--down, down to the fathomless abyss; it\nis impossible to realize that it still exists at the bottom of the\ndeep. These were my reflections as I watched the corpses vanish--the\nremains of those brave fighting-men, so full of life only the day\nbefore--the pride of their country and the joy of all who loved them.\n\nThe sailors were thrown overboard with less ceremony; the regulation\nis that they shall be tied up in their hammocks, but there was no time\nto carry this out. Some indeed were wrapped round as the rules\nrequire, but most of them were thrown into the sea without any shroud\nor ball at their feet, for the simple reason that there were not\nenough for all. There were four hundred of them, more or less, and\nmerely to clear them overboard and out of sight every able-bodied man\nthat was left had to lend a hand, so as to get it done as quickly as\npossible. Much to my horror I saw myself forced to offer my services\nin the dismal duty, and many a dead man dropped over the ship's side\nat a push from my hand helping other and stronger ones.\n\nOne incident--or rather coincidence--occurred which filled me with\nhorror. A body horribly mauled and mutilated had been picked up by two\nsailors, and just as they lifted it one or two of the by-standers\nallowed themselves to utter some of those coarse and grim jests which\nare always offensive, and at such a moment revolting. I know not how\nit was that this poor wretch was the only one which moved them so\ncompletely to lose the sense of reverence due to the dead, but they\nexclaimed: \"He has been paid out for old scores--he will never be at\nhis tricks again,\" and other witticisms of the same kind. For a moment\nmy blood rose, but my indignation suddenly turned to astonishment\nmingled with an indescribable feeling of awe, regret, and aversion,\nwhen, on looking at the mangled features of the corpse, I recognized\nmy uncle. I shut my eyes with a shudder, and did not open them again\ntill the splash of the water in my face told me that he had\ndisappeared forever from mortal ken. This man had been very cruel to\nme, very cruel to his sister; still, he was my own flesh and blood, my\nmother's brother; the blood that flowed in my veins was his, and that\nsecret voice which warns us to be charitable to the faults of our own\nkith and kin could not be silenced after what I had seen, for at the\nmoment when I recognized him I had perceived in those blood-stained\nfeatures some reminder of my mother's face, and this stirred my\ndeepest feelings. I forgot that the man had been a brutal wretch, and\nall his barbarous treatment of me during my hapless childhood. I can\nhonestly declare--and I venture to do so though it is to my own\ncredit--that I forgave him with all my heart and lifted up my soul to\nGod, praying for mercy on him for all his sins.\n\nI learnt afterwards that he had behaved gallantly in the fight, but\neven this had not won him the respect of his comrades who, regarding\nhim as a low sneak, never found a good word for him--not even at that\nsupreme moment when, as a rule, every offence is forgiven on earth in\nthe belief that the sinner is rendering an account to his Maker.\n\nAs the day advanced the _Prince_ attempted once more to take the\n_Sant\u00edsima Trinidad_ in tow, but with no better success than before.\nOur situation was no worse, although the tempest raged with\nundiminished fury, for a good deal of the mischief had been patched\nup, and we thought that if the weather should mend the hulk, at any\nrate, might be saved. The English made a great point of it, for they\nwere very anxious to take the largest man-of-war ever seen afloat into\nGibraltar as a trophy; so they willingly plied the pumps by night and\nby day and allowed us to rest awhile. All through the day of the 22nd\nthe sea continued terrific, tossing the huge and helpless vessel as\nthough it were a little fishing-boat, and the enormous mass of timber\nproved the soundness of her build by not simply falling to pieces\nunder the furious lashing of the waters. At some moments she rolled\nover so completely on her beam ends that it seemed as though she must\ngo to the bottom, but suddenly the wave would fly off in smoke, as it\nwere, before the hurricane, the ship, righting herself, rode over it\nwith a toss of her mighty prow--which displayed the Lion of\nCastile--and we breathed once more with the hope of escaping with our\nlives.\n\nOn all sides we could see the scattered fleets; many of the ships were\nEnglish, severely damaged and striving to gain shelter under the\ncoast. There were Frenchmen and Spaniards too, some dismasted, others\nin tow of the enemy. Marcial recognized the _San Ildefonso_. Floating\nabout were myriads of fragments and masses of wreck--spars, timbers,\nbroken boats, hatches, bulwarks, and doors--besides two unfortunate\nsailors who were clinging to a plank, and who must have been swept off\nand drowned if the English had not hastened to rescue them. They were\nbrought on board more dead than alive, and their resuscitation after\nbeing in the very jaws of death was like a new birth to them.\n\nThat day went by between agonies and hopes--now we thought nothing\ncould save the ship and that we must be taken on board an Englishman\nthen again we hoped to keep her afloat. The idea of being taken into\nGibraltar as prisoners was intolerable, not so much to me perhaps as\nto men of punctilious honor and sensitive dignity like my master whose\nmental anguish at the thought must have been intolerable. However, all\nthe torment of suspense, at any rate, was relieved by the evening when\nit was unanimously agreed that if we were not transferred to an\nEnglish ship at once, to the bottom we must go with the vessel, which\nnow had five feet of water in the hold. Uriarte and Cisneros took the\nannouncement with dignified composure, saying that it mattered little\nto them whether they perished at once or were prisoners in a foreign\nland. The task was at once begun in the doubtful twilight, and as\nthere were above three hundred wounded to be transferred it was no\neasy matter. The available number of hands was about five hundred, all\nthat were left uninjured of the original crew of eleven hundred and\nfifteen before the battle.\n\nWe set to work promptly with the launches of the _Trinidad_ and the\n_Prince_, and three other boats belonging to the English. The wounded\nwere attended to first; but though they were lifted with all possible\ncare they could not be moved without great suffering, and some\nentreated with groans and shrieks to be left in peace, preferring\nimmediate death to anything that could aggravate and prolong their\ntorments. But there was no time for pity, and they were carried to the\nboats as ruthlessly as the cold corpses of their comrades had been\nflung into the sea.\n\nUriarte and Cisneros embarked in the English captain's gig, but when\nthey urged my master to accompany them he obstinately refused, saying\nthat he wished to be last to leave the sinking ship. This I confess\ndisturbed me not a little, for as by this time, the hardy patriotism\nwhich at first had given me courage had evaporated, I thought only of\nsaving my life, and to stay on board a foundering vessel was clearly\nnot the best means to that laudable end. Nor were my fears ill\nfounded, for not more than half the men had been taken off when a dull\nroar of terror echoed through the ship.\n\n\"She is going to the bottom--the boats, to the boats!\" shouted some,\nand there was a rush to the ship's side, all looking out eagerly for\nthe return of the boats. Every attempt at work or order was given up,\nthe wounded were forgotten, and several who had been brought on deck\ndragged themselves to the side in a sort of delirium, to seek an\nopening and throw themselves into the sea. Up through the hatchways\ncame a hideous shriek which I think I can hear as I write, freezing\nthe blood in my veins and setting my hair on end. It came from the\npoor wretches on the lowest deck who already felt the waters rising to\ndrown them and vainly cried for help--to God or men--who can tell!\nVainly indeed to men, for they had enough to do to save themselves.\nThey jumped wildly into the boats, and this confusion in the darkness\nhindered progress. One man alone, quite cool in the midst of the\ndanger, remained in the state cabin, paying no heed to all that was\ngoing on around him, walking up and down sunk in thought, as though\nthe planks he trod were not fast sinking into the gulf below. It was\nmy master. I ran to rouse him from his stupefaction. \"Sir,\" I cried,\n\"we are drowning!\"\n\nDon Alonzo did not heed me, and if I may trust my memory he merely\nsaid without looking round:\n\n\"How Paca will laugh at me, when I go home after such a terrible\ndefeat!\"\n\n\"Sir, the ship is sinking!\" I insisted, not indeed exaggerating the\ndanger, but in vehement entreaty.\n\nMy master looked at the sea, at the boats, at the men who were blindly\nand desperately leaping overboard; I looked anxiously for Marcial and\ncalled him as loudly as I could shout. At the same time I seemed to\nlose all consciousness of where I was and what was happening. I turned\ngiddy and I could see nothing. To tell how I was saved from death I\ncan only trust to the vaguest recollections, like the memory of a\ndream, for in fact I fairly swooned with terror. A sailor, as I fancy,\ncame up to Don Alonso while I was speaking to him; in his strong arms\nI felt myself lifted up and when I somewhat recovered my wits I found\nmyself in one of the boats, propped up against my master's knees,\nwhile he held my head in his hands with fatherly care and kindness.\nMarcial held the tiller and the boat was crowded with men.\n\nLooking up I saw, apparently not more than four or five yards away,\nthe black side of our ship sinking fast; but through the port-holes of\nthe deck that was still above water I could see a dim light--that of\nthe lamp which had been lighted at dusk and which still kept unwearied\nwatch over the wreck of the deserted vessel. I still could hear the\ngroans and cries of the hapless sufferers whom it had been impossible\nto remove and who were within a few feet of the abyss while, by that\ndismal lamp they could see each other's misery and read each other's\nagony in their eyes.\n\nMy fancy reverted to the dreadful scene on board--another inch of\nwater would be enough to overweight her and destroy the little\nbuoyancy that was left her. How far did those poor creatures\nunderstand the nearness of their fate? What were they saying in this\nawful moment? If they could see us safe in our boat--if they could\nhear the splash of our oars, how bitterly must their tortured souls\ncomplain to Heaven! But such agonizing martyrdom must surely avail to\npurify them of all guilt, and the grace of God must fill that hapless\nvessel, now when it was on the point of disappearing for ever!\n\nOur boat moved away; and still I watched the shapeless mass--though I\nconfess that I believe it was my imagination rather than my eyes that\ndiscerned the _Trinidad_ through the darkness, till I believe I saw,\nagainst the black sky, a huge arm reaching down to the tossing\nwaters--the effect no doubt of my imagination on my senses.\n\n\n\n\nCHAPTER XIII.\n\n\nThe boat moved on--but whither? Not Marcial himself knew where he was\nsteering her to. The darkness was so complete that we lost sight of\nthe other boats and the lights on board the _Prince_ were as invisible\nthrough the fog, as though a gust of wind had extinguished them. The\nwaves ran so high and the squalls were so violent that our frail bark\nmade very little way, but thanks to skilful steering she only once\nshipped water. We all sat silent, most of us fixing a melancholy gaze\non the spot where we supposed our deserted comrades were at this\nmoment engaged in an agonizing death-struggle. In the course of this\npassage I could not fail to make, as was my habit, certain reflections\nwhich I may venture to call philosophical. Some may laugh at a\nphilosopher of fourteen; but I will not heed their laughter; I will\ntry to write down the thoughts that occupied me at this juncture.\nChildren too can think great thoughts and at such a moment, in face\nof such a spectacle, what brain but an idiot's could remain unmoved.\n\nThere were both English and Spaniards in our boat--though most\nSpaniards--and it was strange to note how they fraternized, helping\nand encouraging each other in their common danger, and quite\nforgetting that only the day before they had been killing each other\nin hideous fight, more like wild beasts than men. I looked at the\nEnglish who rowed with as good a will as our own sailors, I saw in\ntheir faces the same tokens of fear or of hope, and above all the same\nexpression, sacred to humanity, of kindness and fellowship which was\nthe common motive of all. And as I noted it I said to myself: \"Good\nGod! why are there wars? Why cannot these men be friends under all the\ncircumstances of life as they are in danger? Is not such a scene as\nthis enough to prove that all men are brothers?\"\n\nBut the idea of nationality suddenly occurred to me to cut short these\nspeculations, and my geographical theory of islands. \"To be sure,\"\nsaid I to myself, \"the islands must need want to rob each other of\nsome portion of the land, and that is what spoils everything. And\nindeed there must be a great many bad men there who make wars for\ntheir own advantage, because they are ambitious and wish for power,\nor are avaricious and wish for wealth. It is these bad men who deceive\nthe rest--all the miserable creatures who do the fighting for them;\nand to make the fraud complete, they set them against other nations,\nsow discord and foment envy--and here you see the consequences. I am\ncertain\"--added I to myself, \"that this can never go on; I will bet\ntwo to one that before long the inhabitants of the different Islands\nwill be convinced that they are committing a great folly in making\nsuch tremendous wars, and that a day will come when they will embrace\neach other and all agree to be like one family.\" So I thought then;\nand now, after sixty years of life, I have not seen that day dawn.\n\nThe launch labored on through the heavy sea. I believe that if only my\nmaster would have consented Marcial would have been quite ready to\npitch the English overboard and steer the boat to Cadiz or the nearest\ncoast, even at the imminent risk of foundering on the way. I fancy he\nhad suggested something of the kind to Don Alonso, speaking in a low\nvoice, and that my master wished to give him a lesson in honor, for I\nheard him say:\n\n\"We are prisoners, Marcial--we are prisoners.\"\n\nThe worst of it was that no vessel came in sight. The _Prince_ had\nmoved off, and no light on either side told us of the existence of an\nEnglish ship. At last, however, we descried one at some distance and a\nfew minutes later the vague outline came in sight of a ship before the\nstorm, to our windward, and on the opposite tack to ours. Some thought\nit was a Frenchman, others said it was English; Marcial was sure she\nwas a Spaniard. We pulled hard to meet her and were soon within\nspeaking distance. Our men hailed her and the answer was in Spanish.\n\n\"It is the _San Agustin_\" said Marcial.\n\n\"The _San Agustin_ was sunk,\" said Don Alonso; \"I believe it is the\n_Santa Ana_ which was also captured.\" In fact, as we got close, we all\nrecognized the _Santa Ana_ which had gone into action under the\ncommand of Alava. The English officers in charge immediately prepared\nto take us on board, and before long we were all safe and sound on\ndeck.\n\nThe _Santa Ana_, 112 guns, had suffered severely, though not to such\nan extent as the _Sant\u00edsima Trinidad_; for, though she had lost all\nher masts and her rudder, the hull was fairly sound. The _Santa Ana_\nsurvived the battle of Trafalgar eleven years, and would have lived\nmuch longer if she had not gone to the bottom for want of repairs in\nthe bay of Havana, in 1816. She had behaved splendidly in the fight.\nShe was commanded, as I have said, by Vice-admiral Alava leader of the\nvan which, as the order of battle was altered, became the rear. As the\nreader knows, the line of English ships led by Collingwood attacked\nthe Spanish rear while Nelson took the centre. The _Santa Ana_, only\nsupported by the _Fougueux_, a Frenchman, had to fight the _Royal\nSovereign_ and four other English ships; and in spite of their unequal\nstrength one side suffered as much as the other, for Collingwood's\nship was the first to retire and the _Euryalus_ took her place. By all\naccounts the fighting was terrific, and the two great ships, whose\nmasts were almost entangled, fired into each other for six hours until\nAlava and Gardoqui, both being wounded (Alava subsequently died), five\nofficers and ninety-seven sailors being killed, besides more than 150\nwounded, the _Santa Ana_ was forced to surrender. The English took\npossession of her, but it was impossible to work her on account of her\nshattered condition, and the dreadful storm that rose during the night\nof the 21st; so when we went on board she was in a very critical,\nthough not a desperate situation, floating at the mercy of the wind\nand waves and unable to make any course. From that moment I was\ngreatly comforted by seeing that every face on board betrayed a dread\nof approaching death. They were all very sad and quiet, enduring with\na solemn mien the disgrace of defeat and the sense of being prisoners.\nOne circumstance I could not help observing, and that was that the\nEnglish officers in charge of the ship were not by a great deal so\npolite or so kind as those sent on board the _Trinidad_; on the\ncontrary, among those on the _Santa Ana_ were some who were both stern\nand repellent, doing all they could to mortify us, exaggerating their\nown dignity and authority, and interfering in everything with the\nrudest impertinence. This greatly annoyed the captured crew,\nparticularly the sailors; and I fancied I overheard many alarming\nmurmurs of rebellion which would have been highly disquieting to the\nEnglish if they had come to their ears.\n\nBeyond this there is nothing to tell of our progress that night--if\nprogress it can be called when we were driven at the will of the wind\nand waves, sailless and rudderless. Nor do I wish to weary the reader\nwith a repetition of the scenes we had witnessed on board the\n_Trinidad_, so I will go on to other and newer incidents which will\nsurprise him as much as they did me.\n\nI had lost my liking for hanging about the deck and poop, and as soon\nas we got on board the _Santa Ana_ I took shelter in the cabin with my\nmaster, hoping to get food and rest, both of which I needed sorely.\nHowever, I found there many wounded who required constant attention\nand this duty, which I gladly fulfilled, prevented my getting the\nsleep which my wearied frame required. I was engaged in placing a\nbandage on Don Alonso's arm when a hand was laid on my shoulder. I\nturned round and saw a tall young officer wrapped in a large blue\ncloak whom I did not immediately recognize; but after gazing at him\nfor a few seconds, I exclaimed aloud with surprise; it was Don Rafael\nMalespina, my young mistress's lover.\n\nMy master embraced him affectionately and he sat down by us. He had\nbeen wounded in the shoulder, and was so pale from fatigue and loss of\nblood that his face looked quite altered. His presence here filled me\nwith strange sensations--some of which I am fain to own were anything\nrather than pleasing. At first I felt glad enough indeed to see any\none I knew and who had come out alive from those scenes of horror, but\nthe next moment my old aversion for this man rose up, as strong as\never in my breast, like some dormant pain reviving to torment me after\nan interval of respite. I confess with shame that I was sorry to see\nhim safe and sound, but I must do myself the justice to add that the\nregret was but momentary, as brief as a lightning flash--a flash of\nblackness, as I may say, darkening my soul; or rather a transient\neclipse of the light of conscience which shone clearly again in the\nnext instant. The evil side of my nature for a moment came uppermost;\nbut I was able to suppress it at once and drive it down again to the\ndepths whence it had come. Can every one say as much?\n\nAfter this brief mental struggle I could look at Malespina, glad that\nhe was alive and sorry that he was hurt; and I remember, not without\npride, that I did all I could to show him my feelings. Poor little\nmistress! How terrible must her anguish have been all this time. My\nheart overflowed with pitiful kindness at the thought--I could have\nrun all the way to Vejer to say: \"Se\u00f1orita Do\u00f1a Rosa, your Don Rafael\nis safe and sound.\"\n\nThe luckless Malespina had been brought on board the _Santa Ana_ from\nthe _Nepomuceno_, which had also been captured, and with so many\nwounded on board that it had been necessary, as we learnt, to\ndistribute them or they must have perished of neglect. When the father\nand his daughter's _fianc\u00e9_ had exchanged the first greetings and\nspoken of the absent ones on shore, the conversation turned on the\ndetails of the battle. My master related all that had occurred on\nboard the _Trinidad_ and then he added: \"But no one has told me\nexactly what has become of Gravina. Was he taken prisoner, or has he\ngot off to Cadiz?\"\n\n\"The Admiral,\" said Malespina, \"stood a terrific fire from the\n_Defiance_ and the _Revenge_. The _Neptune_, a Frenchman, came to her\nassistance with the _San Ildefonso_ and the _San Justo_; but our\nenemies were reinforced by the _Dreadnought_, the _Thunderer_, and the\n_Polyphemus_; so that resistance was hopeless. Seeing the _Pr\u00edncipe de\nAst\u00farias_ with all her tackle cut, her masts overboard and her sides\nriddled with balls, while Gravina himself and Esca\u00f1o, his second in\ncommand, were both wounded, they resolved on giving up the struggle\nwhich was quite in vain for the battle was lost. Gravina hoisted the\nsignal to retire on the stump of a mast and sailed off for Cadiz,\nfollowed by the _San Justo_, the _San Leandro_, the _Monta\u00f1es_ and\nthree others; only regretting their inability to rescue the _San\nIldefonso_ which had fallen into the hands of the enemy.\"\n\n\"But tell us what happened on board the _Nepomuceno_,\" said my\nmaster, deeply interested. \"I can hardly believe that Churruca can be\ndead; and, though every one tells me that he is, I cannot help\nfancying that that wonderful man must still be alive somewhere on\nearth.\"\n\nBut Malespina told him that it had been his misfortune to see Churruca\nkilled and said he would relate every detail. A few officers gathered\nround him while I, as curious as they could be, was all ears in order\nnot to lose a syllable.\n\n\"Even as we came out of Cadiz,\" said Malespina, \"Churruca had a\npresentiment of disaster. He had voted against sailing out to sea, for\nhe knew the inferiority of our armament, and he also had little\nconfidence in Villeneuve's skill and judgment. All his predictions\nwere verified--all, even to his own death: for there is no doubt that\nhe had foreseen it as surely as he did our defeat. On the 19th he had\nsaid to Apodaca, his brother-in-law, before going on board: 'Sooner\nthan surrender my ship, I will blow her up or go to the bottom. That\nis the duty of every man who serves his king and country.' And the\nsame day, writing to a friend, he said: 'If you hear that my ship is\ntaken you will know that I am dead.'\n\n\"Indeed it was legible in his sad grave face that he looked forward\nto nothing but a catastrophe. I believe that this conviction, and the\nabsolute impossibility of avoiding defeat while feeling himself strong\nenough for his own part, seriously weighed upon his mind, for he was\nas capable of great deeds as he was of noble thoughts.\n\n\"Churruca's was a religious as well as a superior mind. On the 21st,\nat eleven in the morning, he called up all the soldiers and crew; he\nbid them all kneel and said to his chaplain in solemn tones: 'Fulfil\nyour function, holy Father, and absolve these brave souls that know\nnot what this fight may have in store for them.' When the priest had\npronounced absolution Churruca desired them to stand up, and speaking\nin friendly but audible tones he added: 'My children all:--In God's\nname I promise heavenly bliss to all who die doing their duty. If one\nof you shirks it he shall be shot on the spot; or, if he escapes my\nnotice or that of the gallant officers I have the honor to command,\nhis remorse shall pursue him so long as he crawls through the rest of\nhis miserable and dishonored days.'\n\n\"This harangue, as eloquent as it was wise, combining the ideas of\nreligion and of military duty, filled every man on board with\nenthusiasm. Alas for all these brave hearts!--wasted like gold sunk\nat the bottom of the ocean! Face to face with the English, Churruca\nwatched Villeneuve's preliminary man\u0153uvres with entire disapproval,\nand when the signal was given for the whole fleet to turn about--a\nman\u0153uvre which, as we know, reversed the order of battle--he told his\ncaptain in so many words that this blunder had lost us the day. He\nimmediately understood the masterly plan struck out by Nelson of\ncutting our line through the centre from the rear, and engaging the\nwhole fleet at once, dealing with our ships in separate divisions so\nthat they could not assist each other.\n\n\"The _Nepomuceno_ was at the end of the line. The _Royal Sovereign_\nand the _Santa Ana_ opened fire and then all the ships in turn came\ninto action. Five English vessels under Collingwood attacked our ship;\ntwo, however, passed on and Churruca had only three to deal with.\n\n\"We held out bravely against these odds till two in the afternoon,\nsuffering terribly, however, though we dealt double havoc on the foe.\nOur Admiral seemed to have infused his heroic spirit into the crew and\nsoldiers, and the ship was handled and the broadsides delivered with\nterrible promptitude and accuracy. The new recruits had learnt their\nlesson in courage in no more than a couple of hours' apprenticeship,\nand our defence struck the English not merely with dismay but with\nastonishment.\n\n\"They were in fact forced to get assistance and bring up no less than\nsix against one. The two ships that had at first sailed past now\nreturned, and the _Dreadnought_ came alongside of us, with not more\nthan half a pistol-shot between her and our stern. You may imagine the\nfire of these six giants pouring balls and small shot into a vessel of\n74 guns. But our ship seemed positively to grow bigger in proportion\nto the desperate bravery of her defenders. They themselves seemed to\ngrow in strength as their courage mounted, and seeing the dismay we\ncreated in an enemy six times as strong, we could have believed\nourselves something more than men.\n\n\"Churruca, meanwhile, who was the brain of us all, directed the action\nwith gloomy calmness. Knowing that only care and skill could supply\nthe place of strength he economized our fire, trusting entirely to\ncareful aim, and the consequence was that each ball did terrible havoc\non the foe. He saw to everything, settled everything, and the shot\nflew round him and over his head without his ever once changing color\neven. That frail and delicate man, whose beautiful and melancholy\nfeatures looked so little fitted to dare such scenes of terror,\ninspired us all with unheard-of courage, simply by a glance of his\neye.\n\n\"However, it was not the will of God that he should escape alive from\nthat storm of fire. Seeing that no one could hit one of the enemy's\nships which was battering us with impunity, he went down himself to\njudge of the line of fire and succeeded in dismasting her. He was\nreturning to the quarter-deck when a cannon ball hit his right leg\nwith such violence as almost to take it off, tearing it across the\nthigh in the most frightful manner. We rushed to support him and our\nhero sank into my arms. It was a fearful moment. I still fancy I can\nfeel his heart beating under my hand--a heart which, even at that\nterrible moment, beat only for his country. He sank rapidly. I saw him\nmake an effort to raise his head, which had fallen forward on his\nbreast; I saw him try to force a smile while his face was as white as\ndeath, and he said, in a voice that was scarcely weaker than usual:\n'It is nothing--go on firing.'\n\n\"His spirit revolted against death and he did all he could to conceal\nthe terrible sufferings of his mutilated frame, while his heart beat\nmore feebly every instant. We wanted to carry him down into the\ncabin, but nothing would persuade him to quit the quarter-deck. At\nlast he yielded to our entreaties and understood that he must give up\nthe command. He called for Moyna, his lieutenant, and was told that he\nwas dead; then he called for the officer in command of the first\nbattery, and the latter though himself seriously wounded at once\nmounted the quarter-deck and assumed the command.\n\n\"But from that moment the men lost heart; from giants they shrank to\npigmies; their courage was worn out and it was plain that we must\nsurrender. The consternation that had possessed me from the instant\nwhen our hero fell into my arms had not prevented my observing the\nterrible effect that this disaster had produced in the minds of all. A\nsudden paralysis of soul and body seemed to have fallen on the crew;\nthey all stood petrified and speechless and the grief of losing their\nbeloved leader quite overpowered the disgrace of surrender.\n\n\"Quite half of the men were dead or wounded; most of the guns were\npast serving; all the masts except the main-mast were gone by the\nboard and the rudder could not be used. Even in this deplorable plight\nwe made an attempt to follow the _Pr\u00edncipe de Ast\u00farias_ which had\ngiven the signal to retreat, but the _Nepomuceno_ was mortally\nwounded and could not move nor steer. Even then, in spite of the\nwrecked state of the ship, in spite of the dismayed condition of the\nmen, in spite of a concurrence of circumstances to render our case\nhopeless, not one of the six English captains attempted to board us.\nThey respected our ship even when she was at their mercy.\n\n\"Churruca, in the midst of his agony, ordered that the flag should be\nnailed to the mast, for the ship should never surrender so long as he\nbreathed. The delay alas! could be but brief, for Churruca was going\nrapidly, and we who supported him only wondered that a body so mangled\ncould still breathe; it was his indomitable spirit that kept him alive\nadded to a resolute determination to live, for he felt it his first\nduty. He never lost consciousness till the very end, nor complained of\nhis sufferings, nor seemed to dread his approaching death; his sole\ncare and anxiety was that the crew should not know how dangerous his\ncondition was, so that no one should fail in his duty. He desired that\nthe men should be thanked for their heroic bravery, spoke a few words\nto his brother-in-law, Ruiz de Apodaca, and, after sending a message\nto his young wife he fixed his thoughts on God, whose name we heard\nfrequently on his parched lips, and died with the calm resignation of\na just man and the fortitude of a hero; bereft of the satisfaction of\nvictory but with no angry sense of defeat. In him duty and dignity\nwere equally combined, and discipline was second only to religion. As\na soldier he was resolute, as a man he was resigned, and without a\nmurmur or an accusing word he died as nobly as he had lived. We looked\nat his body, not yet cold, and it seemed all a delusion--he must\nsurely wake to give us our orders; and we wept with less fortitude\nthan he had shown in dying, for in him we had lost all the valor and\nenthusiasm that had borne us up.\n\n\"Well, the ship struck; and when the officers from the six vessels\nthat had destroyed her came on board each claimed the honor of\nreceiving the sword of our dead hero. Each exclaimed: 'He surrendered\nto me!'--and for a few minutes they eagerly disputed the victory, each\nfor the ship he represented. Then they asked the officer who had taken\nthe command to which of the Englishmen he had struck. 'To all,' he\nreplied. 'The _Nepomuceno_ would never have surrendered to one.'\n\n\"The English gazed with sincere emotion on the body of the hapless\nChurruca, for the fame of his courage and genius was known to them\nand one of them spoke to this effect: 'A man of such illustrious\nqualities ought never to be exposed to the risks of battle; he should\nbe kept to live and serve the interests of science and navigation.'\nThen they prepared for dropping him overboard, the English marines and\nseamen forming a line of honor alongside of the Spaniards; they\nbehaved throughout like noble-minded and magnanimous gentlemen.\n\n\"The number of our wounded was very considerable, and they were\ntransferred on board other English or captured ships. It was my lot to\nbe sent to this one which has suffered worse than most; however, they\ncount more on getting her into Gibraltar than any other, now that they\nhave lost the _Trinidad_ which was the finest and most coveted of our\nships.\"\n\nThus ended Malespina's narrative which was attentively listened to as\nbeing that of an eye-witness. From what I heard I understood that a\ntragedy just as fearful as that I myself had seen had been enacted on\nboard every ship of the fleet. \"Good God!\" said I to myself, \"what\ninfinite misery! and all brought about by the obstinacy of a single\nman!\" And child as I was, I remember thinking: \"One man, however mad\nhe may be, can never commit such extravagant follies as whole nations\nsometimes plunge into at the bidding of a hundred wise ones.\"\n\n\n\n\nCHAPTER XIV.\n\n\nA large part of the night was spent in listening to Malespina's\nnarrative and the experiences of other officers. They were interesting\nenough to keep me awake and I was so excited that I found great\ndifficulty afterwards in going to sleep at all. I could not get the\nimage of Churruca out of my mind as I had seen him, handsome and\nstrong, at Do\u00f1a Flora's house. On that occasion, even, I had been\nstartled by the expression of intense sadness on the hero's features,\nas if he had a sure presentiment of his near and painful death. His\nnoble life had come to an untimely end when he was only forty-four\nyears old, after twenty-nine years of honorable service as a soldier,\na navigator, and a man of science--for Churruca was all of these,\nbesides being a noble and cultivated gentlemen. I was still thinking\nof all these things when, at length, my brain surrendered to fatigue\nand I fell asleep on the morning of the 23rd, my youthful nature\nhaving got the better of my excitement and curiosity. But in my\nsleep, which was long if not quiet, I was still haunted by nightmare\nvisions, as was natural in my overwrought state of mind, hearing the\nroar of cannon, the tumult of battle and the thunder of billows;\nmeanwhile I fancied I was serving out ammunition, climbing the\nrigging, rushing about between decks to encourage the gunners and even\nstanding on the quarter-deck in command of the vessel. I need hardly\nsay that in this curious but visionary battle I routed all the English\npast, present, or to come, with as much ease as though their ships\nwere made of paper and their cannon-balls were bread-pills. I had a\nthousand men-of-war under my command, each larger than the _Trinidad_,\nand they moved before me with as much precision as the toy-ships with\nwhich I and my comrades had been wont to play in the puddles of _la\nCaleta_.\n\nAt last, however, all this glory faded away, which, as it was but a\ndream, is scarcely to be wondered at when we see how even the reality\nvanishes. It was all over when I opened my eyes and remembered how\nsmall a part I had actually played in the stupendous catastrophe I had\nwitnessed. Still--strange to say--even when wide awake I heard cannon\nand the all-dreadful tumult of war, with shouts and a clatter that\ntold of some great turmoil on deck. I thought I must still be\ndreaming; I sat up on the sofa on which I had fallen asleep; I\nlistened with all my ears, and certainly a thundering shout of \"God\nsave the King\" left no doubt in my mind that the _Santa Ana_ was\nfighting once more.\n\nI went out of the cabin and studied the situation. The weather had\nmoderated; to the windward a few battered ships were in sight, and two\nof them, Englishmen, had opened fire on the _Santa Ana_ which was\ndefending herself with the aid of two others, a Frenchman and a\nSpaniard. I could not understand the sudden change in the aspect of\naffairs. Were we no longer prisoners of war? I looked up--our flag was\nflying in the place of the Union Jack. What could have happened?--or\nrather what was happening? For the drama was in progress.\n\nOn the quarter-deck stood a man who, I concluded, must be Alava, and\nthough suffering from several wounds he still had strength enough to\ncommand this second action, which seemed likely enough to recover the\nhonor his good ship had lost in the disaster of the first. The\nofficers were encouraging the sailors who were serving those guns that\ncould still be worked, while a detachment kept guard over the English,\nwho had been disarmed and shut up in the lower deck. Their officers\nwho had been our jailers were now become our prisoners.\n\nI understood it all. The brave commander of the _Santa Ana_, Don\nIgnacio de Alva, seeing that we were within hail of some Spanish\nships, which had come out of Cadiz in hope of rescuing some of our\ncaptured vessels and to take off the survivors from such as might be\nsinking, had addressed a stirring harangue to his disheartened crew\nwho responded to his enthusiasm by a supreme effort. By a sudden rush\nthey had disarmed the English who were in charge and hoisted the\nSpanish flag once more. The _Santa Ana_ was free, but she had to fight\nfor life, a more desperate struggle perhaps than the first had been.\n\nThis bold attempt--one of the most honorable episodes of the battle of\nTrafalgar--was made on board a dismasted ship, that had lost her\nrudder, with half her complement of men killed or wounded, and the\nother half in a wretched condition both moral and physical. However,\nthe deed once done we had to face the consequences; two Englishmen,\nconsiderably battered no doubt, fired on the _Santa Ana_; but the\n_As\u00eds_, the _Monta\u00f1es_, and the _Rayo_--three ships that had got off\nwith Gravina on the 21st--opportunely came to the rescue, having come\nout with a view to recapturing the prizes. The brave <DW36>s rushed\ninto the desperate action, with even more courage perhaps than into\nthe former battle, for their unhealed wounds spurred them to fury and\nthey seemed to fight with greater ardor in proportion as they had less\nlife to lose.\n\nAll the incidents of the dreadful 21st were repeated before my eyes;\nthe enthusiasm was tremendous, but the hands were so few that twice\nthe will and energy were needed. This heroic action fills indeed but a\nbrief page in history, for, by the side of the great event which is\nnow known as the Battle of Trafalgar, such details are dwarfed or\ndisappear altogether like a transient spark in a night of gloom and\nhorror.\n\nThe next thing that happened to me personally cost me some bitter\ntears. Not finding my master at once I felt sure he was in some\ndanger, so I went down to the upper gun-deck and there I found him,\ntraining a cannon. His trembling hand had snatched the linstock from\nthat of a wounded sailor and he was trying, with the feeble sight of\nhis right eye, to discover to what point in the foe he had better send\nthe missile. When the piece went off he turned to me trembling with\nsatisfaction, and said in a scarcely audible voice:\n\n\"Ah ha! Paca need not laugh at me now. We shall return to Cadiz in\ntriumph.\"\n\nFinally we won the fight. The English perceived the impossibility of\nrecapturing the _Santa Ana_ when, besides the three ships already\nmentioned, two other Frenchmen and a frigate came up to her assistance\nin the very thick of the fray.\n\nWe were free, and by a glorious effort; but at the very moment of\nvictory we saw most clearly the peril we were in, for the _Santa Ana_\nwas now so completely disabled that we could only be towed into Cadiz.\nThe French frigate _Themis_ sent a cable on board and put her head to\nthe North, but what could she do with such a deadweight in tow as the\n_Santa Ana_, which could do little enough to help herself with the\nragged sails that still clung to her one remaining mast? The other\nships that had supported her--the _Rayo_, the _Monta\u00f1es_, and the _San\nFrancisco de As\u00eds_, were forced to proceed at full sail to the\nassistance of the _San Juan_ and the _Bahama_, which were also in the\nhands of the English. There we were, alone, with no help but the\nfrigate that was doing her best for us--a child leading a giant. What\nwould become of us if the enemy--as was very probable--recovering\nfrom their repulse, were to fall upon us with renewed energy and\nreinforcements? However, Providence thought good to protect us; the\nwind favored us, and our frigate gently leading the way, we found\nourselves nearing Cadiz.\n\nOnly five leagues from port! What an unspeakable comfort! Our miseries\nseemed ended; ere long we should set foot on _terra firma_, and though\nwe brought news no doubt of a terrible disaster, we were bringing\nrelief and joy to many faithful souls who were suffering mortal\nanguish in the belief that those who were returning alive and well had\nall perished.\n\nThe valor of the Spaniards did not avail to rescue any ships but ours,\nfor they were too late and had to return without being able to give\nchase to the English ships that kept guard over the _San Juan_, the\n_Bahama_, and the _San Ildefonso_. We were still four leagues from\nland when we saw them making towards us. A southerly gale was blowing\nup and it was clear to all on board the _Santa Ana_ that if we did not\nsoon get into port we should have a bad time of it. Once more we were\nfilled with anxiety; once more we lost hope almost in sight of safety,\nand when a few hours more on the cruel sea would have seen us safe\nand sound in harbor. Night was coming on black and angry; the sky was\ncovered with dark clouds which seemed to lie on the face of the ocean,\nand the lurid flashes which lighted them up from time to time added\nterror to the gloom. The sea waxing in fury every instant, as if it\nwere not yet satiated, raved and roared with hungry rage, demanding\nmore and yet more victims. The remnant of the mighty fleet which a\nshort time since had defied its fury combined with that of the foe was\nnot to escape from the wrath of the angry element which, implacable as\nan ancient god and pitiless to the last, was as cruel to the victor as\nto the conquered.\n\nI could read the signs of deep depression in the face not only of my\nmaster but of the Admiral, Alava, who, in spite of his wounds, still\nkept on his feet and signalled to the frigate to make all possible\nspeed; but, instead of responding to his very natural haste, the\n_Themis_ prepared to shorten sail so as to be able to keep before the\ngale. I shared the general dismay and could not help reflecting on the\nirony with which Fate mocks at our surest calculations and best\nfounded hopes, on the swiftness with which she flings us from happy\nsecurity to the depth of misery. Here we were, on the wide ocean, that\nmajestic emblem of human life. A gust of wind and it is completely\ntransformed, the light ripple which gently caressed the vessel's side\nswells into a mountain of water that lashes and beats it, the soft\nmusic of the wavelets in a calm turns to a loud, hoarse voice,\nthreatening the frail bark which flings itself into the waters as\nthough its keel were unable to balance it, to rise the next moment\nbuffeted and tossed by the very wave that has lifted it from the\nabyss. A lovely day ends in a fearful night, or, on the other hand, a\nradiant moon that illumines an infinite sky and soothes the soul,\npales before an angry sun at whose light all nature quakes with\ndismay.\n\nWe had experienced all these viscissitudes, and in addition, those\nwhich are the result of the will of man. We had suffered shipwreck in\nthe midst of defeat; after escaping once we had been compelled to\nfight again, this time with success; and then, when we thought\nourselves out of our troubles, when we hailed Cadiz with delight, we\nwere once more at the mercy of the tempest which had treacherously\ndeluded us only to destroy us outright. Such a succession of adverse\nfortune seemed monstrous--it was like the malignant aberrations of a\ndivinity trying to do all the harm he could devise to us hapless\nmortals--but it was only the natural course of things at sea,\ncombined with the fortune of war. Given a combination of these two\nfearful forces and none but an idiot can be astonished at the\ndisasters that must ensue.\n\nAnother circumstance contributed to my master's distress of mind, and\nto mine too, that evening. Since the rescue of the _Santa Ana_\nMalespina had disappeared. At last, after seeking him everywhere, I\ndiscovered him lying in a heap on a sofa in the cabin. I went up to\nhim and saw that he was very pale; I spoke to him but he could not\nanswer. He tried to move but fell back gasping.\n\n\"Are you wounded?\" I asked. \"I will fetch some one to attend to you.\"\n\n\"It is nothing,\" he said. \"Can you get me some water?\"\n\nI went at once for my master.\n\n\"What is the matter--this wound in your hand?\" said he, examining the\nyoung officer.\n\n\"It is more than that,\" replied Don Rafael sadly, and he put his hand\nto his right side close by his sword-belt. And, then, as if the effort\nof pointing out his wound and speaking those few words had been too\nmuch for his weakened frame, he closed his eyes and neither spoke nor\nmoved for some minutes.\n\n\"This is serious,\" said my master anxiously.\n\n\"It is more than serious,\" said a surgeon who had come to examine\nhim. Malespina, deeply depressed by finding himself in so evil a\nplight, and believing himself past all hope, had not even reported\nhimself as wounded, but had crept away to this corner where he had\ngiven himself up to his reflections and memories. He believed that he\nwas killed and he would not have the wound touched. The surgeon\nassured him that though it was dangerous it need not prove mortal,\nthough he owned that if he did not get into port that night so that he\nmight be properly treated on shore, his life, like that of the rest of\nthe wounded, was in the greatest danger. The _Santa Ana_ had lost\nninety-seven men killed on the 21st, and a hundred and forty wounded;\nall the resources of the surgery were exhausted and many indispensable\narticles were altogether wanting. Malespina's catastrophe was not the\nonly one during the rescue, and it had been the will of Heaven that\nanother man very near and dear to me should share his fate. Marcial\nhad been wounded; though at first his indomitable spirit had kept him\nup and he hardly felt the pain and depression, before long he\nsubmitted to be carried down into the cock-pit, confessing that he was\nvery badly hit. My master sent a surgeon to attend to him, but all he\nwould say was that the wound would have been trifling in a man of\nfive-and-twenty--but Marcial was past sixty.\n\nMeanwhile the _Rayo_ passed to leeward and we hailed her. Alava begged\nher to enquire of the _Themis_ whether the captain thought he could\nget us into Cadiz, and when he roundly said, No, the Admiral asked\nwhether the _Rayo_, which was almost unharmed, expected to get in\nsafely. Her captain thought she might and it was agreed that Gardoqui,\nwho was severely wounded, and several others, should be sent on board\nher, among them Don Rafael Malespina. Don Alonso obtained that Marcial\nshould also be transferred to her in consideration for his age which\ngreatly aggravated his case, and he sent me, too, in charge of them as\npage or sick-nurse, desiring me never to lose sight of them for an\ninstant till I saw them safe in the hands of their family, at Cadiz,\nor even at Vejer. I prepared to obey him, though I tried to persuade\nmy master that he too ought to come on board the _Rayo_ for greater\nsafety, but he would not even listen to such a suggestion.\n\n\"Fate,\" he said, \"has brought me on board this ship, and in it I will\nstay till it shall please God to save us or no. Alava is very bad,\nmost of the officers are more or less hurt, and I may be able to be of\nsome service here. I am not one of those who run away from danger; on\nthe contrary, since the defeat of the 21st I have sought it; I long\nfor the moment when my presence may prove to be of some use. If you\nreach home before me, as I hope you will, tell Paca that a good sailor\nis the slave of his country, that I am very glad that I came--that I\ndo not regret it--on the contrary. Tell her that she is to be glad,\ntoo, when she sees me, and that my comrades would certainly have\nthought badly of me if I had not come. How could I have done\notherwise? You--do you not think that I did well to come?\"\n\n\"Of course, certainly,\" I replied, anxious to soothe his agitation,\n\"who doubts it?\" For his excitement was so great that the absurdity of\nasking the opinion of a page-boy had not even occurred to him.\n\n\"I see you are a reasonable fellow,\" he went on, much comforted by my\nadmission. \"I see you have a noble and patriotic soul. But Paca never\nsees anything excepting through her own selfishness, as she has a very\nodd temper and has taken it into her head that fleets and guns are\nuseless inventions, she cannot understand why I.... In short, I know\nthat she will be furious when she sees me and then--as we have not won\nthe battle, she will say one thing and another--oh! she will drive me\nmad! However, I will not mind her. You--what do you say? Was I not\nright to come?\"\n\n\"Yes, indeed, I think so,\" I said once more: \"You were very right to\ncome. It shows that you are a brave officer.\"\n\n\"Well then go--go to Paca, go and tell her so, and you will see what\nshe will say,\" he went on more excited than ever. \"And tell her that I\nam safe and sound, and my presence here is indispensable. In point of\nfact, I was the principal leader in the rescue of the _Santa Ana_. If\nI had not trained those guns--who knows, who knows? You--what do you\nthink? We may do more yet; if the wind favors us to-morrow morning we\nmay rescue some more ships. Yes sir, for I have a plan in my head....\nWe shall see, we shall see. And so good bye, my boy. Be careful of\nwhat you say to Paca.\"\n\n\"I will not forget,\" said I. \"She shall know that if it had not been\nfor you we should not have recaptured the _Santa Ana_, and that if you\nare lucky you may still bring a couple of dozen ships into Cadiz.\"\n\n\"A couple of dozen!--no man; that is a large number. Two ships, I\nsay--or perhaps three. In short, I am sure I was right to join the\nfleet. She will be furious and will drive me mad when I get home\nagain; but I was right, I say--I am sure I was right.\" With these\nwords he left me and I saw him last sitting in a corner of the cabin.\nHe was praying, but he told his beads with as little display as\npossible, for he did not choose to be detected at his devotions. My\nmaster's last speech had convinced me that he had lost his wits and,\nseeing him pray, I understood how his enfeebled spirit had struggled\nin vain to triumph over the exhaustion of age, and now, beaten in\nstrife, turned to God for support and consolation. Do\u00f1a Francisca was\nright; for many years my master had been past all service but prayer.\n\nWe left the ship according to orders. Don Rafael and Marcial with the\nrest of the wounded officers were carefully let down into the boats by\nthe strong-armed sailors. The violence of the sea made this a long and\ndifficult business, but at last it was done and two boat loads were\npulled off to the _Rayo_. The passage, though short was really\nfrightful; but at last, though there were moments when it seemed to me\nthat we must be swallowed up by the waves, we got alongside of the\n_Rayo_ and with great difficulty clambered on board.\n\n\n\n\nCHAPTER XV.\n\n\n\"Out of the frying-pan into the fire,\" said Marcial, when they laid\nhim down on deck. \"However, when the captain commands the men must\nobey. _Rayo_ is an unlucky name for this cursed ship. They say she\nwill be in Cadiz by midnight, and I say she won't. We shall see what\nwe shall see.\"\n\n\"What do you say, Marcial? we shall not get in?\" I asked in much\nalarm.\n\n\"You, master Gabrielito, you know nothing about such matters,\" said\nhe.\n\n\"But when Don Alonso and the officers of the _Santa Ana_ say that the\n_Rayo_ will get in to-night.... She must get in when they say she\nwill.\"\n\n\"Do not you know, you little landlubber, that the gentlemen of the\nquarter-deck are far more often mistaken than we are in the fo'castle?\nIf not, what was the admiral of the fleet about?--_Mr. Corneta_--devil\ntake him! You see he had not brains enough to work a fleet. Do you\nsuppose that if _Mr. Corneta_ had asked my advice we should have lost\nthe battle?\"\n\n\"And you think we shall not get into Cadiz?\"\n\n\"I say this old ship is as heavy as lead itself and not to be trusted\neither. She rides the sea badly and will not answer her helm. Why, she\nis as lop-sided and crippled as I am! If you try to put her to port\noff she goes to starboard.\"\n\nIn point of fact the _Rayo_ was considered by all as bad a ship as\never sailed. But in spite of that, in spite of her advanced age--for\nshe had been afloat nearly fifty-six years--as she was still sound she\ndid not seem to be in any danger though the gale increased in fury\nevery minute, for we were almost close to port. At any rate, did it\nnot stand to reason that the _Santa Ana_ was in greater jeopardy,\ndismasted and rudderless, in tow of a frigate?\n\nMarcial was carried to the cock-pit and Malespina to the captain's\ncabin. When we had settled him there, with the rest of the wounded\nofficers, I suddenly heard a voice that was familiar to me though for\nthe moment I could not identify it with any one I knew. However, on\ngoing up to the group whence the stentorian accents proceeded,\ndrowning every other voice, what was my surprise at recognizing Don\nJos\u00e9 Maria Malespina! I ran to tell him that his son was on board, and\nthe worthy parent at once broke off the string of rodomontade that he\nwas pouring forth and flew to the wounded man. His delight was great\nat finding him alive; he had come out of Cadiz because he could no\nlonger endure the suspense and he must know what had become of his boy\nat any cost.\n\n\"Why your wound is a mere trifle,\" he said, embracing his son. \"A mere\nscratch! But you are not used to wounds; you are quite a mollycoddle,\nRafael. Oh, if only you had been old enough to go with me to fight in\nRousillon! You would have learnt there what wounds are--something like\nwounds! Do you know a ball hit me in the fleshy part of the arm, ran\nup to my shoulder and then right round the shoulder blade and out by\nthe belt. A most extraordinary case, that was. But in three days I was\nall right again and commanding the artillery at Bellegarde.\" He went\non to give the following account of his presence on board the _Rayo_.\n\n\"We knew the issue of the battle at Cadiz, by the evening of the 21st.\nI tell you, gentlemen--no one would listen to me when I talked of\nreforming our artillery and you see the consequences. Well, as soon as\nI knew the worst and had learnt that Gravina had come in with a few\nships I went to see if the _San Juan Nepomuceno_, on board which you\nwere, was one of them; but they told me she had been captured. You\ncannot imagine my anxiety; I could hardly doubt that you were dead,\nparticularly when I heard how many had been killed on board your ship.\nHowever, I am one of those men who must follow a matter up to the end,\nand knowing that some of the ships in port were preparing to put out\nto sea in hope of picking up derelicts and rescuing captured vessels,\nI determined to set out without a moment's delay and sail in one of\nthem. I explained my wishes to Solano and then to the Admiral in\ncommand, my old friend Esca\u00f1o, and after some hesitation they allowed\nme on board. I embarked this morning, and enquired of every one in the\n_Rayo_ for some news of you and of the _San Juan_, but I could get no\ncomfort; nay, quite the contrary, for I heard that Churruca was killed\nand that his ship, after a glorious defence, had struck to the enemy.\nYou may fancy my anxiety. How far was I from supposing this morning,\nwhen we rescued the _Santa Ana_, that you were on board! If I had but\nknown it for certain I would have redoubled my efforts in the orders I\nissued--by the kind permission of these gentlemen; Alava's ship should\nhave been free in two minutes.\"\n\nThe officers who were standing round us looked at each other with a\nshrug as they heard Don Jos\u00e9's last audacious falsehood. I could\ngather from their smiles and winks that he had afforded them much\ndiversion all day with his vainglorious fictions, for the worthy\ngentleman could put no bridle on his indefatigable tongue, even under\nthe most critical and painful circumstances.\n\nThe surgeon now said that his patients ought to be left to rest and\nthat there must be no conversation in their presence, particularly no\nreference to the recent disaster. Don Jos\u00e9 Maria, however,\ncontradicted him flatly, saying that it was good to keep their spirits\nup by talking to them.\n\n\"In the war in Rousillon,\" he added, \"those who were badly\nwounded--and I was several times--sent for the soldiers to dance and\nplay the guitar in the infirmary; and I am very certain that this\ntreatment did more to cure us than all your plasters and dosing.\"\n\n\"Yes, and in the wars with the French Republic,\" said an Andalusian\nofficer who wanted to trump Don Jos\u00e9's trick, \"it was a regular thing\nthat a _corps de ballet_ should be attached to the ambulance corps,\nand an opera company as well. It left the surgeons and apothecaries\nnothing to do, for a few songs, and a short course of pirouettes and\ncapers set them to rights again, as good as new.\"\n\n\"Come, come!\" cried Malespina, \"this is too much. You do not mean to\nsay that music and dancing can heal a wound?\"\n\n\"You said so.\"\n\n\"Yes, but that was only once and it is not likely to occur again.\nPerhaps you think it not unlikely that we may have such another war as\nthat in Rousillon? The most bloody, the best conducted, the most\nsplendidly planned war since the days of Epaminondas! Certainly not.\nEvery thing about it was exceptional; and you may believe me when I\nsay it, for I was in the thick of it, from the Introit to the last\nblessing. It is to my experience there that I owe my knowledge of\nartillery--did you never hear me spoken of? I am sure you must\nrecognize my name. Well, you must know that I have in my head a\nmagnificent scheme, and if one of these days it is only realized we\nshall hear of no more disasters like that of the 21st. Yes,\ngentlemen,\" he said, looking round at the three or four officers who\nwere listening, with consummate gravity and conceit: \"Something must\nbe done for the country. Something must be devised--something\nstupendous, to recoup us at once for our losses and secure victory to\nour fleets for ever and ever, Amen.\"\n\n\"Let us hear, Don Jos\u00e9,\" said one of the audience. \"Explain your\nscheme to us.\"\n\n\"Well, I am devoting my mind to the construction of 300-pounders.\"\n\n\"Three-hundred-pounders!\" cried the officers with shouts of laughter\nand derision. \"Why, the largest we carry is a 36-pounder.\"\n\n\"Mere toys! Just imagine the ruin that would be dealt by a 300-pound\ngun fired into the enemy's fleet,\" said Malespina. \"But what the devil\nis that?\" he added putting out his hand to keep himself from falling,\nfor the _Rayo_ rolled so heavily that it was very difficult for any\none to keep his feet.\n\n\"The gale is stiffening and I doubt our getting into Cadiz to-night,\"\nsaid one of the officers moving away. The worthy man had now but two\nlisteners, but he proceeded with his mendacious harangue all the same.\n\n\"The first thing must be to build a ship from 95 to 100 yards in\nlength.\"\n\n\"The Devil you will! That would be a snug little craft with a\nvengeance!\" said one of the officers. \"A hundred yards! Why the\n_Trinidad_--God rest her--was but seventy and everybody thought her\ntoo long. She did not sail well you know and was very difficult to\nhandle.\"\n\n\"It does not take much to astonish you I see,\" Malespina went on.\n\"What is a hundred yards? Why, much larger ships than that might be\nbuilt. And you must know that I would build her of iron.\"\n\n\"Of iron!\" and his listeners went into fits of laughter.\n\n\"Yes sir, of iron. Perhaps you are not familiar with the science of\nhydrostatics? There can be no difficulty in building an iron ship of\n7000 tons.\"\n\n\"And the _Trinidad_ was of 4000! and that was too big. But do you not\nsee that in order to move such a monster you would want such gigantic\ntackle that no human power could work it?\"\n\n\"Not a bit of it!--Besides, my good sir, who told you that I was so\nstupid as to think that I could trust to the wind alone to propel my\nship? If you knew--I have an idea.--But I do not care to explain my\nscheme to you for you would not understand me.\"\n\nAt this point of his discourse Don Jos\u00e9 was so severely shaken that he\nfell on all fours. But not even this could stop his tongue. Another of\nhis audience walked away, leaving only one who had to listen and to\nkeep up the conversation.\n\n\"What a pitching and tossing,\" said the old man. \"I should not wonder\nif we were driven on shore.--Well, as I was saying--I should move my\nmonster by an invention of my own--can you guess what?--By steam. To\nthis end I should construct a peculiar kind of machine in which the\nsteam, expanding and contracting alternately inside two cylinders,\nwould put certain wheels in motion; then....\"\n\nThe officer would listen no longer, and though he had no commission on\nboard the ship nor any fixed duty, being one of the rescued, he went\noff to assist in working the ship, which was hard enough to do as the\ntempest increased. Malespina was left alone with me for an audience,\nand at first I thought he would certainly cease talking, not thinking\nme capable of sustaining the conversation. But, for my sins, it would\nseem that he credited me with more merit than I could lay claim to,\nfor he turned to me and went on:\n\n\"You understand what I mean? Seven thousand tons, and steam working\ntwo wheels, and then....\"\n\n\"Yes se\u00f1or, I understand you perfectly,\" I replied, to see if he would\nbe silent, for I did not care to hear him, nor did the violent motion\nof the ship which threatened us with immediate peril at all incline my\nmind to dissertations on the aggrandizement of the Spanish navy.\n\n\"I see,\" he continued, \"that you know how to appreciate me and value\nmy inventions. You see at once that such a ship as I describe would\nbe invincible, and as available for attack as for defence. With four\nor five discharges it could rout thirty of the enemy's ships.\"\n\n\"Would not their cannon do it some damage?\" I asked timidly, and\nspeaking out of civility rather than from any interest I felt in the\nmatter.\n\n\"Your observation is a very shrewd one my little gentleman, and proves\nthat you really appreciate my great invention. But to avoid injury\nfrom the enemy's guns I should cover my ship with thick plates of\nsteel. I should put on it a breastplate, in fact, such as warriors\nwore of old. With this protection it could attack the foe, while their\nprojectiles would have no more effect on its sides than a broadside of\nbread-pills flung by a child. It is a wonderful idea I can tell you,\nthis notion of mine. Just fancy our navy with two or three ships of\nthis kind! What would become of the English fleet then, in spite of\nits Nelsons and Collingwoods?\"\n\n\"But they might make such ships themselves,\" I returned eagerly and\nfeeling the force of this argument. \"The English would do the same,\nand then the conditions of the battle would be equal again.\"\n\nDon Jos\u00e9 was quite dumbfounded by this suggestion and for a minute\ndid not know what to say, but his inexhaustible imagination did not\ndesert him for long and he answered, but somewhat crossly:\n\n\"And who said, impertinent boy, that I should be such a fool as to\ndivulge the secret so that the English might learn it? These ships\nwould be constructed in perfect secrecy without a word being whispered\neven to any one. Suppose a fresh war were to break out. We should defy\nthe English: 'Come on, gentlemen,' we should say, 'we are ready, quite\nready.' The common ships would put out to sea and begin the action\nwhen lo and behold! out come two or three of these iron monsters into\nthe thick of the fight, vomiting steam and smoke and turning here and\nthere without troubling themselves about the wind; they go wherever\nthey are wanted, splintering the wooden sides of the enemy's ships by\nthe blows of their sharp bows, and then with a broadside or two.... It\nwould all be over in a quarter of an hour.\"\n\nI did not care to raise any further difficulties for the conviction\nthat our vessel was in the greatest danger quite kept my mind from\ndwelling on ideas so inappropriate to our critical situation. In fact,\nI never thought again of the monster ship of the old man's fancy till\nthirty years after when we first heard of the application of steam to\npurposes of navigation; and again when, half-way through the century,\nour fine frigate the _Numancia_ actually realized the extravagant\ndreams of the braggart of Trafalgar.\n\nHalf a century later I remembered Don Jos\u00e9 Maria Malespina and I said:\n\"He seemed to us a bombastic liar; but conceptions which are\nextravagant in one place and time, when born in due season become\nmarvellous realities! And since living to see this particular instance\nof the fact, I have ceased to think any Utopia impossible, and the\ngreatest visionaries seem to me possible men of genius.\"\n\nI left Don Jos\u00e9 in the cabin and ascended the companion-way, to see\nwhat was going forward, and as soon as I was on deck I understood the\ndangerous situation of the _Rayo_. The gale not only prevented her\ngetting into Cadiz, but was driving her towards the coast where she\nmust inevitably be wrecked on the rocky shore. Melancholy as was the\nfate of the abandoned _Santa Ana_ it could not be more desperate than\nours. I looked with dismay into the faces of the officers and crew to\nsee if I could read hope in any one of them, but despair was written\nin all. I glanced at the sky--it was black and awful; I gazed at the\nsea--it was raging with fury. God was our only hope--and He had shown\nus no mercy since the fatal 21st!\n\nThe _Rayo_ was running northwards. I could understand, from what I\nheard the men about me saying, that we were driving past the reef of\nMarajotes--past Hazte Afuera--Juan Bola--Torregorda, and at last past\nthe entrance to Cadiz. In vain was every effort made to put her head\nround to enter the bay. The old ship, like a frightened horse, refused\nto obey; the wind and waves carried her on, due north, with\nirresistible fury and science could do nothing to prevent it.\n\nWe flew past the bay, and could make out to our right, Rota, Punta\nCandor, Punta de Meca, Regla and Chipiona. There was not a doubt that\nthe _Rayo_ must be driven on shore, close to the mouth of the\nGuadalquivir. I need hardly say that the sails were close reefed and\nthat as this proved insufficient in such a furious tempest the\ntopmasts were lowered; at last it was even thought necessary to cut\naway the masts to prevent her from foundering. In great storms a ship\nhas to humble herself, to shrink from a stately tree to a lowly plant;\nand as her masts will no more yield than the branches of an oak, she\nis under the sad necessity of seeing them amputated and losing her\nlimbs to save her life.\n\nThe loss of the ship was now inevitable. The main and mizzen-masts\nwere cut through and sent overboard, and our only hope was that we\nmight be able to cast anchor near the coast. The anchors were got\nready and the chains and cables strengthened. We were now running\nright on shore, and two cannon were fired as a signal that we wanted\nhelp; for as we could clearly distinguish fires we kept up our hope\nthat there must be some one to come to our rescue. Some were of\nopinion that a Spanish or English ship had already been wrecked here\nand that the fires we saw had been lighted by the destitute crew. Our\nanxiety increased every instant, and as for myself I firmly believed\nthat I was face to face with a cruel death. I paid no attention to\nwhat was doing on board, being much too agitated to think of anything\nbut my end, which seemed inevitable. If the vessel ran on a rock, what\nman could swim through the breakers that still divided us from the\ncoast? The most dangerous spot in a storm is just where the waves are\nhurled revolving against the shore, as if they were trying to scoop it\nout and drag away whole tracts of earth into the gulfs below. The blow\nof a wave as it dashes forward and its gluttonous fury as it rushes\nback again, is such as no human strength can stand against.\n\nAt last, after some hours of mortal anguish, the keel of the _Rayo_\ncame upon a sand bank and there she stuck. The hull and the remaining\nmasts shivered as she struck; she seemed to be trying to cut her way\nthrough the obstacle; but it was too much for her; after heaving\nviolently for a few moments, her stern went slowly down with fearful\ncreaks and groans, and she remained steady. All was over now, nothing\nremained to be done but to save ourselves by getting across the tract\nof sea which separated us from the land. This seemed almost impossible\nin the boats we had on board; our best hope was that they might send\nus help from the shore, for it was evident that the crew of a lately\nwrecked vessel was encamped there, and one of the government cutters,\nwhich had been placed on the coast by the naval authorities for\nservice in such cases, must surely be in the neighborhood. The _Rayo_\nfired again and again, and we watched with desperate impatience, for\nif some succour did not reach us soon we must all go down in the ship.\nThe hapless crippled mass whose timbers had parted as she struck\nseemed likely to hasten her end by the violence of her throes, and\nthe moment could not be far off when her ribs must fall asunder and we\nshould be left at the mercy of the waves with nothing to cling to but\nthe floating wreck.\n\nThose on shore could do nothing for us, but by God's mercy our signal\nguns were heard by a sloop which had put to sea at Chipiona and which\nnow approached us, keeping, however, at a respectful distance. As soon\nas her broad mainsail came in view we knew that we were saved, and the\ncaptain of the _Rayo_ gave orders to insure our all getting on board\nwithout confusion in such imminent peril. My first idea, when I saw\nthe boats being got out, was to run to the two men who most interested\nme on board: Marcial and young Malespina, both wounded, though\nMarcial's was not a serious case. I found the young officer in a very\nbad way and saying to the men around him: \"I will not be moved--leave\nme to die here.\"\n\nMarcial had crawled up on deck and was lying on the planks so utterly\nprostrate and indifferent that I was really terrified at his\nappearance. He looked up as I went near him and taking my hand said in\npiteous tones: \"Gabrielillo, do not forsake me!\"\n\n\"To land!\" I cried trying to encourage him, \"we are all going on\nshore.\" But he only shook his head sadly as if he foresaw some\nimmediate disaster.\n\nI tried to help him up, but after the first effort he let himself drop\nas if he were dead. \"I cannot,\" he said at length. The bandages had\ncome off his wound and in the confusion of our desperate situation no\none had thought of applying fresh ones. I dressed it as well as I\ncould, comforting him all the time with hopeful words; I even went so\nfar as to laugh at his appearance to see if that would rouse him. But\nthe poor old man could not smile; he let his head droop gloomily on\nhis breast, as insensible to a jest as he was to consolation. Thinking\nonly of him, I did not observe that the boats were putting off. Among\nthe first to be put on board were Don Jos\u00e9 Malespina and his son; my\nfirst impulse had been to follow them in obedience to my master's\norders, but the sight of the wounded sailor was too much for me.\nMalespina could not need me, while Marcial was almost a dead man and\nstill clung to my hand with his cold fingers, saying again and again:\n\"Gabrielillo, do not leave me.\"\n\nThe boats labored hard through the breakers, but notwithstanding, when\nonce the wounded had been moved the embarkation went forward rapidly,\nthe sailors flinging themselves in by a rope or taking a flying leap.\nSeveral jumped into the water and saved themselves by swimming. It\nflashed through my mind as a terrible problem, by which of these means\nI could escape with my life, and there was no time to lose for the\n_Rayo_ was breaking up; the after-part was all under water and the\ncracking of the beams and timbers, which were in many places half\nrotten, warned me that the huge hulk would soon cease to exist. Every\none was rushing to the boats, and the sloop, which kept at a safe\ndistance, very skilfully handled so as to avoid shipping water, took\nthem all on board. The empty boats came back at once and were filled\nagain in no time.\n\nSeeing the helpless state in which Marcial was lying I turned,\nhalf-choked with tears, to some sailors and implored them to pick him\nup and carry him to a boat; but it was as much as they could do to\nsave themselves. In my desperation I tried to lift him and drag him to\nthe ship's side, but my small strength was hardly enough to raise his\nhelpless arms. I ran about the deck, seeking some charitable soul; and\nsome seemed on the point of yielding to my entreaties, but their own\npressing danger choked their kind impulses. To understand such\ncold-blooded cruelty you must have gone through such a scene of\nhorror; every feeling of humanity vanishes before the stronger\ninstinct of self-preservation which becomes a perfect possession, and\nsometimes reduces man to the level of a wild beast.\n\n\"Oh the wretches! they will do nothing to save you, Marcial,\" I cried\nin bitter anguish.\n\n\"Let them be,\" he said. \"They are the same at sea as on shore. But you\nchild, be off, run, or they will leave you behind.\" I do not know\nwhich seemed to me the most horrible alternative--to remain on board\nwith the certainty of death, or to go and leave the miserable man\nalone. At length, however, natural instinct proved the stronger and I\ntook a few steps towards the ship's side; but I turned back to embrace\nthe poor old man once more and then I ran as fast as I could to the\nspot where the last men were getting into the boat. There were but\nfour, and when I reached the spot I saw that all four had jumped into\nthe sea and were swimming to meet the boat which was still a few yards\ndistant.\n\n\"Take me!\" I shrieked, seeing that they were leaving me behind. \"I am\ncoming too!--Take me too!\"\n\nI shouted with all my strength but they either did not hear or did not\nheed me. Dark as it was, I could make out the boat and even knew when\nthey were getting into it, though I could hardly say that I saw them.\nI was on the point of flinging myself overboard to take my chance of\nreaching the boat when, at that very moment, it had vanished--there\nwas nothing to be seen but the black waste of waters. Every hope of\nescape had vanished with it. I looked round in despair--nothing was\nvisible but the waves preying on what was left of the ship; not a star\nin the sky, not a spark on shore--the sloop had sailed away.\n\nBeneath my feet, which I stamped with rage and anguish, the hull of\nthe _Rayo_ was going to pieces, nothing remained indeed but the bows,\nand the deck was covered with wreck; I was actually standing on a sort\nof raft which threatened every moment to float away at the mercy of\nthe waves.\n\nI flew back to Marcial. \"They have left me, they have left us!\" I\ncried. The old man sat up with great difficulty, leaning on one hand\nand his dim eyes scanned the scene and the darkness around us.\n\n\"Nothing....\" he said. \"Nothing to be seen; no boats, no land, no\nlights, no beach.--They are not coming back!\"\n\nAs he spoke a tremendous crash was heard beneath our feet in the\ndepths of the hold under the bows, long since full of water; the deck\ngave a great lurch and we were obliged to clutch at a capstan to save\nourselves from falling into the sea. We could not stand up; the last\nremains of the _Rayo_ were on the point of being engulfed. Still, hope\nnever forsakes us; and I, at any rate, consoled myself with the belief\nthat things might remain as they were now till day-break and with\nobserving that the fore-mast had not yet gone overboard. I looked up\nat the tall mast, round which some tatters of sails and ends of ropes\nstill flapped in the wind, and which stood like a dishevelled giant\npointing heavenward and imploring mercy with the persistency of\ndespair; and I fully determined that if the rest of the hull sank\nunder water I would climb it for a chance of life.\n\nMarcial laid himself down on the deck.\n\n\"There is no hope, Gabrielillo,\" he said. \"They have no idea of coming\nback, nor could they if they tried in such a sea. Well, since it is\nGod's will, we must both die where we are. For me, it matters not; I\nam an old man, and of no use for any earthly thing.--But you, you are\na mere child and you....\" But here his voice broke with emotion.\n\"You,\" he went on, \"have no sins to answer for, you are but a child.\nBut I.... Still, when a man dies like this--what shall I say--like a\ndog or a cat--there is no need, I have heard, for the priest to give\nhim absolution--all that is needed is that he should make his peace\nhimself with God. Have you not heard that said?\"\n\nI do not know what answer I made; I believe I said nothing, but only\ncried miserably.\n\n\"Keep your heart up, Gabrielillo,\" he went on. \"A man must be a man,\nand it is at a time like this that you get to know the stuff you are\nmade of. You have no sins to answer for, but I have. They say that\nwhen a man is dying and there is no priest for him to confess to, he\nought to tell whatever he has on his conscience to any one who will\nlisten to him. Well, I will confess to you Gabrielillo; I will tell\nyou all my sins, and I expect God will hear me through you and then he\nwill forgive me.\"\n\nDumb with terror and awe at the solemnity of his address, I threw my\narms round the old man who went on speaking.\n\n\"Well, I say, I have always been a Christian, a Catholic, Apostolic\nRoman; and that I always was and still am devoted to the Holy Virgin\ndel C\u00e1rmen, to whom I pray for help at this very minute; and I say too\nthat though for twenty years I have never been to confession nor\nreceived the sacrament, it has not been my fault, but that of this\ncursed service, and because one always puts it off from one Sunday to\nthe next. But it is a trouble to me now that I failed to do it, and I\ndeclare and swear that I pray God and the Virgin and all the Saints to\npunish me if it was my fault; for this year, if I have never been to\nconfession or communion, it was all because of those cursed English\nthat forced me to go to sea again just when I really meant to make it\nup with the Church. I never stole so much as a pin's head, and I never\ntold a lie, except for the fun of it now and then. I repent of the\nthrashings I gave my wife thirty years ago--though I think she rightly\ndeserved them, for her temper was more venomous than a scorpion's\nsting. I never failed to obey the captain's order in the least thing;\nI hate no one on earth but the 'great-coats,' and I should have liked\nto see them made mince-meat of. However, they say we are all the\nchildren of the same God, so I forgive them, and I forgive the French\nwho brought us into this war. I will say no more, for I believe I am\ngoing--full sail. I love God and my mind is easy. Gabriel hold me\ntight and stick close to me; you have no sins to answer for, you will\ngo straight away to Heaven to pipe tunes with the angels. Ah well, it\nis better to die so, at your age, than to stay below in this wicked\nworld. Keep up your courage, boy, till the end. The sea is rising and\nthe _Rayo_ will soon be gone. Death by drowning is an easy one; do not\nbe frightened--stick close to me. In less than no time we shall be out\nof it all; I answering to God for all my shortcomings, and you as\nhappy as a fairy, dancing through the star-paved heavens--and they\ntell us happiness never comes to an end up there because it is\neternal, or, as they say, to-morrow and to-morrow and to-morrow, world\nwithout end....\"\n\nHe could say no more. I clung passionately to the poor mutilated body.\nA tremendous sea swept over the bows and I felt the water dash against\nmy shoulder. I shut my eyes and fixed my thoughts on God. Then I lost\nconsciousness and knew no more.\n\n\n\n\nCHAPTER XVI.\n\n\nWhen, I know not how long after, the idea of life dawned once more on\nmy darkened spirit, I was conscious only of being miserably cold;\nindeed, this was the only fact that made me aware of my own existence,\nfor I remembered nothing whatever of all that had happened and had not\nthe slightest idea of where I was. When my mind began to get clearer\nand my senses recovered their functions I found that I was lying on\nthe beach; some men were standing round me and watching me with\ninterest. The first thing I heard was: \"Poor little fellow!--he is\ncoming round.\"\n\nBy degrees I recovered my wits and, with them, my recollection of past\nevents. My first thought was for Marcial, and I believe that the first\nwords I spoke were an enquiry for him. But no one could tell me\nanything about him; I recognized some of the crew of the _Rayo_ among\nthe men on the beach and asked them where he was; they were all agreed\nthat he must have perished. Then I wanted to know how I had been\nsaved, but they would tell me nothing about that either. They gave me\nsome liquor to drink, I know not what, and carried me to a neighboring\nhut, where, warmed by a good fire and cared for by an old woman, I\nsoon felt quite well, though still rather weak. Meanwhile I learnt\nthat another cutter had put out to reconnoitre the wreck of the _Rayo_\nand that of a French ship which had met with the same fate, and that\nthey had picked me up still clinging to Marcial; they found that I\ncould be saved but my companion was dead. I learnt too that a number\nof poor wretches had been drowned in trying to reach the coast. Then I\nwanted to know what had become of Malespina, but no one knew anything\neither of him or of his father. I enquired about the _Santa Ana_\nwhich, it appeared, had reached Cadiz in safety, so I determined to\nset out forthwith to join my master. We were at some distance from\nCadiz, on the coast to the north of the Guadalquivir, I wanted\ntherefore to start at once to make so long a journey. I took two days'\nrest to recover my strength, and then set out for Sanl\u00facar, in the\ncompany of a sailor who was going the same way. We crossed the river\non the morning of the 27th and then continued our walk, keeping along\nthe coast. As my companion was a jolly, friendly fellow the journey\nwas as pleasant as I could expect in the frame of mind I was in,\ngrieved at Marcial's death and depressed by the scenes I had so lately\nwitnessed. As we walked on we discussed the battle and the shipwrecks\nthat had ensued.\n\n\"A very good sailor was that old <DW36>,\" said my companion. \"But\nwhat possessed him to go to sea again with more than sixty years on\nhis shoulders? It served him right to come to a bad end.\"\n\n\"He was a brave seaman,\" said I, \"and had such a passion for fighting\nthat even his infirmities could not keep him quiet when he had made up\nhis mind to join the fleet.\"\n\n\"Well, I have had enough of it for my part,\" said the sailor. \"I do\nnot want to see any more fighting at sea. The King pays us badly, and\nthen, if you are maimed or crippled--good-bye to you--I know nothing\nabout you--I never set eyes on you in my life.--Perhaps you don't\nbelieve me when I tell you the King pays his men so badly? But I can\ntell you this: most of the officers in command of the ships that went\ninto action on the 21st had seen no pay for months. Only last year\nthere was a navy captain at Cadiz who went as waiter in an inn because\nhe had no other way of keeping himself or his children. His friends\nfound him out though he tried to conceal his misery, and they\nsucceeded at last in getting him out of his degrading position. Such\nthings do not happen in any other country in the world; and then we\nare horrified at finding ourselves beaten by the English! As to the\narsenals, I will say nothing about them; they are empty and it is of\nno use to hope for money from Madrid--not a _cuarto_ comes this way.\nAll the King's revenues are spent in paying the court officials, and\nchief among them the Prince of Peace; who gets 40,000 dollars as\nCounsellor of the Realm, Secretary of State, Captain-General, and\nSergeant-Major of the Guards.--No, say I, I have had enough of serving\nthe King. I am going home to my wife and children, for I have served\nmy time and in a few days they must give me my papers.\"\n\n\"But you have nothing to complain of friend,\" said I, \"since you were\non board the _Rayo_ which hardly did any fighting.\"\n\n\"I was not in the _Rayo_ but in the _Bahama_, one of the ships that\nfought hardest and longest.\"\n\n\"She was taken and her captain killed, if I remember rightly.\"\n\n\"Aye, so it was,\" he said. \"I could cry over it when I think of\nhim--Don Dionisio Alcal\u00e1 Galiano, the bravest seaman in the fleet.\nWell, he was a stern commander; he never overlooked the smallest\nfault, and yet his very severity made us love him all the more, for a\ncaptain who is feared for his severity--if his severity is unfailingly\njust--inspires respect and wins the affection of his men. I can\nhonestly say that a more noble and generous gentleman than Don\nDionisio Alcal\u00e1 Galiano was never born. And when he wanted to do a\ncivility to his friends he did not do it by halves; once, out in\nHavana he spent ten thousand dollars on a supper he gave on board\nship.\"\n\n\"He was a first-rate seaman too, I have heard.\"\n\n\"Ah, that he was. And he was more learned than Merlin and all the\nFathers of the Church. He made no end of maps, and discovered Lord\nknows how many countries out there, where it is as hot as hell itself!\nAnd then they send men like these out to fight and to be killed like a\nparcel of cabin-boys. I will just tell you what happened on board the\n_Bahama_. As soon as the fighting began Don Dionisio Alcal\u00e1 Galiano\nknew we must be beaten on account of that infernal trick of turning\nthe ships round--we were in the reserve and had been in the rear.\nNelson, who was certainly no fool, looked along our line, and he said:\n'If we cut them through at two separate points, and keep them between\ntwo fires, hardly a ship will escape me.' And so he did, blast him;\nand as our line was so long the head could never help the tail. He\nfought us in detachments, attacking us in two wedge-shaped columns\nwhich, as I have heard say, were the tactics adopted by the great\nMoorish general, Alexander the Great, and now used by Napoleon. It is\nvery certain, at any rate, that they got round us and cut us in three,\nand fought us ship to ship in such a manner that we could not support\nor help each other; every Spaniard had to deal with three or four\nEnglishmen.\n\n\"Well, so you see the _Bahama_ was one of the first to be under fire.\nGaliano reviewed the crew at noon, went round the gun-decks, and made\nus a speech in which he said: 'Gentlemen, you all know that our flag\nis nailed to the mast.' Yes, we all knew the sort of man our Captain\nwas, and we were not at all surprised to hear it. Then he turned to\nthe captain of the marines, Don Alonso Butron, 'I charge you to defend\nit,' he said. 'No Galiano ever surrenders and no Butron should\neither.'\n\n\"'What a pity it is,' said I, 'that such men should not have had a\nleader worthy of such courage, since they could not themselves conduct\nthe fleet.'\n\n\"Aye, it is a pity, and you shall hear what happened. The battle\nbegan, and you know something of what it was like if you were on\nboard the _Trinidad_. The ships riddled us with broadsides to port and\nstarboard. The wounded fell like flies from the very first, and the\ncaptain first had a bad bruise on his foot and then a splinter struck\nhis head and hurt him badly. But do you think he would give in, or\nsubmit to be plastered with ointment? Not a bit of it; he staid on the\nquarter-deck, just as if nothing had happened, though many a man he\nloved truly fell close to him never to stand up again. Alcal\u00e1 Galiano\ngave his orders and directed his guns as if we had been firing a\nsalute at a review. A spent ball knocked his telescope out of his hand\nand that made him laugh. I fancy I can see him now; the blood from his\nwound stained his uniform and his hands and he cared no more than if\nit had been drops of salt-water splashed up from the sea. He was a man\nof great spirit and a hasty temper; he shouted out his orders so\npositively that if we had not obeyed them because it was our duty, we\nshould have done so out of sheer alarm.--But suddenly it was all over\nwith him.--He was struck in the head by a shot and instantly killed.\n\n\"The fight was not at an end, but all our heart in it was gone. When\nour beloved captain fell the officers covered his body that we men\nmight not see it, but we all knew at once what had happened, and\nafter a short and desperate struggle for the honor of our flag, the\n_Bahama_ surrendered to the English who carried her off to Gibraltar\nif she did not go to the bottom on the way, as I rather suspect she\ndid.\"\n\nAfter giving this history and telling me how he had been transferred\nfrom the _Bahama_ to the _Santa Ana_, my companion sighed deeply and\nwas silent for some time. However, as the way was long and dull I\ntried to reopen the conversation and I began telling him what I myself\nhad seen, and how I had at last been put on board the _Rayo_ with\nyoung Malespina.\n\n\"Ah!\" said he. \"Was he a young artillery officer who was transferred\nto the sloop to be taken to shore on the night of the 23d?\"\n\n\"The very same,\" said I. \"But no one has been able to tell me for\ncertain what became of him.\"\n\n\"He was one of a party in the second boat which could not get to\nshore; some of those who were whole and strong contrived to escape,\nand among them that young officer's father; but all the wounded were\ndrowned, as you may easily suppose, as the poor souls of course could\nnot swim to land.\"\n\nI was shocked to hear of Don Rafael's death, and the thought of the\ngrief it would be to my hapless and adored little mistress quite\novercame me, choking every mean and jealous feeling.\n\n\"What a dreadful thing!\" I exclaimed. \"And is it my misfortune to have\nto carry the news to his sorrowing friends? But, tell me, are you\ncertain of the facts?\"\n\n\"I saw his father with my own eyes, lamenting bitterly and telling all\nthe details of the catastrophe with such distress it was enough to\nbreak your heart. From what he said he seemed to have saved everybody\non board the boat, and he declared that if he had saved his son it\nwould have been at the cost of the lives of all the others, so he\nchose, on the whole, to preserve the lives of the greatest number,\neven in sacrificing that of his son, and he did so. He must be a\nsingularly humane man, and wonderfully brave and dexterous.\" But I was\nso deeply distressed that I could not discuss the subject. Marcial\ndead, Malespina dead! What terrible news to take home to my master's\nhouse. For a moment my mind was almost made up not to return to Cadiz;\nI would leave it to chance or to public rumor to carry the report to\nthe sad hearts that were waiting in such painful suspense. However, I\nwas bound to present myself before Don Alonso and give him some\naccount of my proceedings.\n\nAt length we reached Rota and there embarked for Cadiz. It is\nimpossible to describe the commotion produced by the report of the\ndisaster to our fleet. News of the details had come in by degrees, and\nby this time the fate of most of our ships was known, though what had\nbecome of many men and even whole crews had not been ascertained. The\nstreets were full of distressing scenes at every turn, where some one\nwho had come off scot-free stood telling off the deaths he knew of,\nand the names of those who would be seen no more. The populace crowded\ndown to the quays to see the wounded as they came on shore, hoping to\nrecognize a father, husband, son or brother. There were episodes of\nfrantic joy mingled with shrieks of dismay and bitter cries of\ndisappointment. Too often were hopes deceived and fears confirmed, and\nthe losers in this fearful lottery were far more numerous than the\nwinners. The bodies thrown up on the shore put an end to the suspense\nof many families, while others still hoped to find those they had lost\namong the prisoners taken to Gibraltar.\n\nTo the honor of Cadiz be it said never did a community devote itself\nwith greater willingness to the care of the wounded, making no\ndistinctions between friends and foes but hoisting the standard, as it\nwere, of universal and comprehensive charity. Collingwood, in his\nnarrative, does justice to this generosity on the part of my\nfellow-countrymen. The magnitude of the disaster had deadened all\nresentment, but is it not sad to reflect that it is only in misfortune\nthat men are truly brothers?\n\nIn Cadiz I saw collected in the harbor the whole results of the\nconflict which previously, as an actor in it, I had only partially\nunderstood, since the length of the line and the man\u0153uvring of the\nvessels would not allow me to see everything that happened. As I now\nlearnt--besides the _Trinidad_--the _Argonauta_, 92 guns, Captain Don\nAntonio Pareja, and the _San Augustin_, 80 guns, Don Felipe Cagigal,\nhad been sunk. Gravina had got back into Cadiz with the _Pr\u00edncipe de\nAst\u00farias_, as well as the _Monta\u00f1es_, 80 guns, commanded by Alcedo,\nwho with his second officer Casta\u00f1os, had been killed; the _San\nJusto_, 76 guns, Captain Don Miguel Gaston; the _San Leandro_, 74,\nCaptain Don Jos\u00e9 Quevedo; the _San Francisco_, 74, Don Luis Flores;\nand the _Rayo_, 100, commanded by Macdonell. Four of these had gone\nout again on the 23d to recapture the vessels making for Gibraltar;\nand of these, two, the _San Francisco_ and the _Rayo_ were wrecked on\nthe coast. So, too, was the _Monarca_, 74 guns, under Argumosa, and\nthe _Neptuno_, 80 guns; and her heroic commander, Don Cayetano Vald\u00e9s,\nwho had previously distinguished himself at Cape St. Vincent, narrowly\nescaped with his life. The _Bahama_ had surrendered but went to pieces\nbefore she could be got into Gibraltar; the _San Ildefonso_, 74 guns,\nCaptain Vargas, was taken to England, while the _San Juan Nepomuceno_\nwas left for many years at Gibraltar, where she was regarded as an\nobject of veneration and curiosity. The _Santa Ana_ had come safely\ninto Cadiz the very night we were taken off her.\n\nThe English too lost some fine ships, and not a few of their gallant\nofficers shared Nelson's glorious fate.\n\nWith regard to the French it need not be said that they had suffered\nas severely as we had. With the exception of the four ships that\nwithdrew under Dumanoir without showing fight--a stain which the\nImperial navy could not for a long time wipe out--our allies behaved\nsplendidly. Villeneuve, only caring to efface in one day the\nremembrance of all his mistakes, fought desperately to the last and\nwas carried off a prisoner to Gibraltar. Many of their officers were\ntaken with him, and very many were killed. Their vessels shared all\nour risks and dangers; some got off with Gravina, some were taken and\nseveral were wrecked on the coast. The _Achille_ blew up, as I have\nsaid, in the midst of the action.\n\nBut in spite of all these disasters, Spain had paid dearer for the war\nthan her haughty ally. France had lost the flower of her navy indeed,\nbut at that very time Napoleon had won a glorious victory on land. His\narmy had marched with wonderful rapidity from the shores of the\nEnglish Channel across Europe, and was carrying out his colossal\nschemes in the campaign against Austria. It was on the 20th of\nOctober, the day before Trafalgar, that Napoleon, at the camp at Ulm,\nlooked on as the Austrian troops marched past, while their officers\ndelivered up their swords; only two months later, on the 2d of\nDecember, he won, on the field of Austerlitz, the greatest of his many\nvictories.\n\nThese triumphs consoled France for the defeat of Trafalgar; Napoleon\nsilenced the newspapers, forbidding them to discuss the matter; and\nwhen the victory of his implacable enemies, the English, was reported\nto him he simply shrugged his shoulders and said: \"I cannot be\neverywhere at once.\"\n\n\n\n\nCHAPTER XVII.\n\n\nI postponed the fatal hour when I must face my master as long as\npossible, but at last my destitute condition, without money and\nwithout a home, brought me to the point. As I went to the house of\nDo\u00f1a Flora my heart beat so violently that I had to stop for breath at\nevery step. The terrible shock I was about to give the family by\nannouncing young Malespina's death weighed so terribly on my soul that\nI could not have felt more crushed and guilty if I had myself been the\noccasion of it. At last however, I went in. My presence in the\ncourt-yard caused an immense sensation. I heard heavy steps hurrying\nalong the upper galleries and I had not been able to speak a word\nbefore I felt myself in a close embrace. I at once recognized Do\u00f1a\nFlora, with more paint on her face than if it had been a picture, but\nseriously discomposed in effect by the good old soul's delight at\nseeing me once more. But all the fond names she lavished on me--her\ndear boy, her pet, her little angel--could not make me smile. I went\nup stairs, every one was in a bustle of excitement. I heard my master\nexclaim: \"Oh! thank God! he is safe.\" I went into the drawing-room,\nand there it was Do\u00f1a Francisca who came forward, asking with mortal\nanxiety--\"And Don Rafael?--Where is Don Rafael?\"\n\nBut for some minutes I could not speak; my voice failed me, I had not\ncourage to tell the fatal news. They questioned me eagerly and I saw\nDo\u00f1a Rosita come in from an adjoining room, pale, heavy-eyed, and\naltered by the anguish she had gone through. At the sight of my young\nmistress I burst into tears, and there was then no need for words.\nRosita gave a terrible cry and fell senseless; her father and mother\nflew to her side, smothering their own grief, while Do\u00f1a Flora melted\ninto tears and took me aside to assure herself that I, at any rate,\nhad returned whole in every part.\n\n\"Tell me,\" she said, \"how did he come by his death? I felt sure of\nit--I told Paca so; but she would only say her prayers and believed\nthat so she could save him. As if God could be troubled with such\nmatters.--And you are safe and sound--what a comfort! No damage\nanywhere?\"\n\nIt is impossible to describe the consternation of the whole\nhousehold. For a quarter of an hour nothing was to be heard but\ncrying, lamentation, and sobbing; for Malespina's mother had come to\nCadiz and was also in the house. But how mysterious are the ways of\nProvidence in working out its ends! About a quarter of an hour, as I\nsay, had elapsed since I had told them the news when a loud assertive\nvoice fell on my ear. It was that of Don Jos\u00e9 Maria, shouting in the\ncourt-yard, calling his wife, Don Alonso, and Rosita. That which first\nstruck me was that his tones seemed just as strident and cheerful as\never, which I thought very indecorous after the misfortune that had\nhappened. We all ran to meet him, and I stared to see him radiant and\nsmiling.\n\n\"But poor Don Rafael....\" said my master.\n\n\"Safe and sound,\" replied Don Jos\u00e9. \"That is to say not exactly sound,\nbut out of danger, for his wound is nothing to be anxious about. The\nfool of a surgeon said he would die, but I knew better. What do I care\nfor surgeons! I cured him, gentlemen--I, I myself, by a new treatment\nwhich no one knows of but myself.\"\n\nThese words, which so suddenly and completely altered the aspect of\naffairs, astounded the audience. The greatest joy took the place of\ngrief and dismay; and to wind up, as soon as their agitation allowed\nthem to think of the delusion they had suffered under, they scolded me\nsoundly for the fright I had given them. I excused myself by saying\nthat I had only repeated the tale as it was told to me, and Don Jos\u00e9\nflew into a great rage, calling me a rascal, an imposter, and a\nbusybody.\n\nIt was happily true that Don Rafael was alive and out of danger; he\nhad remained with some friends at Sanl\u00facar while his father had come\nto Cadiz to fetch his mother to see him. My readers will hardly\nbelieve in the origin of the mistake which had led me to announce the\nyoung man's death in such perfect good faith; though a few may have\nbeen led to suspect that some tremendous fib of the old man's must\nhave given rise to the report that reached me. And so it was, neither\nmore nor less. I heard all about it at Sanl\u00facar whither I went with\nthe family. Don Jos\u00e9 Maria had invented a whole romance of devotion\nand skill on his own part, and had related more than once the history\nof his son's death, inventing so many dramatic details that for a few\ndays he figured as a hero, and had been the object of universal\nadmiration for his humanity and courage. His story was that the boat\nhad upset, and that as the choice lay between rescuing his son and\nsaving all the others he had chosen the latter alternative as the most\nmagnanimous and philanthropical. This romance he dressed up in so many\ninteresting, and at the same time probable circumstances that it could\nnot fail to be believed. The falsehood was of course very soon found\nout, and his success was of brief duration, but not before the story\nhad come to my ears and put me under the necessity of reporting it to\nthe family. Though I knew very well how absolutely mendacious old\nMalespina could be, I had never dreamed of his lying about so serious\na matter.\n\n       *       *       *       *       *\n\nWhen all this excitement was over my master sank into deep melancholy;\nhe would scarcely speak and seemed as though his soul, having no\nillusions left, had closed accounts with the world and was only\nwaiting to take its departure. The absence of Marcial was to him the\nloss of the only companion of his childish old age; he had no one now\nto fight mimic battles with, and he gave himself up to dull sorrow.\nNor did Do\u00f1a Francisca spare him any drop of mortification, seeing him\nin this crest-fallen state. I heard her the same day saying\nspitefully:\n\n\"A pretty mess you have made of it! What do you think of yourself\nnow? Now are you satisfied? Go, oh go by all means and join the fleet!\nWas I right or was I wrong? If you would only have listened to me. But\nyou have had a lesson I hope; you see now how God has punished you.\"\n\n\"Woman, leave me in peace,\" said my master sadly.\n\n\"And now we are left without any fleet at all, and without sailors,\nand we shall soon find ourselves ruined out of hand if we keep up our\nalliance with the French.--Please God those gentry may not pay us out\nfor their misfortunes. Se\u00f1or Villeneuve!--he has covered himself with\nglory indeed! And Gravina again! If he had opposed the scheme of\ntaking the fleet out, as Churruca and Alcal\u00e1 Galiano did, he might\nhave prevented this heartbreaking catastrophe.\"\n\n\"Woman, woman--what do you know about it? Do not annoy me,\" said Don\nAlonso quite vexed.\n\n\"What do I know about it? More than you do. Yes--I repeat it: Gravina\nmay be a worthy gentleman and as brave as you please; but in this\ncase, much good he has done!\"\n\n\"He did his duty. You would have liked us all to be set down as\ncowards, I suppose?\"\n\n\"Cowards, no--but prudent. It is as I say and repeat: the fleet ought\nnever to have gone out of Cadiz just to humor the whims and conceit of\nVilleneuve.\n\n\"Every one here knew that Gravina, like the others, was of opinion\nthat it ought to stop in the bay. But Villeneuve had made up his mind\nto it, intending to hit a blow that might restore him to his master's\nfavor, and he worked on our Spanish pride. It seems that one of the\nreasons Gravina gave was the badness of the weather, and that he said,\nlooking at the barometer in the cabin: 'Do you not see that the\nbarometer foretells foul weather? Do you not see how it has gone\ndown?' And then Villeneuve said drily: 'What is gone down here is\ncourage!' At such an insult Gravina stood up, blind with rage, and\nthrew the French Admiral's own conduct at Finisterre in his teeth.\nSome angry words were spoken on both sides and at last our Admiral\nexclaimed: 'To sea then to-morrow morning.'\n\n\"But I say that Gravina ought to have taken no notice of Villeneuve's\ninsolence--none whatever; that prudence is an officer's first duty,\nand particularly when he knew--as we all knew--that the fleet was not\nin a condition to fight the English.\"\n\nThis view, which at the time seemed to me an insult to our national\nhonor, I understood later was well-founded. Do\u00f1a Francisca was right.\nGravina ought not to have given way to Villeneuve's obstinacy, and I\nsay it almost dims the halo of prestige with which the popular voice\ncrowned the leader of the Spanish forces on that disastrous occasion.\nWithout denying Gravina's many merits, in my opinion there was much\nexaggeration in the high-flown praises that were lavished upon him,\nboth after the battle, and again when he died of his wounds a few\nmonths later.[3] Everything he did proved him to be an accomplished\ngentleman and a brave sailor, but he was perhaps too much of a\ncourtier to show the determination which commonly comes of long\nexperience in war; he was deficient too in that complete superiority\nwhich, in so learned a profession as the Navy, can only be acquired by\nassiduous study of the sciences on which it relies. Gravina was a good\ncommander of a division under superior orders, but nothing more. The\nforesight, coolness, and immovable determination, which are\nindispensable elements in the man whose fortune it is to wield such\nmighty forces, he had not; Don Cosme Damian Churruca had--and Don\nDionisio Alcal\u00e1 Galiano.\n\n    [3] March, 1806.\n\nMy master made no reply to Do\u00f1a Francisca's last speech and when she\nleft the room I observed that he was praying as fervently as when I\nhad left him in the cabin of the _Santa Ana_. Indeed, from that day\nDon Alonso did nothing else but pray; he prayed incessantly till the\nday came when he had to sail in the ship that never comes home.\n\nHe did not die till some time after his daughter's marriage with Don\nRafael Malespina, an event which took place two months after the\naction which the Spanish know as \"the 21st,\" and the English as the\nbattle of Trafalgar. My young mistress was married one lovely morning,\nthough it was winter time, and set out at once for Medina-Sidonia\nwhere a house was ready and waiting for the young couple. I might look\non at her happiness during the days preceding the wedding but she did\nnot observe the melancholy that I was suffering under; nor, if she\nhad, would she have guessed the cause. She thought more of herself\nevery day, as I could see, and I felt more and more humiliated by her\nbeauty and her superior position in life. But I had taught myself to\nunderstand that such a sweet vision of all the graces could never be\nmine, and this kept me calm; for resignation--honest renunciation of\nall hope--is a real consolation, though it is a consolation akin to\ndeath.\n\nWell, they were married, and the very day they had left us for\nMedina-Sidonia Do\u00f1a Francisca told me that I was to follow them and\nenter their service. I set out that night, and during my solitary\njourney I tried to fight down my thoughts and my feelings which\nwavered between accepting a place in the house of the bridegroom or\nflying from them forever. I arrived very early in the morning and\nfound out the house. I went into the garden but on the bottom step I\nstopped, for my reflections absorbed all my energies, and I had to\nstand still to think more clearly; I must have stood there for more\nthan half an hour.\n\nPerfect silence prevailed. The young couple were sleeping untroubled\nby a care or a sorrow. I could not help recalling that far-off time\nwhen my young mistress and I had played together. To me she had then\nbeen my first and only thought. To her, though I had not been the\nfirst in her affections, I had been something she loved and that she\nmissed if we were apart for an hour. In so short a time how great a\nchange!\n\nI looked round me and all I saw seemed to symbolize the happiness of\nthe lovers and to mock my forlorn fate. Although it was winter time, I\ncould picture the trees in full leaf, and the porch in front of the\ndoor seemed suddenly overgrown with creepers to shade them when they\nshould come out. The sun was warmer, the air blew softer round this\nnest for which I myself had carried the first straws when I served as\nthe messenger of their loves. I seemed to see the bare rose-bushes\ncovered with roses, the orange-trees with blossoms and fruit pecked by\ncrowds of birds thus sharing in the wedding feast. My dreams and\nreflections were at last interrupted by a fresh young voice breaking\nthe silence of the place and which made me tremble from head to foot\nas I heard it. It thrilled me with an indescribable sensation, that\nclear, happy, happy voice--whether of fear or shame I can hardly say;\nall I am sure of is that a sudden impulse made me turn from the door,\nand fly from the spot like a thief afraid of being caught.\n\nMy mind was made up. I quitted Medina-Sidonia forthwith, quite\ndetermined never to be a servant in that house, nor to return to\nVejer. After a few minutes reflection I set out for Cadiz intending to\nget from thence to Madrid; and this was what I ultimately did, in\nspite of the persuasions of Do\u00f1a Flora who tried to chain me to her\nside with a wreath of the faded flowers of her affection!\n\nBut since that day how much I have gone through; how much I have seen,\nwell worthy of record. My fate, which had taken me to Trafalgar, led\nme subsequently through many glorious and inglorious scenes, all in\ntheir way worth remembering. If the reader cares to hear the story of\nmy life I will tell him more about it at a future opportunity.\n\n\n\n\n      *      *      *      *      *      *\n\n\n\n\nTranscriber's note:\n\nMinor punctuation errors have been corrected.\n\nHyphenation and accent usage has been made consistent.\n\nVariant and archaic spelling is preserved as printed.\n\nOne character is named variously as Alonso, Alonzo and Alfonso. Both\nMedina-Sidonia and Medinasidonia appear, as do both Rousillon and\nRoussillon. These are all preserved as printed.\n\nAssuming typographic errors, the following changes have been made:\n\n    Page 20--converstion amended to conversation--Being now desirous\n    of taking part in the conversation, ...\n\n    Page 68--Veger amended to Vejer--... he was stationed at Cadiz\n    and at Vejer only on leave.\n\n    Page 82--Audulusian amended to Andalusian--... and above all\n    some of our beautiful Andalusian--but he could ...\n\n    Page 113--predominence amended to predominance--... the\n    predominance of vertical over horizontal lines, ...\n\n    Page 143--canon-balls amended to cannon-balls--The cannon-balls,\n    fired at such a short range, ...\n\n    Page 166--feeing amended to feeling--... mingled with an\n    indescribable feeling of awe, ...\n\n    Page 178--Gardogui amended to Gardoqui--... for six hours until\n    Alava and Gardoqui, ...\n\n    Page 196--transcient amended to transient--... like a transient\n    spark in a night of gloom and horror.\n\n    Page 208--ruddeless amended to rudderless--... was in greater\n    jeopardy, dismasted and rudderless, ...\n\nA Table of Contents has been added by the transcriber for the\nconvenience of the reader.\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nFYODOR DOSTOYEVSK\u0130\n\n\u0130NSANCIKLAR\n\nROMAN\n\nRus\u00e7a asl\u0131ndan \u00e7eviren\n\nSabri G\u00fcrses\n\n_Bedniye lyudi_ , Fyodor Dostoyevski\n\n\u00a9 2013, Can Sanat Yay\u0131nlar\u0131 Ltd. \u015eti.\n\nT\u00fcm haklar\u0131 sakl\u0131d\u0131r. Tan\u0131t\u0131m i\u00e7in yap\u0131lacak k\u0131sa al\u0131nt\u0131lar d\u0131\u015f\u0131nda yay\u0131nc\u0131n\u0131n yaz\u0131l\u0131 izni olmaks\u0131z\u0131n hi\u00e7bir yolla \u00e7o\u011falt\u0131lamaz.\n\n1. bas\u0131m: 2013\n\n3. bas\u0131m: Aral\u0131k 2013, \u0130stanbul\n\nE-kitap 1. S\u00fcr\u00fcm Ocak 2014, \u0130stanbul\n\n2013 tarihli 3. Bas\u0131m esas al\u0131narak haz\u0131rlanm\u0131\u015ft\u0131r.\n\nYay\u0131na haz\u0131rlayan: Faruk Duman\n\nKapak tasar\u0131m\u0131: Ay\u015fe \u00c7elem Design\n\nISBN 9789750720284\n\nCAN SANAT YAYINLARI\n\nYAPIM, DA\u011eITIM, T\u0130CARET VE SANAY\u0130 LTD. \u015eT\u0130.\n\nHayriye Caddesi No. 2, 34430 Galatasaray, \u0130stanbul\n\nTelefon: (0212) 252 56 75 \/ 252 59 88 \/ 252 59 89 Faks: (0212) 252 72 33\n\nwww.canyayinlari.com\n\nyayinevi@canyayinlari.com\n\nSertifika No: 10758\nFyodor Dostoyevski'nin Can Yay\u0131nlar\u0131'ndaki di\u011fer kitaplar\u0131:\n\n_Budala_ , 1982\n\n_Amcan\u0131n D\u00fc\u015f\u00fc_ , 1994\n\n_Tats\u0131z Bir Olay_ , 2005\n\n_Beyaz Geceler_ , 2009\n\n_Karamazov Karde\u015fler_ , 2010\n\n_\u0130kiz_ , 2010\n\n_Yeralt\u0131ndan Notlar_ , 2011\n\n_Yufka Y\u00fcrek_ , 2011\n\n_\u00d6l\u00fcler Evinden Notlar_ , 2012\nFYODOR M\u0130HAYLOV\u0130\u00c7 DOSTOYEVSK\u0130, 1821'de Moskova'da do\u011fdu. Petersburg Asker\u00ee M\u00fchendislik Okulu'nu bitirdikten k\u0131sa bir s\u00fcre sonra edebiyatla u\u011fra\u015fmak i\u00e7in askerlikten ayr\u0131ld\u0131. \u0130lk roman\u0131 _\u0130nsanc\u0131klar_ (1846), d\u00f6nemin \u00fcnl\u00fc ele\u015ftirmeni Belinski'den b\u00fcy\u00fck \u00f6vg\u00fc ald\u0131. Hemen ard\u0131ndan _\u0130kiz_ (1846) adl\u0131 k\u0131sa roman\u0131 geldi. Daha sonra art arda _Ev Sahibesi_ (1847) _, Beyaz Geceler_ (1848) ve _Yufka Y\u00fcrek_ 'i (1848) yay\u0131mlayan Dostoyevski, _Neto\u00e7ka Nezvanova_ (1849) adl\u0131 roman\u0131 yay\u0131mland\u0131\u011f\u0131 s\u0131ralar, devlet d\u00fczenini y\u0131kmaya \u00e7al\u0131\u015ft\u0131\u011f\u0131 gerek\u00e7esiyle tutukland\u0131; \u00f6l\u00fcm cezas\u0131 Sibirya'da d\u00f6rt y\u0131l a\u011f\u0131r hapis cezas\u0131na \u00e7evrildi. Serbest b\u0131rak\u0131ld\u0131ktan ve evlendikten sonra _Amcam\u0131n R\u00fcyas\u0131_ (1859) adl\u0131 komik \u00f6yk\u00fcy\u00fc yazd\u0131. Ayn\u0131 y\u0131l kaleme ald\u0131\u011f\u0131 _Stepan\u00e7ikovo K\u00f6y\u00fc_ 'n\u00fcn (1859) ard\u0131ndan karde\u015fiyle birlikte _Vremya_ adl\u0131 bir edebiyat dergisi \u00e7\u0131kard\u0131. Ayn\u0131 dergide tefrika edilen _\u00d6l\u00fcler Evinden An\u0131lar_ (1861-62), Dostoyevski'nin ba\u015flang\u0131\u00e7taki ba\u015far\u0131s\u0131n\u0131 yeniden g\u00fcndeme getirdi. Yine ayn\u0131 dergide yay\u0131mlanan _Ezilenler_ (1861), ele\u015ftirmenlerin tepkilerine hedef olmas\u0131na kar\u015f\u0131n okurlarca be\u011fenildi. 1862 yaz\u0131nda \u00e7\u0131kt\u0131\u011f\u0131 bir Avrupa gezisi, beraberinde _Bat\u0131 Bat\u0131 Dedikleri \/ Yaz \u0130zlenimleri \u00dczerine K\u0131\u015f Notlar\u0131_ (1863) adl\u0131 \u00fcnl\u00fc makalesini getirdi. Ayn\u0131 y\u0131l dergisi kapat\u0131ld\u0131. Yeniden Avrupa'ya gitti. Tek umudu, art\u0131k tutkunu haline geldi\u011fi kumard\u0131. T\u00fcm paras\u0131n\u0131 rulette yitirince g\u00fc\u00e7l\u00fckle Rusya'ya d\u00f6nd\u00fc. Karde\u015fiyle _Epoha_ adl\u0131 yeni bir dergi \u00e7\u0131kard\u0131. Derginin ilk say\u0131s\u0131nda _Yeralt\u0131ndan Notlar_ (1864) roman\u0131 tefrika edilmeye ba\u015flad\u0131. Hayat\u0131nda bir talihsizlik dizisi ba\u015flad\u0131; kar\u0131s\u0131n\u0131 ve karde\u015fini kaybettikten sonra dergi kapand\u0131. Alacakl\u0131lar\u0131n\u0131n tehditleri \u00fczerine, bir yay\u0131mc\u0131dan ald\u0131\u011f\u0131 bor\u00e7la Avrupa'ya ka\u00e7t\u0131. _Su\u00e7 ve Ceza_ 'y\u0131 (1866) yazmaya ba\u015flad\u0131 ve onun i\u00e7in ald\u0131\u011f\u0131 avansla Rusya'ya d\u00f6nd\u00fc. \u00d6nce _Kumarbaz_ (1866) adl\u0131 roman\u0131n\u0131 yay\u0131mlad\u0131. Ayn\u0131 y\u0131l tamamlad\u0131\u011f\u0131 _Su\u00e7 ve Ceza_ , ele\u015ftirmenleri ve okurlar\u0131 b\u00fcy\u00fcledi. 1867'de asistan\u0131 Anna Snitkina'yla evlenerek yeniden yurtd\u0131\u015f\u0131na \u00e7\u0131kt\u0131; d\u00f6rt y\u0131l Rusya'dan uzak kald\u0131. Al\u00e7alt\u0131c\u0131 bir yoksulluk i\u00e7inde ge\u00e7en bu d\u00f6nem boyunca \u00fclkeden \u00fclkeye dola\u015ft\u0131. B\u00fct\u00fcn bu g\u00fc\u00e7l\u00fcklere, sara n\u00f6betlerine, vazge\u00e7emedi\u011fi kumar tutkusuna ra\u011fmen, ilk \u00e7ocuklar\u0131n\u0131n trajik \u00f6l\u00fcm\u00fcne de katlanan gen\u00e7 kar\u0131s\u0131, ba\u011fl\u0131l\u0131\u011f\u0131n\u0131 bir an yitirmeden ona ger\u00e7ek a\u015fk\u0131 tatt\u0131rd\u0131. Dostoyevski, bu a\u011f\u0131r ya\u015fam ko\u015fullar\u0131 alt\u0131nda da sendelemeyerek ikinci ba\u015fyap\u0131t\u0131 _Budala_ 'y\u0131 (1870) yazd\u0131. _Budala_ 'y\u0131, 1870'te _Ebed\u00ee Koca_ , 1872'de _Ecinniler_ , 1875'te _Delikanl\u0131_ izledi. _Karamazov Karde\u015fler_ 'i (1879-1880) yazmaya ba\u015flad\u0131\u011f\u0131nda, art\u0131k \u00fclke \u00e7ap\u0131nda tan\u0131nan bir yazard\u0131. Dostoyevski bu arada, Grajdanin'e yazd\u0131\u011f\u0131 \"Bir Yazar\u0131n G\u00fcnl\u00fc\u011f\u00fc\" (1873-1881) ba\u015fl\u0131kl\u0131 k\u00f6\u015fe yaz\u0131lar\u0131n\u0131, 1876-1877 y\u0131llar\u0131 aras\u0131nda ayl\u0131k bir yay\u0131n olarak \u00e7\u0131kard\u0131 ve daha sonra yaz\u0131lar, ayn\u0131 adla kitapla\u015ft\u0131r\u0131ld\u0131. 1881'de Petersburg'da \u00f6len Dostoyevski, Bat\u0131l\u0131 \u00fclkelerin edebiyat ve d\u00fc\u015f\u00fcn ya\u015fam\u0131nda \u00f6nemli bir rol oynam\u0131\u015f, \u00f6zellikle varolu\u015f\u00e7ulu\u011fun temel kaynaklar\u0131ndan biri say\u0131lm\u0131\u015ft\u0131r.\n\n* * *\n\nSABR\u0130 G\u00dcRSES, 1972 y\u0131l\u0131nda \u0130stanbul'da do\u011fdu. 1989'da Ni\u015fanta\u015f\u0131 Anadolu Lisesi'nden mezun oldu. _Gereksinimler, Elde Edemeyi\u015fler ve \u0130lerlemeler_ (\u015fiir, 1990); _Unutulmu\u015f Ay Alt\u0131nda_ (1992); _Bo\u015fvermi\u015fler: Bir Bilimkurgu \u00dc\u00e7lemesi_ (1995); _Sevi\u015fme_ (1996) adl\u0131 kitaplar\u0131 yay\u0131mland\u0131. 1999 y\u0131l\u0131nda \u0130\u00dc Rus Dili ve Edebiyat\u0131 b\u00f6l\u00fcm\u00fcnden mezun oldu. Y\u00fcksek lisans\u0131n\u0131 \u0130\u00dc \u00c7eviribilim B\u00f6l\u00fcm\u00fc'nde \"Vladimir Nabokov'un Eugenie Onegin \u00c7evirisi ve T\u00fcrk\u00e7e Onegin \u00c7evirileri\" ba\u015fl\u0131kl\u0131 teziyle yapt\u0131. 1999 y\u0131l\u0131ndan beri \u0130ngilizceden, 2000 y\u0131l\u0131ndan beri Rus\u00e7adan kitap \u00e7evirisi yap\u0131yor. G\u00fcrses, 1989 Akademi \u015eiir \u00d6d\u00fcl\u00fc, 1991 _Milliyet Sanat_ A\u015fk \u00f6yk\u00fcleri Yar\u0131\u015fmas\u0131'nda ba\u015far\u0131 \u00f6d\u00fcl\u00fc, 2009 Pu\u015fkin Enstit\u00fcs\u00fc Onur Diplomas\u0131, 2010 _D\u00fcnya Kitap_ Y\u0131l\u0131n \u00c7evirisi \u00d6d\u00fcl\u00fc ve 2011 \u00c7eviri Derne\u011fi Gen\u00e7 Soluk \u00d6d\u00fcl\u00fc'n\u00fc ald\u0131. \u0130ngilizceden Joseph Campbell, Jonathan Lethem, John Smolens, Shusha Guppy, Don DeLillo, William Guthrie, Werner Sombart, Kim Stanley Robinson, Fredric Jameson, Slavoj \u017di\u017eek, Karel \u010capek; Rus\u00e7adan Dostoyevski, Turgenyev, Pu\u015fkin, \u0130van Gon\u00e7arov, Mihail Bahtin, Yuri Lotman, Sultan Galiyev, Andrey Bel\u0131y, Yuri Ole\u015fa, Mihail Bulgakov \u00e7evirileri bulunmaktad\u0131r.\n\n### \u0130\u00e7indekiler\n\nBir b\u00fcy\u00fck yazar do\u011fuyor\n\nNisan\n\nMay\u0131s\n\nHaziran\n\nTemmuz\n\nA\u011fustos\n\nEyl\u00fcl\nAh \u015fu benim masalc\u0131lar\u0131m! Yararl\u0131, ho\u015f, keyif verici bir \u015fey kalmam\u0131\u015f gibi yery\u00fcz\u00fcn\u00fc incik cincik kurcalarlar!.. Onlara yazmay\u0131 yasaklamak laz\u0131m! Yani, bu \u015fimdi neye benziyor, biliyor musunuz? Hani bir \u015feyler okurken... insan ister istemez d\u00fc\u015f\u00fcn\u00fcr ve t\u00fcrl\u00fc t\u00fcrl\u00fc sa\u00e7mal\u0131klar gelir akl\u0131na; cidden, yasaklamak laz\u0131m yazmay\u0131; hi\u00e7 d\u00fc\u015f\u00fcnmeden, hepsine yasaklamak laz\u0131m.\n\nV.F. ODOYEVSK\u01301\n\n1. Rus yazar, tiyatro ve m\u00fczik ele\u015ftirmeni, filozof, pedagog Vladimir Odoyevski'nin (1803-1869), \"Jivoy Myortvets\" (Ya\u015fayan \u00d6l\u00fc, 1838) adl\u0131 \u00f6yk\u00fcs\u00fcnden.\n\n# Bir b\u00fcy\u00fck yazar do\u011fuyor\n\n\"Yoksul insanda gurur olmamal\u0131d\u0131r asla, asla!\"\n\n_\u0130nsanc\u0131klar_ ya da as\u0131l isminin birebir \u00e7evirisiyle s\u00f6ylenirse \"Zavall\u0131, Yoksul \u0130nsanlar\" adl\u0131 roman, edebiyat tarihinin tuhaf zirvelerinden biridir. Hem yirmi \u00fc\u00e7 ya\u015f\u0131ndaki bir gencin, Fyodor Mihaylovi\u00e7 Dostoyevski'nin ya\u015fam\u0131nda bir d\u00f6n\u00fcm noktas\u0131d\u0131r hem de Rus edebiyat\u0131n\u0131n geli\u015fiminde.\n\nT\u00fcm\u00fcyle bir tesad\u00fcf\u00fcn, bir \u00e7\u0131k\u0131\u015f aray\u0131\u015f\u0131n\u0131n, yoksulluktan, paras\u0131zl\u0131ktan \u00e7\u0131k\u0131\u015f aray\u0131\u015f\u0131yla bu aray\u0131\u015f\u0131 edeb\u00ee bir deneyime d\u00f6n\u00fc\u015ft\u00fcrme \u00e7abas\u0131n\u0131n ortak \u00fcr\u00fcn\u00fcd\u00fcr bu roman. Petersburg'da m\u00fchendislik e\u011fitimi alan gen\u00e7 Dostoyevski, memur olmak d\u0131\u015f\u0131nda bir se\u00e7enek aramakta, daha sonra _Beyaz Geceler_ 'de, _Ev Sahibesi_ 'nde, _Yufka Y\u00fcrek_ 'de, _\u0130kiz_ 'de anlataca\u011f\u0131 hayalperest ve buhranl\u0131 bir tipin hayat\u0131n\u0131 s\u00fcrmektedir. Yoksulluk i\u00e7inde ya\u015famaktad\u0131r ve buhranlar\u0131 bazen fiziksel, somut ac\u0131lar, hastal\u0131klar olarak d\u0131\u015fa vurmaktad\u0131r.\n\nEdebiyatla, hem yeni \u00e7i\u00e7eklenen ve do\u011falc\u0131l\u0131kla romantizm e\u011filimlerini i\u00e7 i\u00e7e ya\u015fayan yerli edebiyatla, hem de bu edebiyatla i\u00e7 i\u00e7e ge\u00e7mi\u015f olan yabanc\u0131 edebiyatla, Balzac, Schiller, Hoffmann, Scott'la da ilgilenir. Ve ek gelir kazanmak \u00fczere \u00e7eviriye y\u00f6nelir. 1843 y\u0131l\u0131nda karde\u015fi Mihail'le birlikte Eug\u00e8ne Sue'nun _Mathilde_ adl\u0131 roman\u0131n\u0131 \u00e7evirmeye ba\u015flar, ertesi y\u0131l George Sand'\u0131n _La derni\u00e8re Aldini_ adl\u0131 roman\u0131n\u0131 \u00e7evirir. Bu yazarlar\u0131 tesad\u00fcfen se\u00e7mez: Eug\u00e8ne Sue d\u00f6nemin ele\u015ftirmenlerinin sosyalist e\u011filimleri nedeniyle \u00f6vd\u00fc\u011f\u00fc bir yazard\u0131r, George Sand ise bir \u00e7oksatar. Fakat \u00e7eviri yapman\u0131n ba\u015far\u0131l\u0131 bir ge\u00e7im yolu olmad\u0131\u011f\u0131 k\u0131sa s\u00fcrede anla\u015f\u0131l\u0131r. Karde\u015fiyle birlikte Friedrich Schiller'in t\u00fcm oyunlar\u0131n\u0131 \u00e7evirmek \u00fczere anla\u015f\u0131rlar, karde\u015fi iki oyun \u00e7evirir, fakat bunlar da kazan\u00e7 getirmez. Son olarak, 1843 y\u0131l\u0131nda, bir a\u015fk maceras\u0131 y\u00fcz\u00fcnden, zengin bir dul kontesin pe\u015finden Petersburg'a gelen Honor\u00e9 de Balzac bir \u00e7eviri tasar\u0131s\u0131na esin kayna\u011f\u0131 olur: Dostoyevski onun _Eugenie Grandet_ adl\u0131 roman\u0131n\u0131 \u00e7evirir. 1844 y\u0131l\u0131nda yay\u0131mlanan bu \u00e7eviri eser, ad\u0131n\u0131n ge\u00e7ti\u011fi ilk edebiyat \u00fcr\u00fcn\u00fc ve onu kendi eserini yazmaya s\u00fcr\u00fckleyen \u00e7al\u0131\u015fma olacakt\u0131r.\n\n1844 Eyl\u00fcl\u00fc'nde memurluktan bir s\u00fcre \u00f6nce istifa etmi\u015f olan Dostoyevski karde\u015fine yazd\u0131\u011f\u0131 bir mektupta _\u0130nsanc\u0131klar_ '\u0131n do\u011faca\u011f\u0131n\u0131 haber verir:\n\n> Eugenie Grandet b\u00fcy\u00fckl\u00fc\u011f\u00fcnde bir roman\u0131 tamamlamak \u00fczereyim... Biraz de\u011fi\u015fik bir roman... Onu [Nekrasov'un yay\u0131mlad\u0131\u011f\u0131] _Ote\u00e7estvenniye zapiski_ dergisine vermeyi tasarl\u0131yorum.\n\n1844 Kas\u0131m\u0131'nda roman haz\u0131rd\u0131r, ama Aral\u0131k ay\u0131nda neredeyse yeniden yaz\u0131l\u0131r. 1845 May\u0131s\u0131'nda roman son kez bir daha yaz\u0131l\u0131r:\n\n> Art\u0131k roman tamam, bu d\u00fczeltme son d\u00fczeltme oldu. Bir daha ele almamak i\u00e7in s\u00f6z verdim.\n\nDostoyevski, roman\u0131 ilk olarak ev arkada\u015f\u0131 ve yazar Grigorovi\u00e7'e okutur. Grigorovi\u00e7 duygular\u0131n\u0131 \u015f\u00f6yle anlatacakt\u0131r daha sonra:\n\n> _\u0130nsanc\u0131klar_ '\u0131 okumaya ba\u015flay\u0131nca daha ilk sayfalarda Dostoyevski'nin yazd\u0131\u011f\u0131 \u015feyin benim yazd\u0131klar\u0131mdan \u00e7ok daha iyi oldu\u011funu anlad\u0131m; okuduk\u00e7a g\u00fc\u00e7lendi bu izlenimim. O kadar heyecanlanm\u0131\u015ft\u0131m ki birka\u00e7 kez kalk\u0131p boynuna sar\u0131lmak istedim; fakat onun \u015famatal\u0131, a\u015f\u0131r\u0131 duygu g\u00f6sterilerinden ho\u015flanmayan bir insan olmas\u0131 engel oldu bana.\n\nGrigorovi\u00e7 ertesi g\u00fcn roman\u0131 al\u0131p yazar ve yay\u0131mc\u0131 Nekrasov'a g\u00f6t\u00fcr\u00fcr:\n\n> Roman\u0131 ba\u015ftan sona okudum ona. Son sayfada, ya\u015fl\u0131 Devu\u015fkin'in Varenka'ya veda etti\u011fi yerde kendimi daha fazla tutamay\u0131p h\u0131\u00e7k\u0131rmaya ba\u015flad\u0131m; yava\u015f\u00e7a Nekrasov'a bakt\u0131m; onun da g\u00f6z\u00fcnden ya\u015flar ak\u0131yordu. Art\u0131k daha da inan\u0131yordum ona... Hemen ko\u015fup Dostoyevski'ye, saat ge\u00e7 oldu\u011fu halde (gecenin d\u00f6rd\u00fcyd\u00fc) ba\u015far\u0131s\u0131n\u0131 bildirmek istedim...\n> \n> Nekrasov heyecandan \u015fa\u015fk\u0131n bir halde kabul etti... Ev arkada\u015f\u0131m\u0131n karakterini, onun yabanili\u011fini, a\u015f\u0131r\u0131 hassasl\u0131\u011f\u0131n\u0131, i\u00e7ine kapan\u0131kl\u0131\u011f\u0131n\u0131 bildi\u011fim i\u00e7in, olanlar\u0131 ona ertesi g\u00fcn, daha sakince, onu tedirgin etmeden anlatmam daha do\u011fru olacakt\u0131...\n> \n> Ertesi g\u00fcn kap\u0131y\u0131 Dostoyevski a\u00e7t\u0131; yan\u0131mda tan\u0131mad\u0131\u011f\u0131 birini g\u00f6r\u00fcnce \u00e7ekindi, bembeyaz oldu ve uzun s\u00fcre Nekrasov'un ona s\u00f6yledikleri kar\u015f\u0131s\u0131nda sessiz kald\u0131.\n\nNekrasov oradan ayr\u0131l\u0131p hemen ele\u015ftirmen Belinski'ye gider:\n\n\"Yeni Gogol do\u011fdu!\" der Nekrasov daha kap\u0131 a\u011fz\u0131ndan.\n\n\"Bizde mantar gibi Gogol yeti\u015fir!\" diye yan\u0131t verir ciddi bir tav\u0131rla Belinski. Ama roman\u0131 al\u0131r.\n\nAyn\u0131 g\u00fcn, Belinski'ye tekrar u\u011frayan Nekrasov onu heyecan i\u00e7inde bulur:\n\n\"Nerede kald\u0131n\u0131z? Nerede bu Dostoyevski'niz? Gen\u00e7 mi? Ka\u00e7 ya\u015f\u0131nda? Hemen getirin bana onu!..\"\n\nBelinski'nin evine getirilen yirmi \u00fc\u00e7 ya\u015f\u0131ndaki gen\u00e7 yazar, daha sonra orada olanlar\u0131 \u015f\u00f6yle hat\u0131rl\u0131yor:\n\n> Ve i\u015fte... beni onun yan\u0131na g\u00f6t\u00fcrd\u00fcler. Belinski'yi birka\u00e7 y\u0131l \u00f6nce heyecanla okumu\u015ftum, ama bana \u00fcrk\u00fct\u00fcc\u00fc ve sert gelmi\u015fti ve benim _\u0130nsanc\u0131klar_ '\u0131mla alay edecek, diye d\u00fc\u015f\u00fcn\u00fcyordum. Beni \u00e7ok sayg\u0131l\u0131 ve a\u011f\u0131rba\u015fl\u0131 bir \u015fekilde kar\u015f\u0131lad\u0131, ama daha bir dakika bile ge\u00e7meden her \u015fey bamba\u015fka oldu... Ate\u015fli ate\u015fli, alevli g\u00f6zlerle konu\u015fuyordu. \"Siz kendiniz anl\u0131yor musunuz?\" diyordu bana tekrar tekrar, al\u0131\u015fkanl\u0131\u011f\u0131 oldu\u011fu \u00fczere ba\u011f\u0131rarak, \"Ne yazm\u0131\u015f oldu\u011funuzu anl\u0131yor musunuz!.. B\u00fct\u00fcn bu korkun\u00e7 ger\u00e7e\u011fi, bizlere g\u00f6stermi\u015f oldu\u011funuz bu ger\u00e7e\u011fi siz mi d\u00fc\u015f\u00fcnd\u00fcn\u00fcz? Olamaz, sizin gibi yirmi ya\u015f\u0131nda birinin b\u00fct\u00fcn bunlar\u0131 anlam\u0131\u015f olmas\u0131na imk\u00e2n yok... Ger\u00e7e\u011fi ke\u015ffetmi\u015f ve bir sanat\u00e7\u0131 olarak ilan etmi\u015fsiniz, size bir yetenek verilmi\u015f, yetene\u011finizin de\u011ferini bilin ve emin olun, siz b\u00fcy\u00fck bir yazar olacaks\u0131n\u0131z!\"\n\n* * *\n\nBug\u00fcn, yaz\u0131ld\u0131ktan y\u00fcz elli y\u0131l\u0131 a\u015fk\u0131n bir zaman sonra, _\u0130nsanc\u0131klar_ neden h\u00e2l\u00e2 okunmaktad\u0131r?\n\n\u0130ki temel nedeni vard\u0131r bunun: Birincisi, bu roman Dostoyevski'nin ruhunu ve evrenini, eserlerinde anlatt\u0131\u011f\u0131 Petersburg'un d\u00fcnyas\u0131n\u0131 ilk kez ortaya \u00e7\u0131karan bir kitap olmas\u0131 nedeniyle okunmaktad\u0131r: Ya\u015fl\u0131 Devu\u015fkin ile gen\u00e7 Varvara'n\u0131n bir avlunun kar\u015f\u0131l\u0131kl\u0131 iki odas\u0131nda oturdu\u011fu ve birbiriyle yaz\u0131\u015ft\u0131\u011f\u0131 bina, Dostoyevski'nin \u00f6zellikle ilk d\u00f6nem eserlerinin neredeyse b\u00fct\u00fcn kadrosunu, b\u00fct\u00fcn o yoksul, dar gelirli insanlar\u0131, memurlar\u0131, ev sahibelerini, subaylar\u0131 vb. bar\u0131nd\u0131rmaktad\u0131r. Di\u011fer b\u00fct\u00fcn eserlerin, \u00f6rne\u011fin _Beyaz Geceler_ 'in, _Yeralt\u0131ndan Notlar_ '\u0131n, _Ezilenler_ 'in ve _Delikanl\u0131_ 'n\u0131n kadrosu buradad\u0131r \u2013 hatta Raskolnikov'un bile o hezeyanl\u0131 planlar\u0131n\u0131 Makar Devu\u015fkin'in ve b\u00fcy\u00fck olas\u0131l\u0131kla Bay Prohar\u00e7in'in de oturdu\u011fu bu binan\u0131n \u00e7at\u0131 kat\u0131nda yapt\u0131\u011f\u0131n\u0131 hayal etmek zor de\u011fildir.\n\nBu roman, Dostoyevski'nin d\u00fcnyas\u0131n\u0131, onu bir yazar yapan d\u00fcnyay\u0131 neredeyse tam kadrosuyla ve Dostoyevski'nin hayalindeki ilk haliyle sergilemektedir.\n\nDi\u011fer yandan, roman\u0131n kal\u0131c\u0131 bir etki b\u0131rakmas\u0131n\u0131n ikinci bir nedeni de, Baudelaire'le ayn\u0131 y\u0131l do\u011fmu\u015f olan Dostoyevski'nin bu romanla bir bak\u0131ma Rus modernizminin \u00f6nc\u00fcleri aras\u0131nda yer alm\u0131\u015f olmas\u0131, en az\u0131ndan bu modernizmin temel karakterini ortaya \u00e7\u0131karm\u0131\u015f olmas\u0131d\u0131r. Bu romanda XIX. y\u00fczy\u0131l sonu Rus roman\u0131n\u0131n ele alaca\u011f\u0131 temel konular\u0131n hemen hepsi vard\u0131r neredeyse: Yoksulluk, paran\u0131n g\u00fcc\u00fc, de\u011fi\u015fen toplumsal ili\u015fkiler, entelijensiya, yani e\u011fitimli, ayd\u0131n kesim ile halk aras\u0131ndaki u\u00e7urum, entelijensiya'n\u0131n hayalperestli\u011fi, edebiyat ile ger\u00e7ek hayat\u0131n \u00e7eli\u015fkisi, yirmi y\u0131l sonra Gon\u00e7arov'un Oblomov karakteriyle tiple\u015fecek olan bir l\u00fczumsuz adaml\u0131k hali, \u015fehir ile k\u00f6y kar\u015f\u0131tl\u0131\u011f\u0131 ve elbette, ironi, mizah. B\u00fcy\u00fck olas\u0131l\u0131kla, Belinski'yi co\u015fkuland\u0131ran \u015fey, bir ilk eserde b\u00fct\u00fcn bu konular\u0131n dengeli bir \u00f6l\u00e7\u00fcyle bir araya getirilmi\u015f olmas\u0131yd\u0131. Belinski bu romandaki do\u011falc\u0131, ger\u00e7ek\u00e7i \u00f6\u011feleri be\u011fenmi\u015f, ama bu roman\u0131n i\u00e7inde k\u0131v\u0131lc\u0131mlanmaya ba\u015flayan ve Dostoyevski'nin _\u0130kiz_ 'de ( _\u0130kiz_ , yani Rus\u00e7as\u0131yla _Dvoynik_ s\u00f6zc\u00fc\u011f\u00fcn\u00fcn Hoffmann'\u0131n, Poe'nun kulland\u0131\u011f\u0131 _doppelg\u00e4nger_ fig\u00fcr\u00fcn\u00fcn bir \u00e7evirisi oldu\u011funu hat\u0131rlayal\u0131m) dizginlerini serbest b\u0131rakaca\u011f\u0131 modernist sesi tan\u0131mam\u0131\u015f ya da yabanc\u0131 bulmu\u015ftur.\n\nN.M. \u00c7irkov, _\u0130nsanc\u0131klar_ 'daki \u015fehir-k\u00f6y kar\u015f\u0131tl\u0131\u011f\u0131na, _O stile Dostoyevskova_ (Dostoyevski'nin \u00dcslubu) adl\u0131 kitab\u0131nda \u015f\u00f6yle dikkat \u00e7ekiyor:\n\n> Bu roman\u0131n ve ayn\u0131 d\u00f6nemdeki eserlerin temel eylem alan\u0131 \u2013 kalabal\u0131k \u015fehir, g\u00fcr\u00fclt\u00fc, ama en \u00f6nemlisi s\u0131k\u0131\u015f\u0131kl\u0131k ve s\u0131k\u0131c\u0131l\u0131k, \u015fehrin ve \u015fehir sakinlerinin kasvetli g\u00f6r\u00fcn\u00fcm\u00fc, \u00e7evrede bulunan e\u015fyalar\u0131n gri, \u00e7amurlu, koyu renkli boyalar\u0131d\u0131r. Dostoyevski'nin ilk d\u00f6nem eserlerindeki kahramanlar\u0131n kaderini belirleyen ve \u00e7evrelerini saran d\u00fcnya budur. Ge\u00e7mi\u015f, k\u00f6y d\u00fcnyas\u0131d\u0131r, kahramanlar\u0131n hat\u0131ralar\u0131nda her \u015feyin parlak renklerle dolu oldu\u011fu ve \"her \u015feyin dingin, herkesin mutlu\" oldu\u011fu bir d\u00fcnyad\u0131r. Bu y\u00fczden, insanlar\u0131n eski, g\u00fczel bir d\u00fcnyadan kopmu\u015f olmalar\u0131 motifi \u00f6nce _\u0130nsanc\u0131klar_ 'da duyulmaya ba\u015flar.\n\nRomanda bu eski, g\u00fczel d\u00fcnya 3 Eyl\u00fcl g\u00fcn\u00fc kad\u0131n karakter taraf\u0131ndan \u015f\u00f6yle anlat\u0131l\u0131r (Gon\u00e7arov, _Oblomov_ 'da buna benzer bir tasvire \"Oblomov'un D\u00fc\u015f\u00fc\" adl\u0131 b\u00f6l\u00fcmdeki Oblomovka k\u00f6y\u00fc tasvirinde yer verecektir):\n\n> K\u00f6ydeyken sonbahar\u0131 ne \u00e7ok severdim! Daha k\u00fc\u00e7\u00fcc\u00fck \u00e7ocuktum, ama o zaman da bir\u00e7ok \u015fey hissederdim. Sonbahar ak\u015fam\u0131n\u0131 sabah\u0131ndan daha \u00e7ok severdim. Evimizden iki ad\u0131m \u00f6tede, da\u011f\u0131n k\u0131y\u0131s\u0131nda, g\u00f6l vard\u0131. Bu g\u00f6l \u2013\u015fimdi g\u00f6zlerimin \u00f6n\u00fcne geliyor\u2013 bu g\u00f6l \u00f6yle b\u00fcy\u00fck, \u00f6yle ayd\u0131nl\u0131k, \u00f6yle temizdi ki, kristal gibiydi!\n\n5 Eyl\u00fcl'de erkek karakter, Petersburg'un yoksul ve zengin insanlar\u0131n\u0131n ya\u015fad\u0131\u011f\u0131 semtlerin, Fontanka ile Gorohovaya'n\u0131n kar\u015f\u0131tl\u0131\u011f\u0131n\u0131 anlat\u0131r:\n\n> Biraz hava almak i\u00e7in Fontanka k\u0131y\u0131s\u0131nda gezmeye \u00e7\u0131kt\u0131m. \u00c7ok s\u0131cak, nemli bir ak\u015famd\u0131. Saat alt\u0131da ortal\u0131k kararm\u0131\u015ft\u0131 \u2013 t\u0131pk\u0131 \u015fimdiki gibi! Ya\u011fmur ya\u011fm\u0131yordu ama sis vard\u0131; s\u0131k\u0131 bir ya\u011fmurdan geri kalmayan bir sis. G\u00f6kte uzun, geni\u015f bulutlar \u015ferit \u015ferit ilerliyordu. Halk ak\u0131n ak\u0131n y\u00fcr\u00fcyordu k\u0131y\u0131da ve hepsinin de y\u00fcz\u00fc sanki kastenmi\u015f gibi korkun\u00e7, kasvetliydi; sarho\u015f adamlar, kalk\u0131k burunlu kad\u0131nlar \u2013 _\u00e7uhonka_ 'lar, \u00e7izme giymi\u015f ba\u015f\u0131 ba\u011fs\u0131z bek\u00e2r kad\u0131nlar, artel i\u015f\u00e7ileri, arabac\u0131lar, t\u00fcrl\u00fc t\u00fcrl\u00fc ihtiya\u00e7lar\u0131n pe\u015findeki karde\u015flerimiz; \u00e7ubuklu \u00f6nl\u00fc\u011f\u00fcn\u00fc giymi\u015f, s\u0131ska, hastal\u0131kl\u0131, y\u00fcz\u00fc ya\u011f kaplanm\u0131\u015f, eline bir kilit alm\u0131\u015f \u00e7ilingir \u00e7\u0131ra\u011f\u0131; iki metre boyunda bir emekli asker \u2013 i\u015fte b\u00f6yle bir halk vard\u0131. O saatte de anla\u015f\u0131lan ba\u015fka bir halk olamazd\u0131. Gemi kanal\u0131 Fontanka!..\n> \n> (...)\n> \n> Gorohovaya'ya sapt\u0131\u011f\u0131mda, ortal\u0131k iyice kararm\u0131\u015f ve gaz lambalar\u0131 yanm\u0131\u015ft\u0131. Uzun zamand\u0131r gitmemi\u015ftim Gorohovaya'ya \u2013 gidememi\u015ftim. Ne g\u00fcr\u00fclt\u00fcl\u00fc bir cadde! Ne kadar \u00e7ok zengin d\u00fckk\u00e2nlar, ma\u011fazalar var; her \u015fey \u0131\u015f\u0131l \u0131\u015f\u0131l yan\u0131yor, camlar\u0131n arkas\u0131nda kuma\u015flar, \u00e7i\u00e7ekler, kurdeleli bir s\u00fcr\u00fc \u015fapkalar var. B\u00fct\u00fcn bunlar\u0131n s\u00fcs diye konuldu\u011funu d\u00fc\u015f\u00fcn\u00fcyor insan \u2013 ama \u00f6yle de\u011fil, \u00e7\u00fcnk\u00fc adamlar var, b\u00fct\u00fcn bunlar\u0131 sat\u0131n al\u0131yor, kar\u0131lar\u0131na hediye ediyorlar. Zengin bir cadde! Gorohovaya'da bir s\u00fcr\u00fc Alman f\u0131r\u0131nc\u0131 ya\u015f\u0131yor; cadde halk\u0131n\u0131n da hali vakti yerinde olmal\u0131. Her an bir s\u00fcr\u00fc araba ge\u00e7iyor caddeden; k\u00f6pr\u00fc b\u00fct\u00fcn hepsini nas\u0131l ta\u015f\u0131yabiliyor! \u015eatafatl\u0131 yayl\u0131 arabalar var, camlar\u0131 ayna gibi, i\u00e7leri kadife ve k\u00fcrk kapl\u0131; soylular\u0131n u\u015faklar\u0131 apoletli, k\u0131l\u0131\u00e7l\u0131. B\u00fct\u00fcn arabalar\u0131 seyrettim, i\u00e7lerinde oturan kad\u0131nlar\u0131n hepsi de prenses ve kontesler gibi giyinmi\u015fti. Herhalde, herkesin balolara ve toplant\u0131lara tela\u015f etti\u011fi bir saatti...\n\nParis pasajlar\u0131n\u0131 anlat\u0131yor sanki Baudelaire.\n\n* * *\n\nDostoyevski'nin s\u0131rad\u0131\u015f\u0131 yetene\u011fini fark eden Belinski, 1846 y\u0131l\u0131nda, _\u0130nsanc\u0131klar_ yay\u0131mland\u0131ktan sonra, _Peterburgski Sbornik_ adl\u0131 dergiye yazd\u0131\u011f\u0131 ele\u015ftirilerden birine, Dostoyevski' nin bu s\u0131rad\u0131\u015f\u0131 yetene\u011fini a\u00e7\u0131klamak \u00fczere Pu\u015fkin'i \u00f6rnek vererek ba\u015flar:\n\n> S\u0131rad\u0131\u015f\u0131 bir yetene\u011fin ortaya \u00e7\u0131kmas\u0131 okuyan ve yazanlar aras\u0131nda tart\u0131\u015fma ve anla\u015fmazl\u0131klara yol a\u00e7ar. E\u011fer bu yetenek daha kurulu\u015funu tamamlamam\u0131\u015f bir edebiyat\u0131n erken d\u00f6neminde ortaya \u00e7\u0131km\u0131\u015fsa, bir yandan heyecan dolu \u00e7\u0131\u011fl\u0131klarla, \u00f6l\u00e7\u00fcs\u00fcz \u00f6vg\u00fclerle, di\u011fer yandan ko\u015fulsuz bir yarg\u0131yla, ko\u015fulsuz bir olumsuzlamayla kar\u015f\u0131la\u015f\u0131r. Pu\u015fkin'de b\u00f6yle oldu. Onda \"Kuzeyli bir Byron\" g\u00f6rd\u00fcler (Sanki G\u00fcneyli bir Byron varm\u0131\u015f gibi!), \"\u00e7a\u011fda\u015f insanl\u0131\u011f\u0131n temsilcisi\"ni g\u00f6rd\u00fcler, \u00fcstelik onun daha ilk eserleri i\u00e7in, \u00f6zellikle de di\u011ferlerinden daha zay\u0131f olan ve art\u0131k benzersiz de\u011ferini kesinlikle kaybetmi\u015f olanlar i\u00e7in s\u00f6yleniyordu b\u00fct\u00fcn bunlar; ba\u015fkalar\u0131 da onun eserlerini \u00e7ocukluktan beri al\u0131\u015ft\u0131klar\u0131 t\u00f6rensel gevezeliklere k\u0131yasla \u015fiirin seviyesizle\u015ftirilmesi, s\u0131radanla\u015ft\u0131r\u0131lmas\u0131 olarak g\u00f6r\u00fcyorlard\u0131. Pu\u015fkin'i anlamak ba\u015fka bir ku\u015fa\u011fa kald\u0131 ve bu da \u00f6l\u00fcm\u00fcnden \u00f6nce ger\u00e7ekle\u015fmedi.\n\nArd\u0131ndan Pu\u015fkin ile Gogol ili\u015fkisini de\u011ferlendirmeye ge\u00e7er:\n\n> Ama Pu\u015fkin'in \u00f6nc\u00fcleri i\u00e7in de onun \u00fczerinde az \u00e7ok etki b\u0131rakm\u0131\u015f olduklar\u0131 ve onlar\u0131n \u015fiirinin az \u00e7ok onun \u015fiirini, \u00f6zellikle de ilk denemelerini haber verdi\u011fi s\u00f6ylenebilir. Pu\u015fkin'in \u00fczerinde \u00e7a\u011fda\u015f\u0131 olan Avrupal\u0131 \u015fairlerin etkisi daha a\u00e7\u0131k ve do\u011frudan olmu\u015ftu. E\u011fer Pu\u015fkin'in bu ilk eserleri baz\u0131lar\u0131na tats\u0131z gelmi\u015f, baz\u0131lar\u0131nda da sadece yenili\u011fiyle de\u011fil, \u00f6zg\u00fcnl\u00fc\u011f\u00fc ve kendine \u00f6zg\u00fcl\u00fc\u011f\u00fcyle de tam bir ho\u015fnutluk ve heyecan yaratm\u0131\u015fsa bu, onun yetene\u011finin ne kadar d\u00e2hiyane oldu\u011funu g\u00f6sterir. Ama her \u015feye ra\u011fmen onun ilk eserleri uzaktan da olsa Rus edebiyat\u0131ndaki bir\u00e7ok kimseyi, ama yak\u0131n bir \u015fekilde, yabanc\u0131 edebiyattaki \u00e7ok daha fazla kimseyi hat\u0131rlat\u0131yordu; bu da ona ba\u015far\u0131s\u0131z ve yersiz bir \u015fekilde verilen Rus Byron unvan\u0131n\u0131n bir kan\u0131t\u0131 olarak durmaktad\u0131r. Gogol'\u00fcn Rus edebiyat\u0131nda bir \u00f6nc\u00fcl\u00fc yoktu, yabanc\u0131 edebiyatta da benzerleri yoktu (ve olamazd\u0131). Onun \u015fiirinin soyu, ortaya \u00e7\u0131kmadan \u00f6nce, ima bile edilmemi\u015fti. \u015eiiri birdenbire, beklenmedik \u015fekilde, ba\u015fka hi\u00e7bir \u015fiire benzemeden ortaya \u00e7\u0131kt\u0131. Elbette, Gogol'\u00fcn, s\u00f6zgelimi Pu\u015fkin'den etkilendi\u011fi reddedilemez; ama bu etki do\u011frudan de\u011fildi: Gogol'\u00fcn eserine yans\u0131m\u0131\u015ft\u0131 ama deyim yerindeyse, Gogol'\u00fcn eserinin y\u00fcz hatlar\u0131na, \u00e7ehresine yans\u0131mam\u0131\u015ft\u0131. Bu etki daha \u00e7ok Pu\u015fkin'in, Pu\u015fkin'in kendisinden bile \u00f6ne \u00e7\u0131kt\u0131\u011f\u0131 bir d\u00f6nemin etkisiydi. Tabii e\u011fer Gogol, Pu\u015fkin'den \u00f6nce gelmi\u015f olsayd\u0131, \u015fu anda durdu\u011fu y\u00fcksekli\u011fe eri\u015femezdi. Ama do\u011frudan etki, yani Pu\u015fkin \u00fczerinde ondan \u00f6nceki Rus ve \u00e7a\u011fda\u015f\u0131 olan Avrupal\u0131 \u015fairlerin (az ya da \u00e7ok \u00f6l\u00e7\u00fcde, yak\u0131ndan ya da uzaktan) sahip oldu\u011fu etki d\u00fc\u015f\u00fcn\u00fcl\u00fcrse belli ki Pu\u015fkin taraf\u0131ndan Gogol'e y\u00f6nelik b\u00f6yle bir etki, Gogol'\u00fcn eserlerinde herhangi bir \u015fekilde bulunamaz.\n\nDostoyevski'yi ve _\u0130nsanc\u0131klar_ '\u0131 bu \u00e7er\u00e7evede Pu\u015fkin-Gogol \u00e7izgisinin haber verdi\u011fi bir yetenek olarak de\u011ferlendirir:\n\n> _\u0130nsanc\u0131klar_ ve yeni, s\u0131rad\u0131\u015f\u0131 bir yetenekle, Rus edebiyat\u0131 sahnesine \u00e7\u0131kmaya haz\u0131rlanan bir yetenekle ilgili s\u00f6ylentiler, uzun zamand\u0131r bu roman\u0131n ortaya \u00e7\u0131k\u0131\u015f\u0131n\u0131 haber veriyordu.(...) _\u0130nsanc\u0131klar_ ger\u00e7ek bir ba\u015far\u0131 elde etti. E\u011fer bu roman herkes taraf\u0131ndan tart\u0131\u015fmas\u0131z \u00f6vg\u00fclerle ve tart\u0131\u015fmas\u0131z heyecanla kar\u015f\u0131land\u0131ysa, bu onda kesinlikle yetenek oldu\u011fu, ama s\u0131rad\u0131\u015f\u0131 bir \u015fey olmad\u0131\u011f\u0131 konusunda \u00e7\u00fcr\u00fct\u00fclmesi imk\u00e2ns\u0131z bir kan\u0131t say\u0131l\u0131rd\u0131. B\u00f6yle bir \u00e7\u0131k\u0131\u015f \u00fcz\u00fcc\u00fc olurdu. Ama \u00e7ok daha iyi bir \u015fey oldu: \u015eiiri anlama yetene\u011finden kesinlikle yoksun olan insanlar ve iki-\u00fc\u00e7 yazar bozuntusu d\u0131\u015f\u0131nda, herkes bu roman\u0131n hi\u00e7 de s\u0131radan bir yetene\u011fe ait olmad\u0131\u011f\u0131n\u0131 kabul etti. \u0130lk sefer i\u00e7in daha fazlas\u0131 istenemezdi.\n> \n> (...)\n> \n> En b\u00fcy\u00fck, en dahi, en hatas\u0131z ele\u015ftirmen zamand\u0131r. Fakat, Dostoyevski'nin roman\u0131n\u0131n Petersburg'daki herkes taraf\u0131ndan okundu\u011funu ve sadece Petersburg'un bu yeni \u015fairin yetene\u011fiyle ilgili g\u00f6r\u00fc\u015f\u00fcn\u00fc sergiledi\u011fini unutmamak gerekir. Moskova'da onun _\u0130nsanc\u0131klar_ ve _\u0130kiz_ adl\u0131 romanlar\u0131 daha yeni okunuyor, ta\u015fradaysa daha okunmad\u0131 bunlar. Dostoyevski'nin Moskova ve ta\u015frada ne gibi izlenimler b\u0131rakaca\u011f\u0131n\u0131 \u00f6\u011frenmek \u00e7ok ilgin\u00e7 olurdu. Ama bunu beklerken, kendi izlenimlerimizi de\u011ferlendirmekte gecikmeyelim.\n\nArd\u0131ndan, Dostoyevski \u00fczerindeki Gogol etkisinin, Dostoyevski'nin \u00f6zg\u00fcrl\u00fc\u011f\u00fcnden ve ba\u011f\u0131ms\u0131zl\u0131\u011f\u0131ndan geri planda kald\u0131\u011f\u0131n\u0131 savunur:\n\n> \u0130lk bak\u0131\u015fta Dostoyevski'nin yetene\u011finin satirik, tasvire dayal\u0131 olmad\u0131\u011f\u0131, y\u00fcksek bir yarat\u0131c\u0131l\u0131k seviyesinde oldu\u011fu ve yetene\u011finin h\u00e2kim karakterinin mizah oldu\u011fu yazd\u0131klar\u0131na, ilk bak\u0131ld\u0131\u011f\u0131 anda belli oluyor.\n> \n> Hayat\u0131 ve insan kalbini tan\u0131mas\u0131yla, yani deneyim ve g\u00f6zlemle gelen bir \u015feyle etkilemiyor; hay\u0131r, bunlar\u0131 biliyor, hem de derinden biliyor, ama a _priori_ , \u00f6ncelikle, saf \u015fiirsel olarak, sanatsal olarak biliyor. Onun bilgisi yetenektir, esindir. Onu kimseyle k\u0131yaslamak istemiyoruz, \u00e7\u00fcnk\u00fc bu t\u00fcr k\u0131yaslar\u0131 \u00e7ocuklar yapar ve bunlar hi\u00e7bir \u015feyi a\u00e7\u0131klamaz. Sadece bu yetene\u011fin s\u0131rad\u0131\u015f\u0131 ve kendine \u00f6zg\u00fc oldu\u011funu, daha ilk eseriyle, e\u011filimi ve karakteriyle az ya da \u00e7ok derecede Gogol'e bor\u00e7lu olan o yazar kalabal\u0131\u011f\u0131m\u0131zdan keskin bir \u015fekilde ayr\u0131lm\u0131\u015f oldu\u011funu, yetene\u011finin ba\u015far\u0131s\u0131n\u0131n da buradan geldi\u011fini s\u00f6ylemeliyiz.\n> \n> Onun Gogol'le ili\u015fkisine gelince, g\u00fc\u00e7l\u00fc ve ba\u011f\u0131ms\u0131z bir yetene\u011fe sahip bir yazar olarak onu bir Gogol taklit\u00e7isi olarak adland\u0131ramad\u0131\u011f\u0131m\u0131z gibi, onun Gogol'e olan borcunun Lermontov'un Pu\u015fkin'e olan borcundan daha fazla oldu\u011funu da s\u00f6yleyemeyiz. Dostoyevski'nin iki roman\u0131nda da ( _\u0130nsanc\u0131klar_ ve _\u0130kiz_ ) g\u00fc\u00e7l\u00fc bir Gogol etkisi g\u00f6r\u00fclmektedir, devrik c\u00fcmlelerde bile; ama Bay Dostoyevski b\u00fct\u00fcn yetene\u011fiyle o kadar ba\u011f\u0131ms\u0131zd\u0131r ki \u015fimdi onun \u00fczerinde Gogol etkisi g\u00f6r\u00fcl\u00fcyor olsa bile, herhalde, bu etki devam etmeyecek ve k\u0131sa s\u00fcrede ba\u015fka, sadece ona ait olan kusurlarla birlikte silinip gidecektir, yani Gogol sonsuza dek onun, deyim yerindeyse, sanatsal a\u00e7\u0131dan babas\u0131 olarak kalsa da... Bay Dostoyevski'nin yetene\u011finin, bireyselli\u011finin ve ki\u015filik \u00f6zelliklerinin ne oldu\u011funu kesin olarak belirlemek h\u00e2l\u00e2 g\u00fc\u00e7t\u00fcr, ama b\u00fct\u00fcn bunlara sahip oldu\u011funda hi\u00e7bir \u015f\u00fcpheye yer yok.\n\nYayg\u0131n bir kan\u0131ya g\u00f6re Belinski, _\u0130nsanc\u0131klar_ '\u0131 \u00f6vm\u00fc\u015f fakat k\u0131sa s\u00fcre sonra yay\u0131mlanan _\u0130kiz_ adl\u0131 roman\u0131 be\u011fenmeyerek Dostoyevski'den uzakla\u015fm\u0131\u015ft\u0131r. Bu kan\u0131 genel olarak yanl\u0131\u015ft\u0131r. Belinski _\u0130kiz_ 'i _\u0130nsanc\u0131klar_ kadar iyi bulmam\u0131\u015f, _\u0130kiz_ 'de mizah \u00f6\u011fesinin dengeli kullan\u0131lmad\u0131\u011f\u0131n\u0131 savunmu\u015ftur; as\u0131l be\u011fenmedi\u011fi kitap _Ev Sahibesi_ olmu\u015ftur. Yukar\u0131daki ele\u015ftirinin devam\u0131nda _\u0130nsanc\u0131klar_ ile _\u0130kiz_ aras\u0131nda temel fark\u0131 \u015f\u00f6yle dile getirir:\n\n> _\u0130nsanc\u0131klar_ 'a bakarak derin insani ve patetik \u00f6\u011fenin, mizahi \u00f6\u011feyle kayna\u015farak onun yetene\u011finin temel karakterini olu\u015fturdu\u011fu sonucuna varm\u0131\u015ft\u0131k; ama, _\u0130kiz_ 'i okurken, bu t\u00fcr bir \u00e7\u0131kar\u0131m\u0131n \u00e7ok acelecilik oldu\u011funu g\u00f6rd\u00fck. Do\u011fru, sadece ahlaken k\u00f6r ve sa\u011f\u0131r olanlar _\u0130kiz_ 'de derin bir patetik, derin bir trajik renk ve ton bulundu\u011funu g\u00f6remez; ama birincisi, bu renk ve ton \u0130kiz'de, mizah\u0131n arkas\u0131na, deyim yerindeyse, saklanm\u0131\u015ft\u0131r, t\u0131pk\u0131 Gogol'\u00fcn _Bir Delinin G\u00fcncesi_ 'ndeki gibi saklanm\u0131\u015ft\u0131r. (...) Genel olarak Bay Dostoyevski'nin yetene\u011fi, b\u00fct\u00fcn b\u00fcy\u00fckl\u00fc\u011f\u00fcne ra\u011fmen, h\u00e2l\u00e2 o kadar gen\u00e7tir ki kendini ifade edemez. Bu da do\u011fald\u0131r; ilk eseriyle her \u015feyi ifade etmi\u015f olan bir yazardan \u00e7ok \u015fey beklenemez. _Zaman\u0131m\u0131z\u0131n Bir Kahraman\u0131_ ne kadar g\u00fczeldi, ama e\u011fer biri \u00e7\u0131k\u0131p da Lermontov'un daha sonra ondan daha iyi bir \u015fey yazamayaca\u011f\u0131n\u0131 d\u00fc\u015f\u00fcnecek olsa bu, onun Lermontov'un yetene\u011fi hakk\u0131nda \u00e7ok iyi \u015feyler d\u00fc\u015f\u00fcnmedi\u011fini g\u00f6sterirdi.\n\nBelinski, belki de bir bak\u0131ma, otuz y\u0131l kadar sonra Paul Verlaine'in Avrupa modernizminin \u00f6nc\u00fclerinden Arthur Rimbaud kar\u015f\u0131s\u0131nda ya\u015fayaca\u011f\u0131 co\u015fku ve hayalk\u0131r\u0131kl\u0131\u011f\u0131n\u0131 Dostoyevski kar\u015f\u0131s\u0131nda ya\u015famaktad\u0131r, \u00e7\u00fcnk\u00fc onda onun gelece\u011fini g\u00f6rm\u00fc\u015ft\u00fcr:\n\n> B\u00fct\u00fcn bu korkun\u00e7 ger\u00e7e\u011fi, bizlere g\u00f6stermi\u015f oldu\u011funuz bu ger\u00e7e\u011fi siz mi d\u00fc\u015f\u00fcnd\u00fcn\u00fcz? Olamaz, sizin gibi yirmi ya\u015f\u0131nda birinin b\u00fct\u00fcn bunlar\u0131 anlam\u0131\u015f olmas\u0131na imk\u00e2n yok... Ger\u00e7e\u011fi ke\u015ffetmi\u015f ve bir sanat\u00e7\u0131 olarak ilan etmi\u015fsiniz, size bir yetenek verilmi\u015f, yetene\u011finizin de\u011ferini bilin ve emin olun, siz b\u00fcy\u00fck bir yazar olacaks\u0131n\u0131z!\n\nBelinski hakl\u0131 \u00e7\u0131kt\u0131.\n\n* * *\n\n_\u0130nsanc\u0131klar_ , Rus\u00e7adan T\u00fcrk\u00e7eye ilk kez, yay\u0131mland\u0131ktan y\u00fcz sekiz y\u0131l sonra, 1954 y\u0131l\u0131nda, Nihal Yalaza Taluy taraf\u0131ndan, _\u0130nsanc\u0131klar_ ismiyle \u00e7evrildi. Rus\u00e7adan ikinci \u00e7evirisi 1970 y\u0131l\u0131nda Ergin Altay taraf\u0131ndan, Nihal Han\u0131m'\u0131n benimsedi\u011fi isimle yap\u0131ld\u0131. Roman\u0131n bu \u00fc\u00e7\u00fcnc\u00fc Rus\u00e7adan \u00e7evirisinde de, Dostoyevski'nin koydu\u011fu as\u0131l isme en yak\u0131n bi\u00e7imi veren \"Zavall\u0131 \u0130nsanlar\", \"Yoksul \u0130nsanlar\" gibi isimler kullanmak yerine, 1954 y\u0131l\u0131nda ya \u00e7evirmen Nihal Yalaza Taluy'un ya da yay\u0131mc\u0131 Ya\u015far Nabi Nay\u0131r'\u0131n bulmu\u015f oldu\u011fu _\u0130nsanc\u0131klar_ ismini, yoksulluk vurgusunu hafifletse de se\u00e7kin bir bulu\u015f olan bu ismi korumay\u0131 uygun g\u00f6rd\u00fck. Bu isim, Nihal Han\u0131m'\u0131n romantikle\u015ftirici \u00e7eviri \u00fcslubunun bir yank\u0131s\u0131 olsa da, hem benimsenmi\u015f hem de zarif bir bulu\u015f olarak korunmay\u0131 hak ediyor.\n\nSABR\u0130 G\u00dcRSES\n\n# KAYNAK\u00c7A\n\nVissarion Belinski, \"'Bedniye Lyudi,' roman g. Dastayeskova,\" _Peterburgski Sbornik_ , 1846, 1848; _Sobranie So\u00e7ineni v Treh Tomah_ , 1948.\n\nN.M. \u00c7irkov, _O stile Dostoyevskova_ , Nauka, 1967.\n\nJoseph Frank, _Dastaevsky: A Writer in His Time_ , Princeton University Press, 2009.\n\nYuri Seleznev, _Dastayevski_ , Molodaya Gvardiya, 1981.\n\n# Nisan\n\n8 Nisan\n\nE\u015fsiz Varvara Alekseyevna,\n\nD\u00fcn mutlu oldum, a\u015f\u0131r\u0131 mutlu oldum, ak\u0131l almaz derecede mutlu oldum! \u0130nat\u00e7\u0131s\u0131n\u0131zd\u0131r, ama hayatta bir kez olsun beni dinlediniz. D\u00fcn ak\u015fam, saat sekizde, uyand\u0131m (biliyorsunuz can\u0131m, i\u015ften geldikten sonra bir saat kadar uyumay\u0131 severim), mumu ald\u0131m, k\u00e2\u011f\u0131tlar\u0131 haz\u0131rlad\u0131m, kalemi \u00e7\u0131kard\u0131m, sonra birden, tesad\u00fcfen bak\u0131\u015flar\u0131m\u0131 yukar\u0131 \u00e7evirdim... Yemin ederim, y\u00fcre\u011fim hop etti! Benim i\u00e7imden ne ge\u00e7ti\u011fini, k\u00fc\u00e7\u00fck kalbimin neyi arzu etti\u011fini anlam\u0131\u015fs\u0131n\u0131z! Pencerenizdeki perdenin ucunun tam da benim size s\u00f6yledi\u011fim gibi k\u0131vr\u0131l\u0131p k\u0131na\u00e7i\u00e7e\u011fi saks\u0131s\u0131na ili\u015ftirildi\u011fini g\u00f6rd\u00fcm; o s\u0131rada pencerede zarif y\u00fcz\u00fcn\u00fcz\u00fc g\u00f6r\u00fcr gibi oldum, sanki odan\u0131zdan bana bak\u0131yor, beni d\u00fc\u015f\u00fcn\u00fcyordunuz. Ama o sevimli zarif y\u00fcz\u00fcn\u00fcz\u00fc do\u011fru d\u00fcr\u00fcst g\u00f6remedi\u011fim i\u00e7in can\u0131m \u00e7ok s\u0131k\u0131ld\u0131 g\u00fcvercinim!\n\nBir zamanlar billur gibi g\u00f6r\u00fcrd\u00fck can\u0131m. Ya\u015fl\u0131l\u0131k mutluluk getirmiyor bir tanem! Daha \u015fimdiden ya\u015fard\u0131 g\u00f6zlerim; geceleyin \u00e7al\u0131\u015f\u0131yorum, bir \u015feyler yaz\u0131yorum, sabahleyin bir bak\u0131yorum g\u00f6zlerim k\u0131pk\u0131rm\u0131z\u0131, sanki ba\u015fkalar\u0131ndan utanm\u0131\u015f\u0131m gibi g\u00f6zya\u015flar\u0131m ak\u0131yor. Fakat hayalimde sizin g\u00fcl\u00fc\u015f\u00fcn\u00fcz canlan\u0131yor mele\u011fim, sizin iyi y\u00fcrekli, davetk\u00e2r g\u00fcl\u00fc\u015f\u00fcn\u00fcz; kalbimde de sizi \u00f6pt\u00fc\u011f\u00fcm zamanki duygular var, Varenka2 , hat\u0131rlar m\u0131s\u0131n\u0131z, mele\u011fim? Biliyor musunuz g\u00fcvercinim, beni bu arada parma\u011f\u0131n\u0131zla tehdit etmi\u015fsiniz gibi geldi? \u00d6yle mi oldu, yaramaz\u0131m? Bunlar\u0131 mektubunuzda ayr\u0131nt\u0131s\u0131yla yazmal\u0131s\u0131n\u0131z muhakkak.\n\nPeki, ama \u015fu sizin perdenizle ilgili icad\u0131m\u0131za ne diyorsunuz, Varenka? \u00c7ok sevimli, de\u011fil mi? \u0130\u015ften d\u00f6nd\u00fckten sonra otururken, yatarken, uyand\u0131\u011f\u0131mda, sizin beni d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcn\u00fcz\u00fc, beni hat\u0131rlad\u0131\u011f\u0131n\u0131z\u0131, hem de sa\u011fl\u0131kl\u0131 ve ne\u015feli oldu\u011funuzu biliyorum. Perdeyi kapat\u0131nca, \"Ho\u015f\u00e7a kal\u0131n, Makar Alekseyevi\u00e7,\" demi\u015f oluyorsunuz, \"uyku vakti!\" Perdeyi a\u00e7\u0131nca da, \"G\u00fcnayd\u0131n Makar Alekseyevi\u00e7\", demi\u015f oluyorsunuz, \"Nas\u0131l uyudunuz?\" ya da, \"Sa\u011fl\u0131\u011f\u0131n\u0131z nas\u0131l Makar Alekseyevi\u00e7?\" Ne diyeyim i\u015fte, rabbimiz sayesinde sa\u011fl\u0131kl\u0131 ve sapasa\u011flam\u0131m! G\u00f6r\u00fcyor musunuz can\u0131mc\u0131\u011f\u0131m, ne kadar hince bir fikir bu; mektuba da hi\u00e7 gerek yok! Kurnazca, \u00f6yle de\u011fil mi? Ama zaten benim icad\u0131m! Peki, iyi yapm\u0131\u015f m\u0131y\u0131m Varvara Alekseyevna?\n\nSize \u015funu s\u00f6yleyeyim, can\u0131m, Varvara Alekseyevna: Bu gece hi\u00e7 beklemedi\u011fim halde \u00e7ok iyi uyudum, bundan da \u00e7ok ho\u015fnutum; oysa yeni ta\u015f\u0131nd\u0131\u011f\u0131 bir evde, yeni bir yerde insan asla uyuyamaz; bir s\u00fcr\u00fc \u015fey d\u00f6n\u00fcp durur kafas\u0131nda! Bug\u00fcn bir \u015fahin gibi ferah kalkt\u0131m, ne\u015fe i\u00e7inde uyand\u0131m! Bu sabahtan daha g\u00fczel ne var ki can\u0131m! Bizim evde k\u00fc\u00e7\u00fck bir pencere a\u00e7m\u0131\u015flar; g\u00fcne\u015fcik \u0131\u015f\u0131l \u0131\u015f\u0131l, ku\u015flar c\u0131v\u0131ld\u0131yor, hava bahar kokular\u0131 sa\u00e7\u0131yor ve b\u00fct\u00fcn do\u011fa canlanm\u0131\u015f... geri kalan her \u015fey de buna ayak uyduruyor; her \u015fey yerli yerinde, baharda nas\u0131l olursa i\u015fte \u00f6yle. Ben de bug\u00fcn bol bol hayal kurdum ve hayallerimde hep siz vard\u0131n\u0131z Varenka. Sizi g\u00f6kteki ku\u015fla, insanlar\u0131 e\u011flendirmek ve do\u011fay\u0131 g\u00fczelle\u015ftirmek i\u00e7in yarat\u0131lm\u0131\u015f ku\u015fla k\u0131yaslad\u0131m. Hep \u015funu d\u00fc\u015f\u00fcnd\u00fcm, Varenka, biz insanlar, kayg\u0131 ve tela\u015f i\u00e7inde ya\u015fayan biz insanlar, g\u00f6kteki ku\u015flar\u0131n kayg\u0131s\u0131z ve masum mutlulu\u011funu da k\u0131skanmal\u0131y\u0131z \u2013 sonra yine buna benzer bir s\u00fcr\u00fc \u015fey geldi akl\u0131ma; yani hep bu t\u00fcr soyut kar\u015f\u0131la\u015ft\u0131rmalar yapt\u0131m. Bende bir kitap var, Varenka, onda da ayn\u0131s\u0131 s\u00f6yleniyor, her \u015fey \u00e7ok ayr\u0131nt\u0131l\u0131 olarak yaz\u0131lm\u0131\u015f. Ben size t\u00fcrl\u00fc t\u00fcrl\u00fc hayal var diye yaz\u0131yorum, can\u0131m. \u0130\u015fte bahar geldi, b\u00fct\u00fcn d\u00fc\u015f\u00fcnceler art\u0131k tatl\u0131, sert, a\u011fdal\u0131, hayaller de sevecen; her \u015fey pembe. B\u00fct\u00fcn bunlar\u0131 ben yazd\u0131m size; fakat hepsini bir kitaptan ald\u0131m. Orada yazar ayn\u0131 arzuyu \u015fiirle dile getiriyor ve \u015f\u00f6yle yaz\u0131yor:\n\n> Neden bir ku\u015f de\u011filim, ke\u015fke bir y\u0131rt\u0131c\u0131 ku\u015f olsam!\n\nFalan filan. T\u00fcrl\u00fc t\u00fcrl\u00fc fikirleri var b\u00f6yle, Tanr\u0131 yard\u0131mc\u0131s\u0131 olsun! Ama siz bu sabah nereye gittiniz Varvara Alekseyevna? Ben i\u015fe gitmeye haz\u0131rlan\u0131yordum daha, o s\u0131rada siz, bir ilkbahar ku\u015fu gibi odan\u0131zdan \u00e7\u0131kt\u0131n\u0131z, avludan ne\u015feyle ge\u00e7ip gittiniz. Size bakarken nas\u0131l da ne\u015felendim! Ah Varenka, Varenka! Kederlenmeyin hi\u00e7; g\u00f6zya\u015flar\u0131 ac\u0131ya yard\u0131mc\u0131 olmaz; bilirim bunu can\u0131m, tecr\u00fcbeyle bilirim. Art\u0131k siz daha sakinsiniz neyse ki, sa\u011fl\u0131\u011f\u0131n\u0131z da biraz d\u00fczeldi. Peki sizin Fedora nas\u0131l? Ah, ne kadar iyi bir kad\u0131n o! Varenka, bana onunla \u015fimdi nas\u0131l ya\u015fad\u0131\u011f\u0131n\u0131z\u0131, her \u015feyden ho\u015fnut olup olmad\u0131\u011f\u0131n\u0131z\u0131 yazar m\u0131s\u0131n\u0131z? Fedora biraz homurdan\u0131r; ama siz ona bakmay\u0131n, Varenka. Tanr\u0131 yard\u0131mc\u0131s\u0131 olsun! \u00c7ok iyidir.\n\nSize buradaki Tereza'y\u0131 yazm\u0131\u015ft\u0131m, o da iyi ve g\u00fcvenilir bir kad\u0131nd\u0131r. Oysa ben nas\u0131l tela\u015flanm\u0131\u015ft\u0131m mektuplar\u0131m\u0131z i\u00e7in! Onlar nas\u0131l gidip gelecek diye. Ama sonra i\u015fte Tanr\u0131 mutlulu\u011fumuz i\u00e7in Tereza'y\u0131 g\u00f6nderdi. Bu kad\u0131n iyi, ketum, a\u011fz\u0131 var dili yok sanki. Fakat ev sahibemiz \u00e7ok ac\u0131mas\u0131z. Onu pa\u00e7avra gibi hoyrat\u00e7a \u00e7al\u0131\u015ft\u0131r\u0131yor.\n\nBen de nas\u0131l bir gecekonduya d\u00fc\u015fm\u00fc\u015f\u00fcm, Varvara Alekseyevna! B\u00f6yle ev mi olur! Eskiden sa\u011f\u0131r gibi ya\u015fard\u0131m, siz de bilirsiniz: sessiz, sakin; bir sinek u\u00e7sa, sesi duyulurdu. Ama burada g\u00fcr\u00fclt\u00fc, ba\u011f\u0131rt\u0131, g\u00fcmb\u00fcrt\u00fc! Tabii siz buran\u0131n nas\u0131l bir yer oldu\u011funu bilmiyorsunuz. \u015eimdi, upuzun bir koridor d\u00fc\u015f\u00fcn\u00fcn, kapkara ve pis bir koridor. Sa\u011f taraf\u0131nda \u00e7\u0131plak bir duvar, sol taraf\u0131nda da bir s\u00fcr\u00fc kap\u0131 var, tam bir otel gibi, yanyana s\u0131ralanm\u0131\u015flar. Tabii, otel odalar\u0131na benziyorlar, zaten her biri k\u00fc\u00e7\u00fck birer odadan ibaret; her odada iki ki\u015fi, \u00fc\u00e7 ki\u015fi kal\u0131yor. D\u00fczen falan beklemeyin, buras\u0131 Nuh'un gemisi! Fakat, herhalde, iyi insanlar, hepsi de e\u011fitimli, bilgili insanlar. Biri memur (edebiyat mezunu), okumu\u015f bir adam; hem Homeros'u hem Brambeus'u3 biliyor, onlar\u0131n de\u011fi\u015fik de\u011fi\u015fik eserlerinden bahsediyor, her konuda konu\u015fuyor zeki bir adam! \u0130ki tane subay var, onlar hep iskambil oynuyorlar. Bir ba\u015f\u00e7avu\u015f var; bir de \u0130ngiliz \u00f6\u011fretmen. Durun, sizi e\u011flendireyim, can\u0131m; bir sonraki mektubumda onlar\u0131 satirik bir \u015fekilde, yani tam da olduklar\u0131 gibi, b\u00fct\u00fcn ayr\u0131nt\u0131s\u0131yla anlatay\u0131m size. Ev sahibemiz \u2013 ufak tefek ve pek temiz olmayan bir ihtiyar \u2013 b\u00fct\u00fcn g\u00fcn terlikle ve _schlafrock_ 'la4 gezer ve yine b\u00fct\u00fcn g\u00fcn Tereza'ya ba\u011f\u0131r\u0131r. Ben mutfakta ya\u015f\u0131yorum ya da daha do\u011frusu \u015f\u00f6yle s\u00f6ylemeli: Mutfa\u011f\u0131n hemen yan\u0131nda bir oda var (\u015funu s\u00f6ylemeden ge\u00e7memeli, bizim mutfak temiz, ayd\u0131nl\u0131k, \u00e7ok g\u00fczel bir yerdir), ufak bir oda, m\u00fctevaz\u0131 bir k\u00f6\u015fecik... yani, daha do\u011frusu \u00fc\u00e7 pencereli b\u00fcy\u00fck bir mutfak buras\u0131, duvar taraf\u0131na benim i\u00e7in boydan boya bir paravan kondu, b\u00f6ylece oda gibi bir \u015fey oldu, fazladan bir otel odas\u0131; her \u015feyiyle sade, rahat, penceresi de var, her \u015feyiyle \u2013 kesinlikle, her \u015feyiyle rahat bir yer. \u0130\u015fte b\u00f6yle benim k\u00f6\u015feci\u011fim. Ama can\u0131m, siz sak\u0131n, bunun arkas\u0131nda ba\u015fka, gizemli bir \u015fey aramay\u0131n; mutfak i\u015fte buras\u0131, hepsi bu! Ben de, ne yaz\u0131k ki, paravan\u0131n arkas\u0131ndaki bu odada ya\u015f\u0131yorum, ama bu \u00f6nemli de\u011fil; herkesten ayr\u0131y\u0131m, azla ya\u015f\u0131yorum, sessizce ya\u015f\u0131yorum. Bu k\u00f6\u015feme bir yatak, bir masa, bir komodin, bir \u00e7ift sandalye koydum, bir de ikona ast\u0131m. Do\u011fru, daha iyi evler de var \u2013belki \u00e7ok daha iyisi de vard\u0131r\u2013 ama rahatl\u0131k en \u00f6nemlisi; sonu\u00e7ta ben bunu s\u0131rf rahatl\u0131k i\u00e7in yapt\u0131m, ba\u015fka bir \u015fey i\u00e7in oldu\u011funu sanmay\u0131n. Sizin pencereci\u011finiz de kar\u015f\u0131da, avlunun \u00f6te yan\u0131nda duruyor; avlu da darac\u0131k, gelip giderken sizi seyretmek m\u00fcmk\u00fcn... her \u015fey benim i\u00e7in, benim gibi bir talihsiz i\u00e7in eskisinden daha ne\u015feli, hem de daha ucuz. Burada son bir oda kald\u0131, i\u00e7inde masas\u0131 da var, otuz be\u015f ruble kira istiyorlar. Ate\u015f pahas\u0131! Benim odamsa bana yedi ruble kiraya mal oluyor, masaysa be\u015f g\u00fcm\u00fc\u015f ruble; toplam yirmi d\u00f6rt ruble elli kopek ediyor, eskiden otuz ruble \u00f6d\u00fcyordum, o y\u00fczden de kendimi bir\u00e7ok \u015feyden mahrum b\u0131rakt\u0131m; \u00e7ay\u0131 her zaman i\u00e7mezdim zaten, ama \u015fimdi \u00e7ay i\u00e7in de \u015feker i\u00e7in de para biriktiriyorum. Canca\u011f\u0131z\u0131m, biliyor musunuz, \u00e7ay i\u00e7memek ay\u0131p oluyor; burada herkesin durumu iyi, o y\u00fczden de ay\u0131p oluyor. Ba\u015fkalar\u0131 y\u00fcz\u00fcnden onu da i\u00e7iyorum Varenka, g\u00f6r\u00fcnt\u00fc olsun, hava olsun diye; yoksa benim i\u00e7in fark etmez, tiryakisi de\u011filim. \u0130nsan\u0131n nakit paraya da ihtiyac\u0131 oluyor, \u00e7izme, k\u0131l\u0131k k\u0131yafet i\u00e7in. Geriye de bir \u015fey kalm\u0131yor. Sonu\u00e7ta maa\u015f\u0131m belli. S\u0131zlan\u0131p homurdanm\u0131yorum, ho\u015fnutum. Bu da yeter. Zaten birka\u00e7 y\u0131l yeter; ikramiye de geliyor. Neyse, ho\u015f\u00e7a kal\u0131n, mele\u011fim. Ben iki saks\u0131 k\u0131na\u00e7i\u00e7e\u011fi ve sardunya ald\u0131m \u2013 pahal\u0131 bir \u015fey de\u011fil. Siz \u0131t\u0131r sever miydiniz? It\u0131r da var, isterseniz yaz\u0131n bana; evet, biliyorsunuz, her \u015feyi ayr\u0131nt\u0131s\u0131yla yazabilirsiniz. Fakat bir \u015fey gelmesin akl\u0131n\u0131za, benim b\u00f6yle bir oda tutmu\u015f olmamdan \u015f\u00fcphe etmeyin can\u0131m. Hay\u0131r, rahat oldu\u011fu i\u00e7in yapt\u0131m ve bir tek rahatl\u0131k \u00e7ekti beni. Can\u0131m, zaten para sakl\u0131yorum, biriktiriyorum; param var. Benim b\u00f6yle sessiz oldu\u011fuma, bir sine\u011fin kanad\u0131ndan bile zarar g\u00f6rebilecekmi\u015f gibi olmama ald\u0131rmay\u0131n. Hay\u0131r can\u0131m, hakk\u0131mda yan\u0131lm\u0131yorum, karakter olarak \u00e7ok kararl\u0131 ve ruhu huzurlu bir insan\u0131m ben. Ho\u015f\u00e7a kal\u0131n, mele\u011fim! Size neredeyse iki sayfa yazm\u0131\u015f\u0131m, ama i\u015fe de ge\u00e7 kalm\u0131\u015f\u0131m. Can\u0131m, \u00f6p\u00fcyorum parmaklar\u0131n\u0131z\u0131.\n\nSizin en al\u00e7akg\u00f6n\u00fcll\u00fc u\u015fa\u011f\u0131n\u0131z ve sad\u0131k dostunuz,\n\nMakar Devu\u015fkin\n\nHami\u015f: Bir tek ricam var: Bana olabildi\u011fince ayr\u0131nt\u0131l\u0131 yaz\u0131n mele\u011fim. Bu arada size bir paket bonbon \u015fekeri yolluyorum Varenka; afiyetle yiyin onlar\u0131 ve Tanr\u0131 a\u015fk\u0131na, benim i\u00e7in kayg\u0131lanmay\u0131n, \u015fik\u00e2yet etmeyin. Neyse, ho\u015f\u00e7a kal\u0131n, can\u0131m.\n\n8 Nisan\n\nSayg\u0131de\u011fer Beyefendi, Makar Alekseyevi\u00e7,\n\nBiliyor musunuz, sonunda sizinle ciddi ciddi tart\u0131\u015faca\u011f\u0131z herhalde? \u0130yi kalpli Makar Alekseyevi\u00e7, size yemin ederim, bana sizden hediye almak \u00e7ok a\u011f\u0131r geliyor. Onlar\u0131n sizin i\u00e7in neye mal oldu\u011funu, ihtiyac\u0131n\u0131z olan \u015feylerden vazge\u00e7erek yoksun kald\u0131\u011f\u0131n\u0131z\u0131 biliyorum. Size ka\u00e7 kere hi\u00e7bir \u015feye, kesinlikle hi\u00e7bir \u015feye ihtiyac\u0131m olmad\u0131\u011f\u0131n\u0131 s\u00f6yledim; bu vakte kadar bana yapt\u0131\u011f\u0131n\u0131z iyiliklerin bedelini \u00f6deyecek g\u00fcc\u00fcm de yok. Hem ne yapay\u0131m ben bu saks\u0131lar\u0131? Yani, k\u0131na\u00e7i\u00e7e\u011fi saks\u0131lar\u0131 neyse de, sardunya ne i\u00e7in? Size dikkatsizlik edip bir kere sardunya demek yetiyor, hemen gidip al\u0131yorsunuz; \u00fcstelik, herhalde, pahal\u0131 bir \u015fey bu? \u00dczerindeki renkler de o kadar \u00e7ekici ki! Ta\u00e7yapraklar\u0131 alev rengi. Siz bu g\u00fczel sardunyay\u0131 nereden buldunuz? Onu pencerenin ortas\u0131na koydum, en g\u00f6r\u00fcn\u00fcr yere; yerdeyken bir sedir koyuyordum, sedirin \u00fcst\u00fcne de \u00e7i\u00e7ekleri koyuyordum; bana b\u00fcy\u00fck bir hazine veriyorsunuz! Fedora sevin\u00e7 i\u00e7inde; odam\u0131z tam bir cennet \u015fimdi; tertemiz, \u0131\u015f\u0131l \u0131\u015f\u0131l! Ama \u015fekerler ne i\u00e7in? Do\u011fru, mektubunuza bakar bakmaz tahmin ettim sizde bir haller oldu\u011funu: Cennet, bahar, u\u00e7u\u015fan g\u00fczel kokular, c\u0131v\u0131ldayan ku\u015flar. Ne oluyor, diye d\u00fc\u015f\u00fcnd\u00fcm, \u015fiir de olmas\u0131n burada? Zaten, do\u011frusu, mektubunuza bir dize yetmemi\u015f Makar Alekseyevi\u00e7! Tatl\u0131 duygular, pembe renkli hayaller, hepsi var burada! Perdeyi hi\u00e7 d\u00fc\u015f\u00fcnmemi\u015ftim; herhalde, kendi kendine tak\u0131lm\u0131\u015f oraya, saks\u0131y\u0131 yerle\u015ftirdi\u011fim s\u0131rada; i\u015fe bak\u0131n!\n\nAh, Makar Alekseyevi\u00e7! Siz beni kand\u0131rmak i\u00e7in, her \u015feyi kendinize harcad\u0131\u011f\u0131n\u0131z\u0131 kan\u0131tlamak i\u00e7in ne s\u00f6ylerseniz s\u00f6yleyin, ne hesaplar yaparsan\u0131z yap\u0131n, benden bir \u015fey saklayamazs\u0131n\u0131z. Besbelli, benim y\u00fcz\u00fcmden bo\u015f yere yoksunluk \u00e7ekiyorsunuz. \u00d6rne\u011fin, kimin akl\u0131na gelirdi sizin b\u00f6yle bir ev tutaca\u011f\u0131n\u0131z? Sizi rahats\u0131z ediyorlar, kayg\u0131land\u0131r\u0131yorlar; s\u0131k\u0131lm\u0131\u015fs\u0131n\u0131z, rahats\u0131zs\u0131n\u0131z. Siz yaln\u0131zl\u0131\u011f\u0131 seversiniz, ama orada \u00e7evrenizde kim bilir neler neler var! Gelirinize bak\u0131l\u0131rsa, \u00e7ok daha iyi ya\u015fayabilirsiniz oysa. Fedora sizin eskiden de \u015fimdikinden daha iyi ya\u015famad\u0131\u011f\u0131n\u0131z\u0131 s\u00f6yl\u00fcyor. Ger\u00e7ekten hayat\u0131n\u0131z boyunca b\u00f6yle, tek ba\u015f\u0131n\u0131za, yoksunluk i\u00e7inde, mutluluktan, dost\u00e7a davetk\u00e2r s\u00f6zlerden yoksun, yabanc\u0131 k\u00f6\u015felerde kirada m\u0131 ya\u015fad\u0131n\u0131z? Ah, sevgili dostum, \u00e7ok \u00fcz\u00fcld\u00fcm sizin i\u00e7in! En az\u0131ndan sa\u011fl\u0131\u011f\u0131n\u0131za ac\u0131y\u0131n, Makar Alekseyevi\u00e7! G\u00f6zlerinizin zay\u0131flad\u0131\u011f\u0131n\u0131, mum \u0131\u015f\u0131\u011f\u0131nda yazamad\u0131\u011f\u0131n\u0131z\u0131 s\u00f6yl\u00fcyorsunuz; neden yaz\u0131yorsunuz? Mesle\u011fe olan tutkunuzu amirleriniz, siz bunun i\u00e7in \u00e7aba harcamasan\u0131z da, biliyordur herhalde.\n\nSizden bir kez daha rica ediyorum, benim i\u00e7in para harcamay\u0131n. Biliyorum, beni seviyorsunuz ama siz de zengin de\u011filsiniz... Bug\u00fcn ben de ne\u015feyle kalkt\u0131m. \u00c7ok iyiydim; Fedora zaten \u00e7oktan kalkm\u0131\u015f \u00e7al\u0131\u015f\u0131yordu, bana da i\u015f kalm\u0131\u015ft\u0131. \u00c7ok sevindim; sadece ipek almak gerekiyordu, al\u0131p i\u015fe koyuldum. B\u00fct\u00fcn sabah i\u00e7im o kadar feraht\u0131, o kadar ne\u015feliydim ki! Ama \u015fimdi yine kara d\u00fc\u015f\u00fcnceler geldi, kederliyim; kalbim s\u0131zl\u0131yor.\n\nAh, ne yapaca\u011f\u0131m, ne olacak benim kaderim? \u00c7ok a\u011f\u0131r geliyor benim b\u00f6yle bir bilinmezlikte olmam, bir gelece\u011fimin olmamas\u0131, ba\u015f\u0131ma ne gelece\u011fini tahmin edememek. Geriye bakmak da korkutucu. Orada hep ac\u0131 var, bir hat\u0131rayla bile kalbim iki par\u00e7aya ayr\u0131l\u0131yor. Beni mahveden k\u00f6t\u00fc insanlar y\u00fcz\u00fcnden sonsuza dek a\u011flayaca\u011f\u0131m!\n\nHava karar\u0131yor. \u00c7al\u0131\u015fma vakti. Size \u00e7ok \u015fey yazmak isterdim, ama vakit yok, \u00e7al\u0131\u015fmaya oturmal\u0131y\u0131m. Acele etmem laz\u0131m. Elbette, mektuplar g\u00fczel \u015feyler; asla s\u0131k\u0131c\u0131 de\u011fil. Ama neden siz hi\u00e7 kalk\u0131p gelmiyorsunuz? Neden Makar Alekseyevi\u00e7? Gelin, l\u00fctfen! Sizin Tereza'y\u0131 g\u00f6rd\u00fcm. O da hasta sanki; \u00fczg\u00fcnd\u00fc; ona yirmi kopek verdim. Evet! Unutmadan: Hemen yaz\u0131n her \u015feyi, g\u00fcnl\u00fck hayat\u0131n\u0131z\u0131 olabildi\u011fince ayr\u0131nt\u0131l\u0131 bir \u015fekilde yaz\u0131n. \u00c7evrenizde nas\u0131l insanlar var, onlarla ya\u015famaktan ho\u015fnut musunuz? B\u00fct\u00fcn bunlar\u0131 bilmeyi \u00e7ok istiyorum. Aman, hemen yaz\u0131n! Bug\u00fcn kasten perdeyi k\u0131v\u0131rd\u0131m. Erken yat\u0131n; d\u00fcn g\u00f6rd\u00fcm, gece yar\u0131s\u0131na kadar yand\u0131 \u0131\u015f\u0131\u011f\u0131n\u0131z. Neyse, ho\u015f\u00e7a kal\u0131n. Bug\u00fcn de keder, s\u0131k\u0131nt\u0131, h\u00fcz\u00fcn! Yani, bu da b\u00f6yle bir g\u00fcn i\u015fte! Ho\u015f\u00e7a kal\u0131n.\n\nSizin olan,  \nVarvara Dobroselova\n\n8 Nisan\n\nSayg\u0131de\u011fer Han\u0131mefendi, Varvara Alekseyevna,\n\nEvet can\u0131m, evet bir tanem, yani, benim talihsiz pay\u0131ma b\u00f6yle bir g\u00fcn d\u00fc\u015fm\u00fc\u015f! Evet, siz alay etmi\u015fsiniz benim gibi bir ihtiyarla. Varvara Alekseyevna! Fakat, benim su\u00e7um, hepsi benim su\u00e7um! Ya\u015fl\u0131l\u0131k \u00e7a\u011f\u0131mda bir tutam sa\u00e7la a\u015fka ve kinayelere kalk\u0131\u015fmak da neyin nesi... Ama \u015f\u00f6yle diyeyim, can\u0131m: Bazen insan tuhaf olur, \u00e7ok tuhaf. Siz benim i\u00e7in kutsals\u0131n\u0131z! Kalkm\u0131\u015f nelerden bahsediyorum, kap\u0131l\u0131p gidiyorum bazen! Ama ne olacak, ne \u00e7\u0131kacak ki bundan? Tabii hi\u00e7bir \u015fey \u00e7\u0131kmayacak, beni mezara g\u00f6t\u00fcrecek bir sa\u00e7mal\u0131k \u00e7\u0131kacak, Tanr\u0131m! Can\u0131m, ben, ben g\u00fccenmedim, her \u015feyi ho\u015fnutlukla hat\u0131rl\u0131yorum, size \u00f6yle s\u00fcsl\u00fc p\u00fcsl\u00fc ve aptalca yazd\u0131\u011f\u0131m i\u00e7in de ho\u015fnutum. \u0130\u015fe de bug\u00fcn pek bir cicili bicili gittim; kalbim c\u0131v\u0131l c\u0131v\u0131ld\u0131. Ruhum sanki bayram yeriydi; ne\u015fe i\u00e7indeydim! Evraklar\u0131 dalg\u0131n dalg\u0131n kayda ge\u00e7irdim... acaba neye benzedi o i\u015fler! Sonra \u00e7evreme bak\u0131nd\u0131m, her \u015fey her zamanki gibiydi... gri ve koyu renkli. Hep ayn\u0131 m\u00fcrekkep lekeleri, hep ayn\u0131 masa ve evraklar, ben bile ayn\u0131yd\u0131m; nas\u0131l \u00f6yle ayn\u0131 kalabilmi\u015fti her \u015fey... Pegasos'un s\u0131rt\u0131na binip gitmek varken burada i\u015fim neydi? Nereden \u00e7\u0131km\u0131\u015ft\u0131 b\u00fct\u00fcn bunlar? G\u00f6kte duran g\u00fcne\u015fin ayartmas\u0131 bundan m\u0131yd\u0131? Pencerenin ard\u0131ndaki avlumuzda bir \u015feyler olmuyorsa, bu kokular da neredendi? Yani, bana her \u015fey aptall\u0131k gibi g\u00f6r\u00fcnd\u00fc. \u0130nsan bazen duygular\u0131n\u0131 aptall\u0131k derecesine vard\u0131r\u0131r da yolunu \u015fa\u015f\u0131r\u0131r ya, \u00f6yle bir \u015fey. Ba\u015fka bir \u015feyden de olmaz, kalbin a\u015f\u0131r\u0131 gereksiz, budalaca ate\u015finden olur hep. Eve gelmedim de s\u00fcr\u00fcklendim sanki; hi\u00e7bir \u015fey yokken ba\u015f\u0131m a\u011fr\u0131yordu; yani bir \u015fey ba\u015fka bir \u015feye yol a\u00e7\u0131yordu. (S\u0131rt\u0131mdan r\u00fczg\u00e2r yemi\u015ftim.) Bahara sevinmi\u015ftim budala gibi, hafif paltomla \u00e7\u0131km\u0131\u015ft\u0131m. Ve duygular\u0131m konusunda yan\u0131lm\u0131\u015fs\u0131n\u0131z karde\u015fim! Onlar\u0131n a\u015f\u0131r\u0131l\u0131\u011f\u0131n\u0131 kesinlikle ba\u015fka bir y\u00f6ne \u00e7ekmi\u015fsiniz. Babacan bir muhabbet kaplam\u0131\u015ft\u0131 beni, sadece saf babacan bir muhabbet, Varvara Alekseyevna; \u00e7\u00fcnk\u00fc ben sizin gibi ac\u0131l\u0131 bir yetim kar\u015f\u0131s\u0131nda kendimi \u00f6z baban\u0131z yerine koyuyorum; bunu ruhumun derinliklerinden, tertemiz bir kalple, bir akraba gibi s\u00f6yl\u00fcyorum. O da ba\u015fka bir mesele, ben sizin i\u00e7in uzak bir akraba olsam da, hani \"d\u0131\u015f kap\u0131n\u0131n mandal\u0131\" derler, \u00f6yle olsam da her \u015feye ra\u011fmen akraban\u0131z\u0131m ve art\u0131k en yak\u0131n akraban\u0131z ve haminizim; \u00e7\u00fcnk\u00fc siz hamilik ve yak\u0131nl\u0131k g\u00f6stermeleri gereken yak\u0131nlar\u0131n\u0131zdan hainlik ve hakaret g\u00f6rd\u00fcn\u00fcz.\n\n\u015eiir konusunda da, can\u0131m, size \u015funu s\u00f6yleyece\u011fim: Ya\u015fl\u0131l\u0131k \u00e7a\u011f\u0131mda \u015fiirle oyalanmak bana yak\u0131\u015fmaz. Sa\u00e7mal\u0131kt\u0131r \u015fiir! \u015eiir denen \u015feyi art\u0131k okul \u00e7ocuklar\u0131 \u015fak\u0131yor... \u0130\u015fte b\u00f6yle, bir tanem.\n\nRahattan, huzurdan ve t\u00fcrl\u00fc \u015feylerden bahsederek bana neler yaz\u0131yorsunuz \u00f6yle Varvara Alekseyevna? Can\u0131m benim, ben ne s\u0131zlan\u0131yorum ne de kayg\u0131l\u0131y\u0131m, asla \u015fimdikinden daha iyi ya\u015famad\u0131m; bu ya\u015fta da ne kaprisi yapaca\u011f\u0131m? Yiyorum, i\u00e7iyorum, giyiniyorum; daha ne isterim! Kont s\u0131n\u0131f\u0131ndan falan da de\u011filim! Anne babam\u0131n soyluluk unvan\u0131 yoktu ve b\u00fct\u00fcn ailemiz benden daha az gelirle ya\u015fard\u0131. Han\u0131m evlad\u0131 de\u011filim! Fakat, do\u011fruyu s\u00f6ylemek gerekirse, eski evimde durum daha parlak de\u011fildi diyemem; daha rahatt\u0131 can\u0131m. Elbette, \u015fimdiki evim g\u00fczel, hatta belli a\u00e7\u0131lardan daha ne\u015feli ve \u00e7ok daha renkli de denebilir; bunu bir \u015feye itiraz i\u00e7in s\u00f6ylemiyorum, ya\u015fl\u0131lar her \u015feyden yak\u0131n\u0131r. Biz ya\u015fl\u0131, yani g\u00f6rm\u00fc\u015f ge\u00e7irmi\u015f insanlar, eski e\u015fyalara, bir akrabam\u0131za al\u0131\u015ft\u0131\u011f\u0131m\u0131z gibi al\u0131\u015f\u0131r\u0131z. Ev, biliyor musunuz, \u00e7ok k\u00fc\u00e7\u00fck; duvarlar... aman, neler diyorum! Duvarlar, s\u0131radan duvarlar, bir \u00f6zellikleri yok, ama ge\u00e7mi\u015fimle ilgili hat\u0131ralarla birlikte \u00fczerime kasvet \u00e7\u00f6k\u00fcyor... Tuhaf \u015fey \u2013 ama hat\u0131ralar ho\u015f sanki. Hatta k\u00f6t\u00fc olan, o s\u0131rada can s\u0131kan bir \u015fey, hat\u0131ralarda nas\u0131lsa k\u00f6t\u00fcl\u00fc\u011f\u00fcnden ar\u0131n\u0131yor ve hayalimde harika bir g\u00f6r\u00fcn\u00fcm kazan\u0131yor. Sessiz ya\u015fard\u0131k biz Varenka; ben ve ev sahibem, merhume ihtiyar. Baksan\u0131za benim ihtiyar\u0131 bile h\u00fcz\u00fcnle hat\u0131rl\u0131yorum art\u0131k! \u0130yi bir kad\u0131nd\u0131 ve ev i\u00e7in de \u00e7ok \u015fey almazd\u0131. O zamanlar, \u00e7e\u015fitli battaniyelerin par\u00e7alar\u0131ndan uzun \u015fi\u015flerle \u00f6rg\u00fc yapard\u0131; bir tek bununla u\u011fra\u015f\u0131rd\u0131. Birlikte ate\u015fi beslerdik, tek masan\u0131n arkas\u0131na oturur \u00e7al\u0131\u015f\u0131rd\u0131k. Ma\u015fa adl\u0131 bir torunu vard\u0131 \u2013 bebekli\u011fini bile hat\u0131rlar\u0131m \u2013 on \u00fc\u00e7 ya\u015f\u0131nda bir gen\u00e7 k\u0131z olmal\u0131 \u015fimdi. \u00d6yle yaramaz, ne\u015feli bir k\u0131zd\u0131 ki hepimizi g\u00fcld\u00fcr\u00fcrd\u00fc; \u00fc\u00e7\u00fcm\u00fcz b\u00f6ylece ya\u015fay\u0131p giderdik. O zamanlar, uzun k\u0131\u015f ak\u015fam\u0131nda yuvarlak masan\u0131n \u00e7evresine oturur, \u00e7ay i\u00e7er, sonra da i\u015fimize bakard\u0131k. Ya\u015fl\u0131 kad\u0131n da, Ma\u015fa s\u0131k\u0131l\u0131p yaramazl\u0131k yapmas\u0131n diye masallar anlatmaya ba\u015flard\u0131. Hem de ne masallar anlat\u0131rd\u0131! \u00c7ocuk bir yana, akl\u0131 ba\u015f\u0131nda ve zeki bir adam bile otursun dinlesin. Ne demek! Ben de, o zamanlar, \u00e7ubu\u011fumu t\u00fctt\u00fcr\u00fcr, olay nereye var\u0131yor diye dinlerdim. \u00c7ocuk, yani bizim yaramaz k\u0131z d\u00fc\u015f\u00fcncelere dalard\u0131; elini pembe yana\u011f\u0131na yaslar, k\u00fc\u00e7\u00fck a\u011fz\u0131n\u0131 ho\u015f bir \u015fekilde aralar ve korkun\u00e7 bir masal anlat\u0131l\u0131yorsa ya\u015fl\u0131 kad\u0131na sokulur, iyice sokulurdu. Onu seyretmek en sevdi\u011fimiz \u015feydi; mumun yan\u0131p s\u00f6nd\u00fc\u011f\u00fcn\u00fc g\u00f6rmezsin, avluda tipi delirmi\u015f, esip \u00fcf\u00fcr\u00fcyor, duymazs\u0131n. Hayat\u0131m\u0131z \u00e7ok iyiydi, Varenka; b\u00f6yle sanki yirmi y\u0131l birlikte ya\u015fad\u0131k. Ama ben ne sa\u00e7mal\u0131yorum! Herhalde siz b\u00f6yle bir konudan ho\u015flanmazs\u0131n\u0131z, benim i\u00e7in de hat\u0131rlamak kolay de\u011fil, hele \u015fimdi, alacakaranl\u0131k vakti. Tereza bir \u015feyler ta\u015f\u0131yor, benim ba\u015f\u0131m a\u011fr\u0131yor, s\u0131rt\u0131m da a\u011fr\u0131yor, ama bunlar tuhaf d\u00fc\u015f\u00fcnceler, sanki onlar da a\u011fr\u0131yorlar; h\u00fcz\u00fcnlendim ben bug\u00fcn Varenka! Neler yazm\u0131\u015fs\u0131n\u0131z \u00f6yle, karde\u015fim? Ben size nas\u0131l geleyim? G\u00fcvercinim, elalem buna ne der? Sonu\u00e7ta avludan ge\u00e7mek gerekecek, bizimkiler fark\u0131na var\u0131r, soru sorar, yorum yaparlar, dedikodular ba\u015flar, olay ba\u015fka bir mana kazan\u0131r. Hay\u0131r, mele\u011fim, ben sizi en iyisi sabah kilisede g\u00f6reyim; bu ikimiz i\u00e7in de daha ak\u0131ll\u0131ca ve zarars\u0131z olur. Size b\u00f6yle bir mektup yazd\u0131\u011f\u0131m i\u00e7in de ho\u015f g\u00f6r\u00fcn beni can\u0131m; okuyunca ben de her yerinin ba\u011flant\u0131s\u0131z oldu\u011funu g\u00f6rd\u00fcm. Varenka, ben, ya\u015fl\u0131, e\u011fitimsiz biriyim; gen\u00e7ken okumad\u0131m, \u015fimdi de akl\u0131m t\u00fckendi t\u00fckenecek, yeniden e\u011fitim almam gerekiyor. \u0130tiraf edeyim, can\u0131m, yazman\u0131n \u00fcstad\u0131 de\u011filim ve yabanc\u0131 birinin \u00f6\u011f\u00fct ve uyar\u0131lar\u0131na ihtiya\u00e7 duymadan biliyorum ki, daha maharetli bir \u015feyler yazmak isteseydim, sa\u00e7malamaya ba\u015flard\u0131m. Bug\u00fcn sizi pencerenin k\u0131y\u0131s\u0131nda g\u00f6rd\u00fcm, perdeyi nas\u0131l indirdi\u011finizi g\u00f6rd\u00fcm. Ho\u015f\u00e7a kal\u0131n, ho\u015f\u00e7a kal\u0131n, Tanr\u0131 sizi korusun. Ho\u015f\u00e7a kal\u0131n, Varvara Alekseyevna.\n\nMasum dostunuz,  \nMakar Devu\u015fkin\n\nHami\u015f: Can\u0131m akrabam, art\u0131k kimse hakk\u0131nda sat\u0131r yazmayaca\u011f\u0131m. \u00d6fkelenmek i\u00e7in \u00e7ok ya\u015fland\u0131m art\u0131k, can\u0131m! Rus atas\u00f6z\u00fcndeki gibi g\u00fcl\u00fcyorlar bana: \"El i\u00e7in kuyu kazan\u0131n,\" diyorlar, \"evvela kendisi d\u00fc\u015fer.\"\n\n9 Nisan\n\nSayg\u0131de\u011fer Beyefendi, Makar Alekseyevi\u00e7,\n\nB\u00f6yle kederlenip kapris yapmaya nas\u0131l utanm\u0131yorsunuz dostum ve velinimetim Makar Alekseyevi\u00e7. Siz ciddi ciddi g\u00fccenmi\u015fsiniz! Ah, s\u0131k s\u0131k dikkatsizlik yapar\u0131m, ama benim s\u00f6zlerimi i\u011fneleyici bir \u015faka sayaca\u011f\u0131n\u0131z\u0131 hi\u00e7 d\u00fc\u015f\u00fcnmemi\u015ftim. Emin olun, ben asla sizin ya\u015f\u0131n\u0131z\u0131 ve karakterinizi \u015faka konusu yapmaya kalk\u0131\u015fmad\u0131m. Hep benim bo\u015fbo\u011fazl\u0131\u011f\u0131mdan oldu bu, ama daha da \u00f6nemlisi, \u00e7ok s\u0131k\u0131ld\u0131\u011f\u0131mdan oldu, zaten ne olursa s\u0131k\u0131nt\u0131dan olmaz m\u0131? Sizin kendi mektubunuzla alay etmek isteyece\u011finiz gelmemi\u015fti akl\u0131ma. Sizin benden ho\u015fnut olmad\u0131\u011f\u0131n\u0131z\u0131 g\u00f6r\u00fcnce, korkun\u00e7 bir kedere kap\u0131ld\u0131m. Hay\u0131r, sevgili dostum ve velinimetim, yan\u0131l\u0131yorsunuz, e\u011fer benim duygusuz ve minnetsiz oldu\u011fumdan \u015f\u00fcphe ediyorsan\u0131z, yan\u0131l\u0131yorsunuz. Bana yapt\u0131\u011f\u0131n\u0131z her \u015feyi, beni k\u00f6t\u00fc insanlardan, onlar\u0131n \u00f6fke ve nefretinden beni koruman\u0131za kalbimde hak etti\u011fi de\u011feri vermesini biliyorum. Sonsuza dek dua edece\u011fim Tanr\u0131'ya sizin i\u00e7in ve e\u011fer benim duam\u0131n Tanr\u0131 i\u00e7in bir de\u011feri varsa ve g\u00f6kler ona ilgi g\u00f6sterirse, o zaman siz mutlu olacaks\u0131n\u0131z.\n\nBug\u00fcn kendimi \u00e7ok sa\u011fl\u0131ks\u0131z hissediyorum. \u00dczerime ikide bir ate\u015f ve titreme geliyor. Fedora kayg\u0131lan\u0131yor benim i\u00e7in. Bo\u015f yere \u00e7ekiniyorsunuz bize gelmeye Makar Alekseyevi\u00e7. Daha neler! Siz bizim ahbab\u0131m\u0131zs\u0131n\u0131z, hepsi bu!.. Ho\u015f\u00e7a kal\u0131n, Makar Alekseyevi\u00e7. Daha fazla yazmayaca\u011f\u0131m, yazam\u0131yorum; \u00e7ok k\u00f6t\u00fc bozuldu sa\u011fl\u0131\u011f\u0131m. Rica ederim yine \u00f6fkelenmeyin bana ve sizin en sad\u0131k ve uysal hizmetk\u00e2r\u0131n\u0131z olma \u015ferefine nail olmam\u0131 sa\u011flayan o sonsuz sayg\u0131 ve sadakatime inan\u0131n.\n\nVarvara Dobroselova\n\n12 Nisan\n\nSayg\u0131de\u011fer Han\u0131mefendi, Varvara Alekseyevna,\n\nAh, can\u0131m benim, ne oldu size! Zaten her seferinde beni \u00fcrk\u00fct\u00fcyorsunuz. Size her mektupta kendinizi koruman\u0131z\u0131, kal\u0131n giyinmenizi, k\u00f6t\u00fc havada d\u0131\u015far\u0131 \u00e7\u0131kmaman\u0131z\u0131, her \u015feye dikkat etmenizi s\u00f6yl\u00fcyorum zaten, ama siz, mele\u011fim, beni dinlemiyorsunuz. Ah, g\u00fcvercinim, sanki \u00e7ocuk gibisiniz! Zaten zay\u0131fs\u0131n\u0131z, saman sap\u0131 kadar zay\u0131fs\u0131n\u0131z, bunu biliyorum. Bir r\u00fczg\u00e2r esse siz hemen hasta olursunuz. \u0130nsan\u0131n kendini sak\u0131nmas\u0131 laz\u0131m, kendine \u00f6zen g\u00f6stermesi, tehlikelerden ka\u00e7\u0131nmas\u0131 ve dostlar\u0131n\u0131 ac\u0131 ve kedere s\u00fcr\u00fcklememesi laz\u0131m.\n\nCan\u0131m, benim g\u00fcnl\u00fck hayat\u0131m\u0131 ve \u00e7evremdeki her \u015feyi ayr\u0131nt\u0131s\u0131yla \u00f6\u011frenmek istedi\u011finizi s\u00f6yl\u00fcyorsunuz. Bu arzunuzu yerine getirmekten mutluluk duyar\u0131m, bir tanem. En ba\u015ftan ba\u015flayay\u0131m, can\u0131m, b\u00f6yle daha d\u00fczg\u00fcn olur. Birincisi, bizim evimizde, temiz hol\u00fcm\u00fczde, \u00e7ok uyduruk merdivenler vard\u0131r; antre d\u00fczg\u00fcnd\u00fcr; temiz, ayd\u0131nl\u0131k, geni\u015f, her yeri maun kaplamad\u0131r. Fakat o bula\u015f\u0131khaneyi hi\u00e7 sormay\u0131n: koku sa\u00e7an, \u0131slak, ya\u011fl\u0131, k\u0131r\u0131k d\u00f6k\u00fck basamaklar, ya\u011f i\u00e7inde duvarlar, \u00f6yle ki dokundu\u011fu zaman insan\u0131n eli yap\u0131\u015f\u0131r duvara. Her sahanl\u0131kta k\u0131r\u0131k d\u00f6k\u00fck sand\u0131klar, sandalyeler ve dolaplar durur, pa\u00e7avralar as\u0131lm\u0131\u015f, pencereler k\u0131r\u0131lm\u0131\u015f; ya\u011f, \u00e7er\u00e7\u00f6p, yumurta kabu\u011fu ve bal\u0131k art\u0131klar\u0131 dolu pis su birikintileri vard\u0131r yerlerde; i\u011fren\u00e7 kokar... k\u0131sacas\u0131, berbatt\u0131r.\n\nOdalar\u0131n durumunu yazm\u0131\u015ft\u0131m size; diyecek bir \u015fey yok, do\u011frusu rahat, ama biraz bo\u011fucudur, yani k\u00f6t\u00fc koktu\u011fundan de\u011fil, \u00fcst\u00fcn\u00fcze afiyet, biraz \u00e7\u00fcr\u00fck kokusu, keskin bir koku vard\u0131r i\u00e7eride. \u0130lk seferinde tats\u0131z bir izlenim olur, ama bu \u00f6nemli de\u011fildir; iki dakika kal\u0131r \u00fczerinizde bu izlenim, sonra ge\u00e7er, hissetmezsiniz, her \u015fey gibi ge\u00e7er, \u00e7\u00fcnk\u00fc siz de k\u00f6t\u00fc kokars\u0131n\u0131z, elbiseniz kokar, elleriniz kokar, her \u015fey kokar... b\u00f6ylece, al\u0131\u015f\u0131rs\u0131n\u0131z. Bizdeki isketeler b\u00f6yle can verir. Denizci, be\u015finci ku\u015fu sat\u0131n ald\u0131, ama bizim evin havas\u0131nda ya\u015fayam\u0131yorlar. Mutfa\u011f\u0131m\u0131z b\u00fcy\u00fck, geni\u015f, ayd\u0131nl\u0131k. Do\u011frusu, sabahlar\u0131 bal\u0131k ya da s\u0131\u011f\u0131r eti k\u0131zart\u0131l\u0131nca, biraz bo\u011fucu oluyor, duman alt\u0131nda kal\u0131yor her yer, ama ak\u015famleyin tam bir cennet. Mutfa\u011f\u0131m\u0131zda hep iplere as\u0131l\u0131 eski \u00e7ama\u015f\u0131rlar oluyor; odam da yak\u0131nda oldu\u011fu i\u00e7in, yani neredeyse mutfa\u011f\u0131n yan\u0131ba\u015f\u0131nda oldu\u011fu i\u00e7in, \u00e7ama\u015f\u0131rlardan gelen koku beni biraz rahats\u0131z ediyor; ama \u00f6nemi yok, insan ya\u015fad\u0131k\u00e7a al\u0131\u015f\u0131yor.\n\nBizde sabah\u0131n erken vakitlerinde bir karga\u015fa ba\u015flar, Varenka, herkes girer \u00e7\u0131kar, ortal\u0131kta gezinir, kap\u0131lar\u0131 \u00e7arpar... kalkmas\u0131 gereken gerekmeyen, bir i\u015fi olan olmayan herkesi kald\u0131r\u0131r bunlar; herkes \u00e7ay i\u00e7meye ba\u015flar. Semaverlerin b\u00fcy\u00fck k\u0131sm\u0131 ev sahibesinindir bizde, say\u0131lar\u0131 az oldu\u011fu i\u00e7in, s\u0131rayla hepimiz saklar\u0131z; fincan\u0131yla biri gelip s\u0131raya d\u0131\u015far\u0131dan girmeye kalkarsa, hemen fincan\u0131 elinden al\u0131n\u0131p ba\u015f\u0131ndan a\u015fa\u011f\u0131ya d\u00f6k\u00fcl\u00fcr. \u0130lk seferinde ben de d\u00fc\u015ft\u00fcm bu hale, ya... fakat yazmayay\u0131m en iyisi! B\u00f6ylece herkesle tan\u0131\u015fm\u0131\u015f oldum. \u00d6nce denizciyle tan\u0131\u015ft\u0131k; \u00e7ok samimiydi, bana her \u015feyi anlatt\u0131: Babas\u0131n\u0131, anas\u0131n\u0131, Tulal\u0131 bir vergi memuruyla evlenen k\u0131zkarde\u015fini, Kron\u015ftadt \u015fehrini anlatt\u0131. Beni kay\u0131rmay\u0131 vaat etti ve bana hemen kendi \u00e7ay\u0131n\u0131 ikram etti. Onu hep iskambil oynad\u0131\u011f\u0131m\u0131z odada buldum. Orada bana \u00e7ay verdiler ve hemen onlar\u0131n co\u015fkulu oyunlar\u0131na kat\u0131lmam\u0131 istediler. Alay m\u0131 ettiler, \u00f6ylesine mi s\u00f6ylediler, bilmiyorum; ama kendileri b\u00fct\u00fcn gece ara vermeden oynam\u0131\u015flard\u0131 ve ben yanlar\u0131na gitti\u011fimde h\u00e2l\u00e2 oynuyorlard\u0131. Tebe\u015fir tozu, kartlar, b\u00fct\u00fcn oday\u0131 kaplam\u0131\u015f, g\u00f6zleri yakan bir duman. Oynamaya kalk\u0131\u015fmad\u0131m, bana hemen i\u015fin felsefesini yapmaya kalk\u0131\u015ft\u0131\u011f\u0131m\u0131 s\u00f6ylediler. Sonra kimse benimle konu\u015fmad\u0131 bir daha; ama do\u011frusu ben de, buna memnun oldum. Onlar\u0131n yan\u0131na gitmiyorum art\u0131k; co\u015fku i\u00e7indeydiler, saf co\u015fku! Fakat edebiyat mezunu memurun odas\u0131nda da ak\u015famlar\u0131 toplant\u0131lar oluyor. Neyse ki onun odas\u0131 iyi, m\u00fctevaz\u0131, masum ve zarif bir yer; her \u015fey terbiye s\u0131n\u0131rlar\u0131nda.\n\nAma, Varenka, bu arada ev sahibemizin de \u00e7ok i\u011fren\u00e7 oldu\u011funu, tam bir cad\u0131 oldu\u011funu s\u00f6yleyeyim size. Tereza'y\u0131 g\u00f6rd\u00fcn\u00fcz. Ne halde sizce? Zavall\u0131, c\u0131l\u0131z bir pili\u00e7 gibi s\u0131ska. Binada toplam iki hizmetli var: Tereza ile ev sahibesinin u\u015fa\u011f\u0131, Faldoni.5 Bilmiyorum, belki, ba\u015fka bir ismi de vard\u0131r, ama ona b\u00f6yle sesleniyorlar; herkes onu b\u00f6yle \u00e7a\u011f\u0131r\u0131yor. Ev sahibesi k\u0131pk\u0131rm\u0131z\u0131, bir _\u00e7uhonka_6 , e\u011fri, kalk\u0131k burunlu, kaba saba; s\u00fcrekli Tereza'ya k\u00fcfrediyor, neredeyse d\u00f6vecek. Yani genel olarak, buradaki hayat\u0131m pek de iyi de\u011fil... En az\u0131ndan geceleri bir anda uyuyup sakinle\u015fseler... Bu da hi\u00e7 olmuyor. Ya bir yerde oturup oyuna dal\u0131yorlar ya da bazen \u00f6yle \u015feyler yap\u0131yorlar ki anlatmaya dilim varmaz. Art\u0131k b\u00fct\u00fcn olup bitene al\u0131\u015ft\u0131m, ama b\u00f6yle bir g\u00fcr\u00fclt\u00fc pat\u0131rt\u0131 i\u00e7inde aile sahibi insanlar\u0131n nas\u0131l ya\u015fad\u0131\u011f\u0131na h\u00e2l\u00e2 hayret ediyorum. Yoksul bir aile gelip ev sahibemizden oda kiralad\u0131, ama di\u011fer odalar\u0131n yan\u0131nda de\u011fil, di\u011fer y\u00f6nde, k\u00f6\u015fede, ayr\u0131 bir oda. Sakin insanlar! Onlar hakk\u0131nda kimse bir \u015fey duymuyor, i\u015fitmiyor. Tek bir k\u00fc\u00e7\u00fck odada, oday\u0131 paravanla b\u00f6lerek oturuyorlar. Adam i\u015fsiz bir memur, yedi y\u0131l \u00f6nce bir sebeple i\u015finden \u00e7\u0131kar\u0131lm\u0131\u015f. Soyad\u0131 Gor\u015fkov; ak sa\u00e7l\u0131, ufak tefek biri; ya\u011f i\u00e7inde, y\u0131rt\u0131k p\u0131rt\u0131k bir giysiyle geziyor, insan bakmaya dayanam\u0131yor; benimkinden kat be kat k\u00f6t\u00fc! Zavall\u0131, c\u0131l\u0131z bir adam (onunla s\u0131k s\u0131k koridorda kar\u015f\u0131la\u015f\u0131yoruz); dizleri titriyor, elleri titriyor, ba\u015f\u0131 titriyor, hep hastal\u0131ktan, ama hangi hastal\u0131ktan Tanr\u0131 bilir; \u00fcrkek, her \u015feyden korkuyor, kenardan kenardan geziyor; ben de \u00e7ekinirim zaman zaman, ama bu benden de beter. Ailesi kar\u0131s\u0131yla \u00fc\u00e7 \u00e7ocuktan ibaret. B\u00fcy\u00fck olan \u00e7ocuk babas\u0131n\u0131n t\u0131pk\u0131s\u0131, onun gibi c\u0131l\u0131z. Kar\u0131s\u0131 bir zamanlar \u00e7irkin de\u011filmi\u015f, ama \u015fimdi pek anla\u015f\u0131lm\u0131yor; yoksul bir halde, ac\u0131nas\u0131 d\u00f6k\u00fcnt\u00fclerle geziyor. Ev sahibesine bor\u00e7land\u0131klar\u0131n\u0131 duydum; kad\u0131n onlara hi\u00e7 kibar davranm\u0131yordu. Gor\u015fkov'un kendisinin de birtak\u0131m tats\u0131zl\u0131klar ya\u015fad\u0131\u011f\u0131n\u0131, i\u015fini bu y\u00fczden kaybetti\u011fini de duydum... bir dava m\u0131 olmu\u015f, mahkemeye mi \u00e7\u0131km\u0131\u015f, birileri ona kar\u015f\u0131 tan\u0131kl\u0131k m\u0131 etmi\u015f... size kesin olarak s\u00f6ylememe imk\u00e2n yok. \u00d6yle yoksullar ki, y\u00fcce Tanr\u0131m! Odalar\u0131 hep sessiz sakin olur, sanki i\u00e7eride kimse ya\u015fam\u0131yormu\u015f gibi. \u00c7ocuklar\u0131n bile sesi duyulmaz. \u00c7ocuklar\u0131n \u015famata yap\u0131p oynad\u0131\u011f\u0131 da g\u00f6r\u00fclmez, ama bu da k\u00f6t\u00fcye i\u015faret. Bir ara, bir gece vakti, kap\u0131n\u0131n yan\u0131ndan ge\u00e7erken bir \u015feyler duydum; o s\u0131rada evde nedense al\u0131\u015f\u0131lmad\u0131k bir sessizlik vard\u0131; bir h\u0131\u00e7k\u0131r\u0131k, sonra bir f\u0131s\u0131lt\u0131, sonra yine h\u0131\u00e7k\u0131r\u0131k, sanki biri a\u011fl\u0131yormu\u015f gibi, ama \u00f6yle sessiz, \u00f6yle ac\u0131kl\u0131 bir ses ki kalbim par\u00e7aland\u0131 ve sonra b\u00fct\u00fcn gece bu yoksullar akl\u0131mdan hi\u00e7 \u00e7\u0131kmad\u0131, do\u011fru d\u00fcr\u00fcst uyuyamad\u0131m.\n\nNeyse, ho\u015f\u00e7a kal\u0131n, e\u015fsiz dostum Varenka! Size elimden geldi\u011fince her \u015feyi yazd\u0131m. Bug\u00fcn b\u00fct\u00fcn g\u00fcn hep sizi d\u00fc\u015f\u00fcnece\u011fim. Bir tanem, sizin i\u00e7in kalbim hep s\u0131zl\u0131yor. Yani can\u0131m benim, sizi biliyorum, s\u0131cak tutacak bir mantonuz olmad\u0131\u011f\u0131n\u0131 biliyorum. Petersburg'un bu baharlar\u0131, r\u00fczg\u00e2rlar\u0131, sulusepken, karla kar\u0131\u015f\u0131k ya\u011fmurlar\u0131 da beni peri\u015fan ediyor Varenka! Tanr\u0131 bizi b\u00f6yle havalardan korusun! Yaz\u0131mdan dolay\u0131 mazur g\u00f6r\u00fcn beni can\u0131m; \u00fcslubum yok, Varenka, do\u011fru d\u00fcr\u00fcst bir \u00fcslubum yok. Ke\u015fke olsayd\u0131 biraz! Akl\u0131ma ne gelirse yaz\u0131yorum, s\u0131rf sizi biraz e\u011flendirebilmek i\u00e7in. Zaten biraz e\u011fitim alsayd\u0131m, her \u015fey ba\u015fka t\u00fcrl\u00fc olurdu; ama nas\u0131l okuyacakt\u0131m? Bak\u0131r parayla olmuyor \u00f6yle \u015feyler.\n\nEbediyen ve sad\u0131k dostunuz,  \nMakar Devu\u015fkin\n\n25 Nisan\n\nSayg\u0131de\u011fer Beyefendi, Makar Alekseyevi\u00e7,\n\nBug\u00fcn kuzenim Sa\u015fa'yla kar\u015f\u0131la\u015ft\u0131m! Korkun\u00e7! O da mahvolacak, zavall\u0131! Bir yerlerden Anna Fedorovna'n\u0131n h\u00e2l\u00e2 beni sorup durdu\u011funu \u00f6\u011frendim. Herhalde hi\u00e7 b\u0131rakmayacak beni takip etmeyi. Beni _affetmek istedi\u011fini_ , ya\u015fananlar\u0131 unutmak ve hemen beni ziyarete gelmek istedi\u011fini s\u00f6ylemi\u015f. Sizin kesinlikle benim akrabam olmad\u0131\u011f\u0131n\u0131z\u0131, onun bana bir akrabadan da yak\u0131n oldu\u011funu, sizin aile i\u015flerimize kar\u0131\u015fmaya hi\u00e7 hakk\u0131n\u0131z olmad\u0131\u011f\u0131n\u0131 ve sizin sadakan\u0131z ve sizin gelirinizle ge\u00e7inmemin benim i\u00e7in ay\u0131p ve yak\u0131\u015f\u0131ks\u0131z oldu\u011funu s\u00f6ylemi\u015f... Onun ekme\u011fini yedi\u011fimi unuttu\u011fumu, anneci\u011fimle beni belki de a\u00e7l\u0131ktan kurtarm\u0131\u015f oldu\u011funu, bizi yedirip i\u00e7irdi\u011fini ve iki bu\u00e7uk y\u0131ldan fazla bir s\u00fcre bize bakt\u0131\u011f\u0131n\u0131, b\u00fct\u00fcn bunlara ra\u011fmen borcumuzu affetti\u011fini s\u00f6yl\u00fcyormu\u015f. Anneci\u011fime bile ac\u0131mak istemiyor! Onlar\u0131n bana neler yapt\u0131\u011f\u0131n\u0131 zavall\u0131 yoksul anneci\u011fim bilseydi! Tanr\u0131 g\u00f6r\u00fcyor!.. Anna Fedorovna aptall\u0131k edip talihime sahip \u00e7\u0131kamad\u0131\u011f\u0131m\u0131, bana mutluluk getirdi\u011fini, ba\u015fka da bir \u015fey yapmad\u0131\u011f\u0131n\u0131 ve benim kendi onuruma sahip \u00e7\u0131kmad\u0131\u011f\u0131m\u0131, belki de sahip \u00e7\u0131kmak istemedi\u011fimi s\u00f6yl\u00fcyor. Burada su\u00e7 kimin peki, y\u00fcce Tanr\u0131m! Bay B\u0131kov'un kesinlikle hakl\u0131 oldu\u011funu ve herkesle evlenemeyece\u011fini, yani... yazamayaca\u011f\u0131m! B\u00f6yle bir haks\u0131zl\u0131\u011f\u0131 duymak \u00e7ok a\u011f\u0131r bir \u015fey Makar Alekseyevi\u00e7! \u015eimdi ne yapaca\u011f\u0131m\u0131 bilemiyorum. Titriyorum, a\u011fl\u0131yorum, h\u0131\u00e7k\u0131r\u0131yorum; bu mektubu size iki saatte yazabildim. En az\u0131ndan bana kar\u015f\u0131 su\u00e7lu oldu\u011funu kabul eder, diye ummu\u015ftum; ama hi\u00e7 \u00f6yle de\u011fil! Tanr\u0131 a\u015fk\u0131na, kayg\u0131lanmay\u0131n dostum, iyili\u011fimi isteyen tek ki\u015fi! Fedora hep abart\u0131yor, hasta de\u011filim. Sadece d\u00fcn ak\u015fam annemi kilisede anmak i\u00e7in Volkovo'ya gitti\u011fim zaman biraz \u00fc\u015f\u00fctt\u00fcm. Neden siz de benimle gelmediniz; \u00e7ok rica etmi\u015ftim. Ah, yoksul, yoksul anneci\u011fim, kalksayd\u0131 mezar\u0131ndan, \u00f6\u011frenseydi, g\u00f6rseydi onlar\u0131n bana neler yapt\u0131\u011f\u0131n\u0131!..\n\nV.D.\n\n2. Varvara isminin samimi kullan\u0131m\u0131.\n\n3. Brambeus takma ad\u0131yla yazan, d\u00f6nemin pop\u00fcler gazeteci ve ele\u015ftirmeni J\u00f3zef-Julian S\u1eb9kowski (1800-1858) .\n\n4. (Alm.) Sabahl\u0131k.\n\n5. Nicolas-Germain L\u00e9onard'\u0131n (1744-1793) Rus\u00e7aya 1804'te \u00e7evrilen, _Lettres de deux amants, habitants de Lyon, contenant l'histoire de Th\u00e9r\u00e8se et de Faldoni_ (Lyon'da Oturan \u0130ki \u00c2\u015f\u0131\u011f\u0131n Mektuplar\u0131, Th\u00e9r\u00e8se ile Faldoni'nin Hik\u00e2yesi, 1783) adl\u0131 roman\u0131na g\u00f6nderme.\n\n6. (Rus.) Ruslar\u0131n etnik a\u015fa\u011f\u0131lama amac\u0131yla Finlandiyal\u0131 ve Estonyal\u0131 kad\u0131nlara verdikleri isim.\n\n# May\u0131s\n\n20 May\u0131s\n\nG\u00fcvercinim, Varenka,\n\nSize biraz \u00fcz\u00fcm g\u00f6nderiyorum, can\u0131m; iyile\u015fmeniz i\u00e7in iyi gelirmi\u015f, diyorlar ve doktor da susuzlu\u011fu bast\u0131rmak i\u00e7in bunu tavsiye ediyor, yani bir tek susuzluk i\u00e7in. Siz beyaz ekmek istemi\u015ftiniz ge\u00e7en g\u00fcn, can\u0131m; hemen yolluyorum size. \u0130\u015ftah\u0131n\u0131z nas\u0131l can\u0131m? En \u00f6nemlisi bu. Fakat, Tanr\u0131'ya \u015f\u00fck\u00fcrler olsun, her \u015fey ge\u00e7ti, talihsizli\u011fimiz de art\u0131k kesin olarak sona erdi. \u0130yilik g\u00f6klerden gelir! Kitaba gelince, onu \u015fimdilik hi\u00e7bir yerde bulamad\u0131m. M\u00fckemmel ve \u00e7ok y\u00fcksek \u00fcslupla yaz\u0131lm\u0131\u015f bir kitab\u0131n var oldu\u011funu s\u00f6yl\u00fcyorlar; m\u00fckemmel oldu\u011funu s\u00f6yl\u00fcyorlar, ben kendim okumad\u0131m, ama burada \u00e7ok \u00f6v\u00fcyorlar. Onu kendim i\u00e7in istettim; getirmeye s\u00f6z verdiler. Ama siz de okumak ister miydiniz? Bana g\u00f6re siz bu konuda \u00e7ok nazl\u0131s\u0131n\u0131z; sizin zevkinize hitap etmek zor, sizi art\u0131k tan\u0131yorum g\u00fcvercinim; size, herhalde, hep \u015fairanelik uygun olur, i\u00e7 \u00e7eki\u015fler, a\u015fklar... ama \u015fiir de bulaca\u011f\u0131m, hepsini bulaca\u011f\u0131m; temize \u00e7ekilecek bir defter kald\u0131.\n\nBen \u00f6yle b\u00f6yle ya\u015f\u0131yorum. Siz benim i\u00e7in kayg\u0131lanmay\u0131n, can\u0131m, l\u00fctfen. Fedora'n\u0131n size benim hakk\u0131mda anlatt\u0131klar\u0131 da hepten sa\u00e7mal\u0131k; yalan s\u00f6yledi\u011fini hemen s\u00f6yleyin o dedikoducuya!.. Yeni \u00fcniformam\u0131 kesinlikle satmad\u0131m. D\u00fc\u015f\u00fcnsenize bir, zaten neden, neden satay\u0131m? Dediklerine g\u00f6re, k\u0131rk g\u00fcm\u00fc\u015f ruble ikramiye vereceklermi\u015f bana, neden onu satay\u0131m? Can\u0131m, siz kayg\u0131lanmay\u0131n; ku\u015fkucunun teki Fedora, o tam bir ku\u015fkucu. Ge\u00e7inip gidiyoruz biz, g\u00fcvercinim! Ama siz, mele\u011fim, Tanr\u0131 a\u015fk\u0131na, iyile\u015fin, iyile\u015fin, bu ihtiyar\u0131 \u00fczmeyin. Benim zay\u0131flad\u0131\u011f\u0131m\u0131 size kim s\u00f6ylemi\u015f? \u0130ftira, yine iftira! Turp gibiyim ve \u00f6yle \u015fi\u015fmanlad\u0131m ki yine utanmaya ba\u015flad\u0131m, midem g\u0131rtla\u011f\u0131ma kadar dolu; bir de siz iyile\u015fmi\u015f olsan\u0131z! Neyse, ho\u015f\u00e7a kal\u0131n, mele\u011fim; k\u00fc\u00e7\u00fck parmaklar\u0131n\u0131z\u0131n hepsini \u00f6p\u00fcyor sizin sonsuz, de\u011fi\u015fmez dostunuz.\n\nMakar Devu\u015fkin\n\nHami\u015f: Ah, can\u0131m benim, siz yine b\u00f6yle ger\u00e7ekten ne yazmaya kalkm\u0131\u015fs\u0131n\u0131z?.. Neden kapris yap\u0131yorsunuz! Ben size nas\u0131l \u00f6yle s\u0131k s\u0131k geleyim can\u0131m, nas\u0131l? Size soruyorum. Belki gece karanl\u0131\u011f\u0131ndan yararlan\u0131r\u0131m; ama gece de olmuyor art\u0131k sanki hi\u00e7; bir s\u00fcre b\u00f6yle.7 Ben de, can\u0131m, mele\u011fim, hastal\u0131\u011f\u0131n\u0131z boyunca, kendinizi bilmez durumda oldu\u011funuz s\u0131rada sizi hemen hi\u00e7 b\u0131rakmad\u0131m; ama \u015fimdi hi\u00e7 bilemiyorum b\u00fct\u00fcn bunlar\u0131 nas\u0131l yapt\u0131\u011f\u0131m\u0131; sonra gelip gitmeye ara verdim; \u00e7\u00fcnk\u00fc merak ve sorular ba\u015flam\u0131\u015ft\u0131. Burada zaten t\u00fcrl\u00fc t\u00fcrl\u00fc dedikodu var. Tereza'ya g\u00fcveniyorum ben; o geveze de\u011fildir; ama yine de, kendiniz de takdir edersiniz can\u0131m, bunlar\u0131n hepsi bizim hakk\u0131m\u0131zda her \u015feyi \u00f6\u011frenirse neler olur? Ne d\u00fc\u015f\u00fcnecekler, ne diyecekler o zaman? \u0130\u015fte o y\u00fczden kalbinizi sa\u011flam tutun, can\u0131m, iyile\u015finceye kadar bekleyin; ondan sonra evin d\u0131\u015f\u0131nda bir yerlerde _rendez-vouz_ 'la\u015f\u0131r\u0131z.8\n\n7. Petersburg'da may\u0131s sonu g\u00fcne\u015fin hi\u00e7 batmad\u0131\u011f\u0131 \"beyaz gecelerin\" ba\u015flang\u0131c\u0131d\u0131r.\n\n8. (Fr.) Randevu.\n\n# Haziran\n\n1 Haziran\n\nSevgili Makar Alekseyevi\u00e7,\n\nBenim i\u00e7in ya\u015fad\u0131\u011f\u0131n\u0131z b\u00fct\u00fcn tela\u015f ve \u00fcz\u00fcnt\u00fc i\u00e7in, bana kar\u015f\u0131 g\u00f6sterdi\u011finiz sevgi i\u00e7in, g\u00fczel ve ho\u015f bir \u015fey yapmay\u0131 \u00f6yle \u00e7ok istiyorum ki, sonunda s\u0131k\u0131nt\u0131dan komodinimi kar\u0131\u015ft\u0131rmaya ba\u015flad\u0131m ve \u015fimdi size g\u00f6ndermekte oldu\u011fum defterimi buldum. Ona ba\u015flarken hayat\u0131m\u0131n mutlu d\u00f6nemindeydim daha. Siz s\u0131k s\u0131k merak edip benim eski g\u00fcnl\u00fck hayat\u0131m\u0131, anneci\u011fimi, Pokrovski' yi, benim Anna Fedorovna'n\u0131n yan\u0131nda ge\u00e7irdi\u011fim g\u00fcnleri ve son olarak, k\u0131sa s\u00fcre \u00f6nce ya\u015fad\u0131\u011f\u0131m talihsizlikleri soruyordunuz ve sab\u0131rs\u0131zca bu defteri, i\u00e7inde, Tanr\u0131 bilir ne diye hayat\u0131m\u0131n birtak\u0131m anlar\u0131n\u0131 kaydetmeye kalk\u0131\u015ft\u0131\u011f\u0131m defteri \u00f6yle \u00e7ok okumak istiyordunuz ki, size g\u00f6nderdi\u011fim i\u00e7in b\u00fcy\u00fck bir mutluluk duyaca\u011f\u0131n\u0131zdan hi\u00e7 ku\u015fku duymuyorum. Benim i\u00e7inse kederli bir i\u015f oldu onu yeniden okumak. Bu karalamalara son sat\u0131r\u0131 koydu\u011fum vakitten bu yana iki kat ya\u015flanm\u0131\u015f\u0131m gibi geliyor bana. B\u00fct\u00fcn bunlar\u0131 farkl\u0131 aral\u0131klarla yazm\u0131\u015f\u0131m. Ho\u015f\u00e7a kal\u0131n, Makar Alekseyevi\u00e7! \u015eimdi \u00e7ok s\u0131k\u0131ld\u0131m ve uykusuzluk i\u015fkence ediyor. Nekahat d\u00f6nemi ne kadar s\u0131k\u0131c\u0131!\n\nV.D.\n\n# I\n\nBabac\u0131\u011f\u0131m \u00f6ld\u00fc\u011f\u00fcnde ben sadece on d\u00f6rt ya\u015f\u0131ndayd\u0131m. \u00c7ocuklu\u011fum hayat\u0131m\u0131n en mutlu zamanlar\u0131 oldu. Burada ba\u015flamad\u0131, buradan uzak bir yerde, ta\u015frada, \u0131ss\u0131zlarda ba\u015flad\u0131. Babac\u0131\u011f\u0131m T... eyaletinde, Prens P.nin b\u00fcy\u00fck \u00e7iftli\u011finin k\u00e2hyas\u0131yd\u0131. Prensin k\u00f6ylerinden birinde sessiz sedas\u0131z, sakin ve mutlu ya\u015f\u0131yorduk... \u00c7ok afacan bir \u00e7ocuktum; tek yapt\u0131\u011f\u0131m tarlalarda, korularda, bah\u00e7ede ko\u015fturmakt\u0131, benim i\u00e7in de kimse kayg\u0131lanmazd\u0131. Babac\u0131\u011f\u0131m durmadan bir \u015feylerle me\u015fgul olurdu, anneci\u011fim de ev i\u015fleriyle u\u011fra\u015f\u0131rd\u0131; bana bir \u015fey \u00f6\u011fretmezlerdi, ben de mutluydum bu y\u00fczden. O zamanlar, sabah erkenden ya g\u00f6le, ya koruya, ya harman yerine, ya ekin yerine ko\u015fard\u0131m \u2013 hi\u00e7bir \u015feye ihtiya\u00e7 duymadan g\u00fcne\u015f \u0131s\u0131t\u0131rd\u0131, nereye oldu\u011funu bilmeden k\u00f6yden uzaklara ko\u015far, \u00e7al\u0131lara tak\u0131l\u0131r, elbisemi y\u0131rtard\u0131m \u2013 daha sonra evde azarlarlard\u0131, ama ald\u0131rmazd\u0131m.\n\nB\u00fct\u00fcn hayat\u0131m\u0131 k\u00f6yden \u00e7\u0131kmadan ve tek bir yerde ya\u015fayarak ge\u00e7irecek bile olsam, mutlu olacakm\u0131\u015f\u0131m gibi gelirdi bana. Buna ra\u011fmen daha \u00e7ocuk ya\u015fta do\u011fdu\u011fum yerleri terk etmek zorunda kald\u0131m. Petersburg'a ta\u015f\u0131nd\u0131\u011f\u0131m\u0131z zaman daha on iki ya\u015f\u0131ndayd\u0131m. Ah, kederle topland\u0131\u011f\u0131m\u0131z \u00e2n\u0131 nas\u0131l da h\u00fcz\u00fcnle hat\u0131rl\u0131yorum! Herkesle, sevdiklerimle vedala\u015f\u0131rken nas\u0131l da a\u011flam\u0131\u015ft\u0131m. Babac\u0131\u011f\u0131m\u0131n boynuna at\u0131ld\u0131\u011f\u0131m\u0131 ve g\u00f6zya\u015flar\u0131 i\u00e7inde k\u00f6yde biraz daha kalabilmek i\u00e7in yalvard\u0131\u011f\u0131m\u0131 hat\u0131rl\u0131yorum. Babac\u0131\u011f\u0131m bana ba\u011f\u0131rm\u0131\u015f, annem de a\u011flam\u0131\u015ft\u0131; yapacak bir \u015fey olmad\u0131\u011f\u0131n\u0131, i\u015flerin b\u00f6yle gerektirdi\u011fini s\u00f6yl\u00fcyorlard\u0131. Ya\u015fl\u0131 Prens P. \u00f6lm\u00fc\u015ft\u00fc. Miras\u00e7\u0131lar\u0131 babam\u0131 i\u015ften \u00e7\u0131karm\u0131\u015ft\u0131. Babam\u0131n Petersburg'daki \u00f6zel yat\u0131r\u0131mlarda paras\u0131 vard\u0131. Durumumuzu d\u00fczeltme umuduyla, burada kalmay\u0131 gereksiz say\u0131yordu. B\u00fct\u00fcn bunlar\u0131 daha sonra annemden \u00f6\u011frendim. Burada Petersburg k\u0131y\u0131s\u0131na yerle\u015ftik ve babam \u00f6l\u00fcnceye dek hep ayn\u0131 yerde ya\u015fad\u0131k.\n\nYeni hayata al\u0131\u015fmak benim i\u00e7in \u00e7ok zor oldu! Petersburg'a sonbaharda gelmi\u015ftik. K\u00f6y\u00fc terk etti\u011fimiz s\u0131rada, hava ayd\u0131nl\u0131k, s\u0131cak, a\u00e7\u0131kt\u0131; k\u00f6y i\u015fleri bitmi\u015fti; harman yerlerinde dev tah\u0131l tepeleri y\u00fckselmi\u015fti ve \u00e7\u0131\u011fl\u0131k \u00e7\u0131\u011fl\u0131\u011fa ku\u015flar konup kalk\u0131yordu; her \u015fey ayd\u0131nl\u0131k ve ne\u015feliydi, ama burada, \u015fehre girdi\u011fimiz s\u0131rada ya\u011fmur, kasvetli sonbahar k\u0131ra\u011f\u0131s\u0131, k\u00f6t\u00fc bir hava, c\u0131v\u0131k \u00e7amur ve yeni, yabanc\u0131, ho\u015f kar\u015f\u0131lamayan, ho\u015fnutsuz, \u00f6fkeli y\u00fczlerden olu\u015fan bir kalabal\u0131k vard\u0131. Bir \u015fekilde yerle\u015ftik. Her \u015feye nas\u0131l tela\u015f etti\u011fimizi, her \u015feyi yapmaya \u00e7al\u0131\u015f\u0131p yeni bir ev sahibesi buldu\u011fumuzu hat\u0131rl\u0131yorum. Babam evde olmuyordu, anneme bir dakika huzur yoktu; beni iyiden iyiye unutmu\u015flard\u0131. Yeni evimizdeki ilk geceden sonra sabahlar\u0131 kalkmak h\u00fcz\u00fcn verdi bana. Pencerelerimiz sar\u0131 bir duvara a\u00e7\u0131l\u0131yordu. Caddede s\u00fcrekli \u00e7amur vard\u0131. Gelip ge\u00e7en azd\u0131 ve hepsi de kal\u0131n kal\u0131n giyinmi\u015fti, hepsi de \u00fc\u015f\u00fcyordu.\n\nEvimizse b\u00fct\u00fcn g\u00fcn a\u015f\u0131r\u0131 bo\u011fucu ve s\u0131k\u0131c\u0131 oluyordu. Akrabam\u0131z da, yak\u0131n tan\u0131d\u0131klar\u0131m\u0131z da yoktu neredeyse. Anna Fedorovna'yla babam tart\u0131\u015fm\u0131\u015flard\u0131. (Babam\u0131n ona biraz borcu vard\u0131.) Bize s\u0131k s\u0131k i\u015fyerinden insanlar geliyordu. Tart\u0131\u015f\u0131yor, g\u00fcr\u00fclt\u00fc yap\u0131yor, ba\u011f\u0131r\u0131yorlard\u0131. Her ziyaretten sonra babam ho\u015fnutsuz, \u00f6fkeli bir hal al\u0131yordu; bazen bir saat boyunca surat\u0131n\u0131 as\u0131p odan\u0131n i\u00e7inde bir k\u00f6\u015feden di\u011ferine dolan\u0131yor ve kimseyle tek kelime konu\u015fmuyordu. Annem onunla o zaman konu\u015fmaya cesaret edemiyor ve susuyordu. Bir kitap al\u0131p bir k\u00f6\u015feye oturuyordum; uslu, sessiz duruyor, k\u0131p\u0131rdamaya cesaret edemiyordum.\n\nPetersburg'a geli\u015fimizden \u00fc\u00e7 ay sonra beni yat\u0131l\u0131 okula g\u00f6nderdiler. \u00d6nceleri yabanc\u0131 insanlar\u0131n aras\u0131nda ne kadar kederlendim! Her \u015fey \u00e7ok kat\u0131, sevimsiz g\u00f6r\u00fcn\u00fcyordu; m\u00fcrebbiyeler ba\u011f\u0131r\u0131yordu, k\u0131zlar as\u0131k suratl\u0131yd\u0131, ben de yabaniydim. Bo\u011fucuydu, \u00e7ok bo\u011fucuydu! Her \u015fey i\u00e7in belirlenmi\u015f saatler, ortak masa, s\u0131k\u0131c\u0131 \u00f6\u011fretmenler... b\u00fct\u00fcn bunlar beni \u00f6nce zorlad\u0131, y\u0131pratt\u0131. Orada uyuyam\u0131yordum. B\u00fct\u00fcn gece a\u011fl\u0131yordum; uzun, s\u0131k\u0131c\u0131, so\u011fuk gecelerde. Geceleri herkes dersleri tekrarlar ya da \u00e7al\u0131\u015f\u0131rd\u0131; ben de tek ba\u015f\u0131ma s\u00f6zl\u00fc ya da yaz\u0131l\u0131 derslere haz\u0131rlan\u0131yor, f\u0131s\u0131ldamaya bile cesaret edemiyordum, hep evimizi, babam\u0131, annemi, ya\u015fl\u0131 dad\u0131m\u0131, dad\u0131m\u0131n masallar\u0131n\u0131 d\u00fc\u015f\u00fcn\u00fcyordum... Ah, nas\u0131l da \u00fcz\u00fcl\u00fcyordum! Evdeki bibloyu, ho\u015fnutsuzlukla hat\u0131rl\u0131yor insan. D\u00fc\u015f\u00fcn\u00fcyor, d\u00fc\u015f\u00fcn\u00fcyor: Ne g\u00fczel olurdu \u015fimdi evde olmak! K\u00fc\u00e7\u00fck odam\u0131zda otururdum, semaverin ba\u015f\u0131nda, bizimkilerle birlikte; \u00e7ok s\u0131cak, g\u00fczel, tan\u0131d\u0131k bir yer olurdu. Sanki anneni s\u0131k\u0131 s\u0131k\u0131, ate\u015fli ate\u015fli kucaklam\u0131\u015fs\u0131n gibi! D\u00fc\u015f\u00fcn\u00fcp durursun, kederden sessizce a\u011flars\u0131n, g\u00f6zya\u015flar\u0131 g\u00f6\u011fs\u00fcne damlar ve yaz\u0131l\u0131 dersler akl\u0131ndan u\u00e7up gider. Yar\u0131nki derse haz\u0131rlanamazs\u0131n; b\u00fct\u00fcn gece r\u00fcyanda \u00f6\u011fretmeni, madam\u0131, k\u0131zlar\u0131 g\u00f6r\u00fcrs\u00fcn; b\u00fct\u00fcn gece r\u00fcyanda dersleri tekrarlars\u0131n, ama ertesi g\u00fcn hi\u00e7bir \u015fey bilemezsin. Bana dizlerimin \u00fcst\u00fcne \u00e7\u00f6kme cezas\u0131 ve bir kap yemek verirlerdi. O kadar ne\u015fesiz, b\u0131kk\u0131nd\u0131m ki. Ba\u015flang\u0131\u00e7ta b\u00fct\u00fcn k\u0131zlar bana g\u00fcl\u00fcyor, benimle alay ediyor, beni d\u00f6v\u00fcyordu, ben derslerde konu\u015funca \u00e7imdik atarlard\u0131, yemek ya da \u00e7ay i\u00e7in s\u0131raya girdi\u011fimiz zaman beni yok yere m\u00fcrebbiyeye \u015fik\u00e2yet ederlerdi. Cumartesi ak\u015famlar\u0131 dad\u0131m\u0131n benim i\u00e7in geldi\u011fi anlar cennet gibi olurdu. Bir mutluluk patlamas\u0131yla ya\u015fl\u0131 kad\u0131n\u0131 kucaklard\u0131m. Beni giydirir, sarar, yolda pe\u015fimden ko\u015ftururdu, bense ona s\u00fcrekli gevezelik edip bir \u015feyler anlat\u0131rd\u0131m. Eve ne\u015feyle, mutlulukla gider, bizimkilere sanki on y\u0131l ayr\u0131l\u0131k ya\u015fam\u0131\u015f\u0131z gibi s\u0131k\u0131ca sar\u0131l\u0131rd\u0131m. Yorumlar, sohbetler, hik\u00e2yeler ba\u015flard\u0131; herkesle merhabala\u015f\u0131r, g\u00fcl\u00fc\u015f\u00fcr, kahkahalar atar, ko\u015fturur, hoplar s\u0131\u00e7rard\u0131m. Babac\u0131\u011f\u0131m dersler, \u00f6\u011fretmenlerimiz, Frans\u0131zca hakk\u0131nda, L'Homond9 grameri hakk\u0131nda ciddi konu\u015fmalar yapard\u0131 ve hepimiz \u00e7ok ne\u015feli, \u00e7ok mutlu olurduk. \u015eimdi bile o dakikalar\u0131 ne\u015feyle hat\u0131rl\u0131yorum. Bir \u015feyler \u00f6\u011frenmeye ve babac\u0131\u011f\u0131m\u0131 memnun etmeye var g\u00fcc\u00fcmle \u00e7al\u0131\u015f\u0131rd\u0131m. Kendini sonuna kadar bana adad\u0131\u011f\u0131n\u0131 g\u00f6r\u00fcr, kendim de Tanr\u0131 bilir nas\u0131l \u00e7abalard\u0131m. Her g\u00fcn gitgide daha karamsar, ho\u015fnutsuz, \u00f6fkeli oluyordu; karakteri iyice bozulmu\u015ftu; i\u015fler y\u00fcr\u00fcm\u00fcyor, bor\u00e7lar \u00f6denmiyordu. Anneci\u011fim de babam\u0131 \u00f6fkelendirmemek i\u00e7in a\u011flamaya, bir \u015fey s\u00f6ylemeye korkar olmu\u015ftu; hastalanm\u0131\u015ft\u0131; iyice k\u00f6t\u00fcle\u015fmi\u015f ve k\u00f6t\u00fc bir \u015fekilde \u00f6ks\u00fcrmeye ba\u015flam\u0131\u015ft\u0131. Yat\u0131l\u0131 okuldan gelince kar\u015f\u0131mda hep h\u00fcz\u00fcnl\u00fc y\u00fczler buluyordum; annem sessizce a\u011fl\u0131yor, babam \u00f6fkeleniyordu. Azarlar, sitemler ba\u015fl\u0131yordu. Babam onu hi\u00e7 mutlu etmedi\u011fimi, hi\u00e7 teselli etmedi\u011fimi s\u00f6ylemeye ba\u015fl\u0131yordu; benim y\u00fcz\u00fcmden her \u015feylerini kaybettiklerini, ama benim h\u00e2l\u00e2 Frans\u0131zca konu\u015fmaya ba\u015flamam\u0131\u015f oldu\u011fumu s\u00f6yl\u00fcyordu; k\u0131sacas\u0131, b\u00fct\u00fcn ba\u015far\u0131s\u0131zl\u0131klar, b\u00fct\u00fcn mutsuzluklar, hepsi, hepsi benim ve annemin \u00fczerine y\u0131k\u0131l\u0131yordu. Ama nas\u0131l i\u015fkence edebiliyordu zavall\u0131 anneme? Ona bakarken kalbim par\u00e7alan\u0131rd\u0131; yanaklar\u0131 sarkm\u0131\u015f, g\u00f6zleri \u00e7\u00f6km\u00fc\u015ft\u00fc, y\u00fcz\u00fcnde veremli bir renk vard\u0131. Benim pay\u0131ma hepsinden daha fazlas\u0131 d\u00fc\u015f\u00fcyordu. Hep birtak\u0131m ufak tefek \u015feylerden ba\u015fl\u0131yor, sonra i\u015f Tanr\u0131 bilir nereye kadar gidiyordu; hatta genellikle neden bahsedildi\u011fini anlam\u0131yordum bile. Nelere k\u00f6p\u00fcrm\u00fcyordu ki!.. Frans\u0131zcaya, benim b\u00fcy\u00fck bir aptal olu\u015fuma, yat\u0131l\u0131 okulumuzun sahibinin ilgisiz, aptal bir kad\u0131n olmas\u0131na; benim \u015ferefimizi hi\u00e7 d\u00fc\u015f\u00fcnm\u00fcyor olmama; babam\u0131n o vakte dek bir i\u015f bulamamas\u0131na ve L'Homond gramerinin i\u011fren\u00e7 bir gramer olmas\u0131na, Zapolski gramerinin ise10 \u00e7ok daha iyi olmas\u0131na; bana bo\u015f yere bir s\u00fcr\u00fc para harcam\u0131\u015f olmas\u0131na; benim duygusuz, ta\u015f biri olmama... K\u0131sacas\u0131, zavall\u0131 ben var g\u00fcc\u00fcmle \u00e7al\u0131\u015f\u0131r, _dialogue_ 'lar\u013111 ve kelimeleri ezberlerdim, ama her \u015feyde su\u00e7lu bendim, her \u015feyin hesab\u0131n\u0131 verirdim! B\u00fct\u00fcn bunlar\u0131n sebebi de babam\u0131n beni sevmemesi de\u011fildi; benim i\u00e7in, annem i\u00e7in kimseyi dinlemezdi. Ama ne yaz\u0131k ki karakteri b\u00f6yleydi.\n\nKayg\u0131lar, n\u00f6betler, ba\u015far\u0131s\u0131zl\u0131klar zavall\u0131 babac\u0131\u011f\u0131ma a\u015f\u0131r\u0131 i\u015fkence ediyordu: G\u00fcvensiz, huysuz olmu\u015ftu; s\u0131k s\u0131k umutsuzlu\u011fa kap\u0131l\u0131yordu, sa\u011fl\u0131\u011f\u0131n\u0131 ihmal etmeye ba\u015flam\u0131\u015ft\u0131, \u00fc\u015f\u00fctt\u00fc ve birden hastaland\u0131, k\u0131sa s\u00fcre ac\u0131 \u00e7ekti ve \u00f6yle birdenbire, \u00f6yle k\u0131sa s\u00fcrede vefat etti ki biz birka\u00e7 g\u00fcn bu darbeden kendimize gelemedik. Anneci\u011fim donup kald\u0131; hatta ben onun akl\u0131ndan korktum. Babac\u0131\u011f\u0131m \u00f6l\u00fcr \u00f6lmez, alacakl\u0131lar yerden biter gibi geldi, s\u00fcr\u00fcyle \u00fc\u015f\u00fc\u015ft\u00fcler. Elimizde ne varsa, hepsini verdik. Petersburg taraf\u0131ndaki, babam\u0131n biz Petersburg'a geldikten alt\u0131 ay sonra ald\u0131\u011f\u0131 k\u00fc\u00e7\u00fck evimiz de sat\u0131ld\u0131. Kalanlar nas\u0131l payla\u015f\u0131ld\u0131 bilmiyorum, ama biz yataks\u0131z, bar\u0131naks\u0131z, yiyeceksiz kald\u0131k. Annem i\u00e7ten i\u00e7e hastal\u0131kla m\u00fccadele ediyordu, kendimizi ge\u00e7indiremiyorduk, ya\u015famak imk\u00e2ns\u0131zd\u0131, \u00f6l\u00fcme s\u00fcr\u00fckleniyorduk. On d\u00f6rt ya\u015f\u0131ma daha yeni girmi\u015ftim. \u0130\u015fte o s\u0131rada Anna Fedorovna bizi ziyaret etti. Bir malikane sahibi oldu\u011funu ve bizimle akraba oldu\u011funu s\u00f6yl\u00fcyordu hep. Annem de bizim uzaktan akraba oldu\u011fumuzu do\u011frulad\u0131. Babam hayattayken o bize asla gelmemi\u015fti. G\u00f6zya\u015flar\u0131 i\u00e7inde geldi, bizi yan\u0131na almaktan b\u00fcy\u00fck mutluluk duyaca\u011f\u0131n\u0131 s\u00f6yledi; bizim kayb\u0131m\u0131z\u0131n, yoksul halimizin ac\u0131s\u0131n\u0131 payla\u015f\u0131yordu, o arada babam\u0131n su\u00e7lu oldu\u011funu s\u00f6yledi; gelirine g\u00f6re ya\u015famad\u0131\u011f\u0131n\u0131, fazla ileri gitti\u011fini ve art\u0131k kendi g\u00fcc\u00fcm\u00fcze g\u00f6re ya\u015famam\u0131z gerekti\u011fini s\u00f6yledi. Bizimle yine g\u00f6r\u00fc\u015fmek istedi\u011fini s\u00f6yledi, kar\u015f\u0131l\u0131kl\u0131 tats\u0131zl\u0131klar\u0131 unutmay\u0131 \u00f6nerdi; annem ona kar\u015f\u0131 asla herhangi bir k\u00f6t\u00fc his beslemedi\u011fini s\u00f6yleyince de, g\u00f6zya\u015flar\u0131na bo\u011fuldu, annemi kiliseye \u00e7a\u011f\u0131rd\u0131 ve g\u00fcvercini (babama b\u00f6yle diyordu) i\u00e7in din\u00ee t\u00f6ren yapt\u0131rd\u0131. Sonra da annemle g\u00f6rkemli bir \u015fekilde bar\u0131\u015ft\u0131.\n\nUzun giri\u015flerden ve uyar\u0131lardan sonra Anna Fedorovna, bizim yoksul durumumuzu, \u00f6ks\u00fczl\u00fc\u011f\u00fcm\u00fcz\u00fc, umutsuzlu\u011fumuzu, \u00e7aresizli\u011fimizi parlak renklerle tasvir ederek bizi onun yan\u0131na, kendi ifadesiyle, s\u0131\u011f\u0131nmaya davet etti. Anneci\u011fim te\u015fekk\u00fcr etti, ama uzun s\u00fcre karar veremedi; fakat yapacak bir \u015fey olmad\u0131\u011f\u0131ndan ve ba\u015fka t\u00fcrl\u00fc d\u00fczene girmek imk\u00e2ns\u0131z oldu\u011fundan, sonunda Anna Fedorovna'ya teklifini memnuniyetle kabul edece\u011fimizi s\u00f6yledi. Petersburg taraf\u0131ndan Vasilyevski Adas\u0131'na do\u011fru yola \u00e7\u0131kt\u0131\u011f\u0131m\u0131z sabah\u0131 \u00e7ok iyi hat\u0131rl\u0131yorum. Sonbahar sabah\u0131yd\u0131, ayd\u0131nl\u0131k, kuru, so\u011fuk bir sabaht\u0131. Annem a\u011fl\u0131yordu; ben \u00e7ok h\u00fcz\u00fcnl\u00fcyd\u00fcm; g\u00f6\u011fs\u00fcm s\u0131k\u0131\u015f\u0131yordu, ruhum tarifsiz, korkun\u00e7 bir kederle bo\u011fuluyordu... Zor bir zamand\u0131.\n\n# II\n\n\u00d6nceleri, biz, yani ben ve annem yeni yerimize al\u0131\u015famad\u0131k, ikimiz de biraz tedirgin, yabani olduk Anna Fedorovna'n\u0131n yan\u0131nda. Anna Fedorovna Alt\u0131nc\u0131 Hat'ta, kendi evinde ya\u015f\u0131yordu. Evde be\u015f oda vard\u0131. \u00dc\u00e7 tanesinde Anna Fedorovna ve onun yan\u0131nda yeti\u015fen kuzinim Sa\u015fa oturuyordu; Sa\u015fa annesi babas\u0131 olmayan, kimsesiz bir \u00e7ocuktu. Bir odada da biz kal\u0131yorduk ve son olarak, yan\u0131m\u0131zdaki, dip odada, Anna Fedorovna'n\u0131n kirac\u0131s\u0131 olan, Pokrovski adl\u0131 yoksul bir \u00f6\u011frenci kal\u0131yordu. Anna Fedorovna \u00e7ok iyi, bizim tahmin etti\u011fimizden daha zengin ya\u015f\u0131yordu; ama serveti bilmece gibiydi, ne i\u015fle u\u011fra\u015ft\u0131\u011f\u0131 da \u00f6yle. Hep kayg\u0131l\u0131yd\u0131, hep tela\u015fl\u0131yd\u0131, g\u00fcnde birka\u00e7 kez gidip geliyordu; ama ne yapt\u0131\u011f\u0131n\u0131, neyle u\u011fra\u015ft\u0131\u011f\u0131n\u0131 ve ne i\u00e7in kayg\u0131land\u0131\u011f\u0131n\u0131 asla tahmin edemiyordum. \u00c7ok say\u0131da ve \u00e7ok \u00e7e\u015fitli tan\u0131d\u0131klar\u0131 vard\u0131. Ona zaman zaman misafirler gelirdi ve hepsi de Tanr\u0131 bilir kimdi, hep bir i\u015f i\u00e7in gelir, k\u0131sac\u0131k kal\u0131p giderlerdi. Annem kap\u0131 \u00e7an\u0131 \u00e7alar \u00e7almaz beni odam\u0131za g\u00f6t\u00fcr\u00fcrd\u00fc. Anna Fedorovna bu y\u00fczden anneme \u00e7ok k\u0131zard\u0131 ve durmadan a\u015f\u0131r\u0131 kibirli oldu\u011fumuzu, kibirli olacak halde olmad\u0131\u011f\u0131m\u0131z\u0131, kibirlenecek ba\u015fka \u015feyler oldu\u011funu s\u00f6yler ve saatlerce susmazd\u0131. O zamanlar bu kibirlilik imalar\u0131n\u0131 anlamazd\u0131m; annemin Anna Fedorovna'n\u0131n yan\u0131nda ya\u015famay\u0131 neden istemedi\u011fini de ancak \u015fimdi anl\u0131yor ya da en az\u0131ndan tahmin edebiliyorum zaten. K\u00f6t\u00fc bir kad\u0131nd\u0131 Anna Fedorovna; bize s\u00fcrekli i\u015fkence ediyordu.\n\n\u015eu vakte dek onun bizi neden yan\u0131na davet etti\u011fi bir s\u0131r olarak kald\u0131. \u00d6nceleri bize kar\u015f\u0131 \u00e7ok kibard\u0131, sonra, bizim kesinlikle \u00e7aresiz oldu\u011fumuzu ve gidecek yerimiz olmad\u0131\u011f\u0131n\u0131 g\u00f6r\u00fcnce ger\u00e7ek karakterini a\u00e7\u0131k a\u00e7\u0131k sergiledi. Sonralar\u0131 bana kar\u015f\u0131 \u00e7ok kibar davrand\u0131, hatta kaba bir kibarl\u0131k sergiledi, ama ba\u015flang\u0131\u00e7ta ben de annemle birlikte katlan\u0131yordum. \u0130kide bir azarl\u0131yordu bizi; bir tek kendi iyiliklerinden bahsedip duruyordu. Yabanc\u0131lara bizi yoksul akrabalar\u0131 olarak, yard\u0131ma muhta\u00e7 dul ve yetim olarak tan\u0131t\u0131yor, merhametten, H\u0131ristiyanca sevgi y\u00fcz\u00fcnden yan\u0131na ald\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyordu. Masada yedi\u011fimiz her lokmay\u0131 say\u0131yordu; e\u011fer yemezsek ayn\u0131 hik\u00e2ye ba\u015fl\u0131yordu: Biz tenezz\u00fcl etmiyormu\u015fuz; sanmayaym\u0131\u015f\u0131z zenginli\u011fin huzur getirece\u011fini; en iyisi de bizdeymi\u015f zaten... \u0130kide bir babac\u0131\u011f\u0131ma hakaret ediyordu; ba\u015fkalar\u0131ndan daha iyi olmak istedi\u011fini, ama daha da k\u00f6t\u00fc oldu\u011funu; kar\u0131s\u0131yla k\u0131z\u0131n\u0131 yery\u00fcz\u00fcnde tek ba\u015f\u0131na b\u0131rakt\u0131\u011f\u0131n\u0131 ve hay\u0131rsever, H\u0131ristiyanl\u0131k ruhuna sahip, merhametli bir akraba bulunmam\u0131\u015f olsaym\u0131\u015f, Tanr\u0131 bilir ba\u015f\u0131m\u0131za neler gelece\u011fini, belki bir sokakta a\u00e7l\u0131ktan \u00e7\u00fcr\u00fcy\u00fcp gidece\u011fimizi s\u00f6ylerdi. Neler neler s\u00f6yl\u00fcyordu b\u00f6yle! Onu dinlemek ac\u0131dan \u00e7ok tiksinti veriyordu. Annem durmadan a\u011fl\u0131yordu; sa\u011fl\u0131\u011f\u0131 g\u00fcnden g\u00fcne k\u00f6t\u00fcle\u015fiyordu, belirgin bir \u015fekilde solmu\u015ftu, buna ra\u011fmen sabahtan ak\u015fama kadar \u00e7al\u0131\u015f\u0131yor, sipari\u015f i\u015fler yap\u0131yor, diki\u015f dikiyorduk. Anna Fedorovna bundan da pek ho\u015flanm\u0131yordu; ikide bir evde bir moda ma\u011fazas\u0131 istemedi\u011fini s\u00f6yl\u00fcyordu. Ama giyinmek laz\u0131md\u0131, \u00f6ng\u00f6r\u00fclmeyen harcamalara haz\u0131r olmak laz\u0131md\u0131, hep paraya sahip olmak laz\u0131md\u0131. Her f\u0131rsatta para biriktiriyor, zaman i\u00e7inde bir yere ta\u015f\u0131nabilece\u011fimizi umut ediyorduk. Fakat annem kalan sa\u011fl\u0131\u011f\u0131n\u0131 da \u00e7al\u0131\u015f\u0131rken kaybetti; ge\u00e7en her g\u00fcn g\u00fc\u00e7ten d\u00fc\u015ft\u00fc. Hastal\u0131k, tam bir kurt gibi, g\u00f6r\u00fcn\u00fcr bir \u015fekilde hayat\u0131n\u0131 kemiriyor ve mezara \u00e7ekiyordu onu. Her \u015feyi g\u00f6r\u00fcyor, her \u015feyi hissediyor, hep ac\u0131 \u00e7ekiyordum; b\u00fct\u00fcn bunlar g\u00f6zlerimin \u00f6n\u00fcnde oluyordu!\n\nG\u00fcnler birbirini koval\u0131yor ve her g\u00fcn bir \u00f6ncekine benziyordu. Sessiz sakin, sanki \u015fehirde de\u011filmi\u015f gibi ya\u015f\u0131yorduk. Anna Fedorovna, kendi iktidar\u0131ndan emin olduk\u00e7a, yava\u015f yava\u015f sustu. Fakat kimse asla ona kar\u015f\u0131 \u00e7\u0131kmay\u0131 d\u00fc\u015f\u00fcnm\u00fcyordu. Odam\u0131z, onunkinden yar\u0131m koridor uzaktayd\u0131, yan taraf\u0131m\u0131zda da, daha \u00f6nce s\u00f6yledi\u011fim gibi, Pokrovski oturuyordu. Sa\u015fa'ya Frans\u0131zca ve Almanca, tarih, co\u011frafya \u00f6\u011fretiyordu Anna Fedorovna'n\u0131n dedi\u011fi gibi, b\u00fct\u00fcn bilimleri \u00f6\u011fretiyordu ve buna kar\u015f\u0131l\u0131k ondan yer ve yemek al\u0131yordu; Sa\u015fa afacan ve haylaz olsa da, \u00e7ok anlay\u0131\u015fl\u0131 bir gen\u00e7 k\u0131z olmu\u015ftu; on \u00fc\u00e7 ya\u015f\u0131na basm\u0131\u015ft\u0131. Anna Fedorovna anneme, yat\u0131l\u0131 okulda bana \u00f6\u011fretmedikleri \u015feyleri \u00f6\u011frenmeye ba\u015flamam\u0131n k\u00f6t\u00fc olmayaca\u011f\u0131n\u0131 s\u00f6yledi. Annem bunu mutlulukla kabul etti ve ben b\u00fct\u00fcn bir y\u0131l Sa\u015fa'yla birlikte Pokrovski'den ders ald\u0131m.\n\nPokrovski yoksuldu, \u00e7ok yoksul bir delikanl\u0131yd\u0131; sa\u011fl\u0131\u011f\u0131 s\u00fcrekli ders vermesine izin vermiyordu ve bize s\u0131rf al\u0131\u015fkanl\u0131ktan \u00f6\u011frenci diyordu. M\u00fctevaz\u0131, sakin, sessiz ya\u015f\u0131yordu, odam\u0131zdan sesi hi\u00e7 duyulmazd\u0131. G\u00f6r\u00fcn\u00fc\u015f\u00fc \u00e7ok tuhaft\u0131; o kadar hantal y\u00fcr\u00fcr, o kadar hantal bir \u015fekilde vedala\u015f\u0131r, o kadar tuhaf konu\u015furdu ki \u00f6nceleri ona g\u00fclmeden bakamazd\u0131m. Sa\u015fa s\u00fcrekli ona yaramazl\u0131klar yapard\u0131, \u00f6zellikle de bize ders verdi\u011fi s\u0131rada. Ama onun ayr\u0131ca asabi bir karakteri vard\u0131, bir hi\u00e7 i\u00e7in bile olsa aniden \u00f6fkelenir, bize ba\u011f\u0131r\u0131r, bizden \u015fik\u00e2yet eder ve s\u0131k s\u0131k, dersi bitirmeden, \u00f6fkeyle odas\u0131na giderdi. B\u00fct\u00fcn g\u00fcn odas\u0131nda oturur kitap okurdu. Bir s\u00fcr\u00fc kitab\u0131 vard\u0131 ve hepsi de de\u011ferli, nadir kitaplard\u0131. Ba\u015fka bir yerde daha ders veriyordu, biraz \u00fccret al\u0131yordu, eline para ge\u00e7er ge\u00e7mez de hemen kitap almaya ko\u015fuyordu.\n\nZamanla onu daha iyi, daha yak\u0131ndan tan\u0131d\u0131m. \u0130yi kalpli, de\u011ferli bir insand\u0131, kar\u015f\u0131la\u015fm\u0131\u015f oldu\u011fum ki\u015filer aras\u0131nda en iyisiydi. Annem onu \u00e7ok \u00f6v\u00fcyordu. Sonra benim de en iyi dostum oldu \u2013 tabii, annemden sonra.\n\n\u00d6nceleri ben o kadar b\u00fcy\u00fck bir gen\u00e7 k\u0131z oldu\u011fum halde, Sa\u015fa'yla birlikte haylazl\u0131k ederdim ve o zamanlar biz saatlerce Pokrovski'yi nas\u0131l sinirlendirip \u00e7ileden \u00e7\u0131karaca\u011f\u0131m\u0131z\u0131 d\u00fc\u015f\u00fcnerek kafa yorard\u0131k. A\u015f\u0131r\u0131 g\u00fcl\u00fcn\u00e7 bir \u015fekilde \u00f6fkelenirdi, biz de bunu \u00e7ok e\u011flenceli bulurduk. (Bunu hat\u0131rlarken bile utan\u0131yorum.) Bir keresinde onu a\u011flayacak kadar \u00e7ok k\u0131zd\u0131rm\u0131\u015ft\u0131k ve onun, \"Hain \u00e7ocuklar,\" diye f\u0131s\u0131ldad\u0131\u011f\u0131n\u0131 a\u00e7\u0131k se\u00e7ik duydum. Birdenbire mahcup oldum; utanm\u0131\u015ft\u0131m, k\u0131r\u0131lm\u0131\u015ft\u0131m, ona \u00fcz\u00fclm\u00fc\u015ft\u00fcm. Kulaklar\u0131ma kadar k\u0131zard\u0131\u011f\u0131m\u0131 ve a\u011flayacak gibi bir halde ondan sakin olmas\u0131n\u0131 ve bizim aptalca maskaral\u0131klar\u0131m\u0131za g\u00fccenmemesini rica etmi\u015ftim, ama o kitab\u0131 kapam\u0131\u015f, dersi bitirmeden odas\u0131na gitmi\u015fti. B\u00fct\u00fcn g\u00fcn h\u0131\u00e7k\u0131ra h\u0131\u00e7k\u0131ra a\u011flad\u0131m. Biz \u00e7ocuklar\u0131n ac\u0131mas\u0131zl\u0131\u011f\u0131m\u0131zla onu a\u011flayacak hale getirdi\u011fimizi d\u00fc\u015f\u00fcnmek benim i\u00e7in katlan\u0131lmaz bir \u015fey olmu\u015ftu. Anla\u015f\u0131lan a\u011flamas\u0131n\u0131 bekliyorduk. Anla\u015f\u0131lan bunu istiyorduk; anla\u015f\u0131lan, onun sabr\u0131n\u0131 t\u00fcketmeyi ba\u015farm\u0131\u015ft\u0131k; anla\u015f\u0131lan, zorla onu, o talihsizi, yoksulu, sald\u0131rgan talihini hat\u0131rlamaya zorlam\u0131\u015ft\u0131k! B\u00fct\u00fcn gece s\u0131k\u0131nt\u0131dan, kederden, pi\u015fmanl\u0131ktan uyuyamad\u0131m. Pi\u015fmanl\u0131\u011f\u0131n ruhu rahatlatt\u0131\u011f\u0131 s\u00f6ylenir, oysa tam tersi do\u011frudur. Benim kendi ac\u0131ma ve \u00f6zsevgime ne oldu\u011funu bilmiyorum. Onun beni \u00e7ocuk sanmas\u0131n\u0131 istemiyordum. O s\u0131rada art\u0131k on be\u015fime basm\u0131\u015ft\u0131m.\n\nO g\u00fcnden sonra Pokrovski'nin benim hakk\u0131mdaki fikrini de\u011fi\u015ftirmesini birdenbire nas\u0131l sa\u011flayabilece\u011fime dair binlerce plan yapmaya zorlad\u0131m hayal g\u00fcc\u00fcm\u00fc. Ama birdenbire \u00fcrkek ve \u00e7ekingen olmu\u015ftum; o anki durumumda hi\u00e7bir \u015feye karar veremiyor ve sadece hayal etmekle (Ve Tanr\u0131 bilir neler hayal etmekle!) yetiniyordum. Sa\u015fa'yla birlikte haylazl\u0131k etmeyi b\u0131rakt\u0131m; Pokrovski de bize \u00f6fkelenmeyi kesti; ama benim gururum i\u00e7in bu yeterli de\u011fildi.\n\n\u015eimdi bir zamanlar kar\u015f\u0131la\u015fma f\u0131rsat\u0131 buldu\u011fum insanlar aras\u0131nda en tuhaf, en ilgin\u00e7 ve en zavall\u0131 insanla ilgili birka\u00e7 \u015fey s\u00f6yleyece\u011fim. Ondan \u015fimdi, tam da an\u0131lar\u0131m\u0131n bu k\u0131sm\u0131nda \u015fu noktaya dek hi\u00e7bir ilgi g\u00f6stermedi\u011fim ki\u015fiden bahsetmemin nedeni, Pokrovski'yle ilgili her \u015feyin benim i\u00e7in ans\u0131z\u0131n ilgi \u00e7ekici olmu\u015f olmas\u0131d\u0131r!\n\nEvimize bazen \u00fcst\u00fc ba\u015f\u0131 toprakl\u0131, kirli, k\u00fc\u00e7\u00fck, ak sa\u00e7l\u0131, hantal, b\u00f6n, k\u0131sacas\u0131 a\u015f\u0131r\u0131 tuhaf bir ihtiyar gelirdi. \u0130lk bak\u0131\u015fta onun bir \u015feyden, sanki kendi kendinden utand\u0131\u011f\u0131n\u0131 d\u00fc\u015f\u00fcnebilirdi insan. Bu y\u00fczden de hep biraz iki b\u00fckl\u00fcm y\u00fcr\u00fcrd\u00fc; \u00f6yle kas\u0131lmalar\u0131, yapmac\u0131k tav\u0131rlar\u0131 vard\u0131 ki hi\u00e7 yan\u0131lmadan onun akl\u0131n\u0131n yerinde olmad\u0131\u011f\u0131n\u0131 s\u00f6ylemek m\u00fcmk\u00fcnd\u00fc. Bazen bize gelirdi, caml\u0131 kap\u0131lar\u0131n a\u011fz\u0131nda durur ve eve girmeye cesaret edemezdi. Birimiz \u2013ben ya da Sa\u015fa, ya da ona iyili\u011fi dokunmu\u015f u\u015faklardan biri\u2013 yan\u0131ndan ge\u00e7ecek olsa hemen sallanmaya ba\u015flar, b\u00fcz\u00fcl\u00fcr, t\u00fcrl\u00fc i\u015faretler yapar ve belki de ancak ona do\u011fru ba\u015f selam\u0131 verilip \u00e7a\u011fr\u0131ld\u0131\u011f\u0131 zaman \u2013evde bir yabanc\u0131 olmad\u0131\u011f\u0131n\u0131 ve ona uygun oldu\u011fu zaman gelebilece\u011fini g\u00f6steren bir i\u015faret\u2013 ancak o zaman ihtiyar sessizce kap\u0131y\u0131 a\u00e7ar, mutlulukla g\u00fcl\u00fcmser, mutluluktan ellerini \u00e7\u0131rpar ve parmak ucunda y\u00fcr\u00fcyerek do\u011fruca Pokrovski'nin odas\u0131na y\u00f6nelirdi. Onun babas\u0131yd\u0131.\n\nDaha sonra bu yoksul ihtiyar\u0131n b\u00fct\u00fcn hik\u00e2yesini ayr\u0131nt\u0131l\u0131 olarak \u00f6\u011frendim. Bir zamanlar bir yerde memurmu\u015f, hi\u00e7bir yetene\u011fi yokmu\u015f ve memuriyette en alt seviyede, en \u00f6nemsiz mevkide \u00e7al\u0131\u015f\u0131yormu\u015f. \u0130lk kar\u0131s\u0131 (\u00f6\u011frenci Pokrovski'nin annesi) \u00f6l\u00fcnce, ikinci kez evlenmeyi d\u00fc\u015f\u00fcnm\u00fc\u015f ve bir k\u00fc\u00e7\u00fck burjuvayla evlenmi\u015f. Yeni kar\u0131s\u0131 evde her \u015feyi alt\u00fcst etmi\u015f; onun y\u00fcz\u00fcnden kimse ya\u015fayamaz hale gelmi\u015f; herkesi avcunun i\u00e7ine alm\u0131\u015f. \u00d6\u011frenci Pokrovski o s\u0131rada daha \u00e7ocukmu\u015f, on ya\u015f\u0131ndaym\u0131\u015f. \u00dcvey annesi ondan nefret ediyormu\u015f. Ama k\u00fc\u00e7\u00fck Pokrovski'nin talihi yaver gitmi\u015f. Memur Pokrovski'yi tan\u0131yan ve eskiden onun hamisi olan toprak sahibi B\u0131kov, \u00e7ocu\u011fu korumas\u0131na alm\u0131\u015f ve onu bir okula yerle\u015ftirmi\u015f. Onunla ilgilenmesinin sebebi merhume annesini tan\u0131mas\u0131ym\u0131\u015f, onu gen\u00e7 k\u0131zken evlendiren Anna Fedorovna'ym\u0131\u015f ve onu memur Pokrovski'yle evlendirmi\u015f. Anna Fedorovna'n\u0131n dostu ve yak\u0131n ahbab\u0131 olan Bay B\u0131kov, c\u00f6mertli\u011fini sergileyerek, geline be\u015f bin ruble \u00e7eyiz vermi\u015f. Bu paralar\u0131n nereye gitti\u011fi bilinmiyor. Bana b\u00fct\u00fcn bunlar\u0131 Anna Fedorovna anlatt\u0131; \u00f6\u011frenci Pokrovski'yse asla aile sorunlar\u0131ndan bahsetmezdi. Annesinin \u00e7ok iyi bir insan oldu\u011funu s\u00f6ylerler, bense neden b\u00f6yle ba\u015far\u0131s\u0131z bir evlilik yapt\u0131\u011f\u0131n\u0131, b\u00f6yle \u00f6nemsiz bir insanla nas\u0131l evlendi\u011fini anlam\u0131yorum... Gen\u00e7ken, evlendikten d\u00f6rt y\u0131l sonra \u00f6lm\u00fc\u015f.\n\n\u0130lkokuldan sonra gen\u00e7 Pokrovski bir liseye ve sonra da \u00fcniversiteye gitmi\u015f. Petersburg'a \u00e7ok s\u0131k gelen Bay B\u0131kov da onun hamili\u011fini b\u0131rakmam\u0131\u015f. Bozuk sa\u011fl\u0131\u011f\u0131 y\u00fcz\u00fcnden Pokrovski \u00fcniversitedeki \u00e7al\u0131\u015fmalar\u0131n\u0131 s\u00fcrd\u00fcrememi\u015f. Bay B\u0131kov da onu Anna Fedorovna'yla tan\u0131\u015ft\u0131rm\u0131\u015f, onu tavsiye etmi\u015f ve b\u00f6ylece gen\u00e7 Pokrovski Sa\u015fa'ya ne gerekiyorsa \u00f6\u011fretmek \u00fczere i\u015fe al\u0131nm\u0131\u015f.\n\nYa\u015fl\u0131 Pokrovski'yse, kar\u0131s\u0131n\u0131n ac\u0131mas\u0131zl\u0131\u011f\u0131n\u0131n ac\u0131s\u0131yla, kendini pis bir sefahate vermi\u015f ve s\u00fcrekli sarho\u015f gezmeye ba\u015flam\u0131\u015f. Kar\u0131s\u0131 onu d\u00f6v\u00fcyormu\u015f, mutfakta ya\u015famaya zorluyormu\u015f ve o kadar ileri gidiyormu\u015f ki sonunda dayak giyip k\u00f6t\u00fc muamele g\u00f6rmeye al\u0131\u015fm\u0131\u015f ve yak\u0131nmamaya ba\u015flam\u0131\u015f. Asl\u0131nda \u00e7ok ya\u015fl\u0131 bir adam de\u011fildi, ama k\u00f6t\u00fc e\u011filimleri y\u00fcz\u00fcnden neredeyse akl\u0131n\u0131 yitirmi\u015fti. \u0130nsanca soylu duygular\u0131n tek i\u015fareti o\u011fluna duydu\u011fu s\u0131n\u0131rs\u0131z sevgiydi. Gen\u00e7 Pokorovski'nin merhume annesine t\u0131pat\u0131p benzedi\u011fini s\u00f6yl\u00fcyorlard\u0131. Eski iyi kar\u0131s\u0131n\u0131n hat\u0131ralar\u0131 m\u0131yd\u0131 mahvolmu\u015f ihtiyar\u0131n kalbinde o\u011fluna kar\u015f\u0131 s\u0131n\u0131rs\u0131z bir sevgiye sebep olan \u015fey? \u0130htiyar t\u0131pk\u0131 o\u011flu gibi \u00e7ok konu\u015fmuyordu art\u0131k ve haftada iki kere d\u00fczenli olarak onu ziyarete geliyordu. Daha s\u0131k gelmeye cesaret edemiyordu, \u00e7\u00fcnk\u00fc gen\u00e7 Pokorovski babas\u0131n\u0131n ziyaretlerine katlanam\u0131yordu. B\u00fct\u00fcn kusurlar\u0131 i\u00e7inde, tart\u0131\u015fmas\u0131z olarak, ilki ve en \u00f6nemlisi babas\u0131na kar\u015f\u0131 sayg\u0131s\u0131zl\u0131\u011f\u0131yd\u0131. Fakat, ihtiyar da yery\u00fcz\u00fcn\u00fcn en katlan\u0131lmaz varl\u0131klar\u0131ndan biriydi. Birincisi, korkun\u00e7 merakl\u0131yd\u0131, ikincisi durmaks\u0131z\u0131n o\u011flunu en bo\u015f ve manas\u0131z konu\u015fma ve sorularla rahats\u0131z ederdi; son olarak, bazen sarho\u015f bir halde gelirdi. O\u011flu sefahatten, merakl\u0131l\u0131ktan ve ikide bir sa\u00e7malamaktan ihtiyar\u0131 biraz vazge\u00e7irdi, hatta sonunda i\u015f baban\u0131n o\u011flunu peygamber gibi her konuda dinlemesine ve a\u011fz\u0131n\u0131 o izin vermeden a\u00e7mamas\u0131na vard\u0131.\n\nYoksul ihtiyar Petyonka's\u0131na (o\u011fluna bu ismi takm\u0131\u015ft\u0131) hayranl\u0131k duyuyor ve onu seviyordu. Misafirli\u011fe geldi\u011fi zaman, hemen her zaman herhalde o\u011flunun onu nas\u0131l kar\u015f\u0131layaca\u011f\u0131n\u0131 bilmedi\u011fi i\u00e7in kayg\u0131l\u0131, \u00fcrkek bir hal tak\u0131n\u0131r, genellikle uzun s\u00fcre i\u00e7eri girmeye karar veremez, e\u011fer girerse bana yirmi dakika kadar sorard\u0131: Petyonka nas\u0131l? Sa\u011fl\u0131\u011f\u0131 yerinde mi? Nas\u0131l bir ruh halinde, daha \u00f6nemli bir \u015feyle ilgileniyor olmas\u0131n? Ne yap\u0131yor ger\u00e7ekten? Birtak\u0131m \u00f6nemli fikirleri mi yaz\u0131yor? Yeterince u\u011fra\u015f\u0131p sakinle\u015ftirince, ihtiyar sonunda girmeye karar verirdi ve sessiz sessiz, dikkatle kap\u0131y\u0131 \u00e7alar, \u00f6nce sadece kafas\u0131n\u0131 uzat\u0131r ve o\u011flunun \u00f6fkelenmeyip ba\u015f\u0131n\u0131 ondan \u00e7evirmedi\u011fini g\u00f6r\u00fcnce, sessizce i\u00e7eri girer, paltosunu, y\u0131rt\u0131k p\u0131rt\u0131k, delik, s\u00f6k\u00fck u\u00e7lar\u0131yla hi\u00e7 \u00e7\u0131karmad\u0131\u011f\u0131 \u015fapkas\u0131n\u0131 \u00e7\u0131kar\u0131rd\u0131 \u2013 hepsini kancaya asar, b\u00fct\u00fcn bunlar\u0131 usulca, ses \u00e7\u0131karmadan yapard\u0131; sonra dikkatle bir yerde sandalyeye oturur ve o\u011flundan g\u00f6zlerini ay\u0131rmazd\u0131, her hareketini takip eder, Petyonka's\u0131n\u0131n ruh halini tahmin etmeye \u00e7al\u0131\u015f\u0131rd\u0131. E\u011fer o\u011flunun can\u0131 s\u0131kk\u0131nsa, ihtiyar bunu fark eder, hemen yerinden kalk\u0131p, \"Neyse Petyonka, bir dakikal\u0131\u011f\u0131na gelmi\u015ftim. Uza\u011fa gidiyordum, ge\u00e7erken dinlenmek i\u00e7in u\u011frad\u0131m,\" derdi. Sonra sessizce, uysalca paltosunu, \u015fapkas\u0131n\u0131 al\u0131r, sessizce kap\u0131dan \u00e7\u0131kar ve ruhunda k\u00f6p\u00fcren ac\u0131y\u0131 bast\u0131rmak ve onu o\u011fluna belli etmemek i\u00e7in b\u00fcy\u00fck \u00e7aba harcayarak giderdi.\n\nAma o\u011flu, bazen, babas\u0131n\u0131 iyi kar\u015f\u0131lard\u0131 ve o zaman ihtiyar mutluluktan deliye d\u00f6nerdi. Ho\u015fnutluk belirirdi y\u00fcz\u00fcnde, jestlerinde, hareketlerinde. O\u011flu onunla konu\u015facak olursa, ihtiyar hep sandalyesinde biraz do\u011frulur ve sessizce, alttan alarak, neredeyse h\u00fcrmetle yan\u0131t verir ve hep \u00f6zenli, yani anla\u015f\u0131l\u0131r s\u00f6zler s\u00f6ylemeye \u00e7al\u0131\u015f\u0131rd\u0131. Ama dil yetene\u011fi yoktu onda; her seferinde kafas\u0131 kar\u0131\u015f\u0131r, ellerini nereye koyaca\u011f\u0131n\u0131, nas\u0131l hareket edece\u011fini bilemeden geveler, sonra da uzun s\u00fcre kendi kendine, d\u00fczeltmek ister gibi f\u0131s\u0131ldard\u0131. E\u011fer d\u00fczg\u00fcn bir yan\u0131t vermeyi ba\u015farabilirse, ihtiyar rahatlar, \u00fczerindeki yele\u011fi, kravat\u0131, ceketi d\u00fczeltir ve kendi kendine g\u00fcvenen bir havaya b\u00fcr\u00fcn\u00fcrd\u00fc. Bazen de o kadar canlan\u0131rd\u0131 ki sessizce sandalyesinden kalk\u0131p kitap dolu rafa kadar gidecek, eline birka\u00e7 kitap alacak ve hatta hangi kitap olursa olsun bir \u015feyler okuyacak kadar toplard\u0131 cesaretini. B\u00fct\u00fcn bunlar\u0131 yapmac\u0131k bir ald\u0131r\u0131\u015fs\u0131zl\u0131k ve so\u011fukkanl\u0131l\u0131k havas\u0131yla, sanki o\u011flunun kitaplar\u0131n\u0131 hep b\u00f6yle sahiplenebilirmi\u015f, sanki o\u011flunun \u015fefkatine ba\u011fl\u0131 de\u011filmi\u015f gibi yapard\u0131. Ama ben bir keresinde, Pokrovski'nin ondan kitaplara dokunmamas\u0131n\u0131 istemesi \u00fczerine ihtiyar\u0131n ne kadar \u00fcrkt\u00fc\u011f\u00fcn\u00fc g\u00f6rd\u00fcm. Ne yapaca\u011f\u0131n\u0131 bilemedi, tela\u015fland\u0131, kitab\u0131 ba\u015f a\u015fa\u011f\u0131 b\u0131rakt\u0131, sonra d\u00fczeltmeye \u00e7al\u0131\u015ft\u0131, \u00e7evirdi, kitab\u0131n kenar\u0131 d\u0131\u015far\u0131da kald\u0131, g\u00fcld\u00fc, k\u0131pk\u0131rm\u0131z\u0131 oldu ve su\u00e7unu nas\u0131l \u00f6rtece\u011fini bilemedi. Pokrovski verdi\u011fi \u00f6\u011f\u00fctlerle ihtiyar\u0131 k\u00f6t\u00fc al\u0131\u015fkanl\u0131klar\u0131ndan biraz vazge\u00e7irmi\u015fti ve pe\u015f pe\u015fe \u00fc\u00e7 kez d\u00fczg\u00fcn bir halde g\u00f6r\u00fcnce, \u00e7eyrek ruble, yar\u0131m ruble ya da daha \u00e7ok para veriyordu ona. Bazen ona \u00e7izme, kravat ya da yelek al\u0131rd\u0131. O zaman ihtiyar horoz gibi gururlan\u0131rd\u0131. Bazen bize de gelirdi. Bana ve Sa\u015fa'ya horoz\u015fekerleri, elmalar getiriyor ve bizimle hep Petyonka hakk\u0131nda konu\u015fuyordu. Bizden dersleri dikkatle dinlememizi istiyor, Petyonka'n\u0131n iyi bir o\u011ful, \u00f6rnek bir o\u011ful ve iki kat \u00e2lim bir o\u011ful oldu\u011funu s\u00f6yl\u00fcyordu. Bunlar\u0131 s\u00f6ylerken sol g\u00f6z\u00fcn\u00fc g\u00fcl\u00fcn\u00e7 bir \u015fekilde k\u0131rparak e\u011flenceli bir \u015fekilde y\u00fcz\u00fcn\u00fc e\u011fip b\u00fckerdi, g\u00fclmekten kendimizi alamay\u0131p b\u00fct\u00fcn i\u00e7tenli\u011fimizle ona g\u00fclerdik. Anneci\u011fim onu \u00e7ok severdi. Ama Anna Fedorovna'n\u0131n, kar\u015f\u0131s\u0131nda sudan sessiz, otlardan al\u00e7ak duran ihtiyar, ondan nefret ederdi.\n\nBir s\u00fcre sonra Pokrovski'den ders almay\u0131 b\u0131rakt\u0131m. Beni eskisi gibi \u00e7ocuk, asi bir gen\u00e7 k\u0131z say\u0131yor, Sa\u015fa'yla ayn\u0131 g\u00f6r\u00fcyordu. Bundan hi\u00e7 ho\u015flanm\u0131yordum, \u00e7\u00fcnk\u00fc var g\u00fcc\u00fcmle eski halimi saklamak i\u00e7in \u00e7abal\u0131yordum. Ama beni fark etmiyordu. Bu da beni gitgide yaral\u0131yordu. Pokrovski'yle ders d\u0131\u015f\u0131nda hemen hi\u00e7 konu\u015fmam\u0131\u015ft\u0131m, zaten konu\u015famazd\u0131m. K\u0131zar\u0131yordum, ne diyece\u011fimi \u015fa\u015f\u0131r\u0131yordum ve sonra bir k\u00f6\u015fede utan\u00e7tan a\u011fl\u0131yordum.\n\nE\u011fer yak\u0131nla\u015fmam\u0131za tuhaf bir olay yard\u0131mc\u0131 olmasa, b\u00fct\u00fcn bunlar nereye varacakt\u0131, bilemiyorum. Bir ak\u015fam, annem Anna Fedorovna'yla otururken, ben sessizce Pokrovski'nin odas\u0131na gittim. Onun evde olmad\u0131\u011f\u0131n\u0131 biliyordum ve do\u011frusu, onun odas\u0131na gitmek nereden akl\u0131ma gelmi\u015fti bilmiyorum. O zamana kadar o odaya hi\u00e7 bakmam\u0131\u015ft\u0131m, oysa bir y\u0131l\u0131 a\u015fk\u0131n zamand\u0131r yan yana ya\u015f\u0131yorduk. Bu kez kalbim \u00f6yle g\u00fc\u00e7l\u00fc, \u00f6yle g\u00fc\u00e7l\u00fc at\u0131yordu ki g\u00f6\u011fs\u00fcmden f\u0131rlayacak gibiydi. \u00c7evreme \u00f6zel bir merakla bak\u0131nd\u0131m. Pokrovski'nin odas\u0131 \u00e7ok yoksul d\u00f6\u015fenmi\u015fti; pek d\u00fczen yoktu. Duvarlarda kitap dolu be\u015f uzun raf vard\u0131. Masan\u0131n ve sandalyelerin \u00fcst\u00fc kitap doluydu. Kitap ve k\u00e2\u011f\u0131tla! Akl\u0131ma tuhaf bir fikir geldi ve o s\u0131rada i\u00e7imi tats\u0131z bir ho\u015fnutsuzluk duygusu kaplad\u0131. Benim dostlu\u011fum, benim seven kalbim onun i\u00e7in \u00f6nemsizmi\u015f gibi geldi. O e\u011fitimliydi, bense aptall\u0131kla yan\u0131yordum ve hi\u00e7bir \u015fey bilmiyordum, hi\u00e7bir \u015fey okumam\u0131\u015ft\u0131m, bir tek kitap bile... Kitaplarla dolup ta\u015fan uzun raflara k\u0131skan\u00e7l\u0131kla bakt\u0131m. Beni bir ho\u015fnutsuzluk, bir s\u0131k\u0131nt\u0131, bir \u00e7\u0131lg\u0131nl\u0131k kaplad\u0131. Onun b\u00fct\u00fcn kitaplar\u0131n\u0131 tek tek, olabildi\u011fince \u00e7abuk okumay\u0131 istedim ve buna karar verdim. Bilmiyorum, belki de onun bildi\u011fi her \u015feyi \u00f6\u011frenince, onun dostlu\u011funa lay\u0131k olaca\u011f\u0131ma inand\u0131m. \u0130lk rafa at\u0131ld\u0131m; hi\u00e7 d\u00fc\u015f\u00fcnmeden, hi\u00e7 duraksamadan elime ilk gelen tozlu, eski cildi ald\u0131m ve \u00e7ald\u0131\u011f\u0131m kitab\u0131 k\u0131zararak solarak heyecandan ve korkudan titreyerek odama g\u00f6t\u00fcrd\u00fcm, geceleyin, annem uyurken lamba \u0131\u015f\u0131\u011f\u0131 alt\u0131nda onu okumaya karar vermi\u015ftim.\n\nAma odam\u0131za gidince kitab\u0131 tela\u015fla a\u00e7\u0131p da eski, yar\u0131 yar\u0131ya par\u00e7alanm\u0131\u015f, her yerini kurtlar yemi\u015f Latince bir eserle kar\u015f\u0131la\u015f\u0131nca \u00e7ok \u00fcz\u00fcld\u00fcm. Hi\u00e7 vakit kaybetmeden geri d\u00f6nd\u00fcm. Tam kitab\u0131 rafa yerle\u015ftirmek isterken, koridorda bir g\u00fcr\u00fclt\u00fc oldu ve birinin ayak sesleri duyuldu. Acele ettim, tela\u015f ettim, ama kahrolas\u0131 kitap di\u011ferlerinin aras\u0131na \u00f6yle s\u0131k\u0131 yerle\u015ftirilmi\u015fti ki ben birini \u00e7ekince, di\u011ferleri serbest kal\u0131p \u00fcst \u00fcste y\u0131\u011f\u0131lm\u0131\u015f, eski arkada\u015flar\u0131na yer b\u0131rakmam\u0131\u015flard\u0131. Kitab\u0131 araya sokmaya g\u00fcc\u00fcm yetmedi. Fakat kitaplar\u0131 var g\u00fcc\u00fcmle yana ittim. Raf\u0131 tutan ve herhalde k\u0131r\u0131lmak i\u00e7in kasten o \u00e2n\u0131 bekleyen pasl\u0131 mente\u015fe k\u0131r\u0131ld\u0131. Raf\u0131n bir ucu a\u015fa\u011f\u0131ya indi. Kitaplar g\u00fcr\u00fclt\u00fcyle yere d\u00fc\u015ft\u00fc. Kap\u0131 a\u00e7\u0131ld\u0131 ve Pokrovski odaya girdi.\n\nHerhangi birinin onun e\u015fyalar\u0131n\u0131 kar\u0131\u015ft\u0131rmas\u0131na katlanamad\u0131\u011f\u0131n\u0131 belirtmek gerekiyor. Onun kitaplar\u0131na dokunmaya cesaret edenin vay haline! \u0130rili ufakl\u0131, t\u00fcrl\u00fc t\u00fcrl\u00fc bi\u00e7imdeki, t\u00fcrl\u00fc t\u00fcrl\u00fc b\u00fcy\u00fckl\u00fck ve incelikteki kitaplar raflardan yuvarland\u0131; masan\u0131n, sandalyelerin \u00fcst\u00fcne, b\u00fct\u00fcn odaya sa\u00e7\u0131ld\u0131. Ka\u00e7mak istedim ama art\u0131k \u00e7ok ge\u00e7ti. \"Mahvoldum,\" diye d\u00fc\u015f\u00fcnd\u00fcm, \"mahvoldum! Peri\u015fan oldum, bittim! On ya\u015f\u0131nda \u00e7ocuk gibi \u015f\u0131mard\u0131m, haylazl\u0131k ettim; aptal bir gen\u00e7 k\u0131z\u0131m ben! B\u00fcy\u00fck bir aptal\u0131m!!\" Pokrovski deh\u015fetli \u00f6fkelenmi\u015fti. \"\u0130yi yani, bir bu kalm\u0131\u015ft\u0131!\" diye ba\u011f\u0131rd\u0131. \"Yaramazl\u0131k etmeye utanm\u0131yorsunuz!.. Uslu duram\u0131yor musunuz siz?\" diyerek kitaplar\u0131 toplamak i\u00e7in at\u0131ld\u0131. Ona yard\u0131m etmeye \u00e7al\u0131\u015ft\u0131m. \"Gerek yok, gerek yok,\" diye ba\u011f\u0131rd\u0131. \"Sizin yapman\u0131z gereken \u015fey, istenmedi\u011finiz yere girmemek.\" Fakat, benim uysal hareketlerim y\u00fcz\u00fcnden biraz yumu\u015fayarak daha hafif bir sesle, k\u0131sa s\u00fcre \u00f6nceki e\u011fitmen sesiyle, \u00f6\u011fretmenlik hakk\u0131n\u0131 kullanarak devam etti: \"Ne zaman ya\u015f\u0131n\u0131za yara\u015f\u0131r davranacaks\u0131n\u0131z, ne zaman durup bir d\u00fc\u015f\u00fcneceksiniz? Yani ben size bak\u0131yorum, siz art\u0131k \u00e7ocuk de\u011filsiniz, k\u00fc\u00e7\u00fck k\u0131z de\u011filsiniz, art\u0131k on be\u015f ya\u015f\u0131na bast\u0131n\u0131z!\" Ve bu s\u0131rada, herhalde hakl\u0131 oldu\u011fundan, benim k\u00fc\u00e7\u00fck olmad\u0131\u011f\u0131mdan emin olmak i\u00e7in bana bakt\u0131 ve kulaklar\u0131na kadar k\u0131zard\u0131. Anlayamad\u0131m; kar\u015f\u0131s\u0131nda durmu\u015f g\u00f6zlerimi hayretle a\u00e7m\u0131\u015f bir halde ona bak\u0131yordum. Do\u011fruldu, mahcup bir ifadeyle yan\u0131ma geldi, g\u00fc\u00e7bela bir \u015feyler s\u00f6ylemeye \u00e7al\u0131\u015ft\u0131, sanki bir \u015fey y\u00fcz\u00fcnden \u00f6z\u00fcr dilemeye, belki de, benim b\u00fcy\u00fck bir k\u0131z oldu\u011fumu \u015fimdi fark etti\u011fi i\u00e7in \u00f6z\u00fcr dilemeye \u00e7al\u0131\u015ft\u0131. Sonunda anlad\u0131m. O s\u0131rada bana ne oldu\u011funu hat\u0131rlam\u0131yorum; \u015fa\u015f\u0131rd\u0131m, ne yapaca\u011f\u0131m\u0131 bilemedim, Pokrovski'den de \u00e7ok k\u0131zard\u0131m, y\u00fcz\u00fcm\u00fc ellerimle \u00f6rtt\u00fcm ve odadan ko\u015farak \u00e7\u0131kt\u0131m.\n\nNe yapabilece\u011fimi, utan\u00e7tan nereye saklanaca\u011f\u0131m\u0131 bilemiyordum. Ne olursa olsun, beni odas\u0131nda yakalam\u0131\u015ft\u0131! \u00dc\u00e7 g\u00fcn boyunca ona bakamad\u0131m. A\u011flayacak kadar k\u0131pk\u0131rm\u0131z\u0131 oluyordum. En tuhaf d\u00fc\u015f\u00fcnceler, karman \u00e7orman d\u00fc\u015f\u00fcnceler dolan\u0131yordu kafamda. Bunlar\u0131n biri, en delice olan\u0131, onun yan\u0131na gitmek, ona a\u00e7\u0131klamak, ona her \u015feyi itiraf etmek, a\u00e7\u0131k\u00e7a anlatmak ve onu aptal bir gen\u00e7 k\u0131z gibi de\u011fil, iyi niyetlerle hareket etti\u011fime inand\u0131rmakt\u0131. Az kals\u0131n gidiyordum, ama Tanr\u0131'ya \u015f\u00fck\u00fcr, cesaretimi toplayamad\u0131m. Ya yapsayd\u0131m, diye d\u00fc\u015f\u00fcn\u00fcyorum! \u015eimdi bile bunlar\u0131 hat\u0131rlamak i\u00e7imi s\u0131zlat\u0131yor.\n\nBirka\u00e7 g\u00fcn sonra annem birdenbire a\u011f\u0131r bir hastal\u0131\u011fa yakaland\u0131. \u0130ki g\u00fcn yataktan kalkamam\u0131\u015f ve \u00fc\u00e7\u00fcnc\u00fc g\u00fcn ate\u015fler i\u00e7inde k\u00e2bus g\u00f6rmeye ba\u015flam\u0131\u015ft\u0131. Bir gece anneme bakarak uyumam\u0131\u015ft\u0131m, yata\u011f\u0131n\u0131n yan\u0131nda oturuyor, ona su getirip belli saatlerde ilac\u0131n\u0131 veriyordum.\n\n\u0130kinci gece bitkin d\u00fc\u015ft\u00fcm. Zaman zaman uyku bast\u0131r\u0131yor, g\u00f6zlerim karar\u0131yor, ba\u015f\u0131m d\u00f6n\u00fcyordu ve her an yorgunluktan bay\u0131l\u0131r gibi oluyordum, annemin g\u00fc\u00e7s\u00fcz iniltileri beni uyand\u0131r\u0131yordu, irkiliyor, bir an uyan\u0131k duruyordum, sonra yine uyku bast\u0131r\u0131yordu. Ac\u0131 \u00e7ekiyordum. Bilmiyorum \u2013kendimi hat\u0131rlayam\u0131yorum\u2013 ama k\u00e2buslu uykuyla m\u00fccadele etmenin i\u015fkence dolu \u00e2n\u0131nda korkun\u00e7 bir r\u00fcya, deh\u015fet verici bir hayal belirdi \u00f6n\u00fcmde. Korkuyla uyand\u0131m. Oda karanl\u0131kt\u0131, lamba s\u00f6nm\u00fc\u015ft\u00fc, \u0131\u015f\u0131k huzmeleri birden b\u00fct\u00fcn oday\u0131 kaplad\u0131, duvarda belli belirsiz bir \u015fey yan\u0131p s\u00f6nd\u00fc, sonra t\u00fcmden kayboldu. Nedense deh\u015fete kap\u0131ld\u0131m, bir korku \u00e7\u00f6kt\u00fc \u00fczerime; hayal g\u00fcc\u00fcm korkun\u00e7 bir r\u00fcyayla canlanm\u0131\u015ft\u0131; kalbimi s\u0131k\u0131nt\u0131 sard\u0131... Sandalyeden s\u0131\u00e7ray\u0131p kalkt\u0131m ve ac\u0131 verici, korkun\u00e7 bir \u015fekilde i\u00e7 karart\u0131c\u0131 bir duygu y\u00fcz\u00fcnden kendimi tutamay\u0131p \u00e7\u0131\u011fl\u0131k att\u0131m. O s\u0131rada kap\u0131 araland\u0131 ve odam\u0131za Pokrovski girdi.\n\nSadece onun kollar\u0131nda kendime geldi\u011fimi hat\u0131rl\u0131yorum. Beni \u00f6zenle koltu\u011fa oturttu, bir bardak su verdi ve sorular sordu. Ne yan\u0131t verdi\u011fimi hat\u0131rlam\u0131yorum. \"Siz hastas\u0131n\u0131z, siz \u00e7ok hastas\u0131n\u0131z,\" dedi elimi tutarak, \"ate\u015finiz var, kendinizi mahvediyorsunuz, sa\u011fl\u0131\u011f\u0131n\u0131z\u0131 d\u00fc\u015f\u00fcnm\u00fcyorsunuz; sakin olun, uzan\u0131n, uyuyun. Ben iki saat sonra uyand\u0131r\u0131r\u0131m sizi, dinlenin biraz... Yat\u0131n, yat\u0131n!\" diye devam etti benim kar\u015f\u0131 \u00e7\u0131kmak i\u00e7in bir \u015fey s\u00f6ylememe f\u0131rsat vermeden. Yorgunluk son g\u00fcc\u00fcm\u00fc de alm\u0131\u015ft\u0131; g\u00fc\u00e7s\u00fczl\u00fckten g\u00f6zlerim kapand\u0131. Koltu\u011fa uzand\u0131m, yar\u0131m saat uyumaya karar verdiysem de sabaha kadar uyudum. Pokrovski beni ancak anneme ila\u00e7 verme vakti geldi\u011fi zaman uyand\u0131rd\u0131.\n\nErtesi g\u00fcn, biraz dinlendikten sonra, bu kez uyumamaya kesin olarak karar vererek, annemin yata\u011f\u0131n\u0131n yan\u0131ndaki koltukta yeniden oturmaya haz\u0131rlan\u0131rken, Pokrovski saat on birde odam\u0131z\u0131n kap\u0131s\u0131n\u0131 \u00e7ald\u0131. Kap\u0131y\u0131 a\u00e7t\u0131m. \"Tek ba\u015f\u0131n\u0131za oturmaktan s\u0131k\u0131l\u0131rs\u0131n\u0131z,\" dedi bana, \"i\u015fte size kitap; al\u0131n; s\u0131k\u0131lmazs\u0131n\u0131z b\u00f6ylece.\" Kitab\u0131 ald\u0131m; hangi kitap oldu\u011funu hat\u0131rlam\u0131yorum; b\u00fct\u00fcn gece uyumad\u0131\u011f\u0131m halde o zaman ona hi\u00e7 bakmam\u0131\u015f da olabilirim. Tuhaf bir i\u00e7 heyecan uyumama izin vermedi; yerimde duram\u0131yordum; birka\u00e7 kez koltuktan kalkt\u0131m ve odada dolanmaya ba\u015flad\u0131m. B\u00fct\u00fcn varl\u0131\u011f\u0131m\u0131 bir i\u00e7 ho\u015fnutluk kaplam\u0131\u015ft\u0131. Pokrovski'nin ilgisinden \u00e7ok mutlu olmu\u015ftum. Onun benim i\u00e7in huzursuz olup kayg\u0131lanmas\u0131ndan gurur duymu\u015ftum. B\u00fct\u00fcn gece d\u00fc\u015f\u00fcn\u00fcp hayaller kurdum. Pokrovski bir daha gelmedi; gelmeyece\u011fini biliyordum ve bir sonraki geceyi merak ediyordum.\n\nErtesi gece, evde herkes yatt\u0131ktan sonra, Pokrovski kap\u0131s\u0131n\u0131 a\u00e7t\u0131 ve odan\u0131n e\u015fi\u011finde durarak benimle sohbet etmeye ba\u015flad\u0131. O s\u0131rada kar\u015f\u0131l\u0131kl\u0131 neler konu\u015ftu\u011fumuzun bir kelimesini bile hat\u0131rlam\u0131yorum; sadece \u00e7ekindi\u011fimi, kendimden utand\u0131\u011f\u0131m\u0131, s\u0131k\u0131ld\u0131\u011f\u0131m\u0131 ve sab\u0131rs\u0131zca konu\u015fman\u0131n sonunu bekledi\u011fimi, ama var g\u00fcc\u00fcmle bu konu\u015fmay\u0131 istedi\u011fimi, b\u00fct\u00fcn g\u00fcn bunu hayal etti\u011fimi, soru ve yan\u0131tlar\u0131m\u0131 haz\u0131rlad\u0131\u011f\u0131m\u0131 hat\u0131rl\u0131yorum \u015fimdi... O gece arkada\u015fl\u0131\u011f\u0131m\u0131z\u0131n ilk k\u0131v\u0131lc\u0131m\u0131 oldu. Annemin hastal\u0131\u011f\u0131 boyunca her gece birka\u00e7 saati beraber ge\u00e7irdik. S\u0131k\u0131lganl\u0131\u011f\u0131m\u0131 az \u00e7ok yenmeyi ba\u015fard\u0131m, oysa her konu\u015fmam\u0131zdan sonra kendimden daha da s\u0131k\u0131l\u0131yordum. Fakat, gizli bir mutlulukla ve gururlu bir ho\u015fnutlukla onun o paha bi\u00e7ilmez kitaplar\u0131n\u0131 benim y\u00fcz\u00fcmden unutmu\u015f oldu\u011funu g\u00f6r\u00fcyordum. Laf aras\u0131nda, \u015faka yollu, kitaplar\u0131n raftan d\u00fc\u015fmesi de konu\u015fuldu. Tuhaf bir and\u0131, ben _a\u015f\u0131r\u0131_ samimi ve a\u00e7\u0131k y\u00fcrekliydim; bir ate\u015f, tuhaf bir heyecan kaplam\u0131\u015ft\u0131 beni ve ona her \u015feyi itiraf ettim... e\u011fitim almak, bir \u015feyler \u00f6\u011frenmek istedi\u011fimi, beni bir gen\u00e7 k\u0131z, bir \u00e7ocuk saymas\u0131ndan s\u0131k\u0131ld\u0131\u011f\u0131m\u0131... Tekrar ediyorum, \u00e7ok tuhaf bir ruh hali i\u00e7indeydim; kalbim yumu\u015fam\u0131\u015ft\u0131, g\u00f6zlerimde g\u00f6zya\u015flar\u0131 birikmi\u015fti \u2013 ve hi\u00e7bir \u015fey saklamay\u0131p her \u015feyi anlatt\u0131m, her \u015feyi \u2013 ona duydu\u011fum dostlu\u011fu, onu sevme arzumu, onunla bir g\u00f6n\u00fcl birli\u011fi i\u00e7inde ya\u015fama, onda teselli bulmak, onu sakinle\u015ftirmek arzumu. Bana biraz tuhaf bir \u015fekilde, \u015fa\u015fk\u0131nl\u0131kla, hayretle bakt\u0131 ve bana tek kelime s\u00f6ylemedi. Birdenbire can\u0131m \u00e7ok yand\u0131, h\u00fcz\u00fcnlendim. Beni anlamam\u0131\u015f gibi, belki de benimle alay ediyormu\u015f gibi geldi. Birdenbire bir \u00e7ocuk gibi, h\u0131\u00e7k\u0131ra h\u0131\u00e7k\u0131ra a\u011flad\u0131m, kendimi tutamad\u0131m; sanki n\u00f6bet ge\u00e7iriyor gibiydim. Ellerimi tuttu, onlar\u0131 \u00f6pt\u00fc, beni g\u00f6\u011fs\u00fcne yaslad\u0131, konu\u015ftu, sakinle\u015ftirdi beni; derinden etkilenmi\u015fti; benimle ne konu\u015ftu\u011funu hat\u0131rlam\u0131yorum, ama ben de a\u011flad\u0131m, g\u00fcld\u00fcm ve yine a\u011flad\u0131m, k\u0131zard\u0131m, mutluluktan tek kelime bile s\u00f6yleyemedim. Fakat, heyecan\u0131ma ra\u011fmen, Pokrovski'de yine de belli bir \u00e7ekingenlik ve \u00fcrkeklik kald\u0131\u011f\u0131n\u0131 fark ettim. Herhalde, benim heyecan\u0131ma, benim co\u015fkuma, b\u00f6yle birdenbire gelen ate\u015fli, alevli arkada\u015fl\u0131\u011f\u0131ma hayret ediyordu. Belki de sadece merak ediyordu \u00f6nce; daha sonra karars\u0131zl\u0131\u011f\u0131 kayboldu ve benimle ayn\u0131 sade, a\u00e7\u0131k duyguyla kabul etti ona olan ba\u011fl\u0131l\u0131\u011f\u0131m\u0131, tatl\u0131 s\u00f6zlerimi, ilgimi ve b\u00fct\u00fcn bunlara ayn\u0131 ilgiyle, ayn\u0131 dostlukla ve ilgiyle, i\u00e7ten bir arkada\u015f, bir karde\u015f gibi yan\u0131t verdi. Kalbim \u00f6yle \u0131s\u0131nm\u0131\u015f, \u00f6yle \u015fenlenmi\u015fti ki!.. Hi\u00e7bir \u015feyi saklamad\u0131m, gizlemedim; her \u015feyi g\u00f6r\u00fcyordu ve g\u00fcnden g\u00fcne daha \u00e7ok ba\u011flan\u0131yordu bana.\n\nVe ger\u00e7ekten, hat\u0131rlam\u0131yorum, onunla bulu\u015fmalar\u0131m\u0131z\u0131n bu ac\u0131 dolu ve buna ra\u011fmen tatl\u0131 saatlerinde, geceleyin, gaz lambalar\u0131n\u0131n titrek \u0131\u015f\u0131\u011f\u0131nda ve zavall\u0131 hasta annemin yata\u011f\u0131n\u0131n yan\u0131ba\u015f\u0131nda neler konu\u015fuyorduk?.. Akla gelen, kalpten \u00e7\u0131kan, dile gelen her \u015feyi \u2013 ve neredeyse mutluyduk... Ah, hem kederli hem mutlu bir vakitti bu \u2013 hep beraberdik; ben de bunu hat\u0131rlarken hem kederli hem mutluyum. Hat\u0131ralar mutlu olsun, kederli olsun, hep ac\u0131 verir; en az\u0131ndan benim i\u00e7in \u00f6yle; ama bu ac\u0131 tatl\u0131 bir ac\u0131. Ve kalp a\u011f\u0131rla\u015ft\u0131\u011f\u0131, darald\u0131\u011f\u0131, s\u0131k\u0131ld\u0131\u011f\u0131, kederli oldu\u011fu zaman, o zaman hat\u0131ralar onu t\u0131pk\u0131 s\u0131cak bir g\u00fcn\u00fcn ard\u0131ndan gelen rutubetli bir gecede \u00e7iy damlalar\u0131n\u0131n zavall\u0131, kurumu\u015f, g\u00fcnd\u00fcz vakti s\u0131caktan kavrulmu\u015f \u00e7i\u00e7e\u011fi canland\u0131rmas\u0131 gibi ayd\u0131nlat\u0131p canland\u0131r\u0131r.\n\nAnnem iyile\u015fti, ama ben geceleri onun yata\u011f\u0131n\u0131n ba\u015f\u0131nda oturmaya devam ettim. Pokrovski bana s\u0131k s\u0131k kitap veriyordu; \u00f6nceleri uyumamak i\u00e7in okuyordum, sonra daha dikkatle, sonra da a\u00e7 gibi okumaya ba\u015flad\u0131m; \u00f6n\u00fcmde bir s\u00fcr\u00fc yeni \u015fey, daha \u00f6nceden bilmedi\u011fim, tan\u0131mad\u0131\u011f\u0131m \u015feyler belirmi\u015fti. Yeni d\u00fc\u015f\u00fcnceler, yeni izlenimler bir anda, g\u00fcr bir sel halinde kalbime h\u00fccum etti. Ve yeni izlenimler bana ne kadar \u00e7ok heyecan, ne kadar \u00e7ok mahcupluk ve \u00e7aba getirdiyse, b\u00fct\u00fcn ruhuma da o kadar tatl\u0131 geliyor, o kadar mutlulukla sars\u0131yorlard\u0131 beni. Bir anda, birdenbire, kalbime doluvermi\u015f, kalbime dinlenme f\u0131rsat\u0131 vermemi\u015flerdi. B\u00fct\u00fcn varl\u0131\u011f\u0131m\u0131 tuhaf bir kaos sarsmaya ba\u015flad\u0131. Ama bu ruhsal zorlama beni t\u00fcm\u00fcyle bozamazd\u0131, bozacak kadar g\u00fc\u00e7l\u00fc de\u011fildi. A\u015f\u0131r\u0131 hayalperesttim ve bu beni kurtard\u0131.\n\nAnnemin hastal\u0131\u011f\u0131 ge\u00e7ince, gece bulu\u015fmalar\u0131m\u0131z ve uzun sohbetlerimiz kesildi; bazen \u00e7o\u011fu bo\u015f ve \u00e7ok anlam ta\u015f\u0131mayan birka\u00e7 s\u00f6zc\u00fck al\u0131\u015fveri\u015fi yapabiliyorduk, ama ben hepsine kendi \u00f6nemini, kendine \u00f6zg\u00fc, imal\u0131 de\u011ferini veriyordum. Hayat\u0131m dopdoluydu, mutluydum, huzurluydum, sessiz bir mutluluk i\u00e7indeydim. B\u00f6yle birka\u00e7 hafta ge\u00e7ti...\n\nBir ara ya\u015fl\u0131 Pokrovski bize geldi. Uzun s\u00fcre bizimle gevezelik etti, her zamankinden farkl\u0131 olarak ne\u015feli, rahat, konu\u015fkand\u0131; g\u00fcl\u00fcyor, kendince \u015faka yap\u0131yordu ve sonunda heyecan\u0131n\u0131n s\u0131rr\u0131n\u0131 a\u00e7\u0131klad\u0131 ve bize Petyonka'n\u0131n do\u011fum g\u00fcn\u00fcn\u00fcn bir hafta sonra oldu\u011funu ve bu f\u0131rsatla o\u011fluna mutlaka gelece\u011fini bildirdi; yeni yele\u011fini giyecekti ve kar\u0131s\u0131 da ona yeni \u00e7izmeler almaya s\u00f6z vermi\u015fti. K\u0131sacas\u0131, ihtiyar fazlas\u0131yla mutluydu ve akl\u0131na gelen her \u015fey hakk\u0131nda gevezelik ediyordu.\n\nOnun do\u011fum g\u00fcn\u00fc! Bu do\u011fum g\u00fcn\u00fc bana gece g\u00fcnd\u00fcz huzur vermez oldu. Hemen Pokrovski'ye dostlu\u011fumuzu hat\u0131rlatmak ve ona bir \u015feyler hediye etmek i\u00e7in karar verdim. Ama ne? Sonunda ona kitap almak geldi akl\u0131ma. Onun Pu\u015fkin'in t\u00fcm eserlerinin12 son bask\u0131s\u0131na sahip olmay\u0131 istedi\u011fini biliyordum ve Pu\u015fkin sat\u0131n almaya karar verdim. El i\u015flerinden kazand\u0131\u011f\u0131m otuz ruble param vard\u0131. Bu paray\u0131 yeni elbiseme saklam\u0131\u015ft\u0131m. Hemen a\u015f\u00e7\u0131m\u0131z\u0131n, ya\u015fl\u0131 Matryona'n\u0131n yan\u0131na ko\u015fup b\u00fct\u00fcn Pu\u015fkin'lerin ne kadara mal oldu\u011funu sordum. Eyvah! Toplam on bir kitab\u0131n bedeli, ciltlenip buraya getirilmesi dahil, en az altm\u0131\u015f ruble olacakt\u0131. Nereden para bulmal\u0131? D\u00fc\u015f\u00fcn\u00fcp durdum ve ne yapaca\u011f\u0131m\u0131 bilemedim. Annemden rica etmek istemiyordum. Elbette, annem hemen yard\u0131m ederdi; ama o zaman evde herkes bizim hediyemizden haberdar olurdu; b\u00f6ylece bu hediye, Pokrovski' nin bir y\u0131ll\u0131k eme\u011fi kar\u015f\u0131l\u0131\u011f\u0131ndaki bir te\u015fekk\u00fcre d\u00f6n\u00fc\u015fecekti. Ben herkesten habersiz, tek ba\u015f\u0131ma hediye vermek istiyordum. Onun benim \u00fczerimdeki emekleri i\u00e7in de dostlu\u011fumun d\u0131\u015f\u0131nda bir \u015fey \u00f6demeden sonsuza dek bor\u00e7lu kalmak istiyordum. Sonunda, bu g\u00fc\u00e7l\u00fckten \u00e7\u0131kman\u0131n bir yolunu buldum.\n\nGostiniy Dvor'daki sahaflarda bir kitab\u0131n, ikinci el, biraz h\u0131rpalanm\u0131\u015f ve neredeyse yepyeni olarak bazen fiyat\u0131n\u0131n yar\u0131s\u0131na bulunabilece\u011fini biliyordum. Hemen Gostiniy Dvor'a gitmeye karar verdim. Gittim; ertesi g\u00fcn hem bizim hem Anna Fedorovna'n\u0131n bir \u015feye ihtiyac\u0131 oldu. Annemin sa\u011fl\u0131\u011f\u0131 iyi de\u011fildi, Anna Fedorovna o s\u0131rada \u00e7ok tembellik etti\u011fi i\u00e7in bu g\u00f6revi yerine getirmek bana d\u00fc\u015ft\u00fc ve ben Matryona'yla birlikte yola koyuldum.\n\n\u015eans\u0131ma, Pu\u015fkin'i, hem de \u00e7ok g\u00fczel bir ciltle k\u0131sa s\u00fcrede buldum. Pazarl\u0131k etmeye ba\u015flad\u0131m. \u00d6nce kitap\u00e7\u0131lardakinden bile daha y\u00fcksek fiyat verdiler; ama sonra, fakat epey bir \u00e7abayla, birka\u00e7 kez gidip gelerek t\u00fcccar\u0131 fiyat\u0131 d\u00fc\u015f\u00fcrmeye ve on g\u00fcm\u00fc\u015f rubleyle yetinmeye ikna ettim. \u00c7ok e\u011flenceli olmu\u015ftu pazarl\u0131k yapmak!.. Zavall\u0131 Matriyona ne yapt\u0131\u011f\u0131m\u0131 ve neden bu kadar \u00e7ok kitap almaya kalk\u0131\u015ft\u0131\u011f\u0131m\u0131 anlam\u0131yordu. Fakat felaket! Benim b\u00fct\u00fcn sermayem otuz rubleydi, t\u00fcccarsa daha ucuza vermeye asla yana\u015fm\u0131yordu. Sonunda rica etmeye, \u0131srar etmeye ba\u015flad\u0131m, sonunda yalvard\u0131m. Raz\u0131 oldu, ama sadece iki bu\u00e7uk ruble indi ve bu indirimi s\u0131rf benim hat\u0131r\u0131m i\u00e7in, b\u00f6yle g\u00fczel bir han\u0131mefendi oldu\u011fum i\u00e7in yapt\u0131\u011f\u0131n\u0131, ba\u015fkas\u0131 olsa asla yapmayaca\u011f\u0131n\u0131 s\u00f6yledi. \u0130ki bu\u00e7uk ruble eksik kald\u0131! S\u0131k\u0131nt\u0131dan a\u011flamak \u00fczereydim. Ama bu ac\u0131ma hi\u00e7 beklenmedik bir olay yard\u0131mc\u0131 oldu.\n\nBiraz \u00f6temde, kitap dolu bir ba\u015fka masan\u0131n ba\u015f\u0131nda, ya\u015fl\u0131 Pokrovski'yi g\u00f6rd\u00fcm. \u00c7evresine d\u00f6rt-be\u015f sahaf toplanm\u0131\u015ft\u0131; bir \u015feyler verip duruyor, bo\u011facak gibi s\u0131k\u0131\u015ft\u0131r\u0131yorlard\u0131. Her biri kendi mal\u0131n\u0131 teklif ediyordu, bir s\u00fcr\u00fc \u015fey teklif ediyorlar, o da bir s\u00fcr\u00fc \u015fey almak istiyordu! Zavall\u0131 ihtiyar ortalar\u0131nda dayak yemi\u015f gibi duruyor ve ona teklif edilen \u015feyi neden alaca\u011f\u0131n\u0131 bilemiyordu. Yan\u0131na gidip burada ne arad\u0131\u011f\u0131n\u0131 sordum. \u0130htiyar beni g\u00f6rd\u00fc\u011f\u00fcne \u00e7ok sevindi; beni tarifsiz seviyordu, belki Petyonka kadar seviyordu. \"Kitap alacakt\u0131m Varvara Alekseyevna,\" dedi bana, \"Petyonka'ya kitap al\u0131yorum. Onun do\u011fum g\u00fcn\u00fc yakla\u015f\u0131yor, o da kitap sever, o y\u00fczden kitap al\u0131yorum ona...\" \u0130htiyar hep g\u00fcl\u00fcn\u00e7 bir \u015fekilde a\u00e7\u0131klama yapard\u0131, ama \u015fimdi iki kat deh\u015fetli bir \u015fa\u015fk\u0131nl\u0131k i\u00e7indeydi. Neyin fiyat\u0131n\u0131 sorsa, bir g\u00fcm\u00fc\u015f ruble, iki ruble, \u00fc\u00e7 g\u00fcm\u00fc\u015f ruble; b\u00fcy\u00fck kitaplar\u0131n fiyat\u0131n\u0131 sormuyordu \u00fcstelik, onlar\u0131 k\u0131skan\u00e7l\u0131kla g\u00f6zden ge\u00e7iriyor, sayfalar\u0131n\u0131 kar\u0131\u015ft\u0131r\u0131yor, elinde d\u00f6nd\u00fcr\u00fcyor ve yine yerine b\u0131rak\u0131yordu. \"Hay\u0131r, hay\u0131r, bu pahal\u0131,\" diyordu al\u00e7ak sesle, \"belki \u015furada bir \u015fey vard\u0131r,\" diyerek incecik defterleri, nota defterlerini, y\u0131ll\u0131klar\u0131 ay\u0131klamaya ba\u015fl\u0131yordu; bunlar\u0131n hepsi \u00e7ok ucuzdu. \"Fakat neden bunlar\u0131 al\u0131yorsunuz ki,\" dedim ona, \"bunlar a\u015f\u0131r\u0131 gereksiz \u015feyler.\" \"Ah, hay\u0131r,\" dedi bana, \"hay\u0131r, baksan\u0131za \u015furada g\u00fczel kitap\u00e7\u0131klar var; \u00e7ok, \u00e7ok g\u00fczel kitap\u00e7\u0131klar var!\" Ve bu son s\u00f6zleri \u00f6yle ac\u0131kl\u0131 bir \u015fekilde uzatarak s\u00f6yledi ki bana iyi kitaplar daha pahal\u0131 oldu\u011fu i\u00e7in s\u0131k\u0131nt\u0131dan a\u011flamaya ba\u015flayacakm\u0131\u015f, solgun yana\u011f\u0131ndan k\u0131rm\u0131z\u0131 burnuna bir g\u00f6zya\u015f\u0131 damlas\u0131 yuvarlan\u0131yormu\u015f gibi geldi. Paras\u0131 olup olmad\u0131\u011f\u0131n\u0131 sordum. \"\u0130\u015fte bu,\" diye bir gazete k\u00e2\u011f\u0131d\u0131na sar\u0131lm\u0131\u015f paralar\u0131n\u0131 uzatt\u0131 bana zavall\u0131, \"i\u015fte yar\u0131m g\u00fcm\u00fc\u015f, yirmi kopek, yirmi bak\u0131r kopek.\" Hemen onu benim sahafa s\u00fcr\u00fckledim. \"\u0130\u015fte toplam on bir kitap, otuz iki bu\u00e7uk ruble tutuyor; bende otuz var; siz de iki bu\u00e7uk koyun, b\u00fct\u00fcn bu kitaplar\u0131 al\u0131p birlikte hediye edelim.\" \u0130htiyar sevin\u00e7ten kendini kaybetti, b\u00fct\u00fcn paralar\u0131n\u0131 d\u00f6kt\u00fc ve sahaf ortak k\u00fct\u00fcphanemizi ona y\u00fckledi. \u0130htiyar kitaplar\u0131n hepsini ceplerine koydu, iki koluna, koltuk altlar\u0131na doldurdu ve b\u00fct\u00fcn kitaplar\u0131 ertesi g\u00fcn gizlice bana getirme s\u00f6z\u00fc vererek al\u0131p eve g\u00f6t\u00fcrd\u00fc.\n\nErtesi g\u00fcn ihtiyar o\u011fluna geldi, bir saat her zamanki gibi onun yan\u0131nda oturdu, sonra bize gelip \u00e7ok komik bir esrarl\u0131 havayla yan\u0131ma oturdu. \u00d6nce g\u00fclerek, ellerini bir s\u0131r sahibi olman\u0131n gururlu ho\u015fnutlu\u011fuyla ovu\u015fturarak bana b\u00fct\u00fcn kitaplar\u0131 getirdi\u011fini ve hepsinin mutfakta, Matryona'n\u0131n himayesinde bir k\u00f6\u015fede durdu\u011funu bildirdi. Sonra sohbet do\u011fal olarak beklenen kutlamaya y\u00f6neldi; sonra ihtiyar bizim nas\u0131l hediye verece\u011fimizi uzun uzun anlatt\u0131, hatta bu konuya o kadar dald\u0131 ki o konu\u015ftuk\u00e7a dilinin alt\u0131nda bir \u015feyler oldu\u011funu, ama s\u00f6ylemeye cesaret edemedi\u011fini, hatta korktu\u011funu anlad\u0131m. Yine de susup bekledim. O vakte dek onun tuhaf a\u011f\u0131z hareketlerinden, s\u0131r\u0131tmalar\u0131ndan, sol g\u00f6z\u00fcyle g\u00f6z k\u0131rpmalar\u0131ndan kolayca okunan o gizli mutluluk, gizli ho\u015fnutluk ortadan kaybolmu\u015ftu. Gitgide daha sab\u0131rs\u0131z ve \u00fcz\u00fcnt\u00fcl\u00fc bir hal al\u0131yordu; sonunda kendini tutamad\u0131.\n\n\"Dinleyin,\" diye ba\u015flad\u0131 s\u00f6ze \u00fcrkerek, al\u00e7ak sesle, \"dinleyin Varvara Alekseyevna... Biliyor musunuz ne yapal\u0131m, Varvara Alekseyevna?..\" \u0130htiyar korkun\u00e7 bir bocalama i\u00e7indeydi. \"Bak\u0131n, onun do\u011fum g\u00fcn\u00fc s\u0131ras\u0131nda, bu on kitab\u0131 al\u0131n ve kendiniz hediye edin, yani kendi hediyeniz olarak, kendiniz verin; ben de on birinci cildi al\u0131r kendi hediyem olarak veririm, yani s\u0131rf benim hediyem olarak. Yani, anl\u0131yor musunuz \u2013 hem siz bir \u015fey vermi\u015f olacaks\u0131n\u0131z, hem de ben bir \u015fey vermi\u015f olaca\u011f\u0131m; ikimiz de bir \u015fey vermi\u015f olaca\u011f\u0131z.\" \u0130htiyar tela\u015flan\u0131p sustu. Ona bakt\u0131m; \u00fcrkek bir merakla benim konu\u015fmam\u0131 bekliyordu.\n\n\"Neden birlikte vermemizi istemiyorsunuz Zahar Petrovi\u00e7?\"\n\n\"Yani, Varvara Alekseyevna, bunun nedeni... yani, \u00e7\u00fcnk\u00fc...\"\n\n\u0130htiyar tam anlam\u0131yla bocalad\u0131, k\u0131pk\u0131rm\u0131z\u0131 oldu, ne diyece\u011fini bilemedi ve \u00f6ylece durdu.\n\n\"Bak\u0131n,\" diye a\u00e7\u0131klad\u0131 sonunda. \"Varvara Alekseyevna, ben, zaman zaman savurganl\u0131k yapar\u0131m... yani neredeyse s\u00fcrekli yapar\u0131m, hep savurganl\u0131k yapar\u0131m demek istiyorum.. \u0130yi olmad\u0131\u011f\u0131n\u0131 da biliyorum... yani, anl\u0131yor musunuz, d\u0131\u015far\u0131da so\u011fuk oluyor, bazen de birtak\u0131m tats\u0131zl\u0131klar ya da \u00fcz\u00fcc\u00fc bir \u015feyler oluyor, ben kendimi tutam\u0131yorum, elimdekiyle savurganl\u0131k yap\u0131yorum, bazen gereksiz yere i\u00e7iyorum. Petenka bundan hi\u00e7 ho\u015flanm\u0131yor. Anl\u0131yor musunuz, Varvara Alekseyevna, buna \u00f6fkeleniyor, bana hakaret ediyor ve bana birtak\u0131m ahlak dersleri veriyor. \u0130\u015fte o y\u00fczden ben de \u015fimdi ona kendi hediyemi vermek, d\u00fczeldi\u011fimi, d\u00fczg\u00fcn davrand\u0131\u011f\u0131m\u0131 g\u00f6stermek istiyorum. Ona kitap almak i\u00e7in para biriktirdim, uzun zamand\u0131r biriktiriyordum, \u00e7\u00fcnk\u00fc bende para olmaz pek, Petenka verirse olur. O da biliyor bunu. O y\u00fczden, \u015fimdi param\u0131 nas\u0131l kulland\u0131\u011f\u0131m\u0131 g\u00f6recek ve b\u00fct\u00fcn bunlar\u0131 s\u0131rf onun i\u00e7in yapt\u0131\u011f\u0131m\u0131 anlayacak.\"\n\n\u0130htiyara \u00e7ok \u00fcz\u00fcld\u00fcm. Biraz d\u00fc\u015f\u00fcnd\u00fcm. \u0130htiyar sab\u0131rs\u0131zl\u0131kla bak\u0131yordu bana. \"Dinleyin, Zahar Petrovi\u00e7,\" dedim, \"ona hepsini kendiniz verin!\"\n\n\"Nas\u0131l hepsini? B\u00fct\u00fcn kitaplar\u0131 m\u0131?..\"\n\n\"Tabii evet, b\u00fct\u00fcn kitaplar\u0131.\"\n\n\"Kendimden diye mi?\"\n\n\"Kendinizden.\"\n\n\"Bir tek kendimden mi? Yeni kendi ad\u0131ma m\u0131?\"\n\n\"Evet, kendi ad\u0131n\u0131za verin..\"\n\nSan\u0131r\u0131m \u00e7ok a\u00e7\u0131k s\u00f6ylemi\u015ftim, ama ihtiyar uzun s\u00fcre anlayamad\u0131 s\u00f6ylediklerimi.\n\n\"Peki \u00f6yleyse,\" dedi biraz d\u00fc\u015f\u00fcnd\u00fckten sonra, \"olur! Bu \u00e7ok g\u00fczel olur, bu \u00e7ok \u00e7ok g\u00fczel olur, ama siz ne yapacaks\u0131n\u0131z Varvara Alekseyevna?\" \u2013 \"Bir hediye vermeyece\u011fim.\" \u2013 \"Nas\u0131l!\" diye ba\u011f\u0131rd\u0131 ihtiyar \u00fcrkm\u00fc\u015f gibi, \"Petenka'ya bir hediye vermeyecek misiniz, nas\u0131l istemezsiniz ona bir \u015fey vermek?\" \u0130htiyar \u00fcrkm\u00fc\u015ft\u00fc; o s\u0131rada, san\u0131r\u0131m, s\u0131rf ben onun o\u011fluna bir \u015fey vereyim diye teklifinden vazge\u00e7meye de haz\u0131rd\u0131. \u0130yi kalpliydi bu ihtiyar! Bir hediye vermekten mutlu olaca\u011f\u0131m\u0131, ama onun mutlulu\u011funu elinden almak istemedi\u011fimi s\u00f6yledim ona. \"E\u011fer o\u011flunuz ho\u015fnut olursa,\" dedim, \"siz de mutlu olursan\u0131z, o zaman ben de mutlu olurum, \u00e7\u00fcnk\u00fc gizli gizli, kalbimin derinlerinde, ger\u00e7ekten kendim vermi\u015fim gibi hissederim.\" \u0130htiyar bunu duyunca sakinle\u015fti. Bizde iki saat daha oturdu, ama bu arada yerinde oturam\u0131yor, aya\u011fa kalk\u0131yor, geziniyor, gevezelik ediyor, Sa\u015fa'yla \u015fakala\u015f\u0131yor, yana\u011f\u0131m\u0131 \u00f6p\u00fcyor, elimi s\u0131k\u0131yor ve Anna Fedorovna'ya sessizce g\u00fcl\u00fcms\u00fcyordu. Anna Fedorovna sonunda onu evden kovdu. K\u0131sacas\u0131, belki de hi\u00e7 olmad\u0131\u011f\u0131 kadar mutlu olmu\u015ftu.\n\nKutlama g\u00fcn\u00fc saat on bire do\u011fru, ayinden \u00e7\u0131k\u0131p geldi, jilet gibi \u00fct\u00fclenmi\u015f bir frak, yepyeni bir yelek ve yeni \u00e7izmeler giymi\u015fti. \u0130ki elinde iple ba\u011flanm\u0131\u015f kitaplar vard\u0131. O s\u0131rada hep birlikte salonda Anna Fedorovna'n\u0131n yan\u0131nda oturmu\u015f kahve i\u00e7iyorduk (pazar g\u00fcn\u00fcyd\u00fc). \u0130htiyar, san\u0131r\u0131m, Pu\u015fkin'in \u00e7ok iyi bir \u015fair oldu\u011funu s\u00f6yleyerek girdi s\u00f6ze; sonra, birdenbire d\u00fczg\u00fcn davranmak gerekti\u011fini ve bir insan d\u00fczg\u00fcn davranm\u0131yorsa, bunun onun aylak biri oldu\u011funu g\u00f6sterdi\u011fini s\u00f6yledi; k\u00f6t\u00fc e\u011filimler insan\u0131 y\u0131prat\u0131r ve mahvederdi; hatta kendine h\u00e2kim olamaman\u0131n birka\u00e7 y\u0131k\u0131c\u0131 \u00f6rne\u011fini s\u0131ralad\u0131 ve s\u00f6z\u00fcn\u00fc bir s\u00fcredir kesinlikle d\u00fczeldi\u011fini ve art\u0131k \u00f6rnek olacak kadar iyi davrand\u0131\u011f\u0131n\u0131 s\u00f6yleyerek tamamlad\u0131. Bunun kan\u0131t\u0131 olarak da uzun zaman biriktirdi\u011fi parayla ald\u0131\u011f\u0131 kitaplar\u0131 hediye ediyordu.\n\nZavall\u0131 ihtiyar\u0131 dinlerken kendimi tutamay\u0131p hem g\u00fcld\u00fcm hem a\u011flad\u0131m; sonu\u00e7ta gerekirse yalan s\u00f6ylemek de laz\u0131m! Kitaplar Pokrovski'nin odas\u0131na g\u00f6t\u00fcr\u00fcld\u00fc ve yere b\u0131rak\u0131ld\u0131. Pokrovski hemen tahmin etti ger\u00e7e\u011fi. \u0130htiyar\u0131 yeme\u011fe davet ettik. O g\u00fcn herkes \u00e7ok ne\u015feliydi. Yemekten sonra oyun oynad\u0131k, iskambil oynad\u0131k; Sa\u015fa hoplay\u0131p z\u0131pl\u0131yordu, ben de ondan geri kalmad\u0131m. Pokrovski benimle ilgilendi ve benimle yaln\u0131z kal\u0131p konu\u015fmak i\u00e7in f\u0131rsat arad\u0131, ama bulamad\u0131. O g\u00fcn son d\u00f6rt y\u0131l boyunca ya\u015fad\u0131\u011f\u0131m en g\u00fczel g\u00fcnd\u00fc.\n\nAma \u015fimdi kederli, a\u011f\u0131r hat\u0131ralar geliyor; kara g\u00fcnlerimin hik\u00e2yesi ba\u015fl\u0131yor. Belki de bu y\u00fczden, kalemim daha a\u011f\u0131r hareket ediyor ve sanki daha fazla yazmaktan ka\u00e7\u0131n\u0131yor gibi. Belki de bu y\u00fczden, mutlu g\u00fcnlerimde ya\u015fad\u0131klar\u0131m\u0131n en k\u00fc\u00e7\u00fck ayr\u0131nt\u0131s\u0131 bile hat\u0131rlay\u0131nca bana b\u00fcy\u00fck bir heyecan ve sevgi veriyor. O g\u00fcnler \u00e7ok k\u0131sa s\u00fcrd\u00fc; onlar\u0131n yerini ac\u0131, ne zaman bitece\u011fini bir tek Tanr\u0131'n\u0131n bildi\u011fi kara ac\u0131 ald\u0131.\n\nPokrovski'nin hastalan\u0131p \u00f6lmesiyle ba\u015flad\u0131 talihsizliklerim.\n\nBurada anlatt\u0131\u011f\u0131m son olaylardan iki ay sonra hastaland\u0131. Bu iki ay i\u00e7inde s\u00fcrekli ge\u00e7im derdindeydi, \u00e7\u00fcnk\u00fc o zamana kadar belli bir mevki sahibi olamam\u0131\u015ft\u0131. B\u00fct\u00fcn veremliler gibi, o da son \u00e2n\u0131na kadar uzun s\u00fcre ya\u015fama umudunu kaybetmedi. Bir yerlerde \u00f6\u011fretmenlik i\u015fi buldu; ama bu mesle\u011fi reddetti. Resm\u00ee bir mevkide \u00e7al\u0131\u015fmas\u0131 sa\u011fl\u0131\u011f\u0131 nedeniyle imk\u00e2ns\u0131zd\u0131. Ayr\u0131ca ilk maa\u015f\u0131n\u0131 almak i\u00e7in beklemek zorunda kalacakt\u0131. K\u0131sacas\u0131, Pokrovski her a\u00e7\u0131dan ba\u015far\u0131s\u0131z oluyordu; karakteri bozuldu. Sa\u011fl\u0131\u011f\u0131 bozuldu; bunu fark etmedi. Sonbahar geldi. Her g\u00fcn ince paltosuyla \u00e7\u0131k\u0131p i\u015f ar\u0131yor, \u00e7e\u015fitli yerlerden i\u015f rica ediyor ve yalvar\u0131yordu ki bu da onu i\u00e7ten i\u00e7e y\u0131prat\u0131yordu; ayaklar\u0131 \u0131slan\u0131yor, ya\u011fmurdan s\u0131r\u0131ls\u0131klam oluyordu ve sonunda yata\u011fa d\u00fc\u015ft\u00fc, bir daha da kalkamad\u0131... Sonbahar sonlar\u0131nda, ekim ay\u0131n\u0131n sonunda \u00f6ld\u00fc.\n\nOnun hastal\u0131\u011f\u0131 boyunca odas\u0131ndan hemen hi\u00e7 \u00e7\u0131kmam\u0131\u015f, ona bakm\u0131\u015f, hizmetini g\u00f6rm\u00fc\u015ft\u00fcm. Genellikle geceleri uyumuyordu. Nadiren kendinde oluyordu; say\u0131kl\u0131yordu; Tanr\u0131 bilir neler s\u00f6yl\u00fcyordu: kendi mevkisini, kitaplar\u0131n\u0131, beni, babas\u0131n\u0131 anlat\u0131yordu... O s\u0131rada daha \u00f6nce hi\u00e7 bilmedi\u011fim, hatta akl\u0131m\u0131n ucundan bile ge\u00e7meyen \u015feyler \u00f6\u011frendim. Hastal\u0131\u011f\u0131n ilk d\u00f6neminde bizimkiler bana tuhaf bir \u015fekilde bak\u0131yordu; Anna Fedorovna ba\u015f\u0131n\u0131 sall\u0131yordu. Ama ben hi\u00e7birine ald\u0131rmad\u0131m ve onlar da Pokrovski'ye yard\u0131mc\u0131 olmam\u0131 k\u0131namaktan vazge\u00e7tiler \u2013 en az\u0131ndan annem vazge\u00e7ti.\n\nPokrovski bazen beni tan\u0131yordu, ama bu nadiren oluyordu. Neredeyse hep kendini kaybetmi\u015f durumda oluyordu. Bazen b\u00fct\u00fcn gece biriyle uzun uzun konu\u015furdu, anla\u015f\u0131lmaz karanl\u0131k s\u00f6zler s\u00f6ylerdi ve h\u0131r\u0131lt\u0131l\u0131 sesi darac\u0131k, mezara benzeyen odas\u0131nda bo\u011fuk bir \u015fekilde yank\u0131lan\u0131rd\u0131; o zaman \u00e7ok korkard\u0131m. \u00d6zellikle son gece \u00e7\u0131ld\u0131rm\u0131\u015f gibiydi; a\u015f\u0131r\u0131 ac\u0131 \u00e7ekti, \u00fcz\u00fcld\u00fc; inlemeleri ruhumu par\u00e7al\u0131yordu. Evdeki herkes \u00fcrkm\u00fc\u015ft\u00fc. Anna Fedorovna Tanr\u0131 onu hemen yan\u0131na als\u0131n, diye dua ediyordu. Doktoru \u00e7a\u011f\u0131rd\u0131lar. Doktor hastan\u0131n sabaha kadar kesinlikle \u00f6lece\u011fini s\u00f6yledi.\n\n\u0130htiyar Pokrovski b\u00fct\u00fcn geceyi koridorda, o\u011flunun odas\u0131n\u0131n \u00f6n\u00fcnde ge\u00e7irmi\u015fti; orada onun i\u00e7in bir \u00f6rt\u00fc serdiler. \u0130kide bir odaya girip \u00e7\u0131k\u0131yordu; korkun\u00e7 g\u00f6r\u00fcn\u00fcyordu. Ac\u0131dan peri\u015fand\u0131, duygusuz ve anlams\u0131z bak\u0131yordu. Korkuyla sallan\u0131yordu ba\u015f\u0131. Ba\u015ftan a\u015fa\u011f\u0131 titriyor ve kendi kendine bir \u015feyler f\u0131s\u0131ld\u0131yor, kendi kendine birtak\u0131m kararlar veriyordu. Ac\u0131dan delirmi\u015f gibi geliyordu bana.\n\nTan vaktinden \u00f6nce, ruhsal ac\u0131dan yorgun d\u00fc\u015fen ihtiyar \u00f6rt\u00fcn\u00fcn \u00fczerinde \u00f6l\u00fc gibi uyuyakald\u0131. Saat sekizde o\u011flu can \u00e7eki\u015fmeye ba\u015flad\u0131; babas\u0131n\u0131 uyand\u0131rd\u0131m. Pokrovski'nin belle\u011fi geri gelmi\u015fti, hepimizle vedala\u015ft\u0131. \u00c7ok tuhaft\u0131! A\u011flayamad\u0131m; ama ruhum parampar\u00e7a olmu\u015ftu.\n\nAma beni en \u00e7ok peri\u015fan eden ve ac\u0131ya bo\u011fan \u015fey onun son anlar\u0131 oldu. Nedense kat\u0131la\u015fm\u0131\u015f diliyle uzun uzun bir \u015feyler rica etti, ama ben s\u00f6ylediklerinden hi\u00e7bir \u015fey anlayamad\u0131m. Kalbim ac\u0131dan a\u011f\u0131rla\u015ft\u0131! Bir saat boyunca huzursuzluk \u00e7ekti, hep bir \u015feyler istedi, so\u011fumu\u015f elleriyle birtak\u0131m i\u015faretler yapmaya \u00e7al\u0131\u015ft\u0131 ve sonra yine ac\u0131kl\u0131, h\u0131r\u0131lt\u0131l\u0131, bo\u011fuk bir sesle bir \u015feyler rica etmeye ba\u015flad\u0131; ama s\u00f6zleri anlams\u0131z seslerden ibaretti ve yine hi\u00e7bir \u015fey anlamad\u0131m. Ona ne bulduysam g\u00f6t\u00fcrd\u00fcm, su verdim; ama o hep h\u00fcz\u00fcnle ba\u015f\u0131n\u0131 sallay\u0131p reddediyordu. Sonunda ne istedi\u011fini anlad\u0131m. Perdeyi a\u00e7\u0131p pancurlar\u0131 a\u00e7mam\u0131 istiyordu. Herhalde, son bir kez g\u00fcn \u0131\u015f\u0131\u011f\u0131n\u0131, Tanr\u0131'n\u0131n \u0131\u015f\u0131\u011f\u0131n\u0131, g\u00fcne\u015fi seyretmek istiyordu. Perdeyi a\u00e7t\u0131m; ama ba\u015flayan g\u00fcn h\u00fcz\u00fcnl\u00fc ve kederliydi, \u00f6len adam\u0131n s\u00f6nmekte olan zavall\u0131 hayat\u0131 gibi. G\u00fcne\u015f yoktu. Bulutlar g\u00f6\u011f\u00fc dumandan bir perdeyle \u00f6rtm\u00fc\u015ft\u00fc; ya\u011fmurlu, kapal\u0131, kederli bir hava vard\u0131. \u0130nce bir ya\u011fmur s\u00fcz\u00fcl\u00fcyordu camlardan ve onlar\u0131 so\u011fuk, kirli su ak\u0131nt\u0131lar\u0131yla siliyordu; donuk ve karanl\u0131kt\u0131 hava. Odaya solgun g\u00fcn\u00fcn \u0131\u015f\u0131nlar\u0131 hafif hafif geliyordu ve ikonan\u0131n \u00f6n\u00fcnde yanan lamban\u0131n titrek \u0131\u015f\u0131\u011f\u0131ndan pek de fazla de\u011fildi. \u00d6len adam h\u00fcz\u00fcnl\u00fc h\u00fcz\u00fcnl\u00fc bakt\u0131 bana ve ba\u015f\u0131n\u0131 sallad\u0131. Bir dakika sonra \u00f6lm\u00fc\u015ft\u00fc.\n\nCenaze t\u00f6renini Anna Fedorovna haz\u0131rlad\u0131. Sade bir tabut sat\u0131n al\u0131nd\u0131 ve bir y\u00fck arabas\u0131 kiraland\u0131. Anna Fedorovna masraflar kar\u015f\u0131l\u0131\u011f\u0131nda merhumun kitaplar\u0131na ve di\u011fer e\u015fyalar\u0131na el koydu. \u0130htiyar onunla tart\u0131\u015ft\u0131, ba\u011f\u0131rd\u0131, alabildi\u011fi kadar kitab\u0131 elinden ald\u0131, ceplerine doldurdu, \u015fapkas\u0131na doldurdu, \u00fc\u00e7 g\u00fcn boyunca yan\u0131nda ta\u015f\u0131d\u0131, hatta kiliseye giderken bile yan\u0131ndan ay\u0131rmad\u0131. O g\u00fcnlerde kendini kaybetmi\u015f gibiydi, sersemlemi\u015fti ve tuhaf bir dalg\u0131nl\u0131kla tabutun \u00e7evresinde u\u011fra\u015f\u0131p duruyordu: Bazen merhumun aln\u0131ndaki _ven\u00e7ik_ 'i13 d\u00fczeltiyor, bazen mumlar\u0131 yak\u0131p al\u0131yordu. Do\u011fru d\u00fcr\u00fcst d\u00fc\u015f\u00fcnemedi\u011fi belliydi. Annem de Anna Fedorovna da kilisedeki ayine kat\u0131lmad\u0131lar. Annem hastayd\u0131, Anna Fedorovna da her \u015feyi toplam\u0131\u015f, ihtiyar Pokrovski'yle tart\u0131\u015fm\u0131\u015f ve evde kalm\u0131\u015ft\u0131. Bir tek benle ihtiyar oradayd\u0131k. T\u00f6ren s\u0131ras\u0131nda bir korku d\u00fc\u015ft\u00fc i\u00e7ime; sanki olacaklar\u0131 sezmi\u015ftim. Kiliseden ka\u00e7mamak i\u00e7in zor tuttum kendimi. Sonunda tabut kapat\u0131ld\u0131, \u00e7ivisi \u00e7ak\u0131ld\u0131, arabaya kondu ve yola \u00e7\u0131kt\u0131. Ona caddenin sonuna kadar e\u015flik ettim. Arabac\u0131 t\u0131r\u0131s gidiyordu. \u0130htiyar pe\u015finden ko\u015fuyor ve ba\u011f\u0131ra \u00e7a\u011f\u0131ra a\u011fl\u0131yordu; a\u011flay\u0131\u015f\u0131 ko\u015ftu\u011fu i\u00e7in titrek, kesik kesik bir sesle duyuluyordu. Zavall\u0131 \u015fapkas\u0131n\u0131 d\u00fc\u015f\u00fcrd\u00fc ve onu kald\u0131rmak i\u00e7in durmad\u0131. Ba\u015f\u0131 ya\u011fmurdan s\u0131r\u0131ls\u0131klam oldu; r\u00fczg\u00e2r \u00e7\u0131kt\u0131; y\u00fcz\u00fc k\u0131ra\u011f\u0131yla kaplan\u0131p \u0131sland\u0131. \u0130htiyar k\u00f6t\u00fc havan\u0131n fark\u0131nda de\u011filmi\u015f gibiydi ve a\u011flayarak araban\u0131n bir sa\u011f\u0131nda bir solunda ko\u015fuyordu. Eski redingotunun etekleri r\u00fczg\u00e2rda kanat gibi havalan\u0131yordu. B\u00fct\u00fcn ceplerinden kitaplar d\u00f6k\u00fcl\u00fcyordu; iki eliyle dev bir kitap tutuyordu, s\u0131k\u0131ca sar\u0131lm\u0131\u015ft\u0131 ona. Yoldan gelip ge\u00e7enler \u015fapkalar\u0131n\u0131 \u00e7\u0131kar\u0131p ha\u00e7 \u00e7\u0131kar\u0131yorlard\u0131. Baz\u0131lar\u0131 da durmu\u015f \u015fa\u015fk\u0131n \u015fa\u015fk\u0131n ihtiyar\u0131 seyrediyordu. Tek tek \u00e7amura d\u00fc\u015f\u00fcyordu cebinden kitaplar. Onu durduruyor, d\u00fc\u015f\u00fcrd\u00fc\u011f\u00fc \u015feyi g\u00f6steriyorlard\u0131; onu yerden al\u0131p yine tabutun pe\u015finden ko\u015fuyordu. Caddenin k\u00f6\u015fesinde yoksul, ya\u015fl\u0131 bir kad\u0131n yana\u015ft\u0131 tabuta, ona e\u015flik etmeye ba\u015flad\u0131. Eve d\u00f6nd\u00fcm. Korkun\u00e7 bir kederle annemin g\u00f6\u011fs\u00fcne at\u0131ld\u0131m. Kollar\u0131mla onu s\u0131k\u0131 s\u0131k\u0131 sard\u0131m, \u00f6pt\u00fcm ve h\u0131\u00e7k\u0131ra h\u0131\u00e7k\u0131ra a\u011flad\u0131m, \u00fcrkerek sokuldum ona sanki son dostumu kollar\u0131mda korumak ve onu \u00f6l\u00fcme teslim etmemek ister gibi... Ama \u00f6l\u00fcm \u00e7oktan \u00e7\u00f6km\u00fc\u015ft\u00fc zavall\u0131 annemin \u00fczerine!\n\n11 Haziran\n\nD\u00fcn ak\u015famki ada gezimiz i\u00e7in size nas\u0131l minnettar\u0131m, Makar Alekseyevi\u00e7! Ne kadar ayd\u0131nl\u0131k, g\u00fczeldi oras\u0131, ne kadar ye\u015fildi! Uzun zamand\u0131r ye\u015fillik g\u00f6rmemi\u015ftim; hastaland\u0131\u011f\u0131m zaman, \u00f6lmem gerekiyor ve hemen \u00f6lece\u011fim gibi gelmi\u015fti; bir d\u00fc\u015f\u00fcn\u00fcn d\u00fcn neler hissetmi\u015f, nas\u0131l hissetmi\u015f olabilirim! D\u00fcn o kadar h\u00fcz\u00fcnl\u00fc oldu\u011fum i\u00e7in bana k\u0131zmay\u0131n; \u00e7ok iyiydim, \u00e7ok rahatt\u0131m, ama en iyi anlar\u0131mda bile, ben hep h\u00fcz\u00fcnlenirim. A\u011flam\u0131\u015f olmam\u0131n da hi\u00e7 \u00f6nemi yok; ben bile bilmiyorum neden a\u011flad\u0131\u011f\u0131m\u0131. Hastay\u0131m, sinir bozucu hisler i\u00e7indeyim; izlenimlerim hastal\u0131kl\u0131. Bulutsuz, solgun g\u00f6k, g\u00fcne\u015fin bat\u0131\u015f\u0131, d\u00fcn ak\u015famki sessizlik \u2013b\u00fct\u00fcn bunlar\u2013 bilemiyorum ama d\u00fcn b\u00fct\u00fcn bu izlenimleri a\u011f\u0131r ve ac\u0131 verici bulacak bir haldeydim, kalbim kabard\u0131 ve ruhum g\u00f6zya\u015flar\u0131n\u0131 koyuverdi. Ama neden yaz\u0131yorum b\u00fct\u00fcn bunlar\u0131 size? B\u00fct\u00fcn bunlar\u0131 s\u00f6ylemek kendine itiraf etmek zor, bir ba\u015fkas\u0131na anlatmak daha da zor. Ama siz, herhalde beni anlars\u0131n\u0131z. Hem h\u00fcz\u00fcn hem kahkaha! Ger\u00e7ekten, ne kadar iyisiniz Makar Alekseyevi\u00e7! D\u00fcn g\u00f6zlerimin i\u00e7ine, neler hissetti\u011fimi g\u00f6zlerimde okumak i\u00e7in nas\u0131l bakt\u0131n\u0131z \u00f6yle, benim heyecan\u0131ma nas\u0131l hayran oldunuz. Bir \u00e7al\u0131l\u0131k, bir a\u011fa\u00e7l\u0131kl\u0131 yol, bir su k\u0131y\u0131s\u0131 buluyordunuz; bulunca hemen kar\u015f\u0131ma ge\u00e7ip g\u00f6steriyor ve sanki bana kendi m\u00fclk\u00fcn\u00fcz\u00fc g\u00f6steriyormu\u015f gibi bak\u0131yordunuz g\u00f6zlerimin i\u00e7ine. Sizin iyi bir kalbiniz oldu\u011funu kan\u0131tl\u0131yor bu Makar Alekseyevi\u00e7. Sizi bu y\u00fczden seviyorum. Neyse, ho\u015f\u00e7a kal\u0131n. Bug\u00fcn yine hastaland\u0131m; d\u00fcn ayaklar\u0131m \u0131sland\u0131 ve \u00fc\u015f\u00fctm\u00fc\u015f\u00fcm bu y\u00fczden; Fedora da biraz hasta, o y\u00fczden ikimiz de hasta yat\u0131yoruz. Unutmay\u0131n beni, daha s\u0131k gelin.\n\nSizin olan,  \nV.D.\n\n12 Haziran\n\nG\u00fcvercinim, Varvara Alekseyevna,\n\nCan\u0131m, ben de d\u00fcn ak\u015famdan sonra bana ger\u00e7ek \u015fiirler yazars\u0131n\u0131z diye d\u00fc\u015f\u00fcn\u00fcyordum, oysa sizden \u00e7\u0131ka \u00e7\u0131ka bir sayfa \u00e7\u0131km\u0131\u015f. Bunu s\u00f6ylememin nedeni \u015fu: Siz bana bir sayfadan da az yazard\u0131n\u0131z, ama o zaman da s\u0131rad\u0131\u015f\u0131 g\u00fczellik ve tatl\u0131l\u0131kla yazard\u0131n\u0131z. Bir yandan do\u011fa, bir yandan \u00e7e\u015fitli k\u00f6y manzaralar\u0131, bir yandan o duygular; k\u0131sacas\u0131, b\u00fct\u00fcn bunlar\u0131 \u00e7ok g\u00fczel yazm\u0131\u015fs\u0131n\u0131z. Ama benim b\u00f6yle bir yetene\u011fim yok. On sayfa yazmaya niyetlensem bile, hi\u00e7bir \u015fey \u00e7\u0131kmaz, hi\u00e7bir \u015fey yazamam. Denedim.\n\nCan\u0131m\u0131n i\u00e7i, siz benim iyi, k\u00f6t\u00fcl\u00fckten uzak, yak\u0131n\u0131na bir fenal\u0131k yapamayan ve iyilik d\u00fc\u015fk\u00fcn\u00fc, do\u011fay\u0131 bilen, anlayan biri oldu\u011fumu yaz\u0131yor, dahas\u0131, beni \u00f6vg\u00fclere bo\u011fuyorsunuz. B\u00fct\u00fcn bunlar do\u011fru, can\u0131m, b\u00fct\u00fcn bunlar ger\u00e7ekten do\u011fru; ben ger\u00e7ekten de b\u00f6yleyim, dedi\u011finiz gibiyim ve ben de biliyorum bunu; ama sizin yazd\u0131klar\u0131n\u0131z\u0131 okurken, ister istemez duygulan\u0131yorum, sonra da t\u00fcrl\u00fc fikirler geliyor akl\u0131ma. \u015eimdi dinleyin beni can\u0131m, size bir \u015feyler anlataca\u011f\u0131m.\n\nMemuriyete girdi\u011fimde on yedi ya\u015f\u0131mda oldu\u011fumu s\u00f6yleyerek ba\u015flayay\u0131m, \u015fimdi otuz y\u0131l ge\u00e7mi\u015f oldu bu mesle\u011fe girdi\u011fimden beri. Bir s\u00fcr\u00fc \u00fcniforma eskitti\u011fimi s\u00f6ylemeye gerek bile yok; olgunla\u015ft\u0131m, ak\u0131lland\u0131m, insanlar\u0131 tan\u0131d\u0131m; g\u00f6rd\u00fcm ge\u00e7irdim, diyebilirim, hatta bir keresinde bana ha\u00e7 ni\u015fan\u0131 takmak isteyecekleri kadar \u015fey g\u00f6rd\u00fcm. Belki inanmazs\u0131n\u0131z ama emin olun, yalan s\u00f6ylemiyorum size.\n\nNeyse can\u0131m, k\u00f6t\u00fc insanlar bunu da \u00e7ok g\u00f6rd\u00fcler! Ama size \u015funu s\u00f6yleyeyim can\u0131m, karanl\u0131k bir insan, aptal bir insan olsam da, ne yaz\u0131k ki benim de ba\u015fkalar\u0131ndaki gibi bir kalbim var. Biliyor musunuz Varenka, k\u00f6t\u00fc bir adam bana neler yapt\u0131? Yapt\u0131\u011f\u0131n\u0131 s\u00f6ylemek bile ay\u0131p; sorun bir bakal\u0131m, neden yapt\u0131? S\u0131rf ben uysal oldu\u011fum i\u00e7in, sessiz oldu\u011fum i\u00e7in, iyi kalpli oldu\u011fum i\u00e7in! Onlar\u0131n huyuna gitmedim, zamanla onlar benim huyuma geldi. \u00d6nceleri, \"Ya, Makar Alekseyevi\u00e7, siz \u015f\u00f6yle \u015f\u00f6yle,\" diye ba\u015flad\u0131lar; sonra, \"Ya, Makar Alekseyevi\u00e7'e sormay\u0131n,\" haline geldi bu. Sonra da, \"Ya, tabii, Makar Alekseyevi\u00e7 b\u00f6yledir,\" diye karar verdiler. \u0130\u015fte can\u0131m, g\u00f6r\u00fcyorsunuz i\u015flerin nereye vard\u0131\u011f\u0131n\u0131. Her \u015fey Makar Alekseyevi\u00e7'ten bilinir oldu; Makar Alekseyevi\u00e7'i bizim idarede a\u011fza sak\u0131z ettiler. \u00dcstelik benim ad\u0131ma atas\u00f6zleri, hatta k\u00fcf\u00fcrl\u00fc s\u00f6zler \u00e7\u0131kard\u0131lar, \u00fcniformama, sa\u00e7lar\u0131ma, \u015feklime \u015femalime kar\u0131\u015ft\u0131lar; onlara g\u00f6re olmayan ne varsa, hepsini de\u011fi\u015ftirmek gerekiyormu\u015f! Hem de b\u00fct\u00fcn bunlar, hemen hemen her g\u00fcn tekrar tekrar s\u00f6yleniyordu. Al\u0131\u015ft\u0131m, \u00e7\u00fcnk\u00fc her \u015feye al\u0131\u015f\u0131yorum, \u00e7\u00fcnk\u00fc uysal bir insan\u0131m, \u00e7\u00fcnk\u00fc ben k\u00fc\u00e7\u00fck insan\u0131m; fakat, b\u00fct\u00fcn bunlar ne i\u00e7in? Kime ne k\u00f6t\u00fcl\u00fc\u011f\u00fcm olmu\u015f? Birinin r\u00fctbesine mi engel oldum? \u00dcstlerimin \u00f6n\u00fcnde birini mi karalad\u0131m? \u00d6d\u00fcl m\u00fc istedim? Birine k\u00f6lelik mi yapt\u0131m? B\u00f6yle bir \u015feyi d\u00fc\u015f\u00fcnmeniz bile g\u00fcnaht\u0131r, can\u0131m! Ne diye yapay\u0131m b\u00fct\u00fcn bunlar\u0131?\n\nBaksan\u0131za, can\u0131m\u0131n i\u00e7i, korkakl\u0131k ve ikbal d\u00fc\u015fk\u00fcnl\u00fc\u011f\u00fcne yetene\u011fim var m\u0131, yapabilir miyim? Peki nedir bana yap\u0131lan bu sald\u0131r\u0131lar, s\u00f6ylesenize Tanr\u0131 a\u015fk\u0131na? Siz \u015fimdi beni onurlu bir insan say\u0131yorsunuz, ama siz di\u011ferlerinden farkl\u0131s\u0131n\u0131z, can\u0131m. Bir yurtta\u015f i\u00e7in en b\u00fcy\u00fck iyilik nedir? Evstafi \u0130vanovi\u00e7'le yapt\u0131\u011f\u0131m\u0131z \u00f6zel bir g\u00f6r\u00fc\u015fmede, bir yurtta\u015f\u0131n en b\u00fcy\u00fck iyili\u011finin para saklayabilmek oldu\u011funu konu\u015ftuk. \u015eaka yollu \u015f\u00f6yle dedi kendisi (\u015faka oldu\u011funu biliyorum), kimseye y\u00fck olmamak bir ahlak dersidir; ben kimseye y\u00fck olmuyorum! Ben kendi ekme\u011fimi kazan\u0131p yiyorum; do\u011fru, kuru bir par\u00e7a ekmek, hatta kararm\u0131\u015f ekmek; ama \u00e7al\u0131\u015farak kazan\u0131lm\u0131\u015f, yasal ve hile hurda yapmadan elde edilmi\u015f bir ekmek. Ne yapay\u0131m! Yaz\u0131lar\u0131 temize \u00e7ekme i\u015fiyle az kazand\u0131\u011f\u0131m\u0131 ben de biliyorum; yine de gurur duyuyorum bundan; \u00e7al\u0131\u015f\u0131yorum, ter d\u00f6k\u00fcyorum. Ama sonu\u00e7ta ger\u00e7ekten b\u00f6yle; yaz\u0131 temize \u00e7ekiyorum! G\u00fcnah de\u011fil ya yaz\u0131 temize \u00e7ekmek? \"Zaten o yaz\u0131lar\u0131 temize \u00e7eker!\", \"Bu memur m\u00fcsveddesi var ya,\" derler, \"yaz\u0131 temize \u00e7eker!\" Bunda onursuz bir \u015fey var m\u0131? \u015e\u00f6yle daha okunakl\u0131, daha g\u00fczel, g\u00f6ze ho\u015f g\u00f6r\u00fcnen ve a\u015f\u0131r\u0131 ho\u015fnutluk verici bir yaz\u0131; ben onlar i\u00e7in \u00f6nemli evraklar\u0131 temize \u00e7ekiyorum. Ama \u00fcslubum yok, bunu ben de biliyorum, olmad\u0131\u011f\u0131n\u0131, k\u00f6t\u00fc oldu\u011funu; i\u015fte bu y\u00fczden de meslekte yer edinemedim, hatta i\u015fte size bile \u015fimdi, sevgili akrabam, \u00f6ylece, e\u011flenmeden, i\u00e7imden nas\u0131l ge\u00e7erse \u00f6yle yaz\u0131yorum... Fakat herkes eser verecek olsayd\u0131, kim yaz\u0131lar\u0131 temize \u00e7ekecekti? \u0130\u015fte b\u00f6yle sorup yan\u0131t\u0131n\u0131 da kendim veriyorum, can\u0131m. Yani biliyorum bana gerek oldu\u011funu, vazge\u00e7ilmez oldu\u011fumu ve insanlar\u0131n sa\u00e7mal\u0131klar\u0131na kulak asmamak gerekti\u011fini. Ama ne yaz\u0131k ki m\u00fcsvedde diyerek bir benzerlik buluyorlar! Tamam da bu m\u00fcsvedde laz\u0131m size, bu m\u00fcsvedde i\u015finize yar\u0131yor, bu m\u00fcsvedde sayesinde i\u015fler yolunda gidiyor, bu m\u00fcsveddeye ikramiye vermeniz laz\u0131m; m\u00fcsvedde olsa bile!\n\nFakat, bu konu uzad\u0131 iyice can\u0131m; bundan bahsetmek istemiyordum, ama kendimi tutamad\u0131m. Zaman zaman insan\u0131n hakk\u0131n\u0131 aramas\u0131 da ho\u015f her \u015feye ra\u011fmen. Ho\u015f\u00e7a kal\u0131n, can\u0131m, g\u00fcvercinim, benim iyi y\u00fcrekli tesellim! Gelece\u011fim, kesinlikle gelece\u011fim size, s\u00f6z veriyorum bir tanem. Siz s\u0131k\u0131lmay\u0131n bu arada. Kitap da getirece\u011fim size. Ho\u015f\u00e7a kal\u0131n, Varenka.\n\n\u0130yili\u011finizi can\u0131 g\u00f6n\u00fclden dileyen,  \nMakar Devu\u015fkin\n\n20 Haziran\n\nSayg\u0131de\u011fer Beyefendi Makar Alekseyevi\u00e7,\n\nSize k\u0131sa yazaca\u011f\u0131m, acele ediyorum, elimdeki i\u015fi bitirmem gerekiyor. Bak\u0131n \u015f\u00f6yle bir durum var: iyi bir al\u0131\u015fveri\u015f yapabilirsiniz. Fedora bir tan\u0131d\u0131\u011f\u0131n\u0131n yepyeni bir resm\u00ee redingot, i\u00e7 \u00e7ama\u015f\u0131r\u0131, yelek ve g\u00f6mlek satt\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyor, fiyat\u0131 da \u00e7ok ucuzmu\u015f; siz alabilirsiniz. Zaten \u015fimdi bir ihtiyac\u0131n\u0131z yok, paran\u0131z da var; \u00f6yle demi\u015ftiniz. Yeter art\u0131k, l\u00fctfen, cimrilik etmeyin; bunlar\u0131n hepsi gerekli \u015feyler. Bir baksan\u0131za kendinize, ne kadar eski giysilerle geziyorsunuz. Ay\u0131p! Hep yamalar i\u00e7inde. Yeni bir \u015feyiniz yok; siz ne kadar var, diye \u0131srar etseniz de, olmad\u0131\u011f\u0131n\u0131 biliyorum. Sizin onu nereye verdi\u011finizi Tanr\u0131 bilir. Beni dinleyin, al\u0131n l\u00fctfen. Bunu benim i\u00e7in yap\u0131n; beni seviyorsan\u0131z, sat\u0131n al\u0131n.\n\nBana hediye olarak i\u00e7 \u00e7ama\u015f\u0131r\u0131 g\u00f6ndermi\u015fsiniz; ama Makar Alekseyevi\u00e7, iflas edeceksiniz. Benim i\u00e7in harcad\u0131\u011f\u0131n\u0131z para \u015faka gibi, \u00e7ok fazla! Ah, sa\u00e7\u0131p savurmay\u0131 nas\u0131l da seviyorsunuz! Benim ihtiyac\u0131m yok ki; b\u00fct\u00fcn bunlar \u00e7ok gereksiz. Beni sevdi\u011finizi biliyorum, eminim buna; do\u011frusu bunu bana hediyelerle hat\u0131rlatman\u0131z gereksiz; onlar\u0131 sizden kabul etmem de zor oluyor; size neye mal oldu\u011funu biliyorum. Son kez s\u00f6yl\u00fcyorum, yeter; duyuyor musunuz? Rica ediyorum, yalvar\u0131yorum size. Makar Alekyesevi\u00e7, benden hat\u0131ralar\u0131m\u0131n devam\u0131n\u0131 g\u00f6ndermemi istemi\u015fsiniz; onlar\u0131 bitirmemi istiyorsunuz. B\u00fct\u00fcn onlar\u0131 nas\u0131l yazd\u0131\u011f\u0131m\u0131, ne yazd\u0131\u011f\u0131m\u0131 bile bilmiyorum! Fakat art\u0131k ya\u015fad\u0131klar\u0131m\u0131 anlatmaya g\u00fcc\u00fcm yetmiyor; onlar\u0131 d\u00fc\u015f\u00fcnmek bile istemiyorum; bu hat\u0131ralar \u00e7ok rahats\u0131z ediyor beni. Zavall\u0131 \u00e7ocu\u011funu bu canavarlar\u0131n eline b\u0131rakan zavall\u0131 anneci\u011fimden bahsetmek beni peri\u015fan ediyor. Bir tek olay\u0131 hat\u0131rlamak bile kalbimi parampar\u00e7a ediyor. B\u00fct\u00fcn bunlar o kadar yeni ki; b\u00fct\u00fcn bunlar\u0131n \u00fcst\u00fcnden bir y\u0131ldan fazla zaman ge\u00e7mi\u015f olsa bile, ne sakinle\u015febildim ne de kendimi toplayabildim. Ama siz her \u015feyi biliyorsunuz zaten.\n\nSize art\u0131k Anna Fedorovna hakk\u0131nda ne d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcm\u00fc anlatm\u0131\u015ft\u0131m; beni nank\u00f6rl\u00fckle su\u00e7luyor ve Bay B\u0131kov ile herhangi bir i\u015fbirli\u011fi yapt\u0131\u011f\u0131n\u0131 reddediyor! Beni yan\u0131na \u00e7a\u011f\u0131r\u0131yor; dilenci oldu\u011fumu, k\u00f6t\u00fc yola d\u00fc\u015ft\u00fc\u011f\u00fcm\u00fc s\u00f6yl\u00fcyor. E\u011fer ona geri d\u00f6nersem, Bay B\u0131kov'la ya\u015fananlar\u0131 yoluna koyaca\u011f\u0131n\u0131 ve onu bana kar\u015f\u0131 i\u015fledi\u011fi su\u00e7u kabul etmeye zorlayaca\u011f\u0131n\u0131 s\u00f6yl\u00fcyor. Bay B\u0131kov'un bana \u00e7eyiz vermek istedi\u011fini s\u00f6yl\u00fcyor. Tanr\u0131 yolunu a\u00e7\u0131k etsin onlar\u0131n! Ben burada sizinle, bana olan ba\u011fl\u0131l\u0131\u011f\u0131yla bana merhum dad\u0131m\u0131 hat\u0131rlatan iyi kalpli Fedora'mla mutluyum. Siz uzak bir akrabam oldu\u011funuz halde beni ad\u0131n\u0131zla koruyorsunuz. Ama onlar\u0131 tan\u0131m\u0131yorum; yapabilsem unutaca\u011f\u0131m onlar\u0131. Ne istiyorlar ki benden? Fedora bunlar\u0131n hep dedikodu oldu\u011funu, sonunda beni rahat b\u0131rakacaklar\u0131n\u0131 s\u00f6yl\u00fcyor. Tanr\u0131'ya \u015f\u00fck\u00fcr!\n\nV.D.\n\n21 Haziran\n\nG\u00fcvercinim benim, can\u0131m,\n\nYazmak istiyorum, ama nas\u0131l ba\u015flayaca\u011f\u0131m\u0131 bilemiyorum. Bizim sizinle ya\u015fad\u0131klar\u0131m\u0131z zaten \u00e7ok tuhaf, can\u0131m. Size s\u00f6yl\u00fcyorum, hi\u00e7 b\u00f6yle mutluluk ya\u015famam\u0131\u015ft\u0131m. Tanr\u0131 bana bir ev ve bir aile verdi sanki! Siz k\u0131z\u0131ms\u0131n\u0131z benim, g\u00fczelim! Kalkm\u0131\u015f size g\u00f6nderdi\u011fim d\u00f6rt \u00e7ama\u015f\u0131rdan bahsediyorsunuz. Size onlar\u0131n laz\u0131m oldu\u011funu da Fedora'dan \u00f6\u011frenmi\u015ftim. Can\u0131m, benim ki\u015fisel mutlulu\u011fum da sizi mutlu etmektir; beni bu mutlulu\u011fumdan al\u0131koymay\u0131n\u0131z, can\u0131m; ili\u015fmeyin, engel olmay\u0131n bana. Ba\u015f\u0131ma hi\u00e7 b\u00f6yle bir \u015fey gelmemi\u015fti, can\u0131m. Art\u0131k d\u00fcnyaya kar\u0131\u015ft\u0131m i\u015fte. Birincisi, iki kat ya\u015f\u0131yorum, \u00e7\u00fcnk\u00fc bana yak\u0131n oturuyor ve co\u015fkuland\u0131r\u0131yorsunuz beni; ikincisi, bug\u00fcn beni kirac\u0131lardan biri, kom\u015fum Ratazyayev, o edeb\u00ee toplant\u0131lar\u0131 yapan memur \u00e7aya davet etti. Bug\u00fcn toplant\u0131 var; edebiyat okumalar\u0131 yapaca\u011f\u0131z. \u0130\u015fte g\u00f6r\u00fcyor musunuz halimizi can\u0131m, i\u015fte! Neyse, ho\u015f\u00e7a kal\u0131n. B\u00fct\u00fcn bunlar\u0131 bir maksat olmadan, s\u0131rf sizi saadetimden haberdar etmek i\u00e7in yazd\u0131m. Bir tanem, Tereza'dan nak\u0131\u015f i\u00e7in renkli ipli\u011fe ihtiyac\u0131n\u0131z oldu\u011funu s\u00f6ylemesini istemi\u015fsiniz; al\u0131r\u0131m, can\u0131m, ipek de al\u0131r\u0131m. Yar\u0131n b\u00fct\u00fcn isteklerinizi kar\u015f\u0131lamay\u0131 umut ediyorum. Tan\u0131d\u0131\u011f\u0131m yerden alaca\u011f\u0131m. Ama \u015fimdi kesmek zorunday\u0131m\n\nEn i\u00e7ten dostunuz,  \nMakar Devu\u015fkin\n\n22 Haziran\n\nSevgili Han\u0131mefendi, Varvara Alekseyevna,\n\nSize, bir taneme, evimizde \u00e7ok \u00fcz\u00fcc\u00fc, ger\u00e7ekten ger\u00e7ekten \u00fcz\u00fclmeyi gerektiren bir olay ya\u015fand\u0131\u011f\u0131n\u0131 bildirece\u011fim! Bug\u00fcn, sabah\u0131n be\u015finde, Gor\u015fkov'un k\u00fc\u00e7\u00fck \u00e7ocu\u011fu \u00f6ld\u00fc. Neden oldu\u011funu, k\u0131z\u0131ldan m\u0131 oldu\u011funu bilmiyorum, ne oldu\u011funu Tanr\u0131 bilir! Ziyarete gittim bu Gor\u015fkov'lar\u0131. Can\u0131m, ne kadar yoksuldular! \u00dcstelik nas\u0131l bir da\u011f\u0131n\u0131kl\u0131k! \u015ea\u015facak bir \u015fey de yok; b\u00fct\u00fcn aile bir odada ya\u015f\u0131yor, terbiye gere\u011fi paravanlarla b\u00f6lm\u00fc\u015fler. Tabutu da haz\u0131rlam\u0131\u015flar, basit ama \u00e7ok g\u00fczel bir tabut; haz\u0131r alm\u0131\u015flar, \u00e7ocuk da dokuz ya\u015f\u0131ndaym\u0131\u015f; \u00e7ok umut vaat ediyordu, dediler. Onlar\u0131 seyretmek ne kadar \u00fcz\u00fcc\u00fcyd\u00fc, Varenka! Anne a\u011flam\u0131yor, ama o kadar kederli, o kadar zavall\u0131 ki. Belki omuzlar\u0131ndan bir y\u00fck eksildi\u011fi i\u00e7in rahatlam\u0131\u015f olabilirler; ama iki \u00e7ocuklar\u0131 daha var, daha meme \u00e7a\u011f\u0131nda bir \u00e7ocukla alt\u0131s\u0131na basmak \u00fczere olan bir k\u00fc\u00e7\u00fck k\u0131z. Bir \u00e7ocu\u011fun ac\u0131 \u00e7ekti\u011fini g\u00f6rmek, hem de \u00f6z \u00e7ocu\u011funun ac\u0131 \u00e7ekti\u011fini g\u00f6rmek ama yard\u0131m edememek ne kadar tats\u0131z bir durum! Baba eski p\u00fcsk\u00fc, yamal\u0131 bir ceket giymi\u015f, k\u0131r\u0131k bir sandalyede oturuyor. G\u00f6zya\u015flar\u0131 bo\u015fan\u0131yor g\u00f6zlerinden, belki de \u00fcz\u00fcnt\u00fcden de\u011fil, al\u0131\u015fkanl\u0131ktan g\u00f6zleri doldu\u011fu i\u00e7in a\u011fl\u0131yor. \u00c7ok acayip biri! Onunla konu\u015furken k\u0131pk\u0131rm\u0131z\u0131 oluyor, ne s\u00f6yleyece\u011fini, nas\u0131l yan\u0131t verece\u011fini bilemiyor. K\u00fc\u00e7\u00fck k\u0131z da tabuta yaslanm\u0131\u015f duruyor, peri\u015fan, s\u0131k\u0131nt\u0131l\u0131, dalg\u0131n bir halde! Bir \u00e7ocu\u011fun dalg\u0131n dalg\u0131n d\u00fc\u015f\u00fcnmesinden ho\u015flanm\u0131yorum Varenka, can\u0131m; seyretmesi ho\u015f de\u011fil! Yan\u0131nda, yerde bir pa\u00e7avra kukla duruyor \u2013onunla oynam\u0131yor; dudaklar\u0131na g\u00f6t\u00fcrd\u00fc\u011f\u00fc parmak titriyor; \u00f6ylece duruyor\u2013 k\u0131p\u0131rdam\u0131yor. Ev sahibesi bir \u015feker verdi; \u00e7ocuk ald\u0131, ama yemedi. Kederli, de\u011fil mi, Varenka?\n\nMakar Devu\u015fkin\n\n25 Haziran\n\nSayg\u0131de\u011fer Makar Alekseyevi\u00e7,\n\nSize kitab\u0131 geri g\u00f6nderiyorum. A\u015f\u0131r\u0131 k\u00f6t\u00fc bir kitap bu! \u0130nsan\u0131n eline almamas\u0131 gerekir. Nereden buldunuz bu paha bi\u00e7ilmez \u015feyi? \u015eaka bir yana, ger\u00e7ekten be\u011feniyor musunuz b\u00f6yle kitaplar\u0131 Makar Alekseyevi\u00e7? Bana birka\u00e7 g\u00fcn i\u00e7inde okuyacak bir \u015feyler temin edeceklerini s\u00f6ylediler. Sizinle de payla\u015f\u0131r\u0131m isterseniz. \u015eimdilik ho\u015f\u00e7a kal\u0131n. Ger\u00e7ekten, yazmaya vakit yok.\n\nV.D.\n\n26 Haziran\n\nSevgili Varenka,\n\n\u0130\u015fin asl\u0131, o kitab\u0131 okumam\u0131\u015ft\u0131m, can\u0131m. Do\u011fru, biraz bak\u0131nca insanlar\u0131 g\u00fcld\u00fcrmek i\u00e7in, komiklik olsun, diye yaz\u0131lm\u0131\u015f bir s\u00fcr\u00fc \u015fey oldu\u011funu g\u00f6rd\u00fcm; niye olmas\u0131n, bence, ger\u00e7ekten komik olabilir; belki Varenka'n\u0131n ho\u015funa gider, diyerek al\u0131p size g\u00f6nderdim.\n\nNeyse, Ratazyayev okumak i\u00e7in bana hakikaten edeb\u00ee bir \u015feyler vermeye s\u00f6z verdi, o kitaplar\u0131 size de getiririm, can\u0131m. Ratazyayev bu i\u015ften anl\u0131yor, i\u015fin uzman\u0131; kendisi de yaz\u0131yor, hem de nas\u0131l yaz\u0131yor! Kalemi cesur ve \u00fcslubu engin; yani her s\u00f6zc\u00fc\u011f\u00fcnde \u2013ne olursa olsun\u2013 en bo\u015f, en s\u0131radan, seviyesiz s\u00f6zc\u00fc\u011f\u00fcnde, benim bazen Faldona ya da Tereza'ya s\u00f6yledi\u011fim t\u00fcrden s\u00f6zc\u00fcklerinde bile, kendine \u00f6zg\u00fc bir \u00fcslup var. Onun ak\u015fam toplant\u0131lar\u0131na da kat\u0131ld\u0131m. Biz t\u00fct\u00fcn i\u00e7iyoruz, o da bize okuyor, be\u015f saat boyunca okuyor, \u00f6ylece dinliyoruz. Edebiyat de\u011fil, tam bir ziyafet! \u00d6yle \u00e7ekici ki, renkler, sadece renkler; her sayfadan buket buket derleyebilir insan! \u00d6yle nazik, iyi kalpli, kibar ki kendisi. Fakat onun kar\u015f\u0131s\u0131nda ben neyim? Hi\u00e7. O \u015f\u00f6hret sahibi bir insan, ya ben? Sadece \u2013 hayata gelmi\u015fim; ama o benimle konu\u015fmaya l\u00fctfediyor. Onun i\u00e7in bir \u015feyleri temize \u00e7ekiyorum. Ama Varenka, siz bunun bir marifet oldu\u011funu, onun ben bunlar\u0131 temize \u00e7ekti\u011fim i\u00e7in bana h\u00fcrmet etti\u011fini sanmay\u0131n. Dedikodulara inanmay\u0131n can\u0131m, al\u00e7ak\u00e7a dedikodulara inanmay\u0131n! Hay\u0131r, bunu kendi iste\u011fimle, kendi irademle, onu ho\u015fnut etmek i\u00e7in yap\u0131yorum, o da beni ho\u015fnut etmek i\u00e7in l\u00fctfediyor benimle konu\u015fmaya. Bu davran\u0131\u015f\u0131n zarafetini anl\u0131yorum, can\u0131m. \u0130yi kalpli, \u00e7ok iyi kalpli bir insan ve benzersiz bir yazar.\n\nEdebiyat da \u00e7ok iyi bir \u015fey, Varenka, \u00e7ok iyi bir \u015fey; bunu onlarda ge\u00e7en \u00fc\u00e7\u00fcnc\u00fc g\u00fcn\u00fcmde anlad\u0131m. Derin bir \u015fey! \u0130nsanlar\u0131n kalplerini g\u00fc\u00e7lendiren, e\u011fiten bir \u015fey ve onlar\u0131n elindeki kitapta da bu konuda bir\u00e7ok \u015fey yaz\u0131lm\u0131\u015f. \u00c7ok g\u00fczel yaz\u0131lm\u0131\u015f! Edebiyat bir tablo, yani bir t\u00fcr tablo ve ayna; ifade tutkusu, ince bir ele\u015ftiri, edebe y\u00f6nelik bir e\u011fitim ve bir belge. Onlarda b\u00fct\u00fcn bunlar\u0131 g\u00f6rd\u00fcm. Can\u0131m, size a\u00e7\u0131k\u00e7a s\u00f6yl\u00fcyorum, aralar\u0131nda oturup (ne yaz\u0131k ki, onlar gibi pipo da i\u00e7ip) dinliyorum; ama onlar \u00e7e\u015fitli konularda yar\u0131\u015f eder gibi tart\u0131\u015fmaya ba\u015flad\u0131lar m\u0131, o zaman tam anlam\u0131yla susup kal\u0131yorum, can\u0131m, sizle ben b\u00f6yle ortamda hemen susup kal\u0131r\u0131z. O anda tam bir k\u00fct\u00fck gibi oluyorum, kendimden utan\u0131yorum, b\u00fct\u00fcn gece sa\u011fa sola bak\u0131n\u0131yorum, genel bir konuya iki-\u00fc\u00e7 s\u00f6zc\u00fck katmak istiyorum, ama iki-\u00fc\u00e7 s\u00f6zc\u00fck bile \u00e7\u0131km\u0131yor! Ve kendime ac\u0131yorum Varenka, b\u00f6yle biri olmad\u0131\u011f\u0131ma \u00fcz\u00fcl\u00fcyorum; atas\u00f6z\u00fcnde dedikleri gibi, b\u00fcy\u00fcm\u00fc\u015f\u00fcm, ama adam olamam\u0131\u015f\u0131m. Zaten \u015fimdi de bo\u015f vaktimde ne yap\u0131yorum? Aptal aptal uyuyorum. Bo\u015f yere uyumak yerine iyi bir \u015feyle u\u011fra\u015fabilirdim; oturup yaz\u0131 yazard\u0131m. Hem bana yarar\u0131 olurdu, hem ba\u015fkalar\u0131na. Can\u0131m, \u00fcstelik, ne kadar kazand\u0131klar\u0131n\u0131 bir g\u00f6rseniz, Tanr\u0131 affetsin onlar\u0131! Misal, Ratazyayev, sizce ne kadar kazan\u0131yor olabilir? Bir sayfa yazmak onun i\u00e7in ne ki? Her g\u00fcn be\u015f sayfa yaz\u0131yor, dediklerine g\u00f6re de sayfa ba\u015f\u0131na \u00fc\u00e7 y\u00fcz ruble al\u0131yor. Sayfaya bir anekdot ya da merakl\u0131 bir \u015fey kondurdu mu, tamam, i\u015fte be\u015f y\u00fcz, ister ver ister verme, sen bilirsin, istersen verme! Ama o bile de\u011fil, her seferinde binli\u011fi cebe indiriyoruz! Nas\u0131lm\u0131\u015f, Varvara Alekseyevna! Ya! Adam\u0131n bir \u015fiir defteri var, hem de k\u0131sa k\u0131sa \u015fiirler; yedi bin, can\u0131m, yedi bin istiyor, d\u00fc\u015f\u00fcnsenize. Bu da sabit m\u00fclk, kocaman bir bina demektir! Be\u015f bin veren olmu\u015f, diyorlar, ama vermemi\u015f kitab\u0131. Onu kand\u0131rmaya \u00e7al\u0131\u015f\u0131yorum, al\u0131n i\u015fte, canca\u011f\u0131z\u0131m, be\u015f bin al\u0131n onlardan sonra da t\u00fck\u00fcr\u00fcrs\u00fcn\u00fcz suratlar\u0131na diyorum, sonu\u00e7ta s\u00f6zkonusu para be\u015f bin! Hay\u0131r, diyor, yedi verecek o d\u00fczenbazlar. Bu kadar a\u00e7\u0131kg\u00f6z, yani!\n\nCan\u0131m, laf\u0131 uzatmay\u0131p size _\u0130talyan Tutkular\u0131_ 'ndan bir yer aktaray\u0131m. Eserine bu ismi vermi\u015f. \u0130\u015fte okuyun Varenka, siz karar verin.\n\n> ...Vladimir irkildi ve i\u00e7indeki tutkular delice k\u00f6p\u00fcrd\u00fc, kan\u0131 alev alev oldu.\n> \n> \"Kontes!\" diye ba\u011f\u0131rd\u0131, \"Kontes! Bunun ne korkun\u00e7 bir tutku oldu\u011funu, bu delili\u011fin ne s\u0131n\u0131rs\u0131z oldu\u011funu biliyor musunuz? Hay\u0131r, hayallerim beni kand\u0131rmad\u0131! Seviyorum, seviyorum dizginsizce, \u00e7\u0131lg\u0131nca, delice! Kocan\u0131z\u0131n b\u00fct\u00fcn kan\u0131 aksa bile benim ruhumun bu \u00e7\u0131lg\u0131n, kaynayan heyecan\u0131n\u0131 s\u00f6nd\u00fcremez! K\u00fcl olmu\u015f g\u00f6nl\u00fcm\u00fc yak\u0131p kavuran cehennem ate\u015fini hi\u00e7bir \u015fey durduramaz. Ah Zinayda, Zinayda!..\"\n> \n> \"Vladimir,\" diye f\u0131s\u0131ldad\u0131 kendini kaybeden Kontes ve omzuna yasland\u0131...\n> \n> \"Zinayda!\" diye ba\u011f\u0131rd\u0131 co\u015fku i\u00e7indeki Smelski. Derin bir i\u00e7 \u00e7ekti. A\u015fk suna\u011f\u0131ndaki yang\u0131n parlak bir alevle canland\u0131 ve talihsiz kurbanlar\u0131n\u0131n g\u00f6\u011fs\u00fcn\u00fc yalad\u0131.\n> \n> \"Vladimir!\" diye f\u0131s\u0131ldad\u0131 kendinden ge\u00e7en Kontes. G\u00f6\u011fs\u00fc inip inip kalk\u0131yordu, yanaklar\u0131 k\u0131pk\u0131rm\u0131z\u0131yd\u0131, g\u00f6zleri yan\u0131yordu...\n> \n> Yeni ve korkun\u00e7 bir d\u00fc\u011f\u00fcn ger\u00e7ekle\u015fti!\n> \n> Yar\u0131m saat sonra ya\u015fl\u0131 kont kar\u0131s\u0131n\u0131n odas\u0131na girdi.\n> \n> \"Bir tanem, sayg\u0131de\u011fer misafirine bir semaver haz\u0131rlanmas\u0131n\u0131 emretmeyecek misin?\" diye sordu kar\u0131s\u0131n\u0131n yana\u011f\u0131ndan bir makas alarak.\n\n\u0130\u015fte b\u00f6yle, \u015fimdi soray\u0131m size can\u0131m, nas\u0131l buldunuz? Do\u011frusu biraz a\u00e7\u0131k sa\u00e7\u0131k, buna diyecek yok, ama ona ra\u011fmen iyi. \u0130yi oldu\u011funa laf yok! Ama durun, izin verin size _Yermak ile Z\u00fcleyha_ hik\u00e2yesinden bir yer aktaray\u0131m.\n\nSibirya'n\u0131n vah\u015fi ve \u00fcrk\u00fct\u00fcc\u00fc sava\u015f\u00e7\u0131s\u0131 olan Kazak \u00f6nder Yermak'\u0131n, esir ald\u0131\u011f\u0131 Sibir Han\u0131 K\u00fc\u00e7\u00fcm'\u00fcn k\u0131z\u0131 Z\u00fcleyha'ya \u00e2\u015f\u0131k oldu\u011funu d\u00fc\u015f\u00fcn\u00fcn, can\u0131m. Anlam\u0131\u015fs\u0131n\u0131zd\u0131r, bu olay tam Korkun\u00e7 \u0130van zaman\u0131nda ge\u00e7iyor. Yermak ile Z\u00fcleyha \u015f\u00f6yle konu\u015fuyorlar:\n\n> \"Beni seviyorsun Z\u00fcleyha! Ah, s\u00f6yle, s\u00f6yle!..\"\n> \n> \"Seni seviyorum Yermak,\" diye f\u0131s\u0131ldad\u0131 Z\u00fcleyha.\n> \n> \"Ey g\u00f6ky\u00fcz\u00fc, ey yery\u00fcz\u00fc, te\u015fekk\u00fcrler size! Mutluyum!.. Co\u015fkun ruhumun gen\u00e7lik \u00e7a\u011f\u0131ndan beri arad\u0131\u011f\u0131 her \u015feyi, her \u015feyi verdiniz bana. Rehber y\u0131ld\u0131z\u0131m sen getirdin beni buraya; bak sen beni ne i\u00e7in getirmi\u015fsin buraya, ey \u00c7obany\u0131ld\u0131z\u0131! B\u00fct\u00fcn d\u00fcnyaya g\u00f6sterece\u011fim Z\u00fcleyha'm\u0131 ve insanlar, o kudurmu\u015f canavarlar cesaret edemeyecek beni su\u00e7lamaya! Ah, ke\u015fke onun zarif ruhunun bu gizli ac\u0131lar\u0131n\u0131 anlayabilselerdi, ke\u015fke benim Z\u00fcleyha'm\u0131n tek bir g\u00f6zya\u015f\u0131nda b\u00fct\u00fcn bir \u015fiiri g\u00f6rebilselerdi! Ah, izin ver \u00f6p\u00fcc\u00fcklerle kurutay\u0131m o g\u00f6zya\u015f\u0131n\u0131, izin ver i\u00e7eyim onu, bu ilah\u00ee... bu do\u011fa\u00fcst\u00fc g\u00f6zya\u015f\u0131n\u0131!\"\n> \n> \"Yermak!\" dedi Z\u00fcleyha, \"D\u00fcnya k\u00f6t\u00fc, insanlar adaletsiz! Kovacaklar bizi, yarg\u0131layacaklar bizi sevgili Yermak'\u0131m! Sibirya'n\u0131n karlar\u0131 aras\u0131nda yeti\u015fmi\u015f zavall\u0131 bir gen\u00e7 k\u0131z ne yapacak senin baban\u0131n yurdunda, sizin so\u011fuk, dondurucu, ruhsuz, onurlu d\u00fcnyan\u0131zda? \u0130nsanlar istemeyecek beni, can\u0131m\u0131n i\u00e7i, sevgilim benim!\"\n> \n> \"O zaman Kazak k\u0131l\u0131c\u0131 onlar\u0131 ikiye kesip \u0131sl\u0131k \u00e7alacak havada!\" diye hayk\u0131rd\u0131 Yermak g\u00f6zlerini devirerek.\n\nYa\u015fl\u0131 k\u00f6r K\u00fc\u00e7\u00fcm Han, bir gece karanl\u0131\u011f\u0131ndan yararlanarak Yermak'\u0131n \u00e7ad\u0131r\u0131na s\u00fcz\u00fcl\u00fcyor; elinden asas\u0131n\u0131 ve tac\u0131n\u0131 alan Yermak'a \u00f6l\u00fcmc\u00fcl bir darbe indirmek iste\u011fiyle kendi k\u0131z\u0131n\u0131 b\u0131\u00e7akl\u0131yor. \u015eimdi Z\u00fcleyha'n\u0131n b\u0131\u00e7akland\u0131\u011f\u0131n\u0131 \u00f6\u011frendi\u011fi zaman Yermak'\u0131n ne yapt\u0131\u011f\u0131na bak\u0131n, Varenka:\n\n> \"Demiri ta\u015fta bilemeye bay\u0131l\u0131yorum!\" diye hayk\u0131rd\u0131 Yermak vah\u015fi bir azg\u0131nl\u0131kla, Bulat b\u0131\u00e7a\u011f\u0131n\u0131 \u015faman ta\u015f\u0131na s\u00fcrterken. \"Bana kan laz\u0131m, kan! Kesip bi\u00e7ece\u011fim, kesip bi\u00e7ece\u011fim, kesip bi\u00e7ece\u011fim!!!\"\n\nB\u00fct\u00fcn bunlar\u0131n ard\u0131ndan, Z\u00fcleyha's\u0131n\u0131n ard\u0131ndan ya\u015famaya g\u00fc\u00e7 bulamayan Yermak, kendini \u0130rti\u015f Irma\u011f\u0131'na atar ve b\u00f6ylece her \u015fey sona erer.\n\nTabii, \u00f6rne\u011fin, k\u00fc\u00e7\u00fck bir al\u0131nt\u0131 vereyim, \u015fakac\u0131 bir anlat\u0131m tarz\u0131 var, kasten g\u00fcld\u00fcrmek \u00fczere yaz\u0131lm\u0131\u015f:\n\n> \"Bay \u0130van Prokofyevi\u00e7 Jyoltopuz'u bilir misiniz? Bay Prokofi \u0130vanovi\u00e7'i aya\u011f\u0131ndan \u0131s\u0131ran hani. Bay \u0130van Prokofyevi\u00e7 sert karakterli bir insand\u0131r, ama buna ra\u011fmen nadir bulunan hay\u0131rseverlerdendir; buna kar\u015f\u0131l\u0131k, Bay Prokofi \u0130vanovi\u00e7 ball\u0131 turbu \u00e7ok sever. \u0130\u015fte, Bayan Pelageya Antonovna'yla ahbapl\u0131k etti\u011fi zamanlarda... Peki siz Bayan Pelageya Antonovna'y\u0131 tan\u0131r m\u0131s\u0131n\u0131z? Hani ete\u011fini hep ters giyen han\u0131m\u0131.\"\n\nBu tabii mizah Varenka, saf mizah! Bunu bize okudu\u011fu zaman g\u00fclmekten k\u0131r\u0131ld\u0131k. B\u00f6yledir i\u015fte o, Tanr\u0131 affetsin! Tabii bu yaz\u0131 biraz matrak, fazlas\u0131yla da yaramaz, can\u0131m, ama buna ra\u011fmen masum bir yaz\u0131, en k\u00fc\u00e7\u00fck bir serbest fikir ve liberal d\u00fc\u015f\u00fcnce yok i\u00e7inde. Can\u0131m, Ratazyayev'in \u00e7ok terbiyeli oldu\u011funu ve bu y\u00fczden de ba\u015fka yazarlara benzemedi\u011fini, muhte\u015fem bir yazar oldu\u011funu s\u00f6ylemek laz\u0131m. Asl\u0131nda, insan\u0131n akl\u0131na bazen b\u00f6yle bir fikir gelir... Yani ben de bir \u015feyler yazsayd\u0131m, ne olacakt\u0131 o zaman? Varsayal\u0131m, durup dururken \"Makar Devu\u015fkin'in \u015eiirleri\" diye bir kitap \u00e7\u0131kt\u0131 g\u00fcn \u0131\u015f\u0131\u011f\u0131na! Ne derdiniz o zaman mele\u011fim? Bunu nas\u0131l kar\u015f\u0131lard\u0131n\u0131z, ne d\u00fc\u015f\u00fcn\u00fcrd\u00fcn\u00fcz? Kendi ad\u0131ma \u015f\u00f6yle s\u00f6yleyeyim can\u0131m, ben kitab\u0131m \u00e7\u0131kt\u0131ktan sonra kalk\u0131p Neva Bulvar\u0131'nda g\u00f6r\u00fcnmeye cesaret edemezdim. Orada herkes birdenbire, \"\u0130\u015fte bir edebiyat eseri sahibi ve \u015fair Devu\u015fkin geliyor, i\u015fte bu gelen Devu\u015fkin'in ta kendisi!\" dese ne yapard\u0131m! \u00d6rne\u011fin, \u00e7izmelerimi ne yapard\u0131m o zaman? Laf aras\u0131nda s\u00f6yleyeyim, can\u0131m, \u00e7izmelerim ba\u015ftan a\u015fa\u011f\u0131 yama kapl\u0131, pen\u00e7e kapl\u0131, do\u011frusunu s\u00f6ylemek gerekirse, genellikle \u00e7ok tats\u0131z bir durumda oluyorlar. \u0130\u015fte o zaman, herkes edebiyat\u00e7\u0131 Devu\u015fkin'in \u00e7izmelerinin pen\u00e7e pen\u00e7e oldu\u011funu \u00f6\u011frenirdi! Oradaki bir kontes, bir d\u00fc\u015fes bunu \u00f6\u011frenir \u00f6\u011frenmez kim bilir neler s\u00f6ylerdi? Belki de fark etmezdi bile; \u00e7\u00fcnk\u00fc, tahminimce, kontesler \u00e7izmelerle ilgilenmez, hele memurlar\u0131n \u00e7izmeleriyle hi\u00e7 ilgilenmez (sonu\u00e7ta \u00e7izme var, \u00e7izme var), ama ona her \u015feyi anlat\u0131rlard\u0131, onun arkada\u015flar\u0131 beni g\u00f6r\u00fcrlerdi. Hatta ilk olarak Ratazyayev g\u00f6r\u00fcrd\u00fc; Kontes V.ye gidip geliyor o; ona ne zaman istese gitti\u011fini s\u00f6yl\u00fcyor. \u00c7ok tatl\u0131 oldu\u011funu, edebiyattan anlad\u0131\u011f\u0131n\u0131, tam bir kad\u0131n oldu\u011funu anlat\u0131yor. Yap\u0131\u015fkand\u0131r bu Razatyayev!\n\nEvet, fakat, bu konuda bu kadar gevezelik yeter; zaten b\u00fct\u00fcn bunlar\u0131 sizi e\u011flendirmek i\u00e7in, laf olsun diye yaz\u0131yorum mele\u011fim. Ho\u015f\u00e7a kal\u0131n, g\u00fcvercinim! Burada bir s\u00fcr\u00fc \u015fey karalad\u0131m size, ama s\u0131rf bug\u00fcn \u00e7ok ne\u015feli bir ruh halinde oldu\u011fum i\u00e7in yapt\u0131m bunu. Bug\u00fcn hep birlikte Ratazyayev'de yemek yedik, derken (\u0130nsan\u0131 kand\u0131r\u0131yorlar, can\u0131m!) kaliteli bir \u015farap \u00e7\u0131kard\u0131lar... Bunu size ne diye yazd\u0131m! L\u00fctfen benim hakk\u0131mda yanl\u0131\u015f bir \u015fey d\u00fc\u015f\u00fcnmeyin Varenka. B\u00fct\u00fcn bunlar \u00f6ylesine \u015feyler. Kitap g\u00f6nderece\u011fim, hemen g\u00f6nderece\u011fim... Burada elime Paul de Kock'un14 bir eseri ge\u00e7ti, ama Paul de Kock size g\u00f6re de\u011fil can\u0131m, asla olamaz... Yo-yo! Size g\u00f6re olamaz Paul de Kock. Can\u0131m, Petersburglu ele\u015ftirmenlerin hepsinde soylu bir ho\u015fnutsuzluk uyand\u0131rd\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyorlar onun. Size bir kutucuk \u015feker yolluyorum, sadece sizin i\u00e7in ald\u0131m. Ama bonbon \u015fekerini kemirmeyin, onu emin, yoksa di\u015fleriniz a\u011fr\u0131r. Ama siz belki de meyve \u015fekerlemesi seviyorsunuzdur; yaz\u0131n bunu. Neyse, ho\u015f\u00e7a kal\u0131n, ho\u015f\u00e7a kal\u0131n. \u0130sa sizinle olsun, g\u00fcvercinim.\n\nSonsuza dek sad\u0131k dostunuz,  \nMakar Devu\u015fkin\n\n27 Haziran\n\nSevgili Beyefendi Makar Alekseyevi\u00e7,\n\nFedora e\u011fer istersem baz\u0131 kimselerin bana yard\u0131mc\u0131 olmaktan mutluluk duyaca\u011f\u0131n\u0131 ve bana bir evde, m\u00fcrebbiye olarak \u00e7ok daha iyi bir yer bulabilece\u011fini s\u00f6yl\u00fcyor. Ne dersiniz, dostum \u2013 gitsem mi, gitmesem mi? Elbette, o zaman size y\u00fck olmayaca\u011f\u0131m ve yer de, herhalde, uygun bir yer olacak; ama, di\u011fer yandan, yabanc\u0131 bir eve gitmek biraz zor bir \u015fey. Toprak sahibiymi\u015fler. Benim hakk\u0131mda bilgi almaya, soru sormaya, merak etmeye ba\u015flayacaklar; ne s\u00f6yleyece\u011fim o zaman? Ben hep b\u00f6yle \u00fcrkek, yabani biriyim; al\u0131\u015ft\u0131\u011f\u0131m bir yerde uzun s\u00fcre ya\u015famay\u0131 severim. \u0130nsan\u0131n al\u0131\u015ft\u0131\u011f\u0131 yer iyidir; ac\u0131yla ya\u015fasan da, her \u015feye ra\u011fmen daha iyidir. Kim bir yerden gitmek ister; Tanr\u0131 bilir ne i\u015fler \u00e7\u0131kaca\u011f\u0131n\u0131; belki, sadece \u00e7ocuklara dad\u0131l\u0131k yapt\u0131r\u0131rlar. Bu insanlar b\u00f6yle: \u0130ki y\u0131lda \u00fc\u00e7 m\u00fcrebbiye de\u011fi\u015ftirmi\u015fler.\n\nTanr\u0131 a\u015fk\u0131na, ne tavsiye edersiniz Makar Alekseyevi\u00e7, gitmeli mi, gitmemeli mi? Siz neden hi\u00e7 gelmiyorsunuz bana? Nadiren y\u00fcz y\u00fcze geliyoruz. Neredeyse sadece pazarlar\u0131 ayinde g\u00f6r\u00fc\u015f\u00fcyoruz. Ne kadar \u00fcrkeksiniz b\u00f6yle! T\u0131pk\u0131 benim gibisiniz! Fakat akraban\u0131z say\u0131l\u0131r\u0131m. Siz beni sevmiyorsunuz Makar Alekseyevi\u00e7, ben bazen \u00e7ok kederleniyorum. Bazen, \u00f6zellikle de hava karar\u0131nca, tek ba\u015f\u0131ma otururken, Fedora bir yere gitmi\u015fken. Oturuyorum, d\u00fc\u015f\u00fcn\u00fcyorum, d\u00fc\u015f\u00fcn\u00fcyorum \u2013eskiden olan her \u015feyi, mutlu olanlar\u0131, kederli olanlar\u0131\u2013 hepsi ge\u00e7iyor g\u00f6zlerimin \u00f6n\u00fcnden, hepsi sislerin aras\u0131ndaym\u0131\u015f gibi yan\u0131p s\u00f6n\u00fcyor. Tan\u0131d\u0131k y\u00fczler beliriyor (sanki uyan\u0131kken r\u00fcya g\u00f6rmeye ba\u015fl\u0131yorum) \u2013 annemi g\u00f6r\u00fcyorum en \u00e7ok... Ne r\u00fcyalar g\u00f6r\u00fcyorum b\u00f6yle!\n\nSa\u011fl\u0131\u011f\u0131m\u0131n bozuldu\u011funu hissediyorum; g\u00fc\u00e7s\u00fczle\u015ftim; i\u015fte bug\u00fcn, sabahleyin yataktan kalk\u0131nca, bir fena oldum; \u00fcstelik, \u00f6yle k\u00f6t\u00fc bir \u00f6ks\u00fcr\u00fck var ki bende! Yak\u0131nda \u00f6lece\u011fimi hissediyorum. Beni kim g\u00f6mecek? Kim tabutumun pe\u015finden gelecek? Kim benim i\u00e7in \u00fcz\u00fclecek?.. Ve i\u015fte, belki de, yabanc\u0131 bir yerde, yabanc\u0131 bir evde, yabanc\u0131 bir k\u00f6\u015fede \u00f6lece\u011fim!.. Tanr\u0131m, ya\u015famak ne kederli \u015fey, Makar Alekseyevi\u00e7! Dostum, ne diye beni hep \u015fekerle besliyorsunuz? Do\u011frusu, ben paray\u0131 nereden buldu\u011funuzu anlam\u0131yorum? Ah, dostum, saklay\u0131n paran\u0131z\u0131, Tanr\u0131 a\u015fk\u0131na, saklay\u0131n onlar\u0131. Fedora benim \u00f6rd\u00fc\u011f\u00fcm hal\u0131y\u0131 sat\u0131yor; elli ruble veriyorlar. \u00c7ok iyi bu; ben daha az verirler san\u0131yordum. Fedora'ya \u00fc\u00e7 ruble verece\u011fim, kendime de sade, daha s\u0131cak tutan bir elbise dikece\u011fim. Size yelek yapaca\u011f\u0131m, ben yapaca\u011f\u0131m ve iyi malzeme se\u00e7ece\u011fim.\n\nFedora bana bir kitap getirdi \u2013 _Biyelkin'in Hik\u00e2yeleri_15 , okumak isterseniz size de g\u00f6ndereyim. L\u00fctfen, kirletmeyin, k\u0131v\u0131rmay\u0131n: ba\u015fkas\u0131n\u0131n kitab\u0131; Pu\u015fkin'in bir eseri bu. Bundan iki y\u0131l \u00f6nce bu hik\u00e2yeleri annemle birlikte okumu\u015ftuk, \u015fimdi onlar\u0131 yeniden okumak beni kederlendiriyor. E\u011fer sizde de kitap varsa g\u00f6nderin bana, tabii Ratazyayev'den almad\u0131ysan\u0131z. Herhalde, e\u011fer bir \u015fey yay\u0131mlad\u0131ysa, kendi eserlerini verecektir o. Onun eserlerini nas\u0131l sevebiliyorsunuz Makar Alekseyevi\u00e7? Bu bo\u015f... neyse, ho\u015f\u00e7a kal\u0131n! \u00c7ok gevezelik ettim! Kederlendi\u011fim zaman, mutluymu\u015f gibi ondan bundan gevezelik ediyorum. \u0130la\u00e7 bu; hemen ferahl\u0131yorum, hele kalbimde yatan her \u015feyi s\u00f6yleyebilirsem, iyice ferahl\u0131yorum. Ho\u015f\u00e7a kal\u0131n, ho\u015f\u00e7a kal\u0131n dostum!\n\nSizin olan,  \nV.D.\n\n28 Haziran\n\nCan\u0131m, Varvara Alekseyevna,\n\nYeter kendinize eziyet etti\u011finiz! Nas\u0131l ay\u0131p gelmiyor bu size! Yeter ama, mele\u011fim; b\u00f6yle fikirler nereden geliyor akl\u0131n\u0131za? Hasta de\u011filsiniz bir tanem, hi\u00e7 hasta de\u011filsiniz; \u00e7i\u00e7ek a\u00e7\u0131yorsunuz, ger\u00e7ekten \u00e7i\u00e7ek a\u00e7\u0131yorsunuz; biraz soldunuz, ama yine de \u00e7i\u00e7ek a\u00e7\u0131yorsunuz. Peki o r\u00fcyalar, hayaller ne \u00f6yle! Ay\u0131p ama g\u00fcvercinim, yeter; b\u0131rak\u0131n bu r\u00fcyalar\u0131, \u00f6ylece b\u0131rak\u0131n. Neden ben iyi uyuyorum? Yapacak bir i\u015fim niye yok? Bana baksan\u0131za, can\u0131m. B\u00f6yle ya\u015f\u0131yorum, rahat uyuyorum, sa\u011fl\u0131\u011f\u0131m iyi, dipdiriyim, g\u00f6ren bir daha bak\u0131yor. Yeter, yeter,bir tanem, ay\u0131p. Kendinize gelin. Sizin kafan\u0131z\u0131n i\u00e7ini biliyorum zaten, can\u0131m, bir \u015fey buldunuz mu onu hayal edip ona \u00fcz\u00fcl\u00fcp durursunuz siz. Birinin yan\u0131na girmek mi? Asla! Hay\u0131r, hay\u0131r, hay\u0131r! Neler d\u00fc\u015f\u00fcn\u00fcyorsunuz b\u00f6yle, nereden \u00e7\u0131k\u0131yor bu? Bir de \u00fcstelik ta\u015fraya! Asla olmaz, can\u0131m, izin vermem, bu niyetinize kar\u015f\u0131 b\u00fct\u00fcn silahlar\u0131m\u0131 ku\u015fan\u0131r\u0131m. Eski fra\u011f\u0131m\u0131 satar, sokakta g\u00f6mlekle dola\u015f\u0131r\u0131m, s\u0131rf siz ihtiya\u00e7 i\u00e7inde olmay\u0131n diye. Hay\u0131r, Varenka, hay\u0131r; sizi tan\u0131yorum art\u0131k! Z\u0131rva bunlar, s\u0131rf z\u0131rva! Ama herhalde, tabii, bundan hep Fedora su\u00e7lu; anla\u015f\u0131lan o aptal kocakar\u0131 size ak\u0131llar vermi\u015f. Ama can\u0131m, siz inanmay\u0131n ona. Yani bunu h\u00e2l\u00e2 anlayamad\u0131n\u0131z m\u0131 birtanem?.. Aptal, \u015firret, huysuz bir kocakar\u0131 o; merhum kocas\u0131n\u0131 da d\u00fcnyadan o g\u00f6\u00e7\u00fcrtt\u00fc. Yoksa sizi bir \u015fekilde g\u00fccendirdi mi? Hay\u0131r, hay\u0131r, can\u0131m, olmaz! Ben ne yapaca\u011f\u0131m o zaman, benim yapacak neyim kalacak? Hay\u0131r, Varenka, bir tanem, siz bunu kafan\u0131zdan \u00e7\u0131kar\u0131n. Bizim neyimiz eksik? Sizsiz biz mutlu olamay\u0131z, siz bizi seviyorsunuz, bu sayede kendimle bar\u0131\u015f\u0131k ya\u015f\u0131yorum; ister diki\u015f yap\u0131n ister kitap okuyun, ama diki\u015f yapmay\u0131 b\u0131rakmay\u0131n, fark etmez, yeter ki bizimle ya\u015fay\u0131n. Bir de d\u00fc\u015f\u00fcnsenize, oras\u0131 kim bilir neye benzer?.. Size kitap getirece\u011fim, sonra yine bir yere gitmek i\u00e7in bulu\u015faca\u011f\u0131z. Yeter ki siz durun, can\u0131m, durun, akl\u0131n\u0131z\u0131 ba\u015f\u0131n\u0131za toplay\u0131n ve bo\u015f yere kayg\u0131lanmay\u0131n! Size gelece\u011fim, hem de \u00e7ok yak\u0131nda, ama siz de benim do\u011frudan ve a\u00e7\u0131k \u00e7a\u011fr\u0131m\u0131 kabul edeceksiniz; iyi de\u011fil, bir tanem, hi\u00e7 iyi de\u011fil! Ben, elbette, e\u011fitimsiz bir insan\u0131m ve e\u011fitimsiz oldu\u011fumu, k\u00f6t\u00fc bir okula gitti\u011fimi biliyorum, bu y\u00fczden dilim iyi de\u011fildir, benim i\u015fim de de\u011fil, ama Ratazyayev'i savunaca\u011f\u0131m, izninizle. Benim dostum, bu y\u00fczden onu savunaca\u011f\u0131m. \u0130yi yaz\u0131yor, \u00e7ok, \u00e7ok ve yine \u00e7ok iyi yaz\u0131yor. S\u00f6yledi\u011finizi kabul etmiyorum ve asla da etmeyece\u011fim. Renkli, kesik kesik, fig\u00fcrlerle yazm\u0131\u015f, t\u00fcrl\u00fc fikirleri var; \u00e7ok iyi! Siz, belki de, duygu katmadan okudunuz Varenka ya da okurken keyfiniz yoktu, Fedora'ya bir \u015fey i\u00e7in k\u0131zm\u0131\u015ft\u0131n\u0131z ya da tats\u0131z bir hal vard\u0131 \u00fczerinizde. Hay\u0131r, bir daha duyguyla okuyun, daha iyi olacak, ho\u015fnut ve ne\u015feliyken, ho\u015f bir ruh hali i\u00e7indeyken, s\u00f6zgelimi, a\u011fz\u0131n\u0131zda bir \u015fekeri emerken, i\u015fte o zaman okuyun. Tart\u0131\u015fm\u0131yorum (kim kar\u015f\u0131 \u00e7\u0131kabilir), Ratazyayev'den iyi yazarlar var, hatta \u00e7ok iyileri var, ama onlar da iyi, Ratazyayev de iyi; iyi yaz\u0131yor ve iyi yazacak. Kendine has biri, kendince yaz\u0131yor ve b\u00f6yle yaparak \u00e7ok iyi yap\u0131yor. Ama, elveda can\u0131m; daha fazla yazamayaca\u011f\u0131m; acele etmem laz\u0131m, i\u015f var. Baksan\u0131za anac\u0131\u011f\u0131m, tatl\u0131 sevdi\u011fim, sakin olun ve Tanr\u0131 ne olaca\u011f\u0131n\u0131 belirlesin, \u015fimdi kesiyorum.\n\nSad\u0131k dostunuz,  \nMakar Devu\u015fkin\n\nHami\u015f: Kitap i\u00e7in sa\u011folun can\u0131m benim, Pu\u015fkin'i de okuyal\u0131m; bug\u00fcn de\u011fil, yar\u0131n gece, dosdo\u011fru size gelece\u011fim.\n\n9. Charles Fran\u00e7oise L'Homond (1792-1794) taraf\u0131ndan haz\u0131rlanan _\u00c9l\u00e9mens de la grammaire fran\u00e7aise_ (1797) adl\u0131 Frans\u0131zca gramer kitab\u0131. 1831 y\u0131l\u0131nda Rusya'da yay\u0131mlanm\u0131\u015ft\u0131.\n\n10. V. Zapolski'nin _Novaya u\u00e7obnaya kniga dlya frantsuzkova yaz\u0131ka_ (Yeni Frans\u0131zca Ders Kitab\u0131, 1817) adl\u0131 kitab\u0131 kastediliyor.\n\n11. (Fr.) Diyalog.\n\n12. Pu\u015fkin'in \u00f6l\u00fcm\u00fcnden sonra, 1838-1841y\u0131llar\u0131 aras\u0131nda, on bir cilt olarak yay\u0131mlanan toplu eserleri kastediliyor.\n\n13. (Rus.) Ortodoks gelene\u011finde \u00f6l\u00fcn\u00fcn aln\u0131na konan, \u00fczerinde Hz. \u0130sa'n\u0131n tasvirinin oldu\u011fu kurdele.\n\n14. Charles-Paul de Kock (1793-1871). Paris'teki ya\u015fam\u0131 yans\u0131tan pornografik romanlar\u0131 o d\u00f6nemde Avrupa'da ilgiyle okunan Frans\u0131z yazar.\n\n15. _Biyelkin'in Hik\u00e2yeleri_ 1831, 1834 ve 1838'de olmak \u00fczere \u00fc\u00e7 bask\u0131 yapm\u0131\u015ft\u0131.\n\n# Temmuz\n\n1 Temmuz\n\nAziz Makar Alekseyevi\u00e7,\n\nHay\u0131r, dostum, hay\u0131r aran\u0131zda ya\u015famam imk\u00e2ns\u0131z. Uzun uzun d\u00fc\u015f\u00fcnd\u00fcm ve b\u00f6yle uygun bir i\u015fi geri \u00e7evirmenin \u00e7ok yanl\u0131\u015f olaca\u011f\u0131na karar verdim. Orada benim i\u00e7in en az\u0131ndan bir lokma ekmek gelece\u011finden emin olaca\u011f\u0131m; \u00e7al\u0131\u015f\u0131p \u00e7abalayacak, yabanc\u0131 insanlar\u0131n \u015fefkatini kazanmaya \u00e7al\u0131\u015faca\u011f\u0131m, hatta \u00f6ylesi uygun olacaksa kendi karakterimi de\u011fi\u015ftirmeye \u00e7al\u0131\u015faca\u011f\u0131m. Yabanc\u0131lar aras\u0131nda ya\u015famak, yabanc\u0131 bir merhamet aramak, kendini gizlemek ve zorlamak zor ve a\u011f\u0131r bir \u015fey olacak elbette, Tanr\u0131 yard\u0131mc\u0131m olsun. Sonsuza dek kimsesiz kalamam. Ba\u015f\u0131mdan b\u00f6yle olaylar ge\u00e7mi\u015fti. Hat\u0131rl\u0131yorum, daha k\u00fc\u00e7\u00fck bir \u00e7ocukken yat\u0131l\u0131 okula gidip gelirdim. Pazar g\u00fcnleri evde b\u00fct\u00fcn g\u00fcn hoplay\u0131p s\u0131\u00e7rard\u0131m, her seferinde de k\u0131zard\u0131 annem, g\u00fccenmezdim; kalbimde her \u015fey iyi, ruhumda her \u015fey ayd\u0131nl\u0131k olurdu. Ak\u015fam olurdu ve \u00f6l\u00fcmc\u00fcl bir keder \u00e7\u00f6kerdi i\u00e7ime, saat dokuzda yat\u0131l\u0131 okula gitmem gerekirdi, orada da her \u015fey yabanc\u0131, so\u011fuk, kat\u0131 olurdu, m\u00fcrebbiyeler pazartesi g\u00fcnleri \u00e7ok ac\u0131mas\u0131z olurdu, kalbim k\u0131r\u0131l\u0131rd\u0131, a\u011flamak isterdim; bir k\u00f6\u015feye ge\u00e7erdim, tek ba\u015f\u0131ma yapayaln\u0131z a\u011flard\u0131m, g\u00f6zya\u015flar\u0131m\u0131 saklard\u0131m, bana tembel derlerdi; ama asla bu y\u00fczden, okumak zorunda oldu\u011fum i\u00e7in a\u011flamazd\u0131m. Peki ne oldu? Al\u0131\u015ft\u0131m ve sonra, yat\u0131l\u0131 okuldan \u00e7\u0131kt\u0131\u011f\u0131m zaman, arkada\u015flar\u0131mla vedala\u015f\u0131rken yine a\u011flad\u0131m. Sizin ikinize y\u00fck olarak ya\u015famakla iyi bir \u015fey yapm\u0131yorum. Bu d\u00fc\u015f\u00fcnce bana azap veriyor. Size b\u00fct\u00fcn bunlar\u0131 a\u00e7\u0131k\u00e7a s\u00f6yl\u00fcyorum, \u00e7\u00fcnk\u00fc size kar\u015f\u0131 a\u00e7\u0131k olmaya al\u0131\u015ft\u0131m. Sanki g\u00f6rm\u00fcyor muyum Fedora'n\u0131n nas\u0131l her sabah erkenden kalkt\u0131\u011f\u0131n\u0131, \u00e7ama\u015f\u0131r y\u0131kamaya ba\u015flay\u0131p gece ge\u00e7 vakitlere kadar \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131. Oysa ya\u015fl\u0131 kemiklere laz\u0131m olan \u015fey huzurdur. Sanki g\u00f6rm\u00fcyor muyum sizin benim i\u00e7in var\u0131n\u0131z\u0131 yo\u011funuzu harcad\u0131\u011f\u0131n\u0131z\u0131, bir kenara ay\u0131rd\u0131\u011f\u0131n\u0131z son kope\u011finize kadar hepsini benim i\u00e7in harcad\u0131\u011f\u0131n\u0131z\u0131. Sizin servetinizle olmaz, dostum. Her \u015feyinizi verece\u011finizi, beni darda b\u0131rakmayaca\u011f\u0131n\u0131z\u0131 yaz\u0131yorsunuz. \u0130nan\u0131yorum dostum, iyi kalbinize inan\u0131yorum; ama siz de \u015fimdi b\u00f6yle s\u00f6yl\u00fcyorsunuz. \u015eimdi elinizde beklenmedik paralar var, ikramiye ald\u0131n\u0131z; ama sonra ne olacak, sonra? Siz de biliyorsunuz, ben hep hastay\u0131md\u0131r; ben sizin gibi \u00e7al\u0131\u015famam, oysa \u00e7ok isterdi ruhum, hem i\u015f de bulunmaz hep. Ne yapaca\u011f\u0131m \u00f6yleyse? \u00dcz\u00fcnt\u00fcyle m\u00fccadele etmek, g\u00f6n\u00fclden sevdi\u011fim ikinize bakmak. K\u00fc\u00e7\u00fcc\u00fck bile olsa size nas\u0131l yararl\u0131 olabilirim? \u00dcstelik ben sizin i\u00e7in neden b\u00f6yle vazge\u00e7ilmezim, dostum? Ben sizin i\u00e7in iyi ne yapt\u0131m? Tek yapt\u0131\u011f\u0131m b\u00fct\u00fcn ruhumla size ba\u011flanmak oldu, sizi sa\u011flam, g\u00fc\u00e7l\u00fc bir \u015fekilde, b\u00fct\u00fcn kalbimle sevmek oldu, ama \u2013Makus talihim!\u2013 sevme yetene\u011fim var ve sevebiliyorum, ama iyilik yapamam, iyiliklerinizin kar\u015f\u0131l\u0131\u011f\u0131n\u0131 size \u00f6deyemem. Beni daha fazla dizginlemeyin, d\u00fc\u015f\u00fcn\u00fcn ve son g\u00f6r\u00fc\u015f\u00fcn\u00fcz\u00fc s\u00f6yleyin. Yan\u0131t bekleyerek burada kesiyorum.\n\nSizi seven,  \nV.D.\n\n1 Temmuz\n\nZ\u0131rva, z\u0131rva Varenka, sadece z\u0131rva! Sizi \u00f6ylece b\u0131raksak o k\u00fc\u00e7\u00fck kafan\u0131zla neler neler uyduracaks\u0131n\u0131z. O \u00f6yleymi\u015f de bu da b\u00f6yleymi\u015f de! Ben art\u0131k iyice g\u00f6r\u00fcyorum b\u00fct\u00fcn bunlar\u0131n heves oldu\u011funu. Bizim neyimiz yetmiyor size, can\u0131m, siz onu s\u00f6yleyin! Sizi seviyoruz, siz bizi seviyorsunuz, hepimiz ho\u015fnutuz, mutluyuz... daha ne olsun? Hem, yabanc\u0131 insanlar\u0131n aras\u0131nda ne yapacaks\u0131n\u0131z? Siz herhalde daha bilmiyorsunuz yabanc\u0131n\u0131n ne demek oldu\u011funu?.. Hay\u0131r, izin verirseniz ben anlatay\u0131m size yabanc\u0131n\u0131n ne demek oldu\u011funu. Onu tan\u0131yorum, can\u0131m, ben \u00e7ok iyi tan\u0131yorum; onun ekme\u011fini yedim ben. K\u00f6t\u00fcd\u00fcr o, Varenka, k\u00f6t\u00fcd\u00fcr, hem de \u00f6yle k\u00f6t\u00fcd\u00fcr ki sizin k\u00fc\u00e7\u00fck kalbiniz katlanamaz, sitemle, yak\u0131nmayla ve bak\u0131\u015flar\u0131yla k\u00f6t\u00fcl\u00fck sa\u00e7ar. Siz \u015fimdi s\u0131caktas\u0131n\u0131z, iyisiniz, yuvan\u0131zda gibisiniz. Bizi de ba\u015fs\u0131z b\u0131rak\u0131rs\u0131n\u0131z siz. Biz sizsiz ne yapar\u0131z; benim gibi bir ihtiyar ne yapar o zaman? Siz bize laz\u0131m de\u011fil misiniz? Yararl\u0131 de\u011fil misiniz? Nas\u0131l de\u011filsiniz? Hay\u0131r, can\u0131m, kendi k\u0131ymetinizi bilin, nas\u0131l yararl\u0131 de\u011filsiniz? Bana \u00e7ok yararl\u0131s\u0131n\u0131z, Varenka. Benim \u00fczerimde \u00e7ok hay\u0131rl\u0131 bir etkiniz var... Misal, \u015fimdi sizi d\u00fc\u015f\u00fcn\u00fcyorum, ne\u015feleniyorum... Size ne zaman mektup yaz\u0131p i\u00e7ine duygular\u0131m\u0131 koysam, sizden ayr\u0131nt\u0131l\u0131 bir yan\u0131t al\u0131yorum. Size giyecek bir \u015feyler al\u0131yorum, \u015fapka yapt\u0131rd\u0131m; sizden ara s\u0131ra sipari\u015fler geliyor, sipari\u015fi de hemen... Hay\u0131r, nas\u0131l yarars\u0131z olursunuz? Yani ben tek ba\u015f\u0131ma ihtiyarl\u0131kta ne yapaca\u011f\u0131m, neye g\u00fcvenece\u011fim? Siz, belki de d\u00fc\u015f\u00fcnmemi\u015fsinizdir bunu, Varenka; hay\u0131r, siz kesinlikle bunu d\u00fc\u015f\u00fcnmediniz mi, yani o bensiz ne yapacak demediniz mi? Ben size al\u0131\u015ft\u0131m, can\u0131m\u0131n i\u00e7i. Ne olacak bu? Neva Irma\u011f\u0131'na gider, hayat\u0131ma son veririm. Evet, ger\u00e7ekten, b\u00f6yle olur Varenka; ben sizsiz ne yapar\u0131m! Ah, bir tanem, Varenka! Anla\u015f\u0131lan siz beni bir arabaya y\u00fckleyip Volkovo Mezarl\u0131\u011f\u0131'na g\u00f6t\u00fcrmelerini istiyorsunuz; orada yoksul bir ihtiyar benim mezar\u0131m\u0131 kazs\u0131n; \u00fczerime kum ats\u0131nlar, gitsinler, orada tek ba\u015f\u0131ma kalay\u0131m istiyorsunuz. G\u00fcnaht\u0131r, g\u00fcnaht\u0131r, can\u0131m! Ger\u00e7ekten, g\u00fcnaht\u0131r, Tanr\u0131 a\u015fk\u0131na, g\u00fcnaht\u0131r! Size kitab\u0131n\u0131z\u0131 g\u00f6nderiyorum, k\u00fc\u00e7\u00fck dostum benim, Varenka, e\u011fer siz, k\u00fc\u00e7\u00fck dostum, kitab\u0131n\u0131z hakk\u0131ndaki g\u00f6r\u00fc\u015f\u00fcm\u00fc soracak olursan\u0131z, hayat\u0131m boyunca b\u00f6yle harika bir kitap okumad\u0131\u011f\u0131m\u0131 s\u00f6yleyece\u011fim. Soruyorum size can\u0131m, bu vakte dek nas\u0131l bir budala gibi ya\u015fam\u0131\u015f\u0131m ben, Tanr\u0131 affetsin? Ne yapm\u0131\u015f\u0131m? Hangi ormandan gelmi\u015fim? Hi\u00e7bir \u015fey bilmiyormu\u015fum, can\u0131m, hemen hi\u00e7bir \u015fey bilmiyormu\u015fum! Kesinlikle hi\u00e7bir \u015fey bilmiyormu\u015fum! Varenka, size a\u00e7\u0131k\u00e7a s\u00f6yl\u00fcyorum, e\u011fitimsiz biriyim ben; bu vakte dek \u00e7ok az \u015fey okudum, \u00e7ok az okudum, neredeyse hi\u00e7bir \u015fey: \"\u0130nsan\u0131n Hali\"16 , bilgi dolu bir eserdi, okudum; \"\u00c7anlarla T\u00fcrl\u00fc \u015eakalar Yapan \u00c7ocuk\"17 ile \"\u0130bykos'un Turnalar\u0131\"n\u013118 okudum \u2013 i\u015fte hepsi bu, ba\u015fka da bir \u015fey okumad\u0131m. \u015eimdi burada, verdi\u011finiz kitaptaki \"Menzil M\u00fcfeti\u015fi\"ni19 de okudum; size \u015funu s\u00f6yleyeyim, can\u0131m, insan ya\u015fay\u0131p gidiyor, ama hemen yan\u0131nda b\u00f6yle bir kitab\u0131n varl\u0131\u011f\u0131n\u0131, b\u00fct\u00fcn hayat\u0131n\u0131n i\u00e7ine ilmek ilmek i\u015flendi\u011fi bir kitab\u0131n varl\u0131\u011f\u0131n\u0131 bilmiyor. Daha \u00f6nce hi\u00e7 fark etmedi\u011fim \u015fey, i\u015fte burada, daha kitab\u0131 okumaya ba\u015flar ba\u015flamaz kar\u015f\u0131s\u0131na \u00e7\u0131k\u0131yor insan\u0131n, yava\u015f yava\u015f hat\u0131rl\u0131yor, d\u00fc\u015f\u00fcn\u00fcyor, anl\u0131yor insan. Dahas\u0131, i\u015fte \u015fu y\u00fczden sevdim sizin kitab\u0131n\u0131z\u0131: i\u00e7inde her \u015fey olan, ba\u015fka bir eseri, okuyorsun okuyorsun, kendini zorluyorsun zorluyorsun; fakat ne kadar kurnaz olursan ol, anlam\u0131yorsun. S\u00f6zgelimi, ben, k\u00fct\u00fck gibiyim, do\u011fu\u015ftan k\u00fct\u00fck gibiyim, hatta \u00e7ok \u00f6nemli eserleri okuyamam; ama bunu okudum, sanki kendim yazm\u0131\u015f\u0131m gibi, sanki bu, misal olarak s\u00f6yl\u00fcyorum, benim kendi g\u00f6nl\u00fcmden \u00e7\u0131km\u0131\u015f gibi, g\u00f6nl\u00fcmde ne varsa alm\u0131\u015f, insan\u0131 ters y\u00fcz etmi\u015f, oturup ince ince her ayr\u0131nt\u0131s\u0131yla yazm\u0131\u015f, kesinlikle b\u00f6yle olmu\u015f! Tanr\u0131m, kolay da bir i\u015f de\u011fil bu; vay vay! Ger\u00e7ekten, ben bile yazabilirdim bunu; neden yazmam\u0131\u015f\u0131m? Sonu\u00e7ta ayn\u0131 hissediyorum, kesinlikle tam kitapta yazd\u0131\u011f\u0131 gibi hissediyorum, tam olarak ayn\u0131 durumday\u0131m \u00fcstelik, misal olarak s\u00f6yl\u00fcyorum, hik\u00e2yedeki zavall\u0131 Samson V\u0131rin gibiyim. Kim bilir ka\u00e7 Samson V\u0131rin dola\u015f\u0131yor aram\u0131zda, yaral\u0131 y\u00fcre\u011fiyle! \u00dcstelik nas\u0131l rahat yazm\u0131\u015f her \u015feyi! O g\u00fcnahk\u00e2r\u0131n i\u00e7meye ba\u015flad\u0131\u011f\u0131n\u0131, haf\u0131zas\u0131n\u0131 kaybetti\u011fini, ac\u0131 \u00e7ekip koyun derisi gocu\u011fuyla b\u00fct\u00fcn g\u00fcn kafay\u0131 \u00e7ekti\u011fini, bir yandan l\u0131k\u0131r l\u0131k\u0131r i\u00e7ti\u011fini, bir yandan ac\u0131 ac\u0131 a\u011flad\u0131\u011f\u0131n\u0131, kay\u0131p koyununu, k\u0131z\u0131 Dunya\u015fa'y\u0131 hat\u0131rlad\u0131k\u00e7a da gocu\u011fun \u00e7amurlu ete\u011fiyle g\u00f6zlerini sildi\u011fini okuyunca az kals\u0131n a\u011fl\u0131yordum, can\u0131m! \u00dcstelik, \u00e7ok do\u011fal! Okuyunca g\u00f6r\u00fcrs\u00fcn\u00fcz; \u00e7ok do\u011fal! Canl\u0131 gibi! Ben bunu g\u00f6rd\u00fcm, \u00e7evremde ya\u015fayan her \u015fey var burada; i\u015fte Tereza \u2013Uza\u011fa gitmeye gerek yok!\u2013 i\u015fte bizim yoksul memur \u2013 sonu\u00e7ta o, belki de, aynen Samson V\u0131rin, sadece k\u00f6t\u00fc bir soyad\u0131 var, Gor\u015fkov. Olay ortak, can\u0131m, sizin de benim de ba\u015f\u0131m\u0131za gelebilir. Neva Bulvar\u0131'nda ya da r\u0131ht\u0131mda ya\u015fayan o kont, o da ayn\u0131 \u015feyi ya\u015fayabilir, ama sadece ba\u015fka t\u00fcrl\u00fc g\u00f6r\u00fcn\u00fcr, \u00e7\u00fcnk\u00fc onun da her \u015feyi kendine g\u00f6redir, y\u00fcksek \u00fcsluplad\u0131r, ama o da ayn\u0131d\u0131r, her \u015fey gelebilir ba\u015f\u0131na, benim de ba\u015f\u0131ma ayn\u0131 \u015fey gelebilir. \u0130\u015fte b\u00f6yle, can\u0131m, bir de siz sizden uzakla\u015fmak istiyorsunuz; g\u00fcnaht\u0131r, Varenka, bir \u015fey olabilir bana. Hem kendinizi hem beni mahvedebilirsiniz, can\u0131m\u0131n i\u00e7i. Ah, yavru ku\u015fumsunuz benim, Tanr\u0131 a\u015fk\u0131na, kafan\u0131zdan at\u0131n bu hastal\u0131kl\u0131 fikirleri ve beni bo\u015f yere y\u0131pratmay\u0131n. Siz benim s\u0131ska, t\u00fcys\u00fcz yavru ku\u015fumsunuz, nerede kendi kendinizi besleyeceksiniz, \u00f6l\u00fcmden, k\u00f6t\u00fcl\u00fcklerden kendinizi nas\u0131l koruyacaks\u0131n\u0131z! Yeter, Varenka, kendinize gelin; sa\u00e7ma \u00f6\u011f\u00fct ve iftiralara kulak asmay\u0131n, kitab\u0131n\u0131z\u0131 da bir kez daha okuyun; dikkatle okuyun; bu sizin yarar\u0131n\u0131za olacak.\n\nRatazyayev'le \"Menzil M\u00fcfetti\u015fi\" hakk\u0131nda konu\u015ftum. Bana b\u00fct\u00fcn bunlar\u0131n eskidi\u011fini ve art\u0131k resimli, i\u00e7inde \u00e7e\u015fitli tasvirler olan kitaplar\u0131n \u00e7\u0131kt\u0131\u011f\u0131n\u0131 s\u00f6yledi; do\u011frusu, bu dediklerini \u00e7ok iyi anlayamad\u0131m.20 Sonunda Pu\u015fkin'in iyi oldu\u011funu ve kutsal Rusya'y\u0131 \u00f6vd\u00fc\u011f\u00fcn\u00fc s\u00f6yledi ve onun hakk\u0131nda daha bir s\u00fcr\u00fc \u015fey anlatt\u0131 bana. Evet, \u00e7ok iyi, Varenka, \u00e7ok iyi; kitab\u0131 bir kez daha dikkatle okuyun, \u00f6\u011f\u00fcd\u00fcm\u00fc dinleyin ve bana, bu ihtiyara kulak verirseniz, sevindirirsiniz. O zaman Tanr\u0131 sizi \u00f6d\u00fcllendirir, can\u0131m\u0131n i\u00e7i, hemen \u00f6d\u00fcllendirir.\n\nSamimi dostunuz,  \nMakar Devu\u015fkin\n\n6 Temmuz\n\nSevgili Beyefendi Makar Alekseyevi\u00e7,\n\nFedora bug\u00fcn bana on be\u015f g\u00fcm\u00fc\u015f ruble getirdi. Zavall\u0131c\u0131k, ona \u00fc\u00e7 ruble verince nas\u0131l mutlu oldu! Size acele yaz\u0131yorum. \u015eimdi sizin i\u00e7in bir yeleklik kuma\u015f kesiyorum \u2013\u00e7ok \u00e7ekici bir malzemeden\u2013 \u00e7i\u00e7ekli, sar\u0131 kuma\u015ftan.\n\nSize bir kitap g\u00f6nderiyorum; i\u00e7inde \u00e7e\u015fitli hik\u00e2yeler var; birka\u00e7\u0131n\u0131 okudum; siz de \"Palto\" adl\u0131 olan\u0131 okuyun.21 Bana birlikte tiyatroya gitmeyi \u00f6neriyorsunuz: pahal\u0131 olmaz m\u0131? Herhalde balkondan bilet almal\u0131. Uzun zamand\u0131r gitmemi\u015ftim tiyatroya, do\u011frusu en son ne zaman gitti\u011fimi hat\u0131rlam\u0131yorum. Fakat yine de korkuyorum, bu giri\u015fim pahal\u0131ya mal olmayacak m\u0131? Fedora ba\u015f\u0131n\u0131 sallamakla yetiniyor. Sizin art\u0131k kazand\u0131\u011f\u0131n\u0131zdan fazla harcamaya ba\u015flad\u0131\u011f\u0131n\u0131z\u0131 s\u00f6yl\u00fcyor; ben de g\u00f6r\u00fcyorum bunu; benim i\u00e7in ne \u00e7ok masrafa giriyorsunuz! Baksan\u0131za dostum, bir tats\u0131zl\u0131k olmas\u0131n. Fedora bana birtak\u0131m dedikodulardan bahsetti \u2013 anla\u015f\u0131lan siz ev sahibenizle ona para \u00f6demedi\u011finiz i\u00e7in tart\u0131\u015f\u0131yormu\u015fsunuz; \u00e7ok korkuyorum sizin i\u00e7in. Neyse ho\u015f\u00e7a kal\u0131n; acelem var. K\u00fc\u00e7\u00fck bir i\u015f; \u015fapkan\u0131n \u00fczerindeki kurdeleyi de\u011fi\u015ftiriyorum.\n\nV.D.\n\nHami\u015f: Biliyor musunuz, e\u011fer tiyatroya gidersek yeni \u015fapkam\u0131 giyece\u011fim, omzuma da siyah \u015fal\u0131m\u0131 alay\u0131m diyorum. G\u00fczel olur mu?\n\n7 Temmuz\n\nSevgili Han\u0131mefendi, Varvara Alekseyevna,\n\nYine d\u00fcnk\u00fc olay hakk\u0131nda konu\u015faca\u011f\u0131m. Evet, can\u0131m, bir ara bir hevese kap\u0131lm\u0131\u015ft\u0131k. Bu aktrise kap\u0131ld\u0131m, hem de delicesine kap\u0131ld\u0131m, hi\u00e7bir \u015feyi g\u00f6z\u00fcm g\u00f6rm\u00fcyordu; en tuhaf olan da onu hemen hi\u00e7 g\u00f6rmemi\u015f olmam, sadece bir kere tiyatroda g\u00f6r\u00fcp hemen kap\u0131lm\u0131\u015f olmamd\u0131. O zamanlar benim paravan\u0131n arkas\u0131ndaki b\u00f6lmede be\u015f gen\u00e7 adam vard\u0131, c\u0131v\u0131l c\u0131v\u0131l be\u015f adam. Onlarla birlikte gittim, istemeden gittim, oysa onlarla hep belli bir edep \u00e7er\u00e7evesini korumu\u015ftum. Onlardan geri kalmamak i\u00e7in her dediklerine tamam dedim. Bana o aktristen bahsedip duruyorlard\u0131! Her ak\u015fam, tiyatro ba\u015flar ba\u015flamaz, b\u00fct\u00fcn ekip \u2013ciddi bir ihtiya\u00e7 i\u00e7in bir kopek bulamad\u0131klar\u0131 halde\u2013 evet, b\u00fct\u00fcn ekip tiyatroya, balkona dolu\u015fuyor ve bu aktrisi deli gibi alk\u0131\u015fl\u0131yor, deli gibi \u0131sl\u0131kl\u0131yordu \u2013 tam bir \u00e7\u0131lg\u0131nl\u0131k! Sonra da hi\u00e7 uyumuyorlard\u0131; b\u00fct\u00fcn gece sabaha kadar ondan bahsediyor, her biri onu kendi Gla\u015fa's\u0131 say\u0131yor, hep bir a\u011f\u0131zdan ona a\u015fk ilan ediyor, kalplerinin kanaryas\u0131 say\u0131yorlard\u0131 onu. Beni bile heyecanland\u0131rm\u0131\u015ft\u0131, savunmas\u0131z b\u0131rakm\u0131\u015flard\u0131; o zamanlar gen\u00e7tim tabii. Onlarla tiyatroya nas\u0131l gitti\u011fimi, kendimi nas\u0131l balkonun d\u00f6rd\u00fcnc\u00fc s\u0131ras\u0131nda buldu\u011fumu hat\u0131rlam\u0131yorum. Sadece perdenin k\u0131y\u0131s\u0131n\u0131 g\u00f6r\u00fcyordum, ama her \u015feyi duyuyordum. Minik aktrisin g\u00fczel bir sesi vard\u0131; tiz, b\u00fclb\u00fcl gibi, bal gibi bir sesi vard\u0131! \u2013 Hepimiz alk\u0131\u015fl\u0131yor, ba\u011f\u0131r\u0131p \u00e7a\u011f\u0131r\u0131yorduk, yani k\u0131sacas\u0131, az kals\u0131n atacaklard\u0131 bizi, hatta birini \u00e7\u0131kard\u0131lar. Eve d\u00f6nerken kafam bulut gibiydi! Cebimde tek bir ruble kalm\u0131\u015ft\u0131, maa\u015fa da en az on g\u00fcn vard\u0131. Siz ne yapard\u0131n\u0131z, can\u0131m? Ertesi g\u00fcn, i\u015fe gitmeden \u00f6nce, Frans\u0131z parf\u00fcmerisine gittim, b\u00fct\u00fcn servetimle ona birtak\u0131m kokular ve \u015f\u0131k bir sabun ald\u0131m; h\u00e2l\u00e2 hi\u00e7 bilmiyorum o zaman b\u00fct\u00fcn bunlar\u0131 niye ald\u0131\u011f\u0131m\u0131? Evde yemek yemedim, onun penceresinin alt\u0131nda gezinmeye gittim. Neva Bulvar\u0131'nda, d\u00f6rd\u00fcnc\u00fc katta oturuyordu. Eve d\u00f6nd\u00fcm, birka\u00e7 saat dinlendim, sonra yine Neva Bulvar\u0131'na penceresinin yan\u0131ndan ge\u00e7meye gittim. \u0130ki hafta boyunca b\u00f6yle dolan\u0131p durdum, pe\u015finden ko\u015ftum; bir atl\u0131 araba kiralad\u0131m ve ikide bir \u00f6n\u00fcnden ge\u00e7meye ba\u015flad\u0131m; iyice kaybetmi\u015ftim kendimi, bor\u00e7land\u0131m, sonra da sevmekten vazge\u00e7tim onu; s\u0131k\u0131ld\u0131m! \u0130\u015fte bir aktris d\u00fcr\u00fcst bir adam\u0131 ne hale getirebiliyor, can\u0131m! Fakat, gen\u00e7tim ben, gen\u00e7tim o zamanlar!..\n\nM.D.\n\n8 Temmuz\n\nSevgili Han\u0131mefendim, Varvara Alekseyevna,\n\nBana bu ay\u0131n 6's\u0131nda g\u00f6nderdi\u011finiz kitab\u0131 size g\u00f6nderiyorum ve bu arada bu mektupla size bir a\u00e7\u0131klama yapmak istiyorum. Beni b\u00f6yle bir a\u015f\u0131r\u0131l\u0131\u011fa zorlaman\u0131z k\u00f6t\u00fc oldu, k\u00f6t\u00fc oldu can\u0131m. L\u00fctfen, can\u0131m; insan\u0131n pay\u0131na d\u00fc\u015fen her olay y\u00fccelerde belirlenir. Birine general apoleti verilir, di\u011ferinin dokuzuncu dereceden memur olmas\u0131 gerekir; b\u00f6yle emredilmi\u015ftir ve buna ses \u00e7\u0131karmadan ve korkarak itaat etmek gerekir. Bunun hesab\u0131 hep insan\u0131n yeteneklerine g\u00f6re yap\u0131lm\u0131\u015ft\u0131r; baz\u0131s\u0131 bir \u015feye yeteneklidir, bir ba\u015fkas\u0131 ba\u015fka bir \u015feye, bu yetenekleri de Tanr\u0131 verir. Yakla\u015f\u0131k otuz y\u0131ld\u0131r memurluk yap\u0131yorum; kusursuz hizmet ediyorum, \u00f6zenli davran\u0131yorum, kurallar\u0131 asla \u00e7i\u011fnemem. Bir yurtta\u015f olarak kendimi, kendine has kusurlar\u0131 olan, ama iyilikleri de olan biri say\u0131yorum. Amirlerime sayg\u0131 g\u00f6steriyorum ve Majesteleri de benden ho\u015fnut; bu vakte dek bana \u00f6zel bir h\u00fcrmet g\u00f6stermediler, ama ho\u015fnut olduklar\u0131n\u0131 biliyorum. Sa\u00e7lar\u0131m\u0131 a\u011fart\u0131ncaya dek ya\u015fad\u0131m; i\u015fledi\u011fim b\u00fcy\u00fck bir g\u00fcnah bilmiyorum. Elbette, kimin k\u00fc\u00e7\u00fck g\u00fcnahlar\u0131 yoktur? Herkes g\u00fcnahk\u00e2r, hatta siz bile g\u00fcnahk\u00e2rs\u0131n\u0131z, can\u0131m! Ama b\u00fcy\u00fck su\u00e7lar ve k\u00fcstahl\u0131klar sergiledi\u011fim, ters birtak\u0131m davran\u0131\u015flar\u0131m ya da toplumsal huzuru bozdu\u011fum g\u00f6r\u00fclmedi asla, b\u00f6yle bir \u015fey hi\u00e7 olmad\u0131; hatta madalya alacakt\u0131m ama neyse! B\u00fct\u00fcn bunlar\u0131 sizin vicdanen biliyor olman\u0131z gerekirdi, can\u0131m, o adam\u0131n da bilmesi gerekirdi; biri bir \u015fey anlatmaya kalk\u0131yorsa, her \u015feyi bilmesi gerekir. Hay\u0131r, bunu hi\u00e7 beklemezdim sizden, can\u0131m; hay\u0131r Varenka! Sizden b\u00f6yle bir \u015feyi asla beklemezdim.\n\nNas\u0131l olur! Bundan b\u00f6yle insan\u0131n sakin sakin ya\u015famas\u0131, kendi k\u00f6\u015fesinde durmas\u0131 imk\u00e2ns\u0131z \u2013sakin olmas\u0131na imk\u00e2n yok\u2013 suya sabuna dokunmadan, hi\u00e7 kayg\u0131lanmadan deyiminde oldu\u011fu gibi, Tanr\u0131 korkusu ta\u015f\u0131y\u0131p kendini bilerek, kimse rahats\u0131z etmeden, kul\u00fcbene girmesinler ve etraf\u0131 yoklamas\u0131nlar diye umut ederek ya\u015famaya imk\u00e2n yok; yani kendi evinde oldu\u011fun gibi ya\u015fayamayacaks\u0131n, hep bilecekler, s\u00f6zgelimi \u00fczerindeki yelek iyi mi, \u00fczerine oturuyor mu, i\u00e7 \u00e7ama\u015f\u0131r\u0131n var m\u0131; \u00e7izmelerin var m\u0131, tabanlar\u0131 neyle kapl\u0131; ne yersin, ne i\u00e7ersin, ne yaz\u0131p duruyorsun? Peki ama bu ne demek can\u0131m; istersem kald\u0131r\u0131m\u0131n \u0131slak oldu\u011fu yerden, belki bir kere de \u00e7\u0131plak ayakla, \u00e7izmelerimi elime al\u0131p ge\u00e7iyorumdur! K\u00f6t\u00fc olan \u015feyleri ne diye yazmal\u0131, her seferinde birinin neye ihtiyac\u0131 oldu\u011funu, \u00e7ay i\u00e7ip i\u00e7medi\u011fini neden yaz\u0131yor? Sanki herkesin de durmadan \u00e7ay i\u00e7mesi laz\u0131m gibi! Acaba ben herkesin a\u011fz\u0131na bak\u0131p ka\u00e7 lokma yuttu\u011funu \u00f6\u011frenmeye \u00e7al\u0131\u015f\u0131yor muyum? Kimi g\u00fccendirdim b\u00f6yle bir \u015fey yaparak? Hay\u0131r, can\u0131m, insan kendisini rahats\u0131z etmeyen birilerini neden g\u00fccendirsin! \u0130\u015fte size \u00f6rnek, Varvara Alekseyevna, bak\u0131n bu ne demek: ate\u015fle, gayretle, samimiyetle \u00e7al\u0131\u015f\u0131p \u00e7abal\u0131yorsun \u2013Ne i\u00e7in!\u2013 amirlerin de sana sayg\u0131 duyuyor (belli etmeseler bile, her \u015feye ra\u011fmen, sayg\u0131 duyarlar). Ve i\u015fte hemen burnunun dibindeki biri, hi\u00e7 sebepsiz yere, ne ondan ne bundan, seninle \u015famata ge\u00e7iyor. Elbette, do\u011fru, bazen sen de yeni bir \u015fey al\u0131nca mutlu oluyor, uyumuyorsun, \u00e7izmelerin yeni, s\u00f6zgelimi b\u00fcy\u00fck bir keyifle giyinmi\u015fsin; bu do\u011fru, bunu hissettim, \u00e7\u00fcnk\u00fc insan\u0131n kendi aya\u011f\u0131n\u0131 incecik ipek \u00e7izmenin i\u00e7inde g\u00f6rmesi bamba\u015fka bir \u015fey oluyor, bu k\u0131sm\u0131 do\u011fru anlat\u0131lm\u0131\u015f! Ama yine de Fedor Fedorovi\u00e7'in b\u00f6yle bir kitaba hi\u00e7 dikkat etmeden nas\u0131l izin verdi\u011fine ve kendini sa\u011flama almad\u0131\u011f\u0131na hayret ediyorum. Do\u011fru, kendisi daha gen\u00e7 bir idareci ve hemen ba\u011f\u0131r\u0131p \u00e7a\u011f\u0131rmay\u0131 seviyor; ama nas\u0131l ba\u011f\u0131r\u0131p \u00e7a\u011f\u0131rmas\u0131n? Nas\u0131l azarlamas\u0131n e\u011fer karde\u015fimizi azarlamak gerekiyorsa. Varsayal\u0131m, \u015f\u00f6yle, s\u00f6zgelimi, hava olsun diye azarlayanlar vard\u0131r \u2013 hava i\u00e7in de yap\u0131labilir; al\u0131\u015ft\u0131rmak gerekir; korkutmak gerekir; \u00e7\u00fcnk\u00fc \u2013aram\u0131zda kals\u0131n Varenka\u2013 karde\u015fimiz azarlanmadan i\u015f yapamaz, herkes bir yerlerde bir i\u015f bulma derdinde, yani, ben \u015furada \u015fuyum, deme derdinde, i\u015fe gelince de yan \u00e7iziyorlar. \u00c7\u00fcnk\u00fc bir s\u00fcr\u00fc r\u00fctbe var ve her r\u00fctbe kesinlikle kendi r\u00fctbesine g\u00f6re bir davran\u0131\u015f bekliyor, do\u011fal olarak, sonra da azarlamalar r\u00fctbe derecesine g\u00f6re de\u011fi\u015fiyor, \u00e7\u00fcnk\u00fc e\u015fyan\u0131n tabiat\u0131 bu! Zaten d\u00fcnyan\u0131n d\u00fczeni bu, can\u0131m, hepimiz birbirimize hava at\u0131yoruz, hepimiz birini azarl\u0131yoruz. Bu tedbir olmasa d\u00fcnya ayakta kalmaz, d\u00fczen diye bir \u015fey olmazd\u0131. Ger\u00e7ekten hayret ediyorum, Fedor Fedorovi\u00e7'in b\u00f6yle bir ay\u0131ba dikkat etmedi\u011fine!\n\nHem b\u00f6yle bir \u015feyi neden yazarlar? Ni\u00e7in gerekliymi\u015f ki bu? Okurlardan biri kalk\u0131p bana palto mu yapt\u0131racak? Yeni \u00e7izmeler mi alacak? Hay\u0131r, Varenka, okuyup yoluna gidecek. Bazen saklan\u0131r insan, saklan\u0131r, yakalanmamak i\u00e7in gizlenir, burnunun ucunu bile g\u00f6stermeye korkar; yerini belli etmez, \u00e7\u00fcnk\u00fc \u00f6nyarg\u0131 kol geziyordur, \u00e7\u00fcnk\u00fc yery\u00fcz\u00fcnde ba\u015fka \u015fey kalmam\u0131\u015f gibi, herkesin aras\u0131ndan seni bulup \u015famataya al\u0131rlar, bir bakars\u0131n senin \u00f6zel hayat\u0131n da, aile hayat\u0131n da edebiyata girmi\u015f, hepsi yay\u0131mlanm\u0131\u015f, okunmu\u015f, alaya al\u0131nm\u0131\u015f, de\u011ferlendirilmi\u015f! O zaman soka\u011fa \u00e7\u0131kacak hali kalmaz insan\u0131n; her \u015fey \u00f6yle bir anlat\u0131lm\u0131\u015ft\u0131r ki, karde\u015fimizi s\u0131rf y\u00fcr\u00fcy\u00fc\u015f\u00fcnden bile tan\u0131r\u0131z art\u0131k. Birazc\u0131k ucundan da olsa d\u00fczeltse, bir \u015feyleri yumu\u015fatsa, ne bileyim, s\u00f6zgelimi adam\u0131n kafas\u0131na evraklar at\u0131ld\u0131ktan sonra olanlar\u0131 de\u011fi\u015ftirse; sonu\u00e7ta bu adamda her \u015feye ra\u011fmen d\u00fcr\u00fcst biri, iyi bir yurtta\u015f, arkada\u015flar\u0131n\u0131n b\u00f6yle davranmas\u0131n\u0131 hak etmiyor, b\u00fcy\u00fcklerini dinliyor (bu bir \u00f6rnek olabilir mesela), kimseye bir fenal\u0131k istemiyor, Tanr\u0131'ya inan\u0131yor ve \u00f6l\u00fcyor (yazar onun hemen \u00f6lmesini istiyorsa e\u011fer), o zaman da arkas\u0131ndan a\u011fl\u0131yorlar. En iyisi onun, bu zavall\u0131n\u0131n \u00f6lmesine izin vermemek, paltosu bulunsun, iyiliklerini ayr\u0131nt\u0131l\u0131 olarak \u00f6\u011frenen general onu kendi kalem dairesine als\u0131n, r\u00fctbesini y\u00fckseltsin ve iyi bir maa\u015f versin, b\u00f6ylece g\u00f6r\u00fcyor musunuz neler olurdu; k\u00f6t\u00fcl\u00fck cezaland\u0131r\u0131lm\u0131\u015f, iyilik de zafer kazanm\u0131\u015f olurdu ve memur arkada\u015flar\u0131n\u0131n da yapt\u0131klar\u0131 yan\u0131na kalmazd\u0131. Mesela, ben b\u00f6yle yapard\u0131m; yoksa bunun \u00f6zel yan\u0131 ne, iyi olan ne var burada? G\u00fcnl\u00fck, rezil hayattan b\u00f6yle bir \u00f6rnek vermek bo\u015f. Yani bana kalkm\u0131\u015f b\u00f6yle bir kitap g\u00f6ndermi\u015fsiniz, bir tanem. Bu kitap k\u00f6t\u00fc niyetli bir kitap, Varenka; k\u0131sacas\u0131 muz\u0131r bir kitap, \u00e7\u00fcnk\u00fc b\u00f6yle bir memurun var olmas\u0131na imk\u00e2n yok. Zaten b\u00f6ylelerini \u015fik\u00e2yet etmek laz\u0131m, Varenka, resm\u00ee olarak \u015fik\u00e2yet etmek.\n\nH\u00fcrmetk\u00e2r u\u015fa\u011f\u0131n\u0131z  \nMakar Devu\u015fkin\n\n27 Temmuz\n\nSevgili Beyefendi, Makar Alekseyevi\u00e7,\n\nSon olaylar ve mektuplar\u0131n\u0131z beni \u00fcrk\u00fctt\u00fc, sarst\u0131 ve hayrete d\u00fc\u015f\u00fcrd\u00fc, ama Fedora'n\u0131n hik\u00e2yeleri her \u015feyi anlamam\u0131 sa\u011flad\u0131. Ama nas\u0131l bu kadar umutsuz oldunuz ve birdenbire b\u00f6yle bir u\u00e7uruma d\u00fc\u015ft\u00fcn\u00fcz Makar Alekseyevi\u00e7? A\u00e7\u0131klamalar\u0131n\u0131z hi\u00e7 ho\u015fnut etmedi beni. Bana teklif ettikleri uygun i\u015fi kabul etmek i\u00e7in \u0131srar ederken hakl\u0131 oldu\u011fumu g\u00f6r\u00fcyor musunuz \u015fimdi? Bu son maceram\u0131n g\u00f6t\u00fcrd\u00fc\u011f\u00fc yer \u00fcrk\u00fct\u00fcyor ger\u00e7ekten beni.\n\nBana y\u00f6nelik sevginizin sizi benden gizlemeye zorlad\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyorsunuz. O zaman da, siz bana fazla paran\u0131zdan, zaten her ko\u015fulda rehin sand\u0131\u011f\u0131nda duran paran\u0131zdan bahsetti\u011finiz zaman da anlam\u0131\u015ft\u0131m, size \u00e7ok \u015fey bor\u00e7luyum. \u015eimdi, hi\u00e7 paran\u0131z olmad\u0131\u011f\u0131n\u0131, \u015fans eseri benim yoksul halimi \u00f6\u011frendi\u011finizi ve bundan etkilenerek maa\u015f\u0131n\u0131zla beni desteklemeye karar verdi\u011finizi, avans ald\u0131\u011f\u0131n\u0131z\u0131 ve ben hastalan\u0131nca elbiselerinizi satt\u0131\u011f\u0131n\u0131z\u0131 \u00f6\u011frenince yani, b\u00fct\u00fcn bunlar\u0131 \u00f6\u011frenince, \u00e7ok ac\u0131 verici, hi\u00e7 ya\u015famad\u0131\u011f\u0131m bir durumda kald\u0131m, bunlar\u0131 nas\u0131l kar\u015f\u0131layaca\u011f\u0131m\u0131 ve bu konuda ne d\u00fc\u015f\u00fcnece\u011fimi hi\u00e7 bilmiyorum. Ah Makar Alekseyevi\u00e7! Bana ac\u0131yarak ve akraba sevgisine dayanarak yapt\u0131\u011f\u0131n\u0131z ilk iyiliklerden \u00f6teye ge\u00e7memeliydiniz, ayr\u0131ca paran\u0131z\u0131 bo\u015f yere savurmamal\u0131yd\u0131n\u0131z. Siz dostlu\u011fumuza ihanet ettiniz, Makar Alekseyevi\u00e7, \u00e7\u00fcnk\u00fc bana kar\u015f\u0131 a\u00e7\u0131k olmad\u0131n\u0131z ve \u015fimdi, sizin bana hediye etti\u011finiz giysileri, \u015fekerleri, gezileri, tiyatrolar\u0131 ve kitaplar\u0131 g\u00f6r\u00fcnce, b\u00fct\u00fcn bunlar\u0131n bedelini \u015fimdi kendi ba\u011f\u0131\u015flanmas\u0131 imk\u00e2ns\u0131z havaili\u011fime \u00fcz\u00fclerek pahal\u0131ya \u00f6d\u00fcyorum (\u00e7\u00fcnk\u00fc ben sizin i\u00e7in hi\u00e7 kayg\u0131 duymadan kabul ettim hepsini); ve sizin benim mutlulu\u011fum i\u00e7in vermek istedi\u011finiz her \u015fey, benim i\u00e7in bir kedere d\u00f6n\u00fc\u015ft\u00fc ve yarars\u0131z bir \u00fcz\u00fcnt\u00fcden ba\u015fka bir \u015fey b\u0131rakmad\u0131 bende. Son zamanlarda s\u0131k\u0131nt\u0131l\u0131 oldu\u011funuzu fark etmi\u015ftim, kendim de s\u0131k\u0131nt\u0131yla bir \u015feyleri bekliyordum, ama bu olup biteni akl\u0131m almad\u0131.\n\nNas\u0131l olur! Nas\u0131l ruhen bu kadar d\u00fc\u015febilirsiniz, Makar Alekseyevi\u00e7! Ama \u015fimdi sizin hakk\u0131n\u0131zda ne d\u00fc\u015f\u00fcn\u00fcrler, ne der hakk\u0131n\u0131zda sizi tan\u0131yan herkes? Ruhunun iyili\u011fi, al\u00e7akg\u00f6n\u00fcll\u00fcl\u00fc\u011f\u00fc ve sa\u011fduyusu y\u00fcz\u00fcnden benim ve herkesin sayg\u0131 duydu\u011fu ki\u015fi, siz, siz \u015fimdi birdenbire i\u011fren\u00e7 bir durumun, anla\u015f\u0131lan daha \u00f6nce hi\u00e7 ya\u015famad\u0131\u011f\u0131n\u0131z bir k\u00f6t\u00fcl\u00fc\u011f\u00fcn i\u00e7ine d\u00fc\u015ft\u00fcn\u00fcz. Fedora bana sizi sokakta sarho\u015f bulduklar\u0131n\u0131 ve evinize polis taraf\u0131ndan getirildi\u011finizi anlat\u0131nca ne hale geldim, bilemezsiniz! Hayretten donakald\u0131m, oysa s\u0131rad\u0131\u015f\u0131 bir \u015feyler olmas\u0131n\u0131 bekliyordum, \u00e7\u00fcnk\u00fc d\u00f6rt g\u00fcnd\u00fcr yoktunuz ortal\u0131kta. Ama hi\u00e7 amirlerinizin size ne diyece\u011fini d\u00fc\u015f\u00fcnd\u00fcn\u00fcz m\u00fc Makar Alekseyevi\u00e7, ortadan kaybolman\u0131z\u0131n ger\u00e7ek sebebini \u00f6\u011frendikleri zaman ne diyecekler? Size herkesin g\u00fcld\u00fc\u011f\u00fcn\u00fc s\u00f6yl\u00fcyorsunuz; herkesin ili\u015fkimizi \u00f6\u011frendi\u011fini ve kom\u015fular\u0131n\u0131z aras\u0131ndaki alayl\u0131 \u015fakalarda ad\u0131m\u0131n ge\u00e7ti\u011fini s\u00f6yl\u00fcyorsunuz. Ald\u0131rmay\u0131n buna, Makar Alekseyevi\u00e7 ve Tanr\u0131 a\u015fk\u0131na, sakin olun. Sizin bu subaylarla ya\u015fad\u0131klar\u0131n\u0131z beni \u00fcrk\u00fct\u00fcyor; karanl\u0131k \u015feyler duydum olay hakk\u0131nda.\n\nAnlatsan\u0131za bana, b\u00fct\u00fcn bunlar\u0131n manas\u0131 nedir? Bana a\u00e7\u0131lmaya korktu\u011funuzu yazm\u0131\u015fs\u0131n\u0131z, dostlu\u011fumu kaybetmekten korktu\u011funuzu, \u00e7aresiz oldu\u011funuzu, hastal\u0131\u011f\u0131mda bana nas\u0131l yard\u0131m edece\u011finizi bilemedi\u011finizi, bana destek olmak ve hastaneye d\u00fc\u015fmeme izin vermemek i\u00e7in her \u015feyi satt\u0131\u011f\u0131n\u0131z\u0131, bor\u00e7lanabilece\u011finiz kadar bor\u00e7land\u0131\u011f\u0131n\u0131z\u0131 ve her g\u00fcn ev sahibesiyle tats\u0131zl\u0131k ya\u015fad\u0131\u011f\u0131n\u0131z\u0131 yazm\u0131\u015fs\u0131n\u0131z, iyi de b\u00fct\u00fcn bunlar\u0131 benden saklayarak daha da k\u00f6t\u00fc bir i\u015f yapm\u0131\u015fs\u0131n\u0131z. Ama zaten art\u0131k her \u015feyi \u00f6\u011frendim. Sizin bu talihsiz durumunuzun sebebinin ben oldu\u011fumu \u00f6\u011frenmemi engellediniz, ama \u015fimdi bu davran\u0131\u015flar\u0131n\u0131zla beni iki kat daha \u00e7ok ac\u0131ya s\u00fcr\u00fcklediniz. B\u00fct\u00fcn bunlar etkiledi beni Makar Alekseyevi\u00e7. Ah, dostum! Mutsuzluk bula\u015f\u0131c\u0131 bir hastal\u0131kt\u0131r. Mutsuz ile yoksulun birbirinden uzak durmas\u0131 laz\u0131m, birbirlerine bula\u015ft\u0131rmamak i\u00e7in. Daha \u00f6nce m\u00fctevaz\u0131 ve sakin hayat\u0131n\u0131zda hi\u00e7 ya\u015famad\u0131\u011f\u0131n\u0131z t\u00fcrden mutsuzluklar ya\u015fatt\u0131m size. B\u00fct\u00fcn bunlar eziyor ve y\u0131prat\u0131yor beni.\n\nBana \u015fimdi her \u015feyi, ba\u015f\u0131n\u0131za ne geldi\u011fini ve b\u00f6yle davranmaya nas\u0131l karar verdi\u011finizi a\u00e7\u0131k a\u00e7\u0131k yaz\u0131n. Sakinle\u015ftirin beni, m\u00fcmk\u00fcnse. Sizden bu huzuru talep etmemi sa\u011flayan \u015fey onurum de\u011fil, size kar\u015f\u0131 duydu\u011fum, kalbimden hi\u00e7 silinmeyecek olan dostlu\u011fum ve sevgim. Ho\u015f\u00e7a kal\u0131n. Sab\u0131rs\u0131zl\u0131kla yan\u0131t\u0131n\u0131z\u0131 bekliyorum. Siz beni yanl\u0131\u015f tan\u0131m\u0131\u015fs\u0131n\u0131z, Makar Alekseyevi\u00e7.\n\nSizi y\u00fcrekten seven,  \nVarvara Dobroselova\n\n28 Temmuz\n\nE\u015fsiz Varvara Alekseyevna,\n\nNeyse, her \u015fey art\u0131k sona erdi ve her \u015fey yava\u015f yava\u015f eski haline d\u00f6n\u00fcyor, o y\u00fczden size \u015funu s\u00f6yleyeyim, can\u0131m; benim hakk\u0131mda ne d\u00fc\u015f\u00fcnecekler diye kayg\u0131lanm\u0131\u015fs\u0131n\u0131z, hemen size \u015funu s\u00f6yleyeyim, Varvara Alekseyevna, gururum her \u015feyden de\u011ferlidir. Bu y\u00fczden de size talihsizliklerim ve b\u00fct\u00fcn bu kar\u0131\u015f\u0131kl\u0131klar hakk\u0131nda, amirlerimin hi\u00e7birinin hi\u00e7bir \u015fey bilmedi\u011fini ve bilmeyece\u011fini, bu y\u00fczden de bana eskisi gibi sayg\u0131 besleyeceklerini bildirmeliyim. Tek korkum var: Dedikodu. Evde ev sahibemiz ba\u011f\u0131r\u0131yordu, ama \u015fimdi, sizin on rublenizle ona borcun bir k\u0131sm\u0131n\u0131 \u00f6dedikten sonra, sadece homurdan\u0131yor, hepsi o kadar. Di\u011ferlerine gelince, onlar\u0131n \u00f6nemi yok; onlardan sadece bor\u00e7 istememek gerekir, yoksa \u00f6nemli de\u011filler. A\u00e7\u0131klamam\u0131n sonunda, can\u0131m, size \u015funu s\u00f6yleyeyim, sizin bana duydu\u011funuz sayg\u0131y\u0131 yery\u00fcz\u00fcnde her \u015feyden y\u00fcksek say\u0131yorum ve art\u0131k ge\u00e7ici s\u0131k\u0131nt\u0131lar\u0131mda bununla teselli buluyorum. Tanr\u0131'ya \u015f\u00fck\u00fcrler olsun, ilk darbe ve ilk sars\u0131nt\u0131lar azald\u0131 ve siz durumu kabullendiniz, size kendimi a\u00e7mad\u0131\u011f\u0131m, sizi kand\u0131rd\u0131\u011f\u0131m, sizden ayr\u0131lma g\u00fcc\u00fcn\u00fc kendimde bulamad\u0131\u011f\u0131m ve sizi k\u00fc\u00e7\u00fck mele\u011fim gibi sevdi\u011fim i\u00e7in beni hain ve bencil biri saymad\u0131n\u0131z. \u00d6zetle i\u015fe koyuldum \u015fimdi ve mesle\u011fimi lay\u0131k\u0131yla yapmaya ba\u015flad\u0131m. Evstafi \u0130vanovi\u00e7 ben \u00f6nlerinden ge\u00e7erken bir s\u00f6z bile s\u00f6ylemedi. Sizden saklayamam, can\u0131m, bor\u00e7lar\u0131m ve giysilerimin k\u00f6t\u00fc durumu beni bunalt\u0131yor, ama \u00f6nemli bir \u015fey yok ve sizden rica ediyorum, umudunuzu yitirmeyin, can\u0131m. Bana bir yar\u0131m g\u00fcm\u00fc\u015f ruble yollam\u0131\u015fs\u0131n\u0131z Varenka, bu yar\u0131m ruble kalbimi par\u00e7alad\u0131. Nereden nereye, bak\u0131n neler oldu! Yani ben, o ya\u015fl\u0131 budala, size, k\u00fc\u00e7\u00fck mele\u011fe yard\u0131m eden ben de\u011filim; siz, \u00f6ks\u00fcz bir yoksul, siz bana yard\u0131m ediyorsunuz! Fedora para g\u00f6ndermekle iyi yapm\u0131\u015f. Bu s\u0131ralar benim herhangi bir yerden para alma umudum yok, e\u011fer b\u00f6yle bir umut belirirse, yazar\u0131m size ayr\u0131nt\u0131s\u0131yla. Ama dedikodular, her \u015feyden \u00e7ok dedikodular huzursuz ediyor beni. Ho\u015f\u00e7a kal\u0131n, k\u00fc\u00e7\u00fck mele\u011fim. K\u00fc\u00e7\u00fck elinizi \u00f6p\u00fcyor ve size iyile\u015fmeniz i\u00e7in yalvar\u0131yorum. \u0130\u015f yeti\u015ftirmem gerekti\u011fi i\u00e7in ayr\u0131nt\u0131l\u0131 yazam\u0131yorum size, \u00e7\u00fcnk\u00fc \u00e7aba ve gayretlerimle a\u00e7\u0131klar\u0131m\u0131 kapatmak istiyorum; b\u00fct\u00fcn olaylar\u0131 ve subaylarla maceralar\u0131m\u0131 daha ayr\u0131nt\u0131l\u0131 anlatmay\u0131 ak\u015fama b\u0131rak\u0131yorum.\n\nSize sayg\u0131 duyan ve sizi y\u00fcrekten seven,  \nMakar Devu\u015fkin\n\n28 Temmuz\n\nAh, Varenka, Varenka!\n\n\u0130\u015fte \u015fimdi g\u00fcnah sizde ve sizin vicdan\u0131n\u0131zda. Son mektubunuzla beni \u015fa\u015f\u0131rtt\u0131n\u0131z, sersemlettiniz ve ancak \u015fimdi, bo\u015f vakit bulup kalbimin derinliklerine g\u00f6m\u00fclm\u00fc\u015fken, hakl\u0131 oldu\u011fumu, kesinlikle hakl\u0131 oldu\u011fumu anlad\u0131m. Ya\u015fad\u0131\u011f\u0131m yumrukla\u015fmadan bahsetmiyorum (Aman, can\u0131m, aman!), sizi sevdi\u011fimden ve sizi sevmemin hi\u00e7 de mant\u0131ks\u0131z bir i\u015f olmad\u0131\u011f\u0131ndan bahsediyorum. Can\u0131m, siz hi\u00e7bir \u015fey bilmiyorsunuz; e\u011fer bilseydiniz, b\u00fct\u00fcn bunlar\u0131n niye oldu\u011funu, sizi niye sevmem gerekti\u011fini bilseydiniz, b\u00f6yle s\u00f6ylemezdiniz. Siz sadece d\u00fc\u015f\u00fcnd\u00fcklerinizi s\u00f6yl\u00fcyorsunuz, ben kalbinizde hi\u00e7 de \u00f6yle ge\u00e7medi\u011fine inan\u0131yorum.\n\nCan\u0131m benim, subaylarla neler ya\u015fad\u0131\u011f\u0131m\u0131 ben de bilmiyorum ve \u00e7ok iyi hat\u0131rlam\u0131yorum. Mele\u011fim, \u015funu belirtmem laz\u0131m ki o vakte kadar korkun\u00e7 bir utan\u00e7 i\u00e7indeydim. Bir d\u00fc\u015f\u00fcn\u00fcn, koca bir ay boyunca, deyim yerindeyse, tek bir ipli\u011fe tutunmu\u015f gibiydim. \u00c7ok yoksul bir haldeydim. Sizden gizledim, evden de gizledim, ama ev sahibem ba\u011f\u0131r\u0131p \u00e7a\u011f\u0131r\u0131yordu. Ona ald\u0131rmazd\u0131m. U\u011fursuz kocakar\u0131 isterse ba\u011f\u0131r\u0131p dursun, ama hem ay\u0131p, hem de, o, Tanr\u0131 bilir nas\u0131l, ili\u015fkimizi \u00f6\u011frenmi\u015f ve b\u00fct\u00fcn eve bunu anlat\u0131p duruyordu, ne yapaca\u011f\u0131m\u0131 bilemeyip kulaklar\u0131m\u0131 t\u0131kad\u0131m. Asl\u0131nda, di\u011ferlerinin de kulaklar\u0131n\u0131 t\u0131kamas\u0131 gerekirdi, ama tersine, daha da a\u00e7t\u0131lar onlar\u0131. Can\u0131m, art\u0131k nereye gidece\u011fimi hi\u00e7 bilmiyorum...\n\nVe i\u015fte, mele\u011fim, b\u00fct\u00fcn bunlar, b\u00fct\u00fcn bu felaketlerin d\u00f6k\u00fcnt\u00fcs\u00fc beni peri\u015fan etti sonunda. Birdenbire tuhaf \u015feyler duydum Fedora'dan, a\u015fa\u011f\u0131l\u0131k bir merakl\u0131n\u0131n sizin evinize geldi\u011fini ve sizi de\u011fersiz bir teklifle incitti\u011fini duydum; sizi incitti\u011fini, \u00e7ok fazla incitti\u011fini, kendi kendime de\u011ferlendiriyorum, can\u0131m, \u00e7\u00fcnk\u00fc ben de derinden incindim. B\u00f6ylece ben, siz benim mele\u011fimsiniz, sars\u0131ld\u0131m, kendimi kaybettim ve tamam\u0131yla peri\u015fan oldum. Ben, dostum, Varenka, duyulmam\u0131\u015f bir delilikle f\u0131rlad\u0131m d\u0131\u015far\u0131, ona, o g\u00fcnahk\u00e2ra gitmek istedim; ne yapmak istedi\u011fimi hi\u00e7 bilmiyordum, \u00e7\u00fcnk\u00fc, mele\u011fim, sizi g\u00fccendirmelerini istemiyordum! Fakat, kederlenmi\u015ftim!\n\nTam o s\u0131rada ya\u011fmur bast\u0131rd\u0131, \u0131sland\u0131m, s\u0131r\u0131ls\u0131klam oldum!.. Geri d\u00f6nmek istedim... O s\u0131rada da yere yuvarland\u0131m, can\u0131m. Emelyan'la kar\u015f\u0131la\u015ft\u0131m, Emelyan \u0130li\u00e7'le, memurdur kendisi, yani memurdu, art\u0131k memur de\u011fil, \u00e7\u00fcnk\u00fc i\u015ften \u00e7\u0131kar\u0131ld\u0131. Art\u0131k ne yap\u0131yor onu da bilmiyordum, bir \u015fekilde ge\u00e7inip gidiyordu; i\u015fte onunla kar\u015f\u0131la\u015ft\u0131k. Bu arada \u2013 ne dersiniz Varenka, e\u011flenceli mi talihsiz dostunuz hakk\u0131nda bir \u015feyler okumak, ya\u015fad\u0131\u011f\u0131 talihsizli\u011fi ve kand\u0131r\u0131l\u0131p azap \u00e7ekmesinin hik\u00e2yesini okumak? \u00dc\u00e7\u00fcnc\u00fc g\u00fcn, ak\u015fam, Emelyan kand\u0131rd\u0131 beni, ben de ona, o subaya gittim. Adresi de bizim kap\u0131c\u0131ya sordum. Can\u0131m, ben, \u015funu unutmadan s\u00f6yleyeyim, bu delikanl\u0131y\u0131 \u00e7oktand\u0131r fark etmi\u015ftim; bizim evde kald\u0131\u011f\u0131 s\u0131rada soru\u015fturmu\u015ftum onu. \u015eimdi g\u00f6r\u00fcyorum ki uygunsuz bir i\u015f yapm\u0131\u015f\u0131m, \u00e7\u00fcnk\u00fc onun hakk\u0131nda bana bir \u015feyler s\u00f6yledikleri s\u0131rada, kendimde de\u011fildim. Varenka, ben, do\u011frusu, hi\u00e7bir \u015fey hat\u0131rlam\u0131yorum; tek hat\u0131rlad\u0131\u011f\u0131m, onun evinde bir s\u00fcr\u00fc subay oldu\u011fu ya da bana herkesin \u00e7ift g\u00f6r\u00fcnd\u00fc\u011f\u00fc. Art\u0131k, Tanr\u0131 bilir. Ayr\u0131ca ne s\u00f6yledi\u011fimi de hat\u0131rlam\u0131yorum, sadece hakl\u0131 \u00f6fkemle bir\u00e7ok \u015fey s\u00f6yledi\u011fimi hat\u0131rl\u0131yorum. Neyse beni d\u0131\u015far\u0131 att\u0131lar, merdivenlerden yuvarlad\u0131lar, yani tam yuvarlamad\u0131lar da ittiler. Geri nas\u0131l d\u00f6nd\u00fc\u011f\u00fcm\u00fc de zaten biliyorsunuz Varenka; i\u015fte hepsi bu.\n\nElbette, kendimi onlar\u0131n seviyesine d\u00fc\u015f\u00fcrd\u00fcm ve gururum incindi, ama yabanc\u0131lar haberdar de\u011fil, sizden ba\u015fka kimse haberdar de\u011fil; o zaman olup olmamas\u0131 fark etmez herhalde. Belki de b\u00f6yledir, Varenka, ne dersiniz? Benim emin olarak bildi\u011fim kadar\u0131yla, ge\u00e7en y\u0131l sizde Aksentiy Osipovi\u00e7 ayn\u0131 \u015fekilde Pyotr Petrovi\u00e7'in \u015fahs\u0131na hakaret etti, ama gizlice, bunu gizlice yapt\u0131. Onu bek\u00e7i odas\u0131na \u00e7a\u011f\u0131rd\u0131, ben de kap\u0131 aral\u0131\u011f\u0131ndan g\u00f6rd\u00fcm her \u015feyi; orada ona bir bak\u0131ma emir verdi, ama soylu bir \u015fekilde yapt\u0131 bunu, \u00e7\u00fcnk\u00fc bunu benim d\u0131\u015f\u0131mda kimse g\u00f6rmedi; beni ilgilendirmez, yani demek istedi\u011fim, kimseye s\u00f6ylemedim bunu. Hem, bundan sonra Pyotr Petrovi\u00e7 ile Aksentiy Osipovi\u00e7 de hi\u00e7bir \u015fey olmam\u0131\u015f gibiydi. Pyotr Petrovi\u00e7, bilirsiniz, kimseye s\u00f6ylemeyecek kadar gururludur, art\u0131k selamla\u015f\u0131yor, el s\u0131k\u0131\u015f\u0131yorlar.\n\nTart\u0131\u015fm\u0131yorum, Varenka, ben sizinle tart\u0131\u015fmaya cesaret edemem, \u00e7ok d\u00fc\u015ft\u00fcm ve daha da deh\u015fet verici olan \u015fey, kendi g\u00f6z\u00fcmde d\u00fc\u015ft\u00fcm, ama bu da art\u0131k, herhalde, kaderime yaz\u0131lm\u0131\u015f bir \u015feydi, herhalde, kader bu \u2013 kaderden de ka\u00e7\u0131lmaz, bildi\u011finiz gibi. Neyse, b\u00f6ylece talihsizliklerimi ve felaketlerimi ayr\u0131nt\u0131l\u0131 olarak a\u00e7\u0131klam\u0131\u015f oldum, Varenka, i\u015fte, hepsi bu, okunmasa da olur, vakti gelince okunur. Ben biraz hastay\u0131m, can\u0131m, duygular\u0131m\u0131n b\u00fct\u00fcn canl\u0131l\u0131\u011f\u0131 kayboldu. Bu y\u00fczden, sevgili han\u0131mefendim, Varvara Alekseyevna, size ba\u011fl\u0131l\u0131\u011f\u0131m\u0131, sevgimi ve sayg\u0131m\u0131 g\u00f6stermek \u00fczere, burada kesiyorum...\n\nH\u00fcrmetk\u00e2r u\u015fa\u011f\u0131n\u0131z  \nMakar Devu\u015fkin\n\n28 Temmuz\n\nSevgili Beyefendi, Makar Alekseyevi\u00e7,\n\n\u0130ki mektubunuzu da okudum, okur okumaz da ah dedim! Dinleyin, dostum, siz ya benden bir \u015feyler sakl\u0131yorsunuz ve bana sadece ya\u015fad\u0131\u011f\u0131n\u0131z tats\u0131zl\u0131klar\u0131n bir k\u0131sm\u0131n\u0131 yaz\u0131yorsunuz ya da... asl\u0131nda, Makar Alekseyevi\u00e7, mektuplar\u0131n\u0131z biraz da\u011f\u0131n\u0131k g\u00f6r\u00fcn\u00fcyor... Bana gelin, Tanr\u0131 a\u015fk\u0131na, bug\u00fcn gelin; bak\u0131n, ne diyece\u011fim, hemen bize yeme\u011fe gelin. Orada nas\u0131l ya\u015f\u0131yorsunuz ve ev sahibenizle i\u015fler nas\u0131l yoluna girdi, hi\u00e7 bilemiyorum. Siz b\u00fct\u00fcn bunlar hakk\u0131nda hi\u00e7bir \u015fey yazm\u0131yorsunuz ve sanki kasten susuyor gibisiniz. Yani ho\u015f\u00e7a kal\u0131n, dostum; hemen gelin bug\u00fcn bize; yemek i\u00e7in hep bize gelmi\u015f olsayd\u0131n\u0131z, en iyisini yapm\u0131\u015f olurdunuz zaten. Fedora \u00e7ok iyi yemek yap\u0131yor. Ho\u015f\u00e7a kal\u0131n.\n\nSizin  \nVarvara Dobroselova'n\u0131z\n\n16. Rus \u00f6\u011fretmen, filozof, \u015fair Aleksandr Gali\u00e7'in 1834'te yay\u0131mlanan bir felsefe kitab\u0131 (Kartina \u00c7eloveka). Psikolog ve filozof olan Gali\u00e7, Pu\u015fkin'in okudu\u011fu lisede Latince, tarih ve mant\u0131k \u00f6\u011fretmeniydi.\n\n17. Fran\u00e7ois Guillaume Ducray-Duminil'in (1761-1819) _Le Petit Carrillonneur_ (K\u00fc\u00e7\u00fck Zango\u00e7, 1809) adl\u0131 kitab\u0131 kastediliyor.\n\n18. Friedrich Schiller'in 1797 tarihli \"Die Kraniche des Ibykus\" adl\u0131 \u015fiirini Rus \u015fair Jukovski 1813 y\u0131l\u0131nda \u00e7evirmi\u015fti.\n\n19. Biyelkin'in Hik\u00e2yeleri'nde yer alan bir \u00f6yk\u00fc.\n\n20. O d\u00f6nemde Rusya'da fizyolojik \u00f6zelliklerin \u00f6ne \u00e7\u0131kar\u0131ld\u0131\u011f\u0131, nat\u00fcralist bir tarz yayg\u0131nla\u015fm\u0131\u015ft\u0131.\n\n21. Nikolay Gogol'\u00fcn 1843'te yay\u0131mlanan kitab\u0131 kastediliyor; bu kitapta \"Palto\" ilk kez yay\u0131mlan\u0131yordu.\n\n# A\u011fustos\n\n1 A\u011fustos\n\nCan\u0131m, Varvara Alekseyevna,\n\nNe mutlu size, can\u0131m, Tanr\u0131 size pe\u015f pe\u015fe iyilikler yapma ve beni mutlu etme f\u0131rsat\u0131 vermi\u015f. Buna inan\u0131yorum Varenka ve melek kalbinizin iyili\u011fine de inan\u0131yorum, bunu da imayla s\u00f6ylemiyorum; ama ne olur bu ya\u015f\u0131mda yapt\u0131\u011f\u0131m hatay\u0131 y\u00fcz\u00fcme vurmay\u0131n. Neyse, g\u00fcnah i\u015flendi art\u0131k, yapacak bir \u015fey yok! Israr ediyorsan\u0131z, g\u00fcnaht\u0131 elbette bu; fakat bunu sizden duymak, bana \u00e7ok a\u011f\u0131r geliyor! B\u00f6yle s\u00f6yledi\u011fim i\u00e7in bana \u00f6fkelenmeyin; benim g\u00f6nl\u00fcmde her \u015fey de\u011fi\u015fti, can\u0131m. Yoksul insanlar do\u011fu\u015ftan kaprisli olur. Bunu daha \u00f6nce de hissetmi\u015ftim, ama art\u0131k daha \u00e7ok hissediyorum. Yoksuldur, kat\u0131d\u0131r; Tanr\u0131'n\u0131n d\u00fcnyas\u0131n\u0131 bir ba\u015fka g\u00f6r\u00fcr ve gelip ge\u00e7en herkese yan yan bakar, \u00e7evresine \u00fcrkek bir bak\u0131\u015f atar, s\u00f6ylenen her s\u00f6z\u00fc dinler: Acaba onun hakk\u0131nda ne konu\u015fuyorlar diye. Yani, o neden b\u00f6yle g\u00f6steri\u015fsiz? Tam olarak ne hissetmesi gerekir? S\u00f6zgelimi, \u015fu yandan nas\u0131l durmal\u0131, bu yandan nas\u0131l durmal\u0131? Ve Varenka'm, yoksul insan pa\u00e7avradan k\u00f6t\u00fcd\u00fcr, kimseden sayg\u0131 g\u00f6remez, yazarlar ne yazarsa yazs\u0131n! Pasakl\u0131n\u0131n tekidir o! Yoksul insan\u0131n ba\u015f\u0131na gelecek olan gelmi\u015ftir. Peki neden b\u00f6yledir? Yoksul insan\u0131n her \u015feyi, onlara g\u00f6re, tersy\u00fcz edilmelidir \u00e7\u00fcnk\u00fc; onun gizli hi\u00e7bir \u015feyi olmamal\u0131d\u0131r, onda gurur olmamal\u0131d\u0131r asla, asla! Zaten Emelyan bir ara s\u00f6ylemi\u015fti, bir yerde ona bir s\u00f6zle\u015fme haz\u0131rlam\u0131\u015flar, her on kopek i\u00e7in adamca\u011f\u0131z\u0131 inceleyip durmu\u015flar. On kopeklerini ona kar\u015f\u0131l\u0131ks\u0131z verdiklerini d\u00fc\u015f\u00fcn\u00fcyorlarm\u0131\u015f, ama \u00f6yle de\u011fil; yoksul insan\u0131 birbirlerine g\u00f6stermek i\u00e7in \u00f6d\u00fcyorlard\u0131 paray\u0131. Can\u0131m, g\u00fcn\u00fcm\u00fczde hay\u0131r i\u015fleri de biraz tuhaf bir \u015fekilde yap\u0131l\u0131yor... Belki de hep b\u00f6yle yap\u0131l\u0131yordu, kim bilir! Ya yapmay\u0131 beceremiyorlar ya da \u00e7ok ustalar bu i\u015fte, ikisinden biri. Herhalde bilmiyordunuz bunu, al\u0131n i\u015fte! Ba\u015fka bir \u015feyi pas ge\u00e7eriz, ama bunu biliriz! Peki yoksul insan neden b\u00fct\u00fcn bunlar\u0131 bilir ve hep b\u00f6yle d\u00fc\u015f\u00fcn\u00fcr? Cevap? Deneyimle! S\u00f6zgelimi, biliyor ki bir beyefendi restorana gider ve kendi kendine \u015f\u00f6yle der: Bug\u00fcn ne yesem? Ben _saut\u00e9 papillote_22 yiyece\u011fim, ama ayn\u0131 esnada zavall\u0131 adam, g\u00fcn\u00fc ya\u011fs\u0131z lapayla ge\u00e7irecektir. \u0130yi de ona ne, benim ya\u011fs\u0131z lapa yememden? B\u00f6yle insanlar vard\u0131r, Varenka, vard\u0131r, s\u0131rf bunu d\u00fc\u015f\u00fcn\u00fcrler. Ve bunlar, bu edepsiz hicivciler, ortal\u0131kta dolan\u0131r, ta\u015fa b\u00fct\u00fcn aya\u011f\u0131n\u0131zla sa\u011flam bas\u0131yor musunuz yoksa sadece parma\u011f\u0131n\u0131z\u0131n ucuyla m\u0131 bas\u0131yorsunuz, bakarlar; bilmem hangi memur, bilmem hangi dairede, alt\u0131nc\u0131 dereceden memurun, \u00e7izmesinden \u00e7\u0131plak ayaklar\u0131n\u0131n f\u0131rlad\u0131\u011f\u0131n\u0131, dirsek yerlerinin y\u0131rt\u0131lm\u0131\u015f oldu\u011funu hep g\u00f6zler; yetmezmi\u015f gibi b\u00fct\u00fcn bunlar\u0131 uzun uzun tasvir edip bu rezilli\u011fi bir de yay\u0131mlarlar... \u0130yi de sana ne benim dirseklerimin y\u0131rt\u0131lm\u0131\u015f olmas\u0131ndan? Benim kaba s\u00f6zlerimi ho\u015f g\u00f6r\u00fcrseniz, Varenka, size \u015funu s\u00f6ylemek istiyorum, yoksul insan da bu s\u0131rada, \u00f6rnek olarak s\u00f6ylersek, t\u0131pk\u0131 sizin gibi, k\u0131z gibi utanmaktad\u0131r. Sonu\u00e7ta siz de herkesin kar\u015f\u0131s\u0131nda \u2013kaba ifademi ba\u011f\u0131\u015flay\u0131n\u2013 soyunmaya kalkmazs\u0131n\u0131z; i\u015fte yoksul insan da ya\u015fad\u0131\u011f\u0131 ine bu \u015fekilde bak\u0131lmas\u0131n\u0131, yani onun oradaki aile ili\u015fkilerinin nas\u0131l oldu\u011funa bak\u0131lmas\u0131n\u0131 istemez, elbette. Ama o s\u0131rada beni inciten \u015fey, Varenka, benim d\u00fc\u015fmanlar\u0131mla, d\u00fcr\u00fcst bir insan\u0131n \u015ferefine ve gururuna el uzatanlarla bir olunmas\u0131yd\u0131!\n\nBug\u00fcn i\u015fyerinde zavall\u0131 bir ay\u0131 yavrusu, bir ser\u00e7e gibi otururken, utanc\u0131mdan yan\u0131p k\u00fcl olacakt\u0131m sanki. Utand\u0131m Varenka! Ceketinden \u00e7\u0131plak dirsekleri f\u0131rl\u0131yor ve d\u00fc\u011fmeleri ipliklerinden sark\u0131yorsa, utan\u0131r insan do\u011fal olarak. Benim de, sanki kastenmi\u015f gibi, benim de \u00fczerimdeki her \u015fey tam bir karma\u015fa i\u00e7indeydi! \u0130ster istemez insan\u0131n cesareti k\u0131r\u0131l\u0131yor. Ne yapars\u0131n!.. Mesela bug\u00fcn bizzat Stepan Karlovi\u00e7 benimle konu\u015fmaya kalkt\u0131, konu\u015fup durdu, sanki hi\u00e7 fark\u0131nda de\u011filmi\u015f gibi \u015f\u00f6yle dedi laf aras\u0131nda: \"Ah siz, can\u0131m Makar Alekseyevi\u00e7!\" Ba\u015fka da bir \u015fey s\u00f6ylemedi, ne d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcn\u00fc s\u00f6ylemedi, ama ben her \u015feyi tahmin ettim ve \u00f6yle k\u0131zard\u0131m ki, tepemdeki kelim bile k\u0131pk\u0131rm\u0131z\u0131 oldu. Asl\u0131nda \u00f6nemsiz bir \u015feydi, ama yine de huzursuzluk veren, akla rahats\u0131zl\u0131k veren bir \u015feydi. Ya bir \u015feyler \u00f6\u011frendilerse! Tanr\u0131 korusun, nas\u0131l \u00f6\u011frenebilirler bir \u015feyleri! \u0130tiraf edeyim, bir ki\u015fiden ku\u015fkulan\u0131yorum, \u00e7ok ku\u015fkulan\u0131yorum. Sonu\u00e7ta bu canavarlar ald\u0131rmazlar! Satarlar! Senin b\u00fct\u00fcn \u00f6zel hayat\u0131n\u0131 \u00fc\u00e7 kuru\u015fa satarlar; kutsal bir \u015fey tan\u0131mazlar.\n\n\u015eimdi bunun bir \u015faka oldu\u011funu biliyorum: Ratazyayev'in \u015fakas\u0131. Bizim dairede arkada\u015flar\u0131 var, evet, herhalde, laf aras\u0131nda her \u015feyi bire bin katarak anlatm\u0131\u015f; ya da, kendi dairesinde anlatm\u0131\u015f, sonra laf bizim daireye ula\u015fm\u0131\u015f. Ama evde herkes b\u00fct\u00fcn her \u015feyi ayr\u0131nt\u0131s\u0131yla biliyor ve sizin pencerenizi g\u00f6steriyorlar parmakla; g\u00f6sterdiklerini biliyorum. D\u00fcn size yeme\u011fe gelirken hepsi pencereden sarkm\u0131\u015ft\u0131, ev sahibesi de \u015feytan yine \u00e7ocu\u011fa musallat oldu, diyerek sizin ad\u0131n\u0131z\u0131 edepsiz bir \u015fekilde and\u0131. Ama b\u00fct\u00fcn bunlar Ratazyayev'in sizinle bana kendi edebiyat\u0131nda yer verme ve ince bir hicivle bizi tasvir etmek gibi i\u011fren\u00e7 niyetlerinin kar\u015f\u0131s\u0131nda hi\u00e7 kal\u0131r; bunu kendisi s\u00f6yledi, aram\u0131zdaki iyi insanlar da anlatt\u0131lar bunu bana. Art\u0131k hi\u00e7bir \u015fey d\u00fc\u015f\u00fcnemiyorum can\u0131m, ne diyece\u011fime karar veremiyorum. G\u00fcnah saklanmaz, y\u00fcce Tanr\u0131'y\u0131 \u00f6fkelendirdik, k\u00fc\u00e7\u00fck mele\u011fim! Can\u0131m, siz bana s\u0131k\u0131l\u0131nca oyalanay\u0131m, diye bir kitap g\u00f6nderece\u011finizi s\u00f6ylemi\u015ftiniz. Ne yapay\u0131m kitab\u0131! Kitap da neymi\u015f? \u0130\u00e7ine ki\u015filer sokulmu\u015f bir masal! Roman da sa\u00e7mal\u0131k olsun diye yaz\u0131lm\u0131\u015ft\u0131r, aylak insanlar okusun diye; inan\u0131n bana, can\u0131m, benim bunca y\u0131ll\u0131k deneyimime inan\u0131n. Ne olacak yani, size Shakespeare falan derlerse, bak\u0131n, edebiyatta bir Shakespeare var, derlerse; Shakespeare de sa\u00e7mal\u0131k, b\u00fct\u00fcn bunlar cidden sa\u00e7mal\u0131k ve hepsi de s\u0131rf hiciv i\u00e7in yap\u0131l\u0131yor!\n\nSizin olan,  \nMakar Devu\u015fkin\n\n2 A\u011fustos\n\nSevgili Beyefendi, Makar Alekseyevi\u00e7,\n\nHi\u00e7bir konuda kayg\u0131lanmay\u0131n; Tanr\u0131 yard\u0131mc\u0131m\u0131z olsun, her \u015fey ge\u00e7er. Fedora kendine ve bana bir y\u0131\u011f\u0131n i\u015f buldu, ne\u015feyle \u00e7al\u0131\u015fmaya koyulduk biz; belki de, her \u015fey d\u00fczelir. Fedora benim ya\u015fad\u0131\u011f\u0131m son tats\u0131zl\u0131klar\u0131 Anna Fedorovna'n\u0131n duymu\u015f oldu\u011fundan ku\u015fkulan\u0131yor; ama art\u0131k bir \u00f6nemi yok. Bug\u00fcn s\u0131rad\u0131\u015f\u0131 bir ne\u015fe i\u00e7indeyim. Siz bor\u00e7 almak istiyorsunuz; Tanr\u0131 korusun! \u00d6deme vakti gelince binbir t\u00fcrl\u00fc felaket olur. En iyisi bir s\u00fcre bizimle ya\u015fay\u0131n, bize daha s\u0131k gelin ve ev sahibenize de ald\u0131rmay\u0131n. Di\u011fer d\u00fc\u015fmanlar ve k\u00f6t\u00fc niyetli ki\u015filere gelince, ben bo\u015f ku\u015fkularla kendinize eziyet etti\u011finize inan\u0131yorum Makar Alekseyevi\u00e7! Bak\u0131n, size ge\u00e7en sefer de s\u00f6ylemi\u015ftim, a\u015f\u0131r\u0131 dengesiz bir \u00fcslubunuz var. Neyse, ho\u015f\u00e7a kal\u0131n, g\u00f6r\u00fc\u015fmek \u00fczere. Sizi hemen bana bekliyorum.\n\nSizin olan,  \nV.D.\n\n3 A\u011fustos\n\nMele\u011fim benim, Varvara Alekseyevna,\n\nSize hemen bildirmeliyim, hayat\u0131ms\u0131n\u0131z benim, baz\u0131 umutlar do\u011fdu benim i\u00e7in. Affedersiniz, ama k\u0131z\u0131m, bana bor\u00e7 almamam\u0131 m\u0131 s\u00f6yl\u00fcyorsunuz, mele\u011fim? G\u00fcvercinim, almadan yapamam; durumum k\u00f6t\u00fc, sizin de \u00f6yle, iyi ama birdenbire bir \u015fey olursa ya! Sonu\u00e7ta g\u00fc\u00e7s\u00fczs\u00fcn\u00fcz siz de; neyse i\u015fte bunun i\u00e7in yaz\u0131yorum, hemen bor\u00e7 almam gerekiyor. Neyse, ben devam edeyim.\n\nVarvara Alekseyevna, i\u015fyerinde Emelyan \u0130vanovi\u00e7'le yan yana oturdu\u011fumu da belirteyim. Bu sizin tan\u0131d\u0131\u011f\u0131n\u0131z Emelyan de\u011fil. Bu, t\u0131pk\u0131 benim gibi, alt\u0131nc\u0131 dereceden bir memur ve biz onunla bizim dairenin en eski, en k\u00f6kl\u00fc memurlar\u0131 say\u0131l\u0131r\u0131z. \u0130yi biridir, temiz bir ruhu vard\u0131r, konu\u015fkan de\u011fildir ve hep ay\u0131 gibi bakar. Fakat \u00e7al\u0131\u015fkand\u0131r, kalemi elinden d\u00fc\u015fmez \u2013tam bir \u0130ngiliz yaz\u0131s\u0131 vard\u0131r ve do\u011frusunu s\u00f6ylemek gerekirse, benim kadar iyi yazar\u2013 de\u011ferli bir adamd\u0131r! K\u0131sacas\u0131 biz onunla hi\u00e7 yak\u0131nla\u015fmad\u0131k, ama al\u0131\u015fkanl\u0131kla vedala\u015f\u0131r ve selamla\u015f\u0131rd\u0131k; e\u011fer bana zarf a\u00e7aca\u011f\u0131 laz\u0131m olursa, o zaman rica eder, \"Emelyan \u0130vanovi\u00e7, b\u0131\u00e7a\u011f\u0131 verir misiniz?\" derdim sadece, yani yak\u0131nl\u0131\u011f\u0131m\u0131z yan yana ya\u015farken ne kadar gerekirse o kadard\u0131. \u0130\u015fte o bana bug\u00fcn \u015f\u00f6yle dedi: \"Makar Alekseyevi\u00e7, siz ne d\u00fc\u015f\u00fcnmeye dald\u0131n\u0131z \u00f6yle?\" Bakt\u0131m, adam iyili\u011fimi istiyor, a\u00e7\u0131ld\u0131m ona; yani, \u015f\u00f6yle \u015f\u00f6yle dedim, Emelyan \u0130vanovi\u00e7, her \u015feyi anlatmad\u0131m tabii, Tanr\u0131 korusun, asla anlatmam, \u00e7\u00fcnk\u00fc anlatmaya cesaret edemem, ama elim s\u0131k\u0131\u015f\u0131k falan gibi bir \u015feyler a\u00e7t\u0131m ona. \"Can\u0131m, siz en iyisi,\" dedi Emelyan \u0130vanovi\u00e7, \"bor\u00e7 al\u0131n; baksan\u0131za Pyotr Petrovi\u00e7'ten al\u0131yorlar, faizle veriyor; ben ald\u0131m; makul de bir faiz veriyorum.\" Yani, Varenka, kalbim \u00fcrperdi. D\u00fc\u015f\u00fcnd\u00fcm d\u00fc\u015f\u00fcnd\u00fcm, belki Tanr\u0131 onun ruhunu yumu\u015fat\u0131r, Pyotr Petrovi\u00e7 hay\u0131rseverdir, bana da bor\u00e7 verir. B\u00f6ylece hesab\u0131ma g\u00f6re hem ev sahibisine paras\u0131n\u0131 \u00f6derim, hem size yard\u0131m ederim hem de \u00fcst\u00fcmdekileri tamir ettiririm, utan\u00e7tan kurtulurum; yerimde oturam\u0131yorum, Tanr\u0131 cezalar\u0131n\u0131 versin, bu s\u0131r\u0131t\u0131klar yine bizimle alay edecekler diye! Majesteleri de bu aralar masam\u0131n yan\u0131ndan s\u0131k s\u0131k ge\u00e7iyor; Tanr\u0131 korusun, ya bana bir bakar da uygunsuz bir \u015fekilde giyindi\u011fimi g\u00f6r\u00fcrse! En \u00f6nem verdi\u011fi \u015fey de temizlik ve d\u00fczendir. Kendileri, belki, hi\u00e7bir \u015fey s\u00f6ylemeyebilirler, ama ben utan\u00e7tan \u00f6l\u00fcr\u00fcm, \u00f6l\u00fcr\u00fcm i\u015fte. Bunun sonucunda ben, utanc\u0131m\u0131 delik cebime koyup Pyotr Petrovi\u00e7'e gittim, hem umut dolu, hem de bilememekten \u00f6l\u00fcmle canl\u0131l\u0131k aras\u0131 bir halde gittim. Neyse, Varenka, sonu\u00e7ta her \u015fey sa\u00e7ma sapan bir son buldu! Bir \u015feyle me\u015fguld\u00fc, Fedosey \u0130vanovi\u00e7'le konu\u015fuyordu. Yan\u0131na yakla\u015ft\u0131m, ellerine yap\u0131\u015ft\u0131m: \"Pyotr Petrovi\u00e7,\" dedim, \"Pyotr Petrovi\u00e7!\" Ba\u015f\u0131n\u0131 \u00e7evirdi, ben de devam ettim: Durum b\u00f6yle b\u00f6yle, dedim, otuz ruble falan filan dedim. \u00d6nce beni anlayamad\u0131, ama sonra, her \u015feyi a\u00e7\u0131klad\u0131\u011f\u0131m zaman da g\u00fcld\u00fc ve hi\u00e7, \u00f6ylece sustu. Ayn\u0131 \u015feyleri tekrar ettim ona. O da bana, \"Rehin verecek bir \u015feyiniz var m\u0131?\" diye sordu. Bu arada evra\u011f\u0131na d\u00f6nm\u00fc\u015ft\u00fc bile, bir \u015feyler yaz\u0131yor, bana bakm\u0131yordu. Biraz \u015fa\u015falad\u0131m. \"Hay\u0131r, Piyotr Petrovi\u00e7, rehinim yok,\" diyerek ona a\u00e7\u0131klad\u0131m; \"ama,\" dedim, \"maa\u015f alaca\u011f\u0131m, o zaman veririm, hemen veririm, \u00f6nce borcumu \u00f6derim.\" Bu s\u0131rada birisi \u00e7a\u011f\u0131rd\u0131 onu, bekledim, geri geldi, kaleminin ucunu a\u00e7maya koyuldu, beni fark etmemi\u015f gibi yapt\u0131. Ben yine de konu\u015ftum, \"Ne dersiniz,\" dedim, \"Pyotr Petrovi\u00e7, bir \u015fey yap\u0131lamaz m\u0131?\" Sustu, duymam\u0131\u015f gibi yapt\u0131, ben durdum bekledim, d\u00fc\u015f\u00fcnd\u00fcm, son kez deneyeyim, dedim; eline yap\u0131\u015ft\u0131m. Bir \u015feyler s\u00f6yleyecek gibi oldu, kalemini a\u00e7may\u0131 bitirdi, yazmaya ba\u015flad\u0131; ben de uzakla\u015ft\u0131m. Can\u0131m, g\u00f6r\u00fcyor musunuz, belki de\u011ferli insanlar olabilir hepsi ama kibirli, \u00e7ok kibirli insanlar... beni kat kat a\u015farlar! Biz kimiz onlar kim, Varenka! Bu y\u00fczden size yazd\u0131m b\u00fct\u00fcn bunlar\u0131. Emelyan \u0130vanovi\u00e7 g\u00fcl\u00fcp ba\u015f\u0131n\u0131 sallad\u0131, ama beni i\u00e7tenlikle cesaretlendirdi. Emelyan \u0130vanovi\u00e7 sayg\u0131n bir insan. Bana birini tavsiye etmeye s\u00f6z verdi; Varenka, bu adam, V\u0131borg \u00fczerinde oturuyor, o da faizle para veriyor, 14. dereceden23 biri. Emelyan \u0130vanovi\u00e7 bu adam\u0131n hemen verece\u011fini s\u00f6yl\u00fcyor; yar\u0131n gidece\u011fim mele\u011fim, iyi mi? Ne dersiniz? Zaten bor\u00e7 almamak bir k\u00e2bus! Ev sahibesi beni evden atmak \u00fczere ve bana yemek vermeyi kabul etmiyor. \u00c7izmelerim de \u00e7ok k\u00f6t\u00fc durumda, can\u0131m, d\u00fc\u011fmem bile yok... Daha neler neler eksik bende! \u0130dareden biri bu t\u00fcr uygunsuzluklar\u0131 fark edecek olursa, ne olur? Felaket, Varenka, felaket, tam bir felaket!\n\nMakar Devu\u015fkin\n\n4 A\u011fustos\n\nSevgili Makar Alekseyevi\u00e7,\n\nTanr\u0131 a\u015fk\u0131na, Makar Alekseyevi\u00e7, olabildi\u011fince \u00e7abuk bir \u015fekilde biraz para bulun; bug\u00fcnk\u00fc ko\u015fullarda sizden hi\u00e7bir \u015fekilde yard\u0131m istemezdim, ama durumumu bir bilseydiniz! Bu dairede kalmam\u0131za imk\u00e2n yok. Korkun\u00e7 tats\u0131zl\u0131klar geldi ba\u015f\u0131ma ve ke\u015fke \u015fu anda ne kadar \u015fa\u015fk\u0131n ve heyecan i\u00e7inde oldu\u011fumu bilseydiniz! D\u00fc\u015f\u00fcnsenize, dostum: Bu sabah bize yabanc\u0131, ya\u015fl\u0131, ihtiyar denebilecek, madalyal\u0131 bir adam geldi. Bizde ne i\u015fi oldu\u011funu anlayamad\u0131\u011f\u0131mdan tedirgin oldum. Fedora o s\u0131rada d\u00fckk\u00e2na gitmi\u015fti. Adam bana nas\u0131l ya\u015fad\u0131\u011f\u0131m\u0131, ne yapt\u0131\u011f\u0131m\u0131 sormaya ba\u015flad\u0131 ve yan\u0131t vermemi bile beklemeden bana o subay\u0131n amcas\u0131 oldu\u011funu s\u00f6yledi; kaba davran\u0131\u015f\u0131 ve bizi b\u00fct\u00fcn binaya k\u00fc\u00e7\u00fck d\u00fc\u015f\u00fcrmesi nedeniyle ye\u011fenine \u00e7ok \u00f6fkelenmi\u015f; ye\u011feninin \u00e7ocuk oldu\u011funu, havai oldu\u011funu ve kendisinin beni korumas\u0131 alt\u0131na almaya haz\u0131r oldu\u011funu s\u00f6yledi; bana gen\u00e7lere kulak asmamam\u0131 \u00f6\u011f\u00fctledi, benim i\u00e7in bir baba gibi kayg\u0131land\u0131\u011f\u0131n\u0131, bana kar\u015f\u0131 babacan duygular besledi\u011fini ve bana yard\u0131m etmeye haz\u0131r oldu\u011funu s\u00f6yledi. K\u0131pk\u0131rm\u0131z\u0131 oldum, ne d\u00fc\u015f\u00fcnece\u011fimi bilemedim, ama hemen kabul etmedim. Zorla elimi tuttu, yana\u011f\u0131m\u0131 ok\u015fad\u0131, \u00e7ok g\u00fczel oldu\u011fumu ve yanaklar\u0131mda gamzeler olmas\u0131ndan \u00e7ok ho\u015fnut oldu\u011funu (Tanr\u0131 bilir ne s\u00f6yledi\u011fini!) s\u00f6yledi ve sonunda, art\u0131k ihtiyar biri oldu\u011funu s\u00f6yleyerek beni \u00f6pmeye kalkt\u0131. (Bu kadar i\u011fren\u00e7ti!) O s\u0131rada Fedora geldi i\u00e7eri. Adam hemen tedirgin oldu ve benim sadeli\u011fime ve ahlak\u0131ma sayg\u0131 duydu\u011funu tekrar s\u00f6yleyip ona kar\u015f\u0131 yabanc\u0131l\u0131k g\u00f6stermememi istedi\u011fini s\u00f6yledi. Sonra da Fedora'y\u0131 bir kenara \u00e7ekti ve tuhaf bir teklifle ona biraz para vermek istedi. Elbette, almad\u0131 Fedora. Sonunda adam evine gitti, gitmeden \u00f6nce bir kez daha b\u00fct\u00fcn s\u00f6zlerini tekrarlad\u0131, bana yine gelece\u011fini ve k\u00fcpe getirece\u011fini s\u00f6yledi (herhalde, \u00e7ok utanm\u0131\u015ft\u0131); bana evimi de\u011fi\u015ftirmemi \u00f6nerdi ve onun elinde olan, bana hi\u00e7bir \u015feye mal olmayacak olan g\u00fczel bir ev tavsiye etti; beni \u00e7ok sevdi\u011fini s\u00f6yledi, onurlu ve ak\u0131ll\u0131 bir gen\u00e7 k\u0131z oldu\u011fumu s\u00f6yledi, yozla\u015fm\u0131\u015f gen\u00e7lerden uzak durmam\u0131 \u00f6\u011f\u00fctledi ve sonunda, Anna Fedorovna'y\u0131 tan\u0131d\u0131\u011f\u0131n\u0131 ve Anna Fedorovna'n\u0131n ondan beni bizzat ziyarete gelece\u011fini bana s\u00f6ylemesini rica etti\u011fini s\u00f6yledi. O zaman her \u015feyi anlad\u0131m. Bana ne oldu, bilmiyorum; hayat\u0131mda ilk kez b\u00f6yle bir \u015fey ya\u015fad\u0131m; kendimi kaybettim; onu ay\u0131plad\u0131m. Fedora bana yard\u0131m etti ve adam\u0131 neredeyse att\u0131k evden. B\u00fct\u00fcn bunlar\u0131n Anna Fedorovna'n\u0131n i\u015fi oldu\u011funa karar verdik; yoksa o nereden bizi bilecekti?\n\n\u015eimdi size sesleniyorum, Makar Alekseyevi\u00e7, yard\u0131m etmeniz i\u00e7in yalvar\u0131yorum. Beni bu durumda b\u0131rakmay\u0131n, Tanr\u0131 a\u015fk\u0131na! Bor\u00e7 al\u0131n, l\u00fctfen, birazc\u0131k olsun para bulun, evden ta\u015f\u0131nmam\u0131z laz\u0131m, burada daha fazla kalmam\u0131za imk\u00e2n yok: Fedora da bunu s\u00f6yl\u00fcyor. En az yirmi be\u015f ruble laz\u0131m bize; bu paray\u0131 size \u00f6deyece\u011fim; \u00e7al\u0131\u015f\u0131p kazanaca\u011f\u0131m; Fedora bana birka\u00e7 g\u00fcnl\u00fck daha i\u015f bulacak, yani e\u011fer sizden daha \u00e7ok faiz isterlerse, ald\u0131rmay\u0131n, kabul edin. Size hepsini \u00f6deyece\u011fim, yeter ki, Tanr\u0131 a\u015fk\u0131na yard\u0131m edin bana. Sizi b\u00f6yle bir durumdayken rahats\u0131z etmek \u00e7ok zor geliyor bana; ama bir tek sizde umudum! Ho\u015f\u00e7a kal\u0131n, Makar Alekseyevi\u00e7, beni d\u00fc\u015f\u00fcn\u00fcn ve Tanr\u0131 yard\u0131mc\u0131n\u0131z olsun!\n\nV.D.\n\n4 A\u011fustos\n\nG\u00fcvercinim, Varvara Alekseyevna,\n\n\u0130\u015fte b\u00fct\u00fcn bu beklenmedik darbeler sars\u0131yor beni! \u0130\u015fte bu t\u00fcr korkun\u00e7 felaketler mahvediyor ruhumu! Bu f\u0131rsat\u00e7\u0131 ve i\u015fe yaramaz pinpon d\u00f6k\u00fcnt\u00fclerinin, mele\u011fim, sizi \u00f6l\u00fcm d\u00f6\u015fe\u011fine s\u00fcr\u00fcklemek istemesi bir yana, b\u00fct\u00fcn bunlar bir yana, bu f\u0131rsat\u00e7\u0131lar beni de oraya s\u00fcr\u00fcklemek istiyorlar. Ve s\u00fcr\u00fckl\u00fcyorlar da, yemin ederim, s\u00fcr\u00fckl\u00fcyorlar! Zaten art\u0131k size yard\u0131m edemezsem \u00f6lmeye haz\u0131r\u0131m! Size yard\u0131m edemezsem, \u00f6l\u00fcm\u00fcm olur benim Varenka, saf, tertemiz bir \u00f6l\u00fcm, ama yard\u0131m edersem o zaman benden u\u00e7up gideceksiniz, yuvadan u\u00e7an bir ku\u015f gibi, bayku\u015flar\u0131n, y\u0131rt\u0131c\u0131 ku\u015flar\u0131n gagalaya gagalaya \u00f6ld\u00fcrmeye haz\u0131rland\u0131\u011f\u0131 bir ku\u015f gibi. \u0130\u015fte bu bana i\u015fkence ediyor, can\u0131m. Ama siz de, Varenka, siz de \u00e7ok ac\u0131mas\u0131zs\u0131n\u0131z! Nas\u0131l yapt\u0131n\u0131z bunu? Size ac\u0131 veriyorlar, sizi incitiyorlar, siz, yavru ku\u015fum, ac\u0131 \u00e7ekiyorsunuz, beni kayg\u0131land\u0131rd\u0131\u011f\u0131n\u0131z i\u00e7in kah\u0131rlan\u0131yorsunuz, hatta borcu \u00f6demeye s\u00f6z veriyorsunuz, yani, do\u011frusunu s\u00f6ylemek gerekirse, zay\u0131f sa\u011fl\u0131\u011f\u0131n\u0131zla bana vaktinde \u00f6demek i\u00e7in \u00f6leceksiniz. Fakat siz, Varenka, bir d\u00fc\u015f\u00fcnsenize neden bahsetti\u011finizi! Neden diki\u015f yapman\u0131z gereksin, neden \u00e7al\u0131\u015fman\u0131z gereksin, zavall\u0131 k\u00fc\u00e7\u00fck kafan\u0131z\u0131 kayg\u0131larla yorman\u0131z, k\u00fc\u00e7\u00fck g\u00fczel g\u00f6zlerinizi y\u0131pratman\u0131z ve sa\u011fl\u0131\u011f\u0131n\u0131z\u0131 bozman\u0131z niye gereksin? Ah, Varenka, Varenka, g\u00f6r\u00fcyor musunuz g\u00fcvercinim, hi\u00e7bir \u015feye yaram\u0131yorum, ben de biliyorum hi\u00e7bir \u015feye yaramad\u0131\u011f\u0131m\u0131; ama sanki yar\u0131yormu\u015fum gibi yap\u0131yorum! Her \u015feyi a\u015faca\u011f\u0131m, i\u015fleri yoluna sokaca\u011f\u0131m, baz\u0131 edebiyat\u00e7\u0131lar\u0131n evraklar\u0131n\u0131 temize \u00e7ekece\u011fim, onlara gidece\u011fim, bizzat gidece\u011fim, i\u015f isteyece\u011fim; \u00e7\u00fcnk\u00fc onlar, can\u0131m, iyi yaz\u0131c\u0131lar ararlar, biliyorum arad\u0131klar\u0131n\u0131, sizin yorgun d\u00fc\u015fmenize izin vermeyece\u011fim; b\u00f6yle y\u0131k\u0131c\u0131 hallerin i\u00e7ine girmenize izin vermeyece\u011fim. K\u00fc\u00e7\u00fck mele\u011fim, ben, hemen i\u015f bulaca\u011f\u0131m, bulamamaktansa \u00f6l\u00fcr\u00fcm. Bir de k\u00fc\u00e7\u00fck g\u00fcvercinim, faizden art\u0131k korkmamam gerekti\u011fini yazm\u0131\u015fs\u0131n\u0131z; korkmuyorum zaten can\u0131m, korkmuyorum, art\u0131k hi\u00e7bir \u015feyden korkmuyorum. Can\u0131m, ben k\u0131rk ruble isteyece\u011fim; bu \u00e7ok de\u011fildir, de\u011fil mi Varenka, ne dersiniz? Benim s\u00f6z\u00fcme g\u00fcvenip k\u0131rk rubleyi verirler mi acaba? Yani, demek istedi\u011fim, beni bir bak\u0131\u015fta g\u00fcvenilir ve emniyetli biri sayarlar m\u0131? \u00c7ehreme ilk bak\u0131\u015fta benim i\u00e7in iyi bir yarg\u0131ya varabilirler mi? D\u00fc\u015f\u00fcnsenize, mele\u011fim, ikna edebilir miyim ben? Siz ne dersiniz bu konuda? Biliyor musunuz, b\u00f6yle bir korku hissetmek... hastal\u0131kl\u0131, tam olarak s\u00f6ylenirse hastal\u0131kl\u0131! K\u0131rk rubleden yirmi be\u015fini size verece\u011fim, Varenka; iki rubleyi ev sahibisine, kalan\u0131 da kendi masraflar\u0131ma ay\u0131r\u0131r\u0131m. Bak\u0131n, ev sahibesine daha \u00e7ok vermek gerekirdi, hatta fazladan vermek laz\u0131md\u0131; ama b\u00fct\u00fcn her \u015feyi de\u011ferlendirir, benim ihtiya\u00e7lar\u0131m\u0131 bir daha hesaplarsan\u0131z, daha fazla veremeyece\u011fimi g\u00f6r\u00fcrs\u00fcn\u00fcz, bu y\u00fczden, bundan bahsetmek imk\u00e2ns\u0131z, d\u00fc\u015f\u00fcnmek bile gereksiz. Bir g\u00fcm\u00fc\u015f rubleye \u00e7izme alaca\u011f\u0131m; art\u0131k bilemiyorum, yar\u0131n bu eskilerle i\u015fe gidip gidemeyece\u011fimi. Boyun fular\u0131 da gerekli bana, \u00e7\u00fcnk\u00fc eskisi bir y\u0131l\u0131 doldurdu art\u0131k; ama siz bana eski \u00f6nl\u00fc\u011f\u00fcn\u00fczden sadece bir fular yapmay\u0131 de\u011fil, g\u00f6\u011f\u00fcsl\u00fck dikmeyi de vaat etmi\u015ftiniz, fular\u0131 d\u00fc\u015f\u00fcnmeyece\u011fim o y\u00fczden. Yani b\u00f6ylece, fular da var, \u00e7izme de. \u015eimdi s\u0131ra d\u00fc\u011fmelerde k\u00fc\u00e7\u00fck dostum! Ama siz kabul edersiniz ki, canca\u011f\u0131z\u0131m, d\u00fc\u011fme olmadan yapamam; ceketimin yar\u0131s\u0131nda d\u00fc\u011fme kalmad\u0131 bile! Majesteleri'nin bendeki bu d\u00fczensizli\u011fi fark edip de bir \u015fey s\u00f6yleyebilece\u011fini ne zaman d\u00fc\u015f\u00fcnsem titriyorum, kim bilir neler s\u00f6yler! Can\u0131m, ben, dinleyemem bile s\u00f6ylediklerini; \u00e7\u00fcnk\u00fc \u00f6l\u00fcr\u00fcm, \u00f6l\u00fcr\u00fcm, orac\u0131kta \u00f6l\u00fcr\u00fcm, \u00e7\u00fcnk\u00fc bu olursa utan\u00e7tan \u00f6l\u00fcr\u00fcm, bir tek d\u00fc\u015f\u00fcncesi bile yeter! Ah, can\u0131m! Evet, \u015fimdi b\u00fct\u00fcn ihtiya\u00e7lardan geriye \u00fc\u00e7 ruble kald\u0131; o da ge\u00e7inmeye ve yar\u0131m k\u00fclah t\u00fct\u00fcne gidiyor; \u00e7\u00fcnk\u00fc, k\u00fc\u00e7\u00fck mele\u011fim, t\u00fct\u00fcns\u00fcz ya\u015fayamam, fakat dokuz g\u00fcnd\u00fcr a\u011fz\u0131ma pipo koyamad\u0131m. Alabilirdim, b\u00fct\u00fcn samimiyetimle s\u00f6yl\u00fcyorum, alabilirdim, ama size de bir \u015fey s\u00f6ylemedim, b\u00fct\u00fcn samimiyetimle. Sizin de ba\u015f\u0131n\u0131zda bela var, siz de en ufak \u015feye ihtiya\u00e7 halindesiniz, bense burada \u00e7e\u015fitli hazlar\u0131n tad\u0131n\u0131 \u00e7\u0131kar\u0131yorum; i\u015fte bu y\u00fczden almad\u0131m ve size bunu s\u00f6ylememin nedeni vicdan azab\u0131 \u00e7ekmemeniz i\u00e7in. Size a\u00e7\u0131k\u00e7a itiraf ediyorum, Varenka, art\u0131k a\u015f\u0131r\u0131 yoksul bir haldeyim, yani buna benzer bir \u015fey asla gelmemi\u015fti ba\u015f\u0131ma. Ev sahibesi beni a\u015fa\u011f\u0131l\u0131yor, kimseden bir sayg\u0131 g\u00f6rm\u00fcyorum; en korkun\u00e7 ihtiya\u00e7lar, bor\u00e7lar i\u00e7indeyim; i\u015fyerinde meslekta\u015flarla daha \u00f6nce de \u015fen \u015fakrak de\u011fildi aram\u0131z, ama can\u0131m, art\u0131k, hi\u00e7 konu\u015fmuyorlar. Sakl\u0131yorum, dikkatle sakl\u0131yorum herkesten, hem kendimi sakl\u0131yorum, hem de i\u015fe gitti\u011fim zaman, yan yan, herkesten uzak durarak gidiyorum. Zaten bunlar\u0131 sadece size itiraf etmek i\u00e7in g\u00fc\u00e7 bulabiliyorum ruhumda... Ya, paray\u0131 vermezlerse?! Neyse, hay\u0131r, en iyisi, Varenka, b\u00f6yle \u015feyleri \u00f6nceden d\u00fc\u015f\u00fcn\u00fcp i\u00e7inizi karartmayay\u0131m. Siz de bunu d\u00fc\u015f\u00fcnmeyin ve k\u00f6t\u00fc d\u00fc\u015f\u00fcncelerle ac\u0131 \u00e7ekmeyin diye uyarmak \u00fczere yaz\u0131yorum bunu da. Ah, Tanr\u0131m, siz ne yapacaks\u0131n\u0131z! Ger\u00e7ekten de, o zaman bu evden \u00e7\u0131kamayacaks\u0131n\u0131z ve ben de sizin yan\u0131n\u0131zda olaca\u011f\u0131m. Ama hay\u0131r, o zaman d\u00f6nemem de, kaybolup giderim bir yere, mahvolurum. Art\u0131k burada bitireyim, t\u0131ra\u015f olmam laz\u0131m, t\u0131ra\u015f daha derli toplu g\u00f6steriyor, derli toplu g\u00f6r\u00fcnmek hep i\u015fe yarar. Neyse, Tanr\u0131 yard\u0131mc\u0131m\u0131z olsun! Dua edip yola koyuluyorum!\n\nM. Devu\u015fkin\n\n5 A\u011fustos\n\nSevgili Makar Alekseyevi\u00e7,\n\nEn az\u0131ndan siz umutsuzlu\u011fa kap\u0131lmasayd\u0131n\u0131z! Zaten yeterince dert var. Size otuz g\u00fcm\u00fc\u015f kopek g\u00f6nderiyorum; daha fazlas\u0131 gelmiyor elimden. Kendinize gerekeni al\u0131n, yar\u0131na kadar bir \u015fekilde ge\u00e7inebilecek kadar\u0131n\u0131. Bizim elimizde hemen hi\u00e7bir \u015fey kalmad\u0131, yar\u0131n art\u0131k ne olur bilemiyorum. H\u00fcz\u00fcn verici, Makar Alekseyevi\u00e7! Fakat, h\u00fcz\u00fcnlenmeyin; ba\u015faramad\u0131k, ne yapars\u0131n\u0131z! Fedora bunun felaket olmad\u0131\u011f\u0131n\u0131, bir s\u00fcre daha bu dairede kal\u0131p gerekirse ta\u015f\u0131nabilece\u011fimizi, zaten isterlerse bizi her yerde bulabileceklerini s\u00f6yl\u00fcyor. Fakat yine de art\u0131k burada kalmak iyi olmayacak. E\u011fer kederlenmeyecekseniz, bir \u015feyler yazacakt\u0131m size.\n\nSizin tuhaf bir karakteriniz var Makar Alekseyevi\u00e7! Her \u015feyden \u00e7ok a\u015f\u0131r\u0131 etkileniyor kalbiniz; bu y\u00fczden de hep mutsuz bir insan oluyorsunuz. B\u00fct\u00fcn mektuplar\u0131n\u0131z\u0131 dikketle okuyorum ve her mektubunuzda benim i\u00e7in ac\u0131 \u00e7ekip kayg\u0131land\u0131\u011f\u0131n\u0131z\u0131, benim bile kayg\u0131lanmad\u0131\u011f\u0131m kadar kayg\u0131land\u0131\u011f\u0131n\u0131z\u0131 g\u00f6r\u00fcyorum. Sizin iyi kalbiniz i\u00e7in her \u015fey s\u00f6ylenecektir elbette, ama ben onun \u00e7ok a\u015f\u0131r\u0131 iyi oldu\u011funu s\u00f6yleyece\u011fim. Size dost\u00e7a bir \u00f6\u011f\u00fct vereyim, Makar Alekseyevi\u00e7. Size minnettar\u0131m, her \u015fey i\u00e7in \u00e7ok minnettar\u0131m, bana yapt\u0131klar\u0131n\u0131z i\u00e7in minnettar\u0131m, b\u00fct\u00fcn bunlar\u0131 \u00e7ok iyi anl\u0131yorum; o y\u00fczden takdir edersiniz, sizi \u015fimdi, b\u00fct\u00fcn bu dertlerinizden, benim ister istemez sebebi oldu\u011fum dertlerinizden dolay\u0131 bu halde g\u00f6rmek beni nas\u0131l etkiliyor; art\u0131k sadece benim ya\u015fay\u0131\u015f\u0131mla \u2013mutluluklar\u0131mla, ac\u0131lar\u0131mla, \u00f6fkelerimle\u2013 ya\u015fad\u0131\u011f\u0131n\u0131z\u0131 g\u00f6rmek beni etkiliyor! E\u011fer ba\u015fkas\u0131n\u0131n olan her \u015feyi insan\u0131n kalbine almas\u0131 ve ayn\u0131 g\u00fc\u00e7te hissetmesi m\u00fcmk\u00fcn olsayd\u0131, do\u011frusu, insan bundan en mutsuz insan olurdu. Bug\u00fcn, siz i\u015ften sonra bize geldi\u011finiz zaman, sizi g\u00f6r\u00fcr g\u00f6rmez \u00fcrkt\u00fcm. O kadar solgun, tedirgin, umutsuzdunuz ki, y\u00fcz\u00fcn\u00fczden d\u00fc\u015fen bin par\u00e7a olmu\u015ftu, bana ba\u015far\u0131s\u0131z oldu\u011funuzu anlatmaktan korktunuz, beni \u00fczmekten, tedirgin etmekten korktunuz, kalbinizden her \u015fey u\u00e7up gitti. Makar Alekseyevi\u00e7! H\u00fcz\u00fcnlenmeyin, kederlenmeyin, daha sa\u011fduyulu olun, rica ediyorum, yalvar\u0131yorum size bunun i\u00e7in. G\u00f6receksiniz, her \u015fey iyi olacak, her \u015fey daha iyi olacak; yoksa sizin ya\u015faman\u0131z g\u00fc\u00e7le\u015fecek, ba\u015fkas\u0131n\u0131n ac\u0131lar\u0131yla sonsuz kederlenir ac\u0131 \u00e7ekersiniz. Ho\u015f\u00e7a kal\u0131n, dostum; yalvar\u0131r\u0131m size, benim i\u00e7in \u00e7ok kayg\u0131lanmay\u0131n.\n\nV.D.\n\n5 A\u011fustos\n\nG\u00fcvercinim, Varenka,\n\nPeki, iyi, k\u00fc\u00e7\u00fck mele\u011fim, iyi! Para bulamamam\u0131n bir felaket olmad\u0131\u011f\u0131na karar vermi\u015fsiniz. Peki, iyi, huzurluyum, mutluyum sizin ad\u0131n\u0131za. Hatta beni, bu ihtiyar\u0131 terk etmeyip o evde kald\u0131\u011f\u0131n\u0131z i\u00e7in de mutluyum. Hatta dahas\u0131n\u0131 s\u00f6yleyeyim, kalbim sizin mektubunuzda benim hakk\u0131mda iyi \u015feyler yazd\u0131\u011f\u0131n\u0131z\u0131 ve duygular\u0131m\u0131 \u00f6vd\u00fc\u011f\u00fcn\u00fcz\u00fc g\u00f6r\u00fcnce b\u00fcy\u00fck bir mutlulukla doldu. Bunu gururdan s\u00f6ylemiyorum, benim kalbim i\u00e7in kayg\u0131land\u0131\u011f\u0131n\u0131z\u0131 g\u00f6rerek beni sevdi\u011finizi anlam\u0131\u015f oldu\u011fum i\u00e7in s\u00f6yl\u00fcyorum. Peki, iyi; art\u0131k kendi kalbimden bahsedece\u011fim! Kalbim kendince bir havada; siz de cesaretsiz olmamam i\u00e7in uyar\u0131yorsunuz, can\u0131m. Evet, k\u00fc\u00e7\u00fck mele\u011fim, ne yaz\u0131k ki, ben de s\u00f6yl\u00fcyorum buna, cesaretsizli\u011fe gerek olmad\u0131\u011f\u0131n\u0131; ama \u00f6yleyse, can\u0131m benim, siz karar verin yar\u0131n hangi \u00e7izmelerle gideyim i\u015fe! Durum bu, can\u0131m; bu t\u00fcr bir fikir bile insan\u0131 mahvedebilir, kesinlikle mahveder. En \u00f6nemlisi, canca\u011f\u0131z\u0131m, kendim i\u00e7in \u00fcz\u00fclm\u00fcyor, kendim i\u00e7in ac\u0131 \u00e7ekmiyorum; bana g\u00f6re hi\u00e7bir \u015fey fark etmez, dondurucu ayazda paltosuz, \u00e7izmesiz gezerim, her \u015feye katlan\u0131r, dayan\u0131r\u0131m, bana v\u0131z gelir; sade, k\u00fc\u00e7\u00fck bir insan\u0131m ben, ama insanlar ne der? D\u00fc\u015fmanlar\u0131m, k\u00f6t\u00fc dillilerin hepsi ne der paltosuz gidersem? Zaten insan paltoyla geziyor, \u00e7izme giyiyorsa ne yaz\u0131k ki hep ba\u015fkalar\u0131 i\u00e7in yap\u0131yor bunu. Bu durumda \u00e7izmeler de, can\u0131m, ruhum benim, isminin onurunu ve de\u011ferini korumak i\u00e7in gerekli; delik \u00e7izmeler i\u00e7indeyken ya ruhum ya onurum zarar g\u00f6r\u00fcr, inan\u0131n, can\u0131m, bunca y\u0131ll\u0131k deneyimime inan\u0131n; benim gibi d\u00fcnyay\u0131 ve insanlar\u0131 tan\u0131yan bir ihtiyar\u0131 dinleyin, birtak\u0131m acemi ve k\u00f6t\u00fc yazarlar\u0131 dinlemeyin.\n\nBen size ayr\u0131nt\u0131s\u0131yla anlatamad\u0131m, can\u0131m, bug\u00fcn asl\u0131nda neler oldu\u011funu, bug\u00fcn neler \u00e7ekti\u011fimi. Neler \u00e7ektim, tek bir sabahta ne kadar \u00e7ok ruhsal zorlu\u011fa katland\u0131m, b\u00f6ylesini b\u00fct\u00fcn y\u0131l hi\u00e7 \u00e7ekmemi\u015ftim. \u015e\u00f6yle oldu: Gittim, \u00f6nce, ondan bor\u00e7 para almak ve i\u015fe gidebilmek i\u00e7in erkenden gittim. Bug\u00fcn de \u00f6yle bir ya\u011fmur vard\u0131, \u00f6yle \u015fiddetli ya\u011fd\u0131 ki! Bir tanem, ben paltoma sar\u0131nd\u0131m, giderken hep d\u00fc\u015f\u00fcn\u00fcyorum: \"Tanr\u0131m! Ba\u011f\u0131\u015fla,\" diyorum, \"g\u00fcnahlar\u0131m\u0131 ve arzular\u0131m\u0131 yerine getir.\"... Kilisesi'nin yan\u0131ndan ge\u00e7tim, ha\u00e7 \u00e7\u0131kard\u0131m, b\u00fct\u00fcn g\u00fcnahlar\u0131m\u0131 hat\u0131rlad\u0131m, y\u00fcce Tanr\u0131'yla anla\u015fmam\u0131n uygunsuz oldu\u011funu hat\u0131rlad\u0131m. Kendi kendime k\u0131zd\u0131m ve hi\u00e7bir \u015feyi g\u00f6rmek istemedim; \u00f6ylece, yolu takip edip gittim. Caddeler bo\u015ftu, kime rastlasam me\u015fguld\u00fc, dalg\u0131nd\u0131, \u015fa\u015facak bir \u015fey de yoktu; o kadar erken vakitte, b\u00f6yle bir havada kim gezer! Kir pas i\u00e7inde bir i\u015f\u00e7i toplulu\u011fu \u00e7\u0131kt\u0131 kar\u015f\u0131ma; \u00e7arpa \u00e7arpa ge\u00e7tiler, k\u00f6yl\u00fcler! \u00dcrkeklik geldi \u00fczerime, tedirgin oldum, paray\u0131 d\u00fc\u015f\u00fcnmek bile istemiyordum a\u00e7\u0131k\u00e7as\u0131 \u2013 art\u0131k ne olacaksa olsun! Voskresenski K\u00f6pr\u00fcs\u00fc'nde tabanlar\u0131m iflas etti, art\u0131k neyin \u00fczerinde gitti\u011fimi bilmez haldeydim. O s\u0131rada bizim katip Ermolayev kar\u015f\u0131ma \u00e7\u0131kt\u0131, selam verdi, durdu, g\u00f6zleriyle takip etti sanki votka \u0131smarlamam\u0131 ister gibi; ah, karde\u015fim, diye d\u00fc\u015f\u00fcnd\u00fcm, votkaym\u0131\u015f, nereden \u00e7\u0131kt\u0131 bu votka! Korkun\u00e7 yorulmu\u015ftum, durdum, dinlendim biraz, devam ettim biraz daha. Kasten \u00e7evreme bak\u0131nd\u0131m, birtak\u0131m fikirler gelsin, oyalanay\u0131m, kendime geleyim diye; olmad\u0131 \u2013 tek bir fikir bile gelmedi akl\u0131ma, bir de iki kat \u00e7amurlanm\u0131\u015ft\u0131m, utan\u0131lacak haldeydim. Sonunda uzakta ah\u015fap, sar\u0131, tara\u00e7a gibi bir \u00e7at\u0131 kat\u0131 olan evi g\u00f6rd\u00fcm; tam \u00f6yleymi\u015f, tam Emelyan \u0130vanovi\u00e7'in dedi\u011fi gibiymi\u015f Markov'un evi. (Faizle para veren buydu, can\u0131m, Markov.) Art\u0131k kendimi bilmiyordum ve Markov'un evi oldu\u011funu anlad\u0131\u011f\u0131m halde, bir bek\u00e7iye sordum, \"Karde\u015f,\" dedim, \"bu kimin evi.\" Bek\u00e7i kaba sabayd\u0131, isteksizce konu\u015ftu, sanki birine sinirlenmi\u015f de, s\u00f6zc\u00fckleri di\u015flerinin aras\u0131ndan t\u0131slar gibi, \"Budur, budur,\" dedi, \"Markov'un evi.\" Bek\u00e7iler hep b\u00f6yledir ama bana ne bek\u00e7iden? Hepsi kaba ve tats\u0131z bir his b\u0131rak\u0131r insanda, k\u0131sacas\u0131, hepsi ayn\u0131d\u0131r; insan hep kendi durumuna uygun bir \u015feyle kar\u015f\u0131la\u015f\u0131r, hep b\u00f6yle olur bu. Evin yan\u0131ndan caddede \u00fc\u00e7 kez gidip geldim, gidip geldik\u00e7e de iyice k\u00f6t\u00fcle\u015ftim, \"Hay\u0131r,\" diye d\u00fc\u015f\u00fcnd\u00fcm, \"vermez, hi\u00e7bir \u015fekilde vermez!\" Tan\u0131mad\u0131\u011f\u0131m bir adam, i\u015fim de nazik bir i\u015f, tipim de d\u00fczg\u00fcn de\u011fil, \"Bakal\u0131m kader ne diyecek,\" diye d\u00fc\u015f\u00fcnd\u00fcm; daha sonra denemedim diye pi\u015fman olmayay\u0131m, diyerek sessizce ittim kap\u0131y\u0131. O s\u0131rada da tats\u0131z bir bela geldi; kirli, budala bir bah\u00e7e k\u00f6pe\u011fi tak\u0131ld\u0131 pe\u015fime; \u00fczerime t\u0131rman\u0131yor, yal\u0131yor! \u0130\u015fte b\u00f6yle rezil, sefil olaylar insan\u0131n pe\u015fini b\u0131rakmaz, can\u0131m, insana \u00fcrkeklik getirir, daha \u00f6nceden haz\u0131rlad\u0131\u011f\u0131n b\u00fct\u00fcn kararl\u0131l\u0131k yerle bir olur; neyse yar\u0131 diri yar\u0131 \u00f6l\u00fc bir halde girdim binaya, dosdo\u011fru belan\u0131n i\u00e7ine dald\u0131m; e\u015fikte, karanl\u0131kta, a\u015fa\u011f\u0131da ne oldu\u011funa bakmad\u0131m, t\u00f6kezledim ve bir kocakar\u0131ya \u00e7arpt\u0131m, kad\u0131n da bir testiden s\u00fct i\u00e7iyordu, b\u00fct\u00fcn s\u00fct d\u00f6k\u00fcl\u00fcverdi. Ciyak ciyak ba\u011f\u0131rd\u0131, esip \u00fcf\u00fcrd\u00fc aptal kocakar\u0131, \"Nereye gidiyorsun, moruk, ne i\u015fin var burada?\" Ba\u015flad\u0131 a\u011f\u0131r a\u011f\u0131r konu\u015fmaya. Can\u0131m, bu t\u00fcr olaylar\u0131n hep benim ba\u015f\u0131ma geldi\u011fini fark ettim; demek kaderime b\u00f6yle yaz\u0131lm\u0131\u015f; durmadan hi\u00e7 alakas\u0131z bir \u015feyler y\u00fcz\u00fcnden bocal\u0131yorum. G\u00fcr\u00fclt\u00fcye ya\u015fl\u0131, cadaloz ve _\u00e7uhonka_ ev sahibesi geldi, ben de ona hemen, \"Markov burada m\u0131 ya\u015f\u0131yor?\" diye sordum. \"Hay\u0131r,\" dedi; durdu, bana iyice bakt\u0131. \"Ne i\u015finiz var onunla?\" Ona a\u00e7\u0131klad\u0131m, Emelyan \u0130vanovi\u00e7 dedim, sonras\u0131n\u0131 anlatt\u0131m, bir i\u015fim var dedim. \u0130htiyar k\u0131z\u0131na seslendi, k\u0131z\u0131 \u00e7\u0131k\u0131p geldi, \u00e7\u0131plak ayakl\u0131, b\u00fcy\u00fck\u00e7e bir k\u0131z. \"Baban\u0131 \u00e7a\u011f\u0131r; yukar\u0131da kirac\u0131n\u0131n yan\u0131nda.\" Bana da, \"Buyurun,\" dedi. \u0130\u00e7eri girdim. Odada hi\u00e7bir \u015fey yoktu, duvarlarda tablolar as\u0131l\u0131yd\u0131, hep birtak\u0131m general portreleri, bir divan duruyordu, yuvarlak bir masa, bir muhabbet \u00e7i\u00e7e\u011fi, bir k\u0131na\u00e7i\u00e7e\u011fi; d\u00fc\u015f\u00fcnd\u00fcm, d\u00fc\u015f\u00fcnd\u00fcm, \u00e7\u0131k\u0131p gidemedim, yeter, \u00f6yle mi, b\u00f6ylemi, gitsem mi kalsam m\u0131? Yani, can\u0131m, az kals\u0131n ka\u00e7acakt\u0131m! En iyisi diye d\u00fc\u015f\u00fcnd\u00fcm, yar\u0131n geleyim; hava da daha iyi olur, gelir beklerim; bug\u00fcn s\u00fct d\u00f6k\u00fcld\u00fc i\u015fte, bu generaller de \u00f6fkeyle bak\u0131yor... Tam kap\u0131ya do\u011fru y\u00fcr\u00fcrken adam i\u00e7eri girdi; kendi halinde, sa\u00e7lar\u0131 ak, g\u00f6zleri h\u0131rs\u0131z\u0131ms\u0131, beli iple ba\u011flanm\u0131\u015f ya\u011fl\u0131 bir sabahl\u0131k var \u00fczerinde. Ne istedi\u011fimi sordu, ben de anlatt\u0131m: \"\u015e\u00f6yle \u015f\u00f6yle,\" dedim, \"i\u015fte Emelyan \u0130vanovi\u00e7, k\u0131rk ruble,\" dedim; \"durum bu.\" Ba\u015fka da bir \u015fey s\u00f6ylemedim. G\u00f6zlerinden davay\u0131 kaybetti\u011fimi g\u00f6rd\u00fcm. \"Hay\u0131r, yaz\u0131k ki,\" dedi, \"konu \u015fu ki, param yok; sizin rehin verebilece\u011finiz bir \u015fey var m\u0131yd\u0131?\" A\u00e7\u0131klamaya \u00e7al\u0131\u015ft\u0131m rehinlik bir \u015fey olmad\u0131\u011f\u0131n\u0131, ama Emelyon \u0130vanovi\u00e7, a\u00e7\u0131klamaya gerek olmad\u0131\u011f\u0131n\u0131 belli etti. Sonuna kadar dinleyip, \"Hay\u0131r,\" dedi. \"Emelyan \u0130vanovi\u00e7 bilmem! Param yok benim.\" Neyse i\u015fte, neler dedi\u011fini ona anlatt\u0131m. \"Hep b\u00f6yledir zaten,\" diye d\u00fc\u015f\u00fcnd\u00fcm; bunu biliyordum, hissetmi\u015ftim; k\u0131sacas\u0131, Varenka, yer yar\u0131l\u0131p beni i\u00e7ine alsa \u00e7ok iyi olurdu; hava so\u011fuk, ayaklar\u0131m donmu\u015f, s\u0131rt\u0131m kar\u0131ncalanm\u0131\u015f. Ben adama bak\u0131yorum, o da bana bak\u0131yor ve hi\u00e7bir \u015fey s\u00f6ylemiyor \u2013gitsene, karde\u015fim, diyor, burada yapacak bir \u015fey yok\u2013 yani bu da olsa, iyice yerin dibine ge\u00e7ecektim. \"Nedir peki, ne i\u00e7in para laz\u0131md\u0131?\" (Durmu\u015f neyi soruyor, can\u0131m!) A\u011fz\u0131m\u0131 a\u00e7t\u0131m \u00f6yle bo\u015f bo\u015f durmamak i\u00e7in, ama dinlemedi bile, \"Hay\u0131r,\" dedi, \"para yok; olsayd\u0131,\" dedi, \"memnuniyetle verirdim.\" Ona anlatt\u0131m, anlatt\u0131m, \"Az bir para dedim zaten, size \u00f6deyece\u011fim, vaktinde \u00f6deyece\u011fim,\" vaktinden bile evvel verece\u011fim, ne kadar isterseniz al\u0131n faizi, Tanr\u0131 a\u015fk\u0131na, \u00f6deyece\u011fim. Can\u0131m, o anda sizi hat\u0131rlad\u0131m, hep sizin talihsizli\u011finizi ve ihtiya\u00e7lar\u0131n\u0131z\u0131 hat\u0131rlad\u0131m, sizin yar\u0131m rublenizi hat\u0131rlad\u0131m. \"Hay\u0131r,\" dedi, \"faiz \u00f6nemli de\u011fil, ke\u015fke rehin olsayd\u0131! Ama zaten param yok, Tanr\u0131 a\u015fk\u0131na yok; olsayd\u0131,\" dedi, \"memnuniyetle,\" b\u00f6yle yemin de etti, haydut!\n\nNeyse, sonra, bir tanem, oradan nas\u0131l \u00e7\u0131kt\u0131\u011f\u0131m\u0131, V\u0131borg'da nas\u0131l y\u00fcr\u00fcd\u00fc\u011f\u00fcm\u00fc, Voskresenski K\u00f6pr\u00fcs\u00fc'n\u00fc nas\u0131l ge\u00e7ti\u011fimi hi\u00e7 hat\u0131rlam\u0131yorum, \u00e7ok yoruldum, dondum, kemiklerim dondu ve i\u015fe ancak saat onda varabildim. \u00c7amurlar\u0131 temizlemek istedim, ama kap\u0131c\u0131, Snegriyev yapamayaca\u011f\u0131m\u0131, f\u0131r\u00e7ay\u0131 kirletece\u011fimi, f\u0131r\u00e7an\u0131n devlete ait, z\u0131mmetli f\u0131r\u00e7a oldu\u011funu s\u00f6yledi. \u0130\u015fte b\u00f6yleler, can\u0131m, bu adamlar\u0131n ayaklar\u0131n\u0131 sildi\u011fi pa\u00e7avradan daha k\u00f6t\u00fc bir durumda de\u011fil miyim? Bu beni \u00f6ld\u00fcrmez mi, Varenka? Para de\u011fil beni \u00f6ld\u00fcren, b\u00fct\u00fcn bu ya\u015fam kayg\u0131lar\u0131, b\u00fct\u00fcn bu f\u0131s\u0131lt\u0131lar, g\u00fcl\u00fc\u015fler, \u015fakalar. Majesteleri hi\u00e7 haber vermeden i\u015fime son verebilir; ah, can\u0131m, benim alt\u0131n \u00e7a\u011f\u0131m ge\u00e7ti! Bug\u00fcn sizin b\u00fct\u00fcn mektuplar\u0131n\u0131z\u0131 okudum yeniden; h\u00fcz\u00fcnlendim, can\u0131m! Ho\u015f\u00e7a kal\u0131n, bir tanem, Tanr\u0131 sizi korusun!\n\nM. Devu\u015fkin\n\nHami\u015f: Varenka, bu kederimi, size yar\u0131 yar\u0131ya \u015faka yollu anlatmak istedim, fakat, anla\u015f\u0131lan, \u015faka benim i\u015fim de\u011fil. G\u00f6nl\u00fcn\u00fcz\u00fc almak istemi\u015ftim. Size gelece\u011fim, can\u0131m, hemen gelece\u011fim, yar\u0131n gelece\u011fim.\n\n11 A\u011fustos\n\nVarvara Alekseyevna,\n\nG\u00fcvercinim benim, can\u0131m! Bittim ben, bittik ikimiz de, ikimiz birden, onar\u0131lmas\u0131 imk\u00e2ns\u0131z bir \u015fekilde bittik. \u015eerefim, gururum... hepsini kaybettim! Ben \u00f6ld\u00fcm, siz de \u00f6ld\u00fcn\u00fcz can\u0131m, siz de benimle birlikte, onar\u0131lmas\u0131 imk\u00e2ns\u0131z bir \u015fekilde \u00f6ld\u00fcn\u00fcz! Ben yapt\u0131m, sizi \u00f6l\u00fcme ben s\u00fcr\u00fckledim! Beni kovdular, can\u0131m, a\u015fa\u011f\u0131lad\u0131lar, alay ettiler, ev sahibesi de hakaret etmeye ba\u015flad\u0131; ba\u011f\u0131rd\u0131, ba\u011f\u0131rd\u0131 bana bug\u00fcn, azarlad\u0131, azarlad\u0131 beni, paspastan beter yapt\u0131. Ak\u015famleyin Ratazyayev'deyken, oradakilerden biri bir mektup tasla\u011f\u0131n\u0131 y\u00fcksek sesle okumaya ba\u015flad\u0131, benim size yazd\u0131\u011f\u0131m, cebimden gizlice \u00e7ekip ald\u0131\u011f\u0131 bir mektubu okumaya ba\u015flad\u0131. Anac\u0131\u011f\u0131m, ne bi\u00e7im alay ettiler! K\u00fc\u00e7\u00fcltt\u00fcler, k\u00fc\u00e7\u00fcltt\u00fcler bizi, kahkahalar att\u0131lar, al\u00e7aklar! \u00dczerlerine y\u00fcr\u00fcy\u00fcp Ratazyayev'i hainlikle su\u00e7lad\u0131m; ona al\u00e7ak oldu\u011funu s\u00f6yledim! Ratazyayev de as\u0131l benim al\u00e7ak oldu\u011fumu, \u00e7e\u015fit \u00e7e\u015fit _coquette_ 'lerle24 dola\u015f\u0131p g\u00f6n\u00fcl e\u011flendirdi\u011fimi; \"Siz bizden sakland\u0131n\u0131z,\" dedi, \"siz bir Lovelace's\u0131n\u0131z,25 \" dedi; \u015fimdi de beni Lovelace diye \u00e7a\u011f\u0131r\u0131yorlar, ba\u015fka ad\u0131m yokmu\u015f gibi! Duyuyor musunuz, k\u00fc\u00e7\u00fck mele\u011fim, duyuyor musunuz... art\u0131k her \u015feyi biliyorlar, her \u015feyden haberdarlar ve sizi, bir tanem, sizi biliyorlar, size ait ne var ne yok her \u015feyi, her \u015feyi biliyorlar! Olacak \u015fey de\u011fil! Faldoni de oradayd\u0131, onlar\u0131n taraf\u0131n\u0131 tuttu; bug\u00fcn onu kasaptan bir \u015feyler almaya g\u00f6nderdim; gitmedi, i\u015fim var dedi! \"\u0130yi de sen mecbursun,\" dedim. \"Hi\u00e7 de de\u011fil,\" dedi, \"benim han\u0131m\u0131ma para \u00f6demiyorsunuz, size bir mecburiyetim yok benim,\" dedi. Dayanamad\u0131m onun gibi birinin, e\u011fitimsiz bir k\u00f6yl\u00fcn\u00fcn yapt\u0131\u011f\u0131 bu hakarete ve bir aptal oldu\u011funu s\u00f6yledim; o da bana, \"As\u0131l bunu s\u00f6yleyen aptal,\" dedi. Sarho\u015f oldu\u011fu i\u00e7in bana kar\u015f\u0131 b\u00f6yle kaba konu\u015fuyor, diye d\u00fc\u015f\u00fcnd\u00fcm, \"Sarho\u015fsun, dedim, k\u00f6yl\u00fc herif,\" \u015f\u00f6yle yan\u0131t verdi: \"Siz mi i\u00e7irmi\u015ftiniz beni? Ay\u0131lmak i\u00e7in bile bir damlan\u0131z yok; kuru\u015f i\u00e7in dileniyorsunuz,\" diyerek ekledi: \"Ah, buna bir de efendi diyorlar!\" \u0130\u015fte, can\u0131m, olay nerelere vard\u0131! Ya\u015famak utan\u00e7 veriyor, Varenka! Tam anlam\u0131yla dile d\u00fc\u015ft\u00fck; kimliksiz bir berdu\u015ftan beter olduk. \u00c7ok a\u011f\u0131r felaketler! Bittim ben, kesinlikle bittim ben! Onar\u0131lmas\u0131 imk\u00e2ns\u0131z bir \u015fekilde bittim.\n\nM.D.\n\n13 A\u011fustos\n\nSevgili Makar Alekseyevi\u00e7,\n\nBa\u015f\u0131m\u0131za pe\u015f pe\u015fe felaketler geliyor, art\u0131k ben de bilmiyorum ne yapaca\u011f\u0131m\u0131! \u015eimdi siz ne yapacaks\u0131n\u0131z, benim de umudum t\u00fckendi; bug\u00fcn sol elimi \u00fct\u00fcyle yakt\u0131m; kazara d\u00fc\u015f\u00fcrd\u00fcm \u00fct\u00fcy\u00fc, elim hem incindi, hem yand\u0131, bir anda. \u00c7al\u0131\u015fmama imk\u00e2n yok, Fedora da \u00fc\u00e7 g\u00fcnd\u00fcr hasta yat\u0131yor. Ac\u0131 verici bir huzursuzluk i\u00e7indeyim. Size otuz g\u00fcm\u00fc\u015f kopek g\u00f6nderiyorum; bu bizim son varl\u0131\u011f\u0131m\u0131z say\u0131l\u0131r, yoksa, Tanr\u0131 biliyor, sizin ihtiyac\u0131n\u0131za yard\u0131m etmeyi \u00e7ok isterdim \u015fimdi. A\u011flayacak kadar \u00fczg\u00fcn\u00fcm! Ho\u015f\u00e7a kal\u0131n, dostum! Bug\u00fcn bize gelseydiniz, beni \u00e7ok teselli etmi\u015f olurdunuz.\n\nV.D.\n\n14 A\u011fustos\n\nMakar Alekseyevi\u00e7,\n\nNeyiniz var sizin? Tanr\u0131 korkunuz yok, herhalde! Beni kesinlikle delirteceksiniz. Utanm\u0131yor musunuz siz! Kendinizi mahvediyorsunuz, sadece kendi itibar\u0131n\u0131z\u0131 d\u00fc\u015f\u00fcn\u00fcyorsunuz! \u015eerefli, soylu, gururlu bir insans\u0131n\u0131z, herkes b\u00f6yle biliyor sizi! Ama utan\u00e7tan \u00f6lmeniz laz\u0131m sizin! Yoksa yaz\u0131k de\u011fil mi a\u011farm\u0131\u015f sa\u00e7lar\u0131n\u0131za? Ama, Tanr\u0131'dan korkun! Fedora, art\u0131k size yard\u0131m etmeyece\u011fini s\u00f6yledi, ben de size bir daha para vermeyece\u011fim. Beni ne hale getirdiniz, Makar Alekseyevi\u00e7! Herhalde, bana bir \u015fey olmayaca\u011f\u0131n\u0131, b\u00f6yle \u00e7irkin davranman\u0131z\u0131n beni etkilemedi\u011fini san\u0131yorsunuz; siz h\u00e2l\u00e2 benim sizin y\u00fcz\u00fcn\u00fczden nelere katland\u0131\u011f\u0131m\u0131 bilmiyorsunuz! Merdivenlere bile \u00e7\u0131kam\u0131yorum: herkes bana bak\u0131yor, parma\u011f\u0131yla g\u00f6sterip tuhaf tuhaf \u015feyler s\u00f6yl\u00fcyor; evet, \"benim bir ayya\u015fla ili\u015fkim oldu\u011funu\" a\u00e7\u0131k\u00e7a s\u00f6yl\u00fcyorlar. Bunu duymak ne a\u011f\u0131r \u015fey! Sizi eve ta\u015f\u0131d\u0131klar\u0131 zaman b\u00fct\u00fcn kirac\u0131lar tiksinerek g\u00f6steriyor: \u0130\u015fte, diyorlar, o memuru getirdiler. Sizin y\u00fcz\u00fcn\u00fczden nas\u0131l utand\u0131\u011f\u0131m\u0131 s\u00f6ylemem imk\u00e2ns\u0131z. Yemin ederim, buradan ta\u015f\u0131naca\u011f\u0131m. Bir yere gidip oda hizmet\u00e7isi, \u00e7ama\u015f\u0131rc\u0131 olaca\u011f\u0131m, burada durmayaca\u011f\u0131m. Bana gelmenizi yazm\u0131\u015ft\u0131m, gelmediniz. Demek, benim g\u00f6zya\u015f\u0131 ve ricalar\u0131m sizin i\u00e7in \u00f6nemsiz, Makar Alekseyevi\u00e7! Peki paray\u0131 nereden buldunuz? Tanr\u0131 a\u015fk\u0131na, kendinize gelin. Mahvolacaks\u0131n\u0131z, bir hi\u00e7 i\u00e7in mahvolacaks\u0131n\u0131z! Hem rezalet, hem ay\u0131p bu yapt\u0131\u011f\u0131n\u0131z! Ev sahibeniz de ak\u015fam eve almak istemedi, sokakta yatt\u0131n\u0131z: Biliyorum bunlar\u0131. Ke\u015fke bunu \u00f6\u011frenmenin bana ne kadar a\u011f\u0131r geldi\u011fini bilseydiniz. Bana gelin, bizde ne\u015feniz yerine gelir; birlikte kitap okuruz, eskileri hat\u0131rlar\u0131z. Fedora din\u00ee ziyaretlerini anlatacak. Benim i\u00e7in, g\u00fcvercinim, peri\u015fan etmeyin kendinizi, beni peri\u015fan etmeyin. Ben bir tek sizin i\u00e7in ya\u015f\u0131yorum, sizin i\u00e7in yan\u0131n\u0131zda kal\u0131yorum. Ne yap\u0131yorsunuz \u015fimdi b\u00f6yle! Soylu bir insan, talihsizlikler kar\u015f\u0131s\u0131nda kararl\u0131 bir insan olun; unutmay\u0131n, yoksulluk kusur de\u011fil. Ne diye umudunuzu yitiriyorsunuz: B\u00fct\u00fcn bunlar ge\u00e7ici! Tanr\u0131 verir \u2013 her \u015fey d\u00fczelir, yeter ki siz \u015fimdi kendinizi toplay\u0131n. Size yirmi kopek g\u00f6nderiyorum, kendinize t\u00fct\u00fcn ya da istedi\u011finiz bir \u015fey al\u0131n, yeter ki, Tanr\u0131 a\u015fk\u0131na, k\u00f6t\u00fc bir \u015feye harcamay\u0131n. Bize gelin, hemen gelin. Belki, eskisi gibi, utan\u0131rs\u0131n\u0131z, ama utanmay\u0131n; sahte bir utanga\u00e7l\u0131k bu. Yeter ki siz en samimi bir \u015fekilde pi\u015fman olun. Tanr\u0131'ya g\u00fcvenin. O her \u015feyi iyi eder.\n\nV.D.\n\n19 A\u011fustos\n\nVarvara Alekseyevna, can\u0131m,\n\nUtan\u0131yorum, can\u0131m\u0131n i\u00e7i, Varvara Alekseyevna, \u00e7ok utan\u0131yorum. Fakat, bunun nesi de\u011fi\u015fik, can\u0131m? Neden kalbimi e\u011flendirmeyeyim? O zaman ayakkab\u0131m\u0131n taban\u0131n\u0131 d\u00fc\u015f\u00fcnm\u00fcyorum, \u00e7\u00fcnk\u00fc sa\u00e7ma geliyor ve o da hep sade, sefil, kirli bir taban olarak kal\u0131yor. \u00c7izmeler de bir sa\u00e7mal\u0131k! Yunan bilgeler de \u00e7izmesiz gezerdi, o zaman bizim karde\u015fimiz niye b\u00f6yle de\u011fersiz meselelerle u\u011fra\u015f\u0131yor? Ne diye g\u00fccendiriyor, ne diye k\u00fc\u00e7\u00fcms\u00fcyor beni bu durumda? Ah! Can\u0131m, can\u0131m, neler yazm\u0131\u015fs\u0131n\u0131z! Fedora'ya s\u00f6yleyin, sa\u00e7malayan, huzursuz, asi bir kocakar\u0131 kendisi ve iki kat aptal, hem de tarifsiz aptal! Benim ak sa\u00e7lar\u0131ma gelince, bunda yan\u0131l\u0131yorsunuz, bir tanem, \u00e7\u00fcnk\u00fc hi\u00e7 de sizin sand\u0131\u011f\u0131n\u0131z kadar ihtiyar de\u011filim. Emelya size selam s\u00f6yl\u00fcyor. \u00dcz\u00fcl\u00fcp a\u011flad\u0131\u011f\u0131n\u0131z\u0131 yaz\u0131yorsunuz; ben de size, ben de \u00fcz\u00fcld\u00fcm, a\u011flad\u0131m diye yaz\u0131yorum. Sonu\u00e7ta size her t\u00fcr sa\u011fl\u0131k ve refah dilerim, bana gelince, ben de hem sa\u011fl\u0131kl\u0131 hem mutluyum k\u00fc\u00e7\u00fck mele\u011fim.\n\nDostunuz,  \nMakar Devu\u015fkin\n\n21 A\u011fustos\n\nSayg\u0131de\u011fer Han\u0131mefendi ve Sevgili Dostum,\n\nVarvara Alekseyevna!\n\nSu\u00e7lu oldu\u011fumu hissediyorum, size kar\u015f\u0131 kabal\u0131k etti\u011fimi hissediyorum ve bana g\u00f6re, bunun hi\u00e7bir yarar\u0131 yok, can\u0131m, yani benim b\u00fct\u00fcn bunlar\u0131 hissederken sizin hi\u00e7bir \u015fey s\u00f6ylememeniz kar\u015f\u0131s\u0131nda. Su\u00e7 i\u015flemeden \u00f6nce de b\u00fct\u00fcn bunlar\u0131 hissetmi\u015ftim, ama i\u015fte ruhen d\u00fc\u015ft\u00fcm, g\u00fcnah\u0131m\u0131 bilerek d\u00fc\u015ft\u00fcm. Can\u0131m benim, k\u00f6t\u00fc de\u011filim, ac\u0131mas\u0131z de\u011filim; sizin kalbinizi par\u00e7alamak i\u00e7in de, g\u00fcvercinim, en az\u0131ndan kana susam\u0131\u015f bir kaplan olmak gerekir, ama bende bir koyun kalbi var. Ve ben, bildi\u011finiz gibi, kan d\u00f6kmeye arzu duymuyorum; bu y\u00fczden, k\u00fc\u00e7\u00fck mele\u011fim, hi\u00e7 de su\u00e7lu de\u011filim davran\u0131\u015f\u0131mda, \u00e7\u00fcnk\u00fc ne kalbimde ne de d\u00fc\u015f\u00fcncelerimde su\u00e7 var; \u00fcstelik neyle su\u00e7land\u0131\u011f\u0131m\u0131 da bilmiyorum. Yani bu i\u015f karanl\u0131k bir i\u015f, can\u0131m! Otuz g\u00fcm\u00fc\u015f kopek g\u00f6ndermi\u015fsiniz, sonra da yirmi kopek g\u00f6nderdiniz; sizin \u00f6ks\u00fcz paralar\u0131n\u0131za bakarken kalbim karard\u0131. Eliniz yanm\u0131\u015f, yak\u0131nda a\u00e7l\u0131k \u00e7ekeceksiniz, ama bana t\u00fct\u00fcn almam\u0131 s\u00f6yl\u00fcyorsunuz. Ama nas\u0131l hareket etmeliyim b\u00f6yle bir durumda? Yoksa, hi\u00e7 vicdan azab\u0131 \u00e7ekmeden, bir haydut gibi, sizi, bir \u00f6ks\u00fcz\u00fc soymaya m\u0131 ba\u015flayay\u0131m! Bunun kar\u015f\u0131s\u0131nda ruhen d\u00fc\u015ft\u00fcm, can\u0131m, yani ba\u015fta, hi\u00e7bir \u015feye yaramad\u0131\u011f\u0131m\u0131 ve ayakkab\u0131m\u0131n tabanlar\u0131ndan sadece biraz daha de\u011ferli oldu\u011fumu hissederken, kendimi daha anlaml\u0131 bir \u015fey saymak edepsizlik gibi geldi, tersine, kendi kendimi edepsiz ve belli \u00f6l\u00e7\u00fcde terbiyesiz biri olarak g\u00f6rmeye ba\u015flad\u0131m. Ama, kendi kendime sayg\u0131m\u0131 nas\u0131l kaybettim, nas\u0131l iyi niteliklerimi ve kendi de\u011ferimi reddetmeye kalk\u0131\u015ft\u0131m, her \u015feyi kaybeder etmez de d\u00fc\u015fmeye ba\u015flad\u0131m! Bu da kaderin yaz\u0131s\u0131, ben bundan su\u00e7lu de\u011filim. \u00d6nceleri biraz hava almaya \u00e7\u0131k\u0131yordum. O s\u0131rada her \u015fey birbirini izledi: do\u011fa o kadar a\u011flamakl\u0131yd\u0131, hava da o kadar so\u011fuktu, ya\u011fmur ya\u011f\u0131yordu, o s\u0131rada Emelya'yla kar\u015f\u0131la\u015ft\u0131k yine. Varenka, elinde ne varsa rehin vermi\u015f adam, evinde ne varsa vermi\u015f ve ona rastlad\u0131\u011f\u0131m s\u0131rada, iki g\u00fcnd\u00fcr \u00e7iy damlas\u0131 bile girmemi\u015fti a\u011fz\u0131na, yeni bir \u015fey rehin vermek istiyordu, ama bulam\u0131yordu, verecek bir \u015feyi kalmam\u0131\u015ft\u0131. Yani, Varenka, bende meyil oldu\u011fundan de\u011fil, adama merhamet etti\u011fim i\u00e7in kar\u0131\u015ft\u0131m bu i\u015fe. \u0130\u015fte b\u00f6ylece g\u00fcnah i\u015fledik, can\u0131m! Onunla birlikte nas\u0131l a\u011flad\u0131k! Sizi and\u0131k. O \u00e7ok iyi, \u00e7ok a\u015f\u0131r\u0131 iyi bir insan ve \u00e7ok hassas biri. Can\u0131m, ben b\u00fct\u00fcn bunlar\u0131 anl\u0131yorum; benim de ba\u015f\u0131ma gelen \u015feyler b\u00fct\u00fcn bunlar, \u00e7ok iyi hissediyorum bunlar\u0131. Size ne kadar bor\u00e7land\u0131\u011f\u0131m\u0131 biliyorum, g\u00fcvercinimsiniz benim! Sizi tan\u0131y\u0131nca, birincisi, kendimi daha iyi tan\u0131d\u0131m ve sizi sevmeye ba\u015flad\u0131m; k\u00fc\u00e7\u00fck mele\u011fim, sizden \u00f6nce, tek ba\u015f\u0131mayd\u0131m ve yery\u00fcz\u00fcnde uyuyor, ya\u015fam\u0131yordum sanki. Onlar, bana zalimlik edenler, \u00e7ehremin bile uygunsuz oldu\u011funu s\u00f6yl\u00fcyorlard\u0131 ve hor g\u00f6r\u00fcyorlard\u0131 beni, kendi kendimi hor g\u00f6r\u00fcr olmu\u015ftum; k\u00fct\u00fck gibi oldu\u011fumu s\u00f6yl\u00fcyorlard\u0131, ben de ger\u00e7ekten k\u00fct\u00fc\u011f\u00fcm ben, diye d\u00fc\u015f\u00fcn\u00fcyordum, ama siz kar\u015f\u0131ma \u00e7\u0131k\u0131nca, siz karanl\u0131k hayat\u0131m\u0131 ayd\u0131nlatt\u0131n\u0131z kalbimi ve ruhumu ayd\u0131nlatt\u0131n\u0131z ve ben ruhsal huzur bularak di\u011ferlerinden daha k\u00f6t\u00fc olmad\u0131\u011f\u0131m\u0131 anlad\u0131m; belki par\u0131lt\u0131 yok, \u0131\u015f\u0131lt\u0131 yok, renk yok, ama yine de insan\u0131m, kalbiyle ve d\u00fc\u015f\u00fcnceleriyle bir insan\u0131m ben. \u015eimdi de, kaderin kovalad\u0131\u011f\u0131n\u0131 hissederek, onun taraf\u0131ndan ezilerek, kendi kendimin de\u011ferini reddetmeye kalk\u0131\u015ft\u0131m, ben felaketlerimle i\u00e7im kararm\u0131\u015f ve ruhen d\u00fc\u015fm\u00fc\u015f bir halde bunu yapt\u0131m. \u0130\u015fte siz de her \u015feyi bildi\u011finize g\u00f6re, can\u0131m, size g\u00f6zya\u015flar\u0131yla yalvar\u0131yorum bu konuyu daha fazla kurcalamaman\u0131z i\u00e7in yalvar\u0131yorum, \u00e7\u00fcnk\u00fc kalbim par\u00e7alan\u0131yor, ac\u0131 geliyor, a\u011f\u0131r geliyor.\n\nSize sayg\u0131lar\u0131m\u0131 sunuyorum.\n\nSad\u0131k dostunuz,  \nMakar Devu\u015fkin\n\n22. (Fr.) K\u00e2\u011f\u0131da sararak pi\u015firilen et ve sebze yeme\u011fi.\n\n23. Memuriyetteki en d\u00fc\u015f\u00fck derece.\n\n24. (Fr.) G\u00fczel g\u00f6r\u00fcnmeye \u00f6zenen kad\u0131n.\n\n25. Samuel Richardson'\u0131n (1689-1761) _Clarissa_ (1748) adl\u0131 roman\u0131n\u0131n ba\u015fkarakteri olan Robert Lovelace, Clarissa Harlowe'u serveti i\u00e7in ba\u015ftan \u00e7\u0131karmaya \u00e7al\u0131\u015fmaktad\u0131r. Bu t\u00fcr durumlar i\u00e7in simge isim halini alm\u0131\u015ft\u0131r. Roman, Rusya'da XIX. y\u00fczy\u0131l ba\u015f\u0131nda pop\u00fcler romanlar aras\u0131ndayd\u0131.\n\n# Eyl\u00fcl\n\n3 Eyl\u00fcl\n\nBir \u00f6nceki mektubumu tamamlamad\u0131m, Makar Alekseyevi\u00e7, \u00e7\u00fcnk\u00fc benim i\u00e7in yazmak zor oldu. Bazen \u00f6yle dakikalar oluyor ki tek ba\u015f\u0131ma kalmaktan, tek ba\u015f\u0131ma h\u00fcz\u00fcnlenip tek ba\u015f\u0131ma kesintisiz kederlenmekten mutlu oluyorum ve b\u00f6yle hallerim gitgide s\u0131kla\u015f\u0131yor art\u0131k. Hat\u0131ralar\u0131m\u0131n a\u00e7\u0131klanamayan bir yan\u0131 var, beni dizginsizce \u00e7ekiyor, \u00f6yle g\u00fc\u00e7l\u00fc \u00e7ekiyor ki birka\u00e7 saat \u00e7evremdeki her \u015feye kar\u015f\u0131 duygusuz kal\u0131p her \u015feyi, ger\u00e7ek her \u015feyi unutuyorum. Ve bug\u00fcn ya\u015fad\u0131\u011f\u0131m her \u015fey, ac\u0131 olsun, kederli olsun, tatl\u0131 olsun, her \u015fey bana ge\u00e7mi\u015fimdeki benzer bir \u015feyi, genellikle de \u00e7ocuklu\u011fumda, \u00e7ocuklu\u011fumun alt\u0131n \u00e7a\u011flar\u0131nda olan bir \u015feyi hat\u0131rlat\u0131yor. Ama bu t\u00fcr anlardan sonra fenala\u015f\u0131yorum. Biraz g\u00fc\u00e7s\u00fczle\u015fiyorum, hayalperestli\u011fim y\u0131prat\u0131yor beni, sa\u011fl\u0131\u011f\u0131m da zaten gitgide k\u00f6t\u00fcle\u015fiyor.\n\nAma bug\u00fcn sonbaharda az bulunan t\u00fcrden daha ayd\u0131nl\u0131k, a\u00e7\u0131k, parlak bir sabah var, beni canland\u0131rd\u0131 ve mutlulukla kar\u015f\u0131lad\u0131m onu. Demek, sonbahar oldu art\u0131k! K\u00f6ydeyken sonbahar\u0131 ne \u00e7ok severdim! Daha k\u00fc\u00e7\u00fcc\u00fck \u00e7ocuktum, ama o zaman da bir\u00e7ok \u015fey hissederdim. Sonbahar ak\u015fam\u0131n\u0131 sabah\u0131ndan daha \u00e7ok severdim. Evimizden iki ad\u0131m \u00f6tede, da\u011f\u0131n k\u0131y\u0131s\u0131nda, g\u00f6l vard\u0131. Bu g\u00f6l \u2013\u015fimdi g\u00f6zlerimin \u00f6n\u00fcne geliyor\u2013 bu g\u00f6l \u00f6yle b\u00fcy\u00fck, \u00f6yle ayd\u0131nl\u0131k, \u00f6yle temizdi ki, kristal gibiydi! Bazen, e\u011fer sessiz bir ak\u015famsa g\u00f6l sakin olurdu; k\u0131y\u0131ya toplanm\u0131\u015f k\u00f6ylerde k\u0131p\u0131rt\u0131 olmazd\u0131, su d\u00fcmd\u00fcz dururdu, ayna gibi. Ayd\u0131nl\u0131k! So\u011fuk! Otun \u00fczerine \u00e7iy d\u00fc\u015fer, k\u0131y\u0131daki izbelerde mumlar yanar, s\u00fcr\u00fc toplan\u0131r; bu s\u0131rada ben de sessizce evden ka\u00e7ar, g\u00f6l\u00fcme bakmaya giderdim ve oturup seyrederdim, bazen. Suyun k\u0131y\u0131s\u0131ndaki bal\u0131k\u00e7\u0131lar \u00e7al\u0131 \u00e7\u0131rp\u0131 toplay\u0131p yakard\u0131 ve \u0131\u015f\u0131k suyun \u00fczerinde uzaklara uzan\u0131rd\u0131. G\u00f6k \u00e7ok so\u011fuk, mavi ve k\u0131z\u0131l, alevli ku\u015faklarla \u00e7evrelenmi\u015f olurdu ve bu ku\u015faklar gitgide daha soluk renge kayard\u0131; ay \u00e7\u0131kard\u0131; havada \u00f6yle bir sessizlik olurdu ki, \u00fcrken bir k\u00fc\u00e7\u00fck ku\u015f kanat \u00e7\u0131rpt\u0131 m\u0131, bir kam\u0131\u015f hafif r\u00fczg\u00e2rdan salland\u0131 m\u0131, bir bal\u0131k suda s\u00fcz\u00fcld\u00fc m\u00fc; hepsi, hepsi duyulurdu. Mavi suyun \u00fczerinde beyaz, ince, saydam bir sis y\u00fckselir, uzaklar karar\u0131r; her \u015fey sanki g\u00f6m\u00fcl\u00fcr dumana, yak\u0131nlarda da her \u015feyin hatlar\u0131 belirginle\u015fmi\u015f, sanki kalemle \u00e7izilmi\u015f gibi olur... kay\u0131k, k\u0131y\u0131, adalar; savrulmu\u015f, k\u0131y\u0131n\u0131n kenar\u0131nda unutulmu\u015f bir f\u0131\u00e7\u0131, hafif hafif sal\u0131n\u0131r suda, salk\u0131ms\u00f6\u011f\u00fct sar\u0131 yapraklar\u0131n\u0131 kam\u0131\u015flara d\u00f6ker, ge\u00e7 kalm\u0131\u015f bir mart\u0131 havalan\u0131r, dalar so\u011fuk suya, sonra yine kanat \u00e7\u0131rpar ve kaybolur siste. Seyrettim, dinledim; mucizevi bir iyilik halindeydim! \u00dcstelik bebektim, \u00e7ocuktum daha!..\n\nSonbahar\u0131 \u00f6yle severdim ki ekinlerin art\u0131k topland\u0131\u011f\u0131, b\u00fct\u00fcn i\u015flerin sona erdi\u011fi, kl\u00fcbelerde toplanmalar\u0131n ba\u015flad\u0131\u011f\u0131, herkesin art\u0131k k\u0131\u015f\u0131 bekledi\u011fi o sonbahar sonunu. O zaman etraf daha da karamsar olurdu, g\u00f6k bulutlarla kaplan\u0131r, sar\u0131 yapraklar \u00e7\u0131plak orman\u0131n k\u0131y\u0131lar\u0131ndaki patikalara toplan\u0131rd\u0131, orman da \u2013\u00f6zellikle de ak\u015famleyin, nemli bir sis \u00e7\u00f6kt\u00fc\u011f\u00fc ve k\u00f6yler sisin i\u00e7inde devler gibi, \u00e7irkin, korkun\u00e7 hayaller gibi g\u00f6r\u00fcn\u00fcp kayboldu\u011fu zaman\u2013 mavile\u015fir, karar\u0131rd\u0131. Y\u00fcr\u00fcrken ge\u00e7 kal\u0131rd\u0131m bazen, di\u011ferlerinden geri kal\u0131r, tek ba\u015f\u0131ma y\u00fcr\u00fcrd\u00fcm, sonra da korkup acele ederdim! Yaprak gibi titrerdim; b\u00f6yle giderken ya \u015fu kovuktan korkun\u00e7 biri \u00e7\u0131karsa, diye d\u00fc\u015f\u00fcn\u00fcrd\u00fcm; bu arada r\u00fczg\u00e2r esip gelir ormandan, u\u011fuldar, k\u00fckrer, ac\u0131kl\u0131 bir \u015fekilde inler, c\u0131l\u0131z dallardan y\u0131\u011f\u0131n y\u0131\u011f\u0131n yaprak kopar\u0131r, havada d\u00f6nd\u00fcr\u00fcr onlar\u0131 ve arkalar\u0131ndan uzun, geni\u015f, g\u00fcr\u00fclt\u00fcc\u00fc bir s\u00fcr\u00fc, y\u00fcrek par\u00e7alay\u0131c\u0131 vah\u015fi bir \u00e7\u0131\u011fl\u0131kla, bir ku\u015f s\u00fcr\u00fcs\u00fc s\u00f6k\u00fcn eder, g\u00f6k karar\u0131r ve hepsini \u00f6rter. Korkun\u00e7 olur ortal\u0131k, bu s\u0131rada \u2013sanki birini duymu\u015f gibi\u2013 bir ses, sanki birisi f\u0131s\u0131ldar: \"Ko\u015f, ko\u015f, \u00e7ocuk, ge\u00e7 kalma; korkun\u00e7 bir \u015fey olacak az sonra, ko\u015f, \u00e7ocuk!\" \u2013 korku d\u00fc\u015fer kalbime ve nefesim alev alm\u0131\u015f gibi ko\u015far\u0131m, ko\u015far\u0131m. Ko\u015farak, nefes nefese var\u0131r\u0131m eve; ev g\u00fcr\u00fclt\u00fcl\u00fc, ne\u015feli; b\u00fct\u00fcn \u00e7ocuklara bir i\u015f vermi\u015fler; bezelye ya da ha\u015fha\u015f ay\u0131klan\u0131r. Islak k\u00fct\u00fckler sobada \u00e7\u0131t\u0131rdar; annem ne\u015feyle seyreder bizim ne\u015feyle \u00e7al\u0131\u015fmam\u0131z\u0131; ya\u015fl\u0131 dad\u0131 Ulyana eski zamanlar\u0131 ya da b\u00fcy\u00fcc\u00fcler ve \u00f6l\u00fclerle ilgili korkun\u00e7 masallar anlat\u0131r. Biz, \u00e7ocuklar da, bir k\u00f6\u015feye b\u00fcz\u00fcl\u00fcr\u00fcz, hepimizin dudaklar\u0131nda bir g\u00fcl\u00fcmseme. Sonra birden bir anda susar\u0131z... Ne o! G\u00fcr\u00fclt\u00fc! Sanki biri vuruyor kap\u0131ya! Hi\u00e7bir \u015fey de\u011filmi\u015f me\u011fer; ya\u015fl\u0131 Frolovna'n\u0131n \u00e7\u0131kr\u0131\u011f\u0131 g\u0131c\u0131rd\u0131yor; ne \u00e7ok g\u00fclerim! Sonra geceleyin korkudan uyuyamam; korkun\u00e7 r\u00fcyalar g\u00f6r\u00fcr\u00fcm. Uyan\u0131r\u0131m, bazen, k\u0131p\u0131rdamaya cesaret edemeden \u015fafak vaktine kadar yorgan\u0131n alt\u0131nda titrerim. Sabahleyin \u00e7i\u00e7ek gibi taze kalkar\u0131m. Pencereden bakar\u0131m; b\u00fct\u00fcn tarla ayaz olmu\u015f; ince sonbahar k\u0131ra\u011f\u0131s\u0131yla kaplanm\u0131\u015f \u00e7\u0131plak dallar; yaprak gibi ince bir buzla \u00f6rt\u00fclm\u00fc\u015f g\u00f6l; g\u00f6l\u00fcn \u00fczerinden beyaz bir duman y\u00fckseliyor; ne\u015feli ku\u015flar ba\u011fr\u0131\u015f\u0131yor. G\u00fcne\u015f parlak \u0131\u015f\u0131nlar sa\u00e7\u0131yor d\u00f6rt bir yana ve \u0131\u015f\u0131nlar cam gibi ince buzda k\u0131r\u0131l\u0131yor. Ayd\u0131nl\u0131k, parlak, ne\u015feli! Sobada yine ate\u015f titre\u015fiyor; semaverin ba\u015f\u0131na oturuyoruz, geceleyin \u00fc\u015f\u00fcm\u00fc\u015f kara k\u00f6pe\u011fimiz Polkan bize pencereden bak\u0131yor ve davetk\u00e2r bir \u015fekilde kuyruk sall\u0131yor. Bir k\u00f6yl\u00fc ge\u00e7iyor pencerenin k\u0131y\u0131s\u0131ndan din\u00e7 bir at\u0131n \u00fcst\u00fcnde, ormana odun toplamaya gidiyor. Herkes \u00f6yle ho\u015fnut, \u00f6yle ne\u015feli ki!.. Ah, ne kadar alt\u0131n bir \u00e7ocukluktu benimki!..\n\n\u0130\u015fte a\u011flad\u0131m \u015fimdi, \u00e7ocuk gibi, hat\u0131ralara kap\u0131l\u0131p gidince. Her \u015feyi \u00f6yle canl\u0131, \u00f6yle canl\u0131 hat\u0131rlad\u0131m, \u00f6yle parlak bir \u015fekilde belirdi ki b\u00fct\u00fcn ge\u00e7mi\u015f \u00f6n\u00fcmde ve \u015fu an \u00f6yle bulan\u0131k, \u00f6yle karanl\u0131k ki!.. Nas\u0131l sona erecek bu, nas\u0131l sona erecek b\u00fct\u00fcn bunlar? Biliyor musunuz, benim bir inanc\u0131m var, bu sonbahar \u00f6lece\u011fime inan\u0131yorum. \u00c7ok, \u00e7ok hastay\u0131m. S\u0131k s\u0131k \u00f6lece\u011fimi d\u00fc\u015f\u00fcn\u00fcyorum, ama yine de b\u00f6yle \u00f6lmek istemiyorum, buralarda yatmak istemiyorum. Belki de, yine yat\u0131p kalaca\u011f\u0131m, o zamanki, bahar vakti gibi, h\u00e2l\u00e2 iyile\u015femedim. \u0130\u015fte \u015fimdi de \u00e7ok k\u00f6t\u00fcy\u00fcm. Fedora bug\u00fcn b\u00fct\u00fcn g\u00fcn bir yerlere gidecekti, tek ba\u015f\u0131ma oturuyorum. Bir s\u00fcredir tek kalmaktan korkuyorum; odada benimle birlikte ba\u015fkas\u0131 varm\u0131\u015f, biri benimle konu\u015fuyormu\u015f gibi geliyor; \u00f6zellikle de bir \u015feylerle u\u011fra\u015f\u0131rken aniden dalg\u0131nl\u0131ktan \u00e7\u0131k\u0131p kendime gelince, korkuya kap\u0131l\u0131yorum. \u0130\u015fte bu y\u00fczden size b\u00f6yle uzun bir mektup yazd\u0131m; yazd\u0131\u011f\u0131m zaman, ge\u00e7iyor bu. Ho\u015f\u00e7a kal\u0131n, mektubu bitiriyorum, \u00e7\u00fcnk\u00fc k\u00e2\u011f\u0131t da vakit de kalmad\u0131. Yapt\u0131\u011f\u0131m elbiseden ve \u015fapkadan gelen paradan sadece bir g\u00fcm\u00fc\u015f ruble kald\u0131. Siz ev sahibesine iki g\u00fcm\u00fc\u015f ruble verdiniz; bu \u00e7ok iyi; bir s\u00fcre susar \u015fimdi.\n\nKendinize bir elbise uydurun. Ho\u015f\u00e7a kal\u0131n; \u00e7ok yoruldum; neden bu kadar g\u00fc\u00e7s\u00fczle\u015fti\u011fimi anlam\u0131yorum; en k\u00fc\u00e7\u00fck u\u011fra\u015f beni yoruyor. \u0130\u015f olsa bile, nas\u0131l \u00e7al\u0131\u015faca\u011f\u0131m? \u0130\u015fte bu mahvediyor beni.\n\nV.D.\n\n5 Eyl\u00fcl\n\nG\u00fcvercinim, Varenka,\n\nMele\u011fim, bug\u00fcn bir s\u00fcr\u00fc olay ge\u00e7ti ba\u015f\u0131mdan. Birincisi, b\u00fct\u00fcn g\u00fcn ba\u015f\u0131m a\u011fr\u0131d\u0131. Biraz hava almak i\u00e7in Fontanka k\u0131y\u0131s\u0131nda gezmeye \u00e7\u0131kt\u0131m. \u00c7ok s\u0131cak, nemli bir ak\u015famd\u0131. Saat alt\u0131da ortal\u0131k kararm\u0131\u015ft\u0131 \u2013 t\u0131pk\u0131 \u015fimdiki gibi! Ya\u011fmur ya\u011fm\u0131yordu ama sis vard\u0131; s\u0131k\u0131 bir ya\u011fmurdan geri kalmayan bir sis. G\u00f6kte uzun, ipliksi bulutlar ilerliyordu. Halk ak\u0131n ak\u0131n y\u00fcr\u00fcyordu k\u0131y\u0131da ve hepsinin de y\u00fcz\u00fc sanki kastenmi\u015f gibi korkun\u00e7, kasvetliydi; sarho\u015f adamlar, kalk\u0131k burunlu kad\u0131nlar \u2013 _\u00e7uhonka_ 'lar, \u00e7izme giymi\u015f ba\u015f\u0131 ba\u011fs\u0131z bek\u00e2r kad\u0131nlar, artel i\u015f\u00e7ileri, arabac\u0131lar, t\u00fcrl\u00fc t\u00fcrl\u00fc ihtiya\u00e7lar\u0131n pe\u015findeki karde\u015flerimiz; \u00e7ubuklu \u00f6nl\u00fc\u011f\u00fcn\u00fc giymi\u015f, s\u0131ska, hastal\u0131kl\u0131, y\u00fcz\u00fc ya\u011f kaplanm\u0131\u015f, eline bir kilit alm\u0131\u015f \u00e7ilingir \u00e7\u0131ra\u011f\u0131; iki metre boyunda bir emekli asker \u2013 i\u015fte b\u00f6yle bir halk vard\u0131. O saatte de herhalde ba\u015fka bir halk olamazd\u0131. Gemi kanal\u0131 Fontanka! Mavnalar\u0131n sonu gelmiyor, insan hepsinin nas\u0131l yer bulabildi\u011fine \u015fa\u015f\u0131yor. K\u00f6pr\u00fclerde \u00e7\u00fcr\u00fck elma ve \u0131slak kurabiyeler satan kocakar\u0131lar oturuyor ve hepsi de kirli, \u0131slak bu kad\u0131nlar\u0131n. Fontanka'da gezmek ne s\u0131k\u0131c\u0131! Ayaklar\u0131n alt\u0131nda \u0131slak mermer, iki yanda y\u00fcksek, kara, isten kararm\u0131\u015f evler; ayaklar\u0131n alt\u0131nda sis, ba\u015f\u0131n tepesinde sis. Bu ak\u015fam b\u00f6yle h\u00fcz\u00fcnl\u00fc, b\u00f6yle karanl\u0131kt\u0131 ortal\u0131k.\n\nGorohovaya'ya sapt\u0131\u011f\u0131mda, ortal\u0131k iyice kararm\u0131\u015f ve gaz lambalar\u0131 yanm\u0131\u015ft\u0131. Uzun zamand\u0131r gitmemi\u015ftim Gorohovaya'ya, gidememi\u015ftim. Ne g\u00fcr\u00fclt\u00fcl\u00fc bir cadde! Ne kadar \u00e7ok zengin d\u00fckk\u00e2nlar, ma\u011fazalar var; her \u015fey \u0131\u015f\u0131l \u0131\u015f\u0131l yan\u0131yor, camlar\u0131n arkas\u0131nda kuma\u015flar, \u00e7i\u00e7ekler, kurdeleli bir s\u00fcr\u00fc \u015fapkalar var. B\u00fct\u00fcn bunlar\u0131n s\u00fcs diye konuldu\u011funu d\u00fc\u015f\u00fcn\u00fcyor insan ama \u00f6yle de\u011fil bunlar, \u00e7\u00fcnk\u00fc adamlar var, b\u00fct\u00fcn bunlar\u0131 sat\u0131n al\u0131yor, kar\u0131lar\u0131na hediye ediyorlar. Zengin bir cadde! Gorohovaya'da bir s\u00fcr\u00fc Alman f\u0131r\u0131nc\u0131 ya\u015f\u0131yor; cadde halk\u0131n\u0131n da hali vakti yerinde olmal\u0131. Her an bir s\u00fcr\u00fc araba ge\u00e7iyor caddeden; k\u00f6pr\u00fc b\u00fct\u00fcn hepsini nas\u0131l ta\u015f\u0131yabiliyor! \u015eatafatl\u0131 yayl\u0131 arabalar var, camlar\u0131 ayna gibi, i\u00e7leri kadife ve k\u00fcrk kapl\u0131; soylular\u0131n u\u015faklar\u0131 apoletli, k\u0131l\u0131\u00e7l\u0131. B\u00fct\u00fcn arabalar\u0131 seyrettim, i\u00e7lerinde oturan kad\u0131nlar\u0131n hepsi de prenses ve kontesler gibi giyinmi\u015fti. Herhalde, herkesin balolara ve toplant\u0131lara tela\u015f etti\u011fi bir saatti. Prensesleri ve \u00fcnl\u00fc kad\u0131nlar\u0131 yak\u0131ndan g\u00f6rmek ilgin\u00e7 bir \u015fey; herhalde \u00e7ok iyi bir \u015fey olmal\u0131; hi\u00e7 g\u00f6rmedim ben; belki de \u015fimdiki gibi, arabayla ge\u00e7iyorlard\u0131r. O s\u0131rada sizi hat\u0131rlad\u0131m. Ah, g\u00fcvercinim, bir tanem! Nas\u0131l hat\u0131rl\u0131yorum \u015fimdi sizi, kalbim nas\u0131l da bunal\u0131yor! Neden siz bu kadar mutsuzsunuz Varenka? Mele\u011fim! Neyiniz eksik ki onlardan? Siz benim i\u00e7in iyi kalpli, harika, okumu\u015f birisiniz; neden sizin pay\u0131n\u0131za k\u00f6t\u00fc bir talih d\u00fc\u015fm\u00fc\u015f? Neden her \u015fey b\u00f6yle oluyor, iyi bir insan karanl\u0131kta kal\u0131yor, bir ba\u015fkas\u0131naysa mutluluk kendili\u011finden geliyor? Biliyorum, biliyorum, can\u0131m, b\u00f6yle d\u00fc\u015f\u00fcnmek iyi de\u011fil, serbest fikircilik bu; ama i\u00e7tenlikle, hakikat ad\u0131na, neden baz\u0131lar\u0131 i\u00e7in kader, yavrusu daha annesinin karn\u0131ndayken mutluluk m\u00fcjdesi verir; ama bir ba\u015fkas\u0131 yetimhaneden \u00e7\u0131k\u0131p do\u011fruca sokaklara d\u00fc\u015fer? Zaten mutluluk genellikle Aptal \u0130van'c\u0131\u011fa26 gelir. Yani Aptal \u0130van'c\u0131\u011f\u0131m, dededen kalma \u00e7uvallar\u0131 kar\u0131\u015ft\u0131r, ye, i\u00e7, e\u011flen, fakat sen, oradaki, i\u015fte sen, yutkun, o sana yeter; yani, sana da b\u00f6ylesi lay\u0131kt\u0131r, karde\u015fim, i\u015fte \u00f6yle!\" G\u00fcnah, can\u0131m, g\u00fcnah b\u00f6yle d\u00fc\u015f\u00fcnmek, ama ister istemez gelip giriyor ruha g\u00fcnah. Ke\u015fke siz b\u00f6yle bir araba i\u00e7inde gezseydiniz, bir tanem, can\u0131m\u0131n i\u00e7i. Kibar bak\u0131\u015flar\u0131n\u0131z generalleri esir al\u0131rd\u0131, bizim gibileri de\u011fil; eski bir basma elbiseyle de\u011fil, alt\u0131n s\u0131rmal\u0131 ipekler i\u00e7inde gezerdiniz. \u015eimdiki gibi zay\u0131f, \u00e7\u00f6p gibi olmazd\u0131n\u0131z, \u015feker gibi, ayd\u0131nl\u0131k, k\u0131rm\u0131z\u0131, dolgun bir \u00e7ehreye sahip olurdunuz. Ben de o zaman sokaktan sizin g\u00fc\u00e7l\u00fc bir \u015fekilde ayd\u0131nlat\u0131lm\u0131\u015f pencerelerinize bak\u0131p g\u00f6lgenizden ba\u015fka bir \u015fey g\u00f6remesem bile mutlu olurdum; sadece sizin mutlu ve ne\u015feli oldu\u011funuzu d\u00fc\u015f\u00fcnmek bile mutlu ederdi beni, g\u00fczel ku\u015fum benim, ne\u015felenirdim ben de. Ama \u015fimdi! K\u00f6t\u00fc insanlar\u0131n sizi y\u0131pratmalar\u0131 yetmiyormu\u015f gibi, oraya sefil, ahlaks\u0131z bir adam gelip sizi g\u00fccendiriyor. \u00dczerindeki frakla hindi gibi dikiliyor, alt\u0131n kelebekg\u00f6zl\u00fc\u011f\u00fcyle size bakan o utanmaz, her \u015fey onun elindeymi\u015f gibi, edepsiz a\u011fz\u0131ndan \u00e7\u0131kanlar\u0131 dinlemek gerekliymi\u015f gibi davran\u0131yor! Yeter ama art\u0131k, de\u011fil mi g\u00fcvercinim! Peki neden oluyor b\u00fct\u00fcn bunlar? Siz yetim oldu\u011funuz i\u00e7in, siz savunmas\u0131z oldu\u011funuz i\u00e7in, yan\u0131n\u0131zda kimse olmad\u0131\u011f\u0131 i\u00e7in; s\u0131rt\u0131n\u0131z\u0131 dayayaca\u011f\u0131n\u0131z g\u00fc\u00e7l\u00fc bir dost olmad\u0131\u011f\u0131 i\u00e7in. Peki bir yetimi bir hi\u00e7 i\u00e7in g\u00fccendiren bu insanlara insan m\u0131 dememiz gerekir? Sefil bunlar, insan de\u011fil, sadece sefil; onlar insan de\u011fil, g\u00f6r\u00fcn\u00fc\u015fte \u00f6yleler sadece, buna inan\u0131yorum ben. \u0130\u015fte b\u00f6yle bu insanlar! Ve bence, bir tanem, i\u015fte bug\u00fcn Gorohovaya'da rastlad\u0131\u011f\u0131m laternac\u0131, onlardan daha \u00e7ok hak ediyor sayg\u0131y\u0131. B\u00fct\u00fcn g\u00fcn dolan\u0131p yoruluyor, bo\u011faz\u0131na girecek bayat, kalitesiz bir lokmay\u0131 bekliyor, ama b\u00f6ylece kendisinin efendisi oluyor, kendi kendine bak\u0131yor. Sadaka istemiyor; insanlar\u0131n keyfi i\u00e7in, kurulmu\u015f makine gibi \u00e7al\u0131\u015f\u0131yor... \u0130\u015fte, \"Elimden geldi\u011fi kadar\u0131yla, mutluluk getireyim,\" diyor. Dilenci, bir dilenci o, a\u00e7\u0131k\u00e7as\u0131, bildi\u011fimiz dilenci; ama yine de soylu bir dilenci; yorgun, \u00fc\u015f\u00fcm\u00fc\u015f, ama yine de \u00e7al\u0131\u015f\u0131yor, kendince de olsa her \u015feye ra\u011fmen \u00e7al\u0131\u015f\u0131yor. Bir s\u00fcr\u00fc \u015ferefli insan var, can\u0131m, emeklerinin kar\u015f\u0131l\u0131\u011f\u0131n\u0131 alamasalar da, kimseye boyun e\u011fmeyen, kimseden ekmek istemeyen insanlar var. \u0130\u015fte ben de onun gibiyim, bu laternac\u0131 gibi, o de\u011filim tabii, hi\u00e7 onunla ayn\u0131 de\u011filim, ama kendi fikrimce, t\u0131pk\u0131 onun gibi soylu, terbiyeli bir tav\u0131r i\u00e7indeyim, g\u00fcc\u00fcm yetti\u011fince, yani yapabildi\u011fimce \u00e7al\u0131\u015f\u0131yorum. Daha fazlas\u0131 gelmiyor elimden; ama yapacak bir \u015fey yok.\n\nBu laternac\u0131dan bahsetmemin nedeni, bug\u00fcn yoksullu\u011fumu iki kat hissetmi\u015f olmam. Laternac\u0131ya bakmak i\u00e7in durmu\u015ftum. Bir \u015feyler d\u00fc\u015f\u00fcn\u00fcyordum, akl\u0131m\u0131 da\u011f\u0131tmak i\u00e7in durmu\u015ftum. Benden ba\u015fka arabac\u0131lar, bir gen\u00e7 k\u0131z bir de k\u00fc\u00e7\u00fck bir k\u0131z vard\u0131, hepsi de kir pas i\u00e7indeydi. Laternac\u0131 bir pencerenin \u00f6n\u00fcnde duruyordu. Bir \u00e7ocuk, on ya\u015flar\u0131nda olabilecek k\u00fc\u00e7\u00fck bir \u00e7ocuk g\u00f6z\u00fcme \u00e7arpt\u0131; g\u00fczel bir \u00e7ocuktu, ama \u00e7ok hastal\u0131kl\u0131, \u00e7ok c\u0131l\u0131z bir g\u00f6r\u00fcn\u00fc\u015f\u00fc vard\u0131, \u00fczerinde de bir g\u00f6mlekle bir \u015fey daha vard\u0131, ayaklar\u0131 \u00e7\u0131plak say\u0131l\u0131rd\u0131, a\u011fz\u0131 iki kar\u0131\u015f a\u00e7\u0131k m\u00fczi\u011fi dinliyordu, \u00e7ocuk i\u015fte! Hayran hayran seyrediyordu Alman\u0131n kuklalar\u0131 dans ettirmesini, ama kendi eli aya\u011f\u0131 buz kesmi\u015fti, titriyor, g\u00f6mle\u011finin kolunu kemiriyordu. Elinde bir k\u00e2\u011f\u0131t oldu\u011funu fark ettim. Bir adam geldi ve laternac\u0131ya biraz para att\u0131; para do\u011fruca gitti, i\u00e7inde kad\u0131nlarla dans eden bir Frans\u0131z kuklas\u0131n\u0131n durdu\u011fu k\u00fc\u00e7\u00fck sahnenin kutusunun i\u00e7ine d\u00fc\u015ft\u00fc. Para \u00e7\u0131nlad\u0131\u011f\u0131 anda, benim \u00e7ocuk silkindi, \u00fcrkek \u00fcrkek \u00e7evresine bak\u0131nd\u0131 ve o s\u0131rada paray\u0131 benim verdi\u011fimi sand\u0131. Bana do\u011fru ko\u015ftu, elleri titriyordu, sesi titriyordu, bana bir k\u00e2\u011f\u0131t uzat\u0131p \u015f\u00f6yle dedi: Okuyun! K\u00e2\u011f\u0131d\u0131 a\u00e7t\u0131m, hep bildik \u015feyler; i\u015fte, \"Hay\u0131rsever insanlar, bu \u00e7ocuklar\u0131n annesi \u00f6l\u00fcyor, \u00fc\u00e7 \u00e7ocuk a\u00e7, \u015fimdi yard\u0131m edin, yoksa \u00f6lece\u011fim, yavrular\u0131m\u0131 unutmay\u0131n, hay\u0131rsever insanlar, ben de sizi \u00f6teki d\u00fcnyada unutmam.\" Ee, ne olacak; durum belli, hayati bir durum, peki ben ne verece\u011fim ona? Ona hi\u00e7bir \u015fey vermedim. Ama nas\u0131l \u00fcz\u00fcld\u00fcm! Yoksul \u00e7ocuk, so\u011fuktan, belki de a\u00e7l\u0131ktan bitmi\u015f durumda ve yalan s\u00f6ylemiyor, hay\u0131r hay\u0131r, yalan s\u00f6ylemiyor; bu durumu bilirim. Ama \u00e7ocuklar\u0131n bu i\u011fren\u00e7 annesinin ne diye sakland\u0131\u011f\u0131n\u0131 ve onlar\u0131 yar\u0131 \u00e7\u0131plak halde bu so\u011fukta mektuplarla etrafa g\u00f6nderdi\u011fini anlamak m\u00fcmk\u00fcn de\u011fil. Belki aptal bir kocakar\u0131d\u0131r, karaktersizdir; belki onun i\u00e7in kayg\u0131lanan kimse yoktur, ayaklar\u0131n\u0131 k\u0131v\u0131rm\u0131\u015f oturuyordur, belki ger\u00e7ekten hastad\u0131r. Neyse, yine de ilgili yerlere ba\u015fvurmak gerekir; fakat, belki de, sadece \u00fc\u00e7k\u00e2\u011f\u0131t\u00e7\u0131d\u0131r, kasten a\u00e7 ve s\u0131ska \u00e7ocuklar\u0131 hasta ediyor, halk\u0131 kand\u0131rmak i\u00e7in g\u00f6nderiyordur. Bu mektuplar\u0131 ta\u015f\u0131yan yoksul \u00e7ocuk ne \u00f6\u011freniyor? Sadece kalbi kat\u0131la\u015f\u0131yor; dolan\u0131yor, ko\u015fuyor, yalvar\u0131yor. \u0130nsanlar gelip ge\u00e7iyor, ona hi\u00e7 ald\u0131rm\u0131yor. Kalpleri ta\u015f olmu\u015f; dilleri ac\u0131mas\u0131z. \"Defol! \u00c7ekil! Yaramaz!\" \u0130\u015fte bunlar\u0131 duyuyor herkesten ve kalbi kat\u0131la\u015f\u0131yor \u00e7ocu\u011fun, zavall\u0131, \u00fcrkek \u00e7ocuk so\u011fukta, yuvas\u0131ndan d\u00fc\u015fm\u00fc\u015f yavru ku\u015f gibi bo\u015f yere titriyor. Bir bak\u0131yorsun, \u00f6ks\u00fcr\u00fcyor; \u00fczerine \u00e7ullanmak i\u00e7in bekliyor hastal\u0131k, t\u0131pk\u0131 al\u00e7ak bir s\u00fcr\u00fcngen gibi, g\u00f6\u011fs\u00fcne h\u00fccum ediyor, oraya yerle\u015fiyor, bir bak\u0131yorsun \u00f6l\u00fcm tepesine dikilmi\u015f, karanl\u0131k bir k\u00f6\u015fede durduruyor, \u00e7\u0131k\u0131\u015f yok, yard\u0131m eden yok, i\u015fte b\u00fct\u00fcn hayat\u0131 bu! \u0130\u015fte b\u00f6yle, bu da onun ya\u015fam\u0131 oluyor! Ah, Varenka, \"\u0130sa a\u015fk\u0131na\" dendi\u011fini duymak ve yan\u0131ndan ge\u00e7mek, hi\u00e7bir \u015fey vermeyip, \"Tanr\u0131 versin,\" demek \u00e7ok ac\u0131 verici bir \u015fey. Baz\u0131 \"\u0130sa a\u015fk\u0131na\"lar\u0131n etkisi olmuyor. (\"\u0130sa a\u015fk\u0131na\"lar da \u00e7e\u015fit \u00e7e\u015fit olur, can\u0131m.) Baz\u0131lar\u0131 uzun, a\u011f\u0131r, s\u0131radan, ezberden, tam dilenci s\u0131zlanmas\u0131d\u0131r; ona bir \u015fey vermemek ac\u0131 vermez, o uzun zamand\u0131r dilencidir, eski, meslekten dilencidir, bu al\u0131\u015f\u0131k, diye d\u00fc\u015f\u00fcn\u00fcrs\u00fcn, bu bir yolunu bulur, yolunu nas\u0131l bulaca\u011f\u0131n\u0131 bilir. Ama baz\u0131 \"\u0130sa a\u015fk\u0131na\"lar al\u0131\u015f\u0131lmad\u0131k olur, kaba, korkun\u00e7 olur; i\u015fte bug\u00fcn, \u00e7ocuktan mektubu ald\u0131\u011f\u0131m zaman, \u00e7itin hemen dibinde biri duruyordu, herkesten de istemiyordu, bana, \"Bir kuru\u015f versene beyefendi, \u0130sa a\u015fk\u0131na!\" dedi, ama \u00f6yle kesik kesik, kaba bir sesle s\u00f6yledi ki, korkun\u00e7 bir duyguyla irkildim, yine de para vermedim; yoktu. Zaten zengin insanlar yoksullar\u0131n k\u00f6t\u00fc kaderlerinden seslice yak\u0131nmas\u0131n\u0131 sevmez \u2013yani, birileri rahats\u0131z olur, birileri de s\u0131zlan\u0131r! Yoksullar hep s\u0131zlan\u0131r\u2013 uyumaya kalkars\u0131n, onlar\u0131n a\u00e7 iniltileri uykuna engel olur!\n\nBir tanem, size itiraf edeyim, b\u00fct\u00fcn bunlar\u0131 size anlatmaya ba\u015flamam\u0131n sebebi k\u0131smen kalbimi ferahlatmak, ama daha da \u00f6nemlisi size yarat\u0131c\u0131l\u0131\u011f\u0131m\u0131n y\u00fcksek \u00fcslubuna bir \u00f6rnek sunabilmek. \u00c7\u00fcnk\u00fc siz, herhalde, kendiniz de kabul edersiniz ki can\u0131m, bir s\u00fcre \u00f6nce kendime has bir \u00fcslubum olu\u015ftu. Ama \u015fimdi \u00fczerime \u00f6yle bir s\u0131k\u0131nt\u0131 \u00e7\u00f6kt\u00fc ki, d\u00fc\u015f\u00fcncelerimi can\u0131 g\u00f6n\u00fclden hissetmeye ba\u015flad\u0131m ve can\u0131m, bununla bir yere varamaz insan, ama yine de bir par\u00e7a kendi kendine de\u011fer vermeli. Ger\u00e7ekten, bir tanem, s\u0131k s\u0131k kendi kendimi hi\u00e7 neden yokken peri\u015fan ediyorum, kendime hi\u00e7 de\u011fer vermiyorum ve bir ta\u015f par\u00e7as\u0131ndan de\u011fersiz say\u0131yorum. Bunu bir benzetmeyle dile getirirsek, belki de bunun nedeni, benim benden sadaka isteyen \u015fu yoksul \u00e7ocuk gibi itilip kak\u0131lm\u0131\u015f olmamd\u0131r. \u00d6rne\u011fin, \u015fimdi size alegorik bir \u015fekilde s\u00f6yleyeyim can\u0131m; beni bir dinleyin; dinleyin beni, bir tanem, sabah\u0131n erken vaktinde, i\u015fe tela\u015f ederken, \u015fehre bakar\u0131m, onun nas\u0131l uyand\u0131\u011f\u0131n\u0131, kalkt\u0131\u011f\u0131n\u0131, duman \u00e7\u0131kard\u0131\u011f\u0131n\u0131, kaynad\u0131\u011f\u0131n\u0131, k\u00fckredi\u011fini seyrederim, bazen b\u00f6yle bir manzaran\u0131n kar\u015f\u0131s\u0131nda insan k\u00fc\u00e7\u00fcl\u00fcr, sanki merakl\u0131 burnuna fiske yemi\u015f gibi olur ve elini sallar, kendi yolunda sudan sessiz, otlardan al\u00e7ak bir tav\u0131rla ilerler! \u015eimdi bak\u0131n\u0131z etraf\u0131n\u0131za, bu kara, isli, b\u00fcy\u00fck, kocaman binalarda neler oluyor? Bunu anlarsan\u0131z e\u011fer o zaman kendiniz karar verirsiniz, anlams\u0131zca kendinizi s\u0131n\u0131fland\u0131rman\u0131n ve de\u011fersiz bir utanga\u00e7l\u0131\u011fa kap\u0131lman\u0131n hakl\u0131 olup olmad\u0131\u011f\u0131n\u0131. \u015euna dikkat edin Varenka, d\u00fcz anlamda s\u00f6ylemiyorum, mecazi bir \u015fekilde konu\u015fuyorum. Haydi, bir bakal\u0131m, ne var bu binalarda? \u015eu dumanl\u0131 k\u00f6\u015fede, ihtiya\u00e7 y\u00fcz\u00fcnden ev niyetine say\u0131lan \u0131slak bir kul\u00fcbede, bir i\u015f\u00e7i r\u00fcyas\u0131ndan uyan\u0131yor; r\u00fcyas\u0131nda da, mesela, b\u00fct\u00fcn gece \u00e7izme g\u00f6rm\u00fc\u015f, d\u00fcn kazara kesti\u011fi \u00e7izmeyi g\u00f6rm\u00fc\u015f, sanki insan r\u00fcyas\u0131nda b\u00f6yle bir sa\u00e7mal\u0131k g\u00f6rmek zorundaym\u0131\u015f gibi! Fakat sonu\u00e7ta o bir i\u015f\u00e7i, bir \u00e7izmeci; hep ayn\u0131 \u015feyi d\u00fc\u015f\u00fcnmesi ho\u015f g\u00f6r\u00fclebilir. C\u0131v\u0131l c\u0131v\u0131l \u00e7ocuklar\u0131 ve a\u00e7 bir kar\u0131s\u0131 da var orada; ve bir tek \u00e7izmeciler b\u00f6yle kalkmaz, bir tanem. \u00d6yle olsa \u00f6nemsiz olurdu, bunu yazmaya de\u011fmezdi, ama can\u0131m, orada durum bak\u0131n nas\u0131ld\u0131r: Orada, ayn\u0131 binada, bir \u00fcst ya da alt katta, yald\u0131zl\u0131 odalarda zengin bir \u015fah\u0131s da \u00e7izmeler g\u00f6r\u00fcr belki b\u00fct\u00fcn gece r\u00fcyas\u0131nda, yani ba\u015fka t\u00fcr bir \u00e7izme g\u00f6r\u00fcr, ba\u015fka model bir \u00e7izmedir bunlar, ama yine de \u00e7izmedir; \u00e7\u00fcnk\u00fc benim burada kastetmek istedi\u011fim \u015fu, can\u0131m, hepimiz, bir tanem, biraz \u00e7izmeci gibiyiz. Ama b\u00fct\u00fcn bunlar \u00f6nemsiz, ama bu zengin \u015fahs\u0131n yan\u0131nda onun kula\u011f\u0131na f\u0131s\u0131ldayacak, \"Yeter b\u00f6yle d\u00fc\u015f\u00fcnmek, bir tek kendini d\u00fc\u015f\u00fcnmek, bir tek kendin i\u00e7in ya\u015famak yeter, sen \u00e7izmeci de\u011filsin, senin \u00e7ocuklar\u0131n sa\u011fl\u0131kl\u0131, kar\u0131n\u0131n da bir ihtiyac\u0131 yok; \u00e7evrene bir bak, kayg\u0131lanmak i\u00e7in kendi \u00e7izmelerinden daha soylu bir \u015fey g\u00f6rm\u00fcyor musun etraf\u0131nda!\" diyecek biri olmamas\u0131 k\u00f6t\u00fc as\u0131l. \u0130\u015fte size mecazi olarak demek istedi\u011fim buydu, Varenka. Belki bu a\u015f\u0131r\u0131 serbest fikirdir, bir tanem, ama bu fikir bazen geliyor, bazen ortaya \u00e7\u0131k\u0131yor ve o zaman ister istemez kalpten ate\u015fli bir dil y\u00fckseliyor. S\u0131rf g\u00fcr\u00fclt\u00fc ve g\u00fcmb\u00fcrt\u00fcden \u00e7ekinip insan\u0131n kendisini bu y\u00fczden de \u00fc\u00e7 kuru\u015fluk saymas\u0131 kabul edilemez! Can\u0131m, bitirirken akl\u0131ma geldi, belki de size yalan s\u00f6yledi\u011fimi d\u00fc\u015f\u00fcn\u00fcyorsunuzdur ya da karamsarl\u0131\u011fa kap\u0131ld\u0131\u011f\u0131m\u0131 ya da bunu bir kitaptan al\u0131p yazd\u0131\u011f\u0131m\u0131 d\u00fc\u015f\u00fcn\u00fcyorsunuzdur? Hay\u0131r, can\u0131m, g\u00fcvensizlik ediyorsunuz \u2013 \u00f6yle de\u011fil; yalan\u0131 hor g\u00f6r\u00fcr\u00fcm, karamsarl\u0131\u011fa kap\u0131lmad\u0131m ve hi\u00e7bir kitaptan al\u0131p yazmad\u0131m bunlar\u0131 \u2013 i\u015fte o kadar!\n\nEve kasvetli bir ruh haliyle geldim, masan\u0131n ba\u015f\u0131na oturdum, kendime \u00e7ay demledim, bir fincan \u00e7ay i\u00e7meye haz\u0131rland\u0131m. Sonra, birden, bir bakt\u0131m, odama Gor\u015fkov gelmi\u015f, bizim yoksul kirac\u0131. Sabahleyin onun kirac\u0131lar\u0131n \u00e7evresinde doland\u0131\u011f\u0131n\u0131 ve yan\u0131ma gelmek istedi\u011fini fark etmi\u015ftim. Bu arada \u015funu s\u00f6yleyeyim, can\u0131m, onlar\u0131n ya\u015fay\u0131\u015f\u0131 benimkiyle k\u0131yaslan\u0131r gibi de\u011fil. M\u00fcmk\u00fcn de\u011fil! Bir e\u015f, bir de \u00e7ocuklar! Ben Gor\u015fkov olsayd\u0131m, onun yerinde olsayd\u0131m ne yapaca\u011f\u0131m\u0131 hi\u00e7 bilmiyorum! Neyse i\u015fte Gor\u015fkov geldi, her zamanki gibi g\u00f6z\u00fc ya\u015fl\u0131, burnunu \u00e7ekiyor, ama bir \u015fey s\u00f6ylemiyor. Onu sandalyeye oturttum, k\u0131r\u0131kt\u0131 sandalye, ama ba\u015fka da yoktu. Bir fincan \u00e7ay teklif ettim. Geri \u00e7evirdi, uzun s\u00fcre geri \u00e7evirdi, sonunda ald\u0131 yine de barda\u011f\u0131. \u015eekersiz i\u00e7mek istedi, ben ona \u015feker almas\u0131 gerekti\u011fini anlat\u0131rken yine \u00f6z\u00fcrler dilemeye ba\u015flad\u0131, uzun s\u00fcre itiraz etti, geri \u00e7evirdi, sonunda barda\u011f\u0131na k\u00fc\u00e7\u00fcc\u00fck bir par\u00e7a \u015feker koydu ve \u00e7ay\u0131n muhte\u015fem bir \u015fekilde lezzetli oldu\u011funu s\u00f6ylemeye ba\u015flad\u0131. Ah, sefalet insanlar\u0131 ne kadar k\u00fc\u00e7\u00fclmeye s\u00fcr\u00fckl\u00fcyor! \"Peki ne var, nedir istedi\u011finiz can\u0131m?\" dedim ona. \"\u0130\u015fte b\u00f6yle b\u00f6yle,\" dedi, \"siz velinimetimsiniz, Makar Alekseyevi\u00e7, bana bir efendilik edin, talihsiz aileme yard\u0131m edin; \u00e7ocuklar\u0131m ve kar\u0131m\u0131n elinde hi\u00e7bir \u015fey yok; baba diyorlar bana, ama nerede!\" Bir \u015fey s\u00f6ylemek istedim, ama s\u00f6z\u00fcm\u00fc kesti: \"Zaten buradaki herkesten korkuyorum Makar Alekseyevi\u00e7, yani korkmuyorum da, anl\u0131yor musunuz, utan\u0131yorum; hepsi de kibirli ve ma\u011frur insanlar. Sizi rahats\u0131z etmek istemezdim babac\u0131\u011f\u0131m, velinimetim; biliyorum, sizin de ba\u015f\u0131n\u0131zda tats\u0131zl\u0131klar var, biliyorum \u00e7ok veremezsiniz, ama biraz olsun bor\u00e7 verseniz; size sormaya cesaret ettim, \u00e7\u00fcnk\u00fc iyi kalbinizi biliyorum, sizin de ihtiya\u00e7tan anlad\u0131\u011f\u0131n\u0131z\u0131, yoksulluk nedir bildi\u011finizi... ve kalbinizde bu y\u00fczden merhamet duygusu olaca\u011f\u0131n\u0131 biliyorum.\" Sonunda da, \"Benim c\u00fcretimi ve edepsizli\u011fimi ho\u015f g\u00f6r\u00fcn Makar Alekseyevi\u00e7,\" diyerek bitirdi. Ona ruhen mutlu oldu\u011fumu, ama benim de elimde bir \u015fey olmad\u0131\u011f\u0131n\u0131, hemen hemen hi\u00e7bir \u015fey olmad\u0131\u011f\u0131n\u0131 s\u00f6yledim. \"Babac\u0131\u011f\u0131m, Makar Alekseyevi\u00e7,\" dedi bana, \"sizden \u00e7ok istemiyorum, yani b\u00f6yle b\u00f6yle,\" (o s\u0131rada k\u0131pk\u0131rm\u0131z\u0131 olmu\u015ftu) \"kar\u0131m,\" dedi, \"\u00e7ocuklar\u0131m a\u00e7, birka\u00e7 kuru\u015f olsun yeter.\" Benim kalbim alabildi\u011fine s\u0131k\u0131\u015ft\u0131. \"Vay,\" dedim, \"bu benden bask\u0131n \u00e7\u0131kt\u0131!\" \u00dczerimde sadece yirmi kopek kalm\u0131\u015ft\u0131, ama hesab\u0131m\u0131 ona g\u00f6re yapm\u0131\u015ft\u0131m; ertesi g\u00fcn kendi acil ihtiya\u00e7lar\u0131m\u0131 almak istiyordum. \"Hay\u0131r, g\u00fcvercinim, yapamam; durum ortada,\" dedim. \"Babac\u0131\u011f\u0131m, Makar Alekseyevi\u00e7, ne koparsa g\u00f6nl\u00fcn\u00fczden,\" dedi, \"on kopek de yeter.\" Yani, \u00e7ekmeceyi a\u00e7\u0131p yirmi kope\u011fimi ona verdim, can\u0131m, iyi bir i\u015f u\u011fruna! Ah, tam sefalet! Ona \u015f\u00f6yle dedim: \"Peki, can\u0131m\u0131n i\u00e7i,\" diye sordum, \"bu kadar dara d\u00fc\u015fm\u00fc\u015fs\u00fcn\u00fcz, peki be\u015f g\u00fcm\u00fc\u015f rublelik oday\u0131 nas\u0131l kiral\u0131yorsunuz?\" Alt\u0131 ay \u00f6nce kiralay\u0131p \u00fc\u00e7 ayl\u0131k paray\u0131 pe\u015fin \u00f6dedi\u011fini anlatt\u0131 bana; sonra durum k\u00f6t\u00fcye gitmi\u015f, oradan buradan i\u015f bulamam\u0131\u015f, zavall\u0131. Davas\u0131n\u0131n bu s\u0131ralar sonu\u00e7lanmas\u0131n\u0131 bekliyormu\u015f. Davas\u0131 da tats\u0131z bir \u015fey. Mahkemeye bir konuda hesap vermesi gerekiyormu\u015f, Varenka. Hazineye ihale yapan bir t\u00fcccarla davas\u0131 varm\u0131\u015f; bir hile ortaya \u00e7\u0131km\u0131\u015f, t\u00fcccar mahkemeye d\u00fc\u015fm\u00fc\u015f, ama t\u00fcccar yapt\u0131\u011f\u0131 haydutlu\u011fu Gor\u015fkov'un \u00fcst\u00fcne y\u0131km\u0131\u015f, o da i\u015fe b\u00f6yle bula\u015fm\u0131\u015f. Asl\u0131nda Gor\u015fkov sadece ihmal, dikkatsizlik ve devlet \u00e7\u0131kar\u0131n\u0131 g\u00f6zetmemekten su\u00e7lan\u0131yor. Birka\u00e7 y\u0131ld\u0131r dava devam ediyormu\u015f: Gor\u015fkov'un kar\u015f\u0131s\u0131na t\u00fcrl\u00fc t\u00fcrl\u00fc engel \u00e7\u0131km\u0131\u015f. \"\u00dczerime y\u0131kt\u0131klar\u0131 bu \u015fey tam bir onursuzluk,\" diyor bana Gor\u015fkov, \"su\u00e7suzum, hi\u00e7bir su\u00e7um yok, hile ve ya\u011fmaya kar\u0131\u015fmad\u0131m.\" Bu dava ona biraz leke s\u00fcrm\u00fc\u015f; onu i\u015ften \u00e7\u0131karm\u0131\u015flar ve resmen su\u00e7lu bulunmasa da, masumiyetini ispat edinceye kadar, hakk\u0131 olan ve mahkeme taraf\u0131ndan onaylanm\u0131\u015f olan ciddi bir para toplam\u0131n\u0131 t\u00fcccardan alam\u0131yor. Ona inan\u0131yorum, ama mahkeme onun s\u00f6z\u00fcne inanm\u0131yor; dava da her yerinden dolan\u0131p karman \u00e7orman olmu\u015f, y\u00fcz y\u0131l \u00e7\u00f6zmenin imk\u00e2n\u0131 yok. Biraz \u00e7\u00f6z\u00fclecek gibi oluyor, ama t\u00fcccar bir d\u00fc\u011f\u00fcm daha at\u0131yor. Gor\u015fkov'a vicdanen kat\u0131l\u0131yorum, bir tanem, ac\u0131s\u0131n\u0131 payla\u015f\u0131yorum. \u0130\u015fsiz bir adam; hi\u00e7bir yerden umudu yok; birikim harcanm\u0131\u015f; dava s\u00fcr\u00fcyor, ama bu arada ya\u015famak da laz\u0131m; buna ra\u011fmen hi\u00e7 beklenmedik bir \u015fekilde bir bebek do\u011fmu\u015f, masraf; o\u011flan hastaland\u0131 masraf; \u00f6ld\u00fc ayr\u0131 masraf; kar\u0131s\u0131 hasta; o da m\u00fczmin bir hastal\u0131\u011fa sahip; k\u0131sacas\u0131, \u00e7ekiyor, \u00e7ok \u00e7ekiyor.\n\nFakat, dedi\u011fine g\u00f6re, birka\u00e7 g\u00fcn i\u00e7inde davadan hay\u0131rl\u0131 bir karar bekliyormu\u015f ve art\u0131k bundan hi\u00e7 \u015f\u00fcphesi yok. Yaz\u0131k, yaz\u0131k, \u00e7ok yaz\u0131k ona, can\u0131m! \u015eefkatli davrand\u0131m ona. Bitkin, \u015fa\u015fk\u0131n bir adam; bir koruyucu ar\u0131yor, ben de yak\u0131nl\u0131k g\u00f6sterdim ona. Neyse, ho\u015f\u00e7a kal\u0131n can\u0131m, \u0130sa sizinle olsun, sa\u011fl\u0131\u011f\u0131n\u0131z bozulmas\u0131n. G\u00fcvercinim benim! Sizi hat\u0131rlamak benim hasta ruhuma ila\u00e7 gibi geliyor ve sizin i\u00e7in \u0131st\u0131rap \u00e7ekiyorum, ama bu \u0131st\u0131rap bana hafif geliyor.\n\nSamimi dostunuz,  \nMakar Devu\u015fkin\n\n9 Eyl\u00fcl\n\nCan\u0131m, Varvara Alekseyevna,\n\nSize kendimi kaybetmi\u015f bir halde yaz\u0131yorum. Korkun\u00e7 olaylarla heyecan i\u00e7indeyim. Ba\u015f\u0131m d\u00f6n\u00fcyor. \u00c7evremdeki her \u015fey d\u00f6n\u00fcyormu\u015f gibi. Ah, bir tanem, size \u015fimdi neler anlataca\u011f\u0131m! Biz bunu \u00f6ng\u00f6remedik. Hay\u0131r, inanm\u0131yorum bunu \u00f6ng\u00f6remedi\u011fime; bunlar\u0131n tahmin etmi\u015ftim. B\u00fct\u00fcn bunlar\u0131 daha \u00f6nceden hissetmi\u015fti kalbim! Hatta uykumda bile buna benzer \u015feyler g\u00f6rm\u00fc\u015ft\u00fcm.\n\n\u0130\u015fte oldu! Size \u00fcsluba ka\u00e7madan anlatay\u0131m, Tanr\u0131 ruhuma nas\u0131l \u00fcflerse \u00f6yle. Bug\u00fcn i\u015fe gittim. Gittim, oturdum, \u00e7al\u0131\u015f\u0131yorum. Ama \u015funu bilmeniz laz\u0131m, can\u0131m, d\u00fcn ak\u015fam da \u00e7al\u0131\u015fm\u0131\u015ft\u0131m. Yani, ak\u015famleyin Timofey \u0130vanovi\u00e7 geldi ve \u00f6nemli, acil bir evrak haz\u0131rlamak gerekti\u011fini bizzat emretti. \"Temize \u00e7ekin,\" dedi Makar Alekseyevi\u00e7, \"temiz, acele ve dikkatle yaz\u0131n; hemen imzaya gidecek.\" Size \u015funu da belirtmeliyim, mele\u011fim, d\u00fcn ak\u015fam kendimde de\u011fildim, hi\u00e7bir \u015feye bakmak istemiyordum; h\u00fcz\u00fcn, kasvet \u00e7\u00f6km\u00fc\u015ft\u00fc \u00fczerime! Kalbim so\u011fuk, ruhum karanl\u0131kt\u0131; akl\u0131mda hep siz vard\u0131n\u0131z, zavall\u0131 g\u00fczelim. Neyse, temize \u00e7ekmeye koyuldum; temiz, g\u00fczel bir \u015fekilde yazd\u0131m, ama size nas\u0131l anlatmal\u0131, bilmiyorum, karanl\u0131k bir \u015fey bula\u015ft\u0131 bana ya da gizli bir kaderin belirlemesiyle ya da sadece b\u00f6yle olaca\u011f\u0131 i\u00e7in... tam bir sat\u0131r atlam\u0131\u015f\u0131m; anlam da Tanr\u0131 bilir ne hale gelmi\u015f, kalmam\u0131\u015f yani. Evra\u011f\u0131 d\u00fcn ak\u015fam yeti\u015ftiremediler ve onu imza i\u00e7in ancak bug\u00fcn \u00e7\u0131karabildiler Majesteleri'ne. Ben de hi\u00e7bir \u015fey olmam\u0131\u015f gibi i\u015fe her zamanki saatte gittim ve Emelyan \u0130vanovi\u00e7'in yan\u0131ndaki yere yerle\u015ftim. Bir tanem, bir s\u00fcredir eskisinden iki kat daha \u00e7ok utan\u0131yor, \u00e7ekiniyordum gitmeye. Son zamanlarda da kimseye bakam\u0131yordum bile. Birinin sandalyesi g\u0131c\u0131rdasa, korkudan kendimi kaybediyordum. \u0130\u015fte bug\u00fcn de \u00f6yleydi, sinmi\u015ftim, sessizdim, diken \u00fcst\u00fcnde oturuyordum ki Efim Akimovi\u00e7 (b\u00f6yle bir sata\u015fma yery\u00fcz\u00fcnde g\u00f6r\u00fclmemi\u015ftir) y\u00fcksek sesle \"Ne o, Makar Alekseyevi\u00e7, niye \u00f6yle \u00e7\u0131tk\u0131r\u0131ld\u0131m oturuyorsunuz?\" diyerek \u00f6yle bir s\u0131r\u0131tt\u0131 ki herkes, onun ve benim \u00e7evremdeki herkes kahkahadan k\u0131r\u0131ld\u0131, tabii bana g\u00fcl\u00fcyorlard\u0131. Ve durmad\u0131lar, hi\u00e7 durmad\u0131lar! Kulaklar\u0131m\u0131 t\u0131kad\u0131m, g\u00f6zlerimi kapad\u0131m, kimseye kar\u0131\u015fmadan kendi ba\u015f\u0131ma oturdum. B\u00f6yle bir al\u0131\u015fkanl\u0131\u011f\u0131m var; o zaman hemen rahat b\u0131rak\u0131yorlar. Birden bir g\u00fcr\u00fclt\u00fc, ko\u015fu\u015fturma, tela\u015f duydum; duydum ama inanamad\u0131m, kulaklar\u0131m beni yan\u0131ltmas\u0131n? Bana sesleniyorlar, beni \u00e7a\u011f\u0131r\u0131yorlar, Devu\u015fkin diyorlar. Kalbim h\u0131zl\u0131 h\u0131zl\u0131 atmaya ba\u015flad\u0131 ve neden \u00fcrkt\u00fc\u011f\u00fcm\u00fc bilmiyordum bile; sadece \u00fcrkt\u00fc\u011f\u00fcm\u00fc, hayat\u0131mda hi\u00e7 \u00fcrkmedi\u011fim kadar \u00e7ok \u00fcrkt\u00fc\u011f\u00fcm\u00fc biliyordum. Oturdu\u011fum sandalyede b\u00fcz\u00fcld\u00fcm, sanki hi\u00e7bir \u015fey olmam\u0131\u015f, sanki ben de\u011filmi\u015fim gibi. Ama yine y\u00fckseldi sesler, gitgide yakla\u015ft\u0131. \u0130\u015fte tam kula\u011f\u0131m\u0131n dibinde, \"Devu\u015fkin! Devu\u015fkin! Devu\u015fkin nerede?\" dediler. G\u00f6zlerimi kald\u0131rd\u0131m: kar\u015f\u0131mda Evstafi \u0130vanovi\u00e7 duruyor; konu\u015fuyor: \"Makar Alekseyevi\u00e7, hemen Majesteleri'nin yan\u0131na, hemen! Evra\u011f\u0131 rezil etmi\u015fsiniz!\" Bir tek bunu s\u00f6yledi, ama yetti, do\u011fru de\u011fil mi, can\u0131m, s\u00f6ylenmesi bile yetmez mi? \u00d6ld\u00fcm, buz kestim, hissiz kald\u0131m, y\u00fcr\u00fcmeye ba\u015flam\u0131\u015ft\u0131m ama ne ya\u015f\u0131yordum ne de \u00f6l\u00fcyd\u00fcm o anda. Beni bir odadan ge\u00e7irdiler, sonra bir ba\u015fkas\u0131ndan, \u00fc\u00e7\u00fcnc\u00fc bir odadan daha ge\u00e7erek \u00e7al\u0131\u015fma odas\u0131na ula\u015ft\u0131m! O s\u0131rada akl\u0131mdan ne ge\u00e7ti\u011fi konusunda do\u011fru d\u00fcr\u00fcst bir rapor vermeme imk\u00e2n yok. Bakt\u0131m, Majesteleri orada duruyor, herkes \u00e7evresine toplanm\u0131\u015f. San\u0131r\u0131m, selam vermeyi de unuttum. \u015ea\u015fk\u0131na d\u00f6nm\u00fc\u015ft\u00fcm; dudaklar\u0131m, ayaklar\u0131m titriyordu. Neden derseniz: Birincisi, utan\u00e7; sa\u011fdaki aynaya bakt\u0131m, orada g\u00f6rd\u00fc\u011f\u00fcm \u015fey akl\u0131m\u0131 ka\u00e7\u0131rmaya yeterdi. \u0130kincisi, hep b\u00f6yle davran\u0131rd\u0131m, yery\u00fcz\u00fcnde sanki hi\u00e7 yokmu\u015fum gibi yapard\u0131m. Majesteleri de varl\u0131\u011f\u0131mdan haberdar de\u011fildi belki. Belki, duymu\u015ftu, ofisinde bir Devu\u015fkin oldu\u011funu belli belirsiz s\u00f6ylemi\u015flerdi; ama onunla k\u00fc\u00e7\u00fcc\u00fck bir ili\u015fkisi bile olmam\u0131\u015ft\u0131 hi\u00e7.\n\n\u00d6fkeyle s\u00f6ze ba\u015flad\u0131. \"Bunu nas\u0131l yapt\u0131n\u0131z, bay\u0131m! G\u00f6z\u00fcn\u00fcz g\u00f6rm\u00fcyor mu? \u00d6nemli evrak, acele haz\u0131rlanmas\u0131 gerekli, ama siz rezil etmi\u015fsiniz. Nas\u0131l yapabildiniz bunu,\" derken Majesteleri Evstafi \u0130vanovi\u00e7'e d\u00f6nd\u00fc. Ben sadece bana kadar ula\u015fan sesleri dinliyordum: \"Yeteneksizlik! Dikkatsizlik! Tats\u0131zl\u0131k \u00e7\u0131kar\u0131yorsunuz!\" Bir \u015fey i\u00e7in a\u011fz\u0131m\u0131 a\u00e7t\u0131m. Af dilemek istiyordum, ama yapamad\u0131m, ka\u00e7ay\u0131m dedim, denemeye bile cesaret edemedim ve o s\u0131rada... o s\u0131rada, olan oldu, \u015fimdi bile utan\u00e7tan kalemi elimde zar zor tutuyorum. Benim d\u00fc\u011fmem \u2013\u015eeytan als\u0131n onu\u2013 bir iplik ucunda duran d\u00fc\u011fmem... birdenbire koptu, sekti, s\u0131\u00e7rad\u0131 (anla\u015f\u0131lan kazara \u00e7arpm\u0131\u015ft\u0131m ona), \u00e7\u0131nlad\u0131, yuvarland\u0131 ve do\u011fruca, \u00f6yle do\u011fruca, lanetli d\u00fc\u011fme, Majesteleri'nin ayaklar\u0131n\u0131n dibine gitti ve b\u00fct\u00fcn bunlar tam bir sessizli\u011fin ortas\u0131nda oldu! \u0130\u015fte b\u00fct\u00fcn a\u00e7\u0131klamam, b\u00fct\u00fcn \u00f6z\u00fcr\u00fcm, b\u00fct\u00fcn yan\u0131t\u0131m, Majesteleri'ne s\u00f6ylemeye haz\u0131rland\u0131\u011f\u0131m her \u015fey buydu! Sonu\u00e7lar\u0131 korkun\u00e7 oldu! Majesteleri hemen benim \u015fahs\u0131ma ve k\u0131l\u0131\u011f\u0131ma y\u00f6neltti ilgisini. Aynada ne g\u00f6rd\u00fc\u011f\u00fcm\u00fc hat\u0131rlad\u0131m; d\u00fc\u011fmeyi tutmak \u00fczere \u00f6ne at\u0131ld\u0131m! Aptalla\u015fm\u0131\u015ft\u0131m! D\u00fc\u011fmeyi almak i\u00e7in uzand\u0131m ama kayd\u0131m, yuvarland\u0131m, yakalayamad\u0131m, k\u0131sacas\u0131, becerilerimden yoksundum. O s\u0131rada kalan son g\u00fcc\u00fcm\u00fcn de beni terk etti\u011fini hissettim, art\u0131k her \u015fey, her \u015fey bitmi\u015fti! B\u00fct\u00fcn itibar\u0131m\u0131 kaybetmi\u015ftim, b\u00fct\u00fcn insanl\u0131\u011f\u0131m yerle bir olmu\u015ftu! Bir yandan da iki kula\u011f\u0131mda hem Tereza hem Faldoni \u00e7\u0131nl\u0131yordu. Sonunda d\u00fc\u011fmeyi yakalad\u0131m, kalkt\u0131m, kenara \u00e7ekildim, iyi de, aptal, bari sakin sakin dur, elim b\u00f6\u011fr\u00fcnde olsun! Olmaz; d\u00fc\u011fmeyi kopan ipli\u011fe ba\u011flamaya ba\u015flad\u0131m, sanki orada duracakm\u0131\u015f gibi; bir yandan da g\u00fcl\u00fcms\u00fcyorum. Hatta s\u0131r\u0131t\u0131yorum. Majesteleri \u00f6nce ba\u015f\u0131n\u0131 \u00e7evirdi, sonra tekrar bakt\u0131 bana, sonra Evstafi \u0130vanovi\u00e7'le konu\u015ftu\u011funu duydum: \"Nas\u0131l olur?.. Baksan\u0131za, bu ne k\u0131l\u0131k!.. Nas\u0131l olur!.. Ne bu b\u00f6yle!..\" Ah bir tanem, bu da ne b\u00f6yle, nas\u0131l olur? Evet, ne bu b\u00f6yle? Yakaland\u0131n! Evstafi \u0130vanovi\u00e7'in konu\u015ftu\u011funu duyuyorum: \"Zarars\u0131zd\u0131r, kimseye zarar\u0131 olmaz, d\u00fczg\u00fcn davran\u0131r, maa\u015f\u0131 yeterli, derecesine g\u00f6re...\" \"Onu biraz rahatlatal\u0131m,\" dedi Majesteleri. \"Ona avans verelim...\" \"Ama ald\u0131,\" dediler, \"ald\u0131, bir s\u00fcre \u00f6nce alm\u0131\u015ft\u0131. Bir s\u0131k\u0131nt\u0131s\u0131 var herhalde, yoksa davran\u0131\u015flar\u0131 d\u00fczg\u00fcnd\u00fcr, zarars\u0131zd\u0131r, hi\u00e7 zarar vermez.\" Mele\u011fim, ben, yan\u0131yordum, cehennem ate\u015finde yan\u0131yordum! \u00d6ld\u00fcm! \"Peki,\" dedi Majesteleri y\u00fcksek sesle, \"hemen bir daha yaz\u0131ls\u0131n; Devu\u015fkin, gelin buraya, bu kez hatas\u0131z bir daha yaz\u0131n; ama bir dakika...\" Bu s\u0131rada Majesteleri di\u011ferlerine d\u00f6nd\u00fc, \u00e7e\u015fitli emirler verdi hepsi da\u011f\u0131l\u0131p gitti. Onlar gidince Majesteleri tela\u015fla c\u00fczdan\u0131ndan y\u00fcz ruble \u00e7\u0131kard\u0131. \"Al\u0131n,\" dedi, \"elimden gelen, nas\u0131l isterseniz \u00f6yle harcay\u0131n...\" ve elime s\u0131k\u0131\u015ft\u0131rd\u0131. Mele\u011fim, sars\u0131ld\u0131m, b\u00fct\u00fcn ruhum sars\u0131ld\u0131; bana ne oldu bilmiyorum; kendisinin elini tutmak istedim. Ama o da k\u0131pk\u0131rm\u0131z\u0131 olmu\u015ftu, g\u00fcvercinim, evet, do\u011fruluktan \u015fa\u015fm\u0131\u015f bir tek sa\u00e7 telim bile yoktur, bir tanem; benim de\u011fersiz elimi tuttu, onu s\u0131kt\u0131, b\u00f6yle eliyle s\u0131kt\u0131, sanki akran\u0131ym\u0131\u015f\u0131m gibi, sanki onun gibi bir generalmi\u015fim gibi. \"Gidin,\" dedi, \"elimden gelen bu. Hata yapmay\u0131n, bu kez ucuz atlatt\u0131k.\"\n\nCan\u0131m, art\u0131k \u015f\u00f6yle karar verdim: Sizden ve Fedora'dan rica edece\u011fim \u2013\u00e7ocuklar\u0131m olsayd\u0131, onlara emrederdim\u2013 Tanr\u0131'ya, \u00f6z baban\u0131z i\u00e7in de\u011fil, her g\u00fcn ve sonsuza dek Majesteleri i\u00e7in dua edin. Yine s\u00f6yl\u00fcyorum, can\u0131m ve abart\u0131l\u0131 bir tav\u0131rla s\u00f6yl\u00fcyorum \u2013dinleyin beni, can\u0131m, iyi dinleyin\u2013 yemin ederim, talihsizli\u011fimizin ac\u0131mas\u0131z g\u00fcnlerinde size, yoksullu\u011funuza ve kendime, kendi ezikli\u011fime ve yeteneksizli\u011fime bakarken, ruhsal utan\u00e7tan \u00f6l\u00fcp \u00f6l\u00fcp dirilirdim, buna ra\u011fmen, size yemin ederim, benim i\u00e7in de\u011ferli olan \u015fey y\u00fcz ruble de\u011fil, Majesteleri'nin bizzat kendisinin bir saman tanesinin, bir sarho\u015fun, benim gibi de\u011fersiz birinin elini s\u0131kmaya tenezz\u00fcl etmesi oldu! Bunu yaparak beni kendime getirdi. Bu davran\u0131\u015flar\u0131yla ruhumu dirilttiler, hayat\u0131m\u0131 sonsuza dek daha tatl\u0131 k\u0131ld\u0131lar ve kesin olarak inan\u0131yorum ki Y\u00fccelerin Y\u00fccesi kar\u015f\u0131s\u0131nda ne kadar g\u00fcnahk\u00e2r olsam da, Majesteleri'nin mutlulu\u011fu ve refah\u0131 i\u00e7in etti\u011fim dualar onun taht\u0131na kadar ula\u015f\u0131yordur!..\n\nCan\u0131m! \u015eimdi deh\u015fetli bir ruh kar\u0131\u015f\u0131kl\u0131\u011f\u0131, deh\u015fetli bir heyecan i\u00e7indeyim! Kalbim \u00e7arp\u0131yor, g\u00f6\u011fs\u00fcmden f\u0131rlamak istiyor. B\u00fct\u00fcn g\u00fcc\u00fcm eridi gitti sanki. Size o paradan k\u0131rk be\u015f ruble g\u00f6nderiyorum, yirmi rubleyi ev sahibesine verece\u011fim, geriye otuz be\u015f kal\u0131yor: yirmisi elbiseyi tamire gidiyor, on be\u015fini de ge\u00e7inmeye b\u0131rak\u0131yorum. B\u00fct\u00fcn bu sabahki izlenimler benim varl\u0131\u011f\u0131m\u0131 derinden sarst\u0131. Uzanaca\u011f\u0131m. Fakat iyiyim, \u00e7ok iyiyim. Sadece ruhum s\u0131zl\u0131yor ve orada, derinlerde, ruhumun titredi\u011fi, \u00fcrperdi\u011fi, k\u0131p\u0131rdand\u0131\u011f\u0131 duyuluyor. Size gelece\u011fim; ama \u015fimdi b\u00fct\u00fcn bu duygulardan sarho\u015f gibiyim... Tanr\u0131 her \u015feyi g\u00f6r\u00fcyor, can\u0131m benim, paha bi\u00e7ilmez g\u00fcvercinim!\n\nSayg\u0131n dostunuz,  \nMakar Devu\u015fkin\n\n10 Eyl\u00fcl\n\nSevgili Makar Alekseyevi\u00e7,\n\nSizin mutlulu\u011funuza tarifsiz sevindim ve sizin amirinizin iyiliklerini takdir ediyorum, dostum. Art\u0131k ac\u0131lar\u0131n\u0131zdan kurtulacaks\u0131n\u0131z! Ama tabii, Tanr\u0131 a\u015fk\u0131na, bo\u015f yere para harcamay\u0131n. Tutumlu ya\u015fay\u0131n, olabildi\u011fince m\u00fctevaz\u0131 ya\u015fay\u0131n ve bug\u00fcnden itibaren hep biraz olsun bir \u015feyleri bir kenara koyun ki talihsizlik sizi yine birdenbire yakalamas\u0131n. Bizim i\u00e7in, Tanr\u0131 a\u015fk\u0131na, kayg\u0131lanmay\u0131n. Fedora'yla ben bir \u015fekilde ya\u015far\u0131z. Ne i\u00e7in bize para g\u00f6nderdiniz, Makar Alekseyevi\u00e7? Bize hi\u00e7 de gerekli de\u011fil. Elimizdekiyle yetiniyoruz. Do\u011fru, bu daireden ta\u015f\u0131nmak i\u00e7in acilen paraya ihtiyac\u0131m\u0131z var, ama Fedora eski bir alaca\u011f\u0131n\u0131 almay\u0131 umut ediyor. Fakat, kendime acil ihtiya\u00e7lar i\u00e7in yirmi ruble ay\u0131r\u0131yorum. Geri kalan\u0131 da size g\u00f6nderiyorum. Paray\u0131 al\u0131n l\u00fctfen Makar Alekseyevi\u00e7. Ho\u015f\u00e7a kal\u0131n. Art\u0131k huzurla ya\u015fay\u0131n, sa\u011fl\u0131kl\u0131 ve ne\u015feli olun. Size daha \u00e7ok yazard\u0131m, ama korkun\u00e7 bir yorgunluk hissediyorum, d\u00fcn b\u00fct\u00fcn g\u00fcn yataktan kalkamad\u0131m. Gelmeye s\u00f6z vererek iyi yapt\u0131n\u0131z. Beni ziyaret edin l\u00fctfen Makar Alekseyevi\u00e7.\n\nV.D.\n\n11 Eyl\u00fcl\n\nSevgili Varvara Alekseyevna,\n\nYalvar\u0131r\u0131m size, bir tanem, benden uzakla\u015fmay\u0131n \u015fimdi, \u015fimdi, tam da kesinlikle mutlu ve her \u015feyden ho\u015fnut oldu\u011fum s\u0131rada. G\u00fcvercinim! Fedora'y\u0131 dinlemeyin, ben de sizin emretti\u011finiz her \u015feyi yapaca\u011f\u0131m; d\u00fczg\u00fcn davranaca\u011f\u0131m, s\u0131rf Majesteleri'ne olan sayg\u0131mdan da kendime d\u00fczg\u00fcn ve sayg\u0131n bakaca\u011f\u0131m; biz yine birbirimize mutlu mektuplar yazaca\u011f\u0131z, birbirimize d\u00fc\u015f\u00fcncelerimizi a\u00e7aca\u011f\u0131z, mutluluklar\u0131m\u0131z\u0131, kayg\u0131lar\u0131m\u0131z\u0131 yazaca\u011f\u0131z, e\u011fer kayg\u0131m\u0131z olursa; birlikte ya\u015fayaca\u011f\u0131z; uyumlu ve mutlu. Edebiyatla ilgilenece\u011fiz... K\u00fc\u00e7\u00fck mele\u011fim! Benim kaderimde her \u015fey de\u011fi\u015fti ve hep iyiye do\u011fru de\u011fi\u015fti. Ev sahibesi daha s\u0131cak davran\u0131yor, Tereza daha ak\u0131ll\u0131, hatta Faldoni de biraz konu\u015fkan oldu. Ratazyayev'le bar\u0131\u015ft\u0131k. Mutluluktan, ben bizzat gittim ona. Do\u011frusu, biraz iyi oldu, can\u0131m, onun hakk\u0131nda s\u00f6ylenen k\u00f6t\u00fc \u015feyler, hepsi sa\u00e7mal\u0131kt\u0131. \u015eimdi anlad\u0131m bunlar\u0131n hep i\u011fren\u00e7 iftiralar oldu\u011funu. Bizi yazmay\u0131 hi\u00e7 d\u00fc\u015f\u00fcnmemi\u015f: bana kendisi s\u00f6yledi. Yeni eserini okudu bana. Bana Lovelace demesine gelince; bu da bir hakaret ya da edepsiz bir \u015fey de\u011filmi\u015f, bunu da a\u00e7\u0131klad\u0131 bana. Bu s\u00f6z birebir olarak yabanc\u0131 dilden aktar\u0131lm\u0131\u015f ve \"\u00e7apk\u0131n \u00e7ocuk\" anlam\u0131na geliyor ve daha g\u00fczel, daha edeb\u00ee s\u00f6ylemek gerekirse, \"yere bakan, y\u00fcrek yakan\". \u0130\u015fte, hepsi bu, ba\u015fka bir \u015fey de\u011fil. Masum bir \u015fakaym\u0131\u015f, k\u00fc\u00e7\u00fck mele\u011fim. Cahillik ettim, aptall\u0131k edip g\u00fccendim. Ama art\u0131k onun \u00f6n\u00fcnde ben \u00f6z\u00fcr dileyece\u011fim... Hava da bug\u00fcn harika, Varenka, \u00f6yle g\u00fczel ki. Do\u011frusu, sabahleyin biraz de\u011fi\u015fti, so\u011fuk oldu. \u00c7izme almaya gittim ve harika \u00e7izmeler ald\u0131m. Neva Bulvar\u0131'nda gezdim. \"Ar\u0131c\u0131k\"27 okudum. Ah! En \u00f6nemli \u015feyi de anlatmay\u0131 unutacakt\u0131m size.\n\nBak\u0131n ne olmu\u015f:\n\nBu sabah Emelyan \u0130vanovi\u00e7 ve Aksent Mihaylovi\u00e7'le Majesteleri hakk\u0131nda konu\u015ftum. Evet, Varenka, kendileri bir tek bana \u015fefkatli davranmam\u0131\u015f. Sadece bana iyilik etmemi\u015f ve kalplerinin iyili\u011fini b\u00fct\u00fcn d\u00fcnyaya ilan etmi\u015fler. Bir\u00e7ok mevkiden onun \u015ferefine \u00f6vg\u00fcler y\u00fckseliyor, minnettarl\u0131k g\u00f6zya\u015flar\u0131 d\u00f6k\u00fcl\u00fcyor. Zat\u0131\u00e2lileri yetim bir k\u0131z yeti\u015ftirmi\u015fler. Onun bir yere yerle\u015ftirilmesini emretmi\u015fler; \u00fcnl\u00fc biriyle, \u00f6zel bir g\u00f6revle Majesteleri'ne ba\u011fl\u0131 olarak \u00e7al\u0131\u015fan bir memurla evlendirmi\u015fler. Bir dul kad\u0131n\u0131n o\u011flunu da bir \u015fubeye yerle\u015ftirmi\u015fler ve daha bir\u00e7ok hay\u0131r ger\u00e7ekle\u015ftirmi\u015fler. Can\u0131m, ben de buna katk\u0131 yapmay\u0131 bir sorumluluk sayd\u0131m, orada bulunan herkese Majesteleri'nin davran\u0131\u015f\u0131n\u0131 anlatt\u0131m; her \u015feyi anlatt\u0131m ve hi\u00e7bir \u015fey saklamad\u0131m. Utanc\u0131 cebime saklad\u0131m. Ne utanc\u0131 olacak burada, b\u00f6yle bir durumda gurur mu d\u00fc\u015f\u00fcn\u00fcl\u00fcr! B\u00f6yle b\u00f6yle diye anlatt\u0131m ba\u011f\u0131ra \u00e7a\u011f\u0131ra, Majesteleri'nin \u015fan\u0131na \u015fan kat\u0131ls\u0131n! S\u00fcr\u00fckleyici bir \u015fekilde konu\u015ftum, ate\u015fli bir \u015fekilde konu\u015ftum ve k\u0131zarmad\u0131m, tersine b\u00f6yle anlatt\u0131\u011f\u0131m i\u00e7in gurur duydum. Her \u015feyi anlatt\u0131m (sizin hakk\u0131n\u0131zda sa\u011fduyulu davran\u0131p sustum, can\u0131m), ev sahibemi, Faldona'y\u0131, Ratazyayev'i, \u00e7izmeleri, Markov'u... hepsini anlatt\u0131m. Birileri k\u0131s k\u0131s g\u00fcld\u00fc, evet, do\u011fru, hepsi k\u0131s k\u0131s g\u00fcld\u00fc. Fakat bu benim g\u00f6r\u00fcn\u00fc\u015f\u00fcm y\u00fcz\u00fcndendi, herhalde, g\u00fcl\u00fcn\u00e7 bir \u015fey buldular g\u00f6r\u00fcn\u00fc\u015f\u00fcmde ya da \u00e7izmelerimde \u2013 tabii \u00e7izmelerimde. Ama \u00e7irkin bir niyetle yapm\u0131\u015f olamazlard\u0131 bunu. Gen\u00e7 olduklar\u0131, belki de zengin olduklar\u0131 i\u00e7in yapt\u0131lar; ama k\u00f6t\u00fc, \u00e7irkin bir niyetle alay etmediler konu\u015fmamla. Yani Majesteleri'ni d\u00fc\u015f\u00fcn\u00fcrsek, asla bunu yapm\u0131\u015f olamazlar. Do\u011fru de\u011fil mi, Varenka?\n\n\u015eu vakte dek h\u00e2l\u00e2 kendime gelemedim, can\u0131m. B\u00fct\u00fcn bu olaylar beni serseme \u00e7evirdi! Sizin odununuz var m\u0131? \u00dc\u015f\u00fcmeyin, Varenka; uzun zamand\u0131r hastas\u0131n\u0131z. Ah, can\u0131m, kederli d\u00fc\u015f\u00fcncelerinizle beni mahvediyorsunuz. Tanr\u0131'ya dua ediyorum, sizin i\u00e7in ona ne \u00e7ok dua ediyorum, can\u0131m! \u00d6rne\u011fin, y\u00fcn \u00e7oraplar\u0131n\u0131z var m\u0131, ya da daha s\u0131cak tutacak elbiseleriniz var m\u0131? Bak\u0131n, g\u00fcvercinim. E\u011fer bir \u015feye ihtiyac\u0131n\u0131z varsa, hemen s\u00f6yleyin, Tanr\u0131 a\u015fk\u0131na, g\u00fccendirmeyin bu ihtiyar\u0131. A\u00e7\u0131k ve rahat olun bana kar\u015f\u0131. Art\u0131k k\u00f6t\u00fc zamanlar ge\u00e7ti. Benim ad\u0131ma kayg\u0131lanmay\u0131n. \u0130leride her \u015fey \u00e7ok ayd\u0131nl\u0131k olacak, g\u00fczel olacak!\n\nKederli zamanlar oldu, Varenka! Ama art\u0131k fark etmez, ge\u00e7ti art\u0131k! Y\u0131llar ge\u00e7iyor, bu zaman\u0131 i\u00e7 \u00e7ekip hat\u0131rlayaca\u011f\u0131z. Kendi gen\u00e7lik y\u0131llar\u0131m\u0131 hat\u0131rl\u0131yorum. Heyhat! Bazen bir kopek olmazd\u0131. So\u011fuk olurdu, a\u00e7l\u0131k olurdu, ama ne\u015feli olurdum, yeterdi. Sabahleyin Neva Bulvar\u0131'nda gezerdik, g\u00fczel k\u00fc\u00e7\u00fck bir y\u00fczle kar\u015f\u0131la\u015f\u0131r, b\u00fct\u00fcn g\u00fcn mutlu olurdum. Enfes, enfes zamanlard\u0131, can\u0131m! Yery\u00fcz\u00fcnde ya\u015famak g\u00fczeldir, Varenka! \u00d6zellikle de Petersburg'da. D\u00fcn g\u00f6zya\u015flar\u0131 i\u00e7inde itiraf ettim y\u00fcce Tanr\u0131'ya, bu kederli zamanda i\u015fledi\u011fim g\u00fcnahlar\u0131m\u0131 affetmesi i\u00e7in yalvard\u0131m; ho\u015fnutsuzluk, liberal fikirler, iti\u015fip kak\u0131\u015fmalar ve hakaretler. Dua ederken sizi and\u0131m co\u015fkulanarak. Bir tek siz, mele\u011fim, korudunuz beni, bir tek siz teselli ettiniz, hay\u0131rl\u0131 \u00f6\u011f\u00fctlerle ve emirlerle cesaretlendirdiniz beni. Can\u0131m, bunu asla unutamam. Sizin mektuplar\u0131n\u0131z\u0131n hepsini \u00f6pt\u00fcm bug\u00fcn, g\u00fcvercinim! Neyse, ho\u015f\u00e7a kal\u0131n, can\u0131m. Buraya yak\u0131n bir yerde giyecek sat\u0131ld\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyorlar. Gidip bakaca\u011f\u0131m. Ho\u015f\u00e7a kal\u0131n, mele\u011fim. Ho\u015f\u00e7a kal\u0131n.\n\nRuhunu size adam\u0131\u015f olan,  \nMakar Devu\u015fkin\n\n15 Eyl\u00fcl\n\nSevgili Beyefendi Makar Alekseyevi\u00e7,\n\nKorkun\u00e7 bir heyecan i\u00e7indeyim. Dinler misiniz, ba\u015f\u0131m\u0131za neler geldi. Biraz k\u00f6t\u00fc bir \u015feyler seziyorum. Ama siz karar verin, de\u011ferli dostum: Bay B\u0131kov, Petersburg'da. Fedora kar\u015f\u0131la\u015fm\u0131\u015f onunla. Arabayla gidiyormu\u015f, arabaya durmas\u0131n\u0131 emretmi\u015f, Fedora'n\u0131n yan\u0131na gelmi\u015f ve nerede oturdu\u011funu sorgulamaya ba\u015flam\u0131\u015f. \u00d6nce s\u00f6ylememi\u015f o. Sonra B\u0131kov g\u00fclerek kimin yan\u0131nda ya\u015fad\u0131\u011f\u0131n\u0131 zaten bildi\u011fini s\u00f6ylemi\u015f. (Anla\u015f\u0131lan, Anna Fedorovna ona her \u015feyi anlatm\u0131\u015f.) O zaman Fedora kendini tutamam\u0131\u015f ve caddenin ortas\u0131nda onu azarlamaya, ona sitem etmeye ba\u015flam\u0131\u015f, onun ahlaks\u0131z bir adam oldu\u011funu, benim ya\u015fad\u0131\u011f\u0131m talihsizliklerin sebebinin o oldu\u011funu s\u00f6ylemi\u015f. B\u0131kov da para olmay\u0131nca, do\u011fal olarak, insan mutsuz olur diye yan\u0131t vermi\u015f. Fedora ona benim \u00e7al\u0131\u015farak ya\u015famaya cesaret edebilece\u011fimi, evlenebilece\u011fimi, ama olmazsa bir evde i\u015f bulabilece\u011fimi, ama art\u0131k mutlulu\u011fu sonsuza dek kaybetti\u011fimi, hasta oldu\u011fumu ve yak\u0131nda \u00f6lece\u011fimi s\u00f6ylemi\u015f. O da buna kar\u015f\u0131l\u0131k benim h\u00e2l\u00e2 \u00e7ok gen\u00e7 oldu\u011fumu, akl\u0131m\u0131n h\u00e2l\u00e2 bulutlarda oldu\u011funu ve bizim erdemimizin soldu\u011funu (onun ifadesi bu) s\u00f6ylemi\u015f. Fedora'yla ben onun bizim evi bilmedi\u011fini d\u00fc\u015f\u00fcn\u00fcyorduk, ama d\u00fcn, ben bir \u015feyler almak i\u00e7in Gostiniy Dvor'a gitti\u011fim s\u0131rada, odam\u0131za gelmi\u015f; anla\u015f\u0131lan, beni evde bulmak istememi\u015f. Uzun s\u00fcre Fedora'ya nas\u0131l ge\u00e7indi\u011fimizi sormu\u015f; evdeki her \u015feyi g\u00f6zden ge\u00e7irmi\u015f; benim yapt\u0131\u011f\u0131m i\u015flere bakm\u0131\u015f, sonunda da sormu\u015f: \"Sizinle arkada\u015f olan \u015fu memur nas\u0131l biridir?\" O s\u0131rada siz avludan ge\u00e7iyormu\u015fsunuz; Fedora ona sizi g\u00f6stermi\u015f; o da bak\u0131p g\u00fclm\u00fc\u015f; Fedora ondan gitmesini rica etmi\u015f, ona benim art\u0131k \u00fcz\u00fcnt\u00fcden sa\u011fl\u0131\u011f\u0131m\u0131 yitirdi\u011fimi ve onu evde g\u00f6rmenin bana iyi gelmeyece\u011fini s\u00f6ylemi\u015f. Ses \u00e7\u0131karmam\u0131\u015f B\u0131kov; \u00f6ylesine geldi\u011fini, bir \u015fey yapmaya gelmedi\u011fini s\u00f6ylemi\u015f ve Fedora'ya yirmi be\u015f ruble vermek istemi\u015f; o da, do\u011fal olarak, almam\u0131\u015f. Ne demek b\u00fct\u00fcn bunlar? Neden geldi bize? Anlayam\u0131yorum, bizi nereden \u00f6\u011frenmi\u015f olabilir, anlayam\u0131yorum! Tahminlerimden sonu\u00e7 \u00e7\u0131karamad\u0131m. Fedora bize gelip giden, \u00e7ama\u015f\u0131rc\u0131 Nastasya'n\u0131n ahbab\u0131 olan g\u00f6r\u00fcmcesi Aksinya olabilir diyor, Nastasya'n\u0131n amca taraf\u0131ndan ye\u011feni de Anna Fedorovna'n\u0131n ye\u011feninin bir arkada\u015f\u0131n\u0131n \u00e7al\u0131\u015ft\u0131\u011f\u0131 dairede bek\u00e7ilik yap\u0131yormu\u015f. B\u00f6ylece yay\u0131lm\u0131\u015f olmas\u0131n bir dedikodu? Fakat, Fedora yan\u0131l\u0131yor da olabilir; ne d\u00fc\u015f\u00fcnece\u011fimizi bilemiyoruz. Acaba bize gelir mi yeniden? D\u00fc\u015f\u00fcncesi bile korkutuyor beni! Fedora d\u00fcn b\u00fct\u00fcn bunlar\u0131 anlatt\u0131\u011f\u0131 zaman, o kadar korktum ki korkudan d\u00fc\u015f\u00fcp kalacakt\u0131m sanki. H\u00e2l\u00e2 ne istiyorlar benim gibi bir yoksuldan? Ah! Korku i\u00e7indeyim \u015fimdi; her an B\u0131kov i\u00e7eri girecekmi\u015f gibi geliyor. Ne yapaca\u011f\u0131m! Kader daha neler haz\u0131rl\u0131yor bana? \u0130sa a\u015fk\u0131na, hemen bana gelin Makar Alekseyevi\u00e7. Gelin, Tanr\u0131 a\u015fk\u0131na, gelin.\n\nV.D.\n\n18 Eyl\u00fcl\n\nCan\u0131m, Varvara Alekseyevna,\n\nBug\u00fcn evimizde katlan\u0131lmas\u0131 imk\u00e2ns\u0131z derecede ac\u0131, hi\u00e7bir \u015feyle a\u00e7\u0131klanamaz ve beklenmedik bir olay ya\u015fand\u0131. Bizim yoksul Gor\u015fkov (size belirtmem gerekiyor, can\u0131m) kesin olarak akland\u0131. Karar daha \u00f6nce \u00e7\u0131km\u0131\u015f, ama bug\u00fcn Gor\u015fkov kesin karar\u0131 dinlemeye gitmi\u015fti. Dava onun i\u00e7in \u00e7ok mutlu bir \u015fekilde sona erdi. Dikkatsizlik ve ihmal y\u00fcz\u00fcnden ona isnat edilen su\u00e7lar\u0131n hepsi i\u00e7in kesin beraat karar\u0131 verildi. T\u00fcccar\u0131n ona ciddi miktarda zarar \u00f6demesi karara ba\u011fland\u0131, b\u00f6ylece durumunu ciddi bir \u015fekilde yoluna koyacak, onuru \u00fczerindeki lekeden de kurtulacak ve her \u015fey yoluna girecek... k\u0131sacas\u0131, her istedi\u011fi eksiksiz olarak ger\u00e7ekle\u015fti. Eve saat \u00fc\u00e7te geldi. Y\u00fcz\u00fc karman \u00e7ormand\u0131, kire\u00e7 gibi bembeyazd\u0131, dudaklar\u0131 titriyordu ama g\u00fcl\u00fcyordu; kar\u0131s\u0131n\u0131, \u00e7ocuklar\u0131n\u0131 kucaklad\u0131. Hepimiz onun odas\u0131nda toplan\u0131p onu tebrik ettik. Bizim davran\u0131\u015f\u0131m\u0131zdan \u00e7ok heyecanland\u0131, herkese e\u011filerek te\u015fekk\u00fcr etti, her birimizin elini birka\u00e7 kez s\u0131kt\u0131. Hatta bana o b\u00fcy\u00fcm\u00fc\u015f, dikle\u015fmi\u015f ve art\u0131k g\u00f6zlerinde g\u00f6zya\u015flar\u0131 kalmam\u0131\u015f gibi geldi. Heyecan i\u00e7indeydi, zavall\u0131. \u0130ki dakika yerinde duram\u0131yordu; her \u015feyi tutup b\u0131rak\u0131yor, sonra yine al\u0131yor, Tanr\u0131 bilir neler s\u00f6yl\u00fcyordu \u2013 \"Onurum, onur, ismimin itibar\u0131, \u00e7ocuklar\u0131m,\" diyordu \u2013 tekrar tekrar bunlar\u0131 s\u00f6yl\u00fcyordu! Hatta a\u011flad\u0131. \u00c7o\u011fumuz a\u011flamakl\u0131 olduk. Ratazyayev elbette onu cesaretlendirmek i\u00e7in \u015f\u00f6yle dedi: \"Ama canca\u011f\u0131z\u0131m, elde bir \u015fey olmay\u0131nca onur neye yarar; para, canca\u011f\u0131z\u0131m, en \u00f6nemlisi para; i\u015fte bunun i\u00e7in \u015f\u00fck\u00fcr edin Tanr\u0131'ya!\" B\u00f6yle derken de omzunu s\u0131vazlad\u0131. Gor\u015fkov g\u00fccenmi\u015f gibi geldi bana, yani a\u00e7\u0131k\u00e7a ho\u015fnutsuzlu\u011funu belli etmedi, ama Ratazyayev'e tuhaf bir \u015fekilde bak\u0131p elini omzundan indirdi. Daha \u00f6nce b\u00f6yle bir \u015fey olmazd\u0131, can\u0131m! Fakat, t\u00fcrl\u00fc t\u00fcrl\u00fc karakter var. S\u00f6zgelimi, i\u015fte ben, b\u00f6yle mutluluklar kar\u015f\u0131s\u0131nda kibirli davranmazd\u0131m; sonu\u00e7ta, bir tanem, a\u015f\u0131r\u0131 tevazu ve ezilip b\u00fcz\u00fclme sadece ruhsal iyilik n\u00f6betine yakalanmaktan ve kalbin a\u015f\u0131r\u0131 yumu\u015fak olmas\u0131ndan da kaynaklanabilir... ama, fakat, ben ne kar\u0131\u015f\u0131r\u0131m! \"Evet, dedi, para da iyidir; Tanr\u0131'ya \u015f\u00fck\u00fcr, Tanr\u0131'ya \u015f\u00fck\u00fcr!\" Onun odas\u0131nda bulundu\u011fumuz s\u00fcrece hep, \"Tanr\u0131'ya \u015f\u00fck\u00fcr, Tanr\u0131'ya \u015f\u00fck\u00fcr!..\" deyip durdu. Kar\u0131s\u0131 lezzetli ve bereketli bir yemek sipari\u015f etti. Ev sahibemiz bizzat onlar i\u00e7in yapt\u0131 yeme\u011fi. Yemek vaktine kadar oturamad\u0131 yerinde Gor\u015fkov. Ona seslensinler, seslenmesinler, b\u00fct\u00fcn odalar\u0131 dola\u015ft\u0131. Kendi odas\u0131na giriyor, g\u00fcl\u00fcms\u00fcyor, sandalyeye oturuyor, bir \u015feyler s\u00f6yl\u00fcyor, bazen de hi\u00e7 konu\u015fmuyor, sonra \u00e7\u0131k\u0131yordu. Ba\u015f\u00e7avu\u015fun odas\u0131nda oyun kartlar\u0131n\u0131 ald\u0131 eline; onu oturtup d\u00f6rd\u00fcnc\u00fc olarak oynatt\u0131lar. Oynad\u0131, oynad\u0131, oyunda birtak\u0131m sa\u00e7mal\u0131klar yapt\u0131, \u00fc\u00e7-d\u00f6rt el oynad\u0131 ve oynamay\u0131 b\u0131rakt\u0131. \"Hay\u0131r,\" dedi, \"yani ben,\" dedi, \"bu b\u00f6yle olmaz,\" diyerek oradan \u00e7\u0131kt\u0131. Koridorda benimle kar\u015f\u0131la\u015ft\u0131, iki elimi tuttu, dosdo\u011fru g\u00f6zlerimin i\u00e7ine bakt\u0131, tuhaf bir \u015fey g\u00f6r\u00fcyormu\u015f gibiydi; elimi s\u0131k\u0131p gitti, hep g\u00fcl\u00fcms\u00fcyordu, ama a\u011f\u0131r, tuhaf bir \u015fekilde, sanki \u00f6lm\u00fc\u015f gibi g\u00fcl\u00fcms\u00fcyordu. Kar\u0131s\u0131 mutluluktan a\u011fl\u0131yordu; hepsi bayram gibi ne\u015feliydiler. H\u0131zl\u0131 h\u0131zl\u0131 yemi\u015fler yeme\u011fi. Yemekten sonra da kar\u0131s\u0131na, \"Ruhum, izninle, ben biraz uzanaca\u011f\u0131m,\" demi\u015f ve gidip yata\u011fa yatm\u0131\u015f. K\u0131z\u0131n\u0131 yan\u0131na \u00e7a\u011f\u0131rm\u0131\u015f, elini ba\u015f\u0131na koymu\u015f ve uzun uzun \u00e7ocu\u011fun ba\u015f\u0131n\u0131 ok\u015fam\u0131\u015f. Sonra yine kar\u0131s\u0131na seslenmi\u015f: \"Petyonka nerede? Bizim Petyonka nerede?..\" Kar\u0131s\u0131 ha\u00e7 \u00e7\u0131kar\u0131p \u00e7ocu\u011fun \u00f6lm\u00fc\u015f oldu\u011funu s\u00f6ylemi\u015f. \"Evet, evet, biliyorum, hepsini biliyorum, Petyonka art\u0131k Cennet'te.\" Kar\u0131s\u0131 onun kendinde olmad\u0131\u011f\u0131n\u0131, olay\u0131n onu sarst\u0131\u011f\u0131n\u0131 anlayarak ona \u015f\u00f6yle demi\u015f: \"Ruhum, biraz uyusan.\", \"Evet, iyi olur, \u015fimdi... ben biraz,\" derken s\u0131rt\u0131n\u0131 d\u00f6n\u00fcp yatm\u0131\u015f, bir s\u00fcre sonra yan d\u00f6nm\u00fc\u015f, bir \u015fey s\u00f6ylemek istemi\u015f. Kar\u0131s\u0131 anlayamay\u0131p sormu\u015f: \"Ne dedin, hayat\u0131m?\" Ama Gor\u015fkov yan\u0131t vermemi\u015f. Kad\u0131n biraz daha beklemi\u015f, uyuyakald\u0131\u011f\u0131n\u0131 d\u00fc\u015f\u00fcn\u00fcp bir saatli\u011fine ev sahibesinin yan\u0131na gitmi\u015f. Bir saat sonra d\u00f6nm\u00fc\u015f, bakm\u0131\u015f, kocas\u0131 h\u00e2l\u00e2 uyanmam\u0131\u015f, hi\u00e7 ses \u00e7\u0131karmadan yat\u0131yor. Uyudu\u011funu d\u00fc\u015f\u00fcn\u00fcp bir k\u00f6\u015feye oturmu\u015f ve i\u015f yapmaya ba\u015flam\u0131\u015f. Yar\u0131m saat \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131 ve birtak\u0131m d\u00fc\u015f\u00fcncelere dald\u0131\u011f\u0131n\u0131, ne d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcn\u00fc bile hat\u0131rlamad\u0131\u011f\u0131n\u0131, ama kocas\u0131n\u0131 unutup gitti\u011fini s\u00f6yl\u00fcyor. Sonra birden kayg\u0131l\u0131 bir hisle kendine gelmi\u015f ve ilk olarak odadaki mezar sessizli\u011fi dikkatini \u00e7ekmi\u015f. Yata\u011fa bak\u0131nca kocas\u0131n\u0131n hep ayn\u0131 halde yatt\u0131\u011f\u0131n\u0131 g\u00f6rm\u00fc\u015f. Yan\u0131na gitmi\u015f, battaniyeyi \u00e7ekip bakm\u0131\u015f \u2013adam art\u0131k buz gibiymi\u015f\u2013 \u00f6lm\u00fc\u015f can\u0131m; Gor\u015fkov \u00f6lm\u00fc\u015f, aniden \u00f6lm\u00fc\u015f, sanki \u015fim\u015fek \u00e7arpm\u0131\u015f gibi! Peki neden \u00f6ld\u00fc, Tanr\u0131 bilir. Bu beni \u00e7ok etkiledi Varenka, bu vakte dek kendime gelemedim. Bir insan\u0131n b\u00f6yle sessiz sedas\u0131z \u00f6lebilece\u011fine inanmak zor. Ne zavall\u0131, ne talihsiz bir adamm\u0131\u015f bu Gor\u015fkov! Ah, kader, b\u00f6yle kader! Kar\u0131s\u0131 \u00fcrkm\u00fc\u015f, g\u00f6zya\u015flar\u0131 i\u00e7indeydi. K\u00fc\u00e7\u00fck k\u0131z bir k\u00f6\u015feye saklanm\u0131\u015f. Orada tam bir ko\u015fu\u015fturma ya\u015fan\u0131yor; t\u0131bbi tahkikat yap\u0131lacak... Herhalde size anlatmam g\u00fc\u00e7. Ama yaz\u0131k, ah \u00e7ok yaz\u0131k! \u00d6l\u00fcm\u00fcn ne g\u00fcn ne de saat bildi\u011fini d\u00fc\u015f\u00fcnmek \u00e7ok h\u00fcz\u00fcnl\u00fc... Hi\u00e7 yoktan \u00f6l\u00fcveriyor insan...\n\nSizin,  \nMakar Devu\u015fkin\n\n19 Eyl\u00fcl\n\nSevgili Bayan Varvara Alekseyevna,\n\nDostum, Ratazyayev'in bana bir yazardan i\u015f ald\u0131\u011f\u0131n\u0131 bildirmek istiyorum hemen. Biri gelmi\u015f ona, kal\u0131n bir elyazmas\u0131n\u0131 getirmi\u015f, Tanr\u0131'ya \u015f\u00fck\u00fcr, bir s\u00fcr\u00fc i\u015f demek bu. Fakat \u00f6yle \u00f6zensiz yaz\u0131lm\u0131\u015f ki, bu i\u015fin alt\u0131ndan nas\u0131l kalkaca\u011f\u0131m\u0131 bilemiyorum; hemen de istiyorlar. Neredeyse hi\u00e7 anlamad\u0131\u011f\u0131m bir \u015feyler yazm\u0131\u015flar... Yapra\u011f\u0131 k\u0131rk kope\u011fe anla\u015ft\u0131k. B\u00fct\u00fcn bunlar\u0131 size art\u0131k fazladan param olaca\u011f\u0131n\u0131 s\u00f6ylemek i\u00e7in yaz\u0131yorum, bir tanem. Neyse, \u015fimdi ho\u015f\u00e7a kal\u0131n, can\u0131m. Hemen i\u015fe koyulmam laz\u0131m.\n\nSad\u0131k dostunuz,  \nMakar Devu\u015fkin\n\n23 Eyl\u00fcl\n\nSevgili Dostum, Makar Alekseyevi\u00e7,\n\n\u00dc\u00e7 g\u00fcnd\u00fcr size bir \u015fey yazamad\u0131m, dostum, bir s\u00fcr\u00fc, bir s\u00fcr\u00fc kayg\u0131m, bir s\u00fcr\u00fc tedirginli\u011fim vard\u0131.\n\n\u00dc\u00e7 g\u00fcn \u00f6nce B\u0131kov geldi. Tek ba\u015f\u0131mayd\u0131m. Fedora bir yere gitmi\u015fti. Kap\u0131y\u0131 a\u00e7t\u0131m ve onu g\u00f6r\u00fcnce o kadar \u00fcrkt\u00fcm ki yerimden k\u0131p\u0131rdayamad\u0131m. Bembeyaz oldu\u011fumu hissettim. Her zamanki g\u00f6steri\u015fli g\u00fcl\u00fc\u015f\u00fcyle i\u00e7eri girdi, bir sandalye \u00e7ekip oturdu. Uzun s\u00fcre kendime gelemedim, sonunda bir k\u00f6\u015feye i\u015fimin ba\u015f\u0131na oturdum. Bir s\u00fcre sonra g\u00fclmeyi kesti. Herhalde, halim onu etkilemi\u015fti. Son zamanlarda \u00e7ok k\u00f6t\u00fcle\u015ftim; yanaklar\u0131m ve g\u00f6zlerim \u00e7\u00f6kt\u00fc, mendil gibi bembeyaz oldum... Ger\u00e7ekten, beni bir y\u0131l \u00f6nce tan\u0131yanlar \u015fimdi tan\u0131makta g\u00fc\u00e7l\u00fck \u00e7ekiyor. Uzun s\u00fcre ve dimdik bakt\u0131 bana, sonunda yine ne\u015felendi. Bir \u015feyler s\u00f6yledi; ne yan\u0131t verdi\u011fimi hat\u0131rlam\u0131yorum, o tekrar g\u00fcld\u00fc. Bir saat oturdu benim yan\u0131mda; uzun s\u00fcre konu\u015ftu benimle; \u015funu bunu sordu. Sonunda, veda etmeden \u00f6nce, elimi tuttu ve \u015f\u00f6yle dedi (kelimesi kelimesine yaz\u0131yorum size): \"Varvara Alekseyevna! Aram\u0131zda kals\u0131n, akrabam\u0131z olan, benim de yak\u0131n bir tan\u0131d\u0131\u011f\u0131m ve arkada\u015f\u0131m olan Anna Fedorovna, al\u00e7ak bir kad\u0131n.\" (Bu s\u0131rada onun i\u00e7in edepsiz bir s\u00f6zc\u00fck kulland\u0131.) \"Sizin kuzeninizi yoldan \u00e7\u0131kard\u0131 ve sizi mahvetti. Kendi pay\u0131ma ben de bu durumda al\u00e7ak bir rol ald\u0131m, ama bu da, ya\u015fam\u0131n bir cilvesi.\"\n\nBunu derken c\u0131rtlak bir kahkaha att\u0131. Sonra g\u00fczel konu\u015fman\u0131n \u00fcstad\u0131 olmad\u0131\u011f\u0131n\u0131 ve en \u00f6nemlisi de bir a\u00e7\u0131klama yapmak gerekti\u011fini ve asil davranmak ad\u0131na susmamas\u0131 gerekti\u011fini belirterek kalan \u015feyleri de k\u0131sa k\u0131sa s\u00f6yleyece\u011fini bildirdi. Bunu diyerek ellerimi arand\u0131, bana benim onurumu yeniden kazand\u0131rmay\u0131 bir bor\u00e7 bildi\u011fini, zengin oldu\u011funu, beni d\u00fc\u011f\u00fcnden sonra bozk\u0131rdaki k\u00f6y\u00fcne g\u00f6t\u00fcrece\u011fini, orada tav\u015fan avlayaca\u011f\u0131n\u0131; art\u0131k Petersburg'a bir daha gelmeyece\u011fini, \u00e7\u00fcnk\u00fc Petersburg'un i\u011fren\u00e7 oldu\u011funu, burada, Petersburg'da, kendi ifadesiyle, haydut bir ye\u011feni oldu\u011funu, onu miras\u0131ndan mahrum etmeye yemin etti\u011fini ve tam da bunun i\u00e7in, yani yasal bir v\u00e2ris sahibi olabilmek i\u00e7in benim elimi istedi\u011fini, evlilik teklifinin ba\u015fl\u0131ca sebebinin bu oldu\u011funu s\u00f6yledi. Sonra benim \u00e7ok yoksul ya\u015fad\u0131\u011f\u0131m\u0131, b\u00f6yle bir barakada ya\u015farken hasta olmam\u0131n \u015fa\u015f\u0131rt\u0131c\u0131 olmad\u0131\u011f\u0131n\u0131, e\u011fer bir ay daha b\u00f6yle kal\u0131rsam benim i\u00e7in \u00f6l\u00fcm\u00fcn ka\u00e7\u0131n\u0131lmaz oldu\u011funu belirtti, Petersburg'daki evlerin i\u011fren\u00e7 oldu\u011funu s\u00f6yledi ve en sonunda da bir \u015feye ihtiyac\u0131m olup olmad\u0131\u011f\u0131n\u0131 sordu.\n\nTeklifinden o kadar etkilenmi\u015ftim ki, neden oldu\u011funu anlayamadan a\u011flad\u0131m. G\u00f6zya\u015flar\u0131m\u0131 minnettarl\u0131\u011f\u0131ma verdi ve bana benim iyi, duygulu ve e\u011fitimli bir gen\u00e7 k\u0131z oldu\u011fuma hep inand\u0131\u011f\u0131n\u0131, ama daha \u00f6nce b\u00f6yle bir \u00f6nlem almaya, benim bug\u00fcnk\u00fc davran\u0131\u015flar\u0131m\u0131, ayr\u0131nt\u0131s\u0131yla \u00f6\u011frendikten sonra karar verdi\u011fini s\u00f6yledi. Bu s\u0131rada sizi sordu, her \u015feyi i\u015fitti\u011fini, sizin soylu ilkelere sahip bir insan oldu\u011funuzu, kendi ad\u0131na size bor\u00e7lu kalmak istemedi\u011fini s\u00f6yleyerek benim i\u00e7in yapt\u0131\u011f\u0131n\u0131z her \u015fey i\u00e7in size be\u015f y\u00fcz ruble vermenin yeterli olup olmad\u0131\u011f\u0131n\u0131 sordu. Ben ona sizin benim i\u00e7in yapt\u0131klar\u0131n\u0131z\u0131n parayla \u00f6denemeyece\u011fini a\u00e7\u0131klay\u0131nca, bana bunlar\u0131n sa\u00e7mal\u0131k oldu\u011funu, b\u00fct\u00fcn bunlar\u0131n romanlarda ve \u015fiirlerde oldu\u011funu, gen\u00e7 oldu\u011fumu ve romanlar\u0131n gen\u00e7 k\u0131zlar\u0131 bozdu\u011funu, kitaplar\u0131n terbiyeyi bozmaktan ba\u015fka bir i\u015fe yaramad\u0131\u011f\u0131n\u0131 ve kitap g\u00f6rmeye dayanamad\u0131\u011f\u0131n\u0131 s\u00f6yledi; onun kadar ya\u015fay\u0131p ondan sonra insanlardan bahsetmeyi tavsiye etti; \"O zaman,\" diye ekledi, \"insanlar\u0131 tan\u0131m\u0131\u015f olacaks\u0131n\u0131z.\" Sonra onun teklifini iyice bir d\u00fc\u015f\u00fcnmem gerekti\u011fini s\u00f6yledi, b\u00f6yle \u00f6nemli bir ad\u0131m\u0131 d\u00fc\u015f\u00fcncesizce atmam\u0131n \u00e7ok tats\u0131z olaca\u011f\u0131n\u0131 s\u00f6yledi, d\u00fc\u015f\u00fcncesizlik ve u\u00e7ar\u0131l\u0131\u011f\u0131n deneyimsiz gen\u00e7leri mahvetti\u011fini, ama benim taraf\u0131mdan olumlu yan\u0131t almay\u0131 \u00e7ok istedi\u011fini, son olarak da, tersi durumda, Moskova'da bir t\u00fcccar kad\u0131nla evlenmek zorunda kalaca\u011f\u0131n\u0131 s\u00f6yledi, \"... \u00e7\u00fcnk\u00fc,\" dedi, \"haydut ye\u011fenimi mirastan reddetmeye kararl\u0131y\u0131m.\"\n\nGergefimin \u00fczerine zorla be\u015f y\u00fcz ruble b\u0131rakt\u0131, \u015feker almak i\u00e7in dedi; k\u00f6yde b\u00f6rek gibi \u015fi\u015fmanlayaca\u011f\u0131m\u0131, orada ya\u011f\u0131n i\u00e7inde y\u00fczen peynir gibi olaca\u011f\u0131m\u0131, art\u0131k \u00e7ok tela\u015f\u0131 oldu\u011funu, b\u00fct\u00fcn g\u00fcn birtak\u0131m i\u015flerle u\u011fra\u015fmas\u0131 gerekti\u011fini ve iki i\u015f aras\u0131nda bana ko\u015fup geldi\u011fini s\u00f6yledi. Sonra gitti. Uzun zaman d\u00fc\u015f\u00fcnd\u00fcm, tekrar tekrar d\u00fc\u015f\u00fcnd\u00fcm, d\u00fc\u015f\u00fcnmekten peri\u015fan oldum, dostum, sonunda karar\u0131m\u0131 verdim. Dostum, onunla evlenece\u011fim, teklifini kabul etmek zorunday\u0131m. E\u011fer beni rezaletten kurtarabilecek biri varsa, onurlu ismimi bana geri verecek, yoksulluktan, yoksunluktan ve gelecekteki mutsuzluktan \u00e7\u0131karacak biri varsa, bir tek o. Gelecekten ne bekleyebilirim, kaderden ne isteyebilirim? Fedora kendi mutlulu\u011fumu kaybetmeme gerek olmad\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyor; bu durumda neye mutluluk diyece\u011fiz, diyor. En az\u0131ndan kendim i\u00e7in ba\u015fka bir yol bulam\u0131yorum, de\u011ferli dostum. Ne yapaca\u011f\u0131m? \u00c7al\u0131\u015f\u0131yorum ve b\u00fct\u00fcn sa\u011fl\u0131\u011f\u0131m bozuldu; s\u00fcrekli \u00e7al\u0131\u015fmam imk\u00e2ns\u0131z. Birinin yan\u0131nda i\u015fe girmek? Kederden t\u00fckenirim, kimsenin yan\u0131na gidemem. Do\u011fu\u015ftan hastal\u0131kl\u0131y\u0131m ve bu y\u00fczden hep ba\u015fkas\u0131na y\u00fck olaca\u011f\u0131m. Elbette, art\u0131k Cennet'e de gidemem, ama ne yapaca\u011f\u0131m, dostum, ne yapaca\u011f\u0131m? Neyi se\u00e7ece\u011fim ben?\n\nSizden \u00f6\u011f\u00fct vermenizi rica etmiyorum. Tek ba\u015f\u0131ma d\u00fc\u015f\u00fcnmek istedim. Sizin \u015fu anda okudu\u011funuz karar de\u011fi\u015fmeyecek ve zaten kesin karar\u0131m\u0131 vermem i\u00e7in beni s\u0131k\u0131\u015ft\u0131ran B\u0131kov'a karar\u0131m\u0131 hemen a\u00e7\u0131klayaca\u011f\u0131m. \u0130\u015flerinin beklemeyece\u011fini, gitmesi gerekti\u011fini ve onlar\u0131 bo\u015f i\u015fler y\u00fcz\u00fcnden erteleyemeyece\u011fini s\u00f6yledi. Mutlu olup olmayaca\u011f\u0131m\u0131 Tanr\u0131 bilir. Onun, o aziz, s\u0131rr\u0131na eri\u015filmez kudretinin elinde benim kaderim, ama karar\u0131m\u0131 verdim. B\u0131kov'un iyi biri oldu\u011funu s\u00f6yl\u00fcyorlar; bana sayg\u0131 g\u00f6sterecek; belki de ben de ona sayg\u0131 g\u00f6sterece\u011fim. Evlili\u011fimizden ba\u015fka ne beklenir ki?\n\nSizi her konuda bilgilendiriyorum, Makar Alekseyevi\u00e7. Benim kederimi anlayaca\u011f\u0131n\u0131za inan\u0131yorum. Beni niyetimden al\u0131koymay\u0131n. Bo\u015fa \u00e7aba harcars\u0131n\u0131z. Beni b\u00f6yle davranmaya s\u00fcr\u00fckleyen \u015feyleri kalbinizde bir tart\u0131n. \u00d6nce \u00e7ok kayg\u0131 duyuyordum, ama \u015fimdi sakinim. \u0130leride ne olacak, bilmiyorum. Ne olacaksa olacak; Tanr\u0131 nas\u0131l yazd\u0131ysa!..\n\nB\u0131kov geldi; mektubu bitirmeden b\u0131rak\u0131yorum. Size daha bir s\u00fcr\u00fc \u015fey anlatmak istiyordum. B\u0131kov geldi art\u0131k!\n\nV.D.\n\n23 Eyl\u00fcl\n\nCan\u0131m, Varvara Alekseyevna,\n\nCan\u0131m, ben, hemen yan\u0131t veriyorum; can\u0131m, ben, size \u015fa\u015fk\u0131nl\u0131k i\u00e7inde oldu\u011fumu bildirece\u011fim hemen. B\u00fct\u00fcn bunlar b\u00f6yle... D\u00fcn Gor\u015fkov'u g\u00f6mm\u00fc\u015ft\u00fck. Evet, b\u00f6yledir, Varenka, b\u00f6yledir; B\u0131kov soylu davranm\u0131\u015f; ama i\u015fte, g\u00f6r\u00fcyor musunuz, bir tanem, siz de kabul etmi\u015fsiniz. Elbette, hepsi Tanr\u0131'n\u0131n iradesi; bu b\u00f6yledir, bunun muhakkak b\u00f6yle olmas\u0131 gerekir, yani burada muhakkak Tanr\u0131'n\u0131n iradesinin olmu\u015f olmas\u0131 gerekir; ve g\u00f6klerdeki Yarat\u0131c\u0131'n\u0131n tasar\u0131s\u0131 elbette hay\u0131rl\u0131 ve tart\u0131\u015f\u0131lmazd\u0131r, kaderler de \u00f6yle, onlar da hay\u0131rl\u0131 ve tart\u0131\u015f\u0131lmazd\u0131r. Fedora da size kat\u0131l\u0131yor. Elbette, art\u0131k mutlu olacaks\u0131n\u0131z, can\u0131m, ho\u015fnut olacaks\u0131n\u0131z g\u00fcvercinim, bir tanem, sevgilimsiniz, mele\u011fimsiniz benim; i\u015fte, g\u00f6r\u00fcyor musunuz Varenka, bu nas\u0131l \u00e7abuk oldu?.. Evet, i\u015fler... Bay B\u0131kov'un i\u015fleri var; elbette, kimin i\u015fi yok, onun da i\u015fleri olabilir... Onu sizden \u00e7\u0131karken g\u00f6rd\u00fcm. G\u00f6z al\u0131c\u0131, g\u00f6z al\u0131c\u0131 bir adam; hatta \u00e7ok g\u00f6z al\u0131c\u0131 bir adam. Ama b\u00fct\u00fcn bunlar biraz, konu pek de onun g\u00f6z al\u0131c\u0131 bir adam olmas\u0131 de\u011fil, ben \u015fimdi kendimde de\u011filim pek. Fakat biz \u015fimdi birbirimize nas\u0131l mektup yazaca\u011f\u0131z? Ben, ben nas\u0131l tek ba\u015f\u0131ma kalaca\u011f\u0131m? Mele\u011fim, ben, her \u015feyi tart\u0131yorum, sizin bana yazd\u0131\u011f\u0131n\u0131z gibi her \u015feyi tart\u0131yorum kalbimde, bu nedenleri tart\u0131yorum. Yirmi yapra\u011f\u0131 yazmay\u0131 bitirdim ben, ama bu arada bu olaylar olmu\u015f! Can\u0131m, madem siz \u015fimdi gidiyorsunuz, sizin bir s\u00fcr\u00fc al\u0131\u015fveri\u015f yapman\u0131z gerekir, \u00e7e\u015fitli ayakkab\u0131lar, elbiseler, bak\u0131n benim Gorohovaya'da tan\u0131d\u0131\u011f\u0131m bir d\u00fckk\u00e2n var; hat\u0131rlay\u0131n, size ondan bahsetmi\u015ftim. Ama olacak \u015fey de\u011fil! Nas\u0131l yapars\u0131n\u0131z, can\u0131m, nas\u0131l? \u015eimdi gitmemeniz gerekir, kesinlikle olmaz, asla olmaz. B\u00fcy\u00fck al\u0131\u015fveri\u015fler yapman\u0131z laz\u0131m, araba da tutmak laz\u0131m. Bu arada hava da k\u00f6t\u00fc; baksan\u0131za, bardaktan bo\u015fan\u0131rcas\u0131na ya\u011fmur ya\u011f\u0131yor ve b\u00f6yle bir ya\u011fmurda, yani... \u00fcstelik, \u00fc\u015f\u00fct\u00fcrs\u00fcn\u00fcz, mele\u011fim; kalbiniz \u00fc\u015f\u00fcr! Siz korkuyordunuz yabanc\u0131 bir insandan, ama gidiyorsunuz. Peki ben burada nas\u0131l tek ba\u015f\u0131ma kalaca\u011f\u0131m? \u0130\u015fte Fedora sizi b\u00fcy\u00fck mutluluk bekledi\u011fini s\u00f6ylemi\u015f... zaten tam bir kocakar\u0131d\u0131r ve benim \u00f6lmemi ister. Bug\u00fcn ak\u015fam ayinine gidecek misiniz, can\u0131m? Sizi g\u00f6rm\u00fc\u015f olurdum. Can\u0131m, kesinlikle do\u011fru, do\u011fru sizin e\u011fitimli, iyilik\u00e7i ve duygulu bir gen\u00e7 k\u0131z oldu\u011funuz; ama ke\u015fke o gidip t\u00fcccar kad\u0131nla evlenseydi! Ne dersiniz? Ke\u015fke t\u00fcccar kad\u0131nla evlense! Varenka'c\u0131\u011f\u0131m, hava karar\u0131nca ben size bir saatli\u011fine gelirim hemen. \u015eimdi erken karar\u0131yor hava, ben de gelece\u011fim. Can\u0131m, kesinlikle bir saatli\u011fine gelece\u011fim bug\u00fcn. \u015eimdi siz B\u0131kov'u bekliyorsunuz, o gidince, o zaman... Ama bekleyin, can\u0131m, gelece\u011fim...\n\nMakar Devu\u015fkin\n\n27 Eyl\u00fcl\n\nDostum, Makar Alekseyevi\u00e7,\n\nBay B\u0131kov, bana hemen \u00fc\u00e7 d\u00fczine Hollanda kuma\u015f\u0131ndan giysiye sahip olmam gerekti\u011fini s\u00f6yledi. \u0130ki d\u00fczinesi i\u00e7in terzi aramam\u0131z laz\u0131m hemen, zaman\u0131m\u0131z da \u00e7ok az. Bay B\u0131kov \u00f6fkelendi, bu \u0131v\u0131r z\u0131v\u0131r\u0131n \u00e7ok ayak ba\u011f\u0131 oldu\u011funu s\u00f6yledi. D\u00fc\u011f\u00fcn\u00fcm\u00fcz be\u015f g\u00fcn sonra, d\u00fc\u011f\u00fcnden sonraki g\u00fcn de gidiyoruz. Bay B\u0131kov tela\u015f ediyor, sa\u00e7mal\u0131klarla \u00e7ok vakit kaybetmemek gerekti\u011fini s\u00f6yl\u00fcyor. Ko\u015fu\u015fturmadan peri\u015fan oldum ve ayakta zor duruyorum. Korkun\u00e7 i\u015f var, do\u011frusu, bu hi\u00e7 olmasa en iyisi olacakt\u0131. Hatta, ipekli ve ipeksiz dantellerimiz yetmiyor, daha \u00e7ok almak gerekecek, \u00e7\u00fcnk\u00fc Bay B\u0131kov kar\u0131s\u0131n\u0131n a\u015f\u00e7\u0131 kad\u0131n gibi gezmesini istemedi\u011fini s\u00f6yl\u00fcyor, benim kesinlikle, \"b\u00fct\u00fcn toprak sahibi e\u015flerini \u00e7atlatmam\" gerekiyormu\u015f. Bunu kendisi s\u00f6yledi. \u0130\u015fte b\u00f6yle, Makar Alekseyevi\u00e7, l\u00fctfen Gorohovaya'daki Madam \u015eifon'a u\u011fray\u0131n ve rica edin, birincisi bize terzi g\u00f6ndersin, ikincisi, bir zahmet kendisi de gelsin. Bug\u00fcn hastay\u0131m. Yeni evimiz so\u011fuk ve her \u015fey darmada\u011f\u0131n. Bay B\u0131kov'un teyzesi ya\u015fl\u0131l\u0131ktan zor nefes al\u0131yor. Biz gitmeden \u00f6nce \u00f6lecek diye korkuyorum, ama Bay B\u0131kov \u00f6nemsiz bu, kendine gelir diyor. Evimizde korkun\u00e7 bir da\u011f\u0131n\u0131kl\u0131k var. Bay B\u0131kov bizimle oturmuyor, hep birilerine gidiyor, Tanr\u0131 bilir nereye. Bu y\u00fczden bir tek Fedora i\u015flere bak\u0131yor: Bay B\u0131kov'un her i\u015fe bakan oda u\u015fa\u011f\u0131 daha \u00fc\u00e7\u00fcnc\u00fc g\u00fcn kim bilir nerelere kayboldu. Bay B\u0131kov her sabah geliyor, hep \u00f6fkeli ve ak\u015famleyin evin k\u00e2hyas\u0131n\u0131 d\u00f6vd\u00fc, bu y\u00fczden polisle bir tats\u0131zl\u0131k ya\u015fand\u0131... Size mektup getirecek kimse de yok. \u015eehir postanesinden yaz\u0131yorum. Evet! Az kals\u0131n en \u00f6nemlisini unutuyordum. Madam \u015eifon'a s\u00f6yleyin, _blonde_ 'lar\u013128 hemen tamir etsin, d\u00fcnk\u00fc kal\u0131ba g\u00f6re yaps\u0131n ve yeni se\u00e7imleri g\u00f6stermek i\u00e7in bizzat gelsin bana. Ona bir de s\u00f6yleyin, _canezou_29 konusunu bir daha d\u00fc\u015f\u00fcnd\u00fcm; onu _crochet_30 ile \u00f6rmek gerekecek. Hatta, mendillerdeki armalar\u0131n harfleri _tambour_ 'la31 i\u015flenecek; anl\u0131yor musunuz? _Tambour_ 'la yap\u0131lacak, d\u00fcz olmayacak. Bak\u0131n unutmay\u0131n, _tambour_ 'la! Bak\u0131n az kals\u0131n unutuyordum! Pelerinin \u00fczerindeki yapraklar kabart\u0131lm\u0131\u015f, anten ve halkalar _cordonnet_32 i\u015flensin, sonra yakaya dantel ya da geni\u015f farbala dikilsin. L\u00fctfen s\u00f6yleyin bunlar\u0131, Makar Alekseyevi\u00e7.\n\nSizin,  \nV.D.\n\nHami\u015f: Sizi hep sipari\u015flerimle b\u0131kt\u0131rd\u0131\u011f\u0131m i\u00e7in vicdan azab\u0131 \u00e7ekiyorum. \u00dc\u00e7 g\u00fcnd\u00fcr sabahtan ak\u015fama ko\u015fturuyorsunuz. Ama ne yapars\u0131n\u0131z! Evimizde hi\u00e7bir d\u00fczen yok, ben de sa\u011fl\u0131ks\u0131z\u0131m. Yani bana k\u0131zmay\u0131n Makar Alekseyevi\u00e7. Ne keder! Ah, ne olacak b\u00f6yle, dostum, tatl\u0131m, iyi y\u00fcreklim Makar Alekseyevi\u00e7? Ben gelece\u011fime bakmaya korkuyorum. Hep bir \u015feyler seziyorum ve sanki bir duman\u0131n i\u00e7inde ya\u015f\u0131yor gibiyim.\n\nHami\u015f: Tanr\u0131 a\u015fk\u0131na, dostum, size \u015fimdi s\u00f6ylediklerimi unutmay\u0131n. Hata yapars\u0131n\u0131z diye \u00e7ok korkuyorum. Unutmay\u0131n, d\u00fcz olmayacak, _tambour_ 'la yap\u0131lacak.\n\nV.D.\n\n27 Eyl\u00fcl\n\nSevgili Han\u0131mefendi Varvara Alekseyevna,\n\n\u0130steklerinizin hepsini dikkatle yerine getirdim. Madam \u015eifon kendisinin de _tambour_ 'la dikmeyi d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcn\u00fc s\u00f6yledi; bu daha sayg\u0131n olur dedi, bilmiyorum, oras\u0131n\u0131 \u00e7ok iyi anlayamad\u0131m. Bir de, siz orada farbala yazm\u0131\u015fs\u0131n\u0131z, o da farbaladan bahsetti. Fakat ben unuttum can\u0131m, onun bana farbala hakk\u0131nda ne dedi\u011fini. Tek hat\u0131rlad\u0131\u011f\u0131m, bir s\u00fcr\u00fc \u015fey s\u00f6yledi\u011fi; ne kadar i\u011fren\u00e7 bir kocakar\u0131! Ne diyor b\u00f6yle? Zaten kendisi size anlatacak. Can\u0131m, ben iyice sersemledim. Bug\u00fcn i\u015fe de gitmedim. Fakat siz, bir tanem, bence bo\u015f yere kayg\u0131lan\u0131yorsunuz. Sizin huzurunuz i\u00e7in b\u00fct\u00fcn d\u00fckk\u00e2nlar\u0131 dola\u015fmaya haz\u0131r\u0131m. Gelecekte neler olaca\u011f\u0131ndan korktu\u011funuzu yaz\u0131yorsunuz. Zaten bug\u00fcn saat yedide hepsini \u00f6\u011freneceksiniz. Madam \u015eifon bizzat size gelecek. Kayg\u0131lanmay\u0131n yani; umutlu olun, can\u0131m; belki her \u015fey daha iyi olur, yani. Ama \u015funu diyeyim, bu farbela denen \u015feye lanet olsun; ah, nedir bu farbala, farbala! Size ko\u015fup gelecektim, mele\u011fim, ko\u015fup gelecektim, hemen ko\u015fup gelecektim; sizin binan\u0131z\u0131n kap\u0131s\u0131na iki kez geldim. Ama hep B\u0131kov, yani, demek istedi\u011fim, Bay B\u0131kov hep \u00f6yle sinirli ki, o oradayken yani... Neyse, aman bo\u015fverin!\n\nMakar Devu\u015fkin\n\n28 Eyl\u00fcl\n\nSevgili Beyefendi, Makar Alekseyevi\u00e7,\n\nTanr\u0131 a\u015fk\u0131na, hemen kuyumcuya ko\u015fun. Ona inci ve z\u00fcmr\u00fct ta\u015fl\u0131 k\u00fcpe yapmas\u0131na gerek olmad\u0131\u011f\u0131n\u0131 s\u00f6yleyin. Bay B\u0131kov bunun \u00e7ok pahal\u0131 oldu\u011funu, el yakt\u0131\u011f\u0131n\u0131 s\u00f6yl\u00fcyor. \u00d6fkelendi; cebine dadand\u0131\u011f\u0131m\u0131z\u0131, onu soydu\u011fumuzu s\u00f6yl\u00fcyor, d\u00fcn de e\u011fer \u00f6nceden b\u00f6yle masraflar \u00e7\u0131kaca\u011f\u0131n\u0131 bilseymi\u015f, hi\u00e7 temas kurmam\u0131\u015f olaca\u011f\u0131n\u0131 s\u00f6yledi. Bizi nik\u00e2hlad\u0131klar\u0131 zaman kalk\u0131p gidece\u011fimizi, konuk falan olmayaca\u011f\u0131n\u0131 ve benim hoplay\u0131p z\u0131playarak dans etmeyi en az\u0131ndan bayrama kadar hi\u00e7 umut etmememi s\u00f6yledi. \u0130\u015fte b\u00f6yle diyor! Tanr\u0131 biliyor bana b\u00fct\u00fcn bunlar\u0131n laz\u0131m olup olmad\u0131\u011f\u0131n\u0131! Bay B\u0131kov kendisi sipari\u015f etmi\u015fti hepsini. Ona yan\u0131t vermeye de cesaret edemiyorum; b\u00f6yleyken \u00e7ok sinirli oluyor. Ne yapaca\u011f\u0131m ben?\n\nV.D.\n\n28 Eyl\u00fcl\n\nG\u00fcvercinim, Varvara Alekseyevna,\n\nKuyumcu, tamam, diyor; ama ben kendi ad\u0131ma hasta oldu\u011fumu ve yataktan kalkacak halde olmad\u0131\u011f\u0131m\u0131 s\u00f6ylemeliyim. Tam da b\u00f6yle tela\u015fl\u0131, zaruri bir vakitte so\u011fuk alg\u0131nl\u0131\u011f\u0131 \u00e7ulland\u0131, olacak \u015fey de\u011fil! Bir de size bildirmek zorunday\u0131m ki talihsizliklerim tam olsun diye Majesteleri sert olmaya karar verdiler ve Emelyan \u0130vanovi\u00e7'e uzun uzun \u00f6fkelenip ba\u011f\u0131rd\u0131lar, zavall\u0131 adam neredeyse korkudan \u00f6lecekti. \u0130\u015fte size her \u015feyi haber veriyorum. Size bir \u015feyler daha yazmak isterdim, ama sizi yormaktan korkuyorum. Zaten ben, can\u0131m, aptal, s\u0131radan bir insan\u0131m, akl\u0131ma ne gelirse yaz\u0131yorum, belki de, siz orada bir \u015feye ve b\u00f6yle... neyse, aman bo\u015fverin!\n\nSizin olan,  \nMakar Devu\u015fkin\n\n29 Eyl\u00fcl\n\nVarvara Alekseyevna, Bir Tanem,\n\nBug\u00fcn Fedora'y\u0131 g\u00f6rd\u00fcm, g\u00fcvercinim. Sizin yar\u0131n evlenece\u011finizi s\u00f6yledi, \u00f6b\u00fcr g\u00fcn de gidiyormu\u015fsunuz, Bay B\u0131kov atlar\u0131 kiralam\u0131\u015f bile. Majesteleri'nden bahsetmi\u015ftim size, can\u0131m. Bir de, Gorohovaya'daki d\u00fckk\u00e2n\u0131n faturas\u0131n\u0131 g\u00f6zden ge\u00e7irdim; hepsi do\u011fru, ama \u00e7ok pahal\u0131. Fakat Bay B\u0131kov ne diye size \u00f6fkeleniyor? Neyse, mutlu olun, can\u0131m! Ben mutluyum; evet, mutlu olaca\u011f\u0131m, e\u011fer siz de mutlu olacaksan\u0131z. Kiliseye gelirdim, can\u0131m, ama gelemem, belim a\u011fr\u0131yor. Ama ben de mektuplar\u0131m\u0131z\u0131 d\u00fc\u015f\u00fcn\u00fcyorum: \u015fimdi kim onlar\u0131 getirip g\u00f6t\u00fcrecek, can\u0131m? Evet! Fedora'ya \u00e7ok iyilik yapm\u0131\u015fs\u0131n\u0131z, bir tanem! \u0130yi bir \u015fey yapm\u0131\u015fs\u0131n\u0131z, dostum; bunu \u00e7ok iyi yapm\u0131\u015fs\u0131n\u0131z. \u0130yi bir \u015fey! Yapt\u0131\u011f\u0131n\u0131z her iyi i\u015f i\u00e7in Tanr\u0131 sizi m\u00fck\u00e2fatland\u0131racakt\u0131r. \u0130yi i\u015fler \u00f6d\u00fcls\u00fcz kalmaz, iyilik yapanlar hep Tanr\u0131'n\u0131n adaletinin tac\u0131n\u0131 takar, er ya da ge\u00e7. Can\u0131m! Size \u00e7ok \u015fey yazmak isterdim, yani her saat, her dakika her \u015feyi yazard\u0131m, her \u015feyi yazard\u0131m! Bende bir kitab\u0131n\u0131z kald\u0131, \"Biyelkin'in Hik\u00e2yeleri\", ama siz, can\u0131m, almay\u0131n onu benden, hediye edin bana, g\u00fcvercinim. Kitab\u0131 yine okumak istedi\u011fimden de\u011fil. Ama siz de biliyorsunuz, can\u0131m, k\u0131\u015f geliyor; geceler uzun olacak; h\u00fcz\u00fcnl\u00fc olacak, i\u015fte o zaman okunur. Can\u0131m, ben evimden \u00e7\u0131k\u0131p sizin eski eve ta\u015f\u0131naca\u011f\u0131m ve Fedora'dan yer kiralayaca\u011f\u0131m. Bu onurlu kad\u0131ndan art\u0131k asla ayr\u0131lmam; bu kadar \u00e7al\u0131\u015fkan bir kad\u0131ndan. Sizin bo\u015faltt\u0131\u011f\u0131n\u0131z eve d\u00fcn ayr\u0131nt\u0131l\u0131 bir \u015fekilde bakt\u0131m. Sizin gergefiniz, \u00fczerinde i\u015flemesiyle bir yerde dururdu, hi\u00e7 el de\u011fmemi\u015f bir halde duruyor: k\u00f6\u015fede. \u0130\u015flemenize bakt\u0131m. Orada \u0131v\u0131r z\u0131v\u0131r bir \u015feyler daha kalm\u0131\u015f. Benim bir mektubuma iplik dolamaya ba\u015flam\u0131\u015fs\u0131n\u0131z. Sehpada bir k\u00e2\u011f\u0131t buldum, \u00fczerinde \u015f\u00f6yle yaz\u0131yordu: \"Sevgili Bay, Makar Alekseyevi\u00e7, aceleyle\"...Hepsi bu. Anla\u015f\u0131lan, en ilgin\u00e7 yerde birisi size engel olmu\u015f. K\u00f6\u015fede paravan\u0131n arkas\u0131nda k\u00fc\u00e7\u00fck yata\u011f\u0131n\u0131z duruyor... K\u00fc\u00e7\u00fck g\u00fcvercinim!!! Neyse, ho\u015f\u00e7a kal\u0131n, ho\u015f\u00e7a kal\u0131n; Tanr\u0131 a\u015fk\u0131na, bu mektuba acilen bir yan\u0131t verin.\n\nMakar Devu\u015fkin\n\n30 Eyl\u00fcl\n\nPaha Bi\u00e7ilmez Dostum, Makar Alekseyevi\u00e7,\n\nHer \u015fey bitti! Benim kaderim ba\u011fland\u0131; nas\u0131l olacak bilmiyorum, ama Tanr\u0131 ne buyurursa \u00f6yle olacak. Yar\u0131n yola \u00e7\u0131k\u0131yoruz. Sizinle son kez vedala\u015f\u0131yorum, paha bi\u00e7ilmez dostum, velinimetim, bir tanem! Benim i\u00e7in ac\u0131 \u00e7ekmeyin, mutlu ya\u015fay\u0131n, hat\u0131rlay\u0131n beni ve Tanr\u0131'n\u0131n hay\u0131r duas\u0131 \u00fczerinizde olsun! D\u00fc\u015f\u00fcncelerimde, dualar\u0131mda sizi s\u0131k s\u0131k hat\u0131rlayaca\u011f\u0131m. \u0130\u015fte b\u00f6yle bitti bu vakit! Ge\u00e7mi\u015f hat\u0131ralardan yeni hayat\u0131ma ta\u015f\u0131d\u0131\u011f\u0131m kayda de\u011fer \u00e7ok az \u015fey var; en de\u011ferlileri sizinle ilgili hat\u0131ralar olacak, kalbimdeki en de\u011ferli \u015fey siz olacaks\u0131n\u0131z. Biricik dostumsunuz benim; bir tek siz sevdiniz beni. Her \u015feyi g\u00f6rd\u00fcm, sizin beni nas\u0131l sevdi\u011finizi de biliyordum! Benim bir tek g\u00fcl\u00fc\u015f\u00fcmle mutlu oluyordunuz, bir tek sat\u0131r mektubumla. \u015eimdi benden vazge\u00e7meniz gerekecek! Tek ba\u015f\u0131n\u0131za burada nas\u0131l kalacaks\u0131n\u0131z! Kiminle kalacaks\u0131n\u0131z, iyi, paha bi\u00e7ilmez, biricik dostum benim! Size kitab\u0131 b\u0131rak\u0131yorum, i\u015flemeyi, yar\u0131m kalm\u0131\u015f mektubu b\u0131rak\u0131yorum; bu yar\u0131m kalm\u0131\u015f sat\u0131ra bakt\u0131\u011f\u0131n\u0131z zaman, akl\u0131n\u0131zdan ge\u00e7ireceksiniz sonraki sat\u0131rlar\u0131, benden duymak ya da okumak istediklerinizi akl\u0131n\u0131zdan ge\u00e7ireceksiniz, sanki ben yazm\u0131\u015f\u0131m gibi; neyi yazmad\u0131ysam \u015fimdi! Hat\u0131rlay\u0131n zavall\u0131 Varenka'n\u0131z\u0131, b\u00f6yle g\u00fc\u00e7l\u00fc bir \u015fekilde sevdi\u011finiz k\u0131z\u0131. B\u00fct\u00fcn mektuplar\u0131n\u0131z Fedora'daki komodinde, \u00fcst \u00e7ekmecede kald\u0131. Hasta oldu\u011funuzu yazm\u0131\u015fs\u0131n\u0131z, Bay B\u0131kov da bug\u00fcn beni hi\u00e7bir yere b\u0131rakmaz. Size yazaca\u011f\u0131m, dostum, s\u00f6z veriyorum, ama neler olabilece\u011fini bir tek Tanr\u0131 bilir. \u0130\u015fte, sonsuza dek vedala\u015fal\u0131m \u015fimdi, dostum, g\u00fcvercinim, bir tanem, sonsuza dek!.. Ah, \u015fimdi nas\u0131l sar\u0131l\u0131rd\u0131m size! Ho\u015f\u00e7a kal\u0131n, dostum, ho\u015f\u00e7a kal\u0131n, ho\u015f\u00e7a kal\u0131n. Mutlu ya\u015fay\u0131n; sa\u011fl\u0131kl\u0131 olun. Benim dualar\u0131m sonsuza dek size dair olacak. Ah! Ne kadar h\u00fcz\u00fcnl\u00fcy\u00fcm, nas\u0131l eziliyor b\u00fct\u00fcn ruhum. Bay B\u0131kov \u00e7a\u011f\u0131r\u0131yor beni.\n\nSizi sonsuzca seven,  \nV.\n\nHami\u015f: Ruhum \u00f6yle dolu, \u00f6yle dolu ki g\u00f6zya\u015flar\u0131yla... G\u00f6zya\u015flar\u0131 eziyor, par\u00e7al\u0131yor beni. Ho\u015f\u00e7a kal\u0131n. Tanr\u0131m! Ne kadar h\u00fcz\u00fcnl\u00fc! Hat\u0131rlay\u0131n, hat\u0131rlay\u0131n yoksul Varenka'n\u0131z\u0131!\n\n* * *\n\nCan\u0131m Varenka, G\u00fcvercinim, Paha Bi\u00e7ilmezim,\n\nSizi \u00e7a\u011f\u0131r\u0131yorlar, siz de gidiyorsunuz! Ama \u015fimdi sizi benden alacaklar\u0131na, g\u00f6\u011fs\u00fcmden kalbimi s\u00f6k\u00fcp alsalard\u0131 daha iyi olurdu! Nas\u0131l yap\u0131yorsunuz bunu! Hem a\u011fl\u0131yorsunuz, hem de gidiyorsunuz?! \u0130\u015fte sizden mektubu az \u00f6nce ald\u0131m, g\u00f6zya\u015f\u0131ndan s\u0131r\u0131ls\u0131klam. Demek, gitmek istemiyorsunuz; demek, sizi zorla g\u00f6t\u00fcr\u00fcyorlar, demek, bana \u00fcz\u00fcld\u00fcn\u00fcz, demek, beni seviyorsunuz! Nas\u0131l olacak, kiminle olacaks\u0131n\u0131z art\u0131k? Sizin k\u00fc\u00e7\u00fck kalbiniz h\u00fcz\u00fcnl\u00fc, bulan\u0131k ve so\u011fuk olacak. Keder yiyip bitirecek onu, h\u00fcz\u00fcn par\u00e7alara ay\u0131racak. \u00d6leceksiniz orada, orada \u0131slak topra\u011fa koyacaklar sizi; sizin i\u00e7in a\u011flayacak kimse olmayacak! Bay B\u0131kov hep tav\u015fan kovalayacak... Ah, can\u0131m, can\u0131m! Ne diye bu karar\u0131 verdiniz, nas\u0131l b\u00f6yle bir \u00f6nlem almaya karar verebildiniz? Ne yapt\u0131n\u0131z, ne yapt\u0131n\u0131z, kendinize ne yapt\u0131n\u0131z! Orada sizi mezara g\u00f6t\u00fcr\u00fcyorlar; sizi peri\u015fan edecekler orada, k\u00fc\u00e7\u00fck mele\u011fim. Can\u0131m, siz zaten t\u00fcy gibi zay\u0131fs\u0131n\u0131z! Peki ben neredeydim? Ne diye burada durup aptal gibi, seyrettim! Nas\u0131l oluyor da ate\u015flenmi\u015f bir \u00e7ocu\u011fun say\u0131klamalar\u0131na kulak asmad\u0131m. Ama ben aptal\u0131m. Hi\u00e7bir \u015fey d\u00fc\u015f\u00fcnemedim. Hi\u00e7bir \u015fey g\u00f6remedim. Sanki b\u00fct\u00fcn bunlar olmas\u0131 gereken \u015feylerdi; sanki benimle ilgisi yoktu; b\u0131rakm\u0131\u015f farbala pe\u015finde ko\u015fuyorum!.. Hay\u0131r, ben, Varenka, kar\u015f\u0131 koyaca\u011f\u0131m; yar\u0131n, belki de, iyile\u015fece\u011fim, o zaman kar\u015f\u0131 koyaca\u011f\u0131m!.. Can\u0131m, ben, tekerle\u011fin alt\u0131na ataca\u011f\u0131m; sizin gitmenize izin vermeyece\u011fim! Hay\u0131r, ger\u00e7ekten de bu nedir b\u00f6yle? Ne hakla oluyor b\u00fct\u00fcn bunlar? Ben de sizinle gelece\u011fim; araban\u0131z\u0131n arkas\u0131ndan ko\u015faca\u011f\u0131m, e\u011fer beni almazsan\u0131z g\u00fcc\u00fcmle ko\u015faca\u011f\u0131m, ruhum bedenden ayr\u0131l\u0131ncaya dek. Ama nereye gitti\u011finizi biliyor musunuz, nereye gidiyorsunuz, can\u0131m? Belki de bilmiyorsunuz bunu, bana sorsan\u0131za! Oras\u0131 bozk\u0131r, bir tanem, oras\u0131 bozk\u0131r, \u00e7\u0131r\u0131l\u00e7\u0131plak bozk\u0131r; i\u015fte t\u0131pk\u0131 benim \u00e7\u0131plak avucum gibi! Orada duygusuz kocakar\u0131yla okumam\u0131\u015f adam, sarho\u015f sarho\u015f gezer. Orada art\u0131k a\u011fa\u00e7lar\u0131n yapraklar\u0131 d\u00f6k\u00fclm\u00fc\u015ft\u00fcr, ya\u011fmur vard\u0131r, so\u011fuktur; ama siz oraya gidiyorsunuz! Tabii, Bay B\u0131kov'un orada i\u015fi varm\u0131\u015f; orada tav\u015fan pe\u015finde ko\u015facak o; peki ya siz? Siz bey kar\u0131s\u0131 olmak istiyor musunuz, can\u0131m? Ama, siz \u00e7ocuk mele\u011fimsiniz benim! Bir baksan\u0131za kendinize, bir bey kar\u0131s\u0131na benziyor musunuz?.. B\u00f6yle bir \u015fey nas\u0131l olur, Varenka! Kime mektup yazaca\u011f\u0131m ben, can\u0131m? Evet! Siz bir d\u00fc\u015f\u00fcnsenize bakal\u0131m, can\u0131m, \u015fimdi bu adam kime mektup yazacak? Kime can\u0131m diyece\u011fim ben; b\u00f6yle sevgi dolu bir isimle kime seslenece\u011fim? Sonra sizi ben nerede bulaca\u011f\u0131m, k\u00fc\u00e7\u00fck mele\u011fim? Ben \u00f6lece\u011fim, Varenka, hemen \u00f6lece\u011fim; benim kalbim kald\u0131rmaz b\u00f6yle mutsuzlu\u011fu! Ben sizi, Tanr\u0131'n\u0131n g\u00fcn \u0131\u015f\u0131\u011f\u0131 gibi sevdim, \u00f6z k\u0131z\u0131m gibi sevdim, sizi hep sevdim, can\u0131m, bir tanem! Ve ben bir tek sizin i\u00e7in ya\u015f\u0131yordum! \u00c7al\u0131\u015f\u0131yordum, evrak yaz\u0131yordum, gidip geliyordum, geziyordum, g\u00f6zlemlerimi k\u00e2\u011f\u0131da aktar\u0131yordum dost\u00e7a mektuplar \u015feklinde, b\u00fct\u00fcn bunlar, can\u0131m, siz burada, kar\u015f\u0131mda, yak\u0131nda ya\u015fad\u0131\u011f\u0131n\u0131z i\u00e7indi. Siz, belki de bilmiyordunuz bunu, ama b\u00fct\u00fcn bunlar b\u00f6yle oldu! Evet, dinleyin can\u0131m, siz bir tart\u0131n bakal\u0131m, sevgili g\u00fcvercinim, nas\u0131l olur, siz nas\u0131l bizden ka\u00e7\u0131p gidersiniz? Bir tanem, gidemezsiniz, olmaz; basit\u00e7e, kesinlikle, olamaz b\u00f6yle bir \u015fey! Zaten bak\u0131n ya\u011fmur ya\u011f\u0131yor, siz de zay\u0131fs\u0131n\u0131z, \u00fc\u015f\u00fct\u00fcrs\u00fcn\u00fcz. Araban\u0131z su al\u0131r; kesinlikle su al\u0131r. \u015eehir kap\u0131s\u0131ndan \u00e7\u0131kt\u0131\u011f\u0131n\u0131z anda, k\u0131r\u0131l\u0131r; kasten k\u0131r\u0131l\u0131r. Zaten bu Petersburg'da rezil arabalar yaparlar! B\u00fct\u00fcn bu arabac\u0131lar\u0131 bilirim; hepsi fason \u00e7al\u0131\u015f\u0131r, oyuncak gibi s\u00fcslerler arabalar\u0131, ama dayan\u0131ks\u0131z yaparlar! Yemin ederim, kasten yaparlar! Can\u0131m, ben, Bay B\u0131kov'un \u00f6n\u00fcnde dizlerimin \u00fcst\u00fcne \u00e7\u00f6kerim; ona kan\u0131tlayaca\u011f\u0131m, her \u015feyi kan\u0131tlayaca\u011f\u0131m! Siz de can\u0131m, siz de kan\u0131tlay\u0131n; mant\u0131kl\u0131 bir \u015fekilde kan\u0131tlay\u0131n ona! Burada kalaca\u011f\u0131n\u0131z\u0131 ve gidemeyece\u011finizi s\u00f6yleyin!.. Ah, neden bu adam Moskova'daki t\u00fcccar kad\u0131nla evlenmedi? Gitsin orada onunla evlensin! T\u00fcccar kad\u0131n ona daha \u00e7ok yak\u0131\u015f\u0131r; bunu da \u00e7ok iyi biliyorum! Ben de sizi burada tutard\u0131m. Ondan size ne can\u0131m, B\u0131kov'dan size ne? Birdenbire nas\u0131l o sizin i\u00e7in de\u011ferli oldu? Belki de, siz size hep farbala al\u0131yor diye, belki de s\u0131rf bu y\u00fczden ilgi duydunuz! \u0130yi de nedir bu farbala? Ne i\u015fe yarar farbala? Tam bir sa\u00e7mal\u0131k bu, can\u0131m! Burada insan hayat\u0131ndan bahsediyoruz, ama can\u0131m, siz kalkm\u0131\u015f, \u0131v\u0131r z\u0131v\u0131ra tak\u0131l\u0131yorsunuz \u2013 farbala'ym\u0131\u015f; farbala budur, can\u0131m \u2013 \u0131v\u0131r z\u0131v\u0131rd\u0131r. Hem ben size bizzat, maa\u015f\u0131m\u0131 al\u0131r almaz, farbala al\u0131r\u0131m; size al\u0131r\u0131m onu, can\u0131m; o d\u00fckk\u00e2n ahbab\u0131md\u0131r zaten; maa\u015f\u0131 al\u0131ncaya kadar bekleyin beni, \u00e7ocuk mele\u011fim benim, Varenka! Ah, Tanr\u0131m, Tanr\u0131m! Ama siz kesinlikle bozk\u0131ra gidiyorsunuz Bay B\u0131kov'la, d\u00f6nmemek \u00fczere gidiyorsunuz! Ah, can\u0131m!.. Hay\u0131r, bana yine yaz\u0131n, bana yine mektupla yaz\u0131n her \u015feyi ve gitti\u011finiz zaman, oradan mektup yaz\u0131n. Yoksa, yoksa g\u00f6ksel mele\u011fim benim, bu en son mektup olacak; fakat asla olamaz, bu en son mektup olamaz. Yani siz, nas\u0131l olur, b\u00f6yle birdenbire, sahiden, kesinlikle en son mektup bu! Ama olamaz, ben yazaca\u011f\u0131m, siz de yazacaks\u0131n\u0131z... Benim \u00fcslubum da daha yeni \u015fekillenmi\u015fti... Ah, bir tanem, ne \u00fcslubu! Zaten art\u0131k ne yazd\u0131\u011f\u0131m\u0131 bile bilmiyorum i\u015fte, hi\u00e7 bilmiyorum, hi\u00e7bir \u015fey bilmiyorum, tekrar okumuyorum da, \u00fcsluba da \u00f6zen g\u00f6stermiyorum, yaz\u0131yorum s\u0131rf yazmak i\u00e7in, s\u0131rf size daha \u00e7ok yazmak i\u00e7in... G\u00fcvercinim benim, bir tanem, can\u0131ms\u0131n\u0131z benim!\n\n1844-1846\n\n26. Aptal \u0130van, Rus halk masallar\u0131n\u0131n, bizdeki Kelo\u011flan benzeri, saf bir gen\u00e7 kahraman\u0131d\u0131r.\n\n27. Petersburg'da 1825-1864 aras\u0131nda yay\u0131mlanan _Severnaya P\u00e7ola_ (Kuzey Ar\u0131s\u0131) adl\u0131 tutucu dergi kastediliyor. Gogol, _Bir Delinin G\u00fcncesi_ adl\u0131 kitab\u0131nda, bu gazeteyle \"klasik memur gazetesi\" diyerek alay etmi\u015fti.\n\n28. (Fr.) \u0130pek dantel.\n\n29. (Fr.) Kolsuz bluz.\n\n30. (Fr.) T\u0131\u011f.\n\n31. (Fr.) Gergef, kasnak.\n\n32. (Fr.) Kordone.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThank you for downloading this Simon & Schuster eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## Contents\n\nForeword by Debbie Reynolds and Carrie Fisher\n\nINTRODUCTION\n\nA GALLERY OF GOLDEN AGE MOMS\n\nMotherhood in the Movies by Sam Robards\n\nSILENT MOMS\n\nTHE MOTHER MARTYRS OF PRE-CODE\n\nTHE GREAT AMERICAN MOM\n\nOn Jane Darwell in The Grapes of Wrath by Eva Marie Saint\n\nSERIAL MOMS\n\nPERENNIAL MOMS\n\nOn Movie Motherhood by Jane Powell\n\nAUNTS\n\nMAMMIES AND NANNIES\n\nA MOM BY ANY OTHER NAME\n\nWHEN MOMS COLLIDE WITH THEIR KIDS\n\nBAD SEEDS\n\nMALEVOLENT MOMS\n\nCRIME AND HORROR MOMS\n\nOn Being a Mother and Actress by Tippi Hedren\n\nSCI-FI MOMS\n\nSHOWBIZ MOMS\n\nThe Greatest Mama of Them All by Illeana Douglas\n\nA GALLERY OF MODERN MOMS\n\nSEMPER MOM!\n\nCredits\n\nAbout Turner Classic Movies and Richard Corliss\n\n## Foreword by Debbie Reynolds and Carrie Fisher\n\nMovie mothers are a topic near and dear to me for many reasons. I've been a movie admirer since I was a little girl and I have always loved my mother. I had the immense honor of working with marvelous actresses who played my mother from age seventeen, until I portrayed a mother myself in the movie Mother. And of course, I am a mother of two wonderful children, one of whom is likewise an actress and a mother.\n\nI raised my children on the great old movies that always had delightful actors\u2014great character actors, and especially great mothers. These movie mothers made my children happy, as they have so many children. If you had a difficult childhood, movie mothers depicted understanding, aided you through those hard years and gave you hope. How many of us wanted our real mothers to be more like our reel mothers, the ones we grew up with in the movies? I will guess all of us!\n\nGreer Garson in Mrs. Miniver and Maureen O'Hara in How Green Was My Valley, Jane Darwell in The Grapes of Wrath and Margaret Wycherly in Sergeant York\u2014to name just a few. We loved them all because of their wisdom and understanding, which always saved the day (and, more importantly, the children).\n\nI was lucky when it came to both kinds of mothers, the real and the reel. My own mother instilled in me the desire to accomplish every project that I started and follow it through to its completion. Both of my parents were in agreement on that point. Also, always to follow \"The Golden Rule\" and be kind to everyone\u2014that was her major influence on shaping me as a person, which I, in turn, have imparted to my own children. Her ideas and teachings have followed me throughout my life. I am very grateful to both of my parents for their love and guidance, which have served me well in both my career and life. But it's also fun for anyone to imagine being hugged and completely understood by a movie mother, who happens to be a marvelous actress and so loved by millions\u2014and I had plenty of those experiences as well.\n\nWhen I moved to MGM Studios, I got to work with Bette Davis as my mother in The Catered Affair. She was fabulous\u2014powerful, yet helpful. Scary too, because she was always helping to teach you how to properly act, and she made every point a strong one. How I grew to adore her. She was a wonderful \"second mother.\" MGM had all the beautiful actresses under contract, including the supporting actresses who never played the glamorous mothers, but the more supportive, loving, funny mothers.\n\nMy next movie mother was a lady by the name of Una Merkel. She was the sweetest and most loving of all. We remained great friends for many years afterward. Later, Lilli Palmer played my mother in the movie The Pleasure of His Company, with Fred Astaire and Tab Hunter. See? Just like that I had three new mothers!\n\nOf course, I enjoyed watching movie mothers as much as I did working with them. My favorite was Spring Byington\u2014so cute and cuddly, always understanding when others were critical. In Presenting Lily Mars she, or rather, her role, let you believe you could realize your dreams. And remember Irene Dunne in I Remember Mama? Among my other favorites were Myrna Loy, Agnes Moorehead, Dorothy McGuire, Sophia Loren, Jane Wyatt, Donna Reed and Norma Shearer.\n\nThe movie studios, of course, made a point of distinguishing between reel mothers and real mothers. The Breen Office and the Production Code forced married couples into separate beds on the set (how did they think these women became mothers?), and films could rarely depict motherhood in any way that was nearly so messy as it is in real life. Movie mothers were neat, organized, energetic, and seemed always ready to be a perfect spouse and parent. They made most of us forget the areas in our life we weren't so happy with, for a movie moment.\n\n\u2014Debbie Reynolds\n\nThe first awareness I had of what a family was came from the television show Father Knows Best. Robert Young played the role of the father, Mr. Anderson\u2014the man who presumably knew best. Jane Wyatt portrayed the iconic 1960s television mother, always in the kitchen wearing her apron and making dinner, beaming proudly over her beloved brood. Somehow my memories of these characters are far more vivid than a lot of my recollections of my own relatives. The TV father would come home each evening dressed in a suit and tie, and pull his daughter onto his lap, giving her a hug and a kiss. He lovingly referred to his daughter as \"Princess,\" a title that would become synonymous with me, but more on that another time. Their Princess, their kitten, their baby girl would grow up to be just like her mother but do more charity work and play tennis. She would never be too big to be Mommy and Daddy's favorite little girl.\n\nI dreamt of having a family like this. Never mind that I had a beautiful mother all my own at home, who beamed proudly at my brother and me. My mother was nothing like Margaret Anderson, the mother character on Father Knows Best. She didn't wear an apron or serve homemade meals\u2014well, until recently, but there is no apron involved when she makes dinner for her adored dog, Dwight. Not only that, I didn't have a father who knew best; he refused to hoist me onto the safe harbor of his lap and make it all better than okay. I found out later that many of my schoolmates lacked their own Father Knows Best families\u2014we all lived in this media-made world where everyone grinned and lived happily ever after every day. Is it any wonder we all grew up to be serial killers or failed anorexics (if we were lucky) before the age of thirty-five?\n\nIt must have conveniently slipped my rebellious adolescent mind that I had my very own media mother at home. I really didn't need Margaret Anderson. I mean, let's face it, Margaret wasn't prettier than my mom, and both were loving and kind. What Margaret was that Debbie wasn't was edited. In addition, Margaret could cook, whereas my mother had a cook. My mother could spell apron; Mrs. Anderson just wore one. But kindness and aprons aside, I think the biggest difference is the editing. Editing rescues us from the tedium and upset of everyday life. I don't mean to suggest that my mother is boring and upsetting\u2014beyond quite the opposite\u2014but let's face it, everyone shares that potential. I know I do. Ask around.\n\nParents are rarely strangers, although my father was frequently the exception to that rule. It was Father Knows Best that taught me that my relationship with my male parent was many miles from ideal. So, media can teach us something, the welcome lesson of what we have versus the punishing ideal that whispers what we lack\u2014what we endure, as opposed to what we desire. The odd thing was, because my mother, for the most part, could be found wandering through \"Ideal Land,\" I was envied, as most kids my age only knew the edited, filmed version of Debbie Reynolds. I envied others, as I assumed they lived with the beaming, apron-clad Mrs. Anderson.\n\nAs far as I know, I've only played mothers a few times in my nuanced career as an actress. I truly wasn't certain of this fact until I looked myself up on IMDb, something I have to do these days in order to reminisce. The difficulty is that I try not to get too far into character when acting, as there's a chance I might be unable to easily extract myself later. Unfortunately, I can only say with absolute certainty that I was anyone's parent when I played Carol Peterson. My husband and I had a nine-year-old son named Dave. (I thought I had had two children, though according to IMDb, I had one. So you see, as the saying goes, you never know for certain how many children you have until you verify it online.) So, judging from my casting record, I wasn't really the maternal type. Which might explain why I only had one child in Actual Unedited Life. A fine example of life imitating art (or thereabouts), depending on whether or not you consider the movie The 'Burbs art, in which I played a Mrs. Anderson-type mother as Tom Hanks's wife.\n\nAccording to my source at IMDb, I was in something called From Here to Maternity. Inspired by one of Anton Chekhov's early short stories (I'm lying), I'm assuming the piece related to women waiting to welcome their little stranger; so, though I wasn't a bona fide parent \u00e0 la The 'Burbs, my guess is that I played a mother-to-be (that, or a gynecologist). Though I discovered it on IMDb as a credit of mine, the main things I found out were that it was a short and Richard Simmons was also in it, hopefully playing a pregnant person himself.\n\nIt turns out, not only have I not played a host of screen moms, I haven't had a lot of screen mothers. But then, how many do you really need? Lee Grant was my mother in Shampoo, though we didn't have any scenes together. Anne Bancroft played my mother-in-law in Garbo Talks, and last but far, far away from least, Natalie Portman portrayed the mother-to-be of my Star Wars character Princess Leia (alluded to earlier) three times.\n\nBut, though this was a fine assortment of moms, I must've felt cheated in some way, because once I began writing, the very first screenplay I wrote, Postcards from the Edge, included the mother of all mothers. I'm not bragging or anything. I'm just stating a fact. If you saw it, you'd know what I mean. It was the story of a mother and daughter. A mother and daughter in show business. As I recall, a certain amount of the dialogue in Postcards was actually taken from my relationship with my mother, from our Actual Unedited Life. Go figure.\n\nAll my life I'd watched mothers and daughters in films and either wished that my mother and I were at least a little more like them, as their characters trended toward the ideal, or was thrilled that we were as unlike them as we were.\n\nObviously, the relationships that appealed to me most could be found in black and white films (both speaking and silent), but they were also in Technicolor films and occasionally even modern-day color films\u2014of the sentimental and\/or old-fashioned variety.\n\nOf course, the film relationships that made me feel even more grateful in the extreme for having Debbie Reynolds as a parent were in films such as Mommie Dearest, Carrie, and, of course, the classic mother-daughter tribute, Nice Girls Don't Explode, to name but a few.\n\nEither way\u2014whether you want to feel beat up and\/or inspired by the seemingly ideal, filmed relationship, or thrilled and relieved by those cinematic torture-filled, murderous relationships\u2014it's always interesting to shake out the cobwebs of your Actual Familial Female Intertwining (AFFI). It's never boring to check out someone else's idea of family life and see where you fall on the chaotic scale of art imitating life. See if by treating yourself to a mother-daughter film you can maybe even tip that scale, change your place on it or revel in the wonder of where you fell. Take leave of your Actual Unedited Life for a few hours and be anointed in entertainments of every style and type. You can never go wrong, even if you have to go wrong in a self-righteous way. My plan is to do a documentary of my mother and my life. That way we can document our actual life and pop in the editing later on\u2014a kind of combination of real life and life of the artistic variety. Call it Wash Out That Grey Gardens! or The Over Fifty Shades of Grey Gardens.\n\n\u2014Carrie Fisher\n\n## INTRODUCTION\n\n* * *\n\n## Where Have All the Mothers Gone?\n\nI remember Mama. We called her Mom or Mother. Elizabeth McCluskey Corliss lived her entire long life in Philadelphia. For forty-one years Elizabeth taught first grade in the city's public school system; for thirty years she was married to the love of her life, my father, Paul, until his death in 1963; and for many more years she raised\u2014uplifted\u2014my brother, Paul Jr., and me, teaching us less by command than by example, by her easy charm and uncommon sense. I hardly exaggerate when I say that everyone who knew Elizabeth adored her, and that she repaid that affection a thousandfold. She loved her siblings, her sons, her grandchildren, her whole extended family and the friends she had accumulated over a gracious lifetime.\n\nHer life, which lasted one hundred years, two months and twenty-two days\u2014exactly the span of Gloria Stuart, the actress who played the older Rose in James Cameron's Titanic\u2014wasn't all good luck: Within a year and a half in her mid-fifties she lost her husband and a breast to cancer. Her dear sister Margaret, who lived with us, died of cancer a few years later. But nothing dimmed Elizabeth's dimpled optimism, her belief in the decency of the people she knew. I knew most of them, and I agree they were good. Just not as wonderful as my mom.\n\nGrowing up in the 1950s, I imagined that my mother was the inspiration for television moms played by Donna Reed and Harriet Nelson, and that cheerful family programs like Father Knows Best were documentaries filmed with hidden cameras in our neighborhood. TV shows of that decade did what my parents did: teach, comfort and reassure. But who would make a movie of Elizabeth Corliss's life? In fifties films, I saw few women who reminded me of my mom. James Dean, who I thought nearly matched me in faunlike sensitivity, had a clueless mother in Rebel Without a Cause and a brothel-owning mother in East of Eden. A slew of serious films sprang from novels and plays that ripped the Band-Aid off generational wounds. No movie family convened as we did each evening in the living room, eating tuna salad on table trays and watching the evening news on a twelve-inch Philco set. The Corliss home life was surely worth living, I'll testify to that, but Hollywood might say it was not worth filming.\n\nI sing the movie mother, proud and strong. Also warm and gentle and, on occasion, misguided or downright nasty. Motherhood has embraced multitudes across the history of the medium, from the earliest peep shows\u2014when that regal mom Mary Queen of Scots was shown being beheaded in an 1895 trick film\u2014through the 1930s-'40s golden age of both Hollywood and mothers, and up to the most modern films, when . . . Wait a minute, what happened? Where have all the mothers gone?\n\nThey should still be around, because they always have been. Mothers have driven dramatic literature from Euripides's Medea in 431 B.C. to Tyler Perry's Madea today. Ever since the Greeks, the theater has honored the compact between what any audience has experienced\u2014the joys and tensions of the maternal bond\u2014and what it can see onstage. Shakespeare knew that the actions of mothers could stoke classic tragedy: If Gertrude hadn't married her late husband's brother, her son, Hamlet, would have been just another morbid teen instead of a palace insurrectionist. In some of the great plays of the mid-twentieth century\u2014Death of a Salesman. Long Day's Journey into Night, A Raisin in the Sun and Who's Afraid of Virginia Woolf?\u2014women address their own or their husbands' infirmities and the urges of their children, real or imagined. In the last decade, Perry's touring plays about the sassy Mabel \"Madea\" Simpson, a character he says is based on his own mother and aunt, may not win drama awards, but they have extended into the twenty-first century the theater's fascination with mother love.\n\nAmerican television, for its entire lifespan, has teemed with mothers, from Lucy Ricardo and June Cleaver to Marge Simpson and Breaking Bad's Skyler White. All have served as anchors of love, loyalty and common sense. Mostly, they serve at home, where the hearth is, and the TV set. The media room is near the kitchen, traditionally portrayed as Mom's domain, and the typical small-screen drama\u2014a weekly habit, not an event\u2014affirms her time-honored values: nurturing, reassurance, continuity, commitment. Learning and hugging, as the old TV rule goes. Mother knows best.\n\nIn films of the last half century, though, mothers have nearly become an endangered species. We grant that a modern superhero may have a mom: Thor (Chris Hemsworth) reveres his Fridda (Ren\u00e9 Russo); and in Man of Steel, Superman (Henry Cavill) has a birth mother (Ayelet Zurer) and an Earth mother (Diane Lane), both of them selflessly devoted to their boy. A villain, like the drug-cartel prince played by Ryan Gosling in Only God Forgives, may also have a mother: the cartel queen Kristin Scott Thomas, with a vicious mouth and a malicious will. But mainstream Hollywood has essentially consigned mother stories to foreign films (the Korean Mother and Pieta, for example) and to indie movies of the Sundance stripe (Precious, The Kids Are All Right), which have inherited the domestic-drama genre that long sustained American cinema. In real life, mothers far outnumber superheroes or serial killers in this country\u2014but not on this country's multiplex screens.\n\nWhy is the mother movie near extinction? Let us suggest a few reasons.\n\n1. Movies are about movement. People watch TV but go to a movie. The big screen doesn't duplicate the cozy home viewing of TV; it's a giant wall for communal dreams. Having got out of the house, filmgoers presumably wanted to see the characters on that wall do the same: break out of the status quo to court danger and achieve physical triumph. The convention of womanly acceptance gave way to the depiction of manly quests. Unlike the theater and TV, media that from their beginnings embraced the power of the voice, cinema began in silence; it was all show, no talk. On a movie set, a crew member shouts, \"Lights, camera, action!\"\u2014not \"Lights, camera, chatting.\"\n\nWhen sound films replaced silents in the late 1920s, they did become more like theater: filmed plays, in which people talked out their conflicts, often brilliantly. These were sophisticated wars of words, in an arena where women could battle men on equal terms, and win. But most of today's big movies are all show, little tell; they offer world-threatening scenarios resolved by rockets, space suits, muscles\u2014in a word, manpower. Hollywood replaced weighty subjects with heavy lifting. Superheroes may have mothers, but they aren't mothers.\n\n2. The role of women has evolved. For millennia, mothers have superbly performed crucial tasks\u2014keeping the brood together, cooking and cleaning, instilling the values of civility and civilization\u2014that were not seen as intrinsically cinematic; they were routine duties, drudge work, homework. In the last half century, the number of women in the workforce has risen sharply until it is now close to the percentage of employed men. That change should have cued a new era of movies about working women, including working mothers. But the demographic trend coincided with the ebbing of movie realism and the blooming of fantasy; we're talking about you, Star Wars. The mammoth success of that sci-fi epic, surely the single most influential movie of the past forty years, certified Hollywood's flight away from women (and mothers) and toward heroic sons (with father issues).\n\nThe increased economic power of women meant only that they had more money to spend on movies about science-fiction and pulp-fiction men. If women wanted to see the drama and fun of a working mother's life in fictional form, they watched TV. In the rare cases when a woman is at the center of some big action picture, she is usually the solitary warrior, as unencumbered by children as any Marvel man. Katniss Everdeen couldn't win the Hunger Games if she had to go home at sundown to cook the kids' dinner.\n\n3. The audience has evolved. For generations, when moviegoing was a family activity, Hollywood designed its films to appeal to all and offend few. In the half century of feature filmmaking before the industry introduced its rating system in 1968, anyone of any age could see any movie; that fact imposed a self-censorship on filmmakers and a modicum of good behavior on film content. Even in the early years of G-, PG-, R- and X-rated movies, Hollywood believed that, when a dating couple was considering what to see at the Bijou, the woman usually made the choice. Today, that rule doesn't apply so much. Movies are still a relatively cheap date, but women tend to go in groups to see a women's movie (Sex and the City, Mamma Mia!) and men go in packs to see guy movies (everything else). Since men, young men, are the crucial demographic in a hit franchise, Hollywood produces stories to appeal to their fantasy self-image.\n\nAnd that's just in North America. In the rest of the world\u2014a huge market that accounts for more than half of the business for pictures produced by the big studios\u2014audiences don't care about the niceties of dialogue, especially as spoken by women with children. The global box office seems to validate this gender prejudice. Of the seventeen films that had earned more than a billion dollars at the worldwide box office through the end of 2013, the top two\u2014James Cameron's Avatar and Titanic\u2014were, yes, love stories of the Romeo-and-Juliet stripe, on the Atlantic or in outer space, but the heroine's mother in Titanic was the merest marplot. And Titanic, unlike Star Wars, was seen as a fluke, not a blueprint for a new spate of fateful romances. In the other billion-dollar movies, mothers were usually poignantly distant memories (as with Harry Potter's late mum) or nonexistent. Only the James Bond film Skyfall boasted a significant maternal figure: Judi Dench's M, the stern surrogate mom to both 007 and the movie's villain, Silva. Otherwise, it's all superheroes, pirates, wizards and Mad Hatters.\n\nThat list is tabulated by dollars, which have inflated as ticket prices soared over the decades. But the true measure of a film's success is tickets sold, and by this standard, five of the all-time top ten films contain significant maternal figures. In tenth place, Snow White and the Seven Dwarfs: a wicked stepmother, scheming to kill a girl who learns to mother seven small bachelors. At nine, The Exorcist: a single mother battles her child's possession by demons. At eight, Doctor Zhivago: the search for the lost child of the good doctor and his vibrant love Lara. At three, The Sound of Music: the nun postulant who becomes the wonderful, singing stepmother of seven starchy Austrian kids. And at number one, in what is still the most popular of all films, Gone With the Wind, with three prime maternal roles: Scarlett, the willful belle whose motherhood is creased by tragedy; Melanie, the saintly one, whose baby Scarlett midwifes; and Mammy, the house slave and Earth Mother of them all.\n\nTurner Classic Movies knows the importance of GWTW. It was the first film shown on the network, as well as on its predecessor, TNT, when that channel was the flagship of the Ted Turner's MGM-RKO-Warner Bros. library. Moms still live in old movies, especially on TCM\u2014which, if it shows a series of Irene Dunne or Fay Bainter films, can seem like Turner Classic Mothers. To look at the movies of Hollywood's golden age and beyond is to chart both the progression of women's power and the devolution of motherhood as the defining aspect of womanhood. Recalling the movies' saintly or sinful women, we hope to bring them back to life in all their heroism and travail, their wit and their passion.\n\nThis book will trace the course of movie mothers from their birth in silent films, through their flourishing in the golden age and into the thinning of the herd in later decades. These pages are graced by good moms and bad, stepmothers and surrogate mothers, maternal figures in crime movies, horror films and sci-fi epics. We don't guarantee that all your favorite movie mothers are included, but we offer more than a hundred for your edification and fond remembrance. You will revisit old friends; you may meet new ones. Where have the movie mothers gone? Here.\n\nA word of warning: Any mother's story spans her whole life, and we sometimes tell what happens at the end, without the black flag of a SPOILER ALERT. But the appeal of these stories is less in their resolutions than in the details that give mood and meaning to any film. Mothers may die, but they'll never die out\u2014not on TCM and not in this book, where we remember Mamas: mine, yours and the movies'.\n\nThe Human Comedy, 1943: Widowed mother Fay Bainter with her children Donna Reed and Mickey Rooney and family friend Marsha Hunt.\n\n## A GALLERY OF GOLDEN AGE MOMS\n\n* * *\n\n## These actresses played women who were heart strong and hearth warm.\n\nGolden Age Hollywood, which reveled in good women passing their finest qualities on to their young charges, boasted an informal stock company of actresses suitable for all manner of dear ladies; in memory they blend into a composite portrait of classic movie moms. Finding Fay Bainter, Beulah Bondi or Anne Revere in a film, audiences of the 1930s and '40s were instantly assured that maternal warmth awaited. At a higher level of marquee glamour, Myrna Loy and Irene Dunne proved ideal movie mothers: poised and caring, with a salutary leavening of worldly humor.\n\n### FAY BAINTER\n\nOnstage from childhood, she trod the boards for thirty-five years before getting her first movie role, at forty, as Lionel Barrymore's wife and the mother of two grown children in the 1934 film This Side of Heaven. Bainter executed a diversionary maneuver as Bondi's impatient daughter-in-law in 1937's Make Way for Tomorrow (see the chapter \"Bad Seeds\") before solidifying her primacy as Hollywood's most motherly mother, with large eyes that cooed with empathy and the fret lines that were a concerned mom's war scars. She nurtured William Holden in Our Town (1940); twice served as Mickey Rooney's mother, in Young Tom Edison (1940) and The Human Comedy (1943); and played an Iowa farm woman shepherding her children Jeanne Crain and Dick Haymes in Rodgers and Hammerstein's State Fair (1945). As if testing the tensile strength of Bainter's characters, the studios often multiplied her children and amped up her woes. Warners transferred the Lane sisters and Gale Page from Claude Rains's care in Four Daughters to Bainter's in Daughters Courageous (1939). And in Mrs. Wiggs of the Cabbage Patch (1942) she is a shantytown mother with more kids than Ma Kettle and as many domestic challenges as Ma Joad. And she still managed to play it as comedy.\n\nThe essential Bainter films are two from 1938. Mother Carey's Chickens casts her as a Rhode Island \"hen\" who sustains her \"chicks\"\u2014primarily, daughters Anne Shirley and Ruby Keeler\u2014after the death of her sea captain husband. She imparts a mother's lore about first love to Shirley and scares away prospective buyers of the family's overmortgaged home by claiming the place is haunted. Offered the chance to live with their rich aunt, the daughters naturally stick with Mom. Anyone would\u2014anyone who had seen White Banners, in which Bainter is Hannah, a woman of mystery who brightens the Indiana lives of Paul and Marcia Ward (Rains and Kay Johnson) and their children with her good cooking, spot-on housekeeping and inspired hints for the inventions that Paul hopes to patent. Being a more genial Mary Poppins is only incidental to Hannah's secret mission: monitoring the progress of the Wards' neighbor boy Peter (Jackie Cooper), the child she had out of wedlock and was forced to abandon.\n\nWhite Banners, 1938: Fay Bainter intercedes between Claude Rains and the \"neighbor boy\" Jackie Cooper.\nThe performance earned Bainter a Best Actress Oscar nomination in the same year she won for Best Supporting Actress in a more worldly role as Bette Davis's aunt in Jezebel. Davis took Best Actress, and posterity wouldn't take it from her. But Bainter's work in White Banners deserves some award, for bringing shivering heartache and beneficence to Hannah, the constant mother and secular saint.\n\n### BEULAH BONDI\n\nIf Fay Bainter's face was a velvet pillow that welcomed a child's cuddles and cries, Beulah Bondi's was sharp, beaky, with small eyes and a thin mouth. Yet her features could crinkle with a memory of passion as Victor Moore's wife on their fiftieth wedding anniversary in Make Way for Tomorrow, or with rabid joy at the sight of a dear son, like Fred MacMurray in Remember the Night (1940). In that Preston Sturges script, directed by Mitchell Leisen, MacMurray is a Manhattan DA who has brought a shoplifting sharpie (Barbara Stanwyck) to his Indiana home for Christmas.\n\nLike Bainter, Bondi came to movies in her forties, a ready-made mom. (In her 1931 film debut, Street Scene, she reprised her role as a tenement gossip in Elmer Rice's Broadway hit.) Often playing women far older than she, Bondi certified her golden age Mom status by serving as James Stewart's mother in four films, including three released within twenty months. In Of Human Hearts (1938) she is a minister's pioneer wife, Mary, selling precious silver spoons so her son Jason (Gene Reynolds as a boy) can read Harper's Monthly, and, a decade later, selling the family horse to buy a uniform for the grown Jason (Stewart) to wear as a Civil War doctor; even Abraham Lincoln intercedes to return a young man to his mother. Bondi and Stewart were granted a comic respite in Vivacious Lady (1938), with the mother a friendly negotiator in the war between Stewart's jazzy bride Ginger Rogers and his stuffy father (Charles Coburn). The following year Bondi shone in the small, pivotal role of Ma Smith, the source of Senator Jimmy's humane grit in Mr. Smith Goes to Washington.\n\nRemember the Night, 1940: Fred MacMurray with his best girls, mother Beulah Bondi and aunt Elizabeth Patterson.\n\nIt's a Wonderful Life, 1946: James Stewart with Beulah Bondi as Ma Bailey and Samuel S. Hinds as his father, Peter.\nBondi's last work with Stewart was her finest: spanning a quarter century as Ma Bailey to his George in Frank Capra's It's a Wonderful Life (1946). For most of the film she is the perfect mom, kissing George and saying, \"That's for nothing,\" informing him of the crush his future wife Mary (Donna Reed) has on him and, when the Bailey Building & Loan fails, praying for the restoration of her boy's emotional equilibrium. Toward the end, in George's glimpse of Bedford Falls as the noir-Dickensian Pottersville, Bondi metamorphoses into the crone figure, suspicious and bitter, telling the son she never had that he should be in a lunatic asylum. A sepulchral vision of a town missing one small, crucial soul, this section of It's a Wonderful Life also suggests that the first requisite of a wonderful mother is the child into whom she can pour her love.\n\n### ANNE REVERE\n\nBefore Beulah Bondi got the part of Ma Bailey, Frank Capra had considered casting Anne Revere. But the actress, a direct descendant of Revolutionary War hero Paul Revere, was never short of mother roles\u2014until she lost a later revolutionary war, when Hollywood blacklisted her in 1951. Some called her \"Anne Severe\" for her strong features worth chiseling on Mount Rushmore and for a glance that could penetrate platinum. And she played her share of forbidding biddies: the widow in Sunday Dinner for a Soldier, and Alice Faye's older sister in Fallen Angel (both 1945). Robert Rossen's Body and Soul (1947) casts her as Anna Davis, the proud, defiant mother of John Garfield's Charley, who, after his father dies, turns to boxing to support his urban-poor family. Abraham Polonsky's script sizzles with elemental mother-son exchanges. \"I want money, money, MONEY!\" Charlie tells Anna. And when she spits out, \"Better you go buy a gun and shoot yourself,\" he snaps back, \"You need money to buy a gun!\"\n\nNational Velvet, 1943: Anne Revere encourages her horse-loving daughter Elizabeth Taylor.\n\nShe mothered a saint\u2014Bernadette of Lourdes (Jennifer Jones) in 1943's The Song of Bernadette\u2014and an angel: Elizabeth Taylor, then twelve, in National Velvet (1944). As Mrs. Brown, she teaches Velvet a species of Hollywood moral relativism (\"What's the meaning of goodness if there isn't a little badness to overcome?\") and the value of pursuing unlikely goals. \"We're alike,\" she tells her daughter. \"I, too, believe that everyone should have a chance at a breathtaking piece of folly once in his life. . . . You're twelve; you think a horse of yours can win the Grand National. Your dream has come early; but remember, Velvet, it will have to last you all the rest of your life.\" And as the mother of Gregory Peck, crusading journalist, in Gentleman's Agreement (1947), she both encourages her son to expose anti-Semitism and gently chides him for his own moderate sexism. All three of the roles earned Revere Oscar nominations for Best Supporting Actress. She won for National Velvet.\n\nGentleman's Agreement, 1947: Anne Revere counsels her son, Gregory Peck, on the stain of ethnic bigotry.\nAs Montgomery Clift's mother in A Place in the Sun (1951), her last film role before being blacklisted, Revere cedes pride of place to Taylor, now a ravishing eighteen-year-old, as the posh girl who purrs to her new beau, \"Tell Mama. Tell Mama more.\"\n\n### MYRNA LOY\n\nOne of Hollywood's minor mysteries is why producers thought that Montana's very Anglo Myrna Williams\u2014screen name, Myrna Loy\u2014was suitable for roles of exotic mischief early in her career. In 1932 she played a malevolent Eurasian with hypnotic powers in Thirteen Women and Boris Karloff's sadistic daughter (\"The whip! The whip!\") in The Mask of Fu Manchu. Lucky for Loy, MGM's Irving Thalberg saw the affable modernity in the actress's screen presence; he teamed her with William Powell and Clark Gable in 1934's Manhattan Melodrama. She proved the perfect complement to each of these strong men, making seven films in all with Gable, fourteen with Powell, including the six Thin Man movies that stand today as the definitively breezy fantasy of modern marriage (see the \"Serial Moms\" chapter). Rarely top-billed, Loy shone as the wise mate with a bubbling intelligence and a high emotional IQ. In the words of the Cole Porter song that Richard Schickel used for the title of his TCM Loy tribute film in 1991, she was \"so nice to come home to.\"\n\nAs Milly Stephenson in William Wyler's The Best Years of Our Lives (1946), she is the wife whom Fredric March's Al comes home to\u2014beautifully visualized in the embrace of a reluctant warrior and the woman who has shared his ordeal from a continent away. During World War II, Loy had virtually retired from the screen to work for the Red Cross and other relief agencies. So the film, written by Robert E. Sherwood from MacKinley Kantor's verse novel, was a homecoming for Loy as well. Now forty-one, and having segued into mother roles, Loy is not different, simply more mature; Milly must counsel her daughter Peggy (Teresa Wright), who loves another veteran, the unhappily married Fred Derry (Dana Andrews). When Peggy, believing herself in a unique romantic dilemma, tells her parents they never had any trouble, Milly turns to Al and snaps back: \" 'We never had any trouble!' How many times have I told you I hated you, and believed it in my heart? How many times have you said you were sick and tired of me\u2014that we were all washed up? How many times have we had to fall in love all over again?\" She is describing a married life that Peggy never knew, and that audiences of that time had rarely seen in the movies.\n\nA year later, in The Bachelor and the Bobby Soxer, Loy played Shirley Temple's older sister\u2014older by twenty-three years, in real life, so it may as well have been a mother role. In this comedy about a teenager with a crush on a playboy, Loy winds up with Cary Grant, who would be her husband in Mr. Blandings Builds His Dream House (1948). Like The Egg and I, which spawned the Ma and Pa Kettle series, Blandings is a warning against city folk's dreams of a country home. For a change, Loy's Muriel Blandings is the source of exasperation rather than wifely wisdom. She drives husband Jim and the renovators daft with minute instructions on the color scheme (\"Not just yellow\u2014a very gay yellow, something bright and sunshiny\"). And she expresses her concern for her two teenage daughters by insisting, \"I refuse to endanger the lives of my children in a house with less than four bathrooms!\"\n\nThe more familiar Loy\u2014the lovely, flinty American mom\u2014returned in The Red Pony (1949), John Steinbeck's story of a Salinas County ranch couple (Loy and Shepperd Strudwick) and their young son (Peter Miles) who yearns for, and finally gets, his pony. Loy's genial unflappability was tested\u2014and she aced the test\u2014as Lillian Gilbreth in Cheaper by the Dozen (1950). Like Meet Me in St. Louis, this movie was based on a memoir of a happy family and was set in the same house that MGM built for the Judy Garland film. (In a TV version of Meet Me in St. Louis, in 1959, Loy took the Mary Astor role, with Walter Pidgeon as her husband and Jeanne Crain, Jane Powell and Patty Duke as their daughters.)\n\nMr. Blandings Builds His Dream House, 1948: Myrna Loy leads husband Cary Grant and daughters Connie Marshall and Sharyn Moffett to their new country estate.\nCheaper by the Dozen\u2014which shares only its title with the two Steve Martin comedies of 2003 and 2005\u2014mines its humor from the efforts of Lillian's husband Frank (Clifton Webb) to apply his theories of time-and-motion efficiency to his large brood. Lillian was Frank's partner in these studies, and continued his work after his death (shown in the sequel Belles on Their Toes, 1952). Given Myrna's light touch in the film, she is also the smiling referee between her husband's conservative edicts of dress and behavior and her children's pleas for access to the twentieth century. Lillian manages to assuage both factions, the way Loy so often found a golden mean in her film roles. Her chiding felt like a caress; her voice suggested common sense after a nice bath. She was the wife as good pal, the mother as best friend, the home any moviegoer would be grateful to share.\n\n### IRENE DUNNE\n\nShe could play lover, friend or mother, in cutting comedy or ennobling weepie. Her drawling smile semaphored a you've-got-to-be-kidding undercut to her dashing leading men. Earning renown in 1920s musicals\u2014including the Jerome Kern-Oscar Hammerstein II-Edna Ferber classic Show Boat\u2014Dunne entered films in her early thirties and quickly hit the mother lode. She earned her first of five Best Actress Oscar nominations as Sabra Cravat in Cimarron, the 1931 Best Picture winner taken from another of Ferber's multigenerational novels.\n\nJoining her journalist husband Yancey (Richard Dix) in the Oklahoma territory in 1889, Sabra outlasts his male pride (\"You shouldn't interfere when men are having a little friendly shootin' \"), edits his newspaper under his name, raises a daughter (Nancy Dover) and eventually wins public office. In effect, she drags or lifts Oklahoma and feminism into the twentieth century. Yancey shows his pride and gratitude with this florid testament: \"Wife and mother, stainless woman, hide me, hide me in your love.\" That might have been the epigram for The Secret of Madame Blanche (1933), a Madame X-ish story of a good woman wronged and persevering to raise her son (Douglas Walton) to become less like his vanished father and more like her. Such was the march of civility in women's films of the 1930s, or any Hollywood decade that took mother love seriously.\n\nDunne starred in three Kern movie musicals: Roberta (1935), the 1936 Show Boat remake\u2014in which she plays Magnolia, the showstopper and single mom\u2014and High, Wide and Handsome (1937). She also put her lilting laugh to sparkling use in such comedies as Theodora Goes Wild (1936), The Awful Truth (1937) and My Favorite Wife (1940), all of which come close to defining screwball comedy at its most raucous and refined. Comedy is a genre more concerned with courtship than with motherhood. But occasionally a lighthearted tryst took an abrupt left turn toward catastrophe\u2014with Charles Boyer in Love Affair or Cary Grant in the all-time weepie Penny Serenade.\n\nDunne's Julie and Grant's Roger yearn for kids, but on a New Year's Eve in Tokyo, disaster strikes; as critic Donald Ritchie caustically put it, \"hundreds are destroyed in an earthquake so that Irene Dunne's miscarriage may be successfully accomplished.\" With Julie unable to bear children, the couple must adopt, and they find an infant that kindly Beulah Bondi says is \"like no other in the world.\" Bondi means that the girl, Trina, is unique because she is theirs. As the clueless Julie and Roger cope with raising a child, and the story continues to pirouette between slapstick and tragedy, Penny Serenade confronts issues that can drive a loving couple apart. Cementing Julie's resolve are vintage pop songs that remind her of episodes in Peter's and her life and that underline No\u00ebl Coward's tart, truthful observation in Private Lives: \"Extraordinary how potent cheap music is.\" Dunne's performance is equally potent, and not cheap but priceless.\n\nIn a movie career that lasted little more than twenty years, ending with the minor comedy It Grows on Trees in 1952, Dunne played good mothers with the same lithe determination she had brought to her portrayal of a feisty daughter-in-law in The Silver Cord (see the chapter \"Malevolent Moms\"). She was the American nurse in wartime London, hoping that her soldier son did not meet the fate of her World War I husband, in The White Cliffs of Dover (1944). She served as teacher and surrogate mom to the children of monarch Rex Harrison in Anna and the King of Siam (1946). She deflated William Powell's bluster in Life with Father (1947) and a year later incarnated the beatific Norwegian matriarch in I Remember Mama (see the chapter \"The Great American Mom\"). All three of these films were based on biographies of real women who held their families\u2014and, in Anna Leonowens's case, an Asian country\u2014together with the gifts of tact and stubbornness. But could any of them have radiated Irene Dunne's easy, enduring grace? And who could sing her babies a lovelier lullaby?\n\nPenny Serenade, 1941: Beulah Bondi makes Irene Dunne and Cary Grant the new parents of the infant Trina (Baby Biffle).\n\nThe White Cliffs of Dover, 1944: World War II nurse Irene Dunne tends to her soldier son, Peter Lawford.\n\n### Motherhood in the Movies\n\nGrowing up, the films of my mother's that made an impression on me at that time were mainly from the Warner Bros. years\u2014a time when the parts she was playing were about as far from motherhood as I could imagine. As a result, I simply couldn't identify the person whom I called \"Mom\" as the same person I saw on-screen. And because the sixties and seventies were such a time of change in the business of movies, yet still very much dominated by men on-screen, I was confronted by many more screen fathers than mothers.\n\nIt was the discovery of Beulah Bondi in It's a Wonderful Life (1946) that helped me to understand more fully these difficult constraints placed on movie mothers. While she was the very thread of motherhood that ran through the thirties and forties, what struck me about her character in It's a Wonderful Life was not her time onscreen as a mom\u2014despite her success as an actor at injecting humanity into a stock \"mom\" character\u2014but rather her appearance in George Bailey's nightmare Bedford Falls, had he not been born. Mary Bailey as a spinster was closed, pinched, afraid. Unhappy in capital letters, without the joy of motherhood, we were told. Reinforcing this theme was Violet, played by Gloria Grahame, who was decidedly not a mother either in the actual Bedford Falls or in George's nightmarish vision. Her character's only option was, short of full emancipation, to be a fallen woman. While there is no denying Frank Capra's unparalleled ability to open up the inner recesses of an audience's emotions, this was a decidedly male point of view on motherhood, and the lack thereof.\n\nMy godmother, Katie Hepburn, a paragon of an independent woman as Eleanor of Aquitaine in The Lion in Winter (1968), jumped out at me as a newer type of mother. Sure of herself, with an unquenchable ambition not only of her own but for her favorite son as well. Yet a prisoner of her ambition, her upbringing and, quite literally, her husband the king. Her own woman, but again, only to a point\u2014ultimately defined by men.\n\nIt would be years before the art form would give us the two screen mothers who, for me, stood out the most: Dianne Wiest in Parenthood (1989) and Emma Thompson in Love Actually (2003) were actual, fully formed women. Wiest had recently played a character who was so emotionally needy, in Hannah and Her Sisters (1986), that it was jarring and thrilling to see her as a mother who was there for her kids. Not just there, but able and willing to talk to them on their terms and not afraid to not know the answers. Emma Thompson, whose performance gave Love Actually a real emotional foundation, was so completely present and fully committed to her marriage and children that I couldn't understand Alan Rickman's infidelity to her, despite his secretary's attributes. But I completely understood her willingness to remain in that marriage, doing the dirty work that men are sometimes unable to. That, it seems, is motherhood.\n\nAnd yet, screen parents carry just as much power in their absence as well. After seeing Cecil Kellaway's character in Harvey (1950) talk about his wish to return to the Akron of his youth, I was fascinated about his character's upbringing there, so golden and safe that he would willingly return. What created that warm memory? How did he grow up? Clearly in the arms of a loving mother and father.\n\n\u2014Sam Robards\n\nMadame Butterfly, 1915: America's Sweetheart, Mary Pickford, as a suffering young Japanese mother.\n\n## SILENT MOMS\n\n* * *\n\n## The cinema's first female stars brought a girlish poignancy to maternal love.\n\nImagine, at the dawn of the twentieth century, entering an arcade or music hall for the first glimpse of that new mechanical marvel: the motion picture. Or, in the 1920s, a visit to that true cathedral of commerce, the movie palace. In a lavish auditorium seating up to four thousand people, the lights would dim, and on a huge screen, magic would ensue. To a generation of Americans, home-bred or immigrant, silent movies provided the kind of iridescent spectacles that a mother gifted in storytelling might enthrall her child with at bedtime. Those theaters were the worldwide audience's womb, nursery, babysitter and mom.\n\nThe first American films, created in the 1890s by Thomas Edison's company to be shown as \"peep shows\" in arcades, were documentaries, lasting only a few seconds each, that showed the manly spectacles of blacksmiths hammering, athletes flexing their muscles, firemen at a blazing house, combatants in a boxing ring. Women appeared infrequently and sensationally in Edison's early pictures. The 1895 film The Execution of Mary, Queen of Scots employed trick photography to portray the beheading of the monarch (and mother of the future king of England); it may be the first mother movie and the first splatter film. In the notorious Fatima, Muscle Dancer, the skimpily dressed performer whirls and shimmies, and when the shaking of bosom and booty reaches its climax, two censorious rows of fence posts are painted on the image to obscure the action.\n\nIn 1896 Edison released its most popular picture of the decade, and one of its first to be viewed not through a peephole but on a large screen. A twenty-second excerpt from the Broadway play The Widow Jones, it featured two heavyset, middle-aged stage actors, May Irwin and John C. Rice, embracing, silently chatting and finally smooching. The movie, known as The Kiss, stoked a furor because of its intimacy: the two figures in medium close-up (instead of the standard long shot) engaged in a traditionally private moment. Here was the forerunner of every love story, romantic comedy and by extension\u2014if the kiss were to lead to kids\u2014motherhood drama.\n\nAs the form attracted larger audiences by expanding into music halls, then storefront nickelodeons and finally theaters across the land\u2014and as filmmakers like D. W. Griffith enlarged the cinematic vocabulary\u2014movies might have acknowledged that half of their constituency was female, and that 100 percent had mothers. Yet the masters of this new medium, invented after most of its customers had been born, sensed that they should offer dreams of escape: leaving home to discover the manifest destiny of a happy ending.\n\nThe early film entrepreneurs also realized that the old would rather watch the young than the young the old. The harsh lighting of the first motion picture cameras punished the faces of middle-aged stage stars (like Rice and Irwin) while idealizing the features of girleens and college-age men. Beauty was youth, youth beauty; that twist on Keats's adage is all the movies knew on earth, and all they needed to know.\n\n### GRIFFITH'S GIRLS\n\nThe first movie-bred performer to become famous was Florence Lawrence. Born Florence Bridgwood in Hamilton, Ontario, she was barely eighteen when she joined Biograph Studios, where D. W. Griffith was the house director. (For mother roles\u2014in nearly 200 shorts and features covering the entire span of his twenty-three-year film career\u2014he cast the flinty, saintly Kate Bruce.) Lawrence appeared with John R. Cumpson, then more than twice her age, in eleven \"Mr. and Mrs. Jones\" shorts, directed by Griffith, about the comic misadventures of a young married couple. In one film, The Peachbasket Hat, a Jones baby shows up, but only as the prop in a farcical confusion involving the couple's maid and her gypsy friends.\n\nAt the time, film actors were not publicly identified (neither were the directors or crew). Carl Laemmle would change that. He signed Lawrence to his Independent Moving Pictures Company (IMP) and promoted her into celebrity with an outrageously effective publicity stunt: spreading the rumor that she had died, then announcing, \"We nail the lie,\" and producing his new star in the flesh. Lawrence moved to two other studios but by 1916 had been supplanted by real actors, real stars: Lillian Gish and Mary Pickford, both discovered by Griffith while still in their teens. Lawrence sank into bit parts\u2014one of her last was a townsperson in the Barbara Stanwyck film So Big! (1932)\u2014and died a suicide in 1938 at age fifty-two.\n\nLaemmle had made Lawrence a famous name, but Mary Pickford virtually invented movie stardom. Tutored on Broadway and at Biograph, where she made 132 short films directed by Griffith between 1909 and 1912, Pickford was the first to kindle the wildfire of the film public's ardor; Charlie Chaplin, who would achieve even greater renown, made his mark slightly later. Fans mobbed Pickford; they fetishized her luxuriant curls; they bought massage creams and calendars with her face on them. And though she was known for playing cute or pathetic little girls, it was the five-foot pixie who first made the moguls pay huge sums for talent. \"No, I really cannot afford to work for only $10,000 a week,\" she coyly told Adolph Zukor of Famous Players in 1916, back when that was real money. Although she never took a director's credit, she supervised every aspect of her films. And in 1919, as the cofounder of United Artists with Griffith, Chaplin and her soon-to-be husband Douglas Fairbanks, Pickford showed the canniest business sense.\n\n\"America's Sweetheart,\" like her predecessor Florence Lawrence, was Canadian. Pickford was born in Toronto in 1892, as Gladys Smith, to a father who died when she was five and a stage mother, Charlotte, determined to raise her three children out of poverty. Mary and her siblings, Jack and Lottie, all became actors, and Charlotte remained Pickford's closest adviser. (Mary faced the moneymen without an agent or manager.) Hidden under the celebrated ringlets, Pickford's adding-machine brain calculated that the worldwide audience would pay to see moral tales of girls\u2014Stella Maris, Amarilly of Clothes-Line Alley, Pollyanna, Little Annie Rooney\u2014who grew up poor, as she had, colliding with the rich and ultimately instructing them that the lower class is not a lower species. Far from being lace valentines, many of her films exposed the crimes of orphanages run like penitentiaries and the tragedies of children allowed to die from inhumanity or neglect. Like Dickens, Pickford wed sentiment to social passion and created enduring popular art.\n\nFrozen in a simulated preadolescence for most of her silent-film career, \"Little Mary\" specialized in sweet, spunky teens and tweens. (Four of the milder literary properties she filmed when she was in her twenties\u2014The Poor Little Rich Girl, Rebecca of Sunnybrook Farm, The Little Princess and Daddy-Long-Legs\u2014were remade as talkies for the child star Shirley Temple.) Occasionally Pickford would wander from her prototype to play a young mother. She was not quite convincing as the Japanese heroine in the 1915 Madame Butterfly, but the film did honor the ending of the original story by having Cho-Cho-San kill herself when she realizes her Western lover will not stay with her and their infant.\n\nIn Little Lord Fauntleroy (1921), Pickford essayed two roles: Cedric Errol, the poor American boy who becomes an English aristocrat, and his beloved widowed mother. (One shot, a miracle of double exposure, shows Pickford the boy leaning in to kiss Pickford the mother.) And in Tess of the Storm Country, which she made in 1914 and again in 1922, the Pickford character is a poor squatter who selflessly claims as her own the illegitimate baby of the landowner's daughter. Refused milk for the dying infant, Tess sneaks into a church to perform an emergency baptism\u2014a surrogate mother's final brave gesture.\n\nPickford is again an impromptu mom in director William Beaudine's Sparrows (1926), perhaps the finest of her surviving films. On a baby farm, where cash-strapped parents have sent the children they can't afford to raise, Molly protects nearly a dozen younger unfortunates from the owner, an evil, limping hunchback named Grimes (Gustav von Seyffertitz). She is the only mother these waifs know\u2014also their cheerleader, their teacher and scoutmistress and, when they must straddle a rickety tree branch across an alligator-infested swamp, their Indiana Jones. Too young for sexual fervor, Molly invests all her energy in mothering her forlorn charges. Yet Grimes will not give her enough food to keep them healthy: one infant dies of malnutrition, and Molly dreams that Jesus has come to take him to heaven. Fate finally intervenes in the guise of a wealthy widower whose baby Grimes had kidnapped and Molly has saved. When the baby cries out for Molly, the widower lets her and the whole orphan gang stay in his mansion. \"You win, Mama Molly,\" he says. Another victory for Pickford, the mother of movie stardom.\n\nIf Pickford was the great female star of silent Hollywood, Lillian Gish was its greatest actress. Another child of a deserted mother who saw the theater as a way to support her family, Lillian moved in 1912 from Ohio to New York City, where Pickford introduced Lillian and her younger sister, Dorothy, to Griffith. Both Gish girls became popular performers, appearing together in Griffith's Orphans of the Storm (1921), but Dorothy was more the saucy soubrette, Lillian the elfin tragedian. Her wispy frame and doll face, dominated by soulful blue eyes, made Lillian ideal for demure young women who rise to heroism to battle life's disasters. She played the female lead in The Birth of a Nation (1915), though the men carried that Civil War story to its controversial climax in the founding of the Ku Klux Klan. In Griffith's Intolerance (1916), a film figure that interwove four stories across three millennia\u2014think Cloud Atlas, but more wildly ambitious\u2014Gish was the recurring figure of Eternal Motherhood, sitting next to a cradle, as Walt Whitman's lines from Leaves of Grass (\"Out of the cradle endlessly rocking . . . uniter of here and hereafter\") crept across the screen.\n\nSparrows, 1926: Mary Pickford as the impromptu mom to the abandoned children in a Southern \"baby farm.\"\nGriffith's Way Down East (1920) was famous for its climax: Richard Barthelmess running across the ice floes on a frozen river to save Gish (who spent so much time collapsed on the sheet of ice that her right hand never fully regained its feeling). But Gish is the real elemental sensation as the poor, saintly Anna, seduced into a fake marriage by the rich rou\u00e9 Lennox Sanderson (Lowell Sherman), who abandons her when he learns she is pregnant. Scorned as a fallen woman at the moment her own loving mother has died, Anna is left alone to tend both the ailing baby and her social shame. \"Maternity\u2014Woman's Gethsemane\" reads one of the intertitles that link Anna's plight with Jesus on the way to Calvary. As she realizes her infant son is dying, Anna pours a lifetime of devotion and desperation into her few moments of motherhood. Like Pickford's Tess, she baptizes the child (with the garishly ironic name Trust Lennox) and breathes on its cold hands in a futile fight to sustain its life. This is a performance of exquisitely balanced frenzy and subtlety; John Barrymore, the preeminent romantic classical actor of his day, called it \"the most superlative exquisite and poignantly enchanting thing I have ever seen in my life.\"\n\nIntolerance, 1916: Lillian Gish as the figure of Eternal Motherhood.\n\nGish stayed with Griffith longer than his other top stars, leaving in 1921 for the company that would become Metro-Goldwyn-Mayer, where she continued to suffer triumphantly in The White Sister, La Boh\u00e8me and The Wind. She was able to make The Scarlet Letter (1926) only after convincing balky Protestant groups that the Nathaniel Hawthorne novel, about a fiery Puritan minister who sires a child with a member of his congregation, would not give religious offense. Gish and Frances Marion, the eminent scenarist who wrote twenty Pickford films, solved or sidestepped this problem by focusing on Hester Prynne (Gish), not the Reverend Dimmesdale (Lars Hanson), and by making her another in the actress's long line of wronged women. Marked with a scarlet \"A\" for \"adulteress,\" cursed and besmirched by the sanctimonious locals, Hester is every bit as unjustly ostracized as Anna was in Way Down East.\n\nThis time, the love child, Pearl, gets a proper baptism\u2014from her father the reverend\u2014and grows into healthy girlhood. Eight years after Pearl's birth, Hester wants to flee to Europe with her daughter and Dimmesdale; she tears off her embroidered \"A\" and removes her bonnet to let her long hair flow free as a signifier of her sexual liberation. But the minister must pay for his \"sin.\" He denounces himself before the congregation, revealing an \"A\" branded on his chest. In one of many Piet\u00e0 images in Gish films, he dies in Hester's arms, asking, \"Is this not a better freedom than any we have dreamed of?\" Well, no, but, under Victor Sj\u00f6str\u00f6m's acute directorial hand, the ending emphasizes the silent-film law that a lover's survival is less important than a mother's.\n\nThe Scarlet Letter, 1926: Lillian Gish as Hester Prynne, with Joyce Coad as her daughter, Pearl.\n\n### SACRIFICING IN SILENCE, AND SACRIFICING IN THE SILENTS\n\nLess important, that is, unless the mother's death makes for better melodrama, as in that warhorse of self-sacrifice, Madame Butterfly. One version of that story, 1922's The Toll of the Sea, which Frances Marion adapted and Chester M. Franklin directed, was the first Technicolor feature shot in the U.S.\u2014and the rare American silent film with a non-Caucasian female lead. Anna May Wong, born in Los Angeles's Chinatown, would later costar with Douglas Fairbanks (The Thief of Bagdad) and Marlene Dietrich (Shanghai Express). She was just seventeen when she played the Asian child-woman in love with a white man.\n\nThe sea throws Allen Carver (Kenneth Harlan) onto the rocks near a Chinese village, where he is rescued and cared for by young Lotus Flower (Wong). He leaves her with child and returns four years later with a Caucasian wife. Lotus Flower sees her duty and hands the child over to his new family, after which she throws herself into the sea that brought her a two-edged gift. The climax is the standard act of noble renunciation\u2014bowing to convention while showing how it destroys decent people\u2014that the movies frequently imposed on heroic outsiders. Wong, who cries real, fat tears when Lotus Flower is heartbroken, shares her most intense scene with the wonderfully poised child performer known as Baby Moran. Taken from his mother and held by Carver's wife, the little boy touches Wong's face, leans in and, with great tenderness, kisses her twice. A good child always knows its rightful mother.\n\nSo often, for the virgin goddesses of the silent cinema, the purest emotion was not between woman and man but mother and child. Greta Garbo usually played women of the world whose attachment to any man could be a kiss of death for him or her. A goddess from a distant, cooler planet, the Garbo woman rarely bothered to marry, let alone bear a child\u2014because, in any screen affair, she was the far more mature one, a figurative mother to her callow courtiers. (Garbo's two films based on Anna Karenina, in which her Anna makes a rapturous connection with her young sons, are addressed in the \"Perennial Moms\" chapter.)\n\nImported from Sweden in 1925, after supporting roles in just three feature films, Garbo was a relative novice in movies. Yet she intuitively understood the unique power of silent-screen acting: how the fervid glance, the slumped shoulders or the gentle caressing of a child could express emotion as eloquently as any stirring oration. By 1926, with Pickford in Sparrows, Gish in The Scarlet Letter and Garbo at the beginning of her sixteen-year reign at MGM, Hollywood had refined the silent film into a unique and elevated art form. As Pickford dismissively said, \"Adding sound to movies would be like putting lipstick on the Venus de Milo.\" In fact, the Warner brothers were about to apply exactly that cosmetic\u2014giving rise to the talkies and, within three years, effectively slaughtering the silents.\n\nThe Toll of the Sea, 1922: Will Anna May Wong forfeit her child (Baby Moran) to his father's Caucasian wife (Beatrice Bentley)?\nThe birth of talking pictures was long in gestation. As early as 1889, Edison and Dickson had devised a machine that could record sound with film. When the medium reached its early maturity in the years after The Birth of a Nation, silent films were rarely silent. Indeed, every movie was also a musical: in smaller houses a pianist seated to one side of the screen would accompany the images, and in the movie palaces a full orchestra performed elaborate pastiche scores. By the mid-twenties, technology had advanced to permit sound to be recorded on film (the system that would be used for the first eighty years of talkies) or on records played in sync with the drama. The Warners employed the latter process, known as Vitaphone, for The Jazz Singer: the first feature film to include singing and some talking. Al Jolson's cry to the audience\u2014\"Wait a minute, wait a minute, you ain't heard nothin' yet!\"\u2014was truly the shout heard 'round the world.\n\nThe Jazz Singer, 1927: Al Jolson sings to his \"mammy\" (Eugenie Besserer).\n\nJolson, the Lithuanian rabbi's son known as the World's Greatest Entertainer, plays Jakie Rabinowitz, whose father the cantor (Warner Oland, the Swede who later impersonated Charlie Chan in sixteen movies) insists he go into the family business. He condemns the boy's love of Negro-inflected secular music, while Jakie's mother, Sara (Eugenie Besserer), encourages him to follow his heart and art. With this maternal blessing, Jakie changes his name to Jack Robin and becomes a vaudeville star. As Sara proudly says (in intertitles), \"He's not my boy anymore\u2014he belongs to the whole world now.\" Jakie returns briefly to the congregation, replacing his dying father to sing the \"Kol Nidre.\" The cantor addresses his last words to his wife: \"Mama, we have our son again.\" In the final scene, Jakie is back onstage, in blackface, belting the maternal torch song \"My Mammy.\" \"Doncha know me?\" he cries out to a tearful, smiling Sara in the front row. \"It's your little baby!\"\n\nLadies and gentlemen, mothers and sons: the talkies!\n\nThe Sin of Madelon Claudet, 1931: Helen Hayes with Neil Hamilton, in the happy prelude to decades of suffering.\n\n## THE MOTHER MARTYRS OF PRE-CODE\n\n* * *\n\n## Condemned by society or hounded by fate, these mothers heroically protected their children in the early Depression years.\n\nThe \"Pre-Code era\"\u2014actually, the period between the introduction of a moderately liberal Production Code in 1930 and the establishment of a much stricter set of rules that went into effect July 1, 1934\u2014is fondly recalled for saucy movies that spoke in the tart cadences of fast-talking men and faster women. This freedom created fresh stars, including James Cagney, Barbara Stanwyck, Mae West and Jean Harlow, and teased audiences with a sexual impudence that riled the burghers of propriety. Watch Baby Face or Employees' Entrance or The Story of Temple Drake and feel the sizzle.\n\nIn Hollywood movies and American life, though, other codes of behavior still applied: that a man could desert a pregnant woman, or terminate a marriage if the girl wasn't of his class, or, in the depths of the Great Depression, sell his wife's child to a rich couple. Under those strictures, a quartet of strong movie mothers dared to assert rights to their children, against all the odds that a patriarchal, puritanical society could stack against them. Spanning years, often decades, these Pre-Code weepies put mothers on-screen, and their avatars in the audience, through the emotional wringer. They may not have the contemporary sass of the Cagney films, but they can still touch hearts, because their themes of love and loss, sacrifice and redemption, are timeless.\n\n\"I wish Spaniards wouldn't put so much tragedy in their music,\" says the bon vivant Carlo (Lewis Stone) to his mistress Madelon (Helen Hayes) one night at the theater. \"They ask for love with so many tears.\" Watching The Sin of Madelon Claudet in theaters in 1931, moviegoers shed buckets of tears at this portrayal of the purest love: Madelon's for Larry, the son she had to abandon soon after he was born. A French country girl who falls for an American painter (Neil Hamilton) and is left pregnant when he returns home, Madelon falls much farther: into a liaison with the cool Carlo, a jewel thief masquerading as a count; into a ten-year jail sentence, when Carlo is arrested and she convicted for his crimes; and onto the streets, where she is reduced to getting men drunk and stealing their money. And all in support of the son she has rarely seen. Put baldly, this is the story of a woman who becomes a prostitute to put her boy through med school. But can a harlot also be a saint? Kindly Dr. Dulac (Jean Hersholt) thinks so, saying of Madelon and Larry that her \"life has been one long martyrdom for him.\"\n\nAdapting Edward Knoblock's 1923 play The Lullaby, screenwriter Charles MacArthur (Hayes's husband) tightened the dramatic strings by dispensing with a murder trial, having Dr. Dulac narrate the story to Larry's wife and elevating the young man's vocation from sailor to physician, which affords him a professional as well as personal interest in the woman who crosses his path only twice. MacArthur also gave his wife, a Broadway star in her talking-film debut, a role to display the range of emotions that would win her a Best Actress Oscar. Hayes ages seamlessly from teen farm girl to courtesan to a pretty petty thief and then to an \"old hag\" whose company no man would pay for\u2014and who, when an innkeeper reaches for her purse, screams, \"Keep off of me or I'll cut your face open!\" Yet in Hayes's performance, under Edgar Selwyn's acute direction, Madelon betrays no shame for her actions, no regret or self-pity for her sorry status. Money is life for this woman, because every penny she earns or steals goes to Dr. Dulac to support Larry's studies. \"Tell him he had a good mother,\" she says to Dulac, who replies, \"A great mother.\"\n\nFrom Madelon's tragedy, Hayes makes plangent music. Her emotional arias come in four magnificent scenes:\n\n1. When Madelon gives birth, she speaks in gaunt delirium: \"I wish it was dead. I wish we were both dead. Take it away!\" But when handed the baby, she becomes instantly attached and addicted to motherhood, smiling through tears at this social burden for her to endure, this gift for her to redeem.\n\n2. In prison, she leans into the mesh screen separating her from her friend Rosalie (Marie Prevost), who has kept contact with the boy. Through the mesh, she kisses Rosalie once, then again, saying of the second kiss, \"That was for him.\"\n\n3. After her release from prison, Madelon meets Larry (Frankie Darro), now about thirteen and a ward of the state. He confesses that he fought with another boy who called his mother a jailbird\u2014\" 'cause anybody as beautiful as my mother wouldn't be in jail.\" Identifying herself as someone who knew Madelon, and realizing that Larry's devotion for his missing mother is as innocent and intense as hers for him, she decides to protect the boy's reputation by telling him his mother is dead. Then she asks, \"Could you give an old friend a kiss?\" As he pecks her on the lips, she clasps his face in her hands like a lifeline. As he leaves, she begs for a hug and devours him in her embrace. \"That's from your mother.\"\n\n4. In her only scene with the grown Larry (Robert Young), she has entered his home for one last meeting. Still without revealing her identity, she tells him that she is a mother who lost contact with her child. Larry diagnoses her physical condition as dire, adding, \"I guess mothers are hard to kill.\" She holds his hands, gazing up with both a spaniel's adoration and a mother's pride in her caring son, then retreats into the night. Larry's wife (Karen Morley), to whom Dr. Dulac relates Madelon's story, says, \"She should have told him.\" \"Oh,\" Dulac says, \"she's too great a woman for that.\" So everyone keeps the secret from Larry. Missing the big reconciliation is frustrating for audiences, but the moral must be: Where's the sanctity in maternal martyrdom if you brag about it?\n\n\"I'm taking the kid with me,\" reads the farewell note from the rotten, barely literate husband in the 1930 film Sarah and Son. \"You're not a fit wife, and a woman who is not a fit wife isn't a fit mother, neither. He'll be all right, don worry. Better than with you. But you will never see I or him again.\" Sarah Storm (Ruth Chatterton), an Austrian immigrant who is the fittest wife and mother imaginable, has long wished for the departure of the shiftless, abusive Jim (Fuller Mellish Jr.), her partner in a vaudeville act. \"Maybe I'd be sorry like nothing,\" she says. \"I'd be glad to get rid of such a big loafer.\" He's gone soon enough, as malingering impregnators often are in Pre-Code weepies. But to abscond with their child is to rob Sarah of one of her two reasons for living: motherhood and show business. She also learns that her younger sister Rosel, whom she raised from infancy, has died back in Austria. As Sarah hones her singing skills and American accent by entertaining World War I troops in hospitals, she encounters Jim again, who speaks one word: \"Ashmore\"\u2014the name of the Texas oil baron to whom, years before, he sold Bobby.\n\nIn ten more years, Sarah has become a \"prima donna soprano\" with enough fortune to engage lawyer Howard Vanning (Fredric March) to search for her son. Mrs. Ash-more (Doris Lloyd) refuses to allow Sarah a visit with the boy and, when she insists, spitefully sends an imposter\u2014a deaf-mute. Finally meeting the real Bobby (Philippe De Lacy, Greta Garbo's son in the 1927 Anna Karenina adaptation Love) on a dock at Vanning's country home, Sarah befriends him and, on a boat ride, saves him from drowning. As Bobby lies in bed, delirious, both \"mothers\" enter. He cries out, \"Mother! It's cold! I'm afraid\"\u2014and hugs Sarah. \"He knows, somehow,\" the chastened Mrs. Ashmore acknowledges. And Sarah sings the tune with which she used to lull Bobby to sleep: Brahms's \"Lullaby\" (the theme of this movie, and of The Sin of Madelon Claudet). The boy has his mother and, with Vanning, perhaps a good father as well. Under Dorothy Arzner's direction, Chatterton\u2014diva of the soulful, reproachful glance\u2014wrestles her early mittel-European accent to a draw, and by the end nearly matches her work in 1929's Madame X. This time, the boy she loves gets to know who his mom is.\n\nSarah and Son, 1930: Ruth Chatterton fights for years to reclaim her son (Philippe De Lacy).\n\nA wronged woman seeking reunion with her son needed twenty, twenty-five years to accomplish the job: that was the dictum that governed Madelon Claudet, Madame X, Sarah Storm and Ellen Holmes, whom Jean Arthur embodies in director Lambert Hillyer's The Most Precious Thing in Life (1934) as a sweet teen aging into spinsterhood. In their courtship at Eastmore College, the richboy football hero Bob Kelsey (Donald Cook) is keen on Ellen, who works in the college laundry, as her mother did. But once they're married and back home with his snooty mother (Mary Forbes), the difference in class clashes like cymbals or symbols. Ellen wants to call their new son Francis. \"Such a common name,\" says the elder Mrs. K, and the boy is christened Christopher. When Bob insists on his mother raising the boy, Ellen leaves, and the Kelseys tell the child his mother is dead.\n\nFast-forward to 1930, when Ellen is now a washerwoman at Eastmore, which Chris (Richard Cromwell) has entered as a freshman. A pampered bully, as befits his upbringing, Chris needs an education\u2014mothering\u2014from the old woman he calls Biddy. She offers tips on how to impress the football coach (Ward Bond) and steers him toward pretty Patty O'Day (Anita Louise), a younger version of herself. When his father objects to Chris's romantic alliance with a splendid girl of the lower class, Ellen confronts him. \"You want him to marry that scrubwoman's daughter?\" he demands, leaving him defenseless against her zinger, \"He's a scrubwoman's son.\" Now Ellen spits out the truth she so long withheld: \"For twenty years I've kept still, so he could have what I thought was the best. Do you think I'm going to let you cheat him now? You've had your chance and you've ruined it. Now it's my turn. I'm not going to let you wreck my boy's happiness.\"\n\nA few years before she became Frank Capra's all-American glamour girl in Mr. Deeds Goes to Town, You Can't Take It with You and Mr. Smith Goes to Washington, Arthur plays the elder Ellen, who's barely forty, in gray hair pulled back in a bun and with cobweb lines of age on her pallid face. (Hayes as Madelon Claudet achieved a similar makeunder.) Also, Arthur's honeyed voice assumes a high-pitched, wizened tone, like the Cumaean Sybil with strep throat. Her Ellen speaks with an authority she has earned in two decades of suffering at the soft hands of class prejudice, and all to claim the most precious thing in life: a mother's right to guide her spoiled son into being a loving, responsible man.\n\nThe Most Precious Thing in Life, 1934: Jean Arthur as Biddy, whose son (Richard Cromwell) doesn't realize that his college-dorm cleaning lady is his mother.\nClass differences become cultural chasms in Madame Butterfly, Paramount's 1932 adaptation of the John Luther Long story that David Belasco turned into a Broadway hit and Giacomo Puccini into an opera warhorse. Sylvia Sidney might seem an odd choice to play the highborn Japanese woman who marries an American navy officer and is left to cope with their child and hope for his return. Like Barbara Stanwyck, Sidney parlayed New York grit and gravity into a tantalizing screen glamour. Yet she brings pathos and humor to Cho-Cho San, who brags of her pidgin English, \"I learned from a visiting scholar. She teach me very high-class Brooklyn accent.\" (Sidney was Bronx-born.) A preternaturally mature star by the age of twenty, in the 1931 films City Streets, Street Scene and An American Tragedy, Sidney was dubbed \"the Saddest Eyes in Hollywood\"; her tears could sparkle like emeralds. She sheds a noble shower of them here, under Marion Gering's direction, and in the general direction of Cary Grant's Lieutenant Pinkerton. Simultaneously hoping and grieving, Sidney fills those eyes with the solemn, soaring music of a woman whose son is the only living remnant of a vanished love.\n\nMadame Butterfly, 1932: Sylvia Sidney as Cho-Cho San, from ancient Tokyo by way of the Bronx, with Cary Grant as Lieutenant Pinkerton.\n\nAt the start, Cho-Cho San is already mourning a death: her mother's. \"May I do nothing to bring dishonor to my departed mother's honorable name,\" she intones to her gods. \"Make it possible for me to be of assistance to my illustrious family from now on, even though I'm only a woman.\" However subservient the words, Sidney stares heavenward like a haughty headmistress. Then, after she allows Pinkerton to court her, Cho-Cho San indirectly causes her father's suicide, being told, \"Your father died with honor when he could no longer live with honor.\" Pinkerton weds her, in what he believes is a ceremony valid only in Japan, and sails away. \"I will wait for always,\" Cho-Cho San avers. \"I know he will come. He will not forget.\" She gives her child (played by Philip Horomato, the only Asian in the cast) the name Trouble, saying to the absent Pinkerton, \"We wait till you come back to call him Joy.\" Butterfly sustains her wan hope for four years, until the lieutenant returns with his \"real\" (American) wife. After handing her child over to the couple, she follows her father into the afterlife by committing suicide.\n\nThree women\u2014Madelon, Sarah and Ellen\u2014lose their children, then metaphorically reclaim them. One, Cho-Cho San, raises her son and gives him away. All four of these films' leading actresses locate every nugget of histrionic gold. Sidney especially brings a power, both demure and defiant, to the handover of her child. It is not a gesture of surrender but a gift from a goddess who sinned by loving a mortal. Her Butterfly is a deity who dies.\n\nThe Grapes of Wrath, 1940: Ma Joad (Jane Darwell) leads her family (Henry Fonda, Russell Simpson, Doris Bowdon seated in the rear) to California, the land of broken promises.\n\n## THE GREAT AMERICAN MOM\n\n* * *\n\n## As warm and nourishing as apple pie.\n\nThey endured hardship in the silent films and in the early talkies, then flourished in the 1940s: the decade of the great American movie mom. Back when going to the pictures was a happy national habit\u2014when, in 1946, 150 million Americans bought four billion movie tickets, or one every two weeks for every man, woman and child in the country\u2014Hollywood made most of its product for the family and made sure Mom was part of the audience's on-screen family. Some of the most acclaimed and popular films of the forties portrayed the heroism of mothers. Poor women led their broods across the country for a better life, and Ma Joad was there in The Grapes of Wrath. Women whose husbands had gone to war sacrificed to maintain civility at home: Anne Hilton in Since You Went Away and Kay Miniver in Mrs. Miniver. Women used tact and music to restore harmony to the household: Mrs. Smith in Meet Me in St. Louis. Women struggled to raise their children above their own status: Katie Nolan in A Tree Grows in Brooklyn and Marta Hanson in I Remember Mama.\n\nThese and a hundred other female characters in forties films fused into the Great American Movie Mom. The sextet of moms in this chapter represents a small but instructive glimpse at how Hollywood made good films about good women. Ma Joad and Katie Nolan might hide their love under a veneer of toughness, but they believed their guiding mission was to elevate their offspring. And the other four lavished affection as their children's confessor, coach and cheerleader. Call them the tip of the nice-berg.\n\nThe Joad family in The Grapes of Wrath (1940) ought to be called Job, so many plagues do they suffer. In the Oklahoma dust bowl, the sharecroppers forfeit their land and their home, which Ma Joad (Jane Darwell) is ready to leave without a farewell glance: \"I never had my house pushed over before. Never had my family stuck out on the road. Never had to lose everything I had in life.\" The sprawling family crowds into an old truck to seek an employment bonanza promised in California. Isn't it the Golden State? But both grandparents die en route. The child of Ma's daughter Rose of Sharon is stillborn. The family can earn only five cents a box for picked peaches, and the farm they work is effectively a prison, with barbed wire and armed guards. Son Tom (Henry Fonda) gets involved in labor agitation and kills a man. Directed by John Ford and scripted by Nunnally Johnson from John Steinbeck's novel, the movie wrings every drop of rage and pathos out of the plight of desperate farmers, ten years into the Depression.\n\nTom is the firebrand of the family, Ma the hearth. She sees the Joads' woes as the middle chapter in a story as old as humankind. \"Woman can change better than a man,\" she says. \"A man lives sorta, well, in jerks. Baby's born or somebody dies, and that's a jerk. He gets a farm or loses it, and that's a jerk. With a woman, it's all in one flow, like a stream. Little eddies and waterfalls. But the river, it goes right on.\" She might be describing the difference between male- and female-oriented Hollywood films, between combustion and continuity. In the movie's famous peroration, Ma proclaims that the underclass will triumph because women like her have the power to replenish America with their children, millions of them. \"Rich fellas come up and they die,\" she says, \"and their kids ain't no good and they die out. But we keep a-comin'. We're the people that live. They can't wipe us out; they can't lick us. We'll go on forever, Pa, 'cause we're the people.\" As men kill, so women give life. Their bodies are the arsenals of future generations.\n\nMa Joad is on the front lines of a labor battle. Anne Hilton (Claudette Colbert) in Since You Went Away (1944) mans\u2014or womans\u2014the home front when her husband goes to war. Producer David O. Selznick's metaphorical follow-up to Gone With the Wind, with World War II replacing the Civil War, again ignores the men risking their lives in the field to concentrate on the women coping without them. But unlike Scarlett O'Hara, whose ambition drove her into business to save her plantation, Anne exhibits the more passive heroism of waiting and worrying, her smile a rictus of dread, her poise a plug for the scream lodged in her heart.\n\nIn this nearly three-hour movie, the drama comes in the silence that may be punctuated at any moment by a phone call, or the doorbell ring of a Western Union telegram, announcing a death in the family. Anne's two daughters\u2014college-age Jane (Jennifer Jones) and teenage Bridget (Shirley Temple)\u2014try pursuing the adventures of girls their age; they also volunteer for the war effort, as does Anne eventually, by working in a shipyard. The Hiltons do nothing so indecorous as suffer; they abide and endure. Only Jones, under John Cromwell's direction, gets to emote in sync with women in the audience whose beaux never came home. And to both parental roles, Anne must add that of chaplain, when she comforts Jane by urging catharsis: \"Cry, darling. Cry your heart out. I won't try to tell you that you'll get over it soon, because it will take time. Maybe a long time.\"\n\nSince You Went Away, 1944: Jennifer Jones (left) and Shirley Temple listen as mother Claudette Colbert reads a letter from their father away at war.\nGreer Garson in Mrs. Miniver (1942) had something to cry about. World War II left English cities, and Kay Miniver's decorous village of Balham, cratered by Nazi bomb raids; the local church lost its roof in those sorties. She suffers the loss of family; she confronts and disarms a downed German aviator. She is determined to maintain a semblance of normality by reading her youngest son Alice in Wonderland during an air raid. Mrs. Miniver taught that there's no crying in wartime, except in the audience. She kept a stiff upper lip even as moviegoers' lips were tremulous from her shining example.\n\nThe United States was never in danger of Nazi planes strafing its cities, but American boys were risking their lives in the European Theater. The movie meant to show what they were fighting for; as film historian David Thomson has written, it \"conditioned American public opinion to the sentimental explanation for a necessary war.\" Directed by William Wyler at MGM's Culver City studios, the movie featured Canadian Walter Pidgeon as husband Clem and New Yorkers Richard Ney and Teresa Wright as their grown son and his wife. (Not long after the filming, Garson the mother married Ney the son.) Studio boss Louis B. Mayer made sure that the Minivers approximated his ideal of small-town Americana, but with a starchier accent; Andrew Sarris called the Minivers a \"British version of the Hardy family.\" Because Kay Miniver is Yankee by adoption and adaptation, she qualifies as a Great American Mom.\n\nOut of twelve Oscar nominations, Mrs. Miniver won six: Best Picture, Director, Actress (Garson), Supporting Actress (Wright), Screenplay and Black and White Cinematography. Everyone loved the movie; Winston Churchill called it \"propaganda worth a hundred battleships,\" and Nazi propaganda minister Joseph Goebbels urged the German film industry to study and imitate it. Churchill was speaking from gratitude, Goebbels from envy\u2014and all at a mother's resolve to celebrate the small gifts of life as the skies rain death.\n\nForties films could create dramatic sparks without resorting to political outrage or the background noise of the war against Hitler. Mothers didn't have to domineer and dominate to be persuasive. Sometimes a woman's gentlest touch could push her husband's buttons and coax him toward the decision that she and her children hoped for. A single subtle scene can reveal it. In Meet Me in St. Louis (1944), set in December 1903 in MGM's dream of the urban Midwest, businessman Alonzo Smith (Leon Ames) has been offered a better job in New York. His announcement of the family's imminent departure triggers heartache and resentment for his girls, Rose (Lucille Bremer), Esther (Judy Garland), Agnes (Joan Carroll) and Tootie (Margaret O'Brien), who bolt from the dinner table to their rooms. Only his wife, Anna (Mary Astor), remains, attending his sputtering with a gaze that scrupulously avoids reproach. Following him into the living room and handing him his cake, she says gravely, \"If you think it's best for us to go away, why, that's what we'll do.\"\n\nMrs. Miniver, 1942: Greer Garson and Walter Pidgeon clutch their children (Christopher Severn and Clare Sanders) during an air raid.\nShe sits down at the piano to play the love ballad \"You and I\" (cowritten by the film's producer, Arthur Freed). \"It's good to hear you play, Anna,\" Mr. Smith muses. \"It's been a long time.\" When he tries singing in a scratchy tenor, his accompanist obliges: \"I'll put it down in your key.\" She prompts a few words, then sings in harmony. The four girls creep downstairs, eat their cake and listen to the last verse: \"Time goes by, but we'll be together, you and I.\" Accompanying these musical sentiments of lasting love, director Vincente Minnelli's tableau of father, mother and daughters establishes that the clan will not crack, because who they are is more important than where they are. Anna's ostensible compromises, indicated by Astor with not even a secret smile, have secured the notion of a permanent home. The genius of motherhood has restored the family.\n\nMeet Me in St. Louis, 1944: Mary Astor and husband, Leon Ames, sing a song at Christmas, attended by daughter Margaret O'Brien.\n\nSome mothers get outshone by their husbands, at least in their children's eyes. That is the burden that Katie Nolan (Dorothy McGuire) finds most difficult to bear in A Tree Grows in Brooklyn (1945). She works tirelessly scrubbing floors to keep her singing-waiter husband Johnny (James Dunn, who won the Supporting Actor Oscar for the role) and their preteen children Francie (Peggy Ann Garner) and Neeley (Ted Donaldson) in their turn-of-the-century tenement flat. Yet Francie, the soul and narrator of director Elia Kazan's beautiful debut film, loves her irresponsible father as a vital force, while only tolerating her mother as the family's grim judge and drudge. Johnny provides the entertainment and Katie the essentials, which she thinks he and the kids take for granted; they appreciate his gaiety, not her utility. No one feels better knowing that Katie's hard work pays the rent; but when Johnny sings \"Annie Laurie,\" even Katie's adamantine heart can soften. It reminds her of a million years ago, when she was young and in love with him. Since they had children they have grown into two mismatched boarders in the same apartment: Johnny the alcoholic charmer, Katie weary and severe. He can't quit drinking and she can't get bubbly.\n\nFrancie has the aspirations and aptitude to become a writer. (She grew up to become Betty Smith, whose memories of her Brooklyn family became the bestseller the movie was based on.) A helpful teacher asks her to understand the difference between imagination and pipe dreams, but for most of the film Francie's guiding pipe dream is her father. And because a child has to take sides, she views her mother with resentment, for not being Johnny. \"Some of its best scenes,\" wrote Time critic James Agee when the movie was released, \"are those in which the long-estranged parents try vainly to make up, and in which the father and daughter realize, beyond any chance of forgetting, that she is growing up and that he never will.\" A Tree Grows in Brooklyn traces the growing up of both Katie, who realizes that Johnny's blarney is a healing gift, and Francie, who learns to like and love her mother. McGuire, who was often cast as an ideal mom in Disney films (Old Yeller, Swiss Family Robinson) and was Jesus's mother in The Greatest Story Ever Told, finds the perfect register for Katie, a woman who can't express the devotion she feels for her dear, bright daughter. Katie is a mother whose mute heroism only the sympathetic viewer can see.\n\nA Tree Grows in Brooklyn, 1945: Dorothy McGuire (center) with Peggy Ann Garner and Ted Donaldson.\n\nMarta Hanson (Irene Dunne) in I Remember Mama (1948) is a secular saint, worth praying to for special favors\u2014because she always gives. Like the Sally Benson book that spawned Meet Me in St. Louis, and Smith's A Tree Grows in Brooklyn, Kathryn Forbes's Mama's Bank Account was a fictionalized memoir, this one looking back in rapture at her Norwegian mother's grit and gentility. Kindness seeps through Marta's formal manner as she does the weekly accounts, adjudicates among her three contentious sisters, and scrimps to educate her four children. The one boy, Nels (Steve Brown), wants to attend high school; at Mama's urging, the three daughters chip in from their meager savings. Katrin (Barbara Bel Geddes), the story's narrator and rememberer, has literary dreams that she achieves with the publication of the lightly fictionalized story that became Mama's Bank Account. Mama, so industrious that her one nervous tic is to scrub the floor, impersonates a hospital cleaning woman to sneak in to see her youngest daughter. Other than that, sensational incidents are few.\n\nThis may seem an impossibly idealized family portrait, yet the film, directed by George Stevens, provides an acute, organic view of the Hanson home\u2014its ebb and flow, as Ma Joad might call it. So chipper is the mood of the piece that, when it ran for nearly two years on Broadway (where Nels was played by the young Marlon Brando), it was described as a comedy. Dunne often brought a coloratura comic lilt to her films. That larkish impulse is subdued here, in a movie role first offered to Greta Garbo, but Dunne's Marta is reserved, not grave. (And her resemblance to Bel Geddes, in their soft, rhyming faces, makes them look not just kindred but kin.) Her generous spirit sets a high standard for the other family members; it lifts them to meet her. An extraordinary ordinary woman, Marta is a nexus of maternal virtues: the great Norwegian-American mom.\n\nI Remember Mama, 1948: Irene Dunne as the Northern nurturer who lovingly provides the best for her family, including daughter Katrin (Barbara Bel Geddes, far right).\n\n### On Jane Darwell in The Grapes of Wrath\n\nBeing an actress, wife, mother and grandmother has been the best of all worlds for me. All these roles have given me an understanding of what it means to have close and important family relationships.\n\nThis kind of family love and devotion was beautifully depicted in the 1940 John Ford film The Grapes of Wrath, based on the novel by John Steinbeck. It's the story of a poor sharecropper family making its way west in the 1930s, looking for work. They journey from the dust bowl of Oklahoma to California in a broken-down truck during the Great Depression.\n\nThe actress Jane Darwell plays the mother who keeps the family all together. She's a devoted wife, mother and grandmother who cleans, washes and cooks without a break; she's everything to everyone in the family. There's not one false moment in her performance, a magnificent portrayal of a true farm lady who loves and is devoted to her family. In a wrenching good-bye scene, she knows one of her sons (played by Henry Fonda) must leave the family and sends him off with all her love. Of course, her heart is breaking, but she'll cry after he leaves.\n\nThese are life lessons to be learned and Jane Darwell was a fine teacher. I saw The Grapes of Wrath in the late fifties, when I was a young mother, and it made a lasting and indelible impression on me. Inspired by Darwell's closeness to her family, I turned down several films because I didn't want to be away from home, and I chose to take our children with me on two distant film shoots that didn't interfere with their schooling: Exodus and Grand Prix. I admired Darwell's combination of tenderness and strength, and she became a role model for me.\n\nThere's no better example of motherhood in a film that I have ever seen. It's no wonder she won an Academy Award for her performance. The picture was made in 1940, but Darwell's beautiful portrayal lives on.\n\n\u2014Eva Marie Saint\n\nA Hardy clan: Fay Holden as Emily Hardy shares a laugh with Mickey Rooney as Andy.\n\n## SERIAL MOMS\n\n* * *\n\n## The wife-mother flourished in family-friendly film series.\n\nThe movie landscape was so different, almost lunar, before the James Bond films legitimized big-budget sequels in the 1960s. Since then, the sequel has become the norm. In nine of the past ten years, the top-grossing picture has been an extension of some earlier blockbuster. (The one exception: Avatar.) Pre-Bond, though, the most successful Hollywood movies were unique and induplicable; nobody made a Gone With the Wind II or The Even Better Years of Our Lives. In the 1940s, the only sequels to be among their years' ten most popular films were four of the Bob Hope-Bing Crosby Road movies, The Bells of St. Mary's (with Bing reprising his Father Chuck O'Malley character from Going My Way) and Jolson Sings Again (following The Jolson Story).\n\nFar below these, in budget and popular esteem, were the family-films series: thrifty B movies designed to fill the bottom half of a studio's double-feature bill. In these modest, amiable entertainments, such wife-mother types as Louise Jones, Blondie Bumstead and Phoebe Kettle flourished or at least persisted from the mid-1930s to the mid-'50s. They sketched the domestic grit of the middle- or working-class woman coping with the comic shortcomings of her kin. Her motto: not divide and conquer, but abide and endure.\n\nThese movies have to be distinguished from the postwar \"A\" film series Father of the Bride and Father's Little Dividend (with Joan Bennett and Spencer Tracy as the parents) and Cheaper By the Dozen and Belles on Their Toes (with Myrna Loy and Clifton Webb), each of which lasted only two episodes. The Jones family, Blondie and Ma and Pa Kettle pictures stretched over seventeen, twenty-eight and ten features, respectively. They also focused far more acutely on the burden of motherhood than did MGM's two \"family\" series of higher esteem: The Thin Man and the Hardys. Nick and Nora Charles (William Powell and Myrna Loy) suaved their way through six MGM features from 1934 to 1947\u2014a fantasy of the chic married couple beloved by swells and thugs alike. By the fourth entry, Shadow of the Thin Man, they had acquired a son, Nick Jr. (Dickie Hall), who had sprouted into a teenager (Dean Stockwell) by the 1947 finale, Song of the Thin Man, but little Nicky was a less prominent and useful member of the clan than the Charles's wire fox terrier, Asta.\n\nAnd though the fifteen Hardy films from 1937 to 1946 surely fit MGM boss Louis B. Mayer's dewy vision of the American family, the moral authority on view was utterly patriarchal. In nearly every episode, impulsive Andy (Mickey Rooney) would commit some minor breach of propriety, think himself a big shot in small-town Carvel, and receive a third-act lesson from his father, Judge James Hardy (Lewis Stone); his study is Andy's courtroom. And the judge takes his counsel less from his wife, Emily (Fay Holden), than from maiden aunt Milly (Sara Haden). Emily would disappear for a film to tend her ailing mother, and in Judge Hardy and Son she risks death from pneumonia\u2014not giving care but needing it. Andy relies on his mom for favors, his father for wisdom. The route of sagacity is more man-to-boy than mother-to-child. Sister Marian (Cecilia Parker) might get maternal advice from Emily, but since the stories depended on Andy's mischief, and the series on Rooney's star quality, girl talk was mostly a background whisper.\n\nThe Jones Family series, at 20th Century-Fox, preceded the MGM Hardys by a year: the first installment, 1936's Every Saturday Night (in which the family name was Evers), launched a seasonal skein, four films a year, until On Their Own concluded the project in 1940. In small-town Maryville, pharmacist John Jones (Jed Prouty) and his wife, Louise (Spring Byington, who played Andy's mom in the first Hardy film, A Family Affair), share a crowded home with their five children\u2014sensible eldest daughter Bonnie (Shirley Deane), egotistical Jack (Kenneth Howell), studious Roger (George Ernest), histrionic Lucy (June Carlson) and baby Bobby (Billy Mahan)\u2014and John's mother (Florence Roberts). In the sixth installment, Hot Water, John runs for Maryville mayor and wins. In the next film, Borrowing Trouble, John becomes a Big Brother to the orphan Tommy McGuire (Marvin Stephens), who moves in with the Joneses for the rest of the series. Occasionally the family travels: Off to the Races, Down on the Farm, A Trip to Paris, The Jones Family in Hollywood.\n\nOn the road or at home, the family dynamic rarely varies. John blusters as his children challenge the authority he tries to assert. Louise (billed in the screen credits as \"Mrs. John Jones\") knits and purrs, defusing family feuds with a smiling \"Yes, dear\" that could signal either approval or mild exasperation. The emotional anchor of the series, she also urges financial restraint in a money-strapped household with many mouths to feed; the Great Depression, still endemic, was the unspoken subtext in many Jones movies. But she frequently allows Granny\u2014more tolerant of the kids than John, and more assertive than Louise\u2014to fire the salvos of flinty common sense. \"You're not bringing them up,\" Granny warns her son of his children in the first film, \"you're always knocking them down.\" In the Jones family, it was often Grandmother who knew best.\n\nThe Jones Family films: parents Spring Byington and Jed Prouty at the table with his mother (Florence Roberts) and the sprawling clan.\n\nBarely nosed out in family-film obscurity by the 1938-41 Republic Pictures series The Higgins Family (starring real-life husband, wife and son James, Lucile and Russell Gleason as the nuclear unit), the Joneses got little respect even from their own studio. A 1936 Fox pressbook described the films as \"energetic hokum . . . tolerably amusing in an entirely inconsequential way . . . They strive, and rather successfully, to catch the spirit of a small-town tribe; but that, in itself, is not too important an enterprise.\" Important, perhaps not; salutary, for sure. The Jones movies allotted sympathy to every one of the family members, including Father; over the series they all got to step out of their mild caricatures to reveal a clever brain and a giving heart. In a film frame that could be clogged with eight or more Joneses fighting for attention, the noise could be antiphonal but the lilting melody rang through\u2014a chorus of all for one and one for all.\n\nA daily comic strip from 1930 to today\u2014and over the decades a radio show, live-action TV sitcom and two animated cartoon specials\u2014Blondie came to movies in 1938. In the early days of Chic Young's strip, Blondie Boopadoop had been a ditsy flapper and her beau Dagwood Bumstead a scion of wealth whose father disinherited him when he married. By the time of the Columbia film series, Blondie (Penny Singleton) had matured into an ideal of the American wife-mother: pert, sweet-tempered and ferociously competent\u2014qualities needed to manage the household on a tight budget, and to save her amiably inept spouse (Arthur Lake) from catastrophes at home and at work with his splenetic boss J. C. Dithers (Jonathan Hale). Film historian Jeanine Basinger synopsized each plot of the series this way: \"Dagwood screws everything up, and Blondie sorts everything out.\"\n\nGiven the youthful demands of their boy, Alexander, a.k.a. Baby Dumpling (Larry Simms), and their daughter, Cookie (Marjorie Ann Mutchie), who joined the family in 1942 in Blondie's Blessed Event, Blondie can be seen as a single mom with three kids. Dagwood is the perpetually addled adolescent, the good-hearted but hopeless teenage son whose appetites don't extend much beyond devouring the triple-decker sandwich that popular culture named after him. In the series debut, Blondie!, when Dagwood asks about the dinner menu, Blondie parries, \"What difference does it make? You cover everything with ketchup anyway.\" In an argument, the two parents use the three-year-old Baby Dumpling as a marriage mediator; Blondie tells the child, \"He forgets that we have a happy home because I planned and schemed.\" Dragged into court at the film's climax, she explains to the judge, \"Most men have a lot of little boy in them.\" She turns the Bumstead bumbling into triumph when she wheedles a raise and a bonus out of Mr. Dithers for Dagwood. (In later films, Dithers will directly employ Blondie for the more complex deals.) Her last words: \"Sometimes I think it's harder to raise a husband than a baby.\"\n\nLest Blondie come across as a scold, her tongue sharpened by domestic misery, Singleton delivers her lines with generous cheer. On Broadway from her teens as an eccentric dancer (under the name Dorothy McNulty) and in movies since 1930, when she led her college classmates through the rousing \"Varsity Drag\" in MGM's musical Good News, and much later the voice of space-age wife Jane on the cartoon series The Jetsons, Singleton radiated an unquenchable effervescence\u2014think Ginger Rogers, cubed\u2014and a good wife's tolerance for her husband's failings and flailings. She provided the perfect partner for Lake, a kind of Fred MacMurray lite, who softened Dagwood's doltishness with his blithely befuddled innocence. Though they endured a brief marital separation in 1939's Blondie Meets the Boss (she got custody of Baby Dumpling, Dagwood of the Bumstead dog, Daisy), the two loved and sustained each other; they and their children grew up together. The final film, Beware of Blondie in 1950, opens with a placid living room tableau: Mother knitting, Father reading, the teenage Alexander doing homework and grade-schooler Cookie playing the piano. The girl pauses in her practice to glance at her older brother's math problem and correct it. In the Bumstead family, heredity and Hollywood gave the girls all the brains.\n\nMa and Pa Kettle were real people\u2014anyway, realish. Neighbors of Betty MacDonald when she and her husband moved to Cape Flattery on Washington's Olympic Peninsula to become chicken farmers, the \"Kettles\" provided rural color to MacDonald's bestselling 1945 memoir The Egg and I. They also graced the 1947 movie version, starring Claudette Colbert and Fred MacMurray as the MacDonalds, and stole the show with their raucous backwoods wit. Recognizing the star quality of this supporting couple, Universal International gave the Kettles their own franchise, which lasted nine films, from 1949 to 1957.\n\nAs played by Marjorie Main (fifty-nine at the start of the series) and Percy Kilbride (sixty-one), Phoebe and Franklin Kettle might seem unlikely parents of fifteen kids who ranged from college age to infancy, but probability was the least of the screenwriters' concerns. This was flat-out, deadpan farce: a live-action cartoo of rubes who managed to survive with gumption and luck\u2014and the popular success of their low-budget series. Each episode cost about $500,000 to produce and usually earned five times that amount at the box office. The guess here is that the peninsular couple who inspired the Ma and Pa characters never saw a penny of that largesse.\n\nThat scruple aside, as well as the derisive portrayal of two Native Americans (Oliver Blake and Teddy Hart) who helped out on the farm, the Ma and Pa films provide a lot of fun, in their oddly abrasive way. Though natives of the Pacific Northwest, the Kettles exhibit an inbred indolence that would be at home in Faulkner's Yoknapatawpha County, Mississippi. Ma, in a sack dress that Tyler Perry's Madea would consider infra dig, presides over a brood so large that she often refers to her kids by the wrong names. (\"Now hang on there, Billy.\" \"I'm Danny!\") Ma's dinner table is a pigsty with plates, and her kitchen closet holds more junk than a full season of Storage Wars; \"Might as well be one place as another,\" she opines. Pa, with a lower IQ than Ma's and a longer fuse, is a man of few words\u2014his prayer before dinner is \"Much obliged\"\u2014and compact actions: he changes radio stations by slamming his chair onto the floor. With children running around like field mice, Pa hardly gets the chance to relieve himself, drawling, \"Maybe we should have had less kids and more bathrooms.\"\n\nIn the series debut, Ma and Pa Kettle, Pa wins a model home that promptly becomes a horror house. Then the Kettles get infected with wanderlust, in . . . at the Fair, . . . on Vacation, . . . at Waikiki, and . . . in the Ozarks. But with the whole family along, only the backdrop changes; the constant is Ma's gruffly doting, mostly un-ruffleable attitude toward her children. A few of the older kids are housebroken, but the rest seem feral. The \"Be-ware of childrun\" sign outside the farmhouse is no idle admonition: trespassers will be discouraged with slingshot rocks and peashooter pellets. In fact, the whole family is heavily armed. Suspicious of any stranger, they could be the Kettle militia, the Minute-Mas.\n\nWhat's the appeal of these movies about a backwoods, backward family of borderline sociopaths? That the Kettles are presented without explanation or apology; they barge onto the screen, occupy it for an hour or so, then retreat until the next incursion, leaving the viewer to decide if they're nice, nuts or both. Sometimes they seem like yokel geniuses, as in . . . Back on the Farm's classic scene of Ma and Pa \"proving\" on a blackboard that twenty-five divided by five is fourteen. (Watch it on YouTube: \"Ma and Pa Kettle Math.\") The Kettle films are the low-rent equivalents of those amazingly cluttered Preston Sturges comedies\u2014The Miracle of Morgan's Creek or Hail the Conquering Hero\u2014except that the supporting cast of small-town eccentrics has seized control of the narrative. Kilbride, channeling the comic solemnity of Buster Keaton, serves as the ideal opposite to Main, with her Brillo-pad voice and hulking bonhomie; together they exhibited a kind of bleak hope on an endless road, as if Vladimir and Estragon, in Samuel Beckett's Waiting for Godot, had been a married couple. Marjorie's Ma didn't have to lavish maternal sagacity on the kids, just a kind of authority by default. Besides, they were all fellow lifers in the only prison they knew and accepted: the Kettle family.\n\nThe Ma and Pa Kettle films: Marjorie Main and Percy Kilbride rode herd on their fourteen, fifteen (nobody knew how many) kids.\n\nThe B movie family series insisted not only on a continuity of talent (the actors were under contract) but continuity in writers and directors. If the Blondie series had auteurs, they would be Frank R. Strayer, director of the first dozen features, and Richard Flournoy, who wrote nine of those twelve. Charles Lamont helmed Ma and Pa Kettle and four of the eight sequels. Often, the craftsmen of one series moved on to others. Strayer came to Blondie after directing four of the Jones films. Karen DeWolf contributed scripts to both series. Jack Henley wrote six of the later Blondie movies, then four of the Kettles. These anonymous artisans probably considered this work not an adventure but a job: maneuvering stock domestic characters from Point A to Point B and back home again for supper.\n\nFor unlike the majority of golden age movies that pursued one story to its conclusion, but quite like the sitcoms then on radio and later on television, these family-film series didn't so much resolve as organically evolve: parents got older and children grew up. Andy Hardy graduated from high school, entered college and, in World War II, joined the military; in a 1958 postscript to the series, Andy Hardy Comes Home, he returned to Carvel with a family of his own. The movies' Blondie and Dagwood welcomed their second child, daughter Cookie, shortly after she was introduced in the comic strip. The Kettles' eldest son, Tom (Richard Long), studied animal husbandry at college and married the newspaper gal (Meg Randall) who had reported on the model home won by Ma and Pa. And when Bonnie Jones married her beau Herbert Thompson (The Higgins Family's Russell Gleason), the family expanded to four generations, with episodes on newlywed economics (Love on a Budget) and the perils of young motherhood under the gaze of a buttinsky brood (Everybody's Baby). By the end of the series, three tiers of mothers\u2014Grandma, Louise and Bonnie\u2014were speaking womanly truth to male power, more or less simultaneously.\n\nAging with their audiences, encountering minor domestic trials and triumphs, the Joneses and the other film families came as close to a middle-class audience's rosy doubles as fictional characters could get. And since they showed up in new films three or four times a year, they might have been more familiar to a moviegoing woman than some of her own relatives. They were Ma, Pa and the rest of us kids, on the big screen.\n\nMadame X, 1929: Young lawyer Raymond Hackett defends the mystery lady Ruth Chatterton, who is\u2014had he but known!\u2014his mother.\n\n## PERENNIAL MOMS\n\n* * *\n\n## Mother stories retold across the decades.\n\nOn the stage, actresses test their mettle by playing a Shakespeare or Ibsen heroine, Blanche DuBois or Edward Albee's Martha. On-screen, classic novels may give birth to enduring female icons, as Leo Tolstoy's Anna Karenina has done for more than a century, but often the source is a French melodrama (Madame X) or the less elevated species of literature known as women's fiction (for Stella Dallas and Imitation of Life).\n\nWhatever the provenance, these stories paint indelible portraits of women stigmatized by the sin of adultery, or by the accident of class or race. They lash loving mothers to the rack of noble suffering. And they reach their apogee with some form of convulsive public event: a trial, a wedding, a funeral or a train-station suicide. Crucially, these properties allow prominent stars to tear a passion to tatters. Here you will find Ruth Chatterton, Barbara Stanwyck, Greta Garbo, Vivien Leigh, Lana Turner (twice) and two other actresses\u2014Fredi Washington and Susan Kohner\u2014whose daring performances place them in that august company. That's a bonus of movies about Perennial Moms: they transcend stereotype to achieve archetype.\n\n### MADAME X\n\nAlexandre Bisson's play, first staged in Paris in 1908, spread around the world like news of a royal scandal. In 1910 Broadway mounted two productions, one in French starring Sarah Bernhardt. The same year, the first movie version appeared in Denmark. Pauline Frederick headed a Hollywood silent feature adaptation in 1920, but talkies were the true medium for Madame X: all that suffering meted out to the hapless Jacqueline Floriot deserves to be declaimed rather than mimed. In all, Bisson's play spawned about a dozen screen versions, including an overwrought, undernourished 1966 version with Lana Turner and a more crackling 1981 TV movie starring Tuesday Weld.\n\nThe plot traces a woman's journey of degeneration and regeneration, sin and redemption. When Jacqueline commits adultery, her husband, a distinguished judge, bans her from their house and severs her ties to her beloved infant son Raymond, whom he tells that his mother is dead. As if to prove her husband's dim view of her actions, she leaves Paris and sinks into sin as the globe-trotting Madame X. Decades later, a drunken Jacqueline lets her real identity slip to the bounder Laroque. He takes her back to Paris, with plans to blackmail the judge, and she shoots him dead. When she goes on trial for murder, the young lawyer assigned to defend her is . . . Raymond! He doesn't know she is his mother, and she won't tell him.\n\nThe 1929 Madame X, directed by Lionel Barrymore, snagged a nomination for Best Actress in the second year of the Academy Awards and launched Ruth Chatterton on a triumphant four-year run of movie agony at MGM and Warner Bros. All well earned, since she hits the proper high notes of emotion\u2014full, not shrill\u2014in each of Jacqueline's three big scenes. At the beginning, when her husband (Lewis Stone) exiles her from their home, she plants herself against the door she will vanish through and utters a sibylline curse. \"Remember this,\" she intones, \"that if it ever occurs to you again to tell that boy his mother is dead, you tell him too that her last request was to be allowed to look at him, just once. And you refused it. You tell him that, Louis.\" In her act 2 confrontation with the venomous Laroque (Ullrich Haupt), Chatterton's Jacqueline proves her maternal mettle by shooting him dead; the pain and resolution play subtly, starkly, across her face.\n\nThe climax, in which Jacqueline takes the witness stand to justify the killing of Laroque as the act of shielding her son from learning what his mother had become, propels Raymond (Raymond Hackett) into spasms of sympathy for her and denunciations of the unjust husband who sent her away\u2014his own father, had he but known. Why is this mother's sacrifice kept secret, when every rule of melodrama screams for revelation, forgiveness and reconciliation? Perhaps because Jacqueline, having sinned and confessed, must pay her penance: saving her son's good name while denying herself a final healing embrace. The harboring of her secret makes Chatterton a screen sister to Helen Hayes in The Sin of Madelon Claudet and Jean Arthur in The Most Precious Thing in Life.\n\nThe 1937 Madame X, directed by Sam Wood, executes a few plot handsprings to satisfy a stricter Production Code and tilt some sympathy toward the husband. At the start, Jacqueline (Gladys George) is involved in a tryst that leads to a crime of passion. She returns home and finds that her son, Raymond, is ill; because she has evaded her duties as a mother and a wife, her husband Bernard (Warren William) is almost justified in throwing her out and in telling their son that his mother is dead. To hide from the law, she becomes a governess, not a harlot, while Bernard shows signs of kindness by searching for her. The climax is the same, with Jacqueline choosing death before her son's dishonor and the grown Raymond (John Beal) defending her as passionately as if he knows she is his own mother. \"She was a wonderful woman, Father,\" he says at the end. \"Whoever she was.\"\n\nIn the eight years between the 1929 and 1937 Madame X films, between the birth of talkies and their luxurious flowering, Hollywood had imposed a more naturalistic style of acting on sound films; George doesn't scale the histrionic Alps that Chatterton did, and the makeup that aged each actress twenty years is more nuanced here. But both performances are splendid, just in different registers\u2014the difference between George's sotto voce and Chatterton's fortissima.\n\n### STELLA DALLAS\n\nIn some venerable films about sacrificing mothers, a murder can be easier to forgive than bad manners. And any poor woman deserves a rich husband, unless her demeanor and couture are loud enough to wake the neighbors\u2014and embarrass her dear daughter. The 1923 novel Stella Dallas by Olive Higgins Prouty, who also wrote Now, Voyager, sees issues of class (upper vs. lower) as matters of tone in speaking (dulcet vs. braying) and dress. The gulf of social acceptance is nearly as wide as that between a black woman and a woman who can pass for white in the 1934 film Imitation of Life. Of course, a semblance of breeding is more easily achieved than a modification in skin color. If Henry Higgins had taught Stella Dallas the fine points of elocution, as he did for Eliza Doolittle, she might have kept her high-born husband, Stephen, and lived comfortably ever after. But then she would not have created a dilemma for her sweet Laurel: to choose between a birth mother and, as society and even Stella see it, a better one.\n\nPeople who have never read Stella Dallas, or seen any of the films, know the ending: a mother standing in the rain, smiling through tears, as she peers into a townhouse window to glimpse her daughter's wedding. Here's the rest of the story. In a weak moment, Stephen Dallas, the mill owner's son, marries Stella Martin, a millworker's daughter. By the time their daughter, Laurel, is ten, Stephen is far away at his New York City job, and Stella is keeping time\u2014not explicitly sexual, more for laughs\u2014with the boisterous, borderline-repellent Ed Munn. The Munn connection further befouls the odor of Stella's reputation among the small-town gentry, and Laurel is denied access to a private school for young ladies of good background.\n\nNow a teenager with an unblemished soul and impeccable demeanor, Laurel has fallen in love with Richard, a plutocrat, and he with her. When Stella overhears mocking comments about her by Laurel's friends, who say that Richard would never marry a girl with \"that mother,\" she leaps into expiatory action. She visits Helen Morrison, the refined widow whom Stephen has been courting in New York, saying she will agree to divorce him if Helen raises Laurel as her own. Yet the girl, still ferociously faithful to her mother, refuses to leave home. How can Stella push her daughter into the destiny she deserves in New York? By turning Laurel against her. By announcing she's about to marry Ed Munn. The trick works. Stephen and Helen marry, and then Laurel and Richard, in a wedding Stella doesn't feel worthy of joining. She stands outside, in a final gesture of sanctified masochism, to see that her little girl has found happiness in the upper class.\n\nMadame X, 1937: In the remake just eight years later, Gladys George is the notorious defendant, John Beal her unknowing son.\nThe challenge of the three Stella Dallas films, which spanned two-thirds of a century and ended with Bette Midler's endearing if anachronistic Stella in 1990, is to portray a woman who is crass in her behavior but harbors class in her heart. Gaudy and gauche, Stella is the sort of woman whose voice rises rudely above the murmurs on a train, as the more decorous riders flee to other cars. She is the \"loathly lady\" from Chaucer's Wife of Bath tale, and for viewers to sympathize with her, they must be shown how much she loves her daughter and how fully that love is reciprocated. (The plot might have been more piquant if Laurel, as she matured into a young lady, had been tempted to renounce her mother as a route to the social acceptance she craved\u2014if she had said, \"I'm ashamed of the mother I love.\" But that's another story: Imitation of Life.)\n\nScreenwriter Frances Marion establishes the Stella-Laurel bond early in the 1925 Stella Dallas, directed by Henry King and starring Belle Bennett. Of the infant Laurel, Stella tells Stephen (Ronald Colman), \"Maybe I haven't been much of a mother, but I love my baby\u2014and I'd die if she were taken away from me.\" Whatever kind of mother she may be, she's quite a sight. Strutting through town in her mismatched faux finery, she stokes an almost Technicolor riot in the black and white film, and, though the movie is silent, when Stella is shown sleeping you can practically hear her snore. But Laurel (persuasively impersonated from the ages of ten to twenty by sixteen-year-old Lois Moran) can see the character behind this cartoon. And just as important as Laurel's love is Stella's snooty dismissal by \"nice people.\" They shake her hand as if it were a decayed fish, boycott the birthday party she has thrown for Laurel and libel her within her hearing: \"To think that that dreadful creature we saw today is Laurel's mother! . . . A mother like that is a millstone around her neck.\" Rather than exposing Stella's deficiencies, these rejections serve to solder Laurel's loyalty. Stella becomes her daughter's great lost cause.\n\nStella Dallas, 1925: Belle Bennett as the frowsy Stella, Ronald Colman as her husband, Stephen, and Jean Hersholt as the ungentlemanly caller.\nFor Stella to convince both Laurel and the audience of her noble renunciation, she needs a confederate: Helen (Alice Joyce), the woman her husband wants to marry, and the mother of three boys. \"Why no,\" she protests to Stella, \"I couldn't rob a mother of her only little girl.\" Stella presses her point: \"I'd like people to think she's yours. You're the kind of a mother she could be proud of; I\u2014I ain't. . . . She'll never be nobody with me shackled around one foot.\" The girl will be moving up and not lacking the love of a good woman. In fact, Helen has already prepared Laurel's bedroom, which is about the size of the old Penn Station concourse; on the bed pillow, Stella plants a moist kiss. An intertitle describes Stella and Helen as \"two mothers, united by an understanding beyond words.\" Their benign conspiracy doesn't change Laurel's mind until Stella's letter announcing her alliance with Ed Munn (Jean Hersholt). Helen is not fooled, saying to Stephen, \"That pitiful letter\u2014couldn't you read between the lines?\" Indulgent but myopic, he replies, \"Helen dear, you see with the eyes of an angel.\" And, in a line that reveals a woman's empathy in reading another woman's soul, Helen says, \"No, Stephen\u2014with the eyes of a mother.\"\n\n(Here the writer pauses to compose himself, wipe his own eyes and continue.)\n\nBarbara Stanwyck was no dewy damsel. Like James Cagney, her male equivalent at Warners in the early thirties, she carried the flag of the urban poor, with all their hard-won wisdom, their skepticism, their knowing which rules to play by and which to disregard. Many of her films were fables of the underclass fighting the privileged class, and winning. So Stanwyck seemed an odd choice for the leading role in the 1937 Stella Dallas\u2014where the war of the classes was the film's subject, not just its subtext, and where the screen's toughest cookie would impersonate a self-sacrificing back-street mother. How could a character played by brassy Barbara Stanwyck be willing to forfeit her husband (John Boles) and her daughter (Anne Shirley) to another woman (Barbara O'Neil, who two years later played Scarlett's mother, the matriarch of Tara, in Gone With the Wind) and consign herself to the company of that belligerent oaf Ed Munn (Alan Hale)?\n\nStella Dallas, 1937: Barbara Stanwyck in the frock with the marshmallow sleeves and Anne Shirley as her loving daughter, Laurel.\n\nIn this King Vidor version, Stanwyck plays Stella hard and broad, almost taunting the audience to dismiss her. The former Broadway dancer strides across the room, cranking her arms like a fast-walk marathoner. The sleek clotheshorse sports a dress that resembles a giant banana split, with oversize marshmallows for triceps. She often shouts her lines, as if trying to drown out the voices in Stella's head telling her she's not good enough. At times, Stanwyck seems outside her character, commenting on and subverting the sentiment that the 1925 film embraced; she will cue the audience to pity Stella here, despise her there. But that is her genius. She pushes viewers' sympathy away, then draws it in like a master angler. At the end, resplendent in the serious actress's \"no-makeup\" makeup look, Stanwyck has transformed skeptics\u2014the ones she created in the first half of the film\u2014into true believers. For once, this classy underclass dame told an audience that the poor should leave their best dream, their children, in the care of the upper class. And nothing in Stanwyck's moist eyes and triumphant smile, at the final fadeout, says that she doesn't believe it herself.\n\n### ANNA KARENINA\n\n\"All happy families are alike,\" begins Leo Tolstoy's 1875-77 novel. \"Each unhappy family is unhappy in its own way.\" Brought to the big or small screen a couple dozen times, in Germany, Italy, Argentina, and of course Russia, Anna Karenina offered the domestic tragedy of a dutiful wife betraying her stuffy husband and precious son for the unreliable affection of Count Vronsky, in a liaison that leads to ostracism and death. The character gave meaty roles to Greta Garbo\u2014twice, in the silent Love and a talkie remake\u2014and to Vivien Leigh and Keira Knightley. The last might seem a distant third in competition with a pair of cinema's most incandescent actresses, but Knightley gives the boldest interpretation of the woman who, as generations of schoolchildren and moviegoers know, threw herself under a train after her adulterous passion turned to ashes.\n\nLove (1927), directed in lambent, late-silent-film fashion by Edmund Goulding, cast Garbo opposite the Vronsky of John Gilbert, MGM's silent dreamboat and, at the time, the actress's off-screen paramour. \"John Gilbert and Greta Garbo in Love,\" the posters proclaimed, as if the movie were a gossip headline. The studio might also have been acknowledging that this wasn't exactly Tolstoy's story, since screenwriter Frances Marion contrived to give Anna a happy ending. (Three years after they part, Karenin dies and the lovers are reunited.) Of all the Vronskys, Gilbert's is the rare one to court Anna without realizing that she is married; he is also the most smitten by, and loyal to, the woman on whom he would brand a social stigma. And Garbo was playing the femme fatale established in her earlier movies. To her lovers she was an obsession; to her husbands, only an ornament. Karenin (Brandon Hurst), with dead eyes and a sewn-on scowl, cares not about her faithlessness but about his position. \"I may find it convenient not to know,\" he says in the intertitles. \"I shall do nothing [until] you have made public your guilt.\" That happens soon enough, followed by Karenin's painful penalty: forbidding Anna to see their son, Serezha.\n\nLove, 1927: Garbo as Anna Karenina with Philippe De Lacy as her seraphic son.\n\nIn any Garbo film, the camera is her true mirror and lover. But both of her Anna Kareninas place her son, not Vronsky, near the center of Anna's heart. Played in Love by the seraphic nine-year-old Philippe De Lacy, Serezha exudes a pre-Raphaelite sensuality that makes him the ideal love object for a repressed and doting mother. When Anna and Serezha meet after a long separation, he takes her head in his hands as he kisses it from above, in patented Garbo style. She gives him many loving pats on all four cheeks. Not that the relationship is creepily incestuous\u2014only that Anna is more suited to being a mother than a mistress, and that Serezha, as her son, will be ever faithful to the woman who bore him. Because, after all, she's Garbo.\n\nThe difference between Freddie Bartholomew, who played Garbo's son (here called Sergei) in the 1935 Anna Karenina, and De Lacy is the difference between an asexual fawn and an androgynous satyr, between Peter Pan and pagan Pan. Anyway, Bartholomew needn't have radiated preadolescent warmth in competition with Vronsky, since Fredric March is starchy and sexless in the role. (Perhaps Garbo was in part to blame for that. Director Clarence Brown later recalled that, when March \"showed signs of wanting to get romantic . . . before each love scene Garbo put a small piece of garlic in her mouth.\") The more powerful character is Basil Rathbone's Karenin, a figure of preening righteousness who is jealous less of her passion for her lover than of her devotion to her son\u2014a bond that excludes him. When Karenin surprises Anna during her clandestine visit to Sergei, he orders her out of the house. As Garbo walks down the grand staircase and out the door, her posture becomes more stooped, with all the albatrosses of a disapproving Russian society on her lustrous shoulders. She knows she will never see her son again. A rendezvous with the train awaits.\n\nBetween her signature Hollywood roles as Scarlett O'Hara in Gone With the Wind and Blanche DuBois in A Streetcar Named Desire, Leigh played Emma Hamilton to her husband Laurence Olivier's Lord Nelson in Lady Hamilton (That Hamilton Woman in the United States), Cleopatra to Claude Rains's Julius Caesar and, in 1948, Anna Karenina. Director Julien Duvivier surrounded Leigh with one towering presence\u2014Ralph Richardson as Karenin\u2014and two actors who made little impression in their important roles: twenty-three-year-old Kieron Moore as Vronsky and Patrick Skipwith (in his only film appearance) as Sergei. Both are, in effect, her doomed boys\u2014the charges to whom she imparts her tragic wisdom\u2014even as Karenin treats her less as a wife than as a misbehaving daughter. From the first scene, in a train to Moscow, where Anna shows a picture of her adored son to a stranger who we soon learn is Vronsky's mother, to her final farewell to Sergei, Leigh summons a fiery maternal love. But the movie produces more visual smoke than emotional steam in chugging toward its tragic destination.\n\nJoe Wright's brazen, exhilarating Anna Karenina (2012), scripted by Tom Stoppard, reimagines the Tolstoy story as comic opera that blooms into grand opera. Wright stages most of the action in a reproduction of a nineteenth-century theater, while Sidi Larbi Cherkaoui's virtuoso choreography sets dozens of characters awhirl and aghast at the reckless affair that Anna (Keira Knightley), wife of a respected judge (Jude Law), pursues with the dashing Vronsky (Aaron Taylor-Johnson). All the world, the world of the Russian aristocracy, becomes the stage on which the lovers play out their scandalous affair.\n\nAnna Karenina, 1935: Basil Rathbone as Karenin, Greta Garbo as Anna and Freddie Bartholomew as their son prove that this unhappy family is different.\nKnightley's Anna is the girlish wife of a good, gray man for whom she feels respect; her love is poured into their beautiful son, Serhoza (Oskar McNamara). Once smitten with the precocious Vronsky, who might be Serhoza a dozen years older, she is raised to heavenly ecstasy before tumbling into the abyss of shame. It's a nervy performance, acutely attuned to the volcanic changes that a na\u00efve creature must enjoy and endure on her first leap into mad passion. Wright allows more sympathy for Law's Karenin, a decent fellow stunned by his wife's betrayal, than for Taylor-Johnson's Vronsky, a primping soldier somewhat startled to realize that his lovely conquest thinks the tryst is eternal love. Why would Anna risk not just her marriage but access to her son? It's just not sensible. But that is Wright's\u2014perhaps Tolstoy's\u2014point: that motherhood is a mature commitment and adulterous passion a meteorological accident.\n\n### IMITATION OF LIFE\n\n\"Hope for the best, expect the worst. \/ You could be Tolstoy or Fannie Hurst.\" Mel Brooks's lyric for a song from his 1970 film The Twelve Chairs expressed the received opinion of the novelists' relative literary merits. Hollywood had a different opinion. From Tolstoy's novels it made expensive adaptations only of Anna Karenina and Resurrection, filmed as 1934's We Live Again, with Fredric March this time seducing Anna Sten. (The 1956 War and Peace was Italian producer Dino De Laurentiis's big idea.)\n\nHurst, though, was movie gold. Back Street, detailing the decades-long affair of a working-class woman and the banker who loves her but will never leave his wife and kids, got spun into three Universal weepies: with Irene Dunne in 1932, Margaret Sullavan in 1941 and Susan Hayward in 1961. Hurst's novel Sister Act, about a widowed father and his musical family, birthed Warner Bros.' Four Daughters (with Claude Rains, Gale Page and the three Lane sisters) in 1938, plus two sequels and the 1954 musical remake Young at Heart. A Hurst short story became Warners' Humoresque (1946), in which a mother (Ruth Nelson) encourages her son (John Garfield) to pursue his career as a violinist and warns him against entanglement with his benefactress (Joan Crawford). But all these were the merest lace valentines compared with the films made from Hurst's 1933 novel Imitation of Life: director John M. Stahl's in 1934 and Douglas Sirk's in 1959. These two hit the jackpot, both as mother-daughter tales and as potent parables for an America mired in racial confusion.\n\nThe star billing for the two movies\u2014Claudette Colbert, Warren William and Rochelle Hudson in 1934, Lana Turner, John Gavin and Sandra Dee in 1959\u2014suggests that the story is about a widowed woman whose stalwart beau becomes the object of her daughter's affections. Mother and child fighting over the same man? Can't-miss melodrama. Actually, that is only the docile subplot in a much more volatile generational tangle: between the leading lady's African-American maid and her daughter, who is so light-skinned she can pass for white. She does so, breaking both the color barrier and her mother's heart.\n\nThe 1959 Imitation of Life, described by Dave Kehr in The New York Times as \"a Lana Turner soap opera turned into an exercise in metaphysical formalism by Sirk's finely textured and densely layered images,\" earned Oscar nominations for both Juanita Moore (as Annie, the mother) and Susan Kohner (as Sarah Jane, the daughter). The climactic scene of Annie's funeral, when Sarah Jane throws herself on her mother's coffin to beg forgiveness, sticks in the viewer's gut as one of the signature movie moments of repentance and catharsis.\n\nKohner was born of mixed ethnic legacy. Her father was the Czech-Jewish Paul Kohner, the renowned agent who had worked as a producer at Universal on late silent and early sound films, including the 1931 Spanish-language version of Dracula. Her mother was Lupita Tovar, the Mexican actress who played the female lead in Dracula's Spanish-language \"remake,\" which was shot at night on the same sets used for the Bela Lugosi picture. (Susan's sons became directors: Paul with American Pie, Chris with About a Boy.) Kohner pours a nervy sensuousness into Sarah Jane; she radiates light in her attempts to enter posh society by seeming Caucasian, and heat in her verbal takedown of the clearly outmatched Lana. Annie sympathizes with Sarah Jane's dilemma, asking, \"How do you tell a child that she was born to be hurt?\" But the young woman crushes her mother's spirit by refusing to attend \"a colored teachers' college,\" by denying the legacy of her mother and her race, and by proclaiming, \"I'm somebody else. I'm white . . . white . . . white!\"\n\nImitation of Life, 1959: Lana Turner and her daughter Susie (Terry Burnham) with Juanita Hall and her light-skinned Sarah Jane (Karin Dicker); the grown-up girls would later be played by Sandra Dee and Susan Kohner.\nThe social and racial vectors of the 1934 Imitation of Life are even more wrenching, both on-screen and in the two actresses' biographies. The black mother, Delilah (Louise Beavers), has devised a pancake recipe that makes a fortune for her employer Bea (Colbert). Delilah is now a millionaire, sharing a mansion with Bea. The two women share an affection and respect unusual for films of that period; they are partners in business and in the mutual raising of their two kids. Yet Delilah's wealth never gives her a feeling of equality with whites. She insists on sleeping in the basement, as she always did, with her rebellious daughter Peola (Fredi Washington).\n\nPeola's wonderful mother is exactly the wrong mother for her, so far apart are their respective ideas of what is possible and proper. When the girl leaves home and finds a job as a restaurant cashier, Delilah begs her to come home, and Peola denies her mom, as Peter did Jesus. \"Even if you pass me on the street,\" she later says, \"you'll have to pass me by.\" Bea is shocked by this callousness, but Peola tells her, \"You don't know what it's like to look white and be black. You don't know. I can't go on this way any longer!\" Washington's plangent contralto makes this line a heartbreaker.\n\nThe stocky, Christmas-angel-faced Beavers, who had worked as a maid to silent screen star Leatrice Joy (Mrs. John Gilbert), went on to play maids in many movies; she also followed Ethel Waters and Hattie McDaniel as the problem-solving maid in the early-fifties sitcom Beulah. But the true revelation and pathos come from Washington, an elegant, light-skinned beauty born of black parents in 1903. Getting work in Dudley Murphy's indie films of the early sound era, she played Duke Ellington's dancer girlfriend in 1929's Black and Tan and a prostitute opposite Paul Robeson in 1933's The Emperor Jones (where makeup darkened her skin so viewers would not think Robeson was consorting with a white woman). Washington's exotic outsider status pursued her, defined her, wherever she went. She occasionally accompanied her husband, trombonist Lawrence Brown, with the Duke Ellington orchestra on dates in the American South. Josephine Baker's adopted son Jean-Claude has said that the black musicians \"could not go into ice cream parlors, so she would go in and buy the ice cream, then go outside and give it to Ellington and the band. Whites screamed at her, 'N\u2014\u2014 lover!' \" To feed her friends, Washington really did pass for white.\n\nImitation of Life, 1934: Louise Beavers and Sebie Hendricks (later Fredi Washington), with Claudette Colbert and Baby Jane Quigley (later Rochelle Hudson).\nIn those days the word \"passing\" had an almost tragic heft. It meant not only secretly renouncing one's race but becoming a \"real\" American\u2014to enjoy the privileges of equality and anonymity, at a time when a tenth of all citizens were denied access to voting booths, hotels, restaurants, restrooms and a fair shake from the white majority. \"America\" was a club whose membership numbered 120 million or so, all of them white. Why shouldn't someone who looked as if she belonged in the club try to join it, if only as a kind of double agent? So Peola decides that, to escape second-class citizenry and be accepted by whites, she must convince them that she is \"one of them.\" Gliding on the edges of white society, she wonders why, if her mother's employer could ascend to its elevated center through money, a light-skinned young woman couldn't get there with prettiness.\n\nAt least, in the two versions of Imitation of Life, Hollywood addressed the subject of passing\u2014a clear metaphor for color-blindness. \"I know it's asking a lot,\" Peola says. \"But I've got to live my own life.\" On the movie's terms, this is only an imitation of life, a sham citizenship. Peola is a tragic figure whose quest is seen as a betrayal of mother, roots, self. And Washington, who lived to be ninety, got only one more role in a Hollywood film: One Mile from Heaven (1937), where she plays a \"black\" woman who claims that the \"white\" girl she cares for is her natural daughter.\n\n### IMITATION OF IMITATION OF LIFE\n\nLike the United States, Mexico had a race problem. With much intermarrying between the indigenous population and the descendants of Spanish \u00e9migr\u00e9s, the country was its own rainbow coalition, or contradiction. Most of the movie stars were light-skinned; those who weren't often played comic or villainous relief. And at least once, in its own golden age, the Mexico film industry didn't ignore the race issue. Director Joselito Rodr\u00edguez's 1948 film Little Black Angels (Angelitos Negros) is officially based on a play by the Cuban F\u00e9lix B. Caignet, but it's a transparent gloss on Imitation of Life: the story of a light-skinned woman who disdains all people of color\u2014including her dark-skinned mother\u2014without realizing that she is a member of the race she hates. Available with English subtitles in the Pedro Infante Collection from Warner Home Video, the movie is crucial, touching and bizarre.\n\nInfante, the actor-singer megastar who was the Crosby, Sinatra and Elvis of 1940s-'50s Mexico, plays the nightclub entertainer Jos\u00e9 Carlos, deeply in love with Ana Luisa (Emilia Gui\u00fa), a blond schoolteacher who's snooty to those of darker hue. She's downright rude to Jos\u00e9 Carlos's Afro-Cuban bandmate, Fernando (Chimi Monterrey) and thinks that Jos\u00e9 Carlos's performing in blackface is demeaning, but he shrugs that off with a laugh: \"You're very lucky: You're being courted by two men. A white one by day and a black one by night.\" As boy and girl get closer to marriage, unexpected opposition emerges from Ana Luisa's black maid Merc\u00e9 (Rita Montaner), who has raised her selflessly since infancy; Ana Luisa believes she is an orphan. Merc\u00e9's reason for fighting the betrothal: she is Ana Luisa's mother, though she has never told the girl. Love has made Merc\u00e9 endure both her maid status and the contempt her daughter sometimes shows her\u2014as when she denies Merc\u00e9's plea to attend the wedding.\n\nA year later, a child, Bel\u00e9n, is born, and her skin is dark. Ana Luisa, horrified, and fearful of being ostracized by her rich friends, blames Jos\u00e9 Carlos for having had black ancestors. Now would be the time to tell Ana Luisa where she came from, but the doctor says her heart is too weak to take a severe shock. Four years later, Bel\u00e9n (Titina Romay) is a darling child with the loving disposition of her father\u2014and her grandmother. But Ana Luisa still resents and recoils from her: \"Why didn't God send me a white, blond girl? I would have loved her so much.\" Desperate for affection from her mother, Bel\u00e9n puts pancake makeup on her face, saying, \"I want to be white so Mama will love me.\" Touched by this poignant travesty, Ana Luisa kisses Bel\u00e9n for the first time.\n\nNow the tensions boil over and the ironic dialogue gets hotter. Ana Luisa exclaims, in agony, \"Oh, if only my mother were alive!\" and, in anger, \"I'd kill myself if my mother were black.\" (Hearing that, Jos\u00e9 Carlos slaps her.) Merc\u00e9, ground down by the ill feelings, is getting weaker. And when Ana Luisa calls her a \"damn black woman\" and strikes her, Jos\u00e9 Carlos shouts the words that were denied to Ruth Chatterton and Gladys George in their Madame X films, but which this movie has been galloping toward: \"Don't! She is your mother!\" Shock; guilt; death; sensation! Cue tears that would flood the Rio Grande.\n\nLittle Black Angels, 1948: Pedro Infante cradles his dark-skinned daughter, flanked by the local priest (Nicholas Rodr\u00edguez) and the child's grandmother (Rita Montaner).\n\nPretty amazing, eh? The movie extends the subterfuge of Imitation of Life's Peola, who knew her mother was black but needed to pass for white, to Ana Luisa, so at ease in her dismissal of blacks because she believes she is white. Finally learning the true color of her mother and herself, Ana Luisa must now accept all those of color or despise herself forever. But as unusual as the Little Black Angels narrative is, the casting is even stranger. Except for Monterrey, all the actors playing African or mixed-race characters, including Merc\u00e9 and Bel\u00e9n, were white. The movie is a tale of mother love submerged in race hatred\u2014a plea for harmony among people of all colors\u2014done in blackface.\n\n### On Movie Motherhood\n\nThere are movie mothers, moms, mamas, maws and mums. Most of them were sweet, loving and caring, and taught us lessons to live by. But there were bad ones, too\u2014like Mildred Pierce, played by Joan Crawford\u2014who could put us on the wrong track.\n\nThere were many actresses who made wonderful movie moms: Barbara Stanwyck, Marlene Dietrich, Selena Royle, Ann Harding and more. To me, Fay Bainter was the perfect mom. She was gentle but strong and understanding\u2014a complete mom. I don't know if she had any children or not, but she obviously liked children, or was a fantastic actress.\n\nI played a mother once in Seven Brides for Seven Brothers, though that wasn't the focus of the film. It was a Technicolor musical, and like most musicals and most color films, it was not a \"mother movie\" per se (the black and white films focused strongly on character, whereas color seems to dissipate the drama). In a musical, mothering is rarely that important. You know it will be a short scene with lots and lots of music. But I had a couple of children of my own when I did this film, and because I wanted the children and adored the children, I didn't have to work at being a mother for this role.\n\nAbove all, I really miss the tearjerkers, but they are not in vogue today. I'll always remember my favorite, Irene Dunne in I Remember Mama. Irene had a great sense of whatever was needed, and an intelligent sense of humor that was catching and on the mark. I have always loved everything she did.\n\n\u2014Jane Powell\n\nGigi, 1958: Isabel Jeans instructs her niece Gigi (Leslie Caron) in the art of being a courtesan.\n\n## AUNTS\n\n* * *\n\n## Sometimes another relative can step into the role.\n\nMovie aunts come in two basic packages. One is the woman who may not be suited for motherhood but is obliged to assume the rearing of her absent sister's or brother's child. She tends to be the disciplinarian, almost a prison matron, imposing structure and strictures on a young, intrusive presence. If she has a child of her own, as Aunt Petunia Dursley does in the Harry Potter movies, she spoils her offspring and torments the foundling. The second type is the free spirit, a believer in inspiration over obedience. When a child is entrusted to her care, she teaches her most important, joyous lesson: to embrace life with a capital L.\n\nIn Auntie Mame (1958), Manhattan diva Mame Dennis (Rosalind Russell) offers her ten-year-old nephew Patrick (Jan Handzlik) a most liberal education, including enrollment in a Greenwich Village progressive school, where the children \"romp around naked\" playing Fish Families, and all the finest adult literature. When Patrick shows Mame a list of words he doesn't understand\u2014\"neurotic, heterosexual\"\u2014she says, \"You won't need some of these words for months and months.\" As Patrick (now played by Roger Smith) grows stolid in young manhood in his twenties and gets engaged to a snooty \"Aryan from Darien,\" Mame expertly scotches the alliance with one of her scandalous cocktail parties; as she serves flaming cocktails, she asks, \"There now, are we all lit?\" Plenty of men we know wish they'd been raised by Mame, who famously said, \"Life's a banquet, and most poor suckers are starving to death!\"\n\nAuntie Mame, 1958: Mame (Rosalind Russell) defends her nephew Patrick (Jan Handzlik) against the boy's stuffy lawyer (Fred Clark).\nPatrick Dennis's education was downright Amish compared with the lessons that Leslie Caron in Gigi (1958) receives from her aunt Alicia (Isabel Jeans): how to be a courtesan. In Alan Jay Lerner and Frederick Loewe's original movie musical, based on a Colette novella and directed by Vincente Minnelli, teenage Gigi learns how to please a man, win his patronage and, when the game becomes tiresome, end it. \"Marriage is not forbidden to us,\" Alicia tells the girl, \"but instead of getting married at once, it sometimes happens we get married at last.\" Alicia's ethos is purely monetary: \"Wait for the first-class jewels, Gigi. Hold on to your ideals.\" But this marvelous movie is a romance. When Gigi grows up and conquers the aristocrat Gaston (Louis Jourdan), they both shock their jaded relatives by falling into that bourgeois trap called love. Alicia has taught her student the price of everything; Gigi, on her own, learns the value of values.\n\nMore often, aunts are the stuffy beneficiaries of a child's fresh perspective. In Disney's 1960 version of Pollyanna, the Eleanor Porter novel that Mary Pickford had filmed forty years before, Hayley Mills is the orphan who revives the spirit of rich, crabby Aunt Polly (Jane Wyman). While Polly teaches the girl good posture, Pollyanna helps her aunt relax, enjoy life and let the town build a new orphanage. Even a brief case of paralyzed legs can't wipe the smile permanently from this Miss Sunshine, who manages to implant a smile in Aunt Polly's Grinchy heart.\n\nAny adaptation of Mark Twain's Tom Sawyer, like David O. Selznick's 1938 film, will put his guardian at a disadvantage. May Robson's Aunt Polly must be the disciplinarian whose strategies backfire\u2014as when she punishes Tom (Tommy Kelly) for his playing hooky by insisting he whitewash the fence and he persuades other boys to do the job\u2014and who, for all her sensible strictures, can never wrest sympathy from the spirit of impish, impious youth that Twain created. Robson clucks at Tom and all but winks at the audience, to show her, if not Polly's, complicity in the fun.\n\nRobson was at one time set for the role of Auntie Em in The Wizard of Oz (1939). The part went instead to Clara Blandick, who had played Aunt Polly in the 1938 Twain sequel Tom Sawyer, Detective. Her casting would make Em perhaps the most perplexing figure in the canon of movie aunts. She is a woman worn from running the Gale farm; her husband Henry (Charley Grapewin) and the three farmhands (Ray Bolger, Jack Haley and Bert Lahr), who will later show up in Oz, are clearly her subordinates. When her niece Dorothy (Judy Garland) begs for attention\u2014evil Miss Gulch (Margaret Hamilton) wants to dognap Toto\u2014Em, preoccupied with the urgency of counting chickens, dismisses her with a \"Find yourself a place where you won't get into any trouble.\" That would be a corner of the farm where Dorothy sings \"Over the Rainbow.\" With the aid of a cyclone and a bump on the head, Dorothy is in Oz. Black and white becomes ravishing Technicolor, and a solitary girl defeats the Wicked Witch and saves the kingdom. She has escaped prosaic Kansas for the poetry of Oz, the mundane dialogue of the farm for the immortal songs of the Emerald City. In the scarecrow, the Tin Man and the Cowardly Lion, Dorothy has found the family she dreamed of.\n\nFrank Morgan's Professor Marvel, later the Wizard, must teach the runaway Dorothy\u2014and the audience\u2014that Em's stern ways conceal a love for her niece. He says that her aunt is near death from worry, because \"Someone has just about broken her heart . . . someone she loves very much.\" The movie hasn't exactly shown us this; the Em on screen looks like a Dust Bowl refugee in a Dorothea Lange photo and behaves like Ma Joad without the fiery optimism. She may well be a woman, like the mother in A Tree Grows in Brooklyn, who feels love for the girl but hasn't the gift of expressing it\u2014of taking time from her duties to caress and console a lonely niece. But when the Professor intuits (really, invents) Em's tender feelings, Dorothy believes it.\n\nThat is why, throughout her stay in the Merry Old Land of Oz, the orphan girl keeps yearning to return to Kansas and Auntie Em. She decides that the mother figure she has thought of as an oppressor is really her guardian angel\u2014that the rainbow is in Auntie Em's heart. Dorothy has convinced herself that, whatever the seductions of Oz, she can find happiness back in her own farmyard.\n\nThe Wizard of Oz, 1939: Dorothy (Judy Garland) petitions her Auntie Em (Clara Blandick) and Uncle Henry (Charley Grapewin) to save Toto from the clutches of the wicked old Miss Gulch (Margaret Hamilton).\n\nGone With the Wind, 1939: Who pulls the strings at Tara\u2014Scarlett (Vivien Leigh) or Mammy (Hattie McDaniel)?\n\n## MAMMIES AND NANNIES\n\n* * *\n\n## They pour their love or pain into other people's children.\n\nThe very word \"mammy\" is antique, anachronistic and possibly offensive. It summons a vision of the Old South that is, to modern sensibilities, a magnolia-scented gulag where black slaves toiled in the fields to sustain a cotton economy or worked in the owners' houses as a permanent indentured servant class. Director Steve McQueen's 12 Years a Slave, based on the memoir of a free Northern black man sold into Southern bondage, serves as scalding testimony to a stained era.\n\nYet in those houses, and in the less ornate homes of the South for a century after the Confederacy collapsed, women of color could speak the truth of the powerless to whites, share the affectionate kinship of sisterhood with the women in charge and infuse their values in the de facto raising of white children. If Mammy in Gone With the Wind has no name but her job, she still is a brave and formidable woman\u2014an elemental force recognized by the Academy of Motion Picture Arts and Sciences when, in 1940, it made Hattie McDaniel the first African-American to win an Oscar for Best Supporting Actress.\n\nGone With the Wind (1939), which in terms of tickets sold is still the most popular movie of all time, is remembered as the love story of Scarlett O'Hara (Vivien Leigh) and Rhett Butler (Clark Gable). Misremembered, really, since the object of Scarlett's true and unrequited ardor is the prim aristocrat Ashley Wilkes (Leslie Howard). She marries Rhett not for his sex appeal, which was evident to every female in the movie audience, but for his money, which will restore her social standing after the Civil War and maintain her estate. Indeed, Scarlett's abiding beau\u2014her constant lover and the spur to her ambition\u2014is the land, Tara. Producer David O. Selznick's film, no less than the Margaret Mitchell novel that spawned it, is really about real estate, the eternal obsession of field marshals and homemakers alike. Less a war movie than a woman's picture, GWTW stints on the men's battlefield exploits and sticks with the home-front females: Scarlett, Ashley's demure bride Melanie (Olivia de Havilland) and the Tara house servants Mammy (Hattie McDaniel) and Prissy (Butterfly McQueen).\n\nScarlett, for all her Southern-belle airs, is a modern businesswoman, as ingenious and ruthless as any man in rebuilding her wealth. Trained to dominate by charm, she is unschooled in all the skills Mammy possesses\u2014including bossing Scarlett to uphold the honor of Tara. \"If you don't care what folks says about dis family, I does!\" she thunders, making sure her famished employer eats some food. It is Mammy who summons Melanie to Tara after the death of Scarlett's and Rhett's daughter, Bonnie. She stares smoldering disapproval at Rhett, chides Scarlett for continuing to moon after Ashley and carries much of the narrative in her capacious bosom. Scarlett may be Tara's mistress, but Mammy is its muscle and heart.\n\nThe Member of the Wedding, 1952: Ethel Waters comforts Julie Harris and Brandon deWilde.\n\nThe black servant class persisted long after the official end of segregation; en masse, it has raised several hundred years of white children. In The Member of the Wedding (1952), from Carson McCullers's novel and play, the housekeeper Berenice Sadie Brown (Ethel Waters) takes care of Frankie Addams, a twelve-year-old tomboy (played by twenty-seven-year-old Julie Harris). In a Southern backwater in late summer, the impossible dreams are the most fervent: Frankie wants to accompany her older brother and his bride to Alaska. With her mother dead and her father absent, Frankie begrudges Berenice's attempts to impose discipline but accepts her moral authority. A black woman is the only friend, the only mother figure, this white girl has.\n\nAnother Southern story: In New Orleans in 1918, a child is born who looks like a tiny old man\u2014a wrinkled thing with cataracts, arthritis and a murderer's curse on his head; his mother died giving birth to him. Thus begins The Curious Case of Benjamin Button (2008), the expansion by screenwriter Eric Roth and director David Fincher of F. Scott Fitzgerald's 1922 fable about a person who was born an old man and got younger each day for the rest of his long life. His father leaves Benjamin (Brad Pitt) on the steps of an old folks' home, where he is cared for by a black maid, Queenie (Taraji P. Henson), who cradles him with a mother's love, saying, \"You as ugly as an old pot, but you still a child of God.\" In this nursery for the aged, death is the biggest part of life; that suits Benjamin, who was born with an old man's wisdom\u2014the knowledge that every story has a beginning and an end, even if they run in opposite directions. Benjamin never forgets Queenie through his Forrest Gump-ish series of adventures, though of course he outlives her. Courting the beautiful Daisy (Cate Blanchett), he becomes her husband and, as he spirals into his teens, tweens and infancy, her most beloved child.\n\nThe Help, 2011: Cicely Tyson, the matriarch of maids, with the young Skeeter (Lila Rogers).\n\nFinally, Jackson, Mississippi, in the 1960s, a full century after the Civil War. White women live in fancy homes, not plantation mansions, but they still have women darker than themselves to cook the food and mind the babies. An ensemble morality play, The Help (2011), directed by Tate Taylor from Kathryn Stockett's novel, draws its characters in broad, swift strokes, so that the heroines and harridans are instantly distinguishable. Black is beautiful and white is either hateful or na\u00efve\u2014unless the white person is the Stockett stand-in Skeeter (Emma Stone), herself raised by a woman of color (Cicely Tyson) and now planning a book of testimony by the town's black maids. In itemizing the small-mindedness of the ruling class, the film raises admiration for the maids and outrage at the contempt with which their employers treat them. There is ferocious moral power in the saintly stoicism of Viola Davis's Aibileen and a stoic warmth in Octavia Spencer's Minny, for whom revenge is a dish best served as a chocolate pie. Davis was nominated for a Best Actress Oscar, and Spencer won for Supporting Actress. Hattie McDaniel would be pleased.\n\nBlossoms in the Dust, 1941: Foundling-home founder Greer Garson and her brood of \"illegitimate\" children.\n\n## A MOM BY ANY OTHER NAME\n\n* * *\n\n## Because it's the right thing to do.\n\nThe misanthropic wit Oscar Levant told this brief fable: \"A son murders his mother and cuts her heart out to present to his sweetheart. . . . In his hurry he stumbles, and the disembodied heart that he clutches in his hand cries out, 'Did you hurt yourself, son?' \" Levant's explanation, that \"the mother's solicitude for her son was unimpaired,\" assumes a bond that extends from womb to tomb\u2014and, in this case, beyond it. But the maternal spirit can also fill a woman who didn't give birth to the objects of her love. Sometimes she inherits a child from the wife of a previous marriage. Sometimes kids just show up, and instinct or ethics or bravado compels a woman to let them stay. She may find a lonely child on a foundling-home doorstep, on the mean streets or on a deserted hilltop mansion. Whatever the circumstances, these women become mothers because it is the right thing to do. Their Bible lesson might be: Suffer the little children, that they suffer no more.\n\n### MOTHERS OF LOST CHILDREN\n\nA woman's kindness toward children can be indiscriminate; she opens her arms to all wounded creatures. Edna Kahly Gladney (Greer Garson) in Blossoms in the Dust (1941) founded the Texas Children's Home and Aid Society of Fort Worth, which gave shelter, education and, crucially, legal legitimacy to children of unwed parents. Gladney, herself born out of wedlock, persuaded the Texas state senate to strike the word \"illegitimate\" from birth records, declaring in a gallery speech that galvanized listeners (and earned Garson her first of five consecutive Best Actress Oscar nominations) that \"there are no illegitimate children. There are only illegitimate parents.\"\n\n\"It's a hard world for little things,\" says Rachel Cooper (Lillian Gish) in The Night of the Hunter (1955), the miracle of mood, depredation and redemption directed by Charles Laughton and scripted by James Agee from Davis Grubb's novel. In the West Virginia swamps of the 1930s, Rachel has no government approval, only a selfless impulse to collect unwanted children. John Harper (Billy Chapin) and his younger sister, Pearl (Sally Jane Bruce), need her intervention. Their father has been hanged, after first hiding his stolen money; their mother, Willa (Shelley Winters), has fallen for the Reverend Harry Powell (Robert Mitchum), who wants that money and will threaten the children or marry Willa\u2014and worse\u2014to get it. As seductively malevolent as Powell is, with his baritone parables and the words L-O-V-E and H-A-T-E tattooed on his knuckles, so beatifically empowered is Rachel, describing herself as \"a strong tree with branches for many birds.\"\n\nEdna and Rachel give homes to the young that society has discarded. Christine Collins, played by Angelina Jolie in Clint Eastwood's Changeling (2008), loses the person she loves most: her nine-year-old boy Walter (Gattlin Griffith, later Kate Winslet's son in Labor Day). Several weeks after his disappearance, which triggers a highly publicized manhunt, word comes that the boy has been located. Instantly Christine sees that this \"Walter\" (Devon Conti) is not her son; but the police insist that he is. Case closed. When she points out the obvious differences in height and dental records, the officer in charge (Jeffrey Donovan) orders her confined to a psychiatric ward.\n\nThe Night of the Hunter, 1955: Lillian Gish protects Ruby (Gloria Castilo) and her other foundling children with love and a rifle.\nBased on the notorious Wineville Chicken Coop case, Changeling is an indictment of the corruption rampant in the municipal government, the police department and the medical establishment of Los Angeles in the 1920s; it's almost Chinatown, for real. As a heartbroken mother determined to find her son, dead or alive, Jolie fights on through a stream of tears; her sighs come with shrugs worthy of Atlas. Sleuthing at great personal risk and in a mire of depravity, Jolie's Christine is a reallife hero of the old, urban West, with no weapon but her inexhaustible love.\n\n### STEPMOTHERS\n\nMoviegoers would have to wait until 1965 for a luminous portrait of a stepmother\u2014first, granted, a nun postulant and a governess\u2014who charmed seven children and their stern father by singing to them in the Austrian hills. A Best Picture Oscar winner and, in real dollars, still the third-biggest hit in American box-office history (after Gone With the Wind and Star Wars), Robert Wise's The Sound of Music (1965) built on Julie Andrews's skill with children as the nanny in the previous year's Mary Poppins. Her character, Maria, served as an antidote to all rotten stepmoms, an inspiration to children with a \"new\" mother and, in more recent years, camp fodder for the revival engagements of the Sing-a-Long-a Sound of Music, at which the audience dressed in convent or Tyrolean garb and warbled the Rodgers and Hammerstein secular hymns in harmony with Maria.\n\nBy that time the traditional family structure had been sundered into postnuclear shards; more than a third of all kids had stepmothers, and these second-class citizens of golden age movies saw Hollywood toss some affirmative-action love their way. (Though they might not always have loved the results.) Director Chris Columbus's Stepmom (1998), in which Isabel Kelly (Julia Roberts) marries the ex-husband of Jackie Harrison (Susan Sarandon) and must try to control her two new children, gaudily mixed the zingers of sassy sitcom kids with the emotional blackmail of Jackie's cancer. All this banter and badgering nearly obscures Jackie's sensible homily about two-mother families: \"They don't have to choose. They can have us both. I have their past, you can have their future.\" But please: tens of millions stepmothering in domestic America, and this is how Hollywood addresses the issue?\n\nThe Sound of Music, 1965: Julie Andrews, in her transformation from governess to stepmother of the von Trapp children.\nNow we move on to another death, this time of a child, in a much better movie, Don Roos's The Other Woman (made in 2008, released in 2011), from Ayelet Waldman's novel Love and Other Impossible Pursuits. Emilia (Natalie Portman) had set her sights on Jack (Scott Cohen), her law firm's new boss, and pretty much seduced him away from his obstetrician wife Carolyn (Lisa Kudrow). Soon after they were married they had a daughter, Isabel, who died just after coming home from the maternity ward. A few years later, Emilia is still adjusting to losing Isabel while being a mother, kind of, to Jack and Carolyn's son, Will (Charlie Tahan).\n\nAs a second wife who in many eyes still wears a home-wrecker tag, Emilia has two missions: to win Will over from his efficient, stern mom, and to earn membership in the boys' club at home. Jack needs constant reassurance that he hasn't broken his first family for a failed second time around. Emilia must connect with Will because it might prove her marriage a success. Eventually they build an intimacy of equals, which for Will is a different sort of bond from the one he shares with his father, but warmer and just as strong. The relationship isn't at all romantic, but by the end she's become Will's first girl friend. If The Other Woman turns Wife No. 1 into a caricature and the husband into a cipher, it eventually ingratiates by the connection of young Tahan and Portman, in what is arguably her subtlest and most natural performance. Her Emilia is a woman wrestling with demons who overcomes them when she discovers that one of her opponents\u2014her balky stepson\u2014just showed up in her corner.\n\nThe Other Woman, 2011: Natalie Portman tries making peace with her stepson, Charlie Tahan.\n\n### SURROGATE MOMS\n\nIn the decades when an unwed woman with a child could stoke scandal and banishment from polite society, Hollywood occasionally played the story as a comedy of mistaken identity. In Bachelor Mother (1939), Ginger Rogers is a salesgirl who notices an abandoned newborn on the steps of a foundling home on Christmas Eve and takes it inside to hand to the authorities. They think she's the mother, and so do the owner of her department store (Charles Coburn) and his playboy son (David Niven). She must declare the child her own, tapping her own unexpected yen for motherhood and winning, under the rules of romantic comedy, a rich husband and a richer father-in-law, who overcomes curmudgeonly scruples to declare, \"I don't care who's the father\u2014I'm the grandfather!\" In similar style, two young masters of romantic comedy movies, screenwriter Norman Krasna and director Garson Kanin, evaded Production Code prudence and created this effervescent charmer of a movie about the unwed nonmother who became a wed mother.\n\nEverything's proper in The King and I (1956), director Walter Lang's film of the Rodgers and Hammerstein Broadway hit. The widow Anna Leonowens (Deborah Kerr) has come with her young son to Siam as governess to the children of the King (Yul Brynner) in the mid-nineteenth century. Anna and the King are each emotionally isolated: she as a widow and a foreigner, he as a monarch bred to believe in his own infallibility. When they finally, lightly touch in the majestic \"Shall We Dance?\" it has the thrilling impact of two worlds colliding in harmony. The King is weaned from his stubbornness by seeing Anna's deft connection with his kids. She teaches not just their schoolwork but also lessons in independence, even rebellion against their master. And he learns that a woman is more than a concubine; she is a professor of the civilizing impulse.\n\nBachelor Mother, 1939: David Niven suspects that the infant Ginger Rogers is feeding is her own baby.\n\nIn its casting, the movie may rasp against modern ethnic sensitivity, let alone visual sense; none of the Siamese courtiers and only a few of the children are played by Asians. Yet this remains the most satisfying of all the story's film versions, which include Irene Dunne and Rex Harrison in Anna and the King of Siam in 1948, Jodie Foster and Chow Yun-fat in Anna and the King in 1999 and, the same year, an animated feature called The King and I. Brynner, in the role that defined his career, and Kerr, with her appealing blend of gentility and moral passion, make a lovely mismatched couple\u2014stern father, warm mother\u2014who tiptoe or polka to the brink of breaking convention. Shall they dance? Divinely. Will they consummate their affections? Impossible, impossible!\n\nAt the beginning of Tim Burton's Edward Scissorhands (1990), Peg the Avon lady (Dianne Wiest) wanders into a forbidding castle on a hill and discovers Edward (Johnny Depp), a grim automaton with shears at the ends of his arms. With equal parts salesmanship and motherly determination, she starts chattering: \"Do you live up here all by yourself? What happened to your face? No, I won't hurt you. But at the very least, let me give you a good astringent.\" In a trice she has told the creature, \"I think you should just come home with me.\" There, amid a quiet riot of coordinated pastels, she shows Edward pictures of her husband and kids, offers him the run of the refrigerator and gets him new clothes. Edward is the innocent other, a literary type that stretches from Kaspar Hauser to Being There's Chauncey Gardiner, and Peg is the gentle mom of fifties sitcoms, approached for once with love, not derision. Despite her best efforts, this story of the Frankenstein monster as dreamboat will not end well, but it will end beautifully. Burton's career-long theme\u2014of the misfit as gentle genius\u2014gets its most bittersweet rendition here, perhaps because this time, the creature has a loving surrogate mom.\n\nEdward Scissorhands, 1990: Dianne Wiest invites the boy with the cutlery cuticles (Johnny Depp) to come live at her place.\n\nThe Blind Side, 2009: Sandra Bullock reads to her son (Jae Head) and her adopted son (Quinton Aaron).\nThe Blind Side (2009) earned Sandra Bullock a Best Actress Oscar as Leigh Anne Tuohy, the Memphis mom who took in the homeless black teen Michael Oher (Quinton Aaron), prodding and coaching him toward football renown. In writerdirector John Lee Hancock's film of the Michael Lewis nonfiction book, Leigh Anne struts through life with a scary assurance; she's a blond tornado, looking for people to challenge, causes to champion. Finding Michael gives her a goal that unites home, school and her preternatural ambition. It's as if Erin Brockovich had been given charge of E.T. What Leigh Anne sees in Michael is a gentle stillness; he's like a Buddha statue transplanted to Tennessee, and his only impediment to gridiron stardom is that he doesn't like to hurt people. Leigh Anne can solve that problem too. She has just the energy to push this soft boulder up the hill of achievement. No Miss Daisy was ever so driving.\n\nThe Blind Side has its cheesy moments, and its fable of a black child saved from his addict mother by a noble white woman allows for indelicate racial generalizations. But it offers a heroine rarely seen in modern movies (including Bullock movies). Too often, women with career drive are portrayed as rudderless and incomplete; they must end up being educated and tamed by men. But Bullock's Leigh Anne is a take-charge gal who pursues her vision and achieves it with very little help from men\u2014and with no apologies. She has the holy mission, the elbows-out attitude and, let's say, the balls of a tough male film hero. Bullock could have dedicated her Oscar to all strong women, on-screen or off.\n\nA Raisin in the Sun, 1961: A Chicago mother (Claudia McNeil) is torn between helping her two grown children (Sidney Poitier and Diana Sands) and realizing her own dream of a move to the suburbs.\n\n## WHEN MOMS COLLIDE WITH THEIR KIDS\n\n* * *\n\n## Sometimes it's tough sharing space with a grown child.\n\nOn the \"Babysitting\" episode of NPR's This American Life, first broadcast in 2001, Myron Jones and his sister Carol Jones Bove, then near seventy, described being raised in 1940s Buffalo, New York, by their strict widowed mother\u2014a woman so deficient in giving love or accepting it that her teenage kids invented a family for whom they said they babysat, just to get free of Mom for a few hours a night. Carol recalled, \"When I was thirty-five, I lashed out at her . . . told her how I felt about her. And she sat in a chair in the kitchen and she was crying. And I had never even seen her cry before. And when I finally stopped talking, she said, 'I did the very best I could.' And I thought, Oh my God, she did. Her best was so bad. Her best was so empty.\"\n\nIn the centuries before women's financial liberation, most men, even those of limited means, had some choice in their line of work: bootblack or bank teller, street sweeper or salesman. Women had many jobs\u2014cook, housekeeper, child producer, teacher, nurse\u2014under a single compound title: wife-mother. And aside from enduring this 24\/7 grind, they were expected to seem to be enjoying it, as if the ordeal were a blessing, as if they were to whistle, and smile and nurture, while they worked. The law of averages would indicate that not all women were physically or emotionally suited to these demanding chores. Even if they did the very best they could, their children might judge the effort as bad or empty. Were the mothers at fault? The children? Or was it all a question of perspective? Maybe they just weren't suitable companions; husbands and wives sometimes take decades to come to that conclusion.\n\nHere are a few examples of mothers and children whom personalities or circumstances put at emotional odds. Whether the figures are heroic or pathetic or comic, or a mixture of all three, they have trouble sharing the same space, especially when grown children live with their mothers. These psychodramas and psychological comedies recognize the truth of Amy Tan's observation, in The Kitchen God's Wife, \"Whenever I'm with my mother, I feel as though I have to spend the whole time avoiding land mines.\"\n\n\"Seems like God didn't see fit to give the black man nothin' but dreams,\" says Lena Younger (Claudia McNeil), quoting her late husband, Big Walter, in A Raisin in the Sun (1961). \"But He did give us children to make them dreams seem worthwhile.\" Those dreams may soon explode like fatal fireworks in a two-bedroom Chicago apartment that also houses Lena's thirty-five-year-old son, Walter Lee (Sidney Poitier); her college-age daughter, Beneatha (Diana Sands); and Walter Lee's wife, Ruth (Ruby Dee), and their son, Travis (Stephen Perry). Beneatha, hoping to become a doctor, needs money to pursue her studies. Walter Lee, chafing at his chauffeur's job, hopes to start a business of his own, a liquor store, with some friends. Ruth, strapped for funds in a marriage grown cold, is pregnant with a second child she plans to abort. They all could use a chunk of Big Walter's $10,000 insurance policy. But Lena has her own mission: to move her family out of a ghetto flat and into a nice house in a white neighborhood. That means no investment in a business she disapproves of: Walter Lee's liquor store. \"You butchered up a dream of mine,\" he cries. \"You, mama, who's always talkin' 'bout her children's dreams.\"\n\nWinner of the New York Drama Critics Circle award in 1959, A Raisin in the Sun made Lorraine Hansberry, twenty-eight, the first black woman to have a play on Broadway. She took her title from a Langston Hughes poem\u2014which asks whether \"a dream deferred\" dries up \"like a raisin in the sun?\"\u2014and the kernel of her plot from a lawsuit that her realtor father brought against racially segregated housing in Chicago (Hansberry v. Lee), which the Supreme Court decided in his favor in 1940. But the play and its faithful film version, with the original cast under Daniel Petrie's astute direction, contain echoes of other classic family parables and other powerful, burdened wives and mothers. Lena is like Ma Joad in The Grapes of Wrath, determined to move from the meager home she cherished to a future of uncertain hope. She could also be Linda Loman in Death of a Salesman, if Linda had believed in Willy's dream and tried to impose it on her grown sons after he died. Lena hasn't raised her family through war's battles and blitzes, but when the Youngers get a glimpse of their possible home in the suburbs, her children give her a straw hat with the dedication: \"To our own Mrs. Miniver.\"\n\nAs much as Walter Lee and Beneatha love their mother, they also resent her clinging to the ways of her generation. Beneatha harbors all manner of advanced ideas, from black nationalism to modern dance, but when she declares herself an atheist, Lena slaps her hard and thunders, \"Now you say after me: In my mother's house there is still God..\" Walter Lee, the chauffeur-servant who wants to be a boss, bitterly proclaims, \"I'm a volcano, I'm a giant, and I'm surrounded by ants!\" Smoldering with a black man's aspirations, he feels he's treated like a child by the mother who often summons the memory of Big Walter to demean her son. (\"I'm waiting to hear how you'd be like your father, be the man that he was.\") Haunted and taunted, frustrated and emasculated, he finally assails Lena: \"You're the head of this family! You run our lives the way you want, you spend the money the way you want.\" In this titanic war of wills, only Ruth can apply the balm of perspective. \"You just got some strong-willed children,\" she tells her mother-in-law, \"and it takes a strong woman like you to keep 'em in hand.\"\n\nJuggling a half-dozen hot-button issues in a finely crafted problem play, Hansberry (who died of pancreatic cancer at thirty-four) never let agitprop overwhelm her flair for family drama; she gives each character compelling reasons for speaking up in pride and pain. The entire cast rises to the splendid occasion she created for them. Poitier, who took a year off from movie stardom to appear in the play, is the marquee name, but McNeil\u2014just six and a half years older than the actor playing her son\u2014is the dominant force. With an ample frame suggesting an urbanized, updated Hattie McDaniel, McNeil plants herself at the center of the screen and rarely yields the spatial foreground or moral high ground. In his memoir, The Measure of an Actor: A Spiritual Autobiography, Poitier recalls that McNeil \"was in complete dominance over most of the other members of the cast.\" On the matter of whether the mother or the daughter should be the focus of the piece, \"we argued constantly.\" Poitier might have skipped the debate and ceded preeminence to McNeil's grand, grounded performance. She makes Lena a mother with whom it would be a challenge and an honor to collide.\n\nThe first TV play to be expanded into a movie, Marty (1955), won four Oscars: Best Motion Picture, Director (Delbert Mann), Actor (Ernest Borgnine) and Screenplay (Paddy Chayefsky). In a decade when Hollywood was fighting the incursion of TV with, to quote the Cole Porter song from Silk Stockings, \"glorious Technicolor, breathtaking CinemaScope and stereophonic sound,\" Marty signaled both a return to the black-and-white intimacy of thirties movies and a harbinger of the Sundance genre of indie films. It was a relationship dramedy about an ordinary guy\u2014the thirty-four-year-old Bronx butcher Marty Piletti\u2014with nothing but a friend (Joe Mantell), an eventual girlfriend (Betsy Blair) and a mother (Esther Minciotti), whom he lives with. He spends his days chopping meat and his weeknights with his pal Angie. Saturdays, at his mother's insistence, he goes to a local dance, looks for a nice girl and comes home alone, with defeat that nags like an impacted molar.\n\nMarty, 1955: Esther Minciotti as Ernest Borgnine's comforting, confining mother.\nMarty, whose \"charm lies in his almost indestructible good humor\" (says Chayefsky in the script), and his mother are decent people with irreconcilable agitations; they could be a long-married couple whose arguments have become rituals. She wants him to find a nice girl; he thinks he can't. One Saturday evening at the kitchen table, Marty tries to explain his situation: \"Sooner or later there comes a point in a man's life when he's gotta face some facts. And one fact I gotta face is that whatever it is that women like, I ain't got it. I chased enough girls in my life, I went to enough dances. I got hurt enough. I don't wanna get hurt no more. . . . I'm gonna stay home tonight and watch the Hit Parade.\" When his mother keeps pressing her case, the quiet man explodes: \"I'm ugly, I'm ugly, I'm ugly\u2014Ma, leave me alone!\" Just as quickly, he subsides back into resignation. \"All right,\" he says, \"so I'll go to the Stardust Ballroom, I'll put on a blue suit and I'll go. And you know what I'm gonna get for my trouble? Heartache. A big night of heartache.\"\n\nThe scene's wellspring is excellent writing from Chayefsky, but it flows through Borgnine's precise and intuitive orchestration of emotion. He recognized that Marty has accepted his failure\u2014a disease he's lived with so long he can diagnose it without getting upset\u2014and that only when his mother pushes the point does he trip-hammer into rage. After his outburst, he sits down; pats his mother's hand twice, as if to say it's all right; and eats his spaghetti. (That was a Borgnine touch. As he told Robert Osborne during a TCM interview in 2009, \"It's one of those gestures: I understand, Mom, that's how I feel.\") So Marty won by making his mom realize he thinks he's a loser. But guess what? He's wrong, she's right, because later in the film, Miss Wonderful\u2014or, in the Chayefsky universe of livable compromises, Miss Okay\u2014is waiting for Mr. Ugly.\n\nHollywood quickly got the Marty message. Within three months of its Oscar victory, MGM released The Catered Affair (1956), another Bronx tale from a Chayefsky teleplay (adapted for movies by Gore Vidal) and starring Borgnine. This time, instead of Esther Minciotti as the woman of his house, the actor had Bette Davis. Borgnine is Tom Hurley, a cab driver who has nearly saved enough money to buy his own hack; Davis is his wife, Agnes Hurley, whom the decades have left about as chipper as a DMV drudge at the end of a horrible day. One morning, their daughter, Jane (Debbie Reynolds), tells them that she will be marrying her beau Ralph (Rod Taylor). Nothing fancy, just a quick service with the immediate family and no reception. But when Ralph's middle-class parents start bragging about the big weddings they gave their daughters, Aggie gets competitive, even as the guest list rises toward two hundred for what Tom calls \"this criminal breakfast we're serving a bunch of strangers.\" Ready to plunder his taxi savings, Aggie tells Jane, \"You're going to have a big wedding whether you like it or not. And if you don't like it, you don't have to come.\"\n\nIn the TV version, Thelma Ritter played Aggie. It requires a leap of faith, and fondness for the magnificent Bette, to believe that Davis's grand gestures can be shoehorned into a lumpen character in a tiny apartment. But under Richard Brooks's direction, she pushes Aggie past parody into a snapshot of bluster and defeat. As she makes the bed, Aggie offers Jane a mother's wisdom on wedded bliss: \"Marriage is a big thing like some girls don't know nowadays. You gotta make sacrifices, and when the children start comin' you gotta put 'em first, ahead of everybody. . . . And then one day you'll find out a lotta time's gone by, and you wake up knowin' this is the way it's always gonna be, just like this, day after day, year after year, just the same. . . . And don't get married thinkin' it's only a good time. It's not a bad time but it's not a good time, livin' all your life with one man and strugglin' to raise the children decent.\" In a single spouting of autobiography masquerading as advice, Aggie has escalated from acidic realism to primal scream.\n\nThe Catered Affair, 1956: Bette Davis eyes her bride-to-be daughter, Debbie Reynolds.\n\"If you've never been hated by your child,\" the real Davis once said, \"you've never been a parent.\" (Her daughter B. D. Hyman published a memoir, My Mother's Keeper, after which the star disinherited her and her children.) As a mother, on-screen or off, Davis might have been a handful, and in The Catered Affair Aggie's bitter whims test Jane's patience. Reynolds's smart tactic is to glide through the turmoil by ignoring it; she might be closing her eyes and picturing Mom as played by Mary Astor. Jane knows that the wedding is something she doesn't need or want but her mother does. Decades earlier, Aggie's father had offered Tom $300 to marry her. She has held up her end of the connubial contract dutifully but not warmly; when she slips out of her dress, she tells her husband to \"turn around.\" And now, with Jane leaving, she and Tom will be alone together for the first time. No wonder the sour, stoic Aggie collapses in sobs\u2014and why, when discovered, she shrugs and says, \"Well, that was a dumb thing to do.\" A lady, especially a movie queen in a Bronx tenement, has to salvage her dignity.\n\nHaving imagined Debbie Reynolds as the child of Ernest Borgnine and Bette Davis, you must now picture Reynolds, forty years later, as the mother of Albert Brooks. Who could the father have been\u2014Albert Einstein? (Which happens to be Brooks's birth name.) In Mother (1996), the actor-director-cowriter plays John Henderson, devastated and impoverished by the failure of his second marriage. Years before the trend of America's adult children moving back in with their folks exploded into a societal phenomenon, John decides to return home. By staying with his widowed mother Beatrice (Reynolds), he thinks he will be able to unearth the root of his problems with women. It's as if he figured that therapy would be more successful if he lived with his shrink\u2014or, rather, with the person who made him feel he needed a shrink.\n\nBeatrice isn't sold on the idea; she's accommodated herself to the single life, and to a man friend. (\"We're not intimate, dear,\" she tells her son. \"We just have sex occasionally.\") And she pretty clearly is disappointed in John: she introduces him to her friends as \"my other son.\" But he moves back into his old bedroom, with the furnishings of his youth, to promote his DIY psychotherapy. Often he acts as Beatrice's analyst. She says, \"I love you, John\"; he replies, \"I know you think you do, Mother.\" The odd couple argues over food: his organic and fresh, hers generic and years old. Scanning her refrigerator, which he calls \"a food museum,\" he tries some frozen sherbet in what she says is its \"protective ice coating.\" One bite, and he observes, \"This tastes like an orange foot.\" All right, so culinary habits evolved from Beatrice's prime to John's. That's how they eventually find solace: by understanding that two people are different, even or especially if one gave birth to the other. In her first starring role in a quarter century, Reynolds infused Beatrice with a delicate comic underplaying\u2014organic, not generic\u2014that lets the viewer decide if this mother was right or wrong, or just a woman trying to remember what she loved about her amiably neurotic son.\n\nMother, 1996: Debbie Reynolds has a maternal issue\u2014son Albert Brooks.\n\"She respects me, but she doesn't want to become like me,\" says Jane Fonda of her daughter in Neil Simon's California Suite (1978). \"We have a perfectly normal mother-daughter relationship.\" In the 1976 Freaky Friday, mother Ellen (Barbara Harris) and daughter Annabel (Jodie Foster) did become each other, for a day, by simultaneously shouting a magic curse. A progenitor of body-switch comedies, Mary Rodgers's book and screenplay also gave pert prominence to the distinction of family roles: the mother burdened by domestic chores, the tween girl with a carefree life she thinks is indentured servitude. The humor, befitting a Disney film, is carefree and laundered: Annabel as Ellen calls her father\/husband \"Daddy,\" and, since she's in charge of cooking, prepares a rum-raisin banana-split breakfast. She should move in with Debbie Reynolds's Beatrice.\n\nThe 2003 remake, in a snider time for movie comedy, manages the same easy, mostly sexless fantasy, this time with Jamie Lee Curtis as Tess, the mother, and the momentarily adorable Lindsay Lohan as Anna, the kid. One generation's scandal is the next one's rite of passage, as when Tess, in Anna's body, realizes her daughter had her navel pierced. And Anna is shocked to pinch her flesh and realize she's old, \"like the Crypt Keeper.\" The mild satirical point is that we are what we are even when we look like someone else. Says Tess-Anna to Anna-Tess, \"When you get your body back, you're grounded.\"\n\nFreaky Friday, 1976: Jodie Foster and Barbara Harris endure a one-day body switch.\n\nA woman with a teenage or adult child has to acknowledge, at least implicitly, a challenge to her own youth and sex appeal. Some mothers get the message early, like Charlotte Haze in Stanley Kubrick's 1962 film of the Vladimir Nabokov novel Lolita. Charlotte may thrust her charms under the nose of visiting scholar Humbert Humbert (James Mason), but one look at the fourteenyear-old Lo (Sue Lyon) in her skimpy bikini, with her downy skin and a vixen's smile, obliterates all thoughts of mature womanhood in Humbert the besotted nympholept. A telling satire on, among many topics, the American worship of youth as it blossomed in the 1950s, Lolita pushes its point by encasing Charlotte, whom the Nabokov novel describes as \"a weak solution of Marlene Dietrich,\" in the zaftig shrillness of Shelley Winters. Viewers may not share Humbert's taste for underage flesh, but they may be as glad to see Winters meet her watery end here (thunderstorm, speeding car) as they were to see Montgomery Clift drown her in A Place in the Sun.\n\nMother-daughter resentment over good looks is even pricklier when the older woman is the dish, the younger the dishrag. In The Mirror Has Two Faces (1996), Columbia professor Rose Morgan (Barbra Streisand) feels totally out-glammed by her pretty sister Claire (Mimi Rogers) and their haughty, resplendent mother, Hannah (Lauren Bacall), who presses her advantage by telling Rose to put on some makeup. Rose: \"I am wearing makeup,\" adding, \"What's the point? I'll still look like me, only in color.\" Rose soon gets her Bette Davis-in-Now, Voyager makeover and her all-but-ideal husband (Jeff Bridges). She also gets to make up with her mother. In a lovely scene shared by two generations of Jewish movie queens\u2014the forties cover girl Bacall and the sixties sensation Streisand\u2014Rose asks, \"How did it feel to be beautiful?\" Hannah's reply: \"It was wonderful!\" At that moment, the mean parent is humanized, as Rose realizes that most mothers, like most people, have their own memories, dreams and regrets.\n\nFreaky Friday, 2003: Lindsay Lohan and Jamie Lee Curtis take on the roles of Foster and Harris\u2014and Harris and Foster.\n\nLolita, 1962: Shelley Winters concentrates on ignoring the affection her daughter, Sue Lyon, betrays for the new boarder, James Mason.\n\nMildred Pierce, 1945: Joan Crawford slaves her way to success for her thankless child, Ann Blyth.\n\n## BAD SEEDS\n\n* * *\n\n## What's a mother to do when she senses something terribly wrong in the child she loves?\n\n\"How sharper than a serpent's tooth,\" keened King Lear, \"it is to have a thankless child.\" He was speaking of Cordelia, the one of his three daughters who loved him enough to tell him the bleak truth, and the only one to accompany the monarch on his journey into madness. Whereas Lear misread his good daughter, some movie mothers are blinded by love from realizing, until too late, the devious minds behind their children's seraphic faces.\n\nThe trickle of evil-kid movies became a flood after the success of 1978's Halloween, which begins with a mother discovering that her adorable eight-year-old son has just slashed his sister to bits. But even in its more decorous golden age, Hollywood occasionally made films about children who were thankless, soulless, rotten to the core or simply unable to do the right thing by a mother's love. Here be monsters\u2014underage sociopaths. And\u2014if we look in the mirror of our last example, Make Way for Tomorrow\u2014here we may be.\n\nOn his Campbell Playhouse broadcast of October 29, 1939, Orson Welles described Booth Tarkington's 1918 novel The Magnificent Ambersons as \"the truest, cruelest picture of the growth of the Middle West, and the liveliest portrait left to us of the people who made it grow.\" In its tale of two intertwined dynasties, the upstart Morgans and the old-money Ambersons, Welles's 1942 film version contrasted the go-getter grace of early-twentieth-century entrepreneurs with the dissolution, the spread of ethical and emotional flab, of a young man born to wealth. \"True\" would describe this story of a son who sabotages his mother's one chance for happiness; \"cruel\" the film's fate at the scissors of the RKO executives who had sponsored Welles's previous effort, Citizen Kane; and \"lively\" Ambersons' sustained grip, even in its mutilated form, on audiences today.\n\n\"Old money\" in Indianapolis meant three generations. Major Amberson (Richard Bennett) had created the family fortune by \"buying and building and trading and banking,\" by fighting and conniving for it, employing brains and elbows and possibly fists. Was the major a robber baron? His grandson George Amberson Minafer only assumes he was a baron, an American monarch, which must make George a royal\u2014if only a royal pain. His mother, Isabel (Dolores Costello), a sad beauty who loved the upstart Eugene Morgan (Joseph Cotten), instead married the middle-class cipher Wilbur Minafer (Don Dillaway). George, the only child of that drab marriage, is unlovely and unloving but not unloved. Indeed, his mother has lavished all her devotion on him\u2014too much\u2014and in the scorching sun of her indulgence, he spoils.\n\nA \"girlie-cutie\" in ringlets, young George had been a spiteful prig: the towns people couldn't wait for the day \"when that boy would get his comeuppance.\" At first, Eugene tried to overlook George's faults: \"Oh, he's still only a boy. Plenty of fine stuff in him. Can't help but be. He's Isabel Amberson's son.\" That's ardor trumping evidence. In fact, George more closely resembles Wilbur's spinster sister Fanny (Agnes Moorehead), who secretly loved Eugene and, when ignored, poured her poison into her nephew. By his college years, George's unchecked privilege has calcified into contempt for everyone less lucky (or, as he thinks, less worthy) than he. It will fester into envy of Eugene: of the older man's success as a manufacturer of automobiles, and of the love that Eugene and Isabel share\u2014the film's one pure connection. Eugene's daughter, Lucy (Anne Baxter), who inherited her father's good sense and pacific poise, might have offered George the lifeline of her affection, if only his vision weren't occluded by moral myopia, if only he would stop insulting her father and his business. \"Automobiles!\" he spits out as an expletive to the startled Lucy. \"People aren't going to spend their lives lying on their backs in the road letting grease drip in their faces.\"\n\nAs the mechanical marvel of motion pictures displaced the stage in bringing narrative myths to the public, so did cars replace carriages; they signaled the freedom of Americans to go where and when they wanted. Economically, that mobility was also vertical, spurring Eugene's wealth while the Amberson fortune is sapped by George's profligacy. The older man is the gleaming technological future; the younger, the crumbling aristocratic past. George, who disdains all professions, from lawyers and bankers to tinker-mechanics like Eugene, tells Lucy he plans to be \"a yachtsman.\" But his real career, his one mission, will be to keep his mother for himself and away from Eugene. Raised as the sole object of her love, he won't admit that anyone else can occupy her heart.\n\nIn a letter to Isabel, Eugene finally dares to address her misshapen maternal bond. \"Oh, dearest woman in the world, I know what your son is to you and it frightens me. . . . Will you live your life your way, or George's way? Dear, it breaks my heart for you, but what you have to oppose now is your own selfless and perfect motherhood. Are you strong enough, Isabel? Can you make a fight?\" She cannot. She takes a European holiday with George, which he prolongs as her health ebbs. By the time they return, she is near death, and again George refuses Eugene's request to visit his beloved. (Her dying words: \"I'd like to have seen him. Just once.\") Let Eugene be the inventor; George has a unique gift for destruction. By the end he might agree that he killed his mother.\n\nOn the Campbell Playhouse radiocast of Ambersons, Welles had played George. Perhaps Welles thought he would squeeze too much of his grand, wastrel charm into the George role on film; the twenty-six-year-old actor-director's outsize showmanship might have won some sympathy for this agent of malice and misery. He gave the part to Tim Holt, a twenty-two-year-old cowboy star with a thin voice and no hint of screen charisma. Holt drove home the theme that a small personality, pursuing narrow vindictiveness, can bring down a great house. In some of the film's many elegant long-take long shots, Welles plants Holt just off center screen in a family debate; yet George's meanness dominates the action, because his mother and the other characters are too weak to challenge his viperous will.\n\nDeparting for Latin America to research his next project, It's All True, Welles left behind his 132-minute Ambersons. Then another family crime occurred. After a poorly received preview screening, the George Minafers at RKO cut fully a third of the film, down to 88 minutes. They then destroyed the excised footage, confounding all attempts at restoration. (In 2002, director Alfonso Arau shot Welles's screenplay for an Ambersons TV movie. Starring Madeleine Stowe as Isabel, Bruce Greenwood as Eugene and Jonathan Rhys Meyers as George, it ran 139 minutes.) From a mistaken form of love for her son, Isabel had criminally spoiled him. From the frantic desperation to carve a marketable commodity out of Welles's second masterpiece, the RKO executives turned the Amberson property into its own ruined, haunted mansion. Like the demolition crew at the end of Citizen Kane, they consigned precious footage to the flames, saying, in effect, \"Throw that junk in.\"\n\nMildred Pierce (Joan Crawford) had one nice daughter: Kay (Jo Ann Marlowe). But this pert tomboy, who wanted to be a dancer, died of pneumonia at ten. In the 1945 Mildred Pierce, from the James M. Cain novel, Kay's death leaves the bereaved mother with her elder daughter, Veda (Ann Blyth), the very definition of thanklessness. Veda might have admired Mildred\u2014the family's sole support after the departure of her shiftless husband Bert (Bruce Bennett)\u2014for working hard to give her the material bounty she never had. That is the American ethos: parents strive so their children can thrive. Veda, though, has the snobbish airs of a girl born to wealth; she could be George Minafer transplanted from turn-of-the-century Indianapolis to 1940s Glendale, California. Mildred, in her voice-over narration, calls her \"a young lady with expensive tastes.\" Translation: ungrateful bitch.\n\nThe mild contempt that Veda ladled on her kid sister (\"You act like a peasant,\" she told Kay) ripens to venom against her mother. When she learns that Mildred's job is waitressing, she explodes\u2014\"Did you have to degrade us?\"\u2014ignoring the elementary math that the cash Mildred earns buys the new home, clothes and status that Veda thinks she deserves. Mildred's keen entrepreneurship\u2014buying a roadside property from playboy Monte Beragon (Zachary Scott) and expanding it to a flourishing chain of restaurants\u2014brings them wealth, but Veda wants to be more than moneyed. She wants to be old money, with its attendant luxe and dissolution. Another American story: children take for granted the gifts their parents struggled to lavish on them; they resent the grimy grind that produced the largesse. To Veda, the poloplaying idler Monte represents wealth unsullied by labor. \"I want the kind of life that Monte taught me,\" she tells Mildred, \"and you won't give it to me.\"\n\nVeda can never forgive her mother for making money the old-fashioned way: by earning it. The girl has a simpler route to her own wealth: she becomes engaged to a rich young man, then takes money from his family to walk away. That check finances Veda's dreams of escape. \"With this money I can get away from you,\" she tells Mildred in a memorably serpentine soliloquy. \"From you and your chickens and your pies and your kitchens and everything that smells of grease. I can get away from this shack with its cheap furniture, and this town and its dollar days, and its women that wear uniforms and its men that wear overalls.\" She hates the working class, but not as much as she despises the mother who rose from it. \"You think just because you made a little money you can get a new hairdo and some expensive clothes and turn yourself into a lady. But you can't, because you'll never be anything but a common frump whose father lived over a grocery store and whose mother took in washing. With this money I can get away from every rotten, stinking thing that makes me think of this place or you.\" Mildred tears up the check, slaps Veda and throws her out.\n\nBroke but independent, Veda takes a job singing at a nightclub dive; when Mildred sees this, her face freezes in shame\u2014in a generational flip of the scene in Applause where Helen Morgan performs in a burlesque house with her abashed daughter in the audience. (In 1957, Blyth would star in the Warners biopic The Helen Morgan Story.) And still Mildred wants Veda's love and respect. In her most cynical transaction, Mildred offers Monte one-third of the restaurant business to marry her\u2014\"Sold, one Beragon\"\u2014and buys a new house with a picture of the young Veda atop the grand piano. Veda's brief return and honeyed words (\"I'll never say mean things to you again\") brings tears to Mildred's eyes. She will cry again when she discovers that Veda wants one more thing Mildred owns: Monte.\n\nRanald MacDougall's script adds the framing device of a murder to the Cain novel (to whose plot particulars Todd Haynes adhered much more faithfully in his five-and-a-half-hour HBO miniseries in 2011), and director Michael Curtiz imprisons the characters in the rhomboidal architecture of film noir shadows. Crawford, who won the Best Actress Oscar in her first Warners movie after sixteen years at MGM, should be the dominant figure, sporting career-woman jackets with the shoulder pads of an NFL lineman and radiating the nobility of an ardent mother who, like another of Shakespeare's tragic heroes, \"loved not wisely, but too well.\" Yet it's Blyth who corrals the viewer's appalled fascination. From her doll's face come the most heinous words a mother could hear, including her climactic plea: \"Give me another chance. It's your fault I'm the way I am.\" Veda has decided that, if she's bad, her mother must be the seed.\n\nThe Bad Seed. And Christine Penmark (Nancy Kelly) has the nastiest\n\"Oh, I've got the prettiest mother,\" eight-year-old Rhoda Penmark (Patty McCormack) purrs in a metallic singsong at the start of the 1956 film daughter. An icon of midcentury wholesomeness in blond pigtails with a chirpy voice to charm grownups, Rhoda seems miffed that her classmate Claude won the penmanship medal she coveted but, later, is unfazed at the news of Claude's death at a school picnic. Christine is mystified, but the creepy janitor Leroy (Henry Jones) knows a vicious kid when he sees one. Taunting Rhoda about the missing medal, he says that the state executes children who murder: \"They got a little blue chair for little boys and a little pink chair for little gals.\" (Leroy's intuitive sleuthing will earn him a fiery demise in the basement furnace.) When Christine finds the medal, she confronts Rhoda. \"Do you realize that you murdered him?\" \"But it was his fault!\" Psychopathy always has an excuse.\n\nTrailers for the movie implored audiences to keep its secret: not that Rhoda is elfin evil in a pinafore dress, which is evident from reel one, but that her malignancy may be inherited. (Christine is horrified to learn that her actual mother was a serial killer, Bessie Denker, and infers that the bad seed skipped a generation, from Bessie to Rhoda.) William March's 1954 novel and the play adaptation of the same year by Maxwell Anderson both ended with Christine's suicide and Rhoda's survival. The Hollywood Production Code allowed murder but insisted on retribution, so in director Mervyn LeRoy's studiously stagey film version, which reunited virtually all of the original Broadway cast, Christine survives, and Rhoda returns to the site of Claude's death, where God's lightning strikes her down. (In a 2004 interview, McCormack recalls that, as a good Catholic girl, she had prayed to get the Rhoda role. God answered her prayers too.) For good measure, the movie summons the actors for a curtain call. Kelly, the last to appear, bends McCormack over her knee and gives her a severe spanking, as if a good mother could beat the devil out of a killer child.\n\nBad kids: where do they come from? Hollywood films had invested so much narrative capital in the idealizing of the American family\u2014in ancestry as destiny\u2014that they did handsprings when they had to justify the source of childhood monstrosity. George Minafer? The product of a loveless marriage and a mother's thoughtless worship. Veda Pierce? Her lazy father, Bert, was just as dependent on, and dismissive of, Mildred's efforts to support the family. Rhoda Penmark? Like grandmother, like granddaughter.\n\nMelodrama resides in the far corners of most people's experience. Audiences can distance themselves from genre excesses with a shiver or a shrug, saying, \"Well, my mother wasn't a serial killer.\" It's the rare film, touching issues closer to home, that burrows into the collective conscience and gives a moviegoer more chills and squirms than any homicidal child could. Perhaps that is why so few of these films get made. The 1937 film Make Way for Tomorrow, directed by Leo McCarey from Vi\u00f1a Delmar's screenplay, addresses two uncomfortable, unavoidable topics\u2014the struggles of the middle class in the Great Depression, and the responsibility that adult children owe to their parents\u2014and does so with judicious grace. What Welles said of Tarkington's Ambersons applies here: the movie manages to be simultaneously cruel, lively and true.\n\nThe title sounds chipper; it could be the billboard slogan for some lustrous planned community. But to Lucy (Beulah Bondi) and Barkley Cooper (Victor Moore), married for a half century, \"Make way for tomorrow\" has a dictatorial air, as in: get out of the way for the next generation\u2014you old folks are yesterday. Bark, a bookkeeper, has been out of work for four years, and now the bank has foreclosed on the house where he and Lucy raised their five children. At a family conference, the eldest son, George (Thomas Mitchell), and his wife, Anita (Fay Bainter), agree to take Lucy into their Manhattan apartment, sharing a room with their teenage daughter Rhoda (Barbara Read), while Bark will sleep on the couch in the smalltown home of daughter Cora (Elizabeth Risdon) and her husband Bill (Ralph Remley), who is also out of work. Though it might seem thoughtless, separating the single organism that Barkley and Lucy have grown into, their children have limited space and means, and their own lives to lead. The chain of family obligations goes down the generations, not back up. As Bark says, \"I sometimes think that children should never grow past the age when you have to tuck 'em into bed every night.\"\n\nIn short order, Lucy gets on the nerves of her hosts as much as the squeaks from her rocking chair. Her nattering presence drives Rhoda's young friends away from visiting the apartment. She disrupts a bridge class Anita teaches in the living room and upsets the players with her side of a phone call from Bark, which ends with Lucy promising, \"We'll soon be together always.\" She already suspects that is a fantasy; George has been receiving mail from the Idylwild Home for Aged Women. In a beautiful scene, Lucy anticipates her son's plan by insisting that she go to the home. But George mustn't tell Bark: \"It'll be the first secret I've ever had from him.\" Bark is also being moved, to another daughter out in California\u2014for health reasons, the exasperated Cora proclaims. He and Lucy get one last day together, in Manhattan, where they rekindle their love, far from the children to whom they have become such a bother. \"I guess this is the first time away from home together since our honeymoon,\" Bark notes. And their last.\n\nThe movie validates the testimony of the great French auteur Jean Renoir that \"Leo McCarey understood people better than any other Hollywood director.\" The Japanese screenwriter Kogo Noda recognized the knotty human elements in Make Way for Tomorrow and wrote the script for a similarly poignant tale of aged parents and busy children: Yasujiro Ozu's 1953 masterpiece Tokyo Story. Some critics have charged McCarey, who won Best Director Oscars for screwball comedy (The Awful Truth) and uplifting priestly drama (Going My Way), with ruthless sentimentalism. Make Way for Tomorrow, though, is a spectacular balancing act of humor and dread. Yes, the movie opens with a rendition of \"Mother\"\u2014\" 'M' is for the million things she gave me; \/ 'O' means only that she's growing old\"\u2014and closes with \"Let Me Call You Sweetheart.\" But the first song is meant sarcastically and the second as an elegy for two people, together forever, now parting. Earlier, Lucy says of a movie she saw that it's \"a little sad in places, but it had a happy ending.\" Make Way for Tomorrow is a little sad and a little funny, but it dares to follow its narrative logic to a wrenching conclusion.\n\nTokyo Story, 1953: two aging parents (Chish\u00fa Ry\u00fc, Chieko Higashiyama), one dutiful daughter (Setsuko Hara).\n\nMcCarey allows some jokes at the grown kids' expense, as when a friendly stranger hears that Bark and Lucy have five children, saying, \"I'll bet they brought you a lot of pleasure,\" and Bark replies, \"I bet you haven't any children.\" Toward the end, Robert (Ray Mayer), the mild scoundrel of the siblings, pronounces stern judgment on the grown children when he observes, \"We've known all along that we're probably the most good-for-nothing bunch of kids that were ever raised. But it didn't bother us much until we found out that Pop knew it too.\" The children have behaved selfishly, or at least clumsily, toward their parents, but this is a film of nuances, not extremes; of no real villains, only victims. And no final catharsis. Moviegoers may feel a momentary cleansing when a wicked child gets her comeuppance in Mildred Pierce or The Bad Seed. But some real-life dilemmas elude easy solutions. Such as: Where do people go between old age and death? And: Should grown children disrupt their lives for the parents who sacrificed theirs?\n\nNow Voyager, 1942: A malicious Boston matriarch (Gladys Cooper) schemes to ensure that the life of her daughter (Bette Davis) is a Vale of tears.\n\n## MALEVOLENT MOMS\n\n* * *\n\n## An actress's fast route to Oscar nominations: playing rotten mothers. Throw these mommas from the train!\n\nBad mothers: you know they're out there. Not you, dear reader, or your mom. Sweet and spotless, we're sure. But the law of averages would argue that, among the billions of women who have given birth and raised kids, a few must be really rotten. From misguided notions of the maternal bond, a woman may become a dominating, domineering mother. Or avarice may drive her to propose a marriage between her daughter and her first cousin, and to allow her husband to die without treatment (Bette Davis's Regina Giddens in The Little Foxes). Perhaps she wishes to assert her own youthfulness by initiating an affair with the young neighbor who will become her daughter's boyfriend (Anne Bancroft as Mrs. Robinson in The Graduate). Or she exercises power, not affection, over her child, until and after she dies (Judi Dench in J. Edgar).\n\nSome of these sins stain the four mothers in this chapter. Mrs. Vale in Now, Voyager may be imposing a literal meaning on the old verse \"A son's a son till he takes a wife; a daughter's a daughter the rest of her life\"; she treats her daughter as an unloved maid. Mrs. Phelps in The Silver Cord extends that truism to her two sons. Needing love, she takes it, drains it from them, and cripples them by refusing, when they're grown, to treat them as adults with their own wills and passions. Mrs. Iselin in The Manchurian Candidate uses her son to terrorist political ends. (Note that the mothers in all three films are not referred to by their first names; each is a \"Mrs.,\" a widow concentrating her twisted energies on her children.) In Precious, Mary Johnston prostitutes her daughter to hold on to her man\u2014the girl's father.\n\nAs goodness can reveal nobility, evil can display grandeur. Medea, the queen who killed her children, was a tragic figure and an object lesson in Euripides's fifthcentury B.C. play. Played by Maria Callas in Pier Paolo Pasolini's 1969 film, she is an otherworldly eminence\u2014a ghost whose nature and duty it is to bring the living to death. Twenty-five hundred years of drama have verified the lasting lure of wicked women. And by tracing the descent of \"bad mothers\" into ignominy, the actresses who play them rise in stature. \"Some of my most interesting roles have been completely unsympathetic,\" Barbara Stanwyck said in the late 1940s. \"Actresses welcome such parts, knowing that vitriol makes a stronger impression than syrup.\" Audiences know that too. We may deplore bad movie mothers, but they lure us into their webs.\n\n### NOW, VOYAGER . . .\n\nThis is the movie where a man lights two cigarettes and gives one to the woman he loves\u2014back when smoking was cool. It's the film where Bette Davis's Charlotte Vale transforms herself from spinster to bombshell by dropping twenty-five pounds, choosing a livelier couture, trading sensible shoes for stylish ones, taking off her glasses and tweezing her eyebrows. Casey Robinson's screenplay, from the third in Olive Higgins Prouty's quintet of Vale novels, packs in mother hatred, adoptive-mother love, three nervous breakdowns, two ill-starred engagements, an adulterous affair, a messily platonic relationship and a huge, rich, hostile family. This 1942 tightrope walk above a tub of scalding bathos is also, arguably, the best movie directed by anyone named Irving. Last name: Rapper.\n\nBut as potent as is the smoldering, erotic glow of passion and nicotine between Charlotte and Paul Henreid's Jerry Durrance, and a Max Steiner score that samples Tchaikovsky's grandly romantic Symphony No. 6, the film is also a scathing portrait of maternal possessiveness. Mrs. Henry Vale holds her only daughter's life and potential happiness in a death grip. And Gladys Cooper (who, like Davis, was Oscar-nominated for her performance) embodies her as a severe, sepulchral creature with gray hair in a bun\u2014like a better-dressed, slightly more animated but no less vindictive Mother Bates. Mother Vale has kept Charlotte a virtual prisoner in their Boston mansion since the girl was twenty and dared to fall in love with a man not of her class. Now Charlotte is nearly twice that age, and still, she tells the amiable psychiatrist Dr. Jaquith (Claude Rains), \"I am my mother's well-loved daughter [this with sarcasm]. I am her companion. I am my mother's servant. My mother! My mother!\" Charlotte won't challenge the Vale virago, but Jaquith will: \"[Charlotte] is most seriously ill, thanks to you. My dear Mrs. Vale, if you had deliberately and maliciously planned to destroy your daughter's life, you couldn't have done it more completely.\"\n\nIn believing that the frumpy, withdrawn Charlotte could never attract a husband, Mrs. Vale could be a New England cousin of another plutocrat with a homely daughter, Austin Sloper in Henry James's Washington Square (filmed in 1949 as The Heiress, with Ralph Richardson and Olivia de Havilland). A parent's contempt is reflected in her child's shattered self-image. But Mrs. Vale's true meanness resides in the grim pleasure she takes in imposing her steel will on her only daughter. She has ruined Charlotte out of rancor and for sport.\n\nAfter her recovery in Jaquith's sanatorium and her shipboard affair with Jerry, the transformed Charlotte is prepared to live at home as her mother's equal. (\"Remember that whatever she may have done,\" Jaquith advises, \"she's your mother.\") To Mrs. Vale, though, the new Charlotte is a threat to her domestic despotism. She insists that Charlotte doff the Orry-Kelly gowns and don the old frump attire. \"As to your hair and eyebrows,\" she piquantly adds, \"you can say that often after a severe illness one loses one's hair, but you're letting yours grow as quickly as possible.\" Charlotte cannot be surprised that her mother demands the status quo. Yet something has changed: Charlotte is no longer afraid of her.\n\nBad mothers in movies often say they wish their child had died. Mrs. Vale pushes that curse into derision: \"I should think you'd be ashamed to be born and live all your life as Charlotte Vale. Miss Charlotte Vale.\" By now, Charlotte's external and internal makeover gives her confidence to fight back: \"Dr. Jaquith says that tyranny is sometimes an expression of the maternal instinct. If that's a mother's love, I want no part of it. I didn't want to be born. You didn't want me to be born either. It's been a calamity on both sides.\" Stricken by her daughter's bold veracity, Mrs. Vale has a heart attack and dies. Charlotte blames herself: \"We quarreled. I did it.\" Only a good daughter would feel so guilty. Yet after an earlier argument her mother had tumbled down the Vale staircase, risking killing herself just to spite Charlotte. Now she is dead, and with the added malediction of leaving her substantial fortune to Charlotte, she means to haunt her decent daughter forever. Mrs. Vale's bony hands claw at her from the grave.\n\n### THE SILVER CORD\n\n\"You'll love Mother, she's marvelous,\" David Phelps (Joel McCrea) tells his bride Christina (Irene Dunne). In fact, Mrs. Phelps (Laura Hope Crews) is a marvel of scheming, suffocating possessiveness; she makes Mrs. Vale seem liberal by comparison. But whereas Charlotte's mother ruled with a tyrant's icy hauteur, Mrs. Phelps in The Silver Cord (1933) is jovial, when it suits her, and doting, to a fault. She employs flutter and bluster to shackle her grown sons David and Robert (Eric Linden) and divert outsiders. Christina is a gifted scientist who has returned from Germany to the States to work at the Rockefeller Institute\u2014an appointment Mrs. Phelps dismisses by observing airily that \"science is hardly a profession, is it? It's more of a hobby.\" She then assigns the new husband and wife to separate bedrooms.\n\nIn her close adaptation of Sidney Howard's 1926 play, screenwriter Jane Murfin focuses on how the apron strings of a strong woman can strangle her weak sons. Telling Robert, \"Sit down\u2014no, no, head in my lap,\" for an oddly erotic Piet\u00e0, Mrs. Phelps persuades him to break off his engagement to Hester (Frances Dee) and seals the deal with the malevolent-mother tactic of kissing him on the mouth. (See The Manchurian Candidate.) Later she tucks David into his solitary bed, sits on it, holds his hand and continues her dulcet plotting against Christina. Neither mama's boy\u2014the effeminate Robert or the rugged-looking David\u2014is a match for her. Robert declares that he broke up with Hester because of \"the ideal of womankind that Mother gave us both by being the great woman that she is.\" Christina, who is pregnant, probes David about his mother's hold on him, and \"what she may not do with you\u2014to me, and the baby.\" Yet he remains myopic, saying, \"A man's mother's his mother.\" (As in: the rest of his life.) \"And what's his wife?\" asks Christina, who comes to the sickening belief that she's \"going to have a baby by a man who belongs to another woman.\"\n\nLet us consider Mrs. Phelps's side for a moment. Her battle with Christina may in part be one of their respective generations; the cozy ideals of one age often seem antediluvian to the next. Christina is a modern, indeed pioneering, career woman; she seconds Hester's notion of raising children: \"Have 'em, love 'em and then leave 'em be.\" Mrs. Phelps, raised in an earlier century, makes an eloquent plea for stay-at-home moms. \"Give us our due, Christina. We weren't altogether bustles and smelling salts, we girls who did not go out into the world. We made a great profession, which I fear is in some danger of vanishing from the face of the earth. We made a profession of motherhood. That may sound old-fashioned to you. Believe me, it had its value.\"\n\nIt had, and has. But Mrs. Phelps distorted that value when she transferred the passion missing from her marriage to her sons. Widowed at twenty-five, she effectively became her boys' mother and wife. Any woman to whom David and Robert might feel romantic attraction earned the designation of a home wrecker, the Other Woman. \"And I do not deny,\" she tells Christina, \"that I'd cut off my right hand and burn the sight out of my eyes to rid my son of you.\" In this singeing five-character drama, which deserves to be rescued from obscurity, the main combatants are three strong women; the men are their puny pawns. Crews, reprising her Broadway role under John Cromwell's direction (he staged both the play and the film), reveals Mrs. Phelps's depredation artfully, gradually, like the slow opening of a Venus flytrap. But the movie allows the two modern women to render stern judgment on maternal love turned rancid. Christina: \"You're not fit to be anyone's mother.\" And Hester, as she walks out of this nest of near-incest: \"I'm going to marry an orphan!\"\n\n### THE MANCHURIAN CANDIDATE\n\n\"My boys, my two boys,\" she says with a smile, hugging her son and her husband. \"That's all I've ever cared about.\" Her second husband is Johnny Iselin (James Gregory), a U.S. senator in the 1950s Red-baiting mold, who uses a Heinz ketchup label to determine the number of Communists he insists are in the State Department. Her son, Johnny's stepson, is Raymond Shaw (Laurence Harvey), a Cold War cold-fish misfit who quite systematically kills eight people and, in the twisted political and domestic world of The Manchurian Candidate (1962), turns out to be the hero. And Mrs. Iselin (Angela Lansbury), whom Johnny calls \"Babe\" and Raymond \"Mother\"\u2014which he pronounces as if expectorating a toad\u2014is the complete villainous package: Joe McCarthy's puppeteer, the Red menace and the would-be assassin of a presidential candidate. Frank Sinatra and Janet Leigh also appear in the movie, as two of the three top-billed stars, but Lansbury and Harvey provide the central tension of two exceptional, odious creatures: a mother and her son whom genetics and geopolitics have consigned to a death match.\n\nWriter George Axelrod and director John Frankenheimer's surreal, pertinent, acerbically comic film version of Richard Condon's multi-conspiracy novel proposes the view that America's most rabid anti-Communists were in cahoots with the Kremlin. Johnny is a secret Soviet agent, and Raymond, during his service in the Korean War, was brainwashed by Moscow and Peking to follow their murderous instructions whenever he hears a code phrase about card playing. The next time he hears it, his mission will be to shoot and kill this year's presidential nominee and thus sweep Senator Iselin\u2014the Manchurian candidate\u2014into the White House with, as Johnny's boss and Raymond's controller prophesizes, \"powers that will make martial law seem like anarchy.\" We don't know the identity of the American Soviet boss until Mrs. Iselin says to Raymond, \"Why don't you pass the time by playing a little solitaire?\"\n\nThe Manchurian Candidate, 1962: Mrs. Iselin (Angela Lansbury), the mother of war hero Raymond Shaw (Laurence Harvey). Is she Little Bo Peep or the Red Queen?\n\nIn Lansbury's exquisite, audacious performance, which earned her an Oscar nomination and deserved more, Mrs. Iselin becomes both a ruthless, ingenious manipulator of realpolitik and, in her way, a victim. For her allies abroad also control her; they turned the son she may somehow love into the instrument of death she needs to use. In her last speech to Raymond she says, \"I told them to build me an assassin. I wanted a killer from a world filled with killers, and they chose you because they thought it would bind me closer to them.\" She cradles Raymond's face in her hands. \"But now, we have come almost to the end. One last step. And then when I take power, they will be pulled down and ground into dirt for what they did to you\u2014and what they did in so contemptuously underestimating me.\" Then she gives her son a big slimy smooch on the lips; mother love never seemed so despotic or desperate.\n\nLansbury was just thirty-six during the filming, Harvey thirty-three (his first two wives were, respectively, six and seventeen years older than he). The actor's reptilian sangfroid makes him the perfect mom-hating son\u2014and until nearly the end he doesn't know half of her capacity for evil. In the Mrs. Iselin character, Lansbury goes farther, darker and deeper: she invests a mother's love, a lover's passion and a killer's calculation in the Raymond relationship. Whatever Mrs. Iselin's defining political motives, that kiss is a satanic promise that no one, no nation or ideology, can ruin her son without tasting this mother's vengeance.\n\n### PRECIOUS: BASED ON THE NOVEL \"PUSH\" BY SAPPHIRE\n\nClaireece Precious Jones (Gabourey Sidibe) is an illiterate, grossly obese sixteen-year-old with a wildly abusive mother, Mary (Mo'Nique), and a vagrant father who comes home only to rape his daughter. He has just impregnated the girl for the second time; her first child\u2014dubbed Mongo, for Mongoloid\u2014is brain-damaged and stays with Precious's grandmother. Word of the girl's pregnancy has led to her suspension from junior high school and threatens the family's eligibility for the welfare checks that are their only support. Precious's life is an unrelenting hell, and when a well-meaning soul suggests that things may be looking up, she tartly replies, \"I'm lookin' up. Lookin' for a piano to fall. A desk, a couch, TV. My mom, maybe.\" Mary Johnston is the all-time most vile, deplorable, eye-magnetizing monster movie mother: Momzilla.\n\nThe novel Push by Sapphire (Ramona Lofton) updated the woeful early life of Celie, protagonist of Alice Walker's The Color Purple, from the early-twentieth-century American South to modern Harlem. Like the book, the film made in 2009, scripted by Geoffrey Fletcher and directed by Lee Daniels, is a horror story that becomes a hymn to the transformative effects of schooling. In this nearly all-female saga, Precious finds someone, Ms. Rain (Paula Patton), who gives her time, her passion and, best of all, a sense of hope to the poor kids in her care; she incarnates the view that a good teacher can be the best mother. As demonized as Ms. Rain is idealized, Mary Johnston bends her own talents for abuse to verbal and physical torrents against her daughter. In one tirade (with the obscenities excised here), she roars, \"I shoulda aborted your ass! . . . I knew it when the doctor put you in my goddam hand you wasn't a goddam thing! You wear that smirk on your face, bitch?\" She heaves a glass that lands at her daughter's feet. \"Now smile about that, you fat bitch!\"\n\nVenturing inside Mary Johnston's mind, one finds a disturbed but poignant rationalization for her misbehavior. \"Real women sacrifice!\" she tells Precious. And Mary sacrificed Precious: she allowed her to be raped by her father so that he would continue to have sex with her. To the social worker Mrs. Weiss (Mariah Carey), Mary cries, \"Who else was going to love me? Who else was going to touch me? Who else was going to make me feel good about myself?\" In scenes like this one, audiences must bow, or cringe, before the explosion of malice, cunning and finally self-pity that Mo'Nique displays as Precious's brutal mom. A stand-up comedian by trade, Mo'Nique commands the screen even in those infrequent moments when her character's not doing something awful. She does more than bring what could be a cartoon of malice to vivid screen life; she makes a dreadful emotional sense of the Worst Mom Ever.\n\n### . . . AND NOW, VOYAGER\n\nWe can't leave it at this. Too many exposed nerves, too much genetic detritus in these stories of women whose motherly legacy is the pain of their children. Old movies aren't the only form of entertainment that requires a happy ending. Some book chapters do as well. So let us return to our first bad-mothers example and see if Charlotte Vale managed to crawl from under the rock of her mother's evil and find the sunlight. At night.\n\nCharlotte, the heiress, has checked herself back into Dr. Jaquith's spa-clinic, where she meets Tina (Janis Wilson), the troubled twelve-year-old daughter of Charlotte's erstwhile lover Jerry. Fleeing from conflicts with her own mother, Tina is a next-generation image of Charlotte's dismal life before Jerry. Charlotte takes the girl out for ice cream and camping trips and at night comforts the crying Tina. \"This is Jerry's child in my arms,\" she says in ecstatic voice-over. \"This is Jerry's child clinging to me.\" Propriety, and Dr. Jaquith's strict instructions, may keep Charlotte from resuming her affair with an unhappily married man. But she can be his daughter's mother, in a way, by bringing the child to live with her in the Vale mansion. Jerry gets visitation rights, and Charlotte says, \"We can talk about your child.\" \"Our child,\" he says, correcting her. Their love is expressed through joint custody of Tina and by sharing the cigarettes he lights, two at a time. \"Oh, Jerry, don't let's ask for the moon,\" says the woman who has freed herself from the grasp of a bad mother by becoming a wonderful one. \"We have the stars.\"\n\nWhite Heat, 1949: James Cagney as Cody Jarrett and Margaret Wycherly as the mother who spurred him to the top of the world.\n\n## CRIME AND HORROR MOMS\n\n* * *\n\nWhether breaking the law or battling demons, they never forget their kids.\n\n\"Don't know what I'd do without ya, Ma,\" the gangster Cody Jarrett (James Cagney) tells his enabling mom as he sits on her lap and she pours him a shot of whiskey. \"Well,\" Norman Bates (Anthony Perkins) primly observes, \"a boy's best friend is his mother.\" In two violent genres\u2014crime movies like White Heat and the horror films inspired by Psycho\u2014a mother would seize center screen by working her will on her offspring. That influence was often malignant, occasionally benign. But both varieties tended to illustrate the ferocious devotion of a mother to her child. They also gave actresses the opportunity to embody women who did things, heroic or egregious, rather than simply stand by and watch men escalate the body count.\n\n### CRIME MOMS\n\nIn The Public Enemy, the 1931 gangster film that made him a star, Cagney played Tom Powers, a tough guy with a na\u00efve, doting mother and an abusive father, a cop, who dealt out punishment with a leather strop. (At the end, Tom's enemies deliver his body, wrapped like a parcel, to his mother's doorstep, as she fluffs his bedroom pillows, awaiting her prodigal son's return.) Cagney felt straitjacketed by hoodlum roles and played none in the decade after 1939's The Roaring Twenties. But when he revisited the genre to play Cody, the career criminal in director Raoul Walsh's 1949 classic White Heat, Cagney lent full force to his portrait of a sociopath with an Oedipus complex.\n\nMa Jarrett (Margaret Wycherly, who was Gary Cooper's mother in Sergeant York) pours all her ardor and craftiness into Cody; she massages his roiling skull, offers canny advice on eluding the Feds, runs the gang when he's in prison and goes out to buy his favorite strawberries\u2014inadvertently tipping off the law to the gang's hideout. Cody treats her as mother and spouse, and ignores his slatternly wife, Verna (Virginia Mayo), as if she were a tropical disease. Verna responds by betraying Cody, as any woman but his mother would. Cody recognizes that Ma was also bred to a life of crime. \"Some life!\" he sneers. \"First there was my old man, died kickin' and screamin' in a nuthouse. Then my brother. And after that, it was takin' care of me. Always tryin' to put me on top. 'Top of the world,' she used to say.\" At the end, cornered atop a huge gas-storage tank, he pumps bullets into the facility and shouts, \"Made it, Ma! Top of the world!\" Boooom! Now he can join her.\n\nThis primal modern crime movie\u2014written by Ivan Goff and Ben Roberts, who later created the pinup-detective TV series Charlie's Angels\u2014establishes a Freudian basis for Cody's behavior, exactly as Alfred Hitchcock's Psycho, the primal modern horror film, would do for Norman Bates eleven years later. \"The only person he's ever cared about or trusted is his mother,\" a Treasury Department man (John Archer) tells Hank Fallon (Edmond O'Brien), the undercover agent assigned to infiltrate Cody's gang. \"No one else has ever made a dent, not even his wife. His mother's been the prop that's held him up. He's got a fierce, psychopathic devotion for her. All his life, whenever he got in a spot, he just put out his hand and there was Ma Jarrett.\" Fallon's job is to worm his way into Cody's trust and \"take Mama's place. . . . I'll practice up on my lullabies.\" In another Psycho premonition, the shot of Cody's face dissolving into his mother's anticipates the climactic dissolve of Norman's face into his mother's skull. And after Ma's death, Cody has a Norman-like conversation with her ghost. \"That was a good feelin' out there,\" he says, \"talkin' to her, just me and Ma. Good feelin'. Liked it.\" He pauses to reflect: \"Maybe I am nuts.\"\n\nCriminals of a more ambitious sort\u2014Nazis in South America, just after the war, with a uranium cache\u2014tiptoe Teutonically through Notorious (1946), the Hitchcock thriller famous for a wine bottle filled with uranium dust and the record-length kissing scene between Cary Grant and Ingrid Bergman. Grant, as U.S. secret agent Devlin, must convince Bergman's Alicia Huberman, the promiscuous daughter of a Nazi bigwig, to fly down to Rio and cozy up to another Nazi, Alexander Sebastian (Claude Rains). Her undercover assignment extends to marrying Sebastian, who has fallen in love with her, while taking orders from Devlin, who insults and mistreats Alicia and puts her life in jeopardy; he loves her too, but duty comes first. Indeed, the Nazi, not the Fed, is the only male character with a purchase on audience sympathy: whatever his war crimes, Sebastian is the smitten dupe. The only member of his entourage who sees through Alicia's scheme is his mother (Leopoldine Konstantin, billed as Madame Konstantin). Like Ma Jarrett, the sepulchral Frau Sebastian can smell betrayal in her son's wife.\n\nHitchcock cleverly cast Rains, fifty-six when the movie was shot, as a paternal figure to Bergman, then thirty\u2014a suaver, more affectionate version of Alicia's own father. Konstantin, just three years older than Rains, serves as Sebastian's grasping mother and early warning system for all suspicious outsiders. His sharp instincts occluded by love, he attributes her misgivings to spite: \"You've always been jealous of any woman I've shown any interest in.\" When he discovers the wine bottle Devlin has tampered with, and realizes that Alicia gave Devlin the keys to the cellar, the little man sinks into a cuckolded husband's despair and goes to the only person who might save him. When he begs his mother for help, she tartly replies, \"We are protected by the enormity of your stupidity.\" Their wills now merged in his peril, they begin poisoning Alicia. As the drug takes effect and her senses blur, she sees two figures fuse into a single image: mother and son, Rio's first family of the Third Reich.\n\nArizona \"Arrie\" Clark, known to the law and the tabloid press as Kate \"Ma\" Barker, cared for her kids, at least as played by Shelley Winters in Roger Corman's 1970 film Bloody Mama. The real Ma Barker may have been only an accomplice to the crimes of her four sons, but the script, by Robert Thom and Don Peters, promotes her to mastermind. The 1967 Bonnie and Clyde cued a spate of biopics on the most wanted criminals of the 1930s. Using the same mixture of bloodshed and broad comedy, the Corman film had its own theme song: \"Ma Barker taught her boys \/ To play with guns like toys . . .\"\n\nBloody Mama, like White Heat before it, provides a psychological backstory to explain its protagonist's errant ways: As a girl, Kate was brutalized by her father and brothers. Turning an excess of abuse into an excess of love, Kate dominates her grown sons, Herman (Don Stroud), Arthur (Clint Kimbrough), Fred (Robert Walden) and Lloyd (the young Robert De Niro). What might be tender affection from another mother\u2014washing Herman's back and chest as he takes a bath, flirting with Lloyd to get him to sample her homemade cookies, leading the lads in a group sing of the antiwar anthem \"I Didn't Raise My Boy to Be a Soldier\"\u2014is for Ma the hothouse flowering of derangement. She raised her boys to rob banks and shoot cops, and when the sons develop a fondness for a businessman (Pat Hingle) they have kidnapped, Ma recognizes the threat of an appealing father figure and orders them to kill him. (They don't.) The sons may be Cody Jarrett quadrupled\u2014four kinds of poisonous nuts\u2014but Ma, as Winters plays her, is a villain of grandeur. The actress shoots out so much demonic energy that the machine gun she carries is a redundant appliance.\n\nSleek and sinewy where Winters was buxom, Angie Dickinson played the fictional Wilma McClatchie in another Depression-era drama, Big Bad Mama (1974), produced by Corman and directed by Steve Carver from a script by William W. Norton and Frances Doel. Instead of four grown sons, Wilma has two teenage daughters, blond Billy Jean (Susan Sennett) and brunette Polly (Robbie Lee), both as man-hungry as Mom. While dispensing maternal mots\u2014\"Keep your legs together, Billie Jean, and shut up\"\u2014Wilma also runs their lives in a fashion that merges the protective with the exploitative. She bursts into a men's-club smoker and wrests her girls from the shame of stripping, which they were eager to do. \"A mother's still got rights in this country,\" Wilma announces as she fleeces the club of its treasury. With the FBI in pursuit, she takes over a bootlegging operation and falls in with a bank robber (Tom Skerritt) and a gambler (William Shatner), whose favors she occasionally shares with her randy kids.\n\nBloody Mama, 1970: Shelley Winters as Ma Barker; one of her sons was the young Robert De Niro.\n\nLike a Dust Bowl Scarlett O'Hara, Wilma keeps saying, \"We ain't ever gonna be poor again.\" These are her dying words\u2014though Dickinson miraculously reappeared, thirteen years later, in the Corman-produced Big Bad Mama II, where a slightly more righteous Wilma indulges in criminal behavior to expose the venality of the land baron who foreclosed on her house. As she tells one of her kids: \"Your mama may not always be right, but Mama will always be Mama.\" Well, she's right about that.\n\nThey don't call firebugs \"arson auteurs\" or serial stabbers \"knife surgeons.\" Yet the underworldlings who separate marks from their money are known as con artists, perhaps because the tools of their trade are the gifts of a great actor's performance: glamour, intimacy, cool patter, the steady stare that conceals the lie. Doesn't a mother rely on similar subterfuges to get her child to eat his kale or stop crying? Con artists might deserve the name \"gifters,\" but Jim Thompson labeled them The Grifters in his 1963 paperback original about three dissemblers who play their deadliest tricks on one another. It's only fitting that, in the book and the 1990 film adaptation written by Donald E. Westlake and directed by Stephen Frears, two of these should be mother and son.\n\nBig Bad Mama, 1974: Angie Dickinson with her daughters, Susan Sennett and Robbie Lee, and their escorts, William Shatner and Tom Skerritt.\nRoy Dillon (John Cusack) works the \"short con,\" using loaded dice and legerdemain to skin cashiers and sailors. Roy's girlfriend Myra (Annette Bening) is cheaper and perkier, aiming for the \"long con\"\u2014the elaborate scheme that takes suckers for big stakes. And Roy's mother Lilly (Anjelica Huston) is the con woman supreme. Abused and abusing since girlhood, she can pull off a motel-room kill and do it with a hard smirk. She can also stand up for Roy, at least when he's been seriously wounded by a baseball bat after a grift went wrong. \"My son is going to be all right,\" she tells the hospital doctor. \"If not, I'll have you killed.\" Lilly means she'll call on the muscle of her bookmaker boss Bobo Justus (Pat Hingle, having survived Bloody Mama), a sadist who imposes discipline by burning Lilly's hand with a lighted cigar when she ducks out on a job. She explains she did it to tend to Roy, and Bobo, nonplussed, asks, \"What the f\u2014are you doing with a son?\"\n\nRoy must wonder that too. He is grateful for Lilly's fast work in the hospital. \"I guess I owe you my life, Lilly,\" he says, and she, imposing the bond of motherhood, replies, \"You always did, Roy.\" But this typical Thompson antihero\u2014a smart guy getting outsmarted by fate, fast company or himself\u2014doesn't stand much of a chance with a mother who is barely fifteen years older than he is and who treats him, in her friendlier moments, less as a boy than as a beau. He can rebuff her seductions, but he can't duck her wrath. Neither can Myra, who's also a bit older than Roy and plies her sexuality to pay the rent or pawn a jewel. They might be battling mother figures at war over custody of Roy's sooty soul. Huston (in a role once meant for Cher) and Bening make for two splendidly sexy carnivores, with Cusack as the would-be lion tamer whose destiny is to be devoured. Guess who finally dines on Roy? The one crime-movie mother who isn't ready to die for her son but can find a reason to kill him.\n\nNot every crime mother is hard as nails. One is as soft as Charmin tissue, until she's pushed to extremes because the lurid mores of modern America are not the ones she has tried to instill in her children. In Serial Mom, Beverly Sutphin (Kathleen Turner) is a suburban matron with a caring dentist husband, Eugene (Sam Waterston), and two fairly normal teenage kids named Misty (Ricki Lake) and Chip (Matthew Lillard). She also has an urge to kill anyone who affronts her antique notion of decency. If someone should chastise her children or date a fast girl or wear white after Labor Day, Mom goes maniacal.\n\nThe Grifters, 1990: Mother Anjelica Huston and girlfriend Annette Bening contend over which one of them gets to destroy John Cusack.\nYou could say that the fifties made her do it. Anyway, fifties family sitcoms. Forty years later, Beverly wants to live\u2014does live, poor deranged dear\u2014in the white-bread, Populuxe world of Ozzie and Harriet Nelson. Modern movies? \"They're so violent.\" When Chip lets slip the word \"shit,\" Mother reproves him: \"You know how I hate the brown word.\" As her crimes escalate, and with them her tabloid renown, Chip gently asks his mother if she's a serial killer. Beverly responds, with an indulgent smile, \"The only serial I know anything about is Rice Krispies.\" But the cops don't buy her pristine propriety. \"All units,\" the radio bulletin sounds. \"Serial Mom is headed south on Keswick. Proceed with caution. She is armed and f\u2014in' nuts.\"\n\nShould we have mentioned earlier that John Waters's movie is a comedy? And not so much about a compulsive murderer as about the modern mix of celebrity and notoriety? (Patty Hearst played the juror who gets whacked for wearing white.) \"My movies aren't about violence,\" Waters has said, \"but about how America is so confused about fame.\" Serial Mom, which came out a few months before the O. J. Simpson killings and trial, offers further evidence that in the modern United States, the wages of sin are stardom. Beverly has earned role-model rep when a friend of Misty's asks, \"Can I borrow your mother? My aunt is coming for dinner, and she's always getting on my nerves.\" In Waters's world, even a killer's family stays true to her. \"Is it menopausal?\" asks a worried but sympathetic Eugene, who tells the kids, \"No matter what your mother is, we'll love her anyway.\"\n\nJanine \"Smurf\" Cody (Jacki Weaver) seems every inch the sassy Aussie mom, brandishing a steely-cheery attitude gleaned from some Carry On film or Benny Hill sketch. But in writer-director David Mich\u00f4d's 2010 film Animal Kingdom, Smurf is the leader of a Melbourne crime family. Family, literally, for the gang comprises Smurf's three sons: Andrew (Ben Mendelsohn), whom everyone, from reverence or fear, calls Pope; Craig (Sullivan Stapleton), a hyperactive coke addict who sells drugs; and Darren (Luke Ford), the youngest, who tags along on his brothers' sprees. They are joined by their seventeen-year-old nephew J (James Frecheville) after his mother dies of a heroin overdose. Turns out she was the renegade white sheep of the family, keeping J away from her toxic clan\u2014and especially from the smiling Smurf, who sees every malicious act as a moral imperative. Her own eleventh commandment: \"We do what we have to do, we do what we must. Just because we don't want to do something doesn't mean it can't be done.\"\n\nSerial Mom, 1994: Beverly Sutphin (Kathleen Turner) has a loving husband (Sam Waterston), two nice teen kids (Ricki Lake and Matthew Lillard) and an urge to kill the people whose lifestyles she disapproves of.\nBased loosely on the 1980s exploits of Kath Pettingill, whose sons were convicted of armed robbery and acquitted of murder, Animal Kingdom also implicitly cites White Heat: Cody Jarrett, meet Janine Cody. Smurf's sons are just as dependent on her as Cody was on Ma. More than a comforter and abettor to her sons' misbehavior, she is their spiritual guide\u2014the petite, purring, bleach-blond fountainhead of the family disease. A twisted parody of maternal devotion, Smurf gives a full-mouth kiss to one son, snuggles on the lap of another, counsels that \"it's all right to cry\" when hearing bad news and suggests that Pope might want to go back on his meds. When she learns that J could be persuaded by a Melbourne detective (Guy Pearce) to give state's evidence against his uncles, Smurf pinwheels into action, hatching a plot to get rid of a bad seed who might turn good. Weaver earned an Oscar nomination for playing this pert agent of iniquity. Hers is an ordinary suburban, dead-thrilling kind of evil. She makes the best bad mama of twenty-first-century crime movies.\n\nHonorable mention in this category goes to Kristin Scott Thomas, that aristocratic icon of English and French cinema, who has wicked fun as Crystal in Nicolas Winding Refn's 2013 film Only God Forgives. Spitting obscenities at the world, depraved by greed and mother love, Crystal runs a drugsmuggling syndicate; her elder son, Julian (Ryan Gosling), manages the Bangkok branch. She flies in from the States on learning of the death of her younger son, Billy (Tom Burke). Told that Billy raped and slaughtered a sixteen-year-old girl, Crystal shrugs: \"I'm sure he had his reasons.\" No wonder both her sons are Oedipal wrecks.\n\nSome crime-family movie mothers\u2014like Morgana King as Mama Corleone in the first two Godfather films\u2014knew their place: in the kitchen, away from the dirty business. Profiting from the blood money that sustained their family, they could pretend that hubby made his stash in the olive oil business they took no part in. Others, like Ma Jarrett, Ma Barker, Wilma McClatchie and Smurf Cody, ran the family crime business with the ruthless brio of working-class women elbowing into a men-only job. Scott Thomas's Crystal, dolled up in a blond hairdo and slinky clothes, represents the next generation, the new breed of crime mother: a career woman who chose to make her millions through drug smuggling instead of, say, investment banking.\n\nOne thing never changed, though. Crystal, Mrs. Sebastian, the Mas Jarrett and Barker, Wilma McClatchie and Smurf Cody\u2014they did it for their kids. Beverly Sutphin did it in a desperate rear-guard belief that her children should grow up in the same antiseptic environment she did. Only The Grifters' Lilly stands outside this group. Not protective but predatory, she consumes her young for the same reason a mama scorpion does: she's still hungry.\n\n### HORROR MOMS\n\nSometimes, not seeing is believing. In Alfred Hitchcock's Psycho (1960), Norman Bates's mother is a spectral presence: glimpsed behind a window shade or a shower curtain, heard chastising her son (\"Huh, boy? You have the guts, boy?\"), or viewed indistinctly from above, carried by Norman from her bedroom to the fruit cellar. Only at the end do we get a quick hard look at Mrs. Bates: a mummified corpse, her hollow eyes seeming to move as a light swings above her. But she has dominated the film, because of the postmortem influence she holds on her son. Norma Bates lives in Norman (Anthony Perkins); her fury stoked his madness. \"He tried to be his mother,\" says the psychiatrist (Simon Oakland) at the end of the movie. \"And now he is.\"\n\nPsycho, scripted by Joseph Stefano from Robert Bloch's novel, set new rules for the horror genre. Rather, it said the old rules no longer applied. A movie could kill off the major character in the second act; it could turn a nice guy into a psychopath who is also a victim; it could be both clinically realistic and artistically adventurous in the depiction of violence. The first major slasher movie, Psycho established the incestuous connection of prurience and puritanism: show a pretty woman undressed, then punish her display, and the viewer's voyeurism, in the shower. Virtually all modern horror films flow from this exalted source.\n\nModern horror, like ancient Greek tragedy from Jocasta and Oedipus to Medea and her unlucky offspring, often resides in the relationship between mother and son. Psycho came out at the popular crest of Freudian psychoanalysis, which traced an adult's emotional wounds to an infantile trauma involving the mother. In the 2012 film Hitchcock, a fictional retelling of the making of Psycho, Stefano (Ralph Macchio) tells Hitch (Anthony Hopkins) that he spends a lot of time in therapy talking to his shrink about \"the usual things: sex, rage, my mother . . .\" Click. Whether or not that conversation took place, Stefano and, no less, Perkins crept inside Norman to provide a superb study of derangement, of a dual, dueling personality, of a grown child strangled by his umbilical tether to his mother\u2014a theme of which succeeding horror films would take extensive note.\n\nPsycho, 1960: \"A boy's best friend is his mother.\"\n\nTo Marion Crane (Janet Leigh), his guest for the evening at the Bates Motel, Norman can perceptively and charmingly discuss his widowed mother's failings and his loyalty to her: \"If you love somebody, you wouldn't leave them even if they treat you badly. Do you understand? I don't hate my mother. I hate what she's become. I hate her illness.\" When Marion suggests putting Mrs. Bates \"someplace,\" meaning an asylum, Norman lashes out: \"Have you ever seen the inside of one of those places? The laughing, and the tears, and those cruel eyes studying you?\" (Apparently he has.) He sees his mother's disease as a facet of her quirky humanity: \"It's not like my mother is a maniac or a raving thing. She just goes a little mad sometimes. We all go a little mad sometimes,\" he says, and asks his visitor, \"Haven't you?\"\n\nNorman went a little\u2014a lot\u2014mad long ago, when he killed his mother after she told him she was marrying another man, effectively ending her exclusive relationship with her son. (He also murdered the interloping man\u2014a casual casualty of his envious wrath.) Now, briefly, he has connected with Marion, who, on the run after stealing $40,000 from her boss, privately resolves to return home. She won't get there because, having shared a sandwich and a soulful chat with Norman, she just graduated to number one on Mother Bates's to-kill list. (She is collateral damage of the jealous-mother side of Norman.) But this man still loved the thing he killed. To keep Mother near, he husbanded her corpse and taxidermized it, wore her dress, spoke in her voice and, when Marion took a shower, became his vengeful mom. The psychiatrist: \"He was never all Norman, but he was often only Mother.\"\n\nPsycho eventually devolved into a franchise: three sequels, all starring Perkins, then Gus Van Sant's shot-by-shot 1998 remake with Vince Vaughn, and now an A&E series, Bates Motel. In this origins story, which imagines how mother and son constructed the cage they can never crawl out of, Norma (Vera Farmiga) fiercely shelters and dotes on the seventeen-year-old Norman (Freddie Highmore), doing her best to annihilate all pretty girls who might distract him from her love. The Hitchcock film, until its last scene, explains little about Mrs. Bates and shows less. The Master of Suspense would never tip his hand about a movie mystery. But the audience learned all it needed to know about Mother from her nagging and knifing, and from the fear, love and madness in Norman's eyes.\n\nA mean mother and the weak son who takes postmortem power from her: that was Psycho. All the crimes could be attributed to natural, if creepy and twisted, motives. But in the Vietnam years\u2014when, not to overstate the case, everything in popular culture changed\u2014upscale horror films found a new explanation for a son's or daughter's infernal streak. The root of all evil shifted from the natural to the supernatural, from the secular to the religious. Could it be . . . Satan?\n\n\"What Jesus was to the 1950s movie epic,\" wrote Roger Ebert in his review of The Omen, \"the devil is to the 1970s.\" In that film and two important predecessors, a child from hell had a decent earthly mother. Lucifer, or rather his near relation the antichrist, was the force behind 1968's Rosemary's Baby, filmed by Roman Polanski from Ira Levin's bestseller. Frustrated actor Guy (John Cassavetes) makes a deal with the devil: get me the starring role in a play and you can have Satan impregnate my wife, Rosemary (Mia Farrow). By the merest chance, some of Guy's neighbors in their new Central Park West apartment house are members of a coven who drug Rosemary on the night the Horned One will debauch her. She had planned dreamily for her first child, but not This One. At the culmination of her arduous pregnancy, when Rosemary first sees her demon baby, a small smile wreathes her face, as if she believes that a good mother can handle a difficult child. Adrian may be Satan's son, but he will always be Rosemary's baby.\n\nRosemary's Baby, 1968: Mia Farrow, impregnated by the devil, ponders weighty issues.\n\nPregnancy and delivery are just the first two traumas mothers endure. A dozen or so years after, they must monitor the child's volcanic changes of body and mood, otherwise known as puberty. In the 1973 film The Exorcist, directed by William Friedkin from William Peter Blatty's novel and screenplay, single mom Chris MacNeil (Ellen Burstyn) notices that her daughter Regan (Linda Blair) is suffering from even more troubling symptoms: she spits out obscenities, causes her bed to shake and has traded in Captain Howdy, her imaginary friend from childhood, for a demon named Pazuzu. Chris can't be the first fretful mother to say of her twelve-year-old, \"That thing upstairs isn't my daughter.\" Regan's, though, is a rarer case: not of rambunctious hormones but of satanic possession. The agnostic mom must brush away sensible skepticism and employ two Catholic priests (Jason Miller and Max von Sydow) who specialize in exorcism.\n\nThe first horror movie to be nominated for the Best Picture Oscar, with Burstyn, Miller and Blair nominees in the acting categories, The Exorcist was a popular smash\u2014ninth on the all-time list of real-dollar box-office grossers. The movie earned its notoriety from its high fright quotient and from the obscenities that the demon expels through Regan (euphemized in the first season of Saturday Night Live as \"Your mama sews socks that smell!\"). But at its tremulous heart, this is a love story about a mother who will try anything, beyond science or sense, to rid her child of the attitudes, the disease, the demons that have seized her.\n\nLee Remick, that glistening American beauty, and movie patriarch Gregory Peck: how could they be the parents of the antichrist? Through some strenuous plot exertions by David Seltzer in his original screenplay for the 1976 film The Omen. Katherine Thorn (Remick) gives birth in Rome to a son who soon dies. To secure her emotional stability, Katherine's wealthy diplomat husband Robert (Peck) follows the suggestion of a local priest and adopts the child of a woman who gave birth that same night. Bad idea for Robert, smart for the movie, since the boy, Damien (Harvey Spencer Stephens), is the Book of Revelation's jackal-born antichrist, complete with a retinue of enabling, infernal nannies. One of these handmaidens hangs herself from the Thorn roof; another, Mrs. Baylock (Billie Whitelaw), shows up to comfort the parents and raise the little one to his prophesized stature: \"Have no fear, little one. I am here to protect Thee.\"\n\nThe Exorcist, 1973: Ellen Burstyn tries to soothe her sweet, Satan-possessed daughter, Linda Blair.\nAnd Damien is there to rid the world of Robert and Katherine\u2014his first step to ascending the apocalyptic throne. He bangs his tricycle against a stepladder Katherine is perched on, causing her to miscarry the child who might have been a nuisance or a rival. As Katherine recuperates, she is pushed out of a high window by Mrs. Baylock, who must think of herself as the true mother Damien never had. And when Robert resolves to kill his adopted son, the son wins. The surprise of director Richard Donner's stolid, solid film was the demise of the parents who served as audience surrogates. That innovation proved a prophecy: from then on, stars would be less important than franchises. With The Omen on its way to becoming a trilogy, Peck and Remick, and William Holden and Lee Grant in 1978's Damien: Omen II, were disposable commodities, as was the time-honored notion of heroic mothers who went through hell to nurture a better world. In sequels, only the antichild need survive.\n\nStephen King is the Oliver Sacks of horror fiction. As Sacks has documented unusual cases of people with neurological disorders that can seem like gifts, so King creates ordinary people blessed or cursed with extraordinary abilities: the girl with pyrokinetic powers in Firestarter, the death-row inmate with the gift of healing in The Green Mile. In his first published novel he related the story of a teen girl's telekinesis, triggered like a mega-explosion of puberty in the high school shower. The book already bore kid-sister similarities to Norman Bates and Psycho. And when Brian De Palma, Hitchcock's most persistent imitator, directed the 1976 movie of Carrie, he underlined the connection by changing the girl's Thomas Ewen Consolidated School to . . . Bates High School.\n\nPoor Carrie (Sissy Spacek) suffers vicious taunting from her classmates, but her first and final torturer is her mother, Margaret (Piper Laurie), an evangelical kook as virulent as Mrs. Bates. Foregrounding the mother-daughter antagonism, and pushing their final confrontation to the film's climax, De Palma and screenwriter Lawrence D. Cohen fashioned the most vivid specimens of a possessive parent with murderous intent and a meek child driven to revenge.\n\nCarrie has always been a reproach to her mother: Margaret's late husband, with whom she had \"lived sinlessly,\" got drunk one night and had sex with her, and, even worse, Margaret says, \"I liked it!\" Carrie's existence is a constant reminder of that moral crime to a woman who believes that religion is \"spreading the gospel of God's salvation through Christ's blood\" but that \"the Lord visited Eve with the curse, and the curse was the curse of blood.\" Carrie's first menstrual period (which Margaret never prepared her for) marks the visitation of that curse on the next generation. Margaret regrets having brought Carrie into the world and, at the end, tries to take her out of it, which triggers the girl's last violent eruption of telekinesis. Yet Carrie feels a tenderness for her mother; her final impulse is to save her. This important gesture is missing in the inferior 2013 remake starring Chlo\u00eb Grace Moretz and Julianne Moore, where horror-film vengeance overcomes the love for a deranged mother.\n\nCarrie, 1976: Blaming her daughter, Carrie (Sissy Spacek), for all sins of the flesh, Margaret White (Piper Laurie) prepares to exact some Old Testament retribution.\n\nMothers may lurch into madness from hating their children or from loving and losing them. In Wes Craven's The Last House on the Left (1972), a kind of slasher remake of Ingmar Bergman's The Virgin Spring, a husband and wife, upon learning that the four strangers staying the night have slaughtered their daughter, determine to take grisly revenge. Sean S. Cunningham, the producer of Last House, then borrowed from Psycho for his Friday the 13th franchise. Jason Voorhees, the hockey-masked angel of death, was not the killer of the Crystal Lake camp counselors in the original 1980 Friday film. His mother, Pamela (Betsy Palmer), had blamed the camp for the death by drowning of her teenage son twenty-three years before and came back to massacre the next generation of counselors. In the sequels, Jason, revived by the miraculous science of Hollywood greed, did his own dirty work. But the 2009 reboot begins with the boy witnessing his mother's beheading after her murder spree. Decades later, Jason (Derek Mears) returns to kidnap a girl, Whitney (Amanda Righetti), who looks just like Pamela\u2014as if Norman Bates had found a living replica of Norma. Whitney, no fool, pretends to be Pamela, distracting the smitten Jason long enough to eviscerate him with his own machete. Son and \"mother\" lock one last time in intimate enmity.\n\nAt the heart of these films is a parent's grief at the loss of a child. The old are meant to die, the young to live. A child's death upends and outrages that natural order, leaving its gravest mark on the woman who bore and raised the child. A mature mother is expected to deal with it, tend the emotional wound until, with time, scar tissue forms. She may even take pride in her children's sacrifice, as Alleta Sullivan (Selena Royle) did in The Fighting Sullivans (1944), after all five of her sons were killed together at Guadalcanal. But an unstable mother may never recover from the signal atrocity of her child's death and may choose a more drastic remedy: evening the score through the catharsis of violence. If she did not love and grieve so fiercely, she would not act so furiously. Such a mother is the patron demon of horror films\u2014though they usually spend less time probing the inconsolable ache than reveling in the repellent spectacle. Melodrama reduces sad moms to mad moms, and over the movie decades their march has grown to parade proportions.\n\nTwo of the most popular, profitable movies of 2013 unleashed the ancient spirits of killer moms to haunt modern families, and in each film, an outside mother figure intervenes, confronting evil with her own surprising strength. In Andr\u00e9s Muschietti's richly atmospheric Mama, Annabel (Jessica Chastain) realizes that a maleficent force has claimed the two young nieces of her boyfriend. The ghost, whom the girls call Mama, was a nineteenth-century woman, Edith Brennan, sent to an insane asylum, her only child given to nuns. She escaped the asylum, retrieved her baby and, pursued by the law, jumped off a cliff to her death. Not realizing her infant was snagged on a tree branch and saved, Edith has taken Annabel's nieces as living avatars of the child she thought dead. Annabel must summon her own protective maternal instincts to try prying the girls from Mama's clutches.\n\nThe malignant force in director James Wan's The Conjuring is another nineteenth-century spirit: the witch Bathsheba, who slaughtered her children and, before killing herself, laid a curse on anyone who would occupy her Rhode Island home. In the 1970s, when the Perrons, Carolyn (Lili Taylor) and Roger (Ron Livingston), unwittingly move in with their five daughters, they sense Bathsheba's wrath and call on two real-life ghostbusters, Lorraine (Vera Farmiga) and Ed Warren (Patrick Wilson), famed for their exorcism of a Long Island house that inspired the 1979 film The Amityville Horror. Bathsheba soon co-opts Carolyn's will, forcing her to attempt killing her daughters, and Lorraine must find a way around the witch to reach the good mother she hopes Carolyn still is.\n\nA huge critical and popular hit, The Conjuring plays it straight. Like Mama, it abjures CGI gimmicks and instead weaves an ominous mood out of long takes, astute performances from splendid actors\u2014especially Farmiga, taking a break from Norma Bates to play a mother figure who is savior, not corrupter\u2014and that most terrifying of aural effects: silence. Both Mama and The Conjuring ignore the more lurid Psycho imitations of the past half century and return to the pristine Hitchcock model: dramatizing the spell that the dead cast on the living, and a clinging mother on the children she left behind. \n\n### On Being a Mother and Actress\n\n\"Ahhh . . . a daughter of great fortune!\" said the mysterious tea leaf reader at the Russian Tea Room in New York City as she looked into my eyes. I will never forget that statement, made several years prior to the birth of my beautiful daughter, Melanie Griffith, at Doctors Hospital in New York.\n\nWhen Melanie was put into my arms, I told her she was beautiful and that she was going to have a very special life. That phrase from the tea leaf reader has often proven to be true. Even as a child, Melanie was giving, thoughtful, funny, sensitive\u2014loved animals and was always so kind and loving to them. She began displaying her independence and self-assuredness around age one, liked to mimic me and was always interested in what I was doing. She loved books; she still does. We loved shopping; we still do. She loved playing with makeup; we still do.\n\nHer father, the late Peter Griffith, and I divorced when she was two years old. I was a working mom\u2014as a fashion model with the Eileen Ford Agency, later doing TV commercials, the residuals from which made our lives financially secure. It was difficult at times, but we learned to make it work. Melanie's nanny, the late Josephine Milton, was a good teacher to Melanie and they liked each other, which made it much easier for me as I would so have loved to be with Melanie all the time. All of those feelings strongly made decisions for me. By the time Melanie was four, I felt we should have a house where she could just say, \"Mommy, I'm going out to play,\" which wasn't possible living in a New York City apartment. So all of us packed up\u2014Melanie, me, Josephine, the puppy, the kitty\u2014and off we flew to Los Angeles, where I rented a wonderful home in Westwood.\n\nI thought my career as a fashion model and in commercials would go on as it had in New York. But it didn't. I was worried. Then, on Friday the thirteenth of October, 1961, along came the call from an executive at Universal Studios asking if I was the woman in the Sego Pet Milk commercial. I was. A few days later I was under contract to Alfred Hitchcock, though he had seen only my reel of about fifteen commercials and my book of fashion photos. I was asked by the agent at MCA to read the contract carefully, and, if I approved of it, to sign it, and we would then go to the studio to meet Mr. Hitchcock! I could hardly wait to get home and tell Melanie we were going to be all right. Our lives had changed.\n\nAt the time, children weren't as welcome on film sets as they are today, but Melanie was allowed to be on the set of The Birds (1963) to meet Mr. Hitchcock, Rod Taylor, Suzanne Pleshette and Jessica Tandy. Then came Marnie (1964), another Hitchcock film, so Melanie met Sean Connery, Diane Baker and Louise Latham.\n\nMy real-life role as Melanie's mom had made me more aware of the complexities of Marnie and the complicated life she had experienced. All life situations come to aid you in the roles you play.\n\nIt was all very exciting, on the surface. I never talked to anyone but my sister Patty about the difficult situation I was having at the studio. By the end of Marnie I ended the relationship with Hitchcock. I demanded to be let out of my contract, but he refused to let that happen. He said he would ruin my career, and he did. He continued to pay me for almost two years, turning down roles offered to me, saying, \"She isn't available.\"\n\nLuckily, I received a call from Charlie Chaplin to join Marlon Brando, Sophia Loren and Charlie's son Sydney in A Countess from Hong Kong (1967). So off we went to London. Melanie had a really wonderful time with Sophia. That amazing lady took Melanie under her wing and they were inseparable. I could see Sophia would make a really caring, devoted mother, and later she did. We love Sophia.\n\nAs a mother, I probably should have suspected that all of these experiences would encourage Melanie to become an actress. As years went by, she became more beautiful by the minute. Soon she was cast in big films, one after the other. The most important in my eyes was Working Girl (1988), for which she earned an Academy Award nomination for Best Actress. I believe Melanie's role as a real-life mother gave her so many acting tools to excel in scenes with children, as is clear in her films Milk Money (1994) and Crazy in Alabama (1999), featuring her real-life children. Melanie is the most creative woman I have ever known, and I am so proud of her for her honesty, her bravery and her capacity to love. She is also the best mother to my three beautiful grandchildren!\n\nBless my \"daughter of great fortune\"! I love her more than my next breath. And a great tip of my hat to that mystical tea leaf reader. She had it so right!\n\n\u2014Tippi Hedren\n\nClose Encounters of the Third Kind, 1977: Melinda Dillon and her four-year-old son (Cary Guffey) who talks to aliens.\n\n## SCI-FI MOMS\n\n* * *\n\nThey face down aliens, benign or devouring, to protect their kids.\n\n\"Oh, come on,\" says Sarah Connor, a lonely L.A. waitress who's just been told she is destined to give birth to a child who will save the world. \"Do I look like the mother of the future?\" She does if she is the heroine of a science-fiction film, a genre that turns the impossible into the plausible. Sci-fi typically launches muscular guys on a mission to conquer the solar system through superior weaponry and battle skills. But in some of the finest examples, the hero is a woman: a mother-to-be like Sarah in The Terminator, a single mom in Close Encounters of the Third Kind, a woman warrior who discovers the mother within her when she defends an orphan girl in Aliens. In the Superman reboot Man of Steel, she is the caring adoptive mom who instills the finest American values in an alien boy named Clark Kent.\n\nIn several of the films, humanity makes contact with the metaphorical divine; science-fiction writers are as fond of Jesus analogies as horror writers are of Satan figures. But all of these mothers show how caring for children can contribute to saving the planet.\n\nTwo events in the late 1940s cued science fiction's graduation from the pulp magazines to the movie screen: the onset of the Cold War, in which the U.S. and the USSR became global rivals armed with atomic and hydrogen bombs, and the spreading stories of unidentified flying objects\u2014flying saucers\u2014in America's skies. If we weren't going to die in a nuclear Armageddon, or if our small towns and coastal cities didn't face destruction by insects or lizards that A-bomb fallout had irradiated into giant monsters, then we might be invaded or annihilated by evil spacemen! Dozens of no-budget movies throughout the fifties energetically exploited this nexus of popular fear. But among the earliest A-budget sci-fi films was director Robert Wise's The Day the Earth Stood Still (1951), which imposed both a liberal and a biblical agenda on this nervous planet, while leavening the doomsday message with the healing touch of a mother and son.\n\nFrom a spaceship that lands in Washington, DC's President's Park comes Klaatu (Michael Rennie), a human-looking emissary from a more advanced species than ours, and his oversize robot Gort. He says he has come in peace, but a nervous U.S. soldier shoots him. Escaping from a military hospital, Klaatu assumes the pseudonym \"Mr. Carpenter\" and takes lodging at a boardinghouse, where he befriends war widow Helen Benson (Patricia Neal) and her son, Bobby (Billy Gray). Bobby takes his stately guest to the Lincoln Memorial and to the home of a world-renowned scientist the boy happens to know. During a global shutdown of electrical power\u2014the half hour the Earth stands still\u2014Klaatu confides his identity to Helen and says that, should he be apprehended, she must sneak onto the spaceship and give Gort this message: \"Klaatu barada nikto.\" To save the world, he trusts not in government officials but in a mother with a kind face and an open mind.\n\nBehind Bernard Herrmann's eerie theremin music and the pulp poetry of a robot carrying a pretty woman into a spaceship, The Day the Earth Stood Still is a parable of Christ returning to Earth with an apocalyptic warning. Adapting Harry Bates's 1940 story \"Farewell to the Master,\" screenwriter Edmund H. North turned Klaatu into the alien Messiah. His Earth name is Carpenter (Jesus's trade); he dies and rises again; and like so many movie Christs, he has an elegant bone structure and a posh English accent. He also befriends a woman whom others would ignore: Helen is a sinless Mary Magdalene. When her boyfriend Tom (Hugh Marlowe) gets jealous and informs the government, Helen breaks off their relationship. She presumably kept Tom around to be a father figure to Bobby. Now the boy has the ultimate role model: God the Son. But Klaatu's final message carries some of the Old Testament God's severity. All nations must live in peace, he proclaims. But if our military belligerence extends into outer space, then robots like Gort will destroy the Earth. \"The decision rests with you.\" In other words, try to be as peaceful as we, your superiors, are\u2014or we'll kill you.\n\nSci-fi mothers often found themselves drawn to celestial powers. In Steven Spielberg's Close Encounters of the Third Kind (1977), the mission of single mom Jillian Guiler (Melinda Dillon) is simply to find her missing four-year-old Barry (Cary Guffey). First he wanders to a hillside where strangers have convened to hear a plangent astral melody; then he is abducted\u2014in fact, briefly adopted\u2014by aliens, propelling Jillian and fellow searcher Roy Neary (Richard Dreyfuss) on a trip that culminates in the first official meeting of humankind and extraterrestrials. Barry has hitched a ride on a starship, to be returned unharmed to his mother. This vision of alien benignity pays heartfelt homage to the spirit of early Disney features, not only in its use of the Pinocchio song \"When You Wish Upon a Star\" but also in its childlike belief in the magic of movies and in the precious link between mother and son. Spielberg built on this domestic innocence in 1982's E.T. the Extra-Terrestrial, where Dee Wallace as the divorced mother is the top-billed performer. The story, though, is really about her boy Elliott (Henry Thomas) and his very special pal\u2014another godlike being who befriends an anxious mother's sensitive son.\n\nE.T. the Extra-Terrestrial., 1982: Suburban mom Dee Wallace and daughter Drew Barrymore are witnesses to the closest encounter of all.\nThe New Woman meets the Man of Iron in The Terminator. A humanoid robot (who but Arnold Schwarzenegger?) has taken a trip back from the mid-twenty-first century to the movie's present time (when but 1984?) to try to reverse the history of the future. A man named John Connor is destined to lead the survivors of a nuclear war to victory over evil machines like Arnold\u2014if a woman named Sarah Connor (Linda Hamilton) lives long enough to give birth. While the automaton Arnold goes on his mission to perform a \"retroactive abortion,\" another time traveler, Kyle Reese (Michael Biehn), follows him to save John's prospective mother and terminate the Terminator. Aside from its kick-ass thriller energy, James Cameron's movie is a hip retelling of the Annunciation: Sarah is a secular Virgin Mary, John is her divine son and Reese is the messenger angel sent to impregnate Sarah with the holy word. Hamilton imbues Sarah with an indomitable maternal protectiveness of the child she may not as yet have conceived. The New Testament vibe continues in Cameron's 1991 sequel Terminator 2: Judgment Day, which sends Sarah and the teenage John (Edward Furlong) on the sci-fi equivalent of the Holy Family's flight into Egypt.\n\nThe Terminator, 1984: Waitress Sarah Connor (Linda Hamilton) asks, \"Do I look like the mother of the future?\"\n\nAliens, 1986: A devoted surrogate mom (Sigourney Weaver) defends little Newt (Carrie Henn) against one big bad mother.\nHowever odd it seems to propose the five-times-married Cameron as a feminist filmmaker, he has filled almost all his stories (The Abyss, True Lies, Titanic, Avatar) with strong, or at least headstrong, women. In Aliens, the 1986 sequel to Ridley Scott's 1979 sci-fi horror thriller, Cameron set up the ultimate fantasy confrontation between two righteous single mothers: the earthling heroine Ripley (Sigourney Weaver) and a mammoth bug\u2014the matriarch of the creatures killing off the Sulaco's crew members. In the first film, Ripley had shown protective affection only for her cat, Jones. Now she has taken maternal responsibility for Newt (Carrie Henn), an orphan who tells Ripley, \"My mommy always said there were no monsters, no real ones. But there are.\" The main monster is the alien queen, on a mission to protect her slavering, devouring and, as she sees it, endangered young. When the queen goes for Newt, Ripley slips inside the exoskeleton of a forklift power loader to wage war, one mother against another. When she saves and hugs Newt, the girl calls her \"Mommy.\"\n\nRipley, a character that Weaver puckishly nicknamed \"Rambolina,\" applies mother wit as well as a warrior's will to defeating her giant-insect rival. That combination makes her the definitive heroine in science-fiction action pictures. In an age that rewards strength over grace, let there be women as strong as Ripley. May homeless children have no less determined an adoptive mother; may extraterrestrial predators meet no less wily an antagonist. Trust moviegoers to detect the glamorous resolve beneath the smudges and sweat on Ripley's face and the feral humor in her challenge to Big Mama Alien: \"Get away from her, you bitch!\"\n\nIn director Zack Snyder's Man of Steel (2013), the twelve-year-old Clark Kent (Dylan Sprayberry) saves a school bus from disaster and, more horrifying, discovers he has X-ray vision that gives him an inside look at death; it's like the most violent onset of puberty. Understandably traumatized, he locks himself in a Smallville school closet. His mother Martha (Diane Lane) arrives, but the boy is still empowered by panic: he turns the outside doorknob glowing hot, until Martha's calming words about his special nature reduce his fears. For Clark, this episode provides a revelatory lightning flash of his Otherness.\n\nBut the parents who found him as an infant in a spaceship have known from the start that he's different: he is Kal-El, a superior, indeed supreme being, sent from another planet by his loving parents Jor-El (Russell Crowe) and Lara (Ayelet Zurer). Martha and her husband, Jonathan (Kevin Costner), have decided that the best way to raise this superman is as a normal American kid. Under their gentle tutelage, Clark (played as an adult by Henry Cavill) will learn to reconcile the Krypton divinity of his nature with the Kansas humanity of his nurture.\n\nThis movie takes its cue from Bryan Singer's 2006 film Superman Returns, which posited our hero as the Christian God come to Earth to save humankind: Jesus Christ Superman. As Man of Steel screenwriter David S. Goyer noted, \"We didn't come up with these allusions of Superman being Christ-like. That's something that's been embedded in the character from the beginning\" (since his 1938 debut in Action Comics). Goyer goes further, giving the character a backstory reminiscent of the Gospels: the all-seeing father from afar (plus a mother); the Earth parents; the ascetic wandering in his early maturity (forty days in the desert for Jesus, a dozen years in odd jobs for Kal-El); his public life, in which he performs a series of miracles; and then, at age thirty-three, the ultimate test of his divinely human nature. \"The fate of your planet rests in your hands,\" the Holy Ghostly Jor-El, in one of his frequent postmortem visits, tells his only begotten son. You could call Man of Steel the psychoanalytical case study of a god-man with a two-father complex.\n\nBut whereas the Gospels spent little space on Jesus's early years, Man of Steel, being an origins story, devotes its first hour or so to demonstrating that Kal-El\/ Clark is the sum of the love of his two sets of parents. His birth father and mother sacrificed themselves to give him life away from their dying planet; his Earth mother and father fill him with sensible wisdom. When the nine-year-old Clark worries that \"the world's too big,\" Martha expertly comforts him: \"Then make it small. Focus on my voice. Pretend it's an island out in the ocean. Can you see it?\" And the boy, soothed and enlightened, says, \"I see it.\" Lane, an uplifting presence in movies since her debut at fourteen in 1979's A Little Romance, plays Martha as grave, generous and ferociously protective of her boy, this otherworldly treasure and responsibility. At the climax, after Clark has singlehandedly repelled an attack on Metropolis, Martha displays a mother's abiding concern for her son when she asks, \"What are you going to do when you're not saving the world?\" (Answer: Go to work at the Daily Planet.) He may be Superman, but he will always be her baby.\n\nMan of Steel, 2013: Even Superman (Henry Cavill) needs an earth mother (Diane Lane).\n\nMother Wore Tights, 1947: Bette Grable and Dan Dailey as the showbiz parents of Mona Freeman.\n\n## SHOWBIZ MOMS\n\n* * *\n\nTensions at home for ladies in the limelight.\n\nIt's said that everyone has two jobs: her own and show business. Such is the proprietary spell that popular entertainment casts on the mass audience. But when a woman's main job is performing under the spotlight, managing her family can be a burden for all concerned. Movies about showbiz moms detail the grind and the glory, while touching ever so lightly on the social significance of these pioneer working women.\n\nIn the 1920s, for example, only one in four women was employed outside the home\u2014usually either in subordinate (secretarial) roles or in positions (nurse, teacher, maid, waitress) that extended the traditional role of motherhood into the workplace. If any job allowed women a chance at equality, even supremacy, it was acting. Ladies of the stage and then the screen became among the world's most famous, loved and desired figures.\n\nBeing adored, and sustaining that popularity, was a full-time job. A movie actress might have a six a.m. makeup call; she'd be on the set when she could be driving her daughter to school, and exhausted by early evening, when the parent-teacher conference was scheduled. The life of a stage performer often required living out of suitcases; home was a series of hotels or boardinghouses; showtime for her was bedtime for her kids. It's appropriate that the name of the definitive musical about a showbiz mother and daughter was Gypsy\u2014not just because Gypsy Rose Lee was the stripper nom de plume taken by Mama Rose's daughter Louise, but also because women in vaudeville and stock companies were forever on the move. Like it or not, they had the gypsy in their souls.\n\nSo Mother Wore Tights in a vaudeville act with her husband, and when she went back on the road with him after twice giving birth, she left her daughters in the care of their grandmother. Desertion? No: gamely, and gamly, supporting her family the only way she knows\u2014for Myrtle McKinley Burt is played by all-American pinup girl Betty Grable (with chipper Dan Dailey as her husband, Frank) in the 1947 Fox musical, set in the early twentieth century and based on Miriam Young's rosy reminiscence of her song-and-dance family. The only agitation arises when the elder daughter, Iris (Mona Freeman), falls in love with a well-bred collegiate and is briefly embarrassed that her parents will be performing nearby. Chagrin is fleeting: the young man loves Myrtle as much as her audiences do, and Iris ultimately follows Mother's example by getting married and going onstage.\n\nApplause, 1929: Burlesque matron Helen Morgan (far right) titillates the crowd but shocks her prim daughter.\nA teenage girl could accept a mother in vaudeville. What of a mother in burlesque? In Rouben Mamoulian's Applause (1929), one of the first talking pictures to plunder the full cinematic resources of silent films, heavyset chorines shake their stuff onstage for the lowlifes in the seats. The headliner, Kitty Darling (Helen Morgan), is a true trouper: she gave birth to her daughter April one night directly after a performance. Then she sent the child to a convent school, far from the seedy luminescence of Manhattan fleshpots. When April (Joan Peers) is seventeen, she visits New York and is ushered into Kitty's theater, where she sees what her mother does for a living, and mortification streaks her face like a tragic mask. Can it get worse? It can and does. The man Kitty has married to give April a father figure assaults the girl with kisses and groping, and insists that she join her mother's tawdry act. April refuses, until the night Kitty dies from taking poison, and April goes on in her place\u2014briefly, before running off with a decent guy who will rescue her from the grimy business of show.\n\nMorgan, the torch singer who starred as Julie in the original 1927 Broadway production of Show Boat (and would reprise her role in James Whale's 1936 film version), was only twenty-eight, nine years older than Peers, when she made Applause. Yet the star's alcohol abuse added a couple of decades to her appearance. She is no pert twenties flapper; she's all flapped out. When Kitty struts onstage, in a spangled costume open at the front to reveal a tiny brassiere, she almost solicits the insults\u2014\"They oughta auction off that faded old blonde!\"\u2014shouted by the customers. She endures these leers and Bronx cheers to give the daughter she cherishes a better life than hers. If only you could realize that I'm abasing myself for you, dear: that was the message of so many showbiz-mother films, from Applause to Blonde Venus . . .\n\n. . . where Mother wears a gorilla costume. Marlene Dietrich was an icon of Teutonic glamour, whether she performed in tuxedos, tights or, as in this 1932 drama, ape couture, to the tune of \"Hot Voodoo,\" backed by a dozen African-American chorines. As with Kitty in Applause, Helen Faraday is working onstage less for the glory than for her family: she must support her chemist husband Ned (Herbert Marshall), who has suffered radium poisoning, and their young son, Johnny (six-year-old Dickie Moore). To raise more money for Ned's treatments she rents herself out to millionaire Nick Townsend (the young Cary Grant), and when the cured Ned returns to discover her infidelity, she flees with her son. The boy loves his mother, though their fugitive life propels them into a Depression-era social spiral, from swank dives to farmhouse to flophouse. They are apprehended, and the boy is taken by his father, who tells Helen, \"Stay away from Johnny, for good. Give him a chance to forget you. That's the only way you can be a good mother to him now.\"\n\nBlonde Venus, 1932: Marlene Dietrich dallies with Cary Grant to support her son, Dickie Moore.\n\nDietrich's seven films for director Josef von Sternberg established her cool allure as a smiling goddess who made her body available to many men, her heart to few. Blonde Venus domesticated Dietrich, making her a loving wife and mother\u2014a woman who betrayed her husband to save his life, and who kidnapped her son to nurture him; between her nightclub shows, she teaches him the alphabet. Oddly, erotic passion informs neither Helen's marriage to Ned nor her affair with Nick. (We repeat: the young Cary Grant!) Her most intense and playful emotional relationship is with Johnny, whom she cuddles in a hayloft, bathes and kisses . . . bathes with kisses. In a life on the run with his mother, the lad finds adventure: \"I hope they never find us.\" And when Helen finally comes home to Ned, Johnny stage-manages their reconciliation. \"Aw, you're not doing it right at all,\" he says, playfully pouting. \"You're supposed to kiss each other.\" Like a kindergarten Sternberg, Johnny is directing his parents to their happy ending.\n\nMothers, the true auteurs of the children to whom they give birth, tend also to be their directors. And if a woman isn't a stage or screen star, perhaps she can guide her child to the fame that life denied her. \"You are my daughter and you can become an actress,\" says mama Maddalena in Luchino Visconti's Bellissima (1952), adding, \"I could have done it myself if I'd wanted to.\" No matter that the open casting call at the Cinecitt\u00e0 studio is for a girl eight to ten, and that tiny Maria (Tina Apicella) is barely six\u2014for Maddalena is played by Anna Magnani, the great Earth Mother of Italian cinema in Roberto Rossellini's Open City and The Miracle, and Pier Paolo Pasolini's Mamma Roma. Who can stop this cyclonic force of nature from securing the ing\u00e9nue role for her daughter? Not her exasperated working-class husband, nor anyone connected with the film project\u2014nor even Maria, a shy sweetie with no perceptible ambition to see her name in lights. For Maddalena, stardom seems as close as the outdoor movie theater next to her Rome tenement. Some night, she hopes, little Maria will be a giant light on that screen.\n\nAt Cinecitt\u00e0, Maddalena meets Iris, a lovely young woman who appeared in a few films but now can get movie work only as an assistant editor. (Liliana Mancini, the actress playing this role, endured the same career trajectory in her own life.) \"Give up your dreams,\" Iris advises all aspiring performers, \"and stick to a real job.\" A flicker of resignation briefly creases Maddalena's face when she sneaks into a projection booth and overhears the crew making jokes about her daughter's screen test\u2014before she assaults the director with an impassioned aria about the great little actress he is ignoring. In Cesare Zavattini's original script, Maria was denied the role. But such is Magnani's magnificence, as one of the all-time indomitable movie moms, that Visconti decided the only plausible ending was a happy one. Bellissima finally reconciles stardom in movies with a mother's belated understanding of her child's needs. Maria gets the part, and Maddalena turns it down. It's enough that the girl is a star in her mother's eyes.\n\nBellissima, 1952: Soundstage mom Anna Magnani wants her young daughter (Tina Apicella) to be a star, but it's mother who has the star quality.\n\"History,\" Winston Churchill said, \"is written by the victors.\" But the history of showbiz mothers is written by the survivors: their daughters. Occasionally, those recollections are as sunny as in Mother Wore Tights. More often, they brandish an exasperation tinged with affection (Postcards from the Edge) and respect (Gypsy). And sometimes they serve as acts of revenge for a miserable childhood. Thank you, Mommie Dearest.\n\nJoan Crawford won an Oscar for Mildred Pierce as the selfless mother of a rancid child. But at the B-movie end of a legendary career, she devolved from screen queen to scream queen by headlining a series of tawdry psychological thrillers. In the first of these, What Ever Happened to Baby Jane? (1962), she got to swap torments with another Warner Bros. leading lady, Bette Davis. In two others, Strait-Jacket (1964) and Berserk! (1967), the Crawford character is suspected of violent murders that turn out to have been committed by her deranged daughter. The reason, in Berserk!: years ago, Mother had ignored her little girl, so the grown daughter (Judy Geeson) is getting close to Mom by killing off the competition.\n\nChristina Crawford, whom Joan adopted in 1940 after her divorce from Franchot Tone, certainly wished she had received less attention from Mommie dearest. In the 1981 movie of Christina's memoir, Joan (Faye Dunaway) is both the apotheosis and the parody of the wicked mother figure. She treats the girl (played by Mara Hobel as a child and Diana Scarwid as an adult) like a scullery maid; she demeans her, beats her, nearly chokes her to death. No less hurtful is Joan's trick of seeming the victim\u2014\"You love to make me hit you!\"\u2014to a daughter whose fear and devotion she demands in equal doses. When she insisted that Christina call her \"mommie dearest,\" Faye's Joan says, \"I wanted you to mean it.\"\n\nMommie Dearest, 1981: Faye Dunaway as Joan Crawford and Mara Hobel as the actress's adopted daughter, Christina.\nWhether the charges are Bible-true or exaggerated from spite (Christina, who also became an actress, published the book a year after she learned she had been cut out of Joan's will), director Frank Perry's movie lurches between inside-Hollywood expos\u00e9 and a catalog of child abuse, between operatic high camp and a Joan Crawford horror movie. (Joan, in the rose garden: \"Tina, bring me the axe!\") In the infamous scene of Joan whupping the young Christina with a wire coat hanger, Dunaway is photographed from below and in moody lighting as a classic monster from the closet. Unlike many over-the-top performances, this one is not a pleasure but an ordeal to watch\u2014a scary-great turn, up there with Anthony Perkins's in the monster-mother film Psycho. So indelible was the connection of these stars to these roles that they were instantly defined by them, and their careers never quite recovered.\n\nGypsy from its Broadway premiere in 1959, had a more benign effect on starring actresses: it extended their \u00e9clat through middle age and beyond. Ethel Merman created the role of Mama Rose, who shepherds her two daughters through vaudeville until Louise emerges as Gypsy Rose Lee and June as actress June Havoc. Angela Lansbury, Tyne Daly, Bernadette Peters and Patti LuPone headlined Broadway revivals of the classic show by author Arthur Laurents, composer Jule Styne and lyricist Stephen Sondheim.\n\nThis is the primal tale of showbiz striving\u2014bad breaks, no breaks, heartbreak, then the big break\u2014that is also a kind of backstage Gone With the Wind: the period drama of a woman whose hurricane will triumphs over all obstacles and all those she wants to love her. Like a shark, Rose has a single mission: to push her daughters toward the stardom that fate or circumstance denied her. \"I was born too soon,\" she says, \"and started too late.\" And like Charles Foster Kane, she conflates her function as promoter with her prot\u00e9g\u00e9's role onstage. \"We're going to be a great opera star,\" Kane boasted of Susan Alexander. Susan's career flopped, while Gypsy's flourished, allowing Rose to proclaim, \"I always promised my daughter we'd be a star.\"\n\nThe starchy 1962 movie replaced Merman with actual Hollywood star Rosalind Russell, who played Rose as an equally bombastic, slightly more acidic cousin of Russell's Auntie Mame (with Natalie Wood as Louise). It remained for Bette Midler, in the 1993 CBS TV film, to put the definitive balls and verve into the part. In the climactic aria \"Rose's Turn,\" Midler summons the ferocity of an oak-strong mother whose daughters have learned from her and left her. \"Give 'em love and what does it get you? . . . One quick look as each of 'em leaves you!\" Caroming toward madness, Midler's Rose rises to majesty as she gives each line a caress or a curse. She underscores the moral of this \"musical fable\": that in a family of two girls who became famous performers, Mama really deserved top billing.\n\nGypsy 1962: Rosalind Russell as Mama Rose and Natalie Wood as Gypsy Rose Lee.\n\nGypsy, 1993: Everything's coming up Bette Midler!\n\"I was such an awful mother,\" Doris Mann (Shirley MacLaine) says sarcastically to her daughter Suzanne Vale (Meryl Streep), adding, \"What if you had a mother like Joan Crawford or Lana Turner?\" \"These are the options?\" Suzanne replies. \"You, Joan or Lana?\" In Hollywood, kind of. Because Postcards from the Edge (1990) was based on a novel by Carrie Fisher, the writer and actress whose parents were Debbie Reynolds and Eddie Fisher, audiences were welcome to infer it was the inside dish on her movie-star mom. In fact, this is the tale of two people, who happen to be mother and daughter, learning to accept their vast differences in personality. Think of Gypsy siring Terms of Endearment.\n\nThe novel, written in epistolary form, concentrated more on the dark laughter of the rehab clinic. The movie, scripted by Fisher and directed by Mike Nichols, drops the postcards but keeps the edge in its sympathetic, evenhanded portrayal of two women who must fight their addictions: the daughter to hard drugs, the mother to alcohol (though Doris claims, \"Now I just drink like an Irish person\") and both to each other. Neither is the villain or the heroine. Suzanne might love to blame all her woes on her mother, but a wise director (Gene Hackman) gives her some advice that is applicable to all children. \"I don't know your mother,\" he says, \"but I'll tell you something. She did it to you and her mother did it to her and back and back and back all the way to Eve, and at some point you just say, 'F\u2014it, I start with me.' \"\n\nPostcards starts and ends with the family ties of women who get on each other's nerves, like some feuding vaudeville duo, and, despite or because of that abrasion, are great company. Streep, in one of her few daughter roles, catches Suzanne's exhaustion and impatience with a mother who's too much of a kooky character to hate. MacLaine, in a role that Reynolds wanted to play, softens her Terms of Endearment mom with a plethora of frailties. She also brandishes her musical chops with a wonderfully fulsome rendition of Sondheim's Follies song \"I'm Still Here\": \"First you're another sloe-eyed vamp, \/ Then someone's mother, then you're camp.\" In Postcards she is all of these. Better still, she finds an aging woman's tenacious grimace under decades of gamine makeup.\n\nDoris and Suzanne, and MacLaine and Streep, duel to a draw in the titanic battle of mother and daughter\u2014not just in show business clans but often in the lives of people we know\u2014to determine which one is the family's true, incandescent presence. And at the end they realize that a family, like a movie, can have two great stars.\n\nPostcards from the Edge, 1990: Shirley MacLaine and Meryl Streep as a most theatrical mother and daughter.\n\n### The Greatest Mama of Them All\n\nMama Rose inspires all of us whose hearts lie in performing. You don't quit. You don't whine. You just go on, and pretty soon, \"Everything's Coming Up Roses.\"\n\nGypsy is my favorite musical. Every song by Jule Styne and Stephen Sondheim is memorable. It has also given us numerous great tour de force performances. There have been many versions of this musical mother-daughter story, starring the likes of Bernadette Peters, Tyne Daly, Bette Midler. Each has brought varied nuances to the showbiz tyrant who will stop at nothing to achieve stardom for her children. I was always enthralled with Rosalind Russell's brittle film performance. Sure, Ethel Merman owned the Broadway production. She was the original\u2014the barking mama who drove her children to succeed at all costs. We have all come to love and hate her. After all, she was a monster, right? Or was she?\n\nI saw the 1962 film Gypsy (directed by Mervyn LeRoy) many times on television. It seemed to play in heavy rotation with Auntie Mame (1958). I often got the films confused because they both starred Rosalind Russell playing a dysfunctional, kooky showbiz mom or surrogate mom. I loved the theatrical mother she played in Mame and I loved the vulnerability and sacrifice she brought to Gypsy. Rosalind Russell's Mama Rose is the driving force, the beating heart, the blood and sweat of showbizness. She instilled in her children, Baby June and Gypsy Rose Lee, that someday their ship would come in, but before it did they had eight shows a week and two on Sunday, so they had to \"sing out, Louise! Sing out!\"\n\nMama Rose\u2014both the historical woman and the many fictional portrayals\u2014was the ultimate stage mother because she sacrificed the love of her children for the love of the audience. I like to think that a child's responsibility is to come through for his or her parents, to see their frailties and supplement their dreams. Mama Rose loved her children, but she simply loved show business more. Couldn't they forgive her for that? Sure, she did make them do that lousy act with the cow, but to me those two kids just seemed ungrateful. After all, she got them out of burlesque! They were playing the Orpheum circuit! What were they carping about in \"If Mama Was Married\"? If Mama was married, they would never have learned the time step or that \"You Gotta Have a Gimmick\" with those interesting stripper ladies! In short, they wouldn't have made it without their mother's drive and ambition.\n\nMaybe that's what they resented. The symbiotic relationship that drew them together and pulled them apart. Fame.\n\nThe culminating moment in Gypsy is the confrontation between Mama Rose and her elder daughter, now a star, Gypsy Rose Lee. In the film, daughter Natalie Wood, clad in white fur, accuses her mother of driving her and sister Baby June to be successful because she\u2014Mama Rose\u2014never could be. It's the first time you actually see Mama Rose cry. Her years of toil as a stage mother have left her haggard, unsure of who she really is. In the end, she is neither rewarded nor loved for her sacrifice. Both of her children have rejected her. Left alone onstage, she contemplates if it was worth it. And then what does Mama Rose do? She dries her tears and sings the showstopper of all time. Curtain up. Light the lights. It's \"Rose's Turn.\"\n\n\u2014Illeana Douglas\n\nTerms of Endearment, 1983: Shirley MacLaine and Debra Winger swim in the turbulent flow of mother-daughter love.\n\n## A GALLERY OF MODERN MOMS\n\n* * *\n\nIn the last forty years, these very contemporary stars kept the torch of motherhood burning.\n\nForty is a dangerous stage in a film actress's career; fifty could be a death knell. Once Hollywood stardom was gynocentric, but for the last few decades it has been guy-ocentric, allowing male stars to go on exuding their macho musk forever. For actresses, \"mid\" often means \"the end\"; they are discarded, while the latest vixen is promoted. If they find a reprieve, it is usually playing mother roles. That's how Shirley MacLaine has extended her movie career to nearly a half century, how Ellen Burstyn keeps honorably and Emmy-winningly employed forty years after The Exorcist. Sally Field and Meryl Streep have been playing mothers, and earning Oscars for them, since the late seventies\u2014around the same time that Susan Sarandon reaped shock and admiration for her work as a prostitute mom in Pretty Baby. Let the Stallones and Schwarzeneggers go on impersonating surly studs into their Medicare years. These great ladies will take roles that connect with real people on the screen and in the audience.\n\n### SHIRLEY MACLAINE\n\nA Broadway dancer in her teens and a movie star at twenty-one in Alfred Hitchcock's The Trouble with Harry MacLaine spread the charm of a soiled but resolute gamine through a decade of memorable films\u2014Some Came Running, The Apartment, Irma La Douce (with an Oscar nomination for each of these three). Radiating too much independence to be a submissive-daughter type in Hollywood films, she proved in her forties to be a dynamite movie mom. Her facility with both light comedy and dark drama prepared her for the emotional teeterboard of Deedee in The Turning Point (1977), a woman whose daughter's rise in the ballet world tests her decision, when she was young, to give up her career for marriage. MacLaine's delicate equipoise between a mother's pride and a dancer's regret resulted in her fourth Best Actress nomination.\n\nFifth time was the charm, with a win as Aurora Greenway in Terms of Endearment (1983), James L. Brooks's adaptation of the Larry McMurtry novel about a woman who nearly crushes her daughter Emma (Debra Winger) with maternal love. Spanning three decades, this warmly episodic film saunters through Aurora's widowhood, Emma's marriage and the turbulent flow of mother-daughter bonding and bondage. Epic emotions play out on an intimate scale as the two women get in each other's way and heart. Opposing forces of nature, they naturally dominate their men: Jack Nicholson as Aurora's astronaut beau (\"I was just inches from a clean getaway\") and Jeff Daniels as Emma's unworthy husband (\"I mean, who am I if I'm not the man who's failing Emma?\").\n\nAs in 1990's Postcards from the Edge (see the chapter \"Showbiz Moms\"),\n\nMacLaine got the showier role and was showered in acclaim, all deserved. Terms of Endearment delivers an emotional body blow toward the end, when the actress nimbly pirouettes from outsize comedy to subtle pathos and reveals the pain behind Aurora's grand gestures. MacLaine would find use for them thirty years later, as the brassy American mother of Countess Crawley (Elizabeth McGovern) in the British serial drama Downton Abbey and, deglamorized, as Ben Stiller's earthy mom in The Secret Life of Walter Mitty. You needn't be a believer in reincarnation to suspect that MacLaine will go on dancing forever.\n\n### SALLY FIELD\n\nEven as TV's Gidget and the Flying Nun in her teens and early twenties, she was great mother material. The apple cheeks framing an incandescent smile suggested a generosity that Field, in her mature roles, could pass on to her movie children. From her Emmy-winning role in Sybil (1976), where she played a woman who created more than a dozen personalities to erase an abused childhood, she graduated to film stardom, and her first Best Actress Oscar, as a North Carolina textile worker in Norma Rae (1979). A widowed mother of two who cares for her own mother, going deaf from the noise in the mill, Norma Rae realizes, thanks to a New York labor organizer (Ron Liebman), that she harbors the spirit of a heroic maternal figure: Mary Harris \"Mother\" Jones, who helped win livable working conditions for Pennsylvania miners. The adorably feisty Field shows how a woman who never thought herself more than ordinary, and whose prime weekend recreation is soaking her tired feet, can give birth to a passion for social justice.\n\nField earned her second Oscar for Places in the Heart (1984), writer-director Robert Benton's poignant memory film of his own mother in Depression-era Waxahachie, Texas. Widowed when a wandering drunk kills her sheriff husband, Edna Spalding must raise her two young children and bring the family cotton crop to market, helped only by a blind man (John Malkovich) and a black man (Danny Glover). Lending a bit of her autobiography to Edna, Field engenders an aura of communal resolve that prepares the viewer for the final scene: an amazing grace of reconciliation. As calmly resolute as she was in Places in the Heart Field adroitly switched to empathetic agitation in Steel Magnolias (1989) as a woman who must watch and worry as her diabetic daughter (Julia Roberts) risks her life to bear a child. She played another woman with an unusual child as the loving, supportive mother in Forrest Gump (1994). If the IQ-challenged Forrest (Tom Hanks) had the good luck to stumble into every headline event in forty years of American history, it is because his mother honed and home-cooked that Alabama boy's preternaturally sweet soul.\n\nPlaces in the Heart 1984: Widowed mother Sally Field fights to save her farm and raise her two kids (Gennie James and Yankton Hatten).\nThe core of her Mary Todd Lincoln in Steven Spielberg's Lincoln (2012) was less nurturing than nudging: advising her president husband (Daniel Day-Lewis) on political tactics and pleading with him to keep their son Robert (Joseph Gordon-Levitt) from joining the Union Army. \"She was complicated and brilliant,\" Field has said of Mrs. Lincoln, \"and she would not be looked at fondly.\" The actress has given brilliant, often complicated performances over her half-century career. Really, though, what viewer doesn't look at Sally Field fondly?\n\n### ELLEN BURSTYN\n\nAfter purging the devil from your young daughter, in The Exorcist (see the chapter \"Crime and Horror Moms\"), what's a mother to do? Leave your New Mexico home and drive your eleven-year-old kid with the R-rated mouth toward California to pursue your unlikely dream as a singer\u2014and get stuck slinging hash at Mel's Diner. In Martin Scorsese's mother-son road movie Alice Doesn't Live Here Anymore (1974), Burstyn's Alice tried balancing threats with affection, often in a single tirade. When Tommy (Alfred Lutter III) wonders for the zillionth time, \"Are we in Arizona yet?\" Alice explodes into \"If you ask me that one more time, I'm gonna beat you to death,\" then, as if instantly exorcised, adds, \"Just sit back there and relax and enjoy life, huh?\" Monitoring this exchange, any parent (or child) could say, \"Yep, that's me.\" Robert Getchell's screenplay, while nailing the bonhomie of waitresses and acknowledging the violent eruptions of which quiet men are capable, dawdles in a few clich\u00e9s, like the dreamboat (Kris Kristofferson) waiting for Alice in the final reels. Yes, he loves Alice, and he even likes her kid.\n\nAlice Doesn't Live Here Anymore 1974: Ellen Burstyn gets a cold one from her son, Alfred Lutter III.\nAn adventurous actress, Burstyn appeared in a naughty movie version of Henry Miller's Tropic of Cancer the year before breaking out in The Last Picture Show (1971), the Peter Bogdanovich film of another Larry McMurtry novel. She is Lois Farrow, once the reigning beauty of Anarene, Texas, who married a poor boy and \"scared [him] into getting rich.\" Her daughter, Jacy (Cybill Shepherd), who has assumed the crown, figures she can prod the same success out of her beau Duane (Jeff Bridges), but Mom is skeptical: \"You're not scary enough.\" They share something, though: the bed of a local oilman (Clu Gulager). Like mother, like daughter, perhaps even on the same night.\n\nLois was a nun compared to Sara Goldfarb in Darren Aronofsky's Requiem for a Dream (2000). The first line of Hubert Selby Jr.'s source novel\u2014\"Harry locked his mother in the closet\"\u2014gets to the essence of two warring addicts: Sara, the Blanche DuBois or Mary Tyrone of Brighton Beach, and her son (Jared Leto). They're made for each other: Mom swears by amphetamines and TV hucksters; Harry loves heroin and, to buy it, steals her TV set. By the time the crash-dieting Sara starts hallucinating that her refrigerator is clambering through the kitchen to devour her, Burstyn has woven strands of madness into a brave and mesmerizing distillation of paranoia. She takes the viewer on a jolting trip through the theme park called hell.\n\n### SUSAN SARANDON\n\nUtterly, sumptuously at ease under the eye of the camera, Sarandon finds a star's equilibrium between the personality she is and the person she's bringing to the screen. No one else in films manages to be so regal and so egalitarian. From her movie debut in Joe through her early prime in The Rocky Horror Picture Show, Atlantic City, The Hunger and The Witches of Eastwick and up to the four roles across the millennia she played in the Wachowskis' Cloud Atlas Sarandon has been the American cinema's beacon of erotic intelligence: our own Statue of Libertine. She's got the advantage of those expressive orbs; young moviegoers, asked to conjure up the memory of Bette Davis, might say she had Susan Sarandon eyes. But it's her mouth that sets the tone; it seems ever on the verge of a smile, as if she understands that our lives may end in tragedy, but we're myopic if we don't see the fun in it.\n\nThe Last Picture Show, 1971: Ellen Burstyn and Cybill Shepherd in a Prom Queen battle of the generations.\nGravity mixes with the comedy in Sarandon's attitude. She has the range to be big sister (Thelma & Louise) both manager and Madame (Bull Durham) and doting, crackpot mom (Anywhere But Here). She was Ariel, a very earthy sprite, in Paul Ma-zursky's Shakespeare update Tempest. In the past dozen years, older but no less alluring, she has movie-mothered Jake Gyllenhaal (Moonlight Mile) Shia LaBeouf (Wall Street: Money Never Sleeps) Orlando Bloom (Elizabethtown) Rachel Weisz (The Lovely Bones) Liv Tyler (Robot & Frank) Jason Segal and Ed Helms (Jeff, Who Lives at Home) and her own daughter Eva Amurri (Middle of Nowhere). Yet she earned her Oscar playing a nun (in Dead Man Walking) whose faith and spirituality glow like God's love light.\n\nShe boldly redefined the movies' notion of motherhood in Louis Malle's Pretty Baby (1978). Her Hattie was a New Orleans prostitute with a twelve-year-old daughter\u2014Brooke Shields in the flush of her nymphancy\u2014and an ethical obtuseness more shocking than the silk and sizzle of her exposed flesh. (The same year, Sarandon and Shields reunited as American-Romany mother and daughter in King of the Gypsies.) In George Miller's Lorenzo's Oil from 1992, a Sarandon character faces one of her gravest challenges. She and Nick Nolte play a married couple suckerpunched by fate: their son has a dreadful disease, whose incurability the wife is loath to accept. The parents share an intimate closeup, nearly three minutes long, whose focus gradually shifts from Nolte describing the disease to Sarandon's dawning dread as she realizes the consequences. Tears drop simultaneously from both eyes, as if the last of this mother's illusions had been squeezed out of her. It is a devastating scene, played with spectacular subtlety.\n\nWe should also remember Sarandon for an overlooked performance in an underappreciated film: the Wachowskis' Speed Racer (2008). In this technologically precocious fable of the little man fighting the big corporation, Pops Racer (John Goodman) is a mechanic turned car designer and his son Speed (Emile Hirsch) is the youngster ready to win the big rallies against a rigged system. Sarandon plays\n\nMom, the family's emotional center and a font of dewy sagacity. Mom seems utterly at home in the Wachowskis' hallucinogenic reimagining of the 1950s Father Knows Best as a Tex Avery cartoon. And like a good '50s mother, she is her son's biggest fan and the source of his cheerful determination. Of Speed's mastery behind the wheel, she says it is \"inspiring and beautiful, and everything art should be.\" We could say the same about the forty-plus-year career of Susan Sarandon.\n\nLorenzo's Oil 1992 : Susan Sarandon helps research a cure for the rare disease afflicting her son (Zack O'Malley Greenburg).\n\nThe Manchurian Candidate, 2004: The credenza gives bat wings to Eleanor Shaw (Meryl Streep), manipulative mother of Raymond Shaw (Liev Schreiber).\n\n### MERYL STREEP\n\nThe most honored actress\u2014or actor\u2014of her generation, Dame Meryl has also been Mama Meryl, in a dozen or so movies, almost since the beginning of her film career. In 1979 she costarred with Dustin Hoffman in Kramer vs. Kramer and won her first Oscar in the chilly role of a mother who walks out on her husband and eight-year-old son. Streep's second Oscar came for Sophie's Choice (1982), in which William Styron's Polish woman, trapped in the Nazi Holocaust, faces the awful decision of which of her children she must consign to death. Streep's blond, aristocratic affect and the hauteur of her cheekbones have allowed her to play all manner of mothers who are suspect because she looks so darned smart. She was Lindy Chamberlain, the Australian woman unjustly convicted of murdering the baby she said was abducted by a dingo, in A Cry in the Dark (1988). In Marvin's Room (1996) she was the selfish mother of a troubled child\u2014a definition that, if raised to nuclear proportions, might fit her Eleanor Shaw in the 2004 remake of The Manchurian Candidate. No son, no domestic or foreign government, is safe from the toxic touch of Eleanor, who, in Streep's satirical turn, oddly resembled the then junior senator from New York: Hillary Clinton.\n\nTackling any role in the belief she can wrestle behavioral truth out of it\u2014for example, the lesbian mother of Claire Danes in The Hours (2002)\u2014Streep has also submerged herself in many genres. She became an action figure, an expert rafter protecting her young son against two killers in The River Wild (1994). She sang country music as Shirley MacLaine's daughter in Postcards from the Edge; played Lindsay Lohan's country-singing mom in the Robert Altman movie based on Garrison Keillor's radio show A Prairie Home Companion (2006); and, in Mamma Mia! (2008), sang ABBA hits while juggling three suitors, arranging her daughter's wedding and nearly exploding with manic energy. Over-the-top is just the launch pad for her turn as the cancer-ridden matriarch in the 2013 film of the Pulitzer-winning play August: Osage County. Hers is the loudest performance but not the most acute: Julia Roberts and Julianne Nicholson as her daughters, and Margo Martindale as her sister, all do work more grounded and less histrionic.\n\nA taut, naturalistic Streep is on view in . . . First Do No Harm (1997), a fact-based TV movie that she produced as well as starred in. Playing the mother of an epileptic child, Streep searches for treatments her doctor will not prescribe and juggles questions of aching financial need, the humiliations of the newly poor, the sight of kids crying while the parents argue over huge hospital bills. She has the small gestures and tight, hectoring voice of a woman untrained for conflict, and, finally, the exhaustion of a longtime caregiver. Here she shows that great acting can be the sum of the most minute calibrations, in a character who overcomes emotional exhaustion to attain domestic heroism.\n\nMamma Mia! 2008: Meryl Streep, with Amanda Seyfried, sings her way into her daughter's heart.\n\nTyler Perry's Madea: an old-fashioned matriarch for the twenty-first century.\n\n## SEMPER MOM!\n\n* * *\n\n## Why movies will always have mothers.\n\nHas Hollywood pushed movie moms out of the calendar? The studios usually schedule some romantic comedy or drama for Valentine's Day, and every Friday the thirteenth, which occurs one to three times a year, a horror film will open. A mothers' movie on Mother's Day would be an annual treat; alas, that celebration falls on the second weekend of what Hollywood calls its summer season. So instead of a cinematic tribute to the women who gave us life and raised us, we get a would-be blockbuster aimed at the all-important fourteen-to-twenty-four guy market. Indeed, whole summers can pass without any big American picture that prominently features a cheery or, for that matter, a scary mom. Independent movies may try to fill the gap; Woody Allen's Blue Jasmine starring Cate Blanchett as a Manhattan widow who locates her elusive stepson in San Francisco, was the most popular indie film of summer 2013. But for most of the year, to see mothers in the popular media, you have to watch TV: HBO, Showtime, TCM.\n\nGood news! The suspicion that movies have rendered mothers extinct is only a seasonal depression. Toward the end of the year, the producers gear up their campaigns for critics' and industry awards, culminating in Oscar night. In effect, Hollywood has relocated Mother's Day from May to December. And since the average member of the Academy of Motion Picture Arts and Sciences is nearly twice as old (fifty-eight) as the typical moviegoer (mean age: thirty), films with Oscar ambitions resemble TV biopics more than big-screen action franchises and are more likely to contain juicy roles for actresses of a certain age\u2014i.e., mother movies. Mind you, this return to the maternal cannot be attributed to studio bosses getting sentimental late in the year. It's all about the money: An Oscar nomination brings untold millions in free publicity to the films and performers receiving them.\n\nAnd often the gamble on movie moms, especially those taken from real life, pays off in statuettes. Reese Witherspoon earned the Best Actress Oscar in 2006 as June Carter Cash in Walk the Line and a year later Helen Mirren won as Elizabeth II, the imperious, helpless mother-in-law of the late Diana, in The Queen. But it was Julia Roberts who inaugurated the new millennium of Academy Awards for true-life mothers by taking Best Actress as the star of Steven Soderbergh's Erin Brockovich.\n\nA twice-divorced, unemployed mother of three, Erin lacks money for her kids' day care or her own meal at a diner. Brassy of couture and character, she talks her way into a job for a lawyer (Albert Finney) and begins digging into a lawsuit until she uncovers a public health scandal: Pacific Gas and Electric had covered up the fact that it had improperly disposed of hexavalent chromium, poisoning the local water supply. A double underdog as an uneducated single mother, this paralegal helped win one of the largest civil judgments in California history: $333 million to her grateful plaintiffs. (The real Brockovich has a cameo as a waitress.) Academy voters responded to Roberts's performance, which plays this inspirational figure at a near-comic level of intensity and volume. So did audiences: the film earned $256.3 million worldwide, or nearly $400 million in today's dollars. Those are blockbuster numbers for a little movie about a single mom on a mission.\n\nMore recently, mother roles have become the surest route to Oscar gold. In the four years of Academy Awards from 2010 to 2013, women who played mothers won the Best Actress twice: Sandra Bullock in The Blind Side as Leigh Anne Tuohy, the Memphis mom who adopted a homeless black teen and spurred him to football renown and emotional stability, and Meryl Streep as Margaret Thatcher in The Iron Lady. In all four of those years, the Best Supporting Actress award went to movie mothers: Mo'Nique in Precious Melissa Leo in The Fighter Octavia Spencer in The Help Anne Hathaway in Les Mis\u00e9rables. These films gave audiences the chance to engage in an unfamiliar activity\u2014to cry without shame at the movies\u2014and, because they struck that atavistic chord, most were box-office hits. Moviegoers, get out your handkerchiefs; moguls, stock your bank accounts and trophy cases.\n\nLate 2013 saw the theatrical debuts of several contending, contentious mother movies\u2014all starring Oscar laureates for Best Actress. In August: Osage County John Wells's film of Tracy Letts's Pulitzer Prize-winning play, Streep and Julia Roberts faced off as the mother-daughter team in a family cage match whose prime saints and monsters were all women. Labor Day adapted by director Jason Reitman from Joyce Maynard's novel, handed Kate Winslet the sultry role of a depressive mother whose encounter with an escaped convict (Josh Brolin) may help her reclaim her sexuality and provide a father figure for her thirteen-year-old son (Gattlin Griffith). Philomena directed by Stephen Frears and inspired by a true story, casts Judi Dench as a working-class Irish woman searching for the son whom, fifty years earlier, she was forced to give up for adoption. The movie stokes both outrage at the mistreatment of unwed mothers by Catholic nuns and delight at the sweet nature of Dench's Philomena, a mother who never gives up.\n\nThe mother-movie genre is just as tenacious in the face of daunting odds, for two reasons: to earn Oscar prestige for studios and to give distinguished actresses great roles. And perhaps there's a third reason: Tyler Perry. In nine feature films, all based on his plays, the Atlanta-based multimedia mogul has incarnated the outsize character of Madea (a.k.a. Mabel Simmons), a mother, grandmother and trash-talking, Bible-quoting earth mother. Produced on the cheap and aimed at African-American audiences, specifically older women, Perry's movies have earned some $700 million at the domestic box office. These are tales of class as much as race, setting wealthy people of color who mimic the white lifestyle against poor folks who are black and proud. Standing above them in sassy judgment\u2014and resembling, as one of her relatives describes her, \"a fat Energizer bunny\"\u2014is Madea, a truth teller with nothing to lose.\n\nA black actor playing a fat woman is not exactly an innovation, as Martin Lawrence and Eddie Murphy have shown. But the six-foot-five-inch Perry, who in civvies has the smooth good looks of a Will Smith, cuts a startling figure. Outfitted in a purple print dress, giant glasses and sandbag bosoms, carrying a purse with three handguns and often punctuating her comments with the wave of a cigarette, Madea shouts out advice and ridicules the supporting players for being too short, too fat or insufficiently black. With titles that recall those of the old Blondie or Ma and Pa Kettle movie series\u2014Madea's Family Reunion, Madea Goes to Jail, Madea's Big Happy Family and Madea's Witness Protection\u2014the films are calculated combustions of domestic melodrama and low comedy, Saturday-night insult humor climaxing in a Sunday-morning sermon. In the 2013 A Madea Christmas she negotiates a truce between a stubborn black lady and her lovely daughter, married to a white man. You're left thinking that Madea's tough love could solve any Middle East crisis in no time.\n\nA long shot to win mainstream critical acclaim\u2014or any Oscar, ever\u2014Tyler Perry offers a salutary reminder of the breadth of mother roles. From the era of the nickelodeons to the age of Netflix, these women, their interpreters and their films have given audiences the finest excuses for laughing and crying at the movies. Mary Pick-ford a century ago achieved what Streep and her sister actors create today: portrayals of motherhood in all its faces, heroic or troubled, horrifying or edifying.\n\nWhatever the financial restraints in an industry focused on the young male audience, and of an art form that wants to look forward, mother movies should always be around. Children must be shown where they came from, and from whom. Working mothers\u2014and, as the National Organization for Women pointed out decades ago, \"Every mother is a working mother\"\u2014deserve the affirmation of popular culture for their important, impossible job. All of us need images in films to reflect and enrich, challenge and validate, the ones in our hearts. On that big screen, we demand to see our better selves: our mothers.\n\n## Credits\n\nThe following photographs are courtesy of Everett Collection: pages ii (Terms of Endearment, Bloody Mama, Mommie Dearest), , , , , , , , , , , , , , , , , , , .\n\nThe following photographs are courtesy of Turner Theatrical Library: pages ii (Madame X, Gone With the Wind, Mildred Pierce, I Remember Mama) xxii, , , , , , , , , , , , \u201399, , , , .\n\nThe following photographs are courtesy of Photofest: pages ii (Mamma Mia!, A Tree Grows in Brooklyn, The Help, The Grapes of Wrath, The Night of the Hunter, Gypsy), , , , , , , , , , , , , , , , , , \u201395, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .\n\nPhoto courtesy of Debbie Reynolds: page xi.\n\nPhoto courtesy of Sam Robards: page 19.\n\nPhoto \u00a9 Barry King: page 57.\n\nPhoto courtesy of Jane Powell: page 91.\n\nPhoto \u00a9 2013 by Bill Dow: page 181.\n\nPhoto courtesy of Illeana Douglas: page 209.\nTURNER CLASSIC MOVIES (TCM) is a Peabody Award\u2013winning network that presents great films, uncut and commercial free, from the largest film libraries in the world. Currently seen in more than 85 million homes, TCM is the foremost authority in classic films.\n\nRICHARD CORLISS is the movie critic for Time magazine, whose staff he joined in 1980. For fifteen years the editor of Film Comment, he wrote the book Talking Pictures: Screenwriters in the American Cinema and monographs on Greta Garbo and Lolita.\n\nMEET THE AUTHORS, WATCH VIDEOS AND MORE AT\n\nSimonandSchuster.com\n\nauthors.simonandschuster.com\/Richard-Corliss\nWe hope you enjoyed reading this Simon & Schuster eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\nThis book is for Mary Corliss, its inspirer and first reader, who helped choose the films and photographs and poured her loving critical acuity into the entire process; and for Priscilla Painton, Shannon Clute and Sydney Tanigawa\u2014the three glorious godparents of Mom in the Movies.\n\n\u2014Richard Corliss\n\nSimon & Schuster\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nwww.SimonandSchuster.com\n\nCopyright \u00a9 2014 by Simon & Schuster, Inc., and Turner Entertainment Networks, Inc., a Time Warner Company. All Rights Reserved.\n\nTCM is the trademark of Turner Entertainment Networks, Inc.\n\nForeword copyright \u00a9 2014 by Debbie Reynolds\n\nForeword copyright \u00a9 2014 by Carrie Fisher\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever.\n\nFor information address Simon & Schuster Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Simon & Schuster hardcover edition April 2014\n\nSIMON & SCHUSTER and colophon are registered trademarks of Simon & Schuster, Inc.\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event.\n\nFor more information or to book an event, contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nInterior design by Ruth Lee-Mui\n\nJacket design by Marc Cohen\n\nJacket photographs: Front (left to right), Changeling, Erin Brockovich, The King and I courtesy of Photofest; A Raisin in the Sun courtesy of Everett Collection. Back (left to right), Aliens courtesy of Photofest; Mildred Pierce courtesy of Turner Theatrical Library; Forrest Gump, Lorenzo's Oil courtesy of Everett Collection.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nCorliss, Richard.\n\nMom in the movies : the iconic screen mothers you love (and a few you love to hate) \/ by Richard Corliss ; foreword by Debbie Reynolds and Carrie Fisher; [with editorial assistance from Turner Classic Movies].\u2014First Simon & Schuster hardcover edition.\n\npages cm\n\n1. Mothers in motion pictures. I. Turner Classic Movies (Firm) II. Title.\n\nPN1995.9.M63C68 2014\n\n791.43'65252\u2014dc23 2013044850\n\nISBN 978-1-4767-3826-0\n\nISBN 978-1-4767-3828-4 (ebook)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Buy products related to gold ribbon for christmas tree products and see what customers say about gold ribbon for christmas tree products on Amazon. com.\nRed Velvet with Gold Jeweled Center Ribbon FREE SHIP TO STORE \u00b7 Christmas Cheer Bows, Set of Four FREE SHIP TO STORE 5. 0 Stars (6 Reviews). Explore Jennifer Moreland's board\" Christmas Tree Gold Ribbon\" on Pinterest. | See more ideas about Xmas trees, Christmas trees and Christmas crafts. Christmas Mesh Ribbons. Showing 40 of 1341 results that match your query. Search Product Result.\nProduct - BalsaCircle 2. 5-Inch x 25 yards Deco Mesh Ribbon by the Roll - Wedding Party Favors Decorations DIY Craft. Vickerman 4\" Gold Glitter Mesh Christmas Ribbon. Product - Vickerman 4\" Emerald Metallic Mesh Christmas and. Browse our selection of christmas at Trendy Tree - RAZ, Christmas tree ribbon gold, Halloween, Easter decor, ornaments; Deco Poly Mesh rolls, wreath supplies, ribbons, burlap. Color: Gold. An gold organza ribbon with a crisp glittered diagonal gold decoration Offray Wired Edge Encore Sheer Craft Ribbon, 2-1\/2-Inch Wide by 25-Yard Spool, Gold by Offray Find Christmas ribbon, holiday ribbon, winter ribbon, Christmas tree ribbon and more at Lion Ribbon.\nQuick Look \u00b7 Analise, Wired Edge - Gold. Find great prices on christmas tree ribbons and other christmas tree ribbons deals on Shop. Metallic Green and Gold Christmas Trees and Stars Wired Craft Ribbon 2. Find great deals on eBay for gold christmas tree bows. Shop with confidence. Decorative Christmas Ribbons. Vickerman 2. 5\" Cream and Gold Zebra Christmas and Craft Ribbon.\nProduct Image. Aisle Decorative Trees Wired Christmas Craft. Shop for gold christmas ribbon online at Target. Free shipping on purchases over $35 and save 5% every day with your Target REDcard. Browse our selection of christmas at Trendy Tree - RAZ, Christmas, Halloween, Easter decor, ornaments; Deco Poly Mesh rolls, wreath supplies, ribbons, burlap.\nColor: Gold. Aug 26, 2018. Our Gold and Champagne Christmas Tree Ribbons boast of rich colors that complement your holiday d\u00e9cor. Shop Christmas accents on. Shop for gold christmas ribbon online at Target. Free shipping on purchases over $35 and save 5% every day with your Target REDcard.\nHipgirl 25 Yard 1. 5\" Gold Glitter Metallic Sparkle Fabric Ribbon For Christmas Holiday, Birthday Gift Wrapping, Hair Bow Clips& Card Making, Sewing, Wedding Decor, Boy Girl Baby Shower, Floral Design Explore Jennifer Moreland's board\" Christmas Tree Gold Ribbon\" on Pinterest. | See more ideas about Xmas trees, Christmas trees and Christmas crafts. Shop Frontgate's assortment of decorative Christmas tree ribbon and bows. Find Christmas wreath bows and other holiday trim for the added touch to your Christmas greenery.\nAdd a layer of elegance to your Christmas tree or wreath with Balsam Hill Australia's line of exquisite Christmas Tree Ribbons and Garlands. Gold Fine Mesh. Browse our selection of christmas at Trendy Tree - RAZ, Christmas, Halloween, Easter decor, ornaments; Deco Poly Mesh rolls, wreath supplies, ribbons, burlap.\nIdeal for cake decorations, gift bows, Christmas wreaths and wrapping around your tree instead of the traditional tinsel. 2. 5\" \/ 63mm Wired Gold Ribbon with a Glittery Pattern.\nQuality wired ribbon ma. How to Decorate your Christmas Tree with Ribbon (Achieve a High End New York Department Store Look). Christmas tree decoration blue, silver& Gold - Duration:. How To Decorate A Christmas. Buy\" Gold Ribbon\" products like Sirena\u00ae Collection 10K Rose Gold.\n10 cttw Diamond 18-Inch Chain Ribbon Pendant Necklace, Quatrefoil Gold Ribbon Throw Blanket in Black\/White, Christmas Ribbons 52-Inch x 70-Inch Tablecloth in Gold, Sirena\u00ae Collection 10K Rose Gold. 10 cttw Diamond 18-Inch Chain Swirling Ribbon Pendant Necklace An gold organza ribbon with a crisp glittered diagonal gold decoration Offray Wired Edge Encore Sheer Craft Ribbon, 2-1\/2-Inch Wide by 25-Yard Spool, Gold by Offray Hipgirl 25 Yard 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"See and hear our new Steinways and get a behind-the-scenes tour of the Steinway factory.\nUConn's most successful football coach is back on the UConn sideline.\nThe Class of 2021 is breaking records academically and physically.\nAt CRT, UConn's dramatic arts students get the chance to act alongside professionals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Domenico Codispoti is an Italian man under house arrest for a variety of crimes, including petty theft and drug dealing. Signor Codispoti, 48, is homeless.\nEvery night at 9pm, he settles down in front of number 22 Via Vittor Pisani in Milan and arranges his blankets and sleeping bag. He isn't allowed to move until 7am the next morning. Police patrols check after sunset to make sure he's \"at home.\"\n\"I have always done my stealing at night,\" he says. \"That's why the court gave me this sentence. Since I don't have a house, there was no other solution left. During the night I can't move, I have to stay here, stuck on this sidewalk.\"\nThe unusual punishment was first imposed on Codispoti in 2006, when he was sentenced to two years of surveillance and house arrest. After a few more run-ins with the law, Codispoti's sentence was extended. He must now sleep in this particular spot near Milan's Central Station until April 2014.\nHe should just start building a structure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Toribio Mar\u00edn, Carmen (2016). The Fourfold Water Garden, a Renaissance Invention. \"Gardens and Landscapes of Portugal\", v. 4 (n. 1); pp. 1-15. ISSN 2182-942X. https:\/\/doi.org\/10.1515\/glp-2016-0001.\nThe different combinations between the \"classic\" fourfold pattern and water in the garden have produced a high number of varied solutions since the distant past. However, during the Renaissance a new model emerges: a crossaxial garden with four basins arranged symmetrically around its center. The composite analysis of the related examples is addressed in this paper, which attempts to find an explanation for the different models as for the appearance of the contrasting solution at the same time in two different locations: the Villa Lante (Bagnaia, Italy) and the Royal Monastery of San Lorenzo in El Escorial (Madrid, Spain).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's Monday. Politics is more vicious than ever. So here's something a little lighter.\nYou can check it out, here.\nHere's something that's been happening since the beginning of gay time. Not big deal.\nOver 3,000 people participated in the survey, which was conducted earlier this month. One if four respondents admitted they had lied about their age at least once while chatting with a potential new hookup.\nHere's the link. Lucille Ball once said that she couldn't lie about her age because she told too many people her real age when she first went to Hollywood. But she wished she hadn't told the truth back then.\nThis one is for all the transphobes in Scotland.\n'Dear transphobes, do you think it's right to harass people on the street?' the poster reads.\n'Right to push transgender people around in clubs? Right to humiliate, intimidate and threaten them online?\nHere's more. Good for them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Anger is a natural emotion, all humans express it, either silently or verbally. Anger is a messenger; it tells you where your beliefs are not working and where your next steps are in our journey home to God. A spiritual approach to anger includes the awareness of where your anger starts within you and a recovery process of Self-Forgiveness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"El opositor venezolano ingres\u00f3 por tierra a trav\u00e9s del Puente Internacional Sim\u00f3n Bol\u00edvar, de Villa del Rosario, C\u00facuta y se espera que en breve parta rumbo a Europa.\nVestido con camisa clara y chaqueta oscura, Ledezma asegur\u00f3 por su parte que burl\u00f3 el arresto sin consultarle a su \"esposa e hijas\" y no dio pistas claras sobre sus pr\u00f3ximos pasos.\nEl dirigente opositor, Antonio Ledezma, huy\u00f3 a Colombia desde Venezuela, donde permanec\u00eda en arresto domiciliario, en Caracas.\nLedezma, de 62 a\u00f1os, tras su detenci\u00f3n, estuvo preso en el penal militar de Ramo Verde, en el estado Miranda, vecino a Caracas, y a los dos meses fue trasladado a su domicilio por motivos de salud.\nEn breves declaraciones a la prensa en la capital colombiana, Ledezma ratific\u00f3 sus observaciones a ese proceso y reclam\u00f3 al gobernante venezolano que libere \"de inmediato\" a los presos pol\u00edticos si quiere un di\u00e1logo con la oposici\u00f3n.\nThere are 1,820 adults with wealth of over $50 million (Rs325 crore), while 760 have more than $100 million (Rs650 crore). The rise in global wealth reflected widespread gains in equity markets and similar rises in non-financial assets.\nBidaskClub lowered Chevron Corporation from a \"sell\" rating to a \"strong sell\" rating in a report on Monday, July 24th. Prelude Mgmt Limited Liability Corporation stated it has 0.01% of its portfolio in Williams Companies Inc (NYSE:WMB).\n\"Esa llamada y esa palabra no son una solidaridad con la oposici\u00f3n, son una solidaridad con la democracia venezolana\", dijo Ledezma al canal NTN 24, donde apareci\u00f3 con camisa clara y una bandera venezolana colgada al hombro.\nDijo adem\u00e1s que Ledezma hizo su correspondiente tr\u00e1mite migratorio ante las autoridades colombianas.\nRecognition for recent economic reforms in India & 5000 year track record of no default. This is unedited, unformatted feed from the Press Trust of India wire.\nIf you are reading this story on another domain, it was stolen and reposted in violation of United States & global copyright law. Commonwealth of Pennsylvania Public School Empls Retrmt SYS raised its position in Henry Schein by 0.3% in the second quarter.\nBut on Friday, a meal at the Methuen restaurant took on a special meaning for the now blended and bigger Lawrence family. The adoption of 40 infants, children and teens were finalized at Miami Children's Museum as part of the celebration.\nDefence minister Niramala Sitharaman addressing a press conference on the Rafale deal. It was not financially prudent to negotiate ToT for just 36 fighters, she said.\nIn the interview, Eisenkot called Iran \"the biggest threat to the region\" and said Saudi Arabia was in \"complete agreement\". We will not intervene in the fighting in Syria, unless we see an attempt to harm our Druze brothers.\nIt was created using data compiled from satellites orbiting the Earth and it shows how life on our planet has changed. Ivona Cetenic joined FOX10 News for an interview Friday morning to tell us more about it.\nNovember 3 investment analysts at SunTrust Banks maintained a company rating of \"Buy\" with a current price target of $210.00. Finally, Partnervest Advisory Services LLC grew its stake in Alibaba Group Holding Limited by 1.8% in the second quarter.\nHe described it as a dissociative episode: He woke up from a nap, and \"it just felt like I was on the outside looking in at a conversation\".\nHe said assistance could also be sought from the Rangers and Frontier Constabulary to disperse the protesters . However, Justice Siddiqui said that the application will only be heard if the parties ended their sit-in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've Hustled During Dozens of Races\u2026 But on Sunday, I Hustled Up The Hancock for the Very First Time! | Keep It Up, David!\nWith 100 stories, the Hancock is the 8th tallest building in the US. When it was finished in 1969, it was the second-tallest building in the world, after the Empire State Building.\nHustle Up The Hancock is the annual stair race here. This was the 20th year. It attracts over 4,000 athletes every year, and is a fundraiser for the Respiratory Health Association, bringing in over a million smackers. Whoa.\nThis race has been on my radar for a couple years, but two things have prevented me from coming in the past: scheduling conflicts, and the fact that it's so popular it sells out, and quickly! This year it all came together, and I've been excited to tackle this landmark, so I can add it to my list of iconic structures I've raced in, alongside Willis Tower, One World Trade Center, the Space Needle, and the Stratosphere.\nWhile the building has 100 stories, the race is only to the the 94th floor, home to the 360 Chicago observation deck. Only \u2013 ha! 94 stories is a lot!\nI was scheduled to start at 11:45am, and that was fantastic news. I had raced up 217 stories in Minneapolis just the day before, and then flown to Chicago, so I would be able to get a good night's sleep without having to get up before dawn to make it to a 7am start time.\nHustle Up The Hancock is a well-oiled machine. They distribute lots of information in advance so you know where to go and how things will work, because the crowds on race day are so large that it can make finding your way around confusing.\nMy 11:45 wave consisted of a couple hundred people, and I positioned myself well, so when they opened the line, I ended up being first. That was strategic on my part. If there was any gap between my wave and the prior wave, than the stairwell would be a little less crowded, and that was something I could take advantage of as I climbed.\nWhen I entered the line, I passed a sign saying to expect a 30-minute wait, because the line snaked around the basement floor, then went up to the street level lobby, then down another hall, around the corner, and over to the elevator bank. It turns out there was a huge gap, so much so that a race organizer frantically ushered us straight to the starting line. My 30-minute wait ended up being about five minutes. I was caught off guard by this, and let a couple others go before me, so I had a few seconds to adjust my earbuds and gloves and prep my heart rate monitor and watch.\nAnd then\u2026. it was go time. I had been worried that I'd be achy and sore from Saturday's hour-long climbing fiesta, but I was surprisingly limber and not sore, and I started the 94-story climb feeling really good. I quickly passed the three people that I let start before me, and then had the stairwell all to myself for the next 15 stories.\nThat's when I started catching up to the slowest people in the wave before me, and passing them. This was my first race in a long time where I wasn't in a competitive heat, and I'm not gonna lie \u2013 passing people provides a mental boost that helps fuel me to keep going.\nWhen I started slowing down, around the 35th floor or so, when the fatigue from Saturday's race really set in, it was seeing these racers that kept me going. I thought about my own journey, about how fear of failure had stagnated me at 402 pounds, and how scared I was to take the first steps towards building a healthier life.\nAnd here I was, seven years later, 160 pounds lighter, and a veteran in a sport so utterly, back-breakingly difficult that many people I explain it to can't believe it's real. I've been those people I passed in the stairwell. I've walked in their shoes. Except I haven't\u2026 because they're much braver than I was at that weight.\nSo I kept climbing. I kept climbing because my first steps ultimately led me here, led me to the belly of this American landmark, led me to all sorts of amazing places I never could have predicted, and the only way I knew how to honor that journey was to not give up, never give up, keep climbing, keep pushing until I had nothing more to give, until I left every last watt of energy from my body in that stairwell.\nI kept climbing for ME, because I'm worth all the hard work, the devotion, the sacrifices I've made to get and stay healthy. And I climbed for YOU, to show you that the fears, self-doubts and reservations that we share aren't permanent. They don't need to be anchors. You may not end up becoming a tower runner, but taking those first steps can led you to amazing places, even if you don't know yet where they are.\nAfter 40-something repetitive floors, the stairwell changed. The floors got shorter \u2013 from 20 steps per floor to 15. This was a welcome change for a tired stair climber, because the floor numbers flew by faster. To be honest, much of this race is a blur, even though I remember my body being ablaze with fatigue, and my legs feeling heavier after every flight.\nI remember thinking I could muster a sprint up the final five floors, so I tried turning up the juice at the 89th floor. Then, the 90th floor ended up being twice as tall as the ones that preceded it, and my sprint dissipated in a cloud of complete weariness and sweat.\nI reclaimed my sprint as I turned on the 93rd floor landing. There, a few steps in front of me, was another climber, and I decided in that instant that I needed to pass him. I flew up the final two flights, passing the guy four steps from the top. I blasted across the finish line, and staggered around until I found a place to sit down.\nAnd then it hit me. It all hit me. I was slammed with so many overwhelming thoughts, concurrently, that I broke down. I just climbed 311 floors in one weekend. I'm 94 stories above the sidewalk, and MY FEET brought me here. I got here in 19 minutes and 22 seconds \u2013 much faster that I ever could have imagined. But most of all, I'm here, in this moment, right now, because I'm strong, I work hard, and I did it. The tears mixed with the sweat; my body shuddered from emotion and twitched with pain.\nAnd then I looked up, and saw the view. Man, the view. It was a crystal clear day, and from that height, beyond the forest of skyscrapers, you could see the curvature of the earth.\nI stayed on the observation deck for nearly an hour, soaking up the experience, and promising myself not to forget. Not to forget what I did this morning, what I've done with my life, how I've taken control of my health. I left that John Hancock Center feeling as strong and unwavering as the building I had just climbed.\nA huge thank you to Bowflex, my sponsor for this race. Their products help me train, and they've proven to be invaluable, and means so much to have this incredible company in my corner.\nAnother thank you goes to my friend Anne, who made a very generous donation to the Respiratory Health Association on my behalf. You're the best, Anne!\nThis entry was posted on Monday, February 27th, 2017 at 10:49 pm and is filed under Uncategorized.\tYou can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.\n8 Responses to I've Hustled During Dozens of Races\u2026 But on Sunday, I Hustled Up The Hancock for the Very First Time!\nWow! Great race recap David and so inspiring! Congratulations and Keep It Up David!\nKeep it up Amy! Hope the drive went great and gracias.\nYour father stands with pride and in awe of your accomplishments, David ! Well done!\nI think this is my favorite write up of yours ever! Or maybe it's just because I read it while on the step mill training for the vertical mile.\nyou are awesome dear friend!\nThanks for these re-caps, David! Amazing back-to-back ELITE performances by you! Very impressive accomplishments! So happy for you! Glad you stayed on the deck for an hour to soak it in!\nThank you George! Hope all your stairs are going well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some one was complaining about her plastic containers not smelling very nice, so I told her to put some newspaper in and reattach the lid, leaving overnight to see if it would draw the smell out. She said it did. Works with suitcases and bags, just make sure that the lining will not be marked by newsprint, so save the plain paper for that job.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"December 22, 2018 . Tags: 3d sculpture, bust, Churchill.\n3D model for manufacturing of bronze sculpture.\nOctober 25, 2018 . Tags: 3d sculpture, sculptural portrait.\n3D models for manufacturing of bronze busts.\nJanuary 17, 2018 . Tags: 3d sculpture, bust.\nBust 3d model for milling out of marble.\nMay 16, 2017 . Tags: 3d sculpture, bust, Newton.\n3D model for milling out of granite.\nOctober 5, 2016 . Tags: 3d sculpture, cow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you had to buy an offroad vehicle that would never see pavement, What would you buy?\nAnd I mean \"Never see pavement\". After my wife started a discussion about the state of the JEEP brand, I began wondering what my answer to this question would be. The vehicle has to remain in stock form (not one modification). It has to be no older than 2016. I would say a Range Rover, but if it breaks... It can not touch pavement. It doesn't matter how good the warranty is, you can't use it.\nI think I'd choose a Rubicon Wrangler, but I'm not sure FCA is the right answer. Any ideas????","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Psal. 8:1 For the musician on the gittith, a psalm of David's.\nThe meaning of the term gittith is not well understood. Some believe it is a harp-like instrument.\nas You set Your majesty over the heavens!\nto insure failure of the enemy and the avenger.\nPsal. 8:5 what is humankind that You should remember us?\nand of glory and honor You would crown him?\nDavid reminds us how small we are compared to the immensity and grandeur of the heavens. Yet God is mindful of us. Something we have probably pondered since our ancient ancestors became aware of their individuality.\nbeyond the lanes of the sea.\nPsal. 8:10 Lord, our Master, How glorious is Your name in all the earth!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Thunderbird 3 Extension project!\nA vtiger extension for Mozilla's Thunderbird 3 (and beyond) email platform. This project has built and improved on the original vtiger extension which was only available for earlier versions of Thunderbird.\nVersion 0.4.2 Released - Automatic updates from now on!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Much Love Is In It?\nThis is true whether I am looking at a manmade object, a pet, a person or a plant.\nA thing, a dog, a human or even a beautiful rose could be a little on the low side, love-speaking.\nI am explaining because this goes to the core of how I think.\nI will take my hobby, jewelry making. Because I make necklaces, earrings and bracelets for a hobby, I spend a lot of time looking. Handmade jewelry. Store bought jewelry. High end, low end. Craft stores. Antique stores.\nMany store-bought objects are the equivalent of fast food. They have little to no chi in them \u2013 even very expensive items. They are so perfectly uniform they look almost ugly. The less energy a human hand put into them, the more lifeless and unattractive they are to me.\nMy tai chi teacher, Jeff Cook, taught me a great phrase, wabi-sabi, which means, simply put, the beauty of imperfect things. Like nature. Nature is unpredictable but magnificent. It's a Japanese term and goes to the heart of the artistic process \u2013 nothing lasts, nothing is finished and nothing is perfect.\nI remember once making a rose quartz necklace for one of my clients. I put great care and thought into it.\nWhenever I make jewelry for any person, I tune in to their energy and meditate about what exactly it is that they need at that time. I have learned the qualities of each stone and take that information into account as I meditate. Then I choose the colors, the stones, the length and start beading.\nI looked at that necklace many times and at the end it seemed just right for her.\nThen one day she wore it into my office. I noticed that one of the beads on one side was smaller than its mate on the other side. I was aghast! How did I not see this the first time? Or even the second, third or tenth time I looked? Even though it had been a gift, I offered to take the necklace back and repair it.\nThis same lady hired me one year to make a necklace for her mother. I tuned into her mother's energy and made a gorgeous, elaborate statement piece of green pearls. The necklace was considerable, but I heard back that because she had arthritis, the mother was having difficulty with the clasp and could not put it on all by herself.\nSo I made another necklace for free. I sat down and tuned into my client's mother's energy. I decided she needed to heal her heart, so I chose all deep green stones that heal the heart, principally malachite and aventurine.\nThe mother accompanied my client to a session, so I was able to present it to her myself as a gift. I had not seen the mother during this time or heard any news of what was happening with her.\n\"How did you know?\" the mother exclaimed.\nThe clasp was very simple. She could do it all by herself. But more to the point, there was a lot of love in the object and she could feel it.\nMy gardener, Gabe Horrisberger, has a green thumb. That might seem like a basic requirement for someone who is in his profession, but trust me, if Gabe puts a living thing into the ground, it will thrive. Sometimes I have run scientific experiments where I will put pansies in one part of my garden and Gabe will plant the others. I love my flowers very much, but whatever Gabe puts into the ground will do the best. He just loves gardens.\nI think to myself, \"How hard can this be? You dig a hole and put the thing in the ground.\" I use fertilizer and everything.\nEarlier this year, I went to one garden center. They had all the basic flora, but everything looked loveless. Then I went to ACE Hardware, around the corner from me. It's a small place, unpretentious and slightly disorganized, but the cashiers are friendly, even the guys who put the bags of cedar chips in the back of my car are helpful and all the plants look like someone is actually interested in them. I buy flowers from ACE Hardware all the time, and whatever I get from that particular store will always thrive.\nDr. David Hawkins, M.D., author of Power Versus Force, is my favorite writer. I have read virtually every book he has ever written, usually multiple times. Dr. Hawkins says whether or not a patient has side effects from any medication has to do with the intention of the doctor who prescribed the drug. He said that some doctors can prescribe virtually anything and their patients will have no side effects. That is because they have only love for their patients.\nA long time ago I remember being very touched by the words of Emilie Conrad, an off-the-wall genius healer, who, to my mind, has always been 30 years ahead of her time. She said, \"YOU are the technique.\" In other words, we can study all these techniques, but at the end of the day, it's not the technique that makes the shift, it is our vibration, our intention and how much love we put into whatever it is that we are doing.\nWhen I visited Don Wetsel in Virginia, I was very touched that he, the teacher, asked me to work on his neck. He had taught the techniques for releasing pain in the neck and then he came to me to practice on him. He told everybody in the class afterwards that it was the first time in his life he felt he had his head on straight.\nWhen we touch another person with great love in our heart, everything is transformed. It doesn't matter whether it's homemade jewelry, homemade cooking or a garden.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200bThe internet is like a shark's tank. It's all because people unconsciously still live in the past. They are carrying around large amounts of baggage. All that baggage and pain gets dumped on you instead. The bullies, rather than taking responsibility for their own life and feeling their feelings so they can heal from the past, act them out passively, sarcastically or sideways on innocent people who have not harmed them. Gaslighting others is also a form of passive aggression. This covert bullying behavior you see all over the internet is normalized and even applauded. We might not even realize we're doing it because it is so normal. People are so roughened up by life and this is how they discharge their repressed shame and rage so they don't have to feel it themselves or remember its source from the past. They might also think bullying others can prevent them from being bullied. Because if you attack someone else, you feel in control of your life and your pain is \u2026. GONE.\nWhat pain? I have pain? Ha. How would I know if I'm the one causing it?\nI used to be bullied as a kid because I was sensitive, different, small, fragile, and lacked the strength to beat a gang of 8th graders who towered over me. Also, when you don't realize your own importance, you might let people pick on you. The only way for me to come out of that was to start being a bit mean myself. Then all that pain \"goes away\". It's like magic. I used to be very capable of hurting the vulnerable. This is why I can see this in others; I used to do it. It's the ultimate pain-reliever. It was how I felt powerful \u2014 albeit false power. But I was powerful enough to forget how powerLESS I actually felt.\nNow, being on the other side of it all, while I can understand what motivates bullies, it still stings to watch it going on online. I still get stung myself. The flashbacks and the worthlessness inadvertently come right back up to the surface. Then I remember how I survived it. And I see others behaving similarly. It's a vicious sad cycle.\nYou can't do anything about the sting other than to let it hurt and feel the unworthiness it might bring up. It's old stuff that needs to be released before you can appreciate how great you really are. Once you KNOW how great you are, sometimes you need to remind yourself you are worthy, important, special and loved. Hold on to that to get by if you truly believe it is so. That's the higher vibrational way of dealing with bullies. Sure it feels good to give them a spoonful of their own medicine, but at the expensive of your own energy, that is. You might die a little every time you bite back. What feels even better is realizing you don't deserve it and that you ARE worthy.\nLove yourself and be your own bestie. Nobody can ever hurt you again when you really love yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvrmx b/data_all_eng_slimpj/shuffled/split2/finalzzvrmx
new file mode 100644
index 0000000000000000000000000000000000000000..361ce123b2cb1b41ca5bc461c4d8170c5ae5e19c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"1. Preheat oven to 400\u00b0. Grease a large baking sheet with olive oil or nonstick cooking spray.\n2. Add flour to a medium shallow bowl and eggs to a separate medium shallow bowl. In a third shallow bowl, whisk together panko bread crumbs, Parmesan cheese, olive oil, salt, garlic powder, and cayenne.\n3. Working in batches, toss Brussels sprouts in the flour until fully coated, then dunk in eggs. Dredge in panko mixture, then place on baking sheet. Bake until golden and crispy, about 25 minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the request of medical specialists, AMEND is pleased to be partnering with the NET Patient Foundation to expand services and to provide information and support to those affected by Adrenocortical Cancer (ACC) in the UK.\nTo learn more and to join ACC Support UK, please click here to visit the dedicated website for ACC Support UK which was fully launched in December 2015.\nIf you are interested in helping AMEND's new ACC Group, please contact Jo Grey for further information on opportunities available.\nMD, DSc, FRCP, FMedSci \u2013 Wiebke Arlt is the William Withering Chair of Medicine and Director of the Institute of Metabolism and Systems Research, IMSR, at the University of Birmingham.\nDr. Bancos graduated from the Univ De Med Si Farm Iuliu Hatieganu, Fac De Med, Cluj Napoca, Romania in 2003. She works in Rochester, MN and specializes in Endocrinology, Diabetes & Metabolism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In his role as executive vice president of human resources, Woody is responsible for all of the company's human resources functions, including talent acquisition, training and development, associate relations, compensation and benefit administration, risk management, policies and procedures, legal compliance and internal communications. He is accountable for nurturing the fundamental cultural environment of the organization, the balanced scorecard business model and establishing metrics that measure performance and progress. As a member of the company's executive committee, he contributes to the strategic planning that ensures that PM Hotel Group remains one of the best operators in the industry. With more than 25 years of human resources leadership experience in the hospitality industry with an emphasis on building HR systems in high growth dynamic organizations, he helped build one of the nation's largest independent management companies. He has guided the HR processes in mergers & acquisitions and an IPO process and established foundational human resources systems in entrepreneurial companies. In addition to hospitality, he previously held human resources leadership positions in the financial services industry with such prestigious organizations as Deloitte and Touche. He served as an officer of the Council of Personnel Officers and is a graduate of Davidson College.\nAs Senior Vice President of Sales & Marketing Lovell Casiero brings more than 25 years of experience in the hospitality industry to PM Hotel Group. Most recently she partnered with hotels and management companies to drive sales performance as Founder and President of FLC Consulting Group. Previously, as Senior Vice President of Sales & Marketing for Crescent Hotels & Resorts, she was responsible for the direction and oversight of revenue enhancement strategies for the company's diverse portfolio, and worked closely with the corporate operations and business development teams to ensure profitability exceeded client expectations.\nLovell has extensive hotel sales experience in developing and leading sales teams affiliated with Marriott, Hilton, Hyatt and Intercontinental Hotel Group, including senior leadership roles with Merritt Hospitality, Interstate Hotels & Resorts and MeriStar Hotels & Resorts. Her experience includes direction of direct sales and catering strategies, developing hospitality marketing campaigns, sales training, task force, troubleshooting and transitions. Lovell has also served as President and COO for an international non-profit organization, overseeing national and international media campaigns, event planning and fundraising.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He's recorded five studio albums, has a distinguished career and now a Best Of album! Full Circle is releasing on October 5, Full Circle isn't your mainstream Greatest Hits album and features the brand new single 'Drowning'.\nMitchell was in the Australian music spotlight as singer\/guitarist for Jebediah when in 1998 he found a t-shirt in a Perth op-shop with the number 15 on the back, and the name 'Bob Evans' printed on the front. Coincidentally at this time, he was embarking on his first solo shows in Perth and the random name on someone's discarded basketball shirt took hold and the rest is history.\nTo celebrate the release, Bob Evans will hit the road for a national capital cities tour.\nPosted in Music, Uncategorized, What's on in Adelaide and tagged Bob Evans, Kevin Mitchell. Bookmark the permalink.\nROCKWIZ IS COMING BACK TO TOWN!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Months ago, I posted an idea. Little did I know that a short while later, I would be using that idea to keep my own wedding within budget. M and I have decided to use potted plants as our centerpieces. We started early (buying small plants and growing them for 10 months before the event) and were able to cut costs significantly. We spent a mere $250 on centerpieces for 20 tables including the terracotta pots and saucers, various plants, and potting soil.\nConsider your talent, your vision of your event, and your own DIY spirit before undertaking this project. Also consider that a plant may die in the time between purchase and display at the event, so you could incur additional costs. That said, it's so fun to see them everyday and be able to watch them grow!\ni just killed like infinity affids!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwcat b/data_all_eng_slimpj/shuffled/split2/finalzzwcat
new file mode 100644
index 0000000000000000000000000000000000000000..65fc7072200c02d4f7ab6b365373a4ae3392f49d
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@@ -0,0 +1,5 @@
+{"text":"The Montagne de Reims boasts some of the best Pinot Noir in the region, and Bouzy is its capital. The key to Bouzy's inherent greatness lies in its deep, chalky subsoil which imparts intense expression of fruit and great mineral complexity in its grand cru wines. The village of Bouzy and Champagne Paul Bara are practically synonymous. As the published village historian, Paul is indelibly linked to the lore of his hometown. Many argue that he is their most renowned producer, being one of the rare r\u00e9coltants-manipulants in a region inundated with the mass-produced wines of the large, corporate champagne houses. These r\u00e9coltants-manipulants, or R.M.s as they are known, are of the few that still grow their own grapes and make their own wines. Champagne Paul Bara is the quintessential example, where everything is done with a personalized touch.\n\u2022\tThe Annonciade bottling is an hommage to an ancestor of the Bara family. In 1833, Annonciade, daughter of a winemaker from Bouzy, married Auguste Fran\u00e7ois Bara, a young cooper from a neighboring village. Jules Bara, their son, constructed the first cellar at Champagne Paul Bara. This cuv\u00e9e symbolizes the perfect marriage between vigneronne and cooper.\nAuthor of a \"History of Bouzy,\" Paul Bara is the memory and the great specialist of the cru, the fruit of which he knows how to preserve so marvelously without adding weight to the palate. That his eldest daughter is succeeding him does not at all change the house style. In particular, their Brut R\u00e9serve and the Cuv\u00e9e Club, two wines of rare aromatic distinction, can be compared to the best, most prestigious cuv\u00e9es from negociants at a half or a third of their price.\nBara epitomizes all that is wonderful about old world, hand-crafted Champagne. Their vineyards are all rated 100% Grand Cru; the bottles are all processed one at a time, using techniques developed over 100 years ago; the wines are seamless, at once delicate, and at the same time powerful. There's simply nothing to compare them to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An exciting and unique day conference exploring the important role that sport and exercise plays in the relationship between our faith and our bodies. Through a number a keynote speakers, and through workshops we will explore the ways in which we can improve connections between mind, body and spirit and how that can contribute to health and healing.\nThe Revd. Dr Rob Ellis has been Principal of Regent Park College, Oxford since 2007, having been Tutorial Fellow in Pastoral Theology since 2001. Previously, he has served as a Baptist minister in Milton Keynes and Bristol, and was Tutor in Christian Doctrine at Bristol Baptist College. Rob's research interests are currently in the interface of belief and practice. He has published books on the theology of intercessory prayer, and in 2014 on theology and sport, in addition to articles on a wide range of subjects including theology and film.\nThe Revd. Graham Buckle, Vicar of St Stephen's, Rochester Row and English Channel swimmer. Graham is a highly experienced community leader, specialising in mental health and pastoral ministry. He is also a wild swimmer, marathon runner and cyclist. He will be speaking on mental health and fitness, and leading a workshop based on his experience of swimming the channel.\nThere will also be workshops on Yoga and Keep Fit, and the chance to reflect on your fitness and wellbeing on the run up to the event.\nThis event is an ecumenical event, open to all ages and fitness levels, from super athlete to coach potato \u2013 ALL WELCOME!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"temp0rary were asked to give a presentation on working methods for CRUX. On the evening, the Photosynthesiser light-to-MIDI flowers and EEG to MIDI equipment were explained and demonstrated. Audience members were then invited to jam with the band using these input devices, and 2 further songs were then performed with synchrinosed DMX rig and 3 projectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feeling refreshed on returning to work after the long summer holidays? Some shiny new stationery and box fresh workwear could certainly help ease you in \u2013 we've got it all at Wurlin. Our ability to handle almost any request is just one of the many great benefits of ordering promotional products through us.\nWurlin offers a wide range of premium branded pens, pencils and notepads. As well as looking super smart in the office, they can be specially packaged for use as corporate gifts. Check out these designer Moleskine diaries we produced for one of our clients, complete with bespoke bookplates.\nAlso in our repertoire are branded mugs and coasters \u2013 every office needs some of these! We love the embossed leatherette coasters we were asked to make earlier this year. They'll go with absolutely anything. Simply send us your spec with details of the colours and materials you want.\nWurlin can even supply your company workwear, printing or embroidering any items of uniform, including: caps, polo shorts, fleeces, etc. And for company away days or even just the daily commute, there's a vast array of business type bags and corporate umbrellas for branding.\nMarketing collateral is another speciality, so don't hesitate to ask us about anything you need to support special events like trade shows, product launches and conferences. Printed tote bags are quick and easy to produce and custom edible items like sweet treats make great giveaways.\nIf you'd like us to take care of all this and more for your company, simply send us an email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"4. Where a business case can be made, work with your BDM to develop a technology transfer plan that includes intellectual property protection, the market and potential partners, and development plan, including funding.\nThe assessment consists of three key elements.\nThe IP assessment seeks to determine whether the research results are patentable. It will analyse the prior art (past publications, patents, presentations, posters, and products on the market) to determine if the results satisfy the key criteria for patenting - novelty, obviousness, and usefulness.\nOnce you understand the market opportunity, you can develop a work plan to develop the technology. ISS will work to identify applied research and development opportunities to help you bring the technology to market. ISS assists the research team in accessing funds from partners, as well as from funding programs such as NSERC CRD, NSERC I2I, CIHR, Genome Canada, Government of Ontario funding, and others.\nThe market assessment identifies the market need for the proposed products or services. It will look at the end users for a technology, the market size, the potential products, the competition, and the companies that may want to license the technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyflj b/data_all_eng_slimpj/shuffled/split2/finalzzyflj
new file mode 100644
index 0000000000000000000000000000000000000000..33f7062626e6e1d21c0a6b69b352d7b290830836
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyflj
@@ -0,0 +1,5 @@
+{"text":"Aluminium oxide blasting is a process used to clean surfaces back to the base material.\nThis removes all contamination such as rust and corrosion and leaves a good clean surface for the application of powder coating or wet paint.\nThe process involves a nozzle with a high-pressure airline to deliver the aluminium oxide particles at the object. This is a particularly effective method for removing scale, rust, paint and minor surface flaws from metal objects.\nN.B. Products that have been previously powder coated or plastic dipped will take longer to remove and will be charged at a higher rate.\nWe will let you know if certain parts are not suitable for blasting.\nN.B. Once blasted it is very important to stop rust forming on steel and cast iron parts and therefore to collect and protect these parts as soon as possible.\nThis process involves the complete immersion of the painted part in the stripping chemical. It can take from 15 minutes to several hours for the process to be complete.\nChemical stripping is ideally suited for aluminium parts and old partly rusted panel work as it is both non abrasive and less aggressive than media blasting.\nWe will let you know if certain parts are not suitable for chemical stripping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each of the seven modern double rooms has a terrace or balcony facing towards the south. The room terraces and balconies ensure absolutely privacy beause of the offset construction of the Quinta do Atl\u00e2ntico.\nThe rooms are individually decorated and brightly furnished with wild cherry. Each room has a table with chairs, CD-player, Sat-TV and its own bathroom with bidet and shower or bath.\nOn Satellite TV (51 cm Stereo) you can watch international television programs in your room.\nRelax with a cool drink on the broad terraces of the villa as the sun goes down.\nTake a swim in our large sweet water pool or bathe in the sun.\nParking places are available in front of the house.\nTake a rest in our comfortable lounge. Here we serve you a most varied buffet breakfast with different kinds of bread, sausage, cheese, jam, muesli, yoghurt and fresh fruits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A driver or transformer must be purchased separately.\nStep and wall lights are manufactured with a stainless steel, aluminum, or brass construction available in a variety of finishes. Brass and Stainless Steel may provide enhanced protection against corrosive elements and is recommended in coastal and marine grade environments.\nHigh Powered LED, Input: 9-15V AC, Power: 2W (3VA), Brightness: up to 68Lm, CRI: 90, Rated Hours: 60,000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SSC's vision is to emerge as a leading engineering company through Innovative Technology and Techniques with commitment to deliver in Time, Quality, Safety and Cost Effective Solutions to the satisfaction of our customers.\nIntelligent, creative solutions from sound engineering, science and judgment.\nBeing responsible brings more responsibility authority and creates future leaders.\nThis value-driven approach and unwavering commitment to excellence consistently results in award-winning projects, delivered on schedule and within budget, for satisfied clients. We proudly continue to stand as one company, driven by one vision.\nWe believe that developing strong relationships with partners, employees and clients yields personal satisfaction, as well as business success.\nEarning and keeping our clients trust is the foundation for our success \u2013 and the building block for our future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cosham Court Nursing Home is registered to provide accommodation for up to 47 older people, including those with dementia, who require nursing or personal care. It is located in the town centre of Cosham, Hampshire.\nAlexandra Rose Residential Care Home is registered to provide accommodation and personal care for up to 32 people. At the time of the inspection the home accommodated 25 people.\nEdinburgh House is registered to provide nursing and personal care for up to 32 older people. The service is registered to provide the regulated activity of accommodation for persons who require nursing or personal care.\nThe Haven Rest Home is a care home without nursing. The home is registered to accommodate 20 people. It provides personal care for older people.\nLiving Plus Healthcare Limited trading as Queen Anne Lodge is registered to provide accommodation with personal care and nursing for up to 40 people including people who have dementia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyjsw b/data_all_eng_slimpj/shuffled/split2/finalzzyjsw
new file mode 100644
index 0000000000000000000000000000000000000000..bb3abb5d985ae74955449e2cc100dbb13efa2661
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyjsw
@@ -0,0 +1,5 @@
+{"text":"To solve difficulties with searchability together with assessment of information, it's essential to understand something about the content. In the event that the info isn't right, or perhaps low-quality, the manager could earn a decision that has a negative effect on the organization. Use the hyperlink below to see the academic catalog that could supply all that info. Become the information supply, and allow the knowledge be yours. Detailed information can be gotten from your nearest charge or consulate of the region you would like to visit. Incomplete info can result in bad decisions, due to the fact if a choice is created with simply a number of the information chances are it will not be the right choice. You will end up supplied access to a diff for each edited article that can demonstrate the changes which were made.\nYou concerned if you can't locate a individuality remedy straightaway, though. The real key issue is to choose therapies for your existing emotions. In the same way, in regards to selecting Bach remedies you should ignore any physical symptoms.\nOn the BBB's internet site, you may file a problem against a neighborhood firm, view national complaint statistics, and find out the complaint history and BETTER BUSINESS BUREAU standing of a particular business enterprise. There are not any decrease case f's. It can be hard to supply a positive business case for business intelligence (bi) initiatives, and frequently the tasks have to be prioritized through proper initiatives.\nWhen gathering certain requirements from the company users, typically the regional IT department should also be consulted as a way to figure out to which degree it's possible to satisfy the business's needs dependent on the particular available data. Fantastic administrators must be in a position to separate the suitable information from the sounds. Moreover, editors may email you with clarification questions in connection with info in your articles. Not all the articles in your accounts is going to be edited at exactly the same time, but you can have more compared to 1 article selected over time. Your article is going to be locked while it's being modified. An article that's normally component to a helpful article could itself be a practical document, for instance, an ornamental wheel cover on a car. Remember to remember that you're absolutely free to change a report to your liking once coach anyone how to unlocked.\nOur life is basically full of useful lessons that we ought to uncover. If you've done similar work before, the contact information associated with an earlier employer who will will give you a reference is beneficial. It is essential of which personnel who take part in the particular project have a vision plus a notion of the advantages and drawbacks of implementing a BI program.\nWhether the answer offers information depends upon the informed particular person. When you fully grasp the reaction to that question, you'll be within a better place to establish what exactly data to collect and the way to transform it into information you ought to generate decisions. If you concentrate on understanding them, you will appear to be more interesting and dynamic. If you would like to show that there's a huge quantity, you will say there are several of them. If you would like to express that there's a massive volume, you would say there's very much. The needs and advantages of typically the implementation are occasionally driven by simply competition and the need to acquire an advantage on the market. If you need help with a physical problem you should discuss an experienced medical advisor and taking Bach remedies.\nUser help can be incorporated in lots of ways, for example by developing a site. Even greater, helpdesk support may be used. Offering user support is essential to keep the BI system and resolve https:\/\/fjco.hadd.world user issues.\nThe person who paid the most attention was a 3 year old boy. Keep in mind that you don't over it. Keep this question at the back of your mind while you browse on.\nIdeally, make something you have zero idea how to complete. There are some things we now have to review real quick. Small things and big things are often more interesting together than all smaller things or all tremendous things. As you are in charge of a bookmarking website think of merchandise which those who visit a poetry blog would be considering. In fact, it's not possible to really write down everything that someone says. In spite of the insecurity and also some minor political instabilities facing the state, there are always quite a couple of fascinating truth about Nigeria which are worth noting.\nNow internet has turned into a world in a room. With all the help of the web and internet sites, the net has really ever come to be rather useful in a variety of manners for the regular individual. Since it is now very popular, it's being used for most purposes. Folks also use the internet to auction goods.\nDebate topics do not always need to be serious they're able to be amusing also! Debate topics broadly speaking are assumed to carry attention of listeners, and we have some which are certain to keep anybody's interest! In order to really have a fascinating debate, you first have to get a fascinating debate topic. The discussion of its future as a networking company on the opposite hand is all about its capacity to monetise its position for an worldwide scale. Thus, answer the questions that you recognize well, first. Actually you'll be able to look at our interesting questions to ask a guy if you would like more interesting questions. If there is any question out of syllabus, then you simply attend it.\nPresently aday's all of the folks are using online classified web sites to sell or purchase or sell their own goods or solutions. Lots of people will give anything to be able to state what they mean. Before you begin doing this, a small bit of research about indoor gardening is equally crucial.\nAny sort of information on almost any topic is found online. Such advice regards for example the price of residing in a town, the quality of studies in a university . Hopefully, you have enough info to begin earning profits from your poetry. Also some simple information is going to be provided on the website and will not be required to create a query. Imagine you have a web site in which you have editors producing content, for example, for instance, a flow https:\/\/duct.zephyr.website of information shared in just a local domain. It is possible to also prevent by all our societal networking websites and see what the excitement is all about.\nWhenever you have anything to grow the article then I wish to know in the comments! It's pretty obvious that no body is likely to stay unless they locate your site interesting. For instance, if your own blog or weblog post is about love poems you may like to market romantic present solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Network for Employment & Social Care [DAKM] was founded at 2011 as a non profit organisation which offers counseling and supporting services mainly to vulnerable groups that are exposed to the risk of unemployment and social exclusion. Career counseling lies at the heart of DAKM activities along with entrepreneurial counseling, employer mentoring, support of young, social and female entrepreneurship. Additionally, DAKM activates in the area of the social care and well-being for those at risk of poverty and social exclusion. Our organisation considers of utmost interest the use of new technologies in counseling and in that direction it has developed applications such as digital psychometric systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mission: To build a digital marketing company that is universally known for its personal customer approach and is a market leader in creating dynamic digital spaces for any brand that it represents. Our company, Radiata Solutions LLC, was established in 2012 as an experiment to help one of our family members build a better experience on their website.\nThat project turned into a passion for us. We became enthralled with helping people develop their messages and stories that they wanted to tell others, in the digital space, about their brand. We were also astonished at the variety of ways in which those messages were conventionally conveyed, because they were being lost, crowded and fragmented across the ever changing technological landscape.\n1. More simplistic designs with better \"calls to action\" led more interactions across mobile technology.\n2. Cross-platform development, led by the website, alongside AdWords campaigns brought significant returns to our clients.\n3. It became clear that the old practices of information dumps had cluttered the messages of businesses and these websites needed to be given life and function to be dynamic.\n\u2022 Or \u2013 the full package, along with service and support to run an entire \"digital facelift\" for your business.\nAs an owner, I think what makes us different is that we truly believe in transparent practices, ensuring that if our customers decide that they no longer need our services, they will have access to what we have done in order to continue building their brand. Also, we invest time in working with our clients because one of our core values as a business is Healthy Relationships. We strive to do this with our own company culture, and with our clients.\nAnother is Open Partnership. We take our mission and values very seriously in our company. We believe that nothing sustainable can be built without establishing a meaningful mission, values, and principles. With open partnerships, we ensure our clients understand our focus, our lead-in time and our operation before signing up with us.\nThe last is Ethical and Social Responsibility. We also decide which businesses fit within our social values. Our company is partially owned and invested in by Radial Holdings LLC, which strives to build a better world through social responsibility. I hope that you decide to give us a call and we can decide if we will be a good fit \u2013 together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During the third round of the Waste Management Phoenix Open at TPC Scottsdale on February 6, 2016 in Scottsdale, Arizona.\nWaste Management Phoenix Open chairman Andy Markham contends that Scottsdale is a great golf town. His tournament builds on that belief by bringing tourists and Arizonans alike to enjoy golf, food and music Monday, January 30 to Monday, February 6.\nNow in its 31st year at TPC, the Waste Management Phoenix Open has a long and storied history of phenomenal golf dating back to the 1930s.\nThat is an understatement, as Markham anticipates that the 2017 event, dubbed the \"The Greatest Show on Grass,\" will draw record crowds of more than 600,000\u2014despite a conflict.\nThe roster of past winners is a who's who of golfing legends including Arnold Palmer, Ben Hogan, Jack Nicklaus and Johnny Miller. In the TPC era, greats such as Phil Mickelson, Vijay Singh and Tom Lehman have prevailed. In recent years, the next generation of talent has hoisted the championship trophy, with last year's victor 24-year-old Hideki Matsuyama.\nThat great field includes last year's sixth-place finisher Will Wilcox.\nAttendees understand that there is more to the Phoenix Open than golf; the atmosphere is a compelling draw. Whether it is the Coors Light Birds Nest, the event's music series, or the electrifying Par 3 16th hole that defines energy and passion for raucous and perfervid fans, the emotion is palpable throughout tournament week.\nThe environment has become more vibrant since Tiger Woods' famous 16th hole ace in 1997. Incredible noise and energy flowed down from the stands.\nIt seems that there is no stopping the momentum of the Waste Management Phoenix Open. Each year the Thunderbirds, sponsors and partners outdo themselves to provide the world's best golfers with the canvas to dazzle patrons, while also catering to guests who want the entire entertainment experience.\nOne such interesting evolution could potentially be the addition of the adjacent Champions Course to the tournament to broaden the player field and enliven the Pro-Am.\nIt is not just the fans who win at the annual event. In 2016, the Thunderbirds raised $9.3 million for charities, making the total amount of $111 million over the life of the tournament, according to its website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your FAFSA application is selected for Verification, make sure you understand what documents and whose information you need to submit. Not all applicants will be asked to submit the same documents. Always check your CCC student portal.\nVerification is the process used to confirm the data you provided on the FAFSA. It is to your benefit to submit everything promptly. If there are any differences found between information reported on the FAFSA and the actual figures provided in the verification documents, CCC will submit the corrections on your behalf. Financial aid awards will be based on the corrected information.\nYour FAFSA may have been selected for verification randomly by the federal processor, or because it may appear to have errors or conflicting information.\nIf selected for Verification, CCC will send a request for documentation by mail. In addition, your CCC student portal will list any required documents. Click \"My Financial Aid\".\nNot all applicants will be asked to submit the same documents. You may be asked to provide different documents than you have submitted in previous years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Document of Metal Gear Solid 2 Plant Chapter Script .\nThe stair area where the Bravo team made its final stand has been honeycombed by the fire during the Indian Head times. Peter reminisces about Fatman nostalgically. Peter: I know Fatman well. I know how into his own aesthetics he is .\nBusiness listings of Staircase manufacturers, suppliers and exporters in India along with their contact details & address. Find here Staircase suppliers, manufacturers, wholesalers, traders with Staircase prices for buying.\nThe Moon Staircase and the 1m2 staircase are two possibilities. Perhaps the Cells Design or the Transparancy 1-03 series is more to your liking. An EeStairs? Standard Design brings designer staircases within the reach of even a limited budget.\nIndia United Kingdom United States ZDNet around the globe: . Having effective architectural plans are the staircase to reach the rooms of a dream home. In an age when retail business is booming, construction firms have been witnessing rapid growth. . there is one problem in opting for readymade home plans. Suppose you have purchased an .\nEp. 1455: Steve Jobs to you: shut up and eat your hummus .\nOn today's show, the iPhone is apparently storing your location data even when you've turned off the location tracking services. And law enforcement agencies and a cottage industry of iOS .\nFind your metal frame staircase easily amongst the 1,425 products from the leading brands on ArchiExpo, the architecture and design specialist for your professional purchases.\nCheck out the range of Railings & Stairs available in India. View designs, pricing, availability, delivery info and special offers.\nWooden Stairs - ????? ?? ?????, Wooden Stairs Manufacturer .\nSudhakar Engineering - offering Readymade Steel Staircase, Ms Staircases at Rs 170 \/kilogram in Coimbatore, Tamil Nadu. Read about company and get contact details and address.\nWooden stairs - Complement your floor with wooden stair .\nWooden stairs can also be used as a decorative natural element in a space characterised by concrete, stone and glass. Please note that we supply planks, not a ready made staircase. Available in Classic in standard measurements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I need to update the links as I moved from gDrive to OneDrive.\nHe put quite some rathena links in both, grf and msgstring. Nearly all links are leading to rA. I can provide you my grf. It works with the latest clients (I need to find some time to upload it).\nThis grf is not compatible with browedit. Try the ones from my other topic.\nSorry but I have no idea what you are trying to do nor what this script is supposed to do.\nYou build your map in Revision 620. Afterwards (due to problems with missing objects) you load it with Revision 586 and save it again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Keep it simple with the FlipOut changing mat, the essential compact item every parent needs.\nIt's a super-sized changing mat which folds up nice and neat \u2013 perfect for popping in your handbag, beach bag or stashing in the glovebox for that trip into town.\nThe FlipOut Changing Mat Pack is a perfect product that allows you the parent to leave the house with the essentials required to change baby without having the bulk of a changing bag! The FlipOut is an ideal product to leave in the boot of the car for those emergencies when the changing bag has been forgotten and baby needs to be changed quickly and efficiently!\nAs the FlipOut is a practical yet compact product, it allows those parents to use their own handbags whilst stashing the FlipOut inside their own handbags!\nThe FlipOut Changing Mat Pack has been designed to allow at least 5 nappies and a quantity of baby wipes to fit inside the changing mat pack, as well as allowing the required creams and other sundries to be located at the large pocket at the back of the product that is easily accessible for the parent!\nThe large 'super-size' changing mat can be easily removed allowing this to be used on it's own in the nursery, living room or bathroom when changing your child!\nThe FlipOut Changing Mat Pack also comes with a simple Velcro loop to allow the FlipOut to attach to any pushchair\/stroller with ease so that instant access can be gained to the essential items that are stored within the product with minimal fuss!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A superb & extremely rare Plaque commemmorating the expected coronation of King Edward VIII. This magnificent item is modelled in relief with beautiful detailing & gilding, it is signed within the cameo of the Kings head by Lucien Boullemier (Maling's famous designer). It is larger than the usual Maling plaques & very imposing at 12 5\/8\" diameter. Very few of these items ever become available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My name is Affairing and I'm a beautiful Uni. Already I'm 90960 hours old!\nI am not just a pretty face however, I read books regularly and once a week manchestersontonberg takes me to the Neopian Battledome to watch the tournament. I would like to enter, it looks such fun but I am afraid I might chip a hoof.\nI really love my owner and I think I am the luckiest pet in all Neopia. If you would like to play with me or chat with my owner, NeoMessage manchestersontonberg!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Europe introduced its own set of sanctions against Iran in what represented a small shift in policy towards the Iranian regime. Not since before the Iran nuclear deal has Europe employed sanctions against the Iranian Regime. The move signifies that European leaders may finally be awakening to the fact that the Iran nuclear deal has \u2026 Continue reading Europe's Sanctions Are Welcome, Now A Change In Policies Is Required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working the ECO ways can be exhausting\u2026 or so we've heard, because we're just getting started.\nThis year, the ECO Spot at the Electric Castle Festival will have its' own stage, powered and supported by Lidl Romania. And as you already know, the stage will be 100% friendly to our environment, using the Solar Sound System which runs on solar and kinetic energy. If the sun will shine high and bright and if the festival people don't stop pedalling, you're in for a party! But who exactly are we going to fuel?\nA line-up that combines Romania's most noticeable rising artists and unique international DJ's. The result is a perfect synergy of up-beat sounds raging from house, to electro, tribal and groove. They are the ECO fighters that will animate and give more life to the camping area of the festival. The stage will be ready for you on Wednesday, the 18th of July, with its' first guest Julian M. starting 14:00. From Tuesday you can see your favourite artists perform here daily starting 12:00.\nPut your dancing shoes on and meet us at the castle. Or in this case your sneakers, you'll definitely need sneakers for pedalling!\nSummer has already made its full presence since about a month now. We can stay out late and we constantly feel the need to eat ice cream.\nBut most importantly, our minds wander more. We think about dancing our shoes off while our faces are covered in sunshine. Thinking about those blissful moments which we're gonna experience at Electric Castle 2018.\nOf course, we're all preparing our festival outfits in advance. We're also thinking ahead of what artists we're gonna see and making fun \"to do\" lists.\nHowever, the mindset we bring to Bon\u021bida is most decisive when it comes to the amount of fulfillment we're gonna get at the end. It's dandier than any brand new shoes we're gonna wear. Why?\n'Cause taking care of the Banffy castle and its surroundings is important. Supporting the ECO cause which affects all of us is crucial. Basically, treating the dance-floor with respect and driving eco-responsibility through music is uber cool.\nIn 2018, MAINOI's ECO Program of Electric Castle proudly celebrates 5 years of sustained progress towards implementing sustainability standards at the festival.\nWe're still riding along the journey towards creating the most ECO-friendly festival in Romania. Together with Lidl, we are continuing the #ZeroWaste mission, our long-term objective to reduce, reuse and recycle all the waste generated during the festival.\nLast year, when we launched the #ZeroWaste mission, we managed to recycle 65% of the waste produced. This year we are raising the bar to 75-80% recycling rate.\nBut, there is a catch: we can only do it with all of you onboard!\nEverywhere in the fe stival, you will find recycling stations for recyclables (aluminum, plastic, cardboard, paper) and general trash. Thus, recyclable materials are not being discarded together with the other garbage but will get a chance to a second life.\nIf you camp, you will receive a disposable green bag for your recyclables and a black bag for your trash; please use the bags accordingly and bring the green bags, full with recyclables, at the ECO Spot, near the Lidl store, to get cool incentives in return.\nIn addition, as of this year, we will be running the awesome NO STRAW policy, therefore you won't receive plastic straws with your drink. However, if you insist, biodegradable straws made from 100% compostable materials, will be available for you, at the bars, upon request.\nWe have a contest that might really spark your interest!\nHelp us raise awareness on our eco activities by showing the whole wide online world your perspective about the ECO program. Use either your lens or your writing skills in the most unique and surprising way, and get the chance to win great eco prizes!\nWe're expecting images, videos or articles that showcase the sustainability practices you observe at the festival, but we don't mind if your imagination goes further than that!\nYou only have to post your creative content on your social network (Facebook, Instagram, blog, etc.) with #ecoec and #zerowaste hashtags, send us the link at electriccastle@mainoi.ro, until 24th of July, and you can win a bicycle, a solar backpack, and other nice props to help you become a legendary EcoCitizen.\nThis year you're going to simultaneously pedal and dance at our ECO Spot, as we are going to set-up the first solar stage ever at a Romanian music festival, by bringing the unique Solar Sound System to the Banffy Castle for your musical delight.\nOne of the most engaging alternative transportation with zero carbon footprint awareness campaign is the ECO Ambassadors, now at its forth edition. 40 cyclists who will hop on their saddles and pedal for more than 250 km to get to the Electric Banffy Castle to promote the idea that cycling should be considered a national objective, being a zero carbon footprint means of transportation.. Greet them at arrival, on Wednesday, at 18h00, at the festival's entry gate and celebrate together afterwards at the Beer Garden.\nElectric Castle 2018 is incredibly close and the time to kick off the recruitment for the ECO Team has come!\nL.E: Deadline extended on some roles, check below! You will join the waiting list & if you are the kind of people with flexible schedule there are many chances you'll get a call.\nEvery year, wonderful people have joined our team with the belief that Electric Castle is the place where their contribution to change is guaranteed.\nThis summer we continue the #ZeroWaste campaign, which is our long-term goal of reducing, reusing and recycling all the waste generated at the festival. Last year we recycled, with the efforts and enthusiasm of the entire team, 60% of the waste created during the festival.\nIn 2018 we want to aim even higher, who's with us?\nEvery year there is always a positive energy around the ECO Spots. Here only good music is playing and anyone's welcome to join!\ncoherence and expressiveness in speaking Romanian and English (advanced level).\n>>>APPLY HERE<<< until Friday, June 22nd.\n\/ 8 hour shift during the day.\ninvolve festival goers in an ECO questionnaire about their recycling habits and their level of satisfaction for the program.\n\/ 6 hour shift, 16:00-22:00 and 22:00-04:00.\nwear the ECO team uniform (overalls, t-shirt \u2013 don't be afraid, it will be very cool \ud83d\ude42 ).\nhave experience in photography at outdoor events.\n>>>APPLY HERE<<< until Friday, June 15th.\nElectric Castle 2018 este incredibil de aproape \u0219i a venit momentul s\u0103 d\u0103m START-ul recrut\u0103rilor pentru echipa ECO!\n\u00cen fiecare an, oameni minuna\u021bi s-au al\u0103turat echipei Mainoi din convingerea c\u0103 Electric Castle este locul \u00een care au garan\u021bia c\u0103 pot contribui la schimbare.\nPentru edi\u021bia aniversar\u0103 \u2013 5 ani de program ECO \u2013 punem la b\u0103taie zeci de roluri de voluntari, promoteri, reciclatori \u0219i fotografi, pentru a sus\u021bine principalul scop comun: crearea celui mai ECO festival din Rom\u00e2nia!\nVara aceasta continu\u0103m campania #ZeroWaste, care constituie obiectivul nostru pe termen lung de reducere, reutilizare \u0219i reciclare a tuturor de\u0219eurilor generate \u00een cadrul festivalului. Anul trecut am reciclat, cu eforturile \u0219i entuziasmul \u00eentregii echipe, 60% din de\u0219eurile create pe parcursul festivalului.\n\u00cen 2018 vrem s\u0103 \u021bintim \u0219i mai sus, cine e cu noi?\n\u00cen jurul ECO Spot-urilor din fiecare an se adun\u0103 mereu o energie pozitiv\u0103 aparte, se leag\u0103 constant prietenii, se ascult\u0103 muzic\u0103 bun\u0103 \u0219i oricine e bine-venit.\najute echipa ECO la orice este necesar si tine de specificul organizarii unui eveniment de outdoor.\ncoeren\u021b\u0103 \u0219i expresivitate \u00een vorbirea limbii rom\u00e2ne \u0219i a limbii engleze (nivel avansat).\n>>>APLIC\u0102 AICI<<< p\u00e2n\u0103 vineri, 15 iunie.\n\/ tur\u0103 de 8 ore pe timpul zilei.\n>>>APLIC\u0102 AICI<<< p\u00e2n\u0103 vineri, 22 iunie.\n\/ tur\u0103 de 6 ore, \u00een intervalele orare 16:00-22:00 \u0219i 22:00-04:00.\npoarte echipamentul de echip\u0103 (salopet\u0103, tricou \u2013 nu v\u0103 fie team\u0103, vor fi unele foarte cool \ud83d\ude42 ).\nUPDATE 13th July \u2013 The Eco Game is up and running! Come visit us at the ECO Spot in the camping!\nUPDATE 12th July, 19:37 \u2013 The ECO Game will start tomorrow, Thursday, 13th July. We will announce the exact starting time through another update on the Regulation's webpage. Stay tuned!\nAre you a camper of Electric Castle 2017? See you soon!\nFind us in the camping, sign up at the ECO Spot and start the journey of an Eco Citizen. We're looking for nature lovers, people who want to take action and lead by example. You are most welcomed in our cosy spot. \ud83d\ude42 Pedal the best out of you at the charging point and have fun playing the ECO Twister!\nYou can become Eco Guardian, Angel or Legend of Electric Castle. What you need to do is to complete between three to five activities and get the chance to win some cool merchandise, a 2018 Premium EC Pass and a Iute Bike. Read the FULL regulation below.\nWhat ECO Citizen title can you gain at Electric Castle? An ECO-Guardian (for 3 eco-actions completed), an ECO-Angel (4 eco-actions completed), or maybe even an ECO-Legend (all five eco-actions completed).\nCite\u0219te \u00een limba rom\u00e2n\u0103: Regulament ECO GameContinue Reading..\nIf you and your friends are cycling enthusiasts and got the rhythm in your veins, hop on your bike and have a blast at Electric Castle Festival. It's on us! Free full pass, camping & other goodies await you at the gate of the Banffy Castle (Bon\u021bida, Cluj).\nRadu, a 2016's ECO Ambassador, arrived at the Electric Castle after travelling 500 km by bicycle.\nWhat do you have to do to become an ECO-Ambassador?\nChoose an eco cause to promote before, during and after the festival. Any cause from deforestation to ecological education. Are you already working for a cause and you're motivated to support it in the long run? Even better, your chances of being selected increase.\nPedal to the Banffy Castle from a distance greater than 250km.\nBe visible online, a good communicator and active on social media (Facebook, Instagram, Twitter). The stronger your ability to reach people, the greater the chance of being selected.\nOwn a smartphone\/tablet with mobile data. It is very important to have a device with Internet connection to share media content (photos, videos, live streaming) during your journey.\nHave a public Strava account \/ other biking app. It is useful for proving your experience in pedalling on long distances. You will have to use this account to track your journey by bicycle towards Electric Castle.\nHave the determination to convince as many festival participants as possible to join the ECO program at Electric Castle.\nOwn suitable cycling equipments and bicycle for this kind of journey.\nBe experienced and physically capable to complete this kind of journey in safe conditions.\nPlan your journey ahead. We need you to be in Cluj-Napoca in the morning on the 12th of July (place and time TBA).\nWe greatly promote team working, but we also accept lone wolfs. Continue Reading..\nUPDATE: Regulamentul s-a modificat la articolul 6 \u2013 Premiile. Toate categoriile de premii vor fi acordate prin tragere la sorti, cu random.org, in ctombola fiind inscrisi toti participantii care au facut cel putin o actiune ECO.\nECO Challenge este provocarea pe care \u021bi-o arunc\u0103m la Electric Castle 2016: fii ECO, distreaz\u0103-te \u0219i \u00eenva\u021b\u0103 s\u0103 protejezi mediul! Concursul este un joc prin care \u00eencercam s\u0103 g\u0103sim cei mai ECO participan\u021bi la festival \u0219i s\u0103 \u00eei premiem pe m\u0103sur\u0103.\nTot ce trebuie s\u0103 faci este s\u0103 \u00eendepline\u0219ti cu succes c\u00e2t mai multe activit\u0103\u021bi ECO din cele cinci categorii: Energie, Ap\u0103, Transport, De\u0219euri \u0219i Social, \u0219i astfel s\u0103 aduni c\u00e2t mai multe puncte. \u00censcrierile se fac la ECO Spot-ul din festival, amenjat fix la intrarea in hide-out. Pentru ac\u021biunile care au loc \u00een camping (du\u0219uri sau cinema) voluntarii v\u0103 ofer\u0103 cartona\u0219e pe loc, pe care ulterior trebuie s\u0103 le preschimba\u021bi \u00een puncte la ECO Spot din festival.\nAc\u021biunile ECO se valideaz\u0103 de c\u0103tre un voluntar ECO-Crew prin \u00eenscrierea direct\u0103 \u00een baza de date la ECO Spot.\nFiecare ac\u021biune se puncteaz\u0103 o singur\u0103 dat\u0103.\nParticipan\u021bii trebuie s\u0103 ofere datele de contact corecte \u0219i s\u0103 fie de acord cu Regulamentul concursului.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WHERE WILL PHOTOGRAPHY TAKE YOU?\nJoin us on a photo journey you'll never forget! 2018 and 2019 departures are coming soon!\nWe love photography and we love travel.\nWe're a tour operator that specializes in photo tours and photo workshops. We create and operate our own unique photo trips & we also build and operate custom photo trips for others in the industry.\nYou deserve to take a photo trip.\nTravel enriches the mind in ways that few other experiences can. When you travel, you'll be inspired to capture the beautiful scenes that you're experiencing in person. You take photos because you find beauty in scenes you look at. When you take a photo trip, you get to explore new places or see places you've been in a new way.\nBeing in the right place at the right time with great light is crucial for any photographer. With our expert planning and guidance from professional instructors, you'll have the perfect setup to create images that will make your portfolio pop. Not only will you increase your chances of capturing great images, but you'll learn new techniques to help you process those images and make them look amazing.\nYou've heard the saying, \"it's not what you know but who you know\". Taking a photo trip provides a unique opportunity to rub shoulders with industry professionals and spend quality time with other like-minded photographers. Often, some of the most valuable experiences of your trip are networking with others in the industry.\nOn every trip you'll have industry professionals with you sharing photography knowledge to help you improve your skills at every stop along the way.\nPhotography is the focus of our trips, but a trip encompasses many other aspects besides taking photos. On each trip you'll have a staff member who is knowledgeable in the areas you're traveling to and trained in operating trips who will make sure you're well taken care of throughout your journey. Sometimes, your guide and photo instructor are one in the same!\nWe're committed to continually offering top level trips and experiences for our clients. Whether you're coming along for one of our own trips or hiring us to operate a trip for you - we're committed to making sure your trip is everything you hoped for. We continually strive to lead the industry, improving and making our trips the best out there is our passion.\nOur trips are crafted by photographers and travel industry pros. Photography is always the priority, but experiencing culture, history, food and fun activities is also a key part of traveling. When these experiences are available, we weave them into our trips during times where taking photos is less than ideal. This allows us to provide a well rounded travel photography experience for our guests.\nNo matter where you travel with us, expect to be greeted by friendly, fun, helpful staff. There's nothing worse than booking a trip and showing up only to find out your instructor feels like you are the lucky one to be traveling with them. We're grateful that you've made the decision to travel with us. You are the priority and we'll work tirelessly to make sure you have an enjoyable, safe and productive travel experience with us.\nYour experience is what matters most to us.\nOver the years I have taken multiple photo trips. This trip to Banff & Jasper was one of the best that I have been on... All the accommodations were excellent & the same applied to all the meals provided. I would go on another trip without any hesitation. Thanks for a great time.\nThe trip to Havasupai and the Grand Canyon was excellent - amazing photo opportunities. David was very free with advice on how to make the most of the photographic opportunities that were available during the trip. David took great care of the group and the food was wonderful too. The other photographers on the trip were extremely nice and the chance to exchange photo information was a hit - overall a great experience.\nI'd like to thank David for organising such a seamless trip. He, along with our driver Dwayne, got us to all the locations on time and adjusted the itinerary to take advantage of changing conditions - and with early snows, we had lots of changing conditions! However, it was the friendliness and ease with which Dave made everything happen. Okay, so I was one of the instructors, but I was just along for the ride. I've travelled with many photography groups and leaders over the years and David is up there with the very best! I'd have no hesitation going on another trip with Photo Nomads - hell, I reckon we need to organise something for next year!!\nI would recommend Photo Nomads to anyone. Great knowledge of both logistical stuff but also good photography knowledge... just amazing support. Dave was awesome & easy to get on with. Hope to catch up with him in the future.\nDave made every aspect of the adventure memorable. He was accommodating to everyone and knew every area throughly that we visited. Dave made the trip effortless. I couldn't recommend Photo Nomads any higher!\nDuring my trip to the Canadian Rockies with Photo Nomads, I was shown a photographer's paradise with stunning peaks, emerald lakes and majestic wildlife. I have found all of this thanks to David Gaiz (my guide and photography instructor). David knows where and when to go in order to take the best pictures. The Photo Nomads \"Oh Canada!\" trip was wonderful. Thanks David!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Troutville, Virginia is the 14,004th largest city in the United States as of 2017.\nTroutville, Virginia is a town.\nTroutville, Virginia is currently 5.5% smaller than it was in 1990, the year of its peak recorded population.\nTroutville, Virginia is growing somewhat quickly. It is growing faster than 70% of similarly sized cities since 2000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was within hours, that Georgia would become one of our favourite countries visited on this trip. And after spending 6 nights throughout the country it became everyone's absolute favourite to date. Must have been a perfect mix of the wine, people, variety of landscapes, food and costs that won us all over.\nIn Batumi, on the Black Sea coast on a 'day off' from driving we all went about our own ways, till regrouping at the beach for a swim. That night we quickly discovered the variety of Georgian foods; chef hat dumplings full of different types of meat and white 'porch pounder' wine.\nAfter meeting with the Subaru Motors in Tbilisi for photos and an oil change, we all met up and dined near Freedom Square, sampling more fine Georgian wine.\nAs the boys were still on the hunt to get the skid plate, us ladies took off up into the mountains. We first stopped at Makheti, the birthplace of Georgia. Then we drove up to 2300m where there were ski lifts and upwards to Kazbegi, near the Russian border. We kind of stumbled THE ROOM, a magnificent hotel\/casino with views to die for. We went so far to look at a room and were really tempted, but, NO, we wanted to camp. Luckily upon asking about camping on the hotel's grounds and they said yes for $25.00 for a 10 star camping spot!. We had a 5star lunch on the massive deck overlooking the mountains, lounging, feeling relaxed. After setting up the teepee we sipped our own cocktails in the hotel's beanbag chairs. That night we checked out the bedroom-like casino, where Corinne won a whopping $20 on roulette.\nWoke up to a magnificent sunrise hitting the teepee. We managed to get a couple of free breakfasts and some excellent coffee at the posh hotel. Took a long detour, aka the backroads to Signaghi, with a stop in Tavali for lunch. En route, a German owned vineyard gave us an awesome tour and we did some sampling and purchased a few bottles of the Saparvi for $6\/bottle. Signaghi is a stone walled town of 2000 and we found rooms at Nato&Lato's Guest House where we were greeted with homemade Georgian cha cha, a national potent spirit, and traditional folk music. The boys met us there with a skid plate-HURRAY!!!!!\nWe took the following day off, wandering the small, visiting Pheasant Tears, a local winery where they age their wine in clay cask that are covered in honeycomb and sunk into the ground. The cat back at our Guest House finally showed us her kittens and so of course we had to go out and buy them some dried fish. We had a roaring game of blackjack where the forms of money were everything from corks to a lighter to random coins to a bottle cap. Andy cleaned house!\nIt was tough saying goodbye to Georgia, but also welcoming as we knew Azerbaijan onwards was going to be challenging\u2026first step, getting the cars across the border.\nTo everyone's surprise crossing the border only took an hour. The drivers and went separately, while us passengers walked across. Smooth sailing and no issue with the car not being in Corinne's name, though wearing the Subaru Rally Team Canada shirt doesn't hurt. We drove to Sheki a very old town in Azerbaijan and we ended up staying at a caravanserai which was over 300 years old with really thick walls, so no AC needed. The street side front door was massive but closed as there was a tiny door that opened up into a massive ancient lobby, then beautiful courtyard. Definitely one of the more unique places we've stayed, and at a good price.\nBest of week four: All things Georgian: wine, food, people, variety of landscapes, selection of beers, affordability. Most unexpected country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaxqp b/data_all_eng_slimpj/shuffled/split2/finalzzzaaxqp
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@@ -0,0 +1,5 @@
+{"text":"Sassy Astray turned in one of the most professional performances of the new Gulfstream Park meeting Friday, breaking her maiden at 7 1\/2 furlongs on the grass while becoming Gone Astray's 46th winner of the year. With Paco Lopez aboard, the 3-year-old filly broke well and settled in comfortably on the rail, just behind the two pacesetters. Lopez kept her there until they neared the stretch, where he swung her out three wide and Sassy Astray took over from there, galloping away from the field and winning by 4 3\/4 widening lengths. The victory was worth $12,200.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What will we do when the temple falls?\nWhat will we do when they breach the walls?\nWhat will we do when the walls come down?\nWhat will we do when they burn the ground?\nWhere will we go when the crops all die?\nWhere will we go when the birds won't fly?\nWe got it wrong, just a pack of lies?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want to check out Mallorca this summer, but afraid of being bored during your time there? If this is your situation, then you should take time out to explore Palma de Mallorca.\nBeing a city of 400,000, there is plenty of cultural attractions and big city amenities to keep you busy on days where the beach just won't do it for you.\nBegin to discover Palma de Mallorca by dropping by Castell de Bellver. Serving as the royal residence of King James II back when Majorca was its own self-sufficient fiefdom, as well as a military prison from the 1700's straight through to the 1950's, this imposing circular castle is now home to the history museum that chronicles the life and times of the island of Mallorca.\nWhile its contents are certainly interesting, the structure itself is well worth seeing on its own, as many castles that were built when Castell de Bellver was have long since succumbed to the ravages of war and time, while it has maintained its Spanish style grandeur.\nThose that are fascinated by Spanish architecture owe it to themselves to check out Pueblo Espa\u00f1ol, as it contains reconstructions of major structures that are essential to the history of design in this part of the world.\nModeled like a village, it it easy to stroll through and appreciate the structures that this nation has spawned over the ages, and you can do so without having to stray far from your beach resort!\nWhile many that visit Palma during their holiday on Mallorca seek out its urban amenities, this city has plenty of amazing beaches within city limits that are well worth visiting. The best of the bunch is Ciudad Jard\u00edn, which offers sands that are just as tantalizing as most other beaches are elsewhere on the island, but with services that include incredible restaurants and cafes, it has all the perks that are often lacking at rural seaside spots.\nIf the idea of jumping in the ocean doesn't appeal to you, but you still need to cool off in the worst way, then heading to Aqualand, Palma's premiere water park, is the perfect way to go about this in a fun manner. While this place is perfect for families, even those that are young at heart can enjoy the many exciting slides and wave pools here, as they are far more engaging than your static pool back at the resort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IRS Announces 2018 Tax Filing Season Opens with April 17 Deadline; 155 Million Tax Returns Projected, 70 Percent Expect Refunds.\nMany Hurricane Victims Qualify for Earned Income Tax Credit; Special Method Can Aid Workers Whose Income Dropped.\nIRS Issues Additional Guidance on Transition Tax on Foreign Earnings.\nGet Ready for Taxes: Plan Ahead to Avoid Tax Refund Delays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 16th annual Florida Bull Test Sale was held on January 16, 2016 at the conclusion of the 2015\u20132016 Florida Bull Test. The test evaluated the performance potential and breeding soundness of bulls consigned to the program at the UF\/IFAS North Florida Research and Education Center (NFREC). In an ongoing effort to better serve consignors and bull buyers, the 2015\u20132016 test provided data on individual feed efficiency for each bull, making this the fifth year that this information has been included. After the test, consignors with bulls that met the established benchmarks of the test were then given the opportunity to sell their consignments in the Florida Bull Test Sale. In order to qualify, bulls needed to meet growth performance, structural soundness, and disposition standards. Sale bulls also must have successfully passed a breeding soundness exam to qualify for the auction. Potential buyers and consignors were able to fully evaluate the animals on all parameters including actual performance data, expected progeny differences (EPDs), and carcass ultrasound data.\nThe 112-day test began on July 28, 2015, with bulls being sorted into contemporary groups based on consignor and breed (8 to 12 bulls per pen). Bulls were then housed in the NFREC Feed Efficiency Facility where they received free-choice access to a total mixed ration and water to achieve a target rate of gain of 3.5 lb per day. On a dry matter (DM) basis, the diet consisted of 42% pelleted soy hulls, 41% pelleted corn gluten feed, 12% loose peanut hulls, and 5% molasses-based liquid supplement containing vitamins, minerals, and ionophore (monensin). The diet was formulated to contain 16.3% crude protein and 0.51 Mcal\/lb of DM net energy of gain (NEg).\nAfter a three-week adaptation period, bulls were weighed on two consecutive days to obtain an accurate average unshrunk weight, which was used as the on-test starting weight. Throughout the test, bulls were inspected daily for any potential health problems. An intermediate unshrunk weight was obtained after 28 days. Bulls were then weighed on two consecutive days for an accurate 56-day weight, completing the feed efficiency portion of the test. On the same day, bulls were removed from the feed efficiency facility and then housed in 3.25-acre pastures where they stayed for the remainder of the test. While pasture grazing, bulls remained in the contemporary groups assigned in the feed efficiency facility pens. In addition to grazing and free-choice bermudagrass hay, bulls were offered free-choice access to the same total mixed ration which was provided in the feed efficiency facility. On day 84 of the test, an additional intermediate unshrunk weight was obtained. At the conclusion of the 112-day feeding period, bulls were weighed again on two consecutive days, December 8 and 9, to determine their final test weights. Animal performance by average daily gain (ADG) was assessed through calculation using only the starting and finishing test weights. Bulls were monitored throughout the test for overall health and were also observed and screened for structural soundness and disposition. Any bull deemed structurally unsound with poor conformation or chronic lameness or with poor disposition was disqualified from the sale.\nFor the second year, the Florida Bull Test offered remote bidding for the sale of bulls through a verified online bidding company. This service allowed consignors to market their bulls to potential buyers who were unable to attend the sale in person. Additionally, a video clip of each bull was made available online as a means of visual evaluation for those who could not visit the NFREC facility before the sale. On the day of the sale, 18 of the 29 registered bidders actively participated in the live auction.\nUpon arrival at the feed efficiency facility, bulls were tagged with electronic identification (EID) tags to monitor daily feed intake using the GrowSafe system. ADG was calculated for the 56-day feed efficiency portion of the test. Residual feed intake (RFI), calculated as the difference between actual feed intake and expected feed intake, was the measure of feed efficiency used to rank the bulls in the test. Daily feed intake was measured for each bull, and RFI was calculated as described by Maddock and Lamb (2009).\nBulls must have been born between August 15 and December 31, 2014.\nAll consignors' herds must have been enrolled in their respective breed association performance program. State BCIA programs were accepted for herds whose breed association did not have a performance record program.\nBulls must have completed the weaning phase of the performance record program with their contemporary group, and this information must have been presented upon delivery. If data had not yet been returned from the consignor's respective association, a copy of the weight data with the number of contemporaries was requested.\nBulls could either be purebred or fullblood, but must have been registered with their breed association. Composite bulls must have had both sire and dam registered in an acknowledged beef breed association. A registration certificate and pedigree were required at time of bull delivery to the test station. Otherwise, bulls were not accepted.\nEach bull was required to weigh 2.5 pounds per day of age when delivered to the test station. A transit shrink of 1% per hour of transit time was allowed.\nBulls must have been weaned a minimum of three weeks prior to delivery.\nBulls were required to be structurally sound and show evidence of good growth potential.\nBirth weights of bulls were required at time of delivery.\nConsignments were limited to ten bulls per owner or operation. Additional consignments would have been considered on the basis of space availability.\nThere was a limit of 132 bulls for the Florida Bull Test. More than 132 bulls were nominated. As a result, the following selection criteria were used to determine which bulls were accepted.\nFirst preference was given to breeders or consignors who were members of the Florida Cattlemen's Association.\nSecondly, bulls were accepted based on the order in which nominations were received.\nBulls originating from embryo transfer must have been designated as such, and the breed of the recipient cow must have been identified.\nBulls must have had legible and permanent identification (tattoo or brand) corresponding to the registration paper at delivery.\nConsignors were informed that horned bulls may be grouped separately. Additionally, it was recommended that bulls be dehorned and healed by time of delivery.\nAll bulls were required to be in good health. They needed to be accompanied by a health certificate bearing the herd number that indicated they were from an accredited or certified brucellosis-free state or herd or had a negative test for brucellosis no more than 30 days before delivery. Bulls originating from a state that is not tuberculosis (TB)-free were required to be accompanied by a health certificate showing they were from a certified TB-free herd or have had a negative test result for TB no more than 30 days prior to time of delivery.\nBulls must have been vaccinated twice with at least 21 days between vaccinations for the following: 5-way leptospirosis, 7- or 8-way clostridium with Haemophilus somnus, and IBR\/PI3\/BVD\/BRSV, with the last vaccination occurring at least three weeks prior to delivery. Vaccination for Pasteurella was optional. The use of intranasal vaccination against IBR\/PI3 was recommended.\nConsignors were responsible for all examination and treatment costs if veterinary attention was required.\nConsignors were encouraged to contact their local or state veterinarian for interstate permit and health requirements. An official certificate of veterinary inspection (health paper) was required for each bull.\nAt the conclusion of the test, an overall ranking assessing the parameters of ADG and the weight per day of age (WDA) of each bull was compiled, and an index ratio was generated. The top performing bull as well as top performing Charolais bull was owned by Rogers Bar HR of Collins, MS, indexing 136 with an ADG of 4.95 and WDA of 3.22 lb\/day. The top SimAngus bull, BF Catalyst B59, was owned by Boyd Farm of New Brockton, AL, ranking 2nd overall and indexing 131 with an ADG of 4.24 and WDA of 3.64 lb\/day.\nThe top Simmental bull, WF Powerball, was owned by Wells Farm in Selma, AL and ranked 5th overall, indexing 127 with an ADG of 4.22 and WDA of 3.40 lb\/day. The top Angus bull, Tates CC&7 Blackberry B14, owned by Windy Hill Angus Farm in Boaz, AL, was ranked 9th overall and indexed 118 with an ADG of 3.83 and WDA of 3.29 lb\/day. The top Hereford bull, JTN Rev 4R 9114 B20, was owned by J. Taylor Neighbors of Americus, GA and ranked 36th overall, indexing 107 with an ADG of 3.45 and WDA of 2.96 lb\/day.\nThe top Brahman bull, FF MR BAR COYO, owned by Ford Farms of Malone, FL, was ranked 45th overall and indexed 105 with an ADG of 3.82 and WDA of 2.52 lb\/day. The top composite bull owned by Running M Ranch from Tallahassee, FL was ranked 86th overall and indexed 97 with an ADG of 3.00 and WDA of 2.84 lb\/day. The top Braford bull was also owned by Running M Ranch and was ranked 95th overall, indexing 94 with an ADG of 2.98 and WDA of 2.66 lb\/day. Table 1 summarizes feed efficiency data, and Table 2 summarizes individual feed intake and feed efficiency. Table 3 summarizes individual animal performance.\nThe Florida Bull Test Sale was held at the NFREC Beef Unit in Marianna, FL. Initially, 136 bulls entered the test. Eighty-five head of the 108 qualified bulls walked through the sale ring for offer. The sale grossed $350,778 with an average of $4,078 per lot. The one and only Brahman bull offered was sold for $4,600. Angus bulls averaged $4,586 on 18 lots while SimAngus bulls averaged $4,020 on 44 lots. Simmental bulls averaged $4,005 on 17 lots. Hereford bulls averaged $3,620 on five lots, and Charolais bulls averaged $2,833 on three lots. The high-selling bull was lot 9, Tates CC&7 Blackberry B14, selling for $6,300. Tates CC&7 Blackberry B14 was consigned by Windy Hill Angus Farm of Boaz, AL and purchased by Herman Walker of DeFuniak Springs, FL.\nCharles and Laura Flythe of C & L Farm receiving their award for their consignment, CLS Moving Forward14, that concluded the test as the Florida Bull Test Feed Efficiency Winner. Pictured from front to back: Charles and Laura Flythe of C & L Farm, Dr. Nick Comerford (UF\/IFAS NFREC director), and David Thomas (UF\/IFAS NFREC Beef Unit supervisor).\nSummary of feed efficiency data for bulls in the 2015\u20132016 Florida Bull Test.\n2015\u20132016 Florida Bull Test individual bull feed efficiency and feed intake data.\n2015\u20132016 Florida Bull Test individual bull performance in order of final test index.\nThis document is AN328, one of a series of the Department of Animal Sciences, UF\/IFAS Extension. Original publication date November 2016. Visit the EDIS website at http:\/\/edis.ifas.ufl.edu.\nCarla D. Sanford, graduate assistant; G. Cliff Lamb, professor; and Nicolas DiLorenzo, assistant professor; Department of Animal Sciences, UF\/IFAS North Florida Research and Education Center, Marianna, FL 32446.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Combined with the Koyo 7314DT bearing intech in Japan, Koyo 7314DT bearing intech in Japan has the most complete product outline in rolling bearing industry Koyo 7314DT bearing intech in Japan provide customers with a series of comprehensive and perfect service and technical support.\nThe main application areas of Koyo 7314DT bearing intech in Japan bearings: aerospace engineering, metal cutting machine tools, steel processing equipment,paper making machinery, cement machinery, engineering machinery and mechanical vibration, environmental protection equipment, packaging machinery and so on.The main types of Koyo 7314DT bearing intech in Japan bearings : deep groove ball bearings, angular contact ball bearings, cylindrical roller bearings, thrust ball bearings, thrust roller bearings, joint bearings, etc.\nKoyo 7314DT bearing size and details i. Details :Thrust Ball RNA4900 NSK 7314DT NTN 51272 Koyo CFE12-1VUU IKO How to determine KOYO bearing size parameters.\nGreat service and quality bearings KOYO 7314DT Bearing don't have to mean high prices. Find it cheaper online and we will agree to match it with our Price Match.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are You a Good Blepharoplasty Candidate?\nThere are many benefits to lower eyelid rejuvenation surgery. The key is to determine exactly what the problem is.\nExample of how elevating the lateral canthus in addition to lower blepharoplasty can create an attractive, almond-shape appearance.\nTightens and smoothes out bags underneath the eyes.\nRepositions lower eyelids into a more elegant, almond-shaped appearance.\nRepositions underlying fat to get rid of the sunken, aged appearance.\nRepositions skin and underlying muscles to smooth out wrinkles.\nMarkedly fills out the deep crease known as the tear-trough between the eye and nose.\nEliminates circles underneath the eyes.\nImproves dark discoloration within the lower eyelid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"why not try \"crazy baseball mitt on your head\" guy?\nit worked for eddie murray on this 1992 baseball card magazine card. a costume like that should net you at least a box of raisins.\nit's taking me longer than i first expected to get through my steve garvey oddball binder. i've got most of it scanned so now it's just a matter of getting things into posts. this particular post has a few interesting items that don't fit into standard 9-pocket sheets. that's why they are in the oddball binder.\ni thought at one time about trimming this panel, but then i would be left with a lee mazzilli card. and i wouldn't want that. so, i've kept the panel intact.\nthe back of the linnett card featured an itala vehicle.\ni do have a complete team set of the linnett 'cards' that is uncut, but found a single garvey online a few years ago.\njohnny bench was on the cover of this booklet that had something to do with the 1985 midsummer classic. garvey was featured on this page that is all about being a good sport, although there's nothing here about not sticking your hip out during the world series to deflect a ball and get away with interference. yes, reggie jackson, i am talking to you.\ngarvey represents the dodgers on the sheet - the dodgers that had a 2 games to none lead in the series, and were still up 2 games to 1 with a 3-1 lead in game 4 when reggie interfered with what would have been an inning ending double play relay. the umps missed the call, the yankees tied the series, and the rest is history.\napparently, there was an old-timer's all-star game a couple days before the actual all-star game in san diego, and this sheet was distributed to fans in attendance. marcus sent a couple more sheets that i'll show a different time since they aren't in the garvey binder.\nmore specifically, hail to this 2014 upper deck goodwin champions steve garvey autographed card that i recently picked up for the garvey collection.\nwait. what did you think i was referring to?\nbill russell's career ended in 1986, but continues today. sort of.\njust as i did for steve garvey a few days ago, here is a look at who has carried bill russell's torch forward to today, through final plate or pitching appearances.\nin his final game, russell was called upon to pinch-hit for dennis powell in the 8th inning of a game that the dodgers were losing 3-2. padres' reliever goose gossage struck russell out, unfortunately.\ngossage was pitching for the mariners when he earned a 3-inning save on august 8, 1994. pitching against the rangers, gossage set down all 9 players who faced him, including jose canseco who flied out to left to end the game and the goose's career.\nhe flew out in the 9th inning of the white sox game against the twins' eddie guardado on october 6, 2001 in what would be his final appearance as a major leaguer. he later went on to write a couple of books.\neddie guardado pitched in his last game as a member of the texas rangers in 2009. here's his 2010 upper deck card with angel coach alfredo griffin hanging out in the background.\nthe last batter guardado faced was the angels' robb quinlan, but it was on a wednesday night (september 30, 2009 to be exact), so the card doesn't feature a photo from that game. quinlan hit into a 5-4-3 double play to end the 8th inning, as well as guardado's career.\nquinlan was announced as a pinch-hitter on june 25, 2010 against the rockies, but he did not make it to the plate as rockies' manager jim tracy switched pitchers and erick aybar replaced quinlan for mike scioscia's angels. furthermore, quinlan appeared in a couple of games against the dodgers just prior to that, but was only used as a defensive replacement. so, quinlan's final plate appearance came on june 20, 2010 against the cubs. the last pitcher he faced was andrew cashner, who got quinlan to ground out.\nis currently pitching for the padres, and is hopefully the proud torchbearer of bill russell's (and goose gossage's and even jose canseco's) career!\nin 2011, it was dee gordon. in 2012, adam kennedy. last year, nick punto's topps update card was my favorite. this year, i am getting a kick out of justin turner's card.\nthat's some good double play turning right there from one of the best free agent signings of last offseason (the best after phil hughes and nelson cruz? or better, since he signed as a minor league free agent?). oh, there are better cards in update - kurt suzuki has a couple, and i do like pat neshek's all-star card, too - but if there were a card i would chase a rainbow of this year, it's turner's.\nafter finding only chone figgins and scott vanslyke cards in the update stuff i bought, i went ahead and ordered (most of) the other dodger base cards from sportlots. have a look.\nhis name didn't fit on the card properly. still seems weird to see a dodger other than manny mota wear number 11.\nif not for jose altuve, i am guessing he would have been the littlest all-star. nice gatorade cameo, by the way.\nnot a fan of the all-star caps, except for the teams like the twins and padres who actually have worn caps like that during their existence.\nalmost reminds me of the 1978 topps rick monday card where mo seems to be wiping mustard off of his face.\nnow i realize that i forgot to order a paul maholm card. anyone have an extra?\nel toro was 3-1 in 5 starts in the 1981 postseason, but made only one start in the world series - he threw a complete game in the dodgers' game 3 5-4 win over the yankees. i had hoped to see a ron cey\/pedro guerrero\/steve yeager card in this insert set instead of fernando, if i'm being honest.\nmeanwhile, justin turner parallels and paul maholm have taken over the nefarious nine (see list at right on the web version of the blog). please, help a blogger out!\nbill lucas was a minor league player in the braves' system before joining the team's front office in the mid-1960's. he worked his way up from public relations to player development, becoming the director of the braves' farm system in 1972. in 1976, he became the club's vice president and director of player personnel and performed the duties of a general manager. as such, he is widely considered to be the first african-american gm in baseball history, although bob watson was the first to have the actual gm title.\nthat features the players and coaches wearing black armbands in honor of lucas who died on may 5, 1979. shortly after watching the braves on tv on may 1, lucas suffered a brain hemorrhage resulting from an aneurysm. he was taken to the hospital where he died of cardiac arrest. the team wore the armbands for the remainder of the season.\nin the memorials binder as a result. it shows the black armband front and center, and provides a good link to lucas's work as a de facto general manager. lucas was just 43 years old when he died.\n68-year old former braves player, coach, and then-current minor league outfield instructor jim beauchamp passed away on christmas day, 2007 from leukemia. the braves added a \"beach\" patch to their uniforms for the 2008 season in tribute.\nthat represents this particular memorial in my collection.\nbeauchamp followed up a 10-year big league career as a player with 16 years as a minor league manager. the last six of those years (1985-1990) were spent managing in the braves' system, and he was added to bobby cox's big league staff for the 1991 season. beauchamp spent the next 10 years or so as cox's bench coach before taking on the outfield coordinator role.\nberg was an ivy-leaguer (he attended princeton and columbia) who spent his first year in the majors (1923) with the brooklyn robins. he hit only .186 as a 21-year old infielder, but later found a niche in the big leagues with other teams as a backup catcher. he also improved at the plate, and was able to carve out a 14-year major league career with four american league clubs following his one year in brooklyn. the most interesting part of berg's story, however, is that he was a spy. berg was fluent in several languages, and was added to the connie mack all-star team that was to make a tour of japan in 1934. berg may have seemed out of place on such a team that featured babe ruth, lou gehrig, and others, but he was there to spend some time doing some recon for the us government. other missions berg completed included a trip to switzerland to assess the germans' likelihood of developing the atomic bomb.\ni don't know that i've seen camilli's name spelled with an 'f' instead of the 'ph' before. anyway, camilli (father of future dodger doug camilli) played for the dodgers from 1938 until july of 1943 when he was traded to the giants (in true rivalry fashion, he refused to report to the club). during his dodger days, camilli was a two-time all-star and led the league in home runs and rbi in 1941, earning the national league's mvp award - the first dodger to win the award since dazzy vance in 1924.\nessegian had the good fortune to be traded to the dodgers during the 1959 season. as a result, he was able to appear in the 1959 world series against the white sox, and he made the most of his appearances. after striking out as a pinch-hitter in the dodgers' game 1 loss, he hit a game-tying solo home run in the 7th inning of game 2 which the dodgers went on to win. essegian pinch-hit again in game 5, drawing a walk in a losing effort, but came up big with his second pinch-hit home run of the series (hitting for duke snider of all people) in the 9th inning of the dodgers' 9-3 series clinching game 6 victory.\nfernandez was signed by the dodgers in 1951 and worked his way through the minors to the point where some thought the good fielding shortstop would eventually replace pee wee reese at the position for the dodgers. fernandez was called up and debuted in july of 1956 and finished the season with a .227 batting average in 34 games. the dodgers decided to deal fernandez following the season, and he became the phillies regular shortstop after they acquired him for elmer valo and a few other players. he went on to play for the tigers, too, before they traded him to the braves on may 8, 1963. later that same day, the braves dealt fernandez to the mets and he finished out his big league career that year with new york.\ngarvin joined the superbas late in the 1902 season, earning a win and a loss in his two starts for the club that year. in 1903, he was 15-18 for brooklyn, and in 1904 he had a record of 5-15 (despite a 1.68 era) before he was picked up on waivers by the new york highlanders with whom he finished out the season. bill james recognizes garvin as perhaps the \"hard-luck pitcher of all-time\" due to his consistently poor record and low era.\nlocal boy hopkins attended pepperdine and ucla in the 1960's but was signed by the white sox rather than the dodgers in 1965. following a total of six seasons spent in the majors with the chisox and kansas city a's, hopkins joined the dodgers during the 1974 campaign. he only appeared in 15 games for the 1974 national league champs, hitting .222 with 4 singles in 18 at bats. he went on to play in japan for a couple of seasons and returned to the states to earn an m.d. and ph.d and practiced medicine in california and illinois for many years. his card features the left field pavilion of dodger stadium in the background, which is nice.\nlindsey had pitched for the indians, cardinals (in two different stints), and reds before joining the dodgers in 1937. he pitched in 20 games for brooklyn, amassing a record of 0-1 with a 3.52 era. that was the end of his big league career, although he continued to pitch in the minors for many more seasons.\norosco had two stints with the dodgers, including that glorious 1988 season. he is the all-time leader in games pitched with 1,252. the only active pitcher with at least 1,000 games pitched is latroy hawkins who is right at the 1,000 mark.\nregan pitched for the tigers, often as a starter, from 1960 through 1965 when he joined the dodgers via trade prior to the 1966 season. walter alston moved regan to the bullpen, and the pitcher responded with a tremendous season. regan pitched exclusively in relief, appearing in 65 games and earning a record of 14-1 with what would have been a league-leading 21 saves had that been an official statistic back then. regan was an all-star that year, as well as the comeback player of the year and reliever of the year, and he wound up pitching in two games of the world series as well. regan's 1967 season was not quite as spectacular, and he was traded to the cubs just a few games into the 1968 campaign. he did return to the dodger organization following his retirement as a scout and later as manager of the triple-a dukes in 1996.\nrogers hit .189 in 23 games for the 1938 dodgers which was the only time he appeared in the major leagues. he returned to the minors and eventually managed there for a few seasons following his service during world war ii.\nsmith had a record of 69-70 over his 7 seasons with the brooklyn robins spanning from 1915 into the 1922 campaign (he did not pitch in 1918 for some reason), but his era as a robin was 2.91 and he twice won 14 games in a season. he was a big part of the robins' pennant winning teams of 1916 and 1920, and he threw 13.1 innings in game 3 of the 1916 fall classic only to surrender the winning run and take the loss opposite the bosox starting pitcher, babe ruth. smith was 1-1 in the 1920 series against cleveland, giving up just one earned run in 17 innings of work in the robins' losing effort.\ni think of strahler as one of the other guys on charlie hough's 1972 rookie card. he pitched in parts of three seasons for the dodgers - 1970 through 1972. in total, he appeared in 31 games as a dodger, earning a record of 2-3 with a save and an era of 2.76. he made 2 starts for los angeles (both in 1972), one of which turned out to be a complete game victory for strahler in his final dodger appearance. after the '72 season ended, strahler was traded to the angels in the frank robinson for andy messersmith deal, and the halos traded him soon after that to the tigers.\nstuart could hit. he hit .272 with 192 home runs in his first seven seasons, which were spent with the pirates and red sox between 1958 and 1964. in fact, stuart led the american league in rbi in 1963 while with the red sox, and drove in 114 the following year. unfortunately, he was not the best fielder. in fact, stuart led his league in errors by a first baseman in each of those same seven seasons mentioned above. after a down year with the phillies in 1965 (in which he finished second with 17 errors and hit just .234), stuart joined the mets for the start of the 1966 season, which is where the photo used on his card above comes from. he was ultimately released by the mets during the season and was signed by the dodgers. stuart hit .264 with 3 homers in 38 at bats for the dodgers, and even managed a couple of plate appearances for them in the 1966 world series.\nwalls joined the dodgers following the 1961 season in a deal that sent charlie neal to the expansion mets. he spent three non-descript seasons with the dodgers as a utility player\/pinch-hitter, although i found his final big league appearance to be fairly interesting. on the final day of the 1964 season, walter alston replaced his starting catcher (john roseboro) following the first inning with rookie jeff torborg. roseboro had reached base, stolen third, and scored on a wild pitch prior to being removed from the game, but i don't know if his removal had to do with an injury or if it was just because it was the last game of the year and the dodgers were leading the colt .45's 5-0 after one inning of play. anyway, torborg played through the eighth inning, but was replaced at catcher by walls for the 9th. it was the first (and only) time walls had caught in a big league game, and it was also the last time he wore a big league uniform - the dodgers released him 10 days later.\nwilson joined the dodger organization in 1950, but spent only two days in 1958 on the field in the big leagues. he may have been up with the dodgers for longer than that, but his actual playing career consists of 3 games over 2 days in saint louis in may of 1958. on saturday, may 17, wilson made his debut when he pinch-hit for stan williams against the cardinals' larry jackson and stroked a single. he was replaced at the start of the following inning in the lineup by don drysdale. the next day, in the first game of a doubleheader, wilson again pinch-hit (this time for clem labine), but struck out against the cards' billy muffett. in the nightcap, wilson got the start in right field, but was replaced late in the game by gino cimoli following an 0 for 3 performance at the plate. and that was that for wilson's big league career.\ni recently completed my first trade with the overly generous and world cup of trading champion matthew from bob walk the plank. there was so much to enjoy in the cards that he sent.\nit's only the second autograph card of a guy who at one time was my favorite non-dodger and then my favorite dodger that i have in my collection. the back of the card mentions dodger broadcasting legend and hall of famer vin scully and green's monster 4-home run game in 2002 - a game which i attended. good stuff.\ntopps went all-in with nolasco in their 2014 releases, even though he signed with the twins in the offseason. the result is an abundance of signed nolasco dodger cards, of which i now have two. nice.\nwho will hopefully anchor the back end of the rotation in 2015.\nwith andrew friedman in the loop, i wonder if ethier will lose his fourth outfielder spot in favor of joc pederson. ethier's got 3 years and over $50 million left on his contract, so he may very well stay.\nkemp's got 5 years and over $100 million left on his deal. his agent is now the gm of the diamondbacks, which might make me nervous if his contract weren't so large. kemp is the kind of guy that i would really like to see wear dodger blue until he retires.\nspeaking of retirement, furky finally made it back from injury to appear in a handful of games with the marlins last season but is a free agent and may be looking at the end of the line.\nloney is not doing too badly in tampa - he hit .290 last season and played decent defense - and i still hold him in high regard thanks to that grand slam he hit in the 2008 nlds against the cubs.\nlast, but certainly not least, here is the first card in my collection from the 2013 leaf memories buyback set.\nit's a 1991 leaf darryl strawberry buyback stamped 3\/5. 3\/5 is appropriate because 3 is the number of seasons that darryl played for the dodgers out of the 5-year contract to which they signed him. following a fairly productive season in 1991, he saw limited playing time in 1992 and 1993, and was released without playing for the team in 1994. still, a nice card to cap a great trade.\nthanks for the cards matthew!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Share the post \"This Music Is Very Unique For One Amazing Reason. It's Kind Of Hard To Believe\u2026\"\nArtist Bartholom\u00e4us Traubeck once noticed that the lines in a cross-section of a tree bore a striking resemblance to a vinyl record. In his art piece entitled, \"Years,\" Traubeck uses this to his advantage, creating a beautiful, but very eerie song with a custom record player and a slice of a tree trunk. This definitely made me shiver a little.\nUsing a Playstation Eye Camera, the record player takes in data from the rings of the tree trunk and relays it to a computer. The computer then uses a program called Ableton Live to translate the rings into music, with the instrument of choice being a piano. Although the tree itself isn't musical, the process involved in creating \"Years\" is like a translation, taking something physical and turning it into sound.\nWhile you can't chop down a tree and slap it on your dad's old record player, \"Years\" is an incredible way to see nature in a whole new way. Hearing something so beautiful come out of a plant that we see every day really shows us that there is more to everything that meets the eye\u2026 even if we have to dig a little deeper to find it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Also called South Africa's \"mountain desert\", this arid Park is home to the Nama people. Traditionally pastoralists, they followed the rain with their herds of goat and sheep staying in igloo-like huts made of Wag-n-bietjie saplings. The framework of these huts is covered with \"biesiematjies\" when they are used, and remained behind when they moved on.\nThis age-old way of roofing inspired the design of the tourism facilities developed in the Park. Various interpretations thereof made for interesting structures. To make the mats waterproof (for the rare times that rain fell in this desert) they were painted with a waterproofing paint or covered with canvas.\nWalls were either dry stack stone or reeds from the Gariep River, with a double layer being used with shade netting in between the layers. This keeps out insects and breaks the speed of the cold winter winds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Situated in Simola Golf & Country Estate, Knysna, this luxury three-bedroom house has sweeping views and is marketed by Pam Golding Properties at R19.32 million.\nThe stresses of work, city living and simply the ongoing 'daily grind' sees many people looking to realise their dream of owning a holiday home where they can kick back and unwind with family and friends on weekends or longer getaways \u2013 or ultimately, even relocate.\nAccording to FNB, house prices in towns typically considered holiday towns continue to accelerate quite strongly \u2013 possibly because a growing number of young people are turning to these smaller towns for their permanent homes as they offer more affordable accommodation, less congestion and increasingly, educational facilities such as private Curro schools, says Sandra Gordon, Pam Golding Properties senior research analyst.\nTo ensure you make a wise acquisition that will bring you years of leisure enjoyment and a sound investment, and perhaps even a home to retire to in the future, there are a number of useful aspects to consider.\nFirstly, make a list of you and your family's own requirements, say Stef Nel and Mohammed Amra, Pam Golding Properties area principals in Scottburgh and Port Shepstone respectively, on the tranquil KwaZulu-Natal South Coast, an affordable and sought-after location for leisure getaways, including homes to retire to. \"Ensure there are activities the whole family can enjoy. Keep a look out for restaurants, animal parks, beaches, hiking routes, fishing spots, adventure sports and other fun holiday activities, as well as convenient shopping malls,\" says Amra.\nThen there is location. Says Nel: \"Is the property within easy reach and in a destination you would want to visit regularly and is the cost of the transport to and from within your budget? This location is increasingly sought after among families wanting to be near parents residing in the ever-developing retirement facilities on this coastline.\nFor example, if it's within walking distance of the beach, river or lagoon, with shops and restaurants nearby, this will add to the appeal not only for you but also for a future investor or buyer, ultimately adding to the value of the property. If it's not too far from where you live you can have short breaks more often and invite relatives and friends for occasional visits, while teenage children will be prepared to come along if they can stay for short periods and return easily to their friends and activities.\nLocation and position of the property within the area of your choice is particularly important if the home needs renovation, which requires additional capital investment, says Erika van der Westhuizen, Pam Golding Properties area principal in Hartbeespoort Dam.\nSecurity is important, especially if the home will remain vacant for lengthy periods, so it's advisable to look for a lock-up-and-go property which is easy to maintain, such as within a sectional title complex, as then security and external maintenance are taken care of via the Body Corporate.\nBefore you buy check that the complex has effective management in place, a healthy track record and sound financial records.\nIs the house easy to maintain, asks Amra. When looking for the ideal holiday home, a big plus is a lock-up-and-go type home. Be sure to choose a home that does not require a huge amount of upkeep, especially if the property may stand empty for a large portion of the year.\nAlternatively, if you need a more spacious leisure property for a larger family or extended family, you could consider a lifestyle estate which also offers excellent security, says Dobson. These are increasingly popular and usually include a number of leisure facilities, such as golf, gym, swimming and walking.\nFor resale purposes, ensure you purchase a home that suits most families, and rather spend a little more in order to have a fantastic view, which again adds value should you decide to disinvest later on, advises van der Westhuizen.\nIf buying a coastal property, bear in mind that buying a home, ideally a three-bedroom, two-bathroom unit with a sea view or in close proximity to the beach, will optimise the investment return should you need to sell the property later on.\nIf you are in your 40's, you may be looking ahead to buying a property to retire to. This makes good sense as by the time you reach 60 you could have a home on the coast that is then fully paid for. At the same time your primary residence has increased in value, the sale of which will provide you with cash flow or funds for further investment.\nAn added benefit is that you have most likely built relationships in the local community and at the local golf, tennis or bowling clubs and you already know your neighbours. This makes relocation when you retire less stressful.\nAlways ensure you invest in a healthy complex where the body corporate is well run and the complex is well-maintained. Your holiday home investment should be as low maintenance and hassle-free as possible, you don't want to become embroiled in a poorly managed apartment block with weak financials and insufficient funds for ongoing maintenance.\nImage: Situated in Simola Golf & Country Estate, Knysna, this luxury three-bedroom house has sweeping views and is marketed by Pam Golding Properties at R19.32 million. The undercover north-west facing patio is fully equipped with an outside bar area and wood burning BBQ flowing to a heated infinity rim-flow pool, while the property includes a lower deck pergola area, sunken private inner courtyard and a Zen-like roof garden off the master suite, among numerous other features.\nIf you are considering purchasing a holiday home, we recommend contacting your local Pam Golding Properties office for professional advice and to view the properties for sale in that area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's better than a guest post? A returning blogger! Enjoy - we sure did!\nI hear my band director yell across the field, 'REEEEEEESSSSSSSSEETTTT!!\" - that means run across the field back to your original starting position. The first thing that goes through my mind after I hear that is I hope my ostomy bag isn't full. If any of you have ever experienced running with your ostomy bag sloshing \u2026. Well let's just leave that to your imagination. I am starting on my third year in marching band and Grade 11 or my junior year in High School. The last two years of marching band, Crohn's hasn't allowed me to march a whole season. My freshmen year I didn't have the energy to do all the activity that marching band takes from a person. I would spend two hours at practice and go home and sleep for four. Last year, I was in the hospital for resection surgery during all three weeks of band camp. When I was cleared to practice with my instrument, I broke both heels because of my osteoporosis during one of my first rehearsals.\nThis year\u2026 I am hopeful! I have gotten through two weeks of band camp which has included two 12 hour days and two 8 hour days. My feet are sore, I am shaking off my sore muscles, ignoring my swollen ostomy, but I am proud and really pleased with myself knowing this year I am fully participating in band camp \u2013 for the first time.\nWhat's different this year versus the last two? I am not in the hospital \u2013 that helps. One of the things I hear a lot at band camp is \"how are you feeling?\" My band camp answer this year \u2026 \"I am here, aren't I?\" The primary reason is with careful attention to my disease and my diet and a lot of effort and perseverance I am doing what I love doing with my friends. I hurry back home after a 12 hour day to get hooked up to TPN and lipids so I can get my 10 hour window of nutrition through my PICC line before returning the next morning. I am careful to take a few extra breaks to rehydrate, rest, and get some extra calories where I can. I bring a cooler with extra food and the band moms are looking after me making arrangements to meet my semi-vegetarian diet at meal time. Sounds like a lot of effort and a lot of adjustments to try and live a normal life. Some might ask, is this worth it? Duh, of course it is! Friends with common interests, striving to accomplish something, being part of a group, heading towards a goal \u2026 it doesn't get any better than this.\nCrohn's does not limit your dreams, you do. I have always known that even with Crohn's and its complications I could still march. Last year after I broke both heels, I came back and marched in semi-finals at Grand Nationals with my band. It may not have been a perfect show for me, but I have proven to myself once again that no matter what I encounter I can overcome it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had seen a few versions of this on the internet, so I decided to do my own take on this card using our Up and Away Thinlits. We made this for my March Stamp Club, however this particular card I elaborated on slightly and I took it to our On Stage Local in Perth on 8th April and popped it on the display board. Like my previously posted Up and Away cards, I glued the coloured card behind the die cut but for this one I trimmed the top section from the balloon and turned it upside down to resemble the base of the perfume bottle. I also cut a smaller balloon and embossed it to resemble the atomiser bulb, and glued it behind the \"neck of the bottle\". The gold tassel is non SU! and the cord ends I glued behind the \"bulb\" \u2013 the bottle and bulb is popped up with Stampin' Dimensionals. I hastily took this photo the morning I had to drive to Perth, so I'm afraid it's not a very elaborate shot (I don't think the dark green cutting mat background really does it justice!) I'm so saddened that these gorgeous doileys have been discontinued and are no longer available \u2013 I wasn't quick enough to order more before stocks were exhausted from the retirement list. Stay tuned for another card I made for swaps at On Stage Local using the same Lace Doileys and you'll see why I am sad I missed out on stocking up on them!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Andrew is a system administrator working at ScaleEngine in Hamilton, Canada. He oversees a fleet of over 100 globally distributed servers and performs day to day management of the ScaleEngine video streaming CDN. Andrew has extensive experience in monitoring and automation of FreeBSD systems and in his free time enjoys playing with older computers and networking systems.\nMichael W Lucas has been a BSD user since the late 80s and a sysadmin since 1995. He's written a bunch of the standard FreeBSD and OpenBSD reference books and shows no sign of stopping any time soon. Learn more at https:\/\/mwl.io.\nKris Moore is the founder of the PC-BSD (now TrueOS) project and heads up the engineering teams for FreeNAS, TrueNAS and related iXsystems efforts into FreeBSD-based products. He resides in the Knoxville, TN area with his wife and 5 kids and enjoys hacking Makefiles and shell in his limited spare time.\nNick Principe is a Technical Marketing Engineer at iXsystems working on customer-focused performance engineering and marketing collateral. Nick has worked in the performance engineering field for over 12 years, focusing on the performance measurement and analysis of NAS systems. Nick has also contributed to the development of two SPEC SFS benchmarks, the SNIA Emerald v3.0 specification, and was one of the first co-chairs of the SNIA Workload TWG.\nDevin has used FreeBSD since the 4.4 release and has been a committer since 2012. She has worked extensively on bsdinstall\/bsdconfig and the bootloader. Devin works at Smule, Inc. and is an avid runner and dog lover.\nBen Widawsky is a senior software architect and developer for the Open Source Technology Center at Intel Corporation. In 2018, he received a vendor commit bit for src and is currently under the mentorship of Ed Maste. His primary focus has been to improve power management in FreeBSD. Prior to that, he led the Linux Graphics organization as the software architect where contributed over 500 patches in the Linux kernel and 200 patches in the Mesa 3D project.\nG. Clifford Williams is a restless geek who wears many hats from DevOps Practice Manager at 8ions, Inc. to Technical Advisor for several startups, longtime advocate of open source software, maintainer of the wempy template system and contributor to such projects as Cherokee, SaltStack, and Web2py. Currently obsessed with large scale service orchestration, application packaging\/delivery, and the current crop of high performance languages, he still finds time to play video games and write things in AWK that most people would never consider.\nMariusz Zaborski is a lead software developer at Wheel Systems. Mariusz's main areas of interest are OS security and low-level programming. At Wheel Systems, Mariusz is developing a solution to monitor, record and control traffic in an IT infrastructure. He has been involved in the development of Capsicum and Casper since the Google Summer of Code 2013, which he successfully passed under the mentorship of Pawe\u0142 Jakub Dawidek. Mariusz has been a FreeBSD project committer since 2015. In 2018 Mariusz organized the Polish BSD User Group.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Product prices and availability are accurate as of 2019-04-17 19:59:41 EDT and are subject to change. Any price and availability information displayed on http:\/\/www.amazon.com\/ at the time of purchase will apply to the purchase of this product.\nMy Home Barista are delighted to offer the fantastic UNEGO Stainless Steel Milk Cup.\nWith so many available these days, it is great to have a make you can recognise. The UNEGO Stainless Steel Milk Cup is certainly that and will be a excellent acquisition.\nFor this great price, the UNEGO Stainless Steel Milk Cup comes highly recommended and is always a popular choice with many people. UNEGO have provided some great touches and this results in great value for money.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Here is a brief look at the underlying philosophy that influences SurveyGold product design decisions and a look at where we are going in the future.\nSurveying is not an \"every day\" activity for most people. SurveyGold solutions need to leverage what people know and provide a user experience that is intuitive and easy to understand. The price point should not be punitive as the occasional surveyor cannot always predict how many responses they may collect for a given survey.\nPeople want to get a survey out there without having to deal with a lot of technology barriers. SurveyGold solutions need to reliably leverage technology while hiding the hard parts from the user experience. For institutional customers, need to make it so that IT technical staff rarely, if ever, need to get involved in deploying SurveyGold solutions.\nWeb surveys are all the rage. But, they're not always the most effective means of engaging and collecting survey responses. SurveyGold solutions need to provide options to the surveyor by making it easy to conduct and collect responses for the same survey in a variety of ways (e.g., web, offline, kiosk, tablet, paper, interview).\nPeople helping people is a good thing. SurveyGold solutions need to make it possible to crowd-source surveys from our user base. In addition, feedback from our user community input should help inform product decisions.\nWe exist to solve problems. Sometimes in the course of solving problems, new problems or challenges occur. SurveyGold support solutions need to be personal, accessible and adaptive to new circumstances that emerge in the lives of our buyers and users.\nWe know that being cash-strapped is a way of life for many in this sector. So, we endeavor to provide an affordable, effective solution for all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Don't drive the Volkswagen in the fish tank.\" Jessica forced her fingers through her short hair. She wished it were still long so she could wrap it around her hands and yank it all out. Maybe then she'd feel better. But the long hair was gone, an early victim of her 'predicament', as her husband called it. And the biggest part of that predicament refused to obey Jessica's orders.\n\"You can have it back if you'll go in your room and leave me alone to finish my homework. Okay?\" With a shrug Laura agreed and disappeared into the bedroom making beep-beep noises. Jessica looked at her watch. She had to review the whole chapter today before class. She would never get it done.\nMaybe it was a mistake, trying to take a class. She'd hardly been able to tend to her own child because of the lupus. How could she expect to go to college? What about all those times she'd forgotten to change Laura's diaper? Or spent an hour searching for her keys only to find them still in the front door. Or passed out on the floor from sheer exhaustion, to wake with a shock half an hour later, praying the baby was still alive. She didn't even know her own phone number half the time. What was she doing trying to learn computers?\nHer husband appeared in the doorway, almost wearing a suit. His necktie dangled loose around his neck, jacket in his hand.\nJess sighed. A little voice shrieked, \"Daddy!\" and feet thundered down the hallway. For a moment Jessica was alone. Leaning over, she scrounged around on the floor for her pen.\nJess made it halfway through the next paragraph before her husband came back. \"Having fun?\" he asked.\nOne cup of apple juice, one search for a lost puzzle piece, and two cookies later, Jessica closed her book and prayed. This was terrible. She couldn't remember anything. Between her atrophied brain and Laura's constant interruptions, she simply could not concentrate. Perhaps it would be better to quit now, rather than fail. If she quit, at least she could still believe in the possibility of success. Maybe after Laura started school and she had time to sit down alone with her books and study, maybe then she could convince her brain to function.\nBut what if she couldn't, even then? What if her brain was too far gone to come back again?\n\"Now put them on.\" Slowly Jess made her way to the bathroom to wash her face. Her knees resisted her desire to walk and her numb hand fumbled with the doorknob. Looking in the mirror, she sneered at the splotchy reflection, the ruddy lupus flush over her nose, the vague age spots beginning to show on her cheeks. Even four years after giving birth, she still looked puffy and tired.\nMakeup might help, she realized. Make her look alive, at least, instead of washed-out, which was too much like washed-up. Well, she decided, it couldn't hurt.\nIt was a disaster. Jess hadn't worn makeup in so long, she'd forgotten how to put it on, and ended up looking like an adolescent just learning to paint her face \u2013 except for the crow's feet and the shards of gray in her hair. Once she'd gotten Laura to the car and settled in her booster seat, Jess surveyed her face in the rearview mirror and nearly went back in the house. She looked absurd.\nShe dropped her keys twice trying to lock the front door but finally made it to the car again. Throwing the books in the back seat, she climbed in. How sad to see a grown woman fall apart like this. If she told the teacher about her lupus, perhaps he would give her a break. But no, she couldn't ask for favors. She wanted to drag herself back into the real would on her own two feet, even if she did limp a little, and see what she could accomplish on her own. After all, she must learn to live with this. The lupus wasn't going to go away. True, she didn't need a cane anymore, and she'd stopped passing out. But there were no tests that could assure her that her common sense and intelligence had survived.\nFor a moment the car's whine distracted her. Jess' left foot stomped down on the floorboards but the clutch wasn't there. Another concession, when the disease weakened her hands. How she missed her old Mustang! Driving a standard gave her such pride. An unmistakable sense of power: mash the pedal and yank the gearshift into place when she wanted to. When she knew it was time. When the engine reached just the right pitch. Driving an automatic was no fun at all. You just sat, passive, and let some computer-engineered machinery decide your fate.\nFinally the car shifted into second, just in time to reach the intersection. Jessica stepped on the brake too hard and threw them both forward. Good Lord! Would she never get the hang of driving this car?\nJess touched the accelerator. \"Doesn't,\" she corrected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you ever wondered how the North builds, this exhibition is for you. The \"Nordic Urban Spaces\" shows examples of Nordic architecture and urban planning already implemented that make life in the city better. The Nordic weather tends to be quite extreme, being either a bright and hot summer or a very long and cold winter, so weather particularly affects architecture and urban planning.\nThis exhibition shows successful, innovative and participatory examples of Nordic construction and planning. The projects not only bring summer into the city (for example, with urban swimming spots) but also provide light and color in winter (for example with brightly colored subway stations). They also try to demonstrate that functionality and sustainability, consideration and elegance are not mutually exclusive.\nWe are happy to announce that our project Dreamhamar is displayed in the exhibition until the 28th of September. The redesign of the Stortorget Square in Hamar, Norway, through participation and a network design process, took place during Fall 2011. Citizens took part in a collective creative process that helped us shape the future of the square. Our approach was supported by workshops, lectures, urban actions, communication, and participation tools. Dreamhamar was awarded as BEST PRACTICE by the United Nations-HABITAT program in 2014.\nIf you are in Berlin and want to check it out, go to The Felleshus. This building is the cultural center and event venue of the five Nordic embassies (Denmark, Finland, Iceland, Norway and Sweden).\nBelinda Tato and Jos\u00e9 Luis Vallejo will be participating in the On Cities Workshop, organised by the Norman Foster Foundation, which will take place this week (18 to 22 June 2018) in Madrid. The workshop will focus on Autonomous Innovative Communities, selecting a district in Madrid as a case-study for a research project that will be developed throughout the week. The On Cities Workshop will include seminars, lectures, one-to-one tutoring and urban architectural tours to learn more about the context of Madrid and it's districts. During the course of the workshops, participants will have the opportunity to engage with the Norman Foster Foundation's archive and research projects.\nCan each community locally produce all of the energy, food, and clean water needed for basic living\u2014requiring no centralised infrastructure? Can humans transition from ownership to sharing, while living and working in compact, agile, supportive environments? This workshop explores the premise that emerging urban innovations can dramatically reduce resources consumed by cities while simultaneously creating more livable, entrepreneurial communities.\nAfter reviewing applications submitted by hundreds of candidates from around the world, the selection committee awarded ten scholarships to students from the following universities and institutions: American University of Dubai, Dubai, United Arab Emirates; Harvard Graduate School of Design, Cambridge, United States; London School of Economics and Political Science, London, United Kingdom; Pontificia Universidad Cat\u00f3lica de Chile, Santiago de Chile, Chile; Royal Danish Academy of Fine Arts, Copenhagen, Denmark; Technische Universiteit Delft, Delft, the Netherlands; Tongji University, Shanghai, China; Tsinghua University, Beijing, China; Universidad Polit\u00e9cnica de Catalu\u00f1a, Barcelona, Spain and University of the Witwatersrand, Johannesburg, South Africa.\nThese ten students will engage with a group of specialists through a series of seminars and lectures culminating in a five day workshop led by the Atelier mentor, Kent Larson, Director of MIT Media Lab City Science Group and Initiative, and his team. Nicholas Negroponte, Co-Founder and former Director of MIT Media Lab, Cambridge, United States will act as the Chief Advisor of the workshop tutoring the students through the research process.\nThe Academic Body spans a wide range of practitioners working in different fields interrelated with the City, including: Beatriz Colomina, Director of Graduate Studies, School of Architecture, Princeton University, Princeton, United States; Luis Cueto, General Coordinator for the Mayor in Madrid, Madrid City Hall, Madrid, Spain; Anupama Kundoo, Principal, Anupama Kundoo Architects, Madrid, Spain\/Auroville, India; Winy Maas, Co-Founder and Director of MVRDV and Director of the Why Factory, Delft, the Netherlands; Tim Stonor, Managing Director of Space Syntax, London, United Kingdom; Leonor Tarras\u00f3n, Director of Environmental Solutions, Norwegian Institute for Air Research, Oslo, Norway; Belinda Tato and Jos\u00e9 Luis Vallejo, Founders and Directors of Ecosistema Urbano, Madrid, Spain\/Miami, United States.\nNext Sunday 27th May Jos\u00e9 Luis Vallejo will be giving a lecture at the Milano Arch Week 2018.\nThis important event, this year under the title \"Urbania\", is promoted by the Comune di Milano, the Politecnico di Milano and the Triennale di Milano in collaboration with Fondazione Giangiacomo Feltrinelli, and includes a rich program of conferences, workshops, installations, exhibitions, performances, events open to citizens to reflect together on the future of the city and the dynamics of contemporary architecture.\nDate and time: May 27th, 19:00 h.\nNext Thursday 23rd and Friday 24th Jos\u00e9 Luis Vallejo will be giving an open lecture and leading a workshop at the Aalto University Department of Architecture.\nThe workshop will develop the topic \"Atmospheres for Social Interaction\". How can we, as architects or urban planners, support the development of the social aspect of urban life?\nTomorrow Jos\u00e9 Luis Vallejo will be giving a lecture at the Utzon Centre in Aalborg, Denmark, as part of the Utzon(x) Lecture Series).\nSince 2010 the international Utzon(x) Lecture Series have arranged lectures and symposiums with the participation of Jacob van Rijs (MVRDV), Juhani Pallasmaa, Mathias Kohler (ETH Z\u00fcrich), Sigrid Adriaenssens (Princeton University), Dirk van Gameren (TU Delft\/Mecanoo), Yoshiharu Tsukamoto (Bow-Wow,9 Jenny Osludsen (Sn\u00f8hetta) and Nanne de Ru (Berlage Institute) among others.\nNext Tuesday September 26, Jose Luis Vallejo will be lecturing as keynote speaker in Rome on the occasion of the Fourth Edition of New Generation Festival.\nAmong other experts and architects that will take part to the event, there are: ENORME Studio (ES), Fosbury Architecture (IT), Luca Montuori (Deputy Mayor for Urban Planning at the Municipality of Rome, IT), Orizzontale (IT), Olga Polishuk (Chief Operating Officer of Strelka Institute, RU), SET Architects (IT), Jean-Beno\u00eet Vetillard (FR), U67 (DK), and WikiSpazioPubblico (IT). Moreover, the second day of the festival on Monday, September 25, will be closed by the keynote speech of Koert van Mensvoort (Next Nature, NL).\nThe 4th edition of the New Generations Festival \u2013 Architects VS Rest of the World \u2013 proposes an intense programme of discussions, workshops and cultural activities, involving numerous international guests, gathered to reflect on the profession of the architect from multidisciplinary points of view. After successful events in Milan (2013), Florence (2014) and Genoa (2015), Rome will host the fourth edition of the New Generations festival with the aim of creating a community of architects and experts from different disciplines, in order to redefine the role of architecture in contemporary society.\nThe relation between the new generations of architects and other disciplines is a broad field of discussion that will be addressed via three umbrella topics: (a) Urban Vocabulary & Public Space, (b) New Economies & Values, and (c) Digital Infrastructure & New Media. These 3 topics will be at the center of the debate during the Festival, inviting young architects, city makers, sociologists, economists, public & private institutions, startups, communication and digital media experts, web developers, programmers and many others professionals to discuss and exchange ideas. The discussion will see the participation of representatives from the wide network of the New Generations Platform, which counts more than 80 young practices and more than 500 international experts.\nURBAN VOCABULARY & PUBLIC SPACE: cities are changing at a fast pace, is the profession and the way we face urban problems keeping up with this rapid developments? New Urban Vocabulary looks at the way architects are more and more becoming mediators in complex urban processes, proposing new ways of thinking not just in terms of urban planning methods, but ways of working with communities, re-activating and re-claiming public space.\nNEW ECONOMIES & VALUES: how did the economic crisis of 2008 affect the profession? What do we mean when we talk about successful, sustainable and collaborative economic models in relation to the design field? Collaborative Economies proposes an investigation of the economic models behind successful, emerging and innovative architectural practices. The aim is to analyse the contemporary workscapes with a specific attention to economic sustainability, unveiling challenges and opportunities of the contemporary workfield.\nDIGITAL INFRASTRUCTURES & NEW MEDIA: New Technologies and Digital Media are changing the way we work in an unprecedented way, but what is the impact of those changes on the architectural profession? Which new figures and experts need to be involved in this highly complex process? Digital infrastructures looks at how seamlessly integrated technological systems run in the background of our cities but have the power to fundamentally change both the approach to architecture and the way we experience spaces, creating a base for innovation.\nWe are honored to announce that Ecosistema Urbano has been recognized as a 2017 Social Design Circle Honoree by the Curry Stone Design Prize.\nWhat is the Curry Stone Design Prize?\nThe Curry Stone Design Prize is awarded each year to honor innovative projects that use design to address pressing social justice issues. Supported by the Curry Stone Foundation, the Prize highlights and rewards projects that improve daily living conditions of people in communities around the world. The Prize acknowledges work that is considered emerging in the professional and public consciousness.\nWhat is the Social Design Cirle?\nShould designers be outlaws? Is the right to housing real? Can design challenge inequality? Can design prevent disaster? Can we design community engagement?\nCan design reclaim public space?\nEcosistema Urbano has been included in the category \"Can design reclaim public space?\" of the Circle, together with other colleagues and collectives as Asiye eTafuleni, Basurama, Collectif Etc., EXYZT, Interboro, Interbreeding Field, Studio Basar, Kounkuey Design Initiative, Y A + K and Raumlabor Berlin.\nWe honor Ecosistema Urbano particularly for their progressive ideas on community participation. The group has worked to update the very notion of \"community participation\" through the development of online tools which encourage global participation on local projects. The group has developed several apps to collect community input throughout the design process. New technologies work to break down barriers which traditionally inhibited the full participation of community. Many of our 'communities' today are in fact digital, so the idea of community participation must be updated as well.\nIn a physical space, the group is best known for their green projects like Ecobulevar \u2013 a project of 'air trees' in the Madrid suburb of Vallecas. The project is intended to be temporary, but creates the same sort of community space that one would find in an old growth all\u00e9e.\nThe air trees are made from repurposed industrial materials such as recycled plastic, greenhouse fabric, rubber tires. They contain rooting vegetation and atomizers that cool and moisten the air in the cylinder and around it (8oC to 10oC cooler than the rest of the street in summer). The cylinders can be used for public gatherings, and solar panels provide electricity for lighting when needed (excess energy is sold back to the grid and helps fund the maintenance of the structures).\nThis and other sustainability projects like Ecopolis in Madrid speak to a shared sense of community responsibility and interaction.\nMoreover, an interview we gave for the occasion together with our colleagues of Interboro constitute the episode 24 and 25 \"Tools for urban action\" of the Social Design Insight podcast. You can listen to episode 24 here, while the episode 25 will be shared on Thursday June 8 on Curry Stone Design Prize webpage.\nAs we mentioned in our previous post, Ecosistema Urbano is working with the Joint Research Center of the European Commission in Seville for a research project focused on the topic of maintenance of public space, aiming to define the EU Green Public Procurement (GPP) Criteria for Public Space Maintenance.\nIn this framework we are currently seeking a civil engineer, construction engineer, building engineer or architect, with proven expertise in the field of construction and maintenance of public space to start a collaboration in this research project. Chosen candidate will work on the topic of maintenance of public space side by side with ecosistema urbano. The official language of the project is English and all the documents to be produced should be in English.\nCandidates should submit their cv to am@ecosistemaurbano.com.\nEcosistema Urbano est\u00e1 buscando un ingeniero civil o de construcci\u00f3n, arquitecto o arquitecto t\u00e9cnico con experiencia probada en el campo de la construcci\u00f3n y mantenimiento del espacio p\u00fablico para iniciar una colaboraci\u00f3n en un proyecto de investigaci\u00f3n que se est\u00e1 desarrollando dentro del marco de pol\u00edticas p\u00fablicas sostenibles de la Comisi\u00f3n Europea. El candidato elegido trabajar\u00e1 sobre el tema del mantenimiento del espacio p\u00fablico junto con ecosistema urbano. El idioma oficial del proyecto es el ingl\u00e9s y todos los documentos de proyecto se producir\u00e1n en ingl\u00e9s.\nLas entrevistas ser\u00e1n en ingl\u00e9s.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200bBOOM!! The terrifying sound of a Blown out tire! Getting a flat tire can be the one of the most frightening things that can happen to you when driving your car. Roseville Tow Truck Company can help you change that tire and get you back on the road. Our tow truck drivers are highly trained, certified and extremely qualified. Roseville Tow Truck Company will install your spare tire, bring it in to be repaired or tow you to a tire shop to have it replace for you. We have a great relationship with every tire store in town including Les Schwab and Tire Factory. Roseville Tow Truck Company will help you get back on the road as soon as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If there is one virtue that is most sought after it is happiness. However in today's times it is also the most elusive. Unlocking the secret to being happy however is in your hands.\nIf there is one question that everyone seeks out an answer it has to be 'What is the secret to be happy?' Well, to put it simply there probably isn't a single person who will say they do not want to be happy.\nEach one of us wants to be happy and we do not have to go anywhere to find it. Happiness is within us and there are many ways to liven up. Out of all the approaches, the most important is to have a positive outlook. An optimistic attitude will not let any bad circumstances creep into your mind and you will be able to keep your head high. You have to keep telling yourself that life is full of hopes and whatever happens it is for good. Secondly, live in today and stop worrying about the past or the future. Live in the present and enjoy, it will help you gain the happiness in that moment. Socialize with people who makes you happy,\" says Dr.Madhu Kotiya, Tarot Mentor and Spiritual healer. It has been proved that we are happy when we are around those who are happy too. This also means that we should spread smiles to those in need. You feel a sense of contentment when you know that you are the reason for someone's happiness. And sometimes listen to your heart.\nCircumstances play a very important role in happiness. But it is not a linear relationship. These are all variable factors. Happiness is a concept. In the last 30 or 40 years, a lot of research has been done on this topic. \"In Western countries, courses are conducted on happiness. People believe that happiness increases productivity and reduces tension. However, as a doctor, I can say that good physical health is in our hand. Sincerity in the role which has been thrown at us is in our control. We have to be careful not to fall into the trap of drugs, alcohol and smoking. Recreation and meditation can be of a great help and a balance in life is important. Losing this balance can make a person unhappy,\" says Dr. P D Lakdawala, Psychiatrist, Bhatia Hospital. Happiness is an overall feeling and experience which results from a positive and desirable outcome of life goals desires, actions and experiences in tune with the overall personal objectives of a person during their lifetime. \"For some people it emanates from acquisition of tangible things like material goods, money, career, relationships, good health while some derive happiness from more spiritual pursuits not linked to material gains or losses. We have to set short, medium and long term life goals so that once we achieve them we have small doses of happiness and satisfaction coming our way. Linking overall and total happiness to fulfilment of large goals or milestones can be long drawn out and in case the final milestone is not achieved, leads to disappointment and despair,\" avers Arindam Seen Founder, Zyego, an emotional wellness service.\nHow many times we have seen or met two people, in the same profession, same state of life, same financial obligations. One of them seem to be complaining and the other one seems to be enjoying his life. Singer Shahid Mallya says, \"for me, gratitude leads to happiness and determines our attitude. As we practice gratitude, it becomes our second nature and we become able to find the beauty in the diminutive things and appreciate all life has to offer. Expressing gratitude turns our mental focus to the positive way of life. By doing this we can fully experience the moment and learn to engage with each moment on its terms, taking things as they come. We can be way happier this way.\" Happiness is generally confused with the absence of struggles and inabilities in life. For example, the son of a rich man may put in the category of generally happy on a social scale. And that's where the confusion begins. \"Happiness is the human ability to realise that everyone is equally capable of achieving it and The lack of happiness is the human inability to find it within and rather search for it among the stuff outside. We must remember that a daily wage labourer, who is finding it difficult to make his ends meet can be equally happy or unhappy as compared to a business man who is under loan of crores to start a new venture and has hundreds of families to feed,\" says Abhinav Singh, author of 'The Last Attractor of Chaos'.\nLearn to meditate. It is the sure-shot way of listening to your heart's voice.\nThink positive always. Access the situation you are in and try to know the pitfalls and keep your eyes on what can go right!\nConstantly brooding over past mistakes can rob one of happiness. Let go the past is gone. Get over it and move ahead.\nWorrying about what the future entails also robs us of joy! Do the right thing and the future will take care of itself.\nThe only guarantee we have for our future is good karma. So plant good seeds as you go along with it. Knowingly and deliberately work on planting good deeds good words and good thoughts.\nAyesha Kapur, Co-owner, Ayesha Accessories recollects her journey to finding happiness.\nWriting a list for clarity is something my mother taught me a long time ago and writing down the things that make you happy and getting it done also gives you an amazing sense of achievement. Ticking them off is extremely satisfying.\nRoutine makes me feel happy whether it is the same brand of coffee in the morning or fruits with slightly sour unsweetened yoghurt makes me feel healthy and in control.\nStop making excuses and tackle the things that you know would make you happy but you have not for whatever reason. Start seeking it and make it part of your routine. Then start adding more \"healthy lifestyle choices\" to your routine. I was an avid reader when I was a child but slowly grew out of the habit but knew that reading was something that I really have benefited from and starting again would truly make me happier and more successful in everything I'm doing.\nBe aware of your diet as it can affect your mood. Do your research, but also listen to your body and how you feel. I find that meals with a large number of carbohydrates make me feel sluggish and less energetic or too much sugar makes me feel anxious.\nExercise not just to get fit or look a certain way, but because starting consistent exercise can change your life, your mind, your state of being. It really does release the happy hormones.\nIt does not matter what is the nature of your activity, essentially you are in pursuit of happiness. If you look at your own life's experience, whenever happiness happened to you, it does not matter what was the stimulus \u2013 maybe you watched the sun rise and you became very joyful, maybe you heard some music and you became very joyful, maybe you achieved some success and you were joyful \u2013 it does not matter what was the stimulus, but whenever happiness or joy happened to you it always bubbled up from within you. It never rained upon you from somewhere else. So the very source of happiness is within you. Right now, the stimulus is outside. Now the choice is just this whether you want to keep the stimulus outside or you want to keep it inside. The source of happiness is within you but the switch for it is in somebody else's hands. They can turn it on or turn it off. Anybody can make you happy, anybody can make you unhappy; any situation can hijack your happiness at any moment. All those people, who depend on external situations to be happy, will never know true joy in their lives. No matter what kind of a person you are, how powerful you are, even if you are a super human, you do not have absolute control over the external situation. Even if there are two people in the family, they do not have total control over the situation. You can manage the external situation only to a certain extent. But your interior can be taken into absolute control.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"NATIONAL CARBIDE CORP. v. COMMISSIONER OF INT. REV.\nMr. Erwin N. Griswold, of Cambridge, Mass., for petitioners.\nMr. Hilbert P. Zarky, of Washington, D.C., for respondent.\nThe contracts between Airco and its subsidiaries provided, in substance, that the latter were employed as agents to manage and operate plants designed for the production of the products assigned to each, and as agents to sell the output of the plants. Airco was to furnish working capital, executive management and office facilities for its subsidiaries. They in turn agreed to pay Airco all profits in excess of six percent on their outstanding capital stock, which in each case was nominal in amount. 3 Title to the assets utilized by the subsidiaries was held by them, and amounts advanced by Airco for the purchase of assets and working capital were shown on the books of the subsidiaries as accounts payable to Airco. The value of the assets of each company thus approximated the amount owed to Airco. No interest ran on these accounts. [336 U.S. 422 , 426] Airco and its subsidiaries were organized horizontally into six overriding divisions: corporate, operations, sales, financial, distribution, and research. Officers heading each division were, in turn, officers of the subsidiaries. Top officials of Airco held similar positions in the subsidiary companies. Directors of the subsidiaries met only to ratify the actions of the directors and officers of Airco.\nAirco considered the profits turned over to it by the subsidiaries pursuant to the contracts as its own income and reported it as such. Petitioners reported as income only the six per cent return on capital that each was entitled to retain. Similarly, in declaring the value of their capital stock for declared value excess profits tax purposes, the subsidiaries reported only the nominal amounts at which the stock was carried on the books of each. The Commissioner notified petitioners of substantial income and excess profits tax deficiencies in their 1938 returns, having taken the position that they are taxable on the income turned over to Airco as well as the nominal amounts retained. The Tax Court held, however, that the income from petitioners' operations in excess of six per cent of their capital stock was income and property of Airco. 8 T.C. 594. Three judges dissented. The Court of Appeals for the Second Circuit reversed. 167 F.2d 304. We granted the petition for a writ of certiorari, 335 U.S. 810 , because of this conflict of opinion and the disagreement between courts as to the continuing vitality of Southern Pacific Co. v. Lowe, 1918, 247 U.S. 330 .\nPetitioners' contention is, in substance, that our decision in Moline Properties, Inc., v. Commissioner of Internal Revenue, 1943, 319 U.S. 436 , 1135, which held that the tax laws require taxation of the corporate entity if it engages in 'business activity,' expressly excepted the situation in which the corporation is the agent of its owner; that Southern Pacific Co. v. [336 U.S. 422 , 427] Lowe, supra, defines the content of 'agency' for tax purposes; and that, as the Tax Court found, this Court's characterization of the relationship between the corporations in the Southern Pacific case is 'aptly descriptive' of the relationship between Airco and petitioners. It must follow, according to petitioners, that income received by them and transmitted to Airco is taxable only to Airco.\nRespondent does not quarrel with the first and third propositions. The collision occurs at the second. The issue as presented by petitioners is, therefore, whether the principal-agent relationship described in the Southern Pacific case-and the similar arrangement between Airco and petitioners-contains the 'usual incidents of an agency relationship,' as that phrase was used in Moline roperties, Inc., v. Commissioner, supra.\n'* * * the Central Pacific and the Southern Pacific were in substance identical because of the complete ownership and control which the latter possessed over the former, as stockholder and in other capacities. While the two companies were separate legal entities, yet in fact, and for all practical purposes they were merged, the former being but a part of the latter, acting merely as its agent and subject in all things to its proper direction and control.' 247 U.S. at page 337, 38 S.Ct. at page 542. It is thus clear beyond doubt that the subsidiary was not referred to as an agent of the parent in the usual or technical sense. 'Agency' and 'practical identity,' as those words are used in the Southern Pacific case, are unquestionably opposite sides of the same coin. 6 The close relationship between corporations because of com- [336 U.S. 422 , 429] plete ownership and control of one by the other was the basis for the result reached, whatever its articulation.\nWe think, therefore, that petitioners' argument is without merit because based on an erroneous interpretation of Southern Pacific Co. v. Lowe, supra. The agency argument, to quote the opinion in Moline Properties, 'is basically the same argument of identity in a different form * * * the question of agency or not depends upon the same legal issues as does the question of identity previously discussed.'9 Ownership of a corporation and the control incident thereto can have no different tax consequences when clothed in the garb of agency than when worn as a removable corporate veil.\nThe result reached by the Court of Appeals is clearly required by our later decisions. Our reluctance to erase Southern Pacific from the books has been due not to any belief that it lays down a correct rule for tax purposes generally, but to the fact that it concerns 'very peculiar facts' which make it clearly distinguishable from later cases involving the tax status of a subsidiary or other wholly owned corporation. 12 For that reason, we have, instead, held that it lays down no rule for tax purposes. [336 U.S. 422 , 433] Burnet v. Commonwealth Improvement Co., supra, 287 U.S. at page 419, 53 S. Ct. at page 199; Moline Properties, Inc., v. Commissioner of Internal Revenue, supra, 319 U.S. at page 439, 63 S.Ct. at page 1134. That the concept of identity of the corporation with its owner set out in the Southern Pacific case is incompatible with later decisions of this Court may be demonstrated by a consideration of the facts enumerated and relied upon by the Tax Court, which based such reliance on the emphasis placed upon similar facts in the Southern Pacific case. These facts relate to the ownership, control, and right to income reserved by the parent.\nNot do the contracts between Airco and petitioners by which the latter agreed to pay all profits above a [336 U.S. 422 , 436] nominal return to the former, on that account, become 'agency' contracts within the meaning of our decisions. The Tax Court felt that the fact that Airco was entitled to the profits by contract shows that the income 'belonged to Airco' and should not, for that reason, be taxed to petitioners. Our decisions requiring that income be taxed to those who earn it, despite anticipatory agreements designed to prevent vesting of the income in the earners, foreclose this result. Lucas v. Earl, 1930, 281 U.S. 111 ; Helvering v. Clifford, 1940, 309 U.S. 331 ; United States v. Joliet & Chicago R. Co., 1942, 315 U.S. 44 ; Commissioner v. Sunnen, 1948, 333 U.S. 591 . Of course one of the duties of a collection agent is to transmit the money he receives to his principal according to their agreement. 17 But the fact that petitioners were required by contract to turn over the money received by them to Airco, after deducting expenses and nominal profits, is no sure indication that they were mere collection agents. Such an agreement is entirely consistent with the corporation-sole stockholder relationship whether or not any agency exists, and with other relationships as well. 18 [336 U.S. 422 , 437] What we have said does not foreclose a true corporate agent or trustee from handling the property and income of its owner-principal without being taxable therefor. Whether the corporation operates in the name and for the account of the principal, binds the principal, by its actions, transmits money received to the principal, and whether receipt of income is attributable to the services of employees of the principal and to assets belonging to the principal19 are some of the relevant considerations in determining whether a true agency exists. If the corporation is a true agent, its relations with its principal must not be dependent upon the fact that it is owned by the principal, if such is the case. Its business purpose must be the carrying on of the normal duties of an agent. 20 Absence of the factors mentioned above, and the essentiality of ownership of the corporation to the existence [336 U.S. 422 , 438] of any 'agency' relationship in the Moline Properties, Commonwealth Improvement Co., and Southern Pacific cases indicates the fallacy of the agency argument made in those case.\nThe same fallacy is apparent in the contention that petitioners are agents of Airco. They claim that they should be taxable on net income aggregating only $1,350, despite the fact that during the tax year (1938) they owned assets worth nearly 20 million dollars, had net sales of approximately 22 million dollars, and earned nearly four and one-half million dollars net. Their employees number in the thousands. We have passed the question whether Airco's interest in these assets is that of owner of the subsidiaries or lender, but whatever the answer, they do not belong to Airco as principal. The entire earnings of petitioners, except for trifling amounts, are turned over to Airco not because the latter could command this income if petitioners were owned by third persons, but because it owns, and thus completely dominates the subsidiaries. Airco, for sufficient reasons of its own, wished to avoid the burdens of principalship. 21 See Moline Properties, Inc., v. Commissioner of Internal Revenue, supra; Sheldon [336 U.S. 422 , 439] Building Corporation v. Commissioner of Internal Revenue, 7 Cir., 1941, 118 F.2d 855. Compare Forshay v. Commissioner of Internal Revenue, 1930, 20 B.T.A. 537. It cannot now escape the tax consequences of that choice, no matter how bona fide its motives or longstanding its arrangements. When we referred to the 'usual incidents of an agency relationship' in the Moline Properties case, we meant just that-not the identity of ownership and control disclo ed by the facts of this case.\nWe have considered the other arguments made by petitioners and find them to be without merit. The judgment of the Court of Appeals is affirmed.\n[ Footnote 2 ] Wilson Welder had a net deficit during the year here involved and is not a petitioner in this action.\n[ Footnote 3 ] Sales had outstanding 125 shares of stock of $100 par value; Carbide's outstanding capital stock was 50 shares of $100 par value; Carbonic also had 50 shares of $100 par value.\n[ Footnote 4 ] Mertens, Law of Federal Income Taxation (1948 ed.), Vol. 10A, p. 237.\n[ Footnote 5 ] Finkelstein, The Corporate Entity and the Income Tax, 44 Yale L.J. 436, 448.\n[ Footnote 7 ] The case other than Southern Pacific relied upon by the Tax Court was Munson Steamship Line v. Commissioner of Internal Revenue, 2 Cir., 77 F.2d 849. That case w explained in Moline Properties, Inc., v. Commissioner of Internal Revenue, supra, as depending upon a particular legislative purpose which justified disregarding the separate entity.\n[ Footnote 8 ] Plaintiff's Exhibit P, No. 452, October Term 1917, is an income tax statement of the subsidiary, central pacific Co., showing payment of income taxes on $3,333,846.18, its total net income for 1913 less one- sixth (i.e., making an allowance for the two months before the income tax law went into effect March 1).\n[ Footnote 9 ] 319 U.S. at pages 440-441, 63 S.Ct. at page 1135.\n[ Footnote 10 ] 8 T.C. 594, 611. After enumerating the facts considered pertinent, the court concluded: 'It is true, of course, that taken separately some of the foregoing facts would not be sufficient in themselves to make inoperative the general rule that corporations are separate juristic persons and are to be so treated for tax purposes. We think, however, that when all these facts are viewed together they bring petitioners within the rule announced by the Supreme Court in Southern Pacific Co. v. Lowe, supra.' 8 T.C. at page 614.\nIt should be added that the Court of Appeals, whose opinion was written by its Chief Judge, did not so much as mention the agency argument now made by petitioners. Its only references to agency were isolated quotations from the Tax Court's opinion and the contracts quoted in footnote 1. The court's opinion phrases the question as 'when a wholly owned subsidiary of a parent corporation shall be treated for purposes of income taxation as a separate taxable person, and when as merely a part of the corporate activities of the parent'. 167 F.2d 304, 305.\n[ Footnote 11 ] 167 F.2d at page 307.\n[ Footnote 12 ] Two basic distinctions between the Southern Pacific case and subsequent cases, except Gulf Oil Corporation v. Lewellyn, 1918, 248 U.S. 71 , which followed Southern Pacific, are immediately apparent. First, the Southern Pacific case involved taxation of the parent-owner rather than the subsidiary corporation; second, the question was when the income had been earned, rather than who had earned it. The importance of these distinctions is indicated by the fact that the subsidiary paid income taxes upon income received subsequent to March 1, 1913, see footnote 8, supra, and that the parent did not dispute its tax liability for dividends from post-1913 earnings of the subsidiary. The decision is based on an interpretation of the Income Tax Act of 1913. The Court felt that it was not the intent of the Act to tax earnings prior to the effective date of the Act, and that the Central Pacific's pre-1913 income had actually accrued to the parent before the effective date of the Act. The opinion states that 'The case turns upon its very peculiar facts, and is distinguishable from others in which the question of the identity of a controlling stockholder with his corporation has been raised.' 247 U. S. at pages 338-339, 38 S.Ct. at page 543. By its very terms, the decision is limited to its precise facts.\n[ Footnote 13 ] Much of the testimony introduced by petitioners had to do with the intercorporate relationship between Airco and its subsidiaries, the use of certain facilities by two or more of the subsidiaries, the duties of various officers who held positions with Airco and its subsidiaries, and the services performed for all of the subsidiaries by certain departments of Airco. So far as this testimony shows the integration of the corporate system and its direction by Airco, it is, as we have indicated, immaterial. So far as it indicates that the subsidiaries received the use of equipment and services for which they were not charged, it is relevant as showing that their income was distorted to that extent, but it does not indicate that the income received 'belonged' to Airco at the time of its receipt. The Commissioner made allowance for this distortion by allocating over $ 400,000 of the expenses reported by Airco to petitioners under the authority given him by 45 of the Revenue Act of 1938, 26 U.S.C. 45, 26 U.S.C.A. 45.\n[ Footnote 14 ] 45 B.T.A. 647, 650.\n[ Footnote 15 ] As a practical matter, a considerable part of the assets of petitioners was supplied out of profits from their operations. Even though assets were purchased directly out of the earnings of a subsidiary, however, the amount withdrawn was entered in the accounts payable by the subsidiary and in the accounts receivable of Airco, since substantially all profits of the subsidiaries were, by contract, payable only to the parent.\n[ Footnote 16 ] Since petitioners were required to pay all profits except very small amounts to Airco each year, it was obviously impossible for them to pay the accounts payable to Airco. See note 15. Mr. C. E. Adams, Chairman of Air Reduction Corporation, testified that the assets of the subsidiaries represented by the accounts payable could be realized by Airco only upon dissolution of the subsidiaries. In other words, there was never any expectation that the accounts would be paid prior to dissolution. Since no interest ran on these accounts, the 'loans' were identical, except in name, with contributions of capital. See American Cigar Co. v. Commissioner of Internal Revenue, 2 Cir., 33 F.2d 425; Hoyt v. Commissioner of Internal Revenue, 2 Cir., 145 F.2d 634; Van Clief v. Helvering, 77 U.S.App.D.C. 337, 135 F.2d 254; Reading Co. v. Commissioner of Internal Revenue, 3 Cir., 132 F.2d 306. Levy and Simonds, Stockholder Advances to Corporations . . . Are They Loans or Capital Contributions? 25 Taxes 127, 128, state that 'intention to lend and expectation of repayment are necessary to the existence of a valid debt.' The fact that no interest ran on these 'loans' is, of course, further indication that they are capital contributions. Berry Motor Car Co., 43 B.T.A. Memo. Op. 1209, Jan. 25, 1941.\n[ Footnote 17 ] Restatement o Agency, 427.\n[ Footnote 18 ] In United States v. Joliet & Chicago R. Co., 1942, 315 U.S. 44 , a lessee railroad agreed to pay rental payments to the lessor's stockholders directly. The lessor thereafter carried on no active business. It was nevertheless held taxable on the income received by its stockholders, since they received the payments only because they held its stock.\n'Art. 22(a)-1. What included in gross income.\u00c4Gross income includes in general compensation for personal and professional services, business income, profits from sales of and dealings in property, interest, rent, dividends, and gains, profits, and income derived from any source whatever, unless exempt from tax by law. (See sections 22(b) and 116.) In general, income is the gain derived from capital, from labor, or from both combined, provided it be understood to include profit gained through a sale or conversion of capital assets. * * *' See Eisner v. Macomber, 1920, 252 U.S. 189, 207 , 9 A.L.R. 1570; Merchants Loan & Trust Co. v. Smietanka, 1921, 255 U.S. 509, 519 , 15 A.L.R. 1305.\n[ Footnote 20 ] Of course even a corporation which satisfies the usual tests of agency may be disregarded by the Commissioner if it is a sham or unreal. Higgins v. Smith, 1940, 308 U.S. 473 ; Gregory v. Helvering, 1935, 293 U.S. 465, 97 A.L.R. 1355. Escaping taxation is not a 'business' activity. See National Investors Corporation v. Hoey, 2 Cir., 144 F.2d 466.\n'Frankly, in 1918 and still, Air Reduction, Inc., was and is a New York corporation. Even at that early date it became evident, as I already said, we were going to have plants scattered all over the United States. We didn't want to domicile the parent company in 48 states of the Union and have us subject to service in all those states, that is, have the parent company subject to service in all those states, and that was distinctly a reason for using this corporate setup in connection with operations to be run as divisions, just as the contract sets forth.\nIt is thus apparent that Airco was attempting to avoid the status of principal vis-a-vis its subsidiaries. As principal it would have been subject to service of process through its agents; as owner of the subsidiary it was not. See Peterson v. Chicago, R.I. & P.R. Co., 1907, 205 U.S. 364, 391 ; Cannon Mfg. Co. v. Cudahy Packing Co., 1925, 267 U.S. 333 . The purpose of having officers of subsidiaries who could deal directly with customers does not indicate an agency relationship. On the contrary, the very purpose of the organization adopted was to lead customers to believe that they were dealing with top men in the company actually manufacturing and selling the products they purchased.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Passaretti F, Tia M, D'Esposito V, Pascale MD, Corso MD, Sepulveres R, Liguoro D, Valentino R, Beguinot F, Formisano P, Sammartino G.\nEffects of Choukroun's platelet-rich fibrin on human gingival fibroblasts proliferation, migration and type I collagen secretion.\nWhole blood clots are more resistant to lysis than plasma clots\u2013greater efficacy of rivaroxaban.\nThe primacy of platelet-rich fibrin on bone regeneration of various grafts in rabbit's calvarial defects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"fake bonsai tree artificial bonsai tree simulation plant flower with pot fake plant with pot artificial bonsai.\nfake bonsai tree mini plastic artificial bonsai trees art pots tray fake small rose flowers for home indoor.\nfake bonsai tree artificial bonsai trees single preserved juniper bonsai tree single preserved juniper bonsai tree artificial bonsai trees.\nfake bonsai tree fake bonsai tree elm bonsai artificial w decorative planter.\nfake bonsai tree artificial bonsai trees 6 foot preserved bonsai tree artificial bonsai trees.\nfake bonsai tree 5 artificial bonsai trees.\nfake bonsai tree artificial mini palm trees plant ornaments artificial garden plant artificial bonsai tree.\nfake bonsai tree artificial bonsai tree welcoming plant potted bonsai fake mini flower green plant pine pot vase wedding home decoration from china smoke.\nfake bonsai tree fake bonsai tree artificial topiary boxwood spiral artificial bonsai trees fake bonsai tree fake bonsai tree.\nfake bonsai tree fake bonsai tree fake bonsai tree how to make an artificial bonsai tree artificial bonsai tree.\nfake bonsai tree inch tall vintage bonsai artificial bonsai trees.\nfake bonsai tree bonsai tree.\nfake bonsai tree artificial bonsai tree plant potted bonsai vase home decor.\nfake bonsai tree double bonsai tree artificial bonsai trees.\nfake bonsai tree artificial bonsai tree creating real ambience silk plants vertical wall garden azalea flowering bonsai artificial bonsai china artificial bonsai tree.\nfake bonsai tree picture of artificial bonsai tree.\nfake bonsai tree wholesale mini decorative artificial ornamental bonsai plants.\nfake bonsai tree double double bonsai tree artificial bonsai trees.\nfake bonsai tree artificial bonsai tree for sale shop artificial luxury panda bonsai tree line from silk blooms artificial artificial bonsai tree.\nfake bonsai tree fake pine bonsai cm in shiny black.\nfake bonsai tree real new artificial pine bonsai tree for sale floral decor simulation desktop display.\nfake bonsai tree artificial bonsai tree fake indoor ficus bonsai tree plant for decoration desktop display.\nfake bonsai tree inch preserved bonsai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your opinion matters. Leave your review of Little Sisters of Assumption Family Health Service.\nDo you have items to donate? Contact Little Sisters of Assumption Family Health Service at the phone number provided above to see if they can use any items you may have to donate.\nIf you are looking to volunteer for Thanksgiving or any other time, please contact Little Sisters of Assumption Family Health Service at the phone number provided above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disco Toilet 4 \u2013 Still at the Vicky Tee. As you may have expected, a new release from the ever-popular YouTube Series 'Disco Toilet'. This one is a cracker!\nIf you've never been here before, then you might want to know where the toilets are\u2026 Well, they are\u2026 Everywhere.\nWildwood, sort it out, what the hell is going on?\nThis screaming Jesus is a remnant of a toilet roll holder sticker.\nDoes Terrance Stamp think that Andrew Collins is a bummer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our platform is available with a range of automotive connectors to suit your requirements. Our standard enclosures are rated for extreme conditions, including under-bonnet and on-engine mounting.\nR2G2 has in-depth experience covering diesel injection systems of many types: in-line pumps, rotary pumps, jerk pumps, unit injectors, direct injection and common rail.\nOur track record on embedded vehicle systems includes EGR control, air filtration control, ride-height control, drive-by-wire steering, drive-by-wire braking and many other embedded vehicle systems.\nOur automotive common rail ECUs are suitable for 4-8 cylinder engines. Our flexible platform can be adapted to support a huge range of sensors, outputs and actuators. We can produce units in quantities as small as 5 pieces.\nOur dual fuel systems combine diesel with natural gas (LNG, CNG or LPG) to reduce running costs. The system can be retrofitted to existing vehicles, keeping the original ECU and seamlessly replacing a proportion of diesel power with gas. Depending on the engine, natural gas can replace up to 80% of the diesel consumption.\nIn addition, the natural gas improves the combustion and so reduces the particulates. Together with the reduction in diesel burnt, this gives a major improvement in exhaust emissions.\nFor OEM applications, a combined diesel and natural gas controller can be used. This system can use a high proportion of natural gas to minimise running cost, with the security of diesel-only operation when natural gas is unavailable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When we arrived at the airport in Dublin, PJ, Liz, Claire, and Anthony were all at the gate waiting for us. It was so great seeing such friendly faces after ten hours in the air. Heather was also there with a trailer so she could transport our luggage to the hotel we were staying at. After we got our rental vans, we drove to Avoca Restaurant for lunch. After a tasty meal of seafood chowder, our group went to our hotel, The Osprey, to check in, go to our rooms, get settled, maybe shower and then leave so we could head out to the church. Once we arrived at the church, PJ Booth, who is the pastor over the church, gave our team some background information on Open Arms and Ireland itself. After that, we toured the church and saw what we would be painting and got a run down of how things work for them on the weekend. Our team at that point was exhausted from traveling and the loss of time so we were then able to go back to the hotel where everybody crashed and slept like logs for an early morning.\nThat morning, when we went to the church, our Executive Children's Pastor, Ken Jackson, preached both services at Open Arms. For the first service, our team sat in the service and let me just say, Pastor Ken did an amazing job. As for the second service, our team had the option to either sit in the sermon again or volunteer in the children's ministry. I served in Cresch, which ranges from children who are walking up to age three. In that area, the kids played for around an hour and a half until their parents came and picked them up. There were a few kiddos who I instantly connected and played with: Teagan, Omega, Jess, and David. I also had the opportunity to work with an older lady named Breda who had so much love and compassion for the kids in her room and Leona who was fifteen and brought an incredible amount of happiness into the children. It was all incredibly inspiring to see. After church, our team, along with PJ, Liz, Claire, Anthony, Alex, and Rosie, went to dinner at Fallon's Restaurant where we had incredible food. That night, for whoever was hungry, went to the bar at our hotel for fish and chips and we sat around and enjoyed each other's company and fellowship.\nThis was the day that our projects began at OAC. The reason that our team was in Ireland was because we had children leaders from Gateway Church preparing to speak at the first international Amazing Kids Leadership Conference. At the hotel after breakfast, we gathered our team and had a morning prayer, preparing for the day ahead of us. Once we arrived at Open Arms, we immediately began working in the auditorium. Our project for that day was to begin painting the main auditorium and we started by sanding. Part of our team from Texas partnered up with the team from Ireland. This was the time that we were really able to meet everybody and start friendships. Another part of our team and a few from the church, began the project that was outside \u2013 constructing the playground. We were served lunch and then once we had finished, we began working again. After working for eight hours, we were done for the day and headed back to the hotel were we, once again, had dinner at the bar with our team and PJ, Liz, and Claire.\nThis day consisted of our project work again, getting ready for the conference. We really began painting the auditorium and started on the main children's area. We were once again able to work alongside people from Open Arms; painting, talking, laughing, and singing along to the music that played in the background. Because of the days we had with them in such close quarters working, we were able to become close and that it the reason I made all of the friends that I did. We broke for lunch a little bit after working and like the day before, once we were done eating, we went right back to work. Our goal was to finish painting the auditorium ahead of schedule so we could have the paint smell aired out before the conference. Dinner that night was at Lawlor's in Naas where a small portion of our group (myself, Ellen, Mark, and Ryan), talked with Claire about all things Texas and what's different in Ireland. That night we were able to really hang with Claire when she wasn't working and prepping for the conference herself. That day also happened to be one of our team members, Terri's, birthday. Once we finished eating, another member of our team, Tyler Bates, did his interpretation of an Irish jig outside the restaurant (and there is video to prove it). All of our stomachs hurt from laughing and our faces were sore from smiling so hard. It was a great end to a productive and successful day.\nAfter we finished breakfast, our team gathered together again in the hotel lobby for another prayer time and to share what the Lord had done in our life in just the few days we had been there. Just like the days before this, we worked on the auditorium and kids area, finishing up painting and we started on the check-in desks that were going to go in the lobby of the children's room. Once we were all done for the day, we were able to go back to the hotel for a quick clean up and then went back to Open Arms for a kids camp reunion where we were able to meet a lot of people who volunteered in the children's ministry there. We had pizza and chips (french fries), played intense games like Mafia (a card game), spoons, and fruit basket which is a VERY intense game of musical chairs and spoons combined. I don't think I've had as much fun as I did that night. In the games of spoons and fruit basket, both times two people from our team, Ryan and Paul, one the games, making the Americans reigning champions over the Irish. \ud83d\ude09 After a long night, we went back to our hotel to get some much needed rest.\nOn this day, we only had to work half a day because we were able to get all of the project work done. The auditorium had been completely painted with a second coat added on to it, and the children's area was painted with a second coat on it as well. One of the members from our team, Ellen, REALLY wanted to see sheep while we were in Ireland. So, after all of our work was done and we were relieved to go, one of our leaders, Mark, along with A.J. and Ch\u00e9, took the small part of the team that wasn't speaking at the conference to a place called the Curragh. The Curragh is a massive stretch of land were sheep roam free for miles upon miles. It was absolutely amazing. We saw thousands of sheep (Ellen squealed), and we also were able to see illegal Irish gypsies which made for a good story once we arrived back at the church. We hiked up an immensely steep hill where we saw the miles of green land. After we finished with the Curragh, we then went to the Wicklow Mountains. We saw old and abandoned castles, drank from a REAL Irish spring, saw were two lakes met into one, and watched the sun set over old castle ruins. Our legs were dead tired and our hearts were racing from all of the physical exertion but it was totally worth it. When we arrived back to the church, we went to Naas where we had dinner at GG's Restaurant with our team and a few members from Open Arms. Myself, Ryan, Ellen, Tabitha, Kayla, A.J. and Ch\u00e9 all sat together where we told our life stories basically and we laughed and in that moment, I realized that these people weren't just my friends anymore; they were my family.\nLET THE CONFERENCE BEGIN!! After breakfast we had a morning prayer, praying that the Lord let this conference be an amazing thing and people would get so much insight for their children's ministry. When we arrived at OAC, everyone was running around trying to get the last bits of stuff finalized for the next two days. Ellen and myself were stationed at the front desk where we welcomed and check people in. We were able to meet so many people from all around Ireland and got them even more excited for the conference. The conference started on time and from the first speaker of the morning session to the last speaker of the evening session, things ran smoothly. I was excited because I was able to sit in on a lot of the conference and since I work in children's ministry at Gateway, I was able to learn a lot that I now apply in the classrooms here at home.\nWaking up at 5:30 in the morning to get ready and eat breakfast made us in not so happy moods but we put smiles on our faces and braced ourselves for another amazing day of the conference. While the conference was going on, the small portion of our team that wasn't speaking got to hang out with the teens of Open Arms and got to know them even better. Because of the days we had with them previously, I was already really close with a lot of them. It was good to sit and talk while drinking tea getting to know about them, where they came from and their interests. We in turn were able to share about ourselves and it was a really grand time being able to become so close with people that we had met not even a week before. The conference ended that day and it was extremely powerful. So much information and insight was shared in just two days and people left feeling refreshed and encouraged. To say that I participated and witnessed the first international Amazing Kids Leadership Conference is something I will always be grateful for.\nThis was our last day at Open Arms. We attended service and dreadfully awaited the time we had to depart and say goodbye to everyone. After the service was over, we painfully started hugging the amazing people we had met and I was sort of okay until I hugged this one girl goodbye. Her name is Jenny and we met the first day we started our project. Her and I became very close and talked about anything and everything. As I hugged her, I started crying. I couldn't leave. I didn't want to leave. It was so hard to say goodbye to everybody that in the pictures we took, my face was red and swollen from crying so hard. I've never met so many amazing people in such a short amount of time. Each person that I met in Ireland has a very special place in my heart and will forever. Once we were dragged into the vans, we left for Dublin. We toured Trinity College and got a first-hand look at the Book of Kells (if you have no idea what that is, Google it and it's history. It's amazing.). We were then set free to explore Dublin and were given a time to meet back at the vans. Ryan, Brittany, Ellen, and myself got together and went at Dublin. We shopped, had lunch in a small bistro, got coffee from the Irish Starbucks, and saw street performers. It was great to be able to explore a town so rich in history and really experience Ireland like a tourist.\nThis was another early morning. At 7:30 in the morning, we departed for the West of Ireland. Our first stop was at Bunratty Castle. We were set loose on the grounds and the castle where Ellen and myself explored. The castle is still active in which the have dinner parties like they did back when it was thriving. We saw all the rooms, walked through the old halls, and traveled up and down the small spiral stairs to the very top where we had a gorgeous view. After Bunratty, we all met for lunch at Durty Nelly's (you have to say it with your best Irish accent so it's authentic), and then departed for the . . . wait for it . . . CLIFFS. OF. MOHER. Talk about the most breathtaking view you will ever see in your entire life. If you have not been to Ireland, or have been and not seen these cliffs, it is a must on your bucket list. The cliffs stretch for a couple miles and drop a few hundred feet directly into the Atlantic Ocean. We went to a small castle and climbed to the top where you can see the ocean stretching out for miles upon miles upon miles, and saw the cliffs stretch out for miles itself. We then walked until we were standing directly on top of the cliffs themselves. I can't begin to try and explain the excitement I had being there on the place my favorite movie was shot (if you've never seen The Princess Bride, you need to). Pictures were taken, I sat on the edge of the cliff, and I recorded a video of myself to my mom saying if I fell off, I loved her and my family very much. It was all exhilarating and most definitely something I will never forget. We were all upset that we had to leave but we had dinner on the coast at a pizza place and we sat on the sea wall overlooking the Atlantic. Once we arrived back at the hotel, we had to pack our bags because the next day was our departure date back to the states.\nAfter a small breakfast at the hotel, we gathered our luggage in the lobby and sad goodbye to Claire who brought departing presents for us all. Although I didn't cry as hard as I thought I would, I had tears in my eye because the thought of leaving was heartbreaking. When we got to the airport in Dublin, we took a group picture and said goodbye to PJ and Liz. About an hour later, we boarded our flight and began the eight hours to Atlanta. I was so tired from staying up the night before I tried to fall asleep. I now know that I'm unable to fall asleep on an airplane. I was determined to get some kind of sleep so I watched Tangled and The Best of Me because I knew that both would make me cry. Even though I cried, hard, I wasn't able to get any sleep. Upon arrival in Atlanta, we were given ten dollars for lunch and my first meal back home was the one and only, Chipotle. We had a three hour layover so at our gate, we ate and talked and waited to board. Our flight was delayed another hour because of bad weather so our team took over the gate with us and our luggage, and I was finally able to lay on the floor and sleep. We were then able to board the flight and before take off, I got another thirty minutes of sleep. At that point, I was tired, irritable, and jet lagged. I was ready to be home and sleep in my own bed. Upon arrival in DFW, we went to baggage claim where I saw my parents, sister, and niece and cried tears of immense joy to see them again. In the truck on the way home, and even when we got home, I managed somehow to put eleven days in a different country together in about thirty minutes.\nI am so thankful, blessed, and grateful that I was given the opportunity to go to Ireland. As my first mission trip, I had no idea what to expect. But, the Lord put amazing people on my team to guide me along the way and I can't begin to express my gratitude towards the people in Ireland. I knew from the very start that this trip was something God had planned but I couldn't pinpoint what it was. While in Ireland, I came to find that my calling was ministry, especially children's. God knew where I needed to be and who I needed to be with. Claire Hearty (now Afolabi), is the children's pastor over Open Arms. She has given me the once in a lifetime opportunity to intern under her next summer in Ireland. When that opportunity was brought up to me, I immediately took it with tears in my eyes. The internship was the reason that God put me on this trip. Although I miss Ireland and it's people like crazy, I know that within the year I'll be back to the place that feels like home. I wouldn't have been able to go on this trip without the support I had from friends and family. I want to thank all of them. Another huge thank you goes to Reed Public Relations who gave me a scholarship to go on this trip. Without Lauren Reed and her company, this trip would definitely would not have happened. Thank you a million Lauren for giving me this once in a lifetime opportunity to go and experience Ireland they way I did. This trip changed my life in so many ways and because of it, I know now exactly what God has planned for my life and the best ways to accomplish those plans. Ireland has a huge place in my heart and I cannot wait to be back. Thank you all for reading this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Released on August 11, 1966: Everyone wants to kill Johnny Yuma, but the fastest gun in the West has other ideas.\nCowboy Western movies are among the best loved and most filmed adventures in the history of motion pictures, and there is a rich heritage of cowboys versus the bad guys from the first decades of motion pictures, but I gotta tell ya, nobody did music for Cowboy movies better than the Italians in the 1960's and 1970's. I can still think of the tense music accompanying the grizzled face of Clint Eastwood riding across the western desert in search of the bad guys. Well, he wasn't the only leading man that filmed Italian Westerns, and this one features Mark Damon, Golden Globe winner who has almost 70 acting credits and over 50 producing credits, including the 1981 Das Boot. Wealthy rancher Felton tells his eye-candy young wife that he is leaving his wealth to his nephew Johnny Yuma instead of to her and her brother Pedro. Before we know it, Pedro is killing the rancher and now the only thing between wife Samantha and the wealth of the ranch is nephew Johnny Yuma, coming at the call of his Uncle after many years. Now, you gotta know that today we are a 'gentle and kind' society and there are unwritten rules about how many killings should be shown on-screen for general audiences, but back in 1966 watching the bad guys go down in a hail of bullets was an awesome and exciting part of good Cowboy Westerns. As we watch Johnny Yuma travel to his Uncle's ranch the bodies pile up, but pale by comparison with the shoot-out ending. Widow Samantha hires the fastest gunman she can find, Lawrence Jerome Carradine, played by Lawrence Dobkin, to kill Johnny Yuma before he can get to the ranch and claim the estate, but like I told you, although the bodies will pile up on the way to the final shoot-out, the climax is as rootin-tootin hard-boiled cowboy fight to the finish as you will ever experience in a 1960's Western . . . bad guys will die . . . . LOTS of bad guys will die . . . nobody could escape the guns of Johnny Yuma . . . with the best music in the history of Cowboy Westerns in the background. Pop a big bowl of white kernel popcorn with plenty of warm melted butter on it and enjoy the show. Ohh . . . and the eye-patch hombre that I used next to the show description . . . that isn't Johnny Yuma, but his face and expressing tell us perfectly what the tone of this adventure will be like.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Queenstown Airport is hiking commercial pick-up and drop-off fees \u2013 but doesn't want to talk about it.\nRight now the airport charges $3.45 for a pick-up.\nBut from next Thursday, Mountain Scene understands the airport will charge $8.05 for pick-ups as well as drop-offs.\nThe corporation \u2013 mostly owned by Queenstown ratepayers \u2013 won't confirm anything, saying its commercial vehicle deals are \"sensitive and confidential\".\nBut several sources confirm the fee hikes.\nLike the airport, taxi operators are similarly shy to comment.\nFounder of app-based private hire business TakeMe, Luis Kreische, reckons the airport's standardising fees as \"revenge\" for taxi companies clipping the ticket.\nHe says taxis are charging $6 to pick up when they're only being charged $3.45.\nMountain Scene rang three taxi companies and confirmed the $6 charge.\nBut the issue's an old chestnut.\nTaxi companies have said for years the $6 charge is a way to try and recover the cost of leasing space at the airport's taxi rank.\nKreische criticises the airport for the \"absolutely stupid, silly decision\" to hike pick-up and drop-off fees \u2013 which he says was taken without consultation.\nThe result will be higher prices and unhappy passengers, he says, although he mentions TakeMe has recently lowered its rates.\nThe airport's taxi rank will now become free-flowing, Kreische says.\nThat means companies won't be charged leases, just the one-off fees.\nThe airport's property and commercial boss Mark Samways echoes statements made by the company in 2011.\nThe airport's fee hike shouldn't be surprising. In its latest annual report, the airport corporation lists one objective as \"Grow non-aero revenue\".\nBetween 2010 \u2013 when Auckland International Airport controversially took a 24.9 per cent stake \u2013 and this year the airport's total revenue has increased from $13.3 million to $31.5m, as total passengers have soared from 812,000 to 1.65 million.\nEarlier this year it paid a record dividend of $6.3m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What an amazing! I have been ridden for 2,000 miles already, but still great. That's why I am going to buy another one. Ideal for ama racing (I am on Cat 3) and long cruising. No pinch flat of course but fast rolling which gives me a comfort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafiyc b/data_all_eng_slimpj/shuffled/split2/finalzzzafiyc
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The Democratic Alliance said on Sunday it was \"extraordinary\" that Deputy President Jacob Zuma had been given new responsibility, as watchdog over the \"second economy\", before judgment in the Shaik trial had been delivered.\nThe Sunday Times reported on Sunday that President Thabo Mbeki had tasked Zuma with overseeing government programmes to grow the second economy and integrate the poor into the economic mainstream.\nHe had assigned the role to Zuma last year, but the decision had taken time to implement.\nChief government strategist Joel Netshitenzhe's policy co-ordination and advisory services unit was preparing its first report for Zuma and expected to present it in about two weeks, the newspaper said.\nHowever Democratic Alliance finance spokesman Ian Davidson told Sapa he was surprised that the president had \"almost cocked a snook\" at the fact that judgment was still to be delivered in the trial of Durban businessman Schabir Shaik, who the prosecution claims had a corrupt relationship with Zuma.\nZuma has not been charged, but a conviction in the case, now drawing to a close, will severely dent his image.\nDavidson said it was important that attention was paid to the second economy.\n\"One needs a champion. But the fact that he [Zuma] has been given this role is extraordinary. It just shows an arrogance, a disrespect for a process that is going on that can fundamentally change the status of this person.\n\"One would think out of respect they would hold off till judgment has been delivered.\"\nZuma, former economic affairs MEC in KwaZulu-Natal, is currently also a member of the African National Congress's national executive, leader of government business in the National Assembly, peace negotiator in the Great Lakes region and head of both the government's Moral Regeneration Movement and the SA National Aids Council.\nZuma's responsibilities in the \"second economy\" portfolio would be to oversee skills development, basic training under the expanded public works programme, financial services for the poor, and access to information about economic opportunities and community development.\nThe government uses the term \"second economy\" to include the country's rural and urban poor, informal traders, and micro entrepreneurs, whose participation in mainstream economic activities is either minimal or peripheral.\nIt also embraces, in this definition, a large number of uneducated young people unable to find employment, and employees who have been made redundant.\nIn February this year Mbeki told Parliament that success in the growth of the economy should be measured not merely in terms of the returns that accrued to investors or job opportunities for those with skills.\n\"Rather, it should also manifest in the extent to which the marginalised in the wilderness of the second economy are included and are at least afforded sustainable livelihoods,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Positions at this level are skilled in making independent judgements in complex situations. May serve as charge nurse or team leader by delegating assignments and reviewing the work of staff. Regular contacts are with individuals and\/or their families, staff nurses, supervisory nurses, physicians, health care facilities, and other administrative staff.\n\u2022 At least one-year experience as a Registered Nurse.\n\u2022 Knowledge of the principles, practices, and procedures of registered nursing.\n\u2022 Knowledge of quality assurance techniques.\n\u2022 Knowledge of the specific program area of practice.\n\u2022 Knowledge of individual\/group dynamics.\n\u2022 Skill in providing health care instruction and guidance to individuals, families, community groups and\/or nursing home facilities.\n\u2022 Skill in accurate documentation.\n\u2022 Ability to plan, implement and evaluate nursing care plans.\n\u2022 Ability to complete records and reports in a timely manner.\n\u2022 Ability to establish and maintain effective relationships with individuals, families and co-workers.\n\u2022 Ability to interpret data and apply the appropriate problem-solving techniques.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The FPOS course will teach you the skills associated with rapid assessment, stabilisation and transfer of both trauma victim and those suffering a medical emergency. You will develop and gain a foundation knowledge enabling you to work individually or as part of a team in a variety of situations and locations in any part of the world.\nThis 5 day FPOS course is approved by Edexcel and is IHCD (Institute of Health and Care Development) managed. The training includes both theoretical lectures and realistic practical sessions designed to offer students the opportunity to practice new-found skills.\nInstructors host a variety of experience with backgrounds including; military medics, civilian paramedics, CP Operators; all with hostile environment operational experience. This is what sets us apart from our local competitors who have no hostile environment experience.\nThe course can be delivered in-house or at any of our training venues throughout the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LITTLE TENNIS (4 \u2013 6 yrs.)This specialized program is designed to stimulate, challenge and appeal to children age 4 to 6 years old. Our team of enthusiastic and friendly coaches create a wonderful environment where students develop the footwork, motor skills and basic strokes needed to play tennis. Students and parents alike will be impressed by the dynamic range of equipment, games and activities used throughout this program.\nThis program is for graduates of the Intro to Bronze level or those who have the ability to rally consistently off both sides and get a serve in the box consistently on a 60' court or larger. Knowledge of basic technique, scoring and etiquette are required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There was a time when it was so sorry looking that it didn't deserve a sideways glance. But its exterior has been rejuvenated to proudly display its historic heritage now.\nThis house, at 407 University, had become the home of Charles and Roena Hutton by 1872. They may have had it built that year, though some estimates are that it was built around 1870. Either way, it is definitely one of Laramie's oldest.\nThe Huttons had been living on the Laramie Plains since at least 1865, making them among the earliest people to put down roots here.\nCharles Hutton (1832-1899) had probably been here even earlier. Canadian-born, he migrated to the western frontier when it still was a frontier, making his living as a \"freighter\" driving ox teams for an Omaha entrepreneur named Ed Creighton.\nCharles Hutton and Roena Wilcox (1835-1888) had married in 1859 at Roena's home town of Anamosa, Iowa in the east-central part of the state. Being a foreman for Creighton no doubt took Hutton far from Anamosa often. Creighton's crews built the telegraph line through the Laramie Plains in 1861. Later they did grading work for the Union Pacific Railroad, in addition to Creighton's regular freighting service along the Overland Trail. Very likely Charles Hutton was part of Creighton's work force from 1861 on.\nThe stories Hutton told Roena and her family about the fertile Laramie Plains must have been persuasive because in 1865 her parents joined the Huttons for the trek west. They traveled by ox team across all of Nebraska to the Big Laramie River. They settled on what they called the \"Old Home Ranch\" nine miles north of what is now Laramie.\nThere was no one here to buy the land from, so they were essentially squatters. It must have been a somewhat primitive and isolated existence. We don't know what became of her parents, though ads for a shop operated by \"L.T. Wilcox\" do turn up in the Laramie newspapers from 1870-1872, and there is an early station stop on the UPRR in Albany County by that name.\nIn 1866, the year after the Huttons arrived, Fort Sanders was built three miles south of where Laramie City would be located (in 1868). Hutton had a contract to supply meat to the many hungry soldiers.\nAs Hutton's cattle ranching paid off, he tried operating a meat market in Laramie. \"Hutton & Co.\" is listed in the Laramie City Directory for 1870. However, by June 14 of that year the newspaper announced the dissolution of the partnership of C. Hutton and Ora Haley, with the firm to continue in business with Haley alone.\nIt may be that Roena demanded a house in town, or Hutton realized that they couldn't continue squatting on land that didn't belong to them. So, by 1872, Hutton arranged for the house in Laramie where Roena would live the rest of her life. But Charles Hutton was determined to have a ranch. In a partnership with his former employer, Creighton, he was eventually able to purchase land south and west of Laramie. He could usually be found at his ranch, though he spent enough time in Laramie to be very well known in town.\nWhen he was at the house in town with his wife and child (George L. Hutton is listed as the surviving son in Roena's 1888 obituary), their house was not located where it is now. Instead, the address was 110 South 4th St. where Hunter Hall on the Cathedral block is now located.\nBy April of 1920, the Episcopal Church of Laramie had gained ownership of all the properties on the whole city block where St. Matthew's Cathedral now stands. The church successfully petitioned to have the city council vacate the city-owned alley that had been platted on that block, and set about moving all the buildings that remained, including the former Hutton house (both Huttons had passed away by then).\nOriginally, the Hutton house had its long axis parallel to the street. But when the Morris company of Denver moved it a block away to a narrow lot on the north side of University St., they couldn't reorient it. So now what had been the side of the house became the front, and the partial front porch goes along the alley instead of along the street.\nFor many years the house was covered with gray siding that may have been asbestos shingles. Tenants of the apartments the home had been converted to didn't always show the kind of respect a proud old lady deserves. It did obtain a metal roof at some point, but it had none of the pizzazz that typifies Victorian architecture.\nHowever, recently Advantage Realty of Laramie spruced it up, it has a new owner, and it is wearing its age well with painted stucco siding and bright red trim. The multi-gable retro-Victorian home design is back in style, but this one is the real thing. It sports eight gables (counting the wall dormers and porch pediment), a steep pointed roof, two prominent bay windows and the tall narrow windows that are hallmarks of Victorian design.\nCaption: A view of Laramie looking northwest, probably taken from the scaffolding when the Albany County Courthouse was under construction.. The date someone wrote on the negative is wrong\u2014it is late 1871 at the earliest because the courthouse opened in spring 1872. The street going diagonally is Ivinson Ave. (then called South \"A\" St.). The Hutton house is at the center, when it was located at 110 S. 4th St., with the first wooden Episcopal Church west of it.Photo courtesy of the Laramie Plains Museum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagkzs b/data_all_eng_slimpj/shuffled/split2/finalzzzagkzs
new file mode 100644
index 0000000000000000000000000000000000000000..27ea778a2c94a553331e2654161b722e5cb521f7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"to get the node type of an entity. However, entity-wrappers can wrap other types of entities, e.g. users. When calling getBundle() on a user entity, 'user' is returned. What should I call to return the type of entity (should return 'node' in the example above)? I can see when I print_r($entity) that $entity->type is a protected field of the wrapper object.\nUsing Entity API, if given an entity in a function scope, how do you get the entity type?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two RAF fighter jets had to make an emergency landing in Liverpool during the night.\nResidents reported hearing a loud noise overhead around 1am today as the two RAF Typhoon jets were forced to land at Liverpool John Lennon Airport for an emergency fuel stop.\nAccording to Liverpool Echo, it is thought the jets had set off from the RAF military airbase in Lossiemouth, Scotland and were on a routine operation but had to divert to Liverpool for urgent refueling possibly due to bad weather.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CA IPCC RTP & CA Final RTP for Nov 2017 are being made available for fee on online. These RTPs are available in PDF Format for Download. Revision Test Papers (RTPs) are official books released by ICAI for each examination attempt. These RTPs will also contain all the relevant Amendments, Circulars and Notifications applicable for that particular examination. RTPs are very important for exam because, it is estimated that, almost 10-15 marks in each exam will be asked from these RTPs only. RTPs also helps students to revise their subject.\nRTPs are exclusive to each attempt. This means, RTPs released for May 2017 exam may not be relevant for Nov 2017 exam. Similarly, RTPs released for Nov 2017 exam will not in any way be useful for the next May 2018 exams. The price of CA IPCC RTP and CA Final RTP for Nov 2017 is the same (Rs.100).\nStudents can get these RTPs from all local branch offices of ICAI, and can also download RTP in PDF for free here. You can also place an order for physical books from ICAI Online Book Store. Once you order the RTPs online, ICAI will dispatch the books in week and you will receive the RTPs in 4 more days.\nFew students are facing a problem where, after they download RTP, Practice Manual or Study Material from ICAI Website, the file keeps asking for a password. To solve this, first install a pdf viewer software from Google PlayStore and then open the file using that software.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Warm summers are melting large amounts of snow on the Devon Ice Cap, in the Canadian Arctic. As this melt water gathers into rivers, it carves deep channels in the ice, like the one shown here.\nMartin Sharp vividly recalls his first slush flow. He spent that June, in 2007, camping on the Devon Ice Cap. This gently curving dome of ice is 140 kilometers (87 miles) across and rises to a height of 1,900 meters (6,200 feet). It sits atop an island in the Canadian high Arctic.\nSharp was riding a snowmobile when he heard a roar. It sounded like the rumble of a subway train. The entire snow slope in front of him was moving: A slow-motion river of waterlogged slush was slurping down the mountainside. A spate of hot, sunny days had melted so much snow that the slope could no longer support itself.\nSlush flows didn't used to happen here. The summers weren't warm enough, notes this glaciologist at the University of Alberta in Canada. But by the time he saw this one in 2007, he had been hearing about them more and more. In one memorable case, a lake of melt water sitting on top an ice cap suddenly drained. This sent a flood of water, slush and ice rampaging 10 kilometers (6 miles) down a valley. It nearly wiped out a camp where scientists had been staying.\nThis lake was formed by melt water on Belcher Glacier. It's part of the Devon Ice Cap. Such lakes can spill their water out suddenly, triggering dangerous floods or slush flows.\nFor Sharp, it prompted a decision. He and his colleagues had visited Devon Ice Cap nearly every summer for the last few years. Going forward, they would now come in April or May, when the days were still cool enough to avoid such dangers. \"The world has changed,\" he says.\nDevon is one of thousands of glacial ice masses that dot the world. Some are narrow streams of ice, called glaciers. These ooze down mountain canyons. Others, like Devon, are broad, blobby ice caps that cover entire islands, spreading outward at the edges.\nThese icy areas are tiny compared to the world's three great ice sheets, which cover Greenland, and East and West Antarctica. Altogether, these smaller icy areas hold only a hundredth of the world's ice. But because they occupy parts of the Earth that are warmer than the ice sheets, they are melting much more quickly. For now, they also are a major source of sea-level rise. Collectively, they are losing some 230 cubic kilometers (55 cubic miles) of ice per year. Dump all of that ice into one big pile, and it could form its own mountain range.\nIt started when a belt of strong winds shifted that year. Called the polar jet stream, it bent northward like a slithering snake. Its winds carried warm air from the Atlantic Ocean up along the west side of Greenland, into the islands of Arctic Canada.\nThe rate of melting on Canada's northern ice caps rose dramatically between 2004 and 2009. In these maps, red depicts areas losing ice; areas gaining ice are blue. The map on left shows ice changes from 2004 to 2006; map on right shows changes from 2007 to 2009. Devon Ice Cap appears in the lower-right corner of each map.\nAlex Gardner was a PhD student at the University of Alberta during the late 2000s. He worked with Sharp. He used a satellite to measure how the glaciers and ice caps of Arctic Canada were changing.\nCalled IceSat, it carried an instrument called a laser altimeter. As the satellite orbited, it shined a laser beam directly down onto the Earth. By measuring exactly how long the beam took to bounce off Earth's surface and return, IceSat could measure the precise height of the terrain it was passing over.\nIceSat made hundreds of passes over Canada's glaciers and ice caps. Its data showed Gardner that the ice had begun thinning \u2014 and quickly.\nFrom 2004 to 2006, these glaciers and ice caps were losing around 30 billion tons of ice per year. Over time, Gardner found, the rate of ice loss was speeding up. Between 2007 and 2009, for instance, these same glaciers were losing roughly 90 billion tons of ice per year.\nBy comparing the summer temperatures to the amount of ice lost each year, Gardner discovered a staggering fact. For every 1 degree Celsius (1.8 degree Fahrenheit) of added summer warming, the region was losing and extra 64 billion tons of ice per year. That showed this region had become \"incredibly sensitive to a small change in temperature,\" he says. He finished his PhD in 2010 and is now continuing this work at the Jet Propulsion Laboratory. It's at the California Institute of Technology in Pasadena.\nThe Arctic is warming several times more quickly than the world as a whole. And Gardner's data suggest that as the Canadian high Arctic continues to warm over the next 70 years, its rate of ice loss could easily double or triple.\nThe Karakoram (Kaer-uh-KOR-am) Range spans the borders of Pakistan, India and western China. It is home to hundreds of glaciers and the world's second-highest mountain: K2. Its peak reaches up 8,611 meters (28,251 feet). Despite rising temperatures and rapid melting, Karakoram's glaciers don't seem to be shrinking. Gardner suspects that higher temperatures may be evaporating more water and sending more clouds over the mountains. These increased clouds are dropping more snow on the glaciers. And this could compensate for the effects of local melting \u2014 at least for now.\nBut mountains farther north, the Tien Shan of western China, have been losing ice quickly. Between 1961 and 2012, their glaciers collectively lost a sixth of their area \u2014 and one-fourth of their total ice!\nGlaciers don't seem to be shrinking here, in the Karakoram Range (a northwestern extension of the Himalayas). These mountains sit at the borders separating Pakistan, India and China.\nEven during dry seasons, these glaciers feed rivers. Millions of people across the region depend on those rivers to irrigate their crops when rain isn't falling.\nThough very important to downstream communities, such small mountain glaciers are, in some ways, harder to monitor than the vast ice sheets of Greenland and Antarctica. Many of the satellites that measure changes in the ice sheets don't work well on small glaciers. Gardner relied on IceSat to keep track of these glaciers. Its narrow laser beam was perfect for measuring the height of small areas of ice. Alas, in 2009 the satellite stopped working.\nAtlantic One of the world's five oceans, it is second in size only to the Pacific. It separates Europe and Africa to the east from North and South America to the west.\ndownstream Further on in the direction in which a stream is flowing or the path at which stream water will flow in its trek to towards the oceans.\nGrand Canyon A natural canyon in northwest Arizona that formed as the Colorado River cut through the rock here over the past 5 million to 6 million years. This is one of many canyons on the river, which drains water from seven states. The Grand Canyon is 446 kilometers (277 miles) long. Its depth varies. At its deepest point, the river is 1,829 vertical meters (6,000 feet) below the upper rim. From rim to rim, the canyon's width also varies \u2014 from about 16 kilometers (10 miles) to 29 kilometers. Of this impressive national land formation, more than a million acres (4,931 square kilometers, to be exact) was turned into a U.S. national park in 1919.\nGreenland The world's largest island, Greenland sits between the Arctic Ocean and North Atlantic. Ice covers roughly 80 percent of Greenland. Indeed, this island's ice sheet is the world's largest. If its frozen water were to melt, it could raise sea levels around the world by 6 meters (about 20 feet).\nirrigate The supply of water to land or crops to help plant growth.\njet stream A fast-flowing, high-altitude air current. On Earth, the major jet streams flow from west to east in the mid-latitude regions of the Northern and Southern Hemispheres.\nrange The full extent or distribution of something, such as the full length and breadth of a line of mountains.\nslope (in geology) The steeply pitched side of a cliff, hill or mountain. (in mathematics) The degree to which some line rises or falls from a strictly horizontal direction. A line that appears to rise as it moves to the right has a positive slope. One that appears to fall as runs to the right has a negative slope. Vertical lines have neither. Their slope is described as undefined.\nterrain The land in a particular area and whatever covers it. The term might refer to anything from a smooth, flat and dry landscape to a mountainous region covered with boulders, bogs and forest cover.\nJournal: F. Brun et al. A spatially resolved estimate of High Mountain Asia glacier mass balances from 2000 to 2016. Nature Geoscience. Vol. 10, August 7, 2017, p. 668, doi: 10.1038\/ngeo2999.\nJournal: C. Mortimer et al. Glacier surface temperatures in the Canadian High Arctic, 2000\u201315. Journal of Glaciology. Vol. 62, July 18, 2016, p. 963, doi: 10.1017\/jog.2016.80.\nJournal: D. Farinotti et al. Substantial glacier mass loss in the Tien Shan over the past 50 years. Nature Geoscience. Vol. 8, August 17, 2015, p. 716, doi: 10.1038\/ngeo2513.\nJournal: S. Kapnick et al. Snowfall less sensitive to warming in Karakoram than in Himalayas due to a unique seasonal cycle. Nature Geoscience. Vol. 7, October 12, 2014, p. 834, doi: 10.1038\/ngeo2269.\nJournal: A. Gardner et al. A Reconciled Estimate of Glacier Contributions to Sea Level Rise: 2003 to 2009. Science. Vol. 340, May 17, 2013, p. 852, doi: 10.1126\/science.1234532.\nJournal: A. Gardner et al. Sharply increased mass loss from glaciers and ice caps in the Canadian Arctic Archipelago. Nature. Vol. 473, May 19, 2011, p. 357, doi: doi:10.1038\/nature10089.\nJournal: A. Arendt et al. Rapid Wastage of Alaska Glaciers and Their Contribution to Rising Sea Level. Science. Vol. 297, July 19, 2002, p. 382, doi: 10.1126\/science.1072497.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Is CBD? Is CBD Safe? Why Global CBD?\nCBD is short for Cannabidiol and belongs to the class of 113+ known cannabinoid compounds found amongst a total of over 400 currently identified compounds in the cannabis plant. Although CBD is non-psychoactive, meaning it doesn't make people feel high like THC does, it is causing excitement among scientists, health professionals, and medical marijuana patients who are adopting pure CBD products to treat a wide range of conditions.\nOur CBD Oils and Extracts come from organically grown, male agriculture hemp stocks and stems, derived from the raw product and 99% pure with absolutely NO THC. The hemp then undergoes supercritical CO2 extraction in order to organically isolate the CBD from the plant.\nWe are best known for our Nano CBD product line. Since CBD is a multi-molecule cluster, our Nano products come from separating the CBD clusters and spinning them into individual Nano molecules. This makes the product much more Bioavailable and thus more quickly and easily absorbed. A Nano product of 150 mg is equal to 1,500 mg of a clustered molocule product. This means the suggested serving size is much less and a single bottle lasts much longer.\nAll of our CBD Products at GlobalCBD.com are 3rd Party Tested. Our CBD options are products we use ourselves, whether it is a product for wholesale CBD vendors, CBD distributor activity, or for our own Global CBD and Hemp online store.\nWe do it all. Wholesale, Distributor, & CBD Affiliate programs. We feature CBD oil, hemp oil, edibles, tinctures, bottles, and CBD training. We also offer CBD Massage oils and balms.\nWe only feature the highest quality CBD and Hemp Skincare, Edibles, Beverages, Isolates, Distillates, Oils & of course CBD honey!\nOne of our favorite products is CBD Pet Care for dogs, cats, lions, tigers and bears. We use the highest quality Nano CBD isolate for animals.\nWe love that we have the highest pure CBD Nano Hemp Oil available, ensuring maximum bioavailability. You'll find our Nano CBD in our tinctures, bottles, and as we said, inside our CBD pet line.\nFinally, we always end on CBD Chocolate and honey because it's just so darn good!\nWHAT IS GLOBALCBD.com's AFFILIATE PROGRAM? WHOLESALE PROGRAM?\nWHOLESALE PROGRAM FEATURING ISOLATE, DISTILLATE, & WATER SOLUBLE CBD PRODUCTS!\nGlobal CBD is a leader in bulk wholesale isolate, distillate, and water soluble products. We also have wholesale and distributor pricing for companies\/individuals looking to expand or enter into the CBD industry and are looking for cutting edge products. We have multiple channels for white label and private labeling.\n10-100 kilos is a beautiful place as well, as we have a special pricing structure and can negotiate, ensuring you can get started or maintain your supply of the highest quality CBD isolate, distillate or water soluble CBD. We look forward to getting to know you. Once set up, we can also get you your own negotiated ordering portal to make everything easy for all your CBD wholesale needs.\nOn all bulk wholesale orders a Proof of Funds (POF) is required prior to the order.\nGlobal CBD INC is a full service wholesale distributor and Isolate distributor. We will work with companies to create or build premium products from pure 99% CBD isolates. We choose to work with THC free isolate or full spectrum without THC to protect against liability of failed employment drug tests and a consistency of product quality. Our affiliate program offers extensive training, marketing materials, continued education and full time support with your own personal custom portal access.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ralph Nader: United States an \"Oligarchy-Plutocracy\"\nNader detailed a number of relatively recent co-sponsored conservative\/liberal initiatives including the Ron Paul\/Barney Frank call for defense cuts, the Rand Paul\/Ron Wyden filibuster demanding transparency on the drone program, and the proposed Walter Jones\/Dennis Kucinich legislation to amend the War Powers Act. Further evidence of right-left coalitions include the successful avoidance of pre-emptive airstrikes on Syria, the bipartisan rebellion against NSA blanket surveillance on the telephone\/email communications of American citizens, and the joining of conservative and liberal minds to push comprehensive prison reform and drug policy overhaul.\n\"Unstoppable\" introduces readers to an incredible group of Depression-era concerned citizens who likewise recognized and wrote of the malignant potential of power centralization and resulting undue corporate influence. Nader describes the enlightened insight of this eclectic group of author-activists: \"They were artists, writers, poets, workers, farmers, and they called themselves decentralists\u2026 I featured them in my book because I have never come across a more powerful critique of the corporate state\u2026they weren't armchair philosophers\u2026they weren't part of cushy think tanks in D.C. or New York\u2026they lived among the dispossessed and the deprived in the horrible depression and poverty of those years.. The amazing thing about it is they weren't taken in by government regulation. They thought government regulation would be taken over by the corporate lobbyists and turned against the people- which has happened.\" In the same spirit that brought together such a diverse group of decentralists, Nader believes that a coalition of left and right can, and must, unify today in opposition to corporate appropriation.\nGiven the increasing mistrust of American leadership by the public, Ralph Nader's message of bipartisan cooperation is more relevant than ever. Encouraged by cooperative left-right victories, Nader looks to the 2016 election and possible independent-thinking contenders such as Elizabeth Warren, Bernie Sanders and Sherrod Brown to spur meaningful discourse and embolden politicians to act on behalf of constituency over loyalty to corporation or party affiliation.\nKevin Patrick Kelly is a university student majoring in History and Political Science. Previously, he was a columnist for Washington Times Communities. Follow him on Twitter @TheKevinPKelly.\nUntold History does not subscribe to any specific agenda and seeks to promote independent commentary about history and current affairs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Austin is a broadly experienced project manager having previously worked for a tier one mechanical services contractor and as an assistant contracts administrator for a tier 2 builder, as well as roles as a drill hand for an HDD rig and roles for commercial fit outs. Similarly, Austin has been a Quality Assurance Officer where scoping, budgeting and management for roll outs of business improvement initiatives nationally across Australia has meant his experience extends beyond the construction industry. This site based experience, combined with a Masters in Project Management, Bachelor of Construction Management, Diploma of Project Management and Cert III of OH&S creates the core grounding for a project manager that can work across a variety of sectors and industries with organisations that have unique requirements.\nAustin's site based experience is invaluable when it comes to handing over extremely complex building services with staged handovers of priority areas including data centres and commercial kitchens for a base hospital. Furthermore, handover of mechanical plantrooms, general hospital wards and even a psychiatric ward prove to be important contribution to the company's diverse knowledge base.\nAustin's attention to detail, approachability and determined attitude as well as his site based, hands-on experience and academic credentials make him well placed to add value to client projects across multiple sectors with varying size and requirements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tennessee Property Management Companies directory (www.trexglobal.com\/property-management\/property-management-companies\/tennessee-property-management-companies) is a one stop shop for all your property management needs. Are you looking for a property management company in Tennessee for any of these services: single family property management, condo property management, townhouse property management, vacation property management, multi-family property management, association property management, homeowner association property management, condo association property management, commercial property management, warehouse property management, retail property management, office property management, or hotel property management? Find a property management company in Tennessee that exclusively serves your type of property in Tennessee. Our comprehensive database of property managers and association managers is used by thousands of property owners and associations every day. Tennessee Property Management companies that want to be found online by property owners advertise their services at www.trexglobal.com\/property-management\/property-management-companies. www.trexglobal.com offers the simplest property management company search. Property owners and associations in Tennessee can browse from a comprehensive list of property management companies in their city for free. A detailed list of Tennessee property management companies with their website and contact details can be found at www.trexglobal.com\/property-management\/property-management-companies\/tennessee-property. Owning a property is time consuming and comes with the responsibilites of renting it out, maintaining it, paying utilities and taxes. Access our free property management resources at www.trexglobal.com\/property-management to help you understand how property management companies can help you save money. Hire your property management company in Tennessee using www.trexglobal.com.\nThe VehicleManager random vehicle \/ car module help you show random vehicle \/ car by different parameters.\nManual Software Submission services provide secure way to improve ranking of commercial websites in less time. Data doctor offers advance services of Software Submission at reasonable price for enhancing website popularity among visitor clients.\nHotel Reservation System Lite Version is an easy-to-use reservation management software that can manage your hotel or motel. Ideal for reservations, hotels and guest houses. Track Customers and avoid Double Bookings.\nSearch engine optimization technique raises your website rank in search engine results. Professional SEO expert services helps in growing your business with specific set of keywords to divert higher amount of traffic towards your software website.\nComputer Fillable Medical PDF Forms. Patient records can be typed, e-mailed, saved and stored on your computer. Easy access and retrieval. Customized forms are available. E-mail us yours for a free quote today. Newest technology and easy to use.\nThis site does not contain serial number, crack, keygen Tennessee Property Management Companies serial number : serial number is the simplest type of crack. A serial number(cdkey,product key,etc) is register the program as you may already know. a crack or keygen may contain harmful software. If you cannot find the exact version you want, just give it a try. for more information read .nfo\/.txt\/.diz file include in zipped file.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Point of sale marketing materials are the final stage in a marketing campaign and an effective way of prompting purchase. They are an essential marketing tool for any brand, reinforcing campaign and brand messages right at the point of purchase.\nInnovative, eye catching point of sale print displays make the most of in-store opportunities for brands, and create an impactful retail experience for shoppers.\nOur team has years of experience in creating quality point of sale marketing materials and creative packaging to capture the attention of your target buyers.\nUsing specialist cardboard engineering combined with excellent print, we offer POS display solutions such as; standees, floor units, pallet units, half pallet, pallet unit with die-cut end caps, table top displays and more.\nComplementary products such as box mailers, leaflet holders, die cut stand up characters, paper carrier bags, even apps and NFC communications can be incorporated into our project management.\nWhatever your point of sale needs, we can bring your brand to life just where and when you need it!\nContact us for an informal discussion, whatever your requirements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lindsey Mann is strongly influenced by her childhood, and with her fascination for old tools produces these imaginative wearable tools.\n\"I have created my own series of tools for coping with everyday scenarios and to help turn an awkward social situation on its head. The idea of spreading a little joy and happiness rather appeals.\"\nI absolutely love that extractor fan brooch.\nMe too,Her work is lovely .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahdge b/data_all_eng_slimpj/shuffled/split2/finalzzzahdge
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+{"text":"Being key members of the family, your pets are more than any trusted fellow companions you can ever have. It is a sad fact that they are no longer with you, but you need to put a stop at this suffering. So it is important to understand that as much as we human requires a cremation ceremony, it is necessary to do the hold the same for the pets. Most of the families believe that cremating a dear pet is a basic human need to memorialise the beloved for a lifetime. High-quality dog and cat cremation urns for ashes is an ideal kind of memorial for someone who has ever owned a pet in their life and wants to memorialise their pet's last cremated remains. These act as a true gentle reminder of the love and bond you have shared with the pet.\nUltimately, dog urns and cat urns for ashes is a final resting place of your beloved pet for eternity now. Being one of the most comforting ways of giving a tribute towards a special long lost friend, the urns used to commemorate the life of a pet can help in making the grieving part easy. Dog urns for ashes are getting very popular among those who wish to keep a small and handy reminder of their pet with them all the time. The urns for pets are available in an assortment of designs, colours and shapes, to accommodate almost all kinds of pet breeds weighing different. Finding the right urn according to your needs is really important because most of the urns made for pets vary in their capacity, depending on the pets they are usually made for.\nCelebrate all the loving years spent with your dear pet with these pet urns that will make up for that one lasting memorial to be appreciated forever. DELUXE PAINTED CREMATION ASHES PET COFFIN and CORBY BRONZE CAT CREMATION ASHES SCULPTURE URN are two intricately designed cremation urns for pet ashes that are perfect to be used in the remembrance of the pet. They are suitable to be displayed at your homes because clearly, they are representing the pet in memorial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Explore the fragrances and eau de colognes of Johnny Was. A good fragrance takes you back to a place you once were. It doesn't overpower the people around you, it accentuates your natural scent and essence. Be sure to spray on your skin or hair, not your clothes as many of our fragrances are made with 100% natural plant ingredients and may stain. These extracts are combined with a harmonious accompaniment of Essential Oils, Resins, and Absolutes to form a simple, sophisticated finished fragrance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PCP is a synthetic drug sold as tablets, capsules, or white or colored powder. It can be snorted, smoked, or eaten. Developed in the 1950s as an IV anesthetic, PCP was never approved for human use because of problems during clinical studies, including intensely negative psychological effects.\nHow Does PCP Affect The Brain?\nThe use of PCP as an approved anesthetic in humans was discontinued in 1965 because patients often became agitated, delusional, and irrational while recovering from its anesthetic effects. PCP is a \"dissociative drug,\" meaning that it distorts perceptions of sight and sound and produces feelings of detachment (dissociation) from the environment and self.\nMood disturbances: Approximately 50 percent of individuals brought to emergency rooms because of PCP-induced problems (related to use within the past 48 hours), report significant elevations in anxiety symptoms.\nPCP, developed in the 1950s as an intravenous surgical anesthetic, is classified as a dissociative anesthetic: Its sedative and anesthetic effects are trance-like, and patients experience a feeling of being \"out of body\" and detached from their environment. PCP has been used in veterinary medicine but was never approved for human use because of problems that arose during clinical studies, including delirium and extreme agitation experienced by patients emerging from anesthesia.\nAt low-to-moderate doses, physiological effects of PCP include a slight increase in breathing rate and a pronounced rise in blood pressure and pulse rate. Breathing becomes shallow; flushing and profuse sweating, generalized numbness of the extremities, and loss of muscular coordination may occur.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This diagram shows the percentages of websites using Sun-shop broken down by site elements. See technologies overview for explanations on the methodologies used in the surveys.\nSun-shop is used by less than 0.1% of all the websites whose content management system we know and that use CSS as site element.\nYou can find ranking, usage and market share data for Sun-shop within all site elements in our Sun-shop market report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"<\"Eileen\" At The Victor Herbert Renaissance Projekt Live!\nCommissioned by producers Julian Mitchell and Fred Hamlin to follow up their 1902 success with the fairytale spectacular The Wizard of Oz, Babes in Toyland was purposefully constructed on similar lines to the earlier show. It premiered at the Grand Opera House, Chicago, on 17 June 1903. The Broadway premiere was at the Majestic Theater, on 13 October 1903.\nGlen MacDonough's libretto followed little Jane (Mabel Barrison) and Alan (top-billed William Norris) through the colourful realms of Toyland as they struggled, with the help of the familiar characters of children's storybooks, to outwit wicked, miserly Uncle Barnaby (George W Denham) and his allies, the Master Toymaker (Mark Smith) and Contrary Mary (Amy Ricard), and win their way to a happy ending. The production, like its predecessor, was staged with lavish scenic effects, from the prologue's shipwreck (paralleling The Wizard of Oz's tornado) through all kinds of picturesque fairytale venues including a country f\u00eate in Contrary Mary's Garden, the Spider's Forest, the Floral Palace of the Moth Queen, Toyland's Christmas Tree Grove and the Master Toymaker's workshop and castle, to the final Palace of Toyland and its Court of Justice.\nAlongside its spectacle, the other principal attraction of Babes in Toyland was its music. Victor Herbert provided a score which was very much in advance of the one which the producers had cobbled together for The Wizard of Oz. It had plenty of very fine large-scale and orchestral numbers \u2013 notably the enduring 'March of the Toys' and its succeeding 'The Military Ball' \u2013 but also some charming and delicate musical moments ranging from the lovely trio 'Go to Sleep, Slumber Deep' (Alan, Jane, and a soprano fairy w chorus) and the sweet 'Never Mind, Bo Peep (we will find your sheep)' sung by Tom Tom (Bessie Wynn) and the widow Piper's other children to the culpable little shepherdess (Nella Webb), to the chirpily childish 'I Can't Do That Sum', added to the score after opening for Miss Barrison, and Tom Tom's dreamy hymn to 'Toyland'. There were occasional more obvious spots \u2013 if nothing quite as tacked-in as Lotta Faust singing 'Sammy' in The Wizard of Oz \u2013 notably when Contrary Mary sang fairly irrelevantly about 'Barney O'Flynn', but on the whole Herbert's score hit the medium line between cultured fairy play and pantomime jollity to perfection.\nBabes in Toyland had a highly successful Chicago season of 117 performances (13 weeks) and played through 192 performances at New York's Majestic Theater before going on the road, establishing itself as an enduring favourite of its kind throughout America \u2013 sufficient of a favourite, indeed, that it became the subject of a Broadway lawsuit. This one, however, was settled with full logic. In the face of defence testimony from Mitchell, the Hamlin brothers, Ben Teal, MacDonough and others, one loopy Mrs Riley who had claimed that she was the show's real author got short legal shrift.\nBabes in Toyland returned briefly to the Majestic in 1905 (2 January, 25 performances) and in 1929 (Jolson Theater 23 December) the Shubert brothers brought the show back to Broadway for a Christmas season, and although it has not had a career outside America, versions of Babes in Toyland (sometimes textually quite remote from the original, but always staged with 'grand spectacle') have been played regularly throughout the country since the first production.\nIt has also been filmed twice, once with a cast including Laurel and Hardy (1934) and once by Walt Disney (1961) in widely variant versions, and was also televised in variously unfaithful forms 1950, 1954, 1960, 1968 and in a 1986 NBC version in which only 'Toyland' and 'The March of the Toys' of Herbert's score survived.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"World Class Customer ServiceJGBM offers a 'Premium' and proactive distribution model. 'Stress free' purchasing - your calls answered in 3 rings, product advice from the industry's recognised experts, next day delivery, returns collected next day. Nothing is too much trouble, really!\nThe Industry's Best Product DataJGBM Product Data feed will maximise your Machines sales. The industry's most up to date datafeed ensures you're back office is always quoting the latest products at the latest prices. We upload data every single day!\nAward winning innovation.Ask about your own 2020Pro dealer branded website http:\/\/2020prosoftware.com. Sell more machines with our range of Six of the Best end-user flyers, our TechStuff! e-catalogue and our CD-ROM product selling tool!\nStock AvailabilityJGBM enjoys considerable and preferential stock supply from its 30 manufacturer partners, available with next day delivery on orders placed by 5pm - delivery free of charge for orders over \u00a3100. Plain-label end user delivery available too!\n\"You have amazing customer service, always friendly and always there to help. You can even hear the smile through the line :)\"\n\"The service we receive is fantastic! the team are very friendly, efficient and extremely knowledgeable.\"\n\"Just very happy with everything, sales team are great, deliveries are spot on and your marketing is second to none!\"\n\"JGBM is one of my favourite companies to call. Friendly and playful even, but you are also very well informed. Its seems that you always have the answers to my questions, in fact offer me expanded information that my customer would benefit from, which I have forgotten to ask.\"\n\"Out of all the suppliers I deal with on a day to day basis, JGBM are by far the most straightforward company to work with. Orders are processed straight away, notifications and invoices sent without fail, always next day delivery and never any problems. On the odd occasion where we have had a faulty product I used their online RMA system, it was simple and effective and someone even give me a courtesy call to make sure everything was OK with the return. Tops marks to every one at JGBM, I couldn't recommend you highly enough.\"\nOur World Cup Caption Competition \u2013 Final Week!\nOur World Cup Caption Competition \u2013 Week 3!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"San Francisco, CA, January 12, 2018 --(PR.com)-- Innovaccer Inc., a San Francisco-based healthcare data platform company, will be launching InAssist, a new point-of-care alerts solution at HIMSS 2018 this March. The solution will stream in information from EMRs, recognize patients that need attention, and offer actionable insights to healthcare providers - all without requiring physicians to leave their EMR experience. The solution will be compatible with all browser-accessible EMRs.\nAs healthcare becomes increasingly data-centric, providers are seeking to integrate data-driven insights into their practice while avoiding further complicating operations. Pulling data together from different sources to identify the best treatment regimens can often be time-consuming, resulting in physician burnout and limiting patient interaction time. Additionally, integrating disparate data sources manually can often create substantial gaps in care as EMR information at times does not align with patient priorities and clinical workflow.\nInAssist leverages advanced algorithms to integrate EMR data with other data sources in order to assess and surface care gaps, risk scores, disease registries, clinical documentation gaps, and other relevant information available for a patient. InAssist presents this crucial information on-screen in simple, real-time alerts as care team members work within EMRs. It also synchronizes with the providers' calendar and presents upcoming appointments via timely reminders. The application runs with all the browser-accessible EMRs with no prior configuration and recognizes patients in need of immediate attention.\nThe alerts solution can also be integrated with other data systems using RESTful APIs leveraging Innovaccer's InData platform to gain access to more information.\n\"With the growing role of data and technology in healthcare, we understand the continued importance of providers spending more time with patients rather than spending half their day sorting through static files and millions of records,\" says Abhinav Shashank, CEO at Innovaccer. \"InAssist is a significant step in our commitment to making health data transparent, accessible, and impactful. We are optimistic that it will not only deliver crucial point-of-care alerts but also save a great deal of time for care teams,\" he concludes.\nInnovaccer will be launching InAssist at HIMSS 2018, being held from March 5-9, 2018 at the Sands Expo Center in Las Vegas. Join Team Innovaccer at HIMSS 2018 at booth #12752 and get a first hand demo of InAssist.\nInnovaccer Inc is the healthcare data platform company focused on delivering a more efficient and effective healthcare through the use of pioneering analytics and transparent, clean, and accurate data. Innovaccer's aim is to simplify complex data from all points of care, streamline the information, and help organizations make powerful decisions and realize strategic goals based on key insights and predictions from their data. Its products have been deployed across more than 500 locations with over 10,000 providers leveraging it at institutions, governmental organizations, and several corporate enterprises such as Mercy ACO, StratiFi Health, Catalyst Health Network, Osler Health Network, and PHIX HIE. Innovaccer is based in San Francisco with offices around the United States and Asia.\nFor more information, please visit innovaccer.com or follow us on Twitter @innovaccer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BRITS \u2013 The family of a 28-year-old man in Mothotlung in the Brits area have confirmed their relative died after service delivery protests in the area.\nLerato Seema is believed to be the third victim of clashes with police over water shortages.\nColonel Sabata Mokgwabone said the man and his friend were arrested on Tuesday for public violence in the area during a protest about water shortages.\n\"The men were put in the Nyala that was headed to the police station; one man managed to escape and the other one tried to escape, but fell from a moving Nyala,\" said Mokgwabone.\nHe said the situation in the area remained calm and police were patrolling.\nOn Monday, two people were killed allegedly by the police, during the protest.\nTheir innocent request [for water] was met with brutal murder. This is what this current government has become known for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The first time we met Dominic and Dorothy it was obvious that they are truly in love. The way they talk and treat each other made us feel like we were watching a scene from an Asian love story. We can just feel their affection for each other. From the start we understood it is their dream to have a perfect wedding day, a day to share with their family and friends who were flying in to Auckland from overseas. The wedding was meticulously planned and everyone knew their roles and timings. They were very organised, they wanted a perfect garden wedding. Just as the ceremony was about to begin and everyone was settled in their seats, the unexpected happened\u2014it rained. The rain drove everyone indoors, away from the garden. As wedding filmmakers, it is our duty to be observant during the wedding day and we noticed guests were starting to worry about everything\u2014the weather, the ceremony, the rest of the day. Naturally, we expected the bride and groom to worry the most. But out of all the people on that day, these two never showed any sign of anxiousness, not even the slightest hint of stress. We saw in their eyes that the rain didn't matter, that to them what mattered most is they're marrying each other on that day. And so everyone took their place with Dominic standing firm, the music played and no one could have walked down the aisle of that cramped little room as elegantly as Dorothy. It was a very beautiful wedding and no amount of rain could have stopped it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This function first operates the same as a call to ifDraw2D.Finish(), completing all queued drawing operations on the back buffer (draw surface).\nIf the screen is single buffered, SwapBuffers() returns immediately after this operation.\nIf the screen is double buffered, SwapBuffers swaps the back buffer with the front buffer, so the back buffer is now visible. The new back buffer should be assumed to be in a garbage state after this call is complete, which means you will need to re-render the entire frame before a subsequent call to SwapBuffers. This call will not return until the back buffer is ready to be drawn on to. Depending on the implementation, it may take up to a single video frame period for the new front buffer to become visible.\nThis operation is extremely fast (that is, it never copies a bitmap from one location to another), and is guaranteed not to \"tear\" the visible image.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahkvp b/data_all_eng_slimpj/shuffled/split2/finalzzzahkvp
new file mode 100644
index 0000000000000000000000000000000000000000..212a17748a5abada5e32c490671ba15fef77f4ef
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahkvp
@@ -0,0 +1,5 @@
+{"text":"If you're in the market for a new or used car, you've come to the right place. The staff at Maguire Volvo Cars of Ithaca is committed to helping you find the right vehicle for your needs. What's more, they're also dedicated to helping you maintain it long after you drive it home for the first time.\nSo where do we begin? Most like to take a look at the latest Volvo models to hit our showroom, including the Volvo XC60, XC90, S60 and V60.\nOf course, whether you choose new or used, Maguire Volvo Cars of Ithaca staffs a team of expert auto finance specialists who can help you find the right loan or lease for your needs. From start to finish, we're by your side to make the buying process as easy as possible.\nAnd after you've purchased your next vehicle, Maguire Volvo Cars of Ithaca doesn't go anywhere. Instead, we continue to support you with an onsite service department, which operates using only certified Volvo parts - a combination that's sure to result in many worry-free miles in your vehicle's future.\nWhatever your automotive needs may be, Maguire Volvo Cars of Ithaca is here to serve you. Located at 370 Elmira Road in Ithaca, New York, we're a quick drive away from the surrounding Dryden, Cortland, Elmira and Corning, NY, areas. Come see what we can do for you today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quick, hassle-free pest and termite control for your home and yard.\nYour family and pets are our top priority. We use family-friendly products that pass our strict internal safety measures.\nWe're locally-owned and we've been providing quality pest control in Maryland & Virginia for over 25 years.\nGet pest control service you can trust! We're backed by Angie's List, the BBB, and great customer reviews.\nMy neighbors recommended B.O.G. Pest Control, and they've exceeded my expectations. Not only did they get the job done - their customer service blew me away!\nWe love calling Maryland and Northern Virginia home, and we're proud to offer quality pest control to local homeowners. We've been bringing fast, expert extermination services to homeowners just like you for over 20 years. From effective preventative treatments to extermination - we have a pest professional ready to take care of your needs! Your safety and the safety of the environment is always our top priority, explore our family and eco-friendly services below.\nLooking to get rid of your current pest problem? We have the solution! At B.O.G. Pest Control, pests have no place in your home, but neither do harmful pesticides. We create individualized plans designed to get rid of pests from your property, while also using the safest products available in the industry.\nDid you know termites cause $5 billion worth of home repair each year in the U.S. alone? If you suspect termites or want to prevent them in the future, we will immediately treat your property to keep your home termite free, guaranteed.\nTired of pesky pests ruining valuable time you spend in your yard? Don't let mosquitoes, fleas & ticks come to a party they weren't invited to- let us manage them for you so your yard and life can be free of pesky outdoor pests. At B.O.G. Pest Control, we provide treatments that protect your yard against mosquitoes, fleas & ticks quickly and effectively!\nDid you know fleas and ticks can carry deadly diseases? These pests have no place in your life, lawn or bothering your children or pets.\nOur trained technicians tailor a treatment specific to your property, identifying risk areas for breeding and local harboring sites so you can be worry free and protected all year long!\nThe last guest you want to see scurrying around your kitchen is an ant! Let B.O.G. Pest Control completely solve your ant problem so you can spend more time in the kitchen, and less time worrying. We create a custom design to effectively eliminate ants from invading your home.\nMoles and voles can cause serious damage to your property. It is time to protect your lawn from unwanted guests! At B.O.G. Pest Control, we understand every mole\/vole infestation requires a proper, personalized treatment to ensure elimination. Let us give your property a detailed inspection so that you can say goodbye to these rodents for good!\nInsect stings can be painful and dangerous to the safety of your home. Don't put you or your family at risk by trying to remove nests yourself. Instead, let our highly-trained team safely remove these stingers' homes from your property for you. We also provide fast, effective pest control treatment to your property to keep wasps, bees, hornets, yellow jackets and more away from your home for good.\nDreaming of having the best looking lawn in the neighborhood? Let us turn your dream into a reality! At BOG Pest Control, we work with you to design a unique program to meet all of your lawn's needs. Our professional team treats your lawn like their own and are dedicated to helping your lawn reach its full potential.\nFirst impressions matter, especially when it comes to your business. Let us create a safe and welcoming place for your business by eliminating unwanted pests for good. Our team works hard to take care of all your pest needs, so you can work hard to focus on your business.\nStink bugs hide in curtains, bedding, along windows, in the attic and walls. They gain entry from cracks and crevices found around the exterior of your home. To avoid severe infestation, a thorough inspection of the interior and exterior is recommended. Let us help remove stink bugs before it gets worse!\nWhen you purchase a complete mole or vole program, you will get $50.00 off your first treatment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(EDIT: Ha, made the runners up for submissions for the day, cheers guys)..\nFast paced and frantic gem stacking game with a twist! Rotate the field to form groups of same colored gems, and click them to remove them! Play in Open Play, Obstruction mode or Timed mode.\nEdit: Don't forget to use the powerups when you reach 50 of each gem type by clicking on the \"ready\" icon.\nI really haven't enjoyed a simple game like this in a long time. The only problem is that the graphics are stale and there aren't any levels or music changes. It's also a little too random. Sometimes you just cant make any moves even if you turn it 50 times.\nIt was good, but sometimes later in the game a glitch would occur where some lines wouldn't fall down all the way after I turned it. Then when I turned it again they would all fall. I was in free play, not obstruction mode. Also, the time between lines falling stopped completely (which was good because it let me clear the gems) and I had to hit the drop lines button....other then that I would suggest putting a picture of which gems are coming next in the game. Fun though.\nI have had this very rarely on the game .. I thought it was fixed after the beta testers pointed it out .. Not sure if it is a flash version issue or not ... Will look at it again.\nVery chill and relaxing, with an innovative design. Loved the sound and overall was a good game. Keep it up!\nIt was very basic but the concept was amazing! It was very fun to play and I'm sure that it will become a future front-pager or my name isn't Demnoobshow! Again great game.\nOoooh I do hope it makes the front page, I know I got runner up status today.\ni feel good when i play a future front pager!\n\/me keeps fingers crossed ..\nThe match 3 game with a real twist is back!\nBeat your opponent by swapping the tile colors.\nThe blobs are back and it's your job to remove them from the island! With bombs of course.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As has become tradition, the Christian Dior couture show took place at Paris's Mus\u00e9e Rodin, with the collection itself shown inside a large, white cube within the grounds. Guests entered through the museum, walking past Rodin's The Kiss and beds of roses crisping up in the July heat.\nInside, bright lights lit up the space, while large display units containing various white tailored garments stretched from floor to ceiling. The overall effect was the intimate feeling of an atelier \u2013 and one which casted us, the guests, as clients at a fitting. If last season's interior (and collection) was themed around fantasy and nature, this season's has been designed to be more real, more behind-the-scenes \u2013 and more grounded.\nTwo years ago Maria Grazia Chiuri, the artistic director of Dior, created a T-shirt emblazoned with the message \"We should all be feminists\" for her first ready-to-wear show. If fashion echoes the mood of the times, then she was up to speed again with this collection, albeit within the confines of couture.\nThe question hanging over this collection's concept \u2013 \"Is it possible to celebrate haute couture while also offering a critical reading of it?\" according to the show notes \u2013 was arguably controversial given the astronomical prices of couture. Still, she concentrated on answering it. First by suggesting couture is \"a concept\" and by suggesting it could be modernised, either using lighter, more wearable fabrics, all the while using the clothing exhibited on the walls as reminders of how the pieces came to be.\nAnd so under a mirrored ceiling that stretched the length of the room \u2013 which conversely became well utilised on social media \u2013 she sent out a collection that started with a classic couture structure and ended with newness and fluidity.\nThe first look, a midnight blue three-piece, herringbone suit, with a spin on the famous Dior bar jacket (cinched waist, padded hips and \u2013 update \u2013 batwing sleeves) set the tone. Tweed coats and leather berets with tightly wound ponytails peeking out, alongside form-fitting strapless dresses in chiffon and crepe, which came in blue and taupe. Sultrier greens and blacks and gold gowns and coats followed, arriving in different textures, some covered with tapestry fabric, others with flowers in relief.\nAs the show continued, the skirts grew fuller and the fabrics became more modern, with gowns switching from chiffon to lace, to scuba. The final three dresses were jewel-coloured, strapless dresses \u2013 Gallic evening wear as imagined by Dior \u2013 except made from modern-sounding silk scuba. \"The clothes maintain the shape but actually, they are different. It is hard to see,\" she said. For one, these final three pieces were light, made from one piece of fabric and built without lining.\nThe collection tallies with a new exhibition opening next spring at the V&A \u2013 Christian Dior: Designer of Dreams \u2013 which traces the history of the couturier and those that followed \u2013 Chiuri included \u2013 from 1947 to the present day. A fan of fashion exhibitions (\"I wept at Frida Kahlo!\" she said), Chiuri is heavily involved in the exhibition which is moving from a stint at the Museum of Decorative Arts in Paris.\nShe described herself as \"not from the satin generation\" of her predecessors and discussed the way women wear couture. As is tradition, the collection was inspired by a woman \u2013 this time the cultural critic Alison Bancroft, who views fashion as art and explores what style says about identification and femininity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where students gather to perform, study, play, improve, prepare for auditions, hang out, and hear the finest musicians.\nTo see the schedule for the year, go to Calendar.\nTo sign up for courses, go to the Join Us page for instructions and then print out and complete an application form.\nFlute Lessons \u2013available on site (Saturdays and weekdays) for students who are not currently studying weekly with one of Pittsburgh's wonderful, numerous teachers. Private lesson calendars, fees, and policies are set by the INDEPENDENT teacher.\nPiano Lessons \u2013available on site (Saturdays) for flutists who plan to major or minor in music at the college level. Also available for siblings. Tuition, policies and calendar are determined by the INDEPENDENT teacher. Beginning August 27, 2011 (2nd semester: January 7, 2012).\nChamber Ensembles \u2013 Flutists are placed into a small chamber ensemble for coaching and public performances. Held weekly by appointment. Flute Academy students may also elect to play in small groups on their own without formal coaching. However, you must register for a room.\nMusic Theory Beginning through Advanced \u2013Sections for beginners through advanced placement, offered each semester and includes ear training. Designed for any musically literate student to prepare for college auditions and placement exams. Held weekly if enough students register.\nFlutations \u2013 A large, high-energy flute choir for high school students with weekly rehearsals from 11:00 \u2013 12:00, public performances, and semester-end concerts. Bass, Alto, and Contrabass flutes provided. Concert attire: All black plus Flute Academy Shirt (order during registration). Bring a MUSIC STAND! Busy marching schedule? The Flute Academy will work around it. JOIN US! Field Trips and guest artists will be announced throughout the year. Past performances include: Fiddlesticks and the Pittsburgh Symphony, world premier piece with Sir James Galway and the PSO for the PSO 2006 opening concert, National Flute Association convention Showcase Concert in 2006 and the NFA convention in NYC in August 2009. We will play in Heinz Hall lobby on October 28-29 prior to the Galway, PSO concerts. December 10: HOLIDAY CONCERTS at 11:00 a.m. AND 7:00 p.m.; SPRING CONCERT: March 26.\nFlute Loops \u2013 Middle School (grades 4-8) flute choir. No audition necessary. Two years playing experience recommended. Weekly rehearsals are on Saturdays from 11:00 \u2013 12:00. Concert attire: All Black plus Flute Academy Shirt (order during registration). Bring a MUSIC STAND! Field Trips and guest artists will be announced throughout the year. Past performances include: Mozart Room at Heinz Hall, First Presbyterian Church in Pittsburgh, and the Cultural District on PSO opening night 2006, and the NFA convention in NYC in August 2009, PSO Fiddlesticks in 2010, and we will play in Heinz Hall lobby on October 28-29 prior to the Galway, PSO concerts. December 10: HOLIDAY CONCERTS at 11:00 a.m. AND 7:00 p.m.; SPRING CONCERT: March 26.\nThe City Flutes \u2013 This highly skilled, friendly adult flute choir plays challenging repertoire (5 years of private study is the recommended minimum) and meets 6 \u2013 7 times per semester on Saturdays from 10:45 \u2013 12:30. Reasonable tuition rates encourage members to stay for years. Bring a MUSIC STAND. Concert attire is determined before each performance (includes all black and\/or Flute Academy Shirt. Past performances include: on the Red Carpet in front of Heinz Hall - PSO opening concert 2006, National Flute Association convention Showcase Concert in 2006, Tuesday Musical Solo Recital series 2008, NFA convention in NYC in August 2009.\nFall Term: September 24, October 8, 22, 29, November 5, 19, December 3 (dress). December 10: HOLIDAY CONCERTS at 11:00 a.m. AND 7:00 p.m.\nWinter term: January 14, 28, February 11, 25, March 10, 17 (dress). SPRING CONCERT: March 26. Field Trips and guest artists will be announced throughout the year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahykm b/data_all_eng_slimpj/shuffled/split2/finalzzzahykm
new file mode 100644
index 0000000000000000000000000000000000000000..d1dd24f3cc0dc1fe952ad7691e9312b33417fc53
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahykm
@@ -0,0 +1,5 @@
+{"text":"The ASNC Nominating Committee is now seeking recommendations for members to serve as a director of the Society. The Nominating Committee will submit a slate of nominations for approval by the Board, and the slate will be voted upon by the membership at the annual business meeting. Directors serve four-year terms.\nApplications are due April 19. Apply today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That is the cast of Skunk Attack.\nIn 2010, we won trivia twice.\nAnd came in second place four times.\nWho would have thought we'd be really good kinda?\nFrom it, we've gained knowledge.\nWe've participated in healthy competition.\nWe've drank three bottles of white wine.\nWe've tried Spanish tapas outside of the North End.\nAnd, most importantly, we had fun with our friends.\nHere's to Skunk Attack spraying 2011 out of the water.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dec 15, by Phil Lesh and Friends. Lazy River Road Sugaree Encore: Oct 15, by Phil Lesh and Friends. Oct 27, by Phil Lesh and Friends. Mountains of the Moon Disc 2: \u0424\u043e\u0440\u043c\u0430\u0442 \u043f\u043e\u043a\u0443\u043f\u043a\u0438 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u044c .\nLovelight Set 2: Built To Last Topic: Oct 11, by Phil Lesh and Friends. Lazy River Road Bertha Set 2: Deal Topic: Transfer by Marc Pujol: Deal Source: Oct 10, by Phil Lesh and Friends. Blue Sky Source: Soulshine, The Eleven Midnight Hour Blue Sky Topic: Oct 9, by Phil Lesh and Friends. Imagine Bird Song All Along The Watchtower Just a Little Light Tomorrow Never Knows I Know You Rider Oct 7, by Phil Lesh and Friends.\nSet 1 Disc 1 Cumberland Mines Mirror of Thalassa Bertha 8: Ship of Fools 6: Passenger 7: Front Row Mezzanine - Seat Cumberland Blues 6. Ship of Fools Disk 1 Set I: Cumberland 4. Mirror of Thalasa 6. Bertha 7. Passenger Set II: Stephen 4. Lovelite Encore: Scarlet favorite favorite favorite favorite favorite 2 reviews Topic: Neumann KM balcony, dfc. Oct 6, by Phil Lesh and Friends. DISC1 Set 1: Official Phillesh.\nCasey Jones 4. Pride of Cucamonga 5. Dire Wolf 8. Low Spark of the High Heeled Ray Ackerman, Tim Burke Source: Casey Jones 3. Pride of Cucamonga 4. Dire Wolf 6. Black Peter 3. Into The Mystic 6.\nDonor Rap 5. We Bid You Goodnight Topic: Oct 5, by Phil Lesh and Friends. Crowd 2. Viola Lee Blues V3 7. Tambourine Man 8. Viola Lee 5. Tambourine Man 7. Little Light 8. Blue Sky Set II: Playin 4. Cold Rain 6. Unbroken Chain 2.\nSugaree Encore: Phil speach 6. Disc 1 1. Viola Lee Blues 5. Tambourine Man 6. Just A Little Light Disc 2 1. Bird Song 2. Blue Sky Set 2 3. Milestones 4. In the late s, Skyy signed to Salsoul Records. Featured on this album was the single \" Call Me \", which gave the group their first and only, to date top 40 hit on the pop charts , peaking at number 26 in In the mids, the group signed with Capitol Records and released their next album, From the Left Side in Things were looking rather bleak for the band by the late s.\nHowever, after signing to Atlantic Records , Skyy launched a major comeback in with the release of their successful Start of a Romance album. By the release of the Nearer to You album in , the hits had again dried up, and the band has not released a new studio album since then. Although Skyy is one of very few bands to release 12 studio albums in 12 years two compilation albums , which turned out multiple hits, they often go unnoticed and unsung.\nUnsourced material may be challenged and removed. Limited Edition \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. Psychedelic Rock Speed: Rock Speed: \u041f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0430\u044f \u0446\u0435\u043d\u0430 ,57 \u0440\u0443\u0431. Pop Style: Britpop Speed: Pop Speed: Jazz Style: Contemporary Jazz Speed: First Pressing \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. Soul Jazz Speed: \u041f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0430\u044f \u0446\u0435\u043d\u0430 ,30 \u0440\u0443\u0431. \u041f\u0440\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. Promo Decade: English \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. Pop \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. New Wave Speed: Jazz \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430.\nRock Duration: LP Speed: Reissue \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. Rock Edition: \u041e\u0441\u0442\u0430\u043b\u0441\u044f 1 \u0442\u043e\u0432.! Greece \u0414\u0440\u0443\u0433\u0438\u0435 \u0442\u043e\u0432\u0430\u0440\u044b \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0430\u0432\u0446\u0430. \u041f\u0440\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 Speed: Experimental Rock Speed: Limited Edition Language: \u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u043f\u043e \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u044f\u043c. \u0420\u0430\u0437\u043c\u0435\u0440 \u0433\u0440\u0430\u043c\u043f\u043b\u0430\u0441\u0442\u0438\u043d\u043a\u0438 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u0432\u0441\u0435.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Commercial Driver School provide Training Programs & Services and are situated in Lakewood, Washington. For more information about the services Commercial Driver School provide please call 253-983-0200.\nDoes this listing for Commercial Driver School, Employee Training in Lakewood, WA need to be updated or deleted? Click here and tell us what needs to be changed.\nIs Commercial Driver School not what you're looking for? Use How ToCorp's B2B services to find more Employee Training in Lakewood\tand throughout Washington, as well as to find other Human Resources related businesses and organizations nationwide.\nHow To Corp's Human Resources directory aims to promote B2B sales, networking and partnerships in Lakewood, Washington. The Employee Training listings are of businesses and organizations that typically offer least several, if not all, of the following products and services: sales training, new employee training, customer service training, team building, leadership development & employee development.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HMD Global, the parent company, is expected to unveil Nokia phones in the month of June. Ahead of the official unveiling of their phones, HMD has sent out media invites for explaining the roadmap of their product lineup in India.\nThe press invite states \"HMD Global, the home of Nokia phones invites you to a conversation with Juho Sarvikas, Chief Product Officer, HMD Global.\" It's pretty much evident with the invite itself that the company is not gonna launch any phone, but instead, they will be discussing the plans for the Indian market.\nThe journey won't be an easy one for HMD Global as there are way too many smartphone brands in India right now and especially the market is dominated by Chinese brands such as Xiaomi, Coolpad, Lenovo, etc.\nIt is rumored that HMDG Global will pull the wraps of its Nokia 3310 first by the end of May. Later on, other smartphones such as the Nokia 6, Nokia 5, and Nokia 3 will be announced. There might be some details about the pricing as well on May 8.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaigpg b/data_all_eng_slimpj/shuffled/split2/finalzzzaigpg
new file mode 100644
index 0000000000000000000000000000000000000000..5e53b1fb8ac88e79e2e972a4b2e811a145e12e7b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaigpg
@@ -0,0 +1,5 @@
+{"text":"Interlaw's representative firm in Singapore, Colin Ng & Partners LLP (\"CNP\"), is advising Fund Singapore on a pre-IPO investment round of medical technology business Biolidics Limited.\nThe IPO is set to raise approximately S$6.1m in net proceeds, some of which will be used to develop Biolidics' clinical services applications and the customer segment of the company's products.\nThe fundraising will also advance Biolidics' pipeline of products through in-house development, investments, mergers and acquisitions, joint ventures and\/or strategic collaborations.\nThe CNP team acting for Fund Singapore comprised Christopher Huang (partner), Randall Perera (senior associate) and Rajkumar Mannar (associate) (all pictured above).\nBiolidics, which was incorporate in 2009, successfully listed on the Catalist Board of the Singapore Exchange on 19 December 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's clear from a promenade around the Internet that people are confused in the chaos of war. They are stuck in the WYSIWYG mentality despite proof that is faulty logic.\nThe current strategies in play by all sides of the war have been very effective in this information war. Many reading the news and updates in the alt media are confused and taken in by the subterfuge and players put in place to mislead, obstruct and distract.\nI don't think there's anything confusing about this measure, but I doubt most people bothered to look.\nI'm confused, too. Why isn't this on every blog, website, video platform, Twitter, Facebook, MeWe, Reddit, and everywhere else?\nWho does and does not promote this in the alt media will be very telling.\nTo promote freedom, fairness, and economic opportunity by repealing the income tax and other taxes, abolishing the Internal Revenue Service, and enacting a national sales tax to be administered primarily by the States. Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, SECTION 1. Short title; table of contents.\n(a) Short Title.\u2014This Act may be cited as the \"FairTax Act of 2019\".\n(b) Table of Contents.\u2014The table of contents for this Act is as follows:Sec.\u20021.\u2002Short title; table of contents.\nSec.\u2002101.\u2002Income taxes repealed.\nSec.\u2002102.\u2002Payroll taxes repealed.\nSec.\u2002103.\u2002Estate and gift taxes repealed.\nSec.\u2002301.\u2002Phase-out of administration of repealed Federal taxes.\nSec.\u2002302.\u2002Administration of other Federal taxes.\nSec.\u2002401.\u2002Elimination of sales tax if Sixteenth Amendment not repealed.\n(3) have a disproportionately adverse impact on lower income Americans.\n(4) impose unacceptably high tax planning costs on small businesses and farms.\n(10) will respect the privacy interests and civil rights of taxpayers.\n(4) businesses that must collect and remit taxes should receive reasonable compensation for the cost of doing so.\n(f) Findings Relating to Repeal of Present Federal Tax System.\u2014Congress further finds that the 16th Amendment to the United States Constitution should be repealed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whenever anyone goes to New York City, one of the things they want to do is go up in the Empire State Building. And why not? It is an iconic building and it offers great view of the city.\nBut there is a problem with the view from the Empire State Building for photographers. That problem is that you don't have the Empire State Building in your shot.\nThe solution is to go to the top of 30 Rockefeller Center, also known as Top of the Rock.\nThis is an observation deck that sits atop this 70 story building (850 feet\/260 meters tall). After being closed for several years, it underwent a $75 million renovation and reopened in 2005. There are actually two stories to the deck and you can see in every direction. To the south you will see downtown, to the north is Central Park. It is surrounded by thick plexi-glass for safety (which also cuts down on the wind), but there are gaps between the panels large enough to stick your lens through.\nIt is a pricey ticket at $32 for adults ($26 for children 6-12 and $30 for seniors), but it is worth it. The entrance is on 50th Street between 5th and 6th Avenues. Fore more logistical information, check out their website.\nTop of the Rock is open daily from 8:00 am through 12:00 midnight (last trip up at 11 pm), so you have plenty of options for when to visit. The best time to visit is about an hour before sunset. This will give you time to get tickets, go through the displays, and take the elevator to the top before sunset. Once at the top, you will be able to take daylight pictures before the sun goes down. Then you will get sunset photos. Then you can hang around for a few night shots as well.\nIf you want to see the view during the day and at night you can also purchase their Sun & Stars pass, which lets you go up twice in the same day at different times. It costs a little more though ($45 for adults).\nExposure settings at Top of the Rock might not be what you expect. To start with, while you might think that view would require a deep depth of field, it really does not. Everything in the shot is really far away from you (over 50 feet\/30 meters). All that really needs to be in focus is \"infinity.\" Therefore with your camera set to focus very far away, you can use an aperture setting like f\/5.6 or f\/8 and still have everything sharp. This has the benefit of letting a lot of light into your camera and also eliminating any concerns about diffraction.\nMonopods are allowed. Put your camera on the monopod and then find a way to stabilize it as you shoot between the plexi-glass panels or from the top deck.\nUse your tripod as a monopod. They will allow you to bring your tripod to the top, and you can use it by extending one leg only and using it as a monopod. Then use it in the same way as mentioned for the monopod.\nFind a place to set your camera. There are concrete\/stone shelves where you can place your camera for stability. I've done this but it makes me nervous. Keep your strap wrapped around your wrist a few times to make sure it doesn't blow off.\nThis is one place where it makes a lot of sense to bracket your photos. You will likely be dealing with a dynamic range problem. During the day, the sky will be much brighter than the foreground. At night, the city lights will be much brighter than everything else around them. To capture all the tones, bracketing and using HDR later might be the only way to keep from having blown highlights.\nSunset view from Top of the Rock (yes, there are other views, but I like this one the best).\nWhen setting up your shot, think about two things: a center of interest and the foreground. As to the center of interest, the Empire State Building is an obvious place to start. I have taken hundreds of photos from Top of the Rock, and no matter how creative I tried to be, almost all the photos I liked when I got home were straight-ahead shots of the Empire State Building (see Take the Obvious Shot). Then just try to set up the shot so that the foreground makes sense to you. It is difficult to explain what makes a good foreground in this context, just make sure you pay attention to it.\nDon't forget to shoot a panorama while you are up there as well!\nEven for non-photographers, Top of the Rock is often the highlight of their trip to New York. Check out the reviews on Trip Advisor, where it is currently rated as the #3 attraction in all of New York. For photographers, it is even better. Do not miss this place next time you head to New York.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remarkable opportunity to build in a million dollar subdivision on a view property with little site prep. Gentle sloping easy build 30,000+ square foot lot. Amazing and highly sought after Moon Valley neighborhood with low traffic. Viewing with prior agent approval from guard gate. Very much worth the effort!! You will not be disappointed! NOTE: No HOA fees till building and no requirement to build!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rewind back to a \"few\" years ago, I was a 17-year-old girl who moved out on her own. I was clueless but brave, young but confident. I reminisce about and envy that girl. Where did she go?\nShe became a wife, a mother, and in the mix of all that \u2014 she became fearful.\nWhy do we feel fearful instead of fearless? There are many contributing factors. Perhaps, it's our environment or maybe it's our past experiences. For me, it's a combination of being stuck in the \"what ifs\" of life and comparing myself to everyone else. Why do we do that to ourselves?\nMy goal this year is to let those paralyzing feeling go. Of course, I might not be able to do it perfectly, but I'll give it my best shot. I don't want to live in the future, I want to embrace the present \u2014 fearlessly.\nlt's time to stop fearing that I don't measure up by comparing myself to others. I want to celebrate my accomplishments without weighing them against someone else's.\nWe will start by believing in ourselves! Continue to grow your weaknesses until they become your strengths. Pursue what ignites the fire inside your bones.\nWe will stop letting opportunities pass us by because fear took us as its prisoner. Try turning fear into gratitude. Whenever you start to feel fearful about what you're facing, remember to be grateful that this situation is in front of you. I'm fearful to publish this article, but I'm grateful that I have the opportunity to hopefully inspire someone.\nThe bridge between fearful and fearless is built with honesty. Have you ever been limited by your fear of having a conversation? When we have the opportunity to honestly open up about how we feel, often times, our fears naturally dissipate. Real talk can never be underestimated. Sometimes the hardest conversations are the ones we need to have with ourselves.\nDo you remember what it was like learning to ride a bike? You rode around everywhere until you built up the skill and confidence to take off the training wheels. No doubt, you might have felt apprehensive at first. However, you took the next step and you started to ride on your own. I know you didn't become a professional bike rider overnight. No, you probably wiped out many times.\nMuch like that experience, we become fearless by taking risks. We might reach out for something and \"wipe out\". DO NOT let this hinder you from moving forward. Get right back up and ride again. Challenge yourself once a week to take a risk on something that you normally would shy away from. Make it a habit to stare fear in the face.\nYour true friends and family want to see you succeed. Be open with them about your goals and how you hope to achieve them. Having a support system can help build your confidence. Find people who have similar goals as you do and discuss ways in which you can learn and grow from each other's support. There is always someone out there who needs your support as much as you need theirs. Search for that person, don't give up until you find them.\nI sincerely hope that this has helped motivated you.\nFeel free to leave a comment & share. Together we can become fearless and confident!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaincy b/data_all_eng_slimpj/shuffled/split2/finalzzzaincy
new file mode 100644
index 0000000000000000000000000000000000000000..900008a2f18fc01d586f76d67bdfa763d4dfddbb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Bislr \u2013 A marketing automation tool. Users can draw collaborative automated campaigns on a whiteboard, listen to prospect behavior, and create database prospects through social media hashtags and keywords.\nHey guys! My name is Mac Bruce from England,UK. I am using this medium to announce to the general public on how Andrew Wills Investments Company restored happiness to my home again, through the help of Mr Wills, When all hope was lost they granted me a loan sum of \u00a3157,000GBP on the 29th of October 2017 to settle my debts,pay my tax and start up a business of my own. They saved me and my family from loosing our home as we were not able to pay out tax bills. I am writing this message cause I feel it might be important to you out there seriously in need of a genuine loan in other for you not to fall into the wrong hands in search of a liable loan,I strongly advice you contact this company via email: aw391loanagency@gmail.com. Andrew Wills Loan Finance service include; Refinance,Home Improvement,Investment Loan,Auto loans,Debt Consolidation,Line of Credit,Second Mortgages,Business Loans, Personal Loans,Car Loan and Xmas Loan.\nPPC advertising is about running multiple tests, selecting the top performers and then doing it again. Once you find an ad group that is achieving your margin targets, you can begin to scale up. Start slowly at first and never make a sudden, massive leap in your spending.\nWhen you first get started with internet marketing, it might like there's a hundred different ways to do everything that matters. Consider the enormous number of different strategies and approaches, and the ideas you could implement quickly multiply out of control.\nCharles Crawford is a cofounder of Crawford and O'Brien, a Phoenix-based digital marketing agency that applies high-level dental internet marketing strategies to get dentists more new patients. Follow us on twitter @crawfordobrien.\nAnother mistake many businesses make is to pull away from content marketing all together. This likely results from a trend against guest blogging before the recent Google updates. Aggressive marketers were creating guest blog posts, and then submitting them to anyone who would publish them in return for a backlink to their site. These backlinks were thought to be the perfect marriage of content marketing and good SEO.\n\u00a9 2017 Intuit Inc. All rights reserved. Intuit and QuickBooks are registered trademarks of Intuit Inc. Terms and conditions, features, support, pricing, and service options subject to change without notice.\nThis gives a marketer an unprecedented number of customers to reach with product and service offerings, available 24 hours a day, 7 days a week. The interactive nature of the internet facilitates immediate communication between businesses and consumers, allowing businesses to respond quickly to the needs of consumers and changes in the marketplace.\nAs a digital marketer, you don't need to be able to build a web page from scratch. It does help, though, to have a basic understanding of the fundamentals that make up the pixels and interactions you see on every screen. This free course will teach you how to think like a designer and a developer, which can give you deeper context around your marketing initiatives.\nA crucial task in your business is analyzing your data. What was the response to your last email? How many people liked, shared and\/or commented on your social media post? What resource is sending the most traffic to your website? This information is then used to maximize what's working or tweak what isn't.\nOf these, Majestic and Moz Open Site Explorer have to be the \"must use\" tools. I think for non-SEOs, the Backlink History is great for basic benchmarking of competitor success in gaining backlinks although link quality isn't shown.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm always thought that what makes or breaks any ORM layer for use in any but the simplest of applications is the underlying SQL code that's generated. In fact, I'm a proponent of using stored procedures with any ORM for best performance. So it was with great interest that I came across the blog posting \"Improvements to the Generated SQL in .NET 4.0 Beta1\" by the folks that are working on the Entity Framework SQL Server provider at Microsoft. These improvements, along with some others I wrote about in May, show they're serious about improving TSQL performance in the next release of Entity Framework.\nI'd really like to see support in the EDM provider for SQL Server (and LINQ to SQL) for the spatial and hierarchyid data types (and UDTs in general), table-valued parameters, filestream through the streaming APIs, and better support for stored procedures (beyong the ComplexType generation that's in EDM 4.0 beta) and UDFs. This would complete and enhance the SQL Server support in future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The epidemic of drug overdose deaths is a national disaster. It claimed more than 64,000 lives in 2016, many of them by opioid overdoses. That's far more than the number of deaths from HIV\/AIDS in the peak year of 1995.\nMy comparing the opioid and HIV\/AIDS epidemics isn't just a matter of statistics. Among those of us who were involved in responding to the HIV\/AIDS crisis, there is a sense of dismay. We are once again facing a public health crisis for which there is inadequate research, limited funding, and much public stigma. And just as happened in the early years of HIV\/AIDS, the impact on children of the opioid epidemic is not getting the attention it deserves.\nAbout half of opioid overdose deaths occur among men and women ages 25 to 44; it's reasonable to assume that many are parents. Imagine the impact on a child when a parent overdoses at home or in a grocery store. Statistics can't tally the trauma felt by a seven-year-old who calls 911 to get help for an unconscious parent, or the responsibility undertaken by a twelve-year-old to feed and diaper a toddler sibling, or the impact of school absences and poor grades on a formerly successful high school student.\nParental overdoses have an immediate impact on children. There's also a cumulative impact as these children become adults and are themselves at risk from the same influences that drove their parents to drugs, overdoses, and early deaths.\nWho are these children and adolescents?\nNewborns whose mothers are addicted to opioids. These babies may undergo withdrawal themselves and need special treatment.\nChildren of all ages at risk for accidental ingestion or inhalation of toxic substances.\nChildren living with an addicted parent, dealing with constant uncertainty and fear.\nNo one knows how many of these vulnerable children there are in the U.S. because no one is counting. As a point of comparison, an advisory group to the British government estimated that there are between 250,000 and 350,000 children of drug abusers in the U.K. \u2014 about one for every drug user. The title of its report, Hidden Harm, applies equally well to American children. They remain hidden in families with addiction until a crisis erupts and law enforcement or child welfare agencies get involved.\nMany of these children are taken in by grandparents, who may themselves be struggling with illness and poverty. According to U.S. census data, more than 2.4 million grandparents are currently raising grandchildren. Unless they become formal foster parents, subject to an agency's monitoring, these relatives are not eligible for financial or other resources to help them deal with their own or the child's emotional distress and basic needs.\nWhen relatives are unable to take in these children, foster care is the next option. In 2016, about 274,000 children entered the foster care system, 22,000 more than in 2012. One-third of those youngsters were removed from their homes because at least one parent had a drug abuse issue.\nResponding to the opioid crisis requires action on many fronts. Prevention, treatment, and control of prescription opioids and illegal substances are already on the agenda for adults. But children are rarely the focus of concerted planning and action.\nIntegrating child-centered policies into prevention and treatment programs is essential. We need targeted research that draws from the fields of addiction treatment, child development, family therapy, mental health, child welfare, law enforcement, and others to determine the best evidence-based solutions.\nAmong the many lessons we learned from the HIV\/AIDS epidemic is that it affects more than just newborns or children entering foster or kinship care. Any child whose family life is disrupted by addiction or illness carries a heavy weight \u2014 sometimes because they are acting as caregivers to siblings and parents, and sometimes because their peers, teachers, and coaches know nothing about their home lives. Understanding what these young people need requires listening to them, including them in their parents' treatment process, and planning interventions for their own mental health.\nIn many ways, HIV\/AIDS changed our conceptions of family by incorporating nontraditional relationships based on commitment. That needs to be extended to the opioid epidemic. Children whose parents have overdosed, or died from overdoses, need support from grandparents and other relatives as well as from their educators, religious leaders, and community agencies.\nSome legal precedents established during the HIV\/AIDS epidemic should also come into play today, such as standby guardianships to give relatives or friends legal standing in case a parent is unable to take care of a child. And as in HIV\/AIDS, grandparents and other relatives who take on responsibility for these children need financial and other kinds of support.\nPerhaps the most powerful lesson from the HIV\/AIDS epidemic is the emphasis on linking prevention and treatment to human rights. In the era of opioid overdoses, the rights of children to be protected, nurtured, and educated are fundamental to ensuring their futures.\nWe must start immediately. With each day that passes, more children lose their parents to addiction \u2014 physically or emotionally \u2014 and suffer severe mental trauma or become overwhelmed by anxiety. This too is part of the national disaster that we must work to reverse.\nCarol Levine directs the Families and Health Care Project for United Hospital Fund, a non-profit based in New York. She was named a MacArthur Fellow in 1993 for her groundbreaking work on the legal, ethical, and public policy aspects of the AIDS epidemic.\nWe do need to help these children! I often wish there was an agency that could keep tabs on these children. Keep a directory of names so that other agencies\/non-profits could contribute to their well being. Is there a way to identify where these kids are being placed, or who need placement? We have to start somewhere!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saturday and Sunday training are at the regular hours (10:30am to 12pm) at the pool this weekend.\nBeware though \u2013 Saturday is Australia Day and as such there are many activities going in the Harbour. Thus, there is a clearway on the street parallel to Bradfield Park on the Kirribilli side. Expect parking to be at a premium \u2013 you would be best to opt for your feet, bicycle or public transport. There is a good view of harbour from the sun deck at the pool, so it's a good place to be. Don't let the signs for Family Fun Day throw you \u2013 our lanes are reserved as normal, but there will be some large inflatables on the other side of the pool open for business starting at 11am.\nMonday will be a bit different. Instead of public holiday hours at the pool, we will be meeting at Balmoral Baths (near the south end of the beach) at 9am for a fun session with Mickaela. Parking can be difficult here as well, so the bus &\/or carpooling is always a good option. Some will be having breakfast \/ coffee afterwards, so come along and have some fun.\nSchedule returns to normal on Wednesday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to uninstall \/ remove Ruby Gems (on Ubuntu)?\nFirst we need to find out the exact location that the Ruby installs it's libraries. In the terminal program, You can type the following command to find out the exact location.\nContinue reading How to uninstall \/ remove Ruby Gems (on Ubuntu)?\nScreencast: How to Install Ruby on Rails on Ubuntu?\nThis screencast will teach you to set up the Ruby on Rails development environment on Ubuntu.\nContinue reading Screencast: How to Install Ruby on Rails on Ubuntu?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajtrx b/data_all_eng_slimpj/shuffled/split2/finalzzzajtrx
new file mode 100644
index 0000000000000000000000000000000000000000..d40048e8fd1fac1a70cd2f8e71ae413e04f5dd6a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Good afternoon, my friends. Firstly, I will like to thank y'all for the memorable day yesterday. For that brief afternoon I was there, we forgot about the hate, bullying and bigotry that was tearing through the outside world. Instead, we spent our time enjoying the one thing we had in common: Chess. You guys have showed me that love truly trumps hate (*cough cough*).\nBut let's get back to business. As promised, here is the analysis from Game 3 of our final activity. I have tried my best to provide comments while keeping a straight face.\nWhite gives up his bishop pair to give Black doubled pawns, a common strategy in many openings. The question is, are the doubled pawns a major weakness in this position? For now they control vital squares on the queenside, open the b-file, and allow the light-squared bishop access to a6; the only vulnerability being the weak square on c4.\nThe threat is to take on c5. After the exchange, Black's isolated doubled pawns will become vulnerable to attack.\nWhile well-intended (developing the bishop and disrupting White's chances of kingside castling), Black overlooks the earlier threat. 6... cxd4 had to be played, 7. Nxd4 c5 8. Nf3 Ba6 gaining tempo for Black.\nLet us analyze the position. White is far ahead in development: His immediate threat is dxc4, opening up the d-file for his heavy pieces to attack Black's exposed king. Yet he needs to be careful of Black's b-file control, and the opening of the long diagonal for the enemy dark-squared bishop should he capture on c5.\nBetter was 11... cxd4 12. Bxd4 Nf6... You get the idea. After ... 0-0 Black will have a very comfortable position.\nThe purpose of 11... e6. But this wasn't the soundest plan.\n13. exd5? is tempting; 13... cxd6 gives White a passed pawn, while 13... exd6 further exposes Black's king and creates a weakness on c6. But Black has a cunning way to slip out: Qxa4! 14. bxa4 Bb2+ 15. Kb1 Bg7+ with a draw by perpetual.\nFritz suggested the intermezzo 13. Nd4! Qb7 upon which White can safely take on c5 without fear of the enemy queen interfering. 14. exd5: Now, Black is in trouble whichever pawn he recaptures with.\nOnce again, the silicon monster Deep Fritz conjures up a winning variation for White: 14. Ng5! f6 15. f4! upon which we see that Black's rook is pretty much useless, since the adjacent knight cannot move out of the way without compromising kingside safety. But admittedly, spotting the winning combination would be no mean feat!\nAt first glance this looks ludicrous... why would one accept doubled pawns so willingly when it was perfectly alright to recapture with the queen? But by leaving the queen on d2, it allows White to retain control of the d-file, threatening the exposed Black king as well as the possibility of Qd4 securing the long diagonal.\nOtherwise 16. exd5 is detrimental to Black's position.\nLocking up the centre only makes the Black king safer. I still feel it was better to undouble the pawns asap: 18. exd5 Nxd5 19. Nxd5+ cxd5 20. Ne5 with the passed pawn playing to White's advantage. 21. c4 opening the d-file could follow next.\nSince White can't make progress in the centre, he turns to the semi-open f-file.\nAttempting to trade pieces to relieve the pressure on the king.\nFinally bringing the knight back to hit f7, but is it too late?\nBlack successfully simplifies and reduces the strength of the attack.\nAttempting a Cheap tactic, but White can step over it easily.\n29. Rf7+! This powerful intermezzo is what both sides missed. 29... Ke8 30. cxb3 Black can resign.\nBlack has equalized comfortably in the endgame, although his pawn structure is still slightly inferior.\nThe correct decision. 33. Nxf6 Kxf6 helps Black centralize his king.\nThe final mistake of this crazy game. This trade benefits Black in no way, and only harms his pawn structure by creating weak, isolated doubled pawns.\nBlack lost on time, but the endgame was already better for White.\nI understand that this was not the most legit of games, played under non-tournament conditions with much discussion and disagreement. My hope is that our playing skills in the tournament hall will not be as comical as what we have just seen!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vintage Rabari Textile Leather Strap Handbag. One of a Kind!\nVintage Rabari Textile Leather Strap Bag. One of a Kind!\nVintage Rabari Embroidered Textile Leather Fringe Bag. One of a Kind!\nVintage Rabari Embroidered Textile Leather Bag. One of a Kind!\nVintage Rabari Embroidered Textile Fringe Leather Bag. One of a Kind!\nVintage Rabari Embroidered Textile Front Leather Fringe Bag. One of a Kind!\nVintage Rabari Embroidered Textile Cross Body Bag. One of a Kind!\nEmbroidered Guatemalan Hoop Handle Bag. One of a Kind!\nVintage Rabari Textile Handbag. One of a Kind!\nPony Skin Leather Rabari Medallion Clutch Handbag. One of a Kind!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's easier to say, I do video stuff.\nI was born and raised in Oklahoma where I played school sports for the majority of my life. During high school, I made a decision to focus less on sports and more on creating videos and becoming a rapper. So naturally, I skipped college and moved to Atlanta where I could work on both of those things.\nBecause I was too white to rap, I spent most of my time learning video production. And I really enjoyed it!\nStarting out as a freelance videographer, most of my work revolved around marketing for bands and musicians.\nAs time went on, I began to do video work with larger corporations, creating culture-shaping content to influence and architect an organization's culture.\nMy entire goal is to help shape a culture that values people and empowers creatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tampa, FL (WGR 550) - This season, the Sabres have beaten Tampa Bay 2-1 and lost 5-4 and 4-3. It's pretty strange they've all been one-goal games because the Lightning are 15 points better than the No. 2 team in the NHL, which is Calgary. They average 3.90 goals per-game, which also leads the league. At an average 2.64 goals-against average, they are fifth in goaltending.\nTampa Bay has won seven straight games, scoring 35 goals. The NHL's leading scorer Nikita Kucherov has seven goals and 11 assists for 18 points in his last six games, and that includes no points in his last game at Philadelphia.\nIn the seven-game winning streak, Steven Stamkos has five goals and seven assists for 12 points.\nBrayden Point was suspended last game by John Cooper for missing a meeting, but he has five goals and eight assists for 13 points in eight games.\nAndrei Vasilevskiy has played in 24 out of 29 games since coming back from injury. The All-Star goaltender is 26-7-4 this season with a 2.28 goals-against and .930 save percentage.\nWith all the talent this Lightning team has, it's not surprising that they're 10-for-24 on the power play in their last six games.\nThe Sabres aren't that aggressive and don't take a lot of penalties, and they must stay out of the box in this one. Buffalo killed four penalties against Florida, but only had to kill three penalties total in the four games before that.\nBuffalo has lost three straight to 29th-ranked New Jersey, the 20th-ranked New York Rangers and 19th-ranked Florida. The Sabres have four losses in five games and are 3-11-2 on the road since the 10-game winning streak ended. They're 2-4-0 in their last six on the road.\nBuffalo only has five goals in its last three games, while giving up 14.\nLinus Ullmark has given up nine goals in his last two starts and was supposed to start his fourth game out of five, but he suffered a minor injury in the morning skate and will serve as the backup to Carter Hutton. Hutton is 15-17-3 with a 2.84 goals-against and .909 save percentage.\nIt seems like both goaltenders have been letting in that one bad goal on a nightly basis. Phil Housley said, \"We haven't really given our goalies a lot of run support, so that's one area we're going to focus on because our 5-on-5 differential hasn't been very good our last three games.\nDespite all the losing, the Sabres are still six points behind Columbus for the Eastern Conference's final Wild Card spot. Montreal is seven ahead, while Carolina leads Buffalo by five points. The Flyers are tied with the Sabres and the Panthers are three back.\nOn Thursday, Carolina is at Florida, while San Jose visits Pittsburgh and Philadelphia is at Montreal.\nJoin Schopp and the Bulldog for the pre-game starting at 6:30 when they'll be joined by Phil Housley, Jack Eichel, Evan Rodrigues and Vladimir Sobotka.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Butterfly Dawn is committed to providing you with the best possible customer service experience. Butterfly Dawn is bound by the Privacy Act 1988 (Crh), which sets out a number of principles concerning the privacy of individuals.\nThere are many aspects of the site which can be viewed without providing personal information. We do collect personal information, for example your name, email address and contact telephone number, where you have provided it for subscription services.\nYou have a right to access your personal information, subject to exceptions allowed by law. We encourage you to contact us about your personal information if you find that it is not accurate or complete. If you would like to do so, please let us know. You may be required to put your request in writing for security reasons. Butterfly Dawn reserves the right to charge a fee for searching for, and providing access to, your information on a per request basis.\nThis website is bound by the Privacy Act 1988 (Cth).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakdig b/data_all_eng_slimpj/shuffled/split2/finalzzzakdig
new file mode 100644
index 0000000000000000000000000000000000000000..3bb7aa55d914e6d0fcf8d1496a7c2dc0f0388a66
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"If you register one instance via a server, its address will be stored in the .key file (https:\/\/intellij-support.jetbrains.com\/hc\/articles\/206544609). You can then copy this file to the IDE config folders on other machines (per user).\n@Serge, thank you for your answer.\nBut, I need to configure this within a company, so more than one instance via server.\nPlease let me know if there is a different solution in my case.\nYou can also set up automatic server discovery, but in this case user will have to select the server activation option manually and click discover button to the the server address: https:\/\/www.jetbrains.com\/help\/license_server\/configure_automatic_server_discovery.html.\nSo only the DNS server should be configured and all the users don't need to change anything locally, right?\nYes, users will need to choose a server option when registering, then click Discover button that will fill in the server URL automatically, then click OK and accept the license agreement.\nBut the goal of my question is how to avoid this.\nIn this case the .key file is the only option.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We meet and exceed the high standard for wastewater treatment set by the state and federal government. Each day, our system collects and processes about 12 million gallons of wastewater from Jackson, TN homes and businesses. Our process is state of the art, using the latest techniques to return wastewater to a clean state. This same purification process occurs naturally in streams, but takes days to complete depending on water temperature, microorganism population, and etc. Our two treatment plants treat wastewater, controlling the environment to clean our water efficiently.\nThe water from your home or business travels to our treatment plant. Storm water (or rainwater) that goes through drains in the street doesn't enter the wastewater drainage system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you want to strut around in some new vibrant feathers, I suggest you start with the Chanel Spring 2011 collection.\nA stunning line, the Chanel Spring 2011 collection is all about feminine flair with lace, feathers, prints and shiny touches of gold and silver. Flitting between structured and flowing pieces, the collection shows \"The Power of Karl,\" as Fab Sugar notes. Check out the different pieces in the featured gallery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let me guess, you have been playing Universal crossword and got stuck on the clue Soul go-with. Well, you have come to the right place to find the answer to this clue.\nThe girl from downstairs with the bad knee?\n\"Edith Hamilton called him \"\"a bully at heart... unable to bear what he brought upon unnumbered multitudes of men\"\"\"\nTropical fruit that grows underground?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nummular dermatitis is identified by its unique coin-shaped lesions. See more pictures of skin problems.\nThe name \"nummular dermatitis\" may seem like a mouthful of a medical condition. With a name like that, it may sound like a rare, complex disease, but it's actually a form of dermatitis, one of the most common skin conditions. Nummular dermatitis, also known as nummular eczema, is a condition that causes itchy, red oval- or coin-shaped lesions to form on the skin, usually on the legs, arms, feet, hands and torso.\nNummular dermatitis affects about two in every 1,000 people, and it's more common in men than in women. The first outbreak usually occurs between the ages of 55 and 65, but in rare cases, the condition can develop between the ages of 15 and 25 [source: Miller]. Nummular dermatitis is very rare in children.\nHeightened sensitivity to certain substances, including rubber, nickel, formaldehyde and mercury, may also play a role in triggering an outbreak. Nummular dermatitis isn't caused by food allergies, and the condition isn't genetic [source: American Academy of Dermatology].\nAlthough the exact cause of nummular dermatitis is unknown, its diagnosis is usually straightforward. Typically, a dermatologist will visually examine the skin and, if necessary, take a small biopsy. If this isn't enough to make a diagnosis, or if the dermatologist thinks that allergies may be the cause, patch testing, a method of testing the skin for allergies, may be used [source: American Academy of Dermatology].\nBut how do you know if you should be tested for nummular dermatitis? Read on to learn more about its symptoms.\nNummular dermatitis often begins with extremely dry skin. Patches of blister-like spots develop, and eventually the rash forms into patches. The coin-shaped patches, which are the defining symptom of the condition, are usually well-defined circles. This causes the rash to form a ring shape that looks similar to a ringworm infection. The rash eventually may form a yellowish crust, especially if it becomes infected. After a few days, the lesions and surrounding areas may become dry and scaly [source: Miller].\nThe patches of lesions, which can be red, pink or brown, are usually itchy and sometimes cause a burning or stinging sensation that worsens at night. The number and size of the patches and the duration of the outbreak can vary [source: American Academy of Dermatology]. After the first appearance of lesions, nummular dermatitis can become chronic and recur frequently. The lesions may last a few weeks and then disappear, but they can come back several weeks or months later [source: American Osteopathic College of Dermatology]. The lesions often reappear in the same places as the old ones, and even though the lesions may heal, they often leave scars [source: Miller].\nIt can be difficult to distinguish nummular dermatitis from other types of dermatitis, but similar treatments are used to treat various types of the condition. Read on to learn more about these treatments.\nThe unique coin-shaped lesions are the most distinctive factor in distinguishing nummular dermatitis from other forms of eczema. In fact, that's what gives it its name. The word \"nummular\" comes from the Latin \"nummus,\" meaning \"small coin\" [source: MedicineNet].\nWhile there's no cure for nummular dermatitis, there are several ways to treat and control its symptoms. Once an outbreak occurs, prescription-strength cortisone ointments can be applied to the area to help heal the skin and soothe the itchiness. Coal tar ointments may also be recommended because coal tar reduces inflammation, scaling and itching [source: New Zealand Dermatological Society]. In severe cases, doses of cortisone may be taken, or phototherapy -- ultraviolet light treatment -- may be used [source: American Academy of Dermatology]. These treatments can help heal nummular dermatitis outbreaks; however, lesions on the legs -- especially below the knees -- may take longer to heal and are more likely to scar [source: American Academy of Dermatology].\nBut there's good news for people with nummular dermatitis: Simple preventive measures may be the key to avoiding future outbreaks. Extremely dry skin is often a trigger, so it's important to hydrate the skin using concentrated moisturizers or lotions. Limit your showers to one five-minute shower a day and don't use extremely hot water because this can dry out the skin [source: Mayo Clinic]. Use a humidifier in your home to keep moisture in the air. And wear loose-fitting cotton clothing washed in unscented detergents to avoid unnecessary skin irritation [source: American Osteopathic College of Dermatology].\nTo read more about dermatitis, check out the links on the next page.\nTopical ointments are helpful in treating skin irritations, but sometimes a skin treatment isn't enough. If the itchiness interrupts or prevents sleep, oral antihistamines can be taken to treat the discomfort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakeel b/data_all_eng_slimpj/shuffled/split2/finalzzzakeel
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+{"text":"The St-Laurent Penthouse suite is located on the 4th floor of the Royal Dalhousie. This corner unit offers a spectacular view of the Saint Lawrence River. Original ceiling beams and superb hardwood floors accent this high-end unit.\nFloor-to-ceiling windows looking out onto the Saint Lawrence River.\nWith its 3 rooms and spacious common areas, this suite is ideal for a family travelling with children that wishes to stay near city attractions without sacrificing comfort. The kitchen and dining room allow guests of this suite to get together for a good meal.\nBest time ever and we were so glad to have spent it at your condo's.\nVery spacious, nice views, good location near shops and great restaurants. Also, within close proximity of skiing at Mont-St-Anne and sledding at Village Vacance Valcartier.\nAnyways, your place was very accommodating and comfortable. Our crew of 12 rented out 2 condo's on the same floor so there was plenty of room.\nThanks so much for making our stay feel like home away from home !\nFrom lots of folks from Texas, thanks so much! Until next time\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How can the creative arts help our NHS? Nicky Goulder from charity Create explains.\nStudies have shown that the creative arts have a measurable impact on decreasing hospital admissions. So why are there not more programmes throughout the health and social care system?\nDelivering creative arts workshops is often a hard sell for NHS trusts and local authorities as the accepted forms of medicine are doctors and pills, not artists and paint brushes. It's easy to placate critics by announcing extra funding or more doctors without any real thought given to other working options. However, we need to take advantage of new, effective methods of treatment.\nCreate is a UK charity empowering lives through the creative arts. We engage the most marginalised participants in inspiring, sustainable arts programmes in areas where provision is poor and engagement in the arts is low. We prioritise our work with seven participant groups: young patients, disabled children and adults, young and adult carers, schoolchildren in areas of deprivation, vulnerable older people, young and adult offenders, and marginalised children and adults.\nFor now, charities such as Create rely on voluntary donations to run projects and deliver rigorously evaluated programmes. But we are paving the way for a future where the creative arts are seen as not just beneficial, but vital to a healthy lifestyle and wellbeing.\nNicky Goulder is co-founder and chief executive of Create.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Half Price Books is hosting its huge annual warehouse sale from 9 a.m. to 6 p.m. Saturday and Sunday at 9241 Greenwood Ave. N., with more than 50,000 books, movies and CDs. If you really want to stock up, you can fill up a big red reusable Half Price Books tote bag for $30.\nThey've also enlarged the interior sales area so more people can fit inside instead of waiting in the usual long line outside.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"15-year-old Margaret Novobilsky was stabbed to death in 1984. As her ex-boyfriend's parole hearing nears, a petition has garnered 1,600 signatures.\nThe news arrived by mail, and it opened old wounds for a family devastated by tragedy.\nThe man convicted of killing 15-year-old Margaret Novobilsky in 1984 is coming up for parole in a few months.\nOn Feb. 21, 1984, 15-year-old Margaret was stabbed to death in her Ocean Township home. Her father, Joseph, came home from work and found her lying in a pool of blood in her bedroom. Ex-boyfriend Anthony Neapolitano was sentenced to 30 years to life in prison.\nThis will be Neapolitano's second appearance before the parole board. He was denied in 2014. At the time, Margaret's parents and two sisters wrote letters to the board opposing his release. They were confident he would remain in jail, and he did.\nThey're less confident this time, so they took an extra step: an online petition.\n(To hear an excerpt of what Reed wrote to the parole board, watch the video atop this story).\n\"I thought a couple hundred people would sign it,\" said Reed's daughter, Alexandra Bolton, who posted the petition at www.Change.org Tuesday.\nMore than 1,600 have added their names, some with pained comments that echo just how hard Margaret's death rocked their community.\nMargaret Novobilsky was a sophomore at Ocean Township High School and \"a friend to everyone,\" Fletcher recalled. But she was having a tough time getting past her breakup with Neapolitano, who was 19.\nAccording to Asbury Park Press reports from the time, prosecutors said Neapolitano \u2014 who was quoted as having said that if Novobilsky ever dated another man \"it would be all over\" \u2014 committed the murder shortly after he saw her kissing someone else.\nMargaret's parents still live in the home where the crime occurred. The bedroom where she suffered 13 stab wounds is a television room now.\nOver the years society has become more aware of the perils of relationships gone bad. Things like stalking and harassment hardly were discussed 1984. Did Margaret think her problems were unique? Could a more vigilant culture have prevented her death? It's a sad what-if.\nThe day of the murder, Reed was working as a hairdresser in Fair Haven. Police called the salon and asked her to come home. Reed's boss drove her.\n\"I remember the ride like it was yesterday,\" Reed said.\nHer boss didn't forget, either. She signed the petition. So did a long-forgotten acquaintance who consoled Reed that day. So did many of Margaret's classmates.\nJoseph Novobilsky declined to be interviewed for this, but through Reed he sent word publicly thanking everyone who signed the petition.\nIt's a measure of balm. So is an amazing coincidence. Bolton's daughter \u2014 Reed's granddaughter and the great-granddaughter of Margaret's parents \u2014 was born on Feb. 21, 2014. She arrived by surprise nearly two months premature, exactly 30 years after Margaret's death.\n\"I felt like God was trying to help us heal,\" Reed said.\nAs Reed held Hailey Bolton for the first time, something happened.\nHailey's middle name is Margaret. The family keeps the candle burning.\nRegardless of the parole hearing's outcome, the Novobilskys learned something meaningful over the past week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BlenderArt Magazine is the free periodic Blender Magazine that covers one or two specific themes in each issue. This issue's theme is Modeling Techniques and Python Scripts. If you're looking for character-, low-poly- o treemodeling tutorials, be sure to check it out!\nWelcome to Issue #5, Modeling Techniques and Python Scripts.\nIn this issue we have focused on different methods of modeling, trying to explain why any given method is used, as well as some insightful articles on python scripts, UV mapping and how to set up blueprints for use in blender.\nSetting up blue prints in Blender.\nI must say that Sandra and her team are really building up an outstanding collection of high-quality tutorials here. The downside of publishing them in one PDF though is that the individual tutorials will be harder to find for Blender users. I'd think that making the tutorials available either as webpages or as individual PDFs (say, after the issue has been out for two months) could benefit many users.\nThe only critique I want to give the team is that the quality of some of the artwork in the Galleria section is really low. It should be quality over quantity, guys!\nDunno what to say other than great efford and great reading.\nIt's a great mag and a great contribution to the Blender community.\nI really enjoy this mixture of OpenSource-News, Blender-News, Tutorials and Techniques. I agree with bart about the quality of the artwork section, sometimes less is more.\nP.S. I never heard of 'Artweaver' before - just downloaded it and it looks like a good alternative to the Windows ports of GIMP.\nWell, it's bigger, it's better, it's a whopping 79 pages long!\nPersonal Favourite (or favorite)? 'Urchopia' by Ramon cerrols. I don't know, I just seem to like buildings that seem high up and in perilous positions.\nI was also interested how the innocent looking photos of Rupert the vaguely cute monkey were turned into an evil looking monkey, although with a bit of editing on the supplied .blend file it didn't take much to stop him looking so... mad.\nAnyway, overall, it's huge, looks interesting and certainly a contribution to the community well worth congratulating.\nThanks very much to all who Release this Great \"Blender Art Magazine\"!\nWe can learn alot from this...very Helpfull!!!\nI've been downloading all Blender Art Magazines since the number one. I think they are reaching the main proposal of this important work. By the way, we have many high quality galleries in the net to stock them. Let Blender Magazine members work in their best. When they suppose to gave a pictures gallery is just to give something to enjoy the community no to make posters. Sorry its just my oppinion. About the breakdown of tutorials in small files I agree with Bart.\nIt's like creative fuel that keeps us running.\nbart: \"The only critique I want to give the team is that the quality of some of the artwork in the Galleria section is really low. It should be quality over quantity, guys!\"\nOtherwise, another great issue. Thanks to the publishers and contributors!\nThanks again to everyone at the magazine for another amazing issue.\nThese tutorials are really helpful and give others the impetus to try things that they normally wouldn't.\nI too think the idea of seperate tuts in HTML using old articles is a great idea.\nis it me or the images shown in the art gallery got low? Im not a great modeller, but i think if they want to show a good work made with blender, they should publish works better than the last issues, dont you think? Sorry about the autors of the images..\nThe thing is that we have art submitted to us. And usually, whatever comes in is what we put in the magazine. I don't know if Gaurav (blenderart editor) actively seeks them out, so we need more people submitting us their best work. If you guys have some high quality artwork, then please submit them to us and help us build a better magazine for everyone. We're going to be calling for content soon, so when we do, just submit some cool stuff!!\nI would be happy to discuss breaking the tutorials into into individual files. We will have to decide how and where to upload them, perhaps the wiki book would be a good place. Something we will have to look into.\nAs for the artwork, nam is right, we include all submissions at this point into the current issue. Perhaps we should look into having the perment staff vote on the best images submitted and only those go in.\nAs usual, every issue we do try and improve our look and message. I do find it reassuring that this issue there haven't been alot of comments about typos :P Which up to date has been our biggest problem.\nI agree I put up all artwork that comes to the magazine, that is for various reasons, first I do not wish to dishearten any newbie, IMHO having a image posted some where, where it is visible to lots of eyes can make it more encouraging for the newbie artist. Remember It's not about competition here. Secondly putting all sort of images kinda shows the range of thoughts that goes behind making an artwork, considering the variety of Blender users it is interesting in its own way, but again thats my personal thought.\nSo personally I do not think it in terms of low quality or quality vs quantity, remember my main intent of magazine was not to show off, rather help new users with good quality learning material. So its have been same with the current issue.\nGiven that now a days we are getting quite a lot of gallery submissions, and on the other hand the magazine is getting quite popular, I was already thinking of having a mechanism to filter out outstanding works. So yeah we will be cutting down on the number of images possibly from the next issue.\nI agree with Gaurav, I think that newbies also deserve to have there works shown, and is a pdf magazine, so why not have a bunch of images, if someone does'nt want to see teorical \"bad work\" just need to skip the last pages.\nI think I speek for everyone when I say that everyone did is best with the knowlodge and the tools that they had avaible at the creation time.\nI hope have many artworker join it!\nwell sad Rogper! I agree with you. Thank you all who contributed to this great mag.\nI hardly waiting every issue...! Thanx a lot!\nwhy not a top10 of images selected by the permanent staff and then all the other images in a random order, that way I think that everyone would be happy and would give artists motivation to try to be the nunber1, the man (or women)of the issue, the best of the best... therefore rising the avarege image quality level sky high.\nwell isn't that cool? That issue rocks! Thank you Sandra for adding the lowpoly tut!\n:) We aim to please! Glad you liked the tutorial.\nWhile I understand it would maybe be asking a lot, I personally would like to see tutorials for complete beginners, who have never even touched a 3D programme before and something for more advanced users.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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+{"text":"Footy mad in Munich? Check out Allianz Arena, home to FC Bayern Munich, the winningest club in Bundesliga. The Allianz Arena also hosted the semifinal between Portugal and France at the 2006 FIFA World Cup. Get in and up-close to some serious sporting action!\nThe Allianz Arena boasts a capacity of 75,000, and die-hard arena observers will also know it was the first stadium in the world with a full color-changing exterior!\nSee some of Munich's best sights on a coach with a guide, then explore the epic stadium, including the middle tier, the main stand lower tier, the press conference area, and the dressing rooms.\nStep inside the player's tunnel and imagine the crowds calling your name! Explore the Promenade\/stadium exterior and then get more kicks on the FC Bayern training grounds.\nIt's a must-see for football fans everywhere.\nDeparture is at 10:00. Please board 20 minutes before departure!\nAs Allianz isn't a sponsor of the Champions League, the stadium name is covered up during Champions League matches!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are an organization using an on-premise instance of Sugar, you've probably heard the word \"end-of-life\" thrown around quite a bit within your world. These announcements sound dire for a reason--it's because the End-of-Life (EOL) of a software platform is something to seriously consider in your CRM strategy each year.\nWhen a Sugar version reaches End-of-Life, it means that SugarCRM no longer provides customer support, maintenance patches, or bug fixes for that version of the platform. Because all Sugar Cloud customers are upgraded automatically, this only effects on-premise customers.\nWhat should my organization do in response to End-of-Life?\nMigrate to Sugar Cloud prior to the End-of-Life date. All requests to migrate to Sugar Cloud must be submitted by November 15th, 2018. Being on Sugar Cloud will allow you to constantly stay on the latest version of Sugar.\nUpgrade to version 8.0 (with the current patch being 8.0.1 at the time of this post), which will be supported until October 31, 2019, with an optional 12 months of extended support available for purchase. Sugar 9.0 will be released in Spring 2019.\nPurchase extended support for version 7.9, which extends support through November 15, 2019. This will ensure customers running 7.9 beyond November 15th, 2018 will be eligible to receive critical security fixes.\nWe recommend that you start planning for EOL six months before the EOL date. This allows you and your software partners plenty of time to develop a safe and smooth upgrade plan. If your system is highly customized, it's even more important to start planning at least six months in advance because your system is more delicate when it comes to instance migration.\nWhat if I don't upgrade?\nOnce the EOL date passes, you and your partner will no longer be able to call Sugar support to request help or report bugs. You also will not be eligible for security maintenance on your instance. You will certainly still have access to your instance, but we would never recommend that you stay on your version longer than it is supported by SugarCRM. As your software partner, we will do everything we can to support you and your CRM decisions, but if you remain on a version of Sugar that is no longer supported, there is usually very little we can do when you run into issues.\nWe understand that decisions around EOL can be overwhelming. That's why we are here! Reach out to us to begin the process of planning for EOL of your version. We can help you make a decision, plan an upgrade, and even complete a health check to estimate what will break in the process of upgrading.\nSign up for our free personalized demo and find out why Act-On is the right marketing automation for your business. Register Now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is surreal. How delightful it must be to walk in the mists. You have such great sites.\nLet's hope that in a far away future, the real thing will still be available, although less nearby and more of a few times in a lifetime experience. This wonderful 'park' brings every thing nearby and within reach, whenever we'd like to be there.\nYes, I hope too! Hope is always the best choice to get thing work out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Giving this a go... I am an honest, caring, positive people person who likes to have a giggle. I'd like to find someone to take good care of me!\nAn honest, caring and genuine person. Someone to have fun with, enjoy life with and make the most time together.\nDaydreambeliever86 hasn't asked any friends to write a recommendation yet.\nDaydreambeliever86 has not yet answered any of the optional questions that would tell you more about her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A fine needle aspiration biopsy of a thyroid nodule is a simple and safe procedure performed in the doctor's office. Typically, the biopsy is performed under ultrasound guidance to ensure accurate placement of the needle within the thyroid nodule. You will be asked to lie down on your back with your head tipped backwards, so that your neck is extended. Sometimes, a pillow is placed under your shoulders to help you get in best position for the biopsy.\nDuring the procedure you may feel some neck pressure from the ultrasound probe and from the needle. You will be asked to remain as still as possible and avoid coughing, talking and swallowing during the biopsy.\nMost medications can be continued. However, anticoagulants, also called \"blood thinners\", often need to be stopped temporarily in anticipation of your thyroid biopsy. These medications can increase the risk of bleeding. It is common to receive specific instructions regarding when to stop taking medications from your doctor's office prior to the procedure. If you have any questions about taking your medications prior to the thyroid biopsy, be sure to talk to your doctor.\nThe neck will first be cleaned with an antiseptic. A local or topical anesthetic may be applied. For the biopsy, your doctor will use a very thin needle to withdraw cells from the thyroid nodule. The needle used is smaller in diameter than those used in most blood draws. Your doctor will insert the needle through the skin and into the thyroid nodule. After the sampling, which only takes several seconds, the needle will be removed. New needles are used for additional samples. Several samples of cells will be obtained, by sticking a fine needle in various parts of the nodule usually between two and six times . This assures a better chance to find cancerous cells if they are present. If there is fluid in the nodule, a syringe may be used to drain it.\nThe procedure is usually performed using a local anesthetic and no medications are used that affect consciousness or thinking. After the procedure, you may be asked to sit up slowly to prevent you from getting lightheaded. Most patients typically leave feeling well. There are very few, if any, restrictions on what you can do after a thyroid biopsy. Because of this, it is not generally necessary to bring a companion to help or drive you home.\nThe biopsy samples may be used to make slides immediately and\/or collected in a solution to wash excess blood. Specially trained doctors, cytopathologists, then make slides from the material and examine them under a microscope to make a diagnosis.\nGenerally, it can take anywhere from a few days to 2 weeks for the result to return.\nPlease note that different institutions and centers will have different rates of results depending on their specific populations.\nFor more detailed information about FNA results, please see the Thyroid Nodules brochure.\nWHAT GENETIC TESTS ARE USED FOR THYROID BIOPSIES OF NODULES?\nWithin the past few years, several molecular tests have become available to help determine whether some nodules are cancerous or benign. These tests look at many genes within the thyroid nodule's genetic make-up (DNA). They are being used when a nodule biopsy comes back with a diagnosis of 'indeterminate'. Sometimes, the person doing the biopsy will perform an additional pass of the needle to obtain material for such a test. This may be done on the first biopsy or at the time of a repeat biopsy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..ce26105b1aece79cd072045a205f6e9892a4a0b8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Most Lake County residents are looking at next week`s election with a mixture of anticipation toward the outcome and relief that the process will end. At least for a while.\nBut Lake Forest residents will have less time than most to relax before democracy again intrudes. A year from this Election Day, for the first time since the town was founded in 1861, voters will elect school board members. Some folks think the process won`t be easy.\nUpon its founding, the Lake Forest City Council ran the schools. In 1879, the mayor began appointing a school board. But in recent years Lake Forest has grown. Like other established North Shore towns, it has begun atttracting people who lack generational roots in the community.\nSuch familial roots can provide ties to the existing power structure, links new residents often lack. Feeling disadvantaged under a who-you-know system of government, some residents pushed for change.\nSo now Lake Forest will join the ranks of more than 900 districts in Illinois that elect school board members. After next year, Chicago will be alone in the state in appointing school board members.\nThe pending change already is apparent in a town whose schools have had a rough time lately. Board member J.K. MacKendree Day announced at a board meeting this month that he will resign effective Nov. 1.\nBy resigning early, Day said during the meeting, he could free up a spot for someone who wants to gain experience on the board before running next year, when seven seats will be open.\nSchool board member Wendy Stotts said other members may follow Day`s lead.\nAnd Stotts, the board`s spokeswoman, may be among that group.\nIt is too early to tell how many of the nine appointed school board members will seek election. But not all want to endure the process, Stotts said.\n''If it could be a process in which people were civil and stuck to the issues and didn`t have names hurled at them'' more people may be willing to run, she said.\nThe Caucus Party, for years the only party in town, is now being challenged by a new group, the Heritage Party. The new group draws much of its strength from the 4th Ward, where many of the newer residents live.\nAdd to the rise of a new party the school district`s recent woes, including last year`s test-cheating scandal at Cherokee Elementary School, and you have a tense atmosphere in which to run an election.\nNext year`s race may well be a contest between the newcomer and the established resident, as candidates backed by the new group face opponents drawn from the Caucus Party.\nCurrent school board members, though, say they hope it will be a test of something else: whether a community that never had to elect a school board can do so and still live, work and play together after the ballots are counted.\nsaid School Board President Melanie Rummel.\nIn your wildest dreams: If children daydream about the end of the school year, extracurricular activities and that cute classmate in the next row, what kinds of fantasies do school superintendents have?\nDavid Duffy, superintendent of Grayslake Elementary District 46, provides the answer.\nDuffy was sitting at home on a recent weekend when Charles Cardella, a Republican candidate for the state House of Representatives, came to the superintendent`s door.\nCardella probably expected to do the campaign thing: shake a hand, leave a pamphlet and move on. Duffy had other ideas.\nThe school superintendent delivered a lecture on impact fees, the money schools can collect from developers to pay for educating the additional students that new subdivisions draw. A proposal wending its way through the General Assembly would ''seriously erode'' a district`s ability to collect that money, Duffy believes.\nAnd he wanted Cardella to know that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A police officer bitten by a man who told him that he had HIV spoke of his nightmare six month wait to be cleared of the disease by doctors.\nPC Richard Terry was hurt as he arrested Ryan Pearce on suspicion of assaulting his father.\nDuring his sentencing hearing Kirklees magistrates heard details of the profound impact his selfish behaviour has had on the hurt officer.\nHe said in a victim impact statement that his wedding anniversary plans were ruined and he had been suffering from sleepless nights.\nPearce was found guilty after a trial of assaulting a police constable in the execution of his duty.\nPolice were scrambled to his home in Third Avenue, Liversedge, on May 9. They were responding to a dropped 999 call after Pearce's father was allegedly assaulted by him.\nThis led to Pearce's arrest in the living room where he was placed in handcuffs but he started to struggle.\nThe trial was told that someone then shouted \"lock the front door\" to prevent police and Pearce from leaving.\nMatters escalated and as Pearce's brother Aaron tried to interfere PC Terry was forced to use PAVA (pepper) spray on him.\nHe then pushed Pearce into the corner of the living room to bring him under control. Pearce responded by biting the officer on his finger and laughing: \"Good, I've got HIV!\".\nPC Terry, who is stationed at Dewsbury Police Station , was taken to hospital where he was given a hepatitis B and tetanus shots. He was told that it would be six months before he was officially given the all-clear of the disease.\nHe said in a personal victim impact statement read out to the Huddersfield court: \"Since the incident I can't stop thinking I have HIV and have been suffering from sleeplessness.\n\"I'm 48-years-old and nothing has ever affected me like this. The male that bit me did exactly what he wanted to do which is cause long-term suffering.\"\nThe officer said that he would rather have a broken leg than be bitten by someone who said that he had HIV and his ordeal had put a strain on his relationship.\nHe commented: \"It's our 25th wedding anniversary and we had events planned to celebrate but this has spoilt everything.\"\nPC Terry said that he couldn't help thinking that he had been infected and that it might spread to his wife.\nThe incident has also affected his employment as he feels that \"it's not worth putting myself in danger for people to get away with it\" and dealing with people who \"think it's funny to assault police and make them think they've infected them with HIV.\"\nSajid Majeed, mitigating, said that his client was sprayed at \"point blank range\" by the officer and things escalated from there.\nHe told magistrates that Pearce, of Carlinghow Lane in Batley , had suffered from difficulties throughout his life and never had any stability due to going through the care system.\nHe admitted: \"This was not a pleasant situation and he should have done things differently.\"\nMagistrates sentenced Pearce to a community order with 120 hours of unpaid work. He was told to pay PC Terry \u00a3300 compensation as well as \u00a3100 prosecution costs and \u00a385 victim surcharge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I am Yohane, the fallen angel! Descended from above!!\"\nFrom the anime series 'LoveLive!Sunshine!!' comes a Nendoroid of the Aqours member Yoshiko Tsushima wearing her outfit from the song 'Aozora Jumping Heart'! She comes with both a cheerful standard expression as well as her signature winking pose completely in 'fallen angel' mood!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was once called up at 4am by someone in our China office telling me that something was broken. I was not best pleased, my role does not include out of hours support, and I spent about 5 minutes just shouting down the phone at her \"Do you know what time it is in London?\" over and over again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Masterful in their design, the art deco hotels that line the avenues of South Beach are a true inspiration. Wouldn't you love to park your vintage Vette under the palm trees out front of one of these beauties? Yes, of course you would! JOOLcity loves these retro city t-shirts with famous Miami landmarks for tourists as well as city proud people who love travel t-shirts. This Miami t-shirt with a cool art deco hotel is everything you've dreamed of and more. It feels soft and lightweight, with the right amount of stretch. It's comfortable and flattering for both men and women.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanzdb b/data_all_eng_slimpj/shuffled/split2/finalzzzanzdb
new file mode 100644
index 0000000000000000000000000000000000000000..c052ed441921f7de6968f3b80f3371d6f82783ee
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The Panda Hair-do game is under the android game, dress up, gamepix, girl, girls, html5, ipad, iphone, make up, makeover, mobile, touchscreen category. Do gamepix you love panda?\nThe Bouncy Dunk game is under the android game, ball, basketball, bounce, gamepix, ipad, iphone, mobile, mouse skill, sports, timing game, touchscreen category. Get ready, shoot, score and WIN: Bouncy Dunk will make your excitement grow!\nThe Birdy Rush game is under the android game, bird, collecting games, evade, gamepix, html5, ipad, iphone, mobile, running games, touchscreen category. Bird's gamepix life is never easy.\nThe Iceberg game is under the android game, gamepix, ice, ipad, iphone, mobile, physics, puzzle, thinking, touchscreen category. Frozen shapes hang in the air and will not move until you unfreeze them.\nThe Bomber Friends game is under the android game, arcade, blocks, bomb, bomberman, gamepix, ipad, iphone, mobile, touchscreen category. Bomb the enemies and be the last to survive to win the match!\nThe Tentrix game is under the arcade, blocks, gamepix, puzzle, tetris, thinking category. TenTrix is easy to play and a pleasurable game for all ages!\nThe Fantasy Star Pinball game is under the arcade, bounce, faerie, fantasy, gamepix, girl, kids, pinball category. Fantasy Star Pinball 3D is an incredible arcade pinball game with a fantasy theme.\nThe Doodle Creatures game is under the android game, animal, gamepix, ipad, iphone, mobile, mutant, science, simulation, touchscreen category. What happens if you combine an elephant with a flamingo?\nThe Paper Craft Wars game is under the android game, animal, gamepix, ipad, iphone, mobile, real time, strategy, touchscreen category. Paper Craft Wars is a fast thinking RTS\/RPG game, with deep tactic capabilities and amazing graphic in unique style.\nThe Boss Level Shootout game is under the android game, arcade, boy, evade, gamepix, ipad, iphone, mobile, purchase equipment upgrades, retro, shooting, touchscreen category. Gamer boy Billy loved to play video games but one day his retro console ate him.\nThe 1941 Frozen Front game is under the android game, army games, gamepix, ipad, iphone, mobile, snow, strategy, tank, touchscreen, turn based, war category. Lead German forces on the rise towards the East or defend mother Russia on the Soviet side.\nWe found 57 matches about Gamepix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Planning to enjoy salmon for dinner? A nutritional scientist might suggest seasoning your sockeye with a healthy dose of curcumin. Here's why: Omega-3 fats and curcumin make up just one of many food and nutrient pairings that stand to maximize your nutrient absorption and work together to boost your overall health.\nAnd while some nutrient combos work together to increase the absorption of one another, others simply complement each other to provide incredible protection and an awesome boost for your health. Here are eight complementary nutrient combos you should look to consume together.\nCurcumin holds extraordinary potential for protecting your brain and reducing cerebral inflammation. In fact, many experts are calling curcumin the 'King of All Spices\" for its role in preventing the formation of beta amyloid plaque in the brain, in addition to the many other health benefits it provides. Add some brain-boosting omega-3s to your intake of curcumin and you have a powerful brain-protecting duo that should keep you sharp for years to come.\nWe've gotten a few emails about whether or not one should continue taking CoQ-10 once they start taking astaxanthin, and our response is always an enthusiastic YES! Here's why: different antioxidants concentrate in different parts of the body, benefiting our cells in very different ways. While CoQ-10 plays a huge role in optimizing cellular energy for brain and heart health, astaxanthin best targets free radicals linked to inflammation, making it a powerhouse when protecting joint, muscle and skin health.\nWhile the iron found in meat sources is pretty readily absorbed, the iron found in plant sources, such as broccoli and spinach, needs a little bit of assistance. That's where vitamin C comes in. Vitamin C helps convert plant-sourced iron into the form that's more easily absorbed by the body. It's the perfect excuse to squeeze some lemon onto your next spinach salad.\nInulin is a type of soluble fiber that's loaded with fructooligosaccharides, sugars that feed friendly bacteria and boost your digestive health. Pairing inulin-rich foods, such as bananas, artichokes and dandelion greens, with probiotic-rich foods, such a kimchi and yogurt, can give your gut the tools it needs to keep your digestion on track.\nTurns out two nutrients that boost the immune system independently are even more powerful when combined. The immune boosting sulfur compounds found in foods like garlic and onions can help increase the absorption of zinc, a cold and flu fighting mineral found in foods like whole grains and legumes.\nA magical duo for anyone looking to maximize weight-loss potential, EGCG and cinnamon join together to tackle fat in a unique way: the EGCG from green tea helps boost metabolism, while cinnamon keeps blood sugar in check.\nOne of the main reasons dairy products are fortified with vitamin D is that vitamin D helps usher the bone-protecting mineral calcium into the bone matrix. However, one often overlooked player in this process is magnesium, which is involved in every step of vitamin D metabolism. Making sure all three of these nutrients are available in abundance when consumed provides a trifecta of bone and heart health protection.\nMany studies have been conducted on curcumin's ability to clear amyloid plaques from the brain, but a recent study took it one step further by exploring the relationship between vitamin D and curcumin for brain health. Their research found that vitamin D greatly enhanced the plaque clearing strength of curcumin, and that the two in combination are now considered possible therapies for Alzheimer's disease. Another recent study indicates this duo also works wonders for the immune system.\nAll you need now is a product with all these naturals, at research levels, and make it available.\nis vitamin d and turmeric as effective in combination as vitamin d and curcumin?\nThe main thing we are looking for here is how quickly a customer's problem was resolved.\nHiernaast vind u een overzicht van de verschillende opslagmogelijkheden die de website verhuur(dot)nl aanbiedt op de verschillende locaties in Amsterdam.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For press inquiries, or to be added to the Media list, please contact Bobette Banks, Director of Communications, at 301.590.8764 or via email.\nGCAAR is committed to keeping members of the press and various news media in the know about the ever-changing metro DC real estate environment.\nZillow recently announced a new initiative, Zillow Instant Offers. GCAAR encourages the Office of the Attorney General in Maryland and the District of Columbia to work with Zillow and others to comply with local and state real estate licensing laws.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To ensure the highest quality of local services to customers, LabVantage forms business partnerships with some of the top consulting, validation, software, hardware, and service firms around the globe. To optimize the value of our laboratory solutions, LabVantage forms technology partnerships with premier hardware, platform, software and service organizations. We work closely with these regional partners to ensure that they are well-informed and well-trained so that they can deliver, implement, and support top quality systems that meet the unique requirements of customers' laboratory environments.\nOn-site product support and service.\nTo optimize the value of our laboratory solutions, LABVANTAGE forms technology partnerships with premier hardware platform, software, and service organizations. Our technology partners provide platforms, databases, integration tools, application servers, analytical software, and validation services that help improve our customers' productivity and efficiency.\nIf you would like to discuss becoming a LABVANTAGE Partner, please contact our Business Development Department here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a placeholder page for Alexis Call, which means this person is not currently on this site. We do suggest using the tools below to find Alexis Call.\nYou are visiting the placeholder page for Alexis Call. This page is here because someone used our placeholder utility to look for Alexis Call. We created this page automatically in hopes Alexis Call would find it. If you are not Alexis Call, but are an alumni of Churchville Chili High School, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaobkt b/data_all_eng_slimpj/shuffled/split2/finalzzzaobkt
new file mode 100644
index 0000000000000000000000000000000000000000..a28334889266c437668479eff20c5353cd8d9aa7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaobkt
@@ -0,0 +1,5 @@
+{"text":"Have you ever been to a popular restaurant during brunch on a Sunday? The place is crowded, the staff is overwhelmed, and the experience just falls flat. The food isn't getting there fast enough, the servers are frantic, we've all seen this happen. This is a sign of poor labor deployment.\nLabor deployment is the ability to use the correct amount of staff, in the right place, at the right time. It allows you to bring all of your resources into effective action. This means you get the correct staff for the job and you have enough people to make the most profit during peak times and all hours. It's the right staff at the right time, making your restaurant into a well-oiled machine.\nWhen you look at a restaurant, there are so many different areas that need to run smoothly for a successful day. Management, BOH, FOH, and everything in between. You need the right staff at the right time for every position, using efficient and effective labor deployment is so necessary.\nDuring peak times if the restaurant is busy, you can't have service fall flat. Using staffing models can show you where those peak times are and where you might be short-staffed.\nThis doesn't just mean you need more people during peak hours. It also means you cut back when you don't need them, so that you aren't overspending on staff when you aren't bringing in profits. It's all about maximizing customer service and revenue opportunities.\nOverstaffing can be a difficult problem. If you're just playing it safe by bringing in too many people and the cost of the staff is drastically outweighing your profits, that is equally detrimental to your restaurant as when you're understaffed. It's all about balance.\nLook at your team as individuals. What are their best hours? You want to be placing your most experienced employees at the high-impact times and making sure they have the right hours that's best for them and your restaurant.\nOften labor deployment is based off of intuition, but managers don't always have the access to the correct information. You need to be able to analyze your top performing employees, the factors that affect service, the skill sets for the shift, and the peak hours. All of these factors are constantly changing.\nBenefits of a good staffing model can reduce cost, increase revenue, and drive profits by balancing demand with available labor. It's important to remember that as your restaurant grows, the conditions may change and it's important to keep it updated and pay attention.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Visual Importer Enterprise is a platform for building high performance data integration solutions, Including extraction, transformation, and load (ETL) packages for data warehousing and business automation. Visual Importer Enterprise includes graphical tools and wizards for building and debugging automation packages; Plus automation of business processes such as FTP, Email: POP3 and SMTP, File operations, SQL scripts, Zip and Unzip, ETL Task scheduler and comprehensive logging. This comprehensive application provides support for numerous databases: MS SQL Server, SQLite, Oracle, XML, MySQL and many more.Here is simple usage example: Download files from an FTP server, load them into the database, run stored procedure or SQL script and send email to the admin, once the package is completed. Unlike Oracle SQL loader, BCP, DTS or SSIS, Visual Importer Enterprise can also add new and update old records based on user defined primary key. Let's say, for example, you work at university and must publish results of exams from different departments. Visual Importer Enterprise can help you automate data processing without the need for more staff. It will enable you to process data sent via email and FTP without user intervention. Simply create a dedicated email account, design a package and that's it - start processing data! Another unique benefit of using Visual Importer Enterprise its ability to load data from multiple files or tables using mask. If there is a problem it is very easy to resolve: Every package execution is logged. In case of loading errors, Visual Importer Enterprise writes a detailed message into the log and writes a record into the rejected records file. The Enterprise version includes execution agent, which can be run as a Windows service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Open, light-filled residences have been thoughtfully designed with quality features and modern appointments that allow your personal style to take center stage.\nFrom the penthouse level fitness center, mindfulness meditation room and chef's kitchen to a rooftop deck with bbq grills and lawn games, you'll always have plenty of options to exercise your mind, body and spirit.\nGrab a cup of coffee with friends, take a leisurely stroll along the Charles River or catch an exciting Bruin's practice game. Your new apartment provides the ideal mix of essential conveniences, dining and entertainment options, natural beauty and a close proximity to all that the city has to offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This revolutionary process uses intensive pulsating light to permanently remove unwanted hair from any area of the body. The process is quick, gentle and safe.\nHighly controlled flashes of light energy are selectively absorbed by hair-filled follicles. The melanin absorbs the light, heats the hair shaft damaging hair re-growth but not the surrounding skin.\nThe number of treatments a particular patient requires will vary due to a variety of factors including hair density, colour of skin and hair, and on location of hair on the body. At Natural Contours we use the latest high specification equipment that requires the use of skilled practitioners who optimise the settings for each patient and at each subsequent visit.\nAs well as hair removal IPL is used to remove the effect of thread veins. The light energy is absorbed by the haemoglobin and converted to heat energy. In time the vessel is broken down and absorbed by the body.\nPhoto rejuvenation is now increasingly popular the effects of the pulsating light stimulate the skin's natural collagen production toning the skin and improving texture and vitality. A course of four to five treatments is usually required with four monthly follow-ups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Craftsman 950845 18 oz. Flex claw hammer whether you need to hang a picture frame or are trying to pry out an old nail, this 18 oz. Flex claw hammer delivers versatility and power to the palm of your hand. This Combo flex claw features 2 great tools in 1 with both an 18 oz. Hammer and an adjustable pry bar. The claw has 4 different locking positions, so you can customize your leverage depending on the job at hand.\nEstwing's Solid Steel Framing Hammer's feature a longer handle and larger face resulting in more striking power and increase leverage for pulling and prying nails or boards. The head and handle are forged in one piece for maximum durability and hand polished to a beautiful finish. This hammer features our patented Shock Reduction Grip the most comfortable, durable grip on the market. The Shock Reduction Grip reduces impact vibration by over 70%. It is molded and bonded directly to the tools steel core, meaning it will never slip, even under the harshest conditions. Estwing striking tools are proudly made in Rockford, IL.\nHeavy duty casting steel construction exclusive vinyl deep cushion safety grip is steel handle for lasting comfort and durability.\nHeavy duty cast steel construction.\nBright pink handle so its easy to identify your tools.\non the market, you can't loose with our product guarantee! A must have addition to any tool box.\nThe STANLEY STHT51246 20oz 1pc Steel Hammer is an excellent heavy duty hammer, constructed of 1-piece forged high-carbon steel for strength and durability. It has a high visibility comfort grip that features a ribbed surface and flared end that helps to reduce slipping. The exclusive rim temper helps reduce incidences of chipping and spalling, and is also precision balanced for efficient nail driving.\n\"Made from durable, lightweight, metal construction, this Stalwart Stubby Claw 8 Ounce Hammer gives you the precision and control you need for a wide range of jobs! The high-strength fiberglass handle includes a comfortable cushion grip for added accuracy and helps to absorb shock, while the curved claw helps provide needed leverage for removing nails. The ideal compact tool for small jobs, house repair, home improvement, do-it-yourself projects, minor maintenance, this is an essential addition to your home tool kit collection. IMPORTANT: Avoid buying counterfeit products and transacting with unauthorized sellers. Look for our logo on the packaging for every one of our products. Stalwart is committed to providing the consumer with the absolute best price and value on our entire line of products, which we ensure by applying a rigorous Quality Control process.\"\nDURABLE- The Stalwart Fiberglass Hammer is built to last. Made with a high-strength fiberglass handle and high carbon steel, this tool will help you get the job done with ease and ultimate precision.\nESSENTIAL MULTIFUNCTIONAL TOOL- With the comfort grip handle and curved claw design, this hammer is a great compact and easy to use tool for minor maintenances, building, woodwork, home repair, extracting or removing nails, hanging household items and home DIY projects. With the wide range of functions, this is an ideal tool for your home improvement needs and a perfect addition to any tool set!\nCOMFORT GRIP HANDLE- The cushion grip on the handle provides a soft barrier for tackling tough jobs. Easily clasp the handle for comfortable control, accuracy, and shock absorption while you work!\nStiletto finishing hammer, straight claw, 10 oz head, 14-1\/2 in overall length, milled face, 1-1\/4 in face diameter, titanium head, curved Axe handle, hickory handle, Brown\/tan handle.\nEdward Tools Oak Claw Hammer 16 oz is heavy duty all purpose claw hammer for nailing removing nails or pounding. The solid oak handle is contoured for anti vibration while the wood better absorbs nailing than plastic handles. Etched wood handle design makes more greater grip than standard wood handle hammers. Made of forged carbon steel the head is more durable. Edward Tools Oak Claw Hammer 16 oz comes with a lifetime warranty and we will replace it if for any reason this claw hammer breaks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoidc b/data_all_eng_slimpj/shuffled/split2/finalzzzaoidc
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index 0000000000000000000000000000000000000000..d03055ed5ea843dc3ebcc94871cb0a4af6873c2f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoidc
@@ -0,0 +1,5 @@
+{"text":"While we should never expect much in the way of genius or understanding of political science from the people we pay to entertain us, they're still entitled to their opinion and using their platform to promote it. Even if it's woefully, dreadfully, sometimes hilariously wrong.\nAnyone else reminded of the punctuation and netiquette used by an elderly relative who just discovered social media? Well, with Cher being as old as the hills, I suppose that's to be expected, but it's no less amusing to see her try to offer political commentary and end up sounding like a nincompoop.\nDon't you get SJW points docked for failing to use \"body positive\" language? Oh, wait, I forgot it's okay to mock someone's size if they're \"literally Hitler\". Noted.\nTruthfully, those were the tamest of Cher's \"best\" anti-Trump tweets that could be shared among mixed company. The rest of them demonstrated nothing but vulgar, mindless hatred toward a fellow human being who, whether she likes it or not, is her president.\nCher stumbled her way into the headlines over the weekend, though not for her heavily pitch-corrected pipes or her obnoxious fashion sense. Instead, Cher tried to throw another anti-Trump tweet on the pile in response to his idea that, well, if so many of these welfare state sanctuary cities are opposed to lifting a finger against the all-too-real illegal immigration crisis, maybe they wouldn't mind housing and feeding any of the hundreds of thousands of people that flow through the border each month.\nOf course, President Trump's idea terrified and enraged the leftists and gave them something to squawk about over the weekend, but Cher's response takes the cake.\nNow, I don't know if anyone close to Cher is available to do a wellness check and make sure she hasn't suffered any recent head injuries, but that tweet is a precious gold nugget among a mountain of leftist verbal diarrhea.\nWhether or not she realizes it\u2014and, based on her older tweets, I honestly have to wonder\u2014Cher succinctly illustrated why we can't handle porous borders a moment longer. If she could think outside her L.A. bubble, she'd see that much of the nation is experiencing the same inability to handle any more extraneous personnel. Logic would demand her to at least find middle ground with the president, and his supporters, on this issue.\nBut, then, we all know how scarce logic can be these days.\nPrevious articleWhose Bright Idea Was It To Do An Anti-2nd Amendment Version Of \"Oklahoma!\"?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Republican Peter Flores wins historic Texas Senate special election in a district Hillary Clinton won by 12 percent.\nFlores had run four years earlier and lost by 20 points. Now he is the first Hispanic Republican state senator in Texas (yet another reason the liberal media is avoiding talking about the race). Since Flores defeated former Democratic U.S. Rep. Pete Gallego, his victory has some extra punch to it.\nPatrick said this race was a classic example of a big-choice campaign. He had read my recent paper, The Republican Choice for 2018: Win or Lose, which outlines the importance of big- choice campaigns over small-choice campaigns and said this was a perfect example of the potential for a big-choice campaign to overwhelm the media bias (a point Brooke Rollins had also made to me).\n\u00ab Is the president really seething over Sessions and other setbacks?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The visual direction of content is a less widely discussed, but crucial aspect of high-converting web design. Every visitor \"soaks up\" a new site on the first page load \u2014 whether they do it consciously or not.\nAesthetics do play a role, but it's more about the overall feel of the design. This feeling can be affected by whitespace, typography, symmetry, but mostly relationships between page elements.\nDesigners want visitors to stay on the page and keep scrolling by capturing their attention and keeping them interested in the site. Design principles should always focus on function before form. This means the design should complement the content, not shove it in as an afterthought.\nIn this post, I'd like to show you some tips on how you can improve your layouts and visual content flows on your site.\nEvery single piece of a website builds upon the overall layout. This overall layout creates a composition that follows the rules of Gestalt theory which states that the whole is always greater than the sum of the parts.\nIndividual areas of a page come together to form a whole. Design elements need to build a gravitational pull on the content; everything on the page should naturally guide visitors down further until they reach the bottom of the page.\nThis is why relationships between different parts of the content (visuals, text, buttons, etc.) matter so much to design.\nYour goal should be encouraging people to browse the site from their own inclination. It's easier said than done, but you can learn a lot by studying real examples.\nThe home page for Monkop is a great example of visual hierarchy with both text and visuals. Plenty of space is used between elements, and the typography complements the branded vector illustrations.\nAs you scroll, you'll notice straight horizontal page blocks divided by colors, borders and graphics. These are built with design patterns in mind to offer consistency throughout the page.\nTowards the bottom, you'll find a two-column split with images on one side, text on the other. The images also swap sides in a pattern of right-left-right-left. This draws attention, and breaks up the monotony of the typical page while still keeping a natural flow in the content.\nA similar design aesthetic can be found on the website of Picjumbo, a landing page for a photo addon for Photoshop and Sketch users.\nThe home page places focus on the logo and the preview video. As you scroll, you'll notice custom animations that move throughout the page. This animation really captures attention, and gets the viewer interested to keep scrolling.\nOverall, the page feels open and easy to browse. The content is divided into horizontal blocks with crisp typography and clean icons.\nConsider the way different page elements balance together, the space between elements, contrast between colors, and differing shapes. All of these things play a role in the overall composition. Every site naturally draws a certain weight onto the content.\nThere's no absolute answer because it's different for every site. For example, some navigation links look better when they're big and oversized. Others fit better when they're small with uppercase letters.\nI suggest you study other websites in your niche. Really analyze how they're put together. Even try rebuilding layouts to see which elements finally make the design \"come together\".\nThe way you design your typography will affect content direction on your site. This has to do with type hierarchy and the design styles of different page elements like paragraphs, headers, bulleted lists, quotes, and special layout elements like columns or tables.\nVisuals can affect the layout too, so it's a good idea to design content with a natural progression. Write content in a manner that flows down the page, and keeps people reading through each paragraph.\nThe greatest tool you have at your disposal is your eye for design. Learn to recognize the differences in typographic elements, and how they relate to other page elements. Create relationships between page sections to distinguish areas of content.\nFor example, look at the homepage of Campaign Monitor. The top navigation links use all caps with small lettering. Other headers on the page follow this same all caps design which creates a sense of uniformity.\nOther larger headers on the site are much more prominent, and they really jump off the page. Just by looking at a typical header design, it should be easy to tell the difference between a header and its paired body copy.\nThe typographic design styles on Campaign Monitor are exquisite, and they blend naturally into the layout. It takes practice to achieve a result like this, but the more you try, the easier it'll be.\nUnderstand that different types of websites have different methods for guiding visitors through the site. For example landing pages want to guide visitors with tidbits of information, small icons, screenshots, and testimonials.\nOther sites like blogs don't usually bring in people to the home page at once. Most people land on an article page, so blog post layouts are meant to highlight the headline, and draw people further into the content. This is where quality copywriting comes into play because you want readers hanging off every word.\nSocial networks and web apps need quality user experience, so that's a slightly different topic, but consider how the Facebook feed is designed to encourage scrolling and user interaction.\nThe design methods you employ to keep people browsing the site will change over time. But generally, your goal is to guide visitors with a visual content direction.\nLet's take a look at a landing page and a blog design to spot the differences.\nCactus is a static site generator for OS X. Their home page follows closely Apple's design style \u2014 plenty of whitespace and thin sans-serif fonts.\nContent is organized into columns, blocks, and chunks of text with simple graphics. These same aesthetics are common with Apple products, so Mac users will enjoy this design style.\nInformation about the product \u2014 including features and setup \u2014 are listed right on the home page. The page itself encourages scrolling through unique content, basic icons, and an alternating column pattern of left\/right content blocks.\nThe goal here is to provide information to existing users, and to sell new users the idea of Cactus.\nNow compare that design to the home page for The Next Web. The content is much more sporadic on a blog home page, because there's a lot of different post topics.\nRectangles create a grid system that encapsulates multiple posts into a single layout. The goal here is to get users reading content on the site. It doesn't matter if visitors download anything, but it does matter if they stick around to read something.\nThe way to get people reading is with great photos and catchy headlines. TNW does a great job of this, and their layout is built to keep people browsing with related post thumbnails in the sidebar and after-content area.\nGuiding visitors to a particular action is different on every site. But you can learn a lot by studying what other successful sites do, and learning how to copy.\nIndividual design properties can be explained analytically, but the implementation changes for each site. A hero image with a \"Scroll further\" link doesn't perform the same on all websites.\nLearning to design is very much a visual process. Your eye for design is the most important aspect. You need to see things properly to identify this visual hierarchy. If you can see it on other websites then you'll be able to replicate it on your own sites.\nThe best advice I have is to just trust your eyes. Create a list of your favorite websites, and spend 5 minutes browsing each one. Write down your favorite elements on the page, and how they affect the design. This will help you internalize these concepts from a UI\/UX perspective, rather than a user's perspective.\nAlso don't be afraid to try stuff! Nobody got good at design just by reading articles about design. Yes, they help \u2014 they can actually help a lot. But you need to create stuff from scratch to learn what works and what doesn't.\nTrain your eye by studying website layouts you like, and recreating them. Over time, you'll build up a pattern library in your mind which makes designing new sites much easier.\nHopefully, these tips will get you started and give you a basic roadmap to follow. It's not easy becoming a web designer, but the world does need talent, and it's never been easier to teach yourself these fundamental concepts.\nStudy the best examples of websites with page elements you enjoy. Train your eye to recognize relationships, and you'll quickly develop the skills necessary to replicate those relationships in your own work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of my favorite things to do during the summer time is grab a huge bag of padron peppers, marinate them in olive oil, then fry 'em up in a pan and dash them with kosher salt. I think I could eat hundreds of them in that format, just continuously lifting them towards my mouth\u2014one by one\u2014and biting into that green bit of savoriness.\nBut once I start doing that I'm going to need something cold and refreshing. Like the new 2014 Alkoomi Frankland River riesling from Western Australia. This wine is a like cold, crisp dream of fresh acidity and steely minerality, but with a core of ripe stone fruit that haunts you from within. Like the peppers, I think I could probably drink hundreds of bottles of this wine.\nPlus, it pairs with just about anything. Spinach tortellini? Why not. Just about anything works with dry riesling.\nFor the beef course, I decided to keep the \"down under\" theme going and reached for the new 2013 Te Mata \"Awatea\" Bordeaux Blend from New Zealand. I didn't do much decanting, but it didn't matter. The wine is so tasty right from the bottle. I did mention to my parents, however: \"It's only going to taste better tomorrow, so you two might be fighting over who gets what's left in the bottle.\" With subtle hints of dark fruit, plush tannins, and soft layers of richness, the 2013 Awatea is an understated wine, but one that brings a diminutive and delicious pairing for basic dishes like steak and vegetables. I'll probably buy at least a half-case of this before we run out. It'll hold up over the next decade and only get better with time.\nThe great wines from Australia and New Zealand are so underpriced right now it's insane. They're where Bourbon was in 2006\u2014the quality and availability are at an all time high, but the prices are still relatively quite low. I love that we're getting in on the ground floor here, pounding bottles of amazing quality one after the other like it's no big deal. It's exciting and reminiscent of the old days of liquor!\nSpeaking of liquor, my family is still old school when it comes to the after-dinner course: grappa and coffee will always rule the post-meal protocol. Our most-popular (and best, in my opinion) Tuscan producer\u2014Sesta de Sopra\u2014makes a straight sangiovese grappa, and a second one that sees a bit of barrel maturation. Neither are sold in the U.S. at the moment, but much like we did with their Rosso di Montalcino and Brunello wines, we'll be fixing that problem very shortly. Watch for these lovely little bottles before the end of the year. Just because no one else is drinking grappa doesn't mean that I'm going to starve! I'll drink it all myself if I have to!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Variable voltage or variable wattage mod is a battery powered vaping gear which can adjust output power in two ways: changing the voltage, and the wattage. The mod features safety protection circuitry. Our VV\/VW MODs are high-quality vaping devices which come with tremendous maximum output power and most of them have an adjustable lightning-shaped knob with LED indicator. Almost every vapers have one vv\/vw mod, get yours now at buybest.com.\nCurrently, most mods are VV\/VW mods that support a wider range of atomizers and enable you to choose your own power setting. This represents a significant improvement in performance. However, for newer vapers, VV\/VW mods can also be a bit overwhelming. So here is a basic introduction to VV\/VW mods and how to use them.\nWhat is a VV\/VW mod?\nA VV\/VW Mod is a vaping device using circuitry to regulate its output power and allows users to adjust its output power by adjusting the voltage or wattage. In addition, to protect users and prevent damage to the vaping device, VV\/VW Mods utilize protection circuitry to provide a wide range of safety features, which makes them an ideal device for both new and experienced vapers.\nWhy do I use a VV\/VW mod?\nThe VV\/VW mod enables its users to customize the output, which is helpful in many ways, allowing the flexibility to change fast and to use atomizers with various resistances, coil types and wire sizes. The ability to adjust the output power also enables the user to control the heat produced by the coils, which can change the flavor or burn certain e-liquids and help adjust if the user wants to take shorter or longer pulls to produce the same amount of vapor.\nThe VV\/VW mod is currently the safest vape device on the market as they have tons of safety features. It is usually equipped with thermal protection, overcharge\/discharge protection, min\/max resistance protection, short circuit protection and reverse polarity protection. For instance, if the batteries are put in the wrong direction, the VV\/VW mod will refuse to fire, or if you try to use a depleted battery, it will show a \"low battery\" and refuse to work, so you must recharge. These types of safety features means that you can vape without worrying about the safety.\nNowadays, most VV\/VW mods are box mods with display screens, which can show all the key information such as current settings, coil resistance and remaining battery life. In addition to the fire button you can press to vape, almost all of the VV\/VW mods have \"up\" and \"down\" buttons to adjust settings to your preferences.\nBesides, many VV\/VW mods also provide temp control, mechanical\/bypass mode, lock\/unlock, puff counter, USB passthrough and other functions. The temp control prevents the coils from exceeding a pre-specified temperature, allowing you to avoid dry puffs completely; the mechanical\/bypass mode allows output to be determined by the charge state of the battery and the resistance of the atomizer. The lock\/unlock function can prevent firing the device accidentally; the puff counter shows the number of pulls that have been taken; the USB passthrough enables the user to charge a mod while vaping on it.\nThough VV\/VW Mods have tons of advantages, we cannot deny that they still have some disadvantages.\nFirst of all, the size. Compared with fixed voltage regulated mods or mechanical mods, VV\/VW mods are usually larger because VV\/VW mods need space for their circuitry and wiring to perform the same or exceed the performance of mechanical devices. Typically, a smaller VV\/VW mod has a shorter battery life, a lower maximum output power and a smaller range of usable resistances.\nSecondly, the lifespan. VV\/VW mods use many electrical components such as wiring, circuitry, LED or OLED displays, microprocessors and tactile switches. Compared with mechanical mods without circuitry, VV\/VW mods are easy to be damaged if dropped or worn out after extended use.\nWhen using a VV\/VW mod, start with a low setting and gradually increase it until you find what works best for you. To prevent damage to the device, you need to keep your VV\/VW mod away from water and extreme temperatures because they use delicate circuitry. Typically, batteries have a shorter life than the devices they power. Some VV\/VW mods have non-replaceable internal batteries while others use replaceable rechargeable batteries so you need to take this factor into account when buying a VV\/VW mod.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Maple Lake Academy would like to Welcome the newest addition to the Education Department, Ms.Julie Smith! Julie is our new English teacher!\nJulie Smith graduated from BYU, with degrees in English Teaching and History Teaching. Since then, she has taught public school and private school, but her favorite place to work is at residential centers like Maple Lake! Julie is married and has five kids. Her family has lived in Utah, Missouri, and Nevada. She loves to read, write, and travel \u2013 especially to historical sites. Some of the fun things she has done include walking up The Statue of Liberty and down The Washington Monument, sleeping over in a zoo, and camping in Alaska. She also interned in Washington DC when she was in college, and was an summer exchange student in Japan when she was in high school. She is super excited to work with the students at Maple Lake Academy!\nWe are excited to have her a part of the team!\nA lot has been going on here at Maple Lake Academy! From long days with RT to class field trips and fun project assignments.\nTwo field trips this year have been to a Teen Author Boot Camp and our annual Eco Challenge at Thanksgiving Point.\nThe Teen Author Boot Camp was a day long youth conference where over 1,000 teen writers came to receive training from the best writers out there. We went to five workshops that focused on how to write an action packed scene, how to create character depth, taking risks as an author, had lunch and met professional authors.\nDuring our annual Eco Challenge with Thanksgiving Point in Lehi, our students competed in different events from creating designer clothes and accessories with recycled items, wrote songs about the environment, drew and painted pictures about pollution and endangered species and wrote and acted out a skit they had written about being good to our earth. During our break, we walked around the beautiful tulip fields which are a local highlight at Thanksgiving Point.\nDuring the summer, we have enjoyed weekly trips with the Recreational Therapy group and look forward to the new term's school year and the projects and trips it will bring. Until then! Enjoy the rest of your summer!\nStudents enjoying the summer day researching for their Children's Book project.\nRecently, both schools participated in a Lego Competition sponsored by Techie For Life. They were challenged to create and build their own unique Lego designs. The categories were: functional, artistic and scale-model.\nTechie for Life sent each school, individual packs of Legos to use and document.\nThe students were excited about the hands-on challenge and the freedom to design their own creations.\nFrom living room scale models, animals, cellos, spin-tops and light rays, the Lego creations were endless.\nSeveral received honorable mentions and received a myriad of prizes based on technology and robotics. Maple Lake Academy was well represented and made a lasting impression on the judges.\nAll students at the Girls School at Maple Lake completed an Innovations project this past summer. Each girl was able to choose a topic that they had interest in and then researched that topic in depth.\nAt the end of the summer, a ll of the girls presented their findings to the school. Each presentation lasted at least ten minutes and several lasted longer.\nEach girl did a great job in communicating what they had learned during the summer. Several subjects were covered during the student presentations.\nThese subjects included: Espionage, Vet Science, Egyptology, Music Therapy, Snakes, Types of Riding Styles on Horses, Becoming A Doctor, Decorating Cakes, History of Soccer, and many other topics.\nStudents could use a variety of delivery methods. This picture is a story board that a student created. Many students used a power point or prezi but other students had creative ways that they shared their topic with the school.\nOne student had a box, and on each side of the box were several facts about snakes.\nOther students used You Tube Videos to help them illustrate their presentation.\nOverall, the quality of the presentations were very high and well done. The teachers and Ed. Director were very proud of the girls and the efforts that they gave in presenting these projects.\nLast Thursday and Friday Maple Lake took the students on a snowmobiling trip. The boys school went on Thursday and the Girls School went on Friday.\nMaple Lake Staff drove the snowmobiles and gave every student a chance to ride for around an hour.\nSince this was an outdoor activity and all day, the school provided snacks and a hot lunch for the students.\nIt was also quite cold so everyone dressed in warm snow clothes. And for safety, everyone wore helmets while on the snowmobile.\nWhile they waited, other students built snow sculptures or just played around in the snow.\nEven though it was cold, it was a beautiful sunny day out in the mountains. This is the type of winter weather that Utah is famous for.\nIn addition to building stuff. The students had different kinds of races and snow activities.\nOverall, it was a very successful day for both schools.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thiol Reactive Acrylate Linear PEGs from JenKem Technology. Methoxy PEG Acrylate is available with MW 5,000 (M-ACLT-5000) and MW 20,000 Da (M-ACLT-20K), in 1g and 10g packing sizes. JenKem Technology provides repackaging services for an additional fee, please contact us if you require a different package size than our catalog selection.\nDifferent MW of Methoxy PEG Acrylate products may be available by custom synthesis, please email us at tech@jenkemusa.com for details on custom PEGs.\nJenKem Technology provides GMP grade PEG derivatives and bulk orders via custom synthesis, offering the opportunity to match customers' special quality requirements. JenKem Technology is capable of development and synthesis of a wide range of GMP PEG derivatives starting at 200g up to 40 kg or greater batches, under ISO 9001 and ISO 13485 certified quality management system, following ICH Q7A guidelines. For inquiries on cGMP production of PEG derivatives please contact us at tech@jenkemusa.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Email marketing has been popular since the days of dial-up, but much like how the Internet has changed, so has how we market to customers. The key to steady conversion is a newsletter template that engages your subscribers and leads them back to your products. If you've noticed your email marketing isn't working like in the past, then you probably need to update your newsletter template for today's savvy customer.\nThere are a handful of great email softwares on the market including Mailchimp, Constant Contact, and AWeber. Each of these options has their own unique pros and cons \u2013 user experience, features, price, etc. There's also a handful of great newsletter template designs you can download or pay for from sources like Themeforest, Litmus, and Fiverr.\nWhether you are just getting started with your newsletter or you are looking to get better results from the one you have, you'll need to know some important factors that ensure more opens and clicks. Here are the important newsletter template best practices to ensure email marketing success.\nMany businesses try to cram every aspect of their business into their newsletter. You'll have sales information next to an article about an industry trend, along with product information. The newsletter feels cramped and disjointed. If you want a newsletter that converts, then narrow your focus. You don't need to sell an entire store in a single newsletter. Put the focus on a single sale and why your customers need to take advantage of it. You can have an article about an industry trend and connect it products or services.\nThe last thing customers want is for their inboxes to be inundated with product offers. They want to be entertained and educated. When your newsletter is filled with nothing but products and prices, they get turned off. They'll either ignore most of the email or not open it all. You need to give them a reason to open your email. If you focus your newsletter on quality content, then the sales will follow naturally. Provide articles that educate them on your business or products, but don't make it a sales pitch.\nWhen most people check their personal email, it's not on a desktop at home. It's on their phone in the subway train on the way to work or in the doctor's office waiting room. When they hear the notification sound or see the icon on their phone screen, they're first instinct is to check what it is. If your email newsletter looks great on a desktop, but not on a smartphone, you're wasting your money. A responsive newsletter template makes sure your newsletter looks good on a smartphone, tablet and desktop. A newsletter can't convert if customers are confused at what they're seeing. It's important that any graphics are resized appropriately and calls to action, headers and footers look good too.\nWe all like to think that when our email arrives, customers spend hours looking at every article and word, but in today's digital world, you're lucky to get 30 seconds of their time. Most people will skim emails for items they're interested in, and you need to prepare for that. Use colors and type face to emphasize the important aspects of the newsletter. Use images with bright colors, bold, important words in the text, and make sure your call to action is attention grabbing. You only have a few seconds to make a sale before your customers move on to the next email.\nYou can spend hours and thousands of dollars on a gorgeous email newsletter template, but if your subject line is drab and boring, then customers may never get to see it. The eye-catching subject line is an artform and the Achilles heel of many business owners. First, you need to know your audience and the tone of your newsletter. If your newsletter is more business-oriented, then the subject line should match the tone, but still be creative. If the newsletter is fun and lighthearted, then do the same with your subject line. You can even include emojis in the subject line to really grab attention.\nJust like when writing an advertorial, every newsletter should have a call to action. This is the button, link or phrase that leads them to the next stage of the buying process. It could be a link to a product, blog post, or specially created landing page. You can have multiple CTAs in a newsletter, but there should be one primary. It's the one you want everyone to use. Put it in a place of prominence and make sure it stands out with a bright contrasting color. Be creative with the CTA just like the subject line. The goal is to grab their attention and get them to click to the next area of the sales funnel.\nAre your customers ignoring your emails or newsletter templates just not converting? Leave your thoughts and comments below and make sure to subscribe to our newsletter for weekly digital marketing insights!\n\u00a9 2019 Presto Media. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This video was made in 1993. At that time Professor Cowan was Chair of the Scottish Studies program at the University of Guelph, a position now held by Professor James Fraser.\nThis portion of the video includes footage of the McLaughlin Library's Scottish collection, which is now the largest collection of Scottish materials outside the UK.\nThe Scottish Studies Foundation has agreed to fund the installation of a digital archive centre at the University of Guelph which will allow this unique collection of rare and unique Scottish books and manuscripts to be digitized and placed online. Help us to complete this project by donating here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lifehacker's Better State of Living Conversation series is brought to you with the support of State FarmTM. Today, time management ninja Craig Jarrow is live in the comments to answer all your questions about traveling drama-free.\nDo thoughts of trains, planes and automobiles fill your dreams the night before a trip? Do you inevitably get to your destination and discover that you have forgotten to pack something?\nWell, traveling doesn't have to be a stressful. In fact, with a little preparation, you'll be able to relax and kick back to enjoy the ride. That's why I'm here to offer strategies to make your experience as efficient as possible. Check out a couple of quick tips below, and then let me help you solve your travel worries in the the comments.\nPlanning \u2014 Ever been traveling and realize that you didn't have all the details about your trip? A few extra minutes of planning can prevent a travel mess once you have hit the road. Make sure you have all your itinerary details in one, easy-to-access location.\nAvoid the Morning Scramble \u2014 The last thing you need is added stress before you get out the door for your trip. Don't do the morning scramble, and ensure that you get ready the night before. Packing the morning of a trip inevitably leads to forgotten items. Pack early enough that you have time to remedy any unforeseen issues.\nBe Early \u2014 Have you ever missed a flight? Do you run through the airport like an Olympic sprinter dragging carry-on luggage? You never know what may happen on the way to your flight (or train). Always be early enough that bad traffic, or a long security line, won't derail your plans.\nUse a Pack-list \u2014 How do you ensure that you've packed everything including your toothbrush? Use a personal pack-list to pack in less time and with more confidence. It will also help you avoid over-packing.\nFind Your Way Home \u2014 Getting to your destination is only half the trip. You don't want to have to stress on your way home either. Always put important items like car keys and parking stubs in the same location, so that you know exactly where they are upon your return.\nI'm looking forward to sharing some of my best travel efficiency tips and answering your questions. So head below, and ask away!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"2014 has been an amazing year for mobile. Mobile audiences have been increasing faster than ever before, websites are becoming more mobile-optimized. Your followers' demands for high quality content on their smartphones continue to soar as their time becomes more and more valuable to marketers.\nIt really is an amazing revelation.\nHow do you expect to reach customers where they are spending most of their time: on the mobile device?\nIn this post we're highlighting some of the most insightful articles we've read on mobile marketing this year that will help us improve our mobile strategy in 2015.\nSometimes we like to look back at the year's predictions and see if they are accurate with the present. Let's look at Ryan Blake's 20 mobile industry predictions for 2014.\nYou can easily analyze your mobile SEO performance using SEMrush and Google Webmaster. Aleyda Solis gives us step-by-step instructions on going through the process.\nIf it's not obvious to you yet \u2014 mobile matters. Now! Structural changes are happening with Google Search for mobile devices, and people are using this medium more than ever. Dr. Peter J. Meyers delves into an article to give us more information.\nWhen crafting your mobile strategy, you should be able to map out the possible touchpoints at every step of the way. Devika Girish offers strong mobile advice for us to learn from.\nWhen performing a mobile SEO audit, you need to focus on the more technical aspects of your mobile frameworks. Alex Moss sees these first steps as some of the most important.\nDo you want your mobile questions answered by one of the top marketers in the space? Andrew Warren-Payne takes the time to answer your burning mobile questions.\nThink of the mobile device as a completely separate device entirely. its screens are much smaller and their users spend less time on any one particular topic. Nicolette Beard showcases some of the best and worst mobile content examples.\nHow much do you know about your own mobile-optimized SEO? Learn from the masters (14 to be exact) in order to cultivate better practices on your mobile presence by Gary Viray.\nMobile applications or mobile websites; which one of the two is a better decision in the long-run? Adam Torkildson gives us the answer in a terrific post.\nJoseph Kerschbaum shows us five critical factors for mobile-optimized PPC targeting.\nEver heard of mobile app SEO? It's what allows you to position your mobile application within search using a combination of on-page and off-page meta data. Alan Smith provides a superb breakdown of app search optimization.\nThere are also several ways you can promote your mobile application. Rahul Varshneya lists the top three most effective ones to try out.\nIf you aren't knowledgeable on which metrics to track or how to report mobile metrics at all, you've come to the right post. Avinash Kaushik writes a highly valuable guide on how to find the right metrics to make note of, as well as how to create reports for them.\nHere are some more interesting mobile application metrics to consider adding to your reports by Bryn Adler.\nWe agree that people need to take mobile much more seriously. It's not just a channel, it's a lifestyle behavior. And you have to treat the content you share via mobile with more consideration. Jeff Fagel delivers an interesting conclusion in regards to mobile marketing.\nOf course, optimizing for the mobile device isn't a simple task. Mobile performance limits a lot of what you can do on the platform (although this should continue to improve with better smartphone iterations). Isaac Rothstein lists eight limitations that can affect mobile design.\nGet a refresher on the 'how' and 'why' of targeting mobile Internet users in this post from PromozSEO.\nSusan Waldes sheds light on mobile call conversions and how SEMs can properly tackle these calls with seven robust tips!\nKeep in mind the concept of deep linking with regards to directing your visitors closer to your content, product, or service. Ina Toncheva gives us more detail in her article.\nThink deeply about how you plan to structure your mobile design. Here's a special guide to mobile-first design by the folks at MDG Blog.\nWe hope you gain a lot of insight from these posts. May they help your marketing practices from now and into 2015!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The American Heart Association no longer recommends daily low-dose aspirin to prevent heart attacks in healthy patients over the age of 70, according to new guidelines published Sunday.\nSince the 1970s, low-dose aspirin, under 100 milligrams or less, has been recommended by doctors for patients at risk of heart attack, as well as for those that have already experienced a heart attack. Aspirin can prevent blood clots, reduce the risk of heart attack, reduce damage to the heart, and the risk of death during an attack.\nBut today, doctors have improved their ability to treat the risk factors of heart disease, such as hypertension, diabetes, and high cholesterol. And recent research has suggested that the increased risk of gastrointestinal bleeding likely outweighs the benefits of aspirin \u2013 especially for those at high risk for internal bleeding. This risk increases with aging, and with kidney or heart disease, diabetes, a history of ulcers, and high blood pressure. Medications like steroids, nonsteroidal anti-inflammatory drugs, and oral anticoagulants can also increase the risk of bleeding.\nOther research has raised questions as to whether aspirin offers any benefits at all for patients with a lower risk of heart disease. The new guidelines promote lifestyle changes and statins as a preferable treatment, as well as statins for those with high cholesterol.\nAspirin may still be recommended for some high-risk patients that struggle to lower cholesterol or manage blood sugar, in cases where there is no otherwise increased risk for internal bleeding. It can also benefit patients that have already had a heart attack, stroke, open-heart surgery, or that have had stents inserted to open clogged arteries. The new guidelines specifically change recommendations for patients without heart disease, and with risk factors for internal bleeding.\nFor younger age groups, aspirin is now considered a 2b recommendation, indicating that it's not an ideal choice, and that there is substantial disagreement among experts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here at The Paper Review, we are obsessed with graphic design. We love it for web design, on merchandise and especially stationery!\nWe had the honor of interviewing a really talented designer by the name of Kristen Newman. Based in California, Kristen has an extensive client list including Target, Ross, HomeGoods and more!\nTo learn more about Kristen's journey and insight on how it is to work as a graphic designer, keep reading!\nTPR: Tell us a little about yourself and where you're from.\nKN: I'm a freelance Graphic Designer who was born and raised in San Diego but now resides in Huntington Beach, CA. I got my BA in Graphic Design from Long Beach State in 2012.\nTPR: What were you doing before Kristen Newman Designs?\nKN: Before I started my business, I worked as a Graphic Designer at several various companies, which taught me a lot about my own personal style and abilities. Each company that I worked for was in a completely different industry, which allowed me to gain experience and knowledge working with many different types of design.\nTPR: Tell us about your brand and how you got started.\nKN: I got started with my brand while I was still in school. I was in the Greek system, where we had an event of some sort almost every other day, and I had a lot of friends that would need help designing t-shirts or flyers. I began my career by freelancing for my fellow students that would come to me for help.\nTPR: Where does your inspiration come from?\nKN: My inspiration mostly comes from current trends that I see throughout my day. Anything from fashion, to architecture, and even nature can lead to inspiration.\nTPR: How do you define your design style?\nKN: My personal style is clean and simple with a feminine touch. I love black and white as a predominant color, with a secondary soft, pastel color.I also like a lot of white space to keep things simple and easy to read.\nHowever, the most important part of working with different clients is to design to the clients' needs and taste, and not my own. I have to be flexible and realize that I may not always love what someone else loves, or vice versa.\nTPR: What's the most important thing you've learned on your journey so far?\nKN: The most important thing I've learned so far, is to put pride aside when working on any project. If a client doesn't like something that you created for them, it doesn't mean you're a bad designer. It only means that you have different tastes, and that's okay!\nEveryone is different, and you learn more and more about your client and their preferences as you go. So if you're ever told to scratch everything and start over, you have permission to be disappointed for 30 seconds, and then you open a new document and start over. Design isn't personal.\nTPR: Is there any advice you would give to other creatives?\nKN: As far as advice goes, I would have to say just not to get discouraged if you don't start receiving a ton of work as soon as you start your business. Building a brand takes time! A LOT of time! Most of the clients you first get will be friends and family.\nYour business will slowly build up over time through referrals and word of mouth. Before you know it, you'll have people you've never met contacting you saying that they saw your work somewhere and loved it, and would like to hire you. But that all takes patience and it can be several years before you start getting clients willing to pay what you're worth.\nTPR: Where can people view your work and collaborate\/shop with you?\nWe'd like to thank Kristen Newman so much for doing this interview with The Paper Review. It's always so inspiring hearing from amazing creatives in the industry and we love bringing you insight to something they are so passionate about!\nBe sure to check out Kristen's work via the links above and stay up to date with her amazing designs by following her on Instagram!\nThank's so much for reading and sharing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Bachelor's in Real Estate Development is a professional degree program focusing on the acquisition, development and sale or rent of commercial and residential properties. Many universities offer this, and similar programs for aspiring realty moguls. The core concepts imparted to students typically entail professional communication, property law, commercial business and tenancy. These areas of knowledge train graduates to be well-rounded professionals in the real estate world.\nWhat is a Bachelor's in Real Estate Development?\nThis focused curriculum readies students for the challenging landscape of real estate and the best practices for successful investments.\nWhat Are the Benefits of Earning a Bachelor's in Real Estate Development?\nDevelop the knowledge that is included in tests required for real estate agent licensure.\nLearn to connect with clients and provide meaningful assistance in finding a home.\nLearn about both the sale and purchase of properties.\nUnderstand best practices for developing properties for commercial and residential use.\nMaximize potential returns for both income properties and flip properties.\nThese skills and abilities make a Bachelor's in Real Estate Development a benefit to anybody seeking success in realty.\nWhat Kind of Career Can You Expect With a Bachelor's in Real Estate Development?\nShow homes to prospective buyers.\nPartner with sellers to close quickly.\nThese standard duties are ones that are typically covered in a Bachelor's of Real Estate Development degree program.\nAssess initial investment that must be made.\nConnect with potential renters or buyers.\nDevelop properties to optimal value.\nThere are plenty of schools in the US, UK and around the world where you can learn these skills and earn a Bachelor's in Real Estate Development. If you would like to join the realty game with the know-how to succeed, start researching schools today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saturday's EPP 60 competition was completed after numerous showers. John Bennett won with a perfect 100, with myself second.\n21 pilots entered for Sunday's F3F. Unfortunately the showers were persistent, and only two rounds were flown before the event was abandoned.\nMany thanks to Jon Edison and NYSA for hosting under difficult conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"**LC's Adventures in Libraryland** A YA Book Blog: Follow Me Friday (16) & Book Blogger Hop!\nI've been really loving The Iron King series though, so I'm loving your answer!\nHopping through. Hard not to pick Harry Potter. I did. Iron Fey series is a great choice. I agree.\nI feel like I'm one of the few that wasn't so thrilled with the Iron Fey series. I do like Ash though. I'll continue for more of him!\nI've heard a lot of people say Harry Potter, too. The Iron King, though? That looks interesting. I wonder how I managed to miss reading it? I might have to correct that. :) New follower.\nI love Harry Potter and that was one of my answers too! So was Iron Fey though! :] Both of them are amazing and I like that you can switch between the human side and the magic side! Happy Friday!\nI would love to live in the world of Harry Potter!\nI've read book one and two in the Iron Fey series, and loved them! The Iron Daughter was fantastic!\nI still need to read the Iron Fey series. It is on that ever growing list!\nAmazing choice. The world of the Winter Court would be fantastic to see.\nOld follower hopping by!I haven't read the Iron Fey series but I'll bet it's beautiful.\nNew follower, I wanted to pick harry potter but i figured everyone would.\nI also picked The Iron Fey world, I agree with your answers! This is definitely a great world.\nI recently changed my profile to reflect the fact I'm almost done with library school. Please come and read my new introduction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Look out for a new poster on the London Underground this week. It's Anne Wilson's 'Winding Through the City' which won first prize in the Serco Prize for Illustration on Monday evening. London Transport Museum is home to an exhibition of the best 50 entries to the competition run in partnership with the Association of Illustrators. The brief was to \"submit images that feature the River Thames as an exciting and varied place for Londoners and visitors alike.\"\nThe exhibition runs until 3rd June 2011 and has some wonderful illustrations on display. The Thames is such a part London's iconic images that there was an outcry when it was temporarily removed from the Tube map.\nMatt Oxborrow's \"An Average Day on the Thames\" was my personal favourite, as I loved the idea of people floating down the river on roundels. He's also illustrated a whole range of Londoners who might use the Thames for work or pleasure.\nIt was interesting to see so many interpretations of the Thames many of them showing a vertical West\/East divide, rather than the more familiar horizontal division.\nI liked Anne Wilson's winning entry and it was good to see how well it worked on London Transport Museum merchandise, that was on display. The detail on some of the other illustrations was amazing, but I'm not sure how well they would have transferred to mugs, mousemats, bag and posters.\nI'm a sucker for retro looking things and particularly liked Matthew Bromley's Explore The Thames for this reason.\nAndy Potts' London Flow worked really well with a simple colour scheme and various modes of transport literally flowing through the river.\nI loved the fairytale detail in Madalina Andronic's 5 o'clock Thames. She was standing next to me when I was taking a picture and I asked her how long it took. \"Three days\", she said. I don't think I could complete something like that in three years!\nThere's a few more pictures on my Flickr set of the exhibition. If you look closely in one you'll see the reflection of IanVisits who also came along the opening and has written a blog post with his impressions too.\nTo support the exhibition on Saturday 21 May at 3pm London Transport Museum Senior Curator, Claire Dobbin, will give an illustrated talk \"Poster Art and the River Thames\" about how the River Thames has featured in over a century of London Transport posters. The hour long talk is \u00a38 \u2013 or free to annual season ticket holders (you're advised to phone 020 7565 7298 in advance to reserve your place).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the City of Balashar, the trademasters are in dire need of adventurers to seek out glory and a small gold to help them gain the benefit for the local merchants and their superiors. But within each task, will always be a 2nd guessing towards the greater evil.\nThis game is located in the event of N\u00f8rdmarked in DGI byen the 2\/3-2019 and will be held between 12 and 16. The price for this event will be 30 dkr.. and will allow you entrance towards the whole marked before and after the game. Sign ups will be open from 11 towards 12.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AA Media was started in 2006 with the grand idea of offering a Complete Design, Print, Promotional and Media Solution for our clients.\nWe didn't want our clients to waste their valuable time trying to source different products from different supplier's.The idea was to simplify the sourcing process to one point of contact \u2013 AA Media.\nAs we can manage all your print and marketing needs it makes it very easy for us to keep close control of your brand identity and company message ensuring your image is consistent across all your physical and on-line material.Of course, you may have a team in house who manages this for you and you may just want to get some brochures or pens printed. That's no problem, many of our clients work with us to supply their print or promotional requirements without the print management element and we are happy to accept your own print ready artwork if you already have a designer working on it for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The software does not properly manage a user within its environment.\nUsers can be assigned to the wrong group (class) of permissions resulting in unintended access rights to sensitive objects.\nThis item needs more work. Possible sub-categories include: user in wrong group, and user with insecure profile or \"configuration\". It also might be better expressed as a category than a weakness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Sterling Silver Earrings Collection \u2013 Tagged \"baltic earrings\" \u2013 Paz Creations Ltd.\nThese shapely earrings are made from sterling silver and pure amber. No matter how small a piece of amber is, it still looks amazing in the light, just like these lovely earrings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"explains the issue and proposes new wording to clean it up.\n| storage duration associated with the enclosing block.\naddress is taken, there is a problem.\nthe wording should be altered to clarify it.\nThe function f() always returns the value 1.\nlonger guaranteed to exist, which is undefined behaviour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Bruins are bound for success this season! Gear up to support UCLA with UCLA Bruins clothing from CBS Sports. Grab UCLA Bruins apparel, UCLA basketball merchandise and Bruins gear for the whole family, including UCLA T-Shirts, Hats, Jerseys, Sweatshirts and more from CBS Sports Shop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Curry maybe born in India but it's found a permanent home in the heart of the British. The Brits have embraced curry culture stupendously, in fact, London has more curry houses than New Delhi or Mumbai and this industry raking in more than a billion pounds a year. Mostly Friday night is a curry night and most of the Brits love to have curries and going out for a curry or simmer one or two at home in their own comfort zone over the weekend.\nThe British put the curry under the spotlight pairing it with their own beloved beer, because chilled and crisp beer can embrace a range of curries and certainly beer offers more complex flavours and aromas that go so wonderfully with all things spicy. Curry and chilled bear is as synonymous as fish and chips. Friends, a curry and a bottle of chilled beer is perfect way to have good times!\nRecently the lovely folks from Kingfisher sent me an incredibly generous case of Kingfisher Beer, The real taste of India.\n\"Brewed for over 150 years to an authentic and most venerable recipe, Kingfisher is the first choice of the nation, from Delhi to Doddanapudi. Established in Mysore, Bangalore in 1857. Kingfisher has risen from a soldier's tipple to a nation's pride and joy. Kingfisher is India's No.1 beer and is one of the country's most heralded exports. Now brewed in Britain, this triumphant tipple can trace its roots all the way back to 1857 when Castle Breweries first started brewing in the majestic city of Mysore. As it has grown, the beer has become synonymous with the exciting side of Indian life. It is a sponsor of the Bangalore Royal Challengers, the Force India Formula 1 team and countless music festivals.\"\nBecause I myself don't drink any alcohol, few friends and family happily volunteered in tasting this top choice premium Kingfisher beer for me. They all had a great fun pairing this beer with array of curries, veg and non veg and these are the feedback that I received from them.\n\" It's cool, crisp and light and goes perfectly with a spicy curry and best served ice cold\"\n\" We had kingfisher beer with spicy chicken curry, slightly sharp but not too bitter or too sweet, just right and not too gassy\"\n\"Very crisp and fresh, definitely I would drink this at the cricket while I munch on spicy hot samosa\"\n\"A well made, cool and crisp beer, good stuff for the hot days and great even with barbecue, it's the ultimate summer drink\"\nSo hey, let's make it an Indian summer with Kingfisher and celebrate in the sunshine with the real taste of India.\nSpice it up or cool it down this summer with Kingfisher, the No. 1 Indian beer the world-over. Crisp, clean and unfailingly refreshing, it's a top choice for summer, whether you're having a BBQ with the friends and family, enjoying a summer of sport or looking for a perfect curry companion.\nIf you WATCH THIS SPACE, then I will also be doing a giveaway of a case of Kingfisher beer soon.\nTo pair it with a traditional Indian recipe, today I am sharing Matar Paneer also known as Matar Paneer Masala. Matar Paneer is a incredible delicious combination of soft and velvety paneer (the mild Indian cheese) and sweet green peas, which works a charm when left to absorb all the flavours of your thick and mildly spiced tomato gravy. It is one of the most popular vegetarian curries in every Indian household and restaurant. A perfect recipe for you to enjoy when you are surrounded by friends and family. It's an incredibly tasty, finger licking good dish, simply phenomenal.\nI can vouch for this recipe as you won't forget the taste in a hurry, this can beat any restaurant matar paneer, best to pair it with Peshwari Naan.\nPaneer is readily available in all the Asian store and major British supermarket.\nIf you are using fresh peas, steam them few minutes.\nIf you don't want to use cream in this recipe, use full fat milk.\nIf not using fresh tomato use canned tomatoes.\nFor a vegan recipe use tofu and coconut or cashew cream.\nMatar Paneer is incredibly and irresistible delicious and popular main course for vegetarians, prepared with sweet green peas, paneer in creamy and mildly spiced tomato gravy.\nIf you are using shop bought paneer, soak in a hot water for 15-20 minutes.\nRemove it from the water, pat dry with the clean kitchen towel and cut into cubes.\nHeat very little oil in a griddle or frying pan , and saute paneer cubes until light brown.\nPlace blanched tomatoes, garlic and ginger in a grinder and make smooth puree.In a kadai, heat oil and add all the whole spices, once they splutter add finely chopped onion, saute until it turns light brown.\nNow add red chilli powder, turmeric powder and ground cumin and coriander.\nCook the masala for few seconds, then add tomato puree.Keep heat on medium heat, keep stirring the mixture until you see oil separates masala, ( This indicates that your masala cooked properly ) Add frozen peas and salt, cook couple of minutes and add water ( 3\/4 cup )Bring it to a boil ( it will take 3-4 minutes ).\nAdd kasoori methi, garam masala and kitchen king masala if using.\nNow add fried paneer and cream ( If using ) and simmer for 4-5 minutes.Switch off the heat, serve in a serving bowl.\nGarnish it with the fresh coriander. Serve hot with roti, paratha, puri or naan and chilled beer.Enjoy !\nDisclaimer :- Thank you Kingfisher for sending me a case of beer bottles. I was not told or paid to post a positive review. All the opinions are my own.\nI love matar paneer and Kishfisher is my first choice when it comes beer with Indian food!\nYou're making me hungry and thirsty this morning! I'll have to look for KingFisher at the market -- and I definitely want a bowl of that paneer!\nI'm a huge fan of muttar paneer! YUM!\nThat is a lovely looking curry! And I agree that a nice bottle beer is great way to wash down a curry. I'll have to look for that beer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.5 What to Look for When Buying a Kayak?\nWhether you want to spend a leisurely afternoon outdoors, or you want a fun, action-packed adventure you can enjoy with friends, kayaking is an excellent activity to pursue. This is a very versatile way to enjoy the outdoor scenery, get the best views from rivers, lakes, and the ocean, and even get a good workout. There are so many different ways to kayak, from a slow-paced, relaxing ride, to hitting some heavy water with big waves and a lot of adventure.\nIn recent years, kayaking has continued to increase in popularity and for good reason \u2013 it is an affordable, easy, and unique way to experience your local waterways and nature offerings like never before. It is a great way to stay active and in shape, as well as to spend quality time with your friends and loved ones.\nYour body type \u2013 height, size, weight. Deep hull kayaks are for taller and heavier paddlers.\nShorter length kayaks are easier to navigate, and long kayaks are better with speed.\nThere is an infographic and set of FAQS at the end of the article to help you.\nThere are so many different options on the market when it comes to kayaks, and there are tons of different kinds of kayaks that are specially designed for use in a certain environment or for a particular purpose. In this article of kayak reviews, we will briefly discuss the main types of kayaks, and then we will provide a brief review of our pick for the best choice within each subcategory of the kayak. This takes a lot of the work and effort out of your hands, allowing you to choose from a preselected range of best kayaks that have been specially picked by experts who have combed through the myriad of available options on the market. Another top kayak guide on our website is best fishing kayak under 1000 which shares all the popular fishing kayaks in that budget.\nWhat follows is a buyer's guide to all the types of kayak available on the market \u2013 there are niche and specialty types that we have discussed in detailed articles on each category on our website. What follows is a short description of popular and commonly-used types of kayak, as well as a brief description of the best value and quality kayak we found within that category.\nAn inflatable kayak is a popular type of kayak for those who are looking for a lightweight, space-saving kayak, that is often a lot cheaper than a hard-hulled kayak. In the course of our research, we found the Advanced Element Inflatable Kayak to be the best choice of value and quality in this category.\nThe kayak takes up very little space and is incredibly light, making it very easy to store and transport. It can be inflated in about ten minutes, meaning you can hit the water without a ton of prep time. It is a stable craft that is well suited for leisurely rides.\nHowever, it should be noted that this is a very shallow-hulled craft, so it might not be very comfortable for taller riders, and it cannot handle more than 235 lbs., making it unsuitable for larger riders.\nStrong material Not for paddlers above 235 lbs.\nFor the avid fisherman, kayak fishing is a fun and unique way to enjoy a bit of angling in your free time. There are actually a ton of benefits to using a kayak for fishing, including the fact that, due to their low profile, it is a lot easier to stealthily sneak up on a school of fish than with a traditional fishing boat.\nA fishing kayak is also quite a bit less expensive than a fishing boat. You can get a high quality, top-of-the-line fishing kayak for between $500-$3,000, depending on the features you are looking for.\nWe determined in the course of our research that the best combination of performance and value is the Malibu Kayaks X-Factor Fish and Dive Package Kayak. It is a high-performance fishing kayak that has a ton of added features with the fisherman in mind. It offers a stable ride and can handle a maximum weight capacity of 625 lbs.\nIt should be noted that, if you have trouble with seasickness, this fishing kayak is probably not going to be the best choice for you, as you feel a lot more of the waves and movement than you would in a traditional fishing boat. For people who are taller or heavier, the size of the hull may be insufficient for comfort.\nA sit-on-top kayak differs from a traditional kayak in a way that it is implied by the very name of this type of kayak. In many kayaks, you actually sit inside the hull of the craft, whereas with a sit-on-top kayak you, well, sit on top. One of the biggest benefits of this type of kayak is that it provides for much great stability and has a shorter learning curve than a sit-inside kayak.\nThey are easy to get in and out of, which is great for people with limited mobility, or if you want to be able to easily take a dip in the water when you so choose. When it comes to the best value of sit-on-top kayaks, we selected the Perception R15 Pescadors 120 Kayak as our top choice.\nThis is a lightweight sit-on-top kayak that is easy to carry and transport. It features a tank well with a bungee cord and a keepers foot brace that keeps the paddler comfortably secure during the ride. It also boasts proprietary Comfort Seating System design to provide for enhanced comfort to the rider.\nIt should be noted that the biggest complaint about this kayak is that, especially for taller people, extended riding can lead to back pain as you do not get the support with a sit-on-top kayak that you do with a sit-inside kayak. This is also not the kayak for you if you want to avoid getting splashed a ton during the ride. You will not stay dry on a sit-on-top kayak.\nWhen choosing a good ocean kayak, durability and comfort are going to be two of the most important concerns you have. Most commonly, when it comes to an ocean-going kayak, we recommend a sit-on-top kayak as opposed to a sit-inside kayak, as you get enhanced stability and don't have to worry about a sit-on-top kayak taking on water.\nDepending on what type of ocean activity you are taking part in will determine how long, deep, and wide of a kayak you want, but good quality, general purpose ocean-safe kayak will be suitable for a range of different activities. Our top choice for a versatile, ocean-going kayak, is the Ocean Kayak Prowler 13 Sit-On-Top Fishing Kayak. We would also recommend you to check Ocean Kayak Malibu series. If you have a dog or pet, we rate Malibu series as the best kayak for dogs and pets.\nThis is a kayak that is designed with performance in mind and is even suitable for operation in rougher waters. The craft features a couple of different storage options, whether you are bringing fishing gear or just some refreshments. It can handle a max weight of about 400 lbs., making it suitable even for heavier paddlers.\nSome of the biggest complaints associated with this kayak are about the placement of the pre-installed fishing rod holders. There are some fishermen who have noted that the rod holder does not have ideal placement for hands-free fishing.\nNot everyone is looking for a top-of-the-line, performance kayak that is rated highly for those who are keen on extreme sports. Some of us are beginners that are just looking for an inexpensive way to (pun intended) dip our toes in a new pursuit. In short, looking for cheap kayaks. We want something that has a reasonably easy learning curve and that provides a stable and comfortable ride and doesn't have a huge price tag.\nWhen it came time to pick a high quality, inexpensive kayak for beginners who just want to get a feel for the activity, we chose the Sun Dolphin sit-inside kayak. It features a sturdy, well-built design and comfortable seating. It also has an array of storage options if you want to bring refreshments, electronics, and more.\nThe biggest complaint associated with this basic kayak is that it isn't the most maneuverable kayak available, so it probably isn't suited to water terrain that has a lot of curves to contend with.\nA tandem kayak is one that is designed to hold more than one person. Most often, these are designed for two or three people and can be sit-on-top or sit-inside in design. These are popular as they allow you to actually ride in the same watercraft as other people, making it a more communal activity.\nWhen it comes to the best tandem bike for the money, we chose the Lifetime 10 Foot Sport Fisher Tandem Kayak. It can seat up to three people and is designed to be able to handle a max weight of about 500 lbs. It is sit-inside in style, making it a lot more comfortable and stable, which is a good thing when multiple people are trying to operate a craft.\nThe biggest complaints associated with this kayak are that they can take on a bit of water, so don't expect to stay dry while paddling this craft. There are also some complaints about the placement of fishing rod holders.\nA recreational kayak is a more basic and general-purpose kayak, as opposed to one that is designed with a specific activity in mind (such as fishing or hitting rough water). These are great for beginners and those who want to be able to use the same kayak for a range of different activities.\nFor the best recreational kayak, we chose the Wilderness Systems Aspire 105 Sonar Kayak. It is a great value and is a lightweight, sit-inside kayak that is well suited for use on lakes, rivers, and the ocean and is a great craft for more leisurely pursuits.\nIt can hold up to 400 lbs. and has knee and thigh padding for increased comfort for the paddler. It also features a range of storage options, allowing you to bring refreshments and other necessary gear without taking up a ton of space in the craft.\nThe biggest complaint associated with this kayak is that it is not the best kayak for those who are really tall, as it is a fairly shallow-hulled craft.\nThe advanced elements, Malibu Kayaks, Ocean Kayak, Perception, Old Town Canoes & Kayaks, Sun Dolphin and Lifetime are some of the best kayak manufacturers.\nAgain, you have to assess your requirements and activity before choosing a kayak. Some kayaks are for calmers waters and fishing and some are designed and know for tracking and handling in rough waters. The size and space is also something you should consider before buying a kayak.\nWhat kind of kayak is best for beginners?\nWhich Kayak is the most stable?\nOne of the biggest differences in the design of kayaks will have to do with the profile and construction of the kayak. If you want a great kayak for slicing through the still waters of a lake, you will want something with a very different aerodynamic body profile and construction than if you want the perfect canoe to hit some white water on a river. It might not seem like there would be a whole lot of difference, I mean, a kayak is a kayak, right? Well, the answer is, of course not. Otherwise, there wouldn't be so many different designs.\nWhen it comes to kayak design, the differences typically fall within the following categories: width, length, depth, and hull (or body) shape. Each of these things will make a difference in what kind of activity it is best suited for and we will briefly describe why you might choose to emphasize one of these factors over another, depending on what you plan to do in your kayak.\nThe width of a kayak really correlates to the overall stability of the watercraft. The wider a kayak is, the more stable it will be in the water. However, due to the wide profile, it is less efficient to use a wider kayak than a narrow one.\nThis means that a wider kayak is better suited to those who are looking for a more leisurely ride. It is also a better choice for heavier or taller passengers as it gives you more leg room and gives you a higher center of gravity, which will reduce your likelihood of tipping the kayak.\nNext, you will want to consider the length of the kayak. Kayaks that are shorter will be easier to navigate and will be more agile in the water than a longer craft. However, the longer the boat, the more stable the craft will be overall. Also, a longer length kayak will track straight and travel faster and more efficiently than a shorter boat.\nDepth is the next variation in kayaks. This is basically about comfort and fit. If you are a taller or heavier rider, you may want a deeper hull, as it gives you more leg and body room, while also increasing the amount of weight the craft can safely carry. For smaller riders, and those who are really looking to maximize performance, the shallower the hull is the less drag it will face from the wind.\nFinally, is the design of the hull, or body, of the craft. This refers to the shape of the hull, which will be tailored towards emphasizing different qualities. A shallow arch style will have good stability, so you are less likely to tip upon entry, and they are very easy to maneuver.\nA shallow \"v\" shape has a lot of the same characteristics of the shallow arch, except that the hull comes to a \"v\" shape, which improves the tracking of the boat and can also improve the feel of stability.\nWith the hard chine type of kayak, there is edge and angled shape where the hull comes to a point on the bottom. This gives the paddler a much higher degree of stability. A soft chine has a less angular, smoother shape, and is one of the most popular and common shapes of kayak hull that you will find.\nNot only is kayaking a lot of fun and a great hobby for people of all walks of life, whether you live in rural or urban area, but it is also incredibly good exercise too. A lot of people agree that the best kind of exercise is that which you don't do for the express purpose of exercising. What we mean by this is that the point of kayaking is that it is fun, and it also happens to be great cardiovascular exercise. Whereas, when we go to the gym, it is just exercise for exercise's sake, and most of us don't find it particularly enjoyable. Another great thing about the fitness benefits of kayaking is that it is a full-body workout, working your upper body, core, and lower body, as well as your arms and shoulders, all at the same time.\nThere are many reasons that so many people love kayaking. It is a unique and unforgettable way to experience your area's lakes, rivers, and oceans, in a way that is unlike any other outdoor pursuit. In a kayak, you can enjoy a relaxed, leisurely ride around a calm, placid lake, or you can rip the rapids in a sport-style kayak for an exciting, more extreme-type of afternoon fun. Kayaking is a great activity to bring together friends and family, and it provides a full-body workout that is, above all else, a lot of fun.\nIn this guide, we briefly discuss the benefits of kayaking, as well as some of the differences between different types of kayaks and what they are best suited for. Then, we take a look at some of the most popular types of kayak brands and models available today, along with a short, comprehensive review of the kayak we found to be the best value and quality for the money within each subcategory. We chose to cover the most common types of kayak and to choose a model within that category that was the best combination of value, performance, and quality.\nWhether you choose one of the kayaks we review here or use this as a starting point for further research and shopping, we are sure that you will love the varied ways you can enjoy the water and outdoors with a kayak. With the information we provide in this guide, you are now armed with the information you need to choose the right kayak type for the activity you are looking to participate in.\nI love what you said about having a solid understanding of different kayak designs and how that plays a role in the water. I would say that it's best to contact a reputable sporting goods store in order to obtain professional opinions. My friend wants to buy a kayak for fishing, so I'll be sure to help him find a reliable sporting goods store in his respective area that has a wide range of kayak models.\nHey Michael, great article with lots of useful info about starting up in the kayak world. Thanks for sharing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Found one file extension association related to 18 Wheels of Steel: Pedal to the Metal and 1 file format developed specifically for use with 18 Wheels of Steel: Pedal to the Metal.\n18 Wheels of Steel: Pedal to the Metal is truck simulation computer game for Windows. Take your show on the road. Drive your rig to make it big and build your business. Money talks - when you're in charge, let the drivers to their job while you count your cash. You're not only a trucker, you're a truck tycoon.\nSTEP ON IT Stay one step ahead of the law as you rumble through crowded cities, past bustling sea ports and across rolling plains.\nThe 18 Wheels of Steel: Pedal to the Metal software seems to be old or discontinued.\nIf you need more information please contact the developers of 18 Wheels of Steel: Pedal to the Metal (ValuSoft), or check out their product website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for an experienced tutor for my 9 year old son ( Aspergers \/ ASD). Experience with ASD essential. Looking to work on social skills and teaching of maths concepts such as time, money etc in real life practical environments aswell as actual home tutoring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Richard Li Tzar Kai is a Hong Kong businessman and philanthropist. He is the younger son of businessman Li Ka-Shing and brother of Victor Li.Li was 26th in the Forbes List of Hong Kong's 40 Richest people for 2010. The same publication named Li as 773rd in the list of the world's billionaires, with an estimated fortune of $1.3 billion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: Purpose: Multifocal contact lens use is not extended in presbyopic population in Spain. The aim of this work is to develop a questionnaire to investigate the reasons of this lack of use among the target population of these contact lenses. Methods: A bibliographic review was performed in order to analyse the scientific paper published related to presbyopia and contact lenses, as well as the design of questionnaires about multifocal contact lenses. With this information, it was created a group of experts which developed the items to ask in the questionnaire. Results: A questionnaire was developed selecting by brainstorming the items that should be present on it. Then, it was held a meeting with a small sample of the target population to take into account their point of view. Finally, the questionnaire was validated following Delphi method. Conclusions: an 85 items questionnaire has been developed to investigate the reasons why multifocal contact lenses are not used among the presbyopic population.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are a few hip hop beats before and after mastering samples from a group an artist I worked with. If you need audio mastering check out my low rates. Visit this page for more mastering samples.\n\"I just loaded it through my protools. I got it to play and it sounds amazing. I am glad that you assisted me with this project and I don't think I could have ever got it to sound this good without your service. Good Job. \"\nThis CD represents the future of music. Buy a Hip Hop beat online. Record and mix in your vocals at home. Get them professionally mastered by JR mastering (for a fraction of the usual price). And the finished product sounds like a major label ($15,000 per song) production!\nIn this day and age, your music can sound just as good (or better) than a major label, for 99% less than what they're paying!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavvbs b/data_all_eng_slimpj/shuffled/split2/finalzzzavvbs
new file mode 100644
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+{"text":"\"Leif was recommend via a friend. Engagement was light, yet we got really fast feedback and he got the concept out of the box, so much so that we will look to engage on client projects.\n\"I recently worked with Leif on a pitch for a video production project for the Red Cross. Leif was a pleasure to work with and understood exactly what was required in order to construct a successful pitch.\nHe did everything on time, professionally and with enthusiasm. I was very impressed with the amount of thought he put into his work and he was always thinking about my project whilst bringing new ideas to the table.\n\"Leif Kendall was introduced to 2fifty.com at a fairly late stage which meant he had to work with an established project. Although it can be difficult to bring new ideas without input from the beginning, he adapted to the situation and proved he was able to understand our concept. Leif interviewed us to ascertain our requirements and carried out independent research to support his ideas, he also worked closely with our graphic designer to ensure continuity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hayrides \u2013 includes 3 acre Corn Maze & PYO Pumpkins!\nPetting Zoo \u2013 Free. Need to pay for feeding animals.\nPlease purchase tickets prior to getting in line for the activity!\nTickets can be purchased at ticket booths. NO ATM \u2013 Cash back up to $50 with purchase at registers. We accept all major credit cards!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This standard is issued under the fixed designation E92; the number immediately 1 These test methods are under the jurisdiction of ASTM Committee E28 on. Astm e92 Vickers Hardness of Matallic Materials \u2013 Free download as PDF File . pdf), Text File .txt) or read online for free. priate safety and health practices and determine the applica- bility of regulatory limitations prior to use. 2. Referenced Documents 2. 1 ASTM Standards.\nHardness, although empirical in nature, can be correlated to tensile strength for many metals, and is an indicator of wear resistance and ductility.\nRecommendations for microindentation testing can be found in Test Method E Vickers and Knoop Hardness Standardizing Asttm.\nHowever, in practice, the most commonly used force units are kilogram-force kgf and gram-force gf. This rise in hardness number with lower test forces is often more significant when testing higher hardness materials, and is increasingly more significant when using test forces below 50 gf see Test Method E Link to Active This link will always route to the current Active version of the asgm.\nWhen Newton units of force are used, the force must be divided by the conversion factor 9. Today, the hardness numbers are internationally defined in terms of SI units, that is, 9e2 test force in Newtons N.\nThe significant differences between the two tests are the geometries of the respective indenters, s92 method of calculation of the hardness numbers, and that Vickers hardness may be used at higher force levels than Knoop hardness.\nReferenced Documents purchase separately The documents listed below are referenced within the subject standard but are not provided as part of the standard. Historical Version s \u2013 view previous versions of ast. This standard provides the requirements for Vickers and Knoop hardness machines and the procedures for performing Vickers and Knoop hardness tests.\nHowever, because of the historical precedent and continued common usage, force values in gf and kgf units are provided for information and much of the discussion in this standard as well as the method of reporting the test results refers to these units. Hence, the Knoop hardness test is very useful for evaluating hardness gradients since Knoop indentations can be made closer together than Qstm indentations by orienting the Knoop indentations with the short diagonals in the direction of the hardness gradient.\nMicroindentation hardness tests also allow specific phases or constituents and regions or asstm too small for macroindentation hardness testing to be evaluated.\nWhile Committee E28 is primarily concerned with metallic materials, the test procedures described are applicable to other materials. Standardization of Vickers and Knoop F92.\nFor isotropic materials, the ashm diagonals of a Vickers indentation are equal in length. Other materials ast, require special considerations, for example see C and C for ceramic testing. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use. The Vickers and Knoop hardness numbers were originally defined in terms of the test force in kilogram-force kgf and the surface area or projected area in millimetres squared mm 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description Safe online shopping is all about doing your homework and if you've never searched online then start with some recommendations from your family and friends and try to find out more about the organization you intend to make deal with and pass on your credit card information. Be taught more on an affiliated link by visiting linklicious spidered never.\nAll that is impor-tant because if something goes wrong with your purchase, then the extra information you have the greater place you will your hands on. Always search for the company street address, contact number and other data if it's a nearby site. If you're unsure of a companys track record you have to do some research online when you can seek out complaints and the company name into google boards. Dig up more on our affiliated essay - Hit this website: linklicious youtube.\n1. Examine the description of the merchandise, price like the cost of taxes, currency and distribution and other guarantee facts. Great shopping websites will give an opportunity to you to confirm or reject your order before you purchase it.\n2. Check for the method of payment including credit-card, money order, cheque and so forth. In most of the circumstances it is always safe to pay through credit card including Visa, Master card that have chargeback schemes and wont hold you accountable for any undelivered things or unauthorized transactions.\n3. Discover further on linklicious by navigating to our thrilling URL. You need to confirm that the web site has a secure checkout because Secure Socket Layer (SSL) is the most typical technology used to secure shopping websites. Learn more on this partner wiki - Click here: linklicious free account. This is because it encrypts your private information as it passes on the internet.\n4. Check always the place where your facts will be located later o-n a secure server because some online businesses store them or destroy them after the transaction is processed.\nFor further information, visit website entertainment-coupon-book-2006.info.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi there, this attachment is about Plenty High, We Weigh All Our Deer And Have Plenty Of Room For That. Four Stations Total. Got This From Outback Feeders. (awesome Deer Cleaning Rack #5). This post is a image\/jpeg and the resolution of this image is 707 x 943. It's file size is just 119 KB. If You want to save This photo to Your PC, you may Click here. You could also download more attachments by clicking the following image or see more at this post: Deer Cleaning Rack.\nAs among the rooms to the homes inside the West around the properties in Plenty High, We Weigh All Our Deer And Have Plenty Of Room For That. Four Stations Total. Got This From Outback Feeders. (awesome Deer Cleaning Rack #5) continues to be regarded in contrast that ought to be there. Commensurate with the lifestyle of the country that wants to socialize and visit one another between relatives this is actually. Although a lot of modern homes which have a idea due to property that is limited but together with the interior design minimalist livingroom, a particular place to get visits the folks closest to you may also seem wonderful and elegant.\nYou are able to for the professionals send the interior style of contemporary minimalist family room needless to say, because it is likely to be carry fulfillment but some people choose to do-it myself. In this room you also can show your preferences at the same time for you to give your guests. The family room can also be viewed as an expression of the type of manager or home as this really is where you could offer a first-impression for the visitors. Pursuing some motivation not merely can make you into a look wonderful but in addition makes it look sophisticated.\n1. Use a reflection. Setting a big reflection in the family room also gives the effect be treated.\n2. Choose proportionally sized furniture. In the variety of furniture within the inside of the livingroom minimalist type 45 should be retained healthy with all the dimension of your livingroom minimalist. Should pick small coffee-table and a chair were in and comfy equilibrium together with the space.\n3. Utilize low- lasting bulkhead. It is possible to select drapes or any portable timber bulkhead as being a screen involving the living-room to some other area inside your home. While it has offered stunning decorations to various kinds of bulkhead that could meet a decorative purpose.\n4. Use carpet. In some houses you will not even find a chair but gentle carpet to get visitors while sitting cross-legged with pillows sit not small as Japanese-style houses.\nTags: Plenty High, We Weigh All Our Deer And Have Plenty Of Room For That. Four Stations Total. Got This From Outback Feeders., Plenty, High,, We, Weigh, All, Our, Deer, And, Have, Plenty, Of, Room, For, That., Four, , Stations, Total., Got, This, From, Outback, Feeders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Proudly serving the Greater Los Angeles area, Baron Construction has the experience and equipment needed to handle projects both big and small. We can tear down houses, concrete, pools, sheds, and so much more.\nWhen it comes to demolition, excavation, and backhoe services, nobody does it better than I.N. Backhoe Service Inc. They're experts at removing houses, pools, sheds, and more safely and efficiently.\nSince 1988, Big K Concrete Coring & Cutting, Inc. has been serving Southern California in all their coring, cutting, and demolition needs. They are interior removal specialists and promise free estimates and fast service.\nSince 1985, Saied Contractors Co., Inc. has been serving the Sun Valley, CA area. Their main lines of business include comprehensive demolition services. They also offer site preparation, heavy construction work, and much more.\nBeckco, Inc. offers expertise in concrete breaking-and-removal for larger projects. We work with residential and commercial clients in and around Pomona, CA. We've proudly served the area since 1971. Get the job done right by hiring Beckco for driveway demo, parking lot removal, patio removal and more. Please note, we do not offer swimming pool removal services.\nMagnum Land Clearing specializes in demolition and land clearing services. They also provide earthwork, excavation, soil transport, material hauling, trucking, grading, backfilling, disaster recovery & cleanups, sewer & utility work, and more.\nLocated in Los Angeles and serving communities in surrounding areas, Oscar Trash Dumpster Rentals is a leader in residential and commercial clean up services, as well as a wide variety of demolition services.\nStar Stone Inc offers full-service demolition, including barn demolition, concrete removal, and more. We have the experience needed to complete the toughest and most complex jobs in the Los Angeles area.\nAt Gama Contracting, service always comes first. They offer over 100 years of combined experience in the demolition and environmental remediation industry. Their training is extensive in asbestos abatement, structural and interior demo, and lead and mold remediation.\nDepending on the state of your concrete, you may be better off repairing the concrete instead of replacing it; and sometimes, it's the opposite.\nIn general, complete concrete removal and replacement is your best choice when the concrete has several wide, deep cracks that are starting to settle on one side or are uneven, if part of the concrete is pushed up thanks to cold weather, or if the concrete is settling due to improper grading preparation. In these instances, it's best to replace the concrete rather than repair it.\nOn the other hand, if there are just thin, hairline cracks with no evidence of settling, or the concrete is sunken because of excess weight being placed on the concrete, you should just repair the concrete, as it will be the fastest and most cost-effective option.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home\/ Gerard J. Chiche, D.D.S.\nDr. Chiche is the Thomas P. Hinman Endowed Chair in Restorative Dentistry, and the Director of the Center for Esthetic and Implant Dentistry at Georgia Regents University College of Dental Medicine in Augusta, GA.\nHe is a Past President of the American Academy of Esthetic Dentistry and is also, respectively with Alain Pinault and Hitoshi Aoshima the author of the textbooks: Esthetics of Anterior Fixed Restorations, and Smile Design \u2013 A guide for Clinician, Ceramist and Patient both published by Quintessence Pub. Co.\nHe became in 2009 the first recipient of the Endowed Chair sponsored by the Thomas P. Hinman Dental Society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I was attacked by a shark.\nMy partner and I had spent the morning fishing from our double kayak in the Bay of Islands in the far north of New Zealand. All the fish seemed to have gone on holiday as I had not been getting any bites at all. That was until now.\nBut first, let me tell you about the lead up to the events. It was our last full day, housesitting in the idyllic location. We should have been packing up, but I'd promised my father I'd bring home some fresh snapper, and well\u2026 the boat ramp is just across the road.\nThe sea was calm as we headed out, excited by the prospect of what we might catch.\n'Hope we catch a Kahawai,' I said to Mike.\nAs we paddled past the boats moored in the bay, I noticed one called Amityville. I shrugged it off, refusing to see it as an omen.\nMy shoulder ached so I stopped paddling and let Mike do all the work. I sat back and relaxed, taking in the scenery. We are so lucky to be invited to housesit in this stunning location; fresh fish on our doorstep, kiwi living in the surrounding bush, and of course Buddy, the dog we look after.\nMike's paddle splished and splashed as he propelled us along, serenading me with Just One Cornetto.\nI scanned the area for a good spot to fish from when something caught my attention. In the channel, a bright orange boat cruised along slowly, as if looking for something.\n'Is that the coast guard?' I asked.\n'Oh yeah, I forgot about that,' he said.\nTwo days earlier we were out fishing in the kayak, when a 45ft yacht sailed directly towards us, beating into the wind. I kept a close eye on him, hoping he'd seen us. We were probably sitting in the prime position for him to tack, so I wondered whether we should move out of his way just in case. I mentioned this to Mike, but he was snagged on the rocks and didn't want to cut his line. Meanwhile the yacht got closer and closer.\nThe brown yacht continued to encroach on our space. An old man on board raced around the bow of the boat, yelling and cursing, trying to take his sails down. I know a little bit about sailing and I'm pretty sure you can't steer the boat from the front. By this stage the yacht was almost upon us.\n'Okay, cut your line, we need to move. Now!' Mike continued to ignore me as he tried to de-snag, no sense of the danger.\n'Just a little,' I yelled back, silently cursing him under my breath.\nAnyway, that was two days ago. And someone phoned looking for a brown yacht\u2026 Hmmm, dead bodies were even more on my mind now.\n'You have an overactive imagination,' Mike said.\nIt's true, I do. But I'm an author, it's my job.\nWe bobbed around a while longer, but weren't getting any bites. I picked up my oar and paddled closer to the yellow buoy where we've had success catching fish in the past. I put my line back in and waited.\nWe have a system. I catch a fish and Mike hauls it into the net, then does all the gory stuff behind me.\nI slowly reeled it in, being careful as I didn't want to lose him. I could tell it was a snapper by his tell-tale bites and the way he tugged at the end of the line. Okay, I had no idea, but all we ever catch are snapper, so it was an educated guess.\nAs the fish neared the surface, I turned my rod back towards the net to make it easier for Mike to scoop it in. What happened next was totally unexpected.\n'Holy f**k,' I screamed as a huge shark appeared beneath the kayak. Immediately I turfed my rod and reel into the sea as far as I could.\nThe shark's tail came into the kayak and whacked me, before disappearing after my rod. I wasn't mucking around. I grabbed my oar and paddled like my life depended on it. Mike still had his line in the water.\n'Get rid of it, get rid of it,' I yelled, convinced the shark was going to take his bait.\n'Like f**k,' I said and paddled faster, keen to get to the safe hard ground, as far away from the huge beast as possible.\nMike took his time pulling in his line, leaving me to do all the paddling. By the time he'd reeled it in, we were back at the shore.\nIf anyone in the Bay of Islands catches a 12 foot bronze whaler with a pink rod attached, it's mine and I'd like it back please. The rod that is, you can keep the shark.\nThis entry was posted in Blog, HouseSitting and tagged House Sitting, Kayak fishing, New Zealand. Bay of Islands, shark. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Because the democratic process operates apart from the Church, it possesses no corrective against corrupted human nature beyond its own equally corrupt judiciary, which ends by judging not only points of law but morality itself.\nThere is no mention of the family in the U.S. Constitution, nor in any others patterned on its principles, and that in itself is ominous. With the onset of democracy, the family's juridical standing disappeared, along with that of God and His Church. Nor is there any mention of marriage, upon which the family rests as a social institution above all others. From the moment all men were declared equal, the family was legally replaced by the individual as basic cell and archetype of society. In other words, family rights simply disappeared into \"the Rights of Man.\"\nIn a radio broadcast in 1949 Pius XII said, \"Counterfeiting God's designs in social matters has taken place at the very root, by deforming the divine image in man. For his true figure as a creature with an origin and destiny in God, has been substituted the false portrait of a man autonomous in conscience, uncontrolled master of himself, irresponsible towards his neighbor and the social group, with no destiny beyond the earth, with no purpose beyond enjoying finite goods, with no law beyond the 'fait accompli' and the undisciplined satisfaction of his desires.\"\nIn addition to the previously mentioned topics, Hertz also addresses the clashes between science and faith, the Secret of La Salette, papal politics, and the apocalyptic \"abomination of desolation.\" Whether you agree with her or not, Solange Hertz causes us to reconsider basic assumptions of the Brave New World that we live in; assumptions we might otherwise have never questioned in our lives. Because of its thought-provoking nature, Beyond Politics is a must-read for all Catholics who are seeking to preserve their Faith.\nHistory which deals only with visible events cannot be real history, for it takes into account only one part of reality, and the less important part of reality at that. The chapters which follow might therefore be described as sallies into meta-history, tentative probes into that indeterminate region lying on the crust of the material world just below the vast invisible realm above it and just above that other realm below it. There, in juncture between time and eternity, the apparent causes of human events can be glimpsed momentarily conjoined with the real causes which lie beyond and just out of sight. The terrain remains the same, but in the half-light thrown upon it by divine revelation, secrets both loathsome and glorious may be suddenly disclosed to the beholder.\nSuch is the theme running through this text, which is a compilation of articles and talks which have been published in one form or another in The Remnant or other Catholic periodicals. The original versions of \"Lucifer's Primacy,\" \"Our Lady's Apocalypse,\" \"Recanting Galileo,\" and \"Hell's Amazing Grace\" figured among the Big Rock Papers published by the author in the decade after 1973. The chapter on Galileo also appeared in the scientific quarterly once known as the Bulletin of the Tychonian Society now become The Biblical Astronomer. The author confides the dissemination of these pages, along with the souls of her readers, to the Holy Souls in Purgatory.\nI knew little about Solange's work until I read this book and I can honestly say that this book stimulated thoughts like few other books I have ever read. An idea will be presented that, at first, you think sounds crazy. However, the more you read, the more you start to entertain these ideas and come to realize that you agree with more than what you thought you would. She may even change your mind completely on a subject that you thought settled. Few times do I come back repeatedly to subjects presented in books to think about them time and again, but this is one work that will do it.\nReaders beware. This book should come with a warning label on it because in it are radical ideas, many of which will offend people with strong values to the contrary. In Beyond Politics, you will hear opinions on issues that you didn't even realize were issues.\nBut it should be clarified what is meant by \"radical ideas.\" Normally the word \"radical\" has a very negative connotation to it, implying that the person or idea is extreme, crazy, and ultimately wrong. In this instance it merely means \"very different.\" No doubt, there will be some ideas you will vehemently disagree with, while others you're more inclined to agree with, all of them being radical in nature.\nHertz's writing is filled with such passion and conviction that it will both energize and wear you down at the same time. Ultimately the most valuable aspects of Beyond Politics aren't the answers she gives, but rather the lingering questions she poses that won't go away in your mind. And that's what makes Beyond Politics a must-read.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now that you're set up on your working branch, you can start developing on your website by making code changes and pushing those changes to Platform.sh to test them live.\nEach time you push a commit, Platform.sh will rebuild your environment and run the Composer command if a proper composer.json file has been found.\nThat will update your composer.json and composer.lock files, which you can then commit.\nIf you're using Composer, 3rd party PHP libraries can be added in the exact same way as Drupal modules.\nTo commit your custom modules, themes, or libraries, add those to the web\/modules\/custom or web\/themes\/custom directory and commit them to Git as normal.\n$ git commit -m \"Made changes to my files.\"\nAn alternate method that is appropriate for larger databases is to use the pipe | to stream the data, instead of making a copy of the dump file.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbblkl b/data_all_eng_slimpj/shuffled/split2/finalzzzbblkl
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+{"text":"I have to give some credit to my MIL on this one, she is the first to introduced me to the Kernel Seasons cheddar popcorn flavor. You can't go wrong with 100 calorie popcorn as a snack and adding flavors helps to change it up. My kids love the kettle corn flavor and the Husband likes the chili lime which is why it's not in the picture, because it's gone. He also uses that flavor on cucumber for a super low in calorie snack.\nThe majority of the flavors are around 8 calories per teaspoon.\nWe bought ours online and at Wal-Mart.\nHow About a 41 Calorie Snack?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Devri Vannalat is chief shaman of the Anouki tribe, who live in the frigid wastes of the distant north. His role in the tribe is to manage the tribe's oral histories, provide healing (both supernatural and otherwise), and help locate the richest hunting\/fishing grounds. He also uses ice magic to construct the tribe's large dwellings and buildings, and provides other magical support. He is relaxed and confident, and tries to be diplomatic in his dealings with others. He is attempting to discover a way of animating ice statues to provide simple labour for the village.\nThis might be about as close as you get to a 'cleric' or 'druid'\u2026 and it looks close enough to me. Devri Vannalat will be able to call on ice magic and life magic pretty reliably, and will be able to brew potions that literally let him 'bottle spells' for later use. I haven't worked out the details of how magic works, let alone item creation, but I'm pretty sure Devri will be able to do more or less what GreyKnight expects him to be able to do.\nThis is probably a good opportunity to point out how the rolls shown above are only there for convenience, and don't necessarily represent everything the character is good at. For instance, Devri Vannalat is probably really quite good at building igloos: he gets d8 as his tier die, but I'm quite certain I'd grant another d8 on the expectation that he would use his Ice Magic while building the igloo, and I'd have to think that a Snow Shaman would have the sort of skill and experience that would apply as well. This comes to 3d8, he should be able to build quite a comfortable igloo pretty easily.\nCome to that, if he's willing to spend a bit of magic he could probably whip up a spell to do it entirely, rather than depend on skill, and roll 4d8 instead of the 3d8. It's not quite an Elsa-grade palace, and costs some casting resources, but it'll be quicker than packing and cutting snow.\nConversely, if I were feeling a bit crowded for space I might drop the Architecture entry under skills, especially if the character is higher level and Architecture checks didn't come up very often.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CCE's Marine Program has been giving Back to the Bays since 1985, and we are committed to offering communities, individuals, and businesses opportunities to help us expand our efforts! Our goal is to increase stakeholder engagement in our: Water Quality Protection Initiatives, Habitat Improvement Projects, Shellfish and Fin Fish Research and Restoration, and Youth and Community Education Experiences.\nWhether you are a concerned citizen, an aspiring citizen scientist or a business owner looking for an opportunity to support our local maritime heritage, we will be able to effectively link you to an appropriate project, sponsorship opportunity, or volunteer effort! By providing time, energy, and monetary support, local communities and individuals alike can play a significant part in the success and expansion of the many important initiatives of CCE's Marine Program. One of the best ways that you can support these initiatives is by becoming an official Back to the Bays member!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are not qualified to work on a fueling system then consult a qualified service technician. \u00b7 Apps on Windows 10; Windows; Search. with an update which resolve the issue of camera which im reporting to being crashing ive got lumia 640xl it is. Find cutting- edge convenience store technology and fuel dispensers by Gilbarco Veeder- Root to enhance your gas station' s customer experience! Pump Codes Pro V3. 0: Android app ( 4. 3 \u2605, 500+ downloads) \u2192 The Pro version is the same as the free version but, without the advertisements. This handy little app does ONE. DISCLAIMER - This app IS NOT a substitute for a qualified service technician! What' s new in version 3.\n0 This update is signed with Apple' s latest signing certificate. Gilbarco Gas Pump Error Codes Highly secure, Gilbarco' s gas pumps, with powerful CRIND\u00ae electronics, build a flexible and innovative platform for your changing. List of all errors in pump dispensers Gilbarco Encore 500 S and 700 S in numerical order. Error Codes Encore 500 S,. Codigos de Error Gilbarco Encore 500 S y. He writes troubleshooting content and is the General Manager of Lifewire. \u00b7 Petroleum Equipment Forum. gilbarco 500s error code 20 and i. Post by crscyn717 \u00bb Wed Jun 03, 1: 54 am What power supply does it have? Founded in 1983, ANGI Energy Systems is a North American company that designs and manufactures systems for compressed natural gas ( CNG) vehicle fueling and tube trailer transport in applications around the world. The Socialist is the latest app which allows everyone to connect with the Samajwadi Party.\nIt allows like- minded individuals that believe in a secular, democratic & equally developed India to come together and spread these values. Gilbarco challenges the merits of Plaintiff' s disability discrimination claim, arguing that \" 1) Plaintiff' s violation of the attendance policy is a legitimate non- discriminatory reason for his discipline and discharge, and 2) Gilbarco had no knowledge of Plaintiff' s purported disability. TST 8 That darn Gilbarco Code 34 Page 26 to 27 code 31 or 35\" Page 28. IT SHOULD ALSO BE NOTED THAT THE \" ERROR CODE\" WILL BE FLASHING BACK AND. \u00b7 Learn how to fix the Code 29 error in Device. Software & Apps; Do More. New & Next; Fix \u203a Windows How to Fix Code 29 Errors Share Pin Email Print. Ingenico Group is the global leader in seamless payment, providing smart, trusted and secure solutions to empower commerce across all channels, in- store, online and mobile. To avoid this situation, get essentially the most recent and updated version administrator is webmaster. Dimension\u00ae Series The Advantage\u00ae Series Eclipse\u00ae Gilbarco. \u2022 For administrator is web. You select the type of dispenser from the drop down box, enter the error code then click on \" Lookup\". The meaning of the error code will be displayed. \u00b7 My Microsoft app store won' t open and constantly sends the Code: 0xerro. I did try every suggestion but no results have come.\nThis is becoming. What' s Changed The Pro version removes all of the advertisements. 0 allows update files to be downloaded to add new service codes. Gilbarco Error Code 44. on your other device. Get the full title to continueGet the full title to continue reading from where you Gilbarco Error Code App ERROR. Find great deals on eBay for gilbarco gas pump. Shop with confidence. Gilbarco Support: Service when you need it. Gilbarco offers service and support around the clock, around the globe. - based help desk to our wide variety of software partners, Gilbarco ensures that our customers are never alone. \u00b7 Resolve app installation errors in Google Play Store. Google Play Store error code 18.\nGo to Apps or Application Manager. HOME; CONTACT; Search. Categories Encore Printers Card Readers. NHome > The Advantage > Valves > N. Family Sharing With Family Sharing set up, up to six family members can use this app. \u00b7 Describes when you run automation code to control Word, you receive \" Run- time error ' ' ba) \" or \" Run- time error ' 462' \" message. Memory Error Code ROM Error 1 RAM Error 2 Unit Type Code Highline, 1 Salesmaker- 2. Level Two Command Codes 13 Changing PIN Number 14 For Gilbarco use only. Gilbarco - PA025800T1EMV - Smart Gilbarco Security Module ( GSM) For Exxon\/ Mobil. Gilbarco E500 Error Code 4322. Look here for answers to your technical questions. Links to my apps. \u00b7 Error Codes & Fixes contains guides and descriptions on how to solve the most annoying and common Android errors like error code 495, 505, 492 etc. A Gilbarco model T923A8ND SK700- 2 fuel dispenser ( Figures 1 and 2) for motor vehicles is approved to dispense LPG and various grades of liquid fuels, in attendant- operated mode, or in attended self- service mode using any compatible.\nGilbarco\u00ae Encore\u00ae Dispenser Family. \u2022 7300 West Friendly Ave. \/ PO Box 2 Greensboro NC. debit, RFID, bar code E15, E85. Build an app in seconds,. Gilbarco Passport Cookie. A Passport Tech will be able to quickly get a cookie code or the weekly password with out pulling laptop. 8 Yes, and another nice work could be a way to Support tag the secrets in peopleAC- C Tombstone system32 PerfStringBackup. Gilbarco Encore 500s Error Codes Download Gilbarco Encore 500 Service Manual PDF \u00b7 Gilbarco Encore\u2122 Series Dispenser Retrofit Common Error Codes. Gilbarco gas pump parts and accessories keypads overlays circuit boards, hydraulics, printers, displays and everything else for the maintenance of Gilbarco Encore 500. \u00b7 This vid helps Fix error code 0x80073d0a while trying to install some app on Windows Store. \u00b7 I am Unable to connect to desktop computer using Windows phone Remote Desktop preview App. I get an error \" Inquiring minds may find this error code helpful. The keyboard layout will change slightly depending on if you have selected a Gilbarco, Tokheim, or Dresser- Wayne dispenser. DISCLAIMER - This app IS NOT a substitute for a qualified service.\nWhen using iTunes, you might see alert messages that include specific error numbers. To get more help, find the error code. Extremism in the defense of liberty is no vice. And moderation in the pursuit of justice is no virtue. Barry Goldwater \" An armed society is a polite society. This End User Agreement is a contract between Gilbarco Inc. and the Veeder- Root Company Inc. ( the Host Company), which includes various business names and brands, including Gilbarco, Veeder- Root, Red Jacket, or Gasboy International. This feature is not available right now. Please try again later. About Gilbarco Veeder- Root Whether it' s outside, inside, underground or in the cloud, Gilbarco has the solution for your site' s needs. Our systems and solutions are designed and tested to work together seamlessly to deliver the lowest cost of ownership and best integration possible. Post a question or share your expertise with others. Codes for CT Payment ( formerly Telus & Emergis) Codes for Moneris Royal Bank Merchant Link ( Western) Codes for TSYS ( Vital) Codes for.\nThe splash screen will stay up for 10. Hard- to- find Dispenser Decals and Keypad Overlays for Gilbarco, Schlumberger, Tokheim, and Wayne dispensers as well as many miscellaneous decals. Gilbarco Advantage. If you are the developer of this app and would like your information removed, please send a request to [ email protected] and your information will be removed.\nSimilar Apps to Pump Codes Pro V3. Gilbarco Veeder- Root is the worldwide technology leader for retail and commercial fueling operations. We offer the broadest range of integrated solutions fro.\nBy logging in to Extranet, the user agrees to be bound by the Gilbarco Inc. Electronic End- User Agreement for Extranet. The WOLF Service App is the ideal partner for the heating technician. The app offers a digital spare parts catalog for the heating sector with over 6800 spare parts, more than 4600 associated photos and over 950 clickable exploded views.\nand any mismatch will cause improper operation or an error code to be. com Command Code 91. Common Error Codes. Is the Advantage Error Code 4329 message popping up frequently?\nWould you like to safely and quickly eliminate Error Code which additionally can lead to a blue screen. , doing business as Gilbarco Veeder- Root, is a supplier of fuel dispensers, point of sale systems, payment systems,. Gilbarco Fuel Dispensers, Wholesale Various High Quality Gilbarco Fuel Dispensers Products from Global Gilbarco Fuel Dispensers Suppliers and Gilbarco Fuel Dispensers Factory, Importer, Exporter at Alibaba.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is gray, gloomy, and raining here in Lasarte, but cozy clothes and cafe con leche on a day off make everything better.\nYesterday I woke up and hopped on the train to Saint Jean de Luz, a very small, very old town in Southern France. I knew nothing about the town, what there was to do, or how to get there, but the more time I spend over here, the more these little adventures seem to appeal to me. I got some info from the guy at the train station and in less than an hour, I was there.\nThe town itself was very quaint: lots of old, old buildings, churches, and streets, but with the contemporary touch of copious boutiques, bars, restaurants, etc. The town clearly realized the tourist appeal of its antiquity and built as many opportunities as possible to take tourists' money. I was starving when I arrived, so I scoped out a few cafes and found one that seemed to call out to me. They served me the best croque monsieur I have ever tasted in my entire life. The cheese atop the sandwich was the absolute perfect balance of crispy, bubbly, and melty, and the mornay, ham, and bread beneath just melted together into creamy bites of heaven. Washed down by a glass of Bordeaux, my day was off to a good start.\nFrom there, I spent a few hours wandering around the town. I window shopped in what were actually some cool little stores, walked along the water, snapped photos of old buildings, etc. Grabbed a chocolate croissant from a little bakery, and good God, it was like eating a chocolate filled cloud. Tried exploring some streets off the beaten path and found a cemetery outside town that totally blew my mind. It had clearly been around for at least a couple hundred years (based on the inscriptions on some of the headstones), and some of the markers were beyond impressive. Pyramids, giant crosses, obelisks, mausoleums, this place was like the Cadillac showroom of graveyards. I think I snapped as many photos within its walls as I did the entire rest of the day.\nIt started to get dark, so I headed back into town, found a wine shop, and bought a bottle of Bordeaux to drink upon my return home. Then for dinner I knew I had to do something nice and French... obviously, duck confit was the way to go. Very yummy, with frites and a salad, watching some tennis tournament live from Paris. I made it back to the train station, only to find that despite what the arrivals\/departures screen said, trains in Saint Jean de Luz show up whenever they please. I watched three different listed departure times come and go and started to panic a little, thinking that I might be stranded for the night; thankfully, after 90 minutes of waiting, my train arrived. The night ended nicely with the Bordeaux (though to my unfortunate surprise, it was only so-so) and a movie.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeucb b/data_all_eng_slimpj/shuffled/split2/finalzzzbeucb
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Everyone has done a Christmas gift Guide this year and I must admit it has helped me with my Christmas shopping! Most things on this list I have had the pleasure of trying and these are my honest reviews on the products I think would make great Pressies!\nThese three Products from Grace and Stella would make great stocking fillers for friends and family and they are all reasonably priced.\nI had seen so many good reviews about Stella and Graces Facial Cupping set, So I decided that as they were such a reasonable price I would give it a try.\nThe set comes with a face oil, but I have been using a serum of my skin by skinkissed for a few weeks now and I chose to use that instead as I didn't want to overload my skin with products.\nI wasn't sure how to use them at first but the guide that came in my pack was very useful and told me all the information I needed to know!\nIt was easy to glide the cups over my skin in the motions recommended in the pack.\nI had an uneven skin tone but using these cups my skin is looking much brighter and fresh.\nI have also noticed that using the smaller cups has given me much less noticeable bags and dark circles under my eyes.\nIt takes a little while to get used to using the cups on a regular basis and getting a technique that works for you, but it is well worth the venture.\nThese will definitely stay part of my daily skin care routine.\nI first tried this while having a bath, so I could follow the full instructions and apply it to my damp hair and leave it on for 10-15 minutes.\nFirst of all, I have to mention the smell, it was amazing I could smell it for days after and every time I got a whiff I was in heaven.\nI have average hair, it's not thin but it's not thick either so I was worried this mask might weigh it down and make it look greasy.\nI was pleasantly surprised as once I washed it out my hair was like silk.\nI didn't even use conditioner because my hair already felt so clean and soft.\nI didn't weigh my hair down at all and it was super shiny for a couple of days.\nI would only use this once a week or so as it lasts such a long time after use.\nI have noticed after a few uses my hair seems to have less frizzy bits and to me seems stronger.\nAnother great point about this product is its Paraben free and completely Vegan.\nI don't often have baths as I never have the time, but I really enjoyed using these bath rocks.\nAt first, I thought the smell was a little strange and I wasn't sure about putting them in the bath, but I popped three in anyway and let them fizz away.\nThe smell was very different once I was in my nice warm bath, it was very refreshing with sweet hits to it.\nThe rocks dissolved completely so I didn't have to worry about little bits in the bottom of the bath.\nI thoroughly enjoyed them and can't wait to try some of the different ones Grace & Stella have to offer.\nI am so glad I stumbled across this company, they are based in the countryside and the idea is driven by the passion to help everyone get a beautiful night's sleep. Sleep is something a lot of people struggle with having full and busy lives! Jo Foster the founder of the company struggled herself with sleep and is now giving us the gift of a good night's rest back with her range of kiss the moon products.\nKiss the moon were kind enough to send me lots of sample sized products to try out for this Gift Guide. The first thing I must mention is the amazing aroma that came out of the package as soon as I opened it! I was I was lost for a few seconds, caught up in the heavenly smell drifting up my nostrils.\nI lit a few tea lights and used the Deep Sleep Bath oils on a week day as I just couldn't shut off from work. I laid back enjoying every aspect of my bath afterwards my skin felt so soft and smooth. I decided to try the night cream for hands as well as I was feeling refreshed after my bath. I went right to bed and I did have a great night's rest. In the morning I just felt relaxed when I woke up and my entire body was silky soft I didn't even need to moisturise.\nI am so looking forward to trying the bath salts and the deep sleep balm!\nI would also love to try a couple of the different Candles they have.\nIroha products are available in Superdrug and they will make the perfect stocking fillers this Christmas.\nI got the opportunity to try out three different products they have to offer, the charcoal peel off mask (my mum also got to try this one.) The charcoal nose cleansing Strips and finally (this one I was excited about) the charcoal black tissue mask!\nSo far, my mum and I had a great time trying the Peel Off Mask, there is enough in the packet to split between two maybe even three to be honest. I was slightly disappointed I could not reseal the pack to use it again, but that's the only bad point I have to say on the product! It left my skin soft and not dried out I also noticed my pours looked smaller right away and a spot my mum was worried about had gone right down by the morning.\nLooking for the perfect gift for any blogger? Try the 'Blogging sweater' by coconut Lane its cute, quirky and totally accurate (although I might trade my coffee in for tea most of the time!) They also do loads of other little gifts perfect for stocking fillers like the marble effect coaster or the beautifully designed phone skins.\nThe beauties are great little gifts and have some really good offers on for Christmas. I have been using them for a week and my teeth look significantly whiter already. It's a little beauty necessity that sometimes can just boost your confidence for a night out or a hot date??\nThese are similar to the 'bubble tea' accept they are for your Alcohol!\nYou can get Shimmers for Prosecco and Gin as well as little balls of flavour that pop in your mouth! I bought Raspberry and cherry as I thought these would go well in my prosecco over Christmas!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you sure you want to know?: What did you do this weekend?\nA friend of ours celebrated a birthday this weekend. He is part of the Chi-Town Squares Square Dancing Troupe, which was having it's Friends & Family event so he invited all of his friends to come to the event instead of having a party.\nThe Boyfriend and I had no idea what to expect, we didn't know what to wear, I just got rid of all my cowboy shirts so I went Dickie's Butch and The Boyfriend went Cowboy Butch!\nWell we had a blast, we danced for two hours, and we even broke a sweat. And considering that I'm \"dance challenged\" that means a lot!\nThey had a member team up with a non-member and they taught us how to dance.\nWe do-see-doed and did an alamand right and did a grand left something or other, we also did a swing and a four ladies in the center with a courtesy turn.\nThe great thing about going to a Gay Square Dancing event is you can either be the boy or the girl.....considering we had no idea what we were doing, we ended up both being \"girls\" because they follow.\nThe last time I square danced professionally was in the 5th Grade when we did a May Day performance. It brought back a lot of memories and we had a blast, we laughed and laughed.\nWell Saturday we had dance class and then not one, but TWO birthday parties. The first one was family, so I had to go to, but the second one I was able to just drop her off.\nSunday I rested. A lot.\nI slept most of Friday, and laid on the couch most of Saturday.\nThe last time I square danced was at GayCamp! YAY! It's fun, but alot of running around.\nI did nothing exciting on Saturday. Groceries, got my eyes checked and ordered new contacts and glasses. Then sat outside in the sun a worked a bit.\nSunday was a rainy day, we we did very little.\nSeems like you had a really good time.\nDrove to Detroit but when I saw the signs for Detroit right next to the signs for Chicago, I was all \"Oooooh, which one? which one?\"\nWe went to Detroit but you know my heart was with you in Chi-town dancing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Centrecoat Cushion Grip Anti Slip Tape provides a protective and comfortable way of ensuring handrails are not slippery and can be used in all common commercial areas.\nCentrecoat Cushion Grip Anti Slip Tape has an aesthetically pleasing design that is regularly specified by boat and yacht users in the marine environment for application to interior appliances.\nCentrecoat Cushion Grip Anti Slip Tape is a premium quality product that is self adhesive and can applied directly onto most sound, clean and dry surfaces to ensure a non slip surface. Centrecoat Cushion Grip Anti Slip Tape can be used in conjunction with Edge Fix Adhesive to ensure all the edges are sealed appropriately with a small application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Humor - Jokes - Games and Diversions' started by CRC, Jul 26, 2006.\nold one..but cracks me up...can't remember if I posted it here before or not??\nAn Irishman in a wheelchair entered a restaurant one afternoon and asked the waitress for a cup of coffee. The Irishman looked across the restaurant and asked, \"Is that Jesus sitting over there?\"\nThe waitress nodded \"yes,\" so the Irishman told her to give Jesus a cup of coffee on him.\nThe next patron was an Englishman with a hunched back. He shuffled over to a booth, painfully sat down, and asked the waitress for a cup of hot tea. He also glanced across the restaurant and asked, \"Is that Jesus over there?\"\nThe waitress nodded, so the Englishman said to give Jesus a Cup of hot tea, \"My treat.\nThe third patron was a Redneck on crutches. He hobbled over to a booth, sat down and hollered, \"Hey there, sweet thang! How's about gettin' me a cold glass of Coke!\"\nHe, too, looked across the restaurant and asked, \"Is that God's boy over there?\"\nThe waitress once more nodded, so the Redneck said to give Jesus a cold glass of Coke, \"On my bill.\"\nAs Jesus got up to leave, he passed by the Irishman, touched him and said, \"For your kindness, you are healed.\" The Irishman felt the strength come back into his legs, got up, and danced a jig out the door.\nThen Jesus walked towards the Redneck.\nThe Redneck jumped up and yelled, \"Don't touch me.. I'm drawin' disability!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Duluth's own Duluth Pack will be showcased on Monday's edition (3\/4\/19) of NBC's TODAY with Kathy Lee and Hoda. Products from Duluth Pack will be getting airtime during the 4th segment of the program, with Duluth Pack's 15 oz cotton duck canvas Sportsman's Duffel and Leather Logo Journal being highlighted.\nThe items being featured, like most of Duluth Pack's products, put an emphasis on both rugged fashion and function, \"intended for rugged adventures, including outdoor activities and everyday life\". The duffel and leather journal to be featured range in price from $65.00 - $435.00.\nAward show season and the various fashions tied to celebrities attending these award shows inspired TODAY to spotlight a number of different fashion and lifestyle items, including the Duluth Pack products set to appear on Monday.\nIf you're interested in getting these featured items for yourself, they are available both through the Duluth Pack website, as well as at their retail location in Canal Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhhlz b/data_all_eng_slimpj/shuffled/split2/finalzzzbhhlz
new file mode 100644
index 0000000000000000000000000000000000000000..b22ecf9a5d3b904137a80c5ea1b1f6e50ac11bfc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Following my prior post Play2Train: The Virtual Elk's Rehabilitation Hospital Construction, the teaser trailer of the recently built hospital replica is now available.\nThe Play2Train Open Content Alliance (POCA) represents collaborative efforts to build a permanent self-sustaining archive of open source virtual worlds, including their content and applications, to support public domain emergency preparedness training and exercises.\nThe construction of the virtual Idaho Elk's Rehabilitation Hospital aims at supporting the planning of full scale exercises that will take place in May\/June next year.\nThey will also run exercises in the virtual environment and compare it with full scale exercises in the real world. This effort differs from previous ones in that, this time, they have actually replicated a virtual version of a real hospital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The End PJ Paralysis campaign has taken centre stage as part of Occupational Therapy Week, as staff at Harrogate District Hospital donned pyjamas to support the initiative.\nStaff decided to wear their pyjamas to raise awareness of the End PJ Paralysis campaign, and occupational therapists across the Trust have been promoting of Occupational Therapy Week with a stand at Harrogate District Hospital across the week. The stand contains information on what occupational therapy is, what occupational therapists do and recent developments within the Trust that have been led by Occupational Therapy.\nEnd PJ Paralysis is a global social movement embraced by nurses, therapists and medical colleagues, to get patients up, dressed and moving. Having patients in their day clothes while in hospital, rather than in pyjamas (PJs) or gowns, enhances dignity and encourages independence, and, in many instances, shortens a patient's length of stay.\nOccupational Therapy Week 2018 is led by the Royal College of Occupational Therapists, and offers an opportunity for occupational therapists to promote their profession; and also encourage other people who have benefited from occupational therapy to tell the difference it has made in their lives. It takes place from 5-11 November.\nEmma Havercroft, Professional Lead for Occupational Therapy at Harrogate and District NHS Foundation Trust, said: \"We are delighted to be supporting the End PJ Paralysis campaign as part of Occupational Therapy Week 2018. It is great that so many of our occupational therapy colleagues are supporting the initiative and even donning their own pyjamas to raise awareness.\nOccupational therapists at the Trust work with people of all ages to help them overcome the effects of illness, injury, disability or accident and enable them to lead full and satisfying lives as independently as possible. The Occupational Therapy Team works in many areas across the Trust and offers a variety of services. For more information on Occupational Therapy at Harrogate and District NHS Foundation Trust, visit www.hdft.nhs.uk\/services\/therapy-services\/occupational-therapy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The London Borough of Hackney has produced a new guide called 'What you need to know about renting from a private landlord'.\nThe guide explains how the Council can help privately-renting tenants, as well as information on the legal rights and also responsibilities of tenants and landlords. Read and download the guide HERE.\nThe Money Advice Service is an independent service set up by the government to help people manage their money better. It has just published new information about the Universal Credit benefit.\nOn Monday 24th February 2014, Hackney Community Law Centre was delighted to have Maurice Wren address our Annual General Meeting as its Guest Speaker.\nOn Wednesday 19th February 2014, HCLC Housing Solicitors Wendy Pettifer (pictured right) and Tayyabah Ahmed joined other charities and housing campaigners working in Waltham Forest to examine the issue of 'Street' Homelessness in the borough.\nHCLC Annual Report 2013 now published!\nPlease find an electronic version of the 2013 Annual Report of the Hackney Community Law Centre available to read and download HERE.\nOn Thursday 23rd January 2014, representatives of Hackney's advice agencies, the local council, corporate partners, sister law centres and advice agency volunteers, attended the official launch of Hackney's Big Lottery Funded 'Sustainable Advice in Hackney' project.\nHackney Community Law Centre's next Annual General Meeting will take place at : The Salvation Army, 122-124 Lower Clapton Road, London E5 0QR On Monday 24 February 2014 between 7pm and 9pm.\nHCLC is delighted to welcome James Krumrey-Quinn as a new intern.\nOn Monday 6th January 2014, HCLC staff and interns joined barristers, solicitors and campaigners from all over the country to protest at Government proposals to make even more cuts to criminal legal aid.\nSEASONS GREETINGS FROM EVERYONE AT HACKNEY COMMUNITY LAW CENTRE!\nThank you for your support in 2013 and looking forward to continuing to work with you in 2014! We are closed from 1pm on Tuesday 24th December 2013 until 10am on Thursday 2nd January 2014.\nHCLC is recruiting for a Housing and Welfare Benefits Caseworker!\nBarrister Tunde Okewale becomes HCLC Patron!\nHCLC is delighted to announce that Tunde Okewale, currently a barrister at Doughty Street Chambers, has accepted our invitation to become a Patron of Hackney Community Law Centre.\nHCLC is pleased to announce that we are leading the Hackney Big Lottery Advice Services Transition Fund (ASTF).\nDelicious Support for the Hackney Migrant Centre!\nHCLC is really pleased to support the publication of a new cookbook aimed at raising funds for our friends and colleagues at the Hackney Migrant Centre and North London Action for the Homeless.\nThanks to the London Borough of Hackney who have provided us with special funding, HCLC is delighted to welcome Marie Froysa Hole (pictured right) to the Centre as our new Benefits Caseworker.\nHCLC is very pleased to welcome three new members to our Board of Directors.\nHCLC is very pleased to have teamed up with the Big Voice London project to respond to the Home Office's latest consultation on police 'stop and search' powers.\nIf your rent is not met in full by housing benefit and you have a temporary situation which makes it difficult to pay your rent, you can apply to Hackney Council for extra help in the form of discretionary housing payments (DHPs).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Community STEM Badge Ecosystem developing a shared, validated, digital badge ecosystem that recognizes out-of-school STEM learning across institutions and bridges the gap between formal and informal learning environments. C-STEMBE is expanding on the Hive-supported work of the Chicago STEM Museum Digital Badge Working Group, which developed and pilot tested an initial set of twelve badges representing four critical cross-disciplinary STEM skills (investigation, building knowledge, building models, and communicating science) at three levels of achievement.\nIn this new phase of the project the build out of the badge ecosystem will be completed, creating a total of The work of the group will include completing the build out of the badges and STEM skill areas included in the ecosystem, addressing challenges that became apparent during the development of the pilot badging system, and pilot-testing solutions in a national context, with the end goal of creating a useable, useful, and educationally powerful STEM badge ecosystem. Working groups will focus on four challenge areas.\nExplore ways to make technology-based badges accessible to all learners.\nIncrease student access to opportunities, helping them create pathways through their areas of interest.\nUnderstand how to integrate badges into learning activities so that students' intrinsically value them as representations of their learning.\nEngage the education and career communities around the value of badging and create a concrete benefits structure (internships, service learning hours, college admissions\/job opportunities).\nExplore and define the effective relationship between institutional level (content) badges and skills badges like C-STEM.\nCollaborate with CCOL to integrate C-STEM badges within the CCOL framework.\nUse the OBI Standard to support interoperability between badging platforms.\nEnable organizations to choose the badging product that works best for each circumstance.\nWork toward providing a seamless experience for organizations and learners.\nAs work progresses, it will be documented, reported back, and integrated into the process and technology development of the ecosystem as a whole, resulting in a system that is useful to learners, documents learning effectively, and is usable across multiple learning contexts.\nThe goal of the Community STEM (digital) Badge Ecosystem is to build on the work of the Chicago STEM Museum Digital Badge Working Group -- further developing the ecosystem, expanding the scope nationally, and addressing the critical challenges of equity and access, including how badges can supports student learning, are valued by youth and adults, and can be integrated across institutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's an eBook package worth $32.70, but for one week only, we're charging $6. BECAUSE IT'S SUMMMMMMMERRRRRRRR!!!\nThis package is extra special because it focuses on simple summer food, gardening, getting your kids to eat fresh summer veggies, and even finding awesome activities to do with your kids that do not involve a screen. It's ridiculously fun, and it's all for just six bucks!\nSimple Summer Suppers is a brand new eBook that we're releasing just in time to include in this offer! (Notice that the full price of this one new book is higher than the cost of the entire package? Whooop!) This book gives you a sneak peek into the 13 great simple summer supper ideas we'll be sharing here during the month of July. Too hot to cook, too tired to care, but the people still need to eat? This book's got you covered!\nThis book is also brand new!! (Yep, we've been busy!) Get dozens of great ideas for fun to have as a family to make the most of your summer days. Get fun recipes, cute printables, oodles of games and activities to do inside and out. Wait till you see all the creative ways to use sidewalk chalk!\nWe pulled out this oldie-but-goodie that I originally wrote in 2008 and re-worked it to add more and better ideas. There's no better time than summer to focus on adding more fruits and veggies to your diet. Learn fun new ways to encourage your kids to eat these gems, and who knows? You might start liking a few new fruits and veggies too!\nI wrote this back in 2009 and just like the book above, we re-worked it a little bit just for this occasion! There are picture of my babies in this eBook, because in 2009, my boys were little guys helping me snap green beans, shuck corn, and plant potatoes. Be still my heart. They still do all of the above, they're just a lot taller than they were in 2009. Learn some simple gardening and preserving basics in this eBook. Get the kids involved!\nGet 33 fabulous ideas and 49 simple recipes to help you beat the heat this summer! From fun ways to use the crock pot (while avoiding the oven) to some awesome summer snack ideas, this eBook is delightfully refreshing!\nAww, this gem is one of the first eBooks I ever wrote. I loved the chicken in 2007, I love it today. This eBook is everything chicken \u2013 basic, simple, delicious recipes you'll love!\nReady to dig in? Get this entire Summer Survival Package for just $6! Offer ends Monday, July 2.\nKeep the kids busy, feed everyone well, and do it all with simplicity! These eBooks equip you with loads of recipes, tips, ideas, and activities for awesome Summer Survival!\nHi! I've enjoyed your recipes and blog and one of the ebooks I have but I'm curious as to whether you have any sort of return policy or if you have sample pages I can look at before purchasing. I've found that many of the ebooks I've purchased have the same content as blogs or have so few recipes that I'd actually use that it hasn't been worth keeping them for me. Much easier to browse through books at bookstores but the times they are a changing! ;) thanks for your blog!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbiqxt b/data_all_eng_slimpj/shuffled/split2/finalzzzbiqxt
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+{"text":"Advertising campaigns that state \"Up to 50% off\" are likely to cause consumers to believe a number of items are discounted by 50 per cent and that all other items within the store will also be discounted to some degree.\nYou should not advertise this type of saving if only a few items are offered at 50 per cent off and the remaining items in the store are offered with little or no discount at all.\nYou must be able to show that the customer is really saving money.\nIf you wish to limit the stock included in this type of campaign, you could use terms like \"Up to X% off all CDs\" or \"Up to X% off selected items\".\nThe test is that it must be clear to the consumer that not all stock is discounted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the top of the limestone bluff at the north end of Grand Turk stands the Imperial Lighthouse. Built in 1852, this lighthouse warns sailors of a shallow reef that extends from the north end of the island. Originally, the lighthouse was lit by eight small oil lamps. But these lamps were not very bright, and were unable to protect the reef on some dark and stormy nights. There are shipwrecks within a mile of the lighthouse! Later, the oil lamps were replaced with brighter kerosene lamps and a powerful lens, providing better light and protection. In 1972, the lighthouse was electrified, and now a bright beacon protects both ships and the reef.\nThis picture shows the Imperial Lighthouse gleaming in the afternoon sun. This picture was taken from the beach along the bottom of the bluff on the west coast near the north end of the island. Notice the long roots of the small Mangrove Trees growing out of the water. These trees help prevent erosion by holding the sand in place as waves wash over the beach. The roots also provide an important habitat for young fish. Young fish hide in the water between these roots and get protection from predators such as shorebirds and bigger fishes.\nThis picture shows the lighhouse keeper's house. In the old days, a lighthouse keeper stayed on guard all night to make sure that the lighthouse oil or kerosene lamps stayed lit. Then the lighthouse keeper would sleep all day in this little building next to the lighthouse. The windows are small, which probably helped keep the house dark inside so that the lighthouse keeper could sleep during the day.\nCopyright \u00a9 2001, ReefNews \u00ae, Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One day back in October, while discussing our next volume of Dorothy, Mark and Anna realized that a Kickstarter level with an Oz Tarot Deck would be REALLY awesome. They set the idea aside for about 12 hours, until Mark's wife, Chaunacey shouted, \"YES, THAT IS AMAZING and should be its own project!\"\nWe contacted one of our favorite comic book artists to draw the deck, but he's booked for the whole next year. So then we thought, \"What's the next best thing? Oh yeah! A BUNCH of our favorite comic book artists!\" So we started sending out emails to people whom Anna has met through signings at her shop, Illusive Comics & Games, friends of ours, and people we have admired. To our utter shock a lot of them replied that they would love to be part of the project.\nThus, we give you an anthology of art, in the form of a Wizard of Oz Tarot Deck, created by some of the most amazing comic book artists and graphic designers that exist. And now we're going to announce one artist a day, in semi-random alphabetical order.\nOh yeah, and the Kickstarter will start February 1st.\nThe first artist we are happy to announce that is working on the Tarot deck is Ben Templesmith.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A medical malpractice case was filed against a hospital in Connecticut for not preventing a patient from committing suicide. The case went to court and a jury ruled that the hospital was not liable. However, an appeal has decided to throw the jury verdict out.\nThe New York Medical Malpractice Attorney handling the case explains that the reason the Jury's verdict is being ignored is because the verdict could have been biased. The judge failed to ask the jury whether or not they read a newspaper article relating to the case, and whether or not it affected their opinions.\nThe newspaper article was published a few weeks before the start of the trial. It is believed that many of the members of the jury read this. It could have quite easily influenced their opinion. Hospitals all over in places like Staten Island and The Bronx must protect themselves against law suits like this one.\nAll of this then leads the court to order a re-trial in the malpractice suit. The NYC Medical Malpractice Lawyer explains that the woman was being treated at the Psychiatric hospital in 2002 for depression. The 41 year old patient died shortly after being admitted as she hung herself.\nThe Silver Hill Hospital is well known because it is so popular with many celebrities and has an excellent reputation.\nThe jury originally found that the hospital and doctor were not liable because they were not negligent. However, a public ruling recently said that the judge should of asked whether or not the members of the jury read the article.\nWhile the article was not accurate in all of the details, the appeals court argued that it could affect a persona emotions relating to the case. The New York Medical Malpractice Lawyer says that this is a valid case for a retrial.\nThe executor of the diseased patient's estate said that he was happy with the ruling as it mad their case much stronger. The argument is whether the jury members will still be fair even after reading the article.\nThe lawyer for the hospital said that it was very likely the hospital would appeal the case. The hospital is sure that the article did not cause any bias because much of the content was mentioned in the trial proceedings.\nIf you believe that medical treatment has left you worse off before then it might be possible to file a Medical malpractice claim. A New York Medical Malpractice Lawyer will be able to help you decide whether or not you have a claim, and how best to file your case.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Launched in 2016, Little Rock ADC is a small group of intelligent people who thrive meet the staffing needs of the great state of Arkansas. Each member of our team brings a different set of skills and strengths to the table \u2014 allowing us to be thorough and incredibly good at what we do.\nOur Nurses and CNAs are all Board Certified with current CPR certification. State Background checks are run on a yearly basis as well as Adult and Child Maltreatment Registries. OSHA and JCAHO Regulations are followed during the hiring process.\n\"Little Rock ADC was an absolute pleasure to deal with. Their staff is not only competent but their commitment and professionalism is fantastic.\"\nContact our offices to speak with us about staffing your company's needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjefz b/data_all_eng_slimpj/shuffled/split2/finalzzzbjefz
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index 0000000000000000000000000000000000000000..22ad3ded284aec62e4db6473562527815c57b1c1
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+{"text":"No expense has been spared to meticulously ensure The Number 1, formerly Southside X in Moorabbin is the most ultra-luxurious 5 Star purpose built boutique bordello in Melbourne.\nOur magnificent decor, and state-of-the art facilities, built within an intimate world of exquisite luxury provides a sensual environment for you to explore your most sensual desires in Melbourne`s most exclusive bayside brothel.\nWe invite you to experience the exceptional quality and elegance we have created in our world-class brothel in Moorabbin, that we can assure you, is second to none.\nWe aim to change the face of the adult industry, and lift the current poor standard and quality of ladies that have become all too common in other massage parlours and brothels in Melbourne.\nWhen you visit The Number 1, formerly Southside X in Moorabbin, you can expect to meet some of the most deliciously sensual and seductive ladies imaginable, (most of which have never ever been seen before in any other massge parlour or brothel in Melbourne).\nIt is our well-earned impeccable reputation for these types of ladies, which is accredited to our success. You can be assured that our ladies will take your experience to a whole new level.\nOur ultra-luxurious Melbourne establishment has been designed to create a world totally devoted to your every sexual pleasure, making it our priority to ensure that your experience with us is truly unforgettable.\nFrom the moment you walk through the door of Melbourne`s leading brothel, you will be captured by the lush surroundings that have a richness that cannot be surpassed.\nMost of our valued clients like to meet our ladies while having a drink in our luxuriously appointed state-of-the-art lounge area.\nThe vibrant decor and party ambience together with funky videos, stylish decoration, pumping sound and lighting system creates the experience of an exclusive Melbourne \"nightclub\".\nHowever, if you prefer a totally discreet atmosphere, you can ask to be taken to a private waiting room, where you may individually meet some of the hottest ladies in any Melbourne brothel.\nOur private ultra-luxurious establishment is in Moorabbin only 10km (approximately 15 minutes by car) from the Melbourne CBD, and is open from 10am daily, to provide only the very best adult services in Melbourne..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200bAs all adult literacy programs know, volunteers are the lifeblood to our programs. We use volunteers to tutor students, to assist in the office, to serve on boards, etc. But the challenge is always the same: where do we find more volunteers? This month we would love to hear some of the things that you do to find and recruit new volunteers. Here are a couple of examples to get you started.\nSocial media. Though the usage of social media by literacy programs is improving, there are still more opportunities to improve usage of the media. It is imperative that programs keep an active presence - whether it is Facebook, Twitter, Instagram, etc. Sharing of stories, or opportunities available, and of program status updates will keep people interested in the program, and will get them involved.\nGet your name out. All too often, literacy programs serve their communities in relative obscurity. More needs to be done to keep programs in the public eye. This means not only using social media, but creating contacts with local print and broadcast media. This means having a presence at local fairs, festivals, and community events. There are also opportunities for your current volunteers to share their stories, and share their reasons for volunteering. People can't volunteer for programs they don't know about, so you need to get your name out.\nThese are just a few ideas we should all keep in mind, but we would love to learn more about what you do! We will follow up with your ideas and feedback next month!\nClick here to send us an idea for recruiting volunteers!\nA lot would not believe this at first but those who have already experienced real joy in their lives would attest that you can only find this if you serve the Lord. There's no way you can serve the Lord without a deep desire to make other people's lives better. This is the true value of volunteer work. You may not be making any money from this but what you gain is something way better No amount of material things can ever give you this kind of satisfaction and the only way you could agree is to experience this yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The final re-exam schedule is now available on the password protected area.\nPlease register for the re-exam by sending an email to Michael with the subject [SL][re-exam registration]. Add your full name and your matriculation number to the e-mail's body. For further information, e.g. about the doodle poll, see the password protected area.\nThe results of the exam are now available on the password protected area.\nThe final exam schedule is now available on the password protected area.\nA preliminary exam schedule is now available in the password protected area. If your name is not on the list, please folilow the instructions below or contact the TA.\nIn order to successfully participate, you need to register for the exam in the LSF\/HISPOS system of Saarland University - this will be possible as soon as the exam date has been entered into the system (this usually happens a few weeks into the semester).\nThis course covers a subject that is relevant for computer scientists in general as well as for other scientists involved in data analysis and modeling. It is not limited to the field of computational biology. The course fulfills the requirements for the curricula of computer science and bioinformatics as special lecture (Spezialvorlesung, 5 credit points).\nThe course will convey the ability, given a data set, to choose an appropriate statistical method for analyzing it, to select the appropriate parameters for the statistical model generated by that method and to assess the quality of the resulting model. Both theoretical and practical aspects will be covered.\nThe course will, by and large, follow the book An Introduction to Statistical Learning with Applications in R (2013). At times the course will take additional material from the book The Elements of Statistical Learning, Springer (second edition, 2009). The former book is the more introductory text, the latter book is more advanced. Both books are available as free PDFs. We encourage you, though, to acquire at least the first book in print. There will be one lecture (90 min) and one tutorial (90 min) per week. The slot for the tutorial will be set after the first lecture.\nThe course is targeted to advanced students in bioinformatics, computer science, math, and general science with mathematical background. Students should know linear algebra and have basic knowledge of statistics.\nYou need a cumulative 50% of the points in the problem sets (in both theoretical and programming exercises) to be admitted to the exam.\nJames, Witten, Hastie, Tibshirani: An Introduction to Statistical Learning with Applications in R (2013). The students of the course are encouraged to acquire this book.\nHastie, Tibshirani, Friedman: The Elements of Statistical Learning, Springer (second edition, 2009).\nBoth books are available online as free PDFs.\nAdditional literature can be found in the library; the reserve list for the lecture can be found here: library reserve list for 'Elements of Statistical Learning 1'.\nProblem sets will cover theoretical proofs and programming exercises with roughly equal weight. In general, they are due before the lecture (10:00 sharp); further details regarding the assignments will be announced in the first lecture.\nThe programming language that will be used is R - a language for statistical computing. It is freely available for Windows, Linux and Mac. As a vectorized programming language, it is ideally suited for the problems we will encounter. There are also many freely available packages (or libraries) to perform a variety of classification and regression tasks, or to visualize the results of statistical analyses in a convenient way.\nThe tutorials focus on the problem sets. A very brief reiteration of parts of the lecture is also given. If you have any questions about the lecture, write an e-mail to stat-learn-staff@mpi-inf.mpg.de. We also have a student mailing list (stat-learn-students@mpi-inf.mpg.de); to register for that, write an email with the subject ESL mailing list to mscherer@mpi-inf.mpg.de.\nWhat can I do to prepare for the lecture?\nRefresh your knowledge on basic statistics. Basic linear algebra will also be useful.\nR for Beginners by Emmanuel Paradis. Especially relevant for us are chapters 1, 2, 3 and 6.\nAn Introduction to R - the standard R introduction. This is a very detailed manual; it is therefore quite lengthy.\nThis course was developed by Thomas Lengauer and we thank him for providing his lecture materials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pokemon Fire Red Cheats : Gameboy PC To use this cheat you must have 1 master ball. After defeating the Elite 4 walk around grassy areas and you might be attacked(for better chance of finding them is super replel). If you start with Squirtle you will be attacked by Ryeikou if you started with bublsaur you will be attacked by Entei if you start with Chamander you will be attacked by Suicune... Just enable one legendary Pokemon at a time and be cautious when mixing this cheat with other Pokemon FireRed Cheats. If you are looking for more cheats for your FireRed game, check our Pokemon FireRed cheats page and for extra fun and excitement I would suggest download any of our listed Pokemon ROM Hacks .\nThis cheat for Pokemon FireRed [Game Boy Advance] has been posted at 10 Sep 2007 by someonedied and is called \"Get infinite ultra balls in island 1\".\nPokemon Sapphire : Master Ball + Rare Candy Cheat Codes Hey guys ,I'm back ! Sorry for not uploading videos these late days If you enjoyed Smash that like button !\nThat's it for Pokemon Ruby video game cheats and walkthroughs this week, but stick around and see what cheats Gary can come up with for next week. He'll get the 411 on all the coolest game codes !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is the Selective Mutism Program?\nOur program assesses children and youth with selective mutism, a form of childhood anxiety. The child is able to speak but does not speak in certain situations, like at school, because of his or her anxiety.\nYour child can speak when he or she is comfortable, like at home, but doesn't speak in certain settings, such as school. This has been happening for at least 3 months, or 6 months if English is not your child's first language.\nWhat does the Selective Mutism Program do?\nWe support you and your child's school in creating a long-term plan to reduce your child's anxiety and generate functional speech.\nTo be accepted for a consultation in our program, your child must be aged 4-18 years, have established conversational language and with no more than a mild developmental disability.\nWhen you are accepted to the program, you will come in for an assessment of your child's mental health and communication abilities. This takes about 3 hours, and you and your child will be together the entire time. We will not try to make your child talk. We would like you to bring in a recorded speech sample, and we can observe through a one-way mirror with your consent.\nBefore you leave, we will give you a package of information to keep and one to share with your child's school. We will mail you a written report of the assessment and we will refer you to other programs at CPRI if needed.\nOur goal is to make a plan to help your child in your home community and school. We will follow up with you and the school by telephone or meetings at the school.\nThe initial assessment is done in a playroom or an interview room at CPRI in London, Ontario. Follow-up support is typically telephone contact with the school and family team, or a school visit.\nIf you choose not to proceed with the assessment, please feel free to access our list of resources. We also provided education and consultation about selective mutism to communities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..f5344815faa6f9ed04751f72a7bcb499065ed145
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+{"text":"D-2462, Copper Color, Shimmery Finish Paper, Single Sheet Cards, Small Size Cards, Light Weight Cards, Designer Multifaith Invitations.\nSingle sheet card with wooly overlay on front. Card and envelope in Copper shimmery finish paper.\nSilk Screen Printing in Silver or white color.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Every year, Congress authorizes billions of dollars of Pentagon pork at the expense of other security needs and other taxpayer priorities.\"\nThe U.S. House on Friday overwhelmingly approved a $696 billion defense policy bill\u2014the 2018 National Defense Authorization Act (NDAA)\u2014that critics were quick to denounce as yet another vote for \"endless wars\" and \"Pentagon boondoggles.\"\n\"As always, only spending that benefits the poor is 'unaffordable.' Larger sums are funneled straight into corporate pockets without protest.\"\n\"Every year, Congress authorizes billions of dollars of Pentagon pork at the expense of other security needs and other taxpayer priorities,\" Paul Kawika Martin, senior director for policy and political affairs at Peace Action, said in a statement. \"While war profiteers rejoice, voters wonder why the government cannot provide for education, job creation, healthcare, and needs in their community.\"\nThe legislation, which passed with a vote of 344-81, far surpasses in cost President Donald Trump's request for $603 billion earlier this year.\nThe House measure would authorize $621.5 billion for national defense programs, including the Pentagon's base budget and nuclear programs under the Energy Department, as well as another $75 billion in war funding.\nIt also would tap $10 billion from the war-related Overseas Contingency Operations account to pay for base budget items, including $6 billion to boost Navy shipbuilding. The NDAA funding levels mirror a budget blueprint being crafted by the House Budget Committee.\nRep. Barbara Lee (D-Calif.), who voted against the measure, said in a statement that \"Congressional Republicans have chosen to bankroll bloated Pentagon spending and funnel billions into the Overseas Contingency Operations slush fund, while refusing to address our fundamental obligation to debate our ongoing military operations.\"\nThe Intercept's Alex Emmons argued that the exorbitant military spending approved year after year underscores the warped priorities of Congress.\n\"As always,\" Emmons wrote, \"only spending that benefits the poor is 'unaffordable.' Larger sums are funneled straight into corporate pockets without protest.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So just how much is owning an Apple iPad 2 worth to you? The old saying an arm and a leg perhaps, or how about a new saying, I'd give my kidney to own an Apple iPad 2? Sound weird doesn't it, but it's apparently true, some Chinese kid sold his kidney to buy an Apple iPad 2.\nAccording to an article over on Newslite by way of The Telegraph, 17-year old Chinese teen only identified by his surname \"Zheng,\" apparently sold one of his kidneys after seeing an advert that offered money for being an organ donor.\nThus Zheng travelled to Chenzhou in Hunan Province, had his kidney removed and was handed the cash, approximately \u00a32000. Word is Zheng has complications after the surgery and this was the reason he told his mother what he had done.\nThe kids must have been desperate for that iOS tablet, would you be prepared to sell off one of your organs if you couldn't afford the Apple iPad 2, or do you believe that is taking things a bit too far?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Formed in 1990, Germany's Disbelief has steadily been releasing full length records since 1997. Protected Hell is already their 8th offering, their 5th for Massacre Records. Let's see what these sludge\/death metal dudes have in store for us.\nWhat you can expect from Disbelief is some slow, mid-tempo death metal with sludgy, doom riffing in between. Think of bands like Jungle Rot, Obituary, Bludgeon and Six Feet Under, only with an occasional emphasis on the slower parts. The overall production and mixing is really nice, the record really goes down like a cold glass of beer. But, the same as with cold glasses of beer, after a number of songs you don't remember anything anymore. Almost everything is written in the same tempo and groove, which makes it sort of dull after a while.\nThe total package is more than alright. A good production, good convincingly tormented vocals, nice songs written by good musicians who know exactly what they're doing and what they're aiming for. Yet a little more variation wouldn't kill them I'd say. If you dig their previous work you can pick this up blindly. Otherwise this is a great way of getting to know this band. If you like fast and brutal, stay away from Disbelief.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Keep your grooming essentials organized with our Hunter Toiletry Bag. A removable zippered compartment and side slip pocket perfectly hold items like a small comb, contact lens case or tissue packet, while elastic loops cradle items like lip balm, razors or travel fragrance. Throw this small case inside any of our 365 bags for life on the go.\nAir Mesh is a hand-washable, tear-resistant textile that lends breathability and versatility to our bags and pouches. Its tactile nature makes it a perfect complement to smooth neoprene, but also allows for a nearly weightless and cushioned carry.\nWith our Hunter Toiletry Bag, you can keep your grooming essentials perfectly organized. In addition to a removable zippered interior pouch for your toothbrush, contact lens case or comb, the larger iteration is complete with a side slip pocket and detachable air mesh brush pouch. Elastic loops cradle items like lip balm, razors or travel fragrance, and the extra volume inside provides enough space to easily contain TSA-sized moisturizers, shampoo and shower gel.\nYou know that annoying moment when you've fit all of your toiletries into one bag...except for one. With the XL Hunter, no more rogue deodorant, lone dry shampoo, or trying to make that awkwardly shaped hair brush fit. Because guess what? Everything fits. With an interior airmesh slip pocket, removable airmesh zip pouch, three elastic loops, and ample space in the main bag, the XL Hunter is the toiletry one-stop you should probably shop. But it's not just about space, with its premium neoprene material and color-plated Zinc-Alloy hardware, it's about waterproof style too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmqmo b/data_all_eng_slimpj/shuffled/split2/finalzzzbmqmo
new file mode 100644
index 0000000000000000000000000000000000000000..e3f68d846f15db450be0d7693b72db1dd6df767a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmqmo
@@ -0,0 +1,5 @@
+{"text":"Now before you get all judgey and grossed out, it isn't as bad as it sounds.\nHave you ever heard of CLEAN SLEEP? My friend Michael Ingle, Denton, Texas entrepreneur, founded a new technology for dry cleaning mattresses. Click the link above and check it out! It's a commercial grade mobile cleaning business right to your front door. Clean Sweep is quick, green and clean. You might have seen Michael and his flourishing CLEAN SLEEP business on Shark Tank several years back. This heated sterilization technique has proven to be a reputable brand and thriving business.\nNeedless to say, Michael cleaned our mattress twice in 20 years. However, it time to quit monkeying around, throw in the towel and buy a new one. Unlike buying a car or house, purchasing a new mattress isn't exactly the sexiest or most interesting agenda item on my purchase list. In my book it's right up there with sticking a needle in my eye.\nWant to know why? Because mattresses are expense y'all!\nBuying a new mattress is equivalent to the purchase of car tires. You've got to do it but you fight it. You grin and bear it until you can't take it anymore or you're sliding off the road or off the bed and finally break down and get the new mattress already!\nSide bar, I don't really have an issue with sleep. I work hard, I play hard, I wash my face, I eat dinner, I climb into bed, turn off the lights and boom, I'm out! Sound a sleep. The END! So what exactly can a new mattress do that tops that!?!?!\nTop it? No. Enhance it? YES! Man, mattress buying has come along way. You should have seen me and Mr. Chicken Fry up in the Mattress Firm.\nChicken Fry, \"we are?????,\" with a questioned look on his face? \"Are you sure\u2026I think I'm FIRM.\" looking at me for blessed reassurance.\n\"Honey, we are talking about mattresses here get your head in the game.\" (rolling my eyes) This was basically the connotation and mood of how the next 15 minutes would play out.\nChicken Fry shot over as fast as he could bob and weave between the beds. \"This is kind of cool don't ya think?\" He immediately hoped in to take a spin. \"Look,\" I said, \"it will elevate your feet and vibrate too.\" It was like we had never seen a TV commercial and come out from underneath a rock. I felt like I was in a sitcom only it was real. By this point the sales man who Mr. Chicken Fry had abandoned comes over and is staring down at us. \"How much is this,\" utters Chicken Fry. Who keeps firing back with more questions.\nIn less than 30 minutes we came, we saw, we took turns and in the end purchased a new remote control base and mattress. What a country.\nThe day it was delivered was sheer excitement. The anticipation was off the chain. We couldn't wait to plop into the soft pillow like mattress and get to sleeping, lol. Adjustable beds are fun! Turns out\u2026it's maybe the best investment we have ever made together in our marriage. It's truly amazing in every way.\nHere are my top 5 reasons why.\nWhen Mr. Chicken Fry begins to (cough cough) SNORE, I can easily elevate his side of the bed just enough that he will stop snoring! Call me geriatric or call me genius, either way works. P.S. We did not purchase the split mattress. I didn't want to fuss with sheets and such.\nThe ability to elevate my feet after a long day of training, running, cooking, errands, and working on my feet all day is like BUTTER! It's great for blood flow and circulation.\nIt's like being in a giant soft Barcalounger \/lounge chair only it's my bed!!!! I can tilt the bed up for drinking coffee! I can write my blog in a soft cozy position and eat my nightly bowl of cereal.\nWhen Chicken Fry comes to bed, I don't even feel it! The mattress absorbs all body movement. It's a full on do not disturb bed. Any tossing and turning that either of us do is amazingly unnoticeable. We had grown immune to the bumping or creaking sounds of our bed and now, total silence. AWWWWW, sleep in heavenly peace!\nOn the off-chance one of us is feeling sick or under the weather,(this week was a doozie) this bed is the money. The ability to tilt upward helps suspend coughing and drainage. It makes watching television easier and feels better for minor body aches and pains associated with the flu or other related sickness. In sickness and in health right folks?!?!\nNow, me and Mr. Chicken Fry can just lay or sit up in bed and talk all night long! (insert heavy eye roll) Nothing better than hunting for remotes and fighting over the perfect setting and tv channel. This is what 19 years of marriage, true love and perfect pillow talk look like.\nLove, snuggle and make time to cuddle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Slater is the most caring Physician anyone could ever ask for.\nDr. Slater: asked several medical questions regarding my past and current health.\nDr. Slater is easy to converse with, efficient, effective, & I am glad she is my new doc.\nExtended hours: Tuesdays & Thursdays to 8 PM.\nSaturdays: Open by appointment only!\nOur Icon\u2122 IPL treatment is a quick and easy way to undo the damage caused by the sun including vessels, sun spots and freckles.\nIn just a few treatment sessions you'll have a clear and glowing complexion. Schedule your consultation today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When mining companies extract minerals from the earth, they leave huge deposits of soil and ore next to the mining site. These deposits are commonly known as tailings. In most instances, tailings contain a considerable amount of valuable mineral resources which cannot be exploited because of a lack of appropriate equipment, or as a result of economic non-viability. However, many mining companies choose to keep and maintain these tailings, in the hope that such minerals can later be exploited when time or technology allows for this. Under common-law the owner of a property is considered to also own any minerals contained on the property, in terms of the principle of cuius est solum. In South African law, however, a practice evolved whereby owners of minerals separated rights to minerals from the surface rights on the property. This created a mining right which was independent from the land and could be transferred to third parties, often in return for compensation. Under the Minerals Act of 1991 the owner of a mining right over a property (be that the owner of the property or a third-party mining right holder) also held the mining right to tailings which were created as a result of mining activities under the right. Thus, if a mining company performed mining activities on a property, the company was also free to exploit the tailings which were left next to its mine, regardless of whether the dump had remained there for a long period of time. Owing to South Africa's long history of mining, some tailings are over a century old and resemble small mountains rather than mining deposits. The Mineral and Petroleum Resources Development Act of 2002 changed the entire mineral legislative regime in South Africa. Whereas owners of land were previously free to separate and sell their rights to minerals to anyone they wished, the MPRDA placed the country\"s mineral and petroleum resources under the state's \"custodianship.\" Where the law talks about custodianship, however, it supposedly refers only to minerals that have not yet been extracted from the earth. It is well established in South African law that, once a mineral is extracted, it becomes the movable property of the person who extracted it \u2013 in other words, that of the mining company. Does this mean that minerals in tailings also fall under the state's custodianship? The Free State High Court did not think so. In the case of De Beers v Ataqua it held that, in terms of the common law principles of acquisition by way of attachment, tailings are clearly movable property and therefore belong to the mining company who created them. For the MPRDA to hold otherwise would amount to expropriation. The state did not wish for some mining activities to be regulated by a different set of legislation, so it amended the MPRDA to try and define \"residue deposits\" (the name by which the MPRDA calls tailings) more clearly. However, due to the legislature's unfortunate choice of wording, tailings created before the enactment of the MPRDA are still, strictly speaking, not regulated by that Act. So the legislature proposed another amendment to the Act, this time making sure that any historical mine dump created at any point in South Africa's history are placed under the Act's regime. The subject matter of this study is whether the above amendments to the MPRDA could be considered to be expropriation. For background purposes, a brief overview of the Ataqua decision as well as the subsequent amendments to the MPRDA will be given. Then the history of mining legislation and the development of a separate mining right will be summarised. The reason for this summary is to establish whether, in terms of constitutional litigation, a clear right has been established for purposes of protection under section 25 of the Constitution. The last phase of the study will look at the particular characteristics of expropriation and ask the question whether acquisition of a right by the state is always a fundamental requirement for expropriation to take place. It is submitted that the destruction of an entire class of property by way of legislation, amounts to so-called \"institutional expropriation,\" which is subject to compensation in terms of section 25.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I can't say I'm a steampunk fan, but I can't deny my love for Shakespeare and Wilde. If I read a book using them that didn't stand up to their work (and it rarely does), I have major sulking issues, but it sounds like this stands the test! I love the fact it is over the top farce because while some people get Wilde is farce, it is hard to convince a lot of people that even Hamlet has farcical elements and Twelfth Night is over the top ridiculous.\nI saw live performances of Tweltfh Night 3 or 4 times plus a bunch of screen adaptation and I have never failed to be enchanted and this novel reproduces well the \"spirit\" of the play.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why choose Spectrum Cleaning and Restoration as your O Fallon water damage restoration company?\nWhen fire or water damage in O Fallon MO strikes your home or business, it is never convenient. We know any kind of disaster can cause havoc on businesses large and small. When there is business interruption, downtime goes up and profitability goes down.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbojws b/data_all_eng_slimpj/shuffled/split2/finalzzzbojws
new file mode 100644
index 0000000000000000000000000000000000000000..e4beee72bb778524c9565d53ef56d6eb49520fe4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbojws
@@ -0,0 +1,5 @@
+{"text":"The Arts Council, Inc. is a non-profit 501(c)3 charitable arts organization located in Gainesville, Georgia. Serving Northeast Georgia, The Arts Council, Inc. was originally founded in 1970, as the keeper and creator of the community's arts calendar and newsletters. Today, it is one of Georgia's leading arts organizations. It promotes over 30 regional arts affiliates, presents a broad spectrum of performing, visual, literary arts and film programs, and provides many other services that also support its mission. The Arts Council is funded by individuals, corporations, foundations, ticket revenue and facility rental income.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For more details view the Weed Management Expo for School Sites Flyer, PDF (132 kb) and Weed Management Expo for School Sites Agenda, PDF (150 kb).\nI gave a presentation (same one I did at the FarWest Show) today and was asked to post it. Unfortunately, I haven't had time to figure out the new website program (sorry DK!) so I'm putting it here.\nIt is called \"Are you spending too much on weed control?\" and it is geared towards container nurseries.\nAre you spending too much on weed control?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We use the rule in IT Brain to send an alert on High CPU usage. One of the servers sends an alert every hour caused by an export. This only takes a few seconds and is fine, but it fills the mailbox. We don't want to completely disable the CPU monitoring though. Is there any way we can set the alert only when CPU usage is exeeding the limit during a longer period of time?\nThanks for the feedback. This is a very good use case.\nWe will improve our check and add a grace period in the near future.\nDo you think this would solve your current situation and will generate less alert spamming?\nCurrently there is no time threshold for CPU alerts from ITbrain Monitoring & Asset Tracking.\nIf possible, I would recommend setting the alert threshold just above what this alerts (if the CPU % is still within the device's standard daily operations).\nI will forward this feeback to our development team; while I can not promise if or when these features could be implemented (as the decision is based on popular demand), your feedback is very important to us!\nYes, this will really solve the issue. Not sure why you want to limit periods to the 3 options and not just let users insert the minutes that work for them, but suggested options seem fine to me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Livraria Brit\u00e2nica :: Quick, Barney, Run!\nWhat can Ruby Roo do on a rainy day? Use her imagination! This jaunty, thrilling rhyming text sees these two intrepid explorers jet off to the jungle in their cardboard seaplane, and swing through the exotic pot plant canopy . . . encountering some strangely familiar characters on their way!\nPacked with humour, colour and adventure - and with tons of 'clues' for children to spot - this is the latest picture book sensation from Pip Jones and superstar illustrator, Laura Hughes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The multi-language discussion paper explores the health risks that may occur when workers are exposed to fumigants and presents procedures and guidelines for safe practices when handling fumigated containers. It also gives an overview on legislation and relevant fumigants and how to measure them.\nEU-OSHA uvodi novu bazu podataka o opasnim tvarima \u2013 pogledajte o \u010demu je rije\u010d!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbozdc b/data_all_eng_slimpj/shuffled/split2/finalzzzbozdc
new file mode 100644
index 0000000000000000000000000000000000000000..bd1dac0e7919664b7b15f2320f3b2ab40105e66d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbozdc
@@ -0,0 +1,5 @@
+{"text":"These brushes were designed to stand the test of time, the biggest feature in my mind is the new synthetic fibers used in the brush head. The 'DuPont Natrafil' fibers mimic the structure of natural hairs. We all know that synthetic brushes hold liquids well, but often fail in their ability to hold powders. With the move to cruelty free products becoming something many makeup lovers are interested in, there was always a compromise on quality if you want vegan brushes. These brushes overcome that.\nDesigned to be used day in day out in the most intensive professional situations, these brushes- if cared for properly, are designed to last.\nAmazingly, in my rather extensive brush collection, I do not have any brush similar to this. It's completely flat, beautifully soft and animal product free!\nAt IMATS, when I caught up with Sara-May, she said the brushes were so soft I'd want to sit there and brush them up and down my face all day\u2026.I raised an eyebrow, but quickly realised she was right. They are super soft.\nIn terms of their ability to hold product as claimed, it's true. They grip powder and they blend well- being black, the powder is very visible, which also means they are super easy to wash as you can see when they're clean.\nIndividual brushes come in a plastic sleeves, sets come in a plastic tray within a larger plastic sleeve.\nI keep the individual sleeves as I find them convenient for traveling.\nAlthough this flat fluff brush was a generous gift from Royal and Langnickel \u2013 I like the [R]evolution range so much I have since bought the 6pc complexion set. I have since ordered more of the collection.\nOne thing worth mentioning in terms of the complexion set it that the large Domed Kabuki (BX-00) takes ages to dry following a deep cleanse, I suspect (and it is not a bad thing) it is purely due to the density of the bristles on this brush.\nOur [R]evolution line pushes the limits of brush performance and durability.\nWish brush heads made from new synthetic fibers that mimic the performance and feel of the finest natural hairs available. Incredibly soft and extremely durable, you'll be amazed to discover the revolution.\nThe highest quality non-corrosive, nickel-plated, solid brass ferrules are utilized in the [Revolution] brushes for its durability and resistance to damage. A necessary addition to protect from daily wear and tear.\nAcrylic handle with Silk Touch comfort coating.\n[R]evolution brushes feature the latest advancements in synthetic brush technology with DuPont Natrafil fibers. Which mimic the performance and feel of the finest natural hair. Each monofilament features a micro-studded surface that mimics the exterior cuticle of the exterior hair, allowing for precise pickup, release and blending of even powder makeup.\nOf the brushes I have tried (so far), I love this entire range. Softness is always a top priority for me, on my own face it's less critical but if I'm going to be using brushes on someone else I want them super soft.\nI also like synthetic brushes for other people because I feel they are easier to clean and therefore more hygienic. This means I usually have to compromise somewhere, with these brushes I don't.\nIf you want reasonably priced, ethical, animal product\/cruelty free, durable brushes, that will perform as well as some of the most expensive natural hair brushes (trust me, I own some of those too) you cannot, and will not find better than these.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Wise One who understood the law that was handed down from God long ago reflected the true meaning 2 his woman every day and she surrendered her discerning of it in2 his care and keeping 4 she trusted he would lead in the right way. Her children in subjection 2 her. She in subjection 2 the Wise One in subjection 2 the only begotten one, all ~ 4ever in subjection 2 God.\nJust like the Sun, the Rainbow Children rise. Flying upon the wings of the New Translation. C them fly, fly. The covenant will b kept this time. Just like the Sun, the Rainbow Children rise.\nRainbow children, it's time 2 rise! Rainbow children, it's time 2 rise!\nAs prophesied, the Wise One and his woman were tempted by the Resistor. He knowing full well the Wise One\u00b4s love 4 God assimilated the woman first and only. Quite naturally, chaos ensued and she and 5 others were banished from the Rainbow\u2026.4ever.\nThe one who came from nothing . And yet from this one everything comes. The one who commands ur momma With the simple phrase \"I am\" And every time that she obeys She gives birth 2 the Son of Man Who is this?\nReproduction of the new breed leader Stand up and organize!\nThe Wise One who understood the law that was handed down from God long ago, held fast in his belief that the Lord would bring him another one who loved him so.\nThe helix be might get between them.\nFilho, voc\u00ea \u00e9 bastante forte e capaz de conseguir muito. Acredite em mim. N\u00e3o foi escolhido sem raz\u00e3o para estar neste momento na Terra. Tem uma vontade forte, uma constitui\u00e7\u00e3o forte e muito amor em seu cora\u00e7\u00e3o.\nMuitas outras criaturas se voluntariaram e tamb\u00e9m foram escolhidas. Isso por si s\u00f3 j\u00e1 \u00e9 um sinal de sua grandeza. Est\u00e1 muito bem equipado para esta exigente, embora gratificante miss\u00e3o.\nPode fazer e conseguir\u00e1 minha crian\u00e7a. Sei disso porque est\u00e1 muito bem a banhar-se no meu Amor para lhe dar ainda mais vigor.\nSophia was the Mother of Creation; her consort and assistant was Jehovah. Her sacred shrine, Hagia Sophia in Istanbul, is one of the seven wonders of the world. Her symbol, the dove, represents spirit; she is crowned by stars, a Middle Eastern icon, to indicate her absolute divinity.\nThe saga of Sophia is the story of the soul's adventure across infinity, as well as the perennial symbol of the triumphant return of the Divine Feminine.\nThese energies, or Aeons manifest sensory worlds by dreaming. The One creates multiple emanations in the Pleroma. Each with a masculine and feminine polarity. Always in movement, these Aeons formed pairs called syzygies. Together these emanations come together to create a thought form.\n\"YOU ARE MISTAKEN BLIND ONE.\nAND WHO WILL APPEAR AMONG YOUR DUPLICATE FORMS IN YOUR SIMULATED WORLD.\nHUMANITY EXISTS AND THE OFFSPRING OF THE HUMAN STRAIN EXISTS.\nA celestial light came to her aid. She aligned her divine power with a new born star. That star is our Sun. In order to redeem her action that created the Archons, according to the Gnostic texts, she took the physical form as Gaia (The Original Earth) and her daughter was called Zoe (the Imagination).\nWhile the Demiurge and the Archons built their replica Earth Matrix, (using other electromagnetic bodies known as Planets and Suns), an illusionary construct, was made to perfectly 'mimic' the original earth reality.\nThis Father God created seven realms, each more material than the last, until finally this human world was formed and the beings of this world rose and walked, crawled or flew through the skies.\nBut the Father God was vain and jealous, angry and forbidding, not knowing the power of the higher Aeons, nor of the Pleroma, nor even of his own Mother, the Divine Sophia. When the Divine Sophia instructed him and opened his mind to Truth, he was amazed and refused to divulge these mysteries to those of his own creation.\nBeing a god of the material, social and psychic order, it was not possible for him to be a teacher of the higher mysteries, and Sophia was dismayed by his wrath, anger and jealousy. So when he created the first human beings, She was there and secretly, without his knowing, She gave to them the gift of the Holy Spirit as a divine spark in every human heart.\nAnd it is said, that in the Garden of Eden, created by the lower Father God, that Eve was the manifestation of the Lower Wisdom and that the serpent or snake of the tree, was actually the Christos who urged Eve to eat the Fruit of the Tree of Knowledge that she might attain the true Gnosis or knowledge of her origins and realize the Higher Sophia in perfect illumination and bliss.\nBut the Father God, discovering that this secret teaching was disturbing his supremacy in the lower realms grew angry and cast them out of the garden and into the suffering of the world. Yet each and every descendent has this spark and the potential to recover the true Gnosis.\nThe Gnostics say that humanity is caught up between two realities, the 'original Earth' and the one created out of what the Gnostic texts call HAL forged by its 'Creator' \u2013 the Demiurge (Father God). On Earth the Archons are the 'genies in a bottle', phantoms that can take many forms through fire and water, so to be let loose or to be commanded. Once loose, they can and do influence the illusionary world for those that 'communicate with them'. See all 'magicians' from Dr John Dee to Crowley, & the Satanic elite.\nRediscovering the Lineage of Sophia becomes an essential key to accomplishing the task of Humanity as Anthropos. (Anthropos is a Greek word meaning, in the ancient mystery schools, the template of divine human potential.) The Gnostics have said that the gift of the imaginal reveals Divinity's greatest gift to humanity, and is one of our most essential tools in expanding awareness. The intention in telling this story, largely based on Gnostic texts, is to spark your own \"direct knowing,\" which will then inform your sacred activism in the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Continental (Mirror Finish) There are 11 products.\n\u2022 Thickness: 2.5 mm. \u2022 18\/0 Stainless steel. \u2022 Mirror finish.\n\u2022 Thickness: 3.0 mm. \u2022 18\/0 Stainless steel. \u2022 Mirror finish.\n\u2022 100 gram. \u2022 18\/8 Stainless steel. \u2022 Mirror finish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oriental flame retardant paper bunting flags, with an elephant, Indian d\u00e9cor. Length is 15ft \/ 4,50m, 8 rectangular flags 30 x 20cm. Perfect to give joy to your private party.\nOriental Cut Out. Printed both sides, circle shape, 30cm diameter. Ideal for oriental themed party.\nIndian Cut Out. This paper card cutout is printed both sides, circle shape, 30cm diameter. Ideal for Indian Themed Party.\nOriental banner, Taj Mahal Bunting, Flame resistant, 15ft \/ 4,50m long 8 rectangular flags 20x30cm printed both sides. Ideal for room decorating. Low price for very high quality.\nIsraeli plastic flag bunting 17ft\/5m lengths 10 flags 8x12inch\/20x30xcm or 33ft\/10m lengths 20 flags 8x12inch\/20x30cm. Indoor and outdoor use. Give any colours for your Patriotic decoration, Jewish Party. Israeli flag is white background with blue star. Our modern Israeli banner will be perfect for your garden party, event, birthday and religious party.\nIsrael paper hand-waving flag 5,5x8inch\/14x21cm plastic stick 37cm, pack of 100. Our top quality Israeli hand wavers are very popular in the UK. If you want to support your country's team while sports game live, this will offer the good colours for sure while your party. Deco Party UK propose four different Israel flag party supplies.\n150 table confetti Isreali flag 1inch\/25mm diameter. This Israeli table decoration is perfect for Bar Mitzvah, Bat Mitzvah and Mazal Tov. All Jewish party will be decorated with great success. For every occasion and cocktail our Israeli flag confetti will give colours to your party decoration. Party supplies made by Deco Party UK.\nDeco Party UK makes plastic oriental country flag banners for indoor or outdoor use and also offers flame-resistant paper Taj Mahal banner, Israeli bunting flags, Egypt Bunting Flags, pyramid banner for indoor decoration. Several lengths are available as 4,50m, 5m and 10m long. We also can make your own lengths and your own designs. Buy online our oriental banners for a very low price !\nVery popular in the UK, we produce original hanging decorations. Easy to hang from the ceiling, from the wall or to stick to the window, our Egypt Pyramid Cut Outs are printed both sides in paper card. We also supply oriental elephant cut out, circle shape, 30 cm diameter, with a hole to hang easilly. Top quality oriental cut out !\nBuy online and sprinkle on the table our unique and original Israeli flag confetti, circle shape, 25mm diameter, printed both sides. Need any Israel party decorations ? We also can create those confetti for other oriental themed party.\nNeed any other oriental theme party decorations ideas ? Deco Party UK makes high quality Israeli cocktail picks, ideal to jazz up your sandwiches, canap\u00e9s, pizza, burgers and much more. Paper flag measures 3 x 5 cm and food wood stick measures 7,5cm long. If you need further Indian Party Decoration Supplies we also can create Indian cocktail sticks with special enquiry.\nWe have supplied Israeli Hand-Waving Flags, also called Israel Hand Held Flags. If you need any for Indian Themed Party, feel free to send us your enquiry and we can produce Indian Handwavers or Egypt Handwavers. Each flag measures 14 x 21 cm and plastic stick 37cm long.\nWould you like to buy from our factory our oriental party decorations in bulk ? You have been a wholesaler, importer, retail shop, communication agency or just a reseller and you need better prices ? Feel free to contact us by email or by phone and we will offer special prices to satisfy your needs. We are able to supply large quantities as we have made high quantity each day of production. If you would like other kind of oriental party decorations as oriental hanging swirls or more choice for a fantastic atmosphere we can create new items to celebrate all your oriental party decorarions ideas !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An appellate court has upheld a lower court ruling, throwing out a case that could have blocked the redevelopment of property owned by the Village of Hempstead.\nMany Long Island law firms experienced growth for the first time in years, hiring and expanding to meet emergent need.\nThese are frightening times for businesses employing one to 100 people, says Greg Lisi, head of employment and labor law at a Uniondale-based law firm.\nOnes to watch in the various areas of commercial real estate on Long Island.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpaea b/data_all_eng_slimpj/shuffled/split2/finalzzzbpaea
new file mode 100644
index 0000000000000000000000000000000000000000..c1483d7d5387cc54b430486db119bcf9192a69b7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"With a waist-accentuating high rise and tapered legs, these vintage-inspired jeans look fresh in nonstretch white denim with raw, jagged hems. Style Name: Madewell Perfect Summer High Waist Pieced Jeans (Tile White). Style Number: 5622475. Available in stores.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the functional diagram of flotation machinery principle. From that, we can have a clear picture of the instruction of the flotation machines. At present, flotation machine mainly includes mechanical flotation machine, inflatable flotation machine, and inflatable flotation machine.\nVertical shaft impact crusher. Sand making machine (vertical shaft impact crusher) is of high-power and low-consumption, which is designed based on several years hard work and study of mechanical mining equipment. Daswell sand maker absorbed the advanced technology of the same products from Barmac Company of America, so now it has achieved the international level.\nA typical process flow diagram of cumene production on a zeolite catalyst is process flow diagram cumene production design and economic analysis of a cumene production plant groupfadi commercial uses\/ applications of cumene . block flow pid diagrams . process flow diagram .\nNIAflow makes it easy to design processes and optimize your profit. Crushing plant. Crushing plant. Crushing Plant. Three stage lime stone crushing plant set up in three individual plants. Secondary and tertiary plant featuring closed crushing circuits. Each object in NIAflow represents an individual type of machine or device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Attending LISA 2012 in San Diego, CA? If so, join us on Wednesday night from 8:00PM to 9:00PM for a Birds of a Feather session!\nThis BoF will discuss the Apache CloudStack project and the complimentary technologies for deploying and managing cloud infrastructure primarily using open source software. This sessions will be led by David Nalley a recovering sysadmin who now works as an evangelist and committer for CloudStack. Attendees should expect to discuss virtualization, networking and storage options as they relate to geographically disbursed cloud computing deployments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: Instruction scheduling and register allocation are two of the most important optimization phases in modern compilers as they have a significant impact on the quality of the generated code. Unfortunately, the objectives of these two optimizations are in conflict with one another. The instruction scheduler attempts to exploit instruction-level parallelism and requires many operands to be available in registers. On the other hand, the register allocator wants register pressure to be kept low so that the amount of spill code can be minimized. Currently these two phases are done separately, typically in three passes: prepass scheduling, register allocation and postpass scheduling. But this separation can lead to poor results. Previous works attempted to solve the phase-ordering problem by combining the instruction scheduler with graph-coloring-based register allocators. The latter tend to be computationally expensive. Linear-scan register allocators, on the other hand, are simple, fast and efficient. In this paper, we describe our effort to integrate instruction scheduling with a linear-scan allocator. Furthermore, our integrated optimizer is able to take advantage of execution frequencies obtained through profiling. Our integrated register allocator and instruction scheduler achieved good code quality with significantly reduced compilation times. On the SPEC2000 benchmarks running on a 900 MHz Itanium II, compared with Open IMPACT, we halved the time spent in instruction scheduling and register allocation with negligible impact on execution times. Copyright \u00a9 2007 John Wiley & Sons, Ltd.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not expecting any miracles, but here's hoping the Dems can make the necessary changes to our health care system. With a little modification, I found the caduceus to be very accommodating as a pogo stick.\nThe CONCEPT KING strikes again. This is like the old mainstay of the gag cartoonists...the desert island. There's always a new way to do it... like the caduceus. This is probably the most creative way I've ever seen it. I'm in humble awe...having done about 14,000 of them myself over the years.\nBoing...Boing... Great piece of art, says it all about this health care mess.\nHave you seen Sicko? It will piss you off. The idea of maximizing profit off of peoples health and well being suddenly seems so wrong. Not sure it will ever fly here in the \"land of the free\" I think the money making insurance companies, HMO's etc are just too entrenched. I hope I'm wrong because what's going on now is a crime. I just paid 2k for a routine exam for my wife. I'm assuming the illo is to make the idea of passing universal health a daunting task indeed-nice work.\nAs Randall said, you are indeed the king of concepts. Love this idea. Like you, I am not holding my breath for health care changes. But I sure hope they happen.\nThe caduceus as a pogo stick?, brilliant. I too, have done a hundred caduceus concepts and this one rocks!.\nHi David, powerful image. I like the colors too.\nHey, David... what Christopher said: your header just knocks me to the floor!\nThanks all! Randy - coming from you, that title is quite a distinction. Christopher & Lou...thanks for the feedback on the header. And Walter - it's taken me awhile to simplify an approach to color that works for me.\nPoignant, as always. Great piece David.\nDavid, really like your octopod header. I think that pogo stick has retail possibilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpnby b/data_all_eng_slimpj/shuffled/split2/finalzzzbpnby
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index 0000000000000000000000000000000000000000..afabb36a463abc5206d1da0799b8f079a36b8aae
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A report issued by the Election Commission of Pakistan (ECP) stated that the majority of the 'Form-15' (ballot paper account) were missing from the constituencies won by the Pakistan Tehreek-e-Insaf (PTI) candidates in the 2013 General Elections.\nThe Judicial Commission (JC), that was assigned the task of investigating the rigging in the 2013 General Elections, had asked the ECP to produce the Form-15 used in different polling stations of the country.\nAccording to the ECP report, the Returning officers were assigned the maintenance of the documents which consisted the 'Form-15' used, remain unused or discarded during the elections.\nAs per report, Form-15 of 43% of polling stations where PTI won are missing; 40% of Form-15 are missing where PPP won and 28% of Form-15 are missing where PML-N won. The report also discloses that Form-15 of 33% National Assembly constituencies are missing. On the other hand, records are missing from 43 percent polling stations in KPK, 45% in Sindh, 29% in Punjab and 40% in Baluchistan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Running a new customer engagement channel doesn't have to be difficult. Integrate your social media accounts within your live chat dashboard to allow your agents to handle social media queries with ease \u2013 included in our Multi-Channel platform.\nOur integration allows you to implement a well-organized social media workflow which cuts down on the number of systems, screens and passwords that your agents have to handle, and gives you a clear view of the messages and replies being sent to and from your teams. It's cleaner and more efficient than working with each social media native application.\nFacebook and Twitter are made for people, not businesses. They lack common tools needed to handle work within a team \u2013 for example the ability to assign cases to other agents, pend awaiting responses, or add notes visible within a back-end interface.\nOur social media integration does all of these things \u2013 allowing you to sort, filter, label and prioritize cases, helping to make social media case work a breeze for you and your team.\nIt's not always easy to see the history of a conversation in each social media application \u2013 different methods of getting in touch (e.g. wall post versus direct message) mean that it can be tough to see how a customer's contact has played out over time.\nIt's easy to see the history of a communication thread within our interface. And wrap-up notes can be added too, to help agents handle future queries from that customer in a more personalized, context-aware way.\nComm100 offers a full suite of reports for our social media integration, giving you access to data about your service which can help you to identify and implement service improvements. With these reports you can view data from different perspectives, display it in easy-to-understand formats, and access real-time insights to make instant improvements to your service.\nRunning social media through their native websites can be tricky for enterprises \u2013 agents need to share logins, keep record of different logins for different accounts, and spend time logging in and out of sites manually.\nNot only does this waste time, but it's a security risk too \u2013 what would happen if an agent carelessly left login details lying around? With integrated social media, agents only need to use their single live chat login to access and respond to social media queries \u2013 no matter how many Facebook and Twitter accounts you handle.\nOnce you've entered your account information into your dashboard, your Facebook and Twitter queries will be pulled into a special tab within your agents' interface. That's it \u2013 no more customization is required to get your social media channel up and running.\nJoin our live webinar to learn how you can build better relationship with your customers and achieve your customer engagement goals on social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The City of Oakland lost a big whistleblower lawsuit yesterday in federal district court.\nThe case, brought by Daryelle Lawana Preston, Oakland's former director of employee relations, included allegations that a top Oakland official pressured her to falsify reports to the city council, and that other Oakland officials improperly negotiated labor contracts with city unions, and willfully violated labor union contracts. Preston said she was punished, and ultimately fired, for speaking up about these activities. A jury agreed with Preston, awarding her $613,302 in damages.\nPreston also alleged that Santana asked her to lie in an oral report to the city council. The lawsuit charged Santana with depriving Preston of her federally protected right to free speech. The judge dismissed this charge against Santana, however, leaving only the city of Oakland as the defendant.\nPreston also alleged that Oakland Fire Chief Teresa Deloach Reed improperly negotiated with the city's firefighters union without authority to do so, and that Santana condoned the activity. Finally, Preston alleged that City Treasurer Katano Kasaine violated the city's contract with the union SEIU 1021 by refusing to collect dues from part-time employee salaries for several years.\nAfter speaking up about these activities and refusing to cooperate with actions she believed to be illegal, according to the lawsuit, Preston was fired by Santana on October 13, 2013.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Performance Bonds cover performance of the terms of a contract. These bonds frequently incorporate payment bond (labor and materials) and maintenance bond liability. This protects the owner from financial loss, should the contractor fail to perform the contract in accordance with its terms and conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The major would increase from 30 credits to 33 credits by including a requirement that one of the following courses to be taken at SUNY Brockport (SOC 210 \u2013 Social Problems, SOC 220 \u2013 Social Psychology, or SOC 230 Social Institutions). In addition, we would institute a minimum of a C in the three core courses: SOC 200 Social Statistics; SOC 300 Social Theory; and SOC 310 Research Methods.\nThe College at Brockport, College Senate, \"Change in major requirements for Sociology\" (2006). College Senate Resolutions. 1291.","meta":{"redpajama_set_name":"RedPajamaC4"}}